LMFDB - Embedded newform 4012.2.b.b.237.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(237,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4012.237");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.b (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:	$32.0359812909$
Analytic rank:	$0$
Dimension:	$46$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			237.17
Character	$\chi$	$=$	4012.237
Dual form			4012.2.b.b.237.30

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.38906i q^{3} -2.35139i q^{5} -0.244801i q^{7} +1.07052 q^{9} +O(q^{10})$	$q-1.38906i q^{3} -2.35139i q^{5} -0.244801i q^{7} +1.07052 q^{9} -6.06593i q^{11} +0.379019 q^{13} -3.26622 q^{15} +(3.68677 - 1.84601i) q^{17} +1.83644 q^{19} -0.340043 q^{21} -3.35137i q^{23} -0.529047 q^{25} -5.65419i q^{27} +0.959029i q^{29} +7.69476i q^{31} -8.42593 q^{33} -0.575623 q^{35} -5.21503i q^{37} -0.526480i q^{39} -9.98774i q^{41} +3.23514 q^{43} -2.51721i q^{45} +2.89151 q^{47} +6.94007 q^{49} +(-2.56421 - 5.12114i) q^{51} +9.23452 q^{53} -14.2634 q^{55} -2.55092i q^{57} +1.00000 q^{59} +5.34838i q^{61} -0.262063i q^{63} -0.891223i q^{65} -3.05101 q^{67} -4.65525 q^{69} +12.1190i q^{71} -4.32377i q^{73} +0.734877i q^{75} -1.48494 q^{77} +9.09865i q^{79} -4.64244 q^{81} -14.4197 q^{83} +(-4.34069 - 8.66904i) q^{85} +1.33215 q^{87} +12.1589 q^{89} -0.0927842i q^{91} +10.6885 q^{93} -4.31819i q^{95} +17.9932i q^{97} -6.49368i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q - 54 q^{9}+O(q^{10}) $ 46 * q - 54 * q^9	$ 46 q - 54 q^{9} + 8 q^{13} - 10 q^{15} + q^{17} - 20 q^{19} - 24 q^{21} - 54 q^{25} + 2 q^{33} + 26 q^{35} - 38 q^{43} + 6 q^{47} - 66 q^{49} + 26 q^{51} + 18 q^{53} - 20 q^{55} + 46 q^{59} + 48 q^{67} + 28 q^{69} + 22 q^{77} + 70 q^{81} - 52 q^{83} - 2 q^{85} + 44 q^{87} - 76 q^{89} - 26 q^{93}+O(q^{100}) $ 46 * q - 54 * q^9 + 8 * q^13 - 10 * q^15 + q^17 - 20 * q^19 - 24 * q^21 - 54 * q^25 + 2 * q^33 + 26 * q^35 - 38 * q^43 + 6 * q^47 - 66 * q^49 + 26 * q^51 + 18 * q^53 - 20 * q^55 + 46 * q^59 + 48 * q^67 + 28 * q^69 + 22 * q^77 + 70 * q^81 - 52 * q^83 - 2 * q^85 + 44 * q^87 - 76 * q^89 - 26 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4012\mathbb{Z}\right)^\times$.

$n$	$2007$	$3129$	$3777$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		−	1.38906i		−	0.801973i	−0.916084	−	0.400987i	$-0.868667\pi$
$3$		−	1.38906i		−	0.801973i	0.916084	−	0.400987i	$-0.131333\pi$
$4$		0			0
$5$		−	2.35139i		−	1.05157i	−0.850616	−	0.525787i	$-0.823771\pi$
$5$		−	2.35139i		−	1.05157i	0.850616	−	0.525787i	$-0.176229\pi$
$6$		0			0
$7$		−	0.244801i		−	0.0925260i	−0.998929	−	0.0462630i	$-0.985269\pi$
$7$		−	0.244801i		−	0.0925260i	0.998929	−	0.0462630i	$-0.0147312\pi$
$8$		0			0
$9$	1.07052			0.356839
$10$		0			0
$11$		−	6.06593i		−	1.82895i	−0.404647	−	0.914473i	$-0.632606\pi$
$11$		−	6.06593i		−	1.82895i	0.404647	−	0.914473i	$-0.367394\pi$
$12$		0			0
$13$	0.379019			0.105121			0.0525605	−	0.998618i	$-0.483262\pi$
$13$	0.379019			0.105121			0.0525605	+	0.998618i	$0.483262\pi$
$14$		0			0
$15$	−3.26622			−0.843335
$16$		0			0
$17$	3.68677	−	1.84601i	0.894173	−	0.447722i
$17$	3.68677	−	1.84601i	0.894173	−	0.447722i
$18$		0			0
$19$	1.83644			0.421308			0.210654	−	0.977561i	$-0.432441\pi$
$19$	1.83644			0.421308			0.210654	+	0.977561i	$0.432441\pi$
$20$		0			0
$21$	−0.340043			−0.0742034
$22$		0			0
$23$		−	3.35137i		−	0.698809i	−0.936972	−	0.349404i	$-0.886384\pi$
$23$		−	3.35137i		−	0.698809i	0.936972	−	0.349404i	$-0.113616\pi$
$24$		0			0
$25$	−0.529047			−0.105809
$26$		0			0
$27$		−	5.65419i		−	1.08815i
$28$		0			0
$29$			0.959029i			0.178087i	0.996028	+	0.0890436i	$0.0283810\pi$
$29$			0.959029i			0.178087i	−0.996028	+	0.0890436i	$0.971619\pi$
$30$		0			0
$31$			7.69476i			1.38202i	0.722845	+	0.691010i	$0.242834\pi$
$31$			7.69476i			1.38202i	−0.722845	+	0.691010i	$0.757166\pi$
$32$		0			0
$33$	−8.42593			−1.46677
$34$		0			0
$35$	−0.575623			−0.0972980
$36$		0			0
$37$		−	5.21503i		−	0.857346i	−0.903460	−	0.428673i	$-0.858981\pi$
$37$		−	5.21503i		−	0.857346i	0.903460	−	0.428673i	$-0.141019\pi$
$38$		0			0
$39$		−	0.526480i		−	0.0843042i
$40$		0			0
$41$		−	9.98774i		−	1.55982i	−0.625889	−	0.779912i	$-0.715264\pi$
$41$		−	9.98774i		−	1.55982i	0.625889	−	0.779912i	$-0.284736\pi$
$42$		0			0
$43$	3.23514			0.493354			0.246677	−	0.969098i	$-0.420661\pi$
$43$	3.23514			0.493354			0.246677	+	0.969098i	$0.420661\pi$
$44$		0			0
$45$		−	2.51721i		−	0.375243i
$46$		0			0
$47$	2.89151			0.421770			0.210885	−	0.977511i	$-0.432365\pi$
$47$	2.89151			0.421770			0.210885	+	0.977511i	$0.432365\pi$
$48$		0			0
$49$	6.94007			0.991439
$50$		0			0
$51$	−2.56421	−	5.12114i	−0.359061	−	0.717103i
$52$		0			0
$53$	9.23452			1.26846			0.634229	−	0.773145i	$-0.281317\pi$
$53$	9.23452			1.26846			0.634229	+	0.773145i	$0.281317\pi$
$54$		0			0
$55$	−14.2634			−1.92327
$56$		0			0
$57$		−	2.55092i		−	0.337878i
$58$		0			0
$59$	1.00000			0.130189
$59$	1.00000			0.130189
$60$		0			0
$61$			5.34838i			0.684789i	0.939556	+	0.342395i	$0.111238\pi$
$61$			5.34838i			0.684789i	−0.939556	+	0.342395i	$0.888762\pi$
$62$		0			0
$63$		−	0.262063i		−	0.0330169i
$64$		0			0
$65$		−	0.891223i		−	0.110543i
$66$		0			0
$67$	−3.05101			−0.372741			−0.186370	−	0.982480i	$-0.559672\pi$
