LMFDB - Embedded newform 336.5.d.b.113.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,5,Mod(113,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("336.113");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	336.d (of order $2$, degree $1$, not 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:	$34.7323075962$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 82x^{6} + 2017x^{4} + 13020x^{2} + 756 $ x^8 + 82x^6 + 2017x^4 + 13020*x^2 + 756
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{3}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.6
Root			$0.242064i$ of defining polynomial
Character	$\chi$	$=$	336.113
Dual form			336.5.d.b.113.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.46977 + 7.81161i) q^{3} -30.4863i q^{5} -18.5203 q^{7} +(-41.0424 + 69.8321i) q^{9} +O(q^{10})$	\(q+(4.46977 + 7.81161i) q^{3} -30.4863i q^{5} -18.5203 q^{7} +(-41.0424 + 69.8321i) q^{9} -88.0887i q^{11} -15.4739 q^{13} +(238.147 - 136.266i) q^{15} +331.481i q^{17} -298.104 q^{19} +(-82.7812 - 144.673i) q^{21} +815.926i q^{23} -304.412 q^{25} +(-728.951 - 8.47391i) q^{27} +1565.67i q^{29} +1379.33 q^{31} +(688.115 - 393.736i) q^{33} +564.613i q^{35} -556.508 q^{37} +(-69.1647 - 120.876i) q^{39} +66.1756i q^{41} +1119.70 q^{43} +(2128.92 + 1251.23i) q^{45} +3163.36i q^{47} +343.000 q^{49} +(-2589.40 + 1481.64i) q^{51} -212.973i q^{53} -2685.50 q^{55} +(-1332.45 - 2328.67i) q^{57} -616.765i q^{59} -3132.18 q^{61} +(760.116 - 1293.31i) q^{63} +471.741i q^{65} -7028.51 q^{67} +(-6373.69 + 3647.00i) q^{69} +2049.88i q^{71} -6134.03 q^{73} +(-1360.65 - 2377.94i) q^{75} +1631.43i q^{77} +7314.45 q^{79} +(-3192.04 - 5732.15i) q^{81} -6100.56i q^{83} +10105.6 q^{85} +(-12230.4 + 6998.18i) q^{87} +6802.07i q^{89} +286.581 q^{91} +(6165.27 + 10774.8i) q^{93} +9088.06i q^{95} -9391.65 q^{97} +(6151.42 + 3615.37i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{3} + 64 q^{9}+O(q^{10}) $ 8 * q + 2 * q^3 + 64 * q^9	$ 8 q + 2 q^{3} + 64 q^{9} + 420 q^{13} - 76 q^{15} + 372 q^{19} + 98 q^{21} + 1056 q^{25} + 1862 q^{27} + 2776 q^{31} + 1396 q^{33} - 2560 q^{37} + 2540 q^{39} - 4720 q^{43} + 9700 q^{45} + 2744 q^{49} - 4764 q^{51} - 184 q^{55} - 14144 q^{57} + 972 q^{61} + 6076 q^{63} - 10200 q^{67} - 5760 q^{69} - 32008 q^{73} - 2114 q^{75} + 23168 q^{79} - 17216 q^{81} + 32016 q^{85} - 50764 q^{87} - 11956 q^{91} + 31848 q^{93} + 28112 q^{97} + 32432 q^{99}+O(q^{100}) $ 8 * q + 2 * q^3 + 64 * q^9 + 420 * q^13 - 76 * q^15 + 372 * q^19 + 98 * q^21 + 1056 * q^25 + 1862 * q^27 + 2776 * q^31 + 1396 * q^33 - 2560 * q^37 + 2540 * q^39 - 4720 * q^43 + 9700 * q^45 + 2744 * q^49 - 4764 * q^51 - 184 * q^55 - 14144 * q^57 + 972 * q^61 + 6076 * q^63 - 10200 * q^67 - 5760 * q^69 - 32008 * q^73 - 2114 * q^75 + 23168 * q^79 - 17216 * q^81 + 32016 * q^85 - 50764 * q^87 - 11956 * q^91 + 31848 * q^93 + 28112 * q^97 + 32432 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$-1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	4.46977	+	7.81161i	0.496641	+	0.867956i
$3$	4.46977	+	7.81161i	0.496641	+	0.867956i
$4$		0			0
$5$		−	30.4863i		−	1.21945i	−0.792613	−	0.609725i	$-0.791280\pi$
$5$		−	30.4863i		−	1.21945i	0.792613	−	0.609725i	$-0.208720\pi$
$6$		0			0
$7$	−18.5203			−0.377964
$7$	−18.5203			−0.377964
$8$		0			0
$9$	−41.0424	+	69.8321i	−0.506696	+	0.862125i
$10$		0			0
$11$		−	88.0887i		−	0.728006i	−0.931398	−	0.364003i	$-0.881410\pi$
$11$		−	88.0887i		−	0.728006i	0.931398	−	0.364003i	$-0.118590\pi$
$12$		0			0
$13$	−15.4739			−0.0915615			−0.0457808	−	0.998952i	$-0.514578\pi$
$13$	−15.4739			−0.0915615			−0.0457808	+	0.998952i	$0.514578\pi$
$14$		0			0
$15$	238.147	−	136.266i	1.05843	−	0.605628i
$16$		0			0
$17$			331.481i			1.14699i	0.819207	+	0.573497i	$0.194414\pi$
$17$			331.481i			1.14699i	−0.819207	+	0.573497i	$0.805586\pi$
$18$		0			0
$19$	−298.104			−0.825772			−0.412886	−	0.910783i	$-0.635479\pi$
$19$	−298.104			−0.825772			−0.412886	+	0.910783i	$0.635479\pi$
$20$		0			0
$21$	−82.7812	−	144.673i	−0.187713	−	0.328057i
$22$		0			0
$23$			815.926i			1.54239i	0.636598	+	0.771196i	$0.280341\pi$
$23$			815.926i			1.54239i	−0.636598	+	0.771196i	$0.719659\pi$
$24$		0			0
$25$	−304.412			−0.487058
$26$		0			0
$27$	−728.951	−	8.47391i	−0.999932	−	0.0116240i
$28$		0			0
$29$			1565.67i			1.86168i	0.365430	+	0.930839i	$0.380922\pi$
$29$			1565.67i			1.86168i	−0.365430	+	0.930839i	$0.619078\pi$
$30$		0			0
$31$	1379.33			1.43530			0.717652	−	0.696402i	$-0.245217\pi$
$31$	1379.33			1.43530			0.717652	+	0.696402i	$0.245217\pi$
$32$		0			0
$33$	688.115	−	393.736i	0.631878	−	0.361557i
$34$		0			0
$35$			564.613i			0.460909i
$36$		0			0
$37$	−556.508			−0.406507			−0.203254	−	0.979126i	$-0.565152\pi$
$37$	−556.508			−0.406507			−0.203254	+	0.979126i	$0.565152\pi$
$38$		0			0
$39$	−69.1647	−	120.876i	−0.0454732	−	0.0794714i
$40$		0			0
$41$			66.1756i			0.0393668i	0.999806	+	0.0196834i	$0.00626583\pi$
$41$			66.1756i			0.0393668i	−0.999806	+	0.0196834i	$0.993734\pi$
$42$		0			0
$43$	1119.70			0.605572			0.302786	−	0.953059i	$-0.402083\pi$
$43$	1119.70			0.605572			0.302786	+	0.953059i	$0.402083\pi$
$44$		0			0
$45$	2128.92	+	1251.23i	1.05132	+	0.617891i
$46$		0			0
$47$			3163.36i			1.43203i	0.698085	+	0.716015i	$0.254036\pi$
$47$			3163.36i			1.43203i	−0.698085	+	0.716015i	$0.745964\pi$
$48$		0			0
$49$	343.000			0.142857
$50$		0			0
$51$	−2589.40	+	1481.64i	−0.995541	+	0.569644i
$52$		0			0
$53$		−	212.973i		−	0.0758181i	−0.999281	−	0.0379090i	$-0.987930\pi$
$53$		−	212.973i		−	0.0758181i	0.999281	−	0.0379090i	$-0.0120697\pi$
$54$		0			0
$55$	−2685.50			−0.887767
$56$		0			0
$57$	−1332.45	−	2328.67i	−0.410112	−	0.716734i
$58$		0			0
$59$		−	616.765i		−	0.177180i	−0.996068	−	0.0885902i	$-0.971764\pi$
$59$		−	616.765i		−	0.177180i	0.996068	−	0.0885902i	$-0.0282362\pi$
$60$		0			0
$61$	−3132.18			−0.841757			−0.420879	−	0.907117i	$-0.638278\pi$
