LMFDB - Embedded newform 2352.3.m.l.1471.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,3,Mod(1471,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2352.1471");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2352.m (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:	$64.0873581775$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.3
Root			$0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	2352.1471
Dual form			2352.3.m.l.1471.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} +4.58579 q^{5} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} +4.58579 q^{5} -3.00000 q^{9} +0.594346i q^{11} +2.10051 q^{13} +7.94282i q^{15} -1.07107 q^{17} -14.6969i q^{19} +21.3790i q^{23} -3.97056 q^{25} -5.19615i q^{27} +15.5147 q^{29} +37.1626i q^{31} -1.02944 q^{33} +33.9411 q^{37} +3.63818i q^{39} -27.8995 q^{41} +28.2052i q^{43} -13.7574 q^{45} -0.840532i q^{47} -1.85514i q^{51} +28.9117 q^{53} +2.72554i q^{55} +25.4558 q^{57} +92.5882i q^{59} +81.9828 q^{61} +9.63247 q^{65} -10.9867i q^{67} -37.0294 q^{69} +89.9647i q^{71} -37.4975 q^{73} -6.87722i q^{75} -7.42058i q^{79} +9.00000 q^{81} +64.0349i q^{83} -4.91169 q^{85} +26.8723i q^{87} -3.12994 q^{89} -64.3675 q^{93} -67.3970i q^{95} +92.4437 q^{97} -1.78304i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 24 q^{5} - 12 q^{9}+O(q^{10}) $ 4 * q + 24 * q^5 - 12 * q^9	$ 4 q + 24 q^{5} - 12 q^{9} + 48 q^{13} + 24 q^{17} + 52 q^{25} + 96 q^{29} - 72 q^{33} - 72 q^{41} - 72 q^{45} - 88 q^{53} + 96 q^{61} + 344 q^{65} - 216 q^{69} + 48 q^{73} + 36 q^{81} + 184 q^{85} - 120 q^{89} + 48 q^{93} + 432 q^{97}+O(q^{100}) $ 4 * q + 24 * q^5 - 12 * q^9 + 48 * q^13 + 24 * q^17 + 52 * q^25 + 96 * q^29 - 72 * q^33 - 72 * q^41 - 72 * q^45 - 88 * q^53 + 96 * q^61 + 344 * q^65 - 216 * q^69 + 48 * q^73 + 36 * q^81 + 184 * q^85 - 120 * q^89 + 48 * q^93 + 432 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	1.73205i	0.577350i
$3$	1.73205i	0.577350i
$4$	0	0
$5$	4.58579	0.917157	0.458579	−	0.888654i	$-0.348359\pi$
$5$	4.58579	0.917157	0.458579	+	0.888654i	$0.348359\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	0.594346i	0.0540314i	0.999635	+	0.0270157i	$0.00860042\pi$
$11$	0.594346i	0.0540314i	−0.999635	+	0.0270157i	$0.991400\pi$
$12$	0	0
$13$	2.10051	0.161577	0.0807887	−	0.996731i	$-0.474256\pi$
$13$	2.10051	0.161577	0.0807887	+	0.996731i	$0.474256\pi$
$14$	0	0
$15$	7.94282i	0.529521i
$16$	0	0
$17$	−1.07107	−0.0630040	−0.0315020	−	0.999504i	$-0.510029\pi$
$17$	−1.07107	−0.0630040	−0.0315020	+	0.999504i	$0.510029\pi$
$18$	0	0
$19$	− 14.6969i	− 0.773523i	−0.922180	−	0.386762i	$-0.873594\pi$
$19$	− 14.6969i	− 0.773523i	0.922180	−	0.386762i	$-0.126406\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	21.3790i	0.929520i	0.885437	+	0.464760i	$0.153859\pi$
$23$	21.3790i	0.929520i	−0.885437	+	0.464760i	$0.846141\pi$
$24$	0	0
$25$	−3.97056	−0.158823
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	15.5147	0.534990	0.267495	−	0.963559i	$-0.413804\pi$
$29$	15.5147	0.534990	0.267495	+	0.963559i	$0.413804\pi$
$30$	0	0
$31$	37.1626i	1.19879i	0.800452	+	0.599397i	$0.204593\pi$
$31$	37.1626i	1.19879i	−0.800452	+	0.599397i	$0.795407\pi$
$32$	0	0
$33$	−1.02944	−0.0311951
$34$	0	0
$35$	0	0
$36$	0	0
$37$	33.9411	0.917328	0.458664	−	0.888610i	$-0.348328\pi$
$37$	33.9411	0.917328	0.458664	+	0.888610i	$0.348328\pi$
$38$	0	0
$39$	3.63818i	0.0932867i
$40$	0	0
$41$	−27.8995	−0.680475	−0.340238	−	0.940339i	$-0.610508\pi$
$41$	−27.8995	−0.680475	−0.340238	+	0.940339i	$0.610508\pi$
$42$	0	0
$43$	28.2052i	0.655935i	0.944689	+	0.327967i	$0.106364\pi$
$43$	28.2052i	0.655935i	−0.944689	+	0.327967i	$0.893636\pi$
$44$	0	0
$45$	−13.7574	−0.305719
$46$	0	0
$47$	− 0.840532i	− 0.0178837i	−0.999960	−	0.00894183i	$-0.997154\pi$
$47$	− 0.840532i	− 0.0178837i	0.999960	−	0.00894183i	$-0.00284631\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 1.85514i	− 0.0363754i
$52$	0	0
$53$	28.9117	0.545504	0.272752	−	0.962084i	$-0.412066\pi$
$53$	28.9117	0.545504	0.272752	+	0.962084i	$0.412066\pi$
$54$	0	0
$55$	2.72554i	0.0495553i
$56$	0	0
$57$	25.4558	0.446594
$58$	0	0
$59$	92.5882i	1.56929i	0.619944	+	0.784646i	$0.287155\pi$
$59$	92.5882i	1.56929i	−0.619944	+	0.784646i	$0.712845\pi$
$60$	0	0
$61$	81.9828	1.34398	0.671990	−	0.740560i	$-0.265440\pi$
$61$	81.9828	1.34398	0.671990	+	0.740560i	$0.265440\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	9.63247	0.148192
$66$	0	0
$67$	− 10.9867i	− 0.163980i	−0.996633	−	0.0819899i	$-0.973872\pi$
$67$	− 10.9867i	− 0.163980i	0.996633	−	0.0819899i	$-0.0261275\pi$
$68$	0	0
$69$	−37.0294	−0.536659
$70$	0	0
$71$	89.9647i	1.26711i	0.773698	+	0.633554i	$0.218405\pi$
$71$	89.9647i	1.26711i	−0.773698	+	0.633554i	$0.781595\pi$
$72$	0	0
