LMFDB - Embedded newform 7600.2.a.bk.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7600 = 2^{4} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7600.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:	$60.6863055362$
Analytic rank:	$1$
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:	$\Z[a_1, a_2, a_3]$
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	$\chi$	$=$	7600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.77339 q^{3} +2.69168 q^{7} +0.144903 q^{9} +O(q^{10})$	$q+1.77339 q^{3} +2.69168 q^{7} +0.144903 q^{9} -5.54677 q^{11} -2.91829 q^{13} +4.91829 q^{17} -1.00000 q^{19} +4.77339 q^{21} -3.60997 q^{23} -5.06319 q^{27} +1.08171 q^{29} -7.54677 q^{31} -9.83658 q^{33} +4.54677 q^{37} -5.17526 q^{39} +9.54677 q^{43} +0.836581 q^{47} +0.245129 q^{49} +8.72203 q^{51} +9.78523 q^{53} -1.77339 q^{57} -12.9933 q^{59} -7.38336 q^{61} +0.390032 q^{63} -2.85510 q^{67} -6.40187 q^{69} -14.4769 q^{71} +5.15674 q^{73} -14.9301 q^{77} +3.09355 q^{79} -9.41371 q^{81} -1.71019 q^{83} +1.91829 q^{87} -5.09355 q^{89} -7.85510 q^{91} -13.3834 q^{93} -17.2570 q^{97} -0.803744 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} - 2 q^{7} + 5 q^{9}+O(q^{10}) $ 3 * q - 2 * q^3 - 2 * q^7 + 5 * q^9	$ 3 q - 2 q^{3} - 2 q^{7} + 5 q^{9} - 2 q^{11} - 6 q^{13} + 12 q^{17} - 3 q^{19} + 7 q^{21} + 2 q^{23} - 17 q^{27} + 6 q^{29} - 8 q^{31} - 24 q^{33} - q^{37} + 11 q^{39} + 14 q^{43} - 3 q^{47} + 9 q^{49} - 15 q^{51} - 10 q^{53} + 2 q^{57} - 6 q^{59} - 2 q^{61} + 14 q^{63} - 4 q^{67} + 6 q^{71} - 12 q^{73} - 10 q^{77} - 20 q^{79} + 23 q^{81} + 4 q^{83} + 3 q^{87} + 14 q^{89} - 19 q^{91} - 20 q^{93} - 28 q^{97} + 36 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 - 2 * q^7 + 5 * q^9 - 2 * q^11 - 6 * q^13 + 12 * q^17 - 3 * q^19 + 7 * q^21 + 2 * q^23 - 17 * q^27 + 6 * q^29 - 8 * q^31 - 24 * q^33 - q^37 + 11 * q^39 + 14 * q^43 - 3 * q^47 + 9 * q^49 - 15 * q^51 - 10 * q^53 + 2 * q^57 - 6 * q^59 - 2 * q^61 + 14 * q^63 - 4 * q^67 + 6 * q^71 - 12 * q^73 - 10 * q^77 - 20 * q^79 + 23 * q^81 + 4 * q^83 + 3 * q^87 + 14 * q^89 - 19 * q^91 - 20 * q^93 - 28 * q^97 + 36 * q^99

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	0
$2$	0	0
$3$	1.77339	1.02387	0.511933	−	0.859025i	$-0.328930\pi$
$3$	1.77339	1.02387	0.511933	+	0.859025i	$0.328930\pi$
$4$	0	0
$5$	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	$0.330134\pi$
$8$	0	0
$9$	0.144903	0.0483010
$10$	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	$-0.632622\pi$
$13$	−2.91829	−0.809388	−0.404694	+	0.914452i	$0.632622\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	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
$21$	4.77339	1.04164
$22$	0	0
$23$	−3.60997	−0.752730	−0.376365	−	0.926471i	$-0.622826\pi$
$23$	−3.60997	−0.752730	−0.376365	+	0.926471i	$0.622826\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.06319	−0.974412
$28$	0	0
$29$	1.08171	0.200868	0.100434	−	0.994944i	$-0.467977\pi$
$29$	1.08171	0.200868	0.100434	+	0.994944i	$0.467977\pi$
$30$	0	0
$31$	−7.54677	−1.35544	−0.677720	−	0.735320i	$-0.737032\pi$
$31$	−7.54677	−1.35544	−0.677720	+	0.735320i	$0.737032\pi$
$32$	0	0
$33$	−9.83658	−1.71233
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.54677	0.747485	0.373743	−	0.927532i	$-0.378074\pi$
$37$	4.54677	0.747485	0.373743	+	0.927532i	$0.378074\pi$
$38$	0	0
$39$	−5.17526	−0.828705
$40$	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	$0.240480\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.836581	0.122028	0.0610139	−	0.998137i	$-0.480567\pi$
$47$	0.836581	0.122028	0.0610139	+	0.998137i	$0.480567\pi$
$48$	0	0
$49$	0.245129	0.0350184
$50$	0	0
$51$	8.72203	1.22133
$52$	0	0
$53$	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
$56$	0	0
$57$	−1.77339	−0.234891
$58$	0	0
