LMFDB - Embedded newform 8550.2.a.cr.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8550,2,Mod(1,8550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8550, 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("8550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8550.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:	$68.2720937282$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 570)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	8550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +4.42864 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +4.42864 q^{7} +1.00000 q^{8} +5.80642 q^{11} -6.42864 q^{13} +4.42864 q^{14} +1.00000 q^{16} +3.37778 q^{17} -1.00000 q^{19} +5.80642 q^{22} +6.42864 q^{23} -6.42864 q^{26} +4.42864 q^{28} -7.80642 q^{29} +9.05086 q^{31} +1.00000 q^{32} +3.37778 q^{34} +3.67307 q^{37} -1.00000 q^{38} -4.42864 q^{41} -1.05086 q^{43} +5.80642 q^{44} +6.42864 q^{46} -5.18421 q^{47} +12.6128 q^{49} -6.42864 q^{52} +4.75557 q^{53} +4.42864 q^{56} -7.80642 q^{58} -4.62222 q^{59} +2.00000 q^{61} +9.05086 q^{62} +1.00000 q^{64} +2.75557 q^{67} +3.37778 q^{68} -7.61285 q^{71} +11.6128 q^{73} +3.67307 q^{74} -1.00000 q^{76} +25.7146 q^{77} +2.94914 q^{79} -4.42864 q^{82} +0.133353 q^{83} -1.05086 q^{86} +5.80642 q^{88} +3.18421 q^{89} -28.4701 q^{91} +6.42864 q^{92} -5.18421 q^{94} -11.4193 q^{97} +12.6128 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} + 3 q^{8} + 4 q^{11} - 6 q^{13} + 3 q^{16} + 10 q^{17} - 3 q^{19} + 4 q^{22} + 6 q^{23} - 6 q^{26} - 10 q^{29} + 14 q^{31} + 3 q^{32} + 10 q^{34} - 2 q^{37} - 3 q^{38} + 10 q^{43} + 4 q^{44} + 6 q^{46} - 2 q^{47} + 11 q^{49} - 6 q^{52} + 14 q^{53} - 10 q^{58} - 14 q^{59} + 6 q^{61} + 14 q^{62} + 3 q^{64} + 8 q^{67} + 10 q^{68} + 4 q^{71} + 8 q^{73} - 2 q^{74} - 3 q^{76} + 24 q^{77} + 22 q^{79} + 10 q^{86} + 4 q^{88} - 4 q^{89} - 32 q^{91} + 6 q^{92} - 2 q^{94} + 6 q^{97} + 11 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^8 + 4 * q^11 - 6 * q^13 + 3 * q^16 + 10 * q^17 - 3 * q^19 + 4 * q^22 + 6 * q^23 - 6 * q^26 - 10 * q^29 + 14 * q^31 + 3 * q^32 + 10 * q^34 - 2 * q^37 - 3 * q^38 + 10 * q^43 + 4 * q^44 + 6 * q^46 - 2 * q^47 + 11 * q^49 - 6 * q^52 + 14 * q^53 - 10 * q^58 - 14 * q^59 + 6 * q^61 + 14 * q^62 + 3 * q^64 + 8 * q^67 + 10 * q^68 + 4 * q^71 + 8 * q^73 - 2 * q^74 - 3 * q^76 + 24 * q^77 + 22 * q^79 + 10 * q^86 + 4 * q^88 - 4 * q^89 - 32 * q^91 + 6 * q^92 - 2 * q^94 + 6 * q^97 + 11 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.42864	1.67387	0.836934	−	0.547304i	$-0.184346\pi$
$7$	4.42864	1.67387	0.836934	+	0.547304i	$0.184346\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	5.80642	1.75070	0.875351	−	0.483487i	$-0.160630\pi$
$11$	5.80642	1.75070	0.875351	+	0.483487i	$0.160630\pi$
$12$	0	0
$13$	−6.42864	−1.78298	−0.891492	−	0.453037i	$-0.850341\pi$
$13$	−6.42864	−1.78298	−0.891492	+	0.453037i	$0.850341\pi$
$14$	4.42864	1.18360
$15$	0	0
$16$	1.00000	0.250000
$17$	3.37778	0.819233	0.409617	−	0.912258i	$-0.365663\pi$
$17$	3.37778	0.819233	0.409617	+	0.912258i	$0.365663\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	5.80642	1.23793
$23$	6.42864	1.34046	0.670232	−	0.742152i	$-0.266195\pi$
$23$	6.42864	1.34046	0.670232	+	0.742152i	$0.266195\pi$
$24$	0	0
$25$	0	0
$26$	−6.42864	−1.26076
$27$	0	0
$28$	4.42864	0.836934
$29$	−7.80642	−1.44962	−0.724808	−	0.688951i	$-0.758072\pi$
$29$	−7.80642	−1.44962	−0.724808	+	0.688951i	$0.758072\pi$
$30$	0	0
$31$	9.05086	1.62558	0.812791	−	0.582556i	$-0.197947\pi$
$31$	9.05086	1.62558	0.812791	+	0.582556i	$0.197947\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.37778	0.579285
$35$	0	0
$36$	0	0
$37$	3.67307	0.603849	0.301925	−	0.953332i	$-0.402371\pi$
$37$	3.67307	0.603849	0.301925	+	0.953332i	$0.402371\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	0	0
$41$	−4.42864	−0.691637	−0.345819	−	0.938301i	$-0.612399\pi$
$41$	−4.42864	−0.691637	−0.345819	+	0.938301i	$0.612399\pi$
$42$	0	0
$43$	−1.05086	−0.160254	−0.0801270	−	0.996785i	$-0.525533\pi$
$43$	−1.05086	−0.160254	−0.0801270	+	0.996785i	$0.525533\pi$
$44$	5.80642	0.875351
$45$	0	0
$46$	6.42864	0.947851
$47$	−5.18421	−0.756194	−0.378097	−	0.925766i	$-0.623422\pi$
$47$	−5.18421	−0.756194	−0.378097	+	0.925766i	$0.623422\pi$
$48$	0	0
$49$	12.6128	1.80184
$50$	0	0
$51$	0	0
$52$	−6.42864	−0.891492
$53$	4.75557	0.653228	0.326614	−	0.945158i	$-0.394092\pi$
$53$	4.75557	0.653228	0.326614	+	0.945158i	$0.394092\pi$
$54$	0	0
$55$	0	0
$56$	4.42864	0.591802
$57$	0	0
$58$	−7.80642	−1.02503
$59$	−4.62222	−0.601761	−0.300881	−	0.953662i	$-0.597281\pi$
$59$	−4.62222	−0.601761	−0.300881	+	0.953662i	$0.597281\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	9.05086	1.14946
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	2.75557	0.336646	0.168323	−	0.985732i	$-0.446165\pi$