$61$	−3132.18			−0.841757			−0.420879	+	0.907117i	$0.638278\pi$
$62$		0			0
$63$	760.116	−	1293.31i	0.191513	−	0.325852i
$64$		0			0
$65$			471.741i			0.111655i
$66$		0			0
$67$	−7028.51			−1.56572			−0.782859	−	0.622199i	$-0.786239\pi$
$67$	−7028.51			−1.56572			−0.782859	+	0.622199i	$0.786239\pi$
$68$		0			0
$69$	−6373.69	+	3647.00i	−1.33873	+	0.766015i
$70$		0			0
$71$			2049.88i			0.406642i	0.979112	+	0.203321i	$0.0651734\pi$
$71$			2049.88i			0.406642i	−0.979112	+	0.203321i	$0.934827\pi$
$72$		0			0
$73$	−6134.03			−1.15107			−0.575533	−	0.817779i	$-0.695205\pi$
$73$	−6134.03			−1.15107			−0.575533	+	0.817779i	$0.695205\pi$
$74$		0			0
$75$	−1360.65	−	2377.94i	−0.241893	−	0.422745i
$76$		0			0
$77$			1631.43i			0.275160i
$78$		0			0
$79$	7314.45			1.17200			0.586000	−	0.810311i	$-0.300702\pi$
$79$	7314.45			1.17200			0.586000	+	0.810311i	$0.300702\pi$
$80$		0			0
$81$	−3192.04	−	5732.15i	−0.486518	−	0.873671i
$82$		0			0
$83$		−	6100.56i		−	0.885551i	−0.896633	−	0.442775i	$-0.853994\pi$
$83$		−	6100.56i		−	0.885551i	0.896633	−	0.442775i	$-0.146006\pi$
$84$		0			0
$85$	10105.6			1.39870
$86$		0			0
$87$	−12230.4	+	6998.18i	−1.61585	+	0.924585i
$88$		0			0
$89$			6802.07i			0.858739i	0.903129	+	0.429369i	$0.141264\pi$
$89$			6802.07i			0.858739i	−0.903129	+	0.429369i	$0.858736\pi$
$90$		0			0
$91$	286.581			0.0346070
$92$		0			0
$93$	6165.27	+	10774.8i	0.712830	+	1.24578i
$94$		0			0
$95$			9088.06i			1.00699i
$96$		0			0
$97$	−9391.65			−0.998156			−0.499078	−	0.866557i	$-0.666328\pi$
$97$	−9391.65			−0.998156			−0.499078	+	0.866557i	$0.666328\pi$
$98$		0			0
$99$	6151.42	+	3615.37i	0.627632	+	0.368878i
$100$		0			0
$101$			8054.19i			0.789549i	0.918778	+	0.394774i	$0.129177\pi$
$101$			8054.19i			0.789549i	−0.918778	+	0.394774i	$0.870823\pi$
$102$		0			0
$103$	3170.08			0.298810			0.149405	−	0.988776i	$-0.452264\pi$
$103$	3170.08			0.298810			0.149405	+	0.988776i	$0.452264\pi$
$104$		0			0
$105$	−4410.54	+	2523.69i	−0.400049	+	0.228906i
$106$		0			0
$107$			3148.48i			0.275000i	0.990502	+	0.137500i	$0.0439067\pi$
$107$			3148.48i			0.275000i	−0.990502	+	0.137500i	$0.956093\pi$
$108$		0			0
$109$	4292.91			0.361326			0.180663	−	0.983545i	$-0.442176\pi$
$109$	4292.91			0.361326			0.180663	+	0.983545i	$0.442176\pi$
$110$		0			0
$111$	−2487.46	−	4347.22i	−0.201888	−	0.352830i
$112$		0			0
$113$			4334.19i			0.339430i	0.985493	+	0.169715i	$0.0542848\pi$
$113$			4334.19i			0.339430i	−0.985493	+	0.169715i	$0.945715\pi$
$114$		0			0
$115$	24874.5			1.88087
$116$		0			0
$117$	635.086	−	1080.57i	0.0463939	−	0.0789375i
$118$		0			0
$119$		−	6139.12i		−	0.433523i
$120$		0			0
$121$	6881.37			0.470007
$122$		0			0
$123$	−516.938	+	295.790i	−0.0341687	+	0.0195512i
$124$		0			0
$125$		−	9773.54i		−	0.625507i
$126$		0			0
$127$	−14744.4			−0.914155			−0.457078	−	0.889427i	$-0.651104\pi$
$127$	−14744.4			−0.914155			−0.457078	+	0.889427i	$0.651104\pi$
$128$		0			0
$129$	5004.81	+	8746.67i	0.300751	+	0.525610i
$130$		0			0
$131$			20831.8i			1.21390i	0.794740	+	0.606950i	$0.207607\pi$
$131$			20831.8i			1.21390i	−0.794740	+	0.606950i	$0.792393\pi$
$132$		0			0
$133$	5520.96			0.312112
$134$		0			0
$135$	−258.338	+	22223.0i	−0.0141749	+	1.21937i
$136$		0			0
$137$		−	24279.2i		−	1.29358i	−0.762669	−	0.646789i	$-0.776111\pi$
$137$		−	24279.2i		−	1.29358i	0.762669	−	0.646789i	$-0.223889\pi$
$138$		0			0
$139$	−11786.6			−0.610042			−0.305021	−	0.952346i	$-0.598663\pi$
$139$	−11786.6			−0.610042			−0.305021	+	0.952346i	$0.598663\pi$
$140$		0			0
$141$	−24710.9	+	14139.5i	−1.24294	+	0.711204i
$142$		0			0
$143$			1363.08i			0.0666574i
$144$		0			0
$145$	47731.4			2.27022
$146$		0			0
$147$	1533.13	+	2679.38i	0.0709487	+	0.123994i
$148$		0			0
$149$			6241.91i			0.281154i	0.990070	+	0.140577i	$0.0448958\pi$
$149$			6241.91i			0.281154i	−0.990070	+	0.140577i	$0.955104\pi$
$150$		0			0
$151$	−9337.45			−0.409519			−0.204760	−	0.978812i	$-0.565641\pi$
$151$	−9337.45			−0.409519			−0.204760	+	0.978812i	$0.565641\pi$
$152$		0			0
$153$	−23148.0	−	13604.8i	−0.988852	−	0.581178i
$154$		0			0
$155$		−	42050.5i		−	1.75028i
$156$		0			0
$157$	8467.95			0.343541			0.171771	−	0.985137i	$-0.445051\pi$
$157$	8467.95			0.343541			0.171771	+	0.985137i	$0.445051\pi$
$158$		0			0
$159$	1663.66	−	951.939i	0.0658068	−	0.0376543i
$160$		0			0
$161$		−	15111.2i		−	0.582970i
$162$		0			0
$163$	7222.49			0.271839			0.135919	−	0.990720i	$-0.456601\pi$
$163$	7222.49			0.271839			0.135919	+	0.990720i	$0.456601\pi$
$164$		0			0
$165$	−12003.5	−	20978.0i	−0.440901	−	0.770543i
$166$		0			0
$167$		−	17419.4i		−	0.624599i	−0.949984	−	0.312300i	$-0.898901\pi$
$167$		−	17419.4i		−	0.624599i	0.949984	−	0.312300i	$-0.101099\pi$
$168$		0			0
$169$	−28321.6			−0.991616
$170$		0			0
$171$	12234.9	−	20817.2i	0.418416	−	0.711918i
$172$		0			0
$173$		−	15954.1i		−	0.533066i	−0.963826	−	0.266533i	$-0.914122\pi$
$173$		−	15954.1i		−	0.533066i	0.963826	−	0.266533i	$-0.0858781\pi$
$174$		0			0
$175$	5637.78			0.184091
$176$		0			0
$177$	4817.93	−	2756.80i	0.153785	−	0.0879950i
$178$		0			0
$179$		−	27975.2i		−	0.873105i	−0.899679	−	0.436553i	$-0.856199\pi$
$179$		−	27975.2i		−	0.873105i	0.899679	−	0.436553i	$-0.143801\pi$
$180$		0			0
$181$	25826.2			0.788321			0.394160	−	0.919042i	$-0.371035\pi$
$181$	25826.2			0.788321			0.394160	+	0.919042i	$0.371035\pi$
$182$		0			0
$183$	−14000.1	−	24467.3i	−0.418051	−	0.730608i
$184$		0			0
$185$			16965.9i			0.495715i
$186$		0			0
$187$	29199.8			0.835019
$188$		0			0
$189$	13500.4	+	156.939i	0.377939	+	0.00439347i
$190$		0			0
$191$			44447.1i			1.21836i	0.793031	+	0.609182i	$0.208502\pi$
$191$			44447.1i			1.21836i	−0.793031	+	0.609182i	$0.791498\pi$
$192$		0			0
$193$	17780.1			0.477330			0.238665	−	0.971102i	$-0.423290\pi$
$193$	17780.1			0.477330			0.238665	+	0.971102i	$0.423290\pi$
$194$		0			0
$195$	−3685.06	+	2108.57i	−0.0969114	+	0.0554523i
