LMFDB - Embedded newform 4020.2.g.c.1609.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(1609,4020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4020, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4020.1609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4020.g (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:	$32.0998616126$
Analytic rank:	$0$
Dimension:	$38$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1609.19
Character	$\chi$	$=$	4020.1609
Dual form			4020.2.g.c.1609.38

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(2.21373 + 0.315287i) q^{5} -0.0918930i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +(2.21373 + 0.315287i) q^{5} -0.0918930i q^{7} -1.00000 q^{9} -3.06268 q^{11} +5.35772i q^{13} +(0.315287 - 2.21373i) q^{15} -2.02960i q^{17} +7.88708 q^{19} -0.0918930 q^{21} -5.50964i q^{23} +(4.80119 + 1.39592i) q^{25} +1.00000i q^{27} -2.34320 q^{29} -1.86797 q^{31} +3.06268i q^{33} +(0.0289727 - 0.203426i) q^{35} +6.88400i q^{37} +5.35772 q^{39} -5.27113 q^{41} -6.19532i q^{43} +(-2.21373 - 0.315287i) q^{45} +3.51696i q^{47} +6.99156 q^{49} -2.02960 q^{51} -1.50798i q^{53} +(-6.77993 - 0.965623i) q^{55} -7.88708i q^{57} +7.85352 q^{59} +12.9188 q^{61} +0.0918930i q^{63} +(-1.68922 + 11.8605i) q^{65} -1.00000i q^{67} -5.50964 q^{69} +12.3591 q^{71} +6.48293i q^{73} +(1.39592 - 4.80119i) q^{75} +0.281438i q^{77} +8.72623 q^{79} +1.00000 q^{81} +3.43081i q^{83} +(0.639909 - 4.49299i) q^{85} +2.34320i q^{87} +4.92101 q^{89} +0.492337 q^{91} +1.86797i q^{93} +(17.4599 + 2.48670i) q^{95} -3.65014i q^{97} +3.06268 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 38 q - 2 q^{5} - 38 q^{9}+O(q^{10}) $ 38 * q - 2 * q^5 - 38 * q^9	$ 38 q - 2 q^{5} - 38 q^{9} + 24 q^{11} + 2 q^{15} - 16 q^{19} + 12 q^{21} + 4 q^{25} - 56 q^{29} - 4 q^{31} - 2 q^{35} + 60 q^{41} + 2 q^{45} - 70 q^{49} + 12 q^{55} - 52 q^{59} + 48 q^{61} + 16 q^{65} - 12 q^{69} + 12 q^{75} - 24 q^{79} + 38 q^{81} + 16 q^{85} - 76 q^{89} + 44 q^{91} + 36 q^{95} - 24 q^{99}+O(q^{100}) $ 38 * q - 2 * q^5 - 38 * q^9 + 24 * q^11 + 2 * q^15 - 16 * q^19 + 12 * q^21 + 4 * q^25 - 56 * q^29 - 4 * q^31 - 2 * q^35 + 60 * q^41 + 2 * q^45 - 70 * q^49 + 12 * q^55 - 52 * q^59 + 48 * q^61 + 16 * q^65 - 12 * q^69 + 12 * q^75 - 24 * q^79 + 38 * q^81 + 16 * q^85 - 76 * q^89 + 44 * q^91 + 36 * q^95 - 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4020\mathbb{Z}\right)^\times$.

$n$	$1141$	$2011$	$2681$	$3217$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	2.21373	+	0.315287i	0.990009	+	0.141001i
$5$	2.21373	+	0.315287i	0.990009	+	0.141001i
$6$		0			0
$7$		−	0.0918930i		−	0.0347323i	−0.999849	−	0.0173661i	$-0.994472\pi$
$7$		−	0.0918930i		−	0.0347323i	0.999849	−	0.0173661i	$-0.00552809\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−3.06268			−0.923432			−0.461716	−	0.887028i	$-0.652766\pi$
$11$	−3.06268			−0.923432			−0.461716	+	0.887028i	$0.652766\pi$
$12$		0			0
$13$			5.35772i			1.48596i	0.669312	+	0.742982i	$0.266589\pi$
$13$			5.35772i			1.48596i	−0.669312	+	0.742982i	$0.733411\pi$
$14$		0			0
$15$	0.315287	−	2.21373i	0.0814068	−	0.571582i
$16$		0			0
$17$		−	2.02960i		−	0.492251i	−0.969238	−	0.246126i	$-0.920842\pi$
$17$		−	2.02960i		−	0.492251i	0.969238	−	0.246126i	$-0.0791576\pi$
$18$		0			0
$19$	7.88708			1.80942			0.904710	−	0.426027i	$-0.140087\pi$
$19$	7.88708			1.80942			0.904710	+	0.426027i	$0.140087\pi$
$20$		0			0
$21$	−0.0918930			−0.0200527
$22$		0			0
$23$		−	5.50964i		−	1.14884i	−0.818561	−	0.574419i	$-0.805228\pi$
$23$		−	5.50964i		−	1.14884i	0.818561	−	0.574419i	$-0.194772\pi$
$24$		0			0
$25$	4.80119	+	1.39592i	0.960238	+	0.279184i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	−2.34320			−0.435122			−0.217561	−	0.976047i	$-0.569810\pi$
$29$	−2.34320			−0.435122			−0.217561	+	0.976047i	$0.569810\pi$
$30$		0			0
$31$	−1.86797			−0.335498			−0.167749	−	0.985830i	$-0.553650\pi$
$31$	−1.86797			−0.335498			−0.167749	+	0.985830i	$0.553650\pi$
$32$		0			0
$33$			3.06268i			0.533144i
$34$		0			0
$35$	0.0289727	−	0.203426i	0.00489728	−	0.0343853i
$36$		0			0
$37$			6.88400i			1.13172i	0.824500	+	0.565861i	$0.191456\pi$
$37$			6.88400i			1.13172i	−0.824500	+	0.565861i	$0.808544\pi$
$38$		0			0
$39$	5.35772			0.857921
$40$		0			0
$41$	−5.27113			−0.823213			−0.411606	−	0.911362i	$-0.635032\pi$
$41$	−5.27113			−0.823213			−0.411606	+	0.911362i	$0.635032\pi$
$42$		0			0
$43$		−	6.19532i		−	0.944777i	−0.881390	−	0.472388i	$-0.843392\pi$
$43$		−	6.19532i		−	0.944777i	0.881390	−	0.472388i	$-0.156608\pi$
$44$		0			0
$45$	−2.21373	−	0.315287i	−0.330003	−	0.0470003i
$46$		0			0
$47$			3.51696i			0.513001i	0.966544	+	0.256500i	$0.0825695\pi$
$47$			3.51696i			0.513001i	−0.966544	+	0.256500i	$0.917431\pi$
$48$		0			0
$49$	6.99156			0.998794
$50$		0			0
$51$	−2.02960			−0.284201
$52$		0			0
$53$		−	1.50798i		−	0.207137i	−0.994622	−	0.103568i	$-0.966974\pi$
$53$		−	1.50798i		−	0.207137i	0.994622	−	0.103568i	$-0.0330261\pi$
$54$		0			0
$55$	−6.77993	−	0.965623i	−0.914206	−	0.130205i
$56$		0			0
$57$		−	7.88708i		−	1.04467i
$58$		0			0
$59$	7.85352			1.02244			0.511221	−	0.859450i	$-0.329194\pi$
$59$	7.85352			1.02244			0.511221	+	0.859450i	$0.329194\pi$
$60$		0			0
$61$	12.9188			1.65408			0.827042	−	0.562141i	$-0.190022\pi$
$61$	12.9188			1.65408			0.827042	+	0.562141i	$0.190022\pi$
$62$		0			0
$63$			0.0918930i			0.0115774i
$64$		0			0
$65$	−1.68922	+	11.8605i	−0.209522	+	1.47112i
$66$		0			0
$67$		−	1.00000i		−	0.122169i
$67$		−	1.00000i		−	0.122169i
$68$		0			0