$67$	−3.05101			−0.372741			−0.186370	+	0.982480i	$0.559672\pi$
$68$		0			0
$69$	−4.65525			−0.560426
$70$		0			0
$71$			12.1190i			1.43826i	0.694874	+	0.719131i	$0.255460\pi$
$71$			12.1190i			1.43826i	−0.694874	+	0.719131i	$0.744540\pi$
$72$		0			0
$73$		−	4.32377i		−	0.506059i	−0.967459	−	0.253029i	$-0.918573\pi$
$73$		−	4.32377i		−	0.506059i	0.967459	−	0.253029i	$-0.0814269\pi$
$74$		0			0
$75$			0.734877i			0.0848563i
$76$		0			0
$77$	−1.48494			−0.169225
$78$		0			0
$79$			9.09865i			1.02368i	0.859081	+	0.511839i	$0.171036\pi$
$79$			9.09865i			1.02368i	−0.859081	+	0.511839i	$0.828964\pi$
$80$		0			0
$81$	−4.64244			−0.515827
$82$		0			0
$83$	−14.4197			−1.58277			−0.791386	−	0.611317i	$-0.790640\pi$
$83$	−14.4197			−1.58277			−0.791386	+	0.611317i	$0.790640\pi$
$84$		0			0
$85$	−4.34069	−	8.66904i	−0.470813	−	0.940289i
$86$		0			0
$87$	1.33215			0.142821
$88$		0			0
$89$	12.1589			1.28884			0.644418	−	0.764673i	$-0.277100\pi$
$89$	12.1589			1.28884			0.644418	+	0.764673i	$0.277100\pi$
$90$		0			0
$91$		−	0.0927842i		−	0.00972643i
$92$		0			0
$93$	10.6885			1.10834
$94$		0			0
$95$		−	4.31819i		−	0.443037i
$96$		0			0
$97$			17.9932i			1.82694i	0.406910	+	0.913468i	$0.366606\pi$
$97$			17.9932i			1.82694i	−0.406910	+	0.913468i	$0.633394\pi$
$98$		0			0
$99$		−	6.49368i		−	0.652639i
$100$		0			0
$101$	−5.98935			−0.595963			−0.297982	−	0.954572i	$-0.596313\pi$
$101$	−5.98935			−0.595963			−0.297982	+	0.954572i	$0.596313\pi$
$102$		0			0
$103$	6.69831			0.660004			0.330002	−	0.943980i	$-0.392951\pi$
$103$	6.69831			0.660004			0.330002	+	0.943980i	$0.392951\pi$
$104$		0			0
$105$			0.799574i			0.0780304i
$106$		0			0
$107$		−	10.7325i		−	1.03755i	−0.854912	−	0.518773i	$-0.826389\pi$
$107$		−	10.7325i		−	1.03755i	0.854912	−	0.518773i	$-0.173611\pi$
$108$		0			0
$109$			5.19178i			0.497282i	0.968596	+	0.248641i	$0.0799839\pi$
$109$			5.19178i			0.497282i	−0.968596	+	0.248641i	$0.920016\pi$
$110$		0			0
$111$	−7.24399			−0.687569
$112$		0			0
$113$		−	4.91246i		−	0.462125i	−0.972939	−	0.231063i	$-0.925780\pi$
$113$		−	4.91246i		−	0.462125i	0.972939	−	0.231063i	$-0.0742203\pi$
$114$		0			0
$115$	−7.88039			−0.734850
$116$		0			0
$117$	0.405746			0.0375113
$118$		0			0
$119$	−0.451904	−	0.902524i	−0.0414260	−	0.0827343i
$120$		0			0
$121$	−25.7955			−2.34504
$122$		0			0
$123$	−13.8736			−1.25094
$124$		0			0
$125$		−	10.5130i		−	0.940308i
$126$		0			0
$127$	−14.5330			−1.28959			−0.644796	−	0.764355i	$-0.723057\pi$
$127$	−14.5330			−1.28959			−0.644796	+	0.764355i	$0.723057\pi$
$128$		0			0
$129$		−	4.49380i		−	0.395657i
$130$		0			0
$131$		−	7.06962i		−	0.617676i	−0.951115	−	0.308838i	$-0.900060\pi$
$131$		−	7.06962i		−	0.617676i	0.951115	−	0.308838i	$-0.0999400\pi$
$132$		0			0
$133$		−	0.449562i		−	0.0389820i
$134$		0			0
$135$	−13.2952			−1.14427
$136$		0			0
$137$	11.3645			0.970936			0.485468	−	0.874255i	$-0.338649\pi$
$137$	11.3645			0.970936			0.485468	+	0.874255i	$0.338649\pi$
$138$		0			0
$139$		−	3.60281i		−	0.305586i	−0.988258	−	0.152793i	$-0.951173\pi$
$139$		−	3.60281i		−	0.305586i	0.988258	−	0.152793i	$-0.0488268\pi$
$140$		0			0
$141$		−	4.01648i		−	0.338248i
$142$		0			0
$143$		−	2.29910i		−	0.192261i
$144$		0			0
$145$	2.25505			0.187272
$146$		0			0
$147$		−	9.64017i		−	0.795107i
$148$		0			0
$149$	−5.97875			−0.489798			−0.244899	−	0.969549i	$-0.578755\pi$
$149$	−5.97875			−0.489798			−0.244899	+	0.969549i	$0.578755\pi$
$150$		0			0
$151$	10.2843			0.836925			0.418462	−	0.908234i	$-0.362569\pi$
$151$	10.2843			0.836925			0.418462	+	0.908234i	$0.362569\pi$
$152$		0			0
$153$	3.94675	−	1.97618i	0.319076	−	0.159765i
$154$		0			0
$155$	18.0934			1.45330
$156$		0			0
$157$	16.1612			1.28981			0.644903	−	0.764265i	$-0.276898\pi$
$157$	16.1612			1.28981			0.644903	+	0.764265i	$0.276898\pi$
$158$		0			0
$159$		−	12.8273i		−	1.01727i
$160$		0			0
$161$	−0.820418			−0.0646580
$162$		0			0
$163$			17.9742i			1.40785i	0.710274	+	0.703925i	$0.248571\pi$
$163$			17.9742i			1.40785i	−0.710274	+	0.703925i	$0.751429\pi$
$164$		0			0
$165$			19.8127i			1.54241i
$166$		0			0
$167$			18.9921i			1.46965i	0.678256	+	0.734826i	$0.262736\pi$
$167$			18.9921i			1.46965i	−0.678256	+	0.734826i	$0.737264\pi$
$168$		0			0
$169$	−12.8563			−0.988950
$170$		0			0
$171$	1.96594			0.150339
$172$		0			0
$173$			18.4506i			1.40277i	0.712782	+	0.701385i	$0.247435\pi$
$173$			18.4506i			1.40277i	−0.712782	+	0.701385i	$0.752565\pi$
$174$		0			0
$175$			0.129511i			0.00979013i
$176$		0			0
$177$		−	1.38906i		−	0.104408i
$178$		0			0
$179$	−11.0649			−0.827031			−0.413515	−	0.910497i	$-0.635699\pi$
$179$	−11.0649			−0.827031			−0.413515	+	0.910497i	$0.635699\pi$
$180$		0			0
$181$			19.9796i			1.48507i	0.669806	+	0.742536i	$0.266377\pi$
$181$			19.9796i			1.48507i	−0.669806	+	0.742536i	$0.733623\pi$
$182$		0			0
$183$	7.42921			0.549183
$184$		0			0
$185$	−12.2626			−0.901564
$186$		0			0
$187$	−11.1977	−	22.3637i	−0.818860	−	1.63539i
$188$		0			0
$189$	−1.38415			−0.100682
$190$		0			0
$191$	−8.50151			−0.615147			−0.307574	−	0.951524i	$-0.599517\pi$
$191$	−8.50151			−0.615147			−0.307574	+	0.951524i	$0.599517\pi$
$192$		0			0
$193$			3.59449i			0.258737i	0.991597	+	0.129369i	$0.0412950\pi$
$193$			3.59449i			0.258737i	−0.991597	+	0.129369i	$0.958705\pi$
$194$		0			0
$195$	−1.23796			−0.0886522
$196$		0			0
