LMFDB - Embedded newform 1900.3.e.g.1101.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,3,Mod(1101,1900)]

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, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1900.1101");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1900.e (of order $2$, degree $1$, 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:	$51.7712502285$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 89x^{12} + 3026x^{10} + 49092x^{8} + 390297x^{6} + 1440495x^{4} + 1994425x^{2} + 151875 $ x^14 + 89x^12 + 3026x^10 + 49092x^8 + 390297x^6 + 1440495x^4 + 1994425x^2 + 151875
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1101.14
Root			$5.26531i$ of defining polynomial
Character	$\chi$	$=$	1900.1101
Dual form			1900.3.e.g.1101.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.26531i q^{3} +12.1735 q^{7} -18.7235 q^{9} +O(q^{10})$	$q+5.26531i q^{3} +12.1735 q^{7} -18.7235 q^{9} -4.59727 q^{11} +19.9019i q^{13} -4.10564 q^{17} +(-17.9241 + 6.30286i) q^{19} +64.0972i q^{21} +35.2013 q^{23} -51.1972i q^{27} -16.4948i q^{29} -4.01062i q^{31} -24.2060i q^{33} +10.6049i q^{37} -104.790 q^{39} +22.4270i q^{41} -51.8177 q^{43} -23.7025 q^{47} +99.1940 q^{49} -21.6174i q^{51} +97.3018i q^{53} +(-33.1865 - 94.3760i) q^{57} -33.1216i q^{59} -75.1967 q^{61} -227.930 q^{63} +6.31874i q^{67} +185.346i q^{69} +118.291i q^{71} +7.02357 q^{73} -55.9648 q^{77} +76.0473i q^{79} +101.058 q^{81} -100.198 q^{83} +86.8500 q^{87} -123.299i q^{89} +242.276i q^{91} +21.1171 q^{93} +101.202i q^{97} +86.0769 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 4 q^{7} - 52 q^{9}+O(q^{10}) $ 14 * q - 4 * q^7 - 52 * q^9	$ 14 q - 4 q^{7} - 52 q^{9} + 4 q^{11} + 6 q^{17} - 29 q^{19} + 28 q^{23} - 56 q^{39} - 22 q^{43} - 84 q^{47} + 138 q^{49} + 5 q^{57} - 50 q^{61} - 234 q^{63} + 204 q^{73} - 68 q^{77} + 66 q^{81} - 256 q^{83} - 18 q^{87} - 118 q^{93} + 92 q^{99}+O(q^{100}) $ 14 * q - 4 * q^7 - 52 * q^9 + 4 * q^11 + 6 * q^17 - 29 * q^19 + 28 * q^23 - 56 * q^39 - 22 * q^43 - 84 * q^47 + 138 * q^49 + 5 * q^57 - 50 * q^61 - 234 * q^63 + 204 * q^73 - 68 * q^77 + 66 * q^81 - 256 * q^83 - 18 * q^87 - 118 * q^93 + 92 * q^99

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
$3$			5.26531i			1.75510i	0.479482	+	0.877552i	$0.340825\pi$
$3$			5.26531i			1.75510i	−0.479482	+	0.877552i	$0.659175\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	12.1735			1.73907			0.869535	−	0.493871i	$-0.164418\pi$
$7$	12.1735			1.73907			0.869535	+	0.493871i	$0.164418\pi$
$8$		0			0
$9$	−18.7235			−2.08039
$10$		0			0
$11$	−4.59727			−0.417933			−0.208967	−	0.977923i	$-0.567010\pi$
$11$	−4.59727			−0.417933			−0.208967	+	0.977923i	$0.567010\pi$
$12$		0			0
$13$			19.9019i			1.53092i	0.643486	+	0.765458i	$0.277488\pi$
$13$			19.9019i			1.53092i	−0.643486	+	0.765458i	$0.722512\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.10564			−0.241508			−0.120754	−	0.992682i	$-0.538531\pi$
$17$	−4.10564			−0.241508			−0.120754	+	0.992682i	$0.538531\pi$
$18$		0			0
$19$	−17.9241	+	6.30286i	−0.943375	+	0.331729i
$19$	−17.9241	+	6.30286i	−0.943375	+	0.331729i
$20$		0			0
$21$			64.0972i			3.05225i
$22$		0			0
$23$	35.2013			1.53049			0.765246	−	0.643738i	$-0.222617\pi$
$23$	35.2013			1.53049			0.765246	+	0.643738i	$0.222617\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	51.1972i		−	1.89619i
$28$		0			0
$29$		−	16.4948i		−	0.568785i	−0.958708	−	0.284392i	$-0.908208\pi$
$29$		−	16.4948i		−	0.568785i	0.958708	−	0.284392i	$-0.0917919\pi$
$30$		0			0
$31$		−	4.01062i		−	0.129375i	−0.997906	−	0.0646873i	$-0.979395\pi$
$31$		−	4.01062i		−	0.129375i	0.997906	−	0.0646873i	$-0.0206050\pi$
$32$		0			0
$33$		−	24.2060i		−	0.733516i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			10.6049i			0.286620i	0.989678	+	0.143310i	$0.0457745\pi$
$37$			10.6049i			0.286620i	−0.989678	+	0.143310i	$0.954225\pi$
$38$		0			0
$39$	−104.790			−2.68692
$40$		0			0
$41$			22.4270i			0.546999i	0.961872	+	0.273500i	$0.0881812\pi$
$41$			22.4270i			0.546999i	−0.961872	+	0.273500i	$0.911819\pi$
$42$		0			0
$43$	−51.8177			−1.20506			−0.602531	−	0.798095i	$-0.705841\pi$
$43$	−51.8177			−1.20506			−0.602531	+	0.798095i	$0.705841\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−23.7025			−0.504308			−0.252154	−	0.967687i	$-0.581139\pi$
$47$	−23.7025			−0.504308			−0.252154	+	0.967687i	$0.581139\pi$
$48$		0			0
$49$	99.1940			2.02437
$50$		0			0
$51$		−	21.6174i		−	0.423871i
$52$		0			0
$53$			97.3018i			1.83588i	0.396715	+	0.917942i	$0.370150\pi$
$53$			97.3018i			1.83588i	−0.396715	+	0.917942i	$0.629850\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−33.1865	−	94.3760i	−0.582219	−	1.65572i
$58$		0			0
$59$		−	33.1216i		−	0.561384i	−0.959798	−	0.280692i	$-0.909436\pi$
$59$		−	33.1216i		−	0.561384i	0.959798	−	0.280692i	$-0.0905639\pi$
$60$		0			0
$61$	−75.1967			−1.23273			−0.616366	−	0.787460i	$-0.711396\pi$
$61$	−75.1967			−1.23273			−0.616366	+	0.787460i	$0.711396\pi$
$62$		0			0
$63$	−227.930			−3.61794
$64$		0			0
$65$		0			0
$66$		0			0
$67$			6.31874i			0.0943095i	0.998888	+	0.0471548i	$0.0150154\pi$
$67$			6.31874i			0.0943095i	−0.998888	+	0.0471548i	$0.984985\pi$
