LMFDB - Embedded newform 320.6.d.d.161.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,6,Mod(161,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("320.161");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	320.d (of order $2$, degree $1$, 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:	$51.3228223402$
Analytic rank:	$0$
Dimension:	$12$
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} - 343 x^{10} - 696 x^{9} + 44406 x^{8} + 179640 x^{7} - 2401691 x^{6} - 15554592 x^{5} + \cdots + 653157349 $ x^12 - 343x^10 - 696x^9 + 44406x^8 + 179640x^7 - 2401691x^6 - 15554592x^5 + 26901210x^4 + 434775816x^3 + 1271335685x^2 + 1475231592x + 653157349
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{30} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.11
Root			$-8.35048 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	320.161
Dual form			320.6.d.d.161.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+20.6292i q^{3} -25.0000i q^{5} -93.5254 q^{7} -182.562 q^{9} +O(q^{10})$	$q+20.6292i q^{3} -25.0000i q^{5} -93.5254 q^{7} -182.562 q^{9} +659.055i q^{11} +183.844i q^{13} +515.729 q^{15} -467.075 q^{17} +225.914i q^{19} -1929.35i q^{21} +1109.57 q^{23} -625.000 q^{25} +1246.78i q^{27} -1738.98i q^{29} -7270.81 q^{31} -13595.7 q^{33} +2338.14i q^{35} -14229.7i q^{37} -3792.54 q^{39} -4929.92 q^{41} +12379.7i q^{43} +4564.05i q^{45} -619.842 q^{47} -8060.00 q^{49} -9635.36i q^{51} +16989.0i q^{53} +16476.4 q^{55} -4660.40 q^{57} -50742.8i q^{59} -20666.3i q^{61} +17074.2 q^{63} +4596.09 q^{65} -41920.3i q^{67} +22889.5i q^{69} +54687.7 q^{71} -25211.5 q^{73} -12893.2i q^{75} -61638.4i q^{77} +29000.4 q^{79} -70082.7 q^{81} -92679.1i q^{83} +11676.9i q^{85} +35873.7 q^{87} +54644.0 q^{89} -17194.1i q^{91} -149991. i q^{93} +5647.84 q^{95} +45542.8 q^{97} -120318. i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 268 q^{7} - 428 q^{9}+O(q^{10}) $ 12 * q + 268 * q^7 - 428 * q^9	$ 12 q + 268 q^{7} - 428 q^{9} - 900 q^{15} + 2400 q^{17} + 8108 q^{23} - 7500 q^{25} + 7976 q^{31} - 20776 q^{33} + 40984 q^{39} - 56408 q^{41} + 21172 q^{47} - 5540 q^{49} + 18400 q^{55} + 39992 q^{57} + 179516 q^{63} + 44000 q^{65} + 367704 q^{71} + 58736 q^{73} + 26192 q^{79} + 411692 q^{81} + 183200 q^{87} + 87672 q^{89} + 121000 q^{95} - 172336 q^{97}+O(q^{100}) $ 12 * q + 268 * q^7 - 428 * q^9 - 900 * q^15 + 2400 * q^17 + 8108 * q^23 - 7500 * q^25 + 7976 * q^31 - 20776 * q^33 + 40984 * q^39 - 56408 * q^41 + 21172 * q^47 - 5540 * q^49 + 18400 * q^55 + 39992 * q^57 + 179516 * q^63 + 44000 * q^65 + 367704 * q^71 + 58736 * q^73 + 26192 * q^79 + 411692 * q^81 + 183200 * q^87 + 87672 * q^89 + 121000 * q^95 - 172336 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	20.6292i	1.32336i	0.749786	+	0.661680i	$0.230156\pi$
$3$	20.6292i	1.32336i	−0.749786	+	0.661680i	$0.769844\pi$
$4$	0	0
$5$	− 25.0000i	− 0.447214i
$5$	− 25.0000i	− 0.447214i
$6$	0	0
$7$	−93.5254	−0.721414	−0.360707	−	0.932679i	$-0.617465\pi$
$7$	−93.5254	−0.721414	−0.360707	+	0.932679i	$0.617465\pi$
$8$	0	0
$9$	−182.562	−0.751284
$10$	0	0
$11$	659.055i	1.64225i	0.570747	+	0.821126i	$0.306654\pi$
$11$	659.055i	1.64225i	−0.570747	+	0.821126i	$0.693346\pi$
$12$	0	0
$13$	183.844i	0.301711i	0.988556	+	0.150855i	$0.0482027\pi$
$13$	183.844i	0.301711i	−0.988556	+	0.150855i	$0.951797\pi$
$14$	0	0
$15$	515.729	0.591825
$16$	0	0
$17$	−467.075	−0.391980	−0.195990	−	0.980606i	$-0.562792\pi$
$17$	−467.075	−0.391980	−0.195990	+	0.980606i	$0.562792\pi$
$18$	0	0
$19$	225.914i	0.143568i	0.997420	+	0.0717841i	$0.0228693\pi$
$19$	225.914i	0.143568i	−0.997420	+	0.0717841i	$0.977131\pi$
$20$	0	0
$21$	− 1929.35i	− 0.954691i
$22$	0	0
$23$	1109.57	0.437357	0.218678	−	0.975797i	$-0.429825\pi$
$23$	1109.57	0.437357	0.218678	+	0.975797i	$0.429825\pi$
$24$	0	0
$25$	−625.000	−0.200000
$26$	0	0
$27$	1246.78i	0.329141i
$28$	0	0
$29$	− 1738.98i	− 0.383972i	−0.981398	−	0.191986i	$-0.938507\pi$
$29$	− 1738.98i	− 0.383972i	0.981398	−	0.191986i	$-0.0614929\pi$
$30$	0	0
$31$	−7270.81	−1.35887	−0.679436	−	0.733735i	$-0.737775\pi$
$31$	−7270.81	−1.35887	−0.679436	+	0.733735i	$0.737775\pi$
$32$	0	0
$33$	−13595.7	−2.17329
$34$	0	0
$35$	2338.14i	0.322626i
$36$	0	0
$37$	− 14229.7i	− 1.70881i	−0.519610	−	0.854403i	$-0.673923\pi$
$37$	− 14229.7i	− 1.70881i	0.519610	−	0.854403i	$-0.326077\pi$
$38$	0	0
$39$	−3792.54	−0.399272
$40$	0	0
$41$	−4929.92	−0.458015	−0.229008	−	0.973425i	$-0.573548\pi$
$41$	−4929.92	−0.458015	−0.229008	+	0.973425i	$0.573548\pi$
$42$	0	0
$43$	12379.7i	1.02103i	0.859870	+	0.510514i	$0.170545\pi$
$43$	12379.7i	1.02103i	−0.859870	+	0.510514i	$0.829455\pi$
$44$	0	0
$45$	4564.05i	0.335984i
$46$	0	0
$47$	−619.842	−0.0409295	−0.0204647	−	0.999791i	$-0.506515\pi$
$47$	−619.842	−0.0409295	−0.0204647	+	0.999791i	$0.506515\pi$
$48$	0	0
$49$	−8060.00	−0.479562
$50$	0	0
$51$	− 9635.36i	− 0.518731i
$52$	0	0
$53$	16989.0i	0.830765i	0.909647	+	0.415382i	$0.136352\pi$
$53$	16989.0i	0.830765i	−0.909647	+	0.415382i	$0.863648\pi$
$54$	0	0