$197$		−	25.0789i		−	1.78680i	−0.449262	−	0.893400i	$-0.648313\pi$
$197$		−	25.0789i		−	1.78680i	0.449262	−	0.893400i	$-0.351687\pi$
$198$		0			0
$199$			9.89913i			0.701731i	0.936426	+	0.350865i	$0.114113\pi$
$199$			9.89913i			0.701731i	−0.936426	+	0.350865i	$0.885887\pi$
$200$		0			0
$201$			4.23804i			0.298928i
$202$		0			0
$203$	0.234771			0.0164777
$204$		0			0
$205$	−23.4851			−1.64027
$206$		0			0
$207$		−	3.58770i		−	0.249362i
$208$		0			0
$209$		−	11.1397i		−	0.770550i
$210$		0			0
$211$			2.26006i			0.155589i	0.996969	+	0.0777944i	$0.0247878\pi$
$211$			2.26006i			0.155589i	−0.996969	+	0.0777944i	$0.975212\pi$
$212$		0			0
$213$	16.8340			1.15345
$214$		0			0
$215$		−	7.60709i		−	0.518799i
$216$		0			0
$217$	1.88368			0.127873
$218$		0			0
$219$	−6.00596			−0.405845
$220$		0			0
$221$	1.39736	−	0.699672i	0.0939963	−	0.0470650i
$222$		0			0
$223$	−0.818467			−0.0548086			−0.0274043	−	0.999624i	$-0.508724\pi$
$223$	−0.818467			−0.0548086			−0.0274043	+	0.999624i	$0.508724\pi$
$224$		0			0
$225$	−0.566354			−0.0377569
$226$		0			0
$227$			18.9195i			1.25573i	0.778322	+	0.627865i	$0.216071\pi$
$227$			18.9195i			1.25573i	−0.778322	+	0.627865i	$0.783929\pi$
$228$		0			0
$229$	−15.8931			−1.05025			−0.525123	−	0.851026i	$-0.675981\pi$
$229$	−15.8931			−1.05025			−0.525123	+	0.851026i	$0.675981\pi$
$230$		0			0
$231$			2.06267i			0.135714i
$232$		0			0
$233$			6.37035i			0.417336i	0.977987	+	0.208668i	$0.0669127\pi$
$233$			6.37035i			0.417336i	−0.977987	+	0.208668i	$0.933087\pi$
$234$		0			0
$235$		−	6.79908i		−	0.443523i
$236$		0			0
$237$	12.6386			0.820963
$238$		0			0
$239$	8.98990			0.581508			0.290754	−	0.956798i	$-0.406094\pi$
$239$	8.98990			0.581508			0.290754	+	0.956798i	$0.406094\pi$
$240$		0			0
$241$			8.75918i			0.564229i	0.959381	+	0.282114i	$0.0910358\pi$
$241$			8.75918i			0.564229i	−0.959381	+	0.282114i	$0.908964\pi$
$242$		0			0
$243$		−	10.5139i		−	0.674469i
$244$		0			0
$245$		−	16.3188i		−	1.04257i
$246$		0			0
$247$	0.696046			0.0442883
$248$		0			0
$249$			20.0299i			1.26934i
$250$		0			0
$251$	5.10871			0.322459			0.161230	−	0.986917i	$-0.448454\pi$
$251$	5.10871			0.322459			0.161230	+	0.986917i	$0.448454\pi$
$252$		0			0
$253$	−20.3292			−1.27808
$254$		0			0
$255$	−12.0418	+	6.02947i	−0.754087	+	0.377580i
$256$		0			0
$257$	−17.0717			−1.06490			−0.532452	−	0.846460i	$-0.678729\pi$
$257$	−17.0717			−1.06490			−0.532452	+	0.846460i	$0.678729\pi$
$258$		0			0
$259$	−1.27665			−0.0793269
$260$		0			0
$261$			1.02666i			0.0635484i
$262$		0			0
$263$	−9.49782			−0.585661			−0.292830	−	0.956164i	$-0.594597\pi$
$263$	−9.49782			−0.585661			−0.292830	+	0.956164i	$0.594597\pi$
$264$		0			0
$265$		−	21.7140i		−	1.33388i
$266$		0			0
$267$		−	16.8894i		−	1.03361i
$268$		0			0
$269$			9.19982i			0.560923i	0.959865	+	0.280461i	$0.0904875\pi$
$269$			9.19982i			0.560923i	−0.959865	+	0.280461i	$0.909513\pi$
$270$		0			0
$271$	3.41812			0.207636			0.103818	−	0.994596i	$-0.466894\pi$
$271$	3.41812			0.207636			0.103818	+	0.994596i	$0.466894\pi$
$272$		0			0
$273$	−0.128883			−0.00780034
$274$		0			0
$275$			3.20916i			0.193520i
$276$		0			0
$277$		−	24.2213i		−	1.45532i	−0.685940	−	0.727658i	$-0.740609\pi$
$277$		−	24.2213i		−	1.45532i	0.685940	−	0.727658i	$-0.259391\pi$
$278$		0			0
$279$			8.23737i			0.493159i
$280$		0			0
$281$	−26.9643			−1.60856			−0.804278	−	0.594254i	$-0.797448\pi$
$281$	−26.9643			−1.60856			−0.804278	+	0.594254i	$0.797448\pi$
$282$		0			0
$283$		−	4.65439i		−	0.276675i	−0.990385	−	0.138337i	$-0.955824\pi$
$283$		−	4.65439i		−	0.276675i	0.990385	−	0.138337i	$-0.0441758\pi$
$284$		0			0
$285$	−5.99822			−0.355304
$286$		0			0
$287$	−2.44501			−0.144324
$288$		0			0
$289$	10.1845	−	13.6116i	0.599090	−	0.800682i
$290$		0			0
$291$	24.9937			1.46515
$292$		0			0
$293$	−4.85974			−0.283909			−0.141955	−	0.989873i	$-0.545339\pi$
$293$	−4.85974			−0.283909			−0.141955	+	0.989873i	$0.545339\pi$
$294$		0			0
$295$		−	2.35139i		−	0.136903i
$296$		0			0
$297$	−34.2979			−1.99016
$298$		0			0
$299$		−	1.27023i		−	0.0734595i
$300$		0			0
$301$		−	0.791965i		−	0.0456481i
$302$		0			0
$303$			8.31956i			0.477946i
$304$		0			0
$305$	12.5761			0.720107
$306$		0			0
$307$	−22.0681			−1.25949			−0.629747	−	0.776800i	$-0.716841\pi$
$307$	−22.0681			−1.25949			−0.629747	+	0.776800i	$0.716841\pi$
$308$		0			0
$309$		−	9.30434i		−	0.529306i
$310$		0			0
$311$		−	3.19178i		−	0.180989i	−0.995897	−	0.0904946i	$-0.971155\pi$
$311$		−	3.19178i		−	0.180989i	0.995897	−	0.0904946i	$-0.0288448\pi$
$312$		0			0
$313$			28.3216i			1.60083i	0.599446	+	0.800415i	$0.295388\pi$
$313$			28.3216i			1.60083i	−0.599446	+	0.800415i	$0.704612\pi$
$314$		0			0
$315$	−0.616214			−0.0347197
$316$		0			0
$317$			22.0626i			1.23916i	0.784935	+	0.619579i	$0.212696\pi$
$317$			22.0626i			1.23916i	−0.784935	+	0.619579i	$0.787304\pi$
$318$		0			0
$319$	5.81740			0.325712
$320$		0			0
$321$	−14.9080			−0.832085
$322$		0			0
$323$	6.77053	−	3.39008i	0.376722	−	0.188629i
$324$		0			0
$325$	−0.200519			−0.0111228
$326$		0			0
$327$	7.21168			0.398807
$328$		0			0
$329$		−	0.707845i		−	0.0390247i
$330$		0			0
$331$	−24.4525			−1.34403			−0.672014	−	0.740538i	$-0.734571\pi$
$331$	−24.4525			−1.34403			−0.672014	+	0.740538i	$0.734571\pi$
$332$		0			0
$333$		−	5.58278i		−	0.305935i
$334$		0			0
