LMFDB - Embedded newform 6240.2.a.cc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6240,2,Mod(1,6240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6240.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:	$49.8266508613$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1849.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 14x - 8 $ x^3 - x^2 - 14*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.615072$ of defining polynomial
Character	$\chi$	$=$	6240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} -2.61507 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} -2.61507 q^{7} +1.00000 q^{9} -2.61507 q^{11} -1.00000 q^{13} +1.00000 q^{15} +4.39154 q^{17} -7.77647 q^{19} -2.61507 q^{21} +6.39154 q^{23} +1.00000 q^{25} +1.00000 q^{27} +3.23014 q^{29} -5.00662 q^{31} -2.61507 q^{33} -2.61507 q^{35} -4.39154 q^{37} -1.00000 q^{39} +11.1614 q^{41} -1.23014 q^{43} +1.00000 q^{45} +9.00662 q^{47} -0.161400 q^{49} +4.39154 q^{51} +12.3915 q^{53} -2.61507 q^{55} -7.77647 q^{57} -9.00662 q^{59} -0.391544 q^{61} -2.61507 q^{63} -1.00000 q^{65} +13.0066 q^{67} +6.39154 q^{69} +5.38493 q^{71} +2.00000 q^{73} +1.00000 q^{75} +6.83860 q^{77} +10.3915 q^{79} +1.00000 q^{81} -13.0066 q^{83} +4.39154 q^{85} +3.23014 q^{87} -1.62169 q^{89} +2.61507 q^{91} -5.00662 q^{93} -7.77647 q^{95} +5.93126 q^{97} -2.61507 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{5} - 5 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 3 * q^5 - 5 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} + 3 q^{5} - 5 q^{7} + 3 q^{9} - 5 q^{11} - 3 q^{13} + 3 q^{15} - 9 q^{17} - 4 q^{19} - 5 q^{21} - 3 q^{23} + 3 q^{25} + 3 q^{27} + 4 q^{29} + 10 q^{31} - 5 q^{33} - 5 q^{35} + 9 q^{37} - 3 q^{39} + 17 q^{41} + 2 q^{43} + 3 q^{45} + 2 q^{47} + 16 q^{49} - 9 q^{51} + 15 q^{53} - 5 q^{55} - 4 q^{57} - 2 q^{59} + 21 q^{61} - 5 q^{63} - 3 q^{65} + 14 q^{67} - 3 q^{69} + 19 q^{71} + 6 q^{73} + 3 q^{75} + 37 q^{77} + 9 q^{79} + 3 q^{81} - 14 q^{83} - 9 q^{85} + 4 q^{87} + 23 q^{89} + 5 q^{91} + 10 q^{93} - 4 q^{95} + 7 q^{97} - 5 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^5 - 5 * q^7 + 3 * q^9 - 5 * q^11 - 3 * q^13 + 3 * q^15 - 9 * q^17 - 4 * q^19 - 5 * q^21 - 3 * q^23 + 3 * q^25 + 3 * q^27 + 4 * q^29 + 10 * q^31 - 5 * q^33 - 5 * q^35 + 9 * q^37 - 3 * q^39 + 17 * q^41 + 2 * q^43 + 3 * q^45 + 2 * q^47 + 16 * q^49 - 9 * q^51 + 15 * q^53 - 5 * q^55 - 4 * q^57 - 2 * q^59 + 21 * q^61 - 5 * q^63 - 3 * q^65 + 14 * q^67 - 3 * q^69 + 19 * q^71 + 6 * q^73 + 3 * q^75 + 37 * q^77 + 9 * q^79 + 3 * q^81 - 14 * q^83 - 9 * q^85 + 4 * q^87 + 23 * q^89 + 5 * q^91 + 10 * q^93 - 4 * q^95 + 7 * q^97 - 5 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.61507	−0.988404	−0.494202	−	0.869347i	$-0.664540\pi$
$7$	−2.61507	−0.988404	−0.494202	+	0.869347i	$0.664540\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.61507	−0.788474	−0.394237	−	0.919009i	$-0.628991\pi$
$11$	−2.61507	−0.788474	−0.394237	+	0.919009i	$0.628991\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	4.39154	1.06511	0.532553	−	0.846397i	$-0.321233\pi$
$17$	4.39154	1.06511	0.532553	+	0.846397i	$0.321233\pi$
$18$	0	0
$19$	−7.77647	−1.78405	−0.892023	−	0.451991i	$-0.850714\pi$
$19$	−7.77647	−1.78405	−0.892023	+	0.451991i	$0.850714\pi$
$20$	0	0
