LMFDB - Embedded newform 4012.2.b.a.237.11

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:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			237.11
Character	$\chi$	$=$	4012.237
Dual form			4012.2.b.a.237.30

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.67902i q^{3} -0.714826i q^{5} +1.81204i q^{7} +0.180890 q^{9} +O(q^{10})$	$q-1.67902i q^{3} -0.714826i q^{5} +1.81204i q^{7} +0.180890 q^{9} -1.90342i q^{11} -0.684932 q^{13} -1.20021 q^{15} +(-2.75528 - 3.06732i) q^{17} -8.58944 q^{19} +3.04245 q^{21} +2.35240i q^{23} +4.48902 q^{25} -5.34078i q^{27} -1.67302i q^{29} +9.13920i q^{31} -3.19588 q^{33} +1.29529 q^{35} +2.45344i q^{37} +1.15001i q^{39} -5.79625i q^{41} -0.853793 q^{43} -0.129305i q^{45} -5.83666 q^{47} +3.71652 q^{49} +(-5.15009 + 4.62617i) q^{51} -11.7493 q^{53} -1.36061 q^{55} +14.4218i q^{57} -1.00000 q^{59} -0.855431i q^{61} +0.327780i q^{63} +0.489607i q^{65} -2.72694 q^{67} +3.94973 q^{69} -1.42200i q^{71} +1.06012i q^{73} -7.53716i q^{75} +3.44907 q^{77} +10.5497i q^{79} -8.42461 q^{81} -8.47593 q^{83} +(-2.19260 + 1.96954i) q^{85} -2.80903 q^{87} -17.7400 q^{89} -1.24112i q^{91} +15.3449 q^{93} +6.13995i q^{95} -14.7739i q^{97} -0.344310i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 36 q^{9}+O(q^{10}) $ 40 * q - 36 * q^9	$ 40 q - 36 q^{9} + 8 q^{13} + 20 q^{15} - 8 q^{17} + 20 q^{19} + 6 q^{21} - 24 q^{25} - 14 q^{33} - 52 q^{35} + 22 q^{43} - 10 q^{47} + 8 q^{49} - 6 q^{51} - 2 q^{53} - 12 q^{55} - 40 q^{59} - 24 q^{67} - 36 q^{69} - 38 q^{77} + 16 q^{81} + 32 q^{83} - 22 q^{85} - 18 q^{87} + 40 q^{89} + 22 q^{93}+O(q^{100}) $ 40 * q - 36 * q^9 + 8 * q^13 + 20 * q^15 - 8 * q^17 + 20 * q^19 + 6 * q^21 - 24 * q^25 - 14 * q^33 - 52 * q^35 + 22 * q^43 - 10 * q^47 + 8 * q^49 - 6 * q^51 - 2 * q^53 - 12 * q^55 - 40 * q^59 - 24 * q^67 - 36 * q^69 - 38 * q^77 + 16 * q^81 + 32 * q^83 - 22 * q^85 - 18 * q^87 + 40 * q^89 + 22 * 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
$3$		−	1.67902i		−	0.969383i	−0.874685	−	0.484691i	$-0.838932\pi$
$3$		−	1.67902i		−	0.969383i	0.874685	−	0.484691i	$-0.161068\pi$
$4$		0			0
$5$		−	0.714826i		−	0.319680i	−0.987143	−	0.159840i	$-0.948902\pi$
$5$		−	0.714826i		−	0.319680i	0.987143	−	0.159840i	$-0.0510978\pi$
$6$		0			0
$7$			1.81204i			0.684886i	0.939539	+	0.342443i	$0.111254\pi$
$7$			1.81204i			0.684886i	−0.939539	+	0.342443i	$0.888746\pi$
$8$		0			0
$9$	0.180890			0.0602967
$10$		0			0
$11$		−	1.90342i		−	0.573902i	−0.957945	−	0.286951i	$-0.907358\pi$
$11$		−	1.90342i		−	0.573902i	0.957945	−	0.286951i	$-0.0926418\pi$
$12$		0			0
$13$	−0.684932			−0.189966			−0.0949830	−	0.995479i	$-0.530280\pi$
$13$	−0.684932			−0.189966			−0.0949830	+	0.995479i	$0.530280\pi$
$14$		0			0
$15$	−1.20021			−0.309892
$16$		0			0
$17$	−2.75528	−	3.06732i	−0.668253	−	0.743934i
$17$	−2.75528	−	3.06732i	−0.668253	−	0.743934i
$18$		0			0
$19$	−8.58944			−1.97055			−0.985276	−	0.170972i	$-0.945309\pi$
$19$	−8.58944			−1.97055			−0.985276	+	0.170972i	$0.945309\pi$
$20$		0			0
$21$	3.04245			0.663917
$22$		0			0
$23$			2.35240i			0.490509i	0.969459	+	0.245255i	$0.0788715\pi$
$23$			2.35240i			0.490509i	−0.969459	+	0.245255i	$0.921128\pi$
$24$		0			0
$25$	4.48902			0.897805
$26$		0			0
$27$		−	5.34078i		−	1.02783i
$28$		0			0
$29$		−	1.67302i		−	0.310671i	−0.987862	−	0.155336i	$-0.950354\pi$
$29$		−	1.67302i		−	0.310671i	0.987862	−	0.155336i	$-0.0496459\pi$
$30$		0			0
$31$			9.13920i			1.64145i	0.571325	+	0.820724i	$0.306430\pi$
$31$			9.13920i			1.64145i	−0.571325	+	0.820724i	$0.693570\pi$
$32$		0			0
$33$	−3.19588			−0.556331
$34$		0			0
$35$	1.29529			0.218944
$36$		0			0
$37$			2.45344i			0.403344i	0.979453	+	0.201672i	$0.0646374\pi$
$37$			2.45344i			0.403344i	−0.979453	+	0.201672i	$0.935363\pi$
$38$		0			0
$39$			1.15001i			0.184150i
$40$		0			0
$41$		−	5.79625i		−	0.905222i	−0.891708	−	0.452611i	$-0.850493\pi$
$41$		−	5.79625i		−	0.905222i	0.891708	−	0.452611i	$-0.149507\pi$
$42$		0			0
$43$	−0.853793			−0.130202			−0.0651011	−	0.997879i	$-0.520737\pi$
$43$	−0.853793			−0.130202			−0.0651011	+	0.997879i	$0.520737\pi$
$44$		0			0
$45$		−	0.129305i		−	0.0192756i
$46$		0			0
$47$	−5.83666			−0.851365			−0.425682	−	0.904873i	$-0.639966\pi$
$47$	−5.83666			−0.851365			−0.425682	+	0.904873i	$0.639966\pi$
$48$		0			0
$49$	3.71652			0.530931
$50$		0			0
$51$	−5.15009	+	4.62617i	−0.721157	+	0.647793i
$52$		0			0
$53$	−11.7493			−1.61389			−0.806943	−	0.590629i	$-0.798879\pi$
$53$	−11.7493			−1.61389			−0.806943	+	0.590629i	$0.798879\pi$
$54$		0			0
$55$	−1.36061			−0.183465
$56$		0			0
$57$			14.4218i			1.91022i
$58$		0			0
$59$	−1.00000			−0.130189
$59$	−1.00000			−0.130189
$60$		0			0
$61$		−	0.855431i		−	0.109527i	−0.998499	−	0.0547634i	$-0.982560\pi$
$61$		−	0.855431i		−	0.109527i	0.998499	−	0.0547634i	$-0.0174404\pi$
$62$		0			0
$63$			0.327780i			0.0412964i
$64$		0			0
$65$			0.489607i			0.0607282i
$66$		0			0
$67$	−2.72694			−0.333148			−0.166574	−	0.986029i	$-0.553270\pi$
$67$	−2.72694			−0.333148			−0.166574	+	0.986029i	$0.553270\pi$
$68$		0			0
$69$	3.94973			0.475491
$70$		0			0
$71$		−	1.42200i		−	0.168760i	−0.996434	−	0.0843802i	$-0.973109\pi$
$71$		−	1.42200i		−	0.168760i	0.996434	−	0.0843802i	$-0.0268910\pi$
$72$		0			0
$73$			1.06012i			0.124077i	0.998074	+	0.0620386i	$0.0197602\pi$
