LMFDB - Embedded newform 4012.2.b.b.237.41

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			237.41
Character	$\chi$	$=$	4012.237
Dual form			4012.2.b.b.237.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.74502i q^{3} -1.20235i q^{5} +3.93843i q^{7} -4.53516 q^{9} +O(q^{10})$	$q+2.74502i q^{3} -1.20235i q^{5} +3.93843i q^{7} -4.53516 q^{9} +1.10522i q^{11} -0.0851767 q^{13} +3.30049 q^{15} +(3.92673 + 1.25729i) q^{17} +7.38549 q^{19} -10.8111 q^{21} -5.67905i q^{23} +3.55435 q^{25} -4.21405i q^{27} +5.59463i q^{29} +5.38593i q^{31} -3.03385 q^{33} +4.73538 q^{35} +7.96819i q^{37} -0.233812i q^{39} +7.91312i q^{41} +4.18238 q^{43} +5.45286i q^{45} +4.24249 q^{47} -8.51122 q^{49} +(-3.45129 + 10.7790i) q^{51} +3.82436 q^{53} +1.32886 q^{55} +20.2733i q^{57} +1.00000 q^{59} +2.00065i q^{61} -17.8614i q^{63} +0.102412i q^{65} -8.91072 q^{67} +15.5891 q^{69} -8.59825i q^{71} -7.38541i q^{73} +9.75677i q^{75} -4.35283 q^{77} -13.5159i q^{79} -2.03780 q^{81} -14.8577 q^{83} +(1.51170 - 4.72132i) q^{85} -15.3574 q^{87} -10.9415 q^{89} -0.335462i q^{91} -14.7845 q^{93} -8.87996i q^{95} +15.6089i q^{97} -5.01235i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$			2.74502i			1.58484i	0.609975	+	0.792420i	$0.291179\pi$
$3$			2.74502i			1.58484i	−0.609975	+	0.792420i	$0.708821\pi$
$4$		0			0
$5$		−	1.20235i		−	0.537708i	−0.963181	−	0.268854i	$-0.913355\pi$
$5$		−	1.20235i		−	0.537708i	0.963181	−	0.268854i	$-0.0866450\pi$
$6$		0			0
$7$			3.93843i			1.48859i	0.667853	+	0.744293i	$0.267213\pi$
$7$			3.93843i			1.48859i	−0.667853	+	0.744293i	$0.732787\pi$
$8$		0			0
$9$	−4.53516			−1.51172
$10$		0			0
$11$			1.10522i			0.333236i	0.986022	+	0.166618i	$0.0532847\pi$
$11$			1.10522i			0.333236i	−0.986022	+	0.166618i	$0.946715\pi$
$12$		0			0
$13$	−0.0851767			−0.0236238			−0.0118119	−	0.999930i	$-0.503760\pi$
$13$	−0.0851767			−0.0236238			−0.0118119	+	0.999930i	$0.503760\pi$
$14$		0			0
$15$	3.30049			0.852182
$16$		0			0
$17$	3.92673	+	1.25729i	0.952372	+	0.304937i
$17$	3.92673	+	1.25729i	0.952372	+	0.304937i
$18$		0			0
$19$	7.38549			1.69435			0.847173	−	0.531317i	$-0.178303\pi$
$19$	7.38549			1.69435			0.847173	+	0.531317i	$0.178303\pi$
$20$		0			0
$21$	−10.8111			−2.35917
$22$		0			0
$23$		−	5.67905i		−	1.18416i	−0.805878	−	0.592081i	$-0.798306\pi$
$23$		−	5.67905i		−	1.18416i	0.805878	−	0.592081i	$-0.201694\pi$
$24$		0			0
$25$	3.55435			0.710870
$26$		0			0
$27$		−	4.21405i		−	0.810995i
$28$		0			0
$29$			5.59463i			1.03890i	0.854502	+	0.519449i	$0.173863\pi$
$29$			5.59463i			1.03890i	−0.854502	+	0.519449i	$0.826137\pi$
$30$		0			0
$31$			5.38593i			0.967342i	0.875250	+	0.483671i	$0.160697\pi$
$31$			5.38593i			0.967342i	−0.875250	+	0.483671i	$0.839303\pi$
$32$		0			0
$33$	−3.03385			−0.528126
$34$		0			0
$35$	4.73538			0.800425
$36$		0			0
$37$			7.96819i			1.30996i	0.755646	+	0.654981i	$0.227323\pi$
$37$			7.96819i			1.30996i	−0.755646	+	0.654981i	$0.772677\pi$
$38$		0			0
$39$		−	0.233812i		−	0.0374399i
$40$		0			0
$41$			7.91312i			1.23582i	0.786248	+	0.617911i	$0.212021\pi$
$41$			7.91312i			1.23582i	−0.786248	+	0.617911i	$0.787979\pi$
$42$		0			0
$43$	4.18238			0.637808			0.318904	−	0.947787i	$-0.396685\pi$
$43$	4.18238			0.637808			0.318904	+	0.947787i	$0.396685\pi$
$44$		0			0
$45$			5.45286i			0.812865i
$46$		0			0
$47$	4.24249			0.618830			0.309415	−	0.950927i	$-0.399867\pi$
$47$	4.24249			0.618830			0.309415	+	0.950927i	$0.399867\pi$
$48$		0			0
$49$	−8.51122			−1.21589
$50$		0			0
$51$	−3.45129	+	10.7790i	−0.483277	+	1.50936i
$52$		0			0
$53$	3.82436			0.525317			0.262658	−	0.964889i	$-0.415401\pi$
$53$	3.82436			0.525317			0.262658	+	0.964889i	$0.415401\pi$
$54$		0			0
$55$	1.32886			0.179184
$56$		0			0
$57$			20.2733i			2.68527i
$58$		0			0
$59$	1.00000			0.130189
$59$	1.00000			0.130189
$60$		0			0
$61$			2.00065i			0.256156i	0.991764	+	0.128078i	$0.0408809\pi$
$61$			2.00065i			0.256156i	−0.991764	+	0.128078i	$0.959119\pi$
$62$		0			0
$63$		−	17.8614i		−	2.25033i
$64$		0			0
$65$			0.102412i			0.0127027i
$66$		0			0
$67$	−8.91072			−1.08862			−0.544309	−	0.838885i	$-0.683208\pi$
$67$	−8.91072			−1.08862			−0.544309	+	0.838885i	$0.683208\pi$
$68$		0			0
$69$	15.5891			1.87671
$70$		0			0
$71$		−	8.59825i		−	1.02042i	−0.860048	−	0.510212i	$-0.829567\pi$
$71$		−	8.59825i		−	1.02042i	0.860048	−	0.510212i	$-0.170433\pi$
$72$		0			0
$73$		−	7.38541i		−	0.864397i	−0.901778	−	0.432199i	$-0.857738\pi$
$73$		−	7.38541i		−	0.864397i	0.901778	−	0.432199i	$-0.142262\pi$
$74$		0			0
$75$			9.75677i			1.12662i
$76$		0			0
$77$	−4.35283			−0.496051
$78$		0			0
$79$		−	13.5159i		−	1.52066i	−0.649536	−	0.760331i	$-0.725037\pi$
$79$		−	13.5159i		−	1.52066i	0.649536	−	0.760331i	$-0.274963\pi$
$80$		0			0
$81$	−2.03780			−0.226422
$82$		0			0
$83$	−14.8577			−1.63085			−0.815424	−	0.578865i	$-0.803496\pi$
$83$	−14.8577			−1.63085			−0.815424	+	0.578865i	$0.803496\pi$
$84$		0			0
$85$	1.51170	−	4.72132i	0.163967	−	0.512099i
$86$		0			0
$87$	−15.3574			−1.64649
$88$		0			0
$89$	−10.9415			−1.15980			−0.579900	−	0.814687i	$-0.696909\pi$
