LMFDB - Embedded newform 656.2.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [656,2,Mod(1,656)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(656, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("656.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 656 = 2^{4} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	656.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$5.23818637260$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.25808.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 10x^{2} - 6x + 9 $ x^4 - 10x^2 - 6x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 164)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.46810$ of defining polynomial
Character	$\chi$	$=$	656.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.21551 q^{3} +3.59669 q^{5} +5.06479 q^{7} +1.90849 q^{9} +O(q^{10})$	$q-2.21551 q^{3} +3.59669 q^{5} +5.06479 q^{7} +1.90849 q^{9} +2.55961 q^{11} -4.93620 q^{13} -7.96851 q^{15} +2.68821 q^{17} -4.72069 q^{19} -11.2211 q^{21} +1.49482 q^{23} +7.93620 q^{25} +2.41826 q^{27} +2.43102 q^{29} -3.19339 q^{31} -5.67085 q^{33} +18.2165 q^{35} +2.90849 q^{37} +10.9362 q^{39} -1.00000 q^{41} +7.62441 q^{43} +6.86424 q^{45} -5.15171 q^{47} +18.6521 q^{49} -5.95575 q^{51} -11.1192 q^{53} +9.20614 q^{55} +10.4587 q^{57} -1.49482 q^{59} +1.06380 q^{61} +9.66608 q^{63} -17.7540 q^{65} -10.5596 q^{67} -3.31179 q^{69} +12.6023 q^{71} +8.28490 q^{73} -17.5827 q^{75} +12.9639 q^{77} +4.28967 q^{79} -11.0831 q^{81} -10.5606 q^{83} +9.66866 q^{85} -5.38595 q^{87} +9.36722 q^{89} -25.0008 q^{91} +7.07498 q^{93} -16.9789 q^{95} -3.37641 q^{97} +4.88499 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 4 q^{5} + 12 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 + 4 * q^5 + 12 * q^9	$ 4 q - 2 q^{3} + 4 q^{5} + 12 q^{9} - 4 q^{11} + 10 q^{15} - 4 q^{17} - 6 q^{19} + 12 q^{23} + 12 q^{25} + 10 q^{27} - 4 q^{29} + 8 q^{31} - 20 q^{33} + 26 q^{35} + 16 q^{37} + 24 q^{39} - 4 q^{41} - 4 q^{43} + 4 q^{45} + 6 q^{47} + 16 q^{49} + 4 q^{51} - 16 q^{53} + 2 q^{55} + 4 q^{57} - 12 q^{59} + 24 q^{61} + 10 q^{63} + 4 q^{65} - 28 q^{67} - 28 q^{69} + 2 q^{71} + 8 q^{73} - 30 q^{75} + 8 q^{77} + 18 q^{79} + 28 q^{81} + 12 q^{83} + 32 q^{85} - 44 q^{87} + 4 q^{89} - 36 q^{91} - 28 q^{93} - 14 q^{95} + 16 q^{97} - 58 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 + 4 * q^5 + 12 * q^9 - 4 * q^11 + 10 * q^15 - 4 * q^17 - 6 * q^19 + 12 * q^23 + 12 * q^25 + 10 * q^27 - 4 * q^29 + 8 * q^31 - 20 * q^33 + 26 * q^35 + 16 * q^37 + 24 * q^39 - 4 * q^41 - 4 * q^43 + 4 * q^45 + 6 * q^47 + 16 * q^49 + 4 * q^51 - 16 * q^53 + 2 * q^55 + 4 * q^57 - 12 * q^59 + 24 * q^61 + 10 * q^63 + 4 * q^65 - 28 * q^67 - 28 * q^69 + 2 * q^71 + 8 * q^73 - 30 * q^75 + 8 * q^77 + 18 * q^79 + 28 * q^81 + 12 * q^83 + 32 * q^85 - 44 * q^87 + 4 * q^89 - 36 * q^91 - 28 * q^93 - 14 * q^95 + 16 * q^97 - 58 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.21551	−1.27913	−0.639563	−	0.768739i	$-0.720885\pi$
$3$	−2.21551	−1.27913	−0.639563	+	0.768739i	$0.720885\pi$
$4$	0	0
$5$	3.59669	1.60849	0.804245	−	0.594298i	$-0.202570\pi$
$5$	3.59669	1.60849	0.804245	+	0.594298i	$0.202570\pi$
$6$	0	0
$7$	5.06479	1.91431	0.957156	−	0.289574i	$-0.0935135\pi$
$7$	5.06479	1.91431	0.957156	+	0.289574i	$0.0935135\pi$
$8$	0	0
$9$	1.90849	0.636162
$10$	0	0
$11$	2.55961	0.771753	0.385876	−	0.922551i	$-0.373899\pi$
$11$	2.55961	0.771753	0.385876	+	0.922551i	$0.373899\pi$
$12$	0	0
$13$	−4.93620	−1.36906	−0.684528	−	0.728987i	$-0.739991\pi$
$13$	−4.93620	−1.36906	−0.684528	+	0.728987i	$0.739991\pi$
$14$	0	0
$15$	−7.96851	−2.05746
$16$	0	0
$17$	2.68821	0.651986	0.325993	−	0.945372i	$-0.394301\pi$
$17$	2.68821	0.651986	0.325993	+	0.945372i	$0.394301\pi$
$18$	0	0
$19$	−4.72069	−1.08300	−0.541500	−	0.840701i	$-0.682143\pi$
$19$	−4.72069	−1.08300	−0.541500	+	0.840701i	$0.682143\pi$
$20$	0	0
$21$	−11.2211	−2.44864
$22$	0	0
$23$	1.49482	0.311692	0.155846	−	0.987781i	$-0.450190\pi$
$23$	1.49482	0.311692	0.155846	+	0.987781i	$0.450190\pi$
$24$	0	0
$25$	7.93620	1.58724
$26$	0	0
$27$	2.41826	0.465395
$28$	0	0
$29$	2.43102	0.451429	0.225715	−	0.974193i	$-0.427528\pi$
$29$	2.43102	0.451429	0.225715	+	0.974193i	$0.427528\pi$
$30$	0	0
$31$	−3.19339	−0.573549	−0.286774	−	0.957998i	$-0.592583\pi$
$31$	−3.19339	−0.573549	−0.286774	+	0.957998i	$0.592583\pi$
$32$	0	0
$33$	−5.67085	−0.987168
$34$	0	0
$35$	18.2165	3.07915
$36$	0	0
$37$	2.90849	0.478152	0.239076	−	0.971001i	$-0.423155\pi$
$37$	2.90849	0.478152	0.239076	+	0.971001i	$0.423155\pi$
$38$	0	0
$39$	10.9362	1.75119
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	7.62441	1.16271	0.581356	−	0.813650i	$-0.302523\pi$
$43$	7.62441	1.16271	0.581356	+	0.813650i	$0.302523\pi$
$44$	0	0
$45$	6.86424	1.02326
$46$	0	0
$47$	−5.15171	−0.751454	−0.375727	−	0.926730i	$-0.622607\pi$
$47$	−5.15171	−0.751454	−0.375727	+	0.926730i	$0.622607\pi$
