LMFDB - Embedded newform 8550.2.a.cp.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8550, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8550.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$68.2720937282$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.993.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x + 3 $ x^3 - x^2 - 6*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 950)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.77339$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +2.69168 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +2.69168 q^{7} +1.00000 q^{8} -5.54677 q^{11} +2.91829 q^{13} +2.69168 q^{14} +1.00000 q^{16} +4.91829 q^{17} +1.00000 q^{19} -5.54677 q^{22} +3.60997 q^{23} +2.91829 q^{26} +2.69168 q^{28} -1.08171 q^{29} +7.54677 q^{31} +1.00000 q^{32} +4.91829 q^{34} -4.54677 q^{37} +1.00000 q^{38} +9.54677 q^{43} -5.54677 q^{44} +3.60997 q^{46} -0.836581 q^{47} +0.245129 q^{49} +2.91829 q^{52} +9.78523 q^{53} +2.69168 q^{56} -1.08171 q^{58} -12.9933 q^{59} -7.38336 q^{61} +7.54677 q^{62} +1.00000 q^{64} -2.85510 q^{67} +4.91829 q^{68} -14.4769 q^{71} -5.15674 q^{73} -4.54677 q^{74} +1.00000 q^{76} -14.9301 q^{77} -3.09355 q^{79} +1.71019 q^{83} +9.54677 q^{86} -5.54677 q^{88} +5.09355 q^{89} +7.85510 q^{91} +3.60997 q^{92} -0.836581 q^{94} +17.2570 q^{97} +0.245129 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8} - 2 q^{11} + 6 q^{13} - 2 q^{14} + 3 q^{16} + 12 q^{17} + 3 q^{19} - 2 q^{22} - 2 q^{23} + 6 q^{26} - 2 q^{28} - 6 q^{29} + 8 q^{31} + 3 q^{32} + 12 q^{34} + q^{37} + 3 q^{38} + 14 q^{43} - 2 q^{44} - 2 q^{46} + 3 q^{47} + 9 q^{49} + 6 q^{52} - 10 q^{53} - 2 q^{56} - 6 q^{58} - 6 q^{59} - 2 q^{61} + 8 q^{62} + 3 q^{64} - 4 q^{67} + 12 q^{68} + 6 q^{71} + 12 q^{73} + q^{74} + 3 q^{76} - 10 q^{77} + 20 q^{79} - 4 q^{83} + 14 q^{86} - 2 q^{88} - 14 q^{89} + 19 q^{91} - 2 q^{92} + 3 q^{94} + 28 q^{97} + 9 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8 - 2 * q^11 + 6 * q^13 - 2 * q^14 + 3 * q^16 + 12 * q^17 + 3 * q^19 - 2 * q^22 - 2 * q^23 + 6 * q^26 - 2 * q^28 - 6 * q^29 + 8 * q^31 + 3 * q^32 + 12 * q^34 + q^37 + 3 * q^38 + 14 * q^43 - 2 * q^44 - 2 * q^46 + 3 * q^47 + 9 * q^49 + 6 * q^52 - 10 * q^53 - 2 * q^56 - 6 * q^58 - 6 * q^59 - 2 * q^61 + 8 * q^62 + 3 * q^64 - 4 * q^67 + 12 * q^68 + 6 * q^71 + 12 * q^73 + q^74 + 3 * q^76 - 10 * q^77 + 20 * q^79 - 4 * q^83 + 14 * q^86 - 2 * q^88 - 14 * q^89 + 19 * q^91 - 2 * q^92 + 3 * q^94 + 28 * q^97 + 9 * 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
$7$	2.69168	1.01736	0.508679	−	0.860956i	$-0.330134\pi$
$7$	2.69168	1.01736	0.508679	+	0.860956i	$0.330134\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	−5.54677	−1.67242	−0.836208	−	0.548413i	$-0.815232\pi$
$11$	−5.54677	−1.67242	−0.836208	+	0.548413i	$0.815232\pi$
$12$	0	0
$13$	2.91829	0.809388	0.404694	−	0.914452i	$-0.367378\pi$
$13$	2.91829	0.809388	0.404694	+	0.914452i	$0.367378\pi$
$14$	2.69168	0.719381
$15$	0	0
$16$	1.00000	0.250000
$17$	4.91829	1.19286	0.596430	−	0.802665i	$-0.296585\pi$
$17$	4.91829	1.19286	0.596430	+	0.802665i	$0.296585\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	−5.54677	−1.18258
$23$	3.60997	0.752730	0.376365	−	0.926471i	$-0.377174\pi$
$23$	3.60997	0.752730	0.376365	+	0.926471i	$0.377174\pi$
$24$	0	0
$25$	0	0
$26$	2.91829	0.572324
$27$	0	0
$28$	2.69168	0.508679
$29$	−1.08171	−0.200868	−0.100434	−	0.994944i	$-0.532023\pi$
$29$	−1.08171	−0.200868	−0.100434	+	0.994944i	$0.532023\pi$
$30$	0	0
$31$	7.54677	1.35544	0.677720	−	0.735320i	$-0.262968\pi$
$31$	7.54677	1.35544	0.677720	+	0.735320i	$0.262968\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	4.91829	0.843480
$35$	0	0
$36$	0	0
$37$	−4.54677	−0.747485	−0.373743	−	0.927532i	$-0.621926\pi$
$37$	−4.54677	−0.747485	−0.373743	+	0.927532i	$0.621926\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	9.54677	1.45587	0.727935	−	0.685646i	$-0.240480\pi$
$43$	9.54677	1.45587	0.727935	+	0.685646i	$0.240480\pi$
$44$	−5.54677	−0.836208
$45$	0	0
$46$	3.60997	0.532261
$47$	−0.836581	−0.122028	−0.0610139	−	0.998137i	$-0.519433\pi$
$47$	−0.836581	−0.122028	−0.0610139	+	0.998137i	$0.519433\pi$
$48$	0	0
$49$	0.245129	0.0350184
$50$	0	0
$51$	0	0
$52$	2.91829	0.404694
$53$	9.78523	1.34410	0.672052	−	0.740504i	$-0.265413\pi$
$53$	9.78523	1.34410	0.672052	+	0.740504i	$0.265413\pi$
$54$	0	0
$55$	0	0
$56$	2.69168	0.359691
$57$	0	0
$58$	−1.08171	−0.142035
$59$	−12.9933	−1.69159	−0.845793	−	0.533511i	$-0.820872\pi$
$59$	−12.9933	−1.69159	−0.845793	+	0.533511i	$0.820872\pi$
$60$	0	0
$61$	−7.38336	−0.945342	−0.472671	−	0.881239i	$-0.656710\pi$
$61$	−7.38336	−0.945342	−0.472671	+	0.881239i	$0.656710\pi$
$62$	7.54677	0.958441