$89$	−10.9415			−1.15980			−0.579900	+	0.814687i	$0.696909\pi$
$90$		0			0
$91$		−	0.335462i		−	0.0351660i
$92$		0			0
$93$	−14.7845			−1.53308
$94$		0			0
$95$		−	8.87996i		−	0.911064i
$96$		0			0
$97$			15.6089i			1.58484i	0.609975	+	0.792421i	$0.291179\pi$
$97$			15.6089i			1.58484i	−0.609975	+	0.792421i	$0.708821\pi$
$98$		0			0
$99$		−	5.01235i		−	0.503760i
$100$		0			0
$101$	14.4865			1.44146			0.720729	−	0.693217i	$-0.243807\pi$
$101$	14.4865			1.44146			0.720729	+	0.693217i	$0.243807\pi$
$102$		0			0
$103$	−2.27955			−0.224611			−0.112306	−	0.993674i	$-0.535824\pi$
$103$	−2.27955			−0.224611			−0.112306	+	0.993674i	$0.535824\pi$
$104$		0			0
$105$			12.9987i			1.26855i
$106$		0			0
$107$			2.26402i			0.218871i	0.993994	+	0.109435i	$0.0349043\pi$
$107$			2.26402i			0.218871i	−0.993994	+	0.109435i	$0.965096\pi$
$108$		0			0
$109$			8.30760i			0.795724i	0.917445	+	0.397862i	$0.130248\pi$
$109$			8.30760i			0.795724i	−0.917445	+	0.397862i	$0.869752\pi$
$110$		0			0
$111$	−21.8729			−2.07608
$112$		0			0
$113$		−	13.7598i		−	1.29441i	−0.762315	−	0.647206i	$-0.775937\pi$
$113$		−	13.7598i		−	1.29441i	0.762315	−	0.647206i	$-0.224063\pi$
$114$		0			0
$115$	−6.82821			−0.636734
$116$		0			0
$117$	0.386290			0.0357125
$118$		0			0
$119$	−4.95174	+	15.4652i	−0.453926	+	1.41769i
$120$		0			0
$121$	9.77849			0.888954
$122$		0			0
$123$	−21.7217			−1.95858
$124$		0			0
$125$		−	10.2853i		−	0.919949i
$126$		0			0
$127$	17.3837			1.54256			0.771278	−	0.636499i	$-0.219618\pi$
$127$	17.3837			1.54256			0.771278	+	0.636499i	$0.219618\pi$
$128$		0			0
$129$			11.4807i			1.01082i
$130$		0			0
$131$		−	15.6197i		−	1.36470i	−0.731025	−	0.682350i	$-0.760958\pi$
$131$		−	15.6197i		−	1.36470i	0.731025	−	0.682350i	$-0.239042\pi$
$132$		0			0
$133$			29.0872i			2.52218i
$134$		0			0
$135$	−5.06678			−0.436079
$136$		0			0
$137$	−2.32537			−0.198670			−0.0993349	−	0.995054i	$-0.531672\pi$
$137$	−2.32537			−0.198670			−0.0993349	+	0.995054i	$0.531672\pi$
$138$		0			0
$139$		−	6.36572i		−	0.539933i	−0.962870	−	0.269967i	$-0.912987\pi$
$139$		−	6.36572i		−	0.539933i	0.962870	−	0.269967i	$-0.0870126\pi$
$140$		0			0
$141$			11.6457i			0.980747i
$142$		0			0
$143$		−	0.0941389i		−	0.00787229i
$144$		0			0
$145$	6.72672			0.558624
$146$		0			0
$147$		−	23.3635i		−	1.92699i
$148$		0			0
$149$	−6.98869			−0.572536			−0.286268	−	0.958150i	$-0.592415\pi$
$149$	−6.98869			−0.572536			−0.286268	+	0.958150i	$0.592415\pi$
$150$		0			0
$151$	−10.7987			−0.878790			−0.439395	−	0.898294i	$-0.644807\pi$
$151$	−10.7987			−0.878790			−0.439395	+	0.898294i	$0.644807\pi$
$152$		0			0
$153$	−17.8084	−	5.70201i	−1.43972	−	0.460980i
$154$		0			0
$155$	6.47579			0.520148
$156$		0			0
$157$	−18.8562			−1.50489			−0.752444	−	0.658656i	$-0.771125\pi$
$157$	−18.8562			−1.50489			−0.752444	+	0.658656i	$0.771125\pi$
$158$		0			0
$159$			10.4980i			0.832544i
$160$		0			0
$161$	22.3665			1.76273
$162$		0			0
$163$		−	3.55300i		−	0.278292i	−0.990272	−	0.139146i	$-0.955564\pi$
$163$		−	3.55300i		−	0.278292i	0.990272	−	0.139146i	$-0.0444358\pi$
$164$		0			0
$165$			3.64776i			0.283978i
$166$		0			0
$167$			7.67778i			0.594124i	0.954858	+	0.297062i	$0.0960069\pi$
$167$			7.67778i			0.594124i	−0.954858	+	0.297062i	$0.903993\pi$
$168$		0			0
$169$	−12.9927			−0.999442
$170$		0			0
$171$	−33.4944			−2.56138
$172$		0			0
$173$		−	21.8061i		−	1.65788i	−0.559335	−	0.828942i	$-0.688943\pi$
$173$		−	21.8061i		−	1.65788i	0.559335	−	0.828942i	$-0.311057\pi$
$174$		0			0
$175$			13.9985i			1.05819i
$176$		0			0
$177$			2.74502i			0.206329i
$178$		0			0
$179$	12.2648			0.916715			0.458357	−	0.888768i	$-0.348438\pi$
$179$	12.2648			0.916715			0.458357	+	0.888768i	$0.348438\pi$
$180$		0			0
$181$			19.1303i			1.42194i	0.703220	+	0.710972i	$0.251745\pi$
$181$			19.1303i			1.42194i	−0.703220	+	0.710972i	$0.748255\pi$
$182$		0			0
$183$	−5.49182			−0.405967
$184$		0			0
$185$	9.58057			0.704377
$186$		0			0
$187$	−1.38958	+	4.33990i	−0.101616	+	0.317365i
$188$		0			0
$189$	16.5967			1.20724
$190$		0			0
$191$	8.10126			0.586187			0.293093	−	0.956084i	$-0.405315\pi$
$191$	8.10126			0.586187			0.293093	+	0.956084i	$0.405315\pi$
$192$		0			0
$193$			15.9862i			1.15071i	0.817904	+	0.575355i	$0.195136\pi$
$193$			15.9862i			1.15071i	−0.817904	+	0.575355i	$0.804864\pi$
$194$		0			0
$195$	−0.281125			−0.0201318
$196$		0			0
$197$		−	13.6949i		−	0.975723i	−0.872921	−	0.487862i	$-0.837777\pi$
$197$		−	13.6949i		−	0.975723i	0.872921	−	0.487862i	$-0.162223\pi$
$198$		0			0
$199$		−	26.4946i		−	1.87815i	−0.343710	−	0.939076i	$-0.611684\pi$
$199$		−	26.4946i		−	1.87815i	0.343710	−	0.939076i	$-0.388316\pi$
$200$		0			0
$201$		−	24.4602i		−	1.72529i
$202$		0			0
$203$	−22.0341			−1.54649
$204$		0			0
$205$	9.51436			0.664512
$206$		0			0
$207$			25.7554i			1.79012i
$208$		0			0
$209$			8.16258i			0.564618i
$210$		0			0
$211$		−	22.4564i		−	1.54596i	−0.634428	−	0.772982i	$-0.718764\pi$
$211$		−	22.4564i		−	1.54596i	0.634428	−	0.772982i	$-0.281236\pi$
$212$		0			0
$213$	23.6024			1.61721
$214$		0			0
$215$		−	5.02870i		−	0.342955i
