LMFDB - Embedded newform 6525.2.a.cc.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6525,2,Mod(1,6525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6525, 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("6525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6525 = 3^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6525.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:	$52.1023873189$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 2x^{8} - 12x^{7} + 21x^{6} + 48x^{5} - 68x^{4} - 73x^{3} + 66x^{2} + 40x - 10 $ x^9 - 2x^8 - 12x^7 + 21x^6 + 48x^5 - 68x^4 - 73x^3 + 66x^2 + 40x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.200649$ of defining polynomial
Character	$\chi$	$=$	6525.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.200649 q^{2} -1.95974 q^{4} -0.775135 q^{7} -0.794517 q^{8} +O(q^{10})$	$q+0.200649 q^{2} -1.95974 q^{4} -0.775135 q^{7} -0.794517 q^{8} -4.45447 q^{11} -3.67934 q^{13} -0.155530 q^{14} +3.76006 q^{16} +1.05645 q^{17} +3.76245 q^{19} -0.893784 q^{22} +6.21538 q^{23} -0.738254 q^{26} +1.51906 q^{28} -1.00000 q^{29} -3.96384 q^{31} +2.34348 q^{32} +0.211974 q^{34} -2.64473 q^{37} +0.754931 q^{38} -6.67607 q^{41} -9.96230 q^{43} +8.72961 q^{44} +1.24711 q^{46} -12.6272 q^{47} -6.39917 q^{49} +7.21054 q^{52} -2.87463 q^{53} +0.615858 q^{56} -0.200649 q^{58} +7.92534 q^{59} -2.22149 q^{61} -0.795340 q^{62} -7.04991 q^{64} +4.12604 q^{67} -2.07036 q^{68} -13.6161 q^{71} +8.60138 q^{73} -0.530661 q^{74} -7.37343 q^{76} +3.45282 q^{77} +6.97979 q^{79} -1.33955 q^{82} +2.90876 q^{83} -1.99892 q^{86} +3.53915 q^{88} +4.30975 q^{89} +2.85198 q^{91} -12.1805 q^{92} -2.53364 q^{94} -3.82019 q^{97} -1.28398 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 2 q^{2} + 10 q^{4} - q^{7} + 9 q^{8}+O(q^{10}) $ 9 * q + 2 * q^2 + 10 * q^4 - q^7 + 9 * q^8	$ 9 q + 2 q^{2} + 10 q^{4} - q^{7} + 9 q^{8} - 2 q^{11} - q^{13} + 3 q^{14} + 4 q^{16} + 12 q^{17} - q^{19} - 3 q^{22} + 16 q^{23} - 6 q^{26} + 4 q^{28} - 9 q^{29} + 5 q^{31} + 20 q^{32} + 3 q^{34} + 30 q^{38} + 10 q^{41} - 3 q^{43} + 13 q^{44} + 4 q^{46} + 26 q^{47} - 8 q^{49} + 9 q^{52} + 22 q^{53} - 22 q^{56} - 2 q^{58} - 4 q^{59} + 7 q^{61} + 28 q^{62} + 9 q^{64} - 5 q^{67} + 39 q^{68} + 10 q^{73} + 34 q^{74} - 2 q^{76} + 34 q^{77} + 10 q^{79} + 8 q^{82} + 46 q^{83} - 28 q^{86} - 2 q^{88} - 4 q^{89} - 21 q^{91} + 20 q^{92} + 5 q^{94} - 7 q^{97} + 51 q^{98}+O(q^{100}) $ 9 * q + 2 * q^2 + 10 * q^4 - q^7 + 9 * q^8 - 2 * q^11 - q^13 + 3 * q^14 + 4 * q^16 + 12 * q^17 - q^19 - 3 * q^22 + 16 * q^23 - 6 * q^26 + 4 * q^28 - 9 * q^29 + 5 * q^31 + 20 * q^32 + 3 * q^34 + 30 * q^38 + 10 * q^41 - 3 * q^43 + 13 * q^44 + 4 * q^46 + 26 * q^47 - 8 * q^49 + 9 * q^52 + 22 * q^53 - 22 * q^56 - 2 * q^58 - 4 * q^59 + 7 * q^61 + 28 * q^62 + 9 * q^64 - 5 * q^67 + 39 * q^68 + 10 * q^73 + 34 * q^74 - 2 * q^76 + 34 * q^77 + 10 * q^79 + 8 * q^82 + 46 * q^83 - 28 * q^86 - 2 * q^88 - 4 * q^89 - 21 * q^91 + 20 * q^92 + 5 * q^94 - 7 * q^97 + 51 * 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 $

$2$	0.200649	0.141880	0.0709400	−	0.997481i	$-0.477400\pi$
$2$	0.200649	0.141880	0.0709400	+	0.997481i	$0.477400\pi$
$3$	0	0
$3$	0	0
$4$	−1.95974	−0.979870
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.775135	−0.292974	−0.146487	−	0.989213i	$-0.546797\pi$
$7$	−0.775135	−0.292974	−0.146487	+	0.989213i	$0.546797\pi$
$8$	−0.794517	−0.280904
$9$	0	0
$10$	0	0
$11$	−4.45447	−1.34307	−0.671537	−	0.740971i	$-0.734365\pi$
$11$	−4.45447	−1.34307	−0.671537	+	0.740971i	$0.734365\pi$
$12$	0	0
$13$	−3.67934	−1.02046	−0.510232	−	0.860037i	$-0.670441\pi$
$13$	−3.67934	−1.02046	−0.510232	+	0.860037i	$0.670441\pi$
$14$	−0.155530	−0.0415671
$15$	0	0
$16$	3.76006	0.940015
$17$	1.05645	0.256226	0.128113	−	0.991760i	$-0.459108\pi$
$17$	1.05645	0.256226	0.128113	+	0.991760i	$0.459108\pi$
$18$	0	0
$19$	3.76245	0.863166	0.431583	−	0.902073i	$-0.357955\pi$
$19$	3.76245	0.863166	0.431583	+	0.902073i	$0.357955\pi$
$20$	0	0
$21$	0	0
$22$	−0.893784	−0.190555
$23$	6.21538	1.29600	0.647998	−	0.761642i	$-0.275607\pi$
$23$	6.21538	1.29600	0.647998	+	0.761642i	$0.275607\pi$
$24$	0	0
$25$	0	0
$26$	−0.738254	−0.144784
$27$	0	0
$28$	1.51906	0.287076
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−3.96384	−0.711927	−0.355963	−	0.934500i	$-0.615847\pi$
$31$	−3.96384	−0.711927	−0.355963	+	0.934500i	$0.615847\pi$
$32$	2.34348	0.414273
$33$	0	0
$34$	0.211974	0.0363533
$35$	0	0
$36$	0	0
$37$	−2.64473	−0.434790	−0.217395	−	0.976084i	$-0.569756\pi$
$37$	−2.64473	−0.434790	−0.217395	+	0.976084i	$0.569756\pi$
$38$	0.754931	0.122466
$39$	0	0
$40$	0	0
$41$	−6.67607	−1.04263	−0.521314	−	0.853365i	$-0.674558\pi$
$41$	−6.67607	−1.04263	−0.521314	+	0.853365i	$0.674558\pi$
$42$	0	0
$43$	−9.96230	−1.51924	−0.759618	−	0.650369i	$-0.774614\pi$
$43$	−9.96230	−1.51924	−0.759618	+	0.650369i	$0.774614\pi$
$44$	8.72961	1.31604
$45$	0	0
$46$	1.24711	0.183876
