LMFDB - Embedded newform 4140.3.d.b.2161.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,3,Mod(2161,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4140.2161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	4140.d (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:	$112.806829445$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2161.13
Character	$\chi$	$=$	4140.2161
Dual form			4140.3.d.b.2161.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{5} +7.68094i q^{7} +O(q^{10})$	$q-2.23607i q^{5} +7.68094i q^{7} +8.06491i q^{11} +14.0051 q^{13} +21.5443i q^{17} +5.94674i q^{19} +(16.1796 - 16.3469i) q^{23} -5.00000 q^{25} -9.04988 q^{29} -34.1831 q^{31} +17.1751 q^{35} +55.1713i q^{37} -42.2395 q^{41} +37.9213i q^{43} +70.5192 q^{47} -9.99678 q^{49} +21.2820i q^{53} +18.0337 q^{55} -5.95482 q^{59} -25.1224i q^{61} -31.3163i q^{65} +21.0706i q^{67} -85.4394 q^{71} +75.5625 q^{73} -61.9460 q^{77} +53.4097i q^{79} -80.9986i q^{83} +48.1746 q^{85} -57.7196i q^{89} +107.572i q^{91} +13.2973 q^{95} +56.0293i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q - 24 q^{13} - 160 q^{25} - 28 q^{31} - 260 q^{49} + 120 q^{55} - 296 q^{73} - 60 q^{85}+O(q^{100}) $ 32 * q - 24 * q^13 - 160 * q^25 - 28 * q^31 - 260 * q^49 + 120 * q^55 - 296 * q^73 - 60 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$		0			0
$7$			7.68094i			1.09728i	0.836060	+	0.548638i	$0.184854\pi$
$7$			7.68094i			1.09728i	−0.836060	+	0.548638i	$0.815146\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			8.06491i			0.733173i	0.930384	+	0.366587i	$0.119474\pi$
$11$			8.06491i			0.733173i	−0.930384	+	0.366587i	$0.880526\pi$
$12$		0			0
$13$	14.0051			1.07731			0.538656	−	0.842526i	$-0.318932\pi$
$13$	14.0051			1.07731			0.538656	+	0.842526i	$0.318932\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			21.5443i			1.26731i	0.773615	+	0.633656i	$0.218447\pi$
$17$			21.5443i			1.26731i	−0.773615	+	0.633656i	$0.781553\pi$
$18$		0			0
$19$			5.94674i			0.312987i	0.987679	+	0.156493i	$0.0500190\pi$
$19$			5.94674i			0.312987i	−0.987679	+	0.156493i	$0.949981\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	16.1796	−	16.3469i	0.703460	−	0.710735i
$23$	16.1796	−	16.3469i	0.703460	−	0.710735i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−9.04988			−0.312065			−0.156032	−	0.987752i	$-0.549870\pi$
$29$	−9.04988			−0.312065			−0.156032	+	0.987752i	$0.549870\pi$
$30$		0			0
$31$	−34.1831			−1.10268			−0.551341	−	0.834280i	$-0.685884\pi$
$31$	−34.1831			−1.10268			−0.551341	+	0.834280i	$0.685884\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	17.1751			0.490717
$36$		0			0
$37$			55.1713i			1.49112i	0.666440	+	0.745559i	$0.267817\pi$
$37$			55.1713i			1.49112i	−0.666440	+	0.745559i	$0.732183\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−42.2395			−1.03023			−0.515116	−	0.857120i	$-0.672251\pi$
$41$	−42.2395			−1.03023			−0.515116	+	0.857120i	$0.672251\pi$
$42$		0			0
$43$			37.9213i			0.881891i	0.897534	+	0.440946i	$0.145357\pi$
$43$			37.9213i			0.881891i	−0.897534	+	0.440946i	$0.854643\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	70.5192			1.50041			0.750205	−	0.661206i	$-0.229955\pi$
$47$	70.5192			1.50041			0.750205	+	0.661206i	$0.229955\pi$
$48$		0			0
$49$	−9.99678			−0.204016
$50$		0			0
$51$		0			0
$52$		0			0
$53$			21.2820i			0.401547i	0.979638	+	0.200774i	$0.0643455\pi$
$53$			21.2820i			0.401547i	−0.979638	+	0.200774i	$0.935654\pi$
$54$		0			0
$55$	18.0337			0.327885
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.95482			−0.100929			−0.0504646	−	0.998726i	$-0.516070\pi$
$59$	−5.95482			−0.100929			−0.0504646	+	0.998726i	$0.516070\pi$
$60$		0			0
$61$		−	25.1224i		−	0.411842i	−0.978569	−	0.205921i	$-0.933981\pi$
$61$		−	25.1224i		−	0.411842i	0.978569	−	0.205921i	$-0.0660190\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	31.3163i		−	0.481789i
$66$		0			0
$67$			21.0706i			0.314487i	0.987560	+	0.157244i	$0.0502608\pi$
$67$			21.0706i			0.314487i	−0.987560	+	0.157244i	$0.949739\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−85.4394			−1.20337			−0.601686	−	0.798733i	$-0.705504\pi$
$71$	−85.4394			−1.20337			−0.601686	+	0.798733i	$0.705504\pi$
$72$		0			0
$73$	75.5625			1.03510			0.517552	−	0.855652i	$-0.326844\pi$
$73$	75.5625			1.03510			0.517552	+	0.855652i	$0.326844\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−61.9460			−0.804494
$78$		0			0
$79$			53.4097i			0.676072i	0.941133	+	0.338036i	$0.109762\pi$
$79$			53.4097i			0.676072i	−0.941133	+	0.338036i	$0.890238\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	80.9986i		−	0.975887i	−0.872875	−	0.487944i	$-0.837747\pi$
$83$		−	80.9986i		−	0.975887i	0.872875	−	0.487944i	$-0.162253\pi$
$84$		0			0
$85$	48.1746			0.566759
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	57.7196i		−	0.648534i	−0.945966	−	0.324267i	$-0.894882\pi$
