LMFDB - Embedded newform 3800.2.d.l.3649.5

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:	6.0.3356224.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 8x^{4} + 16x^{2} + 1 $ x^6 + 8x^4 + 16x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 760)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.5
Root			$1.86081i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.l.3649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.32340i q^{3} +1.39821i q^{7} -2.39821 q^{9} +O(q^{10})$	$q+2.32340i q^{3} +1.39821i q^{7} -2.39821 q^{9} +4.32340i q^{13} -0.601793i q^{17} -1.00000 q^{19} -3.24860 q^{21} +6.04502i q^{23} +1.39821i q^{27} -4.60179 q^{29} +2.79641 q^{31} -1.07480i q^{37} -10.0450 q^{39} -5.44322 q^{41} -8.64681i q^{43} +1.85039i q^{47} +5.04502 q^{49} +1.39821 q^{51} -3.11982i q^{53} -2.32340i q^{57} +6.69182 q^{59} -2.64681 q^{61} -3.35319i q^{63} +14.4134i q^{67} -14.0450 q^{69} -5.59283 q^{71} +12.6918i q^{73} +4.64681 q^{79} -10.4432 q^{81} -1.20359i q^{83} -10.6918i q^{87} -9.44322 q^{89} -6.04502 q^{91} +6.49720i q^{93} -4.51803i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 8 q^{9}+O(q^{10}) $ 6 * q - 8 * q^9	$ 6 q - 8 q^{9} - 6 q^{19} + 6 q^{21} - 34 q^{29} + 4 q^{31} - 22 q^{39} + 12 q^{41} - 8 q^{49} + 2 q^{51} - 30 q^{59} + 16 q^{61} - 46 q^{69} - 8 q^{71} - 4 q^{79} - 18 q^{81} - 12 q^{89} + 2 q^{91}+O(q^{100}) $ 6 * q - 8 * q^9 - 6 * q^19 + 6 * q^21 - 34 * q^29 + 4 * q^31 - 22 * q^39 + 12 * q^41 - 8 * q^49 + 2 * q^51 - 30 * q^59 + 16 * q^61 - 46 * q^69 - 8 * q^71 - 4 * q^79 - 18 * q^81 - 12 * q^89 + 2 * q^91

Display 100 coefficients 10 coefficients

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
$3$	2.32340i	1.34142i	0.741721	+	0.670709i	$0.234010\pi$
$3$	2.32340i	1.34142i	−0.741721	+	0.670709i	$0.765990\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.39821i	0.528473i	0.964458	+	0.264236i	$0.0851199\pi$
$7$	1.39821i	0.528473i	−0.964458	+	0.264236i	$0.914880\pi$
$8$	0	0
$9$	−2.39821	−0.799402
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	4.32340i	1.19910i	0.800339	+	0.599548i	$0.204653\pi$
$13$	4.32340i	1.19910i	−0.800339	+	0.599548i	$0.795347\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.601793i	− 0.145956i	−0.997334	−	0.0729781i	$-0.976750\pi$
$17$	− 0.601793i	− 0.145956i	0.997334	−	0.0729781i	$-0.0232503\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−3.24860	−0.708903
$22$	0	0
$23$	6.04502i	1.26047i	0.776403	+	0.630236i	$0.217042\pi$
$23$	6.04502i	1.26047i	−0.776403	+	0.630236i	$0.782958\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.39821i	0.269085i
$28$	0	0
$29$	−4.60179	−0.854531	−0.427266	−	0.904126i	$-0.640523\pi$
$29$	−4.60179	−0.854531	−0.427266	+	0.904126i	$0.640523\pi$
$30$	0	0
$31$	2.79641	0.502251	0.251125	−	0.967955i	$-0.419199\pi$
$31$	2.79641	0.502251	0.251125	+	0.967955i	$0.419199\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 1.07480i	− 0.176697i	−0.996090	−	0.0883483i	$-0.971841\pi$
$37$	− 1.07480i	− 0.176697i	0.996090	−	0.0883483i	$-0.0281588\pi$
$38$	0	0
$39$	−10.0450	−1.60849
$40$	0	0
$41$	−5.44322	−0.850089	−0.425044	−	0.905173i	$-0.639741\pi$
$41$	−5.44322	−0.850089	−0.425044	+	0.905173i	$0.639741\pi$
$42$	0	0
$43$	− 8.64681i	− 1.31863i	−0.751869	−	0.659313i	$-0.770847\pi$
$43$	− 8.64681i	− 1.31863i	0.751869	−	0.659313i	$-0.229153\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.85039i	0.269908i	0.990852	+	0.134954i	$0.0430886\pi$
$47$	1.85039i	0.269908i	−0.990852	+	0.134954i	$0.956911\pi$
$48$	0	0
$49$	5.04502	0.720717
$50$	0	0
$51$	1.39821	0.195788
$52$	0	0
$53$	− 3.11982i	− 0.428540i	−0.976774	−	0.214270i	$-0.931263\pi$
$53$	− 3.11982i	− 0.428540i	0.976774	−	0.214270i	$-0.0687373\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 2.32340i	− 0.307742i
$58$	0	0
$59$	6.69182	0.871201	0.435601	−	0.900140i	$-0.356536\pi$
$59$	6.69182	0.871201	0.435601	+	0.900140i	$0.356536\pi$
$60$	0	0
$61$	−2.64681	−0.338889	−0.169445	−	0.985540i	$-0.554197\pi$
$61$	−2.64681	−0.338889	−0.169445	+	0.985540i	$0.554197\pi$
$62$	0	0
$63$	− 3.35319i	− 0.422462i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	14.4134i	1.76088i	0.474156	+	0.880441i	$0.342753\pi$
$67$	14.4134i	1.76088i	−0.474156	+	0.880441i	$0.657247\pi$
$68$	0	0
$69$	−14.0450	−1.69082
$70$	0	0
$71$	−5.59283	−0.663747	−0.331873	−	0.943324i	$-0.607681\pi$
$71$	−5.59283	−0.663747	−0.331873	+	0.943324i	$0.607681\pi$
$72$	0	0
$73$	12.6918i	1.48547i	0.669588	+	0.742733i	$0.266471\pi$
$73$	12.6918i	1.48547i	−0.669588	+	0.742733i	$0.733529\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	4.64681	0.522807	0.261403	−	0.965230i	$-0.415815\pi$
