LMFDB - Embedded newform 4012.2.b.b.237.39

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			237.39
Character	$\chi$	$=$	4012.237
Dual form			4012.2.b.b.237.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.34087i q^{3} +2.91532i q^{5} +3.43927i q^{7} -2.47969 q^{9} +O(q^{10})$	$q+2.34087i q^{3} +2.91532i q^{5} +3.43927i q^{7} -2.47969 q^{9} +1.22244i q^{11} -4.98028 q^{13} -6.82439 q^{15} +(0.473595 - 4.09582i) q^{17} +3.37891 q^{19} -8.05090 q^{21} +0.446807i q^{23} -3.49908 q^{25} +1.21799i q^{27} +2.96447i q^{29} +1.59177i q^{31} -2.86158 q^{33} -10.0266 q^{35} -3.24673i q^{37} -11.6582i q^{39} -1.29097i q^{41} -7.24180 q^{43} -7.22908i q^{45} +3.49672 q^{47} -4.82860 q^{49} +(9.58779 + 1.10863i) q^{51} -11.1260 q^{53} -3.56380 q^{55} +7.90961i q^{57} +1.00000 q^{59} -12.7963i q^{61} -8.52832i q^{63} -14.5191i q^{65} -10.5687 q^{67} -1.04592 q^{69} -0.203780i q^{71} +13.1593i q^{73} -8.19090i q^{75} -4.20431 q^{77} +10.4001i q^{79} -10.2902 q^{81} +7.96025 q^{83} +(11.9406 + 1.38068i) q^{85} -6.93945 q^{87} -8.89649 q^{89} -17.1285i q^{91} -3.72614 q^{93} +9.85061i q^{95} -11.9595i q^{97} -3.03127i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$			2.34087i			1.35150i	0.737129	+	0.675752i	$0.236181\pi$
$3$			2.34087i			1.35150i	−0.737129	+	0.675752i	$0.763819\pi$
$4$		0			0
$5$			2.91532i			1.30377i	0.758318	+	0.651885i	$0.226021\pi$
$5$			2.91532i			1.30377i	−0.758318	+	0.651885i	$0.773979\pi$
$6$		0			0
$7$			3.43927i			1.29992i	0.759967	+	0.649962i	$0.225215\pi$
$7$			3.43927i			1.29992i	−0.759967	+	0.649962i	$0.774785\pi$
$8$		0			0
$9$	−2.47969			−0.826563
$10$		0			0
$11$			1.22244i			0.368580i	0.982872	+	0.184290i	$0.0589985\pi$
$11$			1.22244i			0.368580i	−0.982872	+	0.184290i	$0.941001\pi$
$12$		0			0
$13$	−4.98028			−1.38128			−0.690640	−	0.723198i	$-0.742671\pi$
$13$	−4.98028			−1.38128			−0.690640	+	0.723198i	$0.742671\pi$
$14$		0			0
$15$	−6.82439			−1.76205
$16$		0			0
$17$	0.473595	−	4.09582i	0.114864	−	0.993381i
$17$	0.473595	−	4.09582i	0.114864	−	0.993381i
$18$		0			0
$19$	3.37891			0.775176			0.387588	−	0.921833i	$-0.373308\pi$
$19$	3.37891			0.775176			0.387588	+	0.921833i	$0.373308\pi$
$20$		0			0
$21$	−8.05090			−1.75685
$22$		0			0
$23$			0.446807i			0.0931656i	0.998914	+	0.0465828i	$0.0148331\pi$
$23$			0.446807i			0.0931656i	−0.998914	+	0.0465828i	$0.985167\pi$
$24$		0			0
$25$	−3.49908			−0.699816
$26$		0			0
$27$			1.21799i			0.234401i
$28$		0			0
$29$			2.96447i			0.550489i	0.961374	+	0.275244i	$0.0887587\pi$
$29$			2.96447i			0.550489i	−0.961374	+	0.275244i	$0.911241\pi$
$30$		0			0
$31$			1.59177i			0.285891i	0.989731	+	0.142945i	$0.0456573\pi$
$31$			1.59177i			0.285891i	−0.989731	+	0.142945i	$0.954343\pi$
$32$		0			0
$33$	−2.86158			−0.498137
$34$		0			0
$35$	−10.0266			−1.69480
$36$		0			0
$37$		−	3.24673i		−	0.533759i	−0.963730	−	0.266880i	$-0.914007\pi$
$37$		−	3.24673i		−	0.533759i	0.963730	−	0.266880i	$-0.0859926\pi$
$38$		0			0
$39$		−	11.6582i		−	1.86681i
$40$		0			0
$41$		−	1.29097i		−	0.201615i	−0.994906	−	0.100807i	$-0.967857\pi$
$41$		−	1.29097i		−	0.201615i	0.994906	−	0.100807i	$-0.0321426\pi$
$42$		0			0
$43$	−7.24180			−1.10436			−0.552182	−	0.833724i	$-0.686205\pi$
$43$	−7.24180			−1.10436			−0.552182	+	0.833724i	$0.686205\pi$
$44$		0			0
$45$		−	7.22908i		−	1.07765i
$46$		0			0
$47$	3.49672			0.510050			0.255025	−	0.966935i	$-0.417916\pi$
$47$	3.49672			0.510050			0.255025	+	0.966935i	$0.417916\pi$
$48$		0			0
$49$	−4.82860			−0.689800
$50$		0			0
$51$	9.58779	+	1.10863i	1.34256	+	0.155239i
$52$		0			0
$53$	−11.1260			−1.52827			−0.764136	−	0.645055i	$-0.776834\pi$
$53$	−11.1260			−1.52827			−0.764136	+	0.645055i	$0.776834\pi$
$54$		0			0
$55$	−3.56380			−0.480543
$56$		0			0
$57$			7.90961i			1.04765i
$58$		0			0
$59$	1.00000			0.130189
$59$	1.00000			0.130189
$60$		0			0
$61$		−	12.7963i		−	1.63840i	−0.573507	−	0.819201i	$-0.694417\pi$
$61$		−	12.7963i		−	1.63840i	0.573507	−	0.819201i	$-0.305583\pi$
$62$		0			0
$63$		−	8.52832i		−	1.07447i
$64$		0			0
$65$		−	14.5191i		−	1.80087i
$66$		0			0
$67$	−10.5687			−1.29117			−0.645586	−	0.763688i	$-0.723387\pi$
$67$	−10.5687			−1.29117			−0.645586	+	0.763688i	$0.723387\pi$
$68$		0			0
$69$	−1.04592			−0.125914
$70$		0			0
$71$		−	0.203780i		−	0.0241842i	−0.999927	−	0.0120921i	$-0.996151\pi$
$71$		−	0.203780i		−	0.0241842i	0.999927	−	0.0120921i	$-0.00384914\pi$
$72$		0			0
$73$			13.1593i			1.54018i	0.637937	+	0.770088i	$0.279788\pi$
$73$			13.1593i			1.54018i	−0.637937	+	0.770088i	$0.720212\pi$
$74$		0			0
$75$		−	8.19090i		−	0.945804i
$76$		0			0
$77$	−4.20431			−0.479125
$78$		0			0
$79$			10.4001i			1.17010i	0.810996	+	0.585051i	$0.198926\pi$
$79$			10.4001i			1.17010i	−0.810996	+	0.585051i	$0.801074\pi$
$80$		0			0
$81$	−10.2902			−1.14336
$82$		0			0
$83$	7.96025			0.873751			0.436875	−	0.899522i	$-0.356085\pi$
$83$	7.96025			0.873751			0.436875	+	0.899522i	$0.356085\pi$
$84$		0			0
$85$	11.9406	+	1.38068i	1.29514	+	0.149756i
$86$		0			0
$87$	−6.93945			−0.743987
$88$		0			0
$89$	−8.89649			−0.943026			−0.471513	−	0.881859i	$-0.656292\pi$
