LMFDB - Embedded newform 4140.2.s.b.737.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(737,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 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("4140.737");

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.s (of order $4$, degree $2$, 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:	$44$
Relative dimension:	$22$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			737.15
Character	$\chi$	$=$	4140.737
Dual form			4140.2.s.b.2393.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.07726 + 1.95947i) q^{5} +(2.36049 + 2.36049i) q^{7} +1.43372i q^{11} +(-4.27764 + 4.27764i) q^{13} +(-0.739991 + 0.739991i) q^{17} -3.16505i q^{19} +(-0.707107 - 0.707107i) q^{23} +(-2.67901 + 4.22172i) q^{25} +8.01732 q^{29} +7.15041 q^{31} +(-2.08243 + 7.16817i) q^{35} +(-6.14782 - 6.14782i) q^{37} +10.2995i q^{41} +(-1.42388 + 1.42388i) q^{43} +(-5.93386 + 5.93386i) q^{47} +4.14383i q^{49} +(4.27375 + 4.27375i) q^{53} +(-2.80933 + 1.54450i) q^{55} -11.8478 q^{59} +10.6358 q^{61} +(-12.9900 - 3.77375i) q^{65} +(-2.37607 - 2.37607i) q^{67} -2.71584i q^{71} +(-9.82434 + 9.82434i) q^{73} +(-3.38429 + 3.38429i) q^{77} -10.7612i q^{79} +(5.86966 + 5.86966i) q^{83} +(-2.24715 - 0.652822i) q^{85} -5.70441 q^{89} -20.1947 q^{91} +(6.20180 - 3.40959i) q^{95} +(-9.11436 - 9.11436i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 4 q^{7} - 4 q^{13} - 24 q^{25} + 32 q^{31} + 40 q^{37} - 8 q^{43} - 24 q^{55} + 64 q^{61} + 12 q^{67} - 84 q^{73} - 104 q^{85} - 48 q^{91} + 44 q^{97}+O(q^{100}) $ 44 * q - 4 * q^7 - 4 * q^13 - 24 * q^25 + 32 * q^31 + 40 * q^37 - 8 * q^43 - 24 * q^55 + 64 * q^61 + 12 * q^67 - 84 * q^73 - 104 * q^85 - 48 * q^91 + 44 * q^97

Character values

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

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$	$1$	$1$

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
$5$	1.07726	+	1.95947i	0.481767	+	0.876300i
$5$	1.07726	+	1.95947i	0.481767	+	0.876300i
$6$		0			0
$7$	2.36049	+	2.36049i	0.892182	+	0.892182i	0.994728	−	0.102547i	$-0.0326991\pi$
$7$	2.36049	+	2.36049i	0.892182	+	0.892182i	−0.102547	+	0.994728i	$0.532699\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			1.43372i			0.432284i	0.976362	+	0.216142i	$0.0693474\pi$
$11$			1.43372i			0.432284i	−0.976362	+	0.216142i	$0.930653\pi$
$12$		0			0
$13$	−4.27764	+	4.27764i	−1.18640	+	1.18640i	−0.208350	+	0.978054i	$0.566809\pi$
$13$	−4.27764	+	4.27764i	−1.18640	+	1.18640i	−0.978054	+	0.208350i	$0.933191\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.739991	+	0.739991i	−0.179474	+	0.179474i	−0.791127	−	0.611652i	$-0.790505\pi$
$17$	−0.739991	+	0.739991i	−0.179474	+	0.179474i	0.611652	+	0.791127i	$0.290505\pi$
$18$		0			0
$19$		−	3.16505i		−	0.726112i	−0.931767	−	0.363056i	$-0.881733\pi$
$19$		−	3.16505i		−	0.726112i	0.931767	−	0.363056i	$-0.118267\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.707107	−	0.707107i	−0.147442	−	0.147442i
$23$	−0.707107	−	0.707107i	−0.147442	−	0.147442i
$24$		0			0
$25$	−2.67901	+	4.22172i	−0.535802	+	0.844344i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	8.01732			1.48878			0.744389	−	0.667746i	$-0.232741\pi$
$29$	8.01732			1.48878			0.744389	+	0.667746i	$0.232741\pi$
$30$		0			0
$31$	7.15041			1.28425			0.642126	−	0.766599i	$-0.278053\pi$
$31$	7.15041			1.28425			0.642126	+	0.766599i	$0.278053\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.08243	+	7.16817i	−0.351995	+	1.21164i
$36$		0			0
$37$	−6.14782	−	6.14782i	−1.01070	−	1.01070i	−0.999942	−	0.0107528i	$-0.996577\pi$
$37$	−6.14782	−	6.14782i	−1.01070	−	1.01070i	−0.0107528	−	0.999942i	$-0.503423\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			10.2995i			1.60851i	0.594283	+	0.804256i	$0.297436\pi$
$41$			10.2995i			1.60851i	−0.594283	+	0.804256i	$0.702564\pi$
$42$		0			0
$43$	−1.42388	+	1.42388i	−0.217140	+	0.217140i	−0.807292	−	0.590152i	$-0.799068\pi$
$43$	−1.42388	+	1.42388i	−0.217140	+	0.217140i	0.590152	+	0.807292i	$0.299068\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.93386	+	5.93386i	−0.865542	+	0.865542i	−0.991975	−	0.126433i	$-0.959647\pi$
$47$	−5.93386	+	5.93386i	−0.865542	+	0.865542i	0.126433	+	0.991975i	$0.459647\pi$
$48$		0			0
$49$			4.14383i			0.591976i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.27375	+	4.27375i	0.587045	+	0.587045i	0.936830	−	0.349785i	$-0.113745\pi$
$53$	4.27375	+	4.27375i	0.587045	+	0.587045i	−0.349785	+	0.936830i	$0.613745\pi$
$54$		0			0
$55$	−2.80933	+	1.54450i	−0.378810	+	0.208260i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−11.8478			−1.54245			−0.771226	−	0.636562i	$-0.780356\pi$
$59$	−11.8478			−1.54245			−0.771226	+	0.636562i	$0.780356\pi$
$60$		0			0
$61$	10.6358			1.36177			0.680886	−	0.732390i	$-0.261595\pi$
$61$	10.6358			1.36177			0.680886	+	0.732390i	$0.261595\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−12.9900	−	3.77375i	−1.61122	−	0.468076i
$66$		0			0
$67$	−2.37607	−	2.37607i	−0.290284	−	0.290284i	0.546909	−	0.837192i	$-0.315804\pi$
$67$	−2.37607	−	2.37607i	−0.290284	−	0.290284i	−0.837192	+	0.546909i	$0.815804\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	2.71584i		−	0.322311i	−0.986929	−	0.161155i	$-0.948478\pi$
$71$		−	2.71584i		−	0.322311i	0.986929	−	0.161155i	$-0.0515220\pi$
$72$		0			0
$73$	−9.82434	+	9.82434i	−1.14985	+	1.14985i	−0.163270	+	0.986581i	$0.552204\pi$
$73$	−9.82434	+	9.82434i	−1.14985	+	1.14985i	−0.986581	+	0.163270i	$0.947796\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.38429	+	3.38429i	−0.385676	+	0.385676i
$78$		0			0