$79$	4.64681	0.522807	0.261403	+	0.965230i	$0.415815\pi$
$80$	0	0
$81$	−10.4432	−1.16036
$82$	0	0
$83$	− 1.20359i	− 0.132111i	−0.997816	−	0.0660553i	$-0.978959\pi$
$83$	− 1.20359i	− 0.132111i	0.997816	−	0.0660553i	$-0.0210414\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 10.6918i	− 1.14628i
$88$	0	0
$89$	−9.44322	−1.00098	−0.500490	−	0.865742i	$-0.666847\pi$
$89$	−9.44322	−1.00098	−0.500490	+	0.865742i	$0.666847\pi$
$90$	0	0
$91$	−6.04502	−0.633690
$92$	0	0
$93$	6.49720i	0.673728i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 4.51803i	− 0.458736i	−0.973340	−	0.229368i	$-0.926334\pi$
$97$	− 4.51803i	− 0.458736i	0.973340	−	0.229368i	$-0.0736659\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−5.05398	−0.502890	−0.251445	−	0.967872i	$-0.580906\pi$
$101$	−5.05398	−0.502890	−0.251445	+	0.967872i	$0.580906\pi$
$102$	0	0
$103$	− 11.0748i	− 1.09123i	−0.838035	−	0.545616i	$-0.816296\pi$
$103$	− 11.0748i	− 1.09123i	0.838035	−	0.545616i	$-0.183704\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.17380i	0.403496i	0.979437	+	0.201748i	$0.0646622\pi$
$107$	4.17380i	0.403496i	−0.979437	+	0.201748i	$0.935338\pi$
$108$	0	0
$109$	−14.8414	−1.42155	−0.710776	−	0.703419i	$-0.751656\pi$
$109$	−14.8414	−1.42155	−0.710776	+	0.703419i	$0.751656\pi$
$110$	0	0
$111$	2.49720	0.237024
$112$	0	0
$113$	1.72161i	0.161956i	0.996716	+	0.0809778i	$0.0258043\pi$
$113$	1.72161i	0.161956i	−0.996716	+	0.0809778i	$0.974196\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 10.3684i	− 0.958561i
$118$	0	0
$119$	0.841431	0.0771338
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	− 12.6468i	− 1.14032i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2.12878i	0.188899i	0.995530	+	0.0944494i	$0.0301091\pi$
$127$	2.12878i	0.188899i	−0.995530	+	0.0944494i	$0.969891\pi$
$128$	0	0
$129$	20.0900	1.76883
$130$	0	0
$131$	1.59283	0.139166	0.0695831	−	0.997576i	$-0.477833\pi$
$131$	1.59283	0.139166	0.0695831	+	0.997576i	$0.477833\pi$
$132$	0	0
$133$	− 1.39821i	− 0.121240i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 9.89541i	− 0.845422i	−0.906265	−	0.422711i	$-0.861079\pi$
$137$	− 9.89541i	− 0.845422i	0.906265	−	0.422711i	$-0.138921\pi$
$138$	0	0
$139$	−3.70079	−0.313897	−0.156948	−	0.987607i	$-0.550166\pi$
$139$	−3.70079	−0.313897	−0.156948	+	0.987607i	$0.550166\pi$
$140$	0	0
$141$	−4.29921	−0.362059
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	11.7216i	0.966782i
$148$	0	0
$149$	−18.3476	−1.50309	−0.751547	−	0.659680i	$-0.770692\pi$
$149$	−18.3476	−1.50309	−0.751547	+	0.659680i	$0.770692\pi$
$150$	0	0
$151$	13.9404	1.13446	0.567228	−	0.823561i	$-0.308016\pi$
$151$	13.9404	1.13446	0.567228	+	0.823561i	$0.308016\pi$
$152$	0	0
$153$	1.44322i	0.116678i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0.149606i	0.0119399i	0.999982	+	0.00596994i	$0.00190030\pi$
$157$	0.149606i	0.0119399i	−0.999982	+	0.00596994i	$0.998100\pi$
$158$	0	0
$159$	7.24860	0.574851
$160$	0	0
$161$	−8.45219	−0.666126
$162$	0	0
$163$	− 24.6468i	− 1.93049i	−0.261352	−	0.965244i	$-0.584168\pi$
$163$	− 24.6468i	− 1.93049i	0.261352	−	0.965244i	$-0.415832\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.8656i	1.45987i	0.683519	+	0.729933i	$0.260449\pi$
$167$	18.8656i	1.45987i	−0.683519	+	0.729933i	$0.739551\pi$
$168$	0	0
$169$	−5.69182	−0.437833
$170$	0	0
$171$	2.39821	0.183396
$172$	0	0
$173$	− 6.36842i	− 0.484182i	−0.970254	−	0.242091i	$-0.922167\pi$
$173$	− 6.36842i	− 0.484182i	0.970254	−	0.242091i	$-0.0778333\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	15.5478i	1.16865i
$178$	0	0
$179$	−11.4432	−0.855307	−0.427653	−	0.903943i	$-0.640660\pi$
$179$	−11.4432	−0.855307	−0.427653	+	0.903943i	$0.640660\pi$
$180$	0	0
$181$	−3.29362	−0.244813	−0.122406	−	0.992480i	$-0.539061\pi$
$181$	−3.29362	−0.244813	−0.122406	+	0.992480i	$0.539061\pi$
$182$	0	0
$183$	− 6.14961i	− 0.454592i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1.95498	−0.142204
$190$	0	0
$191$	20.5422	1.48638	0.743191	−	0.669079i	$-0.233311\pi$
$191$	20.5422	1.48638	0.743191	+	0.669079i	$0.233311\pi$
$192$	0	0
$193$	− 19.6620i	− 1.41530i	−0.706561	−	0.707652i	$-0.749754\pi$
$193$	− 19.6620i	− 1.41530i	0.706561	−	0.707652i	$-0.250246\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	0	0
$199$	9.74580	0.690862	0.345431	−	0.938444i	$-0.387733\pi$
$199$	9.74580	0.690862	0.345431	+	0.938444i	$0.387733\pi$
$200$	0	0
$201$	−33.4882	−2.36208
$202$	0	0
$203$	− 6.43426i	− 0.451597i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 14.4972i	− 1.00763i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−9.09899	−0.626401	−0.313200	−	0.949687i	$-0.601401\pi$
