LMFDB - Embedded newform 9680.2.a.bh.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9680,2,Mod(1,9680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9680.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9680 = 2^{4} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9680.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,-2,0,-3,0,6,0,0,0,-1,0,2,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$77.2951891566$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 110)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	9680.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.23607 q^{3} -1.00000 q^{5} -0.381966 q^{7} -1.47214 q^{9} +0.618034 q^{13} -1.23607 q^{15} -0.763932 q^{17} -7.09017 q^{19} -0.472136 q^{21} -6.85410 q^{23} +1.00000 q^{25} -5.52786 q^{27} +3.23607 q^{29} +2.76393 q^{31} +0.381966 q^{35} +5.85410 q^{37} +0.763932 q^{39} +4.85410 q^{41} +4.76393 q^{43} +1.47214 q^{45} +4.32624 q^{47} -6.85410 q^{49} -0.944272 q^{51} +12.0902 q^{53} -8.76393 q^{57} -4.61803 q^{59} -8.94427 q^{61} +0.562306 q^{63} -0.618034 q^{65} -5.23607 q^{67} -8.47214 q^{69} -8.76393 q^{71} +11.7082 q^{73} +1.23607 q^{75} +10.4721 q^{79} -2.41641 q^{81} -2.29180 q^{83} +0.763932 q^{85} +4.00000 q^{87} +17.0344 q^{89} -0.236068 q^{91} +3.41641 q^{93} +7.09017 q^{95} -12.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} - 3 q^{7} + 6 q^{9} - q^{13} + 2 q^{15} - 6 q^{17} - 3 q^{19} + 8 q^{21} - 7 q^{23} + 2 q^{25} - 20 q^{27} + 2 q^{29} + 10 q^{31} + 3 q^{35} + 5 q^{37} + 6 q^{39} + 3 q^{41} + 14 q^{43}+ \cdots - 24 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 - 3 * q^7 + 6 * q^9 - q^13 + 2 * q^15 - 6 * q^17 - 3 * q^19 + 8 * q^21 - 7 * q^23 + 2 * q^25 - 20 * q^27 + 2 * q^29 + 10 * q^31 + 3 * q^35 + 5 * q^37 + 6 * q^39 + 3 * q^41 + 14 * q^43 - 6 * q^45 - 7 * q^47 - 7 * q^49 + 16 * q^51 + 13 * q^53 - 22 * q^57 - 7 * q^59 - 19 * q^63 + q^65 - 6 * q^67 - 8 * q^69 - 22 * q^71 + 10 * q^73 - 2 * q^75 + 12 * q^79 + 22 * q^81 - 18 * q^83 + 6 * q^85 + 8 * q^87 + 5 * q^89 + 4 * q^91 - 20 * q^93 + 3 * q^95 - 24 * q^97

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
$3$	1.23607	0.713644	0.356822	−	0.934172i	$-0.383860\pi$
$3$	1.23607	0.713644	0.356822	+	0.934172i	$0.383860\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.381966	−0.144370	−0.0721848	−	0.997391i	$-0.522997\pi$
$7$	−0.381966	−0.144370	−0.0721848	+	0.997391i	$0.522997\pi$
$8$	0	0
$9$	−1.47214	−0.490712
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	0.618034	0.171412	0.0857059	−	0.996320i	$-0.472685\pi$
$13$	0.618034	0.171412	0.0857059	+	0.996320i	$0.472685\pi$
$14$	0	0
$15$	−1.23607	−0.319151
$16$	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
$19$	−7.09017	−1.62660	−0.813298	−	0.581847i	$-0.802330\pi$
$19$	−7.09017	−1.62660	−0.813298	+	0.581847i	$0.802330\pi$
$20$	0	0
$21$	−0.472136	−0.103029
$22$	0	0
$23$	−6.85410	−1.42918	−0.714590	−	0.699544i	$-0.753386\pi$
$23$	−6.85410	−1.42918	−0.714590	+	0.699544i	$0.753386\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.52786	−1.06384
$28$	0	0
$29$	3.23607	0.600923	0.300461	−	0.953794i	$-0.402859\pi$
$29$	3.23607	0.600923	0.300461	+	0.953794i	$0.402859\pi$
$30$	0	0
$31$	2.76393	0.496417	0.248208	−	0.968707i	$-0.420158\pi$
$31$	2.76393	0.496417	0.248208	+	0.968707i	$0.420158\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.381966	0.0645640
$36$	0	0
$37$	5.85410	0.962408	0.481204	−	0.876609i	$-0.340200\pi$
$37$	5.85410	0.962408	0.481204	+	0.876609i	$0.340200\pi$
$38$	0	0
$39$	0.763932	0.122327
$40$	0	0
$41$	4.85410	0.758083	0.379042	−	0.925380i	$-0.376254\pi$
$41$	4.85410	0.758083	0.379042	+	0.925380i	$0.376254\pi$
$42$	0	0
$43$	4.76393	0.726493	0.363246	−	0.931693i	$-0.381668\pi$
$43$	4.76393	0.726493	0.363246	+	0.931693i	$0.381668\pi$
$44$	0	0
$45$	1.47214	0.219453
$46$	0	0
$47$	4.32624	0.631047	0.315523	−	0.948918i	$-0.397820\pi$
$47$	4.32624	0.631047	0.315523	+	0.948918i	$0.397820\pi$
$48$	0	0
$49$	−6.85410	−0.979157
$50$	0	0
$51$	−0.944272	−0.132225
$52$	0	0
$53$	12.0902	1.66071	0.830356	−	0.557233i	$-0.188137\pi$
$53$	12.0902	1.66071	0.830356	+	0.557233i	$0.188137\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−8.76393	−1.16081
$58$	0	0
$59$	−4.61803	−0.601217	−0.300608	−	0.953748i	$-0.597190\pi$
$59$	−4.61803	−0.601217	−0.300608	+	0.953748i	$0.597190\pi$
$60$	0	0
$61$	−8.94427	−1.14520	−0.572598	−	0.819836i	$-0.694065\pi$
$61$	−8.94427	−1.14520	−0.572598	+	0.819836i	$0.694065\pi$
$62$	0	0
$63$	0.562306	0.0708439
$64$	0	0
$65$	−0.618034	−0.0766577
$66$	0	0
$67$	−5.23607	−0.639688	−0.319844	−	0.947470i	$-0.603630\pi$
$67$	−5.23607	−0.639688	−0.319844	+	0.947470i	$0.603630\pi$
