LMFDB - Embedded newform 3724.2.a.m.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3724,2,Mod(1,3724)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3724, 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("3724.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3724 = 2^{2} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3724.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:	$29.7362897127$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 18x^{5} + 84x^{3} - 4x^{2} - 44x + 8 $ x^7 - 18x^5 + 84x^3 - 4x^2 - 44x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 532)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.678078$ of defining polynomial
Character	$\chi$	$=$	3724.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.678078 q^{3} +0.873900 q^{5} -2.54021 q^{9} -1.23451 q^{11} +1.27278 q^{13} -0.592573 q^{15} +0.716798 q^{17} -1.00000 q^{19} +0.537193 q^{23} -4.23630 q^{25} +3.75670 q^{27} +6.82520 q^{29} -2.68697 q^{31} +0.837092 q^{33} -0.287650 q^{37} -0.863046 q^{39} -11.3686 q^{41} +10.5302 q^{43} -2.21989 q^{45} +9.95523 q^{47} -0.486045 q^{51} +9.81923 q^{53} -1.07883 q^{55} +0.678078 q^{57} -5.80977 q^{59} -3.59903 q^{61} +1.11229 q^{65} +7.43360 q^{67} -0.364259 q^{69} -7.77009 q^{71} +9.09066 q^{73} +2.87254 q^{75} +12.9038 q^{79} +5.07329 q^{81} +2.56549 q^{83} +0.626410 q^{85} -4.62802 q^{87} -4.29123 q^{89} +1.82198 q^{93} -0.873900 q^{95} -11.0676 q^{97} +3.13590 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 2 q^{5} + 15 q^{9} + 14 q^{11} + 12 q^{15} - 10 q^{17} - 7 q^{19} + 7 q^{23} + 21 q^{25} + 2 q^{29} - 4 q^{31} + 24 q^{33} - 12 q^{37} + 18 q^{39} - 4 q^{41} - 6 q^{45} + 16 q^{47} + 2 q^{51} - 6 q^{53}+ \cdots + 40 q^{99}+O(q^{100}) $ 7 * q - 2 * q^5 + 15 * q^9 + 14 * q^11 + 12 * q^15 - 10 * q^17 - 7 * q^19 + 7 * q^23 + 21 * q^25 + 2 * q^29 - 4 * q^31 + 24 * q^33 - 12 * q^37 + 18 * q^39 - 4 * q^41 - 6 * q^45 + 16 * q^47 + 2 * q^51 - 6 * q^53 - 9 * q^55 + 4 * q^59 + 28 * q^61 - 8 * q^65 + 22 * q^67 - 34 * q^69 + 34 * q^71 - 16 * q^73 + 32 * q^75 + 14 * q^79 + 51 * q^81 - 13 * q^83 - 9 * q^85 - 10 * q^87 + 16 * q^89 - 36 * q^93 + 2 * q^95 + 4 * q^97 + 40 * 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
$3$	−0.678078	−0.391489	−0.195744	−	0.980655i	$-0.562712\pi$
$3$	−0.678078	−0.391489	−0.195744	+	0.980655i	$0.562712\pi$
$4$	0	0
$5$	0.873900	0.390820	0.195410	−	0.980722i	$-0.437396\pi$
$5$	0.873900	0.390820	0.195410	+	0.980722i	$0.437396\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.54021	−0.846737
$10$	0	0
$11$	−1.23451	−0.372217	−0.186109	−	0.982529i	$-0.559588\pi$
$11$	−1.23451	−0.372217	−0.186109	+	0.982529i	$0.559588\pi$
$12$	0	0
$13$	1.27278	0.353006	0.176503	−	0.984300i	$-0.443521\pi$
$13$	1.27278	0.353006	0.176503	+	0.984300i	$0.443521\pi$
$14$	0	0
$15$	−0.592573	−0.153002
$16$	0	0
$17$	0.716798	0.173849	0.0869245	−	0.996215i	$-0.472296\pi$
$17$	0.716798	0.173849	0.0869245	+	0.996215i	$0.472296\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.537193	0.112013	0.0560063	−	0.998430i	$-0.482163\pi$
$23$	0.537193	0.112013	0.0560063	+	0.998430i	$0.482163\pi$
$24$	0	0
$25$	−4.23630	−0.847260
$26$	0	0
$27$	3.75670	0.722977
$28$	0	0
$29$	6.82520	1.26741	0.633704	−	0.773575i	$-0.281534\pi$
$29$	6.82520	1.26741	0.633704	+	0.773575i	$0.281534\pi$
$30$	0	0
$31$	−2.68697	−0.482595	−0.241297	−	0.970451i	$-0.577573\pi$
$31$	−2.68697	−0.482595	−0.241297	+	0.970451i	$0.577573\pi$
$32$	0	0
$33$	0.837092	0.145719
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.287650	−0.0472894	−0.0236447	−	0.999720i	$-0.507527\pi$
$37$	−0.287650	−0.0472894	−0.0236447	+	0.999720i	$0.507527\pi$
$38$	0	0
$39$	−0.863046	−0.138198
$40$	0	0
$41$	−11.3686	−1.77548	−0.887741	−	0.460342i	$-0.847727\pi$
$41$	−11.3686	−1.77548	−0.887741	+	0.460342i	$0.847727\pi$
$42$	0	0
$43$	10.5302	1.60584	0.802919	−	0.596088i	$-0.203279\pi$
$43$	10.5302	1.60584	0.802919	+	0.596088i	$0.203279\pi$
$44$	0	0
$45$	−2.21989	−0.330922
$46$	0	0
$47$	9.95523	1.45212	0.726060	−	0.687632i	$-0.241350\pi$
$47$	9.95523	1.45212	0.726060	+	0.687632i	$0.241350\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−0.486045	−0.0680600
$52$	0	0
$53$	9.81923	1.34878	0.674388	−	0.738377i	$-0.264408\pi$
$53$	9.81923	1.34878	0.674388	+	0.738377i	$0.264408\pi$
$54$	0	0
$55$	−1.07883	−0.145470
$56$	0	0
$57$	0.678078	0.0898137
$58$	0	0
$59$	−5.80977	−0.756367	−0.378184	−	0.925731i	$-0.623451\pi$
$59$	−5.80977	−0.756367	−0.378184	+	0.925731i	$0.623451\pi$
$60$	0	0
$61$	−3.59903	−0.460809	−0.230405	−	0.973095i	$-0.574005\pi$
$61$	−3.59903	−0.460809	−0.230405	+	0.973095i	$0.574005\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.11229	0.137962