$216$		0			0
$217$	−21.2121			−1.43997
$218$		0			0
$219$	20.2731			1.36993
$220$		0			0
$221$	−0.334466	−	0.107092i	−0.0224986	−	0.00720377i
$222$		0			0
$223$	−13.9282			−0.932699			−0.466349	−	0.884601i	$-0.654431\pi$
$223$	−13.9282			−0.932699			−0.466349	+	0.884601i	$0.654431\pi$
$224$		0			0
$225$	−16.1195			−1.07464
$226$		0			0
$227$			12.4714i			0.827757i	0.910332	+	0.413879i	$0.135826\pi$
$227$			12.4714i			0.827757i	−0.910332	+	0.413879i	$0.864174\pi$
$228$		0			0
$229$	24.1944			1.59881			0.799404	−	0.600794i	$-0.205149\pi$
$229$	24.1944			1.59881			0.799404	+	0.600794i	$0.205149\pi$
$230$		0			0
$231$		−	11.9486i		−	0.786161i
$232$		0			0
$233$			14.8016i			0.969683i	0.874602	+	0.484842i	$0.161123\pi$
$233$			14.8016i			0.969683i	−0.874602	+	0.484842i	$0.838877\pi$
$234$		0			0
$235$		−	5.10096i		−	0.332750i
$236$		0			0
$237$	37.1016			2.41001
$238$		0			0
$239$	22.6552			1.46544			0.732721	−	0.680529i	$-0.238250\pi$
$239$	22.6552			1.46544			0.732721	+	0.680529i	$0.238250\pi$
$240$		0			0
$241$			21.8877i			1.40991i	0.709252	+	0.704955i	$0.249033\pi$
$241$			21.8877i			1.40991i	−0.709252	+	0.704955i	$0.750967\pi$
$242$		0			0
$243$		−	18.2360i		−	1.16984i
$244$		0			0
$245$			10.2335i			0.653794i
$246$		0			0
$247$	−0.629071			−0.0400269
$248$		0			0
$249$		−	40.7848i		−	2.58463i
$250$		0			0
$251$	−3.71941			−0.234767			−0.117384	−	0.993087i	$-0.537451\pi$
$251$	−3.71941			−0.234767			−0.117384	+	0.993087i	$0.537451\pi$
$252$		0			0
$253$	6.27659			0.394606
$254$		0			0
$255$	12.9601	+	4.14967i	0.811595	+	0.259862i
$256$		0			0
$257$	−10.4585			−0.652382			−0.326191	−	0.945304i	$-0.605765\pi$
$257$	−10.4585			−0.652382			−0.326191	+	0.945304i	$0.605765\pi$
$258$		0			0
$259$	−31.3821			−1.94999
$260$		0			0
$261$		−	25.3726i		−	1.57052i
$262$		0			0
$263$	4.33686			0.267422			0.133711	−	0.991020i	$-0.457311\pi$
$263$	4.33686			0.267422			0.133711	+	0.991020i	$0.457311\pi$
$264$		0			0
$265$		−	4.59823i		−	0.282467i
$266$		0			0
$267$		−	30.0348i		−	1.83810i
$268$		0			0
$269$			8.89174i			0.542139i	0.962560	+	0.271069i	$0.0873773\pi$
$269$			8.89174i			0.542139i	−0.962560	+	0.271069i	$0.912623\pi$
$270$		0			0
$271$	11.0797			0.673043			0.336521	−	0.941676i	$-0.390750\pi$
$271$	11.0797			0.673043			0.336521	+	0.941676i	$0.390750\pi$
$272$		0			0
$273$	0.920853			0.0557325
$274$		0			0
$275$			3.92833i			0.236887i
$276$		0			0
$277$		−	16.8618i		−	1.01313i	−0.862202	−	0.506564i	$-0.830915\pi$
$277$		−	16.8618i		−	1.01313i	0.862202	−	0.506564i	$-0.169085\pi$
$278$		0			0
$279$		−	24.4261i		−	1.46235i
$280$		0			0
$281$	−23.0983			−1.37793			−0.688966	−	0.724794i	$-0.741935\pi$
$281$	−23.0983			−1.37793			−0.688966	+	0.724794i	$0.741935\pi$
$282$		0			0
$283$			10.3712i			0.616504i	0.951305	+	0.308252i	$0.0997439\pi$
$283$			10.3712i			0.616504i	−0.951305	+	0.308252i	$0.900256\pi$
$284$		0			0
$285$	24.3757			1.44389
$286$		0			0
$287$	−31.1653			−1.83963
$288$		0			0
$289$	13.8384	+	9.87408i	0.814026	+	0.580828i
$290$		0			0
$291$	−42.8468			−2.51172
$292$		0			0
$293$	−23.4558			−1.37030			−0.685152	−	0.728400i	$-0.740264\pi$
$293$	−23.4558			−1.37030			−0.685152	+	0.728400i	$0.740264\pi$
$294$		0			0
$295$		−	1.20235i		−	0.0700037i
$296$		0			0
$297$	4.65745			0.270253
$298$		0			0
$299$			0.483722i			0.0279744i
$300$		0			0
$301$			16.4720i			0.949432i
$302$		0			0
$303$			39.7657i			2.28448i
$304$		0			0
$305$	2.40548			0.137737
$306$		0			0
$307$	−10.6235			−0.606315			−0.303157	−	0.952941i	$-0.598041\pi$
$307$	−10.6235			−0.606315			−0.303157	+	0.952941i	$0.598041\pi$
$308$		0			0
$309$		−	6.25743i		−	0.355973i
$310$		0			0
$311$		−	9.27315i		−	0.525832i	−0.964819	−	0.262916i	$-0.915316\pi$
$311$		−	9.27315i		−	0.525832i	0.964819	−	0.262916i	$-0.0846842\pi$
$312$		0			0
$313$			2.10767i			0.119133i	0.998224	+	0.0595663i	$0.0189718\pi$
$313$			2.10767i			0.119133i	−0.998224	+	0.0595663i	$0.981028\pi$
$314$		0			0
$315$	−21.4757			−1.21002
$316$		0			0
$317$		−	10.9507i		−	0.615054i	−0.951539	−	0.307527i	$-0.900499\pi$
$317$		−	10.9507i		−	0.615054i	0.951539	−	0.307527i	$-0.0995014\pi$
$318$		0			0
$319$	−6.18330			−0.346198
$320$		0			0
$321$	−6.21479			−0.346875
$322$		0			0
$323$	29.0008	+	9.28569i	1.61365	+	0.516670i
$324$		0			0
$325$	−0.302748			−0.0167934
$326$		0			0
$327$	−22.8046			−1.26110
$328$		0			0
$329$			16.7087i			0.921182i
$330$		0			0
$331$	29.7366			1.63447			0.817235	−	0.576304i	$-0.195506\pi$
$331$	29.7366			1.63447			0.817235	+	0.576304i	$0.195506\pi$
$332$		0			0
$333$		−	36.1370i		−	1.98030i
$334$		0			0
$335$			10.7138i			0.585359i
$336$		0			0
$337$			27.4455i			1.49505i	0.664232	+	0.747527i	$0.268759\pi$
$337$			27.4455i			1.49505i	−0.664232	+	0.747527i	$0.731241\pi$
$338$		0			0
$339$	37.7710			2.05144
$340$		0			0
$341$	−5.95264			−0.322353
$342$		0			0
$343$		−	5.95185i		−	0.321370i
$344$		0			0
$345$		−	18.7436i		−	1.00912i
$346$		0			0
$347$		−	10.1963i		−	0.547366i	−0.961820	−	0.273683i	$-0.911758\pi$
$347$		−	10.1963i		−	0.547366i	0.961820	−	0.273683i	$-0.0882419\pi$
$348$		0			0
