LMFDB - Embedded newform 450.6.f.e.107.5

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:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 3457x^{8} + 2937456x^{4} + 12960000 $ x^12 + 3457x^8 + 2937456x^4 + 12960000
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{4}\cdot 5^{8} $
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			107.5
Root			$4.38999 - 4.38999i$ of defining polynomial
Character	$\chi$	$=$	450.107
Dual form			450.6.f.e.143.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.82843 - 2.82843i) q^{2} -16.0000i q^{4} +(9.67825 + 9.67825i) q^{7} +(-45.2548 - 45.2548i) q^{8} +O(q^{10})$	\(q+(2.82843 - 2.82843i) q^{2} -16.0000i q^{4} +(9.67825 + 9.67825i) q^{7} +(-45.2548 - 45.2548i) q^{8} -479.171i q^{11} +(177.573 - 177.573i) q^{13} +54.7484 q^{14} -256.000 q^{16} +(-1380.00 + 1380.00i) q^{17} +121.286i q^{19} +(-1355.30 - 1355.30i) q^{22} +(-1694.88 - 1694.88i) q^{23} -1004.51i q^{26} +(154.852 - 154.852i) q^{28} +7043.95 q^{29} +443.600 q^{31} +(-724.077 + 724.077i) q^{32} +7806.47i q^{34} +(-4792.67 - 4792.67i) q^{37} +(343.047 + 343.047i) q^{38} +1025.44i q^{41} +(-6318.40 + 6318.40i) q^{43} -7666.73 q^{44} -9587.68 q^{46} +(-12832.1 + 12832.1i) q^{47} -16619.7i q^{49} +(-2841.17 - 2841.17i) q^{52} +(-16707.1 - 16707.1i) q^{53} -875.975i q^{56} +(19923.3 - 19923.3i) q^{58} -41366.2 q^{59} -35702.6 q^{61} +(1254.69 - 1254.69i) q^{62} +4096.00i q^{64} +(40179.3 + 40179.3i) q^{67} +(22080.0 + 22080.0i) q^{68} +32974.0i q^{71} +(-20366.1 + 20366.1i) q^{73} -27111.4 q^{74} +1940.57 q^{76} +(4637.53 - 4637.53i) q^{77} +27180.7i q^{79} +(2900.40 + 2900.40i) q^{82} +(-32286.4 - 32286.4i) q^{83} +35742.3i q^{86} +(-21684.8 + 21684.8i) q^{88} -64897.3 q^{89} +3437.20 q^{91} +(-27118.1 + 27118.1i) q^{92} +72589.4i q^{94} +(-22306.4 - 22306.4i) q^{97} +(-47007.5 - 47007.5i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 144 q^{7}+O(q^{10}) $ 12 * q - 144 * q^7	$ 12 q - 144 q^{7} + 276 q^{13} - 3072 q^{16} - 384 q^{22} - 2304 q^{28} - 58512 q^{31} - 25764 q^{37} - 16080 q^{43} - 60672 q^{46} - 4416 q^{52} - 23952 q^{58} - 145200 q^{61} - 33552 q^{67} + 158988 q^{73} - 86016 q^{76} + 75024 q^{82} - 6144 q^{88} - 465024 q^{91} + 631116 q^{97}+O(q^{100}) $ 12 * q - 144 * q^7 + 276 * q^13 - 3072 * q^16 - 384 * q^22 - 2304 * q^28 - 58512 * q^31 - 25764 * q^37 - 16080 * q^43 - 60672 * q^46 - 4416 * q^52 - 23952 * q^58 - 145200 * q^61 - 33552 * q^67 + 158988 * q^73 - 86016 * q^76 + 75024 * q^82 - 6144 * q^88 - 465024 * q^91 + 631116 * 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
$7$	9.67825	+	9.67825i	0.0746538	+	0.0746538i	0.743448	−	0.668794i	$-0.233189\pi$
$7$	9.67825	+	9.67825i	0.0746538	+	0.0746538i	−0.668794	+	0.743448i	$0.733189\pi$
$8$	−45.2548	−	45.2548i	−0.250000	−	0.250000i
$9$		0			0
$10$		0			0
$11$		−	479.171i		−	1.19401i	−0.802237	−	0.597006i	$-0.796357\pi$
$11$		−	479.171i		−	1.19401i	0.802237	−	0.597006i	$-0.203643\pi$
$12$		0			0
$13$	177.573	−	177.573i	0.291420	−	0.291420i	−0.546221	−	0.837641i	$-0.683934\pi$
$13$	177.573	−	177.573i	0.291420	−	0.291420i	0.837641	+	0.546221i	$0.183934\pi$
$14$	54.7484			0.0746538
$15$		0			0
$16$	−256.000			−0.250000
$17$	−1380.00	+	1380.00i	−1.15813	+	1.15813i	−0.173254	+	0.984877i	$0.555428\pi$
$17$	−1380.00	+	1380.00i	−1.15813	+	1.15813i	−0.984877	+	0.173254i	$0.944572\pi$
$18$		0			0
$19$			121.286i			0.0770770i	0.999257	+	0.0385385i	$0.0122702\pi$
$19$			121.286i			0.0770770i	−0.999257	+	0.0385385i	$0.987730\pi$
$20$		0			0
$21$		0			0
$22$	−1355.30	−	1355.30i	−0.597006	−	0.597006i
$23$	−1694.88	−	1694.88i	−0.668065	−	0.668065i	0.289203	−	0.957268i	$-0.406610\pi$
$23$	−1694.88	−	1694.88i	−0.668065	−	0.668065i	−0.957268	+	0.289203i	$0.906610\pi$
$24$		0			0
$25$		0			0
$26$		−	1004.51i		−	0.291420i
$27$		0			0
$28$	154.852	−	154.852i	0.0373269	−	0.0373269i
$29$	7043.95			1.55533			0.777663	−	0.628682i	$-0.216405\pi$
$29$	7043.95			1.55533			0.777663	+	0.628682i	$0.216405\pi$
$30$		0			0
$31$	443.600			0.0829063			0.0414531	−	0.999140i	$-0.486801\pi$
$31$	443.600			0.0829063			0.0414531	+	0.999140i	$0.486801\pi$
$32$	−724.077	+	724.077i	−0.125000	+	0.125000i
$33$		0			0
$34$			7806.47i			1.15813i
$35$		0			0
$36$		0			0
$37$	−4792.67	−	4792.67i	−0.575537	−	0.575537i	0.358133	−	0.933670i	$-0.383413\pi$
$37$	−4792.67	−	4792.67i	−0.575537	−	0.575537i	−0.933670	+	0.358133i	$0.883413\pi$
$38$	343.047	+	343.047i	0.0385385	+	0.0385385i
$39$		0			0
$40$		0			0
$41$			1025.44i			0.0952692i	0.998865	+	0.0476346i	$0.0151683\pi$
$41$			1025.44i			0.0952692i	−0.998865	+	0.0476346i	$0.984832\pi$
$42$		0			0
$43$	−6318.40	+	6318.40i	−0.521118	+	0.521118i	−0.917909	−	0.396791i	$-0.870124\pi$
$43$	−6318.40	+	6318.40i	−0.521118	+	0.521118i	0.396791	+	0.917909i	$0.370124\pi$
$44$	−7666.73			−0.597006
$45$		0			0
$46$	−9587.68			−0.668065
$47$	−12832.1	+	12832.1i	−0.847332	+	0.847332i	−0.989799	−	0.142468i	$-0.954496\pi$
$47$	−12832.1	+	12832.1i	−0.847332	+	0.847332i	0.142468	+	0.989799i	$0.454496\pi$
$48$		0			0
$49$		−	16619.7i		−	0.988854i
$50$		0			0
$51$		0			0
$52$	−2841.17	−	2841.17i	−0.145710	−	0.145710i
$53$	−16707.1	−	16707.1i	−0.816980	−	0.816980i	0.168689	−	0.985669i	$-0.446047\pi$
$53$	−16707.1	−	16707.1i	−0.816980	−	0.816980i	−0.985669	+	0.168689i	$0.946047\pi$
$54$		0			0
$55$		0			0
$56$		−	875.975i		−	0.0373269i
$57$		0			0
$58$	19923.3	−	19923.3i	0.777663	−	0.777663i
$59$	−41366.2			−1.54709			−0.773545	−	0.633741i	$-0.781518\pi$
$59$	−41366.2			−1.54709			−0.773545	+	0.633741i	$0.781518\pi$
$60$		0			0
$61$	−35702.6			−1.22850			−0.614250	−	0.789111i	$-0.710541\pi$
$61$	−35702.6			−1.22850			−0.614250	+	0.789111i	$0.710541\pi$
$62$	1254.69	−	1254.69i	0.0414531	−	0.0414531i
$63$		0			0
$64$			4096.00i			0.125000i
$65$		0			0
$66$		0			0
$67$	40179.3	+	40179.3i	1.09349	+	1.09349i	0.995153	+	0.0983373i	$0.0313524\pi$
$67$	40179.3	+	40179.3i	1.09349	+	1.09349i	0.0983373	+	0.995153i	$0.468648\pi$
$68$	22080.0	+	22080.0i	0.579065	+	0.579065i
$69$		0			0
$70$		0			0
$71$			32974.0i			0.776293i	0.921598	+	0.388147i	$0.126885\pi$
$71$			32974.0i			0.776293i	−0.921598	+	0.388147i	$0.873115\pi$
$72$		0			0
$73$	−20366.1	+	20366.1i	−0.447302	+	0.447302i	−0.894457	−	0.447154i	$-0.852437\pi$
$73$	−20366.1	+	20366.1i	−0.447302	+	0.447302i	0.447154	+	0.894457i	$0.352437\pi$
$74$	−27111.4			−0.575537
$75$		0			0
$76$	1940.57			0.0385385
