LMFDB - Embedded newform 616.2.a.g.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [616,2,Mod(1,616)] 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("616.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(616, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 616 = 2^{3} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	616.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,1,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$4.91878476451$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.254102$ of defining polynomial
Character	$\chi$	$=$	616.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.68133 q^{3} +2.68133 q^{5} +1.00000 q^{7} +4.18953 q^{9} +1.00000 q^{11} -2.50820 q^{13} +7.18953 q^{15} -6.37907 q^{17} -3.87086 q^{19} +2.68133 q^{21} -4.55220 q^{23} +2.18953 q^{25} +3.18953 q^{27} +3.01641 q^{29} +5.18953 q^{31} +2.68133 q^{33} +2.68133 q^{35} -6.55220 q^{37} -6.72532 q^{39} -4.34625 q^{41} -1.01641 q^{43} +11.2335 q^{45} +0.637339 q^{47} +1.00000 q^{49} -17.1044 q^{51} -3.01641 q^{53} +2.68133 q^{55} -10.3791 q^{57} +12.0440 q^{59} +15.6126 q^{61} +4.18953 q^{63} -6.72532 q^{65} +5.56860 q^{67} -12.2059 q^{69} +11.5358 q^{71} -13.3627 q^{73} +5.87086 q^{75} +1.00000 q^{77} +6.37907 q^{79} -4.01641 q^{81} +2.50820 q^{83} -17.1044 q^{85} +8.08798 q^{87} -12.9313 q^{89} -2.50820 q^{91} +13.9149 q^{93} -10.3791 q^{95} -10.8105 q^{97} +4.18953 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + q^{5} + 3 q^{7} + 4 q^{9} + 3 q^{11} - 6 q^{13} + 13 q^{15} - 2 q^{17} + 4 q^{19} + q^{21} + 9 q^{23} - 2 q^{25} + q^{27} + 6 q^{29} + 7 q^{31} + q^{33} + q^{35} + 3 q^{37} + 8 q^{39}+ \cdots + 4 q^{99}+O(q^{100}) $ 3 * q + q^3 + q^5 + 3 * q^7 + 4 * q^9 + 3 * q^11 - 6 * q^13 + 13 * q^15 - 2 * q^17 + 4 * q^19 + q^21 + 9 * q^23 - 2 * q^25 + q^27 + 6 * q^29 + 7 * q^31 + q^33 + q^35 + 3 * q^37 + 8 * q^39 - 2 * q^41 + 4 * q^45 + 16 * q^47 + 3 * q^49 - 6 * q^51 - 6 * q^53 + q^55 - 14 * q^57 + 15 * q^59 + 4 * q^63 + 8 * q^65 - 9 * q^67 - 25 * q^69 + 15 * q^71 - 26 * q^73 + 2 * q^75 + 3 * q^77 + 2 * q^79 - 9 * q^81 + 6 * q^83 - 6 * q^85 - 18 * q^87 + q^89 - 6 * q^91 + 5 * q^93 - 14 * q^95 - 41 * q^97 + 4 * q^99

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
$3$	2.68133	1.54807	0.774033	−	0.633145i	$-0.218236\pi$
$3$	2.68133	1.54807	0.774033	+	0.633145i	$0.218236\pi$
$4$	0	0
$5$	2.68133	1.19913	0.599564	−	0.800327i	$-0.295341\pi$
$5$	2.68133	1.19913	0.599564	+	0.800327i	$0.295341\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	4.18953	1.39651
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−2.50820	−0.695650	−0.347825	−	0.937559i	$-0.613080\pi$
$13$	−2.50820	−0.695650	−0.347825	+	0.937559i	$0.613080\pi$
$14$	0	0
$15$	7.18953	1.85633
$16$	0	0
$17$	−6.37907	−1.54715	−0.773576	−	0.633704i	$-0.781534\pi$
$17$	−6.37907	−1.54715	−0.773576	+	0.633704i	$0.781534\pi$
$18$	0	0
$19$	−3.87086	−0.888037	−0.444019	−	0.896018i	$-0.646448\pi$
$19$	−3.87086	−0.888037	−0.444019	+	0.896018i	$0.646448\pi$
$20$	0	0
$21$	2.68133	0.585114
$22$	0	0
$23$	−4.55220	−0.949198	−0.474599	−	0.880202i	$-0.657407\pi$
$23$	−4.55220	−0.949198	−0.474599	+	0.880202i	$0.657407\pi$
$24$	0	0
$25$	2.18953	0.437907
$26$	0	0
$27$	3.18953	0.613826
$28$	0	0
$29$	3.01641	0.560133	0.280066	−	0.959981i	$-0.409644\pi$
$29$	3.01641	0.560133	0.280066	+	0.959981i	$0.409644\pi$
$30$	0	0
$31$	5.18953	0.932068	0.466034	−	0.884767i	$-0.345682\pi$
$31$	5.18953	0.932068	0.466034	+	0.884767i	$0.345682\pi$
$32$	0	0
$33$	2.68133	0.466760
$34$	0	0
$35$	2.68133	0.453228
$36$	0	0
$37$	−6.55220	−1.07717	−0.538587	−	0.842570i	$-0.681042\pi$
$37$	−6.55220	−1.07717	−0.538587	+	0.842570i	$0.681042\pi$
$38$	0	0
$39$	−6.72532	−1.07691
$40$	0	0
$41$	−4.34625	−0.678771	−0.339385	−	0.940647i	$-0.610219\pi$
$41$	−4.34625	−0.678771	−0.339385	+	0.940647i	$0.610219\pi$
$42$	0	0
$43$	−1.01641	−0.155001	−0.0775003	−	0.996992i	$-0.524694\pi$
$43$	−1.01641	−0.155001	−0.0775003	+	0.996992i	$0.524694\pi$
$44$	0	0
$45$	11.2335	1.67460
$46$	0	0
$47$	0.637339	0.0929654	0.0464827	−	0.998919i	$-0.485199\pi$
$47$	0.637339	0.0929654	0.0464827	+	0.998919i	$0.485199\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−17.1044	−2.39509
$52$	0	0
$53$	−3.01641	−0.414335	−0.207168	−	0.978305i	$-0.566425\pi$
$53$	−3.01641	−0.414335	−0.207168	+	0.978305i	$0.566425\pi$
$54$	0	0
$55$	2.68133	0.361551
$56$	0	0
$57$	−10.3791	−1.37474
$58$	0	0
$59$	12.0440	1.56799	0.783997	−	0.620764i	$-0.213178\pi$
$59$	12.0440	1.56799	0.783997	+	0.620764i	$0.213178\pi$
$60$	0	0
$61$	15.6126	1.99899	0.999494	−	0.0318107i	$-0.0101274\pi$
$61$	15.6126	1.99899	0.999494	+	0.0318107i	$0.0101274\pi$
$62$	0	0
