LMFDB - Embedded newform 3680.2.f.b.1841.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3680,2,Mod(1841,3680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3680, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3680.1841");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3680 = 2^{5} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3680.f (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:	$29.3849479438$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1871773696.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 31x^{4} + 81 $ x^8 + 31*x^4 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1841.6
Root			$-0.921201 + 0.921201i$ of defining polynomial
Character	$\chi$	$=$	3680.1841
Dual form			3680.2.f.b.1841.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.30278i q^{3} -1.00000i q^{5} -2.30278 q^{7} +1.30278 q^{9} +O(q^{10})$	$q+1.30278i q^{3} -1.00000i q^{5} -2.30278 q^{7} +1.30278 q^{9} +3.36805i q^{11} -4.73097i q^{13} +1.30278 q^{15} +1.14518 q^{17} -1.55939i q^{19} -3.00000i q^{21} +1.00000 q^{23} -1.00000 q^{25} +5.60555i q^{27} +5.43398i q^{29} +1.17672 q^{31} -4.38782 q^{33} +2.30278i q^{35} -11.4340i q^{37} +6.16339 q^{39} -0.382030 q^{41} -6.29036i q^{43} -1.30278i q^{45} +3.87973 q^{47} -1.69722 q^{49} +1.49191i q^{51} +7.74917i q^{53} +3.36805 q^{55} +2.03154 q^{57} -6.51323i q^{59} -2.38203i q^{61} -3.00000 q^{63} -4.73097 q^{65} -6.51323i q^{67} +1.30278i q^{69} -1.14518 q^{71} +10.8959 q^{73} -1.30278i q^{75} -7.75587i q^{77} +15.0908 q^{79} -3.39445 q^{81} -3.34166i q^{83} -1.14518i q^{85} -7.07925 q^{87} +4.80048 q^{89} +10.8944i q^{91} +1.53300i q^{93} -1.55939 q^{95} +8.82998 q^{97} +4.38782i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{7} - 4 q^{9}+O(q^{10}) $ 8 * q - 4 * q^7 - 4 * q^9	$ 8 q - 4 q^{7} - 4 q^{9} - 4 q^{15} - 20 q^{17} + 8 q^{23} - 8 q^{25} + 4 q^{31} + 28 q^{33} + 16 q^{39} + 12 q^{41} - 8 q^{47} - 28 q^{49} + 4 q^{55} + 40 q^{57} - 24 q^{63} - 20 q^{65} + 20 q^{71} + 24 q^{79} - 56 q^{81} - 56 q^{87} + 28 q^{95} + 12 q^{97}+O(q^{100}) $ 8 * q - 4 * q^7 - 4 * q^9 - 4 * q^15 - 20 * q^17 + 8 * q^23 - 8 * q^25 + 4 * q^31 + 28 * q^33 + 16 * q^39 + 12 * q^41 - 8 * q^47 - 28 * q^49 + 4 * q^55 + 40 * q^57 - 24 * q^63 - 20 * q^65 + 20 * q^71 + 24 * q^79 - 56 * q^81 - 56 * q^87 + 28 * q^95 + 12 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3680\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1381$	$3041$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	1.30278i	0.752158i	0.926588	+	0.376079i	$0.122728\pi$
$3$	1.30278i	0.752158i	−0.926588	+	0.376079i	$0.877272\pi$
$4$	0	0
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	−2.30278	−0.870367	−0.435184	−	0.900342i	$-0.643317\pi$
$7$	−2.30278	−0.870367	−0.435184	+	0.900342i	$0.643317\pi$
$8$	0	0
$9$	1.30278	0.434259
$10$	0	0
$11$	3.36805i	1.01551i	0.861503	+	0.507753i	$0.169524\pi$
$11$	3.36805i	1.01551i	−0.861503	+	0.507753i	$0.830476\pi$
$12$	0	0
$13$	− 4.73097i	− 1.31213i	−0.754703	−	0.656067i	$-0.772219\pi$
$13$	− 4.73097i	− 1.31213i	0.754703	−	0.656067i	$-0.227781\pi$
$14$	0	0
$15$	1.30278	0.336375
$16$	0	0
$17$	1.14518	0.277747	0.138873	−	0.990310i	$-0.455652\pi$
$17$	1.14518	0.277747	0.138873	+	0.990310i	$0.455652\pi$
$18$	0	0
$19$	− 1.55939i	− 0.357749i	−0.983872	−	0.178875i	$-0.942754\pi$
$19$	− 1.55939i	− 0.357749i	0.983872	−	0.178875i	$-0.0572456\pi$
$20$	0	0
$21$	− 3.00000i	− 0.654654i
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	5.60555i	1.07879i
$28$	0	0
$29$	5.43398i	1.00906i	0.863393	+	0.504532i	$0.168335\pi$
$29$	5.43398i	1.00906i	−0.863393	+	0.504532i	$0.831665\pi$
$30$	0	0
$31$	1.17672	0.211345	0.105672	−	0.994401i	$-0.466301\pi$
$31$	1.17672	0.211345	0.105672	+	0.994401i	$0.466301\pi$
$32$	0	0
$33$	−4.38782	−0.763821
$34$	0	0
$35$	2.30278i	0.389240i
$36$	0	0
$37$	− 11.4340i	− 1.87973i	−0.341540	−	0.939867i	$-0.610948\pi$
$37$	− 11.4340i	− 1.87973i	0.341540	−	0.939867i	$-0.389052\pi$
$38$	0	0
$39$	6.16339	0.986932
$40$	0	0
$41$	−0.382030	−0.0596631	−0.0298316	−	0.999555i	$-0.509497\pi$
$41$	−0.382030	−0.0596631	−0.0298316	+	0.999555i	$0.509497\pi$
$42$	0	0
$43$	− 6.29036i	− 0.959270i	−0.877468	−	0.479635i	$-0.840769\pi$
$43$	− 6.29036i	− 0.959270i	0.877468	−	0.479635i	$-0.159231\pi$
$44$	0	0
$45$	− 1.30278i	− 0.194206i
$46$	0	0
$47$	3.87973	0.565917	0.282958	−	0.959132i	$-0.408684\pi$
$47$	3.87973	0.565917	0.282958	+	0.959132i	$0.408684\pi$
$48$	0	0
$49$	−1.69722	−0.242461
$50$	0	0
$51$	1.49191i	0.208909i
$52$	0	0
$53$	7.74917i	1.06443i	0.846609	+	0.532215i	$0.178640\pi$
$53$	7.74917i	1.06443i	−0.846609	+	0.532215i	$0.821360\pi$
$54$	0	0
$55$	3.36805	0.454148
$56$	0	0
$57$	2.03154	0.269084
$58$	0	0
$59$	− 6.51323i	− 0.847951i	−0.905674	−	0.423975i	$-0.860634\pi$
$59$	− 6.51323i	− 0.847951i	0.905674	−	0.423975i	$-0.139366\pi$
$60$	0	0