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−2.85510	−0.348806	−0.174403	−	0.984674i	$-0.555799\pi$
$67$	−2.85510	−0.348806	−0.174403	+	0.984674i	$0.555799\pi$
$68$	4.91829	0.596430
$69$	0	0
$70$	0	0
$71$	−14.4769	−1.71809	−0.859046	−	0.511898i	$-0.828943\pi$
$71$	−14.4769	−1.71809	−0.859046	+	0.511898i	$0.828943\pi$
$72$	0	0
$73$	−5.15674	−0.603551	−0.301776	−	0.953379i	$-0.597579\pi$
$73$	−5.15674	−0.603551	−0.301776	+	0.953379i	$0.597579\pi$
$74$	−4.54677	−0.528552
$75$	0	0
$76$	1.00000	0.114708
$77$	−14.9301	−1.70145
$78$	0	0
$79$	−3.09355	−0.348052	−0.174026	−	0.984741i	$-0.555678\pi$
$79$	−3.09355	−0.348052	−0.174026	+	0.984741i	$0.555678\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.71019	0.187718	0.0938591	−	0.995585i	$-0.470080\pi$
$83$	1.71019	0.187718	0.0938591	+	0.995585i	$0.470080\pi$
$84$	0	0
$85$	0	0
$86$	9.54677	1.02946
$87$	0	0
$88$	−5.54677	−0.591288
$89$	5.09355	0.539915	0.269958	−	0.962872i	$-0.412990\pi$
$89$	5.09355	0.539915	0.269958	+	0.962872i	$0.412990\pi$
$90$	0	0
$91$	7.85510	0.823438
$92$	3.60997	0.376365
$93$	0	0
$94$	−0.836581	−0.0862867
$95$	0	0
$96$	0	0
$97$	17.2570	1.75218	0.876090	−	0.482148i	$-0.160143\pi$
$97$	17.2570	1.75218	0.876090	+	0.482148i	$0.160143\pi$
$98$	0.245129	0.0247618
$99$	0	0
$100$	0	0
$101$	9.38336	0.933679	0.466839	−	0.884342i	$-0.345393\pi$
$101$	9.38336	0.933679	0.466839	+	0.884342i	$0.345393\pi$
$102$	0	0
$103$	12.9301	1.27404	0.637022	−	0.770846i	$-0.280166\pi$
$103$	12.9301	1.27404	0.637022	+	0.770846i	$0.280166\pi$
$104$	2.91829	0.286162
$105$	0	0
$106$	9.78523	0.950425
$107$	18.9420	1.83119	0.915595	−	0.402102i	$-0.131720\pi$
$107$	18.9420	1.83119	0.915595	+	0.402102i	$0.131720\pi$
$108$	0	0
$109$	−2.69168	−0.257816	−0.128908	−	0.991657i	$-0.541147\pi$
$109$	−2.69168	−0.257816	−0.128908	+	0.991657i	$0.541147\pi$
$110$	0	0
$111$	0	0
$112$	2.69168	0.254340
$113$	−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
$116$	−1.08171	−0.100434
$117$	0	0
$118$	−12.9933	−1.19613
$119$	13.2385	1.21357
$120$	0	0
$121$	19.7667	1.79697
$122$	−7.38336	−0.668458
$123$	0	0
$124$	7.54677	0.677720
$125$	0	0
$126$	0	0
$127$	1.87361	0.166256	0.0831282	−	0.996539i	$-0.473509\pi$
$127$	1.87361	0.166256	0.0831282	+	0.996539i	$0.473509\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	11.5468	1.00885	0.504423	−	0.863457i	$-0.331705\pi$
$131$	11.5468	1.00885	0.504423	+	0.863457i	$0.331705\pi$
$132$	0	0
$133$	2.69168	0.233398
$134$	−2.85510	−0.246643
$135$	0	0
$136$	4.91829	0.421740
$137$	4.23845	0.362115	0.181058	−	0.983472i	$-0.442048\pi$
$137$	4.23845	0.362115	0.181058	+	0.983472i	$0.442048\pi$
$138$	0	0
$139$	9.21994	0.782025	0.391012	−	0.920385i	$-0.372125\pi$
$139$	9.21994	0.782025	0.391012	+	0.920385i	$0.372125\pi$
$140$	0	0
$141$	0	0
$142$	−14.4769	−1.21487
$143$	−16.1871	−1.35363
$144$	0	0
$145$	0	0
$146$	−5.15674	−0.426775
$147$	0	0
$148$	−4.54677	−0.373743
$149$	7.25697	0.594514	0.297257	−	0.954797i	$-0.403928\pi$
$149$	7.25697	0.594514	0.297257	+	0.954797i	$0.403928\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	−14.9301	−1.20310
$155$	0	0
$156$	0	0
$157$	9.54677	0.761916	0.380958	−	0.924592i	$-0.375594\pi$
$157$	9.54677	0.761916	0.380958	+	0.924592i	$0.375594\pi$
$158$	−3.09355	−0.246110
$159$	0	0
$160$	0	0
$161$	9.71687	0.765797
$162$	0	0
$163$	−15.7102	−1.23052	−0.615259	−	0.788325i	$-0.710948\pi$
$163$	−15.7102	−1.23052	−0.615259	+	0.788325i	$0.710948\pi$
$164$	0	0
$165$	0	0
$166$	1.71019	0.132737
$167$	6.45323	0.499366	0.249683	−	0.968328i	$-0.419674\pi$
$167$	6.45323	0.499366	0.249683	+	0.968328i	$0.419674\pi$
$168$	0	0
$169$	−4.48358	−0.344891
$170$	0	0
$171$	0	0
$172$	9.54677	0.727935
$173$	−0.873614	−0.0664196	−0.0332098	−	0.999448i	$-0.510573\pi$
$173$	−0.873614	−0.0664196	−0.0332098	+	0.999448i	$0.510573\pi$
$174$	0	0
$175$	0	0
$176$	−5.54677	−0.418104
$177$	0	0
$178$	5.09355	0.381778
$179$	12.3834	0.925575	0.462788	−	0.886469i	$-0.346849\pi$
$179$	12.3834	0.925575	0.462788	+	0.886469i	$0.346849\pi$
$180$	0	0
$181$	−17.6403	−1.31119	−0.655597	−	0.755111i	$-0.727583\pi$
$181$	−17.6403	−1.31119	−0.655597	+	0.755111i	$0.727583\pi$
$182$	7.85510	0.582259
$183$	0	0
$184$	3.60997	0.266130
$185$	0	0
$186$	0	0
$187$	−27.2806	−1.99496
$188$	−0.836581	−0.0610139
$189$	0	0
$190$	0	0
$191$	15.1567	1.09670	0.548352	−	0.836248i	$-0.315256\pi$
$191$	15.1567	1.09670	0.548352	+	0.836248i	$0.315256\pi$
$192$	0	0
$193$	−23.3834	−1.68317	−0.841585	−	0.540124i	$-0.818377\pi$
$193$	−23.3834	−1.68317	−0.841585	+	0.540124i	$0.818377\pi$
$194$	17.2570	1.23898
$195$	0	0
$196$	0.245129	0.0175092
$197$	−23.0935	−1.64535	−0.822674	−	0.568514i	$-0.807519\pi$
$197$	−23.0935	−1.64535	−0.822674	+	0.568514i	$0.807519\pi$
$198$	0	0
$199$	−25.7852	−1.82787	−0.913933	−	0.405865i	$-0.866970\pi$