$335$			7.17413i			0.391965i
$336$		0			0
$337$			29.4804i			1.60590i	0.596047	+	0.802950i	$0.296737\pi$
$337$			29.4804i			1.60590i	−0.596047	+	0.802950i	$0.703263\pi$
$338$		0			0
$339$	−6.82369			−0.370612
$340$		0			0
$341$	46.6759			2.52764
$342$		0			0
$343$		−	3.41254i		−	0.184260i
$344$		0			0
$345$			10.9463i			0.589330i
$346$		0			0
$347$		−	7.50267i		−	0.402764i	−0.979513	−	0.201382i	$-0.935457\pi$
$347$		−	7.50267i		−	0.402764i	0.979513	−	0.201382i	$-0.0645433\pi$
$348$		0			0
$349$	15.6923			0.839987			0.419993	−	0.907527i	$-0.362032\pi$
$349$	15.6923			0.839987			0.419993	+	0.907527i	$0.362032\pi$
$350$		0			0
$351$		−	2.14305i		−	0.114387i
$352$		0			0
$353$	6.21440			0.330759			0.165380	−	0.986230i	$-0.447115\pi$
$353$	6.21440			0.330759			0.165380	+	0.986230i	$0.447115\pi$
$354$		0			0
$355$	28.4966			1.51244
$356$		0			0
$357$	−1.25366	+	0.627721i	−0.0663507	+	0.0332225i
$358$		0			0
$359$	−21.6016			−1.14009			−0.570045	−	0.821613i	$-0.693074\pi$
$359$	−21.6016			−1.14009			−0.570045	+	0.821613i	$0.693074\pi$
$360$		0			0
$361$	−15.6275			−0.822499
$362$		0			0
$363$			35.8314i			1.88066i
$364$		0			0
$365$	−10.1669			−0.532158
$366$		0			0
$367$			6.27674i			0.327643i	0.986490	+	0.163821i	$0.0523821\pi$
$367$			6.27674i			0.327643i	−0.986490	+	0.163821i	$0.947618\pi$
$368$		0			0
$369$		−	10.6920i		−	0.556606i
$370$		0			0
$371$		−	2.26062i		−	0.117365i
$372$		0			0
$373$	25.6148			1.32628			0.663141	−	0.748494i	$-0.269223\pi$
$373$	25.6148			1.32628			0.663141	+	0.748494i	$0.269223\pi$
$374$		0			0
$375$	−14.6031			−0.754102
$376$		0			0
$377$			0.363490i			0.0187207i
$378$		0			0
$379$		−	19.5785i		−	1.00568i	−0.864379	−	0.502841i	$-0.832288\pi$
$379$		−	19.5785i		−	1.00568i	0.864379	−	0.502841i	$-0.167712\pi$
$380$		0			0
$381$			20.1871i			1.03422i
$382$		0			0
$383$	−23.3826			−1.19480			−0.597399	−	0.801944i	$-0.703799\pi$
$383$	−23.3826			−1.19480			−0.597399	+	0.801944i	$0.703799\pi$
$384$		0			0
$385$			3.49169i			0.177953i
$386$		0			0
$387$	3.46327			0.176048
$388$		0			0
$389$	19.5733			0.992406			0.496203	−	0.868206i	$-0.334727\pi$
$389$	19.5733			0.992406			0.496203	+	0.868206i	$0.334727\pi$
$390$		0			0
$391$	−6.18665	−	12.3557i	−0.312872	−	0.624856i
$392$		0			0
$393$	−9.82011			−0.495359
$394$		0			0
$395$	21.3945			1.07647
$396$		0			0
$397$		−	12.5936i		−	0.632053i	−0.948750	−	0.316026i	$-0.897651\pi$
$397$		−	12.5936i		−	0.632053i	0.948750	−	0.316026i	$-0.102349\pi$
$398$		0			0
$399$	−0.624468			−0.0312625
$400$		0			0
$401$			5.51985i			0.275648i	0.990457	+	0.137824i	$0.0440109\pi$
$401$			5.51985i			0.275648i	−0.990457	+	0.137824i	$0.955989\pi$
$402$		0			0
$403$			2.91646i			0.145279i
$404$		0			0
$405$			10.9162i			0.542431i
$406$		0			0
$407$	−31.6340			−1.56804
$408$		0			0
$409$	8.49200			0.419902			0.209951	−	0.977712i	$-0.432670\pi$
$409$	8.49200			0.419902			0.209951	+	0.977712i	$0.432670\pi$
$410$		0			0
$411$		−	15.7860i		−	0.778664i
$412$		0			0
$413$		−	0.244801i		−	0.0120459i
$414$		0			0
$415$			33.9065i			1.66440i
$416$		0			0
$417$	−5.00451			−0.245072
$418$		0			0
$419$			20.7166i			1.01207i	0.862512	+	0.506036i	$0.168890\pi$
$419$			20.7166i			1.01207i	−0.862512	+	0.506036i	$0.831110\pi$
$420$		0			0
$421$	19.6449			0.957434			0.478717	−	0.877969i	$-0.341102\pi$
$421$	19.6449			0.957434			0.478717	+	0.877969i	$0.341102\pi$
$422$		0			0
$423$	3.09541			0.150504
$424$		0			0
$425$	−1.95047	+	0.976624i	−0.0946119	+	0.0473732i
$426$		0			0
$427$	1.30929			0.0633608
$428$		0			0
$429$	−3.19359			−0.154188
$430$		0			0
$431$			15.2837i			0.736190i	0.929788	+	0.368095i	$0.119990\pi$
$431$			15.2837i			0.736190i	−0.929788	+	0.368095i	$0.880010\pi$
$432$		0			0
$433$	−9.63445			−0.463002			−0.231501	−	0.972835i	$-0.574364\pi$
$433$	−9.63445			−0.463002			−0.231501	+	0.972835i	$0.574364\pi$
$434$		0			0
$435$		−	3.13240i		−	0.150187i
$436$		0			0
$437$		−	6.15459i		−	0.294414i
$438$		0			0
$439$		−	18.0746i		−	0.862653i	−0.902196	−	0.431326i	$-0.858046\pi$
$439$		−	18.0746i		−	0.862653i	0.902196	−	0.431326i	$-0.141954\pi$
$440$		0			0
$441$	7.42946			0.353784
$442$		0			0
$443$	6.17615			0.293438			0.146719	−	0.989178i	$-0.453129\pi$
$443$	6.17615			0.293438			0.146719	+	0.989178i	$0.453129\pi$
$444$		0			0
$445$		−	28.5902i		−	1.35531i
$446$		0			0
$447$			8.30483i			0.392805i
$448$		0			0
$449$		−	9.40253i		−	0.443733i	−0.975077	−	0.221866i	$-0.928785\pi$
$449$		−	9.40253i		−	0.443733i	0.975077	−	0.221866i	$-0.0712149\pi$
$450$		0			0
$451$	−60.5849			−2.85283
$452$		0			0
$453$		−	14.2855i		−	0.671191i
$454$		0			0
$455$	−0.218172			−0.0102281
$456$		0			0
$457$	36.4982			1.70731			0.853657	−	0.520835i	$-0.174380\pi$
$457$	36.4982			1.70731			0.853657	+	0.520835i	$0.174380\pi$
$458$		0			0
$459$	−10.4377	−	20.8457i	−0.487188	−	0.972993i
$460$		0			0
$461$	29.4996			1.37393			0.686967	−	0.726688i	$-0.258942\pi$
$461$	29.4996			1.37393			0.686967	+	0.726688i	$0.258942\pi$
$462$		0			0
$463$	7.94349			0.369165			0.184583	−	0.982817i	$-0.440907\pi$
$463$	7.94349			0.369165			0.184583	+	0.982817i	$0.440907\pi$
$464$		0			0
$465$		−	25.1328i		−	1.16551i
$466$		0			0
$467$	25.3103			1.17122			0.585611	−	0.810593i	$-0.300855\pi$
$467$	25.3103			1.17122			0.585611	+	0.810593i	$0.300855\pi$
$468$		0			0
