LMFDB - Embedded newform 450.6.f.g.107.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,6,Mod(107,450)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("450.107");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	450.f (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$72.1727189158$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 252 x^{14} + 27174 x^{12} - 1635700 x^{10} + 60061815 x^{8} - 1376564028 x^{6} + \cdots + 498214340649 $ x^16 - 252x^14 + 27174x^12 - 1635700x^10 + 60061815x^8 - 1376564028x^6 + 19220200150x^4 - 149540021784*x^2 + 498214340649
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{8}\cdot 5^{16} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			107.8
Root			$-5.79464 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	450.107
Dual form			450.6.f.g.143.8

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.82843 - 2.82843i) q^{2} -16.0000i q^{4} +(158.675 + 158.675i) q^{7} +(-45.2548 - 45.2548i) q^{8} +O(q^{10})$	\(q+(2.82843 - 2.82843i) q^{2} -16.0000i q^{4} +(158.675 + 158.675i) q^{7} +(-45.2548 - 45.2548i) q^{8} +147.071i q^{11} +(516.434 - 516.434i) q^{13} +897.604 q^{14} -256.000 q^{16} +(-1093.40 + 1093.40i) q^{17} -1175.84i q^{19} +(415.980 + 415.980i) q^{22} +(3134.18 + 3134.18i) q^{23} -2921.39i q^{26} +(2538.81 - 2538.81i) q^{28} -1863.75 q^{29} -3287.55 q^{31} +(-724.077 + 724.077i) q^{32} +6185.20i q^{34} +(1680.28 + 1680.28i) q^{37} +(-3325.78 - 3325.78i) q^{38} +9099.78i q^{41} +(-13737.1 + 13737.1i) q^{43} +2353.14 q^{44} +17729.6 q^{46} +(887.609 - 887.609i) q^{47} +33548.8i q^{49} +(-8262.94 - 8262.94i) q^{52} +(16360.3 + 16360.3i) q^{53} -14361.7i q^{56} +(-5271.49 + 5271.49i) q^{58} +957.597 q^{59} +19303.4 q^{61} +(-9298.59 + 9298.59i) q^{62} +4096.00i q^{64} +(-35783.9 - 35783.9i) q^{67} +(17494.4 + 17494.4i) q^{68} +25982.7i q^{71} +(52858.1 - 52858.1i) q^{73} +9505.07 q^{74} -18813.5 q^{76} +(-23336.6 + 23336.6i) q^{77} -16371.2i q^{79} +(25738.1 + 25738.1i) q^{82} +(69565.0 + 69565.0i) q^{83} +77708.8i q^{86} +(6655.67 - 6655.67i) q^{88} +133670. q^{89} +163891. q^{91} +(50146.9 - 50146.9i) q^{92} -5021.08i q^{94} +(-8350.12 - 8350.12i) q^{97} +(94890.3 + 94890.3i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 528 q^{7}+O(q^{10}) $ 16 * q + 528 * q^7	$ 16 q + 528 q^{7} - 192 q^{13} - 4096 q^{16} + 2688 q^{22} + 8448 q^{28} + 13024 q^{31} + 47328 q^{37} + 55440 q^{43} + 44544 q^{46} + 3072 q^{52} + 101184 q^{58} + 28400 q^{61} - 242256 q^{67} + 430944 q^{73} - 7168 q^{76} - 158208 q^{82} + 43008 q^{88} - 185472 q^{91} - 457152 q^{97}+O(q^{100}) $ 16 * q + 528 * q^7 - 192 * q^13 - 4096 * q^16 + 2688 * q^22 + 8448 * q^28 + 13024 * q^31 + 47328 * q^37 + 55440 * q^43 + 44544 * q^46 + 3072 * q^52 + 101184 * q^58 + 28400 * q^61 - 242256 * q^67 + 430944 * q^73 - 7168 * q^76 - 158208 * q^82 + 43008 * q^88 - 185472 * q^91 - 457152 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	2.82843	−	2.82843i	0.500000	−	0.500000i
$2$	2.82843	−	2.82843i	0.500000	−	0.500000i
$3$		0			0
$3$		0			0
$4$		−	16.0000i		−	0.500000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	158.675	+	158.675i	1.22395	+	1.22395i	0.966215	+	0.257737i	$0.0829770\pi$
$7$	158.675	+	158.675i	1.22395	+	1.22395i	0.257737	+	0.966215i	$0.417023\pi$
$8$	−45.2548	−	45.2548i	−0.250000	−	0.250000i
$9$		0			0
$10$		0			0
$11$			147.071i			0.366476i	0.983069	+	0.183238i	$0.0586579\pi$
$11$			147.071i			0.366476i	−0.983069	+	0.183238i	$0.941342\pi$
$12$		0			0
$13$	516.434	−	516.434i	0.847532	−	0.847532i	−0.142292	−	0.989825i	$-0.545447\pi$
$13$	516.434	−	516.434i	0.847532	−	0.847532i	0.989825	+	0.142292i	$0.0454473\pi$
$14$	897.604			1.22395
$15$		0			0
$16$	−256.000			−0.250000
$17$	−1093.40	+	1093.40i	−0.917607	+	0.917607i	−0.996855	−	0.0792478i	$-0.974748\pi$
$17$	−1093.40	+	1093.40i	−0.917607	+	0.917607i	0.0792478	+	0.996855i	$0.474748\pi$
$18$		0			0
$19$		−	1175.84i		−	0.747248i	−0.927580	−	0.373624i	$-0.878115\pi$
$19$		−	1175.84i		−	0.747248i	0.927580	−	0.373624i	$-0.121885\pi$
$20$		0			0
$21$		0			0
$22$	415.980	+	415.980i	0.183238	+	0.183238i
$23$	3134.18	+	3134.18i	1.23539	+	1.23539i	0.961865	+	0.273525i	$0.0881897\pi$
$23$	3134.18	+	3134.18i	1.23539	+	1.23539i	0.273525	+	0.961865i	$0.411810\pi$
$24$		0			0
$25$		0			0
$26$		−	2921.39i		−	0.847532i
$27$		0			0
$28$	2538.81	−	2538.81i	0.611976	−	0.611976i
$29$	−1863.75			−0.411522			−0.205761	−	0.978602i	$-0.565967\pi$
$29$	−1863.75			−0.411522			−0.205761	+	0.978602i	$0.565967\pi$
$30$		0			0
$31$	−3287.55			−0.614424			−0.307212	−	0.951641i	$-0.599396\pi$
$31$	−3287.55			−0.614424			−0.307212	+	0.951641i	$0.599396\pi$
$32$	−724.077	+	724.077i	−0.125000	+	0.125000i
$33$		0			0
$34$			6185.20i			0.917607i
$35$		0			0
$36$		0			0
$37$	1680.28	+	1680.28i	0.201779	+	0.201779i	0.800762	−	0.598983i	$-0.204428\pi$
$37$	1680.28	+	1680.28i	0.201779	+	0.201779i	−0.598983	+	0.800762i	$0.704428\pi$
$38$	−3325.78	−	3325.78i	−0.373624	−	0.373624i
$39$		0			0
$40$		0			0
$41$			9099.78i			0.845418i	0.906266	+	0.422709i	$0.138921\pi$
$41$			9099.78i			0.845418i	−0.906266	+	0.422709i	$0.861079\pi$
$42$		0			0
$43$	−13737.1	+	13737.1i	−1.13298	+	1.13298i	−0.143306	+	0.989678i	$0.545773\pi$
$43$	−13737.1	+	13737.1i	−1.13298	+	1.13298i	−0.989678	+	0.143306i	$0.954227\pi$
$44$	2353.14			0.183238
$45$		0			0
$46$	17729.6			1.23539
$47$	887.609	−	887.609i	0.0586107	−	0.0586107i	−0.677194	−	0.735805i	$-0.736804\pi$
$47$	887.609	−	887.609i	0.0586107	−	0.0586107i	0.735805	+	0.677194i	$0.236804\pi$
$48$		0			0
$49$			33548.8i			1.99612i
$50$		0			0
$51$		0			0
$52$	−8262.94	−	8262.94i	−0.423766	−	0.423766i
$53$	16360.3	+	16360.3i	0.800020	+	0.800020i	0.983098	−	0.183078i	$-0.0586061\pi$
$53$	16360.3	+	16360.3i	0.800020	+	0.800020i	−0.183078	+	0.983098i	$0.558606\pi$
$54$		0			0
$55$		0			0
$56$		−	14361.7i		−	0.611976i
$57$		0			0
$58$	−5271.49	+	5271.49i	−0.205761	+	0.205761i
$59$	957.597			0.0358140			0.0179070	−	0.999840i	$-0.494300\pi$
$59$	957.597			0.0358140			0.0179070	+	0.999840i	$0.494300\pi$
$60$		0			0
$61$	19303.4			0.664215			0.332107	−	0.943242i	$-0.392240\pi$
$61$	19303.4			0.664215			0.332107	+	0.943242i	$0.392240\pi$
$62$	−9298.59	+	9298.59i	−0.307212	+	0.307212i
$63$		0			0
$64$			4096.00i			0.125000i
$65$		0			0
$66$		0			0
$67$	−35783.9	−	35783.9i	−0.973868	−	0.973868i	0.0257987	−	0.999667i	$-0.491787\pi$
$67$	−35783.9	−	35783.9i	−0.973868	−	0.973868i	−0.999667	+	0.0257987i	$0.991787\pi$
