LMFDB - Embedded newform 3072.2.d.f.1537.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3072,2,Mod(1537,3072)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3072, 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("3072.1537");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3072 = 2^{10} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3072.d (of order $2$, degree $1$, not 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:	$24.5300435009$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.18939904.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 $ x^8 - 4x^7 + 14x^6 - 28x^5 + 43x^4 - 44x^3 + 30x^2 - 12*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1537.8
Root			$0.500000 + 2.10607i$ of defining polynomial
Character	$\chi$	$=$	3072.1537
Dual form			3072.2.d.f.1537.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +1.79793i q^{5} +0.158942 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +1.79793i q^{5} +0.158942 q^{7} -1.00000 q^{9} +5.37109i q^{11} +5.95687i q^{13} -1.79793 q^{15} -3.05320 q^{17} +3.05320i q^{19} +0.158942i q^{21} +2.82843 q^{23} +1.76744 q^{25} -1.00000i q^{27} +2.96951i q^{29} +4.15894 q^{31} -5.37109 q^{33} +0.285766i q^{35} -8.46742i q^{37} -5.95687 q^{39} -2.60365 q^{41} -8.13853i q^{43} -1.79793i q^{45} +2.82843 q^{47} -6.97474 q^{49} -3.05320i q^{51} -5.03049i q^{53} -9.65685 q^{55} -3.05320 q^{57} +5.65685i q^{59} +5.18944i q^{61} -0.158942 q^{63} -10.7101 q^{65} +1.08532i q^{67} +2.82843i q^{69} -0.317883 q^{71} -1.33897 q^{73} +1.76744i q^{75} +0.853690i q^{77} +9.69382 q^{79} +1.00000 q^{81} -0.163788i q^{83} -5.48946i q^{85} -2.96951 q^{87} -14.3990 q^{89} +0.946795i q^{91} +4.15894i q^{93} -5.48946 q^{95} -0.571533 q^{97} -5.37109i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{7} - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^7 - 8 * q^9	$ 8 q - 8 q^{7} - 8 q^{9} + 8 q^{15} - 8 q^{25} + 24 q^{31} - 16 q^{39} + 8 q^{49} - 32 q^{55} + 8 q^{63} - 16 q^{65} + 16 q^{71} + 16 q^{73} + 24 q^{79} + 8 q^{81} - 24 q^{87} + 16 q^{89} - 48 q^{95}+O(q^{100}) $ 8 * q - 8 * q^7 - 8 * q^9 + 8 * q^15 - 8 * q^25 + 24 * q^31 - 16 * q^39 + 8 * q^49 - 32 * q^55 + 8 * q^63 - 16 * q^65 + 16 * q^71 + 16 * q^73 + 24 * q^79 + 8 * q^81 - 24 * q^87 + 16 * q^89 - 48 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3072\mathbb{Z}\right)^\times$.

$n$	$1025$	$2047$	$2053$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	1.79793i	0.804060i	0.915627	+	0.402030i	$0.131695\pi$
$5$	1.79793i	0.804060i	−0.915627	+	0.402030i	$0.868305\pi$
$6$	0	0
$7$	0.158942	0.0600743	0.0300371	−	0.999549i	$-0.490437\pi$
$7$	0.158942	0.0600743	0.0300371	+	0.999549i	$0.490437\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	5.37109i	1.61944i	0.586814	+	0.809722i	$0.300382\pi$
$11$	5.37109i	1.61944i	−0.586814	+	0.809722i	$0.699618\pi$
$12$	0	0
$13$	5.95687i	1.65214i	0.563568	+	0.826070i	$0.309428\pi$
$13$	5.95687i	1.65214i	−0.563568	+	0.826070i	$0.690572\pi$
$14$	0	0
$15$	−1.79793	−0.464224
$16$	0	0
$17$	−3.05320	−0.740511	−0.370255	−	0.928930i	$-0.620730\pi$
$17$	−3.05320	−0.740511	−0.370255	+	0.928930i	$0.620730\pi$
$18$	0	0
$19$	3.05320i	0.700453i	0.936665	+	0.350227i	$0.113895\pi$
$19$	3.05320i	0.700453i	−0.936665	+	0.350227i	$0.886105\pi$
$20$	0	0
$21$	0.158942i	0.0346839i
$22$	0	0
$23$	2.82843	0.589768	0.294884	−	0.955533i	$-0.404719\pi$
$23$	2.82843	0.589768	0.294884	+	0.955533i	$0.404719\pi$
$24$	0	0
$25$	1.76744	0.353488
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	2.96951i	0.551423i	0.961240	+	0.275712i	$0.0889135\pi$
$29$	2.96951i	0.551423i	−0.961240	+	0.275712i	$0.911087\pi$
$30$	0	0
$31$	4.15894	0.746968	0.373484	−	0.927637i	$-0.378163\pi$
$31$	4.15894	0.746968	0.373484	+	0.927637i	$0.378163\pi$
$32$	0	0
$33$	−5.37109	−0.934986
$34$	0	0
$35$	0.285766i	0.0483033i
$36$	0	0
$37$	− 8.46742i	− 1.39203i	−0.718025	−	0.696017i	$-0.754954\pi$
$37$	− 8.46742i	− 1.39203i	0.718025	−	0.696017i	$-0.245046\pi$
$38$	0	0
$39$	−5.95687	−0.953863
$40$	0	0
$41$	−2.60365	−0.406622	−0.203311	−	0.979114i	$-0.565170\pi$
$41$	−2.60365	−0.406622	−0.203311	+	0.979114i	$0.565170\pi$
$42$	0	0
$43$	− 8.13853i	− 1.24111i	−0.784162	−	0.620557i	$-0.786907\pi$
$43$	− 8.13853i	− 1.24111i	0.784162	−	0.620557i	$-0.213093\pi$
$44$	0	0
$45$	− 1.79793i	− 0.268020i
$46$	0	0
$47$	2.82843	0.412568	0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843	0.412568	0.206284	+	0.978492i	$0.433863\pi$
$48$	0	0
$49$	−6.97474	−0.996391
$50$	0	0
$51$	− 3.05320i	− 0.427534i
$52$	0	0
$53$	− 5.03049i	− 0.690992i	−0.938420	−	0.345496i	$-0.887711\pi$
$53$	− 5.03049i	− 0.690992i	0.938420	−	0.345496i	$-0.112289\pi$
$54$	0	0
$55$	−9.65685	−1.30213
$56$	0	0
$57$	−3.05320	−0.404407
$58$	0	0
$59$	5.65685i	0.736460i	0.929735	+	0.368230i	$0.120036\pi$
$59$	5.65685i	0.736460i	−0.929735	+	0.368230i	$0.879964\pi$
$60$	0	0
$61$	5.18944i	0.664439i	0.943202	+	0.332220i	$0.107798\pi$
$61$	5.18944i	0.664439i	−0.943202	+	0.332220i	$0.892202\pi$
$62$	0	0
$63$	−0.158942	−0.0200248
$64$	0	0