$89$	−8.89649			−0.943026			−0.471513	+	0.881859i	$0.656292\pi$
$90$		0			0
$91$		−	17.1285i		−	1.79556i
$92$		0			0
$93$	−3.72614			−0.386382
$94$		0			0
$95$			9.85061i			1.01065i
$96$		0			0
$97$		−	11.9595i		−	1.21430i	−0.794586	−	0.607152i	$-0.792312\pi$
$97$		−	11.9595i		−	1.21430i	0.794586	−	0.607152i	$-0.207688\pi$
$98$		0			0
$99$		−	3.03127i		−	0.304654i
$100$		0			0
$101$	−4.07128			−0.405108			−0.202554	−	0.979271i	$-0.564924\pi$
$101$	−4.07128			−0.405108			−0.202554	+	0.979271i	$0.564924\pi$
$102$		0			0
$103$	18.0364			1.77718			0.888592	−	0.458699i	$-0.151684\pi$
$103$	18.0364			1.77718			0.888592	+	0.458699i	$0.151684\pi$
$104$		0			0
$105$		−	23.4709i		−	2.29053i
$106$		0			0
$107$		−	14.2888i		−	1.38135i	−0.723164	−	0.690677i	$-0.757313\pi$
$107$		−	14.2888i		−	1.38135i	0.723164	−	0.690677i	$-0.242687\pi$
$108$		0			0
$109$			15.5752i			1.49183i	0.666040	+	0.745916i	$0.267988\pi$
$109$			15.5752i			1.49183i	−0.666040	+	0.745916i	$0.732012\pi$
$110$		0			0
$111$	7.60018			0.721378
$112$		0			0
$113$		−	0.667011i		−	0.0627471i	−0.999508	−	0.0313736i	$-0.990012\pi$
$113$		−	0.667011i		−	0.0627471i	0.999508	−	0.0313736i	$-0.00998816\pi$
$114$		0			0
$115$	−1.30258			−0.121467
$116$		0			0
$117$	12.3495			1.14171
$118$		0			0
$119$	14.0866	+	1.62882i	1.29132	+	0.149314i
$120$		0			0
$121$	9.50564			0.864149
$122$		0			0
$123$	3.02199			0.272483
$124$		0			0
$125$			4.37566i			0.391371i
$126$		0			0
$127$	−6.95451			−0.617113			−0.308556	−	0.951206i	$-0.599846\pi$
$127$	−6.95451			−0.617113			−0.308556	+	0.951206i	$0.599846\pi$
$128$		0			0
$129$		−	16.9521i		−	1.49255i
$130$		0			0
$131$			7.68462i			0.671408i	0.941967	+	0.335704i	$0.108974\pi$
$131$			7.68462i			0.671408i	−0.941967	+	0.335704i	$0.891026\pi$
$132$		0			0
$133$			11.6210i			1.00767i
$134$		0			0
$135$	−3.55081			−0.305605
$136$		0			0
$137$	19.9196			1.70185			0.850924	−	0.525290i	$-0.176043\pi$
$137$	19.9196			1.70185			0.850924	+	0.525290i	$0.176043\pi$
$138$		0			0
$139$			8.29999i			0.703996i	0.936001	+	0.351998i	$0.114498\pi$
$139$			8.29999i			0.703996i	−0.936001	+	0.351998i	$0.885502\pi$
$140$		0			0
$141$			8.18539i			0.689334i
$142$		0			0
$143$		−	6.08810i		−	0.509112i
$144$		0			0
$145$	−8.64238			−0.717710
$146$		0			0
$147$		−	11.3031i		−	0.932268i
$148$		0			0
$149$	16.5265			1.35391			0.676953	−	0.736026i	$-0.263300\pi$
$149$	16.5265			1.35391			0.676953	+	0.736026i	$0.263300\pi$
$150$		0			0
$151$	21.0461			1.71271			0.856353	−	0.516391i	$-0.172725\pi$
$151$	21.0461			1.71271			0.856353	+	0.516391i	$0.172725\pi$
$152$		0			0
$153$	−1.17437	+	10.1563i	−0.0949420	+	0.821092i
$154$		0			0
$155$	−4.64052			−0.372736
$156$		0			0
$157$	−9.33570			−0.745070			−0.372535	−	0.928018i	$-0.621511\pi$
$157$	−9.33570			−0.745070			−0.372535	+	0.928018i	$0.621511\pi$
$158$		0			0
$159$		−	26.0445i		−	2.06547i
$160$		0			0
$161$	−1.53669			−0.121108
$162$		0			0
$163$		−	8.27257i		−	0.647957i	−0.946064	−	0.323979i	$-0.894979\pi$
$163$		−	8.27257i		−	0.647957i	0.946064	−	0.323979i	$-0.105021\pi$
$164$		0			0
$165$		−	8.34241i		−	0.649456i
$166$		0			0
$167$			20.2252i			1.56508i	0.622603	+	0.782538i	$0.286075\pi$
$167$			20.2252i			1.56508i	−0.622603	+	0.782538i	$0.713925\pi$
$168$		0			0
$169$	11.8032			0.907936
$170$		0			0
$171$	−8.37865			−0.640731
$172$		0			0
$173$			20.2456i			1.53925i	0.638498	+	0.769623i	$0.279556\pi$
$173$			20.2456i			1.53925i	−0.638498	+	0.769623i	$0.720444\pi$
$174$		0			0
$175$		−	12.0343i		−	0.909707i
$176$		0			0
$177$			2.34087i			0.175951i
$178$		0			0
$179$	0.893467			0.0667808			0.0333904	−	0.999442i	$-0.489370\pi$
$179$	0.893467			0.0667808			0.0333904	+	0.999442i	$0.489370\pi$
$180$		0			0
$181$		−	9.58982i		−	0.712806i	−0.934332	−	0.356403i	$-0.884003\pi$
$181$		−	9.58982i		−	0.712806i	0.934332	−	0.356403i	$-0.115997\pi$
$182$		0			0
$183$	29.9546			2.21431
$184$		0			0
$185$	9.46525			0.695899
$186$		0			0
$187$	5.00689	+	0.578942i	0.366140	+	0.0423364i
$188$		0			0
$189$	−4.18898			−0.304704
$190$		0			0
$191$	−14.4637			−1.04655			−0.523277	−	0.852163i	$-0.675291\pi$
$191$	−14.4637			−1.04655			−0.523277	+	0.852163i	$0.675291\pi$
$192$		0			0
$193$			26.2572i			1.89003i	0.327024	+	0.945016i	$0.393954\pi$
$193$			26.2572i			1.89003i	−0.327024	+	0.945016i	$0.606046\pi$
$194$		0			0
$195$	33.9874			2.43389
$196$		0			0
$197$		−	8.98094i		−	0.639865i	−0.947440	−	0.319933i	$-0.896340\pi$
$197$		−	8.98094i		−	0.639865i	0.947440	−	0.319933i	$-0.103660\pi$
$198$		0			0
$199$			2.21702i			0.157160i	0.996908	+	0.0785801i	$0.0250386\pi$
$199$			2.21702i			0.157160i	−0.996908	+	0.0785801i	$0.974961\pi$
$200$		0			0
$201$		−	24.7400i		−	1.74502i
$202$		0			0
$203$	−10.1956			−0.715593
$204$		0			0
$205$	3.76358			0.262859
$206$		0			0
$207$		−	1.10794i		−	0.0770072i
$208$		0			0
$209$			4.13052i			0.285714i
$210$		0			0
$211$		−	7.70593i		−	0.530498i	−0.964180	−	0.265249i	$-0.914546\pi$
$211$		−	7.70593i		−	0.530498i	0.964180	−	0.265249i	$-0.0854542\pi$
$212$		0			0
$213$	0.477023			0.0326851
$214$		0			0
$215$		−	21.1121i		−	1.43984i
