LMFDB - Embedded newform 8550.2.a.cv.1.4

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:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1710)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.93185$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.86370 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +3.86370 q^{7} +1.00000 q^{8} -1.03528 q^{11} -3.86370 q^{13} +3.86370 q^{14} +1.00000 q^{16} -0.535898 q^{17} -1.00000 q^{19} -1.03528 q^{22} -5.46410 q^{23} -3.86370 q^{26} +3.86370 q^{28} -4.62158 q^{29} -10.9282 q^{31} +1.00000 q^{32} -0.535898 q^{34} -1.79315 q^{37} -1.00000 q^{38} -1.79315 q^{41} -6.69213 q^{43} -1.03528 q^{44} -5.46410 q^{46} -2.53590 q^{47} +7.92820 q^{49} -3.86370 q^{52} -6.00000 q^{53} +3.86370 q^{56} -4.62158 q^{58} -6.96953 q^{59} -2.00000 q^{61} -10.9282 q^{62} +1.00000 q^{64} +13.3843 q^{67} -0.535898 q^{68} -4.14110 q^{71} -3.58630 q^{73} -1.79315 q^{74} -1.00000 q^{76} -4.00000 q^{77} -4.00000 q^{79} -1.79315 q^{82} -16.3923 q^{83} -6.69213 q^{86} -1.03528 q^{88} +3.86370 q^{89} -14.9282 q^{91} -5.46410 q^{92} -2.53590 q^{94} +2.55103 q^{97} +7.92820 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} + 4 q^{16} - 16 q^{17} - 4 q^{19} - 8 q^{23} - 16 q^{31} + 4 q^{32} - 16 q^{34} - 4 q^{38} - 8 q^{46} - 24 q^{47} + 4 q^{49} - 24 q^{53} - 8 q^{61} - 16 q^{62} + 4 q^{64} - 16 q^{68} - 4 q^{76} - 16 q^{77} - 16 q^{79} - 24 q^{83} - 32 q^{91} - 8 q^{92} - 24 q^{94} + 4 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 + 4 * q^16 - 16 * q^17 - 4 * q^19 - 8 * q^23 - 16 * q^31 + 4 * q^32 - 16 * q^34 - 4 * q^38 - 8 * q^46 - 24 * q^47 + 4 * q^49 - 24 * q^53 - 8 * q^61 - 16 * q^62 + 4 * q^64 - 16 * q^68 - 4 * q^76 - 16 * q^77 - 16 * q^79 - 24 * q^83 - 32 * q^91 - 8 * q^92 - 24 * q^94 + 4 * 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
$7$	3.86370	1.46034	0.730171	−	0.683264i	$-0.239440\pi$
$7$	3.86370	1.46034	0.730171	+	0.683264i	$0.239440\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	−1.03528	−0.312148	−0.156074	−	0.987745i	$-0.549884\pi$
$11$	−1.03528	−0.312148	−0.156074	+	0.987745i	$0.549884\pi$
$12$	0	0
$13$	−3.86370	−1.07160	−0.535799	−	0.844345i	$-0.679990\pi$
$13$	−3.86370	−1.07160	−0.535799	+	0.844345i	$0.679990\pi$
$14$	3.86370	1.03262
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.535898	−0.129974	−0.0649872	−	0.997886i	$-0.520701\pi$
$17$	−0.535898	−0.129974	−0.0649872	+	0.997886i	$0.520701\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	−1.03528	−0.220722
$23$	−5.46410	−1.13934	−0.569672	−	0.821872i	$-0.692930\pi$
$23$	−5.46410	−1.13934	−0.569672	+	0.821872i	$0.692930\pi$
$24$	0	0
$25$	0	0
$26$	−3.86370	−0.757735
$27$	0	0
$28$	3.86370	0.730171
$29$	−4.62158	−0.858206	−0.429103	−	0.903256i	$-0.641170\pi$
$29$	−4.62158	−0.858206	−0.429103	+	0.903256i	$0.641170\pi$
$30$	0	0
$31$	−10.9282	−1.96276	−0.981382	−	0.192068i	$-0.938481\pi$
$31$	−10.9282	−1.96276	−0.981382	+	0.192068i	$0.938481\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−0.535898	−0.0919058
$35$	0	0
$36$	0	0
$37$	−1.79315	−0.294792	−0.147396	−	0.989078i	$-0.547089\pi$
$37$	−1.79315	−0.294792	−0.147396	+	0.989078i	$0.547089\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	0	0
$41$	−1.79315	−0.280043	−0.140022	−	0.990148i	$-0.544717\pi$
$41$	−1.79315	−0.280043	−0.140022	+	0.990148i	$0.544717\pi$
$42$	0	0
$43$	−6.69213	−1.02054	−0.510270	−	0.860014i	$-0.670455\pi$
$43$	−6.69213	−1.02054	−0.510270	+	0.860014i	$0.670455\pi$
$44$	−1.03528	−0.156074
$45$	0	0
$46$	−5.46410	−0.805638
$47$	−2.53590	−0.369899	−0.184949	−	0.982748i	$-0.559212\pi$
$47$	−2.53590	−0.369899	−0.184949	+	0.982748i	$0.559212\pi$
$48$	0	0
$49$	7.92820	1.13260
$50$	0	0
$51$	0	0
$52$	−3.86370	−0.535799
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0	0
$56$	3.86370	0.516309
$57$	0	0
$58$	−4.62158	−0.606843
$59$	−6.96953	−0.907356	−0.453678	−	0.891166i	$-0.649888\pi$
$59$	−6.96953	−0.907356	−0.453678	+	0.891166i	$0.649888\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−10.9282	−1.38788
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	13.3843	1.63515	0.817574	−	0.575824i	$-0.195319\pi$
$67$	13.3843	1.63515	0.817574	+	0.575824i	$0.195319\pi$
$68$	−0.535898	−0.0649872
$69$	0	0
$70$	0	0
$71$	−4.14110	−0.491459	−0.245729	−	0.969338i	$-0.579027\pi$
$71$	−4.14110	−0.491459	−0.245729	+	0.969338i	$0.579027\pi$
$72$	0	0
$73$	−3.58630	−0.419745	−0.209872	−	0.977729i	$-0.567305\pi$
$73$	−3.58630	−0.419745	−0.209872	+	0.977729i	$0.567305\pi$
$74$	−1.79315	−0.208450
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−4.00000	−0.455842
