LMFDB - Embedded newform 8550.2.a.cr.1.1

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			1.1
Root			$-1.48119$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -3.35026 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -3.35026 q^{7} +1.00000 q^{8} +1.61213 q^{11} +1.35026 q^{13} -3.35026 q^{14} +1.00000 q^{16} +6.96239 q^{17} -1.00000 q^{19} +1.61213 q^{22} -1.35026 q^{23} +1.35026 q^{26} -3.35026 q^{28} -3.61213 q^{29} -2.31265 q^{31} +1.00000 q^{32} +6.96239 q^{34} -11.2750 q^{37} -1.00000 q^{38} +3.35026 q^{41} +10.3127 q^{43} +1.61213 q^{44} -1.35026 q^{46} -4.57452 q^{47} +4.22425 q^{49} +1.35026 q^{52} +11.9248 q^{53} -3.35026 q^{56} -3.61213 q^{58} -1.03761 q^{59} +2.00000 q^{61} -2.31265 q^{62} +1.00000 q^{64} +9.92478 q^{67} +6.96239 q^{68} +0.775746 q^{71} +3.22425 q^{73} -11.2750 q^{74} -1.00000 q^{76} -5.40105 q^{77} +14.3127 q^{79} +3.35026 q^{82} +10.8872 q^{83} +10.3127 q^{86} +1.61213 q^{88} +2.57452 q^{89} -4.52373 q^{91} -1.35026 q^{92} -4.57452 q^{94} +1.16362 q^{97} +4.22425 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−3.35026	−1.26628	−0.633140	−	0.774037i	$-0.718234\pi$
$7$	−3.35026	−1.26628	−0.633140	+	0.774037i	$0.718234\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	1.61213	0.486075	0.243037	−	0.970017i	$-0.421856\pi$
$11$	1.61213	0.486075	0.243037	+	0.970017i	$0.421856\pi$
$12$	0	0
$13$	1.35026	0.374495	0.187248	−	0.982313i	$-0.440043\pi$
$13$	1.35026	0.374495	0.187248	+	0.982313i	$0.440043\pi$
$14$	−3.35026	−0.895395
$15$	0	0
$16$	1.00000	0.250000
$17$	6.96239	1.68863	0.844314	−	0.535849i	$-0.180008\pi$
$17$	6.96239	1.68863	0.844314	+	0.535849i	$0.180008\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	1.61213	0.343707
$23$	−1.35026	−0.281549	−0.140775	−	0.990042i	$-0.544959\pi$
$23$	−1.35026	−0.281549	−0.140775	+	0.990042i	$0.544959\pi$
$24$	0	0
$25$	0	0
$26$	1.35026	0.264808
$27$	0	0
$28$	−3.35026	−0.633140
$29$	−3.61213	−0.670755	−0.335378	−	0.942084i	$-0.608864\pi$
$29$	−3.61213	−0.670755	−0.335378	+	0.942084i	$0.608864\pi$
$30$	0	0
$31$	−2.31265	−0.415364	−0.207682	−	0.978196i	$-0.566592\pi$
$31$	−2.31265	−0.415364	−0.207682	+	0.978196i	$0.566592\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	6.96239	1.19404
$35$	0	0
$36$	0	0
$37$	−11.2750	−1.85360	−0.926802	−	0.375549i	$-0.877454\pi$
$37$	−11.2750	−1.85360	−0.926802	+	0.375549i	$0.877454\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	0	0
$41$	3.35026	0.523223	0.261611	−	0.965173i	$-0.415746\pi$
$41$	3.35026	0.523223	0.261611	+	0.965173i	$0.415746\pi$
$42$	0	0
$43$	10.3127	1.57266	0.786332	−	0.617804i	$-0.211977\pi$
$43$	10.3127	1.57266	0.786332	+	0.617804i	$0.211977\pi$
$44$	1.61213	0.243037
$45$	0	0
$46$	−1.35026	−0.199085
$47$	−4.57452	−0.667262	−0.333631	−	0.942704i	$-0.608274\pi$
$47$	−4.57452	−0.667262	−0.333631	+	0.942704i	$0.608274\pi$
$48$	0	0
$49$	4.22425	0.603465
$50$	0	0
$51$	0	0
$52$	1.35026	0.187248
$53$	11.9248	1.63799	0.818997	−	0.573798i	$-0.194530\pi$
$53$	11.9248	1.63799	0.818997	+	0.573798i	$0.194530\pi$
$54$	0	0
$55$	0	0
$56$	−3.35026	−0.447698
$57$	0	0
$58$	−3.61213	−0.474295
$59$	−1.03761	−0.135085	−0.0675427	−	0.997716i	$-0.521516\pi$
$59$	−1.03761	−0.135085	−0.0675427	+	0.997716i	$0.521516\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−2.31265	−0.293707
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	9.92478	1.21250	0.606252	−	0.795272i	$-0.292672\pi$
$67$	9.92478	1.21250	0.606252	+	0.795272i	$0.292672\pi$
$68$	6.96239	0.844314
$69$	0	0
$70$	0	0
$71$	0.775746	0.0920641	0.0460321	−	0.998940i	$-0.485342\pi$
$71$	0.775746	0.0920641	0.0460321	+	0.998940i	$0.485342\pi$
$72$	0	0
$73$	3.22425	0.377370	0.188685	−	0.982038i	$-0.439577\pi$
$73$	3.22425	0.377370	0.188685	+	0.982038i	$0.439577\pi$
$74$	−11.2750	−1.31070
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−5.40105	−0.615506
$78$	0	0
$79$	14.3127	1.61030	0.805149	−	0.593072i	$-0.202085\pi$
$79$	14.3127	1.61030	0.805149	+	0.593072i	$0.202085\pi$
$80$	0	0
$81$	0	0
$82$	3.35026	0.369975
$83$	10.8872	1.19502	0.597511	−	0.801861i	$-0.296156\pi$
$83$	10.8872	1.19502	0.597511	+	0.801861i	$0.296156\pi$
$84$	0	0
$85$	0	0
$86$	10.3127	1.11204
$87$	0	0
$88$	1.61213	0.171853
$89$	2.57452	0.272898	0.136449	−	0.990647i	$-0.456431\pi$
$89$	2.57452	0.272898	0.136449	+	0.990647i	$0.456431\pi$
$90$	0	0
$91$	−4.52373	−0.474216
$92$	−1.35026	−0.140775
$93$	0	0
$94$	−4.57452	−0.471825
$95$	0	0
$96$	0	0
$97$	1.16362	0.118148	0.0590738	−	0.998254i	$-0.481185\pi$