$79$		−	10.7612i		−	1.21073i	−0.795948	−	0.605365i	$-0.793027\pi$
$79$		−	10.7612i		−	1.21073i	0.795948	−	0.605365i	$-0.206973\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	5.86966	+	5.86966i	0.644279	+	0.644279i	0.951605	−	0.307325i	$-0.0994339\pi$
$83$	5.86966	+	5.86966i	0.644279	+	0.644279i	−0.307325	+	0.951605i	$0.599434\pi$
$84$		0			0
$85$	−2.24715	−	0.652822i	−0.243738	−	0.0708085i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5.70441			−0.604666			−0.302333	−	0.953202i	$-0.597765\pi$
$89$	−5.70441			−0.604666			−0.302333	+	0.953202i	$0.597765\pi$
$90$		0			0
$91$	−20.1947			−2.11698
$92$		0			0
$93$		0			0
$94$		0			0
$95$	6.20180	−	3.40959i	0.636291	−	0.349816i
$96$		0			0
$97$	−9.11436	−	9.11436i	−0.925423	−	0.925423i	0.0719825	−	0.997406i	$-0.477067\pi$
$97$	−9.11436	−	9.11436i	−0.925423	−	0.925423i	−0.997406	+	0.0719825i	$0.977067\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			15.3613i			1.52851i	0.644914	+	0.764255i	$0.276893\pi$
$101$			15.3613i			1.52851i	−0.644914	+	0.764255i	$0.723107\pi$
$102$		0			0
$103$	5.37979	−	5.37979i	0.530087	−	0.530087i	−0.390511	−	0.920598i	$-0.627702\pi$
$103$	5.37979	−	5.37979i	0.530087	−	0.530087i	0.920598	+	0.390511i	$0.127702\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	11.5770	−	11.5770i	1.11919	−	1.11919i	0.127329	−	0.991861i	$-0.459360\pi$
$107$	11.5770	−	11.5770i	1.11919	−	1.11919i	0.991861	−	0.127329i	$-0.0406404\pi$
$108$		0			0
$109$		−	4.96184i		−	0.475259i	−0.971356	−	0.237629i	$-0.923630\pi$
$109$		−	4.96184i		−	0.475259i	0.971356	−	0.237629i	$-0.0763703\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−7.02913	−	7.02913i	−0.661245	−	0.661245i	0.294428	−	0.955674i	$-0.404871\pi$
$113$	−7.02913	−	7.02913i	−0.661245	−	0.661245i	−0.955674	+	0.294428i	$0.904871\pi$
$114$		0			0
$115$	0.623811	−	2.14729i	0.0581707	−	0.200236i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3.49348			−0.320247
$120$		0			0
$121$	8.94444			0.813131
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−11.1583	−	0.701523i	−0.998030	−	0.0627461i
$126$		0			0
$127$	−11.2616	−	11.2616i	−0.999308	−	0.999308i	0.000691828	−	1.00000i	$-0.499780\pi$
$127$	−11.2616	−	11.2616i	−0.999308	−	0.999308i	−1.00000		0.000691828i	$0.999780\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			15.0880i			1.31825i	0.752034	+	0.659124i	$0.229073\pi$
$131$			15.0880i			1.31825i	−0.752034	+	0.659124i	$0.770927\pi$
$132$		0			0
$133$	7.47107	−	7.47107i	0.647824	−	0.647824i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	13.6557	−	13.6557i	1.16668	−	1.16668i	0.183702	−	0.982982i	$-0.441192\pi$
$137$	13.6557	−	13.6557i	1.16668	−	1.16668i	0.982982	−	0.183702i	$-0.0588082\pi$
$138$		0			0
$139$			17.7226i			1.50321i	0.659615	+	0.751604i	$0.270719\pi$
$139$			17.7226i			1.50321i	−0.659615	+	0.751604i	$0.729281\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−6.13296	−	6.13296i	−0.512864	−	0.512864i
$144$		0			0
$145$	8.63676	+	15.7097i	0.717244	+	1.30462i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.47737			0.448723			0.224362	−	0.974506i	$-0.427970\pi$
$149$	5.47737			0.448723			0.224362	+	0.974506i	$0.427970\pi$
$150$		0			0
$151$	19.5207			1.58857			0.794285	−	0.607546i	$-0.207846\pi$
$151$	19.5207			1.58857			0.794285	+	0.607546i	$0.207846\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	7.70287	+	14.0110i	0.618710	+	1.12539i
$156$		0			0
$157$	−8.60289	−	8.60289i	−0.686586	−	0.686586i	0.274890	−	0.961476i	$-0.411359\pi$
$157$	−8.60289	−	8.60289i	−0.686586	−	0.686586i	−0.961476	+	0.274890i	$0.911359\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	3.33824i		−	0.263090i
$162$		0			0
$163$	7.10226	−	7.10226i	0.556292	−	0.556292i	−0.371958	−	0.928250i	$-0.621313\pi$
$163$	7.10226	−	7.10226i	0.556292	−	0.556292i	0.928250	+	0.371958i	$0.121313\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−9.27820	+	9.27820i	−0.717969	+	0.717969i	−0.968189	−	0.250220i	$-0.919497\pi$
$167$	−9.27820	+	9.27820i	−0.717969	+	0.717969i	0.250220	+	0.968189i	$0.419497\pi$
$168$		0			0
$169$		−	23.5965i		−	1.81511i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−10.5243	−	10.5243i	−0.800146	−	0.800146i	0.182972	−	0.983118i	$-0.441428\pi$
$173$	−10.5243	−	10.5243i	−0.800146	−	0.800146i	−0.983118	+	0.182972i	$0.941428\pi$
$174$		0			0
$175$	−16.2891	+	3.64155i	−1.23134	+	0.275276i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	9.28105			0.693698			0.346849	−	0.937921i	$-0.387252\pi$
$179$	9.28105			0.693698			0.346849	+	0.937921i	$0.387252\pi$
$180$		0			0
$181$	−18.6502			−1.38626			−0.693129	−	0.720814i	$-0.743768\pi$
$181$	−18.6502			−1.38626			−0.693129	+	0.720814i	$0.743768\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	5.42362	−	18.6693i	0.398752	−	1.37259i
$186$		0			0
$187$	−1.06094	−	1.06094i	−0.0775838	−	0.0775838i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			21.3782i			1.54687i	0.633875	+	0.773436i	$0.281464\pi$
$191$			21.3782i			1.54687i	−0.633875	+	0.773436i	$0.718536\pi$
$192$		0			0
$193$	0.487819	−	0.487819i	0.0351140	−	0.0351140i	−0.689332	−	0.724446i	$-0.742096\pi$
$193$	0.487819	−	0.487819i	0.0351140	−	0.0351140i	0.724446	+	0.689332i	$0.242096\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−13.8271	+	13.8271i	−0.985139	+	0.985139i	−0.999891	−	0.0147517i	$-0.995304\pi$
$197$	−13.8271	+	13.8271i	−0.985139	+	0.985139i	0.0147517	+	0.999891i	$0.495304\pi$
$198$		0			0
$199$		−	4.28566i		−	0.303803i	−0.988396	−	0.151901i	$-0.951460\pi$
$199$		−	4.28566i		−	0.303803i	0.988396	−	0.151901i	$-0.0485396\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	18.9248	+	18.9248i	1.32826	+	1.32826i