$59$	−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	$0.656710\pi$
$62$	0	0
$63$	0.390032	0.0491394
$64$	0	0
$65$	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	$0.555799\pi$
$68$	0	0
$69$	−6.40187	−0.770695
$70$	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	$-0.402421\pi$
$73$	5.15674	0.603551	0.301776	+	0.953379i	$0.402421\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−14.9301	−1.70145
$78$	0	0
$79$	3.09355	0.348052	0.174026	−	0.984741i	$-0.444322\pi$
$79$	3.09355	0.348052	0.174026	+	0.984741i	$0.444322\pi$
$80$	0	0
$81$	−9.41371	−1.04597
$82$	0	0
$83$	−1.71019	−0.187718	−0.0938591	−	0.995585i	$-0.529920\pi$
$83$	−1.71019	−0.187718	−0.0938591	+	0.995585i	$0.529920\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.91829	0.205662
$88$	0	0
$89$	−5.09355	−0.539915	−0.269958	−	0.962872i	$-0.587010\pi$
$89$	−5.09355	−0.539915	−0.269958	+	0.962872i	$0.587010\pi$
$90$	0	0
$91$	−7.85510	−0.823438
$92$	0	0
$93$	−13.3834	−1.38779
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−17.2570	−1.75218	−0.876090	−	0.482148i	$-0.839857\pi$
$97$	−17.2570	−1.75218	−0.876090	+	0.482148i	$0.839857\pi$
$98$	0	0
$99$	−0.803744	−0.0807793
$100$	0	0
$101$	−9.38336	−0.933679	−0.466839	−	0.884342i	$-0.654607\pi$
$101$	−9.38336	−0.933679	−0.466839	+	0.884342i	$0.654607\pi$
$102$	0	0
$103$	12.9301	1.27404	0.637022	−	0.770846i	$-0.280166\pi$
$103$	12.9301	1.27404	0.637022	+	0.770846i	$0.280166\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−18.9420	−1.83119	−0.915595	−	0.402102i	$-0.868280\pi$
$107$	−18.9420	−1.83119	−0.915595	+	0.402102i	$0.868280\pi$
$108$	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
$111$	8.06319	0.765324
$112$	0	0
$113$	−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
$116$	0	0
$117$	−0.422869	−0.0390942
$118$	0	0
$119$	13.2385	1.21357
$120$	0	0
$121$	19.7667	1.79697
$122$	0	0
$123$	0	0
$124$	0	0
$125$	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	$0.473509\pi$
$128$	0	0
$129$	16.9301	1.49061
$130$	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
$134$	0	0
$135$	0	0
$136$	0	0
$137$	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	$-0.627875\pi$
$139$	−9.21994	−0.782025	−0.391012	+	0.920385i	$0.627875\pi$
$140$	0	0
$141$	1.48358	0.124940
$142$	0	0
$143$	16.1871	1.35363
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0.434709	0.0358542
$148$	0	0
$149$	−7.25697	−0.594514	−0.297257	−	0.954797i	$-0.596072\pi$
$149$	−7.25697	−0.594514	−0.297257	+	0.954797i	$0.596072\pi$
$150$	0	0
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	0	0
$153$	0.712675	0.0576163
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−9.54677	−0.761916	−0.380958	−	0.924592i	$-0.624406\pi$
$157$	−9.54677	−0.761916	−0.380958	+	0.924592i	$0.624406\pi$
$158$	0	0
$159$	17.3530	1.37618
$160$	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
$166$	0	0
$167$	−6.45323	−0.499366	−0.249683	−	0.968328i	$-0.580326\pi$
$167$	−6.45323	−0.499366	−0.249683	+	0.968328i	$0.580326\pi$
$168$	0	0
$169$	−4.48358	−0.344891
$170$	0	0
$171$	−0.144903	−0.0110810
$172$	0	0
$173$	−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
$176$	0	0
$177$	−23.0422	−1.73196
$178$	0	0
$179$	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	$0.727583\pi$
$182$	0	0
$183$	−13.0935	−0.967903
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−27.2806	−1.99496
$188$	0	0
$189$	−13.6285	−0.991326
$190$	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	$-0.181623\pi$
$193$	23.3834	1.68317	0.841585	+	0.540124i	$0.181623\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−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	$-0.133030\pi$
$199$	25.7852	1.82787	0.913933	+	0.405865i	$0.133030\pi$
$200$	0	0
$201$	−5.06319	−0.357130
$202$	0	0