$211$	−9.09899	−0.626401	−0.313200	+	0.949687i	$0.601401\pi$
$212$	0	0
$213$	− 12.9944i	− 0.890362i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.90997i	0.265426i
$218$	0	0
$219$	−29.4882	−1.99263
$220$	0	0
$221$	2.60179	0.175016
$222$	0	0
$223$	− 13.8712i	− 0.928885i	−0.885603	−	0.464443i	$-0.846255\pi$
$223$	− 13.8712i	− 0.928885i	0.885603	−	0.464443i	$-0.153745\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 10.3234i	− 0.685188i	−0.939483	−	0.342594i	$-0.888694\pi$
$227$	− 10.3234i	− 0.685188i	0.939483	−	0.342594i	$-0.111306\pi$
$228$	0	0
$229$	14.7368	0.973838	0.486919	−	0.873447i	$-0.338121\pi$
$229$	14.7368	0.973838	0.486919	+	0.873447i	$0.338121\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	13.1857i	0.863821i	0.901916	+	0.431911i	$0.142160\pi$
$233$	13.1857i	0.863821i	−0.901916	+	0.431911i	$0.857840\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	10.7964i	0.701303i
$238$	0	0
$239$	0.452186	0.0292495	0.0146247	−	0.999893i	$-0.495345\pi$
$239$	0.452186	0.0292495	0.0146247	+	0.999893i	$0.495345\pi$
$240$	0	0
$241$	13.5333	0.871754	0.435877	−	0.900006i	$-0.356438\pi$
$241$	13.5333	0.871754	0.435877	+	0.900006i	$0.356438\pi$
$242$	0	0
$243$	− 20.0692i	− 1.28744i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 4.32340i	− 0.275092i
$248$	0	0
$249$	2.79641	0.177216
$250$	0	0
$251$	21.2936	1.34404	0.672021	−	0.740532i	$-0.265427\pi$
$251$	21.2936	1.34404	0.672021	+	0.740532i	$0.265427\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	28.3989i	1.77147i	0.464188	+	0.885737i	$0.346346\pi$
$257$	28.3989i	1.77147i	−0.464188	+	0.885737i	$0.653654\pi$
$258$	0	0
$259$	1.50280	0.0933793
$260$	0	0
$261$	11.0361	0.683115
$262$	0	0
$263$	11.1857i	0.689737i	0.938651	+	0.344869i	$0.112077\pi$
$263$	11.1857i	0.689737i	−0.938651	+	0.344869i	$0.887923\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 21.9404i	− 1.34273i
$268$	0	0
$269$	−19.5928	−1.19460	−0.597298	−	0.802019i	$-0.703759\pi$
$269$	−19.5928	−1.19460	−0.597298	+	0.802019i	$0.703759\pi$
$270$	0	0
$271$	5.95498	0.361740	0.180870	−	0.983507i	$-0.442109\pi$
$271$	5.95498	0.361740	0.180870	+	0.983507i	$0.442109\pi$
$272$	0	0
$273$	− 14.0450i	− 0.850043i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	23.9404i	1.43844i	0.694782	+	0.719220i	$0.255501\pi$
$277$	23.9404i	1.43844i	−0.694782	+	0.719220i	$0.744499\pi$
$278$	0	0
$279$	−6.70638	−0.401501
$280$	0	0
$281$	10.7368	0.640506	0.320253	−	0.947332i	$-0.396232\pi$
$281$	10.7368	0.640506	0.320253	+	0.947332i	$0.396232\pi$
$282$	0	0
$283$	− 16.0900i	− 0.956453i	−0.878237	−	0.478227i	$-0.841280\pi$
$283$	− 16.0900i	− 0.956453i	0.878237	−	0.478227i	$-0.158720\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 7.61076i	− 0.449249i
$288$	0	0
$289$	16.6378	0.978697
$290$	0	0
$291$	10.4972	0.615357
$292$	0	0
$293$	14.5630i	0.850782i	0.905010	+	0.425391i	$0.139863\pi$
$293$	14.5630i	0.850782i	−0.905010	+	0.425391i	$0.860137\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−26.1350	−1.51143
$300$	0	0
$301$	12.0900	0.696858
$302$	0	0
$303$	− 11.7424i	− 0.674585i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 2.77559i	− 0.158411i	−0.996858	−	0.0792057i	$-0.974762\pi$
$307$	− 2.77559i	− 0.158411i	0.996858	−	0.0792057i	$-0.0252384\pi$
$308$	0	0
$309$	25.7312	1.46380
$310$	0	0
$311$	−7.63785	−0.433102	−0.216551	−	0.976271i	$-0.569481\pi$
$311$	−7.63785	−0.433102	−0.216551	+	0.976271i	$0.569481\pi$
$312$	0	0
$313$	− 32.5243i	− 1.83838i	−0.393812	−	0.919191i	$-0.628844\pi$
$313$	− 32.5243i	− 1.83838i	0.393812	−	0.919191i	$-0.371156\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 10.8206i	− 0.607746i	−0.952713	−	0.303873i	$-0.901720\pi$
$317$	− 10.8206i	− 0.607746i	0.952713	−	0.303873i	$-0.0982798\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−9.69742	−0.541257
$322$	0	0
$323$	0.601793i	0.0334846i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 34.4826i	− 1.90689i
$328$	0	0
$329$	−2.58723	−0.142639
$330$	0	0
$331$	14.6918	0.807536	0.403768	−	0.914861i	$-0.367700\pi$
$331$	14.6918	0.807536	0.403768	+	0.914861i	$0.367700\pi$
$332$	0	0
$333$	2.57760i	0.141252i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.07480i	0.0585483i	0.999571	+	0.0292741i	$0.00931958\pi$
$337$	1.07480i	0.0585483i	−0.999571	+	0.0292741i	$0.990680\pi$
$338$	0	0
$339$	−4.00000	−0.217250
$340$	0	0
$341$	0	0
$342$	0	0
$343$	16.8414i	0.909352i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 32.0900i	− 1.72268i	−0.508026	−	0.861342i	$-0.669625\pi$