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	0	0
$82$	−1.79315	−0.198020
$83$	−16.3923	−1.79929	−0.899645	−	0.436623i	$-0.856174\pi$
$83$	−16.3923	−1.79929	−0.899645	+	0.436623i	$0.856174\pi$
$84$	0	0
$85$	0	0
$86$	−6.69213	−0.721631
$87$	0	0
$88$	−1.03528	−0.110361
$89$	3.86370	0.409552	0.204776	−	0.978809i	$-0.434353\pi$
$89$	3.86370	0.409552	0.204776	+	0.978809i	$0.434353\pi$
$90$	0	0
$91$	−14.9282	−1.56490
$92$	−5.46410	−0.569672
$93$	0	0
$94$	−2.53590	−0.261558
$95$	0	0
$96$	0	0
$97$	2.55103	0.259017	0.129509	−	0.991578i	$-0.458660\pi$
$97$	2.55103	0.259017	0.129509	+	0.991578i	$0.458660\pi$
$98$	7.92820	0.800869
$99$	0	0
$100$	0	0
$101$	10.5558	1.05034	0.525172	−	0.850996i	$-0.324001\pi$
$101$	10.5558	1.05034	0.525172	+	0.850996i	$0.324001\pi$
$102$	0	0
$103$	4.89898	0.482711	0.241355	−	0.970437i	$-0.422408\pi$
$103$	4.89898	0.482711	0.241355	+	0.970437i	$0.422408\pi$
$104$	−3.86370	−0.378867
$105$	0	0
$106$	−6.00000	−0.582772
$107$	2.92820	0.283080	0.141540	−	0.989933i	$-0.454795\pi$
$107$	2.92820	0.283080	0.141540	+	0.989933i	$0.454795\pi$
$108$	0	0
$109$	8.92820	0.855167	0.427583	−	0.903976i	$-0.359365\pi$
$109$	8.92820	0.855167	0.427583	+	0.903976i	$0.359365\pi$
$110$	0	0
$111$	0	0
$112$	3.86370	0.365086
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	−4.62158	−0.429103
$117$	0	0
$118$	−6.96953	−0.641597
$119$	−2.07055	−0.189807
$120$	0	0
$121$	−9.92820	−0.902564
$122$	−2.00000	−0.181071
$123$	0	0
$124$	−10.9282	−0.981382
$125$	0	0
$126$	0	0
$127$	8.48528	0.752947	0.376473	−	0.926427i	$-0.377137\pi$
$127$	8.48528	0.752947	0.376473	+	0.926427i	$0.377137\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	−3.10583	−0.271357	−0.135679	−	0.990753i	$-0.543322\pi$
$131$	−3.10583	−0.271357	−0.135679	+	0.990753i	$0.543322\pi$
$132$	0	0
$133$	−3.86370	−0.335026
$134$	13.3843	1.15622
$135$	0	0
$136$	−0.535898	−0.0459529
$137$	−0.535898	−0.0457849	−0.0228924	−	0.999738i	$-0.507288\pi$
$137$	−0.535898	−0.0457849	−0.0228924	+	0.999738i	$0.507288\pi$
$138$	0	0
$139$	6.92820	0.587643	0.293821	−	0.955860i	$-0.405073\pi$
$139$	6.92820	0.587643	0.293821	+	0.955860i	$0.405073\pi$
$140$	0	0
$141$	0	0
$142$	−4.14110	−0.347514
$143$	4.00000	0.334497
$144$	0	0
$145$	0	0
$146$	−3.58630	−0.296804
$147$	0	0
$148$	−1.79315	−0.147396
$149$	−20.3538	−1.66745	−0.833724	−	0.552182i	$-0.813795\pi$
$149$	−20.3538	−1.66745	−0.833724	+	0.552182i	$0.813795\pi$
$150$	0	0
$151$	5.85641	0.476588	0.238294	−	0.971193i	$-0.423412\pi$
$151$	5.85641	0.476588	0.238294	+	0.971193i	$0.423412\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	−4.00000	−0.322329
$155$	0	0
$156$	0	0
$157$	14.1421	1.12867	0.564333	−	0.825547i	$-0.309134\pi$
$157$	14.1421	1.12867	0.564333	+	0.825547i	$0.309134\pi$
$158$	−4.00000	−0.318223
$159$	0	0
$160$	0	0
$161$	−21.1117	−1.66383
$162$	0	0
$163$	23.6627	1.85341	0.926703	−	0.375796i	$-0.122631\pi$
$163$	23.6627	1.85341	0.926703	+	0.375796i	$0.122631\pi$
$164$	−1.79315	−0.140022
$165$	0	0
$166$	−16.3923	−1.27229
$167$	−10.9282	−0.845650	−0.422825	−	0.906211i	$-0.638961\pi$
$167$	−10.9282	−0.845650	−0.422825	+	0.906211i	$0.638961\pi$
$168$	0	0
$169$	1.92820	0.148323
$170$	0	0
$171$	0	0
$172$	−6.69213	−0.510270
$173$	22.7846	1.73228	0.866141	−	0.499800i	$-0.166593\pi$
$173$	22.7846	1.73228	0.866141	+	0.499800i	$0.166593\pi$
$174$	0	0
$175$	0	0
$176$	−1.03528	−0.0780369
$177$	0	0
$178$	3.86370	0.289597
$179$	16.7675	1.25326	0.626631	−	0.779316i	$-0.284434\pi$
$179$	16.7675	1.25326	0.626631	+	0.779316i	$0.284434\pi$
$180$	0	0
$181$	19.8564	1.47592	0.737958	−	0.674847i	$-0.235790\pi$
$181$	19.8564	1.47592	0.737958	+	0.674847i	$0.235790\pi$
$182$	−14.9282	−1.10655
$183$	0	0
$184$	−5.46410	−0.402819
$185$	0	0
$186$	0	0
$187$	0.554803	0.0405712
$188$	−2.53590	−0.184949
$189$	0	0
$190$	0	0
$191$	17.2480	1.24802	0.624009	−	0.781417i	$-0.285503\pi$
$191$	17.2480	1.24802	0.624009	+	0.781417i	$0.285503\pi$
$192$	0	0
$193$	8.76268	0.630752	0.315376	−	0.948967i	$-0.397869\pi$
$193$	8.76268	0.630752	0.315376	+	0.948967i	$0.397869\pi$
$194$	2.55103	0.183153
$195$	0	0
$196$	7.92820	0.566300
$197$	−23.4641	−1.67175	−0.835874	−	0.548921i	$-0.815039\pi$
$197$	−23.4641	−1.67175	−0.835874	+	0.548921i	$0.815039\pi$
$198$	0	0
$199$	13.0718	0.926635	0.463318	−	0.886192i	$-0.346659\pi$
$199$	13.0718	0.926635	0.463318	+	0.886192i	$0.346659\pi$
$200$	0	0
$201$	0	0
$202$	10.5558	0.742706
$203$	−17.8564	−1.25327
$204$	0	0
$205$	0	0
$206$	4.89898	0.341328
$207$	0	0
$208$	−3.86370	−0.267900
$209$	1.03528	0.0716116
$210$	0	0
$211$	−17.8564	−1.22929	−0.614643	−	0.788806i	$-0.710700\pi$
