LMFDB - Embedded newform 819.2.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [819,2,Mod(1,819)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(819, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("819.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 819 = 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	819.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:	$6.53974792554$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	819.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} -1.58579 q^{5} +1.00000 q^{7} -2.82843 q^{8} -2.24264 q^{10} -4.24264 q^{11} -1.00000 q^{13} +1.41421 q^{14} -4.00000 q^{16} -1.41421 q^{17} -7.24264 q^{19} -6.00000 q^{22} +5.82843 q^{23} -2.48528 q^{25} -1.41421 q^{26} -0.171573 q^{29} +3.24264 q^{31} -2.00000 q^{34} -1.58579 q^{35} +2.24264 q^{37} -10.2426 q^{38} +4.48528 q^{40} -8.82843 q^{41} -5.00000 q^{43} +8.24264 q^{46} -1.58579 q^{47} +1.00000 q^{49} -3.51472 q^{50} +0.171573 q^{53} +6.72792 q^{55} -2.82843 q^{56} -0.242641 q^{58} -0.343146 q^{59} +6.00000 q^{61} +4.58579 q^{62} +8.00000 q^{64} +1.58579 q^{65} -14.4853 q^{67} -2.24264 q^{70} +13.0711 q^{71} -9.24264 q^{73} +3.17157 q^{74} -4.24264 q^{77} +15.4853 q^{79} +6.34315 q^{80} -12.4853 q^{82} -13.2426 q^{83} +2.24264 q^{85} -7.07107 q^{86} +12.0000 q^{88} -1.58579 q^{89} -1.00000 q^{91} -2.24264 q^{94} +11.4853 q^{95} +11.7279 q^{97} +1.41421 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{5} + 2 q^{7} + 4 q^{10} - 2 q^{13} - 8 q^{16} - 6 q^{19} - 12 q^{22} + 6 q^{23} + 12 q^{25} - 6 q^{29} - 2 q^{31} - 4 q^{34} - 6 q^{35} - 4 q^{37} - 12 q^{38} - 8 q^{40} - 12 q^{41} - 10 q^{43}+ \cdots - 2 q^{97}+O(q^{100}) $ 2 * q - 6 * q^5 + 2 * q^7 + 4 * q^10 - 2 * q^13 - 8 * q^16 - 6 * q^19 - 12 * q^22 + 6 * q^23 + 12 * q^25 - 6 * q^29 - 2 * q^31 - 4 * q^34 - 6 * q^35 - 4 * q^37 - 12 * q^38 - 8 * q^40 - 12 * q^41 - 10 * q^43 + 8 * q^46 - 6 * q^47 + 2 * q^49 - 24 * q^50 + 6 * q^53 - 12 * q^55 + 8 * q^58 - 12 * q^59 + 12 * q^61 + 12 * q^62 + 16 * q^64 + 6 * q^65 - 12 * q^67 + 4 * q^70 + 12 * q^71 - 10 * q^73 + 12 * q^74 + 14 * q^79 + 24 * q^80 - 8 * q^82 - 18 * q^83 - 4 * q^85 + 24 * q^88 - 6 * q^89 - 2 * q^91 + 4 * q^94 + 6 * q^95 - 2 * q^97

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 $

$2$	1.41421	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$2$	1.41421	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.58579	−0.709185	−0.354593	−	0.935021i	$-0.615380\pi$
$5$	−1.58579	−0.709185	−0.354593	+	0.935021i	$0.615380\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.82843	−1.00000
$9$	0	0
$10$	−2.24264	−0.709185
$11$	−4.24264	−1.27920	−0.639602	−	0.768706i	$-0.720901\pi$
$11$	−4.24264	−1.27920	−0.639602	+	0.768706i	$0.720901\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	1.41421	0.377964
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−1.41421	−0.342997	−0.171499	−	0.985184i	$-0.554861\pi$
$17$	−1.41421	−0.342997	−0.171499	+	0.985184i	$0.554861\pi$
$18$	0	0
$19$	−7.24264	−1.66158	−0.830788	−	0.556589i	$-0.812110\pi$
$19$	−7.24264	−1.66158	−0.830788	+	0.556589i	$0.812110\pi$
$20$	0	0
$21$	0	0
$22$	−6.00000	−1.27920
$23$	5.82843	1.21531	0.607656	−	0.794201i	$-0.292110\pi$
$23$	5.82843	1.21531	0.607656	+	0.794201i	$0.292110\pi$
$24$	0	0
$25$	−2.48528	−0.497056
$26$	−1.41421	−0.277350
$27$	0	0
$28$	0	0
$29$	−0.171573	−0.0318603	−0.0159301	−	0.999873i	$-0.505071\pi$
$29$	−0.171573	−0.0318603	−0.0159301	+	0.999873i	$0.505071\pi$
$30$	0	0
$31$	3.24264	0.582395	0.291198	−	0.956663i	$-0.405946\pi$
$31$	3.24264	0.582395	0.291198	+	0.956663i	$0.405946\pi$
$32$	0	0
$33$	0	0
$34$	−2.00000	−0.342997
$35$	−1.58579	−0.268047
$36$	0	0
$37$	2.24264	0.368688	0.184344	−	0.982862i	$-0.440984\pi$
$37$	2.24264	0.368688	0.184344	+	0.982862i	$0.440984\pi$
$38$	−10.2426	−1.66158
$39$	0	0
$40$	4.48528	0.709185
$41$	−8.82843	−1.37877	−0.689384	−	0.724396i	$-0.742119\pi$
$41$	−8.82843	−1.37877	−0.689384	+	0.724396i	$0.742119\pi$
$42$	0	0
$43$	−5.00000	−0.762493	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	−5.00000	−0.762493	−0.381246	+	0.924473i	$0.624505\pi$
$44$	0	0
$45$	0	0
$46$	8.24264	1.21531
$47$	−1.58579	−0.231311	−0.115655	−	0.993289i	$-0.536897\pi$
$47$	−1.58579	−0.231311	−0.115655	+	0.993289i	$0.536897\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−3.51472	−0.497056
$51$	0	0
$52$	0	0
$53$	0.171573	0.0235673	0.0117837	−	0.999931i	$-0.496249\pi$
$53$	0.171573	0.0235673	0.0117837	+	0.999931i	$0.496249\pi$
$54$	0	0
$55$	6.72792	0.907193
$56$	−2.82843	−0.377964
$57$	0	0
$58$	−0.242641	−0.0318603
$59$	−0.343146	−0.0446738	−0.0223369	−	0.999751i	$-0.507111\pi$
$59$	−0.343146	−0.0446738	−0.0223369	+	0.999751i	$0.507111\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	4.58579	0.582395
$63$	0	0
$64$	8.00000	1.00000
$65$	1.58579	0.196693
$66$	0	0
$67$	−14.4853	−1.76966	−0.884829	−	0.465915i	$-0.845725\pi$
