LMFDB - Embedded newform 8016.2.a.x.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.a (trivial)

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:	yes
Analytic conductor:	$64.0080822603$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 28x^{5} + 9x^{4} - 64x^{3} + 17x^{2} + 23x + 1 $ x^8 - 3x^7 - 8x^6 + 28x^5 + 9x^4 - 64x^3 + 17x^2 + 23*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 501)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.71120$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.45529 q^{5} -2.43610 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.45529 q^{5} -2.43610 q^{7} +1.00000 q^{9} -2.22944 q^{11} -1.05815 q^{13} -2.45529 q^{15} +0.227451 q^{17} +2.91787 q^{19} +2.43610 q^{21} -2.55355 q^{23} +1.02846 q^{25} -1.00000 q^{27} -0.104461 q^{29} +3.77043 q^{31} +2.22944 q^{33} -5.98134 q^{35} -0.875391 q^{37} +1.05815 q^{39} +7.02942 q^{41} +1.10345 q^{43} +2.45529 q^{45} -0.498974 q^{47} -1.06541 q^{49} -0.227451 q^{51} +5.62512 q^{53} -5.47392 q^{55} -2.91787 q^{57} +2.62950 q^{59} +4.56007 q^{61} -2.43610 q^{63} -2.59807 q^{65} -9.29821 q^{67} +2.55355 q^{69} -7.68037 q^{71} -9.24479 q^{73} -1.02846 q^{75} +5.43113 q^{77} +1.46616 q^{79} +1.00000 q^{81} +7.40427 q^{83} +0.558459 q^{85} +0.104461 q^{87} -1.00920 q^{89} +2.57776 q^{91} -3.77043 q^{93} +7.16422 q^{95} +4.09308 q^{97} -2.22944 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} + 7 q^{5} + 4 q^{7} + 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^3 + 7 * q^5 + 4 * q^7 + 8 * q^9	$ 8 q - 8 q^{3} + 7 q^{5} + 4 q^{7} + 8 q^{9} - 13 q^{11} - 7 q^{15} + 11 q^{17} - 12 q^{19} - 4 q^{21} - 7 q^{23} - 5 q^{25} - 8 q^{27} + q^{29} + 2 q^{31} + 13 q^{33} + 4 q^{35} - 9 q^{37} + 4 q^{41} - 2 q^{43} + 7 q^{45} - 17 q^{47} - 2 q^{49} - 11 q^{51} + 9 q^{53} - 7 q^{55} + 12 q^{57} - 29 q^{59} - 12 q^{61} + 4 q^{63} + 8 q^{65} + 7 q^{69} - 13 q^{71} - 20 q^{73} + 5 q^{75} - 22 q^{77} - 8 q^{79} + 8 q^{81} - 33 q^{83} - 31 q^{85} - q^{87} + 4 q^{89} - q^{91} - 2 q^{93} - 3 q^{95} - 31 q^{97} - 13 q^{99}+O(q^{100}) $ 8 * q - 8 * q^3 + 7 * q^5 + 4 * q^7 + 8 * q^9 - 13 * q^11 - 7 * q^15 + 11 * q^17 - 12 * q^19 - 4 * q^21 - 7 * q^23 - 5 * q^25 - 8 * q^27 + q^29 + 2 * q^31 + 13 * q^33 + 4 * q^35 - 9 * q^37 + 4 * q^41 - 2 * q^43 + 7 * q^45 - 17 * q^47 - 2 * q^49 - 11 * q^51 + 9 * q^53 - 7 * q^55 + 12 * q^57 - 29 * q^59 - 12 * q^61 + 4 * q^63 + 8 * q^65 + 7 * q^69 - 13 * q^71 - 20 * q^73 + 5 * q^75 - 22 * q^77 - 8 * q^79 + 8 * q^81 - 33 * q^83 - 31 * q^85 - q^87 + 4 * q^89 - q^91 - 2 * q^93 - 3 * q^95 - 31 * q^97 - 13 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	2.45529	1.09804	0.549020	−	0.835809i	$-0.315001\pi$
$5$	2.45529	1.09804	0.549020	+	0.835809i	$0.315001\pi$
$6$	0	0
$7$	−2.43610	−0.920759	−0.460380	−	0.887722i	$-0.652287\pi$
$7$	−2.43610	−0.920759	−0.460380	+	0.887722i	$0.652287\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.22944	−0.672200	−0.336100	−	0.941826i	$-0.609108\pi$
$11$	−2.22944	−0.672200	−0.336100	+	0.941826i	$0.609108\pi$
$12$	0	0
$13$	−1.05815	−0.293478	−0.146739	−	0.989175i	$-0.546878\pi$
$13$	−1.05815	−0.293478	−0.146739	+	0.989175i	$0.546878\pi$
$14$	0	0
$15$	−2.45529	−0.633954
$16$	0	0
$17$	0.227451	0.0551650	0.0275825	−	0.999620i	$-0.491219\pi$
$17$	0.227451	0.0551650	0.0275825	+	0.999620i	$0.491219\pi$
$18$	0	0
$19$	2.91787	0.669405	0.334702	−	0.942324i	$-0.391364\pi$
$19$	2.91787	0.669405	0.334702	+	0.942324i	$0.391364\pi$
$20$	0	0
$21$	2.43610	0.531601
$22$	0	0
$23$	−2.55355	−0.532451	−0.266226	−	0.963911i	$-0.585777\pi$
$23$	−2.55355	−0.532451	−0.266226	+	0.963911i	$0.585777\pi$
$24$	0	0
$25$	1.02846	0.205692
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−0.104461	−0.0193979	−0.00969894	−	0.999953i	$-0.503087\pi$
$29$	−0.104461	−0.0193979	−0.00969894	+	0.999953i	$0.503087\pi$
$30$	0	0
$31$	3.77043	0.677189	0.338594	−	0.940932i	$-0.390049\pi$
$31$	3.77043	0.677189	0.338594	+	0.940932i	$0.390049\pi$
$32$	0	0
$33$	2.22944	0.388095
$34$	0	0
$35$	−5.98134	−1.01103
$36$	0	0
$37$	−0.875391	−0.143913	−0.0719567	−	0.997408i	$-0.522924\pi$
$37$	−0.875391	−0.143913	−0.0719567	+	0.997408i	$0.522924\pi$
$38$	0	0
$39$	1.05815	0.169440
$40$	0	0
$41$	7.02942	1.09781	0.548906	−	0.835884i	$-0.315045\pi$
$41$	7.02942	1.09781	0.548906	+	0.835884i	$0.315045\pi$
$42$	0	0
$43$	1.10345	0.168275	0.0841376	−	0.996454i	$-0.473186\pi$
$43$	1.10345	0.168275	0.0841376	+	0.996454i	$0.473186\pi$
$44$	0	0
$45$	2.45529	0.366013
$46$	0	0
$47$	−0.498974	−0.0727828	−0.0363914	−	0.999338i	$-0.511586\pi$
$47$	−0.498974	−0.0727828	−0.0363914	+	0.999338i	$0.511586\pi$
$48$	0	0
$49$	−1.06541	−0.152202
$50$	0	0
$51$	−0.227451	−0.0318495
$52$	0	0
$53$	5.62512	0.772670	0.386335	−	0.922358i	$-0.373741\pi$
$53$	5.62512	0.772670	0.386335	+	0.922358i	$0.373741\pi$