$97$	1.16362	0.118148	0.0590738	+	0.998254i	$0.481185\pi$
$98$	4.22425	0.426714
$99$	0	0
$100$	0	0
$101$	−8.88717	−0.884306	−0.442153	−	0.896940i	$-0.645785\pi$
$101$	−8.88717	−0.884306	−0.442153	+	0.896940i	$0.645785\pi$
$102$	0	0
$103$	7.03761	0.693436	0.346718	−	0.937969i	$-0.387296\pi$
$103$	7.03761	0.693436	0.346718	+	0.937969i	$0.387296\pi$
$104$	1.35026	0.132404
$105$	0	0
$106$	11.9248	1.15824
$107$	0.775746	0.0749942	0.0374971	−	0.999297i	$-0.488062\pi$
$107$	0.775746	0.0749942	0.0374971	+	0.999297i	$0.488062\pi$
$108$	0	0
$109$	−20.1622	−1.93119	−0.965594	−	0.260052i	$-0.916260\pi$
$109$	−20.1622	−1.93119	−0.965594	+	0.260052i	$0.916260\pi$
$110$	0	0
$111$	0	0
$112$	−3.35026	−0.316570
$113$	−11.1490	−1.04881	−0.524406	−	0.851468i	$-0.675713\pi$
$113$	−11.1490	−1.04881	−0.524406	+	0.851468i	$0.675713\pi$
$114$	0	0
$115$	0	0
$116$	−3.61213	−0.335378
$117$	0	0
$118$	−1.03761	−0.0955199
$119$	−23.3258	−2.13827
$120$	0	0
$121$	−8.40105	−0.763732
$122$	2.00000	0.181071
$123$	0	0
$124$	−2.31265	−0.207682
$125$	0	0
$126$	0	0
$127$	13.7381	1.21906	0.609531	−	0.792762i	$-0.291358\pi$
$127$	13.7381	1.21906	0.609531	+	0.792762i	$0.291358\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	−5.61213	−0.490334	−0.245167	−	0.969481i	$-0.578843\pi$
$131$	−5.61213	−0.490334	−0.245167	+	0.969481i	$0.578843\pi$
$132$	0	0
$133$	3.35026	0.290505
$134$	9.92478	0.857370
$135$	0	0
$136$	6.96239	0.597020
$137$	17.6629	1.50904	0.754522	−	0.656275i	$-0.227869\pi$
$137$	17.6629	1.50904	0.754522	+	0.656275i	$0.227869\pi$
$138$	0	0
$139$	10.7005	0.907607	0.453803	−	0.891102i	$-0.350067\pi$
$139$	10.7005	0.907607	0.453803	+	0.891102i	$0.350067\pi$
$140$	0	0
$141$	0	0
$142$	0.775746	0.0650992
$143$	2.17679	0.182033
$144$	0	0
$145$	0	0
$146$	3.22425	0.266841
$147$	0	0
$148$	−11.2750	−0.926802
$149$	−1.03761	−0.0850044	−0.0425022	−	0.999096i	$-0.513533\pi$
$149$	−1.03761	−0.0850044	−0.0425022	+	0.999096i	$0.513533\pi$
$150$	0	0
$151$	1.16362	0.0946940	0.0473470	−	0.998879i	$-0.484923\pi$
$151$	1.16362	0.0946940	0.0473470	+	0.998879i	$0.484923\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	−5.40105	−0.435229
$155$	0	0
$156$	0	0
$157$	10.9624	0.874894	0.437447	−	0.899244i	$-0.355883\pi$
$157$	10.9624	0.874894	0.437447	+	0.899244i	$0.355883\pi$
$158$	14.3127	1.13865
$159$	0	0
$160$	0	0
$161$	4.52373	0.356520
$162$	0	0
$163$	21.0132	1.64588	0.822939	−	0.568129i	$-0.192333\pi$
$163$	21.0132	1.64588	0.822939	+	0.568129i	$0.192333\pi$
$164$	3.35026	0.261611
$165$	0	0
$166$	10.8872	0.845008
$167$	−9.92478	−0.768002	−0.384001	−	0.923333i	$-0.625454\pi$
$167$	−9.92478	−0.768002	−0.384001	+	0.923333i	$0.625454\pi$
$168$	0	0
$169$	−11.1768	−0.859753
$170$	0	0
$171$	0	0
$172$	10.3127	0.786332
$173$	−14.6253	−1.11194	−0.555971	−	0.831202i	$-0.687653\pi$
$173$	−14.6253	−1.11194	−0.555971	+	0.831202i	$0.687653\pi$
$174$	0	0
$175$	0	0
$176$	1.61213	0.121519
$177$	0	0
$178$	2.57452	0.192968
$179$	−11.7381	−0.877349	−0.438675	−	0.898646i	$-0.644552\pi$
$179$	−11.7381	−0.877349	−0.438675	+	0.898646i	$0.644552\pi$
$180$	0	0
$181$	21.4617	1.59523	0.797617	−	0.603164i	$-0.206094\pi$
$181$	21.4617	1.59523	0.797617	+	0.603164i	$0.206094\pi$
$182$	−4.52373	−0.335321
$183$	0	0
$184$	−1.35026	−0.0995426
$185$	0	0
$186$	0	0
$187$	11.2243	0.820799
$188$	−4.57452	−0.333631
$189$	0	0
$190$	0	0
$191$	21.2750	1.53941	0.769704	−	0.638401i	$-0.220404\pi$
$191$	21.2750	1.53941	0.769704	+	0.638401i	$0.220404\pi$
$192$	0	0
$193$	−14.1622	−1.01942	−0.509709	−	0.860347i	$-0.670247\pi$
$193$	−14.1622	−1.01942	−0.509709	+	0.860347i	$0.670247\pi$
$194$	1.16362	0.0835430
$195$	0	0
$196$	4.22425	0.301732
$197$	−3.87399	−0.276011	−0.138005	−	0.990431i	$-0.544069\pi$
$197$	−3.87399	−0.276011	−0.138005	+	0.990431i	$0.544069\pi$
$198$	0	0
$199$	3.47627	0.246426	0.123213	−	0.992380i	$-0.460680\pi$
$199$	3.47627	0.246426	0.123213	+	0.992380i	$0.460680\pi$
$200$	0	0
$201$	0	0
$202$	−8.88717	−0.625299
$203$	12.1016	0.849364
$204$	0	0
$205$	0	0
$206$	7.03761	0.490334
$207$	0	0
$208$	1.35026	0.0936238
$209$	−1.61213	−0.111513
$210$	0	0
$211$	9.92478	0.683250	0.341625	−	0.939836i	$-0.389023\pi$
$211$	9.92478	0.683250	0.341625	+	0.939836i	$0.389023\pi$
$212$	11.9248	0.818997
$213$	0	0
$214$	0.775746	0.0530289
$215$	0	0
$216$	0	0
$217$	7.74798	0.525967
$218$	−20.1622	−1.36556
$219$	0	0
$220$	0	0
$221$	9.40105	0.632383
$222$	0	0
$223$	7.03761	0.471273	0.235637	−	0.971841i	$-0.424282\pi$
$223$	7.03761	0.471273	0.235637	+	0.971841i	$0.424282\pi$