$211$	−17.8564	−1.22929	−0.614643	+	0.788806i	$0.710700\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	2.92820	0.200168
$215$	0	0
$216$	0	0
$217$	−42.2233	−2.86631
$218$	8.92820	0.604694
$219$	0	0
$220$	0	0
$221$	2.07055	0.139280
$222$	0	0
$223$	−10.5558	−0.706871	−0.353435	−	0.935459i	$-0.614987\pi$
$223$	−10.5558	−0.706871	−0.353435	+	0.935459i	$0.614987\pi$
$224$	3.86370	0.258155
$225$	0	0
$226$	−6.00000	−0.399114
$227$	1.85641	0.123214	0.0616070	−	0.998100i	$-0.480377\pi$
$227$	1.85641	0.123214	0.0616070	+	0.998100i	$0.480377\pi$
$228$	0	0
$229$	−3.85641	−0.254839	−0.127419	−	0.991849i	$-0.540669\pi$
$229$	−3.85641	−0.254839	−0.127419	+	0.991849i	$0.540669\pi$
$230$	0	0
$231$	0	0
$232$	−4.62158	−0.303421
$233$	−12.5359	−0.821254	−0.410627	−	0.911803i	$-0.634690\pi$
$233$	−12.5359	−0.821254	−0.410627	+	0.911803i	$0.634690\pi$
$234$	0	0
$235$	0	0
$236$	−6.96953	−0.453678
$237$	0	0
$238$	−2.07055	−0.134214
$239$	−13.1069	−0.847812	−0.423906	−	0.905706i	$-0.639341\pi$
$239$	−13.1069	−0.847812	−0.423906	+	0.905706i	$0.639341\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	−9.92820	−0.638209
$243$	0	0
$244$	−2.00000	−0.128037
$245$	0	0
$246$	0	0
$247$	3.86370	0.245842
$248$	−10.9282	−0.693942
$249$	0	0
$250$	0	0
$251$	28.3586	1.78998	0.894990	−	0.446087i	$-0.147183\pi$
$251$	28.3586	1.78998	0.894990	+	0.446087i	$0.147183\pi$
$252$	0	0
$253$	5.65685	0.355643
$254$	8.48528	0.532414
$255$	0	0
$256$	1.00000	0.0625000
$257$	−22.7846	−1.42126	−0.710632	−	0.703563i	$-0.751591\pi$
$257$	−22.7846	−1.42126	−0.710632	+	0.703563i	$0.751591\pi$
$258$	0	0
$259$	−6.92820	−0.430498
$260$	0	0
$261$	0	0
$262$	−3.10583	−0.191879
$263$	−19.3205	−1.19135	−0.595677	−	0.803224i	$-0.703116\pi$
$263$	−19.3205	−1.19135	−0.595677	+	0.803224i	$0.703116\pi$
$264$	0	0
$265$	0	0
$266$	−3.86370	−0.236899
$267$	0	0
$268$	13.3843	0.817574
$269$	31.9449	1.94772	0.973858	−	0.227160i	$-0.0729439\pi$
$269$	31.9449	1.94772	0.973858	+	0.227160i	$0.0729439\pi$
$270$	0	0
$271$	10.9282	0.663841	0.331921	−	0.943307i	$-0.392303\pi$
$271$	10.9282	0.663841	0.331921	+	0.943307i	$0.392303\pi$
$272$	−0.535898	−0.0324936
$273$	0	0
$274$	−0.535898	−0.0323748
$275$	0	0
$276$	0	0
$277$	−18.2832	−1.09853	−0.549267	−	0.835647i	$-0.685093\pi$
$277$	−18.2832	−1.09853	−0.549267	+	0.835647i	$0.685093\pi$
$278$	6.92820	0.415526
$279$	0	0
$280$	0	0
$281$	31.1870	1.86046	0.930231	−	0.366974i	$-0.119606\pi$
$281$	31.1870	1.86046	0.930231	+	0.366974i	$0.119606\pi$
$282$	0	0
$283$	5.17638	0.307704	0.153852	−	0.988094i	$-0.450832\pi$
$283$	5.17638	0.307704	0.153852	+	0.988094i	$0.450832\pi$
$284$	−4.14110	−0.245729
$285$	0	0
$286$	4.00000	0.236525
$287$	−6.92820	−0.408959
$288$	0	0
$289$	−16.7128	−0.983107
$290$	0	0
$291$	0	0
$292$	−3.58630	−0.209872
$293$	19.8564	1.16002	0.580012	−	0.814608i	$-0.303048\pi$
$293$	19.8564	1.16002	0.580012	+	0.814608i	$0.303048\pi$
$294$	0	0
$295$	0	0
$296$	−1.79315	−0.104225
$297$	0	0
$298$	−20.3538	−1.17906
$299$	21.1117	1.22092
$300$	0	0
$301$	−25.8564	−1.49034
$302$	5.85641	0.336998
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	0	0
$307$	17.5254	1.00023	0.500113	−	0.865960i	$-0.333292\pi$
$307$	17.5254	1.00023	0.500113	+	0.865960i	$0.333292\pi$
$308$	−4.00000	−0.227921
$309$	0	0
$310$	0	0
$311$	1.79315	0.101680	0.0508401	−	0.998707i	$-0.483810\pi$
$311$	1.79315	0.101680	0.0508401	+	0.998707i	$0.483810\pi$
$312$	0	0
$313$	−19.5959	−1.10763	−0.553813	−	0.832641i	$-0.686828\pi$
$313$	−19.5959	−1.10763	−0.553813	+	0.832641i	$0.686828\pi$
$314$	14.1421	0.798087
$315$	0	0
$316$	−4.00000	−0.225018
$317$	−14.0000	−0.786318	−0.393159	−	0.919470i	$-0.628618\pi$
$317$	−14.0000	−0.786318	−0.393159	+	0.919470i	$0.628618\pi$
$318$	0	0
$319$	4.78461	0.267887
$320$	0	0
$321$	0	0
$322$	−21.1117	−1.17651
$323$	0.535898	0.0298182
$324$	0	0
$325$	0	0
$326$	23.6627	1.31056
$327$	0	0
$328$	−1.79315	−0.0990102
$329$	−9.79796	−0.540179
$330$	0	0
$331$	13.8564	0.761617	0.380808	−	0.924654i	$-0.375646\pi$
$331$	13.8564	0.761617	0.380808	+	0.924654i	$0.375646\pi$
$332$	−16.3923	−0.899645
$333$	0	0
$334$	−10.9282	−0.597965
$335$	0	0
$336$	0	0
$337$	1.03528	0.0563951	0.0281975	−	0.999602i	$-0.491023\pi$
$337$	1.03528	0.0563951	0.0281975	+	0.999602i	$0.491023\pi$
$338$	1.92820	0.104880
$339$	0	0
$340$	0	0
$341$	11.3137	0.612672
$342$	0	0
$343$	3.58630	0.193642
$344$	−6.69213	−0.360815
$345$	0	0
$346$	22.7846	1.22491
$347$	−18.5359	−0.995059	−0.497530	−	0.867447i	$-0.665759\pi$
$347$	−18.5359	−0.995059	−0.497530	+	0.867447i	$0.665759\pi$
$348$	0	0
$349$	4.14359	0.221801	0.110901	−	0.993831i	$-0.464626\pi$
$349$	4.14359	0.221801	0.110901	+	0.993831i	$0.464626\pi$
$350$	0	0
$351$	0	0
