LMFDB - Embedded newform 1856.2.a.z.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1856,2,Mod(1,1856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1856, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1856.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1856 = 2^{6} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1856.a (trivial)

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:	yes
Analytic conductor:	$14.8202346151$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 928)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.90211$ of defining polynomial
Character	$\chi$	$=$	1856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.726543 q^{3} -0.236068 q^{5} +3.80423 q^{7} -2.47214 q^{9} +O(q^{10})$	$q+0.726543 q^{3} -0.236068 q^{5} +3.80423 q^{7} -2.47214 q^{9} +5.42882 q^{11} +6.23607 q^{13} -0.171513 q^{15} -0.763932 q^{17} -1.45309 q^{19} +2.76393 q^{21} +2.35114 q^{23} -4.94427 q^{25} -3.97574 q^{27} -1.00000 q^{29} -3.07768 q^{31} +3.94427 q^{33} -0.898056 q^{35} +6.47214 q^{37} +4.53077 q^{39} -2.76393 q^{41} -11.5842 q^{43} +0.583592 q^{45} -4.53077 q^{47} +7.47214 q^{49} -0.555029 q^{51} +8.23607 q^{53} -1.28157 q^{55} -1.05573 q^{57} +14.6619 q^{59} +7.23607 q^{61} -9.40456 q^{63} -1.47214 q^{65} +2.90617 q^{67} +1.70820 q^{69} -13.7638 q^{71} +12.9443 q^{73} -3.59222 q^{75} +20.6525 q^{77} +9.23305 q^{79} +4.52786 q^{81} -0.555029 q^{83} +0.180340 q^{85} -0.726543 q^{87} -11.7082 q^{89} +23.7234 q^{91} -2.23607 q^{93} +0.343027 q^{95} -3.70820 q^{97} -13.4208 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{5} + 8 q^{9}+O(q^{10}) $ 4 * q + 8 * q^5 + 8 * q^9	$ 4 q + 8 q^{5} + 8 q^{9} + 16 q^{13} - 12 q^{17} + 20 q^{21} + 16 q^{25} - 4 q^{29} - 20 q^{33} + 8 q^{37} - 20 q^{41} + 56 q^{45} + 12 q^{49} + 24 q^{53} - 40 q^{57} + 20 q^{61} + 12 q^{65} - 20 q^{69} + 16 q^{73} + 20 q^{77} + 36 q^{81} - 44 q^{85} - 20 q^{89} + 12 q^{97}+O(q^{100}) $ 4 * q + 8 * q^5 + 8 * q^9 + 16 * q^13 - 12 * q^17 + 20 * q^21 + 16 * q^25 - 4 * q^29 - 20 * q^33 + 8 * q^37 - 20 * q^41 + 56 * q^45 + 12 * q^49 + 24 * q^53 - 40 * q^57 + 20 * q^61 + 12 * q^65 - 20 * q^69 + 16 * q^73 + 20 * q^77 + 36 * q^81 - 44 * q^85 - 20 * q^89 + 12 * q^97

Display 100 coefficients 10 coefficients

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
$3$	0.726543	0.419470	0.209735	−	0.977758i	$-0.432740\pi$
$3$	0.726543	0.419470	0.209735	+	0.977758i	$0.432740\pi$
$4$	0	0
$5$	−0.236068	−0.105573	−0.0527864	−	0.998606i	$-0.516810\pi$
$5$	−0.236068	−0.105573	−0.0527864	+	0.998606i	$0.516810\pi$
$6$	0	0
$7$	3.80423	1.43786	0.718931	−	0.695081i	$-0.244632\pi$
$7$	3.80423	1.43786	0.718931	+	0.695081i	$0.244632\pi$
$8$	0	0
$9$	−2.47214	−0.824045
$10$	0	0
$11$	5.42882	1.63685	0.818426	−	0.574612i	$-0.194847\pi$
$11$	5.42882	1.63685	0.818426	+	0.574612i	$0.194847\pi$
$12$	0	0
$13$	6.23607	1.72957	0.864787	−	0.502139i	$-0.167453\pi$
$13$	6.23607	1.72957	0.864787	+	0.502139i	$0.167453\pi$
$14$	0	0
$15$	−0.171513	−0.0442846
$16$	0	0
$17$	−0.763932	−0.185281	−0.0926404	−	0.995700i	$-0.529531\pi$
$17$	−0.763932	−0.185281	−0.0926404	+	0.995700i	$0.529531\pi$
$18$	0	0
$19$	−1.45309	−0.333361	−0.166680	−	0.986011i	$-0.553305\pi$
$19$	−1.45309	−0.333361	−0.166680	+	0.986011i	$0.553305\pi$
$20$	0	0
$21$	2.76393	0.603139
$22$	0	0
$23$	2.35114	0.490247	0.245123	−	0.969492i	$-0.421172\pi$
$23$	2.35114	0.490247	0.245123	+	0.969492i	$0.421172\pi$
$24$	0	0
$25$	−4.94427	−0.988854
$26$	0	0
$27$	−3.97574	−0.765131
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−3.07768	−0.552768	−0.276384	−	0.961047i	$-0.589136\pi$
$31$	−3.07768	−0.552768	−0.276384	+	0.961047i	$0.589136\pi$
$32$	0	0
$33$	3.94427	0.686610
$34$	0	0
$35$	−0.898056	−0.151799
$36$	0	0
$37$	6.47214	1.06401	0.532006	−	0.846740i	$-0.321438\pi$
$37$	6.47214	1.06401	0.532006	+	0.846740i	$0.321438\pi$
$38$	0	0
$39$	4.53077	0.725504
$40$	0	0
$41$	−2.76393	−0.431654	−0.215827	−	0.976432i	$-0.569245\pi$
$41$	−2.76393	−0.431654	−0.215827	+	0.976432i	$0.569245\pi$
$42$	0	0
$43$	−11.5842	−1.76657	−0.883286	−	0.468834i	$-0.844674\pi$
$43$	−11.5842	−1.76657	−0.883286	+	0.468834i	$0.844674\pi$
$44$	0	0
$45$	0.583592	0.0869968
$46$	0	0
$47$	−4.53077	−0.660881	−0.330440	−	0.943827i	$-0.607197\pi$
$47$	−4.53077	−0.660881	−0.330440	+	0.943827i	$0.607197\pi$
$48$	0	0
$49$	7.47214	1.06745
$50$	0	0
$51$	−0.555029	−0.0777196
$52$	0	0
$53$	8.23607	1.13131	0.565655	−	0.824642i	$-0.308623\pi$
$53$	8.23607	1.13131	0.565655	+	0.824642i	$0.308623\pi$
$54$	0	0
$55$	−1.28157	−0.172807
$56$	0	0
$57$	−1.05573	−0.139835
$58$	0	0
$59$	14.6619	1.90881	0.954407	−	0.298509i	$-0.0964893\pi$
$59$	14.6619	1.90881	0.954407	+	0.298509i	$0.0964893\pi$
$60$	0	0