$67$	2.75557	0.336646	0.168323	+	0.985732i	$0.446165\pi$
$68$	3.37778	0.409617
$69$	0	0
$70$	0	0
$71$	−7.61285	−0.903479	−0.451739	−	0.892150i	$-0.649196\pi$
$71$	−7.61285	−0.903479	−0.451739	+	0.892150i	$0.649196\pi$
$72$	0	0
$73$	11.6128	1.35918	0.679591	−	0.733592i	$-0.262157\pi$
$73$	11.6128	1.35918	0.679591	+	0.733592i	$0.262157\pi$
$74$	3.67307	0.426986
$75$	0	0
$76$	−1.00000	−0.114708
$77$	25.7146	2.93045
$78$	0	0
$79$	2.94914	0.331805	0.165902	−	0.986142i	$-0.446946\pi$
$79$	2.94914	0.331805	0.165902	+	0.986142i	$0.446946\pi$
$80$	0	0
$81$	0	0
$82$	−4.42864	−0.489061
$83$	0.133353	0.0146374	0.00731870	−	0.999973i	$-0.497670\pi$
$83$	0.133353	0.0146374	0.00731870	+	0.999973i	$0.497670\pi$
$84$	0	0
$85$	0	0
$86$	−1.05086	−0.113317
$87$	0	0
$88$	5.80642	0.618967
$89$	3.18421	0.337525	0.168763	−	0.985657i	$-0.446023\pi$
$89$	3.18421	0.337525	0.168763	+	0.985657i	$0.446023\pi$
$90$	0	0
$91$	−28.4701	−2.98448
$92$	6.42864	0.670232
$93$	0	0
$94$	−5.18421	−0.534710
$95$	0	0
$96$	0	0
$97$	−11.4193	−1.15945	−0.579726	−	0.814812i	$-0.696840\pi$
$97$	−11.4193	−1.15945	−0.579726	+	0.814812i	$0.696840\pi$
$98$	12.6128	1.27409
$99$	0	0
$100$	0	0
$101$	1.86665	0.185738	0.0928692	−	0.995678i	$-0.470396\pi$
$101$	1.86665	0.185738	0.0928692	+	0.995678i	$0.470396\pi$
$102$	0	0
$103$	10.6222	1.04664	0.523319	−	0.852137i	$-0.324694\pi$
$103$	10.6222	1.04664	0.523319	+	0.852137i	$0.324694\pi$
$104$	−6.42864	−0.630380
$105$	0	0
$106$	4.75557	0.461902
$107$	−7.61285	−0.735962	−0.367981	−	0.929833i	$-0.619951\pi$
$107$	−7.61285	−0.735962	−0.367981	+	0.929833i	$0.619951\pi$
$108$	0	0
$109$	5.53972	0.530609	0.265304	−	0.964165i	$-0.414528\pi$
$109$	5.53972	0.530609	0.265304	+	0.964165i	$0.414528\pi$
$110$	0	0
$111$	0	0
$112$	4.42864	0.418467
$113$	−12.3684	−1.16352	−0.581761	−	0.813360i	$-0.697636\pi$
$113$	−12.3684	−1.16352	−0.581761	+	0.813360i	$0.697636\pi$
$114$	0	0
$115$	0	0
$116$	−7.80642	−0.724808
$117$	0	0
$118$	−4.62222	−0.425509
$119$	14.9590	1.37129
$120$	0	0
$121$	22.7146	2.06496
$122$	2.00000	0.181071
$123$	0	0
$124$	9.05086	0.812791
$125$	0	0
$126$	0	0
$127$	1.76494	0.156613	0.0783064	−	0.996929i	$-0.475049\pi$
$127$	1.76494	0.156613	0.0783064	+	0.996929i	$0.475049\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	−9.80642	−0.856791	−0.428396	−	0.903591i	$-0.640921\pi$
$131$	−9.80642	−0.856791	−0.428396	+	0.903591i	$0.640921\pi$
$132$	0	0
$133$	−4.42864	−0.384012
$134$	2.75557	0.238045
$135$	0	0
$136$	3.37778	0.289643
$137$	−1.47949	−0.126402	−0.0632009	−	0.998001i	$-0.520131\pi$
$137$	−1.47949	−0.126402	−0.0632009	+	0.998001i	$0.520131\pi$
$138$	0	0
$139$	−4.85728	−0.411989	−0.205995	−	0.978553i	$-0.566043\pi$
$139$	−4.85728	−0.411989	−0.205995	+	0.978553i	$0.566043\pi$
$140$	0	0
$141$	0	0
$142$	−7.61285	−0.638856
$143$	−37.3274	−3.12147
$144$	0	0
$145$	0	0
$146$	11.6128	0.961086
$147$	0	0
$148$	3.67307	0.301925
$149$	−4.62222	−0.378667	−0.189333	−	0.981913i	$-0.560633\pi$
$149$	−4.62222	−0.378667	−0.189333	+	0.981913i	$0.560633\pi$
$150$	0	0
$151$	−11.4193	−0.929287	−0.464644	−	0.885498i	$-0.653818\pi$
$151$	−11.4193	−0.929287	−0.464644	+	0.885498i	$0.653818\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	25.7146	2.07214
$155$	0	0
$156$	0	0
$157$	7.37778	0.588811	0.294406	−	0.955681i	$-0.404878\pi$
$157$	7.37778	0.588811	0.294406	+	0.955681i	$0.404878\pi$
$158$	2.94914	0.234621
$159$	0	0
$160$	0	0
$161$	28.4701	2.24376
$162$	0	0
$163$	−5.90813	−0.462761	−0.231380	−	0.972863i	$-0.574324\pi$
$163$	−5.90813	−0.462761	−0.231380	+	0.972863i	$0.574324\pi$
$164$	−4.42864	−0.345819
$165$	0	0
$166$	0.133353	0.0103502
$167$	−2.75557	−0.213232	−0.106616	−	0.994300i	$-0.534002\pi$
$167$	−2.75557	−0.213232	−0.106616	+	0.994300i	$0.534002\pi$
$168$	0	0
$169$	28.3274	2.17903
$170$	0	0
$171$	0	0
$172$	−1.05086	−0.0801270
$173$	8.10171	0.615962	0.307981	−	0.951393i	$-0.400347\pi$
$173$	8.10171	0.615962	0.307981	+	0.951393i	$0.400347\pi$
$174$	0	0
$175$	0	0
$176$	5.80642	0.437676
$177$	0	0
$178$	3.18421	0.238666
$179$	0.235063	0.0175695	0.00878473	−	0.999961i	$-0.497204\pi$
$179$	0.235063	0.0175695	0.00878473	+	0.999961i	$0.497204\pi$
$180$	0	0
$181$	11.3176	0.841228	0.420614	−	0.907240i	$-0.361815\pi$
$181$	11.3176	0.841228	0.420614	+	0.907240i	$0.361815\pi$
$182$	−28.4701	−2.11035
$183$	0	0
$184$	6.42864	0.473926
$185$	0	0
$186$	0	0
$187$	19.6128	1.43423
$188$	−5.18421	−0.378097
$189$	0	0
$190$	0	0
$191$	6.32693	0.457801	0.228900	−	0.973450i	$-0.426487\pi$
$191$	6.32693	0.457801	0.228900	+	0.973450i	$0.426487\pi$
$192$	0	0
$193$	11.5397	0.830647	0.415324	−	0.909674i	$-0.363668\pi$
$193$	11.5397	0.830647	0.415324	+	0.909674i	$0.363668\pi$
$194$	−11.4193	−0.819856
$195$	0	0
$196$	12.6128	0.900918