$68$	17494.4	+	17494.4i	0.458804	+	0.458804i
$69$		0			0
$70$		0			0
$71$			25982.7i			0.611700i	0.952080	+	0.305850i	$0.0989405\pi$
$71$			25982.7i			0.611700i	−0.952080	+	0.305850i	$0.901059\pi$
$72$		0			0
$73$	52858.1	−	52858.1i	1.16093	−	1.16093i	0.176652	−	0.984273i	$-0.443473\pi$
$73$	52858.1	−	52858.1i	1.16093	−	1.16093i	0.984273	−	0.176652i	$-0.0565266\pi$
$74$	9505.07			0.201779
$75$		0			0
$76$	−18813.5			−0.373624
$77$	−23336.6	+	23336.6i	−0.448549	+	0.448549i
$78$		0			0
$79$		−	16371.2i		−	0.295130i	−0.989052	−	0.147565i	$-0.952856\pi$
$79$		−	16371.2i		−	0.295130i	0.989052	−	0.147565i	$-0.0471436\pi$
$80$		0			0
$81$		0			0
$82$	25738.1	+	25738.1i	0.422709	+	0.422709i
$83$	69565.0	+	69565.0i	1.10840	+	1.10840i	0.993361	+	0.115036i	$0.0366983\pi$
$83$	69565.0	+	69565.0i	1.10840	+	1.10840i	0.115036	+	0.993361i	$0.463302\pi$
$84$		0			0
$85$		0			0
$86$			77708.8i			1.13298i
$87$		0			0
$88$	6655.67	−	6655.67i	0.0916190	−	0.0916190i
$89$	133670.			1.78879			0.894395	−	0.447278i	$-0.147606\pi$
$89$	133670.			1.78879			0.894395	+	0.447278i	$0.147606\pi$
$90$		0			0
$91$	163891.			2.07468
$92$	50146.9	−	50146.9i	0.617695	−	0.617695i
$93$		0			0
$94$		−	5021.08i		−	0.0586107i
$95$		0			0
$96$		0			0
$97$	−8350.12	−	8350.12i	−0.0901080	−	0.0901080i	0.660616	−	0.750724i	$-0.270295\pi$
$97$	−8350.12	−	8350.12i	−0.0901080	−	0.0901080i	−0.750724	+	0.660616i	$0.770295\pi$
$98$	94890.3	+	94890.3i	0.998060	+	0.998060i
$99$		0			0
$100$		0			0
$101$			15700.1i			0.153144i	0.997064	+	0.0765718i	$0.0243975\pi$
$101$			15700.1i			0.153144i	−0.997064	+	0.0765718i	$0.975603\pi$
$102$		0			0
$103$	71583.6	−	71583.6i	0.664845	−	0.664845i	−0.291673	−	0.956518i	$-0.594212\pi$
$103$	71583.6	−	71583.6i	0.664845	−	0.664845i	0.956518	+	0.291673i	$0.0942119\pi$
$104$	−46742.2			−0.423766
$105$		0			0
$106$	92547.7			0.800020
$107$	−71776.6	+	71776.6i	−0.606071	+	0.606071i	−0.941917	−	0.335846i	$-0.890978\pi$
$107$	−71776.6	+	71776.6i	−0.606071	+	0.606071i	0.335846	+	0.941917i	$0.390978\pi$
$108$		0			0
$109$			214412.i			1.72855i	0.503020	+	0.864275i	$0.332222\pi$
$109$			214412.i			1.72855i	−0.503020	+	0.864275i	$0.667778\pi$
$110$		0			0
$111$		0			0
$112$	−40620.9	−	40620.9i	−0.305988	−	0.305988i
$113$	−60197.8	−	60197.8i	−0.443491	−	0.443491i	0.449693	−	0.893183i	$-0.351534\pi$
$113$	−60197.8	−	60197.8i	−0.443491	−	0.443491i	−0.893183	+	0.449693i	$0.851534\pi$
$114$		0			0
$115$		0			0
$116$			29820.0i			0.205761i
$117$		0			0
$118$	2708.49	−	2708.49i	0.0179070	−	0.0179070i
$119$	−346991.			−2.24622
$120$		0			0
$121$	139421.			0.865695
$122$	54598.2	−	54598.2i	0.332107	−	0.332107i
$123$		0			0
$124$			52600.8i			0.307212i
$125$		0			0
$126$		0			0
$127$	−149979.	−	149979.i	−0.825125	−	0.825125i	0.161713	−	0.986838i	$-0.448298\pi$
$127$	−149979.	−	149979.i	−0.825125	−	0.825125i	−0.986838	+	0.161713i	$0.948298\pi$
$128$	11585.2	+	11585.2i	0.0625000	+	0.0625000i
$129$		0			0
$130$		0			0
$131$			296250.i			1.50827i	0.656717	+	0.754137i	$0.271945\pi$
$131$			296250.i			1.50827i	−0.656717	+	0.754137i	$0.728055\pi$
$132$		0			0
$133$	186577.	−	186577.i	0.914596	−	0.914596i
$134$	−202424.			−0.973868
$135$		0			0
$136$	98963.3			0.458804
$137$	279939.	−	279939.i	1.27427	−	1.27427i	0.330448	−	0.943824i	$-0.392800\pi$
$137$	279939.	−	279939.i	1.27427	−	1.27427i	0.943824	−	0.330448i	$-0.107200\pi$
$138$		0			0
$139$			271547.i			1.19209i	0.802951	+	0.596045i	$0.203262\pi$
$139$			271547.i			1.19209i	−0.802951	+	0.596045i	$0.796738\pi$
$140$		0			0
$141$		0			0
$142$	73490.1	+	73490.1i	0.305850	+	0.305850i
$143$	75952.4	+	75952.4i	0.310600	+	0.310600i
$144$		0			0
$145$		0			0
$146$		−	299010.i		−	1.16093i
$147$		0			0
$148$	26884.4	−	26884.4i	0.100890	−	0.100890i
$149$	−74640.5			−0.275429			−0.137714	−	0.990472i	$-0.543976\pi$
$149$	−74640.5			−0.275429			−0.137714	+	0.990472i	$0.543976\pi$
$150$		0			0
$151$	101065.			0.360711			0.180356	−	0.983601i	$-0.442275\pi$
$151$	101065.			0.360711			0.180356	+	0.983601i	$0.442275\pi$
$152$	−53212.5	+	53212.5i	−0.186812	+	0.186812i
$153$		0			0
$154$			132011.i			0.448549i
$155$		0			0
$156$		0			0
$157$	179927.	+	179927.i	0.582569	+	0.582569i	0.935608	−	0.353040i	$-0.114852\pi$
$157$	179927.	+	179927.i	0.582569	+	0.582569i	−0.353040	+	0.935608i	$0.614852\pi$
$158$	−46304.9	−	46304.9i	−0.147565	−	0.147565i
$159$		0			0
$160$		0			0
$161$			994634.i			3.02412i
$162$		0			0
$163$	451895.	−	451895.i	1.33220	−	1.33220i	0.428796	−	0.903402i	$-0.358938\pi$
$163$	451895.	−	451895.i	1.33220	−	1.33220i	0.903402	−	0.428796i	$-0.141062\pi$
$164$	145597.			0.422709
$165$		0			0
$166$	393519.			1.10840
$167$	−485733.	+	485733.i	−1.34774	+	1.34774i	−0.459634	+	0.888109i	$0.652019\pi$
$167$	−485733.	+	485733.i	−1.34774	+	1.34774i	−0.888109	+	0.459634i	$0.847981\pi$
$168$		0			0
$169$		−	162115.i		−	0.436622i
$170$		0			0
$171$		0			0
$172$	219794.	+	219794.i	0.566492	+	0.566492i
$173$	258825.	+	258825.i	0.657493	+	0.657493i	0.954786	−	0.297293i	$-0.0960839\pi$
$173$	258825.	+	258825.i	0.657493	+	0.657493i	−0.297293	+	0.954786i	$0.596084\pi$
$174$		0			0
$175$		0			0
$176$		−	37650.2i		−	0.0916190i
$177$		0			0
$178$	378076.	−	378076.i	0.894395	−	0.894395i
$179$	−643315.			−1.50069			−0.750345	−	0.661046i	$-0.770113\pi$
$179$	−643315.			−1.50069			−0.750345	+	0.661046i	$0.770113\pi$
$180$		0			0
$181$	−253825.			−0.575887			−0.287944	−	0.957647i	$-0.592972\pi$
$181$	−253825.			−0.575887			−0.287944	+	0.957647i	$0.592972\pi$
$182$	463553.	−	463553.i	1.03734	−	1.03734i
$183$		0			0
$184$		−	283673.i		−	0.617695i
$185$		0			0
$186$		0			0
$187$	−160807.	−	160807.i	−0.336281	−	0.336281i
$188$	−14201.7	−	14201.7i	−0.0293054	−	0.0293054i
$189$		0			0
$190$		0			0
$191$		−	452729.i		−	0.897956i	−0.893543	−	0.448978i	$-0.851788\pi$
$191$		−	452729.i		−	0.897956i	0.893543	−	0.448978i	$-0.148212\pi$
$192$		0			0
$193$	449759.	−	449759.i	0.869133	−	0.869133i	−0.123244	−	0.992376i	$-0.539330\pi$
$193$	449759.	−	449759.i	0.869133	−	0.869133i	0.992376	+	0.123244i	$0.0393297\pi$
$194$	−47235.4			−0.0901080
$195$		0			0
$196$	536780.			0.998060
$197$	142034.	−	142034.i	0.260751	−	0.260751i	−0.564608	−	0.825359i	$-0.690973\pi$
$197$	142034.	−	142034.i	0.260751	−	0.260751i	0.825359	+	0.564608i	$0.190973\pi$
$198$		0			0
$199$		−	721560.i		−	1.29163i	−0.763492	−	0.645817i	$-0.776517\pi$
$199$		−	721560.i		−	1.29163i	0.763492	−	0.645817i	$-0.223483\pi$
$200$		0			0
$201$		0			0
$202$	44406.6	+	44406.6i	0.0765718	+	0.0765718i