$204$		0			0
$205$	−20.1815	+	11.0953i	−1.40954	+	0.774927i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.53780			0.313886
$210$		0			0
$211$	−20.5870			−1.41727			−0.708635	−	0.705575i	$-0.750689\pi$
$211$	−20.5870			−1.41727			−0.708635	+	0.705575i	$0.750689\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−4.32394	−	1.25615i	−0.294890	−	0.0856688i
$216$		0			0
$217$	16.8785	+	16.8785i	1.14579	+	1.14579i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	6.33084i		−	0.425858i
$222$		0			0
$223$	−12.6753	+	12.6753i	−0.848804	+	0.848804i	−0.989984	−	0.141180i	$-0.954910\pi$
$223$	−12.6753	+	12.6753i	−0.848804	+	0.848804i	0.141180	+	0.989984i	$0.454910\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−5.36572	+	5.36572i	−0.356135	+	0.356135i	−0.862386	−	0.506251i	$-0.831031\pi$
$227$	−5.36572	+	5.36572i	−0.356135	+	0.356135i	0.506251	+	0.862386i	$0.331031\pi$
$228$		0			0
$229$		−	7.36542i		−	0.486721i	−0.969936	−	0.243360i	$-0.921750\pi$
$229$		−	7.36542i		−	0.486721i	0.969936	−	0.243360i	$-0.0782498\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	4.29910	+	4.29910i	0.281644	+	0.281644i	0.833764	−	0.552121i	$-0.186181\pi$
$233$	4.29910	+	4.29910i	0.281644	+	0.281644i	−0.552121	+	0.833764i	$0.686181\pi$
$234$		0			0
$235$	−18.0195	−	5.23486i	−1.17546	−	0.341485i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	6.87398			0.444641			0.222320	−	0.974974i	$-0.428637\pi$
$239$	6.87398			0.444641			0.222320	+	0.974974i	$0.428637\pi$
$240$		0			0
$241$	−9.52062			−0.613277			−0.306638	−	0.951826i	$-0.599204\pi$
$241$	−9.52062			−0.613277			−0.306638	+	0.951826i	$0.599204\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−8.11970	+	4.46400i	−0.518748	+	0.285194i
$246$		0			0
$247$	13.5389	+	13.5389i	0.861462	+	0.861462i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			4.30367i			0.271645i	0.990733	+	0.135822i	$0.0433677\pi$
$251$			4.30367i			0.271645i	−0.990733	+	0.135822i	$0.956632\pi$
$252$		0			0
$253$	1.01380	−	1.01380i	0.0637368	−	0.0637368i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	7.15928	−	7.15928i	0.446584	−	0.446584i	−0.447634	−	0.894217i	$-0.647733\pi$
$257$	7.15928	−	7.15928i	0.446584	−	0.446584i	0.894217	+	0.447634i	$0.147733\pi$
$258$		0			0
$259$		−	29.0237i		−	1.80345i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	19.1505	+	19.1505i	1.18087	+	1.18087i	0.979519	+	0.201354i	$0.0645342\pi$
$263$	19.1505	+	19.1505i	1.18087	+	1.18087i	0.201354	+	0.979519i	$0.435466\pi$
$264$		0			0
$265$	−3.77031	+	12.9782i	−0.231609	+	0.797246i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−13.4817			−0.821993			−0.410997	−	0.911637i	$-0.634819\pi$
$269$	−13.4817			−0.821993			−0.410997	+	0.911637i	$0.634819\pi$
$270$		0			0
$271$	1.05275			0.0639498			0.0319749	−	0.999489i	$-0.489820\pi$
$271$	1.05275			0.0639498			0.0319749	+	0.999489i	$0.489820\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−6.05278	−	3.84096i	−0.364996	−	0.231618i
$276$		0			0
$277$	−9.67191	−	9.67191i	−0.581129	−	0.581129i	0.354085	−	0.935213i	$-0.384792\pi$
$277$	−9.67191	−	9.67191i	−0.581129	−	0.581129i	−0.935213	+	0.354085i	$0.884792\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			11.3868i			0.679278i	0.940556	+	0.339639i	$0.110305\pi$
$281$			11.3868i			0.679278i	−0.940556	+	0.339639i	$0.889695\pi$
$282$		0			0
$283$	−9.97976	+	9.97976i	−0.593235	+	0.593235i	−0.938504	−	0.345269i	$-0.887788\pi$
$283$	−9.97976	+	9.97976i	−0.593235	+	0.593235i	0.345269	+	0.938504i	$0.387788\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−24.3119	+	24.3119i	−1.43508	+	1.43508i
$288$		0			0
$289$			15.9048i			0.935578i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	4.79618	+	4.79618i	0.280196	+	0.280196i	0.833187	−	0.552991i	$-0.186514\pi$
$293$	4.79618	+	4.79618i	0.280196	+	0.280196i	−0.552991	+	0.833187i	$0.686514\pi$
$294$		0			0
$295$	−12.7632	−	23.2153i	−0.743102	−	1.35165i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.04950			0.349852
$300$		0			0
$301$	−6.72211			−0.387456
$302$		0			0
$303$		0			0
$304$		0			0
$305$	11.4575	+	20.8404i	0.656056	+	1.19332i
$306$		0			0
$307$	8.60169	+	8.60169i	0.490925	+	0.490925i	0.908597	−	0.417673i	$-0.137154\pi$
$307$	8.60169	+	8.60169i	0.490925	+	0.490925i	−0.417673	+	0.908597i	$0.637154\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	26.7964i		−	1.51949i	−0.650223	−	0.759743i	$-0.725325\pi$
$311$		−	26.7964i		−	1.51949i	0.650223	−	0.759743i	$-0.274675\pi$
$312$		0			0
$313$	21.9766	−	21.9766i	1.24219	−	1.24219i	0.283103	−	0.959089i	$-0.408636\pi$
$313$	21.9766	−	21.9766i	1.24219	−	1.24219i	0.959089	−	0.283103i	$-0.0913639\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.273414	+	0.273414i	−0.0153565	+	0.0153565i	−0.714743	−	0.699387i	$-0.753456\pi$
$317$	−0.273414	+	0.273414i	−0.0153565	+	0.0153565i	0.699387	+	0.714743i	$0.253456\pi$
$318$		0			0
$319$			11.4946i			0.643575i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2.34211	+	2.34211i	0.130318	+	0.130318i
$324$		0			0
$325$	−6.59916	−	29.5188i	−0.366056	−	1.63741i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−28.0136			−1.54444
$330$		0			0
$331$	6.82665			0.375227			0.187613	−	0.982243i	$-0.439925\pi$
$331$	6.82665			0.375227			0.187613	+	0.982243i	$0.439925\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2.09618	−	7.21549i	0.114526	−	0.394224i
$336$		0			0
$337$	−8.47778	−	8.47778i	−0.461814	−	0.461814i	0.437436	−	0.899250i	$-0.355887\pi$
$337$	−8.47778	−	8.47778i	−0.461814	−	0.461814i	−0.899250	+	0.437436i	$0.855887\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			10.2517i			0.555161i
