LMFDB - Embedded newform 7616.2.a.bp.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7616,2,Mod(1,7616)] 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("7616.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7616, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7616 = 2^{6} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7616.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,3,0,1,0,4,0,7,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$60.8140661794$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.5225.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + x + 11 $ x^4 - x^3 - 8*x^2 + x + 11
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 952)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.26498$ of defining polynomial
Character	$\chi$	$=$	7616.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.264977 q^{3} -0.163765 q^{5} +1.00000 q^{7} -2.92979 q^{9} +3.23607 q^{11} -2.47214 q^{13} +0.0433939 q^{15} +1.00000 q^{17} +1.47005 q^{19} -0.264977 q^{21} -4.85748 q^{23} -4.97318 q^{25} +1.57126 q^{27} -0.378584 q^{29} -8.78727 q^{31} -0.857484 q^{33} -0.163765 q^{35} -8.56569 q^{37} +0.655059 q^{39} +5.97318 q^{41} +10.0310 q^{43} +0.479796 q^{45} +8.18710 q^{47} +1.00000 q^{49} -0.264977 q^{51} +8.72945 q^{53} -0.529954 q^{55} -0.389529 q^{57} +2.85748 q^{59} +13.2671 q^{61} -2.92979 q^{63} +0.404849 q^{65} +14.9319 q^{67} +1.28712 q^{69} -14.3317 q^{71} -4.91327 q^{73} +1.31778 q^{75} +3.23607 q^{77} +16.1871 q^{79} +8.37301 q^{81} -5.65715 q^{83} -0.163765 q^{85} +0.100316 q^{87} +13.5167 q^{89} -2.47214 q^{91} +2.32843 q^{93} -0.240742 q^{95} -12.0744 q^{97} -9.48099 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{3} + q^{5} + 4 q^{7} + 7 q^{9} + 4 q^{11} + 8 q^{13} - 12 q^{15} + 4 q^{17} + 14 q^{19} + 3 q^{21} - 8 q^{23} + 11 q^{25} + 12 q^{27} - 4 q^{29} - 5 q^{31} + 8 q^{33} + q^{35} + 4 q^{37} - 4 q^{39}+ \cdots + 2 q^{99}+O(q^{100}) $ 4 * q + 3 * q^3 + q^5 + 4 * q^7 + 7 * q^9 + 4 * q^11 + 8 * q^13 - 12 * q^15 + 4 * q^17 + 14 * q^19 + 3 * q^21 - 8 * q^23 + 11 * q^25 + 12 * q^27 - 4 * q^29 - 5 * q^31 + 8 * q^33 + q^35 + 4 * q^37 - 4 * q^39 - 7 * q^41 + 19 * q^43 + 2 * q^45 - 8 * q^47 + 4 * q^49 + 3 * q^51 - 5 * q^53 + 6 * q^55 + 44 * q^57 + 23 * q^61 + 7 * q^63 - 8 * q^65 + 15 * q^67 + 2 * q^69 - 2 * q^71 - 5 * q^73 + 10 * q^75 + 4 * q^77 + 24 * q^79 - 8 * q^81 + 10 * q^83 + q^85 - 16 * q^87 - 16 * q^89 + 8 * q^91 + 20 * q^93 - 22 * q^95 - 15 * q^97 + 2 * 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
$3$	−0.264977	−0.152985	−0.0764923	−	0.997070i	$-0.524372\pi$
$3$	−0.264977	−0.152985	−0.0764923	+	0.997070i	$0.524372\pi$
$4$	0	0
$5$	−0.163765	−0.0732379	−0.0366189	−	0.999329i	$-0.511659\pi$
$5$	−0.163765	−0.0732379	−0.0366189	+	0.999329i	$0.511659\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.92979	−0.976596
$10$	0	0
$11$	3.23607	0.975711	0.487856	−	0.872924i	$-0.337779\pi$
$11$	3.23607	0.975711	0.487856	+	0.872924i	$0.337779\pi$
$12$	0	0
$13$	−2.47214	−0.685647	−0.342824	−	0.939400i	$-0.611383\pi$
$13$	−2.47214	−0.685647	−0.342824	+	0.939400i	$0.611383\pi$
$14$	0	0
$15$	0.0433939	0.0112043
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	1.47005	0.337252	0.168626	−	0.985680i	$-0.446067\pi$
$19$	1.47005	0.337252	0.168626	+	0.985680i	$0.446067\pi$
$20$	0	0
$21$	−0.264977	−0.0578228
$22$	0	0
$23$	−4.85748	−1.01286	−0.506428	−	0.862282i	$-0.669034\pi$
$23$	−4.85748	−1.01286	−0.506428	+	0.862282i	$0.669034\pi$
$24$	0	0
$25$	−4.97318	−0.994636
$26$	0	0
$27$	1.57126	0.302389
$28$	0	0
$29$	−0.378584	−0.0703013	−0.0351506	−	0.999382i	$-0.511191\pi$
$29$	−0.378584	−0.0703013	−0.0351506	+	0.999382i	$0.511191\pi$
$30$	0	0
$31$	−8.78727	−1.57824	−0.789120	−	0.614239i	$-0.789463\pi$
$31$	−8.78727	−1.57824	−0.789120	+	0.614239i	$0.789463\pi$
$32$	0	0
$33$	−0.857484	−0.149269
$34$	0	0
$35$	−0.163765	−0.0276813
$36$	0	0
$37$	−8.56569	−1.40819	−0.704095	−	0.710106i	$-0.748647\pi$
$37$	−8.56569	−1.40819	−0.704095	+	0.710106i	$0.748647\pi$
$38$	0	0
$39$	0.655059	0.104893
$40$	0	0
$41$	5.97318	0.932854	0.466427	−	0.884560i	$-0.345541\pi$
$41$	5.97318	0.932854	0.466427	+	0.884560i	$0.345541\pi$
$42$	0	0
$43$	10.0310	1.52971	0.764857	−	0.644201i	$-0.222810\pi$
$43$	10.0310	1.52971	0.764857	+	0.644201i	$0.222810\pi$
$44$	0	0
$45$	0.479796	0.0715238
$46$	0	0
$47$	8.18710	1.19421	0.597106	−	0.802162i	$-0.296317\pi$
$47$	8.18710	1.19421	0.597106	+	0.802162i	$0.296317\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.264977	−0.0371042
$52$	0	0
$53$	8.72945	1.19908	0.599541	−	0.800344i	$-0.295350\pi$
$53$	8.72945	1.19908	0.599541	+	0.800344i	$0.295350\pi$
$54$	0	0
$55$	−0.529954	−0.0714590
$56$	0	0
$57$	−0.389529	−0.0515943
$58$	0	0
$59$	2.85748	0.372013	0.186006	−	0.982549i	$-0.440445\pi$