$203$	−295732.	−	295732.i	−0.503683	−	0.503683i
$204$		0			0
$205$		0			0
$206$		−	404938.i		−	0.664845i
$207$		0			0
$208$	−132207.	+	132207.i	−0.211883	+	0.211883i
$209$	172932.			0.273848
$210$		0			0
$211$	252159.			0.389913			0.194956	−	0.980812i	$-0.437543\pi$
$211$	252159.			0.389913			0.194956	+	0.980812i	$0.437543\pi$
$212$	261764.	−	261764.i	0.400010	−	0.400010i
$213$		0			0
$214$			406030.i			0.606071i
$215$		0			0
$216$		0			0
$217$	−521653.	−	521653.i	−0.752025	−	0.752025i
$218$	606447.	+	606447.i	0.864275	+	0.864275i
$219$		0			0
$220$		0			0
$221$			1.12934e6i			1.55540i
$222$		0			0
$223$	−334048.	+	334048.i	−0.449828	+	0.449828i	−0.895297	−	0.445469i	$-0.853037\pi$
$223$	−334048.	+	334048.i	−0.449828	+	0.449828i	0.445469	+	0.895297i	$0.353037\pi$
$224$	−229787.			−0.305988
$225$		0			0
$226$	−340530.			−0.443491
$227$	−42990.2	+	42990.2i	−0.0553738	+	0.0553738i	−0.734251	−	0.678878i	$-0.762467\pi$
$227$	−42990.2	+	42990.2i	−0.0553738	+	0.0553738i	0.678878	+	0.734251i	$0.262467\pi$
$228$		0			0
$229$			1.31576e6i			1.65801i	0.559240	+	0.829006i	$0.311093\pi$
$229$			1.31576e6i			1.65801i	−0.559240	+	0.829006i	$0.688907\pi$
$230$		0			0
$231$		0			0
$232$	84343.8	+	84343.8i	0.102880	+	0.102880i
$233$	−842302.	−	842302.i	−1.01643	−	1.01643i	−0.999863	−	0.0165689i	$-0.994726\pi$
$233$	−842302.	−	842302.i	−1.01643	−	1.01643i	−0.0165689	−	0.999863i	$-0.505274\pi$
$234$		0			0
$235$		0			0
$236$		−	15321.6i		−	0.0179070i
$237$		0			0
$238$	−981440.	+	981440.i	−1.12311	+	1.12311i
$239$	1.06608e6			1.20725			0.603624	−	0.797269i	$-0.293723\pi$
$239$	1.06608e6			1.20725			0.603624	+	0.797269i	$0.293723\pi$
$240$		0			0
$241$	51058.8			0.0566275			0.0283138	−	0.999599i	$-0.490986\pi$
$241$	51058.8			0.0566275			0.0283138	+	0.999599i	$0.490986\pi$
$242$	394342.	−	394342.i	0.432848	−	0.432848i
$243$		0			0
$244$		−	308854.i		−	0.332107i
$245$		0			0
$246$		0			0
$247$	−607244.	−	607244.i	−0.633317	−	0.633317i
$248$	148777.	+	148777.i	0.153606	+	0.153606i
$249$		0			0
$250$		0			0
$251$			516422.i			0.517393i	0.965959	+	0.258697i	$0.0832930\pi$
$251$			516422.i			0.517393i	−0.965959	+	0.258697i	$0.916707\pi$
$252$		0			0
$253$	−460947.	+	460947.i	−0.452741	+	0.452741i
$254$	−848407.			−0.825125
$255$		0			0
$256$	65536.0			0.0625000
$257$	−887334.	+	887334.i	−0.838020	+	0.838020i	−0.988598	−	0.150578i	$-0.951887\pi$
$257$	−887334.	+	887334.i	−0.838020	+	0.838020i	0.150578	+	0.988598i	$0.451887\pi$
$258$		0			0
$259$			533237.i			0.493936i
$260$		0			0
$261$		0			0
$262$	837922.	+	837922.i	0.754137	+	0.754137i
$263$	−624251.	−	624251.i	−0.556506	−	0.556506i	0.371805	−	0.928311i	$-0.378739\pi$
$263$	−624251.	−	624251.i	−0.556506	−	0.556506i	−0.928311	+	0.371805i	$0.878739\pi$
$264$		0			0
$265$		0			0
$266$		−	1.05544e6i		−	0.914596i
$267$		0			0
$268$	−572542.	+	572542.i	−0.486934	+	0.486934i
$269$	341395.			0.287658			0.143829	−	0.989603i	$-0.454058\pi$
$269$	341395.			0.287658			0.143829	+	0.989603i	$0.454058\pi$
$270$		0			0
$271$	−706610.			−0.584463			−0.292231	−	0.956348i	$-0.594398\pi$
$271$	−706610.			−0.584463			−0.292231	+	0.956348i	$0.594398\pi$
$272$	279910.	−	279910.i	0.229402	−	0.229402i
$273$		0			0
$274$		−	1.58358e6i		−	1.27427i
$275$		0			0
$276$		0			0
$277$	10908.2	+	10908.2i	0.00854186	+	0.00854186i	0.711365	−	0.702823i	$-0.248077\pi$
$277$	10908.2	+	10908.2i	0.00854186	+	0.00854186i	−0.702823	+	0.711365i	$0.748077\pi$
$278$	768052.	+	768052.i	0.596045	+	0.596045i
$279$		0			0
$280$		0			0
$281$		−	2.16368e6i		−	1.63466i	−0.576171	−	0.817329i	$-0.695454\pi$
$281$		−	2.16368e6i		−	1.63466i	0.576171	−	0.817329i	$-0.304546\pi$
$282$		0			0
$283$	407593.	−	407593.i	0.302524	−	0.302524i	−0.539476	−	0.842001i	$-0.681378\pi$
$283$	407593.	−	407593.i	0.302524	−	0.302524i	0.842001	+	0.539476i	$0.181378\pi$
$284$	415723.			0.305850
$285$		0			0
$286$	429652.			0.310600
$287$	−1.44391e6	+	1.44391e6i	−1.03475	+	1.03475i
$288$		0			0
$289$		−	971190.i		−	0.684006i
$290$		0			0
$291$		0			0
$292$	−845729.	−	845729.i	−0.580463	−	0.580463i
$293$	60301.5	+	60301.5i	0.0410354	+	0.0410354i	0.727327	−	0.686291i	$-0.240762\pi$
$293$	60301.5	+	60301.5i	0.0410354	+	0.0410354i	−0.686291	+	0.727327i	$0.740762\pi$
$294$		0			0
$295$		0			0
$296$		−	152081.i		−	0.100890i
$297$		0			0
$298$	−211115.	+	211115.i	−0.137714	+	0.137714i
$299$	3.23719e6			2.09407
$300$		0			0
$301$	−4.35948e6			−2.77344
$302$	285856.	−	285856.i	0.180356	−	0.180356i
$303$		0			0
$304$			301015.i			0.186812i
$305$		0			0
$306$		0			0
$307$	−132817.	−	132817.i	−0.0804284	−	0.0804284i	0.665748	−	0.746177i	$-0.268113\pi$
$307$	−132817.	−	132817.i	−0.0804284	−	0.0804284i	−0.746177	+	0.665748i	$0.768113\pi$
$308$	373385.	+	373385.i	0.224275	+	0.224275i
$309$		0			0
$310$		0			0
$311$		−	310276.i		−	0.181906i	−0.995855	−	0.0909530i	$-0.971009\pi$
$311$		−	310276.i		−	0.181906i	0.995855	−	0.0909530i	$-0.0289913\pi$
$312$		0			0
$313$	−889368.	+	889368.i	−0.513122	+	0.513122i	−0.915482	−	0.402359i	$-0.868190\pi$
$313$	−889368.	+	889368.i	−0.513122	+	0.513122i	0.402359	+	0.915482i	$0.368190\pi$
$314$	1.01782e6			0.582569
$315$		0			0
$316$	−261940.			−0.147565
$317$	−2.29681e6	+	2.29681e6i	−1.28374	+	1.28374i	−0.345213	+	0.938524i	$0.612193\pi$
$317$	−2.29681e6	+	2.29681e6i	−1.28374	+	1.28374i	−0.938524	+	0.345213i	$0.887807\pi$
$318$		0			0
$319$		−	274104.i		−	0.150813i
$320$		0			0
$321$		0			0
$322$	2.81325e6	+	2.81325e6i	1.51206	+	1.51206i
$323$	1.28566e6	+	1.28566e6i	0.685680	+	0.685680i
$324$		0			0
$325$		0			0
$326$		−	2.55630e6i		−	1.33220i
$327$		0			0
$328$	411809.	−	411809.i	0.211354	−	0.211354i
$329$	281683.			0.143473
$330$		0			0
$331$	−2.74088e6			−1.37505			−0.687527	−	0.726159i	$-0.741304\pi$
$331$	−2.74088e6			−1.37505			−0.687527	+	0.726159i	$0.741304\pi$
$332$	1.11304e6	−	1.11304e6i	0.554199	−	0.554199i
$333$		0			0
$334$			2.74772e6i			1.34774i
$335$		0			0
$336$		0			0
$337$	277606.	+	277606.i	0.133154	+	0.133154i	0.770542	−	0.637389i	$-0.219985\pi$
$337$	277606.	+	277606.i	0.133154	+	0.133154i	−0.637389	+	0.770542i	$0.719985\pi$
$338$	−458530.	−	458530.i	−0.218311	−	0.218311i
$339$		0			0
$340$		0			0
$341$		−	483503.i		−	0.225171i
$342$		0			0
$343$	−2.65651e6	+	2.65651e6i	−1.21920	+	1.21920i
$344$	1.24334e6			0.566492
$345$		0			0
$346$	1.46414e6			0.657493
$347$	1.35770e6	−	1.35770e6i	0.605313	−	0.605313i	−0.336405	−	0.941717i	$-0.609211\pi$