$73$			1.06012i			0.124077i	−0.998074	+	0.0620386i	$0.980240\pi$
$74$		0			0
$75$		−	7.53716i		−	0.870317i
$76$		0			0
$77$	3.44907			0.393058
$78$		0			0
$79$			10.5497i			1.18693i	0.804859	+	0.593466i	$0.202241\pi$
$79$			10.5497i			1.18693i	−0.804859	+	0.593466i	$0.797759\pi$
$80$		0			0
$81$	−8.42461			−0.936068
$82$		0			0
$83$	−8.47593			−0.930355			−0.465177	−	0.885218i	$-0.654009\pi$
$83$	−8.47593			−0.930355			−0.465177	+	0.885218i	$0.654009\pi$
$84$		0			0
$85$	−2.19260	+	1.96954i	−0.237821	+	0.213627i
$86$		0			0
$87$	−2.80903			−0.301159
$88$		0			0
$89$	−17.7400			−1.88044			−0.940219	−	0.340570i	$-0.889380\pi$
$89$	−17.7400			−1.88044			−0.940219	+	0.340570i	$0.889380\pi$
$90$		0			0
$91$		−	1.24112i		−	0.130105i
$92$		0			0
$93$	15.3449			1.59119
$94$		0			0
$95$			6.13995i			0.629945i
$96$		0			0
$97$		−	14.7739i		−	1.50006i	−0.661404	−	0.750030i	$-0.730039\pi$
$97$		−	14.7739i		−	1.50006i	0.661404	−	0.750030i	$-0.269961\pi$
$98$		0			0
$99$		−	0.344310i		−	0.0346044i
$100$		0			0
$101$	−18.8039			−1.87106			−0.935531	−	0.353243i	$-0.885079\pi$
$101$	−18.8039			−1.87106			−0.935531	+	0.353243i	$0.885079\pi$
$102$		0			0
$103$	14.8495			1.46317			0.731584	−	0.681751i	$-0.238781\pi$
$103$	14.8495			1.46317			0.731584	+	0.681751i	$0.238781\pi$
$104$		0			0
$105$		−	2.17482i		−	0.212241i
$106$		0			0
$107$			11.0844i			1.07157i	0.844354	+	0.535785i	$0.179984\pi$
$107$			11.0844i			1.07157i	−0.844354	+	0.535785i	$0.820016\pi$
$108$		0			0
$109$			1.82064i			0.174386i	0.996191	+	0.0871928i	$0.0277896\pi$
$109$			1.82064i			0.174386i	−0.996191	+	0.0871928i	$0.972210\pi$
$110$		0			0
$111$	4.11938			0.390994
$112$		0			0
$113$			5.40577i			0.508533i	0.967134	+	0.254266i	$0.0818340\pi$
$113$			5.40577i			0.508533i	−0.967134	+	0.254266i	$0.918166\pi$
$114$		0			0
$115$	1.68155			0.156806
$116$		0			0
$117$	−0.123897			−0.0114543
$118$		0			0
$119$	5.55810	−	4.99266i	0.509510	−	0.457677i
$120$		0			0
$121$	7.37700			0.670636
$122$		0			0
$123$	−9.73202			−0.877506
$124$		0			0
$125$		−	6.78300i		−	0.606690i
$126$		0			0
$127$	−3.65149			−0.324017			−0.162009	−	0.986789i	$-0.551797\pi$
$127$	−3.65149			−0.324017			−0.162009	+	0.986789i	$0.551797\pi$
$128$		0			0
$129$			1.43354i			0.126216i
$130$		0			0
$131$		−	18.1807i		−	1.58845i	−0.607622	−	0.794227i	$-0.707876\pi$
$131$		−	18.1807i		−	1.58845i	0.607622	−	0.794227i	$-0.292124\pi$
$132$		0			0
$133$		−	15.5644i		−	1.34960i
$134$		0			0
$135$	−3.81773			−0.328578
$136$		0			0
$137$	5.63208			0.481181			0.240591	−	0.970627i	$-0.422659\pi$
$137$	5.63208			0.481181			0.240591	+	0.970627i	$0.422659\pi$
$138$		0			0
$139$		−	2.43441i		−	0.206484i	−0.994656	−	0.103242i	$-0.967078\pi$
$139$		−	2.43441i		−	0.206484i	0.994656	−	0.103242i	$-0.0329217\pi$
$140$		0			0
$141$			9.79988i			0.825299i
$142$		0			0
$143$			1.30371i			0.109022i
$144$		0			0
$145$	−1.19591			−0.0993153
$146$		0			0
$147$		−	6.24011i		−	0.514676i
$148$		0			0
$149$	−4.93265			−0.404099			−0.202049	−	0.979375i	$-0.564760\pi$
$149$	−4.93265			−0.404099			−0.202049	+	0.979375i	$0.564760\pi$
$150$		0			0
$151$	−5.18723			−0.422130			−0.211065	−	0.977472i	$-0.567693\pi$
$151$	−5.18723			−0.422130			−0.211065	+	0.977472i	$0.567693\pi$
$152$		0			0
$153$	−0.498402	−	0.554848i	−0.0402934	−	0.0448568i
$154$		0			0
$155$	6.53293			0.524738
$156$		0			0
$157$	−13.3947			−1.06901			−0.534506	−	0.845165i	$-0.679502\pi$
$157$	−13.3947			−1.06901			−0.534506	+	0.845165i	$0.679502\pi$
$158$		0			0
$159$			19.7273i			1.56447i
$160$		0			0
$161$	−4.26263			−0.335943
$162$		0			0
$163$			12.1596i			0.952415i	0.879333	+	0.476207i	$0.157989\pi$
$163$			12.1596i			0.952415i	−0.879333	+	0.476207i	$0.842011\pi$
$164$		0			0
$165$			2.28450i			0.177848i
$166$		0			0
$167$			4.46607i			0.345595i	0.984957	+	0.172798i	$0.0552806\pi$
$167$			4.46607i			0.345595i	−0.984957	+	0.172798i	$0.944719\pi$
$168$		0			0
$169$	−12.5309			−0.963913
$170$		0			0
$171$	−1.55374			−0.118818
$172$		0			0
$173$		−	1.37146i		−	0.104270i	−0.998640	−	0.0521351i	$-0.983397\pi$
$173$		−	1.37146i		−	0.104270i	0.998640	−	0.0521351i	$-0.0166026\pi$
$174$		0			0
$175$			8.13428i			0.614894i
$176$		0			0
$177$			1.67902i			0.126203i
$178$		0			0
$179$	−12.0666			−0.901900			−0.450950	−	0.892549i	$-0.648915\pi$
$179$	−12.0666			−0.901900			−0.450950	+	0.892549i	$0.648915\pi$
$180$		0			0
$181$			2.51028i			0.186587i	0.995639	+	0.0932936i	$0.0297395\pi$
$181$			2.51028i			0.186587i	−0.995639	+	0.0932936i	$0.970260\pi$
$182$		0			0
$183$	−1.43629			−0.106173
$184$		0			0
$185$	1.75378			0.128941
$186$		0			0
$187$	−5.83840	+	5.24445i	−0.426946	+	0.383512i
$188$		0			0
$189$	9.67769			0.703949
$190$		0			0
$191$	21.9908			1.59120			0.795600	−	0.605823i	$-0.207156\pi$
$191$	21.9908			1.59120			0.795600	+	0.605823i	$0.207156\pi$
$192$		0			0
$193$		−	11.1248i		−	0.800782i	−0.916344	−	0.400391i	$-0.868874\pi$
$193$		−	11.1248i		−	0.800782i	0.916344	−	0.400391i	$-0.131126\pi$
$194$		0			0
$195$	0.822060			0.0588689
$196$		0			0
$197$			25.9702i			1.85030i	0.379601	+	0.925150i	$0.376061\pi$
$197$			25.9702i			1.85030i	−0.379601	+	0.925150i	$0.623939\pi$
$198$		0			0
$199$			25.6470i			1.81807i	0.416719	+	0.909035i	$0.363180\pi$
$199$			25.6470i			1.81807i	−0.416719	+	0.909035i	$0.636820\pi$