$59$	2.85748	0.372013	0.186006	+	0.982549i	$0.440445\pi$
$60$	0	0
$61$	13.2671	1.69867	0.849337	−	0.527851i	$-0.177002\pi$
$61$	13.2671	1.69867	0.849337	+	0.527851i	$0.177002\pi$
$62$	0	0
$63$	−2.92979	−0.369118
$64$	0	0
$65$	0.404849	0.0502153
$66$	0	0
$67$	14.9319	1.82422	0.912110	−	0.409946i	$-0.134453\pi$
$67$	14.9319	1.82422	0.912110	+	0.409946i	$0.134453\pi$
$68$	0	0
$69$	1.28712	0.154951
$70$	0	0
$71$	−14.3317	−1.70086	−0.850431	−	0.526087i	$-0.823658\pi$
$71$	−14.3317	−1.70086	−0.850431	+	0.526087i	$0.823658\pi$
$72$	0	0
$73$	−4.91327	−0.575055	−0.287528	−	0.957772i	$-0.592833\pi$
$73$	−4.91327	−0.575055	−0.287528	+	0.957772i	$0.592833\pi$
$74$	0	0
$75$	1.31778	0.152164
$76$	0	0
$77$	3.23607	0.368784
$78$	0	0
$79$	16.1871	1.82119	0.910596	−	0.413298i	$-0.135623\pi$
$79$	16.1871	1.82119	0.910596	+	0.413298i	$0.135623\pi$
$80$	0	0
$81$	8.37301	0.930335
$82$	0	0
$83$	−5.65715	−0.620953	−0.310476	−	0.950581i	$-0.600489\pi$
$83$	−5.65715	−0.620953	−0.310476	+	0.950581i	$0.600489\pi$
$84$	0	0
$85$	−0.163765	−0.0177628
$86$	0	0
$87$	0.100316	0.0107550
$88$	0	0
$89$	13.5167	1.43277	0.716385	−	0.697705i	$-0.245795\pi$
$89$	13.5167	1.43277	0.716385	+	0.697705i	$0.245795\pi$
$90$	0	0
$91$	−2.47214	−0.259150
$92$	0	0
$93$	2.32843	0.241447
$94$	0	0
$95$	−0.240742	−0.0246996
$96$	0	0
$97$	−12.0744	−1.22597	−0.612984	−	0.790095i	$-0.710031\pi$
$97$	−12.0744	−1.22597	−0.612984	+	0.790095i	$0.710031\pi$
$98$	0	0
$99$	−9.48099	−0.952875
$100$	0	0
$101$	−15.3916	−1.53152	−0.765762	−	0.643124i	$-0.777638\pi$
$101$	−15.3916	−1.53152	−0.765762	+	0.643124i	$0.777638\pi$
$102$	0	0
$103$	−10.7997	−1.06412	−0.532061	−	0.846706i	$-0.678582\pi$
$103$	−10.7997	−1.06412	−0.532061	+	0.846706i	$0.678582\pi$
$104$	0	0
$105$	0.0433939	0.00423482
$106$	0	0
$107$	9.27857	0.896993	0.448496	−	0.893785i	$-0.351960\pi$
$107$	9.27857	0.896993	0.448496	+	0.893785i	$0.351960\pi$
$108$	0	0
$109$	2.90436	0.278187	0.139094	−	0.990279i	$-0.455581\pi$
$109$	2.90436	0.278187	0.139094	+	0.990279i	$0.455581\pi$
$110$	0	0
$111$	2.26971	0.215431
$112$	0	0
$113$	8.67456	0.816034	0.408017	−	0.912974i	$-0.366220\pi$
$113$	8.67456	0.816034	0.408017	+	0.912974i	$0.366220\pi$
$114$	0	0
$115$	0.795485	0.0741794
$116$	0	0
$117$	7.24283	0.669600
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	−0.527864	−0.0479876
$122$	0	0
$123$	−1.58276	−0.142712
$124$	0	0
$125$	1.63326	0.146083
$126$	0	0
$127$	−6.19950	−0.550117	−0.275058	−	0.961428i	$-0.588697\pi$
$127$	−6.19950	−0.550117	−0.275058	+	0.961428i	$0.588697\pi$
$128$	0	0
$129$	−2.65799	−0.234023
$130$	0	0
$131$	−8.23816	−0.719771	−0.359886	−	0.932996i	$-0.617184\pi$
$131$	−8.23816	−0.719771	−0.359886	+	0.932996i	$0.617184\pi$
$132$	0	0
$133$	1.47005	0.127469
$134$	0	0
$135$	−0.257317	−0.0221463
$136$	0	0
$137$	2.02682	0.173163	0.0865814	−	0.996245i	$-0.472406\pi$
$137$	2.02682	0.173163	0.0865814	+	0.996245i	$0.472406\pi$
$138$	0	0
$139$	−16.6979	−1.41630	−0.708149	−	0.706063i	$-0.750469\pi$
$139$	−16.6979	−1.41630	−0.708149	+	0.706063i	$0.750469\pi$
$140$	0	0
$141$	−2.16940	−0.182696
$142$	0	0
$143$	−8.00000	−0.668994
$144$	0	0
$145$	0.0619988	0.00514872
$146$	0	0
$147$	−0.264977	−0.0218549
$148$	0	0
$149$	13.1004	1.07322	0.536612	−	0.843829i	$-0.319704\pi$
$149$	13.1004	1.07322	0.536612	+	0.843829i	$0.319704\pi$
$150$	0	0
$151$	−12.5031	−1.01749	−0.508745	−	0.860917i	$-0.669891\pi$
$151$	−12.5031	−1.01749	−0.508745	+	0.860917i	$0.669891\pi$
$152$	0	0
$153$	−2.92979	−0.236859
$154$	0	0
$155$	1.43905	0.115587
$156$	0	0
$157$	16.7171	1.33417	0.667083	−	0.744983i	$-0.267543\pi$
$157$	16.7171	1.33417	0.667083	+	0.744983i	$0.267543\pi$
$158$	0	0
$159$	−2.31311	−0.183441
$160$	0	0
$161$	−4.85748	−0.382823
$162$	0	0
$163$	−11.0089	−0.862280	−0.431140	−	0.902285i	$-0.641888\pi$
$163$	−11.0089	−0.862280	−0.431140	+	0.902285i	$0.641888\pi$
$164$	0	0
$165$	0.140426	0.0109321
$166$	0	0
$167$	4.97109	0.384675	0.192337	−	0.981329i	$-0.438393\pi$
$167$	4.97109	0.384675	0.192337	+	0.981329i	$0.438393\pi$
$168$	0	0
$169$	−6.88854	−0.529888
$170$	0	0
$171$	−4.30692	−0.329358
$172$	0	0
$173$	17.4542	1.32702	0.663508	−	0.748169i	$-0.269067\pi$
$173$	17.4542	1.32702	0.663508	+	0.748169i	$0.269067\pi$
$174$	0	0
$175$	−4.97318	−0.375937
$176$	0	0
$177$	−0.757168	−0.0569122
$178$	0	0
$179$	−2.38332	−0.178138	−0.0890688	−	0.996025i	$-0.528389\pi$
$179$	−2.38332	−0.178138	−0.0890688	+	0.996025i	$0.528389\pi$
$180$	0	0
$181$	24.7103	1.83670	0.918351	−	0.395767i	$-0.129521\pi$
$181$	24.7103	1.83670	0.918351	+	0.395767i	$0.129521\pi$
$182$	0	0
$183$	−3.51547	−0.259871
$184$	0	0
$185$	1.40276	0.103133
$186$	0	0
$187$	3.23607	0.236645
$188$	0	0
$189$	1.57126	0.114292
$190$	0	0
$191$	20.3193	1.47025	0.735127	−	0.677929i	$-0.237122\pi$