$21$	−2.61507	−0.570655
$22$	0	0
$23$	6.39154	1.33273	0.666364	−	0.745626i	$-0.267849\pi$
$23$	6.39154	1.33273	0.666364	+	0.745626i	$0.267849\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.23014	0.599823	0.299911	−	0.953967i	$-0.403043\pi$
$29$	3.23014	0.599823	0.299911	+	0.953967i	$0.403043\pi$
$30$	0	0
$31$	−5.00662	−0.899215	−0.449607	−	0.893226i	$-0.648436\pi$
$31$	−5.00662	−0.899215	−0.449607	+	0.893226i	$0.648436\pi$
$32$	0	0
$33$	−2.61507	−0.455226
$34$	0	0
$35$	−2.61507	−0.442028
$36$	0	0
$37$	−4.39154	−0.721965	−0.360983	−	0.932573i	$-0.617559\pi$
$37$	−4.39154	−0.721965	−0.360983	+	0.932573i	$0.617559\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	11.1614	1.74312	0.871559	−	0.490291i	$-0.163109\pi$
$41$	11.1614	1.74312	0.871559	+	0.490291i	$0.163109\pi$
$42$	0	0
$43$	−1.23014	−0.187595	−0.0937975	−	0.995591i	$-0.529901\pi$
$43$	−1.23014	−0.187595	−0.0937975	+	0.995591i	$0.529901\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	9.00662	1.31375	0.656875	−	0.754000i	$-0.271878\pi$
$47$	9.00662	1.31375	0.656875	+	0.754000i	$0.271878\pi$
$48$	0	0
$49$	−0.161400	−0.0230572
$50$	0	0
$51$	4.39154	0.614939
$52$	0	0
$53$	12.3915	1.70211	0.851055	−	0.525077i	$-0.175963\pi$
$53$	12.3915	1.70211	0.851055	+	0.525077i	$0.175963\pi$
$54$	0	0
$55$	−2.61507	−0.352616
$56$	0	0
$57$	−7.77647	−1.03002
$58$	0	0
$59$	−9.00662	−1.17256	−0.586281	−	0.810108i	$-0.699408\pi$
$59$	−9.00662	−1.17256	−0.586281	+	0.810108i	$0.699408\pi$
$60$	0	0
$61$	−0.391544	−0.0501320	−0.0250660	−	0.999686i	$-0.507980\pi$
$61$	−0.391544	−0.0501320	−0.0250660	+	0.999686i	$0.507980\pi$
$62$	0	0
$63$	−2.61507	−0.329468
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	13.0066	1.58901	0.794505	−	0.607257i	$-0.207730\pi$
$67$	13.0066	1.58901	0.794505	+	0.607257i	$0.207730\pi$
$68$	0	0
$69$	6.39154	0.769451
$70$	0	0
$71$	5.38493	0.639073	0.319537	−	0.947574i	$-0.396473\pi$
$71$	5.38493	0.639073	0.319537	+	0.947574i	$0.396473\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	6.83860	0.779331
$78$	0	0
$79$	10.3915	1.16914	0.584570	−	0.811343i	$-0.301263\pi$
$79$	10.3915	1.16914	0.584570	+	0.811343i	$0.301263\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−13.0066	−1.42766	−0.713831	−	0.700318i	$-0.753041\pi$
$83$	−13.0066	−1.42766	−0.713831	+	0.700318i	$0.753041\pi$
$84$	0	0
$85$	4.39154	0.476330
$86$	0	0
$87$	3.23014	0.346308
$88$	0	0
$89$	−1.62169	−0.171898	−0.0859492	−	0.996300i	$-0.527392\pi$
$89$	−1.62169	−0.171898	−0.0859492	+	0.996300i	$0.527392\pi$
$90$	0	0
$91$	2.61507	0.274134
$92$	0	0
$93$	−5.00662	−0.519162
$94$	0	0
$95$	−7.77647	−0.797849
$96$	0	0
$97$	5.93126	0.602228	0.301114	−	0.953588i	$-0.402642\pi$
$97$	5.93126	0.602228	0.301114	+	0.953588i	$0.402642\pi$
$98$	0	0
$99$	−2.61507	−0.262825
$100$	0	0
$101$	−6.78309	−0.674942	−0.337471	−	0.941336i	$-0.609572\pi$
$101$	−6.78309	−0.674942	−0.337471	+	0.941336i	$0.609572\pi$
$102$	0	0
$103$	−2.46029	−0.242419	−0.121210	−	0.992627i	$-0.538677\pi$
$103$	−2.46029	−0.242419	−0.121210	+	0.992627i	$0.538677\pi$
$104$	0	0
$105$	−2.61507	−0.255205
$106$	0	0
$107$	18.3915	1.77798	0.888989	−	0.457929i	$-0.151409\pi$
$107$	18.3915	1.77798	0.888989	+	0.457929i	$0.151409\pi$
$108$	0	0
$109$	11.2301	1.07565	0.537826	−	0.843056i	$-0.319246\pi$