$349$	−31.5314			−1.68784			−0.843918	−	0.536472i	$-0.819757\pi$
$349$	−31.5314			−1.68784			−0.843918	+	0.536472i	$0.819757\pi$
$350$		0			0
$351$			0.358939i			0.0191588i
$352$		0			0
$353$	14.8932			0.792683			0.396342	−	0.918103i	$-0.370280\pi$
$353$	14.8932			0.792683			0.396342	+	0.918103i	$0.370280\pi$
$354$		0			0
$355$	−10.3381			−0.548691
$356$		0			0
$357$	−42.4522	−	13.5927i	−2.24681	−	0.719400i
$358$		0			0
$359$	−16.3543			−0.863145			−0.431572	−	0.902078i	$-0.642041\pi$
$359$	−16.3543			−0.863145			−0.431572	+	0.902078i	$0.642041\pi$
$360$		0			0
$361$	35.5454			1.87081
$362$		0			0
$363$			26.8422i			1.40885i
$364$		0			0
$365$	−8.87987			−0.464794
$366$		0			0
$367$			6.28407i			0.328026i	0.986458	+	0.164013i	$0.0524439\pi$
$367$			6.28407i			0.328026i	−0.986458	+	0.164013i	$0.947556\pi$
$368$		0			0
$369$		−	35.8873i		−	1.86822i
$370$		0			0
$371$			15.0620i			0.781980i
$372$		0			0
$373$	12.5989			0.652348			0.326174	−	0.945310i	$-0.394240\pi$
$373$	12.5989			0.652348			0.326174	+	0.945310i	$0.394240\pi$
$374$		0			0
$375$	28.2335			1.45797
$376$		0			0
$377$		−	0.476532i		−	0.0245427i
$378$		0			0
$379$			32.3527i			1.66184i	0.556389	+	0.830922i	$0.312187\pi$
$379$			32.3527i			1.66184i	−0.556389	+	0.830922i	$0.687813\pi$
$380$		0			0
$381$			47.7187i			2.44470i
$382$		0			0
$383$	−33.7881			−1.72649			−0.863246	−	0.504783i	$-0.831572\pi$
$383$	−33.7881			−1.72649			−0.863246	+	0.504783i	$0.831572\pi$
$384$		0			0
$385$			5.23363i			0.266731i
$386$		0			0
$387$	−18.9678			−0.964187
$388$		0			0
$389$	−12.2883			−0.623041			−0.311520	−	0.950239i	$-0.600838\pi$
$389$	−12.2883			−0.623041			−0.311520	+	0.950239i	$0.600838\pi$
$390$		0			0
$391$	7.14020	−	22.3001i	0.361096	−	1.12776i
$392$		0			0
$393$	42.8765			2.16283
$394$		0			0
$395$	−16.2509			−0.817672
$396$		0			0
$397$		−	12.6943i		−	0.637107i	−0.947905	−	0.318554i	$-0.896803\pi$
$397$		−	12.6943i		−	0.637107i	0.947905	−	0.318554i	$-0.103197\pi$
$398$		0			0
$399$	−79.8451			−3.99726
$400$		0			0
$401$		−	11.2890i		−	0.563746i	−0.959452	−	0.281873i	$-0.909044\pi$
$401$		−	11.2890i		−	0.563746i	0.959452	−	0.281873i	$-0.0909556\pi$
$402$		0			0
$403$		−	0.458756i		−	0.0228523i
$404$		0			0
$405$			2.45016i			0.121749i
$406$		0			0
$407$	−8.80659			−0.436527
$408$		0			0
$409$	−19.2092			−0.949835			−0.474918	−	0.880030i	$-0.657522\pi$
$409$	−19.2092			−0.949835			−0.474918	+	0.880030i	$0.657522\pi$
$410$		0			0
$411$		−	6.38320i		−	0.314860i
$412$		0			0
$413$			3.93843i			0.193797i
$414$		0			0
$415$			17.8642i			0.876920i
$416$		0			0
$417$	17.4740			0.855708
$418$		0			0
$419$			33.3995i			1.63167i	0.578282	+	0.815837i	$0.303724\pi$
$419$			33.3995i			1.63167i	−0.578282	+	0.815837i	$0.696276\pi$
$420$		0			0
$421$	−22.4111			−1.09225			−0.546125	−	0.837704i	$-0.683898\pi$
$421$	−22.4111			−1.09225			−0.546125	+	0.837704i	$0.683898\pi$
$422$		0			0
$423$	−19.2404			−0.935498
$424$		0			0
$425$	13.9570	+	4.46884i	0.677013	+	0.216771i
$426$		0			0
$427$	−7.87940			−0.381311
$428$		0			0
$429$	0.258414			0.0124763
$430$		0			0
$431$		−	39.1271i		−	1.88468i	−0.334651	−	0.942342i	$-0.608618\pi$
$431$		−	39.1271i		−	1.88468i	0.334651	−	0.942342i	$-0.391382\pi$
$432$		0			0
$433$	−12.4682			−0.599182			−0.299591	−	0.954068i	$-0.596850\pi$
$433$	−12.4682			−0.599182			−0.299591	+	0.954068i	$0.596850\pi$
$434$		0			0
$435$			18.4650i			0.885330i
$436$		0			0
$437$		−	41.9425i		−	2.00638i
$438$		0			0
$439$		−	9.33016i		−	0.445304i	−0.974898	−	0.222652i	$-0.928529\pi$
$439$		−	9.33016i		−	0.445304i	0.974898	−	0.222652i	$-0.0714714\pi$
$440$		0			0
$441$	38.5998			1.83808
$442$		0			0
$443$	18.4886			0.878420			0.439210	−	0.898384i	$-0.355258\pi$
$443$	18.4886			0.878420			0.439210	+	0.898384i	$0.355258\pi$
$444$		0			0
$445$			13.1556i			0.623635i
$446$		0			0
$447$		−	19.1841i		−	0.907378i
$448$		0			0
$449$			0.117579i			0.00554888i	0.999996	+	0.00277444i	$0.000883133\pi$
$449$			0.117579i			0.00554888i	−0.999996	+	0.00277444i	$0.999117\pi$
$450$		0			0
$451$	−8.74574			−0.411821
$452$		0			0
$453$		−	29.6428i		−	1.39274i
$454$		0			0
$455$	−0.403344			−0.0189091
$456$		0			0
$457$	−38.3980			−1.79618			−0.898091	−	0.439809i	$-0.855046\pi$
$457$	−38.3980			−1.79618			−0.898091	+	0.439809i	$0.855046\pi$
$458$		0			0
$459$	5.29828	−	16.5475i	0.247303	−	0.772369i
$460$		0			0
$461$	−34.9206			−1.62642			−0.813208	−	0.581973i	$-0.802281\pi$
$461$	−34.9206			−1.62642			−0.813208	+	0.581973i	$0.802281\pi$
$462$		0			0
$463$	13.6149			0.632736			0.316368	−	0.948636i	$-0.397536\pi$
$463$	13.6149			0.632736			0.316368	+	0.948636i	$0.397536\pi$
$464$		0			0
$465$			17.7762i			0.824352i
$466$		0			0
$467$	−2.25425			−0.104314			−0.0521571	−	0.998639i	$-0.516610\pi$
$467$	−2.25425			−0.104314			−0.0521571	+	0.998639i	$0.516610\pi$
$468$		0			0
$469$		−	35.0942i		−	1.62050i
$470$		0			0
$471$		−	51.7607i		−	2.38501i
$472$		0			0
$473$			4.62245i			0.212541i
$474$		0			0
$475$	26.2506			1.20446
$476$		0			0
$477$	−17.3441			−0.794132
$478$		0			0
$479$		−	41.2559i		−	1.88503i	−0.334162	−	0.942516i	$-0.608453\pi$