$77$	4637.53	−	4637.53i	0.0891374	−	0.0891374i
$78$		0			0
$79$			27180.7i			0.489997i	0.969523	+	0.244999i	$0.0787875\pi$
$79$			27180.7i			0.489997i	−0.969523	+	0.244999i	$0.921212\pi$
$80$		0			0
$81$		0			0
$82$	2900.40	+	2900.40i	0.0476346	+	0.0476346i
$83$	−32286.4	−	32286.4i	−0.514428	−	0.514428i	0.401452	−	0.915880i	$-0.368506\pi$
$83$	−32286.4	−	32286.4i	−0.514428	−	0.514428i	−0.915880	+	0.401452i	$0.868506\pi$
$84$		0			0
$85$		0			0
$86$			35742.3i			0.521118i
$87$		0			0
$88$	−21684.8	+	21684.8i	−0.298503	+	0.298503i
$89$	−64897.3			−0.868464			−0.434232	−	0.900801i	$-0.642980\pi$
$89$	−64897.3			−0.868464			−0.434232	+	0.900801i	$0.642980\pi$
$90$		0			0
$91$	3437.20			0.0435112
$92$	−27118.1	+	27118.1i	−0.334033	+	0.334033i
$93$		0			0
$94$			72589.4i			0.847332i
$95$		0			0
$96$		0			0
$97$	−22306.4	−	22306.4i	−0.240713	−	0.240713i	0.576432	−	0.817145i	$-0.304445\pi$
$97$	−22306.4	−	22306.4i	−0.240713	−	0.240713i	−0.817145	+	0.576432i	$0.804445\pi$
$98$	−47007.5	−	47007.5i	−0.494427	−	0.494427i
$99$		0			0
$100$		0			0
$101$		−	43450.0i		−	0.423825i	−0.977289	−	0.211913i	$-0.932031\pi$
$101$		−	43450.0i		−	0.423825i	0.977289	−	0.211913i	$-0.0679692\pi$
$102$		0			0
$103$	60897.1	−	60897.1i	0.565592	−	0.565592i	−0.365298	−	0.930891i	$-0.619033\pi$
$103$	60897.1	−	60897.1i	0.565592	−	0.565592i	0.930891	+	0.365298i	$0.119033\pi$
$104$	−16072.1			−0.145710
$105$		0			0
$106$	−94509.6			−0.816980
$107$	−163755.	+	163755.i	−1.38273	+	1.38273i	−0.542983	+	0.839744i	$0.682705\pi$
$107$	−163755.	+	163755.i	−1.38273	+	1.38273i	−0.839744	+	0.542983i	$0.817295\pi$
$108$		0			0
$109$			196077.i			1.58074i	0.612629	+	0.790371i	$0.290112\pi$
$109$			196077.i			1.58074i	−0.612629	+	0.790371i	$0.709888\pi$
$110$		0			0
$111$		0			0
$112$	−2477.63	−	2477.63i	−0.0186634	−	0.0186634i
$113$	−6940.19	−	6940.19i	−0.0511299	−	0.0511299i	0.681080	−	0.732209i	$-0.261511\pi$
$113$	−6940.19	−	6940.19i	−0.0511299	−	0.0511299i	−0.732209	+	0.681080i	$0.761511\pi$
$114$		0			0
$115$		0			0
$116$		−	112703.i		−	0.777663i
$117$		0			0
$118$	−117001.	+	117001.i	−0.773545	+	0.773545i
$119$	−26712.0			−0.172918
$120$		0			0
$121$	−68553.5			−0.425663
$122$	−100982.	+	100982.i	−0.614250	+	0.614250i
$123$		0			0
$124$		−	7097.60i		−	0.0414531i
$125$		0			0
$126$		0			0
$127$	138770.	+	138770.i	0.763458	+	0.763458i	0.976946	−	0.213488i	$-0.0684824\pi$
$127$	138770.	+	138770.i	0.763458	+	0.763458i	−0.213488	+	0.976946i	$0.568482\pi$
$128$	11585.2	+	11585.2i	0.0625000	+	0.0625000i
$129$		0			0
$130$		0			0
$131$		−	377088.i		−	1.91984i	−0.280275	−	0.959920i	$-0.590426\pi$
$131$		−	377088.i		−	1.91984i	0.280275	−	0.959920i	$-0.409574\pi$
$132$		0			0
$133$	−1173.83	+	1173.83i	−0.00575409	+	0.00575409i
$134$	227288.			1.09349
$135$		0			0
$136$	124904.			0.579065
$137$	96793.5	−	96793.5i	0.440600	−	0.440600i	−0.451613	−	0.892214i	$-0.649151\pi$
$137$	96793.5	−	96793.5i	0.440600	−	0.440600i	0.892214	+	0.451613i	$0.149151\pi$
$138$		0			0
$139$			338833.i			1.48747i	0.668475	+	0.743735i	$0.266948\pi$
$139$			338833.i			1.48747i	−0.668475	+	0.743735i	$0.733052\pi$
$140$		0			0
$141$		0			0
$142$	93264.6	+	93264.6i	0.388147	+	0.388147i
$143$	−85087.9	−	85087.9i	−0.347959	−	0.347959i
$144$		0			0
$145$		0			0
$146$			115208.i			0.447302i
$147$		0			0
$148$	−76682.8	+	76682.8i	−0.287769	+	0.287769i
$149$	−421623.			−1.55582			−0.777909	−	0.628377i	$-0.783720\pi$
$149$	−421623.			−1.55582			−0.777909	+	0.628377i	$0.783720\pi$
$150$		0			0
$151$	−529841.			−1.89105			−0.945525	−	0.325550i	$-0.894451\pi$
$151$	−529841.			−1.89105			−0.945525	+	0.325550i	$0.894451\pi$
$152$	5488.76	−	5488.76i	0.0192693	−	0.0192693i
$153$		0			0
$154$		−	26233.8i		−	0.0891374i
$155$		0			0
$156$		0			0
$157$	110529.	+	110529.i	0.357871	+	0.357871i	0.863028	−	0.505157i	$-0.168565\pi$
$157$	110529.	+	110529.i	0.357871	+	0.357871i	−0.505157	+	0.863028i	$0.668565\pi$
$158$	76878.8	+	76878.8i	0.244999	+	0.244999i
$159$		0			0
$160$		0			0
$161$		−	32806.9i		−	0.0997472i
$162$		0			0
$163$	−245678.	+	245678.i	−0.724265	+	0.724265i	−0.969471	−	0.245206i	$-0.921144\pi$
$163$	−245678.	+	245678.i	−0.724265	+	0.724265i	0.245206	+	0.969471i	$0.421144\pi$
$164$	16407.1			0.0476346
$165$		0			0
$166$	−182639.			−0.514428
$167$	349737.	−	349737.i	0.970400	−	0.970400i	−0.0291739	−	0.999574i	$-0.509288\pi$
$167$	349737.	−	349737.i	0.970400	−	0.970400i	0.999574	+	0.0291739i	$0.00928767\pi$
$168$		0			0
$169$			308228.i			0.830149i
$170$		0			0
$171$		0			0
$172$	101094.	+	101094.i	0.260559	+	0.260559i
$173$	−143942.	−	143942.i	−0.365655	−	0.365655i	0.500235	−	0.865890i	$-0.333247\pi$
$173$	−143942.	−	143942.i	−0.365655	−	0.365655i	−0.865890	+	0.500235i	$0.833247\pi$
$174$		0			0
$175$		0			0
$176$			122668.i			0.298503i
$177$		0			0
$178$	−183557.	+	183557.i	−0.434232	+	0.434232i
$179$	−583232.			−1.36053			−0.680267	−	0.732965i	$-0.738136\pi$
$179$	−583232.			−1.36053			−0.680267	+	0.732965i	$0.738136\pi$
$180$		0			0
$181$	753570.			1.70973			0.854864	−	0.518853i	$-0.173641\pi$
$181$	753570.			1.70973			0.854864	+	0.518853i	$0.173641\pi$
$182$	9721.86	−	9721.86i	0.0217556	−	0.0217556i
$183$		0			0
$184$			153403.i			0.334033i
$185$		0			0
$186$		0			0
$187$	661257.	+	661257.i	1.38282	+	1.38282i
$188$	205314.	+	205314.i	0.423666	+	0.423666i
$189$		0			0
$190$		0			0
$191$		−	511363.i		−	1.01425i	−0.861872	−	0.507126i	$-0.830708\pi$
$191$		−	511363.i		−	1.01425i	0.861872	−	0.507126i	$-0.169292\pi$
$192$		0			0
$193$	−9207.24	+	9207.24i	−0.0177925	+	0.0177925i	−0.715947	−	0.698155i	$-0.754005\pi$
$193$	−9207.24	+	9207.24i	−0.0177925	+	0.0177925i	0.698155	+	0.715947i	$0.254005\pi$
$194$	−126184.			−0.240713
$195$		0			0
$196$	−265915.			−0.494427
$197$	227109.	−	227109.i	0.416935	−	0.416935i	−0.467211	−	0.884146i	$-0.654741\pi$
$197$	227109.	−	227109.i	0.416935	−	0.416935i	0.884146	+	0.467211i	$0.154741\pi$
$198$		0			0
$199$		−	738531.i		−	1.32201i	−0.750380	−	0.661007i	$-0.770130\pi$
$199$		−	738531.i		−	1.32201i	0.750380	−	0.661007i	$-0.229870\pi$
$200$		0			0
$201$		0			0
$202$	−122895.	−	122895.i	−0.211913	−	0.211913i
$203$	68173.1	+	68173.1i	0.116111	+	0.116111i
$204$		0			0
$205$		0			0
$206$		−	344486.i		−	0.565592i
$207$		0			0
$208$	−45458.8	+	45458.8i	−0.0728550	+	0.0728550i
$209$	58116.5			0.0920309
$210$		0			0
$211$	−579506.			−0.896091			−0.448045	−	0.894011i	$-0.647880\pi$