$469$			0.746891i			0.0344882i
$470$		0			0
$471$		−	22.4489i		−	1.03439i
$472$		0			0
$473$		−	19.6241i		−	0.902319i
$474$		0			0
$475$	−0.971563			−0.0445784
$476$		0			0
$477$	9.88571			0.452635
$478$		0			0
$479$		−	26.5313i		−	1.21225i	−0.795371	−	0.606124i	$-0.792724\pi$
$479$		−	26.5313i		−	1.21225i	0.795371	−	0.606124i	$-0.207276\pi$
$480$		0			0
$481$		−	1.97660i		−	0.0901251i
$482$		0			0
$483$			1.13961i			0.0518540i
$484$		0			0
$485$	42.3092			1.92116
$486$		0			0
$487$			26.6480i			1.20754i	0.797160	+	0.603768i	$0.206335\pi$
$487$			26.6480i			1.20754i	−0.797160	+	0.603768i	$0.793665\pi$
$488$		0			0
$489$	24.9673			1.12906
$490$		0			0
$491$	−15.8809			−0.716697			−0.358349	−	0.933588i	$-0.616660\pi$
$491$	−15.8809			−0.716697			−0.358349	+	0.933588i	$0.616660\pi$
$492$		0			0
$493$	1.77037	+	3.53572i	0.0797336	+	0.159241i
$494$		0			0
$495$	−15.2692			−0.686299
$496$		0			0
$497$	2.96675			0.133077
$498$		0			0
$499$			3.65268i			0.163516i	0.996652	+	0.0817581i	$0.0260535\pi$
$499$			3.65268i			0.163516i	−0.996652	+	0.0817581i	$0.973947\pi$
$500$		0			0
$501$	26.3811			1.17862
$502$		0			0
$503$		−	30.2692i		−	1.34964i	−0.737985	−	0.674818i	$-0.764222\pi$
$503$		−	30.2692i		−	1.34964i	0.737985	−	0.674818i	$-0.235778\pi$
$504$		0			0
$505$			14.0833i			0.626700i
$506$		0			0
$507$			17.8582i			0.793111i
$508$		0			0
$509$	5.49284			0.243466			0.121733	−	0.992563i	$-0.461155\pi$
$509$	5.49284			0.243466			0.121733	+	0.992563i	$0.461155\pi$
$510$		0			0
$511$	−1.05846			−0.0468236
$512$		0			0
$513$		−	10.3836i		−	0.458446i
$514$		0			0
$515$		−	15.7504i		−	0.694044i
$516$		0			0
$517$		−	17.5397i		−	0.771395i
$518$		0			0
$519$	25.6289			1.12498
$520$		0			0
$521$		−	25.4861i		−	1.11657i	−0.829651	−	0.558283i	$-0.811461\pi$
$521$		−	25.4861i		−	1.11657i	0.829651	−	0.558283i	$-0.188539\pi$
$522$		0			0
$523$	9.75471			0.426544			0.213272	−	0.976993i	$-0.431588\pi$
$523$	9.75471			0.426544			0.213272	+	0.976993i	$0.431588\pi$
$524$		0			0
$525$	0.179899			0.00785142
$526$		0			0
$527$	14.2046	+	28.3688i	0.618761	+	1.23576i
$528$		0			0
$529$	11.7683			0.511666
$530$		0			0
$531$	1.07052			0.0464565
$532$		0			0
$533$		−	3.78555i		−	0.163970i
$534$		0			0
$535$	−25.2362			−1.09106
$536$		0			0
$537$			15.3698i			0.663257i
$538$		0			0
$539$		−	42.0980i		−	1.81329i
$540$		0			0
$541$			32.3643i			1.39145i	0.718309	+	0.695724i	$0.244916\pi$
$541$			32.3643i			1.39145i	−0.718309	+	0.695724i	$0.755084\pi$
$542$		0			0
$543$	27.7528			1.19099
$544$		0			0
$545$	12.2079			0.522929
$546$		0			0
$547$		−	25.2865i		−	1.08117i	−0.841289	−	0.540586i	$-0.818203\pi$
$547$		−	25.2865i		−	1.08117i	0.841289	−	0.540586i	$-0.181797\pi$
$548$		0			0
$549$			5.72553i			0.244359i
$550$		0			0
$551$			1.76120i			0.0750296i
$552$		0			0
$553$	2.22736			0.0947169
$554$		0			0
$555$			17.0335i			0.723030i
$556$		0			0
$557$	−12.4140			−0.525998			−0.262999	−	0.964796i	$-0.584712\pi$
$557$	−12.4140			−0.525998			−0.262999	+	0.964796i	$0.584712\pi$
$558$		0			0
$559$	1.22618			0.0518619
$560$		0			0
$561$	−31.0644	+	15.5543i	−1.31154	+	0.656704i
$562$		0			0
$563$	2.45827			0.103604			0.0518019	−	0.998657i	$-0.483504\pi$
$563$	2.45827			0.103604			0.0518019	+	0.998657i	$0.483504\pi$
$564$		0			0
$565$	−11.5511			−0.485959
$566$		0			0
$567$			1.13647i			0.0477274i
$568$		0			0
$569$	39.5479			1.65794			0.828968	−	0.559296i	$-0.188929\pi$
$569$	39.5479			1.65794			0.828968	+	0.559296i	$0.188929\pi$
$570$		0			0
$571$		−	0.210260i		−	0.00879912i	−0.999990	−	0.00439956i	$-0.998600\pi$
$571$		−	0.210260i		−	0.00879912i	0.999990	−	0.00439956i	$-0.00140043\pi$
$572$		0			0
$573$			11.8091i			0.493332i
$574$		0			0
$575$			1.77303i			0.0739405i
$576$		0			0
$577$	−15.4221			−0.642032			−0.321016	−	0.947074i	$-0.604024\pi$
$577$	−15.4221			−0.642032			−0.321016	+	0.947074i	$0.604024\pi$
$578$		0			0
$579$	4.99296			0.207500
$580$		0			0
$581$			3.52996i			0.146448i
$582$		0			0
$583$		−	56.0159i		−	2.31994i
$584$		0			0
$585$		−	0.954069i		−	0.0394459i
$586$		0			0
$587$	25.0617			1.03441			0.517203	−	0.855863i	$-0.326973\pi$
$587$	25.0617			1.03441			0.517203	+	0.855863i	$0.326973\pi$
$588$		0			0
$589$			14.1310i			0.582256i
$590$		0			0
$591$	−34.8361			−1.43297
$592$		0			0
$593$	29.0880			1.19450			0.597251	−	0.802054i	$-0.296259\pi$
$593$	29.0880			1.19450			0.597251	+	0.802054i	$0.296259\pi$
$594$		0			0
$595$	−2.12219	+	1.06260i	−0.0870013	+	0.0435625i
$596$		0			0
$597$	13.7505			0.562769
$598$		0			0
$599$	−23.0386			−0.941331			−0.470666	−	0.882312i	$-0.655986\pi$
$599$	−23.0386			−0.941331			−0.470666	+	0.882312i	$0.655986\pi$
$600$		0			0
$601$		−	26.3979i		−	1.07679i	−0.842692	−	0.538396i	$-0.819031\pi$
$601$		−	26.3979i		−	1.07679i	0.842692	−	0.538396i	$-0.180969\pi$
$602$		0			0
$603$	−3.26616			−0.133008
$604$		0			0
$605$			60.6553i			2.46599i
$606$		0			0
$607$		−	22.5729i		−	0.916204i	−0.888900	−	0.458102i	$-0.848529\pi$
$607$		−	22.5729i		−	0.916204i	0.888900	−	0.458102i	$-0.151471\pi$
$608$		0			0
$609$		−	0.326111i		−	0.0132147i
$610$		0			0
$611$	1.09594			0.0443369
$612$		0			0
$613$	3.12681			0.126291			0.0631454	−	0.998004i	$-0.479887\pi$
$613$	3.12681			0.126291			0.0631454	+	0.998004i	$0.479887\pi$
$614$		0			0
$615$			32.6222i			1.31545i
$616$		0			0
