LMFDB - Embedded newform 4140.2.f.d.829.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(829,4140)]

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, 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("4140.829");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.f (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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			829.6
Character	$\chi$	$=$	4140.829
Dual form			4140.2.f.d.829.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.79970 + 1.32706i) q^{5} -0.476731i q^{7} +O(q^{10})$	$q+(-1.79970 + 1.32706i) q^{5} -0.476731i q^{7} -3.39788 q^{11} +2.28658i q^{13} +2.61642i q^{17} -5.08093 q^{19} -1.00000i q^{23} +(1.47785 - 4.77661i) q^{25} -3.19050 q^{29} +3.65066 q^{31} +(0.632648 + 0.857973i) q^{35} -8.44368i q^{37} -5.25142 q^{41} -0.269348i q^{43} +8.26483i q^{47} +6.77273 q^{49} -7.77590i q^{53} +(6.11517 - 4.50918i) q^{55} +8.35367 q^{59} +5.45789 q^{61} +(-3.03442 - 4.11517i) q^{65} +6.83523i q^{67} -6.28891 q^{71} -4.90839i q^{73} +1.61987i q^{77} +9.29907 q^{79} +6.19955i q^{83} +(-3.47214 - 4.70878i) q^{85} -0.423349 q^{89} +1.09008 q^{91} +(9.14415 - 6.74268i) q^{95} -8.67355i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 8 q^{19} - 8 q^{25} - 12 q^{31} - 28 q^{49} - 16 q^{55} - 16 q^{61} + 8 q^{79} + 12 q^{85} - 16 q^{91}+O(q^{100}) $ 24 * q + 8 * q^19 - 8 * q^25 - 12 * q^31 - 28 * q^49 - 16 * q^55 - 16 * q^61 + 8 * q^79 + 12 * q^85 - 16 * q^91

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
$5$	−1.79970	+	1.32706i	−0.804851	+	0.593477i
$5$	−1.79970	+	1.32706i	−0.804851	+	0.593477i
$6$		0			0
$7$		−	0.476731i		−	0.180187i	−0.995933	−	0.0900936i	$-0.971283\pi$
$7$		−	0.476731i		−	0.180187i	0.995933	−	0.0900936i	$-0.0287166\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.39788			−1.02450			−0.512250	−	0.858837i	$-0.671188\pi$
$11$	−3.39788			−1.02450			−0.512250	+	0.858837i	$0.671188\pi$
$12$		0			0
$13$			2.28658i			0.634184i	0.948395	+	0.317092i	$0.102706\pi$
$13$			2.28658i			0.634184i	−0.948395	+	0.317092i	$0.897294\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			2.61642i			0.634576i	0.948329	+	0.317288i	$0.102772\pi$
$17$			2.61642i			0.634576i	−0.948329	+	0.317288i	$0.897228\pi$
$18$		0			0
$19$	−5.08093			−1.16565			−0.582823	−	0.812599i	$-0.698052\pi$
$19$	−5.08093			−1.16565			−0.582823	+	0.812599i	$0.698052\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.00000i		−	0.208514i
$23$		−	1.00000i		−	0.208514i
$24$		0			0
$25$	1.47785	−	4.77661i	0.295569	−	0.955321i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.19050			−0.592460			−0.296230	−	0.955117i	$-0.595730\pi$
$29$	−3.19050			−0.592460			−0.296230	+	0.955117i	$0.595730\pi$
$30$		0			0
$31$	3.65066			0.655678			0.327839	−	0.944734i	$-0.393680\pi$
$31$	3.65066			0.655678			0.327839	+	0.944734i	$0.393680\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.632648	+	0.857973i	0.106937	+	0.145024i
$36$		0			0
$37$		−	8.44368i		−	1.38813i	−0.719911	−	0.694066i	$-0.755818\pi$
$37$		−	8.44368i		−	1.38813i	0.719911	−	0.694066i	$-0.244182\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−5.25142			−0.820134			−0.410067	−	0.912055i	$-0.634495\pi$
$41$	−5.25142			−0.820134			−0.410067	+	0.912055i	$0.634495\pi$
$42$		0			0
$43$		−	0.269348i		−	0.0410752i	−0.999789	−	0.0205376i	$-0.993462\pi$
$43$		−	0.269348i		−	0.0410752i	0.999789	−	0.0205376i	$-0.00653778\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			8.26483i			1.20555i	0.797911	+	0.602775i	$0.205938\pi$
$47$			8.26483i			1.20555i	−0.797911	+	0.602775i	$0.794062\pi$
$48$		0			0
$49$	6.77273			0.967533
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	7.77590i		−	1.06810i	−0.845452	−	0.534051i	$-0.820669\pi$
$53$		−	7.77590i		−	1.06810i	0.845452	−	0.534051i	$-0.179331\pi$
$54$		0			0
$55$	6.11517	−	4.50918i	0.824569	−	0.608017i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	8.35367			1.08756			0.543778	−	0.839229i	$-0.316994\pi$
$59$	8.35367			1.08756			0.543778	+	0.839229i	$0.316994\pi$
$60$		0			0
$61$	5.45789			0.698811			0.349405	−	0.936972i	$-0.386384\pi$
$61$	5.45789			0.698811			0.349405	+	0.936972i	$0.386384\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.03442	−	4.11517i	−0.376374	−	0.510424i
$66$		0			0
$67$			6.83523i			0.835056i	0.908664	+	0.417528i	$0.137103\pi$
$67$			6.83523i			0.835056i	−0.908664	+	0.417528i	$0.862897\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−6.28891			−0.746357			−0.373178	−	0.927760i	$-0.621732\pi$
$71$	−6.28891			−0.746357			−0.373178	+	0.927760i	$0.621732\pi$
$72$		0			0
$73$		−	4.90839i		−	0.574483i	−0.957858	−	0.287242i	$-0.907262\pi$
$73$		−	4.90839i		−	0.574483i	0.957858	−	0.287242i	$-0.0927383\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			1.61987i			0.184602i
$78$		0			0
$79$	9.29907			1.04623			0.523113	−	0.852263i	$-0.324770\pi$
$79$	9.29907			1.04623			0.523113	+	0.852263i	$0.324770\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			6.19955i			0.680489i	0.940337	+	0.340244i	$0.110510\pi$