$196$		0			0
$197$		−	51169.6i		−	1.31850i	−0.751924	−	0.659250i	$-0.770874\pi$
$197$		−	51169.6i		−	1.31850i	0.751924	−	0.659250i	$-0.229126\pi$
$198$		0			0
$199$	7247.46			0.183012			0.0915061	−	0.995805i	$-0.470832\pi$
$199$	7247.46			0.183012			0.0915061	+	0.995805i	$0.470832\pi$
$200$		0			0
$201$	−31415.8	−	54903.9i	−0.777599	−	1.35897i
$202$		0			0
$203$		−	28996.6i		−	0.703648i
$204$		0			0
$205$	2017.45			0.0480059
$206$		0			0
$207$	−56977.8	−	33487.5i	−1.32973	−	0.781524i
$208$		0			0
$209$			26259.6i			0.601167i
$210$		0			0
$211$	32606.5			0.732385			0.366192	−	0.930539i	$-0.380661\pi$
$211$	32606.5			0.732385			0.366192	+	0.930539i	$0.380661\pi$
$212$		0			0
$213$	−16012.9	+	9162.48i	−0.352947	+	0.201955i
$214$		0			0
$215$		−	34135.5i		−	0.738464i
$216$		0			0
$217$	−25545.5			−0.542494
$218$		0			0
$219$	−27417.7	−	47916.6i	−0.571666	−	0.999075i
$220$		0			0
$221$		−	5129.31i		−	0.105021i
$222$		0			0
$223$	55353.6			1.11311			0.556553	−	0.830812i	$-0.312124\pi$
$223$	55353.6			1.11311			0.556553	+	0.830812i	$0.312124\pi$
$224$		0			0
$225$	12493.8	−	21257.7i	0.246791	−	0.419905i
$226$		0			0
$227$		−	5840.79i		−	0.113349i	−0.998393	−	0.0566747i	$-0.981950\pi$
$227$		−	5840.79i		−	0.113349i	0.998393	−	0.0566747i	$-0.0180498\pi$
$228$		0			0
$229$	−2103.16			−0.0401053			−0.0200526	−	0.999799i	$-0.506383\pi$
$229$	−2103.16			−0.0401053			−0.0200526	+	0.999799i	$0.506383\pi$
$230$		0			0
$231$	−12744.1	+	7292.09i	−0.238827	+	0.136656i
$232$		0			0
$233$			61482.6i			1.13251i	0.824232	+	0.566253i	$0.191607\pi$
$233$			61482.6i			1.13251i	−0.824232	+	0.566253i	$0.808393\pi$
$234$		0			0
$235$	96438.8			1.74629
$236$		0			0
$237$	32693.9	+	57137.6i	0.582063	+	1.01724i
$238$		0			0
$239$			74258.2i			1.30001i	0.759928	+	0.650007i	$0.225234\pi$
$239$			74258.2i			1.30001i	−0.759928	+	0.650007i	$0.774766\pi$
$240$		0			0
$241$	111286.			1.91605			0.958026	−	0.286683i	$-0.0925525\pi$
$241$	111286.			1.91605			0.958026	+	0.286683i	$0.0925525\pi$
$242$		0			0
$243$	30509.6	−	50556.4i	0.516683	−	0.856177i
$244$		0			0
$245$		−	10456.8i		−	0.174207i
$246$		0			0
$247$	4612.83			0.0756090
$248$		0			0
$249$	47655.2	−	27268.1i	0.768619	−	0.439800i
$250$		0			0
$251$		−	41512.0i		−	0.658910i	−0.944171	−	0.329455i	$-0.893135\pi$
$251$		−	41512.0i		−	0.658910i	0.944171	−	0.329455i	$-0.106865\pi$
$252$		0			0
$253$	71873.9			1.12287
$254$		0			0
$255$	45169.8	+	78941.2i	0.694653	+	1.21401i
$256$		0			0
$257$			53152.3i			0.804741i	0.915477	+	0.402370i	$0.131814\pi$
$257$			53152.3i			0.804741i	−0.915477	+	0.402370i	$0.868186\pi$
$258$		0			0
$259$	10306.7			0.153645
$260$		0			0
$261$	−109334.	−	64258.9i	−1.60500	−	0.943305i
$262$		0			0
$263$			64112.0i			0.926890i	0.886126	+	0.463445i	$0.153387\pi$
$263$			64112.0i			0.926890i	−0.886126	+	0.463445i	$0.846613\pi$
$264$		0			0
$265$	−6492.75			−0.0924564
$266$		0			0
$267$	−53135.1	+	30403.7i	−0.745348	+	0.426484i
$268$		0			0
$269$		−	115300.i		−	1.59341i	−0.604371	−	0.796703i	$-0.706576\pi$
$269$		−	115300.i		−	1.59341i	0.604371	−	0.796703i	$-0.293424\pi$
$270$		0			0
$271$	−108645.			−1.47936			−0.739678	−	0.672961i	$-0.765022\pi$
$271$	−108645.			−1.47936			−0.739678	+	0.672961i	$0.765022\pi$
$272$		0			0
$273$	1280.95	+	2238.66i	0.0171872	+	0.0300374i
$274$		0			0
$275$			26815.2i			0.354582i
$276$		0			0
$277$	−45058.9			−0.587248			−0.293624	−	0.955921i	$-0.594861\pi$
$277$	−45058.9			−0.587248			−0.293624	+	0.955921i	$0.594861\pi$
$278$		0			0
$279$	−56610.9	+	96321.3i	−0.727263	+	1.23741i
$280$		0			0
$281$		−	19372.2i		−	0.245339i	−0.992448	−	0.122670i	$-0.960854\pi$
$281$		−	19372.2i		−	0.245339i	0.992448	−	0.122670i	$-0.0391455\pi$
$282$		0			0
$283$	12147.5			0.151675			0.0758375	−	0.997120i	$-0.475837\pi$
$283$	12147.5			0.151675			0.0758375	+	0.997120i	$0.475837\pi$
$284$		0			0
$285$	−70992.4	+	40621.5i	−0.874021	+	0.500111i
$286$		0			0
$287$		−	1225.59i		−	0.0148793i
$288$		0			0
$289$	−26359.0			−0.315597
$290$		0			0
$291$	−41978.5	−	73363.9i	−0.495725	−	0.866356i
$292$		0			0
$293$		−	39539.1i		−	0.460566i	−0.973124	−	0.230283i	$-0.926035\pi$
$293$		−	39539.1i		−	0.460566i	0.973124	−	0.230283i	$-0.0739651\pi$
$294$		0			0
$295$	−18802.9			−0.216063
$296$		0			0
$297$	−746.456	+	64212.4i	−0.00846236	+	0.727957i
$298$		0			0
$299$		−	12625.6i		−	0.141224i
$300$		0			0
$301$	−20737.2			−0.228885
$302$		0			0
$303$	−62916.2	+	36000.3i	−0.685294	+	0.392122i
$304$		0			0
$305$			95488.4i			1.02648i
$306$		0			0
$307$	51639.4			0.547904			0.273952	−	0.961743i	$-0.411669\pi$
$307$	51639.4			0.547904			0.273952	+	0.961743i	$0.411669\pi$
$308$		0			0
$309$	14169.5	+	24763.4i	0.148401	+	0.259354i
$310$		0			0
$311$		−	65566.1i		−	0.677889i	−0.940806	−	0.338944i	$-0.889930\pi$
$311$		−	65566.1i		−	0.677889i	0.940806	−	0.338944i	$-0.110070\pi$
$312$		0			0
$313$	58295.3			0.595038			0.297519	−	0.954716i	$-0.403841\pi$
$313$	58295.3			0.595038			0.297519	+	0.954716i	$0.403841\pi$
$314$		0			0
$315$	−39428.1	−	23173.1i	−0.397361	−	0.233541i
$316$		0			0
$317$			125449.i			1.24839i	0.781271	+	0.624193i	$0.214572\pi$
$317$			125449.i			1.24839i	−0.781271	+	0.624193i	$0.785428\pi$
$318$		0			0
$319$	137918.			1.35531
$320$		0			0
$321$	−24594.6	+	14072.9i	−0.238688	+	0.136576i
$322$		0			0
$323$		−	98815.8i		−	0.947156i
$324$		0			0
$325$	4710.43			0.0445958
$326$		0			0
$327$	19188.3	+	33534.5i	0.179449	+	0.313615i
$328$		0			0
$329$		−	58586.2i		−	0.541257i
$330$		0			0
$331$	−84438.8			−0.770701			−0.385351	−	0.922770i	$-0.625920\pi$
$331$	−84438.8			−0.770701			−0.385351	+	0.922770i	$0.625920\pi$
$332$		0			0
$333$	22840.4	−	38862.1i	0.205976	−	0.350460i
$334$		0			0
$335$			214273.i			1.90931i
$336$		0			0
$337$	−185242.			−1.63110			−0.815549	−	0.578689i	$-0.803565\pi$
$337$	−185242.			−1.63110			−0.815549	+	0.578689i	$0.803565\pi$
$338$		0			0
$339$	−33856.9	+	19372.8i	−0.294611	+	0.168575i