$200$		0			0
$201$			4.57858i			0.322948i
$202$		0			0
$203$	3.03157			0.212774
$204$		0			0
$205$	−4.14331			−0.289381
$206$		0			0
$207$			0.425526i			0.0295761i
$208$		0			0
$209$			16.3493i			1.13090i
$210$		0			0
$211$			11.5468i			0.794916i	0.917620	+	0.397458i	$0.130108\pi$
$211$			11.5468i			0.794916i	−0.917620	+	0.397458i	$0.869892\pi$
$212$		0			0
$213$	−2.38757			−0.163593
$214$		0			0
$215$			0.610313i			0.0416230i
$216$		0			0
$217$	−16.5606			−1.12420
$218$		0			0
$219$	1.77996			0.120278
$220$		0			0
$221$	1.88718	+	2.10091i	0.126945	+	0.141322i
$222$		0			0
$223$	−5.76265			−0.385896			−0.192948	−	0.981209i	$-0.561805\pi$
$223$	−5.76265			−0.385896			−0.192948	+	0.981209i	$0.561805\pi$
$224$		0			0
$225$	0.812020			0.0541347
$226$		0			0
$227$		−	19.7345i		−	1.30982i	−0.755706	−	0.654911i	$-0.772706\pi$
$227$		−	19.7345i		−	1.30982i	0.755706	−	0.654911i	$-0.227294\pi$
$228$		0			0
$229$	10.4423			0.690050			0.345025	−	0.938594i	$-0.387870\pi$
$229$	10.4423			0.690050			0.345025	+	0.938594i	$0.387870\pi$
$230$		0			0
$231$		−	5.79105i		−	0.381023i
$232$		0			0
$233$		−	6.84443i		−	0.448393i	−0.974544	−	0.224197i	$-0.928024\pi$
$233$		−	6.84443i		−	0.448393i	0.974544	−	0.224197i	$-0.0719758\pi$
$234$		0			0
$235$			4.17220i			0.272164i
$236$		0			0
$237$	17.7131			1.15059
$238$		0			0
$239$	−16.5947			−1.07342			−0.536712	−	0.843766i	$-0.680334\pi$
$239$	−16.5947			−1.07342			−0.536712	+	0.843766i	$0.680334\pi$
$240$		0			0
$241$			12.8462i			0.827493i	0.910392	+	0.413747i	$0.135780\pi$
$241$			12.8462i			0.827493i	−0.910392	+	0.413747i	$0.864220\pi$
$242$		0			0
$243$		−	1.87725i		−	0.120426i
$244$		0			0
$245$		−	2.65666i		−	0.169728i
$246$		0			0
$247$	5.88318			0.374338
$248$		0			0
$249$			14.2313i			0.901870i
$250$		0			0
$251$	−10.9566			−0.691575			−0.345788	−	0.938313i	$-0.612388\pi$
$251$	−10.9566			−0.691575			−0.345788	+	0.938313i	$0.612388\pi$
$252$		0			0
$253$	4.47760			0.281504
$254$		0			0
$255$	3.30690	+	3.68142i	0.207086	+	0.230539i
$256$		0			0
$257$	2.60680			0.162608			0.0813039	−	0.996689i	$-0.474092\pi$
$257$	2.60680			0.162608			0.0813039	+	0.996689i	$0.474092\pi$
$258$		0			0
$259$	−4.44573			−0.276244
$260$		0			0
$261$		−	0.302632i		−	0.0187324i
$262$		0			0
$263$	5.73132			0.353408			0.176704	−	0.984264i	$-0.443456\pi$
$263$	5.73132			0.353408			0.176704	+	0.984264i	$0.443456\pi$
$264$		0			0
$265$			8.39868i			0.515927i
$266$		0			0
$267$			29.7859i			1.82286i
$268$		0			0
$269$			18.2781i			1.11444i	0.830366	+	0.557219i	$0.188132\pi$
$269$			18.2781i			1.11444i	−0.830366	+	0.557219i	$0.811868\pi$
$270$		0			0
$271$	−0.348261			−0.0211553			−0.0105777	−	0.999944i	$-0.503367\pi$
$271$	−0.348261			−0.0211553			−0.0105777	+	0.999944i	$0.503367\pi$
$272$		0			0
$273$	−2.08387			−0.126122
$274$		0			0
$275$		−	8.54449i		−	0.515252i
$276$		0			0
$277$			14.4270i			0.866832i	0.901194	+	0.433416i	$0.142692\pi$
$277$			14.4270i			0.866832i	−0.901194	+	0.433416i	$0.857308\pi$
$278$		0			0
$279$			1.65319i			0.0989739i
$280$		0			0
$281$	7.83791			0.467571			0.233785	−	0.972288i	$-0.424889\pi$
$281$	7.83791			0.467571			0.233785	+	0.972288i	$0.424889\pi$
$282$		0			0
$283$		−	23.8602i		−	1.41834i	−0.705036	−	0.709172i	$-0.749069\pi$
$283$		−	23.8602i		−	1.41834i	0.705036	−	0.709172i	$-0.250931\pi$
$284$		0			0
$285$	10.3091			0.610658
$286$		0			0
$287$	10.5030			0.619974
$288$		0			0
$289$	−1.81691	+	16.9026i	−0.106877	+	0.994272i
$290$		0			0
$291$	−24.8056			−1.45413
$292$		0			0
$293$	−25.2399			−1.47453			−0.737264	−	0.675604i	$-0.763883\pi$
$293$	−25.2399			−1.47453			−0.737264	+	0.675604i	$0.763883\pi$
$294$		0			0
$295$			0.714826i			0.0416188i
$296$		0			0
$297$	−10.1657			−0.589876
$298$		0			0
$299$		−	1.61123i		−	0.0931800i
$300$		0			0
$301$		−	1.54711i		−	0.0891737i
$302$		0			0
$303$			31.5722i			1.81378i
$304$		0			0
$305$	−0.611484			−0.0350135
$306$		0			0
$307$	−8.84254			−0.504671			−0.252335	−	0.967640i	$-0.581199\pi$
$307$	−8.84254			−0.504671			−0.252335	+	0.967640i	$0.581199\pi$
$308$		0			0
$309$		−	24.9327i		−	1.41837i
$310$		0			0
$311$			4.44412i			0.252003i	0.992030	+	0.126002i	$0.0402144\pi$
$311$			4.44412i			0.252003i	−0.992030	+	0.126002i	$0.959786\pi$
$312$		0			0
$313$		−	21.1314i		−	1.19442i	−0.802085	−	0.597209i	$-0.796276\pi$
$313$		−	21.1314i		−	1.19442i	0.802085	−	0.597209i	$-0.203724\pi$
$314$		0			0
$315$	0.234305			0.0132016
$316$		0			0
$317$		−	31.1234i		−	1.74806i	−0.485869	−	0.874031i	$-0.661497\pi$
$317$		−	31.1234i		−	1.74806i	0.485869	−	0.874031i	$-0.338503\pi$
$318$		0			0
$319$	−3.18445			−0.178295
$320$		0			0
$321$	18.6109			1.03876
$322$		0			0
$323$	23.6663	+	26.3465i	1.31683	+	1.46596i
$324$		0			0
$325$	−3.07468			−0.170552
$326$		0			0
$327$	3.05689			0.169047
$328$		0			0
$329$		−	10.5763i		−	0.583088i
$330$		0			0
$331$	22.6276			1.24373			0.621863	−	0.783126i	$-0.286376\pi$
$331$	22.6276			1.24373			0.621863	+	0.783126i	$0.286376\pi$
$332$		0			0
$333$			0.443804i			0.0243203i
$334$		0			0
$335$			1.94928i			0.106501i
$336$		0			0
$337$		−	16.8415i		−	0.917415i	−0.888587	−	0.458707i	$-0.848313\pi$
$337$		−	16.8415i		−	0.917415i	0.888587	−	0.458707i	$-0.151687\pi$
$338$		0			0
$339$	9.07641			0.492963
$340$		0			0