$83$			6.19955i			0.680489i	−0.940337	+	0.340244i	$0.889490\pi$
$84$		0			0
$85$	−3.47214	−	4.70878i	−0.376607	−	0.510739i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.423349			−0.0448749			−0.0224374	−	0.999748i	$-0.507143\pi$
$89$	−0.423349			−0.0448749			−0.0224374	+	0.999748i	$0.507143\pi$
$90$		0			0
$91$	1.09008			0.114272
$92$		0			0
$93$		0			0
$94$		0			0
$95$	9.14415	−	6.74268i	0.938170	−	0.691784i
$96$		0			0
$97$		−	8.67355i		−	0.880666i	−0.897835	−	0.440333i	$-0.854860\pi$
$97$		−	8.67355i		−	0.880666i	0.897835	−	0.440333i	$-0.145140\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	9.41335			0.936663			0.468331	−	0.883553i	$-0.344855\pi$
$101$	9.41335			0.936663			0.468331	+	0.883553i	$0.344855\pi$
$102$		0			0
$103$		−	5.24556i		−	0.516860i	−0.966030	−	0.258430i	$-0.916795\pi$
$103$		−	5.24556i		−	0.516860i	0.966030	−	0.258430i	$-0.0832052\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	1.80032i		−	0.174044i	−0.996206	−	0.0870220i	$-0.972265\pi$
$107$		−	1.80032i		−	0.174044i	0.996206	−	0.0870220i	$-0.0277350\pi$
$108$		0			0
$109$	−2.14276			−0.205240			−0.102620	−	0.994721i	$-0.532722\pi$
$109$	−2.14276			−0.205240			−0.102620	+	0.994721i	$0.532722\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	16.6232i		−	1.56378i	−0.623417	−	0.781889i	$-0.714256\pi$
$113$		−	16.6232i		−	1.56378i	0.623417	−	0.781889i	$-0.285744\pi$
$114$		0			0
$115$	1.32706	+	1.79970i	0.123749	+	0.167823i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.24733			0.114343
$120$		0			0
$121$	0.545586			0.0495987
$122$		0			0
$123$		0			0
$124$		0			0
$125$	3.67914	+	10.5576i	0.329072	+	0.944305i
$126$		0			0
$127$		−	10.1594i		−	0.901503i	−0.892649	−	0.450752i	$-0.851156\pi$
$127$		−	10.1594i		−	0.901503i	0.892649	−	0.450752i	$-0.148844\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.12091			0.272675			0.136338	−	0.990662i	$-0.456467\pi$
$131$	3.12091			0.272675			0.136338	+	0.990662i	$0.456467\pi$
$132$		0			0
$133$			2.42224i			0.210034i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	1.11542i		−	0.0952972i	−0.998864	−	0.0476486i	$-0.984827\pi$
$137$		−	1.11542i		−	0.0952972i	0.998864	−	0.0476486i	$-0.0151728\pi$
$138$		0			0
$139$	1.12777			0.0956564			0.0478282	−	0.998856i	$-0.484770\pi$
$139$	1.12777			0.0956564			0.0478282	+	0.998856i	$0.484770\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	7.76954i		−	0.649721i
$144$		0			0
$145$	5.74194	−	4.23397i	0.476842	−	0.351612i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−13.5098			−1.10676			−0.553382	−	0.832927i	$-0.686663\pi$
$149$	−13.5098			−1.10676			−0.553382	+	0.832927i	$0.686663\pi$
$150$		0			0
$151$	−4.39922			−0.358004			−0.179002	−	0.983849i	$-0.557287\pi$
$151$	−4.39922			−0.358004			−0.179002	+	0.983849i	$0.557287\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−6.57010	+	4.84463i	−0.527723	+	0.389130i
$156$		0			0
$157$		−	20.3773i		−	1.62629i	−0.582063	−	0.813143i	$-0.697754\pi$
$157$		−	20.3773i		−	1.62629i	0.582063	−	0.813143i	$-0.302246\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−0.476731			−0.0375716
$162$		0			0
$163$			2.88927i			0.226305i	0.993578	+	0.113153i	$0.0360949\pi$
$163$			2.88927i			0.226305i	−0.993578	+	0.113153i	$0.963905\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	12.5218i		−	0.968970i	−0.874800	−	0.484485i	$-0.839007\pi$
$167$		−	12.5218i		−	0.968970i	0.874800	−	0.484485i	$-0.160993\pi$
$168$		0			0
$169$	7.77153			0.597810
$170$		0			0
$171$		0			0
$172$		0			0
$173$			14.1150i			1.07315i	0.843854	+	0.536573i	$0.180281\pi$
$173$			14.1150i			1.07315i	−0.843854	+	0.536573i	$0.819719\pi$
$174$		0			0
$175$	−2.27716	−	0.704535i	−0.172137	−	0.0532578i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	0.879658			0.0657487			0.0328744	−	0.999459i	$-0.489534\pi$
$179$	0.879658			0.0657487			0.0328744	+	0.999459i	$0.489534\pi$
$180$		0			0
$181$	−4.29243			−0.319054			−0.159527	−	0.987194i	$-0.550997\pi$
$181$	−4.29243			−0.319054			−0.159527	+	0.987194i	$0.550997\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	11.2052	+	15.1961i	0.823825	+	1.11724i
$186$		0			0
$187$		−	8.89029i		−	0.650123i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	14.5786			1.05487			0.527435	−	0.849595i	$-0.323154\pi$
$191$	14.5786			1.05487			0.527435	+	0.849595i	$0.323154\pi$
$192$		0			0
$193$		−	21.7282i		−	1.56403i	−0.623258	−	0.782016i	$-0.714191\pi$
$193$		−	21.7282i		−	1.56403i	0.623258	−	0.782016i	$-0.285809\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			6.01527i			0.428570i	0.976771	+	0.214285i	$0.0687422\pi$
$197$			6.01527i			0.428570i	−0.976771	+	0.214285i	$0.931258\pi$
$198$		0			0
$199$	7.90913			0.560663			0.280332	−	0.959903i	$-0.409556\pi$
$199$	7.90913			0.560663			0.280332	+	0.959903i	$0.409556\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			1.52101i			0.106754i
$204$		0			0
$205$	9.45099	−	6.96893i	0.660086	−	0.486731i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	17.2644			1.19420
$210$		0			0