$68$	0	0
$69$	−8.47214	−1.01993
$70$	0	0
$71$	−8.76393	−1.04009	−0.520044	−	0.854140i	$-0.674084\pi$
$71$	−8.76393	−1.04009	−0.520044	+	0.854140i	$0.674084\pi$
$72$	0	0
$73$	11.7082	1.37034	0.685171	−	0.728382i	$-0.259728\pi$
$73$	11.7082	1.37034	0.685171	+	0.728382i	$0.259728\pi$
$74$	0	0
$75$	1.23607	0.142729
$76$	0	0
$77$	0	0
$78$	0	0
$79$	10.4721	1.17821	0.589104	−	0.808057i	$-0.299481\pi$
$79$	10.4721	1.17821	0.589104	+	0.808057i	$0.299481\pi$
$80$	0	0
$81$	−2.41641	−0.268490
$82$	0	0
$83$	−2.29180	−0.251557	−0.125779	−	0.992058i	$-0.540143\pi$
$83$	−2.29180	−0.251557	−0.125779	+	0.992058i	$0.540143\pi$
$84$	0	0
$85$	0.763932	0.0828601
$86$	0	0
$87$	4.00000	0.428845
$88$	0	0
$89$	17.0344	1.80565	0.902824	−	0.430011i	$-0.141490\pi$
$89$	17.0344	1.80565	0.902824	+	0.430011i	$0.141490\pi$
$90$	0	0
$91$	−0.236068	−0.0247466
$92$	0	0
$93$	3.41641	0.354265
$94$	0	0
$95$	7.09017	0.727436
$96$	0	0
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−9.70820	−0.966002	−0.483001	−	0.875620i	$-0.660453\pi$
$101$	−9.70820	−0.966002	−0.483001	+	0.875620i	$0.660453\pi$
$102$	0	0
$103$	−4.32624	−0.426277	−0.213138	−	0.977022i	$-0.568369\pi$
$103$	−4.32624	−0.426277	−0.213138	+	0.977022i	$0.568369\pi$
$104$	0	0
$105$	0.472136	0.0460758
$106$	0	0
$107$	16.4721	1.59242	0.796211	−	0.605019i	$-0.206835\pi$
$107$	16.4721	1.59242	0.796211	+	0.605019i	$0.206835\pi$
$108$	0	0
$109$	16.4721	1.57774	0.788872	−	0.614557i	$-0.210665\pi$
$109$	16.4721	1.57774	0.788872	+	0.614557i	$0.210665\pi$
$110$	0	0
$111$	7.23607	0.686817
$112$	0	0
$113$	19.7082	1.85399	0.926996	−	0.375071i	$-0.122382\pi$
$113$	19.7082	1.85399	0.926996	+	0.375071i	$0.122382\pi$
$114$	0	0
$115$	6.85410	0.639148
$116$	0	0
$117$	−0.909830	−0.0841138
$118$	0	0
$119$	0.291796	0.0267489
$120$	0	0
$121$	0	0
$122$	0	0
$123$	6.00000	0.541002
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	4.61803	0.409784	0.204892	−	0.978785i	$-0.434316\pi$
$127$	4.61803	0.409784	0.204892	+	0.978785i	$0.434316\pi$
$128$	0	0
$129$	5.88854	0.518457
$130$	0	0
$131$	15.4164	1.34694	0.673469	−	0.739216i	$-0.264804\pi$
$131$	15.4164	1.34694	0.673469	+	0.739216i	$0.264804\pi$
$132$	0	0
$133$	2.70820	0.234831
$134$	0	0
$135$	5.52786	0.475763
$136$	0	0
$137$	−0.763932	−0.0652671	−0.0326336	−	0.999467i	$-0.510389\pi$
$137$	−0.763932	−0.0652671	−0.0326336	+	0.999467i	$0.510389\pi$
$138$	0	0
$139$	14.8541	1.25991	0.629954	−	0.776632i	$-0.283074\pi$
$139$	14.8541	1.25991	0.629954	+	0.776632i	$0.283074\pi$
$140$	0	0
$141$	5.34752	0.450343
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−3.23607	−0.268741
$146$	0	0
$147$	−8.47214	−0.698770
$148$	0	0
$149$	−4.00000	−0.327693	−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	0	0
$151$	18.6525	1.51792	0.758958	−	0.651139i	$-0.225709\pi$
$151$	18.6525	1.51792	0.758958	+	0.651139i	$0.225709\pi$
$152$	0	0
$153$	1.12461	0.0909195
$154$	0	0
$155$	−2.76393	−0.222004
$156$	0	0
$157$	6.38197	0.509336	0.254668	−	0.967029i	$-0.418034\pi$
$157$	6.38197	0.509336	0.254668	+	0.967029i	$0.418034\pi$
$158$	0	0
$159$	14.9443	1.18516
$160$	0	0
$161$	2.61803	0.206330
$162$	0	0
$163$	2.76393	0.216488	0.108244	−	0.994124i	$-0.465477\pi$
$163$	2.76393	0.216488	0.108244	+	0.994124i	$0.465477\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−19.0902	−1.47724	−0.738621	−	0.674121i	$-0.764523\pi$
$167$	−19.0902	−1.47724	−0.738621	+	0.674121i	$0.764523\pi$
$168$	0	0
$169$	−12.6180	−0.970618
$170$	0	0
$171$	10.4377	0.798190
$172$	0	0
$173$	−10.3820	−0.789326	−0.394663	−	0.918826i	$-0.629139\pi$
$173$	−10.3820	−0.789326	−0.394663	+	0.918826i	$0.629139\pi$
$174$	0	0
$175$	−0.381966	−0.0288739
$176$	0	0
$177$	−5.70820	−0.429055
$178$	0	0
$179$	14.0902	1.05315	0.526574	−	0.850129i	$-0.323476\pi$
$179$	14.0902	1.05315	0.526574	+	0.850129i	$0.323476\pi$
$180$	0	0
$181$	−11.5279	−0.856859	−0.428430	−	0.903575i	$-0.640933\pi$
$181$	−11.5279	−0.856859	−0.428430	+	0.903575i	$0.640933\pi$
$182$	0	0
$183$	−11.0557	−0.817263
$184$	0	0
$185$	−5.85410	−0.430402
$186$	0	0
$187$	0	0
$188$	0	0
$189$	2.11146	0.153586
$190$	0	0
$191$	1.41641	0.102488	0.0512438	−	0.998686i	$-0.483681\pi$
$191$	1.41641	0.102488	0.0512438	+	0.998686i	$0.483681\pi$
$192$	0	0
$193$	21.4164	1.54159	0.770793	−	0.637085i	$-0.219860\pi$
$193$	21.4164	1.54159	0.770793	+	0.637085i	$0.219860\pi$
$194$	0	0
$195$	−0.763932	−0.0547063
$196$	0	0
$197$	−16.0902	−1.14638	−0.573189	−	0.819423i	$-0.694294\pi$
$197$	−16.0902	−1.14638	−0.573189	+	0.819423i	$0.694294\pi$
$198$	0	0
$199$	11.7082	0.829973	0.414986	−	0.909828i	$-0.363786\pi$
$199$	11.7082	0.829973	0.414986	+	0.909828i	$0.363786\pi$
$200$	0	0
$201$	−6.47214	−0.456509
$202$	0	0
$203$	−1.23607	−0.0867550