$65$	−10.7101	−1.32842
$66$	0	0
$67$	1.08532i	0.132593i	0.997800	+	0.0662966i	$0.0211183\pi$
$67$	1.08532i	0.132593i	−0.997800	+	0.0662966i	$0.978882\pi$
$68$	0	0
$69$	2.82843i	0.340503i
$70$	0	0
$71$	−0.317883	−0.0377258	−0.0188629	−	0.999822i	$-0.506005\pi$
$71$	−0.317883	−0.0377258	−0.0188629	+	0.999822i	$0.506005\pi$
$72$	0	0
$73$	−1.33897	−0.156715	−0.0783573	−	0.996925i	$-0.524968\pi$
$73$	−1.33897	−0.156715	−0.0783573	+	0.996925i	$0.524968\pi$
$74$	0	0
$75$	1.76744i	0.204086i
$76$	0	0
$77$	0.853690i	0.0972870i
$78$	0	0
$79$	9.69382	1.09064	0.545320	−	0.838228i	$-0.316408\pi$
$79$	9.69382	1.09064	0.545320	+	0.838228i	$0.316408\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 0.163788i	− 0.0179781i	−0.999960	−	0.00898906i	$-0.997139\pi$
$83$	− 0.163788i	− 0.0179781i	0.999960	−	0.00898906i	$-0.00286135\pi$
$84$	0	0
$85$	− 5.48946i	− 0.595415i
$86$	0	0
$87$	−2.96951	−0.318364
$88$	0	0
$89$	−14.3990	−1.52629	−0.763147	−	0.646225i	$-0.776347\pi$
$89$	−14.3990	−1.52629	−0.763147	+	0.646225i	$0.776347\pi$
$90$	0	0
$91$	0.946795i	0.0992511i
$92$	0	0
$93$	4.15894i	0.431262i
$94$	0	0
$95$	−5.48946	−0.563206
$96$	0	0
$97$	−0.571533	−0.0580304	−0.0290152	−	0.999579i	$-0.509237\pi$
$97$	−0.571533	−0.0580304	−0.0290152	+	0.999579i	$0.509237\pi$
$98$	0	0
$99$	− 5.37109i	− 0.539815i
$100$	0	0
$101$	− 10.1158i	− 1.00656i	−0.864123	−	0.503281i	$-0.832126\pi$
$101$	− 10.1158i	− 1.00656i	0.864123	−	0.503281i	$-0.167874\pi$
$102$	0	0
$103$	11.3507	1.11841	0.559207	−	0.829028i	$-0.311106\pi$
$103$	11.3507	1.11841	0.559207	+	0.829028i	$0.311106\pi$
$104$	0	0
$105$	−0.285766	−0.0278879
$106$	0	0
$107$	1.02109i	0.0987123i	0.998781	+	0.0493561i	$0.0157169\pi$
$107$	1.02109i	0.0987123i	−0.998781	+	0.0493561i	$0.984283\pi$
$108$	0	0
$109$	2.04313i	0.195696i	0.995201	+	0.0978480i	$0.0311959\pi$
$109$	2.04313i	0.195696i	−0.995201	+	0.0978480i	$0.968804\pi$
$110$	0	0
$111$	8.46742	0.803692
$112$	0	0
$113$	3.53488	0.332533	0.166267	−	0.986081i	$-0.446829\pi$
$113$	3.53488	0.332533	0.166267	+	0.986081i	$0.446829\pi$
$114$	0	0
$115$	5.08532i	0.474209i
$116$	0	0
$117$	− 5.95687i	− 0.550713i
$118$	0	0
$119$	−0.485281	−0.0444857
$120$	0	0
$121$	−17.8486	−1.62260
$122$	0	0
$123$	− 2.60365i	− 0.234763i
$124$	0	0
$125$	12.1674i	1.08829i
$126$	0	0
$127$	−1.49791	−0.132918	−0.0664591	−	0.997789i	$-0.521170\pi$
$127$	−1.49791	−0.132918	−0.0664591	+	0.997789i	$0.521170\pi$
$128$	0	0
$129$	8.13853	0.716557
$130$	0	0
$131$	14.7422i	1.28803i	0.765013	+	0.644015i	$0.222733\pi$
$131$	14.7422i	1.28803i	−0.765013	+	0.644015i	$0.777267\pi$
$132$	0	0
$133$	0.485281i	0.0420792i
$134$	0	0
$135$	1.79793	0.154741
$136$	0	0
$137$	−13.7954	−1.17862	−0.589309	−	0.807907i	$-0.700600\pi$
$137$	−13.7954	−1.17862	−0.589309	+	0.807907i	$0.700600\pi$
$138$	0	0
$139$	3.42847i	0.290799i	0.989373	+	0.145399i	$0.0464467\pi$
$139$	3.42847i	0.290799i	−0.989373	+	0.145399i	$0.953553\pi$
$140$	0	0
$141$	2.82843i	0.238197i
$142$	0	0
$143$	−31.9949	−2.67555
$144$	0	0
$145$	−5.33897	−0.443377
$146$	0	0
$147$	− 6.97474i	− 0.575267i
$148$	0	0
$149$	− 4.14108i	− 0.339250i	−0.985509	−	0.169625i	$-0.945744\pi$
$149$	− 4.14108i	− 0.339250i	0.985509	−	0.169625i	$-0.0542557\pi$
$150$	0	0
$151$	22.6644	1.84440	0.922201	−	0.386712i	$-0.126389\pi$
$151$	22.6644	1.84440	0.922201	+	0.386712i	$0.126389\pi$
$152$	0	0
$153$	3.05320	0.246837
$154$	0	0
$155$	7.47750i	0.600607i
$156$	0	0
$157$	− 3.93161i	− 0.313777i	−0.987616	−	0.156888i	$-0.949854\pi$
$157$	− 3.93161i	− 0.313777i	0.987616	−	0.156888i	$-0.0501463\pi$
$158$	0	0
$159$	5.03049	0.398944
$160$	0	0
$161$	0.449555	0.0354299
$162$	0	0
$163$	− 7.68897i	− 0.602247i	−0.953585	−	0.301123i	$-0.902638\pi$
$163$	− 7.68897i	− 0.602247i	0.953585	−	0.301123i	$-0.0973616\pi$
$164$	0	0
$165$	− 9.65685i	− 0.751785i
$166$	0	0
$167$	3.95458	0.306015	0.153007	−	0.988225i	$-0.451104\pi$
$167$	3.95458	0.306015	0.153007	+	0.988225i	$0.451104\pi$
$168$	0	0
$169$	−22.4844	−1.72957
$170$	0	0
$171$	− 3.05320i	− 0.233484i
$172$	0	0
$173$	22.6011i	1.71833i	0.511699	+	0.859165i	$0.329016\pi$
$173$	22.6011i	1.71833i	−0.511699	+	0.859165i	$0.670984\pi$
$174$	0	0
$175$	0.280920	0.0212355
$176$	0	0
$177$	−5.65685	−0.425195
$178$	0	0
$179$	17.2981i	1.29292i	0.762946	+	0.646462i	$0.223752\pi$
$179$	17.2981i	1.29292i	−0.762946	+	0.646462i	$0.776248\pi$
$180$	0	0
$181$	− 8.14953i	− 0.605750i	−0.953030	−	0.302875i	$-0.902054\pi$
$181$	− 8.14953i	− 0.605750i	0.953030	−	0.302875i	$-0.0979465\pi$
$182$	0	0
$183$	−5.18944	−0.383614
$184$	0	0
$185$	15.2238	1.11928
$186$	0	0
$187$	− 16.3990i	− 1.19922i
$188$	0	0
$189$	− 0.158942i	− 0.0115613i
$190$	0	0
$191$	−16.1674	−1.16983	−0.584916	−	0.811094i	$-0.698873\pi$
$191$	−16.1674	−1.16983	−0.584916	+	0.811094i	$0.698873\pi$
$192$	0	0
$193$	−22.1454	−1.59406	−0.797030	−	0.603940i	$-0.793597\pi$
$193$	−22.1454	−1.59406	−0.797030	+	0.603940i	$0.793597\pi$
$194$	0	0
$195$	− 10.7101i	− 0.766963i
$196$	0	0
$197$	− 20.2222i	− 1.44077i	−0.693572	−	0.720387i	$-0.743964\pi$
$197$	− 20.2222i	− 1.44077i	0.693572	−	0.720387i	$-0.256036\pi$