$73$	−37.4975	−0.513664	−0.256832	−	0.966456i	$-0.582679\pi$
$73$	−37.4975	−0.513664	−0.256832	+	0.966456i	$0.582679\pi$
$74$	0	0
$75$	− 6.87722i	− 0.0916962i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 7.42058i	− 0.0939313i	−0.998897	−	0.0469657i	$-0.985045\pi$
$79$	− 7.42058i	− 0.0939313i	0.998897	−	0.0469657i	$-0.0149551\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	64.0349i	0.771505i	0.922602	+	0.385752i	$0.126058\pi$
$83$	64.0349i	0.771505i	−0.922602	+	0.385752i	$0.873942\pi$
$84$	0	0
$85$	−4.91169	−0.0577846
$86$	0	0
$87$	26.8723i	0.308877i
$88$	0	0
$89$	−3.12994	−0.0351679	−0.0175839	−	0.999845i	$-0.505597\pi$
$89$	−3.12994	−0.0351679	−0.0175839	+	0.999845i	$0.505597\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−64.3675	−0.692124
$94$	0	0
$95$	− 67.3970i	− 0.709442i
$96$	0	0
$97$	92.4437	0.953027	0.476514	−	0.879167i	$-0.341900\pi$
$97$	92.4437	0.953027	0.476514	+	0.879167i	$0.341900\pi$
$98$	0	0
$99$	− 1.78304i	− 0.0180105i
$100$	0	0
$101$	34.3259	0.339860	0.169930	−	0.985456i	$-0.445646\pi$
$101$	34.3259	0.339860	0.169930	+	0.985456i	$0.445646\pi$
$102$	0	0
$103$	95.9504i	0.931557i	0.884901	+	0.465778i	$0.154226\pi$
$103$	95.9504i	0.931557i	−0.884901	+	0.465778i	$0.845774\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 171.914i	− 1.60668i	−0.595523	−	0.803338i	$-0.703055\pi$
$107$	− 171.914i	− 1.60668i	0.595523	−	0.803338i	$-0.296945\pi$
$108$	0	0
$109$	−96.8528	−0.888558	−0.444279	−	0.895888i	$-0.646540\pi$
$109$	−96.8528	−0.888558	−0.444279	+	0.895888i	$0.646540\pi$
$110$	0	0
$111$	58.7878i	0.529619i
$112$	0	0
$113$	115.823	1.02499	0.512493	−	0.858692i	$-0.328722\pi$
$113$	115.823	1.02499	0.512493	+	0.858692i	$0.328722\pi$
$114$	0	0
$115$	98.0393i	0.852516i
$116$	0	0
$117$	−6.30152	−0.0538591
$118$	0	0
$119$	0	0
$120$	0	0
$121$	120.647	0.997081
$122$	0	0
$123$	− 48.3233i	− 0.392873i
$124$	0	0
$125$	−132.853	−1.06282
$126$	0	0
$127$	228.919i	1.80251i	0.433286	+	0.901256i	$0.357354\pi$
$127$	228.919i	1.80251i	−0.433286	+	0.901256i	$0.642646\pi$
$128$	0	0
$129$	−48.8528	−0.378704
$130$	0	0
$131$	32.9600i	0.251603i	0.992055	+	0.125801i	$0.0401502\pi$
$131$	32.9600i	0.251603i	−0.992055	+	0.125801i	$0.959850\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 23.8284i	− 0.176507i
$136$	0	0
$137$	101.574	0.741413	0.370707	−	0.928750i	$-0.379116\pi$
$137$	101.574	0.741413	0.370707	+	0.928750i	$0.379116\pi$
$138$	0	0
$139$	211.004i	1.51802i	0.651081	+	0.759008i	$0.274316\pi$
$139$	211.004i	1.51802i	−0.651081	+	0.759008i	$0.725684\pi$
$140$	0	0
$141$	1.45584	0.0103251
$142$	0	0
$143$	1.24843i	0.00873026i
$144$	0	0
$145$	71.1472	0.490670
$146$	0	0
$147$	0	0
$148$	0	0
$149$	52.9117	0.355112	0.177556	−	0.984111i	$-0.443181\pi$
$149$	52.9117	0.355112	0.177556	+	0.984111i	$0.443181\pi$
$150$	0	0
$151$	− 255.936i	− 1.69494i	−0.530845	−	0.847469i	$-0.678125\pi$
$151$	− 255.936i	− 1.69494i	0.530845	−	0.847469i	$-0.321875\pi$
$152$	0	0
$153$	3.21320	0.0210013
$154$	0	0
$155$	170.420i	1.09948i
$156$	0	0
$157$	40.9289	0.260694	0.130347	−	0.991468i	$-0.458391\pi$
$157$	40.9289	0.260694	0.130347	+	0.991468i	$0.458391\pi$
$158$	0	0
$159$	50.0765i	0.314947i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 245.849i	− 1.50828i	−0.656715	−	0.754139i	$-0.728055\pi$
$163$	− 245.849i	− 1.50828i	0.656715	−	0.754139i	$-0.271945\pi$
$164$	0	0
$165$	−4.72078	−0.0286108
$166$	0	0
$167$	92.7922i	0.555642i	0.960633	+	0.277821i	$0.0896122\pi$
$167$	92.7922i	0.555642i	−0.960633	+	0.277821i	$0.910388\pi$
$168$	0	0
$169$	−164.588	−0.973893
$170$	0	0
$171$	44.0908i	0.257841i
$172$	0	0
$173$	−188.267	−1.08825	−0.544124	−	0.839005i	$-0.683138\pi$
$173$	−188.267	−1.08825	−0.544124	+	0.839005i	$0.683138\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−160.368	−0.906031
$178$	0	0
$179$	− 174.292i	− 0.973697i	−0.873486	−	0.486849i	$-0.838146\pi$
$179$	− 174.292i	− 0.973697i	0.873486	−	0.486849i	$-0.161854\pi$
$180$	0	0
$181$	138.988	0.767888	0.383944	−	0.923356i	$-0.374565\pi$
$181$	138.988	0.767888	0.383944	+	0.923356i	$0.374565\pi$
$182$	0	0
$183$	141.998i	0.775947i
$184$	0	0
$185$	155.647	0.841334
$186$	0	0
$187$	− 0.636585i	− 0.00340420i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 51.9615i	− 0.272050i	−0.990705	−	0.136025i	$-0.956567\pi$
$191$	− 51.9615i	− 0.272050i	0.990705	−	0.136025i	$-0.0434327\pi$
$192$	0	0
$193$	15.8234	0.0819864	0.0409932	−	0.999159i	$-0.486948\pi$
$193$	15.8234	0.0819864	0.0409932	+	0.999159i	$0.486948\pi$
$194$	0	0
$195$	16.6839i	0.0855586i
$196$	0	0
$197$	−148.912	−0.755897	−0.377948	−	0.925827i	$-0.623370\pi$
$197$	−148.912	−0.755897	−0.377948	+	0.925827i	$0.623370\pi$
$198$	0	0
$199$	135.202i	0.679407i	0.940533	+	0.339703i	$0.110327\pi$