$341$	17.3957			0.942031
$342$		0			0
$343$			19.4187i			1.04851i
$344$		0			0
$345$		−	2.82336i		−	0.152005i
$346$		0			0
$347$			22.9163i			1.23021i	0.788445	+	0.615105i	$0.210886\pi$
$347$			22.9163i			1.23021i	−0.788445	+	0.615105i	$0.789114\pi$
$348$		0			0
$349$	13.6322			0.729717			0.364858	−	0.931063i	$-0.381117\pi$
$349$	13.6322			0.729717			0.364858	+	0.931063i	$0.381117\pi$
$350$		0			0
$351$			3.65807i			0.195253i
$352$		0			0
$353$	29.8834			1.59053			0.795266	−	0.606261i	$-0.207331\pi$
$353$	29.8834			1.59053			0.795266	+	0.606261i	$0.207331\pi$
$354$		0			0
$355$	−1.01648			−0.0539493
$356$		0			0
$357$	−8.38279	−	9.33216i	−0.443664	−	0.493910i
$358$		0			0
$359$	−6.81223			−0.359536			−0.179768	−	0.983709i	$-0.557535\pi$
$359$	−6.81223			−0.359536			−0.179768	+	0.983709i	$0.557535\pi$
$360$		0			0
$361$	54.7784			2.88307
$362$		0			0
$363$		−	12.3861i		−	0.650103i
$364$		0			0
$365$	0.757798			0.0396650
$366$		0			0
$367$			21.3505i			1.11449i	0.830349	+	0.557244i	$0.188141\pi$
$367$			21.3505i			1.11449i	−0.830349	+	0.557244i	$0.811859\pi$
$368$		0			0
$369$		−	1.04848i		−	0.0545819i
$370$		0			0
$371$		−	21.2901i		−	1.10533i
$372$		0			0
$373$	11.8017			0.611069			0.305534	−	0.952181i	$-0.401165\pi$
$373$	11.8017			0.611069			0.305534	+	0.952181i	$0.401165\pi$
$374$		0			0
$375$	−11.3888			−0.588115
$376$		0			0
$377$			1.14590i			0.0590169i
$378$		0			0
$379$			21.6634i			1.11277i	0.830923	+	0.556387i	$0.187813\pi$
$379$			21.6634i			1.11277i	−0.830923	+	0.556387i	$0.812187\pi$
$380$		0			0
$381$			6.13092i			0.314097i
$382$		0			0
$383$	23.2516			1.18810			0.594049	−	0.804429i	$-0.297528\pi$
$383$	23.2516			1.18810			0.594049	+	0.804429i	$0.297528\pi$
$384$		0			0
$385$		−	2.46548i		−	0.125653i
$386$		0			0
$387$	−0.154443			−0.00785077
$388$		0			0
$389$	−16.5158			−0.837386			−0.418693	−	0.908128i	$-0.637512\pi$
$389$	−16.5158			−0.837386			−0.418693	+	0.908128i	$0.637512\pi$
$390$		0			0
$391$	7.21556	−	6.48151i	0.364907	−	0.327784i
$392$		0			0
$393$	−30.5257			−1.53982
$394$		0			0
$395$	7.54118			0.379438
$396$		0			0
$397$		−	13.4740i		−	0.676238i	−0.941103	−	0.338119i	$-0.890209\pi$
$397$		−	13.4740i		−	0.676238i	0.941103	−	0.338119i	$-0.109791\pi$
$398$		0			0
$399$	−26.1329			−1.30828
$400$		0			0
$401$		−	23.9126i		−	1.19414i	−0.802190	−	0.597069i	$-0.796332\pi$
$401$		−	23.9126i		−	1.19414i	0.802190	−	0.597069i	$-0.203668\pi$
$402$		0			0
$403$		−	6.25973i		−	0.311819i
$404$		0			0
$405$			6.02213i			0.299242i
$406$		0			0
$407$	4.66993			0.231480
$408$		0			0
$409$	−10.3100			−0.509795			−0.254897	−	0.966968i	$-0.582042\pi$
$409$	−10.3100			−0.509795			−0.254897	+	0.966968i	$0.582042\pi$
$410$		0			0
$411$		−	9.45638i		−	0.466449i
$412$		0			0
$413$		−	1.81204i		−	0.0891645i
$414$		0			0
$415$			6.05881i			0.297415i
$416$		0			0
$417$	−4.08743			−0.200162
$418$		0			0
$419$			24.1650i			1.18054i	0.807206	+	0.590270i	$0.200979\pi$
$419$			24.1650i			1.18054i	−0.807206	+	0.590270i	$0.799021\pi$
$420$		0			0
$421$	−25.6330			−1.24928			−0.624638	−	0.780914i	$-0.714754\pi$
$421$	−25.6330			−1.24928			−0.624638	+	0.780914i	$0.714754\pi$
$422$		0			0
$423$	−1.05579			−0.0513345
$424$		0			0
$425$	−12.3685	−	13.7693i	−0.599960	−	0.667908i
$426$		0			0
$427$	1.55007			0.0750133
$428$		0			0
$429$	2.18896			0.105684
$430$		0			0
$431$		−	24.8004i		−	1.19459i	−0.802021	−	0.597296i	$-0.796242\pi$
$431$		−	24.8004i		−	1.19459i	0.802021	−	0.597296i	$-0.203758\pi$
$432$		0			0
$433$	10.5330			0.506184			0.253092	−	0.967442i	$-0.418552\pi$
$433$	10.5330			0.506184			0.253092	+	0.967442i	$0.418552\pi$
$434$		0			0
$435$			2.00796i			0.0962745i
$436$		0			0
$437$		−	20.2058i		−	0.966573i
$438$		0			0
$439$			13.7852i			0.657930i	0.944342	+	0.328965i	$0.106700\pi$
$439$			13.7852i			0.657930i	−0.944342	+	0.328965i	$0.893300\pi$
$440$		0			0
$441$	0.672282			0.0320134
$442$		0			0
$443$	−18.5650			−0.882048			−0.441024	−	0.897495i	$-0.645385\pi$
$443$	−18.5650			−0.882048			−0.441024	+	0.897495i	$0.645385\pi$
$444$		0			0
$445$			12.6810i			0.601138i
$446$		0			0
$447$			8.28202i			0.391726i
$448$		0			0
$449$		−	25.2206i		−	1.19023i	−0.803639	−	0.595117i	$-0.797106\pi$
$449$		−	25.2206i		−	1.19023i	0.803639	−	0.595117i	$-0.202894\pi$
$450$		0			0
$451$	−11.0327			−0.519509
$452$		0			0
$453$			8.70946i			0.409206i
$454$		0			0
$455$	−0.887186			−0.0415919
$456$		0			0
$457$	−19.7281			−0.922843			−0.461422	−	0.887181i	$-0.652660\pi$
$457$	−19.7281			−0.922843			−0.461422	+	0.887181i	$0.652660\pi$
$458$		0			0
$459$	−16.3819	+	14.7153i	−0.764641	+	0.686852i
$460$		0			0
$461$	8.73070			0.406629			0.203315	−	0.979113i	$-0.434829\pi$
$461$	8.73070			0.406629			0.203315	+	0.979113i	$0.434829\pi$
$462$		0			0
$463$	−36.8487			−1.71250			−0.856252	−	0.516559i	$-0.827213\pi$
$463$	−36.8487			−1.71250			−0.856252	+	0.516559i	$0.827213\pi$
$464$		0			0
$465$		−	10.9689i		−	0.508672i
$466$		0			0
$467$	24.3898			1.12863			0.564314	−	0.825560i	$-0.309141\pi$
$467$	24.3898			1.12863			0.564314	+	0.825560i	$0.309141\pi$
$468$		0			0
$469$		−	4.94131i		−	0.228168i
$470$		0			0
$471$			22.4899i			1.03628i
$472$		0			0
$473$			1.62513i			0.0747234i
$474$		0			0
$475$	−38.5582			−1.76917
$476$		0			0
$477$	−2.12533			−0.0973120
$478$		0			0