$211$	24.5795			1.69213			0.846063	−	0.533083i	$-0.178967\pi$
$211$	24.5795			1.69213			0.846063	+	0.533083i	$0.178967\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	0.357440	+	0.484746i	0.0243772	+	0.0330594i
$216$		0			0
$217$		−	1.74038i		−	0.118145i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−5.98267			−0.402438
$222$		0			0
$223$		−	2.35824i		−	0.157920i	−0.996878	−	0.0789598i	$-0.974840\pi$
$223$		−	2.35824i		−	0.157920i	0.996878	−	0.0789598i	$-0.0251599\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	9.40161i		−	0.624006i	−0.950081	−	0.312003i	$-0.899000\pi$
$227$		−	9.40161i		−	0.624006i	0.950081	−	0.312003i	$-0.101000\pi$
$228$		0			0
$229$	2.37921			0.157223			0.0786114	−	0.996905i	$-0.474951\pi$
$229$	2.37921			0.157223			0.0786114	+	0.996905i	$0.474951\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	16.6381i		−	1.09000i	−0.838438	−	0.544998i	$-0.816531\pi$
$233$		−	16.6381i		−	1.09000i	0.838438	−	0.544998i	$-0.183469\pi$
$234$		0			0
$235$	−10.9679	−	14.8742i	−0.715466	−	0.970287i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	17.3942			1.12514			0.562569	−	0.826751i	$-0.309813\pi$
$239$	17.3942			1.12514			0.562569	+	0.826751i	$0.309813\pi$
$240$		0			0
$241$	24.6857			1.59015			0.795073	−	0.606514i	$-0.207432\pi$
$241$	24.6857			1.59015			0.795073	+	0.606514i	$0.207432\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−12.1889	+	8.98779i	−0.778719	+	0.574209i
$246$		0			0
$247$		−	11.6180i		−	0.739234i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−5.29188			−0.334021			−0.167010	−	0.985955i	$-0.553411\pi$
$251$	−5.29188			−0.334021			−0.167010	+	0.985955i	$0.553411\pi$
$252$		0			0
$253$			3.39788i			0.213623i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			15.2623i			0.952037i	0.879435	+	0.476019i	$0.157920\pi$
$257$			15.2623i			0.952037i	−0.879435	+	0.476019i	$0.842080\pi$
$258$		0			0
$259$	−4.02536			−0.250124
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	7.50790i		−	0.462957i	−0.972840	−	0.231478i	$-0.925644\pi$
$263$		−	7.50790i		−	0.462957i	0.972840	−	0.231478i	$-0.0743563\pi$
$264$		0			0
$265$	10.3190	+	13.9943i	0.633894	+	0.859662i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−7.76366			−0.473359			−0.236679	−	0.971588i	$-0.576059\pi$
$269$	−7.76366			−0.473359			−0.236679	+	0.971588i	$0.576059\pi$
$270$		0			0
$271$	−1.50881			−0.0916538			−0.0458269	−	0.998949i	$-0.514592\pi$
$271$	−1.50881			−0.0916538			−0.0458269	+	0.998949i	$0.514592\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−5.02154	+	16.2303i	−0.302811	+	0.978726i
$276$		0			0
$277$			1.93933i			0.116523i	0.998301	+	0.0582616i	$0.0185558\pi$
$277$			1.93933i			0.116523i	−0.998301	+	0.0582616i	$0.981444\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−4.46254			−0.266213			−0.133107	−	0.991102i	$-0.542495\pi$
$281$	−4.46254			−0.266213			−0.133107	+	0.991102i	$0.542495\pi$
$282$		0			0
$283$		−	18.0910i		−	1.07540i	−0.843137	−	0.537699i	$-0.819294\pi$
$283$		−	18.0910i		−	1.07540i	0.843137	−	0.537699i	$-0.180706\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			2.50351i			0.147778i
$288$		0			0
$289$	10.1543			0.597313
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	15.2461i		−	0.890687i	−0.895360	−	0.445344i	$-0.853082\pi$
$293$		−	15.2461i		−	0.890687i	0.895360	−	0.445344i	$-0.146918\pi$
$294$		0			0
$295$	−15.0341	+	11.0858i	−0.875320	+	0.645439i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.28658			0.132237
$300$		0			0
$301$	−0.128406			−0.00740122
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−9.82256	+	7.24292i	−0.562438	+	0.414728i
$306$		0			0
$307$			32.4536i			1.85223i	0.377246	+	0.926113i	$0.376871\pi$
$307$			32.4536i			1.85223i	−0.377246	+	0.926113i	$0.623129\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	25.6744			1.45586			0.727930	−	0.685651i	$-0.240482\pi$
$311$	25.6744			1.45586			0.727930	+	0.685651i	$0.240482\pi$
$312$		0			0
$313$			3.45980i			0.195559i	0.995208	+	0.0977797i	$0.0311741\pi$
$313$			3.45980i			0.195559i	−0.995208	+	0.0977797i	$0.968826\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	13.2692i		−	0.745273i	−0.927977	−	0.372637i	$-0.878454\pi$
$317$		−	13.2692i		−	0.745273i	0.927977	−	0.372637i	$-0.121546\pi$
$318$		0			0
$319$	10.8409			0.606975
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	13.2939i		−	0.739690i
$324$		0			0
$325$	10.9221	+	3.37922i	0.605850	+	0.187445i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3.94010			0.217225
$330$		0			0
$331$	−34.8901			−1.91774			−0.958868	−	0.283852i	$-0.908387\pi$
$331$	−34.8901			−1.91774			−0.958868	+	0.283852i	$0.908387\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−9.07073	−	12.3014i	−0.495587	−	0.672096i
$336$		0			0
$337$		−	4.89570i		−	0.266686i	−0.991070	−	0.133343i	$-0.957429\pi$
$337$		−	4.89570i		−	0.266686i	0.991070	−	0.133343i	$-0.0425712\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−12.4045			−0.671742
$342$		0			0