$66$	0	0
$67$	7.43360	0.908158	0.454079	−	0.890961i	$-0.349968\pi$
$67$	7.43360	0.908158	0.454079	+	0.890961i	$0.349968\pi$
$68$	0	0
$69$	−0.364259	−0.0438517
$70$	0	0
$71$	−7.77009	−0.922140	−0.461070	−	0.887364i	$-0.652534\pi$
$71$	−7.77009	−0.922140	−0.461070	+	0.887364i	$0.652534\pi$
$72$	0	0
$73$	9.09066	1.06398	0.531991	−	0.846750i	$-0.321444\pi$
$73$	9.09066	1.06398	0.531991	+	0.846750i	$0.321444\pi$
$74$	0	0
$75$	2.87254	0.331693
$76$	0	0
$77$	0	0
$78$	0	0
$79$	12.9038	1.45179	0.725895	−	0.687806i	$-0.241426\pi$
$79$	12.9038	1.45179	0.725895	+	0.687806i	$0.241426\pi$
$80$	0	0
$81$	5.07329	0.563699
$82$	0	0
$83$	2.56549	0.281599	0.140800	−	0.990038i	$-0.455033\pi$
$83$	2.56549	0.281599	0.140800	+	0.990038i	$0.455033\pi$
$84$	0	0
$85$	0.626410	0.0679437
$86$	0	0
$87$	−4.62802	−0.496176
$88$	0	0
$89$	−4.29123	−0.454870	−0.227435	−	0.973793i	$-0.573034\pi$
$89$	−4.29123	−0.454870	−0.227435	+	0.973793i	$0.573034\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.82198	0.188930
$94$	0	0
$95$	−0.873900	−0.0896603
$96$	0	0
$97$	−11.0676	−1.12375	−0.561875	−	0.827222i	$-0.689920\pi$
$97$	−11.0676	−1.12375	−0.561875	+	0.827222i	$0.689920\pi$
$98$	0	0
$99$	3.13590	0.315170
$100$	0	0
$101$	0.307018	0.0305494	0.0152747	−	0.999883i	$-0.495138\pi$
$101$	0.307018	0.0305494	0.0152747	+	0.999883i	$0.495138\pi$
$102$	0	0
$103$	13.1010	1.29088	0.645438	−	0.763812i	$-0.276675\pi$
$103$	13.1010	1.29088	0.645438	+	0.763812i	$0.276675\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.56223	0.731069	0.365534	−	0.930798i	$-0.380886\pi$
$107$	7.56223	0.731069	0.365534	+	0.930798i	$0.380886\pi$
$108$	0	0
$109$	10.7986	1.03432	0.517161	−	0.855888i	$-0.326989\pi$
$109$	10.7986	1.03432	0.517161	+	0.855888i	$0.326989\pi$
$110$	0	0
$111$	0.195049	0.0185133
$112$	0	0
$113$	0.184353	0.0173424	0.00867122	−	0.999962i	$-0.497240\pi$
$113$	0.184353	0.0173424	0.00867122	+	0.999962i	$0.497240\pi$
$114$	0	0
$115$	0.469454	0.0437768
$116$	0	0
$117$	−3.23313	−0.298903
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−9.47600	−0.861454
$122$	0	0
$123$	7.70883	0.695082
$124$	0	0
$125$	−8.07161	−0.721946
$126$	0	0
$127$	7.02826	0.623658	0.311829	−	0.950138i	$-0.399058\pi$
$127$	7.02826	0.623658	0.311829	+	0.950138i	$0.399058\pi$
$128$	0	0
$129$	−7.14029	−0.628668
$130$	0	0
$131$	7.80197	0.681661	0.340831	−	0.940125i	$-0.389292\pi$
$131$	7.80197	0.681661	0.340831	+	0.940125i	$0.389292\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	3.28298	0.282554
$136$	0	0
$137$	9.41554	0.804424	0.402212	−	0.915547i	$-0.368241\pi$
$137$	9.41554	0.804424	0.402212	+	0.915547i	$0.368241\pi$
$138$	0	0
$139$	−4.83030	−0.409701	−0.204850	−	0.978793i	$-0.565671\pi$
$139$	−4.83030	−0.409701	−0.204850	+	0.978793i	$0.565671\pi$
$140$	0	0
$141$	−6.75043	−0.568488
$142$	0	0
$143$	−1.57126	−0.131395
$144$	0	0
$145$	5.96455	0.495329
$146$	0	0
$147$	0	0
$148$	0	0
$149$	9.37121	0.767720	0.383860	−	0.923391i	$-0.374595\pi$
$149$	9.37121	0.767720	0.383860	+	0.923391i	$0.374595\pi$
$150$	0	0
$151$	12.7329	1.03619	0.518096	−	0.855323i	$-0.326641\pi$
$151$	12.7329	1.03619	0.518096	+	0.855323i	$0.326641\pi$
$152$	0	0
$153$	−1.82082	−0.147204
$154$	0	0
$155$	−2.34815	−0.188608
$156$	0	0
$157$	6.34875	0.506685	0.253343	−	0.967377i	$-0.418470\pi$
$157$	6.34875	0.506685	0.253343	+	0.967377i	$0.418470\pi$
$158$	0	0
$159$	−6.65821	−0.528030
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−15.1170	−1.18406	−0.592028	−	0.805917i	$-0.701673\pi$
$163$	−15.1170	−1.18406	−0.592028	+	0.805917i	$0.701673\pi$
$164$	0	0
$165$	0.731535	0.0569499
$166$	0	0
$167$	−9.47298	−0.733041	−0.366521	−	0.930410i	$-0.619451\pi$
$167$	−9.47298	−0.733041	−0.366521	+	0.930410i	$0.619451\pi$
$168$	0	0
$169$	−11.3800	−0.875387
$170$	0	0
$171$	2.54021	0.194255
$172$	0	0
$173$	20.1842	1.53458	0.767290	−	0.641300i	$-0.221605\pi$
$173$	20.1842	1.53458	0.767290	+	0.641300i	$0.221605\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.93948	0.296109
$178$	0	0
$179$	1.21682	0.0909497	0.0454748	−	0.998965i	$-0.485520\pi$
$179$	1.21682	0.0909497	0.0454748	+	0.998965i	$0.485520\pi$
$180$	0	0
$181$	−0.720713	−0.0535702	−0.0267851	−	0.999641i	$-0.508527\pi$
$181$	−0.720713	−0.0535702	−0.0267851	+	0.999641i	$0.508527\pi$
$182$	0	0
$183$	2.44043	0.180402
$184$	0	0
$185$	−0.251377	−0.0184816
$186$	0	0
$187$	−0.884891	−0.0647096
$188$	0	0
$189$	0	0
$190$	0	0
$191$	19.7974	1.43249	0.716246	−	0.697848i	$-0.245859\pi$
$191$	19.7974	1.43249	0.716246	+	0.697848i	$0.245859\pi$
$192$	0	0
$193$	−4.27583	−0.307781	−0.153891	−	0.988088i	$-0.549180\pi$
