LMFDB - Embedded newform 4140.2.n.b.2069.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(2069,4140)]

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, 1, 1, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4140.2069");

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.n (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:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2069.11
Character	$\chi$	$=$	4140.2069
Dual form			4140.2.n.b.2069.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.64510 + 1.51448i) q^{5} -2.05693 q^{7} +O(q^{10})$	$q+(-1.64510 + 1.51448i) q^{5} -2.05693 q^{7} +5.60242 q^{11} -2.63195i q^{13} -2.12862i q^{17} +4.87642i q^{19} +(-4.47683 - 1.71989i) q^{23} +(0.412717 - 4.98294i) q^{25} +1.54999i q^{29} -6.05943 q^{31} +(3.38385 - 3.11517i) q^{35} -0.289016 q^{37} +2.93952i q^{41} +12.2419 q^{43} +0.475461 q^{47} -2.76905 q^{49} -11.3863i q^{53} +(-9.21654 + 8.48473i) q^{55} -12.6211i q^{59} +0.760675i q^{61} +(3.98602 + 4.32982i) q^{65} -0.897901 q^{67} -1.57467i q^{71} -3.69924i q^{73} -11.5238 q^{77} +13.5601i q^{79} -6.38796i q^{83} +(3.22375 + 3.50180i) q^{85} +4.36596 q^{89} +5.41372i q^{91} +(-7.38522 - 8.02220i) q^{95} +1.66527 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q + 56 q^{25} + 16 q^{31} - 96 q^{49} - 16 q^{55} - 40 q^{85}+O(q^{100}) $ 32 * q + 56 * q^25 + 16 * q^31 - 96 * q^49 - 16 * q^55 - 40 * 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
$5$	−1.64510	+	1.51448i	−0.735712	+	0.677295i
$5$	−1.64510	+	1.51448i	−0.735712	+	0.677295i
$6$		0			0
$7$	−2.05693			−0.777445			−0.388723	−	0.921355i	$-0.627084\pi$
$7$	−2.05693			−0.777445			−0.388723	+	0.921355i	$0.627084\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	5.60242			1.68919			0.844596	−	0.535404i	$-0.179841\pi$
$11$	5.60242			1.68919			0.844596	+	0.535404i	$0.179841\pi$
$12$		0			0
$13$		−	2.63195i		−	0.729971i	−0.931013	−	0.364985i	$-0.881074\pi$
$13$		−	2.63195i		−	0.729971i	0.931013	−	0.364985i	$-0.118926\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	2.12862i		−	0.516266i	−0.966109	−	0.258133i	$-0.916893\pi$
$17$		−	2.12862i		−	0.516266i	0.966109	−	0.258133i	$-0.0831073\pi$
$18$		0			0
$19$			4.87642i			1.11873i	0.828923	+	0.559363i	$0.188954\pi$
$19$			4.87642i			1.11873i	−0.828923	+	0.559363i	$0.811046\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.47683	−	1.71989i	−0.933483	−	0.358622i
$23$	−4.47683	−	1.71989i	−0.933483	−	0.358622i
$24$		0			0
$25$	0.412717	−	4.98294i	0.0825435	−	0.996587i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			1.54999i			0.287826i	0.989590	+	0.143913i	$0.0459685\pi$
$29$			1.54999i			0.287826i	−0.989590	+	0.143913i	$0.954031\pi$
$30$		0			0
$31$	−6.05943			−1.08831			−0.544153	−	0.838986i	$-0.683149\pi$
$31$	−6.05943			−1.08831			−0.544153	+	0.838986i	$0.683149\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.38385	−	3.11517i	0.571976	−	0.526560i
$36$		0			0
$37$	−0.289016			−0.0475140			−0.0237570	−	0.999718i	$-0.507563\pi$
$37$	−0.289016			−0.0475140			−0.0237570	+	0.999718i	$0.507563\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			2.93952i			0.459076i	0.973300	+	0.229538i	$0.0737216\pi$
$41$			2.93952i			0.459076i	−0.973300	+	0.229538i	$0.926278\pi$
$42$		0			0
$43$	12.2419			1.86687			0.933435	−	0.358747i	$-0.116796\pi$
$43$	12.2419			1.86687			0.933435	+	0.358747i	$0.116796\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.475461			0.0693531			0.0346765	−	0.999399i	$-0.488960\pi$
$47$	0.475461			0.0693531			0.0346765	+	0.999399i	$0.488960\pi$
$48$		0			0
$49$	−2.76905			−0.395579
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	11.3863i		−	1.56402i	−0.623263	−	0.782012i	$-0.714193\pi$
$53$		−	11.3863i		−	1.56402i	0.623263	−	0.782012i	$-0.285807\pi$
$54$		0			0
$55$	−9.21654	+	8.48473i	−1.24276	+	1.14408i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	12.6211i		−	1.64313i	−0.570114	−	0.821566i	$-0.693101\pi$
$59$		−	12.6211i		−	1.64313i	0.570114	−	0.821566i	$-0.306899\pi$
$60$		0			0
$61$			0.760675i			0.0973945i	0.998814	+	0.0486972i	$0.0155069\pi$
$61$			0.760675i			0.0973945i	−0.998814	+	0.0486972i	$0.984493\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.98602	+	4.32982i	0.494406	+	0.537048i
$66$		0			0
$67$	−0.897901			−0.109696			−0.0548480	−	0.998495i	$-0.517467\pi$
$67$	−0.897901			−0.109696			−0.0548480	+	0.998495i	$0.517467\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	1.57467i		−	0.186879i	−0.995625	−	0.0934394i	$-0.970214\pi$
$71$		−	1.57467i		−	0.186879i	0.995625	−	0.0934394i	$-0.0297862\pi$
$72$		0			0
$73$		−	3.69924i		−	0.432963i	−0.976287	−	0.216482i	$-0.930542\pi$
$73$		−	3.69924i		−	0.432963i	0.976287	−	0.216482i	$-0.0694582\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−11.5238			−1.31325
$78$		0			0
$79$			13.5601i			1.52563i	0.646616	+	0.762816i	$0.276184\pi$
$79$			13.5601i			1.52563i	−0.646616	+	0.762816i	$0.723816\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	6.38796i		−	0.701169i	−0.936531	−	0.350585i	$-0.885983\pi$
$83$		−	6.38796i		−	0.701169i	0.936531	−	0.350585i	$-0.114017\pi$