$61$	7.23607	0.926484	0.463242	−	0.886232i	$-0.346686\pi$
$61$	7.23607	0.926484	0.463242	+	0.886232i	$0.346686\pi$
$62$	0	0
$63$	−9.40456	−1.18486
$64$	0	0
$65$	−1.47214	−0.182596
$66$	0	0
$67$	2.90617	0.355045	0.177523	−	0.984117i	$-0.443192\pi$
$67$	2.90617	0.355045	0.177523	+	0.984117i	$0.443192\pi$
$68$	0	0
$69$	1.70820	0.205644
$70$	0	0
$71$	−13.7638	−1.63346	−0.816732	−	0.577017i	$-0.804217\pi$
$71$	−13.7638	−1.63346	−0.816732	+	0.577017i	$0.804217\pi$
$72$	0	0
$73$	12.9443	1.51501	0.757506	−	0.652828i	$-0.226418\pi$
$73$	12.9443	1.51501	0.757506	+	0.652828i	$0.226418\pi$
$74$	0	0
$75$	−3.59222	−0.414794
$76$	0	0
$77$	20.6525	2.35357
$78$	0	0
$79$	9.23305	1.03880	0.519400	−	0.854531i	$-0.326156\pi$
$79$	9.23305	1.03880	0.519400	+	0.854531i	$0.326156\pi$
$80$	0	0
$81$	4.52786	0.503096
$82$	0	0
$83$	−0.555029	−0.0609224	−0.0304612	−	0.999536i	$-0.509698\pi$
$83$	−0.555029	−0.0609224	−0.0304612	+	0.999536i	$0.509698\pi$
$84$	0	0
$85$	0.180340	0.0195606
$86$	0	0
$87$	−0.726543	−0.0778935
$88$	0	0
$89$	−11.7082	−1.24107	−0.620534	−	0.784180i	$-0.713084\pi$
$89$	−11.7082	−1.24107	−0.620534	+	0.784180i	$0.713084\pi$
$90$	0	0
$91$	23.7234	2.48689
$92$	0	0
$93$	−2.23607	−0.231869
$94$	0	0
$95$	0.343027	0.0351938
$96$	0	0
$97$	−3.70820	−0.376511	−0.188256	−	0.982120i	$-0.560283\pi$
$97$	−3.70820	−0.376511	−0.188256	+	0.982120i	$0.560283\pi$
$98$	0	0
$99$	−13.4208	−1.34884
$100$	0	0
$101$	−11.4164	−1.13598	−0.567988	−	0.823037i	$-0.692278\pi$
$101$	−11.4164	−1.13598	−0.567988	+	0.823037i	$0.692278\pi$
$102$	0	0
$103$	13.7638	1.35619	0.678095	−	0.734975i	$-0.262806\pi$
$103$	13.7638	1.35619	0.678095	+	0.734975i	$0.262806\pi$
$104$	0	0
$105$	−0.652476	−0.0636751
$106$	0	0
$107$	−3.80423	−0.367768	−0.183884	−	0.982948i	$-0.558867\pi$
$107$	−3.80423	−0.367768	−0.183884	+	0.982948i	$0.558867\pi$
$108$	0	0
$109$	6.23607	0.597307	0.298653	−	0.954362i	$-0.403463\pi$
$109$	6.23607	0.597307	0.298653	+	0.954362i	$0.403463\pi$
$110$	0	0
$111$	4.70228	0.446321
$112$	0	0
$113$	14.9443	1.40584	0.702919	−	0.711269i	$-0.251879\pi$
$113$	14.9443	1.40584	0.702919	+	0.711269i	$0.251879\pi$
$114$	0	0
$115$	−0.555029	−0.0517567
$116$	0	0
$117$	−15.4164	−1.42525
$118$	0	0
$119$	−2.90617	−0.266408
$120$	0	0
$121$	18.4721	1.67929
$122$	0	0
$123$	−2.00811	−0.181066
$124$	0	0
$125$	2.34752	0.209969
$126$	0	0
$127$	1.45309	0.128940	0.0644702	−	0.997920i	$-0.479464\pi$
$127$	1.45309	0.128940	0.0644702	+	0.997920i	$0.479464\pi$
$128$	0	0
$129$	−8.41641	−0.741023
$130$	0	0
$131$	−18.4661	−1.61339	−0.806695	−	0.590967i	$-0.798746\pi$
$131$	−18.4661	−1.61339	−0.806695	+	0.590967i	$0.798746\pi$
$132$	0	0
$133$	−5.52786	−0.479327
$134$	0	0
$135$	0.938545	0.0807771
$136$	0	0
$137$	−15.4164	−1.31711	−0.658556	−	0.752531i	$-0.728833\pi$
$137$	−15.4164	−1.31711	−0.658556	+	0.752531i	$0.728833\pi$
$138$	0	0
$139$	−9.95959	−0.844762	−0.422381	−	0.906418i	$-0.638806\pi$
$139$	−9.95959	−0.844762	−0.422381	+	0.906418i	$0.638806\pi$
$140$	0	0
$141$	−3.29180	−0.277219
$142$	0	0
$143$	33.8545	2.83106
$144$	0	0
$145$	0.236068	0.0196044
$146$	0	0
$147$	5.42882	0.447762
$148$	0	0
$149$	3.76393	0.308353	0.154177	−	0.988043i	$-0.450728\pi$
$149$	3.76393	0.308353	0.154177	+	0.988043i	$0.450728\pi$
$150$	0	0
$151$	−21.9273	−1.78442	−0.892209	−	0.451622i	$-0.850845\pi$
$151$	−21.9273	−1.78442	−0.892209	+	0.451622i	$0.850845\pi$
$152$	0	0
$153$	1.88854	0.152680
$154$	0	0
$155$	0.726543	0.0583573
$156$	0	0
$157$	−3.70820	−0.295947	−0.147973	−	0.988991i	$-0.547275\pi$
$157$	−3.70820	−0.295947	−0.147973	+	0.988991i	$0.547275\pi$
$158$	0	0
$159$	5.98385	0.474550
$160$	0	0
$161$	8.94427	0.704907
$162$	0	0
$163$	19.1926	1.50328	0.751642	−	0.659571i	$-0.229262\pi$
$163$	19.1926	1.50328	0.751642	+	0.659571i	$0.229262\pi$
$164$	0	0
$165$	−0.931116	−0.0724873
$166$	0	0
$167$	−9.06154	−0.701203	−0.350601	−	0.936525i	$-0.614023\pi$
$167$	−9.06154	−0.701203	−0.350601	+	0.936525i	$0.614023\pi$
$168$	0	0
$169$	25.8885	1.99143
$170$	0	0
$171$	3.59222	0.274704
$172$	0	0
$173$	11.5279	0.876447	0.438224	−	0.898866i	$-0.355608\pi$
$173$	11.5279	0.876447	0.438224	+	0.898866i	$0.355608\pi$
$174$	0	0
$175$	−18.8091	−1.42184
$176$	0	0
$177$	10.6525	0.800689
$178$	0	0
$179$	−7.95148	−0.594321	−0.297161	−	0.954827i	$-0.596040\pi$
$179$	−7.95148	−0.594321	−0.297161	+	0.954827i	$0.596040\pi$
$180$	0	0
$181$	−3.18034	−0.236393	−0.118196	−	0.992990i	$-0.537711\pi$
$181$	−3.18034	−0.236393	−0.118196	+	0.992990i	$0.537711\pi$
$182$	0	0
$183$	5.25731	0.388632
$184$	0	0
$185$	−1.52786	−0.112331
$186$	0	0
$187$	−4.14725	−0.303277
$188$	0	0
$189$	−15.1246	−1.10015
$190$	0	0
$191$	−20.2622	−1.46612	−0.733061	−	0.680163i	$-0.761909\pi$
$191$	−20.2622	−1.46612	−0.733061	+	0.680163i	$0.761909\pi$