$54$	0	0
$55$	−5.47392	−0.738103
$56$	0	0
$57$	−2.91787	−0.386481
$58$	0	0
$59$	2.62950	0.342332	0.171166	−	0.985242i	$-0.445247\pi$
$59$	2.62950	0.342332	0.171166	+	0.985242i	$0.445247\pi$
$60$	0	0
$61$	4.56007	0.583857	0.291929	−	0.956440i	$-0.405703\pi$
$61$	4.56007	0.583857	0.291929	+	0.956440i	$0.405703\pi$
$62$	0	0
$63$	−2.43610	−0.306920
$64$	0	0
$65$	−2.59807	−0.322251
$66$	0	0
$67$	−9.29821	−1.13596	−0.567979	−	0.823043i	$-0.692274\pi$
$67$	−9.29821	−1.13596	−0.567979	+	0.823043i	$0.692274\pi$
$68$	0	0
$69$	2.55355	0.307411
$70$	0	0
$71$	−7.68037	−0.911492	−0.455746	−	0.890110i	$-0.650627\pi$
$71$	−7.68037	−0.911492	−0.455746	+	0.890110i	$0.650627\pi$
$72$	0	0
$73$	−9.24479	−1.08202	−0.541010	−	0.841016i	$-0.681958\pi$
$73$	−9.24479	−1.08202	−0.541010	+	0.841016i	$0.681958\pi$
$74$	0	0
$75$	−1.02846	−0.118756
$76$	0	0
$77$	5.43113	0.618935
$78$	0	0
$79$	1.46616	0.164956	0.0824782	−	0.996593i	$-0.473717\pi$
$79$	1.46616	0.164956	0.0824782	+	0.996593i	$0.473717\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.40427	0.812724	0.406362	−	0.913712i	$-0.366797\pi$
$83$	7.40427	0.812724	0.406362	+	0.913712i	$0.366797\pi$
$84$	0	0
$85$	0.558459	0.0605734
$86$	0	0
$87$	0.104461	0.0111994
$88$	0	0
$89$	−1.00920	−0.106975	−0.0534876	−	0.998569i	$-0.517034\pi$
$89$	−1.00920	−0.106975	−0.0534876	+	0.998569i	$0.517034\pi$
$90$	0	0
$91$	2.57776	0.270223
$92$	0	0
$93$	−3.77043	−0.390975
$94$	0	0
$95$	7.16422	0.735033
$96$	0	0
$97$	4.09308	0.415590	0.207795	−	0.978172i	$-0.433371\pi$
$97$	4.09308	0.415590	0.207795	+	0.978172i	$0.433371\pi$
$98$	0	0
$99$	−2.22944	−0.224067
$100$	0	0
$101$	−10.8700	−1.08161	−0.540803	−	0.841149i	$-0.681880\pi$
$101$	−10.8700	−1.08161	−0.540803	+	0.841149i	$0.681880\pi$
$102$	0	0
$103$	−2.77593	−0.273520	−0.136760	−	0.990604i	$-0.543669\pi$
$103$	−2.77593	−0.273520	−0.136760	+	0.990604i	$0.543669\pi$
$104$	0	0
$105$	5.98134	0.583719
$106$	0	0
$107$	3.73609	0.361181	0.180591	−	0.983558i	$-0.442199\pi$
$107$	3.73609	0.361181	0.180591	+	0.983558i	$0.442199\pi$
$108$	0	0
$109$	−7.19079	−0.688753	−0.344377	−	0.938832i	$-0.611910\pi$
$109$	−7.19079	−0.688753	−0.344377	+	0.938832i	$0.611910\pi$
$110$	0	0
$111$	0.875391	0.0830884
$112$	0	0
$113$	−8.87609	−0.834992	−0.417496	−	0.908679i	$-0.637092\pi$
$113$	−8.87609	−0.834992	−0.417496	+	0.908679i	$0.637092\pi$
$114$	0	0
$115$	−6.26971	−0.584653
$116$	0	0
$117$	−1.05815	−0.0978260
$118$	0	0
$119$	−0.554094	−0.0507937
$120$	0	0
$121$	−6.02961	−0.548147
$122$	0	0
$123$	−7.02942	−0.633822
$124$	0	0
$125$	−9.75129	−0.872182
$126$	0	0
$127$	20.1608	1.78898	0.894490	−	0.447088i	$-0.147539\pi$
$127$	20.1608	1.78898	0.894490	+	0.447088i	$0.147539\pi$
$128$	0	0
$129$	−1.10345	−0.0971538
$130$	0	0
$131$	−19.1565	−1.67371	−0.836855	−	0.547425i	$-0.815608\pi$
$131$	−19.1565	−1.67371	−0.836855	+	0.547425i	$0.815608\pi$
$132$	0	0
$133$	−7.10822	−0.616361
$134$	0	0
$135$	−2.45529	−0.211318
$136$	0	0
$137$	−7.68627	−0.656683	−0.328341	−	0.944559i	$-0.606490\pi$
$137$	−7.68627	−0.656683	−0.328341	+	0.944559i	$0.606490\pi$
$138$	0	0
$139$	−13.4734	−1.14280	−0.571400	−	0.820672i	$-0.693600\pi$
$139$	−13.4734	−1.14280	−0.571400	+	0.820672i	$0.693600\pi$
$140$	0	0
$141$	0.498974	0.0420212
$142$	0	0
$143$	2.35908	0.197276
$144$	0	0
$145$	−0.256482	−0.0212996
$146$	0	0
$147$	1.06541	0.0878739
$148$	0	0
$149$	−23.0126	−1.88527	−0.942634	−	0.333827i	$-0.891660\pi$
$149$	−23.0126	−1.88527	−0.942634	+	0.333827i	$0.891660\pi$
$150$	0	0
$151$	−0.0519514	−0.00422774	−0.00211387	−	0.999998i	$-0.500673\pi$
$151$	−0.0519514	−0.00422774	−0.00211387	+	0.999998i	$0.500673\pi$
$152$	0	0
$153$	0.227451	0.0183883
$154$	0	0
$155$	9.25750	0.743580
$156$	0	0
$157$	11.8201	0.943349	0.471674	−	0.881773i	$-0.343650\pi$
$157$	11.8201	0.943349	0.471674	+	0.881773i	$0.343650\pi$
$158$	0	0
$159$	−5.62512	−0.446101
$160$	0	0
$161$	6.22070	0.490260
$162$	0	0
$163$	21.3895	1.67535	0.837676	−	0.546168i	$-0.183914\pi$
$163$	21.3895	1.67535	0.837676	+	0.546168i	$0.183914\pi$
$164$	0	0
$165$	5.47392	0.426144
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−11.8803	−0.913871
$170$	0	0
$171$	2.91787	0.223135
$172$	0	0
$173$	6.00927	0.456877	0.228438	−	0.973558i	$-0.426638\pi$
$173$	6.00927	0.456877	0.228438	+	0.973558i	$0.426638\pi$
$174$	0	0
$175$	−2.50543	−0.189393
$176$	0	0
$177$	−2.62950	−0.197645
$178$	0	0
$179$	−3.93208	−0.293898	−0.146949	−	0.989144i	$-0.546945\pi$
$179$	−3.93208	−0.293898	−0.146949	+	0.989144i	$0.546945\pi$
$180$	0	0
$181$	4.82100	0.358342	0.179171	−	0.983818i	$-0.442658\pi$
$181$	4.82100	0.358342	0.179171	+	0.983818i	$0.442658\pi$
$182$	0	0
$183$	−4.56007	−0.337090
$184$	0	0
$185$	−2.14934	−0.158023
$186$	0	0
$187$	−0.507088	−0.0370819
$188$	0	0
$189$	2.43610	0.177200
$190$	0	0
$191$	−21.5786	−1.56137	−0.780685	−	0.624925i	$-0.785130\pi$