$204$	0	0
$205$	−4.85410	−0.339025
$206$	0	0
$207$	10.0902	0.701315
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−3.05573	−0.210365	−0.105182	−	0.994453i	$-0.533543\pi$
$211$	−3.05573	−0.210365	−0.105182	+	0.994453i	$0.533543\pi$
$212$	0	0
$213$	−10.8328	−0.742252
$214$	0	0
$215$	−4.76393	−0.324897
$216$	0	0
$217$	−1.05573	−0.0716675
$218$	0	0
$219$	14.4721	0.977936
$220$	0	0
$221$	−0.472136	−0.0317593
$222$	0	0
$223$	−25.3262	−1.69597	−0.847985	−	0.530020i	$-0.822184\pi$
$223$	−25.3262	−1.69597	−0.847985	+	0.530020i	$0.822184\pi$
$224$	0	0
$225$	−1.47214	−0.0981424
$226$	0	0
$227$	7.52786	0.499642	0.249821	−	0.968292i	$-0.419628\pi$
$227$	7.52786	0.499642	0.249821	+	0.968292i	$0.419628\pi$
$228$	0	0
$229$	−5.81966	−0.384574	−0.192287	−	0.981339i	$-0.561590\pi$
$229$	−5.81966	−0.384574	−0.192287	+	0.981339i	$0.561590\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.7639	0.705169	0.352584	−	0.935780i	$-0.385303\pi$
$233$	10.7639	0.705169	0.352584	+	0.935780i	$0.385303\pi$
$234$	0	0
$235$	−4.32624	−0.282213
$236$	0	0
$237$	12.9443	0.840821
$238$	0	0
$239$	−18.4721	−1.19486	−0.597432	−	0.801920i	$-0.703812\pi$
$239$	−18.4721	−1.19486	−0.597432	+	0.801920i	$0.703812\pi$
$240$	0	0
$241$	4.85410	0.312680	0.156340	−	0.987703i	$-0.450030\pi$
$241$	4.85410	0.312680	0.156340	+	0.987703i	$0.450030\pi$
$242$	0	0
$243$	13.5967	0.872232
$244$	0	0
$245$	6.85410	0.437893
$246$	0	0
$247$	−4.38197	−0.278818
$248$	0	0
$249$	−2.83282	−0.179522
$250$	0	0
$251$	−19.0344	−1.20144	−0.600722	−	0.799458i	$-0.705120\pi$
$251$	−19.0344	−1.20144	−0.600722	+	0.799458i	$0.705120\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0.944272	0.0591326
$256$	0	0
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	0	0
$259$	−2.23607	−0.138943
$260$	0	0
$261$	−4.76393	−0.294880
$262$	0	0
$263$	−1.20163	−0.0740954	−0.0370477	−	0.999313i	$-0.511795\pi$
$263$	−1.20163	−0.0740954	−0.0370477	+	0.999313i	$0.511795\pi$
$264$	0	0
$265$	−12.0902	−0.742693
$266$	0	0
$267$	21.0557	1.28859
$268$	0	0
$269$	−31.5967	−1.92649	−0.963244	−	0.268629i	$-0.913430\pi$
$269$	−31.5967	−1.92649	−0.963244	+	0.268629i	$0.913430\pi$
$270$	0	0
$271$	13.4164	0.814989	0.407494	−	0.913208i	$-0.366403\pi$
$271$	13.4164	0.814989	0.407494	+	0.913208i	$0.366403\pi$
$272$	0	0
$273$	−0.291796	−0.0176603
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.61803	0.0972182	0.0486091	−	0.998818i	$-0.484521\pi$
$277$	1.61803	0.0972182	0.0486091	+	0.998818i	$0.484521\pi$
$278$	0	0
$279$	−4.06888	−0.243598
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	28.8328	1.71393	0.856966	−	0.515372i	$-0.172346\pi$
$283$	28.8328	1.71393	0.856966	+	0.515372i	$0.172346\pi$
$284$	0	0
$285$	8.76393	0.519131
$286$	0	0
$287$	−1.85410	−0.109444
$288$	0	0
$289$	−16.4164	−0.965671
$290$	0	0
$291$	−14.8328	−0.869515
$292$	0	0
$293$	18.7426	1.09496	0.547479	−	0.836820i	$-0.315588\pi$
$293$	18.7426	1.09496	0.547479	+	0.836820i	$0.315588\pi$
$294$	0	0
$295$	4.61803	0.268872
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.23607	−0.244978
$300$	0	0
$301$	−1.81966	−0.104883
$302$	0	0
$303$	−12.0000	−0.689382
$304$	0	0
$305$	8.94427	0.512148
$306$	0	0
$307$	9.05573	0.516838	0.258419	−	0.966033i	$-0.416799\pi$
$307$	9.05573	0.516838	0.258419	+	0.966033i	$0.416799\pi$
$308$	0	0
$309$	−5.34752	−0.304210
$310$	0	0
$311$	−6.47214	−0.367001	−0.183501	−	0.983020i	$-0.558743\pi$
$311$	−6.47214	−0.367001	−0.183501	+	0.983020i	$0.558743\pi$
$312$	0	0
$313$	0.583592	0.0329866	0.0164933	−	0.999864i	$-0.494750\pi$
$313$	0.583592	0.0329866	0.0164933	+	0.999864i	$0.494750\pi$
$314$	0	0
$315$	−0.562306	−0.0316823
$316$	0	0
$317$	30.7984	1.72981	0.864905	−	0.501936i	$-0.167379\pi$
$317$	30.7984	1.72981	0.864905	+	0.501936i	$0.167379\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	20.3607	1.13642
$322$	0	0
$323$	5.41641	0.301377
$324$	0	0
$325$	0.618034	0.0342824
$326$	0	0
$327$	20.3607	1.12595
$328$	0	0
$329$	−1.65248	−0.0911039
$330$	0	0
$331$	16.6180	0.913410	0.456705	−	0.889618i	$-0.349030\pi$
$331$	16.6180	0.913410	0.456705	+	0.889618i	$0.349030\pi$
$332$	0	0
$333$	−8.61803	−0.472265
$334$	0	0
$335$	5.23607	0.286077
$336$	0	0
$337$	8.58359	0.467578	0.233789	−	0.972287i	$-0.424887\pi$
$337$	8.58359	0.467578	0.233789	+	0.972287i	$0.424887\pi$
$338$	0	0
$339$	24.3607	1.32309
$340$	0	0
$341$	0	0
$342$	0	0
$343$	5.29180	0.285730
$344$	0	0
$345$	8.47214	0.456124
$346$	0	0
$347$	−13.1246	−0.704566	−0.352283	−	0.935894i	$-0.614595\pi$
$347$	−13.1246	−0.704566	−0.352283	+	0.935894i	$0.614595\pi$
$348$	0	0
$349$	13.0557	0.698857	0.349429	−	0.936963i	$-0.386376\pi$
$349$	13.0557	0.698857	0.349429	+	0.936963i	$0.386376\pi$
