LMFDB - Embedded newform 8016.2.a.z.1.3

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} - 2x^{7} - 8x^{6} + 15x^{5} + 19x^{4} - 31x^{3} - 13x^{2} + 14x + 1 $ x^8 - 2x^7 - 8x^6 + 15x^5 + 19x^4 - 31x^3 - 13x^2 + 14*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			1.3
Root			$-0.853788$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -0.621044 q^{5} +0.794861 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.621044 q^{5} +0.794861 q^{7} +1.00000 q^{9} -2.84218 q^{11} +2.63384 q^{13} -0.621044 q^{15} -3.05189 q^{17} +4.38219 q^{19} +0.794861 q^{21} -2.45857 q^{23} -4.61430 q^{25} +1.00000 q^{27} -3.48971 q^{29} -6.29991 q^{31} -2.84218 q^{33} -0.493644 q^{35} +5.11215 q^{37} +2.63384 q^{39} +9.70703 q^{41} -1.24812 q^{43} -0.621044 q^{45} -10.2334 q^{47} -6.36820 q^{49} -3.05189 q^{51} +4.00998 q^{53} +1.76512 q^{55} +4.38219 q^{57} -8.96524 q^{59} +11.8647 q^{61} +0.794861 q^{63} -1.63573 q^{65} +2.20793 q^{67} -2.45857 q^{69} -6.41628 q^{71} -11.3210 q^{73} -4.61430 q^{75} -2.25914 q^{77} -14.4099 q^{79} +1.00000 q^{81} -1.13222 q^{83} +1.89536 q^{85} -3.48971 q^{87} +5.89284 q^{89} +2.09353 q^{91} -6.29991 q^{93} -2.72153 q^{95} +14.5178 q^{97} -2.84218 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} + q^{5} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^3 + q^5 + 8 * q^9	$ 8 q + 8 q^{3} + q^{5} + 8 q^{9} - 5 q^{11} + q^{15} - 7 q^{17} - 24 q^{19} - q^{23} + 3 q^{25} + 8 q^{27} - 11 q^{29} - 30 q^{31} - 5 q^{33} - 26 q^{35} + 11 q^{37} + 10 q^{41} - 24 q^{43} + q^{45} + 3 q^{47} + 6 q^{49} - 7 q^{51} - 25 q^{53} - 25 q^{55} - 24 q^{57} - 45 q^{59} + 16 q^{61} - 10 q^{65} - 18 q^{67} - q^{69} - 21 q^{71} - 8 q^{73} + 3 q^{75} - 18 q^{77} - 10 q^{79} + 8 q^{81} - 7 q^{83} - 11 q^{85} - 11 q^{87} + 26 q^{89} - 15 q^{91} - 30 q^{93} - q^{95} - 3 q^{97} - 5 q^{99}+O(q^{100}) $ 8 * q + 8 * q^3 + q^5 + 8 * q^9 - 5 * q^11 + q^15 - 7 * q^17 - 24 * q^19 - q^23 + 3 * q^25 + 8 * q^27 - 11 * q^29 - 30 * q^31 - 5 * q^33 - 26 * q^35 + 11 * q^37 + 10 * q^41 - 24 * q^43 + q^45 + 3 * q^47 + 6 * q^49 - 7 * q^51 - 25 * q^53 - 25 * q^55 - 24 * q^57 - 45 * q^59 + 16 * q^61 - 10 * q^65 - 18 * q^67 - q^69 - 21 * q^71 - 8 * q^73 + 3 * q^75 - 18 * q^77 - 10 * q^79 + 8 * q^81 - 7 * q^83 - 11 * q^85 - 11 * q^87 + 26 * q^89 - 15 * q^91 - 30 * q^93 - q^95 - 3 * q^97 - 5 * 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
$5$	−0.621044	−0.277739	−0.138870	−	0.990311i	$-0.544347\pi$
$5$	−0.621044	−0.277739	−0.138870	+	0.990311i	$0.544347\pi$
$6$	0	0
$7$	0.794861	0.300429	0.150215	−	0.988653i	$-0.452004\pi$
$7$	0.794861	0.300429	0.150215	+	0.988653i	$0.452004\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.84218	−0.856948	−0.428474	−	0.903554i	$-0.640949\pi$
$11$	−2.84218	−0.856948	−0.428474	+	0.903554i	$0.640949\pi$
$12$	0	0
$13$	2.63384	0.730495	0.365247	−	0.930911i	$-0.380984\pi$
$13$	2.63384	0.730495	0.365247	+	0.930911i	$0.380984\pi$
$14$	0	0
$15$	−0.621044	−0.160353
$16$	0	0
$17$	−3.05189	−0.740193	−0.370096	−	0.928993i	$-0.620675\pi$
$17$	−3.05189	−0.740193	−0.370096	+	0.928993i	$0.620675\pi$
$18$	0	0
$19$	4.38219	1.00534	0.502671	−	0.864478i	$-0.332351\pi$
$19$	4.38219	1.00534	0.502671	+	0.864478i	$0.332351\pi$
$20$	0	0
$21$	0.794861	0.173453
$22$	0	0
$23$	−2.45857	−0.512647	−0.256324	−	0.966591i	$-0.582511\pi$
$23$	−2.45857	−0.512647	−0.256324	+	0.966591i	$0.582511\pi$
$24$	0	0
$25$	−4.61430	−0.922861
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.48971	−0.648022	−0.324011	−	0.946053i	$-0.605031\pi$
$29$	−3.48971	−0.648022	−0.324011	+	0.946053i	$0.605031\pi$
$30$	0	0
$31$	−6.29991	−1.13150	−0.565749	−	0.824578i	$-0.691413\pi$
$31$	−6.29991	−1.13150	−0.565749	+	0.824578i	$0.691413\pi$
$32$	0	0
$33$	−2.84218	−0.494759
$34$	0	0
$35$	−0.493644	−0.0834410
$36$	0	0
$37$	5.11215	0.840432	0.420216	−	0.907424i	$-0.361954\pi$
$37$	5.11215	0.840432	0.420216	+	0.907424i	$0.361954\pi$
$38$	0	0
$39$	2.63384	0.421751
$40$	0	0
$41$	9.70703	1.51598	0.757992	−	0.652264i	$-0.226181\pi$
$41$	9.70703	1.51598	0.757992	+	0.652264i	$0.226181\pi$
$42$	0	0
$43$	−1.24812	−0.190337	−0.0951683	−	0.995461i	$-0.530339\pi$
$43$	−1.24812	−0.190337	−0.0951683	+	0.995461i	$0.530339\pi$
$44$	0	0
$45$	−0.621044	−0.0925798
$46$	0	0
$47$	−10.2334	−1.49269	−0.746346	−	0.665558i	$-0.768193\pi$
$47$	−10.2334	−1.49269	−0.746346	+	0.665558i	$0.768193\pi$
$48$	0	0
$49$	−6.36820	−0.909742
$50$	0	0
$51$	−3.05189	−0.427351
$52$	0	0
$53$	4.00998	0.550814	0.275407	−	0.961328i	$-0.411187\pi$
$53$	4.00998	0.550814	0.275407	+	0.961328i	$0.411187\pi$
$54$	0	0
$55$	1.76512	0.238008
$56$	0	0
$57$	4.38219	0.580435
$58$	0	0
$59$	−8.96524	−1.16718	−0.583588	−	0.812050i	$-0.698352\pi$
$59$	−8.96524	−1.16718	−0.583588	+	0.812050i	$0.698352\pi$