$191$	−21.5786	−1.56137	−0.780685	+	0.624925i	$0.785130\pi$
$192$	0	0
$193$	17.8687	1.28622	0.643108	−	0.765775i	$-0.277644\pi$
$193$	17.8687	1.28622	0.643108	+	0.765775i	$0.277644\pi$
$194$	0	0
$195$	2.59807	0.186052
$196$	0	0
$197$	−11.7311	−0.835805	−0.417902	−	0.908492i	$-0.637235\pi$
$197$	−11.7311	−0.835805	−0.417902	+	0.908492i	$0.637235\pi$
$198$	0	0
$199$	1.40194	0.0993807	0.0496903	−	0.998765i	$-0.484177\pi$
$199$	1.40194	0.0993807	0.0496903	+	0.998765i	$0.484177\pi$
$200$	0	0
$201$	9.29821	0.655845
$202$	0	0
$203$	0.254477	0.0178608
$204$	0	0
$205$	17.2593	1.20544
$206$	0	0
$207$	−2.55355	−0.177484
$208$	0	0
$209$	−6.50520	−0.449974
$210$	0	0
$211$	−25.7907	−1.77550	−0.887751	−	0.460324i	$-0.847733\pi$
$211$	−25.7907	−1.77550	−0.887751	+	0.460324i	$0.847733\pi$
$212$	0	0
$213$	7.68037	0.526250
$214$	0	0
$215$	2.70930	0.184773
$216$	0	0
$217$	−9.18514	−0.623528
$218$	0	0
$219$	9.24479	0.624705
$220$	0	0
$221$	−0.240677	−0.0161897
$222$	0	0
$223$	−8.89491	−0.595647	−0.297824	−	0.954621i	$-0.596261\pi$
$223$	−8.89491	−0.595647	−0.297824	+	0.954621i	$0.596261\pi$
$224$	0	0
$225$	1.02846	0.0685641
$226$	0	0
$227$	15.3041	1.01577	0.507884	−	0.861425i	$-0.330428\pi$
$227$	15.3041	1.01577	0.507884	+	0.861425i	$0.330428\pi$
$228$	0	0
$229$	7.40079	0.489058	0.244529	−	0.969642i	$-0.421367\pi$
$229$	7.40079	0.489058	0.244529	+	0.969642i	$0.421367\pi$
$230$	0	0
$231$	−5.43113	−0.357342
$232$	0	0
$233$	−24.6200	−1.61291	−0.806456	−	0.591295i	$-0.798617\pi$
$233$	−24.6200	−1.61291	−0.806456	+	0.591295i	$0.798617\pi$
$234$	0	0
$235$	−1.22513	−0.0799185
$236$	0	0
$237$	−1.46616	−0.0952376
$238$	0	0
$239$	−11.7362	−0.759149	−0.379574	−	0.925161i	$-0.623930\pi$
$239$	−11.7362	−0.759149	−0.379574	+	0.925161i	$0.623930\pi$
$240$	0	0
$241$	−18.1406	−1.16854	−0.584268	−	0.811561i	$-0.698618\pi$
$241$	−18.1406	−1.16854	−0.584268	+	0.811561i	$0.698618\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−2.61590	−0.167124
$246$	0	0
$247$	−3.08754	−0.196456
$248$	0	0
$249$	−7.40427	−0.469227
$250$	0	0
$251$	−6.88907	−0.434834	−0.217417	−	0.976079i	$-0.569763\pi$
$251$	−6.88907	−0.434834	−0.217417	+	0.976079i	$0.569763\pi$
$252$	0	0
$253$	5.69297	0.357914
$254$	0	0
$255$	−0.558459	−0.0349721
$256$	0	0
$257$	−0.160088	−0.00998602	−0.00499301	−	0.999988i	$-0.501589\pi$
$257$	−0.160088	−0.00998602	−0.00499301	+	0.999988i	$0.501589\pi$
$258$	0	0
$259$	2.13254	0.132510
$260$	0	0
$261$	−0.104461	−0.00646596
$262$	0	0
$263$	−7.63433	−0.470753	−0.235376	−	0.971904i	$-0.575632\pi$
$263$	−7.63433	−0.470753	−0.235376	+	0.971904i	$0.575632\pi$
$264$	0	0
$265$	13.8113	0.848423
$266$	0	0
$267$	1.00920	0.0617622
$268$	0	0
$269$	−7.21293	−0.439780	−0.219890	−	0.975525i	$-0.570570\pi$
$269$	−7.21293	−0.439780	−0.219890	+	0.975525i	$0.570570\pi$
$270$	0	0
$271$	−0.0760078	−0.00461715	−0.00230857	−	0.999997i	$-0.500735\pi$
$271$	−0.0760078	−0.00461715	−0.00230857	+	0.999997i	$0.500735\pi$
$272$	0	0
$273$	−2.57776	−0.156013
$274$	0	0
$275$	−2.29289	−0.138266
$276$	0	0
$277$	−27.9255	−1.67788	−0.838939	−	0.544225i	$-0.816824\pi$
$277$	−27.9255	−1.67788	−0.838939	+	0.544225i	$0.816824\pi$
$278$	0	0
$279$	3.77043	0.225730
$280$	0	0
$281$	16.5725	0.988633	0.494317	−	0.869282i	$-0.335418\pi$
$281$	16.5725	0.988633	0.494317	+	0.869282i	$0.335418\pi$
$282$	0	0
$283$	−1.06715	−0.0634357	−0.0317178	−	0.999497i	$-0.510098\pi$
$283$	−1.06715	−0.0634357	−0.0317178	+	0.999497i	$0.510098\pi$
$284$	0	0
$285$	−7.16422	−0.424372
$286$	0	0
$287$	−17.1244	−1.01082
$288$	0	0
$289$	−16.9483	−0.996957
$290$	0	0
$291$	−4.09308	−0.239941
$292$	0	0
$293$	14.3187	0.836509	0.418255	−	0.908330i	$-0.362642\pi$
$293$	14.3187	0.836509	0.418255	+	0.908330i	$0.362642\pi$
$294$	0	0
$295$	6.45620	0.375894
$296$	0	0
$297$	2.22944	0.129365
$298$	0	0
$299$	2.70204	0.156263
$300$	0	0
$301$	−2.68813	−0.154941
$302$	0	0
$303$	10.8700	0.624465
$304$	0	0
$305$	11.1963	0.641099
$306$	0	0
$307$	32.7561	1.86949	0.934744	−	0.355322i	$-0.115629\pi$
$307$	32.7561	1.86949	0.934744	+	0.355322i	$0.115629\pi$
$308$	0	0
$309$	2.77593	0.157917
$310$	0	0
$311$	−10.4071	−0.590133	−0.295067	−	0.955477i	$-0.595342\pi$
$311$	−10.4071	−0.590133	−0.295067	+	0.955477i	$0.595342\pi$
$312$	0	0
$313$	6.83190	0.386162	0.193081	−	0.981183i	$-0.438152\pi$
$313$	6.83190	0.386162	0.193081	+	0.981183i	$0.438152\pi$
$314$	0	0
$315$	−5.98134	−0.337010
$316$	0	0
$317$	25.1071	1.41015	0.705077	−	0.709131i	$-0.250913\pi$
$317$	25.1071	1.41015	0.705077	+	0.709131i	$0.250913\pi$
$318$	0	0
$319$	0.232889	0.0130393
$320$	0	0
$321$	−3.73609	−0.208528
$322$	0	0
$323$	0.663672	0.0369277
$324$	0	0
$325$	−1.08827	−0.0603661
$326$	0	0
$327$	7.19079	0.397652
$328$	0	0
$329$	1.21555	0.0670155
$330$	0	0
$331$	−12.6726	−0.696547	−0.348274	−	0.937393i	$-0.613232\pi$
