LMFDB - Embedded newform 3800.2.d.q.3649.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3800,2,Mod(3649,3800)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [12,0,0,0,0,0,0,0,-12,0,6,0,0,0,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$30.3431527681$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 24x^{10} + 194x^{8} + 618x^{6} + 733x^{4} + 286x^{2} + 9 $ x^12 + 24x^10 + 194x^8 + 618x^6 + 733x^4 + 286*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.10
Root			$1.93590i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.q.3649.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.93590i q^{3} +1.24708i q^{7} -0.747704 q^{9} -0.513860 q^{11} -6.15670i q^{13} -4.51986i q^{17} +1.00000 q^{19} -2.41421 q^{21} +5.86084i q^{23} +4.36022i q^{27} -6.62700 q^{29} +6.41995 q^{31} -0.994780i q^{33} -1.40671i q^{37} +11.9187 q^{39} +10.6870 q^{41} +3.04878i q^{43} -1.99478i q^{47} +5.44480 q^{49} +8.74998 q^{51} +14.0848i q^{53} +1.93590i q^{57} +4.34261 q^{59} +10.7173 q^{61} -0.932444i q^{63} -9.89978i q^{67} -11.3460 q^{69} +7.42517 q^{71} -12.8079i q^{73} -0.640822i q^{77} -2.56138 q^{79} -10.6841 q^{81} +7.50864i q^{83} -12.8292i q^{87} +7.85353 q^{89} +7.67787 q^{91} +12.4284i q^{93} +6.74797i q^{97} +0.384215 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{9} + 6 q^{11} + 12 q^{19} + 30 q^{21} - 18 q^{29} + 10 q^{31} - 24 q^{39} + 6 q^{41} - 44 q^{49} + 66 q^{51} + 18 q^{61} + 22 q^{69} + 38 q^{71} + 32 q^{79} + 52 q^{81} - 28 q^{89} + 84 q^{91}+ \cdots + 12 q^{99}+O(q^{100}) $ 12 * q - 12 * q^9 + 6 * q^11 + 12 * q^19 + 30 * q^21 - 18 * q^29 + 10 * q^31 - 24 * q^39 + 6 * q^41 - 44 * q^49 + 66 * q^51 + 18 * q^61 + 22 * q^69 + 38 * q^71 + 32 * q^79 + 52 * q^81 - 28 * q^89 + 84 * 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
$3$	1.93590i	1.11769i	0.829272	+	0.558846i	$0.188756\pi$
$3$	1.93590i	1.11769i	−0.829272	+	0.558846i	$0.811244\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.24708i	0.471350i	0.971832	+	0.235675i	$0.0757302\pi$
$7$	1.24708i	0.471350i	−0.971832	+	0.235675i	$0.924270\pi$
$8$	0	0
$9$	−0.747704	−0.249235
$10$	0	0
$11$	−0.513860	−0.154935	−0.0774673	−	0.996995i	$-0.524683\pi$
$11$	−0.513860	−0.154935	−0.0774673	+	0.996995i	$0.524683\pi$
$12$	0	0
$13$	− 6.15670i	− 1.70756i	−0.520633	−	0.853780i	$-0.674304\pi$
$13$	− 6.15670i	− 1.70756i	0.520633	−	0.853780i	$-0.325696\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 4.51986i	− 1.09623i	−0.836404	−	0.548113i	$-0.815346\pi$
$17$	− 4.51986i	− 1.09623i	0.836404	−	0.548113i	$-0.184654\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−2.41421	−0.526824
$22$	0	0
$23$	5.86084i	1.22207i	0.791603	+	0.611035i	$0.209247\pi$
$23$	5.86084i	1.22207i	−0.791603	+	0.611035i	$0.790753\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.36022i	0.839124i
$28$	0	0
$29$	−6.62700	−1.23060	−0.615302	−	0.788292i	$-0.710966\pi$
$29$	−6.62700	−1.23060	−0.615302	+	0.788292i	$0.710966\pi$
$30$	0	0
$31$	6.41995	1.15306	0.576528	−	0.817077i	$-0.304407\pi$
$31$	6.41995	1.15306	0.576528	+	0.817077i	$0.304407\pi$
$32$	0	0
$33$	− 0.994780i	− 0.173169i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 1.40671i	− 0.231263i	−0.993292	−	0.115631i	$-0.963111\pi$
$37$	− 1.40671i	− 0.231263i	0.993292	−	0.115631i	$-0.0368891\pi$
$38$	0	0
$39$	11.9187	1.90853
$40$	0	0
$41$	10.6870	1.66903	0.834514	−	0.550987i	$-0.185749\pi$
$41$	10.6870	1.66903	0.834514	+	0.550987i	$0.185749\pi$
$42$	0	0
$43$	3.04878i	0.464934i	0.972604	+	0.232467i	$0.0746798\pi$
$43$	3.04878i	0.464934i	−0.972604	+	0.232467i	$0.925320\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 1.99478i	− 0.290969i	−0.989361	−	0.145484i	$-0.953526\pi$
$47$	− 1.99478i	− 0.290969i	0.989361	−	0.145484i	$-0.0464740\pi$
$48$	0	0
$49$	5.44480	0.777829
$50$	0	0
$51$	8.74998	1.22524
$52$	0	0
$53$	14.0848i	1.93469i	0.253459	+	0.967346i	$0.418432\pi$
$53$	14.0848i	1.93469i	−0.253459	+	0.967346i	$0.581568\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.93590i	0.256416i
$58$	0	0
$59$	4.34261	0.565360	0.282680	−	0.959214i	$-0.408777\pi$
$59$	4.34261	0.565360	0.282680	+	0.959214i	$0.408777\pi$
$60$	0	0
$61$	10.7173	1.37221	0.686106	−	0.727502i	$-0.259319\pi$
$61$	10.7173	1.37221	0.686106	+	0.727502i	$0.259319\pi$
$62$	0	0
$63$	− 0.932444i	− 0.117477i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 9.89978i	− 1.20945i	−0.796434	−	0.604725i	$-0.793283\pi$
$67$	− 9.89978i	− 1.20945i	0.796434	−	0.604725i	$-0.206717\pi$
$68$	0	0
$69$	−11.3460	−1.36590
$70$	0	0
$71$	7.42517	0.881205	0.440603	−	0.897702i	$-0.354765\pi$
$71$	7.42517	0.881205	0.440603	+	0.897702i	$0.354765\pi$
$72$	0	0
$73$	− 12.8079i	− 1.49905i	−0.661976	−	0.749525i	$-0.730282\pi$