$84$		0			0
$85$	3.22375	+	3.50180i	0.349664	+	0.379823i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	4.36596			0.462791			0.231395	−	0.972860i	$-0.425671\pi$
$89$	4.36596			0.462791			0.231395	+	0.972860i	$0.425671\pi$
$90$		0			0
$91$			5.41372i			0.567513i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−7.38522	−	8.02220i	−0.757708	−	0.823060i
$96$		0			0
$97$	1.66527			0.169082			0.0845411	−	0.996420i	$-0.473058\pi$
$97$	1.66527			0.169082			0.0845411	+	0.996420i	$0.473058\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			3.86430i			0.384512i	0.981345	+	0.192256i	$0.0615804\pi$
$101$			3.86430i			0.384512i	−0.981345	+	0.192256i	$0.938420\pi$
$102$		0			0
$103$	8.03008			0.791228			0.395614	−	0.918417i	$-0.370532\pi$
$103$	8.03008			0.791228			0.395614	+	0.918417i	$0.370532\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	4.24572i		−	0.410449i	−0.978715	−	0.205225i	$-0.934208\pi$
$107$		−	4.24572i		−	0.410449i	0.978715	−	0.205225i	$-0.0657925\pi$
$108$		0			0
$109$			2.50929i			0.240346i	0.992753	+	0.120173i	$0.0383450\pi$
$109$			2.50929i			0.240346i	−0.992753	+	0.120173i	$0.961655\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	9.36615i		−	0.881093i	−0.897730	−	0.440546i	$-0.854785\pi$
$113$		−	9.36615i		−	0.881093i	0.897730	−	0.440546i	$-0.145215\pi$
$114$		0			0
$115$	9.96957	−	3.95066i	0.929667	−	0.368401i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			4.37842i			0.401369i
$120$		0			0
$121$	20.3871			1.85337
$122$		0			0
$123$		0			0
$124$		0			0
$125$	6.86758	+	8.82249i	0.614255	+	0.789107i
$126$		0			0
$127$			13.2702i			1.17754i	0.808300	+	0.588771i	$0.200388\pi$
$127$			13.2702i			1.17754i	−0.808300	+	0.588771i	$0.799612\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	14.7199i		−	1.28609i	−0.765830	−	0.643043i	$-0.777672\pi$
$131$		−	14.7199i		−	1.28609i	0.765830	−	0.643043i	$-0.222328\pi$
$132$		0			0
$133$		−	10.0304i		−	0.869749i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	6.75954i		−	0.577506i	−0.957404	−	0.288753i	$-0.906759\pi$
$137$		−	6.75954i		−	0.577506i	0.957404	−	0.288753i	$-0.0932407\pi$
$138$		0			0
$139$	16.1387			1.36886			0.684432	−	0.729076i	$-0.260050\pi$
$139$	16.1387			1.36886			0.684432	+	0.729076i	$0.260050\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	14.7453i		−	1.23306i
$144$		0			0
$145$	−2.34742	−	2.54989i	−0.194943	−	0.211757i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	8.89262			0.728512			0.364256	−	0.931299i	$-0.381323\pi$
$149$	8.89262			0.728512			0.364256	+	0.931299i	$0.381323\pi$
$150$		0			0
$151$	3.08091			0.250721			0.125361	−	0.992111i	$-0.459991\pi$
$151$	3.08091			0.250721			0.125361	+	0.992111i	$0.459991\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	9.96838	−	9.17687i	0.800679	−	0.737104i
$156$		0			0
$157$	13.7926			1.10077			0.550383	−	0.834912i	$-0.314482\pi$
$157$	13.7926			1.10077			0.550383	+	0.834912i	$0.314482\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	9.20851	+	3.53769i	0.725732	+	0.278809i
$162$		0			0
$163$		−	21.5610i		−	1.68878i	−0.535725	−	0.844392i	$-0.679962\pi$
$163$		−	21.5610i		−	1.68878i	0.535725	−	0.844392i	$-0.320038\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3.52103			0.272466			0.136233	−	0.990677i	$-0.456500\pi$
$167$	3.52103			0.272466			0.136233	+	0.990677i	$0.456500\pi$
$168$		0			0
$169$	6.07285			0.467142
$170$		0			0
$171$		0			0
$172$		0			0
$173$	13.5086			1.02704			0.513520	−	0.858078i	$-0.328341\pi$
$173$	13.5086			1.02704			0.513520	+	0.858078i	$0.328341\pi$
$174$		0			0
$175$	−0.848929	+	10.2495i	−0.0641730	+	0.774792i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			1.22671i			0.0916885i	0.998949	+	0.0458442i	$0.0145978\pi$
$179$			1.22671i			0.0916885i	−0.998949	+	0.0458442i	$0.985402\pi$
$180$		0			0
$181$		−	6.17440i		−	0.458940i	−0.973316	−	0.229470i	$-0.926301\pi$
$181$		−	6.17440i		−	0.458940i	0.973316	−	0.229470i	$-0.0736992\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.475461	−	0.437708i	0.0349566	−	0.0321810i
$186$		0			0
$187$		−	11.9254i		−	0.872073i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	20.4708			1.48121			0.740606	−	0.671939i	$-0.234538\pi$
$191$	20.4708			1.48121			0.740606	+	0.671939i	$0.234538\pi$
$192$		0			0
$193$			3.23979i			0.233206i	0.993179	+	0.116603i	$0.0372004\pi$
$193$			3.23979i			0.233206i	−0.993179	+	0.116603i	$0.962800\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	20.8616			1.48633			0.743163	−	0.669110i	$-0.233325\pi$
$197$	20.8616			1.48633			0.743163	+	0.669110i	$0.233325\pi$
$198$		0			0
$199$			0.984041i			0.0697568i	0.999392	+	0.0348784i	$0.0111044\pi$
$199$			0.984041i			0.0697568i	−0.999392	+	0.0348784i	$0.988896\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	3.18822i		−	0.223769i
$204$		0			0
$205$	−4.45184	−	4.83581i	−0.310930	−	0.337748i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			27.3197i			1.88974i
$210$		0			0
$211$	14.6355			1.00755			0.503774	−	0.863835i	$-0.331944\pi$
