LMFDB - Embedded newform 1900.2.c.f.1749.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,2,Mod(1749,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1900.1749");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1900.c (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:	$15.1715763840$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.6594624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 18x^{2} + 1 $ x^6 + 9x^4 + 18x^2 + 1
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			1749.5
Root			$-0.239123i$ of defining polynomial
Character	$\chi$	$=$	1900.1749
Dual form			1900.2.c.f.1749.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.18194i q^{3} +3.70370i q^{7} -1.76088 q^{9} +O(q^{10})$	$q+2.18194i q^{3} +3.70370i q^{7} -1.76088 q^{9} -3.18194 q^{11} +1.94282i q^{13} +4.66019i q^{17} -1.00000 q^{19} -8.08126 q^{21} -5.23912i q^{23} +2.70370i q^{27} +1.66019 q^{29} -3.46457 q^{31} -6.94282i q^{33} -7.18194i q^{37} -4.23912 q^{39} -0.717370 q^{41} +1.36389i q^{43} +5.37756i q^{47} -6.71737 q^{49} -10.1683 q^{51} -0.0435069i q^{53} -2.18194i q^{57} +7.88564 q^{59} -4.98633 q^{61} -6.52175i q^{63} +14.5023i q^{67} +11.4315 q^{69} +4.88564 q^{71} -10.6030i q^{73} -11.7850i q^{77} -14.9097 q^{79} -11.1819 q^{81} -9.76088i q^{83} +3.62244i q^{87} -10.0813 q^{89} -7.19562 q^{91} -7.55950i q^{93} -14.6706i q^{97} +5.60301 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 10 q^{9}+O(q^{10}) $ 6 * q - 10 * q^9	$ 6 q - 10 q^{9} - 2 q^{11} - 6 q^{19} - 16 q^{21} - 6 q^{29} - 2 q^{31} - 26 q^{39} - 6 q^{41} - 42 q^{49} - 24 q^{51} + 12 q^{59} - 10 q^{61} - 18 q^{69} - 6 q^{71} - 4 q^{79} - 50 q^{81} - 28 q^{89} - 46 q^{91}+O(q^{100}) $ 6 * q - 10 * q^9 - 2 * q^11 - 6 * q^19 - 16 * q^21 - 6 * q^29 - 2 * q^31 - 26 * q^39 - 6 * q^41 - 42 * q^49 - 24 * q^51 + 12 * q^59 - 10 * q^61 - 18 * q^69 - 6 * q^71 - 4 * q^79 - 50 * q^81 - 28 * q^89 - 46 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	2.18194i	1.25975i	0.776698	+	0.629873i	$0.216893\pi$
$3$	2.18194i	1.25975i	−0.776698	+	0.629873i	$0.783107\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.70370i	1.39987i	0.714209	+	0.699933i	$0.246787\pi$
$7$	3.70370i	1.39987i	−0.714209	+	0.699933i	$0.753213\pi$
$8$	0	0
$9$	−1.76088	−0.586959
$10$	0	0
$11$	−3.18194	−0.959392	−0.479696	−	0.877435i	$-0.659253\pi$
$11$	−3.18194	−0.959392	−0.479696	+	0.877435i	$0.659253\pi$
$12$	0	0
$13$	1.94282i	0.538841i	0.963023	+	0.269421i	$0.0868322\pi$
$13$	1.94282i	0.538841i	−0.963023	+	0.269421i	$0.913168\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.66019i	1.13026i	0.825001	+	0.565131i	$0.191174\pi$
$17$	4.66019i	1.13026i	−0.825001	+	0.565131i	$0.808826\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−8.08126	−1.76347
$22$	0	0
$23$	− 5.23912i	− 1.09243i	−0.837644	−	0.546216i	$-0.816068\pi$
$23$	− 5.23912i	− 1.09243i	0.837644	−	0.546216i	$-0.183932\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.70370i	0.520327i
$28$	0	0
$29$	1.66019	0.308290	0.154145	−	0.988048i	$-0.450738\pi$
$29$	1.66019	0.308290	0.154145	+	0.988048i	$0.450738\pi$
$30$	0	0
$31$	−3.46457	−0.622256	−0.311128	−	0.950368i	$-0.600707\pi$
$31$	−3.46457	−0.622256	−0.311128	+	0.950368i	$0.600707\pi$
$32$	0	0
$33$	− 6.94282i	− 1.20859i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 7.18194i	− 1.18070i	−0.807146	−	0.590352i	$-0.798989\pi$
$37$	− 7.18194i	− 1.18070i	0.807146	−	0.590352i	$-0.201011\pi$
$38$	0	0
$39$	−4.23912	−0.678803
$40$	0	0
$41$	−0.717370	−0.112034	−0.0560172	−	0.998430i	$-0.517840\pi$
$41$	−0.717370	−0.112034	−0.0560172	+	0.998430i	$0.517840\pi$
$42$	0	0
$43$	1.36389i	0.207991i	0.994578	+	0.103995i	$0.0331627\pi$
$43$	1.36389i	0.207991i	−0.994578	+	0.103995i	$0.966837\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.37756i	0.784398i	0.919880	+	0.392199i	$0.128285\pi$
$47$	5.37756i	0.784398i	−0.919880	+	0.392199i	$0.871715\pi$
$48$	0	0
$49$	−6.71737	−0.959624
$50$	0	0
$51$	−10.1683	−1.42384
$52$	0	0
$53$	− 0.0435069i	− 0.00597613i	−0.999996	−	0.00298807i	$-0.999049\pi$
$53$	− 0.0435069i	− 0.00597613i	0.999996	−	0.00298807i	$-0.000951132\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 2.18194i	− 0.289005i
$58$	0	0
$59$	7.88564	1.02662	0.513311	−	0.858202i	$-0.328419\pi$
$59$	7.88564	1.02662	0.513311	+	0.858202i	$0.328419\pi$
$60$	0	0
$61$	−4.98633	−0.638434	−0.319217	−	0.947682i	$-0.603420\pi$
$61$	−4.98633	−0.638434	−0.319217	+	0.947682i	$0.603420\pi$
$62$	0	0
$63$	− 6.52175i	− 0.821664i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	14.5023i	1.77174i	0.463933	+	0.885870i	$0.346438\pi$
$67$	14.5023i	1.77174i	−0.463933	+	0.885870i	$0.653562\pi$
$68$	0	0
$69$	11.4315	1.37619
$70$	0	0
$71$	4.88564	0.579819	0.289909	−	0.957054i	$-0.406375\pi$