$216$		0			0
$217$	−5.47454			−0.371636
$218$		0			0
$219$	−30.8042			−2.08155
$220$		0			0
$221$	−2.35864	+	20.3983i	−0.158659	+	1.37214i
$222$		0			0
$223$	−16.7478			−1.12152			−0.560758	−	0.827980i	$-0.689490\pi$
$223$	−16.7478			−1.12152			−0.560758	+	0.827980i	$0.689490\pi$
$224$		0			0
$225$	8.67662			0.578442
$226$		0			0
$227$		−	12.1224i		−	0.804591i	−0.915510	−	0.402296i	$-0.868212\pi$
$227$		−	12.1224i		−	0.804591i	0.915510	−	0.402296i	$-0.131788\pi$
$228$		0			0
$229$	−15.6583			−1.03473			−0.517364	−	0.855766i	$-0.673087\pi$
$229$	−15.6583			−1.03473			−0.517364	+	0.855766i	$0.673087\pi$
$230$		0			0
$231$		−	9.84175i		−	0.647540i
$232$		0			0
$233$		−	2.53784i		−	0.166259i	−0.996539	−	0.0831297i	$-0.973508\pi$
$233$		−	2.53784i		−	0.166259i	0.996539	−	0.0831297i	$-0.0264916\pi$
$234$		0			0
$235$			10.1941i			0.664987i
$236$		0			0
$237$	−24.3453			−1.58140
$238$		0			0
$239$	16.2543			1.05140			0.525702	−	0.850669i	$-0.323803\pi$
$239$	16.2543			1.05140			0.525702	+	0.850669i	$0.323803\pi$
$240$		0			0
$241$		−	3.83534i		−	0.247056i	−0.992341	−	0.123528i	$-0.960579\pi$
$241$		−	3.83534i		−	0.247056i	0.992341	−	0.123528i	$-0.0394208\pi$
$242$		0			0
$243$		−	20.4341i		−	1.31085i
$244$		0			0
$245$		−	14.0769i		−	0.899341i
$246$		0			0
$247$	−16.8279			−1.07074
$248$		0			0
$249$			18.6339i			1.18088i
$250$		0			0
$251$	−17.2399			−1.08817			−0.544087	−	0.839029i	$-0.683124\pi$
$251$	−17.2399			−1.08817			−0.544087	+	0.839029i	$0.683124\pi$
$252$		0			0
$253$	−0.546195			−0.0343390
$254$		0			0
$255$	−3.23200	+	27.9514i	−0.202396	+	1.75039i
$256$		0			0
$257$	18.8519			1.17595			0.587974	−	0.808880i	$-0.299926\pi$
$257$	18.8519			1.17595			0.587974	+	0.808880i	$0.299926\pi$
$258$		0			0
$259$	11.1664			0.693846
$260$		0			0
$261$		−	7.35096i		−	0.455013i
$262$		0			0
$263$	−19.0906			−1.17717			−0.588587	−	0.808434i	$-0.700316\pi$
$263$	−19.0906			−1.17717			−0.588587	+	0.808434i	$0.700316\pi$
$264$		0			0
$265$		−	32.4358i		−	1.99252i
$266$		0			0
$267$		−	20.8255i		−	1.27450i
$268$		0			0
$269$		−	1.57492i		−	0.0960246i	−0.998847	−	0.0480123i	$-0.984711\pi$
$269$		−	1.57492i		−	0.0960246i	0.998847	−	0.0480123i	$-0.0152887\pi$
$270$		0			0
$271$	27.1833			1.65127			0.825634	−	0.564206i	$-0.190818\pi$
$271$	27.1833			1.65127			0.825634	+	0.564206i	$0.190818\pi$
$272$		0			0
$273$	40.0957			2.42670
$274$		0			0
$275$		−	4.27742i		−	0.257938i
$276$		0			0
$277$			2.94937i			0.177211i	0.996067	+	0.0886053i	$0.0282410\pi$
$277$			2.94937i			0.177211i	−0.996067	+	0.0886053i	$0.971759\pi$
$278$		0			0
$279$		−	3.94710i		−	0.236307i
$280$		0			0
$281$	17.6640			1.05375			0.526873	−	0.849944i	$-0.323364\pi$
$281$	17.6640			1.05375			0.526873	+	0.849944i	$0.323364\pi$
$282$		0			0
$283$		−	18.3900i		−	1.09317i	−0.837403	−	0.546587i	$-0.815927\pi$
$283$		−	18.3900i		−	1.09317i	0.837403	−	0.546587i	$-0.184073\pi$
$284$		0			0
$285$	−23.0590			−1.36590
$286$		0			0
$287$	4.43998			0.262084
$288$		0			0
$289$	−16.5514	−	3.87952i	−0.973613	−	0.228207i
$290$		0			0
$291$	27.9957			1.64114
$292$		0			0
$293$	−23.6641			−1.38247			−0.691235	−	0.722630i	$-0.742933\pi$
$293$	−23.6641			−1.38247			−0.691235	+	0.722630i	$0.742933\pi$
$294$		0			0
$295$			2.91532i			0.169736i
$296$		0			0
$297$	−1.48892			−0.0863956
$298$		0			0
$299$		−	2.22522i		−	0.128688i
$300$		0			0
$301$		−	24.9065i		−	1.43559i
$302$		0			0
$303$		−	9.53035i		−	0.547505i
$304$		0			0
$305$	37.3054			2.13610
$306$		0			0
$307$	11.3394			0.647171			0.323586	−	0.946199i	$-0.395112\pi$
$307$	11.3394			0.647171			0.323586	+	0.946199i	$0.395112\pi$
$308$		0			0
$309$			42.2210i			2.40187i
$310$		0			0
$311$			17.0112i			0.964616i	0.876002	+	0.482308i	$0.160201\pi$
$311$			17.0112i			0.964616i	−0.876002	+	0.482308i	$0.839799\pi$
$312$		0			0
$313$			2.37527i			0.134258i	0.997744	+	0.0671291i	$0.0213839\pi$
$313$			2.37527i			0.134258i	−0.997744	+	0.0671291i	$0.978616\pi$
$314$		0			0
$315$	24.8628			1.40086
$316$		0			0
$317$			11.7504i			0.659969i	0.943986	+	0.329985i	$0.107044\pi$
$317$			11.7504i			0.659969i	−0.943986	+	0.329985i	$0.892956\pi$
$318$		0			0
$319$	−3.62389			−0.202899
$320$		0			0
$321$	33.4483			1.86690
$322$		0			0
$323$	1.60024	−	13.8394i	0.0890395	−	0.770045i
$324$		0			0
$325$	17.4264			0.966642
$326$		0			0
$327$	−36.4595			−2.01622
$328$		0			0
$329$			12.0262i			0.663025i
$330$		0			0
$331$	−4.77937			−0.262698			−0.131349	−	0.991336i	$-0.541931\pi$
$331$	−4.77937			−0.262698			−0.131349	+	0.991336i	$0.541931\pi$
$332$		0			0
$333$			8.05088i			0.441185i
$334$		0			0
$335$		−	30.8111i		−	1.68339i
$336$		0			0
$337$		−	14.3814i		−	0.783406i	−0.920092	−	0.391703i	$-0.871886\pi$
$337$		−	14.3814i		−	0.783406i	0.920092	−	0.391703i	$-0.128114\pi$
$338$		0			0
$339$	1.56139			0.0848030
$340$		0			0
$341$	−1.94585			−0.105374
$342$		0			0
$343$			7.46803i			0.403236i
$344$		0			0
$345$		−	3.04918i		−	0.164163i
$346$		0			0
$347$		−	16.1054i		−	0.864585i	−0.901733	−	0.432293i	$-0.857705\pi$
$347$		−	16.1054i		−	0.864585i	0.901733	−	0.432293i	$-0.142295\pi$