$89$		−	57.7196i		−	0.648534i	0.945966	−	0.324267i	$-0.105118\pi$
$90$		0			0
$91$			107.572i			1.18211i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	13.2973			0.139972
$96$		0			0
$97$			56.0293i			0.577622i	0.957386	+	0.288811i	$0.0932599\pi$
$97$			56.0293i			0.577622i	−0.957386	+	0.288811i	$0.906740\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	34.4784			0.341371			0.170685	−	0.985326i	$-0.445402\pi$
$101$	34.4784			0.341371			0.170685	+	0.985326i	$0.445402\pi$
$102$		0			0
$103$		−	154.157i		−	1.49667i	−0.663322	−	0.748334i	$-0.730854\pi$
$103$		−	154.157i		−	1.49667i	0.663322	−	0.748334i	$-0.269146\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	2.65838i		−	0.0248447i	−0.999923	−	0.0124223i	$-0.996046\pi$
$107$		−	2.65838i		−	0.0248447i	0.999923	−	0.0124223i	$-0.00395426\pi$
$108$		0			0
$109$		−	111.490i		−	1.02284i	−0.859330	−	0.511421i	$-0.829119\pi$
$109$		−	111.490i		−	1.02284i	0.859330	−	0.511421i	$-0.170881\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	162.391i		−	1.43709i	−0.695481	−	0.718545i	$-0.744809\pi$
$113$		−	162.391i		−	1.43709i	0.695481	−	0.718545i	$-0.255191\pi$
$114$		0			0
$115$	−36.5528	−	36.1786i	−0.317850	−	0.314597i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−165.481			−1.39059
$120$		0			0
$121$	55.9573			0.462457
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	−232.969			−1.83440			−0.917202	−	0.398424i	$-0.869557\pi$
$127$	−232.969			−1.83440			−0.917202	+	0.398424i	$0.869557\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−50.9609			−0.389015			−0.194507	−	0.980901i	$-0.562311\pi$
$131$	−50.9609			−0.389015			−0.194507	+	0.980901i	$0.562311\pi$
$132$		0			0
$133$	−45.6766			−0.343433
$134$		0			0
$135$		0			0
$136$		0			0
$137$			204.968i			1.49612i	0.663631	+	0.748060i	$0.269015\pi$
$137$			204.968i			1.49612i	−0.663631	+	0.748060i	$0.730985\pi$
$138$		0			0
$139$	−186.525			−1.34191			−0.670954	−	0.741499i	$-0.734115\pi$
$139$	−186.525			−1.34191			−0.670954	+	0.741499i	$0.734115\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			112.950i			0.789857i
$144$		0			0
$145$			20.2361i			0.139560i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	155.250i		−	1.04195i	−0.853573	−	0.520973i	$-0.825569\pi$
$149$		−	155.250i		−	1.04195i	0.853573	−	0.520973i	$-0.174431\pi$
$150$		0			0
$151$	−4.55804			−0.0301857			−0.0150928	−	0.999886i	$-0.504804\pi$
$151$	−4.55804			−0.0301857			−0.0150928	+	0.999886i	$0.504804\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			76.4358i			0.493134i
$156$		0			0
$157$			284.371i			1.81128i	0.424046	+	0.905641i	$0.360610\pi$
$157$			284.371i			1.81128i	−0.424046	+	0.905641i	$0.639390\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	125.560	+	124.274i	0.779873	+	0.771890i
$162$		0			0
$163$	42.7701			0.262393			0.131196	−	0.991356i	$-0.458118\pi$
$163$	42.7701			0.262393			0.131196	+	0.991356i	$0.458118\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−161.636			−0.967883			−0.483941	−	0.875100i	$-0.660795\pi$
$167$	−161.636			−0.967883			−0.483941	+	0.875100i	$0.660795\pi$
$168$		0			0
$169$	27.1417			0.160602
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−55.1693			−0.318898			−0.159449	−	0.987206i	$-0.550972\pi$
$173$	−55.1693			−0.318898			−0.159449	+	0.987206i	$0.550972\pi$
$174$		0			0
$175$		−	38.4047i		−	0.219455i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	277.877			1.55239			0.776194	−	0.630495i	$-0.217148\pi$
$179$	277.877			1.55239			0.776194	+	0.630495i	$0.217148\pi$
$180$		0			0
$181$			158.292i			0.874543i	0.899330	+	0.437271i	$0.144055\pi$
$181$			158.292i			0.874543i	−0.899330	+	0.437271i	$0.855945\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	123.367			0.666848
$186$		0			0
$187$	−173.753			−0.929160
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	155.616i		−	0.814743i	−0.913262	−	0.407372i	$-0.866445\pi$
$191$		−	155.616i		−	0.814743i	0.913262	−	0.407372i	$-0.133555\pi$
$192$		0			0
$193$	−359.612			−1.86328			−0.931638	−	0.363389i	$-0.881620\pi$
$193$	−359.612			−1.86328			−0.931638	+	0.363389i	$0.881620\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	214.628			1.08948			0.544742	−	0.838604i	$-0.316628\pi$
$197$	214.628			1.08948			0.544742	+	0.838604i	$0.316628\pi$
$198$		0			0
$199$			259.477i			1.30390i	0.758260	+	0.651952i	$0.226050\pi$
$199$			259.477i			1.30390i	−0.758260	+	0.651952i	$0.773950\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	69.5116i		−	0.342421i
$204$		0			0
$205$			94.4504i			0.460734i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−47.9599			−0.229473
$210$		0			0
$211$	−249.799			−1.18388			−0.591940	−	0.805982i	$-0.701638\pi$
$211$	−249.799			−1.18388			−0.591940	+	0.805982i	$0.701638\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	84.7947			0.394394