$47$	−12.6272	−1.84187	−0.920936	−	0.389715i	$-0.872574\pi$
$47$	−12.6272	−1.84187	−0.920936	+	0.389715i	$0.872574\pi$
$48$	0	0
$49$	−6.39917	−0.914166
$50$	0	0
$51$	0	0
$52$	7.21054	0.999922
$53$	−2.87463	−0.394860	−0.197430	−	0.980317i	$-0.563260\pi$
$53$	−2.87463	−0.394860	−0.197430	+	0.980317i	$0.563260\pi$
$54$	0	0
$55$	0	0
$56$	0.615858	0.0822975
$57$	0	0
$58$	−0.200649	−0.0263465
$59$	7.92534	1.03179	0.515896	−	0.856651i	$-0.327459\pi$
$59$	7.92534	1.03179	0.515896	+	0.856651i	$0.327459\pi$
$60$	0	0
$61$	−2.22149	−0.284432	−0.142216	−	0.989836i	$-0.545423\pi$
$61$	−2.22149	−0.284432	−0.142216	+	0.989836i	$0.545423\pi$
$62$	−0.795340	−0.101008
$63$	0	0
$64$	−7.04991	−0.881238
$65$	0	0
$66$	0	0
$67$	4.12604	0.504077	0.252038	−	0.967717i	$-0.418899\pi$
$67$	4.12604	0.504077	0.252038	+	0.967717i	$0.418899\pi$
$68$	−2.07036	−0.251068
$69$	0	0
$70$	0	0
$71$	−13.6161	−1.61593	−0.807966	−	0.589229i	$-0.799432\pi$
$71$	−13.6161	−1.61593	−0.807966	+	0.589229i	$0.799432\pi$
$72$	0	0
$73$	8.60138	1.00672	0.503358	−	0.864078i	$-0.332098\pi$
$73$	8.60138	1.00672	0.503358	+	0.864078i	$0.332098\pi$
$74$	−0.530661	−0.0616881
$75$	0	0
$76$	−7.37343	−0.845790
$77$	3.45282	0.393485
$78$	0	0
$79$	6.97979	0.785288	0.392644	−	0.919691i	$-0.371560\pi$
$79$	6.97979	0.785288	0.392644	+	0.919691i	$0.371560\pi$
$80$	0	0
$81$	0	0
$82$	−1.33955	−0.147928
$83$	2.90876	0.319278	0.159639	−	0.987175i	$-0.448967\pi$
$83$	2.90876	0.319278	0.159639	+	0.987175i	$0.448967\pi$
$84$	0	0
$85$	0	0
$86$	−1.99892	−0.215549
$87$	0	0
$88$	3.53915	0.377275
$89$	4.30975	0.456833	0.228416	−	0.973564i	$-0.426645\pi$
$89$	4.30975	0.456833	0.228416	+	0.973564i	$0.426645\pi$
$90$	0	0
$91$	2.85198	0.298969
$92$	−12.1805	−1.26991
$93$	0	0
$94$	−2.53364	−0.261325
$95$	0	0
$96$	0	0
$97$	−3.82019	−0.387881	−0.193941	−	0.981013i	$-0.562127\pi$
$97$	−3.82019	−0.387881	−0.193941	+	0.981013i	$0.562127\pi$
$98$	−1.28398	−0.129702
$99$	0	0
$100$	0	0
$101$	1.79941	0.179048	0.0895241	−	0.995985i	$-0.471465\pi$
$101$	1.79941	0.179048	0.0895241	+	0.995985i	$0.471465\pi$
$102$	0	0
$103$	0.658227	0.0648570	0.0324285	−	0.999474i	$-0.489676\pi$
$103$	0.658227	0.0648570	0.0324285	+	0.999474i	$0.489676\pi$
$104$	2.92329	0.286653
$105$	0	0
$106$	−0.576790	−0.0560228
$107$	−6.24882	−0.604097	−0.302048	−	0.953293i	$-0.597670\pi$
$107$	−6.24882	−0.604097	−0.302048	+	0.953293i	$0.597670\pi$
$108$	0	0
$109$	7.44434	0.713039	0.356519	−	0.934288i	$-0.383963\pi$
$109$	7.44434	0.713039	0.356519	+	0.934288i	$0.383963\pi$
$110$	0	0
$111$	0	0
$112$	−2.91456	−0.275400
$113$	21.0265	1.97801	0.989003	−	0.147896i	$-0.0472500\pi$
$113$	21.0265	1.97801	0.989003	+	0.147896i	$0.0472500\pi$
$114$	0	0
$115$	0	0
$116$	1.95974	0.181957
$117$	0	0
$118$	1.59021	0.146391
$119$	−0.818888	−0.0750673
$120$	0	0
$121$	8.84231	0.803847
$122$	−0.445738	−0.0403552
$123$	0	0
$124$	7.76810	0.697596
$125$	0	0
$126$	0	0
$127$	11.9855	1.06354	0.531771	−	0.846888i	$-0.321527\pi$
$127$	11.9855	1.06354	0.531771	+	0.846888i	$0.321527\pi$
$128$	−6.10152	−0.539304
$129$	0	0
$130$	0	0
$131$	13.6387	1.19162	0.595810	−	0.803126i	$-0.296831\pi$
$131$	13.6387	1.19162	0.595810	+	0.803126i	$0.296831\pi$
$132$	0	0
$133$	−2.91641	−0.252885
$134$	0.827885	0.0715184
$135$	0	0
$136$	−0.839363	−0.0719748
$137$	5.55942	0.474974	0.237487	−	0.971391i	$-0.423676\pi$
$137$	5.55942	0.474974	0.237487	+	0.971391i	$0.423676\pi$
$138$	0	0
$139$	22.0649	1.87152	0.935759	−	0.352639i	$-0.114716\pi$
$139$	22.0649	1.87152	0.935759	+	0.352639i	$0.114716\pi$
$140$	0	0
$141$	0	0
$142$	−2.73205	−0.229269
$143$	16.3895	1.37056
$144$	0	0
$145$	0	0
$146$	1.72586	0.142833
$147$	0	0
$148$	5.18298	0.426038
$149$	8.92269	0.730975	0.365488	−	0.930816i	$-0.380902\pi$
$149$	8.92269	0.730975	0.365488	+	0.930816i	$0.380902\pi$
$150$	0	0
$151$	−7.84172	−0.638150	−0.319075	−	0.947729i	$-0.603372\pi$
$151$	−7.84172	−0.638150	−0.319075	+	0.947729i	$0.603372\pi$
$152$	−2.98933	−0.242467
$153$	0	0
$154$	0.692803	0.0558277
$155$	0	0
$156$	0	0
$157$	−17.8114	−1.42150	−0.710751	−	0.703444i	$-0.751645\pi$
$157$	−17.8114	−1.42150	−0.710751	+	0.703444i	$0.751645\pi$
$158$	1.40049	0.111417
$159$	0	0
$160$	0	0
$161$	−4.81776	−0.379693
$162$	0	0
$163$	12.3040	0.963724	0.481862	−	0.876247i	$-0.339961\pi$
$163$	12.3040	0.963724	0.481862	+	0.876247i	$0.339961\pi$
$164$	13.0834	1.02164
$165$	0	0
$166$	0.583640	0.0452992
$167$	21.6643	1.67643	0.838216	−	0.545338i	$-0.183599\pi$
$167$	21.6643	1.67643	0.838216	+	0.545338i	$0.183599\pi$
$168$	0	0
$169$	0.537513	0.0413471
$170$	0	0
$171$	0	0
$172$	19.5235	1.48865
$173$	4.25059	0.323167	0.161583	−	0.986859i	$-0.448340\pi$
$173$	4.25059	0.323167	0.161583	+	0.986859i	$0.448340\pi$
$174$	0	0
$175$	0	0
$176$	−16.7491	−1.26251
$177$	0	0
$178$	0.864746	0.0648154
$179$	23.2124	1.73497	0.867487	−	0.497460i	$-0.165734\pi$
$179$	23.2124	1.73497	0.867487	+	0.497460i	$0.165734\pi$
$180$	0	0
$181$	9.68131	0.719606	0.359803	−	0.933028i	$-0.382844\pi$
