LMFDB - Embedded newform 6525.2.a.ca.1.7

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:	$1$
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			1.7
Root			$-1.10241$ 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)

$f(q)$	$=$	$q+1.10241 q^{2} -0.784687 q^{4} -0.259703 q^{7} -3.06987 q^{8} +O(q^{10})$	$q+1.10241 q^{2} -0.784687 q^{4} -0.259703 q^{7} -3.06987 q^{8} -2.44014 q^{11} +2.69984 q^{13} -0.286300 q^{14} -1.81489 q^{16} -3.91523 q^{17} +6.90606 q^{19} -2.69004 q^{22} -2.05376 q^{23} +2.97634 q^{26} +0.203785 q^{28} +1.00000 q^{29} +7.37721 q^{31} +4.13899 q^{32} -4.31620 q^{34} +0.743697 q^{37} +7.61333 q^{38} -1.04589 q^{41} +9.90608 q^{43} +1.91474 q^{44} -2.26409 q^{46} -2.77675 q^{47} -6.93255 q^{49} -2.11853 q^{52} -12.9332 q^{53} +0.797254 q^{56} +1.10241 q^{58} -14.0907 q^{59} +1.19165 q^{61} +8.13273 q^{62} +8.19266 q^{64} -7.08814 q^{67} +3.07223 q^{68} +10.3985 q^{71} +2.04508 q^{73} +0.819861 q^{74} -5.41909 q^{76} +0.633710 q^{77} -14.7942 q^{79} -1.15300 q^{82} +2.61712 q^{83} +10.9206 q^{86} +7.49091 q^{88} -14.1257 q^{89} -0.701156 q^{91} +1.61156 q^{92} -3.06112 q^{94} -3.01317 q^{97} -7.64254 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.10241	0.779523	0.389762	−	0.920916i	$-0.372557\pi$
$2$	1.10241	0.779523	0.389762	+	0.920916i	$0.372557\pi$
$3$	0	0
$3$	0	0
$4$	−0.784687	−0.392343
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.259703	−0.0981584	−0.0490792	−	0.998795i	$-0.515629\pi$
$7$	−0.259703	−0.0981584	−0.0490792	+	0.998795i	$0.515629\pi$
$8$	−3.06987	−1.08536
$9$	0	0
$10$	0	0
$11$	−2.44014	−0.735729	−0.367865	−	0.929879i	$-0.619911\pi$
$11$	−2.44014	−0.735729	−0.367865	+	0.929879i	$0.619911\pi$
$12$	0	0
$13$	2.69984	0.748801	0.374401	−	0.927267i	$-0.377849\pi$
$13$	2.69984	0.748801	0.374401	+	0.927267i	$0.377849\pi$
$14$	−0.286300	−0.0765168
$15$	0	0
$16$	−1.81489	−0.453723
$17$	−3.91523	−0.949582	−0.474791	−	0.880099i	$-0.657476\pi$
$17$	−3.91523	−0.949582	−0.474791	+	0.880099i	$0.657476\pi$
$18$	0	0
$19$	6.90606	1.58436	0.792179	−	0.610288i	$-0.208946\pi$
$19$	6.90606	1.58436	0.792179	+	0.610288i	$0.208946\pi$
$20$	0	0
$21$	0	0
$22$	−2.69004	−0.573518
$23$	−2.05376	−0.428238	−0.214119	−	0.976808i	$-0.568688\pi$
$23$	−2.05376	−0.428238	−0.214119	+	0.976808i	$0.568688\pi$
$24$	0	0
$25$	0	0
$26$	2.97634	0.583708
$27$	0	0
$28$	0.203785	0.0385118
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	7.37721	1.32499	0.662493	−	0.749068i	$-0.269498\pi$
$31$	7.37721	1.32499	0.662493	+	0.749068i	$0.269498\pi$
$32$	4.13899	0.731676
$33$	0	0
$34$	−4.31620	−0.740221
$35$	0	0
$36$	0	0
$37$	0.743697	0.122263	0.0611315	−	0.998130i	$-0.480529\pi$
$37$	0.743697	0.122263	0.0611315	+	0.998130i	$0.480529\pi$
$38$	7.61333	1.23504
$39$	0	0
$40$	0	0
$41$	−1.04589	−0.163340	−0.0816702	−	0.996659i	$-0.526025\pi$
$41$	−1.04589	−0.163340	−0.0816702	+	0.996659i	$0.526025\pi$
$42$	0	0
$43$	9.90608	1.51066	0.755331	−	0.655343i	$-0.227476\pi$
$43$	9.90608	1.51066	0.755331	+	0.655343i	$0.227476\pi$
$44$	1.91474	0.288658
$45$	0	0
$46$	−2.26409	−0.333822
$47$	−2.77675	−0.405030	−0.202515	−	0.979279i	$-0.564911\pi$
$47$	−2.77675	−0.405030	−0.202515	+	0.979279i	$0.564911\pi$
$48$	0	0
$49$	−6.93255	−0.990365
$50$	0	0
$51$	0	0
$52$	−2.11853	−0.293787
$53$	−12.9332	−1.77651	−0.888257	−	0.459348i	$-0.848083\pi$
$53$	−12.9332	−1.77651	−0.888257	+	0.459348i	$0.848083\pi$
$54$	0	0
$55$	0	0
$56$	0.797254	0.106538
$57$	0	0
$58$	1.10241	0.144754
$59$	−14.0907	−1.83445	−0.917226	−	0.398367i	$-0.869577\pi$
$59$	−14.0907	−1.83445	−0.917226	+	0.398367i	$0.869577\pi$
$60$	0	0
$61$	1.19165	0.152575	0.0762877	−	0.997086i	$-0.475693\pi$
$61$	1.19165	0.152575	0.0762877	+	0.997086i	$0.475693\pi$
$62$	8.13273	1.03286
$63$	0	0
$64$	8.19266	1.02408
$65$	0	0
$66$	0	0
$67$	−7.08814	−0.865954	−0.432977	−	0.901405i	$-0.642537\pi$
$67$	−7.08814	−0.865954	−0.432977	+	0.901405i	$0.642537\pi$
$68$	3.07223	0.372562
$69$	0	0
$70$	0	0
$71$	10.3985	1.23407	0.617036	−	0.786935i	$-0.288333\pi$
$71$	10.3985	1.23407	0.617036	+	0.786935i	$0.288333\pi$
$72$	0	0
$73$	2.04508	0.239359	0.119679	−	0.992813i	$-0.461813\pi$
$73$	2.04508	0.239359	0.119679	+	0.992813i	$0.461813\pi$
$74$	0.819861	0.0953069
$75$	0	0
$76$	−5.41909	−0.621612
$77$	0.633710	0.0722180
$78$	0	0
$79$	−14.7942	−1.66448	−0.832240	−	0.554416i	$-0.812942\pi$
$79$	−14.7942	−1.66448	−0.832240	+	0.554416i	$0.812942\pi$
$80$	0	0
$81$	0	0
$82$	−1.15300	−0.127328
$83$	2.61712	0.287266	0.143633	−	0.989631i	$-0.454122\pi$
$83$	2.61712	0.287266	0.143633	+	0.989631i	$0.454122\pi$