$216$		0			0
$217$		−	262.558i		−	1.20995i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			301.729i			1.36529i
$222$		0			0
$223$	−195.564			−0.876970			−0.438485	−	0.898738i	$-0.644485\pi$
$223$	−195.564			−0.876970			−0.438485	+	0.898738i	$0.644485\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			36.0648i			0.158876i	0.996840	+	0.0794378i	$0.0253125\pi$
$227$			36.0648i			0.158876i	−0.996840	+	0.0794378i	$0.974687\pi$
$228$		0			0
$229$			302.027i			1.31889i	0.751751	+	0.659447i	$0.229209\pi$
$229$			302.027i			1.31889i	−0.751751	+	0.659447i	$0.770791\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	342.214			1.46873			0.734365	−	0.678755i	$-0.237480\pi$
$233$	342.214			1.46873			0.734365	+	0.678755i	$0.237480\pi$
$234$		0			0
$235$		−	157.686i		−	0.671003i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−340.746			−1.42572			−0.712858	−	0.701308i	$-0.752600\pi$
$239$	−340.746			−1.42572			−0.712858	+	0.701308i	$0.752600\pi$
$240$		0			0
$241$		−	200.742i		−	0.832953i	−0.909146	−	0.416477i	$-0.863265\pi$
$241$		−	200.742i		−	0.832953i	0.909146	−	0.416477i	$-0.136735\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			22.3535i			0.0912387i
$246$		0			0
$247$			83.2845i			0.337184i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			195.399i			0.778481i	0.921136	+	0.389241i	$0.127263\pi$
$251$			195.399i			0.778481i	−0.921136	+	0.389241i	$0.872737\pi$
$252$		0			0
$253$	131.836	+	130.487i	0.521092	+	0.515758i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	93.1111			0.362300			0.181150	−	0.983455i	$-0.442018\pi$
$257$	93.1111			0.362300			0.181150	+	0.983455i	$0.442018\pi$
$258$		0			0
$259$	−423.768			−1.63617
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	126.258i		−	0.480069i	−0.970764	−	0.240035i	$-0.922841\pi$
$263$		−	126.258i		−	0.480069i	0.970764	−	0.240035i	$-0.0771588\pi$
$264$		0			0
$265$	47.5880			0.179577
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−217.271			−0.807699			−0.403849	−	0.914826i	$-0.632328\pi$
$269$	−217.271			−0.807699			−0.403849	+	0.914826i	$0.632328\pi$
$270$		0			0
$271$	52.4947			0.193707			0.0968537	−	0.995299i	$-0.469122\pi$
$271$	52.4947			0.193707			0.0968537	+	0.995299i	$0.469122\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	40.3245i		−	0.146635i
$276$		0			0
$277$	−381.366			−1.37677			−0.688387	−	0.725344i	$-0.741681\pi$
$277$	−381.366			−1.37677			−0.688387	+	0.725344i	$0.741681\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			469.964i			1.67247i	0.548370	+	0.836236i	$0.315248\pi$
$281$			469.964i			1.67247i	−0.548370	+	0.836236i	$0.684752\pi$
$282$		0			0
$283$			291.521i			1.03011i	0.857157	+	0.515055i	$0.172228\pi$
$283$			291.521i			1.03011i	−0.857157	+	0.515055i	$0.827772\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	324.439i		−	1.13045i
$288$		0			0
$289$	−175.158			−0.606081
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	204.631i		−	0.698399i	−0.937048	−	0.349200i	$-0.886453\pi$
$293$		−	204.631i		−	0.698399i	0.937048	−	0.349200i	$-0.113547\pi$
$294$		0			0
$295$			13.3154i			0.0451369i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	226.596	−	228.939i	0.757846	−	0.765684i
$300$		0			0
$301$	−291.271			−0.967679
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−56.1753			−0.184181
$306$		0			0
$307$	310.699			1.01205			0.506024	−	0.862519i	$-0.331115\pi$
$307$	310.699			1.01205			0.506024	+	0.862519i	$0.331115\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−10.6546			−0.0342591			−0.0171296	−	0.999853i	$-0.505453\pi$
$311$	−10.6546			−0.0342591			−0.0171296	+	0.999853i	$0.505453\pi$
$312$		0			0
$313$			502.170i			1.60438i	0.597070	+	0.802189i	$0.296331\pi$
$313$			502.170i			1.60438i	−0.597070	+	0.802189i	$0.703669\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−435.237			−1.37299			−0.686494	−	0.727136i	$-0.740851\pi$
$317$	−435.237			−1.37299			−0.686494	+	0.727136i	$0.740851\pi$
$318$		0			0
$319$		−	72.9864i		−	0.228798i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−128.119			−0.396652
$324$		0			0
$325$	−70.0253			−0.215462
$326$		0			0
$327$		0			0
$328$		0			0
$329$			541.654i			1.64636i
$330$		0			0
$331$	−270.985			−0.818686			−0.409343	−	0.912381i	$-0.634242\pi$
$331$	−270.985			−0.818686			−0.409343	+	0.912381i	$0.634242\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	47.1154			0.140643
$336$		0			0
$337$		−	8.79415i		−	0.0260954i	−0.999915	−	0.0130477i	$-0.995847\pi$
$337$		−	8.79415i		−	0.0260954i	0.999915	−	0.0130477i	$-0.00415333\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	275.684i		−	0.808457i
$342$		0			0
$343$			299.581i			0.873415i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−80.9876			−0.233394			−0.116697	−	0.993168i	$-0.537231\pi$
$347$	−80.9876			−0.233394			−0.116697	+	0.993168i	$0.537231\pi$