$197$	−20.0415	−1.42790	−0.713948	−	0.700198i	$-0.753095\pi$
$197$	−20.0415	−1.42790	−0.713948	+	0.700198i	$0.753095\pi$
$198$	0	0
$199$	−20.4701	−1.45109	−0.725544	−	0.688175i	$-0.758412\pi$
$199$	−20.4701	−1.45109	−0.725544	+	0.688175i	$0.758412\pi$
$200$	0	0
$201$	0	0
$202$	1.86665	0.131337
$203$	−34.5718	−2.42647
$204$	0	0
$205$	0	0
$206$	10.6222	0.740085
$207$	0	0
$208$	−6.42864	−0.445746
$209$	−5.80642	−0.401639
$210$	0	0
$211$	2.75557	0.189701	0.0948506	−	0.995492i	$-0.469763\pi$
$211$	2.75557	0.189701	0.0948506	+	0.995492i	$0.469763\pi$
$212$	4.75557	0.326614
$213$	0	0
$214$	−7.61285	−0.520404
$215$	0	0
$216$	0	0
$217$	40.0830	2.72101
$218$	5.53972	0.375197
$219$	0	0
$220$	0	0
$221$	−21.7146	−1.46068
$222$	0	0
$223$	10.6222	0.711316	0.355658	−	0.934616i	$-0.384257\pi$
$223$	10.6222	0.711316	0.355658	+	0.934616i	$0.384257\pi$
$224$	4.42864	0.295901
$225$	0	0
$226$	−12.3684	−0.822735
$227$	−15.3461	−1.01856	−0.509280	−	0.860601i	$-0.670088\pi$
$227$	−15.3461	−1.01856	−0.509280	+	0.860601i	$0.670088\pi$
$228$	0	0
$229$	−19.7146	−1.30277	−0.651387	−	0.758745i	$-0.725813\pi$
$229$	−19.7146	−1.30277	−0.651387	+	0.758745i	$0.725813\pi$
$230$	0	0
$231$	0	0
$232$	−7.80642	−0.512517
$233$	29.5625	1.93670	0.968351	−	0.249593i	$-0.0802968\pi$
$233$	29.5625	1.93670	0.968351	+	0.249593i	$0.0802968\pi$
$234$	0	0
$235$	0	0
$236$	−4.62222	−0.300881
$237$	0	0
$238$	14.9590	0.969647
$239$	−1.67307	−0.108222	−0.0541110	−	0.998535i	$-0.517232\pi$
$239$	−1.67307	−0.108222	−0.0541110	+	0.998535i	$0.517232\pi$
$240$	0	0
$241$	−16.9590	−1.09242	−0.546212	−	0.837647i	$-0.683931\pi$
$241$	−16.9590	−1.09242	−0.546212	+	0.837647i	$0.683931\pi$
$242$	22.7146	1.46015
$243$	0	0
$244$	2.00000	0.128037
$245$	0	0
$246$	0	0
$247$	6.42864	0.409045
$248$	9.05086	0.574730
$249$	0	0
$250$	0	0
$251$	−4.94914	−0.312387	−0.156194	−	0.987726i	$-0.549922\pi$
$251$	−4.94914	−0.312387	−0.156194	+	0.987726i	$0.549922\pi$
$252$	0	0
$253$	37.3274	2.34675
$254$	1.76494	0.110742
$255$	0	0
$256$	1.00000	0.0625000
$257$	−5.34614	−0.333483	−0.166742	−	0.986001i	$-0.553325\pi$
$257$	−5.34614	−0.333483	−0.166742	+	0.986001i	$0.553325\pi$
$258$	0	0
$259$	16.2667	1.01076
$260$	0	0
$261$	0	0
$262$	−9.80642	−0.605843
$263$	9.45091	0.582768	0.291384	−	0.956606i	$-0.405884\pi$
$263$	9.45091	0.582768	0.291384	+	0.956606i	$0.405884\pi$
$264$	0	0
$265$	0	0
$266$	−4.42864	−0.271537
$267$	0	0
$268$	2.75557	0.168323
$269$	−2.94914	−0.179813	−0.0899063	−	0.995950i	$-0.528657\pi$
$269$	−2.94914	−0.179813	−0.0899063	+	0.995950i	$0.528657\pi$
$270$	0	0
$271$	30.9590	1.88062	0.940312	−	0.340313i	$-0.110533\pi$
$271$	30.9590	1.88062	0.940312	+	0.340313i	$0.110533\pi$
$272$	3.37778	0.204808
$273$	0	0
$274$	−1.47949	−0.0893795
$275$	0	0
$276$	0	0
$277$	−13.0923	−0.786643	−0.393321	−	0.919401i	$-0.628674\pi$
$277$	−13.0923	−0.786643	−0.393321	+	0.919401i	$0.628674\pi$
$278$	−4.85728	−0.291320
$279$	0	0
$280$	0	0
$281$	−18.5303	−1.10543	−0.552714	−	0.833371i	$-0.686408\pi$
$281$	−18.5303	−1.10543	−0.552714	+	0.833371i	$0.686408\pi$
$282$	0	0
$283$	−26.3783	−1.56802	−0.784012	−	0.620745i	$-0.786830\pi$
$283$	−26.3783	−1.56802	−0.784012	+	0.620745i	$0.786830\pi$
$284$	−7.61285	−0.451739
$285$	0	0
$286$	−37.3274	−2.20722
$287$	−19.6128	−1.15771
$288$	0	0
$289$	−5.59057	−0.328857
$290$	0	0
$291$	0	0
$292$	11.6128	0.679591
$293$	28.5718	1.66918	0.834592	−	0.550868i	$-0.185703\pi$
$293$	28.5718	1.66918	0.834592	+	0.550868i	$0.185703\pi$
$294$	0	0
$295$	0	0
$296$	3.67307	0.213493
$297$	0	0
$298$	−4.62222	−0.267758
$299$	−41.3274	−2.39003
$300$	0	0
$301$	−4.65386	−0.268244
$302$	−11.4193	−0.657105
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	0	0
$307$	22.7556	1.29873	0.649364	−	0.760477i	$-0.275035\pi$
$307$	22.7556	1.29873	0.649364	+	0.760477i	$0.275035\pi$
$308$	25.7146	1.46522
$309$	0	0
$310$	0	0
$311$	6.32693	0.358767	0.179384	−	0.983779i	$-0.442590\pi$
$311$	6.32693	0.358767	0.179384	+	0.983779i	$0.442590\pi$
$312$	0	0
$313$	−13.7146	−0.775193	−0.387596	−	0.921829i	$-0.626695\pi$
$313$	−13.7146	−0.775193	−0.387596	+	0.921829i	$0.626695\pi$
$314$	7.37778	0.416352
$315$	0	0
$316$	2.94914	0.165902
$317$	32.5718	1.82942	0.914708	−	0.404115i	$-0.132420\pi$
$317$	32.5718	1.82942	0.914708	+	0.404115i	$0.132420\pi$
$318$	0	0
$319$	−45.3274	−2.53785
$320$	0	0
$321$	0	0
$322$	28.4701	1.58658
$323$	−3.37778	−0.187945
$324$	0	0
$325$	0	0
$326$	−5.90813	−0.327221
$327$	0	0
$328$	−4.42864	−0.244531
$329$	−22.9590	−1.26577
$330$	0	0
$331$	5.63158	0.309540	0.154770	−	0.987951i	$-0.450536\pi$
$331$	5.63158	0.309540	0.154770	+	0.987951i	$0.450536\pi$
$332$	0.133353	0.00731870
$333$	0	0
$334$	−2.75557	−0.150778
$335$	0	0
$336$	0	0
$337$	5.70471	0.310756	0.155378	−	0.987855i	$-0.450341\pi$
$337$	5.70471	0.310756	0.155378	+	0.987855i	$0.450341\pi$
$338$	28.3274	1.54081