$211$	−579506.			−0.896091			−0.448045	+	0.894011i	$0.647880\pi$
$212$	−267314.	+	267314.i	−0.408490	+	0.408490i
$213$		0			0
$214$			926341.i			1.38273i
$215$		0			0
$216$		0			0
$217$	4293.27	+	4293.27i	0.00618926	+	0.00618926i
$218$	554590.	+	554590.i	0.790371	+	0.790371i
$219$		0			0
$220$		0			0
$221$			490103.i			0.675005i
$222$		0			0
$223$	768537.	−	768537.i	1.03491	−	1.03491i	0.0355417	−	0.999368i	$-0.488684\pi$
$223$	768537.	−	768537.i	1.03491	−	1.03491i	0.999368	−	0.0355417i	$-0.0113157\pi$
$224$	−14015.6			−0.0186634
$225$		0			0
$226$	−39259.6			−0.0511299
$227$	−701571.	+	701571.i	−0.903664	+	0.903664i	−0.995751	−	0.0920868i	$-0.970646\pi$
$227$	−701571.	+	701571.i	−0.903664	+	0.903664i	0.0920868	+	0.995751i	$0.470646\pi$
$228$		0			0
$229$		−	34830.3i		−	0.0438902i	−0.999759	−	0.0219451i	$-0.993014\pi$
$229$		−	34830.3i		−	0.0438902i	0.999759	−	0.0219451i	$-0.00698591\pi$
$230$		0			0
$231$		0			0
$232$	−318773.	−	318773.i	−0.388831	−	0.388831i
$233$	−803689.	−	803689.i	−0.969836	−	0.969836i	0.0297224	−	0.999558i	$-0.490538\pi$
$233$	−803689.	−	803689.i	−0.969836	−	0.969836i	−0.999558	+	0.0297224i	$0.990538\pi$
$234$		0			0
$235$		0			0
$236$			661859.i			0.773545i
$237$		0			0
$238$	−75553.0	+	75553.0i	−0.0864588	+	0.0864588i
$239$	1.11216e6			1.25943			0.629715	−	0.776826i	$-0.283172\pi$
$239$	1.11216e6			1.25943			0.629715	+	0.776826i	$0.283172\pi$
$240$		0			0
$241$	967616.			1.07315			0.536575	−	0.843853i	$-0.319718\pi$
$241$	967616.			1.07315			0.536575	+	0.843853i	$0.319718\pi$
$242$	−193899.	+	193899.i	−0.212832	+	0.212832i
$243$		0			0
$244$			571242.i			0.614250i
$245$		0			0
$246$		0			0
$247$	21537.1	+	21537.1i	0.0224618	+	0.0224618i
$248$	−20075.0	−	20075.0i	−0.0207266	−	0.0207266i
$249$		0			0
$250$		0			0
$251$		−	419861.i		−	0.420651i	−0.977631	−	0.210325i	$-0.932548\pi$
$251$		−	419861.i		−	0.420651i	0.977631	−	0.210325i	$-0.0674523\pi$
$252$		0			0
$253$	−812136.	+	812136.i	−0.797678	+	0.797678i
$254$	784999.			0.763458
$255$		0			0
$256$	65536.0			0.0625000
$257$	165167.	−	165167.i	0.155988	−	0.155988i	−0.624798	−	0.780786i	$-0.714819\pi$
$257$	165167.	−	165167.i	0.155988	−	0.155988i	0.780786	+	0.624798i	$0.214819\pi$
$258$		0			0
$259$		−	92769.3i		−	0.0859320i
$260$		0			0
$261$		0			0
$262$	−1.06657e6	−	1.06657e6i	−0.959920	−	0.959920i
$263$	−433781.	−	433781.i	−0.386706	−	0.386706i	0.486805	−	0.873511i	$-0.338162\pi$
$263$	−433781.	−	433781.i	−0.386706	−	0.386706i	−0.873511	+	0.486805i	$0.838162\pi$
$264$		0			0
$265$		0			0
$266$			6640.19i			0.00575409i
$267$		0			0
$268$	642868.	−	642868.i	0.546745	−	0.546745i
$269$	−643503.			−0.542213			−0.271106	−	0.962549i	$-0.587390\pi$
$269$	−643503.			−0.542213			−0.271106	+	0.962549i	$0.587390\pi$
$270$		0			0
$271$	981970.			0.812223			0.406111	−	0.913824i	$-0.366885\pi$
$271$	981970.			0.812223			0.406111	+	0.913824i	$0.366885\pi$
$272$	353281.	−	353281.i	0.289533	−	0.289533i
$273$		0			0
$274$		−	547547.i		−	0.440600i
$275$		0			0
$276$		0			0
$277$	−1.09057e6	−	1.09057e6i	−0.853991	−	0.853991i	0.136631	−	0.990622i	$-0.456373\pi$
$277$	−1.09057e6	−	1.09057e6i	−0.853991	−	0.853991i	−0.990622	+	0.136631i	$0.956373\pi$
$278$	958363.	+	958363.i	0.743735	+	0.743735i
$279$		0			0
$280$		0			0
$281$			272068.i			0.205548i	0.994705	+	0.102774i	$0.0327718\pi$
$281$			272068.i			0.205548i	−0.994705	+	0.102774i	$0.967228\pi$
$282$		0			0
$283$	1.62568e6	−	1.62568e6i	1.20661	−	1.20661i	0.234497	−	0.972117i	$-0.424656\pi$
$283$	1.62568e6	−	1.62568e6i	1.20661	−	1.20661i	0.972117	−	0.234497i	$-0.0753443\pi$
$284$	527584.			0.388147
$285$		0			0
$286$	−481330.			−0.347959
$287$	−9924.51	+	9924.51i	−0.00711220	+	0.00711220i
$288$		0			0
$289$		−	2.38896e6i		−	1.68253i
$290$		0			0
$291$		0			0
$292$	325858.	+	325858.i	0.223651	+	0.223651i
$293$	−1.02737e6	−	1.02737e6i	−0.699127	−	0.699127i	0.265095	−	0.964222i	$-0.414597\pi$
$293$	−1.02737e6	−	1.02737e6i	−0.699127	−	0.699127i	−0.964222	+	0.265095i	$0.914597\pi$
$294$		0			0
$295$		0			0
$296$			433783.i			0.287769i
$297$		0			0
$298$	−1.19253e6	+	1.19253e6i	−0.777909	+	0.777909i
$299$	−601930.			−0.389375
$300$		0			0
$301$	−122302.			−0.0778068
$302$	−1.49862e6	+	1.49862e6i	−0.945525	+	0.945525i
$303$		0			0
$304$		−	31049.1i		−	0.0192693i
$305$		0			0
$306$		0			0
$307$	−891251.	−	891251.i	−0.539702	−	0.539702i	0.383739	−	0.923441i	$-0.374636\pi$
$307$	−891251.	−	891251.i	−0.539702	−	0.539702i	−0.923441	+	0.383739i	$0.874636\pi$
$308$	−74200.5	−	74200.5i	−0.0445687	−	0.0445687i
$309$		0			0
$310$		0			0
$311$			1.42354e6i			0.834579i	0.908774	+	0.417289i	$0.137020\pi$
$311$			1.42354e6i			0.834579i	−0.908774	+	0.417289i	$0.862980\pi$
$312$		0			0
$313$	854680.	−	854680.i	0.493109	−	0.493109i	−0.416176	−	0.909284i	$-0.636630\pi$
$313$	854680.	−	854680.i	0.493109	−	0.493109i	0.909284	+	0.416176i	$0.136630\pi$
$314$	625245.			0.357871
$315$		0			0
$316$	434892.			0.244999
$317$	1.68305e6	−	1.68305e6i	0.940697	−	0.940697i	−0.0576401	−	0.998337i	$-0.518358\pi$
$317$	1.68305e6	−	1.68305e6i	0.940697	−	0.940697i	0.998337	+	0.0576401i	$0.0183576\pi$
$318$		0			0
$319$		−	3.37525e6i		−	1.85708i
$320$		0			0
$321$		0			0
$322$	−92791.9	−	92791.9i	−0.0498736	−	0.0498736i
$323$	−167374.	−	167374.i	−0.0892653	−	0.0892653i
$324$		0			0
$325$		0			0
$326$			1.38977e6i			0.724265i
$327$		0			0
$328$	46406.3	−	46406.3i	0.0238173	−	0.0238173i
$329$	−248385.			−0.126513
$330$		0			0
$331$	−2.20398e6			−1.10570			−0.552850	−	0.833280i	$-0.686460\pi$
$331$	−2.20398e6			−1.10570			−0.552850	+	0.833280i	$0.686460\pi$
$332$	−516582.	+	516582.i	−0.257214	+	0.257214i
$333$		0			0
$334$		−	1.97841e6i		−	0.970400i
$335$		0			0
$336$		0			0
$337$	468533.	+	468533.i	0.224732	+	0.224732i	0.810488	−	0.585755i	$-0.199202\pi$
$337$	468533.	+	468533.i	0.224732	+	0.224732i	−0.585755	+	0.810488i	$0.699202\pi$
$338$	871802.	+	871802.i	0.415074	+	0.415074i
$339$		0			0
$340$		0			0
$341$		−	212560.i		−	0.0989910i
$342$		0			0
$343$	323512.	−	323512.i	0.148475	−	0.148475i
$344$	571876.			0.260559
$345$		0			0
$346$	−814257.			−0.365655
$347$	2.03791e6	−	2.03791e6i	0.908575	−	0.908575i	−0.0875826	−	0.996157i	$-0.527914\pi$
$347$	2.03791e6	−	2.03791e6i	0.908575	−	0.908575i	0.996157	+	0.0875826i	$0.0279142\pi$
$348$		0			0
$349$			407443.i			0.179062i	0.995984	+	0.0895310i	$0.0285368\pi$
$349$			407443.i			0.179062i	−0.995984	+	0.0895310i	$0.971463\pi$