$331$	−12.6726	−0.696547	−0.348274	+	0.937393i	$0.613232\pi$
$332$	0	0
$333$	−0.875391	−0.0479711
$334$	0	0
$335$	−22.8298	−1.24733
$336$	0	0
$337$	−12.4032	−0.675645	−0.337822	−	0.941210i	$-0.609690\pi$
$337$	−12.4032	−0.675645	−0.337822	+	0.941210i	$0.609690\pi$
$338$	0	0
$339$	8.87609	0.482083
$340$	0	0
$341$	−8.40593	−0.455207
$342$	0	0
$343$	19.6482	1.06090
$344$	0	0
$345$	6.26971	0.337550
$346$	0	0
$347$	26.8602	1.44193	0.720967	−	0.692970i	$-0.243698\pi$
$347$	26.8602	1.44193	0.720967	+	0.692970i	$0.243698\pi$
$348$	0	0
$349$	8.59105	0.459869	0.229934	−	0.973206i	$-0.426149\pi$
$349$	8.59105	0.459869	0.229934	+	0.973206i	$0.426149\pi$
$350$	0	0
$351$	1.05815	0.0564799
$352$	0	0
$353$	−2.80330	−0.149205	−0.0746024	−	0.997213i	$-0.523769\pi$
$353$	−2.80330	−0.149205	−0.0746024	+	0.997213i	$0.523769\pi$
$354$	0	0
$355$	−18.8575	−1.00085
$356$	0	0
$357$	0.554094	0.0293258
$358$	0	0
$359$	18.6314	0.983328	0.491664	−	0.870785i	$-0.336389\pi$
$359$	18.6314	0.983328	0.491664	+	0.870785i	$0.336389\pi$
$360$	0	0
$361$	−10.4860	−0.551897
$362$	0	0
$363$	6.02961	0.316473
$364$	0	0
$365$	−22.6987	−1.18810
$366$	0	0
$367$	12.7297	0.664484	0.332242	−	0.943194i	$-0.392195\pi$
$367$	12.7297	0.664484	0.332242	+	0.943194i	$0.392195\pi$
$368$	0	0
$369$	7.02942	0.365937
$370$	0	0
$371$	−13.7034	−0.711443
$372$	0	0
$373$	−35.4324	−1.83462	−0.917310	−	0.398173i	$-0.869644\pi$
$373$	−35.4324	−1.83462	−0.917310	+	0.398173i	$0.869644\pi$
$374$	0	0
$375$	9.75129	0.503554
$376$	0	0
$377$	0.110535	0.00569285
$378$	0	0
$379$	3.04025	0.156167	0.0780835	−	0.996947i	$-0.475120\pi$
$379$	3.04025	0.156167	0.0780835	+	0.996947i	$0.475120\pi$
$380$	0	0
$381$	−20.1608	−1.03287
$382$	0	0
$383$	27.3117	1.39556	0.697782	−	0.716310i	$-0.254170\pi$
$383$	27.3117	1.39556	0.697782	+	0.716310i	$0.254170\pi$
$384$	0	0
$385$	13.3350	0.679615
$386$	0	0
$387$	1.10345	0.0560918
$388$	0	0
$389$	−12.3862	−0.628008	−0.314004	−	0.949422i	$-0.601670\pi$
$389$	−12.3862	−0.628008	−0.314004	+	0.949422i	$0.601670\pi$
$390$	0	0
$391$	−0.580807	−0.0293727
$392$	0	0
$393$	19.1565	0.966317
$394$	0	0
$395$	3.59986	0.181129
$396$	0	0
$397$	−23.5051	−1.17969	−0.589843	−	0.807518i	$-0.700810\pi$
$397$	−23.5051	−1.17969	−0.589843	+	0.807518i	$0.700810\pi$
$398$	0	0
$399$	7.10822	0.355856
$400$	0	0
$401$	32.0591	1.60096	0.800479	−	0.599361i	$-0.204579\pi$
$401$	32.0591	1.60096	0.800479	+	0.599361i	$0.204579\pi$
$402$	0	0
$403$	−3.98968	−0.198740
$404$	0	0
$405$	2.45529	0.122004
$406$	0	0
$407$	1.95163	0.0967386
$408$	0	0
$409$	−35.2311	−1.74207	−0.871033	−	0.491225i	$-0.836549\pi$
$409$	−35.2311	−1.74207	−0.871033	+	0.491225i	$0.836549\pi$
$410$	0	0
$411$	7.68627	0.379136
$412$	0	0
$413$	−6.40573	−0.315205
$414$	0	0
$415$	18.1796	0.892404
$416$	0	0
$417$	13.4734	0.659796
$418$	0	0
$419$	−32.9143	−1.60797	−0.803984	−	0.594651i	$-0.797290\pi$
$419$	−32.9143	−1.60797	−0.803984	+	0.594651i	$0.797290\pi$
$420$	0	0
$421$	0.654744	0.0319103	0.0159551	−	0.999873i	$-0.494921\pi$
$421$	0.654744	0.0319103	0.0159551	+	0.999873i	$0.494921\pi$
$422$	0	0
$423$	−0.498974	−0.0242609
$424$	0	0
$425$	0.233925	0.0113470
$426$	0	0
$427$	−11.1088	−0.537592
$428$	0	0
$429$	−2.35908	−0.113897
$430$	0	0
$431$	−12.9285	−0.622746	−0.311373	−	0.950288i	$-0.600789\pi$
$431$	−12.9285	−0.622746	−0.311373	+	0.950288i	$0.600789\pi$
$432$	0	0
$433$	20.8493	1.00195	0.500976	−	0.865461i	$-0.332974\pi$
$433$	20.8493	1.00195	0.500976	+	0.865461i	$0.332974\pi$
$434$	0	0
$435$	0.256482	0.0122974
$436$	0	0
$437$	−7.45092	−0.356426
$438$	0	0
$439$	0.618186	0.0295044	0.0147522	−	0.999891i	$-0.495304\pi$
$439$	0.618186	0.0295044	0.0147522	+	0.999891i	$0.495304\pi$
$440$	0	0
$441$	−1.06541	−0.0507340
$442$	0	0
$443$	−2.14180	−0.101760	−0.0508800	−	0.998705i	$-0.516203\pi$
$443$	−2.14180	−0.101760	−0.0508800	+	0.998705i	$0.516203\pi$
$444$	0	0
$445$	−2.47789	−0.117463
$446$	0	0
$447$	23.0126	1.08846
$448$	0	0
$449$	29.8628	1.40931	0.704657	−	0.709548i	$-0.251101\pi$
$449$	29.8628	1.40931	0.704657	+	0.709548i	$0.251101\pi$
$450$	0	0
$451$	−15.6716	−0.737949
$452$	0	0
$453$	0.0519514	0.00244089
$454$	0	0
$455$	6.32915	0.296715
$456$	0	0
$457$	−26.8492	−1.25595	−0.627977	−	0.778232i	$-0.716117\pi$
$457$	−26.8492	−1.25595	−0.627977	+	0.778232i	$0.716117\pi$
$458$	0	0
$459$	−0.227451	−0.0106165
$460$	0	0
$461$	−35.5814	−1.65719	−0.828595	−	0.559848i	$-0.810860\pi$
$461$	−35.5814	−1.65719	−0.828595	+	0.559848i	$0.810860\pi$
$462$	0	0
$463$	−27.8198	−1.29290	−0.646448	−	0.762958i	$-0.723746\pi$
$463$	−27.8198	−1.29290	−0.646448	+	0.762958i	$0.723746\pi$
$464$	0	0
$465$	−9.25750	−0.429306
$466$	0	0
$467$	36.8683	1.70606	0.853030	−	0.521861i	$-0.174762\pi$
$467$	36.8683	1.70606	0.853030	+	0.521861i	$0.174762\pi$
$468$	0	0
$469$	22.6514	1.04594
$470$	0	0
$471$	−11.8201	−0.544643