$181$	9.68131	0.719606	0.359803	+	0.933028i	$0.382844\pi$
$182$	0.572247	0.0424177
$183$	0	0
$184$	−4.93822	−0.364050
$185$	0	0
$186$	0	0
$187$	−4.70590	−0.344130
$188$	24.7461	1.80479
$189$	0	0
$190$	0	0
$191$	−21.1664	−1.53155	−0.765775	−	0.643109i	$-0.777644\pi$
$191$	−21.1664	−1.53155	−0.765775	+	0.643109i	$0.777644\pi$
$192$	0	0
$193$	17.6231	1.26854	0.634269	−	0.773112i	$-0.281301\pi$
$193$	17.6231	1.26854	0.634269	+	0.773112i	$0.281301\pi$
$194$	−0.766516	−0.0550326
$195$	0	0
$196$	12.5407	0.895764
$197$	12.6217	0.899257	0.449628	−	0.893216i	$-0.351556\pi$
$197$	12.6217	0.899257	0.449628	+	0.893216i	$0.351556\pi$
$198$	0	0
$199$	−20.9741	−1.48681	−0.743406	−	0.668841i	$-0.766791\pi$
$199$	−20.9741	−1.48681	−0.743406	+	0.668841i	$0.766791\pi$
$200$	0	0
$201$	0	0
$202$	0.361050	0.0254034
$203$	0.775135	0.0544038
$204$	0	0
$205$	0	0
$206$	0.132072	0.00920191
$207$	0	0
$208$	−13.8345	−0.959252
$209$	−16.7597	−1.15930
$210$	0	0
$211$	−25.1750	−1.73312	−0.866558	−	0.499077i	$-0.833672\pi$
$211$	−25.1750	−1.73312	−0.866558	+	0.499077i	$0.833672\pi$
$212$	5.63352	0.386912
$213$	0	0
$214$	−1.25382	−0.0857093
$215$	0	0
$216$	0	0
$217$	3.07251	0.208576
$218$	1.49370	0.101166
$219$	0	0
$220$	0	0
$221$	−3.88702	−0.261469
$222$	0	0
$223$	−0.225647	−0.0151104	−0.00755521	−	0.999971i	$-0.502405\pi$
$223$	−0.225647	−0.0151104	−0.00755521	+	0.999971i	$0.502405\pi$
$224$	−1.81652	−0.121371
$225$	0	0
$226$	4.21894	0.280640
$227$	26.3772	1.75072	0.875359	−	0.483474i	$-0.160625\pi$
$227$	26.3772	1.75072	0.875359	+	0.483474i	$0.160625\pi$
$228$	0	0
$229$	−3.12235	−0.206331	−0.103165	−	0.994664i	$-0.532897\pi$
$229$	−3.12235	−0.206331	−0.103165	+	0.994664i	$0.532897\pi$
$230$	0	0
$231$	0	0
$232$	0.794517	0.0521626
$233$	12.4808	0.817644	0.408822	−	0.912614i	$-0.365940\pi$
$233$	12.4808	0.817644	0.408822	+	0.912614i	$0.365940\pi$
$234$	0	0
$235$	0	0
$236$	−15.5316	−1.01102
$237$	0	0
$238$	−0.164309	−0.0106506
$239$	12.6664	0.819321	0.409660	−	0.912238i	$-0.365647\pi$
$239$	12.6664	0.819321	0.409660	+	0.912238i	$0.365647\pi$
$240$	0	0
$241$	−0.738241	−0.0475543	−0.0237771	−	0.999717i	$-0.507569\pi$
$241$	−0.738241	−0.0475543	−0.0237771	+	0.999717i	$0.507569\pi$
$242$	1.77420	0.114050
$243$	0	0
$244$	4.35354	0.278707
$245$	0	0
$246$	0	0
$247$	−13.8433	−0.880830
$248$	3.14934	0.199983
$249$	0	0
$250$	0	0
$251$	−12.3549	−0.779832	−0.389916	−	0.920850i	$-0.627496\pi$
$251$	−12.3549	−0.779832	−0.389916	+	0.920850i	$0.627496\pi$
$252$	0	0
$253$	−27.6862	−1.74062
$254$	2.40488	0.150895
$255$	0	0
$256$	12.8755	0.804722
$257$	17.5823	1.09675	0.548377	−	0.836231i	$-0.315246\pi$
$257$	17.5823	1.09675	0.548377	+	0.836231i	$0.315246\pi$
$258$	0	0
$259$	2.05002	0.127382
$260$	0	0
$261$	0	0
$262$	2.73659	0.169067
$263$	11.3491	0.699816	0.349908	−	0.936784i	$-0.386213\pi$
$263$	11.3491	0.699816	0.349908	+	0.936784i	$0.386213\pi$
$264$	0	0
$265$	0	0
$266$	−0.585174	−0.0358793
$267$	0	0
$268$	−8.08597	−0.493930
$269$	−14.3872	−0.877201	−0.438601	−	0.898682i	$-0.644526\pi$
$269$	−14.3872	−0.877201	−0.438601	+	0.898682i	$0.644526\pi$
$270$	0	0
$271$	6.32554	0.384249	0.192124	−	0.981371i	$-0.438462\pi$
$271$	6.32554	0.384249	0.192124	+	0.981371i	$0.438462\pi$
$272$	3.97230	0.240856
$273$	0	0
$274$	1.11549	0.0673893
$275$	0	0
$276$	0	0
$277$	18.3726	1.10390	0.551952	−	0.833876i	$-0.313883\pi$
$277$	18.3726	1.10390	0.551952	+	0.833876i	$0.313883\pi$
$278$	4.42729	0.265531
$279$	0	0
$280$	0	0
$281$	17.1207	1.02133	0.510667	−	0.859778i	$-0.329398\pi$
$281$	17.1207	1.02133	0.510667	+	0.859778i	$0.329398\pi$
$282$	0	0
$283$	−28.6804	−1.70487	−0.852435	−	0.522833i	$-0.824875\pi$
$283$	−28.6804	−1.70487	−0.852435	+	0.522833i	$0.824875\pi$
$284$	26.6840	1.58340
$285$	0	0
$286$	3.28853	0.194455
$287$	5.17486	0.305462
$288$	0	0
$289$	−15.8839	−0.934348
$290$	0	0
$291$	0	0
$292$	−16.8565	−0.986450
$293$	9.71695	0.567670	0.283835	−	0.958873i	$-0.408393\pi$
$293$	9.71695	0.567670	0.283835	+	0.958873i	$0.408393\pi$
$294$	0	0
$295$	0	0
$296$	2.10128	0.122134
$297$	0	0
$298$	1.79033	0.103711
$299$	−22.8685	−1.32252
$300$	0	0
$301$	7.72213	0.445096
$302$	−1.57343	−0.0905408
$303$	0	0
$304$	14.1471	0.811389
$305$	0	0
$306$	0	0
$307$	8.56349	0.488744	0.244372	−	0.969682i	$-0.421418\pi$
$307$	8.56349	0.488744	0.244372	+	0.969682i	$0.421418\pi$
$308$	−6.76663	−0.385564
$309$	0	0
$310$	0	0
$311$	−17.1647	−0.973324	−0.486662	−	0.873590i	$-0.661786\pi$
$311$	−17.1647	−0.973324	−0.486662	+	0.873590i	$0.661786\pi$
$312$	0	0
$313$	−3.40045	−0.192205	−0.0961025	−	0.995371i	$-0.530638\pi$
$313$	−3.40045	−0.192205	−0.0961025	+	0.995371i	$0.530638\pi$
$314$	−3.57383	−0.201683
$315$	0	0
$316$	−13.6786	−0.769480
$317$	−23.0876	−1.29673	−0.648365	−	0.761329i	$-0.724547\pi$
$317$	−23.0876	−1.29673	−0.648365	+	0.761329i	$0.724547\pi$
$318$	0	0
$319$	4.45447	0.249403
$320$	0	0
$321$	0	0
$322$	−0.966677	−0.0538708
$323$	3.97482	0.221165