$84$	0	0
$85$	0	0
$86$	10.9206	1.17760
$87$	0	0
$88$	7.49091	0.798534
$89$	−14.1257	−1.49732	−0.748659	−	0.662955i	$-0.769302\pi$
$89$	−14.1257	−1.49732	−0.748659	+	0.662955i	$0.769302\pi$
$90$	0	0
$91$	−0.701156	−0.0735011
$92$	1.61156	0.168016
$93$	0	0
$94$	−3.06112	−0.315730
$95$	0	0
$96$	0	0
$97$	−3.01317	−0.305941	−0.152971	−	0.988231i	$-0.548884\pi$
$97$	−3.01317	−0.305941	−0.152971	+	0.988231i	$0.548884\pi$
$98$	−7.64254	−0.772013
$99$	0	0
$100$	0	0
$101$	7.03538	0.700046	0.350023	−	0.936741i	$-0.386174\pi$
$101$	7.03538	0.700046	0.350023	+	0.936741i	$0.386174\pi$
$102$	0	0
$103$	−4.79874	−0.472834	−0.236417	−	0.971652i	$-0.575973\pi$
$103$	−4.79874	−0.472834	−0.236417	+	0.971652i	$0.575973\pi$
$104$	−8.28817	−0.812722
$105$	0	0
$106$	−14.2577	−1.38483
$107$	−3.46700	−0.335167	−0.167584	−	0.985858i	$-0.553596\pi$
$107$	−3.46700	−0.335167	−0.167584	+	0.985858i	$0.553596\pi$
$108$	0	0
$109$	−13.3130	−1.27515	−0.637575	−	0.770388i	$-0.720063\pi$
$109$	−13.3130	−1.27515	−0.637575	+	0.770388i	$0.720063\pi$
$110$	0	0
$111$	0	0
$112$	0.471333	0.0445368
$113$	−14.8958	−1.40128	−0.700642	−	0.713513i	$-0.747103\pi$
$113$	−14.8958	−1.40128	−0.700642	+	0.713513i	$0.747103\pi$
$114$	0	0
$115$	0	0
$116$	−0.784687	−0.0728563
$117$	0	0
$118$	−15.5338	−1.43000
$119$	1.01680	0.0932094
$120$	0	0
$121$	−5.04573	−0.458703
$122$	1.31369	0.118936
$123$	0	0
$124$	−5.78880	−0.519850
$125$	0	0
$126$	0	0
$127$	−17.6125	−1.56285	−0.781426	−	0.623997i	$-0.785508\pi$
$127$	−17.6125	−1.56285	−0.781426	+	0.623997i	$0.785508\pi$
$128$	0.753718	0.0666199
$129$	0	0
$130$	0	0
$131$	14.9500	1.30619	0.653094	−	0.757277i	$-0.273471\pi$
$131$	14.9500	1.30619	0.653094	+	0.757277i	$0.273471\pi$
$132$	0	0
$133$	−1.79352	−0.155518
$134$	−7.81406	−0.675032
$135$	0	0
$136$	12.0193	1.03064
$137$	−14.0371	−1.19927	−0.599637	−	0.800272i	$-0.704688\pi$
$137$	−14.0371	−1.19927	−0.599637	+	0.800272i	$0.704688\pi$
$138$	0	0
$139$	−12.2747	−1.04112	−0.520562	−	0.853824i	$-0.674277\pi$
$139$	−12.2747	−1.04112	−0.520562	+	0.853824i	$0.674277\pi$
$140$	0	0
$141$	0	0
$142$	11.4634	0.961988
$143$	−6.58798	−0.550915
$144$	0	0
$145$	0	0
$146$	2.25453	0.186586
$147$	0	0
$148$	−0.583569	−0.0479691
$149$	21.1707	1.73437	0.867184	−	0.497987i	$-0.165927\pi$
$149$	21.1707	1.73437	0.867184	+	0.497987i	$0.165927\pi$
$150$	0	0
$151$	−7.24726	−0.589774	−0.294887	−	0.955532i	$-0.595282\pi$
$151$	−7.24726	−0.589774	−0.294887	+	0.955532i	$0.595282\pi$
$152$	−21.2007	−1.71961
$153$	0	0
$154$	0.698610	0.0562956
$155$	0	0
$156$	0	0
$157$	0.639386	0.0510286	0.0255143	−	0.999674i	$-0.491878\pi$
$157$	0.639386	0.0510286	0.0255143	+	0.999674i	$0.491878\pi$
$158$	−16.3093	−1.29750
$159$	0	0
$160$	0	0
$161$	0.533366	0.0420352
$162$	0	0
$163$	16.1593	1.26569	0.632847	−	0.774277i	$-0.281886\pi$
$163$	16.1593	1.26569	0.632847	+	0.774277i	$0.281886\pi$
$164$	0.820695	0.0640855
$165$	0	0
$166$	2.88514	0.223930
$167$	3.58496	0.277413	0.138706	−	0.990334i	$-0.455706\pi$
$167$	3.58496	0.277413	0.138706	+	0.990334i	$0.455706\pi$
$168$	0	0
$169$	−5.71086	−0.439297
$170$	0	0
$171$	0	0
$172$	−7.77317	−0.592698
$173$	−1.44927	−0.110186	−0.0550930	−	0.998481i	$-0.517546\pi$
$173$	−1.44927	−0.110186	−0.0550930	+	0.998481i	$0.517546\pi$
$174$	0	0
$175$	0	0
$176$	4.42859	0.333818
$177$	0	0
$178$	−15.5723	−1.16719
$179$	6.89047	0.515018	0.257509	−	0.966276i	$-0.417098\pi$
$179$	6.89047	0.515018	0.257509	+	0.966276i	$0.417098\pi$
$180$	0	0
$181$	8.43599	0.627042	0.313521	−	0.949581i	$-0.398491\pi$
$181$	8.43599	0.627042	0.313521	+	0.949581i	$0.398491\pi$
$182$	−0.772963	−0.0572958
$183$	0	0
$184$	6.30478	0.464794
$185$	0	0
$186$	0	0
$187$	9.55369	0.698635
$188$	2.17887	0.158911
$189$	0	0
$190$	0	0
$191$	−19.1119	−1.38289	−0.691443	−	0.722431i	$-0.743025\pi$
$191$	−19.1119	−1.38289	−0.691443	+	0.722431i	$0.743025\pi$
$192$	0	0
$193$	26.4643	1.90494	0.952472	−	0.304628i	$-0.0985320\pi$
$193$	26.4643	1.90494	0.952472	+	0.304628i	$0.0985320\pi$
$194$	−3.32176	−0.238489
$195$	0	0
$196$	5.43988	0.388563
$197$	−16.8537	−1.20077	−0.600387	−	0.799710i	$-0.704987\pi$
$197$	−16.8537	−1.20077	−0.600387	+	0.799710i	$0.704987\pi$
$198$	0	0
$199$	13.8664	0.982966	0.491483	−	0.870887i	$-0.336455\pi$
$199$	13.8664	0.982966	0.491483	+	0.870887i	$0.336455\pi$
$200$	0	0
$201$	0	0
$202$	7.75589	0.545703
$203$	−0.259703	−0.0182276
$204$	0	0
$205$	0	0
$206$	−5.29020	−0.368585
$207$	0	0
$208$	−4.89992	−0.339749
$209$	−16.8517	−1.16566
$210$	0	0
$211$	−3.65686	−0.251748	−0.125874	−	0.992046i	$-0.540174\pi$