$339$	0	0
$340$	0	0
$341$	52.5531	2.84591
$342$	0	0
$343$	24.8573	1.34217
$344$	−1.05086	−0.0566583
$345$	0	0
$346$	8.10171	0.435551
$347$	2.62222	0.140768	0.0703840	−	0.997520i	$-0.477578\pi$
$347$	2.62222	0.140768	0.0703840	+	0.997520i	$0.477578\pi$
$348$	0	0
$349$	−24.1017	−1.29013	−0.645067	−	0.764126i	$-0.723171\pi$
$349$	−24.1017	−1.29013	−0.645067	+	0.764126i	$0.723171\pi$
$350$	0	0
$351$	0	0
$352$	5.80642	0.309483
$353$	−3.64449	−0.193977	−0.0969883	−	0.995286i	$-0.530921\pi$
$353$	−3.64449	−0.193977	−0.0969883	+	0.995286i	$0.530921\pi$
$354$	0	0
$355$	0	0
$356$	3.18421	0.168763
$357$	0	0
$358$	0.235063	0.0124235
$359$	−9.08250	−0.479356	−0.239678	−	0.970852i	$-0.577042\pi$
$359$	−9.08250	−0.479356	−0.239678	+	0.970852i	$0.577042\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	11.3176	0.594838
$363$	0	0
$364$	−28.4701	−1.49224
$365$	0	0
$366$	0	0
$367$	3.95851	0.206633	0.103316	−	0.994649i	$-0.467055\pi$
$367$	3.95851	0.206633	0.103316	+	0.994649i	$0.467055\pi$
$368$	6.42864	0.335116
$369$	0	0
$370$	0	0
$371$	21.0607	1.09342
$372$	0	0
$373$	12.5303	0.648797	0.324398	−	0.945921i	$-0.394838\pi$
$373$	12.5303	0.648797	0.324398	+	0.945921i	$0.394838\pi$
$374$	19.6128	1.01416
$375$	0	0
$376$	−5.18421	−0.267355
$377$	50.1847	2.58464
$378$	0	0
$379$	26.8385	1.37860	0.689302	−	0.724474i	$-0.257917\pi$
$379$	26.8385	1.37860	0.689302	+	0.724474i	$0.257917\pi$
$380$	0	0
$381$	0	0
$382$	6.32693	0.323714
$383$	−17.5111	−0.894777	−0.447389	−	0.894340i	$-0.647646\pi$
$383$	−17.5111	−0.894777	−0.447389	+	0.894340i	$0.647646\pi$
$384$	0	0
$385$	0	0
$386$	11.5397	0.587356
$387$	0	0
$388$	−11.4193	−0.579726
$389$	29.4795	1.49467	0.747335	−	0.664448i	$-0.231333\pi$
$389$	29.4795	1.49467	0.747335	+	0.664448i	$0.231333\pi$
$390$	0	0
$391$	21.7146	1.09815
$392$	12.6128	0.637045
$393$	0	0
$394$	−20.0415	−1.00968
$395$	0	0
$396$	0	0
$397$	9.21279	0.462377	0.231188	−	0.972909i	$-0.425739\pi$
$397$	9.21279	0.462377	0.231188	+	0.972909i	$0.425739\pi$
$398$	−20.4701	−1.02607
$399$	0	0
$400$	0	0
$401$	−29.2859	−1.46247	−0.731234	−	0.682126i	$-0.761055\pi$
$401$	−29.2859	−1.46247	−0.731234	+	0.682126i	$0.761055\pi$
$402$	0	0
$403$	−58.1847	−2.89839
$404$	1.86665	0.0928692
$405$	0	0
$406$	−34.5718	−1.71577
$407$	21.3274	1.05716
$408$	0	0
$409$	−17.8796	−0.884087	−0.442044	−	0.896994i	$-0.645746\pi$
$409$	−17.8796	−0.884087	−0.442044	+	0.896994i	$0.645746\pi$
$410$	0	0
$411$	0	0
$412$	10.6222	0.523319
$413$	−20.4701	−1.00727
$414$	0	0
$415$	0	0
$416$	−6.42864	−0.315190
$417$	0	0
$418$	−5.80642	−0.284001
$419$	−30.8671	−1.50796	−0.753979	−	0.656899i	$-0.771868\pi$
$419$	−30.8671	−1.50796	−0.753979	+	0.656899i	$0.771868\pi$
$420$	0	0
$421$	−18.3970	−0.896615	−0.448307	−	0.893879i	$-0.647973\pi$
$421$	−18.3970	−0.896615	−0.448307	+	0.893879i	$0.647973\pi$
$422$	2.75557	0.134139
$423$	0	0
$424$	4.75557	0.230951
$425$	0	0
$426$	0	0
$427$	8.85728	0.428634
$428$	−7.61285	−0.367981
$429$	0	0
$430$	0	0
$431$	−29.5941	−1.42550	−0.712749	−	0.701419i	$-0.752550\pi$
$431$	−29.5941	−1.42550	−0.712749	+	0.701419i	$0.752550\pi$
$432$	0	0
$433$	27.2988	1.31190	0.655949	−	0.754805i	$-0.272269\pi$
$433$	27.2988	1.31190	0.655949	+	0.754805i	$0.272269\pi$
$434$	40.0830	1.92404
$435$	0	0
$436$	5.53972	0.265304
$437$	−6.42864	−0.307524
$438$	0	0
$439$	9.64143	0.460160	0.230080	−	0.973172i	$-0.426101\pi$
$439$	9.64143	0.460160	0.230080	+	0.973172i	$0.426101\pi$
$440$	0	0
$441$	0	0
$442$	−21.7146	−1.03286
$443$	−6.70519	−0.318573	−0.159287	−	0.987232i	$-0.550919\pi$
$443$	−6.70519	−0.318573	−0.159287	+	0.987232i	$0.550919\pi$
$444$	0	0
$445$	0	0
$446$	10.6222	0.502976
$447$	0	0
$448$	4.42864	0.209234
$449$	13.9398	0.657859	0.328929	−	0.944355i	$-0.393312\pi$
$449$	13.9398	0.657859	0.328929	+	0.944355i	$0.393312\pi$
$450$	0	0
$451$	−25.7146	−1.21085
$452$	−12.3684	−0.581761
$453$	0	0
$454$	−15.3461	−0.720230
$455$	0	0
$456$	0	0
$457$	23.2257	1.08645	0.543226	−	0.839586i	$-0.317203\pi$
$457$	23.2257	1.08645	0.543226	+	0.839586i	$0.317203\pi$
$458$	−19.7146	−0.921201
$459$	0	0
$460$	0	0
$461$	−19.9684	−0.930019	−0.465010	−	0.885306i	$-0.653949\pi$
$461$	−19.9684	−0.930019	−0.465010	+	0.885306i	$0.653949\pi$
$462$	0	0
$463$	−2.79706	−0.129990	−0.0649951	−	0.997886i	$-0.520703\pi$
$463$	−2.79706	−0.129990	−0.0649951	+	0.997886i	$0.520703\pi$
$464$	−7.80642	−0.362404
$465$	0	0
$466$	29.5625	1.36945
$467$	30.9719	1.43321	0.716604	−	0.697480i	$-0.245695\pi$
$467$	30.9719	1.43321	0.716604	+	0.697480i	$0.245695\pi$
$468$	0	0
$469$	12.2034	0.563502
$470$	0	0
$471$	0	0
$472$	−4.62222	−0.212755
$473$	−6.10171	−0.280557
$474$	0	0
$475$	0	0
$476$	14.9590	0.685644
$477$	0	0
$478$	−1.67307	−0.0765245
$479$	−29.8765	−1.36509	−0.682546	−	0.730843i	$-0.739127\pi$
$479$	−29.8765	−1.36509	−0.682546	+	0.730843i	$0.739127\pi$