$347$	− 32.0900i	− 1.72268i	0.508026	−	0.861342i	$-0.330375\pi$
$348$	0	0
$349$	−31.4737	−1.68475	−0.842374	−	0.538894i	$-0.818842\pi$
$349$	−31.4737	−1.68475	−0.842374	+	0.538894i	$0.818842\pi$
$350$	0	0
$351$	−6.04502	−0.322659
$352$	0	0
$353$	− 11.4882i	− 0.611457i	−0.952119	−	0.305729i	$-0.901100\pi$
$353$	− 11.4882i	− 0.611457i	0.952119	−	0.305729i	$-0.0989000\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.95498i	0.103469i
$358$	0	0
$359$	−26.0450	−1.37460	−0.687302	−	0.726372i	$-0.741205\pi$
$359$	−26.0450	−1.37460	−0.687302	+	0.726372i	$0.741205\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	− 25.5574i	− 1.34142i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	21.5512i	1.12496i	0.826810	+	0.562481i	$0.190153\pi$
$367$	21.5512i	1.12496i	−0.826810	+	0.562481i	$0.809847\pi$
$368$	0	0
$369$	13.0540	0.679563
$370$	0	0
$371$	4.36215	0.226472
$372$	0	0
$373$	24.9702i	1.29291i	0.762953	+	0.646454i	$0.223749\pi$
$373$	24.9702i	1.29291i	−0.762953	+	0.646454i	$0.776251\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 19.8954i	− 1.02467i
$378$	0	0
$379$	−28.6323	−1.47074	−0.735370	−	0.677666i	$-0.762992\pi$
$379$	−28.6323	−1.47074	−0.735370	+	0.677666i	$0.762992\pi$
$380$	0	0
$381$	−4.94602	−0.253392
$382$	0	0
$383$	17.9612i	0.917777i	0.888494	+	0.458889i	$0.151752\pi$
$383$	17.9612i	0.917777i	−0.888494	+	0.458889i	$0.848248\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	20.7368i	1.05411i
$388$	0	0
$389$	6.09003	0.308777	0.154388	−	0.988010i	$-0.450659\pi$
$389$	6.09003	0.308777	0.154388	+	0.988010i	$0.450659\pi$
$390$	0	0
$391$	3.63785	0.183974
$392$	0	0
$393$	3.70079i	0.186680i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 21.8325i	− 1.09574i	−0.836563	−	0.547870i	$-0.815439\pi$
$397$	− 21.8325i	− 1.09574i	0.836563	−	0.547870i	$-0.184561\pi$
$398$	0	0
$399$	3.24860	0.162633
$400$	0	0
$401$	−12.7064	−0.634526	−0.317263	−	0.948338i	$-0.602764\pi$
$401$	−12.7064	−0.634526	−0.317263	+	0.948338i	$0.602764\pi$
$402$	0	0
$403$	12.0900i	0.602247i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−16.8864	−0.834981	−0.417491	−	0.908681i	$-0.637090\pi$
$409$	−16.8864	−0.834981	−0.417491	+	0.908681i	$0.637090\pi$
$410$	0	0
$411$	22.9910	1.13406
$412$	0	0
$413$	9.35656i	0.460406i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 8.59843i	− 0.421067i
$418$	0	0
$419$	−27.7908	−1.35767	−0.678835	−	0.734291i	$-0.737515\pi$
$419$	−27.7908	−1.35767	−0.678835	+	0.734291i	$0.737515\pi$
$420$	0	0
$421$	15.0090	0.731492	0.365746	−	0.930715i	$-0.380814\pi$
$421$	15.0090	0.731492	0.365746	+	0.930715i	$0.380814\pi$
$422$	0	0
$423$	− 4.43763i	− 0.215765i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 3.70079i	− 0.179094i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−0.736841	−0.0354924	−0.0177462	−	0.999843i	$-0.505649\pi$
$431$	−0.736841	−0.0354924	−0.0177462	+	0.999843i	$0.505649\pi$
$432$	0	0
$433$	31.4945i	1.51353i	0.653687	+	0.756765i	$0.273221\pi$
$433$	31.4945i	1.51353i	−0.653687	+	0.756765i	$0.726779\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 6.04502i	− 0.289172i
$438$	0	0
$439$	23.2340	1.10890	0.554450	−	0.832217i	$-0.312929\pi$
$439$	23.2340	1.10890	0.554450	+	0.832217i	$0.312929\pi$
$440$	0	0
$441$	−12.0990	−0.576143
$442$	0	0
$443$	16.4793i	0.782954i	0.920188	+	0.391477i	$0.128036\pi$
$443$	16.4793i	0.782954i	−0.920188	+	0.391477i	$0.871964\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 42.6289i	− 2.01628i
$448$	0	0
$449$	2.29921	0.108507	0.0542533	−	0.998527i	$-0.482722\pi$
$449$	2.29921	0.108507	0.0542533	+	0.998527i	$0.482722\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	32.3892i	1.52178i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	20.0450i	0.937666i	0.883287	+	0.468833i	$0.155325\pi$
$457$	20.0450i	0.937666i	−0.883287	+	0.468833i	$0.844675\pi$
$458$	0	0
$459$	0.841431	0.0392746
$460$	0	0
$461$	32.5872	1.51774	0.758869	−	0.651243i	$-0.225752\pi$
$461$	32.5872	1.51774	0.758869	+	0.651243i	$0.225752\pi$
$462$	0	0
$463$	2.58723i	0.120239i	0.998191	+	0.0601195i	$0.0191482\pi$
$463$	2.58723i	0.120239i	−0.998191	+	0.0601195i	$0.980852\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0.299213i	0.0138459i	0.999976	+	0.00692295i	$0.00220366\pi$
$467$	0.299213i	0.0138459i	−0.999976	+	0.00692295i	$0.997796\pi$
$468$	0	0
$469$	−20.1530	−0.930578
$470$	0	0
$471$	−0.347596	−0.0160164
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	7.48197i	0.342576i
$478$	0	0
$479$	35.0665	1.60223	0.801115	−	0.598511i	$-0.204241\pi$
$479$	35.0665	1.60223	0.801115	+	0.598511i	$0.204241\pi$
$480$	0	0
$481$	4.64681	0.211876