$350$	0	0
$351$	−3.41641	−0.182354
$352$	0	0
$353$	−1.70820	−0.0909185	−0.0454593	−	0.998966i	$-0.514475\pi$
$353$	−1.70820	−0.0909185	−0.0454593	+	0.998966i	$0.514475\pi$
$354$	0	0
$355$	8.76393	0.465141
$356$	0	0
$357$	0.360680	0.0190892
$358$	0	0
$359$	21.5279	1.13620	0.568099	−	0.822960i	$-0.307679\pi$
$359$	21.5279	1.13620	0.568099	+	0.822960i	$0.307679\pi$
$360$	0	0
$361$	31.2705	1.64582
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−11.7082	−0.612835
$366$	0	0
$367$	−1.52786	−0.0797539	−0.0398769	−	0.999205i	$-0.512697\pi$
$367$	−1.52786	−0.0797539	−0.0398769	+	0.999205i	$0.512697\pi$
$368$	0	0
$369$	−7.14590	−0.372001
$370$	0	0
$371$	−4.61803	−0.239756
$372$	0	0
$373$	36.0344	1.86579	0.932896	−	0.360145i	$-0.117273\pi$
$373$	36.0344	1.86579	0.932896	+	0.360145i	$0.117273\pi$
$374$	0	0
$375$	−1.23607	−0.0638303
$376$	0	0
$377$	2.00000	0.103005
$378$	0	0
$379$	−5.32624	−0.273590	−0.136795	−	0.990599i	$-0.543680\pi$
$379$	−5.32624	−0.273590	−0.136795	+	0.990599i	$0.543680\pi$
$380$	0	0
$381$	5.70820	0.292440
$382$	0	0
$383$	9.14590	0.467334	0.233667	−	0.972317i	$-0.424928\pi$
$383$	9.14590	0.467334	0.233667	+	0.972317i	$0.424928\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−7.01316	−0.356499
$388$	0	0
$389$	−10.1803	−0.516164	−0.258082	−	0.966123i	$-0.583090\pi$
$389$	−10.1803	−0.516164	−0.258082	+	0.966123i	$0.583090\pi$
$390$	0	0
$391$	5.23607	0.264799
$392$	0	0
$393$	19.0557	0.961234
$394$	0	0
$395$	−10.4721	−0.526910
$396$	0	0
$397$	−17.7984	−0.893275	−0.446637	−	0.894715i	$-0.647379\pi$
$397$	−17.7984	−0.893275	−0.446637	+	0.894715i	$0.647379\pi$
$398$	0	0
$399$	3.34752	0.167586
$400$	0	0
$401$	31.7426	1.58515	0.792576	−	0.609773i	$-0.208739\pi$
$401$	31.7426	1.58515	0.792576	+	0.609773i	$0.208739\pi$
$402$	0	0
$403$	1.70820	0.0850917
$404$	0	0
$405$	2.41641	0.120072
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−7.67376	−0.379443	−0.189722	−	0.981838i	$-0.560759\pi$
$409$	−7.67376	−0.379443	−0.189722	+	0.981838i	$0.560759\pi$
$410$	0	0
$411$	−0.944272	−0.0465775
$412$	0	0
$413$	1.76393	0.0867974
$414$	0	0
$415$	2.29180	0.112500
$416$	0	0
$417$	18.3607	0.899126
$418$	0	0
$419$	−0.618034	−0.0301929	−0.0150965	−	0.999886i	$-0.504806\pi$
$419$	−0.618034	−0.0301929	−0.0150965	+	0.999886i	$0.504806\pi$
$420$	0	0
$421$	12.6525	0.616644	0.308322	−	0.951282i	$-0.400233\pi$
$421$	12.6525	0.616644	0.308322	+	0.951282i	$0.400233\pi$
$422$	0	0
$423$	−6.36881	−0.309662
$424$	0	0
$425$	−0.763932	−0.0370561
$426$	0	0
$427$	3.41641	0.165332
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.81966	0.183987	0.0919933	−	0.995760i	$-0.470676\pi$
$431$	3.81966	0.183987	0.0919933	+	0.995760i	$0.470676\pi$
$432$	0	0
$433$	18.4721	0.887714	0.443857	−	0.896098i	$-0.353610\pi$
$433$	18.4721	0.887714	0.443857	+	0.896098i	$0.353610\pi$
$434$	0	0
$435$	−4.00000	−0.191785
$436$	0	0
$437$	48.5967	2.32470
$438$	0	0
$439$	−27.1246	−1.29459	−0.647294	−	0.762241i	$-0.724099\pi$
$439$	−27.1246	−1.29459	−0.647294	+	0.762241i	$0.724099\pi$
$440$	0	0
$441$	10.0902	0.480484
$442$	0	0
$443$	−32.6525	−1.55137	−0.775683	−	0.631123i	$-0.782594\pi$
$443$	−32.6525	−1.55137	−0.775683	+	0.631123i	$0.782594\pi$
$444$	0	0
$445$	−17.0344	−0.807510
$446$	0	0
$447$	−4.94427	−0.233856
$448$	0	0
$449$	9.85410	0.465044	0.232522	−	0.972591i	$-0.425302\pi$
$449$	9.85410	0.465044	0.232522	+	0.972591i	$0.425302\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	23.0557	1.08325
$454$	0	0
$455$	0.236068	0.0110670
$456$	0	0
$457$	−34.5410	−1.61576	−0.807880	−	0.589347i	$-0.799385\pi$
$457$	−34.5410	−1.61576	−0.807880	+	0.589347i	$0.799385\pi$
$458$	0	0
$459$	4.22291	0.197109
$460$	0	0
$461$	−38.5410	−1.79503	−0.897517	−	0.440980i	$-0.854631\pi$
$461$	−38.5410	−1.79503	−0.897517	+	0.440980i	$0.854631\pi$
$462$	0	0
$463$	22.5066	1.04597	0.522985	−	0.852342i	$-0.324818\pi$
$463$	22.5066	1.04597	0.522985	+	0.852342i	$0.324818\pi$
$464$	0	0
$465$	−3.41641	−0.158432
$466$	0	0
$467$	−12.7639	−0.590644	−0.295322	−	0.955398i	$-0.595427\pi$
$467$	−12.7639	−0.590644	−0.295322	+	0.955398i	$0.595427\pi$
$468$	0	0
$469$	2.00000	0.0923514
$470$	0	0
$471$	7.88854	0.363485
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−7.09017	−0.325319
$476$	0	0
$477$	−17.7984	−0.814932
$478$	0	0
$479$	−19.5967	−0.895398	−0.447699	−	0.894184i	$-0.647756\pi$
$479$	−19.5967	−0.895398	−0.447699	+	0.894184i	$0.647756\pi$
$480$	0	0
$481$	3.61803	0.164968
$482$	0	0
$483$	3.23607	0.147246
$484$	0	0
$485$	12.0000	0.544892
$486$	0	0
$487$	−19.0557	−0.863497	−0.431749	−	0.901994i	$-0.642103\pi$
$487$	−19.0557	−0.863497	−0.431749	+	0.901994i	$0.642103\pi$
$488$	0	0
$489$	3.41641	0.154495
$490$	0	0