$211$	14.6355			1.00755			0.503774	+	0.863835i	$0.331944\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−20.1391	+	18.5401i	−1.37348	+	1.26442i
$216$		0			0
$217$	12.4638			0.846098
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−5.60242			−0.376859
$222$		0			0
$223$		−	11.9900i		−	0.802908i	−0.915879	−	0.401454i	$-0.868505\pi$
$223$		−	11.9900i		−	0.802908i	0.915879	−	0.401454i	$-0.131495\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	0.0784831i		−	0.00520910i	−0.999997	−	0.00260455i	$-0.999171\pi$
$227$		−	0.0784831i		−	0.00520910i	0.999997	−	0.00260455i	$-0.000829056\pi$
$228$		0			0
$229$			12.5761i			0.831050i	0.909582	+	0.415525i	$0.136402\pi$
$229$			12.5761i			0.831050i	−0.909582	+	0.415525i	$0.863598\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	5.34962			0.350465			0.175233	−	0.984527i	$-0.443932\pi$
$233$	5.34962			0.350465			0.175233	+	0.984527i	$0.443932\pi$
$234$		0			0
$235$	−0.782181	+	0.720075i	−0.0510239	+	0.0469725i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	7.33594i		−	0.474523i	−0.971446	−	0.237261i	$-0.923750\pi$
$239$		−	7.33594i		−	0.474523i	0.971446	−	0.237261i	$-0.0762498\pi$
$240$		0			0
$241$			4.03064i			0.259636i	0.991538	+	0.129818i	$0.0414394\pi$
$241$			4.03064i			0.259636i	−0.991538	+	0.129818i	$0.958561\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4.55537	−	4.19366i	0.291032	−	0.267923i
$246$		0			0
$247$	12.8345			0.816638
$248$		0			0
$249$		0			0
$250$		0			0
$251$	12.2806			0.775144			0.387572	−	0.921839i	$-0.373314\pi$
$251$	12.2806			0.775144			0.387572	+	0.921839i	$0.373314\pi$
$252$		0			0
$253$	−25.0811	−	9.63554i	−1.57683	−	0.605781i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−27.2451			−1.69950			−0.849752	−	0.527182i	$-0.823248\pi$
$257$	−27.2451			−1.69950			−0.849752	+	0.527182i	$0.823248\pi$
$258$		0			0
$259$	0.594485			0.0369395
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	20.4501i		−	1.26101i	−0.776187	−	0.630503i	$-0.782849\pi$
$263$		−	20.4501i		−	1.26101i	0.776187	−	0.630503i	$-0.217151\pi$
$264$		0			0
$265$	17.2442	+	18.7316i	1.05931	+	1.15067i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	22.8022i		−	1.39027i	−0.718877	−	0.695137i	$-0.755344\pi$
$269$		−	22.8022i		−	1.39027i	0.718877	−	0.695137i	$-0.244656\pi$
$270$		0			0
$271$	−11.9467			−0.725708			−0.362854	−	0.931846i	$-0.618198\pi$
$271$	−11.9467			−0.725708			−0.362854	+	0.931846i	$0.618198\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.31221	−	27.9165i	0.139432	−	1.68343i
$276$		0			0
$277$			13.3348i			0.801209i	0.916251	+	0.400605i	$0.131200\pi$
$277$			13.3348i			0.801209i	−0.916251	+	0.400605i	$0.868800\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	12.5561			0.749036			0.374518	−	0.927220i	$-0.377808\pi$
$281$	12.5561			0.749036			0.374518	+	0.927220i	$0.377808\pi$
$282$		0			0
$283$	−10.3039			−0.612503			−0.306251	−	0.951951i	$-0.599075\pi$
$283$	−10.3039			−0.612503			−0.306251	+	0.951951i	$0.599075\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	6.04639i		−	0.356907i
$288$		0			0
$289$	12.4690			0.733469
$290$		0			0
$291$		0			0
$292$		0			0
$293$			22.0594i			1.28872i	0.764721	+	0.644361i	$0.222877\pi$
$293$			22.0594i			1.28872i	−0.764721	+	0.644361i	$0.777123\pi$
$294$		0			0
$295$	19.1144	+	20.7630i	1.11288	+	1.20887i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4.52666	+	11.7828i	−0.261784	+	0.681415i
$300$		0			0
$301$	−25.1807			−1.45139
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.15203	−	1.25139i	−0.0659648	−	0.0716543i
$306$		0			0
$307$		−	24.3034i		−	1.38707i	−0.720424	−	0.693534i	$-0.756053\pi$
$307$		−	24.3034i		−	1.38707i	0.720424	−	0.693534i	$-0.243947\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	32.9172i		−	1.86656i	−0.359146	−	0.933282i	$-0.616932\pi$
$311$		−	32.9172i		−	1.86656i	0.359146	−	0.933282i	$-0.383068\pi$
$312$		0			0
$313$	7.24346			0.409425			0.204712	−	0.978822i	$-0.434374\pi$
$313$	7.24346			0.409425			0.204712	+	0.978822i	$0.434374\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−28.1960			−1.58365			−0.791824	−	0.610749i	$-0.790868\pi$
$317$	−28.1960			−1.58365			−0.791824	+	0.610749i	$0.790868\pi$
$318$		0			0
$319$			8.68369i			0.486193i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	10.3800			0.577561
$324$		0			0
$325$	−13.1148	−	1.08625i	−0.727480	−	0.0602543i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−0.977988			−0.0539182
$330$		0			0
$331$	−10.9927			−0.604212			−0.302106	−	0.953274i	$-0.597690\pi$
$331$	−10.9927			−0.604212			−0.302106	+	0.953274i	$0.597690\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.47714	−	1.35985i	0.0807046	−	0.0742965i
$336$		0			0
$337$	22.5345			1.22753			0.613767	−	0.789487i	$-0.289653\pi$
$337$	22.5345			1.22753			0.613767	+	0.789487i	$0.289653\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−33.9475			−1.83836
$342$		0			0
$343$	20.0942			1.08499
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−12.5577			−0.674132			−0.337066	−	0.941481i	$-0.609435\pi$
$347$	−12.5577			−0.674132			−0.337066	+	0.941481i	$0.609435\pi$
$348$		0			0