$482$	0	0
$483$	− 19.6378i	− 0.893553i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	41.7521i	1.89197i	0.324215	+	0.945983i	$0.394900\pi$
$487$	41.7521i	1.89197i	−0.324215	+	0.945983i	$0.605100\pi$
$488$	0	0
$489$	57.2645	2.58959
$490$	0	0
$491$	−30.9765	−1.39795	−0.698974	−	0.715147i	$-0.746360\pi$
$491$	−30.9765	−1.39795	−0.698974	+	0.715147i	$0.746360\pi$
$492$	0	0
$493$	2.76932i	0.124724i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 7.81994i	− 0.350772i
$498$	0	0
$499$	3.90997	0.175034	0.0875171	−	0.996163i	$-0.472107\pi$
$499$	3.90997	0.175034	0.0875171	+	0.996163i	$0.472107\pi$
$500$	0	0
$501$	−43.8325	−1.95829
$502$	0	0
$503$	16.4522i	0.733567i	0.930306	+	0.366783i	$0.119541\pi$
$503$	16.4522i	0.733567i	−0.930306	+	0.366783i	$0.880459\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 13.2244i	− 0.587317i
$508$	0	0
$509$	25.4432	1.12775	0.563876	−	0.825860i	$-0.309310\pi$
$509$	25.4432	1.12775	0.563876	+	0.825860i	$0.309310\pi$
$510$	0	0
$511$	−17.7458	−0.785028
$512$	0	0
$513$	− 1.39821i	− 0.0617324i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	14.7964	0.649491
$520$	0	0
$521$	−27.3836	−1.19970	−0.599850	−	0.800113i	$-0.704773\pi$
$521$	−27.3836	−1.19970	−0.599850	+	0.800113i	$0.704773\pi$
$522$	0	0
$523$	41.5574i	1.81718i	0.417689	+	0.908590i	$0.362840\pi$
$523$	41.5574i	1.81718i	−0.417689	+	0.908590i	$0.637160\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.68286i	− 0.0733066i
$528$	0	0
$529$	−13.5422	−0.588792
$530$	0	0
$531$	−16.0484	−0.696441
$532$	0	0
$533$	− 23.5333i	− 1.01934i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 26.5872i	− 1.14732i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	14.6468	0.629715	0.314858	−	0.949139i	$-0.398043\pi$
$541$	14.6468	0.629715	0.314858	+	0.949139i	$0.398043\pi$
$542$	0	0
$543$	− 7.65240i	− 0.328396i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	24.9252i	1.06572i	0.846202	+	0.532862i	$0.178884\pi$
$547$	24.9252i	1.06572i	−0.846202	+	0.532862i	$0.821116\pi$
$548$	0	0
$549$	6.34760	0.270909
$550$	0	0
$551$	4.60179	0.196043
$552$	0	0
$553$	6.49720i	0.276289i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	44.9348i	1.90395i	0.306176	+	0.951975i	$0.400950\pi$
$557$	44.9348i	1.90395i	−0.306176	+	0.951975i	$0.599050\pi$
$558$	0	0
$559$	37.3836	1.58116
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 10.2605i	− 0.432427i	−0.976346	−	0.216213i	$-0.930629\pi$
$563$	− 10.2605i	− 0.432427i	0.976346	−	0.216213i	$-0.0693707\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 14.6018i	− 0.613218i
$568$	0	0
$569$	−24.6773	−1.03452	−0.517262	−	0.855827i	$-0.673049\pi$
$569$	−24.6773	−1.03452	−0.517262	+	0.855827i	$0.673049\pi$
$570$	0	0
$571$	−10.9765	−0.459351	−0.229676	−	0.973267i	$-0.573767\pi$
$571$	−10.9765	−0.459351	−0.229676	+	0.973267i	$0.573767\pi$
$572$	0	0
$573$	47.7279i	1.99386i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	45.1295i	1.87876i	0.342873	+	0.939382i	$0.388600\pi$
$577$	45.1295i	1.87876i	−0.342873	+	0.939382i	$0.611400\pi$
$578$	0	0
$579$	45.6829	1.89851
$580$	0	0
$581$	1.68286	0.0698169
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 7.90997i	− 0.326479i	−0.986586	−	0.163240i	$-0.947806\pi$
$587$	− 7.90997i	− 0.326479i	0.986586	−	0.163240i	$-0.0521944\pi$
$588$	0	0
$589$	−2.79641	−0.115224
$590$	0	0
$591$	−4.64681	−0.191144
$592$	0	0
$593$	− 7.29362i	− 0.299513i	−0.988723	−	0.149756i	$-0.952151\pi$
$593$	− 7.29362i	− 0.299513i	0.988723	−	0.149756i	$-0.0478490\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	22.6434i	0.926734i
$598$	0	0
$599$	−20.6468	−0.843606	−0.421803	−	0.906688i	$-0.638603\pi$
$599$	−20.6468	−0.843606	−0.421803	+	0.906688i	$0.638603\pi$
$600$	0	0
$601$	45.1440	1.84146	0.920731	−	0.390197i	$-0.127593\pi$
$601$	45.1440	1.84146	0.920731	+	0.390197i	$0.127593\pi$
$602$	0	0
$603$	− 34.5664i	− 1.40765i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	20.7577i	0.842528i	0.906938	+	0.421264i	$0.138413\pi$
$607$	20.7577i	0.842528i	−0.906938	+	0.421264i	$0.861587\pi$
$608$	0	0
$609$	14.9494	0.605780
$610$	0	0
$611$	−8.00000	−0.323645
$612$	0	0
$613$	30.3476i	1.22573i	0.790188	+	0.612864i	$0.209983\pi$
$613$	30.3476i	1.22573i	−0.790188	+	0.612864i	$0.790017\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 47.2161i	− 1.90085i	−0.310955	−	0.950425i	$-0.600649\pi$
$617$	− 47.2161i	− 1.90085i	0.310955	−	0.950425i	$-0.399351\pi$
$618$	0	0
$619$	22.7064	0.912647	0.456323	−	0.889814i	$-0.349166\pi$
$619$	22.7064	0.912647	0.456323	+	0.889814i	$0.349166\pi$
$620$	0	0
$621$	−8.45219	−0.339175
$622$	0	0