$61$	− 2.38203i	− 0.304988i	−0.988304	−	0.152494i	$-0.951270\pi$
$61$	− 2.38203i	− 0.304988i	0.988304	−	0.152494i	$-0.0487304\pi$
$62$	0	0
$63$	−3.00000	−0.377964
$64$	0	0
$65$	−4.73097	−0.586804
$66$	0	0
$67$	− 6.51323i	− 0.795718i	−0.917447	−	0.397859i	$-0.869753\pi$
$67$	− 6.51323i	− 0.795718i	0.917447	−	0.397859i	$-0.130247\pi$
$68$	0	0
$69$	1.30278i	0.156836i
$70$	0	0
$71$	−1.14518	−0.135908	−0.0679538	−	0.997688i	$-0.521647\pi$
$71$	−1.14518	−0.135908	−0.0679538	+	0.997688i	$0.521647\pi$
$72$	0	0
$73$	10.8959	1.27527	0.637635	−	0.770339i	$-0.279913\pi$
$73$	10.8959	1.27527	0.637635	+	0.770339i	$0.279913\pi$
$74$	0	0
$75$	− 1.30278i	− 0.150432i
$76$	0	0
$77$	− 7.75587i	− 0.883864i
$78$	0	0
$79$	15.0908	1.69785	0.848926	−	0.528512i	$-0.177250\pi$
$79$	15.0908	1.69785	0.848926	+	0.528512i	$0.177250\pi$
$80$	0	0
$81$	−3.39445	−0.377161
$82$	0	0
$83$	− 3.34166i	− 0.366795i	−0.983039	−	0.183397i	$-0.941290\pi$
$83$	− 3.34166i	− 0.366795i	0.983039	−	0.183397i	$-0.0587095\pi$
$84$	0	0
$85$	− 1.14518i	− 0.124212i
$86$	0	0
$87$	−7.07925	−0.758976
$88$	0	0
$89$	4.80048	0.508849	0.254425	−	0.967093i	$-0.418114\pi$
$89$	4.80048	0.508849	0.254425	+	0.967093i	$0.418114\pi$
$90$	0	0
$91$	10.8944i	1.14204i
$92$	0	0
$93$	1.53300i	0.158964i
$94$	0	0
$95$	−1.55939	−0.159990
$96$	0	0
$97$	8.82998	0.896549	0.448275	−	0.893896i	$-0.352039\pi$
$97$	8.82998	0.896549	0.448275	+	0.893896i	$0.352039\pi$
$98$	0	0
$99$	4.38782i	0.440992i
$100$	0	0
$101$	6.85638i	0.682235i	0.940021	+	0.341118i	$0.110805\pi$
$101$	6.85638i	0.682235i	−0.940021	+	0.341118i	$0.889195\pi$
$102$	0	0
$103$	5.90833	0.582165	0.291082	−	0.956698i	$-0.405985\pi$
$103$	5.90833	0.582165	0.291082	+	0.956698i	$0.405985\pi$
$104$	0	0
$105$	−3.00000	−0.292770
$106$	0	0
$107$	7.81665i	0.755664i	0.925874	+	0.377832i	$0.123330\pi$
$107$	7.81665i	0.755664i	−0.925874	+	0.377832i	$0.876670\pi$
$108$	0	0
$109$	− 16.8160i	− 1.61068i	−0.592812	−	0.805341i	$-0.701982\pi$
$109$	− 16.8160i	− 1.61068i	0.592812	−	0.805341i	$-0.298018\pi$
$110$	0	0
$111$	14.8959	1.41386
$112$	0	0
$113$	14.6438	1.37757	0.688786	−	0.724965i	$-0.258144\pi$
$113$	14.6438	1.37757	0.688786	+	0.724965i	$0.258144\pi$
$114$	0	0
$115$	− 1.00000i	− 0.0932505i
$116$	0	0
$117$	− 6.16339i	− 0.569805i
$118$	0	0
$119$	−2.63709	−0.241742
$120$	0	0
$121$	−0.343791	−0.0312537
$122$	0	0
$123$	− 0.497700i	− 0.0448761i
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	−3.84149	−0.340877	−0.170439	−	0.985368i	$-0.554518\pi$
$127$	−3.84149	−0.340877	−0.170439	+	0.985368i	$0.554518\pi$
$128$	0	0
$129$	8.19492	0.721523
$130$	0	0
$131$	6.80359i	0.594432i	0.954810	+	0.297216i	$0.0960582\pi$
$131$	6.80359i	0.594432i	−0.954810	+	0.297216i	$0.903942\pi$
$132$	0	0
$133$	3.59093i	0.311373i
$134$	0	0
$135$	5.60555	0.482449
$136$	0	0
$137$	4.09746	0.350070	0.175035	−	0.984562i	$-0.443996\pi$
$137$	4.09746	0.350070	0.175035	+	0.984562i	$0.443996\pi$
$138$	0	0
$139$	19.4722i	1.65161i	0.563954	+	0.825806i	$0.309279\pi$
$139$	19.4722i	1.65161i	−0.563954	+	0.825806i	$0.690721\pi$
$140$	0	0
$141$	5.05442i	0.425659i
$142$	0	0
$143$	15.9341	1.33248
$144$	0	0
$145$	5.43398	0.451267
$146$	0	0
$147$	− 2.21110i	− 0.182369i
$148$	0	0
$149$	11.7088i	0.959222i	0.877481	+	0.479611i	$0.159222\pi$
$149$	11.7088i	0.959222i	−0.877481	+	0.479611i	$0.840778\pi$
$150$	0	0
$151$	14.0411	1.14265	0.571324	−	0.820725i	$-0.306430\pi$
$151$	14.0411	1.14265	0.571324	+	0.820725i	$0.306430\pi$
$152$	0	0
$153$	1.49191	0.120614
$154$	0	0
$155$	− 1.17672i	− 0.0945162i
$156$	0	0
$157$	− 7.02647i	− 0.560773i	−0.959887	−	0.280387i	$-0.909537\pi$
$157$	− 7.02647i	− 0.560773i	0.959887	−	0.280387i	$-0.0904626\pi$
$158$	0	0
$159$	−10.0954	−0.800620
$160$	0	0
$161$	−2.30278	−0.181484
$162$	0	0
$163$	− 3.00065i	− 0.235029i	−0.993071	−	0.117514i	$-0.962507\pi$
$163$	− 3.00065i	− 0.235029i	0.993071	−	0.117514i	$-0.0374926\pi$
$164$	0	0
$165$	4.38782i	0.341591i
$166$	0	0
$167$	−4.47370	−0.346186	−0.173093	−	0.984906i	$-0.555376\pi$
$167$	−4.47370	−0.346186	−0.173093	+	0.984906i	$0.555376\pi$
$168$	0	0
$169$	−9.38203	−0.721695
$170$	0	0
$171$	− 2.03154i	− 0.155356i
$172$	0	0
$173$	6.86640i	0.522043i	0.965333	+	0.261021i	$0.0840593\pi$
$173$	6.86640i	0.522043i	−0.965333	+	0.261021i	$0.915941\pi$
$174$	0	0
$175$	2.30278	0.174073
$176$	0	0
$177$	8.48528	0.637793
$178$	0	0
$179$	− 19.8167i	− 1.48117i	−0.671965	−	0.740583i	$-0.734549\pi$
$179$	− 19.8167i	− 1.48117i	0.671965	−	0.740583i	$-0.265451\pi$
$180$	0	0
$181$	− 15.5139i	− 1.15314i	−0.817049	−	0.576569i	$-0.804391\pi$
$181$	− 15.5139i	− 1.15314i	0.817049	−	0.576569i	$-0.195609\pi$
$182$	0	0
$183$	3.10325	0.229399
$184$	0	0
$185$	−11.4340	−0.840643
$186$	0	0
$187$	3.85702i	0.282054i
$188$	0	0
$189$	− 12.9083i	− 0.938943i
$190$	0	0
$191$	9.87973	0.714872	0.357436	−	0.933938i	$-0.383651\pi$
$191$	9.87973	0.714872	0.357436	+	0.933938i	$0.383651\pi$
