LMFDB - Embedded newform 3800.2.d.o.3649.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(3649,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

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

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

chi := DirichletCharacter("3800.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$30.3431527681$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.419904.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 6x^{4} + 9x^{2} + 1 $ x^6 + 6x^4 + 9x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.4
Root			$0.347296i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.o.3649.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.347296i q^{3} +3.41147i q^{7} +2.87939 q^{9} +2.41147 q^{11} +6.29086i q^{13} +2.34730i q^{17} -1.00000 q^{19} -1.18479 q^{21} +2.49020i q^{23} +2.04189i q^{27} +8.17024 q^{29} -2.77332 q^{31} +0.837496i q^{33} +0.977711i q^{37} -2.18479 q^{39} -3.49020 q^{41} -2.75877i q^{43} -6.29086i q^{47} -4.63816 q^{49} -0.815207 q^{51} +2.38919i q^{53} -0.347296i q^{57} -3.67499 q^{59} -12.7442 q^{61} +9.82295i q^{63} -2.41147i q^{67} -0.864837 q^{69} +4.51754 q^{71} +1.81521i q^{73} +8.22668i q^{77} +5.04189 q^{79} +7.92902 q^{81} -8.07192i q^{83} +2.83750i q^{87} +2.94356 q^{89} -21.4611 q^{91} -0.963163i q^{93} +3.09833i q^{97} +6.94356 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{9} - 6 q^{11} - 6 q^{19} + 6 q^{29} - 30 q^{31} - 6 q^{39} - 18 q^{41} + 6 q^{49} - 12 q^{51} - 12 q^{59} - 18 q^{61} + 42 q^{69} - 18 q^{71} + 24 q^{79} - 18 q^{81} - 12 q^{89} - 54 q^{91} + 12 q^{99}+O(q^{100}) $ 6 * q + 6 * q^9 - 6 * q^11 - 6 * q^19 + 6 * q^29 - 30 * q^31 - 6 * q^39 - 18 * q^41 + 6 * q^49 - 12 * q^51 - 12 * q^59 - 18 * q^61 + 42 * q^69 - 18 * q^71 + 24 * q^79 - 18 * q^81 - 12 * q^89 - 54 * q^91 + 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$.

$n$	$401$	$951$	$1901$	$1977$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	0.347296i	0.200512i	0.994962	+	0.100256i	$0.0319661\pi$
$3$	0.347296i	0.200512i	−0.994962	+	0.100256i	$0.968034\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.41147i	1.28942i	0.764429	+	0.644708i	$0.223021\pi$
$7$	3.41147i	1.28942i	−0.764429	+	0.644708i	$0.776979\pi$
$8$	0	0
$9$	2.87939	0.959795
$10$	0	0
$11$	2.41147	0.727087	0.363543	−	0.931577i	$-0.381567\pi$
$11$	2.41147	0.727087	0.363543	+	0.931577i	$0.381567\pi$
$12$	0	0
$13$	6.29086i	1.74477i	0.488819	+	0.872385i	$0.337428\pi$
$13$	6.29086i	1.74477i	−0.488819	+	0.872385i	$0.662572\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.34730i	0.569303i	0.958631	+	0.284651i	$0.0918779\pi$
$17$	2.34730i	0.569303i	−0.958631	+	0.284651i	$0.908122\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.18479	−0.258543
$22$	0	0
$23$	2.49020i	0.519243i	0.965711	+	0.259621i	$0.0835977\pi$
$23$	2.49020i	0.519243i	−0.965711	+	0.259621i	$0.916402\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.04189i	0.392962i
$28$	0	0
$29$	8.17024	1.51718	0.758588	−	0.651570i	$-0.225889\pi$
$29$	8.17024	1.51718	0.758588	+	0.651570i	$0.225889\pi$
$30$	0	0
$31$	−2.77332	−0.498103	−0.249051	−	0.968490i	$-0.580119\pi$
$31$	−2.77332	−0.498103	−0.249051	+	0.968490i	$0.580119\pi$
$32$	0	0
$33$	0.837496i	0.145789i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.977711i	0.160735i	0.996765	+	0.0803674i	$0.0256093\pi$
$37$	0.977711i	0.160735i	−0.996765	+	0.0803674i	$0.974391\pi$
$38$	0	0
$39$	−2.18479	−0.349847
$40$	0	0
$41$	−3.49020	−0.545078	−0.272539	−	0.962145i	$-0.587863\pi$
$41$	−3.49020	−0.545078	−0.272539	+	0.962145i	$0.587863\pi$
$42$	0	0
$43$	− 2.75877i	− 0.420709i	−0.977625	−	0.210354i	$-0.932538\pi$
$43$	− 2.75877i	− 0.420709i	0.977625	−	0.210354i	$-0.0674617\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 6.29086i	− 0.917616i	−0.888535	−	0.458808i	$-0.848277\pi$
$47$	− 6.29086i	− 0.917616i	0.888535	−	0.458808i	$-0.151723\pi$
$48$	0	0
$49$	−4.63816	−0.662594
$50$	0	0
$51$	−0.815207	−0.114152
$52$	0	0
$53$	2.38919i	0.328180i	0.986445	+	0.164090i	$0.0524687\pi$
$53$	2.38919i	0.328180i	−0.986445	+	0.164090i	$0.947531\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 0.347296i	− 0.0460005i
$58$	0	0
$59$	−3.67499	−0.478443	−0.239222	−	0.970965i	$-0.576892\pi$
$59$	−3.67499	−0.478443	−0.239222	+	0.970965i	$0.576892\pi$
$60$	0	0
$61$	−12.7442	−1.63173	−0.815865	−	0.578242i	$-0.803739\pi$
$61$	−12.7442	−1.63173	−0.815865	+	0.578242i	$0.803739\pi$
$62$	0	0
$63$	9.82295i	1.23758i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 2.41147i	− 0.294608i	−0.989091	−	0.147304i	$-0.952940\pi$
$67$	− 2.41147i	− 0.294608i	0.989091	−	0.147304i	$-0.0470596\pi$
$68$	0	0
$69$	−0.864837	−0.104114
$70$	0	0