$348$		0			0
$349$	396.882			1.13720			0.568598	−	0.822615i	$-0.307486\pi$
$349$	396.882			1.13720			0.568598	+	0.822615i	$0.307486\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	642.057			1.81886			0.909429	−	0.415860i	$-0.136519\pi$
$353$	642.057			1.81886			0.909429	+	0.415860i	$0.136519\pi$
$354$		0			0
$355$			191.048i			0.538164i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			270.936i			0.754696i	0.926072	+	0.377348i	$0.123164\pi$
$359$			270.936i			0.754696i	−0.926072	+	0.377348i	$0.876836\pi$
$360$		0			0
$361$	325.636			0.902039
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	168.963i		−	0.462912i
$366$		0			0
$367$			48.6849i			0.132656i	0.997798	+	0.0663282i	$0.0211284\pi$
$367$			48.6849i			0.132656i	−0.997798	+	0.0663282i	$0.978872\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−163.466			−0.440608
$372$		0			0
$373$			134.229i			0.359864i	0.983679	+	0.179932i	$0.0575879\pi$
$373$			134.229i			0.359864i	−0.983679	+	0.179932i	$0.942412\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−126.744			−0.336191
$378$		0			0
$379$			651.930i			1.72013i	0.510184	+	0.860065i	$0.329577\pi$
$379$			651.930i			1.72013i	−0.510184	+	0.860065i	$0.670423\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	409.393i		−	1.06891i	−0.845197	−	0.534455i	$-0.820517\pi$
$383$		−	409.393i		−	1.06891i	0.845197	−	0.534455i	$-0.179483\pi$
$384$		0			0
$385$			138.516i			0.359781i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	143.926i		−	0.369989i	−0.982740	−	0.184994i	$-0.940773\pi$
$389$		−	143.926i		−	0.369989i	0.982740	−	0.184994i	$-0.0592267\pi$
$390$		0			0
$391$	352.183	+	348.578i	0.900724	+	0.891503i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	119.428			0.302348
$396$		0			0
$397$	−240.842			−0.606654			−0.303327	−	0.952887i	$-0.598097\pi$
$397$	−240.842			−0.606654			−0.303327	+	0.952887i	$0.598097\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			83.5550i			0.208367i	0.994558	+	0.104183i	$0.0332229\pi$
$401$			83.5550i			0.208367i	−0.994558	+	0.104183i	$0.966777\pi$
$402$		0			0
$403$	−478.737			−1.18793
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−444.952			−1.09325
$408$		0			0
$409$	453.483			1.10876			0.554381	−	0.832263i	$-0.312955\pi$
$409$	453.483			1.10876			0.554381	+	0.832263i	$0.312955\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	45.7386i		−	0.110747i
$414$		0			0
$415$	−181.118			−0.436430
$416$		0			0
$417$		0			0
$418$		0			0
$419$			227.563i			0.543110i	0.962423	+	0.271555i	$0.0875379\pi$
$419$			227.563i			0.543110i	−0.962423	+	0.271555i	$0.912462\pi$
$420$		0			0
$421$		−	196.601i		−	0.466986i	−0.972358	−	0.233493i	$-0.924984\pi$
$421$		−	196.601i		−	0.466986i	0.972358	−	0.233493i	$-0.0750157\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	107.722i		−	0.253463i
$426$		0			0
$427$	192.963			0.451905
$428$		0			0
$429$		0			0
$430$		0			0
$431$			481.498i			1.11717i	0.829449	+	0.558583i	$0.188655\pi$
$431$			481.498i			1.11717i	−0.829449	+	0.558583i	$0.811345\pi$
$432$		0			0
$433$		−	67.9413i		−	0.156908i	−0.996918	−	0.0784541i	$-0.975002\pi$
$433$		−	67.9413i		−	0.156908i	0.996918	−	0.0784541i	$-0.0249984\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	97.2109	+	96.2158i	0.222451	+	0.220173i
$438$		0			0
$439$	−496.378			−1.13070			−0.565351	−	0.824851i	$-0.691259\pi$
$439$	−496.378			−1.13070			−0.565351	+	0.824851i	$0.691259\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	726.597			1.64017			0.820087	−	0.572239i	$-0.193925\pi$
$443$	726.597			1.64017			0.820087	+	0.572239i	$0.193925\pi$
$444$		0			0
$445$	−129.065			−0.290033
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−35.9997			−0.0801774			−0.0400887	−	0.999196i	$-0.512764\pi$
$449$	−35.9997			−0.0801774			−0.0400887	+	0.999196i	$0.512764\pi$
$450$		0			0
$451$		−	340.658i		−	0.755339i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	240.538			0.528655
$456$		0			0
$457$			364.981i			0.798645i	0.916811	+	0.399323i	$0.130755\pi$
$457$			364.981i			0.798645i	−0.916811	+	0.399323i	$0.869245\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	144.108			0.312599			0.156299	−	0.987710i	$-0.450044\pi$
$461$	144.108			0.312599			0.156299	+	0.987710i	$0.450044\pi$
$462$		0			0
$463$	−298.690			−0.645119			−0.322560	−	0.946549i	$-0.604543\pi$
$463$	−298.690			−0.645119			−0.322560	+	0.946549i	$0.604543\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			535.919i			1.14758i	0.819003	+	0.573789i	$0.194527\pi$
$467$			535.919i			1.14758i	−0.819003	+	0.573789i	$0.805473\pi$
$468$		0			0
$469$	−161.842			−0.345079
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−305.832			−0.646579
$474$		0			0
$475$		−	29.7337i		−	0.0625973i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	406.765i		−	0.849196i	−0.905382	−	0.424598i	$-0.860415\pi$