$60$	0	0
$61$	11.8647	1.51911	0.759557	−	0.650441i	$-0.225416\pi$
$61$	11.8647	1.51911	0.759557	+	0.650441i	$0.225416\pi$
$62$	0	0
$63$	0.794861	0.100143
$64$	0	0
$65$	−1.63573	−0.202887
$66$	0	0
$67$	2.20793	0.269742	0.134871	−	0.990863i	$-0.456938\pi$
$67$	2.20793	0.269742	0.134871	+	0.990863i	$0.456938\pi$
$68$	0	0
$69$	−2.45857	−0.295977
$70$	0	0
$71$	−6.41628	−0.761472	−0.380736	−	0.924684i	$-0.624329\pi$
$71$	−6.41628	−0.761472	−0.380736	+	0.924684i	$0.624329\pi$
$72$	0	0
$73$	−11.3210	−1.32502	−0.662512	−	0.749051i	$-0.730510\pi$
$73$	−11.3210	−1.32502	−0.662512	+	0.749051i	$0.730510\pi$
$74$	0	0
$75$	−4.61430	−0.532814
$76$	0	0
$77$	−2.25914	−0.257452
$78$	0	0
$79$	−14.4099	−1.62124	−0.810622	−	0.585570i	$-0.800871\pi$
$79$	−14.4099	−1.62124	−0.810622	+	0.585570i	$0.800871\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.13222	−0.124278	−0.0621388	−	0.998068i	$-0.519792\pi$
$83$	−1.13222	−0.124278	−0.0621388	+	0.998068i	$0.519792\pi$
$84$	0	0
$85$	1.89536	0.205581
$86$	0	0
$87$	−3.48971	−0.374136
$88$	0	0
$89$	5.89284	0.624639	0.312320	−	0.949977i	$-0.398894\pi$
$89$	5.89284	0.624639	0.312320	+	0.949977i	$0.398894\pi$
$90$	0	0
$91$	2.09353	0.219462
$92$	0	0
$93$	−6.29991	−0.653270
$94$	0	0
$95$	−2.72153	−0.279223
$96$	0	0
$97$	14.5178	1.47406	0.737030	−	0.675860i	$-0.236228\pi$
$97$	14.5178	1.47406	0.737030	+	0.675860i	$0.236228\pi$
$98$	0	0
$99$	−2.84218	−0.285649
$100$	0	0
$101$	−13.9233	−1.38542	−0.692712	−	0.721214i	$-0.743584\pi$
$101$	−13.9233	−1.38542	−0.692712	+	0.721214i	$0.743584\pi$
$102$	0	0
$103$	6.37017	0.627671	0.313836	−	0.949477i	$-0.398386\pi$
$103$	6.37017	0.627671	0.313836	+	0.949477i	$0.398386\pi$
$104$	0	0
$105$	−0.493644	−0.0481747
$106$	0	0
$107$	−17.6646	−1.70770	−0.853852	−	0.520517i	$-0.825739\pi$
$107$	−17.6646	−1.70770	−0.853852	+	0.520517i	$0.825739\pi$
$108$	0	0
$109$	8.22481	0.787794	0.393897	−	0.919155i	$-0.371127\pi$
$109$	8.22481	0.787794	0.393897	+	0.919155i	$0.371127\pi$
$110$	0	0
$111$	5.11215	0.485224
$112$	0	0
$113$	−9.93200	−0.934324	−0.467162	−	0.884172i	$-0.654724\pi$
$113$	−9.93200	−0.934324	−0.467162	+	0.884172i	$0.654724\pi$
$114$	0	0
$115$	1.52688	0.142382
$116$	0	0
$117$	2.63384	0.243498
$118$	0	0
$119$	−2.42583	−0.222376
$120$	0	0
$121$	−2.92203	−0.265639
$122$	0	0
$123$	9.70703	0.875253
$124$	0	0
$125$	5.97091	0.534054
$126$	0	0
$127$	−6.60003	−0.585658	−0.292829	−	0.956165i	$-0.594597\pi$
$127$	−6.60003	−0.585658	−0.292829	+	0.956165i	$0.594597\pi$
$128$	0	0
$129$	−1.24812	−0.109891
$130$	0	0
$131$	−16.0837	−1.40524	−0.702620	−	0.711565i	$-0.747987\pi$
$131$	−16.0837	−1.40524	−0.702620	+	0.711565i	$0.747987\pi$
$132$	0	0
$133$	3.48323	0.302034
$134$	0	0
$135$	−0.621044	−0.0534510
$136$	0	0
$137$	10.4861	0.895892	0.447946	−	0.894061i	$-0.352156\pi$
$137$	10.4861	0.895892	0.447946	+	0.894061i	$0.352156\pi$
$138$	0	0
$139$	12.6126	1.06978	0.534892	−	0.844920i	$-0.320352\pi$
$139$	12.6126	1.06978	0.534892	+	0.844920i	$0.320352\pi$
$140$	0	0
$141$	−10.2334	−0.861806
$142$	0	0
$143$	−7.48583	−0.625996
$144$	0	0
$145$	2.16726	0.179981
$146$	0	0
$147$	−6.36820	−0.525240
$148$	0	0
$149$	−16.6639	−1.36516	−0.682580	−	0.730811i	$-0.739142\pi$
$149$	−16.6639	−1.36516	−0.682580	+	0.730811i	$0.739142\pi$
$150$	0	0
$151$	24.0206	1.95477	0.977385	−	0.211469i	$-0.0678246\pi$
$151$	24.0206	1.95477	0.977385	+	0.211469i	$0.0678246\pi$
$152$	0	0
$153$	−3.05189	−0.246731
$154$	0	0
$155$	3.91252	0.314261
$156$	0	0
$157$	3.81635	0.304578	0.152289	−	0.988336i	$-0.451336\pi$
$157$	3.81635	0.304578	0.152289	+	0.988336i	$0.451336\pi$
$158$	0	0
$159$	4.00998	0.318012
$160$	0	0
$161$	−1.95422	−0.154014
$162$	0	0
$163$	−21.2874	−1.66736	−0.833680	−	0.552247i	$-0.813770\pi$
$163$	−21.2874	−1.66736	−0.833680	+	0.552247i	$0.813770\pi$
$164$	0	0
$165$	1.76512	0.137414
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−6.06291	−0.466378
$170$	0	0
$171$	4.38219	0.335114
$172$	0	0
$173$	10.4729	0.796242	0.398121	−	0.917333i	$-0.369662\pi$
$173$	10.4729	0.796242	0.398121	+	0.917333i	$0.369662\pi$
$174$	0	0
$175$	−3.66773	−0.277254
$176$	0	0
$177$	−8.96524	−0.673869
$178$	0	0
$179$	18.7361	1.40040	0.700200	−	0.713946i	$-0.253094\pi$
$179$	18.7361	1.40040	0.700200	+	0.713946i	$0.253094\pi$
$180$	0	0
$181$	−11.6895	−0.868871	−0.434435	−	0.900703i	$-0.643052\pi$
$181$	−11.6895	−0.868871	−0.434435	+	0.900703i	$0.643052\pi$
$182$	0	0
$183$	11.8647	0.877061
$184$	0	0
$185$	−3.17487	−0.233421
$186$	0	0
$187$	8.67402	0.634307
$188$	0	0
$189$	0.794861	0.0578176
$190$	0	0
$191$	20.4506	1.47975	0.739877	−	0.672742i	$-0.234884\pi$
$191$	20.4506	1.47975	0.739877	+	0.672742i	$0.234884\pi$
$192$	0	0
$193$	−5.21519	−0.375398	−0.187699	−	0.982227i	$-0.560103\pi$