$340$		0			0
$341$		−	121503.i		−	1.04491i
$342$		0			0
$343$	−6352.45			−0.0539949
$344$		0			0
$345$	111183.	+	194310.i	0.934117	+	1.63251i
$346$		0			0
$347$		−	83517.4i		−	0.693614i	−0.937937	−	0.346807i	$-0.887266\pi$
$347$		−	83517.4i		−	0.693614i	0.937937	−	0.346807i	$-0.112734\pi$
$348$		0			0
$349$	100015.			0.821134			0.410567	−	0.911831i	$-0.365331\pi$
$349$	100015.			0.821134			0.410567	+	0.911831i	$0.365331\pi$
$350$		0			0
$351$	11279.7	+	131.124i	0.0915554	+	0.00106431i
$352$		0			0
$353$		−	57357.4i		−	0.460299i	−0.973155	−	0.230149i	$-0.926078\pi$
$353$		−	57357.4i		−	0.460299i	0.973155	−	0.230149i	$-0.0739215\pi$
$354$		0			0
$355$	62493.2			0.495879
$356$		0			0
$357$	47956.4	−	27440.4i	0.376279	−	0.215305i
$358$		0			0
$359$			165789.i			1.28637i	0.765709	+	0.643187i	$0.222388\pi$
$359$			165789.i			1.28637i	−0.765709	+	0.643187i	$0.777612\pi$
$360$		0			0
$361$	−41455.2			−0.318101
$362$		0			0
$363$	30758.1	+	53754.6i	0.233425	+	0.407946i
$364$		0			0
$365$			187004.i			1.40367i
$366$		0			0
$367$	78946.7			0.586141			0.293070	−	0.956091i	$-0.405323\pi$
$367$	78946.7			0.586141			0.293070	+	0.956091i	$0.405323\pi$
$368$		0			0
$369$	−4621.18	−	2716.01i	−0.0339391	−	0.0199470i
$370$		0			0
$371$			3944.32i			0.0286565i
$372$		0			0
$373$	−236493.			−1.69981			−0.849907	−	0.526932i	$-0.823342\pi$
$373$	−236493.			−1.69981			−0.849907	+	0.526932i	$0.823342\pi$
$374$		0			0
$375$	76347.1	−	43685.4i	0.542912	−	0.310652i
$376$		0			0
$377$		−	24227.0i		−	0.170458i
$378$		0			0
$379$	−111501.			−0.776245			−0.388123	−	0.921608i	$-0.626876\pi$
$379$	−111501.			−0.776245			−0.388123	+	0.921608i	$0.626876\pi$
$380$		0			0
$381$	−65904.1	−	115178.i	−0.454007	−	0.793447i
$382$		0			0
$383$		−	118675.i		−	0.809024i	−0.914533	−	0.404512i	$-0.867441\pi$
$383$		−	118675.i		−	0.809024i	0.914533	−	0.404512i	$-0.132559\pi$
$384$		0			0
$385$	49736.1			0.335544
$386$		0			0
$387$	−45955.2	+	78191.1i	−0.306841	+	0.522078i
$388$		0			0
$389$			278244.i			1.83877i	0.393361	+	0.919384i	$0.371312\pi$
$389$			278244.i			1.83877i	−0.393361	+	0.919384i	$0.628688\pi$
$390$		0			0
$391$	−270464.			−1.76912
$392$		0			0
$393$	−162729.	+	93113.1i	−1.05361	+	0.602873i
$394$		0			0
$395$		−	222990.i		−	1.42919i
$396$		0			0
$397$	108561.			0.688798			0.344399	−	0.938823i	$-0.388083\pi$
$397$	108561.			0.688798			0.344399	+	0.938823i	$0.388083\pi$
$398$		0			0
$399$	24677.4	+	43127.5i	0.155008	+	0.270900i
$400$		0			0
$401$		−	15517.8i		−	0.0965029i	−0.998835	−	0.0482514i	$-0.984635\pi$
$401$		−	15517.8i		−	0.0965029i	0.998835	−	0.0482514i	$-0.0153649\pi$
$402$		0			0
$403$	−21343.6			−0.131419
$404$		0			0
$405$	−174752.	+	97313.5i	−1.06540	+	0.593284i
$406$		0			0
$407$			49022.1i			0.295940i
$408$		0			0
$409$	−3909.15			−0.0233688			−0.0116844	−	0.999932i	$-0.503719\pi$
$409$	−3909.15			−0.0233688			−0.0116844	+	0.999932i	$0.503719\pi$
$410$		0			0
$411$	189659.	−	108522.i	1.12277	−	0.642443i
$412$		0			0
$413$			11422.7i			0.0669679i
$414$		0			0
$415$	−185983.			−1.07988
$416$		0			0
$417$	−52683.4	−	92072.4i	−0.302971	−	0.529489i
$418$		0			0
$419$			131470.i			0.748859i	0.927255	+	0.374429i	$0.122161\pi$
$419$			131470.i			0.748859i	−0.927255	+	0.374429i	$0.877839\pi$
$420$		0			0
$421$	264054.			1.48980			0.744901	−	0.667175i	$-0.232497\pi$
$421$	264054.			1.48980			0.744901	+	0.667175i	$0.232497\pi$
$422$		0			0
$423$	−220904.	−	129832.i	−1.23459	−	0.725604i
$424$		0			0
$425$		−	100907.i		−	0.558653i
$426$		0			0
$427$	58008.8			0.318154
$428$		0			0
$429$	−10647.8	+	6092.63i	−0.0578557	+	0.0331048i
$430$		0			0
$431$			229273.i			1.23424i	0.786870	+	0.617119i	$0.211700\pi$
$431$			229273.i			1.23424i	−0.786870	+	0.617119i	$0.788300\pi$
$432$		0			0
$433$	−110235.			−0.587953			−0.293977	−	0.955813i	$-0.594979\pi$
$433$	−110235.			−0.587953			−0.293977	+	0.955813i	$0.594979\pi$
$434$		0			0
$435$	213348.	+	372859.i	1.12748	+	1.97045i
$436$		0			0
$437$		−	243230.i		−	1.27366i
$438$		0			0
$439$	−61671.8			−0.320005			−0.160003	−	0.987117i	$-0.551150\pi$
$439$	−61671.8			−0.320005			−0.160003	+	0.987117i	$0.551150\pi$
$440$		0			0
$441$	−14077.5	+	23952.4i	−0.0723852	+	0.123161i
$442$		0			0
$443$		−	54702.9i		−	0.278742i	−0.990240	−	0.139371i	$-0.955492\pi$
$443$		−	54702.9i		−	0.278742i	0.990240	−	0.139371i	$-0.0445081\pi$
$444$		0			0
$445$	207370.			1.04719
$446$		0			0
$447$	−48759.3	+	27899.9i	−0.244030	+	0.139633i
$448$		0			0
$449$		−	110150.i		−	0.546375i	−0.961961	−	0.273188i	$-0.911922\pi$
$449$		−	110150.i		−	0.546375i	0.961961	−	0.273188i	$-0.0880780\pi$
$450$		0			0
$451$	5829.33			0.0286593
$452$		0			0
$453$	−41736.2	−	72940.5i	−0.203384	−	0.355445i
$454$		0			0
$455$		−	8736.77i		−	0.0422015i
$456$		0			0
$457$	−247865.			−1.18681			−0.593407	−	0.804902i	$-0.702218\pi$
$457$	−247865.			−1.18681			−0.593407	+	0.804902i	$0.702218\pi$
$458$		0			0
$459$	2808.94	−	241634.i	0.0133327	−	1.14692i
$460$		0			0
$461$			250553.i			1.17896i	0.807784	+	0.589478i	$0.200667\pi$
$461$			250553.i			1.17896i	−0.807784	+	0.589478i	$0.799333\pi$
$462$		0			0
$463$	−192820.			−0.899478			−0.449739	−	0.893160i	$-0.648483\pi$
$463$	−192820.			−0.899478			−0.449739	+	0.893160i	$0.648483\pi$
$464$		0			0
$465$	328482.	−	187956.i	1.51917	−	0.869260i
$466$		0			0
$467$			86179.2i			0.395156i	0.980287	+	0.197578i	$0.0633076\pi$
$467$			86179.2i			0.395156i	−0.980287	+	0.197578i	$0.936692\pi$
$468$		0			0
$469$	130170.			0.591786
$470$		0			0
$471$	37849.8	+	66148.3i	0.170617	+	0.298179i
$472$		0			0
$473$		−	98633.1i		−	0.440860i
$474$		0			0
$475$	90746.2			0.402199
$476$		0			0
$477$	14872.4	+	8740.92i	0.0653646	+	0.0384167i
$478$		0			0
$479$			121298.i			0.528667i	0.964431	+	0.264334i	$0.0851520\pi$
$479$			121298.i			0.528667i	−0.964431	+	0.264334i	$0.914848\pi$
$480$		0			0
$481$	8611.35			0.0372204
$482$		0			0