$48$	0	0
$49$	18.6521	2.66459
$50$	0	0
$51$	−5.95575	−0.833972
$52$	0	0
$53$	−11.1192	−1.52734	−0.763672	−	0.645605i	$-0.776605\pi$
$53$	−11.1192	−1.52734	−0.763672	+	0.645605i	$0.776605\pi$
$54$	0	0
$55$	9.20614	1.24136
$56$	0	0
$57$	10.4587	1.38529
$58$	0	0
$59$	−1.49482	−0.194609	−0.0973046	−	0.995255i	$-0.531022\pi$
$59$	−1.49482	−0.194609	−0.0973046	+	0.995255i	$0.531022\pi$
$60$	0	0
$61$	1.06380	0.136206	0.0681029	−	0.997678i	$-0.478305\pi$
$61$	1.06380	0.136206	0.0681029	+	0.997678i	$0.478305\pi$
$62$	0	0
$63$	9.66608	1.21781
$64$	0	0
$65$	−17.7540	−2.20211
$66$	0	0
$67$	−10.5596	−1.29006	−0.645031	−	0.764156i	$-0.723156\pi$
$67$	−10.5596	−1.29006	−0.645031	+	0.764156i	$0.723156\pi$
$68$	0	0
$69$	−3.31179	−0.398693
$70$	0	0
$71$	12.6023	1.49562	0.747808	−	0.663915i	$-0.231106\pi$
$71$	12.6023	1.49562	0.747808	+	0.663915i	$0.231106\pi$
$72$	0	0
$73$	8.28490	0.969674	0.484837	−	0.874604i	$-0.338879\pi$
$73$	8.28490	0.969674	0.484837	+	0.874604i	$0.338879\pi$
$74$	0	0
$75$	−17.5827	−2.03028
$76$	0	0
$77$	12.9639	1.47737
$78$	0	0
$79$	4.28967	0.482625	0.241313	−	0.970447i	$-0.422422\pi$
$79$	4.28967	0.482625	0.241313	+	0.970447i	$0.422422\pi$
$80$	0	0
$81$	−11.0831	−1.23146
$82$	0	0
$83$	−10.5606	−1.15918	−0.579588	−	0.814909i	$-0.696787\pi$
$83$	−10.5606	−1.15918	−0.579588	+	0.814909i	$0.696787\pi$
$84$	0	0
$85$	9.66866	1.04871
$86$	0	0
$87$	−5.38595	−0.577435
$88$	0	0
$89$	9.36722	0.992923	0.496462	−	0.868059i	$-0.334632\pi$
$89$	9.36722	0.992923	0.496462	+	0.868059i	$0.334632\pi$
$90$	0	0
$91$	−25.0008	−2.62080
$92$	0	0
$93$	7.07498	0.733641
$94$	0	0
$95$	−16.9789	−1.74199
$96$	0	0
$97$	−3.37641	−0.342823	−0.171411	−	0.985200i	$-0.554833\pi$
$97$	−3.37641	−0.342823	−0.171411	+	0.985200i	$0.554833\pi$
$98$	0	0
$99$	4.88499	0.490960
$100$	0	0
$101$	−4.24799	−0.422691	−0.211345	−	0.977411i	$-0.567785\pi$
$101$	−4.24799	−0.422691	−0.211345	+	0.977411i	$0.567785\pi$
$102$	0	0
$103$	−1.11923	−0.110281	−0.0551404	−	0.998479i	$-0.517561\pi$
$103$	−1.11923	−0.110281	−0.0551404	+	0.998479i	$0.517561\pi$
$104$	0	0
$105$	−40.3588	−3.93862
$106$	0	0
$107$	0.322149	0.0311434	0.0155717	−	0.999879i	$-0.495043\pi$
$107$	0.322149	0.0311434	0.0155717	+	0.999879i	$0.495043\pi$
$108$	0	0
$109$	3.23763	0.310109	0.155055	−	0.987906i	$-0.450445\pi$
$109$	3.23763	0.310109	0.155055	+	0.987906i	$0.450445\pi$
$110$	0	0
$111$	−6.44378	−0.611616
$112$	0	0
$113$	2.10187	0.197727	0.0988637	−	0.995101i	$-0.468479\pi$
$113$	2.10187	0.197727	0.0988637	+	0.995101i	$0.468479\pi$
$114$	0	0
$115$	5.37641	0.501353
$116$	0	0
$117$	−9.42066	−0.870941
$118$	0	0
$119$	13.6152	1.24810
$120$	0	0
$121$	−4.44838	−0.404398
$122$	0	0
$123$	2.21551	0.199766
$124$	0	0
$125$	10.5606	0.944569
$126$	0	0
$127$	13.0658	1.15940	0.579700	−	0.814830i	$-0.303170\pi$
$127$	13.0658	1.15940	0.579700	+	0.814830i	$0.303170\pi$
$128$	0	0
$129$	−16.8919	−1.48725
$130$	0	0
$131$	−8.43102	−0.736622	−0.368311	−	0.929703i	$-0.620064\pi$
$131$	−8.43102	−0.736622	−0.368311	+	0.929703i	$0.620064\pi$
$132$	0	0
$133$	−23.9093	−2.07320
$134$	0	0
$135$	8.69774	0.748583
$136$	0	0
$137$	−16.5606	−1.41487	−0.707434	−	0.706779i	$-0.750147\pi$
$137$	−16.5606	−1.41487	−0.707434	+	0.706779i	$0.750147\pi$
$138$	0	0
$139$	2.18303	0.185162	0.0925810	−	0.995705i	$-0.470488\pi$
$139$	2.18303	0.185162	0.0925810	+	0.995705i	$0.470488\pi$
$140$	0	0
$141$	11.4137	0.961204
$142$	0	0
$143$	−12.6348	−1.05657
$144$	0	0
$145$	8.74363	0.726119
$146$	0	0
$147$	−41.3240	−3.40834
$148$	0	0
$149$	2.43102	0.199157	0.0995785	−	0.995030i	$-0.468251\pi$
$149$	2.43102	0.199157	0.0995785	+	0.995030i	$0.468251\pi$
$150$	0	0
$151$	0.343113	0.0279221	0.0139611	−	0.999903i	$-0.495556\pi$
$151$	0.343113	0.0279221	0.0139611	+	0.999903i	$0.495556\pi$
$152$	0	0
$153$	5.13040	0.414769
$154$	0	0
$155$	−11.4856	−0.922548
$156$	0	0
$157$	−10.6348	−0.848746	−0.424373	−	0.905487i	$-0.639506\pi$
$157$	−10.6348	−0.848746	−0.424373	+	0.905487i	$0.639506\pi$
$158$	0	0
$159$	24.6348	1.95366
$160$	0	0
$161$	7.57096	0.596675
$162$	0	0
$163$	0.173834	0.0136157	0.00680785	−	0.999977i	$-0.497833\pi$
$163$	0.173834	0.0136157	0.00680785	+	0.999977i	$0.497833\pi$
$164$	0	0
$165$	−20.3963	−1.58785
$166$	0	0
$167$	−14.5596	−1.12666	−0.563328	−	0.826233i	$-0.690479\pi$
$167$	−14.5596	−1.12666	−0.563328	+	0.826233i	$0.690479\pi$
$168$	0	0
$169$	11.3661	0.874312
$170$	0	0
$171$	−9.00937	−0.688963
$172$	0	0
$173$	−22.2592	−1.69233	−0.846167	−	0.532918i	$-0.821095\pi$
$173$	−22.2592	−1.69233	−0.846167	+	0.532918i	$0.821095\pi$
$174$	0	0
$175$	40.1952	3.03847
$176$	0	0
$177$	3.31179	0.248930
$178$	0	0
$179$	17.9268	1.33991	0.669957	−	0.742400i	$-0.266312\pi$
$179$	17.9268	1.33991	0.669957	+	0.742400i	$0.266312\pi$
$180$	0	0
$181$	20.9916	1.56030	0.780148	−	0.625595i	$-0.215144\pi$
$181$	20.9916	1.56030	0.780148	+	0.625595i	$0.215144\pi$
$182$	0	0
$183$	−2.35686	−0.174224
