LMFDB - Embedded newform 2368.2.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2368,2,Mod(1,2368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2368, 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("2368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2368 = 2^{6} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2368.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:	$18.9085751986$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 74)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	2368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.618034 q^{3} +2.85410 q^{5} +1.23607 q^{7} -2.61803 q^{9} +O(q^{10})$	$q-0.618034 q^{3} +2.85410 q^{5} +1.23607 q^{7} -2.61803 q^{9} +3.61803 q^{11} -3.85410 q^{13} -1.76393 q^{15} +4.47214 q^{17} +4.47214 q^{19} -0.763932 q^{21} -3.85410 q^{23} +3.14590 q^{25} +3.47214 q^{27} -6.32624 q^{29} +9.61803 q^{31} -2.23607 q^{33} +3.52786 q^{35} +1.00000 q^{37} +2.38197 q^{39} +7.38197 q^{41} +0.763932 q^{43} -7.47214 q^{45} +3.23607 q^{47} -5.47214 q^{49} -2.76393 q^{51} +8.47214 q^{53} +10.3262 q^{55} -2.76393 q^{57} +9.23607 q^{59} -8.38197 q^{61} -3.23607 q^{63} -11.0000 q^{65} +10.0902 q^{67} +2.38197 q^{69} -14.9443 q^{71} -4.09017 q^{73} -1.94427 q^{75} +4.47214 q^{77} +11.5623 q^{79} +5.70820 q^{81} +5.52786 q^{83} +12.7639 q^{85} +3.90983 q^{87} -10.4721 q^{89} -4.76393 q^{91} -5.94427 q^{93} +12.7639 q^{95} +8.47214 q^{97} -9.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10}) $ 2 * q + q^3 - q^5 - 2 * q^7 - 3 * q^9	$ 2 q + q^{3} - q^{5} - 2 q^{7} - 3 q^{9} + 5 q^{11} - q^{13} - 8 q^{15} - 6 q^{21} - q^{23} + 13 q^{25} - 2 q^{27} + 3 q^{29} + 17 q^{31} + 16 q^{35} + 2 q^{37} + 7 q^{39} + 17 q^{41} + 6 q^{43} - 6 q^{45} + 2 q^{47} - 2 q^{49} - 10 q^{51} + 8 q^{53} + 5 q^{55} - 10 q^{57} + 14 q^{59} - 19 q^{61} - 2 q^{63} - 22 q^{65} + 9 q^{67} + 7 q^{69} - 12 q^{71} + 3 q^{73} + 14 q^{75} + 3 q^{79} - 2 q^{81} + 20 q^{83} + 30 q^{85} + 19 q^{87} - 12 q^{89} - 14 q^{91} + 6 q^{93} + 30 q^{95} + 8 q^{97} - 10 q^{99}+O(q^{100}) $ 2 * q + q^3 - q^5 - 2 * q^7 - 3 * q^9 + 5 * q^11 - q^13 - 8 * q^15 - 6 * q^21 - q^23 + 13 * q^25 - 2 * q^27 + 3 * q^29 + 17 * q^31 + 16 * q^35 + 2 * q^37 + 7 * q^39 + 17 * q^41 + 6 * q^43 - 6 * q^45 + 2 * q^47 - 2 * q^49 - 10 * q^51 + 8 * q^53 + 5 * q^55 - 10 * q^57 + 14 * q^59 - 19 * q^61 - 2 * q^63 - 22 * q^65 + 9 * q^67 + 7 * q^69 - 12 * q^71 + 3 * q^73 + 14 * q^75 + 3 * q^79 - 2 * q^81 + 20 * q^83 + 30 * q^85 + 19 * q^87 - 12 * q^89 - 14 * q^91 + 6 * q^93 + 30 * q^95 + 8 * q^97 - 10 * 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
$3$	−0.618034	−0.356822	−0.178411	−	0.983956i	$-0.557096\pi$
$3$	−0.618034	−0.356822	−0.178411	+	0.983956i	$0.557096\pi$
$4$	0	0
$5$	2.85410	1.27639	0.638197	−	0.769873i	$-0.279681\pi$
$5$	2.85410	1.27639	0.638197	+	0.769873i	$0.279681\pi$
$6$	0	0
$7$	1.23607	0.467190	0.233595	−	0.972334i	$-0.424951\pi$
$7$	1.23607	0.467190	0.233595	+	0.972334i	$0.424951\pi$
$8$	0	0
$9$	−2.61803	−0.872678
$10$	0	0
$11$	3.61803	1.09088	0.545439	−	0.838150i	$-0.316363\pi$
$11$	3.61803	1.09088	0.545439	+	0.838150i	$0.316363\pi$
$12$	0	0
$13$	−3.85410	−1.06894	−0.534468	−	0.845189i	$-0.679488\pi$
$13$	−3.85410	−1.06894	−0.534468	+	0.845189i	$0.679488\pi$
$14$	0	0
$15$	−1.76393	−0.455445
$16$	0	0
$17$	4.47214	1.08465	0.542326	−	0.840168i	$-0.317544\pi$
$17$	4.47214	1.08465	0.542326	+	0.840168i	$0.317544\pi$
$18$	0	0
$19$	4.47214	1.02598	0.512989	−	0.858395i	$-0.328538\pi$
$19$	4.47214	1.02598	0.512989	+	0.858395i	$0.328538\pi$
$20$	0	0
$21$	−0.763932	−0.166704
$22$	0	0
$23$	−3.85410	−0.803636	−0.401818	−	0.915720i	$-0.631622\pi$
$23$	−3.85410	−0.803636	−0.401818	+	0.915720i	$0.631622\pi$
$24$	0	0
$25$	3.14590	0.629180
$26$	0	0
$27$	3.47214	0.668213
$28$	0	0
$29$	−6.32624	−1.17475	−0.587376	−	0.809314i	$-0.699839\pi$
$29$	−6.32624	−1.17475	−0.587376	+	0.809314i	$0.699839\pi$
$30$	0	0
$31$	9.61803	1.72745	0.863725	−	0.503964i	$-0.168125\pi$
$31$	9.61803	1.72745	0.863725	+	0.503964i	$0.168125\pi$
$32$	0	0
$33$	−2.23607	−0.389249
$34$	0	0
$35$	3.52786	0.596318
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	2.38197	0.381420
$40$	0	0
$41$	7.38197	1.15287	0.576435	−	0.817143i	$-0.304444\pi$
$41$	7.38197	1.15287	0.576435	+	0.817143i	$0.304444\pi$
$42$	0	0
$43$	0.763932	0.116499	0.0582493	−	0.998302i	$-0.481448\pi$
$43$	0.763932	0.116499	0.0582493	+	0.998302i	$0.481448\pi$
$44$	0	0
$45$	−7.47214	−1.11388
$46$	0	0
$47$	3.23607	0.472029	0.236015	−	0.971750i	$-0.424159\pi$
$47$	3.23607	0.472029	0.236015	+	0.971750i	$0.424159\pi$
$48$	0	0
$49$	−5.47214	−0.781734
$50$	0	0
$51$	−2.76393	−0.387028
$52$	0	0
$53$	8.47214	1.16374	0.581869	−	0.813283i	$-0.302322\pi$
$53$	8.47214	1.16374	0.581869	+	0.813283i	$0.302322\pi$
$54$	0	0
$55$	10.3262	1.39239
$56$	0	0