$479$			1.42067i			0.0649119i	0.999473	+	0.0324560i	$0.0103329\pi$
$479$			1.42067i			0.0649119i	−0.999473	+	0.0324560i	$0.989667\pi$
$480$		0			0
$481$		−	1.68044i		−	0.0766215i
$482$		0			0
$483$			7.15705i			0.325657i
$484$		0			0
$485$	−10.5607			−0.479539
$486$		0			0
$487$		−	9.32131i		−	0.422389i	−0.977444	−	0.211194i	$-0.932265\pi$
$487$		−	9.32131i		−	0.422389i	0.977444	−	0.211194i	$-0.0677353\pi$
$488$		0			0
$489$	20.4163			0.923255
$490$		0			0
$491$	−15.5285			−0.700792			−0.350396	−	0.936602i	$-0.613953\pi$
$491$	−15.5285			−0.700792			−0.350396	+	0.936602i	$0.613953\pi$
$492$		0			0
$493$	−5.13167	+	4.60962i	−0.231119	+	0.207607i
$494$		0			0
$495$	−0.246121			−0.0110623
$496$		0			0
$497$	2.57672			0.115582
$498$		0			0
$499$		−	27.5006i		−	1.23110i	−0.788099	−	0.615549i	$-0.788935\pi$
$499$		−	27.5006i		−	1.23110i	0.788099	−	0.615549i	$-0.211065\pi$
$500$		0			0
$501$	7.49863			0.335014
$502$		0			0
$503$		−	30.3703i		−	1.35414i	−0.735916	−	0.677072i	$-0.763248\pi$
$503$		−	30.3703i		−	1.35414i	0.735916	−	0.677072i	$-0.236752\pi$
$504$		0			0
$505$			13.4415i			0.598141i
$506$		0			0
$507$			21.0396i			0.934401i
$508$		0			0
$509$	10.7281			0.475515			0.237758	−	0.971325i	$-0.423588\pi$
$509$	10.7281			0.475515			0.237758	+	0.971325i	$0.423588\pi$
$510$		0			0
$511$	−1.92097			−0.0849787
$512$		0			0
$513$			45.8743i			2.02540i
$514$		0			0
$515$		−	10.6148i		−	0.467745i
$516$		0			0
$517$			11.1096i			0.488600i
$518$		0			0
$519$	−2.30271			−0.101078
$520$		0			0
$521$		−	23.6576i		−	1.03646i	−0.855242	−	0.518228i	$-0.826592\pi$
$521$		−	23.6576i		−	1.03646i	0.855242	−	0.518228i	$-0.173408\pi$
$522$		0			0
$523$	−3.12461			−0.136630			−0.0683149	−	0.997664i	$-0.521762\pi$
$523$	−3.12461			−0.136630			−0.0683149	+	0.997664i	$0.521762\pi$
$524$		0			0
$525$	13.6576			0.596068
$526$		0			0
$527$	28.0328	−	25.1810i	1.22113	−	1.09690i
$528$		0			0
$529$	17.4662			0.759401
$530$		0			0
$531$	−0.180890			−0.00784996
$532$		0			0
$533$			3.97003i			0.171961i
$534$		0			0
$535$	7.92342			0.342559
$536$		0			0
$537$			20.2601i			0.874286i
$538$		0			0
$539$		−	7.07409i		−	0.304703i
$540$		0			0
$541$		−	38.4275i		−	1.65213i	−0.563578	−	0.826063i	$-0.690576\pi$
$541$		−	38.4275i		−	1.65213i	0.563578	−	0.826063i	$-0.309424\pi$
$542$		0			0
$543$	4.21480			0.180875
$544$		0			0
$545$	1.30144			0.0557476
$546$		0			0
$547$			12.8085i			0.547653i	0.961779	+	0.273826i	$0.0882894\pi$
$547$			12.8085i			0.547653i	−0.961779	+	0.273826i	$0.911711\pi$
$548$		0			0
$549$		−	0.154739i		−	0.00660410i
$550$		0			0
$551$			14.3703i			0.612194i
$552$		0			0
$553$	−19.1164			−0.812913
$554$		0			0
$555$		−	2.94464i		−	0.124993i
$556$		0			0
$557$	−29.1887			−1.23676			−0.618382	−	0.785878i	$-0.712211\pi$
$557$	−29.1887			−1.23676			−0.618382	+	0.785878i	$0.712211\pi$
$558$		0			0
$559$	0.584790			0.0247340
$560$		0			0
$561$	8.80553	+	9.80279i	0.371770	+	0.413874i
$562$		0			0
$563$	−2.08510			−0.0878763			−0.0439381	−	0.999034i	$-0.513990\pi$
$563$	−2.08510			−0.0878763			−0.0439381	+	0.999034i	$0.513990\pi$
$564$		0			0
$565$	3.86419			0.162568
$566$		0			0
$567$		−	15.2657i		−	0.641099i
$568$		0			0
$569$	−6.69663			−0.280737			−0.140369	−	0.990099i	$-0.544829\pi$
$569$	−6.69663			−0.280737			−0.140369	+	0.990099i	$0.544829\pi$
$570$		0			0
$571$			20.8675i			0.873277i	0.899637	+	0.436638i	$0.143831\pi$
$571$			20.8675i			0.873277i	−0.899637	+	0.436638i	$0.856169\pi$
$572$		0			0
$573$		−	36.9230i		−	1.54248i
$574$		0			0
$575$			10.5600i			0.440381i
$576$		0			0
$577$	−37.5740			−1.56423			−0.782114	−	0.623136i	$-0.785858\pi$
$577$	−37.5740			−1.56423			−0.782114	+	0.623136i	$0.785858\pi$
$578$		0			0
$579$	−18.6788			−0.776264
$580$		0			0
$581$		−	15.3587i		−	0.637187i
$582$		0			0
$583$			22.3638i			0.926213i
$584$		0			0
$585$			0.0885650i			0.00366171i
$586$		0			0
$587$	−11.0842			−0.457493			−0.228747	−	0.973486i	$-0.573463\pi$
$587$	−11.0842			−0.457493			−0.228747	+	0.973486i	$0.573463\pi$
$588$		0			0
$589$		−	78.5006i		−	3.23456i
$590$		0			0
$591$	43.6045			1.79365
$592$		0			0
$593$	16.2320			0.666569			0.333284	−	0.942826i	$-0.391843\pi$
$593$	16.2320			0.666569			0.333284	+	0.942826i	$0.391843\pi$
$594$		0			0
$595$	−3.56888	−	3.97307i	−0.146310	−	0.162880i
$596$		0			0
$597$	43.0619			1.76241
$598$		0			0
$599$	−2.43229			−0.0993805			−0.0496903	−	0.998765i	$-0.515823\pi$
$599$	−2.43229			−0.0993805			−0.0496903	+	0.998765i	$0.515823\pi$
$600$		0			0
$601$			2.18895i			0.0892890i	0.999003	+	0.0446445i	$0.0142155\pi$
$601$			2.18895i			0.0892890i	−0.999003	+	0.0446445i	$0.985784\pi$
$602$		0			0
$603$	−0.493276			−0.0200877
$604$		0			0
$605$		−	5.27327i		−	0.214389i
$606$		0			0
$607$		−	9.02178i		−	0.366183i	−0.983096	−	0.183091i	$-0.941390\pi$
$607$		−	9.02178i		−	0.366183i	0.983096	−	0.183091i	$-0.0586104\pi$
$608$		0			0
$609$		−	5.09006i		−	0.206260i
$610$		0			0
$611$	3.99772			0.161730
$612$		0			0
$613$	−20.0435			−0.809550			−0.404775	−	0.914416i	$-0.632650\pi$
$613$	−20.0435			−0.809550			−0.404775	+	0.914416i	$0.632650\pi$
$614$		0			0
$615$			6.95669i			0.280521i
$616$		0			0
$617$			8.56346i			0.344752i	0.985031	+	0.172376i	$0.0551444\pi$
$617$			8.56346i			0.344752i	−0.985031	+	0.172376i	$0.944856\pi$