$55$	16476.4	0.734437
$56$	0	0
$57$	−4660.40	−0.189993
$58$	0	0
$59$	− 50742.8i	− 1.89777i	−0.315615	−	0.948887i	$-0.602211\pi$
$59$	− 50742.8i	− 1.89777i	0.315615	−	0.948887i	$-0.397789\pi$
$60$	0	0
$61$	− 20666.3i	− 0.711111i	−0.934655	−	0.355555i	$-0.884292\pi$
$61$	− 20666.3i	− 0.711111i	0.934655	−	0.355555i	$-0.115708\pi$
$62$	0	0
$63$	17074.2	0.541987
$64$	0	0
$65$	4596.09	0.134929
$66$	0	0
$67$	− 41920.3i	− 1.14087i	−0.821341	−	0.570437i	$-0.806774\pi$
$67$	− 41920.3i	− 1.14087i	0.821341	−	0.570437i	$-0.193226\pi$
$68$	0	0
$69$	22889.5i	0.578781i
$70$	0	0
$71$	54687.7	1.28749	0.643744	−	0.765241i	$-0.277380\pi$
$71$	54687.7	1.28749	0.643744	+	0.765241i	$0.277380\pi$
$72$	0	0
$73$	−25211.5	−0.553722	−0.276861	−	0.960910i	$-0.589294\pi$
$73$	−25211.5	−0.553722	−0.276861	+	0.960910i	$0.589294\pi$
$74$	0	0
$75$	− 12893.2i	− 0.264672i
$76$	0	0
$77$	− 61638.4i	− 1.18474i
$78$	0	0
$79$	29000.4	0.522802	0.261401	−	0.965230i	$-0.415816\pi$
$79$	29000.4	0.522802	0.261401	+	0.965230i	$0.415816\pi$
$80$	0	0
$81$	−70082.7	−1.18686
$82$	0	0
$83$	− 92679.1i	− 1.47668i	−0.674428	−	0.738341i	$-0.735610\pi$
$83$	− 92679.1i	− 1.47668i	0.674428	−	0.738341i	$-0.264390\pi$
$84$	0	0
$85$	11676.9i	0.175299i
$86$	0	0
$87$	35873.7	0.508134
$88$	0	0
$89$	54644.0	0.731253	0.365627	−	0.930762i	$-0.380855\pi$
$89$	54644.0	0.731253	0.365627	+	0.930762i	$0.380855\pi$
$90$	0	0
$91$	− 17194.1i	− 0.217658i
$92$	0	0
$93$	− 149991.i	− 1.79828i
$94$	0	0
$95$	5647.84	0.0642056
$96$	0	0
$97$	45542.8	0.491463	0.245731	−	0.969338i	$-0.420972\pi$
$97$	45542.8	0.491463	0.245731	+	0.969338i	$0.420972\pi$
$98$	0	0
$99$	− 120318.i	− 1.23380i
$100$	0	0
$101$	195492.i	1.90689i	0.301571	+	0.953444i	$0.402489\pi$
$101$	195492.i	1.90689i	−0.301571	+	0.953444i	$0.597511\pi$
$102$	0	0
$103$	−81681.1	−0.758627	−0.379314	−	0.925268i	$-0.623840\pi$
$103$	−81681.1	−0.758627	−0.379314	+	0.925268i	$0.623840\pi$
$104$	0	0
$105$	−48233.8	−0.426951
$106$	0	0
$107$	− 107507.i	− 0.907775i	−0.891059	−	0.453888i	$-0.850037\pi$
$107$	− 107507.i	− 0.907775i	0.891059	−	0.453888i	$-0.149963\pi$
$108$	0	0
$109$	− 170366.i	− 1.37346i	−0.726912	−	0.686731i	$-0.759045\pi$
$109$	− 170366.i	− 1.37346i	0.726912	−	0.686731i	$-0.240955\pi$
$110$	0	0
$111$	293548.	2.26137
$112$	0	0
$113$	−81363.4	−0.599423	−0.299711	−	0.954030i	$-0.596890\pi$
$113$	−81363.4	−0.599423	−0.299711	+	0.954030i	$0.596890\pi$
$114$	0	0
$115$	− 27739.3i	− 0.195592i
$116$	0	0
$117$	− 33562.9i	− 0.226670i
$118$	0	0
$119$	43683.4	0.282780
$120$	0	0
$121$	−273302.	−1.69699
$122$	0	0
$123$	− 101700.i	− 0.606119i
$124$	0	0
$125$	15625.0i	0.0894427i
$126$	0	0
$127$	−291463.	−1.60352	−0.801759	−	0.597647i	$-0.796102\pi$
$127$	−291463.	−1.60352	−0.801759	+	0.597647i	$0.796102\pi$
$128$	0	0
$129$	−255382.	−1.35119
$130$	0	0
$131$	276390.i	1.40716i	0.710616	+	0.703580i	$0.248416\pi$
$131$	276390.i	1.40716i	−0.710616	+	0.703580i	$0.751584\pi$
$132$	0	0
$133$	− 21128.7i	− 0.103572i
$134$	0	0
$135$	31169.6	0.147196
$136$	0	0
$137$	−366347.	−1.66760	−0.833798	−	0.552069i	$-0.813839\pi$
$137$	−366347.	−1.66760	−0.833798	+	0.552069i	$0.813839\pi$
$138$	0	0
$139$	162496.i	0.713353i	0.934228	+	0.356676i	$0.116090\pi$
$139$	162496.i	0.713353i	−0.934228	+	0.356676i	$0.883910\pi$
$140$	0	0
$141$	− 12786.8i	− 0.0541645i
$142$	0	0
$143$	−121163.	−0.495485
$144$	0	0
$145$	−43474.5	−0.171718
$146$	0	0
$147$	− 166271.i	− 0.634633i
$148$	0	0
$149$	− 301589.i	− 1.11288i	−0.830887	−	0.556441i	$-0.812166\pi$
$149$	− 301589.i	− 1.11288i	0.830887	−	0.556441i	$-0.187834\pi$
$150$	0	0
$151$	375519.	1.34026	0.670131	−	0.742243i	$-0.266238\pi$
$151$	375519.	1.34026	0.670131	+	0.742243i	$0.266238\pi$
$152$	0	0
$153$	85270.1	0.294488
$154$	0	0
$155$	181770.i	0.607706i
$156$	0	0
$157$	− 548149.i	− 1.77480i	−0.460999	−	0.887400i	$-0.652509\pi$
$157$	− 548149.i	− 1.77480i	0.460999	−	0.887400i	$-0.347491\pi$
$158$	0	0
$159$	−350469.	−1.09940
$160$	0	0
$161$	−103773.	−0.315515
$162$	0	0
$163$	− 19042.7i	− 0.0561383i	−0.999606	−	0.0280692i	$-0.991064\pi$
$163$	− 19042.7i	− 0.0561383i	0.999606	−	0.0280692i	$-0.00893587\pi$
$164$	0	0
$165$	339894.i	0.971926i
$166$	0	0
$167$	−287366.	−0.797340	−0.398670	−	0.917094i	$-0.630528\pi$
$167$	−287366.	−0.797340	−0.398670	+	0.917094i	$0.630528\pi$
$168$	0	0
$169$	337494.	0.908971
$170$	0	0
$171$	− 41243.2i	− 0.107860i
$172$	0	0
$173$	352199.i	0.894692i	0.894361	+	0.447346i	$0.147631\pi$
$173$	352199.i	0.894692i	−0.894361	+	0.447346i	$0.852369\pi$
$174$	0	0
$175$	58453.4	0.144283
$176$	0	0
$177$	1.04678e6	2.51144
$178$	0	0
$179$	87611.7i	0.204376i	0.994765	+	0.102188i	$0.0325843\pi$
$179$	87611.7i	0.204376i	−0.994765	+	0.102188i	$0.967416\pi$
$180$	0	0
$181$	534314.i	1.21227i	0.795361	+	0.606136i	$0.207281\pi$
$181$	534314.i	1.21227i	−0.795361	+	0.606136i	$0.792719\pi$
$182$	0	0
$183$	426328.	0.941056
$184$	0	0
$185$	−355744.	−0.764202
$186$	0	0
$187$	− 307828.i	− 0.643730i
$188$	0	0
$189$	− 116606.i	− 0.237447i
$190$	0	0
$191$	45622.1	0.0904882	0.0452441	−	0.998976i	$-0.485593\pi$