$352$	−1.03528	−0.0551804
$353$	−21.3205	−1.13478	−0.567388	−	0.823451i	$-0.692046\pi$
$353$	−21.3205	−1.13478	−0.567388	+	0.823451i	$0.692046\pi$
$354$	0	0
$355$	0	0
$356$	3.86370	0.204776
$357$	0	0
$358$	16.7675	0.886189
$359$	−34.7733	−1.83527	−0.917633	−	0.397429i	$-0.869903\pi$
$359$	−34.7733	−1.83527	−0.917633	+	0.397429i	$0.869903\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	19.8564	1.04363
$363$	0	0
$364$	−14.9282	−0.782450
$365$	0	0
$366$	0	0
$367$	−5.37945	−0.280805	−0.140403	−	0.990094i	$-0.544840\pi$
$367$	−5.37945	−0.280805	−0.140403	+	0.990094i	$0.544840\pi$
$368$	−5.46410	−0.284836
$369$	0	0
$370$	0	0
$371$	−23.1822	−1.20356
$372$	0	0
$373$	−24.9754	−1.29318	−0.646588	−	0.762840i	$-0.723805\pi$
$373$	−24.9754	−1.29318	−0.646588	+	0.762840i	$0.723805\pi$
$374$	0.554803	0.0286882
$375$	0	0
$376$	−2.53590	−0.130779
$377$	17.8564	0.919652
$378$	0	0
$379$	−35.7128	−1.83444	−0.917222	−	0.398376i	$-0.869574\pi$
$379$	−35.7128	−1.83444	−0.917222	+	0.398376i	$0.869574\pi$
$380$	0	0
$381$	0	0
$382$	17.2480	0.882483
$383$	22.9282	1.17158	0.585788	−	0.810464i	$-0.300785\pi$
$383$	22.9282	1.17158	0.585788	+	0.810464i	$0.300785\pi$
$384$	0	0
$385$	0	0
$386$	8.76268	0.446009
$387$	0	0
$388$	2.55103	0.129509
$389$	19.7990	1.00385	0.501924	−	0.864912i	$-0.332626\pi$
$389$	19.7990	1.00385	0.501924	+	0.864912i	$0.332626\pi$
$390$	0	0
$391$	2.92820	0.148086
$392$	7.92820	0.400435
$393$	0	0
$394$	−23.4641	−1.18210
$395$	0	0
$396$	0	0
$397$	18.2832	0.917610	0.458805	−	0.888537i	$-0.348278\pi$
$397$	18.2832	0.917610	0.458805	+	0.888537i	$0.348278\pi$
$398$	13.0718	0.655230
$399$	0	0
$400$	0	0
$401$	−9.52056	−0.475434	−0.237717	−	0.971334i	$-0.576399\pi$
$401$	−9.52056	−0.475434	−0.237717	+	0.971334i	$0.576399\pi$
$402$	0	0
$403$	42.2233	2.10329
$404$	10.5558	0.525172
$405$	0	0
$406$	−17.8564	−0.886199
$407$	1.85641	0.0920187
$408$	0	0
$409$	−20.9282	−1.03483	−0.517417	−	0.855734i	$-0.673106\pi$
$409$	−20.9282	−1.03483	−0.517417	+	0.855734i	$0.673106\pi$
$410$	0	0
$411$	0	0
$412$	4.89898	0.241355
$413$	−26.9282	−1.32505
$414$	0	0
$415$	0	0
$416$	−3.86370	−0.189434
$417$	0	0
$418$	1.03528	0.0506370
$419$	−29.3195	−1.43235	−0.716177	−	0.697919i	$-0.754110\pi$
$419$	−29.3195	−1.43235	−0.716177	+	0.697919i	$0.754110\pi$
$420$	0	0
$421$	29.7128	1.44811	0.724057	−	0.689740i	$-0.242275\pi$
$421$	29.7128	1.44811	0.724057	+	0.689740i	$0.242275\pi$
$422$	−17.8564	−0.869236
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	−7.72741	−0.373955
$428$	2.92820	0.141540
$429$	0	0
$430$	0	0
$431$	26.2137	1.26267	0.631335	−	0.775510i	$-0.282507\pi$
$431$	26.2137	1.26267	0.631335	+	0.775510i	$0.282507\pi$
$432$	0	0
$433$	−10.8332	−0.520612	−0.260306	−	0.965526i	$-0.583823\pi$
$433$	−10.8332	−0.520612	−0.260306	+	0.965526i	$0.583823\pi$
$434$	−42.2233	−2.02678
$435$	0	0
$436$	8.92820	0.427583
$437$	5.46410	0.261383
$438$	0	0
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	0	0
$441$	0	0
$442$	2.07055	0.0984861
$443$	−16.3923	−0.778822	−0.389411	−	0.921064i	$-0.627321\pi$
$443$	−16.3923	−0.778822	−0.389411	+	0.921064i	$0.627321\pi$
$444$	0	0
$445$	0	0
$446$	−10.5558	−0.499833
$447$	0	0
$448$	3.86370	0.182543
$449$	40.4302	1.90802	0.954009	−	0.299777i	$-0.0969124\pi$
$449$	40.4302	1.90802	0.954009	+	0.299777i	$0.0969124\pi$
$450$	0	0
$451$	1.85641	0.0874148
$452$	−6.00000	−0.282216
$453$	0	0
$454$	1.85641	0.0871255
$455$	0	0
$456$	0	0
$457$	−33.9411	−1.58770	−0.793849	−	0.608114i	$-0.791926\pi$
$457$	−33.9411	−1.58770	−0.793849	+	0.608114i	$0.791926\pi$
$458$	−3.85641	−0.180198
$459$	0	0
$460$	0	0
$461$	−27.5264	−1.28203	−0.641016	−	0.767527i	$-0.721487\pi$
$461$	−27.5264	−1.28203	−0.641016	+	0.767527i	$0.721487\pi$
$462$	0	0
$463$	19.8733	0.923591	0.461796	−	0.886986i	$-0.347205\pi$
$463$	19.8733	0.923591	0.461796	+	0.886986i	$0.347205\pi$
$464$	−4.62158	−0.214551
$465$	0	0
$466$	−12.5359	−0.580714
$467$	−2.53590	−0.117347	−0.0586737	−	0.998277i	$-0.518687\pi$
$467$	−2.53590	−0.117347	−0.0586737	+	0.998277i	$0.518687\pi$
$468$	0	0
$469$	51.7128	2.38788
$470$	0	0
$471$	0	0
$472$	−6.96953	−0.320799
$473$	6.92820	0.318559
$474$	0	0
$475$	0	0
$476$	−2.07055	−0.0949036
$477$	0	0
$478$	−13.1069	−0.599494
$479$	−16.6932	−0.762730	−0.381365	−	0.924425i	$-0.624546\pi$
$479$	−16.6932	−0.762730	−0.381365	+	0.924425i	$0.624546\pi$
$480$	0	0
$481$	6.92820	0.315899
$482$	10.0000	0.455488
$483$	0	0
$484$	−9.92820	−0.451282
$485$	0	0
$486$	0	0
$487$	−3.38323	−0.153309	−0.0766544	−	0.997058i	$-0.524424\pi$
$487$	−3.38323	−0.153309	−0.0766544	+	0.997058i	$0.524424\pi$
$488$	−2.00000	−0.0905357
$489$	0	0
$490$	0	0