$618$		0			0
$619$		−	27.8960i		−	1.12123i	−0.828075	−	0.560617i	$-0.810564\pi$
$619$		−	27.8960i		−	1.12123i	0.828075	−	0.560617i	$-0.189436\pi$
$620$		0			0
$621$	12.5636			0.504162
$622$		0			0
$623$		−	32.1456i		−	1.28789i
$624$		0			0
$625$	17.5965			0.703858
$626$		0			0
$627$	27.4508			1.09628
$628$		0			0
$629$	7.52550	−	6.75991i	0.300061	−	0.269535i
$630$		0			0
$631$	−24.6028			−0.979424			−0.489712	−	0.871884i	$-0.662898\pi$
$631$	−24.6028			−0.979424			−0.489712	+	0.871884i	$0.662898\pi$
$632$		0			0
$633$	19.3874			0.770578
$634$		0			0
$635$			2.61018i			0.103582i
$636$		0			0
$637$	−2.54556			−0.100859
$638$		0			0
$639$		−	0.257226i		−	0.0101757i
$640$		0			0
$641$			1.99523i			0.0788067i	0.999223	+	0.0394033i	$0.0125457\pi$
$641$			1.99523i			0.0788067i	−0.999223	+	0.0394033i	$0.987454\pi$
$642$		0			0
$643$		−	6.32250i		−	0.249335i	−0.992199	−	0.124667i	$-0.960214\pi$
$643$		−	6.32250i		−	0.249335i	0.992199	−	0.124667i	$-0.0397864\pi$
$644$		0			0
$645$	1.02473			0.0403486
$646$		0			0
$647$	1.97828			0.0777742			0.0388871	−	0.999244i	$-0.487619\pi$
$647$	1.97828			0.0777742			0.0388871	+	0.999244i	$0.487619\pi$
$648$		0			0
$649$			1.90342i			0.0747157i
$650$		0			0
$651$			27.8055i			1.08978i
$652$		0			0
$653$		−	32.4357i		−	1.26931i	−0.772797	−	0.634653i	$-0.781143\pi$
$653$		−	32.4357i		−	1.26931i	0.772797	−	0.634653i	$-0.218857\pi$
$654$		0			0
$655$	−12.9960			−0.507796
$656$		0			0
$657$			0.191765i			0.00748145i
$658$		0			0
$659$	−11.4691			−0.446773			−0.223386	−	0.974730i	$-0.571711\pi$
$659$	−11.4691			−0.446773			−0.223386	+	0.974730i	$0.571711\pi$
$660$		0			0
$661$	45.6241			1.77457			0.887287	−	0.461218i	$-0.152588\pi$
$661$	45.6241			1.77457			0.887287	+	0.461218i	$0.152588\pi$
$662$		0			0
$663$	3.52746	−	3.16861i	0.136995	−	0.123059i
$664$		0			0
$665$	−11.1258			−0.431441
$666$		0			0
$667$	3.93560			0.152387
$668$		0			0
$669$			9.67561i			0.374081i
$670$		0			0
$671$	−1.62824			−0.0628576
$672$		0			0
$673$		−	15.8074i		−	0.609330i	−0.952460	−	0.304665i	$-0.901455\pi$
$673$		−	15.8074i		−	0.609330i	0.952460	−	0.304665i	$-0.0985445\pi$
$674$		0			0
$675$		−	23.9749i		−	0.922794i
$676$		0			0
$677$		−	26.6689i		−	1.02497i	−0.858697	−	0.512484i	$-0.828725\pi$
$677$		−	26.6689i		−	1.02497i	0.858697	−	0.512484i	$-0.171275\pi$
$678$		0			0
$679$	26.7708			1.02737
$680$		0			0
$681$	−33.1345			−1.26972
$682$		0			0
$683$		−	11.4593i		−	0.438477i	−0.975671	−	0.219239i	$-0.929643\pi$
$683$		−	11.4593i		−	0.438477i	0.975671	−	0.219239i	$-0.0703573\pi$
$684$		0			0
$685$		−	4.02596i		−	0.153824i
$686$		0			0
$687$		−	17.5329i		−	0.668922i
$688$		0			0
$689$	8.04745			0.306583
$690$		0			0
$691$			19.1703i			0.729274i	0.931150	+	0.364637i	$0.118807\pi$
$691$			19.1703i			0.729274i	−0.931150	+	0.364637i	$0.881193\pi$
$692$		0			0
$693$	0.623902			0.0237001
$694$		0			0
$695$	−1.74018			−0.0660088
$696$		0			0
$697$	−17.7789	+	15.9703i	−0.673426	+	0.604917i
$698$		0			0
$699$	−11.4919			−0.434665
$700$		0			0
$701$	17.3576			0.655589			0.327794	−	0.944749i	$-0.393695\pi$
$701$	17.3576			0.655589			0.327794	+	0.944749i	$0.393695\pi$
$702$		0			0
$703$		−	21.0737i		−	0.794809i
$704$		0			0
$705$	7.00520			0.263831
$706$		0			0
$707$		−	34.0735i		−	1.28146i
$708$		0			0
$709$		−	43.6802i		−	1.64044i	−0.572045	−	0.820222i	$-0.693850\pi$
$709$		−	43.6802i		−	1.64044i	0.572045	−	0.820222i	$-0.306150\pi$
$710$		0			0
$711$			1.90833i			0.0715681i
$712$		0			0
$713$	−21.4990			−0.805145
$714$		0			0
$715$	0.931927			0.0348521
$716$		0			0
$717$			27.8629i			1.04056i
$718$		0			0
$719$		−	33.6142i		−	1.25360i	−0.779180	−	0.626800i	$-0.784364\pi$
$719$		−	33.6142i		−	1.25360i	0.779180	−	0.626800i	$-0.215636\pi$
$720$		0			0
$721$			26.9079i			1.00210i
$722$		0			0
$723$	21.5690			0.802158
$724$		0			0
$725$		−	7.51021i		−	0.278922i
$726$		0			0
$727$	29.2869			1.08619			0.543096	−	0.839671i	$-0.317252\pi$
$727$	29.2869			1.08619			0.543096	+	0.839671i	$0.317252\pi$
$728$		0			0
$729$	−28.4258			−1.05281
$730$		0			0
$731$	2.35244	+	2.61886i	0.0870080	+	0.0968619i
$732$		0			0
$733$	−15.0622			−0.556334			−0.278167	−	0.960533i	$-0.589727\pi$
$733$	−15.0622			−0.556334			−0.278167	+	0.960533i	$0.589727\pi$
$734$		0			0
$735$	−4.46059			−0.164531
$736$		0			0
$737$			5.19050i			0.191195i
$738$		0			0
$739$	−21.2418			−0.781392			−0.390696	−	0.920520i	$-0.627766\pi$
$739$	−21.2418			−0.781392			−0.390696	+	0.920520i	$0.627766\pi$
$740$		0			0
$741$		−	9.87798i		−	0.362877i
$742$		0			0
$743$		−	12.2816i		−	0.450568i	−0.974293	−	0.225284i	$-0.927669\pi$
$743$		−	12.2816i		−	0.450568i	0.974293	−	0.225284i	$-0.0723310\pi$
$744$		0			0
$745$			3.52599i			0.129182i
$746$		0			0
$747$	−1.53321			−0.0560973
$748$		0			0
$749$	−20.0854			−0.733903
$750$		0			0
$751$		−	34.8789i		−	1.27275i	−0.771380	−	0.636375i	$-0.780433\pi$
$751$		−	34.8789i		−	1.27275i	0.771380	−	0.636375i	$-0.219567\pi$
$752$		0			0
$753$			18.3964i			0.670401i
$754$		0			0
$755$			3.70796i			0.134947i
$756$		0			0
$757$	−0.740264			−0.0269053			−0.0134527	−	0.999910i	$-0.504282\pi$
$757$	−0.740264			−0.0269053			−0.0134527	+	0.999910i	$0.504282\pi$
$758$		0			0
$759$		−	7.51798i		−	0.272885i
$760$		0			0
$761$	43.0592			1.56089			0.780447	−	0.625222i	$-0.214992\pi$