$184$	0	0
$185$	10.4609	0.769103
$186$	0	0
$187$	6.88077	0.503172
$188$	0	0
$189$	12.2480	0.890910
$190$	0	0
$191$	−6.45074	−0.466759	−0.233380	−	0.972386i	$-0.574978\pi$
$191$	−6.45074	−0.466759	−0.233380	+	0.972386i	$0.574978\pi$
$192$	0	0
$193$	6.20374	0.446555	0.223278	−	0.974755i	$-0.428324\pi$
$193$	6.20374	0.446555	0.223278	+	0.974755i	$0.428324\pi$
$194$	0	0
$195$	39.3341	2.81678
$196$	0	0
$197$	14.1850	1.01064	0.505320	−	0.862932i	$-0.331374\pi$
$197$	14.1850	1.01064	0.505320	+	0.862932i	$0.331374\pi$
$198$	0	0
$199$	−18.7099	−1.32631	−0.663155	−	0.748482i	$-0.730783\pi$
$199$	−18.7099	−1.32631	−0.663155	+	0.748482i	$0.730783\pi$
$200$	0	0
$201$	23.3949	1.65015
$202$	0	0
$203$	12.3126	0.864176
$204$	0	0
$205$	−3.59669	−0.251204
$206$	0	0
$207$	2.85285	0.198286
$208$	0	0
$209$	−12.0831	−0.835808
$210$	0	0
$211$	−14.6984	−1.01188	−0.505940	−	0.862569i	$-0.668854\pi$
$211$	−14.6984	−1.01188	−0.505940	+	0.862569i	$0.668854\pi$
$212$	0	0
$213$	−27.9205	−1.91308
$214$	0	0
$215$	27.4226	1.87021
$216$	0	0
$217$	−16.1738	−1.09795
$218$	0	0
$219$	−18.3553	−1.24033
$220$	0	0
$221$	−13.2695	−0.892605
$222$	0	0
$223$	−20.1109	−1.34672	−0.673361	−	0.739314i	$-0.735150\pi$
$223$	−20.1109	−1.34672	−0.673361	+	0.739314i	$0.735150\pi$
$224$	0	0
$225$	15.1461	1.00974
$226$	0	0
$227$	6.71149	0.445457	0.222729	−	0.974880i	$-0.428504\pi$
$227$	6.71149	0.445457	0.222729	+	0.974880i	$0.428504\pi$
$228$	0	0
$229$	12.7624	0.843361	0.421680	−	0.906745i	$-0.361440\pi$
$229$	12.7624	0.843361	0.421680	+	0.906745i	$0.361440\pi$
$230$	0	0
$231$	−28.7217	−1.88975
$232$	0	0
$233$	−29.0750	−1.90477	−0.952383	−	0.304906i	$-0.901375\pi$
$233$	−29.0750	−1.90477	−0.952383	+	0.304906i	$0.901375\pi$
$234$	0	0
$235$	−18.5291	−1.20871
$236$	0	0
$237$	−9.50380	−0.617338
$238$	0	0
$239$	−26.0971	−1.68808	−0.844041	−	0.536279i	$-0.819829\pi$
$239$	−26.0971	−1.68808	−0.844041	+	0.536279i	$0.819829\pi$
$240$	0	0
$241$	1.06380	0.0685255	0.0342627	−	0.999413i	$-0.489092\pi$
$241$	1.06380	0.0685255	0.0342627	+	0.999413i	$0.489092\pi$
$242$	0	0
$243$	17.3000	1.10980
$244$	0	0
$245$	67.0859	4.28596
$246$	0	0
$247$	23.3023	1.48269
$248$	0	0
$249$	23.3971	1.48273
$250$	0	0
$251$	7.82815	0.494108	0.247054	−	0.969002i	$-0.420537\pi$
$251$	7.82815	0.494108	0.247054	+	0.969002i	$0.420537\pi$
$252$	0	0
$253$	3.82617	0.240549
$254$	0	0
$255$	−21.4210	−1.34144
$256$	0	0
$257$	7.49482	0.467514	0.233757	−	0.972295i	$-0.424898\pi$
$257$	7.49482	0.467514	0.233757	+	0.972295i	$0.424898\pi$
$258$	0	0
$259$	14.7309	0.915332
$260$	0	0
$261$	4.63957	0.287182
$262$	0	0
$263$	−15.5919	−0.961439	−0.480720	−	0.876874i	$-0.659625\pi$
$263$	−15.5919	−0.961439	−0.480720	+	0.876874i	$0.659625\pi$
$264$	0	0
$265$	−39.9924	−2.45672
$266$	0	0
$267$	−20.7532	−1.27007
$268$	0	0
$269$	8.80860	0.537070	0.268535	−	0.963270i	$-0.413461\pi$
$269$	8.80860	0.537070	0.268535	+	0.963270i	$0.413461\pi$
$270$	0	0
$271$	19.8816	1.20772	0.603860	−	0.797090i	$-0.293628\pi$
$271$	19.8816	1.20772	0.603860	+	0.797090i	$0.293628\pi$
$272$	0	0
$273$	55.3896	3.35233
$274$	0	0
$275$	20.3136	1.22496
$276$	0	0
$277$	−7.08232	−0.425535	−0.212768	−	0.977103i	$-0.568248\pi$
$277$	−7.08232	−0.425535	−0.212768	+	0.977103i	$0.568248\pi$
$278$	0	0
$279$	−6.09453	−0.364870
$280$	0	0
$281$	−31.2695	−1.86538	−0.932692	−	0.360675i	$-0.882547\pi$
$281$	−31.2695	−1.86538	−0.932692	+	0.360675i	$0.882547\pi$
$282$	0	0
$283$	2.18303	0.129768	0.0648838	−	0.997893i	$-0.479332\pi$
$283$	2.18303	0.129768	0.0648838	+	0.997893i	$0.479332\pi$
$284$	0	0
$285$	37.6169	2.22823
$286$	0	0
$287$	−5.06479	−0.298965
$288$	0	0
$289$	−9.77354	−0.574914
$290$	0	0
$291$	7.48048	0.438514
$292$	0	0
$293$	−33.6798	−1.96760	−0.983798	−	0.179278i	$-0.942624\pi$
$293$	−33.6798	−1.96760	−0.983798	+	0.179278i	$0.942624\pi$
$294$	0	0
$295$	−5.37641	−0.313027
$296$	0	0
$297$	6.18982	0.359170
$298$	0	0
$299$	−7.37874	−0.426723
$300$	0	0
$301$	38.6160	2.22579
$302$	0	0
$303$	9.41147	0.540675
$304$	0	0
$305$	3.82617	0.219086
$306$	0	0
$307$	12.7532	0.727862	0.363931	−	0.931426i	$-0.381434\pi$
$307$	12.7532	0.727862	0.363931	+	0.931426i	$0.381434\pi$
$308$	0	0
$309$	2.47966	0.141063
$310$	0	0
$311$	−2.18204	−0.123732	−0.0618660	−	0.998084i	$-0.519705\pi$
$311$	−2.18204	−0.123732	−0.0618660	+	0.998084i	$0.519705\pi$
$312$	0	0
$313$	−29.3042	−1.65637	−0.828187	−	0.560452i	$-0.810627\pi$
$313$	−29.3042	−1.65637	−0.828187	+	0.560452i	$0.810627\pi$
$314$	0	0
$315$	34.7659	1.95884
$316$	0	0
$317$	−28.5511	−1.60359	−0.801794	−	0.597601i	$-0.796121\pi$
$317$	−28.5511	−1.60359	−0.801794	+	0.597601i	$0.796121\pi$
$318$	0	0
$319$	6.22247	0.348392
$320$	0	0
$321$	−0.713725	−0.0398363
$322$	0	0
$323$	−12.6902	−0.706101
$324$	0	0
$325$	−39.1747	−2.17302
$326$	0	0
$327$	−7.17301	−0.396669
$328$	0	0
$329$	−26.0923	−1.43852
$330$	0	0
$331$	4.80761	0.264250	0.132125	−	0.991233i	$-0.457820\pi$