$617$			16.7263i			0.673377i	0.941616	+	0.336688i	$0.109307\pi$
$617$			16.7263i			0.673377i	−0.941616	+	0.336688i	$0.890693\pi$
$618$		0			0
$619$		−	32.9319i		−	1.32364i	−0.749661	−	0.661822i	$-0.769783\pi$
$619$		−	32.9319i		−	1.32364i	0.749661	−	0.661822i	$-0.230217\pi$
$620$		0			0
$621$	−18.9493			−0.760408
$622$		0			0
$623$		−	2.97650i		−	0.119251i
$624$		0			0
$625$	−27.3653			−1.09461
$626$		0			0
$627$	−15.4737			−0.617960
$628$		0			0
$629$	−9.62699	−	19.2266i	−0.383853	−	0.766616i
$630$		0			0
$631$	31.8339			1.26729			0.633645	−	0.773624i	$-0.281558\pi$
$631$	31.8339			1.26729			0.633645	+	0.773624i	$0.281558\pi$
$632$		0			0
$633$	3.13935			0.124778
$634$		0			0
$635$			34.1727i			1.35610i
$636$		0			0
$637$	2.63042			0.104221
$638$		0			0
$639$			12.9736i			0.513228i
$640$		0			0
$641$			50.2859i			1.98617i	0.117382	+	0.993087i	$0.462550\pi$
$641$			50.2859i			1.98617i	−0.117382	+	0.993087i	$0.537450\pi$
$642$		0			0
$643$		−	31.7570i		−	1.25238i	−0.779672	−	0.626188i	$-0.784614\pi$
$643$		−	31.7570i		−	1.25238i	0.779672	−	0.626188i	$-0.215386\pi$
$644$		0			0
$645$	−10.5667			−0.416063
$646$		0			0
$647$	23.9490			0.941533			0.470767	−	0.882258i	$-0.343977\pi$
$647$	23.9490			0.941533			0.470767	+	0.882258i	$0.343977\pi$
$648$		0			0
$649$		−	6.06593i		−	0.238108i
$650$		0			0
$651$		−	2.61655i		−	0.102551i
$652$		0			0
$653$		−	10.7631i		−	0.421191i	−0.977573	−	0.210596i	$-0.932460\pi$
$653$		−	10.7631i		−	0.421191i	0.977573	−	0.210596i	$-0.0675403\pi$
$654$		0			0
$655$	−16.6235			−0.649532
$656$		0			0
$657$		−	4.62866i		−	0.180581i
$658$		0			0
$659$	−21.0210			−0.818862			−0.409431	−	0.912341i	$-0.634273\pi$
$659$	−21.0210			−0.818862			−0.409431	+	0.912341i	$0.634273\pi$
$660$		0			0
$661$	−31.8955			−1.24059			−0.620295	−	0.784369i	$-0.712987\pi$
$661$	−31.8955			−1.24059			−0.620295	+	0.784369i	$0.712987\pi$
$662$		0			0
$663$	−0.971885	−	1.94101i	−0.0377449	−	0.0753826i
$664$		0			0
$665$	−1.05710			−0.0409925
$666$		0			0
$667$	3.21406			0.124449
$668$		0			0
$669$			1.13690i			0.0439550i
$670$		0			0
$671$	32.4429			1.25244
$672$		0			0
$673$			17.1993i			0.662984i	0.943458	+	0.331492i	$0.107552\pi$
$673$			17.1993i			0.662984i	−0.943458	+	0.331492i	$0.892448\pi$
$674$		0			0
$675$			2.99133i			0.115136i
$676$		0			0
$677$			7.57787i			0.291241i	0.989341	+	0.145621i	$0.0465179\pi$
$677$			7.57787i			0.291241i	−0.989341	+	0.145621i	$0.953482\pi$
$678$		0			0
$679$	4.40476			0.169039
$680$		0			0
$681$	26.2803			1.00706
$682$		0			0
$683$			35.2582i			1.34912i	0.738221	+	0.674558i	$0.235666\pi$
$683$			35.2582i			1.34912i	−0.738221	+	0.674558i	$0.764334\pi$
$684$		0			0
$685$		−	26.7224i		−	1.02101i
$686$		0			0
$687$			22.0764i			0.842269i
$688$		0			0
$689$	3.50006			0.133342
$690$		0			0
$691$			17.3942i			0.661706i	0.943682	+	0.330853i	$0.107336\pi$
$691$			17.3942i			0.661706i	−0.943682	+	0.330853i	$0.892664\pi$
$692$		0			0
$693$	−1.58966			−0.0603861
$694$		0			0
$695$	−8.47161			−0.321347
$696$		0			0
$697$	−18.4374	−	36.8225i	−0.698368	−	1.39475i
$698$		0			0
$699$	8.84879			0.334692
$700$		0			0
$701$	32.0840			1.21180			0.605898	−	0.795542i	$-0.292814\pi$
$701$	32.0840			1.21180			0.605898	+	0.795542i	$0.292814\pi$
$702$		0			0
$703$		−	9.57710i		−	0.361207i
$704$		0			0
$705$	−9.44432			−0.355694
$706$		0			0
$707$			1.46620i			0.0551421i
$708$		0			0
$709$		−	14.3393i		−	0.538524i	−0.963067	−	0.269262i	$-0.913220\pi$
$709$		−	14.3393i		−	0.538524i	0.963067	−	0.269262i	$-0.0867798\pi$
$710$		0			0
$711$			9.74026i			0.365288i
$712$		0			0
$713$	25.7880			0.965768
$714$		0			0
$715$	−5.40609			−0.202176
$716$		0			0
$717$		−	12.4875i		−	0.466354i
$718$		0			0
$719$			8.06288i			0.300695i	0.988633	+	0.150347i	$0.0480392\pi$
$719$			8.06288i			0.300695i	−0.988633	+	0.150347i	$0.951961\pi$
$720$		0			0
$721$		−	1.63975i		−	0.0610676i
$722$		0			0
$723$	12.1670			0.452496
$724$		0			0
$725$		−	0.507371i		−	0.0188433i
$726$		0			0
$727$	6.25393			0.231946			0.115973	−	0.993252i	$-0.463001\pi$
$727$	6.25393			0.231946			0.115973	+	0.993252i	$0.463001\pi$
$728$		0			0
$729$	−28.5318			−1.05673
$730$		0			0
$731$	11.9272	−	5.97209i	0.441144	−	0.220886i
$732$		0			0
$733$	23.9489			0.884574			0.442287	−	0.896874i	$-0.354167\pi$
$733$	23.9489			0.884574			0.442287	+	0.896874i	$0.354167\pi$
$734$		0			0
$735$	−22.6678			−0.836115
$736$		0			0
$737$			18.5072i			0.681722i
$738$		0			0
$739$	21.6470			0.796299			0.398149	−	0.917321i	$-0.369653\pi$
$739$	21.6470			0.796299			0.398149	+	0.917321i	$0.369653\pi$
$740$		0			0
$741$		−	0.966849i		−	0.0355181i
$742$		0			0
$743$			21.1026i			0.774178i	0.922042	+	0.387089i	$0.126519\pi$
$743$			21.1026i			0.774178i	−0.922042	+	0.387089i	$0.873481\pi$
$744$		0			0
$745$			14.0584i			0.515059i
$746$		0			0
$747$	−15.4366			−0.564794
$748$		0			0
$749$	−2.62732			−0.0960001
$750$		0			0
$751$			23.6334i			0.862394i	0.902258	+	0.431197i	$0.141909\pi$
$751$			23.6334i			0.862394i	−0.902258	+	0.431197i	$0.858091\pi$
$752$		0			0
$753$		−	7.09630i		−	0.258604i
$754$		0			0
$755$		−	24.1824i		−	0.880089i
$756$		0			0
$757$	48.8655			1.77605			0.888024	−	0.459797i	$-0.152078\pi$
$757$	48.8655			1.77605			0.888024	+	0.459797i	$0.152078\pi$
$758$		0			0
$759$			28.2384i			1.02499i
$760$		0			0
$761$	24.2939			0.880651			0.440326	−	0.897838i	$-0.354863\pi$