$491$	19.4508	0.877805	0.438902	−	0.898535i	$-0.355367\pi$
$491$	19.4508	0.877805	0.438902	+	0.898535i	$0.355367\pi$
$492$	0	0
$493$	−2.47214	−0.111339
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.34752	0.150157
$498$	0	0
$499$	−11.7984	−0.528168	−0.264084	−	0.964500i	$-0.585070\pi$
$499$	−11.7984	−0.528168	−0.264084	+	0.964500i	$0.585070\pi$
$500$	0	0
$501$	−23.5967	−1.05422
$502$	0	0
$503$	30.5066	1.36022	0.680111	−	0.733110i	$-0.261932\pi$
$503$	30.5066	1.36022	0.680111	+	0.733110i	$0.261932\pi$
$504$	0	0
$505$	9.70820	0.432009
$506$	0	0
$507$	−15.5967	−0.692676
$508$	0	0
$509$	9.12461	0.404441	0.202221	−	0.979340i	$-0.435184\pi$
$509$	9.12461	0.404441	0.202221	+	0.979340i	$0.435184\pi$
$510$	0	0
$511$	−4.47214	−0.197836
$512$	0	0
$513$	39.1935	1.73044
$514$	0	0
$515$	4.32624	0.190637
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−12.8328	−0.563298
$520$	0	0
$521$	−6.38197	−0.279599	−0.139800	−	0.990180i	$-0.544646\pi$
$521$	−6.38197	−0.279599	−0.139800	+	0.990180i	$0.544646\pi$
$522$	0	0
$523$	38.6525	1.69015	0.845077	−	0.534644i	$-0.179554\pi$
$523$	38.6525	1.69015	0.845077	+	0.534644i	$0.179554\pi$
$524$	0	0
$525$	−0.472136	−0.0206057
$526$	0	0
$527$	−2.11146	−0.0919765
$528$	0	0
$529$	23.9787	1.04255
$530$	0	0
$531$	6.79837	0.295024
$532$	0	0
$533$	3.00000	0.129944
$534$	0	0
$535$	−16.4721	−0.712153
$536$	0	0
$537$	17.4164	0.751573
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	0	0
$543$	−14.2492	−0.611493
$544$	0	0
$545$	−16.4721	−0.705589
$546$	0	0
$547$	−33.5967	−1.43649	−0.718247	−	0.695789i	$-0.755055\pi$
$547$	−33.5967	−1.43649	−0.718247	+	0.695789i	$0.755055\pi$
$548$	0	0
$549$	13.1672	0.561962
$550$	0	0
$551$	−22.9443	−0.977459
$552$	0	0
$553$	−4.00000	−0.170097
$554$	0	0
$555$	−7.23607	−0.307154
$556$	0	0
$557$	−18.2148	−0.771785	−0.385893	−	0.922544i	$-0.626106\pi$
$557$	−18.2148	−0.771785	−0.385893	+	0.922544i	$0.626106\pi$
$558$	0	0
$559$	2.94427	0.124529
$560$	0	0
$561$	0	0
$562$	0	0
$563$	11.8885	0.501042	0.250521	−	0.968111i	$-0.419398\pi$
$563$	11.8885	0.501042	0.250521	+	0.968111i	$0.419398\pi$
$564$	0	0
$565$	−19.7082	−0.829130
$566$	0	0
$567$	0.922986	0.0387618
$568$	0	0
$569$	−7.32624	−0.307132	−0.153566	−	0.988138i	$-0.549076\pi$
$569$	−7.32624	−0.307132	−0.153566	+	0.988138i	$0.549076\pi$
$570$	0	0
$571$	17.9098	0.749503	0.374752	−	0.927125i	$-0.377728\pi$
$571$	17.9098	0.749503	0.374752	+	0.927125i	$0.377728\pi$
$572$	0	0
$573$	1.75078	0.0731397
$574$	0	0
$575$	−6.85410	−0.285836
$576$	0	0
$577$	−2.94427	−0.122572	−0.0612858	−	0.998120i	$-0.519520\pi$
$577$	−2.94427	−0.122572	−0.0612858	+	0.998120i	$0.519520\pi$
$578$	0	0
$579$	26.4721	1.10014
$580$	0	0
$581$	0.875388	0.0363172
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0.909830	0.0376168
$586$	0	0
$587$	25.2361	1.04160	0.520802	−	0.853678i	$-0.325633\pi$
$587$	25.2361	1.04160	0.520802	+	0.853678i	$0.325633\pi$
$588$	0	0
$589$	−19.5967	−0.807470
$590$	0	0
$591$	−19.8885	−0.818105
$592$	0	0
$593$	−6.29180	−0.258373	−0.129187	−	0.991620i	$-0.541237\pi$
$593$	−6.29180	−0.258373	−0.129187	+	0.991620i	$0.541237\pi$
$594$	0	0
$595$	−0.291796	−0.0119625
$596$	0	0
$597$	14.4721	0.592305
$598$	0	0
$599$	−7.52786	−0.307580	−0.153790	−	0.988104i	$-0.549148\pi$
$599$	−7.52786	−0.307580	−0.153790	+	0.988104i	$0.549148\pi$
$600$	0	0
$601$	2.43769	0.0994356	0.0497178	−	0.998763i	$-0.484168\pi$
$601$	2.43769	0.0994356	0.0497178	+	0.998763i	$0.484168\pi$
$602$	0	0
$603$	7.70820	0.313902
$604$	0	0
$605$	0	0
$606$	0	0
$607$	12.3607	0.501705	0.250852	−	0.968025i	$-0.419289\pi$
$607$	12.3607	0.501705	0.250852	+	0.968025i	$0.419289\pi$
$608$	0	0
$609$	−1.52786	−0.0619122
$610$	0	0
$611$	2.67376	0.108169
$612$	0	0
$613$	20.4721	0.826862	0.413431	−	0.910536i	$-0.364330\pi$
$613$	20.4721	0.826862	0.413431	+	0.910536i	$0.364330\pi$
$614$	0	0
$615$	−6.00000	−0.241943
$616$	0	0
$617$	15.0557	0.606121	0.303060	−	0.952971i	$-0.401992\pi$
$617$	15.0557	0.606121	0.303060	+	0.952971i	$0.401992\pi$
$618$	0	0
$619$	19.7426	0.793524	0.396762	−	0.917922i	$-0.370134\pi$
$619$	19.7426	0.793524	0.396762	+	0.917922i	$0.370134\pi$
$620$	0	0
$621$	37.8885	1.52041
$622$	0	0
$623$	−6.50658	−0.260681
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.47214	−0.178316
$630$	0	0
$631$	19.4164	0.772955	0.386477	−	0.922299i	$-0.373692\pi$
$631$	19.4164	0.772955	0.386477	+	0.922299i	$0.373692\pi$
$632$	0	0
$633$	−3.77709	−0.150126
$634$	0	0
$635$	−4.61803	−0.183261
$636$	0	0
$637$	−4.23607	−0.167839
$638$	0	0
$639$	12.9017	0.510383
$640$	0	0
$641$	−31.1591	−1.23071	−0.615354	−	0.788251i	$-0.710987\pi$
$641$	−31.1591	−1.23071	−0.615354	+	0.788251i	$0.710987\pi$
$642$	0	0