$211$	−3.65686	−0.251748	−0.125874	+	0.992046i	$0.540174\pi$
$212$	10.1485	0.697003
$213$	0	0
$214$	−3.82206	−0.261271
$215$	0	0
$216$	0	0
$217$	−1.91588	−0.130059
$218$	−14.6764	−0.994010
$219$	0	0
$220$	0	0
$221$	−10.5705	−0.711048
$222$	0	0
$223$	−28.4822	−1.90731	−0.953653	−	0.300908i	$-0.902710\pi$
$223$	−28.4822	−1.90731	−0.953653	+	0.300908i	$0.902710\pi$
$224$	−1.07491	−0.0718202
$225$	0	0
$226$	−16.4214	−1.09233
$227$	21.4599	1.42434	0.712171	−	0.702006i	$-0.247712\pi$
$227$	21.4599	1.42434	0.712171	+	0.702006i	$0.247712\pi$
$228$	0	0
$229$	2.45267	0.162077	0.0810386	−	0.996711i	$-0.474176\pi$
$229$	2.45267	0.162077	0.0810386	+	0.996711i	$0.474176\pi$
$230$	0	0
$231$	0	0
$232$	−3.06987	−0.201547
$233$	−0.132148	−0.00865732	−0.00432866	−	0.999991i	$-0.501378\pi$
$233$	−0.132148	−0.00865732	−0.00432866	+	0.999991i	$0.501378\pi$
$234$	0	0
$235$	0	0
$236$	11.0568	0.719735
$237$	0	0
$238$	1.12093	0.0726589
$239$	−14.0167	−0.906664	−0.453332	−	0.891342i	$-0.649765\pi$
$239$	−14.0167	−0.906664	−0.453332	+	0.891342i	$0.649765\pi$
$240$	0	0
$241$	−21.8801	−1.40942	−0.704710	−	0.709495i	$-0.748923\pi$
$241$	−21.8801	−1.40942	−0.704710	+	0.709495i	$0.748923\pi$
$242$	−5.56247	−0.357569
$243$	0	0
$244$	−0.935073	−0.0598619
$245$	0	0
$246$	0	0
$247$	18.6453	1.18637
$248$	−22.6471	−1.43809
$249$	0	0
$250$	0	0
$251$	3.58156	0.226066	0.113033	−	0.993591i	$-0.463943\pi$
$251$	3.58156	0.226066	0.113033	+	0.993591i	$0.463943\pi$
$252$	0	0
$253$	5.01145	0.315067
$254$	−19.4162	−1.21828
$255$	0	0
$256$	−15.5544	−0.972150
$257$	−18.1721	−1.13354	−0.566772	−	0.823874i	$-0.691808\pi$
$257$	−18.1721	−1.13354	−0.566772	+	0.823874i	$0.691808\pi$
$258$	0	0
$259$	−0.193140	−0.0120011
$260$	0	0
$261$	0	0
$262$	16.4811	1.01820
$263$	1.04165	0.0642310	0.0321155	−	0.999484i	$-0.489776\pi$
$263$	1.04165	0.0642310	0.0321155	+	0.999484i	$0.489776\pi$
$264$	0	0
$265$	0	0
$266$	−1.97720	−0.121230
$267$	0	0
$268$	5.56197	0.339751
$269$	7.41914	0.452353	0.226177	−	0.974086i	$-0.427377\pi$
$269$	7.41914	0.452353	0.226177	+	0.974086i	$0.427377\pi$
$270$	0	0
$271$	−4.59362	−0.279042	−0.139521	−	0.990219i	$-0.544556\pi$
$271$	−4.59362	−0.279042	−0.139521	+	0.990219i	$0.544556\pi$
$272$	7.10572	0.430848
$273$	0	0
$274$	−15.4747	−0.934861
$275$	0	0
$276$	0	0
$277$	−28.8324	−1.73237	−0.866187	−	0.499721i	$-0.833436\pi$
$277$	−28.8324	−1.73237	−0.866187	+	0.499721i	$0.833436\pi$
$278$	−13.5317	−0.811580
$279$	0	0
$280$	0	0
$281$	16.0972	0.960279	0.480139	−	0.877192i	$-0.340586\pi$
$281$	16.0972	0.960279	0.480139	+	0.877192i	$0.340586\pi$
$282$	0	0
$283$	−14.7487	−0.876718	−0.438359	−	0.898800i	$-0.644440\pi$
$283$	−14.7487	−0.876718	−0.438359	+	0.898800i	$0.644440\pi$
$284$	−8.15954	−0.484180
$285$	0	0
$286$	−7.26267	−0.429451
$287$	0.271620	0.0160332
$288$	0	0
$289$	−1.67100	−0.0982940
$290$	0	0
$291$	0	0
$292$	−1.60475	−0.0939109
$293$	3.95227	0.230894	0.115447	−	0.993314i	$-0.463170\pi$
$293$	3.95227	0.230894	0.115447	+	0.993314i	$0.463170\pi$
$294$	0	0
$295$	0	0
$296$	−2.28306	−0.132700
$297$	0	0
$298$	23.3388	1.35198
$299$	−5.54482	−0.320665
$300$	0	0
$301$	−2.57264	−0.148284
$302$	−7.98947	−0.459742
$303$	0	0
$304$	−12.5338	−0.718861
$305$	0	0
$306$	0	0
$307$	−13.6414	−0.778558	−0.389279	−	0.921120i	$-0.627276\pi$
$307$	−13.6414	−0.778558	−0.389279	+	0.921120i	$0.627276\pi$
$308$	−0.497264	−0.0283342
$309$	0	0
$310$	0	0
$311$	23.3238	1.32257	0.661287	−	0.750133i	$-0.270011\pi$
$311$	23.3238	1.32257	0.661287	+	0.750133i	$0.270011\pi$
$312$	0	0
$313$	32.2649	1.82372	0.911861	−	0.410499i	$-0.134645\pi$
$313$	32.2649	1.82372	0.911861	+	0.410499i	$0.134645\pi$
$314$	0.704868	0.0397780
$315$	0	0
$316$	11.6088	0.653047
$317$	−12.8351	−0.720889	−0.360445	−	0.932781i	$-0.617375\pi$
$317$	−12.8351	−0.720889	−0.360445	+	0.932781i	$0.617375\pi$
$318$	0	0
$319$	−2.44014	−0.136621
$320$	0	0
$321$	0	0
$322$	0.587990	0.0327674
$323$	−27.0388	−1.50448
$324$	0	0
$325$	0	0
$326$	17.8142	0.986638
$327$	0	0
$328$	3.21075	0.177284
$329$	0.721128	0.0397571
$330$	0	0
$331$	24.8409	1.36538	0.682690	−	0.730708i	$-0.260810\pi$
$331$	24.8409	1.36538	0.682690	+	0.730708i	$0.260810\pi$
$332$	−2.05362	−0.112707
$333$	0	0
$334$	3.95211	0.216250
$335$	0	0
$336$	0	0
$337$	20.0660	1.09306	0.546532	−	0.837438i	$-0.315948\pi$
$337$	20.0660	1.09306	0.546532	+	0.837438i	$0.315948\pi$
$338$	−6.29573	−0.342442
$339$	0	0
$340$	0	0
$341$	−18.0014	−0.974831
$342$	0	0
$343$	3.61832	0.195371
$344$	−30.4104	−1.63962
$345$	0	0
$346$	−1.59769	−0.0858925
$347$	−20.5190	−1.10152	−0.550758	−	0.834665i	$-0.685661\pi$