$203$	2.91161	0.204355
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−0.523095	−0.0363576
$208$	0	0
$209$	5.54677	0.383678
$210$	0	0
$211$	−14.2266	−0.979400	−0.489700	−	0.871891i	$-0.662894\pi$
$211$	−14.2266	−0.979400	−0.489700	+	0.871891i	$0.662894\pi$
$212$	0	0
$213$	−25.6732	−1.75910
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−20.3135	−1.37897
$218$	0	0
$219$	9.14490	0.617955
$220$	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	$0.452361\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5.94865	−0.394826	−0.197413	−	0.980320i	$-0.563254\pi$
$227$	−5.94865	−0.394826	−0.197413	+	0.980320i	$0.563254\pi$
$228$	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
$231$	−26.4769	−1.74205
$232$	0	0
$233$	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
$236$	0	0
$237$	5.48606	0.356358
$238$	0	0
$239$	−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	$0.553771\pi$
$242$	0	0
$243$	−1.50458	−0.0965188
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.91829	0.185686
$248$	0	0
$249$	−3.03284	−0.192198
$250$	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
$254$	0	0
$255$	0	0
$256$	0	0
$257$	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
$261$	0.156743	0.00970214
$262$	0	0
$263$	−22.1871	−1.36812	−0.684058	−	0.729428i	$-0.739786\pi$
$263$	−22.1871	−1.36812	−0.684058	+	0.729428i	$0.739786\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−9.03284	−0.552801
$268$	0	0
$269$	−20.4070	−1.24424	−0.622119	−	0.782922i	$-0.713728\pi$
$269$	−20.4070	−1.24424	−0.622119	+	0.782922i	$0.713728\pi$
$270$	0	0
$271$	−15.9368	−0.968092	−0.484046	−	0.875043i	$-0.660833\pi$
$271$	−15.9368	−0.968092	−0.484046	+	0.875043i	$0.660833\pi$
$272$	0	0
$273$	−13.9301	−0.843090
$274$	0	0
$275$	0	0
$276$	0	0
$277$	4.02368	0.241759	0.120880	−	0.992667i	$-0.461428\pi$
$277$	4.02368	0.241759	0.120880	+	0.992667i	$0.461428\pi$
$278$	0	0
$279$	−1.09355	−0.0654691
$280$	0	0
$281$	1.67316	0.0998124	0.0499062	−	0.998754i	$-0.484108\pi$
$281$	1.67316	0.0998124	0.0499062	+	0.998754i	$0.484108\pi$
$282$	0	0
$283$	−16.8931	−1.00419	−0.502095	−	0.864812i	$-0.667437\pi$
$283$	−16.8931	−1.00419	−0.502095	+	0.864812i	$0.667437\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	7.18958	0.422916
$290$	0	0
$291$	−30.6033	−1.79400
$292$	0	0
$293$	−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
$296$	0	0
$297$	28.0844	1.62962
$298$	0	0
$299$	10.5349	0.609251
$300$	0	0
$301$	25.6968	1.48114
$302$	0	0
$303$	−16.6403	−0.955962
$304$	0	0
$305$	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	$0.521666\pi$
$308$	0	0
$309$	22.9301	1.30445
$310$	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	$-0.139588\pi$
$313$	32.0355	1.81075	0.905377	+	0.424608i	$0.139588\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−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
$321$	−33.5915	−1.87489
$322$	0	0
$323$	−4.91829	−0.273661
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−4.77339	−0.263969
$328$	0	0
$329$	2.25181	0.124146
$330$	0	0
$331$	19.8721	1.09227	0.546135	−	0.837697i	$-0.316099\pi$
$331$	19.8721	1.09227	0.546135	+	0.837697i	$0.316099\pi$
$332$	0	0
$333$	0.658841	0.0361043
$334$	0	0
$335$	0	0
$336$	0	0
$337$	27.5705	1.50186	0.750929	−	0.660383i	$-0.229606\pi$
$337$	27.5705	1.50186	0.750929	+	0.660383i	$0.229606\pi$
$338$	0	0
$339$	−10.6403	−0.577903
$340$	0	0
$341$	41.8603	2.26686
$342$	0	0
$343$	−18.1819	−0.981732
$344$	0	0
$345$	0	0
$346$	0	0
$347$	27.2806	1.46450	0.732251	−	0.681035i	$-0.238470\pi$
$347$	27.2806	1.46450	0.732251	+	0.681035i	$0.238470\pi$
$348$	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
$351$	14.7759	0.788677
$352$	0	0
$353$	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
$356$	0	0
$357$	23.4769	1.24253
$358$	0	0