$349$	−18.5418			−0.992521			−0.496261	−	0.868174i	$-0.665294\pi$
$349$	−18.5418			−0.992521			−0.496261	+	0.868174i	$0.665294\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−27.1538			−1.44525			−0.722626	−	0.691239i	$-0.757065\pi$
$353$	−27.1538			−1.44525			−0.722626	+	0.691239i	$0.757065\pi$
$354$		0			0
$355$	2.38480	+	2.59049i	0.126572	+	0.137489i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	11.3026			0.596529			0.298264	−	0.954483i	$-0.403592\pi$
$359$	11.3026			0.596529			0.298264	+	0.954483i	$0.403592\pi$
$360$		0			0
$361$	−4.77943			−0.251549
$362$		0			0
$363$		0			0
$364$		0			0
$365$	5.60242	+	6.08563i	0.293244	+	0.318536i
$366$		0			0
$367$	−8.13632			−0.424712			−0.212356	−	0.977192i	$-0.568114\pi$
$367$	−8.13632			−0.424712			−0.212356	+	0.977192i	$0.568114\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			23.4207i			1.21594i
$372$		0			0
$373$	−14.4928			−0.750411			−0.375206	−	0.926942i	$-0.622428\pi$
$373$	−14.4928			−0.750411			−0.375206	+	0.926942i	$0.622428\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4.07949			0.210105
$378$		0			0
$379$		−	22.7850i		−	1.17039i	−0.810894	−	0.585193i	$-0.801019\pi$
$379$		−	22.7850i		−	1.17039i	0.810894	−	0.585193i	$-0.198981\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			18.0909i			0.924399i	0.886776	+	0.462200i	$0.152940\pi$
$383$			18.0909i			0.924399i	−0.886776	+	0.462200i	$0.847060\pi$
$384$		0			0
$385$	18.9578	−	17.4525i	0.966177	−	0.889461i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−16.6466			−0.844014			−0.422007	−	0.906593i	$-0.638674\pi$
$389$	−16.6466			−0.844014			−0.422007	+	0.906593i	$0.638674\pi$
$390$		0			0
$391$	−3.66099	+	9.52946i	−0.185144	+	0.481926i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−20.5365	−	22.3077i	−1.03330	−	1.12242i
$396$		0			0
$397$			22.3507i			1.12175i	0.827900	+	0.560876i	$0.189535\pi$
$397$			22.3507i			1.12175i	−0.827900	+	0.560876i	$0.810465\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−17.3987			−0.868848			−0.434424	−	0.900708i	$-0.643048\pi$
$401$	−17.3987			−0.868848			−0.434424	+	0.900708i	$0.643048\pi$
$402$		0			0
$403$			15.9481i			0.794432i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.61919			−0.0802602
$408$		0			0
$409$	−31.4751			−1.55634			−0.778171	−	0.628052i	$-0.783852\pi$
$409$	−31.4751			−1.55634			−0.778171	+	0.628052i	$0.783852\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			25.9608i			1.27745i
$414$		0			0
$415$	9.67441	+	10.5088i	0.474898	+	0.515858i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	23.3877			1.14256			0.571281	−	0.820755i	$-0.306447\pi$
$419$	23.3877			1.14256			0.571281	+	0.820755i	$0.306447\pi$
$420$		0			0
$421$			21.5682i			1.05117i	0.850741	+	0.525585i	$0.176154\pi$
$421$			21.5682i			1.05117i	−0.850741	+	0.525585i	$0.823846\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−10.6068	−	0.878518i	−0.514504	−	0.0426144i
$426$		0			0
$427$		−	1.56465i		−	0.0757189i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	20.3730			0.981332			0.490666	−	0.871348i	$-0.336753\pi$
$431$	20.3730			0.981332			0.490666	+	0.871348i	$0.336753\pi$
$432$		0			0
$433$	−35.3639			−1.69948			−0.849741	−	0.527200i	$-0.823242\pi$
$433$	−35.3639			−1.69948			−0.849741	+	0.527200i	$0.823242\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	8.38690	−	21.8309i	0.401200	−	1.04431i
$438$		0			0
$439$	−17.1219			−0.817184			−0.408592	−	0.912717i	$-0.633980\pi$
$439$	−17.1219			−0.817184			−0.408592	+	0.912717i	$0.633980\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	41.4571			1.96969			0.984843	−	0.173447i	$-0.0554904\pi$
$443$	41.4571			1.96969			0.984843	+	0.173447i	$0.0554904\pi$
$444$		0			0
$445$	−7.18245	+	6.61215i	−0.340481	+	0.313446i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	22.5306i		−	1.06329i	−0.846968	−	0.531643i	$-0.821575\pi$
$449$		−	22.5306i		−	1.06329i	0.846968	−	0.531643i	$-0.178425\pi$
$450$		0			0
$451$			16.4684i			0.775468i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−8.19896	−	8.90613i	−0.384373	−	0.417526i
$456$		0			0
$457$	−23.9241			−1.11912			−0.559561	−	0.828789i	$-0.689030\pi$
$457$	−23.9241			−1.11912			−0.559561	+	0.828789i	$0.689030\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	23.1255i		−	1.07706i	−0.842606	−	0.538530i	$-0.818980\pi$
$461$		−	23.1255i		−	1.07706i	0.842606	−	0.538530i	$-0.181020\pi$
$462$		0			0
$463$		−	31.0295i		−	1.44206i	−0.692903	−	0.721031i	$-0.743669\pi$
$463$		−	31.0295i		−	1.44206i	0.692903	−	0.721031i	$-0.256331\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	6.94185i		−	0.321230i	−0.987017	−	0.160615i	$-0.948652\pi$
$467$		−	6.94185i		−	0.321230i	0.987017	−	0.160615i	$-0.0513478\pi$
$468$		0			0
$469$	1.84692			0.0852827
$470$		0			0
$471$		0			0
$472$		0			0
$473$	68.5841			3.15350
$474$		0			0
$475$	24.2989	+	2.01258i	1.11491	+	0.0923435i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	16.0474			0.733224			0.366612	−	0.930374i	$-0.380518\pi$
$479$	16.0474			0.733224			0.366612	+	0.930374i	$0.380518\pi$
$480$		0			0