$199$	−25.7852	−1.82787	−0.913933	+	0.405865i	$0.866970\pi$
$200$	0	0
$201$	0	0
$202$	9.38336	0.660211
$203$	−2.91161	−0.204355
$204$	0	0
$205$	0	0
$206$	12.9301	0.900885
$207$	0	0
$208$	2.91829	0.202347
$209$	−5.54677	−0.383678
$210$	0	0
$211$	14.2266	0.979400	0.489700	−	0.871891i	$-0.337106\pi$
$211$	14.2266	0.979400	0.489700	+	0.871891i	$0.337106\pi$
$212$	9.78523	0.672052
$213$	0	0
$214$	18.9420	1.29485
$215$	0	0
$216$	0	0
$217$	20.3135	1.37897
$218$	−2.69168	−0.182303
$219$	0	0
$220$	0	0
$221$	14.3530	0.965487
$222$	0	0
$223$	4.45323	0.298210	0.149105	−	0.988821i	$-0.452361\pi$
$223$	4.45323	0.298210	0.149105	+	0.988821i	$0.452361\pi$
$224$	2.69168	0.179845
$225$	0	0
$226$	−6.00000	−0.399114
$227$	5.94865	0.394826	0.197413	−	0.980320i	$-0.436746\pi$
$227$	5.94865	0.394826	0.197413	+	0.980320i	$0.436746\pi$
$228$	0	0
$229$	1.09355	0.0722638	0.0361319	−	0.999347i	$-0.488496\pi$
$229$	1.09355	0.0722638	0.0361319	+	0.999347i	$0.488496\pi$
$230$	0	0
$231$	0	0
$232$	−1.08171	−0.0710177
$233$	14.0935	0.923299	0.461650	−	0.887062i	$-0.347258\pi$
$233$	14.0935	0.923299	0.461650	+	0.887062i	$0.347258\pi$
$234$	0	0
$235$	0	0
$236$	−12.9933	−0.845793
$237$	0	0
$238$	13.2385	0.858121
$239$	−15.6100	−1.00972	−0.504862	−	0.863200i	$-0.668457\pi$
$239$	−15.6100	−1.00972	−0.504862	+	0.863200i	$0.668457\pi$
$240$	0	0
$241$	−5.21994	−0.336246	−0.168123	−	0.985766i	$-0.553771\pi$
$241$	−5.21994	−0.336246	−0.168123	+	0.985766i	$0.553771\pi$
$242$	19.7667	1.27065
$243$	0	0
$244$	−7.38336	−0.472671
$245$	0	0
$246$	0	0
$247$	2.91829	0.185686
$248$	7.54677	0.479221
$249$	0	0
$250$	0	0
$251$	28.6403	1.80776	0.903881	−	0.427785i	$-0.140706\pi$
$251$	28.6403	1.80776	0.903881	+	0.427785i	$0.140706\pi$
$252$	0	0
$253$	−20.0237	−1.25888
$254$	1.87361	0.117561
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.45323	0.402541	0.201271	−	0.979536i	$-0.435493\pi$
$257$	6.45323	0.402541	0.201271	+	0.979536i	$0.435493\pi$
$258$	0	0
$259$	−12.2385	−0.760460
$260$	0	0
$261$	0	0
$262$	11.5468	0.713362
$263$	22.1871	1.36812	0.684058	−	0.729428i	$-0.260214\pi$
$263$	22.1871	1.36812	0.684058	+	0.729428i	$0.260214\pi$
$264$	0	0
$265$	0	0
$266$	2.69168	0.165037
$267$	0	0
$268$	−2.85510	−0.174403
$269$	20.4070	1.24424	0.622119	−	0.782922i	$-0.286272\pi$
$269$	20.4070	1.24424	0.622119	+	0.782922i	$0.286272\pi$
$270$	0	0
$271$	15.9368	0.968092	0.484046	−	0.875043i	$-0.339167\pi$
$271$	15.9368	0.968092	0.484046	+	0.875043i	$0.339167\pi$
$272$	4.91829	0.298215
$273$	0	0
$274$	4.23845	0.256054
$275$	0	0
$276$	0	0
$277$	−4.02368	−0.241759	−0.120880	−	0.992667i	$-0.538572\pi$
$277$	−4.02368	−0.241759	−0.120880	+	0.992667i	$0.538572\pi$
$278$	9.21994	0.552975
$279$	0	0
$280$	0	0
$281$	−1.67316	−0.0998124	−0.0499062	−	0.998754i	$-0.515892\pi$
$281$	−1.67316	−0.0998124	−0.0499062	+	0.998754i	$0.515892\pi$
$282$	0	0
$283$	−16.8931	−1.00419	−0.502095	−	0.864812i	$-0.667437\pi$
$283$	−16.8931	−1.00419	−0.502095	+	0.864812i	$0.667437\pi$
$284$	−14.4769	−0.859046
$285$	0	0
$286$	−16.1871	−0.957163
$287$	0	0
$288$	0	0
$289$	7.18958	0.422916
$290$	0	0
$291$	0	0
$292$	−5.15674	−0.301776
$293$	−1.08171	−0.0631942	−0.0315971	−	0.999501i	$-0.510059\pi$
$293$	−1.08171	−0.0631942	−0.0315971	+	0.999501i	$0.510059\pi$
$294$	0	0
$295$	0	0
$296$	−4.54677	−0.264276
$297$	0	0
$298$	7.25697	0.420385
$299$	10.5349	0.609251
$300$	0	0
$301$	25.6968	1.48114
$302$	20.0000	1.15087
$303$	0	0
$304$	1.00000	0.0573539
$305$	0	0
$306$	0	0
$307$	−2.38336	−0.136025	−0.0680126	−	0.997684i	$-0.521666\pi$
$307$	−2.38336	−0.136025	−0.0680126	+	0.997684i	$0.521666\pi$
$308$	−14.9301	−0.850723
$309$	0	0
$310$	0	0
$311$	29.4584	1.67043	0.835216	−	0.549922i	$-0.185343\pi$
$311$	29.4584	1.67043	0.835216	+	0.549922i	$0.185343\pi$
$312$	0	0
$313$	−32.0355	−1.81075	−0.905377	−	0.424608i	$-0.860412\pi$
$313$	−32.0355	−1.81075	−0.905377	+	0.424608i	$0.860412\pi$
$314$	9.54677	0.538756
$315$	0	0
$316$	−3.09355	−0.174026
$317$	−26.6665	−1.49774	−0.748870	−	0.662717i	$-0.769403\pi$
$317$	−26.6665	−1.49774	−0.748870	+	0.662717i	$0.769403\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	0	0
$322$	9.71687	0.541500
$323$	4.91829	0.273661
$324$	0	0
$325$	0	0
$326$	−15.7102	−0.870107
$327$	0	0
$328$	0	0
$329$	−2.25181	−0.124146
$330$	0	0
$331$	−19.8721	−1.09227	−0.546135	−	0.837697i	$-0.683901\pi$
$331$	−19.8721	−1.09227	−0.546135	+	0.837697i	$0.683901\pi$
$332$	1.71019	0.0938591
$333$	0	0
$334$	6.45323	0.353105
$335$	0	0
$336$	0	0
$337$	−27.5705	−1.50186	−0.750929	−	0.660383i	$-0.770394\pi$
$337$	−27.5705	−1.50186	−0.750929	+	0.660383i	$0.770394\pi$
$338$	−4.48358	−0.243875
$339$	0	0
$340$	0	0
$341$	−41.8603	−2.26686
$342$	0	0
$343$	−18.1819	−0.981732
$344$	9.54677	0.514728
$345$	0	0
$346$	−0.873614	−0.0469658