$224$	−3.35026	−0.223849
$225$	0	0
$226$	−11.1490	−0.741623
$227$	14.5501	0.965723	0.482861	−	0.875697i	$-0.339598\pi$
$227$	14.5501	0.965723	0.482861	+	0.875697i	$0.339598\pi$
$228$	0	0
$229$	11.4010	0.753402	0.376701	−	0.926335i	$-0.377058\pi$
$229$	11.4010	0.753402	0.376701	+	0.926335i	$0.377058\pi$
$230$	0	0
$231$	0	0
$232$	−3.61213	−0.237148
$233$	−21.9149	−1.43569	−0.717847	−	0.696201i	$-0.754872\pi$
$233$	−21.9149	−1.43569	−0.717847	+	0.696201i	$0.754872\pi$
$234$	0	0
$235$	0	0
$236$	−1.03761	−0.0675427
$237$	0	0
$238$	−23.3258	−1.51199
$239$	13.2750	0.858691	0.429345	−	0.903140i	$-0.358744\pi$
$239$	13.2750	0.858691	0.429345	+	0.903140i	$0.358744\pi$
$240$	0	0
$241$	21.3258	1.37372	0.686859	−	0.726791i	$-0.258989\pi$
$241$	21.3258	1.37372	0.686859	+	0.726791i	$0.258989\pi$
$242$	−8.40105	−0.540040
$243$	0	0
$244$	2.00000	0.128037
$245$	0	0
$246$	0	0
$247$	−1.35026	−0.0859151
$248$	−2.31265	−0.146853
$249$	0	0
$250$	0	0
$251$	−16.3127	−1.02965	−0.514823	−	0.857297i	$-0.672142\pi$
$251$	−16.3127	−1.02965	−0.514823	+	0.857297i	$0.672142\pi$
$252$	0	0
$253$	−2.17679	−0.136854
$254$	13.7381	0.862007
$255$	0	0
$256$	1.00000	0.0625000
$257$	24.5501	1.53139	0.765696	−	0.643203i	$-0.222395\pi$
$257$	24.5501	1.53139	0.765696	+	0.643203i	$0.222395\pi$
$258$	0	0
$259$	37.7743	2.34718
$260$	0	0
$261$	0	0
$262$	−5.61213	−0.346718
$263$	30.3488	1.87139	0.935695	−	0.352810i	$-0.114774\pi$
$263$	30.3488	1.87139	0.935695	+	0.352810i	$0.114774\pi$
$264$	0	0
$265$	0	0
$266$	3.35026	0.205418
$267$	0	0
$268$	9.92478	0.606252
$269$	−14.3127	−0.872658	−0.436329	−	0.899787i	$-0.643722\pi$
$269$	−14.3127	−0.872658	−0.436329	+	0.899787i	$0.643722\pi$
$270$	0	0
$271$	−7.32582	−0.445012	−0.222506	−	0.974931i	$-0.571424\pi$
$271$	−7.32582	−0.445012	−0.222506	+	0.974931i	$0.571424\pi$
$272$	6.96239	0.422157
$273$	0	0
$274$	17.6629	1.06706
$275$	0	0
$276$	0	0
$277$	14.4387	0.867535	0.433767	−	0.901025i	$-0.357184\pi$
$277$	14.4387	0.867535	0.433767	+	0.901025i	$0.357184\pi$
$278$	10.7005	0.641775
$279$	0	0
$280$	0	0
$281$	11.9756	0.714402	0.357201	−	0.934028i	$-0.383731\pi$
$281$	11.9756	0.714402	0.357201	+	0.934028i	$0.383731\pi$
$282$	0	0
$283$	24.4894	1.45575	0.727873	−	0.685712i	$-0.240509\pi$
$283$	24.4894	1.45575	0.727873	+	0.685712i	$0.240509\pi$
$284$	0.775746	0.0460321
$285$	0	0
$286$	2.17679	0.128716
$287$	−11.2243	−0.662547
$288$	0	0
$289$	31.4749	1.85146
$290$	0	0
$291$	0	0
$292$	3.22425	0.188685
$293$	−18.1016	−1.05751	−0.528753	−	0.848776i	$-0.677340\pi$
$293$	−18.1016	−1.05751	−0.528753	+	0.848776i	$0.677340\pi$
$294$	0	0
$295$	0	0
$296$	−11.2750	−0.655348
$297$	0	0
$298$	−1.03761	−0.0601072
$299$	−1.82321	−0.105439
$300$	0	0
$301$	−34.5501	−1.99143
$302$	1.16362	0.0669588
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	0	0
$307$	29.9248	1.70790	0.853949	−	0.520357i	$-0.174201\pi$
$307$	29.9248	1.70790	0.853949	+	0.520357i	$0.174201\pi$
$308$	−5.40105	−0.307753
$309$	0	0
$310$	0	0
$311$	21.2750	1.20640	0.603198	−	0.797591i	$-0.293893\pi$
$311$	21.2750	1.20640	0.603198	+	0.797591i	$0.293893\pi$
$312$	0	0
$313$	17.4010	0.983565	0.491783	−	0.870718i	$-0.336345\pi$
$313$	17.4010	0.983565	0.491783	+	0.870718i	$0.336345\pi$
$314$	10.9624	0.618643
$315$	0	0
$316$	14.3127	0.805149
$317$	−14.1016	−0.792023	−0.396012	−	0.918246i	$-0.629606\pi$
$317$	−14.1016	−0.792023	−0.396012	+	0.918246i	$0.629606\pi$
$318$	0	0
$319$	−5.82321	−0.326037
$320$	0	0
$321$	0	0
$322$	4.52373	0.252098
$323$	−6.96239	−0.387398
$324$	0	0
$325$	0	0
$326$	21.0132	1.16381
$327$	0	0
$328$	3.35026	0.184987
$329$	15.3258	0.844940
$330$	0	0
$331$	6.85097	0.376563	0.188282	−	0.982115i	$-0.439708\pi$
$331$	6.85097	0.376563	0.188282	+	0.982115i	$0.439708\pi$
$332$	10.8872	0.597511
$333$	0	0
$334$	−9.92478	−0.543060
$335$	0	0
$336$	0	0
$337$	24.2374	1.32030	0.660148	−	0.751135i	$-0.270493\pi$
$337$	24.2374	1.32030	0.660148	+	0.751135i	$0.270493\pi$
$338$	−11.1768	−0.607937
$339$	0	0
$340$	0	0
$341$	−3.72829	−0.201898
$342$	0	0
$343$	9.29948	0.502125
$344$	10.3127	0.556021
$345$	0	0
$346$	−14.6253	−0.786261
$347$	−0.962389	−0.0516637	−0.0258319	−	0.999666i	$-0.508223\pi$
$347$	−0.962389	−0.0516637	−0.0258319	+	0.999666i	$0.508223\pi$
$348$	0	0
$349$	−1.37470	−0.0735860	−0.0367930	−	0.999323i	$-0.511714\pi$
$349$	−1.37470	−0.0735860	−0.0367930	+	0.999323i	$0.511714\pi$
$350$	0	0
$351$	0	0
$352$	1.61213	0.0859267
$353$	−28.7367	−1.52950	−0.764751	−	0.644326i	$-0.777138\pi$