$348$		0			0
$349$	10.3672			0.554945			0.277473	−	0.960734i	$-0.410503\pi$
$349$	10.3672			0.554945			0.277473	+	0.960734i	$0.410503\pi$
$350$		0			0
$351$		−	6.06591i		−	0.323774i
$352$		0			0
$353$	−17.2322			−0.917177			−0.458588	−	0.888649i	$-0.651645\pi$
$353$	−17.2322			−0.917177			−0.458588	+	0.888649i	$0.651645\pi$
$354$		0			0
$355$	0.594084			0.0315307
$356$		0			0
$357$	−3.81287	+	32.9750i	−0.201798	+	1.74522i
$358$		0			0
$359$	−19.6027			−1.03459			−0.517297	−	0.855806i	$-0.673062\pi$
$359$	−19.6027			−1.03459			−0.517297	+	0.855806i	$0.673062\pi$
$360$		0			0
$361$	−7.58295			−0.399102
$362$		0			0
$363$			22.2515i			1.16790i
$364$		0			0
$365$	−38.3635			−2.00804
$366$		0			0
$367$		−	5.25915i		−	0.274525i	−0.990535	−	0.137263i	$-0.956170\pi$
$367$		−	5.25915i		−	0.274525i	0.990535	−	0.137263i	$-0.0438304\pi$
$368$		0			0
$369$			3.20119i			0.166647i
$370$		0			0
$371$		−	38.2653i		−	1.98664i
$372$		0			0
$373$	−3.75747			−0.194554			−0.0972772	−	0.995257i	$-0.531013\pi$
$373$	−3.75747			−0.194554			−0.0972772	+	0.995257i	$0.531013\pi$
$374$		0			0
$375$	−10.2429			−0.528939
$376$		0			0
$377$		−	14.7639i		−	0.760379i
$378$		0			0
$379$			26.7616i			1.37465i	0.726349	+	0.687326i	$0.241216\pi$
$379$			26.7616i			1.37465i	−0.726349	+	0.687326i	$0.758784\pi$
$380$		0			0
$381$		−	16.2796i		−	0.834030i
$382$		0			0
$383$	13.4615			0.687853			0.343926	−	0.938997i	$-0.388243\pi$
$383$	13.4615			0.687853			0.343926	+	0.938997i	$0.388243\pi$
$384$		0			0
$385$		−	12.2569i		−	0.624669i
$386$		0			0
$387$	17.9574			0.912826
$388$		0			0
$389$	−13.3494			−0.676839			−0.338420	−	0.940995i	$-0.609892\pi$
$389$	−13.3494			−0.676839			−0.338420	+	0.940995i	$0.609892\pi$
$390$		0			0
$391$	1.83004	+	0.211605i	0.0925490	+	0.0107013i
$392$		0			0
$393$	−17.9887			−0.907411
$394$		0			0
$395$	−30.3196			−1.52554
$396$		0			0
$397$			0.760906i			0.0381888i	0.999818	+	0.0190944i	$0.00607831\pi$
$397$			0.760906i			0.0381888i	−0.999818	+	0.0190944i	$0.993922\pi$
$398$		0			0
$399$	−27.2033			−1.36187
$400$		0			0
$401$			28.2339i			1.40993i	0.709241	+	0.704966i	$0.249038\pi$
$401$			28.2339i			1.40993i	−0.709241	+	0.704966i	$0.750962\pi$
$402$		0			0
$403$		−	7.92747i		−	0.394895i
$404$		0			0
$405$		−	29.9992i		−	1.49067i
$406$		0			0
$407$	3.96894			0.196733
$408$		0			0
$409$	0.346970			0.0171565			0.00857827	−	0.999963i	$-0.497269\pi$
$409$	0.346970			0.0171565			0.00857827	+	0.999963i	$0.497269\pi$
$410$		0			0
$411$			46.6293i			2.30005i
$412$		0			0
$413$			3.43927i			0.169236i
$414$		0			0
$415$			23.2067i			1.13917i
$416$		0			0
$417$	−19.4292			−0.951453
$418$		0			0
$419$		−	35.4146i		−	1.73012i	−0.501671	−	0.865059i	$-0.667281\pi$
$419$		−	35.4146i		−	1.73012i	0.501671	−	0.865059i	$-0.332719\pi$
$420$		0			0
$421$	2.57944			0.125714			0.0628572	−	0.998023i	$-0.479979\pi$
$421$	2.57944			0.125714			0.0628572	+	0.998023i	$0.479979\pi$
$422$		0			0
$423$	−8.67078			−0.421588
$424$		0			0
$425$	−1.65715	+	14.3316i	−0.0803834	+	0.695184i
$426$		0			0
$427$	44.0101			2.12980
$428$		0			0
$429$	14.2515			0.688067
$430$		0			0
$431$			23.1050i			1.11293i	0.830872	+	0.556464i	$0.187842\pi$
$431$			23.1050i			1.11293i	−0.830872	+	0.556464i	$0.812158\pi$
$432$		0			0
$433$	−33.8999			−1.62913			−0.814563	−	0.580075i	$-0.803023\pi$
$433$	−33.8999			−1.62913			−0.814563	+	0.580075i	$0.803023\pi$
$434$		0			0
$435$		−	20.2307i		−	0.969988i
$436$		0			0
$437$			1.50972i			0.0722198i
$438$		0			0
$439$		−	3.45855i		−	0.165067i	−0.996588	−	0.0825337i	$-0.973699\pi$
$439$		−	3.45855i		−	0.165067i	0.996588	−	0.0825337i	$-0.0263012\pi$
$440$		0			0
$441$	11.9734			0.570163
$442$		0			0
$443$	14.5483			0.691210			0.345605	−	0.938380i	$-0.387674\pi$
$443$	14.5483			0.691210			0.345605	+	0.938380i	$0.387674\pi$
$444$		0			0
$445$		−	25.9361i		−	1.22949i
$446$		0			0
$447$			38.6865i			1.82981i
$448$		0			0
$449$			11.4645i			0.541045i	0.962714	+	0.270523i	$0.0871965\pi$
$449$			11.4645i			0.541045i	−0.962714	+	0.270523i	$0.912803\pi$
$450$		0			0
$451$	1.57813			0.0743112
$452$		0			0
$453$			49.2662i			2.31473i
$454$		0			0
$455$	49.9351			2.34100
$456$		0			0
$457$	−26.4125			−1.23553			−0.617763	−	0.786364i	$-0.711961\pi$
$457$	−26.4125			−1.23553			−0.617763	+	0.786364i	$0.711961\pi$
$458$		0			0
$459$	4.98864	+	0.576832i	0.232850	+	0.0269242i
$460$		0			0
$461$	−9.35500			−0.435706			−0.217853	−	0.975982i	$-0.569905\pi$
$461$	−9.35500			−0.435706			−0.217853	+	0.975982i	$0.569905\pi$
$462$		0			0
$463$	10.5912			0.492216			0.246108	−	0.969242i	$-0.420848\pi$
$463$	10.5912			0.492216			0.246108	+	0.969242i	$0.420848\pi$
$464$		0			0
$465$		−	10.8629i		−	0.503754i
$466$		0			0
$467$	2.63760			0.122053			0.0610267	−	0.998136i	$-0.480563\pi$
$467$	2.63760			0.122053			0.0610267	+	0.998136i	$0.480563\pi$
$468$		0			0
$469$		−	36.3486i		−	1.67842i
$470$		0			0
$471$		−	21.8537i		−	1.00696i
$472$		0			0
$473$		−	8.85267i		−	0.407046i
$474$		0			0
$475$	−11.8231			−0.542480
$476$		0			0
$477$	27.5890			1.26321
$478$		0			0
$479$			19.9390i			0.911034i	0.890227	+	0.455517i	$0.150546\pi$