$71$	4.51754	0.536133	0.268067	−	0.963400i	$-0.413615\pi$
$71$	4.51754	0.536133	0.268067	+	0.963400i	$0.413615\pi$
$72$	0	0
$73$	1.81521i	0.212454i	0.994342	+	0.106227i	$0.0338770\pi$
$73$	1.81521i	0.212454i	−0.994342	+	0.106227i	$0.966123\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.22668i	0.937517i
$78$	0	0
$79$	5.04189	0.567257	0.283628	−	0.958934i	$-0.408462\pi$
$79$	5.04189	0.567257	0.283628	+	0.958934i	$0.408462\pi$
$80$	0	0
$81$	7.92902	0.881002
$82$	0	0
$83$	− 8.07192i	− 0.886008i	−0.896520	−	0.443004i	$-0.853913\pi$
$83$	− 8.07192i	− 0.886008i	0.896520	−	0.443004i	$-0.146087\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.83750i	0.304212i
$88$	0	0
$89$	2.94356	0.312017	0.156009	−	0.987756i	$-0.450137\pi$
$89$	2.94356	0.312017	0.156009	+	0.987756i	$0.450137\pi$
$90$	0	0
$91$	−21.4611	−2.24973
$92$	0	0
$93$	− 0.963163i	− 0.0998754i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	3.09833i	0.314587i	0.987552	+	0.157294i	$0.0502769\pi$
$97$	3.09833i	0.314587i	−0.987552	+	0.157294i	$0.949723\pi$
$98$	0	0
$99$	6.94356	0.697854
$100$	0	0
$101$	−6.02229	−0.599240	−0.299620	−	0.954059i	$-0.596860\pi$
$101$	−6.02229	−0.599240	−0.299620	+	0.954059i	$0.596860\pi$
$102$	0	0
$103$	− 7.25402i	− 0.714760i	−0.933959	−	0.357380i	$-0.883670\pi$
$103$	− 7.25402i	− 0.714760i	0.933959	−	0.357380i	$-0.116330\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 8.94356i	− 0.864607i	−0.901728	−	0.432303i	$-0.857701\pi$
$107$	− 8.94356i	− 0.864607i	0.901728	−	0.432303i	$-0.142299\pi$
$108$	0	0
$109$	9.81521	0.940126	0.470063	−	0.882633i	$-0.344231\pi$
$109$	9.81521	0.940126	0.470063	+	0.882633i	$0.344231\pi$
$110$	0	0
$111$	−0.339556	−0.0322292
$112$	0	0
$113$	− 7.57667i	− 0.712753i	−0.934342	−	0.356376i	$-0.884012\pi$
$113$	− 7.57667i	− 0.712753i	0.934342	−	0.356376i	$-0.115988\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	18.1138i	1.67462i
$118$	0	0
$119$	−8.00774	−0.734068
$120$	0	0
$121$	−5.18479	−0.471345
$122$	0	0
$123$	− 1.21213i	− 0.109294i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 8.33275i	− 0.739412i	−0.929149	−	0.369706i	$-0.879458\pi$
$127$	− 8.33275i	− 0.739412i	0.929149	−	0.369706i	$-0.120542\pi$
$128$	0	0
$129$	0.958111	0.0843570
$130$	0	0
$131$	−13.0915	−1.14381	−0.571906	−	0.820319i	$-0.693796\pi$
$131$	−13.0915	−1.14381	−0.571906	+	0.820319i	$0.693796\pi$
$132$	0	0
$133$	− 3.41147i	− 0.295812i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15.3327i	1.30997i	0.755644	+	0.654983i	$0.227324\pi$
$137$	15.3327i	1.30997i	−0.755644	+	0.654983i	$0.772676\pi$
$138$	0	0
$139$	7.77837	0.659753	0.329876	−	0.944024i	$-0.392993\pi$
$139$	7.77837	0.659753	0.329876	+	0.944024i	$0.392993\pi$
$140$	0	0
$141$	2.18479	0.183993
$142$	0	0
$143$	15.1702i	1.26860i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1.61081i	− 0.132858i
$148$	0	0
$149$	17.1361	1.40384	0.701922	−	0.712254i	$-0.252326\pi$
$149$	17.1361	1.40384	0.701922	+	0.712254i	$0.252326\pi$
$150$	0	0
$151$	−21.4638	−1.74670	−0.873349	−	0.487094i	$-0.838057\pi$
$151$	−21.4638	−1.74670	−0.873349	+	0.487094i	$0.838057\pi$
$152$	0	0
$153$	6.75877i	0.546414i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	19.5253i	1.55829i	0.626846	+	0.779144i	$0.284346\pi$
$157$	19.5253i	1.55829i	−0.626846	+	0.779144i	$0.715654\pi$
$158$	0	0
$159$	−0.829755	−0.0658039
$160$	0	0
$161$	−8.49525	−0.669520
$162$	0	0
$163$	− 3.59627i	− 0.281681i	−0.990032	−	0.140841i	$-0.955019\pi$
$163$	− 3.59627i	− 0.281681i	0.990032	−	0.140841i	$-0.0449805\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.1361i	1.01650i	0.861209	+	0.508251i	$0.169708\pi$
$167$	13.1361i	1.01650i	−0.861209	+	0.508251i	$0.830292\pi$
$168$	0	0
$169$	−26.5749	−2.04422
$170$	0	0
$171$	−2.87939	−0.220192
$172$	0	0
$173$	16.4757i	1.25262i	0.779574	+	0.626310i	$0.215436\pi$
$173$	16.4757i	1.25262i	−0.779574	+	0.626310i	$0.784564\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 1.27631i	− 0.0959334i
$178$	0	0
$179$	23.1438	1.72985	0.864926	−	0.501900i	$-0.167365\pi$
$179$	23.1438	1.72985	0.864926	+	0.501900i	$0.167365\pi$
$180$	0	0
$181$	24.1634	1.79605	0.898027	−	0.439940i	$-0.145000\pi$
$181$	24.1634	1.79605	0.898027	+	0.439940i	$0.145000\pi$
$182$	0	0
$183$	− 4.42602i	− 0.327181i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5.66044i	0.413933i
$188$	0	0
$189$	−6.96585	−0.506691
$190$	0	0
$191$	2.62361	0.189838	0.0949188	−	0.995485i	$-0.469741\pi$
$191$	2.62361	0.189838	0.0949188	+	0.995485i	$0.469741\pi$
$192$	0	0
$193$	− 12.5544i	− 0.903684i	−0.892098	−	0.451842i	$-0.850767\pi$
$193$	− 12.5544i	− 0.903684i	0.892098	−	0.451842i	$-0.149233\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.9094i	1.20475i	0.798215	+	0.602373i	$0.205778\pi$