$198$	0	0
$199$	−25.0075	−1.77274	−0.886368	−	0.462981i	$-0.846780\pi$
$199$	−25.0075	−1.77274	−0.886368	+	0.462981i	$0.846780\pi$
$200$	0	0
$201$	−1.08532	−0.0765527
$202$	0	0
$203$	0.471978i	0.0331264i
$204$	0	0
$205$	− 4.68119i	− 0.326948i
$206$	0	0
$207$	−2.82843	−0.196589
$208$	0	0
$209$	−16.3990	−1.13434
$210$	0	0
$211$	− 26.0559i	− 1.79376i	−0.442273	−	0.896881i	$-0.645828\pi$
$211$	− 26.0559i	− 1.79376i	0.442273	−	0.896881i	$-0.354172\pi$
$212$	0	0
$213$	− 0.317883i	− 0.0217810i
$214$	0	0
$215$	14.6325	0.997930
$216$	0	0
$217$	0.661029	0.0448736
$218$	0	0
$219$	− 1.33897i	− 0.0904793i
$220$	0	0
$221$	− 18.1876i	− 1.22343i
$222$	0	0
$223$	18.3465	1.22857	0.614286	−	0.789083i	$-0.289444\pi$
$223$	18.3465	1.22857	0.614286	+	0.789083i	$0.289444\pi$
$224$	0	0
$225$	−1.76744	−0.117829
$226$	0	0
$227$	− 0.163788i	− 0.0108710i	−0.999985	−	0.00543551i	$-0.998270\pi$
$227$	− 0.163788i	− 0.0108710i	0.999985	−	0.00543551i	$-0.00173019\pi$
$228$	0	0
$229$	4.02756i	0.266148i	0.991106	+	0.133074i	$0.0424849\pi$
$229$	4.02756i	0.266148i	−0.991106	+	0.133074i	$0.957515\pi$
$230$	0	0
$231$	−0.853690	−0.0561687
$232$	0	0
$233$	11.7211	0.767874	0.383937	−	0.923359i	$-0.374568\pi$
$233$	11.7211	0.767874	0.383937	+	0.923359i	$0.374568\pi$
$234$	0	0
$235$	5.08532i	0.331730i
$236$	0	0
$237$	9.69382i	0.629681i
$238$	0	0
$239$	13.6517	0.883058	0.441529	−	0.897247i	$-0.354436\pi$
$239$	13.6517	0.883058	0.441529	+	0.897247i	$0.354436\pi$
$240$	0	0
$241$	−2.13167	−0.137313	−0.0686565	−	0.997640i	$-0.521871\pi$
$241$	−2.13167	−0.137313	−0.0686565	+	0.997640i	$0.521871\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	− 12.5401i	− 0.801158i
$246$	0	0
$247$	−18.1876	−1.15725
$248$	0	0
$249$	0.163788	0.0103797
$250$	0	0
$251$	− 6.27020i	− 0.395771i	−0.980225	−	0.197886i	$-0.936593\pi$
$251$	− 6.27020i	− 0.395771i	0.980225	−	0.197886i	$-0.0634075\pi$
$252$	0	0
$253$	15.1917i	0.955096i
$254$	0	0
$255$	5.48946	0.343763
$256$	0	0
$257$	15.0853	0.940997	0.470498	−	0.882401i	$-0.344074\pi$
$257$	15.0853	0.940997	0.470498	+	0.882401i	$0.344074\pi$
$258$	0	0
$259$	− 1.34583i	− 0.0836255i
$260$	0	0
$261$	− 2.96951i	− 0.183808i
$262$	0	0
$263$	26.1706	1.61375	0.806875	−	0.590722i	$-0.201157\pi$
$263$	26.1706	1.61375	0.806875	+	0.590722i	$0.201157\pi$
$264$	0	0
$265$	9.04449	0.555599
$266$	0	0
$267$	− 14.3990i	− 0.881206i
$268$	0	0
$269$	− 12.1580i	− 0.741286i	−0.928775	−	0.370643i	$-0.879137\pi$
$269$	− 12.1580i	− 0.741286i	0.928775	−	0.370643i	$-0.120863\pi$
$270$	0	0
$271$	−10.6644	−0.647815	−0.323907	−	0.946089i	$-0.604997\pi$
$271$	−10.6644	−0.647815	−0.323907	+	0.946089i	$0.604997\pi$
$272$	0	0
$273$	−0.946795	−0.0573027
$274$	0	0
$275$	9.49307i	0.572453i
$276$	0	0
$277$	3.76421i	0.226170i	0.993585	+	0.113085i	$0.0360732\pi$
$277$	3.76421i	0.226170i	−0.993585	+	0.113085i	$0.963927\pi$
$278$	0	0
$279$	−4.15894	−0.248989
$280$	0	0
$281$	−10.4496	−0.623368	−0.311684	−	0.950186i	$-0.600893\pi$
$281$	−10.4496	−0.623368	−0.311684	+	0.950186i	$0.600893\pi$
$282$	0	0
$283$	− 17.6569i	− 1.04959i	−0.851228	−	0.524796i	$-0.824142\pi$
$283$	− 17.6569i	− 1.04959i	0.851228	−	0.524796i	$-0.175858\pi$
$284$	0	0
$285$	− 5.48946i	− 0.325167i
$286$	0	0
$287$	−0.413828	−0.0244275
$288$	0	0
$289$	−7.67794	−0.451644
$290$	0	0
$291$	− 0.571533i	− 0.0335038i
$292$	0	0
$293$	30.7465i	1.79623i	0.439762	+	0.898114i	$0.355063\pi$
$293$	30.7465i	1.79623i	−0.439762	+	0.898114i	$0.644937\pi$
$294$	0	0
$295$	−10.1706	−0.592158
$296$	0	0
$297$	5.37109	0.311662
$298$	0	0
$299$	16.8486i	0.974379i
$300$	0	0
$301$	− 1.29355i	− 0.0745590i
$302$	0	0
$303$	10.1158	0.581138
$304$	0	0
$305$	−9.33026	−0.534249
$306$	0	0
$307$	− 21.2981i	− 1.21555i	−0.794110	−	0.607775i	$-0.792062\pi$
$307$	− 21.2981i	− 1.21555i	0.794110	−	0.607775i	$-0.207938\pi$
$308$	0	0
$309$	11.3507i	0.645717i
$310$	0	0
$311$	−1.77883	−0.100868	−0.0504342	−	0.998727i	$-0.516061\pi$
$311$	−1.77883	−0.100868	−0.0504342	+	0.998727i	$0.516061\pi$
$312$	0	0
$313$	2.70320	0.152794	0.0763971	−	0.997077i	$-0.475658\pi$
$313$	2.70320	0.152794	0.0763971	+	0.997077i	$0.475658\pi$
$314$	0	0
$315$	− 0.285766i	− 0.0161011i
$316$	0	0
$317$	22.0653i	1.23931i	0.784874	+	0.619655i	$0.212728\pi$
$317$	22.0653i	1.23931i	−0.784874	+	0.619655i	$0.787272\pi$
$318$	0	0
$319$	−15.9495	−0.892999
$320$	0	0
$321$	−1.02109	−0.0569916
$322$	0	0
$323$	− 9.32206i	− 0.518693i
$324$	0	0
$325$	10.5284i	0.584011i
$326$	0	0
$327$	−2.04313	−0.112985
$328$	0	0
$329$	0.449555	0.0247848
$330$	0	0
$331$	− 21.8431i	− 1.20060i	−0.799774	−	0.600302i	$-0.795047\pi$
$331$	− 21.8431i	− 1.20060i	0.799774	−	0.600302i	$-0.204953\pi$
$332$	0	0
$333$	8.46742i	0.464012i
$334$	0	0
$335$	−1.95133	−0.106613
$336$	0	0
$337$	18.8738	1.02812	0.514062	−	0.857753i	$-0.328140\pi$
$337$	18.8738	1.02812	0.514062	+	0.857753i	$0.328140\pi$
$338$	0	0
$339$	3.53488i	0.191988i
$340$	0	0
$341$	22.3380i	1.20967i
$342$	0	0
$343$	−2.22117	−0.119932
$344$	0	0
$345$	−5.08532	−0.273785
$346$	0	0
$347$	28.0490i	1.50575i	0.658163	+	0.752875i	$0.271334\pi$