$57$	−2.76393	−0.366092
$58$	0	0
$59$	9.23607	1.20243	0.601217	−	0.799086i	$-0.294683\pi$
$59$	9.23607	1.20243	0.601217	+	0.799086i	$0.294683\pi$
$60$	0	0
$61$	−8.38197	−1.07320	−0.536600	−	0.843836i	$-0.680292\pi$
$61$	−8.38197	−1.07320	−0.536600	+	0.843836i	$0.680292\pi$
$62$	0	0
$63$	−3.23607	−0.407706
$64$	0	0
$65$	−11.0000	−1.36438
$66$	0	0
$67$	10.0902	1.23271	0.616355	−	0.787468i	$-0.288609\pi$
$67$	10.0902	1.23271	0.616355	+	0.787468i	$0.288609\pi$
$68$	0	0
$69$	2.38197	0.286755
$70$	0	0
$71$	−14.9443	−1.77356	−0.886779	−	0.462193i	$-0.847063\pi$
$71$	−14.9443	−1.77356	−0.886779	+	0.462193i	$0.847063\pi$
$72$	0	0
$73$	−4.09017	−0.478718	−0.239359	−	0.970931i	$-0.576937\pi$
$73$	−4.09017	−0.478718	−0.239359	+	0.970931i	$0.576937\pi$
$74$	0	0
$75$	−1.94427	−0.224505
$76$	0	0
$77$	4.47214	0.509647
$78$	0	0
$79$	11.5623	1.30086	0.650431	−	0.759566i	$-0.274589\pi$
$79$	11.5623	1.30086	0.650431	+	0.759566i	$0.274589\pi$
$80$	0	0
$81$	5.70820	0.634245
$82$	0	0
$83$	5.52786	0.606762	0.303381	−	0.952869i	$-0.401885\pi$
$83$	5.52786	0.606762	0.303381	+	0.952869i	$0.401885\pi$
$84$	0	0
$85$	12.7639	1.38444
$86$	0	0
$87$	3.90983	0.419178
$88$	0	0
$89$	−10.4721	−1.11004	−0.555022	−	0.831836i	$-0.687290\pi$
$89$	−10.4721	−1.11004	−0.555022	+	0.831836i	$0.687290\pi$
$90$	0	0
$91$	−4.76393	−0.499396
$92$	0	0
$93$	−5.94427	−0.616392
$94$	0	0
$95$	12.7639	1.30955
$96$	0	0
$97$	8.47214	0.860215	0.430108	−	0.902778i	$-0.358476\pi$
$97$	8.47214	0.860215	0.430108	+	0.902778i	$0.358476\pi$
$98$	0	0
$99$	−9.47214	−0.951985
$100$	0	0
$101$	−12.4721	−1.24102	−0.620512	−	0.784197i	$-0.713075\pi$
$101$	−12.4721	−1.24102	−0.620512	+	0.784197i	$0.713075\pi$
$102$	0	0
$103$	−16.2705	−1.60318	−0.801590	−	0.597873i	$-0.796013\pi$
$103$	−16.2705	−1.60318	−0.801590	+	0.597873i	$0.796013\pi$
$104$	0	0
$105$	−2.18034	−0.212779
$106$	0	0
$107$	−8.32624	−0.804928	−0.402464	−	0.915436i	$-0.631846\pi$
$107$	−8.32624	−0.804928	−0.402464	+	0.915436i	$0.631846\pi$
$108$	0	0
$109$	14.9443	1.43140	0.715701	−	0.698407i	$-0.246107\pi$
$109$	14.9443	1.43140	0.715701	+	0.698407i	$0.246107\pi$
$110$	0	0
$111$	−0.618034	−0.0586612
$112$	0	0
$113$	10.9443	1.02955	0.514775	−	0.857325i	$-0.327875\pi$
$113$	10.9443	1.02955	0.514775	+	0.857325i	$0.327875\pi$
$114$	0	0
$115$	−11.0000	−1.02576
$116$	0	0
$117$	10.0902	0.932837
$118$	0	0
$119$	5.52786	0.506738
$120$	0	0
$121$	2.09017	0.190015
$122$	0	0
$123$	−4.56231	−0.411369
$124$	0	0
$125$	−5.29180	−0.473313
$126$	0	0
$127$	0.472136	0.0418953	0.0209476	−	0.999781i	$-0.493332\pi$
$127$	0.472136	0.0418953	0.0209476	+	0.999781i	$0.493332\pi$
$128$	0	0
$129$	−0.472136	−0.0415693
$130$	0	0
$131$	−8.65248	−0.755970	−0.377985	−	0.925812i	$-0.623383\pi$
$131$	−8.65248	−0.755970	−0.377985	+	0.925812i	$0.623383\pi$
$132$	0	0
$133$	5.52786	0.479327
$134$	0	0
$135$	9.90983	0.852902
$136$	0	0
$137$	19.3262	1.65115	0.825576	−	0.564291i	$-0.190850\pi$
$137$	19.3262	1.65115	0.825576	+	0.564291i	$0.190850\pi$
$138$	0	0
$139$	1.85410	0.157263	0.0786314	−	0.996904i	$-0.474945\pi$
$139$	1.85410	0.157263	0.0786314	+	0.996904i	$0.474945\pi$
$140$	0	0
$141$	−2.00000	−0.168430
$142$	0	0
$143$	−13.9443	−1.16608
$144$	0	0
$145$	−18.0557	−1.49945
$146$	0	0
$147$	3.38197	0.278940
$148$	0	0
$149$	6.18034	0.506313	0.253157	−	0.967425i	$-0.418531\pi$
$149$	6.18034	0.506313	0.253157	+	0.967425i	$0.418531\pi$
$150$	0	0
$151$	17.7082	1.44107	0.720537	−	0.693417i	$-0.243896\pi$
$151$	17.7082	1.44107	0.720537	+	0.693417i	$0.243896\pi$
$152$	0	0
$153$	−11.7082	−0.946552
$154$	0	0
$155$	27.4508	2.20491
$156$	0	0
$157$	7.52786	0.600789	0.300394	−	0.953815i	$-0.402882\pi$
$157$	7.52786	0.600789	0.300394	+	0.953815i	$0.402882\pi$
$158$	0	0
$159$	−5.23607	−0.415247
$160$	0	0
$161$	−4.76393	−0.375450
$162$	0	0
$163$	12.4721	0.976893	0.488447	−	0.872594i	$-0.337564\pi$
$163$	12.4721	0.976893	0.488447	+	0.872594i	$0.337564\pi$
$164$	0	0
$165$	−6.38197	−0.496835
$166$	0	0
$167$	7.14590	0.552966	0.276483	−	0.961019i	$-0.410831\pi$
$167$	7.14590	0.552966	0.276483	+	0.961019i	$0.410831\pi$
$168$	0	0
$169$	1.85410	0.142623
$170$	0	0
$171$	−11.7082	−0.895349
$172$	0	0
$173$	−8.47214	−0.644125	−0.322062	−	0.946718i	$-0.604376\pi$
$173$	−8.47214	−0.644125	−0.322062	+	0.946718i	$0.604376\pi$
$174$	0	0
$175$	3.88854	0.293946
$176$	0	0
$177$	−5.70820	−0.429055
$178$	0	0
$179$	−18.6525	−1.39415	−0.697076	−	0.716997i	$-0.745516\pi$
$179$	−18.6525	−1.39415	−0.697076	+	0.716997i	$0.745516\pi$
$180$	0	0
$181$	−5.52786	−0.410883	−0.205441	−	0.978669i	$-0.565863\pi$
$181$	−5.52786	−0.410883	−0.205441	+	0.978669i	$0.565863\pi$
$182$	0	0
$183$	5.18034	0.382942
$184$	0	0
$185$	2.85410	0.209838
$186$	0	0
$187$	16.1803	1.18322
$188$	0	0
$189$	4.29180	0.312182
$190$	0	0
$191$	4.09017	0.295954	0.147977	−	0.988991i	$-0.452724\pi$