$71$	4.88564	0.579819	0.289909	+	0.957054i	$0.406375\pi$
$72$	0	0
$73$	− 10.6030i	− 1.24099i	−0.784211	−	0.620494i	$-0.786932\pi$
$73$	− 10.6030i	− 1.24099i	0.784211	−	0.620494i	$-0.213068\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 11.7850i	− 1.34302i
$78$	0	0
$79$	−14.9097	−1.67747	−0.838737	−	0.544537i	$-0.816706\pi$
$79$	−14.9097	−1.67747	−0.838737	+	0.544537i	$0.816706\pi$
$80$	0	0
$81$	−11.1819	−1.24244
$82$	0	0
$83$	− 9.76088i	− 1.07140i	−0.844410	−	0.535698i	$-0.820049\pi$
$83$	− 9.76088i	− 1.07140i	0.844410	−	0.535698i	$-0.179951\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.62244i	0.388366i
$88$	0	0
$89$	−10.0813	−1.06861	−0.534306	−	0.845291i	$-0.679427\pi$
$89$	−10.0813	−1.06861	−0.534306	+	0.845291i	$0.679427\pi$
$90$	0	0
$91$	−7.19562	−0.754306
$92$	0	0
$93$	− 7.55950i	− 0.783884i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 14.6706i	− 1.48957i	−0.667303	−	0.744787i	$-0.732551\pi$
$97$	− 14.6706i	− 1.48957i	0.667303	−	0.744787i	$-0.267449\pi$
$98$	0	0
$99$	5.60301	0.563124
$100$	0	0
$101$	15.9532	1.58741	0.793703	−	0.608306i	$-0.208151\pi$
$101$	15.9532	1.58741	0.793703	+	0.608306i	$0.208151\pi$
$102$	0	0
$103$	10.6328i	1.04769i	0.851815	+	0.523843i	$0.175502\pi$
$103$	10.6328i	1.04769i	−0.851815	+	0.523843i	$0.824498\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.6465i	1.41593i	0.706246	+	0.707966i	$0.250387\pi$
$107$	14.6465i	1.41593i	−0.706246	+	0.707966i	$0.749613\pi$
$108$	0	0
$109$	3.55950	0.340939	0.170469	−	0.985363i	$-0.445472\pi$
$109$	3.55950	0.340939	0.170469	+	0.985363i	$0.445472\pi$
$110$	0	0
$111$	15.6706	1.48739
$112$	0	0
$113$	− 14.8285i	− 1.39494i	−0.716612	−	0.697472i	$-0.754308\pi$
$113$	− 14.8285i	− 1.39494i	0.716612	−	0.697472i	$-0.245692\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 3.42107i	− 0.316278i
$118$	0	0
$119$	−17.2599	−1.58222
$120$	0	0
$121$	−0.875237	−0.0795670
$122$	0	0
$123$	− 1.56526i	− 0.141135i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	7.69002i	0.682379i	0.939994	+	0.341190i	$0.110830\pi$
$127$	7.69002i	0.682379i	−0.939994	+	0.341190i	$0.889170\pi$
$128$	0	0
$129$	−2.97592	−0.262015
$130$	0	0
$131$	19.4451	1.69893	0.849465	−	0.527645i	$-0.176925\pi$
$131$	19.4451	1.69893	0.849465	+	0.527645i	$0.176925\pi$
$132$	0	0
$133$	− 3.70370i	− 0.321151i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.39699i	0.290224i	0.989415	+	0.145112i	$0.0463543\pi$
$137$	3.39699i	0.290224i	−0.989415	+	0.145112i	$0.953646\pi$
$138$	0	0
$139$	−10.5699	−0.896528	−0.448264	−	0.893901i	$-0.647958\pi$
$139$	−10.5699	−0.896528	−0.448264	+	0.893901i	$0.647958\pi$
$140$	0	0
$141$	−11.7335	−0.988142
$142$	0	0
$143$	− 6.18194i	− 0.516960i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 14.6569i	− 1.20888i
$148$	0	0
$149$	−19.5322	−1.60014	−0.800068	−	0.599909i	$-0.795203\pi$
$149$	−19.5322	−1.60014	−0.800068	+	0.599909i	$0.795203\pi$
$150$	0	0
$151$	−1.17154	−0.0953386	−0.0476693	−	0.998863i	$-0.515179\pi$
$151$	−1.17154	−0.0953386	−0.0476693	+	0.998863i	$0.515179\pi$
$152$	0	0
$153$	− 8.20602i	− 0.663417i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	5.01040i	0.399874i	0.979809	+	0.199937i	$0.0640737\pi$
$157$	5.01040i	0.399874i	−0.979809	+	0.199937i	$0.935926\pi$
$158$	0	0
$159$	0.0949296	0.00752840
$160$	0	0
$161$	19.4041	1.52926
$162$	0	0
$163$	6.78495i	0.531439i	0.964050	+	0.265719i	$0.0856094\pi$
$163$	6.78495i	0.531439i	−0.964050	+	0.265719i	$0.914391\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 5.64652i	− 0.436941i	−0.975844	−	0.218470i	$-0.929893\pi$
$167$	− 5.64652i	− 0.436941i	0.975844	−	0.218470i	$-0.0701067\pi$
$168$	0	0
$169$	9.22545	0.709650
$170$	0	0
$171$	1.76088	0.134658
$172$	0	0
$173$	− 5.38796i	− 0.409639i	−0.978800	−	0.204820i	$-0.934339\pi$
$173$	− 5.38796i	− 0.409639i	0.978800	−	0.204820i	$-0.0656608\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	17.2060i	1.29328i
$178$	0	0
$179$	−3.27687	−0.244925	−0.122462	−	0.992473i	$-0.539079\pi$
$179$	−3.27687	−0.244925	−0.122462	+	0.992473i	$0.539079\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	0	0
$183$	− 10.8799i	− 0.804264i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 14.8285i	− 1.08436i
$188$	0	0
$189$	−10.0137	−0.728388
$190$	0	0
$191$	0.225450	0.0163130	0.00815650	−	0.999967i	$-0.497404\pi$
$191$	0.225450	0.0163130	0.00815650	+	0.999967i	$0.497404\pi$
$192$	0	0
$193$	8.12476i	0.584833i	0.956291	+	0.292417i	$0.0944594\pi$
$193$	8.12476i	0.584833i	−0.956291	+	0.292417i	$0.905541\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.25280i	0.302999i	0.988457	+	0.151500i	$0.0484102\pi$
$197$	4.25280i	0.302999i	−0.988457	+	0.151500i	$0.951590\pi$
$198$	0	0