$109$	11.2301	1.07565	0.537826	+	0.843056i	$0.319246\pi$
$110$	0	0
$111$	−4.39154	−0.416827
$112$	0	0
$113$	0.460287	0.0433001	0.0216501	−	0.999766i	$-0.493108\pi$
$113$	0.460287	0.0433001	0.0216501	+	0.999766i	$0.493108\pi$
$114$	0	0
$115$	6.39154	0.596015
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−11.4842	−1.05275
$120$	0	0
$121$	−4.16140	−0.378309
$122$	0	0
$123$	11.1614	1.00639
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	10.4603	0.928200	0.464100	−	0.885783i	$-0.346378\pi$
$127$	10.4603	0.928200	0.464100	+	0.885783i	$0.346378\pi$
$128$	0	0
$129$	−1.23014	−0.108308
$130$	0	0
$131$	10.0132	0.874860	0.437430	−	0.899252i	$-0.355889\pi$
$131$	10.0132	0.874860	0.437430	+	0.899252i	$0.355889\pi$
$132$	0	0
$133$	20.3360	1.76336
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	−7.93126	−0.672720	−0.336360	−	0.941733i	$-0.609196\pi$
$139$	−7.93126	−0.672720	−0.336360	+	0.941733i	$0.609196\pi$
$140$	0	0
$141$	9.00662	0.758494
$142$	0	0
$143$	2.61507	0.218683
$144$	0	0
$145$	3.23014	0.268249
$146$	0	0
$147$	−0.161400	−0.0133121
$148$	0	0
$149$	9.93126	0.813600	0.406800	−	0.913517i	$-0.366645\pi$
$149$	9.93126	0.813600	0.406800	+	0.913517i	$0.366645\pi$
$150$	0	0
$151$	13.0066	1.05846	0.529232	−	0.848477i	$-0.322480\pi$
$151$	13.0066	1.05846	0.529232	+	0.848477i	$0.322480\pi$
$152$	0	0
$153$	4.39154	0.355035
$154$	0	0
$155$	−5.00662	−0.402141
$156$	0	0
$157$	4.46029	0.355970	0.177985	−	0.984033i	$-0.443042\pi$
$157$	4.46029	0.355970	0.177985	+	0.984033i	$0.443042\pi$
$158$	0	0
$159$	12.3915	0.982713
$160$	0	0
$161$	−16.7143	−1.31727
$162$	0	0
$163$	−4.15479	−0.325428	−0.162714	−	0.986673i	$-0.552025\pi$
$163$	−4.15479	−0.325428	−0.162714	+	0.986673i	$0.552025\pi$
$164$	0	0
$165$	−2.61507	−0.203583
$166$	0	0
$167$	6.54633	0.506570	0.253285	−	0.967392i	$-0.418489\pi$
$167$	6.54633	0.506570	0.253285	+	0.967392i	$0.418489\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−7.77647	−0.594682
$172$	0	0
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	0	0
$175$	−2.61507	−0.197681
$176$	0	0
$177$	−9.00662	−0.676979
$178$	0	0
$179$	12.7831	0.955453	0.477726	−	0.878509i	$-0.341461\pi$
$179$	12.7831	0.955453	0.477726	+	0.878509i	$0.341461\pi$
$180$	0	0
$181$	7.16140	0.532303	0.266151	−	0.963931i	$-0.414248\pi$
$181$	7.16140	0.532303	0.266151	+	0.963931i	$0.414248\pi$
$182$	0	0
$183$	−0.391544	−0.0289437
$184$	0	0
$185$	−4.39154	−0.322873
$186$	0	0
$187$	−11.4842	−0.839808
$188$	0	0
$189$	−2.61507	−0.190218
$190$	0	0
$191$	−2.76986	−0.200420	−0.100210	−	0.994966i	$-0.531951\pi$
$191$	−2.76986	−0.200420	−0.100210	+	0.994966i	$0.531951\pi$
$192$	0	0
$193$	−14.8518	−1.06906	−0.534529	−	0.845150i	$-0.679511\pi$
$193$	−14.8518	−1.06906	−0.534529	+	0.845150i	$0.679511\pi$
$194$	0	0
$195$	−1.00000	−0.0716115
$196$	0	0
$197$	8.46029	0.602770	0.301385	−	0.953502i	$-0.402551\pi$
$197$	8.46029	0.602770	0.301385	+	0.953502i	$0.402551\pi$
$198$	0	0
$199$	11.5529	0.818966	0.409483	−	0.912318i	$-0.365709\pi$
$199$	11.5529	0.818966	0.409483	+	0.912318i	$0.365709\pi$
$200$	0	0
$201$	13.0066	0.917416
$202$	0	0
$203$	−8.44706	−0.592867
$204$	0	0
$205$	11.1614	0.779546
$206$	0	0
$207$	6.39154	0.444243
$208$	0	0
$209$	20.3360	1.40667
$210$	0	0
$211$	14.4603	0.995487	0.497744	−	0.867324i	$-0.334162\pi$