$68$		0			0
$69$			185.346i			2.68617i
$70$		0			0
$71$			118.291i			1.66607i	0.553224	+	0.833033i	$0.313397\pi$
$71$			118.291i			1.66607i	−0.553224	+	0.833033i	$0.686603\pi$
$72$		0			0
$73$	7.02357			0.0962133			0.0481067	−	0.998842i	$-0.484681\pi$
$73$	7.02357			0.0962133			0.0481067	+	0.998842i	$0.484681\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−55.9648			−0.726815
$78$		0			0
$79$			76.0473i			0.962624i	0.876549	+	0.481312i	$0.159840\pi$
$79$			76.0473i			0.962624i	−0.876549	+	0.481312i	$0.840160\pi$
$80$		0			0
$81$	101.058			1.24763
$82$		0			0
$83$	−100.198			−1.20720			−0.603600	−	0.797287i	$-0.706268\pi$
$83$	−100.198			−1.20720			−0.603600	+	0.797287i	$0.706268\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	86.8500			0.998276
$88$		0			0
$89$		−	123.299i		−	1.38538i	−0.721233	−	0.692692i	$-0.756424\pi$
$89$		−	123.299i		−	1.38538i	0.721233	−	0.692692i	$-0.243576\pi$
$90$		0			0
$91$			242.276i			2.66237i
$92$		0			0
$93$	21.1171			0.227066
$94$		0			0
$95$		0			0
$96$		0			0
$97$			101.202i			1.04332i	0.853154	+	0.521659i	$0.174687\pi$
$97$			101.202i			1.04332i	−0.853154	+	0.521659i	$0.825313\pi$
$98$		0			0
$99$	86.0769			0.869463
$100$		0			0
$101$	150.478			1.48988			0.744939	−	0.667133i	$-0.232479\pi$
$101$	150.478			1.48988			0.744939	+	0.667133i	$0.232479\pi$
$102$		0			0
$103$		−	112.271i		−	1.09001i	−0.838434	−	0.545003i	$-0.816529\pi$
$103$		−	112.271i		−	1.09001i	0.838434	−	0.545003i	$-0.183471\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	144.877i		−	1.35399i	−0.735987	−	0.676996i	$-0.763282\pi$
$107$		−	144.877i		−	1.35399i	0.735987	−	0.676996i	$-0.236718\pi$
$108$		0			0
$109$			50.7929i			0.465990i	0.972478	+	0.232995i	$0.0748526\pi$
$109$			50.7929i			0.465990i	−0.972478	+	0.232995i	$0.925147\pi$
$110$		0			0
$111$	−55.8382			−0.503047
$112$		0			0
$113$		−	84.1325i		−	0.744536i	−0.928125	−	0.372268i	$-0.878580\pi$
$113$		−	84.1325i		−	0.744536i	0.928125	−	0.372268i	$-0.121420\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	372.633i		−	3.18490i
$118$		0			0
$119$	−49.9799			−0.419999
$120$		0			0
$121$	−99.8651			−0.825332
$122$		0			0
$123$	−118.085			−0.960040
$124$		0			0
$125$		0			0
$126$		0			0
$127$			129.368i			1.01864i	0.860576	+	0.509322i	$0.170104\pi$
$127$			129.368i			1.01864i	−0.860576	+	0.509322i	$0.829896\pi$
$128$		0			0
$129$		−	272.836i		−	2.11501i
$130$		0			0
$131$	139.421			1.06428			0.532141	−	0.846656i	$-0.321388\pi$
$131$	139.421			1.06428			0.532141	+	0.846656i	$0.321388\pi$
$132$		0			0
$133$	−218.199	+	76.7278i	−1.64060	+	0.576901i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	64.8931			0.473672			0.236836	−	0.971550i	$-0.423890\pi$
$137$	64.8931			0.473672			0.236836	+	0.971550i	$0.423890\pi$
$138$		0			0
$139$	−164.632			−1.18440			−0.592200	−	0.805791i	$-0.701741\pi$
$139$	−164.632			−1.18440			−0.592200	+	0.805791i	$0.701741\pi$
$140$		0			0
$141$		−	124.801i		−	0.885112i
$142$		0			0
$143$		−	91.4944i		−	0.639821i
$144$		0			0
$145$		0			0
$146$		0			0
$147$			522.287i			3.55297i
$148$		0			0
$149$	13.8027			0.0926354			0.0463177	−	0.998927i	$-0.485251\pi$
$149$	13.8027			0.0926354			0.0463177	+	0.998927i	$0.485251\pi$
$150$		0			0
$151$		−	214.161i		−	1.41829i	−0.705064	−	0.709144i	$-0.749082\pi$
$151$		−	214.161i		−	1.41829i	0.705064	−	0.709144i	$-0.250918\pi$
$152$		0			0
$153$	76.8718			0.502430
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−145.383			−0.926009			−0.463004	−	0.886356i	$-0.653229\pi$
$157$	−145.383			−0.926009			−0.463004	+	0.886356i	$0.653229\pi$
$158$		0			0
$159$	−512.324			−3.22217
$160$		0			0
$161$	428.523			2.66163
$162$		0			0
$163$	172.709			1.05956			0.529781	−	0.848135i	$-0.322274\pi$
$163$	172.709			1.05956			0.529781	+	0.848135i	$0.322274\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			232.500i			1.39222i	0.717937	+	0.696108i	$0.245086\pi$
$167$			232.500i			1.39222i	−0.717937	+	0.696108i	$0.754914\pi$
$168$		0			0
$169$	−227.086			−1.34370
$170$		0			0
$171$	335.602	−	118.012i	1.96259	−	0.690126i
$172$		0			0
$173$			290.870i			1.68133i	0.541556	+	0.840665i	$0.317835\pi$
$173$			290.870i			1.68133i	−0.541556	+	0.840665i	$0.682165\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	174.396			0.985286
$178$		0			0
$179$			172.255i			0.962319i	0.876633	+	0.481159i	$0.159784\pi$
$179$			172.255i			0.962319i	−0.876633	+	0.481159i	$0.840216\pi$
$180$		0			0
$181$		−	343.036i		−	1.89523i	−0.319420	−	0.947613i	$-0.603488\pi$
$181$		−	343.036i		−	1.89523i	0.319420	−	0.947613i	$-0.396512\pi$
$182$		0			0
$183$		−	395.934i		−	2.16357i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	18.8747			0.100934
$188$		0			0
$189$		−	623.249i		−	3.29761i
$190$		0			0
$191$	−184.283			−0.964832			−0.482416	−	0.875942i	$-0.660241\pi$
$191$	−184.283			−0.964832			−0.482416	+	0.875942i	$0.660241\pi$
$192$		0			0
$193$		−	304.518i		−	1.57781i	−0.614513	−	0.788907i	$-0.710647\pi$
$193$		−	304.518i		−	1.57781i	0.614513	−	0.788907i	$-0.289353\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−306.839			−1.55756			−0.778778	−	0.627299i	$-0.784160\pi$