$347$	−20.5190	−1.10152	−0.550758	+	0.834665i	$0.685661\pi$
$348$	0	0
$349$	11.9755	0.641035	0.320518	−	0.947243i	$-0.396143\pi$
$349$	11.9755	0.641035	0.320518	+	0.947243i	$0.396143\pi$
$350$	0	0
$351$	0	0
$352$	−10.0997	−0.538315
$353$	−21.1389	−1.12511	−0.562554	−	0.826761i	$-0.690181\pi$
$353$	−21.1389	−1.12511	−0.562554	+	0.826761i	$0.690181\pi$
$354$	0	0
$355$	0	0
$356$	11.0842	0.587463
$357$	0	0
$358$	7.59614	0.401468
$359$	−28.3132	−1.49432	−0.747158	−	0.664647i	$-0.768582\pi$
$359$	−28.3132	−1.49432	−0.747158	+	0.664647i	$0.768582\pi$
$360$	0	0
$361$	28.6936	1.51019
$362$	9.29994	0.488794
$363$	0	0
$364$	0.550188	0.0288377
$365$	0	0
$366$	0	0
$367$	−1.41897	−0.0740694	−0.0370347	−	0.999314i	$-0.511791\pi$
$367$	−1.41897	−0.0740694	−0.0370347	+	0.999314i	$0.511791\pi$
$368$	3.72735	0.194302
$369$	0	0
$370$	0	0
$371$	3.35879	0.174380
$372$	0	0
$373$	−8.75979	−0.453565	−0.226782	−	0.973945i	$-0.572821\pi$
$373$	−8.75979	−0.453565	−0.226782	+	0.973945i	$0.572821\pi$
$374$	10.5321	0.544602
$375$	0	0
$376$	8.52426	0.439605
$377$	2.69984	0.139049
$378$	0	0
$379$	−1.12506	−0.0577903	−0.0288951	−	0.999582i	$-0.509199\pi$
$379$	−1.12506	−0.0577903	−0.0288951	+	0.999582i	$0.509199\pi$
$380$	0	0
$381$	0	0
$382$	−21.0692	−1.07799
$383$	−24.5642	−1.25517	−0.627586	−	0.778547i	$-0.715957\pi$
$383$	−24.5642	−1.25517	−0.627586	+	0.778547i	$0.715957\pi$
$384$	0	0
$385$	0	0
$386$	29.1746	1.48495
$387$	0	0
$388$	2.36440	0.120034
$389$	23.6020	1.19667	0.598333	−	0.801247i	$-0.295830\pi$
$389$	23.6020	1.19667	0.598333	+	0.801247i	$0.295830\pi$
$390$	0	0
$391$	8.04093	0.406647
$392$	21.2821	1.07491
$393$	0	0
$394$	−18.5797	−0.936031
$395$	0	0
$396$	0	0
$397$	−19.6694	−0.987177	−0.493589	−	0.869696i	$-0.664315\pi$
$397$	−19.6694	−0.987177	−0.493589	+	0.869696i	$0.664315\pi$
$398$	15.2865	0.766245
$399$	0	0
$400$	0	0
$401$	9.67611	0.483202	0.241601	−	0.970376i	$-0.422327\pi$
$401$	9.67611	0.483202	0.241601	+	0.970376i	$0.422327\pi$
$402$	0	0
$403$	19.9173	0.992151
$404$	−5.52057	−0.274659
$405$	0	0
$406$	−0.286300	−0.0142088
$407$	−1.81472	−0.0899525
$408$	0	0
$409$	17.3784	0.859306	0.429653	−	0.902994i	$-0.358636\pi$
$409$	17.3784	0.859306	0.429653	+	0.902994i	$0.358636\pi$
$410$	0	0
$411$	0	0
$412$	3.76551	0.185513
$413$	3.65939	0.180067
$414$	0	0
$415$	0	0
$416$	11.1746	0.547880
$417$	0	0
$418$	−18.5776	−0.908658
$419$	21.0717	1.02942	0.514710	−	0.857365i	$-0.327900\pi$
$419$	21.0717	1.02942	0.514710	+	0.857365i	$0.327900\pi$
$420$	0	0
$421$	−1.50826	−0.0735082	−0.0367541	−	0.999324i	$-0.511702\pi$
$421$	−1.50826	−0.0735082	−0.0367541	+	0.999324i	$0.511702\pi$
$422$	−4.03137	−0.196244
$423$	0	0
$424$	39.7033	1.92816
$425$	0	0
$426$	0	0
$427$	−0.309475	−0.0149766
$428$	2.72051	0.131501
$429$	0	0
$430$	0	0
$431$	−18.3955	−0.886081	−0.443041	−	0.896502i	$-0.646100\pi$
$431$	−18.3955	−0.886081	−0.443041	+	0.896502i	$0.646100\pi$
$432$	0	0
$433$	3.48458	0.167458	0.0837291	−	0.996489i	$-0.473317\pi$
$433$	3.48458	0.167458	0.0837291	+	0.996489i	$0.473317\pi$
$434$	−2.11209	−0.101384
$435$	0	0
$436$	10.4465	0.500297
$437$	−14.1834	−0.678483
$438$	0	0
$439$	11.7983	0.563103	0.281552	−	0.959546i	$-0.409151\pi$
$439$	11.7983	0.563103	0.281552	+	0.959546i	$0.409151\pi$
$440$	0	0
$441$	0	0
$442$	−11.6530	−0.554279
$443$	5.61507	0.266780	0.133390	−	0.991064i	$-0.457414\pi$
$443$	5.61507	0.266780	0.133390	+	0.991064i	$0.457414\pi$
$444$	0	0
$445$	0	0
$446$	−31.3991	−1.48679
$447$	0	0
$448$	−2.12766	−0.100522
$449$	−6.76532	−0.319275	−0.159638	−	0.987176i	$-0.551033\pi$
$449$	−6.76532	−0.319275	−0.159638	+	0.987176i	$0.551033\pi$
$450$	0	0
$451$	2.55211	0.120174
$452$	11.6886	0.549784
$453$	0	0
$454$	23.6576	1.11031
$455$	0	0
$456$	0	0
$457$	−23.7375	−1.11039	−0.555196	−	0.831719i	$-0.687357\pi$
$457$	−23.7375	−1.11039	−0.555196	+	0.831719i	$0.687357\pi$
$458$	2.70386	0.126343
$459$	0	0
$460$	0	0
$461$	−5.34778	−0.249071	−0.124536	−	0.992215i	$-0.539744\pi$
$461$	−5.34778	−0.249071	−0.124536	+	0.992215i	$0.539744\pi$
$462$	0	0
$463$	−35.7780	−1.66275	−0.831373	−	0.555715i	$-0.812444\pi$
$463$	−35.7780	−1.66275	−0.831373	+	0.555715i	$0.812444\pi$
$464$	−1.81489	−0.0842543
$465$	0	0
$466$	−0.145682	−0.00674859
$467$	−21.7018	−1.00424	−0.502119	−	0.864799i	$-0.667446\pi$
$467$	−21.7018	−1.00424	−0.502119	+	0.864799i	$0.667446\pi$
$468$	0	0
$469$	1.84081	0.0850007
$470$	0	0
$471$	0	0
$472$	43.2566	1.99105
$473$	−24.1722	−1.11144
$474$	0	0
$475$	0	0
$476$	−0.797865	−0.0365701
$477$	0	0
$478$	−15.4522	−0.706766
$479$	2.64777	0.120980	0.0604899	−	0.998169i	$-0.480734\pi$