$192$	0	0
$193$	12.4853	0.898710	0.449355	−	0.893353i	$-0.351654\pi$
$193$	12.4853	0.898710	0.449355	+	0.893353i	$0.351654\pi$
$194$	0	0
$195$	− 6.16339i	− 0.441369i
$196$	0	0
$197$	− 26.1832i	− 1.86548i	−0.360553	−	0.932739i	$-0.617412\pi$
$197$	− 26.1832i	− 1.86548i	0.360553	−	0.932739i	$-0.382588\pi$
$198$	0	0
$199$	−7.85489	−0.556819	−0.278409	−	0.960463i	$-0.589807\pi$
$199$	−7.85489	−0.556819	−0.278409	+	0.960463i	$0.589807\pi$
$200$	0	0
$201$	8.48528	0.598506
$202$	0	0
$203$	− 12.5132i	− 0.878257i
$204$	0	0
$205$	0.382030i	0.0266822i
$206$	0	0
$207$	1.30278	0.0905492
$208$	0	0
$209$	5.25212	0.363297
$210$	0	0
$211$	5.84332i	0.402270i	0.979563	+	0.201135i	$0.0644631\pi$
$211$	5.84332i	0.402270i	−0.979563	+	0.201135i	$0.935537\pi$
$212$	0	0
$213$	− 1.49191i	− 0.102224i
$214$	0	0
$215$	−6.29036	−0.428999
$216$	0	0
$217$	−2.70971	−0.183947
$218$	0	0
$219$	14.1949i	0.959204i
$220$	0	0
$221$	− 5.41780i	− 0.364441i
$222$	0	0
$223$	−22.3019	−1.49345	−0.746724	−	0.665134i	$-0.768374\pi$
$223$	−22.3019	−1.49345	−0.746724	+	0.665134i	$0.768374\pi$
$224$	0	0
$225$	−1.30278	−0.0868517
$226$	0	0
$227$	− 27.4091i	− 1.81921i	−0.415476	−	0.909604i	$-0.636385\pi$
$227$	− 27.4091i	− 1.81921i	0.415476	−	0.909604i	$-0.363615\pi$
$228$	0	0
$229$	23.7918i	1.57221i	0.618095	+	0.786104i	$0.287905\pi$
$229$	23.7918i	1.57221i	−0.618095	+	0.786104i	$0.712095\pi$
$230$	0	0
$231$	10.1042	0.664805
$232$	0	0
$233$	19.0660	1.24905	0.624527	−	0.781003i	$-0.285292\pi$
$233$	19.0660	1.24905	0.624527	+	0.781003i	$0.285292\pi$
$234$	0	0
$235$	− 3.87973i	− 0.253086i
$236$	0	0
$237$	19.6600i	1.27705i
$238$	0	0
$239$	−14.5807	−0.943148	−0.471574	−	0.881826i	$-0.656314\pi$
$239$	−14.5807	−0.943148	−0.471574	+	0.881826i	$0.656314\pi$
$240$	0	0
$241$	15.0660	0.970486	0.485243	−	0.874379i	$-0.338731\pi$
$241$	15.0660	0.970486	0.485243	+	0.874379i	$0.338731\pi$
$242$	0	0
$243$	12.3944i	0.795104i
$244$	0	0
$245$	1.69722i	0.108432i
$246$	0	0
$247$	−7.37743	−0.469415
$248$	0	0
$249$	4.35343	0.275888
$250$	0	0
$251$	9.75432i	0.615687i	0.951437	+	0.307843i	$0.0996073\pi$
$251$	9.75432i	0.615687i	−0.951437	+	0.307843i	$0.900393\pi$
$252$	0	0
$253$	3.36805i	0.211748i
$254$	0	0
$255$	1.49191	0.0934271
$256$	0	0
$257$	24.4604	1.52580	0.762900	−	0.646516i	$-0.223775\pi$
$257$	24.4604	1.52580	0.762900	+	0.646516i	$0.223775\pi$
$258$	0	0
$259$	26.3299i	1.63606i
$260$	0	0
$261$	7.07925i	0.438195i
$262$	0	0
$263$	24.8541	1.53257	0.766283	−	0.642503i	$-0.222104\pi$
$263$	24.8541	1.53257	0.766283	+	0.642503i	$0.222104\pi$
$264$	0	0
$265$	7.74917	0.476028
$266$	0	0
$267$	6.25394i	0.382735i
$268$	0	0
$269$	4.69658i	0.286355i	0.989697	+	0.143178i	$0.0457321\pi$
$269$	4.69658i	0.286355i	−0.989697	+	0.143178i	$0.954268\pi$
$270$	0	0
$271$	−14.5579	−0.884330	−0.442165	−	0.896934i	$-0.645789\pi$
$271$	−14.5579	−0.884330	−0.442165	+	0.896934i	$0.645789\pi$
$272$	0	0
$273$	−14.1929	−0.858993
$274$	0	0
$275$	− 3.36805i	− 0.203101i
$276$	0	0
$277$	− 8.72582i	− 0.524284i	−0.965029	−	0.262142i	$-0.915571\pi$
$277$	− 8.72582i	− 0.524284i	0.965029	−	0.262142i	$-0.0844288\pi$
$278$	0	0
$279$	1.53300	0.0917782
$280$	0	0
$281$	−5.72122	−0.341299	−0.170650	−	0.985332i	$-0.554587\pi$
$281$	−5.72122	−0.341299	−0.170650	+	0.985332i	$0.554587\pi$
$282$	0	0
$283$	− 3.51472i	− 0.208928i	−0.994529	−	0.104464i	$-0.966687\pi$
$283$	− 3.51472i	− 0.208928i	0.994529	−	0.104464i	$-0.0333128\pi$
$284$	0	0
$285$	− 2.03154i	− 0.120338i
$286$	0	0
$287$	0.879730	0.0519288
$288$	0	0
$289$	−15.6886	−0.922857
$290$	0	0
$291$	11.5035i	0.674346i
$292$	0	0
$293$	31.9193i	1.86474i	0.361501	+	0.932372i	$0.382264\pi$
$293$	31.9193i	1.86474i	−0.361501	+	0.932372i	$0.617736\pi$
$294$	0	0
$295$	−6.51323	−0.379215
$296$	0	0
$297$	−18.8798	−1.09552
$298$	0	0
$299$	− 4.73097i	− 0.273599i
$300$	0	0
$301$	14.4853i	0.834918i
$302$	0	0
$303$	−8.93232	−0.513149
$304$	0	0
$305$	−2.38203	−0.136395
$306$	0	0
$307$	− 7.35259i	− 0.419635i	−0.977741	−	0.209817i	$-0.932713\pi$
$307$	− 7.35259i	− 0.419635i	0.977741	−	0.209817i	$-0.0672869\pi$
$308$	0	0
$309$	7.69722i	0.437880i
$310$	0	0
$311$	10.6304	0.602794	0.301397	−	0.953499i	$-0.402547\pi$
$311$	10.6304	0.602794	0.301397	+	0.953499i	$0.402547\pi$
$312$	0	0
$313$	11.6372	0.657771	0.328886	−	0.944370i	$-0.393327\pi$
$313$	11.6372	0.657771	0.328886	+	0.944370i	$0.393327\pi$
$314$	0	0
$315$	3.00000i	0.169031i
$316$	0	0
$317$	15.9057i	0.893350i	0.894696	+	0.446675i	$0.147392\pi$
$317$	15.9057i	0.893350i	−0.894696	+	0.446675i	$0.852608\pi$
$318$	0	0
$319$	−18.3019	−1.02471
$320$	0	0
$321$	−10.1833	−0.568379
$322$	0	0
$323$	− 1.78578i	− 0.0993636i
$324$	0	0
$325$	4.73097i	0.262427i
$326$	0	0
$327$	21.9075	1.21149
$328$	0	0
$329$	−8.93415	−0.492556
$330$	0	0
$331$	− 4.64379i	− 0.255246i	−0.991823	−	0.127623i	$-0.959265\pi$