$191$	4.09017	0.295954	0.147977	+	0.988991i	$0.452724\pi$
$192$	0	0
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	0	0
$195$	6.79837	0.486842
$196$	0	0
$197$	16.4721	1.17359	0.586796	−	0.809735i	$-0.300389\pi$
$197$	16.4721	1.17359	0.586796	+	0.809735i	$0.300389\pi$
$198$	0	0
$199$	20.9443	1.48470	0.742350	−	0.670012i	$-0.233711\pi$
$199$	20.9443	1.48470	0.742350	+	0.670012i	$0.233711\pi$
$200$	0	0
$201$	−6.23607	−0.439858
$202$	0	0
$203$	−7.81966	−0.548833
$204$	0	0
$205$	21.0689	1.47151
$206$	0	0
$207$	10.0902	0.701315
$208$	0	0
$209$	16.1803	1.11922
$210$	0	0
$211$	22.2705	1.53317	0.766583	−	0.642146i	$-0.221956\pi$
$211$	22.2705	1.53317	0.766583	+	0.642146i	$0.221956\pi$
$212$	0	0
$213$	9.23607	0.632845
$214$	0	0
$215$	2.18034	0.148698
$216$	0	0
$217$	11.8885	0.807047
$218$	0	0
$219$	2.52786	0.170817
$220$	0	0
$221$	−17.2361	−1.15942
$222$	0	0
$223$	−8.18034	−0.547796	−0.273898	−	0.961759i	$-0.588313\pi$
$223$	−8.18034	−0.547796	−0.273898	+	0.961759i	$0.588313\pi$
$224$	0	0
$225$	−8.23607	−0.549071
$226$	0	0
$227$	17.7082	1.17533	0.587667	−	0.809103i	$-0.300046\pi$
$227$	17.7082	1.17533	0.587667	+	0.809103i	$0.300046\pi$
$228$	0	0
$229$	−17.1246	−1.13163	−0.565813	−	0.824534i	$-0.691438\pi$
$229$	−17.1246	−1.13163	−0.565813	+	0.824534i	$0.691438\pi$
$230$	0	0
$231$	−2.76393	−0.181853
$232$	0	0
$233$	13.5623	0.888496	0.444248	−	0.895904i	$-0.353471\pi$
$233$	13.5623	0.888496	0.444248	+	0.895904i	$0.353471\pi$
$234$	0	0
$235$	9.23607	0.602495
$236$	0	0
$237$	−7.14590	−0.464176
$238$	0	0
$239$	3.14590	0.203491	0.101746	−	0.994810i	$-0.467557\pi$
$239$	3.14590	0.203491	0.101746	+	0.994810i	$0.467557\pi$
$240$	0	0
$241$	−10.4721	−0.674570	−0.337285	−	0.941403i	$-0.609509\pi$
$241$	−10.4721	−0.674570	−0.337285	+	0.941403i	$0.609509\pi$
$242$	0	0
$243$	−13.9443	−0.894525
$244$	0	0
$245$	−15.6180	−0.997800
$246$	0	0
$247$	−17.2361	−1.09670
$248$	0	0
$249$	−3.41641	−0.216506
$250$	0	0
$251$	3.05573	0.192876	0.0964379	−	0.995339i	$-0.469255\pi$
$251$	3.05573	0.192876	0.0964379	+	0.995339i	$0.469255\pi$
$252$	0	0
$253$	−13.9443	−0.876669
$254$	0	0
$255$	−7.88854	−0.494000
$256$	0	0
$257$	−18.9443	−1.18171	−0.590856	−	0.806777i	$-0.701210\pi$
$257$	−18.9443	−1.18171	−0.590856	+	0.806777i	$0.701210\pi$
$258$	0	0
$259$	1.23607	0.0768055
$260$	0	0
$261$	16.5623	1.02518
$262$	0	0
$263$	−8.76393	−0.540407	−0.270204	−	0.962803i	$-0.587091\pi$
$263$	−8.76393	−0.540407	−0.270204	+	0.962803i	$0.587091\pi$
$264$	0	0
$265$	24.1803	1.48539
$266$	0	0
$267$	6.47214	0.396088
$268$	0	0
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	0	0
$271$	−4.94427	−0.300343	−0.150172	−	0.988660i	$-0.547983\pi$
$271$	−4.94427	−0.300343	−0.150172	+	0.988660i	$0.547983\pi$
$272$	0	0
$273$	2.94427	0.178195
$274$	0	0
$275$	11.3820	0.686358
$276$	0	0
$277$	7.79837	0.468559	0.234279	−	0.972169i	$-0.424727\pi$
$277$	7.79837	0.468559	0.234279	+	0.972169i	$0.424727\pi$
$278$	0	0
$279$	−25.1803	−1.50751
$280$	0	0
$281$	−5.88854	−0.351281	−0.175641	−	0.984454i	$-0.556200\pi$
$281$	−5.88854	−0.351281	−0.175641	+	0.984454i	$0.556200\pi$
$282$	0	0
$283$	−11.2361	−0.667915	−0.333957	−	0.942588i	$-0.608384\pi$
$283$	−11.2361	−0.667915	−0.333957	+	0.942588i	$0.608384\pi$
$284$	0	0
$285$	−7.88854	−0.467277
$286$	0	0
$287$	9.12461	0.538609
$288$	0	0
$289$	3.00000	0.176471
$290$	0	0
$291$	−5.23607	−0.306944
$292$	0	0
$293$	−18.6525	−1.08969	−0.544845	−	0.838537i	$-0.683411\pi$
$293$	−18.6525	−1.08969	−0.544845	+	0.838537i	$0.683411\pi$
$294$	0	0
$295$	26.3607	1.53478
$296$	0	0
$297$	12.5623	0.728939
$298$	0	0
$299$	14.8541	0.859035
$300$	0	0
$301$	0.944272	0.0544269
$302$	0	0
$303$	7.70820	0.442825
$304$	0	0
$305$	−23.9230	−1.36983
$306$	0	0
$307$	6.14590	0.350765	0.175382	−	0.984500i	$-0.443884\pi$
$307$	6.14590	0.350765	0.175382	+	0.984500i	$0.443884\pi$
$308$	0	0
$309$	10.0557	0.572050
$310$	0	0
$311$	2.03444	0.115363	0.0576813	−	0.998335i	$-0.481629\pi$
$311$	2.03444	0.115363	0.0576813	+	0.998335i	$0.481629\pi$
$312$	0	0
$313$	5.81966	0.328947	0.164473	−	0.986382i	$-0.447408\pi$
$313$	5.81966	0.328947	0.164473	+	0.986382i	$0.447408\pi$
$314$	0	0
$315$	−9.23607	−0.520393
$316$	0	0
$317$	−3.05573	−0.171627	−0.0858134	−	0.996311i	$-0.527349\pi$
$317$	−3.05573	−0.171627	−0.0858134	+	0.996311i	$0.527349\pi$
$318$	0	0
$319$	−22.8885	−1.28151
$320$	0	0
$321$	5.14590	0.287216
$322$	0	0
$323$	20.0000	1.11283
$324$	0	0
$325$	−12.1246	−0.672552
$326$	0	0
$327$	−9.23607	−0.510756
$328$	0	0
$329$	4.00000	0.220527
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	0	0
$333$	−2.61803	−0.143467
$334$	0	0
$335$	28.7984	1.57342
$336$	0	0
$337$	−17.0344	−0.927925	−0.463963	−	0.885855i	$-0.653573\pi$
$337$	−17.0344	−0.927925	−0.463963	+	0.885855i	$0.653573\pi$
$338$	0	0
$339$	−6.76393	−0.367366
$340$	0	0
$341$	34.7984	1.88444
$342$	0	0