$761$	24.2939			0.880651			0.440326	+	0.897838i	$0.354863\pi$
$762$		0			0
$763$	1.27095			0.0460115
$764$		0			0
$765$	−4.64678	−	9.28035i	−0.168005	−	0.335532i
$766$		0			0
$767$	0.379019			0.0136856
$768$		0			0
$769$	−38.2086			−1.37784			−0.688920	−	0.724838i	$-0.741915\pi$
$769$	−38.2086			−1.37784			−0.688920	+	0.724838i	$0.741915\pi$
$770$		0			0
$771$			23.7136i			0.854024i
$772$		0			0
$773$	−39.0683			−1.40519			−0.702594	−	0.711591i	$-0.747975\pi$
$773$	−39.0683			−1.40519			−0.702594	+	0.711591i	$0.747975\pi$
$774$		0			0
$775$		−	4.07089i		−	0.146231i
$776$		0			0
$777$			1.77333i			0.0636180i
$778$		0			0
$779$		−	18.3419i		−	0.657166i
$780$		0			0
$781$	73.5131			2.63050
$782$		0			0
$783$	5.42253			0.193785
$784$		0			0
$785$		−	38.0014i		−	1.35633i
$786$		0			0
$787$			30.5306i			1.08830i	0.838989	+	0.544149i	$0.183147\pi$
$787$			30.5306i			1.08830i	−0.838989	+	0.544149i	$0.816853\pi$
$788$		0			0
$789$			13.1930i			0.469684i
$790$		0			0
$791$	−1.20257			−0.0427586
$792$		0			0
$793$			2.02714i			0.0719858i
$794$		0			0
$795$	−30.1620			−1.06974
$796$		0			0
$797$	−29.5411			−1.04640			−0.523200	−	0.852210i	$-0.675262\pi$
$797$	−29.5411			−1.04640			−0.523200	+	0.852210i	$0.675262\pi$
$798$		0			0
$799$	10.6603	−	5.33775i	0.377135	−	0.188836i
$800$		0			0
$801$	13.0163			0.459907
$802$		0			0
$803$	−26.2277			−0.925554
$804$		0			0
$805$			1.92913i			0.0679927i
$806$		0			0
$807$	12.7791			0.449845
$808$		0			0
$809$			43.8243i			1.54078i	0.637573	+	0.770390i	$0.279938\pi$
$809$			43.8243i			1.54078i	−0.637573	+	0.770390i	$0.720062\pi$
$810$		0			0
$811$		−	8.80784i		−	0.309285i	−0.987970	−	0.154643i	$-0.950577\pi$
$811$		−	8.80784i		−	0.309285i	0.987970	−	0.154643i	$-0.0494226\pi$
$812$		0			0
$813$		−	4.74797i		−	0.166519i
$814$		0			0
$815$	42.2645			1.48046
$816$		0			0
$817$	5.94114			0.207854
$818$		0			0
$819$		−	0.0993271i		−	0.00347077i
$820$		0			0
$821$		−	42.0668i		−	1.46814i	−0.679074	−	0.734070i	$-0.737618\pi$
$821$		−	42.0668i		−	1.46814i	0.679074	−	0.734070i	$-0.262382\pi$
$822$		0			0
$823$			15.7847i			0.550219i	0.961413	+	0.275110i	$0.0887142\pi$
$823$			15.7847i			0.550219i	−0.961413	+	0.275110i	$0.911286\pi$
$824$		0			0
$825$	4.45771			0.155198
$826$		0			0
$827$			29.6027i			1.02939i	0.857375	+	0.514693i	$0.172094\pi$
$827$			29.6027i			1.02939i	−0.857375	+	0.514693i	$0.827906\pi$
$828$		0			0
$829$	4.24473			0.147426			0.0737128	−	0.997280i	$-0.476515\pi$
$829$	4.24473			0.147426			0.0737128	+	0.997280i	$0.476515\pi$
$830$		0			0
$831$	−33.6448			−1.16712
$832$		0			0
$833$	25.5864	−	12.8114i	0.886518	−	0.443889i
$834$		0			0
$835$	44.6578			1.54545
$836$		0			0
$837$	43.5076			1.50384
$838$		0			0
$839$		−	15.6992i		−	0.541997i	−0.962580	−	0.270998i	$-0.912646\pi$
$839$		−	15.6992i		−	0.541997i	0.962580	−	0.270998i	$-0.0873538\pi$
$840$		0			0
$841$	28.0803			0.968285
$842$		0			0
$843$			37.4550i			1.29002i
$844$		0			0
$845$			30.2303i			1.03995i
$846$		0			0
$847$			6.31476i			0.216978i
$848$		0			0
$849$	−6.46522			−0.221886
$850$		0			0
$851$	−17.4775			−0.599121
$852$		0			0
$853$		−	44.4062i		−	1.52044i	−0.649666	−	0.760219i	$-0.725091\pi$
$853$		−	44.4062i		−	1.52044i	0.649666	−	0.760219i	$-0.274909\pi$
$854$		0			0
$855$		−	4.62270i		−	0.158093i
$856$		0			0
$857$			0.344939i			0.0117829i	0.999983	+	0.00589144i	$0.00187532\pi$
$857$			0.344939i			0.0117829i	−0.999983	+	0.00589144i	$0.998125\pi$
$858$		0			0
$859$	−46.6137			−1.59044			−0.795219	−	0.606323i	$-0.792644\pi$
$859$	−46.6137			−1.59044			−0.795219	+	0.606323i	$0.792644\pi$
$860$		0			0
$861$			3.39626i			0.115744i
$862$		0			0
$863$	−32.3758			−1.10208			−0.551042	−	0.834477i	$-0.685770\pi$
$863$	−32.3758			−1.10208			−0.551042	+	0.834477i	$0.685770\pi$
$864$		0			0
$865$	43.3845			1.47512
$866$		0			0
$867$	−18.9073	−	14.1469i	−0.642126	−	0.480454i
$868$		0			0
$869$	55.1918			1.87225
$870$		0			0
$871$	−1.15639			−0.0391829
$872$		0			0
$873$			19.2621i			0.651922i
$874$		0			0
$875$	−2.57358			−0.0870030
$876$		0			0
$877$			16.0246i			0.541112i	0.962704	+	0.270556i	$0.0872076\pi$
$877$			16.0246i			0.541112i	−0.962704	+	0.270556i	$0.912792\pi$
$878$		0			0
$879$			6.75046i			0.227687i
$880$		0			0
$881$			2.89150i			0.0974171i	0.998813	+	0.0487085i	$0.0155105\pi$
$881$			2.89150i			0.0974171i	−0.998813	+	0.0487085i	$0.984489\pi$
$882$		0			0
$883$	21.2357			0.714638			0.357319	−	0.933982i	$-0.383691\pi$
$883$	21.2357			0.714638			0.357319	+	0.933982i	$0.383691\pi$
$884$		0			0
$885$	−3.26622			−0.109793
$886$		0			0
$887$			4.68977i			0.157467i	0.996896	+	0.0787336i	$0.0250876\pi$
$887$			4.68977i			0.157467i	−0.996896	+	0.0787336i	$0.974912\pi$
$888$		0			0
$889$			3.55768i			0.119321i
$890$		0			0
$891$			28.1607i			0.943420i
$892$		0			0
$893$	5.31009			0.177695
$894$		0			0
$895$			26.0180i			0.869685i
$896$		0			0
$897$	−1.76443			−0.0589126
$898$		0			0
$899$	−7.37950			−0.246120
$900$		0			0
$901$	34.0455	−	17.0470i	1.13422	−	0.567917i
$902$		0			0
$903$	−1.10009			−0.0366086
$904$		0			0
$905$	46.9799			1.56167
$906$		0			0
$907$			31.2908i			1.03899i	0.854473	+	0.519496i	$0.173880\pi$
$907$			31.2908i			1.03899i	−0.854473	+	0.519496i	$0.826120\pi$
$908$		0			0
$909$	−6.41170			−0.212663
$910$		0			0