$479$	2.64777	0.120980	0.0604899	+	0.998169i	$0.480734\pi$
$480$	0	0
$481$	2.00786	0.0915507
$482$	−24.1209	−1.09868
$483$	0	0
$484$	3.95931	0.179969
$485$	0	0
$486$	0	0
$487$	22.7956	1.03297	0.516484	−	0.856297i	$-0.327240\pi$
$487$	22.7956	1.03297	0.516484	+	0.856297i	$0.327240\pi$
$488$	−3.65822	−0.165600
$489$	0	0
$490$	0	0
$491$	−15.3430	−0.692419	−0.346209	−	0.938157i	$-0.612531\pi$
$491$	−15.3430	−0.692419	−0.346209	+	0.938157i	$0.612531\pi$
$492$	0	0
$493$	−3.91523	−0.176333
$494$	20.5548	0.924803
$495$	0	0
$496$	−13.3889	−0.601177
$497$	−2.70051	−0.121135
$498$	0	0
$499$	−15.6891	−0.702338	−0.351169	−	0.936312i	$-0.614216\pi$
$499$	−15.6891	−0.702338	−0.351169	+	0.936312i	$0.614216\pi$
$500$	0	0
$501$	0	0
$502$	3.94836	0.176224
$503$	−23.9984	−1.07003	−0.535017	−	0.844841i	$-0.679695\pi$
$503$	−23.9984	−1.07003	−0.535017	+	0.844841i	$0.679695\pi$
$504$	0	0
$505$	0	0
$506$	5.52469	0.245602
$507$	0	0
$508$	13.8203	0.613175
$509$	36.6038	1.62243	0.811217	−	0.584746i	$-0.198806\pi$
$509$	36.6038	1.62243	0.811217	+	0.584746i	$0.198806\pi$
$510$	0	0
$511$	−0.531114	−0.0234951
$512$	−18.6548	−0.824434
$513$	0	0
$514$	−20.0332	−0.883625
$515$	0	0
$516$	0	0
$517$	6.77564	0.297992
$518$	−0.212920	−0.00935517
$519$	0	0
$520$	0	0
$521$	21.7123	0.951232	0.475616	−	0.879653i	$-0.342225\pi$
$521$	21.7123	0.951232	0.475616	+	0.879653i	$0.342225\pi$
$522$	0	0
$523$	−8.97941	−0.392642	−0.196321	−	0.980540i	$-0.562900\pi$
$523$	−8.97941	−0.392642	−0.196321	+	0.980540i	$0.562900\pi$
$524$	−11.7311	−0.512474
$525$	0	0
$526$	1.14833	0.0500696
$527$	−28.8835	−1.25818
$528$	0	0
$529$	−18.7821	−0.816612
$530$	0	0
$531$	0	0
$532$	1.40735	0.0610165
$533$	−2.82373	−0.122310
$534$	0	0
$535$	0	0
$536$	21.7597	0.939876
$537$	0	0
$538$	8.17896	0.352620
$539$	16.9164	0.728640
$540$	0	0
$541$	14.7615	0.634646	0.317323	−	0.948318i	$-0.397216\pi$
$541$	14.7615	0.634646	0.317323	+	0.948318i	$0.397216\pi$
$542$	−5.06406	−0.217520
$543$	0	0
$544$	−16.2051	−0.694786
$545$	0	0
$546$	0	0
$547$	0.521793	0.0223102	0.0111551	−	0.999938i	$-0.496449\pi$
$547$	0.521793	0.0223102	0.0111551	+	0.999938i	$0.496449\pi$
$548$	11.0147	0.470527
$549$	0	0
$550$	0	0
$551$	6.90606	0.294208
$552$	0	0
$553$	3.84210	0.163383
$554$	−31.7852	−1.35043
$555$	0	0
$556$	9.63176	0.408478
$557$	16.1158	0.682849	0.341425	−	0.939909i	$-0.389091\pi$
$557$	16.1158	0.682849	0.341425	+	0.939909i	$0.389091\pi$
$558$	0	0
$559$	26.7448	1.13119
$560$	0	0
$561$	0	0
$562$	17.7458	0.748560
$563$	−41.5917	−1.75288	−0.876440	−	0.481510i	$-0.840088\pi$
$563$	−41.5917	−1.75288	−0.876440	+	0.481510i	$0.840088\pi$
$564$	0	0
$565$	0	0
$566$	−16.2591	−0.683422
$567$	0	0
$568$	−31.9220	−1.33942
$569$	12.6935	0.532139	0.266069	−	0.963954i	$-0.414275\pi$
$569$	12.6935	0.532139	0.266069	+	0.963954i	$0.414275\pi$
$570$	0	0
$571$	7.51726	0.314588	0.157294	−	0.987552i	$-0.449723\pi$
$571$	7.51726	0.314588	0.157294	+	0.987552i	$0.449723\pi$
$572$	5.16950	0.216148
$573$	0	0
$574$	0.299438	0.0124983
$575$	0	0
$576$	0	0
$577$	20.8973	0.869966	0.434983	−	0.900439i	$-0.356754\pi$
$577$	20.8973	0.869966	0.434983	+	0.900439i	$0.356754\pi$
$578$	−1.84213	−0.0766225
$579$	0	0
$580$	0	0
$581$	−0.679672	−0.0281975
$582$	0	0
$583$	31.5588	1.30703
$584$	−6.27815	−0.259792
$585$	0	0
$586$	4.35703	0.179987
$587$	13.4075	0.553388	0.276694	−	0.960958i	$-0.410761\pi$
$587$	13.4075	0.553388	0.276694	+	0.960958i	$0.410761\pi$
$588$	0	0
$589$	50.9475	2.09925
$590$	0	0
$591$	0	0
$592$	−1.34973	−0.0554736
$593$	45.4648	1.86701	0.933507	−	0.358560i	$-0.116732\pi$
$593$	45.4648	1.86701	0.933507	+	0.358560i	$0.116732\pi$
$594$	0	0
$595$	0	0
$596$	−16.6123	−0.680468
$597$	0	0
$598$	−6.11268	−0.249966
$599$	−39.8389	−1.62777	−0.813886	−	0.581025i	$-0.802652\pi$
$599$	−39.8389	−1.62777	−0.813886	+	0.581025i	$0.802652\pi$
$600$	0	0
$601$	3.93631	0.160566	0.0802828	−	0.996772i	$-0.474418\pi$
$601$	3.93631	0.160566	0.0802828	+	0.996772i	$0.474418\pi$
$602$	−2.83611	−0.115591
$603$	0	0
$604$	5.68683	0.231394
$605$	0	0
$606$	0	0
$607$	−20.9392	−0.849895	−0.424948	−	0.905218i	$-0.639707\pi$
$607$	−20.9392	−0.849895	−0.424948	+	0.905218i	$0.639707\pi$
$608$	28.5841	1.15924
$609$	0	0
$610$	0	0
$611$	−7.49677	−0.303287
$612$	0	0
$613$	−21.7071	−0.876743	−0.438372	−	0.898794i	$-0.644445\pi$
$613$	−21.7071	−0.876743	−0.438372	+	0.898794i	$0.644445\pi$
$614$	−15.0385	−0.606904
$615$	0	0
$616$	−1.94541	−0.0783828
$617$	−41.3694	−1.66547	−0.832735	−	0.553672i	$-0.813226\pi$
$617$	−41.3694	−1.66547	−0.832735	+	0.553672i	$0.813226\pi$