$69$	−5.50964			−0.663282
$70$		0			0
$71$	12.3591			1.46675			0.733376	−	0.679823i	$-0.237944\pi$
$71$	12.3591			1.46675			0.733376	+	0.679823i	$0.237944\pi$
$72$		0			0
$73$			6.48293i			0.758769i	0.925239	+	0.379385i	$0.123864\pi$
$73$			6.48293i			0.758769i	−0.925239	+	0.379385i	$0.876136\pi$
$74$		0			0
$75$	1.39592	−	4.80119i	0.161187	−	0.554393i
$76$		0			0
$77$			0.281438i			0.0320729i
$78$		0			0
$79$	8.72623			0.981778			0.490889	−	0.871222i	$-0.336672\pi$
$79$	8.72623			0.981778			0.490889	+	0.871222i	$0.336672\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			3.43081i			0.376580i	0.982113	+	0.188290i	$0.0602946\pi$
$83$			3.43081i			0.376580i	−0.982113	+	0.188290i	$0.939705\pi$
$84$		0			0
$85$	0.639909	−	4.49299i	0.0694078	−	0.487334i
$86$		0			0
$87$			2.34320i			0.251218i
$88$		0			0
$89$	4.92101			0.521626			0.260813	−	0.965389i	$-0.416009\pi$
$89$	4.92101			0.521626			0.260813	+	0.965389i	$0.416009\pi$
$90$		0			0
$91$	0.492337			0.0516109
$92$		0			0
$93$			1.86797i			0.193700i
$94$		0			0
$95$	17.4599	+	2.48670i	1.79134	+	0.255130i
$96$		0			0
$97$		−	3.65014i		−	0.370616i	−0.982681	−	0.185308i	$-0.940672\pi$
$97$		−	3.65014i		−	0.370616i	0.982681	−	0.185308i	$-0.0593282\pi$
$98$		0			0
$99$	3.06268			0.307811
$100$		0			0
$101$	11.9012			1.18421			0.592107	−	0.805859i	$-0.298296\pi$
$101$	11.9012			1.18421			0.592107	+	0.805859i	$0.298296\pi$
$102$		0			0
$103$			13.2029i			1.30092i	0.759541	+	0.650460i	$0.225424\pi$
$103$			13.2029i			1.30092i	−0.759541	+	0.650460i	$0.774576\pi$
$104$		0			0
$105$	−0.203426	−	0.0289727i	−0.0198524	−	0.00282744i
$106$		0			0
$107$		−	8.18848i		−	0.791610i	−0.918334	−	0.395805i	$-0.870466\pi$
$107$		−	8.18848i		−	0.791610i	0.918334	−	0.395805i	$-0.129534\pi$
$108$		0			0
$109$	−5.52582			−0.529278			−0.264639	−	0.964348i	$-0.585253\pi$
$109$	−5.52582			−0.529278			−0.264639	+	0.964348i	$0.585253\pi$
$110$		0			0
$111$	6.88400			0.653400
$112$		0			0
$113$			15.4323i			1.45175i	0.687827	+	0.725875i	$0.258565\pi$
$113$			15.4323i			1.45175i	−0.687827	+	0.725875i	$0.741435\pi$
$114$		0			0
$115$	1.73712	−	12.1968i	0.161987	−	1.13736i
$116$		0			0
$117$		−	5.35772i		−	0.495321i
$118$		0			0
$119$	−0.186506			−0.0170970
$120$		0			0
$121$	−1.62001			−0.147274
$122$		0			0
$123$			5.27113i			0.475282i
$124$		0			0
$125$	10.1884	+	4.60394i	0.911279	+	0.411789i
$126$		0			0
$127$			18.0509i			1.60176i	0.598825	+	0.800880i	$0.295634\pi$
$127$			18.0509i			1.60176i	−0.598825	+	0.800880i	$0.704366\pi$
$128$		0			0
$129$	−6.19532			−0.545467
$130$		0			0
$131$	6.13343			0.535880			0.267940	−	0.963436i	$-0.413657\pi$
$131$	6.13343			0.535880			0.267940	+	0.963436i	$0.413657\pi$
$132$		0			0
$133$		−	0.724767i		−	0.0628453i
$134$		0			0
$135$	−0.315287	+	2.21373i	−0.0271356	+	0.190527i
$136$		0			0
$137$			0.587337i			0.0501796i	0.999685	+	0.0250898i	$0.00798717\pi$
$137$			0.587337i			0.0501796i	−0.999685	+	0.0250898i	$0.992013\pi$
$138$		0			0
$139$	3.67081			0.311354			0.155677	−	0.987808i	$-0.450244\pi$
$139$	3.67081			0.311354			0.155677	+	0.987808i	$0.450244\pi$
$140$		0			0
$141$	3.51696			0.296181
$142$		0			0
$143$		−	16.4090i		−	1.37219i
$144$		0			0
$145$	−5.18721	−	0.738782i	−0.430775	−	0.0613525i
$146$		0			0
$147$		−	6.99156i		−	0.576654i
$148$		0			0
$149$	−16.9988			−1.39259			−0.696297	−	0.717754i	$-0.745170\pi$
$149$	−16.9988			−1.39259			−0.696297	+	0.717754i	$0.745170\pi$
$150$		0			0
$151$	3.60010			0.292972			0.146486	−	0.989213i	$-0.453204\pi$
$151$	3.60010			0.292972			0.146486	+	0.989213i	$0.453204\pi$
$152$		0			0
$153$			2.02960i			0.164084i
$154$		0			0
$155$	−4.13519	−	0.588949i	−0.332146	−	0.0473055i
$156$		0			0
$157$		−	21.5953i		−	1.72349i	−0.507338	−	0.861747i	$-0.669370\pi$
$157$		−	21.5953i		−	1.72349i	0.507338	−	0.861747i	$-0.330630\pi$
$158$		0			0
$159$	−1.50798			−0.119591
$160$		0			0
$161$	−0.506297			−0.0399018
$162$		0			0
$163$		−	12.2430i		−	0.958943i	−0.877557	−	0.479472i	$-0.840828\pi$
$163$		−	12.2430i		−	0.958943i	0.877557	−	0.479472i	$-0.159172\pi$
$164$		0			0
$165$	−0.965623	+	6.77993i	−0.0751737	+	0.527817i
$166$		0			0
$167$		−	0.932800i		−	0.0721822i	−0.999349	−	0.0360911i	$-0.988509\pi$
$167$		−	0.932800i		−	0.0721822i	0.999349	−	0.0360911i	$-0.0114907\pi$
$168$		0			0
$169$	−15.7051			−1.20809
$170$		0			0
$171$	−7.88708			−0.603140
$172$		0			0
$173$		−	16.3751i		−	1.24498i	−0.782628	−	0.622489i	$-0.786121\pi$
$173$		−	16.3751i		−	1.24498i	0.782628	−	0.622489i	$-0.213879\pi$
$174$		0			0
$175$	0.128275	−	0.441195i	0.00969670	−	0.0333512i
$176$		0			0
$177$		−	7.85352i		−	0.590307i
$178$		0			0
$179$	11.3572			0.848877			0.424438	−	0.905457i	$-0.360472\pi$
$179$	11.3572			0.848877			0.424438	+	0.905457i	$0.360472\pi$
$180$		0			0
$181$	−14.1514			−1.05187			−0.525933	−	0.850526i	$-0.676284\pi$
$181$	−14.1514			−1.05187			−0.525933	+	0.850526i	$0.676284\pi$
$182$		0			0
$183$		−	12.9188i		−	0.954985i
$184$		0			0
$185$	−2.17044	+	15.2393i	−0.159574	+	1.12042i
$186$		0			0
$187$			6.21602i			0.454561i
$188$		0			0
$189$	0.0918930			0.00668423
$190$		0			0
$191$	1.33996			0.0969560			0.0484780	−	0.998824i	$-0.484563\pi$
$191$	1.33996			0.0969560			0.0484780	+	0.998824i	$0.484563\pi$
$192$		0			0
$193$		−	7.05425i		−	0.507776i	−0.967234	−	0.253888i	$-0.918290\pi$
$193$		−	7.05425i		−	0.507776i	0.967234	−	0.253888i	$-0.0817095\pi$
$194$		0			0
$195$	11.8605	+	1.68922i	0.849350	+	0.120968i
$196$		0			0
$197$			15.9934i			1.13949i	0.821823	+	0.569743i	$0.192957\pi$
$197$			15.9934i			1.13949i	−0.821823	+	0.569743i	$0.807043\pi$