$480$	0	0
$481$	−23.6128	−1.07665
$482$	−16.9590	−0.772461
$483$	0	0
$484$	22.7146	1.03248
$485$	0	0
$486$	0	0
$487$	17.8479	0.808766	0.404383	−	0.914590i	$-0.367486\pi$
$487$	17.8479	0.808766	0.404383	+	0.914590i	$0.367486\pi$
$488$	2.00000	0.0905357
$489$	0	0
$490$	0	0
$491$	8.68244	0.391833	0.195916	−	0.980621i	$-0.437232\pi$
$491$	8.68244	0.391833	0.195916	+	0.980621i	$0.437232\pi$
$492$	0	0
$493$	−26.3684	−1.18757
$494$	6.42864	0.289238
$495$	0	0
$496$	9.05086	0.406395
$497$	−33.7146	−1.51230
$498$	0	0
$499$	11.2257	0.502531	0.251266	−	0.967918i	$-0.419153\pi$
$499$	11.2257	0.502531	0.251266	+	0.967918i	$0.419153\pi$
$500$	0	0
$501$	0	0
$502$	−4.94914	−0.220891
$503$	17.6543	0.787168	0.393584	−	0.919289i	$-0.371235\pi$
$503$	17.6543	0.787168	0.393584	+	0.919289i	$0.371235\pi$
$504$	0	0
$505$	0	0
$506$	37.3274	1.65941
$507$	0	0
$508$	1.76494	0.0783064
$509$	−14.1748	−0.628289	−0.314144	−	0.949375i	$-0.601718\pi$
$509$	−14.1748	−0.628289	−0.314144	+	0.949375i	$0.601718\pi$
$510$	0	0
$511$	51.4291	2.27509
$512$	1.00000	0.0441942
$513$	0	0
$514$	−5.34614	−0.235808
$515$	0	0
$516$	0	0
$517$	−30.1017	−1.32387
$518$	16.2667	0.714718
$519$	0	0
$520$	0	0
$521$	15.0005	0.657183	0.328591	−	0.944472i	$-0.393426\pi$
$521$	15.0005	0.657183	0.328591	+	0.944472i	$0.393426\pi$
$522$	0	0
$523$	40.2864	1.76160	0.880801	−	0.473488i	$-0.157005\pi$
$523$	40.2864	1.76160	0.880801	+	0.473488i	$0.157005\pi$
$524$	−9.80642	−0.428396
$525$	0	0
$526$	9.45091	0.412079
$527$	30.5718	1.33173
$528$	0	0
$529$	18.3274	0.796844
$530$	0	0
$531$	0	0
$532$	−4.42864	−0.192006
$533$	28.4701	1.23318
$534$	0	0
$535$	0	0
$536$	2.75557	0.119022
$537$	0	0
$538$	−2.94914	−0.127147
$539$	73.2355	3.15448
$540$	0	0
$541$	−22.3872	−0.962499	−0.481249	−	0.876584i	$-0.659817\pi$
$541$	−22.3872	−0.962499	−0.481249	+	0.876584i	$0.659817\pi$
$542$	30.9590	1.32980
$543$	0	0
$544$	3.37778	0.144821
$545$	0	0
$546$	0	0
$547$	19.7333	0.843735	0.421867	−	0.906658i	$-0.361375\pi$
$547$	19.7333	0.843735	0.421867	+	0.906658i	$0.361375\pi$
$548$	−1.47949	−0.0632009
$549$	0	0
$550$	0	0
$551$	7.80642	0.332565
$552$	0	0
$553$	13.0607	0.555397
$554$	−13.0923	−0.556240
$555$	0	0
$556$	−4.85728	−0.205995
$557$	−15.6543	−0.663295	−0.331648	−	0.943403i	$-0.607605\pi$
$557$	−15.6543	−0.663295	−0.331648	+	0.943403i	$0.607605\pi$
$558$	0	0
$559$	6.75557	0.285730
$560$	0	0
$561$	0	0
$562$	−18.5303	−0.781656
$563$	−36.9403	−1.55685	−0.778423	−	0.627740i	$-0.783980\pi$
$563$	−36.9403	−1.55685	−0.778423	+	0.627740i	$0.783980\pi$
$564$	0	0
$565$	0	0
$566$	−26.3783	−1.10876
$567$	0	0
$568$	−7.61285	−0.319428
$569$	1.20294	0.0504300	0.0252150	−	0.999682i	$-0.491973\pi$
$569$	1.20294	0.0504300	0.0252150	+	0.999682i	$0.491973\pi$
$570$	0	0
$571$	37.7975	1.58178	0.790889	−	0.611960i	$-0.209619\pi$
$571$	37.7975	1.58178	0.790889	+	0.611960i	$0.209619\pi$
$572$	−37.3274	−1.56074
$573$	0	0
$574$	−19.6128	−0.818624
$575$	0	0
$576$	0	0
$577$	23.2257	0.966898	0.483449	−	0.875372i	$-0.339384\pi$
$577$	23.2257	0.966898	0.483449	+	0.875372i	$0.339384\pi$
$578$	−5.59057	−0.232537
$579$	0	0
$580$	0	0
$581$	0.590573	0.0245011
$582$	0	0
$583$	27.6128	1.14361
$584$	11.6128	0.480543
$585$	0	0
$586$	28.5718	1.18029
$587$	−16.8069	−0.693695	−0.346848	−	0.937922i	$-0.612748\pi$
$587$	−16.8069	−0.693695	−0.346848	+	0.937922i	$0.612748\pi$
$588$	0	0
$589$	−9.05086	−0.372934
$590$	0	0
$591$	0	0
$592$	3.67307	0.150962
$593$	16.3555	0.671640	0.335820	−	0.941926i	$-0.390987\pi$
$593$	16.3555	0.671640	0.335820	+	0.941926i	$0.390987\pi$
$594$	0	0
$595$	0	0
$596$	−4.62222	−0.189333
$597$	0	0
$598$	−41.3274	−1.69000
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	7.89829	0.322178	0.161089	−	0.986940i	$-0.448499\pi$
$601$	7.89829	0.322178	0.161089	+	0.986940i	$0.448499\pi$
$602$	−4.65386	−0.189677
$603$	0	0
$604$	−11.4193	−0.464644
$605$	0	0
$606$	0	0
$607$	17.8479	0.724424	0.362212	−	0.932096i	$-0.382022\pi$
$607$	17.8479	0.724424	0.362212	+	0.932096i	$0.382022\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	0	0
$611$	33.3274	1.34828
$612$	0	0
$613$	0.622216	0.0251311	0.0125655	−	0.999921i	$-0.496000\pi$
$613$	0.622216	0.0251311	0.0125655	+	0.999921i	$0.496000\pi$
$614$	22.7556	0.918340
$615$	0	0
$616$	25.7146	1.03607
$617$	37.4795	1.50887	0.754434	−	0.656376i	$-0.227912\pi$
$617$	37.4795	1.50887	0.754434	+	0.656376i	$0.227912\pi$
$618$	0	0
$619$	28.6735	1.15249	0.576244	−	0.817278i	$-0.304518\pi$
$619$	28.6735	1.15249	0.576244	+	0.817278i	$0.304518\pi$
$620$	0	0
$621$	0	0
$622$	6.32693	0.253687
$623$	14.1017	0.564973
$624$	0	0
$625$	0	0
$626$	−13.7146	−0.548144
$627$	0	0
$628$	7.37778	0.294406
$629$	12.4068	0.494693
$630$	0	0
$631$	1.24443	0.0495400	0.0247700	−	0.999693i	$-0.492115\pi$
$631$	1.24443	0.0495400	0.0247700	+	0.999693i	$0.492115\pi$