$199$	135.202i	0.679407i	−0.940533	+	0.339703i	$0.889673\pi$
$200$	0	0
$201$	19.0294	0.0946738
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−127.941	−0.624103
$206$	0	0
$207$	− 64.1369i	− 0.309840i
$208$	0	0
$209$	8.73506	0.0417946
$210$	0	0
$211$	100.645i	0.476992i	0.971143	+	0.238496i	$0.0766545\pi$
$211$	100.645i	0.476992i	−0.971143	+	0.238496i	$0.923346\pi$
$212$	0	0
$213$	−155.823	−0.731565
$214$	0	0
$215$	129.343i	0.601595i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 64.9475i	− 0.296564i
$220$	0	0
$221$	−2.24978	−0.0101800
$222$	0	0
$223$	100.357i	0.450031i	0.974355	+	0.225016i	$0.0722433\pi$
$223$	100.357i	0.450031i	−0.974355	+	0.225016i	$0.927757\pi$
$224$	0	0
$225$	11.9117	0.0529408
$226$	0	0
$227$	379.803i	1.67314i	0.547860	+	0.836570i	$0.315443\pi$
$227$	379.803i	1.67314i	−0.547860	+	0.836570i	$0.684557\pi$
$228$	0	0
$229$	21.8995	0.0956310	0.0478155	−	0.998856i	$-0.484774\pi$
$229$	21.8995	0.0956310	0.0478155	+	0.998856i	$0.484774\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	102.780	0.441114	0.220557	−	0.975374i	$-0.429212\pi$
$233$	102.780	0.441114	0.220557	+	0.975374i	$0.429212\pi$
$234$	0	0
$235$	− 3.85450i	− 0.0164021i
$236$	0	0
$237$	12.8528	0.0542313
$238$	0	0
$239$	376.483i	1.57524i	0.616160	+	0.787621i	$0.288687\pi$
$239$	376.483i	1.57524i	−0.616160	+	0.787621i	$0.711313\pi$
$240$	0	0
$241$	−221.179	−0.917754	−0.458877	−	0.888500i	$-0.651748\pi$
$241$	−221.179	−0.917754	−0.458877	+	0.888500i	$0.651748\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 30.8710i	− 0.124984i
$248$	0	0
$249$	−110.912	−0.445428
$250$	0	0
$251$	298.753i	1.19025i	0.803632	+	0.595126i	$0.202898\pi$
$251$	298.753i	1.19025i	−0.803632	+	0.595126i	$0.797102\pi$
$252$	0	0
$253$	−12.7065	−0.0502233
$254$	0	0
$255$	− 8.50729i	− 0.0333619i
$256$	0	0
$257$	−344.517	−1.34053	−0.670266	−	0.742121i	$-0.733820\pi$
$257$	−344.517	−1.34053	−0.670266	+	0.742121i	$0.733820\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−46.5442	−0.178330
$262$	0	0
$263$	− 227.136i	− 0.863635i	−0.901961	−	0.431818i	$-0.857872\pi$
$263$	− 227.136i	− 0.863635i	0.901961	−	0.431818i	$-0.142128\pi$
$264$	0	0
$265$	132.583	0.500313
$266$	0	0
$267$	− 5.42122i	− 0.0203042i
$268$	0	0
$269$	470.590	1.74941	0.874703	−	0.484660i	$-0.161057\pi$
$269$	470.590	1.74941	0.874703	+	0.484660i	$0.161057\pi$
$270$	0	0
$271$	− 156.419i	− 0.577193i	−0.957451	−	0.288596i	$-0.906811\pi$
$271$	− 156.419i	− 0.577193i	0.957451	−	0.288596i	$-0.0931885\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 2.35989i	− 0.00858141i
$276$	0	0
$277$	−279.823	−1.01019	−0.505096	−	0.863063i	$-0.668543\pi$
$277$	−279.823	−1.01019	−0.505096	+	0.863063i	$0.668543\pi$
$278$	0	0
$279$	− 111.488i	− 0.399598i
$280$	0	0
$281$	134.309	0.477967	0.238983	−	0.971024i	$-0.423186\pi$
$281$	134.309	0.477967	0.238983	+	0.971024i	$0.423186\pi$
$282$	0	0
$283$	490.858i	1.73448i	0.497890	+	0.867240i	$0.334108\pi$
$283$	490.858i	1.73448i	−0.497890	+	0.867240i	$0.665892\pi$
$284$	0	0
$285$	116.735	0.409597
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−287.853	−0.996030
$290$	0	0
$291$	160.117i	0.550231i
$292$	0	0
$293$	−154.056	−0.525788	−0.262894	−	0.964825i	$-0.584677\pi$
$293$	−154.056	−0.525788	−0.262894	+	0.964825i	$0.584677\pi$
$294$	0	0
$295$	424.590i	1.43929i
$296$	0	0
$297$	3.08831	0.0103984
$298$	0	0
$299$	44.9066i	0.150189i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	59.4542i	0.196219i
$304$	0	0
$305$	375.955	1.23264
$306$	0	0
$307$	− 23.1023i	− 0.0752517i	−0.999292	−	0.0376258i	$-0.988020\pi$
$307$	− 23.1023i	− 0.0752517i	0.999292	−	0.0376258i	$-0.0119795\pi$
$308$	0	0
$309$	−166.191	−0.537835
$310$	0	0
$311$	364.061i	1.17062i	0.810811	+	0.585308i	$0.199026\pi$
$311$	364.061i	1.17062i	−0.810811	+	0.585308i	$0.800974\pi$
$312$	0	0
$313$	−194.704	−0.622056	−0.311028	−	0.950401i	$-0.600673\pi$
$313$	−194.704	−0.622056	−0.311028	+	0.950401i	$0.600673\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	414.735	1.30831	0.654156	−	0.756359i	$-0.273024\pi$
$317$	414.735	1.30831	0.654156	+	0.756359i	$0.273024\pi$
$318$	0	0
$319$	9.22111i	0.0289063i
$320$	0	0
$321$	297.765	0.927615
$322$	0	0
$323$	15.7414i	0.0487350i
$324$	0	0
$325$	−8.34019	−0.0256621
$326$	0	0
$327$	− 167.754i	− 0.513009i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	70.6747i	0.213519i	0.994285	+	0.106759i	$0.0340474\pi$
$331$	70.6747i	0.213519i	−0.994285	+	0.106759i	$0.965953\pi$
$332$	0	0
$333$	−101.823	−0.305776
$334$	0	0
$335$	− 50.3824i	− 0.150395i
$336$	0	0
$337$	−134.558	−0.399283	−0.199642	−	0.979869i	$-0.563978\pi$
$337$	−134.558	−0.399283	−0.199642	+	0.979869i	$0.563978\pi$
$338$	0	0
$339$	200.612i	0.591776i
$340$	0	0
$341$	−22.0874	−0.0647726