$324$	0	0
$325$	0	0
$326$	2.46878	0.136733
$327$	0	0
$328$	5.30425	0.292878
$329$	9.78781	0.539620
$330$	0	0
$331$	−15.3051	−0.841245	−0.420623	−	0.907236i	$-0.638188\pi$
$331$	−15.3051	−0.841245	−0.420623	+	0.907236i	$0.638188\pi$
$332$	−5.70042	−0.312851
$333$	0	0
$334$	4.34691	0.237852
$335$	0	0
$336$	0	0
$337$	−2.93351	−0.159799	−0.0798994	−	0.996803i	$-0.525460\pi$
$337$	−2.93351	−0.159799	−0.0798994	+	0.996803i	$0.525460\pi$
$338$	0.107851	0.00586634
$339$	0	0
$340$	0	0
$341$	17.6568	0.956170
$342$	0	0
$343$	10.3862	0.560800
$344$	7.91521	0.426760
$345$	0	0
$346$	0.852876	0.0458509
$347$	31.0176	1.66511	0.832555	−	0.553942i	$-0.186877\pi$
$347$	31.0176	1.66511	0.832555	+	0.553942i	$0.186877\pi$
$348$	0	0
$349$	−8.15838	−0.436708	−0.218354	−	0.975870i	$-0.570069\pi$
$349$	−8.15838	−0.436708	−0.218354	+	0.975870i	$0.570069\pi$
$350$	0	0
$351$	0	0
$352$	−10.4390	−0.556400
$353$	1.25800	0.0669568	0.0334784	−	0.999439i	$-0.489342\pi$
$353$	1.25800	0.0669568	0.0334784	+	0.999439i	$0.489342\pi$
$354$	0	0
$355$	0	0
$356$	−8.44599	−0.447637
$357$	0	0
$358$	4.65753	0.246158
$359$	−3.64864	−0.192568	−0.0962838	−	0.995354i	$-0.530696\pi$
$359$	−3.64864	−0.192568	−0.0962838	+	0.995354i	$0.530696\pi$
$360$	0	0
$361$	−4.84396	−0.254945
$362$	1.94254	0.102098
$363$	0	0
$364$	−5.58915	−0.292951
$365$	0	0
$366$	0	0
$367$	13.1323	0.685503	0.342751	−	0.939426i	$-0.388641\pi$
$367$	13.1323	0.685503	0.342751	+	0.939426i	$0.388641\pi$
$368$	23.3702	1.21826
$369$	0	0
$370$	0	0
$371$	2.22822	0.115684
$372$	0	0
$373$	−7.58444	−0.392707	−0.196354	−	0.980533i	$-0.562910\pi$
$373$	−7.58444	−0.392707	−0.196354	+	0.980533i	$0.562910\pi$
$374$	−0.944233	−0.0488251
$375$	0	0
$376$	10.0325	0.517389
$377$	3.67934	0.189495
$378$	0	0
$379$	−27.8949	−1.43287	−0.716434	−	0.697655i	$-0.754227\pi$
$379$	−27.8949	−1.43287	−0.716434	+	0.697655i	$0.754227\pi$
$380$	0	0
$381$	0	0
$382$	−4.24702	−0.217296
$383$	12.0440	0.615418	0.307709	−	0.951481i	$-0.400438\pi$
$383$	12.0440	0.615418	0.307709	+	0.951481i	$0.400438\pi$
$384$	0	0
$385$	0	0
$386$	3.53605	0.179980
$387$	0	0
$388$	7.48658	0.380073
$389$	29.0575	1.47327	0.736636	−	0.676290i	$-0.236413\pi$
$389$	29.0575	1.47327	0.736636	+	0.676290i	$0.236413\pi$
$390$	0	0
$391$	6.56620	0.332067
$392$	5.08424	0.256793
$393$	0	0
$394$	2.53252	0.127587
$395$	0	0
$396$	0	0
$397$	−11.1050	−0.557346	−0.278673	−	0.960386i	$-0.589895\pi$
$397$	−11.1050	−0.557346	−0.278673	+	0.960386i	$0.589895\pi$
$398$	−4.20842	−0.210949
$399$	0	0
$400$	0	0
$401$	14.9928	0.748703	0.374352	−	0.927287i	$-0.377865\pi$
$401$	14.9928	0.748703	0.374352	+	0.927287i	$0.377865\pi$
$402$	0	0
$403$	14.5843	0.726496
$404$	−3.52638	−0.175444
$405$	0	0
$406$	0.155530	0.00771882
$407$	11.7809	0.583955
$408$	0	0
$409$	26.3994	1.30537	0.652684	−	0.757631i	$-0.273643\pi$
$409$	26.3994	1.30537	0.652684	+	0.757631i	$0.273643\pi$
$410$	0	0
$411$	0	0
$412$	−1.28995	−0.0635514
$413$	−6.14321	−0.302288
$414$	0	0
$415$	0	0
$416$	−8.62247	−0.422751
$417$	0	0
$418$	−3.36282	−0.164481
$419$	−15.1863	−0.741901	−0.370950	−	0.928653i	$-0.620968\pi$
$419$	−15.1863	−0.741901	−0.370950	+	0.928653i	$0.620968\pi$
$420$	0	0
$421$	4.51805	0.220196	0.110098	−	0.993921i	$-0.464883\pi$
$421$	4.51805	0.220196	0.110098	+	0.993921i	$0.464883\pi$
$422$	−5.05132	−0.245894
$423$	0	0
$424$	2.28394	0.110918
$425$	0	0
$426$	0	0
$427$	1.72195	0.0833311
$428$	12.2461	0.591936
$429$	0	0
$430$	0	0
$431$	3.10385	0.149507	0.0747536	−	0.997202i	$-0.476183\pi$
$431$	3.10385	0.149507	0.0747536	+	0.997202i	$0.476183\pi$
$432$	0	0
$433$	−8.92413	−0.428866	−0.214433	−	0.976739i	$-0.568790\pi$
$433$	−8.92413	−0.428866	−0.214433	+	0.976739i	$0.568790\pi$
$434$	0.616496	0.0295927
$435$	0	0
$436$	−14.5890	−0.698685
$437$	23.3851	1.11866
$438$	0	0
$439$	−32.6862	−1.56003	−0.780013	−	0.625764i	$-0.784787\pi$
$439$	−32.6862	−1.56003	−0.780013	+	0.625764i	$0.784787\pi$
$440$	0	0
$441$	0	0
$442$	−0.779925	−0.0370972
$443$	14.9791	0.711677	0.355838	−	0.934548i	$-0.384195\pi$
$443$	14.9791	0.711677	0.355838	+	0.934548i	$0.384195\pi$
$444$	0	0
$445$	0	0
$446$	−0.0452757	−0.00214387
$447$	0	0
$448$	5.46463	0.258180
$449$	−34.4130	−1.62405	−0.812024	−	0.583623i	$-0.801634\pi$
$449$	−34.4130	−1.62405	−0.812024	+	0.583623i	$0.801634\pi$
$450$	0	0
$451$	29.7384	1.40033
$452$	−41.2065	−1.93819
$453$	0	0
$454$	5.29256	0.248392
$455$	0	0
$456$	0	0
$457$	−29.7364	−1.39101	−0.695506	−	0.718520i	$-0.744820\pi$
$457$	−29.7364	−1.39101	−0.695506	+	0.718520i	$0.744820\pi$
$458$	−0.626496	−0.0292742
$459$	0	0
$460$	0	0
$461$	5.01736	0.233682	0.116841	−	0.993151i	$-0.462723\pi$
$461$	5.01736	0.233682	0.116841	+	0.993151i	$0.462723\pi$
$462$	0	0
$463$	16.0818	0.747384	0.373692	−	0.927553i	$-0.378092\pi$
$463$	16.0818	0.747384	0.373692	+	0.927553i	$0.378092\pi$
$464$	−3.76006	−0.174556
$465$	0	0
$466$	2.50425	0.116007
$467$	21.5438	0.996928	0.498464	−	0.866910i	$-0.333898\pi$