$632$	2.94914	0.117311
$633$	0	0
$634$	32.5718	1.29359
$635$	0	0
$636$	0	0
$637$	−81.0835	−3.21264
$638$	−45.3274	−1.79453
$639$	0	0
$640$	0	0
$641$	17.4064	0.687510	0.343755	−	0.939059i	$-0.388301\pi$
$641$	17.4064	0.687510	0.343755	+	0.939059i	$0.388301\pi$
$642$	0	0
$643$	−35.1526	−1.38628	−0.693141	−	0.720802i	$-0.743774\pi$
$643$	−35.1526	−1.38628	−0.693141	+	0.720802i	$0.743774\pi$
$644$	28.4701	1.12188
$645$	0	0
$646$	−3.37778	−0.132897
$647$	−34.4286	−1.35353	−0.676765	−	0.736199i	$-0.736619\pi$
$647$	−34.4286	−1.35353	−0.676765	+	0.736199i	$0.736619\pi$
$648$	0	0
$649$	−26.8385	−1.05350
$650$	0	0
$651$	0	0
$652$	−5.90813	−0.231380
$653$	4.30819	0.168593	0.0842963	−	0.996441i	$-0.473136\pi$
$653$	4.30819	0.168593	0.0842963	+	0.996441i	$0.473136\pi$
$654$	0	0
$655$	0	0
$656$	−4.42864	−0.172909
$657$	0	0
$658$	−22.9590	−0.895035
$659$	27.0736	1.05464	0.527319	−	0.849667i	$-0.323197\pi$
$659$	27.0736	1.05464	0.527319	+	0.849667i	$0.323197\pi$
$660$	0	0
$661$	−43.2543	−1.68240	−0.841198	−	0.540727i	$-0.818149\pi$
$661$	−43.2543	−1.68240	−0.841198	+	0.540727i	$0.818149\pi$
$662$	5.63158	0.218878
$663$	0	0
$664$	0.133353	0.00517510
$665$	0	0
$666$	0	0
$667$	−50.1847	−1.94316
$668$	−2.75557	−0.106616
$669$	0	0
$670$	0	0
$671$	11.6128	0.448309
$672$	0	0
$673$	15.0321	0.579446	0.289723	−	0.957111i	$-0.406437\pi$
$673$	15.0321	0.579446	0.289723	+	0.957111i	$0.406437\pi$
$674$	5.70471	0.219737
$675$	0	0
$676$	28.3274	1.08952
$677$	−22.9403	−0.881666	−0.440833	−	0.897589i	$-0.645317\pi$
$677$	−22.9403	−0.881666	−0.440833	+	0.897589i	$0.645317\pi$
$678$	0	0
$679$	−50.5718	−1.94077
$680$	0	0
$681$	0	0
$682$	52.5531	2.01236
$683$	−9.77784	−0.374139	−0.187069	−	0.982347i	$-0.559899\pi$
$683$	−9.77784	−0.374139	−0.187069	+	0.982347i	$0.559899\pi$
$684$	0	0
$685$	0	0
$686$	24.8573	0.949055
$687$	0	0
$688$	−1.05086	−0.0400635
$689$	−30.5718	−1.16469
$690$	0	0
$691$	−44.0830	−1.67700	−0.838498	−	0.544905i	$-0.816566\pi$
$691$	−44.0830	−1.67700	−0.838498	+	0.544905i	$0.816566\pi$
$692$	8.10171	0.307981
$693$	0	0
$694$	2.62222	0.0995379
$695$	0	0
$696$	0	0
$697$	−14.9590	−0.566612
$698$	−24.1017	−0.912263
$699$	0	0
$700$	0	0
$701$	12.7685	0.482259	0.241129	−	0.970493i	$-0.422482\pi$
$701$	12.7685	0.482259	0.241129	+	0.970493i	$0.422482\pi$
$702$	0	0
$703$	−3.67307	−0.138532
$704$	5.80642	0.218838
$705$	0	0
$706$	−3.64449	−0.137162
$707$	8.26671	0.310901
$708$	0	0
$709$	−47.5941	−1.78743	−0.893717	−	0.448631i	$-0.851912\pi$
$709$	−47.5941	−1.78743	−0.893717	+	0.448631i	$0.851912\pi$
$710$	0	0
$711$	0	0
$712$	3.18421	0.119333
$713$	58.1847	2.17903
$714$	0	0
$715$	0	0
$716$	0.235063	0.00878473
$717$	0	0
$718$	−9.08250	−0.338956
$719$	47.0005	1.75282	0.876411	−	0.481564i	$-0.159931\pi$
$719$	47.0005	1.75282	0.876411	+	0.481564i	$0.159931\pi$
$720$	0	0
$721$	47.0420	1.75193
$722$	1.00000	0.0372161
$723$	0	0
$724$	11.3176	0.420614
$725$	0	0
$726$	0	0
$727$	52.6321	1.95202	0.976008	−	0.217737i	$-0.0698674\pi$
$727$	52.6321	1.95202	0.976008	+	0.217737i	$0.0698674\pi$
$728$	−28.4701	−1.05517
$729$	0	0
$730$	0	0
$731$	−3.54956	−0.131285
$732$	0	0
$733$	27.3145	1.00888	0.504442	−	0.863446i	$-0.331698\pi$
$733$	27.3145	1.00888	0.504442	+	0.863446i	$0.331698\pi$
$734$	3.95851	0.146111
$735$	0	0
$736$	6.42864	0.236963
$737$	16.0000	0.589368
$738$	0	0
$739$	25.3274	0.931684	0.465842	−	0.884868i	$-0.345752\pi$
$739$	25.3274	0.931684	0.465842	+	0.884868i	$0.345752\pi$
$740$	0	0
$741$	0	0
$742$	21.0607	0.773163
$743$	11.8796	0.435819	0.217909	−	0.975969i	$-0.430076\pi$
$743$	11.8796	0.435819	0.217909	+	0.975969i	$0.430076\pi$
$744$	0	0
$745$	0	0
$746$	12.5303	0.458769
$747$	0	0
$748$	19.6128	0.717117
$749$	−33.7146	−1.23190
$750$	0	0
$751$	−45.5210	−1.66108	−0.830542	−	0.556956i	$-0.811969\pi$
$751$	−45.5210	−1.66108	−0.830542	+	0.556956i	$0.811969\pi$
$752$	−5.18421	−0.189049
$753$	0	0
$754$	50.1847	1.82762
$755$	0	0
$756$	0	0
$757$	−51.7275	−1.88007	−0.940033	−	0.341083i	$-0.889206\pi$
$757$	−51.7275	−1.88007	−0.940033	+	0.341083i	$0.889206\pi$
$758$	26.8385	0.974820
$759$	0	0
$760$	0	0
$761$	−28.3684	−1.02835	−0.514177	−	0.857684i	$-0.671903\pi$
$761$	−28.3684	−1.02835	−0.514177	+	0.857684i	$0.671903\pi$
$762$	0	0
$763$	24.5334	0.888169
$764$	6.32693	0.228900
$765$	0	0
$766$	−17.5111	−0.632703
$767$	29.7146	1.07293
$768$	0	0
$769$	−0.285442	−0.0102933	−0.00514665	−	0.999987i	$-0.501638\pi$
$769$	−0.285442	−0.0102933	−0.00514665	+	0.999987i	$0.501638\pi$
$770$	0	0
$771$	0	0
$772$	11.5397	0.415324
$773$	−49.8163	−1.79177	−0.895883	−	0.444289i	$-0.853456\pi$
$773$	−49.8163	−1.79177	−0.895883	+	0.444289i	$0.853456\pi$
$774$	0	0
$775$	0	0
$776$	−11.4193	−0.409928
$777$	0	0
$778$	29.4795	1.05689
$779$	4.42864	0.158672
$780$	0	0
$781$	−44.2034	−1.58172
$782$	21.7146	0.776511