$359$	−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
$362$	0	0
$363$	35.0540	1.83986
$364$	0	0
$365$	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	$0.170409\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	26.3387	1.36744
$372$	0	0
$373$	−10.0869	−0.522278	−0.261139	−	0.965301i	$-0.584098\pi$
$373$	−10.0869	−0.522278	−0.261139	+	0.965301i	$0.584098\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.15674	−0.162581
$378$	0	0
$379$	18.4256	0.946457	0.473229	−	0.880940i	$-0.343088\pi$
$379$	18.4256	0.946457	0.473229	+	0.880940i	$0.343088\pi$
$380$	0	0
$381$	3.32264	0.170224
$382$	0	0
$383$	−28.1871	−1.44029	−0.720147	−	0.693822i	$-0.755926\pi$
$383$	−28.1871	−1.44029	−0.720147	+	0.693822i	$0.755926\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.38336	0.0703199
$388$	0	0
$389$	20.0237	1.01524	0.507620	−	0.861581i	$-0.330525\pi$
$389$	20.0237	1.01524	0.507620	+	0.861581i	$0.330525\pi$
$390$	0	0
$391$	−17.7549	−0.897902
$392$	0	0
$393$	20.4769	1.03292
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−8.32684	−0.417912	−0.208956	−	0.977925i	$-0.567007\pi$
$397$	−8.32684	−0.417912	−0.208956	+	0.977925i	$0.567007\pi$
$398$	0	0
$399$	−4.77339	−0.238968
$400$	0	0
$401$	−22.1501	−1.10612	−0.553061	−	0.833141i	$-0.686540\pi$
$401$	−22.1501	−1.10612	−0.553061	+	0.833141i	$0.686540\pi$
$402$	0	0
$403$	22.0237	1.09708
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−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
$411$	7.51642	0.370758
$412$	0	0
$413$	−34.9738	−1.72095
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−16.3505	−0.800688
$418$	0	0
$419$	−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	$0.191464\pi$
$422$	0	0
$423$	0.121223	0.00589406
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−19.8736	−0.961752
$428$	0	0
$429$	28.7060	1.38594
$430$	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	$-0.487097\pi$
$433$	1.68651	0.0810487	0.0405244	+	0.999179i	$0.487097\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.60997	0.172688
$438$	0	0
$439$	−0.326839	−0.0155992	−0.00779958	−	0.999970i	$-0.502483\pi$
$439$	−0.326839	−0.0155992	−0.00779958	+	0.999970i	$0.502483\pi$
$440$	0	0
$441$	0.0355199	0.00169142
$442$	0	0
$443$	−0.490258	−0.0232929	−0.0116464	−	0.999932i	$-0.503707\pi$
$443$	−0.490258	−0.0232929	−0.0116464	+	0.999932i	$0.503707\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−12.8694	−0.608703
$448$	0	0
$449$	23.5468	1.11124	0.555621	−	0.831436i	$-0.312481\pi$
$449$	23.5468	1.11124	0.555621	+	0.831436i	$0.312481\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−35.4677	−1.66642
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8.01184	−0.374778	−0.187389	−	0.982286i	$-0.560003\pi$
$457$	−8.01184	−0.374778	−0.187389	+	0.982286i	$0.560003\pi$
$458$	0	0
$459$	−24.9023	−1.16234
$460$	0	0
$461$	3.83658	0.178687	0.0893437	−	0.996001i	$-0.471523\pi$
$461$	3.83658	0.178687	0.0893437	+	0.996001i	$0.471523\pi$
$462$	0	0
$463$	6.58381	0.305975	0.152988	−	0.988228i	$-0.451110\pi$
$463$	6.58381	0.305975	0.152988	+	0.988228i	$0.451110\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−22.6033	−1.04596	−0.522978	−	0.852346i	$-0.675179\pi$
$467$	−22.6033	−1.04596	−0.522978	+	0.852346i	$0.675179\pi$
$468$	0	0
$469$	−7.68500	−0.354860
$470$	0	0
$471$	−16.9301	−0.780099
$472$	0	0
$473$	−52.9538	−2.43482
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.41791	0.0649215
$478$	0	0
$479$	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
$482$	0	0
$483$	−17.2318	−0.784073
$484$	0	0
$485$	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	$0.338811\pi$
$488$	0	0
$489$	−27.8603	−1.25988
$490$	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
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−38.9672	−1.74792