$467$	21.5438	0.996928	0.498464	+	0.866910i	$0.333898\pi$
$468$	0	0
$469$	−3.19824	−0.147681
$470$	0	0
$471$	0	0
$472$	−6.29681	−0.289834
$473$	44.3768	2.04045
$474$	0	0
$475$	0	0
$476$	1.60481	0.0735562
$477$	0	0
$478$	2.54149	0.116245
$479$	14.2131	0.649414	0.324707	−	0.945815i	$-0.394734\pi$
$479$	14.2131	0.649414	0.324707	+	0.945815i	$0.394734\pi$
$480$	0	0
$481$	9.73084	0.443688
$482$	−0.148127	−0.00674700
$483$	0	0
$484$	−17.3286	−0.787665
$485$	0	0
$486$	0	0
$487$	19.1443	0.867512	0.433756	−	0.901030i	$-0.357188\pi$
$487$	19.1443	0.867512	0.433756	+	0.901030i	$0.357188\pi$
$488$	1.76501	0.0798981
$489$	0	0
$490$	0	0
$491$	−32.2694	−1.45630	−0.728148	−	0.685420i	$-0.759619\pi$
$491$	−32.2694	−1.45630	−0.728148	+	0.685420i	$0.759619\pi$
$492$	0	0
$493$	−1.05645	−0.0475799
$494$	−2.77764	−0.124972
$495$	0	0
$496$	−14.9043	−0.669222
$497$	10.5543	0.473426
$498$	0	0
$499$	20.4860	0.917078	0.458539	−	0.888674i	$-0.348373\pi$
$499$	20.4860	0.917078	0.458539	+	0.888674i	$0.348373\pi$
$500$	0	0
$501$	0	0
$502$	−2.47899	−0.110643
$503$	−3.42204	−0.152581	−0.0762905	−	0.997086i	$-0.524308\pi$
$503$	−3.42204	−0.152581	−0.0762905	+	0.997086i	$0.524308\pi$
$504$	0	0
$505$	0	0
$506$	−5.55520	−0.246959
$507$	0	0
$508$	−23.4885	−1.04213
$509$	21.5512	0.955238	0.477619	−	0.878567i	$-0.341500\pi$
$509$	21.5512	0.955238	0.477619	+	0.878567i	$0.341500\pi$
$510$	0	0
$511$	−6.66723	−0.294941
$512$	14.7865	0.653478
$513$	0	0
$514$	3.52787	0.155608
$515$	0	0
$516$	0	0
$517$	56.2476	2.47377
$518$	0.411334	0.0180730
$519$	0	0
$520$	0	0
$521$	31.7978	1.39309	0.696544	−	0.717514i	$-0.254720\pi$
$521$	31.7978	1.39309	0.696544	+	0.717514i	$0.254720\pi$
$522$	0	0
$523$	39.5953	1.73138	0.865691	−	0.500579i	$-0.166879\pi$
$523$	39.5953	1.73138	0.865691	+	0.500579i	$0.166879\pi$
$524$	−26.7283	−1.16763
$525$	0	0
$526$	2.27718	0.0992900
$527$	−4.18758	−0.182414
$528$	0	0
$529$	15.6309	0.679605
$530$	0	0
$531$	0	0
$532$	5.71540	0.247794
$533$	24.5635	1.06396
$534$	0	0
$535$	0	0
$536$	−3.27821	−0.141597
$537$	0	0
$538$	−2.88677	−0.124457
$539$	28.5049	1.22779
$540$	0	0
$541$	−25.5906	−1.10023	−0.550113	−	0.835090i	$-0.685415\pi$
$541$	−25.5906	−1.10023	−0.550113	+	0.835090i	$0.685415\pi$
$542$	1.26921	0.0545173
$543$	0	0
$544$	2.47576	0.106147
$545$	0	0
$546$	0	0
$547$	5.96642	0.255106	0.127553	−	0.991832i	$-0.459288\pi$
$547$	5.96642	0.255106	0.127553	+	0.991832i	$0.459288\pi$
$548$	−10.8950	−0.465412
$549$	0	0
$550$	0	0
$551$	−3.76245	−0.160286
$552$	0	0
$553$	−5.41028	−0.230069
$554$	3.68645	0.156622
$555$	0	0
$556$	−43.2414	−1.83385
$557$	15.5091	0.657143	0.328572	−	0.944479i	$-0.393433\pi$
$557$	15.5091	0.657143	0.328572	+	0.944479i	$0.393433\pi$
$558$	0	0
$559$	36.6547	1.55033
$560$	0	0
$561$	0	0
$562$	3.43524	0.144907
$563$	8.67282	0.365516	0.182758	−	0.983158i	$-0.441498\pi$
$563$	8.67282	0.365516	0.182758	+	0.983158i	$0.441498\pi$
$564$	0	0
$565$	0	0
$566$	−5.75468	−0.241887
$567$	0	0
$568$	10.8182	0.453922
$569$	−2.05132	−0.0859959	−0.0429980	−	0.999075i	$-0.513691\pi$
$569$	−2.05132	−0.0859959	−0.0429980	+	0.999075i	$0.513691\pi$
$570$	0	0
$571$	−2.72574	−0.114069	−0.0570344	−	0.998372i	$-0.518164\pi$
$571$	−2.72574	−0.114069	−0.0570344	+	0.998372i	$0.518164\pi$
$572$	−32.1192	−1.34297
$573$	0	0
$574$	1.03833	0.0433390
$575$	0	0
$576$	0	0
$577$	−24.3626	−1.01423	−0.507115	−	0.861879i	$-0.669288\pi$
$577$	−24.3626	−1.01423	−0.507115	+	0.861879i	$0.669288\pi$
$578$	−3.18709	−0.132565
$579$	0	0
$580$	0	0
$581$	−2.25469	−0.0935401
$582$	0	0
$583$	12.8049	0.530326
$584$	−6.83394	−0.282790
$585$	0	0
$586$	1.94969	0.0805411
$587$	4.29753	0.177378	0.0886890	−	0.996059i	$-0.471732\pi$
$587$	4.29753	0.177378	0.0886890	+	0.996059i	$0.471732\pi$
$588$	0	0
$589$	−14.9138	−0.614511
$590$	0	0
$591$	0	0
$592$	−9.94433	−0.408710
$593$	20.7059	0.850291	0.425146	−	0.905125i	$-0.360223\pi$
$593$	20.7059	0.850291	0.425146	+	0.905125i	$0.360223\pi$
$594$	0	0
$595$	0	0
$596$	−17.4862	−0.716261
$597$	0	0
$598$	−4.58853	−0.187639
$599$	−2.58646	−0.105680	−0.0528399	−	0.998603i	$-0.516827\pi$
$599$	−2.58646	−0.105680	−0.0528399	+	0.998603i	$0.516827\pi$
$600$	0	0
$601$	25.6870	1.04779	0.523897	−	0.851782i	$-0.324478\pi$
$601$	25.6870	1.04779	0.523897	+	0.851782i	$0.324478\pi$
$602$	1.54944	0.0631503
$603$	0	0
$604$	15.3677	0.625304
$605$	0	0
$606$	0	0
$607$	18.6725	0.757893	0.378947	−	0.925419i	$-0.376286\pi$
$607$	18.6725	0.757893	0.378947	+	0.925419i	$0.376286\pi$
$608$	8.81725	0.357587
$609$	0	0
$610$	0	0
$611$	46.4598	1.87956
$612$	0	0
$613$	−28.5152	−1.15172	−0.575858	−	0.817549i	$-0.695332\pi$
$613$	−28.5152	−1.15172	−0.575858	+	0.817549i	$0.695332\pi$
$614$	1.71825	0.0693430
$615$	0	0
$616$	−2.74332	−0.110532
$617$	−31.7077	−1.27651	−0.638253	−	0.769827i	$-0.720343\pi$
$617$	−31.7077	−1.27651	−0.638253	+	0.769827i	$0.720343\pi$
$618$	0	0
$619$	33.0039	1.32654	0.663269	−	0.748381i	$-0.269169\pi$