$342$		0			0
$343$	6.74195	−	6.74195i	0.364031	−	0.364031i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−20.2564	+	20.2564i	−1.08742	+	1.08742i	−0.0916254	+	0.995794i	$0.529206\pi$
$347$	−20.2564	+	20.2564i	−1.08742	+	1.08742i	−0.995794	+	0.0916254i	$0.970794\pi$
$348$		0			0
$349$		−	25.8771i		−	1.38517i	−0.721336	−	0.692585i	$-0.756472\pi$
$349$		−	25.8771i		−	1.38517i	0.721336	−	0.692585i	$-0.243528\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2.52342	+	2.52342i	0.134308	+	0.134308i	0.771065	−	0.636757i	$-0.219724\pi$
$353$	2.52342	+	2.52342i	0.134308	+	0.134308i	−0.636757	+	0.771065i	$0.719724\pi$
$354$		0			0
$355$	5.32159	−	2.92567i	0.282441	−	0.155278i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	19.5358			1.03106			0.515530	−	0.856872i	$-0.327595\pi$
$359$	19.5358			1.03106			0.515530	+	0.856872i	$0.327595\pi$
$360$		0			0
$361$	8.98247			0.472762
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−29.8338	−	8.66705i	−1.56157	−	0.453654i
$366$		0			0
$367$	10.4560	+	10.4560i	0.545800	+	0.545800i	0.925223	−	0.379424i	$-0.123878\pi$
$367$	10.4560	+	10.4560i	0.545800	+	0.545800i	−0.379424	+	0.925223i	$0.623878\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			20.1763i			1.04750i
$372$		0			0
$373$	−26.7247	+	26.7247i	−1.38375	+	1.38375i	−0.545911	+	0.837843i	$0.683817\pi$
$373$	−26.7247	+	26.7247i	−1.38375	+	1.38375i	−0.837843	+	0.545911i	$0.816183\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−34.2952	+	34.2952i	−1.76629	+	1.76629i
$378$		0			0
$379$			25.5576i			1.31281i	0.754410	+	0.656404i	$0.227923\pi$
$379$			25.5576i			1.31281i	−0.754410	+	0.656404i	$0.772077\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	5.77385	+	5.77385i	0.295030	+	0.295030i	0.839063	−	0.544034i	$-0.183104\pi$
$383$	5.77385	+	5.77385i	0.295030	+	0.295030i	−0.544034	+	0.839063i	$0.683104\pi$
$384$		0			0
$385$	−10.2772	−	2.98563i	−0.523773	−	0.152162i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2.37661			0.120499			0.0602495	−	0.998183i	$-0.480810\pi$
$389$	2.37661			0.120499			0.0602495	+	0.998183i	$0.480810\pi$
$390$		0			0
$391$	1.04651			0.0529241
$392$		0			0
$393$		0			0
$394$		0			0
$395$	21.0862	−	11.5926i	1.06096	−	0.583289i
$396$		0			0
$397$	−3.01481	−	3.01481i	−0.151309	−	0.151309i	0.627393	−	0.778702i	$-0.284122\pi$
$397$	−3.01481	−	3.01481i	−0.151309	−	0.151309i	−0.778702	+	0.627393i	$0.784122\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			10.6096i			0.529816i	0.964274	+	0.264908i	$0.0853416\pi$
$401$			10.6096i			0.529816i	−0.964274	+	0.264908i	$0.914658\pi$
$402$		0			0
$403$	−30.5869	+	30.5869i	−1.52364	+	1.52364i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	8.81427	−	8.81427i	0.436907	−	0.436907i
$408$		0			0
$409$			1.68918i			0.0835243i	0.999128	+	0.0417622i	$0.0132972\pi$
$409$			1.68918i			0.0835243i	−0.999128	+	0.0417622i	$0.986703\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−27.9666	−	27.9666i	−1.37615	−	1.37615i
$414$		0			0
$415$	−5.17823	+	17.8246i	−0.254189	+	0.874974i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	37.4445			1.82928			0.914641	−	0.404266i	$-0.132473\pi$
$419$	37.4445			1.82928			0.914641	+	0.404266i	$0.132473\pi$
$420$		0			0
$421$	23.9493			1.16722			0.583609	−	0.812035i	$-0.301640\pi$
$421$	23.9493			1.16722			0.583609	+	0.812035i	$0.301640\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.14159	−	5.10648i	−0.0553754	−	0.247701i
$426$		0			0
$427$	25.1057	+	25.1057i	1.21495	+	1.21495i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	8.53602i		−	0.411166i	−0.978640	−	0.205583i	$-0.934091\pi$
$431$		−	8.53602i		−	0.411166i	0.978640	−	0.205583i	$-0.0659090\pi$
$432$		0			0
$433$	−4.04078	+	4.04078i	−0.194187	+	0.194187i	−0.797503	−	0.603315i	$-0.793846\pi$
$433$	−4.04078	+	4.04078i	−0.194187	+	0.194187i	0.603315	+	0.797503i	$0.293846\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2.23803	+	2.23803i	−0.107059	+	0.107059i
$438$		0			0
$439$			17.6291i			0.841389i	0.907202	+	0.420695i	$0.138214\pi$
$439$			17.6291i			0.841389i	−0.907202	+	0.420695i	$0.861786\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	8.60747	+	8.60747i	0.408953	+	0.408953i	0.881373	−	0.472420i	$-0.156620\pi$
$443$	8.60747	+	8.60747i	0.408953	+	0.408953i	−0.472420	+	0.881373i	$0.656620\pi$
$444$		0			0
$445$	−6.14515	−	11.1776i	−0.291308	−	0.529868i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	9.16266			0.432413			0.216206	−	0.976348i	$-0.430632\pi$
$449$	9.16266			0.432413			0.216206	+	0.976348i	$0.430632\pi$
$450$		0			0
$451$	−14.7666			−0.695334
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−21.7550	−	39.5708i	−1.01989	−	1.85511i
$456$		0			0
$457$	22.3611	+	22.3611i	1.04601	+	1.04601i	0.998889	+	0.0471209i	$0.0150046\pi$
$457$	22.3611	+	22.3611i	1.04601	+	1.04601i	0.0471209	+	0.998889i	$0.484995\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			33.8833i			1.57810i	0.614328	+	0.789051i	$0.289427\pi$
$461$			33.8833i			1.57810i	−0.614328	+	0.789051i	$0.710573\pi$
$462$		0			0
$463$	28.7582	−	28.7582i	1.33651	−	1.33651i	0.437090	−	0.899418i	$-0.356009\pi$
$463$	28.7582	−	28.7582i	1.33651	−	1.33651i	0.899418	−	0.437090i	$-0.143991\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	7.54232	−	7.54232i	0.349017	−	0.349017i	−0.510726	−	0.859743i	$-0.670624\pi$
$467$	7.54232	−	7.54232i	0.349017	−	0.349017i	0.859743	+	0.510726i	$0.170624\pi$
$468$		0			0
$469$		−	11.2174i		−	0.517971i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2.04145	−	2.04145i	−0.0938660	−	0.0938660i
$474$		0			0