$63$	4.18953	0.527832
$64$	0	0
$65$	−6.72532	−0.834174
$66$	0	0
$67$	5.56860	0.680313	0.340157	−	0.940369i	$-0.389520\pi$
$67$	5.56860	0.680313	0.340157	+	0.940369i	$0.389520\pi$
$68$	0	0
$69$	−12.2059	−1.46942
$70$	0	0
$71$	11.5358	1.36905	0.684523	−	0.728991i	$-0.260010\pi$
$71$	11.5358	1.36905	0.684523	+	0.728991i	$0.260010\pi$
$72$	0	0
$73$	−13.3627	−1.56398	−0.781991	−	0.623290i	$-0.785796\pi$
$73$	−13.3627	−1.56398	−0.781991	+	0.623290i	$0.785796\pi$
$74$	0	0
$75$	5.87086	0.677909
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	6.37907	0.717701	0.358851	−	0.933395i	$-0.383169\pi$
$79$	6.37907	0.717701	0.358851	+	0.933395i	$0.383169\pi$
$80$	0	0
$81$	−4.01641	−0.446267
$82$	0	0
$83$	2.50820	0.275311	0.137656	−	0.990480i	$-0.456043\pi$
$83$	2.50820	0.275311	0.137656	+	0.990480i	$0.456043\pi$
$84$	0	0
$85$	−17.1044	−1.85523
$86$	0	0
$87$	8.08798	0.867123
$88$	0	0
$89$	−12.9313	−1.37071	−0.685356	−	0.728209i	$-0.740353\pi$
$89$	−12.9313	−1.37071	−0.685356	+	0.728209i	$0.740353\pi$
$90$	0	0
$91$	−2.50820	−0.262931
$92$	0	0
$93$	13.9149	1.44290
$94$	0	0
$95$	−10.3791	−1.06487
$96$	0	0
$97$	−10.8105	−1.09764	−0.548818	−	0.835942i	$-0.684922\pi$
$97$	−10.8105	−1.09764	−0.548818	+	0.835942i	$0.684922\pi$
$98$	0	0
$99$	4.18953	0.421064
$100$	0	0
$101$	15.2663	1.51906	0.759529	−	0.650474i	$-0.225430\pi$
$101$	15.2663	1.51906	0.759529	+	0.650474i	$0.225430\pi$
$102$	0	0
$103$	19.3627	1.90786	0.953930	−	0.300030i	$-0.0969966\pi$
$103$	19.3627	1.90786	0.953930	+	0.300030i	$0.0969966\pi$
$104$	0	0
$105$	7.18953	0.701627
$106$	0	0
$107$	7.39547	0.714948	0.357474	−	0.933923i	$-0.383638\pi$
$107$	7.39547	0.714948	0.357474	+	0.933923i	$0.383638\pi$
$108$	0	0
$109$	4.37907	0.419439	0.209719	−	0.977762i	$-0.432745\pi$
$109$	4.37907	0.419439	0.209719	+	0.977762i	$0.432745\pi$
$110$	0	0
$111$	−17.5686	−1.66754
$112$	0	0
$113$	0.843279	0.0793291	0.0396645	−	0.999213i	$-0.487371\pi$
$113$	0.843279	0.0793291	0.0396645	+	0.999213i	$0.487371\pi$
$114$	0	0
$115$	−12.2059	−1.13821
$116$	0	0
$117$	−10.5082	−0.971484
$118$	0	0
$119$	−6.37907	−0.584768
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−11.6537	−1.05078
$124$	0	0
$125$	−7.53579	−0.674021
$126$	0	0
$127$	−5.10439	−0.452941	−0.226471	−	0.974018i	$-0.572719\pi$
$127$	−5.10439	−0.452941	−0.226471	+	0.974018i	$0.572719\pi$
$128$	0	0
$129$	−2.72532	−0.239951
$130$	0	0
$131$	15.6126	1.36408	0.682039	−	0.731315i	$-0.261093\pi$
$131$	15.6126	1.36408	0.682039	+	0.731315i	$0.261093\pi$
$132$	0	0
$133$	−3.87086	−0.335647
$134$	0	0
$135$	8.55220	0.736056
$136$	0	0
$137$	17.2775	1.47612	0.738059	−	0.674736i	$-0.235743\pi$
$137$	17.2775	1.47612	0.738059	+	0.674736i	$0.235743\pi$
$138$	0	0
$139$	−14.8545	−1.25994	−0.629969	−	0.776620i	$-0.716933\pi$
$139$	−14.8545	−1.25994	−0.629969	+	0.776620i	$0.716933\pi$
$140$	0	0
$141$	1.70892	0.143917
$142$	0	0
$143$	−2.50820	−0.209747
$144$	0	0
$145$	8.08798	0.671671
$146$	0	0
$147$	2.68133	0.221152
$148$	0	0
$149$	2.25827	0.185005	0.0925024	−	0.995712i	$-0.470513\pi$
$149$	2.25827	0.185005	0.0925024	+	0.995712i	$0.470513\pi$
$150$	0	0
$151$	−18.3791	−1.49567	−0.747834	−	0.663886i	$-0.768906\pi$
$151$	−18.3791	−1.49567	−0.747834	+	0.663886i	$0.768906\pi$
$152$	0	0
$153$	−26.7253	−2.16061
$154$	0	0
$155$	13.9149	1.11767
$156$	0	0
$157$	−23.7857	−1.89831	−0.949154	−	0.314813i	$-0.898058\pi$
$157$	−23.7857	−1.89831	−0.949154	+	0.314813i	$0.898058\pi$
$158$	0	0
$159$	−8.08798	−0.641419
$160$	0	0
$161$	−4.55220	−0.358763
$162$	0	0
$163$	−15.7417	−1.23299	−0.616494	−	0.787360i	$-0.711447\pi$
$163$	−15.7417	−1.23299	−0.616494	+	0.787360i	$0.711447\pi$
$164$	0	0
$165$	7.18953	0.559704
$166$	0	0
$167$	−6.63734	−0.513613	−0.256806	−	0.966463i	$-0.582670\pi$
$167$	−6.63734	−0.513613	−0.256806	+	0.966463i	$0.582670\pi$
$168$	0	0
$169$	−6.70892	−0.516070
$170$	0	0
$171$	−16.2171	−1.24015
$172$	0	0
$173$	−15.2663	−1.16068	−0.580339	−	0.814375i	$-0.697080\pi$
$173$	−15.2663	−1.16068	−0.580339	+	0.814375i	$0.697080\pi$
$174$	0	0
$175$	2.18953	0.165513
$176$	0	0
$177$	32.2939	2.42736
$178$	0	0
$179$	−16.8984	−1.26305	−0.631525	−	0.775356i	$-0.717571\pi$
$179$	−16.8984	−1.26305	−0.631525	+	0.775356i	$0.717571\pi$
$180$	0	0
$181$	0.0439919	0.00326989	0.00163495	−	0.999999i	$-0.499480\pi$
$181$	0.0439919	0.00326989	0.00163495	+	0.999999i	$0.499480\pi$
$182$	0	0
$183$	41.8625	3.09457
$184$	0	0
$185$	−17.5686	−1.29167
$186$	0	0
$187$	−6.37907	−0.466484
$188$	0	0
$189$	3.18953	0.232004
$190$	0	0
$191$	23.5358	1.70299	0.851495	−	0.524363i	$-0.175696\pi$
$191$	23.5358	1.70299	0.851495	+	0.524363i	$0.175696\pi$
$192$	0	0
$193$	16.7253	1.20392	0.601958	−	0.798528i	$-0.294388\pi$