$193$	−4.27583	−0.307781	−0.153891	+	0.988088i	$0.549180\pi$
$194$	0	0
$195$	−0.754217	−0.0540106
$196$	0	0
$197$	14.7614	1.05171	0.525853	−	0.850576i	$-0.323746\pi$
$197$	14.7614	1.05171	0.525853	+	0.850576i	$0.323746\pi$
$198$	0	0
$199$	−17.9069	−1.26939	−0.634694	−	0.772763i	$-0.718874\pi$
$199$	−17.9069	−1.26939	−0.634694	+	0.772763i	$0.718874\pi$
$200$	0	0
$201$	−5.04056	−0.355534
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−9.93506	−0.693895
$206$	0	0
$207$	−1.36458	−0.0948451
$208$	0	0
$209$	1.23451	0.0853925
$210$	0	0
$211$	−1.30309	−0.0897082	−0.0448541	−	0.998994i	$-0.514282\pi$
$211$	−1.30309	−0.0897082	−0.0448541	+	0.998994i	$0.514282\pi$
$212$	0	0
$213$	5.26873	0.361008
$214$	0	0
$215$	9.20234	0.627594
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−6.16418	−0.416537
$220$	0	0
$221$	0.912328	0.0613698
$222$	0	0
$223$	14.5276	0.972841	0.486421	−	0.873725i	$-0.338302\pi$
$223$	14.5276	0.972841	0.486421	+	0.873725i	$0.338302\pi$
$224$	0	0
$225$	10.7611	0.717406
$226$	0	0
$227$	−0.363913	−0.0241538	−0.0120769	−	0.999927i	$-0.503844\pi$
$227$	−0.363913	−0.0241538	−0.0120769	+	0.999927i	$0.503844\pi$
$228$	0	0
$229$	25.8508	1.70827	0.854134	−	0.520053i	$-0.174088\pi$
$229$	25.8508	1.70827	0.854134	+	0.520053i	$0.174088\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.7470	−0.704058	−0.352029	−	0.935989i	$-0.614508\pi$
$233$	−10.7470	−0.704058	−0.352029	+	0.935989i	$0.614508\pi$
$234$	0	0
$235$	8.69988	0.567517
$236$	0	0
$237$	−8.74978	−0.568359
$238$	0	0
$239$	13.4092	0.867370	0.433685	−	0.901064i	$-0.357213\pi$
$239$	13.4092	0.867370	0.433685	+	0.901064i	$0.357213\pi$
$240$	0	0
$241$	−2.11044	−0.135946	−0.0679728	−	0.997687i	$-0.521653\pi$
$241$	−2.11044	−0.135946	−0.0679728	+	0.997687i	$0.521653\pi$
$242$	0	0
$243$	−14.7102	−0.943659
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.27278	−0.0809852
$248$	0	0
$249$	−1.73961	−0.110243
$250$	0	0
$251$	20.0022	1.26253	0.631263	−	0.775569i	$-0.282537\pi$
$251$	20.0022	1.26253	0.631263	+	0.775569i	$0.282537\pi$
$252$	0	0
$253$	−0.663168	−0.0416930
$254$	0	0
$255$	−0.424755	−0.0265992
$256$	0	0
$257$	5.52879	0.344877	0.172438	−	0.985020i	$-0.444835\pi$
$257$	5.52879	0.344877	0.172438	+	0.985020i	$0.444835\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−17.3374	−1.07316
$262$	0	0
$263$	11.4509	0.706091	0.353046	−	0.935606i	$-0.385146\pi$
$263$	11.4509	0.706091	0.353046	+	0.935606i	$0.385146\pi$
$264$	0	0
$265$	8.58103	0.527129
$266$	0	0
$267$	2.90979	0.178077
$268$	0	0
$269$	−24.9762	−1.52282	−0.761412	−	0.648269i	$-0.775493\pi$
$269$	−24.9762	−1.52282	−0.761412	+	0.648269i	$0.775493\pi$
$270$	0	0
$271$	−3.55748	−0.216101	−0.108051	−	0.994145i	$-0.534461\pi$
$271$	−3.55748	−0.216101	−0.108051	+	0.994145i	$0.534461\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.22973	0.315365
$276$	0	0
$277$	−2.16246	−0.129930	−0.0649649	−	0.997888i	$-0.520694\pi$
$277$	−2.16246	−0.129930	−0.0649649	+	0.997888i	$0.520694\pi$
$278$	0	0
$279$	6.82548	0.408631
$280$	0	0
$281$	25.2588	1.50681	0.753406	−	0.657556i	$-0.228410\pi$
$281$	25.2588	1.50681	0.753406	+	0.657556i	$0.228410\pi$
$282$	0	0
$283$	−2.36163	−0.140384	−0.0701922	−	0.997533i	$-0.522361\pi$
$283$	−2.36163	−0.140384	−0.0701922	+	0.997533i	$0.522361\pi$
$284$	0	0
$285$	0.592573	0.0351010
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.4862	−0.969777
$290$	0	0
$291$	7.50473	0.439935
$292$	0	0
$293$	−30.9231	−1.80655	−0.903274	−	0.429064i	$-0.858844\pi$
$293$	−30.9231	−1.80655	−0.903274	+	0.429064i	$0.858844\pi$
$294$	0	0
$295$	−5.07716	−0.295604
$296$	0	0
$297$	−4.63766	−0.269104
$298$	0	0
$299$	0.683730	0.0395411
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−0.208182	−0.0119598
$304$	0	0
$305$	−3.14520	−0.180093
$306$	0	0
$307$	12.0542	0.687970	0.343985	−	0.938975i	$-0.388223\pi$
$307$	12.0542	0.687970	0.343985	+	0.938975i	$0.388223\pi$
$308$	0	0
$309$	−8.88348	−0.505364
$310$	0	0
$311$	12.3741	0.701668	0.350834	−	0.936438i	$-0.385898\pi$
$311$	12.3741	0.701668	0.350834	+	0.936438i	$0.385898\pi$
$312$	0	0
$313$	−21.4371	−1.21170	−0.605849	−	0.795580i	$-0.707166\pi$
$313$	−21.4371	−1.21170	−0.605849	+	0.795580i	$0.707166\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4.60452	−0.258615	−0.129308	−	0.991605i	$-0.541275\pi$
$317$	−4.60452	−0.258615	−0.129308	+	0.991605i	$0.541275\pi$
$318$	0	0
$319$	−8.42575	−0.471752
$320$	0	0
$321$	−5.12779	−0.286205
$322$	0	0
$323$	−0.716798	−0.0398837
$324$	0	0
$325$	−5.39189	−0.299088
$326$	0	0
$327$	−7.32233	−0.404926
$328$	0	0
$329$	0	0
$330$	0	0
$331$	29.8050	1.63823	0.819115	−	0.573629i	$-0.194465\pi$
$331$	29.8050	1.63823	0.819115	+	0.573629i	$0.194465\pi$