$197$	−306.839			−1.55756			−0.778778	+	0.627299i	$0.784160\pi$
$198$		0			0
$199$	−89.0393			−0.447434			−0.223717	−	0.974654i	$-0.571819\pi$
$199$	−89.0393			−0.447434			−0.223717	+	0.974654i	$0.571819\pi$
$200$		0			0
$201$	−33.2701			−0.165523
$202$		0			0
$203$		−	200.799i		−	0.989157i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−659.091			−3.18402
$208$		0			0
$209$	82.4019	−	28.9759i	0.394268	−	0.138641i
$210$		0			0
$211$			156.363i			0.741056i	0.928821	+	0.370528i	$0.120823\pi$
$211$			156.363i			0.741056i	−0.928821	+	0.370528i	$0.879177\pi$
$212$		0			0
$213$	−622.837			−2.92412
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	48.8232i		−	0.224992i
$218$		0			0
$219$			36.9813i			0.168864i
$220$		0			0
$221$		−	81.7100i		−	0.369728i
$222$		0			0
$223$		−	312.945i		−	1.40334i	−0.712501	−	0.701671i	$-0.752437\pi$
$223$		−	312.945i		−	1.40334i	0.712501	−	0.701671i	$-0.247563\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	199.435i		−	0.878570i	−0.898348	−	0.439285i	$-0.855232\pi$
$227$		−	199.435i		−	0.878570i	0.898348	−	0.439285i	$-0.144768\pi$
$228$		0			0
$229$	251.971			1.10031			0.550155	−	0.835063i	$-0.314569\pi$
$229$	251.971			1.10031			0.550155	+	0.835063i	$0.314569\pi$
$230$		0			0
$231$		−	294.672i		−	1.27564i
$232$		0			0
$233$	452.694			1.94289			0.971446	−	0.237259i	$-0.0762489\pi$
$233$	452.694			1.94289			0.971446	+	0.237259i	$0.0762489\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−400.413			−1.68950
$238$		0			0
$239$	143.219			0.599245			0.299622	−	0.954058i	$-0.403139\pi$
$239$	143.219			0.599245			0.299622	+	0.954058i	$0.403139\pi$
$240$		0			0
$241$		−	6.63375i		−	0.0275259i	−0.999905	−	0.0137630i	$-0.995619\pi$
$241$		−	6.63375i		−	0.0275259i	0.999905	−	0.0137630i	$-0.00438103\pi$
$242$		0			0
$243$			71.3257i			0.293521i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−125.439	−	356.724i	−0.507850	−	1.44423i
$248$		0			0
$249$		−	527.571i		−	2.11876i
$250$		0			0
$251$	180.383			0.718658			0.359329	−	0.933211i	$-0.383006\pi$
$251$	180.383			0.718658			0.359329	+	0.933211i	$0.383006\pi$
$252$		0			0
$253$	−161.830			−0.639643
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	14.9645i		−	0.0582275i	−0.999576	−	0.0291137i	$-0.990732\pi$
$257$		−	14.9645i		−	0.0582275i	0.999576	−	0.0291137i	$-0.00926850\pi$
$258$		0			0
$259$			129.099i			0.498452i
$260$		0			0
$261$			308.840i			1.18329i
$262$		0			0
$263$	439.986			1.67295			0.836475	−	0.548005i	$-0.184613\pi$
$263$	439.986			1.67295			0.836475	+	0.548005i	$0.184613\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	649.209			2.43149
$268$		0			0
$269$			253.894i			0.943843i	0.881641	+	0.471922i	$0.156439\pi$
$269$			253.894i			0.943843i	−0.881641	+	0.471922i	$0.843561\pi$
$270$		0			0
$271$	107.372			0.396208			0.198104	−	0.980181i	$-0.436522\pi$
$271$	107.372			0.396208			0.198104	+	0.980181i	$0.436522\pi$
$272$		0			0
$273$	−1275.66			−4.67274
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−21.9808			−0.0793531			−0.0396765	−	0.999213i	$-0.512633\pi$
$277$	−21.9808			−0.0793531			−0.0396765	+	0.999213i	$0.512633\pi$
$278$		0			0
$279$			75.0927i			0.269150i
$280$		0			0
$281$		−	223.590i		−	0.795694i	−0.917452	−	0.397847i	$-0.869757\pi$
$281$		−	223.590i		−	0.795694i	0.917452	−	0.397847i	$-0.130243\pi$
$282$		0			0
$283$	−52.7660			−0.186452			−0.0932261	−	0.995645i	$-0.529718\pi$
$283$	−52.7660			−0.186452			−0.0932261	+	0.995645i	$0.529718\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			273.015i			0.951270i
$288$		0			0
$289$	−272.144			−0.941674
$290$		0			0
$291$	−532.859			−1.83113
$292$		0			0
$293$		−	293.865i		−	1.00295i	−0.865172	−	0.501476i	$-0.832791\pi$
$293$		−	293.865i		−	1.00295i	0.865172	−	0.501476i	$-0.167209\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$			235.367i			0.792482i
$298$		0			0
$299$			700.573i			2.34305i
$300$		0			0
$301$	−630.802			−2.09569
$302$		0			0
$303$			792.311i			2.61489i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			392.330i			1.27795i	0.769228	+	0.638975i	$0.220641\pi$
$307$			392.330i			1.27795i	−0.769228	+	0.638975i	$0.779359\pi$
$308$		0			0
$309$	591.139			1.91307
$310$		0			0
$311$	223.375			0.718249			0.359124	−	0.933290i	$-0.383075\pi$
$311$	223.375			0.718249			0.359124	+	0.933290i	$0.383075\pi$
$312$		0			0
$313$	−254.194			−0.812123			−0.406062	−	0.913846i	$-0.633098\pi$
$313$	−254.194			−0.812123			−0.406062	+	0.913846i	$0.633098\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			40.8360i			0.128820i	0.997924	+	0.0644101i	$0.0205166\pi$
$317$			40.8360i			0.128820i	−0.997924	+	0.0644101i	$0.979483\pi$
$318$		0			0
$319$			75.8308i			0.237714i
$320$		0			0
$321$	762.823			2.37640
$322$		0			0
$323$	73.5899	−	25.8772i	0.227832	−	0.0801153i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−267.440			−0.817861
$328$		0			0
$329$	−288.542			−0.877027
$330$		0			0
$331$			108.046i			0.326423i	0.986591	+	0.163211i	$0.0521852\pi$
$331$			108.046i			0.326423i	−0.986591	+	0.163211i	$0.947815\pi$
$332$		0			0
$333$		−	198.561i		−	0.596280i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	185.597i		−	0.550732i	−0.961339	−	0.275366i	$-0.911201\pi$