$193$	−5.21519	−0.375398	−0.187699	+	0.982227i	$0.560103\pi$
$194$	0	0
$195$	−1.63573	−0.117137
$196$	0	0
$197$	−3.40714	−0.242749	−0.121374	−	0.992607i	$-0.538730\pi$
$197$	−3.40714	−0.242749	−0.121374	+	0.992607i	$0.538730\pi$
$198$	0	0
$199$	4.64991	0.329623	0.164812	−	0.986325i	$-0.447298\pi$
$199$	4.64991	0.329623	0.164812	+	0.986325i	$0.447298\pi$
$200$	0	0
$201$	2.20793	0.155736
$202$	0	0
$203$	−2.77383	−0.194685
$204$	0	0
$205$	−6.02849	−0.421048
$206$	0	0
$207$	−2.45857	−0.170882
$208$	0	0
$209$	−12.4549	−0.861527
$210$	0	0
$211$	−21.6206	−1.48842	−0.744212	−	0.667944i	$-0.767175\pi$
$211$	−21.6206	−1.48842	−0.744212	+	0.667944i	$0.767175\pi$
$212$	0	0
$213$	−6.41628	−0.439636
$214$	0	0
$215$	0.775138	0.0528640
$216$	0	0
$217$	−5.00756	−0.339935
$218$	0	0
$219$	−11.3210	−0.765003
$220$	0	0
$221$	−8.03819	−0.540707
$222$	0	0
$223$	−23.5150	−1.57468	−0.787339	−	0.616520i	$-0.788542\pi$
$223$	−23.5150	−1.57468	−0.787339	+	0.616520i	$0.788542\pi$
$224$	0	0
$225$	−4.61430	−0.307620
$226$	0	0
$227$	−7.98906	−0.530252	−0.265126	−	0.964214i	$-0.585414\pi$
$227$	−7.98906	−0.530252	−0.265126	+	0.964214i	$0.585414\pi$
$228$	0	0
$229$	−7.94269	−0.524868	−0.262434	−	0.964950i	$-0.584525\pi$
$229$	−7.94269	−0.524868	−0.262434	+	0.964950i	$0.584525\pi$
$230$	0	0
$231$	−2.25914	−0.148640
$232$	0	0
$233$	−7.10575	−0.465513	−0.232756	−	0.972535i	$-0.574774\pi$
$233$	−7.10575	−0.465513	−0.232756	+	0.972535i	$0.574774\pi$
$234$	0	0
$235$	6.35538	0.414579
$236$	0	0
$237$	−14.4099	−0.936025
$238$	0	0
$239$	−5.04207	−0.326144	−0.163072	−	0.986614i	$-0.552140\pi$
$239$	−5.04207	−0.326144	−0.163072	+	0.986614i	$0.552140\pi$
$240$	0	0
$241$	26.8655	1.73056	0.865280	−	0.501289i	$-0.167141\pi$
$241$	26.8655	1.73056	0.865280	+	0.501289i	$0.167141\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	3.95493	0.252671
$246$	0	0
$247$	11.5420	0.734398
$248$	0	0
$249$	−1.13222	−0.0717518
$250$	0	0
$251$	−20.6873	−1.30577	−0.652885	−	0.757457i	$-0.726442\pi$
$251$	−20.6873	−1.30577	−0.652885	+	0.757457i	$0.726442\pi$
$252$	0	0
$253$	6.98769	0.439312
$254$	0	0
$255$	1.89536	0.118692
$256$	0	0
$257$	1.68043	0.104822	0.0524110	−	0.998626i	$-0.483309\pi$
$257$	1.68043	0.104822	0.0524110	+	0.998626i	$0.483309\pi$
$258$	0	0
$259$	4.06345	0.252490
$260$	0	0
$261$	−3.48971	−0.216007
$262$	0	0
$263$	6.27570	0.386976	0.193488	−	0.981103i	$-0.438020\pi$
$263$	6.27570	0.386976	0.193488	+	0.981103i	$0.438020\pi$
$264$	0	0
$265$	−2.49038	−0.152983
$266$	0	0
$267$	5.89284	0.360636
$268$	0	0
$269$	−11.2734	−0.687349	−0.343674	−	0.939089i	$-0.611672\pi$
$269$	−11.2734	−0.687349	−0.343674	+	0.939089i	$0.611672\pi$
$270$	0	0
$271$	−9.89342	−0.600983	−0.300491	−	0.953785i	$-0.597151\pi$
$271$	−9.89342	−0.600983	−0.300491	+	0.953785i	$0.597151\pi$
$272$	0	0
$273$	2.09353	0.126706
$274$	0	0
$275$	13.1147	0.790844
$276$	0	0
$277$	−10.9863	−0.660104	−0.330052	−	0.943963i	$-0.607066\pi$
$277$	−10.9863	−0.660104	−0.330052	+	0.943963i	$0.607066\pi$
$278$	0	0
$279$	−6.29991	−0.377166
$280$	0	0
$281$	−8.69763	−0.518857	−0.259428	−	0.965762i	$-0.583534\pi$
$281$	−8.69763	−0.518857	−0.259428	+	0.965762i	$0.583534\pi$
$282$	0	0
$283$	15.4308	0.917264	0.458632	−	0.888626i	$-0.348340\pi$
$283$	15.4308	0.917264	0.458632	+	0.888626i	$0.348340\pi$
$284$	0	0
$285$	−2.72153	−0.161210
$286$	0	0
$287$	7.71574	0.455446
$288$	0	0
$289$	−7.68594	−0.452114
$290$	0	0
$291$	14.5178	0.851049
$292$	0	0
$293$	−15.8113	−0.923707	−0.461853	−	0.886956i	$-0.652815\pi$
$293$	−15.8113	−0.923707	−0.461853	+	0.886956i	$0.652815\pi$
$294$	0	0
$295$	5.56781	0.324171
$296$	0	0
$297$	−2.84218	−0.164920
$298$	0	0
$299$	−6.47547	−0.374486
$300$	0	0
$301$	−0.992083	−0.0571827
$302$	0	0
$303$	−13.9233	−0.799875
$304$	0	0
$305$	−7.36848	−0.421918
$306$	0	0
$307$	−7.89018	−0.450316	−0.225158	−	0.974322i	$-0.572290\pi$
$307$	−7.89018	−0.450316	−0.225158	+	0.974322i	$0.572290\pi$
$308$	0	0
$309$	6.37017	0.362386
$310$	0	0
$311$	−1.16429	−0.0660209	−0.0330104	−	0.999455i	$-0.510509\pi$
$311$	−1.16429	−0.0660209	−0.0330104	+	0.999455i	$0.510509\pi$
$312$	0	0
$313$	18.6213	1.05254	0.526270	−	0.850318i	$-0.323590\pi$
$313$	18.6213	1.05254	0.526270	+	0.850318i	$0.323590\pi$
$314$	0	0
$315$	−0.493644	−0.0278137
$316$	0	0
$317$	−13.0080	−0.730604	−0.365302	−	0.930889i	$-0.619034\pi$
$317$	−13.0080	−0.730604	−0.365302	+	0.930889i	$0.619034\pi$
$318$	0	0
$319$	9.91836	0.555321
$320$	0	0
$321$	−17.6646	−0.985943
$322$	0	0
$323$	−13.3740	−0.744148
$324$	0	0
$325$	−12.1533	−0.674145
$326$	0	0
$327$	8.22481	0.454833
$328$	0	0
$329$	−8.13412	−0.448448
$330$	0	0
$331$	−26.7117	−1.46821	−0.734104	−	0.679037i	$-0.762398\pi$
$331$	−26.7117	−1.46821	−0.734104	+	0.679037i	$0.762398\pi$
$332$	0	0
$333$	5.11215	0.280144