$73$	− 12.8079i	− 1.49905i	0.661976	−	0.749525i	$-0.269718\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 0.640822i	− 0.0730285i
$78$	0	0
$79$	−2.56138	−0.288178	−0.144089	−	0.989565i	$-0.546025\pi$
$79$	−2.56138	−0.288178	−0.144089	+	0.989565i	$0.546025\pi$
$80$	0	0
$81$	−10.6841	−1.18712
$82$	0	0
$83$	7.50864i	0.824180i	0.911143	+	0.412090i	$0.135201\pi$
$83$	7.50864i	0.824180i	−0.911143	+	0.412090i	$0.864799\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 12.8292i	− 1.37543i
$88$	0	0
$89$	7.85353	0.832473	0.416236	−	0.909256i	$-0.363349\pi$
$89$	7.85353	0.832473	0.416236	+	0.909256i	$0.363349\pi$
$90$	0	0
$91$	7.67787	0.804859
$92$	0	0
$93$	12.4284i	1.28876i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.74797i	0.685152i	0.939490	+	0.342576i	$0.111299\pi$
$97$	6.74797i	0.685152i	−0.939490	+	0.342576i	$0.888701\pi$
$98$	0	0
$99$	0.384215	0.0386151
$100$	0	0
$101$	17.6503	1.75627	0.878134	−	0.478415i	$-0.158788\pi$
$101$	17.6503	1.75627	0.878134	+	0.478415i	$0.158788\pi$
$102$	0	0
$103$	5.84147i	0.575577i	0.957694	+	0.287789i	$0.0929201\pi$
$103$	5.84147i	0.575577i	−0.957694	+	0.287789i	$0.907080\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4.14743i	− 0.400947i	−0.979699	−	0.200474i	$-0.935752\pi$
$107$	− 4.14743i	− 0.400947i	0.979699	−	0.200474i	$-0.0642481\pi$
$108$	0	0
$109$	−5.69767	−0.545738	−0.272869	−	0.962051i	$-0.587973\pi$
$109$	−5.69767	−0.545738	−0.272869	+	0.962051i	$0.587973\pi$
$110$	0	0
$111$	2.72326	0.258480
$112$	0	0
$113$	5.36073i	0.504295i	0.967689	+	0.252148i	$0.0811368\pi$
$113$	5.36073i	0.504295i	−0.967689	+	0.252148i	$0.918863\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.60339i	0.425584i
$118$	0	0
$119$	5.63660	0.516707
$120$	0	0
$121$	−10.7359	−0.975995
$122$	0	0
$123$	20.6889i	1.86546i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	18.3991i	1.63266i	0.577589	+	0.816328i	$0.303994\pi$
$127$	18.3991i	1.63266i	−0.577589	+	0.816328i	$0.696006\pi$
$128$	0	0
$129$	−5.90212	−0.519653
$130$	0	0
$131$	−14.4958	−1.26650	−0.633252	−	0.773946i	$-0.718280\pi$
$131$	−14.4958	−1.26650	−0.633252	+	0.773946i	$0.718280\pi$
$132$	0	0
$133$	1.24708i	0.108135i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.40281i	0.290722i	0.989379	+	0.145361i	$0.0464343\pi$
$137$	3.40281i	0.290722i	−0.989379	+	0.145361i	$0.953566\pi$
$138$	0	0
$139$	4.01062	0.340176	0.170088	−	0.985429i	$-0.445595\pi$
$139$	4.01062	0.340176	0.170088	+	0.985429i	$0.445595\pi$
$140$	0	0
$141$	3.86169	0.325213
$142$	0	0
$143$	3.16368i	0.264560i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	10.5406i	0.869373i
$148$	0	0
$149$	−9.42255	−0.771926	−0.385963	−	0.922514i	$-0.626131\pi$
$149$	−9.42255	−0.771926	−0.385963	+	0.922514i	$0.626131\pi$
$150$	0	0
$151$	1.60111	0.130297	0.0651483	−	0.997876i	$-0.479248\pi$
$151$	1.60111	0.130297	0.0651483	+	0.997876i	$0.479248\pi$
$152$	0	0
$153$	3.37952i	0.273218i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	18.7241i	1.49434i	0.664631	+	0.747172i	$0.268589\pi$
$157$	18.7241i	1.49434i	−0.664631	+	0.747172i	$0.731411\pi$
$158$	0	0
$159$	−27.2667	−2.16239
$160$	0	0
$161$	−7.30892	−0.576024
$162$	0	0
$163$	− 15.6370i	− 1.22479i	−0.790553	−	0.612394i	$-0.790207\pi$
$163$	− 15.6370i	− 1.22479i	0.790553	−	0.612394i	$-0.209793\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	17.5865i	1.36088i	0.732801	+	0.680442i	$0.238212\pi$
$167$	17.5865i	1.36088i	−0.732801	+	0.680442i	$0.761788\pi$
$168$	0	0
$169$	−24.9049	−1.91576
$170$	0	0
$171$	−0.747704	−0.0571784
$172$	0	0
$173$	− 18.7799i	− 1.42781i	−0.700245	−	0.713903i	$-0.746926\pi$
$173$	− 18.7799i	− 1.42781i	0.700245	−	0.713903i	$-0.253074\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	8.40686i	0.631898i
$178$	0	0
$179$	4.22047	0.315453	0.157726	−	0.987483i	$-0.449584\pi$
$179$	4.22047	0.315453	0.157726	+	0.987483i	$0.449584\pi$
$180$	0	0
$181$	11.8180	0.878424	0.439212	−	0.898383i	$-0.355258\pi$
$181$	11.8180	0.878424	0.439212	+	0.898383i	$0.355258\pi$
$182$	0	0
$183$	20.7476i	1.53371i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.32257i	0.169843i
$188$	0	0
$189$	−5.43752	−0.395521
$190$	0	0
$191$	26.7184	1.93327	0.966636	−	0.256153i	$-0.0824551\pi$
$191$	26.7184	1.93327	0.966636	+	0.256153i	$0.0824551\pi$
$192$	0	0
$193$	− 9.25224i	− 0.665990i	−0.942929	−	0.332995i	$-0.891941\pi$
$193$	− 9.25224i	− 0.665990i	0.942929	−	0.332995i	$-0.108059\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.32896i	0.237179i	0.992943	+	0.118589i	$0.0378372\pi$
$197$	3.32896i	0.237179i	−0.992943	+	0.118589i	$0.962163\pi$
$198$	0	0
$199$	0.0700969	0.00496903	0.00248452	−	0.999997i	$-0.499209\pi$
$199$	0.0700969	0.00496903	0.00248452	+	0.999997i	$0.499209\pi$