$331$	− 4.64379i	− 0.255246i	0.991823	−	0.127623i	$-0.0407347\pi$
$332$	0	0
$333$	− 14.8959i	− 0.816291i
$334$	0	0
$335$	−6.51323	−0.355856
$336$	0	0
$337$	14.1090	0.768568	0.384284	−	0.923215i	$-0.374448\pi$
$337$	14.1090	0.768568	0.384284	+	0.923215i	$0.374448\pi$
$338$	0	0
$339$	19.0776i	1.03615i
$340$	0	0
$341$	3.96325i	0.214622i
$342$	0	0
$343$	20.0278	1.08140
$344$	0	0
$345$	1.30278	0.0701391
$346$	0	0
$347$	− 6.56667i	− 0.352517i	−0.984344	−	0.176259i	$-0.943601\pi$
$347$	− 6.56667i	− 0.352517i	0.984344	−	0.176259i	$-0.0563995\pi$
$348$	0	0
$349$	7.08054i	0.379013i	0.981879	+	0.189506i	$0.0606888\pi$
$349$	7.08054i	0.379013i	−0.981879	+	0.189506i	$0.939311\pi$
$350$	0	0
$351$	26.5197	1.41552
$352$	0	0
$353$	−12.8387	−0.683336	−0.341668	−	0.939821i	$-0.610992\pi$
$353$	−12.8387	−0.683336	−0.341668	+	0.939821i	$0.610992\pi$
$354$	0	0
$355$	1.14518i	0.0607798i
$356$	0	0
$357$	− 3.43554i	− 0.181828i
$358$	0	0
$359$	−13.8415	−0.730526	−0.365263	−	0.930904i	$-0.619021\pi$
$359$	−13.8415	−0.730526	−0.365263	+	0.930904i	$0.619021\pi$
$360$	0	0
$361$	16.5683	0.872016
$362$	0	0
$363$	− 0.447882i	− 0.0235077i
$364$	0	0
$365$	− 10.8959i	− 0.570318i
$366$	0	0
$367$	−2.42221	−0.126438	−0.0632190	−	0.998000i	$-0.520137\pi$
$367$	−2.42221	−0.126438	−0.0632190	+	0.998000i	$0.520137\pi$
$368$	0	0
$369$	−0.497700	−0.0259092
$370$	0	0
$371$	− 17.8446i	− 0.926446i
$372$	0	0
$373$	12.1337i	0.628258i	0.949380	+	0.314129i	$0.101712\pi$
$373$	12.1337i	0.628258i	−0.949380	+	0.314129i	$0.898288\pi$
$374$	0	0
$375$	−1.30278	−0.0672750
$376$	0	0
$377$	25.7080	1.32403
$378$	0	0
$379$	− 35.7654i	− 1.83715i	−0.395251	−	0.918573i	$-0.629342\pi$
$379$	− 35.7654i	− 1.83715i	0.395251	−	0.918573i	$-0.370658\pi$
$380$	0	0
$381$	− 5.00460i	− 0.256393i
$382$	0	0
$383$	−37.7918	−1.93107	−0.965536	−	0.260270i	$-0.916188\pi$
$383$	−37.7918	−1.93107	−0.965536	+	0.260270i	$0.916188\pi$
$384$	0	0
$385$	−7.75587	−0.395276
$386$	0	0
$387$	− 8.19492i	− 0.416571i
$388$	0	0
$389$	24.1694i	1.22544i	0.790300	+	0.612720i	$0.209925\pi$
$389$	24.1694i	1.22544i	−0.790300	+	0.612720i	$0.790075\pi$
$390$	0	0
$391$	1.14518	0.0579142
$392$	0	0
$393$	−8.86355	−0.447107
$394$	0	0
$395$	− 15.0908i	− 0.759302i
$396$	0	0
$397$	− 27.3681i	− 1.37356i	−0.726864	−	0.686782i	$-0.759023\pi$
$397$	− 27.3681i	− 1.37356i	0.726864	−	0.686782i	$-0.240977\pi$
$398$	0	0
$399$	−4.67818	−0.234202
$400$	0	0
$401$	−22.1318	−1.10521	−0.552606	−	0.833443i	$-0.686367\pi$
$401$	−22.1318	−1.10521	−0.552606	+	0.833443i	$0.686367\pi$
$402$	0	0
$403$	− 5.56701i	− 0.277312i
$404$	0	0
$405$	3.39445i	0.168672i
$406$	0	0
$407$	38.5103	1.90888
$408$	0	0
$409$	−9.48904	−0.469203	−0.234601	−	0.972092i	$-0.575379\pi$
$409$	−9.48904	−0.469203	−0.234601	+	0.972092i	$0.575379\pi$
$410$	0	0
$411$	5.33807i	0.263308i
$412$	0	0
$413$	14.9985i	0.738029i
$414$	0	0
$415$	−3.34166	−0.164036
$416$	0	0
$417$	−25.3679	−1.24227
$418$	0	0
$419$	− 2.50166i	− 0.122214i	−0.998131	−	0.0611069i	$-0.980537\pi$
$419$	− 2.50166i	− 0.122214i	0.998131	−	0.0611069i	$-0.0194631\pi$
$420$	0	0
$421$	14.2117i	0.692638i	0.938117	+	0.346319i	$0.112569\pi$
$421$	14.2117i	0.692638i	−0.938117	+	0.346319i	$0.887431\pi$
$422$	0	0
$423$	5.05442	0.245754
$424$	0	0
$425$	−1.14518	−0.0555493
$426$	0	0
$427$	5.48528i	0.265451i
$428$	0	0
$429$	20.7586i	1.00224i
$430$	0	0
$431$	2.43546	0.117312	0.0586561	−	0.998278i	$-0.481318\pi$
$431$	2.43546	0.117312	0.0586561	+	0.998278i	$0.481318\pi$
$432$	0	0
$433$	15.2721	0.733933	0.366966	−	0.930234i	$-0.380396\pi$
$433$	15.2721	0.733933	0.366966	+	0.930234i	$0.380396\pi$
$434$	0	0
$435$	7.07925i	0.339424i
$436$	0	0
$437$	− 1.55939i	− 0.0745958i
$438$	0	0
$439$	−26.9934	−1.28832	−0.644162	−	0.764889i	$-0.722794\pi$
$439$	−26.9934	−1.28832	−0.644162	+	0.764889i	$0.722794\pi$
$440$	0	0
$441$	−2.21110	−0.105291
$442$	0	0
$443$	36.6989i	1.74362i	0.489848	+	0.871808i	$0.337052\pi$
$443$	36.6989i	1.74362i	−0.489848	+	0.871808i	$0.662948\pi$
$444$	0	0
$445$	− 4.80048i	− 0.227564i
$446$	0	0
$447$	−15.2539	−0.721487
$448$	0	0
$449$	−17.8789	−0.843757	−0.421878	−	0.906652i	$-0.638629\pi$
$449$	−17.8789	−0.843757	−0.421878	+	0.906652i	$0.638629\pi$
$450$	0	0
$451$	− 1.28670i	− 0.0605883i
$452$	0	0
$453$	18.2924i	0.859451i
$454$	0	0
$455$	10.8944	0.510735
$456$	0	0
$457$	39.8052	1.86201	0.931005	−	0.365007i	$-0.118933\pi$
$457$	39.8052	1.86201	0.931005	+	0.365007i	$0.118933\pi$
$458$	0	0
$459$	6.41936i	0.299630i
$460$	0	0
$461$	− 31.7626i	− 1.47933i	−0.672975	−	0.739665i	$-0.734984\pi$
$461$	− 31.7626i	− 1.47933i	0.672975	−	0.739665i	$-0.265016\pi$
$462$	0	0
$463$	38.8139	1.80383	0.901917	−	0.431910i	$-0.142160\pi$
$463$	38.8139	1.80383	0.901917	+	0.431910i	$0.142160\pi$
$464$	0	0
$465$	1.53300	0.0710911
$466$	0	0
$467$	− 14.9310i	− 0.690926i	−0.938432	−	0.345463i	$-0.887722\pi$
$467$	− 14.9310i	− 0.690926i	0.938432	−	0.345463i	$-0.112278\pi$