$350$		0			0
$351$		0			0
$352$	346957.	+	346957.i	0.149251	+	0.149251i
$353$	2.07267e6	+	2.07267e6i	0.885306	+	0.885306i	0.994068	−	0.108762i	$-0.0346886\pi$
$353$	2.07267e6	+	2.07267e6i	0.885306	+	0.885306i	−0.108762	+	0.994068i	$0.534689\pi$
$354$		0			0
$355$		0			0
$356$			1.03836e6i			0.434232i
$357$		0			0
$358$	−1.64963e6	+	1.64963e6i	−0.680267	+	0.680267i
$359$	−1.15572e6			−0.473279			−0.236640	−	0.971597i	$-0.576046\pi$
$359$	−1.15572e6			−0.473279			−0.236640	+	0.971597i	$0.576046\pi$
$360$		0			0
$361$	2.46139e6			0.994059
$362$	2.13142e6	−	2.13142e6i	0.854864	−	0.854864i
$363$		0			0
$364$		−	54995.1i		−	0.0217556i
$365$		0			0
$366$		0			0
$367$	−628091.	−	628091.i	−0.243421	−	0.243421i	0.574843	−	0.818264i	$-0.305063\pi$
$367$	−628091.	−	628091.i	−0.243421	−	0.243421i	−0.818264	+	0.574843i	$0.805063\pi$
$368$	433889.	+	433889.i	0.167016	+	0.167016i
$369$		0			0
$370$		0			0
$371$		−	323391.i		−	0.121981i
$372$		0			0
$373$	2.26018e6	−	2.26018e6i	0.841144	−	0.841144i	−0.147863	−	0.989008i	$-0.547240\pi$
$373$	2.26018e6	−	2.26018e6i	0.841144	−	0.841144i	0.989008	+	0.147863i	$0.0472396\pi$
$374$	3.74063e6			1.38282
$375$		0			0
$376$	1.16143e6			0.423666
$377$	1.25082e6	−	1.25082e6i	0.453253	−	0.453253i
$378$		0			0
$379$		−	172752.i		−	0.0617769i	−0.999523	−	0.0308885i	$-0.990166\pi$
$379$		−	172752.i		−	0.0617769i	0.999523	−	0.0308885i	$-0.00983366\pi$
$380$		0			0
$381$		0			0
$382$	−1.44635e6	−	1.44635e6i	−0.507126	−	0.507126i
$383$	−404160.	−	404160.i	−0.140785	−	0.140785i	0.633202	−	0.773987i	$-0.281740\pi$
$383$	−404160.	−	404160.i	−0.140785	−	0.140785i	−0.773987	+	0.633202i	$0.781740\pi$
$384$		0			0
$385$		0			0
$386$			52084.0i			0.0177925i
$387$		0			0
$388$	−356902.	+	356902.i	−0.120357	+	0.120357i
$389$	2.31395e6			0.775319			0.387660	−	0.921803i	$-0.373284\pi$
$389$	2.31395e6			0.775319			0.387660	+	0.921803i	$0.373284\pi$
$390$		0			0
$391$	4.67787e6			1.54741
$392$	−752120.	+	752120.i	−0.247213	+	0.247213i
$393$		0			0
$394$		−	1.28472e6i		−	0.416935i
$395$		0			0
$396$		0			0
$397$	−858541.	−	858541.i	−0.273391	−	0.273391i	0.557072	−	0.830464i	$-0.311925\pi$
$397$	−858541.	−	858541.i	−0.273391	−	0.273391i	−0.830464	+	0.557072i	$0.811925\pi$
$398$	−2.08888e6	−	2.08888e6i	−0.661007	−	0.661007i
$399$		0			0
$400$		0			0
$401$			1.38617e6i			0.430482i	0.976561	+	0.215241i	$0.0690537\pi$
$401$			1.38617e6i			0.430482i	−0.976561	+	0.215241i	$0.930946\pi$
$402$		0			0
$403$	78771.5	−	78771.5i	0.0241605	−	0.0241605i
$404$	−695200.			−0.211913
$405$		0			0
$406$	385645.			0.116111
$407$	−2.29651e6	+	2.29651e6i	−0.687198	+	0.687198i
$408$		0			0
$409$		−	340398.i		−	0.100619i	−0.998734	−	0.0503094i	$-0.983979\pi$
$409$		−	340398.i		−	0.100619i	0.998734	−	0.0503094i	$-0.0160207\pi$
$410$		0			0
$411$		0			0
$412$	−974353.	−	974353.i	−0.282796	−	0.282796i
$413$	−400352.	−	400352.i	−0.115496	−	0.115496i
$414$		0			0
$415$		0			0
$416$			257154.i			0.0728550i
$417$		0			0
$418$	164378.	−	164378.i	0.0460154	−	0.0460154i
$419$	−1.30762e6			−0.363869			−0.181934	−	0.983311i	$-0.558236\pi$
$419$	−1.30762e6			−0.363869			−0.181934	+	0.983311i	$0.558236\pi$
$420$		0			0
$421$	476804.			0.131110			0.0655548	−	0.997849i	$-0.479118\pi$
$421$	476804.			0.131110			0.0655548	+	0.997849i	$0.479118\pi$
$422$	−1.63909e6	+	1.63909e6i	−0.448045	+	0.448045i
$423$		0			0
$424$			1.51215e6i			0.408490i
$425$		0			0
$426$		0			0
$427$	−345539.	−	345539.i	−0.0917122	−	0.0917122i
$428$	2.62009e6	+	2.62009e6i	0.691363	+	0.691363i
$429$		0			0
$430$		0			0
$431$		−	1.30974e6i		−	0.339619i	−0.985477	−	0.169809i	$-0.945685\pi$
$431$		−	1.30974e6i		−	0.339619i	0.985477	−	0.169809i	$-0.0543152\pi$
$432$		0			0
$433$	3.62876e6	−	3.62876e6i	0.930118	−	0.930118i	−0.0675951	−	0.997713i	$-0.521533\pi$
$433$	3.62876e6	−	3.62876e6i	0.930118	−	0.930118i	0.997713	+	0.0675951i	$0.0215326\pi$
$434$	24286.4			0.00618926
$435$		0			0
$436$	3.13724e6			0.790371
$437$	205564.	−	205564.i	0.0514925	−	0.0514925i
$438$		0			0
$439$			883070.i			0.218692i	0.994004	+	0.109346i	$0.0348757\pi$
$439$			883070.i			0.218692i	−0.994004	+	0.109346i	$0.965124\pi$
$440$		0			0
$441$		0			0
$442$	1.38622e6	+	1.38622e6i	0.337502	+	0.337502i
$443$	−70500.7	−	70500.7i	−0.0170681	−	0.0170681i	0.698521	−	0.715589i	$-0.253842\pi$
$443$	−70500.7	−	70500.7i	−0.0170681	−	0.0170681i	−0.715589	+	0.698521i	$0.753842\pi$
$444$		0			0
$445$		0			0
$446$		−	4.34750e6i		−	1.03491i
$447$		0			0
$448$	−39642.1	+	39642.1i	−0.00933172	+	0.00933172i
$449$	5.66675e6			1.32653			0.663267	−	0.748383i	$-0.269169\pi$
$449$	5.66675e6			1.32653			0.663267	+	0.748383i	$0.269169\pi$
$450$		0			0
$451$	491363.			0.113753
$452$	−111043.	+	111043.i	−0.0255650	+	0.0255650i
$453$		0			0
$454$			3.96869e6i			0.903664i
$455$		0			0
$456$		0			0
$457$	1.60863e6	+	1.60863e6i	0.360301	+	0.360301i	0.863924	−	0.503623i	$-0.168000\pi$
$457$	1.60863e6	+	1.60863e6i	0.360301	+	0.360301i	−0.503623	+	0.863924i	$0.668000\pi$
$458$	−98514.9	−	98514.9i	−0.0219451	−	0.0219451i
$459$		0			0
$460$		0			0
$461$		−	8.33450e6i		−	1.82653i	−0.407364	−	0.913266i	$-0.633552\pi$
$461$		−	8.33450e6i		−	1.82653i	0.407364	−	0.913266i	$-0.366448\pi$
$462$		0			0
$463$	−4.82641e6	+	4.82641e6i	−1.04634	+	1.04634i	−0.0474636	+	0.998873i	$0.515114\pi$
$463$	−4.82641e6	+	4.82641e6i	−1.04634	+	1.04634i	−0.998873	+	0.0474636i	$0.984886\pi$
$464$	−1.80325e6			−0.388831
$465$		0			0
$466$	−4.54635e6			−0.969836
$467$	1.57887e6	−	1.57887e6i	0.335007	−	0.335007i	−0.519477	−	0.854484i	$-0.673873\pi$
$467$	1.57887e6	−	1.57887e6i	0.335007	−	0.335007i	0.854484	+	0.519477i	$0.173873\pi$
$468$		0			0
$469$			777730.i			0.163266i
$470$		0			0
$471$		0			0
$472$	1.87202e6	+	1.87202e6i	0.386773	+	0.386773i
$473$	3.02759e6	+	3.02759e6i	0.622221	+	0.622221i
$474$		0			0
$475$		0			0
$476$			427392.i			0.0864588i
$477$		0			0
$478$	3.14568e6	−	3.14568e6i	0.629715	−	0.629715i
$479$	5.33389e6			1.06220			0.531099	−	0.847310i	$-0.321779\pi$
$479$	5.33389e6			1.06220			0.531099	+	0.847310i	$0.321779\pi$
$480$		0			0
$481$	−1.70210e6			−0.335446
$482$	2.73683e6	−	2.73683e6i	0.536575	−	0.536575i
$483$		0			0
$484$			1.09686e6i			0.212832i
$485$		0			0
$486$		0			0
$487$	2.16333e6	+	2.16333e6i	0.413334	+	0.413334i	0.882898	−	0.469565i	$-0.155589\pi$
$487$	2.16333e6	+	2.16333e6i	0.413334	+	0.413334i	−0.469565	+	0.882898i	$0.655589\pi$