$337$		−	185.597i		−	0.550732i	0.961339	−	0.275366i	$-0.0887990\pi$
$338$		0			0
$339$	442.984			1.30674
$340$		0			0
$341$			18.4379i			0.0540700i
$342$		0			0
$343$	611.036			1.78145
$344$		0			0
$345$		0			0
$346$		0			0
$347$	590.178			1.70080			0.850400	−	0.526136i	$-0.176360\pi$
$347$	590.178			1.70080			0.850400	+	0.526136i	$0.176360\pi$
$348$		0			0
$349$	−71.3361			−0.204401			−0.102201	−	0.994764i	$-0.532588\pi$
$349$	−71.3361			−0.204401			−0.102201	+	0.994764i	$0.532588\pi$
$350$		0			0
$351$	1018.92			2.90291
$352$		0			0
$353$	174.929			0.495550			0.247775	−	0.968818i	$-0.420301\pi$
$353$	174.929			0.495550			0.247775	+	0.968818i	$0.420301\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		−	263.160i		−	0.737142i
$358$		0			0
$359$	339.112			0.944600			0.472300	−	0.881438i	$-0.343424\pi$
$359$	339.112			0.944600			0.472300	+	0.881438i	$0.343424\pi$
$360$		0			0
$361$	281.548	−	225.946i	0.779911	−	0.625890i
$362$		0			0
$363$		−	525.821i		−	1.44854i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	407.386			1.11004			0.555022	−	0.831836i	$-0.312710\pi$
$367$	407.386			1.11004			0.555022	+	0.831836i	$0.312710\pi$
$368$		0			0
$369$		−	419.911i		−	1.13797i
$370$		0			0
$371$			1184.50i			3.19273i
$372$		0			0
$373$			369.020i			0.989330i	0.869084	+	0.494665i	$0.164709\pi$
$373$			369.020i			0.989330i	−0.869084	+	0.494665i	$0.835291\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	328.277			0.870762
$378$		0			0
$379$			67.5519i			0.178237i	0.996021	+	0.0891187i	$0.0284050\pi$
$379$			67.5519i			0.178237i	−0.996021	+	0.0891187i	$0.971595\pi$
$380$		0			0
$381$	−681.161			−1.78782
$382$		0			0
$383$			352.331i			0.919923i	0.887939	+	0.459962i	$0.152137\pi$
$383$			352.331i			0.919923i	−0.887939	+	0.459962i	$0.847863\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	970.208			2.50700
$388$		0			0
$389$	−628.198			−1.61491			−0.807453	−	0.589932i	$-0.799154\pi$
$389$	−628.198			−1.61491			−0.807453	+	0.589932i	$0.799154\pi$
$390$		0			0
$391$	−144.524			−0.369626
$392$		0			0
$393$			734.094i			1.86792i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	407.340			1.02604			0.513022	−	0.858375i	$-0.328526\pi$
$397$	407.340			1.02604			0.513022	+	0.858375i	$0.328526\pi$
$398$		0			0
$399$	−403.996	−	1148.89i	−1.01252	−	2.87941i
$400$		0			0
$401$		−	433.712i		−	1.08158i	−0.841159	−	0.540788i	$-0.818126\pi$
$401$		−	433.712i		−	1.08158i	0.841159	−	0.540788i	$-0.181874\pi$
$402$		0			0
$403$	79.8189			0.198062
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	48.7537i		−	0.119788i
$408$		0			0
$409$			744.375i			1.81999i	0.414622	+	0.909994i	$0.363914\pi$
$409$			744.375i			1.81999i	−0.414622	+	0.909994i	$0.636086\pi$
$410$		0			0
$411$			341.682i			0.831343i
$412$		0			0
$413$		−	403.206i		−	0.976286i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	866.837i		−	2.07875i
$418$		0			0
$419$	−271.530			−0.648042			−0.324021	−	0.946050i	$-0.605035\pi$
$419$	−271.530			−0.648042			−0.324021	+	0.946050i	$0.605035\pi$
$420$		0			0
$421$			819.454i			1.94645i	0.229860	+	0.973224i	$0.426173\pi$
$421$			819.454i			1.94645i	−0.229860	+	0.973224i	$0.573827\pi$
$422$		0			0
$423$	443.793			1.04916
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−915.406			−2.14381
$428$		0			0
$429$	481.746			1.12295
$430$		0			0
$431$			723.199i			1.67796i	0.544166	+	0.838978i	$0.316846\pi$
$431$			723.199i			1.67796i	−0.544166	+	0.838978i	$0.683154\pi$
$432$		0			0
$433$		−	228.589i		−	0.527920i	−0.964534	−	0.263960i	$-0.914971\pi$
$433$		−	228.589i		−	0.527920i	0.964534	−	0.263960i	$-0.0850287\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−630.952	+	221.869i	−1.44383	+	0.507709i
$438$		0			0
$439$			207.651i			0.473009i	0.971630	+	0.236504i	$0.0760018\pi$
$439$			207.651i			0.473009i	−0.971630	+	0.236504i	$0.923998\pi$
$440$		0			0
$441$	−1857.26			−4.21147
$442$		0			0
$443$	−573.475			−1.29453			−0.647263	−	0.762267i	$-0.724086\pi$
$443$	−573.475			−1.29453			−0.647263	+	0.762267i	$0.724086\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			72.6753i			0.162585i
$448$		0			0
$449$			479.112i			1.06707i	0.845779	+	0.533533i	$0.179136\pi$
$449$			479.112i			1.06707i	−0.845779	+	0.533533i	$0.820864\pi$
$450$		0			0
$451$		−	103.103i		−	0.228609i
$452$		0			0
$453$	1127.63			2.48924
$454$		0			0
$455$		0			0
$456$		0			0
$457$	88.4835			0.193618			0.0968090	−	0.995303i	$-0.469136\pi$
$457$	88.4835			0.193618			0.0968090	+	0.995303i	$0.469136\pi$
$458$		0			0
$459$			210.197i			0.457946i
$460$		0			0
$461$	241.330			0.523493			0.261746	−	0.965137i	$-0.415702\pi$
$461$	241.330			0.523493			0.261746	+	0.965137i	$0.415702\pi$
$462$		0			0
$463$	−661.506			−1.42874			−0.714370	−	0.699769i	$-0.753286\pi$
$463$	−661.506			−1.42874			−0.714370	+	0.699769i	$0.753286\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−286.229			−0.612910			−0.306455	−	0.951885i	$-0.599143\pi$
$467$	−286.229			−0.612910			−0.306455	+	0.951885i	$0.599143\pi$
$468$		0			0
$469$			76.9211i			0.164011i
$470$		0			0
$471$		−	765.489i		−	1.62524i
$472$		0			0
$473$	238.220			0.503636