$491$	−18.5606	−0.837630	−0.418815	−	0.908072i	$-0.637554\pi$
$491$	−18.5606	−0.837630	−0.418815	+	0.908072i	$0.637554\pi$
$492$	0	0
$493$	2.47670	0.111545
$494$	3.86370	0.173836
$495$	0	0
$496$	−10.9282	−0.490691
$497$	−16.0000	−0.717698
$498$	0	0
$499$	25.8564	1.15749	0.578746	−	0.815508i	$-0.303542\pi$
$499$	25.8564	1.15749	0.578746	+	0.815508i	$0.303542\pi$
$500$	0	0
$501$	0	0
$502$	28.3586	1.26571
$503$	11.3205	0.504757	0.252378	−	0.967629i	$-0.418787\pi$
$503$	11.3205	0.504757	0.252378	+	0.967629i	$0.418787\pi$
$504$	0	0
$505$	0	0
$506$	5.65685	0.251478
$507$	0	0
$508$	8.48528	0.376473
$509$	37.0470	1.64208	0.821039	−	0.570873i	$-0.193395\pi$
$509$	37.0470	1.64208	0.821039	+	0.570873i	$0.193395\pi$
$510$	0	0
$511$	−13.8564	−0.612971
$512$	1.00000	0.0441942
$513$	0	0
$514$	−22.7846	−1.00499
$515$	0	0
$516$	0	0
$517$	2.62536	0.115463
$518$	−6.92820	−0.304408
$519$	0	0
$520$	0	0
$521$	−9.52056	−0.417103	−0.208552	−	0.978011i	$-0.566875\pi$
$521$	−9.52056	−0.417103	−0.208552	+	0.978011i	$0.566875\pi$
$522$	0	0
$523$	26.7685	1.17051	0.585253	−	0.810851i	$-0.300995\pi$
$523$	26.7685	1.17051	0.585253	+	0.810851i	$0.300995\pi$
$524$	−3.10583	−0.135679
$525$	0	0
$526$	−19.3205	−0.842414
$527$	5.85641	0.255109
$528$	0	0
$529$	6.85641	0.298105
$530$	0	0
$531$	0	0
$532$	−3.86370	−0.167513
$533$	6.92820	0.300094
$534$	0	0
$535$	0	0
$536$	13.3843	0.578112
$537$	0	0
$538$	31.9449	1.37724
$539$	−8.20788	−0.353538
$540$	0	0
$541$	26.0000	1.11783	0.558914	−	0.829226i	$-0.311218\pi$
$541$	26.0000	1.11783	0.558914	+	0.829226i	$0.311218\pi$
$542$	10.9282	0.469407
$543$	0	0
$544$	−0.535898	−0.0229765
$545$	0	0
$546$	0	0
$547$	−10.3528	−0.442652	−0.221326	−	0.975200i	$-0.571039\pi$
$547$	−10.3528	−0.442652	−0.221326	+	0.975200i	$0.571039\pi$
$548$	−0.535898	−0.0228924
$549$	0	0
$550$	0	0
$551$	4.62158	0.196886
$552$	0	0
$553$	−15.4548	−0.657206
$554$	−18.2832	−0.776780
$555$	0	0
$556$	6.92820	0.293821
$557$	−31.4641	−1.33318	−0.666588	−	0.745426i	$-0.732246\pi$
$557$	−31.4641	−1.33318	−0.666588	+	0.745426i	$0.732246\pi$
$558$	0	0
$559$	25.8564	1.09361
$560$	0	0
$561$	0	0
$562$	31.1870	1.31555
$563$	−25.8564	−1.08972	−0.544859	−	0.838528i	$-0.683417\pi$
$563$	−25.8564	−1.08972	−0.544859	+	0.838528i	$0.683417\pi$
$564$	0	0
$565$	0	0
$566$	5.17638	0.217580
$567$	0	0
$568$	−4.14110	−0.173757
$569$	−44.0165	−1.84527	−0.922634	−	0.385678i	$-0.873968\pi$
$569$	−44.0165	−1.84527	−0.922634	+	0.385678i	$0.873968\pi$
$570$	0	0
$571$	14.9282	0.624726	0.312363	−	0.949963i	$-0.398880\pi$
$571$	14.9282	0.624726	0.312363	+	0.949963i	$0.398880\pi$
$572$	4.00000	0.167248
$573$	0	0
$574$	−6.92820	−0.289178
$575$	0	0
$576$	0	0
$577$	5.65685	0.235498	0.117749	−	0.993043i	$-0.462432\pi$
$577$	5.65685	0.235498	0.117749	+	0.993043i	$0.462432\pi$
$578$	−16.7128	−0.695161
$579$	0	0
$580$	0	0
$581$	−63.3350	−2.62758
$582$	0	0
$583$	6.21166	0.257261
$584$	−3.58630	−0.148402
$585$	0	0
$586$	19.8564	0.820261
$587$	−24.3923	−1.00678	−0.503389	−	0.864060i	$-0.667914\pi$
$587$	−24.3923	−1.00678	−0.503389	+	0.864060i	$0.667914\pi$
$588$	0	0
$589$	10.9282	0.450289
$590$	0	0
$591$	0	0
$592$	−1.79315	−0.0736980
$593$	4.53590	0.186267	0.0931335	−	0.995654i	$-0.470312\pi$
$593$	4.53590	0.186267	0.0931335	+	0.995654i	$0.470312\pi$
$594$	0	0
$595$	0	0
$596$	−20.3538	−0.833724
$597$	0	0
$598$	21.1117	0.863320
$599$	20.5569	0.839931	0.419965	−	0.907540i	$-0.362042\pi$
$599$	20.5569	0.839931	0.419965	+	0.907540i	$0.362042\pi$
$600$	0	0
$601$	−18.7846	−0.766240	−0.383120	−	0.923699i	$-0.625150\pi$
$601$	−18.7846	−0.766240	−0.383120	+	0.923699i	$0.625150\pi$
$602$	−25.8564	−1.05383
$603$	0	0
$604$	5.85641	0.238294
$605$	0	0
$606$	0	0
$607$	−45.6066	−1.85111	−0.925557	−	0.378609i	$-0.876402\pi$
$607$	−45.6066	−1.85111	−0.925557	+	0.378609i	$0.876402\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	0	0
$611$	9.79796	0.396383
$612$	0	0
$613$	−5.85993	−0.236680	−0.118340	−	0.992973i	$-0.537757\pi$
$613$	−5.85993	−0.236680	−0.118340	+	0.992973i	$0.537757\pi$
$614$	17.5254	0.707266
$615$	0	0
$616$	−4.00000	−0.161165
$617$	−28.2487	−1.13725	−0.568625	−	0.822597i	$-0.692524\pi$
$617$	−28.2487	−1.13725	−0.568625	+	0.822597i	$0.692524\pi$
$618$	0	0
$619$	−34.6410	−1.39234	−0.696170	−	0.717877i	$-0.745114\pi$
$619$	−34.6410	−1.39234	−0.696170	+	0.717877i	$0.745114\pi$
$620$	0	0
$621$	0	0
$622$	1.79315	0.0718988
$623$	14.9282	0.598086
$624$	0	0
$625$	0	0
$626$	−19.5959	−0.783210
$627$	0	0
$628$	14.1421	0.564333
$629$	0.960947	0.0383155
$630$	0	0
$631$	21.8564	0.870090	0.435045	−	0.900409i	$-0.356732\pi$
$631$	21.8564	0.870090	0.435045	+	0.900409i	$0.356732\pi$