$343$		−	6.56588i		−	0.354524i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			1.43042i			0.0767889i	0.999263	+	0.0383945i	$0.0122243\pi$
$347$			1.43042i			0.0767889i	−0.999263	+	0.0383945i	$0.987776\pi$
$348$		0			0
$349$	10.1336			0.542441			0.271220	−	0.962517i	$-0.412573\pi$
$349$	10.1336			0.542441			0.271220	+	0.962517i	$0.412573\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	2.70613i		−	0.144033i	−0.997403	−	0.0720165i	$-0.977057\pi$
$353$		−	2.70613i		−	0.144033i	0.997403	−	0.0720165i	$-0.0229434\pi$
$354$		0			0
$355$	11.3182	−	8.34574i	0.600706	−	0.442946i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	7.33325			0.387034			0.193517	−	0.981097i	$-0.438010\pi$
$359$	7.33325			0.387034			0.193517	+	0.981097i	$0.438010\pi$
$360$		0			0
$361$	6.81584			0.358729
$362$		0			0
$363$		0			0
$364$		0			0
$365$	6.51370	+	8.83363i	0.340943	+	0.462373i
$366$		0			0
$367$			25.9825i			1.35628i	0.734935	+	0.678138i	$0.237213\pi$
$367$			25.9825i			1.35628i	−0.734935	+	0.678138i	$0.762787\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.70701			−0.192458
$372$		0			0
$373$		−	28.8726i		−	1.49497i	−0.664280	−	0.747484i	$-0.731262\pi$
$373$		−	28.8726i		−	1.49497i	0.664280	−	0.747484i	$-0.268738\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	7.29534i		−	0.375729i
$378$		0			0
$379$	−4.28834			−0.220277			−0.110139	−	0.993916i	$-0.535129\pi$
$379$	−4.28834			−0.220277			−0.110139	+	0.993916i	$0.535129\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	24.9529i		−	1.27503i	−0.770436	−	0.637517i	$-0.779962\pi$
$383$		−	24.9529i		−	1.27503i	0.770436	−	0.637517i	$-0.220038\pi$
$384$		0			0
$385$	−2.14966	−	2.91529i	−0.109557	−	0.148577i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−24.7851			−1.25665			−0.628327	−	0.777949i	$-0.716260\pi$
$389$	−24.7851			−1.25665			−0.628327	+	0.777949i	$0.716260\pi$
$390$		0			0
$391$	2.61642			0.132318
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−16.7355	+	12.3404i	−0.842056	+	0.620912i
$396$		0			0
$397$		−	8.28792i		−	0.415959i	−0.978133	−	0.207979i	$-0.933311\pi$
$397$		−	8.28792i		−	0.415959i	0.978133	−	0.207979i	$-0.0666887\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	10.6623			0.532451			0.266225	−	0.963911i	$-0.414223\pi$
$401$	10.6623			0.532451			0.266225	+	0.963911i	$0.414223\pi$
$402$		0			0
$403$			8.34754i			0.415821i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			28.6906i			1.42214i
$408$		0			0
$409$	27.2848			1.34914			0.674572	−	0.738209i	$-0.264328\pi$
$409$	27.2848			1.34914			0.674572	+	0.738209i	$0.264328\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	3.98245i		−	0.195964i
$414$		0			0
$415$	−8.22714	−	11.1573i	−0.403855	−	0.547692i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−7.69546			−0.375948			−0.187974	−	0.982174i	$-0.560192\pi$
$419$	−7.69546			−0.375948			−0.187974	+	0.982174i	$0.560192\pi$
$420$		0			0
$421$	−33.7126			−1.64305			−0.821526	−	0.570171i	$-0.806877\pi$
$421$	−33.7126			−1.64305			−0.821526	+	0.570171i	$0.806877\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	12.4976	+	3.86667i	0.606224	+	0.187561i
$426$		0			0
$427$		−	2.60194i		−	0.125917i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−23.4448			−1.12930			−0.564649	−	0.825331i	$-0.690989\pi$
$431$	−23.4448			−1.12930			−0.564649	+	0.825331i	$0.690989\pi$
$432$		0			0
$433$		−	26.1795i		−	1.25811i	−0.777361	−	0.629054i	$-0.783442\pi$
$433$		−	26.1795i		−	1.25811i	0.777361	−	0.629054i	$-0.216558\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			5.08093i			0.243054i
$438$		0			0
$439$	−24.0063			−1.14576			−0.572878	−	0.819641i	$-0.694173\pi$
$439$	−24.0063			−1.14576			−0.572878	+	0.819641i	$0.694173\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	37.5878i		−	1.78585i	−0.450205	−	0.892925i	$-0.648649\pi$
$443$		−	37.5878i		−	1.78585i	0.450205	−	0.892925i	$-0.351351\pi$
$444$		0			0
$445$	0.761901	−	0.561807i	0.0361176	−	0.0266322i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−21.5817			−1.01850			−0.509252	−	0.860617i	$-0.670078\pi$
$449$	−21.5817			−1.01850			−0.509252	+	0.860617i	$0.670078\pi$
$450$		0			0
$451$	17.8437			0.840227
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.96183	+	1.44660i	−0.0919719	+	0.0678178i
$456$		0			0
$457$		−	35.0161i		−	1.63798i	−0.573806	−	0.818991i	$-0.694534\pi$
$457$		−	35.0161i		−	1.63798i	0.573806	−	0.818991i	$-0.305466\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−11.1807			−0.520738			−0.260369	−	0.965509i	$-0.583844\pi$
$461$	−11.1807			−0.520738			−0.260369	+	0.965509i	$0.583844\pi$
$462$		0			0
$463$			22.4818i			1.04482i	0.852695	+	0.522409i	$0.174967\pi$
$463$			22.4818i			1.04482i	−0.852695	+	0.522409i	$0.825033\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			41.5559i			1.92298i	0.274841	+	0.961490i	$0.411375\pi$
$467$			41.5559i			1.92298i	−0.274841	+	0.961490i	$0.588625\pi$
$468$		0			0
$469$	3.25856			0.150467
$470$		0			0
$471$		0			0
$472$		0			0
$473$			0.915212i			0.0420815i
$474$		0			0