$334$	0	0
$335$	−1.37122	−0.0749180
$336$	0	0
$337$	−22.0790	−1.20272	−0.601361	−	0.798977i	$-0.705375\pi$
$337$	−22.0790	−1.20272	−0.601361	+	0.798977i	$0.705375\pi$
$338$	0	0
$339$	−9.93200	−0.539432
$340$	0	0
$341$	17.9055	0.969635
$342$	0	0
$343$	−10.6259	−0.573743
$344$	0	0
$345$	1.52688	0.0822045
$346$	0	0
$347$	3.69033	0.198107	0.0990537	−	0.995082i	$-0.468418\pi$
$347$	3.69033	0.198107	0.0990537	+	0.995082i	$0.468418\pi$
$348$	0	0
$349$	3.62219	0.193891	0.0969456	−	0.995290i	$-0.469093\pi$
$349$	3.62219	0.193891	0.0969456	+	0.995290i	$0.469093\pi$
$350$	0	0
$351$	2.63384	0.140584
$352$	0	0
$353$	34.2395	1.82239	0.911193	−	0.411980i	$-0.135163\pi$
$353$	34.2395	1.82239	0.911193	+	0.411980i	$0.135163\pi$
$354$	0	0
$355$	3.98479	0.211491
$356$	0	0
$357$	−2.42583	−0.128389
$358$	0	0
$359$	18.8199	0.993277	0.496638	−	0.867958i	$-0.334568\pi$
$359$	18.8199	0.993277	0.496638	+	0.867958i	$0.334568\pi$
$360$	0	0
$361$	0.203565	0.0107139
$362$	0	0
$363$	−2.92203	−0.153367
$364$	0	0
$365$	7.03085	0.368011
$366$	0	0
$367$	23.3544	1.21909	0.609545	−	0.792752i	$-0.291352\pi$
$367$	23.3544	1.21909	0.609545	+	0.792752i	$0.291352\pi$
$368$	0	0
$369$	9.70703	0.505328
$370$	0	0
$371$	3.18738	0.165481
$372$	0	0
$373$	−17.3416	−0.897916	−0.448958	−	0.893553i	$-0.648205\pi$
$373$	−17.3416	−0.897916	−0.448958	+	0.893553i	$0.648205\pi$
$374$	0	0
$375$	5.97091	0.308336
$376$	0	0
$377$	−9.19131	−0.473377
$378$	0	0
$379$	−25.5185	−1.31080	−0.655400	−	0.755282i	$-0.727500\pi$
$379$	−25.5185	−1.31080	−0.655400	+	0.755282i	$0.727500\pi$
$380$	0	0
$381$	−6.60003	−0.338130
$382$	0	0
$383$	2.66526	0.136189	0.0680943	−	0.997679i	$-0.478308\pi$
$383$	2.66526	0.136189	0.0680943	+	0.997679i	$0.478308\pi$
$384$	0	0
$385$	1.40302	0.0715047
$386$	0	0
$387$	−1.24812	−0.0634456
$388$	0	0
$389$	25.4752	1.29164	0.645821	−	0.763489i	$-0.276515\pi$
$389$	25.4752	1.29164	0.645821	+	0.763489i	$0.276515\pi$
$390$	0	0
$391$	7.50329	0.379458
$392$	0	0
$393$	−16.0837	−0.811316
$394$	0	0
$395$	8.94920	0.450283
$396$	0	0
$397$	−7.65386	−0.384136	−0.192068	−	0.981382i	$-0.561519\pi$
$397$	−7.65386	−0.384136	−0.192068	+	0.981382i	$0.561519\pi$
$398$	0	0
$399$	3.48323	0.174380
$400$	0	0
$401$	−30.5494	−1.52556	−0.762782	−	0.646656i	$-0.776167\pi$
$401$	−30.5494	−1.52556	−0.762782	+	0.646656i	$0.776167\pi$
$402$	0	0
$403$	−16.5929	−0.826553
$404$	0	0
$405$	−0.621044	−0.0308599
$406$	0	0
$407$	−14.5296	−0.720207
$408$	0	0
$409$	13.9875	0.691639	0.345820	−	0.938301i	$-0.387601\pi$
$409$	13.9875	0.691639	0.345820	+	0.938301i	$0.387601\pi$
$410$	0	0
$411$	10.4861	0.517244
$412$	0	0
$413$	−7.12612	−0.350654
$414$	0	0
$415$	0.703161	0.0345168
$416$	0	0
$417$	12.6126	0.617641
$418$	0	0
$419$	−15.5127	−0.757845	−0.378923	−	0.925428i	$-0.623705\pi$
$419$	−15.5127	−0.757845	−0.378923	+	0.925428i	$0.623705\pi$
$420$	0	0
$421$	−1.85083	−0.0902041	−0.0451021	−	0.998982i	$-0.514361\pi$
$421$	−1.85083	−0.0902041	−0.0451021	+	0.998982i	$0.514361\pi$
$422$	0	0
$423$	−10.2334	−0.497564
$424$	0	0
$425$	14.0824	0.683095
$426$	0	0
$427$	9.43076	0.456386
$428$	0	0
$429$	−7.48583	−0.361419
$430$	0	0
$431$	27.4008	1.31985	0.659925	−	0.751331i	$-0.270588\pi$
$431$	27.4008	1.31985	0.659925	+	0.751331i	$0.270588\pi$
$432$	0	0
$433$	−18.5212	−0.890072	−0.445036	−	0.895513i	$-0.646809\pi$
$433$	−18.5212	−0.890072	−0.445036	+	0.895513i	$0.646809\pi$
$434$	0	0
$435$	2.16726	0.103912
$436$	0	0
$437$	−10.7739	−0.515386
$438$	0	0
$439$	3.11323	0.148587	0.0742933	−	0.997236i	$-0.476330\pi$
$439$	3.11323	0.148587	0.0742933	+	0.997236i	$0.476330\pi$
$440$	0	0
$441$	−6.36820	−0.303247
$442$	0	0
$443$	−2.21928	−0.105441	−0.0527206	−	0.998609i	$-0.516789\pi$
$443$	−2.21928	−0.105441	−0.0527206	+	0.998609i	$0.516789\pi$
$444$	0	0
$445$	−3.65971	−0.173487
$446$	0	0
$447$	−16.6639	−0.788175
$448$	0	0
$449$	−1.54311	−0.0728239	−0.0364119	−	0.999337i	$-0.511593\pi$
$449$	−1.54311	−0.0728239	−0.0364119	+	0.999337i	$0.511593\pi$
$450$	0	0
$451$	−27.5891	−1.29912
$452$	0	0
$453$	24.0206	1.12859
$454$	0	0
$455$	−1.30018	−0.0609532
$456$	0	0
$457$	36.0950	1.68845	0.844227	−	0.535985i	$-0.180060\pi$
$457$	36.0950	1.68845	0.844227	+	0.535985i	$0.180060\pi$
$458$	0	0
$459$	−3.05189	−0.142450
$460$	0	0
$461$	13.2041	0.614975	0.307488	−	0.951552i	$-0.400512\pi$
$461$	13.2041	0.614975	0.307488	+	0.951552i	$0.400512\pi$
$462$	0	0
$463$	−1.13701	−0.0528413	−0.0264207	−	0.999651i	$-0.508411\pi$
$463$	−1.13701	−0.0528413	−0.0264207	+	0.999651i	$0.508411\pi$
$464$	0	0
$465$	3.91252	0.181439
$466$	0	0
$467$	−3.71519	−0.171918	−0.0859591	−	0.996299i	$-0.527395\pi$
$467$	−3.71519	−0.171918	−0.0859591	+	0.996299i	$0.527395\pi$
$468$	0	0
$469$	1.75500	0.0810385
$470$	0	0
$471$	3.81635	0.175848
$472$	0	0