$331$	4.80761	0.264250	0.132125	+	0.991233i	$0.457820\pi$
$332$	0	0
$333$	5.55080	0.304182
$334$	0	0
$335$	−37.9797	−2.07505
$336$	0	0
$337$	16.3148	0.888724	0.444362	−	0.895847i	$-0.353430\pi$
$337$	16.3148	0.888724	0.444362	+	0.895847i	$0.353430\pi$
$338$	0	0
$339$	−4.65672	−0.252918
$340$	0	0
$341$	−8.17383	−0.442638
$342$	0	0
$343$	59.0156	3.18654
$344$	0	0
$345$	−11.9115	−0.641294
$346$	0	0
$347$	−31.6034	−1.69656	−0.848281	−	0.529547i	$-0.822362\pi$
$347$	−31.6034	−1.69656	−0.848281	+	0.529547i	$0.822362\pi$
$348$	0	0
$349$	4.97311	0.266204	0.133102	−	0.991102i	$-0.457506\pi$
$349$	4.97311	0.266204	0.133102	+	0.991102i	$0.457506\pi$
$350$	0	0
$351$	−11.9370	−0.637151
$352$	0	0
$353$	1.72430	0.0917750	0.0458875	−	0.998947i	$-0.485388\pi$
$353$	1.72430	0.0917750	0.0458875	+	0.998947i	$0.485388\pi$
$354$	0	0
$355$	45.3265	2.40568
$356$	0	0
$357$	−30.1646	−1.59648
$358$	0	0
$359$	7.13796	0.376727	0.188364	−	0.982099i	$-0.439682\pi$
$359$	7.13796	0.376727	0.188364	+	0.982099i	$0.439682\pi$
$360$	0	0
$361$	3.28490	0.172889
$362$	0	0
$363$	9.85542	0.517276
$364$	0	0
$365$	29.7982	1.55971
$366$	0	0
$367$	9.38595	0.489943	0.244971	−	0.969530i	$-0.421221\pi$
$367$	9.38595	0.489943	0.244971	+	0.969530i	$0.421221\pi$
$368$	0	0
$369$	−1.90849	−0.0993518
$370$	0	0
$371$	−56.3166	−2.92381
$372$	0	0
$373$	23.1212	1.19717	0.598585	−	0.801059i	$-0.295730\pi$
$373$	23.1212	1.19717	0.598585	+	0.801059i	$0.295730\pi$
$374$	0	0
$375$	−23.3971	−1.20822
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	−5.99080	−0.307727	−0.153863	−	0.988092i	$-0.549172\pi$
$379$	−5.99080	−0.307727	−0.153863	+	0.988092i	$0.549172\pi$
$380$	0	0
$381$	−28.9474	−1.48302
$382$	0	0
$383$	29.5811	1.51153	0.755763	−	0.654845i	$-0.227266\pi$
$383$	29.5811	1.51153	0.755763	+	0.654845i	$0.227266\pi$
$384$	0	0
$385$	46.6272	2.37634
$386$	0	0
$387$	14.5511	0.739672
$388$	0	0
$389$	22.8620	1.15915	0.579576	−	0.814918i	$-0.303218\pi$
$389$	22.8620	1.15915	0.579576	+	0.814918i	$0.303218\pi$
$390$	0	0
$391$	4.01839	0.203219
$392$	0	0
$393$	18.6790	0.942232
$394$	0	0
$395$	15.4286	0.776298
$396$	0	0
$397$	−0.588531	−0.0295375	−0.0147688	−	0.999891i	$-0.504701\pi$
$397$	−0.588531	−0.0295375	−0.0147688	+	0.999891i	$0.504701\pi$
$398$	0	0
$399$	52.9713	2.65188
$400$	0	0
$401$	−0.0926758	−0.00462801	−0.00231400	−	0.999997i	$-0.500737\pi$
$401$	−0.0926758	−0.00462801	−0.00231400	+	0.999997i	$0.500737\pi$
$402$	0	0
$403$	15.7632	0.785220
$404$	0	0
$405$	−39.8626	−1.98079
$406$	0	0
$407$	7.44460	0.369015
$408$	0	0
$409$	−1.29526	−0.0640463	−0.0320232	−	0.999487i	$-0.510195\pi$
$409$	−1.29526	−0.0640463	−0.0320232	+	0.999487i	$0.510195\pi$
$410$	0	0
$411$	36.6902	1.80979
$412$	0	0
$413$	−7.57096	−0.372543
$414$	0	0
$415$	−37.9833	−1.86452
$416$	0	0
$417$	−4.83652	−0.236846
$418$	0	0
$419$	−1.75201	−0.0855912	−0.0427956	−	0.999084i	$-0.513626\pi$
$419$	−1.75201	−0.0855912	−0.0427956	+	0.999084i	$0.513626\pi$
$420$	0	0
$421$	36.7364	1.79042	0.895212	−	0.445641i	$-0.147024\pi$
$421$	36.7364	1.79042	0.895212	+	0.445641i	$0.147024\pi$
$422$	0	0
$423$	−9.83196	−0.478046
$424$	0	0
$425$	21.3341	1.03486
$426$	0	0
$427$	5.38793	0.260740
$428$	0	0
$429$	27.9924	1.35149
$430$	0	0
$431$	−12.3126	−0.593078	−0.296539	−	0.955021i	$-0.595832\pi$
$431$	−12.3126	−0.593078	−0.296539	+	0.955021i	$0.595832\pi$
$432$	0	0
$433$	−17.7632	−0.853644	−0.426822	−	0.904336i	$-0.640367\pi$
$433$	−17.7632	−0.853644	−0.426822	+	0.904336i	$0.640367\pi$
$434$	0	0
$435$	−19.3716	−0.928798
$436$	0	0
$437$	−7.05659	−0.337562
$438$	0	0
$439$	−3.22230	−0.153792	−0.0768961	−	0.997039i	$-0.524501\pi$
$439$	−3.22230	−0.153792	−0.0768961	+	0.997039i	$0.524501\pi$
$440$	0	0
$441$	35.5973	1.69511
$442$	0	0
$443$	35.8537	1.70346	0.851730	−	0.523982i	$-0.175554\pi$
$443$	35.8537	1.70346	0.851730	+	0.523982i	$0.175554\pi$
$444$	0	0
$445$	33.6910	1.59711
$446$	0	0
$447$	−5.38595	−0.254747
$448$	0	0
$449$	−15.3437	−0.724113	−0.362057	−	0.932156i	$-0.617925\pi$
$449$	−15.3437	−0.724113	−0.362057	+	0.932156i	$0.617925\pi$
$450$	0	0
$451$	−2.55961	−0.120528
$452$	0	0
$453$	−0.760170	−0.0357159
$454$	0	0
$455$	−89.9203	−4.21553
$456$	0	0
$457$	27.5917	1.29068	0.645342	−	0.763894i	$-0.276715\pi$
$457$	27.5917	1.29068	0.645342	+	0.763894i	$0.276715\pi$
$458$	0	0
$459$	6.50079	0.303431
$460$	0	0
$461$	−8.08116	−0.376377	−0.188189	−	0.982133i	$-0.560262\pi$
$461$	−8.08116	−0.376377	−0.188189	+	0.982133i	$0.560262\pi$
$462$	0	0
$463$	8.23622	0.382770	0.191385	−	0.981515i	$-0.438702\pi$
$463$	8.23622	0.382770	0.191385	+	0.981515i	$0.438702\pi$
$464$	0	0
$465$	25.4465	1.18005
$466$	0	0
$467$	23.7333	1.09825	0.549123	−	0.835742i	$-0.314962\pi$
$467$	23.7333	1.09825	0.549123	+	0.835742i	$0.314962\pi$
$468$	0	0
$469$	−53.4822	−2.46958
$470$	0	0
$471$	23.5614	1.08565
$472$	0	0
$473$	19.5155	0.897325
$474$	0	0
$475$	−37.4643	−1.71898
$476$	0	0
$477$	−21.2209	−0.971638