$193$	16.7253	1.20392	0.601958	+	0.798528i	$0.294388\pi$
$194$	0	0
$195$	−18.0328	−1.29136
$196$	0	0
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	2.67015	0.189282	0.0946410	−	0.995511i	$-0.469830\pi$
$199$	2.67015	0.189282	0.0946410	+	0.995511i	$0.469830\pi$
$200$	0	0
$201$	14.9313	1.05317
$202$	0	0
$203$	3.01641	0.211710
$204$	0	0
$205$	−11.6537	−0.813933
$206$	0	0
$207$	−19.0716	−1.32557
$208$	0	0
$209$	−3.87086	−0.267753
$210$	0	0
$211$	−14.0552	−0.967598	−0.483799	−	0.875179i	$-0.660743\pi$
$211$	−14.0552	−0.967598	−0.483799	+	0.875179i	$0.660743\pi$
$212$	0	0
$213$	30.9313	2.11938
$214$	0	0
$215$	−2.72532	−0.185865
$216$	0	0
$217$	5.18953	0.352289
$218$	0	0
$219$	−35.8297	−2.42115
$220$	0	0
$221$	16.0000	1.07628
$222$	0	0
$223$	16.9313	1.13380	0.566901	−	0.823786i	$-0.308142\pi$
$223$	16.9313	1.13380	0.566901	+	0.823786i	$0.308142\pi$
$224$	0	0
$225$	9.17313	0.611542
$226$	0	0
$227$	−9.49180	−0.629993	−0.314996	−	0.949093i	$-0.602003\pi$
$227$	−9.49180	−0.629993	−0.314996	+	0.949093i	$0.602003\pi$
$228$	0	0
$229$	−0.648517	−0.0428552	−0.0214276	−	0.999770i	$-0.506821\pi$
$229$	−0.648517	−0.0428552	−0.0214276	+	0.999770i	$0.506821\pi$
$230$	0	0
$231$	2.68133	0.176419
$232$	0	0
$233$	14.6925	0.962538	0.481269	−	0.876573i	$-0.340176\pi$
$233$	14.6925	0.962538	0.481269	+	0.876573i	$0.340176\pi$
$234$	0	0
$235$	1.70892	0.111477
$236$	0	0
$237$	17.1044	1.11105
$238$	0	0
$239$	−16.7581	−1.08399	−0.541997	−	0.840381i	$-0.682332\pi$
$239$	−16.7581	−1.08399	−0.541997	+	0.840381i	$0.682332\pi$
$240$	0	0
$241$	−20.7581	−1.33715	−0.668575	−	0.743645i	$-0.733095\pi$
$241$	−20.7581	−1.33715	−0.668575	+	0.743645i	$0.733095\pi$
$242$	0	0
$243$	−20.3379	−1.30468
$244$	0	0
$245$	2.68133	0.171304
$246$	0	0
$247$	9.70892	0.617764
$248$	0	0
$249$	6.72532	0.426200
$250$	0	0
$251$	−7.69774	−0.485877	−0.242938	−	0.970042i	$-0.578111\pi$
$251$	−7.69774	−0.485877	−0.242938	+	0.970042i	$0.578111\pi$
$252$	0	0
$253$	−4.55220	−0.286194
$254$	0	0
$255$	−45.8625	−2.87202
$256$	0	0
$257$	25.4835	1.58961	0.794807	−	0.606862i	$-0.207572\pi$
$257$	25.4835	1.58961	0.794807	+	0.606862i	$0.207572\pi$
$258$	0	0
$259$	−6.55220	−0.407134
$260$	0	0
$261$	12.6373	0.782232
$262$	0	0
$263$	27.8297	1.71605	0.858027	−	0.513605i	$-0.171690\pi$
$263$	27.8297	1.71605	0.858027	+	0.513605i	$0.171690\pi$
$264$	0	0
$265$	−8.08798	−0.496841
$266$	0	0
$267$	−34.6730	−2.12195
$268$	0	0
$269$	4.76647	0.290617	0.145309	−	0.989386i	$-0.453583\pi$
$269$	4.76647	0.290617	0.145309	+	0.989386i	$0.453583\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	0	0
$273$	−6.72532	−0.407035
$274$	0	0
$275$	2.18953	0.132034
$276$	0	0
$277$	−18.0880	−1.08680	−0.543401	−	0.839473i	$-0.682864\pi$
$277$	−18.0880	−1.08680	−0.543401	+	0.839473i	$0.682864\pi$
$278$	0	0
$279$	21.7417	1.30164
$280$	0	0
$281$	18.3463	1.09445	0.547223	−	0.836987i	$-0.315685\pi$
$281$	18.3463	1.09445	0.547223	+	0.836987i	$0.315685\pi$
$282$	0	0
$283$	32.7170	1.94482	0.972411	−	0.233272i	$-0.0749434\pi$
$283$	32.7170	1.94482	0.972411	+	0.233272i	$0.0749434\pi$
$284$	0	0
$285$	−27.8297	−1.64849
$286$	0	0
$287$	−4.34625	−0.256551
$288$	0	0
$289$	23.6925	1.39368
$290$	0	0
$291$	−28.9864	−1.69921
$292$	0	0
$293$	10.1619	0.593667	0.296834	−	0.954929i	$-0.404069\pi$
$293$	10.1619	0.593667	0.296834	+	0.954929i	$0.404069\pi$
$294$	0	0
$295$	32.2939	1.88022
$296$	0	0
$297$	3.18953	0.185076
$298$	0	0
$299$	11.4178	0.660310
$300$	0	0
$301$	−1.01641	−0.0585847
$302$	0	0
$303$	40.9341	2.35160
$304$	0	0
$305$	41.8625	2.39704
$306$	0	0
$307$	−12.2171	−0.697268	−0.348634	−	0.937259i	$-0.613354\pi$
$307$	−12.2171	−0.697268	−0.348634	+	0.937259i	$0.613354\pi$
$308$	0	0
$309$	51.9177	2.95349
$310$	0	0
$311$	−17.8297	−1.01103	−0.505515	−	0.862818i	$-0.668698\pi$
$311$	−17.8297	−1.01103	−0.505515	+	0.862818i	$0.668698\pi$
$312$	0	0
$313$	−15.9149	−0.899561	−0.449780	−	0.893139i	$-0.648498\pi$
$313$	−15.9149	−0.899561	−0.449780	+	0.893139i	$0.648498\pi$
$314$	0	0
$315$	11.2335	0.632937
$316$	0	0
$317$	19.3103	1.08458	0.542288	−	0.840193i	$-0.317558\pi$
$317$	19.3103	1.08458	0.542288	+	0.840193i	$0.317558\pi$
$318$	0	0
$319$	3.01641	0.168886
$320$	0	0
$321$	19.8297	1.10679
$322$	0	0
$323$	24.6925	1.37393
$324$	0	0
$325$	−5.49180	−0.304630
$326$	0	0
$327$	11.7417	0.649319
$328$	0	0
$329$	0.637339	0.0351376
$330$	0	0
$331$	8.89845	0.489103	0.244552	−	0.969636i	$-0.421359\pi$
$331$	8.89845	0.489103	0.244552	+	0.969636i	$0.421359\pi$
$332$	0	0
$333$	−27.4506	−1.50429
$334$	0	0
$335$	14.9313	0.815782
$336$	0	0
$337$	−16.0328	−0.873363	−0.436682	−	0.899616i	$-0.643846\pi$
$337$	−16.0328	−0.873363	−0.436682	+	0.899616i	$0.643846\pi$
$338$	0	0
$339$	2.26111	0.122807