$332$	0	0
$333$	0.730691	0.0400416
$334$	0	0
$335$	6.49622	0.354927
$336$	0	0
$337$	13.8155	0.752577	0.376289	−	0.926502i	$-0.377200\pi$
$337$	13.8155	0.752577	0.376289	+	0.926502i	$0.377200\pi$
$338$	0	0
$339$	−0.125006	−0.00678937
$340$	0	0
$341$	3.31708	0.179630
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−0.318326	−0.0171381
$346$	0	0
$347$	14.2790	0.766534	0.383267	−	0.923638i	$-0.374799\pi$
$347$	14.2790	0.766534	0.383267	+	0.923638i	$0.374799\pi$
$348$	0	0
$349$	21.9955	1.17739	0.588697	−	0.808354i	$-0.299641\pi$
$349$	21.9955	1.17739	0.588697	+	0.808354i	$0.299641\pi$
$350$	0	0
$351$	4.78146	0.255215
$352$	0	0
$353$	21.5086	1.14479	0.572393	−	0.819980i	$-0.306015\pi$
$353$	21.5086	1.14479	0.572393	+	0.819980i	$0.306015\pi$
$354$	0	0
$355$	−6.79029	−0.360391
$356$	0	0
$357$	0	0
$358$	0	0
$359$	31.3005	1.65198	0.825990	−	0.563685i	$-0.190617\pi$
$359$	31.3005	1.65198	0.825990	+	0.563685i	$0.190617\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	6.42547	0.337250
$364$	0	0
$365$	7.94434	0.415826
$366$	0	0
$367$	21.9530	1.14594	0.572968	−	0.819577i	$-0.305792\pi$
$367$	21.9530	1.14594	0.572968	+	0.819577i	$0.305792\pi$
$368$	0	0
$369$	28.8787	1.50337
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−1.40136	−0.0725599	−0.0362800	−	0.999342i	$-0.511551\pi$
$373$	−1.40136	−0.0725599	−0.0362800	+	0.999342i	$0.511551\pi$
$374$	0	0
$375$	5.47318	0.282634
$376$	0	0
$377$	8.68700	0.447403
$378$	0	0
$379$	9.35921	0.480750	0.240375	−	0.970680i	$-0.422730\pi$
$379$	9.35921	0.480750	0.240375	+	0.970680i	$0.422730\pi$
$380$	0	0
$381$	−4.76571	−0.244155
$382$	0	0
$383$	17.5977	0.899201	0.449600	−	0.893230i	$-0.351566\pi$
$383$	17.5977	0.899201	0.449600	+	0.893230i	$0.351566\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−26.7489	−1.35972
$388$	0	0
$389$	−25.1622	−1.27577	−0.637887	−	0.770130i	$-0.720191\pi$
$389$	−25.1622	−1.27577	−0.637887	+	0.770130i	$0.720191\pi$
$390$	0	0
$391$	0.385059	0.0194733
$392$	0	0
$393$	−5.29035	−0.266863
$394$	0	0
$395$	11.2766	0.567389
$396$	0	0
$397$	15.5172	0.778788	0.389394	−	0.921071i	$-0.372684\pi$
$397$	15.5172	0.778788	0.389394	+	0.921071i	$0.372684\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	9.44663	0.471742	0.235871	−	0.971784i	$-0.424206\pi$
$401$	9.44663	0.471742	0.235871	+	0.971784i	$0.424206\pi$
$402$	0	0
$403$	−3.41993	−0.170359
$404$	0	0
$405$	4.43355	0.220305
$406$	0	0
$407$	0.355105	0.0176019
$408$	0	0
$409$	−24.5353	−1.21319	−0.606597	−	0.795010i	$-0.707466\pi$
$409$	−24.5353	−1.21319	−0.606597	+	0.795010i	$0.707466\pi$
$410$	0	0
$411$	−6.38448	−0.314923
$412$	0	0
$413$	0	0
$414$	0	0
$415$	2.24199	0.110055
$416$	0	0
$417$	3.27532	0.160393
$418$	0	0
$419$	−21.5605	−1.05330	−0.526650	−	0.850082i	$-0.676552\pi$
$419$	−21.5605	−1.05330	−0.526650	+	0.850082i	$0.676552\pi$
$420$	0	0
$421$	−25.2433	−1.23028	−0.615142	−	0.788417i	$-0.710901\pi$
$421$	−25.2433	−1.23028	−0.615142	+	0.788417i	$0.710901\pi$
$422$	0	0
$423$	−25.2884	−1.22956
$424$	0	0
$425$	−3.03657	−0.147295
$426$	0	0
$427$	0	0
$428$	0	0
$429$	1.06544	0.0514397
$430$	0	0
$431$	20.3722	0.981295	0.490647	−	0.871358i	$-0.336760\pi$
$431$	20.3722	0.981295	0.490647	+	0.871358i	$0.336760\pi$
$432$	0	0
$433$	−21.9359	−1.05417	−0.527087	−	0.849812i	$-0.676716\pi$
$433$	−21.9359	−1.05417	−0.527087	+	0.849812i	$0.676716\pi$
$434$	0	0
$435$	−4.04443	−0.193916
$436$	0	0
$437$	−0.537193	−0.0256974
$438$	0	0
$439$	22.1415	1.05676	0.528378	−	0.849009i	$-0.322800\pi$
$439$	22.1415	1.05676	0.528378	+	0.849009i	$0.322800\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	13.2976	0.631790	0.315895	−	0.948794i	$-0.397695\pi$
$443$	13.2976	0.631790	0.315895	+	0.948794i	$0.397695\pi$
$444$	0	0
$445$	−3.75011	−0.177772
$446$	0	0
$447$	−6.35442	−0.300554
$448$	0	0
$449$	−14.3802	−0.678645	−0.339322	−	0.940670i	$-0.610198\pi$
$449$	−14.3802	−0.678645	−0.339322	+	0.940670i	$0.610198\pi$
$450$	0	0
$451$	14.0346	0.660866
$452$	0	0
$453$	−8.63393	−0.405657
$454$	0	0
$455$	0	0
$456$	0	0
$457$	11.7020	0.547397	0.273698	−	0.961816i	$-0.411753\pi$
$457$	11.7020	0.547397	0.273698	+	0.961816i	$0.411753\pi$
$458$	0	0
$459$	2.69279	0.125689
$460$	0	0
$461$	18.0500	0.840674	0.420337	−	0.907368i	$-0.361912\pi$
$461$	18.0500	0.840674	0.420337	+	0.907368i	$0.361912\pi$
$462$	0	0
$463$	−3.20745	−0.149063	−0.0745315	−	0.997219i	$-0.523746\pi$
$463$	−3.20745	−0.149063	−0.0745315	+	0.997219i	$0.523746\pi$
$464$	0	0
$465$	1.59223	0.0738378
$466$	0	0
$467$	−41.8682	−1.93743	−0.968715	−	0.248175i	$-0.920169\pi$
$467$	−41.8682	−1.93743	−0.968715	+	0.248175i	$0.920169\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−4.30495	−0.198362