$488$	1.61572e6	+	1.61572e6i	0.307125	+	0.307125i
$489$		0			0
$490$		0			0
$491$			8.45785e6i			1.58328i	0.610991	+	0.791638i	$0.290771\pi$
$491$			8.45785e6i			1.58328i	−0.610991	+	0.791638i	$0.709229\pi$
$492$		0			0
$493$	−9.72067e6	+	9.72067e6i	−1.80127	+	1.80127i
$494$	121832.			0.0224618
$495$		0			0
$496$	−113562.			−0.0207266
$497$	−319131.	+	319131.i	−0.0579532	+	0.0579532i
$498$		0			0
$499$			2.50745e6i			0.450797i	0.974267	+	0.225398i	$0.0723684\pi$
$499$			2.50745e6i			0.450797i	−0.974267	+	0.225398i	$0.927632\pi$
$500$		0			0
$501$		0			0
$502$	−1.18755e6	−	1.18755e6i	−0.210325	−	0.210325i
$503$	−4.75292e6	−	4.75292e6i	−0.837607	−	0.837607i	0.150937	−	0.988543i	$-0.451771\pi$
$503$	−4.75292e6	−	4.75292e6i	−0.837607	−	0.837607i	−0.988543	+	0.150937i	$0.951771\pi$
$504$		0			0
$505$		0			0
$506$			4.59413e6i			0.797678i
$507$		0			0
$508$	2.22031e6	−	2.22031e6i	0.381729	−	0.381729i
$509$	−7.64193e6			−1.30740			−0.653700	−	0.756754i	$-0.726784\pi$
$509$	−7.64193e6			−1.30740			−0.653700	+	0.756754i	$0.726784\pi$
$510$		0			0
$511$	−394217.			−0.0667856
$512$	185364.	−	185364.i	0.0312500	−	0.0312500i
$513$		0			0
$514$		−	934325.i		−	0.155988i
$515$		0			0
$516$		0			0
$517$	6.14877e6	+	6.14877e6i	1.01172	+	1.01172i
$518$	−262391.	−	262391.i	−0.0429660	−	0.0429660i
$519$		0			0
$520$		0			0
$521$		−	3.21226e6i		−	0.518461i	−0.965815	−	0.259231i	$-0.916531\pi$
$521$		−	3.21226e6i		−	0.518461i	0.965815	−	0.259231i	$-0.0834689\pi$
$522$		0			0
$523$	2.16299e6	−	2.16299e6i	0.345780	−	0.345780i	−0.512755	−	0.858535i	$-0.671375\pi$
$523$	2.16299e6	−	2.16299e6i	0.345780	−	0.345780i	0.858535	+	0.512755i	$0.171375\pi$
$524$	−6.03341e6			−0.959920
$525$		0			0
$526$	−2.45384e6			−0.386706
$527$	−612169.	+	612169.i	−0.0960163	+	0.0960163i
$528$		0			0
$529$		−	691118.i		−	0.107377i
$530$		0			0
$531$		0			0
$532$	18781.3	+	18781.3i	0.00287704	+	0.00287704i
$533$	182092.	+	182092.i	0.0277633	+	0.0277633i
$534$		0			0
$535$		0			0
$536$		−	3.63661e6i		−	0.546745i
$537$		0			0
$538$	−1.82010e6	+	1.82010e6i	−0.271106	+	0.271106i
$539$	−7.96365e6			−1.18070
$540$		0			0
$541$	−1.21690e7			−1.78756			−0.893780	−	0.448506i	$-0.851956\pi$
$541$	−1.21690e7			−1.78756			−0.893780	+	0.448506i	$0.851956\pi$
$542$	2.77743e6	−	2.77743e6i	0.406111	−	0.406111i
$543$		0			0
$544$		−	1.99846e6i		−	0.289533i
$545$		0			0
$546$		0			0
$547$	−6.29790e6	−	6.29790e6i	−0.899969	−	0.899969i	0.0954641	−	0.995433i	$-0.469567\pi$
$547$	−6.29790e6	−	6.29790e6i	−0.899969	−	0.899969i	−0.995433	+	0.0954641i	$0.969567\pi$
$548$	−1.54870e6	−	1.54870e6i	−0.220300	−	0.220300i
$549$		0			0
$550$		0			0
$551$			854329.i			0.119880i
$552$		0			0
$553$	−263062.	+	263062.i	−0.0365801	+	0.0365801i
$554$	−6.16919e6			−0.853991
$555$		0			0
$556$	5.42132e6			0.743735
$557$	25868.2	−	25868.2i	0.00353287	−	0.00353287i	−0.705338	−	0.708871i	$-0.749205\pi$
$557$	25868.2	−	25868.2i	0.00353287	−	0.00353287i	0.708871	+	0.705338i	$0.249205\pi$
$558$		0			0
$559$			2.24396e6i			0.303728i
$560$		0			0
$561$		0			0
$562$	769526.	+	769526.i	0.102774	+	0.102774i
$563$	587908.	+	587908.i	0.0781697	+	0.0781697i	0.745111	−	0.666941i	$-0.232397\pi$
$563$	587908.	+	587908.i	0.0781697	+	0.0781697i	−0.666941	+	0.745111i	$0.732397\pi$
$564$		0			0
$565$		0			0
$566$		−	9.19622e6i		−	1.20661i
$567$		0			0
$568$	1.49223e6	−	1.49223e6i	0.194073	−	0.194073i
$569$	−1.96394e6			−0.254301			−0.127150	−	0.991883i	$-0.540583\pi$
$569$	−1.96394e6			−0.254301			−0.127150	+	0.991883i	$0.540583\pi$
$570$		0			0
$571$	1.36566e6			0.175289			0.0876443	−	0.996152i	$-0.472066\pi$
$571$	1.36566e6			0.175289			0.0876443	+	0.996152i	$0.472066\pi$
$572$	−1.36141e6	+	1.36141e6i	−0.173979	+	0.173979i
$573$		0			0
$574$			56141.5i			0.00711220i
$575$		0			0
$576$		0			0
$577$	1.00497e7	+	1.00497e7i	1.25665	+	1.25665i	0.952684	+	0.303962i	$0.0983097\pi$
$577$	1.00497e7	+	1.00497e7i	1.25665	+	1.25665i	0.303962	+	0.952684i	$0.401690\pi$
$578$	−6.75700e6	−	6.75700e6i	−0.841267	−	0.841267i
$579$		0			0
$580$		0			0
$581$		−	624951.i		−	0.0768079i
$582$		0			0
$583$	−8.00555e6	+	8.00555e6i	−0.975484	+	0.975484i
$584$	1.84333e6			0.223651
$585$		0			0
$586$	−5.81166e6			−0.699127
$587$	−6.34891e6	+	6.34891e6i	−0.760508	+	0.760508i	−0.976414	−	0.215906i	$-0.930730\pi$
$587$	−6.34891e6	+	6.34891e6i	−0.760508	+	0.760508i	0.215906	+	0.976414i	$0.430730\pi$
$588$		0			0
$589$			53802.3i			0.00639017i
$590$		0			0
$591$		0			0
$592$	1.22692e6	+	1.22692e6i	0.143884	+	0.143884i
$593$	1.63841e6	+	1.63841e6i	0.191331	+	0.191331i	0.796271	−	0.604940i	$-0.206803\pi$
$593$	1.63841e6	+	1.63841e6i	0.191331	+	0.191331i	−0.604940	+	0.796271i	$0.706803\pi$
$594$		0			0
$595$		0			0
$596$			6.74597e6i			0.777909i
$597$		0			0
$598$	−1.70252e6	+	1.70252e6i	−0.194688	+	0.194688i
$599$	−4.20108e6			−0.478403			−0.239202	−	0.970970i	$-0.576886\pi$
$599$	−4.20108e6			−0.478403			−0.239202	+	0.970970i	$0.576886\pi$
$600$		0			0
$601$	−6.19973e6			−0.700142			−0.350071	−	0.936723i	$-0.613843\pi$
$601$	−6.19973e6			−0.700142			−0.350071	+	0.936723i	$0.613843\pi$
$602$	−345922.	+	345922.i	−0.0389034	+	0.0389034i
$603$		0			0
$604$			8.47745e6i			0.945525i
$605$		0			0
$606$		0			0
$607$	−4.38869e6	−	4.38869e6i	−0.483462	−	0.483462i	0.422773	−	0.906236i	$-0.361057\pi$
$607$	−4.38869e6	−	4.38869e6i	−0.483462	−	0.483462i	−0.906236	+	0.422773i	$0.861057\pi$
$608$	−87820.1	−	87820.1i	−0.00963463	−	0.00963463i
$609$		0			0
$610$		0			0
$611$			4.55728e6i			0.493859i
$612$		0			0
$613$	4.75066e6	−	4.75066e6i	0.510626	−	0.510626i	−0.404092	−	0.914718i	$-0.632412\pi$
$613$	4.75066e6	−	4.75066e6i	0.510626	−	0.510626i	0.914718	+	0.404092i	$0.132412\pi$
$614$	−5.04168e6			−0.539702
$615$		0			0
$616$	−419741.			−0.0445687
$617$	−1.24103e7	+	1.24103e7i	−1.31241	+	1.31241i	−0.392772	+	0.919636i	$0.628484\pi$
$617$	−1.24103e7	+	1.24103e7i	−1.31241	+	1.31241i	−0.919636	+	0.392772i	$0.871516\pi$
$618$		0			0
$619$			1.61901e7i			1.69833i	0.528125	+	0.849166i	$0.322895\pi$
$619$			1.61901e7i			1.69833i	−0.528125	+	0.849166i	$0.677105\pi$
$620$		0			0
$621$		0			0
$622$	4.02637e6	+	4.02637e6i	0.417289	+	0.417289i
$623$	−628092.	−	628092.i	−0.0648341	−	0.0648341i
$624$		0			0
$625$		0			0
$626$		−	4.83480e6i		−	0.493109i
$627$		0			0
$628$	1.76846e6	−	1.76846e6i	0.178935	−	0.178935i
$629$	1.32278e7			1.33310
$630$		0			0
$631$	−1.03850e7			−1.03832			−0.519162	−	0.854676i	$-0.673756\pi$