$191$	20.3193	1.47025	0.735127	+	0.677929i	$0.237122\pi$
$192$	0	0
$193$	16.4568	1.18459	0.592294	−	0.805722i	$-0.298223\pi$
$193$	16.4568	1.18459	0.592294	+	0.805722i	$0.298223\pi$
$194$	0	0
$195$	−0.107276	−0.00768218
$196$	0	0
$197$	3.38067	0.240863	0.120432	−	0.992722i	$-0.461572\pi$
$197$	3.38067	0.240863	0.120432	+	0.992722i	$0.461572\pi$
$198$	0	0
$199$	−5.14873	−0.364984	−0.182492	−	0.983207i	$-0.558416\pi$
$199$	−5.14873	−0.364984	−0.182492	+	0.983207i	$0.558416\pi$
$200$	0	0
$201$	−3.95661	−0.279077
$202$	0	0
$203$	−0.378584	−0.0265714
$204$	0	0
$205$	−0.978197	−0.0683203
$206$	0	0
$207$	14.2314	0.989150
$208$	0	0
$209$	4.75717	0.329060
$210$	0	0
$211$	−11.6256	−0.800339	−0.400170	−	0.916441i	$-0.631049\pi$
$211$	−11.6256	−0.800339	−0.400170	+	0.916441i	$0.631049\pi$
$212$	0	0
$213$	3.79758	0.260206
$214$	0	0
$215$	−1.64273	−0.112033
$216$	0	0
$217$	−8.78727	−0.596519
$218$	0	0
$219$	1.30190	0.0879746
$220$	0	0
$221$	−2.47214	−0.166294
$222$	0	0
$223$	11.6725	0.781646	0.390823	−	0.920466i	$-0.372190\pi$
$223$	11.6725	0.781646	0.390823	+	0.920466i	$0.372190\pi$
$224$	0	0
$225$	14.5704	0.971357
$226$	0	0
$227$	14.9056	0.989320	0.494660	−	0.869087i	$-0.335293\pi$
$227$	14.9056	0.989320	0.494660	+	0.869087i	$0.335293\pi$
$228$	0	0
$229$	18.7997	1.24232	0.621158	−	0.783685i	$-0.286662\pi$
$229$	18.7997	1.24232	0.621158	+	0.783685i	$0.286662\pi$
$230$	0	0
$231$	−0.857484	−0.0564183
$232$	0	0
$233$	16.7997	1.10058	0.550291	−	0.834973i	$-0.314517\pi$
$233$	16.7997	1.10058	0.550291	+	0.834973i	$0.314517\pi$
$234$	0	0
$235$	−1.34076	−0.0874615
$236$	0	0
$237$	−4.28921	−0.278614
$238$	0	0
$239$	−16.5031	−1.06750	−0.533750	−	0.845643i	$-0.679217\pi$
$239$	−16.5031	−1.06750	−0.533750	+	0.845643i	$0.679217\pi$
$240$	0	0
$241$	19.1768	1.23529	0.617643	−	0.786458i	$-0.288088\pi$
$241$	19.1768	1.23529	0.617643	+	0.786458i	$0.288088\pi$
$242$	0	0
$243$	−6.93243	−0.444716
$244$	0	0
$245$	−0.163765	−0.0104626
$246$	0	0
$247$	−3.63415	−0.231236
$248$	0	0
$249$	1.49902	0.0949962
$250$	0	0
$251$	21.8307	1.37794	0.688972	−	0.724788i	$-0.258062\pi$
$251$	21.8307	1.37794	0.688972	+	0.724788i	$0.258062\pi$
$252$	0	0
$253$	−15.7191	−0.988254
$254$	0	0
$255$	0.0433939	0.00271743
$256$	0	0
$257$	−21.5167	−1.34218	−0.671088	−	0.741378i	$-0.734173\pi$
$257$	−21.5167	−1.34218	−0.671088	+	0.741378i	$0.734173\pi$
$258$	0	0
$259$	−8.56569	−0.532246
$260$	0	0
$261$	1.10917	0.0686559
$262$	0	0
$263$	5.71497	0.352400	0.176200	−	0.984354i	$-0.443619\pi$
$263$	5.71497	0.352400	0.176200	+	0.984354i	$0.443619\pi$
$264$	0	0
$265$	−1.42958	−0.0878183
$266$	0	0
$267$	−3.58162	−0.219192
$268$	0	0
$269$	23.8375	1.45340	0.726699	−	0.686956i	$-0.241054\pi$
$269$	23.8375	1.45340	0.726699	+	0.686956i	$0.241054\pi$
$270$	0	0
$271$	27.9930	1.70046	0.850228	−	0.526415i	$-0.176464\pi$
$271$	27.9930	1.70046	0.850228	+	0.526415i	$0.176464\pi$
$272$	0	0
$273$	0.655059	0.0396460
$274$	0	0
$275$	−16.0936	−0.970478
$276$	0	0
$277$	−2.03155	−0.122064	−0.0610321	−	0.998136i	$-0.519439\pi$
$277$	−2.03155	−0.122064	−0.0610321	+	0.998136i	$0.519439\pi$
$278$	0	0
$279$	25.7448	1.54130
$280$	0	0
$281$	30.4444	1.81616	0.908081	−	0.418795i	$-0.137547\pi$
$281$	30.4444	1.81616	0.908081	+	0.418795i	$0.137547\pi$
$282$	0	0
$283$	−7.19482	−0.427688	−0.213844	−	0.976868i	$-0.568598\pi$
$283$	−7.19482	−0.427688	−0.213844	+	0.976868i	$0.568598\pi$
$284$	0	0
$285$	0.0637911	0.00377866
$286$	0	0
$287$	5.97318	0.352586
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	3.19944	0.187554
$292$	0	0
$293$	−22.6592	−1.32377	−0.661883	−	0.749607i	$-0.730243\pi$
$293$	−22.6592	−1.32377	−0.661883	+	0.749607i	$0.730243\pi$
$294$	0	0
$295$	−0.467956	−0.0272454
$296$	0	0
$297$	5.08470	0.295044
$298$	0	0
$299$	12.0084	0.694461
$300$	0	0
$301$	10.0310	0.578177
$302$	0	0
$303$	4.07843	0.234300
$304$	0	0
$305$	−2.17268	−0.124407
$306$	0	0
$307$	8.44735	0.482116	0.241058	−	0.970511i	$-0.422506\pi$
$307$	8.44735	0.482116	0.241058	+	0.970511i	$0.422506\pi$
$308$	0	0
$309$	2.86166	0.162794
$310$	0	0
$311$	0.203679	0.0115496	0.00577479	−	0.999983i	$-0.498162\pi$
$311$	0.203679	0.0115496	0.00577479	+	0.999983i	$0.498162\pi$
$312$	0	0
$313$	−2.60964	−0.147505	−0.0737527	−	0.997277i	$-0.523498\pi$
$313$	−2.60964	−0.147505	−0.0737527	+	0.997277i	$0.523498\pi$
$314$	0	0
$315$	0.479796	0.0270335
$316$	0	0
$317$	−28.4253	−1.59652	−0.798261	−	0.602312i	$-0.794246\pi$
$317$	−28.4253	−1.59652	−0.798261	+	0.602312i	$0.794246\pi$
$318$	0	0
$319$	−1.22512	−0.0685937
$320$	0	0
$321$	−2.45861	−0.137226
$322$	0	0
$323$	1.47005	0.0817955
$324$	0	0
$325$	12.2944	0.681969
$326$	0	0
$327$	−0.769588	−0.0425583
$328$	0	0
$329$	8.18710	0.451370
$330$	0	0
$331$	−28.8204	−1.58411	−0.792057	−	0.610447i	$-0.790990\pi$