$632$	−4.00000	−0.159111
$633$	0	0
$634$	−14.0000	−0.556011
$635$	0	0
$636$	0	0
$637$	−30.6322	−1.21369
$638$	4.78461	0.189425
$639$	0	0
$640$	0	0
$641$	−17.2480	−0.681254	−0.340627	−	0.940199i	$-0.610639\pi$
$641$	−17.2480	−0.681254	−0.340627	+	0.940199i	$0.610639\pi$
$642$	0	0
$643$	−37.0470	−1.46099	−0.730495	−	0.682918i	$-0.760710\pi$
$643$	−37.0470	−1.46099	−0.730495	+	0.682918i	$0.760710\pi$
$644$	−21.1117	−0.831916
$645$	0	0
$646$	0.535898	0.0210846
$647$	−13.4641	−0.529328	−0.264664	−	0.964341i	$-0.585261\pi$
$647$	−13.4641	−0.529328	−0.264664	+	0.964341i	$0.585261\pi$
$648$	0	0
$649$	7.21539	0.283229
$650$	0	0
$651$	0	0
$652$	23.6627	0.926703
$653$	24.2487	0.948925	0.474463	−	0.880276i	$-0.342642\pi$
$653$	24.2487	0.948925	0.474463	+	0.880276i	$0.342642\pi$
$654$	0	0
$655$	0	0
$656$	−1.79315	−0.0700108
$657$	0	0
$658$	−9.79796	−0.381964
$659$	−20.3538	−0.792871	−0.396436	−	0.918063i	$-0.629753\pi$
$659$	−20.3538	−0.792871	−0.396436	+	0.918063i	$0.629753\pi$
$660$	0	0
$661$	−10.7846	−0.419473	−0.209736	−	0.977758i	$-0.567261\pi$
$661$	−10.7846	−0.419473	−0.209736	+	0.977758i	$0.567261\pi$
$662$	13.8564	0.538545
$663$	0	0
$664$	−16.3923	−0.636145
$665$	0	0
$666$	0	0
$667$	25.2528	0.977791
$668$	−10.9282	−0.422825
$669$	0	0
$670$	0	0
$671$	2.07055	0.0799328
$672$	0	0
$673$	−27.2490	−1.05037	−0.525186	−	0.850988i	$-0.676004\pi$
$673$	−27.2490	−1.05037	−0.525186	+	0.850988i	$0.676004\pi$
$674$	1.03528	0.0398773
$675$	0	0
$676$	1.92820	0.0741617
$677$	−36.6410	−1.40823	−0.704114	−	0.710087i	$-0.748656\pi$
$677$	−36.6410	−1.40823	−0.704114	+	0.710087i	$0.748656\pi$
$678$	0	0
$679$	9.85641	0.378254
$680$	0	0
$681$	0	0
$682$	11.3137	0.433224
$683$	−18.9282	−0.724268	−0.362134	−	0.932126i	$-0.617952\pi$
$683$	−18.9282	−0.724268	−0.362134	+	0.932126i	$0.617952\pi$
$684$	0	0
$685$	0	0
$686$	3.58630	0.136926
$687$	0	0
$688$	−6.69213	−0.255135
$689$	23.1822	0.883172
$690$	0	0
$691$	38.9282	1.48090	0.740449	−	0.672112i	$-0.234613\pi$
$691$	38.9282	1.48090	0.740449	+	0.672112i	$0.234613\pi$
$692$	22.7846	0.866141
$693$	0	0
$694$	−18.5359	−0.703613
$695$	0	0
$696$	0	0
$697$	0.960947	0.0363985
$698$	4.14359	0.156837
$699$	0	0
$700$	0	0
$701$	7.52433	0.284190	0.142095	−	0.989853i	$-0.454616\pi$
$701$	7.52433	0.284190	0.142095	+	0.989853i	$0.454616\pi$
$702$	0	0
$703$	1.79315	0.0676300
$704$	−1.03528	−0.0390184
$705$	0	0
$706$	−21.3205	−0.802408
$707$	40.7846	1.53386
$708$	0	0
$709$	−11.8564	−0.445277	−0.222638	−	0.974901i	$-0.571467\pi$
$709$	−11.8564	−0.445277	−0.222638	+	0.974901i	$0.571467\pi$
$710$	0	0
$711$	0	0
$712$	3.86370	0.144798
$713$	59.7128	2.23626
$714$	0	0
$715$	0	0
$716$	16.7675	0.626631
$717$	0	0
$718$	−34.7733	−1.29773
$719$	−46.6418	−1.73945	−0.869724	−	0.493539i	$-0.835703\pi$
$719$	−46.6418	−1.73945	−0.869724	+	0.493539i	$0.835703\pi$
$720$	0	0
$721$	18.9282	0.704923
$722$	1.00000	0.0372161
$723$	0	0
$724$	19.8564	0.737958
$725$	0	0
$726$	0	0
$727$	−15.1774	−0.562899	−0.281450	−	0.959576i	$-0.590815\pi$
$727$	−15.1774	−0.562899	−0.281450	+	0.959576i	$0.590815\pi$
$728$	−14.9282	−0.553276
$729$	0	0
$730$	0	0
$731$	3.58630	0.132644
$732$	0	0
$733$	−21.8695	−0.807770	−0.403885	−	0.914810i	$-0.632340\pi$
$733$	−21.8695	−0.807770	−0.403885	+	0.914810i	$0.632340\pi$
$734$	−5.37945	−0.198559
$735$	0	0
$736$	−5.46410	−0.201409
$737$	−13.8564	−0.510407
$738$	0	0
$739$	23.7128	0.872290	0.436145	−	0.899876i	$-0.356343\pi$
$739$	23.7128	0.872290	0.436145	+	0.899876i	$0.356343\pi$
$740$	0	0
$741$	0	0
$742$	−23.1822	−0.851046
$743$	34.6410	1.27086	0.635428	−	0.772160i	$-0.280824\pi$
$743$	34.6410	1.27086	0.635428	+	0.772160i	$0.280824\pi$
$744$	0	0
$745$	0	0
$746$	−24.9754	−0.914413
$747$	0	0
$748$	0.554803	0.0202856
$749$	11.3137	0.413394
$750$	0	0
$751$	5.85641	0.213703	0.106852	−	0.994275i	$-0.465923\pi$
$751$	5.85641	0.213703	0.106852	+	0.994275i	$0.465923\pi$
$752$	−2.53590	−0.0924747
$753$	0	0
$754$	17.8564	0.650292
$755$	0	0
$756$	0	0
$757$	40.3559	1.46676	0.733379	−	0.679820i	$-0.237942\pi$
$757$	40.3559	1.46676	0.733379	+	0.679820i	$0.237942\pi$
$758$	−35.7128	−1.29715
$759$	0	0
$760$	0	0
$761$	−23.1822	−0.840355	−0.420177	−	0.907442i	$-0.638032\pi$
$761$	−23.1822	−0.840355	−0.420177	+	0.907442i	$0.638032\pi$
$762$	0	0
$763$	34.4959	1.24884
$764$	17.2480	0.624009
$765$	0	0
$766$	22.9282	0.828430
$767$	26.9282	0.972321
$768$	0	0
$769$	35.8564	1.29302	0.646508	−	0.762908i	$-0.276229\pi$
$769$	35.8564	1.29302	0.646508	+	0.762908i	$0.276229\pi$
$770$	0	0
$771$	0	0
$772$	8.76268	0.315376
$773$	7.85641	0.282575	0.141288	−	0.989969i	$-0.454876\pi$
$773$	7.85641	0.282575	0.141288	+	0.989969i	$0.454876\pi$