$631$	−1.03850e7			−1.03832			−0.519162	+	0.854676i	$0.673756\pi$
$632$	1.23006e6	−	1.23006e6i	0.122499	−	0.122499i
$633$		0			0
$634$		−	9.52079e6i		−	0.940697i
$635$		0			0
$636$		0			0
$637$	−2.95121e6	−	2.95121e6i	−0.288172	−	0.288172i
$638$	−9.54666e6	−	9.54666e6i	−0.928538	−	0.928538i
$639$		0			0
$640$		0			0
$641$		−	1.19326e7i		−	1.14707i	−0.819180	−	0.573537i	$-0.805571\pi$
$641$		−	1.19326e7i		−	1.14707i	0.819180	−	0.573537i	$-0.194429\pi$
$642$		0			0
$643$	1.03095e7	−	1.03095e7i	0.983359	−	0.983359i	−0.0165048	−	0.999864i	$-0.505254\pi$
$643$	1.03095e7	−	1.03095e7i	0.983359	−	0.983359i	0.999864	+	0.0165048i	$0.00525387\pi$
$644$	−524910.			−0.0498736
$645$		0			0
$646$	−946812.			−0.0892653
$647$	−7.30615e6	+	7.30615e6i	−0.686164	+	0.686164i	−0.961382	−	0.275218i	$-0.911250\pi$
$647$	−7.30615e6	+	7.30615e6i	−0.686164	+	0.686164i	0.275218	+	0.961382i	$0.411250\pi$
$648$		0			0
$649$			1.98215e7i			1.84724i
$650$		0			0
$651$		0			0
$652$	3.93085e6	+	3.93085e6i	0.362133	+	0.362133i
$653$	4.82186e6	+	4.82186e6i	0.442519	+	0.442519i	0.892858	−	0.450339i	$-0.148697\pi$
$653$	4.82186e6	+	4.82186e6i	0.442519	+	0.442519i	−0.450339	+	0.892858i	$0.648697\pi$
$654$		0			0
$655$		0			0
$656$		−	262514.i		−	0.0238173i
$657$		0			0
$658$	−702538.	+	702538.i	−0.0632565	+	0.0632565i
$659$	−2.46721e6			−0.221306			−0.110653	−	0.993859i	$-0.535294\pi$
$659$	−2.46721e6			−0.221306			−0.110653	+	0.993859i	$0.535294\pi$
$660$		0			0
$661$	3.92026e6			0.348989			0.174494	−	0.984658i	$-0.444171\pi$
$661$	3.92026e6			0.348989			0.174494	+	0.984658i	$0.444171\pi$
$662$	−6.23380e6	+	6.23380e6i	−0.552850	+	0.552850i
$663$		0			0
$664$			2.92223e6i			0.257214i
$665$		0			0
$666$		0			0
$667$	−1.19386e7	−	1.19386e7i	−1.03906	−	1.03906i
$668$	−5.59580e6	−	5.59580e6i	−0.485200	−	0.485200i
$669$		0			0
$670$		0			0
$671$			1.71076e7i			1.46684i
$672$		0			0
$673$	−1.04432e7	+	1.04432e7i	−0.888787	+	0.888787i	−0.994407	−	0.105619i	$-0.966318\pi$
$673$	−1.04432e7	+	1.04432e7i	−0.888787	+	0.888787i	0.105619	+	0.994407i	$0.466318\pi$
$674$	2.65043e6			0.224732
$675$		0			0
$676$	4.93166e6			0.415074
$677$	−6.24464e6	+	6.24464e6i	−0.523643	+	0.523643i	−0.918670	−	0.395026i	$-0.870735\pi$
$677$	−6.24464e6	+	6.24464e6i	−0.523643	+	0.523643i	0.395026	+	0.918670i	$0.370735\pi$
$678$		0			0
$679$		−	431774.i		−	0.0359403i
$680$		0			0
$681$		0			0
$682$	−601211.	−	601211.i	−0.0494955	−	0.0494955i
$683$	6.24335e6	+	6.24335e6i	0.512113	+	0.512113i	0.915173	−	0.403060i	$-0.132054\pi$
$683$	6.24335e6	+	6.24335e6i	0.512113	+	0.512113i	−0.403060	+	0.915173i	$0.632054\pi$
$684$		0			0
$685$		0			0
$686$		−	1.83006e6i		−	0.148475i
$687$		0			0
$688$	1.61751e6	−	1.61751e6i	0.130279	−	0.130279i
$689$	−5.93347e6			−0.476169
$690$		0			0
$691$	3.85776e6			0.307355			0.153677	−	0.988121i	$-0.450888\pi$
$691$	3.85776e6			0.307355			0.153677	+	0.988121i	$0.450888\pi$
$692$	−2.30307e6	+	2.30307e6i	−0.182827	+	0.182827i
$693$		0			0
$694$		−	1.15281e7i		−	0.908575i
$695$		0			0
$696$		0			0
$697$	−1.41512e6	−	1.41512e6i	−0.110334	−	0.110334i
$698$	1.15242e6	+	1.15242e6i	0.0895310	+	0.0895310i
$699$		0			0
$700$		0			0
$701$			2.09792e6i			0.161248i	0.996745	+	0.0806240i	$0.0256913\pi$
$701$			2.09792e6i			0.161248i	−0.996745	+	0.0806240i	$0.974309\pi$
$702$		0			0
$703$	581282.	−	581282.i	0.0443607	−	0.0443607i
$704$	1.96268e6			0.149251
$705$		0			0
$706$	1.17248e7			0.885306
$707$	420520.	−	420520.i	0.0316401	−	0.0316401i
$708$		0			0
$709$			1.90774e7i			1.42529i	0.701525	+	0.712645i	$0.252503\pi$
$709$			1.90774e7i			1.42529i	−0.701525	+	0.712645i	$0.747497\pi$
$710$		0			0
$711$		0			0
$712$	2.93692e6	+	2.93692e6i	0.217116	+	0.217116i
$713$	−751848.	−	751848.i	−0.0553868	−	0.0553868i
$714$		0			0
$715$		0			0
$716$			9.33172e6i			0.680267i
$717$		0			0
$718$	−3.26888e6	+	3.26888e6i	−0.236640	+	0.236640i
$719$	−3.36879e6			−0.243025			−0.121513	−	0.992590i	$-0.538774\pi$
$719$	−3.36879e6			−0.243025			−0.121513	+	0.992590i	$0.538774\pi$
$720$		0			0
$721$	1.17875e6			0.0844472
$722$	6.96186e6	−	6.96186e6i	0.497030	−	0.497030i
$723$		0			0
$724$		−	1.20571e7i		−	0.854864i
$725$		0			0
$726$		0			0
$727$	3.67673e6	+	3.67673e6i	0.258004	+	0.258004i	0.824242	−	0.566238i	$-0.191602\pi$
$727$	3.67673e6	+	3.67673e6i	0.258004	+	0.258004i	−0.566238	+	0.824242i	$0.691602\pi$
$728$	−155550.	−	155550.i	−0.0108778	−	0.0108778i
$729$		0			0
$730$		0			0
$731$		−	1.74388e7i		−	1.20705i
$732$		0			0
$733$	−2.83561e6	+	2.83561e6i	−0.194933	+	0.194933i	−0.797824	−	0.602891i	$-0.794016\pi$
$733$	−2.83561e6	+	2.83561e6i	−0.194933	+	0.194933i	0.602891	+	0.797824i	$0.294016\pi$
$734$	−3.55302e6			−0.243421
$735$		0			0
$736$	2.45445e6			0.167016
$737$	1.92527e7	−	1.92527e7i	1.30564	−	1.30564i
$738$		0			0
$739$		−	5.14600e6i		−	0.346624i	−0.984867	−	0.173312i	$-0.944553\pi$
$739$		−	5.14600e6i		−	0.346624i	0.984867	−	0.173312i	$-0.0554469\pi$
$740$		0			0
$741$		0			0
$742$	−914688.	−	914688.i	−0.0609906	−	0.0609906i
$743$	−6.09111e6	−	6.09111e6i	−0.404785	−	0.404785i	0.475130	−	0.879915i	$-0.342401\pi$
$743$	−6.09111e6	−	6.09111e6i	−0.404785	−	0.404785i	−0.879915	+	0.475130i	$0.842401\pi$
$744$		0			0
$745$		0			0
$746$		−	1.27855e7i		−	0.841144i
$747$		0			0
$748$	1.05801e7	−	1.05801e7i	0.691411	−	0.691411i
$749$	−3.16973e6			−0.206451
$750$		0			0
$751$	−1.72672e7			−1.11717			−0.558587	−	0.829446i	$-0.688656\pi$
$751$	−1.72672e7			−1.11717			−0.558587	+	0.829446i	$0.688656\pi$
$752$	3.28502e6	−	3.28502e6i	0.211833	−	0.211833i
$753$		0			0
$754$		−	7.07569e6i		−	0.453253i
$755$		0			0
$756$		0			0
$757$	6.87707e6	+	6.87707e6i	0.436178	+	0.436178i	0.890724	−	0.454545i	$-0.150198\pi$
$757$	6.87707e6	+	6.87707e6i	0.436178	+	0.436178i	−0.454545	+	0.890724i	$0.650198\pi$
$758$	−488618.	−	488618.i	−0.0308885	−	0.0308885i
$759$		0			0
$760$		0			0
$761$			1.68325e7i			1.05363i	0.849980	+	0.526815i	$0.176614\pi$
$761$			1.68325e7i			1.05363i	−0.849980	+	0.526815i	$0.823386\pi$
$762$		0			0
$763$	−1.89768e6	+	1.89768e6i	−0.118008	+	0.118008i
$764$	−8.18181e6			−0.507126
$765$		0			0
$766$	−2.28628e6			−0.140785
$767$	−7.34553e6	+	7.34553e6i	−0.450853	+	0.450853i
$768$		0			0
$769$		−	1.54410e7i		−	0.941587i	−0.882243	−	0.470794i	$-0.843968\pi$
$769$		−	1.54410e7i		−	0.941587i	0.882243	−	0.470794i	$-0.156032\pi$
$770$		0			0
$771$		0			0
$772$	147316.	+	147316.i	0.00889624	+	0.00889624i