$347$	28.0490i	1.50575i	−0.658163	+	0.752875i	$0.728666\pi$
$348$	0	0
$349$	16.9307i	0.906279i	0.891440	+	0.453139i	$0.149696\pi$
$349$	16.9307i	0.906279i	−0.891440	+	0.453139i	$0.850304\pi$
$350$	0	0
$351$	5.95687	0.317954
$352$	0	0
$353$	−12.6202	−0.671705	−0.335853	−	0.941915i	$-0.609024\pi$
$353$	−12.6202	−0.671705	−0.335853	+	0.941915i	$0.609024\pi$
$354$	0	0
$355$	− 0.571533i	− 0.0303338i
$356$	0	0
$357$	− 0.485281i	− 0.0256838i
$358$	0	0
$359$	27.0867	1.42958	0.714790	−	0.699339i	$-0.246522\pi$
$359$	27.0867	1.42958	0.714790	+	0.699339i	$0.246522\pi$
$360$	0	0
$361$	9.67794	0.509365
$362$	0	0
$363$	− 17.8486i	− 0.936808i
$364$	0	0
$365$	− 2.40738i	− 0.126008i
$366$	0	0
$367$	20.4937	1.06976	0.534882	−	0.844927i	$-0.320356\pi$
$367$	20.4937	1.06976	0.534882	+	0.844927i	$0.320356\pi$
$368$	0	0
$369$	2.60365	0.135541
$370$	0	0
$371$	− 0.799555i	− 0.0415108i
$372$	0	0
$373$	− 1.46190i	− 0.0756943i	−0.999284	−	0.0378471i	$-0.987950\pi$
$373$	− 1.46190i	− 0.0756943i	0.999284	−	0.0378471i	$-0.0120500\pi$
$374$	0	0
$375$	−12.1674	−0.628322
$376$	0	0
$377$	−17.6890	−0.911028
$378$	0	0
$379$	24.9871i	1.28350i	0.766913	+	0.641751i	$0.221792\pi$
$379$	24.9871i	1.28350i	−0.766913	+	0.641751i	$0.778208\pi$
$380$	0	0
$381$	− 1.49791i	− 0.0767404i
$382$	0	0
$383$	−31.0958	−1.58892	−0.794460	−	0.607316i	$-0.792246\pi$
$383$	−31.0958	−1.58892	−0.794460	+	0.607316i	$0.792246\pi$
$384$	0	0
$385$	−1.53488	−0.0782245
$386$	0	0
$387$	8.13853i	0.413705i
$388$	0	0
$389$	− 3.62218i	− 0.183652i	−0.995775	−	0.0918260i	$-0.970730\pi$
$389$	− 3.62218i	− 0.183652i	0.995775	−	0.0918260i	$-0.0292704\pi$
$390$	0	0
$391$	−8.63577	−0.436729
$392$	0	0
$393$	−14.7422	−0.743644
$394$	0	0
$395$	17.4288i	0.876940i
$396$	0	0
$397$	7.20959i	0.361839i	0.983498	+	0.180920i	$0.0579074\pi$
$397$	7.20959i	0.361839i	−0.983498	+	0.180920i	$0.942093\pi$
$398$	0	0
$399$	−0.485281	−0.0242945
$400$	0	0
$401$	15.2660	0.762349	0.381174	−	0.924503i	$-0.375520\pi$
$401$	15.2660	0.762349	0.381174	+	0.924503i	$0.375520\pi$
$402$	0	0
$403$	24.7743i	1.23410i
$404$	0	0
$405$	1.79793i	0.0893400i
$406$	0	0
$407$	45.4792	2.25432
$408$	0	0
$409$	11.3779	0.562603	0.281302	−	0.959619i	$-0.409234\pi$
$409$	11.3779	0.562603	0.281302	+	0.959619i	$0.409234\pi$
$410$	0	0
$411$	− 13.7954i	− 0.680476i
$412$	0	0
$413$	0.899110i	0.0442423i
$414$	0	0
$415$	0.294481	0.0144555
$416$	0	0
$417$	−3.42847	−0.167893
$418$	0	0
$419$	32.9618i	1.61029i	0.593077	+	0.805146i	$0.297913\pi$
$419$	32.9618i	1.61029i	−0.593077	+	0.805146i	$0.702087\pi$
$420$	0	0
$421$	− 24.9119i	− 1.21413i	−0.794652	−	0.607065i	$-0.792347\pi$
$421$	− 24.9119i	− 1.21413i	0.794652	−	0.607065i	$-0.207653\pi$
$422$	0	0
$423$	−2.82843	−0.137523
$424$	0	0
$425$	−5.39635	−0.261761
$426$	0	0
$427$	0.824818i	0.0399157i
$428$	0	0
$429$	− 31.9949i	− 1.54473i
$430$	0	0
$431$	−10.3211	−0.497151	−0.248576	−	0.968612i	$-0.579962\pi$
$431$	−10.3211	−0.497151	−0.248576	+	0.968612i	$0.579962\pi$
$432$	0	0
$433$	15.3137	0.735930	0.367965	−	0.929840i	$-0.380055\pi$
$433$	15.3137	0.735930	0.367965	+	0.929840i	$0.380055\pi$
$434$	0	0
$435$	− 5.33897i	− 0.255984i
$436$	0	0
$437$	8.63577i	0.413105i
$438$	0	0
$439$	−22.5735	−1.07738	−0.538688	−	0.842505i	$-0.681080\pi$
$439$	−22.5735	−1.07738	−0.538688	+	0.842505i	$0.681080\pi$
$440$	0	0
$441$	6.97474	0.332130
$442$	0	0
$443$	33.5334i	1.59322i	0.604494	+	0.796610i	$0.293375\pi$
$443$	33.5334i	1.59322i	−0.604494	+	0.796610i	$0.706625\pi$
$444$	0	0
$445$	− 25.8885i	− 1.22723i
$446$	0	0
$447$	4.14108	0.195866
$448$	0	0
$449$	−1.75506	−0.0828266	−0.0414133	−	0.999142i	$-0.513186\pi$
$449$	−1.75506	−0.0828266	−0.0414133	+	0.999142i	$0.513186\pi$
$450$	0	0
$451$	− 13.9844i	− 0.658501i
$452$	0	0
$453$	22.6644i	1.06487i
$454$	0	0
$455$	−1.70227	−0.0798039
$456$	0	0
$457$	26.7422	1.25095	0.625473	−	0.780246i	$-0.284906\pi$
$457$	26.7422	1.25095	0.625473	+	0.780246i	$0.284906\pi$
$458$	0	0
$459$	3.05320i	0.142511i
$460$	0	0
$461$	− 13.0662i	− 0.608555i	−0.952584	−	0.304277i	$-0.901585\pi$
$461$	− 13.0662i	− 0.608555i	0.952584	−	0.304277i	$-0.0984149\pi$
$462$	0	0
$463$	−29.4474	−1.36854	−0.684268	−	0.729231i	$-0.739878\pi$
$463$	−29.4474	−1.36854	−0.684268	+	0.729231i	$0.739878\pi$
$464$	0	0
$465$	−7.47750	−0.346761
$466$	0	0
$467$	27.7040i	1.28199i	0.767545	+	0.640995i	$0.221478\pi$
$467$	27.7040i	1.28199i	−0.767545	+	0.640995i	$0.778522\pi$
$468$	0	0
$469$	0.172503i	0.00796544i
$470$	0	0
$471$	3.93161	0.181159
$472$	0	0
$473$	43.7127	2.00991
$474$	0	0
$475$	5.39635i	0.247602i
$476$	0	0
$477$	5.03049i	0.230331i
$478$	0	0
$479$	35.5499	1.62432	0.812159	−	0.583436i	$-0.198292\pi$
$479$	35.5499	1.62432	0.812159	+	0.583436i	$0.198292\pi$
$480$	0	0
$481$	50.4393	2.29984
$482$	0	0
$483$	0.449555i	0.0204555i
$484$	0	0
$485$	− 1.02758i	− 0.0466599i
$486$	0	0
$487$	−9.86632	−0.447086	−0.223543	−	0.974694i	$-0.571762\pi$
$487$	−9.86632	−0.447086	−0.223543	+	0.974694i	$0.571762\pi$
$488$	0	0
$489$	7.68897	0.347707
$490$	0	0
$491$	0.635767i	0.0286917i	0.999897	+	0.0143459i	$0.00456659\pi$