$479$		−	406.765i		−	0.849196i	0.905382	−	0.424598i	$-0.139585\pi$
$480$		0			0
$481$			772.678i			1.60640i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	125.285			0.258320
$486$		0			0
$487$	585.309			1.20187			0.600933	−	0.799299i	$-0.294796\pi$
$487$	585.309			1.20187			0.600933	+	0.799299i	$0.294796\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−424.239			−0.864031			−0.432016	−	0.901866i	$-0.642197\pi$
$491$	−424.239			−0.864031			−0.432016	+	0.901866i	$0.642197\pi$
$492$		0			0
$493$		−	194.973i		−	0.395484i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	656.254i		−	1.32043i
$498$		0			0
$499$	−233.085			−0.467105			−0.233553	−	0.972344i	$-0.575035\pi$
$499$	−233.085			−0.467105			−0.233553	+	0.972344i	$0.575035\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			355.978i			0.707710i	0.935300	+	0.353855i	$0.115129\pi$
$503$			355.978i			0.707710i	−0.935300	+	0.353855i	$0.884871\pi$
$504$		0			0
$505$		−	77.0961i		−	0.152666i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	272.208			0.534791			0.267395	−	0.963587i	$-0.413837\pi$
$509$	272.208			0.534791			0.267395	+	0.963587i	$0.413837\pi$
$510$		0			0
$511$			580.391i			1.13579i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−344.705			−0.669330
$516$		0			0
$517$			568.731i			1.10006i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	107.930i		−	0.207160i	−0.994621	−	0.103580i	$-0.966970\pi$
$521$		−	107.930i		−	0.207160i	0.994621	−	0.103580i	$-0.0330297\pi$
$522$		0			0
$523$			135.656i			0.259381i	0.991555	+	0.129690i	$0.0413983\pi$
$523$			135.656i			0.259381i	−0.991555	+	0.129690i	$0.958602\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	736.452i		−	1.39744i
$528$		0			0
$529$	−5.44294	−	528.972i	−0.0102891	−	0.999947i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−591.567			−1.10988
$534$		0			0
$535$	−5.94432			−0.0111109
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	80.6231i		−	0.149579i
$540$		0			0
$541$	−781.059			−1.44373			−0.721866	−	0.692032i	$-0.756716\pi$
$541$	−781.059			−1.44373			−0.721866	+	0.692032i	$0.756716\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−249.299			−0.457429
$546$		0			0
$547$	−522.867			−0.955881			−0.477940	−	0.878392i	$-0.658616\pi$
$547$	−522.867			−0.955881			−0.477940	+	0.878392i	$0.658616\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	53.8173i		−	0.0976721i
$552$		0			0
$553$	−410.236			−0.741838
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	298.041i		−	0.535083i	−0.963546	−	0.267541i	$-0.913789\pi$
$557$		−	298.041i		−	0.535083i	0.963546	−	0.267541i	$-0.0862112\pi$
$558$		0			0
$559$			531.090i			0.950072i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	95.6528i		−	0.169898i	−0.996385	−	0.0849492i	$-0.972927\pi$
$563$		−	95.6528i		−	0.169898i	0.996385	−	0.0849492i	$-0.0270728\pi$
$564$		0			0
$565$	−363.118			−0.642686
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	63.0004i		−	0.110721i	−0.998466	−	0.0553607i	$-0.982369\pi$
$569$		−	63.0004i		−	0.110721i	0.998466	−	0.0553607i	$-0.0176309\pi$
$570$		0			0
$571$		−	228.722i		−	0.400563i	−0.979738	−	0.200282i	$-0.935814\pi$
$571$		−	228.722i		−	0.400563i	0.979738	−	0.200282i	$-0.0641857\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−80.8979	+	81.7346i	−0.140692	+	0.142147i
$576$		0			0
$577$	−100.501			−0.174179			−0.0870894	−	0.996200i	$-0.527757\pi$
$577$	−100.501			−0.174179			−0.0870894	+	0.996200i	$0.527757\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	622.145			1.07082
$582$		0			0
$583$	−171.637			−0.294404
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−659.990			−1.12434			−0.562172	−	0.827020i	$-0.690034\pi$
$587$	−659.990			−1.12434			−0.562172	+	0.827020i	$0.690034\pi$
$588$		0			0
$589$		−	203.278i		−	0.345124i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−875.504			−1.47640			−0.738199	−	0.674583i	$-0.764323\pi$
$593$	−875.504			−1.47640			−0.738199	+	0.674583i	$0.764323\pi$
$594$		0			0
$595$			370.026i			0.621892i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	900.098			1.50267			0.751334	−	0.659922i	$-0.229411\pi$
$599$	900.098			1.50267			0.751334	+	0.659922i	$0.229411\pi$
$600$		0			0
$601$	−210.509			−0.350265			−0.175133	−	0.984545i	$-0.556035\pi$
$601$	−210.509			−0.350265			−0.175133	+	0.984545i	$0.556035\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	125.124i		−	0.206817i
$606$		0			0
$607$	−182.030			−0.299884			−0.149942	−	0.988695i	$-0.547909\pi$
$607$	−182.030			−0.299884			−0.149942	+	0.988695i	$0.547909\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	987.626			1.61641
$612$		0			0
$613$			879.781i			1.43521i	0.696452	+	0.717603i	$0.254761\pi$
$613$			879.781i			1.43521i	−0.696452	+	0.717603i	$0.745239\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	317.793i		−	0.515062i	−0.966270	−	0.257531i	$-0.917091\pi$