$347$	1.35770e6	−	1.35770e6i	0.605313	−	0.605313i	0.941717	+	0.336405i	$0.109211\pi$
$348$		0			0
$349$		−	3.94136e6i		−	1.73214i	−0.499924	−	0.866069i	$-0.666639\pi$
$349$		−	3.94136e6i		−	1.73214i	0.499924	−	0.866069i	$-0.333361\pi$
$350$		0			0
$351$		0			0
$352$	−106491.	−	106491.i	−0.0458095	−	0.0458095i
$353$	−3.03961e6	−	3.03961e6i	−1.29832	−	1.29832i	−0.929504	−	0.368812i	$-0.879765\pi$
$353$	−3.03961e6	−	3.03961e6i	−1.29832	−	1.29832i	−0.368812	−	0.929504i	$-0.620235\pi$
$354$		0			0
$355$		0			0
$356$		−	2.13872e6i		−	0.894395i
$357$		0			0
$358$	−1.81957e6	+	1.81957e6i	−0.750345	+	0.750345i
$359$	−1.62458e6			−0.665279			−0.332640	−	0.943054i	$-0.607939\pi$
$359$	−1.62458e6			−0.665279			−0.332640	+	0.943054i	$0.607939\pi$
$360$		0			0
$361$	1.09350e6			0.441621
$362$	−717924.	+	717924.i	−0.287944	+	0.287944i
$363$		0			0
$364$		−	2.62225e6i		−	1.03734i
$365$		0			0
$366$		0			0
$367$	363952.	+	363952.i	0.141052	+	0.141052i	0.774107	−	0.633055i	$-0.218199\pi$
$367$	363952.	+	363952.i	0.141052	+	0.141052i	−0.633055	+	0.774107i	$0.718199\pi$
$368$	−802350.	−	802350.i	−0.308848	−	0.308848i
$369$		0			0
$370$		0			0
$371$			5.19195e6i			1.95837i
$372$		0			0
$373$	−1.54530e6	+	1.54530e6i	−0.575096	+	0.575096i	−0.933548	−	0.358452i	$-0.883305\pi$
$373$	−1.54530e6	+	1.54530e6i	−0.575096	+	0.575096i	0.358452	+	0.933548i	$0.383305\pi$
$374$	−909664.			−0.336281
$375$		0			0
$376$	−80337.2			−0.0293054
$377$	−962504.	+	962504.i	−0.348778	+	0.348778i
$378$		0			0
$379$			2.52657e6i			0.903512i	0.892142	+	0.451756i	$0.149202\pi$
$379$			2.52657e6i			0.903512i	−0.892142	+	0.451756i	$0.850798\pi$
$380$		0			0
$381$		0			0
$382$	−1.28051e6	−	1.28051e6i	−0.448978	−	0.448978i
$383$	−1.81889e6	−	1.81889e6i	−0.633592	−	0.633592i	0.315375	−	0.948967i	$-0.397870\pi$
$383$	−1.81889e6	−	1.81889e6i	−0.633592	−	0.633592i	−0.948967	+	0.315375i	$0.897870\pi$
$384$		0			0
$385$		0			0
$386$		−	2.54422e6i		−	0.869133i
$387$		0			0
$388$	−133602.	+	133602.i	−0.0450540	+	0.0450540i
$389$	−3.17134e6			−1.06260			−0.531299	−	0.847185i	$-0.678296\pi$
$389$	−3.17134e6			−1.06260			−0.531299	+	0.847185i	$0.678296\pi$
$390$		0			0
$391$	−6.85382e6			−2.26721
$392$	1.51824e6	−	1.51824e6i	0.499030	−	0.499030i
$393$		0			0
$394$		−	803465.i		−	0.260751i
$395$		0			0
$396$		0			0
$397$	−3.79311e6	−	3.79311e6i	−1.20787	−	1.20787i	−0.971716	−	0.236150i	$-0.924114\pi$
$397$	−3.79311e6	−	3.79311e6i	−1.20787	−	1.20787i	−0.236150	−	0.971716i	$-0.575886\pi$
$398$	−2.04088e6	−	2.04088e6i	−0.645817	−	0.645817i
$399$		0			0
$400$		0			0
$401$		−	2.97722e6i		−	0.924592i	−0.886726	−	0.462296i	$-0.847026\pi$
$401$		−	2.97722e6i		−	0.924592i	0.886726	−	0.462296i	$-0.152974\pi$
$402$		0			0
$403$	−1.69780e6	+	1.69780e6i	−0.520744	+	0.520744i
$404$	251202.			0.0765718
$405$		0			0
$406$	−1.67291e6			−0.503683
$407$	−247120.	+	247120.i	−0.0739472	+	0.0739472i
$408$		0			0
$409$			149571.i			0.0442118i	0.999756	+	0.0221059i	$0.00703710\pi$
$409$			149571.i			0.0442118i	−0.999756	+	0.0221059i	$0.992963\pi$
$410$		0			0
$411$		0			0
$412$	−1.14534e6	−	1.14534e6i	−0.332423	−	0.332423i
$413$	151947.	+	151947.i	0.0438346	+	0.0438346i
$414$		0			0
$415$		0			0
$416$			747876.i			0.211883i
$417$		0			0
$418$	489126.	−	489126.i	0.136924	−	0.136924i
$419$	6.06070e6			1.68650			0.843252	−	0.537518i	$-0.180638\pi$
$419$	6.06070e6			1.68650			0.843252	+	0.537518i	$0.180638\pi$
$420$		0			0
$421$	3.60024e6			0.989979			0.494989	−	0.868899i	$-0.335172\pi$
$421$	3.60024e6			0.989979			0.494989	+	0.868899i	$0.335172\pi$
$422$	713212.	−	713212.i	0.194956	−	0.194956i
$423$		0			0
$424$		−	1.48076e6i		−	0.400010i
$425$		0			0
$426$		0			0
$427$	3.06297e6	+	3.06297e6i	0.812967	+	0.812967i
$428$	1.14843e6	+	1.14843e6i	0.303035	+	0.303035i
$429$		0			0
$430$		0			0
$431$		−	1.61379e6i		−	0.418460i	−0.977866	−	0.209230i	$-0.932904\pi$
$431$		−	1.61379e6i		−	0.418460i	0.977866	−	0.209230i	$-0.0670958\pi$
$432$		0			0
$433$	4.69226e6	−	4.69226e6i	1.20271	−	1.20271i	0.229376	−	0.973338i	$-0.426331\pi$
$433$	4.69226e6	−	4.69226e6i	1.20271	−	1.20271i	0.973338	−	0.229376i	$-0.0736686\pi$
$434$	−2.95092e6			−0.752025
$435$		0			0
$436$	3.43058e6			0.864275
$437$	3.68530e6	−	3.68530e6i	0.923142	−	0.923142i
$438$		0			0
$439$		−	2.32229e6i		−	0.575117i	−0.957763	−	0.287558i	$-0.907156\pi$
$439$		−	2.32229e6i		−	0.575117i	0.957763	−	0.287558i	$-0.0928435\pi$
$440$		0			0
$441$		0			0
$442$	3.19425e6	+	3.19425e6i	0.777702	+	0.777702i
$443$	−3.43425e6	−	3.43425e6i	−0.831424	−	0.831424i	0.156288	−	0.987712i	$-0.450047\pi$
$443$	−3.43425e6	−	3.43425e6i	−0.831424	−	0.831424i	−0.987712	+	0.156288i	$0.950047\pi$
$444$		0			0
$445$		0			0
$446$			1.88966e6i			0.449828i
$447$		0			0
$448$	−649935.	+	649935.i	−0.152994	+	0.152994i
$449$	−152617.			−0.0357262			−0.0178631	−	0.999840i	$-0.505686\pi$
$449$	−152617.			−0.0357262			−0.0178631	+	0.999840i	$0.505686\pi$
$450$		0			0
$451$	−1.33831e6			−0.309825
$452$	−963165.	+	963165.i	−0.221745	+	0.221745i
$453$		0			0
$454$			243189.i			0.0553738i
$455$		0			0
$456$		0			0
$457$	2.56661e6	+	2.56661e6i	0.574870	+	0.574870i	0.933485	−	0.358616i	$-0.116751\pi$
$457$	2.56661e6	+	2.56661e6i	0.574870	+	0.574870i	−0.358616	+	0.933485i	$0.616751\pi$
$458$	3.72153e6	+	3.72153e6i	0.829006	+	0.829006i
$459$		0			0
$460$		0			0
$461$			4.24935e6i			0.931257i	0.884980	+	0.465629i	$0.154172\pi$
$461$			4.24935e6i			0.931257i	−0.884980	+	0.465629i	$0.845828\pi$
$462$		0			0
$463$	3.95866e6	−	3.95866e6i	0.858215	−	0.858215i	−0.132913	−	0.991128i	$-0.542433\pi$
$463$	3.95866e6	−	3.95866e6i	0.858215	−	0.858215i	0.991128	+	0.132913i	$0.0424330\pi$
$464$	477120.			0.102880
$465$		0			0
$466$	−4.76478e6			−1.01643
$467$	−246422.	+	246422.i	−0.0522862	+	0.0522862i	−0.732766	−	0.680480i	$-0.761771\pi$
$467$	−246422.	+	246422.i	−0.0522862	+	0.0522862i	0.680480	+	0.732766i	$0.261771\pi$
$468$		0			0
$469$		−	1.13560e7i		−	2.38394i
$470$		0			0
$471$		0			0
$472$	−43335.9	−	43335.9i	−0.00895350	−	0.00895350i
$473$	−2.02033e6	−	2.02033e6i	−0.415212	−	0.415212i
$474$		0			0
$475$		0			0
$476$			5.55186e6i			1.12311i
$477$		0			0
$478$	3.01534e6	−	3.01534e6i	0.603624	−	0.603624i
$479$	−311522.			−0.0620369			−0.0310184	−	0.999519i	$-0.509875\pi$
$479$	−311522.			−0.0620369			−0.0310184	+	0.999519i	$0.509875\pi$
$480$		0			0
$481$	1.73550e6			0.342029
$482$	144416.	−	144416.i	0.0283138	−	0.0283138i
$483$		0			0
$484$		−	2.23074e6i		−	0.432848i
$485$		0			0
$486$		0			0