$199$	−26.1592	−1.85438	−0.927190	−	0.374592i	$-0.877783\pi$
$199$	−26.1592	−1.85438	−0.927190	+	0.374592i	$0.877783\pi$
$200$	0	0
$201$	−31.6432	−2.23194
$202$	0	0
$203$	6.14884i	0.431564i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	9.22545i	0.641213i
$208$	0	0
$209$	3.18194	0.220100
$210$	0	0
$211$	4.27223	0.294112	0.147056	−	0.989128i	$-0.453020\pi$
$211$	4.27223	0.294112	0.147056	+	0.989128i	$0.453020\pi$
$212$	0	0
$213$	10.6602i	0.730424i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 12.8317i	− 0.871075i
$218$	0	0
$219$	23.1352	1.56333
$220$	0	0
$221$	−9.05391	−0.609032
$222$	0	0
$223$	− 23.0208i	− 1.54159i	−0.637085	−	0.770794i	$-0.719860\pi$
$223$	− 23.0208i	− 1.54159i	0.637085	−	0.770794i	$-0.280140\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.96690i	0.595154i	0.954698	+	0.297577i	$0.0961785\pi$
$227$	8.96690i	0.595154i	−0.954698	+	0.297577i	$0.903822\pi$
$228$	0	0
$229$	8.62709	0.570094	0.285047	−	0.958514i	$-0.407991\pi$
$229$	8.62709	0.570094	0.285047	+	0.958514i	$0.407991\pi$
$230$	0	0
$231$	25.7141	1.69186
$232$	0	0
$233$	29.0014i	1.89994i	0.312336	+	0.949972i	$0.398889\pi$
$233$	29.0014i	1.89994i	−0.312336	+	0.949972i	$0.601111\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 32.5322i	− 2.11319i
$238$	0	0
$239$	23.7414	1.53571	0.767853	−	0.640626i	$-0.221325\pi$
$239$	23.7414	1.53571	0.767853	+	0.640626i	$0.221325\pi$
$240$	0	0
$241$	−6.32614	−0.407502	−0.203751	−	0.979023i	$-0.565313\pi$
$241$	−6.32614	−0.407502	−0.203751	+	0.979023i	$0.565313\pi$
$242$	0	0
$243$	− 16.2873i	− 1.04483i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1.94282i	− 0.123619i
$248$	0	0
$249$	21.2977	1.34969
$250$	0	0
$251$	−21.9201	−1.38359	−0.691793	−	0.722096i	$-0.743179\pi$
$251$	−21.9201	−1.38359	−0.691793	+	0.722096i	$0.743179\pi$
$252$	0	0
$253$	16.6706i	1.04807i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.31246i	0.206626i	0.994649	+	0.103313i	$0.0329443\pi$
$257$	3.31246i	0.206626i	−0.994649	+	0.103313i	$0.967056\pi$
$258$	0	0
$259$	26.5997	1.65283
$260$	0	0
$261$	−2.92339	−0.180953
$262$	0	0
$263$	2.33405i	0.143924i	0.997407	+	0.0719619i	$0.0229260\pi$
$263$	2.33405i	0.143924i	−0.997407	+	0.0719619i	$0.977074\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 21.9967i	− 1.34618i
$268$	0	0
$269$	24.5789	1.49860	0.749302	−	0.662228i	$-0.230389\pi$
$269$	24.5789	1.49860	0.749302	+	0.662228i	$0.230389\pi$
$270$	0	0
$271$	16.3880	0.995498	0.497749	−	0.867321i	$-0.334160\pi$
$271$	16.3880	0.995498	0.497749	+	0.867321i	$0.334160\pi$
$272$	0	0
$273$	− 15.7004i	− 0.950233i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	8.68427i	0.521787i	0.965368	+	0.260894i	$0.0840171\pi$
$277$	8.68427i	0.521787i	−0.965368	+	0.260894i	$0.915983\pi$
$278$	0	0
$279$	6.10069	0.365239
$280$	0	0
$281$	−0.450900	−0.0268985	−0.0134492	−	0.999910i	$-0.504281\pi$
$281$	−0.450900	−0.0268985	−0.0134492	+	0.999910i	$0.504281\pi$
$282$	0	0
$283$	28.6512i	1.70313i	0.524245	+	0.851567i	$0.324348\pi$
$283$	28.6512i	1.70313i	−0.524245	+	0.851567i	$0.675652\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 2.65692i	− 0.156833i
$288$	0	0
$289$	−4.71737	−0.277492
$290$	0	0
$291$	32.0104	1.87648
$292$	0	0
$293$	20.7335i	1.21127i	0.795744	+	0.605633i	$0.207080\pi$
$293$	20.7335i	1.21127i	−0.795744	+	0.605633i	$0.792920\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 8.60301i	− 0.499197i
$298$	0	0
$299$	10.1787	0.588648
$300$	0	0
$301$	−5.05142	−0.291159
$302$	0	0
$303$	34.8090i	1.99973i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	34.4933i	1.96864i	0.176402	+	0.984318i	$0.443554\pi$
$307$	34.4933i	1.96864i	−0.176402	+	0.984318i	$0.556446\pi$
$308$	0	0
$309$	−23.2003	−1.31982
$310$	0	0
$311$	15.5789	0.883400	0.441700	−	0.897163i	$-0.354376\pi$
$311$	15.5789	0.883400	0.441700	+	0.897163i	$0.354376\pi$
$312$	0	0
$313$	2.98057i	0.168472i	0.996446	+	0.0842359i	$0.0268449\pi$
$313$	2.98057i	0.168472i	−0.996446	+	0.0842359i	$0.973155\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.5803i	0.987409i	0.869630	+	0.493704i	$0.164357\pi$
$317$	17.5803i	0.987409i	−0.869630	+	0.493704i	$0.835643\pi$
$318$	0	0
$319$	−5.28263	−0.295771
$320$	0	0
$321$	−31.9579	−1.78371
$322$	0	0
$323$	− 4.66019i	− 0.259300i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	7.76663i	0.429496i
$328$	0	0
$329$	−19.9169	−1.09805
$330$	0	0
$331$	−35.2405	−1.93699	−0.968497	−	0.249027i	$-0.919889\pi$
$331$	−35.2405	−1.93699	−0.968497	+	0.249027i	$0.919889\pi$
$332$	0	0
$333$	12.6465i	0.693025i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 9.74145i	− 0.530650i	−0.964159	−	0.265325i	$-0.914521\pi$
$337$	− 9.74145i	− 0.530650i	0.964159	−	0.265325i	$-0.0854793\pi$
$338$	0	0
$339$	32.3549	1.75727
$340$	0	0
$341$	11.0241	0.596987