$618$	0	0
$619$	20.2608	0.814350	0.407175	−	0.913350i	$-0.366514\pi$
$619$	20.2608	0.814350	0.407175	+	0.913350i	$0.366514\pi$
$620$	0	0
$621$	0	0
$622$	25.7125	1.03098
$623$	3.66847	0.146974
$624$	0	0
$625$	0	0
$626$	35.5693	1.42163
$627$	0	0
$628$	−0.501718	−0.0200207
$629$	−2.91174	−0.116099
$630$	0	0
$631$	36.5698	1.45582	0.727911	−	0.685671i	$-0.240491\pi$
$631$	36.5698	1.45582	0.727911	+	0.685671i	$0.240491\pi$
$632$	45.4164	1.80657
$633$	0	0
$634$	−14.1495	−0.561950
$635$	0	0
$636$	0	0
$637$	−18.7168	−0.741586
$638$	−2.69004	−0.106500
$639$	0	0
$640$	0	0
$641$	47.7470	1.88589	0.942946	−	0.332946i	$-0.108043\pi$
$641$	47.7470	1.88589	0.942946	+	0.332946i	$0.108043\pi$
$642$	0	0
$643$	−33.8571	−1.33519	−0.667596	−	0.744524i	$-0.732677\pi$
$643$	−33.8571	−1.33519	−0.667596	+	0.744524i	$0.732677\pi$
$644$	−0.418525	−0.0164922
$645$	0	0
$646$	−29.8079	−1.17278
$647$	28.7777	1.13137	0.565683	−	0.824623i	$-0.308613\pi$
$647$	28.7777	1.13137	0.565683	+	0.824623i	$0.308613\pi$
$648$	0	0
$649$	34.3832	1.34966
$650$	0	0
$651$	0	0
$652$	−12.6800	−0.496587
$653$	25.3565	0.992277	0.496138	−	0.868243i	$-0.334751\pi$
$653$	25.3565	0.992277	0.496138	+	0.868243i	$0.334751\pi$
$654$	0	0
$655$	0	0
$656$	1.89818	0.0741114
$657$	0	0
$658$	0.794981	0.0309916
$659$	−7.28954	−0.283960	−0.141980	−	0.989870i	$-0.545347\pi$
$659$	−7.28954	−0.283960	−0.141980	+	0.989870i	$0.545347\pi$
$660$	0	0
$661$	−21.1056	−0.820913	−0.410457	−	0.911880i	$-0.634631\pi$
$661$	−21.1056	−0.820913	−0.410457	+	0.911880i	$0.634631\pi$
$662$	27.3849	1.06435
$663$	0	0
$664$	−8.03421	−0.311788
$665$	0	0
$666$	0	0
$667$	−2.05376	−0.0795218
$668$	−2.81307	−0.108841
$669$	0	0
$670$	0	0
$671$	−2.90779	−0.112254
$672$	0	0
$673$	−16.0938	−0.620369	−0.310184	−	0.950676i	$-0.600391\pi$
$673$	−16.0938	−0.620369	−0.310184	+	0.950676i	$0.600391\pi$
$674$	22.1210	0.852068
$675$	0	0
$676$	4.48124	0.172355
$677$	−8.21364	−0.315676	−0.157838	−	0.987465i	$-0.550452\pi$
$677$	−8.21364	−0.315676	−0.157838	+	0.987465i	$0.550452\pi$
$678$	0	0
$679$	0.782529	0.0300307
$680$	0	0
$681$	0	0
$682$	−19.8450	−0.759904
$683$	−38.0962	−1.45771	−0.728856	−	0.684667i	$-0.759948\pi$
$683$	−38.0962	−1.45771	−0.728856	+	0.684667i	$0.759948\pi$
$684$	0	0
$685$	0	0
$686$	3.98888	0.152296
$687$	0	0
$688$	−17.9785	−0.685423
$689$	−34.9176	−1.33025
$690$	0	0
$691$	33.5040	1.27455	0.637276	−	0.770635i	$-0.280061\pi$
$691$	33.5040	1.27455	0.637276	+	0.770635i	$0.280061\pi$
$692$	1.13722	0.0432307
$693$	0	0
$694$	−22.6204	−0.858657
$695$	0	0
$696$	0	0
$697$	4.09489	0.155105
$698$	13.2020	0.499702
$699$	0	0
$700$	0	0
$701$	−43.0350	−1.62541	−0.812704	−	0.582676i	$-0.802006\pi$
$701$	−43.0350	−1.62541	−0.812704	+	0.582676i	$0.802006\pi$
$702$	0	0
$703$	5.13601	0.193708
$704$	−19.9912	−0.753447
$705$	0	0
$706$	−23.3037	−0.877048
$707$	−1.82711	−0.0687154
$708$	0	0
$709$	18.9752	0.712629	0.356315	−	0.934366i	$-0.384033\pi$
$709$	18.9752	0.712629	0.356315	+	0.934366i	$0.384033\pi$
$710$	0	0
$711$	0	0
$712$	43.3640	1.62514
$713$	−15.1510	−0.567410
$714$	0	0
$715$	0	0
$716$	−5.40686	−0.202064
$717$	0	0
$718$	−31.2129	−1.16485
$719$	45.3657	1.69186	0.845928	−	0.533297i	$-0.179047\pi$
$719$	45.3657	1.69186	0.845928	+	0.533297i	$0.179047\pi$
$720$	0	0
$721$	1.24625	0.0464127
$722$	31.6322	1.17723
$723$	0	0
$724$	−6.61961	−0.246016
$725$	0	0
$726$	0	0
$727$	52.7459	1.95624	0.978118	−	0.208051i	$-0.0667120\pi$
$727$	52.7459	1.95624	0.978118	+	0.208051i	$0.0667120\pi$
$728$	2.15246	0.0797755
$729$	0	0
$730$	0	0
$731$	−38.7845	−1.43450
$732$	0	0
$733$	0.902479	0.0333338	0.0166669	−	0.999861i	$-0.494695\pi$
$733$	0.902479	0.0333338	0.0166669	+	0.999861i	$0.494695\pi$
$734$	−1.56429	−0.0577388
$735$	0	0
$736$	−8.50047	−0.313332
$737$	17.2960	0.637108
$738$	0	0
$739$	2.20398	0.0810748	0.0405374	−	0.999178i	$-0.487093\pi$
$739$	2.20398	0.0810748	0.0405374	+	0.999178i	$0.487093\pi$
$740$	0	0
$741$	0	0
$742$	3.70277	0.135933
$743$	−27.9539	−1.02553	−0.512765	−	0.858529i	$-0.671379\pi$
$743$	−27.9539	−1.02553	−0.512765	+	0.858529i	$0.671379\pi$
$744$	0	0
$745$	0	0
$746$	−9.65691	−0.353564
$747$	0	0
$748$	−7.49665	−0.274105
$749$	0.900389	0.0328995
$750$	0	0
$751$	−15.0939	−0.550785	−0.275392	−	0.961332i	$-0.588808\pi$
$751$	−15.0939	−0.550785	−0.275392	+	0.961332i	$0.588808\pi$
$752$	5.03950	0.183772
$753$	0	0
$754$	2.97634	0.108392
$755$	0	0
$756$	0	0
$757$	−20.9780	−0.762459	−0.381229	−	0.924481i	$-0.624499\pi$
$757$	−20.9780	−0.762459	−0.381229	+	0.924481i	$0.624499\pi$
$758$	−1.24028	−0.0450489