$498$	0	0
$499$	37.6968	1.68754	0.843771	−	0.536703i	$-0.180330\pi$
$499$	37.6968	1.68754	0.843771	+	0.536703i	$0.180330\pi$
$500$	0	0
$501$	−11.4441	−0.511283
$502$	0	0
$503$	16.9183	0.754349	0.377175	−	0.926142i	$-0.376896\pi$
$503$	16.9183	0.754349	0.377175	+	0.926142i	$0.376896\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−7.95113	−0.353122
$508$	0	0
$509$	−6.79955	−0.301385	−0.150692	−	0.988581i	$-0.548150\pi$
$509$	−6.79955	−0.301385	−0.150692	+	0.988581i	$0.548150\pi$
$510$	0	0
$511$	13.8803	0.614028
$512$	0	0
$513$	5.06319	0.223545
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−4.64032	−0.204081
$518$	0	0
$519$	−1.54926	−0.0680048
$520$	0	0
$521$	−1.35968	−0.0595685	−0.0297842	−	0.999556i	$-0.509482\pi$
$521$	−1.35968	−0.0595685	−0.0297842	+	0.999556i	$0.509482\pi$
$522$	0	0
$523$	21.9486	0.959747	0.479874	−	0.877338i	$-0.340682\pi$
$523$	21.9486	0.959747	0.479874	+	0.877338i	$0.340682\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−37.1172	−1.61685
$528$	0	0
$529$	−9.96813	−0.433397
$530$	0	0
$531$	−1.88277	−0.0817053
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	21.9605	0.947665
$538$	0	0
$539$	−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	$0.828082\pi$
$542$	0	0
$543$	−31.2831	−1.34249
$544$	0	0
$545$	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	$0.449801\pi$
$548$	0	0
$549$	−1.06987	−0.0456609
$550$	0	0
$551$	−1.08171	−0.0460824
$552$	0	0
$553$	8.32684	0.354093
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−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
$561$	−48.3792	−2.04257
$562$	0	0
$563$	44.8973	1.89220	0.946098	−	0.323881i	$-0.104988\pi$
$563$	44.8973	1.89220	0.946098	+	0.323881i	$0.104988\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−25.3387	−1.06412
$568$	0	0
$569$	31.1172	1.30450	0.652251	−	0.758003i	$-0.273825\pi$
$569$	31.1172	1.30450	0.652251	+	0.758003i	$0.273825\pi$
$570$	0	0
$571$	−33.2570	−1.39176	−0.695880	−	0.718158i	$-0.744986\pi$
$571$	−33.2570	−1.39176	−0.695880	+	0.718158i	$0.744986\pi$
$572$	0	0
$573$	26.8788	1.12288
$574$	0	0
$575$	0	0
$576$	0	0
$577$	23.4702	0.977078	0.488539	−	0.872542i	$-0.337530\pi$
$577$	23.4702	0.977078	0.488539	+	0.872542i	$0.337530\pi$
$578$	0	0
$579$	41.4677	1.72334
$580$	0	0
$581$	−4.60329	−0.190977
$582$	0	0
$583$	−54.2765	−2.24790
$584$	0	0
$585$	0	0
$586$	0	0
$587$	10.1501	0.418938	0.209469	−	0.977815i	$-0.432826\pi$
$587$	10.1501	0.418938	0.209469	+	0.977815i	$0.432826\pi$
$588$	0	0
$589$	7.54677	0.310959
$590$	0	0
$591$	−40.9538	−1.68461
$592$	0	0
$593$	−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
$596$	0	0
$597$	45.7272	1.87149
$598$	0	0
$599$	−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	$0.448074\pi$
$602$	0	0
$603$	−0.413712	−0.0168477
$604$	0	0
$605$	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	$0.455083\pi$
$608$	0	0
$609$	5.16342	0.209232
$610$	0	0
$611$	−2.44139	−0.0987679
$612$	0	0
$613$	−34.2765	−1.38441	−0.692206	−	0.721700i	$-0.743361\pi$
$613$	−34.2765	−1.38441	−0.692206	+	0.721700i	$0.743361\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	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	$-0.297736\pi$
$619$	29.5334	1.18705	0.593524	+	0.804816i	$0.297736\pi$
$620$	0	0
$621$	18.2780	0.733470
$622$	0	0
$623$	−13.7102	−0.549287
$624$	0	0
$625$	0	0
$626$	0	0
$627$	9.83658	0.392835
$628$	0	0
$629$	22.3624	0.891646
$630$	0	0
$631$	15.1634	0.603646	0.301823	−	0.953364i	$-0.402405\pi$
$631$	15.1634	0.603646	0.301823	+	0.953364i	$0.402405\pi$
$632$	0	0
$633$	−25.2293	−1.00277
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−0.715358	−0.0283435
$638$	0	0
$639$	−2.09775	−0.0829855
$640$	0	0
$641$	−25.0802	−0.990608	−0.495304	−	0.868720i	$-0.664943\pi$