$342$	0	0
$343$	1.04678i	0.0565206i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	33.2028i	1.78242i	0.453594	+	0.891209i	$0.350142\pi$
$347$	33.2028i	1.78242i	−0.453594	+	0.891209i	$0.649858\pi$
$348$	0	0
$349$	14.7141	0.787628	0.393814	−	0.919190i	$-0.371155\pi$
$349$	14.7141	0.787628	0.393814	+	0.919190i	$0.371155\pi$
$350$	0	0
$351$	−5.25280	−0.280374
$352$	0	0
$353$	− 3.44841i	− 0.183540i	−0.995780	−	0.0917702i	$-0.970747\pi$
$353$	− 3.44841i	− 0.183540i	0.995780	−	0.0917702i	$-0.0292525\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 37.6602i	− 1.99319i
$358$	0	0
$359$	−6.55623	−0.346025	−0.173012	−	0.984920i	$-0.555350\pi$
$359$	−6.55623	−0.346025	−0.173012	+	0.984920i	$0.555350\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	− 1.90972i	− 0.100234i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	32.5641i	1.69983i	0.526916	+	0.849917i	$0.323348\pi$
$367$	32.5641i	1.69983i	−0.526916	+	0.849917i	$0.676652\pi$
$368$	0	0
$369$	1.26320	0.0657596
$370$	0	0
$371$	0.161136	0.00836578
$372$	0	0
$373$	15.0137i	0.777379i	0.921369	+	0.388689i	$0.127072\pi$
$373$	15.0137i	0.777379i	−0.921369	+	0.388689i	$0.872928\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.22545i	0.166119i
$378$	0	0
$379$	−12.4807	−0.641092	−0.320546	−	0.947233i	$-0.603866\pi$
$379$	−12.4807	−0.641092	−0.320546	+	0.947233i	$0.603866\pi$
$380$	0	0
$381$	−16.7792	−0.859624
$382$	0	0
$383$	− 12.5354i	− 0.640530i	−0.947328	−	0.320265i	$-0.896228\pi$
$383$	− 12.5354i	− 0.640530i	0.947328	−	0.320265i	$-0.103772\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 2.40164i	− 0.122082i
$388$	0	0
$389$	−37.4328	−1.89792	−0.948960	−	0.315395i	$-0.897863\pi$
$389$	−37.4328	−1.89792	−0.948960	+	0.315395i	$0.897863\pi$
$390$	0	0
$391$	24.4153	1.23474
$392$	0	0
$393$	42.4282i	2.14022i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 3.05718i	− 0.153435i	−0.997053	−	0.0767177i	$-0.975556\pi$
$397$	− 3.05718i	− 0.153435i	0.997053	−	0.0767177i	$-0.0244440\pi$
$398$	0	0
$399$	8.08126	0.404569
$400$	0	0
$401$	18.5458	0.926135	0.463067	−	0.886323i	$-0.346749\pi$
$401$	18.5458	0.926135	0.463067	+	0.886323i	$0.346749\pi$
$402$	0	0
$403$	− 6.73104i	− 0.335297i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	22.8525i	1.13276i
$408$	0	0
$409$	−13.9669	−0.690619	−0.345309	−	0.938489i	$-0.612226\pi$
$409$	−13.9669	−0.690619	−0.345309	+	0.938489i	$0.612226\pi$
$410$	0	0
$411$	−7.41204	−0.365609
$412$	0	0
$413$	29.2060i	1.43713i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 23.0629i	− 1.12940i
$418$	0	0
$419$	32.7623	1.60054	0.800270	−	0.599639i	$-0.204689\pi$
$419$	32.7623	1.60054	0.800270	+	0.599639i	$0.204689\pi$
$420$	0	0
$421$	29.5127	1.43836	0.719181	−	0.694823i	$-0.244517\pi$
$421$	29.5127	1.43836	0.719181	+	0.694823i	$0.244517\pi$
$422$	0	0
$423$	− 9.46922i	− 0.460409i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 18.4678i	− 0.893722i
$428$	0	0
$429$	13.4887	0.651238
$430$	0	0
$431$	−2.90507	−0.139932	−0.0699662	−	0.997549i	$-0.522289\pi$
$431$	−2.90507	−0.139932	−0.0699662	+	0.997549i	$0.522289\pi$
$432$	0	0
$433$	29.4887i	1.41713i	0.705643	+	0.708567i	$0.250658\pi$
$433$	29.4887i	1.41713i	−0.705643	+	0.708567i	$0.749342\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.23912i	0.250621i
$438$	0	0
$439$	30.4088	1.45133	0.725666	−	0.688047i	$-0.241532\pi$
$439$	30.4088	1.45133	0.725666	+	0.688047i	$0.241532\pi$
$440$	0	0
$441$	11.8285	0.563260
$442$	0	0
$443$	1.80903i	0.0859496i	0.999076	+	0.0429748i	$0.0136835\pi$
$443$	1.80903i	0.0859496i	−0.999076	+	0.0429748i	$0.986316\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 42.6181i	− 2.01577i
$448$	0	0
$449$	21.4257	1.01114	0.505571	−	0.862785i	$-0.331282\pi$
$449$	21.4257	1.01114	0.505571	+	0.862785i	$0.331282\pi$
$450$	0	0
$451$	2.28263	0.107485
$452$	0	0
$453$	− 2.55623i	− 0.120102i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 19.5789i	− 0.915864i	−0.888987	−	0.457932i	$-0.848590\pi$
$457$	− 19.5789i	− 0.915864i	0.888987	−	0.457932i	$-0.151410\pi$
$458$	0	0
$459$	−12.5997	−0.588106
$460$	0	0
$461$	7.92666	0.369181	0.184591	−	0.982815i	$-0.440904\pi$
$461$	7.92666	0.369181	0.184591	+	0.982815i	$0.440904\pi$
$462$	0	0
$463$	11.0298i	0.512600i	0.966597	+	0.256300i	$0.0825035\pi$
$463$	11.0298i	0.512600i	−0.966597	+	0.256300i	$0.917497\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.48400i	0.161220i	0.996746	+	0.0806102i	$0.0256869\pi$
$467$	3.48400i	0.161220i	−0.996746	+	0.0806102i	$0.974313\pi$
$468$	0	0
$469$	−53.7122	−2.48020
$470$	0	0
$471$	−10.9324	−0.503739
$472$	0	0
$473$	− 4.33981i	− 0.199545i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.0766103i	0.00350774i
$478$	0	0
$479$	29.6569	1.35506	0.677530	−	0.735495i	$-0.263051\pi$