$200$	0	0
$201$	19.1650	1.35179
$202$	0	0
$203$	− 8.26437i	− 0.580045i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 4.38218i	− 0.304583i
$208$	0	0
$209$	−0.513860	−0.0355444
$210$	0	0
$211$	15.5952	1.07362	0.536810	−	0.843703i	$-0.319629\pi$
$211$	15.5952	1.07362	0.536810	+	0.843703i	$0.319629\pi$
$212$	0	0
$213$	14.3744i	0.984916i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	8.00616i	0.543494i
$218$	0	0
$219$	24.7948	1.67547
$220$	0	0
$221$	−27.8274	−1.87187
$222$	0	0
$223$	− 22.4019i	− 1.50014i	−0.661358	−	0.750071i	$-0.730020\pi$
$223$	− 22.4019i	− 1.50014i	0.661358	−	0.750071i	$-0.269980\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 24.3464i	− 1.61593i	−0.589234	−	0.807963i	$-0.700570\pi$
$227$	− 24.3464i	− 1.61593i	0.589234	−	0.807963i	$-0.299430\pi$
$228$	0	0
$229$	5.27650	0.348681	0.174340	−	0.984685i	$-0.444221\pi$
$229$	5.27650	0.348681	0.174340	+	0.984685i	$0.444221\pi$
$230$	0	0
$231$	1.24057	0.0816233
$232$	0	0
$233$	− 4.30335i	− 0.281922i	−0.990015	−	0.140961i	$-0.954981\pi$
$233$	− 4.30335i	− 0.281922i	0.990015	−	0.140961i	$-0.0450192\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 4.95857i	− 0.322094i
$238$	0	0
$239$	1.44454	0.0934395	0.0467197	−	0.998908i	$-0.485123\pi$
$239$	1.44454	0.0934395	0.0467197	+	0.998908i	$0.485123\pi$
$240$	0	0
$241$	5.20083	0.335015	0.167507	−	0.985871i	$-0.446428\pi$
$241$	5.20083	0.335015	0.167507	+	0.985871i	$0.446428\pi$
$242$	0	0
$243$	− 7.60259i	− 0.487707i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 6.15670i	− 0.391741i
$248$	0	0
$249$	−14.5360	−0.921180
$250$	0	0
$251$	1.57046	0.0991267	0.0495634	−	0.998771i	$-0.484217\pi$
$251$	1.57046	0.0991267	0.0495634	+	0.998771i	$0.484217\pi$
$252$	0	0
$253$	− 3.01165i	− 0.189341i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	24.2831i	1.51474i	0.652988	+	0.757368i	$0.273515\pi$
$257$	24.2831i	1.51474i	−0.652988	+	0.757368i	$0.726485\pi$
$258$	0	0
$259$	1.75428	0.109006
$260$	0	0
$261$	4.95504	0.306709
$262$	0	0
$263$	7.50729i	0.462919i	0.972844	+	0.231460i	$0.0743501\pi$
$263$	7.50729i	0.462919i	−0.972844	+	0.231460i	$0.925650\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	15.2036i	0.930448i
$268$	0	0
$269$	−9.37783	−0.571776	−0.285888	−	0.958263i	$-0.592289\pi$
$269$	−9.37783	−0.571776	−0.285888	+	0.958263i	$0.592289\pi$
$270$	0	0
$271$	16.8103	1.02115	0.510577	−	0.859832i	$-0.329432\pi$
$271$	16.8103	1.02115	0.510577	+	0.859832i	$0.329432\pi$
$272$	0	0
$273$	14.8636i	0.899585i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.55337i	0.0933328i	0.998911	+	0.0466664i	$0.0148598\pi$
$277$	1.55337i	0.0933328i	−0.998911	+	0.0466664i	$0.985140\pi$
$278$	0	0
$279$	−4.80022	−0.287382
$280$	0	0
$281$	−5.68294	−0.339016	−0.169508	−	0.985529i	$-0.554218\pi$
$281$	−5.68294	−0.339016	−0.169508	+	0.985529i	$0.554218\pi$
$282$	0	0
$283$	28.2237i	1.67773i	0.544342	+	0.838864i	$0.316780\pi$
$283$	28.2237i	1.67773i	−0.544342	+	0.838864i	$0.683220\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	13.3275i	0.786697i
$288$	0	0
$289$	−3.42909	−0.201711
$290$	0	0
$291$	−13.0634	−0.765789
$292$	0	0
$293$	− 5.92000i	− 0.345850i	−0.984935	−	0.172925i	$-0.944678\pi$
$293$	− 5.92000i	− 0.345850i	0.984935	−	0.172925i	$-0.0553218\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 2.24054i	− 0.130009i
$298$	0	0
$299$	36.0835	2.08676
$300$	0	0
$301$	−3.80206	−0.219147
$302$	0	0
$303$	34.1691i	1.96297i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 5.37023i	− 0.306495i	−0.988188	−	0.153248i	$-0.951027\pi$
$307$	− 5.37023i	− 0.306495i	0.988188	−	0.153248i	$-0.0489732\pi$
$308$	0	0
$309$	−11.3085	−0.643318
$310$	0	0
$311$	−3.90761	−0.221580	−0.110790	−	0.993844i	$-0.535338\pi$
$311$	−3.90761	−0.221580	−0.110790	+	0.993844i	$0.535338\pi$
$312$	0	0
$313$	− 30.1038i	− 1.70156i	−0.525518	−	0.850782i	$-0.676129\pi$
$313$	− 30.1038i	− 1.70156i	0.525518	−	0.850782i	$-0.323871\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.7722i	0.942019i	0.882128	+	0.471010i	$0.156110\pi$
$317$	16.7722i	0.942019i	−0.882128	+	0.471010i	$0.843890\pi$
$318$	0	0
$319$	3.40535	0.190663
$320$	0	0
$321$	8.02900	0.448135
$322$	0	0
$323$	− 4.51986i	− 0.251491i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 11.0301i	− 0.609967i
$328$	0	0
$329$	2.48764	0.137148
$330$	0	0
$331$	−24.4039	−1.34136	−0.670680	−	0.741747i	$-0.733998\pi$
$331$	−24.4039	−1.34136	−0.670680	+	0.741747i	$0.733998\pi$
$332$	0	0
$333$	1.05181i	0.0576387i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	11.0027i	0.599355i	0.954041	+	0.299678i	$0.0968791\pi$
$337$	11.0027i	0.599355i	−0.954041	+	0.299678i	$0.903121\pi$
$338$	0	0
$339$	−10.3778	−0.563646
$340$	0	0