$643$	−13.4164	−0.529091	−0.264546	−	0.964373i	$-0.585222\pi$
$643$	−13.4164	−0.529091	−0.264546	+	0.964373i	$0.585222\pi$
$644$	0	0
$645$	−5.88854	−0.231861
$646$	0	0
$647$	24.9443	0.980661	0.490330	−	0.871537i	$-0.336876\pi$
$647$	24.9443	0.980661	0.490330	+	0.871537i	$0.336876\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−1.30495	−0.0511451
$652$	0	0
$653$	1.20163	0.0470233	0.0235116	−	0.999724i	$-0.492515\pi$
$653$	1.20163	0.0470233	0.0235116	+	0.999724i	$0.492515\pi$
$654$	0	0
$655$	−15.4164	−0.602369
$656$	0	0
$657$	−17.2361	−0.672443
$658$	0	0
$659$	15.6180	0.608392	0.304196	−	0.952609i	$-0.401612\pi$
$659$	15.6180	0.608392	0.304196	+	0.952609i	$0.401612\pi$
$660$	0	0
$661$	−20.1803	−0.784924	−0.392462	−	0.919768i	$-0.628377\pi$
$661$	−20.1803	−0.784924	−0.392462	+	0.919768i	$0.628377\pi$
$662$	0	0
$663$	−0.583592	−0.0226648
$664$	0	0
$665$	−2.70820	−0.105020
$666$	0	0
$667$	−22.1803	−0.858826
$668$	0	0
$669$	−31.3050	−1.21032
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−4.00000	−0.154189	−0.0770943	−	0.997024i	$-0.524564\pi$
$673$	−4.00000	−0.154189	−0.0770943	+	0.997024i	$0.524564\pi$
$674$	0	0
$675$	−5.52786	−0.212768
$676$	0	0
$677$	−10.0000	−0.384331	−0.192166	−	0.981363i	$-0.561551\pi$
$677$	−10.0000	−0.384331	−0.192166	+	0.981363i	$0.561551\pi$
$678$	0	0
$679$	4.58359	0.175902
$680$	0	0
$681$	9.30495	0.356567
$682$	0	0
$683$	−29.0132	−1.11016	−0.555079	−	0.831798i	$-0.687312\pi$
$683$	−29.0132	−1.11016	−0.555079	+	0.831798i	$0.687312\pi$
$684$	0	0
$685$	0.763932	0.0291883
$686$	0	0
$687$	−7.19350	−0.274449
$688$	0	0
$689$	7.47214	0.284666
$690$	0	0
$691$	28.7426	1.09342	0.546711	−	0.837321i	$-0.315880\pi$
$691$	28.7426	1.09342	0.546711	+	0.837321i	$0.315880\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−14.8541	−0.563448
$696$	0	0
$697$	−3.70820	−0.140458
$698$	0	0
$699$	13.3050	0.503239
$700$	0	0
$701$	28.1803	1.06436	0.532178	−	0.846632i	$-0.321374\pi$
$701$	28.1803	1.06436	0.532178	+	0.846632i	$0.321374\pi$
$702$	0	0
$703$	−41.5066	−1.56545
$704$	0	0
$705$	−5.34752	−0.201399
$706$	0	0
$707$	3.70820	0.139461
$708$	0	0
$709$	51.1246	1.92003	0.960013	−	0.279957i	$-0.0903202\pi$
$709$	51.1246	1.92003	0.960013	+	0.279957i	$0.0903202\pi$
$710$	0	0
$711$	−15.4164	−0.578160
$712$	0	0
$713$	−18.9443	−0.709469
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−22.8328	−0.852707
$718$	0	0
$719$	−46.1803	−1.72224	−0.861118	−	0.508405i	$-0.830235\pi$
$719$	−46.1803	−1.72224	−0.861118	+	0.508405i	$0.830235\pi$
$720$	0	0
$721$	1.65248	0.0615414
$722$	0	0
$723$	6.00000	0.223142
$724$	0	0
$725$	3.23607	0.120185
$726$	0	0
$727$	−15.7984	−0.585929	−0.292965	−	0.956123i	$-0.594642\pi$
$727$	−15.7984	−0.585929	−0.292965	+	0.956123i	$0.594642\pi$
$728$	0	0
$729$	24.0557	0.890953
$730$	0	0
$731$	−3.63932	−0.134605
$732$	0	0
$733$	−44.2492	−1.63438	−0.817191	−	0.576367i	$-0.804470\pi$
$733$	−44.2492	−1.63438	−0.817191	+	0.576367i	$0.804470\pi$
$734$	0	0
$735$	8.47214	0.312499
$736$	0	0
$737$	0	0
$738$	0	0
$739$	25.1459	0.925007	0.462503	−	0.886618i	$-0.346951\pi$
$739$	25.1459	0.925007	0.462503	+	0.886618i	$0.346951\pi$
$740$	0	0
$741$	−5.41641	−0.198977
$742$	0	0
$743$	−24.7426	−0.907720	−0.453860	−	0.891073i	$-0.649953\pi$
$743$	−24.7426	−0.907720	−0.453860	+	0.891073i	$0.649953\pi$
$744$	0	0
$745$	4.00000	0.146549
$746$	0	0
$747$	3.37384	0.123442
$748$	0	0
$749$	−6.29180	−0.229897
$750$	0	0
$751$	−6.94427	−0.253400	−0.126700	−	0.991941i	$-0.540439\pi$
$751$	−6.94427	−0.253400	−0.126700	+	0.991941i	$0.540439\pi$
$752$	0	0
$753$	−23.5279	−0.857403
$754$	0	0
$755$	−18.6525	−0.678833
$756$	0	0
$757$	46.7984	1.70092	0.850458	−	0.526043i	$-0.176325\pi$
$757$	46.7984	1.70092	0.850458	+	0.526043i	$0.176325\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	33.4164	1.21134	0.605672	−	0.795714i	$-0.292904\pi$
$761$	33.4164	1.21134	0.605672	+	0.795714i	$0.292904\pi$
$762$	0	0
$763$	−6.29180	−0.227778
$764$	0	0
$765$	−1.12461	−0.0406604
$766$	0	0
$767$	−2.85410	−0.103056
$768$	0	0
$769$	−20.9098	−0.754028	−0.377014	−	0.926208i	$-0.623049\pi$
$769$	−20.9098	−0.754028	−0.377014	+	0.926208i	$0.623049\pi$
$770$	0	0
$771$	−14.8328	−0.534191
$772$	0	0
$773$	−17.4377	−0.627190	−0.313595	−	0.949557i	$-0.601533\pi$
$773$	−17.4377	−0.627190	−0.313595	+	0.949557i	$0.601533\pi$
$774$	0	0
$775$	2.76393	0.0992834
$776$	0	0
$777$	−2.76393	−0.0991555
$778$	0	0
$779$	−34.4164	−1.23310
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−17.8885	−0.639284
$784$	0	0
$785$	−6.38197	−0.227782
$786$	0	0
$787$	36.0000	1.28326	0.641631	−	0.767014i	$-0.278258\pi$
$787$	36.0000	1.28326	0.641631	+	0.767014i	$0.278258\pi$
$788$	0	0
$789$	−1.48529	−0.0528778
$790$	0	0