$479$	29.6569	1.35506	0.677530	+	0.735495i	$0.263051\pi$
$480$	0	0
$481$	13.9532	0.636212
$482$	0	0
$483$	42.3387i	1.92648i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 4.65116i	− 0.210764i	−0.994432	−	0.105382i	$-0.966393\pi$
$487$	− 4.65116i	− 0.210764i	0.994432	−	0.105382i	$-0.0336066\pi$
$488$	0	0
$489$	−14.8044	−0.669477
$490$	0	0
$491$	−21.6947	−0.979067	−0.489533	−	0.871985i	$-0.662833\pi$
$491$	−21.6947	−0.979067	−0.489533	+	0.871985i	$0.662833\pi$
$492$	0	0
$493$	7.73680i	0.348448i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	18.0949i	0.811669i
$498$	0	0
$499$	−7.83886	−0.350916	−0.175458	−	0.984487i	$-0.556141\pi$
$499$	−7.83886	−0.350916	−0.175458	+	0.984487i	$0.556141\pi$
$500$	0	0
$501$	12.3204	0.550434
$502$	0	0
$503$	32.8824i	1.46615i	0.680146	+	0.733076i	$0.261916\pi$
$503$	32.8824i	1.46615i	−0.680146	+	0.733076i	$0.738084\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	20.1294i	0.893978i
$508$	0	0
$509$	43.5530	1.93045	0.965226	−	0.261418i	$-0.0841902\pi$
$509$	43.5530	1.93045	0.965226	+	0.261418i	$0.0841902\pi$
$510$	0	0
$511$	39.2703	1.73722
$512$	0	0
$513$	− 2.70370i	− 0.119371i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 17.1111i	− 0.752545i
$518$	0	0
$519$	11.7562	0.516041
$520$	0	0
$521$	−6.34308	−0.277895	−0.138948	−	0.990300i	$-0.544372\pi$
$521$	−6.34308	−0.277895	−0.138948	+	0.990300i	$0.544372\pi$
$522$	0	0
$523$	− 31.0449i	− 1.35750i	−0.734370	−	0.678749i	$-0.762522\pi$
$523$	− 31.0449i	− 1.35750i	0.734370	−	0.678749i	$-0.237478\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 16.1456i	− 0.703312i
$528$	0	0
$529$	−4.44841	−0.193409
$530$	0	0
$531$	−13.8856	−0.602585
$532$	0	0
$533$	− 1.39372i	− 0.0603687i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 7.14995i	− 0.308543i
$538$	0	0
$539$	21.3743	0.920656
$540$	0	0
$541$	34.3218	1.47561	0.737804	−	0.675015i	$-0.235863\pi$
$541$	34.3218	1.47561	0.737804	+	0.675015i	$0.235863\pi$
$542$	0	0
$543$	− 15.2736i	− 0.655453i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 23.5836i	− 1.00836i	−0.863598	−	0.504181i	$-0.831795\pi$
$547$	− 23.5836i	− 1.00836i	0.863598	−	0.504181i	$-0.168205\pi$
$548$	0	0
$549$	8.78031	0.374734
$550$	0	0
$551$	−1.66019	−0.0707265
$552$	0	0
$553$	− 55.2211i	− 2.34824i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	44.8824i	1.90173i	0.309610	+	0.950864i	$0.399801\pi$
$557$	44.8824i	1.90173i	−0.309610	+	0.950864i	$0.600199\pi$
$558$	0	0
$559$	−2.64979	−0.112074
$560$	0	0
$561$	32.3549	1.36602
$562$	0	0
$563$	− 6.13844i	− 0.258704i	−0.991599	−	0.129352i	$-0.958710\pi$
$563$	− 6.13844i	− 0.258704i	0.991599	−	0.129352i	$-0.0412898\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 41.4145i	− 1.73925i
$568$	0	0
$569$	−42.9384	−1.80007	−0.900037	−	0.435814i	$-0.856460\pi$
$569$	−42.9384	−1.80007	−0.900037	+	0.435814i	$0.856460\pi$
$570$	0	0
$571$	−3.00576	−0.125787	−0.0628935	−	0.998020i	$-0.520033\pi$
$571$	−3.00576	−0.125787	−0.0628935	+	0.998020i	$0.520033\pi$
$572$	0	0
$573$	0.491920i	0.0205502i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 16.2028i	− 0.674529i	−0.941410	−	0.337265i	$-0.890498\pi$
$577$	− 16.2028i	− 0.674529i	0.941410	−	0.337265i	$-0.109502\pi$
$578$	0	0
$579$	−17.7278	−0.736741
$580$	0	0
$581$	36.1513	1.49981
$582$	0	0
$583$	0.138436i	0.00573345i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	26.4542i	1.09188i	0.837824	+	0.545940i	$0.183827\pi$
$587$	26.4542i	1.09188i	−0.837824	+	0.545940i	$0.816173\pi$
$588$	0	0
$589$	3.46457	0.142755
$590$	0	0
$591$	−9.27936	−0.381702
$592$	0	0
$593$	− 25.3710i	− 1.04186i	−0.853599	−	0.520931i	$-0.825585\pi$
$593$	− 25.3710i	− 1.04186i	0.853599	−	0.520931i	$-0.174415\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 57.0780i	− 2.33605i
$598$	0	0
$599$	3.26896	0.133566	0.0667830	−	0.997768i	$-0.478726\pi$
$599$	3.26896	0.133566	0.0667830	+	0.997768i	$0.478726\pi$
$600$	0	0
$601$	−27.6465	−1.12772	−0.563862	−	0.825869i	$-0.690685\pi$
$601$	−27.6465	−1.12772	−0.563862	+	0.825869i	$0.690685\pi$
$602$	0	0
$603$	− 25.5368i	− 1.03994i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 31.6739i	− 1.28560i	−0.766033	−	0.642801i	$-0.777772\pi$
$607$	− 31.6739i	− 1.28560i	0.766033	−	0.642801i	$-0.222228\pi$
$608$	0	0
$609$	−13.4164	−0.543661
$610$	0	0
$611$	−10.4476	−0.422666
$612$	0	0
$613$	20.1970i	0.815749i	0.913038	+	0.407874i	$0.133730\pi$
$613$	20.1970i	0.815749i	−0.913038	+	0.407874i	$0.866270\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	36.4717i	1.46830i	0.678990	+	0.734148i	$0.262418\pi$
$617$	36.4717i	1.46830i	−0.678990	+	0.734148i	$0.737582\pi$
$618$	0	0
$619$	−2.99424	−0.120349	−0.0601744	−	0.998188i	$-0.519166\pi$