$192$	0	0
$193$	13.2361	0.952753	0.476377	−	0.879241i	$-0.341950\pi$
$193$	13.2361	0.952753	0.476377	+	0.879241i	$0.341950\pi$
$194$	0	0
$195$	−1.06957	−0.0765935
$196$	0	0
$197$	−6.94427	−0.494759	−0.247379	−	0.968919i	$-0.579569\pi$
$197$	−6.94427	−0.494759	−0.247379	+	0.968919i	$0.579569\pi$
$198$	0	0
$199$	−14.3188	−1.01504	−0.507518	−	0.861641i	$-0.669437\pi$
$199$	−14.3188	−1.01504	−0.507518	+	0.861641i	$0.669437\pi$
$200$	0	0
$201$	2.11146	0.148931
$202$	0	0
$203$	−3.80423	−0.267004
$204$	0	0
$205$	0.652476	0.0455709
$206$	0	0
$207$	−5.81234	−0.403986
$208$	0	0
$209$	−7.88854	−0.545662
$210$	0	0
$211$	−5.42882	−0.373736	−0.186868	−	0.982385i	$-0.559834\pi$
$211$	−5.42882	−0.373736	−0.186868	+	0.982385i	$0.559834\pi$
$212$	0	0
$213$	−10.0000	−0.685189
$214$	0	0
$215$	2.73466	0.186502
$216$	0	0
$217$	−11.7082	−0.794805
$218$	0	0
$219$	9.40456	0.635502
$220$	0	0
$221$	−4.76393	−0.320457
$222$	0	0
$223$	19.5762	1.31092	0.655458	−	0.755231i	$-0.272476\pi$
$223$	19.5762	1.31092	0.655458	+	0.755231i	$0.272476\pi$
$224$	0	0
$225$	12.2229	0.814861
$226$	0	0
$227$	9.06154	0.601435	0.300718	−	0.953713i	$-0.402774\pi$
$227$	9.06154	0.601435	0.300718	+	0.953713i	$0.402774\pi$
$228$	0	0
$229$	−8.29180	−0.547937	−0.273969	−	0.961739i	$-0.588336\pi$
$229$	−8.29180	−0.547937	−0.273969	+	0.961739i	$0.588336\pi$
$230$	0	0
$231$	15.0049	0.987250
$232$	0	0
$233$	−0.0557281	−0.00365087	−0.00182543	−	0.999998i	$-0.500581\pi$
$233$	−0.0557281	−0.00365087	−0.00182543	+	0.999998i	$0.500581\pi$
$234$	0	0
$235$	1.06957	0.0697710
$236$	0	0
$237$	6.70820	0.435745
$238$	0	0
$239$	9.95959	0.644233	0.322116	−	0.946700i	$-0.395606\pi$
$239$	9.95959	0.644233	0.322116	+	0.946700i	$0.395606\pi$
$240$	0	0
$241$	13.9443	0.898230	0.449115	−	0.893474i	$-0.351739\pi$
$241$	13.9443	0.898230	0.449115	+	0.893474i	$0.351739\pi$
$242$	0	0
$243$	15.2169	0.976165
$244$	0	0
$245$	−1.76393	−0.112693
$246$	0	0
$247$	−9.06154	−0.576572
$248$	0	0
$249$	−0.403252	−0.0255551
$250$	0	0
$251$	−23.8949	−1.50823	−0.754117	−	0.656740i	$-0.771935\pi$
$251$	−23.8949	−1.50823	−0.754117	+	0.656740i	$0.771935\pi$
$252$	0	0
$253$	12.7639	0.802462
$254$	0	0
$255$	0.131025	0.00820508
$256$	0	0
$257$	−7.47214	−0.466099	−0.233050	−	0.972465i	$-0.574870\pi$
$257$	−7.47214	−0.466099	−0.233050	+	0.972465i	$0.574870\pi$
$258$	0	0
$259$	24.6215	1.52990
$260$	0	0
$261$	2.47214	0.153021
$262$	0	0
$263$	−3.07768	−0.189778	−0.0948890	−	0.995488i	$-0.530250\pi$
$263$	−3.07768	−0.189778	−0.0948890	+	0.995488i	$0.530250\pi$
$264$	0	0
$265$	−1.94427	−0.119436
$266$	0	0
$267$	−8.50651	−0.520590
$268$	0	0
$269$	−5.70820	−0.348035	−0.174018	−	0.984743i	$-0.555675\pi$
$269$	−5.70820	−0.348035	−0.174018	+	0.984743i	$0.555675\pi$
$270$	0	0
$271$	25.9030	1.57350	0.786749	−	0.617273i	$-0.211763\pi$
$271$	25.9030	1.57350	0.786749	+	0.617273i	$0.211763\pi$
$272$	0	0
$273$	17.2361	1.04317
$274$	0	0
$275$	−26.8416	−1.61861
$276$	0	0
$277$	−19.8885	−1.19499	−0.597493	−	0.801874i	$-0.703837\pi$
$277$	−19.8885	−1.19499	−0.597493	+	0.801874i	$0.703837\pi$
$278$	0	0
$279$	7.60845	0.455506
$280$	0	0
$281$	−19.4721	−1.16161	−0.580805	−	0.814043i	$-0.697262\pi$
$281$	−19.4721	−1.16161	−0.580805	+	0.814043i	$0.697262\pi$
$282$	0	0
$283$	−9.95959	−0.592036	−0.296018	−	0.955182i	$-0.595659\pi$
$283$	−9.95959	−0.592036	−0.296018	+	0.955182i	$0.595659\pi$
$284$	0	0
$285$	0.249224	0.0147627
$286$	0	0
$287$	−10.5146	−0.620659
$288$	0	0
$289$	−16.4164	−0.965671
$290$	0	0
$291$	−2.69417	−0.157935
$292$	0	0
$293$	17.4164	1.01748	0.508739	−	0.860921i	$-0.330112\pi$
$293$	17.4164	1.01748	0.508739	+	0.860921i	$0.330112\pi$
$294$	0	0
$295$	−3.46120	−0.201519
$296$	0	0
$297$	−21.5836	−1.25241
$298$	0	0
$299$	14.6619	0.847918
$300$	0	0
$301$	−44.0689	−2.54009
$302$	0	0
$303$	−8.29451	−0.476507
$304$	0	0
$305$	−1.70820	−0.0978115
$306$	0	0
$307$	6.53888	0.373194	0.186597	−	0.982437i	$-0.440254\pi$
$307$	6.53888	0.373194	0.186597	+	0.982437i	$0.440254\pi$
$308$	0	0
$309$	10.0000	0.568880
$310$	0	0
$311$	−16.6700	−0.945268	−0.472634	−	0.881259i	$-0.656697\pi$
$311$	−16.6700	−0.945268	−0.472634	+	0.881259i	$0.656697\pi$
$312$	0	0
$313$	−12.4164	−0.701817	−0.350908	−	0.936410i	$-0.614127\pi$
$313$	−12.4164	−0.701817	−0.350908	+	0.936410i	$0.614127\pi$
$314$	0	0
$315$	2.22012	0.125089
$316$	0	0
$317$	−13.7082	−0.769929	−0.384965	−	0.922931i	$-0.625786\pi$
$317$	−13.7082	−0.769929	−0.384965	+	0.922931i	$0.625786\pi$
$318$	0	0
$319$	−5.42882	−0.303956
$320$	0	0
$321$	−2.76393	−0.154268
$322$	0	0
$323$	1.11006	0.0617653
$324$	0	0
$325$	−30.8328	−1.71030
$326$	0	0
$327$	4.53077	0.250552
$328$	0	0
$329$	−17.2361	−0.950255
$330$	0	0
$331$	−13.0373	−0.716594	−0.358297	−	0.933608i	$-0.616642\pi$