$791$	−7.52786	−0.267660
$792$	0	0
$793$	−5.52786	−0.196300
$794$	0	0
$795$	−14.9443	−0.530019
$796$	0	0
$797$	7.96556	0.282155	0.141077	−	0.989999i	$-0.454943\pi$
$797$	7.96556	0.282155	0.141077	+	0.989999i	$0.454943\pi$
$798$	0	0
$799$	−3.30495	−0.116921
$800$	0	0
$801$	−25.0770	−0.886053
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−2.61803	−0.0922736
$806$	0	0
$807$	−39.0557	−1.37483
$808$	0	0
$809$	20.0344	0.704373	0.352187	−	0.935930i	$-0.385438\pi$
$809$	20.0344	0.704373	0.352187	+	0.935930i	$0.385438\pi$
$810$	0	0
$811$	44.3820	1.55846	0.779231	−	0.626737i	$-0.215610\pi$
$811$	44.3820	1.55846	0.779231	+	0.626737i	$0.215610\pi$
$812$	0	0
$813$	16.5836	0.581612
$814$	0	0
$815$	−2.76393	−0.0968163
$816$	0	0
$817$	−33.7771	−1.18171
$818$	0	0
$819$	0.347524	0.0121435
$820$	0	0
$821$	20.3607	0.710593	0.355296	−	0.934754i	$-0.384380\pi$
$821$	20.3607	0.710593	0.355296	+	0.934754i	$0.384380\pi$
$822$	0	0
$823$	−12.0344	−0.419494	−0.209747	−	0.977756i	$-0.567264\pi$
$823$	−12.0344	−0.419494	−0.209747	+	0.977756i	$0.567264\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−22.0000	−0.765015	−0.382507	−	0.923952i	$-0.624939\pi$
$827$	−22.0000	−0.765015	−0.382507	+	0.923952i	$0.624939\pi$
$828$	0	0
$829$	4.94427	0.171722	0.0858608	−	0.996307i	$-0.472636\pi$
$829$	4.94427	0.171722	0.0858608	+	0.996307i	$0.472636\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	0	0
$833$	5.23607	0.181419
$834$	0	0
$835$	19.0902	0.660643
$836$	0	0
$837$	−15.2786	−0.528107
$838$	0	0
$839$	−14.5410	−0.502012	−0.251006	−	0.967986i	$-0.580761\pi$
$839$	−14.5410	−0.502012	−0.251006	+	0.967986i	$0.580761\pi$
$840$	0	0
$841$	−18.5279	−0.638892
$842$	0	0
$843$	22.2492	0.766304
$844$	0	0
$845$	12.6180	0.434074
$846$	0	0
$847$	0	0
$848$	0	0
$849$	35.6393	1.22314
$850$	0	0
$851$	−40.1246	−1.37545
$852$	0	0
$853$	45.0344	1.54195	0.770975	−	0.636865i	$-0.219769\pi$
$853$	45.0344	1.54195	0.770975	+	0.636865i	$0.219769\pi$
$854$	0	0
$855$	−10.4377	−0.356962
$856$	0	0
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	0	0
$859$	42.0902	1.43610	0.718049	−	0.695993i	$-0.245035\pi$
$859$	42.0902	1.43610	0.718049	+	0.695993i	$0.245035\pi$
$860$	0	0
$861$	−2.29180	−0.0781042
$862$	0	0
$863$	30.7984	1.04839	0.524194	−	0.851599i	$-0.324367\pi$
$863$	30.7984	1.04839	0.524194	+	0.851599i	$0.324367\pi$
$864$	0	0
$865$	10.3820	0.352997
$866$	0	0
$867$	−20.2918	−0.689146
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−3.23607	−0.109650
$872$	0	0
$873$	17.6656	0.597891
$874$	0	0
$875$	0.381966	0.0129128
$876$	0	0
$877$	4.61803	0.155940	0.0779700	−	0.996956i	$-0.475156\pi$
$877$	4.61803	0.155940	0.0779700	+	0.996956i	$0.475156\pi$
$878$	0	0
$879$	23.1672	0.781410
$880$	0	0
$881$	28.4508	0.958533	0.479267	−	0.877669i	$-0.340903\pi$
$881$	28.4508	0.958533	0.479267	+	0.877669i	$0.340903\pi$
$882$	0	0
$883$	0.652476	0.0219576	0.0109788	−	0.999940i	$-0.496505\pi$
$883$	0.652476	0.0219576	0.0109788	+	0.999940i	$0.496505\pi$
$884$	0	0
$885$	5.70820	0.191879
$886$	0	0
$887$	−13.7295	−0.460991	−0.230496	−	0.973073i	$-0.574035\pi$
$887$	−13.7295	−0.460991	−0.230496	+	0.973073i	$0.574035\pi$
$888$	0	0
$889$	−1.76393	−0.0591604
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−30.6738	−1.02646
$894$	0	0
$895$	−14.0902	−0.470982
$896$	0	0
$897$	−5.23607	−0.174827
$898$	0	0
$899$	8.94427	0.298308
$900$	0	0
$901$	−9.23607	−0.307698
$902$	0	0
$903$	−2.24922	−0.0748495
$904$	0	0
$905$	11.5279	0.383199
$906$	0	0
$907$	−18.6525	−0.619345	−0.309673	−	0.950843i	$-0.600219\pi$
$907$	−18.6525	−0.619345	−0.309673	+	0.950843i	$0.600219\pi$
$908$	0	0
$909$	14.2918	0.474029
$910$	0	0
$911$	3.05573	0.101241	0.0506204	−	0.998718i	$-0.483880\pi$
$911$	3.05573	0.101241	0.0506204	+	0.998718i	$0.483880\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	11.0557	0.365491
$916$	0	0
$917$	−5.88854	−0.194457
$918$	0	0
$919$	12.0000	0.395843	0.197922	−	0.980218i	$-0.436581\pi$
$919$	12.0000	0.395843	0.197922	+	0.980218i	$0.436581\pi$
$920$	0	0
$921$	11.1935	0.368838
$922$	0	0
$923$	−5.41641	−0.178283
$924$	0	0
$925$	5.85410	0.192482
$926$	0	0
$927$	6.36881	0.209179
$928$	0	0
$929$	19.7295	0.647304	0.323652	−	0.946176i	$-0.395089\pi$
$929$	19.7295	0.647304	0.323652	+	0.946176i	$0.395089\pi$
$930$	0	0
$931$	48.5967	1.59269
$932$	0	0
$933$	−8.00000	−0.261908
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−25.8197	−0.843492	−0.421746	−	0.906714i	$-0.638583\pi$
$937$	−25.8197	−0.843492	−0.421746	+	0.906714i	$0.638583\pi$
$938$	0	0
$939$	0.721360	0.0235407
$940$	0	0
$941$	−2.06888	−0.0674437	−0.0337218	−	0.999431i	$-0.510736\pi$
$941$	−2.06888	−0.0674437	−0.0337218	+	0.999431i	$0.510736\pi$
$942$	0	0