$623$	− 13.2036i	− 0.528990i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−0.646809	−0.0257899
$630$	0	0
$631$	32.4793	1.29298	0.646490	−	0.762923i	$-0.276236\pi$
$631$	32.4793	1.29298	0.646490	+	0.762923i	$0.276236\pi$
$632$	0	0
$633$	− 21.1406i	− 0.840265i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	21.8116i	0.864209i
$638$	0	0
$639$	13.4128	0.530601
$640$	0	0
$641$	10.9460	0.432342	0.216171	−	0.976356i	$-0.430643\pi$
$641$	10.9460	0.432342	0.216171	+	0.976356i	$0.430643\pi$
$642$	0	0
$643$	4.38924i	0.173095i	0.996248	+	0.0865475i	$0.0275834\pi$
$643$	4.38924i	0.173095i	−0.996248	+	0.0865475i	$0.972417\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 16.4522i	− 0.646802i	−0.946262	−	0.323401i	$-0.895174\pi$
$647$	− 16.4522i	− 0.646802i	0.946262	−	0.323401i	$-0.104826\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−9.08444	−0.356047
$652$	0	0
$653$	1.53326i	0.0600009i	0.999550	+	0.0300005i	$0.00955088\pi$
$653$	1.53326i	0.0600009i	−0.999550	+	0.0300005i	$0.990449\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 30.4376i	− 1.18748i
$658$	0	0
$659$	−22.1767	−0.863881	−0.431941	−	0.901902i	$-0.642171\pi$
$659$	−22.1767	−0.863881	−0.431941	+	0.901902i	$0.642171\pi$
$660$	0	0
$661$	−26.0755	−1.01422	−0.507109	−	0.861882i	$-0.669286\pi$
$661$	−26.0755	−1.01422	−0.507109	+	0.861882i	$0.669286\pi$
$662$	0	0
$663$	6.04502i	0.234769i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 27.8179i	− 1.07711i
$668$	0	0
$669$	32.2284	1.24602
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 30.4168i	− 1.17248i	−0.810137	−	0.586241i	$-0.800607\pi$
$673$	− 30.4168i	− 1.17248i	0.810137	−	0.586241i	$-0.199393\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	44.2043i	1.69891i	0.527663	+	0.849454i	$0.323068\pi$
$677$	44.2043i	1.69891i	−0.527663	+	0.849454i	$0.676932\pi$
$678$	0	0
$679$	6.31714	0.242430
$680$	0	0
$681$	23.9854	0.919124
$682$	0	0
$683$	42.3989i	1.62235i	0.584805	+	0.811174i	$0.301171\pi$
$683$	42.3989i	1.62235i	−0.584805	+	0.811174i	$0.698829\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	34.2396i	1.30632i
$688$	0	0
$689$	13.4882	0.513861
$690$	0	0
$691$	7.79082	0.296377	0.148188	−	0.988959i	$-0.452656\pi$
$691$	7.79082	0.296377	0.148188	+	0.988959i	$0.452656\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	3.27569i	0.124076i
$698$	0	0
$699$	−30.6356	−1.15875
$700$	0	0
$701$	−8.05957	−0.304406	−0.152203	−	0.988349i	$-0.548637\pi$
$701$	−8.05957	−0.304406	−0.152203	+	0.988349i	$0.548637\pi$
$702$	0	0
$703$	1.07480i	0.0405370i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 7.06651i	− 0.265763i
$708$	0	0
$709$	35.2936	1.32548	0.662740	−	0.748850i	$-0.269394\pi$
$709$	35.2936	1.32548	0.662740	+	0.748850i	$0.269394\pi$
$710$	0	0
$711$	−11.1440	−0.417933
$712$	0	0
$713$	16.9044i	0.633074i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.05061i	0.0392358i
$718$	0	0
$719$	48.9315	1.82484	0.912418	−	0.409260i	$-0.134213\pi$
$719$	48.9315	1.82484	0.912418	+	0.409260i	$0.134213\pi$
$720$	0	0
$721$	15.4849	0.576687
$722$	0	0
$723$	31.4432i	1.16939i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	3.29025i	0.122029i	0.998137	+	0.0610143i	$0.0194335\pi$
$727$	3.29025i	0.122029i	−0.998137	+	0.0610143i	$0.980566\pi$
$728$	0	0
$729$	15.2992	0.566638
$730$	0	0
$731$	−5.20359	−0.192462
$732$	0	0
$733$	22.3892i	0.826966i	0.910512	+	0.413483i	$0.135688\pi$
$733$	22.3892i	0.826966i	−0.910512	+	0.413483i	$0.864312\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−4.90437	−0.180410	−0.0902051	−	0.995923i	$-0.528752\pi$
$739$	−4.90437	−0.180410	−0.0902051	+	0.995923i	$0.528752\pi$
$740$	0	0
$741$	10.0450	0.369013
$742$	0	0
$743$	28.1592i	1.03306i	0.856268	+	0.516531i	$0.172777\pi$
$743$	28.1592i	1.03306i	−0.856268	+	0.516531i	$0.827223\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.88645i	0.105610i
$748$	0	0
$749$	−5.83584	−0.213237
$750$	0	0
$751$	42.9348	1.56671	0.783357	−	0.621572i	$-0.213506\pi$
$751$	42.9348	1.56671	0.783357	+	0.621572i	$0.213506\pi$
$752$	0	0
$753$	49.4737i	1.80292i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	25.0665i	0.911058i	0.890221	+	0.455529i	$0.150550\pi$
$757$	25.0665i	0.911058i	−0.890221	+	0.455529i	$0.849450\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−30.8898	−1.11975	−0.559877	−	0.828575i	$-0.689152\pi$
$761$	−30.8898	−1.11975	−0.559877	+	0.828575i	$0.689152\pi$
$762$	0	0
$763$	− 20.7514i	− 0.751251i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	28.9315i	1.04465i
$768$	0	0
$769$	−7.78745	−0.280823	−0.140411	−	0.990093i	$-0.544843\pi$