$197$	16.9094i	1.20475i	−0.798215	+	0.602373i	$0.794222\pi$
$198$	0	0
$199$	−19.1607	−1.35827	−0.679135	−	0.734014i	$-0.737645\pi$
$199$	−19.1607	−1.35827	−0.679135	+	0.734014i	$0.737645\pi$
$200$	0	0
$201$	0.837496	0.0590724
$202$	0	0
$203$	27.8726i	1.95627i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	7.17024i	0.498366i
$208$	0	0
$209$	−2.41147	−0.166805
$210$	0	0
$211$	−13.4534	−0.926168	−0.463084	−	0.886314i	$-0.653257\pi$
$211$	−13.4534	−0.926168	−0.463084	+	0.886314i	$0.653257\pi$
$212$	0	0
$213$	1.56893i	0.107501i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 9.46110i	− 0.642262i
$218$	0	0
$219$	−0.630415	−0.0425995
$220$	0	0
$221$	−14.7665	−0.993303
$222$	0	0
$223$	28.9864i	1.94107i	0.240956	+	0.970536i	$0.422539\pi$
$223$	28.9864i	1.94107i	−0.240956	+	0.970536i	$0.577461\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 10.6091i	− 0.704148i	−0.935972	−	0.352074i	$-0.885477\pi$
$227$	− 10.6091i	− 0.704148i	0.935972	−	0.352074i	$-0.114523\pi$
$228$	0	0
$229$	12.8375	0.848326	0.424163	−	0.905586i	$-0.360568\pi$
$229$	12.8375	0.848326	0.424163	+	0.905586i	$0.360568\pi$
$230$	0	0
$231$	−2.85710	−0.187983
$232$	0	0
$233$	− 10.4953i	− 0.687567i	−0.939049	−	0.343783i	$-0.888291\pi$
$233$	− 10.4953i	− 0.687567i	0.939049	−	0.343783i	$-0.111709\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.75103i	0.113742i
$238$	0	0
$239$	−4.07873	−0.263831	−0.131915	−	0.991261i	$-0.542113\pi$
$239$	−4.07873	−0.263831	−0.131915	+	0.991261i	$0.542113\pi$
$240$	0	0
$241$	−14.6382	−0.942927	−0.471463	−	0.881886i	$-0.656274\pi$
$241$	−14.6382	−0.942927	−0.471463	+	0.881886i	$0.656274\pi$
$242$	0	0
$243$	8.87939i	0.569613i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 6.29086i	− 0.400278i
$248$	0	0
$249$	2.80335	0.177655
$250$	0	0
$251$	−12.0692	−0.761803	−0.380902	−	0.924616i	$-0.624386\pi$
$251$	−12.0692	−0.761803	−0.380902	+	0.924616i	$0.624386\pi$
$252$	0	0
$253$	6.00505i	0.377534i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1.62361i	− 0.101278i	−0.998717	−	0.0506389i	$-0.983874\pi$
$257$	− 1.62361i	− 0.101278i	0.998717	−	0.0506389i	$-0.0161258\pi$
$258$	0	0
$259$	−3.33544	−0.207254
$260$	0	0
$261$	23.5253	1.45618
$262$	0	0
$263$	28.2618i	1.74269i	0.490666	+	0.871347i	$0.336753\pi$
$263$	28.2618i	1.74269i	−0.490666	+	0.871347i	$0.663247\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.02229i	0.0625631i
$268$	0	0
$269$	10.1138	0.616650	0.308325	−	0.951281i	$-0.400232\pi$
$269$	10.1138	0.616650	0.308325	+	0.951281i	$0.400232\pi$
$270$	0	0
$271$	−22.9486	−1.39403	−0.697015	−	0.717057i	$-0.745489\pi$
$271$	−22.9486	−1.39403	−0.697015	+	0.717057i	$0.745489\pi$
$272$	0	0
$273$	− 7.45336i	− 0.451098i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 14.1634i	− 0.850998i	−0.904959	−	0.425499i	$-0.860099\pi$
$277$	− 14.1634i	− 0.850998i	0.904959	−	0.425499i	$-0.139901\pi$
$278$	0	0
$279$	−7.98545	−0.478077
$280$	0	0
$281$	0.822948	0.0490930	0.0245465	−	0.999699i	$-0.492186\pi$
$281$	0.822948	0.0490930	0.0245465	+	0.999699i	$0.492186\pi$
$282$	0	0
$283$	4.83481i	0.287399i	0.989621	+	0.143700i	$0.0459000\pi$
$283$	4.83481i	0.287399i	−0.989621	+	0.143700i	$0.954100\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 11.9067i	− 0.702832i
$288$	0	0
$289$	11.4902	0.675894
$290$	0	0
$291$	−1.07604	−0.0630784
$292$	0	0
$293$	− 17.5699i	− 1.02644i	−0.858256	−	0.513221i	$-0.828452\pi$
$293$	− 17.5699i	− 1.02644i	0.858256	−	0.513221i	$-0.171548\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.92396i	0.285717i
$298$	0	0
$299$	−15.6655	−0.905959
$300$	0	0
$301$	9.41147	0.542468
$302$	0	0
$303$	− 2.09152i	− 0.120155i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 17.9632i	− 1.02521i	−0.858624	−	0.512606i	$-0.828680\pi$
$307$	− 17.9632i	− 1.02521i	0.858624	−	0.512606i	$-0.171320\pi$
$308$	0	0
$309$	2.51930	0.143318
$310$	0	0
$311$	27.4543	1.55679	0.778395	−	0.627775i	$-0.216034\pi$
$311$	27.4543	1.55679	0.778395	+	0.627775i	$0.216034\pi$
$312$	0	0
$313$	2.90941i	0.164450i	0.996614	+	0.0822249i	$0.0262026\pi$
$313$	2.90941i	0.164450i	−0.996614	+	0.0822249i	$0.973797\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 23.1070i	− 1.29782i	−0.760866	−	0.648909i	$-0.775225\pi$
$317$	− 23.1070i	− 1.29782i	0.760866	−	0.648909i	$-0.224775\pi$
$318$	0	0
$319$	19.7023	1.10312
$320$	0	0
$321$	3.10607	0.173364
$322$	0	0
$323$	− 2.34730i	− 0.130607i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3.40879i	0.188506i
$328$	0	0
$329$	21.4611	1.18319
$330$	0	0
$331$	2.89218	0.158969	0.0794843	−	0.996836i	$-0.474673\pi$
$331$	2.89218	0.158969	0.0794843	+	0.996836i	$0.474673\pi$
$332$	0	0
$333$	2.81521i	0.154272i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 28.3901i	− 1.54651i	−0.634096	−	0.773254i	$-0.718628\pi$