$67$	−14.4853	−1.76966	−0.884829	+	0.465915i	$0.845725\pi$
$68$	0	0
$69$	0	0
$70$	−2.24264	−0.268047
$71$	13.0711	1.55125	0.775625	−	0.631194i	$-0.217435\pi$
$71$	13.0711	1.55125	0.775625	+	0.631194i	$0.217435\pi$
$72$	0	0
$73$	−9.24264	−1.08177	−0.540885	−	0.841097i	$-0.681910\pi$
$73$	−9.24264	−1.08177	−0.540885	+	0.841097i	$0.681910\pi$
$74$	3.17157	0.368688
$75$	0	0
$76$	0	0
$77$	−4.24264	−0.483494
$78$	0	0
$79$	15.4853	1.74223	0.871115	−	0.491079i	$-0.163397\pi$
$79$	15.4853	1.74223	0.871115	+	0.491079i	$0.163397\pi$
$80$	6.34315	0.709185
$81$	0	0
$82$	−12.4853	−1.37877
$83$	−13.2426	−1.45357	−0.726784	−	0.686866i	$-0.758986\pi$
$83$	−13.2426	−1.45357	−0.726784	+	0.686866i	$0.758986\pi$
$84$	0	0
$85$	2.24264	0.243249
$86$	−7.07107	−0.762493
$87$	0	0
$88$	12.0000	1.27920
$89$	−1.58579	−0.168093	−0.0840465	−	0.996462i	$-0.526784\pi$
$89$	−1.58579	−0.168093	−0.0840465	+	0.996462i	$0.526784\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	0	0
$94$	−2.24264	−0.231311
$95$	11.4853	1.17837
$96$	0	0
$97$	11.7279	1.19079	0.595395	−	0.803433i	$-0.296996\pi$
$97$	11.7279	1.19079	0.595395	+	0.803433i	$0.296996\pi$
$98$	1.41421	0.142857
$99$	0	0
$100$	0	0
$101$	−10.2426	−1.01918	−0.509590	−	0.860417i	$-0.670203\pi$
$101$	−10.2426	−1.01918	−0.509590	+	0.860417i	$0.670203\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	2.82843	0.277350
$105$	0	0
$106$	0.242641	0.0235673
$107$	20.1421	1.94721	0.973607	−	0.228232i	$-0.0732943\pi$
$107$	20.1421	1.94721	0.973607	+	0.228232i	$0.0732943\pi$
$108$	0	0
$109$	16.7279	1.60224	0.801122	−	0.598501i	$-0.204237\pi$
$109$	16.7279	1.60224	0.801122	+	0.598501i	$0.204237\pi$
$110$	9.51472	0.907193
$111$	0	0
$112$	−4.00000	−0.377964
$113$	−2.31371	−0.217655	−0.108828	−	0.994061i	$-0.534710\pi$
$113$	−2.31371	−0.217655	−0.108828	+	0.994061i	$0.534710\pi$
$114$	0	0
$115$	−9.24264	−0.861881
$116$	0	0
$117$	0	0
$118$	−0.485281	−0.0446738
$119$	−1.41421	−0.129641
$120$	0	0
$121$	7.00000	0.636364
$122$	8.48528	0.768221
$123$	0	0
$124$	0	0
$125$	11.8701	1.06169
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	11.3137	1.00000
$129$	0	0
$130$	2.24264	0.196693
$131$	2.82843	0.247121	0.123560	−	0.992337i	$-0.460569\pi$
$131$	2.82843	0.247121	0.123560	+	0.992337i	$0.460569\pi$
$132$	0	0
$133$	−7.24264	−0.628017
$134$	−20.4853	−1.76966
$135$	0	0
$136$	4.00000	0.342997
$137$	4.58579	0.391790	0.195895	−	0.980625i	$-0.437239\pi$
$137$	4.58579	0.391790	0.195895	+	0.980625i	$0.437239\pi$
$138$	0	0
$139$	6.24264	0.529494	0.264747	−	0.964318i	$-0.414712\pi$
$139$	6.24264	0.529494	0.264747	+	0.964318i	$0.414712\pi$
$140$	0	0
$141$	0	0
$142$	18.4853	1.55125
$143$	4.24264	0.354787
$144$	0	0
$145$	0.272078	0.0225948
$146$	−13.0711	−1.08177
$147$	0	0
$148$	0	0
$149$	−16.2426	−1.33065	−0.665324	−	0.746554i	$-0.731707\pi$
$149$	−16.2426	−1.33065	−0.665324	+	0.746554i	$0.731707\pi$
$150$	0	0
$151$	−9.75736	−0.794043	−0.397021	−	0.917809i	$-0.629956\pi$
$151$	−9.75736	−0.794043	−0.397021	+	0.917809i	$0.629956\pi$
$152$	20.4853	1.66158
$153$	0	0
$154$	−6.00000	−0.483494
$155$	−5.14214	−0.413026
$156$	0	0
$157$	−3.75736	−0.299870	−0.149935	−	0.988696i	$-0.547906\pi$
$157$	−3.75736	−0.299870	−0.149935	+	0.988696i	$0.547906\pi$
$158$	21.8995	1.74223
$159$	0	0
$160$	0	0
$161$	5.82843	0.459344
$162$	0	0
$163$	8.48528	0.664619	0.332309	−	0.943170i	$-0.392172\pi$
$163$	8.48528	0.664619	0.332309	+	0.943170i	$0.392172\pi$
$164$	0	0
$165$	0	0
$166$	−18.7279	−1.45357
$167$	−15.3848	−1.19051	−0.595255	−	0.803537i	$-0.702949\pi$
$167$	−15.3848	−1.19051	−0.595255	+	0.803537i	$0.702949\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	3.17157	0.243249
$171$	0	0
$172$	0	0
$173$	−24.7279	−1.88003	−0.940015	−	0.341134i	$-0.889189\pi$
$173$	−24.7279	−1.88003	−0.940015	+	0.341134i	$0.889189\pi$
$174$	0	0
$175$	−2.48528	−0.187870
$176$	16.9706	1.27920
$177$	0	0
$178$	−2.24264	−0.168093
$179$	9.00000	0.672692	0.336346	−	0.941739i	$-0.390809\pi$
$179$	9.00000	0.672692	0.336346	+	0.941739i	$0.390809\pi$
$180$	0	0
$181$	−18.7279	−1.39204	−0.696018	−	0.718025i	$-0.745047\pi$
$181$	−18.7279	−1.39204	−0.696018	+	0.718025i	$0.745047\pi$
$182$	−1.41421	−0.104828
$183$	0	0
$184$	−16.4853	−1.21531
$185$	−3.55635	−0.261468
$186$	0	0
$187$	6.00000	0.438763
$188$	0	0
$189$	0	0
$190$	16.2426	1.17837
$191$	−15.1716	−1.09778	−0.548888	−	0.835896i	$-0.684949\pi$
$191$	−15.1716	−1.09778	−0.548888	+	0.835896i	$0.684949\pi$
$192$	0	0
$193$	−14.4853	−1.04267	−0.521337	−	0.853351i	$-0.674566\pi$
$193$	−14.4853	−1.04267	−0.521337	+	0.853351i	$0.674566\pi$
$194$	16.5858	1.19079
$195$	0	0
$196$	0	0
$197$	−12.3431	−0.879413	−0.439706	−	0.898142i	$-0.644917\pi$
$197$	−12.3431	−0.879413	−0.439706	+	0.898142i	$0.644917\pi$
$198$	0	0
$199$	−3.75736	−0.266352	−0.133176	−	0.991092i	$-0.542518\pi$
$199$	−3.75736	−0.266352	−0.133176	+	0.991092i	$0.542518\pi$