$479$			19.9390i			0.911034i	−0.890227	+	0.455517i	$0.849454\pi$
$480$		0			0
$481$			16.1696i			0.737271i
$482$		0			0
$483$		−	3.59720i		−	0.163678i
$484$		0			0
$485$	34.8657			1.58317
$486$		0			0
$487$		−	2.36250i		−	0.107055i	−0.998566	−	0.0535276i	$-0.982954\pi$
$487$		−	2.36250i		−	0.107055i	0.998566	−	0.0535276i	$-0.0170465\pi$
$488$		0			0
$489$	19.3650			0.875717
$490$		0			0
$491$	−30.9178			−1.39530			−0.697649	−	0.716439i	$-0.745771\pi$
$491$	−30.9178			−1.39530			−0.697649	+	0.716439i	$0.745771\pi$
$492$		0			0
$493$	12.1419	+	1.40396i	0.546845	+	0.0632311i
$494$		0			0
$495$	8.83712			0.397199
$496$		0			0
$497$	0.700855			0.0314377
$498$		0			0
$499$		−	33.2392i		−	1.48799i	−0.668184	−	0.743996i	$-0.732928\pi$
$499$		−	33.2392i		−	1.48799i	0.668184	−	0.743996i	$-0.267072\pi$
$500$		0			0
$501$	−47.3447			−2.11521
$502$		0			0
$503$		−	17.4723i		−	0.779053i	−0.921015	−	0.389527i	$-0.872639\pi$
$503$		−	17.4723i		−	0.779053i	0.921015	−	0.389527i	$-0.127361\pi$
$504$		0			0
$505$		−	11.8691i		−	0.528167i
$506$		0			0
$507$			27.6297i			1.22708i
$508$		0			0
$509$	−5.67789			−0.251668			−0.125834	−	0.992051i	$-0.540161\pi$
$509$	−5.67789			−0.251668			−0.125834	+	0.992051i	$0.540161\pi$
$510$		0			0
$511$	−45.2583			−2.00211
$512$		0			0
$513$			4.11547i			0.181702i
$514$		0			0
$515$			52.5820i			2.31704i
$516$		0			0
$517$			4.27454i			0.187994i
$518$		0			0
$519$	−47.3925			−2.08030
$520$		0			0
$521$			42.4200i			1.85845i	0.369508	+	0.929227i	$0.379526\pi$
$521$			42.4200i			1.85845i	−0.369508	+	0.929227i	$0.620474\pi$
$522$		0			0
$523$	0.0969511			0.00423938			0.00211969	−	0.999998i	$-0.499325\pi$
$523$	0.0969511			0.00423938			0.00211969	+	0.999998i	$0.499325\pi$
$524$		0			0
$525$	28.1708			1.22947
$526$		0			0
$527$	6.51961	+	0.753856i	0.283999	+	0.0328385i
$528$		0			0
$529$	22.8004			0.991320
$530$		0			0
$531$	−2.47969			−0.107609
$532$		0			0
$533$			6.42937i			0.278487i
$534$		0			0
$535$	41.6565			1.80097
$536$		0			0
$537$			2.09149i			0.0902546i
$538$		0			0
$539$		−	5.90268i		−	0.254246i
$540$		0			0
$541$			14.2023i			0.610603i	0.952256	+	0.305302i	$0.0987573\pi$
$541$			14.2023i			0.610603i	−0.952256	+	0.305302i	$0.901243\pi$
$542$		0			0
$543$	22.4486			0.963360
$544$		0			0
$545$	−45.4066			−1.94500
$546$		0			0
$547$		−	17.5292i		−	0.749494i	−0.927127	−	0.374747i	$-0.877729\pi$
$547$		−	17.5292i		−	0.749494i	0.927127	−	0.374747i	$-0.122271\pi$
$548$		0			0
$549$			31.7309i			1.35424i
$550$		0			0
$551$			10.0167i			0.426725i
$552$		0			0
$553$	−35.7688			−1.52104
$554$		0			0
$555$			22.1570i			0.940510i
$556$		0			0
$557$	2.87630			0.121873			0.0609363	−	0.998142i	$-0.480591\pi$
$557$	2.87630			0.121873			0.0609363	+	0.998142i	$0.480591\pi$
$558$		0			0
$559$	36.0662			1.52544
$560$		0			0
$561$	−1.35523	+	11.7205i	−0.0572179	+	0.494840i
$562$		0			0
$563$	−41.7131			−1.75800			−0.878998	−	0.476825i	$-0.841787\pi$
$563$	−41.7131			−1.75800			−0.878998	+	0.476825i	$0.841787\pi$
$564$		0			0
$565$	1.94455			0.0818078
$566$		0			0
$567$		−	35.3909i		−	1.48628i
$568$		0			0
$569$	21.2573			0.891150			0.445575	−	0.895245i	$-0.352999\pi$
$569$	21.2573			0.891150			0.445575	+	0.895245i	$0.352999\pi$
$570$		0			0
$571$		−	34.3672i		−	1.43822i	−0.694894	−	0.719112i	$-0.744549\pi$
$571$		−	34.3672i		−	1.43822i	0.694894	−	0.719112i	$-0.255451\pi$
$572$		0			0
$573$		−	33.8576i		−	1.41442i
$574$		0			0
$575$		−	1.56341i		−	0.0651988i
$576$		0			0
$577$	16.4478			0.684730			0.342365	−	0.939567i	$-0.388772\pi$
$577$	16.4478			0.684730			0.342365	+	0.939567i	$0.388772\pi$
$578$		0			0
$579$	−61.4647			−2.55439
$580$		0			0
$581$			27.3775i			1.13581i
$582$		0			0
$583$		−	13.6009i		−	0.563290i
$584$		0			0
$585$			36.0028i			1.48853i
$586$		0			0
$587$	−19.3375			−0.798143			−0.399071	−	0.916920i	$-0.630667\pi$
$587$	−19.3375			−0.798143			−0.399071	+	0.916920i	$0.630667\pi$
$588$		0			0
$589$			5.37846i			0.221616i
$590$		0			0
$591$	21.0232			0.864781
$592$		0			0
$593$	−29.0165			−1.19156			−0.595782	−	0.803146i	$-0.703158\pi$
$593$	−29.0165			−1.19156			−0.595782	+	0.803146i	$0.703158\pi$
$594$		0			0
$595$	−4.74854	+	41.0670i	−0.194671	+	1.68358i
$596$		0			0
$597$	−5.18975			−0.212403
$598$		0			0
$599$	11.2839			0.461046			0.230523	−	0.973067i	$-0.425956\pi$
$599$	11.2839			0.461046			0.230523	+	0.973067i	$0.425956\pi$
$600$		0			0
$601$		−	11.9053i		−	0.485628i	−0.970073	−	0.242814i	$-0.921930\pi$
$601$		−	11.9053i		−	0.485628i	0.970073	−	0.242814i	$-0.0780705\pi$
$602$		0			0
$603$	26.2071			1.06723
$604$		0			0
$605$			27.7120i			1.12665i
$606$		0			0
$607$		−	21.2225i		−	0.861393i	−0.902497	−	0.430696i	$-0.858268\pi$
$607$		−	21.2225i		−	0.861393i	0.902497	−	0.430696i	$-0.141732\pi$
$608$		0			0
$609$		−	23.8667i		−	0.967126i
$610$		0			0
$611$	−17.4147			−0.704521
$612$		0			0
$613$	6.73020			0.271830			0.135915	−	0.990720i	$-0.456603\pi$
$613$	6.73020			0.271830			0.135915	+	0.990720i	$0.456603\pi$
$614$		0			0
$615$			8.81005i			0.355256i
$616$		0			0