$337$	− 28.3901i	− 1.54651i	0.634096	−	0.773254i	$-0.281372\pi$
$338$	0	0
$339$	2.63135	0.142915
$340$	0	0
$341$	−6.68779	−0.362164
$342$	0	0
$343$	8.05737i	0.435057i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	23.4175i	1.25712i	0.777763	+	0.628558i	$0.216354\pi$
$347$	23.4175i	1.25712i	−0.777763	+	0.628558i	$0.783646\pi$
$348$	0	0
$349$	5.48751	0.293740	0.146870	−	0.989156i	$-0.453080\pi$
$349$	5.48751	0.293740	0.146870	+	0.989156i	$0.453080\pi$
$350$	0	0
$351$	−12.8452	−0.685628
$352$	0	0
$353$	22.8530i	1.21634i	0.793806	+	0.608171i	$0.208096\pi$
$353$	22.8530i	1.21634i	−0.793806	+	0.608171i	$0.791904\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 2.78106i	− 0.147189i
$358$	0	0
$359$	−20.6604	−1.09042	−0.545208	−	0.838301i	$-0.683549\pi$
$359$	−20.6604	−1.09042	−0.545208	+	0.838301i	$0.683549\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	− 1.80066i	− 0.0945101i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 6.29767i	− 0.328736i	−0.986399	−	0.164368i	$-0.947442\pi$
$367$	− 6.29767i	− 0.328736i	0.986399	−	0.164368i	$-0.0525584\pi$
$368$	0	0
$369$	−10.0496	−0.523163
$370$	0	0
$371$	−8.15064	−0.423160
$372$	0	0
$373$	27.0401i	1.40009i	0.714101	+	0.700043i	$0.246836\pi$
$373$	27.0401i	1.40009i	−0.714101	+	0.700043i	$0.753164\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	51.3979i	2.64712i
$378$	0	0
$379$	−11.8375	−0.608051	−0.304026	−	0.952664i	$-0.598331\pi$
$379$	−11.8375	−0.608051	−0.304026	+	0.952664i	$0.598331\pi$
$380$	0	0
$381$	2.89393	0.148261
$382$	0	0
$383$	− 29.4252i	− 1.50356i	−0.659415	−	0.751779i	$-0.729196\pi$
$383$	− 29.4252i	− 1.50356i	0.659415	−	0.751779i	$-0.270804\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 7.94356i	− 0.403794i
$388$	0	0
$389$	6.75877	0.342683	0.171342	−	0.985212i	$-0.445190\pi$
$389$	6.75877	0.342683	0.171342	+	0.985212i	$0.445190\pi$
$390$	0	0
$391$	−5.84524	−0.295606
$392$	0	0
$393$	− 4.54664i	− 0.229347i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	20.3158i	1.01962i	0.860286	+	0.509811i	$0.170285\pi$
$397$	20.3158i	1.01962i	−0.860286	+	0.509811i	$0.829715\pi$
$398$	0	0
$399$	1.18479	0.0593138
$400$	0	0
$401$	−5.57903	−0.278603	−0.139302	−	0.990250i	$-0.544486\pi$
$401$	−5.57903	−0.278603	−0.139302	+	0.990250i	$0.544486\pi$
$402$	0	0
$403$	− 17.4466i	− 0.869075i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.35773i	0.116868i
$408$	0	0
$409$	37.3387	1.84628	0.923141	−	0.384462i	$-0.125613\pi$
$409$	37.3387	1.84628	0.923141	+	0.384462i	$0.125613\pi$
$410$	0	0
$411$	−5.32501	−0.262663
$412$	0	0
$413$	− 12.5371i	− 0.616912i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2.70140i	0.132288i
$418$	0	0
$419$	−4.80840	−0.234906	−0.117453	−	0.993078i	$-0.537473\pi$
$419$	−4.80840	−0.234906	−0.117453	+	0.993078i	$0.537473\pi$
$420$	0	0
$421$	27.0009	1.31594	0.657972	−	0.753042i	$-0.271414\pi$
$421$	27.0009	1.31594	0.657972	+	0.753042i	$0.271414\pi$
$422$	0	0
$423$	− 18.1138i	− 0.880723i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 43.4766i	− 2.10398i
$428$	0	0
$429$	−5.26857	−0.254369
$430$	0	0
$431$	36.2249	1.74489	0.872447	−	0.488709i	$-0.162532\pi$
$431$	36.2249	1.74489	0.872447	+	0.488709i	$0.162532\pi$
$432$	0	0
$433$	− 31.6382i	− 1.52043i	−0.649670	−	0.760216i	$-0.725093\pi$
$433$	− 31.6382i	− 1.52043i	0.649670	−	0.760216i	$-0.274907\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2.49020i	− 0.119122i
$438$	0	0
$439$	−9.88444	−0.471758	−0.235879	−	0.971782i	$-0.575797\pi$
$439$	−9.88444	−0.471758	−0.235879	+	0.971782i	$0.575797\pi$
$440$	0	0
$441$	−13.3550	−0.635954
$442$	0	0
$443$	29.3191i	1.39299i	0.717560	+	0.696497i	$0.245259\pi$
$443$	29.3191i	1.39299i	−0.717560	+	0.696497i	$0.754741\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.95130i	0.281487i
$448$	0	0
$449$	−2.86247	−0.135088	−0.0675442	−	0.997716i	$-0.521516\pi$
$449$	−2.86247	−0.135088	−0.0675442	+	0.997716i	$0.521516\pi$
$450$	0	0
$451$	−8.41653	−0.396319
$452$	0	0
$453$	− 7.45430i	− 0.350233i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	25.6664i	1.20062i	0.799766	+	0.600312i	$0.204957\pi$
$457$	25.6664i	1.20062i	−0.799766	+	0.600312i	$0.795043\pi$
$458$	0	0
$459$	−4.79292	−0.223714
$460$	0	0
$461$	5.77332	0.268890	0.134445	−	0.990921i	$-0.457075\pi$
$461$	5.77332	0.268890	0.134445	+	0.990921i	$0.457075\pi$
$462$	0	0
$463$	− 31.1097i	− 1.44579i	−0.690958	−	0.722895i	$-0.742811\pi$
$463$	− 31.1097i	− 1.44579i	0.690958	−	0.722895i	$-0.257189\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	37.8408i	1.75106i	0.483161	+	0.875532i	$0.339489\pi$
$467$	37.8408i	1.75106i	−0.483161	+	0.875532i	$0.660511\pi$
$468$	0	0
$469$	8.22668	0.379873
$470$	0	0