$472$	0	0
$473$	−12.9996	−0.597721
$474$	0	0
$475$	4.23630	0.194375
$476$	0	0
$477$	−24.9429	−1.14206
$478$	0	0
$479$	15.4595	0.706361	0.353181	−	0.935555i	$-0.385100\pi$
$479$	15.4595	0.706361	0.353181	+	0.935555i	$0.385100\pi$
$480$	0	0
$481$	−0.366116	−0.0166934
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−9.67202	−0.439184
$486$	0	0
$487$	−1.83061	−0.0829527	−0.0414764	−	0.999139i	$-0.513206\pi$
$487$	−1.83061	−0.0829527	−0.0414764	+	0.999139i	$0.513206\pi$
$488$	0	0
$489$	10.2505	0.463545
$490$	0	0
$491$	−8.79309	−0.396826	−0.198413	−	0.980118i	$-0.563579\pi$
$491$	−8.79309	−0.396826	−0.198413	+	0.980118i	$0.563579\pi$
$492$	0	0
$493$	4.89229	0.220338
$494$	0	0
$495$	2.74047	0.123175
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−14.8452	−0.664562	−0.332281	−	0.943180i	$-0.607818\pi$
$499$	−14.8452	−0.664562	−0.332281	+	0.943180i	$0.607818\pi$
$500$	0	0
$501$	6.42342	0.286977
$502$	0	0
$503$	−40.0927	−1.78765	−0.893823	−	0.448420i	$-0.851987\pi$
$503$	−40.0927	−1.78765	−0.893823	+	0.448420i	$0.851987\pi$
$504$	0	0
$505$	0.268303	0.0119393
$506$	0	0
$507$	7.71655	0.342704
$508$	0	0
$509$	5.68128	0.251818	0.125909	−	0.992042i	$-0.459815\pi$
$509$	5.68128	0.251818	0.125909	+	0.992042i	$0.459815\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−3.75670	−0.165862
$514$	0	0
$515$	11.4489	0.504501
$516$	0	0
$517$	−12.2898	−0.540504
$518$	0	0
$519$	−13.6865	−0.600771
$520$	0	0
$521$	10.3731	0.454453	0.227227	−	0.973842i	$-0.427034\pi$
$521$	10.3731	0.454453	0.227227	+	0.973842i	$0.427034\pi$
$522$	0	0
$523$	−32.0155	−1.39994	−0.699970	−	0.714172i	$-0.746803\pi$
$523$	−32.0155	−1.39994	−0.699970	+	0.714172i	$0.746803\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.92602	−0.0838987
$528$	0	0
$529$	−22.7114	−0.987453
$530$	0	0
$531$	14.7580	0.640444
$532$	0	0
$533$	−14.4698	−0.626757
$534$	0	0
$535$	6.60864	0.285716
$536$	0	0
$537$	−0.825102	−0.0356058
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−7.19493	−0.309334	−0.154667	−	0.987967i	$-0.549431\pi$
$541$	−7.19493	−0.309334	−0.154667	+	0.987967i	$0.549431\pi$
$542$	0	0
$543$	0.488700	0.0209721
$544$	0	0
$545$	9.43694	0.404234
$546$	0	0
$547$	2.89772	0.123897	0.0619487	−	0.998079i	$-0.480268\pi$
$547$	2.89772	0.123897	0.0619487	+	0.998079i	$0.480268\pi$
$548$	0	0
$549$	9.14230	0.390184
$550$	0	0
$551$	−6.82520	−0.290763
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0.170454	0.00723535
$556$	0	0
$557$	−32.8177	−1.39053	−0.695266	−	0.718753i	$-0.744713\pi$
$557$	−32.8177	−1.39053	−0.695266	+	0.718753i	$0.744713\pi$
$558$	0	0
$559$	13.4026	0.566871
$560$	0	0
$561$	0.600026	0.0253331
$562$	0	0
$563$	6.39551	0.269539	0.134769	−	0.990877i	$-0.456971\pi$
$563$	6.39551	0.269539	0.134769	+	0.990877i	$0.456971\pi$
$564$	0	0
$565$	0.161106	0.00677777
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.46848	−0.0615618	−0.0307809	−	0.999526i	$-0.509799\pi$
$569$	−1.46848	−0.0615618	−0.0307809	+	0.999526i	$0.509799\pi$
$570$	0	0
$571$	−14.4747	−0.605747	−0.302873	−	0.953031i	$-0.597946\pi$
$571$	−14.4747	−0.605747	−0.302873	+	0.953031i	$0.597946\pi$
$572$	0	0
$573$	−13.4242	−0.560805
$574$	0	0
$575$	−2.27571	−0.0949037
$576$	0	0
$577$	24.3051	1.01184	0.505918	−	0.862582i	$-0.331154\pi$
$577$	24.3051	1.01184	0.505918	+	0.862582i	$0.331154\pi$
$578$	0	0
$579$	2.89935	0.120493
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−12.1219	−0.502038
$584$	0	0
$585$	−2.82544	−0.116817
$586$	0	0
$587$	−10.1829	−0.420292	−0.210146	−	0.977670i	$-0.567394\pi$
$587$	−10.1829	−0.420292	−0.210146	+	0.977670i	$0.567394\pi$
$588$	0	0
$589$	2.68697	0.110715
$590$	0	0
$591$	−10.0094	−0.411731
$592$	0	0
$593$	−20.8749	−0.857230	−0.428615	−	0.903487i	$-0.640998\pi$
$593$	−20.8749	−0.857230	−0.428615	+	0.903487i	$0.640998\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	12.1423	0.496951
$598$	0	0
$599$	−23.7745	−0.971400	−0.485700	−	0.874126i	$-0.661435\pi$
$599$	−23.7745	−0.971400	−0.485700	+	0.874126i	$0.661435\pi$
$600$	0	0
$601$	−33.0802	−1.34937	−0.674685	−	0.738106i	$-0.735721\pi$
$601$	−33.0802	−1.34937	−0.674685	+	0.738106i	$0.735721\pi$
$602$	0	0
$603$	−18.8829	−0.768971
$604$	0	0
$605$	−8.28108	−0.336674
$606$	0	0
$607$	2.99498	0.121563	0.0607813	−	0.998151i	$-0.480641\pi$
$607$	2.99498	0.121563	0.0607813	+	0.998151i	$0.480641\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12.6708	0.512607
$612$	0	0
$613$	7.88093	0.318308	0.159154	−	0.987254i	$-0.449123\pi$
$613$	7.88093	0.318308	0.159154	+	0.987254i	$0.449123\pi$
$614$	0	0
$615$	6.73675	0.271652
$616$	0	0
$617$	31.7387	1.27775	0.638876	−	0.769310i	$-0.279400\pi$