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	1821.83i		−	3.81935i
$478$		0			0
$479$	−239.881			−0.500796			−0.250398	−	0.968143i	$-0.580562\pi$
$479$	−239.881			−0.500796			−0.250398	+	0.968143i	$0.580562\pi$
$480$		0			0
$481$	−211.058			−0.438791
$482$		0			0
$483$			2256.31i			4.67144i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			566.043i			1.16231i	0.813795	+	0.581153i	$0.197398\pi$
$487$			566.043i			1.16231i	−0.813795	+	0.581153i	$0.802602\pi$
$488$		0			0
$489$			909.364i			1.85964i
$490$		0			0
$491$	−390.830			−0.795988			−0.397994	−	0.917388i	$-0.630294\pi$
$491$	−390.830			−0.795988			−0.397994	+	0.917388i	$0.630294\pi$
$492$		0			0
$493$			67.7215i			0.137366i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			1440.01i			2.89740i
$498$		0			0
$499$	163.998			0.328652			0.164326	−	0.986406i	$-0.447455\pi$
$499$	163.998			0.328652			0.164326	+	0.986406i	$0.447455\pi$
$500$		0			0
$501$	−1224.18			−2.44348
$502$		0			0
$503$	−35.1241			−0.0698291			−0.0349146	−	0.999390i	$-0.511116\pi$
$503$	−35.1241			−0.0698291			−0.0349146	+	0.999390i	$0.511116\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	1195.68i		−	2.35834i
$508$		0			0
$509$			306.832i			0.602813i	0.953496	+	0.301406i	$0.0974561\pi$
$509$			306.832i			0.602813i	−0.953496	+	0.301406i	$0.902544\pi$
$510$		0			0
$511$	85.5014			0.167322
$512$		0			0
$513$	322.689	+	917.665i	0.629023	+	1.78882i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	108.966			0.210767
$518$		0			0
$519$	−1531.52			−2.95091
$520$		0			0
$521$			283.389i			0.543932i	0.962307	+	0.271966i	$0.0876739\pi$
$521$			283.389i			0.543932i	−0.962307	+	0.271966i	$0.912326\pi$
$522$		0			0
$523$		−	187.427i		−	0.358368i	−0.983816	−	0.179184i	$-0.942654\pi$
$523$		−	187.427i		−	0.358368i	0.983816	−	0.179184i	$-0.0573458\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			16.4661i			0.0312450i
$528$		0			0
$529$	710.132			1.34240
$530$		0			0
$531$			620.153i			1.16790i
$532$		0			0
$533$	−446.340			−0.837410
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−906.976			−1.68897
$538$		0			0
$539$	−456.021			−0.846050
$540$		0			0
$541$	282.159			0.521552			0.260776	−	0.965399i	$-0.416022\pi$
$541$	282.159			0.521552			0.260776	+	0.965399i	$0.416022\pi$
$542$		0			0
$543$	1806.19			3.32632
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	342.502i		−	0.626146i	−0.949729	−	0.313073i	$-0.898642\pi$
$547$		−	342.502i		−	0.626146i	0.949729	−	0.313073i	$-0.101358\pi$
$548$		0			0
$549$	1407.94			2.56456
$550$		0			0
$551$	103.964	+	295.654i	0.188683	+	0.536577i
$552$		0			0
$553$			925.761i			1.67407i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	27.8687			0.0500336			0.0250168	−	0.999687i	$-0.492036\pi$
$557$	27.8687			0.0500336			0.0250168	+	0.999687i	$0.492036\pi$
$558$		0			0
$559$		−	1031.27i		−	1.84485i
$560$		0			0
$561$			99.3811i			0.177150i
$562$		0			0
$563$		−	359.430i		−	0.638420i	−0.947684	−	0.319210i	$-0.896583\pi$
$563$		−	359.430i		−	0.638420i	0.947684	−	0.319210i	$-0.103417\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	1230.23			2.16971
$568$		0			0
$569$		−	137.757i		−	0.242103i	−0.992646	−	0.121051i	$-0.961373\pi$
$569$		−	137.757i		−	0.242103i	0.992646	−	0.121051i	$-0.0386266\pi$
$570$		0			0
$571$	383.100			0.670929			0.335464	−	0.942053i	$-0.391107\pi$
$571$	383.100			0.670929			0.335464	+	0.942053i	$0.391107\pi$
$572$		0			0
$573$		−	970.307i		−	1.69338i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−655.282			−1.13567			−0.567836	−	0.823142i	$-0.692219\pi$
$577$	−655.282			−1.13567			−0.567836	+	0.823142i	$0.692219\pi$
$578$		0			0
$579$	1603.38			2.76923
$580$		0			0
$581$	−1219.75			−2.09941
$582$		0			0
$583$		−	447.322i		−	0.767277i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−229.615			−0.391167			−0.195583	−	0.980687i	$-0.562660\pi$
$587$	−229.615			−0.391167			−0.195583	+	0.980687i	$0.562660\pi$
$588$		0			0
$589$	25.2783	+	71.8867i	0.0429174	+	0.122049i
$590$		0			0
$591$		−	1615.60i		−	2.73367i
$592$		0			0
$593$	1014.80			1.71129			0.855645	−	0.517562i	$-0.173160\pi$
$593$	1014.80			1.71129			0.855645	+	0.517562i	$0.173160\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	468.819i		−	0.785292i
$598$		0			0
$599$			848.378i			1.41632i	0.706050	+	0.708162i	$0.250475\pi$
$599$			848.378i			1.41632i	−0.706050	+	0.708162i	$0.749525\pi$
$600$		0			0
$601$			712.920i			1.18622i	0.805121	+	0.593111i	$0.202101\pi$
$601$			712.920i			1.18622i	−0.805121	+	0.593111i	$0.797899\pi$
$602$		0			0
$603$		−	118.309i		−	0.196200i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			291.033i			0.479462i	0.970839	+	0.239731i	$0.0770592\pi$
$607$			291.033i			0.479462i	−0.970839	+	0.239731i	$0.922941\pi$
$608$		0			0
$609$	1057.27			1.73607
$610$		0			0
$611$		−	471.724i		−	0.772053i
$612$		0			0
$613$	872.135			1.42273			0.711366	−	0.702822i	$-0.248077\pi$
$613$	872.135			1.42273			0.711366	+	0.702822i	$0.248077\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	587.897			0.952832			0.476416	−	0.879220i	$-0.341936\pi$
$617$	587.897			0.952832			0.476416	+	0.879220i	$0.341936\pi$
$618$		0			0