$353$	−28.7367	−1.52950	−0.764751	+	0.644326i	$0.777138\pi$
$354$	0	0
$355$	0	0
$356$	2.57452	0.136449
$357$	0	0
$358$	−11.7381	−0.620380
$359$	−31.1998	−1.64666	−0.823332	−	0.567561i	$-0.807887\pi$
$359$	−31.1998	−1.64666	−0.823332	+	0.567561i	$0.807887\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	21.4617	1.12800
$363$	0	0
$364$	−4.52373	−0.237108
$365$	0	0
$366$	0	0
$367$	20.1260	1.05057	0.525285	−	0.850927i	$-0.323959\pi$
$367$	20.1260	1.05057	0.525285	+	0.850927i	$0.323959\pi$
$368$	−1.35026	−0.0703873
$369$	0	0
$370$	0	0
$371$	−39.9511	−2.07416
$372$	0	0
$373$	−17.9756	−0.930739	−0.465370	−	0.885116i	$-0.654079\pi$
$373$	−17.9756	−0.930739	−0.465370	+	0.885116i	$0.654079\pi$
$374$	11.2243	0.580392
$375$	0	0
$376$	−4.57452	−0.235913
$377$	−4.87732	−0.251195
$378$	0	0
$379$	1.67276	0.0859240	0.0429620	−	0.999077i	$-0.486321\pi$
$379$	1.67276	0.0859240	0.0429620	+	0.999077i	$0.486321\pi$
$380$	0	0
$381$	0	0
$382$	21.2750	1.08853
$383$	−31.8496	−1.62744	−0.813718	−	0.581260i	$-0.802560\pi$
$383$	−31.8496	−1.62744	−0.813718	+	0.581260i	$0.802560\pi$
$384$	0	0
$385$	0	0
$386$	−14.1622	−0.720837
$387$	0	0
$388$	1.16362	0.0590738
$389$	10.3371	0.524111	0.262056	−	0.965053i	$-0.415600\pi$
$389$	10.3371	0.524111	0.262056	+	0.965053i	$0.415600\pi$
$390$	0	0
$391$	−9.40105	−0.475431
$392$	4.22425	0.213357
$393$	0	0
$394$	−3.87399	−0.195169
$395$	0	0
$396$	0	0
$397$	−31.4372	−1.57779	−0.788895	−	0.614528i	$-0.789346\pi$
$397$	−31.4372	−1.57779	−0.788895	+	0.614528i	$0.789346\pi$
$398$	3.47627	0.174250
$399$	0	0
$400$	0	0
$401$	−5.94921	−0.297090	−0.148545	−	0.988906i	$-0.547459\pi$
$401$	−5.94921	−0.297090	−0.148545	+	0.988906i	$0.547459\pi$
$402$	0	0
$403$	−3.12268	−0.155552
$404$	−8.88717	−0.442153
$405$	0	0
$406$	12.1016	0.600591
$407$	−18.1768	−0.900990
$408$	0	0
$409$	−30.9986	−1.53278	−0.766391	−	0.642375i	$-0.777949\pi$
$409$	−30.9986	−1.53278	−0.766391	+	0.642375i	$0.777949\pi$
$410$	0	0
$411$	0	0
$412$	7.03761	0.346718
$413$	3.47627	0.171056
$414$	0	0
$415$	0	0
$416$	1.35026	0.0662020
$417$	0	0
$418$	−1.61213	−0.0788517
$419$	34.3390	1.67757	0.838785	−	0.544463i	$-0.183266\pi$
$419$	34.3390	1.67757	0.838785	+	0.544463i	$0.183266\pi$
$420$	0	0
$421$	22.8627	1.11426	0.557131	−	0.830425i	$-0.311902\pi$
$421$	22.8627	1.11426	0.557131	+	0.830425i	$0.311902\pi$
$422$	9.92478	0.483131
$423$	0	0
$424$	11.9248	0.579118
$425$	0	0
$426$	0	0
$427$	−6.70052	−0.324261
$428$	0.775746	0.0374971
$429$	0	0
$430$	0	0
$431$	−11.5975	−0.558634	−0.279317	−	0.960199i	$-0.590108\pi$
$431$	−11.5975	−0.558634	−0.279317	+	0.960199i	$0.590108\pi$
$432$	0	0
$433$	27.8350	1.33766	0.668832	−	0.743414i	$-0.266795\pi$
$433$	27.8350	1.33766	0.668832	+	0.743414i	$0.266795\pi$
$434$	7.74798	0.371915
$435$	0	0
$436$	−20.1622	−0.965594
$437$	1.35026	0.0645918
$438$	0	0
$439$	−38.7875	−1.85123	−0.925613	−	0.378471i	$-0.876450\pi$
$439$	−38.7875	−1.85123	−0.925613	+	0.378471i	$0.876450\pi$
$440$	0	0
$441$	0	0
$442$	9.40105	0.447162
$443$	29.2144	1.38802	0.694009	−	0.719966i	$-0.255843\pi$
$443$	29.2144	1.38802	0.694009	+	0.719966i	$0.255843\pi$
$444$	0	0
$445$	0	0
$446$	7.03761	0.333241
$447$	0	0
$448$	−3.35026	−0.158285
$449$	20.4993	0.967421	0.483711	−	0.875228i	$-0.339289\pi$
$449$	20.4993	0.967421	0.483711	+	0.875228i	$0.339289\pi$
$450$	0	0
$451$	5.40105	0.254325
$452$	−11.1490	−0.524406
$453$	0	0
$454$	14.5501	0.682869
$455$	0	0
$456$	0	0
$457$	6.44851	0.301648	0.150824	−	0.988561i	$-0.451807\pi$
$457$	6.44851	0.301648	0.150824	+	0.988561i	$0.451807\pi$
$458$	11.4010	0.532736
$459$	0	0
$460$	0	0
$461$	13.5125	0.629338	0.314669	−	0.949201i	$-0.398106\pi$
$461$	13.5125	0.629338	0.314669	+	0.949201i	$0.398106\pi$
$462$	0	0
$463$	6.20123	0.288196	0.144098	−	0.989563i	$-0.453972\pi$
$463$	6.20123	0.288196	0.144098	+	0.989563i	$0.453972\pi$
$464$	−3.61213	−0.167689
$465$	0	0
$466$	−21.9149	−1.01519
$467$	16.5599	0.766302	0.383151	−	0.923686i	$-0.374839\pi$
$467$	16.5599	0.766302	0.383151	+	0.923686i	$0.374839\pi$
$468$	0	0
$469$	−33.2506	−1.53537
$470$	0	0
$471$	0	0
$472$	−1.03761	−0.0477599
$473$	16.6253	0.764432
$474$	0	0
$475$	0	0
$476$	−23.3258	−1.06914
$477$	0	0
$478$	13.2750	0.607186
$479$	30.5256	1.39475	0.697376	−	0.716705i	$-0.254351\pi$
$479$	30.5256	1.39475	0.697376	+	0.716705i	$0.254351\pi$
$480$	0	0
$481$	−15.2243	−0.694166
$482$	21.3258	0.971365
$483$	0	0
$484$	−8.40105	−0.381866
$485$	0	0
$486$	0	0
$487$	−2.51388	−0.113915	−0.0569574	−	0.998377i	$-0.518140\pi$
$487$	−2.51388	−0.113915	−0.0569574	+	0.998377i	$0.518140\pi$