$340$	0	0
$341$	5.18953	0.281029
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−32.7282	−1.76203
$346$	0	0
$347$	0.604525	0.0324526	0.0162263	−	0.999868i	$-0.494835\pi$
$347$	0.604525	0.0324526	0.0162263	+	0.999868i	$0.494835\pi$
$348$	0	0
$349$	18.5962	0.995431	0.497716	−	0.867340i	$-0.334172\pi$
$349$	18.5962	0.995431	0.497716	+	0.867340i	$0.334172\pi$
$350$	0	0
$351$	−8.00000	−0.427008
$352$	0	0
$353$	26.5522	1.41323	0.706615	−	0.707598i	$-0.250221\pi$
$353$	26.5522	1.41323	0.706615	+	0.707598i	$0.250221\pi$
$354$	0	0
$355$	30.9313	1.64166
$356$	0	0
$357$	−17.1044	−0.905260
$358$	0	0
$359$	25.4506	1.34323	0.671617	−	0.740899i	$-0.265600\pi$
$359$	25.4506	1.34323	0.671617	+	0.740899i	$0.265600\pi$
$360$	0	0
$361$	−4.01641	−0.211390
$362$	0	0
$363$	2.68133	0.140733
$364$	0	0
$365$	−35.8297	−1.87541
$366$	0	0
$367$	−24.7610	−1.29251	−0.646256	−	0.763120i	$-0.723666\pi$
$367$	−24.7610	−1.29251	−0.646256	+	0.763120i	$0.723666\pi$
$368$	0	0
$369$	−18.2088	−0.947911
$370$	0	0
$371$	−3.01641	−0.156604
$372$	0	0
$373$	18.4119	0.953331	0.476666	−	0.879085i	$-0.341845\pi$
$373$	18.4119	0.953331	0.476666	+	0.879085i	$0.341845\pi$
$374$	0	0
$375$	−20.2059	−1.04343
$376$	0	0
$377$	−7.56576	−0.389657
$378$	0	0
$379$	11.9477	0.613711	0.306855	−	0.951756i	$-0.400723\pi$
$379$	11.9477	0.613711	0.306855	+	0.951756i	$0.400723\pi$
$380$	0	0
$381$	−13.6866	−0.701184
$382$	0	0
$383$	−16.2611	−0.830904	−0.415452	−	0.909615i	$-0.636377\pi$
$383$	−16.2611	−0.830904	−0.415452	+	0.909615i	$0.636377\pi$
$384$	0	0
$385$	2.68133	0.136653
$386$	0	0
$387$	−4.25827	−0.216460
$388$	0	0
$389$	−8.67299	−0.439738	−0.219869	−	0.975529i	$-0.570563\pi$
$389$	−8.67299	−0.439738	−0.219869	+	0.975529i	$0.570563\pi$
$390$	0	0
$391$	29.0388	1.46855
$392$	0	0
$393$	41.8625	2.11169
$394$	0	0
$395$	17.1044	0.860615
$396$	0	0
$397$	−23.2335	−1.16606	−0.583029	−	0.812452i	$-0.698132\pi$
$397$	−23.2335	−1.16606	−0.583029	+	0.812452i	$0.698132\pi$
$398$	0	0
$399$	−10.3791	−0.519603
$400$	0	0
$401$	37.7417	1.88473	0.942366	−	0.334584i	$-0.108596\pi$
$401$	37.7417	1.88473	0.942366	+	0.334584i	$0.108596\pi$
$402$	0	0
$403$	−13.0164	−0.648393
$404$	0	0
$405$	−10.7693	−0.535132
$406$	0	0
$407$	−6.55220	−0.324780
$408$	0	0
$409$	−19.0716	−0.943029	−0.471514	−	0.881858i	$-0.656292\pi$
$409$	−19.0716	−0.943029	−0.471514	+	0.881858i	$0.656292\pi$
$410$	0	0
$411$	46.3267	2.28513
$412$	0	0
$413$	12.0440	0.592646
$414$	0	0
$415$	6.72532	0.330133
$416$	0	0
$417$	−39.8297	−1.95047
$418$	0	0
$419$	−24.5962	−1.20160	−0.600801	−	0.799398i	$-0.705152\pi$
$419$	−24.5962	−1.20160	−0.600801	+	0.799398i	$0.705152\pi$
$420$	0	0
$421$	−19.7089	−0.960554	−0.480277	−	0.877117i	$-0.659464\pi$
$421$	−19.7089	−0.960554	−0.480277	+	0.877117i	$0.659464\pi$
$422$	0	0
$423$	2.67015	0.129827
$424$	0	0
$425$	−13.9672	−0.677508
$426$	0	0
$427$	15.6126	0.755546
$428$	0	0
$429$	−6.72532	−0.324702
$430$	0	0
$431$	8.06563	0.388508	0.194254	−	0.980951i	$-0.437771\pi$
$431$	8.06563	0.388508	0.194254	+	0.980951i	$0.437771\pi$
$432$	0	0
$433$	−23.5686	−1.13263	−0.566317	−	0.824187i	$-0.691632\pi$
$433$	−23.5686	−1.13263	−0.566317	+	0.824187i	$0.691632\pi$
$434$	0	0
$435$	21.6866	1.03979
$436$	0	0
$437$	17.6209	0.842923
$438$	0	0
$439$	−4.41188	−0.210568	−0.105284	−	0.994442i	$-0.533575\pi$
$439$	−4.41188	−0.210568	−0.105284	+	0.994442i	$0.533575\pi$
$440$	0	0
$441$	4.18953	0.199502
$442$	0	0
$443$	20.5522	0.976464	0.488232	−	0.872714i	$-0.337642\pi$
$443$	20.5522	0.976464	0.488232	+	0.872714i	$0.337642\pi$
$444$	0	0
$445$	−34.6730	−1.64366
$446$	0	0
$447$	6.05517	0.286400
$448$	0	0
$449$	10.2059	0.481648	0.240824	−	0.970569i	$-0.422582\pi$
$449$	10.2059	0.481648	0.240824	+	0.970569i	$0.422582\pi$
$450$	0	0
$451$	−4.34625	−0.204657
$452$	0	0
$453$	−49.2804	−2.31539
$454$	0	0
$455$	−6.72532	−0.315288
$456$	0	0
$457$	17.8297	0.834039	0.417019	−	0.908898i	$-0.363075\pi$
$457$	17.8297	0.834039	0.417019	+	0.908898i	$0.363075\pi$
$458$	0	0
$459$	−20.3463	−0.949682
$460$	0	0
$461$	−25.6454	−1.19443	−0.597213	−	0.802083i	$-0.703725\pi$
$461$	−25.6454	−1.19443	−0.597213	+	0.802083i	$0.703725\pi$
$462$	0	0
$463$	−3.25516	−0.151280	−0.0756401	−	0.997135i	$-0.524100\pi$
$463$	−3.25516	−0.151280	−0.0756401	+	0.997135i	$0.524100\pi$
$464$	0	0
$465$	37.3103	1.73023
$466$	0	0
$467$	−24.1320	−1.11669	−0.558347	−	0.829607i	$-0.688564\pi$
$467$	−24.1320	−1.11669	−0.558347	+	0.829607i	$0.688564\pi$
$468$	0	0
$469$	5.56860	0.257134
$470$	0	0
$471$	−63.7774	−2.93871
$472$	0	0
$473$	−1.01641	−0.0467344
$474$	0	0
$475$	−8.47539	−0.388878
$476$	0	0
$477$	−12.6373	−0.578624
$478$	0	0
$479$	−7.67610	−0.350730	−0.175365	−	0.984503i	$-0.556111\pi$