$619$	312.874			0.505450			0.252725	−	0.967538i	$-0.418673\pi$
$619$	312.874			0.505450			0.252725	+	0.967538i	$0.418673\pi$
$620$		0			0
$621$		−	1802.21i		−	2.90211i
$622$		0			0
$623$		−	1500.98i		−	2.40928i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	152.567	+	433.872i	0.243329	+	0.691980i
$628$		0			0
$629$		−	43.5400i		−	0.0692209i
$630$		0			0
$631$	−445.242			−0.705614			−0.352807	−	0.935696i	$-0.614773\pi$
$631$	−445.242			−0.705614			−0.352807	+	0.935696i	$0.614773\pi$
$632$		0			0
$633$	−823.299			−1.30063
$634$		0			0
$635$		0			0
$636$		0			0
$637$			1974.15i			3.09914i
$638$		0			0
$639$		−	2214.81i		−	3.46606i
$640$		0			0
$641$		−	111.139i		−	0.173384i	−0.996235	−	0.0866920i	$-0.972370\pi$
$641$		−	111.139i		−	0.173384i	0.996235	−	0.0866920i	$-0.0276296\pi$
$642$		0			0
$643$	−434.676			−0.676012			−0.338006	−	0.941144i	$-0.609752\pi$
$643$	−434.676			−0.676012			−0.338006	+	0.941144i	$0.609752\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	902.589			1.39504			0.697518	−	0.716567i	$-0.254288\pi$
$647$	902.589			1.39504			0.697518	+	0.716567i	$0.254288\pi$
$648$		0			0
$649$			152.269i			0.234621i
$650$		0			0
$651$	257.069			0.394884
$652$		0			0
$653$	190.902			0.292346			0.146173	−	0.989259i	$-0.453304\pi$
$653$	190.902			0.292346			0.146173	+	0.989259i	$0.453304\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−131.506			−0.200161
$658$		0			0
$659$		−	412.242i		−	0.625558i	−0.949826	−	0.312779i	$-0.898740\pi$
$659$		−	412.242i		−	0.625558i	0.949826	−	0.312779i	$-0.101260\pi$
$660$		0			0
$661$			384.455i			0.581626i	0.956780	+	0.290813i	$0.0939257\pi$
$661$			384.455i			0.581626i	−0.956780	+	0.290813i	$0.906074\pi$
$662$		0			0
$663$	430.228			0.648912
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	580.637i		−	0.870520i
$668$		0			0
$669$	1647.76			2.46301
$670$		0			0
$671$	345.699			0.515200
$672$		0			0
$673$		−	1051.54i		−	1.56247i	−0.624237	−	0.781235i	$-0.714590\pi$
$673$		−	1051.54i		−	1.56247i	0.624237	−	0.781235i	$-0.285410\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			954.223i			1.40949i	0.709462	+	0.704744i	$0.248938\pi$
$677$			954.223i			1.40949i	−0.709462	+	0.704744i	$0.751062\pi$
$678$		0			0
$679$			1231.98i			1.81440i
$680$		0			0
$681$	1050.09			1.54198
$682$		0			0
$683$			564.386i			0.826334i	0.910655	+	0.413167i	$0.135577\pi$
$683$			564.386i			0.826334i	−0.910655	+	0.413167i	$0.864423\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$			1326.70i			1.93116i
$688$		0			0
$689$	−1936.49			−2.81058
$690$		0			0
$691$	−566.388			−0.819664			−0.409832	−	0.912161i	$-0.634413\pi$
$691$	−566.388			−0.819664			−0.409832	+	0.912161i	$0.634413\pi$
$692$		0			0
$693$	1047.86			1.51206
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	92.0770i		−	0.132105i
$698$		0			0
$699$			2383.57i			3.40998i
$700$		0			0
$701$	534.434			0.762388			0.381194	−	0.924495i	$-0.375513\pi$
$701$	534.434			0.762388			0.381194	+	0.924495i	$0.375513\pi$
$702$		0			0
$703$	−66.8414	−	190.084i	−0.0950802	−	0.270390i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1831.84			2.59100
$708$		0			0
$709$	574.352			0.810087			0.405044	−	0.914297i	$-0.367256\pi$
$709$	574.352			0.810087			0.405044	+	0.914297i	$0.367256\pi$
$710$		0			0
$711$		−	1423.87i		−	2.00263i
$712$		0			0
$713$		−	141.179i		−	0.198007i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			754.095i			1.05174i
$718$		0			0
$719$	334.794			0.465638			0.232819	−	0.972520i	$-0.425205\pi$
$719$	334.794			0.465638			0.232819	+	0.972520i	$0.425205\pi$
$720$		0			0
$721$		−	1366.72i		−	1.89560i
$722$		0			0
$723$	34.9287			0.0483108
$724$		0			0
$725$		0			0
$726$		0			0
$727$	269.044			0.370075			0.185037	−	0.982731i	$-0.440759\pi$
$727$	269.044			0.370075			0.185037	+	0.982731i	$0.440759\pi$
$728$		0			0
$729$	533.968			0.732467
$730$		0			0
$731$	212.744			0.291032
$732$		0			0
$733$	677.757			0.924634			0.462317	−	0.886715i	$-0.347018\pi$
$733$	677.757			0.924634			0.462317	+	0.886715i	$0.347018\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	29.0489i		−	0.0394151i
$738$		0			0
$739$	−848.152			−1.14770			−0.573851	−	0.818959i	$-0.694551\pi$
$739$	−848.152			−1.14770			−0.573851	+	0.818959i	$0.694551\pi$
$740$		0			0
$741$	1878.26	−	660.475i	2.53477	−	0.891329i
$742$		0			0
$743$			1211.81i			1.63097i	0.578778	+	0.815485i	$0.303530\pi$
$743$			1211.81i			1.63097i	−0.578778	+	0.815485i	$0.696470\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	1876.05			2.51144
$748$		0			0
$749$		−	1763.66i		−	2.35469i
$750$		0			0
$751$			146.559i			0.195152i	0.995228	+	0.0975758i	$0.0311088\pi$
$751$			146.559i			0.195152i	−0.995228	+	0.0975758i	$0.968891\pi$
$752$		0			0
$753$			949.773i			1.26132i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1421.26			−1.87748			−0.938742	−	0.344621i	$-0.888008\pi$
$757$	−1421.26			−1.87748			−0.938742	+	0.344621i	$0.888008\pi$
$758$		0			0
$759$		−	852.084i		−	1.12264i
$760$		0			0
$761$	−879.279			−1.15543			−0.577713	−	0.816240i	$-0.696055\pi$
$761$	−879.279			−1.15543			−0.577713	+	0.816240i	$0.696055\pi$
$762$		0			0
$763$			618.327i			0.810390i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	659.184			0.859431