$483$	118042.	−	67543.3i	0.505992	−	0.289526i
$484$		0			0
$485$			286316.i			1.21720i
$486$		0			0
$487$	−54998.2			−0.231895			−0.115947	−	0.993255i	$-0.536990\pi$
$487$	−54998.2			−0.231895			−0.115947	+	0.993255i	$0.536990\pi$
$488$		0			0
$489$	32282.8	+	56419.2i	0.135006	+	0.235944i
$490$		0			0
$491$			387549.i			1.60755i	0.594935	+	0.803774i	$0.297178\pi$
$491$			387549.i			1.60755i	−0.594935	+	0.803774i	$0.702822\pi$
$492$		0			0
$493$	−518991.			−2.13533
$494$		0			0
$495$	110219.	−	187534.i	0.449828	−	0.765366i
$496$		0			0
$497$		−	37964.3i		−	0.153696i
$498$		0			0
$499$	292129.			1.17320			0.586602	−	0.809876i	$-0.300465\pi$
$499$	292129.			1.17320			0.586602	+	0.809876i	$0.300465\pi$
$500$		0			0
$501$	136074.	−	77860.8i	0.542125	−	0.310201i
$502$		0			0
$503$		−	456757.i		−	1.80530i	−0.430376	−	0.902650i	$-0.641619\pi$
$503$		−	456757.i		−	1.80530i	0.430376	−	0.902650i	$-0.358381\pi$
$504$		0			0
$505$	245542.			0.962816
$506$		0			0
$507$	−126591.	−	221237.i	−0.492477	−	0.860680i
$508$		0			0
$509$			243339.i			0.939239i	0.882869	+	0.469619i	$0.155609\pi$
$509$			243339.i			0.939239i	−0.882869	+	0.469619i	$0.844391\pi$
$510$		0			0
$511$	113604.			0.435062
$512$		0			0
$513$	217303.	+	2526.10i	0.825716	+	0.00959879i
$514$		0			0
$515$		−	96643.8i		−	0.364384i
$516$		0			0
$517$	278656.			1.04253
$518$		0			0
$519$	124627.	−	71311.2i	0.462678	−	0.264742i
$520$		0			0
$521$		−	83338.8i		−	0.307024i	−0.988147	−	0.153512i	$-0.950942\pi$
$521$		−	83338.8i		−	0.307024i	0.988147	−	0.153512i	$-0.0490583\pi$
$522$		0			0
$523$	−5271.56			−0.0192724			−0.00963620	−	0.999954i	$-0.503067\pi$
$523$	−5271.56			−0.0192724			−0.00963620	+	0.999954i	$0.503067\pi$
$524$		0			0
$525$	25199.6	+	44040.1i	0.0914270	+	0.159783i
$526$		0			0
$527$			457221.i			1.64629i
$528$		0			0
$529$	−385894.			−1.37897
$530$		0			0
$531$	43070.0	+	25313.5i	0.152752	+	0.0897767i
$532$		0			0
$533$		−	1024.00i		−	0.00360449i
$534$		0			0
$535$	95985.2			0.335349
$536$		0			0
$537$	218531.	−	125042.i	0.757817	−	0.433620i
$538$		0			0
$539$		−	30214.4i		−	0.104001i
$540$		0			0
$541$	86693.8			0.296206			0.148103	−	0.988972i	$-0.452683\pi$
$541$	86693.8			0.296206			0.148103	+	0.988972i	$0.452683\pi$
$542$		0			0
$543$	115437.	+	201744.i	0.391512	+	0.684228i
$544$		0			0
$545$		−	130875.i		−	0.440619i
$546$		0			0
$547$	43775.8			0.146305			0.0731526	−	0.997321i	$-0.476694\pi$
$547$	43775.8			0.146305			0.0731526	+	0.997321i	$0.476694\pi$
$548$		0			0
$549$	128552.	−	218727.i	0.426515	−	0.725700i
$550$		0			0
$551$		−	466732.i		−	1.53732i
$552$		0			0
$553$	−135465.			−0.442974
$554$		0			0
$555$	−132531.	+	75833.4i	−0.430259	+	0.246192i
$556$		0			0
$557$		−	449830.i		−	1.44990i	−0.688802	−	0.724950i	$-0.741863\pi$
$557$		−	449830.i		−	1.44990i	0.688802	−	0.724950i	$-0.258137\pi$
$558$		0			0
$559$	−17326.2			−0.0554471
$560$		0			0
$561$	130516.	+	228097.i	0.414704	+	0.724760i
$562$		0			0
$563$		−	611954.i		−	1.93064i	−0.261066	−	0.965321i	$-0.584074\pi$
$563$		−	611954.i		−	1.93064i	0.261066	−	0.965321i	$-0.415926\pi$
$564$		0			0
$565$	132133.			0.413918
$566$		0			0
$567$	59117.5	+	106161.i	0.183886	+	0.330216i
$568$		0			0
$569$			164656.i			0.508573i	0.967129	+	0.254286i	$0.0818406\pi$
$569$			164656.i			0.508573i	−0.967129	+	0.254286i	$0.918159\pi$
$570$		0			0
$571$	−275184.			−0.844017			−0.422009	−	0.906592i	$-0.638675\pi$
$571$	−275184.			−0.844017			−0.422009	+	0.906592i	$0.638675\pi$
$572$		0			0
$573$	−347203.	+	198668.i	−1.05749	+	0.605089i
$574$		0			0
$575$		−	248377.i		−	0.751235i
$576$		0			0
$577$	115231.			0.346112			0.173056	−	0.984912i	$-0.444636\pi$
$577$	115231.			0.346112			0.173056	+	0.984912i	$0.444636\pi$
$578$		0			0
$579$	79472.8	+	138891.i	0.237062	+	0.414302i
$580$		0			0
$581$			112984.i			0.334707i
$582$		0			0
$583$	−18760.5			−0.0551960
$584$		0			0
$585$	−32942.7	−	19361.4i	−0.0962603	−	0.0565750i
$586$		0			0
$587$			233476.i			0.677588i	0.940861	+	0.338794i	$0.110019\pi$
$587$			233476.i			0.677588i	−0.940861	+	0.338794i	$0.889981\pi$
$588$		0			0
$589$	−411182.			−1.18523
$590$		0			0
$591$	399717.	−	228716.i	1.14440	−	0.654820i
$592$		0			0
$593$		−	566032.i		−	1.60965i	−0.593511	−	0.804826i	$-0.702259\pi$
$593$		−	566032.i		−	1.60965i	0.593511	−	0.804826i	$-0.297741\pi$
$594$		0			0
$595$	−187159.			−0.528660
$596$		0			0
$597$	32394.5	+	56614.3i	0.0908912	+	0.158847i
$598$		0			0
$599$		−	461869.i		−	1.28726i	−0.765338	−	0.643629i	$-0.777428\pi$
$599$		−	461869.i		−	1.28726i	0.765338	−	0.643629i	$-0.222572\pi$
$600$		0			0
$601$	−419840.			−1.16235			−0.581173	−	0.813780i	$-0.697406\pi$
$601$	−419840.			−1.16235			−0.581173	+	0.813780i	$0.697406\pi$
$602$		0			0
$603$	288467.	−	490815.i	0.793343	−	1.34984i
$604$		0			0
$605$		−	209787.i		−	0.573150i
$606$		0			0
$607$	−26334.2			−0.0714732			−0.0357366	−	0.999361i	$-0.511378\pi$
$607$	−26334.2			−0.0714732			−0.0357366	+	0.999361i	$0.511378\pi$
$608$		0			0
$609$	226510.	−	129608.i	0.610736	−	0.349460i
$610$		0			0
$611$		−	48949.4i		−	0.131119i
$612$		0			0
$613$	587631.			1.56381			0.781905	−	0.623397i	$-0.214248\pi$
$613$	587631.			1.56381			0.781905	+	0.623397i	$0.214248\pi$
$614$		0			0
$615$	9017.52	+	15759.5i	0.0238417	+	0.0416670i
$616$		0			0
$617$			436700.i			1.14713i	0.819159	+	0.573566i	$0.194440\pi$
$617$			436700.i			1.14713i	−0.819159	+	0.573566i	$0.805560\pi$
$618$		0			0
$619$	198892.			0.519081			0.259541	−	0.965732i	$-0.416429\pi$
$619$	198892.			0.519081			0.259541	+	0.965732i	$0.416429\pi$
$620$		0			0
$621$	6914.08	−	594770.i	0.0179288	−	1.54229i
$622$		0			0
$623$		−	125976.i		−	0.324573i
$624$		0			0
$625$	−488216.			−1.24983
$626$		0			0
$627$	−205129.	+	117374.i	−0.521787	+	0.298564i
$628$		0			0
$629$		−	184472.i		−	0.466262i
$630$		0			0
$631$	99629.6			0.250224			0.125112	−	0.992143i	$-0.460071\pi$