$617$			36.3550i			1.46360i	0.681521	+	0.731798i	$0.261319\pi$
$617$			36.3550i			1.46360i	−0.681521	+	0.731798i	$0.738681\pi$
$618$		0			0
$619$		−	24.5215i		−	0.985604i	−0.870141	−	0.492802i	$-0.835973\pi$
$619$		−	24.5215i		−	0.985604i	0.870141	−	0.492802i	$-0.164027\pi$
$620$		0			0
$621$	−0.544204			−0.0218382
$622$		0			0
$623$		−	30.5975i		−	1.22586i
$624$		0			0
$625$	−30.2518			−1.21007
$626$		0			0
$627$	−9.66903			−0.386144
$628$		0			0
$629$	−13.2980	−	1.53764i	−0.530226	−	0.0613095i
$630$		0			0
$631$	46.8028			1.86319			0.931594	−	0.363500i	$-0.118418\pi$
$631$	46.8028			1.86319			0.931594	+	0.363500i	$0.118418\pi$
$632$		0			0
$633$	18.0386			0.716970
$634$		0			0
$635$		−	20.2746i		−	0.804573i
$636$		0			0
$637$	24.0478			0.952808
$638$		0			0
$639$			0.505311i			0.0199898i
$640$		0			0
$641$			24.5811i			0.970894i	0.874266	+	0.485447i	$0.161343\pi$
$641$			24.5811i			0.970894i	−0.874266	+	0.485447i	$0.838657\pi$
$642$		0			0
$643$		−	14.0089i		−	0.552458i	−0.961092	−	0.276229i	$-0.910915\pi$
$643$		−	14.0089i		−	0.552458i	0.961092	−	0.276229i	$-0.0890848\pi$
$644$		0			0
$645$	49.4208			1.94594
$646$		0			0
$647$	−0.524439			−0.0206178			−0.0103089	−	0.999947i	$-0.503281\pi$
$647$	−0.524439			−0.0206178			−0.0103089	+	0.999947i	$0.503281\pi$
$648$		0			0
$649$			1.22244i			0.0479850i
$650$		0			0
$651$		−	12.8152i		−	0.502267i
$652$		0			0
$653$			42.7492i			1.67290i	0.548040	+	0.836452i	$0.315374\pi$
$653$			42.7492i			1.67290i	−0.548040	+	0.836452i	$0.684626\pi$
$654$		0			0
$655$	−22.4031			−0.875362
$656$		0			0
$657$		−	32.6309i		−	1.27305i
$658$		0			0
$659$	17.5822			0.684906			0.342453	−	0.939535i	$-0.388742\pi$
$659$	17.5822			0.684906			0.342453	+	0.939535i	$0.388742\pi$
$660$		0			0
$661$	−29.1880			−1.13528			−0.567642	−	0.823276i	$-0.692144\pi$
$661$	−29.1880			−1.13528			−0.567642	+	0.823276i	$0.692144\pi$
$662$		0			0
$663$	−47.7498	−	5.52127i	−1.85445	−	0.214428i
$664$		0			0
$665$	−33.8789			−1.31377
$666$		0			0
$667$	−1.32455			−0.0512866
$668$		0			0
$669$		−	39.2045i		−	1.51573i
$670$		0			0
$671$	15.6427			0.603882
$672$		0			0
$673$		−	40.5799i		−	1.56424i	−0.623128	−	0.782120i	$-0.714139\pi$
$673$		−	40.5799i		−	1.56424i	0.623128	−	0.782120i	$-0.285861\pi$
$674$		0			0
$675$		−	4.26183i		−	0.164038i
$676$		0			0
$677$			46.8767i			1.80162i	0.434217	+	0.900808i	$0.357025\pi$
$677$			46.8767i			1.80162i	−0.434217	+	0.900808i	$0.642975\pi$
$678$		0			0
$679$	41.1320			1.57850
$680$		0			0
$681$	28.3770			1.08741
$682$		0			0
$683$			40.3634i			1.54446i	0.635341	+	0.772231i	$0.280859\pi$
$683$			40.3634i			1.54446i	−0.635341	+	0.772231i	$0.719141\pi$
$684$		0			0
$685$			58.0720i			2.21882i
$686$		0			0
$687$		−	36.6540i		−	1.39844i
$688$		0			0
$689$	55.4105			2.11097
$690$		0			0
$691$		−	41.0163i		−	1.56033i	−0.625572	−	0.780166i	$-0.715134\pi$
$691$		−	41.0163i		−	1.56033i	0.625572	−	0.780166i	$-0.284866\pi$
$692$		0			0
$693$	10.4254			0.396027
$694$		0			0
$695$	−24.1971			−0.917848
$696$		0			0
$697$	−5.28756	−	0.611395i	−0.200281	−	0.0231582i
$698$		0			0
$699$	5.94076			0.224700
$700$		0			0
$701$	−12.4313			−0.469523			−0.234761	−	0.972053i	$-0.575431\pi$
$701$	−12.4313			−0.469523			−0.234761	+	0.972053i	$0.575431\pi$
$702$		0			0
$703$		−	10.9704i		−	0.413757i
$704$		0			0
$705$	−23.8630			−0.898733
$706$		0			0
$707$		−	14.0023i		−	0.526609i
$708$		0			0
$709$			3.82521i			0.143659i	0.997417	+	0.0718293i	$0.0228837\pi$
$709$			3.82521i			0.143659i	−0.997417	+	0.0718293i	$0.977116\pi$
$710$		0			0
$711$		−	25.7890i		−	0.967163i
$712$		0			0
$713$	−0.711215			−0.0266352
$714$		0			0
$715$	17.7487			0.663765
$716$		0			0
$717$			38.0493i			1.42098i
$718$		0			0
$719$		−	14.1491i		−	0.527671i	−0.964568	−	0.263835i	$-0.915012\pi$
$719$		−	14.1491i		−	0.527671i	0.964568	−	0.263835i	$-0.0849876\pi$
$720$		0			0
$721$			62.0323i			2.31020i
$722$		0			0
$723$	8.97804			0.333897
$724$		0			0
$725$		−	10.3729i		−	0.385241i
$726$		0			0
$727$	−36.9837			−1.37165			−0.685825	−	0.727766i	$-0.740559\pi$
$727$	−36.9837			−1.37165			−0.685825	+	0.727766i	$0.740559\pi$
$728$		0			0
$729$	16.9631			0.628262
$730$		0			0
$731$	−3.42968	+	29.6611i	−0.126851	+	1.09705i
$732$		0			0
$733$	−8.78570			−0.324507			−0.162254	−	0.986749i	$-0.551876\pi$
$733$	−8.78570			−0.324507			−0.162254	+	0.986749i	$0.551876\pi$
$734$		0			0
$735$	32.9523			1.21546
$736$		0			0
$737$		−	12.9196i		−	0.475900i
$738$		0			0
$739$	−32.3626			−1.19048			−0.595239	−	0.803549i	$-0.702943\pi$
$739$	−32.3626			−1.19048			−0.595239	+	0.803549i	$0.702943\pi$
$740$		0			0
$741$		−	39.3920i		−	1.44710i
$742$		0			0
$743$		−	8.55664i		−	0.313913i	−0.987606	−	0.156956i	$-0.949832\pi$
$743$		−	8.55664i		−	0.313913i	0.987606	−	0.156956i	$-0.0501682\pi$
$744$		0			0
$745$			48.1801i			1.76518i
$746$		0			0
$747$	−19.7389			−0.722210
$748$		0			0
$749$	49.1432			1.79565
$750$		0			0
$751$			14.5093i			0.529451i	0.964324	+	0.264725i	$0.0852813\pi$
$751$			14.5093i			0.529451i	−0.964324	+	0.264725i	$0.914719\pi$
$752$		0			0
$753$		−	40.3565i		−	1.47067i
$754$		0			0
$755$			61.3560i			2.23297i