$211$	14.4603	0.995487	0.497744	+	0.867324i	$0.334162\pi$
$212$	0	0
$213$	5.38493	0.368969
$214$	0	0
$215$	−1.23014	−0.0838951
$216$	0	0
$217$	13.0927	0.888788
$218$	0	0
$219$	2.00000	0.135147
$220$	0	0
$221$	−4.39154	−0.295407
$222$	0	0
$223$	−24.5596	−1.64463	−0.822315	−	0.569033i	$-0.807318\pi$
$223$	−24.5596	−1.64463	−0.822315	+	0.569033i	$0.807318\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−23.3294	−1.54843	−0.774214	−	0.632924i	$-0.781855\pi$
$227$	−23.3294	−1.54843	−0.774214	+	0.632924i	$0.781855\pi$
$228$	0	0
$229$	−15.2301	−1.00644	−0.503218	−	0.864159i	$-0.667851\pi$
$229$	−15.2301	−1.00644	−0.503218	+	0.864159i	$0.667851\pi$
$230$	0	0
$231$	6.83860	0.449947
$232$	0	0
$233$	20.0820	1.31561	0.657807	−	0.753187i	$-0.271484\pi$
$233$	20.0820	1.31561	0.657807	+	0.753187i	$0.271484\pi$
$234$	0	0
$235$	9.00662	0.587527
$236$	0	0
$237$	10.3915	0.675003
$238$	0	0
$239$	5.38493	0.348322	0.174161	−	0.984717i	$-0.444279\pi$
$239$	5.38493	0.348322	0.174161	+	0.984717i	$0.444279\pi$
$240$	0	0
$241$	20.3228	1.30911	0.654553	−	0.756016i	$-0.272857\pi$
$241$	20.3228	1.30911	0.654553	+	0.756016i	$0.272857\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−0.161400	−0.0103115
$246$	0	0
$247$	7.77647	0.494805
$248$	0	0
$249$	−13.0066	−0.824261
$250$	0	0
$251$	−8.00000	−0.504956	−0.252478	−	0.967603i	$-0.581245\pi$
$251$	−8.00000	−0.504956	−0.252478	+	0.967603i	$0.581245\pi$
$252$	0	0
$253$	−16.7143	−1.05082
$254$	0	0
$255$	4.39154	0.275009
$256$	0	0
$257$	−28.0132	−1.74742	−0.873709	−	0.486450i	$-0.838292\pi$
$257$	−28.0132	−1.74742	−0.873709	+	0.486450i	$0.838292\pi$
$258$	0	0
$259$	11.4842	0.713594
$260$	0	0
$261$	3.23014	0.199941
$262$	0	0
$263$	18.0132	1.11074	0.555372	−	0.831602i	$-0.312576\pi$
$263$	18.0132	1.11074	0.555372	+	0.831602i	$0.312576\pi$
$264$	0	0
$265$	12.3915	0.761206
$266$	0	0
$267$	−1.62169	−0.0992456
$268$	0	0
$269$	3.23014	0.196945	0.0984727	−	0.995140i	$-0.468604\pi$
$269$	3.23014	0.196945	0.0984727	+	0.995140i	$0.468604\pi$
$270$	0	0
$271$	−10.2368	−0.621839	−0.310919	−	0.950436i	$-0.600637\pi$
$271$	−10.2368	−0.621839	−0.310919	+	0.950436i	$0.600637\pi$
$272$	0	0
$273$	2.61507	0.158271
$274$	0	0
$275$	−2.61507	−0.157695
$276$	0	0
$277$	−3.23014	−0.194080	−0.0970402	−	0.995280i	$-0.530938\pi$
$277$	−3.23014	−0.194080	−0.0970402	+	0.995280i	$0.530938\pi$
$278$	0	0
$279$	−5.00662	−0.299738
$280$	0	0
$281$	20.0132	1.19389	0.596945	−	0.802282i	$-0.296381\pi$
$281$	20.0132	1.19389	0.596945	+	0.802282i	$0.296381\pi$
$282$	0	0
$283$	−14.0132	−0.833000	−0.416500	−	0.909136i	$-0.636743\pi$
$283$	−14.0132	−0.833000	−0.416500	+	0.909136i	$0.636743\pi$
$284$	0	0
$285$	−7.77647	−0.460638
$286$	0	0
$287$	−29.1879	−1.72290
$288$	0	0
$289$	2.28566	0.134450
$290$	0	0
$291$	5.93126	0.347696
$292$	0	0
$293$	−4.32280	−0.252541	−0.126270	−	0.991996i	$-0.540301\pi$
$293$	−4.32280	−0.252541	−0.126270	+	0.991996i	$0.540301\pi$
$294$	0	0
$295$	−9.00662	−0.524385
$296$	0	0
$297$	−2.61507	−0.151742
$298$	0	0
$299$	−6.39154	−0.369633
$300$	0	0
$301$	3.21691	0.185420
$302$	0	0
$303$	−6.78309	−0.389678
$304$	0	0
$305$	−0.391544	−0.0224197
$306$	0	0
$307$	6.92464	0.395210	0.197605	−	0.980282i	$-0.436684\pi$
$307$	6.92464	0.395210	0.197605	+	0.980282i	$0.436684\pi$
$308$	0	0
$309$	−2.46029	−0.139961
$310$	0	0