$200$	7.02944	0.497056
$201$	0	0
$202$	−14.4853	−1.01918
$203$	−0.171573	−0.0120421
$204$	0	0
$205$	14.0000	0.977802
$206$	−11.3137	−0.788263
$207$	0	0
$208$	4.00000	0.277350
$209$	30.7279	2.12549
$210$	0	0
$211$	−15.9706	−1.09946	−0.549729	−	0.835343i	$-0.685269\pi$
$211$	−15.9706	−1.09946	−0.549729	+	0.835343i	$0.685269\pi$
$212$	0	0
$213$	0	0
$214$	28.4853	1.94721
$215$	7.92893	0.540749
$216$	0	0
$217$	3.24264	0.220125
$218$	23.6569	1.60224
$219$	0	0
$220$	0	0
$221$	1.41421	0.0951303
$222$	0	0
$223$	0.757359	0.0507165	0.0253583	−	0.999678i	$-0.491927\pi$
$223$	0.757359	0.0507165	0.0253583	+	0.999678i	$0.491927\pi$
$224$	0	0
$225$	0	0
$226$	−3.27208	−0.217655
$227$	−26.8284	−1.78067	−0.890333	−	0.455311i	$-0.849528\pi$
$227$	−26.8284	−1.78067	−0.890333	+	0.455311i	$0.849528\pi$
$228$	0	0
$229$	29.4558	1.94650	0.973248	−	0.229755i	$-0.0737925\pi$
$229$	29.4558	1.94650	0.973248	+	0.229755i	$0.0737925\pi$
$230$	−13.0711	−0.861881
$231$	0	0
$232$	0.485281	0.0318603
$233$	14.6569	0.960202	0.480101	−	0.877213i	$-0.340600\pi$
$233$	14.6569	0.960202	0.480101	+	0.877213i	$0.340600\pi$
$234$	0	0
$235$	2.51472	0.164042
$236$	0	0
$237$	0	0
$238$	−2.00000	−0.129641
$239$	−3.51472	−0.227348	−0.113674	−	0.993518i	$-0.536262\pi$
$239$	−3.51472	−0.227348	−0.113674	+	0.993518i	$0.536262\pi$
$240$	0	0
$241$	−20.2132	−1.30205	−0.651023	−	0.759058i	$-0.725660\pi$
$241$	−20.2132	−1.30205	−0.651023	+	0.759058i	$0.725660\pi$
$242$	9.89949	0.636364
$243$	0	0
$244$	0	0
$245$	−1.58579	−0.101312
$246$	0	0
$247$	7.24264	0.460838
$248$	−9.17157	−0.582395
$249$	0	0
$250$	16.7868	1.06169
$251$	−16.5858	−1.04689	−0.523443	−	0.852061i	$-0.675353\pi$
$251$	−16.5858	−1.04689	−0.523443	+	0.852061i	$0.675353\pi$
$252$	0	0
$253$	−24.7279	−1.55463
$254$	2.82843	0.177471
$255$	0	0
$256$	0	0
$257$	19.4142	1.21103	0.605513	−	0.795836i	$-0.292968\pi$
$257$	19.4142	1.21103	0.605513	+	0.795836i	$0.292968\pi$
$258$	0	0
$259$	2.24264	0.139351
$260$	0	0
$261$	0	0
$262$	4.00000	0.247121
$263$	−1.97056	−0.121510	−0.0607551	−	0.998153i	$-0.519351\pi$
$263$	−1.97056	−0.121510	−0.0607551	+	0.998153i	$0.519351\pi$
$264$	0	0
$265$	−0.272078	−0.0167136
$266$	−10.2426	−0.628017
$267$	0	0
$268$	0	0
$269$	−9.17157	−0.559201	−0.279600	−	0.960116i	$-0.590202\pi$
$269$	−9.17157	−0.559201	−0.279600	+	0.960116i	$0.590202\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	5.65685	0.342997
$273$	0	0
$274$	6.48528	0.391790
$275$	10.5442	0.635837
$276$	0	0
$277$	−7.48528	−0.449747	−0.224873	−	0.974388i	$-0.572197\pi$
$277$	−7.48528	−0.449747	−0.224873	+	0.974388i	$0.572197\pi$
$278$	8.82843	0.529494
$279$	0	0
$280$	4.48528	0.268047
$281$	−15.5563	−0.928014	−0.464007	−	0.885832i	$-0.653589\pi$
$281$	−15.5563	−0.928014	−0.464007	+	0.885832i	$0.653589\pi$
$282$	0	0
$283$	−8.48528	−0.504398	−0.252199	−	0.967675i	$-0.581154\pi$
$283$	−8.48528	−0.504398	−0.252199	+	0.967675i	$0.581154\pi$
$284$	0	0
$285$	0	0
$286$	6.00000	0.354787
$287$	−8.82843	−0.521126
$288$	0	0
$289$	−15.0000	−0.882353
$290$	0.384776	0.0225948
$291$	0	0
$292$	0	0
$293$	−15.3848	−0.898788	−0.449394	−	0.893334i	$-0.648360\pi$
$293$	−15.3848	−0.898788	−0.449394	+	0.893334i	$0.648360\pi$
$294$	0	0
$295$	0.544156	0.0316820
$296$	−6.34315	−0.368688
$297$	0	0
$298$	−22.9706	−1.33065
$299$	−5.82843	−0.337067
$300$	0	0
$301$	−5.00000	−0.288195
$302$	−13.7990	−0.794043
$303$	0	0
$304$	28.9706	1.66158
$305$	−9.51472	−0.544811
$306$	0	0
$307$	13.2426	0.755797	0.377899	−	0.925847i	$-0.376647\pi$
$307$	13.2426	0.755797	0.377899	+	0.925847i	$0.376647\pi$
$308$	0	0
$309$	0	0
$310$	−7.27208	−0.413026
$311$	7.41421	0.420421	0.210211	−	0.977656i	$-0.432585\pi$
$311$	7.41421	0.420421	0.210211	+	0.977656i	$0.432585\pi$
$312$	0	0
$313$	23.2132	1.31209	0.656044	−	0.754723i	$-0.272229\pi$
$313$	23.2132	1.31209	0.656044	+	0.754723i	$0.272229\pi$
$314$	−5.31371	−0.299870
$315$	0	0
$316$	0	0
$317$	11.3137	0.635441	0.317721	−	0.948184i	$-0.397083\pi$
$317$	11.3137	0.635441	0.317721	+	0.948184i	$0.397083\pi$
$318$	0	0
$319$	0.727922	0.0407558
$320$	−12.6863	−0.709185
$321$	0	0
$322$	8.24264	0.459344
$323$	10.2426	0.569916
$324$	0	0
$325$	2.48528	0.137859
$326$	12.0000	0.664619
$327$	0	0
$328$	24.9706	1.37877
$329$	−1.58579	−0.0874272
$330$	0	0
$331$	−18.0000	−0.989369	−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000	−0.989369	−0.494685	+	0.869072i	$0.664716\pi$
$332$	0	0
$333$	0	0
$334$	−21.7574	−1.19051
$335$	22.9706	1.25502
$336$	0	0
$337$	−33.0000	−1.79762	−0.898812	−	0.438334i	$-0.855569\pi$
$337$	−33.0000	−1.79762	−0.898812	+	0.438334i	$0.855569\pi$
$338$	1.41421	0.0769231
$339$	0	0
$340$	0	0
$341$	−13.7574	−0.745003
$342$	0	0
$343$	1.00000	0.0539949
$344$	14.1421	0.762493
$345$	0	0
$346$	−34.9706	−1.88003
$347$	5.65685	0.303676	0.151838	−	0.988405i	$-0.451481\pi$
$347$	5.65685	0.303676	0.151838	+	0.988405i	$0.451481\pi$
$348$	0	0