$774$	0	0
$775$	0	0
$776$	2.55103	0.0915765
$777$	0	0
$778$	19.7990	0.709828
$779$	1.79315	0.0642463
$780$	0	0
$781$	4.28719	0.153408
$782$	2.92820	0.104712
$783$	0	0
$784$	7.92820	0.283150
$785$	0	0
$786$	0	0
$787$	7.17260	0.255676	0.127838	−	0.991795i	$-0.459196\pi$
$787$	7.17260	0.255676	0.127838	+	0.991795i	$0.459196\pi$
$788$	−23.4641	−0.835874
$789$	0	0
$790$	0	0
$791$	−23.1822	−0.824265
$792$	0	0
$793$	7.72741	0.274408
$794$	18.2832	0.648848
$795$	0	0
$796$	13.0718	0.463318
$797$	−40.6410	−1.43958	−0.719789	−	0.694193i	$-0.755762\pi$
$797$	−40.6410	−1.43958	−0.719789	+	0.694193i	$0.755762\pi$
$798$	0	0
$799$	1.35898	0.0480774
$800$	0	0
$801$	0	0
$802$	−9.52056	−0.336183
$803$	3.71281	0.131022
$804$	0	0
$805$	0	0
$806$	42.2233	1.48725
$807$	0	0
$808$	10.5558	0.371353
$809$	−44.8487	−1.57680	−0.788398	−	0.615166i	$-0.789089\pi$
$809$	−44.8487	−1.57680	−0.788398	+	0.615166i	$0.789089\pi$
$810$	0	0
$811$	−40.4974	−1.42206	−0.711028	−	0.703163i	$-0.751770\pi$
$811$	−40.4974	−1.42206	−0.711028	+	0.703163i	$0.751770\pi$
$812$	−17.8564	−0.626637
$813$	0	0
$814$	1.85641	0.0650670
$815$	0	0
$816$	0	0
$817$	6.69213	0.234128
$818$	−20.9282	−0.731737
$819$	0	0
$820$	0	0
$821$	7.93048	0.276776	0.138388	−	0.990378i	$-0.455808\pi$
$821$	7.93048	0.276776	0.138388	+	0.990378i	$0.455808\pi$
$822$	0	0
$823$	−47.1966	−1.64517	−0.822586	−	0.568641i	$-0.807469\pi$
$823$	−47.1966	−1.64517	−0.822586	+	0.568641i	$0.807469\pi$
$824$	4.89898	0.170664
$825$	0	0
$826$	−26.9282	−0.936952
$827$	−6.92820	−0.240917	−0.120459	−	0.992718i	$-0.538437\pi$
$827$	−6.92820	−0.240917	−0.120459	+	0.992718i	$0.538437\pi$
$828$	0	0
$829$	14.7846	0.513491	0.256745	−	0.966479i	$-0.417350\pi$
$829$	14.7846	0.513491	0.256745	+	0.966479i	$0.417350\pi$
$830$	0	0
$831$	0	0
$832$	−3.86370	−0.133950
$833$	−4.24871	−0.147209
$834$	0	0
$835$	0	0
$836$	1.03528	0.0358058
$837$	0	0
$838$	−29.3195	−1.01283
$839$	28.2843	0.976481	0.488241	−	0.872709i	$-0.337639\pi$
$839$	28.2843	0.976481	0.488241	+	0.872709i	$0.337639\pi$
$840$	0	0
$841$	−7.64102	−0.263483
$842$	29.7128	1.02397
$843$	0	0
$844$	−17.8564	−0.614643
$845$	0	0
$846$	0	0
$847$	−38.3596	−1.31805
$848$	−6.00000	−0.206041
$849$	0	0
$850$	0	0
$851$	9.79796	0.335870
$852$	0	0
$853$	−14.6969	−0.503214	−0.251607	−	0.967830i	$-0.580959\pi$
$853$	−14.6969	−0.503214	−0.251607	+	0.967830i	$0.580959\pi$
$854$	−7.72741	−0.264426
$855$	0	0
$856$	2.92820	0.100084
$857$	−16.6410	−0.568446	−0.284223	−	0.958758i	$-0.591736\pi$
$857$	−16.6410	−0.568446	−0.284223	+	0.958758i	$0.591736\pi$
$858$	0	0
$859$	−4.78461	−0.163249	−0.0816244	−	0.996663i	$-0.526011\pi$
$859$	−4.78461	−0.163249	−0.0816244	+	0.996663i	$0.526011\pi$
$860$	0	0
$861$	0	0
$862$	26.2137	0.892843
$863$	−20.7846	−0.707516	−0.353758	−	0.935337i	$-0.615096\pi$
$863$	−20.7846	−0.707516	−0.353758	+	0.935337i	$0.615096\pi$
$864$	0	0
$865$	0	0
$866$	−10.8332	−0.368128
$867$	0	0
$868$	−42.2233	−1.43315
$869$	4.14110	0.140477
$870$	0	0
$871$	−51.7128	−1.75222
$872$	8.92820	0.302347
$873$	0	0
$874$	5.46410	0.184826
$875$	0	0
$876$	0	0
$877$	29.6713	1.00193	0.500964	−	0.865468i	$-0.332979\pi$
$877$	29.6713	1.00193	0.500964	+	0.865468i	$0.332979\pi$
$878$	20.0000	0.674967
$879$	0	0
$880$	0	0
$881$	38.6370	1.30171	0.650857	−	0.759200i	$-0.274410\pi$
$881$	38.6370	1.30171	0.650857	+	0.759200i	$0.274410\pi$
$882$	0	0
$883$	−49.4703	−1.66481	−0.832404	−	0.554170i	$-0.813036\pi$
$883$	−49.4703	−1.66481	−0.832404	+	0.554170i	$0.813036\pi$
$884$	2.07055	0.0696402
$885$	0	0
$886$	−16.3923	−0.550710
$887$	15.7128	0.527585	0.263792	−	0.964580i	$-0.415027\pi$
$887$	15.7128	0.527585	0.263792	+	0.964580i	$0.415027\pi$
$888$	0	0
$889$	32.7846	1.09956
$890$	0	0
$891$	0	0
$892$	−10.5558	−0.353435
$893$	2.53590	0.0848606
$894$	0	0
$895$	0	0
$896$	3.86370	0.129077
$897$	0	0
$898$	40.4302	1.34917
$899$	50.5055	1.68445
$900$	0	0
$901$	3.21539	0.107120
$902$	1.85641	0.0618116
$903$	0	0
$904$	−6.00000	−0.199557
$905$	0	0
$906$	0	0
$907$	36.5665	1.21417	0.607085	−	0.794637i	$-0.292339\pi$
$907$	36.5665	1.21417	0.607085	+	0.794637i	$0.292339\pi$
$908$	1.85641	0.0616070
$909$	0	0
$910$	0	0
$911$	−45.2548	−1.49936	−0.749680	−	0.661801i	$-0.769792\pi$
$911$	−45.2548	−1.49936	−0.749680	+	0.661801i	$0.769792\pi$
$912$	0	0
$913$	16.9706	0.561644
$914$	−33.9411	−1.12267
$915$	0	0
$916$	−3.85641	−0.127419
$917$	−12.0000	−0.396275
$918$	0	0
$919$	27.7128	0.914161	0.457081	−	0.889425i	$-0.348895\pi$
$919$	27.7128	0.914161	0.457081	+	0.889425i	$0.348895\pi$
$920$	0	0
$921$	0	0
$922$	−27.5264	−0.906534
$923$	16.0000	0.526646
$924$	0	0
$925$	0	0
$926$	19.8733	0.653078