$479$	−7.67610	−0.350730	−0.175365	+	0.984503i	$0.556111\pi$
$480$	0	0
$481$	16.4342	0.749337
$482$	0	0
$483$	−12.2059	−0.555390
$484$	0	0
$485$	−28.9864	−1.31621
$486$	0	0
$487$	−14.5850	−0.660910	−0.330455	−	0.943822i	$-0.607202\pi$
$487$	−14.5850	−0.660910	−0.330455	+	0.943822i	$0.607202\pi$
$488$	0	0
$489$	−42.2088	−1.90875
$490$	0	0
$491$	−39.3132	−1.77418	−0.887089	−	0.461598i	$-0.847276\pi$
$491$	−39.3132	−1.77418	−0.887089	+	0.461598i	$0.847276\pi$
$492$	0	0
$493$	−19.2419	−0.866610
$494$	0	0
$495$	11.2335	0.504909
$496$	0	0
$497$	11.5358	0.517451
$498$	0	0
$499$	2.29108	0.102563	0.0512815	−	0.998684i	$-0.483669\pi$
$499$	2.29108	0.102563	0.0512815	+	0.998684i	$0.483669\pi$
$500$	0	0
$501$	−17.7969	−0.795107
$502$	0	0
$503$	6.03281	0.268990	0.134495	−	0.990914i	$-0.457059\pi$
$503$	6.03281	0.268990	0.134495	+	0.990914i	$0.457059\pi$
$504$	0	0
$505$	40.9341	1.82154
$506$	0	0
$507$	−17.9888	−0.798912
$508$	0	0
$509$	−5.23069	−0.231846	−0.115923	−	0.993258i	$-0.536983\pi$
$509$	−5.23069	−0.231846	−0.115923	+	0.993258i	$0.536983\pi$
$510$	0	0
$511$	−13.3627	−0.591129
$512$	0	0
$513$	−12.3463	−0.545100
$514$	0	0
$515$	51.9177	2.28777
$516$	0	0
$517$	0.637339	0.0280301
$518$	0	0
$519$	−40.9341	−1.79681
$520$	0	0
$521$	29.3655	1.28653	0.643263	−	0.765645i	$-0.277580\pi$
$521$	29.3655	1.28653	0.643263	+	0.765645i	$0.277580\pi$
$522$	0	0
$523$	25.9260	1.13367	0.566833	−	0.823833i	$-0.308168\pi$
$523$	25.9260	1.13367	0.566833	+	0.823833i	$0.308168\pi$
$524$	0	0
$525$	5.87086	0.256226
$526$	0	0
$527$	−33.1044	−1.44205
$528$	0	0
$529$	−2.27752	−0.0990225
$530$	0	0
$531$	50.4587	2.18972
$532$	0	0
$533$	10.9013	0.472187
$534$	0	0
$535$	19.8297	0.857313
$536$	0	0
$537$	−45.3103	−1.95529
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	5.82971	0.250639	0.125319	−	0.992116i	$-0.460004\pi$
$541$	5.82971	0.250639	0.125319	+	0.992116i	$0.460004\pi$
$542$	0	0
$543$	0.117957	0.00506202
$544$	0	0
$545$	11.7417	0.502960
$546$	0	0
$547$	−17.2747	−0.738612	−0.369306	−	0.929308i	$-0.620405\pi$
$547$	−17.2747	−0.738612	−0.369306	+	0.929308i	$0.620405\pi$
$548$	0	0
$549$	65.4095	2.79161
$550$	0	0
$551$	−11.6761	−0.497419
$552$	0	0
$553$	6.37907	0.271266
$554$	0	0
$555$	−47.1072	−1.99959
$556$	0	0
$557$	12.0328	0.509847	0.254923	−	0.966961i	$-0.417950\pi$
$557$	12.0328	0.509847	0.254923	+	0.966961i	$0.417950\pi$
$558$	0	0
$559$	2.54935	0.107826
$560$	0	0
$561$	−17.1044	−0.722148
$562$	0	0
$563$	−21.2335	−0.894886	−0.447443	−	0.894312i	$-0.647665\pi$
$563$	−21.2335	−0.894886	−0.447443	+	0.894312i	$0.647665\pi$
$564$	0	0
$565$	2.26111	0.0951257
$566$	0	0
$567$	−4.01641	−0.168673
$568$	0	0
$569$	−15.5163	−0.650476	−0.325238	−	0.945632i	$-0.605444\pi$
$569$	−15.5163	−0.650476	−0.325238	+	0.945632i	$0.605444\pi$
$570$	0	0
$571$	17.4506	0.730287	0.365143	−	0.930951i	$-0.381020\pi$
$571$	17.4506	0.730287	0.365143	+	0.930951i	$0.381020\pi$
$572$	0	0
$573$	63.1072	2.63634
$574$	0	0
$575$	−9.96719	−0.415660
$576$	0	0
$577$	32.9969	1.37368	0.686839	−	0.726809i	$-0.258998\pi$
$577$	32.9969	1.37368	0.686839	+	0.726809i	$0.258998\pi$
$578$	0	0
$579$	44.8461	1.86374
$580$	0	0
$581$	2.50820	0.104058
$582$	0	0
$583$	−3.01641	−0.124927
$584$	0	0
$585$	−28.1760	−1.16493
$586$	0	0
$587$	−4.59619	−0.189705	−0.0948525	−	0.995491i	$-0.530238\pi$
$587$	−4.59619	−0.189705	−0.0948525	+	0.995491i	$0.530238\pi$
$588$	0	0
$589$	−20.0880	−0.827711
$590$	0	0
$591$	5.36266	0.220590
$592$	0	0
$593$	−5.87920	−0.241430	−0.120715	−	0.992687i	$-0.538519\pi$
$593$	−5.87920	−0.241430	−0.120715	+	0.992687i	$0.538519\pi$
$594$	0	0
$595$	−17.1044	−0.701212
$596$	0	0
$597$	7.15956	0.293021
$598$	0	0
$599$	8.43424	0.344614	0.172307	−	0.985043i	$-0.444878\pi$
$599$	8.43424	0.344614	0.172307	+	0.985043i	$0.444878\pi$
$600$	0	0
$601$	2.46705	0.100633	0.0503166	−	0.998733i	$-0.483977\pi$
$601$	2.46705	0.100633	0.0503166	+	0.998733i	$0.483977\pi$
$602$	0	0
$603$	23.3298	0.950065
$604$	0	0
$605$	2.68133	0.109012
$606$	0	0
$607$	4.34625	0.176409	0.0882045	−	0.996102i	$-0.471887\pi$
$607$	4.34625	0.176409	0.0882045	+	0.996102i	$0.471887\pi$
$608$	0	0
$609$	8.08798	0.327742
$610$	0	0
$611$	−1.59858	−0.0646714
$612$	0	0
$613$	36.8133	1.48688	0.743438	−	0.668805i	$-0.233194\pi$
$613$	36.8133	1.48688	0.743438	+	0.668805i	$0.233194\pi$
$614$	0	0
$615$	−31.2475	−1.26002
$616$	0	0
$617$	−13.0492	−0.525342	−0.262671	−	0.964885i	$-0.584603\pi$
$617$	−13.0492	−0.525342	−0.262671	+	0.964885i	$0.584603\pi$
$618$	0	0
$619$	12.2143	0.490933	0.245467	−	0.969405i	$-0.421059\pi$
$619$	12.2143	0.490933	0.245467	+	0.969405i	$0.421059\pi$
$620$	0	0
$621$	−14.5194	−0.582643
$622$	0	0
$623$	−12.9313	−0.518080
$624$	0	0