$349$	−25.7279	−1.37718	−0.688592	−	0.725149i	$-0.741771\pi$
$349$	−25.7279	−1.37718	−0.688592	+	0.725149i	$0.741771\pi$
$350$	−3.51472	−0.187870
$351$	0	0
$352$	0	0
$353$	8.48528	0.451626	0.225813	−	0.974171i	$-0.427496\pi$
$353$	8.48528	0.451626	0.225813	+	0.974171i	$0.427496\pi$
$354$	0	0
$355$	−20.7279	−1.10012
$356$	0	0
$357$	0	0
$358$	12.7279	0.672692
$359$	27.8995	1.47248	0.736240	−	0.676721i	$-0.236600\pi$
$359$	27.8995	1.47248	0.736240	+	0.676721i	$0.236600\pi$
$360$	0	0
$361$	33.4558	1.76083
$362$	−26.4853	−1.39204
$363$	0	0
$364$	0	0
$365$	14.6569	0.767175
$366$	0	0
$367$	10.2426	0.534661	0.267331	−	0.963605i	$-0.413858\pi$
$367$	10.2426	0.534661	0.267331	+	0.963605i	$0.413858\pi$
$368$	−23.3137	−1.21531
$369$	0	0
$370$	−5.02944	−0.261468
$371$	0.171573	0.00890762
$372$	0	0
$373$	8.48528	0.439351	0.219676	−	0.975573i	$-0.429500\pi$
$373$	8.48528	0.439351	0.219676	+	0.975573i	$0.429500\pi$
$374$	8.48528	0.438763
$375$	0	0
$376$	4.48528	0.231311
$377$	0.171573	0.00883645
$378$	0	0
$379$	23.7574	1.22033	0.610167	−	0.792273i	$-0.291102\pi$
$379$	23.7574	1.22033	0.610167	+	0.792273i	$0.291102\pi$
$380$	0	0
$381$	0	0
$382$	−21.4558	−1.09778
$383$	20.4853	1.04675	0.523374	−	0.852103i	$-0.324673\pi$
$383$	20.4853	1.04675	0.523374	+	0.852103i	$0.324673\pi$
$384$	0	0
$385$	6.72792	0.342887
$386$	−20.4853	−1.04267
$387$	0	0
$388$	0	0
$389$	−29.6569	−1.50366	−0.751831	−	0.659356i	$-0.770829\pi$
$389$	−29.6569	−1.50366	−0.751831	+	0.659356i	$0.770829\pi$
$390$	0	0
$391$	−8.24264	−0.416848
$392$	−2.82843	−0.142857
$393$	0	0
$394$	−17.4558	−0.879413
$395$	−24.5563	−1.23556
$396$	0	0
$397$	−18.2132	−0.914094	−0.457047	−	0.889442i	$-0.651093\pi$
$397$	−18.2132	−0.914094	−0.457047	+	0.889442i	$0.651093\pi$
$398$	−5.31371	−0.266352
$399$	0	0
$400$	9.94113	0.497056
$401$	−6.34315	−0.316762	−0.158381	−	0.987378i	$-0.550627\pi$
$401$	−6.34315	−0.316762	−0.158381	+	0.987378i	$0.550627\pi$
$402$	0	0
$403$	−3.24264	−0.161527
$404$	0	0
$405$	0	0
$406$	−0.242641	−0.0120421
$407$	−9.51472	−0.471627
$408$	0	0
$409$	3.24264	0.160338	0.0801691	−	0.996781i	$-0.474454\pi$
$409$	3.24264	0.160338	0.0801691	+	0.996781i	$0.474454\pi$
$410$	19.7990	0.977802
$411$	0	0
$412$	0	0
$413$	−0.343146	−0.0168851
$414$	0	0
$415$	21.0000	1.03085
$416$	0	0
$417$	0	0
$418$	43.4558	2.12549
$419$	20.8701	1.01957	0.509785	−	0.860302i	$-0.329725\pi$
$419$	20.8701	1.01957	0.509785	+	0.860302i	$0.329725\pi$
$420$	0	0
$421$	6.72792	0.327899	0.163949	−	0.986469i	$-0.447577\pi$
$421$	6.72792	0.327899	0.163949	+	0.986469i	$0.447577\pi$
$422$	−22.5858	−1.09946
$423$	0	0
$424$	−0.485281	−0.0235673
$425$	3.51472	0.170489
$426$	0	0
$427$	6.00000	0.290360
$428$	0	0
$429$	0	0
$430$	11.2132	0.540749
$431$	12.3431	0.594548	0.297274	−	0.954792i	$-0.403922\pi$
$431$	12.3431	0.594548	0.297274	+	0.954792i	$0.403922\pi$
$432$	0	0
$433$	−24.9706	−1.20001	−0.600004	−	0.799997i	$-0.704834\pi$
$433$	−24.9706	−1.20001	−0.600004	+	0.799997i	$0.704834\pi$
$434$	4.58579	0.220125
$435$	0	0
$436$	0	0
$437$	−42.2132	−2.01933
$438$	0	0
$439$	−34.4853	−1.64589	−0.822946	−	0.568119i	$-0.807671\pi$
$439$	−34.4853	−1.64589	−0.822946	+	0.568119i	$0.807671\pi$
$440$	−19.0294	−0.907193
$441$	0	0
$442$	2.00000	0.0951303
$443$	3.68629	0.175141	0.0875705	−	0.996158i	$-0.472090\pi$
$443$	3.68629	0.175141	0.0875705	+	0.996158i	$0.472090\pi$
$444$	0	0
$445$	2.51472	0.119209
$446$	1.07107	0.0507165
$447$	0	0
$448$	8.00000	0.377964
$449$	−21.1716	−0.999148	−0.499574	−	0.866271i	$-0.666510\pi$
$449$	−21.1716	−0.999148	−0.499574	+	0.866271i	$0.666510\pi$
$450$	0	0
$451$	37.4558	1.76373
$452$	0	0
$453$	0	0
$454$	−37.9411	−1.78067
$455$	1.58579	0.0743428
$456$	0	0
$457$	−7.21320	−0.337419	−0.168710	−	0.985666i	$-0.553960\pi$
$457$	−7.21320	−0.337419	−0.168710	+	0.985666i	$0.553960\pi$
$458$	41.6569	1.94650
$459$	0	0
$460$	0	0
$461$	−8.82843	−0.411181	−0.205590	−	0.978638i	$-0.565911\pi$
$461$	−8.82843	−0.411181	−0.205590	+	0.978638i	$0.565911\pi$
$462$	0	0
$463$	−4.24264	−0.197172	−0.0985861	−	0.995129i	$-0.531432\pi$
$463$	−4.24264	−0.197172	−0.0985861	+	0.995129i	$0.531432\pi$
$464$	0.686292	0.0318603
$465$	0	0
$466$	20.7279	0.960202
$467$	3.89949	0.180447	0.0902236	−	0.995922i	$-0.471242\pi$
$467$	3.89949	0.180447	0.0902236	+	0.995922i	$0.471242\pi$
$468$	0	0
$469$	−14.4853	−0.668868
$470$	3.55635	0.164042
$471$	0	0
$472$	0.970563	0.0446738
$473$	21.2132	0.975384
$474$	0	0
$475$	18.0000	0.825897
$476$	0	0
$477$	0	0
$478$	−4.97056	−0.227348
$479$	6.21320	0.283889	0.141944	−	0.989875i	$-0.454665\pi$
$479$	6.21320	0.283889	0.141944	+	0.989875i	$0.454665\pi$
$480$	0	0
$481$	−2.24264	−0.102256
$482$	−28.5858	−1.30205
$483$	0	0
$484$	0	0
$485$	−18.5980	−0.844491
$486$	0	0
$487$	−11.4558	−0.519114	−0.259557	−	0.965728i	$-0.583577\pi$
$487$	−11.4558	−0.519114	−0.259557	+	0.965728i	$0.583577\pi$
$488$	−16.9706	−0.768221
$489$	0	0
$490$	−2.24264	−0.101312