$472$	0	0
$473$	−2.46008	−0.113115
$474$	0	0
$475$	3.00091	0.137691
$476$	0	0
$477$	5.62512	0.257557
$478$	0	0
$479$	−8.20487	−0.374890	−0.187445	−	0.982275i	$-0.560021\pi$
$479$	−8.20487	−0.374890	−0.187445	+	0.982275i	$0.560021\pi$
$480$	0	0
$481$	0.926295	0.0422354
$482$	0	0
$483$	−6.22070	−0.283052
$484$	0	0
$485$	10.0497	0.456334
$486$	0	0
$487$	−16.8096	−0.761716	−0.380858	−	0.924633i	$-0.624371\pi$
$487$	−16.8096	−0.761716	−0.380858	+	0.924633i	$0.624371\pi$
$488$	0	0
$489$	−21.3895	−0.967265
$490$	0	0
$491$	−35.5650	−1.60503	−0.802513	−	0.596635i	$-0.796504\pi$
$491$	−35.5650	−1.60503	−0.802513	+	0.596635i	$0.796504\pi$
$492$	0	0
$493$	−0.0237597	−0.00107008
$494$	0	0
$495$	−5.47392	−0.246034
$496$	0	0
$497$	18.7101	0.839265
$498$	0	0
$499$	−16.1401	−0.722531	−0.361266	−	0.932463i	$-0.617655\pi$
$499$	−16.1401	−0.722531	−0.361266	+	0.932463i	$0.617655\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	5.79335	0.258313	0.129156	−	0.991624i	$-0.458773\pi$
$503$	5.79335	0.258313	0.129156	+	0.991624i	$0.458773\pi$
$504$	0	0
$505$	−26.6890	−1.18765
$506$	0	0
$507$	11.8803	0.527623
$508$	0	0
$509$	1.08693	0.0481774	0.0240887	−	0.999710i	$-0.492332\pi$
$509$	1.08693	0.0481774	0.0240887	+	0.999710i	$0.492332\pi$
$510$	0	0
$511$	22.5212	0.996281
$512$	0	0
$513$	−2.91787	−0.128827
$514$	0	0
$515$	−6.81571	−0.300336
$516$	0	0
$517$	1.11243	0.0489247
$518$	0	0
$519$	−6.00927	−0.263778
$520$	0	0
$521$	10.1451	0.444466	0.222233	−	0.974994i	$-0.428665\pi$
$521$	10.1451	0.444466	0.222233	+	0.974994i	$0.428665\pi$
$522$	0	0
$523$	−18.5798	−0.812439	−0.406219	−	0.913776i	$-0.633153\pi$
$523$	−18.5798	−0.812439	−0.406219	+	0.913776i	$0.633153\pi$
$524$	0	0
$525$	2.50543	0.109346
$526$	0	0
$527$	0.857588	0.0373571
$528$	0	0
$529$	−16.4794	−0.716495
$530$	0	0
$531$	2.62950	0.114111
$532$	0	0
$533$	−7.43818	−0.322183
$534$	0	0
$535$	9.17319	0.396592
$536$	0	0
$537$	3.93208	0.169682
$538$	0	0
$539$	2.37527	0.102310
$540$	0	0
$541$	−6.48542	−0.278830	−0.139415	−	0.990234i	$-0.544522\pi$
$541$	−6.48542	−0.278830	−0.139415	+	0.990234i	$0.544522\pi$
$542$	0	0
$543$	−4.82100	−0.206889
$544$	0	0
$545$	−17.6555	−0.756279
$546$	0	0
$547$	−45.0069	−1.92436	−0.962178	−	0.272423i	$-0.912175\pi$
$547$	−45.0069	−1.92436	−0.962178	+	0.272423i	$0.912175\pi$
$548$	0	0
$549$	4.56007	0.194619
$550$	0	0
$551$	−0.304803	−0.0129850
$552$	0	0
$553$	−3.57172	−0.151885
$554$	0	0
$555$	2.14934	0.0912344
$556$	0	0
$557$	32.6080	1.38164	0.690822	−	0.723025i	$-0.257249\pi$
$557$	32.6080	1.38164	0.690822	+	0.723025i	$0.257249\pi$
$558$	0	0
$559$	−1.16762	−0.0493851
$560$	0	0
$561$	0.507088	0.0214093
$562$	0	0
$563$	−5.42155	−0.228491	−0.114246	−	0.993453i	$-0.536445\pi$
$563$	−5.42155	−0.228491	−0.114246	+	0.993453i	$0.536445\pi$
$564$	0	0
$565$	−21.7934	−0.916855
$566$	0	0
$567$	−2.43610	−0.102307
$568$	0	0
$569$	26.1426	1.09596	0.547978	−	0.836493i	$-0.315398\pi$
$569$	26.1426	1.09596	0.547978	+	0.836493i	$0.315398\pi$
$570$	0	0
$571$	28.4267	1.18962	0.594811	−	0.803866i	$-0.297227\pi$
$571$	28.4267	1.18962	0.594811	+	0.803866i	$0.297227\pi$
$572$	0	0
$573$	21.5786	0.901457
$574$	0	0
$575$	−2.62622	−0.109521
$576$	0	0
$577$	−14.5776	−0.606873	−0.303436	−	0.952852i	$-0.598134\pi$
$577$	−14.5776	−0.606873	−0.303436	+	0.952852i	$0.598134\pi$
$578$	0	0
$579$	−17.8687	−0.742598
$580$	0	0
$581$	−18.0375	−0.748324
$582$	0	0
$583$	−12.5409	−0.519389
$584$	0	0
$585$	−2.59807	−0.107417
$586$	0	0
$587$	−36.9172	−1.52374	−0.761869	−	0.647731i	$-0.775718\pi$
$587$	−36.9172	−1.52374	−0.761869	+	0.647731i	$0.775718\pi$
$588$	0	0
$589$	11.0016	0.453313
$590$	0	0
$591$	11.7311	0.482552
$592$	0	0
$593$	15.5833	0.639930	0.319965	−	0.947429i	$-0.396329\pi$
$593$	15.5833	0.639930	0.319965	+	0.947429i	$0.396329\pi$
$594$	0	0
$595$	−1.36046	−0.0557735
$596$	0	0
$597$	−1.40194	−0.0573774
$598$	0	0
$599$	30.7874	1.25794	0.628970	−	0.777429i	$-0.283477\pi$
$599$	30.7874	1.25794	0.628970	+	0.777429i	$0.283477\pi$
$600$	0	0
$601$	34.3325	1.40045	0.700227	−	0.713920i	$-0.253082\pi$
$601$	34.3325	1.40045	0.700227	+	0.713920i	$0.253082\pi$
$602$	0	0
$603$	−9.29821	−0.378652
$604$	0	0
$605$	−14.8045	−0.601887
$606$	0	0
$607$	15.2478	0.618889	0.309444	−	0.950918i	$-0.399857\pi$
$607$	15.2478	0.618889	0.309444	+	0.950918i	$0.399857\pi$
$608$	0	0
$609$	−0.254477	−0.0103119
$610$	0	0
$611$	0.527989	0.0213602
$612$	0	0
$613$	−31.0305	−1.25331	−0.626655	−	0.779297i	$-0.715576\pi$
$613$	−31.0305	−1.25331	−0.626655	+	0.779297i	$0.715576\pi$
$614$	0	0
$615$	−17.2593	−0.695961
$616$	0	0
$617$	18.9712	0.763751	0.381876	−	0.924214i	$-0.375278\pi$
$617$	18.9712	0.763751	0.381876	+	0.924214i	$0.375278\pi$
$618$	0	0
$619$	−17.4009	−0.699402	−0.349701	−	0.936861i	$-0.613717\pi$
$619$	−17.4009	−0.699402	−0.349701	+	0.936861i	$0.613717\pi$
$620$	0	0