$478$	0	0
$479$	16.6987	0.762985	0.381492	−	0.924372i	$-0.375410\pi$
$479$	16.6987	0.762985	0.381492	+	0.924372i	$0.375410\pi$
$480$	0	0
$481$	−14.3569	−0.654617
$482$	0	0
$483$	−16.7735	−0.763223
$484$	0	0
$485$	−12.1439	−0.551427
$486$	0	0
$487$	−0.927003	−0.0420065	−0.0210033	−	0.999779i	$-0.506686\pi$
$487$	−0.927003	−0.0420065	−0.0210033	+	0.999779i	$0.506686\pi$
$488$	0	0
$489$	−0.385130	−0.0174162
$490$	0	0
$491$	26.8752	1.21286	0.606430	−	0.795137i	$-0.292601\pi$
$491$	26.8752	1.21286	0.606430	+	0.795137i	$0.292601\pi$
$492$	0	0
$493$	6.53509	0.294326
$494$	0	0
$495$	17.5698	0.789703
$496$	0	0
$497$	63.8279	2.86307
$498$	0	0
$499$	−18.2155	−0.815438	−0.407719	−	0.913107i	$-0.633676\pi$
$499$	−18.2155	−0.815438	−0.407719	+	0.913107i	$0.633676\pi$
$500$	0	0
$501$	32.2570	1.44114
$502$	0	0
$503$	−18.9591	−0.845346	−0.422673	−	0.906282i	$-0.638908\pi$
$503$	−18.9591	−0.845346	−0.422673	+	0.906282i	$0.638908\pi$
$504$	0	0
$505$	−15.2787	−0.679894
$506$	0	0
$507$	−25.1816	−1.11835
$508$	0	0
$509$	12.4218	0.550588	0.275294	−	0.961360i	$-0.411225\pi$
$509$	12.4218	0.550588	0.275294	+	0.961360i	$0.411225\pi$
$510$	0	0
$511$	41.9613	1.85626
$512$	0	0
$513$	−11.4159	−0.504022
$514$	0	0
$515$	−4.02552	−0.177386
$516$	0	0
$517$	−13.1864	−0.579937
$518$	0	0
$519$	49.3154	2.16471
$520$	0	0
$521$	−7.17267	−0.314240	−0.157120	−	0.987579i	$-0.550221\pi$
$521$	−7.17267	−0.314240	−0.157120	+	0.987579i	$0.550221\pi$
$522$	0	0
$523$	−35.6563	−1.55914	−0.779570	−	0.626315i	$-0.784563\pi$
$523$	−35.6563	−1.55914	−0.779570	+	0.626315i	$0.784563\pi$
$524$	0	0
$525$	−89.0529	−3.88659
$526$	0	0
$527$	−8.58448	−0.373946
$528$	0	0
$529$	−20.7655	−0.902848
$530$	0	0
$531$	−2.85285	−0.123803
$532$	0	0
$533$	4.93620	0.213811
$534$	0	0
$535$	1.15867	0.0500938
$536$	0	0
$537$	−39.7171	−1.71392
$538$	0	0
$539$	47.7422	2.05640
$540$	0	0
$541$	28.9731	1.24565	0.622826	−	0.782361i	$-0.285985\pi$
$541$	28.9731	1.24565	0.622826	+	0.782361i	$0.285985\pi$
$542$	0	0
$543$	−46.5072	−1.99581
$544$	0	0
$545$	11.6448	0.498807
$546$	0	0
$547$	−16.1178	−0.689148	−0.344574	−	0.938759i	$-0.611977\pi$
$547$	−16.1178	−0.689148	−0.344574	+	0.938759i	$0.611977\pi$
$548$	0	0
$549$	2.03025	0.0866489
$550$	0	0
$551$	−11.4761	−0.488898
$552$	0	0
$553$	21.7263	0.923895
$554$	0	0
$555$	−23.1763	−0.983779
$556$	0	0
$557$	−27.2675	−1.15536	−0.577681	−	0.816262i	$-0.696042\pi$
$557$	−27.2675	−1.15536	−0.577681	+	0.816262i	$0.696042\pi$
$558$	0	0
$559$	−37.6356	−1.59182
$560$	0	0
$561$	−15.2444	−0.643620
$562$	0	0
$563$	−41.9173	−1.76660	−0.883302	−	0.468805i	$-0.844685\pi$
$563$	−41.9173	−1.76660	−0.883302	+	0.468805i	$0.844685\pi$
$564$	0	0
$565$	7.55978	0.318043
$566$	0	0
$567$	−56.1338	−2.35740
$568$	0	0
$569$	37.3762	1.56689	0.783446	−	0.621460i	$-0.213460\pi$
$569$	37.3762	1.56689	0.783446	+	0.621460i	$0.213460\pi$
$570$	0	0
$571$	26.3228	1.10157	0.550787	−	0.834646i	$-0.314327\pi$
$571$	26.3228	1.10157	0.550787	+	0.834646i	$0.314327\pi$
$572$	0	0
$573$	14.2917	0.597044
$574$	0	0
$575$	11.8632	0.494730
$576$	0	0
$577$	−13.8262	−0.575591	−0.287795	−	0.957692i	$-0.592922\pi$
$577$	−13.8262	−0.575591	−0.287795	+	0.957692i	$0.592922\pi$
$578$	0	0
$579$	−13.7445	−0.571200
$580$	0	0
$581$	−53.4873	−2.21903
$582$	0	0
$583$	−28.4609	−1.17873
$584$	0	0
$585$	−33.8832	−1.40090
$586$	0	0
$587$	−22.7004	−0.936945	−0.468472	−	0.883478i	$-0.655195\pi$
$587$	−22.7004	−0.936945	−0.468472	+	0.883478i	$0.655195\pi$
$588$	0	0
$589$	15.0750	0.621154
$590$	0	0
$591$	−31.4270	−1.29274
$592$	0	0
$593$	16.4167	0.674152	0.337076	−	0.941477i	$-0.390562\pi$
$593$	16.4167	0.674152	0.337076	+	0.941477i	$0.390562\pi$
$594$	0	0
$595$	48.9697	2.00756
$596$	0	0
$597$	41.4520	1.69652
$598$	0	0
$599$	6.54909	0.267588	0.133794	−	0.991009i	$-0.457284\pi$
$599$	6.54909	0.267588	0.133794	+	0.991009i	$0.457284\pi$
$600$	0	0
$601$	−24.7783	−1.01073	−0.505365	−	0.862906i	$-0.668642\pi$
$601$	−24.7783	−1.01073	−0.505365	+	0.862906i	$0.668642\pi$
$602$	0	0
$603$	−20.1529	−0.820688
$604$	0	0
$605$	−15.9994	−0.650470
$606$	0	0
$607$	15.7982	0.641231	0.320615	−	0.947209i	$-0.396110\pi$
$607$	15.7982	0.641231	0.320615	+	0.947209i	$0.396110\pi$
$608$	0	0
$609$	−27.2787	−1.10539
$610$	0	0
$611$	25.4299	1.02878
$612$	0	0
$613$	39.3415	1.58899	0.794494	−	0.607272i	$-0.207736\pi$
$613$	39.3415	1.58899	0.794494	+	0.607272i	$0.207736\pi$
$614$	0	0
$615$	7.96851	0.321321
$616$	0	0
$617$	−26.4075	−1.06313	−0.531563	−	0.847019i	$-0.678395\pi$
$617$	−26.4075	−1.06313	−0.531563	+	0.847019i	$0.678395\pi$
$618$	0	0
$619$	−33.9278	−1.36367	−0.681837	−	0.731504i	$-0.738819\pi$
$619$	−33.9278	−1.36367	−0.681837	+	0.731504i	$0.738819\pi$
$620$	0	0
$621$	3.61487	0.145060
$622$	0	0
$623$	47.4430	1.90076
$624$	0	0
$625$	−1.69774	−0.0679097
$626$	0	0
$627$	26.7703	1.06910
$628$	0	0
$629$	7.81861	0.311748
$630$	0	0
$631$	5.68017	0.226124	0.113062	−	0.993588i	$-0.463934\pi$