$631$	99629.6			0.250224			0.125112	+	0.992143i	$0.460071\pi$
$632$		0			0
$633$	145743.	+	254709.i	0.363732	+	0.635678i
$634$		0			0
$635$			449502.i			1.11477i
$636$		0			0
$637$	−5307.55			−0.0130802
$638$		0			0
$639$	−143147.	−	84132.0i	−0.350576	−	0.206044i
$640$		0			0
$641$		−	568044.i		−	1.38250i	−0.722615	−	0.691251i	$-0.757060\pi$
$641$		−	568044.i		−	1.38250i	0.722615	−	0.691251i	$-0.242940\pi$
$642$		0			0
$643$	−547371.			−1.32391			−0.661957	−	0.749542i	$-0.730274\pi$
$643$	−547371.			−1.32391			−0.661957	+	0.749542i	$0.730274\pi$
$644$		0			0
$645$	266653.	−	152578.i	0.640955	−	0.366751i
$646$		0			0
$647$		−	208631.i		−	0.498390i	−0.968453	−	0.249195i	$-0.919834\pi$
$647$		−	208631.i		−	0.498390i	0.968453	−	0.249195i	$-0.0801661\pi$
$648$		0			0
$649$	−54330.1			−0.128988
$650$		0			0
$651$	−114182.	−	199551.i	−0.269424	−	0.470861i
$652$		0			0
$653$		−	44613.2i		−	0.104625i	−0.998631	−	0.0523127i	$-0.983341\pi$
$653$		−	44613.2i		−	0.104625i	0.998631	−	0.0523127i	$-0.0166592\pi$
$654$		0			0
$655$	635082.			1.48029
$656$		0			0
$657$	251755.	−	428352.i	0.583241	−	0.992362i
$658$		0			0
$659$		−	33171.4i		−	0.0763823i	−0.999270	−	0.0381912i	$-0.987840\pi$
$659$		−	33171.4i		−	0.0763823i	0.999270	−	0.0381912i	$-0.0121596\pi$
$660$		0			0
$661$	456359.			1.04449			0.522244	−	0.852796i	$-0.325095\pi$
$661$	456359.			1.04449			0.522244	+	0.852796i	$0.325095\pi$
$662$		0			0
$663$	40068.2	−	22926.8i	0.0911533	−	0.0521575i
$664$		0			0
$665$		−	168313.i		−	0.380606i
$666$		0			0
$667$	−1.27747e6			−2.87144
$668$		0			0
$669$	247418.	+	432401.i	0.552813	+	0.966127i
$670$		0			0
$671$			275910.i			0.612804i
$672$		0			0
$673$	−555138.			−1.22566			−0.612831	−	0.790214i	$-0.709969\pi$
$673$	−555138.			−1.22566			−0.612831	+	0.790214i	$0.709969\pi$
$674$		0			0
$675$	221901.	+	2579.56i	0.487026	+	0.00566158i
$676$		0			0
$677$		−	290742.i		−	0.634353i	−0.948367	−	0.317176i	$-0.897265\pi$
$677$		−	290742.i		−	0.634353i	0.948367	−	0.317176i	$-0.102735\pi$
$678$		0			0
$679$	173936.			0.377268
$680$		0			0
$681$	45625.9	−	26106.9i	0.0983824	−	0.0562940i
$682$		0			0
$683$			179246.i			0.384244i	0.981371	+	0.192122i	$0.0615370\pi$
$683$			179246.i			0.384244i	−0.981371	+	0.192122i	$0.938463\pi$
$684$		0			0
$685$	−740181.			−1.57745
$686$		0			0
$687$	−9400.64	−	16429.1i	−0.0199179	−	0.0348096i
$688$		0			0
$689$			3295.52i			0.00694202i
$690$		0			0
$691$	−260140.			−0.544818			−0.272409	−	0.962182i	$-0.587820\pi$
$691$	−260140.			−0.544818			−0.272409	+	0.962182i	$0.587820\pi$
$692$		0			0
$693$	−113926.	−	66957.6i	−0.237223	−	0.139423i
$694$		0			0
$695$			359330.i			0.743915i
$696$		0			0
$697$	−21936.0			−0.0451536
$698$		0			0
$699$	−480278.	+	274813.i	−0.982966	+	0.562448i
$700$		0			0
$701$			167363.i			0.340584i	0.985394	+	0.170292i	$0.0544710\pi$
$701$			167363.i			0.340584i	−0.985394	+	0.170292i	$0.945529\pi$
$702$		0			0
$703$	165897.			0.335682
$704$		0			0
$705$	431059.	+	753342.i	0.867278	+	1.51570i
$706$		0			0
$707$		−	149166.i		−	0.298421i
$708$		0			0
$709$	−112915.			−0.224626			−0.112313	−	0.993673i	$-0.535826\pi$
$709$	−112915.			−0.224626			−0.112313	+	0.993673i	$0.535826\pi$
$710$		0			0
$711$	−300202.	+	510783.i	−0.593848	+	1.01041i
$712$		0			0
$713$			1.12543e6i			2.21380i
$714$		0			0
$715$	41555.1			0.0812853
$716$		0			0
$717$	−580075.	+	331917.i	−1.12836	+	0.645640i
$718$		0			0
$719$		−	757578.i		−	1.46545i	−0.680527	−	0.732723i	$-0.738249\pi$
$719$		−	757578.i		−	1.46545i	0.680527	−	0.732723i	$-0.261751\pi$
$720$		0			0
$721$	−58710.6			−0.112940
$722$		0			0
$723$	497423.	+	869324.i	0.951589	+	1.66305i
$724$		0			0
$725$		−	476608.i		−	0.906746i
$726$		0			0
$727$	939062.			1.77675			0.888373	−	0.459123i	$-0.151836\pi$
$727$	939062.			1.77675			0.888373	+	0.459123i	$0.151836\pi$
$728$		0			0
$729$	531297.	+	12354.1i	0.999730	+	0.0232465i
$730$		0			0
$731$			371160.i			0.694588i
$732$		0			0
$733$	−24447.0			−0.0455007			−0.0227504	−	0.999741i	$-0.507242\pi$
$733$	−24447.0			−0.0455007			−0.0227504	+	0.999741i	$0.507242\pi$
$734$		0			0
$735$	81684.3	−	46739.4i	0.151204	−	0.0865183i
$736$		0			0
$737$			619132.i			1.13985i
$738$		0			0
$739$	605915.			1.10949			0.554744	−	0.832021i	$-0.312816\pi$
$739$	605915.			1.10949			0.554744	+	0.832021i	$0.312816\pi$
$740$		0			0
$741$	20618.3	+	36033.6i	0.0375505	+	0.0656253i
$742$		0			0
$743$			207400.i			0.375690i	0.982199	+	0.187845i	$0.0601503\pi$
$743$			207400.i			0.375690i	−0.982199	+	0.187845i	$0.939850\pi$
$744$		0			0
$745$	190292.			0.342854
$746$		0			0
$747$	426015.	+	250381.i	0.763455	+	0.448705i
$748$		0			0
$749$		−	58310.6i		−	0.103940i
$750$		0			0
$751$	6227.67			0.0110419			0.00552097	−	0.999985i	$-0.498243\pi$
$751$	6227.67			0.0110419			0.00552097	+	0.999985i	$0.498243\pi$
$752$		0			0
$753$	324275.	−	185549.i	0.571905	−	0.327242i
$754$		0			0
$755$			284664.i			0.499388i
$756$		0			0
$757$	212971.			0.371645			0.185823	−	0.982583i	$-0.440505\pi$
$757$	212971.			0.371645			0.185823	+	0.982583i	$0.440505\pi$
$758$		0			0
$759$	321259.	+	561450.i	0.557663	+	0.974603i
$760$		0			0
$761$		−	41126.2i		−	0.0710149i	−0.999369	−	0.0355075i	$-0.988695\pi$
$761$		−	41126.2i		−	0.0710149i	0.999369	−	0.0355075i	$-0.0113048\pi$
$762$		0			0
$763$	−79505.8			−0.136568
$764$		0			0
$765$	−414759.	+	705697.i	−0.708717	+	1.20586i
$766$		0			0
$767$			9543.76i			0.0162229i
$768$		0			0
$769$	1.15006e6			1.94477			0.972386	−	0.233380i	$-0.0749785\pi$
$769$	1.15006e6			1.94477			0.972386	+	0.233380i	$0.0749785\pi$
$770$		0			0
$771$	−415205.	+	237578.i	−0.698480	+	0.399667i
$772$		0			0
$773$		−	134095.i		−	0.224416i	−0.993685	−	0.112208i	$-0.964208\pi$
$773$		−	134095.i		−	0.224416i	0.993685	−	0.112208i	$-0.0357924\pi$
$774$		0			0
$775$	−419883.			−0.699077
$776$		0			0
$777$	46068.4	+	80511.7i	0.0763065	+	0.133357i
$778$		0			0
$779$		−	19727.2i		−	0.0325080i
$780$		0			0