$487$	−1.12519e6	−	1.12519e6i	−0.214983	−	0.214983i	0.591397	−	0.806380i	$-0.298577\pi$
$487$	−1.12519e6	−	1.12519e6i	−0.214983	−	0.214983i	−0.806380	+	0.591397i	$0.798577\pi$
$488$	−873570.	−	873570.i	−0.166054	−	0.166054i
$489$		0			0
$490$		0			0
$491$		−	6.48585e6i		−	1.21412i	−0.794654	−	0.607062i	$-0.792348\pi$
$491$		−	6.48585e6i		−	1.21412i	0.794654	−	0.607062i	$-0.207652\pi$
$492$		0			0
$493$	2.03783e6	−	2.03783e6i	0.377615	−	0.377615i
$494$	−3.43509e6			−0.633317
$495$		0			0
$496$	841612.			0.153606
$497$	−4.12281e6	+	4.12281e6i	−0.748691	+	0.748691i
$498$		0			0
$499$			5.91058e6i			1.06262i	0.847177	+	0.531311i	$0.178300\pi$
$499$			5.91058e6i			1.06262i	−0.847177	+	0.531311i	$0.821700\pi$
$500$		0			0
$501$		0			0
$502$	1.46066e6	+	1.46066e6i	0.258697	+	0.258697i
$503$	4.45794e6	+	4.45794e6i	0.785623	+	0.785623i	0.980773	−	0.195150i	$-0.0625194\pi$
$503$	4.45794e6	+	4.45794e6i	0.785623	+	0.785623i	−0.195150	+	0.980773i	$0.562519\pi$
$504$		0			0
$505$		0			0
$506$			2.60751e6i			0.452741i
$507$		0			0
$508$	−2.39966e6	+	2.39966e6i	−0.412563	+	0.412563i
$509$	−1.00967e7			−1.72736			−0.863681	−	0.504038i	$-0.831847\pi$
$509$	−1.00967e7			−1.72736			−0.863681	+	0.504038i	$0.831847\pi$
$510$		0			0
$511$	1.67746e7			2.84183
$512$	185364.	−	185364.i	0.0312500	−	0.0312500i
$513$		0			0
$514$			5.01952e6i			0.838020i
$515$		0			0
$516$		0			0
$517$	130542.	+	130542.i	0.0214794	+	0.0214794i
$518$	1.50822e6	+	1.50822e6i	0.246968	+	0.246968i
$519$		0			0
$520$		0			0
$521$		−	3.57041e6i		−	0.576266i	−0.957590	−	0.288133i	$-0.906965\pi$
$521$		−	3.57041e6i		−	0.576266i	0.957590	−	0.288133i	$-0.0930346\pi$
$522$		0			0
$523$	527646.	−	527646.i	0.0843506	−	0.0843506i	−0.663673	−	0.748023i	$-0.731003\pi$
$523$	527646.	−	527646.i	0.0843506	−	0.0843506i	0.748023	+	0.663673i	$0.231003\pi$
$524$	4.74000e6			0.754137
$525$		0			0
$526$	−3.53130e6			−0.556506
$527$	3.59460e6	−	3.59460e6i	0.563799	−	0.563799i
$528$		0			0
$529$			1.32098e7i			2.05238i
$530$		0			0
$531$		0			0
$532$	−2.98523e6	−	2.98523e6i	−0.457298	−	0.457298i
$533$	4.69944e6	+	4.69944e6i	0.716519	+	0.716519i
$534$		0			0
$535$		0			0
$536$			3.23879e6i			0.486934i
$537$		0			0
$538$	965611.	−	965611.i	0.143829	−	0.143829i
$539$	−4.93405e6			−0.731530
$540$		0			0
$541$	−7.31558e6			−1.07462			−0.537311	−	0.843384i	$-0.680560\pi$
$541$	−7.31558e6			−1.07462			−0.537311	+	0.843384i	$0.680560\pi$
$542$	−1.99860e6	+	1.99860e6i	−0.292231	+	0.292231i
$543$		0			0
$544$		−	1.58341e6i		−	0.229402i
$545$		0			0
$546$		0			0
$547$	−2.05132e6	−	2.05132e6i	−0.293133	−	0.293133i	0.545183	−	0.838317i	$-0.316460\pi$
$547$	−2.05132e6	−	2.05132e6i	−0.293133	−	0.293133i	−0.838317	+	0.545183i	$0.816460\pi$
$548$	−4.47903e6	−	4.47903e6i	−0.637136	−	0.637136i
$549$		0			0
$550$		0			0
$551$			2.19148e6i			0.307509i
$552$		0			0
$553$	2.59771e6	−	2.59771e6i	0.361226	−	0.361226i
$554$	61705.9			0.00854186
$555$		0			0
$556$	4.34476e6			0.596045
$557$	7.01106e6	−	7.01106e6i	0.957515	−	0.957515i	−0.0416185	−	0.999134i	$-0.513251\pi$
$557$	7.01106e6	−	7.01106e6i	0.957515	−	0.957515i	0.999134	+	0.0416185i	$0.0132514\pi$
$558$		0			0
$559$			1.41886e7i			1.92048i
$560$		0			0
$561$		0			0
$562$	−6.11981e6	−	6.11981e6i	−0.817329	−	0.817329i
$563$	5.99030e6	+	5.99030e6i	0.796485	+	0.796485i	0.982539	−	0.186055i	$-0.0595702\pi$
$563$	5.99030e6	+	5.99030e6i	0.796485	+	0.796485i	−0.186055	+	0.982539i	$0.559570\pi$
$564$		0			0
$565$		0			0
$566$		−	2.30569e6i		−	0.302524i
$567$		0			0
$568$	1.17584e6	−	1.17584e6i	0.152925	−	0.152925i
$569$	−1.44748e6			−0.187426			−0.0937132	−	0.995599i	$-0.529874\pi$
$569$	−1.44748e6			−0.187426			−0.0937132	+	0.995599i	$0.529874\pi$
$570$		0			0
$571$	9.36363e6			1.20186			0.600930	−	0.799301i	$-0.294797\pi$
$571$	9.36363e6			1.20186			0.600930	+	0.799301i	$0.294797\pi$
$572$	1.21524e6	−	1.21524e6i	0.155300	−	0.155300i
$573$		0			0
$574$			8.16800e6i			1.03475i
$575$		0			0
$576$		0			0
$577$	3.33073e6	+	3.33073e6i	0.416486	+	0.416486i	0.883991	−	0.467505i	$-0.154847\pi$
$577$	3.33073e6	+	3.33073e6i	0.416486	+	0.416486i	−0.467505	+	0.883991i	$0.654847\pi$
$578$	−2.74694e6	−	2.74694e6i	−0.342003	−	0.342003i
$579$		0			0
$580$		0			0
$581$			2.20765e7i			2.71325i
$582$		0			0
$583$	−2.40612e6	+	2.40612e6i	−0.293188	+	0.293188i
$584$	−4.78417e6			−0.580463
$585$		0			0
$586$	341117.			0.0410354
$587$	−2.42159e6	+	2.42159e6i	−0.290072	+	0.290072i	−0.837109	−	0.547037i	$-0.815756\pi$
$587$	−2.42159e6	+	2.42159e6i	−0.290072	+	0.290072i	0.547037	+	0.837109i	$0.315756\pi$
$588$		0			0
$589$			3.86563e6i			0.459127i
$590$		0			0
$591$		0			0
$592$	−430151.	−	430151.i	−0.0504448	−	0.0504448i
$593$	−3.52839e6	−	3.52839e6i	−0.412041	−	0.412041i	0.470408	−	0.882449i	$-0.344107\pi$
$593$	−3.52839e6	−	3.52839e6i	−0.412041	−	0.412041i	−0.882449	+	0.470408i	$0.844107\pi$
$594$		0			0
$595$		0			0
$596$			1.19425e6i			0.137714i
$597$		0			0
$598$	9.15616e6	−	9.15616e6i	1.04703	−	1.04703i
$599$	8.61219e6			0.980723			0.490362	−	0.871519i	$-0.336865\pi$
$599$	8.61219e6			0.980723			0.490362	+	0.871519i	$0.336865\pi$
$600$		0			0
$601$	1.25949e7			1.42235			0.711176	−	0.703014i	$-0.248163\pi$
$601$	1.25949e7			1.42235			0.711176	+	0.703014i	$0.248163\pi$
$602$	−1.23305e7	+	1.23305e7i	−1.38672	+	1.38672i
$603$		0			0
$604$		−	1.61704e6i		−	0.180356i
$605$		0			0
$606$		0			0
$607$	−7.82263e6	−	7.82263e6i	−0.861749	−	0.861749i	0.129792	−	0.991541i	$-0.458569\pi$
$607$	−7.82263e6	−	7.82263e6i	−0.861749	−	0.861749i	−0.991541	+	0.129792i	$0.958569\pi$
$608$	851400.	+	851400.i	0.0934060	+	0.0934060i
$609$		0			0
$610$		0			0
$611$		−	916783.i		−	0.0993490i
$612$		0			0
$613$	−5.39579e6	+	5.39579e6i	−0.579967	+	0.579967i	−0.934894	−	0.354927i	$-0.884506\pi$
$613$	−5.39579e6	+	5.39579e6i	−0.579967	+	0.579967i	0.354927	+	0.934894i	$0.384506\pi$
$614$	−751329.			−0.0804284
$615$		0			0
$616$	2.11218e6			0.224275
$617$	7.83798e6	−	7.83798e6i	0.828880	−	0.828880i	−0.158482	−	0.987362i	$-0.550660\pi$
$617$	7.83798e6	−	7.83798e6i	0.828880	−	0.828880i	0.987362	+	0.158482i	$0.0506600\pi$
$618$		0			0
$619$		−	1.59378e7i		−	1.67187i	−0.548828	−	0.835935i	$-0.684926\pi$
$619$		−	1.59378e7i		−	1.67187i	0.548828	−	0.835935i	$-0.315074\pi$
$620$		0			0
$621$		0			0
$622$	−877593.	−	877593.i	−0.0909530	−	0.0909530i
$623$	2.12102e7	+	2.12102e7i	2.18939	+	2.18939i
$624$		0			0
$625$		0			0
$626$			5.03103e6i			0.513122i
$627$		0			0
$628$	2.87883e6	−	2.87883e6i	0.291284	−	0.291284i
$629$	−3.67443e6			−0.370308