$198$		0			0
$199$	−1.49303			−0.105838			−0.0529191	−	0.998599i	$-0.516853\pi$
$199$	−1.49303			−0.105838			−0.0529191	+	0.998599i	$0.516853\pi$
$200$		0			0
$201$	−1.00000			−0.0705346
$202$		0			0
$203$			0.215324i			0.0151128i
$204$		0			0
$205$	−11.6689	−	1.66192i	−0.814988	−	0.116074i
$206$		0			0
$207$			5.50964i			0.382946i
$208$		0			0
$209$	−24.1556			−1.67088
$210$		0			0
$211$	5.24402			0.361013			0.180507	−	0.983574i	$-0.442226\pi$
$211$	5.24402			0.361013			0.180507	+	0.983574i	$0.442226\pi$
$212$		0			0
$213$		−	12.3591i		−	0.846830i
$214$		0			0
$215$	1.95330	−	13.7147i	0.133214	−	0.935338i
$216$		0			0
$217$			0.171654i			0.0116526i
$218$		0			0
$219$	6.48293			0.438076
$220$		0			0
$221$	10.8740			0.731468
$222$		0			0
$223$			5.24398i			0.351163i	0.984465	+	0.175582i	$0.0561806\pi$
$223$			5.24398i			0.351163i	−0.984465	+	0.175582i	$0.943819\pi$
$224$		0			0
$225$	−4.80119	−	1.39592i	−0.320079	−	0.0930614i
$226$		0			0
$227$		−	18.0350i		−	1.19702i	−0.801114	−	0.598512i	$-0.795759\pi$
$227$		−	18.0350i		−	1.19702i	0.801114	−	0.598512i	$-0.204241\pi$
$228$		0			0
$229$	6.03776			0.398987			0.199493	−	0.979899i	$-0.436070\pi$
$229$	6.03776			0.398987			0.199493	+	0.979899i	$0.436070\pi$
$230$		0			0
$231$	0.281438			0.0185173
$232$		0			0
$233$		−	1.82102i		−	0.119299i	−0.998219	−	0.0596495i	$-0.981002\pi$
$233$		−	1.82102i		−	0.119299i	0.998219	−	0.0596495i	$-0.0189983\pi$
$234$		0			0
$235$	−1.10885	+	7.78558i	−0.0723335	+	0.507875i
$236$		0			0
$237$		−	8.72623i		−	0.566830i
$238$		0			0
$239$	−13.2797			−0.858990			−0.429495	−	0.903069i	$-0.641308\pi$
$239$	−13.2797			−0.858990			−0.429495	+	0.903069i	$0.641308\pi$
$240$		0			0
$241$	10.7660			0.693497			0.346748	−	0.937958i	$-0.387286\pi$
$241$	10.7660			0.693497			0.346748	+	0.937958i	$0.387286\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	15.4774	+	2.20435i	0.988815	+	0.140831i
$246$		0			0
$247$			42.2568i			2.68873i
$248$		0			0
$249$	3.43081			0.217419
$250$		0			0
$251$	−0.301420			−0.0190255			−0.00951274	−	0.999955i	$-0.503028\pi$
$251$	−0.301420			−0.0190255			−0.00951274	+	0.999955i	$0.503028\pi$
$252$		0			0
$253$			16.8742i			1.06087i
$254$		0			0
$255$	−4.49299	−	0.639909i	−0.281362	−	0.0400726i
$256$		0			0
$257$			12.1267i			0.756440i	0.925716	+	0.378220i	$0.123464\pi$
$257$			12.1267i			0.756440i	−0.925716	+	0.378220i	$0.876536\pi$
$258$		0			0
$259$	0.632591			0.0393073
$260$		0			0
$261$	2.34320			0.145041
$262$		0			0
$263$		−	7.20836i		−	0.444486i	−0.974991	−	0.222243i	$-0.928662\pi$
$263$		−	7.20836i		−	0.444486i	0.974991	−	0.222243i	$-0.0713379\pi$
$264$		0			0
$265$	0.475447	−	3.33826i	0.0292065	−	0.205067i
$266$		0			0
$267$		−	4.92101i		−	0.301161i
$268$		0			0
$269$	−21.1140			−1.28735			−0.643673	−	0.765301i	$-0.722590\pi$
$269$	−21.1140			−1.28735			−0.643673	+	0.765301i	$0.722590\pi$
$270$		0			0
$271$	21.8167			1.32527			0.662635	−	0.748943i	$-0.269438\pi$
$271$	21.8167			1.32527			0.662635	+	0.748943i	$0.269438\pi$
$272$		0			0
$273$		−	0.492337i		−	0.0297976i
$274$		0			0
$275$	−14.7045	−	4.27526i	−0.886714	−	0.257808i
$276$		0			0
$277$			10.4668i			0.628889i	0.949276	+	0.314444i	$0.101818\pi$
$277$			10.4668i			0.628889i	−0.949276	+	0.314444i	$0.898182\pi$
$278$		0			0
$279$	1.86797			0.111833
$280$		0			0
$281$	14.7710			0.881165			0.440582	−	0.897712i	$-0.354772\pi$
$281$	14.7710			0.881165			0.440582	+	0.897712i	$0.354772\pi$
$282$		0			0
$283$		−	17.2160i		−	1.02339i	−0.859168	−	0.511694i	$-0.829018\pi$
$283$		−	17.2160i		−	1.02339i	0.859168	−	0.511694i	$-0.170982\pi$
$284$		0			0
$285$	2.48670	−	17.4599i	0.147299	−	1.03423i
$286$		0			0
$287$			0.484380i			0.0285921i
$288$		0			0
$289$	12.8807			0.757689
$290$		0			0
$291$	−3.65014			−0.213975
$292$		0			0
$293$			5.61952i			0.328296i	0.986436	+	0.164148i	$0.0524874\pi$
$293$			5.61952i			0.328296i	−0.986436	+	0.164148i	$0.947513\pi$
$294$		0			0
$295$	17.3856	+	2.47612i	1.01223	+	0.144165i
$296$		0			0
$297$		−	3.06268i		−	0.177715i
$298$		0			0
$299$	29.5191			1.70713
$300$		0			0
$301$	−0.569306			−0.0328143
$302$		0			0
$303$		−	11.9012i		−	0.683707i
$304$		0			0
$305$	28.5987	+	4.07313i	1.63756	+	0.233227i
$306$		0			0
$307$			7.99448i			0.456269i	0.973630	+	0.228135i	$0.0732626\pi$
$307$			7.99448i			0.456269i	−0.973630	+	0.228135i	$0.926737\pi$
$308$		0			0
$309$	13.2029			0.751086
$310$		0			0
$311$	13.2422			0.750897			0.375448	−	0.926843i	$-0.377489\pi$
$311$	13.2422			0.750897			0.375448	+	0.926843i	$0.377489\pi$
$312$		0			0
$313$		−	0.186937i		−	0.0105663i	−0.999986	−	0.00528316i	$-0.998318\pi$
$313$		−	0.186937i		−	0.0105663i	0.999986	−	0.00528316i	$-0.00168169\pi$
$314$		0			0
$315$	−0.0289727	+	0.203426i	−0.00163243	+	0.0114618i
$316$		0			0
$317$		−	16.9563i		−	0.952360i	−0.879348	−	0.476180i	$-0.842021\pi$
$317$		−	16.9563i		−	0.952360i	0.879348	−	0.476180i	$-0.157979\pi$
$318$		0			0
$319$	7.17647			0.401805
$320$		0			0
$321$	−8.18848			−0.457037
$322$		0			0
$323$		−	16.0077i		−	0.890690i
$324$		0			0
$325$	−7.47895	+	25.7234i	−0.414858	+	1.42688i
$326$		0			0
$327$			5.52582i			0.305579i
$328$		0			0
$329$	0.323183			0.0178177
$330$		0			0
$331$	−0.101500			−0.00557894			−0.00278947	−	0.999996i	$-0.500888\pi$
$331$	−0.101500			−0.00557894			−0.00278947	+	0.999996i	$0.500888\pi$
$332$		0			0
$333$		−	6.88400i		−	0.377241i
$334$		0			0
$335$	0.315287	−	2.21373i	0.0172260	−	0.120949i
$336$		0			0
$337$			3.81660i			0.207904i	0.994582	+	0.103952i	$0.0331488\pi$