$347$	−27.2806	−1.46450	−0.732251	−	0.681035i	$-0.761530\pi$
$347$	−27.2806	−1.46450	−0.732251	+	0.681035i	$0.761530\pi$
$348$	0	0
$349$	30.9538	1.65692	0.828460	−	0.560049i	$-0.189218\pi$
$349$	30.9538	1.65692	0.828460	+	0.560049i	$0.189218\pi$
$350$	0	0
$351$	0	0
$352$	−5.54677	−0.295644
$353$	11.1819	0.595154	0.297577	−	0.954698i	$-0.403821\pi$
$353$	11.1819	0.595154	0.297577	+	0.954698i	$0.403821\pi$
$354$	0	0
$355$	0	0
$356$	5.09355	0.269958
$357$	0	0
$358$	12.3834	0.654481
$359$	−20.3387	−1.07343	−0.536717	−	0.843762i	$-0.680336\pi$
$359$	−20.3387	−1.07343	−0.536717	+	0.843762i	$0.680336\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−17.6403	−0.927155
$363$	0	0
$364$	7.85510	0.411719
$365$	0	0
$366$	0	0
$367$	32.9538	1.72017	0.860087	−	0.510147i	$-0.170409\pi$
$367$	32.9538	1.72017	0.860087	+	0.510147i	$0.170409\pi$
$368$	3.60997	0.188183
$369$	0	0
$370$	0	0
$371$	26.3387	1.36744
$372$	0	0
$373$	10.0869	0.522278	0.261139	−	0.965301i	$-0.415902\pi$
$373$	10.0869	0.522278	0.261139	+	0.965301i	$0.415902\pi$
$374$	−27.2806	−1.41065
$375$	0	0
$376$	−0.836581	−0.0431434
$377$	−3.15674	−0.162581
$378$	0	0
$379$	−18.4256	−0.946457	−0.473229	−	0.880940i	$-0.656912\pi$
$379$	−18.4256	−0.946457	−0.473229	+	0.880940i	$0.656912\pi$
$380$	0	0
$381$	0	0
$382$	15.1567	0.775486
$383$	28.1871	1.44029	0.720147	−	0.693822i	$-0.244074\pi$
$383$	28.1871	1.44029	0.720147	+	0.693822i	$0.244074\pi$
$384$	0	0
$385$	0	0
$386$	−23.3834	−1.19018
$387$	0	0
$388$	17.2570	0.876090
$389$	−20.0237	−1.01524	−0.507620	−	0.861581i	$-0.669475\pi$
$389$	−20.0237	−1.01524	−0.507620	+	0.861581i	$0.669475\pi$
$390$	0	0
$391$	17.7549	0.897902
$392$	0.245129	0.0123809
$393$	0	0
$394$	−23.0935	−1.16344
$395$	0	0
$396$	0	0
$397$	8.32684	0.417912	0.208956	−	0.977925i	$-0.432993\pi$
$397$	8.32684	0.417912	0.208956	+	0.977925i	$0.432993\pi$
$398$	−25.7852	−1.29250
$399$	0	0
$400$	0	0
$401$	22.1501	1.10612	0.553061	−	0.833141i	$-0.313460\pi$
$401$	22.1501	1.10612	0.553061	+	0.833141i	$0.313460\pi$
$402$	0	0
$403$	22.0237	1.09708
$404$	9.38336	0.466839
$405$	0	0
$406$	−2.91161	−0.144501
$407$	25.2199	1.25011
$408$	0	0
$409$	34.0237	1.68236	0.841181	−	0.540753i	$-0.181861\pi$
$409$	34.0237	1.68236	0.841181	+	0.540753i	$0.181861\pi$
$410$	0	0
$411$	0	0
$412$	12.9301	0.637022
$413$	−34.9738	−1.72095
$414$	0	0
$415$	0	0
$416$	2.91829	0.143081
$417$	0	0
$418$	−5.54677	−0.271302
$419$	−37.5334	−1.83363	−0.916814	−	0.399315i	$-0.869248\pi$
$419$	−37.5334	−1.83363	−0.916814	+	0.399315i	$0.869248\pi$
$420$	0	0
$421$	33.8341	1.64897	0.824487	−	0.565882i	$-0.191464\pi$
$421$	33.8341	1.64897	0.824487	+	0.565882i	$0.191464\pi$
$422$	14.2266	0.692541
$423$	0	0
$424$	9.78523	0.475213
$425$	0	0
$426$	0	0
$427$	−19.8736	−0.961752
$428$	18.9420	0.915595
$429$	0	0
$430$	0	0
$431$	−24.8037	−1.19475	−0.597377	−	0.801960i	$-0.703790\pi$
$431$	−24.8037	−1.19475	−0.597377	+	0.801960i	$0.703790\pi$
$432$	0	0
$433$	−1.68651	−0.0810487	−0.0405244	−	0.999179i	$-0.512903\pi$
$433$	−1.68651	−0.0810487	−0.0405244	+	0.999179i	$0.512903\pi$
$434$	20.3135	0.975079
$435$	0	0
$436$	−2.69168	−0.128908
$437$	3.60997	0.172688
$438$	0	0
$439$	0.326839	0.0155992	0.00779958	−	0.999970i	$-0.497517\pi$
$439$	0.326839	0.0155992	0.00779958	+	0.999970i	$0.497517\pi$
$440$	0	0
$441$	0	0
$442$	14.3530	0.682703
$443$	0.490258	0.0232929	0.0116464	−	0.999932i	$-0.496293\pi$
$443$	0.490258	0.0232929	0.0116464	+	0.999932i	$0.496293\pi$
$444$	0	0
$445$	0	0
$446$	4.45323	0.210866
$447$	0	0
$448$	2.69168	0.127170
$449$	−23.5468	−1.11124	−0.555621	−	0.831436i	$-0.687519\pi$
$449$	−23.5468	−1.11124	−0.555621	+	0.831436i	$0.687519\pi$
$450$	0	0
$451$	0	0
$452$	−6.00000	−0.282216
$453$	0	0
$454$	5.94865	0.279184
$455$	0	0
$456$	0	0
$457$	8.01184	0.374778	0.187389	−	0.982286i	$-0.439997\pi$
$457$	8.01184	0.374778	0.187389	+	0.982286i	$0.439997\pi$
$458$	1.09355	0.0510982
$459$	0	0
$460$	0	0
$461$	−3.83658	−0.178687	−0.0893437	−	0.996001i	$-0.528477\pi$
$461$	−3.83658	−0.178687	−0.0893437	+	0.996001i	$0.528477\pi$
$462$	0	0
$463$	6.58381	0.305975	0.152988	−	0.988228i	$-0.451110\pi$
$463$	6.58381	0.305975	0.152988	+	0.988228i	$0.451110\pi$
$464$	−1.08171	−0.0502171
$465$	0	0
$466$	14.0935	0.652871
$467$	22.6033	1.04596	0.522978	−	0.852346i	$-0.324821\pi$
$467$	22.6033	1.04596	0.522978	+	0.852346i	$0.324821\pi$
$468$	0	0
$469$	−7.68500	−0.354860
$470$	0	0
$471$	0	0
$472$	−12.9933	−0.598066
$473$	−52.9538	−2.43482
$474$	0	0
$475$	0	0
$476$	13.2385	0.606783
$477$	0	0
$478$	−15.6100	−0.713983
$479$	35.7904	1.63530	0.817652	−	0.575712i	$-0.195275\pi$
$479$	35.7904	1.63530	0.817652	+	0.575712i	$0.195275\pi$
$480$	0	0
$481$	−13.2688	−0.605006
$482$	−5.21994	−0.237762
$483$	0	0
$484$	19.7667	0.898487
$485$	0	0
$486$	0	0