$641$	−25.0802	−0.990608	−0.495304	+	0.868720i	$0.664943\pi$
$642$	0	0
$643$	−19.1306	−0.754437	−0.377218	−	0.926124i	$-0.623119\pi$
$643$	−19.1306	−0.754437	−0.377218	+	0.926124i	$0.623119\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−12.4019	−0.487568	−0.243784	−	0.969830i	$-0.578389\pi$
$647$	−12.4019	−0.487568	−0.243784	+	0.969830i	$0.578389\pi$
$648$	0	0
$649$	72.0710	2.82904
$650$	0	0
$651$	−36.0237	−1.41188
$652$	0	0
$653$	−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
$657$	0.747227	0.0291521
$658$	0	0
$659$	−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	$0.670817\pi$
$662$	0	0
$663$	−25.4534	−0.988529
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.90494	−0.151200
$668$	0	0
$669$	7.89729	0.305327
$670$	0	0
$671$	40.9538	1.58100
$672$	0	0
$673$	−12.9301	−0.498420	−0.249210	−	0.968449i	$-0.580171\pi$
$673$	−12.9301	−0.498420	−0.249210	+	0.968449i	$0.580171\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−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
$681$	−10.5493	−0.404248
$682$	0	0
$683$	−13.2898	−0.508520	−0.254260	−	0.967136i	$-0.581832\pi$
$683$	−13.2898	−0.508520	−0.254260	+	0.967136i	$0.581832\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.93929	0.0739884
$688$	0	0
$689$	−28.5561	−1.08790
$690$	0	0
$691$	−27.2570	−1.03690	−0.518452	−	0.855107i	$-0.673492\pi$
$691$	−27.2570	−1.03690	−0.518452	+	0.855107i	$0.673492\pi$
$692$	0	0
$693$	−2.16342	−0.0821815
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	24.9933	0.945334
$700$	0	0
$701$	−41.4441	−1.56532	−0.782660	−	0.622449i	$-0.786138\pi$
$701$	−41.4441	−1.56532	−0.782660	+	0.622449i	$0.786138\pi$
$702$	0	0
$703$	−4.54677	−0.171485
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−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
$711$	0.448264	0.0168112
$712$	0	0
$713$	27.2436	1.02028
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−27.6825	−1.03382
$718$	0	0
$719$	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
$722$	0	0
$723$	−9.25697	−0.344270
$724$	0	0
$725$	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	$0.230116\pi$
$728$	0	0
$729$	25.5729	0.947146
$730$	0	0
$731$	46.9538	1.73665
$732$	0	0
$733$	28.1264	1.03887	0.519436	−	0.854509i	$-0.326142\pi$
$733$	28.1264	1.03887	0.519436	+	0.854509i	$0.326142\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	15.8366	0.583348
$738$	0	0
$739$	38.1501	1.40337	0.701686	−	0.712486i	$-0.252431\pi$
$739$	38.1501	1.40337	0.701686	+	0.712486i	$0.252431\pi$
$740$	0	0
$741$	5.17526	0.190118
$742$	0	0
$743$	17.8232	0.653871	0.326935	−	0.945047i	$-0.393984\pi$
$743$	17.8232	0.653871	0.326935	+	0.945047i	$0.393984\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−0.247812	−0.00906697
$748$	0	0
$749$	−50.9857	−1.86298
$750$	0	0
$751$	−23.6968	−0.864710	−0.432355	−	0.901703i	$-0.642317\pi$
$751$	−23.6968	−0.864710	−0.432355	+	0.901703i	$0.642317\pi$
$752$	0	0
$753$	50.7904	1.85090
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−6.51394	−0.236753	−0.118377	−	0.992969i	$-0.537769\pi$
$757$	−6.51394	−0.236753	−0.118377	+	0.992969i	$0.537769\pi$
$758$	0	0
$759$	35.5097	1.28892
$760$	0	0
$761$	34.7786	1.26072	0.630361	−	0.776302i	$-0.282907\pi$
$761$	34.7786	1.26072	0.630361	+	0.776302i	$0.282907\pi$
$762$	0	0
$763$	−7.24513	−0.262291
$764$	0	0
$765$	0	0
$766$	0	0
$767$	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
$771$	11.4441	0.412148
$772$	0	0
$773$	−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
$776$	0	0
$777$	21.7035	0.778609
$778$	0	0
$779$	0	0
$780$	0	0
$781$	80.3001	2.87336
$782$	0	0
$783$	−5.47691	−0.195729
$784$	0	0
$785$	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	$0.769360\pi$
$788$	0	0
$789$	−39.3463	−1.40077
$790$	0	0
$791$	−16.1501	−0.574230
$792$	0	0
$793$	21.5468	0.765148