$619$	33.0039	1.32654	0.663269	+	0.748381i	$0.269169\pi$
$620$	0	0
$621$	0	0
$622$	−3.44408	−0.138095
$623$	−3.34064	−0.133840
$624$	0	0
$625$	0	0
$626$	−0.682297	−0.0272701
$627$	0	0
$628$	34.9057	1.39289
$629$	−2.79401	−0.111404
$630$	0	0
$631$	−11.7586	−0.468102	−0.234051	−	0.972224i	$-0.575198\pi$
$631$	−11.7586	−0.468102	−0.234051	+	0.972224i	$0.575198\pi$
$632$	−5.54556	−0.220591
$633$	0	0
$634$	−4.63250	−0.183980
$635$	0	0
$636$	0	0
$637$	23.5447	0.932874
$638$	0.893784	0.0353852
$639$	0	0
$640$	0	0
$641$	−3.63965	−0.143758	−0.0718788	−	0.997413i	$-0.522899\pi$
$641$	−3.63965	−0.143758	−0.0718788	+	0.997413i	$0.522899\pi$
$642$	0	0
$643$	33.1571	1.30759	0.653794	−	0.756673i	$-0.273176\pi$
$643$	33.1571	1.30759	0.653794	+	0.756673i	$0.273176\pi$
$644$	9.44155	0.372049
$645$	0	0
$646$	0.797543	0.0313789
$647$	47.8411	1.88083	0.940413	−	0.340034i	$-0.110439\pi$
$647$	47.8411	1.88083	0.940413	+	0.340034i	$0.110439\pi$
$648$	0	0
$649$	−35.3032	−1.38577
$650$	0	0
$651$	0	0
$652$	−24.1127	−0.944325
$653$	−10.3879	−0.406512	−0.203256	−	0.979126i	$-0.565152\pi$
$653$	−10.3879	−0.406512	−0.203256	+	0.979126i	$0.565152\pi$
$654$	0	0
$655$	0	0
$656$	−25.1024	−0.980086
$657$	0	0
$658$	1.96391	0.0765612
$659$	33.5433	1.30666	0.653331	−	0.757073i	$-0.273371\pi$
$659$	33.5433	1.30666	0.653331	+	0.757073i	$0.273371\pi$
$660$	0	0
$661$	−18.6291	−0.724589	−0.362294	−	0.932064i	$-0.618006\pi$
$661$	−18.6291	−0.724589	−0.362294	+	0.932064i	$0.618006\pi$
$662$	−3.07095	−0.119356
$663$	0	0
$664$	−2.31106	−0.0896866
$665$	0	0
$666$	0	0
$667$	−6.21538	−0.240660
$668$	−42.4564	−1.64269
$669$	0	0
$670$	0	0
$671$	9.89555	0.382013
$672$	0	0
$673$	−29.3062	−1.12967	−0.564835	−	0.825204i	$-0.691060\pi$
$673$	−29.3062	−1.12967	−0.564835	+	0.825204i	$0.691060\pi$
$674$	−0.588606	−0.0226723
$675$	0	0
$676$	−1.05339	−0.0405148
$677$	−20.8134	−0.799925	−0.399962	−	0.916532i	$-0.630977\pi$
$677$	−20.8134	−0.799925	−0.399962	+	0.916532i	$0.630977\pi$
$678$	0	0
$679$	2.96116	0.113639
$680$	0	0
$681$	0	0
$682$	3.54282	0.135661
$683$	−26.6885	−1.02121	−0.510604	−	0.859816i	$-0.670578\pi$
$683$	−26.6885	−1.02121	−0.510604	+	0.859816i	$0.670578\pi$
$684$	0	0
$685$	0	0
$686$	2.08397	0.0795664
$687$	0	0
$688$	−37.4589	−1.42811
$689$	10.5767	0.402941
$690$	0	0
$691$	−34.2486	−1.30288	−0.651439	−	0.758701i	$-0.725834\pi$
$691$	−34.2486	−1.30288	−0.651439	+	0.758701i	$0.725834\pi$
$692$	−8.33006	−0.316661
$693$	0	0
$694$	6.22363	0.236246
$695$	0	0
$696$	0	0
$697$	−7.05290	−0.267148
$698$	−1.63697	−0.0619601
$699$	0	0
$700$	0	0
$701$	−28.4457	−1.07438	−0.537190	−	0.843461i	$-0.680514\pi$
$701$	−28.4457	−1.07438	−0.537190	+	0.843461i	$0.680514\pi$
$702$	0	0
$703$	−9.95066	−0.375296
$704$	31.4036	1.18357
$705$	0	0
$706$	0.252417	0.00949983
$707$	−1.39479	−0.0524564
$708$	0	0
$709$	−37.4260	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$709$	−37.4260	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$710$	0	0
$711$	0	0
$712$	−3.42417	−0.128326
$713$	−24.6368	−0.922654
$714$	0	0
$715$	0	0
$716$	−45.4902	−1.70005
$717$	0	0
$718$	−0.732094	−0.0273215
$719$	−28.3986	−1.05909	−0.529545	−	0.848282i	$-0.677637\pi$
$719$	−28.3986	−1.05909	−0.529545	+	0.848282i	$0.677637\pi$
$720$	0	0
$721$	−0.510215	−0.0190014
$722$	−0.971933	−0.0361716
$723$	0	0
$724$	−18.9729	−0.705120
$725$	0	0
$726$	0	0
$727$	−3.50646	−0.130047	−0.0650236	−	0.997884i	$-0.520712\pi$
$727$	−3.50646	−0.130047	−0.0650236	+	0.997884i	$0.520712\pi$
$728$	−2.26595	−0.0839816
$729$	0	0
$730$	0	0
$731$	−10.5246	−0.389267
$732$	0	0
$733$	36.8907	1.36259	0.681295	−	0.732009i	$-0.261417\pi$
$733$	36.8907	1.36259	0.681295	+	0.732009i	$0.261417\pi$
$734$	2.63499	0.0972592
$735$	0	0
$736$	14.5656	0.536897
$737$	−18.3793	−0.677012
$738$	0	0
$739$	−12.5756	−0.462603	−0.231301	−	0.972882i	$-0.574298\pi$
$739$	−12.5756	−0.462603	−0.231301	+	0.972882i	$0.574298\pi$
$740$	0	0
$741$	0	0
$742$	0.447090	0.0164132
$743$	36.2579	1.33017	0.665086	−	0.746767i	$-0.268395\pi$
$743$	36.2579	1.33017	0.665086	+	0.746767i	$0.268395\pi$
$744$	0	0
$745$	0	0
$746$	−1.52181	−0.0557173
$747$	0	0
$748$	9.22235	0.337202
$749$	4.84368	0.176984
$750$	0	0
$751$	22.2839	0.813151	0.406576	−	0.913617i	$-0.366723\pi$
$751$	22.2839	0.813151	0.406576	+	0.913617i	$0.366723\pi$
$752$	−47.4792	−1.73139
$753$	0	0
$754$	0.738254	0.0268856
$755$	0	0
$756$	0	0
$757$	22.1631	0.805530	0.402765	−	0.915303i	$-0.368049\pi$
$757$	22.1631	0.805530	0.402765	+	0.915303i	$0.368049\pi$
$758$	−5.59708	−0.203295
$759$	0	0
$760$	0	0
$761$	−43.6078	−1.58078	−0.790391	−	0.612603i	$-0.790123\pi$
$761$	−43.6078	−1.58078	−0.790391	+	0.612603i	$0.790123\pi$
$762$	0	0
$763$	−5.77037	−0.208901
$764$	41.4807	1.50072
$765$	0	0
$766$	2.41660	0.0873155
$767$	−29.1600	−1.05291
$768$	0	0
$769$	−5.84025	−0.210605	−0.105302	−	0.994440i	$-0.533581\pi$
$769$	−5.84025	−0.210605	−0.105302	+	0.994440i	$0.533581\pi$
$770$	0	0
$771$	0	0
$772$	−34.5367	−1.24300