$491$	0.635767i	0.0286917i	−0.999897	+	0.0143459i	$0.995433\pi$
$492$	0	0
$493$	− 9.06651i	− 0.408335i
$494$	0	0
$495$	9.65685	0.434043
$496$	0	0
$497$	−0.0505249	−0.00226635
$498$	0	0
$499$	− 3.82750i	− 0.171342i	−0.996323	−	0.0856712i	$-0.972697\pi$
$499$	− 3.82750i	− 0.171342i	0.996323	−	0.0856712i	$-0.0273034\pi$
$500$	0	0
$501$	3.95458i	0.176678i
$502$	0	0
$503$	−23.6719	−1.05548	−0.527739	−	0.849407i	$-0.676960\pi$
$503$	−23.6719	−1.05548	−0.527739	+	0.849407i	$0.676960\pi$
$504$	0	0
$505$	18.1876	0.809336
$506$	0	0
$507$	− 22.4844i	− 0.998565i
$508$	0	0
$509$	− 34.7970i	− 1.54235i	−0.636623	−	0.771175i	$-0.719669\pi$
$509$	− 34.7970i	− 1.54235i	0.636623	−	0.771175i	$-0.280331\pi$
$510$	0	0
$511$	−0.212818	−0.00941453
$512$	0	0
$513$	3.05320	0.134802
$514$	0	0
$515$	20.4077i	0.899273i
$516$	0	0
$517$	15.1917i	0.668132i
$518$	0	0
$519$	−22.6011	−0.992078
$520$	0	0
$521$	14.4889	0.634770	0.317385	−	0.948297i	$-0.397195\pi$
$521$	14.4889	0.634770	0.317385	+	0.948297i	$0.397195\pi$
$522$	0	0
$523$	27.5742i	1.20574i	0.797841	+	0.602868i	$0.205975\pi$
$523$	27.5742i	1.20574i	−0.797841	+	0.602868i	$0.794025\pi$
$524$	0	0
$525$	0.280920i	0.0122603i
$526$	0	0
$527$	−12.6981	−0.553138
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	− 5.65685i	− 0.245487i
$532$	0	0
$533$	− 15.5096i	− 0.671796i
$534$	0	0
$535$	−1.83585	−0.0793706
$536$	0	0
$537$	−17.2981	−0.746470
$538$	0	0
$539$	− 37.4619i	− 1.61360i
$540$	0	0
$541$	− 14.1982i	− 0.610428i	−0.952284	−	0.305214i	$-0.901272\pi$
$541$	− 14.1982i	− 0.610428i	0.952284	−	0.305214i	$-0.0987280\pi$
$542$	0	0
$543$	8.14953	0.349730
$544$	0	0
$545$	−3.67340	−0.157351
$546$	0	0
$547$	− 10.1807i	− 0.435295i	−0.976027	−	0.217648i	$-0.930162\pi$
$547$	− 10.1807i	− 0.435295i	0.976027	−	0.217648i	$-0.0698384\pi$
$548$	0	0
$549$	− 5.18944i	− 0.221480i
$550$	0	0
$551$	−9.06651	−0.386246
$552$	0	0
$553$	1.54075	0.0655194
$554$	0	0
$555$	15.2238i	0.646216i
$556$	0	0
$557$	− 1.44432i	− 0.0611979i	−0.999532	−	0.0305990i	$-0.990259\pi$
$557$	− 1.44432i	− 0.0611979i	0.999532	−	0.0305990i	$-0.00974147\pi$
$558$	0	0
$559$	48.4802	2.05049
$560$	0	0
$561$	16.3990	0.692368
$562$	0	0
$563$	− 9.48585i	− 0.399781i	−0.979818	−	0.199890i	$-0.935941\pi$
$563$	− 9.48585i	− 0.399781i	0.979818	−	0.199890i	$-0.0640586\pi$
$564$	0	0
$565$	6.35547i	0.267377i
$566$	0	0
$567$	0.158942	0.00667492
$568$	0	0
$569$	−8.98711	−0.376759	−0.188380	−	0.982096i	$-0.560324\pi$
$569$	−8.98711	−0.376759	−0.188380	+	0.982096i	$0.560324\pi$
$570$	0	0
$571$	− 12.9706i	− 0.542801i	−0.962466	−	0.271401i	$-0.912513\pi$
$571$	− 12.9706i	− 0.542801i	0.962466	−	0.271401i	$-0.0874868\pi$
$572$	0	0
$573$	− 16.1674i	− 0.675403i
$574$	0	0
$575$	4.99907	0.208476
$576$	0	0
$577$	29.5013	1.22815	0.614077	−	0.789246i	$-0.289528\pi$
$577$	29.5013	1.22815	0.614077	+	0.789246i	$0.289528\pi$
$578$	0	0
$579$	− 22.1454i	− 0.920331i
$580$	0	0
$581$	− 0.0260328i	− 0.00108002i
$582$	0	0
$583$	27.0192	1.11902
$584$	0	0
$585$	10.7101	0.442806
$586$	0	0
$587$	− 2.57988i	− 0.106483i	−0.998582	−	0.0532416i	$-0.983045\pi$
$587$	− 2.57988i	− 0.106483i	0.998582	−	0.0532416i	$-0.0169553\pi$
$588$	0	0
$589$	12.6981i	0.523216i
$590$	0	0
$591$	20.2222	0.831831
$592$	0	0
$593$	35.4338	1.45509	0.727546	−	0.686058i	$-0.240661\pi$
$593$	35.4338	1.45509	0.727546	+	0.686058i	$0.240661\pi$
$594$	0	0
$595$	− 0.872503i	− 0.0357691i
$596$	0	0
$597$	− 25.0075i	− 1.02349i
$598$	0	0
$599$	27.1632	1.10986	0.554930	−	0.831897i	$-0.312745\pi$
$599$	27.1632	1.10986	0.554930	+	0.831897i	$0.312745\pi$
$600$	0	0
$601$	5.33897	0.217781	0.108891	−	0.994054i	$-0.465270\pi$
$601$	5.33897	0.217781	0.108891	+	0.994054i	$0.465270\pi$
$602$	0	0
$603$	− 1.08532i	− 0.0441977i
$604$	0	0
$605$	− 32.0906i	− 1.30467i
$606$	0	0
$607$	16.1084	0.653820	0.326910	−	0.945055i	$-0.393993\pi$
$607$	16.1084	0.653820	0.326910	+	0.945055i	$0.393993\pi$
$608$	0	0
$609$	−0.471978	−0.0191255
$610$	0	0
$611$	16.8486i	0.681621i
$612$	0	0
$613$	0.617903i	0.0249569i	0.999922	+	0.0124784i	$0.00397211\pi$
$613$	0.617903i	0.0249569i	−0.999922	+	0.0124784i	$0.996028\pi$
$614$	0	0
$615$	4.68119	0.188764
$616$	0	0
$617$	−8.80641	−0.354533	−0.177266	−	0.984163i	$-0.556725\pi$
$617$	−8.80641	−0.354533	−0.177266	+	0.984163i	$0.556725\pi$
$618$	0	0
$619$	− 2.72847i	− 0.109666i	−0.998496	−	0.0548332i	$-0.982537\pi$
$619$	− 2.72847i	− 0.109666i	0.998496	−	0.0548332i	$-0.0174627\pi$
$620$	0	0
$621$	− 2.82843i	− 0.113501i
$622$	0	0
$623$	−2.28861	−0.0916910
$624$	0	0
$625$	−13.0390	−0.521559
$626$	0	0
$627$	− 16.3990i	− 0.654914i
$628$	0	0
$629$	25.8528i	1.03082i
$630$	0	0
$631$	−38.7864	−1.54406	−0.772030	−	0.635586i	$-0.780759\pi$
$631$	−38.7864	−1.54406	−0.772030	+	0.635586i	$0.780759\pi$
$632$	0	0
$633$	26.0559	1.03563
$634$	0	0
$635$	− 2.69315i	− 0.106874i
$636$	0	0
$637$	− 41.5476i	− 1.64618i
$638$	0	0
$639$	0.317883	0.0125753
$640$	0	0
$641$	33.1091	1.30773	0.653865	−	0.756611i	$-0.273146\pi$
$641$	33.1091	1.30773	0.653865	+	0.756611i	$0.273146\pi$
$642$	0	0
$643$	− 27.2797i	− 1.07581i	−0.843006	−	0.537904i	$-0.819216\pi$