$341$	−3.29895	−0.178648
$342$	0	0
$343$	15.5196i	0.837980i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 13.2447i	− 0.711014i	−0.934674	−	0.355507i	$-0.884308\pi$
$347$	− 13.2447i	− 0.711014i	0.934674	−	0.355507i	$-0.115692\pi$
$348$	0	0
$349$	−1.03534	−0.0554206	−0.0277103	−	0.999616i	$-0.508822\pi$
$349$	−1.03534	−0.0554206	−0.0277103	+	0.999616i	$0.508822\pi$
$350$	0	0
$351$	26.8445	1.43286
$352$	0	0
$353$	− 5.37333i	− 0.285994i	−0.989723	−	0.142997i	$-0.954326\pi$
$353$	− 5.37333i	− 0.285994i	0.989723	−	0.142997i	$-0.0456739\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	10.9119i	0.577519i
$358$	0	0
$359$	−23.6189	−1.24656	−0.623280	−	0.781999i	$-0.714200\pi$
$359$	−23.6189	−1.24656	−0.623280	+	0.781999i	$0.714200\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	− 20.7837i	− 1.09086i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 7.75683i	− 0.404903i	−0.979292	−	0.202452i	$-0.935109\pi$
$367$	− 7.75683i	− 0.404903i	0.979292	−	0.202452i	$-0.0648909\pi$
$368$	0	0
$369$	−7.99071	−0.415980
$370$	0	0
$371$	−17.5648	−0.911918
$372$	0	0
$373$	32.2543i	1.67006i	0.550201	+	0.835032i	$0.314551\pi$
$373$	32.2543i	1.67006i	−0.550201	+	0.835032i	$0.685449\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	40.8004i	2.10133i
$378$	0	0
$379$	−28.7810	−1.47838	−0.739191	−	0.673496i	$-0.764792\pi$
$379$	−28.7810	−1.47838	−0.739191	+	0.673496i	$0.764792\pi$
$380$	0	0
$381$	−35.6188	−1.82481
$382$	0	0
$383$	− 11.0718i	− 0.565743i	−0.959158	−	0.282871i	$-0.908713\pi$
$383$	− 11.0718i	− 0.565743i	0.959158	−	0.282871i	$-0.0912869\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 2.27958i	− 0.115878i
$388$	0	0
$389$	4.65488	0.236012	0.118006	−	0.993013i	$-0.462350\pi$
$389$	4.65488	0.236012	0.118006	+	0.993013i	$0.462350\pi$
$390$	0	0
$391$	26.4902	1.33967
$392$	0	0
$393$	− 28.0624i	− 1.41556i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 0.573392i	− 0.0287777i	−0.999896	−	0.0143889i	$-0.995420\pi$
$397$	− 0.573392i	− 0.0287777i	0.999896	−	0.0143889i	$-0.00458027\pi$
$398$	0	0
$399$	−2.41421	−0.120862
$400$	0	0
$401$	35.9965	1.79758	0.898791	−	0.438378i	$-0.144447\pi$
$401$	35.9965	1.79758	0.898791	+	0.438378i	$0.144447\pi$
$402$	0	0
$403$	− 39.5257i	− 1.96891i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.722854i	0.0358305i
$408$	0	0
$409$	−18.3961	−0.909626	−0.454813	−	0.890587i	$-0.650294\pi$
$409$	−18.3961	−0.909626	−0.454813	+	0.890587i	$0.650294\pi$
$410$	0	0
$411$	−6.58750	−0.324937
$412$	0	0
$413$	5.41557i	0.266483i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	7.76415i	0.380212i
$418$	0	0
$419$	−14.0610	−0.686923	−0.343461	−	0.939167i	$-0.611599\pi$
$419$	−14.0610	−0.686923	−0.343461	+	0.939167i	$0.611599\pi$
$420$	0	0
$421$	−36.7041	−1.78885	−0.894425	−	0.447218i	$-0.852415\pi$
$421$	−36.7041	−1.78885	−0.894425	+	0.447218i	$0.852415\pi$
$422$	0	0
$423$	1.49151i	0.0725195i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	13.3653i	0.646793i
$428$	0	0
$429$	−6.12456	−0.295697
$430$	0	0
$431$	21.2682	1.02445	0.512226	−	0.858851i	$-0.328821\pi$
$431$	21.2682	1.02445	0.512226	+	0.858851i	$0.328821\pi$
$432$	0	0
$433$	11.8229i	0.568172i	0.958799	+	0.284086i	$0.0916900\pi$
$433$	11.8229i	0.568172i	−0.958799	+	0.284086i	$0.908310\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.86084i	0.280362i
$438$	0	0
$439$	8.03728	0.383599	0.191799	−	0.981434i	$-0.438568\pi$
$439$	8.03728	0.383599	0.191799	+	0.981434i	$0.438568\pi$
$440$	0	0
$441$	−4.07110	−0.193862
$442$	0	0
$443$	10.3632i	0.492368i	0.969223	+	0.246184i	$0.0791768\pi$
$443$	10.3632i	0.492368i	−0.969223	+	0.246184i	$0.920823\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 18.2411i	− 0.862775i
$448$	0	0
$449$	33.4637	1.57925	0.789625	−	0.613590i	$-0.210275\pi$
$449$	33.4637	1.57925	0.789625	+	0.613590i	$0.210275\pi$
$450$	0	0
$451$	−5.49161	−0.258590
$452$	0	0
$453$	3.09959i	0.145631i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 27.6968i	− 1.29560i	−0.761809	−	0.647801i	$-0.775689\pi$
$457$	− 27.6968i	− 1.29560i	0.761809	−	0.647801i	$-0.224311\pi$
$458$	0	0
$459$	19.7075	0.919870
$460$	0	0
$461$	4.70500	0.219134	0.109567	−	0.993979i	$-0.465054\pi$
$461$	4.70500	0.219134	0.109567	+	0.993979i	$0.465054\pi$
$462$	0	0
$463$	5.44945i	0.253257i	0.991950	+	0.126629i	$0.0404157\pi$
$463$	5.44945i	0.253257i	−0.991950	+	0.126629i	$0.959584\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.9687i	0.507571i	0.967260	+	0.253786i	$0.0816758\pi$
$467$	10.9687i	0.507571i	−0.967260	+	0.253786i	$0.918324\pi$
$468$	0	0
$469$	12.3458	0.570075
$470$	0	0
$471$	−36.2479	−1.67022
$472$	0	0
$473$	− 1.56664i	− 0.0720343i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 10.5312i	− 0.482193i