$191$	45622.1	0.0904882	0.0452441	+	0.998976i	$0.485593\pi$
$192$	0	0
$193$	−26392.3	−0.0510017	−0.0255009	−	0.999675i	$-0.508118\pi$
$193$	−26392.3	−0.0510017	−0.0255009	+	0.999675i	$0.508118\pi$
$194$	0	0
$195$	94813.5i	0.178560i
$196$	0	0
$197$	− 483767.i	− 0.888119i	−0.895997	−	0.444059i	$-0.853538\pi$
$197$	− 483767.i	− 0.888119i	0.895997	−	0.444059i	$-0.146462\pi$
$198$	0	0
$199$	−51497.1	−0.0921828	−0.0460914	−	0.998937i	$-0.514677\pi$
$199$	−51497.1	−0.0921828	−0.0460914	+	0.998937i	$0.514677\pi$
$200$	0	0
$201$	864781.	1.50979
$202$	0	0
$203$	162639.i	0.277003i
$204$	0	0
$205$	123248.i	0.204831i
$206$	0	0
$207$	−202566.	−0.328579
$208$	0	0
$209$	−148889.	−0.235775
$210$	0	0
$211$	716759.i	1.10833i	0.832408	+	0.554163i	$0.186961\pi$
$211$	716759.i	1.10833i	−0.832408	+	0.554163i	$0.813039\pi$
$212$	0	0
$213$	1.12816e6i	1.70381i
$214$	0	0
$215$	309491.	0.456617
$216$	0	0
$217$	680005.	0.980309
$218$	0	0
$219$	− 520092.i	− 0.732774i
$220$	0	0
$221$	− 85868.8i	− 0.118265i
$222$	0	0
$223$	−747997.	−1.00725	−0.503625	−	0.863922i	$-0.668001\pi$
$223$	−747997.	−1.00725	−0.503625	+	0.863922i	$0.668001\pi$
$224$	0	0
$225$	114101.	0.150257
$226$	0	0
$227$	598186.i	0.770498i	0.922813	+	0.385249i	$0.125884\pi$
$227$	598186.i	0.770498i	−0.922813	+	0.385249i	$0.874116\pi$
$228$	0	0
$229$	655203.i	0.825633i	0.910814	+	0.412816i	$0.135455\pi$
$229$	655203.i	0.825633i	−0.910814	+	0.412816i	$0.864545\pi$
$230$	0	0
$231$	1.27155e6	1.56784
$232$	0	0
$233$	170801.	0.206111	0.103055	−	0.994676i	$-0.467138\pi$
$233$	170801.	0.206111	0.103055	+	0.994676i	$0.467138\pi$
$234$	0	0
$235$	15496.0i	0.0183042i
$236$	0	0
$237$	598255.i	0.691855i
$238$	0	0
$239$	−1.26920e6	−1.43726	−0.718629	−	0.695393i	$-0.755230\pi$
$239$	−1.26920e6	−1.43726	−0.718629	+	0.695393i	$0.755230\pi$
$240$	0	0
$241$	274289.	0.304205	0.152103	−	0.988365i	$-0.451396\pi$
$241$	274289.	0.304205	0.152103	+	0.988365i	$0.451396\pi$
$242$	0	0
$243$	− 1.14278e6i	− 1.24150i
$244$	0	0
$245$	201500.i	0.214467i
$246$	0	0
$247$	−41532.8	−0.0433160
$248$	0	0
$249$	1.91189e6	1.95418
$250$	0	0
$251$	861371.i	0.862990i	0.902115	+	0.431495i	$0.142014\pi$
$251$	861371.i	0.862990i	−0.902115	+	0.431495i	$0.857986\pi$
$252$	0	0
$253$	731268.i	0.718250i
$254$	0	0
$255$	−240884.	−0.231984
$256$	0	0
$257$	524792.	0.495626	0.247813	−	0.968808i	$-0.420288\pi$
$257$	524792.	0.495626	0.247813	+	0.968808i	$0.420288\pi$
$258$	0	0
$259$	1.33084e6i	1.23276i
$260$	0	0
$261$	317472.i	0.288472i
$262$	0	0
$263$	−2.03635e6	−1.81536	−0.907680	−	0.419664i	$-0.862148\pi$
$263$	−2.03635e6	−1.81536	−0.907680	+	0.419664i	$0.862148\pi$
$264$	0	0
$265$	424725.	0.371529
$266$	0	0
$267$	1.12726e6i	0.967712i
$268$	0	0
$269$	477001.i	0.401919i	0.979600	+	0.200960i	$0.0644060\pi$
$269$	477001.i	0.401919i	−0.979600	+	0.200960i	$0.935594\pi$
$270$	0	0
$271$	−350310.	−0.289754	−0.144877	−	0.989450i	$-0.546279\pi$
$271$	−350310.	−0.289754	−0.144877	+	0.989450i	$0.546279\pi$
$272$	0	0
$273$	354699.	0.288040
$274$	0	0
$275$	− 411909.i	− 0.328450i
$276$	0	0
$277$	− 2.08816e6i	− 1.63518i	−0.575804	−	0.817588i	$-0.695311\pi$
$277$	− 2.08816e6i	− 1.63518i	0.575804	−	0.817588i	$-0.304689\pi$
$278$	0	0
$279$	1.32737e6	1.02090
$280$	0	0
$281$	−1.24077e6	−0.937399	−0.468699	−	0.883358i	$-0.655277\pi$
$281$	−1.24077e6	−0.937399	−0.468699	+	0.883358i	$0.655277\pi$
$282$	0	0
$283$	1.85643e6i	1.37788i	0.724818	+	0.688940i	$0.241924\pi$
$283$	1.85643e6i	1.37788i	−0.724818	+	0.688940i	$0.758076\pi$
$284$	0	0
$285$	116510.i	0.0849672i
$286$	0	0
$287$	461072.	0.330419
$288$	0	0
$289$	−1.20170e6	−0.846352
$290$	0	0
$291$	939510.i	0.650383i
$292$	0	0
$293$	− 1.64126e6i	− 1.11689i	−0.829543	−	0.558443i	$-0.811399\pi$
$293$	− 1.64126e6i	− 1.11689i	0.829543	−	0.558443i	$-0.188601\pi$
$294$	0	0
$295$	−1.26857e6	−0.848711
$296$	0	0
$297$	−821699.	−0.540532
$298$	0	0
$299$	203988.i	0.131955i
$300$	0	0
$301$	− 1.15781e6i	− 0.736583i
$302$	0	0
$303$	−4.03283e6	−2.52350
$304$	0	0
$305$	−516657.	−0.318018
$306$	0	0
$307$	1.40107e6i	0.848423i	0.905563	+	0.424212i	$0.139449\pi$
$307$	1.40107e6i	0.848423i	−0.905563	+	0.424212i	$0.860551\pi$
$308$	0	0
$309$	− 1.68501e6i	− 1.00394i
$310$	0	0
$311$	1.24137e6	0.727783	0.363891	−	0.931441i	$-0.381448\pi$
$311$	1.24137e6	0.727783	0.363891	+	0.931441i	$0.381448\pi$
$312$	0	0
$313$	−2.66880e6	−1.53977	−0.769885	−	0.638183i	$-0.779686\pi$
$313$	−2.66880e6	−1.53977	−0.769885	+	0.638183i	$0.779686\pi$
$314$	0	0
$315$	− 426855.i	− 0.242384i
$316$	0	0
$317$	2.96026e6i	1.65456i	0.561793	+	0.827278i	$0.310112\pi$
$317$	2.96026e6i	1.65456i	−0.561793	+	0.827278i	$0.689888\pi$
$318$	0	0
$319$	1.14608e6	0.630579
$320$	0	0
$321$	2.21778e6	1.20131
$322$	0	0
$323$	− 105518.i	− 0.0562759i
$324$	0	0
$325$	− 114902.i	− 0.0603421i
$326$	0	0
$327$	3.51450e6	1.81758
$328$	0	0
$329$	57971.0	0.0295271
$330$	0	0
$331$	1.50180e6i	0.753430i	0.926329	+	0.376715i	$0.122946\pi$
$331$	1.50180e6i	0.753430i	−0.926329	+	0.376715i	$0.877054\pi$
$332$	0	0
$333$	2.59781e6i	1.28380i
$334$	0	0