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−169.809	−0.492200
$346$	0	0
$347$	352.744i	1.01655i	0.861194	+	0.508277i	$0.169717\pi$
$347$	352.744i	1.01655i	−0.861194	+	0.508277i	$0.830283\pi$
$348$	0	0
$349$	−168.042	−0.481495	−0.240747	−	0.970588i	$-0.577393\pi$
$349$	−168.042	−0.481495	−0.240747	+	0.970588i	$0.577393\pi$
$350$	0	0
$351$	− 10.9145i	− 0.0310956i
$352$	0	0
$353$	406.659	1.15201	0.576004	−	0.817447i	$-0.304611\pi$
$353$	406.659	1.15201	0.576004	+	0.817447i	$0.304611\pi$
$354$	0	0
$355$	412.559i	1.16214i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	81.3554i	0.226617i	0.993560	+	0.113308i	$0.0361448\pi$
$359$	81.3554i	0.226617i	−0.993560	+	0.113308i	$0.963855\pi$
$360$	0	0
$361$	145.000	0.401662
$362$	0	0
$363$	208.966i	0.575665i
$364$	0	0
$365$	−171.955	−0.471111
$366$	0	0
$367$	− 54.1525i	− 0.147554i	−0.997275	−	0.0737772i	$-0.976495\pi$
$367$	− 54.1525i	− 0.147554i	0.997275	−	0.0737772i	$-0.0235054\pi$
$368$	0	0
$369$	83.6985	0.226825
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−523.470	−1.40341	−0.701703	−	0.712470i	$-0.747576\pi$
$373$	−523.470	−1.40341	−0.701703	+	0.712470i	$0.747576\pi$
$374$	0	0
$375$	− 230.108i	− 0.613621i
$376$	0	0
$377$	32.5887	0.0864423
$378$	0	0
$379$	43.9466i	0.115954i	0.998318	+	0.0579770i	$0.0184650\pi$
$379$	43.9466i	0.115954i	−0.998318	+	0.0579770i	$0.981535\pi$
$380$	0	0
$381$	−396.500	−1.04068
$382$	0	0
$383$	− 456.854i	− 1.19283i	−0.802677	−	0.596415i	$-0.796591\pi$
$383$	− 456.854i	− 1.19283i	0.802677	−	0.596415i	$-0.203409\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 84.6156i	− 0.218645i
$388$	0	0
$389$	175.279	0.450589	0.225295	−	0.974291i	$-0.427666\pi$
$389$	175.279	0.450589	0.225295	+	0.974291i	$0.427666\pi$
$390$	0	0
$391$	− 22.8983i	− 0.0585635i
$392$	0	0
$393$	−57.0883	−0.145263
$394$	0	0
$395$	− 34.0292i	− 0.0861498i
$396$	0	0
$397$	182.600	0.459950	0.229975	−	0.973197i	$-0.426136\pi$
$397$	182.600	0.459950	0.229975	+	0.973197i	$0.426136\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	755.897	1.88503	0.942514	−	0.334166i	$-0.108454\pi$
$401$	755.897	1.88503	0.942514	+	0.334166i	$0.108454\pi$
$402$	0	0
$403$	78.0603i	0.193698i
$404$	0	0
$405$	41.2721	0.101906
$406$	0	0
$407$	20.1728i	0.0495645i
$408$	0	0
$409$	−662.704	−1.62030	−0.810151	−	0.586221i	$-0.800615\pi$
$409$	−662.704	−1.62030	−0.810151	+	0.586221i	$0.800615\pi$
$410$	0	0
$411$	175.931i	0.428055i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	293.650i	0.707591i
$416$	0	0
$417$	−365.470	−0.876427
$418$	0	0
$419$	− 268.951i	− 0.641889i	−0.947098	−	0.320945i	$-0.896000\pi$
$419$	− 268.951i	− 0.641889i	0.947098	−	0.320945i	$-0.104000\pi$
$420$	0	0
$421$	445.294	1.05770	0.528852	−	0.848714i	$-0.322623\pi$
$421$	445.294	1.05770	0.528852	+	0.848714i	$0.322623\pi$
$422$	0	0
$423$	2.52160i	0.00596122i
$424$	0	0
$425$	4.25274	0.0100065
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−2.16234	−0.00504042
$430$	0	0
$431$	− 270.182i	− 0.626873i	−0.949609	−	0.313437i	$-0.898520\pi$
$431$	− 270.182i	− 0.626873i	0.949609	−	0.313437i	$-0.101480\pi$
$432$	0	0
$433$	−283.591	−0.654944	−0.327472	−	0.944861i	$-0.606197\pi$
$433$	−283.591	−0.654944	−0.327472	+	0.944861i	$0.606197\pi$
$434$	0	0
$435$	123.231i	0.283289i
$436$	0	0
$437$	314.205	0.719005
$438$	0	0
$439$	− 426.186i	− 0.970812i	−0.874289	−	0.485406i	$-0.838672\pi$
$439$	− 426.186i	− 0.970812i	0.874289	−	0.485406i	$-0.161328\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 183.801i	− 0.414901i	−0.978245	−	0.207451i	$-0.933483\pi$
$443$	− 183.801i	− 0.414901i	0.978245	−	0.207451i	$-0.0665167\pi$
$444$	0	0
$445$	−14.3532	−0.0322545
$446$	0	0
$447$	91.6457i	0.205024i
$448$	0	0
$449$	290.353	0.646666	0.323333	−	0.946285i	$-0.395197\pi$
$449$	290.353	0.646666	0.323333	+	0.946285i	$0.395197\pi$
$450$	0	0
$451$	− 16.5819i	− 0.0367671i
$452$	0	0
$453$	443.294	0.978573
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−277.647	−0.607542	−0.303771	−	0.952745i	$-0.598246\pi$
$457$	−277.647	−0.607542	−0.303771	+	0.952745i	$0.598246\pi$
$458$	0	0
$459$	5.56543i	0.0121251i
$460$	0	0
$461$	−293.002	−0.635579	−0.317790	−	0.948161i	$-0.602941\pi$
$461$	−293.002	−0.635579	−0.317790	+	0.948161i	$0.602941\pi$
$462$	0	0
$463$	− 240.518i	− 0.519477i	−0.965679	−	0.259738i	$-0.916364\pi$
$463$	− 240.518i	− 0.519477i	0.965679	−	0.259738i	$-0.0836363\pi$
$464$	0	0
$465$	−295.176	−0.634787
$466$	0	0
$467$	370.582i	0.793537i	0.917919	+	0.396768i	$0.129868\pi$
$467$	370.582i	0.793537i	−0.917919	+	0.396768i	$0.870132\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	70.8910i	0.150512i
$472$	0	0
$473$	−16.7636	−0.0354411
$474$	0	0
$475$	58.3551i	0.122853i
$476$	0	0
$477$	−86.7351	−0.181835
$478$	0	0
$479$	− 613.885i	− 1.28160i	−0.767709	−	0.640798i	$-0.778604\pi$