$343$	−15.4164	−0.832408
$344$	0	0
$345$	6.79837	0.366012
$346$	0	0
$347$	−12.7639	−0.685204	−0.342602	−	0.939481i	$-0.611308\pi$
$347$	−12.7639	−0.685204	−0.342602	+	0.939481i	$0.611308\pi$
$348$	0	0
$349$	−12.1803	−0.651999	−0.325999	−	0.945370i	$-0.605701\pi$
$349$	−12.1803	−0.651999	−0.325999	+	0.945370i	$0.605701\pi$
$350$	0	0
$351$	−13.3820	−0.714277
$352$	0	0
$353$	29.7082	1.58121	0.790604	−	0.612328i	$-0.209767\pi$
$353$	29.7082	1.58121	0.790604	+	0.612328i	$0.209767\pi$
$354$	0	0
$355$	−42.6525	−2.26376
$356$	0	0
$357$	−3.41641	−0.180815
$358$	0	0
$359$	4.47214	0.236030	0.118015	−	0.993012i	$-0.462347\pi$
$359$	4.47214	0.236030	0.118015	+	0.993012i	$0.462347\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−1.29180	−0.0678017
$364$	0	0
$365$	−11.6738	−0.611033
$366$	0	0
$367$	−27.1246	−1.41589	−0.707947	−	0.706266i	$-0.750378\pi$
$367$	−27.1246	−1.41589	−0.707947	+	0.706266i	$0.750378\pi$
$368$	0	0
$369$	−19.3262	−1.00608
$370$	0	0
$371$	10.4721	0.543686
$372$	0	0
$373$	−14.2918	−0.740001	−0.370001	−	0.929032i	$-0.620643\pi$
$373$	−14.2918	−0.740001	−0.370001	+	0.929032i	$0.620643\pi$
$374$	0	0
$375$	3.27051	0.168888
$376$	0	0
$377$	24.3820	1.25574
$378$	0	0
$379$	−16.9098	−0.868600	−0.434300	−	0.900768i	$-0.643004\pi$
$379$	−16.9098	−0.868600	−0.434300	+	0.900768i	$0.643004\pi$
$380$	0	0
$381$	−0.291796	−0.0149492
$382$	0	0
$383$	−17.8885	−0.914062	−0.457031	−	0.889451i	$-0.651087\pi$
$383$	−17.8885	−0.914062	−0.457031	+	0.889451i	$0.651087\pi$
$384$	0	0
$385$	12.7639	0.650510
$386$	0	0
$387$	−2.00000	−0.101666
$388$	0	0
$389$	0.145898	0.00739732	0.00369866	−	0.999993i	$-0.498823\pi$
$389$	0.145898	0.00739732	0.00369866	+	0.999993i	$0.498823\pi$
$390$	0	0
$391$	−17.2361	−0.871665
$392$	0	0
$393$	5.34752	0.269747
$394$	0	0
$395$	33.0000	1.66041
$396$	0	0
$397$	10.6525	0.534632	0.267316	−	0.963609i	$-0.413863\pi$
$397$	10.6525	0.534632	0.267316	+	0.963609i	$0.413863\pi$
$398$	0	0
$399$	−3.41641	−0.171034
$400$	0	0
$401$	−9.23607	−0.461227	−0.230614	−	0.973045i	$-0.574073\pi$
$401$	−9.23607	−0.461227	−0.230614	+	0.973045i	$0.574073\pi$
$402$	0	0
$403$	−37.0689	−1.84653
$404$	0	0
$405$	16.2918	0.809546
$406$	0	0
$407$	3.61803	0.179339
$408$	0	0
$409$	−3.81966	−0.188870	−0.0944350	−	0.995531i	$-0.530104\pi$
$409$	−3.81966	−0.188870	−0.0944350	+	0.995531i	$0.530104\pi$
$410$	0	0
$411$	−11.9443	−0.589167
$412$	0	0
$413$	11.4164	0.561765
$414$	0	0
$415$	15.7771	0.774467
$416$	0	0
$417$	−1.14590	−0.0561149
$418$	0	0
$419$	−9.56231	−0.467149	−0.233575	−	0.972339i	$-0.575042\pi$
$419$	−9.56231	−0.467149	−0.233575	+	0.972339i	$0.575042\pi$
$420$	0	0
$421$	1.96556	0.0957954	0.0478977	−	0.998852i	$-0.484748\pi$
$421$	1.96556	0.0957954	0.0478977	+	0.998852i	$0.484748\pi$
$422$	0	0
$423$	−8.47214	−0.411929
$424$	0	0
$425$	14.0689	0.682441
$426$	0	0
$427$	−10.3607	−0.501388
$428$	0	0
$429$	8.61803	0.416083
$430$	0	0
$431$	−32.3607	−1.55876	−0.779380	−	0.626552i	$-0.784466\pi$
$431$	−32.3607	−1.55876	−0.779380	+	0.626552i	$0.784466\pi$
$432$	0	0
$433$	−36.3262	−1.74573	−0.872864	−	0.487964i	$-0.837740\pi$
$433$	−36.3262	−1.74573	−0.872864	+	0.487964i	$0.837740\pi$
$434$	0	0
$435$	11.1591	0.535036
$436$	0	0
$437$	−17.2361	−0.824513
$438$	0	0
$439$	−16.7984	−0.801743	−0.400871	−	0.916134i	$-0.631293\pi$
$439$	−16.7984	−0.801743	−0.400871	+	0.916134i	$0.631293\pi$
$440$	0	0
$441$	14.3262	0.682202
$442$	0	0
$443$	18.2705	0.868058	0.434029	−	0.900899i	$-0.357092\pi$
$443$	18.2705	0.868058	0.434029	+	0.900899i	$0.357092\pi$
$444$	0	0
$445$	−29.8885	−1.41685
$446$	0	0
$447$	−3.81966	−0.180664
$448$	0	0
$449$	−19.5279	−0.921577	−0.460788	−	0.887510i	$-0.652433\pi$
$449$	−19.5279	−0.921577	−0.460788	+	0.887510i	$0.652433\pi$
$450$	0	0
$451$	26.7082	1.25764
$452$	0	0
$453$	−10.9443	−0.514207
$454$	0	0
$455$	−13.5967	−0.637425
$456$	0	0
$457$	−29.2361	−1.36761	−0.683803	−	0.729667i	$-0.739675\pi$
$457$	−29.2361	−1.36761	−0.683803	+	0.729667i	$0.739675\pi$
$458$	0	0
$459$	15.5279	0.724779
$460$	0	0
$461$	21.0557	0.980663	0.490332	−	0.871536i	$-0.336876\pi$
$461$	21.0557	0.980663	0.490332	+	0.871536i	$0.336876\pi$
$462$	0	0
$463$	15.5623	0.723242	0.361621	−	0.932325i	$-0.382223\pi$
$463$	15.5623	0.723242	0.361621	+	0.932325i	$0.382223\pi$
$464$	0	0
$465$	−16.9656	−0.786759
$466$	0	0
$467$	−20.3607	−0.942180	−0.471090	−	0.882085i	$-0.656139\pi$
$467$	−20.3607	−0.942180	−0.471090	+	0.882085i	$0.656139\pi$
$468$	0	0
$469$	12.4721	0.575910
$470$	0	0
$471$	−4.65248	−0.214375
$472$	0	0
$473$	2.76393	0.127086
$474$	0	0
$475$	14.0689	0.645525
$476$	0	0
$477$	−22.1803	−1.01557
$478$	0	0
$479$	16.4377	0.751057	0.375529	−	0.926811i	$-0.377461\pi$
$479$	16.4377	0.751057	0.375529	+	0.926811i	$0.377461\pi$
$480$	0	0
$481$	−3.85410	−0.175732
$482$	0	0
$483$	2.94427	0.133969
$484$	0	0
$485$	24.1803	1.09797