$768$		0			0
$769$	412.091			0.535879			0.267940	−	0.963436i	$-0.413657\pi$
$769$	412.091			0.535879			0.267940	+	0.963436i	$0.413657\pi$
$770$		0			0
$771$	78.7925			0.102195
$772$		0			0
$773$		−	526.855i		−	0.681572i	−0.940141	−	0.340786i	$-0.889307\pi$
$773$		−	526.855i		−	0.681572i	0.940141	−	0.340786i	$-0.110693\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−679.747			−0.874835
$778$		0			0
$779$	−141.354	−	401.984i	−0.181456	−	0.516025i
$780$		0			0
$781$		−	543.813i		−	0.696304i
$782$		0			0
$783$	−844.486			−1.07853
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	306.426i		−	0.389359i	−0.980867	−	0.194680i	$-0.937633\pi$
$787$		−	306.426i		−	0.389359i	0.980867	−	0.194680i	$-0.0623667\pi$
$788$		0			0
$789$			2316.66i			2.93620i
$790$		0			0
$791$		−	1024.19i		−	1.29480i
$792$		0			0
$793$		−	1496.56i		−	1.88721i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			797.902i			1.00113i	0.865698	+	0.500566i	$0.166875\pi$
$797$			797.902i			1.00113i	−0.865698	+	0.500566i	$0.833125\pi$
$798$		0			0
$799$	97.3137			0.121794
$800$		0			0
$801$			2308.59i			2.88214i
$802$		0			0
$803$	−32.2892			−0.0402107
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−1336.83			−1.65654
$808$		0			0
$809$	432.577			0.534706			0.267353	−	0.963599i	$-0.413851\pi$
$809$	432.577			0.534706			0.267353	+	0.963599i	$0.413851\pi$
$810$		0			0
$811$			356.906i			0.440081i	0.975491	+	0.220041i	$0.0706190\pi$
$811$			356.906i			0.440081i	−0.975491	+	0.220041i	$0.929381\pi$
$812$		0			0
$813$			565.349i			0.695387i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	928.786	−	326.599i	1.13683	−	0.399755i
$818$		0			0
$819$		−	4536.25i		−	5.53877i
$820$		0			0
$821$	72.1659			0.0879000			0.0439500	−	0.999034i	$-0.486006\pi$
$821$	72.1659			0.0879000			0.0439500	+	0.999034i	$0.486006\pi$
$822$		0			0
$823$	1322.13			1.60647			0.803236	−	0.595662i	$-0.203110\pi$
$823$	1322.13			1.60647			0.803236	+	0.595662i	$0.203110\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			799.247i			0.966441i	0.875499	+	0.483220i	$0.160533\pi$
$827$			799.247i			0.966441i	−0.875499	+	0.483220i	$0.839467\pi$
$828$		0			0
$829$			1131.58i			1.36499i	0.730890	+	0.682496i	$0.239105\pi$
$829$			1131.58i			1.36499i	−0.730890	+	0.682496i	$0.760895\pi$
$830$		0			0
$831$		−	115.736i		−	0.139273i
$832$		0			0
$833$	−407.254			−0.488901
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−205.332			−0.245319
$838$		0			0
$839$			770.869i			0.918795i	0.888231	+	0.459398i	$0.151935\pi$
$839$			770.869i			0.918795i	−0.888231	+	0.459398i	$0.848065\pi$
$840$		0			0
$841$	568.923			0.676484
$842$		0			0
$843$	1177.27			1.39653
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−1215.71			−1.43531
$848$		0			0
$849$		−	277.829i		−	0.327243i
$850$		0			0
$851$			373.307i			0.438669i
$852$		0			0
$853$	426.548			0.500056			0.250028	−	0.968239i	$-0.419560\pi$
$853$	426.548			0.500056			0.250028	+	0.968239i	$0.419560\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	1400.33i		−	1.63400i	−0.576641	−	0.816998i	$-0.695637\pi$
$857$		−	1400.33i		−	1.63400i	0.576641	−	0.816998i	$-0.304363\pi$
$858$		0			0
$859$	−870.240			−1.01308			−0.506542	−	0.862215i	$-0.669077\pi$
$859$	−870.240			−1.01308			−0.506542	+	0.862215i	$0.669077\pi$
$860$		0			0
$861$	−1437.51			−1.66958
$862$		0			0
$863$			929.692i			1.07728i	0.842536	+	0.538640i	$0.181062\pi$
$863$			929.692i			1.07728i	−0.842536	+	0.538640i	$0.818938\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	1432.92i		−	1.65274i
$868$		0			0
$869$		−	349.610i		−	0.402312i
$870$		0			0
$871$	−125.755			−0.144380
$872$		0			0
$873$		−	1894.85i		−	2.17051i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	1223.42i		−	1.39501i	−0.716580	−	0.697505i	$-0.754294\pi$
$877$		−	1223.42i		−	1.39501i	0.716580	−	0.697505i	$-0.245706\pi$
$878$		0			0
$879$	1547.29			1.76028
$880$		0			0
$881$	−146.489			−0.166276			−0.0831378	−	0.996538i	$-0.526494\pi$
$881$	−146.489			−0.166276			−0.0831378	+	0.996538i	$0.526494\pi$
$882$		0			0
$883$	794.391			0.899650			0.449825	−	0.893117i	$-0.351486\pi$
$883$	794.391			0.899650			0.449825	+	0.893117i	$0.351486\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			944.311i			1.06461i	0.846552	+	0.532306i	$0.178674\pi$
$887$			944.311i			1.06461i	−0.846552	+	0.532306i	$0.821326\pi$
$888$		0			0
$889$			1574.86i			1.77149i
$890$		0			0
$891$	−464.590			−0.521425
$892$		0			0
$893$	424.846	−	149.393i	0.475751	−	0.167294i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−3688.74			−4.11230
$898$		0			0
$899$	−66.1541			−0.0735864
$900$		0			0
$901$		−	399.486i		−	0.443381i
$902$		0			0
$903$		−	3321.37i		−	3.67815i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			449.939i			0.496074i	0.968751	+	0.248037i	$0.0797854\pi$
$907$			449.939i			0.496074i	−0.968751	+	0.248037i	$0.920215\pi$
$908$		0			0
$909$	−2817.47			−3.09952
$910$		0			0
$911$		−	1423.86i		−	1.56297i	−0.623926	−	0.781484i	$-0.714463\pi$
$911$		−	1423.86i		−	1.56297i	0.623926	−	0.781484i	$-0.285537\pi$
$912$		0			0
$913$	460.635			0.504529