$331$	−28.8204	−1.58411	−0.792057	+	0.610447i	$0.790990\pi$
$332$	0	0
$333$	25.0956	1.37523
$334$	0	0
$335$	−2.44532	−0.133602
$336$	0	0
$337$	6.95959	0.379113	0.189557	−	0.981870i	$-0.439295\pi$
$337$	6.95959	0.379113	0.189557	+	0.981870i	$0.439295\pi$
$338$	0	0
$339$	−2.29856	−0.124841
$340$	0	0
$341$	−28.4362	−1.53991
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−0.210785	−0.0113483
$346$	0	0
$347$	−4.06876	−0.218423	−0.109211	−	0.994019i	$-0.534833\pi$
$347$	−4.06876	−0.218423	−0.109211	+	0.994019i	$0.534833\pi$
$348$	0	0
$349$	3.58568	0.191937	0.0959686	−	0.995384i	$-0.469405\pi$
$349$	3.58568	0.191937	0.0959686	+	0.995384i	$0.469405\pi$
$350$	0	0
$351$	−3.88436	−0.207332
$352$	0	0
$353$	19.1219	1.01776	0.508878	−	0.860839i	$-0.330060\pi$
$353$	19.1219	1.01776	0.508878	+	0.860839i	$0.330060\pi$
$354$	0	0
$355$	2.34703	0.124567
$356$	0	0
$357$	−0.264977	−0.0140241
$358$	0	0
$359$	11.8047	0.623027	0.311514	−	0.950242i	$-0.399164\pi$
$359$	11.8047	0.623027	0.311514	+	0.950242i	$0.399164\pi$
$360$	0	0
$361$	−16.8390	−0.886261
$362$	0	0
$363$	0.139872	0.00734137
$364$	0	0
$365$	0.804621	0.0421158
$366$	0	0
$367$	−8.73035	−0.455721	−0.227860	−	0.973694i	$-0.573173\pi$
$367$	−8.73035	−0.455721	−0.227860	+	0.973694i	$0.573173\pi$
$368$	0	0
$369$	−17.5001	−0.911021
$370$	0	0
$371$	8.72945	0.453211
$372$	0	0
$373$	−20.0930	−1.04038	−0.520188	−	0.854052i	$-0.674138\pi$
$373$	−20.0930	−1.04038	−0.520188	+	0.854052i	$0.674138\pi$
$374$	0	0
$375$	−0.432776	−0.0223484
$376$	0	0
$377$	0.935911	0.0482019
$378$	0	0
$379$	16.5852	0.851924	0.425962	−	0.904741i	$-0.359936\pi$
$379$	16.5852	0.851924	0.425962	+	0.904741i	$0.359936\pi$
$380$	0	0
$381$	1.64273	0.0841594
$382$	0	0
$383$	−13.5745	−0.693627	−0.346813	−	0.937934i	$-0.612736\pi$
$383$	−13.5745	−0.693627	−0.346813	+	0.937934i	$0.612736\pi$
$384$	0	0
$385$	−0.529954	−0.0270090
$386$	0	0
$387$	−29.3887	−1.49391
$388$	0	0
$389$	−9.46183	−0.479734	−0.239867	−	0.970806i	$-0.577104\pi$
$389$	−9.46183	−0.479734	−0.239867	+	0.970806i	$0.577104\pi$
$390$	0	0
$391$	−4.85748	−0.245654
$392$	0	0
$393$	2.18292	0.110114
$394$	0	0
$395$	−2.65088	−0.133380
$396$	0	0
$397$	13.7383	0.689506	0.344753	−	0.938693i	$-0.387963\pi$
$397$	13.7383	0.689506	0.344753	+	0.938693i	$0.387963\pi$
$398$	0	0
$399$	−0.389529	−0.0195008
$400$	0	0
$401$	−1.20451	−0.0601506	−0.0300753	−	0.999548i	$-0.509575\pi$
$401$	−1.20451	−0.0601506	−0.0300753	+	0.999548i	$0.509575\pi$
$402$	0	0
$403$	21.7233	1.08212
$404$	0	0
$405$	−1.37121	−0.0681358
$406$	0	0
$407$	−27.7191	−1.37399
$408$	0	0
$409$	33.1739	1.64034	0.820171	−	0.572118i	$-0.193878\pi$
$409$	33.1739	1.64034	0.820171	+	0.572118i	$0.193878\pi$
$410$	0	0
$411$	−0.537061	−0.0264912
$412$	0	0
$413$	2.85748	0.140608
$414$	0	0
$415$	0.926442	0.0454773
$416$	0	0
$417$	4.42456	0.216672
$418$	0	0
$419$	−29.0969	−1.42148	−0.710738	−	0.703457i	$-0.751639\pi$
$419$	−29.0969	−1.42148	−0.710738	+	0.703457i	$0.751639\pi$
$420$	0	0
$421$	−9.31812	−0.454137	−0.227069	−	0.973879i	$-0.572914\pi$
$421$	−9.31812	−0.454137	−0.227069	+	0.973879i	$0.572914\pi$
$422$	0	0
$423$	−23.9865	−1.16626
$424$	0	0
$425$	−4.97318	−0.241235
$426$	0	0
$427$	13.2671	0.642038
$428$	0	0
$429$	2.11982	0.102346
$430$	0	0
$431$	−8.88227	−0.427844	−0.213922	−	0.976851i	$-0.568624\pi$
$431$	−8.88227	−0.427844	−0.213922	+	0.976851i	$0.568624\pi$
$432$	0	0
$433$	16.7749	0.806149	0.403075	−	0.915167i	$-0.367942\pi$
$433$	16.7749	0.806149	0.403075	+	0.915167i	$0.367942\pi$
$434$	0	0
$435$	−0.0164283	−0.000787674 0
$436$	0	0
$437$	−7.14072	−0.341587
$438$	0	0
$439$	38.0488	1.81597	0.907986	−	0.419001i	$-0.137620\pi$
$439$	38.0488	1.81597	0.907986	+	0.419001i	$0.137620\pi$
$440$	0	0
$441$	−2.92979	−0.139514
$442$	0	0
$443$	25.8171	1.22661	0.613303	−	0.789848i	$-0.289840\pi$
$443$	25.8171	1.22661	0.613303	+	0.789848i	$0.289840\pi$
$444$	0	0
$445$	−2.21356	−0.104933
$446$	0	0
$447$	−3.47130	−0.164187
$448$	0	0
$449$	19.2470	0.908323	0.454161	−	0.890919i	$-0.349939\pi$
$449$	19.2470	0.908323	0.454161	+	0.890919i	$0.349939\pi$
$450$	0	0
$451$	19.3296	0.910196
$452$	0	0
$453$	3.31304	0.155660
$454$	0	0
$455$	0.404849	0.0189796
$456$	0	0
$457$	−3.86250	−0.180680	−0.0903401	−	0.995911i	$-0.528795\pi$
$457$	−3.86250	−0.180680	−0.0903401	+	0.995911i	$0.528795\pi$
$458$	0	0
$459$	1.57126	0.0733400
$460$	0	0
$461$	4.55474	0.212136	0.106068	−	0.994359i	$-0.466174\pi$
$461$	4.55474	0.212136	0.106068	+	0.994359i	$0.466174\pi$
$462$	0	0
$463$	16.1603	0.751032	0.375516	−	0.926816i	$-0.377465\pi$
$463$	16.1603	0.751032	0.375516	+	0.926816i	$0.377465\pi$
$464$	0	0
$465$	−0.381314	−0.0176830
$466$	0	0
$467$	−3.29659	−0.152548	−0.0762740	−	0.997087i	$-0.524302\pi$
$467$	−3.29659	−0.152548	−0.0762740	+	0.997087i	$0.524302\pi$
$468$	0	0