$643$	− 27.2797i	− 1.07581i	0.843006	−	0.537904i	$-0.180784\pi$
$644$	0	0
$645$	14.6325i	0.576155i
$646$	0	0
$647$	−41.8477	−1.64520	−0.822601	−	0.568620i	$-0.807478\pi$
$647$	−41.8477	−1.64520	−0.822601	+	0.568620i	$0.807478\pi$
$648$	0	0
$649$	−30.3835	−1.19266
$650$	0	0
$651$	0.661029i	0.0259078i
$652$	0	0
$653$	− 20.8937i	− 0.817634i	−0.912616	−	0.408817i	$-0.865941\pi$
$653$	− 20.8937i	− 0.817634i	0.912616	−	0.408817i	$-0.134059\pi$
$654$	0	0
$655$	−26.5054	−1.03565
$656$	0	0
$657$	1.33897	0.0522382
$658$	0	0
$659$	3.15142i	0.122762i	0.998114	+	0.0613809i	$0.0195504\pi$
$659$	3.15142i	0.122762i	−0.998114	+	0.0613809i	$0.980450\pi$
$660$	0	0
$661$	25.5527i	0.993886i	0.867783	+	0.496943i	$0.165544\pi$
$661$	25.5527i	0.993886i	−0.867783	+	0.496943i	$0.834456\pi$
$662$	0	0
$663$	18.1876	0.706346
$664$	0	0
$665$	−0.872503	−0.0338342
$666$	0	0
$667$	8.39903i	0.325212i
$668$	0	0
$669$	18.3465i	0.709317i
$670$	0	0
$671$	−27.8729	−1.07602
$672$	0	0
$673$	20.7981	0.801706	0.400853	−	0.916142i	$-0.368714\pi$
$673$	20.7981	0.801706	0.400853	+	0.916142i	$0.368714\pi$
$674$	0	0
$675$	− 1.76744i	− 0.0680287i
$676$	0	0
$677$	41.0423i	1.57738i	0.614789	+	0.788692i	$0.289241\pi$
$677$	41.0423i	1.57738i	−0.614789	+	0.788692i	$0.710759\pi$
$678$	0	0
$679$	−0.0908404	−0.00348613
$680$	0	0
$681$	0.163788	0.00627639
$682$	0	0
$683$	26.5784i	1.01699i	0.861064	+	0.508497i	$0.169799\pi$
$683$	26.5784i	1.01699i	−0.861064	+	0.508497i	$0.830201\pi$
$684$	0	0
$685$	− 24.8032i	− 0.947680i
$686$	0	0
$687$	−4.02756	−0.153661
$688$	0	0
$689$	29.9660	1.14161
$690$	0	0
$691$	− 14.7899i	− 0.562633i	−0.959615	−	0.281316i	$-0.909229\pi$
$691$	− 14.7899i	− 0.562633i	0.959615	−	0.281316i	$-0.0907710\pi$
$692$	0	0
$693$	− 0.853690i	− 0.0324290i
$694$	0	0
$695$	−6.16415	−0.233820
$696$	0	0
$697$	7.94948	0.301108
$698$	0	0
$699$	11.7211i	0.443332i
$700$	0	0
$701$	25.9245i	0.979155i	0.871960	+	0.489577i	$0.162849\pi$
$701$	25.9245i	0.979155i	−0.871960	+	0.489577i	$0.837151\pi$
$702$	0	0
$703$	25.8528	0.975055
$704$	0	0
$705$	−5.08532	−0.191524
$706$	0	0
$707$	− 1.60782i	− 0.0604685i
$708$	0	0
$709$	20.6082i	0.773958i	0.922089	+	0.386979i	$0.126481\pi$
$709$	20.6082i	0.773958i	−0.922089	+	0.386979i	$0.873519\pi$
$710$	0	0
$711$	−9.69382	−0.363547
$712$	0	0
$713$	11.7633	0.440538
$714$	0	0
$715$	− 57.5247i	− 2.15130i
$716$	0	0
$717$	13.6517i	0.509834i
$718$	0	0
$719$	−44.0949	−1.64446	−0.822230	−	0.569155i	$-0.807270\pi$
$719$	−44.0949	−1.64446	−0.822230	+	0.569155i	$0.807270\pi$
$720$	0	0
$721$	1.80409	0.0671880
$722$	0	0
$723$	− 2.13167i	− 0.0792777i
$724$	0	0
$725$	5.24842i	0.194921i
$726$	0	0
$727$	9.23457	0.342491	0.171246	−	0.985228i	$-0.445221\pi$
$727$	9.23457	0.342491	0.171246	+	0.985228i	$0.445221\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	24.8486i	0.919058i
$732$	0	0
$733$	25.8467i	0.954670i	0.878721	+	0.477335i	$0.158397\pi$
$733$	25.8467i	0.954670i	−0.878721	+	0.477335i	$0.841603\pi$
$734$	0	0
$735$	12.5401	0.462549
$736$	0	0
$737$	−5.82936	−0.214727
$738$	0	0
$739$	24.0403i	0.884337i	0.896932	+	0.442169i	$0.145791\pi$
$739$	24.0403i	0.884337i	−0.896932	+	0.442169i	$0.854209\pi$
$740$	0	0
$741$	− 18.1876i	− 0.668137i
$742$	0	0
$743$	17.8748	0.655762	0.327881	−	0.944719i	$-0.393665\pi$
$743$	17.8748	0.655762	0.327881	+	0.944719i	$0.393665\pi$
$744$	0	0
$745$	7.44538	0.272778
$746$	0	0
$747$	0.163788i	0.00599271i
$748$	0	0
$749$	0.162293i	0.00593007i
$750$	0	0
$751$	35.0731	1.27984	0.639918	−	0.768443i	$-0.278968\pi$
$751$	35.0731	1.27984	0.639918	+	0.768443i	$0.278968\pi$
$752$	0	0
$753$	6.27020	0.228499
$754$	0	0
$755$	40.7490i	1.48301i
$756$	0	0
$757$	46.3962i	1.68630i	0.537679	+	0.843150i	$0.319301\pi$
$757$	46.3962i	1.68630i	−0.537679	+	0.843150i	$0.680699\pi$
$758$	0	0
$759$	−15.1917	−0.551425
$760$	0	0
$761$	10.5531	0.382550	0.191275	−	0.981536i	$-0.438738\pi$
$761$	10.5531	0.382550	0.191275	+	0.981536i	$0.438738\pi$
$762$	0	0
$763$	0.324738i	0.0117563i
$764$	0	0
$765$	5.48946i	0.198472i
$766$	0	0
$767$	−33.6972	−1.21673
$768$	0	0
$769$	−35.2068	−1.26959	−0.634795	−	0.772681i	$-0.718915\pi$
$769$	−35.2068	−1.26959	−0.634795	+	0.772681i	$0.718915\pi$
$770$	0	0
$771$	15.0853i	0.543285i
$772$	0	0
$773$	− 27.4212i	− 0.986271i	−0.869952	−	0.493136i	$-0.835851\pi$
$773$	− 27.4212i	− 0.986271i	0.869952	−	0.493136i	$-0.164149\pi$
$774$	0	0
$775$	7.35067	0.264044
$776$	0	0
$777$	1.34583	0.0482812
$778$	0	0
$779$	− 7.94948i	− 0.284820i
$780$	0	0
$781$	− 1.70738i	− 0.0610948i
$782$	0	0
$783$	2.96951	0.106121
$784$	0	0
$785$	7.06877	0.252295
$786$	0	0
$787$	9.46058i	0.337233i	0.985682	+	0.168617i	$0.0539300\pi$
$787$	9.46058i	0.337233i	−0.985682	+	0.168617i	$0.946070\pi$
$788$	0	0
$789$	26.1706i	0.931700i
$790$	0	0
$791$	0.561839	0.0199767
$792$	0	0
$793$	−30.9128	−1.09775
$794$	0	0
$795$	9.04449i	0.320775i
$796$	0	0
$797$	19.1791i	0.679359i	0.940541	+	0.339680i	$0.110319\pi$
$797$	19.1791i	0.679359i	−0.940541	+	0.339680i	$0.889681\pi$
$798$	0	0
$799$	−8.63577	−0.305511
$800$	0	0
$801$	14.3990	0.508765