$487$	21.4070	0.970045	0.485023	−	0.874502i	$-0.338811\pi$
$487$	21.4070	0.970045	0.485023	+	0.874502i	$0.338811\pi$
$488$	−7.38336	−0.334229
$489$	0	0
$490$	0	0
$491$	4.32684	0.195267	0.0976337	−	0.995222i	$-0.468873\pi$
$491$	4.32684	0.195267	0.0976337	+	0.995222i	$0.468873\pi$
$492$	0	0
$493$	−5.32016	−0.239608
$494$	2.91829	0.131300
$495$	0	0
$496$	7.54677	0.338860
$497$	−38.9672	−1.74792
$498$	0	0
$499$	−37.6968	−1.68754	−0.843771	−	0.536703i	$-0.819670\pi$
$499$	−37.6968	−1.68754	−0.843771	+	0.536703i	$0.819670\pi$
$500$	0	0
$501$	0	0
$502$	28.6403	1.27828
$503$	−16.9183	−0.754349	−0.377175	−	0.926142i	$-0.623104\pi$
$503$	−16.9183	−0.754349	−0.377175	+	0.926142i	$0.623104\pi$
$504$	0	0
$505$	0	0
$506$	−20.0237	−0.890161
$507$	0	0
$508$	1.87361	0.0831282
$509$	6.79955	0.301385	0.150692	−	0.988581i	$-0.451850\pi$
$509$	6.79955	0.301385	0.150692	+	0.988581i	$0.451850\pi$
$510$	0	0
$511$	−13.8803	−0.614028
$512$	1.00000	0.0441942
$513$	0	0
$514$	6.45323	0.284640
$515$	0	0
$516$	0	0
$517$	4.64032	0.204081
$518$	−12.2385	−0.537727
$519$	0	0
$520$	0	0
$521$	1.35968	0.0595685	0.0297842	−	0.999556i	$-0.490518\pi$
$521$	1.35968	0.0595685	0.0297842	+	0.999556i	$0.490518\pi$
$522$	0	0
$523$	21.9486	0.959747	0.479874	−	0.877338i	$-0.340682\pi$
$523$	21.9486	0.959747	0.479874	+	0.877338i	$0.340682\pi$
$524$	11.5468	0.504423
$525$	0	0
$526$	22.1871	0.967404
$527$	37.1172	1.61685
$528$	0	0
$529$	−9.96813	−0.433397
$530$	0	0
$531$	0	0
$532$	2.69168	0.116699
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−2.85510	−0.123321
$537$	0	0
$538$	20.4070	0.879810
$539$	−1.35968	−0.0585654
$540$	0	0
$541$	−39.8973	−1.71532	−0.857659	−	0.514218i	$-0.828082\pi$
$541$	−39.8973	−1.71532	−0.857659	+	0.514218i	$0.828082\pi$
$542$	15.9368	0.684544
$543$	0	0
$544$	4.91829	0.210870
$545$	0	0
$546$	0	0
$547$	7.34632	0.314106	0.157053	−	0.987590i	$-0.449801\pi$
$547$	7.34632	0.314106	0.157053	+	0.987590i	$0.449801\pi$
$548$	4.23845	0.181058
$549$	0	0
$550$	0	0
$551$	−1.08171	−0.0460824
$552$	0	0
$553$	−8.32684	−0.354093
$554$	−4.02368	−0.170950
$555$	0	0
$556$	9.21994	0.391012
$557$	−26.9672	−1.14264	−0.571318	−	0.820729i	$-0.693568\pi$
$557$	−26.9672	−1.14264	−0.571318	+	0.820729i	$0.693568\pi$
$558$	0	0
$559$	27.8603	1.17836
$560$	0	0
$561$	0	0
$562$	−1.67316	−0.0705780
$563$	−44.8973	−1.89220	−0.946098	−	0.323881i	$-0.895012\pi$
$563$	−44.8973	−1.89220	−0.946098	+	0.323881i	$0.895012\pi$
$564$	0	0
$565$	0	0
$566$	−16.8931	−0.710070
$567$	0	0
$568$	−14.4769	−0.607437
$569$	−31.1172	−1.30450	−0.652251	−	0.758003i	$-0.726175\pi$
$569$	−31.1172	−1.30450	−0.652251	+	0.758003i	$0.726175\pi$
$570$	0	0
$571$	33.2570	1.39176	0.695880	−	0.718158i	$-0.255014\pi$
$571$	33.2570	1.39176	0.695880	+	0.718158i	$0.255014\pi$
$572$	−16.1871	−0.676817
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−23.4702	−0.977078	−0.488539	−	0.872542i	$-0.662470\pi$
$577$	−23.4702	−0.977078	−0.488539	+	0.872542i	$0.662470\pi$
$578$	7.18958	0.299047
$579$	0	0
$580$	0	0
$581$	4.60329	0.190977
$582$	0	0
$583$	−54.2765	−2.24790
$584$	−5.15674	−0.213388
$585$	0	0
$586$	−1.08171	−0.0446850
$587$	−10.1501	−0.418938	−0.209469	−	0.977815i	$-0.567174\pi$
$587$	−10.1501	−0.418938	−0.209469	+	0.977815i	$0.567174\pi$
$588$	0	0
$589$	7.54677	0.310959
$590$	0	0
$591$	0	0
$592$	−4.54677	−0.186871
$593$	−16.6732	−0.684685	−0.342342	−	0.939575i	$-0.611220\pi$
$593$	−16.6732	−0.684685	−0.342342	+	0.939575i	$0.611220\pi$
$594$	0	0
$595$	0	0
$596$	7.25697	0.297257
$597$	0	0
$598$	10.5349	0.430806
$599$	−18.2765	−0.746756	−0.373378	−	0.927679i	$-0.621800\pi$
$599$	−18.2765	−0.746756	−0.373378	+	0.927679i	$0.621800\pi$
$600$	0	0
$601$	7.96297	0.324816	0.162408	−	0.986724i	$-0.448074\pi$
$601$	7.96297	0.324816	0.162408	+	0.986724i	$0.448074\pi$
$602$	25.6968	1.04733
$603$	0	0
$604$	20.0000	0.813788
$605$	0	0
$606$	0	0
$607$	6.93013	0.281285	0.140643	−	0.990060i	$-0.455083\pi$
$607$	6.93013	0.281285	0.140643	+	0.990060i	$0.455083\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	0	0
$611$	−2.44139	−0.0987679
$612$	0	0
$613$	34.2765	1.38441	0.692206	−	0.721700i	$-0.256639\pi$
$613$	34.2765	1.38441	0.692206	+	0.721700i	$0.256639\pi$
$614$	−2.38336	−0.0961844
$615$	0	0
$616$	−14.9301	−0.601552
$617$	35.5139	1.42974	0.714869	−	0.699259i	$-0.246486\pi$
$617$	35.5139	1.42974	0.714869	+	0.699259i	$0.246486\pi$
$618$	0	0
$619$	−29.5334	−1.18705	−0.593524	−	0.804816i	$-0.702264\pi$
$619$	−29.5334	−1.18705	−0.593524	+	0.804816i	$0.702264\pi$
$620$	0	0
$621$	0	0
$622$	29.4584	1.18117
$623$	13.7102	0.549287
$624$	0	0
$625$	0	0
$626$	−32.0355	−1.28040
$627$	0	0
$628$	9.54677	0.380958
$629$	−22.3624	−0.891646
$630$	0	0
$631$	−15.1634	−0.603646	−0.301823	−	0.953364i	$-0.597595\pi$