$481$			0.760675i			0.0346838i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.73953	+	2.52201i	−0.124396	+	0.114519i
$486$		0			0
$487$			12.5978i			0.570862i	0.958399	+	0.285431i	$0.0921367\pi$
$487$			12.5978i			0.570862i	−0.958399	+	0.285431i	$0.907863\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			23.2389i			1.04876i	0.851485	+	0.524379i	$0.175702\pi$
$491$			23.2389i			1.04876i	−0.851485	+	0.524379i	$0.824298\pi$
$492$		0			0
$493$	3.29934			0.148595
$494$		0			0
$495$		0			0
$496$		0			0
$497$			3.23898i			0.145288i
$498$		0			0
$499$	28.2226			1.26342			0.631709	−	0.775205i	$-0.282354\pi$
$499$	28.2226			1.26342			0.631709	+	0.775205i	$0.282354\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	21.0040i		−	0.936521i	−0.883590	−	0.468261i	$-0.844881\pi$
$503$		−	21.0040i		−	0.936521i	0.883590	−	0.468261i	$-0.155119\pi$
$504$		0			0
$505$	−5.85239	−	6.35717i	−0.260428	−	0.282890i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	17.5527i		−	0.778008i	−0.921236	−	0.389004i	$-0.872819\pi$
$509$		−	17.5527i		−	0.778008i	0.921236	−	0.389004i	$-0.127181\pi$
$510$		0			0
$511$			7.60907i			0.336605i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−13.2103	+	12.1614i	−0.582116	+	0.535894i
$516$		0			0
$517$	2.66373			0.117151
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−6.29161			−0.275640			−0.137820	−	0.990457i	$-0.544010\pi$
$521$	−6.29161			−0.275640			−0.137820	+	0.990457i	$0.544010\pi$
$522$		0			0
$523$	−25.0532			−1.09550			−0.547749	−	0.836643i	$-0.684515\pi$
$523$	−25.0532			−1.09550			−0.547749	+	0.836643i	$0.684515\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			12.8982i			0.561856i
$528$		0			0
$529$	17.0840	+	15.3993i	0.742781	+	0.669535i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	7.73667			0.335112
$534$		0			0
$535$	6.43004	+	6.98464i	0.277995	+	0.301972i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−15.5134			−0.668208
$540$		0			0
$541$	−8.29038			−0.356431			−0.178216	−	0.983991i	$-0.557032\pi$
$541$	−8.29038			−0.356431			−0.178216	+	0.983991i	$0.557032\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3.80026	−	4.12804i	−0.162785	−	0.176826i
$546$		0			0
$547$			27.2129i			1.16354i	0.813354	+	0.581769i	$0.197639\pi$
$547$			27.2129i			1.16354i	−0.813354	+	0.581769i	$0.802361\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−7.55839			−0.321998
$552$		0			0
$553$		−	27.8921i		−	1.18609i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			11.8941i			0.503968i	0.967731	+	0.251984i	$0.0810830\pi$
$557$			11.8941i			0.503968i	−0.967731	+	0.251984i	$0.918917\pi$
$558$		0			0
$559$		−	32.2200i		−	1.36276i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			19.9423i			0.840467i	0.907416	+	0.420234i	$0.138052\pi$
$563$			19.9423i			0.840467i	−0.907416	+	0.420234i	$0.861948\pi$
$564$		0			0
$565$	14.1848	+	15.4083i	0.596760	+	0.648230i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−12.5391			−0.525665			−0.262832	−	0.964842i	$-0.584657\pi$
$569$	−12.5391			−0.525665			−0.262832	+	0.964842i	$0.584657\pi$
$570$		0			0
$571$			8.77427i			0.367192i	0.983002	+	0.183596i	$0.0587738\pi$
$571$			8.77427i			0.367192i	−0.983002	+	0.183596i	$0.941226\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−10.4178	+	21.5979i	−0.434451	+	0.900696i
$576$		0			0
$577$		−	8.21929i		−	0.342173i	−0.985256	−	0.171087i	$-0.945272\pi$
$577$		−	8.21929i		−	0.342173i	0.985256	−	0.171087i	$-0.0547278\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			13.1396i			0.545121i
$582$		0			0
$583$		−	63.7906i		−	2.64194i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	37.1414			1.53299			0.766495	−	0.642250i	$-0.221999\pi$
$587$	37.1414			1.53299			0.766495	+	0.642250i	$0.221999\pi$
$588$		0			0
$589$		−	29.5483i		−	1.21752i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−6.56660			−0.269658			−0.134829	−	0.990869i	$-0.543049\pi$
$593$	−6.56660			−0.269658			−0.134829	+	0.990869i	$0.543049\pi$
$594$		0			0
$595$	−6.63101	−	7.20294i	−0.271845	−	0.295292i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			19.5217i			0.797637i	0.917030	+	0.398818i	$0.130580\pi$
$599$			19.5217i			0.797637i	−0.917030	+	0.398818i	$0.869420\pi$
$600$		0			0
$601$	34.6272			1.41247			0.706236	−	0.707977i	$-0.250392\pi$
$601$	34.6272			1.41247			0.706236	+	0.707977i	$0.250392\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−33.5388	+	30.8758i	−1.36355	+	1.25528i
$606$		0			0
$607$			25.1311i			1.02004i	0.860163	+	0.510020i	$0.170362\pi$
$607$			25.1311i			1.02004i	−0.860163	+	0.510020i	$0.829638\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	1.25139i		−	0.0506257i
$612$		0			0
$613$	−23.0829			−0.932311			−0.466155	−	0.884703i	$-0.654361\pi$
$613$	−23.0829			−0.932311			−0.466155	+	0.884703i	$0.654361\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	21.6465i		−	0.871454i	−0.900079	−	0.435727i	$-0.856491\pi$
$617$		−	21.6465i		−	0.871454i	0.900079	−	0.435727i	$-0.143509\pi$
$618$		0			0
$619$			40.4608i			1.62626i	0.582083	+	0.813129i	$0.302238\pi$