$479$	− 613.885i	− 1.28160i	0.767709	−	0.640798i	$-0.221396\pi$
$480$	0	0
$481$	71.2935	0.148219
$482$	0	0
$483$	0	0
$484$	0	0
$485$	423.927	0.874076
$486$	0	0
$487$	− 514.860i	− 1.05721i	−0.848868	−	0.528604i	$-0.822716\pi$
$487$	− 514.860i	− 1.05721i	0.848868	−	0.528604i	$-0.177284\pi$
$488$	0	0
$489$	425.823	0.870804
$490$	0	0
$491$	− 306.097i	− 0.623415i	−0.950178	−	0.311707i	$-0.899099\pi$
$491$	− 306.097i	− 0.623415i	0.950178	−	0.311707i	$-0.100901\pi$
$492$	0	0
$493$	−16.6173	−0.0337065
$494$	0	0
$495$	− 8.17663i	− 0.0165184i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 443.609i	− 0.888996i	−0.895780	−	0.444498i	$-0.853382\pi$
$499$	− 443.609i	− 0.888996i	0.895780	−	0.444498i	$-0.146618\pi$
$500$	0	0
$501$	−160.721	−0.320800
$502$	0	0
$503$	− 95.3138i	− 0.189491i	−0.995502	−	0.0947453i	$-0.969796\pi$
$503$	− 95.3138i	− 0.189491i	0.995502	−	0.0947453i	$-0.0302037\pi$
$504$	0	0
$505$	157.411	0.311705
$506$	0	0
$507$	− 285.075i	− 0.562277i
$508$	0	0
$509$	−641.439	−1.26019	−0.630097	−	0.776517i	$-0.716985\pi$
$509$	−641.439	−1.26019	−0.630097	+	0.776517i	$0.716985\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−76.3675	−0.148865
$514$	0	0
$515$	440.008i	0.854384i
$516$	0	0
$517$	0.499567	0.000966280 0
$518$	0	0
$519$	− 326.088i	− 0.628301i
$520$	0	0
$521$	948.718	1.82096	0.910478	−	0.413558i	$-0.135714\pi$
$521$	948.718	1.82096	0.910478	+	0.413558i	$0.135714\pi$
$522$	0	0
$523$	451.606i	0.863492i	0.901995	+	0.431746i	$0.142102\pi$
$523$	451.606i	0.863492i	−0.901995	+	0.431746i	$0.857898\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 39.8037i	− 0.0755288i
$528$	0	0
$529$	71.9403	0.135993
$530$	0	0
$531$	− 277.765i	− 0.523097i
$532$	0	0
$533$	−58.6030	−0.109949
$534$	0	0
$535$	− 788.363i	− 1.47358i
$536$	0	0
$537$	301.882	0.562164
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−438.764	−0.811023	−0.405512	−	0.914090i	$-0.632907\pi$
$541$	−438.764	−0.811023	−0.405512	+	0.914090i	$0.632907\pi$
$542$	0	0
$543$	240.734i	0.443341i
$544$	0	0
$545$	−444.146	−0.814947
$546$	0	0
$547$	− 651.132i	− 1.19037i	−0.803589	−	0.595184i	$-0.797079\pi$
$547$	− 651.132i	− 1.19037i	0.803589	−	0.595184i	$-0.202921\pi$
$548$	0	0
$549$	−245.948	−0.447993
$550$	0	0
$551$	− 228.019i	− 0.413827i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	269.588i	0.485744i
$556$	0	0
$557$	912.205	1.63771	0.818856	−	0.573999i	$-0.194609\pi$
$557$	912.205	1.63771	0.818856	+	0.573999i	$0.194609\pi$
$558$	0	0
$559$	59.2451i	0.105984i
$560$	0	0
$561$	1.10260	0.00196541
$562$	0	0
$563$	− 523.002i	− 0.928956i	−0.885585	−	0.464478i	$-0.846242\pi$
$563$	− 523.002i	− 0.928956i	0.885585	−	0.464478i	$-0.153758\pi$
$564$	0	0
$565$	531.141	0.940073
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1005.34	1.76685	0.883425	−	0.468572i	$-0.155231\pi$
$569$	1005.34	1.76685	0.883425	+	0.468572i	$0.155231\pi$
$570$	0	0
$571$	722.383i	1.26512i	0.774512	+	0.632560i	$0.217996\pi$
$571$	722.383i	1.26512i	−0.774512	+	0.632560i	$0.782004\pi$
$572$	0	0
$573$	90.0000	0.157068
$574$	0	0
$575$	− 84.8865i	− 0.147629i
$576$	0	0
$577$	−614.060	−1.06423	−0.532114	−	0.846672i	$-0.678602\pi$
$577$	−614.060	−1.06423	−0.532114	+	0.846672i	$0.678602\pi$
$578$	0	0
$579$	27.4069i	0.0473349i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	17.1835i	0.0294743i
$584$	0	0
$585$	−28.8974	−0.0493973
$586$	0	0
$587$	140.066i	0.238613i	0.992857	+	0.119307i	$0.0380671\pi$
$587$	140.066i	0.238613i	−0.992857	+	0.119307i	$0.961933\pi$
$588$	0	0
$589$	546.177	0.927295
$590$	0	0
$591$	− 257.923i	− 0.436417i
$592$	0	0
$593$	649.841	1.09585	0.547926	−	0.836527i	$-0.315417\pi$
$593$	649.841	1.09585	0.547926	+	0.836527i	$0.315417\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−234.177	−0.392256
$598$	0	0
$599$	− 707.560i	− 1.18123i	−0.806952	−	0.590617i	$-0.798884\pi$
$599$	− 707.560i	− 1.18123i	0.806952	−	0.590617i	$-0.201116\pi$
$600$	0	0
$601$	−279.057	−0.464321	−0.232160	−	0.972678i	$-0.574579\pi$
$601$	−279.057	−0.464321	−0.232160	+	0.972678i	$0.574579\pi$
$602$	0	0
$603$	32.9600i	0.0546600i
$604$	0	0
$605$	553.260	0.914480
$606$	0	0
$607$	− 795.724i	− 1.31091i	−0.755233	−	0.655456i	$-0.772476\pi$
$607$	− 795.724i	− 1.31091i	0.755233	−	0.655456i	$-0.227524\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 1.76554i	− 0.00288959i
$612$	0	0
$613$	−557.470	−0.909413	−0.454706	−	0.890641i	$-0.650256\pi$
$613$	−557.470	−0.909413	−0.454706	+	0.890641i	$0.650256\pi$
$614$	0	0
$615$	− 221.601i	− 0.360326i
$616$	0	0
$617$	832.514	1.34929	0.674647	−	0.738141i	$-0.264296\pi$
$617$	832.514	1.34929	0.674647	+	0.738141i	$0.264296\pi$
$618$	0	0
$619$	− 593.125i	− 0.958198i	−0.877761	−	0.479099i	$-0.840963\pi$