$625$	−31.1536	−1.24614
$626$	0	0
$627$	−10.3791	−0.414500
$628$	0	0
$629$	41.7969	1.66655
$630$	0	0
$631$	4.52984	0.180330	0.0901650	−	0.995927i	$-0.471261\pi$
$631$	4.52984	0.180330	0.0901650	+	0.995927i	$0.471261\pi$
$632$	0	0
$633$	−37.6866	−1.49791
$634$	0	0
$635$	−13.6866	−0.543135
$636$	0	0
$637$	−2.50820	−0.0993786
$638$	0	0
$639$	48.3296	1.91189
$640$	0	0
$641$	−6.11796	−0.241645	−0.120822	−	0.992674i	$-0.538553\pi$
$641$	−6.11796	−0.241645	−0.120822	+	0.992674i	$0.538553\pi$
$642$	0	0
$643$	−38.6647	−1.52479	−0.762393	−	0.647115i	$-0.775975\pi$
$643$	−38.6647	−1.52479	−0.762393	+	0.647115i	$0.775975\pi$
$644$	0	0
$645$	−7.30749	−0.287732
$646$	0	0
$647$	−11.7389	−0.461503	−0.230752	−	0.973013i	$-0.574118\pi$
$647$	−11.7389	−0.461503	−0.230752	+	0.973013i	$0.574118\pi$
$648$	0	0
$649$	12.0440	0.472768
$650$	0	0
$651$	13.9149	0.545366
$652$	0	0
$653$	−34.9864	−1.36912	−0.684562	−	0.728954i	$-0.740007\pi$
$653$	−34.9864	−1.36912	−0.684562	+	0.728954i	$0.740007\pi$
$654$	0	0
$655$	41.8625	1.63570
$656$	0	0
$657$	−55.9833	−2.18412
$658$	0	0
$659$	2.12080	0.0826145	0.0413073	−	0.999146i	$-0.486848\pi$
$659$	2.12080	0.0826145	0.0413073	+	0.999146i	$0.486848\pi$
$660$	0	0
$661$	−22.1424	−0.861241	−0.430620	−	0.902533i	$-0.641705\pi$
$661$	−22.1424	−0.861241	−0.430620	+	0.902533i	$0.641705\pi$
$662$	0	0
$663$	42.9013	1.66615
$664$	0	0
$665$	−10.3791	−0.402483
$666$	0	0
$667$	−13.7313	−0.531677
$668$	0	0
$669$	45.3983	1.75520
$670$	0	0
$671$	15.6126	0.602718
$672$	0	0
$673$	−30.5222	−1.17655	−0.588273	−	0.808663i	$-0.700192\pi$
$673$	−30.5222	−1.17655	−0.588273	+	0.808663i	$0.700192\pi$
$674$	0	0
$675$	6.98359	0.268799
$676$	0	0
$677$	1.32151	0.0507897	0.0253949	−	0.999677i	$-0.491916\pi$
$677$	1.32151	0.0507897	0.0253949	+	0.999677i	$0.491916\pi$
$678$	0	0
$679$	−10.8105	−0.414868
$680$	0	0
$681$	−25.4506	−0.975271
$682$	0	0
$683$	48.9341	1.87241	0.936206	−	0.351452i	$-0.114312\pi$
$683$	48.9341	1.87241	0.936206	+	0.351452i	$0.114312\pi$
$684$	0	0
$685$	46.3267	1.77005
$686$	0	0
$687$	−1.73889	−0.0663427
$688$	0	0
$689$	7.56576	0.288233
$690$	0	0
$691$	−0.714144	−0.0271673	−0.0135837	−	0.999908i	$-0.504324\pi$
$691$	−0.714144	−0.0271673	−0.0135837	+	0.999908i	$0.504324\pi$
$692$	0	0
$693$	4.18953	0.159147
$694$	0	0
$695$	−39.8297	−1.51083
$696$	0	0
$697$	27.7251	1.05016
$698$	0	0
$699$	39.3955	1.49007
$700$	0	0
$701$	−23.0388	−0.870162	−0.435081	−	0.900391i	$-0.643280\pi$
$701$	−23.0388	−0.870162	−0.435081	+	0.900391i	$0.643280\pi$
$702$	0	0
$703$	25.3627	0.956571
$704$	0	0
$705$	4.58217	0.172574
$706$	0	0
$707$	15.2663	0.574150
$708$	0	0
$709$	−13.5358	−0.508347	−0.254174	−	0.967159i	$-0.581803\pi$
$709$	−13.5358	−0.508347	−0.254174	+	0.967159i	$0.581803\pi$
$710$	0	0
$711$	26.7253	1.00228
$712$	0	0
$713$	−23.6238	−0.884717
$714$	0	0
$715$	−6.72532	−0.251513
$716$	0	0
$717$	−44.9341	−1.67809
$718$	0	0
$719$	−52.5027	−1.95802	−0.979010	−	0.203811i	$-0.934667\pi$
$719$	−52.5027	−1.95802	−0.979010	+	0.203811i	$0.934667\pi$
$720$	0	0
$721$	19.3627	0.721103
$722$	0	0
$723$	−55.6594	−2.07000
$724$	0	0
$725$	6.60453	0.245286
$726$	0	0
$727$	22.0133	0.816428	0.408214	−	0.912886i	$-0.366152\pi$
$727$	22.0133	0.816428	0.408214	+	0.912886i	$0.366152\pi$
$728$	0	0
$729$	−42.4835	−1.57346
$730$	0	0
$731$	6.48373	0.239809
$732$	0	0
$733$	−30.9424	−1.14289	−0.571443	−	0.820642i	$-0.693616\pi$
$733$	−30.9424	−1.14289	−0.571443	+	0.820642i	$0.693616\pi$
$734$	0	0
$735$	7.18953	0.265190
$736$	0	0
$737$	5.56860	0.205122
$738$	0	0
$739$	17.0388	0.626781	0.313391	−	0.949624i	$-0.398535\pi$
$739$	17.0388	0.626781	0.313391	+	0.949624i	$0.398535\pi$
$740$	0	0
$741$	26.0328	0.956339
$742$	0	0
$743$	22.2088	0.814761	0.407381	−	0.913258i	$-0.366442\pi$
$743$	22.2088	0.814761	0.407381	+	0.913258i	$0.366442\pi$
$744$	0	0
$745$	6.05517	0.221844
$746$	0	0
$747$	10.5082	0.384475
$748$	0	0
$749$	7.39547	0.270225
$750$	0	0
$751$	17.8269	0.650512	0.325256	−	0.945626i	$-0.394550\pi$
$751$	17.8269	0.650512	0.325256	+	0.945626i	$0.394550\pi$
$752$	0	0
$753$	−20.6402	−0.752170
$754$	0	0
$755$	−49.2804	−1.79350
$756$	0	0
$757$	8.98359	0.326514	0.163257	−	0.986584i	$-0.447800\pi$
$757$	8.98359	0.326514	0.163257	+	0.986584i	$0.447800\pi$
$758$	0	0
$759$	−12.2059	−0.443048
$760$	0	0
$761$	42.9669	1.55755	0.778775	−	0.627304i	$-0.215841\pi$
$761$	42.9669	1.55755	0.778775	+	0.627304i	$0.215841\pi$
$762$	0	0
$763$	4.37907	0.158533
$764$	0	0
$765$	−71.6594	−2.59085
$766$	0	0
$767$	−30.2088	−1.09078
$768$	0	0
$769$	42.9669	1.54943	0.774713	−	0.632313i	$-0.217894\pi$
$769$	42.9669	1.54943	0.774713	+	0.632313i	$0.217894\pi$
$770$	0	0
$771$	68.3296	2.46083
$772$	0	0