$619$			40.4608i			1.62626i	−0.582083	+	0.813129i	$0.697762\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−8.98046			−0.359795
$624$		0			0
$625$	−24.6593	−	4.11309i	−0.986373	−	0.164524i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			0.615206i			0.0245299i
$630$		0			0
$631$		−	43.0513i		−	1.71385i	−0.515444	−	0.856923i	$-0.672373\pi$
$631$		−	43.0513i		−	1.71385i	0.515444	−	0.856923i	$-0.327627\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−20.0975	−	21.8309i	−0.797543	−	0.866332i
$636$		0			0
$637$			7.28800i			0.288761i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	10.4046			0.410958			0.205479	−	0.978662i	$-0.434125\pi$
$641$	10.4046			0.410958			0.205479	+	0.978662i	$0.434125\pi$
$642$		0			0
$643$	−14.8226			−0.584545			−0.292272	−	0.956335i	$-0.594411\pi$
$643$	−14.8226			−0.584545			−0.292272	+	0.956335i	$0.594411\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−36.1075			−1.41953			−0.709766	−	0.704438i	$-0.751199\pi$
$647$	−36.1075			−1.41953			−0.709766	+	0.704438i	$0.751199\pi$
$648$		0			0
$649$		−	70.7089i		−	2.77557i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	46.9727			1.83818			0.919092	−	0.394042i	$-0.128924\pi$
$653$	46.9727			1.83818			0.919092	+	0.394042i	$0.128924\pi$
$654$		0			0
$655$	22.2930	+	24.2158i	0.871059	+	0.946189i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	26.7353			1.04146			0.520729	−	0.853722i	$-0.325660\pi$
$659$	26.7353			1.04146			0.520729	+	0.853722i	$0.325660\pi$
$660$		0			0
$661$		−	32.6284i		−	1.26910i	−0.772883	−	0.634549i	$-0.781186\pi$
$661$		−	32.6284i		−	1.26910i	0.772883	−	0.634549i	$-0.218814\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	15.1909	+	16.5011i	0.589076	+	0.639884i
$666$		0			0
$667$	2.66581	−	6.93904i	0.103221	−	0.268681i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			4.26162i			0.164518i
$672$		0			0
$673$		−	18.1013i		−	0.697755i	−0.937168	−	0.348877i	$-0.886563\pi$
$673$		−	18.1013i		−	0.697755i	0.937168	−	0.348877i	$-0.113437\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			34.3158i			1.31886i	0.751765	+	0.659431i	$0.229203\pi$
$677$			34.3158i			1.31886i	−0.751765	+	0.659431i	$0.770797\pi$
$678$		0			0
$679$	−3.42533			−0.131452
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−13.6452			−0.522120			−0.261060	−	0.965323i	$-0.584072\pi$
$683$	−13.6452			−0.522120			−0.261060	+	0.965323i	$0.584072\pi$
$684$		0			0
$685$	10.2372	+	11.1201i	0.391142	+	0.424878i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−29.9681			−1.14169
$690$		0			0
$691$	19.7443			0.751107			0.375554	−	0.926801i	$-0.377453\pi$
$691$	19.7443			0.751107			0.375554	+	0.926801i	$0.377453\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−26.5498	+	24.4417i	−1.00709	+	0.927125i
$696$		0			0
$697$	6.25713			0.237006
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−38.8304			−1.46660			−0.733301	−	0.679904i	$-0.762021\pi$
$701$	−38.8304			−1.46660			−0.733301	+	0.679904i	$0.762021\pi$
$702$		0			0
$703$		−	1.40936i		−	0.0531551i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	7.94858i		−	0.298937i
$708$		0			0
$709$		−	7.66614i		−	0.287908i	−0.989584	−	0.143954i	$-0.954018\pi$
$709$		−	7.66614i		−	0.287908i	0.989584	−	0.143954i	$-0.0459817\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	27.1270	+	10.4216i	1.01591	+	0.390290i
$714$		0			0
$715$	22.3314	+	24.2575i	0.835146	+	0.907178i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	10.9591i		−	0.408705i	−0.978897	−	0.204352i	$-0.934491\pi$
$719$		−	10.9591i		−	0.408705i	0.978897	−	0.204352i	$-0.0655089\pi$
$720$		0			0
$721$	−16.5173			−0.615136
$722$		0			0
$723$		0			0
$724$		0			0
$725$	7.72350	+	0.639708i	0.286844	+	0.0237581i
$726$		0			0
$727$	−37.3342			−1.38465			−0.692324	−	0.721587i	$-0.743413\pi$
$727$	−37.3342			−1.38465			−0.692324	+	0.721587i	$0.743413\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	26.0583i		−	0.963802i
$732$		0			0
$733$	6.41798			0.237053			0.118527	−	0.992951i	$-0.462183\pi$
$733$	6.41798			0.237053			0.118527	+	0.992951i	$0.462183\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.03041			−0.185298
$738$		0			0
$739$	−2.36757			−0.0870923			−0.0435462	−	0.999051i	$-0.513866\pi$
$739$	−2.36757			−0.0870923			−0.0435462	+	0.999051i	$0.513866\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	30.2501i		−	1.10977i	−0.831928	−	0.554884i	$-0.812763\pi$
$743$		−	30.2501i		−	1.10977i	0.831928	−	0.554884i	$-0.187237\pi$
$744$		0			0
$745$	−14.6293	+	13.4677i	−0.535975	+	0.493417i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			8.73313i			0.319102i
$750$		0			0
$751$		−	31.4928i		−	1.14919i	−0.818438	−	0.574595i	$-0.805160\pi$
$751$		−	31.4928i		−	1.14919i	0.818438	−	0.574595i	$-0.194840\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−5.06841	+	4.66597i	−0.184458	+	0.169812i
$756$		0			0
$757$	−39.6576			−1.44138			−0.720690	−	0.693257i	$-0.756175\pi$
$757$	−39.6576			−1.44138			−0.720690	+	0.693257i	$0.756175\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			37.3483i			1.35388i	0.736040	+	0.676938i	$0.236693\pi$
$761$			37.3483i			1.35388i	−0.736040	+	0.676938i	$0.763307\pi$