$337$			3.81660i			0.207904i	−0.994582	+	0.103952i	$0.966851\pi$
$338$		0			0
$339$	15.4323			0.838168
$340$		0			0
$341$	5.72100			0.309810
$342$		0			0
$343$		−	1.28573i		−	0.0694227i
$344$		0			0
$345$	−12.1968	−	1.73712i	−0.656656	−	0.0935233i
$346$		0			0
$347$			16.4215i			0.881555i	0.897616	+	0.440777i	$0.145297\pi$
$347$			16.4215i			0.881555i	−0.897616	+	0.440777i	$0.854703\pi$
$348$		0			0
$349$	−18.7294			−1.00256			−0.501282	−	0.865284i	$-0.667138\pi$
$349$	−18.7294			−1.00256			−0.501282	+	0.865284i	$0.667138\pi$
$350$		0			0
$351$	−5.35772			−0.285974
$352$		0			0
$353$		−	36.9358i		−	1.96589i	−0.183893	−	0.982946i	$-0.558870\pi$
$353$		−	36.9358i		−	1.96589i	0.183893	−	0.982946i	$-0.441130\pi$
$354$		0			0
$355$	27.3596	+	3.89666i	1.45210	+	0.206813i
$356$		0			0
$357$			0.186506i			0.00987096i
$358$		0			0
$359$	−17.2201			−0.908844			−0.454422	−	0.890787i	$-0.650154\pi$
$359$	−17.2201			−0.908844			−0.454422	+	0.890787i	$0.650154\pi$
$360$		0			0
$361$	43.2061			2.27400
$362$		0			0
$363$			1.62001i			0.0850286i
$364$		0			0
$365$	−2.04398	+	14.3514i	−0.106987	+	0.751189i
$366$		0			0
$367$		−	13.9707i		−	0.729266i	−0.931151	−	0.364633i	$-0.881194\pi$
$367$		−	13.9707i		−	0.729266i	0.931151	−	0.364633i	$-0.118806\pi$
$368$		0			0
$369$	5.27113			0.274404
$370$		0			0
$371$	−0.138573			−0.00719433
$372$		0			0
$373$			17.0224i			0.881384i	0.897658	+	0.440692i	$0.145267\pi$
$373$			17.0224i			0.881384i	−0.897658	+	0.440692i	$0.854733\pi$
$374$		0			0
$375$	4.60394	−	10.1884i	0.237747	−	0.526127i
$376$		0			0
$377$		−	12.5542i		−	0.646575i
$378$		0			0
$379$	−13.0321			−0.669413			−0.334706	−	0.942322i	$-0.608637\pi$
$379$	−13.0321			−0.669413			−0.334706	+	0.942322i	$0.608637\pi$
$380$		0			0
$381$	18.0509			0.924776
$382$		0			0
$383$		−	0.572367i		−	0.0292466i	−0.999893	−	0.0146233i	$-0.995345\pi$
$383$		−	0.572367i		−	0.0292466i	0.999893	−	0.0146233i	$-0.00465490\pi$
$384$		0			0
$385$	−0.0887340	+	0.623028i	−0.00452230	+	0.0317525i
$386$		0			0
$387$			6.19532i			0.314926i
$388$		0			0
$389$	−37.7164			−1.91230			−0.956148	−	0.292885i	$-0.905385\pi$
$389$	−37.7164			−1.91230			−0.956148	+	0.292885i	$0.905385\pi$
$390$		0			0
$391$	−11.1824			−0.565518
$392$		0			0
$393$		−	6.13343i		−	0.309391i
$394$		0			0
$395$	19.3175	+	2.75127i	0.971969	+	0.138431i
$396$		0			0
$397$			38.3132i			1.92289i	0.275004	+	0.961443i	$0.411321\pi$
$397$			38.3132i			1.92289i	−0.275004	+	0.961443i	$0.588679\pi$
$398$		0			0
$399$	−0.724767			−0.0362838
$400$		0			0
$401$	−14.7432			−0.736239			−0.368120	−	0.929778i	$-0.619998\pi$
$401$	−14.7432			−0.736239			−0.368120	+	0.929778i	$0.619998\pi$
$402$		0			0
$403$		−	10.0081i		−	0.498538i
$404$		0			0
$405$	2.21373	+	0.315287i	0.110001	+	0.0156668i
$406$		0			0
$407$		−	21.0835i		−	1.04507i
$408$		0			0
$409$	7.58223			0.374917			0.187458	−	0.982273i	$-0.439975\pi$
$409$	7.58223			0.374917			0.187458	+	0.982273i	$0.439975\pi$
$410$		0			0
$411$	0.587337			0.0289712
$412$		0			0
$413$		−	0.721683i		−	0.0355117i
$414$		0			0
$415$	−1.08169	+	7.59488i	−0.0530981	+	0.372818i
$416$		0			0
$417$		−	3.67081i		−	0.179760i
$418$		0			0
$419$	−8.86191			−0.432932			−0.216466	−	0.976290i	$-0.569453\pi$
$419$	−8.86191			−0.432932			−0.216466	+	0.976290i	$0.569453\pi$
$420$		0			0
$421$	13.9006			0.677472			0.338736	−	0.940882i	$-0.390001\pi$
$421$	13.9006			0.677472			0.338736	+	0.940882i	$0.390001\pi$
$422$		0			0
$423$		−	3.51696i		−	0.171000i
$424$		0			0
$425$	2.83317	−	9.74451i	0.137429	−	0.472678i
$426$		0			0
$427$		−	1.18715i		−	0.0574501i
$428$		0			0
$429$	−16.4090			−0.792232
$430$		0			0
$431$	31.4435			1.51458			0.757290	−	0.653078i	$-0.226523\pi$
$431$	31.4435			1.51458			0.757290	+	0.653078i	$0.226523\pi$
$432$		0			0
$433$		−	21.4617i		−	1.03138i	−0.856774	−	0.515692i	$-0.827535\pi$
$433$		−	21.4617i		−	1.03138i	0.856774	−	0.515692i	$-0.172465\pi$
$434$		0			0
$435$	−0.738782	+	5.18721i	−0.0354219	+	0.248708i
$436$		0			0
$437$		−	43.4550i		−	2.07873i
$438$		0			0
$439$	−24.5575			−1.17207			−0.586034	−	0.810287i	$-0.699311\pi$
$439$	−24.5575			−1.17207			−0.586034	+	0.810287i	$0.699311\pi$
$440$		0			0
$441$	−6.99156			−0.332931
$442$		0			0
$443$		−	21.6310i		−	1.02772i	−0.857874	−	0.513860i	$-0.828215\pi$
$443$		−	21.6310i		−	1.02772i	0.857874	−	0.513860i	$-0.171785\pi$
$444$		0			0
$445$	10.8938	+	1.55153i	0.516415	+	0.0735497i
$446$		0			0
$447$			16.9988i			0.804014i
$448$		0			0
$449$	−16.7231			−0.789214			−0.394607	−	0.918850i	$-0.629119\pi$
$449$	−16.7231			−0.789214			−0.394607	+	0.918850i	$0.629119\pi$
$450$		0			0
$451$	16.1438			0.760181
$452$		0			0
$453$		−	3.60010i		−	0.169148i
$454$		0			0
$455$	1.08990	+	0.155227i	0.0510953	+	0.00727718i
$456$		0			0
$457$			12.4524i			0.582500i	0.956647	+	0.291250i	$0.0940711\pi$
$457$			12.4524i			0.582500i	−0.956647	+	0.291250i	$0.905929\pi$
$458$		0			0
$459$	2.02960			0.0947338
$460$		0			0
$461$	−9.19791			−0.428390			−0.214195	−	0.976791i	$-0.568713\pi$
$461$	−9.19791			−0.428390			−0.214195	+	0.976791i	$0.568713\pi$
$462$		0			0
$463$		−	0.630410i		−	0.0292976i	−0.999893	−	0.0146488i	$-0.995337\pi$
$463$		−	0.630410i		−	0.0292976i	0.999893	−	0.0146488i	$-0.00466303\pi$
$464$		0			0
$465$	−0.588949	+	4.13519i	−0.0273118	+	0.191765i
$466$		0			0
$467$		−	33.0380i		−	1.52882i	−0.644732	−	0.764409i	$-0.723031\pi$
$467$		−	33.0380i		−	1.52882i	0.644732	−	0.764409i	$-0.276969\pi$
$468$		0			0
$469$	−0.0918930			−0.00424322
$470$		0			0
$471$	−21.5953			−0.995060
$472$		0			0
$473$			18.9743i			0.872437i