$331$	−13.0373	−0.716594	−0.358297	+	0.933608i	$0.616642\pi$
$332$	0	0
$333$	−16.0000	−0.876795
$334$	0	0
$335$	−0.686054	−0.0374831
$336$	0	0
$337$	13.1246	0.714943	0.357472	−	0.933924i	$-0.383639\pi$
$337$	13.1246	0.714943	0.357472	+	0.933924i	$0.383639\pi$
$338$	0	0
$339$	10.8576	0.589707
$340$	0	0
$341$	−16.7082	−0.904800
$342$	0	0
$343$	1.79611	0.0969809
$344$	0	0
$345$	−0.403252	−0.0217104
$346$	0	0
$347$	4.35926	0.234017	0.117009	−	0.993131i	$-0.462670\pi$
$347$	4.35926	0.234017	0.117009	+	0.993131i	$0.462670\pi$
$348$	0	0
$349$	35.6525	1.90843	0.954217	−	0.299116i	$-0.0966919\pi$
$349$	35.6525	1.90843	0.954217	+	0.299116i	$0.0966919\pi$
$350$	0	0
$351$	−24.7930	−1.32335
$352$	0	0
$353$	−14.9443	−0.795403	−0.397702	−	0.917515i	$-0.630192\pi$
$353$	−14.9443	−0.795403	−0.397702	+	0.917515i	$0.630192\pi$
$354$	0	0
$355$	3.24920	0.172449
$356$	0	0
$357$	−2.11146	−0.111750
$358$	0	0
$359$	29.1522	1.53860	0.769298	−	0.638890i	$-0.220606\pi$
$359$	29.1522	1.53860	0.769298	+	0.638890i	$0.220606\pi$
$360$	0	0
$361$	−16.8885	−0.888871
$362$	0	0
$363$	13.4208	0.704409
$364$	0	0
$365$	−3.05573	−0.159944
$366$	0	0
$367$	9.06154	0.473008	0.236504	−	0.971630i	$-0.423998\pi$
$367$	9.06154	0.473008	0.236504	+	0.971630i	$0.423998\pi$
$368$	0	0
$369$	6.83282	0.355702
$370$	0	0
$371$	31.3319	1.62667
$372$	0	0
$373$	−6.23607	−0.322891	−0.161446	−	0.986882i	$-0.551616\pi$
$373$	−6.23607	−0.322891	−0.161446	+	0.986882i	$0.551616\pi$
$374$	0	0
$375$	1.70558	0.0880756
$376$	0	0
$377$	−6.23607	−0.321174
$378$	0	0
$379$	−19.5762	−1.00556	−0.502780	−	0.864414i	$-0.667689\pi$
$379$	−19.5762	−1.00556	−0.502780	+	0.864414i	$0.667689\pi$
$380$	0	0
$381$	1.05573	0.0540866
$382$	0	0
$383$	−30.2218	−1.54426	−0.772131	−	0.635463i	$-0.780809\pi$
$383$	−30.2218	−1.54426	−0.772131	+	0.635463i	$0.780809\pi$
$384$	0	0
$385$	−4.87539	−0.248473
$386$	0	0
$387$	28.6377	1.45574
$388$	0	0
$389$	−16.0000	−0.811232	−0.405616	−	0.914044i	$-0.632943\pi$
$389$	−16.0000	−0.811232	−0.405616	+	0.914044i	$0.632943\pi$
$390$	0	0
$391$	−1.79611	−0.0908333
$392$	0	0
$393$	−13.4164	−0.676768
$394$	0	0
$395$	−2.17963	−0.109669
$396$	0	0
$397$	24.2361	1.21637	0.608187	−	0.793794i	$-0.291897\pi$
$397$	24.2361	1.21637	0.608187	+	0.793794i	$0.291897\pi$
$398$	0	0
$399$	−4.01623	−0.201063
$400$	0	0
$401$	27.0000	1.34832	0.674158	−	0.738587i	$-0.264507\pi$
$401$	27.0000	1.34832	0.674158	+	0.738587i	$0.264507\pi$
$402$	0	0
$403$	−19.1926	−0.956054
$404$	0	0
$405$	−1.06888	−0.0531133
$406$	0	0
$407$	35.1361	1.74163
$408$	0	0
$409$	−23.8885	−1.18121	−0.590606	−	0.806960i	$-0.701111\pi$
$409$	−23.8885	−1.18121	−0.590606	+	0.806960i	$0.701111\pi$
$410$	0	0
$411$	−11.2007	−0.552489
$412$	0	0
$413$	55.7771	2.74461
$414$	0	0
$415$	0.131025	0.00643174
$416$	0	0
$417$	−7.23607	−0.354352
$418$	0	0
$419$	−14.8739	−0.726636	−0.363318	−	0.931665i	$-0.618356\pi$
$419$	−14.8739	−0.726636	−0.363318	+	0.931665i	$0.618356\pi$
$420$	0	0
$421$	21.7082	1.05799	0.528997	−	0.848624i	$-0.322568\pi$
$421$	21.7082	1.05799	0.528997	+	0.848624i	$0.322568\pi$
$422$	0	0
$423$	11.2007	0.544596
$424$	0	0
$425$	3.77709	0.183216
$426$	0	0
$427$	27.5276	1.33216
$428$	0	0
$429$	24.5967	1.18754
$430$	0	0
$431$	17.5680	0.846223	0.423111	−	0.906078i	$-0.360938\pi$
$431$	17.5680	0.846223	0.423111	+	0.906078i	$0.360938\pi$
$432$	0	0
$433$	−13.8885	−0.667441	−0.333720	−	0.942672i	$-0.608304\pi$
$433$	−13.8885	−0.667441	−0.333720	+	0.942672i	$0.608304\pi$
$434$	0	0
$435$	0.171513	0.00822344
$436$	0	0
$437$	−3.41641	−0.163429
$438$	0	0
$439$	12.8658	0.614049	0.307025	−	0.951702i	$-0.400667\pi$
$439$	12.8658	0.614049	0.307025	+	0.951702i	$0.400667\pi$
$440$	0	0
$441$	−18.4721	−0.879626
$442$	0	0
$443$	−11.9677	−0.568603	−0.284301	−	0.958735i	$-0.591762\pi$
$443$	−11.9677	−0.568603	−0.284301	+	0.958735i	$0.591762\pi$
$444$	0	0
$445$	2.76393	0.131023
$446$	0	0
$447$	2.73466	0.129345
$448$	0	0
$449$	−10.5836	−0.499471	−0.249735	−	0.968314i	$-0.580344\pi$
$449$	−10.5836	−0.499471	−0.249735	+	0.968314i	$0.580344\pi$
$450$	0	0
$451$	−15.0049	−0.706553
$452$	0	0
$453$	−15.9311	−0.748509
$454$	0	0
$455$	−5.60034	−0.262548
$456$	0	0
$457$	28.8328	1.34874	0.674371	−	0.738393i	$-0.264415\pi$
$457$	28.8328	1.34874	0.674371	+	0.738393i	$0.264415\pi$
$458$	0	0
$459$	3.03719	0.141764
$460$	0	0
$461$	−0.111456	−0.00519103	−0.00259552	−	0.999997i	$-0.500826\pi$
$461$	−0.111456	−0.00519103	−0.00259552	+	0.999997i	$0.500826\pi$
$462$	0	0
$463$	−8.16348	−0.379389	−0.189695	−	0.981843i	$-0.560750\pi$
$463$	−8.16348	−0.379389	−0.189695	+	0.981843i	$0.560750\pi$
$464$	0	0
$465$	0.527864	0.0244791
$466$	0	0
$467$	−2.17963	−0.100861	−0.0504306	−	0.998728i	$-0.516059\pi$
$467$	−2.17963	−0.100861	−0.0504306	+	0.998728i	$0.516059\pi$