$756$		0			0
$757$	−14.3336			−0.520963			−0.260482	−	0.965479i	$-0.583881\pi$
$757$	−14.3336			−0.520963			−0.260482	+	0.965479i	$0.583881\pi$
$758$		0			0
$759$		−	1.27857i		−	0.0464093i
$760$		0			0
$761$	13.8935			0.503638			0.251819	−	0.967774i	$-0.418971\pi$
$761$	13.8935			0.503638			0.251819	+	0.967774i	$0.418971\pi$
$762$		0			0
$763$	−53.5673			−1.93927
$764$		0			0
$765$	−29.6090	−	3.42366i	−1.07051	−	0.123783i
$766$		0			0
$767$	−4.98028			−0.179827
$768$		0			0
$769$	−4.68199			−0.168837			−0.0844185	−	0.996430i	$-0.526903\pi$
$769$	−4.68199			−0.168837			−0.0844185	+	0.996430i	$0.526903\pi$
$770$		0			0
$771$			44.1298i			1.58930i
$772$		0			0
$773$	−51.7340			−1.86074			−0.930371	−	0.366619i	$-0.880515\pi$
$773$	−51.7340			−1.86074			−0.930371	+	0.366619i	$0.880515\pi$
$774$		0			0
$775$		−	5.56974i		−	0.200071i
$776$		0			0
$777$			26.1391i			0.937735i
$778$		0			0
$779$		−	4.36206i		−	0.156287i
$780$		0			0
$781$	0.249109			0.00891383
$782$		0			0
$783$	−3.61068			−0.129035
$784$		0			0
$785$		−	27.2165i		−	0.971399i
$786$		0			0
$787$			8.90248i			0.317339i	0.987332	+	0.158670i	$0.0507205\pi$
$787$			8.90248i			0.317339i	−0.987332	+	0.158670i	$0.949280\pi$
$788$		0			0
$789$		−	44.6886i		−	1.59095i
$790$		0			0
$791$	2.29403			0.0815665
$792$		0			0
$793$			63.7292i			2.26309i
$794$		0			0
$795$	75.9281			2.69289
$796$		0			0
$797$	6.23576			0.220882			0.110441	−	0.993883i	$-0.464774\pi$
$797$	6.23576			0.220882			0.110441	+	0.993883i	$0.464774\pi$
$798$		0			0
$799$	1.65603	−	14.3219i	0.0585862	−	0.506674i
$800$		0			0
$801$	22.0605			0.779470
$802$		0			0
$803$	−16.0864			−0.567678
$804$		0			0
$805$		−	4.47994i		−	0.157897i
$806$		0			0
$807$	3.68669			0.129778
$808$		0			0
$809$			2.53175i			0.0890115i	0.999009	+	0.0445058i	$0.0141713\pi$
$809$			2.53175i			0.0890115i	−0.999009	+	0.0445058i	$0.985829\pi$
$810$		0			0
$811$			46.1679i			1.62118i	0.585617	+	0.810588i	$0.300852\pi$
$811$			46.1679i			1.62118i	−0.585617	+	0.810588i	$0.699148\pi$
$812$		0			0
$813$			63.6327i			2.23170i
$814$		0			0
$815$	24.1172			0.844787
$816$		0			0
$817$	−24.4694			−0.856076
$818$		0			0
$819$			42.4734i			1.48414i
$820$		0			0
$821$			1.96594i			0.0686118i	0.999411	+	0.0343059i	$0.0109220\pi$
$821$			1.96594i			0.0686118i	−0.999411	+	0.0343059i	$0.989078\pi$
$822$		0			0
$823$			23.3935i			0.815447i	0.913106	+	0.407723i	$0.133677\pi$
$823$			23.3935i			0.815447i	−0.913106	+	0.407723i	$0.866323\pi$
$824$		0			0
$825$	10.0129			0.348604
$826$		0			0
$827$			16.4830i			0.573171i	0.958055	+	0.286586i	$0.0925203\pi$
$827$			16.4830i			0.573171i	−0.958055	+	0.286586i	$0.907480\pi$
$828$		0			0
$829$	0.981980			0.0341056			0.0170528	−	0.999855i	$-0.494572\pi$
$829$	0.981980			0.0341056			0.0170528	+	0.999855i	$0.494572\pi$
$830$		0			0
$831$	−6.90411			−0.239501
$832$		0			0
$833$	−2.28680	+	19.7771i	−0.0792330	+	0.685235i
$834$		0			0
$835$	−58.9630			−2.04050
$836$		0			0
$837$	−1.93876			−0.0670132
$838$		0			0
$839$			14.6226i			0.504827i	0.967620	+	0.252413i	$0.0812242\pi$
$839$			14.6226i			0.504827i	−0.967620	+	0.252413i	$0.918776\pi$
$840$		0			0
$841$	20.2119			0.696962
$842$		0			0
$843$			41.3492i			1.42414i
$844$		0			0
$845$			34.4100i			1.18374i
$846$		0			0
$847$			32.6925i			1.12333i
$848$		0			0
$849$	43.0487			1.47743
$850$		0			0
$851$	1.45066			0.0497280
$852$		0			0
$853$		−	32.8127i		−	1.12348i	−0.827312	−	0.561742i	$-0.810131\pi$
$853$		−	32.8127i		−	1.12348i	0.827312	−	0.561742i	$-0.189869\pi$
$854$		0			0
$855$		−	24.4264i		−	0.835366i
$856$		0			0
$857$		−	8.34140i		−	0.284937i	−0.989799	−	0.142468i	$-0.954496\pi$
$857$		−	8.34140i		−	0.284937i	0.989799	−	0.142468i	$-0.0455039\pi$
$858$		0			0
$859$	31.2016			1.06459			0.532293	−	0.846560i	$-0.321330\pi$
$859$	31.2016			1.06459			0.532293	+	0.846560i	$0.321330\pi$
$860$		0			0
$861$			10.3934i			0.354207i
$862$		0			0
$863$	24.7617			0.842898			0.421449	−	0.906852i	$-0.361522\pi$
$863$	24.7617			0.842898			0.421449	+	0.906852i	$0.361522\pi$
$864$		0			0
$865$	−59.0225			−2.00682
$866$		0			0
$867$	9.08146	−	38.7448i	0.308422	−	1.31584i
$868$		0			0
$869$	−12.7135			−0.431276
$870$		0			0
$871$	52.6350			1.78347
$872$		0			0
$873$			29.6558i			1.00370i
$874$		0			0
$875$	−15.0491			−0.508752
$876$		0			0
$877$			9.22262i			0.311426i	0.987802	+	0.155713i	$0.0497675\pi$
$877$			9.22262i			0.311426i	−0.987802	+	0.155713i	$0.950233\pi$
$878$		0			0
$879$		−	55.3946i		−	1.86841i
$880$		0			0
$881$		−	23.5180i		−	0.792343i	−0.918176	−	0.396172i	$-0.870339\pi$
$881$		−	23.5180i		−	0.792343i	0.918176	−	0.396172i	$-0.129661\pi$
$882$		0			0
$883$	−8.45763			−0.284622			−0.142311	−	0.989822i	$-0.545453\pi$
$883$	−8.45763			−0.284622			−0.142311	+	0.989822i	$0.545453\pi$
$884$		0			0
$885$	−6.82439			−0.229399
$886$		0			0
$887$		−	28.5802i		−	0.959628i	−0.877370	−	0.479814i	$-0.840704\pi$
$887$		−	28.5802i		−	0.959628i	0.877370	−	0.479814i	$-0.159296\pi$
$888$		0			0
$889$		−	23.9184i		−	0.802199i
$890$		0			0
$891$		−	12.5792i		−	0.421418i
$892$		0			0
$893$	11.8151			0.395378