$474$		0			0
$475$	37.8674	+	11.0097i	1.73747	+	0.505162i
$476$		0			0
$477$			1.50798i			0.0690456i
$478$		0			0
$479$	−5.21941			−0.238481			−0.119240	−	0.992865i	$-0.538046\pi$
$479$	−5.21941			−0.238481			−0.119240	+	0.992865i	$0.538046\pi$
$480$		0			0
$481$	−36.8825			−1.68170
$482$		0			0
$483$			0.506297i			0.0230373i
$484$		0			0
$485$	1.15084	−	8.08042i	0.0522571	−	0.366913i
$486$		0			0
$487$			22.8585i			1.03582i	0.855436	+	0.517908i	$0.173289\pi$
$487$			22.8585i			1.03582i	−0.855436	+	0.517908i	$0.826711\pi$
$488$		0			0
$489$	−12.2430			−0.553646
$490$		0			0
$491$	2.24621			0.101370			0.0506851	−	0.998715i	$-0.483860\pi$
$491$	2.24621			0.101370			0.0506851	+	0.998715i	$0.483860\pi$
$492$		0			0
$493$			4.75577i			0.214189i
$494$		0			0
$495$	6.77993	+	0.965623i	0.304735	+	0.0434015i
$496$		0			0
$497$		−	1.13571i		−	0.0509436i
$498$		0			0
$499$	−7.30734			−0.327122			−0.163561	−	0.986533i	$-0.552298\pi$
$499$	−7.30734			−0.327122			−0.163561	+	0.986533i	$0.552298\pi$
$500$		0			0
$501$	−0.932800			−0.0416744
$502$		0			0
$503$			16.3452i			0.728798i	0.931243	+	0.364399i	$0.118726\pi$
$503$			16.3452i			0.728798i	−0.931243	+	0.364399i	$0.881274\pi$
$504$		0			0
$505$	26.3461	+	3.75230i	1.17238	+	0.166975i
$506$		0			0
$507$			15.7051i			0.697490i
$508$		0			0
$509$	−2.62629			−0.116408			−0.0582042	−	0.998305i	$-0.518537\pi$
$509$	−2.62629			−0.116408			−0.0582042	+	0.998305i	$0.518537\pi$
$510$		0			0
$511$	0.595735			0.0263538
$512$		0			0
$513$			7.88708i			0.348223i
$514$		0			0
$515$	−4.16270	+	29.2276i	−0.183431	+	1.28792i
$516$		0			0
$517$		−	10.7713i		−	0.473721i
$518$		0			0
$519$	−16.3751			−0.718789
$520$		0			0
$521$	17.7191			0.776286			0.388143	−	0.921599i	$-0.373117\pi$
$521$	17.7191			0.776286			0.388143	+	0.921599i	$0.373117\pi$
$522$		0			0
$523$		−	17.6541i		−	0.771960i	−0.922507	−	0.385980i	$-0.873863\pi$
$523$		−	17.6541i		−	0.771960i	0.922507	−	0.385980i	$-0.126137\pi$
$524$		0			0
$525$	−0.441195	−	0.128275i	−0.0192553	−	0.00559839i
$526$		0			0
$527$			3.79125i			0.165149i
$528$		0			0
$529$	−7.35611			−0.319831
$530$		0			0
$531$	−7.85352			−0.340814
$532$		0			0
$533$		−	28.2412i		−	1.22326i
$534$		0			0
$535$	2.58172	−	18.1271i	0.111618	−	0.783702i
$536$		0			0
$537$		−	11.3572i		−	0.490099i
$538$		0			0
$539$	−21.4129			−0.922318
$540$		0			0
$541$	−3.46215			−0.148850			−0.0744248	−	0.997227i	$-0.523712\pi$
$541$	−3.46215			−0.148850			−0.0744248	+	0.997227i	$0.523712\pi$
$542$		0			0
$543$			14.1514i			0.607296i
$544$		0			0
$545$	−12.2327	−	1.74222i	−0.523990	−	0.0746286i
$546$		0			0
$547$		−	22.4199i		−	0.958606i	−0.877649	−	0.479303i	$-0.840889\pi$
$547$		−	22.4199i		−	0.958606i	0.877649	−	0.479303i	$-0.159111\pi$
$548$		0			0
$549$	−12.9188			−0.551361
$550$		0			0
$551$	−18.4810			−0.787318
$552$		0			0
$553$		−	0.801879i		−	0.0340994i
$554$		0			0
$555$	15.2393	+	2.17044i	0.646873	+	0.0921300i
$556$		0			0
$557$		−	6.64133i		−	0.281402i	−0.990052	−	0.140701i	$-0.955064\pi$
$557$		−	6.64133i		−	0.281402i	0.990052	−	0.140701i	$-0.0449357\pi$
$558$		0			0
$559$	33.1928			1.40390
$560$		0			0
$561$	6.21602			0.262441
$562$		0			0
$563$			19.0987i			0.804913i	0.915439	+	0.402456i	$0.131843\pi$
$563$			19.0987i			0.804913i	−0.915439	+	0.402456i	$0.868157\pi$
$564$		0			0
$565$	−4.86561	+	34.1630i	−0.204698	+	1.43725i
$566$		0			0
$567$		−	0.0918930i		−	0.00385914i
$568$		0			0
$569$	19.8773			0.833301			0.416651	−	0.909067i	$-0.363204\pi$
$569$	19.8773			0.833301			0.416651	+	0.909067i	$0.363204\pi$
$570$		0			0
$571$	−39.3393			−1.64630			−0.823149	−	0.567825i	$-0.807785\pi$
$571$	−39.3393			−1.64630			−0.823149	+	0.567825i	$0.807785\pi$
$572$		0			0
$573$		−	1.33996i		−	0.0559776i
$574$		0			0
$575$	7.69102	−	26.4528i	0.320738	−	1.10316i
$576$		0			0
$577$			40.6539i			1.69244i	0.532830	+	0.846222i	$0.321128\pi$
$577$			40.6539i			1.69244i	−0.532830	+	0.846222i	$0.678872\pi$
$578$		0			0
$579$	−7.05425			−0.293165
$580$		0			0
$581$	0.315267			0.0130795
$582$		0			0
$583$			4.61845i			0.191277i
$584$		0			0
$585$	1.68922	−	11.8605i	0.0698407	−	0.490373i
$586$		0			0
$587$		−	27.0107i		−	1.11485i	−0.830227	−	0.557425i	$-0.811789\pi$
$587$		−	27.0107i		−	1.11485i	0.830227	−	0.557425i	$-0.188211\pi$
$588$		0			0
$589$	−14.7329			−0.607057
$590$		0			0
$591$	15.9934			0.657882
$592$		0			0
$593$			10.6111i			0.435747i	0.975977	+	0.217873i	$0.0699119\pi$
$593$			10.6111i			0.435747i	−0.975977	+	0.217873i	$0.930088\pi$
$594$		0			0
$595$	−0.412875	−	0.0588031i	−0.0169262	−	0.00241069i
$596$		0			0
$597$			1.49303i			0.0611057i
$598$		0			0
$599$	−26.5506			−1.08483			−0.542414	−	0.840111i	$-0.682490\pi$
$599$	−26.5506			−1.08483			−0.542414	+	0.840111i	$0.682490\pi$
$600$		0			0
$601$	5.38706			0.219743			0.109871	−	0.993946i	$-0.464956\pi$
$601$	5.38706			0.219743			0.109871	+	0.993946i	$0.464956\pi$
$602$		0			0
$603$			1.00000i			0.0407231i
$604$		0			0
$605$	−3.58627	−	0.510769i	−0.145802	−	0.0207657i
$606$		0			0
$607$		−	20.5802i		−	0.835324i	−0.908602	−	0.417662i	$-0.862850\pi$
$607$		−	20.5802i		−	0.835324i	0.908602	−	0.417662i	$-0.137150\pi$
$608$		0			0
$609$	0.215324			0.00872536
$610$		0			0
$611$	−18.8429			−0.762300
$612$		0			0
$613$			16.4998i			0.666421i	0.942853	+	0.333210i	$0.108132\pi$
$613$			16.4998i			0.666421i	−0.942853	+	0.333210i	$0.891868\pi$
$614$		0			0
$615$	−1.66192	+	11.6689i	−0.0670151	+	0.470534i
$616$		0			0
$617$		−	20.7270i		−	0.834439i	−0.908806	−	0.417220i	$-0.863005\pi$