$479$		−	41.2559i		−	1.88503i	0.334162	−	0.942516i	$-0.391547\pi$
$480$		0			0
$481$		−	0.678704i		−	0.0309462i
$482$		0			0
$483$			61.3966i			2.79364i
$484$		0			0
$485$	18.7674			0.852182
$486$		0			0
$487$			23.8489i			1.08070i	0.841441	+	0.540348i	$0.181708\pi$
$487$			23.8489i			1.08070i	−0.841441	+	0.540348i	$0.818292\pi$
$488$		0			0
$489$	9.75307			0.441049
$490$		0			0
$491$	33.6879			1.52031			0.760156	−	0.649741i	$-0.225123\pi$
$491$	33.6879			1.52031			0.760156	+	0.649741i	$0.225123\pi$
$492$		0			0
$493$	−7.03407	+	21.9686i	−0.316799	+	0.989417i
$494$		0			0
$495$	−6.02661			−0.270876
$496$		0			0
$497$	33.8636			1.51899
$498$		0			0
$499$		−	17.4764i		−	0.782350i	−0.920316	−	0.391175i	$-0.872069\pi$
$499$		−	17.4764i		−	0.782350i	0.920316	−	0.391175i	$-0.127931\pi$
$500$		0			0
$501$	−21.0757			−0.941592
$502$		0			0
$503$		−	4.58844i		−	0.204588i	−0.994754	−	0.102294i	$-0.967382\pi$
$503$		−	4.58844i		−	0.204588i	0.994754	−	0.102294i	$-0.0326183\pi$
$504$		0			0
$505$		−	17.4179i		−	0.775084i
$506$		0			0
$507$		−	35.6654i		−	1.58396i
$508$		0			0
$509$	28.4657			1.26172			0.630861	−	0.775896i	$-0.282702\pi$
$509$	28.4657			1.26172			0.630861	+	0.775896i	$0.282702\pi$
$510$		0			0
$511$	29.0869			1.28673
$512$		0			0
$513$		−	31.1228i		−	1.37411i
$514$		0			0
$515$			2.74083i			0.120775i
$516$		0			0
$517$			4.68888i			0.206217i
$518$		0			0
$519$	59.8582			2.62748
$520$		0			0
$521$			2.63818i			0.115581i	0.998329	+	0.0577905i	$0.0184055\pi$
$521$			2.63818i			0.115581i	−0.998329	+	0.0577905i	$0.981594\pi$
$522$		0			0
$523$	31.7380			1.38781			0.693903	−	0.720068i	$-0.255890\pi$
$523$	31.7380			1.38781			0.693903	+	0.720068i	$0.255890\pi$
$524$		0			0
$525$	−38.4264			−1.67706
$526$		0			0
$527$	−6.77168	+	21.1491i	−0.294979	+	0.921270i
$528$		0			0
$529$	−9.25156			−0.402242
$530$		0			0
$531$	−4.53516			−0.196809
$532$		0			0
$533$		−	0.674014i		−	0.0291948i
$534$		0			0
$535$	2.72215			0.117689
$536$		0			0
$537$			33.6672i			1.45285i
$538$		0			0
$539$		−	9.40677i		−	0.405178i
$540$		0			0
$541$		−	18.6626i		−	0.802368i	−0.915998	−	0.401184i	$-0.868599\pi$
$541$		−	18.6626i		−	0.802368i	0.915998	−	0.401184i	$-0.131401\pi$
$542$		0			0
$543$	−52.5131			−2.25355
$544$		0			0
$545$	9.98867			0.427868
$546$		0			0
$547$			25.0800i			1.07234i	0.844109	+	0.536172i	$0.180130\pi$
$547$			25.0800i			1.07234i	−0.844109	+	0.536172i	$0.819870\pi$
$548$		0			0
$549$		−	9.07325i		−	0.387237i
$550$		0			0
$551$			41.3191i			1.76025i
$552$		0			0
$553$	53.2315			2.26364
$554$		0			0
$555$			26.2989i			1.11633i
$556$		0			0
$557$	45.0667			1.90954			0.954768	−	0.297352i	$-0.0961034\pi$
$557$	45.0667			1.90954			0.954768	+	0.297352i	$0.0961034\pi$
$558$		0			0
$559$	−0.356242			−0.0150674
$560$		0			0
$561$	−11.9131	−	3.81443i	−0.502973	−	0.161045i
$562$		0			0
$563$	0.964781			0.0406607			0.0203303	−	0.999793i	$-0.493528\pi$
$563$	0.964781			0.0406607			0.0203303	+	0.999793i	$0.493528\pi$
$564$		0			0
$565$	−16.5441			−0.696016
$566$		0			0
$567$		−	8.02574i		−	0.337049i
$568$		0			0
$569$	−43.1527			−1.80906			−0.904528	−	0.426415i	$-0.859776\pi$
$569$	−43.1527			−1.80906			−0.904528	+	0.426415i	$0.859776\pi$
$570$		0			0
$571$			10.9017i			0.456222i	0.973635	+	0.228111i	$0.0732549\pi$
$571$			10.9017i			0.456222i	−0.973635	+	0.228111i	$0.926745\pi$
$572$		0			0
$573$			22.2382i			0.929012i
$574$		0			0
$575$		−	20.1853i		−	0.841785i
$576$		0			0
$577$	−10.0756			−0.419454			−0.209727	−	0.977760i	$-0.567258\pi$
$577$	−10.0756			−0.419454			−0.209727	+	0.977760i	$0.567258\pi$
$578$		0			0
$579$	−43.8825			−1.82369
$580$		0			0
$581$		−	58.5161i		−	2.42766i
$582$		0			0
$583$			4.22676i			0.175055i
$584$		0			0
$585$		−	0.464457i		−	0.0192029i
$586$		0			0
$587$	30.9781			1.27860			0.639302	−	0.768956i	$-0.279223\pi$
$587$	30.9781			1.27860			0.639302	+	0.768956i	$0.279223\pi$
$588$		0			0
$589$			39.7777i			1.63901i
$590$		0			0
$591$	37.5929			1.54637
$592$		0			0
$593$	27.3220			1.12198			0.560989	−	0.827823i	$-0.310421\pi$
$593$	27.3220			1.12198			0.560989	+	0.827823i	$0.310421\pi$
$594$		0			0
$595$	18.5946	+	5.95374i	0.762303	+	0.244080i
$596$		0			0
$597$	72.7283			2.97657
$598$		0			0
$599$	9.83617			0.401895			0.200947	−	0.979602i	$-0.435598\pi$
$599$	9.83617			0.401895			0.200947	+	0.979602i	$0.435598\pi$
$600$		0			0
$601$		−	8.88222i		−	0.362313i	−0.983454	−	0.181157i	$-0.942016\pi$
$601$		−	8.88222i		−	0.362313i	0.983454	−	0.181157i	$-0.0579841\pi$
$602$		0			0
$603$	40.4116			1.64569
$604$		0			0
$605$		−	11.7572i		−	0.477998i
$606$		0			0
$607$		−	14.9458i		−	0.606633i	−0.952890	−	0.303317i	$-0.901906\pi$
$607$		−	14.9458i		−	0.606633i	0.952890	−	0.303317i	$-0.0980940\pi$
$608$		0			0
$609$		−	60.4841i		−	2.45094i
$610$		0			0
$611$	−0.361361			−0.0146191
$612$		0			0
$613$	29.1577			1.17767			0.588835	−	0.808253i	$-0.299587\pi$
$613$	29.1577			1.17767			0.588835	+	0.808253i	$0.299587\pi$
$614$		0			0
$615$			26.1172i			1.05315i
$616$		0			0
$617$			13.9156i			0.560219i	0.959968	+	0.280110i	$0.0903709\pi$