$621$	2.55355	0.102470
$622$	0	0
$623$	2.45852	0.0984985
$624$	0	0
$625$	−29.0846	−1.16338
$626$	0	0
$627$	6.50520	0.259793
$628$	0	0
$629$	−0.199109	−0.00793898
$630$	0	0
$631$	−12.7024	−0.505674	−0.252837	−	0.967509i	$-0.581364\pi$
$631$	−12.7024	−0.505674	−0.252837	+	0.967509i	$0.581364\pi$
$632$	0	0
$633$	25.7907	1.02509
$634$	0	0
$635$	49.5006	1.96437
$636$	0	0
$637$	1.12737	0.0446679
$638$	0	0
$639$	−7.68037	−0.303831
$640$	0	0
$641$	−13.6357	−0.538577	−0.269289	−	0.963059i	$-0.586789\pi$
$641$	−13.6357	−0.538577	−0.269289	+	0.963059i	$0.586789\pi$
$642$	0	0
$643$	−11.9928	−0.472951	−0.236476	−	0.971637i	$-0.575992\pi$
$643$	−11.9928	−0.472951	−0.236476	+	0.971637i	$0.575992\pi$
$644$	0	0
$645$	−2.70930	−0.106679
$646$	0	0
$647$	−0.0117322	−0.000461239 0	−0.000230620	−	1.00000i	$-0.500073\pi$
$647$	−0.0117322	−0.000461239 0	−0.000230620		1.00000i	$0.500073\pi$
$648$	0	0
$649$	−5.86231	−0.230116
$650$	0	0
$651$	9.18514	0.359994
$652$	0	0
$653$	19.1426	0.749108	0.374554	−	0.927205i	$-0.377796\pi$
$653$	19.1426	0.749108	0.374554	+	0.927205i	$0.377796\pi$
$654$	0	0
$655$	−47.0348	−1.83780
$656$	0	0
$657$	−9.24479	−0.360674
$658$	0	0
$659$	−12.3408	−0.480729	−0.240364	−	0.970683i	$-0.577267\pi$
$659$	−12.3408	−0.480729	−0.240364	+	0.970683i	$0.577267\pi$
$660$	0	0
$661$	30.3464	1.18034	0.590168	−	0.807280i	$-0.299061\pi$
$661$	30.3464	1.18034	0.590168	+	0.807280i	$0.299061\pi$
$662$	0	0
$663$	0.240677	0.00934714
$664$	0	0
$665$	−17.4528	−0.676789
$666$	0	0
$667$	0.266745	0.0103284
$668$	0	0
$669$	8.89491	0.343897
$670$	0	0
$671$	−10.1664	−0.392469
$672$	0	0
$673$	−10.7780	−0.415460	−0.207730	−	0.978186i	$-0.566607\pi$
$673$	−10.7780	−0.415460	−0.207730	+	0.978186i	$0.566607\pi$
$674$	0	0
$675$	−1.02846	−0.0395855
$676$	0	0
$677$	−48.0272	−1.84584	−0.922918	−	0.384996i	$-0.874203\pi$
$677$	−48.0272	−1.84584	−0.922918	+	0.384996i	$0.874203\pi$
$678$	0	0
$679$	−9.97116	−0.382658
$680$	0	0
$681$	−15.3041	−0.586454
$682$	0	0
$683$	−20.2999	−0.776755	−0.388377	−	0.921500i	$-0.626964\pi$
$683$	−20.2999	−0.776755	−0.388377	+	0.921500i	$0.626964\pi$
$684$	0	0
$685$	−18.8720	−0.721064
$686$	0	0
$687$	−7.40079	−0.282358
$688$	0	0
$689$	−5.95223	−0.226762
$690$	0	0
$691$	−2.31184	−0.0879465	−0.0439733	−	0.999033i	$-0.514002\pi$
$691$	−2.31184	−0.0879465	−0.0439733	+	0.999033i	$0.514002\pi$
$692$	0	0
$693$	5.43113	0.206312
$694$	0	0
$695$	−33.0812	−1.25484
$696$	0	0
$697$	1.59885	0.0605607
$698$	0	0
$699$	24.6200	0.931215
$700$	0	0
$701$	−18.1855	−0.686857	−0.343428	−	0.939179i	$-0.611588\pi$
$701$	−18.1855	−0.686857	−0.343428	+	0.939179i	$0.611588\pi$
$702$	0	0
$703$	−2.55428	−0.0963363
$704$	0	0
$705$	1.22513	0.0461410
$706$	0	0
$707$	26.4804	0.995899
$708$	0	0
$709$	−25.4368	−0.955300	−0.477650	−	0.878550i	$-0.658511\pi$
$709$	−25.4368	−0.955300	−0.477650	+	0.878550i	$0.658511\pi$
$710$	0	0
$711$	1.46616	0.0549855
$712$	0	0
$713$	−9.62796	−0.360570
$714$	0	0
$715$	5.79223	0.216617
$716$	0	0
$717$	11.7362	0.438295
$718$	0	0
$719$	−3.91265	−0.145917	−0.0729586	−	0.997335i	$-0.523244\pi$
$719$	−3.91265	−0.145917	−0.0729586	+	0.997335i	$0.523244\pi$
$720$	0	0
$721$	6.76244	0.251846
$722$	0	0
$723$	18.1406	0.674655
$724$	0	0
$725$	−0.107434	−0.00398999
$726$	0	0
$727$	−1.14621	−0.0425105	−0.0212552	−	0.999774i	$-0.506766\pi$
$727$	−1.14621	−0.0425105	−0.0212552	+	0.999774i	$0.506766\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0.250982	0.00928290
$732$	0	0
$733$	−14.9085	−0.550659	−0.275330	−	0.961350i	$-0.588787\pi$
$733$	−14.9085	−0.550659	−0.275330	+	0.961350i	$0.588787\pi$
$734$	0	0
$735$	2.61590	0.0964890
$736$	0	0
$737$	20.7298	0.763591
$738$	0	0
$739$	−23.1089	−0.850074	−0.425037	−	0.905176i	$-0.639739\pi$
$739$	−23.1089	−0.850074	−0.425037	+	0.905176i	$0.639739\pi$
$740$	0	0
$741$	3.08754	0.113424
$742$	0	0
$743$	18.8312	0.690851	0.345426	−	0.938446i	$-0.387735\pi$
$743$	18.8312	0.690851	0.345426	+	0.938446i	$0.387735\pi$
$744$	0	0
$745$	−56.5027	−2.07010
$746$	0	0
$747$	7.40427	0.270908
$748$	0	0
$749$	−9.10149	−0.332561
$750$	0	0
$751$	10.1634	0.370868	0.185434	−	0.982657i	$-0.440631\pi$
$751$	10.1634	0.370868	0.185434	+	0.982657i	$0.440631\pi$
$752$	0	0
$753$	6.88907	0.251052
$754$	0	0
$755$	−0.127556	−0.00464223
$756$	0	0
$757$	−20.4708	−0.744024	−0.372012	−	0.928228i	$-0.621332\pi$
$757$	−20.4708	−0.744024	−0.372012	+	0.928228i	$0.621332\pi$
$758$	0	0
$759$	−5.69297	−0.206642
$760$	0	0
$761$	12.5691	0.455628	0.227814	−	0.973705i	$-0.426842\pi$
$761$	12.5691	0.455628	0.227814	+	0.973705i	$0.426842\pi$
$762$	0	0
$763$	17.5175	0.634176
$764$	0	0
$765$	0.558459	0.0201911
$766$	0	0
$767$	−2.78241	−0.100467
$768$	0	0
$769$	47.7046	1.72027	0.860136	−	0.510064i	$-0.170378\pi$
$769$	47.7046	1.72027	0.860136	+	0.510064i	$0.170378\pi$
$770$	0	0
$771$	0.160088	0.00576543