$769$	−7.78745	−0.280823	−0.140411	+	0.990093i	$0.544843\pi$
$770$	0	0
$771$	−65.9821	−2.37629
$772$	0	0
$773$	20.7126i	0.744982i	0.928036	+	0.372491i	$0.121496\pi$
$773$	20.7126i	0.744982i	−0.928036	+	0.372491i	$0.878504\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3.49161i	0.125261i
$778$	0	0
$779$	5.44322	0.195024
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 6.43426i	− 0.229942i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 10.5035i	− 0.374408i	−0.982321	−	0.187204i	$-0.940057\pi$
$787$	− 10.5035i	− 0.374408i	0.982321	−	0.187204i	$-0.0599426\pi$
$788$	0	0
$789$	−25.9888	−0.925226
$790$	0	0
$791$	−2.40717	−0.0855891
$792$	0	0
$793$	− 11.4432i	− 0.406361i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	37.8871i	1.34203i	0.741444	+	0.671015i	$0.234142\pi$
$797$	37.8871i	1.34203i	−0.741444	+	0.671015i	$0.765858\pi$
$798$	0	0
$799$	1.11355	0.0393947
$800$	0	0
$801$	22.6468	0.800186
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 45.5221i	− 1.60245i
$808$	0	0
$809$	33.9438	1.19340	0.596700	−	0.802464i	$-0.296478\pi$
$809$	33.9438	1.19340	0.596700	+	0.802464i	$0.296478\pi$
$810$	0	0
$811$	4.66137	0.163683	0.0818414	−	0.996645i	$-0.473920\pi$
$811$	4.66137	0.163683	0.0818414	+	0.996645i	$0.473920\pi$
$812$	0	0
$813$	13.8358i	0.485244i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	8.64681i	0.302514i
$818$	0	0
$819$	14.4972	0.506573
$820$	0	0
$821$	−26.5693	−0.927275	−0.463638	−	0.886025i	$-0.653456\pi$
$821$	−26.5693	−0.927275	−0.463638	+	0.886025i	$0.653456\pi$
$822$	0	0
$823$	24.7998i	0.864466i	0.901762	+	0.432233i	$0.142274\pi$
$823$	24.7998i	0.864466i	−0.901762	+	0.432233i	$0.857726\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.11422i	0.0735188i	0.999324	+	0.0367594i	$0.0117035\pi$
$827$	2.11422i	0.0735188i	−0.999324	+	0.0367594i	$0.988296\pi$
$828$	0	0
$829$	−45.7279	−1.58819	−0.794097	−	0.607790i	$-0.792056\pi$
$829$	−45.7279	−1.58819	−0.794097	+	0.607790i	$0.792056\pi$
$830$	0	0
$831$	−55.6233	−1.92955
$832$	0	0
$833$	− 3.03605i	− 0.105193i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.90997i	0.135148i
$838$	0	0
$839$	−15.4432	−0.533159	−0.266580	−	0.963813i	$-0.585894\pi$
$839$	−15.4432	−0.533159	−0.266580	+	0.963813i	$0.585894\pi$
$840$	0	0
$841$	−7.82351	−0.269776
$842$	0	0
$843$	24.9460i	0.859187i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 15.3803i	− 0.528473i
$848$	0	0
$849$	37.3836	1.28300
$850$	0	0
$851$	6.49720	0.222721
$852$	0	0
$853$	38.1801i	1.30726i	0.756814	+	0.653630i	$0.226755\pi$
$853$	38.1801i	1.30726i	−0.756814	+	0.653630i	$0.773245\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 52.2672i	− 1.78541i	−0.450638	−	0.892707i	$-0.648804\pi$
$857$	− 52.2672i	− 1.78541i	0.450638	−	0.892707i	$-0.351196\pi$
$858$	0	0
$859$	39.8809	1.36072	0.680359	−	0.732879i	$-0.261824\pi$
$859$	39.8809	1.36072	0.680359	+	0.732879i	$0.261824\pi$
$860$	0	0
$861$	17.6829	0.602630
$862$	0	0
$863$	22.9557i	0.781420i	0.920514	+	0.390710i	$0.127770\pi$
$863$	22.9557i	0.781420i	−0.920514	+	0.390710i	$0.872230\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	38.6564i	1.31284i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−62.3151	−2.11147
$872$	0	0
$873$	10.8352i	0.366715i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.73057i	0.0922050i	0.998937	+	0.0461025i	$0.0146801\pi$
$877$	2.73057i	0.0922050i	−0.998937	+	0.0461025i	$0.985320\pi$
$878$	0	0
$879$	−33.8358	−1.14125
$880$	0	0
$881$	9.14401	0.308070	0.154035	−	0.988065i	$-0.450773\pi$
$881$	9.14401	0.308070	0.154035	+	0.988065i	$0.450773\pi$
$882$	0	0
$883$	− 11.0540i	− 0.371996i	−0.982550	−	0.185998i	$-0.940448\pi$
$883$	− 11.0540i	− 0.371996i	0.982550	−	0.185998i	$-0.0595518\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 37.1469i	− 1.24727i	−0.781715	−	0.623636i	$-0.785655\pi$
$887$	− 37.1469i	− 1.24727i	0.781715	−	0.623636i	$-0.214345\pi$
$888$	0	0
$889$	−2.97648	−0.0998279
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 1.85039i	− 0.0619211i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 60.7223i	− 2.02746i
$898$	0	0
$899$	−12.8685	−0.429189
$900$	0	0
$901$	−1.87748	−0.0625481
$902$	0	0
$903$	28.0900i	0.934778i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 4.00627i	− 0.133026i	−0.997786	−	0.0665129i	$-0.978813\pi$
$907$	− 4.00627i	− 0.133026i	0.997786	−	0.0665129i	$-0.0211874\pi$
$908$	0	0
$909$	12.1205	0.402011
$910$	0	0
$911$	21.2161	0.702921	0.351461	−	0.936203i	$-0.385685\pi$
$911$	21.2161	0.702921	0.351461	+	0.936203i	$0.385685\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.22711i	0.0735455i
$918$	0	0