$473$	3.54738	0.163109
$474$	0	0
$475$	−20.2207	−0.927791
$476$	0	0
$477$	4.00998	0.183605
$478$	0	0
$479$	41.8179	1.91071	0.955355	−	0.295461i	$-0.0954733\pi$
$479$	41.8179	1.91071	0.955355	+	0.295461i	$0.0954733\pi$
$480$	0	0
$481$	13.4646	0.613931
$482$	0	0
$483$	−1.95422	−0.0889202
$484$	0	0
$485$	−9.01620	−0.409404
$486$	0	0
$487$	−17.4450	−0.790510	−0.395255	−	0.918572i	$-0.629344\pi$
$487$	−17.4450	−0.790510	−0.395255	+	0.918572i	$0.629344\pi$
$488$	0	0
$489$	−21.2874	−0.962651
$490$	0	0
$491$	−31.5276	−1.42282	−0.711411	−	0.702776i	$-0.751944\pi$
$491$	−31.5276	−1.42282	−0.711411	+	0.702776i	$0.751944\pi$
$492$	0	0
$493$	10.6502	0.479661
$494$	0	0
$495$	1.76512	0.0793361
$496$	0	0
$497$	−5.10005	−0.228768
$498$	0	0
$499$	37.1601	1.66352	0.831758	−	0.555139i	$-0.187335\pi$
$499$	37.1601	1.66352	0.831758	+	0.555139i	$0.187335\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	29.5876	1.31925	0.659623	−	0.751597i	$-0.270716\pi$
$503$	29.5876	1.31925	0.659623	+	0.751597i	$0.270716\pi$
$504$	0	0
$505$	8.64701	0.384787
$506$	0	0
$507$	−6.06291	−0.269263
$508$	0	0
$509$	−9.46639	−0.419590	−0.209795	−	0.977745i	$-0.567280\pi$
$509$	−9.46639	−0.419590	−0.209795	+	0.977745i	$0.567280\pi$
$510$	0	0
$511$	−8.99863	−0.398076
$512$	0	0
$513$	4.38219	0.193478
$514$	0	0
$515$	−3.95615	−0.174329
$516$	0	0
$517$	29.0851	1.27916
$518$	0	0
$519$	10.4729	0.459710
$520$	0	0
$521$	−15.5910	−0.683052	−0.341526	−	0.939872i	$-0.610944\pi$
$521$	−15.5910	−0.683052	−0.341526	+	0.939872i	$0.610944\pi$
$522$	0	0
$523$	18.3369	0.801818	0.400909	−	0.916118i	$-0.368694\pi$
$523$	18.3369	0.801818	0.400909	+	0.916118i	$0.368694\pi$
$524$	0	0
$525$	−3.66773	−0.160073
$526$	0	0
$527$	19.2267	0.837526
$528$	0	0
$529$	−16.9554	−0.737193
$530$	0	0
$531$	−8.96524	−0.389058
$532$	0	0
$533$	25.5667	1.10742
$534$	0	0
$535$	10.9705	0.474296
$536$	0	0
$537$	18.7361	0.808522
$538$	0	0
$539$	18.0995	0.779602
$540$	0	0
$541$	28.9982	1.24673	0.623365	−	0.781931i	$-0.285765\pi$
$541$	28.9982	1.24673	0.623365	+	0.781931i	$0.285765\pi$
$542$	0	0
$543$	−11.6895	−0.501643
$544$	0	0
$545$	−5.10797	−0.218801
$546$	0	0
$547$	−1.08139	−0.0462370	−0.0231185	−	0.999733i	$-0.507360\pi$
$547$	−1.08139	−0.0462370	−0.0231185	+	0.999733i	$0.507360\pi$
$548$	0	0
$549$	11.8647	0.506371
$550$	0	0
$551$	−15.2925	−0.651484
$552$	0	0
$553$	−11.4539	−0.487069
$554$	0	0
$555$	−3.17487	−0.134766
$556$	0	0
$557$	−22.7196	−0.962660	−0.481330	−	0.876539i	$-0.659846\pi$
$557$	−22.7196	−0.962660	−0.481330	+	0.876539i	$0.659846\pi$
$558$	0	0
$559$	−3.28735	−0.139040
$560$	0	0
$561$	8.67402	0.366217
$562$	0	0
$563$	−42.7707	−1.80257	−0.901285	−	0.433226i	$-0.857375\pi$
$563$	−42.7707	−1.80257	−0.901285	+	0.433226i	$0.857375\pi$
$564$	0	0
$565$	6.16821	0.259499
$566$	0	0
$567$	0.794861	0.0333810
$568$	0	0
$569$	−33.6546	−1.41087	−0.705436	−	0.708774i	$-0.749249\pi$
$569$	−33.6546	−1.41087	−0.705436	+	0.708774i	$0.749249\pi$
$570$	0	0
$571$	−28.3486	−1.18635	−0.593176	−	0.805073i	$-0.702126\pi$
$571$	−28.3486	−1.18635	−0.593176	+	0.805073i	$0.702126\pi$
$572$	0	0
$573$	20.4506	0.854337
$574$	0	0
$575$	11.3446	0.473102
$576$	0	0
$577$	−7.89519	−0.328681	−0.164341	−	0.986404i	$-0.552550\pi$
$577$	−7.89519	−0.328681	−0.164341	+	0.986404i	$0.552550\pi$
$578$	0	0
$579$	−5.21519	−0.216736
$580$	0	0
$581$	−0.899961	−0.0373367
$582$	0	0
$583$	−11.3971	−0.472019
$584$	0	0
$585$	−1.63573	−0.0676290
$586$	0	0
$587$	−9.17853	−0.378838	−0.189419	−	0.981896i	$-0.560661\pi$
$587$	−9.17853	−0.378838	−0.189419	+	0.981896i	$0.560661\pi$
$588$	0	0
$589$	−27.6074	−1.13754
$590$	0	0
$591$	−3.40714	−0.140151
$592$	0	0
$593$	−38.5103	−1.58143	−0.790714	−	0.612186i	$-0.790290\pi$
$593$	−38.5103	−1.58143	−0.790714	+	0.612186i	$0.790290\pi$
$594$	0	0
$595$	1.50655	0.0617625
$596$	0	0
$597$	4.64991	0.190308
$598$	0	0
$599$	−1.51583	−0.0619353	−0.0309676	−	0.999520i	$-0.509859\pi$
$599$	−1.51583	−0.0619353	−0.0309676	+	0.999520i	$0.509859\pi$
$600$	0	0
$601$	−4.76821	−0.194499	−0.0972497	−	0.995260i	$-0.531005\pi$
$601$	−4.76821	−0.194499	−0.0972497	+	0.995260i	$0.531005\pi$
$602$	0	0
$603$	2.20793	0.0899141
$604$	0	0
$605$	1.81471	0.0737785
$606$	0	0
$607$	−40.8606	−1.65848	−0.829239	−	0.558894i	$-0.811226\pi$
$607$	−40.8606	−1.65848	−0.829239	+	0.558894i	$0.811226\pi$
$608$	0	0
$609$	−2.77383	−0.112401
$610$	0	0
$611$	−26.9530	−1.09040
$612$	0	0
$613$	14.6991	0.593689	0.296845	−	0.954926i	$-0.404066\pi$
$613$	14.6991	0.593689	0.296845	+	0.954926i	$0.404066\pi$
$614$	0	0
$615$	−6.02849	−0.243092
$616$	0	0
$617$	−41.4180	−1.66742	−0.833712	−	0.552199i	$-0.813789\pi$
$617$	−41.4180	−1.66742	−0.833712	+	0.552199i	$0.813789\pi$
$618$	0	0
$619$	1.05663	0.0424696	0.0212348	−	0.999775i	$-0.493240\pi$