$617$	31.7387	1.27775	0.638876	+	0.769310i	$0.279400\pi$
$618$	0	0
$619$	−26.3549	−1.05929	−0.529646	−	0.848219i	$-0.677675\pi$
$619$	−26.3549	−1.05929	−0.529646	+	0.848219i	$0.677675\pi$
$620$	0	0
$621$	2.01807	0.0809825
$622$	0	0
$623$	0	0
$624$	0	0
$625$	14.1277	0.565108
$626$	0	0
$627$	−0.837092	−0.0334302
$628$	0	0
$629$	−0.206187	−0.00822121
$630$	0	0
$631$	−3.56482	−0.141913	−0.0709567	−	0.997479i	$-0.522605\pi$
$631$	−3.56482	−0.141913	−0.0709567	+	0.997479i	$0.522605\pi$
$632$	0	0
$633$	0.883595	0.0351197
$634$	0	0
$635$	6.14200	0.243738
$636$	0	0
$637$	0	0
$638$	0	0
$639$	19.7377	0.780810
$640$	0	0
$641$	−15.8322	−0.625335	−0.312667	−	0.949863i	$-0.601223\pi$
$641$	−15.8322	−0.625335	−0.312667	+	0.949863i	$0.601223\pi$
$642$	0	0
$643$	0.767349	0.0302613	0.0151307	−	0.999886i	$-0.495184\pi$
$643$	0.767349	0.0302613	0.0151307	+	0.999886i	$0.495184\pi$
$644$	0	0
$645$	−6.23991	−0.245696
$646$	0	0
$647$	8.17080	0.321227	0.160614	−	0.987017i	$-0.448653\pi$
$647$	8.17080	0.321227	0.160614	+	0.987017i	$0.448653\pi$
$648$	0	0
$649$	7.17219	0.281533
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−46.0200	−1.80090	−0.900451	−	0.434958i	$-0.856763\pi$
$653$	−46.0200	−1.80090	−0.900451	+	0.434958i	$0.856763\pi$
$654$	0	0
$655$	6.81814	0.266407
$656$	0	0
$657$	−23.0922	−0.900912
$658$	0	0
$659$	30.4306	1.18541	0.592704	−	0.805420i	$-0.298060\pi$
$659$	30.4306	1.18541	0.592704	+	0.805420i	$0.298060\pi$
$660$	0	0
$661$	−15.0814	−0.586600	−0.293300	−	0.956020i	$-0.594754\pi$
$661$	−15.0814	−0.586600	−0.293300	+	0.956020i	$0.594754\pi$
$662$	0	0
$663$	−0.618630	−0.0240256
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.66645	0.141966
$668$	0	0
$669$	−9.85086	−0.380856
$670$	0	0
$671$	4.44303	0.171521
$672$	0	0
$673$	−42.9968	−1.65741	−0.828703	−	0.559689i	$-0.810921\pi$
$673$	−42.9968	−1.65741	−0.828703	+	0.559689i	$0.810921\pi$
$674$	0	0
$675$	−15.9145	−0.612549
$676$	0	0
$677$	−17.5715	−0.675327	−0.337663	−	0.941267i	$-0.609637\pi$
$677$	−17.5715	−0.675327	−0.337663	+	0.941267i	$0.609637\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0.246762	0.00945592
$682$	0	0
$683$	42.2637	1.61718	0.808588	−	0.588375i	$-0.200232\pi$
$683$	42.2637	1.61718	0.808588	+	0.588375i	$0.200232\pi$
$684$	0	0
$685$	8.22825	0.314385
$686$	0	0
$687$	−17.5289	−0.668768
$688$	0	0
$689$	12.4977	0.476126
$690$	0	0
$691$	−30.5650	−1.16275	−0.581374	−	0.813637i	$-0.697485\pi$
$691$	−30.5650	−1.16275	−0.581374	+	0.813637i	$0.697485\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.22120	−0.160119
$696$	0	0
$697$	−8.14902	−0.308666
$698$	0	0
$699$	7.28730	0.275631
$700$	0	0
$701$	0.454165	0.0171536	0.00857679	−	0.999963i	$-0.497270\pi$
$701$	0.454165	0.0171536	0.00857679	+	0.999963i	$0.497270\pi$
$702$	0	0
$703$	0.287650	0.0108489
$704$	0	0
$705$	−5.89920	−0.222177
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−46.6947	−1.75366	−0.876828	−	0.480804i	$-0.840345\pi$
$709$	−46.6947	−1.75366	−0.876828	+	0.480804i	$0.840345\pi$
$710$	0	0
$711$	−32.7783	−1.22928
$712$	0	0
$713$	−1.44343	−0.0540567
$714$	0	0
$715$	−1.37312	−0.0513519
$716$	0	0
$717$	−9.09250	−0.339566
$718$	0	0
$719$	−31.5759	−1.17758	−0.588792	−	0.808285i	$-0.700396\pi$
$719$	−31.5759	−1.17758	−0.588792	+	0.808285i	$0.700396\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	1.43105	0.0532212
$724$	0	0
$725$	−28.9136	−1.07382
$726$	0	0
$727$	49.9721	1.85336	0.926681	−	0.375850i	$-0.122649\pi$
$727$	49.9721	1.85336	0.926681	+	0.375850i	$0.122649\pi$
$728$	0	0
$729$	−5.24522	−0.194267
$730$	0	0
$731$	7.54802	0.279174
$732$	0	0
$733$	−3.43860	−0.127008	−0.0635038	−	0.997982i	$-0.520227\pi$
$733$	−3.43860	−0.127008	−0.0635038	+	0.997982i	$0.520227\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9.17681	−0.338032
$738$	0	0
$739$	−51.8202	−1.90624	−0.953118	−	0.302599i	$-0.902146\pi$
$739$	−51.8202	−1.90624	−0.953118	+	0.302599i	$0.902146\pi$
$740$	0	0
$741$	0.863046	0.0317048
$742$	0	0
$743$	−0.546552	−0.0200510	−0.0100255	−	0.999950i	$-0.503191\pi$
$743$	−0.546552	−0.0200510	−0.0100255	+	0.999950i	$0.503191\pi$
$744$	0	0
$745$	8.18951	0.300040
$746$	0	0
$747$	−6.51689	−0.238441
$748$	0	0
$749$	0	0
$750$	0	0
$751$	40.0726	1.46227	0.731135	−	0.682233i	$-0.238991\pi$
$751$	40.0726	1.46227	0.731135	+	0.682233i	$0.238991\pi$
$752$	0	0
$753$	−13.5630	−0.494265
$754$	0	0
$755$	11.1273	0.404965
$756$	0	0
$757$	−17.5545	−0.638029	−0.319015	−	0.947750i	$-0.603352\pi$
$757$	−17.5545	−0.638029	−0.319015	+	0.947750i	$0.603352\pi$
$758$	0	0
$759$	0.449680	0.0163224
$760$	0	0
$761$	−40.2270	−1.45823	−0.729114	−	0.684393i	$-0.760067\pi$