$469$	14.9319	0.689490
$470$	0	0
$471$	−4.42964	−0.204107
$472$	0	0
$473$	32.4610	1.49256
$474$	0	0
$475$	−7.31080	−0.335443
$476$	0	0
$477$	−25.5754	−1.17102
$478$	0	0
$479$	15.7996	0.721902	0.360951	−	0.932585i	$-0.382452\pi$
$479$	15.7996	0.721902	0.360951	+	0.932585i	$0.382452\pi$
$480$	0	0
$481$	21.1755	0.965522
$482$	0	0
$483$	1.28712	0.0585661
$484$	0	0
$485$	1.97736	0.0897874
$486$	0	0
$487$	22.8369	1.03484	0.517419	−	0.855732i	$-0.326893\pi$
$487$	22.8369	1.03484	0.517419	+	0.855732i	$0.326893\pi$
$488$	0	0
$489$	2.91709	0.131916
$490$	0	0
$491$	−4.20368	−0.189709	−0.0948547	−	0.995491i	$-0.530239\pi$
$491$	−4.20368	−0.189709	−0.0948547	+	0.995491i	$0.530239\pi$
$492$	0	0
$493$	−0.378584	−0.0170506
$494$	0	0
$495$	1.55265	0.0697866
$496$	0	0
$497$	−14.3317	−0.642865
$498$	0	0
$499$	8.86425	0.396818	0.198409	−	0.980119i	$-0.436423\pi$
$499$	8.86425	0.396818	0.198409	+	0.980119i	$0.436423\pi$
$500$	0	0
$501$	−1.31723	−0.0588493
$502$	0	0
$503$	20.6333	0.919994	0.459997	−	0.887920i	$-0.347850\pi$
$503$	20.6333	0.919994	0.459997	+	0.887920i	$0.347850\pi$
$504$	0	0
$505$	2.52061	0.112166
$506$	0	0
$507$	1.82531	0.0810647
$508$	0	0
$509$	−1.96697	−0.0871844	−0.0435922	−	0.999049i	$-0.513880\pi$
$509$	−1.96697	−0.0871844	−0.0435922	+	0.999049i	$0.513880\pi$
$510$	0	0
$511$	−4.91327	−0.217350
$512$	0	0
$513$	2.30982	0.101981
$514$	0	0
$515$	1.76861	0.0779341
$516$	0	0
$517$	26.4940	1.16521
$518$	0	0
$519$	−4.62496	−0.203013
$520$	0	0
$521$	−40.9435	−1.79377	−0.896884	−	0.442266i	$-0.854175\pi$
$521$	−40.9435	−1.79377	−0.896884	+	0.442266i	$0.854175\pi$
$522$	0	0
$523$	37.4879	1.63923	0.819615	−	0.572914i	$-0.194187\pi$
$523$	37.4879	1.63923	0.819615	+	0.572914i	$0.194187\pi$
$524$	0	0
$525$	1.31778	0.0575126
$526$	0	0
$527$	−8.78727	−0.382780
$528$	0	0
$529$	0.595151	0.0258761
$530$	0	0
$531$	−8.37182	−0.363306
$532$	0	0
$533$	−14.7665	−0.639609
$534$	0	0
$535$	−1.51950	−0.0656939
$536$	0	0
$537$	0.631525	0.0272523
$538$	0	0
$539$	3.23607	0.139387
$540$	0	0
$541$	8.60580	0.369992	0.184996	−	0.982739i	$-0.440773\pi$
$541$	8.60580	0.369992	0.184996	+	0.982739i	$0.440773\pi$
$542$	0	0
$543$	−6.54766	−0.280987
$544$	0	0
$545$	−0.475632	−0.0203738
$546$	0	0
$547$	−15.6887	−0.670800	−0.335400	−	0.942076i	$-0.608872\pi$
$547$	−15.6887	−0.670800	−0.335400	+	0.942076i	$0.608872\pi$
$548$	0	0
$549$	−38.8697	−1.65892
$550$	0	0
$551$	−0.556536	−0.0237092
$552$	0	0
$553$	16.1871	0.688346
$554$	0	0
$555$	−0.371699	−0.0157777
$556$	0	0
$557$	−12.0673	−0.511307	−0.255654	−	0.966768i	$-0.582291\pi$
$557$	−12.0673	−0.511307	−0.255654	+	0.966768i	$0.582291\pi$
$558$	0	0
$559$	−24.7980	−1.04884
$560$	0	0
$561$	−0.857484	−0.0362030
$562$	0	0
$563$	−5.97700	−0.251901	−0.125950	−	0.992037i	$-0.540198\pi$
$563$	−5.97700	−0.251901	−0.125950	+	0.992037i	$0.540198\pi$
$564$	0	0
$565$	−1.42059	−0.0597646
$566$	0	0
$567$	8.37301	0.351634
$568$	0	0
$569$	26.1373	1.09573	0.547866	−	0.836566i	$-0.315440\pi$
$569$	26.1373	1.09573	0.547866	+	0.836566i	$0.315440\pi$
$570$	0	0
$571$	7.33818	0.307093	0.153547	−	0.988141i	$-0.450930\pi$
$571$	7.33818	0.307093	0.153547	+	0.988141i	$0.450930\pi$
$572$	0	0
$573$	−5.38415	−0.224926
$574$	0	0
$575$	24.1571	1.00742
$576$	0	0
$577$	0.169395	0.00705202	0.00352601	−	0.999994i	$-0.498878\pi$
$577$	0.169395	0.00705202	0.00352601	+	0.999994i	$0.498878\pi$
$578$	0	0
$579$	−4.36068	−0.181224
$580$	0	0
$581$	−5.65715	−0.234698
$582$	0	0
$583$	28.2491	1.16996
$584$	0	0
$585$	−1.18612	−0.0490401
$586$	0	0
$587$	10.4184	0.430012	0.215006	−	0.976613i	$-0.431023\pi$
$587$	10.4184	0.430012	0.215006	+	0.976613i	$0.431023\pi$
$588$	0	0
$589$	−12.9177	−0.532264
$590$	0	0
$591$	−0.895801	−0.0368483
$592$	0	0
$593$	21.7811	0.894445	0.447222	−	0.894423i	$-0.352413\pi$
$593$	21.7811	0.894445	0.447222	+	0.894423i	$0.352413\pi$
$594$	0	0
$595$	−0.163765	−0.00671371
$596$	0	0
$597$	1.36429	0.0558369
$598$	0	0
$599$	−18.9847	−0.775695	−0.387848	−	0.921723i	$-0.626781\pi$
$599$	−18.9847	−0.775695	−0.387848	+	0.921723i	$0.626781\pi$
$600$	0	0
$601$	17.1797	0.700776	0.350388	−	0.936605i	$-0.386050\pi$
$601$	17.1797	0.700776	0.350388	+	0.936605i	$0.386050\pi$
$602$	0	0
$603$	−43.7472	−1.78152
$604$	0	0
$605$	0.0864456	0.00351451
$606$	0	0
$607$	−27.7740	−1.12731	−0.563657	−	0.826009i	$-0.690606\pi$
$607$	−27.7740	−1.12731	−0.563657	+	0.826009i	$0.690606\pi$
$608$	0	0
$609$	0.100316	0.00406501
$610$	0	0
$611$	−20.2396	−0.818808
$612$	0	0
$613$	37.0901	1.49806	0.749028	−	0.662538i	$-0.230521\pi$
$613$	37.0901	1.49806	0.749028	+	0.662538i	$0.230521\pi$
$614$	0	0
$615$	0.259200	0.0104519
$616$	0	0
$617$	39.8484	1.60424	0.802119	−	0.597165i	$-0.203706\pi$
$617$	39.8484	1.60424	0.802119	+	0.597165i	$0.203706\pi$
$618$	0	0