$468$	0	0
$469$	14.9985i	0.692567i
$470$	0	0
$471$	9.15391	0.421790
$472$	0	0
$473$	21.1863	0.974146
$474$	0	0
$475$	1.55939i	0.0715498i
$476$	0	0
$477$	10.0954i	0.462238i
$478$	0	0
$479$	11.2609	0.514525	0.257262	−	0.966342i	$-0.417180\pi$
$479$	11.2609	0.514525	0.257262	+	0.966342i	$0.417180\pi$
$480$	0	0
$481$	−54.0938	−2.46646
$482$	0	0
$483$	− 3.00000i	− 0.136505i
$484$	0	0
$485$	− 8.82998i	− 0.400949i
$486$	0	0
$487$	−21.1157	−0.956842	−0.478421	−	0.878131i	$-0.658791\pi$
$487$	−21.1157	−0.956842	−0.478421	+	0.878131i	$0.658791\pi$
$488$	0	0
$489$	3.90917	0.176779
$490$	0	0
$491$	− 20.1598i	− 0.909799i	−0.890543	−	0.454900i	$-0.849675\pi$
$491$	− 20.1598i	− 0.909799i	0.890543	−	0.454900i	$-0.150325\pi$
$492$	0	0
$493$	6.22288i	0.280264i
$494$	0	0
$495$	4.38782	0.197218
$496$	0	0
$497$	2.63709	0.118290
$498$	0	0
$499$	13.9340i	0.623769i	0.950120	+	0.311885i	$0.100960\pi$
$499$	13.9340i	0.623769i	−0.950120	+	0.311885i	$0.899040\pi$
$500$	0	0
$501$	− 5.82823i	− 0.260386i
$502$	0	0
$503$	17.5023	0.780389	0.390194	−	0.920732i	$-0.372408\pi$
$503$	17.5023	0.780389	0.390194	+	0.920732i	$0.372408\pi$
$504$	0	0
$505$	6.85638	0.305105
$506$	0	0
$507$	− 12.2227i	− 0.542828i
$508$	0	0
$509$	− 4.43858i	− 0.196737i	−0.995150	−	0.0983683i	$-0.968638\pi$
$509$	− 4.43858i	− 0.196737i	0.995150	−	0.0983683i	$-0.0313623\pi$
$510$	0	0
$511$	−25.0908	−1.10995
$512$	0	0
$513$	8.74125	0.385936
$514$	0	0
$515$	− 5.90833i	− 0.260352i
$516$	0	0
$517$	13.0671i	0.574692i
$518$	0	0
$519$	−8.94538	−0.392659
$520$	0	0
$521$	−30.3038	−1.32763	−0.663816	−	0.747896i	$-0.731064\pi$
$521$	−30.3038	−1.32763	−0.663816	+	0.747896i	$0.731064\pi$
$522$	0	0
$523$	13.6951i	0.598845i	0.954121	+	0.299422i	$0.0967939\pi$
$523$	13.6951i	0.598845i	−0.954121	+	0.299422i	$0.903206\pi$
$524$	0	0
$525$	3.00000i	0.130931i
$526$	0	0
$527$	1.34755	0.0587002
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	− 8.48528i	− 0.368230i
$532$	0	0
$533$	1.80737i	0.0782860i
$534$	0	0
$535$	7.81665	0.337943
$536$	0	0
$537$	25.8167	1.11407
$538$	0	0
$539$	− 5.71634i	− 0.246220i
$540$	0	0
$541$	− 27.1850i	− 1.16877i	−0.811475	−	0.584387i	$-0.801335\pi$
$541$	− 27.1850i	− 1.16877i	0.811475	−	0.584387i	$-0.198665\pi$
$542$	0	0
$543$	20.2111	0.867341
$544$	0	0
$545$	−16.8160	−0.720319
$546$	0	0
$547$	− 26.4844i	− 1.13239i	−0.824270	−	0.566196i	$-0.808414\pi$
$547$	− 26.4844i	− 1.13239i	0.824270	−	0.566196i	$-0.191586\pi$
$548$	0	0
$549$	− 3.10325i	− 0.132444i
$550$	0	0
$551$	8.47370	0.360992
$552$	0	0
$553$	−34.7508	−1.47775
$554$	0	0
$555$	− 14.8959i	− 0.632296i
$556$	0	0
$557$	− 12.4294i	− 0.526650i	−0.964707	−	0.263325i	$-0.915181\pi$
$557$	− 12.4294i	− 0.526650i	0.964707	−	0.263325i	$-0.0848191\pi$
$558$	0	0
$559$	−29.7595	−1.25869
$560$	0	0
$561$	−5.02484	−0.212149
$562$	0	0
$563$	6.27398i	0.264417i	0.991222	+	0.132208i	$0.0422068\pi$
$563$	6.27398i	0.264417i	−0.991222	+	0.132208i	$0.957793\pi$
$564$	0	0
$565$	− 14.6438i	− 0.616069i
$566$	0	0
$567$	7.81665	0.328269
$568$	0	0
$569$	15.4079	0.645931	0.322965	−	0.946411i	$-0.395320\pi$
$569$	15.4079	0.645931	0.322965	+	0.946411i	$0.395320\pi$
$570$	0	0
$571$	− 37.4241i	− 1.56615i	−0.621927	−	0.783075i	$-0.713650\pi$
$571$	− 37.4241i	− 1.56615i	0.621927	−	0.783075i	$-0.286350\pi$
$572$	0	0
$573$	12.8711i	0.537697i
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−39.7214	−1.65362	−0.826811	−	0.562480i	$-0.809847\pi$
$577$	−39.7214	−1.65362	−0.826811	+	0.562480i	$0.809847\pi$
$578$	0	0
$579$	16.2655i	0.675972i
$580$	0	0
$581$	7.69509i	0.319246i
$582$	0	0
$583$	−26.0996	−1.08094
$584$	0	0
$585$	−6.16339	−0.254825
$586$	0	0
$587$	− 27.9892i	− 1.15524i	−0.816306	−	0.577619i	$-0.803982\pi$
$587$	− 27.9892i	− 1.15524i	0.816306	−	0.577619i	$-0.196018\pi$
$588$	0	0
$589$	− 1.83496i	− 0.0756083i
$590$	0	0
$591$	34.1109	1.40313
$592$	0	0
$593$	−27.2227	−1.11790	−0.558951	−	0.829201i	$-0.688796\pi$
$593$	−27.2227	−1.11790	−0.558951	+	0.829201i	$0.688796\pi$
$594$	0	0
$595$	2.63709i	0.108110i
$596$	0	0
$597$	− 10.2332i	− 0.418816i
$598$	0	0
$599$	37.9896	1.55221	0.776106	−	0.630602i	$-0.217192\pi$
$599$	37.9896	1.55221	0.776106	+	0.630602i	$0.217192\pi$
$600$	0	0
$601$	−15.0212	−0.612728	−0.306364	−	0.951914i	$-0.599113\pi$
$601$	−15.0212	−0.612728	−0.306364	+	0.951914i	$0.599113\pi$
$602$	0	0
$603$	− 8.48528i	− 0.345547i
$604$	0	0
$605$	0.343791i	0.0139771i
$606$	0	0
$607$	−8.30194	−0.336965	−0.168483	−	0.985705i	$-0.553887\pi$
$607$	−8.30194	−0.336965	−0.168483	+	0.985705i	$0.553887\pi$
$608$	0	0
$609$	16.3019	0.660588
$610$	0	0
$611$	− 18.3549i	− 0.742558i
$612$	0	0
$613$	− 15.8536i	− 0.640321i	−0.947363	−	0.320160i	$-0.896263\pi$
$613$	− 15.8536i	− 0.640321i	0.947363	−	0.320160i	$-0.103737\pi$
$614$	0	0
$615$	−0.497700	−0.0200692
$616$	0	0
$617$	−39.1435	−1.57586	−0.787929	−	0.615766i	$-0.788847\pi$