$311$	7.24337	0.410734	0.205367	−	0.978685i	$-0.434161\pi$
$311$	7.24337	0.410734	0.205367	+	0.978685i	$0.434161\pi$
$312$	0	0
$313$	−32.0132	−1.80949	−0.904747	−	0.425949i	$-0.859940\pi$
$313$	−32.0132	−1.80949	−0.904747	+	0.425949i	$0.859940\pi$
$314$	0	0
$315$	−2.61507	−0.147343
$316$	0	0
$317$	31.5662	1.77293	0.886466	−	0.462793i	$-0.153153\pi$
$317$	31.5662	1.77293	0.886466	+	0.462793i	$0.153153\pi$
$318$	0	0
$319$	−8.44706	−0.472944
$320$	0	0
$321$	18.3915	1.02652
$322$	0	0
$323$	−34.1507	−1.90020
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	11.2301	0.621028
$328$	0	0
$329$	−23.5529	−1.29852
$330$	0	0
$331$	1.78970	0.0983709	0.0491855	−	0.998790i	$-0.484337\pi$
$331$	1.78970	0.0983709	0.0491855	+	0.998790i	$0.484337\pi$
$332$	0	0
$333$	−4.39154	−0.240655
$334$	0	0
$335$	13.0066	0.710627
$336$	0	0
$337$	12.0132	0.654402	0.327201	−	0.944955i	$-0.393895\pi$
$337$	12.0132	0.654402	0.327201	+	0.944955i	$0.393895\pi$
$338$	0	0
$339$	0.460287	0.0249993
$340$	0	0
$341$	13.0927	0.709007
$342$	0	0
$343$	18.7276	1.01119
$344$	0	0
$345$	6.39154	0.344109
$346$	0	0
$347$	−3.14817	−0.169003	−0.0845013	−	0.996423i	$-0.526930\pi$
$347$	−3.14817	−0.169003	−0.0845013	+	0.996423i	$0.526930\pi$
$348$	0	0
$349$	−9.55294	−0.511357	−0.255679	−	0.966762i	$-0.582299\pi$
$349$	−9.55294	−0.511357	−0.255679	+	0.966762i	$0.582299\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−9.24337	−0.491975	−0.245988	−	0.969273i	$-0.579112\pi$
$353$	−9.24337	−0.491975	−0.245988	+	0.969273i	$0.579112\pi$
$354$	0	0
$355$	5.38493	0.285802
$356$	0	0
$357$	−11.4842	−0.607808
$358$	0	0
$359$	−24.5596	−1.29620	−0.648102	−	0.761554i	$-0.724437\pi$
$359$	−24.5596	−1.29620	−0.648102	+	0.761554i	$0.724437\pi$
$360$	0	0
$361$	41.4735	2.18282
$362$	0	0
$363$	−4.16140	−0.218417
$364$	0	0
$365$	2.00000	0.104685
$366$	0	0
$367$	13.2301	0.690608	0.345304	−	0.938491i	$-0.387776\pi$
$367$	13.2301	0.690608	0.345304	+	0.938491i	$0.387776\pi$
$368$	0	0
$369$	11.1614	0.581039
$370$	0	0
$371$	−32.4048	−1.68237
$372$	0	0
$373$	−34.0265	−1.76182	−0.880912	−	0.473281i	$-0.843070\pi$
$373$	−34.0265	−1.76182	−0.880912	+	0.473281i	$0.843070\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−3.23014	−0.166361
$378$	0	0
$379$	−5.45367	−0.280136	−0.140068	−	0.990142i	$-0.544732\pi$
$379$	−5.45367	−0.280136	−0.140068	+	0.990142i	$0.544732\pi$
$380$	0	0
$381$	10.4603	0.535897
$382$	0	0
$383$	6.23676	0.318683	0.159342	−	0.987223i	$-0.449063\pi$
$383$	6.23676	0.318683	0.159342	+	0.987223i	$0.449063\pi$
$384$	0	0
$385$	6.83860	0.348527
$386$	0	0
$387$	−1.23014	−0.0625317
$388$	0	0
$389$	−24.7963	−1.25722	−0.628612	−	0.777719i	$-0.716376\pi$
$389$	−24.7963	−1.25722	−0.628612	+	0.777719i	$0.716376\pi$
$390$	0	0
$391$	28.0687	1.41950
$392$	0	0
$393$	10.0132	0.505101
$394$	0	0
$395$	10.3915	0.522855
$396$	0	0
$397$	23.6349	1.18620	0.593101	−	0.805128i	$-0.297903\pi$
$397$	23.6349	1.18620	0.593101	+	0.805128i	$0.297903\pi$
$398$	0	0
$399$	20.3360	1.01807
$400$	0	0
$401$	36.0132	1.79841	0.899207	−	0.437523i	$-0.144144\pi$
$401$	36.0132	1.79841	0.899207	+	0.437523i	$0.144144\pi$
$402$	0	0
$403$	5.00662	0.249397
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	11.4842	0.569251
$408$	0	0
$409$	27.7037	1.36986	0.684929	−	0.728610i	$-0.259833\pi$