$478$	0	0
$479$	14.4663	0.660980	0.330490	−	0.943809i	$-0.392786\pi$
$479$	14.4663	0.660980	0.330490	+	0.943809i	$0.392786\pi$
$480$	0	0
$481$	−8.66072	−0.394895
$482$	0	0
$483$	− 14.1493i	− 0.643817i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	26.1973i	1.18711i	0.804793	+	0.593556i	$0.202276\pi$
$487$	26.1973i	1.18711i	−0.804793	+	0.593556i	$0.797724\pi$
$488$	0	0
$489$	30.2717	1.36894
$490$	0	0
$491$	12.3251	0.556224	0.278112	−	0.960549i	$-0.410291\pi$
$491$	12.3251	0.556224	0.278112	+	0.960549i	$0.410291\pi$
$492$	0	0
$493$	29.9531i	1.34902i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.25975i	0.415356i
$498$	0	0
$499$	−27.4827	−1.23029	−0.615146	−	0.788413i	$-0.710903\pi$
$499$	−27.4827	−1.23029	−0.615146	+	0.788413i	$0.710903\pi$
$500$	0	0
$501$	−34.0457	−1.52105
$502$	0	0
$503$	− 21.0989i	− 0.940756i	−0.882465	−	0.470378i	$-0.844118\pi$
$503$	− 21.0989i	− 0.940756i	0.882465	−	0.470378i	$-0.155882\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 48.2134i	− 2.14123i
$508$	0	0
$509$	14.3975	0.638156	0.319078	−	0.947728i	$-0.396627\pi$
$509$	14.3975	0.638156	0.319078	+	0.947728i	$0.396627\pi$
$510$	0	0
$511$	15.9724	0.706577
$512$	0	0
$513$	4.36022i	0.192508i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.02504i	0.0450811i
$518$	0	0
$519$	36.3559	1.59585
$520$	0	0
$521$	9.60225	0.420682	0.210341	−	0.977628i	$-0.432543\pi$
$521$	9.60225	0.420682	0.210341	+	0.977628i	$0.432543\pi$
$522$	0	0
$523$	− 1.07927i	− 0.0471933i	−0.999722	−	0.0235966i	$-0.992488\pi$
$523$	− 1.07927i	− 0.0471933i	0.999722	−	0.0235966i	$-0.00751174\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 29.0172i	− 1.26401i
$528$	0	0
$529$	−11.3495	−0.493457
$530$	0	0
$531$	−3.24699	−0.140907
$532$	0	0
$533$	− 65.7966i	− 2.84997i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	8.17040i	0.352579i
$538$	0	0
$539$	−2.79786	−0.120513
$540$	0	0
$541$	−3.55082	−0.152662	−0.0763309	−	0.997083i	$-0.524321\pi$
$541$	−3.55082	−0.152662	−0.0763309	+	0.997083i	$0.524321\pi$
$542$	0	0
$543$	22.8784i	0.981807i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 44.5233i	− 1.90368i	−0.306596	−	0.951840i	$-0.599190\pi$
$547$	− 44.5233i	− 1.90368i	0.306596	−	0.951840i	$-0.400810\pi$
$548$	0	0
$549$	−8.01339	−0.342003
$550$	0	0
$551$	−6.62700	−0.282320
$552$	0	0
$553$	− 3.19424i	− 0.135833i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11.2157i	0.475227i	0.971360	+	0.237613i	$0.0763651\pi$
$557$	11.2157i	0.475227i	−0.971360	+	0.237613i	$0.923635\pi$
$558$	0	0
$559$	18.7704	0.793903
$560$	0	0
$561$	−4.49626	−0.189832
$562$	0	0
$563$	36.1900i	1.52523i	0.646854	+	0.762614i	$0.276084\pi$
$563$	36.1900i	1.52523i	−0.646854	+	0.762614i	$0.723916\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 13.3238i	− 0.559548i
$568$	0	0
$569$	9.60022	0.402462	0.201231	−	0.979544i	$-0.435506\pi$
$569$	9.60022	0.402462	0.201231	+	0.979544i	$0.435506\pi$
$570$	0	0
$571$	29.4150	1.23098	0.615490	−	0.788145i	$-0.288958\pi$
$571$	29.4150	1.23098	0.615490	+	0.788145i	$0.288958\pi$
$572$	0	0
$573$	51.7240i	2.16080i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 20.9070i	− 0.870371i	−0.900341	−	0.435186i	$-0.856683\pi$
$577$	− 20.9070i	− 0.870371i	0.900341	−	0.435186i	$-0.143317\pi$
$578$	0	0
$579$	17.9114	0.744372
$580$	0	0
$581$	−9.36384	−0.388478
$582$	0	0
$583$	− 7.23760i	− 0.299751i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 41.2156i	− 1.70115i	−0.525853	−	0.850576i	$-0.676254\pi$
$587$	− 41.2156i	− 1.70115i	0.525853	−	0.850576i	$-0.323746\pi$
$588$	0	0
$589$	6.41995	0.264529
$590$	0	0
$591$	−6.44453	−0.265093
$592$	0	0
$593$	− 32.0201i	− 1.31491i	−0.753495	−	0.657454i	$-0.771633\pi$
$593$	− 32.0201i	− 1.31491i	0.753495	−	0.657454i	$-0.228367\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0.135700i	0.00555385i
$598$	0	0
$599$	−46.7364	−1.90960	−0.954798	−	0.297256i	$-0.903929\pi$
$599$	−46.7364	−1.90960	−0.954798	+	0.297256i	$0.903929\pi$
$600$	0	0
$601$	−22.9230	−0.935050	−0.467525	−	0.883980i	$-0.654854\pi$
$601$	−22.9230	−0.935050	−0.467525	+	0.883980i	$0.654854\pi$
$602$	0	0
$603$	7.40211i	0.301437i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 38.7571i	− 1.57310i	−0.617525	−	0.786552i	$-0.711864\pi$
$607$	− 38.7571i	− 1.57310i	0.617525	−	0.786552i	$-0.288136\pi$
$608$	0	0
$609$	15.9990	0.648312
$610$	0	0
$611$	−12.2813	−0.496847
$612$	0	0
$613$	− 13.4487i	− 0.543187i	−0.962412	−	0.271593i	$-0.912449\pi$
$613$	− 13.4487i	− 0.543187i	0.962412	−	0.271593i	$-0.0875505\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	10.9727i	0.441744i	0.975303	+	0.220872i	$0.0708903\pi$
$617$	10.9727i	0.441744i	−0.975303	+	0.220872i	$0.929110\pi$
$618$	0	0