$486$	0	0
$487$	−25.3050	−1.14668	−0.573338	−	0.819319i	$-0.694352\pi$
$487$	−25.3050	−1.14668	−0.573338	+	0.819319i	$0.694352\pi$
$488$	0	0
$489$	−7.70820	−0.348577
$490$	0	0
$491$	−27.4508	−1.23884	−0.619420	−	0.785060i	$-0.712632\pi$
$491$	−27.4508	−1.23884	−0.619420	+	0.785060i	$0.712632\pi$
$492$	0	0
$493$	−28.2918	−1.27420
$494$	0	0
$495$	−27.0344	−1.21511
$496$	0	0
$497$	−18.4721	−0.828589
$498$	0	0
$499$	−23.7082	−1.06132	−0.530662	−	0.847583i	$-0.678057\pi$
$499$	−23.7082	−1.06132	−0.530662	+	0.847583i	$0.678057\pi$
$500$	0	0
$501$	−4.41641	−0.197311
$502$	0	0
$503$	7.90983	0.352682	0.176341	−	0.984329i	$-0.443574\pi$
$503$	7.90983	0.352682	0.176341	+	0.984329i	$0.443574\pi$
$504$	0	0
$505$	−35.5967	−1.58403
$506$	0	0
$507$	−1.14590	−0.0508911
$508$	0	0
$509$	4.29180	0.190231	0.0951153	−	0.995466i	$-0.469678\pi$
$509$	4.29180	0.190231	0.0951153	+	0.995466i	$0.469678\pi$
$510$	0	0
$511$	−5.05573	−0.223652
$512$	0	0
$513$	15.5279	0.685572
$514$	0	0
$515$	−46.4377	−2.04629
$516$	0	0
$517$	11.7082	0.514926
$518$	0	0
$519$	5.23607	0.229838
$520$	0	0
$521$	25.4164	1.11351	0.556757	−	0.830676i	$-0.312046\pi$
$521$	25.4164	1.11351	0.556757	+	0.830676i	$0.312046\pi$
$522$	0	0
$523$	−34.1803	−1.49460	−0.747301	−	0.664486i	$-0.768651\pi$
$523$	−34.1803	−1.49460	−0.747301	+	0.664486i	$0.768651\pi$
$524$	0	0
$525$	−2.40325	−0.104887
$526$	0	0
$527$	43.0132	1.87368
$528$	0	0
$529$	−8.14590	−0.354169
$530$	0	0
$531$	−24.1803	−1.04934
$532$	0	0
$533$	−28.4508	−1.23234
$534$	0	0
$535$	−23.7639	−1.02740
$536$	0	0
$537$	11.5279	0.497464
$538$	0	0
$539$	−19.7984	−0.852776
$540$	0	0
$541$	−5.32624	−0.228993	−0.114496	−	0.993424i	$-0.536525\pi$
$541$	−5.32624	−0.228993	−0.114496	+	0.993424i	$0.536525\pi$
$542$	0	0
$543$	3.41641	0.146612
$544$	0	0
$545$	42.6525	1.82703
$546$	0	0
$547$	−42.0689	−1.79874	−0.899368	−	0.437193i	$-0.855973\pi$
$547$	−42.0689	−1.79874	−0.899368	+	0.437193i	$0.855973\pi$
$548$	0	0
$549$	21.9443	0.936559
$550$	0	0
$551$	−28.2918	−1.20527
$552$	0	0
$553$	14.2918	0.607749
$554$	0	0
$555$	−1.76393	−0.0748747
$556$	0	0
$557$	0.562306	0.0238257	0.0119128	−	0.999929i	$-0.496208\pi$
$557$	0.562306	0.0238257	0.0119128	+	0.999929i	$0.496208\pi$
$558$	0	0
$559$	−2.94427	−0.124529
$560$	0	0
$561$	−10.0000	−0.422200
$562$	0	0
$563$	27.8885	1.17536	0.587681	−	0.809093i	$-0.300041\pi$
$563$	27.8885	1.17536	0.587681	+	0.809093i	$0.300041\pi$
$564$	0	0
$565$	31.2361	1.31411
$566$	0	0
$567$	7.05573	0.296313
$568$	0	0
$569$	−21.8885	−0.917615	−0.458808	−	0.888536i	$-0.651723\pi$
$569$	−21.8885	−0.917615	−0.458808	+	0.888536i	$0.651723\pi$
$570$	0	0
$571$	−9.56231	−0.400170	−0.200085	−	0.979779i	$-0.564122\pi$
$571$	−9.56231	−0.400170	−0.200085	+	0.979779i	$0.564122\pi$
$572$	0	0
$573$	−2.52786	−0.105603
$574$	0	0
$575$	−12.1246	−0.505631
$576$	0	0
$577$	−20.6525	−0.859774	−0.429887	−	0.902883i	$-0.641447\pi$
$577$	−20.6525	−0.859774	−0.429887	+	0.902883i	$0.641447\pi$
$578$	0	0
$579$	−2.47214	−0.102738
$580$	0	0
$581$	6.83282	0.283473
$582$	0	0
$583$	30.6525	1.26950
$584$	0	0
$585$	28.7984	1.19067
$586$	0	0
$587$	30.9443	1.27721	0.638603	−	0.769536i	$-0.279512\pi$
$587$	30.9443	1.27721	0.638603	+	0.769536i	$0.279512\pi$
$588$	0	0
$589$	43.0132	1.77233
$590$	0	0
$591$	−10.1803	−0.418763
$592$	0	0
$593$	−2.56231	−0.105221	−0.0526106	−	0.998615i	$-0.516754\pi$
$593$	−2.56231	−0.105221	−0.0526106	+	0.998615i	$0.516754\pi$
$594$	0	0
$595$	15.7771	0.646798
$596$	0	0
$597$	−12.9443	−0.529774
$598$	0	0
$599$	38.3607	1.56737	0.783687	−	0.621155i	$-0.213336\pi$
$599$	38.3607	1.56737	0.783687	+	0.621155i	$0.213336\pi$
$600$	0	0
$601$	35.6869	1.45570	0.727850	−	0.685737i	$-0.240520\pi$
$601$	35.6869	1.45570	0.727850	+	0.685737i	$0.240520\pi$
$602$	0	0
$603$	−26.4164	−1.07576
$604$	0	0
$605$	5.96556	0.242534
$606$	0	0
$607$	5.96556	0.242135	0.121067	−	0.992644i	$-0.461368\pi$
$607$	5.96556	0.242135	0.121067	+	0.992644i	$0.461368\pi$
$608$	0	0
$609$	4.83282	0.195836
$610$	0	0
$611$	−12.4721	−0.504569
$612$	0	0
$613$	36.1803	1.46131	0.730655	−	0.682747i	$-0.239215\pi$
$613$	36.1803	1.46131	0.730655	+	0.682747i	$0.239215\pi$
$614$	0	0
$615$	−13.0213	−0.525069
$616$	0	0
$617$	−11.0902	−0.446473	−0.223237	−	0.974764i	$-0.571662\pi$
$617$	−11.0902	−0.446473	−0.223237	+	0.974764i	$0.571662\pi$
$618$	0	0
$619$	−18.2705	−0.734354	−0.367177	−	0.930151i	$-0.619676\pi$
$619$	−18.2705	−0.734354	−0.367177	+	0.930151i	$0.619676\pi$
$620$	0	0
$621$	−13.3820	−0.537000
$622$	0	0
$623$	−12.9443	−0.518601
$624$	0	0
$625$	−30.8328	−1.23331
$626$	0	0
$627$	−10.0000	−0.399362
$628$	0	0
$629$	4.47214	0.178316
$630$	0	0
$631$	26.3951	1.05077	0.525387	−	0.850864i	$-0.323921\pi$
$631$	26.3951	1.05077	0.525387	+	0.850864i	$0.323921\pi$
$632$	0	0
$633$	−13.7639	−0.547067
$634$	0	0
$635$	1.34752	0.0534749
$636$	0	0