$911$			29.9690i			0.992916i	0.868061	+	0.496458i	$0.165366\pi$
$911$			29.9690i			0.992916i	−0.868061	+	0.496458i	$0.834634\pi$
$912$		0			0
$913$			87.4691i			2.89480i
$914$		0			0
$915$		−	17.4690i		−	0.577507i
$916$		0			0
$917$	−1.73065			−0.0571511
$918$		0			0
$919$	−29.6939			−0.979511			−0.489755	−	0.871860i	$-0.662914\pi$
$919$	−29.6939			−0.979511			−0.489755	+	0.871860i	$0.662914\pi$
$920$		0			0
$921$			30.6539i			1.01008i
$922$		0			0
$923$			4.59334i			0.151192i
$924$		0			0
$925$			2.75900i			0.0907153i
$926$		0			0
$927$	7.17065			0.235515
$928$		0			0
$929$		−	23.0453i		−	0.756092i	−0.925787	−	0.378046i	$-0.876596\pi$
$929$		−	23.0453i		−	0.756092i	0.925787	−	0.378046i	$-0.123404\pi$
$930$		0			0
$931$	12.7450			0.417701
$932$		0			0
$933$	−4.43357			−0.145148
$934$		0			0
$935$	−52.5858	+	26.3303i	−1.71974	+	0.861092i
$936$		0			0
$937$	52.5856			1.71790			0.858949	−	0.512061i	$-0.171118\pi$
$937$	52.5856			1.71790			0.858949	+	0.512061i	$0.171118\pi$
$938$		0			0
$939$	39.3403			1.28382
$940$		0			0
$941$			9.33005i			0.304151i	0.988369	+	0.152075i	$0.0485957\pi$
$941$			9.33005i			0.304151i	−0.988369	+	0.152075i	$0.951404\pi$
$942$		0			0
$943$	−33.4726			−1.09002
$944$		0			0
$945$			3.25468i			0.105875i
$946$		0			0
$947$			50.1968i			1.63118i	0.578632	+	0.815589i	$0.303587\pi$
$947$			50.1968i			1.63118i	−0.578632	+	0.815589i	$0.696413\pi$
$948$		0			0
$949$		−	1.63879i		−	0.0531974i
$950$		0			0
$951$	30.6462			0.993771
$952$		0			0
$953$	24.9000			0.806592			0.403296	−	0.915070i	$-0.367865\pi$
$953$	24.9000			0.806592			0.403296	+	0.915070i	$0.367865\pi$
$954$		0			0
$955$			19.9904i			0.646874i
$956$		0			0
$957$		−	8.08071i		−	0.261212i
$958$		0			0
$959$		−	2.78204i		−	0.0898368i
$960$		0			0
$961$	−28.2094			−0.909979
$962$		0			0
$963$		−	11.4893i		−	0.370237i
$964$		0			0
$965$	8.45206			0.272082
$966$		0			0
$967$	−29.0338			−0.933665			−0.466833	−	0.884346i	$-0.654605\pi$
$967$	−29.0338			−0.933665			−0.466833	+	0.884346i	$0.654605\pi$
$968$		0			0
$969$	−4.70902	−	9.40466i	−0.151275	−	0.302121i
$970$		0			0
$971$	3.30992			0.106221			0.0531103	−	0.998589i	$-0.483087\pi$
$971$	3.30992			0.106221			0.0531103	+	0.998589i	$0.483087\pi$
$972$		0			0
$973$	−0.881970			−0.0282747
$974$		0			0
$975$			0.278533i			0.00892018i
$976$		0			0
$977$	−57.6244			−1.84357			−0.921784	−	0.387703i	$-0.873269\pi$
$977$	−57.6244			−1.84357			−0.921784	+	0.387703i	$0.873269\pi$
$978$		0			0
$979$		−	73.7547i		−	2.35721i
$980$		0			0
$981$			5.55788i			0.177450i
$982$		0			0
$983$		−	60.7741i		−	1.93839i	−0.246289	−	0.969196i	$-0.579211\pi$
$983$		−	60.7741i		−	1.93839i	0.246289	−	0.969196i	$-0.420789\pi$
$984$		0			0
$985$	−58.9704			−1.87895
$986$		0			0
$987$	−0.983237			−0.0312968
$988$		0			0
$989$		−	10.8422i		−	0.344760i
$990$		0			0
$991$		−	14.0927i		−	0.447669i	−0.974627	−	0.223834i	$-0.928143\pi$
$991$		−	14.0927i		−	0.447669i	0.974627	−	0.223834i	$-0.0718574\pi$
$992$		0			0
$993$			33.9659i			1.07788i
$994$		0			0
$995$	23.2767			0.737922
$996$		0			0
$997$		−	44.0970i		−	1.39657i	−0.715821	−	0.698283i	$-0.753948\pi$
$997$		−	44.0970i		−	1.39657i	0.715821	−	0.698283i	$-0.246052\pi$
$998$		0			0
$999$	−29.4868			−0.932920

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.b.b.237.17	✓	46
17.16	even	2	inner	4012.2.b.b.237.30	yes	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.b.b.237.17	✓	46	1.1	even	1	trivial
4012.2.b.b.237.30	yes	46	17.16	even	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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.38906i q^{3} -2.35139i q^{5} -0.244801i q^{7} +1.07052 q^{9} +O(q^{10})\)	\(q-1.38906i q^{3} -2.35139i q^{5} -0.244801i q^{7} +1.07052 q^{9} -6.06593i q^{11} +0.379019 q^{13} -3.26622 q^{15} +(3.68677 - 1.84601i) q^{17} +1.83644 q^{19} -0.340043 q^{21} -3.35137i q^{23} -0.529047 q^{25} -5.65419i q^{27} +0.959029i q^{29} +7.69476i q^{31} -8.42593 q^{33} -0.575623 q^{35} -5.21503i q^{37} -0.526480i q^{39} -9.98774i q^{41} +3.23514 q^{43} -2.51721i q^{45} +2.89151 q^{47} +6.94007 q^{49} +(-2.56421 - 5.12114i) q^{51} +9.23452 q^{53} -14.2634 q^{55} -2.55092i q^{57} +1.00000 q^{59} +5.34838i q^{61} -0.262063i q^{63} -0.891223i q^{65} -3.05101 q^{67} -4.65525 q^{69} +12.1190i q^{71} -4.32377i q^{73} +0.734877i q^{75} -1.48494 q^{77} +9.09865i q^{79} -4.64244 q^{81} -14.4197 q^{83} +(-4.34069 - 8.66904i) q^{85} +1.33215 q^{87} +12.1589 q^{89} -0.0927842i q^{91} +10.6885 q^{93} -4.31819i q^{95} +17.9932i q^{97} -6.49368i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 54 q^{9}+O(q^{10}) \) 46 * q - 54 * q^9	\( 46 q - 54 q^{9} + 8 q^{13} - 10 q^{15} + q^{17} - 20 q^{19} - 24 q^{21} - 54 q^{25} + 2 q^{33} + 26 q^{35} - 38 q^{43} + 6 q^{47} - 66 q^{49} + 26 q^{51} + 18 q^{53} - 20 q^{55} + 46 q^{59} + 48 q^{67} + 28 q^{69} + 22 q^{77} + 70 q^{81} - 52 q^{83} - 2 q^{85} + 44 q^{87} - 76 q^{89} - 26 q^{93}+O(q^{100}) \) 46 * q - 54 * q^9 + 8 * q^13 - 10 * q^15 + q^17 - 20 * q^19 - 24 * q^21 - 54 * q^25 + 2 * q^33 + 26 * q^35 - 38 * q^43 + 6 * q^47 - 66 * q^49 + 26 * q^51 + 18 * q^53 - 20 * q^55 + 46 * q^59 + 48 * q^67 + 28 * q^69 + 22 * q^77 + 70 * q^81 - 52 * q^83 - 2 * q^85 + 44 * q^87 - 76 * q^89 - 26 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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