$619$	1.05663	0.0424696	0.0212348	+	0.999775i	$0.493240\pi$
$620$	0	0
$621$	−2.45857	−0.0986590
$622$	0	0
$623$	4.68399	0.187660
$624$	0	0
$625$	19.3633	0.774533
$626$	0	0
$627$	−12.4549	−0.497403
$628$	0	0
$629$	−15.6017	−0.622082
$630$	0	0
$631$	33.6469	1.33946	0.669730	−	0.742604i	$-0.266410\pi$
$631$	33.6469	1.33946	0.669730	+	0.742604i	$0.266410\pi$
$632$	0	0
$633$	−21.6206	−0.859342
$634$	0	0
$635$	4.09891	0.162660
$636$	0	0
$637$	−16.7728	−0.664562
$638$	0	0
$639$	−6.41628	−0.253824
$640$	0	0
$641$	−20.6713	−0.816469	−0.408234	−	0.912877i	$-0.633855\pi$
$641$	−20.6713	−0.816469	−0.408234	+	0.912877i	$0.633855\pi$
$642$	0	0
$643$	−9.86401	−0.388999	−0.194499	−	0.980903i	$-0.562308\pi$
$643$	−9.86401	−0.388999	−0.194499	+	0.980903i	$0.562308\pi$
$644$	0	0
$645$	0.775138	0.0305210
$646$	0	0
$647$	−4.04226	−0.158918	−0.0794588	−	0.996838i	$-0.525319\pi$
$647$	−4.04226	−0.158918	−0.0794588	+	0.996838i	$0.525319\pi$
$648$	0	0
$649$	25.4808	1.00021
$650$	0	0
$651$	−5.00756	−0.196262
$652$	0	0
$653$	41.2332	1.61358	0.806789	−	0.590839i	$-0.201203\pi$
$653$	41.2332	1.61358	0.806789	+	0.590839i	$0.201203\pi$
$654$	0	0
$655$	9.98869	0.390290
$656$	0	0
$657$	−11.3210	−0.441675
$658$	0	0
$659$	−17.3330	−0.675200	−0.337600	−	0.941290i	$-0.609615\pi$
$659$	−17.3330	−0.675200	−0.337600	+	0.941290i	$0.609615\pi$
$660$	0	0
$661$	29.5660	1.14998	0.574992	−	0.818159i	$-0.305005\pi$
$661$	29.5660	1.14998	0.574992	+	0.818159i	$0.305005\pi$
$662$	0	0
$663$	−8.03819	−0.312177
$664$	0	0
$665$	−2.16324	−0.0838868
$666$	0	0
$667$	8.57968	0.332207
$668$	0	0
$669$	−23.5150	−0.909141
$670$	0	0
$671$	−33.7214	−1.30180
$672$	0	0
$673$	14.1439	0.545208	0.272604	−	0.962126i	$-0.412115\pi$
$673$	14.1439	0.545208	0.272604	+	0.962126i	$0.412115\pi$
$674$	0	0
$675$	−4.61430	−0.177605
$676$	0	0
$677$	7.77790	0.298929	0.149465	−	0.988767i	$-0.452245\pi$
$677$	7.77790	0.298929	0.149465	+	0.988767i	$0.452245\pi$
$678$	0	0
$679$	11.5396	0.442851
$680$	0	0
$681$	−7.98906	−0.306141
$682$	0	0
$683$	35.1474	1.34488	0.672438	−	0.740153i	$-0.265247\pi$
$683$	35.1474	1.34488	0.672438	+	0.740153i	$0.265247\pi$
$684$	0	0
$685$	−6.51236	−0.248825
$686$	0	0
$687$	−7.94269	−0.303033
$688$	0	0
$689$	10.5616	0.402366
$690$	0	0
$691$	−20.9594	−0.797334	−0.398667	−	0.917096i	$-0.630527\pi$
$691$	−20.9594	−0.797334	−0.398667	+	0.917096i	$0.630527\pi$
$692$	0	0
$693$	−2.25914	−0.0858175
$694$	0	0
$695$	−7.83297	−0.297121
$696$	0	0
$697$	−29.6248	−1.12212
$698$	0	0
$699$	−7.10575	−0.268764
$700$	0	0
$701$	−31.6708	−1.19619	−0.598095	−	0.801425i	$-0.704076\pi$
$701$	−31.6708	−1.19619	−0.598095	+	0.801425i	$0.704076\pi$
$702$	0	0
$703$	22.4024	0.844922
$704$	0	0
$705$	6.35538	0.239357
$706$	0	0
$707$	−11.0671	−0.416222
$708$	0	0
$709$	18.5157	0.695372	0.347686	−	0.937611i	$-0.386968\pi$
$709$	18.5157	0.695372	0.347686	+	0.937611i	$0.386968\pi$
$710$	0	0
$711$	−14.4099	−0.540414
$712$	0	0
$713$	15.4888	0.580059
$714$	0	0
$715$	4.64903	0.173864
$716$	0	0
$717$	−5.04207	−0.188299
$718$	0	0
$719$	−23.5636	−0.878775	−0.439387	−	0.898298i	$-0.644804\pi$
$719$	−23.5636	−0.878775	−0.439387	+	0.898298i	$0.644804\pi$
$720$	0	0
$721$	5.06340	0.188571
$722$	0	0
$723$	26.8655	0.999139
$724$	0	0
$725$	16.1026	0.598034
$726$	0	0
$727$	0.232584	0.00862605	0.00431303	−	0.999991i	$-0.498627\pi$
$727$	0.232584	0.00862605	0.00431303	+	0.999991i	$0.498627\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	3.80913	0.140886
$732$	0	0
$733$	−13.8530	−0.511671	−0.255835	−	0.966720i	$-0.582350\pi$
$733$	−13.8530	−0.511671	−0.255835	+	0.966720i	$0.582350\pi$
$734$	0	0
$735$	3.95493	0.145880
$736$	0	0
$737$	−6.27534	−0.231155
$738$	0	0
$739$	9.55748	0.351578	0.175789	−	0.984428i	$-0.443752\pi$
$739$	9.55748	0.351578	0.175789	+	0.984428i	$0.443752\pi$
$740$	0	0
$741$	11.5420	0.424005
$742$	0	0
$743$	13.2800	0.487198	0.243599	−	0.969876i	$-0.421672\pi$
$743$	13.2800	0.487198	0.243599	+	0.969876i	$0.421672\pi$
$744$	0	0
$745$	10.3490	0.379159
$746$	0	0
$747$	−1.13222	−0.0414259
$748$	0	0
$749$	−14.0409	−0.513044
$750$	0	0
$751$	0.633410	0.0231135	0.0115567	−	0.999933i	$-0.496321\pi$
$751$	0.633410	0.0231135	0.0115567	+	0.999933i	$0.496321\pi$
$752$	0	0
$753$	−20.6873	−0.753887
$754$	0	0
$755$	−14.9179	−0.542916
$756$	0	0
$757$	25.2180	0.916564	0.458282	−	0.888807i	$-0.348465\pi$
$757$	25.2180	0.916564	0.458282	+	0.888807i	$0.348465\pi$
$758$	0	0
$759$	6.98769	0.253637
$760$	0	0
$761$	−13.5704	−0.491927	−0.245963	−	0.969279i	$-0.579104\pi$
$761$	−13.5704	−0.491927	−0.245963	+	0.969279i	$0.579104\pi$
$762$	0	0
$763$	6.53758	0.236676
$764$	0	0
$765$	1.89536	0.0685269
$766$	0	0
$767$	−23.6130	−0.852615
$768$	0	0
$769$	10.6469	0.383937	0.191968	−	0.981401i	$-0.438513\pi$