$637$	21.0902	0.835623
$638$	0	0
$639$	39.1246	1.54775
$640$	0	0
$641$	−22.5066	−0.888956	−0.444478	−	0.895790i	$-0.646611\pi$
$641$	−22.5066	−0.888956	−0.444478	+	0.895790i	$0.646611\pi$
$642$	0	0
$643$	33.2361	1.31070	0.655351	−	0.755324i	$-0.272521\pi$
$643$	33.2361	1.31070	0.655351	+	0.755324i	$0.272521\pi$
$644$	0	0
$645$	−1.34752	−0.0530587
$646$	0	0
$647$	18.9098	0.743422	0.371711	−	0.928348i	$-0.378771\pi$
$647$	18.9098	0.743422	0.371711	+	0.928348i	$0.378771\pi$
$648$	0	0
$649$	33.4164	1.31171
$650$	0	0
$651$	−7.34752	−0.287972
$652$	0	0
$653$	4.72949	0.185079	0.0925396	−	0.995709i	$-0.470502\pi$
$653$	4.72949	0.185079	0.0925396	+	0.995709i	$0.470502\pi$
$654$	0	0
$655$	−24.6950	−0.964915
$656$	0	0
$657$	10.7082	0.417767
$658$	0	0
$659$	15.4508	0.601880	0.300940	−	0.953643i	$-0.402700\pi$
$659$	15.4508	0.601880	0.300940	+	0.953643i	$0.402700\pi$
$660$	0	0
$661$	−1.67376	−0.0651018	−0.0325509	−	0.999470i	$-0.510363\pi$
$661$	−1.67376	−0.0651018	−0.0325509	+	0.999470i	$0.510363\pi$
$662$	0	0
$663$	10.6525	0.413708
$664$	0	0
$665$	15.7771	0.611809
$666$	0	0
$667$	24.3820	0.944073
$668$	0	0
$669$	5.05573	0.195466
$670$	0	0
$671$	−30.3262	−1.17073
$672$	0	0
$673$	17.8541	0.688225	0.344113	−	0.938928i	$-0.388180\pi$
$673$	17.8541	0.688225	0.344113	+	0.938928i	$0.388180\pi$
$674$	0	0
$675$	10.9230	0.420426
$676$	0	0
$677$	−3.34752	−0.128656	−0.0643279	−	0.997929i	$-0.520490\pi$
$677$	−3.34752	−0.128656	−0.0643279	+	0.997929i	$0.520490\pi$
$678$	0	0
$679$	10.4721	0.401884
$680$	0	0
$681$	−10.9443	−0.419385
$682$	0	0
$683$	35.4164	1.35517	0.677586	−	0.735444i	$-0.263026\pi$
$683$	35.4164	1.35517	0.677586	+	0.735444i	$0.263026\pi$
$684$	0	0
$685$	55.1591	2.10752
$686$	0	0
$687$	10.5836	0.403789
$688$	0	0
$689$	−32.6525	−1.24396
$690$	0	0
$691$	35.7771	1.36102	0.680512	−	0.732737i	$-0.261757\pi$
$691$	35.7771	1.36102	0.680512	+	0.732737i	$0.261757\pi$
$692$	0	0
$693$	−11.7082	−0.444758
$694$	0	0
$695$	5.29180	0.200729
$696$	0	0
$697$	33.0132	1.25046
$698$	0	0
$699$	−8.38197	−0.317035
$700$	0	0
$701$	−3.97871	−0.150274	−0.0751370	−	0.997173i	$-0.523939\pi$
$701$	−3.97871	−0.150274	−0.0751370	+	0.997173i	$0.523939\pi$
$702$	0	0
$703$	4.47214	0.168670
$704$	0	0
$705$	−5.70820	−0.214983
$706$	0	0
$707$	−15.4164	−0.579794
$708$	0	0
$709$	−43.2148	−1.62297	−0.811483	−	0.584377i	$-0.801339\pi$
$709$	−43.2148	−1.62297	−0.811483	+	0.584377i	$0.801339\pi$
$710$	0	0
$711$	−30.2705	−1.13523
$712$	0	0
$713$	−37.0689	−1.38824
$714$	0	0
$715$	−39.7984	−1.48837
$716$	0	0
$717$	−1.94427	−0.0726102
$718$	0	0
$719$	0.583592	0.0217643	0.0108822	−	0.999941i	$-0.496536\pi$
$719$	0.583592	0.0217643	0.0108822	+	0.999941i	$0.496536\pi$
$720$	0	0
$721$	−20.1115	−0.748990
$722$	0	0
$723$	6.47214	0.240701
$724$	0	0
$725$	−19.9017	−0.739131
$726$	0	0
$727$	−23.1459	−0.858434	−0.429217	−	0.903201i	$-0.641210\pi$
$727$	−23.1459	−0.858434	−0.429217	+	0.903201i	$0.641210\pi$
$728$	0	0
$729$	−8.50658	−0.315058
$730$	0	0
$731$	3.41641	0.126360
$732$	0	0
$733$	−36.4721	−1.34713	−0.673565	−	0.739128i	$-0.735238\pi$
$733$	−36.4721	−1.34713	−0.673565	+	0.739128i	$0.735238\pi$
$734$	0	0
$735$	9.65248	0.356037
$736$	0	0
$737$	36.5066	1.34474
$738$	0	0
$739$	22.0902	0.812600	0.406300	−	0.913740i	$-0.366819\pi$
$739$	22.0902	0.812600	0.406300	+	0.913740i	$0.366819\pi$
$740$	0	0
$741$	10.6525	0.391328
$742$	0	0
$743$	10.0689	0.369392	0.184696	−	0.982796i	$-0.440870\pi$
$743$	10.0689	0.369392	0.184696	+	0.982796i	$0.440870\pi$
$744$	0	0
$745$	17.6393	0.646255
$746$	0	0
$747$	−14.4721	−0.529508
$748$	0	0
$749$	−10.2918	−0.376054
$750$	0	0
$751$	−18.9443	−0.691286	−0.345643	−	0.938366i	$-0.612339\pi$
$751$	−18.9443	−0.691286	−0.345643	+	0.938366i	$0.612339\pi$
$752$	0	0
$753$	−1.88854	−0.0688224
$754$	0	0
$755$	50.5410	1.83938
$756$	0	0
$757$	10.8541	0.394499	0.197250	−	0.980353i	$-0.436799\pi$
$757$	10.8541	0.394499	0.197250	+	0.980353i	$0.436799\pi$
$758$	0	0
$759$	8.61803	0.312815
$760$	0	0
$761$	−25.8541	−0.937210	−0.468605	−	0.883408i	$-0.655243\pi$
$761$	−25.8541	−0.937210	−0.468605	+	0.883408i	$0.655243\pi$
$762$	0	0
$763$	18.4721	0.668736
$764$	0	0
$765$	−33.4164	−1.20817
$766$	0	0
$767$	−35.5967	−1.28532
$768$	0	0
$769$	−11.8885	−0.428712	−0.214356	−	0.976756i	$-0.568765\pi$
$769$	−11.8885	−0.428712	−0.214356	+	0.976756i	$0.568765\pi$
$770$	0	0
$771$	11.7082	0.421661
$772$	0	0
$773$	24.0000	0.863220	0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000	0.863220	0.431610	+	0.902060i	$0.357946\pi$
$774$	0	0
$775$	30.2574	1.08688
$776$	0	0
$777$	−0.763932	−0.0274059
$778$	0	0
$779$	33.0132	1.18282
$780$	0	0
$781$	−54.0689	−1.93474
$782$	0	0
$783$	−21.9656	−0.784985
$784$	0	0
$785$	21.4853	0.766843
$786$	0	0
$787$	34.4721	1.22880	0.614399	−	0.788995i	$-0.289398\pi$
$787$	34.4721	1.22880	0.614399	+	0.788995i	$0.289398\pi$