$759$	0	0
$760$	0	0
$761$	−24.5060	−0.888341	−0.444171	−	0.895942i	$-0.646502\pi$
$761$	−24.5060	−0.888341	−0.444171	+	0.895942i	$0.646502\pi$
$762$	0	0
$763$	3.45741	0.125167
$764$	14.9968	0.542566
$765$	0	0
$766$	−27.0799	−0.978436
$767$	−38.0426	−1.37364
$768$	0	0
$769$	−17.4343	−0.628698	−0.314349	−	0.949307i	$-0.601786\pi$
$769$	−17.4343	−0.628698	−0.314349	+	0.949307i	$0.601786\pi$
$770$	0	0
$771$	0	0
$772$	−20.7662	−0.747392
$773$	32.1687	1.15703	0.578514	−	0.815672i	$-0.303633\pi$
$773$	32.1687	1.15703	0.578514	+	0.815672i	$0.303633\pi$
$774$	0	0
$775$	0	0
$776$	9.25006	0.332058
$777$	0	0
$778$	26.0191	0.932830
$779$	−7.22297	−0.258790
$780$	0	0
$781$	−25.3737	−0.907943
$782$	8.86442	0.316991
$783$	0	0
$784$	12.5819	0.449352
$785$	0	0
$786$	0	0
$787$	−10.9274	−0.389519	−0.194760	−	0.980851i	$-0.562393\pi$
$787$	−10.9274	−0.389519	−0.194760	+	0.980851i	$0.562393\pi$
$788$	13.2248	0.471115
$789$	0	0
$790$	0	0
$791$	3.86849	0.137548
$792$	0	0
$793$	3.21727	0.114249
$794$	−21.6838	−0.769528
$795$	0	0
$796$	−10.8808	−0.385660
$797$	36.1564	1.28072	0.640362	−	0.768073i	$-0.278784\pi$
$797$	36.1564	1.28072	0.640362	+	0.768073i	$0.278784\pi$
$798$	0	0
$799$	10.8716	0.384609
$800$	0	0
$801$	0	0
$802$	10.6671	0.376667
$803$	−4.99029	−0.176103
$804$	0	0
$805$	0	0
$806$	21.9571	0.773405
$807$	0	0
$808$	−21.5977	−0.759805
$809$	5.45860	0.191914	0.0959570	−	0.995385i	$-0.469409\pi$
$809$	5.45860	0.191914	0.0959570	+	0.995385i	$0.469409\pi$
$810$	0	0
$811$	9.26041	0.325177	0.162588	−	0.986694i	$-0.448016\pi$
$811$	9.26041	0.325177	0.162588	+	0.986694i	$0.448016\pi$
$812$	0.203785	0.00715146
$813$	0	0
$814$	−2.00057	−0.0701201
$815$	0	0
$816$	0	0
$817$	68.4119	2.39343
$818$	19.1582	0.669849
$819$	0	0
$820$	0	0
$821$	−23.2683	−0.812070	−0.406035	−	0.913857i	$-0.633089\pi$
$821$	−23.2683	−0.812070	−0.406035	+	0.913857i	$0.633089\pi$
$822$	0	0
$823$	−8.10683	−0.282586	−0.141293	−	0.989968i	$-0.545126\pi$
$823$	−8.10683	−0.282586	−0.141293	+	0.989968i	$0.545126\pi$
$824$	14.7315	0.513197
$825$	0	0
$826$	4.03416	0.140366
$827$	−17.6578	−0.614020	−0.307010	−	0.951706i	$-0.599329\pi$
$827$	−17.6578	−0.614020	−0.307010	+	0.951706i	$0.599329\pi$
$828$	0	0
$829$	−18.2623	−0.634277	−0.317139	−	0.948379i	$-0.602722\pi$
$829$	−18.2623	−0.634277	−0.317139	+	0.948379i	$0.602722\pi$
$830$	0	0
$831$	0	0
$832$	22.1189	0.766834
$833$	27.1425	0.940433
$834$	0	0
$835$	0	0
$836$	13.2233	0.457338
$837$	0	0
$838$	23.2297	0.802456
$839$	−54.3175	−1.87525	−0.937625	−	0.347649i	$-0.886980\pi$
$839$	−54.3175	−1.87525	−0.937625	+	0.347649i	$0.886980\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−1.66273	−0.0573014
$843$	0	0
$844$	2.86949	0.0987718
$845$	0	0
$846$	0	0
$847$	1.31039	0.0450255
$848$	23.4724	0.806046
$849$	0	0
$850$	0	0
$851$	−1.52737	−0.0523577
$852$	0	0
$853$	13.7415	0.470501	0.235250	−	0.971935i	$-0.424409\pi$
$853$	13.7415	0.470501	0.235250	+	0.971935i	$0.424409\pi$
$854$	−0.341169	−0.0116746
$855$	0	0
$856$	10.6432	0.363779
$857$	44.3493	1.51494	0.757472	−	0.652868i	$-0.226434\pi$
$857$	44.3493	1.51494	0.757472	+	0.652868i	$0.226434\pi$
$858$	0	0
$859$	−13.0913	−0.446671	−0.223335	−	0.974742i	$-0.571694\pi$
$859$	−13.0913	−0.446671	−0.223335	+	0.974742i	$0.571694\pi$
$860$	0	0
$861$	0	0
$862$	−20.2795	−0.690721
$863$	5.94858	0.202492	0.101246	−	0.994861i	$-0.467717\pi$
$863$	5.94858	0.202492	0.101246	+	0.994861i	$0.467717\pi$
$864$	0	0
$865$	0	0
$866$	3.84145	0.130538
$867$	0	0
$868$	1.50337	0.0510276
$869$	36.0999	1.22461
$870$	0	0
$871$	−19.1369	−0.648428
$872$	40.8691	1.38400
$873$	0	0
$874$	−15.6359	−0.528893
$875$	0	0
$876$	0	0
$877$	26.0586	0.879935	0.439968	−	0.898014i	$-0.354990\pi$
$877$	26.0586	0.879935	0.439968	+	0.898014i	$0.354990\pi$
$878$	13.0066	0.438952
$879$	0	0
$880$	0	0
$881$	−9.01092	−0.303586	−0.151793	−	0.988412i	$-0.548505\pi$
$881$	−9.01092	−0.303586	−0.151793	+	0.988412i	$0.548505\pi$
$882$	0	0
$883$	8.80991	0.296477	0.148239	−	0.988952i	$-0.452640\pi$
$883$	8.80991	0.296477	0.148239	+	0.988952i	$0.452640\pi$
$884$	8.29452	0.278975
$885$	0	0
$886$	6.19013	0.207961
$887$	−12.2685	−0.411936	−0.205968	−	0.978559i	$-0.566034\pi$
$887$	−12.2685	−0.411936	−0.205968	+	0.978559i	$0.566034\pi$
$888$	0	0
$889$	4.57400	0.153407
$890$	0	0
$891$	0	0
$892$	22.3496	0.748319
$893$	−19.1764	−0.641713
$894$	0	0
$895$	0	0
$896$	−0.195743	−0.00653930
$897$	0	0
$898$	−7.45818	−0.248883
$899$	7.37721	0.246044
$900$	0	0
$901$	50.6365	1.68695
$902$	2.81348	0.0936787
$903$	0	0
$904$	45.7284	1.52090
$905$	0	0
$906$	0	0