$471$	−6.78106	−0.312455
$472$	0	0
$473$	− 6.65270i	− 0.305892i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.87939i	0.314985i
$478$	0	0
$479$	−5.10338	−0.233179	−0.116590	−	0.993180i	$-0.537196\pi$
$479$	−5.10338	−0.233179	−0.116590	+	0.993180i	$0.537196\pi$
$480$	0	0
$481$	−6.15064	−0.280445
$482$	0	0
$483$	− 2.95037i	− 0.134246i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 18.8503i	− 0.854188i	−0.904207	−	0.427094i	$-0.859537\pi$
$487$	− 18.8503i	− 0.854188i	0.904207	−	0.427094i	$-0.140463\pi$
$488$	0	0
$489$	1.24897	0.0564804
$490$	0	0
$491$	0.593578	0.0267878	0.0133939	−	0.999910i	$-0.495736\pi$
$491$	0.593578	0.0267878	0.0133939	+	0.999910i	$0.495736\pi$
$492$	0	0
$493$	19.1780i	0.863733i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	15.4115i	0.691299i
$498$	0	0
$499$	29.7965	1.33388	0.666938	−	0.745113i	$-0.267605\pi$
$499$	29.7965	1.33388	0.666938	+	0.745113i	$0.267605\pi$
$500$	0	0
$501$	−4.56212	−0.203820
$502$	0	0
$503$	− 21.8357i	− 0.973608i	−0.873511	−	0.486804i	$-0.838163\pi$
$503$	− 21.8357i	− 0.973608i	0.873511	−	0.486804i	$-0.161837\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 9.22937i	− 0.409891i
$508$	0	0
$509$	−12.4861	−0.553436	−0.276718	−	0.960951i	$-0.589247\pi$
$509$	−12.4861	−0.553436	−0.276718	+	0.960951i	$0.589247\pi$
$510$	0	0
$511$	−6.19253	−0.273942
$512$	0	0
$513$	− 2.04189i	− 0.0901516i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 15.1702i	− 0.667187i
$518$	0	0
$519$	−5.72193	−0.251165
$520$	0	0
$521$	3.37908	0.148040	0.0740201	−	0.997257i	$-0.476417\pi$
$521$	3.37908	0.148040	0.0740201	+	0.997257i	$0.476417\pi$
$522$	0	0
$523$	− 15.7419i	− 0.688343i	−0.938907	−	0.344172i	$-0.888160\pi$
$523$	− 15.7419i	− 0.688343i	0.938907	−	0.344172i	$-0.111840\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 6.50980i	− 0.283571i
$528$	0	0
$529$	16.7989	0.730387
$530$	0	0
$531$	−10.5817	−0.459207
$532$	0	0
$533$	− 21.9564i	− 0.951035i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	8.03777i	0.346855i
$538$	0	0
$539$	−11.1848	−0.481763
$540$	0	0
$541$	−1.31820	−0.0566739	−0.0283369	−	0.999598i	$-0.509021\pi$
$541$	−1.31820	−0.0566739	−0.0283369	+	0.999598i	$0.509021\pi$
$542$	0	0
$543$	8.39187i	0.360130i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 9.68954i	− 0.414295i	−0.978310	−	0.207147i	$-0.933582\pi$
$547$	− 9.68954i	− 0.414295i	0.978310	−	0.207147i	$-0.0664180\pi$
$548$	0	0
$549$	−36.6955	−1.56613
$550$	0	0
$551$	−8.17024	−0.348064
$552$	0	0
$553$	17.2003i	0.731430i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 14.4757i	− 0.613353i	−0.951814	−	0.306677i	$-0.900783\pi$
$557$	− 14.4757i	− 0.613353i	0.951814	−	0.306677i	$-0.0992170\pi$
$558$	0	0
$559$	17.3550	0.734040
$560$	0	0
$561$	−1.96585	−0.0829983
$562$	0	0
$563$	− 25.8898i	− 1.09113i	−0.838070	−	0.545563i	$-0.816316\pi$
$563$	− 25.8898i	− 1.09113i	0.838070	−	0.545563i	$-0.183684\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	27.0496i	1.13598i
$568$	0	0
$569$	−34.8384	−1.46050	−0.730251	−	0.683178i	$-0.760597\pi$
$569$	−34.8384	−1.46050	−0.730251	+	0.683178i	$0.760597\pi$
$570$	0	0
$571$	24.8557	1.04018	0.520089	−	0.854112i	$-0.325899\pi$
$571$	24.8557	1.04018	0.520089	+	0.854112i	$0.325899\pi$
$572$	0	0
$573$	0.911169i	0.0380646i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	36.4620i	1.51793i	0.651129	+	0.758967i	$0.274296\pi$
$577$	36.4620i	1.51793i	−0.651129	+	0.758967i	$0.725704\pi$
$578$	0	0
$579$	4.36009	0.181199
$580$	0	0
$581$	27.5371	1.14243
$582$	0	0
$583$	5.76146i	0.238615i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 30.9249i	− 1.27641i	−0.769868	−	0.638204i	$-0.779678\pi$
$587$	− 30.9249i	− 1.27641i	0.769868	−	0.638204i	$-0.220322\pi$
$588$	0	0
$589$	2.77332	0.114273
$590$	0	0
$591$	−5.87258	−0.241566
$592$	0	0
$593$	− 30.2909i	− 1.24390i	−0.783058	−	0.621948i	$-0.786341\pi$
$593$	− 30.2909i	− 1.24390i	0.783058	−	0.621948i	$-0.213659\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 6.65446i	− 0.272349i
$598$	0	0
$599$	−25.3236	−1.03469	−0.517347	−	0.855776i	$-0.673080\pi$
$599$	−25.3236	−1.03469	−0.517347	+	0.855776i	$0.673080\pi$
$600$	0	0
$601$	−34.2894	−1.39869	−0.699347	−	0.714782i	$-0.746526\pi$
$601$	−34.2894	−1.39869	−0.699347	+	0.714782i	$0.746526\pi$
$602$	0	0
$603$	− 6.94356i	− 0.282764i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 42.2550i	− 1.71508i	−0.514421	−	0.857538i	$-0.671993\pi$
$607$	− 42.2550i	− 1.71508i	0.514421	−	0.857538i	$-0.328007\pi$
$608$	0	0
$609$	−9.68004	−0.392255
$610$	0	0
$611$	39.5749	1.60103
$612$	0	0
$613$	23.6759i	0.956262i	0.878289	+	0.478131i	$0.158686\pi$
$613$	23.6759i	0.956262i	−0.878289	+	0.478131i	$0.841314\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	21.8425i	0.879348i	0.898157	+	0.439674i	$0.144906\pi$