$630$		0			0
$631$	1.00728e7			1.00711			0.503555	−	0.863963i	$-0.332025\pi$
$631$	1.00728e7			1.00711			0.503555	+	0.863963i	$0.332025\pi$
$632$	−740878.	+	740878.i	−0.0737826	+	0.0737826i
$633$		0			0
$634$			1.29927e7i			1.28374i
$635$		0			0
$636$		0			0
$637$	1.73257e7	+	1.73257e7i	1.69178	+	1.69178i
$638$	−775283.	−	775283.i	−0.0754064	−	0.0754064i
$639$		0			0
$640$		0			0
$641$		−	1.57715e7i		−	1.51610i	−0.652196	−	0.758050i	$-0.726152\pi$
$641$		−	1.57715e7i		−	1.51610i	0.652196	−	0.758050i	$-0.273848\pi$
$642$		0			0
$643$	516496.	−	516496.i	0.0492651	−	0.0492651i	−0.682045	−	0.731310i	$-0.738909\pi$
$643$	516496.	−	516496.i	0.0492651	−	0.0492651i	0.731310	+	0.682045i	$0.238909\pi$
$644$	1.59141e7			1.51206
$645$		0			0
$646$	7.27282e6			0.685680
$647$	−9.00208e6	+	9.00208e6i	−0.845439	+	0.845439i	−0.989560	−	0.144121i	$-0.953965\pi$
$647$	−9.00208e6	+	9.00208e6i	−0.845439	+	0.845439i	0.144121	+	0.989560i	$0.453965\pi$
$648$		0			0
$649$			140835.i			0.0131250i
$650$		0			0
$651$		0			0
$652$	−7.23032e6	−	7.23032e6i	−0.666099	−	0.666099i
$653$	1.28977e7	+	1.28977e7i	1.18367	+	1.18367i	0.978786	+	0.204884i	$0.0656816\pi$
$653$	1.28977e7	+	1.28977e7i	1.18367	+	1.18367i	0.204884	+	0.978786i	$0.434318\pi$
$654$		0			0
$655$		0			0
$656$		−	2.32954e6i		−	0.211354i
$657$		0			0
$658$	796721.	−	796721.i	0.0717367	−	0.0717367i
$659$	−4.33804e6			−0.389117			−0.194558	−	0.980891i	$-0.562327\pi$
$659$	−4.33804e6			−0.389117			−0.194558	+	0.980891i	$0.562327\pi$
$660$		0			0
$661$	−6.05066e6			−0.538641			−0.269321	−	0.963051i	$-0.586799\pi$
$661$	−6.05066e6			−0.538641			−0.269321	+	0.963051i	$0.586799\pi$
$662$	−7.75238e6	+	7.75238e6i	−0.687527	+	0.687527i
$663$		0			0
$664$		−	6.29630e6i		−	0.554199i
$665$		0			0
$666$		0			0
$667$	−5.84133e6	−	5.84133e6i	−0.508390	−	0.508390i
$668$	7.77174e6	+	7.77174e6i	0.673871	+	0.673871i
$669$		0			0
$670$		0			0
$671$			2.83897e6i			0.243419i
$672$		0			0
$673$	1.05580e7	−	1.05580e7i	0.898558	−	0.898558i	−0.0967510	−	0.995309i	$-0.530845\pi$
$673$	1.05580e7	−	1.05580e7i	0.898558	−	0.898558i	0.995309	+	0.0967510i	$0.0308451\pi$
$674$	1.57037e6			0.133154
$675$		0			0
$676$	−2.59383e6			−0.218311
$677$	1.56712e7	−	1.56712e7i	1.31410	−	1.31410i	0.395740	−	0.918362i	$-0.370488\pi$
$677$	1.56712e7	−	1.56712e7i	1.31410	−	1.31410i	0.918362	−	0.395740i	$-0.129512\pi$
$678$		0			0
$679$		−	2.64992e6i		−	0.220576i
$680$		0			0
$681$		0			0
$682$	−1.36755e6	−	1.36755e6i	−0.112586	−	0.112586i
$683$	7.35948e6	+	7.35948e6i	0.603664	+	0.603664i	0.941283	−	0.337619i	$-0.109622\pi$
$683$	7.35948e6	+	7.35948e6i	0.603664	+	0.603664i	−0.337619	+	0.941283i	$0.609622\pi$
$684$		0			0
$685$		0			0
$686$			1.50275e7i			1.21920i
$687$		0			0
$688$	3.51670e6	−	3.51670e6i	0.283246	−	0.283246i
$689$	1.68980e7			1.35609
$690$		0			0
$691$	3.57697e6			0.284984			0.142492	−	0.989796i	$-0.454488\pi$
$691$	3.57697e6			0.284984			0.142492	+	0.989796i	$0.454488\pi$
$692$	4.14120e6	−	4.14120e6i	0.328747	−	0.328747i
$693$		0			0
$694$		−	7.68030e6i		−	0.605313i
$695$		0			0
$696$		0			0
$697$	−9.94970e6	−	9.94970e6i	−0.775761	−	0.775761i
$698$	−1.11479e7	−	1.11479e7i	−0.866069	−	0.866069i
$699$		0			0
$700$		0			0
$701$			1.49735e6i			0.115087i	0.998343	+	0.0575437i	$0.0183269\pi$
$701$			1.49735e6i			0.115087i	−0.998343	+	0.0575437i	$0.981673\pi$
$702$		0			0
$703$	1.97574e6	−	1.97574e6i	0.150779	−	0.150779i
$704$	−602403.			−0.0458095
$705$		0			0
$706$	−1.71946e7			−1.29832
$707$	−2.49122e6	+	2.49122e6i	−0.187441	+	0.187441i
$708$		0			0
$709$			5.18203e6i			0.387154i	0.981085	+	0.193577i	$0.0620090\pi$
$709$			5.18203e6i			0.387154i	−0.981085	+	0.193577i	$0.937991\pi$
$710$		0			0
$711$		0			0
$712$	−6.04922e6	−	6.04922e6i	−0.447198	−	0.447198i
$713$	−1.03038e7	−	1.03038e7i	−0.759053	−	0.759053i
$714$		0			0
$715$		0			0
$716$			1.02930e7i			0.750345i
$717$		0			0
$718$	−4.59500e6	+	4.59500e6i	−0.332640	+	0.332640i
$719$	−6.23048e6			−0.449468			−0.224734	−	0.974420i	$-0.572151\pi$
$719$	−6.23048e6			−0.449468			−0.224734	+	0.974420i	$0.572151\pi$
$720$		0			0
$721$	2.27171e7			1.62748
$722$	3.09288e6	−	3.09288e6i	0.220810	−	0.220810i
$723$		0			0
$724$			4.06119e6i			0.287944i
$725$		0			0
$726$		0			0
$727$	1.01394e7	+	1.01394e7i	0.711503	+	0.711503i	0.966850	−	0.255347i	$-0.0821896\pi$
$727$	1.01394e7	+	1.01394e7i	0.711503	+	0.711503i	−0.255347	+	0.966850i	$0.582190\pi$
$728$	−7.41685e6	−	7.41685e6i	−0.518670	−	0.518670i
$729$		0			0
$730$		0			0
$731$		−	3.00403e7i		−	2.07927i
$732$		0			0
$733$	2.81889e6	−	2.81889e6i	0.193784	−	0.193784i	−0.603545	−	0.797329i	$-0.706246\pi$
$733$	2.81889e6	−	2.81889e6i	0.193784	−	0.193784i	0.797329	+	0.603545i	$0.206246\pi$
$734$	2.05882e6			0.141052
$735$		0			0
$736$	−4.53877e6			−0.308848
$737$	5.26277e6	−	5.26277e6i	0.356899	−	0.356899i
$738$		0			0
$739$			1.51555e7i			1.02085i	0.859924	+	0.510423i	$0.170511\pi$
$739$			1.51555e7i			1.02085i	−0.859924	+	0.510423i	$0.829489\pi$
$740$		0			0
$741$		0			0
$742$	1.46850e7	+	1.46850e7i	0.979187	+	0.979187i
$743$	−1.08129e7	−	1.08129e7i	−0.718570	−	0.718570i	0.249742	−	0.968312i	$-0.419654\pi$
$743$	−1.08129e7	−	1.08129e7i	−0.718570	−	0.718570i	−0.968312	+	0.249742i	$0.919654\pi$
$744$		0			0
$745$		0			0
$746$			8.74153e6i			0.575096i
$747$		0			0
$748$	−2.57292e6	+	2.57292e6i	−0.168140	+	0.168140i
$749$	−2.27784e7			−1.48360
$750$		0			0
$751$	4.14862e6			0.268413			0.134207	−	0.990953i	$-0.457151\pi$
$751$	4.14862e6			0.268413			0.134207	+	0.990953i	$0.457151\pi$
$752$	−227228.	+	227228.i	−0.0146527	+	0.0146527i
$753$		0			0
$754$			5.44475e6i			0.348778i
$755$		0			0
$756$		0			0
$757$	3.78607e6	+	3.78607e6i	0.240131	+	0.240131i	0.816904	−	0.576773i	$-0.195688\pi$
$757$	3.78607e6	+	3.78607e6i	0.240131	+	0.240131i	−0.576773	+	0.816904i	$0.695688\pi$
$758$	7.14623e6	+	7.14623e6i	0.451756	+	0.451756i
$759$		0			0
$760$		0			0
$761$			5.97949e6i			0.374285i	0.982333	+	0.187142i	$0.0599226\pi$
$761$			5.97949e6i			0.374285i	−0.982333	+	0.187142i	$0.940077\pi$
$762$		0			0
$763$	−3.40218e7	+	3.40218e7i	−2.11566	+	2.11566i
$764$	−7.24367e6			−0.448978
$765$		0			0
$766$	−1.02892e7			−0.633592
$767$	494536.	−	494536.i	0.0303535	−	0.0303535i
$768$		0			0
$769$		−	9.77362e6i		−	0.595991i	−0.954567	−	0.297995i	$-0.903682\pi$
$769$		−	9.77362e6i		−	0.595991i	0.954567	−	0.297995i	$-0.0963180\pi$
$770$		0			0
$771$		0			0
$772$	−7.19614e6	−	7.19614e6i	−0.434566	−	0.434566i