$802$	0	0
$803$	− 7.19173i	− 0.253791i
$804$	0	0
$805$	0.808269i	0.0284878i
$806$	0	0
$807$	12.1580	0.427982
$808$	0	0
$809$	43.1578	1.51735	0.758673	−	0.651472i	$-0.225848\pi$
$809$	43.1578	1.51735	0.758673	+	0.651472i	$0.225848\pi$
$810$	0	0
$811$	− 3.87518i	− 0.136076i	−0.997683	−	0.0680380i	$-0.978326\pi$
$811$	− 3.87518i	− 0.136076i	0.997683	−	0.0680380i	$-0.0216739\pi$
$812$	0	0
$813$	− 10.6644i	− 0.374016i
$814$	0	0
$815$	13.8243	0.484242
$816$	0	0
$817$	24.8486	0.869342
$818$	0	0
$819$	− 0.946795i	− 0.0330837i
$820$	0	0
$821$	− 5.62084i	− 0.196169i	−0.995178	−	0.0980843i	$-0.968729\pi$
$821$	− 5.62084i	− 0.196169i	0.995178	−	0.0980843i	$-0.0312715\pi$
$822$	0	0
$823$	38.5255	1.34291	0.671457	−	0.741043i	$-0.265669\pi$
$823$	38.5255	1.34291	0.671457	+	0.741043i	$0.265669\pi$
$824$	0	0
$825$	−9.49307	−0.330506
$826$	0	0
$827$	4.23674i	0.147326i	0.997283	+	0.0736629i	$0.0234689\pi$
$827$	4.23674i	0.147326i	−0.997283	+	0.0736629i	$0.976531\pi$
$828$	0	0
$829$	− 35.3128i	− 1.22646i	−0.789903	−	0.613231i	$-0.789869\pi$
$829$	− 35.3128i	− 1.22646i	0.789903	−	0.613231i	$-0.210131\pi$
$830$	0	0
$831$	−3.76421	−0.130579
$832$	0	0
$833$	21.2953	0.737838
$834$	0	0
$835$	7.11007i	0.246054i
$836$	0	0
$837$	− 4.15894i	− 0.143754i
$838$	0	0
$839$	39.6005	1.36716	0.683580	−	0.729876i	$-0.260422\pi$
$839$	39.6005	1.36716	0.683580	+	0.729876i	$0.260422\pi$
$840$	0	0
$841$	20.1820	0.695932
$842$	0	0
$843$	− 10.4496i	− 0.359902i
$844$	0	0
$845$	− 40.4253i	− 1.39067i
$846$	0	0
$847$	−2.83688	−0.0974765
$848$	0	0
$849$	17.6569	0.605982
$850$	0	0
$851$	− 23.9495i	− 0.820977i
$852$	0	0
$853$	− 10.8683i	− 0.372124i	−0.982538	−	0.186062i	$-0.940428\pi$
$853$	− 10.8683i	− 0.372124i	0.982538	−	0.186062i	$-0.0595725\pi$
$854$	0	0
$855$	5.48946	0.187735
$856$	0	0
$857$	34.0082	1.16170	0.580849	−	0.814011i	$-0.302721\pi$
$857$	34.0082	1.16170	0.580849	+	0.814011i	$0.302721\pi$
$858$	0	0
$859$	12.9973i	0.443463i	0.975108	+	0.221731i	$0.0711708\pi$
$859$	12.9973i	0.443463i	−0.975108	+	0.221731i	$0.928829\pi$
$860$	0	0
$861$	− 0.413828i	− 0.0141032i
$862$	0	0
$863$	−25.8307	−0.879289	−0.439644	−	0.898172i	$-0.644896\pi$
$863$	−25.8307	−0.879289	−0.439644	+	0.898172i	$0.644896\pi$
$864$	0	0
$865$	−40.6353	−1.38164
$866$	0	0
$867$	− 7.67794i	− 0.260757i
$868$	0	0
$869$	52.0663i	1.76623i
$870$	0	0
$871$	−6.46512	−0.219062
$872$	0	0
$873$	0.571533	0.0193435
$874$	0	0
$875$	1.93391i	0.0653780i
$876$	0	0
$877$	− 45.2150i	− 1.52680i	−0.645926	−	0.763400i	$-0.723528\pi$
$877$	− 45.2150i	− 1.52680i	0.645926	−	0.763400i	$-0.276472\pi$
$878$	0	0
$879$	−30.7465	−1.03705
$880$	0	0
$881$	−34.7403	−1.17043	−0.585215	−	0.810878i	$-0.698990\pi$
$881$	−34.7403	−1.17043	−0.585215	+	0.810878i	$0.698990\pi$
$882$	0	0
$883$	48.9366i	1.64685i	0.567427	+	0.823424i	$0.307939\pi$
$883$	48.9366i	1.64685i	−0.567427	+	0.823424i	$0.692061\pi$
$884$	0	0
$885$	− 10.1706i	− 0.341882i
$886$	0	0
$887$	−22.9284	−0.769860	−0.384930	−	0.922946i	$-0.625774\pi$
$887$	−22.9284	−0.769860	−0.384930	+	0.922946i	$0.625774\pi$
$888$	0	0
$889$	−0.238081	−0.00798497
$890$	0	0
$891$	5.37109i	0.179938i
$892$	0	0
$893$	8.63577i	0.288985i
$894$	0	0
$895$	−31.1009	−1.03959
$896$	0	0
$897$	−16.8486	−0.562558
$898$	0	0
$899$	12.3500i	0.411896i
$900$	0	0
$901$	15.3591i	0.511687i
$902$	0	0
$903$	1.29355	0.0430467
$904$	0	0
$905$	14.6523	0.487059
$906$	0	0
$907$	23.1680i	0.769280i	0.923067	+	0.384640i	$0.125674\pi$
$907$	23.1680i	0.769280i	−0.923067	+	0.384640i	$0.874326\pi$
$908$	0	0
$909$	10.1158i	0.335520i
$910$	0	0
$911$	29.4078	0.974324	0.487162	−	0.873312i	$-0.338032\pi$
$911$	29.4078	0.974324	0.487162	+	0.873312i	$0.338032\pi$
$912$	0	0
$913$	0.879722	0.0291146
$914$	0	0
$915$	− 9.33026i	− 0.308449i
$916$	0	0
$917$	2.34315i	0.0773775i
$918$	0	0
$919$	6.86029	0.226300	0.113150	−	0.993578i	$-0.463906\pi$
$919$	6.86029	0.226300	0.113150	+	0.993578i	$0.463906\pi$
$920$	0	0
$921$	21.2981	0.701798
$922$	0	0
$923$	− 1.89359i	− 0.0623283i
$924$	0	0
$925$	− 14.9656i	− 0.492067i
$926$	0	0
$927$	−11.3507	−0.372805
$928$	0	0
$929$	16.1385	0.529488	0.264744	−	0.964319i	$-0.414713\pi$
$929$	16.1385	0.529488	0.264744	+	0.964319i	$0.414713\pi$
$930$	0	0
$931$	− 21.2953i	− 0.697925i
$932$	0	0
$933$	− 1.77883i	− 0.0582363i
$934$	0	0
$935$	29.4844	0.964241
$936$	0	0
$937$	34.7669	1.13579	0.567893	−	0.823102i	$-0.307759\pi$
$937$	34.7669	1.13579	0.567893	+	0.823102i	$0.307759\pi$
$938$	0	0
$939$	2.70320i	0.0882157i
$940$	0	0
$941$	52.7024i	1.71805i	0.511934	+	0.859025i	$0.328929\pi$
$941$	52.7024i	1.71805i	−0.511934	+	0.859025i	$0.671071\pi$
$942$	0	0
$943$	−7.36423	−0.239812
$944$	0	0
$945$	0.285766	0.00929598
$946$	0	0
$947$	− 26.1972i	− 0.851296i	−0.904889	−	0.425648i	$-0.860046\pi$
$947$	− 26.1972i	− 0.851296i	0.904889	−	0.425648i	$-0.139954\pi$
$948$	0	0
$949$	− 7.97608i	− 0.258915i
$950$	0	0
$951$	−22.0653	−0.715516
$952$	0	0
$953$	11.1752	0.362000	0.181000	−	0.983483i	$-0.442067\pi$
$953$	11.1752	0.362000	0.181000	+	0.983483i	$0.442067\pi$
$954$	0	0
$955$	− 29.0679i	− 0.940615i