$617$	−39.1435	−1.57586	−0.787929	+	0.615766i	$0.788847\pi$
$618$	0	0
$619$	27.4422i	1.10300i	0.834176	+	0.551498i	$0.185944\pi$
$619$	27.4422i	1.10300i	−0.834176	+	0.551498i	$0.814056\pi$
$620$	0	0
$621$	5.60555i	0.224943i
$622$	0	0
$623$	−11.0544	−0.442886
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	6.84233i	0.273256i
$628$	0	0
$629$	− 13.0939i	− 0.522090i
$630$	0	0
$631$	−5.39445	−0.214750	−0.107375	−	0.994219i	$-0.534244\pi$
$631$	−5.39445	−0.214750	−0.107375	+	0.994219i	$0.534244\pi$
$632$	0	0
$633$	−7.61253	−0.302571
$634$	0	0
$635$	3.84149i	0.152445i
$636$	0	0
$637$	8.02951i	0.318141i
$638$	0	0
$639$	−1.49191	−0.0590191
$640$	0	0
$641$	18.1701	0.717675	0.358838	−	0.933400i	$-0.383173\pi$
$641$	18.1701	0.717675	0.358838	+	0.933400i	$0.383173\pi$
$642$	0	0
$643$	− 28.6572i	− 1.13013i	−0.825047	−	0.565065i	$-0.808851\pi$
$643$	− 28.6572i	− 1.13013i	0.825047	−	0.565065i	$-0.191149\pi$
$644$	0	0
$645$	− 8.19492i	− 0.322675i
$646$	0	0
$647$	−42.2024	−1.65915	−0.829575	−	0.558396i	$-0.811417\pi$
$647$	−42.2024	−1.65915	−0.829575	+	0.558396i	$0.811417\pi$
$648$	0	0
$649$	21.9369	0.861100
$650$	0	0
$651$	− 3.53015i	− 0.138357i
$652$	0	0
$653$	− 36.6502i	− 1.43423i	−0.696953	−	0.717117i	$-0.745461\pi$
$653$	− 36.6502i	− 1.43423i	0.696953	−	0.717117i	$-0.254539\pi$
$654$	0	0
$655$	6.80359	0.265838
$656$	0	0
$657$	14.1949	0.553797
$658$	0	0
$659$	− 0.683320i	− 0.0266184i	−0.999911	−	0.0133092i	$-0.995763\pi$
$659$	− 0.683320i	− 0.0266184i	0.999911	−	0.0133092i	$-0.00423657\pi$
$660$	0	0
$661$	30.0328i	1.16814i	0.811703	+	0.584070i	$0.198541\pi$
$661$	30.0328i	1.16814i	−0.811703	+	0.584070i	$0.801459\pi$
$662$	0	0
$663$	7.05818	0.274117
$664$	0	0
$665$	3.59093	0.139250
$666$	0	0
$667$	5.43398i	0.210404i
$668$	0	0
$669$	− 29.0544i	− 1.12331i
$670$	0	0
$671$	8.02281	0.309717
$672$	0	0
$673$	−42.4472	−1.63622	−0.818109	−	0.575063i	$-0.804978\pi$
$673$	−42.4472	−1.63622	−0.818109	+	0.575063i	$0.804978\pi$
$674$	0	0
$675$	− 5.60555i	− 0.215758i
$676$	0	0
$677$	− 10.7464i	− 0.413018i	−0.978445	−	0.206509i	$-0.933790\pi$
$677$	− 10.7464i	− 0.413018i	0.978445	−	0.206509i	$-0.0662102\pi$
$678$	0	0
$679$	−20.3335	−0.780327
$680$	0	0
$681$	35.7080	1.36833
$682$	0	0
$683$	− 22.0946i	− 0.845426i	−0.906264	−	0.422713i	$-0.861078\pi$
$683$	− 22.0946i	− 0.845426i	0.906264	−	0.422713i	$-0.138922\pi$
$684$	0	0
$685$	− 4.09746i	− 0.156556i
$686$	0	0
$687$	−30.9954	−1.18255
$688$	0	0
$689$	36.6611	1.39668
$690$	0	0
$691$	46.5248i	1.76989i	0.465698	+	0.884944i	$0.345803\pi$
$691$	46.5248i	1.76989i	−0.465698	+	0.884944i	$0.654197\pi$
$692$	0	0
$693$	− 10.1042i	− 0.383825i
$694$	0	0
$695$	19.4722	0.738623
$696$	0	0
$697$	−0.437493	−0.0165712
$698$	0	0
$699$	24.8387i	0.939486i
$700$	0	0
$701$	− 30.7014i	− 1.15958i	−0.814767	−	0.579789i	$-0.803135\pi$
$701$	− 30.7014i	− 1.15958i	0.814767	−	0.579789i	$-0.196865\pi$
$702$	0	0
$703$	−17.8301	−0.672473
$704$	0	0
$705$	5.05442	0.190360
$706$	0	0
$707$	− 15.7887i	− 0.593795i
$708$	0	0
$709$	13.0635i	0.490611i	0.969446	+	0.245306i	$0.0788883\pi$
$709$	13.0635i	0.490611i	−0.969446	+	0.245306i	$0.921112\pi$
$710$	0	0
$711$	19.6600	0.737306
$712$	0	0
$713$	1.17672	0.0440684
$714$	0	0
$715$	− 15.9341i	− 0.595903i
$716$	0	0
$717$	− 18.9954i	− 0.709396i
$718$	0	0
$719$	12.3514	0.460630	0.230315	−	0.973116i	$-0.426024\pi$
$719$	12.3514	0.460630	0.230315	+	0.973116i	$0.426024\pi$
$720$	0	0
$721$	−13.6056	−0.506697
$722$	0	0
$723$	19.6276i	0.729959i
$724$	0	0
$725$	− 5.43398i	− 0.201813i
$726$	0	0
$727$	−48.3555	−1.79341	−0.896703	−	0.442632i	$-0.854045\pi$
$727$	−48.3555	−1.79341	−0.896703	+	0.442632i	$0.854045\pi$
$728$	0	0
$729$	−26.3305	−0.975205
$730$	0	0
$731$	− 7.20358i	− 0.266434i
$732$	0	0
$733$	− 11.1747i	− 0.412747i	−0.978473	−	0.206373i	$-0.933834\pi$
$733$	− 11.1747i	− 0.412747i	0.978473	−	0.206373i	$-0.0661661\pi$
$734$	0	0
$735$	−2.21110	−0.0815577
$736$	0	0
$737$	21.9369	0.808057
$738$	0	0
$739$	16.6815i	0.613639i	0.951768	+	0.306819i	$0.0992648\pi$
$739$	16.6815i	0.613639i	−0.951768	+	0.306819i	$0.900735\pi$
$740$	0	0
$741$	− 9.61114i	− 0.353074i
$742$	0	0
$743$	0.486122	0.0178341	0.00891704	−	0.999960i	$-0.497162\pi$
$743$	0.486122	0.0178341	0.00891704	+	0.999960i	$0.497162\pi$
$744$	0	0
$745$	11.7088	0.428977
$746$	0	0
$747$	− 4.35343i	− 0.159284i
$748$	0	0
$749$	− 18.0000i	− 0.657706i
$750$	0	0
$751$	−49.0892	−1.79129	−0.895644	−	0.444771i	$-0.853285\pi$
$751$	−49.0892	−1.79129	−0.895644	+	0.444771i	$0.853285\pi$
$752$	0	0
$753$	−12.7077	−0.463094
$754$	0	0
$755$	− 14.0411i	− 0.511007i
$756$	0	0
$757$	23.2920i	0.846562i	0.905999	+	0.423281i	$0.139122\pi$
$757$	23.2920i	0.846562i	−0.905999	+	0.423281i	$0.860878\pi$
$758$	0	0
$759$	−4.38782	−0.159268
$760$	0	0
$761$	−18.6314	−0.675387	−0.337693	−	0.941256i	$-0.609647\pi$
$761$	−18.6314	−0.675387	−0.337693	+	0.941256i	$0.609647\pi$