$488$	2.00000	0.0905357
$489$	0	0
$490$	0	0
$491$	−1.46168	−0.0659648	−0.0329824	−	0.999456i	$-0.510501\pi$
$491$	−1.46168	−0.0659648	−0.0329824	+	0.999456i	$0.510501\pi$
$492$	0	0
$493$	−25.1490	−1.13266
$494$	−1.35026	−0.0607511
$495$	0	0
$496$	−2.31265	−0.103841
$497$	−2.59895	−0.116579
$498$	0	0
$499$	−5.55149	−0.248519	−0.124259	−	0.992250i	$-0.539656\pi$
$499$	−5.55149	−0.248519	−0.124259	+	0.992250i	$0.539656\pi$
$500$	0	0
$501$	0	0
$502$	−16.3127	−0.728069
$503$	−6.90175	−0.307734	−0.153867	−	0.988092i	$-0.549173\pi$
$503$	−6.90175	−0.307734	−0.153867	+	0.988092i	$0.549173\pi$
$504$	0	0
$505$	0	0
$506$	−2.17679	−0.0967703
$507$	0	0
$508$	13.7381	0.609531
$509$	−8.76116	−0.388331	−0.194166	−	0.980969i	$-0.562200\pi$
$509$	−8.76116	−0.388331	−0.194166	+	0.980969i	$0.562200\pi$
$510$	0	0
$511$	−10.8021	−0.477857
$512$	1.00000	0.0441942
$513$	0	0
$514$	24.5501	1.08286
$515$	0	0
$516$	0	0
$517$	−7.37470	−0.324339
$518$	37.7743	1.65971
$519$	0	0
$520$	0	0
$521$	−39.4518	−1.72842	−0.864208	−	0.503135i	$-0.832180\pi$
$521$	−39.4518	−1.72842	−0.864208	+	0.503135i	$0.832180\pi$
$522$	0	0
$523$	−37.5026	−1.63987	−0.819937	−	0.572453i	$-0.805992\pi$
$523$	−37.5026	−1.63987	−0.819937	+	0.572453i	$0.805992\pi$
$524$	−5.61213	−0.245167
$525$	0	0
$526$	30.3488	1.32327
$527$	−16.1016	−0.701395
$528$	0	0
$529$	−21.1768	−0.920730
$530$	0	0
$531$	0	0
$532$	3.35026	0.145252
$533$	4.52373	0.195945
$534$	0	0
$535$	0	0
$536$	9.92478	0.428685
$537$	0	0
$538$	−14.3127	−0.617062
$539$	6.81003	0.293329
$540$	0	0
$541$	−30.7757	−1.32315	−0.661576	−	0.749878i	$-0.730112\pi$
$541$	−30.7757	−1.32315	−0.661576	+	0.749878i	$0.730112\pi$
$542$	−7.32582	−0.314671
$543$	0	0
$544$	6.96239	0.298510
$545$	0	0
$546$	0	0
$547$	−1.77433	−0.0758649	−0.0379325	−	0.999280i	$-0.512077\pi$
$547$	−1.77433	−0.0758649	−0.0379325	+	0.999280i	$0.512077\pi$
$548$	17.6629	0.754522
$549$	0	0
$550$	0	0
$551$	3.61213	0.153882
$552$	0	0
$553$	−47.9511	−2.03909
$554$	14.4387	0.613440
$555$	0	0
$556$	10.7005	0.453803
$557$	8.90175	0.377179	0.188590	−	0.982056i	$-0.439608\pi$
$557$	8.90175	0.377179	0.188590	+	0.982056i	$0.439608\pi$
$558$	0	0
$559$	13.9248	0.588955
$560$	0	0
$561$	0	0
$562$	11.9756	0.505159
$563$	10.9525	0.461595	0.230797	−	0.973002i	$-0.425867\pi$
$563$	10.9525	0.461595	0.230797	+	0.973002i	$0.425867\pi$
$564$	0	0
$565$	0	0
$566$	24.4894	1.02937
$567$	0	0
$568$	0.775746	0.0325496
$569$	10.2012	0.427658	0.213829	−	0.976871i	$-0.431406\pi$
$569$	10.2012	0.427658	0.213829	+	0.976871i	$0.431406\pi$
$570$	0	0
$571$	−25.6531	−1.07355	−0.536774	−	0.843726i	$-0.680357\pi$
$571$	−25.6531	−1.07355	−0.536774	+	0.843726i	$0.680357\pi$
$572$	2.17679	0.0910163
$573$	0	0
$574$	−11.2243	−0.468491
$575$	0	0
$576$	0	0
$577$	6.44851	0.268455	0.134227	−	0.990951i	$-0.457145\pi$
$577$	6.44851	0.268455	0.134227	+	0.990951i	$0.457145\pi$
$578$	31.4749	1.30918
$579$	0	0
$580$	0	0
$581$	−36.4749	−1.51323
$582$	0	0
$583$	19.2243	0.796187
$584$	3.22425	0.133421
$585$	0	0
$586$	−18.1016	−0.747769
$587$	41.8397	1.72691	0.863455	−	0.504426i	$-0.168296\pi$
$587$	41.8397	1.72691	0.863455	+	0.504426i	$0.168296\pi$
$588$	0	0
$589$	2.31265	0.0952911
$590$	0	0
$591$	0	0
$592$	−11.2750	−0.463401
$593$	−8.73672	−0.358774	−0.179387	−	0.983779i	$-0.557411\pi$
$593$	−8.73672	−0.358774	−0.179387	+	0.983779i	$0.557411\pi$
$594$	0	0
$595$	0	0
$596$	−1.03761	−0.0425022
$597$	0	0
$598$	−1.82321	−0.0745565
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	30.6253	1.24923	0.624616	−	0.780932i	$-0.285255\pi$
$601$	30.6253	1.24923	0.624616	+	0.780932i	$0.285255\pi$
$602$	−34.5501	−1.40816
$603$	0	0
$604$	1.16362	0.0473470
$605$	0	0
$606$	0	0
$607$	−2.51388	−0.102035	−0.0510176	−	0.998698i	$-0.516246\pi$
$607$	−2.51388	−0.102035	−0.0510176	+	0.998698i	$0.516246\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	0	0
$611$	−6.17679	−0.249886
$612$	0	0
$613$	−2.96239	−0.119650	−0.0598249	−	0.998209i	$-0.519054\pi$
$613$	−2.96239	−0.119650	−0.0598249	+	0.998209i	$0.519054\pi$
$614$	29.9248	1.20767
$615$	0	0
$616$	−5.40105	−0.217614
$617$	18.3371	0.738223	0.369112	−	0.929385i	$-0.379662\pi$
$617$	18.3371	0.738223	0.369112	+	0.929385i	$0.379662\pi$
$618$	0	0
$619$	−40.7269	−1.63695	−0.818476	−	0.574541i	$-0.805180\pi$
$619$	−40.7269	−1.63695	−0.818476	+	0.574541i	$0.805180\pi$
$620$	0	0
$621$	0	0
$622$	21.2750	0.853051
$623$	−8.62530	−0.345565
$624$	0	0
$625$	0	0
$626$	17.4010	0.695486
$627$	0	0
$628$	10.9624	0.437447