$762$		0			0
$763$		−	5.16143i		−	0.186856i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−33.2182			−1.19944
$768$		0			0
$769$			30.9314i			1.11541i	0.830038	+	0.557707i	$0.188319\pi$
$769$			30.9314i			1.11541i	−0.830038	+	0.557707i	$0.811681\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			8.64582i			0.310969i	0.987838	+	0.155484i	$0.0496938\pi$
$773$			8.64582i			0.310969i	−0.987838	+	0.155484i	$0.950306\pi$
$774$		0			0
$775$	−2.50083	+	30.1938i	−0.0898325	+	1.08459i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−14.3343			−0.513581
$780$		0			0
$781$		−	8.82196i		−	0.315674i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−22.6902	+	20.8885i	−0.809847	+	0.745544i
$786$		0			0
$787$	55.2982			1.97117			0.985584	−	0.169188i	$-0.0541147\pi$
$787$	55.2982			1.97117			0.985584	+	0.169188i	$0.0541147\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			19.2655i			0.685002i
$792$		0			0
$793$	2.00206			0.0710952
$794$		0			0
$795$		0			0
$796$		0			0
$797$			46.1614i			1.63512i	0.575844	+	0.817560i	$0.304674\pi$
$797$			46.1614i			1.63512i	−0.575844	+	0.817560i	$0.695326\pi$
$798$		0			0
$799$		−	1.01208i		−	0.0358047i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	20.7247i		−	0.731359i
$804$		0			0
$805$	−20.5067	+	8.12622i	−0.722765	+	0.286412i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			18.8335i			0.662149i	0.943605	+	0.331074i	$0.107411\pi$
$809$			18.8335i			0.662149i	−0.943605	+	0.331074i	$0.892589\pi$
$810$		0			0
$811$	9.34042			0.327986			0.163993	−	0.986461i	$-0.447562\pi$
$811$	9.34042			0.327986			0.163993	+	0.986461i	$0.447562\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	32.6536	+	35.4700i	1.14381	+	1.24246i
$816$		0			0
$817$			59.6965i			2.08852i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			9.86481i			0.344284i	0.985072	+	0.172142i	$0.0550688\pi$
$821$			9.86481i			0.344284i	−0.985072	+	0.172142i	$0.944931\pi$
$822$		0			0
$823$		−	0.129113i		−	0.00450058i	−0.999997	−	0.00225029i	$-0.999284\pi$
$823$		−	0.129113i		−	0.00450058i	0.999997	−	0.00225029i	$-0.000716291\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			32.5083i			1.13043i	0.824945	+	0.565213i	$0.191206\pi$
$827$			32.5083i			1.13043i	−0.824945	+	0.565213i	$0.808794\pi$
$828$		0			0
$829$	−36.0719			−1.25283			−0.626414	−	0.779491i	$-0.715478\pi$
$829$	−36.0719			−1.25283			−0.626414	+	0.779491i	$0.715478\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			5.89426i			0.204224i
$834$		0			0
$835$	−5.79246	+	5.33252i	−0.200456	+	0.184540i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	34.8106			1.20180			0.600898	−	0.799326i	$-0.294810\pi$
$839$	34.8106			1.20180			0.600898	+	0.799326i	$0.294810\pi$
$840$		0			0
$841$	26.5975			0.917156
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−9.99046	+	9.19720i	−0.343682	+	0.316393i
$846$		0			0
$847$	−41.9347			−1.44089
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1.29388	+	0.497076i	0.0443535	+	0.0170395i
$852$		0			0
$853$		−	46.7683i		−	1.60132i	−0.599121	−	0.800659i	$-0.704483\pi$
$853$		−	46.7683i		−	1.60132i	0.599121	−	0.800659i	$-0.295517\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	12.4664			0.425844			0.212922	−	0.977069i	$-0.431702\pi$
$857$	12.4664			0.425844			0.212922	+	0.977069i	$0.431702\pi$
$858$		0			0
$859$	−21.0578			−0.718482			−0.359241	−	0.933245i	$-0.616964\pi$
$859$	−21.0578			−0.718482			−0.359241	+	0.933245i	$0.616964\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	14.5342			0.494751			0.247375	−	0.968920i	$-0.420432\pi$
$863$	14.5342			0.494751			0.247375	+	0.968920i	$0.420432\pi$
$864$		0			0
$865$	−22.2230	+	20.4585i	−0.755606	+	0.695609i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			75.9694i			2.57708i
$870$		0			0
$871$			2.36323i			0.0800749i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−14.1261	−	18.1472i	−0.477550	−	0.613488i
$876$		0			0
$877$		−	0.543291i		−	0.0183456i	−0.999958	−	0.00917281i	$-0.997080\pi$
$877$		−	0.543291i		−	0.0183456i	0.999958	−	0.00917281i	$-0.00291984\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−26.2416			−0.884103			−0.442051	−	0.896990i	$-0.645749\pi$
$881$	−26.2416			−0.884103			−0.442051	+	0.896990i	$0.645749\pi$
$882$		0			0
$883$			34.0297i			1.14519i	0.819839	+	0.572595i	$0.194063\pi$
$883$			34.0297i			1.14519i	−0.819839	+	0.572595i	$0.805937\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−5.34962			−0.179623			−0.0898114	−	0.995959i	$-0.528626\pi$
$887$	−5.34962			−0.179623			−0.0898114	+	0.995959i	$0.528626\pi$
$888$		0			0
$889$		−	27.2959i		−	0.915475i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			2.31854i			0.0775871i
$894$		0			0
$895$	−1.85782	−	2.01806i	−0.0621001	−	0.0674563i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	9.39206i		−	0.313243i
$900$		0			0
$901$	−24.2370			−0.807453
$902$		0			0
$903$		0			0
$904$		0			0
$905$	9.35099	+	10.1575i	0.310837	+	0.337647i
$906$		0			0
$907$	7.55208			0.250763			0.125381	−	0.992109i	$-0.459985\pi$