$617$	21.8425i	0.879348i	−0.898157	+	0.439674i	$0.855094\pi$
$618$	0	0
$619$	−38.2841	−1.53877	−0.769383	−	0.638788i	$-0.779436\pi$
$619$	−38.2841	−1.53877	−0.769383	+	0.638788i	$0.779436\pi$
$620$	0	0
$621$	−5.08471	−0.204042
$622$	0	0
$623$	10.0419i	0.402320i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 0.837496i	− 0.0334464i
$628$	0	0
$629$	−2.29498	−0.0915068
$630$	0	0
$631$	7.28817	0.290138	0.145069	−	0.989422i	$-0.453660\pi$
$631$	7.28817	0.290138	0.145069	+	0.989422i	$0.453660\pi$
$632$	0	0
$633$	− 4.67230i	− 0.185707i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 29.1780i	− 1.15607i
$638$	0	0
$639$	13.0077	0.514578
$640$	0	0
$641$	14.2026	0.560970	0.280485	−	0.959858i	$-0.409505\pi$
$641$	14.2026	0.560970	0.280485	+	0.959858i	$0.409505\pi$
$642$	0	0
$643$	0.867526i	0.0342119i	0.999854	+	0.0171059i	$0.00544525\pi$
$643$	0.867526i	0.0342119i	−0.999854	+	0.0171059i	$0.994555\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 46.5503i	− 1.83008i	−0.403364	−	0.915040i	$-0.632159\pi$
$647$	− 46.5503i	− 1.83008i	0.403364	−	0.915040i	$-0.367841\pi$
$648$	0	0
$649$	−8.86215	−0.347870
$650$	0	0
$651$	3.28581	0.128781
$652$	0	0
$653$	27.9050i	1.09201i	0.837783	+	0.546003i	$0.183851\pi$
$653$	27.9050i	1.09201i	−0.837783	+	0.546003i	$0.816149\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	5.22668i	0.203912i
$658$	0	0
$659$	−49.5340	−1.92957	−0.964784	−	0.263042i	$-0.915274\pi$
$659$	−49.5340	−1.92957	−0.964784	+	0.263042i	$0.915274\pi$
$660$	0	0
$661$	40.0164	1.55646	0.778229	−	0.627980i	$-0.216118\pi$
$661$	40.0164	1.55646	0.778229	+	0.627980i	$0.216118\pi$
$662$	0	0
$663$	− 5.12836i	− 0.199169i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	20.3455i	0.787783i
$668$	0	0
$669$	−10.0669	−0.389208
$670$	0	0
$671$	−30.7324	−1.18641
$672$	0	0
$673$	− 26.5449i	− 1.02323i	−0.859215	−	0.511615i	$-0.829047\pi$
$673$	− 26.5449i	− 1.02323i	0.859215	−	0.511615i	$-0.170953\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 22.5398i	− 0.866276i	−0.901328	−	0.433138i	$-0.857406\pi$
$677$	− 22.5398i	− 0.866276i	0.901328	−	0.433138i	$-0.142594\pi$
$678$	0	0
$679$	−10.5699	−0.405634
$680$	0	0
$681$	3.68449	0.141190
$682$	0	0
$683$	44.2154i	1.69186i	0.533297	+	0.845928i	$0.320953\pi$
$683$	44.2154i	1.69186i	−0.533297	+	0.845928i	$0.679047\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	4.45842i	0.170099i
$688$	0	0
$689$	−15.0300	−0.572599
$690$	0	0
$691$	31.0743	1.18212	0.591061	−	0.806627i	$-0.298709\pi$
$691$	31.0743	1.18212	0.591061	+	0.806627i	$0.298709\pi$
$692$	0	0
$693$	23.6878i	0.899825i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 8.19253i	− 0.310314i
$698$	0	0
$699$	3.64496	0.137865
$700$	0	0
$701$	44.8185	1.69277	0.846386	−	0.532570i	$-0.178774\pi$
$701$	44.8185	1.69277	0.846386	+	0.532570i	$0.178774\pi$
$702$	0	0
$703$	− 0.977711i	− 0.0368751i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 20.5449i	− 0.772670i
$708$	0	0
$709$	40.1334	1.50724	0.753621	−	0.657309i	$-0.228305\pi$
$709$	40.1334	1.50724	0.753621	+	0.657309i	$0.228305\pi$
$710$	0	0
$711$	14.5175	0.544450
$712$	0	0
$713$	− 6.90612i	− 0.258636i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 1.41653i	− 0.0529012i
$718$	0	0
$719$	−20.8571	−0.777838	−0.388919	−	0.921272i	$-0.627151\pi$
$719$	−20.8571	−0.777838	−0.388919	+	0.921272i	$0.627151\pi$
$720$	0	0
$721$	24.7469	0.921623
$722$	0	0
$723$	− 5.08378i	− 0.189068i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	11.2540i	0.417389i	0.977981	+	0.208694i	$0.0669214\pi$
$727$	11.2540i	0.417389i	−0.977981	+	0.208694i	$0.933079\pi$
$728$	0	0
$729$	20.7033	0.766788
$730$	0	0
$731$	6.47565	0.239511
$732$	0	0
$733$	− 11.3952i	− 0.420890i	−0.977606	−	0.210445i	$-0.932509\pi$
$733$	− 11.3952i	− 0.420890i	0.977606	−	0.210445i	$-0.0674913\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 5.81521i	− 0.214206i
$738$	0	0
$739$	45.7015	1.68116	0.840579	−	0.541690i	$-0.182215\pi$
$739$	45.7015	1.68116	0.840579	+	0.541690i	$0.182215\pi$
$740$	0	0
$741$	2.18479	0.0802604
$742$	0	0
$743$	3.82470i	0.140315i	0.997536	+	0.0701574i	$0.0223501\pi$
$743$	3.82470i	0.140315i	−0.997536	+	0.0701574i	$0.977650\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 23.2422i	− 0.850386i
$748$	0	0
$749$	30.5107	1.11484
$750$	0	0
$751$	26.5550	0.969005	0.484503	−	0.874790i	$-0.339001\pi$
$751$	26.5550	0.969005	0.484503	+	0.874790i	$0.339001\pi$
$752$	0	0
$753$	− 4.19160i	− 0.152750i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 4.64765i	− 0.168922i	−0.996427	−	0.0844609i	$-0.973083\pi$
$757$	− 4.64765i	− 0.168922i	0.996427	−	0.0844609i	$-0.0269168\pi$
$758$	0	0
$759$	−2.08553	−0.0757000