$619$	−46.1568	−1.85520	−0.927600	−	0.373575i	$-0.878132\pi$
$619$	−46.1568	−1.85520	−0.927600	+	0.373575i	$0.878132\pi$
$620$	0	0
$621$	−25.5546	−1.02547
$622$	0	0
$623$	9.79395i	0.392386i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 0.994780i	− 0.0397277i
$628$	0	0
$629$	−6.35815	−0.253516
$630$	0	0
$631$	43.3700	1.72653	0.863266	−	0.504749i	$-0.168415\pi$
$631$	43.3700	1.72653	0.863266	+	0.504749i	$0.168415\pi$
$632$	0	0
$633$	30.1908i	1.19998i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 33.5220i	− 1.32819i
$638$	0	0
$639$	−5.55183	−0.219627
$640$	0	0
$641$	16.6952	0.659420	0.329710	−	0.944082i	$-0.393049\pi$
$641$	16.6952	0.659420	0.329710	+	0.944082i	$0.393049\pi$
$642$	0	0
$643$	− 23.4243i	− 0.923765i	−0.886941	−	0.461882i	$-0.847174\pi$
$643$	− 23.4243i	− 0.923765i	0.886941	−	0.461882i	$-0.152826\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17.3397i	0.681694i	0.940119	+	0.340847i	$0.110714\pi$
$647$	17.3397i	0.681694i	−0.940119	+	0.340847i	$0.889286\pi$
$648$	0	0
$649$	−2.23149	−0.0875938
$650$	0	0
$651$	−15.4991	−0.607458
$652$	0	0
$653$	18.3541i	0.718251i	0.933289	+	0.359126i	$0.116925\pi$
$653$	18.3541i	0.718251i	−0.933289	+	0.359126i	$0.883075\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	9.57651i	0.373615i
$658$	0	0
$659$	23.4398	0.913085	0.456543	−	0.889702i	$-0.349088\pi$
$659$	23.4398	0.913085	0.456543	+	0.889702i	$0.349088\pi$
$660$	0	0
$661$	−20.7743	−0.808027	−0.404014	−	0.914753i	$-0.632385\pi$
$661$	−20.7743	−0.808027	−0.404014	+	0.914753i	$0.632385\pi$
$662$	0	0
$663$	− 53.8710i	− 2.09218i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 38.8398i	− 1.50388i
$668$	0	0
$669$	43.3678	1.67670
$670$	0	0
$671$	−5.50720	−0.212603
$672$	0	0
$673$	− 6.74273i	− 0.259913i	−0.991520	−	0.129957i	$-0.958516\pi$
$673$	− 6.74273i	− 0.259913i	0.991520	−	0.129957i	$-0.0414838\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 25.1589i	− 0.966934i	−0.875363	−	0.483467i	$-0.839377\pi$
$677$	− 25.1589i	− 0.966934i	0.875363	−	0.483467i	$-0.160623\pi$
$678$	0	0
$679$	−8.41523	−0.322947
$680$	0	0
$681$	47.1321	1.80611
$682$	0	0
$683$	− 13.7289i	− 0.525321i	−0.964888	−	0.262661i	$-0.915400\pi$
$683$	− 13.7289i	− 0.525321i	0.964888	−	0.262661i	$-0.0845999\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	10.2148i	0.389718i
$688$	0	0
$689$	86.7157	3.30360
$690$	0	0
$691$	26.8531	1.02154	0.510770	−	0.859718i	$-0.329361\pi$
$691$	26.8531	1.02154	0.510770	+	0.859718i	$0.329361\pi$
$692$	0	0
$693$	0.479145i	0.0182012i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 48.3037i	− 1.82963i
$698$	0	0
$699$	8.33085	0.315102
$700$	0	0
$701$	−20.1359	−0.760524	−0.380262	−	0.924879i	$-0.624166\pi$
$701$	−20.1359	−0.760524	−0.380262	+	0.924879i	$0.624166\pi$
$702$	0	0
$703$	− 1.40671i	− 0.0530553i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	22.0112i	0.827818i
$708$	0	0
$709$	−23.4008	−0.878837	−0.439419	−	0.898282i	$-0.644816\pi$
$709$	−23.4008	−0.878837	−0.439419	+	0.898282i	$0.644816\pi$
$710$	0	0
$711$	1.91515	0.0718239
$712$	0	0
$713$	37.6263i	1.40912i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.79648i	0.104437i
$718$	0	0
$719$	−10.5133	−0.392082	−0.196041	−	0.980596i	$-0.562809\pi$
$719$	−10.5133	−0.392082	−0.196041	+	0.980596i	$0.562809\pi$
$720$	0	0
$721$	−7.28476	−0.271299
$722$	0	0
$723$	10.0683i	0.374443i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	35.6031i	1.32044i	0.751070	+	0.660222i	$0.229538\pi$
$727$	35.6031i	1.32044i	−0.751070	+	0.660222i	$0.770462\pi$
$728$	0	0
$729$	−17.3343	−0.642011
$730$	0	0
$731$	13.7800	0.509673
$732$	0	0
$733$	− 37.2476i	− 1.37577i	−0.725818	−	0.687886i	$-0.758539\pi$
$733$	− 37.2476i	− 1.37577i	0.725818	−	0.687886i	$-0.241461\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.08710i	0.187386i
$738$	0	0
$739$	26.1515	0.961998	0.480999	−	0.876721i	$-0.340274\pi$
$739$	26.1515	0.961998	0.480999	+	0.876721i	$0.340274\pi$
$740$	0	0
$741$	11.9187	0.437846
$742$	0	0
$743$	16.4700i	0.604226i	0.953272	+	0.302113i	$0.0976920\pi$
$743$	16.4700i	0.604226i	−0.953272	+	0.302113i	$0.902308\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 5.61424i	− 0.205414i
$748$	0	0
$749$	5.17216	0.188987
$750$	0	0
$751$	−24.0052	−0.875962	−0.437981	−	0.898984i	$-0.644306\pi$
$751$	−24.0052	−0.875962	−0.437981	+	0.898984i	$0.644306\pi$
$752$	0	0
$753$	3.04026i	0.110793i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 34.5457i	− 1.25558i	−0.778381	−	0.627792i	$-0.783959\pi$
$757$	− 34.5457i	− 1.25558i	0.778381	−	0.627792i	$-0.216041\pi$
$758$	0	0
$759$	5.83025	0.211625
$760$	0	0
$761$	−34.6920	−1.25758	−0.628792	−	0.777573i	$-0.716450\pi$
$761$	−34.6920	−1.25758	−0.628792	+	0.777573i	$0.716450\pi$