$761$	43.0592			1.56089			0.780447	+	0.625222i	$0.214992\pi$
$762$		0			0
$763$	−3.29907			−0.119434
$764$		0			0
$765$	−0.396620	+	0.356271i	−0.0143398	+	0.0128810i
$766$		0			0
$767$	0.684932			0.0247315
$768$		0			0
$769$	7.69449			0.277471			0.138735	−	0.990330i	$-0.455696\pi$
$769$	7.69449			0.277471			0.138735	+	0.990330i	$0.455696\pi$
$770$		0			0
$771$		−	4.37687i		−	0.157629i
$772$		0			0
$773$	−37.5095			−1.34912			−0.674561	−	0.738219i	$-0.735667\pi$
$773$	−37.5095			−1.34912			−0.674561	+	0.738219i	$0.735667\pi$
$774$		0			0
$775$			41.0261i			1.47370i
$776$		0			0
$777$			7.46447i			0.267787i
$778$		0			0
$779$			49.7865i			1.78379i
$780$		0			0
$781$	−2.70666			−0.0968520
$782$		0			0
$783$	−8.93521			−0.319318
$784$		0			0
$785$			9.57486i			0.341741i
$786$		0			0
$787$		−	33.9179i		−	1.20904i	−0.796589	−	0.604522i	$-0.793364\pi$
$787$		−	33.9179i		−	1.20904i	0.796589	−	0.604522i	$-0.206636\pi$
$788$		0			0
$789$		−	9.62301i		−	0.342588i
$790$		0			0
$791$	−9.79547			−0.348287
$792$		0			0
$793$			0.585912i			0.0208063i
$794$		0			0
$795$	14.1015			0.500130
$796$		0			0
$797$	11.9010			0.421556			0.210778	−	0.977534i	$-0.432400\pi$
$797$	11.9010			0.421556			0.210778	+	0.977534i	$0.432400\pi$
$798$		0			0
$799$	16.0816	+	17.9029i	0.568927	+	0.633360i
$800$		0			0
$801$	−3.20899			−0.113384
$802$		0			0
$803$	2.01784			0.0712082
$804$		0			0
$805$			3.04704i			0.107394i
$806$		0			0
$807$	30.6894			1.08032
$808$		0			0
$809$		−	14.0152i		−	0.492749i	−0.969175	−	0.246375i	$-0.920761\pi$
$809$		−	14.0152i		−	0.492749i	0.969175	−	0.246375i	$-0.0792393\pi$
$810$		0			0
$811$		−	15.5803i		−	0.547100i	−0.961858	−	0.273550i	$-0.911802\pi$
$811$		−	15.5803i		−	0.547100i	0.961858	−	0.273550i	$-0.0881979\pi$
$812$		0			0
$813$			0.584737i			0.0205076i
$814$		0			0
$815$	8.69201			0.304468
$816$		0			0
$817$	7.33360			0.256570
$818$		0			0
$819$		−	0.224507i		−	0.00784490i
$820$		0			0
$821$			31.1019i			1.08546i	0.839906	+	0.542731i	$0.182610\pi$
$821$			31.1019i			1.08546i	−0.839906	+	0.542731i	$0.817390\pi$
$822$		0			0
$823$		−	22.1044i		−	0.770511i	−0.922810	−	0.385255i	$-0.874113\pi$
$823$		−	22.1044i		−	0.770511i	0.922810	−	0.385255i	$-0.125887\pi$
$824$		0			0
$825$	−14.3464			−0.499477
$826$		0			0
$827$			29.8538i			1.03812i	0.854738	+	0.519060i	$0.173718\pi$
$827$			29.8538i			1.03812i	−0.854738	+	0.519060i	$0.826282\pi$
$828$		0			0
$829$	−7.89857			−0.274329			−0.137164	−	0.990548i	$-0.543799\pi$
$829$	−7.89857			−0.274329			−0.137164	+	0.990548i	$0.543799\pi$
$830$		0			0
$831$	24.2232			0.840292
$832$		0			0
$833$	−10.2400	−	11.3998i	−0.354796	−	0.394978i
$834$		0			0
$835$	3.19246			0.110480
$836$		0			0
$837$	48.8104			1.68714
$838$		0			0
$839$		−	21.8731i		−	0.755143i	−0.925980	−	0.377571i	$-0.876759\pi$
$839$		−	21.8731i		−	0.755143i	0.925980	−	0.377571i	$-0.123241\pi$
$840$		0			0
$841$	26.2010			0.903483
$842$		0			0
$843$		−	13.1600i		−	0.453255i
$844$		0			0
$845$			8.95739i			0.308143i
$846$		0			0
$847$			13.3674i			0.459309i
$848$		0			0
$849$	−40.0618			−1.37492
$850$		0			0
$851$	−5.77148			−0.197844
$852$		0			0
$853$		−	44.5618i		−	1.52577i	−0.646536	−	0.762883i	$-0.723783\pi$
$853$		−	44.5618i		−	1.52577i	0.646536	−	0.762883i	$-0.276217\pi$
$854$		0			0
$855$			1.11066i			0.0379836i
$856$		0			0
$857$			28.8404i			0.985169i	0.870265	+	0.492584i	$0.163948\pi$
$857$			28.8404i			0.985169i	−0.870265	+	0.492584i	$0.836052\pi$
$858$		0			0
$859$	−51.7833			−1.76682			−0.883411	−	0.468599i	$-0.844759\pi$
$859$	−51.7833			−1.76682			−0.883411	+	0.468599i	$0.844759\pi$
$860$		0			0
$861$		−	17.6348i		−	0.600992i
$862$		0			0
$863$	1.35274			0.0460479			0.0230239	−	0.999735i	$-0.492671\pi$
$863$	1.35274			0.0460479			0.0230239	+	0.999735i	$0.492671\pi$
$864$		0			0
$865$	−0.980354			−0.0333330
$866$		0			0
$867$	28.3799	+	3.05062i	0.963831	+	0.103605i
$868$		0			0
$869$	20.0805			0.681183
$870$		0			0
$871$	1.86776			0.0632868
$872$		0			0
$873$		−	2.67245i		−	0.0904486i
$874$		0			0
$875$	12.2910			0.415513
$876$		0			0
$877$		−	26.6641i		−	0.900382i	−0.892932	−	0.450191i	$-0.851356\pi$
$877$		−	26.6641i		−	0.900382i	0.892932	−	0.450191i	$-0.148644\pi$
$878$		0			0
$879$			42.3782i			1.42938i
$880$		0			0
$881$		−	0.413832i		−	0.0139424i	−0.999976	−	0.00697118i	$-0.997781\pi$
$881$		−	0.413832i		−	0.0139424i	0.999976	−	0.00697118i	$-0.00221901\pi$
$882$		0			0
$883$	44.8929			1.51077			0.755384	−	0.655283i	$-0.227451\pi$
$883$	44.8929			1.51077			0.755384	+	0.655283i	$0.227451\pi$
$884$		0			0
$885$	1.20021			0.0403445
$886$		0			0
$887$		−	15.9114i		−	0.534251i	−0.963662	−	0.267126i	$-0.913926\pi$
$887$		−	15.9114i		−	0.534251i	0.963662	−	0.267126i	$-0.0860738\pi$
$888$		0			0
$889$		−	6.61663i		−	0.221915i
$890$		0			0
$891$			16.0356i			0.537211i
$892$		0			0
$893$	50.1336			1.67766
$894$		0			0
$895$			8.62551i			0.288319i
$896$		0			0
$897$	−2.70529			−0.0903271
$898$		0			0
$899$	15.2900			0.509951
$900$		0			0
$901$	32.3725	+	36.0388i	1.07848	+	1.20063i
$902$		0			0
$903$	−2.59762			−0.0864434
$904$		0			0
$905$	1.79441			0.0596482
$906$		0			0
$907$		−	13.5441i		−	0.449725i	−0.974390	−	0.224863i	$-0.927807\pi$
$907$		−	13.5441i		−	0.449725i	0.974390	−	0.224863i	$-0.0721933\pi$
$908$		0			0