$617$		−	20.7270i		−	0.834439i	0.908806	−	0.417220i	$-0.136995\pi$
$618$		0			0
$619$	−24.5239			−0.985698			−0.492849	−	0.870115i	$-0.664045\pi$
$619$	−24.5239			−0.985698			−0.492849	+	0.870115i	$0.664045\pi$
$620$		0			0
$621$	5.50964			0.221094
$622$		0			0
$623$		−	0.452206i		−	0.0181173i
$624$		0			0
$625$	21.1028	+	13.4042i	0.844112	+	0.536166i
$626$		0			0
$627$			24.1556i			0.964681i
$628$		0			0
$629$	13.9718			0.557092
$630$		0			0
$631$	−48.3691			−1.92554			−0.962771	−	0.270318i	$-0.912871\pi$
$631$	−48.3691			−1.92554			−0.962771	+	0.270318i	$0.912871\pi$
$632$		0			0
$633$		−	5.24402i		−	0.208431i
$634$		0			0
$635$	−5.69122	+	39.9598i	−0.225849	+	1.58576i
$636$		0			0
$637$			37.4588i			1.48417i
$638$		0			0
$639$	−12.3591			−0.488917
$640$		0			0
$641$	−43.0013			−1.69845			−0.849225	−	0.528032i	$-0.822930\pi$
$641$	−43.0013			−1.69845			−0.849225	+	0.528032i	$0.822930\pi$
$642$		0			0
$643$			0.722092i			0.0284765i	0.999899	+	0.0142383i	$0.00453233\pi$
$643$			0.722092i			0.0284765i	−0.999899	+	0.0142383i	$0.995468\pi$
$644$		0			0
$645$	−13.7147	−	1.95330i	−0.540018	−	0.0769113i
$646$		0			0
$647$		−	47.1920i		−	1.85531i	−0.373441	−	0.927654i	$-0.621822\pi$
$647$		−	47.1920i		−	1.85531i	0.373441	−	0.927654i	$-0.378178\pi$
$648$		0			0
$649$	−24.0528			−0.944155
$650$		0			0
$651$	0.171654			0.00672764
$652$		0			0
$653$			43.3929i			1.69810i	0.528315	+	0.849048i	$0.322824\pi$
$653$			43.3929i			1.69810i	−0.528315	+	0.849048i	$0.677176\pi$
$654$		0			0
$655$	13.5777	+	1.93379i	0.530527	+	0.0755595i
$656$		0			0
$657$		−	6.48293i		−	0.252923i
$658$		0			0
$659$	26.7097			1.04046			0.520232	−	0.854025i	$-0.325846\pi$
$659$	26.7097			1.04046			0.520232	+	0.854025i	$0.325846\pi$
$660$		0			0
$661$	16.8600			0.655779			0.327890	−	0.944716i	$-0.393663\pi$
$661$	16.8600			0.655779			0.327890	+	0.944716i	$0.393663\pi$
$662$		0			0
$663$		−	10.8740i		−	0.422313i
$664$		0			0
$665$	0.228510	−	1.60444i	0.00886124	−	0.0622174i
$666$		0			0
$667$			12.9102i			0.499885i
$668$		0			0
$669$	5.24398			0.202744
$670$		0			0
$671$	−39.5661			−1.52743
$672$		0			0
$673$		−	35.5964i		−	1.37214i	−0.727535	−	0.686070i	$-0.759334\pi$
$673$		−	35.5964i		−	1.37214i	0.727535	−	0.686070i	$-0.240666\pi$
$674$		0			0
$675$	−1.39592	+	4.80119i	−0.0537290	+	0.184798i
$676$		0			0
$677$			34.6656i			1.33231i	0.745815	+	0.666153i	$0.232060\pi$
$677$			34.6656i			1.33231i	−0.745815	+	0.666153i	$0.767940\pi$
$678$		0			0
$679$	−0.335422			−0.0128723
$680$		0			0
$681$	−18.0350			−0.691102
$682$		0			0
$683$			8.79125i			0.336388i	0.985754	+	0.168194i	$0.0537934\pi$
$683$			8.79125i			0.336388i	−0.985754	+	0.168194i	$0.946207\pi$
$684$		0			0
$685$	−0.185180	+	1.30020i	−0.00707536	+	0.0496783i
$686$		0			0
$687$		−	6.03776i		−	0.230355i
$688$		0			0
$689$	8.07933			0.307798
$690$		0			0
$691$	44.4615			1.69139			0.845697	−	0.533663i	$-0.179185\pi$
$691$	44.4615			1.69139			0.845697	+	0.533663i	$0.179185\pi$
$692$		0			0
$693$		−	0.281438i		−	0.0106910i
$694$		0			0
$695$	8.12618	+	1.15736i	0.308244	+	0.0439012i
$696$		0			0
$697$			10.6983i			0.405228i
$698$		0			0
$699$	−1.82102			−0.0688773
$700$		0			0
$701$	−29.2578			−1.10505			−0.552525	−	0.833496i	$-0.686336\pi$
$701$	−29.2578			−1.10505			−0.552525	+	0.833496i	$0.686336\pi$
$702$		0			0
$703$			54.2947i			2.04776i
$704$		0			0
$705$	7.78558	+	1.10885i	0.293222	+	0.0417618i
$706$		0			0
$707$		−	1.09364i		−	0.0411305i
$708$		0			0
$709$	−37.4677			−1.40713			−0.703565	−	0.710631i	$-0.748409\pi$
$709$	−37.4677			−1.40713			−0.703565	+	0.710631i	$0.748409\pi$
$710$		0			0
$711$	−8.72623			−0.327259
$712$		0			0
$713$			10.2919i			0.385433i
$714$		0			0
$715$	5.17354	−	36.3250i	0.193479	−	1.35848i
$716$		0			0
$717$			13.2797i			0.495938i
$718$		0			0
$719$	27.2086			1.01471			0.507355	−	0.861737i	$-0.330623\pi$
$719$	27.2086			1.01471			0.507355	+	0.861737i	$0.330623\pi$
$720$		0			0
$721$	1.21325			0.0451839
$722$		0			0
$723$		−	10.7660i		−	0.400391i
$724$		0			0
$725$	−11.2502	−	3.27092i	−0.417820	−	0.121479i
$726$		0			0
$727$			33.2546i			1.23334i	0.787220	+	0.616672i	$0.211519\pi$
$727$			33.2546i			1.23334i	−0.787220	+	0.616672i	$0.788481\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−12.5740			−0.465068
$732$		0			0
$733$			51.7375i			1.91097i	0.295043	+	0.955484i	$0.404666\pi$
$733$			51.7375i			1.91097i	−0.295043	+	0.955484i	$0.595334\pi$
$734$		0			0
$735$	2.20435	−	15.4774i	0.0813086	−	0.570893i
$736$		0			0
$737$			3.06268i			0.112815i
$738$		0			0
$739$	−18.5101			−0.680905			−0.340452	−	0.940262i	$-0.610580\pi$
$739$	−18.5101			−0.680905			−0.340452	+	0.940262i	$0.610580\pi$
$740$		0			0
$741$	42.2568			1.55234
$742$		0			0
$743$		−	23.8863i		−	0.876304i	−0.898901	−	0.438152i	$-0.855633\pi$
$743$		−	23.8863i		−	0.876304i	0.898901	−	0.438152i	$-0.144367\pi$
$744$		0			0
$745$	−37.6307	−	5.35950i	−1.37868	−	0.196357i
$746$		0			0
$747$		−	3.43081i		−	0.125527i
$748$		0			0
$749$	−0.752464			−0.0274944
$750$		0			0
$751$	−19.8138			−0.723016			−0.361508	−	0.932369i	$-0.617738\pi$
$751$	−19.8138			−0.723016			−0.361508	+	0.932369i	$0.617738\pi$
$752$		0			0
$753$			0.301420i			0.0109844i
$754$		0			0
$755$	7.96965	+	1.13507i	0.290045	+	0.0413093i
$756$		0			0
$757$			44.1625i			1.60511i	0.596576	+	0.802556i	$0.296527\pi$
$757$			44.1625i			1.60511i	−0.596576	+	0.802556i	$0.703473\pi$
$758$		0			0
$759$	16.8742			0.612496
$760$		0			0
$761$	1.81296			0.0657199			0.0328599	−	0.999460i	$-0.489538\pi$
$761$	1.81296			0.0657199			0.0328599	+	0.999460i	$0.489538\pi$