$619$	−2.99424	−0.120349	−0.0601744	+	0.998188i	$0.519166\pi$
$620$	0	0
$621$	14.1650	0.568422
$622$	0	0
$623$	− 37.3379i	− 1.49591i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	6.94282i	0.277270i
$628$	0	0
$629$	33.4692	1.33451
$630$	0	0
$631$	33.1683	1.32041	0.660204	−	0.751086i	$-0.270470\pi$
$631$	33.1683	1.32041	0.660204	+	0.751086i	$0.270470\pi$
$632$	0	0
$633$	9.32176i	0.370507i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 13.0506i	− 0.517085i
$638$	0	0
$639$	−8.60301	−0.340330
$640$	0	0
$641$	12.0435	0.475690	0.237845	−	0.971303i	$-0.423559\pi$
$641$	12.0435	0.475690	0.237845	+	0.971303i	$0.423559\pi$
$642$	0	0
$643$	22.0000i	0.867595i	0.901010	+	0.433798i	$0.142827\pi$
$643$	22.0000i	0.867595i	−0.901010	+	0.433798i	$0.857173\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 28.0150i	− 1.10139i	−0.834708	−	0.550693i	$-0.814364\pi$
$647$	− 28.0150i	− 1.10139i	0.834708	−	0.550693i	$-0.185636\pi$
$648$	0	0
$649$	−25.0917	−0.984934
$650$	0	0
$651$	27.9981	1.09733
$652$	0	0
$653$	− 7.85254i	− 0.307294i	−0.988126	−	0.153647i	$-0.950898\pi$
$653$	− 7.85254i	− 0.307294i	0.988126	−	0.153647i	$-0.0491018\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	18.6706i	0.728409i
$658$	0	0
$659$	29.9590	1.16704	0.583518	−	0.812100i	$-0.301676\pi$
$659$	29.9590	1.16704	0.583518	+	0.812100i	$0.301676\pi$
$660$	0	0
$661$	−3.42107	−0.133064	−0.0665320	−	0.997784i	$-0.521193\pi$
$661$	−3.42107	−0.133064	−0.0665320	+	0.997784i	$0.521193\pi$
$662$	0	0
$663$	− 19.7551i	− 0.767225i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 8.69794i	− 0.336786i
$668$	0	0
$669$	50.2301	1.94201
$670$	0	0
$671$	15.8662	0.612508
$672$	0	0
$673$	− 45.5095i	− 1.75426i	−0.480252	−	0.877130i	$-0.659455\pi$
$673$	− 45.5095i	− 1.75426i	0.480252	−	0.877130i	$-0.340545\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 16.6375i	− 0.639431i	−0.947514	−	0.319715i	$-0.896413\pi$
$677$	− 16.6375i	− 0.639431i	0.947514	−	0.319715i	$-0.103587\pi$
$678$	0	0
$679$	54.3354	2.08520
$680$	0	0
$681$	−19.5653	−0.749742
$682$	0	0
$683$	15.2463i	0.583382i	0.956513	+	0.291691i	$0.0942179\pi$
$683$	15.2463i	0.583382i	−0.956513	+	0.291691i	$0.905782\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	18.8238i	0.718173i
$688$	0	0
$689$	0.0845261	0.00322019
$690$	0	0
$691$	−45.8676	−1.74489	−0.872443	−	0.488717i	$-0.837465\pi$
$691$	−45.8676	−1.74489	−0.872443	+	0.488717i	$0.837465\pi$
$692$	0	0
$693$	20.7518i	0.788298i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 3.34308i	− 0.126628i
$698$	0	0
$699$	−63.2794	−2.39345
$700$	0	0
$701$	−6.59974	−0.249269	−0.124634	−	0.992203i	$-0.539776\pi$
$701$	−6.59974	−0.249269	−0.124634	+	0.992203i	$0.539776\pi$
$702$	0	0
$703$	7.18194i	0.270872i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	59.0859i	2.22215i
$708$	0	0
$709$	−22.0118	−0.826670	−0.413335	−	0.910579i	$-0.635636\pi$
$709$	−22.0118	−0.826670	−0.413335	+	0.910579i	$0.635636\pi$
$710$	0	0
$711$	26.2542	0.984608
$712$	0	0
$713$	18.1513i	0.679773i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	51.8025i	1.93460i
$718$	0	0
$719$	22.2405	0.829431	0.414715	−	0.909951i	$-0.363881\pi$
$719$	22.2405	0.829431	0.414715	+	0.909951i	$0.363881\pi$
$720$	0	0
$721$	−39.3808	−1.46662
$722$	0	0
$723$	− 13.8033i	− 0.513349i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 37.8662i	− 1.40438i	−0.711990	−	0.702190i	$-0.752206\pi$
$727$	− 37.8662i	− 1.40438i	0.711990	−	0.702190i	$-0.247794\pi$
$728$	0	0
$729$	1.99208	0.0737809
$730$	0	0
$731$	−6.35597	−0.235084
$732$	0	0
$733$	5.02735i	0.185689i	0.995681	+	0.0928446i	$0.0295960\pi$
$733$	5.02735i	0.185689i	−0.995681	+	0.0928446i	$0.970404\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 46.1456i	− 1.69979i
$738$	0	0
$739$	−31.6159	−1.16301	−0.581505	−	0.813543i	$-0.697536\pi$
$739$	−31.6159	−1.16301	−0.581505	+	0.813543i	$0.697536\pi$
$740$	0	0
$741$	4.23912	0.155728
$742$	0	0
$743$	− 33.8252i	− 1.24093i	−0.784236	−	0.620463i	$-0.786945\pi$
$743$	− 33.8252i	− 1.24093i	0.784236	−	0.620463i	$-0.213055\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	17.1877i	0.628865i
$748$	0	0
$749$	−54.2463	−1.98212
$750$	0	0
$751$	−45.5368	−1.66166	−0.830831	−	0.556525i	$-0.812134\pi$
$751$	−45.5368	−1.66166	−0.830831	+	0.556525i	$0.812134\pi$
$752$	0	0
$753$	− 47.8285i	− 1.74297i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	4.55950i	0.165718i	0.996561	+	0.0828590i	$0.0264051\pi$
$757$	4.55950i	0.165718i	−0.996561	+	0.0828590i	$0.973595\pi$
$758$	0	0
$759$	−36.3743	−1.32030
$760$	0	0
$761$	−51.4386	−1.86465	−0.932324	−	0.361624i	$-0.882222\pi$
$761$	−51.4386	−1.86465	−0.932324	+	0.361624i	$0.882222\pi$
$762$	0	0
$763$	13.1833i	0.477268i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	15.3204i	0.553187i