$769$	10.6469	0.383937	0.191968	+	0.981401i	$0.438513\pi$
$770$	0	0
$771$	1.68043	0.0605190
$772$	0	0
$773$	16.5795	0.596323	0.298161	−	0.954515i	$-0.403627\pi$
$773$	16.5795	0.596323	0.298161	+	0.954515i	$0.403627\pi$
$774$	0	0
$775$	29.0697	1.04421
$776$	0	0
$777$	4.06345	0.145775
$778$	0	0
$779$	42.5380	1.52408
$780$	0	0
$781$	18.2362	0.652542
$782$	0	0
$783$	−3.48971	−0.124712
$784$	0	0
$785$	−2.37012	−0.0845932
$786$	0	0
$787$	33.8528	1.20672	0.603361	−	0.797468i	$-0.293828\pi$
$787$	33.8528	1.20672	0.603361	+	0.797468i	$0.293828\pi$
$788$	0	0
$789$	6.27570	0.223421
$790$	0	0
$791$	−7.89456	−0.280698
$792$	0	0
$793$	31.2496	1.10970
$794$	0	0
$795$	−2.49038	−0.0883246
$796$	0	0
$797$	30.2688	1.07217	0.536087	−	0.844163i	$-0.319902\pi$
$797$	30.2688	1.07217	0.536087	+	0.844163i	$0.319902\pi$
$798$	0	0
$799$	31.2312	1.10488
$800$	0	0
$801$	5.89284	0.208213
$802$	0	0
$803$	32.1763	1.13548
$804$	0	0
$805$	1.21366	0.0427758
$806$	0	0
$807$	−11.2734	−0.396841
$808$	0	0
$809$	−12.7724	−0.449053	−0.224527	−	0.974468i	$-0.572084\pi$
$809$	−12.7724	−0.449053	−0.224527	+	0.974468i	$0.572084\pi$
$810$	0	0
$811$	35.8718	1.25963	0.629815	−	0.776745i	$-0.283131\pi$
$811$	35.8718	1.25963	0.629815	+	0.776745i	$0.283131\pi$
$812$	0	0
$813$	−9.89342	−0.346977
$814$	0	0
$815$	13.2204	0.463092
$816$	0	0
$817$	−5.46950	−0.191354
$818$	0	0
$819$	2.09353	0.0731540
$820$	0	0
$821$	12.5343	0.437451	0.218725	−	0.975786i	$-0.429810\pi$
$821$	12.5343	0.437451	0.218725	+	0.975786i	$0.429810\pi$
$822$	0	0
$823$	49.8065	1.73614	0.868072	−	0.496438i	$-0.165359\pi$
$823$	49.8065	1.73614	0.868072	+	0.496438i	$0.165359\pi$
$824$	0	0
$825$	13.1147	0.456594
$826$	0	0
$827$	45.3448	1.57679	0.788396	−	0.615167i	$-0.210912\pi$
$827$	45.3448	1.57679	0.788396	+	0.615167i	$0.210912\pi$
$828$	0	0
$829$	22.0609	0.766208	0.383104	−	0.923705i	$-0.374855\pi$
$829$	22.0609	0.766208	0.383104	+	0.923705i	$0.374855\pi$
$830$	0	0
$831$	−10.9863	−0.381111
$832$	0	0
$833$	19.4351	0.673385
$834$	0	0
$835$	0.621044	0.0214921
$836$	0	0
$837$	−6.29991	−0.217757
$838$	0	0
$839$	−3.87102	−0.133643	−0.0668213	−	0.997765i	$-0.521286\pi$
$839$	−3.87102	−0.133643	−0.0668213	+	0.997765i	$0.521286\pi$
$840$	0	0
$841$	−16.8220	−0.580067
$842$	0	0
$843$	−8.69763	−0.299562
$844$	0	0
$845$	3.76533	0.129531
$846$	0	0
$847$	−2.32261	−0.0798059
$848$	0	0
$849$	15.4308	0.529582
$850$	0	0
$851$	−12.5686	−0.430845
$852$	0	0
$853$	−18.8991	−0.647093	−0.323546	−	0.946212i	$-0.604875\pi$
$853$	−18.8991	−0.647093	−0.323546	+	0.946212i	$0.604875\pi$
$854$	0	0
$855$	−2.72153	−0.0930744
$856$	0	0
$857$	−6.40858	−0.218913	−0.109456	−	0.993992i	$-0.534911\pi$
$857$	−6.40858	−0.218913	−0.109456	+	0.993992i	$0.534911\pi$
$858$	0	0
$859$	23.8218	0.812791	0.406395	−	0.913697i	$-0.366786\pi$
$859$	23.8218	0.812791	0.406395	+	0.913697i	$0.366786\pi$
$860$	0	0
$861$	7.71574	0.262952
$862$	0	0
$863$	−19.1576	−0.652131	−0.326066	−	0.945347i	$-0.605723\pi$
$863$	−19.1576	−0.652131	−0.326066	+	0.945347i	$0.605723\pi$
$864$	0	0
$865$	−6.50415	−0.221148
$866$	0	0
$867$	−7.68594	−0.261028
$868$	0	0
$869$	40.9555	1.38932
$870$	0	0
$871$	5.81534	0.197045
$872$	0	0
$873$	14.5178	0.491353
$874$	0	0
$875$	4.74604	0.160446
$876$	0	0
$877$	5.62216	0.189847	0.0949235	−	0.995485i	$-0.469739\pi$
$877$	5.62216	0.189847	0.0949235	+	0.995485i	$0.469739\pi$
$878$	0	0
$879$	−15.8113	−0.533302
$880$	0	0
$881$	−1.16679	−0.0393102	−0.0196551	−	0.999807i	$-0.506257\pi$
$881$	−1.16679	−0.0393102	−0.0196551	+	0.999807i	$0.506257\pi$
$882$	0	0
$883$	−40.1506	−1.35118	−0.675588	−	0.737280i	$-0.736110\pi$
$883$	−40.1506	−1.35118	−0.675588	+	0.737280i	$0.736110\pi$
$884$	0	0
$885$	5.56781	0.187160
$886$	0	0
$887$	3.66495	0.123057	0.0615284	−	0.998105i	$-0.480403\pi$
$887$	3.66495	0.123057	0.0615284	+	0.998105i	$0.480403\pi$
$888$	0	0
$889$	−5.24611	−0.175949
$890$	0	0
$891$	−2.84218	−0.0952165
$892$	0	0
$893$	−44.8446	−1.50067
$894$	0	0
$895$	−11.6359	−0.388946
$896$	0	0
$897$	−6.47547	−0.216210
$898$	0	0
$899$	21.9848	0.733235
$900$	0	0
$901$	−12.2380	−0.407708
$902$	0	0
$903$	−0.992083	−0.0330145
$904$	0	0
$905$	7.25967	0.241320
$906$	0	0
$907$	−3.46153	−0.114938	−0.0574691	−	0.998347i	$-0.518303\pi$
$907$	−3.46153	−0.114938	−0.0574691	+	0.998347i	$0.518303\pi$
$908$	0	0
$909$	−13.9233	−0.461808
$910$	0	0
$911$	19.0825	0.632230	0.316115	−	0.948721i	$-0.397621\pi$
$911$	19.0825	0.632230	0.316115	+	0.948721i	$0.397621\pi$
$912$	0	0
$913$	3.21798	0.106500
$914$	0	0
$915$	−7.36848	−0.243594
$916$	0	0
$917$	−12.7843	−0.422175
$918$	0	0
$919$	35.9246	1.18504	0.592521	−	0.805555i	$-0.298133\pi$
$919$	35.9246	1.18504	0.592521	+	0.805555i	$0.298133\pi$
$920$	0	0
$921$	−7.89018	−0.259990
$922$	0	0
$923$	−16.8994	−0.556251
$924$	0	0