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−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
$801$	−0.738070	−0.0260784
$802$	0	0
$803$	−28.6033	−1.00939
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−36.1896	−1.27393
$808$	0	0
$809$	6.85510	0.241012	0.120506	−	0.992713i	$-0.461548\pi$
$809$	6.85510	0.241012	0.120506	+	0.992713i	$0.461548\pi$
$810$	0	0
$811$	15.1819	0.533110	0.266555	−	0.963820i	$-0.414115\pi$
$811$	15.1819	0.533110	0.266555	+	0.963820i	$0.414115\pi$
$812$	0	0
$813$	−28.2621	−0.991196
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−9.54677	−0.333999
$818$	0	0
$819$	−1.13823	−0.0397729
$820$	0	0
$821$	22.5006	0.785276	0.392638	−	0.919693i	$-0.371563\pi$
$821$	22.5006	0.785276	0.392638	+	0.919693i	$0.371563\pi$
$822$	0	0
$823$	16.1896	0.564333	0.282167	−	0.959365i	$-0.408947\pi$
$823$	16.1896	0.564333	0.282167	+	0.959365i	$0.408947\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	25.0817	0.872177	0.436088	−	0.899904i	$-0.356364\pi$
$827$	25.0817	0.872177	0.436088	+	0.899904i	$0.356364\pi$
$828$	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
$831$	7.13554	0.247529
$832$	0	0
$833$	1.20562	0.0417721
$834$	0	0
$835$	0	0
$836$	0	0
$837$	38.2108	1.32076
$838$	0	0
$839$	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
$842$	0	0
$843$	2.96716	0.102195
$844$	0	0
$845$	0	0
$846$	0	0
$847$	53.2056	1.82817
$848$	0	0
$849$	−29.9580	−1.02816
$850$	0	0
$851$	−16.4137	−0.562655
$852$	0	0
$853$	−38.9908	−1.33502	−0.667511	−	0.744600i	$-0.732640\pi$
$853$	−38.9908	−1.33502	−0.667511	+	0.744600i	$0.732640\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−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	$-0.464757\pi$
$859$	6.47691	0.220989	0.110495	+	0.993877i	$0.464757\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	5.09355	0.173386	0.0866932	−	0.996235i	$-0.472370\pi$
$863$	5.09355	0.173386	0.0866932	+	0.996235i	$0.472370\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	12.7499	0.433010
$868$	0	0
$869$	−17.1592	−0.582087
$870$	0	0
$871$	8.33200	0.282319
$872$	0	0
$873$	−2.50059	−0.0846320
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−29.8341	−1.00743	−0.503713	−	0.863871i	$-0.668033\pi$
$877$	−29.8341	−1.00743	−0.503713	+	0.863871i	$0.668033\pi$
$878$	0	0
$879$	−1.91829	−0.0647023
$880$	0	0
$881$	−11.5796	−0.390127	−0.195064	−	0.980791i	$-0.562491\pi$
$881$	−11.5796	−0.390127	−0.195064	+	0.980791i	$0.562491\pi$
$882$	0	0
$883$	48.0237	1.61613	0.808063	−	0.589096i	$-0.200516\pi$
$883$	48.0237	1.61613	0.808063	+	0.589096i	$0.200516\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−9.38336	−0.315062	−0.157531	−	0.987514i	$-0.550353\pi$
$887$	−9.38336	−0.315062	−0.157531	+	0.987514i	$0.550353\pi$
$888$	0	0
$889$	5.04316	0.169142
$890$	0	0
$891$	52.2157	1.74929
$892$	0	0
$893$	−0.836581	−0.0279951
$894$	0	0
$895$	0	0
$896$	0	0
$897$	18.6825	0.623791
$898$	0	0
$899$	−8.16342	−0.272265
$900$	0	0
$901$	48.1266	1.60333
$902$	0	0
$903$	45.5705	1.51649
$904$	0	0
$905$	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	$0.395423\pi$
$908$	0	0
$909$	−1.35968	−0.0450976
$910$	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
$914$	0	0
$915$	0	0
$916$	0	0
$917$	31.0802	1.02636
$918$	0	0
$919$	−6.15158	−0.202922	−0.101461	−	0.994840i	$-0.532352\pi$
$919$	−6.15158	−0.202922	−0.101461	+	0.994840i	$0.532352\pi$
$920$	0	0
$921$	−4.22661	−0.139272
$922$	0	0
$923$	42.2478	1.39060
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.87361	0.0615375
$928$	0	0
$929$	−7.17010	−0.235243	−0.117622	−	0.993058i	$-0.537527\pi$
$929$	−7.17010	−0.235243	−0.117622	+	0.993058i	$0.537527\pi$
$930$	0	0
$931$	−0.245129	−0.00803378
$932$	0	0
$933$	52.2411	1.71030
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−24.2646	−0.792690	−0.396345	−	0.918102i	$-0.629722\pi$