$617$		−	317.793i		−	0.515062i	0.966270	−	0.257531i	$-0.0829089\pi$
$618$		0			0
$619$		−	611.109i		−	0.987252i	−0.869674	−	0.493626i	$-0.835671\pi$
$619$		−	611.109i		−	0.987252i	0.869674	−	0.493626i	$-0.164329\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	443.340			0.711622
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1188.63			−1.88971
$630$		0			0
$631$			1084.40i			1.71854i	0.511524	+	0.859269i	$0.329081\pi$
$631$			1084.40i			1.71854i	−0.511524	+	0.859269i	$0.670919\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			520.935i			0.820370i
$636$		0			0
$637$	−140.006			−0.219789
$638$		0			0
$639$		0			0
$640$		0			0
$641$			622.729i			0.971497i	0.874099	+	0.485748i	$0.161453\pi$
$641$			622.729i			0.971497i	−0.874099	+	0.485748i	$0.838547\pi$
$642$		0			0
$643$		−	747.393i		−	1.16235i	−0.813777	−	0.581177i	$-0.802593\pi$
$643$		−	747.393i		−	1.16235i	0.813777	−	0.581177i	$-0.197407\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1040.84			−1.60871			−0.804355	−	0.594148i	$-0.797489\pi$
$647$	−1040.84			−1.60871			−0.804355	+	0.594148i	$0.797489\pi$
$648$		0			0
$649$		−	48.0251i		−	0.0739986i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−638.769			−0.978206			−0.489103	−	0.872226i	$-0.662676\pi$
$653$	−638.769			−0.978206			−0.489103	+	0.872226i	$0.662676\pi$
$654$		0			0
$655$			113.952i			0.173973i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	351.832i		−	0.533888i	−0.963712	−	0.266944i	$-0.913986\pi$
$659$		−	351.832i		−	0.533888i	0.963712	−	0.266944i	$-0.0860139\pi$
$660$		0			0
$661$		−	164.295i		−	0.248556i	−0.992247	−	0.124278i	$-0.960339\pi$
$661$		−	164.295i		−	0.248556i	0.992247	−	0.124278i	$-0.0396614\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			102.136i			0.153588i
$666$		0			0
$667$	−146.423	+	147.938i	−0.219525	+	0.221795i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	202.610			0.301952
$672$		0			0
$673$	281.596			0.418420			0.209210	−	0.977871i	$-0.432911\pi$
$673$	281.596			0.418420			0.209210	+	0.977871i	$0.432911\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	840.097i		−	1.24091i	−0.784241	−	0.620456i	$-0.786948\pi$
$677$		−	840.097i		−	1.24091i	0.784241	−	0.620456i	$-0.213052\pi$
$678$		0			0
$679$	−430.358			−0.633811
$680$		0			0
$681$		0			0
$682$		0			0
$683$	883.707			1.29386			0.646930	−	0.762549i	$-0.276052\pi$
$683$	883.707			1.29386			0.646930	+	0.762549i	$0.276052\pi$
$684$		0			0
$685$	458.323			0.669085
$686$		0			0
$687$		0			0
$688$		0			0
$689$			298.056i			0.432592i
$690$		0			0
$691$	−1106.67			−1.60155			−0.800776	−	0.598964i	$-0.795579\pi$
$691$	−1106.67			−1.60155			−0.800776	+	0.598964i	$0.795579\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			417.083i			0.600120i
$696$		0			0
$697$		−	910.021i		−	1.30563i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	99.2170i		−	0.141536i	−0.997493	−	0.0707682i	$-0.977455\pi$
$701$		−	99.2170i		−	0.141536i	0.997493	−	0.0707682i	$-0.0225451\pi$
$702$		0			0
$703$	−328.090			−0.466700
$704$		0			0
$705$		0			0
$706$		0			0
$707$			264.827i			0.374578i
$708$		0			0
$709$		−	719.212i		−	1.01440i	−0.861827	−	0.507202i	$-0.830680\pi$
$709$		−	719.212i		−	1.01440i	0.861827	−	0.507202i	$-0.169320\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−553.068	+	558.788i	−0.775692	+	0.783715i
$714$		0			0
$715$	252.563			0.353235
$716$		0			0
$717$		0			0
$718$		0			0
$719$	120.811			0.168027			0.0840133	−	0.996465i	$-0.473226\pi$
$719$	120.811			0.168027			0.0840133	+	0.996465i	$0.473226\pi$
$720$		0			0
$721$	1184.07			1.64226
$722$		0			0
$723$		0			0
$724$		0			0
$725$	45.2494			0.0624130
$726$		0			0
$727$			269.719i			0.371002i	0.982644	+	0.185501i	$0.0593909\pi$
$727$			269.719i			0.371002i	−0.982644	+	0.185501i	$0.940609\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−816.989			−1.11763
$732$		0			0
$733$		−	1219.64i		−	1.66390i	−0.554852	−	0.831949i	$-0.687225\pi$
$733$		−	1219.64i		−	1.66390i	0.554852	−	0.831949i	$-0.312775\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−169.933			−0.230574
$738$		0			0
$739$	−321.003			−0.434375			−0.217188	−	0.976130i	$-0.569688\pi$
$739$	−321.003			−0.434375			−0.217188	+	0.976130i	$0.569688\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			467.332i			0.628980i	0.949261	+	0.314490i	$0.101834\pi$
$743$			467.332i			0.628980i	−0.949261	+	0.314490i	$0.898166\pi$
$744$		0			0
$745$	−347.149			−0.465972
$746$		0			0
$747$		0			0
$748$		0			0
$749$	20.4189			0.0272615
$750$		0			0
$751$		−	318.138i		−	0.423619i	−0.977311	−	0.211809i	$-0.932064\pi$
$751$		−	318.138i		−	0.423619i	0.977311	−	0.211809i	$-0.0679356\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			10.1921i			0.0134994i
$756$		0			0