$783$	0	0
$784$	12.6128	0.450459
$785$	0	0
$786$	0	0
$787$	−20.7368	−0.739188	−0.369594	−	0.929193i	$-0.620503\pi$
$787$	−20.7368	−0.739188	−0.369594	+	0.929193i	$0.620503\pi$
$788$	−20.0415	−0.713948
$789$	0	0
$790$	0	0
$791$	−54.7753	−1.94758
$792$	0	0
$793$	−12.8573	−0.456575
$794$	9.21279	0.326950
$795$	0	0
$796$	−20.4701	−0.725544
$797$	21.3461	0.756119	0.378060	−	0.925781i	$-0.376591\pi$
$797$	21.3461	0.756119	0.378060	+	0.925781i	$0.376591\pi$
$798$	0	0
$799$	−17.5111	−0.619500
$800$	0	0
$801$	0	0
$802$	−29.2859	−1.03412
$803$	67.4291	2.37952
$804$	0	0
$805$	0	0
$806$	−58.1847	−2.04947
$807$	0	0
$808$	1.86665	0.0656684
$809$	18.6735	0.656527	0.328263	−	0.944586i	$-0.393537\pi$
$809$	18.6735	0.656527	0.328263	+	0.944586i	$0.393537\pi$
$810$	0	0
$811$	−14.8385	−0.521052	−0.260526	−	0.965467i	$-0.583896\pi$
$811$	−14.8385	−0.521052	−0.260526	+	0.965467i	$0.583896\pi$
$812$	−34.5718	−1.21323
$813$	0	0
$814$	21.3274	0.747525
$815$	0	0
$816$	0	0
$817$	1.05086	0.0367648
$818$	−17.8796	−0.625144
$819$	0	0
$820$	0	0
$821$	−16.1521	−0.563712	−0.281856	−	0.959457i	$-0.590950\pi$
$821$	−16.1521	−0.563712	−0.281856	+	0.959457i	$0.590950\pi$
$822$	0	0
$823$	−47.6543	−1.66113	−0.830563	−	0.556925i	$-0.811981\pi$
$823$	−47.6543	−1.66113	−0.830563	+	0.556925i	$0.811981\pi$
$824$	10.6222	0.370042
$825$	0	0
$826$	−20.4701	−0.712247
$827$	−26.1017	−0.907645	−0.453823	−	0.891092i	$-0.649940\pi$
$827$	−26.1017	−0.907645	−0.453823	+	0.891092i	$0.649940\pi$
$828$	0	0
$829$	−13.8894	−0.482399	−0.241199	−	0.970476i	$-0.577541\pi$
$829$	−13.8894	−0.482399	−0.241199	+	0.970476i	$0.577541\pi$
$830$	0	0
$831$	0	0
$832$	−6.42864	−0.222873
$833$	42.6035	1.47612
$834$	0	0
$835$	0	0
$836$	−5.80642	−0.200819
$837$	0	0
$838$	−30.8671	−1.06629
$839$	−31.6958	−1.09426	−0.547131	−	0.837047i	$-0.684280\pi$
$839$	−31.6958	−1.09426	−0.547131	+	0.837047i	$0.684280\pi$
$840$	0	0
$841$	31.9403	1.10139
$842$	−18.3970	−0.634002
$843$	0	0
$844$	2.75557	0.0948506
$845$	0	0
$846$	0	0
$847$	100.595	3.45647
$848$	4.75557	0.163307
$849$	0	0
$850$	0	0
$851$	23.6128	0.809438
$852$	0	0
$853$	34.8702	1.19393	0.596966	−	0.802266i	$-0.296373\pi$
$853$	34.8702	1.19393	0.596966	+	0.802266i	$0.296373\pi$
$854$	8.85728	0.303090
$855$	0	0
$856$	−7.61285	−0.260202
$857$	33.9367	1.15926	0.579628	−	0.814881i	$-0.303198\pi$
$857$	33.9367	1.15926	0.579628	+	0.814881i	$0.303198\pi$
$858$	0	0
$859$	−37.4479	−1.27770	−0.638852	−	0.769330i	$-0.720590\pi$
$859$	−37.4479	−1.27770	−0.638852	+	0.769330i	$0.720590\pi$
$860$	0	0
$861$	0	0
$862$	−29.5941	−1.00798
$863$	−15.5299	−0.528643	−0.264322	−	0.964435i	$-0.585148\pi$
$863$	−15.5299	−0.528643	−0.264322	+	0.964435i	$0.585148\pi$
$864$	0	0
$865$	0	0
$866$	27.2988	0.927652
$867$	0	0
$868$	40.0830	1.36050
$869$	17.1240	0.580891
$870$	0	0
$871$	−17.7146	−0.600235
$872$	5.53972	0.187599
$873$	0	0
$874$	−6.42864	−0.217452
$875$	0	0
$876$	0	0
$877$	−7.87649	−0.265970	−0.132985	−	0.991118i	$-0.542456\pi$
$877$	−7.87649	−0.265970	−0.132985	+	0.991118i	$0.542456\pi$
$878$	9.64143	0.325382
$879$	0	0
$880$	0	0
$881$	−33.5496	−1.13031	−0.565157	−	0.824984i	$-0.691184\pi$
$881$	−33.5496	−1.13031	−0.565157	+	0.824984i	$0.691184\pi$
$882$	0	0
$883$	50.9688	1.71524	0.857619	−	0.514286i	$-0.171943\pi$
$883$	50.9688	1.71524	0.857619	+	0.514286i	$0.171943\pi$
$884$	−21.7146	−0.730340
$885$	0	0
$886$	−6.70519	−0.225265
$887$	−17.0035	−0.570923	−0.285461	−	0.958390i	$-0.592147\pi$
$887$	−17.0035	−0.570923	−0.285461	+	0.958390i	$0.592147\pi$
$888$	0	0
$889$	7.81627	0.262149
$890$	0	0
$891$	0	0
$892$	10.6222	0.355658
$893$	5.18421	0.173483
$894$	0	0
$895$	0	0
$896$	4.42864	0.147950
$897$	0	0
$898$	13.9398	0.465176
$899$	−70.6548	−2.35647
$900$	0	0
$901$	16.0633	0.535146
$902$	−25.7146	−0.856201
$903$	0	0
$904$	−12.3684	−0.411367
$905$	0	0
$906$	0	0
$907$	−21.7778	−0.723121	−0.361561	−	0.932349i	$-0.617756\pi$
$907$	−21.7778	−0.723121	−0.361561	+	0.932349i	$0.617756\pi$
$908$	−15.3461	−0.509280
$909$	0	0
$910$	0	0
$911$	−10.6351	−0.352357	−0.176179	−	0.984358i	$-0.556374\pi$
$911$	−10.6351	−0.352357	−0.176179	+	0.984358i	$0.556374\pi$
$912$	0	0
$913$	0.774305	0.0256257
$914$	23.2257	0.768238
$915$	0	0
$916$	−19.7146	−0.651387
$917$	−43.4291	−1.43416
$918$	0	0
$919$	12.2667	0.404641	0.202321	−	0.979319i	$-0.435152\pi$
$919$	12.2667	0.404641	0.202321	+	0.979319i	$0.435152\pi$
$920$	0	0
$921$	0	0
$922$	−19.9684	−0.657623
$923$	48.9403	1.61089
$924$	0	0
$925$	0	0
$926$	−2.79706	−0.0919170
$927$	0	0
$928$	−7.80642	−0.256258
$929$	27.9180	0.915959	0.457980	−	0.888963i	$-0.348573\pi$
$929$	27.9180	0.915959	0.457980	+	0.888963i	$0.348573\pi$
$930$	0	0
$931$	−12.6128	−0.413369
$932$	29.5625	0.968351
$933$	0	0
$934$	30.9719	1.01343
$935$	0	0
$936$	0	0
$937$	−3.14272	−0.102668	−0.0513341	−	0.998682i	$-0.516347\pi$