$768$	0	0
$769$	−9.34773	−0.337088	−0.168544	−	0.985694i	$-0.553906\pi$
$769$	−9.34773	−0.337088	−0.168544	+	0.985694i	$0.553906\pi$
$770$	0	0
$771$	−7.22761	−0.260296
$772$	0	0
$773$	19.4887i	0.700958i	0.936571	+	0.350479i	$0.113981\pi$
$773$	19.4887i	0.700958i	−0.936571	+	0.350479i	$0.886019\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	58.0391i	2.08214i
$778$	0	0
$779$	0.717370	0.0257024
$780$	0	0
$781$	−15.5458	−0.556274
$782$	0	0
$783$	4.48865i	0.160411i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 5.32941i	− 0.189973i	−0.995479	−	0.0949864i	$-0.969719\pi$
$787$	− 5.32941i	− 0.189973i	0.995479	−	0.0949864i	$-0.0302808\pi$
$788$	0	0
$789$	−5.09277	−0.181307
$790$	0	0
$791$	54.9201	1.95273
$792$	0	0
$793$	− 9.68754i	− 0.344014i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 37.7220i	− 1.33618i	−0.744079	−	0.668091i	$-0.767112\pi$
$797$	− 37.7220i	− 1.33618i	0.744079	−	0.668091i	$-0.232888\pi$
$798$	0	0
$799$	−25.0604	−0.886575
$800$	0	0
$801$	17.7518	0.627231
$802$	0	0
$803$	33.7382i	1.19059i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	53.6298i	1.88786i
$808$	0	0
$809$	−31.0071	−1.09015	−0.545076	−	0.838386i	$-0.683499\pi$
$809$	−31.0071	−1.09015	−0.545076	+	0.838386i	$0.683499\pi$
$810$	0	0
$811$	26.7738	0.940154	0.470077	−	0.882625i	$-0.344226\pi$
$811$	26.7738	0.940154	0.470077	+	0.882625i	$0.344226\pi$
$812$	0	0
$813$	35.7576i	1.25407i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 1.36389i	− 0.0477164i
$818$	0	0
$819$	12.6706	0.442746
$820$	0	0
$821$	24.2873	0.847632	0.423816	−	0.905748i	$-0.360690\pi$
$821$	24.2873	0.847632	0.423816	+	0.905748i	$0.360690\pi$
$822$	0	0
$823$	0.828460i	0.0288783i	0.999896	+	0.0144392i	$0.00459628\pi$
$823$	0.828460i	0.0288783i	−0.999896	+	0.0144392i	$0.995404\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	42.3387i	1.47226i	0.676840	+	0.736130i	$0.263349\pi$
$827$	42.3387i	1.47226i	−0.676840	+	0.736130i	$0.736651\pi$
$828$	0	0
$829$	3.20602	0.111350	0.0556748	−	0.998449i	$-0.482269\pi$
$829$	3.20602	0.111350	0.0556748	+	0.998449i	$0.482269\pi$
$830$	0	0
$831$	−18.9486	−0.657319
$832$	0	0
$833$	− 31.3042i	− 1.08463i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 9.36716i	− 0.323776i
$838$	0	0
$839$	20.2520	0.699177	0.349589	−	0.936903i	$-0.386321\pi$
$839$	20.2520	0.699177	0.349589	+	0.936903i	$0.386321\pi$
$840$	0	0
$841$	−26.2438	−0.904958
$842$	0	0
$843$	− 0.983839i	− 0.0338852i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 3.24161i	− 0.111383i
$848$	0	0
$849$	−62.5152	−2.14552
$850$	0	0
$851$	−37.6271	−1.28984
$852$	0	0
$853$	− 3.10644i	− 0.106363i	−0.998585	−	0.0531813i	$-0.983064\pi$
$853$	− 3.10644i	− 0.106363i	0.998585	−	0.0531813i	$-0.0169361\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	8.89467i	0.303836i	0.988393	+	0.151918i	$0.0485450\pi$
$857$	8.89467i	0.303836i	−0.988393	+	0.151918i	$0.951455\pi$
$858$	0	0
$859$	17.9989	0.614114	0.307057	−	0.951691i	$-0.400656\pi$
$859$	17.9989	0.614114	0.307057	+	0.951691i	$0.400656\pi$
$860$	0	0
$861$	5.79725	0.197570
$862$	0	0
$863$	− 43.9291i	− 1.49537i	−0.664056	−	0.747683i	$-0.731166\pi$
$863$	− 43.9291i	− 1.49537i	0.664056	−	0.747683i	$-0.268834\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 10.2930i	− 0.349570i
$868$	0	0
$869$	47.4419	1.60936
$870$	0	0
$871$	−28.1754	−0.954687
$872$	0	0
$873$	25.8331i	0.874318i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 4.55375i	− 0.153769i	−0.997040	−	0.0768845i	$-0.975503\pi$
$877$	− 4.55375i	− 0.153769i	0.997040	−	0.0768845i	$-0.0244973\pi$
$878$	0	0
$879$	−45.2394	−1.52589
$880$	0	0
$881$	2.71272	0.0913940	0.0456970	−	0.998955i	$-0.485449\pi$
$881$	2.71272	0.0913940	0.0456970	+	0.998955i	$0.485449\pi$
$882$	0	0
$883$	9.92588i	0.334032i	0.985954	+	0.167016i	$0.0534132\pi$
$883$	9.92588i	0.334032i	−0.985954	+	0.167016i	$0.946587\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 52.7018i	− 1.76955i	−0.466015	−	0.884777i	$-0.654311\pi$
$887$	− 52.7018i	− 1.76955i	0.466015	−	0.884777i	$-0.345689\pi$
$888$	0	0
$889$	−28.4815	−0.955239
$890$	0	0
$891$	35.5803	1.19199
$892$	0	0
$893$	− 5.37756i	− 0.179953i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	22.2093i	0.741547i
$898$	0	0
$899$	−5.75185	−0.191835
$900$	0	0
$901$	0.202750	0.00675459
$902$	0	0
$903$	− 11.0219i	− 0.366786i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 37.3743i	− 1.24099i	−0.784209	−	0.620496i	$-0.786931\pi$
$907$	− 37.3743i	− 1.24099i	0.784209	−	0.620496i	$-0.213069\pi$
$908$	0	0
$909$	−28.0917	−0.931742
$910$	0	0
$911$	−47.6752	−1.57955	−0.789776	−	0.613396i	$-0.789803\pi$
$911$	−47.6752	−1.57955	−0.789776	+	0.613396i	$0.789803\pi$
$912$	0	0
$913$	31.0586i	1.02789i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	72.0189i	2.37827i