$475$	13.3619	+	8.47919i	0.613088	+	0.389052i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	42.6425			1.94839			0.974193	−	0.225716i	$-0.0724721\pi$
$479$	42.6425			1.94839			0.974193	+	0.225716i	$0.0724721\pi$
$480$		0			0
$481$	52.5963			2.39819
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8.04071	−	27.6778i	0.365110	−	1.25679i
$486$		0			0
$487$	16.7925	+	16.7925i	0.760939	+	0.760939i	0.976492	−	0.215553i	$-0.0691554\pi$
$487$	16.7925	+	16.7925i	0.760939	+	0.760939i	−0.215553	+	0.976492i	$0.569155\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	15.6111i		−	0.704519i	−0.935902	−	0.352259i	$-0.885413\pi$
$491$		−	15.6111i		−	0.704519i	0.935902	−	0.352259i	$-0.114587\pi$
$492$		0			0
$493$	−5.93274	+	5.93274i	−0.267197	+	0.267197i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	6.41071	−	6.41071i	0.287560	−	0.287560i
$498$		0			0
$499$		−	28.5955i		−	1.28011i	−0.768328	−	0.640056i	$-0.778911\pi$
$499$		−	28.5955i		−	1.28011i	0.768328	−	0.640056i	$-0.221089\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	13.9948	+	13.9948i	0.623999	+	0.623999i	0.946552	−	0.322552i	$-0.104541\pi$
$503$	13.9948	+	13.9948i	0.623999	+	0.623999i	−0.322552	+	0.946552i	$0.604541\pi$
$504$		0			0
$505$	−30.1000	+	16.5482i	−1.33943	+	0.736385i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1.22752			0.0544087			0.0272043	−	0.999630i	$-0.491340\pi$
$509$	1.22752			0.0544087			0.0272043	+	0.999630i	$0.491340\pi$
$510$		0			0
$511$	−46.3805			−2.05175
$512$		0			0
$513$		0			0
$514$		0			0
$515$	16.3370	+	4.74607i	0.719893	+	0.209137i
$516$		0			0
$517$	−8.50751	−	8.50751i	−0.374160	−	0.374160i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	9.59240i		−	0.420251i	−0.977674	−	0.210125i	$-0.932613\pi$
$521$		−	9.59240i		−	0.420251i	0.977674	−	0.210125i	$-0.0673872\pi$
$522$		0			0
$523$	−5.36681	+	5.36681i	−0.234674	+	0.234674i	−0.814641	−	0.579966i	$-0.803066\pi$
$523$	−5.36681	+	5.36681i	−0.234674	+	0.234674i	0.579966	+	0.814641i	$0.303066\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−5.29124	+	5.29124i	−0.230490	+	0.230490i
$528$		0			0
$529$			1.00000i			0.0434783i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−44.0576	−	44.0576i	−1.90835	−	1.90835i
$534$		0			0
$535$	35.1562	+	10.2132i	1.51993	+	0.441557i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−5.94111			−0.255902
$540$		0			0
$541$	43.4522			1.86815			0.934077	−	0.357071i	$-0.116225\pi$
$541$	43.4522			1.86815			0.934077	+	0.357071i	$0.116225\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	9.72256	−	5.34521i	0.416469	−	0.228964i
$546$		0			0
$547$	26.5763	+	26.5763i	1.13632	+	1.13632i	0.989104	+	0.147216i	$0.0470313\pi$
$547$	26.5763	+	26.5763i	1.13632	+	1.13632i	0.147216	+	0.989104i	$0.452969\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	25.3752i		−	1.08102i
$552$		0			0
$553$	25.4017	−	25.4017i	1.08019	−	1.08019i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2.59700	+	2.59700i	−0.110038	+	0.110038i	−0.759982	−	0.649944i	$-0.774792\pi$
$557$	−2.59700	+	2.59700i	−0.110038	+	0.110038i	0.649944	+	0.759982i	$0.274792\pi$
$558$		0			0
$559$		−	12.1817i		−	0.515231i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−8.12684	−	8.12684i	−0.342506	−	0.342506i	0.514803	−	0.857309i	$-0.327865\pi$
$563$	−8.12684	−	8.12684i	−0.342506	−	0.342506i	−0.857309	+	0.514803i	$0.827865\pi$
$564$		0			0
$565$	6.20112	−	21.3456i	0.260883	−	0.898015i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−32.4711			−1.36126			−0.680630	−	0.732627i	$-0.738294\pi$
$569$	−32.4711			−1.36126			−0.680630	+	0.732627i	$0.738294\pi$
$570$		0			0
$571$	31.3156			1.31052			0.655258	−	0.755405i	$-0.272560\pi$
$571$	31.3156			1.31052			0.655258	+	0.755405i	$0.272560\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.87955	−	1.09086i	0.203491	−	0.0454920i
$576$		0			0
$577$	15.0525	+	15.0525i	0.626645	+	0.626645i	0.947222	−	0.320578i	$-0.103877\pi$
$577$	15.0525	+	15.0525i	0.626645	+	0.626645i	−0.320578	+	0.947222i	$0.603877\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			27.7106i			1.14963i
$582$		0			0
$583$	−6.12738	+	6.12738i	−0.253770	+	0.253770i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	5.09099	−	5.09099i	0.210128	−	0.210128i	−0.594194	−	0.804322i	$-0.702529\pi$
$587$	5.09099	−	5.09099i	0.210128	−	0.210128i	0.804322	+	0.594194i	$0.202529\pi$
$588$		0			0
$589$		−	22.6314i		−	0.932510i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−20.1780	−	20.1780i	−0.828611	−	0.828611i	0.158714	−	0.987325i	$-0.449265\pi$
$593$	−20.1780	−	20.1780i	−0.828611	−	0.828611i	−0.987325	+	0.158714i	$0.949265\pi$
$594$		0			0
$595$	−3.76340	−	6.84536i	−0.154284	−	0.280632i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−6.99628			−0.285860			−0.142930	−	0.989733i	$-0.545652\pi$
$599$	−6.99628			−0.285860			−0.142930	+	0.989733i	$0.545652\pi$
$600$		0			0
$601$	3.66796			0.149619			0.0748096	−	0.997198i	$-0.476165\pi$
$601$	3.66796			0.149619			0.0748096	+	0.997198i	$0.476165\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	9.63551	+	17.5263i	0.391739	+	0.712546i
$606$		0			0
$607$	−0.438034	−	0.438034i	−0.0177793	−	0.0177793i	0.698161	−	0.715941i	$-0.254002\pi$
$607$	−0.438034	−	0.438034i	−0.0177793	−	0.0177793i	−0.715941	+	0.698161i	$0.754002\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	50.7658i		−	2.05377i
$612$		0			0
$613$	−3.34350	+	3.34350i	−0.135043	+	0.135043i	−0.771397	−	0.636354i	$-0.780442\pi$
$613$	−3.34350	+	3.34350i	−0.135043	+	0.135043i	0.636354	+	0.771397i	$0.280442\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	8.22022	−	8.22022i	0.330934	−	0.330934i	−0.522007	−	0.852941i	$-0.674817\pi$