$409$	27.7037	1.36986	0.684929	+	0.728610i	$0.259833\pi$
$410$	0	0
$411$	−10.0000	−0.493264
$412$	0	0
$413$	23.5529	1.15896
$414$	0	0
$415$	−13.0066	−0.638470
$416$	0	0
$417$	−7.93126	−0.388395
$418$	0	0
$419$	−26.7699	−1.30779	−0.653897	−	0.756584i	$-0.726867\pi$
$419$	−26.7699	−1.30779	−0.653897	+	0.756584i	$0.726867\pi$
$420$	0	0
$421$	−9.69043	−0.472283	−0.236141	−	0.971719i	$-0.575883\pi$
$421$	−9.69043	−0.472283	−0.236141	+	0.971719i	$0.575883\pi$
$422$	0	0
$423$	9.00662	0.437917
$424$	0	0
$425$	4.39154	0.213021
$426$	0	0
$427$	1.02391	0.0495507
$428$	0	0
$429$	2.61507	0.126257
$430$	0	0
$431$	16.5596	0.797646	0.398823	−	0.917028i	$-0.369419\pi$
$431$	16.5596	0.797646	0.398823	+	0.917028i	$0.369419\pi$
$432$	0	0
$433$	15.5397	0.746791	0.373395	−	0.927672i	$-0.378193\pi$
$433$	15.5397	0.746791	0.373395	+	0.927672i	$0.378193\pi$
$434$	0	0
$435$	3.23014	0.154874
$436$	0	0
$437$	−49.7037	−2.37765
$438$	0	0
$439$	39.6217	1.89104	0.945520	−	0.325564i	$-0.105554\pi$
$439$	39.6217	1.89104	0.945520	+	0.325564i	$0.105554\pi$
$440$	0	0
$441$	−0.161400	−0.00768573
$442$	0	0
$443$	−3.14817	−0.149574	−0.0747870	−	0.997200i	$-0.523828\pi$
$443$	−3.14817	−0.149574	−0.0747870	+	0.997200i	$0.523828\pi$
$444$	0	0
$445$	−1.62169	−0.0768753
$446$	0	0
$447$	9.93126	0.469732
$448$	0	0
$449$	36.7276	1.73328	0.866641	−	0.498933i	$-0.166275\pi$
$449$	36.7276	1.73328	0.866641	+	0.498933i	$0.166275\pi$
$450$	0	0
$451$	−29.1879	−1.37440
$452$	0	0
$453$	13.0066	0.611104
$454$	0	0
$455$	2.61507	0.122596
$456$	0	0
$457$	−27.6349	−1.29271	−0.646353	−	0.763038i	$-0.723707\pi$
$457$	−27.6349	−1.29271	−0.646353	+	0.763038i	$0.723707\pi$
$458$	0	0
$459$	4.39154	0.204980
$460$	0	0
$461$	20.2541	0.943326	0.471663	−	0.881779i	$-0.343654\pi$
$461$	20.2541	0.943326	0.471663	+	0.881779i	$0.343654\pi$
$462$	0	0
$463$	−30.6415	−1.42403	−0.712016	−	0.702163i	$-0.752218\pi$
$463$	−30.6415	−1.42403	−0.712016	+	0.702163i	$0.752218\pi$
$464$	0	0
$465$	−5.00662	−0.232176
$466$	0	0
$467$	−21.1614	−0.979233	−0.489616	−	0.871938i	$-0.662863\pi$
$467$	−21.1614	−0.979233	−0.489616	+	0.871938i	$0.662863\pi$
$468$	0	0
$469$	−34.0132	−1.57059
$470$	0	0
$471$	4.46029	0.205519
$472$	0	0
$473$	3.21691	0.147914
$474$	0	0
$475$	−7.77647	−0.356809
$476$	0	0
$477$	12.3915	0.567370
$478$	0	0
$479$	−23.8452	−1.08952	−0.544758	−	0.838593i	$-0.683378\pi$
$479$	−23.8452	−1.08952	−0.544758	+	0.838593i	$0.683378\pi$
$480$	0	0
$481$	4.39154	0.200237
$482$	0	0
$483$	−16.7143	−0.760529
$484$	0	0
$485$	5.93126	0.269325
$486$	0	0
$487$	−14.9511	−0.677499	−0.338750	−	0.940877i	$-0.610004\pi$
$487$	−14.9511	−0.677499	−0.338750	+	0.940877i	$0.610004\pi$
$488$	0	0
$489$	−4.15479	−0.187886
$490$	0	0
$491$	−34.3228	−1.54897	−0.774483	−	0.632595i	$-0.781990\pi$
$491$	−34.3228	−1.54897	−0.774483	+	0.632595i	$0.781990\pi$
$492$	0	0
$493$	14.1853	0.638874
$494$	0	0
$495$	−2.61507	−0.117539
$496$	0	0
$497$	−14.0820	−0.631663
$498$	0	0
$499$	−5.31619	−0.237985	−0.118993	−	0.992895i	$-0.537966\pi$
$499$	−5.31619	−0.237985	−0.118993	+	0.992895i	$0.537966\pi$
$500$	0	0
$501$	6.54633	0.292468
$502$	0	0
$503$	−18.0132	−0.803170	−0.401585	−	0.915822i	$-0.631541\pi$
$503$	−18.0132	−0.803170	−0.401585	+	0.915822i	$0.631541\pi$
$504$	0	0
$505$	−6.78309	−0.301843