$631$	5.68017	0.226124	0.113062	+	0.993588i	$0.463934\pi$
$632$	0	0
$633$	32.5644	1.29432
$634$	0	0
$635$	46.9936	1.86488
$636$	0	0
$637$	−92.0706	−3.64797
$638$	0	0
$639$	24.0513	0.951454
$640$	0	0
$641$	15.5802	0.615379	0.307690	−	0.951487i	$-0.400444\pi$
$641$	15.5802	0.615379	0.307690	+	0.951487i	$0.400444\pi$
$642$	0	0
$643$	−22.9372	−0.904554	−0.452277	−	0.891877i	$-0.649388\pi$
$643$	−22.9372	−0.904554	−0.452277	+	0.891877i	$0.649388\pi$
$644$	0	0
$645$	−60.7552	−2.39223
$646$	0	0
$647$	−15.0546	−0.591858	−0.295929	−	0.955210i	$-0.595629\pi$
$647$	−15.0546	−0.591858	−0.295929	+	0.955210i	$0.595629\pi$
$648$	0	0
$649$	−3.82617	−0.150190
$650$	0	0
$651$	35.8333	1.40442
$652$	0	0
$653$	−45.8652	−1.79484	−0.897422	−	0.441174i	$-0.854562\pi$
$653$	−45.8652	−1.79484	−0.897422	+	0.441174i	$0.854562\pi$
$654$	0	0
$655$	−30.3238	−1.18485
$656$	0	0
$657$	15.8116	0.616870
$658$	0	0
$659$	4.37302	0.170349	0.0851744	−	0.996366i	$-0.472855\pi$
$659$	4.37302	0.170349	0.0851744	+	0.996366i	$0.472855\pi$
$660$	0	0
$661$	−25.4946	−0.991625	−0.495813	−	0.868430i	$-0.665130\pi$
$661$	−25.4946	−0.991625	−0.495813	+	0.868430i	$0.665130\pi$
$662$	0	0
$663$	29.3988	1.14175
$664$	0	0
$665$	−85.9944	−3.33472
$666$	0	0
$667$	3.63394	0.140707
$668$	0	0
$669$	44.5558	1.72263
$670$	0	0
$671$	2.72292	0.105117
$672$	0	0
$673$	31.7540	1.22403	0.612013	−	0.790848i	$-0.290360\pi$
$673$	31.7540	1.22403	0.612013	+	0.790848i	$0.290360\pi$
$674$	0	0
$675$	19.1918	0.738693
$676$	0	0
$677$	−27.8998	−1.07228	−0.536138	−	0.844131i	$-0.680117\pi$
$677$	−27.8998	−1.07228	−0.536138	+	0.844131i	$0.680117\pi$
$678$	0	0
$679$	−17.1008	−0.656270
$680$	0	0
$681$	−14.8694	−0.569796
$682$	0	0
$683$	−8.11782	−0.310620	−0.155310	−	0.987866i	$-0.549638\pi$
$683$	−8.11782	−0.310620	−0.155310	+	0.987866i	$0.549638\pi$
$684$	0	0
$685$	−59.5634	−2.27580
$686$	0	0
$687$	−28.2752	−1.07876
$688$	0	0
$689$	54.8867	2.09102
$690$	0	0
$691$	26.6390	1.01339	0.506697	−	0.862124i	$-0.330866\pi$
$691$	26.6390	1.01339	0.506697	+	0.862124i	$0.330866\pi$
$692$	0	0
$693$	24.7414	0.939850
$694$	0	0
$695$	7.85168	0.297831
$696$	0	0
$697$	−2.68821	−0.101823
$698$	0	0
$699$	64.4159	2.43643
$700$	0	0
$701$	43.1302	1.62900	0.814502	−	0.580160i	$-0.197010\pi$
$701$	43.1302	1.62900	0.814502	+	0.580160i	$0.197010\pi$
$702$	0	0
$703$	−13.7301	−0.517839
$704$	0	0
$705$	41.0514	1.54609
$706$	0	0
$707$	−21.5152	−0.809162
$708$	0	0
$709$	49.5175	1.85967	0.929835	−	0.367978i	$-0.119950\pi$
$709$	49.5175	1.85967	0.929835	+	0.367978i	$0.119950\pi$
$710$	0	0
$711$	8.18677	0.307028
$712$	0	0
$713$	−4.77354	−0.178771
$714$	0	0
$715$	−45.4434	−1.69949
$716$	0	0
$717$	57.8184	2.15927
$718$	0	0
$719$	16.4839	0.614745	0.307372	−	0.951589i	$-0.400550\pi$
$719$	16.4839	0.614745	0.307372	+	0.951589i	$0.400550\pi$
$720$	0	0
$721$	−5.66866	−0.211112
$722$	0	0
$723$	−2.35686	−0.0876527
$724$	0	0
$725$	19.2931	0.716526
$726$	0	0
$727$	−48.5947	−1.80228	−0.901139	−	0.433530i	$-0.857268\pi$
$727$	−48.5947	−1.80228	−0.901139	+	0.433530i	$0.857268\pi$
$728$	0	0
$729$	−5.07896	−0.188110
$730$	0	0
$731$	20.4960	0.758071
$732$	0	0
$733$	11.8166	0.436457	0.218229	−	0.975898i	$-0.429972\pi$
$733$	11.8166	0.436457	0.218229	+	0.975898i	$0.429972\pi$
$734$	0	0
$735$	−148.630	−5.48228
$736$	0	0
$737$	−27.0285	−0.995609
$738$	0	0
$739$	5.68017	0.208949	0.104474	−	0.994528i	$-0.466684\pi$
$739$	5.68017	0.208949	0.104474	+	0.994528i	$0.466684\pi$
$740$	0	0
$741$	−51.6264	−1.89654
$742$	0	0
$743$	24.4980	0.898743	0.449372	−	0.893345i	$-0.351648\pi$
$743$	24.4980	0.898743	0.449372	+	0.893345i	$0.351648\pi$
$744$	0	0
$745$	8.74363	0.320342
$746$	0	0
$747$	−20.1548	−0.737424
$748$	0	0
$749$	1.63162	0.0596181
$750$	0	0
$751$	35.7897	1.30598	0.652992	−	0.757365i	$-0.273513\pi$
$751$	35.7897	1.30598	0.652992	+	0.757365i	$0.273513\pi$
$752$	0	0
$753$	−17.3433	−0.632027
$754$	0	0
$755$	1.23407	0.0449124
$756$	0	0
$757$	37.7169	1.37084	0.685421	−	0.728147i	$-0.259618\pi$
$757$	37.7169	1.37084	0.685421	+	0.728147i	$0.259618\pi$
$758$	0	0
$759$	−8.47691	−0.307692
$760$	0	0
$761$	−22.6533	−0.821181	−0.410590	−	0.911820i	$-0.634677\pi$
$761$	−22.6533	−0.821181	−0.410590	+	0.911820i	$0.634677\pi$
$762$	0	0
$763$	16.3979	0.593646
$764$	0	0
$765$	18.4525	0.667151
$766$	0	0
$767$	7.37874	0.266431
$768$	0	0
$769$	−42.6292	−1.53725	−0.768624	−	0.639701i	$-0.779058\pi$
$769$	−42.6292	−1.53725	−0.768624	+	0.639701i	$0.779058\pi$
$770$	0	0
$771$	−16.6049	−0.598009
$772$	0	0
$773$	11.7635	0.423105	0.211552	−	0.977367i	$-0.432148\pi$
$773$	11.7635	0.423105	0.211552	+	0.977367i	$0.432148\pi$
$774$	0	0
$775$	−25.3433	−0.910360
$776$	0	0
$777$	−32.6364	−1.17082
$778$	0	0
$779$	4.72069	0.169136
$780$	0	0
$781$	32.2570	1.15425
$782$	0	0
$783$	5.87884	0.210093
$784$	0	0
$785$	−38.2500	−1.36520
$786$	0	0
$787$	−34.4517	−1.22807	−0.614036	−	0.789278i	$-0.710455\pi$