$617$	8.22022	−	8.22022i	0.330934	−	0.330934i	0.852941	+	0.522007i	$0.174817\pi$
$618$		0			0
$619$			17.5378i			0.704904i	0.935830	+	0.352452i	$0.114652\pi$
$619$			17.5378i			0.704904i	−0.935830	+	0.352452i	$0.885348\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−13.4652	−	13.4652i	−0.539472	−	0.539472i
$624$		0			0
$625$	−10.6458	−	22.6200i	−0.425833	−	0.904802i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	9.09866			0.362787
$630$		0			0
$631$	1.04232			0.0414943			0.0207471	−	0.999785i	$-0.493396\pi$
$631$	1.04232			0.0414943			0.0207471	+	0.999785i	$0.493396\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	9.93503	−	34.1985i	0.394260	−	1.35713i
$636$		0			0
$637$	−17.7258	−	17.7258i	−0.702323	−	0.702323i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			21.4308i			0.846465i	0.906021	+	0.423232i	$0.139105\pi$
$641$			21.4308i			0.846465i	−0.906021	+	0.423232i	$0.860895\pi$
$642$		0			0
$643$	−7.38419	+	7.38419i	−0.291204	+	0.291204i	−0.837556	−	0.546352i	$-0.816016\pi$
$643$	−7.38419	+	7.38419i	−0.291204	+	0.291204i	0.546352	+	0.837556i	$0.316016\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	0.475908	−	0.475908i	0.0187099	−	0.0187099i	−0.697690	−	0.716400i	$-0.745789\pi$
$647$	0.475908	−	0.475908i	0.0187099	−	0.0187099i	0.716400	+	0.697690i	$0.245789\pi$
$648$		0			0
$649$		−	16.9865i		−	0.666777i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	18.8215	+	18.8215i	0.736543	+	0.736543i	0.971907	−	0.235364i	$-0.0756283\pi$
$653$	18.8215	+	18.8215i	0.736543	+	0.736543i	−0.235364	+	0.971907i	$0.575628\pi$
$654$		0			0
$655$	−29.5645	+	16.2538i	−1.15518	+	0.635088i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1.08299			0.0421874			0.0210937	−	0.999778i	$-0.493285\pi$
$659$	1.08299			0.0421874			0.0210937	+	0.999778i	$0.493285\pi$
$660$		0			0
$661$	−34.8950			−1.35726			−0.678628	−	0.734482i	$-0.737425\pi$
$661$	−34.8950			−1.35726			−0.678628	+	0.734482i	$0.737425\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	22.6876	+	6.59099i	0.879787	+	0.255588i
$666$		0			0
$667$	−5.66910	−	5.66910i	−0.219508	−	0.219508i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			15.2488i			0.588672i
$672$		0			0
$673$	31.2082	−	31.2082i	1.20299	−	1.20299i	0.229734	−	0.973254i	$-0.426215\pi$
$673$	31.2082	−	31.2082i	1.20299	−	1.20299i	0.973254	−	0.229734i	$-0.0737855\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−3.73333	+	3.73333i	−0.143484	+	0.143484i	−0.775200	−	0.631716i	$-0.782351\pi$
$677$	−3.73333	+	3.73333i	−0.143484	+	0.143484i	0.631716	+	0.775200i	$0.282351\pi$
$678$		0			0
$679$		−	43.0287i		−	1.65129i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−26.1671	−	26.1671i	−1.00126	−	1.00126i	−0.999999	−	0.00125685i	$-0.999600\pi$
$683$	−26.1671	−	26.1671i	−1.00126	−	1.00126i	−0.00125685	−	0.999999i	$-0.500400\pi$
$684$		0			0
$685$	41.4686	+	12.0471i	1.58443	+	0.460295i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−36.5632			−1.39295
$690$		0			0
$691$	−44.3255			−1.68622			−0.843111	−	0.537739i	$-0.819279\pi$
$691$	−44.3255			−1.68622			−0.843111	+	0.537739i	$0.819279\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−34.7267	+	19.0919i	−1.31726	+	0.724195i
$696$		0			0
$697$	−7.62154	−	7.62154i	−0.288686	−	0.288686i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	20.8805i		−	0.788647i	−0.918972	−	0.394323i	$-0.870979\pi$
$701$		−	20.8805i		−	0.788647i	0.918972	−	0.394323i	$-0.129021\pi$
$702$		0			0
$703$	−19.4581	+	19.4581i	−0.733878	+	0.733878i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−36.2603	+	36.2603i	−1.36371	+	1.36371i
$708$		0			0
$709$		−	13.5595i		−	0.509239i	−0.967041	−	0.254619i	$-0.918050\pi$
$709$		−	13.5595i		−	0.509239i	0.967041	−	0.254619i	$-0.0819502\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−5.05610	−	5.05610i	−0.189353	−	0.189353i
$714$		0			0
$715$	5.41051	−	18.6241i	0.202342	−	0.696503i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	19.2048			0.716217			0.358108	−	0.933680i	$-0.383422\pi$
$719$	19.2048			0.716217			0.358108	+	0.933680i	$0.383422\pi$
$720$		0			0
$721$	25.3979			0.945867
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−21.4785	+	33.8469i	−0.797690	+	1.25704i
$726$		0			0
$727$	−0.0313047	−	0.0313047i	−0.00116103	−	0.00116103i	0.706526	−	0.707687i	$-0.250261\pi$
$727$	−0.0313047	−	0.0313047i	−0.00116103	−	0.00116103i	−0.707687	+	0.706526i	$0.750261\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	2.10732i		−	0.0779420i
$732$		0			0
$733$	−29.8085	+	29.8085i	−1.10100	+	1.10100i	−0.106713	+	0.994290i	$0.534033\pi$
$733$	−29.8085	+	29.8085i	−1.10100	+	1.10100i	−0.994290	+	0.106713i	$0.965967\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3.40663	−	3.40663i	0.125485	−	0.125485i
$738$		0			0
$739$			2.55333i			0.0939256i	0.998897	+	0.0469628i	$0.0149542\pi$
$739$			2.55333i			0.0939256i	−0.998897	+	0.0469628i	$0.985046\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−23.5941	−	23.5941i	−0.865583	−	0.865583i	0.126397	−	0.991980i	$-0.459659\pi$
$743$	−23.5941	−	23.5941i	−0.865583	−	0.865583i	−0.991980	+	0.126397i	$0.959659\pi$
$744$		0			0
$745$	5.90056	+	10.7327i	0.216180	+	0.393216i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	54.6547			1.99704
$750$		0			0
$751$	14.4024			0.525553			0.262776	−	0.964857i	$-0.415362\pi$
$751$	14.4024			0.525553			0.262776	+	0.964857i	$0.415362\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	21.0289	+	38.2501i	0.765320	+	1.39206i
$756$		0			0
$757$	11.5333	+	11.5333i	0.419185	+	0.419185i	0.884923	−	0.465738i	$-0.154211\pi$