$617$			13.9156i			0.560219i	−0.959968	+	0.280110i	$0.909629\pi$
$618$		0			0
$619$			1.51547i			0.0609120i	0.999536	+	0.0304560i	$0.00969594\pi$
$619$			1.51547i			0.0609120i	−0.999536	+	0.0304560i	$0.990304\pi$
$620$		0			0
$621$	−23.9318			−0.960350
$622$		0			0
$623$		−	43.0925i		−	1.72646i
$624$		0			0
$625$	5.40514			0.216205
$626$		0			0
$627$	−22.4065			−0.894829
$628$		0			0
$629$	−10.0183	+	31.2889i	−0.399456	+	1.24757i
$630$		0			0
$631$	−25.3766			−1.01023			−0.505113	−	0.863053i	$-0.668549\pi$
$631$	−25.3766			−1.01023			−0.505113	+	0.863053i	$0.668549\pi$
$632$		0			0
$633$	61.6434			2.45011
$634$		0			0
$635$		−	20.9014i		−	0.829445i
$636$		0			0
$637$	0.724958			0.0287239
$638$		0			0
$639$			38.9945i			1.54260i
$640$		0			0
$641$			40.4994i			1.59963i	0.600248	+	0.799814i	$0.295069\pi$
$641$			40.4994i			1.59963i	−0.600248	+	0.799814i	$0.704931\pi$
$642$		0			0
$643$			26.3431i			1.03887i	0.854510	+	0.519435i	$0.173858\pi$
$643$			26.3431i			1.03887i	−0.854510	+	0.519435i	$0.826142\pi$
$644$		0			0
$645$	13.8039			0.543528
$646$		0			0
$647$	17.4076			0.684363			0.342181	−	0.939634i	$-0.388834\pi$
$647$	17.4076			0.684363			0.342181	+	0.939634i	$0.388834\pi$
$648$		0			0
$649$			1.10522i			0.0433837i
$650$		0			0
$651$		−	58.2278i		−	2.28213i
$652$		0			0
$653$			9.26968i			0.362750i	0.983414	+	0.181375i	$0.0580548\pi$
$653$			9.26968i			0.362750i	−0.983414	+	0.181375i	$0.941945\pi$
$654$		0			0
$655$	−18.7804			−0.733811
$656$		0			0
$657$			33.4940i			1.30673i
$658$		0			0
$659$	37.0162			1.44195			0.720973	−	0.692963i	$-0.243695\pi$
$659$	37.0162			1.44195			0.720973	+	0.692963i	$0.243695\pi$
$660$		0			0
$661$	16.1524			0.628255			0.314128	−	0.949381i	$-0.398288\pi$
$661$	16.1524			0.628255			0.314128	+	0.949381i	$0.398288\pi$
$662$		0			0
$663$	0.293970	−	0.918118i	0.0114168	−	0.0356567i
$664$		0			0
$665$	34.9731			1.35620
$666$		0			0
$667$	31.7722			1.23022
$668$		0			0
$669$		−	38.2332i		−	1.47818i
$670$		0			0
$671$	−2.21115			−0.0853606
$672$		0			0
$673$			5.06734i			0.195331i	0.995219	+	0.0976657i	$0.0311376\pi$
$673$			5.06734i			0.195331i	−0.995219	+	0.0976657i	$0.968862\pi$
$674$		0			0
$675$		−	14.9782i		−	0.576512i
$676$		0			0
$677$		−	3.68162i		−	0.141496i	−0.997494	−	0.0707480i	$-0.977461\pi$
$677$		−	3.68162i		−	0.141496i	0.997494	−	0.0707480i	$-0.0225386\pi$
$678$		0			0
$679$	−61.4745			−2.35917
$680$		0			0
$681$	−34.2344			−1.31186
$682$		0			0
$683$			39.8989i			1.52669i	0.645991	+	0.763345i	$0.276444\pi$
$683$			39.8989i			1.52669i	−0.645991	+	0.763345i	$0.723556\pi$
$684$		0			0
$685$			2.79592i			0.106826i
$686$		0			0
$687$			66.4141i			2.53386i
$688$		0			0
$689$	−0.325747			−0.0124100
$690$		0			0
$691$		−	37.1353i		−	1.41269i	−0.707866	−	0.706346i	$-0.750342\pi$
$691$		−	37.1353i		−	1.41269i	0.707866	−	0.706346i	$-0.249658\pi$
$692$		0			0
$693$	19.7408			0.749890
$694$		0			0
$695$	−7.65384			−0.290327
$696$		0			0
$697$	−9.94908	+	31.0727i	−0.376848	+	1.17696i
$698$		0			0
$699$	−40.6307			−1.53679
$700$		0			0
$701$	7.57706			0.286182			0.143091	−	0.989710i	$-0.454296\pi$
$701$	7.57706			0.286182			0.143091	+	0.989710i	$0.454296\pi$
$702$		0			0
$703$			58.8489i			2.21953i
$704$		0			0
$705$	14.0023			0.527356
$706$		0			0
$707$			57.0540i			2.14574i
$708$		0			0
$709$		−	30.0160i		−	1.12727i	−0.826022	−	0.563637i	$-0.809402\pi$
$709$		−	30.0160i		−	1.12727i	0.826022	−	0.563637i	$-0.190598\pi$
$710$		0			0
$711$			61.2969i			2.29881i
$712$		0			0
$713$	30.5870			1.14549
$714$		0			0
$715$	−0.113188			−0.00423300
$716$		0			0
$717$			62.1890i			2.32249i
$718$		0			0
$719$		−	34.1138i		−	1.27223i	−0.771594	−	0.636115i	$-0.780540\pi$
$719$		−	34.1138i		−	1.27223i	0.771594	−	0.636115i	$-0.219460\pi$
$720$		0			0
$721$		−	8.97786i		−	0.334353i
$722$		0			0
$723$	−60.0822			−2.23448
$724$		0			0
$725$			19.8853i			0.738521i
$726$		0			0
$727$	14.4116			0.534497			0.267249	−	0.963628i	$-0.413885\pi$
$727$	14.4116			0.534497			0.267249	+	0.963628i	$0.413885\pi$
$728$		0			0
$729$	43.9448			1.62759
$730$		0			0
$731$	16.4231	+	5.25847i	0.607430	+	0.194491i
$732$		0			0
$733$	−0.588587			−0.0217400			−0.0108700	−	0.999941i	$-0.503460\pi$
$733$	−0.588587			−0.0217400			−0.0108700	+	0.999941i	$0.503460\pi$
$734$		0			0
$735$	−28.0912			−1.03616
$736$		0			0
$737$		−	9.84830i		−	0.362767i
$738$		0			0
$739$	−43.4714			−1.59912			−0.799561	−	0.600584i	$-0.794935\pi$
$739$	−43.4714			−1.59912			−0.799561	+	0.600584i	$0.794935\pi$
$740$		0			0
$741$		−	1.72682i		−	0.0634362i
$742$		0			0
$743$		−	4.80932i		−	0.176437i	−0.996101	−	0.0882185i	$-0.971883\pi$
$743$		−	4.80932i		−	0.176437i	0.996101	−	0.0882185i	$-0.0281174\pi$
$744$		0			0
$745$			8.40287i			0.307857i
$746$		0			0
$747$	67.3822			2.46538
$748$		0			0
$749$	−8.91667			−0.325808
$750$		0			0
$751$		−	28.9960i		−	1.05808i	−0.848598	−	0.529039i	$-0.822553\pi$
$751$		−	28.9960i		−	1.05808i	0.848598	−	0.529039i	$-0.177447\pi$
$752$		0			0
$753$		−	10.2099i		−	0.372069i
$754$		0			0
$755$			12.9839i			0.472533i
$756$		0			0