$619$	21.4852	0.863562	0.431781	−	0.901978i	$-0.357885\pi$
$619$	21.4852	0.863562	0.431781	+	0.901978i	$0.357885\pi$
$620$	0	0
$621$	−7.63236	−0.306276
$622$	0	0
$623$	13.5167	0.541536
$624$	0	0
$625$	24.5984	0.983937
$626$	0	0
$627$	−1.26054	−0.0503411
$628$	0	0
$629$	−8.56569	−0.341536
$630$	0	0
$631$	−37.2884	−1.48443	−0.742213	−	0.670164i	$-0.766224\pi$
$631$	−37.2884	−1.48443	−0.742213	+	0.670164i	$0.766224\pi$
$632$	0	0
$633$	3.08052	0.122440
$634$	0	0
$635$	1.01526	0.0402894
$636$	0	0
$637$	−2.47214	−0.0979496
$638$	0	0
$639$	41.9889	1.66105
$640$	0	0
$641$	−40.5188	−1.60040	−0.800198	−	0.599735i	$-0.795273\pi$
$641$	−40.5188	−1.60040	−0.800198	+	0.599735i	$0.795273\pi$
$642$	0	0
$643$	−28.2963	−1.11590	−0.557949	−	0.829875i	$-0.688412\pi$
$643$	−28.2963	−1.11590	−0.557949	+	0.829875i	$0.688412\pi$
$644$	0	0
$645$	0.435285	0.0171393
$646$	0	0
$647$	−37.6767	−1.48122	−0.740611	−	0.671934i	$-0.765464\pi$
$647$	−37.6767	−1.48122	−0.740611	+	0.671934i	$0.765464\pi$
$648$	0	0
$649$	9.24701	0.362977
$650$	0	0
$651$	2.32843	0.0912582
$652$	0	0
$653$	−23.0915	−0.903639	−0.451819	−	0.892109i	$-0.649225\pi$
$653$	−23.0915	−0.903639	−0.451819	+	0.892109i	$0.649225\pi$
$654$	0	0
$655$	1.34912	0.0527145
$656$	0	0
$657$	14.3948	0.561596
$658$	0	0
$659$	9.40401	0.366328	0.183164	−	0.983082i	$-0.441366\pi$
$659$	9.40401	0.366328	0.183164	+	0.983082i	$0.441366\pi$
$660$	0	0
$661$	48.9621	1.90441	0.952203	−	0.305467i	$-0.0988127\pi$
$661$	48.9621	1.90441	0.952203	+	0.305467i	$0.0988127\pi$
$662$	0	0
$663$	0.655059	0.0254404
$664$	0	0
$665$	−0.240742	−0.00933557
$666$	0	0
$667$	1.83897	0.0712050
$668$	0	0
$669$	−3.09294	−0.119580
$670$	0	0
$671$	42.9331	1.65742
$672$	0	0
$673$	−28.8959	−1.11386	−0.556928	−	0.830561i	$-0.688020\pi$
$673$	−28.8959	−1.11386	−0.556928	+	0.830561i	$0.688020\pi$
$674$	0	0
$675$	−7.81415	−0.300767
$676$	0	0
$677$	39.7744	1.52865	0.764327	−	0.644829i	$-0.223071\pi$
$677$	39.7744	1.52865	0.764327	+	0.644829i	$0.223071\pi$
$678$	0	0
$679$	−12.0744	−0.463373
$680$	0	0
$681$	−3.94965	−0.151351
$682$	0	0
$683$	16.6165	0.635814	0.317907	−	0.948122i	$-0.397020\pi$
$683$	16.6165	0.635814	0.317907	+	0.948122i	$0.397020\pi$
$684$	0	0
$685$	−0.331922	−0.0126821
$686$	0	0
$687$	−4.98148	−0.190055
$688$	0	0
$689$	−21.5804	−0.822148
$690$	0	0
$691$	46.7768	1.77947	0.889737	−	0.456473i	$-0.150887\pi$
$691$	46.7768	1.77947	0.889737	+	0.456473i	$0.150887\pi$
$692$	0	0
$693$	−9.48099	−0.360153
$694$	0	0
$695$	2.73453	0.103727
$696$	0	0
$697$	5.97318	0.226250
$698$	0	0
$699$	−4.45153	−0.168372
$700$	0	0
$701$	6.59195	0.248975	0.124487	−	0.992221i	$-0.460271\pi$
$701$	6.59195	0.248975	0.124487	+	0.992221i	$0.460271\pi$
$702$	0	0
$703$	−12.5920	−0.474915
$704$	0	0
$705$	0.355271	0.0133803
$706$	0	0
$707$	−15.3916	−0.578861
$708$	0	0
$709$	43.7396	1.64267	0.821337	−	0.570443i	$-0.193229\pi$
$709$	43.7396	1.64267	0.821337	+	0.570443i	$0.193229\pi$
$710$	0	0
$711$	−47.4248	−1.77857
$712$	0	0
$713$	42.6840	1.59853
$714$	0	0
$715$	1.31012	0.0489957
$716$	0	0
$717$	4.37295	0.163311
$718$	0	0
$719$	9.08673	0.338878	0.169439	−	0.985541i	$-0.445804\pi$
$719$	9.08673	0.338878	0.169439	+	0.985541i	$0.445804\pi$
$720$	0	0
$721$	−10.7997	−0.402201
$722$	0	0
$723$	−5.08141	−0.188980
$724$	0	0
$725$	1.88277	0.0699242
$726$	0	0
$727$	−18.7843	−0.696673	−0.348336	−	0.937370i	$-0.613253\pi$
$727$	−18.7843	−0.696673	−0.348336	+	0.937370i	$0.613253\pi$
$728$	0	0
$729$	−23.2821	−0.862300
$730$	0	0
$731$	10.0310	0.371010
$732$	0	0
$733$	25.1602	0.929314	0.464657	−	0.885491i	$-0.346178\pi$
$733$	25.1602	0.929314	0.464657	+	0.885491i	$0.346178\pi$
$734$	0	0
$735$	0.0433939	0.00160061
$736$	0	0
$737$	48.3206	1.77991
$738$	0	0
$739$	−30.5073	−1.12223	−0.561115	−	0.827738i	$-0.689627\pi$
$739$	−30.5073	−1.12223	−0.561115	+	0.827738i	$0.689627\pi$
$740$	0	0
$741$	0.962967	0.0353755
$742$	0	0
$743$	−24.0773	−0.883311	−0.441656	−	0.897185i	$-0.645609\pi$
$743$	−24.0773	−0.883311	−0.441656	+	0.897185i	$0.645609\pi$
$744$	0	0
$745$	−2.14538	−0.0786007
$746$	0	0
$747$	16.5742	0.606420
$748$	0	0
$749$	9.27857	0.339031
$750$	0	0
$751$	10.6840	0.389866	0.194933	−	0.980817i	$-0.437551\pi$
$751$	10.6840	0.389866	0.194933	+	0.980817i	$0.437551\pi$
$752$	0	0
$753$	−5.78464	−0.210804
$754$	0	0
$755$	2.04757	0.0745189
$756$	0	0
$757$	39.0013	1.41753	0.708764	−	0.705446i	$-0.249253\pi$
$757$	39.0013	1.41753	0.708764	+	0.705446i	$0.249253\pi$
$758$	0	0
$759$	4.16521	0.151188
$760$	0	0
$761$	−52.6056	−1.90695	−0.953476	−	0.301470i	$-0.902523\pi$
$761$	−52.6056	−1.90695	−0.953476	+	0.301470i	$0.902523\pi$
$762$	0	0
$763$	2.90436	0.105145
$764$	0	0
$765$	0.479796	0.0173471
$766$	0	0
$767$	−7.06409	−0.255069
$768$	0	0
$769$	−21.8596	−0.788276	−0.394138	−	0.919051i	$-0.628957\pi$