$757$	11.5333	+	11.5333i	0.419185	+	0.419185i	−0.465738	+	0.884923i	$0.654211\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	45.1691i		−	1.63738i	−0.574237	−	0.818689i	$-0.694701\pi$
$761$		−	45.1691i		−	1.63738i	0.574237	−	0.818689i	$-0.305299\pi$
$762$		0			0
$763$	11.7124	−	11.7124i	0.424017	−	0.424017i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	50.6806	−	50.6806i	1.82997	−	1.82997i
$768$		0			0
$769$		−	37.6458i		−	1.35754i	−0.734350	−	0.678771i	$-0.762513\pi$
$769$		−	37.6458i		−	1.35754i	0.734350	−	0.678771i	$-0.237487\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−14.4598	−	14.4598i	−0.520082	−	0.520082i	0.397514	−	0.917596i	$-0.369873\pi$
$773$	−14.4598	−	14.4598i	−0.520082	−	0.520082i	−0.917596	+	0.397514i	$0.869873\pi$
$774$		0			0
$775$	−19.1560	+	30.1870i	−0.688104	+	1.08435i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	32.5984			1.16796
$780$		0			0
$781$	3.89376			0.139330
$782$		0			0
$783$		0			0
$784$		0			0
$785$	7.58949	−	26.1247i	0.270881	−	0.932429i
$786$		0			0
$787$	31.6312	+	31.6312i	1.12753	+	1.12753i	0.990578	+	0.136953i	$0.0437309\pi$
$787$	31.6312	+	31.6312i	1.12753	+	1.12753i	0.136953	+	0.990578i	$0.456269\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	33.1844i		−	1.17990i
$792$		0			0
$793$	−45.4961	+	45.4961i	−1.61561	+	1.61561i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	19.2875	−	19.2875i	0.683199	−	0.683199i	−0.277521	−	0.960720i	$-0.589513\pi$
$797$	19.2875	−	19.2875i	0.683199	−	0.683199i	0.960720	+	0.277521i	$0.0895128\pi$
$798$		0			0
$799$		−	8.78200i		−	0.310685i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−14.0854	−	14.0854i	−0.497062	−	0.497062i
$804$		0			0
$805$	6.54116	−	3.59616i	0.230546	−	0.126748i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	16.2075			0.569825			0.284913	−	0.958553i	$-0.408035\pi$
$809$	16.2075			0.569825			0.284913	+	0.958553i	$0.408035\pi$
$810$		0			0
$811$	26.6360			0.935318			0.467659	−	0.883909i	$-0.345098\pi$
$811$	26.6360			0.935318			0.467659	+	0.883909i	$0.345098\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	21.5676	+	6.26563i	0.755481	+	0.219475i
$816$		0			0
$817$	4.50665	+	4.50665i	0.157668	+	0.157668i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	12.3930i		−	0.432517i	−0.976336	−	0.216259i	$-0.930615\pi$
$821$		−	12.3930i		−	0.432517i	0.976336	−	0.216259i	$-0.0693854\pi$
$822$		0			0
$823$	−1.33177	+	1.33177i	−0.0464227	+	0.0464227i	−0.729937	−	0.683514i	$-0.760451\pi$
$823$	−1.33177	+	1.33177i	−0.0464227	+	0.0464227i	0.683514	+	0.729937i	$0.260451\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	9.44923	−	9.44923i	0.328582	−	0.328582i	−0.523465	−	0.852047i	$-0.675361\pi$
$827$	9.44923	−	9.44923i	0.328582	−	0.328582i	0.852047	+	0.523465i	$0.175361\pi$
$828$		0			0
$829$		−	21.7068i		−	0.753910i	−0.926232	−	0.376955i	$-0.876971\pi$
$829$		−	21.7068i		−	0.753910i	0.926232	−	0.376955i	$-0.123029\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−3.06640	−	3.06640i	−0.106244	−	0.106244i
$834$		0			0
$835$	−28.1754	−	8.18525i	−0.975049	−	0.283262i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	2.65951			0.0918166			0.0459083	−	0.998946i	$-0.485382\pi$
$839$	2.65951			0.0918166			0.0459083	+	0.998946i	$0.485382\pi$
$840$		0			0
$841$	35.2774			1.21646
$842$		0			0
$843$		0			0
$844$		0			0
$845$	46.2364	−	25.4196i	1.59058	−	0.874460i
$846$		0			0
$847$	21.1133	+	21.1133i	0.725460	+	0.725460i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			8.69433i			0.298038i
$852$		0			0
$853$	−25.9284	+	25.9284i	−0.887773	+	0.887773i	−0.994309	−	0.106536i	$-0.966024\pi$
$853$	−25.9284	+	25.9284i	−0.887773	+	0.887773i	0.106536	+	0.994309i	$0.466024\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	25.3097	−	25.3097i	0.864562	−	0.864562i	−0.127302	−	0.991864i	$-0.540632\pi$
$857$	25.3097	−	25.3097i	0.864562	−	0.864562i	0.991864	+	0.127302i	$0.0406317\pi$
$858$		0			0
$859$			5.26398i			0.179605i	0.995960	+	0.0898024i	$0.0286235\pi$
$859$			5.26398i			0.179605i	−0.995960	+	0.0898024i	$0.971376\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	8.41629	+	8.41629i	0.286494	+	0.286494i	0.835692	−	0.549198i	$-0.185067\pi$
$863$	8.41629	+	8.41629i	0.286494	+	0.286494i	−0.549198	+	0.835692i	$0.685067\pi$
$864$		0			0
$865$	9.28454	−	31.9594i	0.315684	−	1.08665i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	15.4286			0.523379
$870$		0			0
$871$	20.3280			0.688788
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−24.6831	−	27.9950i	−0.834443	−	0.946405i
$876$		0			0
$877$	−7.62691	−	7.62691i	−0.257542	−	0.257542i	0.566511	−	0.824054i	$-0.308293\pi$
$877$	−7.62691	−	7.62691i	−0.257542	−	0.257542i	−0.824054	+	0.566511i	$0.808293\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			11.0296i			0.371598i	0.982588	+	0.185799i	$0.0594873\pi$
$881$			11.0296i			0.371598i	−0.982588	+	0.185799i	$0.940513\pi$
$882$		0			0
$883$	−34.0579	+	34.0579i	−1.14614	+	1.14614i	−0.158835	+	0.987305i	$0.550774\pi$
$883$	−34.0579	+	34.0579i	−1.14614	+	1.14614i	−0.987305	+	0.158835i	$0.949226\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−7.44680	+	7.44680i	−0.250039	+	0.250039i	−0.820987	−	0.570948i	$-0.806576\pi$
$887$	−7.44680	+	7.44680i	−0.250039	+	0.250039i	0.570948	+	0.820987i	$0.306576\pi$
$888$		0			0
$889$		−	53.1659i		−	1.78313i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	18.7809	+	18.7809i	0.628480	+	0.628480i
$894$		0			0
$895$	9.99813	+	18.1859i	0.334201	+	0.607887i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	57.3271			1.91197
$900$		0			0