$757$	−2.51423			−0.0913813			−0.0456906	−	0.998956i	$-0.514549\pi$
$757$	−2.51423			−0.0913813			−0.0456906	+	0.998956i	$0.514549\pi$
$758$		0			0
$759$			17.2294i			0.625387i
$760$		0			0
$761$	27.5862			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$761$	27.5862			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$762$		0			0
$763$	−32.7189			−1.18450
$764$		0			0
$765$	−6.85582	+	21.4119i	−0.247873	+	0.774150i
$766$		0			0
$767$	−0.0851767			−0.00307555
$768$		0			0
$769$	−20.9417			−0.755175			−0.377588	−	0.925974i	$-0.623246\pi$
$769$	−20.9417			−0.755175			−0.377588	+	0.925974i	$0.623246\pi$
$770$		0			0
$771$		−	28.7088i		−	1.03392i
$772$		0			0
$773$	−24.6969			−0.888284			−0.444142	−	0.895956i	$-0.646491\pi$
$773$	−24.6969			−0.888284			−0.444142	+	0.895956i	$0.646491\pi$
$774$		0			0
$775$			19.1435i			0.687654i
$776$		0			0
$777$		−	86.1447i		−	3.09042i
$778$		0			0
$779$			58.4423i			2.09391i
$780$		0			0
$781$	9.50295			0.340042
$782$		0			0
$783$	23.5761			0.842540
$784$		0			0
$785$			22.6718i			0.809191i
$786$		0			0
$787$			43.3112i			1.54388i	0.635696	+	0.771939i	$0.280713\pi$
$787$			43.3112i			1.54388i	−0.635696	+	0.771939i	$0.719287\pi$
$788$		0			0
$789$			11.9048i			0.423822i
$790$		0			0
$791$	54.1920			1.92684
$792$		0			0
$793$		−	0.170408i		−	0.00605138i
$794$		0			0
$795$	12.6223			0.447666
$796$		0			0
$797$	−0.417268			−0.0147804			−0.00739019	−	0.999973i	$-0.502352\pi$
$797$	−0.417268			−0.0147804			−0.00739019	+	0.999973i	$0.502352\pi$
$798$		0			0
$799$	16.6591	+	5.33403i	0.589357	+	0.188704i
$800$		0			0
$801$	49.6216			1.75329
$802$		0			0
$803$	8.16250			0.288048
$804$		0			0
$805$		−	26.8924i		−	0.947834i
$806$		0			0
$807$	−24.4080			−0.859203
$808$		0			0
$809$		−	40.7668i		−	1.43328i	−0.697442	−	0.716642i	$-0.745678\pi$
$809$		−	40.7668i		−	1.43328i	0.697442	−	0.716642i	$-0.254322\pi$
$810$		0			0
$811$			19.1370i			0.671992i	0.941863	+	0.335996i	$0.109073\pi$
$811$			19.1370i			0.671992i	−0.941863	+	0.335996i	$0.890927\pi$
$812$		0			0
$813$			30.4140i			1.06667i
$814$		0			0
$815$	−4.27196			−0.149640
$816$		0			0
$817$	30.8889			1.08067
$818$		0			0
$819$			1.52138i			0.0531612i
$820$		0			0
$821$			10.2876i			0.359038i	0.983755	+	0.179519i	$0.0574541\pi$
$821$			10.2876i			0.359038i	−0.983755	+	0.179519i	$0.942546\pi$
$822$		0			0
$823$		−	15.1647i		−	0.528607i	−0.964440	−	0.264304i	$-0.914858\pi$
$823$		−	15.1647i		−	0.528607i	0.964440	−	0.264304i	$-0.0851421\pi$
$824$		0			0
$825$	−10.7834			−0.375429
$826$		0			0
$827$		−	18.5741i		−	0.645884i	−0.946419	−	0.322942i	$-0.895328\pi$
$827$		−	18.5741i		−	0.645884i	0.946419	−	0.322942i	$-0.104672\pi$
$828$		0			0
$829$	−24.5314			−0.852009			−0.426005	−	0.904721i	$-0.640079\pi$
$829$	−24.5314			−0.852009			−0.426005	+	0.904721i	$0.640079\pi$
$830$		0			0
$831$	46.2861			1.60565
$832$		0			0
$833$	−33.4213	−	10.7011i	−1.15798	−	0.370770i
$834$		0			0
$835$	9.23140			0.319466
$836$		0			0
$837$	22.6966			0.784510
$838$		0			0
$839$			33.2969i			1.14954i	0.818316	+	0.574769i	$0.194908\pi$
$839$			33.2969i			1.14954i	−0.818316	+	0.574769i	$0.805092\pi$
$840$		0			0
$841$	−2.29992			−0.0793077
$842$		0			0
$843$		−	63.4055i		−	2.18380i
$844$		0			0
$845$			15.6219i			0.537408i
$846$		0			0
$847$			38.5119i			1.32328i
$848$		0			0
$849$	−28.4692			−0.977060
$850$		0			0
$851$	45.2517			1.55121
$852$		0			0
$853$			26.6324i			0.911876i	0.890012	+	0.455938i	$0.150696\pi$
$853$			26.6324i			0.911876i	−0.890012	+	0.455938i	$0.849304\pi$
$854$		0			0
$855$			40.2720i			1.37727i
$856$		0			0
$857$		−	13.2390i		−	0.452237i	−0.974100	−	0.226119i	$-0.927396\pi$
$857$		−	13.2390i		−	0.452237i	0.974100	−	0.226119i	$-0.0726037\pi$
$858$		0			0
$859$	4.93406			0.168348			0.0841739	−	0.996451i	$-0.473175\pi$
$859$	4.93406			0.168348			0.0841739	+	0.996451i	$0.473175\pi$
$860$		0			0
$861$		−	85.5494i		−	2.91552i
$862$		0			0
$863$	16.7188			0.569113			0.284557	−	0.958659i	$-0.408154\pi$
$863$	16.7188			0.569113			0.284557	+	0.958659i	$0.408154\pi$
$864$		0			0
$865$	−26.2186			−0.891458
$866$		0			0
$867$	−27.1046	+	37.9869i	−0.920520	+	1.29010i
$868$		0			0
$869$	14.9381			0.506739
$870$		0			0
$871$	0.758986			0.0257173
$872$		0			0
$873$		−	70.7888i		−	2.39584i
$874$		0			0
$875$	40.5081			1.36942
$876$		0			0
$877$		−	31.5284i		−	1.06464i	−0.846544	−	0.532319i	$-0.821321\pi$
$877$		−	31.5284i		−	1.06464i	0.846544	−	0.532319i	$-0.178679\pi$
$878$		0			0
$879$		−	64.3868i		−	2.17171i
$880$		0			0
$881$			9.87094i			0.332560i	0.986079	+	0.166280i	$0.0531756\pi$
$881$			9.87094i			0.332560i	−0.986079	+	0.166280i	$0.946824\pi$
$882$		0			0
$883$	22.2115			0.747478			0.373739	−	0.927534i	$-0.378076\pi$
$883$	22.2115			0.747478			0.373739	+	0.927534i	$0.378076\pi$
$884$		0			0
$885$	3.30049			0.110945
$886$		0			0
$887$		−	18.8286i		−	0.632203i	−0.948725	−	0.316102i	$-0.897626\pi$
$887$		−	18.8286i		−	0.632203i	0.948725	−	0.316102i	$-0.102374\pi$
$888$		0			0
$889$			68.4645i			2.29623i
$890$		0			0
$891$		−	2.25222i		−	0.0754521i
$892$		0			0
$893$	31.3328			1.04851