$781$	180571.			0.296038
$782$		0			0
$783$	13267.3	−	1.14130e6i	0.0216402	−	1.86155i
$784$		0			0
$785$		−	258156.i		−	0.418932i
$786$		0			0
$787$	−922036.			−1.48867			−0.744335	−	0.667807i	$-0.767233\pi$
$787$	−922036.			−1.48867			−0.744335	+	0.667807i	$0.767233\pi$
$788$		0			0
$789$	−500818.	+	286566.i	−0.804500	+	0.460331i
$790$		0			0
$791$		−	80270.2i		−	0.128293i
$792$		0			0
$793$	48467.0			0.0770726
$794$		0			0
$795$	−29021.1	−	50718.8i	−0.0459176	−	0.0802481i
$796$		0			0
$797$			737363.i			1.16082i	0.814325	+	0.580409i	$0.197107\pi$
$797$			737363.i			1.16082i	−0.814325	+	0.580409i	$0.802893\pi$
$798$		0			0
$799$	−1.04859e6			−1.64253
$800$		0			0
$801$	−475003.	−	279173.i	−0.740340	−	0.435120i
$802$		0			0
$803$			540339.i			0.837983i
$804$		0			0
$805$	−460682.			−0.710902
$806$		0			0
$807$	900681.	−	515366.i	1.38301	−	0.791350i
$808$		0			0
$809$			1.05292e6i			1.60879i	0.594095	+	0.804395i	$0.297510\pi$
$809$			1.05292e6i			1.60879i	−0.594095	+	0.804395i	$0.702490\pi$
$810$		0			0
$811$	1.10643e6			1.68221			0.841105	−	0.540871i	$-0.181905\pi$
$811$	1.10643e6			1.68221			0.841105	+	0.540871i	$0.181905\pi$
$812$		0			0
$813$	−485619.	−	848695.i	−0.734708	−	1.28402i
$814$		0			0
$815$		−	220187.i		−	0.331494i
$816$		0			0
$817$	−333787.			−0.500064
$818$		0			0
$819$	−11762.0	+	20012.5i	−0.0175352	+	0.0298356i
$820$		0			0
$821$		−	123241.i		−	0.182840i	−0.995812	−	0.0914198i	$-0.970859\pi$
$821$		−	123241.i		−	0.182840i	0.995812	−	0.0914198i	$-0.0291405\pi$
$822$		0			0
$823$	−792158.			−1.16953			−0.584766	−	0.811202i	$-0.698814\pi$
$823$	−792158.			−1.16953			−0.584766	+	0.811202i	$0.698814\pi$
$824$		0			0
$825$	−209470.	+	119858.i	−0.307761	+	0.176100i
$826$		0			0
$827$			699053.i			1.02211i	0.859547	+	0.511057i	$0.170746\pi$
$827$			699053.i			1.02211i	−0.859547	+	0.511057i	$0.829254\pi$
$828$		0			0
$829$	1.22153e6			1.77744			0.888721	−	0.458449i	$-0.151595\pi$
$829$	1.22153e6			1.77744			0.888721	+	0.458449i	$0.151595\pi$
$830$		0			0
$831$	−201403.	−	351983.i	−0.291651	−	0.509705i
$832$		0			0
$833$			113698.i			0.163856i
$834$		0			0
$835$	−531054.			−0.761667
$836$		0			0
$837$	−1.00546e6	−	11688.3i	−1.43521	−	0.0166840i
$838$		0			0
$839$			682312.i			0.969303i	0.874707	+	0.484651i	$0.161053\pi$
$839$			682312.i			0.969303i	−0.874707	+	0.484651i	$0.838947\pi$
$840$		0			0
$841$	−1.74404e6			−2.46584
$842$		0			0
$843$	151328.	−	86589.3i	0.212944	−	0.121845i
$844$		0			0
$845$			863418.i			1.20923i
$846$		0			0
$847$	−127445.			−0.177646
$848$		0			0
$849$	54296.4	+	94891.4i	0.0753279	+	0.131647i
$850$		0			0
$851$		−	454069.i		−	0.626994i
$852$		0			0
$853$	765325.			1.05184			0.525918	−	0.850535i	$-0.323722\pi$
$853$	765325.			1.05184			0.525918	+	0.850535i	$0.323722\pi$
$854$		0			0
$855$	−634639.	−	372996.i	−0.868149	−	0.510237i
$856$		0			0
$857$			1.36272e6i			1.85543i	0.373289	+	0.927715i	$0.378230\pi$
$857$			1.36272e6i			1.85543i	−0.373289	+	0.927715i	$0.621770\pi$
$858$		0			0
$859$	1.03986e6			1.40926			0.704628	−	0.709577i	$-0.251114\pi$
$859$	1.03986e6			1.40926			0.704628	+	0.709577i	$0.251114\pi$
$860$		0			0
$861$	9573.83	−	5478.10i	0.0129146	−	0.00738965i
$862$		0			0
$863$			563532.i			0.756653i	0.925672	+	0.378327i	$0.123500\pi$
$863$			563532.i			0.756653i	−0.925672	+	0.378327i	$0.876500\pi$
$864$		0			0
$865$	−486381.			−0.650047
$866$		0			0
$867$	−117818.	−	205906.i	−0.156738	−	0.273924i
$868$		0			0
$869$		−	644321.i		−	0.853223i
$870$		0			0
$871$	108758.			0.143360
$872$		0			0
$873$	385456.	−	655839.i	0.505762	−	0.860535i
$874$		0			0
$875$			181009.i			0.236419i
$876$		0			0
$877$	−646978.			−0.841183			−0.420592	−	0.907250i	$-0.638177\pi$
$877$	−646978.			−0.841183			−0.420592	+	0.907250i	$0.638177\pi$
$878$		0			0
$879$	308864.	−	176730.i	0.399751	−	0.228736i
$880$		0			0
$881$		−	530305.i		−	0.683242i	−0.939838	−	0.341621i	$-0.889024\pi$
$881$		−	530305.i		−	0.683242i	0.939838	−	0.341621i	$-0.110976\pi$
$882$		0			0
$883$	−485702.			−0.622943			−0.311471	−	0.950256i	$-0.600822\pi$
$883$	−485702.			−0.622943			−0.311471	+	0.950256i	$0.600822\pi$
$884$		0			0
$885$	−84044.4	−	146881.i	−0.107306	−	0.187533i
$886$		0			0
$887$		−	670457.i		−	0.852165i	−0.904684	−	0.426082i	$-0.859893\pi$
$887$		−	670457.i		−	0.852165i	0.904684	−	0.426082i	$-0.140107\pi$
$888$		0			0
$889$	273070.			0.345518
$890$		0			0
$891$	−504938.	+	281183.i	−0.636038	+	0.354188i
$892$		0			0
$893$		−	943008.i		−	1.18253i
$894$		0			0
$895$	−852858.			−1.06471
$896$		0			0
$897$	98625.8	−	56433.3i	0.122576	−	0.0701375i
$898$		0			0
$899$			2.15957e6i			2.67207i
$900$		0			0
$901$	70596.6			0.0869630
$902$		0			0
$903$	−92690.3	−	161991.i	−0.113673	−	0.198662i
$904$		0			0
$905$		−	787343.i		−	0.961318i
$906$		0			0
$907$	−916171.			−1.11368			−0.556842	−	0.830618i	$-0.687987\pi$
$907$	−916171.			−1.11368			−0.556842	+	0.830618i	$0.687987\pi$
$908$		0			0
$909$	−562441.	−	330563.i	−0.680690	−	0.400061i
$910$		0			0
$911$		−	1.30288e6i		−	1.56989i	−0.619567	−	0.784944i	$-0.712692\pi$
$911$		−	1.30288e6i		−	1.56989i	0.619567	−	0.784944i	$-0.287308\pi$
$912$		0			0
$913$	−537390.			−0.644686
$914$		0			0
$915$	−745918.	+	426811.i	−0.890940	+	0.509792i
$916$		0			0
$917$		−	385809.i		−	0.458811i
$918$		0			0
$919$	307397.			0.363972			0.181986	−	0.983301i	$-0.441747\pi$
$919$	307397.			0.363972			0.181986	+	0.983301i	$0.441747\pi$
$920$		0			0
$921$	230816.	+	403387.i	0.272111	+	0.475557i
$922$		0			0
$923$		−	31719.6i		−	0.0372327i
$924$		0			0
$925$	169408.			0.197993
$926$		0			0
$927$	−130108.	+	221373.i	−0.151406	+	0.257612i
$928$		0			0
$929$		−	955189.i		−	1.10677i	−0.832925	−	0.553385i	$-0.813336\pi$
$929$		−	955189.i		−	1.10677i	0.832925	−	0.553385i	$-0.186664\pi$
$930$		0			0
$931$	−102250.			−0.117967
$932$		0			0
$933$	512176.	−	293065.i	0.588378	−	0.336667i
$934$		0			0