$907$	7.55208			0.250763			0.125381	+	0.992109i	$0.459985\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−18.2292			−0.603961			−0.301981	−	0.953314i	$-0.597648\pi$
$911$	−18.2292			−0.603961			−0.301981	+	0.953314i	$0.597648\pi$
$912$		0			0
$913$		−	35.7880i		−	1.18441i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			30.2778i			0.999862i
$918$		0			0
$919$			35.3040i			1.16457i	0.812984	+	0.582286i	$0.197842\pi$
$919$			35.3040i			1.16457i	−0.812984	+	0.582286i	$0.802158\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−4.14445			−0.136416
$924$		0			0
$925$	−0.119282	+	1.44015i	−0.00392197	+	0.0473518i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	45.3812i		−	1.48891i	−0.667673	−	0.744455i	$-0.732710\pi$
$929$		−	45.3812i		−	1.48891i	0.667673	−	0.744455i	$-0.267290\pi$
$930$		0			0
$931$		−	13.5030i		−	0.442544i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	18.0608	+	19.6185i	0.590651	+	0.641594i
$936$		0			0
$937$	−8.47288			−0.276797			−0.138398	−	0.990377i	$-0.544195\pi$
$937$	−8.47288			−0.276797			−0.138398	+	0.990377i	$0.544195\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−42.9378			−1.39973			−0.699867	−	0.714273i	$-0.746757\pi$
$941$	−42.9378			−1.39973			−0.699867	+	0.714273i	$0.746757\pi$
$942$		0			0
$943$	5.05566	−	13.1597i	0.164635	−	0.428540i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	13.5086			0.438971			0.219485	−	0.975616i	$-0.429562\pi$
$947$	13.5086			0.438971			0.219485	+	0.975616i	$0.429562\pi$
$948$		0			0
$949$	−9.73621			−0.316051
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	26.4846i		−	0.857921i	−0.903323	−	0.428961i	$-0.858880\pi$
$953$		−	26.4846i		−	0.857921i	0.903323	−	0.428961i	$-0.141120\pi$
$954$		0			0
$955$	−33.6765	+	31.0025i	−1.08975	+	1.00322i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			13.9039i			0.448980i
$960$		0			0
$961$	5.71670			0.184410
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−4.90660	−	5.32979i	−0.157949	−	0.171572i
$966$		0			0
$967$		−	38.4659i		−	1.23698i	−0.785793	−	0.618489i	$-0.787745\pi$
$967$		−	38.4659i		−	1.23698i	0.785793	−	0.618489i	$-0.212255\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	18.2215			0.584755			0.292377	−	0.956303i	$-0.405554\pi$
$971$	18.2215			0.584755			0.292377	+	0.956303i	$0.405554\pi$
$972$		0			0
$973$	−33.1961			−1.06422
$974$		0			0
$975$		0			0
$976$		0			0
$977$			56.2688i			1.80020i	0.435685	+	0.900099i	$0.356506\pi$
$977$			56.2688i			1.80020i	−0.435685	+	0.900099i	$0.643494\pi$
$978$		0			0
$979$	24.4599			0.781743
$980$		0			0
$981$		0			0
$982$		0			0
$983$			4.59655i			0.146607i	0.997310	+	0.0733036i	$0.0233542\pi$
$983$			4.59655i			0.146607i	−0.997310	+	0.0733036i	$0.976646\pi$
$984$		0			0
$985$	−34.3194	+	31.5944i	−1.09351	+	1.00668i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−54.8048	−	21.0547i	−1.74269	−	0.669500i
$990$		0			0
$991$	53.4238			1.69706			0.848531	−	0.529145i	$-0.177487\pi$
$991$	53.4238			1.69706			0.848531	+	0.529145i	$0.177487\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1.49031	−	1.61885i	−0.0472459	−	0.0513209i
$996$		0			0
$997$			32.3283i			1.02385i	0.859030	+	0.511925i	$0.171067\pi$
$997$			32.3283i			1.02385i	−0.859030	+	0.511925i	$0.828933\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.n.b.2069.11	yes	32
3.2	odd	2	inner	4140.2.n.b.2069.21	yes	32
5.4	even	2	inner	4140.2.n.b.2069.10	yes	32
15.14	odd	2	inner	4140.2.n.b.2069.24	yes	32
23.22	odd	2	inner	4140.2.n.b.2069.22	yes	32
69.68	even	2	inner	4140.2.n.b.2069.12	yes	32
115.114	odd	2	inner	4140.2.n.b.2069.23	yes	32
345.344	even	2	inner	4140.2.n.b.2069.9	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.n.b.2069.9	✓	32	345.344	even	2	inner
4140.2.n.b.2069.10	yes	32	5.4	even	2	inner
4140.2.n.b.2069.11	yes	32	1.1	even	1	trivial
4140.2.n.b.2069.12	yes	32	69.68	even	2	inner
4140.2.n.b.2069.21	yes	32	3.2	odd	2	inner
4140.2.n.b.2069.22	yes	32	23.22	odd	2	inner
4140.2.n.b.2069.23	yes	32	115.114	odd	2	inner
4140.2.n.b.2069.24	yes	32	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.n (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.64510 + 1.51448i) q^{5} -2.05693 q^{7} +O(q^{10})\)	\(q+(-1.64510 + 1.51448i) q^{5} -2.05693 q^{7} +5.60242 q^{11} -2.63195i q^{13} -2.12862i q^{17} +4.87642i q^{19} +(-4.47683 - 1.71989i) q^{23} +(0.412717 - 4.98294i) q^{25} +1.54999i q^{29} -6.05943 q^{31} +(3.38385 - 3.11517i) q^{35} -0.289016 q^{37} +2.93952i q^{41} +12.2419 q^{43} +0.475461 q^{47} -2.76905 q^{49} -11.3863i q^{53} +(-9.21654 + 8.48473i) q^{55} -12.6211i q^{59} +0.760675i q^{61} +(3.98602 + 4.32982i) q^{65} -0.897901 q^{67} -1.57467i q^{71} -3.69924i q^{73} -11.5238 q^{77} +13.5601i q^{79} -6.38796i q^{83} +(3.22375 + 3.50180i) q^{85} +4.36596 q^{89} +5.41372i q^{91} +(-7.38522 - 8.02220i) q^{95} +1.66527 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q + 56 q^{25} + 16 q^{31} - 96 q^{49} - 16 q^{55} - 40 q^{85}+O(q^{100}) \) 32 * q + 56 * q^25 + 16 * q^31 - 96 * q^49 - 16 * q^55 - 40 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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