$475$	−7.50883	+	24.2696i	−0.344529	+	1.11357i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−24.6378			−1.12573			−0.562864	−	0.826549i	$-0.690301\pi$
$479$	−24.6378			−1.12573			−0.562864	+	0.826549i	$0.690301\pi$
$480$		0			0
$481$	19.3072			0.880332
$482$		0			0
$483$		0			0
$484$		0			0
$485$	11.5103	+	15.6098i	0.522655	+	0.708805i
$486$		0			0
$487$			31.6802i			1.43557i	0.696267	+	0.717783i	$0.254843\pi$
$487$			31.6802i			1.43557i	−0.696267	+	0.717783i	$0.745157\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	17.7934			0.803006			0.401503	−	0.915858i	$-0.368488\pi$
$491$	17.7934			0.803006			0.401503	+	0.915858i	$0.368488\pi$
$492$		0			0
$493$		−	8.34769i		−	0.375961i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			2.99812i			0.134484i
$498$		0			0
$499$	−27.1069			−1.21347			−0.606737	−	0.794903i	$-0.707522\pi$
$499$	−27.1069			−1.21347			−0.606737	+	0.794903i	$0.707522\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			2.86428i			0.127712i	0.997959	+	0.0638561i	$0.0203399\pi$
$503$			2.86428i			0.127712i	−0.997959	+	0.0638561i	$0.979660\pi$
$504$		0			0
$505$	−16.9412	+	12.4920i	−0.753874	+	0.555888i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−11.7653			−0.521490			−0.260745	−	0.965408i	$-0.583968\pi$
$509$	−11.7653			−0.521490			−0.260745	+	0.965408i	$0.583968\pi$
$510$		0			0
$511$	−2.33998			−0.103515
$512$		0			0
$513$		0			0
$514$		0			0
$515$	6.96115	+	9.44044i	0.306745	+	0.415995i
$516$		0			0
$517$		−	28.0829i		−	1.23508i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	12.5122			0.548169			0.274084	−	0.961706i	$-0.411625\pi$
$521$	12.5122			0.548169			0.274084	+	0.961706i	$0.411625\pi$
$522$		0			0
$523$		−	15.8344i		−	0.692391i	−0.938162	−	0.346195i	$-0.887473\pi$
$523$		−	15.8344i		−	0.692391i	0.938162	−	0.346195i	$-0.112527\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			9.55168i			0.416078i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	12.0078i		−	0.520116i
$534$		0			0
$535$	2.38913	+	3.24005i	0.103291	+	0.140079i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−23.0129			−0.991236
$540$		0			0
$541$	20.1948			0.868240			0.434120	−	0.900855i	$-0.357059\pi$
$541$	20.1948			0.868240			0.434120	+	0.900855i	$0.357059\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	3.85633	−	2.84357i	0.165187	−	0.121805i
$546$		0			0
$547$			34.3218i			1.46750i	0.679422	+	0.733748i	$0.262231\pi$
$547$			34.3218i			1.46750i	−0.679422	+	0.733748i	$0.737769\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	16.2107			0.690599
$552$		0			0
$553$		−	4.43315i		−	0.188517i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			27.9107i			1.18262i	0.806446	+	0.591308i	$0.201388\pi$
$557$			27.9107i			1.18262i	−0.806446	+	0.591308i	$0.798612\pi$
$558$		0			0
$559$	0.615887			0.0260492
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	13.3640i		−	0.563227i	−0.959528	−	0.281614i	$-0.909130\pi$
$563$		−	13.3640i		−	0.563227i	0.959528	−	0.281614i	$-0.0908697\pi$
$564$		0			0
$565$	22.0599	+	29.9168i	0.928067	+	1.25861i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	39.1983			1.64328			0.821640	−	0.570007i	$-0.193059\pi$
$569$	39.1983			1.64328			0.821640	+	0.570007i	$0.193059\pi$
$570$		0			0
$571$	5.64904			0.236405			0.118202	−	0.992990i	$-0.462287\pi$
$571$	5.64904			0.236405			0.118202	+	0.992990i	$0.462287\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.77661	−	1.47785i	−0.199198	−	0.0616305i
$576$		0			0
$577$			8.82278i			0.367297i	0.982992	+	0.183649i	$0.0587908\pi$
$577$			8.82278i			0.367297i	−0.982992	+	0.183649i	$0.941209\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2.95551			0.122615
$582$		0			0
$583$			26.4216i			1.09427i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			6.74701i			0.278479i	0.990259	+	0.139240i	$0.0444658\pi$
$587$			6.74701i			0.278479i	−0.990259	+	0.139240i	$0.955534\pi$
$588$		0			0
$589$	−18.5488			−0.764288
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	18.8242i		−	0.773018i	−0.922286	−	0.386509i	$-0.873681\pi$
$593$		−	18.8242i		−	0.773018i	0.922286	−	0.386509i	$-0.126319\pi$
$594$		0			0
$595$	−2.24482	+	1.65528i	−0.0920287	+	0.0678597i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−20.6986			−0.845723			−0.422861	−	0.906194i	$-0.638974\pi$
$599$	−20.6986			−0.845723			−0.422861	+	0.906194i	$0.638974\pi$
$600$		0			0
$601$	−4.83720			−0.197314			−0.0986568	−	0.995122i	$-0.531455\pi$
$601$	−4.83720			−0.197314			−0.0986568	+	0.995122i	$0.531455\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.981891	+	0.724022i	−0.0399195	+	0.0294357i
$606$		0			0
$607$		−	8.44486i		−	0.342766i	−0.985204	−	0.171383i	$-0.945176\pi$
$607$		−	8.44486i		−	0.342766i	0.985204	−	0.171383i	$-0.0548236\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−18.8982			−0.764540
$612$		0			0
$613$			24.6765i			0.996674i	0.866984	+	0.498337i	$0.166056\pi$
$613$			24.6765i			0.996674i	−0.866984	+	0.498337i	$0.833944\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	6.37233i		−	0.256541i	−0.991739	−	0.128270i	$-0.959058\pi$