$907$	−43.9602	−1.45968	−0.729838	−	0.683620i	$-0.760405\pi$
$907$	−43.9602	−1.45968	−0.729838	+	0.683620i	$0.760405\pi$
$908$	−16.8393	−0.558831
$909$	0	0
$910$	0	0
$911$	49.1806	1.62943	0.814713	−	0.579865i	$-0.196895\pi$
$911$	49.1806	1.62943	0.814713	+	0.579865i	$0.196895\pi$
$912$	0	0
$913$	−6.38612	−0.211350
$914$	−26.1685	−0.865577
$915$	0	0
$916$	−1.92458	−0.0635899
$917$	−3.88256	−0.128213
$918$	0	0
$919$	24.3120	0.801979	0.400990	−	0.916083i	$-0.368666\pi$
$919$	24.3120	0.801979	0.400990	+	0.916083i	$0.368666\pi$
$920$	0	0
$921$	0	0
$922$	−5.89547	−0.194157
$923$	28.0742	0.924074
$924$	0	0
$925$	0	0
$926$	−39.4421	−1.29615
$927$	0	0
$928$	4.13899	0.135869
$929$	51.4063	1.68659	0.843293	−	0.537453i	$-0.180614\pi$
$929$	51.4063	1.68659	0.843293	+	0.537453i	$0.180614\pi$
$930$	0	0
$931$	−47.8766	−1.56909
$932$	0.103695	0.00339664
$933$	0	0
$934$	−23.9243	−0.782827
$935$	0	0
$936$	0	0
$937$	−17.8156	−0.582010	−0.291005	−	0.956722i	$-0.593990\pi$
$937$	−17.8156	−0.582010	−0.291005	+	0.956722i	$0.593990\pi$
$938$	2.02933	0.0662600
$939$	0	0
$940$	0	0
$941$	6.44862	0.210219	0.105110	−	0.994461i	$-0.466481\pi$
$941$	6.44862	0.210219	0.105110	+	0.994461i	$0.466481\pi$
$942$	0	0
$943$	2.14800	0.0699486
$944$	25.5731	0.832334
$945$	0	0
$946$	−26.6477	−0.866392
$947$	−23.1321	−0.751694	−0.375847	−	0.926682i	$-0.622648\pi$
$947$	−23.1321	−0.751694	−0.375847	+	0.926682i	$0.622648\pi$
$948$	0	0
$949$	5.52140	0.179232
$950$	0	0
$951$	0	0
$952$	−3.12143	−0.101166
$953$	9.35394	0.303004	0.151502	−	0.988457i	$-0.451589\pi$
$953$	9.35394	0.303004	0.151502	+	0.988457i	$0.451589\pi$
$954$	0	0
$955$	0	0
$956$	10.9987	0.355724
$957$	0	0
$958$	2.91894	0.0943067
$959$	3.64548	0.117719
$960$	0	0
$961$	23.4233	0.755589
$962$	2.21349	0.0713659
$963$	0	0
$964$	17.1690	0.552977
$965$	0	0
$966$	0	0
$967$	−21.3156	−0.685463	−0.342732	−	0.939433i	$-0.611352\pi$
$967$	−21.3156	−0.685463	−0.342732	+	0.939433i	$0.611352\pi$
$968$	15.4897	0.497859
$969$	0	0
$970$	0	0
$971$	−0.806232	−0.0258732	−0.0129366	−	0.999916i	$-0.504118\pi$
$971$	−0.806232	−0.0258732	−0.0129366	+	0.999916i	$0.504118\pi$
$972$	0	0
$973$	3.18776	0.102195
$974$	25.1302	0.805223
$975$	0	0
$976$	−2.16272	−0.0692270
$977$	−6.87870	−0.220069	−0.110035	−	0.993928i	$-0.535096\pi$
$977$	−6.87870	−0.220069	−0.110035	+	0.993928i	$0.535096\pi$
$978$	0	0
$979$	34.4686	1.10162
$980$	0	0
$981$	0	0
$982$	−16.9143	−0.539756
$983$	−2.23941	−0.0714262	−0.0357131	−	0.999362i	$-0.511370\pi$
$983$	−2.23941	−0.0714262	−0.0357131	+	0.999362i	$0.511370\pi$
$984$	0	0
$985$	0	0
$986$	−4.31620	−0.137456
$987$	0	0
$988$	−14.6307	−0.465464
$989$	−20.3447	−0.646923
$990$	0	0
$991$	18.8061	0.597394	0.298697	−	0.954348i	$-0.403448\pi$
$991$	18.8061	0.597394	0.298697	+	0.954348i	$0.403448\pi$
$992$	30.5342	0.969461
$993$	0	0
$994$	−2.97708	−0.0944272
$995$	0	0
$996$	0	0
$997$	33.5310	1.06194	0.530968	−	0.847392i	$-0.321828\pi$
$997$	33.5310	1.06194	0.530968	+	0.847392i	$0.321828\pi$
$998$	−17.2958	−0.547489
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6525.2.a.ca.1.7	✓	9
3.2	odd	2		6525.2.a.cc.1.3	yes	9
5.4	even	2		6525.2.a.cd.1.3	yes	9
15.14	odd	2		6525.2.a.cb.1.7	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6525.2.a.ca.1.7	✓	9	1.1	even	1	trivial
6525.2.a.cb.1.7	yes	9	15.14	odd	2
6525.2.a.cc.1.3	yes	9	3.2	odd	2
6525.2.a.cd.1.3	yes	9	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.10241 q^{2} -0.784687 q^{4} -0.259703 q^{7} -3.06987 q^{8} +O(q^{10})\)	\(q+1.10241 q^{2} -0.784687 q^{4} -0.259703 q^{7} -3.06987 q^{8} -2.44014 q^{11} +2.69984 q^{13} -0.286300 q^{14} -1.81489 q^{16} -3.91523 q^{17} +6.90606 q^{19} -2.69004 q^{22} -2.05376 q^{23} +2.97634 q^{26} +0.203785 q^{28} +1.00000 q^{29} +7.37721 q^{31} +4.13899 q^{32} -4.31620 q^{34} +0.743697 q^{37} +7.61333 q^{38} -1.04589 q^{41} +9.90608 q^{43} +1.91474 q^{44} -2.26409 q^{46} -2.77675 q^{47} -6.93255 q^{49} -2.11853 q^{52} -12.9332 q^{53} +0.797254 q^{56} +1.10241 q^{58} -14.0907 q^{59} +1.19165 q^{61} +8.13273 q^{62} +8.19266 q^{64} -7.08814 q^{67} +3.07223 q^{68} +10.3985 q^{71} +2.04508 q^{73} +0.819861 q^{74} -5.41909 q^{76} +0.633710 q^{77} -14.7942 q^{79} -1.15300 q^{82} +2.61712 q^{83} +10.9206 q^{86} +7.49091 q^{88} -14.1257 q^{89} -0.701156 q^{91} +1.61156 q^{92} -3.06112 q^{94} -3.01317 q^{97} -7.64254 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 2 q^{2} + 10 q^{4} - q^{7} - 9 q^{8}+O(q^{10}) \) 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

L-functions

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