$943$	−33.2705	−1.08344
$944$	0	0
$945$	−2.11146	−0.0686857
$946$	0	0
$947$	43.9574	1.42842	0.714212	−	0.699929i	$-0.246785\pi$
$947$	43.9574	1.42842	0.714212	+	0.699929i	$0.246785\pi$
$948$	0	0
$949$	7.23607	0.234893
$950$	0	0
$951$	38.0689	1.23447
$952$	0	0
$953$	2.29180	0.0742386	0.0371193	−	0.999311i	$-0.488182\pi$
$953$	2.29180	0.0742386	0.0371193	+	0.999311i	$0.488182\pi$
$954$	0	0
$955$	−1.41641	−0.0458339
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0.291796	0.00942259
$960$	0	0
$961$	−23.3607	−0.753570
$962$	0	0
$963$	−24.2492	−0.781420
$964$	0	0
$965$	−21.4164	−0.689419
$966$	0	0
$967$	−15.0902	−0.485267	−0.242634	−	0.970118i	$-0.578011\pi$
$967$	−15.0902	−0.485267	−0.242634	+	0.970118i	$0.578011\pi$
$968$	0	0
$969$	6.69505	0.215076
$970$	0	0
$971$	52.3394	1.67965	0.839826	−	0.542856i	$-0.182657\pi$
$971$	52.3394	1.67965	0.839826	+	0.542856i	$0.182657\pi$
$972$	0	0
$973$	−5.67376	−0.181892
$974$	0	0
$975$	0.763932	0.0244654
$976$	0	0
$977$	18.1803	0.581641	0.290820	−	0.956778i	$-0.406072\pi$
$977$	18.1803	0.581641	0.290820	+	0.956778i	$0.406072\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−24.2492	−0.774218
$982$	0	0
$983$	19.0344	0.607104	0.303552	−	0.952815i	$-0.401827\pi$
$983$	19.0344	0.607104	0.303552	+	0.952815i	$0.401827\pi$
$984$	0	0
$985$	16.0902	0.512675
$986$	0	0
$987$	−2.04257	−0.0650158
$988$	0	0
$989$	−32.6525	−1.03829
$990$	0	0
$991$	34.7214	1.10296	0.551480	−	0.834188i	$-0.314063\pi$
$991$	34.7214	1.10296	0.551480	+	0.834188i	$0.314063\pi$
$992$	0	0
$993$	20.5410	0.651850
$994$	0	0
$995$	−11.7082	−0.371175
$996$	0	0
$997$	−43.5279	−1.37854	−0.689271	−	0.724504i	$-0.742069\pi$
$997$	−43.5279	−1.37854	−0.689271	+	0.724504i	$0.742069\pi$
$998$	0	0
$999$	−32.3607	−1.02385

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9680.2.a.bh.1.2		2
4.3	odd	2		1210.2.a.t.1.1		2
11.3	even	5		880.2.bo.a.801.1		4
11.4	even	5		880.2.bo.a.401.1		4
11.10	odd	2		9680.2.a.bi.1.2		2
20.19	odd	2		6050.2.a.bu.1.2		2
44.3	odd	10		110.2.g.a.31.1	✓	4
44.15	odd	10		110.2.g.a.71.1	yes	4
44.43	even	2		1210.2.a.p.1.1		2
132.47	even	10		990.2.n.f.361.1		4
132.59	even	10		990.2.n.f.181.1		4
220.3	even	20		550.2.ba.a.449.1		8
220.47	even	20		550.2.ba.a.449.2		8
220.59	odd	10		550.2.h.f.401.1		4
220.103	even	20		550.2.ba.a.49.2		8
220.147	even	20		550.2.ba.a.49.1		8
220.179	odd	10		550.2.h.f.251.1		4
220.219	even	2		6050.2.a.cm.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.2.g.a.31.1	✓	4	44.3	odd	10
110.2.g.a.71.1	yes	4	44.15	odd	10
550.2.h.f.251.1		4	220.179	odd	10
550.2.h.f.401.1		4	220.59	odd	10
550.2.ba.a.49.1		8	220.147	even	20
550.2.ba.a.49.2		8	220.103	even	20
550.2.ba.a.449.1		8	220.3	even	20
550.2.ba.a.449.2		8	220.47	even	20
880.2.bo.a.401.1		4	11.4	even	5
880.2.bo.a.801.1		4	11.3	even	5
990.2.n.f.181.1		4	132.59	even	10
990.2.n.f.361.1		4	132.47	even	10
1210.2.a.p.1.1		2	44.43	even	2
1210.2.a.t.1.1		2	4.3	odd	2
6050.2.a.bu.1.2		2	20.19	odd	2
6050.2.a.cm.1.2		2	220.219	even	2
9680.2.a.bh.1.2		2	1.1	even	1	trivial
9680.2.a.bi.1.2		2	11.10	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9680.a (trivial)

\(f(q)\)	\(=\)	\(q+1.23607 q^{3} -1.00000 q^{5} -0.381966 q^{7} -1.47214 q^{9} +0.618034 q^{13} -1.23607 q^{15} -0.763932 q^{17} -7.09017 q^{19} -0.472136 q^{21} -6.85410 q^{23} +1.00000 q^{25} -5.52786 q^{27} +3.23607 q^{29} +2.76393 q^{31} +0.381966 q^{35} +5.85410 q^{37} +0.763932 q^{39} +4.85410 q^{41} +4.76393 q^{43} +1.47214 q^{45} +4.32624 q^{47} -6.85410 q^{49} -0.944272 q^{51} +12.0902 q^{53} -8.76393 q^{57} -4.61803 q^{59} -8.94427 q^{61} +0.562306 q^{63} -0.618034 q^{65} -5.23607 q^{67} -8.47214 q^{69} -8.76393 q^{71} +11.7082 q^{73} +1.23607 q^{75} +10.4721 q^{79} -2.41641 q^{81} -2.29180 q^{83} +0.763932 q^{85} +4.00000 q^{87} +17.0344 q^{89} -0.236068 q^{91} +3.41641 q^{93} +7.09017 q^{95} -12.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} - 3 q^{7} + 6 q^{9} - q^{13} + 2 q^{15} - 6 q^{17} - 3 q^{19} + 8 q^{21} - 7 q^{23} + 2 q^{25} - 20 q^{27} + 2 q^{29} + 10 q^{31} + 3 q^{35} + 5 q^{37} + 6 q^{39} + 3 q^{41} + 14 q^{43}+ \cdots - 24 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 - 3 * q^7 + 6 * q^9 - q^13 + 2 * q^15 - 6 * q^17 - 3 * q^19 + 8 * q^21 - 7 * q^23 + 2 * q^25 - 20 * q^27 + 2 * q^29 + 10 * q^31 + 3 * q^35 + 5 * q^37 + 6 * q^39 + 3 * q^41 + 14 * q^43 - 6 * q^45 - 7 * q^47 - 7 * q^49 + 16 * q^51 + 13 * q^53 - 22 * q^57 - 7 * q^59 - 19 * q^63 + q^65 - 6 * q^67 - 8 * q^69 - 22 * q^71 + 10 * q^73 - 2 * q^75 + 12 * q^79 + 22 * q^81 - 18 * q^83 + 6 * q^85 + 8 * q^87 + 5 * q^89 + 4 * q^91 - 20 * q^93 + 3 * q^95 - 24 * q^97

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