$769$	−21.8596	−0.788276	−0.394138	+	0.919051i	$0.628957\pi$
$770$	0	0
$771$	5.70144	0.205332
$772$	0	0
$773$	−16.5478	−0.595182	−0.297591	−	0.954693i	$-0.596183\pi$
$773$	−16.5478	−0.595182	−0.297591	+	0.954693i	$0.596183\pi$
$774$	0	0
$775$	43.7007	1.56978
$776$	0	0
$777$	2.26971	0.0814254
$778$	0	0
$779$	8.78085	0.314607
$780$	0	0
$781$	−46.3784	−1.65955
$782$	0	0
$783$	−0.594853	−0.0212583
$784$	0	0
$785$	−2.73767	−0.0977115
$786$	0	0
$787$	−31.1476	−1.11029	−0.555146	−	0.831753i	$-0.687338\pi$
$787$	−31.1476	−1.11029	−0.555146	+	0.831753i	$0.687338\pi$
$788$	0	0
$789$	−1.51434	−0.0539118
$790$	0	0
$791$	8.67456	0.308432
$792$	0	0
$793$	−32.7980	−1.16469
$794$	0	0
$795$	0.378805	0.0134348
$796$	0	0
$797$	−7.15198	−0.253336	−0.126668	−	0.991945i	$-0.540428\pi$
$797$	−7.15198	−0.253336	−0.126668	+	0.991945i	$0.540428\pi$
$798$	0	0
$799$	8.18710	0.289639
$800$	0	0
$801$	−39.6011	−1.39924
$802$	0	0
$803$	−15.8997	−0.561088
$804$	0	0
$805$	0.795485	0.0280372
$806$	0	0
$807$	−6.31639	−0.222347
$808$	0	0
$809$	−34.4048	−1.20961	−0.604805	−	0.796374i	$-0.706749\pi$
$809$	−34.4048	−1.20961	−0.604805	+	0.796374i	$0.706749\pi$
$810$	0	0
$811$	−29.0670	−1.02068	−0.510341	−	0.859972i	$-0.670481\pi$
$811$	−29.0670	−1.02068	−0.510341	+	0.859972i	$0.670481\pi$
$812$	0	0
$813$	−7.41752	−0.260144
$814$	0	0
$815$	1.80286	0.0631516
$816$	0	0
$817$	14.7460	0.515898
$818$	0	0
$819$	7.24283	0.253085
$820$	0	0
$821$	−19.7064	−0.687759	−0.343879	−	0.939014i	$-0.611741\pi$
$821$	−19.7064	−0.687759	−0.343879	+	0.939014i	$0.611741\pi$
$822$	0	0
$823$	−3.35262	−0.116865	−0.0584324	−	0.998291i	$-0.518610\pi$
$823$	−3.35262	−0.116865	−0.0584324	+	0.998291i	$0.518610\pi$
$824$	0	0
$825$	4.26442	0.148468
$826$	0	0
$827$	−32.8201	−1.14127	−0.570633	−	0.821205i	$-0.693302\pi$
$827$	−32.8201	−1.14127	−0.570633	+	0.821205i	$0.693302\pi$
$828$	0	0
$829$	24.4048	0.847615	0.423808	−	0.905752i	$-0.360693\pi$
$829$	24.4048	0.847615	0.423808	+	0.905752i	$0.360693\pi$
$830$	0	0
$831$	0.538315	0.0186739
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	−0.814090	−0.0281727
$836$	0	0
$837$	−13.8071	−0.477242
$838$	0	0
$839$	22.7307	0.784751	0.392376	−	0.919805i	$-0.371653\pi$
$839$	22.7307	0.784751	0.392376	+	0.919805i	$0.371653\pi$
$840$	0	0
$841$	−28.8567	−0.995058
$842$	0	0
$843$	−8.06708	−0.277845
$844$	0	0
$845$	1.12810	0.0388079
$846$	0	0
$847$	−0.527864	−0.0181376
$848$	0	0
$849$	1.90646	0.0654297
$850$	0	0
$851$	41.6077	1.42629
$852$	0	0
$853$	−14.8218	−0.507487	−0.253744	−	0.967272i	$-0.581662\pi$
$853$	−14.8218	−0.507487	−0.253744	+	0.967272i	$0.581662\pi$
$854$	0	0
$855$	0.705322	0.0241215
$856$	0	0
$857$	0.355245	0.0121349	0.00606747	−	0.999982i	$-0.498069\pi$
$857$	0.355245	0.0121349	0.00606747	+	0.999982i	$0.498069\pi$
$858$	0	0
$859$	37.2665	1.27152	0.635759	−	0.771888i	$-0.280687\pi$
$859$	37.2665	1.27152	0.635759	+	0.771888i	$0.280687\pi$
$860$	0	0
$861$	−1.58276	−0.0539402
$862$	0	0
$863$	0.0548924	0.00186856	0.000934279	−	1.00000i	$-0.499703\pi$
$863$	0.0548924	0.00186856	0.000934279		1.00000i	$0.499703\pi$
$864$	0	0
$865$	−2.85838	−0.0971878
$866$	0	0
$867$	−0.264977	−0.00899910
$868$	0	0
$869$	52.3826	1.77696
$870$	0	0
$871$	−36.9136	−1.25077
$872$	0	0
$873$	35.3754	1.19728
$874$	0	0
$875$	1.63326	0.0552142
$876$	0	0
$877$	11.7330	0.396195	0.198098	−	0.980182i	$-0.436524\pi$
$877$	11.7330	0.396195	0.198098	+	0.980182i	$0.436524\pi$
$878$	0	0
$879$	6.00418	0.202516
$880$	0	0
$881$	42.6705	1.43761	0.718803	−	0.695214i	$-0.244690\pi$
$881$	42.6705	1.43761	0.718803	+	0.695214i	$0.244690\pi$
$882$	0	0
$883$	0.555998	0.0187108	0.00935541	−	0.999956i	$-0.497022\pi$
$883$	0.555998	0.0187108	0.00935541	+	0.999956i	$0.497022\pi$
$884$	0	0
$885$	0.123998	0.00416813
$886$	0	0
$887$	16.2420	0.545353	0.272676	−	0.962106i	$-0.412091\pi$
$887$	16.2420	0.545353	0.272676	+	0.962106i	$0.412091\pi$
$888$	0	0
$889$	−6.19950	−0.207925
$890$	0	0
$891$	27.0956	0.907738
$892$	0	0
$893$	12.0354	0.402750
$894$	0	0
$895$	0.390304	0.0130464
$896$	0	0
$897$	−3.18194	−0.106242
$898$	0	0
$899$	3.32672	0.110952
$900$	0	0
$901$	8.72945	0.290820
$902$	0	0
$903$	−2.65799	−0.0884522
$904$	0	0
$905$	−4.04668	−0.134516
$906$	0	0
$907$	−13.6050	−0.451746	−0.225873	−	0.974157i	$-0.572523\pi$
$907$	−13.6050	−0.451746	−0.225873	+	0.974157i	$0.572523\pi$
$908$	0	0
$909$	45.0942	1.49568
$910$	0	0
$911$	13.4163	0.444501	0.222251	−	0.974990i	$-0.428660\pi$
$911$	13.4163	0.444501	0.222251	+	0.974990i	$0.428660\pi$
$912$	0	0
$913$	−18.3069	−0.605871
$914$	0	0
$915$	0.575710	0.0190324
$916$	0	0
$917$	−8.23816	−0.272048
$918$	0	0
$919$	17.9373	0.591695	0.295848	−	0.955235i	$-0.404398\pi$
$919$	17.9373	0.591695	0.295848	+	0.955235i	$0.404398\pi$
$920$	0	0
$921$	−2.23835	−0.0737563
$922$	0	0
$923$	35.4299	1.16619