$491$	−16.6274	−0.750385	−0.375192	−	0.926947i	$-0.622423\pi$
$491$	−16.6274	−0.750385	−0.375192	+	0.926947i	$0.622423\pi$
$492$	0	0
$493$	0.242641	0.0109280
$494$	10.2426	0.460838
$495$	0	0
$496$	−12.9706	−0.582395
$497$	13.0711	0.586318
$498$	0	0
$499$	−13.2721	−0.594140	−0.297070	−	0.954856i	$-0.596009\pi$
$499$	−13.2721	−0.594140	−0.297070	+	0.954856i	$0.596009\pi$
$500$	0	0
$501$	0	0
$502$	−23.4558	−1.04689
$503$	28.6274	1.27643	0.638217	−	0.769857i	$-0.279672\pi$
$503$	28.6274	1.27643	0.638217	+	0.769857i	$0.279672\pi$
$504$	0	0
$505$	16.2426	0.722788
$506$	−34.9706	−1.55463
$507$	0	0
$508$	0	0
$509$	5.10051	0.226076	0.113038	−	0.993591i	$-0.463942\pi$
$509$	5.10051	0.226076	0.113038	+	0.993591i	$0.463942\pi$
$510$	0	0
$511$	−9.24264	−0.408870
$512$	−22.6274	−1.00000
$513$	0	0
$514$	27.4558	1.21103
$515$	12.6863	0.559025
$516$	0	0
$517$	6.72792	0.295894
$518$	3.17157	0.139351
$519$	0	0
$520$	−4.48528	−0.196693
$521$	6.34315	0.277898	0.138949	−	0.990300i	$-0.455628\pi$
$521$	6.34315	0.277898	0.138949	+	0.990300i	$0.455628\pi$
$522$	0	0
$523$	30.9706	1.35425	0.677124	−	0.735869i	$-0.263226\pi$
$523$	30.9706	1.35425	0.677124	+	0.735869i	$0.263226\pi$
$524$	0	0
$525$	0	0
$526$	−2.78680	−0.121510
$527$	−4.58579	−0.199760
$528$	0	0
$529$	10.9706	0.476981
$530$	−0.384776	−0.0167136
$531$	0	0
$532$	0	0
$533$	8.82843	0.382402
$534$	0	0
$535$	−31.9411	−1.38094
$536$	40.9706	1.76966
$537$	0	0
$538$	−12.9706	−0.559201
$539$	−4.24264	−0.182743
$540$	0	0
$541$	−7.21320	−0.310120	−0.155060	−	0.987905i	$-0.549557\pi$
$541$	−7.21320	−0.310120	−0.155060	+	0.987905i	$0.549557\pi$
$542$	28.2843	1.21491
$543$	0	0
$544$	0	0
$545$	−26.5269	−1.13629
$546$	0	0
$547$	35.4853	1.51724	0.758621	−	0.651533i	$-0.225874\pi$
$547$	35.4853	1.51724	0.758621	+	0.651533i	$0.225874\pi$
$548$	0	0
$549$	0	0
$550$	14.9117	0.635837
$551$	1.24264	0.0529383
$552$	0	0
$553$	15.4853	0.658501
$554$	−10.5858	−0.449747
$555$	0	0
$556$	0	0
$557$	−12.0000	−0.508456	−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000	−0.508456	−0.254228	+	0.967144i	$0.581821\pi$
$558$	0	0
$559$	5.00000	0.211477
$560$	6.34315	0.268047
$561$	0	0
$562$	−22.0000	−0.928014
$563$	6.34315	0.267332	0.133666	−	0.991026i	$-0.457325\pi$
$563$	6.34315	0.267332	0.133666	+	0.991026i	$0.457325\pi$
$564$	0	0
$565$	3.66905	0.154358
$566$	−12.0000	−0.504398
$567$	0	0
$568$	−36.9706	−1.55125
$569$	−5.14214	−0.215570	−0.107785	−	0.994174i	$-0.534376\pi$
$569$	−5.14214	−0.215570	−0.107785	+	0.994174i	$0.534376\pi$
$570$	0	0
$571$	42.4558	1.77672	0.888361	−	0.459146i	$-0.151844\pi$
$571$	42.4558	1.77672	0.888361	+	0.459146i	$0.151844\pi$
$572$	0	0
$573$	0	0
$574$	−12.4853	−0.521126
$575$	−14.4853	−0.604078
$576$	0	0
$577$	−6.97056	−0.290188	−0.145094	−	0.989418i	$-0.546349\pi$
$577$	−6.97056	−0.290188	−0.145094	+	0.989418i	$0.546349\pi$
$578$	−21.2132	−0.882353
$579$	0	0
$580$	0	0
$581$	−13.2426	−0.549397
$582$	0	0
$583$	−0.727922	−0.0301475
$584$	26.1421	1.08177
$585$	0	0
$586$	−21.7574	−0.898788
$587$	6.55635	0.270609	0.135305	−	0.990804i	$-0.456799\pi$
$587$	6.55635	0.270609	0.135305	+	0.990804i	$0.456799\pi$
$588$	0	0
$589$	−23.4853	−0.967694
$590$	0.769553	0.0316820
$591$	0	0
$592$	−8.97056	−0.368688
$593$	0.556349	0.0228465	0.0114233	−	0.999935i	$-0.496364\pi$
$593$	0.556349	0.0228465	0.0114233	+	0.999935i	$0.496364\pi$
$594$	0	0
$595$	2.24264	0.0919393
$596$	0	0
$597$	0	0
$598$	−8.24264	−0.337067
$599$	28.7990	1.17669	0.588347	−	0.808608i	$-0.299779\pi$
$599$	28.7990	1.17669	0.588347	+	0.808608i	$0.299779\pi$
$600$	0	0
$601$	−38.9706	−1.58964	−0.794821	−	0.606844i	$-0.792435\pi$
$601$	−38.9706	−1.58964	−0.794821	+	0.606844i	$0.792435\pi$
$602$	−7.07107	−0.288195
$603$	0	0
$604$	0	0
$605$	−11.1005	−0.451300
$606$	0	0
$607$	−26.7279	−1.08485	−0.542426	−	0.840103i	$-0.682494\pi$
$607$	−26.7279	−1.08485	−0.542426	+	0.840103i	$0.682494\pi$
$608$	0	0
$609$	0	0
$610$	−13.4558	−0.544811
$611$	1.58579	0.0641541
$612$	0	0
$613$	−16.0000	−0.646234	−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000	−0.646234	−0.323117	+	0.946359i	$0.604731\pi$
$614$	18.7279	0.755797
$615$	0	0
$616$	12.0000	0.483494
$617$	−12.3431	−0.496916	−0.248458	−	0.968643i	$-0.579924\pi$
$617$	−12.3431	−0.496916	−0.248458	+	0.968643i	$0.579924\pi$
$618$	0	0
$619$	0.970563	0.0390102	0.0195051	−	0.999810i	$-0.493791\pi$
$619$	0.970563	0.0390102	0.0195051	+	0.999810i	$0.493791\pi$
$620$	0	0
$621$	0	0
$622$	10.4853	0.420421
$623$	−1.58579	−0.0635332
$624$	0	0
$625$	−6.39697	−0.255879
$626$	32.8284	1.31209
$627$	0	0
$628$	0	0
$629$	−3.17157	−0.126459
$630$	0	0
$631$	2.00000	0.0796187	0.0398094	−	0.999207i	$-0.487325\pi$
$631$	2.00000	0.0796187	0.0398094	+	0.999207i	$0.487325\pi$
$632$	−43.7990	−1.74223
$633$	0	0
$634$	16.0000	0.635441
$635$	−3.17157	−0.125860
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	1.02944	0.0407558