$762$	0	0
$763$	38.7235i	1.40188i
$764$	0	0
$765$	− 1.49191i	− 0.0539401i
$766$	0	0
$767$	−30.8139	−1.11262
$768$	0	0
$769$	12.6074	0.454634	0.227317	−	0.973821i	$-0.427005\pi$
$769$	12.6074	0.454634	0.227317	+	0.973821i	$0.427005\pi$
$770$	0	0
$771$	31.8665i	1.14764i
$772$	0	0
$773$	21.6995i	0.780477i	0.920714	+	0.390238i	$0.127607\pi$
$773$	21.6995i	0.780477i	−0.920714	+	0.390238i	$0.872393\pi$
$774$	0	0
$775$	−1.17672	−0.0422689
$776$	0	0
$777$	−34.3019	−1.23058
$778$	0	0
$779$	0.595735i	0.0213444i
$780$	0	0
$781$	− 3.85702i	− 0.138015i
$782$	0	0
$783$	−30.4604	−1.08857
$784$	0	0
$785$	−7.02647	−0.250785
$786$	0	0
$787$	− 15.7887i	− 0.562806i	−0.959590	−	0.281403i	$-0.909200\pi$
$787$	− 15.7887i	− 0.562806i	0.959590	−	0.281403i	$-0.0907998\pi$
$788$	0	0
$789$	32.3793i	1.15273i
$790$	0	0
$791$	−33.7214	−1.19899
$792$	0	0
$793$	−11.2693	−0.400185
$794$	0	0
$795$	10.0954i	0.358048i
$796$	0	0
$797$	− 23.7905i	− 0.842704i	−0.906897	−	0.421352i	$-0.861556\pi$
$797$	− 23.7905i	− 0.842704i	0.906897	−	0.421352i	$-0.138444\pi$
$798$	0	0
$799$	4.44298	0.157181
$800$	0	0
$801$	6.25394	0.220972
$802$	0	0
$803$	36.6980i	1.29504i
$804$	0	0
$805$	2.30278i	0.0811622i
$806$	0	0
$807$	−6.11859	−0.215385
$808$	0	0
$809$	−43.5683	−1.53178	−0.765890	−	0.642972i	$-0.777701\pi$
$809$	−43.5683	−1.53178	−0.765890	+	0.642972i	$0.777701\pi$
$810$	0	0
$811$	− 17.8416i	− 0.626505i	−0.949670	−	0.313252i	$-0.898581\pi$
$811$	− 17.8416i	− 0.626505i	0.949670	−	0.313252i	$-0.101419\pi$
$812$	0	0
$813$	− 18.9657i	− 0.665156i
$814$	0	0
$815$	−3.00065	−0.105108
$816$	0	0
$817$	−9.80913	−0.343178
$818$	0	0
$819$	14.1929i	0.495940i
$820$	0	0
$821$	48.4764i	1.69184i	0.533311	+	0.845919i	$0.320948\pi$
$821$	48.4764i	1.69184i	−0.533311	+	0.845919i	$0.679052\pi$
$822$	0	0
$823$	−4.89773	−0.170724	−0.0853621	−	0.996350i	$-0.527205\pi$
$823$	−4.89773	−0.170724	−0.0853621	+	0.996350i	$0.527205\pi$
$824$	0	0
$825$	4.38782	0.152764
$826$	0	0
$827$	17.7230i	0.616291i	0.951339	+	0.308145i	$0.0997083\pi$
$827$	17.7230i	0.616291i	−0.951339	+	0.308145i	$0.900292\pi$
$828$	0	0
$829$	5.60407i	0.194637i	0.995253	+	0.0973186i	$0.0310266\pi$
$829$	5.60407i	0.194637i	−0.995253	+	0.0973186i	$0.968973\pi$
$830$	0	0
$831$	11.3678	0.394344
$832$	0	0
$833$	−1.94363	−0.0673426
$834$	0	0
$835$	4.47370i	0.154819i
$836$	0	0
$837$	6.59615i	0.227996i
$838$	0	0
$839$	40.2447	1.38940	0.694701	−	0.719298i	$-0.255537\pi$
$839$	40.2447	1.38940	0.694701	+	0.719298i	$0.255537\pi$
$840$	0	0
$841$	−0.528121	−0.0182111
$842$	0	0
$843$	− 7.45347i	− 0.256711i
$844$	0	0
$845$	9.38203i	0.322752i
$846$	0	0
$847$	0.791673	0.0272022
$848$	0	0
$849$	4.57889	0.157147
$850$	0	0
$851$	− 11.4340i	− 0.391952i
$852$	0	0
$853$	− 19.5741i	− 0.670204i	−0.942182	−	0.335102i	$-0.891229\pi$
$853$	− 19.5741i	− 0.670204i	0.942182	−	0.335102i	$-0.108771\pi$
$854$	0	0
$855$	−2.03154	−0.0694771
$856$	0	0
$857$	−21.9369	−0.749351	−0.374676	−	0.927156i	$-0.622246\pi$
$857$	−21.9369	−0.749351	−0.374676	+	0.927156i	$0.622246\pi$
$858$	0	0
$859$	39.6290i	1.35213i	0.736844	+	0.676063i	$0.236315\pi$
$859$	39.6290i	1.35213i	−0.736844	+	0.676063i	$0.763685\pi$
$860$	0	0
$861$	1.14609i	0.0390587i
$862$	0	0
$863$	−31.4443	−1.07038	−0.535188	−	0.844733i	$-0.679759\pi$
$863$	−31.4443	−1.07038	−0.535188	+	0.844733i	$0.679759\pi$
$864$	0	0
$865$	6.86640	0.233465
$866$	0	0
$867$	− 20.4387i	− 0.694134i
$868$	0	0
$869$	50.8267i	1.72418i
$870$	0	0
$871$	−30.8139	−1.04409
$872$	0	0
$873$	11.5035	0.389334
$874$	0	0
$875$	− 2.30278i	− 0.0778480i
$876$	0	0
$877$	− 54.0030i	− 1.82355i	−0.410688	−	0.911776i	$-0.634712\pi$
$877$	− 54.0030i	− 1.82355i	0.410688	−	0.911776i	$-0.365288\pi$
$878$	0	0
$879$	−41.5836	−1.40258
$880$	0	0
$881$	4.19492	0.141331	0.0706653	−	0.997500i	$-0.477488\pi$
$881$	4.19492	0.141331	0.0706653	+	0.997500i	$0.477488\pi$
$882$	0	0
$883$	53.3938i	1.79684i	0.439134	+	0.898421i	$0.355285\pi$
$883$	53.3938i	1.79684i	−0.439134	+	0.898421i	$0.644715\pi$
$884$	0	0
$885$	− 8.48528i	− 0.285230i
$886$	0	0
$887$	0.353433	0.0118671	0.00593357	−	0.999982i	$-0.498111\pi$
$887$	0.353433	0.0118671	0.00593357	+	0.999982i	$0.498111\pi$
$888$	0	0
$889$	8.84609	0.296688
$890$	0	0
$891$	− 11.4327i	− 0.383009i
$892$	0	0
$893$	− 6.05002i	− 0.202456i
$894$	0	0
$895$	−19.8167	−0.662398
$896$	0	0
$897$	6.16339	0.205789
$898$	0	0
$899$	6.39425i	0.213260i
$900$	0	0
$901$	8.87419i	0.295642i
$902$	0	0
$903$	−18.8711	−0.627990
$904$	0	0
$905$	−15.5139	−0.515699
$906$	0	0
$907$	− 21.1699i	− 0.702935i	−0.936200	−	0.351467i	$-0.885683\pi$
$907$	− 21.1699i	− 0.702935i	0.936200	−	0.351467i	$-0.114317\pi$
$908$	0	0
$909$	8.93232i	0.296266i
$910$	0	0
$911$	30.4338	1.00832	0.504158	−	0.863611i	$-0.331803\pi$
$911$	30.4338	1.00832	0.504158	+	0.863611i	$0.331803\pi$
$912$	0	0
$913$	11.2549	0.372483
$914$	0	0
$915$	− 3.10325i	− 0.102590i
$916$	0	0