$619$	− 593.125i	− 0.958198i	0.877761	−	0.479099i	$-0.159037\pi$
$620$	0	0
$621$	111.088	0.178886
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−509.971	−0.815953
$626$	0	0
$627$	15.1296i	0.0241301i
$628$	0	0
$629$	−36.3532	−0.0577953
$630$	0	0
$631$	− 921.620i	− 1.46057i	−0.683142	−	0.730285i	$-0.739387\pi$
$631$	− 921.620i	− 1.46057i	0.683142	−	0.730285i	$-0.260613\pi$
$632$	0	0
$633$	−174.323	−0.275392
$634$	0	0
$635$	1049.77i	1.65319i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 269.894i	− 0.422369i
$640$	0	0
$641$	79.4861	0.124003	0.0620017	−	0.998076i	$-0.480252\pi$
$641$	79.4861	0.124003	0.0620017	+	0.998076i	$0.480252\pi$
$642$	0	0
$643$	− 669.310i	− 1.04092i	−0.853887	−	0.520459i	$-0.825761\pi$
$643$	− 669.310i	− 1.04092i	0.853887	−	0.520459i	$-0.174239\pi$
$644$	0	0
$645$	−224.029	−0.347331
$646$	0	0
$647$	− 396.360i	− 0.612612i	−0.951933	−	0.306306i	$-0.900907\pi$
$647$	− 396.360i	− 0.612612i	0.951933	−	0.306306i	$-0.0990931\pi$
$648$	0	0
$649$	−55.0294	−0.0847911
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1008.04	1.54371	0.771855	−	0.635798i	$-0.219329\pi$
$653$	1008.04	1.54371	0.771855	+	0.635798i	$0.219329\pi$
$654$	0	0
$655$	151.147i	0.230759i
$656$	0	0
$657$	112.492	0.171221
$658$	0	0
$659$	− 897.287i	− 1.36159i	−0.732475	−	0.680794i	$-0.761635\pi$
$659$	− 897.287i	− 1.36159i	0.732475	−	0.680794i	$-0.238365\pi$
$660$	0	0
$661$	−797.658	−1.20674	−0.603372	−	0.797460i	$-0.706177\pi$
$661$	−797.658	−1.20674	−0.603372	+	0.797460i	$0.706177\pi$
$662$	0	0
$663$	− 3.89674i	− 0.00587743i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	331.688i	0.497284i
$668$	0	0
$669$	−173.823	−0.259826
$670$	0	0
$671$	48.7261i	0.0726172i
$672$	0	0
$673$	−961.352	−1.42846	−0.714229	−	0.699912i	$-0.753223\pi$
$673$	−961.352	−1.42846	−0.714229	+	0.699912i	$0.753223\pi$
$674$	0	0
$675$	20.6316i	0.0305654i
$676$	0	0
$677$	−881.855	−1.30259	−0.651296	−	0.758824i	$-0.725774\pi$
$677$	−881.855	−1.30259	−0.651296	+	0.758824i	$0.725774\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−657.838	−0.965988
$682$	0	0
$683$	1299.58i	1.90275i	0.308031	+	0.951376i	$0.400330\pi$
$683$	1299.58i	1.90275i	−0.308031	+	0.951376i	$0.599670\pi$
$684$	0	0
$685$	465.795	0.679992
$686$	0	0
$687$	37.9310i	0.0552126i
$688$	0	0
$689$	60.7291	0.0881410
$690$	0	0
$691$	556.141i	0.804835i	0.915456	+	0.402418i	$0.131830\pi$
$691$	556.141i	0.804835i	−0.915456	+	0.402418i	$0.868170\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	967.621i	1.39226i
$696$	0	0
$697$	29.8823	0.0428727
$698$	0	0
$699$	178.020i	0.254678i
$700$	0	0
$701$	−394.690	−0.563039	−0.281520	−	0.959555i	$-0.590838\pi$
$701$	−394.690	−0.563039	−0.281520	+	0.959555i	$0.590838\pi$
$702$	0	0
$703$	− 498.831i	− 0.709574i
$704$	0	0
$705$	6.67619	0.00946977
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−90.0303	−0.126982	−0.0634910	−	0.997982i	$-0.520223\pi$
$709$	−90.0303	−0.126982	−0.0634910	+	0.997982i	$0.520223\pi$
$710$	0	0
$711$	22.2617i	0.0313104i
$712$	0	0
$713$	−794.498	−1.11430
$714$	0	0
$715$	5.72502i	0.00800702i
$716$	0	0
$717$	−652.087	−0.909466
$718$	0	0
$719$	− 1379.78i	− 1.91903i	−0.281660	−	0.959514i	$-0.590885\pi$
$719$	− 1379.78i	− 1.91903i	0.281660	−	0.959514i	$-0.409115\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 383.093i	− 0.529866i
$724$	0	0
$725$	−61.6022	−0.0849685
$726$	0	0
$727$	− 655.250i	− 0.901306i	−0.892699	−	0.450653i	$-0.851191\pi$
$727$	− 655.250i	− 0.901306i	0.892699	−	0.450653i	$-0.148809\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 30.2097i	− 0.0413265i
$732$	0	0
$733$	354.488	0.483613	0.241806	−	0.970325i	$-0.422260\pi$
$733$	354.488	0.483613	0.241806	+	0.970325i	$0.422260\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.52987	0.00886007
$738$	0	0
$739$	1053.46i	1.42552i	0.701408	+	0.712760i	$0.252555\pi$
$739$	1053.46i	1.42552i	−0.701408	+	0.712760i	$0.747445\pi$
$740$	0	0
$741$	53.4701	0.0721594
$742$	0	0
$743$	− 1248.25i	− 1.68001i	−0.542578	−	0.840005i	$-0.682552\pi$
$743$	− 1248.25i	− 1.68001i	0.542578	−	0.840005i	$-0.317448\pi$
$744$	0	0
$745$	242.642	0.325694
$746$	0	0
$747$	− 192.105i	− 0.257168i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	573.325i	0.763415i	0.924283	+	0.381708i	$0.124664\pi$
$751$	573.325i	0.763415i	−0.924283	+	0.381708i	$0.875336\pi$
$752$	0	0
$753$	−517.456	−0.687192
$754$	0	0
$755$	− 1173.67i	− 1.55452i
$756$	0	0
$757$	86.9117	0.114811	0.0574053	−	0.998351i	$-0.481717\pi$
$757$	86.9117	0.114811	0.0574053	+	0.998351i	$0.481717\pi$
$758$	0	0
$759$	− 22.0083i	− 0.0289964i
$760$	0	0
$761$	−850.659	−1.11782	−0.558909	−	0.829229i	$-0.688780\pi$
$761$	−850.659	−1.11782	−0.558909	+	0.829229i	$0.688780\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	14.7351	0.0192615