$631$	−15.1634	−0.603646	−0.301823	+	0.953364i	$0.597595\pi$
$632$	−3.09355	−0.123055
$633$	0	0
$634$	−26.6665	−1.05906
$635$	0	0
$636$	0	0
$637$	0.715358	0.0283435
$638$	6.00000	0.237542
$639$	0	0
$640$	0	0
$641$	25.0802	0.990608	0.495304	−	0.868720i	$-0.335057\pi$
$641$	25.0802	0.990608	0.495304	+	0.868720i	$0.335057\pi$
$642$	0	0
$643$	−19.1306	−0.754437	−0.377218	−	0.926124i	$-0.623119\pi$
$643$	−19.1306	−0.754437	−0.377218	+	0.926124i	$0.623119\pi$
$644$	9.71687	0.382898
$645$	0	0
$646$	4.91829	0.193508
$647$	12.4019	0.487568	0.243784	−	0.969830i	$-0.421611\pi$
$647$	12.4019	0.487568	0.243784	+	0.969830i	$0.421611\pi$
$648$	0	0
$649$	72.0710	2.82904
$650$	0	0
$651$	0	0
$652$	−15.7102	−0.615259
$653$	−10.7801	−0.421856	−0.210928	−	0.977502i	$-0.567649\pi$
$653$	−10.7801	−0.421856	−0.210928	+	0.977502i	$0.567649\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	−2.25181	−0.0877845
$659$	−8.56529	−0.333656	−0.166828	−	0.985986i	$-0.553353\pi$
$659$	−8.56529	−0.333656	−0.166828	+	0.985986i	$0.553353\pi$
$660$	0	0
$661$	−26.2883	−1.02250	−0.511248	−	0.859433i	$-0.670817\pi$
$661$	−26.2883	−1.02250	−0.511248	+	0.859433i	$0.670817\pi$
$662$	−19.8721	−0.772351
$663$	0	0
$664$	1.71019	0.0663684
$665$	0	0
$666$	0	0
$667$	−3.90494	−0.151200
$668$	6.45323	0.249683
$669$	0	0
$670$	0	0
$671$	40.9538	1.58100
$672$	0	0
$673$	12.9301	0.498420	0.249210	−	0.968449i	$-0.419829\pi$
$673$	12.9301	0.498420	0.249210	+	0.968449i	$0.419829\pi$
$674$	−27.5705	−1.06197
$675$	0	0
$676$	−4.48358	−0.172445
$677$	−15.9250	−0.612046	−0.306023	−	0.952024i	$-0.598998\pi$
$677$	−15.9250	−0.612046	−0.306023	+	0.952024i	$0.598998\pi$
$678$	0	0
$679$	46.4502	1.78260
$680$	0	0
$681$	0	0
$682$	−41.8603	−1.60291
$683$	13.2898	0.508520	0.254260	−	0.967136i	$-0.418168\pi$
$683$	13.2898	0.508520	0.254260	+	0.967136i	$0.418168\pi$
$684$	0	0
$685$	0	0
$686$	−18.1819	−0.694190
$687$	0	0
$688$	9.54677	0.363967
$689$	28.5561	1.08790
$690$	0	0
$691$	27.2570	1.03690	0.518452	−	0.855107i	$-0.326508\pi$
$691$	27.2570	1.03690	0.518452	+	0.855107i	$0.326508\pi$
$692$	−0.873614	−0.0332098
$693$	0	0
$694$	−27.2806	−1.03556
$695$	0	0
$696$	0	0
$697$	0	0
$698$	30.9538	1.17162
$699$	0	0
$700$	0	0
$701$	41.4441	1.56532	0.782660	−	0.622449i	$-0.213862\pi$
$701$	41.4441	1.56532	0.782660	+	0.622449i	$0.213862\pi$
$702$	0	0
$703$	−4.54677	−0.171485
$704$	−5.54677	−0.209052
$705$	0	0
$706$	11.1819	0.420838
$707$	25.2570	0.949886
$708$	0	0
$709$	−36.7904	−1.38169	−0.690846	−	0.723002i	$-0.742762\pi$
$709$	−36.7904	−1.38169	−0.690846	+	0.723002i	$0.742762\pi$
$710$	0	0
$711$	0	0
$712$	5.09355	0.190889
$713$	27.2436	1.02028
$714$	0	0
$715$	0	0
$716$	12.3834	0.462788
$717$	0	0
$718$	−20.3387	−0.759033
$719$	4.13726	0.154294	0.0771469	−	0.997020i	$-0.475419\pi$
$719$	4.13726	0.154294	0.0771469	+	0.997020i	$0.475419\pi$
$720$	0	0
$721$	34.8037	1.29616
$722$	1.00000	0.0372161
$723$	0	0
$724$	−17.6403	−0.655597
$725$	0	0
$726$	0	0
$727$	40.4374	1.49974	0.749870	−	0.661585i	$-0.230116\pi$
$727$	40.4374	1.49974	0.749870	+	0.661585i	$0.230116\pi$
$728$	7.85510	0.291129
$729$	0	0
$730$	0	0
$731$	46.9538	1.73665
$732$	0	0
$733$	−28.1264	−1.03887	−0.519436	−	0.854509i	$-0.673858\pi$
$733$	−28.1264	−1.03887	−0.519436	+	0.854509i	$0.673858\pi$
$734$	32.9538	1.21635
$735$	0	0
$736$	3.60997	0.133065
$737$	15.8366	0.583348
$738$	0	0
$739$	−38.1501	−1.40337	−0.701686	−	0.712486i	$-0.747569\pi$
$739$	−38.1501	−1.40337	−0.701686	+	0.712486i	$0.747569\pi$
$740$	0	0
$741$	0	0
$742$	26.3387	0.966923
$743$	−17.8232	−0.653871	−0.326935	−	0.945047i	$-0.606016\pi$
$743$	−17.8232	−0.653871	−0.326935	+	0.945047i	$0.606016\pi$
$744$	0	0
$745$	0	0
$746$	10.0869	0.369307
$747$	0	0
$748$	−27.2806	−0.997479
$749$	50.9857	1.86298
$750$	0	0
$751$	23.6968	0.864710	0.432355	−	0.901703i	$-0.357683\pi$
$751$	23.6968	0.864710	0.432355	+	0.901703i	$0.357683\pi$
$752$	−0.836581	−0.0305070
$753$	0	0
$754$	−3.15674	−0.114962
$755$	0	0
$756$	0	0
$757$	6.51394	0.236753	0.118377	−	0.992969i	$-0.462231\pi$
$757$	6.51394	0.236753	0.118377	+	0.992969i	$0.462231\pi$
$758$	−18.4256	−0.669246
$759$	0	0
$760$	0	0
$761$	−34.7786	−1.26072	−0.630361	−	0.776302i	$-0.717093\pi$
$761$	−34.7786	−1.26072	−0.630361	+	0.776302i	$0.717093\pi$
$762$	0	0
$763$	−7.24513	−0.262291
$764$	15.1567	0.548352
$765$	0	0
$766$	28.1871	1.01844
$767$	−37.9183	−1.36915
$768$	0	0
$769$	−27.6850	−0.998347	−0.499173	−	0.866502i	$-0.666363\pi$
$769$	−27.6850	−0.998347	−0.499173	+	0.866502i	$0.666363\pi$
$770$	0	0
$771$	0	0
$772$	−23.3834	−0.841585
$773$	−19.9738	−0.718409	−0.359205	−	0.933259i	$-0.616952\pi$
$773$	−19.9738	−0.718409	−0.359205	+	0.933259i	$0.616952\pi$
$774$	0	0
$775$	0	0
$776$	17.2570	0.619489
$777$	0	0
$778$	−20.0237	−0.717884
$779$	0	0
$780$	0	0
$781$	80.3001	2.87336