$773$	−26.4587	−0.951654	−0.475827	−	0.879539i	$-0.657851\pi$
$773$	−26.4587	−0.951654	−0.475827	+	0.879539i	$0.657851\pi$
$774$	0	0
$775$	11.3627	0.408159
$776$	0	0
$777$	−17.5686	−0.630270
$778$	0	0
$779$	16.8238	0.602774
$780$	0	0
$781$	11.5358	0.412783
$782$	0	0
$783$	9.62093	0.343824
$784$	0	0
$785$	−63.7774	−2.27631
$786$	0	0
$787$	−30.5962	−1.09064	−0.545318	−	0.838229i	$-0.683591\pi$
$787$	−30.5962	−1.09064	−0.545318	+	0.838229i	$0.683591\pi$
$788$	0	0
$789$	74.6207	2.65657
$790$	0	0
$791$	0.843279	0.0299836
$792$	0	0
$793$	−39.1596	−1.39060
$794$	0	0
$795$	−21.6866	−0.769143
$796$	0	0
$797$	−29.7529	−1.05390	−0.526951	−	0.849896i	$-0.676665\pi$
$797$	−29.7529	−1.05390	−0.526951	+	0.849896i	$0.676665\pi$
$798$	0	0
$799$	−4.06563	−0.143832
$800$	0	0
$801$	−54.1760	−1.91421
$802$	0	0
$803$	−13.3627	−0.471558
$804$	0	0
$805$	−12.2059	−0.430203
$806$	0	0
$807$	12.7805	0.449895
$808$	0	0
$809$	24.0328	0.844949	0.422474	−	0.906375i	$-0.361162\pi$
$809$	24.0328	0.844949	0.422474	+	0.906375i	$0.361162\pi$
$810$	0	0
$811$	−2.59619	−0.0911645	−0.0455822	−	0.998961i	$-0.514514\pi$
$811$	−2.59619	−0.0911645	−0.0455822	+	0.998961i	$0.514514\pi$
$812$	0	0
$813$	32.1760	1.12846
$814$	0	0
$815$	−42.2088	−1.47851
$816$	0	0
$817$	3.93437	0.137646
$818$	0	0
$819$	−10.5082	−0.367186
$820$	0	0
$821$	−3.01641	−0.105273	−0.0526367	−	0.998614i	$-0.516763\pi$
$821$	−3.01641	−0.105273	−0.0526367	+	0.998614i	$0.516763\pi$
$822$	0	0
$823$	1.67326	0.0583262	0.0291631	−	0.999575i	$-0.490716\pi$
$823$	1.67326	0.0583262	0.0291631	+	0.999575i	$0.490716\pi$
$824$	0	0
$825$	5.87086	0.204397
$826$	0	0
$827$	−20.6925	−0.719549	−0.359775	−	0.933039i	$-0.617146\pi$
$827$	−20.6925	−0.719549	−0.359775	+	0.933039i	$0.617146\pi$
$828$	0	0
$829$	−4.10962	−0.142733	−0.0713665	−	0.997450i	$-0.522736\pi$
$829$	−4.10962	−0.142733	−0.0713665	+	0.997450i	$0.522736\pi$
$830$	0	0
$831$	−48.4999	−1.68244
$832$	0	0
$833$	−6.37907	−0.221022
$834$	0	0
$835$	−17.7969	−0.615887
$836$	0	0
$837$	16.5522	0.572128
$838$	0	0
$839$	−51.1177	−1.76478	−0.882389	−	0.470520i	$-0.844066\pi$
$839$	−51.1177	−1.76478	−0.882389	+	0.470520i	$0.844066\pi$
$840$	0	0
$841$	−19.9013	−0.686251
$842$	0	0
$843$	49.1924	1.69428
$844$	0	0
$845$	−17.9888	−0.618834
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	87.7251	3.01072
$850$	0	0
$851$	29.8269	1.02245
$852$	0	0
$853$	−37.8214	−1.29498	−0.647490	−	0.762074i	$-0.724181\pi$
$853$	−37.8214	−1.29498	−0.647490	+	0.762074i	$0.724181\pi$
$854$	0	0
$855$	−43.4835	−1.48710
$856$	0	0
$857$	36.9997	1.26389	0.631943	−	0.775015i	$-0.282258\pi$
$857$	36.9997	1.26389	0.631943	+	0.775015i	$0.282258\pi$
$858$	0	0
$859$	13.0604	0.445615	0.222808	−	0.974862i	$-0.428478\pi$
$859$	13.0604	0.445615	0.222808	+	0.974862i	$0.428478\pi$
$860$	0	0
$861$	−11.6537	−0.397159
$862$	0	0
$863$	41.8849	1.42578	0.712889	−	0.701277i	$-0.247386\pi$
$863$	41.8849	1.42578	0.712889	+	0.701277i	$0.247386\pi$
$864$	0	0
$865$	−40.9341	−1.39180
$866$	0	0
$867$	63.5275	2.15751
$868$	0	0
$869$	6.37907	0.216395
$870$	0	0
$871$	−13.9672	−0.473260
$872$	0	0
$873$	−45.2908	−1.53286
$874$	0	0
$875$	−7.53579	−0.254756
$876$	0	0
$877$	18.7581	0.633417	0.316709	−	0.948523i	$-0.397422\pi$
$877$	18.7581	0.633417	0.316709	+	0.948523i	$0.397422\pi$
$878$	0	0
$879$	27.2475	0.919037
$880$	0	0
$881$	−18.7282	−0.630968	−0.315484	−	0.948931i	$-0.602167\pi$
$881$	−18.7282	−0.630968	−0.315484	+	0.948931i	$0.602167\pi$
$882$	0	0
$883$	−24.9341	−0.839099	−0.419550	−	0.907732i	$-0.637812\pi$
$883$	−24.9341	−0.839099	−0.419550	+	0.907732i	$0.637812\pi$
$884$	0	0
$885$	86.5907	2.91071
$886$	0	0
$887$	0.0879839	0.00295421	0.00147710	−	0.999999i	$-0.499530\pi$
$887$	0.0879839	0.00295421	0.00147710	+	0.999999i	$0.499530\pi$
$888$	0	0
$889$	−5.10439	−0.171196
$890$	0	0
$891$	−4.01641	−0.134555
$892$	0	0
$893$	−2.46705	−0.0825567
$894$	0	0
$895$	−45.3103	−1.51456
$896$	0	0
$897$	30.6150	1.02220
$898$	0	0
$899$	15.6537	0.522082
$900$	0	0
$901$	19.2419	0.641039
$902$	0	0
$903$	−2.72532	−0.0906931
$904$	0	0
$905$	0.117957	0.00392102
$906$	0	0
$907$	54.0161	1.79358	0.896788	−	0.442460i	$-0.145894\pi$
$907$	54.0161	1.79358	0.896788	+	0.442460i	$0.145894\pi$
$908$	0	0
$909$	63.9588	2.12138
$910$	0	0
$911$	48.2416	1.59832	0.799158	−	0.601121i	$-0.205279\pi$
$911$	48.2416	1.59832	0.799158	+	0.601121i	$0.205279\pi$
$912$	0	0
$913$	2.50820	0.0830094
$914$	0	0
$915$	112.247	3.71078
$916$	0	0
$917$	15.6126	0.515573
$918$	0	0
$919$	18.2088	0.600652	0.300326	−	0.953837i	$-0.402904\pi$
$919$	18.2088	0.600652	0.300326	+	0.953837i	$0.402904\pi$
$920$	0	0
$921$	−32.7581	−1.07942
$922$	0	0
$923$	−28.9341	−0.952378
$924$	0	0
$925$	−14.3463	−0.471702
$926$	0	0