$629$	−78.5012	−3.13005
$630$	0	0
$631$	−5.92478	−0.235862	−0.117931	−	0.993022i	$-0.537626\pi$
$631$	−5.92478	−0.235862	−0.117931	+	0.993022i	$0.537626\pi$
$632$	14.3127	0.569327
$633$	0	0
$634$	−14.1016	−0.560045
$635$	0	0
$636$	0	0
$637$	5.70385	0.225995
$638$	−5.82321	−0.230543
$639$	0	0
$640$	0	0
$641$	−19.0494	−0.752405	−0.376202	−	0.926537i	$-0.622770\pi$
$641$	−19.0494	−0.752405	−0.376202	+	0.926537i	$0.622770\pi$
$642$	0	0
$643$	−1.06205	−0.0418831	−0.0209416	−	0.999781i	$-0.506666\pi$
$643$	−1.06205	−0.0418831	−0.0209416	+	0.999781i	$0.506666\pi$
$644$	4.52373	0.178260
$645$	0	0
$646$	−6.96239	−0.273932
$647$	−26.6497	−1.04771	−0.523855	−	0.851808i	$-0.675507\pi$
$647$	−26.6497	−1.04771	−0.523855	+	0.851808i	$0.675507\pi$
$648$	0	0
$649$	−1.67276	−0.0656616
$650$	0	0
$651$	0	0
$652$	21.0132	0.822939
$653$	9.64832	0.377568	0.188784	−	0.982019i	$-0.439545\pi$
$653$	9.64832	0.377568	0.188784	+	0.982019i	$0.439545\pi$
$654$	0	0
$655$	0	0
$656$	3.35026	0.130806
$657$	0	0
$658$	15.3258	0.597463
$659$	−10.0654	−0.392091	−0.196046	−	0.980595i	$-0.562810\pi$
$659$	−10.0654	−0.392091	−0.196046	+	0.980595i	$0.562810\pi$
$660$	0	0
$661$	13.5633	0.527549	0.263775	−	0.964584i	$-0.415032\pi$
$661$	13.5633	0.527549	0.263775	+	0.964584i	$0.415032\pi$
$662$	6.85097	0.266270
$663$	0	0
$664$	10.8872	0.422504
$665$	0	0
$666$	0	0
$667$	4.87732	0.188850
$668$	−9.92478	−0.384001
$669$	0	0
$670$	0	0
$671$	3.22425	0.124471
$672$	0	0
$673$	−5.93937	−0.228946	−0.114473	−	0.993426i	$-0.536518\pi$
$673$	−5.93937	−0.228946	−0.114473	+	0.993426i	$0.536518\pi$
$674$	24.2374	0.933591
$675$	0	0
$676$	−11.1768	−0.429877
$677$	24.9525	0.959004	0.479502	−	0.877541i	$-0.340817\pi$
$677$	24.9525	0.959004	0.479502	+	0.877541i	$0.340817\pi$
$678$	0	0
$679$	−3.89843	−0.149608
$680$	0	0
$681$	0	0
$682$	−3.72829	−0.142763
$683$	−45.6239	−1.74575	−0.872875	−	0.487944i	$-0.837747\pi$
$683$	−45.6239	−1.74575	−0.872875	+	0.487944i	$0.837747\pi$
$684$	0	0
$685$	0	0
$686$	9.29948	0.355056
$687$	0	0
$688$	10.3127	0.393166
$689$	16.1016	0.613421
$690$	0	0
$691$	−11.7480	−0.446914	−0.223457	−	0.974714i	$-0.571734\pi$
$691$	−11.7480	−0.446914	−0.223457	+	0.974714i	$0.571734\pi$
$692$	−14.6253	−0.555971
$693$	0	0
$694$	−0.962389	−0.0365318
$695$	0	0
$696$	0	0
$697$	23.3258	0.883529
$698$	−1.37470	−0.0520331
$699$	0	0
$700$	0	0
$701$	43.8105	1.65470	0.827350	−	0.561686i	$-0.189847\pi$
$701$	43.8105	1.65470	0.827350	+	0.561686i	$0.189847\pi$
$702$	0	0
$703$	11.2750	0.425246
$704$	1.61213	0.0607593
$705$	0	0
$706$	−28.7367	−1.08152
$707$	29.7743	1.11978
$708$	0	0
$709$	−29.5975	−1.11156	−0.555779	−	0.831330i	$-0.687580\pi$
$709$	−29.5975	−1.11156	−0.555779	+	0.831330i	$0.687580\pi$
$710$	0	0
$711$	0	0
$712$	2.57452	0.0964840
$713$	3.12268	0.116945
$714$	0	0
$715$	0	0
$716$	−11.7381	−0.438675
$717$	0	0
$718$	−31.1998	−1.16437
$719$	−7.45183	−0.277906	−0.138953	−	0.990299i	$-0.544374\pi$
$719$	−7.45183	−0.277906	−0.138953	+	0.990299i	$0.544374\pi$
$720$	0	0
$721$	−23.5778	−0.878085
$722$	1.00000	0.0372161
$723$	0	0
$724$	21.4617	0.797617
$725$	0	0
$726$	0	0
$727$	−0.600863	−0.0222848	−0.0111424	−	0.999938i	$-0.503547\pi$
$727$	−0.600863	−0.0222848	−0.0111424	+	0.999938i	$0.503547\pi$
$728$	−4.52373	−0.167661
$729$	0	0
$730$	0	0
$731$	71.8007	2.65564
$732$	0	0
$733$	−36.0625	−1.33200	−0.666000	−	0.745952i	$-0.731995\pi$
$733$	−36.0625	−1.33200	−0.666000	+	0.745952i	$0.731995\pi$
$734$	20.1260	0.742865
$735$	0	0
$736$	−1.35026	−0.0497713
$737$	16.0000	0.589368
$738$	0	0
$739$	−14.1768	−0.521502	−0.260751	−	0.965406i	$-0.583970\pi$
$739$	−14.1768	−0.521502	−0.260751	+	0.965406i	$0.583970\pi$
$740$	0	0
$741$	0	0
$742$	−39.9511	−1.46665
$743$	24.9986	0.917109	0.458555	−	0.888666i	$-0.348367\pi$
$743$	24.9986	0.917109	0.458555	+	0.888666i	$0.348367\pi$
$744$	0	0
$745$	0	0
$746$	−17.9756	−0.658132
$747$	0	0
$748$	11.2243	0.410399
$749$	−2.59895	−0.0949637
$750$	0	0
$751$	−10.2111	−0.372608	−0.186304	−	0.982492i	$-0.559651\pi$
$751$	−10.2111	−0.372608	−0.186304	+	0.982492i	$0.559651\pi$
$752$	−4.57452	−0.166815
$753$	0	0
$754$	−4.87732	−0.177621
$755$	0	0
$756$	0	0
$757$	−44.4847	−1.61682	−0.808412	−	0.588617i	$-0.799673\pi$
$757$	−44.4847	−1.61682	−0.808412	+	0.588617i	$0.799673\pi$
$758$	1.67276	0.0607574
$759$	0	0
$760$	0	0
$761$	−27.1490	−0.984152	−0.492076	−	0.870552i	$-0.663762\pi$
$761$	−27.1490	−0.984152	−0.492076	+	0.870552i	$0.663762\pi$
$762$	0	0
$763$	67.5487	2.44543
$764$	21.2750	0.769704
$765$	0	0