$956$	0	0
$957$	− 15.9495i	− 0.515573i
$958$	0	0
$959$	−2.19266	−0.0708047
$960$	0	0
$961$	−13.7032	−0.442039
$962$	0	0
$963$	− 1.02109i	− 0.0329041i
$964$	0	0
$965$	− 39.8159i	− 1.28172i
$966$	0	0
$967$	12.8452	0.413075	0.206537	−	0.978439i	$-0.433780\pi$
$967$	12.8452	0.413075	0.206537	+	0.978439i	$0.433780\pi$
$968$	0	0
$969$	9.32206	0.299468
$970$	0	0
$971$	− 8.40774i	− 0.269817i	−0.990858	−	0.134909i	$-0.956926\pi$
$971$	− 8.40774i	− 0.269817i	0.990858	−	0.134909i	$-0.0430741\pi$
$972$	0	0
$973$	0.544926i	0.0174695i
$974$	0	0
$975$	−10.5284	−0.337179
$976$	0	0
$977$	−40.4156	−1.29301	−0.646504	−	0.762910i	$-0.723770\pi$
$977$	−40.4156	−1.29301	−0.646504	+	0.762910i	$0.723770\pi$
$978$	0	0
$979$	− 77.3385i	− 2.47175i
$980$	0	0
$981$	− 2.04313i	− 0.0652320i
$982$	0	0
$983$	13.9202	0.443985	0.221993	−	0.975048i	$-0.428744\pi$
$983$	13.9202	0.443985	0.221993	+	0.975048i	$0.428744\pi$
$984$	0	0
$985$	36.3582	1.15847
$986$	0	0
$987$	0.449555i	0.0143095i
$988$	0	0
$989$	− 23.0192i	− 0.731969i
$990$	0	0
$991$	41.0309	1.30339	0.651695	−	0.758481i	$-0.274058\pi$
$991$	41.0309	1.30339	0.651695	+	0.758481i	$0.274058\pi$
$992$	0	0
$993$	21.8431	0.693169
$994$	0	0
$995$	− 44.9618i	− 1.42539i
$996$	0	0
$997$	38.6427i	1.22383i	0.790925	+	0.611913i	$0.209600\pi$
$997$	38.6427i	1.22383i	−0.790925	+	0.611913i	$0.790400\pi$
$998$	0	0
$999$	−8.46742	−0.267897

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3072.2.d.f.1537.8		8
4.3	odd	2		3072.2.d.i.1537.4		8
8.3	odd	2		3072.2.d.i.1537.5		8
8.5	even	2	inner	3072.2.d.f.1537.1		8
16.3	odd	4		3072.2.a.o.1.4		4
16.5	even	4		3072.2.a.t.1.1		4
16.11	odd	4		3072.2.a.n.1.1		4
16.13	even	4		3072.2.a.i.1.4		4
32.3	odd	8		384.2.j.a.97.3		8
32.5	even	8		384.2.j.b.289.1		8
32.11	odd	8		192.2.j.a.145.2		8
32.13	even	8		48.2.j.a.37.1	yes	8
32.19	odd	8		192.2.j.a.49.2		8
32.21	even	8		48.2.j.a.13.1	✓	8
32.27	odd	8		384.2.j.a.289.3		8
32.29	even	8		384.2.j.b.97.1		8
48.5	odd	4		9216.2.a.y.1.4		4
48.11	even	4		9216.2.a.x.1.4		4
48.29	odd	4		9216.2.a.bo.1.1		4
48.35	even	4		9216.2.a.bn.1.1		4
96.5	odd	8		1152.2.k.c.289.3		8
96.11	even	8		576.2.k.b.145.2		8
96.29	odd	8		1152.2.k.c.865.3		8
96.35	even	8		1152.2.k.f.865.3		8
96.53	odd	8		144.2.k.b.109.4		8
96.59	even	8		1152.2.k.f.289.3		8
96.77	odd	8		144.2.k.b.37.4		8
96.83	even	8		576.2.k.b.433.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.2.j.a.13.1	✓	8	32.21	even	8
48.2.j.a.37.1	yes	8	32.13	even	8
144.2.k.b.37.4		8	96.77	odd	8
144.2.k.b.109.4		8	96.53	odd	8
192.2.j.a.49.2		8	32.19	odd	8
192.2.j.a.145.2		8	32.11	odd	8
384.2.j.a.97.3		8	32.3	odd	8
384.2.j.a.289.3		8	32.27	odd	8
384.2.j.b.97.1		8	32.29	even	8
384.2.j.b.289.1		8	32.5	even	8
576.2.k.b.145.2		8	96.11	even	8
576.2.k.b.433.2		8	96.83	even	8
1152.2.k.c.289.3		8	96.5	odd	8
1152.2.k.c.865.3		8	96.29	odd	8
1152.2.k.f.289.3		8	96.59	even	8
1152.2.k.f.865.3		8	96.35	even	8
3072.2.a.i.1.4		4	16.13	even	4
3072.2.a.n.1.1		4	16.11	odd	4
3072.2.a.o.1.4		4	16.3	odd	4
3072.2.a.t.1.1		4	16.5	even	4
3072.2.d.f.1537.1		8	8.5	even	2	inner
3072.2.d.f.1537.8		8	1.1	even	1	trivial
3072.2.d.i.1537.4		8	4.3	odd	2
3072.2.d.i.1537.5		8	8.3	odd	2
9216.2.a.x.1.4		4	48.11	even	4
9216.2.a.y.1.4		4	48.5	odd	4
9216.2.a.bn.1.1		4	48.35	even	4
9216.2.a.bo.1.1		4	48.29	odd	4

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:	\( N \)	\(=\)	\( 3072 = 2^{10} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3072.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +1.79793i q^{5} +0.158942 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +1.79793i q^{5} +0.158942 q^{7} -1.00000 q^{9} +5.37109i q^{11} +5.95687i q^{13} -1.79793 q^{15} -3.05320 q^{17} +3.05320i q^{19} +0.158942i q^{21} +2.82843 q^{23} +1.76744 q^{25} -1.00000i q^{27} +2.96951i q^{29} +4.15894 q^{31} -5.37109 q^{33} +0.285766i q^{35} -8.46742i q^{37} -5.95687 q^{39} -2.60365 q^{41} -8.13853i q^{43} -1.79793i q^{45} +2.82843 q^{47} -6.97474 q^{49} -3.05320i q^{51} -5.03049i q^{53} -9.65685 q^{55} -3.05320 q^{57} +5.65685i q^{59} +5.18944i q^{61} -0.158942 q^{63} -10.7101 q^{65} +1.08532i q^{67} +2.82843i q^{69} -0.317883 q^{71} -1.33897 q^{73} +1.76744i q^{75} +0.853690i q^{77} +9.69382 q^{79} +1.00000 q^{81} -0.163788i q^{83} -5.48946i q^{85} -2.96951 q^{87} -14.3990 q^{89} +0.946795i q^{91} +4.15894i q^{93} -5.48946 q^{95} -0.571533 q^{97} -5.37109i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{7} - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^7 - 8 * q^9	\( 8 q - 8 q^{7} - 8 q^{9} + 8 q^{15} - 8 q^{25} + 24 q^{31} - 16 q^{39} + 8 q^{49} - 32 q^{55} + 8 q^{63} - 16 q^{65} + 16 q^{71} + 16 q^{73} + 24 q^{79} + 8 q^{81} - 24 q^{87} + 16 q^{89} - 48 q^{95}+O(q^{100}) \) 8 * q - 8 * q^7 - 8 * q^9 + 8 * q^15 - 8 * q^25 + 24 * q^31 - 16 * q^39 + 8 * q^49 - 32 * q^55 + 8 * q^63 - 16 * q^65 + 16 * q^71 + 16 * q^73 + 24 * q^79 + 8 * q^81 - 24 * q^87 + 16 * q^89 - 48 * q^95

Introduction

L-functions

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Database

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