$762$	0	0
$763$	− 7.10543i	− 0.257234i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 26.7362i	− 0.965387i
$768$	0	0
$769$	30.1152	1.08598	0.542991	−	0.839738i	$-0.317292\pi$
$769$	30.1152	1.08598	0.542991	+	0.839738i	$0.317292\pi$
$770$	0	0
$771$	−47.0096	−1.69301
$772$	0	0
$773$	29.9879i	1.07859i	0.842117	+	0.539295i	$0.181309\pi$
$773$	29.9879i	1.07859i	−0.842117	+	0.539295i	$0.818691\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3.39611i	0.121835i
$778$	0	0
$779$	10.6870	0.382901
$780$	0	0
$781$	−3.81549	−0.136529
$782$	0	0
$783$	− 28.8952i	− 1.03263i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 10.3894i	− 0.370342i	−0.982706	−	0.185171i	$-0.940716\pi$
$787$	− 10.3894i	− 0.370342i	0.982706	−	0.185171i	$-0.0592838\pi$
$788$	0	0
$789$	−14.5334	−0.517401
$790$	0	0
$791$	−6.68524	−0.237700
$792$	0	0
$793$	− 65.9833i	− 2.34314i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 32.8854i	− 1.16486i	−0.812881	−	0.582430i	$-0.802102\pi$
$797$	− 32.8854i	− 1.16486i	0.812881	−	0.582430i	$-0.197898\pi$
$798$	0	0
$799$	−9.01612	−0.318967
$800$	0	0
$801$	−5.87212	−0.207481
$802$	0	0
$803$	6.58145i	0.232254i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 18.1545i	− 0.639070i
$808$	0	0
$809$	16.5347	0.581329	0.290664	−	0.956825i	$-0.406124\pi$
$809$	16.5347	0.581329	0.290664	+	0.956825i	$0.406124\pi$
$810$	0	0
$811$	2.88477	0.101298	0.0506489	−	0.998717i	$-0.483871\pi$
$811$	2.88477	0.101298	0.0506489	+	0.998717i	$0.483871\pi$
$812$	0	0
$813$	32.5431i	1.14134i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.04878i	0.106663i
$818$	0	0
$819$	−5.74078	−0.200599
$820$	0	0
$821$	−10.0122	−0.349428	−0.174714	−	0.984619i	$-0.555900\pi$
$821$	−10.0122	−0.349428	−0.174714	+	0.984619i	$0.555900\pi$
$822$	0	0
$823$	26.1520i	0.911602i	0.890082	+	0.455801i	$0.150647\pi$
$823$	26.1520i	0.911602i	−0.890082	+	0.455801i	$0.849353\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	25.1880i	0.875873i	0.899006	+	0.437936i	$0.144290\pi$
$827$	25.1880i	0.875873i	−0.899006	+	0.437936i	$0.855710\pi$
$828$	0	0
$829$	−3.32822	−0.115594	−0.0577969	−	0.998328i	$-0.518408\pi$
$829$	−3.32822	−0.115594	−0.0577969	+	0.998328i	$0.518408\pi$
$830$	0	0
$831$	−3.00716	−0.104317
$832$	0	0
$833$	− 24.6097i	− 0.852676i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	27.9924i	0.967557i
$838$	0	0
$839$	50.5339	1.74462	0.872312	−	0.488949i	$-0.162620\pi$
$839$	50.5339	1.74462	0.872312	+	0.488949i	$0.162620\pi$
$840$	0	0
$841$	14.9171	0.514384
$842$	0	0
$843$	− 11.0016i	− 0.378915i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 13.3885i	− 0.460036i
$848$	0	0
$849$	−54.6383	−1.87518
$850$	0	0
$851$	8.24454	0.282619
$852$	0	0
$853$	− 54.6636i	− 1.87165i	−0.352470	−	0.935823i	$-0.614658\pi$
$853$	− 54.6636i	− 1.87165i	0.352470	−	0.935823i	$-0.385342\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 27.8578i	− 0.951603i	−0.879553	−	0.475802i	$-0.842158\pi$
$857$	− 27.8578i	− 0.951603i	0.879553	−	0.475802i	$-0.157842\pi$
$858$	0	0
$859$	55.2474	1.88502	0.942509	−	0.334181i	$-0.108460\pi$
$859$	55.2474	1.88502	0.942509	+	0.334181i	$0.108460\pi$
$860$	0	0
$861$	−25.8007	−0.879285
$862$	0	0
$863$	− 27.3258i	− 0.930181i	−0.885263	−	0.465090i	$-0.846022\pi$
$863$	− 27.3258i	− 0.930181i	0.885263	−	0.465090i	$-0.153978\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 6.63838i	− 0.225451i
$868$	0	0
$869$	1.31619	0.0446487
$870$	0	0
$871$	−60.9500	−2.06521
$872$	0	0
$873$	− 5.04548i	− 0.170764i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	13.3409i	0.450489i	0.974302	+	0.225245i	$0.0723181\pi$
$877$	13.3409i	0.450489i	−0.974302	+	0.225245i	$0.927682\pi$
$878$	0	0
$879$	11.4605	0.386554
$880$	0	0
$881$	−44.5458	−1.50079	−0.750393	−	0.660992i	$-0.770136\pi$
$881$	−44.5458	−1.50079	−0.750393	+	0.660992i	$0.770136\pi$
$882$	0	0
$883$	− 23.4498i	− 0.789148i	−0.918864	−	0.394574i	$-0.870892\pi$
$883$	− 23.4498i	− 0.789148i	0.918864	−	0.394574i	$-0.129108\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	40.1362i	1.34764i	0.738895	+	0.673821i	$0.235348\pi$
$887$	40.1362i	1.34764i	−0.738895	+	0.673821i	$0.764652\pi$
$888$	0	0
$889$	−22.9451	−0.769553
$890$	0	0
$891$	5.49010	0.183925
$892$	0	0
$893$	− 1.99478i	− 0.0667528i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	69.8539i	2.33235i
$898$	0	0
$899$	−42.5450	−1.41895
$900$	0	0
$901$	63.6611	2.12086
$902$	0	0
$903$	− 7.36040i	− 0.244939i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	35.8062i	1.18893i	0.804123	+	0.594463i	$0.202636\pi$
$907$	35.8062i	1.18893i	−0.804123	+	0.594463i	$0.797364\pi$
$908$	0	0
$909$	−13.1972	−0.437723
$910$	0	0
$911$	−29.5302	−0.978381	−0.489190	−	0.872177i	$-0.662708\pi$