$772$	0	0
$773$	−3.65095	−0.131316	−0.0656578	−	0.997842i	$-0.520915\pi$
$773$	−3.65095	−0.131316	−0.0656578	+	0.997842i	$0.520915\pi$
$774$	0	0
$775$	3.87774	0.139292
$776$	0	0
$777$	−2.13254	−0.0765045
$778$	0	0
$779$	20.5109	0.734880
$780$	0	0
$781$	17.1229	0.612705
$782$	0	0
$783$	0.104461	0.00373312
$784$	0	0
$785$	29.0219	1.03584
$786$	0	0
$787$	−24.4202	−0.870485	−0.435243	−	0.900313i	$-0.643337\pi$
$787$	−24.4202	−0.870485	−0.435243	+	0.900313i	$0.643337\pi$
$788$	0	0
$789$	7.63433	0.271789
$790$	0	0
$791$	21.6230	0.768827
$792$	0	0
$793$	−4.82524	−0.171349
$794$	0	0
$795$	−13.8113	−0.489837
$796$	0	0
$797$	−4.30879	−0.152625	−0.0763126	−	0.997084i	$-0.524315\pi$
$797$	−4.30879	−0.152625	−0.0763126	+	0.997084i	$0.524315\pi$
$798$	0	0
$799$	−0.113492	−0.00401507
$800$	0	0
$801$	−1.00920	−0.0356584
$802$	0	0
$803$	20.6107	0.727335
$804$	0	0
$805$	15.2736	0.538325
$806$	0	0
$807$	7.21293	0.253907
$808$	0	0
$809$	−32.4774	−1.14185	−0.570923	−	0.821004i	$-0.693414\pi$
$809$	−32.4774	−1.14185	−0.570923	+	0.821004i	$0.693414\pi$
$810$	0	0
$811$	−18.5030	−0.649728	−0.324864	−	0.945761i	$-0.605319\pi$
$811$	−18.5030	−0.649728	−0.324864	+	0.945761i	$0.605319\pi$
$812$	0	0
$813$	0.0760078	0.00266571
$814$	0	0
$815$	52.5174	1.83960
$816$	0	0
$817$	3.21974	0.112644
$818$	0	0
$819$	2.57776	0.0900742
$820$	0	0
$821$	15.5684	0.543339	0.271670	−	0.962391i	$-0.412424\pi$
$821$	15.5684	0.543339	0.271670	+	0.962391i	$0.412424\pi$
$822$	0	0
$823$	−19.8359	−0.691435	−0.345717	−	0.938339i	$-0.612364\pi$
$823$	−19.8359	−0.691435	−0.345717	+	0.938339i	$0.612364\pi$
$824$	0	0
$825$	2.29289	0.0798281
$826$	0	0
$827$	−48.1892	−1.67570	−0.837852	−	0.545898i	$-0.816189\pi$
$827$	−48.1892	−1.67570	−0.837852	+	0.545898i	$0.816189\pi$
$828$	0	0
$829$	3.45129	0.119868	0.0599341	−	0.998202i	$-0.480911\pi$
$829$	3.45129	0.119868	0.0599341	+	0.998202i	$0.480911\pi$
$830$	0	0
$831$	27.9255	0.968723
$832$	0	0
$833$	−0.242330	−0.00839622
$834$	0	0
$835$	2.45529	0.0849689
$836$	0	0
$837$	−3.77043	−0.130325
$838$	0	0
$839$	−17.5773	−0.606836	−0.303418	−	0.952858i	$-0.598128\pi$
$839$	−17.5773	−0.606836	−0.303418	+	0.952858i	$0.598128\pi$
$840$	0	0
$841$	−28.9891	−0.999624
$842$	0	0
$843$	−16.5725	−0.570788
$844$	0	0
$845$	−29.1697	−1.00347
$846$	0	0
$847$	14.6887	0.504711
$848$	0	0
$849$	1.06715	0.0366246
$850$	0	0
$851$	2.23535	0.0766269
$852$	0	0
$853$	11.9333	0.408588	0.204294	−	0.978910i	$-0.434510\pi$
$853$	11.9333	0.408588	0.204294	+	0.978910i	$0.434510\pi$
$854$	0	0
$855$	7.16422	0.245011
$856$	0	0
$857$	−32.5389	−1.11151	−0.555753	−	0.831347i	$-0.687570\pi$
$857$	−32.5389	−1.11151	−0.555753	+	0.831347i	$0.687570\pi$
$858$	0	0
$859$	16.4547	0.561426	0.280713	−	0.959792i	$-0.409429\pi$
$859$	16.4547	0.561426	0.280713	+	0.959792i	$0.409429\pi$
$860$	0	0
$861$	17.1244	0.583597
$862$	0	0
$863$	−11.5645	−0.393659	−0.196830	−	0.980438i	$-0.563065\pi$
$863$	−11.5645	−0.393659	−0.196830	+	0.980438i	$0.563065\pi$
$864$	0	0
$865$	14.7545	0.501669
$866$	0	0
$867$	16.9483	0.575593
$868$	0	0
$869$	−3.26872	−0.110884
$870$	0	0
$871$	9.83890	0.333378
$872$	0	0
$873$	4.09308	0.138530
$874$	0	0
$875$	23.7551	0.803070
$876$	0	0
$877$	−13.7734	−0.465095	−0.232547	−	0.972585i	$-0.574706\pi$
$877$	−13.7734	−0.465095	−0.232547	+	0.972585i	$0.574706\pi$
$878$	0	0
$879$	−14.3187	−0.482959
$880$	0	0
$881$	40.7853	1.37409	0.687046	−	0.726614i	$-0.258907\pi$
$881$	40.7853	1.37409	0.687046	+	0.726614i	$0.258907\pi$
$882$	0	0
$883$	−34.5045	−1.16117	−0.580585	−	0.814200i	$-0.697176\pi$
$883$	−34.5045	−1.16117	−0.580585	+	0.814200i	$0.697176\pi$
$884$	0	0
$885$	−6.45620	−0.217023
$886$	0	0
$887$	5.54368	0.186138	0.0930692	−	0.995660i	$-0.470332\pi$
$887$	5.54368	0.186138	0.0930692	+	0.995660i	$0.470332\pi$
$888$	0	0
$889$	−49.1137	−1.64722
$890$	0	0
$891$	−2.22944	−0.0746889
$892$	0	0
$893$	−1.45594	−0.0487212
$894$	0	0
$895$	−9.65441	−0.322711
$896$	0	0
$897$	−2.70204	−0.0902184
$898$	0	0
$899$	−0.393862	−0.0131360
$900$	0	0
$901$	1.27944	0.0426244
$902$	0	0
$903$	2.68813	0.0894553
$904$	0	0
$905$	11.8370	0.393474
$906$	0	0
$907$	−21.8652	−0.726022	−0.363011	−	0.931785i	$-0.618251\pi$
$907$	−21.8652	−0.726022	−0.363011	+	0.931785i	$0.618251\pi$
$908$	0	0
$909$	−10.8700	−0.360535
$910$	0	0
$911$	19.7424	0.654095	0.327048	−	0.945008i	$-0.393946\pi$
$911$	19.7424	0.654095	0.327048	+	0.945008i	$0.393946\pi$
$912$	0	0
$913$	−16.5074	−0.546314
$914$	0	0
$915$	−11.1963	−0.370138
$916$	0	0
$917$	46.6671	1.54108
$918$	0	0
$919$	45.9010	1.51413	0.757067	−	0.653337i	$-0.226632\pi$
$919$	45.9010	1.51413	0.757067	+	0.653337i	$0.226632\pi$
$920$	0	0
$921$	−32.7561	−1.07935
$922$	0	0
$923$	8.12698	0.267503
$924$	0	0
$925$	−0.900305	−0.0296019
$926$	0	0
$927$	−2.77593	−0.0911734
$928$	0	0
$929$	0.903753	0.0296512	0.0148256	−	0.999890i	$-0.495281\pi$