$335$	−1.04801e6	−0.510214
$336$	0	0
$337$	−1.61150e6	−0.772957	−0.386478	−	0.922298i	$-0.626309\pi$
$337$	−1.61150e6	−0.772957	−0.386478	+	0.922298i	$0.626309\pi$
$338$	0	0
$339$	− 1.67846e6i	− 0.793253i
$340$	0	0
$341$	− 4.79186e6i	− 2.23161i
$342$	0	0
$343$	2.32570e6	1.06738
$344$	0	0
$345$	572238.	0.258839
$346$	0	0
$347$	− 497865.i	− 0.221967i	−0.993822	−	0.110983i	$-0.964600\pi$
$347$	− 497865.i	− 0.221967i	0.993822	−	0.110983i	$-0.0354001\pi$
$348$	0	0
$349$	3.24174e6i	1.42467i	0.701839	+	0.712335i	$0.252362\pi$
$349$	3.24174e6i	1.42467i	−0.701839	+	0.712335i	$0.747638\pi$
$350$	0	0
$351$	−229214.	−0.0993053
$352$	0	0
$353$	−436867.	−0.186600	−0.0933001	−	0.995638i	$-0.529742\pi$
$353$	−436867.	−0.186600	−0.0933001	+	0.995638i	$0.529742\pi$
$354$	0	0
$355$	− 1.36719e6i	− 0.575783i
$356$	0	0
$357$	901151.i	0.374220i
$358$	0	0
$359$	1.67096e6	0.684274	0.342137	−	0.939650i	$-0.388849\pi$
$359$	1.67096e6	0.684274	0.342137	+	0.939650i	$0.388849\pi$
$360$	0	0
$361$	2.42506e6	0.979388
$362$	0	0
$363$	− 5.63799e6i	− 2.24573i
$364$	0	0
$365$	630288.i	0.247632i
$366$	0	0
$367$	−3.06532e6	−1.18799	−0.593993	−	0.804471i	$-0.702449\pi$
$367$	−3.06532e6	−1.18799	−0.593993	+	0.804471i	$0.702449\pi$
$368$	0	0
$369$	900015.	0.344099
$370$	0	0
$371$	− 1.58890e6i	− 0.599325i
$372$	0	0
$373$	3.45643e6i	1.28634i	0.765724	+	0.643169i	$0.222381\pi$
$373$	3.45643e6i	1.28634i	−0.765724	+	0.643169i	$0.777619\pi$
$374$	0	0
$375$	−322331.	−0.118365
$376$	0	0
$377$	319701.	0.115849
$378$	0	0
$379$	1.61905e6i	0.578979i	0.957181	+	0.289489i	$0.0934855\pi$
$379$	1.61905e6i	0.578979i	−0.957181	+	0.289489i	$0.906514\pi$
$380$	0	0
$381$	− 6.01263e6i	− 2.12203i
$382$	0	0
$383$	151934.	0.0529248	0.0264624	−	0.999650i	$-0.491576\pi$
$383$	151934.	0.0529248	0.0264624	+	0.999650i	$0.491576\pi$
$384$	0	0
$385$	−1.54096e6	−0.529833
$386$	0	0
$387$	− 2.26005e6i	− 0.767081i
$388$	0	0
$389$	− 2.83414e6i	− 0.949615i	−0.880090	−	0.474807i	$-0.842518\pi$
$389$	− 2.83414e6i	− 0.949615i	0.880090	−	0.474807i	$-0.157482\pi$
$390$	0	0
$391$	−518253.	−0.171435
$392$	0	0
$393$	−5.70168e6	−1.86218
$394$	0	0
$395$	− 725011.i	− 0.233804i
$396$	0	0
$397$	− 1.66354e6i	− 0.529732i	−0.964285	−	0.264866i	$-0.914672\pi$
$397$	− 1.66354e6i	− 0.529732i	0.964285	−	0.264866i	$-0.0853277\pi$
$398$	0	0
$399$	435866.	0.137063
$400$	0	0
$401$	3.72720e6	1.15750	0.578751	−	0.815504i	$-0.303540\pi$
$401$	3.72720e6	1.15750	0.578751	+	0.815504i	$0.303540\pi$
$402$	0	0
$403$	− 1.33669e6i	− 0.409986i
$404$	0	0
$405$	1.75207e6i	0.530778i
$406$	0	0
$407$	9.37818e6	2.80629
$408$	0	0
$409$	−2.06547e6	−0.610536	−0.305268	−	0.952267i	$-0.598746\pi$
$409$	−2.06547e6	−0.610536	−0.305268	+	0.952267i	$0.598746\pi$
$410$	0	0
$411$	− 7.55743e6i	− 2.20683i
$412$	0	0
$413$	4.74574e6i	1.36908i
$414$	0	0
$415$	−2.31698e6	−0.660392
$416$	0	0
$417$	−3.35215e6	−0.944023
$418$	0	0
$419$	− 2.02283e6i	− 0.562892i	−0.959577	−	0.281446i	$-0.909186\pi$
$419$	− 2.02283e6i	− 0.562892i	0.959577	−	0.281446i	$-0.0908141\pi$
$420$	0	0
$421$	− 4.21007e6i	− 1.15767i	−0.815445	−	0.578834i	$-0.803508\pi$
$421$	− 4.21007e6i	− 1.15767i	0.815445	−	0.578834i	$-0.196492\pi$
$422$	0	0
$423$	113160.	0.0307497
$424$	0	0
$425$	291922.	0.0783960
$426$	0	0
$427$	1.93282e6i	0.513005i
$428$	0	0
$429$	− 2.49949e6i	− 0.655705i
$430$	0	0
$431$	−3.35990e6	−0.871231	−0.435616	−	0.900133i	$-0.643469\pi$
$431$	−3.35990e6	−0.871231	−0.435616	+	0.900133i	$0.643469\pi$
$432$	0	0
$433$	1.98488e6	0.508762	0.254381	−	0.967104i	$-0.418128\pi$
$433$	1.98488e6	0.508762	0.254381	+	0.967104i	$0.418128\pi$
$434$	0	0
$435$	− 896843.i	− 0.227244i
$436$	0	0
$437$	250667.i	0.0627905i
$438$	0	0
$439$	5.52991e6	1.36948	0.684741	−	0.728786i	$-0.259915\pi$
$439$	5.52991e6	1.36948	0.684741	+	0.728786i	$0.259915\pi$
$440$	0	0
$441$	1.47145e6	0.360287
$442$	0	0
$443$	− 2.55974e6i	− 0.619708i	−0.950784	−	0.309854i	$-0.899720\pi$
$443$	− 2.55974e6i	− 0.619708i	0.950784	−	0.309854i	$-0.100280\pi$
$444$	0	0
$445$	− 1.36610e6i	− 0.327026i
$446$	0	0
$447$	6.22152e6	1.47275
$448$	0	0
$449$	−6.29746e6	−1.47418	−0.737088	−	0.675796i	$-0.763800\pi$
$449$	−6.29746e6	−1.47418	−0.737088	+	0.675796i	$0.763800\pi$
$450$	0	0
$451$	− 3.24908e6i	− 0.752176i
$452$	0	0
$453$	7.74664e6i	1.77365i
$454$	0	0
$455$	−429852.	−0.0973397
$456$	0	0
$457$	4.50464e6	1.00895	0.504475	−	0.863426i	$-0.331686\pi$
$457$	4.50464e6	1.00895	0.504475	+	0.863426i	$0.331686\pi$
$458$	0	0
$459$	− 582341.i	− 0.129017i
$460$	0	0
$461$	− 8.91962e6i	− 1.95476i	−0.211486	−	0.977381i	$-0.567830\pi$
$461$	− 8.91962e6i	− 1.95476i	0.211486	−	0.977381i	$-0.432170\pi$
$462$	0	0
$463$	−7.68162e6	−1.66533	−0.832665	−	0.553778i	$-0.813186\pi$
$463$	−7.68162e6	−1.66533	−0.832665	+	0.553778i	$0.813186\pi$
$464$	0	0
$465$	−3.74977e6	−0.804214
$466$	0	0
$467$	1.20606e6i	0.255903i	0.991780	+	0.127951i	$0.0408402\pi$
$467$	1.20606e6i	0.255903i	−0.991780	+	0.127951i	$0.959160\pi$
$468$	0	0
$469$	3.92062e6i	0.823042i
$470$	0	0
$471$	1.13079e7	2.34870
$472$	0	0