$639$	0	0
$640$	−17.9411	−0.709185
$641$	−15.3431	−0.606018	−0.303009	−	0.952988i	$-0.597991\pi$
$641$	−15.3431	−0.606018	−0.303009	+	0.952988i	$0.597991\pi$
$642$	0	0
$643$	−12.4853	−0.492371	−0.246186	−	0.969223i	$-0.579177\pi$
$643$	−12.4853	−0.492371	−0.246186	+	0.969223i	$0.579177\pi$
$644$	0	0
$645$	0	0
$646$	14.4853	0.569916
$647$	−14.5269	−0.571112	−0.285556	−	0.958362i	$-0.592178\pi$
$647$	−14.5269	−0.571112	−0.285556	+	0.958362i	$0.592178\pi$
$648$	0	0
$649$	1.45584	0.0571469
$650$	3.51472	0.137859
$651$	0	0
$652$	0	0
$653$	−17.3137	−0.677538	−0.338769	−	0.940870i	$-0.610010\pi$
$653$	−17.3137	−0.677538	−0.338769	+	0.940870i	$0.610010\pi$
$654$	0	0
$655$	−4.48528	−0.175254
$656$	35.3137	1.37877
$657$	0	0
$658$	−2.24264	−0.0874272
$659$	15.3431	0.597684	0.298842	−	0.954303i	$-0.403400\pi$
$659$	15.3431	0.597684	0.298842	+	0.954303i	$0.403400\pi$
$660$	0	0
$661$	−10.7574	−0.418413	−0.209206	−	0.977872i	$-0.567088\pi$
$661$	−10.7574	−0.418413	−0.209206	+	0.977872i	$0.567088\pi$
$662$	−25.4558	−0.989369
$663$	0	0
$664$	37.4558	1.45357
$665$	11.4853	0.445380
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	0	0
$670$	32.4853	1.25502
$671$	−25.4558	−0.982712
$672$	0	0
$673$	−26.9411	−1.03850	−0.519252	−	0.854621i	$-0.673789\pi$
$673$	−26.9411	−1.03850	−0.519252	+	0.854621i	$0.673789\pi$
$674$	−46.6690	−1.79762
$675$	0	0
$676$	0	0
$677$	41.3553	1.58941	0.794707	−	0.606993i	$-0.207624\pi$
$677$	41.3553	1.58941	0.794707	+	0.606993i	$0.207624\pi$
$678$	0	0
$679$	11.7279	0.450076
$680$	−6.34315	−0.243249
$681$	0	0
$682$	−19.4558	−0.745003
$683$	21.1716	0.810108	0.405054	−	0.914293i	$-0.367253\pi$
$683$	21.1716	0.810108	0.405054	+	0.914293i	$0.367253\pi$
$684$	0	0
$685$	−7.27208	−0.277852
$686$	1.41421	0.0539949
$687$	0	0
$688$	20.0000	0.762493
$689$	−0.171573	−0.00653641
$690$	0	0
$691$	28.6985	1.09174	0.545871	−	0.837869i	$-0.316199\pi$
$691$	28.6985	1.09174	0.545871	+	0.837869i	$0.316199\pi$
$692$	0	0
$693$	0	0
$694$	8.00000	0.303676
$695$	−9.89949	−0.375509
$696$	0	0
$697$	12.4853	0.472914
$698$	−36.3848	−1.37718
$699$	0	0
$700$	0	0
$701$	10.7990	0.407872	0.203936	−	0.978984i	$-0.434627\pi$
$701$	10.7990	0.407872	0.203936	+	0.978984i	$0.434627\pi$
$702$	0	0
$703$	−16.2426	−0.612603
$704$	−33.9411	−1.27920
$705$	0	0
$706$	12.0000	0.451626
$707$	−10.2426	−0.385214
$708$	0	0
$709$	0.727922	0.0273377	0.0136688	−	0.999907i	$-0.495649\pi$
$709$	0.727922	0.0273377	0.0136688	+	0.999907i	$0.495649\pi$
$710$	−29.3137	−1.10012
$711$	0	0
$712$	4.48528	0.168093
$713$	18.8995	0.707792
$714$	0	0
$715$	−6.72792	−0.251610
$716$	0	0
$717$	0	0
$718$	39.4558	1.47248
$719$	1.75736	0.0655384	0.0327692	−	0.999463i	$-0.489567\pi$
$719$	1.75736	0.0655384	0.0327692	+	0.999463i	$0.489567\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	47.3137	1.76083
$723$	0	0
$724$	0	0
$725$	0.426407	0.0158364
$726$	0	0
$727$	42.0000	1.55769	0.778847	−	0.627214i	$-0.215805\pi$
$727$	42.0000	1.55769	0.778847	+	0.627214i	$0.215805\pi$
$728$	2.82843	0.104828
$729$	0	0
$730$	20.7279	0.767175
$731$	7.07107	0.261533
$732$	0	0
$733$	−16.6985	−0.616773	−0.308386	−	0.951261i	$-0.599789\pi$
$733$	−16.6985	−0.616773	−0.308386	+	0.951261i	$0.599789\pi$
$734$	14.4853	0.534661
$735$	0	0
$736$	0	0
$737$	61.4558	2.26376
$738$	0	0
$739$	−17.6985	−0.651049	−0.325525	−	0.945534i	$-0.605541\pi$
$739$	−17.6985	−0.651049	−0.325525	+	0.945534i	$0.605541\pi$
$740$	0	0
$741$	0	0
$742$	0.242641	0.00890762
$743$	29.6569	1.08800	0.544002	−	0.839084i	$-0.316908\pi$
$743$	29.6569	1.08800	0.544002	+	0.839084i	$0.316908\pi$
$744$	0	0
$745$	25.7574	0.943677
$746$	12.0000	0.439351
$747$	0	0
$748$	0	0
$749$	20.1421	0.735978
$750$	0	0
$751$	15.4853	0.565066	0.282533	−	0.959258i	$-0.408825\pi$
$751$	15.4853	0.565066	0.282533	+	0.959258i	$0.408825\pi$
$752$	6.34315	0.231311
$753$	0	0
$754$	0.242641	0.00883645
$755$	15.4731	0.563123
$756$	0	0
$757$	21.4853	0.780896	0.390448	−	0.920625i	$-0.372320\pi$
$757$	21.4853	0.780896	0.390448	+	0.920625i	$0.372320\pi$
$758$	33.5980	1.22033
$759$	0	0
$760$	−32.4853	−1.17837
$761$	−34.7574	−1.25995	−0.629977	−	0.776614i	$-0.716936\pi$
$761$	−34.7574	−1.25995	−0.629977	+	0.776614i	$0.716936\pi$
$762$	0	0
$763$	16.7279	0.605591
$764$	0	0
$765$	0	0
$766$	28.9706	1.04675
$767$	0.343146	0.0123903
$768$	0	0
$769$	52.2132	1.88286	0.941428	−	0.337214i	$-0.109484\pi$
$769$	52.2132	1.88286	0.941428	+	0.337214i	$0.109484\pi$
$770$	9.51472	0.342887
$771$	0	0
$772$	0	0
$773$	32.8284	1.18076	0.590378	−	0.807127i	$-0.298979\pi$
$773$	32.8284	1.18076	0.590378	+	0.807127i	$0.298979\pi$
$774$	0	0
$775$	−8.05887	−0.289483
$776$	−33.1716	−1.19079
$777$	0	0
$778$	−41.9411	−1.50366
$779$	63.9411	2.29093
$780$	0	0
$781$	−55.4558	−1.98437
$782$	−11.6569	−0.416848
$783$	0	0
$784$	−4.00000	−0.142857
$785$	5.95837	0.212663
$786$	0	0
$787$	−24.7574	−0.882505	−0.441252	−	0.897383i	$-0.645466\pi$