$761$	−40.2270	−1.45823	−0.729114	+	0.684393i	$0.760067\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−1.59121	−0.0575304
$766$	0	0
$767$	−7.39457	−0.267002
$768$	0	0
$769$	−0.653049	−0.0235495	−0.0117748	−	0.999931i	$-0.503748\pi$
$769$	−0.653049	−0.0235495	−0.0117748	+	0.999931i	$0.503748\pi$
$770$	0	0
$771$	−3.74896	−0.135015
$772$	0	0
$773$	50.2290	1.80661	0.903306	−	0.428996i	$-0.141133\pi$
$773$	50.2290	1.80661	0.903306	+	0.428996i	$0.141133\pi$
$774$	0	0
$775$	11.3828	0.408883
$776$	0	0
$777$	0	0
$778$	0	0
$779$	11.3686	0.407324
$780$	0	0
$781$	9.59222	0.343237
$782$	0	0
$783$	25.6402	0.916307
$784$	0	0
$785$	5.54817	0.198023
$786$	0	0
$787$	−27.5490	−0.982014	−0.491007	−	0.871156i	$-0.663371\pi$
$787$	−27.5490	−0.982014	−0.491007	+	0.871156i	$0.663371\pi$
$788$	0	0
$789$	−7.76459	−0.276427
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−4.58079	−0.162668
$794$	0	0
$795$	−5.81861	−0.206365
$796$	0	0
$797$	47.9140	1.69720	0.848601	−	0.529033i	$-0.177445\pi$
$797$	47.9140	1.69720	0.848601	+	0.529033i	$0.177445\pi$
$798$	0	0
$799$	7.13589	0.252450
$800$	0	0
$801$	10.9006	0.385155
$802$	0	0
$803$	−11.2225	−0.396033
$804$	0	0
$805$	0	0
$806$	0	0
$807$	16.9358	0.596168
$808$	0	0
$809$	−47.4786	−1.66926	−0.834629	−	0.550813i	$-0.814318\pi$
$809$	−47.4786	−1.66926	−0.834629	+	0.550813i	$0.814318\pi$
$810$	0	0
$811$	23.1518	0.812968	0.406484	−	0.913658i	$-0.366755\pi$
$811$	23.1518	0.812968	0.406484	+	0.913658i	$0.366755\pi$
$812$	0	0
$813$	2.41225	0.0846012
$814$	0	0
$815$	−13.2108	−0.462753
$816$	0	0
$817$	−10.5302	−0.368405
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−3.15956	−0.110269	−0.0551347	−	0.998479i	$-0.517559\pi$
$821$	−3.15956	−0.110269	−0.0551347	+	0.998479i	$0.517559\pi$
$822$	0	0
$823$	−32.3487	−1.12761	−0.563803	−	0.825909i	$-0.690662\pi$
$823$	−32.3487	−1.12761	−0.563803	+	0.825909i	$0.690662\pi$
$824$	0	0
$825$	−3.54617	−0.123462
$826$	0	0
$827$	44.2730	1.53952	0.769762	−	0.638331i	$-0.220375\pi$
$827$	44.2730	1.53952	0.769762	+	0.638331i	$0.220375\pi$
$828$	0	0
$829$	4.07390	0.141492	0.0707462	−	0.997494i	$-0.477462\pi$
$829$	4.07390	0.141492	0.0707462	+	0.997494i	$0.477462\pi$
$830$	0	0
$831$	1.46632	0.0508661
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−8.27844	−0.286487
$836$	0	0
$837$	−10.0941	−0.348905
$838$	0	0
$839$	−11.8655	−0.409643	−0.204822	−	0.978799i	$-0.565661\pi$
$839$	−11.8655	−0.409643	−0.204822	+	0.978799i	$0.565661\pi$
$840$	0	0
$841$	17.5834	0.606324
$842$	0	0
$843$	−17.1274	−0.589900
$844$	0	0
$845$	−9.94501	−0.342119
$846$	0	0
$847$	0	0
$848$	0	0
$849$	1.60137	0.0549589
$850$	0	0
$851$	−0.154524	−0.00529700
$852$	0	0
$853$	−2.96092	−0.101380	−0.0506900	−	0.998714i	$-0.516142\pi$
$853$	−2.96092	−0.101380	−0.0506900	+	0.998714i	$0.516142\pi$
$854$	0	0
$855$	2.21989	0.0759186
$856$	0	0
$857$	−12.7096	−0.434152	−0.217076	−	0.976155i	$-0.569652\pi$
$857$	−12.7096	−0.434152	−0.217076	+	0.976155i	$0.569652\pi$
$858$	0	0
$859$	−12.0777	−0.412087	−0.206043	−	0.978543i	$-0.566059\pi$
$859$	−12.0777	−0.412087	−0.206043	+	0.978543i	$0.566059\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−51.2481	−1.74451	−0.872253	−	0.489055i	$-0.837342\pi$
$863$	−51.2481	−1.74451	−0.872253	+	0.489055i	$0.837342\pi$
$864$	0	0
$865$	17.6390	0.599745
$866$	0	0
$867$	11.1789	0.379657
$868$	0	0
$869$	−15.9298	−0.540381
$870$	0	0
$871$	9.46135	0.320586
$872$	0	0
$873$	28.1141	0.951519
$874$	0	0
$875$	0	0
$876$	0	0
$877$	42.6177	1.43910	0.719549	−	0.694442i	$-0.244349\pi$
$877$	42.6177	1.43910	0.719549	+	0.694442i	$0.244349\pi$
$878$	0	0
$879$	20.9683	0.707243
$880$	0	0
$881$	15.2551	0.513956	0.256978	−	0.966417i	$-0.417273\pi$
$881$	15.2551	0.513956	0.256978	+	0.966417i	$0.417273\pi$
$882$	0	0
$883$	−36.0622	−1.21359	−0.606795	−	0.794858i	$-0.707545\pi$
$883$	−36.0622	−1.21359	−0.606795	+	0.794858i	$0.707545\pi$
$884$	0	0
$885$	3.44271	0.115726
$886$	0	0
$887$	−45.0387	−1.51225	−0.756126	−	0.654426i	$-0.772911\pi$
$887$	−45.0387	−1.51225	−0.756126	+	0.654426i	$0.772911\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−6.26301	−0.209819
$892$	0	0
$893$	−9.95523	−0.333139
$894$	0	0
$895$	1.06338	0.0355450
$896$	0	0
$897$	−0.463623	−0.0154799
$898$	0	0
$899$	−18.3391	−0.611645
$900$	0	0
$901$	7.03841	0.234483
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−0.629832	−0.0209363
$906$	0	0
$907$	42.3073	1.40479	0.702396	−	0.711787i	$-0.252114\pi$
$907$	42.3073	1.40479	0.702396	+	0.711787i	$0.252114\pi$
$908$	0	0
$909$	−0.779890	−0.0258673
$910$	0	0
$911$	−12.0152	−0.398080	−0.199040	−	0.979991i	$-0.563782\pi$
$911$	−12.0152	−0.398080	−0.199040	+	0.979991i	$0.563782\pi$
$912$	0	0