$473$	−8.15887e6	−1.67678
$474$	0	0
$475$	− 141196.i	− 0.0287136i
$476$	0	0
$477$	− 3.10155e6i	− 0.624140i
$478$	0	0
$479$	−1.89691e6	−0.377753	−0.188876	−	0.982001i	$-0.560485\pi$
$479$	−1.89691e6	−0.377753	−0.188876	+	0.982001i	$0.560485\pi$
$480$	0	0
$481$	2.61605e6	0.515565
$482$	0	0
$483$	− 2.14075e6i	− 0.417540i
$484$	0	0
$485$	− 1.13857e6i	− 0.219789i
$486$	0	0
$487$	−2.24462e6	−0.428864	−0.214432	−	0.976739i	$-0.568790\pi$
$487$	−2.24462e6	−0.428864	−0.214432	+	0.976739i	$0.568790\pi$
$488$	0	0
$489$	392835.	0.0742913
$490$	0	0
$491$	9.45333e6i	1.76962i	0.465948	+	0.884812i	$0.345714\pi$
$491$	9.45333e6i	1.76962i	−0.465948	+	0.884812i	$0.654286\pi$
$492$	0	0
$493$	812234.i	0.150510i
$494$	0	0
$495$	−3.00796e6	−0.551771
$496$	0	0
$497$	−5.11469e6	−0.928812
$498$	0	0
$499$	9.68957e6i	1.74202i	0.491265	+	0.871010i	$0.336535\pi$
$499$	9.68957e6i	1.74202i	−0.491265	+	0.871010i	$0.663465\pi$
$500$	0	0
$501$	− 5.92811e6i	− 1.05517i
$502$	0	0
$503$	−4.98436e6	−0.878394	−0.439197	−	0.898391i	$-0.644737\pi$
$503$	−4.98436e6	−0.878394	−0.439197	+	0.898391i	$0.644737\pi$
$504$	0	0
$505$	4.88729e6	0.852786
$506$	0	0
$507$	6.96223e6i	1.20290i
$508$	0	0
$509$	− 3.72859e6i	− 0.637896i	−0.947772	−	0.318948i	$-0.896670\pi$
$509$	− 3.72859e6i	− 0.637896i	0.947772	−	0.318948i	$-0.103330\pi$
$510$	0	0
$511$	2.35792e6	0.399463
$512$	0	0
$513$	−281665.	−0.0472542
$514$	0	0
$515$	2.04203e6i	0.339268i
$516$	0	0
$517$	− 408510.i	− 0.0672165i
$518$	0	0
$519$	−7.26558e6	−1.18400
$520$	0	0
$521$	−7.50570e6	−1.21143	−0.605713	−	0.795683i	$-0.707112\pi$
$521$	−7.50570e6	−1.21143	−0.605713	+	0.795683i	$0.707112\pi$
$522$	0	0
$523$	− 8.28837e6i	− 1.32500i	−0.749063	−	0.662498i	$-0.769496\pi$
$523$	− 8.28837e6i	− 1.32500i	0.749063	−	0.662498i	$-0.230504\pi$
$524$	0	0
$525$	1.20584e6i	0.190938i
$526$	0	0
$527$	3.39601e6	0.532651
$528$	0	0
$529$	−5.20519e6	−0.808719
$530$	0	0
$531$	9.26371e6i	1.42577i
$532$	0	0
$533$	− 906334.i	− 0.138188i
$534$	0	0
$535$	−2.68768e6	−0.405970
$536$	0	0
$537$	−1.80735e6	−0.270463
$538$	0	0
$539$	− 5.31198e6i	− 0.787561i
$540$	0	0
$541$	4.42308e6i	0.649728i	0.945761	+	0.324864i	$0.105318\pi$
$541$	4.42308e6i	0.649728i	−0.945761	+	0.324864i	$0.894682\pi$
$542$	0	0
$543$	−1.10224e7	−1.60427
$544$	0	0
$545$	−4.25915e6	−0.614231
$546$	0	0
$547$	1.60000e6i	0.228640i	0.993444	+	0.114320i	$0.0364689\pi$
$547$	1.60000e6i	0.228640i	−0.993444	+	0.114320i	$0.963531\pi$
$548$	0	0
$549$	3.77287e6i	0.534246i
$550$	0	0
$551$	392859.	0.0551262
$552$	0	0
$553$	−2.71228e6	−0.377156
$554$	0	0
$555$	− 7.33869e6i	− 1.01131i
$556$	0	0
$557$	− 5.33213e6i	− 0.728220i	−0.931356	−	0.364110i	$-0.881373\pi$
$557$	− 5.33213e6i	− 0.728220i	0.931356	−	0.364110i	$-0.118627\pi$
$558$	0	0
$559$	−2.27592e6	−0.308055
$560$	0	0
$561$	6.35023e6	0.851887
$562$	0	0
$563$	− 5.39829e6i	− 0.717770i	−0.933382	−	0.358885i	$-0.883157\pi$
$563$	− 5.39829e6i	− 0.717770i	0.933382	−	0.358885i	$-0.116843\pi$
$564$	0	0
$565$	2.03409e6i	0.268070i
$566$	0	0
$567$	6.55451e6	0.856215
$568$	0	0
$569$	1.34873e7	1.74641	0.873204	−	0.487355i	$-0.162038\pi$
$569$	1.34873e7	1.74641	0.873204	+	0.487355i	$0.162038\pi$
$570$	0	0
$571$	6.16933e6i	0.791859i	0.918281	+	0.395929i	$0.129578\pi$
$571$	6.16933e6i	0.791859i	−0.918281	+	0.395929i	$0.870422\pi$
$572$	0	0
$573$	941145.i	0.119748i
$574$	0	0
$575$	−693482.	−0.0874713
$576$	0	0
$577$	1.47140e7	1.83989	0.919943	−	0.392052i	$-0.128235\pi$
$577$	1.47140e7	1.83989	0.919943	+	0.392052i	$0.128235\pi$
$578$	0	0
$579$	− 544452.i	− 0.0674937i
$580$	0	0
$581$	8.66786e6i	1.06530i
$582$	0	0
$583$	−1.11967e7	−1.36432
$584$	0	0
$585$	−839072.	−0.101370
$586$	0	0
$587$	1.19357e6i	0.142972i	0.997442	+	0.0714860i	$0.0227741\pi$
$587$	1.19357e6i	0.142972i	−0.997442	+	0.0714860i	$0.977226\pi$
$588$	0	0
$589$	− 1.64257e6i	− 0.195091i
$590$	0	0
$591$	9.97971e6	1.17530
$592$	0	0
$593$	9.29056e6	1.08494	0.542469	−	0.840076i	$-0.317490\pi$
$593$	9.29056e6	1.08494	0.542469	+	0.840076i	$0.317490\pi$
$594$	0	0
$595$	− 1.09208e6i	− 0.126463i
$596$	0	0
$597$	− 1.06234e6i	− 0.121991i
$598$	0	0
$599$	1.60680e7	1.82976	0.914880	−	0.403725i	$-0.132285\pi$
$599$	1.60680e7	1.82976	0.914880	+	0.403725i	$0.132285\pi$
$600$	0	0
$601$	1.28035e7	1.44591	0.722957	−	0.690893i	$-0.242782\pi$
$601$	1.28035e7	1.44591	0.722957	+	0.690893i	$0.242782\pi$
$602$	0	0
$603$	7.65306e6i	0.857120i
$604$	0	0
$605$	6.83255e6i	0.758918i
$606$	0	0
$607$	−8.81825e6	−0.971428	−0.485714	−	0.874118i	$-0.661440\pi$
$607$	−8.81825e6	−0.971428	−0.485714	+	0.874118i	$0.661440\pi$
$608$	0	0
$609$	−3.35510e6	−0.366575
$610$	0	0
$611$	− 113954.i	− 0.0123489i
$612$	0	0
$613$	1.50578e7i	1.61849i	0.587468	+	0.809247i	$0.300125\pi$
$613$	1.50578e7i	1.61849i	−0.587468	+	0.809247i	$0.699875\pi$
$614$	0	0
$615$	−2.54250e6	−0.271065
$616$	0	0
$617$	5.92373e6	0.626445	0.313222	−	0.949680i	$-0.398592\pi$
$617$	5.92373e6	0.626445	0.313222	+	0.949680i	$0.398592\pi$
$618$	0	0
$619$	1.67289e6i	0.175485i	0.996143	+	0.0877426i	$0.0279653\pi$