$617$		−	6.37233i		−	0.256541i	0.991739	−	0.128270i	$-0.0409425\pi$
$618$		0			0
$619$	−25.6093			−1.02932			−0.514662	−	0.857393i	$-0.672083\pi$
$619$	−25.6093			−1.02932			−0.514662	+	0.857393i	$0.672083\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			0.201823i			0.00808588i
$624$		0			0
$625$	−20.6319	−	14.1182i	−0.825278	−	0.564727i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	22.0922			0.880875
$630$		0			0
$631$	−24.2453			−0.965192			−0.482596	−	0.875843i	$-0.660306\pi$
$631$	−24.2453			−0.965192			−0.482596	+	0.875843i	$0.660306\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	13.4821	+	18.2839i	0.535022	+	0.725575i
$636$		0			0
$637$			15.4864i			0.613594i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−29.3162			−1.15792			−0.578961	−	0.815355i	$-0.696542\pi$
$641$	−29.3162			−1.15792			−0.578961	+	0.815355i	$0.696542\pi$
$642$		0			0
$643$		−	2.16350i		−	0.0853200i	−0.999090	−	0.0426600i	$-0.986417\pi$
$643$		−	2.16350i		−	0.0853200i	0.999090	−	0.0426600i	$-0.0135832\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			41.2320i			1.62100i	0.585740	+	0.810499i	$0.300804\pi$
$647$			41.2320i			1.62100i	−0.585740	+	0.810499i	$0.699196\pi$
$648$		0			0
$649$	−28.3848			−1.11420
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	2.15816i		−	0.0844555i	−0.999108	−	0.0422277i	$-0.986554\pi$
$653$		−	2.15816i		−	0.0844555i	0.999108	−	0.0422277i	$-0.0134455\pi$
$654$		0			0
$655$	−5.61670	+	4.14162i	−0.219463	+	0.161827i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	19.6592			0.765812			0.382906	−	0.923787i	$-0.374923\pi$
$659$	19.6592			0.765812			0.382906	+	0.923787i	$0.374923\pi$
$660$		0			0
$661$	−10.8473			−0.421911			−0.210955	−	0.977496i	$-0.567657\pi$
$661$	−10.8473			−0.421911			−0.210955	+	0.977496i	$0.567657\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−3.21444	−	4.35930i	−0.124651	−	0.169046i
$666$		0			0
$667$			3.19050i			0.123537i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−18.5452			−0.715931
$672$		0			0
$673$		−	13.2407i		−	0.510393i	−0.966889	−	0.255197i	$-0.917860\pi$
$673$		−	13.2407i		−	0.510393i	0.966889	−	0.255197i	$-0.0821402\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	49.0637i		−	1.88567i	−0.333257	−	0.942836i	$-0.608148\pi$
$677$		−	49.0637i		−	1.88567i	0.333257	−	0.942836i	$-0.391852\pi$
$678$		0			0
$679$	−4.13495			−0.158685
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	21.8438i		−	0.835831i	−0.908486	−	0.417915i	$-0.862761\pi$
$683$		−	21.8438i		−	0.835831i	0.908486	−	0.417915i	$-0.137239\pi$
$684$		0			0
$685$	1.48023	+	2.00743i	0.0565567	+	0.0767000i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	17.7802			0.677373
$690$		0			0
$691$	−15.7696			−0.599903			−0.299952	−	0.953954i	$-0.596971\pi$
$691$	−15.7696			−0.599903			−0.299952	+	0.953954i	$0.596971\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−2.02965	+	1.49662i	−0.0769891	+	0.0567699i
$696$		0			0
$697$		−	13.7399i		−	0.520437i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	16.6534			0.628991			0.314496	−	0.949259i	$-0.398165\pi$
$701$	16.6534			0.628991			0.314496	+	0.949259i	$0.398165\pi$
$702$		0			0
$703$			42.9017i			1.61807i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	4.48763i		−	0.168775i
$708$		0			0
$709$	−16.7592			−0.629405			−0.314703	−	0.949190i	$-0.601905\pi$
$709$	−16.7592			−0.629405			−0.314703	+	0.949190i	$0.601905\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	3.65066i		−	0.136718i
$714$		0			0
$715$	10.3106	+	13.9828i	0.385595	+	0.522929i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−20.8196			−0.776440			−0.388220	−	0.921567i	$-0.626910\pi$
$719$	−20.8196			−0.776440			−0.388220	+	0.921567i	$0.626910\pi$
$720$		0			0
$721$	−2.50072			−0.0931317
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4.71506	+	15.2397i	−0.175113	+	0.565990i
$726$		0			0
$727$			27.7272i			1.02835i	0.857687	+	0.514173i	$0.171901\pi$
$727$			27.7272i			1.02835i	−0.857687	+	0.514173i	$0.828099\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	0.704728			0.0260653
$732$		0			0
$733$			30.1678i			1.11427i	0.830420	+	0.557137i	$0.188100\pi$
$733$			30.1678i			1.11427i	−0.830420	+	0.557137i	$0.811900\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	23.2253i		−	0.855515i
$738$		0			0
$739$	−0.443024			−0.0162969			−0.00814845	−	0.999967i	$-0.502594\pi$
$739$	−0.443024			−0.0162969			−0.00814845	+	0.999967i	$0.502594\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	40.3237i		−	1.47933i	−0.672973	−	0.739667i	$-0.734983\pi$
$743$		−	40.3237i		−	1.47933i	0.672973	−	0.739667i	$-0.265017\pi$
$744$		0			0
$745$	24.3136	−	17.9282i	0.890780	−	0.656840i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−0.858270			−0.0313605
$750$		0			0
$751$	−6.10545			−0.222791			−0.111396	−	0.993776i	$-0.535532\pi$
$751$	−6.10545			−0.222791			−0.111396	+	0.993776i	$0.535532\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	7.91728	−	5.83801i	0.288139	−	0.212467i
$756$		0			0