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1697.24			1.85086
$918$		0			0
$919$	441.636			0.480562			0.240281	−	0.970703i	$-0.422760\pi$
$919$	441.636			0.480562			0.240281	+	0.970703i	$0.422760\pi$
$920$		0			0
$921$	−2065.74			−2.24293
$922$		0			0
$923$	−2354.21			−2.55061
$924$		0			0
$925$		0			0
$926$		0			0
$927$			2102.10i			2.26763i
$928$		0			0
$929$	−109.177			−0.117521			−0.0587605	−	0.998272i	$-0.518715\pi$
$929$	−109.177			−0.117521			−0.0587605	+	0.998272i	$0.518715\pi$
$930$		0			0
$931$	−1777.96	+	625.206i	−1.90974	+	0.671542i
$932$		0			0
$933$			1176.14i			1.26060i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1319.18			−1.40787			−0.703937	−	0.710262i	$-0.748576\pi$
$937$	−1319.18			−1.40787			−0.703937	+	0.710262i	$0.748576\pi$
$938$		0			0
$939$		−	1338.41i		−	1.42536i
$940$		0			0
$941$		−	459.707i		−	0.488530i	−0.969708	−	0.244265i	$-0.921453\pi$
$941$		−	459.707i		−	0.488530i	0.969708	−	0.244265i	$-0.0785467\pi$
$942$		0			0
$943$			789.459i			0.837178i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1310.81			1.38418			0.692088	−	0.721813i	$-0.256691\pi$
$947$	1310.81			1.38418			0.692088	+	0.721813i	$0.256691\pi$
$948$		0			0
$949$			139.782i			0.147295i
$950$		0			0
$951$	−215.014			−0.226093
$952$		0			0
$953$		−	1302.16i		−	1.36638i	−0.730238	−	0.683192i	$-0.760591\pi$
$953$		−	1302.16i		−	1.36638i	0.730238	−	0.683192i	$-0.239409\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−399.273			−0.417213
$958$		0			0
$959$	789.975			0.823749
$960$		0			0
$961$	944.915			0.983262
$962$		0			0
$963$			2712.61i			2.81683i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−167.641			−0.173362			−0.0866809	−	0.996236i	$-0.527626\pi$
$967$	−167.641			−0.173362			−0.0866809	+	0.996236i	$0.527626\pi$
$968$		0			0
$969$	136.252	+	387.474i	0.140611	+	0.399870i
$970$		0			0
$971$			866.411i			0.892288i	0.894961	+	0.446144i	$0.147203\pi$
$971$			866.411i			0.892288i	−0.894961	+	0.446144i	$0.852797\pi$
$972$		0			0
$973$	−2004.14			−2.05976
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1247.03i		−	1.27639i	−0.769874	−	0.638195i	$-0.779681\pi$
$977$		−	1247.03i		−	1.27639i	0.769874	−	0.638195i	$-0.220319\pi$
$978$		0			0
$979$			566.839i			0.578998i
$980$		0			0
$981$		−	951.021i		−	0.969440i
$982$		0			0
$983$			657.563i			0.668934i	0.942407	+	0.334467i	$0.108556\pi$
$983$			657.563i			0.668934i	−0.942407	+	0.334467i	$0.891444\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		−	1519.26i		−	1.53927i
$988$		0			0
$989$	−1824.05			−1.84434
$990$		0			0
$991$		−	1045.43i		−	1.05492i	−0.849579	−	0.527461i	$-0.823144\pi$
$991$		−	1045.43i		−	1.05492i	0.849579	−	0.527461i	$-0.176856\pi$
$992$		0			0
$993$	−568.895			−0.572906
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1131.50			−1.13491			−0.567455	−	0.823405i	$-0.692072\pi$
$997$	−1131.50			−1.13491			−0.567455	+	0.823405i	$0.692072\pi$
$998$		0			0
$999$	542.943			0.543486

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.3.e.g.1101.14	yes	14
5.2	odd	4		1900.3.g.d.949.27		28
5.3	odd	4		1900.3.g.d.949.2		28
5.4	even	2		1900.3.e.h.1101.1	yes	14
19.18	odd	2	inner	1900.3.e.g.1101.1	✓	14
95.18	even	4		1900.3.g.d.949.28		28
95.37	even	4		1900.3.g.d.949.1		28
95.94	odd	2		1900.3.e.h.1101.14	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1900.3.e.g.1101.1	✓	14	19.18	odd	2	inner
1900.3.e.g.1101.14	yes	14	1.1	even	1	trivial
1900.3.e.h.1101.1	yes	14	5.4	even	2
1900.3.e.h.1101.14	yes	14	95.94	odd	2
1900.3.g.d.949.1		28	95.37	even	4
1900.3.g.d.949.2		28	5.3	odd	4
1900.3.g.d.949.27		28	5.2	odd	4
1900.3.g.d.949.28		28	95.18	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1900.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.26531i q^{3} +12.1735 q^{7} -18.7235 q^{9} +O(q^{10})\)	\(q+5.26531i q^{3} +12.1735 q^{7} -18.7235 q^{9} -4.59727 q^{11} +19.9019i q^{13} -4.10564 q^{17} +(-17.9241 + 6.30286i) q^{19} +64.0972i q^{21} +35.2013 q^{23} -51.1972i q^{27} -16.4948i q^{29} -4.01062i q^{31} -24.2060i q^{33} +10.6049i q^{37} -104.790 q^{39} +22.4270i q^{41} -51.8177 q^{43} -23.7025 q^{47} +99.1940 q^{49} -21.6174i q^{51} +97.3018i q^{53} +(-33.1865 - 94.3760i) q^{57} -33.1216i q^{59} -75.1967 q^{61} -227.930 q^{63} +6.31874i q^{67} +185.346i q^{69} +118.291i q^{71} +7.02357 q^{73} -55.9648 q^{77} +76.0473i q^{79} +101.058 q^{81} -100.198 q^{83} +86.8500 q^{87} -123.299i q^{89} +242.276i q^{91} +21.1171 q^{93} +101.202i q^{97} +86.0769 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 4 q^{7} - 52 q^{9}+O(q^{10}) \) 14 * q - 4 * q^7 - 52 * q^9	\( 14 q - 4 q^{7} - 52 q^{9} + 4 q^{11} + 6 q^{17} - 29 q^{19} + 28 q^{23} - 56 q^{39} - 22 q^{43} - 84 q^{47} + 138 q^{49} + 5 q^{57} - 50 q^{61} - 234 q^{63} + 204 q^{73} - 68 q^{77} + 66 q^{81} - 256 q^{83} - 18 q^{87} - 118 q^{93} + 92 q^{99}+O(q^{100}) \) 14 * q - 4 * q^7 - 52 * q^9 + 4 * q^11 + 6 * q^17 - 29 * q^19 + 28 * q^23 - 56 * q^39 - 22 * q^43 - 84 * q^47 + 138 * q^49 + 5 * q^57 - 50 * q^61 - 234 * q^63 + 204 * q^73 - 68 * q^77 + 66 * q^81 - 256 * q^83 - 18 * q^87 - 118 * q^93 + 92 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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