$773$	−1.78140e7	−	1.78140e7i	−1.07229	−	1.07229i	−0.997175	−	0.0751134i	$-0.976068\pi$
$773$	−1.78140e7	−	1.78140e7i	−1.07229	−	1.07229i	−0.0751134	−	0.997175i	$-0.523932\pi$
$774$		0			0
$775$		0			0
$776$			755767.i			0.0450540i
$777$		0			0
$778$	−8.96990e6	+	8.96990e6i	−0.531299	+	0.531299i
$779$	1.06999e7			0.631737
$780$		0			0
$781$	−3.82130e6			−0.224173
$782$	−1.93855e7	+	1.93855e7i	−1.13360	+	1.13360i
$783$		0			0
$784$		−	8.58849e6i		−	0.499030i
$785$		0			0
$786$		0			0
$787$	−9.48699e6	−	9.48699e6i	−0.545999	−	0.545999i	0.379282	−	0.925281i	$-0.376171\pi$
$787$	−9.48699e6	−	9.48699e6i	−0.545999	−	0.545999i	−0.925281	+	0.379282i	$0.876171\pi$
$788$	−2.27254e6	−	2.27254e6i	−0.130376	−	0.130376i
$789$		0			0
$790$		0			0
$791$		−	1.91038e7i		−	1.08562i
$792$		0			0
$793$	9.96891e6	−	9.96891e6i	0.562943	−	0.562943i
$794$	−2.14571e7			−1.20787
$795$		0			0
$796$	−1.15450e7			−0.645817
$797$	1.21012e7	−	1.21012e7i	0.674812	−	0.674812i	−0.284010	−	0.958821i	$-0.591665\pi$
$797$	1.21012e7	−	1.21012e7i	0.674812	−	0.674812i	0.958821	+	0.284010i	$0.0916648\pi$
$798$		0			0
$799$			1.94102e6i			0.107563i
$800$		0			0
$801$		0			0
$802$	−8.42085e6	−	8.42085e6i	−0.462296	−	0.462296i
$803$	7.77389e6	+	7.77389e6i	0.425451	+	0.425451i
$804$		0			0
$805$		0			0
$806$			9.60421e6i			0.520744i
$807$		0			0
$808$	710505.	−	710505.i	0.0382859	−	0.0382859i
$809$	−707664.			−0.0380151			−0.0190075	−	0.999819i	$-0.506051\pi$
$809$	−707664.			−0.0380151			−0.0190075	+	0.999819i	$0.506051\pi$
$810$		0			0
$811$	−2.78088e7			−1.48467			−0.742335	−	0.670029i	$-0.766282\pi$
$811$	−2.78088e7			−1.48467			−0.742335	+	0.670029i	$0.766282\pi$
$812$	−4.73170e6	+	4.73170e6i	−0.251842	+	0.251842i
$813$		0			0
$814$			1.39792e6i			0.0739472i
$815$		0			0
$816$		0			0
$817$	1.61527e7	+	1.61527e7i	0.846620	+	0.846620i
$818$	423050.	+	423050.i	0.0221059	+	0.0221059i
$819$		0			0
$820$		0			0
$821$		−	1.29079e7i		−	0.668343i	−0.942512	−	0.334171i	$-0.891544\pi$
$821$		−	1.29079e7i		−	0.668343i	0.942512	−	0.334171i	$-0.108456\pi$
$822$		0			0
$823$	−6.17626e6	+	6.17626e6i	−0.317853	+	0.317853i	−0.847942	−	0.530089i	$-0.822158\pi$
$823$	−6.17626e6	+	6.17626e6i	−0.317853	+	0.317853i	0.530089	+	0.847942i	$0.322158\pi$
$824$	−6.47901e6			−0.332423
$825$		0			0
$826$	859543.			0.0438346
$827$	−1.93860e7	+	1.93860e7i	−0.985656	+	0.985656i	−0.999899	−	0.0142426i	$-0.995466\pi$
$827$	−1.93860e7	+	1.93860e7i	−0.985656	+	0.985656i	0.0142426	+	0.999899i	$0.495466\pi$
$828$		0			0
$829$		−	1.85346e7i		−	0.936693i	−0.883545	−	0.468346i	$-0.844850\pi$
$829$		−	1.85346e7i		−	0.936693i	0.883545	−	0.468346i	$-0.155150\pi$
$830$		0			0
$831$		0			0
$832$	2.11531e6	+	2.11531e6i	0.105942	+	0.105942i
$833$	−3.66822e7	−	3.66822e7i	−1.83165	−	1.83165i
$834$		0			0
$835$		0			0
$836$		−	2.76691e6i		−	0.136924i
$837$		0			0
$838$	1.71422e7	−	1.71422e7i	0.843252	−	0.843252i
$839$	2.29948e6			0.112778			0.0563891	−	0.998409i	$-0.482041\pi$
$839$	2.29948e6			0.112778			0.0563891	+	0.998409i	$0.482041\pi$
$840$		0			0
$841$	−1.70376e7			−0.830650
$842$	1.01830e7	−	1.01830e7i	0.494989	−	0.494989i
$843$		0			0
$844$		−	4.03454e6i		−	0.194956i
$845$		0			0
$846$		0			0
$847$	2.21227e7	+	2.21227e7i	1.05957	+	1.05957i
$848$	−4.18823e6	−	4.18823e6i	−0.200005	−	0.200005i
$849$		0			0
$850$		0			0
$851$			1.05326e7i			0.498552i
$852$		0			0
$853$	6.39924e6	−	6.39924e6i	0.301131	−	0.301131i	−0.540325	−	0.841456i	$-0.681699\pi$
$853$	6.39924e6	−	6.39924e6i	0.301131	−	0.301131i	0.841456	+	0.540325i	$0.181699\pi$
$854$	1.73268e7			0.812967
$855$		0			0
$856$	6.49648e6			0.303035
$857$	2.04306e7	−	2.04306e7i	0.950232	−	0.950232i	−0.0485865	−	0.998819i	$-0.515472\pi$
$857$	2.04306e7	−	2.04306e7i	0.950232	−	0.950232i	0.998819	+	0.0485865i	$0.0154717\pi$
$858$		0			0
$859$		−	2.14389e7i		−	0.991335i	−0.868512	−	0.495668i	$-0.834923\pi$
$859$		−	2.14389e7i		−	0.991335i	0.868512	−	0.495668i	$-0.165077\pi$
$860$		0			0
$861$		0			0
$862$	−4.56449e6	−	4.56449e6i	−0.209230	−	0.209230i
$863$	1.26808e7	+	1.26808e7i	0.579587	+	0.579587i	0.934789	−	0.355203i	$-0.115588\pi$
$863$	1.26808e7	+	1.26808e7i	0.579587	+	0.579587i	−0.355203	+	0.934789i	$0.615588\pi$
$864$		0			0
$865$		0			0
$866$		−	2.65434e7i		−	1.20271i
$867$		0			0
$868$	−8.34645e6	+	8.34645e6i	−0.376013	+	0.376013i
$869$	2.40774e6			0.108158
$870$		0			0
$871$	−3.69600e7			−1.65077
$872$	9.70316e6	−	9.70316e6i	0.432137	−	0.432137i
$873$		0			0
$874$		−	2.08472e7i		−	0.923142i
$875$		0			0
$876$		0			0
$877$	1.90955e7	+	1.90955e7i	0.838361	+	0.838361i	0.988643	−	0.150282i	$-0.0480181\pi$
$877$	1.90955e7	+	1.90955e7i	0.838361	+	0.838361i	−0.150282	+	0.988643i	$0.548018\pi$
$878$	−6.56844e6	−	6.56844e6i	−0.287558	−	0.287558i
$879$		0			0
$880$		0			0
$881$		−	4.44332e7i		−	1.92871i	−0.264605	−	0.964357i	$-0.585241\pi$
$881$		−	4.44332e7i		−	1.92871i	0.264605	−	0.964357i	$-0.414759\pi$
$882$		0			0
$883$	4.91929e6	−	4.91929e6i	0.212325	−	0.212325i	−0.592929	−	0.805254i	$-0.702029\pi$
$883$	4.91929e6	−	4.91929e6i	0.212325	−	0.212325i	0.805254	+	0.592929i	$0.202029\pi$
$884$	1.80694e7			0.777702
$885$		0			0
$886$	−1.94270e7			−0.831424
$887$	2.51992e7	−	2.51992e7i	1.07542	−	1.07542i	0.0785053	−	0.996914i	$-0.474985\pi$
$887$	2.51992e7	−	2.51992e7i	1.07542	−	1.07542i	0.996914	−	0.0785053i	$-0.0250147\pi$
$888$		0			0
$889$		−	4.75958e7i		−	2.01983i
$890$		0			0
$891$		0			0
$892$	5.34477e6	+	5.34477e6i	0.224914	+	0.224914i
$893$	−1.04369e6	−	1.04369e6i	−0.0437967	−	0.0437967i
$894$		0			0
$895$		0			0
$896$			3.67658e6i			0.152994i
$897$		0			0
$898$	−431666.	+	431666.i	−0.0178631	+	0.0178631i
$899$	6.12717e6			0.252849
$900$		0			0
$901$	−3.57767e7			−1.46821
$902$	−3.78532e6	+	3.78532e6i	−0.154913	+	0.154913i
$903$		0			0
$904$			5.44848e6i			0.221745i
$905$		0			0
$906$		0			0
$907$	1.60527e6	+	1.60527e6i	0.0647933	+	0.0647933i	0.738761	−	0.673968i	$-0.235411\pi$
$907$	1.60527e6	+	1.60527e6i	0.0647933	+	0.0647933i	−0.673968	+	0.738761i	$0.735411\pi$
$908$	687843.	+	687843.i	0.0276869	+	0.0276869i
$909$		0			0
$910$		0			0
$911$			1.28125e7i			0.511493i	0.966744	+	0.255746i	$0.0823212\pi$
$911$			1.28125e7i			0.511493i	−0.966744	+	0.255746i	$0.917679\pi$
$912$		0			0
$913$	−1.02310e7	+	1.02310e7i	−0.406201	+	0.406201i
$914$	1.45189e7			0.574870
$915$		0			0
$916$	2.10521e7			0.829006
$917$	−4.70076e7	+	4.70076e7i	−1.84606	+	1.84606i
$918$		0			0
$919$			2.73416e7i			1.06791i	0.845512	+	0.533956i	$0.179295\pi$