$901$	−6.32508			−0.210719
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−20.0912	−	36.5444i	−0.667852	−	1.21478i
$906$		0			0
$907$	−6.91758	−	6.91758i	−0.229694	−	0.229694i	0.582871	−	0.812565i	$-0.301929\pi$
$907$	−6.91758	−	6.91758i	−0.229694	−	0.229694i	−0.812565	+	0.582871i	$0.801929\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			40.8344i			1.35290i	0.736487	+	0.676452i	$0.236483\pi$
$911$			40.8344i			1.35290i	−0.736487	+	0.676452i	$0.763517\pi$
$912$		0			0
$913$	−8.41548	+	8.41548i	−0.278512	+	0.278512i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−35.6152	+	35.6152i	−1.17612	+	1.17612i
$918$		0			0
$919$			1.62017i			0.0534445i	0.999643	+	0.0267223i	$0.00850697\pi$
$919$			1.62017i			0.0534445i	−0.999643	+	0.0267223i	$0.991493\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	11.6174	+	11.6174i	0.382391	+	0.382391i
$924$		0			0
$925$	42.4244	−	9.48430i	1.39491	−	0.311842i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	22.1310			0.726093			0.363047	−	0.931771i	$-0.381737\pi$
$929$	22.1310			0.726093			0.363047	+	0.931771i	$0.381737\pi$
$930$		0			0
$931$	13.1154			0.429841
$932$		0			0
$933$		0			0
$934$		0			0
$935$	0.935966	−	3.22180i	0.0306094	−	0.105364i
$936$		0			0
$937$	22.2383	+	22.2383i	0.726494	+	0.726494i	0.969920	−	0.243425i	$-0.0782711\pi$
$937$	22.2383	+	22.2383i	0.726494	+	0.726494i	−0.243425	+	0.969920i	$0.578271\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	19.2838i		−	0.628633i	−0.949318	−	0.314316i	$-0.898225\pi$
$941$		−	19.2838i		−	0.628633i	0.949318	−	0.314316i	$-0.101775\pi$
$942$		0			0
$943$	7.28285	−	7.28285i	0.237162	−	0.237162i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−0.564137	+	0.564137i	−0.0183320	+	0.0183320i	−0.716213	−	0.697881i	$-0.754126\pi$
$947$	−0.564137	+	0.564137i	−0.0183320	+	0.0183320i	0.697881	+	0.716213i	$0.254126\pi$
$948$		0			0
$949$		−	84.0500i		−	2.72838i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	5.91380	+	5.91380i	0.191567	+	0.191567i	0.796373	−	0.604806i	$-0.206749\pi$
$953$	5.91380	+	5.91380i	0.191567	+	0.191567i	−0.604806	+	0.796373i	$0.706749\pi$
$954$		0			0
$955$	−41.8898	+	23.0299i	−1.35552	+	0.745231i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	64.4683			2.08179
$960$		0			0
$961$	20.1284			0.649302
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.48137	+	0.430355i	0.0476871	+	0.0138536i
$966$		0			0
$967$	3.14135	+	3.14135i	0.101019	+	0.101019i	0.755810	−	0.654791i	$-0.227243\pi$
$967$	3.14135	+	3.14135i	0.101019	+	0.101019i	−0.654791	+	0.755810i	$0.727243\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			50.8627i			1.63226i	0.577867	+	0.816131i	$0.303885\pi$
$971$			50.8627i			1.63226i	−0.577867	+	0.816131i	$0.696115\pi$
$972$		0			0
$973$	−41.8339	+	41.8339i	−1.34113	+	1.34113i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	5.54880	−	5.54880i	0.177522	−	0.177522i	−0.612753	−	0.790275i	$-0.709938\pi$
$977$	5.54880	−	5.54880i	0.177522	−	0.177522i	0.790275	+	0.612753i	$0.209938\pi$
$978$		0			0
$979$		−	8.17854i		−	0.261387i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−7.36966	−	7.36966i	−0.235056	−	0.235056i	0.579743	−	0.814799i	$-0.303153\pi$
$983$	−7.36966	−	7.36966i	−0.235056	−	0.235056i	−0.814799	+	0.579743i	$0.803153\pi$
$984$		0			0
$985$	−41.9891	−	12.1983i	−1.33788	−	0.388670i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2.01367			0.0640310
$990$		0			0
$991$	14.9125			0.473711			0.236855	−	0.971545i	$-0.423883\pi$
$991$	14.9125			0.473711			0.236855	+	0.971545i	$0.423883\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	8.39761	−	4.61679i	0.266222	−	0.146362i
$996$		0			0
$997$	−2.25187	−	2.25187i	−0.0713175	−	0.0713175i	0.670548	−	0.741866i	$-0.266059\pi$
$997$	−2.25187	−	2.25187i	−0.0713175	−	0.0713175i	−0.741866	+	0.670548i	$0.766059\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.s.b.737.15	yes	44
3.2	odd	2	inner	4140.2.s.b.737.8	✓	44
5.3	odd	4	inner	4140.2.s.b.2393.8	yes	44
15.8	even	4	inner	4140.2.s.b.2393.15	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.s.b.737.8	✓	44	3.2	odd	2	inner
4140.2.s.b.737.15	yes	44	1.1	even	1	trivial
4140.2.s.b.2393.8	yes	44	5.3	odd	4	inner
4140.2.s.b.2393.15	yes	44	15.8	even	4	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.s (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.07726 + 1.95947i) q^{5} +(2.36049 + 2.36049i) q^{7} +1.43372i q^{11} +(-4.27764 + 4.27764i) q^{13} +(-0.739991 + 0.739991i) q^{17} -3.16505i q^{19} +(-0.707107 - 0.707107i) q^{23} +(-2.67901 + 4.22172i) q^{25} +8.01732 q^{29} +7.15041 q^{31} +(-2.08243 + 7.16817i) q^{35} +(-6.14782 - 6.14782i) q^{37} +10.2995i q^{41} +(-1.42388 + 1.42388i) q^{43} +(-5.93386 + 5.93386i) q^{47} +4.14383i q^{49} +(4.27375 + 4.27375i) q^{53} +(-2.80933 + 1.54450i) q^{55} -11.8478 q^{59} +10.6358 q^{61} +(-12.9900 - 3.77375i) q^{65} +(-2.37607 - 2.37607i) q^{67} -2.71584i q^{71} +(-9.82434 + 9.82434i) q^{73} +(-3.38429 + 3.38429i) q^{77} -10.7612i q^{79} +(5.86966 + 5.86966i) q^{83} +(-2.24715 - 0.652822i) q^{85} -5.70441 q^{89} -20.1947 q^{91} +(6.20180 - 3.40959i) q^{95} +(-9.11436 - 9.11436i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4 q^{7} - 4 q^{13} - 24 q^{25} + 32 q^{31} + 40 q^{37} - 8 q^{43} - 24 q^{55} + 64 q^{61} + 12 q^{67} - 84 q^{73} - 104 q^{85} - 48 q^{91} + 44 q^{97}+O(q^{100}) \) 44 * q - 4 * q^7 - 4 * q^13 - 24 * q^25 + 32 * q^31 + 40 * q^37 - 8 * q^43 - 24 * q^55 + 64 * q^61 + 12 * q^67 - 84 * q^73 - 104 * q^85 - 48 * q^91 + 44 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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