$911$	−29.5302	−0.978381	−0.489190	+	0.872177i	$0.662708\pi$
$912$	0	0
$913$	− 3.85839i	− 0.127694i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 18.0774i	− 0.596967i
$918$	0	0
$919$	−11.7570	−0.387829	−0.193914	−	0.981018i	$-0.562118\pi$
$919$	−11.7570	−0.387829	−0.193914	+	0.981018i	$0.562118\pi$
$920$	0	0
$921$	10.3962	0.342567
$922$	0	0
$923$	− 45.7145i	− 1.50471i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 4.36769i	− 0.143454i
$928$	0	0
$929$	21.5189	0.706012	0.353006	−	0.935621i	$-0.385159\pi$
$929$	21.5189	0.706012	0.353006	+	0.935621i	$0.385159\pi$
$930$	0	0
$931$	5.44480	0.178446
$932$	0	0
$933$	− 7.56473i	− 0.247658i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 31.7690i	− 1.03785i	−0.854820	−	0.518924i	$-0.826333\pi$
$937$	− 31.7690i	− 1.03785i	0.854820	−	0.518924i	$-0.173667\pi$
$938$	0	0
$939$	58.2778	1.90182
$940$	0	0
$941$	−0.730603	−0.0238170	−0.0119085	−	0.999929i	$-0.503791\pi$
$941$	−0.730603	−0.0238170	−0.0119085	+	0.999929i	$0.503791\pi$
$942$	0	0
$943$	62.6348i	2.03967i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 41.2171i	− 1.33938i	−0.742643	−	0.669688i	$-0.766428\pi$
$947$	− 41.2171i	− 1.33938i	0.742643	−	0.669688i	$-0.233572\pi$
$948$	0	0
$949$	−78.8543	−2.55972
$950$	0	0
$951$	−32.4692	−1.05289
$952$	0	0
$953$	− 39.9343i	− 1.29360i	−0.762660	−	0.646800i	$-0.776107\pi$
$953$	− 39.9343i	− 1.29360i	0.762660	−	0.646800i	$-0.223893\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	6.59241i	0.213102i
$958$	0	0
$959$	−4.24357	−0.137032
$960$	0	0
$961$	10.2157	0.329539
$962$	0	0
$963$	3.10105i	0.0999300i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 8.73989i	− 0.281056i	−0.990077	−	0.140528i	$-0.955120\pi$
$967$	− 8.73989i	− 0.281056i	0.990077	−	0.140528i	$-0.0448800\pi$
$968$	0	0
$969$	8.74998	0.281090
$970$	0	0
$971$	7.49072	0.240389	0.120194	−	0.992750i	$-0.461648\pi$
$971$	7.49072	0.240389	0.120194	+	0.992750i	$0.461648\pi$
$972$	0	0
$973$	5.00155i	0.160342i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	51.6069i	1.65105i	0.564364	+	0.825526i	$0.309121\pi$
$977$	51.6069i	1.65105i	−0.564364	+	0.825526i	$0.690879\pi$
$978$	0	0
$979$	−4.03561	−0.128979
$980$	0	0
$981$	4.26017	0.136017
$982$	0	0
$983$	57.9017i	1.84678i	0.383865	+	0.923389i	$0.374593\pi$
$983$	57.9017i	1.84678i	−0.383865	+	0.923389i	$0.625407\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4.81582i	0.153289i
$988$	0	0
$989$	−17.8684	−0.568182
$990$	0	0
$991$	−18.2369	−0.579314	−0.289657	−	0.957131i	$-0.593541\pi$
$991$	−18.2369	−0.579314	−0.289657	+	0.957131i	$0.593541\pi$
$992$	0	0
$993$	− 47.2435i	− 1.49923i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 10.7461i	− 0.340332i	−0.985415	−	0.170166i	$-0.945570\pi$
$997$	− 10.7461i	− 0.340332i	0.985415	−	0.170166i	$-0.0544305\pi$
$998$	0	0
$999$	6.13358	0.194058

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.d.q.3649.10		12
5.2	odd	4		3800.2.a.bc.1.5	yes	6
5.3	odd	4		3800.2.a.ba.1.2	✓	6
5.4	even	2	inner	3800.2.d.q.3649.3		12
20.3	even	4		7600.2.a.cl.1.5		6
20.7	even	4		7600.2.a.ch.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.ba.1.2	✓	6	5.3	odd	4
3800.2.a.bc.1.5	yes	6	5.2	odd	4
3800.2.d.q.3649.3		12	5.4	even	2	inner
3800.2.d.q.3649.10		12	1.1	even	1	trivial
7600.2.a.ch.1.2		6	20.7	even	4
7600.2.a.cl.1.5		6	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 $\F_{q}$
Belyi maps

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)

\(f(q)\)	\(=\)	\(q+1.93590i q^{3} +1.24708i q^{7} -0.747704 q^{9} -0.513860 q^{11} -6.15670i q^{13} -4.51986i q^{17} +1.00000 q^{19} -2.41421 q^{21} +5.86084i q^{23} +4.36022i q^{27} -6.62700 q^{29} +6.41995 q^{31} -0.994780i q^{33} -1.40671i q^{37} +11.9187 q^{39} +10.6870 q^{41} +3.04878i q^{43} -1.99478i q^{47} +5.44480 q^{49} +8.74998 q^{51} +14.0848i q^{53} +1.93590i q^{57} +4.34261 q^{59} +10.7173 q^{61} -0.932444i q^{63} -9.89978i q^{67} -11.3460 q^{69} +7.42517 q^{71} -12.8079i q^{73} -0.640822i q^{77} -2.56138 q^{79} -10.6841 q^{81} +7.50864i q^{83} -12.8292i q^{87} +7.85353 q^{89} +7.67787 q^{91} +12.4284i q^{93} +6.74797i q^{97} +0.384215 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{9} + 6 q^{11} + 12 q^{19} + 30 q^{21} - 18 q^{29} + 10 q^{31} - 24 q^{39} + 6 q^{41} - 44 q^{49} + 66 q^{51} + 18 q^{61} + 22 q^{69} + 38 q^{71} + 32 q^{79} + 52 q^{81} - 28 q^{89} + 84 q^{91}+ \cdots + 12 q^{99}+O(q^{100}) \) 12 * q - 12 * q^9 + 6 * q^11 + 12 * q^19 + 30 * q^21 - 18 * q^29 + 10 * q^31 - 24 * q^39 + 6 * q^41 - 44 * q^49 + 66 * q^51 + 18 * q^61 + 22 * q^69 + 38 * q^71 + 32 * q^79 + 52 * q^81 - 28 * q^89 + 84 * q^91 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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