$468$	0	0
$469$	11.0557	0.510506
$470$	0	0
$471$	−2.69417	−0.124141
$472$	0	0
$473$	−62.8885	−2.89162
$474$	0	0
$475$	7.18445	0.329645
$476$	0	0
$477$	−20.3607	−0.932251
$478$	0	0
$479$	6.66991	0.304756	0.152378	−	0.988322i	$-0.451307\pi$
$479$	6.66991	0.304756	0.152378	+	0.988322i	$0.451307\pi$
$480$	0	0
$481$	40.3607	1.84029
$482$	0	0
$483$	6.49839	0.295687
$484$	0	0
$485$	0.875388	0.0397493
$486$	0	0
$487$	−25.5195	−1.15640	−0.578200	−	0.815895i	$-0.696245\pi$
$487$	−25.5195	−1.15640	−0.578200	+	0.815895i	$0.696245\pi$
$488$	0	0
$489$	13.9443	0.630582
$490$	0	0
$491$	14.4904	0.653941	0.326970	−	0.945035i	$-0.393972\pi$
$491$	14.4904	0.653941	0.326970	+	0.945035i	$0.393972\pi$
$492$	0	0
$493$	0.763932	0.0344058
$494$	0	0
$495$	3.16822	0.142401
$496$	0	0
$497$	−52.3607	−2.34870
$498$	0	0
$499$	−28.0827	−1.25715	−0.628576	−	0.777748i	$-0.716362\pi$
$499$	−28.0827	−1.25715	−0.628576	+	0.777748i	$0.716362\pi$
$500$	0	0
$501$	−6.58359	−0.294133
$502$	0	0
$503$	−21.5438	−0.960590	−0.480295	−	0.877107i	$-0.659470\pi$
$503$	−21.5438	−0.960590	−0.480295	+	0.877107i	$0.659470\pi$
$504$	0	0
$505$	2.69505	0.119928
$506$	0	0
$507$	18.8091	0.835343
$508$	0	0
$509$	21.1803	0.938802	0.469401	−	0.882985i	$-0.344470\pi$
$509$	21.1803	0.938802	0.469401	+	0.882985i	$0.344470\pi$
$510$	0	0
$511$	49.2429	2.17838
$512$	0	0
$513$	5.77709	0.255065
$514$	0	0
$515$	−3.24920	−0.143177
$516$	0	0
$517$	−24.5967	−1.08176
$518$	0	0
$519$	8.37548	0.367643
$520$	0	0
$521$	−9.11146	−0.399180	−0.199590	−	0.979879i	$-0.563961\pi$
$521$	−9.11146	−0.399180	−0.199590	+	0.979879i	$0.563961\pi$
$522$	0	0
$523$	−26.4176	−1.15516	−0.577580	−	0.816334i	$-0.696003\pi$
$523$	−26.4176	−1.15516	−0.577580	+	0.816334i	$0.696003\pi$
$524$	0	0
$525$	−13.6656	−0.596417
$526$	0	0
$527$	2.35114	0.102417
$528$	0	0
$529$	−17.4721	−0.759658
$530$	0	0
$531$	−36.2461	−1.57295
$532$	0	0
$533$	−17.2361	−0.746577
$534$	0	0
$535$	0.898056	0.0388263
$536$	0	0
$537$	−5.77709	−0.249300
$538$	0	0
$539$	40.5649	1.74725
$540$	0	0
$541$	4.36068	0.187480	0.0937401	−	0.995597i	$-0.470118\pi$
$541$	4.36068	0.187480	0.0937401	+	0.995597i	$0.470118\pi$
$542$	0	0
$543$	−2.31065	−0.0991596
$544$	0	0
$545$	−1.47214	−0.0630594
$546$	0	0
$547$	−2.00811	−0.0858608	−0.0429304	−	0.999078i	$-0.513669\pi$
$547$	−2.00811	−0.0858608	−0.0429304	+	0.999078i	$0.513669\pi$
$548$	0	0
$549$	−17.8885	−0.763464
$550$	0	0
$551$	1.45309	0.0619035
$552$	0	0
$553$	35.1246	1.49365
$554$	0	0
$555$	−1.11006	−0.0471193
$556$	0	0
$557$	14.3607	0.608482	0.304241	−	0.952595i	$-0.401597\pi$
$557$	14.3607	0.608482	0.304241	+	0.952595i	$0.401597\pi$
$558$	0	0
$559$	−72.2398	−3.05542
$560$	0	0
$561$	−3.01316	−0.127216
$562$	0	0
$563$	−0.726543	−0.0306201	−0.0153101	−	0.999883i	$-0.504874\pi$
$563$	−0.726543	−0.0306201	−0.0153101	+	0.999883i	$0.504874\pi$
$564$	0	0
$565$	−3.52786	−0.148418
$566$	0	0
$567$	17.2250	0.723383
$568$	0	0
$569$	−14.0000	−0.586911	−0.293455	−	0.955973i	$-0.594805\pi$
$569$	−14.0000	−0.586911	−0.293455	+	0.955973i	$0.594805\pi$
$570$	0	0
$571$	−38.7283	−1.62073	−0.810365	−	0.585926i	$-0.800731\pi$
$571$	−38.7283	−1.62073	−0.810365	+	0.585926i	$0.800731\pi$
$572$	0	0
$573$	−14.7214	−0.614994
$574$	0	0
$575$	−11.6247	−0.484783
$576$	0	0
$577$	−33.1246	−1.37900	−0.689498	−	0.724288i	$-0.742169\pi$
$577$	−33.1246	−1.37900	−0.689498	+	0.724288i	$0.742169\pi$
$578$	0	0
$579$	9.61657	0.399651
$580$	0	0
$581$	−2.11146	−0.0875980
$582$	0	0
$583$	44.7122	1.85179
$584$	0	0
$585$	3.63932	0.150467
$586$	0	0
$587$	16.4580	0.679294	0.339647	−	0.940553i	$-0.389692\pi$
$587$	16.4580	0.679294	0.339647	+	0.940553i	$0.389692\pi$
$588$	0	0
$589$	4.47214	0.184271
$590$	0	0
$591$	−5.04531	−0.207536
$592$	0	0
$593$	−37.8328	−1.55361	−0.776804	−	0.629743i	$-0.783160\pi$
$593$	−37.8328	−1.55361	−0.776804	+	0.629743i	$0.783160\pi$
$594$	0	0
$595$	0.686054	0.0281255
$596$	0	0
$597$	−10.4033	−0.425777
$598$	0	0
$599$	−7.09391	−0.289849	−0.144925	−	0.989443i	$-0.546294\pi$
$599$	−7.09391	−0.289849	−0.144925	+	0.989443i	$0.546294\pi$
$600$	0	0
$601$	−37.2361	−1.51889	−0.759445	−	0.650571i	$-0.774530\pi$
$601$	−37.2361	−1.51889	−0.759445	+	0.650571i	$0.774530\pi$
$602$	0	0
$603$	−7.18445	−0.292573
$604$	0	0
$605$	−4.36068	−0.177287
$606$	0	0
$607$	−42.5730	−1.72799	−0.863993	−	0.503504i	$-0.832044\pi$
$607$	−42.5730	−1.72799	−0.863993	+	0.503504i	$0.832044\pi$
$608$	0	0
$609$	−2.76393	−0.112000
$610$	0	0
$611$	−28.2542	−1.14304
$612$	0	0
$613$	29.6525	1.19765	0.598826	−	0.800879i	$-0.295634\pi$
$613$	29.6525	1.19765	0.598826	+	0.800879i	$0.295634\pi$
$614$	0	0
$615$	0.474051	0.0191156
$616$	0	0
$617$	−20.3607	−0.819690	−0.409845	−	0.912155i	$-0.634417\pi$