$918$	0	0
$919$	35.8605	1.18293	0.591464	−	0.806332i	$-0.298550\pi$
$919$	35.8605	1.18293	0.591464	+	0.806332i	$0.298550\pi$
$920$	0	0
$921$	−75.2624	−2.47998
$922$	0	0
$923$	9.49192i	0.312430i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 18.7231i	− 0.614948i
$928$	0	0
$929$	55.1229	1.80852	0.904261	−	0.426979i	$-0.140422\pi$
$929$	55.1229	1.80852	0.904261	+	0.426979i	$0.140422\pi$
$930$	0	0
$931$	6.71737	0.220153
$932$	0	0
$933$	33.9923i	1.11286i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	53.3333i	1.74232i	0.490997	+	0.871161i	$0.336632\pi$
$937$	53.3333i	1.74232i	−0.490997	+	0.871161i	$0.663368\pi$
$938$	0	0
$939$	−6.50343	−0.212232
$940$	0	0
$941$	29.7335	0.969285	0.484643	−	0.874712i	$-0.338950\pi$
$941$	29.7335	0.969285	0.484643	+	0.874712i	$0.338950\pi$
$942$	0	0
$943$	3.75839i	0.122390i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	39.4821i	1.28300i	0.767125	+	0.641498i	$0.221687\pi$
$947$	39.4821i	1.28300i	−0.767125	+	0.641498i	$0.778313\pi$
$948$	0	0
$949$	20.5997	0.668696
$950$	0	0
$951$	−38.3592	−1.24388
$952$	0	0
$953$	− 15.8479i	− 0.513364i	−0.966496	−	0.256682i	$-0.917371\pi$
$953$	− 15.8479i	− 0.513364i	0.966496	−	0.256682i	$-0.0826292\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 11.5264i	− 0.372596i
$958$	0	0
$959$	−12.5814	−0.406275
$960$	0	0
$961$	−18.9967	−0.612798
$962$	0	0
$963$	− 25.7907i	− 0.831094i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	20.2150i	0.650072i	0.945702	+	0.325036i	$0.105376\pi$
$967$	20.2150i	0.650072i	−0.945702	+	0.325036i	$0.894624\pi$
$968$	0	0
$969$	10.1683	0.326652
$970$	0	0
$971$	−26.8903	−0.862950	−0.431475	−	0.902125i	$-0.642007\pi$
$971$	−26.8903	−0.862950	−0.431475	+	0.902125i	$0.642007\pi$
$972$	0	0
$973$	− 39.1477i	− 1.25502i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 18.6432i	− 0.596450i	−0.954496	−	0.298225i	$-0.903605\pi$
$977$	− 18.6432i	− 0.596450i	0.954496	−	0.298225i	$-0.0963946\pi$
$978$	0	0
$979$	32.0780	1.02522
$980$	0	0
$981$	−6.26785	−0.200117
$982$	0	0
$983$	− 36.7713i	− 1.17282i	−0.810014	−	0.586411i	$-0.800540\pi$
$983$	− 36.7713i	− 1.17282i	0.810014	−	0.586411i	$-0.199460\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 43.4574i	− 1.38327i
$988$	0	0
$989$	7.14557	0.227216
$990$	0	0
$991$	−60.0197	−1.90659	−0.953294	−	0.302043i	$-0.902331\pi$
$991$	−60.0197	−1.90659	−0.953294	+	0.302043i	$0.902331\pi$
$992$	0	0
$993$	− 76.8928i	− 2.44012i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	28.8364i	0.913257i	0.889657	+	0.456629i	$0.150943\pi$
$997$	28.8364i	0.913257i	−0.889657	+	0.456629i	$0.849057\pi$
$998$	0	0
$999$	19.4178	0.614352

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.2.c.f.1749.5		6
5.2	odd	4		1900.2.a.g.1.3	✓	3
5.3	odd	4		1900.2.a.i.1.1	yes	3
5.4	even	2	inner	1900.2.c.f.1749.2		6
20.3	even	4		7600.2.a.bl.1.3		3
20.7	even	4		7600.2.a.ca.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1900.2.a.g.1.3	✓	3	5.2	odd	4
1900.2.a.i.1.1	yes	3	5.3	odd	4
1900.2.c.f.1749.2		6	5.4	even	2	inner
1900.2.c.f.1749.5		6	1.1	even	1	trivial
7600.2.a.bl.1.3		3	20.3	even	4
7600.2.a.ca.1.1		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:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1900.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.18194i q^{3} +3.70370i q^{7} -1.76088 q^{9} +O(q^{10})\)	\(q+2.18194i q^{3} +3.70370i q^{7} -1.76088 q^{9} -3.18194 q^{11} +1.94282i q^{13} +4.66019i q^{17} -1.00000 q^{19} -8.08126 q^{21} -5.23912i q^{23} +2.70370i q^{27} +1.66019 q^{29} -3.46457 q^{31} -6.94282i q^{33} -7.18194i q^{37} -4.23912 q^{39} -0.717370 q^{41} +1.36389i q^{43} +5.37756i q^{47} -6.71737 q^{49} -10.1683 q^{51} -0.0435069i q^{53} -2.18194i q^{57} +7.88564 q^{59} -4.98633 q^{61} -6.52175i q^{63} +14.5023i q^{67} +11.4315 q^{69} +4.88564 q^{71} -10.6030i q^{73} -11.7850i q^{77} -14.9097 q^{79} -11.1819 q^{81} -9.76088i q^{83} +3.62244i q^{87} -10.0813 q^{89} -7.19562 q^{91} -7.55950i q^{93} -14.6706i q^{97} +5.60301 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 10 q^{9}+O(q^{10}) \) 6 * q - 10 * q^9	\( 6 q - 10 q^{9} - 2 q^{11} - 6 q^{19} - 16 q^{21} - 6 q^{29} - 2 q^{31} - 26 q^{39} - 6 q^{41} - 42 q^{49} - 24 q^{51} + 12 q^{59} - 10 q^{61} - 18 q^{69} - 6 q^{71} - 4 q^{79} - 50 q^{81} - 28 q^{89} - 46 q^{91}+O(q^{100}) \) 6 * q - 10 * q^9 - 2 * q^11 - 6 * q^19 - 16 * q^21 - 6 * q^29 - 2 * q^31 - 26 * q^39 - 6 * q^41 - 42 * q^49 - 24 * q^51 + 12 * q^59 - 10 * q^61 - 18 * q^69 - 6 * q^71 - 4 * q^79 - 50 * q^81 - 28 * q^89 - 46 * q^91

Introduction

L-functions

Modular forms

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Database

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