$787$	−34.4517	−1.22807	−0.614036	+	0.789278i	$0.710455\pi$
$788$	0	0
$789$	34.5441	1.22980
$790$	0	0
$791$	10.6455	0.378512
$792$	0	0
$793$	−5.25113	−0.186473
$794$	0	0
$795$	88.6037	3.14245
$796$	0	0
$797$	41.2161	1.45995	0.729974	−	0.683475i	$-0.239532\pi$
$797$	41.2161	1.45995	0.729974	+	0.683475i	$0.239532\pi$
$798$	0	0
$799$	−13.8489	−0.489937
$800$	0	0
$801$	17.8772	0.631660
$802$	0	0
$803$	21.2061	0.748349
$804$	0	0
$805$	27.2304	0.959746
$806$	0	0
$807$	−19.5155	−0.686979
$808$	0	0
$809$	12.4123	0.436393	0.218196	−	0.975905i	$-0.429983\pi$
$809$	12.4123	0.436393	0.218196	+	0.975905i	$0.429983\pi$
$810$	0	0
$811$	−3.91665	−0.137532	−0.0687660	−	0.997633i	$-0.521906\pi$
$811$	−3.91665	−0.137532	−0.0687660	+	0.997633i	$0.521906\pi$
$812$	0	0
$813$	−44.0479	−1.54483
$814$	0	0
$815$	0.625226	0.0219007
$816$	0	0
$817$	−35.9924	−1.25922
$818$	0	0
$819$	−47.7137	−1.66725
$820$	0	0
$821$	28.6326	0.999283	0.499642	−	0.866232i	$-0.333465\pi$
$821$	28.6326	0.999283	0.499642	+	0.866232i	$0.333465\pi$
$822$	0	0
$823$	−43.8212	−1.52751	−0.763755	−	0.645506i	$-0.776647\pi$
$823$	−43.8212	−1.52751	−0.763755	+	0.645506i	$0.776647\pi$
$824$	0	0
$825$	−45.0050	−1.56687
$826$	0	0
$827$	−11.1924	−0.389198	−0.194599	−	0.980883i	$-0.562341\pi$
$827$	−11.1924	−0.389198	−0.194599	+	0.980883i	$0.562341\pi$
$828$	0	0
$829$	26.6278	0.924820	0.462410	−	0.886666i	$-0.346985\pi$
$829$	26.6278	0.924820	0.462410	+	0.886666i	$0.346985\pi$
$830$	0	0
$831$	15.6910	0.544313
$832$	0	0
$833$	50.1408	1.73727
$834$	0	0
$835$	−52.3665	−1.81222
$836$	0	0
$837$	−7.72244	−0.266927
$838$	0	0
$839$	−9.43085	−0.325589	−0.162795	−	0.986660i	$-0.552051\pi$
$839$	−9.43085	−0.325589	−0.162795	+	0.986660i	$0.552051\pi$
$840$	0	0
$841$	−23.0901	−0.796212
$842$	0	0
$843$	69.2780	2.38606
$844$	0	0
$845$	40.8802	1.40632
$846$	0	0
$847$	−22.5301	−0.774144
$848$	0	0
$849$	−4.83652	−0.165989
$850$	0	0
$851$	4.34767	0.149036
$852$	0	0
$853$	32.6045	1.11636	0.558179	−	0.829721i	$-0.311500\pi$
$853$	32.6045	1.11636	0.558179	+	0.829721i	$0.311500\pi$
$854$	0	0
$855$	−32.4039	−1.10819
$856$	0	0
$857$	−4.25499	−0.145348	−0.0726739	−	0.997356i	$-0.523153\pi$
$857$	−4.25499	−0.145348	−0.0726739	+	0.997356i	$0.523153\pi$
$858$	0	0
$859$	−24.5164	−0.836487	−0.418244	−	0.908335i	$-0.637354\pi$
$859$	−24.5164	−0.836487	−0.418244	+	0.908335i	$0.637354\pi$
$860$	0	0
$861$	11.2211	0.382414
$862$	0	0
$863$	4.11643	0.140125	0.0700624	−	0.997543i	$-0.477680\pi$
$863$	4.11643	0.140125	0.0700624	+	0.997543i	$0.477680\pi$
$864$	0	0
$865$	−80.0594	−2.72210
$866$	0	0
$867$	21.6534	0.735387
$868$	0	0
$869$	10.9799	0.372467
$870$	0	0
$871$	52.1244	1.76617
$872$	0	0
$873$	−6.44384	−0.218091
$874$	0	0
$875$	53.4873	1.80820
$876$	0	0
$877$	−3.17301	−0.107145	−0.0535725	−	0.998564i	$-0.517061\pi$
$877$	−3.17301	−0.107145	−0.0535725	+	0.998564i	$0.517061\pi$
$878$	0	0
$879$	74.6180	2.51680
$880$	0	0
$881$	16.0391	0.540371	0.270186	−	0.962808i	$-0.412915\pi$
$881$	16.0391	0.540371	0.270186	+	0.962808i	$0.412915\pi$
$882$	0	0
$883$	36.5648	1.23050	0.615252	−	0.788330i	$-0.289054\pi$
$883$	36.5648	1.23050	0.615252	+	0.788330i	$0.289054\pi$
$884$	0	0
$885$	11.9115	0.400401
$886$	0	0
$887$	44.6498	1.49919	0.749596	−	0.661896i	$-0.230248\pi$
$887$	44.6498	1.49919	0.749596	+	0.661896i	$0.230248\pi$
$888$	0	0
$889$	66.1755	2.21945
$890$	0	0
$891$	−28.3686	−0.950382
$892$	0	0
$893$	24.3196	0.813825
$894$	0	0
$895$	64.4773	2.15524
$896$	0	0
$897$	16.3477	0.545833
$898$	0	0
$899$	−7.76319	−0.258917
$900$	0	0
$901$	−29.8908	−0.995807
$902$	0	0
$903$	−85.5542	−2.84707
$904$	0	0
$905$	75.5004	2.50972
$906$	0	0
$907$	43.2692	1.43673	0.718365	−	0.695667i	$-0.244891\pi$
$907$	43.2692	1.43673	0.718365	+	0.695667i	$0.244891\pi$
$908$	0	0
$909$	−8.10723	−0.268900
$910$	0	0
$911$	1.64116	0.0543739	0.0271870	−	0.999630i	$-0.491345\pi$
$911$	1.64116	0.0543739	0.0271870	+	0.999630i	$0.491345\pi$
$912$	0	0
$913$	−27.0311	−0.894598
$914$	0	0
$915$	−8.47691	−0.280238
$916$	0	0
$917$	−42.7014	−1.41012
$918$	0	0
$919$	−6.67802	−0.220288	−0.110144	−	0.993916i	$-0.535131\pi$
$919$	−6.67802	−0.220288	−0.110144	+	0.993916i	$0.535131\pi$
$920$	0	0
$921$	−28.2548	−0.931027
$922$	0	0
$923$	−62.2074	−2.04758
$924$	0	0
$925$	23.0823	0.758942
$926$	0	0
$927$	−2.13603	−0.0701564
$928$	0	0
$929$	−23.3505	−0.766104	−0.383052	−	0.923727i	$-0.625127\pi$
$929$	−23.3505	−0.766104	−0.383052	+	0.923727i	$0.625127\pi$
$930$	0	0
$931$	−88.0508	−2.88575
$932$	0	0
$933$	4.83433	0.158269
$934$	0	0
$935$	24.7480	0.809347
$936$	0	0
$937$	45.9295	1.50045	0.750225	−	0.661182i	$-0.229945\pi$
$937$	45.9295	1.50045	0.750225	+	0.661182i	$0.229945\pi$
$938$	0	0
$939$	64.9238	2.11871
$940$	0	0
$941$	−13.2488	−0.431899	−0.215949	−	0.976405i	$-0.569285\pi$
$941$	−13.2488	−0.431899	−0.215949	+	0.976405i	$0.569285\pi$
$942$	0	0
$943$	−1.49482	−0.0486781
$944$	0	0
$945$	44.0523	1.43302