$773$	1.30729e7	+	1.30729e7i	0.786904	+	0.786904i	0.980985	−	0.194081i	$-0.0621725\pi$
$773$	1.30729e7	+	1.30729e7i	0.786904	+	0.786904i	−0.194081	+	0.980985i	$0.562173\pi$
$774$		0			0
$775$		0			0
$776$			2.01894e6i			0.120357i
$777$		0			0
$778$	6.54485e6	−	6.54485e6i	0.387660	−	0.387660i
$779$	−124372.			−0.00734307
$780$		0			0
$781$	1.58002e7			0.926903
$782$	1.32310e7	−	1.32310e7i	0.773707	−	0.773707i
$783$		0			0
$784$			4.25463e6i			0.247213i
$785$		0			0
$786$		0			0
$787$	−417501.	−	417501.i	−0.0240282	−	0.0240282i	0.694991	−	0.719019i	$-0.255409\pi$
$787$	−417501.	−	417501.i	−0.0240282	−	0.0240282i	−0.719019	+	0.694991i	$0.755409\pi$
$788$	−3.63374e6	−	3.63374e6i	−0.208467	−	0.208467i
$789$		0			0
$790$		0			0
$791$		−	134338.i		−	0.00763408i
$792$		0			0
$793$	−6.33983e6	+	6.33983e6i	−0.358009	+	0.358009i
$794$	−4.85664e6			−0.273391
$795$		0			0
$796$	−1.18165e7			−0.661007
$797$	1.52951e7	−	1.52951e7i	0.852918	−	0.852918i	−0.137573	−	0.990492i	$-0.543930\pi$
$797$	1.52951e7	−	1.52951e7i	0.852918	−	0.852918i	0.990492	+	0.137573i	$0.0439303\pi$
$798$		0			0
$799$		−	3.54167e7i		−	1.96264i
$800$		0			0
$801$		0			0
$802$	3.92067e6	+	3.92067e6i	0.215241	+	0.215241i
$803$	9.75885e6	+	9.75885e6i	0.534084	+	0.534084i
$804$		0			0
$805$		0			0
$806$		−	445599.i		−	0.0241605i
$807$		0			0
$808$	−1.96632e6	+	1.96632e6i	−0.105956	+	0.105956i
$809$	3.34054e7			1.79451			0.897254	−	0.441514i	$-0.145558\pi$
$809$	3.34054e7			1.79451			0.897254	+	0.441514i	$0.145558\pi$
$810$		0			0
$811$	−1.81302e7			−0.967945			−0.483973	−	0.875083i	$-0.660807\pi$
$811$	−1.81302e7			−0.967945			−0.483973	+	0.875083i	$0.660807\pi$
$812$	1.09077e6	−	1.09077e6i	0.0580554	−	0.0580554i
$813$		0			0
$814$			1.29910e7i			0.687198i
$815$		0			0
$816$		0			0
$817$	−766330.	−	766330.i	−0.0401662	−	0.0401662i
$818$	−962792.	−	962792.i	−0.0503094	−	0.0503094i
$819$		0			0
$820$		0			0
$821$			3.30929e6i			0.171347i	0.996323	+	0.0856735i	$0.0273042\pi$
$821$			3.30929e6i			0.171347i	−0.996323	+	0.0856735i	$0.972696\pi$
$822$		0			0
$823$	−1.07231e7	+	1.07231e7i	−0.551850	+	0.551850i	−0.926974	−	0.375125i	$-0.877600\pi$
$823$	−1.07231e7	+	1.07231e7i	−0.551850	+	0.551850i	0.375125	+	0.926974i	$0.377600\pi$
$824$	−5.51177e6			−0.282796
$825$		0			0
$826$	−2.26474e6			−0.115496
$827$	2.08247e6	−	2.08247e6i	0.105880	−	0.105880i	−0.652182	−	0.758062i	$-0.726146\pi$
$827$	2.08247e6	−	2.08247e6i	0.105880	−	0.105880i	0.758062	+	0.652182i	$0.226146\pi$
$828$		0			0
$829$		−	2.31856e7i		−	1.17174i	−0.810404	−	0.585872i	$-0.800752\pi$
$829$		−	2.31856e7i		−	1.17174i	0.810404	−	0.585872i	$-0.199248\pi$
$830$		0			0
$831$		0			0
$832$	727340.	+	727340.i	0.0364275	+	0.0364275i
$833$	2.29352e7	+	2.29352e7i	1.14522	+	1.14522i
$834$		0			0
$835$		0			0
$836$		−	929863.i		−	0.0460154i
$837$		0			0
$838$	−3.69850e6	+	3.69850e6i	−0.181934	+	0.181934i
$839$	3.40410e7			1.66954			0.834771	−	0.550598i	$-0.185600\pi$
$839$	3.40410e7			1.66954			0.834771	+	0.550598i	$0.185600\pi$
$840$		0			0
$841$	2.91061e7			1.41904
$842$	1.34861e6	−	1.34861e6i	0.0655548	−	0.0655548i
$843$		0			0
$844$			9.27210e6i			0.448045i
$845$		0			0
$846$		0			0
$847$	−663478.	−	663478.i	−0.0317774	−	0.0317774i
$848$	4.27702e6	+	4.27702e6i	0.204245	+	0.204245i
$849$		0			0
$850$		0			0
$851$			1.62460e7i			0.768993i
$852$		0			0
$853$	5.07882e6	−	5.07882e6i	0.238996	−	0.238996i	−0.577438	−	0.816434i	$-0.695948\pi$
$853$	5.07882e6	−	5.07882e6i	0.238996	−	0.238996i	0.816434	+	0.577438i	$0.195948\pi$
$854$	−1.95466e6			−0.0917122
$855$		0			0
$856$	1.48215e7			0.691363
$857$	2.78606e7	−	2.78606e7i	1.29580	−	1.29580i	0.364663	−	0.931139i	$-0.381184\pi$
$857$	2.78606e7	−	2.78606e7i	1.29580	−	1.29580i	0.931139	−	0.364663i	$-0.118816\pi$
$858$		0			0
$859$			4.83134e6i			0.223401i	0.993742	+	0.111700i	$0.0356297\pi$
$859$			4.83134e6i			0.223401i	−0.993742	+	0.111700i	$0.964370\pi$
$860$		0			0
$861$		0			0
$862$	−3.70450e6	−	3.70450e6i	−0.169809	−	0.169809i
$863$	−2.21932e7	−	2.21932e7i	−1.01436	−	1.01436i	−0.999895	−	0.0144652i	$-0.995395\pi$
$863$	−2.21932e7	−	2.21932e7i	−1.01436	−	1.01436i	−0.0144652	−	0.999895i	$-0.504605\pi$
$864$		0			0
$865$		0			0
$866$		−	2.05273e7i		−	0.930118i
$867$		0			0
$868$	68692.3	−	68692.3i	0.00309463	−	0.00309463i
$869$	1.30242e7			0.585062
$870$		0			0
$871$	1.42695e7			0.637330
$872$	8.87344e6	−	8.87344e6i	0.395185	−	0.395185i
$873$		0			0
$874$		−	1.16285e6i		−	0.0514925i
$875$		0			0
$876$		0			0
$877$	1.41235e6	+	1.41235e6i	0.0620074	+	0.0620074i	0.737430	−	0.675423i	$-0.236039\pi$
$877$	1.41235e6	+	1.41235e6i	0.0620074	+	0.0620074i	−0.675423	+	0.737430i	$0.736039\pi$
$878$	2.49770e6	+	2.49770e6i	0.109346	+	0.109346i
$879$		0			0
$880$		0			0
$881$		−	2.50777e7i		−	1.08855i	−0.838908	−	0.544274i	$-0.816805\pi$
$881$		−	2.50777e7i		−	1.08855i	0.838908	−	0.544274i	$-0.183195\pi$
$882$		0			0
$883$	1.84135e7	−	1.84135e7i	0.794759	−	0.794759i	−0.187505	−	0.982264i	$-0.560040\pi$
$883$	1.84135e7	−	1.84135e7i	0.794759	−	0.794759i	0.982264	+	0.187505i	$0.0600399\pi$
$884$	7.84165e6			0.337502
$885$		0			0
$886$	−398812.			−0.0170681
$887$	−3.11782e7	+	3.11782e7i	−1.33058	+	1.33058i	−0.425733	+	0.904849i	$0.639984\pi$
$887$	−3.11782e7	+	3.11782e7i	−1.33058	+	1.33058i	−0.904849	+	0.425733i	$0.860016\pi$
$888$		0			0
$889$			2.68609e6i			0.113990i
$890$		0			0
$891$		0			0
$892$	−1.22966e7	−	1.22966e7i	−0.517455	−	0.517455i
$893$	−1.55635e6	−	1.55635e6i	−0.0653098	−	0.0653098i
$894$		0			0
$895$		0			0
$896$			224250.i			0.00933172i
$897$		0			0
$898$	1.60280e7	−	1.60280e7i	0.663267	−	0.663267i
$899$	3.12470e6			0.128946
$900$		0			0
$901$	4.61117e7			1.89234
$902$	1.38978e6	−	1.38978e6i	0.0568763	−	0.0568763i
$903$		0			0
$904$			628154.i			0.0255650i
$905$		0			0
$906$		0			0
$907$	−2.01780e7	−	2.01780e7i	−0.814441	−	0.814441i	0.170855	−	0.985296i	$-0.445347\pi$
$907$	−2.01780e7	−	2.01780e7i	−0.814441	−	0.814441i	−0.985296	+	0.170855i	$0.945347\pi$
$908$	1.12251e7	+	1.12251e7i	0.451832	+	0.451832i
$909$		0			0
$910$		0			0
$911$			2.83269e7i			1.13085i	0.824801	+	0.565423i	$0.191287\pi$
$911$			2.83269e7i			1.13085i	−0.824801	+	0.565423i	$0.808713\pi$
$912$		0			0
$913$	−1.54707e7	+	1.54707e7i	−0.614232	+	0.614232i
$914$	9.09979e6			0.360301
$915$		0			0
$916$	−557284.			−0.0219451
$917$	3.64956e6	−	3.64956e6i	0.143323	−	0.143323i
$918$		0			0