$757$			52.4883i			1.90772i	0.300253	+	0.953860i	$0.402929\pi$
$757$			52.4883i			1.90772i	−0.300253	+	0.953860i	$0.597071\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	28.4101			1.02987			0.514933	−	0.857230i	$-0.327817\pi$
$761$	28.4101			1.02987			0.514933	+	0.857230i	$0.327817\pi$
$762$		0			0
$763$			1.02152i			0.0369816i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			19.1014i			0.689710i
$768$		0			0
$769$	27.9501			1.00791			0.503953	−	0.863731i	$-0.331878\pi$
$769$	27.9501			1.00791			0.503953	+	0.863731i	$0.331878\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	37.8554i		−	1.36156i	−0.732486	−	0.680782i	$-0.761640\pi$
$773$		−	37.8554i		−	1.36156i	0.732486	−	0.680782i	$-0.238360\pi$
$774$		0			0
$775$	5.39512	−	17.4378i	0.193798	−	0.626383i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	26.6821			0.955985
$780$		0			0
$781$	21.3690			0.764642
$782$		0			0
$783$		0			0
$784$		0			0
$785$	27.0418	+	36.6731i	0.965164	+	1.30892i
$786$		0			0
$787$		−	24.4460i		−	0.871405i	−0.900091	−	0.435702i	$-0.856500\pi$
$787$		−	24.4460i		−	0.871405i	0.900091	−	0.435702i	$-0.143500\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−7.92479			−0.281773
$792$		0			0
$793$			12.4799i			0.443175i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			6.16201i			0.218270i	0.994027	+	0.109135i	$0.0348080\pi$
$797$			6.16201i			0.218270i	−0.994027	+	0.109135i	$0.965192\pi$
$798$		0			0
$799$	−21.6243			−0.765013
$800$		0			0
$801$		0			0
$802$		0			0
$803$			16.6781i			0.588558i
$804$		0			0
$805$	0.857973	−	0.632648i	0.0302396	−	0.0222979i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	45.7345			1.60794			0.803970	−	0.594670i	$-0.202717\pi$
$809$	45.7345			1.60794			0.803970	+	0.594670i	$0.202717\pi$
$810$		0			0
$811$	−23.0837			−0.810578			−0.405289	−	0.914189i	$-0.632829\pi$
$811$	−23.0837			−0.810578			−0.405289	+	0.914189i	$0.632829\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3.83422	−	5.19983i	−0.134307	−	0.182142i
$816$		0			0
$817$			1.36854i			0.0478791i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	35.3412			1.23342			0.616708	−	0.787192i	$-0.288466\pi$
$821$	35.3412			1.23342			0.616708	+	0.787192i	$0.288466\pi$
$822$		0			0
$823$			53.0658i			1.84976i	0.380261	+	0.924879i	$0.375834\pi$
$823$			53.0658i			1.84976i	−0.380261	+	0.924879i	$0.624166\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			16.2618i			0.565479i	0.959197	+	0.282739i	$0.0912431\pi$
$827$			16.2618i			0.565479i	−0.959197	+	0.282739i	$0.908757\pi$
$828$		0			0
$829$	−16.3521			−0.567933			−0.283966	−	0.958834i	$-0.591650\pi$
$829$	−16.3521			−0.567933			−0.283966	+	0.958834i	$0.591650\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			17.7203i			0.613973i
$834$		0			0
$835$	16.6172	+	22.5356i	0.575061	+	0.779876i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	31.2765			1.07979			0.539893	−	0.841734i	$-0.318465\pi$
$839$	31.2765			1.07979			0.539893	+	0.841734i	$0.318465\pi$
$840$		0			0
$841$	−18.8207			−0.648991
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−13.9864	+	10.3133i	−0.481148	+	0.354787i
$846$		0			0
$847$		−	0.260097i		−	0.00893705i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−8.44368			−0.289446
$852$		0			0
$853$			47.9540i			1.64191i	0.570991	+	0.820956i	$0.306559\pi$
$853$			47.9540i			1.64191i	−0.570991	+	0.820956i	$0.693441\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	32.4303i		−	1.10780i	−0.832584	−	0.553899i	$-0.813139\pi$
$857$		−	32.4303i		−	1.10780i	0.832584	−	0.553899i	$-0.186861\pi$
$858$		0			0
$859$	40.0847			1.36767			0.683836	−	0.729636i	$-0.260310\pi$
$859$	40.0847			1.36767			0.683836	+	0.729636i	$0.260310\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	42.5831i		−	1.44955i	−0.688988	−	0.724773i	$-0.741945\pi$
$863$		−	42.5831i		−	1.44955i	0.688988	−	0.724773i	$-0.258055\pi$
$864$		0			0
$865$	−18.7314	−	25.4028i	−0.636888	−	0.863723i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−31.5971			−1.07186
$870$		0			0
$871$	−15.6293			−0.529580
$872$		0			0
$873$		0			0
$874$		0			0
$875$	5.03316	−	1.75396i	0.170152	−	0.0592947i
$876$		0			0
$877$		−	18.1323i		−	0.612283i	−0.951986	−	0.306142i	$-0.900962\pi$
$877$		−	18.1323i		−	0.612283i	0.951986	−	0.306142i	$-0.0990381\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	5.46099			0.183986			0.0919928	−	0.995760i	$-0.470676\pi$
$881$	5.46099			0.183986			0.0919928	+	0.995760i	$0.470676\pi$
$882$		0			0
$883$		−	7.37316i		−	0.248127i	−0.992274	−	0.124063i	$-0.960407\pi$
$883$		−	7.37316i		−	0.248127i	0.992274	−	0.124063i	$-0.0395926\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	0.364198i		−	0.0122286i	−0.999981	−	0.00611429i	$-0.998054\pi$
$887$		−	0.364198i		−	0.0122286i	0.999981	−	0.00611429i	$-0.00194625\pi$
$888$		0			0
$889$	−4.84331			−0.162439
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	41.9930i		−	1.40524i
$894$		0			0
$895$	−1.58312	+	1.16736i	−0.0529179	+	0.0390204i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−11.6474			−0.388463