$762$		0			0
$763$			0.507784i			0.0183830i
$764$		0			0
$765$	−0.639909	+	4.49299i	−0.0231359	+	0.162445i
$766$		0			0
$767$			42.0769i			1.51931i
$768$		0			0
$769$	−33.2530			−1.19914			−0.599568	−	0.800324i	$-0.704661\pi$
$769$	−33.2530			−1.19914			−0.599568	+	0.800324i	$0.704661\pi$
$770$		0			0
$771$	12.1267			0.436731
$772$		0			0
$773$			26.4170i			0.950152i	0.879945	+	0.475076i	$0.157579\pi$
$773$			26.4170i			0.950152i	−0.879945	+	0.475076i	$0.842421\pi$
$774$		0			0
$775$	−8.96849	−	2.60754i	−0.322158	−	0.0936658i
$776$		0			0
$777$		−	0.632591i		−	0.0226941i
$778$		0			0
$779$	−41.5739			−1.48954
$780$		0			0
$781$	−37.8518			−1.35445
$782$		0			0
$783$		−	2.34320i		−	0.0837392i
$784$		0			0
$785$	6.80873	−	47.8062i	0.243014	−	1.70628i
$786$		0			0
$787$			40.2144i			1.43349i	0.697336	+	0.716745i	$0.254369\pi$
$787$			40.2144i			1.43349i	−0.697336	+	0.716745i	$0.745631\pi$
$788$		0			0
$789$	−7.20836			−0.256624
$790$		0			0
$791$	1.41812			0.0504226
$792$		0			0
$793$			69.2153i			2.45791i
$794$		0			0
$795$	−3.33826	−	0.475447i	−0.118396	−	0.0168624i
$796$		0			0
$797$		−	48.7561i		−	1.72703i	−0.504325	−	0.863514i	$-0.668258\pi$
$797$		−	48.7561i		−	1.72703i	0.504325	−	0.863514i	$-0.331742\pi$
$798$		0			0
$799$	7.13803			0.252525
$800$		0			0
$801$	−4.92101			−0.173875
$802$		0			0
$803$		−	19.8551i		−	0.700672i
$804$		0			0
$805$	−1.12080	−	0.159629i	−0.0395031	−	0.00562618i
$806$		0			0
$807$			21.1140i			0.743249i
$808$		0			0
$809$	−9.48165			−0.333357			−0.166678	−	0.986011i	$-0.553304\pi$
$809$	−9.48165			−0.333357			−0.166678	+	0.986011i	$0.553304\pi$
$810$		0			0
$811$	31.6631			1.11184			0.555920	−	0.831235i	$-0.312366\pi$
$811$	31.6631			1.11184			0.555920	+	0.831235i	$0.312366\pi$
$812$		0			0
$813$		−	21.8167i		−	0.765145i
$814$		0			0
$815$	3.86005	−	27.1026i	0.135212	−	0.949363i
$816$		0			0
$817$		−	48.8630i		−	1.70950i
$818$		0			0
$819$	−0.492337			−0.0172036
$820$		0			0
$821$	−20.1538			−0.703373			−0.351686	−	0.936118i	$-0.614392\pi$
$821$	−20.1538			−0.703373			−0.351686	+	0.936118i	$0.614392\pi$
$822$		0			0
$823$		−	13.3560i		−	0.465560i	−0.972529	−	0.232780i	$-0.925218\pi$
$823$		−	13.3560i		−	0.465560i	0.972529	−	0.232780i	$-0.0747823\pi$
$824$		0			0
$825$	−4.27526	+	14.7045i	−0.148845	+	0.511944i
$826$		0			0
$827$			9.79412i			0.340575i	0.985394	+	0.170288i	$0.0544696\pi$
$827$			9.79412i			0.340575i	−0.985394	+	0.170288i	$0.945530\pi$
$828$		0			0
$829$	−7.75785			−0.269441			−0.134721	−	0.990884i	$-0.543014\pi$
$829$	−7.75785			−0.269441			−0.134721	+	0.990884i	$0.543014\pi$
$830$		0			0
$831$	10.4668			0.363089
$832$		0			0
$833$		−	14.1901i		−	0.491658i
$834$		0			0
$835$	0.294100	−	2.06497i	0.0101778	−	0.0714611i
$836$		0			0
$837$		−	1.86797i		−	0.0645666i
$838$		0			0
$839$	11.8169			0.407965			0.203983	−	0.978974i	$-0.434611\pi$
$839$	11.8169			0.407965			0.203983	+	0.978974i	$0.434611\pi$
$840$		0			0
$841$	−23.5094			−0.810669
$842$		0			0
$843$		−	14.7710i		−	0.508741i
$844$		0			0
$845$	−34.7669	−	4.95163i	−1.19602	−	0.170341i
$846$		0			0
$847$			0.148868i			0.00511516i
$848$		0			0
$849$	−17.2160			−0.590853
$850$		0			0
$851$	37.9283			1.30017
$852$		0			0
$853$		−	17.6518i		−	0.604386i	−0.953247	−	0.302193i	$-0.902281\pi$
$853$		−	17.6518i		−	0.604386i	0.953247	−	0.302193i	$-0.0977187\pi$
$854$		0			0
$855$	−17.4599	−	2.48670i	−0.597115	−	0.0850432i
$856$		0			0
$857$		−	28.1117i		−	0.960277i	−0.877193	−	0.480138i	$-0.840586\pi$
$857$		−	28.1117i		−	0.960277i	0.877193	−	0.480138i	$-0.159414\pi$
$858$		0			0
$859$	−47.5355			−1.62189			−0.810946	−	0.585121i	$-0.801047\pi$
$859$	−47.5355			−1.62189			−0.810946	+	0.585121i	$0.801047\pi$
$860$		0			0
$861$	0.484380			0.0165076
$862$		0			0
$863$			51.1163i			1.74002i	0.493035	+	0.870010i	$0.335888\pi$
$863$			51.1163i			1.74002i	−0.493035	+	0.870010i	$0.664112\pi$
$864$		0			0
$865$	5.16287	−	36.2501i	0.175543	−	1.23254i
$866$		0			0
$867$		−	12.8807i		−	0.437452i
$868$		0			0
$869$	−26.7256			−0.906605
$870$		0			0
$871$	5.35772			0.181539
$872$		0			0
$873$			3.65014i			0.123539i
$874$		0			0
$875$	0.423070	−	0.936243i	0.0143024	−	0.0316508i
$876$		0			0
$877$			20.6300i			0.696625i	0.937378	+	0.348312i	$0.113245\pi$
$877$			20.6300i			0.696625i	−0.937378	+	0.348312i	$0.886755\pi$
$878$		0			0
$879$	5.61952			0.189542
$880$		0			0
$881$	−20.9178			−0.704737			−0.352369	−	0.935861i	$-0.614624\pi$
$881$	−20.9178			−0.704737			−0.352369	+	0.935861i	$0.614624\pi$
$882$		0			0
$883$			38.2209i			1.28624i	0.765767	+	0.643118i	$0.222359\pi$
$883$			38.2209i			1.28624i	−0.765767	+	0.643118i	$0.777641\pi$
$884$		0			0
$885$	2.47612	−	17.3856i	0.0832337	−	0.584409i
$886$		0			0
$887$		−	32.8167i		−	1.10188i	−0.834546	−	0.550938i	$-0.814270\pi$
$887$		−	32.8167i		−	1.10188i	0.834546	−	0.550938i	$-0.185730\pi$
$888$		0			0
$889$	1.65875			0.0556328
$890$		0			0
$891$	−3.06268			−0.102604
$892$		0			0
$893$			27.7385i			0.928234i
$894$		0			0
$895$	25.1417	+	3.58078i	0.840396	+	0.119692i
$896$		0			0
$897$		−	29.5191i		−	0.985613i
$898$		0			0
$899$	4.37704			0.145982
$900$		0			0
$901$	−3.06060			−0.101963
$902$		0			0
$903$			0.569306i			0.0189453i
$904$		0			0
$905$	−31.3274	−	4.46176i	−1.04136	−	0.148314i
$906$		0			0
$907$		−	24.9930i		−	0.829878i	−0.909849	−	0.414939i	$-0.863803\pi$
$907$		−	24.9930i		−	0.829878i	0.909849	−	0.414939i	$-0.136197\pi$
$908$		0			0
$909$	−11.9012			−0.394738
$910$		0			0
$911$	29.6346			0.981837			0.490918	−	0.871206i	$-0.336661\pi$