$787$	−24.7574	−0.882505	−0.441252	+	0.897383i	$0.645466\pi$
$788$	0	0
$789$	0	0
$790$	−34.7279	−1.23556
$791$	−2.31371	−0.0822660
$792$	0	0
$793$	−6.00000	−0.213066
$794$	−25.7574	−0.914094
$795$	0	0
$796$	0	0
$797$	24.3431	0.862278	0.431139	−	0.902285i	$-0.358112\pi$
$797$	24.3431	0.862278	0.431139	+	0.902285i	$0.358112\pi$
$798$	0	0
$799$	2.24264	0.0793389
$800$	0	0
$801$	0	0
$802$	−8.97056	−0.316762
$803$	39.2132	1.38380
$804$	0	0
$805$	−9.24264	−0.325760
$806$	−4.58579	−0.161527
$807$	0	0
$808$	28.9706	1.01918
$809$	−17.4853	−0.614750	−0.307375	−	0.951589i	$-0.599451\pi$
$809$	−17.4853	−0.614750	−0.307375	+	0.951589i	$0.599451\pi$
$810$	0	0
$811$	−21.9411	−0.770457	−0.385229	−	0.922821i	$-0.625877\pi$
$811$	−21.9411	−0.770457	−0.385229	+	0.922821i	$0.625877\pi$
$812$	0	0
$813$	0	0
$814$	−13.4558	−0.471627
$815$	−13.4558	−0.471338
$816$	0	0
$817$	36.2132	1.26694
$818$	4.58579	0.160338
$819$	0	0
$820$	0	0
$821$	20.1421	0.702965	0.351483	−	0.936194i	$-0.385678\pi$
$821$	20.1421	0.702965	0.351483	+	0.936194i	$0.385678\pi$
$822$	0	0
$823$	10.4853	0.365494	0.182747	−	0.983160i	$-0.441501\pi$
$823$	10.4853	0.365494	0.182747	+	0.983160i	$0.441501\pi$
$824$	22.6274	0.788263
$825$	0	0
$826$	−0.485281	−0.0168851
$827$	49.4558	1.71975	0.859874	−	0.510506i	$-0.170542\pi$
$827$	49.4558	1.71975	0.859874	+	0.510506i	$0.170542\pi$
$828$	0	0
$829$	50.7279	1.76185	0.880927	−	0.473253i	$-0.156920\pi$
$829$	50.7279	1.76185	0.880927	+	0.473253i	$0.156920\pi$
$830$	29.6985	1.03085
$831$	0	0
$832$	−8.00000	−0.277350
$833$	−1.41421	−0.0489996
$834$	0	0
$835$	24.3970	0.844292
$836$	0	0
$837$	0	0
$838$	29.5147	1.01957
$839$	33.1716	1.14521	0.572605	−	0.819831i	$-0.305933\pi$
$839$	33.1716	1.14521	0.572605	+	0.819831i	$0.305933\pi$
$840$	0	0
$841$	−28.9706	−0.998985
$842$	9.51472	0.327899
$843$	0	0
$844$	0	0
$845$	−1.58579	−0.0545527
$846$	0	0
$847$	7.00000	0.240523
$848$	−0.686292	−0.0235673
$849$	0	0
$850$	4.97056	0.170489
$851$	13.0711	0.448070
$852$	0	0
$853$	39.7279	1.36026	0.680129	−	0.733092i	$-0.261924\pi$
$853$	39.7279	1.36026	0.680129	+	0.733092i	$0.261924\pi$
$854$	8.48528	0.290360
$855$	0	0
$856$	−56.9706	−1.94721
$857$	54.7696	1.87089	0.935446	−	0.353469i	$-0.114998\pi$
$857$	54.7696	1.87089	0.935446	+	0.353469i	$0.114998\pi$
$858$	0	0
$859$	30.9706	1.05670	0.528351	−	0.849026i	$-0.322811\pi$
$859$	30.9706	1.05670	0.528351	+	0.849026i	$0.322811\pi$
$860$	0	0
$861$	0	0
$862$	17.4558	0.594548
$863$	−28.2843	−0.962808	−0.481404	−	0.876499i	$-0.659873\pi$
$863$	−28.2843	−0.962808	−0.481404	+	0.876499i	$0.659873\pi$
$864$	0	0
$865$	39.2132	1.33329
$866$	−35.3137	−1.20001
$867$	0	0
$868$	0	0
$869$	−65.6985	−2.22867
$870$	0	0
$871$	14.4853	0.490815
$872$	−47.3137	−1.60224
$873$	0	0
$874$	−59.6985	−2.01933
$875$	11.8701	0.401281
$876$	0	0
$877$	−10.2426	−0.345869	−0.172935	−	0.984933i	$-0.555325\pi$
$877$	−10.2426	−0.345869	−0.172935	+	0.984933i	$0.555325\pi$
$878$	−48.7696	−1.64589
$879$	0	0
$880$	−26.9117	−0.907193
$881$	−13.1127	−0.441778	−0.220889	−	0.975299i	$-0.570896\pi$
$881$	−13.1127	−0.441778	−0.220889	+	0.975299i	$0.570896\pi$
$882$	0	0
$883$	2.00000	0.0673054	0.0336527	−	0.999434i	$-0.489286\pi$
$883$	2.00000	0.0673054	0.0336527	+	0.999434i	$0.489286\pi$
$884$	0	0
$885$	0	0
$886$	5.21320	0.175141
$887$	−19.1127	−0.641742	−0.320871	−	0.947123i	$-0.603976\pi$
$887$	−19.1127	−0.641742	−0.320871	+	0.947123i	$0.603976\pi$
$888$	0	0
$889$	2.00000	0.0670778
$890$	3.55635	0.119209
$891$	0	0
$892$	0	0
$893$	11.4853	0.384340
$894$	0	0
$895$	−14.2721	−0.477063
$896$	11.3137	0.377964
$897$	0	0
$898$	−29.9411	−0.999148
$899$	−0.556349	−0.0185553
$900$	0	0
$901$	−0.242641	−0.00808353
$902$	52.9706	1.76373
$903$	0	0
$904$	6.54416	0.217655
$905$	29.6985	0.987211
$906$	0	0
$907$	1.97056	0.0654315	0.0327157	−	0.999465i	$-0.489584\pi$
$907$	1.97056	0.0654315	0.0327157	+	0.999465i	$0.489584\pi$
$908$	0	0
$909$	0	0
$910$	2.24264	0.0743428
$911$	43.9706	1.45681	0.728405	−	0.685147i	$-0.240262\pi$
$911$	43.9706	1.45681	0.728405	+	0.685147i	$0.240262\pi$
$912$	0	0
$913$	56.1838	1.85941
$914$	−10.2010	−0.337419
$915$	0	0
$916$	0	0
$917$	2.82843	0.0934029
$918$	0	0
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	26.1421	0.861881
$921$	0	0
$922$	−12.4853	−0.411181
$923$	−13.0711	−0.430239
$924$	0	0
$925$	−5.57359	−0.183259
$926$	−6.00000	−0.197172
$927$	0	0
$928$	0	0
$929$	−36.8995	−1.21063	−0.605317	−	0.795985i	$-0.706953\pi$
$929$	−36.8995	−1.21063	−0.605317	+	0.795985i	$0.706953\pi$
$930$	0	0
$931$	−7.24264	−0.237368
$932$	0	0
$933$	0	0
$934$	5.51472	0.180447
$935$	−9.51472	−0.311165
$936$	0	0
$937$	45.2132	1.47705	0.738525	−	0.674226i	$-0.235522\pi$
$937$	45.2132	1.47705	0.738525	+	0.674226i	$0.235522\pi$
$938$	−20.4853	−0.668868
$939$	0	0
$940$	0	0
$941$	−40.0711	−1.30628	−0.653140	−	0.757237i	$-0.726549\pi$
$941$	−40.0711	−1.30628	−0.653140	+	0.757237i	$0.726549\pi$