$782$	17.7549	0.634913
$783$	0	0
$784$	0.245129	0.00875461
$785$	0	0
$786$	0	0
$787$	−42.0118	−1.49756	−0.748780	−	0.662818i	$-0.769360\pi$
$787$	−42.0118	−1.49756	−0.748780	+	0.662818i	$0.769360\pi$
$788$	−23.0935	−0.822674
$789$	0	0
$790$	0	0
$791$	−16.1501	−0.574230
$792$	0	0
$793$	−21.5468	−0.765148
$794$	8.32684	0.295508
$795$	0	0
$796$	−25.7852	−0.913933
$797$	−45.1054	−1.59771	−0.798857	−	0.601520i	$-0.794562\pi$
$797$	−45.1054	−1.59771	−0.798857	+	0.601520i	$0.794562\pi$
$798$	0	0
$799$	−4.11455	−0.145562
$800$	0	0
$801$	0	0
$802$	22.1501	0.782146
$803$	28.6033	1.00939
$804$	0	0
$805$	0	0
$806$	22.0237	0.775751
$807$	0	0
$808$	9.38336	0.330105
$809$	−6.85510	−0.241012	−0.120506	−	0.992713i	$-0.538452\pi$
$809$	−6.85510	−0.241012	−0.120506	+	0.992713i	$0.538452\pi$
$810$	0	0
$811$	−15.1819	−0.533110	−0.266555	−	0.963820i	$-0.585885\pi$
$811$	−15.1819	−0.533110	−0.266555	+	0.963820i	$0.585885\pi$
$812$	−2.91161	−0.102178
$813$	0	0
$814$	25.2199	0.883958
$815$	0	0
$816$	0	0
$817$	9.54677	0.333999
$818$	34.0237	1.18961
$819$	0	0
$820$	0	0
$821$	−22.5006	−0.785276	−0.392638	−	0.919693i	$-0.628437\pi$
$821$	−22.5006	−0.785276	−0.392638	+	0.919693i	$0.628437\pi$
$822$	0	0
$823$	16.1896	0.564333	0.282167	−	0.959365i	$-0.408947\pi$
$823$	16.1896	0.564333	0.282167	+	0.959365i	$0.408947\pi$
$824$	12.9301	0.450442
$825$	0	0
$826$	−34.9738	−1.21690
$827$	−25.0817	−0.872177	−0.436088	−	0.899904i	$-0.643636\pi$
$827$	−25.0817	−0.872177	−0.436088	+	0.899904i	$0.643636\pi$
$828$	0	0
$829$	21.3068	0.740016	0.370008	−	0.929029i	$-0.379355\pi$
$829$	21.3068	0.740016	0.370008	+	0.929029i	$0.379355\pi$
$830$	0	0
$831$	0	0
$832$	2.91829	0.101174
$833$	1.20562	0.0417721
$834$	0	0
$835$	0	0
$836$	−5.54677	−0.191839
$837$	0	0
$838$	−37.5334	−1.29657
$839$	11.4727	0.396082	0.198041	−	0.980194i	$-0.436542\pi$
$839$	11.4727	0.396082	0.198041	+	0.980194i	$0.436542\pi$
$840$	0	0
$841$	−27.8299	−0.959652
$842$	33.8341	1.16600
$843$	0	0
$844$	14.2266	0.489700
$845$	0	0
$846$	0	0
$847$	53.2056	1.82817
$848$	9.78523	0.336026
$849$	0	0
$850$	0	0
$851$	−16.4137	−0.562655
$852$	0	0
$853$	38.9908	1.33502	0.667511	−	0.744600i	$-0.267360\pi$
$853$	38.9908	1.33502	0.667511	+	0.744600i	$0.267360\pi$
$854$	−19.8736	−0.680061
$855$	0	0
$856$	18.9420	0.647423
$857$	−29.8973	−1.02127	−0.510636	−	0.859797i	$-0.670590\pi$
$857$	−29.8973	−1.02127	−0.510636	+	0.859797i	$0.670590\pi$
$858$	0	0
$859$	−6.47691	−0.220989	−0.110495	−	0.993877i	$-0.535243\pi$
$859$	−6.47691	−0.220989	−0.110495	+	0.993877i	$0.535243\pi$
$860$	0	0
$861$	0	0
$862$	−24.8037	−0.844819
$863$	−5.09355	−0.173386	−0.0866932	−	0.996235i	$-0.527630\pi$
$863$	−5.09355	−0.173386	−0.0866932	+	0.996235i	$0.527630\pi$
$864$	0	0
$865$	0	0
$866$	−1.68651	−0.0573101
$867$	0	0
$868$	20.3135	0.689485
$869$	17.1592	0.582087
$870$	0	0
$871$	−8.33200	−0.282319
$872$	−2.69168	−0.0911517
$873$	0	0
$874$	3.60997	0.122109
$875$	0	0
$876$	0	0
$877$	29.8341	1.00743	0.503713	−	0.863871i	$-0.331967\pi$
$877$	29.8341	1.00743	0.503713	+	0.863871i	$0.331967\pi$
$878$	0.326839	0.0110303
$879$	0	0
$880$	0	0
$881$	11.5796	0.390127	0.195064	−	0.980791i	$-0.437509\pi$
$881$	11.5796	0.390127	0.195064	+	0.980791i	$0.437509\pi$
$882$	0	0
$883$	48.0237	1.61613	0.808063	−	0.589096i	$-0.200516\pi$
$883$	48.0237	1.61613	0.808063	+	0.589096i	$0.200516\pi$
$884$	14.3530	0.482744
$885$	0	0
$886$	0.490258	0.0164705
$887$	9.38336	0.315062	0.157531	−	0.987514i	$-0.449647\pi$
$887$	9.38336	0.315062	0.157531	+	0.987514i	$0.449647\pi$
$888$	0	0
$889$	5.04316	0.169142
$890$	0	0
$891$	0	0
$892$	4.45323	0.149105
$893$	−0.836581	−0.0279951
$894$	0	0
$895$	0	0
$896$	2.69168	0.0899226
$897$	0	0
$898$	−23.5468	−0.785766
$899$	−8.16342	−0.272265
$900$	0	0
$901$	48.1266	1.60333
$902$	0	0
$903$	0	0
$904$	−6.00000	−0.199557
$905$	0	0
$906$	0	0
$907$	19.4347	0.645319	0.322659	−	0.946515i	$-0.395423\pi$
$907$	19.4347	0.645319	0.322659	+	0.946515i	$0.395423\pi$
$908$	5.94865	0.197413
$909$	0	0
$910$	0	0
$911$	19.9866	0.662187	0.331094	−	0.943598i	$-0.392582\pi$
$911$	19.9866	0.662187	0.331094	+	0.943598i	$0.392582\pi$
$912$	0	0
$913$	−9.48606	−0.313943
$914$	8.01184	0.265008
$915$	0	0
$916$	1.09355	0.0361319
$917$	31.0802	1.02636
$918$	0	0
$919$	6.15158	0.202922	0.101461	−	0.994840i	$-0.467648\pi$
$919$	6.15158	0.202922	0.101461	+	0.994840i	$0.467648\pi$
$920$	0	0
$921$	0	0
$922$	−3.83658	−0.126351
$923$	−42.2478	−1.39060
$924$	0	0
$925$	0	0
$926$	6.58381	0.216357
$927$	0	0
$928$	−1.08171	−0.0355089
$929$	7.17010	0.235243	0.117622	−	0.993058i	$-0.462473\pi$
$929$	7.17010	0.235243	0.117622	+	0.993058i	$0.462473\pi$
$930$	0	0
$931$	0.245129	0.00803378
$932$	14.0935	0.461650
$933$	0	0
$934$	22.6033	0.739602
$935$	0	0
$936$	0	0