$919$		−	3.33527e7i		−	1.30269i	−0.758780	−	0.651346i	$-0.774204\pi$
$919$		−	3.33527e7i		−	1.30269i	0.758780	−	0.651346i	$-0.225796\pi$
$920$		0			0
$921$		0			0
$922$	−2.35735e7	−	2.35735e7i	−0.913266	−	0.913266i
$923$	5.85530e6	+	5.85530e6i	0.226227	+	0.226227i
$924$		0			0
$925$		0			0
$926$			2.73023e7i			1.04634i
$927$		0			0
$928$	−5.10036e6	+	5.10036e6i	−0.194416	+	0.194416i
$929$	6.74711e6			0.256495			0.128247	−	0.991742i	$-0.459065\pi$
$929$	6.74711e6			0.256495			0.128247	+	0.991742i	$0.459065\pi$
$930$		0			0
$931$	2.01572e6			0.0762179
$932$	−1.28590e7	+	1.28590e7i	−0.484918	+	0.484918i
$933$		0			0
$934$		−	8.93144e6i		−	0.335007i
$935$		0			0
$936$		0			0
$937$	−3.35505e7	−	3.35505e7i	−1.24839	−	1.24839i	−0.956431	−	0.291959i	$-0.905693\pi$
$937$	−3.35505e7	−	3.35505e7i	−1.24839	−	1.24839i	−0.291959	−	0.956431i	$-0.594307\pi$
$938$	2.19975e6	+	2.19975e6i	0.0816332	+	0.0816332i
$939$		0			0
$940$		0			0
$941$		−	3.88609e6i		−	0.143067i	−0.997438	−	0.0715334i	$-0.977211\pi$
$941$		−	3.88609e6i		−	0.143067i	0.997438	−	0.0715334i	$-0.0227892\pi$
$942$		0			0
$943$	1.73800e6	−	1.73800e6i	0.0636461	−	0.0636461i
$944$	1.05898e7			0.386773
$945$		0			0
$946$	1.71266e7			0.622221
$947$	−1.38895e7	+	1.38895e7i	−0.503281	+	0.503281i	−0.912456	−	0.409175i	$-0.865817\pi$
$947$	−1.38895e7	+	1.38895e7i	−0.503281	+	0.503281i	0.409175	+	0.912456i	$0.365817\pi$
$948$		0			0
$949$			7.23296e6i			0.260706i
$950$		0			0
$951$		0			0
$952$	1.20885e6	+	1.20885e6i	0.0432294	+	0.0432294i
$953$	−2.32501e7	−	2.32501e7i	−0.829263	−	0.829263i	0.158152	−	0.987415i	$-0.449447\pi$
$953$	−2.32501e7	−	2.32501e7i	−0.829263	−	0.829263i	−0.987415	+	0.158152i	$0.949447\pi$
$954$		0			0
$955$		0			0
$956$		−	1.77946e7i		−	0.629715i
$957$		0			0
$958$	1.50865e7	−	1.50865e7i	0.531099	−	0.531099i
$959$	1.87358e6			0.0657849
$960$		0			0
$961$	−2.84324e7			−0.993127
$962$	−4.81427e6	+	4.81427e6i	−0.167723	+	0.167723i
$963$		0			0
$964$		−	1.54819e7i		−	0.536575i
$965$		0			0
$966$		0			0
$967$	1.98191e7	+	1.98191e7i	0.681583	+	0.681583i	0.960357	−	0.278774i	$-0.0899280\pi$
$967$	1.98191e7	+	1.98191e7i	0.681583	+	0.681583i	−0.278774	+	0.960357i	$0.589928\pi$
$968$	3.10238e6	+	3.10238e6i	0.106416	+	0.106416i
$969$		0			0
$970$		0			0
$971$		−	3.25338e7i		−	1.10735i	−0.832732	−	0.553677i	$-0.813224\pi$
$971$		−	3.25338e7i		−	1.10735i	0.832732	−	0.553677i	$-0.186776\pi$
$972$		0			0
$973$	−3.27931e6	+	3.27931e6i	−0.111045	+	0.111045i
$974$	1.22377e7			0.413334
$975$		0			0
$976$	9.13987e6			0.307125
$977$	1.01626e7	−	1.01626e7i	0.340618	−	0.340618i	−0.515981	−	0.856600i	$-0.672573\pi$
$977$	1.01626e7	−	1.01626e7i	0.340618	−	0.340618i	0.856600	+	0.515981i	$0.172573\pi$
$978$		0			0
$979$			3.10969e7i			1.03696i
$980$		0			0
$981$		0			0
$982$	2.39224e7	+	2.39224e7i	0.791638	+	0.791638i
$983$	−5.47881e6	−	5.47881e6i	−0.180843	−	0.180843i	0.610880	−	0.791723i	$-0.290816\pi$
$983$	−5.47881e6	−	5.47881e6i	−0.180843	−	0.180843i	−0.791723	+	0.610880i	$0.790816\pi$
$984$		0			0
$985$		0			0
$986$			5.49884e7i			1.80127i
$987$		0			0
$988$	344593.	−	344593.i	0.0112309	−	0.0112309i
$989$	2.14178e7			0.696282
$990$		0			0
$991$	1.66714e7			0.539246			0.269623	−	0.962966i	$-0.413101\pi$
$991$	1.66714e7			0.539246			0.269623	+	0.962966i	$0.413101\pi$
$992$	−321201.	+	321201.i	−0.0103633	+	0.0103633i
$993$		0			0
$994$			1.80528e6i			0.0579532i
$995$		0			0
$996$		0			0
$997$	5.58846e6	+	5.58846e6i	0.178055	+	0.178055i	0.790507	−	0.612452i	$-0.209817\pi$
$997$	5.58846e6	+	5.58846e6i	0.178055	+	0.178055i	−0.612452	+	0.790507i	$0.709817\pi$
$998$	7.09214e6	+	7.09214e6i	0.225398	+	0.225398i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.6.f.e.107.5		12
3.2	odd	2	inner	450.6.f.e.107.2		12
5.2	odd	4		90.6.f.c.53.4	yes	12
5.3	odd	4	inner	450.6.f.e.143.2		12
5.4	even	2		90.6.f.c.17.3	✓	12
15.2	even	4		90.6.f.c.53.3	yes	12
15.8	even	4	inner	450.6.f.e.143.5		12
15.14	odd	2		90.6.f.c.17.4	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.6.f.c.17.3	✓	12	5.4	even	2
90.6.f.c.17.4	yes	12	15.14	odd	2
90.6.f.c.53.3	yes	12	15.2	even	4
90.6.f.c.53.4	yes	12	5.2	odd	4
450.6.f.e.107.2		12	3.2	odd	2	inner
450.6.f.e.107.5		12	1.1	even	1	trivial
450.6.f.e.143.2		12	5.3	odd	4	inner
450.6.f.e.143.5		12	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)

\(f(q)\)	\(=\)	\(q+(2.82843 - 2.82843i) q^{2} -16.0000i q^{4} +(9.67825 + 9.67825i) q^{7} +(-45.2548 - 45.2548i) q^{8} +O(q^{10})\)	\(q+(2.82843 - 2.82843i) q^{2} -16.0000i q^{4} +(9.67825 + 9.67825i) q^{7} +(-45.2548 - 45.2548i) q^{8} -479.171i q^{11} +(177.573 - 177.573i) q^{13} +54.7484 q^{14} -256.000 q^{16} +(-1380.00 + 1380.00i) q^{17} +121.286i q^{19} +(-1355.30 - 1355.30i) q^{22} +(-1694.88 - 1694.88i) q^{23} -1004.51i q^{26} +(154.852 - 154.852i) q^{28} +7043.95 q^{29} +443.600 q^{31} +(-724.077 + 724.077i) q^{32} +7806.47i q^{34} +(-4792.67 - 4792.67i) q^{37} +(343.047 + 343.047i) q^{38} +1025.44i q^{41} +(-6318.40 + 6318.40i) q^{43} -7666.73 q^{44} -9587.68 q^{46} +(-12832.1 + 12832.1i) q^{47} -16619.7i q^{49} +(-2841.17 - 2841.17i) q^{52} +(-16707.1 - 16707.1i) q^{53} -875.975i q^{56} +(19923.3 - 19923.3i) q^{58} -41366.2 q^{59} -35702.6 q^{61} +(1254.69 - 1254.69i) q^{62} +4096.00i q^{64} +(40179.3 + 40179.3i) q^{67} +(22080.0 + 22080.0i) q^{68} +32974.0i q^{71} +(-20366.1 + 20366.1i) q^{73} -27111.4 q^{74} +1940.57 q^{76} +(4637.53 - 4637.53i) q^{77} +27180.7i q^{79} +(2900.40 + 2900.40i) q^{82} +(-32286.4 - 32286.4i) q^{83} +35742.3i q^{86} +(-21684.8 + 21684.8i) q^{88} -64897.3 q^{89} +3437.20 q^{91} +(-27118.1 + 27118.1i) q^{92} +72589.4i q^{94} +(-22306.4 - 22306.4i) q^{97} +(-47007.5 - 47007.5i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 144 q^{7}+O(q^{10}) \) 12 * q - 144 * q^7	\( 12 q - 144 q^{7} + 276 q^{13} - 3072 q^{16} - 384 q^{22} - 2304 q^{28} - 58512 q^{31} - 25764 q^{37} - 16080 q^{43} - 60672 q^{46} - 4416 q^{52} - 23952 q^{58} - 145200 q^{61} - 33552 q^{67} + 158988 q^{73} - 86016 q^{76} + 75024 q^{82} - 6144 q^{88} - 465024 q^{91} + 631116 q^{97}+O(q^{100}) \) 12 * q - 144 * q^7 + 276 * q^13 - 3072 * q^16 - 384 * q^22 - 2304 * q^28 - 58512 * q^31 - 25764 * q^37 - 16080 * q^43 - 60672 * q^46 - 4416 * q^52 - 23952 * q^58 - 145200 * q^61 - 33552 * q^67 + 158988 * q^73 - 86016 * q^76 + 75024 * q^82 - 6144 * q^88 - 465024 * q^91 + 631116 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more