$924$	0	0
$925$	42.5987	1.40064
$926$	0	0
$927$	31.6407	1.03922
$928$	0	0
$929$	−19.5156	−0.640286	−0.320143	−	0.947369i	$-0.603731\pi$
$929$	−19.5156	−0.640286	−0.320143	+	0.947369i	$0.603731\pi$
$930$	0	0
$931$	1.47005	0.0481788
$932$	0	0
$933$	−0.0539703	−0.00176691
$934$	0	0
$935$	−0.529954	−0.0173314
$936$	0	0
$937$	−48.4958	−1.58429	−0.792145	−	0.610333i	$-0.791035\pi$
$937$	−48.4958	−1.58429	−0.792145	+	0.610333i	$0.791035\pi$
$938$	0	0
$939$	0.691494	0.0225660
$940$	0	0
$941$	−12.3406	−0.402293	−0.201146	−	0.979561i	$-0.564467\pi$
$941$	−12.3406	−0.402293	−0.201146	+	0.979561i	$0.564467\pi$
$942$	0	0
$943$	−29.0146	−0.944846
$944$	0	0
$945$	−0.257317	−0.00837052
$946$	0	0
$947$	26.3633	0.856691	0.428345	−	0.903615i	$-0.359097\pi$
$947$	26.3633	0.856691	0.428345	+	0.903615i	$0.359097\pi$
$948$	0	0
$949$	12.1463	0.394285
$950$	0	0
$951$	7.53204	0.244243
$952$	0	0
$953$	−3.60644	−0.116824	−0.0584120	−	0.998293i	$-0.518604\pi$
$953$	−3.60644	−0.116824	−0.0584120	+	0.998293i	$0.518604\pi$
$954$	0	0
$955$	−3.32759	−0.107678
$956$	0	0
$957$	0.324630	0.0104938
$958$	0	0
$959$	2.02682	0.0654494
$960$	0	0
$961$	46.2161	1.49084
$962$	0	0
$963$	−27.1842	−0.875999
$964$	0	0
$965$	−2.69505	−0.0867567
$966$	0	0
$967$	−28.2139	−0.907299	−0.453649	−	0.891180i	$-0.649878\pi$
$967$	−28.2139	−0.907299	−0.453649	+	0.891180i	$0.649878\pi$
$968$	0	0
$969$	−0.389529	−0.0125135
$970$	0	0
$971$	−20.0355	−0.642971	−0.321486	−	0.946914i	$-0.604182\pi$
$971$	−20.0355	−0.642971	−0.321486	+	0.946914i	$0.604182\pi$
$972$	0	0
$973$	−16.6979	−0.535310
$974$	0	0
$975$	−3.25773	−0.104331
$976$	0	0
$977$	−44.5259	−1.42451	−0.712255	−	0.701921i	$-0.752326\pi$
$977$	−44.5259	−1.42451	−0.712255	+	0.701921i	$0.752326\pi$
$978$	0	0
$979$	43.7410	1.39797
$980$	0	0
$981$	−8.50915	−0.271676
$982$	0	0
$983$	2.54931	0.0813103	0.0406552	−	0.999173i	$-0.487055\pi$
$983$	2.54931	0.0813103	0.0406552	+	0.999173i	$0.487055\pi$
$984$	0	0
$985$	−0.553636	−0.0176403
$986$	0	0
$987$	−2.16940	−0.0690526
$988$	0	0
$989$	−48.7254	−1.54938
$990$	0	0
$991$	−24.8794	−0.790319	−0.395160	−	0.918612i	$-0.629311\pi$
$991$	−24.8794	−0.790319	−0.395160	+	0.918612i	$0.629311\pi$
$992$	0	0
$993$	7.63675	0.242345
$994$	0	0
$995$	0.843181	0.0267306
$996$	0	0
$997$	−3.68557	−0.116723	−0.0583615	−	0.998296i	$-0.518588\pi$
$997$	−3.68557	−0.116723	−0.0583615	+	0.998296i	$0.518588\pi$
$998$	0	0
$999$	−13.4589	−0.425821

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7616.2.a.bp.1.2		4
4.3	odd	2		7616.2.a.bj.1.3		4
8.3	odd	2		952.2.a.g.1.2	✓	4
8.5	even	2		1904.2.a.q.1.3		4
24.11	even	2		8568.2.a.bj.1.2		4
56.27	even	2		6664.2.a.o.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
952.2.a.g.1.2	✓	4	8.3	odd	2
1904.2.a.q.1.3		4	8.5	even	2
6664.2.a.o.1.3		4	56.27	even	2
7616.2.a.bj.1.3		4	4.3	odd	2
7616.2.a.bp.1.2		4	1.1	even	1	trivial
8568.2.a.bj.1.2		4	24.11	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:	\( N \)	\(=\)	\( 7616 = 2^{6} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7616.a (trivial)

\(f(q)\)	\(=\)	\(q-0.264977 q^{3} -0.163765 q^{5} +1.00000 q^{7} -2.92979 q^{9} +3.23607 q^{11} -2.47214 q^{13} +0.0433939 q^{15} +1.00000 q^{17} +1.47005 q^{19} -0.264977 q^{21} -4.85748 q^{23} -4.97318 q^{25} +1.57126 q^{27} -0.378584 q^{29} -8.78727 q^{31} -0.857484 q^{33} -0.163765 q^{35} -8.56569 q^{37} +0.655059 q^{39} +5.97318 q^{41} +10.0310 q^{43} +0.479796 q^{45} +8.18710 q^{47} +1.00000 q^{49} -0.264977 q^{51} +8.72945 q^{53} -0.529954 q^{55} -0.389529 q^{57} +2.85748 q^{59} +13.2671 q^{61} -2.92979 q^{63} +0.404849 q^{65} +14.9319 q^{67} +1.28712 q^{69} -14.3317 q^{71} -4.91327 q^{73} +1.31778 q^{75} +3.23607 q^{77} +16.1871 q^{79} +8.37301 q^{81} -5.65715 q^{83} -0.163765 q^{85} +0.100316 q^{87} +13.5167 q^{89} -2.47214 q^{91} +2.32843 q^{93} -0.240742 q^{95} -12.0744 q^{97} -9.48099 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{3} + q^{5} + 4 q^{7} + 7 q^{9} + 4 q^{11} + 8 q^{13} - 12 q^{15} + 4 q^{17} + 14 q^{19} + 3 q^{21} - 8 q^{23} + 11 q^{25} + 12 q^{27} - 4 q^{29} - 5 q^{31} + 8 q^{33} + q^{35} + 4 q^{37} - 4 q^{39}+ \cdots + 2 q^{99}+O(q^{100}) \) 4 * q + 3 * q^3 + q^5 + 4 * q^7 + 7 * q^9 + 4 * q^11 + 8 * q^13 - 12 * q^15 + 4 * q^17 + 14 * q^19 + 3 * q^21 - 8 * q^23 + 11 * q^25 + 12 * q^27 - 4 * q^29 - 5 * q^31 + 8 * q^33 + q^35 + 4 * q^37 - 4 * q^39 - 7 * q^41 + 19 * q^43 + 2 * q^45 - 8 * q^47 + 4 * q^49 + 3 * q^51 - 5 * q^53 + 6 * q^55 + 44 * q^57 + 23 * q^61 + 7 * q^63 - 8 * q^65 + 15 * q^67 + 2 * q^69 - 2 * q^71 - 5 * q^73 + 10 * q^75 + 4 * q^77 + 24 * q^79 - 8 * q^81 + 10 * q^83 + q^85 - 16 * q^87 - 16 * q^89 + 8 * q^91 + 20 * q^93 - 22 * q^95 - 15 * q^97 + 2 * q^99

Introduction

L-functions

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