$911$	29.6346			0.981837			0.490918	+	0.871206i	$0.336661\pi$
$912$		0			0
$913$		−	10.5075i		−	0.347746i
$914$		0			0
$915$	4.07313	−	28.5987i	0.134654	−	0.945445i
$916$		0			0
$917$		−	0.563619i		−	0.0186123i
$918$		0			0
$919$	−10.0034			−0.329983			−0.164991	−	0.986295i	$-0.552760\pi$
$919$	−10.0034			−0.329983			−0.164991	+	0.986295i	$0.552760\pi$
$920$		0			0
$921$	7.99448			0.263427
$922$		0			0
$923$			66.2164i			2.17954i
$924$		0			0
$925$	−9.60952	+	33.0514i	−0.315959	+	1.08672i
$926$		0			0
$927$		−	13.2029i		−	0.433640i
$928$		0			0
$929$	−0.0228619			−0.000750074			−0.000375037	−	1.00000i	$-0.500119\pi$
$929$	−0.0228619			−0.000750074			−0.000375037		1.00000i	$0.500119\pi$
$930$		0			0
$931$	55.1430			1.80724
$932$		0			0
$933$		−	13.2422i		−	0.433530i
$934$		0			0
$935$	−1.95983	+	13.7606i	−0.0640934	+	0.450019i
$936$		0			0
$937$		−	29.9475i		−	0.978342i	−0.872188	−	0.489171i	$-0.837299\pi$
$937$		−	29.9475i		−	0.978342i	0.872188	−	0.489171i	$-0.162701\pi$
$938$		0			0
$939$	−0.186937			−0.00610047
$940$		0			0
$941$	34.8722			1.13680			0.568400	−	0.822752i	$-0.307562\pi$
$941$	34.8722			1.13680			0.568400	+	0.822752i	$0.307562\pi$
$942$		0			0
$943$			29.0420i			0.945739i
$944$		0			0
$945$	0.203426	+	0.0289727i	0.00661745	+	0.000942482i
$946$		0			0
$947$			20.9162i			0.679685i	0.940482	+	0.339843i	$0.110374\pi$
$947$			20.9162i			0.679685i	−0.940482	+	0.339843i	$0.889626\pi$
$948$		0			0
$949$	−34.7337			−1.12750
$950$		0			0
$951$	−16.9563			−0.549846
$952$		0			0
$953$			0.514681i			0.0166722i	0.999965	+	0.00833608i	$0.00265349\pi$
$953$			0.514681i			0.0166722i	−0.999965	+	0.00833608i	$0.997347\pi$
$954$		0			0
$955$	2.96630	+	0.422472i	0.0959873	+	0.0136709i
$956$		0			0
$957$		−	7.17647i		−	0.231982i
$958$		0			0
$959$	0.0539721			0.00174285
$960$		0			0
$961$	−27.5107			−0.887441
$962$		0			0
$963$			8.18848i			0.263870i
$964$		0			0
$965$	2.22412	−	15.6162i	0.0715968	−	0.502703i
$966$		0			0
$967$		−	48.8901i		−	1.57220i	−0.618100	−	0.786100i	$-0.712097\pi$
$967$		−	48.8901i		−	1.57220i	0.618100	−	0.786100i	$-0.287903\pi$
$968$		0			0
$969$	−16.0077			−0.514240
$970$		0			0
$971$	−12.4542			−0.399673			−0.199837	−	0.979829i	$-0.564041\pi$
$971$	−12.4542			−0.399673			−0.199837	+	0.979829i	$0.564041\pi$
$972$		0			0
$973$		−	0.337322i		−	0.0108140i
$974$		0			0
$975$	25.7234	+	7.47895i	0.823808	+	0.239518i
$976$		0			0
$977$			20.5452i			0.657300i	0.944452	+	0.328650i	$0.106594\pi$
$977$			20.5452i			0.657300i	−0.944452	+	0.328650i	$0.893406\pi$
$978$		0			0
$979$	−15.0715			−0.481686
$980$		0			0
$981$	5.52582			0.176426
$982$		0			0
$983$			42.1196i			1.34341i	0.740820	+	0.671703i	$0.234437\pi$
$983$			42.1196i			1.34341i	−0.740820	+	0.671703i	$0.765563\pi$
$984$		0			0
$985$	−5.04253	+	35.4051i	−0.160668	+	1.12810i
$986$		0			0
$987$		−	0.323183i		−	0.0102870i
$988$		0			0
$989$	−34.1339			−1.08540
$990$		0			0
$991$	53.2509			1.69157			0.845786	−	0.533522i	$-0.179132\pi$
$991$	53.2509			1.69157			0.845786	+	0.533522i	$0.179132\pi$
$992$		0			0
$993$			0.101500i			0.00322100i
$994$		0			0
$995$	−3.30516	−	0.470734i	−0.104781	−	0.0149233i
$996$		0			0
$997$		−	16.0992i		−	0.509868i	−0.966959	−	0.254934i	$-0.917946\pi$
$997$		−	16.0992i		−	0.509868i	0.966959	−	0.254934i	$-0.0820537\pi$
$998$		0			0
$999$	−6.88400			−0.217800

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.g.c.1609.19	✓	38
5.4	even	2	inner	4020.2.g.c.1609.38	yes	38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.g.c.1609.19	✓	38	1.1	even	1	trivial
4020.2.g.c.1609.38	yes	38	5.4	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:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(2.21373 + 0.315287i) q^{5} -0.0918930i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +(2.21373 + 0.315287i) q^{5} -0.0918930i q^{7} -1.00000 q^{9} -3.06268 q^{11} +5.35772i q^{13} +(0.315287 - 2.21373i) q^{15} -2.02960i q^{17} +7.88708 q^{19} -0.0918930 q^{21} -5.50964i q^{23} +(4.80119 + 1.39592i) q^{25} +1.00000i q^{27} -2.34320 q^{29} -1.86797 q^{31} +3.06268i q^{33} +(0.0289727 - 0.203426i) q^{35} +6.88400i q^{37} +5.35772 q^{39} -5.27113 q^{41} -6.19532i q^{43} +(-2.21373 - 0.315287i) q^{45} +3.51696i q^{47} +6.99156 q^{49} -2.02960 q^{51} -1.50798i q^{53} +(-6.77993 - 0.965623i) q^{55} -7.88708i q^{57} +7.85352 q^{59} +12.9188 q^{61} +0.0918930i q^{63} +(-1.68922 + 11.8605i) q^{65} -1.00000i q^{67} -5.50964 q^{69} +12.3591 q^{71} +6.48293i q^{73} +(1.39592 - 4.80119i) q^{75} +0.281438i q^{77} +8.72623 q^{79} +1.00000 q^{81} +3.43081i q^{83} +(0.639909 - 4.49299i) q^{85} +2.34320i q^{87} +4.92101 q^{89} +0.492337 q^{91} +1.86797i q^{93} +(17.4599 + 2.48670i) q^{95} -3.65014i q^{97} +3.06268 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{5} - 38 q^{9}+O(q^{10}) \) 38 * q - 2 * q^5 - 38 * q^9	\( 38 q - 2 q^{5} - 38 q^{9} + 24 q^{11} + 2 q^{15} - 16 q^{19} + 12 q^{21} + 4 q^{25} - 56 q^{29} - 4 q^{31} - 2 q^{35} + 60 q^{41} + 2 q^{45} - 70 q^{49} + 12 q^{55} - 52 q^{59} + 48 q^{61} + 16 q^{65} - 12 q^{69} + 12 q^{75} - 24 q^{79} + 38 q^{81} + 16 q^{85} - 76 q^{89} + 44 q^{91} + 36 q^{95} - 24 q^{99}+O(q^{100}) \) 38 * q - 2 * q^5 - 38 * q^9 + 24 * q^11 + 2 * q^15 - 16 * q^19 + 12 * q^21 + 4 * q^25 - 56 * q^29 - 4 * q^31 - 2 * q^35 + 60 * q^41 + 2 * q^45 - 70 * q^49 + 12 * q^55 - 52 * q^59 + 48 * q^61 + 16 * q^65 - 12 * q^69 + 12 * q^75 - 24 * q^79 + 38 * q^81 + 16 * q^85 - 76 * q^89 + 44 * q^91 + 36 * q^95 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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