$619$	1.67289e6i	0.175485i	−0.996143	+	0.0877426i	$0.972035\pi$
$620$	0	0
$621$	1.38340e6i	0.143952i
$622$	0	0
$623$	−5.11061e6	−0.527536
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	− 3.07146e6i	− 0.312016i
$628$	0	0
$629$	6.64636e6i	0.669818i
$630$	0	0
$631$	1.34503e7	1.34480	0.672402	−	0.740186i	$-0.265263\pi$
$631$	1.34503e7	1.34480	0.672402	+	0.740186i	$0.265263\pi$
$632$	0	0
$633$	−1.47861e7	−1.46671
$634$	0	0
$635$	7.28657e6i	0.717115i
$636$	0	0
$637$	− 1.48178e6i	− 0.144689i
$638$	0	0
$639$	−9.98389e6	−0.967270
$640$	0	0
$641$	4.66915e6	0.448841	0.224421	−	0.974492i	$-0.427951\pi$
$641$	4.66915e6	0.448841	0.224421	+	0.974492i	$0.427951\pi$
$642$	0	0
$643$	1.53466e7i	1.46381i	0.681405	+	0.731906i	$0.261369\pi$
$643$	1.53466e7i	1.46381i	−0.681405	+	0.731906i	$0.738631\pi$
$644$	0	0
$645$	6.38454e6i	0.604269i
$646$	0	0
$647$	−1.28923e7	−1.21079	−0.605395	−	0.795925i	$-0.706985\pi$
$647$	−1.28923e7	−1.21079	−0.605395	+	0.795925i	$0.706985\pi$
$648$	0	0
$649$	3.34423e7	3.11662
$650$	0	0
$651$	1.40279e7i	1.29730i
$652$	0	0
$653$	6.98908e6i	0.641412i	0.947179	+	0.320706i	$0.103920\pi$
$653$	6.98908e6i	0.641412i	−0.947179	+	0.320706i	$0.896080\pi$
$654$	0	0
$655$	6.90974e6	0.629301
$656$	0	0
$657$	4.60267e6	0.416003
$658$	0	0
$659$	− 5.92616e6i	− 0.531569i	−0.964032	−	0.265785i	$-0.914369\pi$
$659$	− 5.92616e6i	− 0.531569i	0.964032	−	0.265785i	$-0.0856310\pi$
$660$	0	0
$661$	− 6.41040e6i	− 0.570665i	−0.958429	−	0.285333i	$-0.907896\pi$
$661$	− 6.41040e6i	− 0.570665i	0.958429	−	0.285333i	$-0.0921040\pi$
$662$	0	0
$663$	1.77140e6	0.156507
$664$	0	0
$665$	−528216.	−0.0463188
$666$	0	0
$667$	− 1.92952e6i	− 0.167933i
$668$	0	0
$669$	− 1.54305e7i	− 1.33296i
$670$	0	0
$671$	1.36202e7	1.16782
$672$	0	0
$673$	1.44237e7	1.22755	0.613775	−	0.789481i	$-0.289650\pi$
$673$	1.44237e7	1.22755	0.613775	+	0.789481i	$0.289650\pi$
$674$	0	0
$675$	− 779240.i	− 0.0658282i
$676$	0	0
$677$	2.61864e6i	0.219585i	0.993955	+	0.109793i	$0.0350187\pi$
$677$	2.61864e6i	0.219585i	−0.993955	+	0.109793i	$0.964981\pi$
$678$	0	0
$679$	−4.25941e6	−0.354548
$680$	0	0
$681$	−1.23401e7	−1.01965
$682$	0	0
$683$	− 1.73254e7i	− 1.42112i	−0.703635	−	0.710561i	$-0.748441\pi$
$683$	− 1.73254e7i	− 1.42112i	0.703635	−	0.710561i	$-0.251559\pi$
$684$	0	0
$685$	9.15867e6i	0.745772i
$686$	0	0
$687$	−1.35163e7	−1.09261
$688$	0	0
$689$	−3.12332e6	−0.250650
$690$	0	0
$691$	9.93677e6i	0.791681i	0.918319	+	0.395840i	$0.129547\pi$
$691$	9.93677e6i	0.791681i	−0.918319	+	0.395840i	$0.870453\pi$
$692$	0	0
$693$	1.12528e7i	0.890079i
$694$	0	0
$695$	4.06239e6	0.319021
$696$	0	0
$697$	2.30264e6	0.179533
$698$	0	0
$699$	3.52348e6i	0.272759i
$700$	0	0
$701$	− 2.70085e6i	− 0.207590i	−0.994599	−	0.103795i	$-0.966901\pi$
$701$	− 2.70085e6i	− 0.207590i	0.994599	−	0.103795i	$-0.0330985\pi$
$702$	0	0
$703$	3.21469e6	0.245330
$704$	0	0
$705$	−319670.	−0.0242231
$706$	0	0
$707$	− 1.82835e7i	− 1.37566i
$708$	0	0
$709$	− 1.98847e6i	− 0.148561i	−0.997237	−	0.0742804i	$-0.976334\pi$
$709$	− 1.98847e6i	− 0.148561i	0.997237	−	0.0742804i	$-0.0236660\pi$
$710$	0	0
$711$	−5.29438e6	−0.392773
$712$	0	0
$713$	−8.06748e6	−0.594312
$714$	0	0
$715$	3.02908e6i	0.221587i
$716$	0	0
$717$	− 2.61825e7i	− 1.90201i
$718$	0	0
$719$	5.01212e6	0.361576	0.180788	−	0.983522i	$-0.442135\pi$
$719$	5.01212e6	0.361576	0.180788	+	0.983522i	$0.442135\pi$
$720$	0	0
$721$	7.63926e6	0.547284
$722$	0	0
$723$	5.65836e6i	0.402573i
$724$	0	0
$725$	1.08686e6i	0.0767945i
$726$	0	0
$727$	−1.08406e7	−0.760709	−0.380355	−	0.924841i	$-0.624198\pi$
$727$	−1.08406e7	−0.760709	−0.380355	+	0.924841i	$0.624198\pi$
$728$	0	0
$729$	6.54445e6	0.456094
$730$	0	0
$731$	− 5.78222e6i	− 0.400222i
$732$	0	0
$733$	9.18857e6i	0.631667i	0.948815	+	0.315833i	$0.102284\pi$
$733$	9.18857e6i	0.631667i	−0.948815	+	0.315833i	$0.897716\pi$
$734$	0	0
$735$	−4.15677e6	−0.283817
$736$	0	0
$737$	2.76278e7	1.87360
$738$	0	0
$739$	6.19968e6i	0.417598i	0.977959	+	0.208799i	$0.0669554\pi$
$739$	6.19968e6i	0.417598i	−0.977959	+	0.208799i	$0.933045\pi$
$740$	0	0
$741$	− 856786.i	− 0.0573227i
$742$	0	0
$743$	−7.11862e6	−0.473068	−0.236534	−	0.971623i	$-0.576012\pi$
$743$	−7.11862e6	−0.473068	−0.236534	+	0.971623i	$0.576012\pi$
$744$	0	0
$745$	−7.53972e6	−0.497696
$746$	0	0
$747$	1.69197e7i	1.10941i
$748$	0	0
$749$	1.00547e7i	0.654882i
$750$	0	0
$751$	−7.67860e6	−0.496801	−0.248400	−	0.968657i	$-0.579905\pi$
$751$	−7.67860e6	−0.496801	−0.248400	+	0.968657i	$0.579905\pi$
$752$	0	0
$753$	−1.77694e7	−1.14205
$754$	0	0
$755$	− 9.38797e6i	− 0.599383i
$756$	0	0
$757$	− 5.96101e6i	− 0.378077i	−0.981970	−	0.189039i	$-0.939463\pi$
$757$	− 5.96101e6i	− 0.378077i	0.981970	−	0.189039i	$-0.0605371\pi$
$758$	0	0
$759$	−1.50854e7	−0.950504
$760$	0	0
$761$	−2.00820e7	−1.25703	−0.628513	−	0.777799i	$-0.716336\pi$
$761$	−2.00820e7	−1.25703	−0.628513	+	0.777799i	$0.716336\pi$
$762$	0	0
$763$	1.59335e7i	0.990834i
$764$	0	0
$765$	− 2.13175e6i	− 0.131699i
$766$	0	0
$767$	9.32875e6	0.572579
$768$	0	0
$769$	−2.98669e7	−1.82127	−0.910635	−	0.413212i	$-0.864407\pi$