$929$	0.903753	0.0296512	0.0148256	+	0.999890i	$0.495281\pi$
$930$	0	0
$931$	−3.10874	−0.101885
$932$	0	0
$933$	10.4071	0.340713
$934$	0	0
$935$	−1.24505	−0.0407175
$936$	0	0
$937$	30.4374	0.994348	0.497174	−	0.867651i	$-0.334371\pi$
$937$	30.4374	0.994348	0.497174	+	0.867651i	$0.334371\pi$
$938$	0	0
$939$	−6.83190	−0.222951
$940$	0	0
$941$	41.4492	1.35121	0.675603	−	0.737265i	$-0.263883\pi$
$941$	41.4492	1.35121	0.675603	+	0.737265i	$0.263883\pi$
$942$	0	0
$943$	−17.9500	−0.584531
$944$	0	0
$945$	5.98134	0.194573
$946$	0	0
$947$	−49.0917	−1.59527	−0.797633	−	0.603143i	$-0.793915\pi$
$947$	−49.0917	−1.59527	−0.797633	+	0.603143i	$0.793915\pi$
$948$	0	0
$949$	9.78237	0.317549
$950$	0	0
$951$	−25.1071	−0.814152
$952$	0	0
$953$	40.1389	1.30023	0.650114	−	0.759837i	$-0.274721\pi$
$953$	40.1389	1.30023	0.650114	+	0.759837i	$0.274721\pi$
$954$	0	0
$955$	−52.9817	−1.71445
$956$	0	0
$957$	−0.232889	−0.00752822
$958$	0	0
$959$	18.7245	0.604647
$960$	0	0
$961$	−16.7839	−0.541416
$962$	0	0
$963$	3.73609	0.120394
$964$	0	0
$965$	43.8729	1.41232
$966$	0	0
$967$	−36.2113	−1.16448	−0.582239	−	0.813018i	$-0.697823\pi$
$967$	−36.2113	−1.16448	−0.582239	+	0.813018i	$0.697823\pi$
$968$	0	0
$969$	−0.663672	−0.0213202
$970$	0	0
$971$	0.199797	0.00641180	0.00320590	−	0.999995i	$-0.498980\pi$
$971$	0.199797	0.00641180	0.00320590	+	0.999995i	$0.498980\pi$
$972$	0	0
$973$	32.8226	1.05224
$974$	0	0
$975$	1.08827	0.0348524
$976$	0	0
$977$	−7.73646	−0.247511	−0.123756	−	0.992313i	$-0.539494\pi$
$977$	−7.73646	−0.247511	−0.123756	+	0.992313i	$0.539494\pi$
$978$	0	0
$979$	2.24995	0.0719088
$980$	0	0
$981$	−7.19079	−0.229584
$982$	0	0
$983$	−0.856966	−0.0273330	−0.0136665	−	0.999907i	$-0.504350\pi$
$983$	−0.856966	−0.0273330	−0.0136665	+	0.999907i	$0.504350\pi$
$984$	0	0
$985$	−28.8032	−0.917747
$986$	0	0
$987$	−1.21555	−0.0386914
$988$	0	0
$989$	−2.81772	−0.0895984
$990$	0	0
$991$	−7.69515	−0.244445	−0.122222	−	0.992503i	$-0.539002\pi$
$991$	−7.69515	−0.244445	−0.122222	+	0.992503i	$0.539002\pi$
$992$	0	0
$993$	12.6726	0.402152
$994$	0	0
$995$	3.44216	0.109124
$996$	0	0
$997$	18.7602	0.594141	0.297070	−	0.954856i	$-0.403990\pi$
$997$	18.7602	0.594141	0.297070	+	0.954856i	$0.403990\pi$
$998$	0	0
$999$	0.875391	0.0276961

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.x.1.6		8
4.3	odd	2		501.2.a.e.1.2	✓	8
12.11	even	2		1503.2.a.e.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
501.2.a.e.1.2	✓	8	4.3	odd	2
1503.2.a.e.1.7		8	12.11	even	2
8016.2.a.x.1.6		8	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:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.45529 q^{5} -2.43610 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.45529 q^{5} -2.43610 q^{7} +1.00000 q^{9} -2.22944 q^{11} -1.05815 q^{13} -2.45529 q^{15} +0.227451 q^{17} +2.91787 q^{19} +2.43610 q^{21} -2.55355 q^{23} +1.02846 q^{25} -1.00000 q^{27} -0.104461 q^{29} +3.77043 q^{31} +2.22944 q^{33} -5.98134 q^{35} -0.875391 q^{37} +1.05815 q^{39} +7.02942 q^{41} +1.10345 q^{43} +2.45529 q^{45} -0.498974 q^{47} -1.06541 q^{49} -0.227451 q^{51} +5.62512 q^{53} -5.47392 q^{55} -2.91787 q^{57} +2.62950 q^{59} +4.56007 q^{61} -2.43610 q^{63} -2.59807 q^{65} -9.29821 q^{67} +2.55355 q^{69} -7.68037 q^{71} -9.24479 q^{73} -1.02846 q^{75} +5.43113 q^{77} +1.46616 q^{79} +1.00000 q^{81} +7.40427 q^{83} +0.558459 q^{85} +0.104461 q^{87} -1.00920 q^{89} +2.57776 q^{91} -3.77043 q^{93} +7.16422 q^{95} +4.09308 q^{97} -2.22944 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} + 7 q^{5} + 4 q^{7} + 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^3 + 7 * q^5 + 4 * q^7 + 8 * q^9	\( 8 q - 8 q^{3} + 7 q^{5} + 4 q^{7} + 8 q^{9} - 13 q^{11} - 7 q^{15} + 11 q^{17} - 12 q^{19} - 4 q^{21} - 7 q^{23} - 5 q^{25} - 8 q^{27} + q^{29} + 2 q^{31} + 13 q^{33} + 4 q^{35} - 9 q^{37} + 4 q^{41} - 2 q^{43} + 7 q^{45} - 17 q^{47} - 2 q^{49} - 11 q^{51} + 9 q^{53} - 7 q^{55} + 12 q^{57} - 29 q^{59} - 12 q^{61} + 4 q^{63} + 8 q^{65} + 7 q^{69} - 13 q^{71} - 20 q^{73} + 5 q^{75} - 22 q^{77} - 8 q^{79} + 8 q^{81} - 33 q^{83} - 31 q^{85} - q^{87} + 4 q^{89} - q^{91} - 2 q^{93} - 3 q^{95} - 31 q^{97} - 13 q^{99}+O(q^{100}) \) 8 * q - 8 * q^3 + 7 * q^5 + 4 * q^7 + 8 * q^9 - 13 * q^11 - 7 * q^15 + 11 * q^17 - 12 * q^19 - 4 * q^21 - 7 * q^23 - 5 * q^25 - 8 * q^27 + q^29 + 2 * q^31 + 13 * q^33 + 4 * q^35 - 9 * q^37 + 4 * q^41 - 2 * q^43 + 7 * q^45 - 17 * q^47 - 2 * q^49 - 11 * q^51 + 9 * q^53 - 7 * q^55 + 12 * q^57 - 29 * q^59 - 12 * q^61 + 4 * q^63 + 8 * q^65 + 7 * q^69 - 13 * q^71 - 20 * q^73 + 5 * q^75 - 22 * q^77 - 8 * q^79 + 8 * q^81 - 33 * q^83 - 31 * q^85 - q^87 + 4 * q^89 - q^91 - 2 * q^93 - 3 * q^95 - 31 * q^97 - 13 * q^99

Introduction

L-functions

Modular forms

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