$766$	0	0
$767$	194.482i	0.253562i
$768$	0	0
$769$	809.762	1.05301	0.526503	−	0.850173i	$-0.323503\pi$
$769$	809.762	1.05301	0.526503	+	0.850173i	$0.323503\pi$
$770$	0	0
$771$	− 596.721i	− 0.773957i
$772$	0	0
$773$	1155.03	1.49421	0.747106	−	0.664704i	$-0.231443\pi$
$773$	1155.03	1.49421	0.747106	+	0.664704i	$0.231443\pi$
$774$	0	0
$775$	− 147.556i	− 0.190395i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	410.037i	0.526363i
$780$	0	0
$781$	−53.4701	−0.0684637
$782$	0	0
$783$	− 80.6168i	− 0.102959i
$784$	0	0
$785$	187.691	0.239097
$786$	0	0
$787$	− 61.8964i	− 0.0786486i	−0.999227	−	0.0393243i	$-0.987479\pi$
$787$	− 61.8964i	− 0.0786486i	0.999227	−	0.0393243i	$-0.0125205\pi$
$788$	0	0
$789$	393.411	0.498620
$790$	0	0
$791$	0	0
$792$	0	0
$793$	172.205	0.217157
$794$	0	0
$795$	229.640i	0.288856i
$796$	0	0
$797$	−60.6750	−0.0761292	−0.0380646	−	0.999275i	$-0.512119\pi$
$797$	−60.6750	−0.0761292	−0.0380646	+	0.999275i	$0.512119\pi$
$798$	0	0
$799$	0.900267i	0.00112674i
$800$	0	0
$801$	9.38983	0.0117226
$802$	0	0
$803$	− 22.2865i	− 0.0277540i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	815.086i	1.01002i
$808$	0	0
$809$	1363.53	1.68545	0.842724	−	0.538346i	$-0.180951\pi$
$809$	1363.53	1.68545	0.842724	+	0.538346i	$0.180951\pi$
$810$	0	0
$811$	8.60927i	0.0106156i	0.999986	+	0.00530781i	$0.00168954\pi$
$811$	8.60927i	0.0106156i	−0.999986	+	0.00530781i	$0.998310\pi$
$812$	0	0
$813$	270.926	0.333242
$814$	0	0
$815$	− 1127.41i	− 1.38333i
$816$	0	0
$817$	414.530	0.507381
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−322.029	−0.392239	−0.196120	−	0.980580i	$-0.562834\pi$
$821$	−322.029	−0.392239	−0.196120	+	0.980580i	$0.562834\pi$
$822$	0	0
$823$	− 645.800i	− 0.784690i	−0.919818	−	0.392345i	$-0.871664\pi$
$823$	− 645.800i	− 0.784690i	0.919818	−	0.392345i	$-0.128336\pi$
$824$	0	0
$825$	4.08745	0.00495448
$826$	0	0
$827$	− 679.608i	− 0.821775i	−0.911686	−	0.410887i	$-0.865219\pi$
$827$	− 679.608i	− 0.821775i	0.911686	−	0.410887i	$-0.134781\pi$
$828$	0	0
$829$	245.719	0.296404	0.148202	−	0.988957i	$-0.452651\pi$
$829$	245.719	0.296404	0.148202	+	0.988957i	$0.452651\pi$
$830$	0	0
$831$	− 484.668i	− 0.583235i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	425.525i	0.509611i
$836$	0	0
$837$	193.103	0.230708
$838$	0	0
$839$	− 706.701i	− 0.842314i	−0.906988	−	0.421157i	$-0.861624\pi$
$839$	− 706.701i	− 0.842314i	0.906988	−	0.421157i	$-0.138376\pi$
$840$	0	0
$841$	−600.294	−0.713785
$842$	0	0
$843$	232.629i	0.275954i
$844$	0	0
$845$	−754.765	−0.893213
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−850.191	−1.00140
$850$	0	0
$851$	725.626i	0.852674i
$852$	0	0
$853$	272.194	0.319102	0.159551	−	0.987190i	$-0.448995\pi$
$853$	272.194	0.319102	0.159551	+	0.987190i	$0.448995\pi$
$854$	0	0
$855$	202.191i	0.236481i
$856$	0	0
$857$	1005.87	1.17371	0.586855	−	0.809692i	$-0.300366\pi$
$857$	1005.87	1.17371	0.586855	+	0.809692i	$0.300366\pi$
$858$	0	0
$859$	− 323.765i	− 0.376910i	−0.982082	−	0.188455i	$-0.939652\pi$
$859$	− 323.765i	− 0.376910i	0.982082	−	0.188455i	$-0.0603479\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	759.250i	0.879780i	0.898052	+	0.439890i	$0.144983\pi$
$863$	759.250i	0.879780i	−0.898052	+	0.439890i	$0.855017\pi$
$864$	0	0
$865$	−863.352	−0.998095
$866$	0	0
$867$	− 498.576i	− 0.575058i
$868$	0	0
$869$	4.41039	0.00507525
$870$	0	0
$871$	− 23.0775i	− 0.0264954i
$872$	0	0
$873$	−277.331	−0.317676
$874$	0	0
$875$	0	0
$876$	0	0
$877$	529.352	0.603595	0.301797	−	0.953372i	$-0.402413\pi$
$877$	529.352	0.603595	0.301797	+	0.953372i	$0.402413\pi$
$878$	0	0
$879$	− 266.833i	− 0.303564i
$880$	0	0
$881$	−80.1005	−0.0909200	−0.0454600	−	0.998966i	$-0.514475\pi$
$881$	−80.1005	−0.0909200	−0.0454600	+	0.998966i	$0.514475\pi$
$882$	0	0
$883$	− 869.964i	− 0.985237i	−0.870245	−	0.492619i	$-0.836040\pi$
$883$	− 869.964i	− 0.985237i	0.870245	−	0.492619i	$-0.163960\pi$
$884$	0	0
$885$	−735.411	−0.830973
$886$	0	0
$887$	− 639.508i	− 0.720979i	−0.932763	−	0.360490i	$-0.882610\pi$
$887$	− 639.508i	− 0.720979i	0.932763	−	0.360490i	$-0.117390\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	5.34911i	0.00600349i
$892$	0	0
$893$	−12.3532	−0.0138334
$894$	0	0
$895$	− 799.265i	− 0.893033i
$896$	0	0
$897$	−77.7805	−0.0867118
$898$	0	0
$899$	576.567i	0.641343i
$900$	0	0
$901$	−30.9664	−0.0343689
$902$	0	0
$903$	0	0
$904$	0	0
$905$	637.368	0.704274
$906$	0	0
$907$	1329.03i	1.46530i	0.680606	+	0.732650i	$0.261717\pi$
$907$	1329.03i	1.46530i	−0.680606	+	0.732650i	$0.738283\pi$
$908$	0	0
$909$	−102.978	−0.113287
$910$	0	0
$911$	397.879i	0.436750i	0.975865	+	0.218375i	$0.0700756\pi$
$911$	397.879i	0.436750i	−0.975865	+	0.218375i	$0.929924\pi$
$912$	0	0
$913$	−38.0589	−0.0416855
$914$	0	0