$788$	0	0
$789$	5.41641	0.192829
$790$	0	0
$791$	13.5279	0.480995
$792$	0	0
$793$	32.3050	1.14718
$794$	0	0
$795$	−14.9443	−0.530019
$796$	0	0
$797$	−12.7295	−0.450902	−0.225451	−	0.974255i	$-0.572386\pi$
$797$	−12.7295	−0.450902	−0.225451	+	0.974255i	$0.572386\pi$
$798$	0	0
$799$	14.4721	0.511987
$800$	0	0
$801$	27.4164	0.968711
$802$	0	0
$803$	−14.7984	−0.522223
$804$	0	0
$805$	−13.5967	−0.479222
$806$	0	0
$807$	2.47214	0.0870233
$808$	0	0
$809$	27.1246	0.953651	0.476825	−	0.878998i	$-0.341787\pi$
$809$	27.1246	0.953651	0.476825	+	0.878998i	$0.341787\pi$
$810$	0	0
$811$	−53.1033	−1.86471	−0.932355	−	0.361544i	$-0.882250\pi$
$811$	−53.1033	−1.86471	−0.932355	+	0.361544i	$0.882250\pi$
$812$	0	0
$813$	3.05573	0.107169
$814$	0	0
$815$	35.5967	1.24690
$816$	0	0
$817$	3.41641	0.119525
$818$	0	0
$819$	12.4721	0.435812
$820$	0	0
$821$	−21.4164	−0.747438	−0.373719	−	0.927542i	$-0.621918\pi$
$821$	−21.4164	−0.747438	−0.373719	+	0.927542i	$0.621918\pi$
$822$	0	0
$823$	−33.8885	−1.18128	−0.590640	−	0.806935i	$-0.701125\pi$
$823$	−33.8885	−1.18128	−0.590640	+	0.806935i	$0.701125\pi$
$824$	0	0
$825$	−7.03444	−0.244908
$826$	0	0
$827$	−56.0689	−1.94971	−0.974853	−	0.222849i	$-0.928464\pi$
$827$	−56.0689	−1.94971	−0.974853	+	0.222849i	$0.928464\pi$
$828$	0	0
$829$	−31.2016	−1.08368	−0.541839	−	0.840483i	$-0.682272\pi$
$829$	−31.2016	−1.08368	−0.541839	+	0.840483i	$0.682272\pi$
$830$	0	0
$831$	−4.81966	−0.167192
$832$	0	0
$833$	−24.4721	−0.847909
$834$	0	0
$835$	20.3951	0.705802
$836$	0	0
$837$	33.3951	1.15430
$838$	0	0
$839$	−5.34752	−0.184617	−0.0923085	−	0.995730i	$-0.529425\pi$
$839$	−5.34752	−0.184617	−0.0923085	+	0.995730i	$0.529425\pi$
$840$	0	0
$841$	11.0213	0.380044
$842$	0	0
$843$	3.63932	0.125345
$844$	0	0
$845$	5.29180	0.182043
$846$	0	0
$847$	2.58359	0.0887733
$848$	0	0
$849$	6.94427	0.238327
$850$	0	0
$851$	−3.85410	−0.132117
$852$	0	0
$853$	12.7426	0.436300	0.218150	−	0.975915i	$-0.429998\pi$
$853$	12.7426	0.436300	0.218150	+	0.975915i	$0.429998\pi$
$854$	0	0
$855$	−33.4164	−1.14282
$856$	0	0
$857$	−26.9443	−0.920399	−0.460199	−	0.887816i	$-0.652222\pi$
$857$	−26.9443	−0.920399	−0.460199	+	0.887816i	$0.652222\pi$
$858$	0	0
$859$	−26.5836	−0.907020	−0.453510	−	0.891251i	$-0.649828\pi$
$859$	−26.5836	−0.907020	−0.453510	+	0.891251i	$0.649828\pi$
$860$	0	0
$861$	−5.63932	−0.192188
$862$	0	0
$863$	−7.41641	−0.252457	−0.126229	−	0.992001i	$-0.540287\pi$
$863$	−7.41641	−0.252457	−0.126229	+	0.992001i	$0.540287\pi$
$864$	0	0
$865$	−24.1803	−0.822156
$866$	0	0
$867$	−1.85410	−0.0629686
$868$	0	0
$869$	41.8328	1.41908
$870$	0	0
$871$	−38.8885	−1.31769
$872$	0	0
$873$	−22.1803	−0.750691
$874$	0	0
$875$	−6.54102	−0.221127
$876$	0	0
$877$	−36.8328	−1.24376	−0.621878	−	0.783114i	$-0.713630\pi$
$877$	−36.8328	−1.24376	−0.621878	+	0.783114i	$0.713630\pi$
$878$	0	0
$879$	11.5279	0.388825
$880$	0	0
$881$	−22.7426	−0.766219	−0.383110	−	0.923703i	$-0.625147\pi$
$881$	−22.7426	−0.766219	−0.383110	+	0.923703i	$0.625147\pi$
$882$	0	0
$883$	−29.3050	−0.986190	−0.493095	−	0.869975i	$-0.664135\pi$
$883$	−29.3050	−0.986190	−0.493095	+	0.869975i	$0.664135\pi$
$884$	0	0
$885$	−16.2918	−0.547643
$886$	0	0
$887$	−7.88854	−0.264871	−0.132436	−	0.991192i	$-0.542280\pi$
$887$	−7.88854	−0.264871	−0.132436	+	0.991192i	$0.542280\pi$
$888$	0	0
$889$	0.583592	0.0195731
$890$	0	0
$891$	20.6525	0.691884
$892$	0	0
$893$	14.4721	0.484292
$894$	0	0
$895$	−53.2361	−1.77949
$896$	0	0
$897$	−9.18034	−0.306523
$898$	0	0
$899$	−60.8460	−2.02933
$900$	0	0
$901$	37.8885	1.26225
$902$	0	0
$903$	−0.583592	−0.0194207
$904$	0	0
$905$	−15.7771	−0.524448
$906$	0	0
$907$	20.1115	0.667790	0.333895	−	0.942610i	$-0.391637\pi$
$907$	20.1115	0.667790	0.333895	+	0.942610i	$0.391637\pi$
$908$	0	0
$909$	32.6525	1.08301
$910$	0	0
$911$	−21.8885	−0.725200	−0.362600	−	0.931945i	$-0.618111\pi$
$911$	−21.8885	−0.725200	−0.362600	+	0.931945i	$0.618111\pi$
$912$	0	0
$913$	20.0000	0.661903
$914$	0	0
$915$	14.7852	0.488784
$916$	0	0
$917$	−10.6950	−0.353182
$918$	0	0
$919$	29.8885	0.985932	0.492966	−	0.870049i	$-0.335913\pi$
$919$	29.8885	0.985932	0.492966	+	0.870049i	$0.335913\pi$
$920$	0	0
$921$	−3.79837	−0.125161
$922$	0	0
$923$	57.5967	1.89582
$924$	0	0
$925$	3.14590	0.103436
$926$	0	0
$927$	42.5967	1.39906
$928$	0	0
$929$	28.4508	0.933442	0.466721	−	0.884405i	$-0.345435\pi$
$929$	28.4508	0.933442	0.466721	+	0.884405i	$0.345435\pi$
$930$	0	0
$931$	−24.4721	−0.802042
$932$	0	0
$933$	−1.25735	−0.0411639
$934$	0	0
$935$	46.1803	1.51026
$936$	0	0
$937$	57.0476	1.86366	0.931832	−	0.362890i	$-0.118210\pi$
$937$	57.0476	1.86366	0.931832	+	0.362890i	$0.118210\pi$
$938$	0	0
$939$	−3.59675	−0.117375
$940$	0	0
$941$	3.81966	0.124517	0.0622587	−	0.998060i	$-0.480170\pi$
$941$	3.81966	0.124517	0.0622587	+	0.998060i	$0.480170\pi$
$942$	0	0