$769$	−2.98669e7	−1.82127	−0.910635	+	0.413212i	$0.864407\pi$
$770$	0	0
$771$	1.08260e7i	0.655892i
$772$	0	0
$773$	− 1.24588e7i	− 0.749944i	−0.927036	−	0.374972i	$-0.877652\pi$
$773$	− 1.24588e7i	− 0.749944i	0.927036	−	0.374972i	$-0.122348\pi$
$774$	0	0
$775$	4.54425e6	0.271774
$776$	0	0
$777$	−2.74542e7	−1.63138
$778$	0	0
$779$	− 1.11373e6i	− 0.0657564i
$780$	0	0
$781$	3.60422e7i	2.11438i
$782$	0	0
$783$	2.16813e6	0.126381
$784$	0	0
$785$	−1.37037e7	−0.793715
$786$	0	0
$787$	1.73472e7i	0.998372i	0.866495	+	0.499186i	$0.166368\pi$
$787$	1.73472e7i	0.998372i	−0.866495	+	0.499186i	$0.833632\pi$
$788$	0	0
$789$	− 4.20081e7i	− 2.40238i
$790$	0	0
$791$	7.60955e6	0.432432
$792$	0	0
$793$	3.79936e6	0.214550
$794$	0	0
$795$	8.76172e6i	0.491667i
$796$	0	0
$797$	− 6.44517e6i	− 0.359408i	−0.983721	−	0.179704i	$-0.942486\pi$
$797$	− 6.44517e6i	− 0.359408i	0.983721	−	0.179704i	$-0.0575141\pi$
$798$	0	0
$799$	289512.	0.0160435
$800$	0	0
$801$	−9.97592e6	−0.549379
$802$	0	0
$803$	− 1.66158e7i	− 0.909351i
$804$	0	0
$805$	2.59433e6i	0.141103i
$806$	0	0
$807$	−9.84013e6	−0.531884
$808$	0	0
$809$	1.85411e6	0.0996010	0.0498005	−	0.998759i	$-0.484141\pi$
$809$	1.85411e6	0.0996010	0.0498005	+	0.998759i	$0.484141\pi$
$810$	0	0
$811$	1.23842e7i	0.661176i	0.943775	+	0.330588i	$0.107247\pi$
$811$	1.23842e7i	0.661176i	−0.943775	+	0.330588i	$0.892753\pi$
$812$	0	0
$813$	− 7.22659e6i	− 0.383449i
$814$	0	0
$815$	−476067.	−0.0251058
$816$	0	0
$817$	−2.79673e6	−0.146587
$818$	0	0
$819$	3.13898e6i	0.163523i
$820$	0	0
$821$	− 1.24225e7i	− 0.643207i	−0.946874	−	0.321603i	$-0.895778\pi$
$821$	− 1.24225e7i	− 0.643207i	0.946874	−	0.321603i	$-0.104222\pi$
$822$	0	0
$823$	−2.32448e7	−1.19626	−0.598132	−	0.801398i	$-0.704090\pi$
$823$	−2.32448e7	−1.19626	−0.598132	+	0.801398i	$0.704090\pi$
$824$	0	0
$825$	8.49734e6	0.434658
$826$	0	0
$827$	− 2.28900e7i	− 1.16381i	−0.813257	−	0.581904i	$-0.802308\pi$
$827$	− 2.28900e7i	− 1.16381i	0.813257	−	0.581904i	$-0.197692\pi$
$828$	0	0
$829$	3.27454e7i	1.65487i	0.561562	+	0.827435i	$0.310201\pi$
$829$	3.27454e7i	1.65487i	−0.561562	+	0.827435i	$0.689799\pi$
$830$	0	0
$831$	4.30770e7	2.16393
$832$	0	0
$833$	3.76462e6	0.187979
$834$	0	0
$835$	7.18414e6i	0.356581i
$836$	0	0
$837$	− 9.06513e6i	− 0.447260i
$838$	0	0
$839$	−2.43580e7	−1.19464	−0.597319	−	0.802003i	$-0.703768\pi$
$839$	−2.43580e7	−1.19464	−0.597319	+	0.802003i	$0.703768\pi$
$840$	0	0
$841$	1.74871e7	0.852565
$842$	0	0
$843$	− 2.55960e7i	− 1.24052i
$844$	0	0
$845$	− 8.43736e6i	− 0.406504i
$846$	0	0
$847$	2.55607e7	1.22423
$848$	0	0
$849$	−3.82965e7	−1.82343
$850$	0	0
$851$	− 1.57889e7i	− 0.747358i
$852$	0	0
$853$	− 4.14097e7i	− 1.94863i	−0.225192	−	0.974314i	$-0.572301\pi$
$853$	− 4.14097e7i	− 1.94863i	0.225192	−	0.974314i	$-0.427699\pi$
$854$	0	0
$855$	−1.03108e6	−0.0482367
$856$	0	0
$857$	−6.65781e6	−0.309656	−0.154828	−	0.987941i	$-0.549482\pi$
$857$	−6.65781e6	−0.309656	−0.154828	+	0.987941i	$0.549482\pi$
$858$	0	0
$859$	− 1.99635e7i	− 0.923111i	−0.887111	−	0.461555i	$-0.847292\pi$
$859$	− 1.99635e7i	− 0.923111i	0.887111	−	0.461555i	$-0.152708\pi$
$860$	0	0
$861$	9.51153e6i	0.437263i
$862$	0	0
$863$	2.53117e7	1.15690	0.578448	−	0.815719i	$-0.303659\pi$
$863$	2.53117e7	1.15690	0.578448	+	0.815719i	$0.303659\pi$
$864$	0	0
$865$	8.80498e6	0.400118
$866$	0	0
$867$	− 2.47900e7i	− 1.12003i
$868$	0	0
$869$	1.91129e7i	0.858572i
$870$	0	0
$871$	7.70679e6	0.344214
$872$	0	0
$873$	−8.31439e6	−0.369228
$874$	0	0
$875$	− 1.46133e6i	− 0.0645252i
$876$	0	0
$877$	3.34243e7i	1.46745i	0.679446	+	0.733726i	$0.262220\pi$
$877$	3.34243e7i	1.46745i	−0.679446	+	0.733726i	$0.737780\pi$
$878$	0	0
$879$	3.38578e7	1.47804
$880$	0	0
$881$	−1.99218e7	−0.864745	−0.432373	−	0.901695i	$-0.642323\pi$
$881$	−1.99218e7	−0.864745	−0.432373	+	0.901695i	$0.642323\pi$
$882$	0	0
$883$	− 3.47030e7i	− 1.49784i	−0.662661	−	0.748920i	$-0.730573\pi$
$883$	− 3.47030e7i	− 1.49784i	0.662661	−	0.748920i	$-0.269427\pi$
$884$	0	0
$885$	− 2.61695e7i	− 1.12315i
$886$	0	0
$887$	1.55960e7	0.665586	0.332793	−	0.943000i	$-0.392009\pi$
$887$	1.55960e7	0.665586	0.332793	+	0.943000i	$0.392009\pi$
$888$	0	0
$889$	2.72592e7	1.15680
$890$	0	0
$891$	− 4.61883e7i	− 1.94912i
$892$	0	0
$893$	− 140031.i	− 0.00587617i
$894$	0	0
$895$	2.19029e6	0.0913996
$896$	0	0
$897$	−4.20810e6	−0.174624
$898$	0	0
$899$	1.26438e7i	0.521769i
$900$	0	0
$901$	− 7.93513e6i	− 0.325643i
$902$	0	0
$903$	2.38847e7	0.974765
$904$	0	0
$905$	1.33578e7	0.542144
$906$	0	0
$907$	1.98603e7i	0.801620i	0.916161	+	0.400810i	$0.131271\pi$
$907$	1.98603e7i	0.801620i	−0.916161	+	0.400810i	$0.868729\pi$
$908$	0	0
$909$	− 3.56894e7i	− 1.43261i
$910$	0	0
$911$	−1.26907e7	−0.506627	−0.253313	−	0.967384i	$-0.581520\pi$
$911$	−1.26907e7	−0.506627	−0.253313	+	0.967384i	$0.581520\pi$
$912$	0	0
$913$	6.10806e7	2.42508
$914$	0	0
$915$	− 1.06582e7i	− 0.420853i
$916$	0	0
$917$	− 2.58494e7i	− 1.01514i
$918$	0	0
$919$	−1.26560e7	−0.494318	−0.247159	−	0.968975i	$-0.579497\pi$
$919$	−1.26560e7	−0.494318	−0.247159	+	0.968975i	$0.579497\pi$