$915$	651.174i	0.711665i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1282.34i	1.39537i	0.716405	+	0.697684i	$0.245786\pi$
$919$	1282.34i	1.39537i	−0.716405	+	0.697684i	$0.754214\pi$
$920$	0	0
$921$	40.0143	0.0434466
$922$	0	0
$923$	188.971i	0.204736i
$924$	0	0
$925$	−134.765	−0.145692
$926$	0	0
$927$	− 287.851i	− 0.310519i
$928$	0	0
$929$	−148.336	−0.159673	−0.0798364	−	0.996808i	$-0.525440\pi$
$929$	−148.336	−0.159673	−0.0798364	+	0.996808i	$0.525440\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−630.573	−0.675855
$934$	0	0
$935$	− 2.91924i	− 0.00312218i
$936$	0	0
$937$	414.032	0.441869	0.220935	−	0.975289i	$-0.429089\pi$
$937$	414.032	0.441869	0.220935	+	0.975289i	$0.429089\pi$
$938$	0	0
$939$	− 337.236i	− 0.359144i
$940$	0	0
$941$	345.266	0.366914	0.183457	−	0.983028i	$-0.441271\pi$
$941$	345.266	0.366914	0.183457	+	0.983028i	$0.441271\pi$
$942$	0	0
$943$	− 596.462i	− 0.632515i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 966.773i	− 1.02088i	−0.859914	−	0.510440i	$-0.829483\pi$
$947$	− 966.773i	− 1.02088i	0.859914	−	0.510440i	$-0.170517\pi$
$948$	0	0
$949$	−78.7636	−0.0829965
$950$	0	0
$951$	718.342i	0.755355i
$952$	0	0
$953$	−366.764	−0.384852	−0.192426	−	0.981312i	$-0.561635\pi$
$953$	−366.764	−0.384852	−0.192426	+	0.981312i	$0.561635\pi$
$954$	0	0
$955$	− 238.284i	− 0.249513i
$956$	0	0
$957$	−15.9714	−0.0166891
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−420.060	−0.437107
$962$	0	0
$963$	515.743i	0.535559i
$964$	0	0
$965$	72.5626	0.0751944
$966$	0	0
$967$	− 563.850i	− 0.583092i	−0.956557	−	0.291546i	$-0.905830\pi$
$967$	− 563.850i	− 0.583092i	0.956557	−	0.291546i	$-0.0941697\pi$
$968$	0	0
$969$	−27.2649	−0.0281372
$970$	0	0
$971$	257.159i	0.264840i	0.991194	+	0.132420i	$0.0422747\pi$
$971$	257.159i	0.264840i	−0.991194	+	0.132420i	$0.957725\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	− 14.4456i	− 0.0148160i
$976$	0	0
$977$	−741.131	−0.758578	−0.379289	−	0.925278i	$-0.623831\pi$
$977$	−741.131	−0.758578	−0.379289	+	0.925278i	$0.623831\pi$
$978$	0	0
$979$	− 1.86027i	− 0.00190017i
$980$	0	0
$981$	290.558	0.296186
$982$	0	0
$983$	− 512.687i	− 0.521553i	−0.965399	−	0.260777i	$-0.916021\pi$
$983$	− 512.687i	− 0.521553i	0.965399	−	0.260777i	$-0.0839786\pi$
$984$	0	0
$985$	−682.877	−0.693276
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−602.997	−0.609704
$990$	0	0
$991$	− 220.633i	− 0.222637i	−0.993785	−	0.111319i	$-0.964493\pi$
$991$	− 220.633i	− 0.222637i	0.993785	−	0.111319i	$-0.0355074\pi$
$992$	0	0
$993$	−122.412	−0.123275
$994$	0	0
$995$	620.007i	0.623123i
$996$	0	0
$997$	−895.071	−0.897764	−0.448882	−	0.893591i	$-0.648178\pi$
$997$	−895.071	−0.897764	−0.448882	+	0.893591i	$0.648178\pi$
$998$	0	0
$999$	− 176.363i	− 0.176540i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.3.m.l.1471.3	yes	4
4.3	odd	2	inner	2352.3.m.l.1471.1	yes	4
7.6	odd	2		2352.3.m.d.1471.2	✓	4
28.27	even	2		2352.3.m.d.1471.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2352.3.m.d.1471.2	✓	4	7.6	odd	2
2352.3.m.d.1471.4	yes	4	28.27	even	2
2352.3.m.l.1471.1	yes	4	4.3	odd	2	inner
2352.3.m.l.1471.3	yes	4	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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2352.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +4.58579 q^{5} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} +4.58579 q^{5} -3.00000 q^{9} +0.594346i q^{11} +2.10051 q^{13} +7.94282i q^{15} -1.07107 q^{17} -14.6969i q^{19} +21.3790i q^{23} -3.97056 q^{25} -5.19615i q^{27} +15.5147 q^{29} +37.1626i q^{31} -1.02944 q^{33} +33.9411 q^{37} +3.63818i q^{39} -27.8995 q^{41} +28.2052i q^{43} -13.7574 q^{45} -0.840532i q^{47} -1.85514i q^{51} +28.9117 q^{53} +2.72554i q^{55} +25.4558 q^{57} +92.5882i q^{59} +81.9828 q^{61} +9.63247 q^{65} -10.9867i q^{67} -37.0294 q^{69} +89.9647i q^{71} -37.4975 q^{73} -6.87722i q^{75} -7.42058i q^{79} +9.00000 q^{81} +64.0349i q^{83} -4.91169 q^{85} +26.8723i q^{87} -3.12994 q^{89} -64.3675 q^{93} -67.3970i q^{95} +92.4437 q^{97} -1.78304i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 24 q^{5} - 12 q^{9}+O(q^{10}) \) 4 * q + 24 * q^5 - 12 * q^9	\( 4 q + 24 q^{5} - 12 q^{9} + 48 q^{13} + 24 q^{17} + 52 q^{25} + 96 q^{29} - 72 q^{33} - 72 q^{41} - 72 q^{45} - 88 q^{53} + 96 q^{61} + 344 q^{65} - 216 q^{69} + 48 q^{73} + 36 q^{81} + 184 q^{85} - 120 q^{89} + 48 q^{93} + 432 q^{97}+O(q^{100}) \) 4 * q + 24 * q^5 - 12 * q^9 + 48 * q^13 + 24 * q^17 + 52 * q^25 + 96 * q^29 - 72 * q^33 - 72 * q^41 - 72 * q^45 - 88 * q^53 + 96 * q^61 + 344 * q^65 - 216 * q^69 + 48 * q^73 + 36 * q^81 + 184 * q^85 - 120 * q^89 + 48 * q^93 + 432 * q^97

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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