$760$	0	0
$761$	−26.0327	−0.943685	−0.471843	−	0.881683i	$-0.656411\pi$
$761$	−26.0327	−0.943685	−0.471843	+	0.881683i	$0.656411\pi$
$762$	0	0
$763$	33.4843i	1.21221i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 23.1189i	− 0.834774i
$768$	0	0
$769$	−3.10876	−0.112105	−0.0560523	−	0.998428i	$-0.517851\pi$
$769$	−3.10876	−0.112105	−0.0560523	+	0.998428i	$0.517851\pi$
$770$	0	0
$771$	0.563873	0.0203074
$772$	0	0
$773$	− 19.4516i	− 0.699626i	−0.936820	−	0.349813i	$-0.886245\pi$
$773$	− 19.4516i	− 0.699626i	0.936820	−	0.349813i	$-0.113755\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 1.15839i	− 0.0415568i
$778$	0	0
$779$	3.49020	0.125049
$780$	0	0
$781$	10.8939	0.389816
$782$	0	0
$783$	16.6827i	0.596192i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	14.0496i	0.500815i	0.968141	+	0.250408i	$0.0805646\pi$
$787$	14.0496i	0.500815i	−0.968141	+	0.250408i	$0.919435\pi$
$788$	0	0
$789$	−9.81521	−0.349431
$790$	0	0
$791$	25.8476	0.919035
$792$	0	0
$793$	− 80.1721i	− 2.84700i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 2.57810i	− 0.0913209i	−0.998957	−	0.0456604i	$-0.985461\pi$
$797$	− 2.57810i	− 0.0913209i	0.998957	−	0.0456604i	$-0.0145392\pi$
$798$	0	0
$799$	14.7665	0.522402
$800$	0	0
$801$	8.47565	0.299472
$802$	0	0
$803$	4.37733i	0.154472i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3.51249i	0.123645i
$808$	0	0
$809$	−19.8220	−0.696905	−0.348452	−	0.937326i	$-0.613293\pi$
$809$	−19.8220	−0.696905	−0.348452	+	0.937326i	$0.613293\pi$
$810$	0	0
$811$	22.6563	0.795571	0.397786	−	0.917478i	$-0.369779\pi$
$811$	22.6563	0.795571	0.397786	+	0.917478i	$0.369779\pi$
$812$	0	0
$813$	− 7.96997i	− 0.279519i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.75877i	0.0965172i
$818$	0	0
$819$	−61.7948	−2.15928
$820$	0	0
$821$	−23.0570	−0.804696	−0.402348	−	0.915487i	$-0.631806\pi$
$821$	−23.0570	−0.804696	−0.402348	+	0.915487i	$0.631806\pi$
$822$	0	0
$823$	− 44.6563i	− 1.55662i	−0.627879	−	0.778311i	$-0.716077\pi$
$823$	− 44.6563i	− 1.55662i	0.627879	−	0.778311i	$-0.283923\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	52.4306i	1.82319i	0.411090	+	0.911595i	$0.365148\pi$
$827$	52.4306i	1.82319i	−0.411090	+	0.911595i	$0.634852\pi$
$828$	0	0
$829$	40.2412	1.39764	0.698818	−	0.715300i	$-0.253710\pi$
$829$	40.2412	1.39764	0.698818	+	0.715300i	$0.253710\pi$
$830$	0	0
$831$	4.91891	0.170635
$832$	0	0
$833$	− 10.8871i	− 0.377217i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 5.66281i	− 0.195735i
$838$	0	0
$839$	52.5185	1.81314	0.906570	−	0.422056i	$-0.138692\pi$
$839$	52.5185	1.81314	0.906570	+	0.422056i	$0.138692\pi$
$840$	0	0
$841$	37.7529	1.30182
$842$	0	0
$843$	0.285807i	0.00984371i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 17.6878i	− 0.607760i
$848$	0	0
$849$	−1.67911	−0.0576269
$850$	0	0
$851$	−2.43470	−0.0834603
$852$	0	0
$853$	20.2094i	0.691958i	0.938242	+	0.345979i	$0.112453\pi$
$853$	20.2094i	0.691958i	−0.938242	+	0.345979i	$0.887547\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 14.9522i	− 0.510759i	−0.966841	−	0.255379i	$-0.917800\pi$
$857$	− 14.9522i	− 0.510759i	0.966841	−	0.255379i	$-0.0822004\pi$
$858$	0	0
$859$	34.3286	1.17128	0.585639	−	0.810572i	$-0.300844\pi$
$859$	34.3286	1.17128	0.585639	+	0.810572i	$0.300844\pi$
$860$	0	0
$861$	4.13516	0.140926
$862$	0	0
$863$	− 27.0392i	− 0.920425i	−0.887809	−	0.460213i	$-0.847773\pi$
$863$	− 27.0392i	− 0.920425i	0.887809	−	0.460213i	$-0.152227\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.99050i	0.135525i
$868$	0	0
$869$	12.1584	0.412445
$870$	0	0
$871$	15.1702	0.514024
$872$	0	0
$873$	8.92127i	0.301939i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 17.2216i	− 0.581533i	−0.956794	−	0.290767i	$-0.906090\pi$
$877$	− 17.2216i	− 0.581533i	0.956794	−	0.290767i	$-0.0939103\pi$
$878$	0	0
$879$	6.10195	0.205814
$880$	0	0
$881$	51.2312	1.72602	0.863012	−	0.505183i	$-0.168575\pi$
$881$	51.2312	1.72602	0.863012	+	0.505183i	$0.168575\pi$
$882$	0	0
$883$	− 27.8939i	− 0.938706i	−0.883011	−	0.469353i	$-0.844487\pi$
$883$	− 27.8939i	− 0.938706i	0.883011	−	0.469353i	$-0.155513\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32.5749i	1.09376i	0.837211	+	0.546879i	$0.184184\pi$
$887$	32.5749i	1.09376i	−0.837211	+	0.546879i	$0.815816\pi$
$888$	0	0
$889$	28.4270	0.953409
$890$	0	0
$891$	19.1206	0.640565
$892$	0	0
$893$	6.29086i	0.210516i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 5.44057i	− 0.181655i
$898$	0	0
$899$	−22.6587	−0.755710
$900$	0	0
$901$	−5.60813	−0.186834
$902$	0	0
$903$	3.26857i	0.108771i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 23.6673i	− 0.785858i	−0.919569	−	0.392929i	$-0.871462\pi$
$907$	− 23.6673i	− 0.785858i	0.919569	−	0.392929i	$-0.128538\pi$