$894$		0			0
$895$		−	14.7466i		−	0.492925i
$896$		0			0
$897$	−1.32783			−0.0443349
$898$		0			0
$899$	−30.1323			−1.00497
$900$		0			0
$901$	15.0173	+	4.80833i	0.500297	+	0.160189i
$902$		0			0
$903$	−45.2161			−1.50470
$904$		0			0
$905$	23.0014			0.764591
$906$		0			0
$907$			9.96685i			0.330944i	0.986214	+	0.165472i	$0.0529147\pi$
$907$			9.96685i			0.330944i	−0.986214	+	0.165472i	$0.947085\pi$
$908$		0			0
$909$	−65.6985			−2.17908
$910$		0			0
$911$			0.932590i			0.0308981i	0.999881	+	0.0154490i	$0.00491778\pi$
$911$			0.932590i			0.0308981i	−0.999881	+	0.0154490i	$0.995082\pi$
$912$		0			0
$913$		−	16.4210i		−	0.543457i
$914$		0			0
$915$			6.60311i			0.218292i
$916$		0			0
$917$	61.5171			2.03147
$918$		0			0
$919$	31.2076			1.02944			0.514722	−	0.857357i	$-0.327895\pi$
$919$	31.2076			1.02944			0.514722	+	0.857357i	$0.327895\pi$
$920$		0			0
$921$		−	29.1617i		−	0.960912i
$922$		0			0
$923$			0.732371i			0.0241063i
$924$		0			0
$925$			28.3217i			0.931212i
$926$		0			0
$927$	10.3381			0.339549
$928$		0			0
$929$			10.9111i			0.357983i	0.983851	+	0.178991i	$0.0572834\pi$
$929$			10.9111i			0.357983i	−0.983851	+	0.178991i	$0.942717\pi$
$930$		0			0
$931$	−62.8595			−2.06014
$932$		0			0
$933$	25.4550			0.833360
$934$		0			0
$935$	5.21809	+	1.67077i	0.170650	+	0.0546399i
$936$		0			0
$937$	45.6839			1.49243			0.746214	−	0.665706i	$-0.231869\pi$
$937$	45.6839			1.49243			0.746214	+	0.665706i	$0.231869\pi$
$938$		0			0
$939$	−5.78560			−0.188806
$940$		0			0
$941$			19.0048i			0.619540i	0.950812	+	0.309770i	$0.100252\pi$
$941$			19.0048i			0.619540i	−0.950812	+	0.309770i	$0.899748\pi$
$942$		0			0
$943$	44.9390			1.46341
$944$		0			0
$945$		−	19.9551i		−	0.649141i
$946$		0			0
$947$			10.6172i			0.345014i	0.985008	+	0.172507i	$0.0551867\pi$
$947$			10.6172i			0.345014i	−0.985008	+	0.172507i	$0.944813\pi$
$948$		0			0
$949$			0.629065i			0.0204203i
$950$		0			0
$951$	30.0600			0.974762
$952$		0			0
$953$	15.9806			0.517663			0.258831	−	0.965923i	$-0.416663\pi$
$953$	15.9806			0.517663			0.258831	+	0.965923i	$0.416663\pi$
$954$		0			0
$955$		−	9.74057i		−	0.315197i
$956$		0			0
$957$		−	16.9733i		−	0.548669i
$958$		0			0
$959$		−	9.15831i		−	0.295737i
$960$		0			0
$961$	1.99171			0.0642486
$962$		0			0
$963$		−	10.2677i		−	0.330872i
$964$		0			0
$965$	19.2210			0.618747
$966$		0			0
$967$	−17.3959			−0.559415			−0.279707	−	0.960085i	$-0.590237\pi$
$967$	−17.3959			−0.559415			−0.279707	+	0.960085i	$0.590237\pi$
$968$		0			0
$969$	−25.4895	+	79.6080i	−0.818839	+	2.55738i
$970$		0			0
$971$	42.5203			1.36454			0.682271	−	0.731099i	$-0.260992\pi$
$971$	42.5203			1.36454			0.682271	+	0.731099i	$0.260992\pi$
$972$		0			0
$973$	25.0709			0.803737
$974$		0			0
$975$		−	0.831050i		−	0.0266149i
$976$		0			0
$977$	−15.7584			−0.504155			−0.252077	−	0.967707i	$-0.581114\pi$
$977$	−15.7584			−0.504155			−0.252077	+	0.967707i	$0.581114\pi$
$978$		0			0
$979$		−	12.0928i		−	0.386488i
$980$		0			0
$981$		−	37.6763i		−	1.20291i
$982$		0			0
$983$			33.6406i			1.07297i	0.843910	+	0.536484i	$0.180248\pi$
$983$			33.6406i			1.07297i	−0.843910	+	0.536484i	$0.819752\pi$
$984$		0			0
$985$	−16.4661			−0.524655
$986$		0			0
$987$	−45.8659			−1.45993
$988$		0			0
$989$		−	23.7520i		−	0.755268i
$990$		0			0
$991$			8.68907i			0.276017i	0.990431	+	0.138009i	$0.0440702\pi$
$991$			8.68907i			0.276017i	−0.990431	+	0.138009i	$0.955930\pi$
$992$		0			0
$993$			81.6277i			2.59038i
$994$		0			0
$995$	−31.8558			−1.00990
$996$		0			0
$997$		−	2.60611i		−	0.0825364i	−0.999148	−	0.0412682i	$-0.986860\pi$
$997$		−	2.60611i		−	0.0825364i	0.999148	−	0.0412682i	$-0.0131398\pi$
$998$		0			0
$999$	33.5784			1.06237

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.b.b.237.41	yes	46
17.16	even	2	inner	4012.2.b.b.237.6	✓	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.b.b.237.6	✓	46	17.16	even	2	inner
4012.2.b.b.237.41	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)

\(f(q)\)	\(=\)	\(q+2.74502i q^{3} -1.20235i q^{5} +3.93843i q^{7} -4.53516 q^{9} +O(q^{10})\)	\(q+2.74502i q^{3} -1.20235i q^{5} +3.93843i q^{7} -4.53516 q^{9} +1.10522i q^{11} -0.0851767 q^{13} +3.30049 q^{15} +(3.92673 + 1.25729i) q^{17} +7.38549 q^{19} -10.8111 q^{21} -5.67905i q^{23} +3.55435 q^{25} -4.21405i q^{27} +5.59463i q^{29} +5.38593i q^{31} -3.03385 q^{33} +4.73538 q^{35} +7.96819i q^{37} -0.233812i q^{39} +7.91312i q^{41} +4.18238 q^{43} +5.45286i q^{45} +4.24249 q^{47} -8.51122 q^{49} +(-3.45129 + 10.7790i) q^{51} +3.82436 q^{53} +1.32886 q^{55} +20.2733i q^{57} +1.00000 q^{59} +2.00065i q^{61} -17.8614i q^{63} +0.102412i q^{65} -8.91072 q^{67} +15.5891 q^{69} -8.59825i q^{71} -7.38541i q^{73} +9.75677i q^{75} -4.35283 q^{77} -13.5159i q^{79} -2.03780 q^{81} -14.8577 q^{83} +(1.51170 - 4.72132i) q^{85} -15.3574 q^{87} -10.9415 q^{89} -0.335462i q^{91} -14.7845 q^{93} -8.87996i q^{95} +15.6089i q^{97} -5.01235i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 54 q^{9}+O(q^{10}) \) 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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