$617$	−20.3607	−0.819690	−0.409845	+	0.912155i	$0.634417\pi$
$618$	0	0
$619$	−21.7558	−0.874439	−0.437219	−	0.899355i	$-0.644037\pi$
$619$	−21.7558	−0.874439	−0.437219	+	0.899355i	$0.644037\pi$
$620$	0	0
$621$	−9.34752	−0.375103
$622$	0	0
$623$	−44.5407	−1.78448
$624$	0	0
$625$	24.1672	0.966687
$626$	0	0
$627$	−5.73136	−0.228889
$628$	0	0
$629$	−4.94427	−0.197141
$630$	0	0
$631$	−17.3560	−0.690933	−0.345467	−	0.938431i	$-0.612279\pi$
$631$	−17.3560	−0.690933	−0.345467	+	0.938431i	$0.612279\pi$
$632$	0	0
$633$	−3.94427	−0.156771
$634$	0	0
$635$	−0.343027	−0.0136126
$636$	0	0
$637$	46.5967	1.84623
$638$	0	0
$639$	34.0260	1.34605
$640$	0	0
$641$	−12.7639	−0.504145	−0.252073	−	0.967708i	$-0.581112\pi$
$641$	−12.7639	−0.504145	−0.252073	+	0.967708i	$0.581112\pi$
$642$	0	0
$643$	23.7234	0.935560	0.467780	−	0.883845i	$-0.345054\pi$
$643$	23.7234	0.935560	0.467780	+	0.883845i	$0.345054\pi$
$644$	0	0
$645$	1.98684	0.0782319
$646$	0	0
$647$	−18.6781	−0.734312	−0.367156	−	0.930159i	$-0.619668\pi$
$647$	−18.6781	−0.734312	−0.367156	+	0.930159i	$0.619668\pi$
$648$	0	0
$649$	79.5967	3.12445
$650$	0	0
$651$	−8.50651	−0.333396
$652$	0	0
$653$	−6.76393	−0.264693	−0.132347	−	0.991204i	$-0.542251\pi$
$653$	−6.76393	−0.264693	−0.132347	+	0.991204i	$0.542251\pi$
$654$	0	0
$655$	4.35926	0.170330
$656$	0	0
$657$	−32.0000	−1.24844
$658$	0	0
$659$	−13.3803	−0.521223	−0.260611	−	0.965444i	$-0.583924\pi$
$659$	−13.3803	−0.521223	−0.260611	+	0.965444i	$0.583924\pi$
$660$	0	0
$661$	31.3050	1.21762	0.608811	−	0.793315i	$-0.291647\pi$
$661$	31.3050	1.21762	0.608811	+	0.793315i	$0.291647\pi$
$662$	0	0
$663$	−3.46120	−0.134422
$664$	0	0
$665$	1.30495	0.0506039
$666$	0	0
$667$	−2.35114	−0.0910365
$668$	0	0
$669$	14.2229	0.549890
$670$	0	0
$671$	39.2833	1.51652
$672$	0	0
$673$	35.8328	1.38125	0.690627	−	0.723211i	$-0.257335\pi$
$673$	35.8328	1.38125	0.690627	+	0.723211i	$0.257335\pi$
$674$	0	0
$675$	19.6571	0.756604
$676$	0	0
$677$	34.3607	1.32059	0.660294	−	0.751007i	$-0.270432\pi$
$677$	34.3607	1.32059	0.660294	+	0.751007i	$0.270432\pi$
$678$	0	0
$679$	−14.1068	−0.541371
$680$	0	0
$681$	6.58359	0.252284
$682$	0	0
$683$	39.8384	1.52437	0.762186	−	0.647358i	$-0.224126\pi$
$683$	39.8384	1.52437	0.762186	+	0.647358i	$0.224126\pi$
$684$	0	0
$685$	3.63932	0.139051
$686$	0	0
$687$	−6.02434	−0.229843
$688$	0	0
$689$	51.3607	1.95669
$690$	0	0
$691$	30.4338	1.15776	0.578878	−	0.815414i	$-0.303491\pi$
$691$	30.4338	1.15776	0.578878	+	0.815414i	$0.303491\pi$
$692$	0	0
$693$	−51.0557	−1.93945
$694$	0	0
$695$	2.35114	0.0891839
$696$	0	0
$697$	2.11146	0.0799771
$698$	0	0
$699$	−0.0404888	−0.00153143
$700$	0	0
$701$	−28.0132	−1.05804	−0.529021	−	0.848609i	$-0.677441\pi$
$701$	−28.0132	−1.05804	−0.529021	+	0.848609i	$0.677441\pi$
$702$	0	0
$703$	−9.40456	−0.354700
$704$	0	0
$705$	0.777088	0.0292668
$706$	0	0
$707$	−43.4306	−1.63338
$708$	0	0
$709$	4.34752	0.163275	0.0816373	−	0.996662i	$-0.473985\pi$
$709$	4.34752	0.163275	0.0816373	+	0.996662i	$0.473985\pi$
$710$	0	0
$711$	−22.8254	−0.856018
$712$	0	0
$713$	−7.23607	−0.270993
$714$	0	0
$715$	−7.99197	−0.298883
$716$	0	0
$717$	7.23607	0.270236
$718$	0	0
$719$	−26.7606	−0.998002	−0.499001	−	0.866601i	$-0.666300\pi$
$719$	−26.7606	−0.998002	−0.499001	+	0.866601i	$0.666300\pi$
$720$	0	0
$721$	52.3607	1.95001
$722$	0	0
$723$	10.1311	0.376780
$724$	0	0
$725$	4.94427	0.183626
$726$	0	0
$727$	−20.2622	−0.751484	−0.375742	−	0.926724i	$-0.622612\pi$
$727$	−20.2622	−0.751484	−0.375742	+	0.926724i	$0.622612\pi$
$728$	0	0
$729$	−2.52786	−0.0936246
$730$	0	0
$731$	8.84953	0.327312
$732$	0	0
$733$	−12.9443	−0.478108	−0.239054	−	0.971006i	$-0.576837\pi$
$733$	−12.9443	−0.478108	−0.239054	+	0.971006i	$0.576837\pi$
$734$	0	0
$735$	−1.28157	−0.0472715
$736$	0	0
$737$	15.7771	0.581156
$738$	0	0
$739$	16.6295	0.611726	0.305863	−	0.952076i	$-0.401055\pi$
$739$	16.6295	0.611726	0.305863	+	0.952076i	$0.401055\pi$
$740$	0	0
$741$	−6.58359	−0.241854
$742$	0	0
$743$	−34.3691	−1.26088	−0.630439	−	0.776239i	$-0.717125\pi$
$743$	−34.3691	−1.26088	−0.630439	+	0.776239i	$0.717125\pi$
$744$	0	0
$745$	−0.888544	−0.0325537
$746$	0	0
$747$	1.37211	0.0502028
$748$	0	0
$749$	−14.4721	−0.528800
$750$	0	0
$751$	0.343027	0.0125172	0.00625861	−	0.999980i	$-0.498008\pi$
$751$	0.343027	0.0125172	0.00625861	+	0.999980i	$0.498008\pi$
$752$	0	0
$753$	−17.3607	−0.632658
$754$	0	0
$755$	5.17633	0.188386
$756$	0	0
$757$	44.1803	1.60576	0.802881	−	0.596139i	$-0.203299\pi$
$757$	44.1803	1.60576	0.802881	+	0.596139i	$0.203299\pi$
$758$	0	0
$759$	9.27354	0.336608
$760$	0	0
$761$	19.8885	0.720959	0.360480	−	0.932767i	$-0.382613\pi$
$761$	19.8885	0.720959	0.360480	+	0.932767i	$0.382613\pi$
$762$	0	0
$763$	23.7234	0.858845
$764$	0	0
$765$	−0.445825	−0.0161188