$920$	0	0
$921$	−2.89028e7	−1.12277
$922$	0	0
$923$	1.00540e7i	0.388449i
$924$	0	0
$925$	8.89359e6i	0.341761i
$926$	0	0
$927$	1.49119e7	0.569944
$928$	0	0
$929$	−3.77116e7	−1.43363	−0.716813	−	0.697265i	$-0.754400\pi$
$929$	−3.77116e7	−1.43363	−0.716813	+	0.697265i	$0.754400\pi$
$930$	0	0
$931$	− 1.82086e6i	− 0.0688498i
$932$	0	0
$933$	2.56085e7i	0.963119i
$934$	0	0
$935$	−7.69569e6	−0.287885
$936$	0	0
$937$	−1.13200e7	−0.421208	−0.210604	−	0.977572i	$-0.567543\pi$
$937$	−1.13200e7	−0.421208	−0.210604	+	0.977572i	$0.567543\pi$
$938$	0	0
$939$	− 5.50551e7i	− 2.03767i
$940$	0	0
$941$	− 1.30012e7i	− 0.478641i	−0.970941	−	0.239320i	$-0.923075\pi$
$941$	− 1.30012e7i	− 0.478641i	0.970941	−	0.239320i	$-0.0769246\pi$
$942$	0	0
$943$	−5.47009e6	−0.200316
$944$	0	0
$945$	−2.91515e6	−0.106189
$946$	0	0
$947$	− 2.89411e7i	− 1.04867i	−0.851511	−	0.524337i	$-0.824313\pi$
$947$	− 2.89411e7i	− 1.04867i	0.851511	−	0.524337i	$-0.175687\pi$
$948$	0	0
$949$	− 4.63498e6i	− 0.167064i
$950$	0	0
$951$	−6.10676e7	−2.18957
$952$	0	0
$953$	−2.70255e6	−0.0963922	−0.0481961	−	0.998838i	$-0.515347\pi$
$953$	−2.70255e6	−0.0963922	−0.0481961	+	0.998838i	$0.515347\pi$
$954$	0	0
$955$	− 1.14055e6i	− 0.0404675i
$956$	0	0
$957$	2.36427e7i	0.834484i
$958$	0	0
$959$	3.42627e7	1.20303
$960$	0	0
$961$	2.42355e7	0.846532
$962$	0	0
$963$	1.96267e7i	0.681997i
$964$	0	0
$965$	659809.i	0.0228087i
$966$	0	0
$967$	2.39559e7	0.823847	0.411923	−	0.911218i	$-0.364857\pi$
$967$	2.39559e7	0.823847	0.411923	+	0.911218i	$0.364857\pi$
$968$	0	0
$969$	2.17676e6	0.0744733
$970$	0	0
$971$	− 1.62085e7i	− 0.551689i	−0.961202	−	0.275844i	$-0.911043\pi$
$971$	− 1.62085e7i	− 0.551689i	0.961202	−	0.275844i	$-0.0889574\pi$
$972$	0	0
$973$	− 1.51975e7i	− 0.514623i
$974$	0	0
$975$	2.37034e6	0.0798544
$976$	0	0
$977$	−2.57236e7	−0.862175	−0.431087	−	0.902310i	$-0.641870\pi$
$977$	−2.57236e7	−0.862175	−0.431087	+	0.902310i	$0.641870\pi$
$978$	0	0
$979$	3.60134e7i	1.20090i
$980$	0	0
$981$	3.11023e7i	1.03186i
$982$	0	0
$983$	5.83343e6	0.192548	0.0962742	−	0.995355i	$-0.469307\pi$
$983$	5.83343e6	0.192548	0.0962742	+	0.995355i	$0.469307\pi$
$984$	0	0
$985$	−1.20942e7	−0.397179
$986$	0	0
$987$	1.19589e6i	0.0390750i
$988$	0	0
$989$	1.37361e7i	0.446553i
$990$	0	0
$991$	−4.31418e7	−1.39545	−0.697725	−	0.716366i	$-0.745804\pi$
$991$	−4.31418e7	−1.39545	−0.697725	+	0.716366i	$0.745804\pi$
$992$	0	0
$993$	−3.09809e7	−0.997059
$994$	0	0
$995$	1.28743e6i	0.0412254i
$996$	0	0
$997$	4.95272e7i	1.57800i	0.614396	+	0.788998i	$0.289400\pi$
$997$	4.95272e7i	1.57800i	−0.614396	+	0.788998i	$0.710600\pi$
$998$	0	0
$999$	1.77414e7	0.562438

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.6.d.d.161.11	yes	12
4.3	odd	2		320.6.d.c.161.2	✓	12
8.3	odd	2		320.6.d.c.161.11	yes	12
8.5	even	2	inner	320.6.d.d.161.2	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.6.d.c.161.2	✓	12	4.3	odd	2
320.6.d.c.161.11	yes	12	8.3	odd	2
320.6.d.d.161.2	yes	12	8.5	even	2	inner
320.6.d.d.161.11	yes	12	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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	320.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+20.6292i q^{3} -25.0000i q^{5} -93.5254 q^{7} -182.562 q^{9} +O(q^{10})\)	\(q+20.6292i q^{3} -25.0000i q^{5} -93.5254 q^{7} -182.562 q^{9} +659.055i q^{11} +183.844i q^{13} +515.729 q^{15} -467.075 q^{17} +225.914i q^{19} -1929.35i q^{21} +1109.57 q^{23} -625.000 q^{25} +1246.78i q^{27} -1738.98i q^{29} -7270.81 q^{31} -13595.7 q^{33} +2338.14i q^{35} -14229.7i q^{37} -3792.54 q^{39} -4929.92 q^{41} +12379.7i q^{43} +4564.05i q^{45} -619.842 q^{47} -8060.00 q^{49} -9635.36i q^{51} +16989.0i q^{53} +16476.4 q^{55} -4660.40 q^{57} -50742.8i q^{59} -20666.3i q^{61} +17074.2 q^{63} +4596.09 q^{65} -41920.3i q^{67} +22889.5i q^{69} +54687.7 q^{71} -25211.5 q^{73} -12893.2i q^{75} -61638.4i q^{77} +29000.4 q^{79} -70082.7 q^{81} -92679.1i q^{83} +11676.9i q^{85} +35873.7 q^{87} +54644.0 q^{89} -17194.1i q^{91} -149991. i q^{93} +5647.84 q^{95} +45542.8 q^{97} -120318. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 268 q^{7} - 428 q^{9}+O(q^{10}) \) 12 * q + 268 * q^7 - 428 * q^9	\( 12 q + 268 q^{7} - 428 q^{9} - 900 q^{15} + 2400 q^{17} + 8108 q^{23} - 7500 q^{25} + 7976 q^{31} - 20776 q^{33} + 40984 q^{39} - 56408 q^{41} + 21172 q^{47} - 5540 q^{49} + 18400 q^{55} + 39992 q^{57} + 179516 q^{63} + 44000 q^{65} + 367704 q^{71} + 58736 q^{73} + 26192 q^{79} + 411692 q^{81} + 183200 q^{87} + 87672 q^{89} + 121000 q^{95} - 172336 q^{97}+O(q^{100}) \) 12 * q + 268 * q^7 - 428 * q^9 - 900 * q^15 + 2400 * q^17 + 8108 * q^23 - 7500 * q^25 + 7976 * q^31 - 20776 * q^33 + 40984 * q^39 - 56408 * q^41 + 21172 * q^47 - 5540 * q^49 + 18400 * q^55 + 39992 * q^57 + 179516 * q^63 + 44000 * q^65 + 367704 * q^71 + 58736 * q^73 + 26192 * q^79 + 411692 * q^81 + 183200 * q^87 + 87672 * q^89 + 121000 * q^95 - 172336 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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