$773$	−19.9643	−0.718067	−0.359034	−	0.933325i	$-0.616894\pi$
$773$	−19.9643	−0.718067	−0.359034	+	0.933325i	$0.616894\pi$
$774$	0	0
$775$	0	0
$776$	3.03520	0.108957
$777$	0	0
$778$	5.83034	0.209028
$779$	−25.1184	−0.899960
$780$	0	0
$781$	60.6525	2.17032
$782$	1.31750	0.0471137
$783$	0	0
$784$	−24.0613	−0.859331
$785$	0	0
$786$	0	0
$787$	−26.1482	−0.932081	−0.466041	−	0.884763i	$-0.654320\pi$
$787$	−26.1482	−0.932081	−0.466041	+	0.884763i	$0.654320\pi$
$788$	−24.7352	−0.881155
$789$	0	0
$790$	0	0
$791$	−16.2984	−0.579503
$792$	0	0
$793$	8.17359	0.290253
$794$	−2.22821	−0.0790763
$795$	0	0
$796$	41.1037	1.45688
$797$	43.5293	1.54189	0.770944	−	0.636903i	$-0.219785\pi$
$797$	43.5293	1.54189	0.770944	+	0.636903i	$0.219785\pi$
$798$	0	0
$799$	−13.3400	−0.471934
$800$	0	0
$801$	0	0
$802$	3.00828	0.106226
$803$	−38.3146	−1.35209
$804$	0	0
$805$	0	0
$806$	2.92632	0.103075
$807$	0	0
$808$	−1.42966	−0.0502954
$809$	−43.8436	−1.54146	−0.770729	−	0.637164i	$-0.780108\pi$
$809$	−43.8436	−1.54146	−0.770729	+	0.637164i	$0.780108\pi$
$810$	0	0
$811$	15.2295	0.534778	0.267389	−	0.963589i	$-0.413839\pi$
$811$	15.2295	0.534778	0.267389	+	0.963589i	$0.413839\pi$
$812$	−1.51906	−0.0533087
$813$	0	0
$814$	2.36381	0.0828516
$815$	0	0
$816$	0	0
$817$	−37.4827	−1.31135
$818$	5.29701	0.185206
$819$	0	0
$820$	0	0
$821$	14.3792	0.501837	0.250919	−	0.968008i	$-0.419267\pi$
$821$	14.3792	0.501837	0.250919	+	0.968008i	$0.419267\pi$
$822$	0	0
$823$	9.68581	0.337626	0.168813	−	0.985648i	$-0.446007\pi$
$823$	9.68581	0.337626	0.168813	+	0.985648i	$0.446007\pi$
$824$	−0.522972	−0.0182186
$825$	0	0
$826$	−1.23263	−0.0428886
$827$	30.2089	1.05047	0.525233	−	0.850958i	$-0.323978\pi$
$827$	30.2089	1.05047	0.525233	+	0.850958i	$0.323978\pi$
$828$	0	0
$829$	38.9917	1.35424	0.677119	−	0.735874i	$-0.263228\pi$
$829$	38.9917	1.35424	0.677119	+	0.735874i	$0.263228\pi$
$830$	0	0
$831$	0	0
$832$	25.9390	0.899272
$833$	−6.76037	−0.234233
$834$	0	0
$835$	0	0
$836$	32.8447	1.13596
$837$	0	0
$838$	−3.04712	−0.105261
$839$	−9.74245	−0.336347	−0.168173	−	0.985757i	$-0.553787\pi$
$839$	−9.74245	−0.336347	−0.168173	+	0.985757i	$0.553787\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0.906541	0.0312415
$843$	0	0
$844$	49.3364	1.69823
$845$	0	0
$846$	0	0
$847$	−6.85399	−0.235506
$848$	−10.8088	−0.371175
$849$	0	0
$850$	0	0
$851$	−16.4380	−0.563486
$852$	0	0
$853$	−51.4478	−1.76154	−0.880769	−	0.473545i	$-0.842974\pi$
$853$	−51.4478	−1.76154	−0.880769	+	0.473545i	$0.842974\pi$
$854$	0.345507	0.0118230
$855$	0	0
$856$	4.96479	0.169693
$857$	−48.8308	−1.66803	−0.834014	−	0.551743i	$-0.813963\pi$
$857$	−48.8308	−1.66803	−0.834014	+	0.551743i	$0.813963\pi$
$858$	0	0
$859$	−16.6720	−0.568841	−0.284421	−	0.958700i	$-0.591801\pi$
$859$	−16.6720	−0.568841	−0.284421	+	0.958700i	$0.591801\pi$
$860$	0	0
$861$	0	0
$862$	0.622783	0.0212121
$863$	2.90194	0.0987831	0.0493915	−	0.998779i	$-0.484272\pi$
$863$	2.90194	0.0987831	0.0493915	+	0.998779i	$0.484272\pi$
$864$	0	0
$865$	0	0
$866$	−1.79061	−0.0608476
$867$	0	0
$868$	−6.02133	−0.204377
$869$	−31.0913	−1.05470
$870$	0	0
$871$	−15.1811	−0.514392
$872$	−5.91465	−0.200295
$873$	0	0
$874$	4.69218	0.158715
$875$	0	0
$876$	0	0
$877$	37.4418	1.26432	0.632160	−	0.774838i	$-0.282169\pi$
$877$	37.4418	1.26432	0.632160	+	0.774838i	$0.282169\pi$
$878$	−6.55843	−0.221336
$879$	0	0
$880$	0	0
$881$	31.8351	1.07255	0.536275	−	0.844043i	$-0.319831\pi$
$881$	31.8351	1.07255	0.536275	+	0.844043i	$0.319831\pi$
$882$	0	0
$883$	16.4903	0.554942	0.277471	−	0.960734i	$-0.410504\pi$
$883$	16.4903	0.554942	0.277471	+	0.960734i	$0.410504\pi$
$884$	7.61754	0.256206
$885$	0	0
$886$	3.00553	0.100973
$887$	−8.78785	−0.295067	−0.147534	−	0.989057i	$-0.547133\pi$
$887$	−8.78785	−0.295067	−0.147534	+	0.989057i	$0.547133\pi$
$888$	0	0
$889$	−9.29039	−0.311590
$890$	0	0
$891$	0	0
$892$	0.442209	0.0148062
$893$	−47.5094	−1.58984
$894$	0	0
$895$	0	0
$896$	4.72951	0.158002
$897$	0	0
$898$	−6.90492	−0.230420
$899$	3.96384	0.132202
$900$	0	0
$901$	−3.03688	−0.101173
$902$	5.96697	0.198678
$903$	0	0
$904$	−16.7059	−0.555630
$905$	0	0
$906$	0	0
$907$	−6.51166	−0.216216	−0.108108	−	0.994139i	$-0.534479\pi$
$907$	−6.51166	−0.216216	−0.108108	+	0.994139i	$0.534479\pi$
$908$	−51.6925	−1.71548
$909$	0	0
$910$	0	0
$911$	22.3746	0.741304	0.370652	−	0.928772i	$-0.379134\pi$
$911$	22.3746	0.741304	0.370652	+	0.928772i	$0.379134\pi$
$912$	0	0
$913$	−12.9570	−0.428814
$914$	−5.96658	−0.197357
$915$	0	0
$916$	6.11900	0.202178
$917$	−10.5718	−0.349113
$918$	0	0
$919$	22.2078	0.732568	0.366284	−	0.930503i	$-0.380630\pi$
$919$	22.2078	0.732568	0.366284	+	0.930503i	$0.380630\pi$
$920$	0	0
$921$	0	0
$922$	1.00673	0.0331548
$923$	50.0982	1.64900
$924$	0	0
$925$	0	0
$926$	3.22679	0.106039
$927$	0	0
$928$	−2.34348	−0.0769287
$929$	35.0239	1.14910	0.574548	−	0.818471i	$-0.305178\pi$
$929$	35.0239	1.14910	0.574548	+	0.818471i	$0.305178\pi$
$930$	0	0