$909$	−3.40145			−0.112819
$910$		0			0
$911$			24.6409i			0.816389i	0.912895	+	0.408195i	$0.133841\pi$
$911$			24.6409i			0.816389i	−0.912895	+	0.408195i	$0.866159\pi$
$912$		0			0
$913$			16.1333i			0.533933i
$914$		0			0
$915$			1.02669i			0.0339415i
$916$		0			0
$917$	32.9441			1.08791
$918$		0			0
$919$	−36.4100			−1.20106			−0.600528	−	0.799604i	$-0.705043\pi$
$919$	−36.4100			−1.20106			−0.600528	+	0.799604i	$0.705043\pi$
$920$		0			0
$921$			14.8468i			0.489219i
$922$		0			0
$923$			0.973973i			0.0320587i
$924$		0			0
$925$			11.0136i			0.362124i
$926$		0			0
$927$	2.68614			0.0882243
$928$		0			0
$929$			6.16222i			0.202176i	0.994877	+	0.101088i	$0.0322323\pi$
$929$			6.16222i			0.202176i	−0.994877	+	0.101088i	$0.967768\pi$
$930$		0			0
$931$	−31.9228			−1.04623
$932$		0			0
$933$	7.46178			0.244288
$934$		0			0
$935$	3.74886	+	4.17343i	0.122601	+	0.136486i
$936$		0			0
$937$	12.9081			0.421688			0.210844	−	0.977520i	$-0.432379\pi$
$937$	12.9081			0.421688			0.210844	+	0.977520i	$0.432379\pi$
$938$		0			0
$939$	−35.4801			−1.15785
$940$		0			0
$941$		−	31.8436i		−	1.03807i	−0.854752	−	0.519037i	$-0.826291\pi$
$941$		−	31.8436i		−	1.03807i	0.854752	−	0.519037i	$-0.173709\pi$
$942$		0			0
$943$	13.6351			0.444019
$944$		0			0
$945$		−	6.91786i		−	0.225038i
$946$		0			0
$947$			29.1627i			0.947661i	0.880616	+	0.473830i	$0.157129\pi$
$947$			29.1627i			0.947661i	−0.880616	+	0.473830i	$0.842871\pi$
$948$		0			0
$949$		−	0.726107i		−	0.0235704i
$950$		0			0
$951$	−52.2568			−1.69454
$952$		0			0
$953$	43.0098			1.39322			0.696611	−	0.717449i	$-0.254690\pi$
$953$	43.0098			1.39322			0.696611	+	0.717449i	$0.254690\pi$
$954$		0			0
$955$		−	15.7196i		−	0.508674i
$956$		0			0
$957$			5.34676i			0.172836i
$958$		0			0
$959$			10.2055i			0.329554i
$960$		0			0
$961$	−52.5249			−1.69435
$962$		0			0
$963$			2.00506i			0.0646121i
$964$		0			0
$965$	−7.95230			−0.255994
$966$		0			0
$967$	20.6234			0.663204			0.331602	−	0.943419i	$-0.392411\pi$
$967$	20.6234			0.663204			0.331602	+	0.943419i	$0.392411\pi$
$968$		0			0
$969$	44.2364	−	39.7361i	1.42108	−	1.27651i
$970$		0			0
$971$	−30.6485			−0.983559			−0.491779	−	0.870720i	$-0.663653\pi$
$971$	−30.6485			−0.983559			−0.491779	+	0.870720i	$0.663653\pi$
$972$		0			0
$973$	4.41125			0.141418
$974$		0			0
$975$			5.16244i			0.165331i
$976$		0			0
$977$	36.5236			1.16849			0.584247	−	0.811576i	$-0.301390\pi$
$977$	36.5236			1.16849			0.584247	+	0.811576i	$0.301390\pi$
$978$		0			0
$979$			33.7667i			1.07919i
$980$		0			0
$981$			0.329336i			0.0105149i
$982$		0			0
$983$		−	12.0107i		−	0.383083i	−0.981485	−	0.191541i	$-0.938651\pi$
$983$		−	12.0107i		−	0.383083i	0.981485	−	0.191541i	$-0.0613486\pi$
$984$		0			0
$985$	18.5642			0.591504
$986$		0			0
$987$	−17.7577			−0.565235
$988$		0			0
$989$		−	2.00846i		−	0.0638654i
$990$		0			0
$991$			33.8831i			1.07633i	0.842839	+	0.538166i	$0.180883\pi$
$991$			33.8831i			1.07633i	−0.842839	+	0.538166i	$0.819117\pi$
$992$		0			0
$993$		−	37.9922i		−	1.20565i
$994$		0			0
$995$	18.3332			0.581200
$996$		0			0
$997$			22.4339i			0.710489i	0.934773	+	0.355245i	$0.115602\pi$
$997$			22.4339i			0.710489i	−0.934773	+	0.355245i	$0.884398\pi$
$998$		0			0
$999$	13.1033			0.414570

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.b.a.237.11	✓	40
17.16	even	2	inner	4012.2.b.a.237.30	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.b.a.237.11	✓	40	1.1	even	1	trivial
4012.2.b.a.237.30	yes	40	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)

\(f(q)\)	\(=\)	\(q-1.67902i q^{3} -0.714826i q^{5} +1.81204i q^{7} +0.180890 q^{9} +O(q^{10})\)	\(q-1.67902i q^{3} -0.714826i q^{5} +1.81204i q^{7} +0.180890 q^{9} -1.90342i q^{11} -0.684932 q^{13} -1.20021 q^{15} +(-2.75528 - 3.06732i) q^{17} -8.58944 q^{19} +3.04245 q^{21} +2.35240i q^{23} +4.48902 q^{25} -5.34078i q^{27} -1.67302i q^{29} +9.13920i q^{31} -3.19588 q^{33} +1.29529 q^{35} +2.45344i q^{37} +1.15001i q^{39} -5.79625i q^{41} -0.853793 q^{43} -0.129305i q^{45} -5.83666 q^{47} +3.71652 q^{49} +(-5.15009 + 4.62617i) q^{51} -11.7493 q^{53} -1.36061 q^{55} +14.4218i q^{57} -1.00000 q^{59} -0.855431i q^{61} +0.327780i q^{63} +0.489607i q^{65} -2.72694 q^{67} +3.94973 q^{69} -1.42200i q^{71} +1.06012i q^{73} -7.53716i q^{75} +3.44907 q^{77} +10.5497i q^{79} -8.42461 q^{81} -8.47593 q^{83} +(-2.19260 + 1.96954i) q^{85} -2.80903 q^{87} -17.7400 q^{89} -1.24112i q^{91} +15.3449 q^{93} +6.13995i q^{95} -14.7739i q^{97} -0.344310i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 36 q^{9}+O(q^{10}) \) 40 * q - 36 * q^9	\( 40 q - 36 q^{9} + 8 q^{13} + 20 q^{15} - 8 q^{17} + 20 q^{19} + 6 q^{21} - 24 q^{25} - 14 q^{33} - 52 q^{35} + 22 q^{43} - 10 q^{47} + 8 q^{49} - 6 q^{51} - 2 q^{53} - 12 q^{55} - 40 q^{59} - 24 q^{67} - 36 q^{69} - 38 q^{77} + 16 q^{81} + 32 q^{83} - 22 q^{85} - 18 q^{87} + 40 q^{89} + 22 q^{93}+O(q^{100}) \) 40 * q - 36 * q^9 + 8 * q^13 + 20 * q^15 - 8 * q^17 + 20 * q^19 + 6 * q^21 - 24 * q^25 - 14 * q^33 - 52 * q^35 + 22 * q^43 - 10 * q^47 + 8 * q^49 - 6 * q^51 - 2 * q^53 - 12 * q^55 - 40 * q^59 - 24 * q^67 - 36 * q^69 - 38 * q^77 + 16 * q^81 + 32 * q^83 - 22 * q^85 - 18 * q^87 + 40 * q^89 + 22 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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