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	21.9577	0.973259	0.486629	−	0.873609i	$-0.338226\pi$
$509$	21.9577	0.973259	0.486629	+	0.873609i	$0.338226\pi$
$510$	0	0
$511$	−5.23014	−0.231368
$512$	0	0
$513$	−7.77647	−0.343340
$514$	0	0
$515$	−2.46029	−0.108413
$516$	0	0
$517$	−23.5529	−1.03586
$518$	0	0
$519$	14.0000	0.614532
$520$	0	0
$521$	−13.5529	−0.593765	−0.296883	−	0.954914i	$-0.595947\pi$
$521$	−13.5529	−0.593765	−0.296883	+	0.954914i	$0.595947\pi$
$522$	0	0
$523$	−22.3228	−0.976108	−0.488054	−	0.872813i	$-0.662293\pi$
$523$	−22.3228	−0.976108	−0.488054	+	0.872813i	$0.662293\pi$
$524$	0	0
$525$	−2.61507	−0.114131
$526$	0	0
$527$	−21.9868	−0.957759
$528$	0	0
$529$	17.8518	0.776167
$530$	0	0
$531$	−9.00662	−0.390854
$532$	0	0
$533$	−11.1614	−0.483454
$534$	0	0
$535$	18.3915	0.795136
$536$	0	0
$537$	12.7831	0.551631
$538$	0	0
$539$	0.422074	0.0181800
$540$	0	0
$541$	−2.44706	−0.105207	−0.0526036	−	0.998615i	$-0.516752\pi$
$541$	−2.44706	−0.105207	−0.0526036	+	0.998615i	$0.516752\pi$
$542$	0	0
$543$	7.16140	0.307325
$544$	0	0
$545$	11.2301	0.481046
$546$	0	0
$547$	6.46029	0.276222	0.138111	−	0.990417i	$-0.455897\pi$
$547$	6.46029	0.276222	0.138111	+	0.990417i	$0.455897\pi$
$548$	0	0
$549$	−0.391544	−0.0167107
$550$	0	0
$551$	−25.1191	−1.07011
$552$	0	0
$553$	−27.1746	−1.15558
$554$	0	0
$555$	−4.39154	−0.186411
$556$	0	0
$557$	31.7037	1.34333	0.671664	−	0.740856i	$-0.265580\pi$
$557$	31.7037	1.34333	0.671664	+	0.740856i	$0.265580\pi$
$558$	0	0
$559$	1.23014	0.0520295
$560$	0	0
$561$	−11.4842	−0.484863
$562$	0	0
$563$	−22.7276	−0.957853	−0.478927	−	0.877855i	$-0.658974\pi$
$563$	−22.7276	−0.957853	−0.478927	+	0.877855i	$0.658974\pi$
$564$	0	0
$565$	0.460287	0.0193644
$566$	0	0
$567$	−2.61507	−0.109823
$568$	0	0
$569$	−26.0265	−1.09109	−0.545543	−	0.838083i	$-0.683677\pi$
$569$	−26.0265	−1.09109	−0.545543	+	0.838083i	$0.683677\pi$
$570$	0	0
$571$	10.8386	0.453581	0.226791	−	0.973944i	$-0.427177\pi$
$571$	10.8386	0.453581	0.226791	+	0.973944i	$0.427177\pi$
$572$	0	0
$573$	−2.76986	−0.115712
$574$	0	0
$575$	6.39154	0.266546
$576$	0	0
$577$	−41.1746	−1.71412	−0.857061	−	0.515215i	$-0.827712\pi$
$577$	−41.1746	−1.71412	−0.857061	+	0.515215i	$0.827712\pi$
$578$	0	0
$579$	−14.8518	−0.617221
$580$	0	0
$581$	34.0132	1.41111
$582$	0	0
$583$	−32.4048	−1.34207
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	0	0
$587$	10.2368	0.422516	0.211258	−	0.977430i	$-0.432244\pi$
$587$	10.2368	0.422516	0.211258	+	0.977430i	$0.432244\pi$
$588$	0	0
$589$	38.9338	1.60424
$590$	0	0
$591$	8.46029	0.348010
$592$	0	0
$593$	23.7037	0.973393	0.486696	−	0.873571i	$-0.338202\pi$
$593$	23.7037	0.973393	0.486696	+	0.873571i	$0.338202\pi$
$594$	0	0
$595$	−11.4842	−0.470806
$596$	0	0
$597$	11.5529	0.472831
$598$	0	0
$599$	13.0927	0.534951	0.267476	−	0.963565i	$-0.413810\pi$
$599$	13.0927	0.534951	0.267476	+	0.963565i	$0.413810\pi$
$600$	0	0
$601$	36.8651	1.50376	0.751879	−	0.659302i	$-0.229148\pi$
$601$	36.8651	1.50376	0.751879	+	0.659302i	$0.229148\pi$
$602$	0	0
$603$	13.0066	0.529670
$604$	0	0
$605$	−4.16140	−0.169185
$606$	0	0
$607$	23.6904	0.961565	0.480782	−	0.876840i	$-0.340353\pi$
$607$	23.6904	0.961565	0.480782	+	0.876840i	$0.340353\pi$
$608$	0	0
$609$	−8.44706	−0.342292
$610$	0	0
$611$	−9.00662	−0.364369
$612$	0	0
$613$	29.3121	1.18391	0.591953	−	0.805973i	$-0.298357\pi$