$943$	−28.4508	−0.926487
$944$	0	0
$945$	12.2492	0.398467
$946$	0	0
$947$	34.8328	1.13191	0.565957	−	0.824435i	$-0.308507\pi$
$947$	34.8328	1.13191	0.565957	+	0.824435i	$0.308507\pi$
$948$	0	0
$949$	15.7639	0.511719
$950$	0	0
$951$	1.88854	0.0612402
$952$	0	0
$953$	−44.4508	−1.43990	−0.719952	−	0.694024i	$-0.755836\pi$
$953$	−44.4508	−1.43990	−0.719952	+	0.694024i	$0.755836\pi$
$954$	0	0
$955$	11.6738	0.377754
$956$	0	0
$957$	14.1459	0.457272
$958$	0	0
$959$	23.8885	0.771401
$960$	0	0
$961$	61.5066	1.98408
$962$	0	0
$963$	21.7984	0.702443
$964$	0	0
$965$	11.4164	0.367507
$966$	0	0
$967$	11.7295	0.377195	0.188597	−	0.982054i	$-0.439606\pi$
$967$	11.7295	0.377195	0.188597	+	0.982054i	$0.439606\pi$
$968$	0	0
$969$	−12.3607	−0.397082
$970$	0	0
$971$	−26.6738	−0.856002	−0.428001	−	0.903778i	$-0.640782\pi$
$971$	−26.6738	−0.856002	−0.428001	+	0.903778i	$0.640782\pi$
$972$	0	0
$973$	2.29180	0.0734716
$974$	0	0
$975$	7.49342	0.239982
$976$	0	0
$977$	52.4721	1.67873	0.839366	−	0.543566i	$-0.182926\pi$
$977$	52.4721	1.67873	0.839366	+	0.543566i	$0.182926\pi$
$978$	0	0
$979$	−37.8885	−1.21092
$980$	0	0
$981$	−39.1246	−1.24915
$982$	0	0
$983$	39.7771	1.26869	0.634346	−	0.773049i	$-0.281269\pi$
$983$	39.7771	1.26869	0.634346	+	0.773049i	$0.281269\pi$
$984$	0	0
$985$	47.0132	1.49796
$986$	0	0
$987$	−2.47214	−0.0786890
$988$	0	0
$989$	−2.94427	−0.0936224
$990$	0	0
$991$	−54.1033	−1.71865	−0.859324	−	0.511431i	$-0.829116\pi$
$991$	−54.1033	−1.71865	−0.859324	+	0.511431i	$0.829116\pi$
$992$	0	0
$993$	17.3050	0.549156
$994$	0	0
$995$	59.7771	1.89506
$996$	0	0
$997$	−53.7771	−1.70314	−0.851569	−	0.524243i	$-0.824348\pi$
$997$	−53.7771	−1.70314	−0.851569	+	0.524243i	$0.824348\pi$
$998$	0	0
$999$	3.47214	0.109854

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2368.2.a.y.1.1		2
4.3	odd	2		2368.2.a.u.1.2		2
8.3	odd	2		592.2.a.g.1.1		2
8.5	even	2		74.2.a.b.1.2	✓	2
24.5	odd	2		666.2.a.i.1.2		2
24.11	even	2		5328.2.a.bc.1.2		2
40.13	odd	4		1850.2.b.j.149.2		4
40.29	even	2		1850.2.a.t.1.1		2
40.37	odd	4		1850.2.b.j.149.3		4
56.13	odd	2		3626.2.a.s.1.1		2
88.21	odd	2		8954.2.a.j.1.2		2
296.221	even	2		2738.2.a.g.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.2.a.b.1.2	✓	2	8.5	even	2
592.2.a.g.1.1		2	8.3	odd	2
666.2.a.i.1.2		2	24.5	odd	2
1850.2.a.t.1.1		2	40.29	even	2
1850.2.b.j.149.2		4	40.13	odd	4
1850.2.b.j.149.3		4	40.37	odd	4
2368.2.a.u.1.2		2	4.3	odd	2
2368.2.a.y.1.1		2	1.1	even	1	trivial
2738.2.a.g.1.2		2	296.221	even	2
3626.2.a.s.1.1		2	56.13	odd	2
5328.2.a.bc.1.2		2	24.11	even	2
8954.2.a.j.1.2		2	88.21	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2368 = 2^{6} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2368.a (trivial)

\(f(q)\)	\(=\)	\(q-0.618034 q^{3} +2.85410 q^{5} +1.23607 q^{7} -2.61803 q^{9} +O(q^{10})\)	\(q-0.618034 q^{3} +2.85410 q^{5} +1.23607 q^{7} -2.61803 q^{9} +3.61803 q^{11} -3.85410 q^{13} -1.76393 q^{15} +4.47214 q^{17} +4.47214 q^{19} -0.763932 q^{21} -3.85410 q^{23} +3.14590 q^{25} +3.47214 q^{27} -6.32624 q^{29} +9.61803 q^{31} -2.23607 q^{33} +3.52786 q^{35} +1.00000 q^{37} +2.38197 q^{39} +7.38197 q^{41} +0.763932 q^{43} -7.47214 q^{45} +3.23607 q^{47} -5.47214 q^{49} -2.76393 q^{51} +8.47214 q^{53} +10.3262 q^{55} -2.76393 q^{57} +9.23607 q^{59} -8.38197 q^{61} -3.23607 q^{63} -11.0000 q^{65} +10.0902 q^{67} +2.38197 q^{69} -14.9443 q^{71} -4.09017 q^{73} -1.94427 q^{75} +4.47214 q^{77} +11.5623 q^{79} +5.70820 q^{81} +5.52786 q^{83} +12.7639 q^{85} +3.90983 q^{87} -10.4721 q^{89} -4.76393 q^{91} -5.94427 q^{93} +12.7639 q^{95} +8.47214 q^{97} -9.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10}) \) 2 * q + q^3 - q^5 - 2 * q^7 - 3 * q^9	\( 2 q + q^{3} - q^{5} - 2 q^{7} - 3 q^{9} + 5 q^{11} - q^{13} - 8 q^{15} - 6 q^{21} - q^{23} + 13 q^{25} - 2 q^{27} + 3 q^{29} + 17 q^{31} + 16 q^{35} + 2 q^{37} + 7 q^{39} + 17 q^{41} + 6 q^{43} - 6 q^{45} + 2 q^{47} - 2 q^{49} - 10 q^{51} + 8 q^{53} + 5 q^{55} - 10 q^{57} + 14 q^{59} - 19 q^{61} - 2 q^{63} - 22 q^{65} + 9 q^{67} + 7 q^{69} - 12 q^{71} + 3 q^{73} + 14 q^{75} + 3 q^{79} - 2 q^{81} + 20 q^{83} + 30 q^{85} + 19 q^{87} - 12 q^{89} - 14 q^{91} + 6 q^{93} + 30 q^{95} + 8 q^{97} - 10 q^{99}+O(q^{100}) \) 2 * q + q^3 - q^5 - 2 * q^7 - 3 * q^9 + 5 * q^11 - q^13 - 8 * q^15 - 6 * q^21 - q^23 + 13 * q^25 - 2 * q^27 + 3 * q^29 + 17 * q^31 + 16 * q^35 + 2 * q^37 + 7 * q^39 + 17 * q^41 + 6 * q^43 - 6 * q^45 + 2 * q^47 - 2 * q^49 - 10 * q^51 + 8 * q^53 + 5 * q^55 - 10 * q^57 + 14 * q^59 - 19 * q^61 - 2 * q^63 - 22 * q^65 + 9 * q^67 + 7 * q^69 - 12 * q^71 + 3 * q^73 + 14 * q^75 + 3 * q^79 - 2 * q^81 + 20 * q^83 + 30 * q^85 + 19 * q^87 - 12 * q^89 - 14 * q^91 + 6 * q^93 + 30 * q^95 + 8 * q^97 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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