$919$	23.5478	0.776771	0.388385	−	0.921497i	$-0.373033\pi$
$919$	23.5478	0.776771	0.388385	+	0.921497i	$0.373033\pi$
$920$	0	0
$921$	6.44882	0.212496
$922$	0	0
$923$	− 24.1801i	− 0.795896i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	26.5597i	0.872334i
$928$	0	0
$929$	42.7223	1.40167	0.700836	−	0.713322i	$-0.252810\pi$
$929$	42.7223	1.40167	0.700836	+	0.713322i	$0.252810\pi$
$930$	0	0
$931$	−5.04502	−0.165344
$932$	0	0
$933$	− 17.7458i	− 0.580972i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	31.1890i	1.01890i	0.860500	+	0.509451i	$0.170151\pi$
$937$	31.1890i	1.01890i	−0.860500	+	0.509451i	$0.829849\pi$
$938$	0	0
$939$	75.5671	2.46604
$940$	0	0
$941$	20.2251	0.659319	0.329659	−	0.944100i	$-0.393066\pi$
$941$	20.2251	0.659319	0.329659	+	0.944100i	$0.393066\pi$
$942$	0	0
$943$	− 32.9044i	− 1.07151i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	20.1801i	0.655764i	0.944719	+	0.327882i	$0.106335\pi$
$947$	20.1801i	0.655764i	−0.944719	+	0.327882i	$0.893665\pi$
$948$	0	0
$949$	−54.8719	−1.78122
$950$	0	0
$951$	25.1406	0.815241
$952$	0	0
$953$	20.6856i	0.670071i	0.942206	+	0.335035i	$0.108748\pi$
$953$	20.6856i	0.670071i	−0.942206	+	0.335035i	$0.891252\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	13.8358	0.446782
$960$	0	0
$961$	−23.1801	−0.747744
$962$	0	0
$963$	− 10.0096i	− 0.322556i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	53.9530i	1.73501i	0.497428	+	0.867505i	$0.334278\pi$
$967$	53.9530i	1.73501i	−0.497428	+	0.867505i	$0.665722\pi$
$968$	0	0
$969$	−1.39821	−0.0449169
$970$	0	0
$971$	−9.41277	−0.302070	−0.151035	−	0.988528i	$-0.548261\pi$
$971$	−9.41277	−0.302070	−0.151035	+	0.988528i	$0.548261\pi$
$972$	0	0
$973$	− 5.17447i	− 0.165886i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.21881i	0.134972i	0.997720	+	0.0674859i	$0.0214978\pi$
$977$	4.21881i	0.134972i	−0.997720	+	0.0674859i	$0.978502\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	35.5928	1.13639
$982$	0	0
$983$	− 10.3088i	− 0.328801i	−0.986394	−	0.164401i	$-0.947431\pi$
$983$	− 10.3088i	− 0.328801i	0.986394	−	0.164401i	$-0.0525690\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 6.01119i	− 0.191338i
$988$	0	0
$989$	52.2701	1.66209
$990$	0	0
$991$	54.6356	1.73556	0.867779	−	0.496951i	$-0.165547\pi$
$991$	54.6356	1.73556	0.867779	+	0.496951i	$0.165547\pi$
$992$	0	0
$993$	34.1350i	1.08324i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 22.9169i	− 0.725786i	−0.931831	−	0.362893i	$-0.881789\pi$
$997$	− 22.9169i	− 0.725786i	0.931831	−	0.362893i	$-0.118211\pi$
$998$	0	0
$999$	1.50280	0.0475464

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.d.l.3649.5		6
5.2	odd	4		760.2.a.j.1.3	✓	3
5.3	odd	4		3800.2.a.x.1.1		3
5.4	even	2	inner	3800.2.d.l.3649.2		6
15.2	even	4		6840.2.a.bg.1.2		3
20.3	even	4		7600.2.a.bq.1.3		3
20.7	even	4		1520.2.a.s.1.1		3
40.27	even	4		6080.2.a.bq.1.3		3
40.37	odd	4		6080.2.a.bv.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.j.1.3	✓	3	5.2	odd	4
1520.2.a.s.1.1		3	20.7	even	4
3800.2.a.x.1.1		3	5.3	odd	4
3800.2.d.l.3649.2		6	5.4	even	2	inner
3800.2.d.l.3649.5		6	1.1	even	1	trivial
6080.2.a.bq.1.3		3	40.27	even	4
6080.2.a.bv.1.1		3	40.37	odd	4
6840.2.a.bg.1.2		3	15.2	even	4
7600.2.a.bq.1.3		3	20.3	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)

\(f(q)\)	\(=\)	\(q+2.32340i q^{3} +1.39821i q^{7} -2.39821 q^{9} +O(q^{10})\)	\(q+2.32340i q^{3} +1.39821i q^{7} -2.39821 q^{9} +4.32340i q^{13} -0.601793i q^{17} -1.00000 q^{19} -3.24860 q^{21} +6.04502i q^{23} +1.39821i q^{27} -4.60179 q^{29} +2.79641 q^{31} -1.07480i q^{37} -10.0450 q^{39} -5.44322 q^{41} -8.64681i q^{43} +1.85039i q^{47} +5.04502 q^{49} +1.39821 q^{51} -3.11982i q^{53} -2.32340i q^{57} +6.69182 q^{59} -2.64681 q^{61} -3.35319i q^{63} +14.4134i q^{67} -14.0450 q^{69} -5.59283 q^{71} +12.6918i q^{73} +4.64681 q^{79} -10.4432 q^{81} -1.20359i q^{83} -10.6918i q^{87} -9.44322 q^{89} -6.04502 q^{91} +6.49720i q^{93} -4.51803i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 8 q^{9}+O(q^{10}) \) 6 * q - 8 * q^9	\( 6 q - 8 q^{9} - 6 q^{19} + 6 q^{21} - 34 q^{29} + 4 q^{31} - 22 q^{39} + 12 q^{41} - 8 q^{49} + 2 q^{51} - 30 q^{59} + 16 q^{61} - 46 q^{69} - 8 q^{71} - 4 q^{79} - 18 q^{81} - 12 q^{89} + 2 q^{91}+O(q^{100}) \) 6 * q - 8 * q^9 - 6 * q^19 + 6 * q^21 - 34 * q^29 + 4 * q^31 - 22 * q^39 + 12 * q^41 - 8 * q^49 + 2 * q^51 - 30 * q^59 + 16 * q^61 - 46 * q^69 - 8 * q^71 - 4 * q^79 - 18 * q^81 - 12 * q^89 + 2 * q^91

Introduction

L-functions

Modular forms

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Database

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