$913$	−3.16712	−0.104816
$914$	0	0
$915$	2.13269	0.0705046
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1.46634	−0.0483702	−0.0241851	−	0.999707i	$-0.507699\pi$
$919$	−1.46634	−0.0483702	−0.0241851	+	0.999707i	$0.507699\pi$
$920$	0	0
$921$	−8.17370	−0.269333
$922$	0	0
$923$	−9.88964	−0.325521
$924$	0	0
$925$	1.21857	0.0400664
$926$	0	0
$927$	−33.2792	−1.09303
$928$	0	0
$929$	−40.1951	−1.31876	−0.659380	−	0.751810i	$-0.729181\pi$
$929$	−40.1951	−1.31876	−0.659380	+	0.751810i	$0.729181\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−8.39058	−0.274695
$934$	0	0
$935$	−0.773307	−0.0252898
$936$	0	0
$937$	−9.59816	−0.313558	−0.156779	−	0.987634i	$-0.550111\pi$
$937$	−9.59816	−0.313558	−0.156779	+	0.987634i	$0.550111\pi$
$938$	0	0
$939$	14.5360	0.474366
$940$	0	0
$941$	40.8639	1.33213	0.666063	−	0.745895i	$-0.267978\pi$
$941$	40.8639	1.33213	0.666063	+	0.745895i	$0.267978\pi$
$942$	0	0
$943$	−6.10716	−0.198876
$944$	0	0
$945$	0	0
$946$	0	0
$947$	33.6475	1.09340	0.546698	−	0.837330i	$-0.315885\pi$
$947$	33.6475	1.09340	0.546698	+	0.837330i	$0.315885\pi$
$948$	0	0
$949$	11.5704	0.375592
$950$	0	0
$951$	3.12222	0.101245
$952$	0	0
$953$	53.7762	1.74198	0.870991	−	0.491300i	$-0.163478\pi$
$953$	53.7762	1.74198	0.870991	+	0.491300i	$0.163478\pi$
$954$	0	0
$955$	17.3010	0.559847
$956$	0	0
$957$	5.71332	0.184685
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−23.7802	−0.767102
$962$	0	0
$963$	−19.2097	−0.619023
$964$	0	0
$965$	−3.73665	−0.120287
$966$	0	0
$967$	−1.56796	−0.0504220	−0.0252110	−	0.999682i	$-0.508026\pi$
$967$	−1.56796	−0.0504220	−0.0252110	+	0.999682i	$0.508026\pi$
$968$	0	0
$969$	0.486045	0.0156140
$970$	0	0
$971$	25.7135	0.825185	0.412593	−	0.910916i	$-0.364623\pi$
$971$	25.7135	0.825185	0.412593	+	0.910916i	$0.364623\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	3.65612	0.117090
$976$	0	0
$977$	36.6885	1.17377	0.586885	−	0.809670i	$-0.300354\pi$
$977$	36.6885	1.17377	0.586885	+	0.809670i	$0.300354\pi$
$978$	0	0
$979$	5.29755	0.169311
$980$	0	0
$981$	−27.4308	−0.875799
$982$	0	0
$983$	20.8920	0.666350	0.333175	−	0.942865i	$-0.391880\pi$
$983$	20.8920	0.666350	0.333175	+	0.942865i	$0.391880\pi$
$984$	0	0
$985$	12.9000	0.411028
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5.65675	0.179874
$990$	0	0
$991$	−16.0500	−0.509846	−0.254923	−	0.966961i	$-0.582050\pi$
$991$	−16.0500	−0.509846	−0.254923	+	0.966961i	$0.582050\pi$
$992$	0	0
$993$	−20.2101	−0.641349
$994$	0	0
$995$	−15.6489	−0.496103
$996$	0	0
$997$	−8.03821	−0.254573	−0.127286	−	0.991866i	$-0.540627\pi$
$997$	−8.03821	−0.254573	−0.127286	+	0.991866i	$0.540627\pi$
$998$	0	0
$999$	−1.08061	−0.0341891

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3724.2.a.m.1.3		7
7.2	even	3		532.2.i.c.305.5	✓	14
7.4	even	3		532.2.i.c.457.5	yes	14
7.6	odd	2		3724.2.a.n.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
532.2.i.c.305.5	✓	14	7.2	even	3
532.2.i.c.457.5	yes	14	7.4	even	3
3724.2.a.m.1.3		7	1.1	even	1	trivial
3724.2.a.n.1.5		7	7.6	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:	\( N \)	\(=\)	\( 3724 = 2^{2} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3724.a (trivial)

\(f(q)\)	\(=\)	\(q-0.678078 q^{3} +0.873900 q^{5} -2.54021 q^{9} -1.23451 q^{11} +1.27278 q^{13} -0.592573 q^{15} +0.716798 q^{17} -1.00000 q^{19} +0.537193 q^{23} -4.23630 q^{25} +3.75670 q^{27} +6.82520 q^{29} -2.68697 q^{31} +0.837092 q^{33} -0.287650 q^{37} -0.863046 q^{39} -11.3686 q^{41} +10.5302 q^{43} -2.21989 q^{45} +9.95523 q^{47} -0.486045 q^{51} +9.81923 q^{53} -1.07883 q^{55} +0.678078 q^{57} -5.80977 q^{59} -3.59903 q^{61} +1.11229 q^{65} +7.43360 q^{67} -0.364259 q^{69} -7.77009 q^{71} +9.09066 q^{73} +2.87254 q^{75} +12.9038 q^{79} +5.07329 q^{81} +2.56549 q^{83} +0.626410 q^{85} -4.62802 q^{87} -4.29123 q^{89} +1.82198 q^{93} -0.873900 q^{95} -11.0676 q^{97} +3.13590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{5} + 15 q^{9} + 14 q^{11} + 12 q^{15} - 10 q^{17} - 7 q^{19} + 7 q^{23} + 21 q^{25} + 2 q^{29} - 4 q^{31} + 24 q^{33} - 12 q^{37} + 18 q^{39} - 4 q^{41} - 6 q^{45} + 16 q^{47} + 2 q^{51} - 6 q^{53}+ \cdots + 40 q^{99}+O(q^{100}) \) 7 * q - 2 * q^5 + 15 * q^9 + 14 * q^11 + 12 * q^15 - 10 * q^17 - 7 * q^19 + 7 * q^23 + 21 * q^25 + 2 * q^29 - 4 * q^31 + 24 * q^33 - 12 * q^37 + 18 * q^39 - 4 * q^41 - 6 * q^45 + 16 * q^47 + 2 * q^51 - 6 * q^53 - 9 * q^55 + 4 * q^59 + 28 * q^61 - 8 * q^65 + 22 * q^67 - 34 * q^69 + 34 * q^71 - 16 * q^73 + 32 * q^75 + 14 * q^79 + 51 * q^81 - 13 * q^83 - 9 * q^85 - 10 * q^87 + 16 * q^89 - 36 * q^93 + 2 * q^95 + 4 * q^97 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more