$942$	0	0
$943$	−51.4558	−1.67563
$944$	1.37258	0.0446738
$945$	0	0
$946$	30.0000	0.975384
$947$	−15.1716	−0.493010	−0.246505	−	0.969142i	$-0.579282\pi$
$947$	−15.1716	−0.493010	−0.246505	+	0.969142i	$0.579282\pi$
$948$	0	0
$949$	9.24264	0.300029
$950$	25.4558	0.825897
$951$	0	0
$952$	4.00000	0.129641
$953$	−11.1421	−0.360929	−0.180465	−	0.983581i	$-0.557760\pi$
$953$	−11.1421	−0.360929	−0.180465	+	0.983581i	$0.557760\pi$
$954$	0	0
$955$	24.0589	0.778527
$956$	0	0
$957$	0	0
$958$	8.78680	0.283889
$959$	4.58579	0.148083
$960$	0	0
$961$	−20.4853	−0.660816
$962$	−3.17157	−0.102256
$963$	0	0
$964$	0	0
$965$	22.9706	0.739449
$966$	0	0
$967$	17.6985	0.569145	0.284572	−	0.958655i	$-0.408148\pi$
$967$	17.6985	0.569145	0.284572	+	0.958655i	$0.408148\pi$
$968$	−19.7990	−0.636364
$969$	0	0
$970$	−26.3015	−0.844491
$971$	−43.4558	−1.39456	−0.697282	−	0.716797i	$-0.745608\pi$
$971$	−43.4558	−1.39456	−0.697282	+	0.716797i	$0.745608\pi$
$972$	0	0
$973$	6.24264	0.200130
$974$	−16.2010	−0.519114
$975$	0	0
$976$	−24.0000	−0.768221
$977$	−24.0416	−0.769160	−0.384580	−	0.923092i	$-0.625654\pi$
$977$	−24.0416	−0.769160	−0.384580	+	0.923092i	$0.625654\pi$
$978$	0	0
$979$	6.72792	0.215025
$980$	0	0
$981$	0	0
$982$	−23.5147	−0.750385
$983$	15.0416	0.479754	0.239877	−	0.970803i	$-0.422893\pi$
$983$	15.0416	0.479754	0.239877	+	0.970803i	$0.422893\pi$
$984$	0	0
$985$	19.5736	0.623667
$986$	0.343146	0.0109280
$987$	0	0
$988$	0	0
$989$	−29.1421	−0.926666
$990$	0	0
$991$	1.02944	0.0327012	0.0163506	−	0.999866i	$-0.494795\pi$
$991$	1.02944	0.0327012	0.0163506	+	0.999866i	$0.494795\pi$
$992$	0	0
$993$	0	0
$994$	18.4853	0.586318
$995$	5.95837	0.188893
$996$	0	0
$997$	−11.5147	−0.364675	−0.182337	−	0.983236i	$-0.558366\pi$
$997$	−11.5147	−0.364675	−0.182337	+	0.983236i	$0.558366\pi$
$998$	−18.7696	−0.594140
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.a.h.1.2		2
3.2	odd	2		91.2.a.c.1.1	✓	2
7.6	odd	2		5733.2.a.s.1.2		2
12.11	even	2		1456.2.a.q.1.1		2
15.14	odd	2		2275.2.a.j.1.2		2
21.2	odd	6		637.2.e.f.508.2		4
21.5	even	6		637.2.e.g.508.2		4
21.11	odd	6		637.2.e.f.79.2		4
21.17	even	6		637.2.e.g.79.2		4
21.20	even	2		637.2.a.g.1.1		2
24.5	odd	2		5824.2.a.bl.1.1		2
24.11	even	2		5824.2.a.bk.1.2		2
39.5	even	4		1183.2.c.d.337.4		4
39.8	even	4		1183.2.c.d.337.2		4
39.38	odd	2		1183.2.a.d.1.2		2
273.272	even	2		8281.2.a.v.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.a.c.1.1	✓	2	3.2	odd	2
637.2.a.g.1.1		2	21.20	even	2
637.2.e.f.79.2		4	21.11	odd	6
637.2.e.f.508.2		4	21.2	odd	6
637.2.e.g.79.2		4	21.17	even	6
637.2.e.g.508.2		4	21.5	even	6
819.2.a.h.1.2		2	1.1	even	1	trivial
1183.2.a.d.1.2		2	39.38	odd	2
1183.2.c.d.337.2		4	39.8	even	4
1183.2.c.d.337.4		4	39.5	even	4
1456.2.a.q.1.1		2	12.11	even	2
2275.2.a.j.1.2		2	15.14	odd	2
5733.2.a.s.1.2		2	7.6	odd	2
5824.2.a.bk.1.2		2	24.11	even	2
5824.2.a.bl.1.1		2	24.5	odd	2
8281.2.a.v.1.2		2	273.272	even	2

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:	\( N \)	\(=\)	\( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	819.a (trivial)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} -1.58579 q^{5} +1.00000 q^{7} -2.82843 q^{8} -2.24264 q^{10} -4.24264 q^{11} -1.00000 q^{13} +1.41421 q^{14} -4.00000 q^{16} -1.41421 q^{17} -7.24264 q^{19} -6.00000 q^{22} +5.82843 q^{23} -2.48528 q^{25} -1.41421 q^{26} -0.171573 q^{29} +3.24264 q^{31} -2.00000 q^{34} -1.58579 q^{35} +2.24264 q^{37} -10.2426 q^{38} +4.48528 q^{40} -8.82843 q^{41} -5.00000 q^{43} +8.24264 q^{46} -1.58579 q^{47} +1.00000 q^{49} -3.51472 q^{50} +0.171573 q^{53} +6.72792 q^{55} -2.82843 q^{56} -0.242641 q^{58} -0.343146 q^{59} +6.00000 q^{61} +4.58579 q^{62} +8.00000 q^{64} +1.58579 q^{65} -14.4853 q^{67} -2.24264 q^{70} +13.0711 q^{71} -9.24264 q^{73} +3.17157 q^{74} -4.24264 q^{77} +15.4853 q^{79} +6.34315 q^{80} -12.4853 q^{82} -13.2426 q^{83} +2.24264 q^{85} -7.07107 q^{86} +12.0000 q^{88} -1.58579 q^{89} -1.00000 q^{91} -2.24264 q^{94} +11.4853 q^{95} +11.7279 q^{97} +1.41421 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{5} + 2 q^{7} + 4 q^{10} - 2 q^{13} - 8 q^{16} - 6 q^{19} - 12 q^{22} + 6 q^{23} + 12 q^{25} - 6 q^{29} - 2 q^{31} - 4 q^{34} - 6 q^{35} - 4 q^{37} - 12 q^{38} - 8 q^{40} - 12 q^{41} - 10 q^{43}+ \cdots - 2 q^{97}+O(q^{100}) \) 2 * q - 6 * q^5 + 2 * q^7 + 4 * q^10 - 2 * q^13 - 8 * q^16 - 6 * q^19 - 12 * q^22 + 6 * q^23 + 12 * q^25 - 6 * q^29 - 2 * q^31 - 4 * q^34 - 6 * q^35 - 4 * q^37 - 12 * q^38 - 8 * q^40 - 12 * q^41 - 10 * q^43 + 8 * q^46 - 6 * q^47 + 2 * q^49 - 24 * q^50 + 6 * q^53 - 12 * q^55 + 8 * q^58 - 12 * q^59 + 12 * q^61 + 12 * q^62 + 16 * q^64 + 6 * q^65 - 12 * q^67 + 4 * q^70 + 12 * q^71 - 10 * q^73 + 12 * q^74 + 14 * q^79 + 24 * q^80 - 8 * q^82 - 18 * q^83 - 4 * q^85 + 24 * q^88 - 6 * q^89 - 2 * q^91 + 4 * q^94 + 6 * q^95 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more