$937$	−3.14272	−0.102668	−0.0513341	+	0.998682i	$0.516347\pi$
$938$	12.2034	0.398456
$939$	0	0
$940$	0	0
$941$	19.6227	0.639681	0.319841	−	0.947471i	$-0.396371\pi$
$941$	19.6227	0.639681	0.319841	+	0.947471i	$0.396371\pi$
$942$	0	0
$943$	−28.4701	−0.927115
$944$	−4.62222	−0.150440
$945$	0	0
$946$	−6.10171	−0.198384
$947$	−34.5018	−1.12116	−0.560578	−	0.828101i	$-0.689421\pi$
$947$	−34.5018	−1.12116	−0.560578	+	0.828101i	$0.689421\pi$
$948$	0	0
$949$	−74.6548	−2.42340
$950$	0	0
$951$	0	0
$952$	14.9590	0.484824
$953$	19.0035	0.615585	0.307793	−	0.951454i	$-0.400410\pi$
$953$	19.0035	0.615585	0.307793	+	0.951454i	$0.400410\pi$
$954$	0	0
$955$	0	0
$956$	−1.67307	−0.0541110
$957$	0	0
$958$	−29.8765	−0.965266
$959$	−6.55215	−0.211580
$960$	0	0
$961$	50.9180	1.64252
$962$	−23.6128	−0.761309
$963$	0	0
$964$	−16.9590	−0.546212
$965$	0	0
$966$	0	0
$967$	−17.8765	−0.574869	−0.287435	−	0.957800i	$-0.592802\pi$
$967$	−17.8765	−0.574869	−0.287435	+	0.957800i	$0.592802\pi$
$968$	22.7146	0.730074
$969$	0	0
$970$	0	0
$971$	−34.0701	−1.09336	−0.546680	−	0.837341i	$-0.684109\pi$
$971$	−34.0701	−1.09336	−0.546680	+	0.837341i	$0.684109\pi$
$972$	0	0
$973$	−21.5111	−0.689615
$974$	17.8479	0.571884
$975$	0	0
$976$	2.00000	0.0640184
$977$	54.0830	1.73027	0.865134	−	0.501541i	$-0.167233\pi$
$977$	54.0830	1.73027	0.865134	+	0.501541i	$0.167233\pi$
$978$	0	0
$979$	18.4889	0.590907
$980$	0	0
$981$	0	0
$982$	8.68244	0.277068
$983$	−56.9403	−1.81611	−0.908056	−	0.418849i	$-0.862434\pi$
$983$	−56.9403	−1.81611	−0.908056	+	0.418849i	$0.862434\pi$
$984$	0	0
$985$	0	0
$986$	−26.3684	−0.839741
$987$	0	0
$988$	6.42864	0.204522
$989$	−6.75557	−0.214815
$990$	0	0
$991$	−47.0321	−1.49402	−0.747012	−	0.664810i	$-0.768512\pi$
$991$	−47.0321	−1.49402	−0.747012	+	0.664810i	$0.768512\pi$
$992$	9.05086	0.287365
$993$	0	0
$994$	−33.7146	−1.06936
$995$	0	0
$996$	0	0
$997$	−20.8256	−0.659555	−0.329777	−	0.944059i	$-0.606974\pi$
$997$	−20.8256	−0.659555	−0.329777	+	0.944059i	$0.606974\pi$
$998$	11.2257	0.355343
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.cr.1.3		3
3.2	odd	2		2850.2.a.bk.1.3		3
5.2	odd	4		1710.2.d.e.1369.5		6
5.3	odd	4		1710.2.d.e.1369.2		6
5.4	even	2		8550.2.a.cf.1.1		3
15.2	even	4		570.2.d.d.229.2	✓	6
15.8	even	4		570.2.d.d.229.5	yes	6
15.14	odd	2		2850.2.a.bn.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
570.2.d.d.229.2	✓	6	15.2	even	4
570.2.d.d.229.5	yes	6	15.8	even	4
1710.2.d.e.1369.2		6	5.3	odd	4
1710.2.d.e.1369.5		6	5.2	odd	4
2850.2.a.bk.1.3		3	3.2	odd	2
2850.2.a.bn.1.1		3	15.14	odd	2
8550.2.a.cf.1.1		3	5.4	even	2
8550.2.a.cr.1.3		3	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:	\( N \)	\(=\)	\( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8550.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +4.42864 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +4.42864 q^{7} +1.00000 q^{8} +5.80642 q^{11} -6.42864 q^{13} +4.42864 q^{14} +1.00000 q^{16} +3.37778 q^{17} -1.00000 q^{19} +5.80642 q^{22} +6.42864 q^{23} -6.42864 q^{26} +4.42864 q^{28} -7.80642 q^{29} +9.05086 q^{31} +1.00000 q^{32} +3.37778 q^{34} +3.67307 q^{37} -1.00000 q^{38} -4.42864 q^{41} -1.05086 q^{43} +5.80642 q^{44} +6.42864 q^{46} -5.18421 q^{47} +12.6128 q^{49} -6.42864 q^{52} +4.75557 q^{53} +4.42864 q^{56} -7.80642 q^{58} -4.62222 q^{59} +2.00000 q^{61} +9.05086 q^{62} +1.00000 q^{64} +2.75557 q^{67} +3.37778 q^{68} -7.61285 q^{71} +11.6128 q^{73} +3.67307 q^{74} -1.00000 q^{76} +25.7146 q^{77} +2.94914 q^{79} -4.42864 q^{82} +0.133353 q^{83} -1.05086 q^{86} +5.80642 q^{88} +3.18421 q^{89} -28.4701 q^{91} +6.42864 q^{92} -5.18421 q^{94} -11.4193 q^{97} +12.6128 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{8} + 4 q^{11} - 6 q^{13} + 3 q^{16} + 10 q^{17} - 3 q^{19} + 4 q^{22} + 6 q^{23} - 6 q^{26} - 10 q^{29} + 14 q^{31} + 3 q^{32} + 10 q^{34} - 2 q^{37} - 3 q^{38} + 10 q^{43} + 4 q^{44} + 6 q^{46} - 2 q^{47} + 11 q^{49} - 6 q^{52} + 14 q^{53} - 10 q^{58} - 14 q^{59} + 6 q^{61} + 14 q^{62} + 3 q^{64} + 8 q^{67} + 10 q^{68} + 4 q^{71} + 8 q^{73} - 2 q^{74} - 3 q^{76} + 24 q^{77} + 22 q^{79} + 10 q^{86} + 4 q^{88} - 4 q^{89} - 32 q^{91} + 6 q^{92} - 2 q^{94} + 6 q^{97} + 11 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^8 + 4 * q^11 - 6 * q^13 + 3 * q^16 + 10 * q^17 - 3 * q^19 + 4 * q^22 + 6 * q^23 - 6 * q^26 - 10 * q^29 + 14 * q^31 + 3 * q^32 + 10 * q^34 - 2 * q^37 - 3 * q^38 + 10 * q^43 + 4 * q^44 + 6 * q^46 - 2 * q^47 + 11 * q^49 - 6 * q^52 + 14 * q^53 - 10 * q^58 - 14 * q^59 + 6 * q^61 + 14 * q^62 + 3 * q^64 + 8 * q^67 + 10 * q^68 + 4 * q^71 + 8 * q^73 - 2 * q^74 - 3 * q^76 + 24 * q^77 + 22 * q^79 + 10 * q^86 + 4 * q^88 - 4 * q^89 - 32 * q^91 + 6 * q^92 - 2 * q^94 + 6 * q^97 + 11 * q^98

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