$766$	−31.8496	−1.15077
$767$	−1.40105	−0.0505889
$768$	0	0
$769$	−31.4010	−1.13235	−0.566175	−	0.824285i	$-0.691578\pi$
$769$	−31.4010	−1.13235	−0.566175	+	0.824285i	$0.691578\pi$
$770$	0	0
$771$	0	0
$772$	−14.1622	−0.509709
$773$	4.02635	0.144818	0.0724088	−	0.997375i	$-0.476931\pi$
$773$	4.02635	0.144818	0.0724088	+	0.997375i	$0.476931\pi$
$774$	0	0
$775$	0	0
$776$	1.16362	0.0417715
$777$	0	0
$778$	10.3371	0.370603
$779$	−3.35026	−0.120036
$780$	0	0
$781$	1.25060	0.0447500
$782$	−9.40105	−0.336181
$783$	0	0
$784$	4.22425	0.150866
$785$	0	0
$786$	0	0
$787$	−18.2981	−0.652255	−0.326128	−	0.945326i	$-0.605744\pi$
$787$	−18.2981	−0.652255	−0.326128	+	0.945326i	$0.605744\pi$
$788$	−3.87399	−0.138005
$789$	0	0
$790$	0	0
$791$	37.3522	1.32809
$792$	0	0
$793$	2.70052	0.0958984
$794$	−31.4372	−1.11567
$795$	0	0
$796$	3.47627	0.123213
$797$	−8.55008	−0.302859	−0.151430	−	0.988468i	$-0.548388\pi$
$797$	−8.55008	−0.302859	−0.151430	+	0.988468i	$0.548388\pi$
$798$	0	0
$799$	−31.8496	−1.12676
$800$	0	0
$801$	0	0
$802$	−5.94921	−0.210074
$803$	5.19791	0.183430
$804$	0	0
$805$	0	0
$806$	−3.12268	−0.109992
$807$	0	0
$808$	−8.88717	−0.312649
$809$	−50.7269	−1.78346	−0.891731	−	0.452566i	$-0.850509\pi$
$809$	−50.7269	−1.78346	−0.891731	+	0.452566i	$0.850509\pi$
$810$	0	0
$811$	10.3272	0.362638	0.181319	−	0.983424i	$-0.441963\pi$
$811$	10.3272	0.362638	0.181319	+	0.983424i	$0.441963\pi$
$812$	12.1016	0.424682
$813$	0	0
$814$	−18.1768	−0.637096
$815$	0	0
$816$	0	0
$817$	−10.3127	−0.360794
$818$	−30.9986	−1.08384
$819$	0	0
$820$	0	0
$821$	−36.5139	−1.27434	−0.637172	−	0.770722i	$-0.719896\pi$
$821$	−36.5139	−1.27434	−0.637172	+	0.770722i	$0.719896\pi$
$822$	0	0
$823$	−23.0982	−0.805154	−0.402577	−	0.915386i	$-0.631885\pi$
$823$	−23.0982	−0.805154	−0.402577	+	0.915386i	$0.631885\pi$
$824$	7.03761	0.245167
$825$	0	0
$826$	3.47627	0.120955
$827$	−3.37470	−0.117350	−0.0586749	−	0.998277i	$-0.518688\pi$
$827$	−3.37470	−0.117350	−0.0586749	+	0.998277i	$0.518688\pi$
$828$	0	0
$829$	22.6399	0.786316	0.393158	−	0.919471i	$-0.371383\pi$
$829$	22.6399	0.786316	0.393158	+	0.919471i	$0.371383\pi$
$830$	0	0
$831$	0	0
$832$	1.35026	0.0468119
$833$	29.4109	1.01903
$834$	0	0
$835$	0	0
$836$	−1.61213	−0.0557566
$837$	0	0
$838$	34.3390	1.18622
$839$	9.02776	0.311673	0.155836	−	0.987783i	$-0.450193\pi$
$839$	9.02776	0.311673	0.155836	+	0.987783i	$0.450193\pi$
$840$	0	0
$841$	−15.9525	−0.550088
$842$	22.8627	0.787902
$843$	0	0
$844$	9.92478	0.341625
$845$	0	0
$846$	0	0
$847$	28.1457	0.967098
$848$	11.9248	0.409499
$849$	0	0
$850$	0	0
$851$	15.2243	0.521881
$852$	0	0
$853$	43.1852	1.47863	0.739317	−	0.673358i	$-0.235149\pi$
$853$	43.1852	1.47863	0.739317	+	0.673358i	$0.235149\pi$
$854$	−6.70052	−0.229287
$855$	0	0
$856$	0.775746	0.0265145
$857$	−33.0249	−1.12811	−0.564055	−	0.825737i	$-0.690759\pi$
$857$	−33.0249	−1.12811	−0.564055	+	0.825737i	$0.690759\pi$
$858$	0	0
$859$	15.1754	0.517777	0.258889	−	0.965907i	$-0.416644\pi$
$859$	15.1754	0.517777	0.258889	+	0.965907i	$0.416644\pi$
$860$	0	0
$861$	0	0
$862$	−11.5975	−0.395014
$863$	−39.4763	−1.34379	−0.671894	−	0.740647i	$-0.734519\pi$
$863$	−39.4763	−1.34379	−0.671894	+	0.740647i	$0.734519\pi$
$864$	0	0
$865$	0	0
$866$	27.8350	0.945871
$867$	0	0
$868$	7.74798	0.262984
$869$	23.0738	0.782725
$870$	0	0
$871$	13.4010	0.454077
$872$	−20.1622	−0.682778
$873$	0	0
$874$	1.35026	0.0456733
$875$	0	0
$876$	0	0
$877$	52.5256	1.77366	0.886832	−	0.462091i	$-0.152901\pi$
$877$	52.5256	1.77366	0.886832	+	0.462091i	$0.152901\pi$
$878$	−38.7875	−1.30901
$879$	0	0
$880$	0	0
$881$	41.8007	1.40830	0.704150	−	0.710051i	$-0.251328\pi$
$881$	41.8007	1.40830	0.704150	+	0.710051i	$0.251328\pi$
$882$	0	0
$883$	−36.9643	−1.24395	−0.621974	−	0.783038i	$-0.713669\pi$
$883$	−36.9643	−1.24395	−0.621974	+	0.783038i	$0.713669\pi$
$884$	9.40105	0.316191
$885$	0	0
$886$	29.2144	0.981477
$887$	−36.0724	−1.21119	−0.605596	−	0.795772i	$-0.707065\pi$
$887$	−36.0724	−1.21119	−0.605596	+	0.795772i	$0.707065\pi$
$888$	0	0
$889$	−46.0263	−1.54367
$890$	0	0
$891$	0	0
$892$	7.03761	0.235637
$893$	4.57452	0.153080
$894$	0	0
$895$	0	0
$896$	−3.35026	−0.111924
$897$	0	0
$898$	20.4993	0.684070
$899$	8.35359	0.278608
$900$	0	0
$901$	83.0249	2.76596
$902$	5.40105	0.179835
$903$	0	0
$904$	−11.1490	−0.370811
$905$	0	0
$906$	0	0
$907$	−57.6239	−1.91337	−0.956685	−	0.291126i	$-0.905970\pi$
$907$	−57.6239	−1.91337	−0.956685	+	0.291126i	$0.905970\pi$
$908$	14.5501	0.482861
$909$	0	0
$910$	0	0