$935$		−	890192.i		−	1.01826i
$936$		0			0
$937$	93998.6			0.107064			0.0535318	−	0.998566i	$-0.482952\pi$
$937$	93998.6			0.107064			0.0535318	+	0.998566i	$0.482952\pi$
$938$		0			0
$939$	260566.	+	455380.i	0.295520	+	0.516467i
$940$		0			0
$941$			904009.i			1.02092i	0.859900	+	0.510462i	$0.170526\pi$
$941$			904009.i			1.02092i	−0.859900	+	0.510462i	$0.829474\pi$
$942$		0			0
$943$	−53994.4			−0.0607191
$944$		0			0
$945$	4784.48	−	411575.i	0.00535761	−	0.460878i
$946$		0			0
$947$		−	1.03257e6i		−	1.15138i	−0.817669	−	0.575689i	$-0.804734\pi$
$947$		−	1.03257e6i		−	1.15138i	0.817669	−	0.575689i	$-0.195266\pi$
$948$		0			0
$949$	94917.4			0.105393
$950$		0			0
$951$	−979958.	+	560728.i	−1.08354	+	0.619999i
$952$		0			0
$953$			1.24014e6i			1.36548i	0.730664	+	0.682738i	$0.239211\pi$
$953$			1.24014e6i			1.36548i	−0.730664	+	0.682738i	$0.760789\pi$
$954$		0			0
$955$	1.35503e6			1.48573
$956$		0			0
$957$	616461.	+	1.07736e6i	0.673103	+	1.17635i
$958$		0			0
$959$			449656.i			0.488926i
$960$		0			0
$961$	979020.			1.06010
$962$		0			0
$963$	−219865.	−	129221.i	−0.237084	−	0.139341i
$964$		0			0
$965$		−	542048.i		−	0.582081i
$966$		0			0
$967$	1.40510e6			1.50264			0.751319	−	0.659940i	$-0.229418\pi$
$967$	1.40510e6			1.50264			0.751319	+	0.659940i	$0.229418\pi$
$968$		0			0
$969$	771911.	−	441684.i	0.822090	−	0.470396i
$970$		0			0
$971$			1.48138e6i			1.57119i	0.618742	+	0.785594i	$0.287643\pi$
$971$			1.48138e6i			1.57119i	−0.618742	+	0.785594i	$0.712357\pi$
$972$		0			0
$973$	218291.			0.230574
$974$		0			0
$975$	21054.5	+	36796.1i	0.0221481	+	0.0387072i
$976$		0			0
$977$			94526.4i			0.0990293i	0.998773	+	0.0495147i	$0.0157675\pi$
$977$			94526.4i			0.0990293i	−0.998773	+	0.0495147i	$0.984233\pi$
$978$		0			0
$979$	599186.			0.625167
$980$		0			0
$981$	−176191.	+	299783.i	−0.183082	+	0.311508i
$982$		0			0
$983$		−	338813.i		−	0.350633i	−0.984512	−	0.175316i	$-0.943905\pi$
$983$		−	338813.i		−	0.350633i	0.984512	−	0.175316i	$-0.0560949\pi$
$984$		0			0
$985$	−1.55997e6			−1.60784
$986$		0			0
$987$	457652.	−	261866.i	0.469787	−	0.268810i
$988$		0			0
$989$			913594.i			0.934029i
$990$		0			0
$991$	1.43200e6			1.45813			0.729066	−	0.684443i	$-0.239955\pi$
$991$	1.43200e6			1.45813			0.729066	+	0.684443i	$0.239955\pi$
$992$		0			0
$993$	−377422.	−	659603.i	−0.382762	−	0.668935i
$994$		0			0
$995$		−	220948.i		−	0.223174i
$996$		0			0
$997$	1.20016e6			1.20739			0.603697	−	0.797214i	$-0.293694\pi$
$997$	1.20016e6			1.20739			0.603697	+	0.797214i	$0.293694\pi$
$998$		0			0
$999$	405667.	+	4715.80i	0.406480	+	0.00472525i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.5.d.b.113.6		8
3.2	odd	2	inner	336.5.d.b.113.5		8
4.3	odd	2		21.5.b.a.8.5	yes	8
12.11	even	2		21.5.b.a.8.4	✓	8
28.3	even	6		147.5.h.c.128.4		16
28.11	odd	6		147.5.h.e.128.4		16
28.19	even	6		147.5.h.c.116.5		16
28.23	odd	6		147.5.h.e.116.5		16
28.27	even	2		147.5.b.e.50.5		8
84.11	even	6		147.5.h.e.128.5		16
84.23	even	6		147.5.h.e.116.4		16
84.47	odd	6		147.5.h.c.116.4		16
84.59	odd	6		147.5.h.c.128.5		16
84.83	odd	2		147.5.b.e.50.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.5.b.a.8.4	✓	8	12.11	even	2
21.5.b.a.8.5	yes	8	4.3	odd	2
147.5.b.e.50.4		8	84.83	odd	2
147.5.b.e.50.5		8	28.27	even	2
147.5.h.c.116.4		16	84.47	odd	6
147.5.h.c.116.5		16	28.19	even	6
147.5.h.c.128.4		16	28.3	even	6
147.5.h.c.128.5		16	84.59	odd	6
147.5.h.e.116.4		16	84.23	even	6
147.5.h.e.116.5		16	28.23	odd	6
147.5.h.e.128.4		16	28.11	odd	6
147.5.h.e.128.5		16	84.11	even	6
336.5.d.b.113.5		8	3.2	odd	2	inner
336.5.d.b.113.6		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	336.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(4.46977 + 7.81161i) q^{3} -30.4863i q^{5} -18.5203 q^{7} +(-41.0424 + 69.8321i) q^{9} +O(q^{10})\)	\(q+(4.46977 + 7.81161i) q^{3} -30.4863i q^{5} -18.5203 q^{7} +(-41.0424 + 69.8321i) q^{9} -88.0887i q^{11} -15.4739 q^{13} +(238.147 - 136.266i) q^{15} +331.481i q^{17} -298.104 q^{19} +(-82.7812 - 144.673i) q^{21} +815.926i q^{23} -304.412 q^{25} +(-728.951 - 8.47391i) q^{27} +1565.67i q^{29} +1379.33 q^{31} +(688.115 - 393.736i) q^{33} +564.613i q^{35} -556.508 q^{37} +(-69.1647 - 120.876i) q^{39} +66.1756i q^{41} +1119.70 q^{43} +(2128.92 + 1251.23i) q^{45} +3163.36i q^{47} +343.000 q^{49} +(-2589.40 + 1481.64i) q^{51} -212.973i q^{53} -2685.50 q^{55} +(-1332.45 - 2328.67i) q^{57} -616.765i q^{59} -3132.18 q^{61} +(760.116 - 1293.31i) q^{63} +471.741i q^{65} -7028.51 q^{67} +(-6373.69 + 3647.00i) q^{69} +2049.88i q^{71} -6134.03 q^{73} +(-1360.65 - 2377.94i) q^{75} +1631.43i q^{77} +7314.45 q^{79} +(-3192.04 - 5732.15i) q^{81} -6100.56i q^{83} +10105.6 q^{85} +(-12230.4 + 6998.18i) q^{87} +6802.07i q^{89} +286.581 q^{91} +(6165.27 + 10774.8i) q^{93} +9088.06i q^{95} -9391.65 q^{97} +(6151.42 + 3615.37i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{3} + 64 q^{9}+O(q^{10}) \) 8 * q + 2 * q^3 + 64 * q^9	\( 8 q + 2 q^{3} + 64 q^{9} + 420 q^{13} - 76 q^{15} + 372 q^{19} + 98 q^{21} + 1056 q^{25} + 1862 q^{27} + 2776 q^{31} + 1396 q^{33} - 2560 q^{37} + 2540 q^{39} - 4720 q^{43} + 9700 q^{45} + 2744 q^{49} - 4764 q^{51} - 184 q^{55} - 14144 q^{57} + 972 q^{61} + 6076 q^{63} - 10200 q^{67} - 5760 q^{69} - 32008 q^{73} - 2114 q^{75} + 23168 q^{79} - 17216 q^{81} + 32016 q^{85} - 50764 q^{87} - 11956 q^{91} + 31848 q^{93} + 28112 q^{97} + 32432 q^{99}+O(q^{100}) \) 8 * q + 2 * q^3 + 64 * q^9 + 420 * q^13 - 76 * q^15 + 372 * q^19 + 98 * q^21 + 1056 * q^25 + 1862 * q^27 + 2776 * q^31 + 1396 * q^33 - 2560 * q^37 + 2540 * q^39 - 4720 * q^43 + 9700 * q^45 + 2744 * q^49 - 4764 * q^51 - 184 * q^55 - 14144 * q^57 + 972 * q^61 + 6076 * q^63 - 10200 * q^67 - 5760 * q^69 - 32008 * q^73 - 2114 * q^75 + 23168 * q^79 - 17216 * q^81 + 32016 * q^85 - 50764 * q^87 - 11956 * q^91 + 31848 * q^93 + 28112 * q^97 + 32432 * q^99

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