$757$		−	1014.80i		−	1.34056i	−0.742108	−	0.670280i	$-0.766174\pi$
$757$		−	1014.80i		−	1.34056i	0.742108	−	0.670280i	$-0.233826\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1424.50			1.87188			0.935938	−	0.352164i	$-0.114554\pi$
$761$	1424.50			1.87188			0.935938	+	0.352164i	$0.114554\pi$
$762$		0			0
$763$	856.345			1.12234
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−83.3976			−0.108732
$768$		0			0
$769$		−	168.748i		−	0.219438i	−0.993963	−	0.109719i	$-0.965005\pi$
$769$		−	168.748i		−	0.219438i	0.993963	−	0.109719i	$-0.0349951\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			676.347i			0.874964i	0.899227	+	0.437482i	$0.144130\pi$
$773$			676.347i			0.874964i	−0.899227	+	0.437482i	$0.855870\pi$
$774$		0			0
$775$	170.916			0.220536
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	251.188i		−	0.322449i
$780$		0			0
$781$		−	689.061i		−	0.882280i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	635.873			0.810030
$786$		0			0
$787$			133.423i			0.169534i	0.996401	+	0.0847668i	$0.0270145\pi$
$787$			133.423i			0.169534i	−0.996401	+	0.0847668i	$0.972985\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1247.32			1.57688
$792$		0			0
$793$		−	351.840i		−	0.443682i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	1048.99i		−	1.31617i	−0.752943	−	0.658086i	$-0.771366\pi$
$797$		−	1048.99i		−	1.31617i	0.752943	−	0.658086i	$-0.228634\pi$
$798$		0			0
$799$			1519.29i			1.90149i
$800$		0			0
$801$		0			0
$802$		0			0
$803$			609.405i			0.758910i
$804$		0			0
$805$	277.886	−	280.760i	0.345200	−	0.348770i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	540.268			0.667822			0.333911	−	0.942605i	$-0.391631\pi$
$809$	540.268			0.667822			0.333911	+	0.942605i	$0.391631\pi$
$810$		0			0
$811$	−1209.15			−1.49094			−0.745471	−	0.666538i	$-0.767775\pi$
$811$	−1209.15			−1.49094			−0.745471	+	0.666538i	$0.767775\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	95.6368i		−	0.117346i
$816$		0			0
$817$	−225.508			−0.276020
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1253.34			1.52660			0.763301	−	0.646042i	$-0.223577\pi$
$821$	1253.34			1.52660			0.763301	+	0.646042i	$0.223577\pi$
$822$		0			0
$823$	1342.23			1.63090			0.815448	−	0.578831i	$-0.196491\pi$
$823$	1342.23			1.63090			0.815448	+	0.578831i	$0.196491\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	490.373i		−	0.592955i	−0.955040	−	0.296477i	$-0.904188\pi$
$827$		−	490.373i		−	0.592955i	0.955040	−	0.296477i	$-0.0958119\pi$
$828$		0			0
$829$	−32.2951			−0.0389567			−0.0194784	−	0.999810i	$-0.506201\pi$
$829$	−32.2951			−0.0389567			−0.0194784	+	0.999810i	$0.506201\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	215.374i		−	0.258552i
$834$		0			0
$835$			361.430i			0.432850i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	1067.79i		−	1.27269i	−0.771404	−	0.636346i	$-0.780445\pi$
$839$		−	1067.79i		−	1.27269i	0.771404	−	0.636346i	$-0.219555\pi$
$840$		0			0
$841$	−759.100			−0.902616
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	60.6906i		−	0.0718232i
$846$		0			0
$847$			429.804i			0.507443i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	901.881	+	892.649i	1.05979	+	1.04894i
$852$		0			0
$853$	876.052			1.02702			0.513512	−	0.858082i	$-0.328344\pi$
$853$	876.052			1.02702			0.513512	+	0.858082i	$0.328344\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1185.03			1.38276			0.691381	−	0.722491i	$-0.257003\pi$
$857$	1185.03			1.38276			0.691381	+	0.722491i	$0.257003\pi$
$858$		0			0
$859$	1152.46			1.34163			0.670816	−	0.741624i	$-0.265944\pi$
$859$	1152.46			1.34163			0.670816	+	0.741624i	$0.265944\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1311.33			−1.51950			−0.759748	−	0.650217i	$-0.774678\pi$
$863$	−1311.33			−1.51950			−0.759748	+	0.650217i	$0.774678\pi$
$864$		0			0
$865$			123.362i			0.142615i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−430.744			−0.495678
$870$		0			0
$871$			295.096i			0.338801i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−85.8755			−0.0981434
$876$		0			0
$877$	1128.81			1.28712			0.643562	−	0.765394i	$-0.277456\pi$
$877$	1128.81			1.28712			0.643562	+	0.765394i	$0.277456\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			793.592i			0.900785i	0.892831	+	0.450393i	$0.148716\pi$
$881$			793.592i			0.900785i	−0.892831	+	0.450393i	$0.851284\pi$
$882$		0			0
$883$	−665.234			−0.753380			−0.376690	−	0.926339i	$-0.622938\pi$
$883$	−665.234			−0.753380			−0.376690	+	0.926339i	$0.622938\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	115.058			0.129716			0.0648580	−	0.997895i	$-0.479341\pi$
$887$	115.058			0.129716			0.0648580	+	0.997895i	$0.479341\pi$
$888$		0			0
$889$		−	1789.42i		−	2.01285i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			419.360i			0.469608i
$894$		0			0