$911$	−30.9234	−1.02454	−0.512268	−	0.858825i	$-0.671195\pi$
$911$	−30.9234	−1.02454	−0.512268	+	0.858825i	$0.671195\pi$
$912$	0	0
$913$	17.5515	0.580870
$914$	6.44851	0.213298
$915$	0	0
$916$	11.4010	0.376701
$917$	18.8021	0.620900
$918$	0	0
$919$	33.7743	1.11411	0.557056	−	0.830475i	$-0.311931\pi$
$919$	33.7743	1.11411	0.557056	+	0.830475i	$0.311931\pi$
$920$	0	0
$921$	0	0
$922$	13.5125	0.445009
$923$	1.04746	0.0344776
$924$	0	0
$925$	0	0
$926$	6.20123	0.203785
$927$	0	0
$928$	−3.61213	−0.118574
$929$	−48.6516	−1.59621	−0.798104	−	0.602519i	$-0.794164\pi$
$929$	−48.6516	−1.59621	−0.798104	+	0.602519i	$0.794164\pi$
$930$	0	0
$931$	−4.22425	−0.138444
$932$	−21.9149	−0.717847
$933$	0	0
$934$	16.5599	0.541857
$935$	0	0
$936$	0	0
$937$	−18.7005	−0.610919	−0.305460	−	0.952205i	$-0.598810\pi$
$937$	−18.7005	−0.610919	−0.305460	+	0.952205i	$0.598810\pi$
$938$	−33.2506	−1.08567
$939$	0	0
$940$	0	0
$941$	−38.4142	−1.25227	−0.626134	−	0.779716i	$-0.715364\pi$
$941$	−38.4142	−1.25227	−0.626134	+	0.779716i	$0.715364\pi$
$942$	0	0
$943$	−4.52373	−0.147313
$944$	−1.03761	−0.0337714
$945$	0	0
$946$	16.6253	0.540535
$947$	−44.0362	−1.43098	−0.715492	−	0.698621i	$-0.753797\pi$
$947$	−44.0362	−1.43098	−0.715492	+	0.698621i	$0.753797\pi$
$948$	0	0
$949$	4.35359	0.141323
$950$	0	0
$951$	0	0
$952$	−23.3258	−0.755994
$953$	38.0724	1.23329	0.616643	−	0.787243i	$-0.288492\pi$
$953$	38.0724	1.23329	0.616643	+	0.787243i	$0.288492\pi$
$954$	0	0
$955$	0	0
$956$	13.2750	0.429345
$957$	0	0
$958$	30.5256	0.986239
$959$	−59.1754	−1.91087
$960$	0	0
$961$	−25.6516	−0.827473
$962$	−15.2243	−0.490850
$963$	0	0
$964$	21.3258	0.686859
$965$	0	0
$966$	0	0
$967$	42.5256	1.36753	0.683766	−	0.729701i	$-0.260341\pi$
$967$	42.5256	1.36753	0.683766	+	0.729701i	$0.260341\pi$
$968$	−8.40105	−0.270020
$969$	0	0
$970$	0	0
$971$	22.1378	0.710435	0.355217	−	0.934784i	$-0.384407\pi$
$971$	22.1378	0.710435	0.355217	+	0.934784i	$0.384407\pi$
$972$	0	0
$973$	−35.8496	−1.14928
$974$	−2.51388	−0.0805499
$975$	0	0
$976$	2.00000	0.0640184
$977$	21.7480	0.695780	0.347890	−	0.937535i	$-0.386898\pi$
$977$	21.7480	0.695780	0.347890	+	0.937535i	$0.386898\pi$
$978$	0	0
$979$	4.15045	0.132649
$980$	0	0
$981$	0	0
$982$	−1.46168	−0.0466441
$983$	−9.04746	−0.288569	−0.144285	−	0.989536i	$-0.546088\pi$
$983$	−9.04746	−0.288569	−0.144285	+	0.989536i	$0.546088\pi$
$984$	0	0
$985$	0	0
$986$	−25.1490	−0.800908
$987$	0	0
$988$	−1.35026	−0.0429575
$989$	−13.9248	−0.442782
$990$	0	0
$991$	−26.0606	−0.827843	−0.413922	−	0.910313i	$-0.635841\pi$
$991$	−26.0606	−0.827843	−0.413922	+	0.910313i	$0.635841\pi$
$992$	−2.31265	−0.0734267
$993$	0	0
$994$	−2.59895	−0.0824338
$995$	0	0
$996$	0	0
$997$	28.2130	0.893514	0.446757	−	0.894655i	$-0.352579\pi$
$997$	28.2130	0.893514	0.446757	+	0.894655i	$0.352579\pi$
$998$	−5.55149	−0.175729
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.cr.1.1		3
3.2	odd	2		2850.2.a.bk.1.1		3
5.2	odd	4		1710.2.d.e.1369.6		6
5.3	odd	4		1710.2.d.e.1369.3		6
5.4	even	2		8550.2.a.cf.1.3		3
15.2	even	4		570.2.d.d.229.1	✓	6
15.8	even	4		570.2.d.d.229.4	yes	6
15.14	odd	2		2850.2.a.bn.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
570.2.d.d.229.1	✓	6	15.2	even	4
570.2.d.d.229.4	yes	6	15.8	even	4
1710.2.d.e.1369.3		6	5.3	odd	4
1710.2.d.e.1369.6		6	5.2	odd	4
2850.2.a.bk.1.1		3	3.2	odd	2
2850.2.a.bn.1.3		3	15.14	odd	2
8550.2.a.cf.1.3		3	5.4	even	2
8550.2.a.cr.1.1		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.35026 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.35026 q^{7} +1.00000 q^{8} +1.61213 q^{11} +1.35026 q^{13} -3.35026 q^{14} +1.00000 q^{16} +6.96239 q^{17} -1.00000 q^{19} +1.61213 q^{22} -1.35026 q^{23} +1.35026 q^{26} -3.35026 q^{28} -3.61213 q^{29} -2.31265 q^{31} +1.00000 q^{32} +6.96239 q^{34} -11.2750 q^{37} -1.00000 q^{38} +3.35026 q^{41} +10.3127 q^{43} +1.61213 q^{44} -1.35026 q^{46} -4.57452 q^{47} +4.22425 q^{49} +1.35026 q^{52} +11.9248 q^{53} -3.35026 q^{56} -3.61213 q^{58} -1.03761 q^{59} +2.00000 q^{61} -2.31265 q^{62} +1.00000 q^{64} +9.92478 q^{67} +6.96239 q^{68} +0.775746 q^{71} +3.22425 q^{73} -11.2750 q^{74} -1.00000 q^{76} -5.40105 q^{77} +14.3127 q^{79} +3.35026 q^{82} +10.8872 q^{83} +10.3127 q^{86} +1.61213 q^{88} +2.57452 q^{89} -4.52373 q^{91} -1.35026 q^{92} -4.57452 q^{94} +1.16362 q^{97} +4.22425 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{8}+O(q^{10}) \) 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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