$919$			2.73416e7i			1.06791i	−0.845512	+	0.533956i	$0.820705\pi$
$920$		0			0
$921$		0			0
$922$	1.20190e7	+	1.20190e7i	0.465629	+	0.465629i
$923$	1.34183e7	+	1.34183e7i	0.518435	+	0.518435i
$924$		0			0
$925$		0			0
$926$		−	2.23936e7i		−	0.858215i
$927$		0			0
$928$	1.34950e6	−	1.34950e6i	0.0514402	−	0.0514402i
$929$	−2.92776e7			−1.11300			−0.556502	−	0.830846i	$-0.687857\pi$
$929$	−2.92776e7			−1.11300			−0.556502	+	0.830846i	$0.687857\pi$
$930$		0			0
$931$	3.94480e7			1.49160
$932$	−1.34768e7	+	1.34768e7i	−0.508216	+	0.508216i
$933$		0			0
$934$			1.39397e6i			0.0522862i
$935$		0			0
$936$		0			0
$937$	−1.60245e7	−	1.60245e7i	−0.596260	−	0.596260i	0.343055	−	0.939315i	$-0.388538\pi$
$937$	−1.60245e7	−	1.60245e7i	−0.596260	−	0.596260i	−0.939315	+	0.343055i	$0.888538\pi$
$938$	−3.21197e7	−	3.21197e7i	−1.19197	−	1.19197i
$939$		0			0
$940$		0			0
$941$			3.40359e7i			1.25304i	0.779407	+	0.626518i	$0.215521\pi$
$941$			3.40359e7i			1.25304i	−0.779407	+	0.626518i	$0.784479\pi$
$942$		0			0
$943$	−2.85203e7	+	2.85203e7i	−1.04442	+	1.04442i
$944$	−245145.			−0.00895350
$945$		0			0
$946$	−1.14287e7			−0.415212
$947$	−1.66968e7	+	1.66968e7i	−0.605003	+	0.605003i	−0.941636	−	0.336633i	$-0.890712\pi$
$947$	−1.66968e7	+	1.66968e7i	−0.605003	+	0.605003i	0.336633	+	0.941636i	$0.390712\pi$
$948$		0			0
$949$		−	5.45954e7i		−	1.96784i
$950$		0			0
$951$		0			0
$952$	1.57030e7	+	1.57030e7i	0.561554	+	0.561554i
$953$	−8.06226e6	−	8.06226e6i	−0.287558	−	0.287558i	0.548556	−	0.836114i	$-0.315178\pi$
$953$	−8.06226e6	−	8.06226e6i	−0.287558	−	0.287558i	−0.836114	+	0.548556i	$0.815178\pi$
$954$		0			0
$955$		0			0
$956$		−	1.70573e7i		−	0.603624i
$957$		0			0
$958$	−881117.	+	881117.i	−0.0310184	+	0.0310184i
$959$	8.88389e7			3.11930
$960$		0			0
$961$	−1.78212e7			−0.622484
$962$	4.90874e6	−	4.90874e6i	0.171014	−	0.171014i
$963$		0			0
$964$		−	816940.i		−	0.0283138i
$965$		0			0
$966$		0			0
$967$	−3.67667e7	−	3.67667e7i	−1.26441	−	1.26441i	−0.948932	−	0.315481i	$-0.897834\pi$
$967$	−3.67667e7	−	3.67667e7i	−1.26441	−	1.26441i	−0.315481	−	0.948932i	$-0.602166\pi$
$968$	−6.30948e6	−	6.30948e6i	−0.216424	−	0.216424i
$969$		0			0
$970$		0			0
$971$		−	1.24102e7i		−	0.422408i	−0.977442	−	0.211204i	$-0.932262\pi$
$971$		−	1.24102e7i		−	0.422408i	0.977442	−	0.211204i	$-0.0677384\pi$
$972$		0			0
$973$	−4.30879e7	+	4.30879e7i	−1.45906	+	1.45906i
$974$	−6.36506e6			−0.214983
$975$		0			0
$976$	−4.94166e6			−0.166054
$977$	1.37859e7	−	1.37859e7i	0.462059	−	0.462059i	−0.437271	−	0.899330i	$-0.644055\pi$
$977$	1.37859e7	−	1.37859e7i	0.462059	−	0.462059i	0.899330	+	0.437271i	$0.144055\pi$
$978$		0			0
$979$			1.96590e7i			0.655548i
$980$		0			0
$981$		0			0
$982$	−1.83448e7	−	1.83448e7i	−0.607062	−	0.607062i
$983$	−1.44300e7	−	1.44300e7i	−0.476303	−	0.476303i	0.427644	−	0.903947i	$-0.359344\pi$
$983$	−1.44300e7	−	1.44300e7i	−0.476303	−	0.476303i	−0.903947	+	0.427644i	$0.859344\pi$
$984$		0			0
$985$		0			0
$986$		−	1.15277e7i		−	0.377615i
$987$		0			0
$988$	−9.71590e6	+	9.71590e6i	−0.316658	+	0.316658i
$989$	−8.61091e7			−2.79936
$990$		0			0
$991$	−3.26137e6			−0.105491			−0.0527455	−	0.998608i	$-0.516797\pi$
$991$	−3.26137e6			−0.105491			−0.0527455	+	0.998608i	$0.516797\pi$
$992$	2.38044e6	−	2.38044e6i	0.0768029	−	0.0768029i
$993$		0			0
$994$			2.33222e7i			0.748691i
$995$		0			0
$996$		0			0
$997$	−1.93241e7	−	1.93241e7i	−0.615689	−	0.615689i	0.328733	−	0.944423i	$-0.393378\pi$
$997$	−1.93241e7	−	1.93241e7i	−0.615689	−	0.615689i	−0.944423	+	0.328733i	$0.893378\pi$
$998$	1.67177e7	+	1.67177e7i	0.531311	+	0.531311i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.6.f.g.107.8	yes	16
3.2	odd	2	inner	450.6.f.g.107.4	yes	16
5.2	odd	4		450.6.f.f.143.5	yes	16
5.3	odd	4	inner	450.6.f.g.143.4	yes	16
5.4	even	2		450.6.f.f.107.1	✓	16
15.2	even	4		450.6.f.f.143.1	yes	16
15.8	even	4	inner	450.6.f.g.143.8	yes	16
15.14	odd	2		450.6.f.f.107.5	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.6.f.f.107.1	✓	16	5.4	even	2
450.6.f.f.107.5	yes	16	15.14	odd	2
450.6.f.f.143.1	yes	16	15.2	even	4
450.6.f.f.143.5	yes	16	5.2	odd	4
450.6.f.g.107.4	yes	16	3.2	odd	2	inner
450.6.f.g.107.8	yes	16	1.1	even	1	trivial
450.6.f.g.143.4	yes	16	5.3	odd	4	inner
450.6.f.g.143.8	yes	16	15.8	even	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	450.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.82843 - 2.82843i) q^{2} -16.0000i q^{4} +(158.675 + 158.675i) q^{7} +(-45.2548 - 45.2548i) q^{8} +O(q^{10})\)	\(q+(2.82843 - 2.82843i) q^{2} -16.0000i q^{4} +(158.675 + 158.675i) q^{7} +(-45.2548 - 45.2548i) q^{8} +147.071i q^{11} +(516.434 - 516.434i) q^{13} +897.604 q^{14} -256.000 q^{16} +(-1093.40 + 1093.40i) q^{17} -1175.84i q^{19} +(415.980 + 415.980i) q^{22} +(3134.18 + 3134.18i) q^{23} -2921.39i q^{26} +(2538.81 - 2538.81i) q^{28} -1863.75 q^{29} -3287.55 q^{31} +(-724.077 + 724.077i) q^{32} +6185.20i q^{34} +(1680.28 + 1680.28i) q^{37} +(-3325.78 - 3325.78i) q^{38} +9099.78i q^{41} +(-13737.1 + 13737.1i) q^{43} +2353.14 q^{44} +17729.6 q^{46} +(887.609 - 887.609i) q^{47} +33548.8i q^{49} +(-8262.94 - 8262.94i) q^{52} +(16360.3 + 16360.3i) q^{53} -14361.7i q^{56} +(-5271.49 + 5271.49i) q^{58} +957.597 q^{59} +19303.4 q^{61} +(-9298.59 + 9298.59i) q^{62} +4096.00i q^{64} +(-35783.9 - 35783.9i) q^{67} +(17494.4 + 17494.4i) q^{68} +25982.7i q^{71} +(52858.1 - 52858.1i) q^{73} +9505.07 q^{74} -18813.5 q^{76} +(-23336.6 + 23336.6i) q^{77} -16371.2i q^{79} +(25738.1 + 25738.1i) q^{82} +(69565.0 + 69565.0i) q^{83} +77708.8i q^{86} +(6655.67 - 6655.67i) q^{88} +133670. q^{89} +163891. q^{91} +(50146.9 - 50146.9i) q^{92} -5021.08i q^{94} +(-8350.12 - 8350.12i) q^{97} +(94890.3 + 94890.3i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 528 q^{7}+O(q^{10}) \) 16 * q + 528 * q^7	\( 16 q + 528 q^{7} - 192 q^{13} - 4096 q^{16} + 2688 q^{22} + 8448 q^{28} + 13024 q^{31} + 47328 q^{37} + 55440 q^{43} + 44544 q^{46} + 3072 q^{52} + 101184 q^{58} + 28400 q^{61} - 242256 q^{67} + 430944 q^{73} - 7168 q^{76} - 158208 q^{82} + 43008 q^{88} - 185472 q^{91} - 457152 q^{97}+O(q^{100}) \) 16 * q + 528 * q^7 - 192 * q^13 - 4096 * q^16 + 2688 * q^22 + 8448 * q^28 + 13024 * q^31 + 47328 * q^37 + 55440 * q^43 + 44544 * q^46 + 3072 * q^52 + 101184 * q^58 + 28400 * q^61 - 242256 * q^67 + 430944 * q^73 - 7168 * q^76 - 158208 * q^82 + 43008 * q^88 - 185472 * q^91 - 457152 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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