$908$	0	0
$909$	−17.3405	−0.575148
$910$	0	0
$911$	11.4233	0.378472	0.189236	−	0.981932i	$-0.439399\pi$
$911$	11.4233	0.378472	0.189236	+	0.981932i	$0.439399\pi$
$912$	0	0
$913$	− 19.4652i	− 0.644205i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 44.6614i	− 1.47485i
$918$	0	0
$919$	34.7401	1.14597	0.572985	−	0.819566i	$-0.305785\pi$
$919$	34.7401	1.14597	0.572985	+	0.819566i	$0.305785\pi$
$920$	0	0
$921$	6.23854	0.205567
$922$	0	0
$923$	28.4192i	0.935430i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 20.8871i	− 0.686023i
$928$	0	0
$929$	−50.1848	−1.64651	−0.823255	−	0.567672i	$-0.807844\pi$
$929$	−50.1848	−1.64651	−0.823255	+	0.567672i	$0.807844\pi$
$930$	0	0
$931$	4.63816	0.152009
$932$	0	0
$933$	9.53478i	0.312155i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 4.48927i	− 0.146658i	−0.997308	−	0.0733290i	$-0.976638\pi$
$937$	− 4.48927i	− 0.146658i	0.997308	−	0.0733290i	$-0.0233623\pi$
$938$	0	0
$939$	−1.01043	−0.0329741
$940$	0	0
$941$	−7.70233	−0.251089	−0.125544	−	0.992088i	$-0.540068\pi$
$941$	−7.70233	−0.251089	−0.125544	+	0.992088i	$0.540068\pi$
$942$	0	0
$943$	− 8.69129i	− 0.283028i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	13.4415i	0.436790i	0.975860	+	0.218395i	$0.0700821\pi$
$947$	13.4415i	0.436790i	−0.975860	+	0.218395i	$0.929918\pi$
$948$	0	0
$949$	−11.4192	−0.370683
$950$	0	0
$951$	8.02498	0.260228
$952$	0	0
$953$	− 23.1943i	− 0.751337i	−0.926754	−	0.375668i	$-0.877413\pi$
$953$	− 23.1943i	− 0.751337i	0.926754	−	0.375668i	$-0.122587\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	6.84255i	0.221188i
$958$	0	0
$959$	−52.3073	−1.68909
$960$	0	0
$961$	−23.3087	−0.751894
$962$	0	0
$963$	− 25.7520i	− 0.829845i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	22.1429i	0.712068i	0.934473	+	0.356034i	$0.115871\pi$
$967$	22.1429i	0.712068i	−0.934473	+	0.356034i	$0.884129\pi$
$968$	0	0
$969$	0.815207	0.0261882
$970$	0	0
$971$	26.9804	0.865842	0.432921	−	0.901432i	$-0.357483\pi$
$971$	26.9804	0.865842	0.432921	+	0.901432i	$0.357483\pi$
$972$	0	0
$973$	26.5357i	0.850696i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.64496i	0.0846199i	0.999105	+	0.0423099i	$0.0134717\pi$
$977$	2.64496i	0.0846199i	−0.999105	+	0.0423099i	$0.986528\pi$
$978$	0	0
$979$	7.09833	0.226863
$980$	0	0
$981$	28.2618	0.902329
$982$	0	0
$983$	22.4492i	0.716020i	0.933718	+	0.358010i	$0.116545\pi$
$983$	22.4492i	0.716020i	−0.933718	+	0.358010i	$0.883455\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	7.45336i	0.237243i
$988$	0	0
$989$	6.86989	0.218450
$990$	0	0
$991$	−26.4570	−0.840434	−0.420217	−	0.907424i	$-0.638046\pi$
$991$	−26.4570	−0.840434	−0.420217	+	0.907424i	$0.638046\pi$
$992$	0	0
$993$	1.00444i	0.0318750i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 2.17881i	− 0.0690035i	−0.999405	−	0.0345017i	$-0.989016\pi$
$997$	− 2.17881i	− 0.0690035i	0.999405	−	0.0345017i	$-0.0109844\pi$
$998$	0	0
$999$	−1.99638	−0.0631626

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.d.o.3649.4		6
5.2	odd	4		3800.2.a.t.1.2	yes	3
5.3	odd	4		3800.2.a.s.1.2	✓	3
5.4	even	2	inner	3800.2.d.o.3649.3		6
20.3	even	4		7600.2.a.br.1.2		3
20.7	even	4		7600.2.a.bs.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.s.1.2	✓	3	5.3	odd	4
3800.2.a.t.1.2	yes	3	5.2	odd	4
3800.2.d.o.3649.3		6	5.4	even	2	inner
3800.2.d.o.3649.4		6	1.1	even	1	trivial
7600.2.a.br.1.2		3	20.3	even	4
7600.2.a.bs.1.2		3	20.7	even	4

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:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.347296i q^{3} +3.41147i q^{7} +2.87939 q^{9} +2.41147 q^{11} +6.29086i q^{13} +2.34730i q^{17} -1.00000 q^{19} -1.18479 q^{21} +2.49020i q^{23} +2.04189i q^{27} +8.17024 q^{29} -2.77332 q^{31} +0.837496i q^{33} +0.977711i q^{37} -2.18479 q^{39} -3.49020 q^{41} -2.75877i q^{43} -6.29086i q^{47} -4.63816 q^{49} -0.815207 q^{51} +2.38919i q^{53} -0.347296i q^{57} -3.67499 q^{59} -12.7442 q^{61} +9.82295i q^{63} -2.41147i q^{67} -0.864837 q^{69} +4.51754 q^{71} +1.81521i q^{73} +8.22668i q^{77} +5.04189 q^{79} +7.92902 q^{81} -8.07192i q^{83} +2.83750i q^{87} +2.94356 q^{89} -21.4611 q^{91} -0.963163i q^{93} +3.09833i q^{97} +6.94356 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{9} - 6 q^{11} - 6 q^{19} + 6 q^{29} - 30 q^{31} - 6 q^{39} - 18 q^{41} + 6 q^{49} - 12 q^{51} - 12 q^{59} - 18 q^{61} + 42 q^{69} - 18 q^{71} + 24 q^{79} - 18 q^{81} - 12 q^{89} - 54 q^{91} + 12 q^{99}+O(q^{100}) \) 6 * q + 6 * q^9 - 6 * q^11 - 6 * q^19 + 6 * q^29 - 30 * q^31 - 6 * q^39 - 18 * q^41 + 6 * q^49 - 12 * q^51 - 12 * q^59 - 18 * q^61 + 42 * q^69 - 18 * q^71 + 24 * q^79 - 18 * q^81 - 12 * q^89 - 54 * q^91 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more