$925$	−23.5890	−0.775602
$926$	0	0
$927$	6.37017	0.209224
$928$	0	0
$929$	14.5106	0.476076	0.238038	−	0.971256i	$-0.423496\pi$
$929$	14.5106	0.476076	0.238038	+	0.971256i	$0.423496\pi$
$930$	0	0
$931$	−27.9066	−0.914603
$932$	0	0
$933$	−1.16429	−0.0381172
$934$	0	0
$935$	−5.38695	−0.176172
$936$	0	0
$937$	−50.0606	−1.63541	−0.817705	−	0.575638i	$-0.804754\pi$
$937$	−50.0606	−1.63541	−0.817705	+	0.575638i	$0.804754\pi$
$938$	0	0
$939$	18.6213	0.607684
$940$	0	0
$941$	−19.9458	−0.650215	−0.325108	−	0.945677i	$-0.605401\pi$
$941$	−19.9458	−0.650215	−0.325108	+	0.945677i	$0.605401\pi$
$942$	0	0
$943$	−23.8654	−0.777164
$944$	0	0
$945$	−0.493644	−0.0160582
$946$	0	0
$947$	25.2352	0.820034	0.410017	−	0.912078i	$-0.365523\pi$
$947$	25.2352	0.820034	0.410017	+	0.912078i	$0.365523\pi$
$948$	0	0
$949$	−29.8177	−0.967923
$950$	0	0
$951$	−13.0080	−0.421814
$952$	0	0
$953$	52.5222	1.70136	0.850680	−	0.525683i	$-0.176190\pi$
$953$	52.5222	1.70136	0.850680	+	0.525683i	$0.176190\pi$
$954$	0	0
$955$	−12.7007	−0.410986
$956$	0	0
$957$	9.91836	0.320615
$958$	0	0
$959$	8.33503	0.269152
$960$	0	0
$961$	8.68888	0.280287
$962$	0	0
$963$	−17.6646	−0.569234
$964$	0	0
$965$	3.23887	0.104263
$966$	0	0
$967$	−44.1438	−1.41957	−0.709785	−	0.704419i	$-0.751208\pi$
$967$	−44.1438	−1.41957	−0.709785	+	0.704419i	$0.751208\pi$
$968$	0	0
$969$	−13.3740	−0.429634
$970$	0	0
$971$	−25.4750	−0.817530	−0.408765	−	0.912640i	$-0.634040\pi$
$971$	−25.4750	−0.817530	−0.408765	+	0.912640i	$0.634040\pi$
$972$	0	0
$973$	10.0252	0.321395
$974$	0	0
$975$	−12.1533	−0.389218
$976$	0	0
$977$	29.5203	0.944439	0.472220	−	0.881481i	$-0.343453\pi$
$977$	29.5203	0.944439	0.472220	+	0.881481i	$0.343453\pi$
$978$	0	0
$979$	−16.7485	−0.535284
$980$	0	0
$981$	8.22481	0.262598
$982$	0	0
$983$	27.3850	0.873447	0.436724	−	0.899596i	$-0.356139\pi$
$983$	27.3850	0.873447	0.436724	+	0.899596i	$0.356139\pi$
$984$	0	0
$985$	2.11599	0.0674209
$986$	0	0
$987$	−8.13412	−0.258912
$988$	0	0
$989$	3.06859	0.0975756
$990$	0	0
$991$	47.0384	1.49422	0.747112	−	0.664698i	$-0.231440\pi$
$991$	47.0384	1.49422	0.747112	+	0.664698i	$0.231440\pi$
$992$	0	0
$993$	−26.7117	−0.847670
$994$	0	0
$995$	−2.88780	−0.0915494
$996$	0	0
$997$	30.0374	0.951295	0.475647	−	0.879636i	$-0.342214\pi$
$997$	30.0374	0.951295	0.475647	+	0.879636i	$0.342214\pi$
$998$	0	0
$999$	5.11215	0.161741

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.z.1.3		8
4.3	odd	2		501.2.a.d.1.1	✓	8
12.11	even	2		1503.2.a.f.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
501.2.a.d.1.1	✓	8	4.3	odd	2
1503.2.a.f.1.8		8	12.11	even	2
8016.2.a.z.1.3		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.621044 q^{5} +0.794861 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.621044 q^{5} +0.794861 q^{7} +1.00000 q^{9} -2.84218 q^{11} +2.63384 q^{13} -0.621044 q^{15} -3.05189 q^{17} +4.38219 q^{19} +0.794861 q^{21} -2.45857 q^{23} -4.61430 q^{25} +1.00000 q^{27} -3.48971 q^{29} -6.29991 q^{31} -2.84218 q^{33} -0.493644 q^{35} +5.11215 q^{37} +2.63384 q^{39} +9.70703 q^{41} -1.24812 q^{43} -0.621044 q^{45} -10.2334 q^{47} -6.36820 q^{49} -3.05189 q^{51} +4.00998 q^{53} +1.76512 q^{55} +4.38219 q^{57} -8.96524 q^{59} +11.8647 q^{61} +0.794861 q^{63} -1.63573 q^{65} +2.20793 q^{67} -2.45857 q^{69} -6.41628 q^{71} -11.3210 q^{73} -4.61430 q^{75} -2.25914 q^{77} -14.4099 q^{79} +1.00000 q^{81} -1.13222 q^{83} +1.89536 q^{85} -3.48971 q^{87} +5.89284 q^{89} +2.09353 q^{91} -6.29991 q^{93} -2.72153 q^{95} +14.5178 q^{97} -2.84218 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} + q^{5} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^3 + q^5 + 8 * q^9	\( 8 q + 8 q^{3} + q^{5} + 8 q^{9} - 5 q^{11} + q^{15} - 7 q^{17} - 24 q^{19} - q^{23} + 3 q^{25} + 8 q^{27} - 11 q^{29} - 30 q^{31} - 5 q^{33} - 26 q^{35} + 11 q^{37} + 10 q^{41} - 24 q^{43} + q^{45} + 3 q^{47} + 6 q^{49} - 7 q^{51} - 25 q^{53} - 25 q^{55} - 24 q^{57} - 45 q^{59} + 16 q^{61} - 10 q^{65} - 18 q^{67} - q^{69} - 21 q^{71} - 8 q^{73} + 3 q^{75} - 18 q^{77} - 10 q^{79} + 8 q^{81} - 7 q^{83} - 11 q^{85} - 11 q^{87} + 26 q^{89} - 15 q^{91} - 30 q^{93} - q^{95} - 3 q^{97} - 5 q^{99}+O(q^{100}) \) 8 * q + 8 * q^3 + q^5 + 8 * q^9 - 5 * q^11 + q^15 - 7 * q^17 - 24 * q^19 - q^23 + 3 * q^25 + 8 * q^27 - 11 * q^29 - 30 * q^31 - 5 * q^33 - 26 * q^35 + 11 * q^37 + 10 * q^41 - 24 * q^43 + q^45 + 3 * q^47 + 6 * q^49 - 7 * q^51 - 25 * q^53 - 25 * q^55 - 24 * q^57 - 45 * q^59 + 16 * q^61 - 10 * q^65 - 18 * q^67 - q^69 - 21 * q^71 - 8 * q^73 + 3 * q^75 - 18 * q^77 - 10 * q^79 + 8 * q^81 - 7 * q^83 - 11 * q^85 - 11 * q^87 + 26 * q^89 - 15 * q^91 - 30 * q^93 - q^95 - 3 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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