$927$	81.1205	2.66435
$928$	0	0
$929$	27.7089	0.909100	0.454550	−	0.890721i	$-0.349800\pi$
$929$	27.7089	0.909100	0.454550	+	0.890721i	$0.349800\pi$
$930$	0	0
$931$	−3.87086	−0.126862
$932$	0	0
$933$	−47.8074	−1.56514
$934$	0	0
$935$	−17.1044	−0.559373
$936$	0	0
$937$	37.1267	1.21288	0.606439	−	0.795130i	$-0.292597\pi$
$937$	37.1267	1.21288	0.606439	+	0.795130i	$0.292597\pi$
$938$	0	0
$939$	−42.6730	−1.39258
$940$	0	0
$941$	23.0961	0.752910	0.376455	−	0.926435i	$-0.377143\pi$
$941$	23.0961	0.752910	0.376455	+	0.926435i	$0.377143\pi$
$942$	0	0
$943$	19.7850	0.644288
$944$	0	0
$945$	8.55220	0.278203
$946$	0	0
$947$	−41.7222	−1.35579	−0.677895	−	0.735159i	$-0.737108\pi$
$947$	−41.7222	−1.35579	−0.677895	+	0.735159i	$0.737108\pi$
$948$	0	0
$949$	33.5163	1.08798
$950$	0	0
$951$	51.7774	1.67900
$952$	0	0
$953$	13.8297	0.447988	0.223994	−	0.974590i	$-0.428090\pi$
$953$	13.8297	0.447988	0.223994	+	0.974590i	$0.428090\pi$
$954$	0	0
$955$	63.1072	2.04210
$956$	0	0
$957$	8.08798	0.261447
$958$	0	0
$959$	17.2775	0.557920
$960$	0	0
$961$	−4.06874	−0.131250
$962$	0	0
$963$	30.9836	0.998432
$964$	0	0
$965$	44.8461	1.44365
$966$	0	0
$967$	18.3791	0.591031	0.295515	−	0.955338i	$-0.404509\pi$
$967$	18.3791	0.591031	0.295515	+	0.955338i	$0.404509\pi$
$968$	0	0
$969$	66.2088	2.12693
$970$	0	0
$971$	−32.5439	−1.04438	−0.522191	−	0.852829i	$-0.674885\pi$
$971$	−32.5439	−1.04438	−0.522191	+	0.852829i	$0.674885\pi$
$972$	0	0
$973$	−14.8545	−0.476212
$974$	0	0
$975$	−14.7253	−0.471588
$976$	0	0
$977$	21.7941	0.697254	0.348627	−	0.937262i	$-0.386648\pi$
$977$	21.7941	0.697254	0.348627	+	0.937262i	$0.386648\pi$
$978$	0	0
$979$	−12.9313	−0.413285
$980$	0	0
$981$	18.3463	0.585751
$982$	0	0
$983$	24.6730	0.786946	0.393473	−	0.919336i	$-0.371273\pi$
$983$	24.6730	0.786946	0.393473	+	0.919336i	$0.371273\pi$
$984$	0	0
$985$	5.36266	0.170869
$986$	0	0
$987$	1.70892	0.0543954
$988$	0	0
$989$	4.62688	0.147126
$990$	0	0
$991$	−38.5327	−1.22403	−0.612015	−	0.790846i	$-0.709641\pi$
$991$	−38.5327	−1.22403	−0.612015	+	0.790846i	$0.709641\pi$
$992$	0	0
$993$	23.8597	0.757164
$994$	0	0
$995$	7.15956	0.226973
$996$	0	0
$997$	−59.6350	−1.88866	−0.944329	−	0.329003i	$-0.893287\pi$
$997$	−59.6350	−1.88866	−0.944329	+	0.329003i	$0.893287\pi$
$998$	0	0
$999$	−20.8984	−0.661198

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	616.2.a.g.1.3	✓	3
3.2	odd	2		5544.2.a.bj.1.1		3
4.3	odd	2		1232.2.a.q.1.1		3
7.6	odd	2		4312.2.a.w.1.1		3
8.3	odd	2		4928.2.a.bz.1.3		3
8.5	even	2		4928.2.a.bw.1.1		3
11.10	odd	2		6776.2.a.y.1.3		3
28.27	even	2		8624.2.a.cm.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.2.a.g.1.3	✓	3	1.1	even	1	trivial
1232.2.a.q.1.1		3	4.3	odd	2
4312.2.a.w.1.1		3	7.6	odd	2
4928.2.a.bw.1.1		3	8.5	even	2
4928.2.a.bz.1.3		3	8.3	odd	2
5544.2.a.bj.1.1		3	3.2	odd	2
6776.2.a.y.1.3		3	11.10	odd	2
8624.2.a.cm.1.3		3	28.27	even	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:	\( N \)	\(=\)	\( 616 = 2^{3} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	616.a (trivial)

\(f(q)\)	\(=\)	\(q+2.68133 q^{3} +2.68133 q^{5} +1.00000 q^{7} +4.18953 q^{9} +1.00000 q^{11} -2.50820 q^{13} +7.18953 q^{15} -6.37907 q^{17} -3.87086 q^{19} +2.68133 q^{21} -4.55220 q^{23} +2.18953 q^{25} +3.18953 q^{27} +3.01641 q^{29} +5.18953 q^{31} +2.68133 q^{33} +2.68133 q^{35} -6.55220 q^{37} -6.72532 q^{39} -4.34625 q^{41} -1.01641 q^{43} +11.2335 q^{45} +0.637339 q^{47} +1.00000 q^{49} -17.1044 q^{51} -3.01641 q^{53} +2.68133 q^{55} -10.3791 q^{57} +12.0440 q^{59} +15.6126 q^{61} +4.18953 q^{63} -6.72532 q^{65} +5.56860 q^{67} -12.2059 q^{69} +11.5358 q^{71} -13.3627 q^{73} +5.87086 q^{75} +1.00000 q^{77} +6.37907 q^{79} -4.01641 q^{81} +2.50820 q^{83} -17.1044 q^{85} +8.08798 q^{87} -12.9313 q^{89} -2.50820 q^{91} +13.9149 q^{93} -10.3791 q^{95} -10.8105 q^{97} +4.18953 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + q^{5} + 3 q^{7} + 4 q^{9} + 3 q^{11} - 6 q^{13} + 13 q^{15} - 2 q^{17} + 4 q^{19} + q^{21} + 9 q^{23} - 2 q^{25} + q^{27} + 6 q^{29} + 7 q^{31} + q^{33} + q^{35} + 3 q^{37} + 8 q^{39}+ \cdots + 4 q^{99}+O(q^{100}) \) 3 * q + q^3 + q^5 + 3 * q^7 + 4 * q^9 + 3 * q^11 - 6 * q^13 + 13 * q^15 - 2 * q^17 + 4 * q^19 + q^21 + 9 * q^23 - 2 * q^25 + q^27 + 6 * q^29 + 7 * q^31 + q^33 + q^35 + 3 * q^37 + 8 * q^39 - 2 * q^41 + 4 * q^45 + 16 * q^47 + 3 * q^49 - 6 * q^51 - 6 * q^53 + q^55 - 14 * q^57 + 15 * q^59 + 4 * q^63 + 8 * q^65 - 9 * q^67 - 25 * q^69 + 15 * q^71 - 26 * q^73 + 2 * q^75 + 3 * q^77 + 2 * q^79 - 9 * q^81 + 6 * q^83 - 6 * q^85 - 18 * q^87 + q^89 - 6 * q^91 + 5 * q^93 - 14 * q^95 - 41 * q^97 + 4 * q^99

Introduction

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