$894$		0			0
$895$			2.60474i			0.0870669i
$896$		0			0
$897$	5.20896			0.173922
$898$		0			0
$899$	−4.71876			−0.157380
$900$		0			0
$901$	−5.26921	+	45.5700i	−0.175543	+	1.51816i
$902$		0			0
$903$	58.3030			1.94020
$904$		0			0
$905$	27.9574			0.929335
$906$		0			0
$907$			14.7963i			0.491303i	0.969358	+	0.245651i	$0.0790019\pi$
$907$			14.7963i			0.491303i	−0.969358	+	0.245651i	$0.920998\pi$
$908$		0			0
$909$	10.0955			0.334847
$910$		0			0
$911$			20.5096i			0.679514i	0.940513	+	0.339757i	$0.110345\pi$
$911$			20.5096i			0.679514i	−0.940513	+	0.339757i	$0.889655\pi$
$912$		0			0
$913$			9.73093i			0.322047i
$914$		0			0
$915$			87.3271i			2.88694i
$916$		0			0
$917$	−26.4295			−0.872779
$918$		0			0
$919$	−45.0148			−1.48490			−0.742451	−	0.669900i	$-0.766337\pi$
$919$	−45.0148			−1.48490			−0.742451	+	0.669900i	$0.766337\pi$
$920$		0			0
$921$			26.5440i			0.874654i
$922$		0			0
$923$			1.01488i			0.0334052i
$924$		0			0
$925$			11.3606i			0.373533i
$926$		0			0
$927$	−44.7247			−1.46895
$928$		0			0
$929$		−	4.84074i		−	0.158819i	−0.996842	−	0.0794097i	$-0.974696\pi$
$929$		−	4.84074i		−	0.158819i	0.996842	−	0.0794097i	$-0.0253035\pi$
$930$		0			0
$931$	−16.3154			−0.534716
$932$		0			0
$933$	−39.8210			−1.30368
$934$		0			0
$935$	−1.68780	+	14.5967i	−0.0551970	+	0.477363i
$936$		0			0
$937$	−41.8056			−1.36573			−0.682865	−	0.730545i	$-0.739266\pi$
$937$	−41.8056			−1.36573			−0.682865	+	0.730545i	$0.739266\pi$
$938$		0			0
$939$	−5.56021			−0.181450
$940$		0			0
$941$		−	28.4399i		−	0.927113i	−0.886067	−	0.463557i	$-0.846573\pi$
$941$		−	28.4399i		−	0.927113i	0.886067	−	0.463557i	$-0.153427\pi$
$942$		0			0
$943$	0.576812			0.0187836
$944$		0			0
$945$		−	12.2122i		−	0.397264i
$946$		0			0
$947$			49.6399i			1.61308i	0.591179	+	0.806541i	$0.298663\pi$
$947$			49.6399i			1.61308i	−0.591179	+	0.806541i	$0.701337\pi$
$948$		0			0
$949$		−	65.5368i		−	2.12742i
$950$		0			0
$951$	−27.5062			−0.891951
$952$		0			0
$953$	−26.1348			−0.846590			−0.423295	−	0.905992i	$-0.639127\pi$
$953$	−26.1348			−0.846590			−0.423295	+	0.905992i	$0.639127\pi$
$954$		0			0
$955$		−	42.1662i		−	1.36447i
$956$		0			0
$957$		−	8.48307i		−	0.274219i
$958$		0			0
$959$			68.5090i			2.21227i
$960$		0			0
$961$	28.4663			0.918266
$962$		0			0
$963$			35.4318i			1.14177i
$964$		0			0
$965$	−76.5480			−2.46417
$966$		0			0
$967$	6.38775			0.205416			0.102708	−	0.994712i	$-0.467249\pi$
$967$	6.38775			0.205416			0.102708	+	0.994712i	$0.467249\pi$
$968$		0			0
$969$	32.3963	+	3.74595i	1.04072	+	0.120337i
$970$		0			0
$971$	−6.23450			−0.200075			−0.100037	−	0.994984i	$-0.531896\pi$
$971$	−6.23450			−0.200075			−0.100037	+	0.994984i	$0.531896\pi$
$972$		0			0
$973$	−28.5459			−0.915140
$974$		0			0
$975$			40.7930i			1.30642i
$976$		0			0
$977$	13.3660			0.427617			0.213809	−	0.976876i	$-0.431413\pi$
$977$	13.3660			0.427617			0.213809	+	0.976876i	$0.431413\pi$
$978$		0			0
$979$		−	10.8754i		−	0.347580i
$980$		0			0
$981$		−	38.6216i		−	1.23309i
$982$		0			0
$983$			38.8224i			1.23824i	0.785296	+	0.619121i	$0.212511\pi$
$983$			38.8224i			1.23824i	−0.785296	+	0.619121i	$0.787489\pi$
$984$		0			0
$985$	26.1823			0.834237
$986$		0			0
$987$	−28.1518			−0.896081
$988$		0			0
$989$		−	3.23568i		−	0.102889i
$990$		0			0
$991$		−	26.3976i		−	0.838549i	−0.907860	−	0.419274i	$-0.862285\pi$
$991$		−	26.3976i		−	0.838549i	0.907860	−	0.419274i	$-0.137715\pi$
$992$		0			0
$993$		−	11.1879i		−	0.355037i
$994$		0			0
$995$	−6.46331			−0.204901
$996$		0			0
$997$			11.0351i			0.349484i	0.984614	+	0.174742i	$0.0559092\pi$
$997$			11.0351i			0.349484i	−0.984614	+	0.174742i	$0.944091\pi$
$998$		0			0
$999$	3.95447			0.125114

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.b.b.237.39	yes	46
17.16	even	2	inner	4012.2.b.b.237.8	✓	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.b.b.237.8	✓	46	17.16	even	2	inner
4012.2.b.b.237.39	yes	46	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+2.34087i q^{3} +2.91532i q^{5} +3.43927i q^{7} -2.47969 q^{9} +O(q^{10})\)	\(q+2.34087i q^{3} +2.91532i q^{5} +3.43927i q^{7} -2.47969 q^{9} +1.22244i q^{11} -4.98028 q^{13} -6.82439 q^{15} +(0.473595 - 4.09582i) q^{17} +3.37891 q^{19} -8.05090 q^{21} +0.446807i q^{23} -3.49908 q^{25} +1.21799i q^{27} +2.96447i q^{29} +1.59177i q^{31} -2.86158 q^{33} -10.0266 q^{35} -3.24673i q^{37} -11.6582i q^{39} -1.29097i q^{41} -7.24180 q^{43} -7.22908i q^{45} +3.49672 q^{47} -4.82860 q^{49} +(9.58779 + 1.10863i) q^{51} -11.1260 q^{53} -3.56380 q^{55} +7.90961i q^{57} +1.00000 q^{59} -12.7963i q^{61} -8.52832i q^{63} -14.5191i q^{65} -10.5687 q^{67} -1.04592 q^{69} -0.203780i q^{71} +13.1593i q^{73} -8.19090i q^{75} -4.20431 q^{77} +10.4001i q^{79} -10.2902 q^{81} +7.96025 q^{83} +(11.9406 + 1.38068i) q^{85} -6.93945 q^{87} -8.89649 q^{89} -17.1285i q^{91} -3.72614 q^{93} +9.85061i q^{95} -11.9595i q^{97} -3.03127i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 54 q^{9}+O(q^{10}) \) 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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