$900$		0			0
$901$	20.3450			0.677792
$902$		0			0
$903$		0			0
$904$		0			0
$905$	7.72508	−	5.69629i	0.256790	−	0.189351i
$906$		0			0
$907$			24.3106i			0.807219i	0.914931	+	0.403609i	$0.132245\pi$
$907$			24.3106i			0.807219i	−0.914931	+	0.403609i	$0.867755\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−17.7464			−0.587964			−0.293982	−	0.955811i	$-0.594981\pi$
$911$	−17.7464			−0.587964			−0.293982	+	0.955811i	$0.594981\pi$
$912$		0			0
$913$		−	21.0653i		−	0.697160i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	1.48783i		−	0.0491326i
$918$		0			0
$919$	2.35677			0.0777428			0.0388714	−	0.999244i	$-0.487624\pi$
$919$	2.35677			0.0777428			0.0388714	+	0.999244i	$0.487624\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	14.3801i		−	0.473328i
$924$		0			0
$925$	−40.3321	−	12.4785i	−1.32611	−	0.410289i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	35.3906			1.16113			0.580564	−	0.814215i	$-0.302832\pi$
$929$	35.3906			1.16113			0.580564	+	0.814215i	$0.302832\pi$
$930$		0			0
$931$	−34.4118			−1.12780
$932$		0			0
$933$		0			0
$934$		0			0
$935$	11.7979	+	15.9999i	0.385833	+	0.523252i
$936$		0			0
$937$		−	2.87779i		−	0.0940133i	−0.998895	−	0.0470067i	$-0.985032\pi$
$937$		−	2.87779i		−	0.0940133i	0.998895	−	0.0470067i	$-0.0149682\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−33.4825			−1.09150			−0.545750	−	0.837948i	$-0.683755\pi$
$941$	−33.4825			−1.09150			−0.545750	+	0.837948i	$0.683755\pi$
$942$		0			0
$943$			5.25142i			0.171010i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	9.66496i		−	0.314069i	−0.987593	−	0.157035i	$-0.949807\pi$
$947$		−	9.66496i		−	0.314069i	0.987593	−	0.157035i	$-0.0501934\pi$
$948$		0			0
$949$	11.2234			0.364328
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	21.4627i		−	0.695245i	−0.937635	−	0.347623i	$-0.886989\pi$
$953$		−	21.4627i		−	0.695245i	0.937635	−	0.347623i	$-0.113011\pi$
$954$		0			0
$955$	−26.2371	+	19.3466i	−0.849013	+	0.626041i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−0.531757			−0.0171713
$960$		0			0
$961$	−17.6727			−0.570086
$962$		0			0
$963$		0			0
$964$		0			0
$965$	28.8346	+	39.1043i	0.928218	+	1.25881i
$966$		0			0
$967$		−	10.6289i		−	0.341803i	−0.985288	−	0.170902i	$-0.945332\pi$
$967$		−	10.6289i		−	0.341803i	0.985288	−	0.170902i	$-0.0546680\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	36.6717			1.17685			0.588425	−	0.808552i	$-0.299748\pi$
$971$	36.6717			1.17685			0.588425	+	0.808552i	$0.299748\pi$
$972$		0			0
$973$		−	0.537644i		−	0.0172361i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			33.6532i			1.07666i	0.842734	+	0.538331i	$0.180945\pi$
$977$			33.6532i			1.07666i	−0.842734	+	0.538331i	$0.819055\pi$
$978$		0			0
$979$	1.43849			0.0459743
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	36.3596i		−	1.15969i	−0.814727	−	0.579845i	$-0.803113\pi$
$983$		−	36.3596i		−	1.15969i	0.814727	−	0.579845i	$-0.196887\pi$
$984$		0			0
$985$	−7.98260	−	10.8257i	−0.254347	−	0.344935i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−0.269348			−0.00856477
$990$		0			0
$991$	−9.75294			−0.309812			−0.154906	−	0.987929i	$-0.549508\pi$
$991$	−9.75294			−0.309812			−0.154906	+	0.987929i	$0.549508\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−14.2341	+	10.4959i	−0.451250	+	0.332741i
$996$		0			0
$997$			1.13205i			0.0358523i	0.999839	+	0.0179261i	$0.00570637\pi$
$997$			1.13205i			0.0358523i	−0.999839	+	0.0179261i	$0.994294\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.f.d.829.6	yes	24
3.2	odd	2	inner	4140.2.f.d.829.19	yes	24
5.4	even	2	inner	4140.2.f.d.829.5	✓	24
15.14	odd	2	inner	4140.2.f.d.829.20	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.f.d.829.5	✓	24	5.4	even	2	inner
4140.2.f.d.829.6	yes	24	1.1	even	1	trivial
4140.2.f.d.829.19	yes	24	3.2	odd	2	inner
4140.2.f.d.829.20	yes	24	15.14	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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.79970 + 1.32706i) q^{5} -0.476731i q^{7} +O(q^{10})\)	\(q+(-1.79970 + 1.32706i) q^{5} -0.476731i q^{7} -3.39788 q^{11} +2.28658i q^{13} +2.61642i q^{17} -5.08093 q^{19} -1.00000i q^{23} +(1.47785 - 4.77661i) q^{25} -3.19050 q^{29} +3.65066 q^{31} +(0.632648 + 0.857973i) q^{35} -8.44368i q^{37} -5.25142 q^{41} -0.269348i q^{43} +8.26483i q^{47} +6.77273 q^{49} -7.77590i q^{53} +(6.11517 - 4.50918i) q^{55} +8.35367 q^{59} +5.45789 q^{61} +(-3.03442 - 4.11517i) q^{65} +6.83523i q^{67} -6.28891 q^{71} -4.90839i q^{73} +1.61987i q^{77} +9.29907 q^{79} +6.19955i q^{83} +(-3.47214 - 4.70878i) q^{85} -0.423349 q^{89} +1.09008 q^{91} +(9.14415 - 6.74268i) q^{95} -8.67355i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 8 q^{19} - 8 q^{25} - 12 q^{31} - 28 q^{49} - 16 q^{55} - 16 q^{61} + 8 q^{79} + 12 q^{85} - 16 q^{91}+O(q^{100}) \) 24 * q + 8 * q^19 - 8 * q^25 - 12 * q^31 - 28 * q^49 - 16 * q^55 - 16 * q^61 + 8 * q^79 + 12 * q^85 - 16 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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