$937$	−24.2646	−0.792690	−0.396345	+	0.918102i	$0.629722\pi$
$938$	0	0
$939$	56.8114	1.85397
$940$	0	0
$941$	−7.44806	−0.242800	−0.121400	−	0.992604i	$-0.538738\pi$
$941$	−7.44806	−0.242800	−0.121400	+	0.992604i	$0.538738\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	52.9538	1.72077	0.860384	−	0.509647i	$-0.170224\pi$
$947$	52.9538	1.72077	0.860384	+	0.509647i	$0.170224\pi$
$948$	0	0
$949$	−15.0489	−0.488507
$950$	0	0
$951$	−47.2900	−1.53348
$952$	0	0
$953$	−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
$956$	0	0
$957$	−10.6403	−0.343953
$958$	0	0
$959$	11.4085	0.368401
$960$	0	0
$961$	25.9538	0.837220
$962$	0	0
$963$	−2.74475	−0.0884482
$964$	0	0
$965$	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	$0.636500\pi$
$968$	0	0
$969$	−8.72203	−0.280192
$970$	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
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−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
$981$	−0.390032	−0.0124528
$982$	0	0
$983$	42.6270	1.35959	0.679795	−	0.733403i	$-0.262069\pi$
$983$	42.6270	1.35959	0.679795	+	0.733403i	$0.262069\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	3.99332	0.127109
$988$	0	0
$989$	−34.4636	−1.09588
$990$	0	0
$991$	34.1737	1.08556	0.542782	−	0.839873i	$-0.317371\pi$
$991$	34.1737	1.08556	0.542782	+	0.839873i	$0.317371\pi$
$992$	0	0
$993$	35.2409	1.11834
$994$	0	0
$995$	0	0
$996$	0	0
$997$	29.3834	0.930580	0.465290	−	0.885158i	$-0.345950\pi$
$997$	29.3834	0.930580	0.465290	+	0.885158i	$0.345950\pi$
$998$	0	0
$999$	−23.0212	−0.728359

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.a.j.1.3	✓	3	20.19	odd	2
950.2.a.l.1.1	yes	3	4.3	odd	2
950.2.b.h.799.1		6	20.3	even	4
950.2.b.h.799.6		6	20.7	even	4
7600.2.a.bk.1.3		3	1.1	even	1	trivial
7600.2.a.bz.1.1		3	5.4	even	2
8550.2.a.ci.1.1		3	12.11	even	2
8550.2.a.cp.1.3		3	60.59	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:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)

\(f(q)\)	\(=\)	\(q+1.77339 q^{3} +2.69168 q^{7} +0.144903 q^{9} +O(q^{10})\)	\(q+1.77339 q^{3} +2.69168 q^{7} +0.144903 q^{9} -5.54677 q^{11} -2.91829 q^{13} +4.91829 q^{17} -1.00000 q^{19} +4.77339 q^{21} -3.60997 q^{23} -5.06319 q^{27} +1.08171 q^{29} -7.54677 q^{31} -9.83658 q^{33} +4.54677 q^{37} -5.17526 q^{39} +9.54677 q^{43} +0.836581 q^{47} +0.245129 q^{49} +8.72203 q^{51} +9.78523 q^{53} -1.77339 q^{57} -12.9933 q^{59} -7.38336 q^{61} +0.390032 q^{63} -2.85510 q^{67} -6.40187 q^{69} -14.4769 q^{71} +5.15674 q^{73} -14.9301 q^{77} +3.09355 q^{79} -9.41371 q^{81} -1.71019 q^{83} +1.91829 q^{87} -5.09355 q^{89} -7.85510 q^{91} -13.3834 q^{93} -17.2570 q^{97} -0.803744 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} - 2 q^{7} + 5 q^{9}+O(q^{10}) \) 3 * q - 2 * q^3 - 2 * q^7 + 5 * q^9	\( 3 q - 2 q^{3} - 2 q^{7} + 5 q^{9} - 2 q^{11} - 6 q^{13} + 12 q^{17} - 3 q^{19} + 7 q^{21} + 2 q^{23} - 17 q^{27} + 6 q^{29} - 8 q^{31} - 24 q^{33} - q^{37} + 11 q^{39} + 14 q^{43} - 3 q^{47} + 9 q^{49} - 15 q^{51} - 10 q^{53} + 2 q^{57} - 6 q^{59} - 2 q^{61} + 14 q^{63} - 4 q^{67} + 6 q^{71} - 12 q^{73} - 10 q^{77} - 20 q^{79} + 23 q^{81} + 4 q^{83} + 3 q^{87} + 14 q^{89} - 19 q^{91} - 20 q^{93} - 28 q^{97} + 36 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 - 2 * q^7 + 5 * q^9 - 2 * q^11 - 6 * q^13 + 12 * q^17 - 3 * q^19 + 7 * q^21 + 2 * q^23 - 17 * q^27 + 6 * q^29 - 8 * q^31 - 24 * q^33 - q^37 + 11 * q^39 + 14 * q^43 - 3 * q^47 + 9 * q^49 - 15 * q^51 - 10 * q^53 + 2 * q^57 - 6 * q^59 - 2 * q^61 + 14 * q^63 - 4 * q^67 + 6 * q^71 - 12 * q^73 - 10 * q^77 - 20 * q^79 + 23 * q^81 + 4 * q^83 + 3 * q^87 + 14 * q^89 - 19 * q^91 - 20 * q^93 - 28 * q^97 + 36 * q^99

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