$946$	0	0
$947$	22.1017	0.718207	0.359104	−	0.933298i	$-0.383082\pi$
$947$	22.1017	0.718207	0.359104	+	0.933298i	$0.383082\pi$
$948$	0	0
$949$	−40.8959	−1.32754
$950$	0	0
$951$	63.2552	2.05119
$952$	0	0
$953$	−18.2642	−0.591635	−0.295818	−	0.955244i	$-0.595592\pi$
$953$	−18.2642	−0.591635	−0.295818	+	0.955244i	$0.595592\pi$
$954$	0	0
$955$	−23.2013	−0.750778
$956$	0	0
$957$	−13.7860	−0.445637
$958$	0	0
$959$	−83.8760	−2.70850
$960$	0	0
$961$	−20.8023	−0.671042
$962$	0	0
$963$	0.614817	0.0198122
$964$	0	0
$965$	22.3130	0.718279
$966$	0	0
$967$	28.0843	0.903132	0.451566	−	0.892238i	$-0.350866\pi$
$967$	28.0843	0.903132	0.451566	+	0.892238i	$0.350866\pi$
$968$	0	0
$969$	28.1152	0.903192
$970$	0	0
$971$	34.1505	1.09594	0.547972	−	0.836497i	$-0.315400\pi$
$971$	34.1505	1.09594	0.547972	+	0.836497i	$0.315400\pi$
$972$	0	0
$973$	11.0566	0.354458
$974$	0	0
$975$	86.7918	2.77956
$976$	0	0
$977$	41.4597	1.32641	0.663206	−	0.748437i	$-0.269195\pi$
$977$	41.4597	1.32641	0.663206	+	0.748437i	$0.269195\pi$
$978$	0	0
$979$	23.9765	0.766291
$980$	0	0
$981$	6.17898	0.197280
$982$	0	0
$983$	34.1240	1.08839	0.544193	−	0.838960i	$-0.316836\pi$
$983$	34.1240	1.08839	0.544193	+	0.838960i	$0.316836\pi$
$984$	0	0
$985$	51.0191	1.62560
$986$	0	0
$987$	57.8078	1.84004
$988$	0	0
$989$	11.3971	0.362408
$990$	0	0
$991$	21.3535	0.678315	0.339158	−	0.940730i	$-0.389858\pi$
$991$	21.3535	0.678315	0.339158	+	0.940730i	$0.389858\pi$
$992$	0	0
$993$	−10.6513	−0.338009
$994$	0	0
$995$	−67.2938	−2.13336
$996$	0	0
$997$	−19.3560	−0.613012	−0.306506	−	0.951869i	$-0.599160\pi$
$997$	−19.3560	−0.613012	−0.306506	+	0.951869i	$0.599160\pi$
$998$	0	0
$999$	7.03348	0.222529

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	656.2.a.i.1.2		4
3.2	odd	2		5904.2.a.bp.1.1		4
4.3	odd	2		164.2.a.a.1.3	✓	4
8.3	odd	2		2624.2.a.v.1.2		4
8.5	even	2		2624.2.a.y.1.3		4
12.11	even	2		1476.2.a.g.1.1		4
20.3	even	4		4100.2.d.c.1149.6		8
20.7	even	4		4100.2.d.c.1149.3		8
20.19	odd	2		4100.2.a.c.1.2		4
28.27	even	2		8036.2.a.i.1.2		4
164.163	odd	2		6724.2.a.c.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.a.a.1.3	✓	4	4.3	odd	2
656.2.a.i.1.2		4	1.1	even	1	trivial
1476.2.a.g.1.1		4	12.11	even	2
2624.2.a.v.1.2		4	8.3	odd	2
2624.2.a.y.1.3		4	8.5	even	2
4100.2.a.c.1.2		4	20.19	odd	2
4100.2.d.c.1149.3		8	20.7	even	4
4100.2.d.c.1149.6		8	20.3	even	4
5904.2.a.bp.1.1		4	3.2	odd	2
6724.2.a.c.1.2		4	164.163	odd	2
8036.2.a.i.1.2		4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 656 = 2^{4} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	656.a (trivial)

\(f(q)\)	\(=\)	\(q-2.21551 q^{3} +3.59669 q^{5} +5.06479 q^{7} +1.90849 q^{9} +O(q^{10})\)	\(q-2.21551 q^{3} +3.59669 q^{5} +5.06479 q^{7} +1.90849 q^{9} +2.55961 q^{11} -4.93620 q^{13} -7.96851 q^{15} +2.68821 q^{17} -4.72069 q^{19} -11.2211 q^{21} +1.49482 q^{23} +7.93620 q^{25} +2.41826 q^{27} +2.43102 q^{29} -3.19339 q^{31} -5.67085 q^{33} +18.2165 q^{35} +2.90849 q^{37} +10.9362 q^{39} -1.00000 q^{41} +7.62441 q^{43} +6.86424 q^{45} -5.15171 q^{47} +18.6521 q^{49} -5.95575 q^{51} -11.1192 q^{53} +9.20614 q^{55} +10.4587 q^{57} -1.49482 q^{59} +1.06380 q^{61} +9.66608 q^{63} -17.7540 q^{65} -10.5596 q^{67} -3.31179 q^{69} +12.6023 q^{71} +8.28490 q^{73} -17.5827 q^{75} +12.9639 q^{77} +4.28967 q^{79} -11.0831 q^{81} -10.5606 q^{83} +9.66866 q^{85} -5.38595 q^{87} +9.36722 q^{89} -25.0008 q^{91} +7.07498 q^{93} -16.9789 q^{95} -3.37641 q^{97} +4.88499 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 4 q^{5} + 12 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 + 4 * q^5 + 12 * q^9	\( 4 q - 2 q^{3} + 4 q^{5} + 12 q^{9} - 4 q^{11} + 10 q^{15} - 4 q^{17} - 6 q^{19} + 12 q^{23} + 12 q^{25} + 10 q^{27} - 4 q^{29} + 8 q^{31} - 20 q^{33} + 26 q^{35} + 16 q^{37} + 24 q^{39} - 4 q^{41} - 4 q^{43} + 4 q^{45} + 6 q^{47} + 16 q^{49} + 4 q^{51} - 16 q^{53} + 2 q^{55} + 4 q^{57} - 12 q^{59} + 24 q^{61} + 10 q^{63} + 4 q^{65} - 28 q^{67} - 28 q^{69} + 2 q^{71} + 8 q^{73} - 30 q^{75} + 8 q^{77} + 18 q^{79} + 28 q^{81} + 12 q^{83} + 32 q^{85} - 44 q^{87} + 4 q^{89} - 36 q^{91} - 28 q^{93} - 14 q^{95} + 16 q^{97} - 58 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 + 4 * q^5 + 12 * q^9 - 4 * q^11 + 10 * q^15 - 4 * q^17 - 6 * q^19 + 12 * q^23 + 12 * q^25 + 10 * q^27 - 4 * q^29 + 8 * q^31 - 20 * q^33 + 26 * q^35 + 16 * q^37 + 24 * q^39 - 4 * q^41 - 4 * q^43 + 4 * q^45 + 6 * q^47 + 16 * q^49 + 4 * q^51 - 16 * q^53 + 2 * q^55 + 4 * q^57 - 12 * q^59 + 24 * q^61 + 10 * q^63 + 4 * q^65 - 28 * q^67 - 28 * q^69 + 2 * q^71 + 8 * q^73 - 30 * q^75 + 8 * q^77 + 18 * q^79 + 28 * q^81 + 12 * q^83 + 32 * q^85 - 44 * q^87 + 4 * q^89 - 36 * q^91 - 28 * q^93 - 14 * q^95 + 16 * q^97 - 58 * q^99

Introduction

L-functions

Modular forms

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Database

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