$766$	0	0
$767$	91.4325	3.30143
$768$	0	0
$769$	12.7639	0.460279	0.230140	−	0.973158i	$-0.426082\pi$
$769$	12.7639	0.460279	0.230140	+	0.973158i	$0.426082\pi$
$770$	0	0
$771$	−5.42882	−0.195514
$772$	0	0
$773$	26.7639	0.962632	0.481316	−	0.876547i	$-0.340159\pi$
$773$	26.7639	0.962632	0.481316	+	0.876547i	$0.340159\pi$
$774$	0	0
$775$	15.2169	0.546607
$776$	0	0
$777$	17.8885	0.641748
$778$	0	0
$779$	4.01623	0.143896
$780$	0	0
$781$	−74.7214	−2.67374
$782$	0	0
$783$	3.97574	0.142081
$784$	0	0
$785$	0.875388	0.0312439
$786$	0	0
$787$	−29.6668	−1.05751	−0.528753	−	0.848776i	$-0.677340\pi$
$787$	−29.6668	−1.05751	−0.528753	+	0.848776i	$0.677340\pi$
$788$	0	0
$789$	−2.23607	−0.0796061
$790$	0	0
$791$	56.8514	2.02140
$792$	0	0
$793$	45.1246	1.60242
$794$	0	0
$795$	−1.41260	−0.0500996
$796$	0	0
$797$	−24.7639	−0.877183	−0.438592	−	0.898686i	$-0.644523\pi$
$797$	−24.7639	−0.877183	−0.438592	+	0.898686i	$0.644523\pi$
$798$	0	0
$799$	3.46120	0.122448
$800$	0	0
$801$	28.9443	1.02270
$802$	0	0
$803$	70.2722	2.47985
$804$	0	0
$805$	−2.11146	−0.0744191
$806$	0	0
$807$	−4.14725	−0.145990
$808$	0	0
$809$	28.9443	1.01763	0.508813	−	0.860877i	$-0.330084\pi$
$809$	28.9443	1.01763	0.508813	+	0.860877i	$0.330084\pi$
$810$	0	0
$811$	17.2250	0.604852	0.302426	−	0.953173i	$-0.402203\pi$
$811$	17.2250	0.604852	0.302426	+	0.953173i	$0.402203\pi$
$812$	0	0
$813$	18.8197	0.660034
$814$	0	0
$815$	−4.53077	−0.158706
$816$	0	0
$817$	16.8328	0.588906
$818$	0	0
$819$	−58.6475	−2.04931
$820$	0	0
$821$	−37.7639	−1.31797	−0.658985	−	0.752156i	$-0.729014\pi$
$821$	−37.7639	−1.31797	−0.658985	+	0.752156i	$0.729014\pi$
$822$	0	0
$823$	10.8576	0.378474	0.189237	−	0.981931i	$-0.439399\pi$
$823$	10.8576	0.378474	0.189237	+	0.981931i	$0.439399\pi$
$824$	0	0
$825$	−19.5016	−0.678957
$826$	0	0
$827$	7.56796	0.263164	0.131582	−	0.991305i	$-0.457994\pi$
$827$	7.56796	0.263164	0.131582	+	0.991305i	$0.457994\pi$
$828$	0	0
$829$	−2.29180	−0.0795974	−0.0397987	−	0.999208i	$-0.512672\pi$
$829$	−2.29180	−0.0795974	−0.0397987	+	0.999208i	$0.512672\pi$
$830$	0	0
$831$	−14.4499	−0.501261
$832$	0	0
$833$	−5.70820	−0.197778
$834$	0	0
$835$	2.13914	0.0740279
$836$	0	0
$837$	12.2361	0.422940
$838$	0	0
$839$	17.1845	0.593276	0.296638	−	0.954990i	$-0.404135\pi$
$839$	17.1845	0.593276	0.296638	+	0.954990i	$0.404135\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−14.1473	−0.487260
$844$	0	0
$845$	−6.11146	−0.210240
$846$	0	0
$847$	70.2722	2.41458
$848$	0	0
$849$	−7.23607	−0.248341
$850$	0	0
$851$	15.2169	0.521629
$852$	0	0
$853$	37.4164	1.28111	0.640557	−	0.767911i	$-0.278704\pi$
$853$	37.4164	1.28111	0.640557	+	0.767911i	$0.278704\pi$
$854$	0	0
$855$	−0.848009	−0.0290013
$856$	0	0
$857$	−45.9443	−1.56943	−0.784713	−	0.619859i	$-0.787190\pi$
$857$	−45.9443	−1.56943	−0.784713	+	0.619859i	$0.787190\pi$
$858$	0	0
$859$	−19.5357	−0.666548	−0.333274	−	0.942830i	$-0.608153\pi$
$859$	−19.5357	−0.666548	−0.333274	+	0.942830i	$0.608153\pi$
$860$	0	0
$861$	−7.63932	−0.260347
$862$	0	0
$863$	27.1846	0.925375	0.462687	−	0.886521i	$-0.346885\pi$
$863$	27.1846	0.925375	0.462687	+	0.886521i	$0.346885\pi$
$864$	0	0
$865$	−2.72136	−0.0925290
$866$	0	0
$867$	−11.9272	−0.405070
$868$	0	0
$869$	50.1246	1.70036
$870$	0	0
$871$	18.1231	0.614077
$872$	0	0
$873$	9.16718	0.310262
$874$	0	0
$875$	8.93051	0.301906
$876$	0	0
$877$	29.1803	0.985350	0.492675	−	0.870213i	$-0.336019\pi$
$877$	29.1803	0.985350	0.492675	+	0.870213i	$0.336019\pi$
$878$	0	0
$879$	12.6538	0.426801
$880$	0	0
$881$	22.3607	0.753350	0.376675	−	0.926345i	$-0.377067\pi$
$881$	22.3607	0.753350	0.376675	+	0.926345i	$0.377067\pi$
$882$	0	0
$883$	45.4387	1.52913	0.764567	−	0.644544i	$-0.222953\pi$
$883$	45.4387	1.52913	0.764567	+	0.644544i	$0.222953\pi$
$884$	0	0
$885$	−2.51471	−0.0845310
$886$	0	0
$887$	47.2753	1.58735	0.793675	−	0.608342i	$-0.208165\pi$
$887$	47.2753	1.58735	0.793675	+	0.608342i	$0.208165\pi$
$888$	0	0
$889$	5.52786	0.185399
$890$	0	0
$891$	24.5810	0.823494
$892$	0	0
$893$	6.58359	0.220312
$894$	0	0
$895$	1.87709	0.0627442
$896$	0	0
$897$	10.6525	0.355676
$898$	0	0
$899$	3.07768	0.102646
$900$	0	0
$901$	−6.29180	−0.209610
$902$	0	0
$903$	−32.0179	−1.06549
$904$	0	0
$905$	0.750776	0.0249567
$906$	0	0
$907$	14.8739	0.493879	0.246939	−	0.969031i	$-0.420575\pi$
$907$	14.8739	0.493879	0.246939	+	0.969031i	$0.420575\pi$
$908$	0	0
$909$	28.2229	0.936095
$910$	0	0
$911$	7.09391	0.235032	0.117516	−	0.993071i	$-0.462507\pi$
$911$	7.09391	0.235032	0.117516	+	0.993071i	$0.462507\pi$
$912$	0	0
$913$	−3.01316	−0.0997209
$914$	0	0
$915$	−1.24108	−0.0410289
$916$	0	0
$917$	−70.2492	−2.31983
$918$	0	0
$919$	−21.3723	−0.705006	−0.352503	−	0.935811i	$-0.614669\pi$