$917$	− 15.6671i	− 0.517375i
$918$	0	0
$919$	8.99722	0.296791	0.148396	−	0.988928i	$-0.452589\pi$
$919$	8.99722	0.296791	0.148396	+	0.988928i	$0.452589\pi$
$920$	0	0
$921$	9.57878	0.315632
$922$	0	0
$923$	5.41780i	0.178329i
$924$	0	0
$925$	11.4340i	0.375947i
$926$	0	0
$927$	7.69722	0.252810
$928$	0	0
$929$	59.0702	1.93803	0.969015	−	0.247001i	$-0.0794452\pi$
$929$	59.0702	1.93803	0.969015	+	0.247001i	$0.0794452\pi$
$930$	0	0
$931$	2.64664i	0.0867401i
$932$	0	0
$933$	13.8490i	0.453396i
$934$	0	0
$935$	3.85702	0.126138
$936$	0	0
$937$	−41.1120	−1.34307	−0.671535	−	0.740973i	$-0.734365\pi$
$937$	−41.1120	−1.34307	−0.671535	+	0.740973i	$0.734365\pi$
$938$	0	0
$939$	15.1606i	0.494748i
$940$	0	0
$941$	− 19.4054i	− 0.632597i	−0.948660	−	0.316299i	$-0.897560\pi$
$941$	− 19.4054i	− 0.632597i	0.948660	−	0.316299i	$-0.102440\pi$
$942$	0	0
$943$	−0.382030	−0.0124406
$944$	0	0
$945$	−12.9083	−0.419908
$946$	0	0
$947$	28.9186i	0.939729i	0.882739	+	0.469864i	$0.155697\pi$
$947$	28.9186i	0.939729i	−0.882739	+	0.469864i	$0.844303\pi$
$948$	0	0
$949$	− 51.5482i	− 1.67332i
$950$	0	0
$951$	−20.7215	−0.671941
$952$	0	0
$953$	−6.28163	−0.203482	−0.101741	−	0.994811i	$-0.532441\pi$
$953$	−6.28163	−0.203482	−0.101741	+	0.994811i	$0.532441\pi$
$954$	0	0
$955$	− 9.87973i	− 0.319701i
$956$	0	0
$957$	− 23.8433i	− 0.770745i
$958$	0	0
$959$	−9.43554	−0.304689
$960$	0	0
$961$	−29.6153	−0.955333
$962$	0	0
$963$	10.1833i	0.328154i
$964$	0	0
$965$	− 12.4853i	− 0.401915i
$966$	0	0
$967$	−2.85029	−0.0916592	−0.0458296	−	0.998949i	$-0.514593\pi$
$967$	−2.85029	−0.0916592	−0.0458296	+	0.998949i	$0.514593\pi$
$968$	0	0
$969$	2.32647	0.0747371
$970$	0	0
$971$	− 2.10234i	− 0.0674673i	−0.999431	−	0.0337336i	$-0.989260\pi$
$971$	− 2.10234i	− 0.0674673i	0.999431	−	0.0337336i	$-0.0107398\pi$
$972$	0	0
$973$	− 44.8401i	− 1.43751i
$974$	0	0
$975$	−6.16339	−0.197386
$976$	0	0
$977$	35.0772	1.12222	0.561110	−	0.827741i	$-0.310374\pi$
$977$	35.0772	1.12222	0.561110	+	0.827741i	$0.310374\pi$
$978$	0	0
$979$	16.1683i	0.516740i
$980$	0	0
$981$	− 21.9075i	− 0.699452i
$982$	0	0
$983$	−53.3526	−1.70168	−0.850842	−	0.525422i	$-0.823907\pi$
$983$	−53.3526	−1.70168	−0.850842	+	0.525422i	$0.823907\pi$
$984$	0	0
$985$	−26.1832	−0.834267
$986$	0	0
$987$	− 11.6392i	− 0.370480i
$988$	0	0
$989$	− 6.29036i	− 0.200022i
$990$	0	0
$991$	6.74795	0.214356	0.107178	−	0.994240i	$-0.465819\pi$
$991$	6.74795	0.214356	0.107178	+	0.994240i	$0.465819\pi$
$992$	0	0
$993$	6.04982	0.191985
$994$	0	0
$995$	7.85489i	0.249017i
$996$	0	0
$997$	46.5382i	1.47388i	0.675958	+	0.736940i	$0.263730\pi$
$997$	46.5382i	1.47388i	−0.675958	+	0.736940i	$0.736270\pi$
$998$	0	0
$999$	64.0938	2.02784

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3680.2.f.b.1841.6		8
4.3	odd	2		920.2.f.b.461.2	✓	8
8.3	odd	2		920.2.f.b.461.7	yes	8
8.5	even	2	inner	3680.2.f.b.1841.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.f.b.461.2	✓	8	4.3	odd	2
920.2.f.b.461.7	yes	8	8.3	odd	2
3680.2.f.b.1841.3		8	8.5	even	2	inner
3680.2.f.b.1841.6		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:	\( N \)	\(=\)	\( 3680 = 2^{5} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3680.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.30278i q^{3} -1.00000i q^{5} -2.30278 q^{7} +1.30278 q^{9} +O(q^{10})\)	\(q+1.30278i q^{3} -1.00000i q^{5} -2.30278 q^{7} +1.30278 q^{9} +3.36805i q^{11} -4.73097i q^{13} +1.30278 q^{15} +1.14518 q^{17} -1.55939i q^{19} -3.00000i q^{21} +1.00000 q^{23} -1.00000 q^{25} +5.60555i q^{27} +5.43398i q^{29} +1.17672 q^{31} -4.38782 q^{33} +2.30278i q^{35} -11.4340i q^{37} +6.16339 q^{39} -0.382030 q^{41} -6.29036i q^{43} -1.30278i q^{45} +3.87973 q^{47} -1.69722 q^{49} +1.49191i q^{51} +7.74917i q^{53} +3.36805 q^{55} +2.03154 q^{57} -6.51323i q^{59} -2.38203i q^{61} -3.00000 q^{63} -4.73097 q^{65} -6.51323i q^{67} +1.30278i q^{69} -1.14518 q^{71} +10.8959 q^{73} -1.30278i q^{75} -7.75587i q^{77} +15.0908 q^{79} -3.39445 q^{81} -3.34166i q^{83} -1.14518i q^{85} -7.07925 q^{87} +4.80048 q^{89} +10.8944i q^{91} +1.53300i q^{93} -1.55939 q^{95} +8.82998 q^{97} +4.38782i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{7} - 4 q^{9}+O(q^{10}) \) 8 * q - 4 * q^7 - 4 * q^9	\( 8 q - 4 q^{7} - 4 q^{9} - 4 q^{15} - 20 q^{17} + 8 q^{23} - 8 q^{25} + 4 q^{31} + 28 q^{33} + 16 q^{39} + 12 q^{41} - 8 q^{47} - 28 q^{49} + 4 q^{55} + 40 q^{57} - 24 q^{63} - 20 q^{65} + 20 q^{71} + 24 q^{79} - 56 q^{81} - 56 q^{87} + 28 q^{95} + 12 q^{97}+O(q^{100}) \) 8 * q - 4 * q^7 - 4 * q^9 - 4 * q^15 - 20 * q^17 + 8 * q^23 - 8 * q^25 + 4 * q^31 + 28 * q^33 + 16 * q^39 + 12 * q^41 - 8 * q^47 - 28 * q^49 + 4 * q^55 + 40 * q^57 - 24 * q^63 - 20 * q^65 + 20 * q^71 + 24 * q^79 - 56 * q^81 - 56 * q^87 + 28 * q^95 + 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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