$895$		−	621.353i		−	0.694249i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	309.353			0.344108
$900$		0			0
$901$	−458.506			−0.508886
$902$		0			0
$903$		0			0
$904$		0			0
$905$	353.952			0.391107
$906$		0			0
$907$			254.682i			0.280796i	0.990095	+	0.140398i	$0.0448382\pi$
$907$			254.682i			0.280796i	−0.990095	+	0.140398i	$0.955162\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			241.548i			0.265146i	0.991173	+	0.132573i	$0.0423238\pi$
$911$			241.548i			0.265146i	−0.991173	+	0.132573i	$0.957676\pi$
$912$		0			0
$913$	653.246			0.715494
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	391.428i		−	0.426857i
$918$		0			0
$919$			1132.25i			1.23204i	0.787729	+	0.616022i	$0.211257\pi$
$919$			1132.25i			1.23204i	−0.787729	+	0.616022i	$0.788743\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1196.58			−1.29641
$924$		0			0
$925$		−	275.857i		−	0.298223i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1764.25			−1.89909			−0.949544	−	0.313633i	$-0.898454\pi$
$929$	−1764.25			−1.89909			−0.949544	+	0.313633i	$0.898454\pi$
$930$		0			0
$931$		−	59.4483i		−	0.0638543i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			388.523i			0.415533i
$936$		0			0
$937$			121.517i			0.129687i	0.997895	+	0.0648436i	$0.0206548\pi$
$937$			121.517i			0.129687i	−0.997895	+	0.0648436i	$0.979345\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			1017.93i			1.08175i	0.841102	+	0.540877i	$0.181907\pi$
$941$			1017.93i			1.08175i	−0.841102	+	0.540877i	$0.818093\pi$
$942$		0			0
$943$	−683.417	+	690.486i	−0.724727	+	0.732222i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1192.59			−1.25934			−0.629669	−	0.776864i	$-0.716809\pi$
$947$	−1192.59			−1.25934			−0.629669	+	0.776864i	$0.716809\pi$
$948$		0			0
$949$	1058.26			1.11513
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1687.69i		−	1.77093i	−0.464708	−	0.885464i	$-0.653841\pi$
$953$		−	1687.69i		−	1.77093i	0.464708	−	0.885464i	$-0.346159\pi$
$954$		0			0
$955$	−347.968			−0.364364
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1574.35			−1.64166
$960$		0			0
$961$	207.486			0.215906
$962$		0			0
$963$		0			0
$964$		0			0
$965$			804.117i			0.833282i
$966$		0			0
$967$	28.3212			0.0292876			0.0146438	−	0.999893i	$-0.495339\pi$
$967$	28.3212			0.0292876			0.0146438	+	0.999893i	$0.495339\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1235.23i		−	1.27212i	−0.771641	−	0.636059i	$-0.780564\pi$
$971$		−	1235.23i		−	1.27212i	0.771641	−	0.636059i	$-0.219436\pi$
$972$		0			0
$973$		−	1432.69i		−	1.47244i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			282.089i			0.288729i	0.989525	+	0.144365i	$0.0461139\pi$
$977$			282.089i			0.288729i	−0.989525	+	0.144365i	$0.953886\pi$
$978$		0			0
$979$	465.503			0.475488
$980$		0			0
$981$		0			0
$982$		0			0
$983$			815.107i			0.829204i	0.910003	+	0.414602i	$0.136079\pi$
$983$			815.107i			0.829204i	−0.910003	+	0.414602i	$0.863921\pi$
$984$		0			0
$985$		−	479.923i		−	0.487232i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	619.896	+	613.551i	0.626791	+	0.620375i
$990$		0			0
$991$	843.796			0.851459			0.425729	−	0.904851i	$-0.360018\pi$
$991$	843.796			0.851459			0.425729	+	0.904851i	$0.360018\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	580.208			0.583124
$996$		0			0
$997$	231.923			0.232621			0.116311	−	0.993213i	$-0.462893\pi$
$997$	231.923			0.232621			0.116311	+	0.993213i	$0.462893\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.3.d.b.2161.13	yes	32
3.2	odd	2	inner	4140.3.d.b.2161.29	yes	32
23.22	odd	2	inner	4140.3.d.b.2161.20	yes	32
69.68	even	2	inner	4140.3.d.b.2161.4	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.3.d.b.2161.4	✓	32	69.68	even	2	inner
4140.3.d.b.2161.13	yes	32	1.1	even	1	trivial
4140.3.d.b.2161.20	yes	32	23.22	odd	2	inner
4140.3.d.b.2161.29	yes	32	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	4140.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.23607i q^{5} +7.68094i q^{7} +O(q^{10})\)	\(q-2.23607i q^{5} +7.68094i q^{7} +8.06491i q^{11} +14.0051 q^{13} +21.5443i q^{17} +5.94674i q^{19} +(16.1796 - 16.3469i) q^{23} -5.00000 q^{25} -9.04988 q^{29} -34.1831 q^{31} +17.1751 q^{35} +55.1713i q^{37} -42.2395 q^{41} +37.9213i q^{43} +70.5192 q^{47} -9.99678 q^{49} +21.2820i q^{53} +18.0337 q^{55} -5.95482 q^{59} -25.1224i q^{61} -31.3163i q^{65} +21.0706i q^{67} -85.4394 q^{71} +75.5625 q^{73} -61.9460 q^{77} +53.4097i q^{79} -80.9986i q^{83} +48.1746 q^{85} -57.7196i q^{89} +107.572i q^{91} +13.2973 q^{95} +56.0293i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q - 24 q^{13} - 160 q^{25} - 28 q^{31} - 260 q^{49} + 120 q^{55} - 296 q^{73} - 60 q^{85}+O(q^{100}) \) 32 * q - 24 * q^13 - 160 * q^25 - 28 * q^31 - 260 * q^49 + 120 * q^55 - 296 * q^73 - 60 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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