$919$	−21.3723	−0.705006	−0.352503	+	0.935811i	$0.614669\pi$
$920$	0	0
$921$	4.75078	0.156543
$922$	0	0
$923$	−85.8321	−2.82520
$924$	0	0
$925$	−32.0000	−1.05215
$926$	0	0
$927$	−34.0260	−1.11756
$928$	0	0
$929$	54.0000	1.77168	0.885841	−	0.463988i	$-0.153582\pi$
$929$	54.0000	1.77168	0.885841	+	0.463988i	$0.153582\pi$
$930$	0	0
$931$	−10.8576	−0.355845
$932$	0	0
$933$	−12.1115	−0.396511
$934$	0	0
$935$	0.979034	0.0320178
$936$	0	0
$937$	−10.3607	−0.338469	−0.169234	−	0.985576i	$-0.554129\pi$
$937$	−10.3607	−0.338469	−0.169234	+	0.985576i	$0.554129\pi$
$938$	0	0
$939$	−9.02105	−0.294391
$940$	0	0
$941$	26.0132	0.848005	0.424002	−	0.905661i	$-0.360625\pi$
$941$	26.0132	0.848005	0.424002	+	0.905661i	$0.360625\pi$
$942$	0	0
$943$	−6.49839	−0.211617
$944$	0	0
$945$	3.57044	0.116146
$946$	0	0
$947$	39.4549	1.28211	0.641055	−	0.767495i	$-0.278497\pi$
$947$	39.4549	1.28211	0.641055	+	0.767495i	$0.278497\pi$
$948$	0	0
$949$	80.7214	2.62033
$950$	0	0
$951$	−9.95959	−0.322962
$952$	0	0
$953$	14.4164	0.466993	0.233497	−	0.972358i	$-0.424983\pi$
$953$	14.4164	0.466993	0.233497	+	0.972358i	$0.424983\pi$
$954$	0	0
$955$	4.78326	0.154783
$956$	0	0
$957$	−3.94427	−0.127500
$958$	0	0
$959$	−58.6475	−1.89383
$960$	0	0
$961$	−21.5279	−0.694447
$962$	0	0
$963$	9.40456	0.303058
$964$	0	0
$965$	−3.12461	−0.100585
$966$	0	0
$967$	−32.7445	−1.05299	−0.526495	−	0.850178i	$-0.676494\pi$
$967$	−32.7445	−1.05299	−0.526495	+	0.850178i	$0.676494\pi$
$968$	0	0
$969$	0.806504	0.0259087
$970$	0	0
$971$	6.15537	0.197535	0.0987676	−	0.995111i	$-0.468510\pi$
$971$	6.15537	0.197535	0.0987676	+	0.995111i	$0.468510\pi$
$972$	0	0
$973$	−37.8885	−1.21465
$974$	0	0
$975$	−22.4014	−0.717417
$976$	0	0
$977$	9.83282	0.314580	0.157290	−	0.987552i	$-0.449724\pi$
$977$	9.83282	0.314580	0.157290	+	0.987552i	$0.449724\pi$
$978$	0	0
$979$	−63.5618	−2.03144
$980$	0	0
$981$	−15.4164	−0.492208
$982$	0	0
$983$	−16.8415	−0.537161	−0.268580	−	0.963257i	$-0.586554\pi$
$983$	−16.8415	−0.537161	−0.268580	+	0.963257i	$0.586554\pi$
$984$	0	0
$985$	1.63932	0.0522331
$986$	0	0
$987$	−12.5227	−0.398603
$988$	0	0
$989$	−27.2361	−0.866057
$990$	0	0
$991$	9.06154	0.287849	0.143925	−	0.989589i	$-0.454028\pi$
$991$	9.06154	0.287849	0.143925	+	0.989589i	$0.454028\pi$
$992$	0	0
$993$	−9.47214	−0.300589
$994$	0	0
$995$	3.38022	0.107160
$996$	0	0
$997$	16.9443	0.536630	0.268315	−	0.963331i	$-0.413533\pi$
$997$	16.9443	0.536630	0.268315	+	0.963331i	$0.413533\pi$
$998$	0	0
$999$	−25.7315	−0.814109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.2.a.z.1.3		4
4.3	odd	2	inner	1856.2.a.z.1.2		4
8.3	odd	2		928.2.a.f.1.3	yes	4
8.5	even	2		928.2.a.f.1.2	✓	4
24.5	odd	2		8352.2.a.ba.1.2		4
24.11	even	2		8352.2.a.ba.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.2.a.f.1.2	✓	4	8.5	even	2
928.2.a.f.1.3	yes	4	8.3	odd	2
1856.2.a.z.1.2		4	4.3	odd	2	inner
1856.2.a.z.1.3		4	1.1	even	1	trivial
8352.2.a.ba.1.1		4	24.11	even	2
8352.2.a.ba.1.2		4	24.5	odd	2

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:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)

\(f(q)\)	\(=\)	\(q+0.726543 q^{3} -0.236068 q^{5} +3.80423 q^{7} -2.47214 q^{9} +O(q^{10})\)	\(q+0.726543 q^{3} -0.236068 q^{5} +3.80423 q^{7} -2.47214 q^{9} +5.42882 q^{11} +6.23607 q^{13} -0.171513 q^{15} -0.763932 q^{17} -1.45309 q^{19} +2.76393 q^{21} +2.35114 q^{23} -4.94427 q^{25} -3.97574 q^{27} -1.00000 q^{29} -3.07768 q^{31} +3.94427 q^{33} -0.898056 q^{35} +6.47214 q^{37} +4.53077 q^{39} -2.76393 q^{41} -11.5842 q^{43} +0.583592 q^{45} -4.53077 q^{47} +7.47214 q^{49} -0.555029 q^{51} +8.23607 q^{53} -1.28157 q^{55} -1.05573 q^{57} +14.6619 q^{59} +7.23607 q^{61} -9.40456 q^{63} -1.47214 q^{65} +2.90617 q^{67} +1.70820 q^{69} -13.7638 q^{71} +12.9443 q^{73} -3.59222 q^{75} +20.6525 q^{77} +9.23305 q^{79} +4.52786 q^{81} -0.555029 q^{83} +0.180340 q^{85} -0.726543 q^{87} -11.7082 q^{89} +23.7234 q^{91} -2.23607 q^{93} +0.343027 q^{95} -3.70820 q^{97} -13.4208 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{5} + 8 q^{9}+O(q^{10}) \) 4 * q + 8 * q^5 + 8 * q^9	\( 4 q + 8 q^{5} + 8 q^{9} + 16 q^{13} - 12 q^{17} + 20 q^{21} + 16 q^{25} - 4 q^{29} - 20 q^{33} + 8 q^{37} - 20 q^{41} + 56 q^{45} + 12 q^{49} + 24 q^{53} - 40 q^{57} + 20 q^{61} + 12 q^{65} - 20 q^{69} + 16 q^{73} + 20 q^{77} + 36 q^{81} - 44 q^{85} - 20 q^{89} + 12 q^{97}+O(q^{100}) \) 4 * q + 8 * q^5 + 8 * q^9 + 16 * q^13 - 12 * q^17 + 20 * q^21 + 16 * q^25 - 4 * q^29 - 20 * q^33 + 8 * q^37 - 20 * q^41 + 56 * q^45 + 12 * q^49 + 24 * q^53 - 40 * q^57 + 20 * q^61 + 12 * q^65 - 20 * q^69 + 16 * q^73 + 20 * q^77 + 36 * q^81 - 44 * q^85 - 20 * q^89 + 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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