$927$	0	0
$928$	−4.62158	−0.151711
$929$	−11.3137	−0.371191	−0.185595	−	0.982626i	$-0.559421\pi$
$929$	−11.3137	−0.371191	−0.185595	+	0.982626i	$0.559421\pi$
$930$	0	0
$931$	−7.92820	−0.259836
$932$	−12.5359	−0.410627
$933$	0	0
$934$	−2.53590	−0.0829771
$935$	0	0
$936$	0	0
$937$	20.5569	0.671563	0.335782	−	0.941940i	$-0.391000\pi$
$937$	20.5569	0.671563	0.335782	+	0.941940i	$0.391000\pi$
$938$	51.7128	1.68848
$939$	0	0
$940$	0	0
$941$	−15.9353	−0.519475	−0.259738	−	0.965679i	$-0.583636\pi$
$941$	−15.9353	−0.519475	−0.259738	+	0.965679i	$0.583636\pi$
$942$	0	0
$943$	9.79796	0.319065
$944$	−6.96953	−0.226839
$945$	0	0
$946$	6.92820	0.225255
$947$	25.1769	0.818140	0.409070	−	0.912503i	$-0.365853\pi$
$947$	25.1769	0.818140	0.409070	+	0.912503i	$0.365853\pi$
$948$	0	0
$949$	13.8564	0.449798
$950$	0	0
$951$	0	0
$952$	−2.07055	−0.0671070
$953$	−30.0000	−0.971795	−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000	−0.971795	−0.485898	+	0.874016i	$0.661507\pi$
$954$	0	0
$955$	0	0
$956$	−13.1069	−0.423906
$957$	0	0
$958$	−16.6932	−0.539332
$959$	−2.07055	−0.0668616
$960$	0	0
$961$	88.4256	2.85244
$962$	6.92820	0.223374
$963$	0	0
$964$	10.0000	0.322078
$965$	0	0
$966$	0	0
$967$	17.2480	0.554657	0.277329	−	0.960775i	$-0.410551\pi$
$967$	17.2480	0.554657	0.277329	+	0.960775i	$0.410551\pi$
$968$	−9.92820	−0.319105
$969$	0	0
$970$	0	0
$971$	−54.2949	−1.74241	−0.871203	−	0.490922i	$-0.836660\pi$
$971$	−54.2949	−1.74241	−0.871203	+	0.490922i	$0.836660\pi$
$972$	0	0
$973$	26.7685	0.858159
$974$	−3.38323	−0.108406
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	−5.71281	−0.182769	−0.0913845	−	0.995816i	$-0.529129\pi$
$977$	−5.71281	−0.182769	−0.0913845	+	0.995816i	$0.529129\pi$
$978$	0	0
$979$	−4.00000	−0.127841
$980$	0	0
$981$	0	0
$982$	−18.5606	−0.592294
$983$	−7.21539	−0.230135	−0.115068	−	0.993358i	$-0.536708\pi$
$983$	−7.21539	−0.230135	−0.115068	+	0.993358i	$0.536708\pi$
$984$	0	0
$985$	0	0
$986$	2.47670	0.0788741
$987$	0	0
$988$	3.86370	0.122921
$989$	36.5665	1.16275
$990$	0	0
$991$	−20.7846	−0.660245	−0.330122	−	0.943938i	$-0.607090\pi$
$991$	−20.7846	−0.660245	−0.330122	+	0.943938i	$0.607090\pi$
$992$	−10.9282	−0.346971
$993$	0	0
$994$	−16.0000	−0.507489
$995$	0	0
$996$	0	0
$997$	18.2832	0.579036	0.289518	−	0.957173i	$-0.406505\pi$
$997$	18.2832	0.579036	0.289518	+	0.957173i	$0.406505\pi$
$998$	25.8564	0.818470
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.cv.1.4		4
3.2	odd	2		8550.2.a.cu.1.4		4
5.2	odd	4		1710.2.d.g.1369.7	yes	8
5.3	odd	4		1710.2.d.g.1369.4	yes	8
5.4	even	2		8550.2.a.cu.1.1		4
15.2	even	4		1710.2.d.g.1369.2	✓	8
15.8	even	4		1710.2.d.g.1369.5	yes	8
15.14	odd	2	inner	8550.2.a.cv.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1710.2.d.g.1369.2	✓	8	15.2	even	4
1710.2.d.g.1369.4	yes	8	5.3	odd	4
1710.2.d.g.1369.5	yes	8	15.8	even	4
1710.2.d.g.1369.7	yes	8	5.2	odd	4
8550.2.a.cu.1.1		4	5.4	even	2
8550.2.a.cu.1.4		4	3.2	odd	2
8550.2.a.cv.1.1		4	15.14	odd	2	inner
8550.2.a.cv.1.4		4	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.86370 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.86370 q^{7} +1.00000 q^{8} -1.03528 q^{11} -3.86370 q^{13} +3.86370 q^{14} +1.00000 q^{16} -0.535898 q^{17} -1.00000 q^{19} -1.03528 q^{22} -5.46410 q^{23} -3.86370 q^{26} +3.86370 q^{28} -4.62158 q^{29} -10.9282 q^{31} +1.00000 q^{32} -0.535898 q^{34} -1.79315 q^{37} -1.00000 q^{38} -1.79315 q^{41} -6.69213 q^{43} -1.03528 q^{44} -5.46410 q^{46} -2.53590 q^{47} +7.92820 q^{49} -3.86370 q^{52} -6.00000 q^{53} +3.86370 q^{56} -4.62158 q^{58} -6.96953 q^{59} -2.00000 q^{61} -10.9282 q^{62} +1.00000 q^{64} +13.3843 q^{67} -0.535898 q^{68} -4.14110 q^{71} -3.58630 q^{73} -1.79315 q^{74} -1.00000 q^{76} -4.00000 q^{77} -4.00000 q^{79} -1.79315 q^{82} -16.3923 q^{83} -6.69213 q^{86} -1.03528 q^{88} +3.86370 q^{89} -14.9282 q^{91} -5.46410 q^{92} -2.53590 q^{94} +2.55103 q^{97} +7.92820 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} + 4 q^{16} - 16 q^{17} - 4 q^{19} - 8 q^{23} - 16 q^{31} + 4 q^{32} - 16 q^{34} - 4 q^{38} - 8 q^{46} - 24 q^{47} + 4 q^{49} - 24 q^{53} - 8 q^{61} - 16 q^{62} + 4 q^{64} - 16 q^{68} - 4 q^{76} - 16 q^{77} - 16 q^{79} - 24 q^{83} - 32 q^{91} - 8 q^{92} - 24 q^{94} + 4 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 + 4 * q^16 - 16 * q^17 - 4 * q^19 - 8 * q^23 - 16 * q^31 + 4 * q^32 - 16 * q^34 - 4 * q^38 - 8 * q^46 - 24 * q^47 + 4 * q^49 - 24 * q^53 - 8 * q^61 - 16 * q^62 + 4 * q^64 - 16 * q^68 - 4 * q^76 - 16 * q^77 - 16 * q^79 - 24 * q^83 - 32 * q^91 - 8 * q^92 - 24 * q^94 + 4 * q^98

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