$931$	−24.0766	−0.789077
$932$	−24.4591	−0.801184
$933$	0	0
$934$	4.32274	0.141444
$935$	0	0
$936$	0	0
$937$	−56.4983	−1.84572	−0.922860	−	0.385136i	$-0.874155\pi$
$937$	−56.4983	−1.84572	−0.922860	+	0.385136i	$0.874155\pi$
$938$	−0.641723	−0.0209530
$939$	0	0
$940$	0	0
$941$	39.3736	1.28354	0.641771	−	0.766896i	$-0.278200\pi$
$941$	39.3736	1.28354	0.641771	+	0.766896i	$0.278200\pi$
$942$	0	0
$943$	−41.4943	−1.35124
$944$	29.7998	0.969900
$945$	0	0
$946$	8.90414	0.289499
$947$	−58.9377	−1.91522	−0.957610	−	0.288069i	$-0.906987\pi$
$947$	−58.9377	−1.91522	−0.957610	+	0.288069i	$0.906987\pi$
$948$	0	0
$949$	−31.6474	−1.02732
$950$	0	0
$951$	0	0
$952$	0.650620	0.0210867
$953$	23.3111	0.755120	0.377560	−	0.925985i	$-0.376763\pi$
$953$	23.3111	0.755120	0.377560	+	0.925985i	$0.376763\pi$
$954$	0	0
$955$	0	0
$956$	−24.8228	−0.802828
$957$	0	0
$958$	2.85184	0.0921389
$959$	−4.30930	−0.139155
$960$	0	0
$961$	−15.2880	−0.493160
$962$	1.95248	0.0629505
$963$	0	0
$964$	1.44676	0.0465970
$965$	0	0
$966$	0	0
$967$	−13.7227	−0.441293	−0.220646	−	0.975354i	$-0.570817\pi$
$967$	−13.7227	−0.441293	−0.220646	+	0.975354i	$0.570817\pi$
$968$	−7.02537	−0.225804
$969$	0	0
$970$	0	0
$971$	59.1078	1.89686	0.948430	−	0.316985i	$-0.102671\pi$
$971$	59.1078	1.89686	0.948430	+	0.316985i	$0.102671\pi$
$972$	0	0
$973$	−17.1033	−0.548306
$974$	3.84128	0.123083
$975$	0	0
$976$	−8.35292	−0.267371
$977$	25.0687	0.802018	0.401009	−	0.916074i	$-0.368660\pi$
$977$	25.0687	0.802018	0.401009	+	0.916074i	$0.368660\pi$
$978$	0	0
$979$	−19.1977	−0.613560
$980$	0	0
$981$	0	0
$982$	−6.47480	−0.206619
$983$	−28.3475	−0.904146	−0.452073	−	0.891981i	$-0.649315\pi$
$983$	−28.3475	−0.904146	−0.452073	+	0.891981i	$0.649315\pi$
$984$	0	0
$985$	0	0
$986$	−0.211974	−0.00675064
$987$	0	0
$988$	27.1293	0.863099
$989$	−61.9195	−1.96892
$990$	0	0
$991$	−4.81196	−0.152857	−0.0764285	−	0.997075i	$-0.524352\pi$
$991$	−4.81196	−0.152857	−0.0764285	+	0.997075i	$0.524352\pi$
$992$	−9.28920	−0.294932
$993$	0	0
$994$	2.11771	0.0671696
$995$	0	0
$996$	0	0
$997$	−18.9930	−0.601514	−0.300757	−	0.953701i	$-0.597239\pi$
$997$	−18.9930	−0.601514	−0.300757	+	0.953701i	$0.597239\pi$
$998$	4.11048	0.130115
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6525.2.a.cc.1.5	yes	9
3.2	odd	2		6525.2.a.ca.1.5	✓	9
5.4	even	2		6525.2.a.cb.1.5	yes	9
15.14	odd	2		6525.2.a.cd.1.5	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6525.2.a.ca.1.5	✓	9	3.2	odd	2
6525.2.a.cb.1.5	yes	9	5.4	even	2
6525.2.a.cc.1.5	yes	9	1.1	even	1	trivial
6525.2.a.cd.1.5	yes	9	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6525.a (trivial)

\(f(q)\)	\(=\)	\(q+0.200649 q^{2} -1.95974 q^{4} -0.775135 q^{7} -0.794517 q^{8} +O(q^{10})\)	\(q+0.200649 q^{2} -1.95974 q^{4} -0.775135 q^{7} -0.794517 q^{8} -4.45447 q^{11} -3.67934 q^{13} -0.155530 q^{14} +3.76006 q^{16} +1.05645 q^{17} +3.76245 q^{19} -0.893784 q^{22} +6.21538 q^{23} -0.738254 q^{26} +1.51906 q^{28} -1.00000 q^{29} -3.96384 q^{31} +2.34348 q^{32} +0.211974 q^{34} -2.64473 q^{37} +0.754931 q^{38} -6.67607 q^{41} -9.96230 q^{43} +8.72961 q^{44} +1.24711 q^{46} -12.6272 q^{47} -6.39917 q^{49} +7.21054 q^{52} -2.87463 q^{53} +0.615858 q^{56} -0.200649 q^{58} +7.92534 q^{59} -2.22149 q^{61} -0.795340 q^{62} -7.04991 q^{64} +4.12604 q^{67} -2.07036 q^{68} -13.6161 q^{71} +8.60138 q^{73} -0.530661 q^{74} -7.37343 q^{76} +3.45282 q^{77} +6.97979 q^{79} -1.33955 q^{82} +2.90876 q^{83} -1.99892 q^{86} +3.53915 q^{88} +4.30975 q^{89} +2.85198 q^{91} -12.1805 q^{92} -2.53364 q^{94} -3.82019 q^{97} -1.28398 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 2 q^{2} + 10 q^{4} - q^{7} + 9 q^{8}+O(q^{10}) \) 9 * q + 2 * q^2 + 10 * q^4 - q^7 + 9 * q^8	\( 9 q + 2 q^{2} + 10 q^{4} - q^{7} + 9 q^{8} - 2 q^{11} - q^{13} + 3 q^{14} + 4 q^{16} + 12 q^{17} - q^{19} - 3 q^{22} + 16 q^{23} - 6 q^{26} + 4 q^{28} - 9 q^{29} + 5 q^{31} + 20 q^{32} + 3 q^{34} + 30 q^{38} + 10 q^{41} - 3 q^{43} + 13 q^{44} + 4 q^{46} + 26 q^{47} - 8 q^{49} + 9 q^{52} + 22 q^{53} - 22 q^{56} - 2 q^{58} - 4 q^{59} + 7 q^{61} + 28 q^{62} + 9 q^{64} - 5 q^{67} + 39 q^{68} + 10 q^{73} + 34 q^{74} - 2 q^{76} + 34 q^{77} + 10 q^{79} + 8 q^{82} + 46 q^{83} - 28 q^{86} - 2 q^{88} - 4 q^{89} - 21 q^{91} + 20 q^{92} + 5 q^{94} - 7 q^{97} + 51 q^{98}+O(q^{100}) \) 9 * q + 2 * q^2 + 10 * q^4 - q^7 + 9 * q^8 - 2 * q^11 - q^13 + 3 * q^14 + 4 * q^16 + 12 * q^17 - q^19 - 3 * q^22 + 16 * q^23 - 6 * q^26 + 4 * q^28 - 9 * q^29 + 5 * q^31 + 20 * q^32 + 3 * q^34 + 30 * q^38 + 10 * q^41 - 3 * q^43 + 13 * q^44 + 4 * q^46 + 26 * q^47 - 8 * q^49 + 9 * q^52 + 22 * q^53 - 22 * q^56 - 2 * q^58 - 4 * q^59 + 7 * q^61 + 28 * q^62 + 9 * q^64 - 5 * q^67 + 39 * q^68 + 10 * q^73 + 34 * q^74 - 2 * q^76 + 34 * q^77 + 10 * q^79 + 8 * q^82 + 46 * q^83 - 28 * q^86 - 2 * q^88 - 4 * q^89 - 21 * q^91 + 20 * q^92 + 5 * q^94 - 7 * q^97 + 51 * q^98

Introduction

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