$937$	24.2646	0.792690	0.396345	−	0.918102i	$-0.370278\pi$
$937$	24.2646	0.792690	0.396345	+	0.918102i	$0.370278\pi$
$938$	−7.68500	−0.250924
$939$	0	0
$940$	0	0
$941$	7.44806	0.242800	0.121400	−	0.992604i	$-0.461262\pi$
$941$	7.44806	0.242800	0.121400	+	0.992604i	$0.461262\pi$
$942$	0	0
$943$	0	0
$944$	−12.9933	−0.422897
$945$	0	0
$946$	−52.9538	−1.72168
$947$	−52.9538	−1.72077	−0.860384	−	0.509647i	$-0.829776\pi$
$947$	−52.9538	−1.72077	−0.860384	+	0.509647i	$0.829776\pi$
$948$	0	0
$949$	−15.0489	−0.488507
$950$	0	0
$951$	0	0
$952$	13.2385	0.429061
$953$	−12.8694	−0.416881	−0.208441	−	0.978035i	$-0.566839\pi$
$953$	−12.8694	−0.416881	−0.208441	+	0.978035i	$0.566839\pi$
$954$	0	0
$955$	0	0
$956$	−15.6100	−0.504862
$957$	0	0
$958$	35.7904	1.15634
$959$	11.4085	0.368401
$960$	0	0
$961$	25.9538	0.837220
$962$	−13.2688	−0.427804
$963$	0	0
$964$	−5.21994	−0.168123
$965$	0	0
$966$	0	0
$967$	−25.8603	−0.831610	−0.415805	−	0.909454i	$-0.636500\pi$
$967$	−25.8603	−0.831610	−0.415805	+	0.909454i	$0.636500\pi$
$968$	19.7667	0.635326
$969$	0	0
$970$	0	0
$971$	−35.0935	−1.12621	−0.563103	−	0.826387i	$-0.690392\pi$
$971$	−35.0935	−1.12621	−0.563103	+	0.826387i	$0.690392\pi$
$972$	0	0
$973$	24.8171	0.795600
$974$	21.4070	0.685926
$975$	0	0
$976$	−7.38336	−0.236335
$977$	−42.1737	−1.34926	−0.674629	−	0.738157i	$-0.735696\pi$
$977$	−42.1737	−1.34926	−0.674629	+	0.738157i	$0.735696\pi$
$978$	0	0
$979$	−28.2528	−0.902963
$980$	0	0
$981$	0	0
$982$	4.32684	0.138075
$983$	−42.6270	−1.35959	−0.679795	−	0.733403i	$-0.737931\pi$
$983$	−42.6270	−1.35959	−0.679795	+	0.733403i	$0.737931\pi$
$984$	0	0
$985$	0	0
$986$	−5.32016	−0.169428
$987$	0	0
$988$	2.91829	0.0928432
$989$	34.4636	1.09588
$990$	0	0
$991$	−34.1737	−1.08556	−0.542782	−	0.839873i	$-0.682629\pi$
$991$	−34.1737	−1.08556	−0.542782	+	0.839873i	$0.682629\pi$
$992$	7.54677	0.239610
$993$	0	0
$994$	−38.9672	−1.23596
$995$	0	0
$996$	0	0
$997$	−29.3834	−0.930580	−0.465290	−	0.885158i	$-0.654050\pi$
$997$	−29.3834	−0.930580	−0.465290	+	0.885158i	$0.654050\pi$
$998$	−37.6968	−1.19327
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.cp.1.3		3
3.2	odd	2		950.2.a.j.1.3	✓	3
5.4	even	2		8550.2.a.ci.1.1		3
12.11	even	2		7600.2.a.bz.1.1		3
15.2	even	4		950.2.b.h.799.1		6
15.8	even	4		950.2.b.h.799.6		6
15.14	odd	2		950.2.a.l.1.1	yes	3
60.59	even	2		7600.2.a.bk.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.a.j.1.3	✓	3	3.2	odd	2
950.2.a.l.1.1	yes	3	15.14	odd	2
950.2.b.h.799.1		6	15.2	even	4
950.2.b.h.799.6		6	15.8	even	4
7600.2.a.bk.1.3		3	60.59	even	2
7600.2.a.bz.1.1		3	12.11	even	2
8550.2.a.ci.1.1		3	5.4	even	2
8550.2.a.cp.1.3		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8550.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.69168 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.69168 q^{7} +1.00000 q^{8} -5.54677 q^{11} +2.91829 q^{13} +2.69168 q^{14} +1.00000 q^{16} +4.91829 q^{17} +1.00000 q^{19} -5.54677 q^{22} +3.60997 q^{23} +2.91829 q^{26} +2.69168 q^{28} -1.08171 q^{29} +7.54677 q^{31} +1.00000 q^{32} +4.91829 q^{34} -4.54677 q^{37} +1.00000 q^{38} +9.54677 q^{43} -5.54677 q^{44} +3.60997 q^{46} -0.836581 q^{47} +0.245129 q^{49} +2.91829 q^{52} +9.78523 q^{53} +2.69168 q^{56} -1.08171 q^{58} -12.9933 q^{59} -7.38336 q^{61} +7.54677 q^{62} +1.00000 q^{64} -2.85510 q^{67} +4.91829 q^{68} -14.4769 q^{71} -5.15674 q^{73} -4.54677 q^{74} +1.00000 q^{76} -14.9301 q^{77} -3.09355 q^{79} +1.71019 q^{83} +9.54677 q^{86} -5.54677 q^{88} +5.09355 q^{89} +7.85510 q^{91} +3.60997 q^{92} -0.836581 q^{94} +17.2570 q^{97} +0.245129 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8} - 2 q^{11} + 6 q^{13} - 2 q^{14} + 3 q^{16} + 12 q^{17} + 3 q^{19} - 2 q^{22} - 2 q^{23} + 6 q^{26} - 2 q^{28} - 6 q^{29} + 8 q^{31} + 3 q^{32} + 12 q^{34} + q^{37} + 3 q^{38} + 14 q^{43} - 2 q^{44} - 2 q^{46} + 3 q^{47} + 9 q^{49} + 6 q^{52} - 10 q^{53} - 2 q^{56} - 6 q^{58} - 6 q^{59} - 2 q^{61} + 8 q^{62} + 3 q^{64} - 4 q^{67} + 12 q^{68} + 6 q^{71} + 12 q^{73} + q^{74} + 3 q^{76} - 10 q^{77} + 20 q^{79} - 4 q^{83} + 14 q^{86} - 2 q^{88} - 14 q^{89} + 19 q^{91} - 2 q^{92} + 3 q^{94} + 28 q^{97} + 9 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8 - 2 * q^11 + 6 * q^13 - 2 * q^14 + 3 * q^16 + 12 * q^17 + 3 * q^19 - 2 * q^22 - 2 * q^23 + 6 * q^26 - 2 * q^28 - 6 * q^29 + 8 * q^31 + 3 * q^32 + 12 * q^34 + q^37 + 3 * q^38 + 14 * q^43 - 2 * q^44 - 2 * q^46 + 3 * q^47 + 9 * q^49 + 6 * q^52 - 10 * q^53 - 2 * q^56 - 6 * q^58 - 6 * q^59 - 2 * q^61 + 8 * q^62 + 3 * q^64 - 4 * q^67 + 12 * q^68 + 6 * q^71 + 12 * q^73 + q^74 + 3 * q^76 - 10 * q^77 + 20 * q^79 - 4 * q^83 + 14 * q^86 - 2 * q^88 - 14 * q^89 + 19 * q^91 - 2 * q^92 + 3 * q^94 + 28 * q^97 + 9 * q^98

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