$613$	29.3121	1.18391	0.591953	+	0.805973i	$0.298357\pi$
$614$	0	0
$615$	11.1614	0.450071
$616$	0	0
$617$	−4.46029	−0.179564	−0.0897822	−	0.995961i	$-0.528617\pi$
$617$	−4.46029	−0.179564	−0.0897822	+	0.995961i	$0.528617\pi$
$618$	0	0
$619$	−46.5728	−1.87192	−0.935959	−	0.352108i	$-0.885465\pi$
$619$	−46.5728	−1.87192	−0.935959	+	0.352108i	$0.885465\pi$
$620$	0	0
$621$	6.39154	0.256484
$622$	0	0
$623$	4.24083	0.169905
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	20.3360	0.812143
$628$	0	0
$629$	−19.2857	−0.768969
$630$	0	0
$631$	26.5463	1.05679	0.528396	−	0.848998i	$-0.322794\pi$
$631$	26.5463	1.05679	0.528396	+	0.848998i	$0.322794\pi$
$632$	0	0
$633$	14.4603	0.574745
$634$	0	0
$635$	10.4603	0.415104
$636$	0	0
$637$	0.161400	0.00639492
$638$	0	0
$639$	5.38493	0.213024
$640$	0	0
$641$	−28.9338	−1.14282	−0.571408	−	0.820666i	$-0.693603\pi$
$641$	−28.9338	−1.14282	−0.571408	+	0.820666i	$0.693603\pi$
$642$	0	0
$643$	39.4246	1.55476	0.777378	−	0.629034i	$-0.216549\pi$
$643$	39.4246	1.55476	0.777378	+	0.629034i	$0.216549\pi$
$644$	0	0
$645$	−1.23014	−0.0484368
$646$	0	0
$647$	6.52903	0.256683	0.128341	−	0.991730i	$-0.459035\pi$
$647$	6.52903	0.256683	0.128341	+	0.991730i	$0.459035\pi$
$648$	0	0
$649$	23.5529	0.924534
$650$	0	0
$651$	13.0927	0.513142
$652$	0	0
$653$	−9.55294	−0.373836	−0.186918	−	0.982376i	$-0.559850\pi$
$653$	−9.55294	−0.373836	−0.186918	+	0.982376i	$0.559850\pi$
$654$	0	0
$655$	10.0132	0.391249
$656$	0	0
$657$	2.00000	0.0780274
$658$	0	0
$659$	10.0132	0.390060	0.195030	−	0.980797i	$-0.437520\pi$
$659$	10.0132	0.390060	0.195030	+	0.980797i	$0.437520\pi$
$660$	0	0
$661$	−1.55294	−0.0604025	−0.0302013	−	0.999544i	$-0.509615\pi$
$661$	−1.55294	−0.0604025	−0.0302013	+	0.999544i	$0.509615\pi$
$662$	0	0
$663$	−4.39154	−0.170553
$664$	0	0
$665$	20.3360	0.788597
$666$	0	0
$667$	20.6456	0.799401
$668$	0	0
$669$	−24.5596	−0.949527
$670$	0	0
$671$	1.02391	0.0395278
$672$	0	0
$673$	10.3096	0.397405	0.198702	−	0.980060i	$-0.436327\pi$
$673$	10.3096	0.397405	0.198702	+	0.980060i	$0.436327\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−6.06874	−0.233241	−0.116620	−	0.993177i	$-0.537206\pi$
$677$	−6.06874	−0.233241	−0.116620	+	0.993177i	$0.537206\pi$
$678$	0	0
$679$	−15.5107	−0.595245
$680$	0	0
$681$	−23.3294	−0.893985
$682$	0	0
$683$	30.7103	1.17510	0.587548	−	0.809189i	$-0.300093\pi$
$683$	30.7103	1.17510	0.587548	+	0.809189i	$0.300093\pi$
$684$	0	0
$685$	−10.0000	−0.382080
$686$	0	0
$687$	−15.2301	−0.581066
$688$	0	0
$689$	−12.3915	−0.472080
$690$	0	0
$691$	43.3559	1.64934	0.824668	−	0.565618i	$-0.191362\pi$
$691$	43.3559	1.64934	0.824668	+	0.565618i	$0.191362\pi$
$692$	0	0
$693$	6.83860	0.259777
$694$	0	0
$695$	−7.93126	−0.300850
$696$	0	0
$697$	49.0158	1.85660
$698$	0	0
$699$	20.0820	0.759570
$700$	0	0
$701$	24.7699	0.935545	0.467772	−	0.883849i	$-0.345057\pi$
$701$	24.7699	0.935545	0.467772	+	0.883849i	$0.345057\pi$
$702$	0	0
$703$	34.1507	1.28802
$704$	0	0
$705$	9.00662	0.339209
$706$	0	0
$707$	17.7383	0.667116
$708$	0	0
$709$	−20.7699	−0.780028	−0.390014	−	0.920809i	$-0.627530\pi$
$709$	−20.7699	−0.780028	−0.390014	+	0.920809i	$0.627530\pi$
$710$	0	0
$711$	10.3915	0.389713
$712$	0	0
$713$	−32.0000	−1.19841
$714$	0	0
$715$	2.61507	0.0977981
$716$	0	0
$717$	5.38493	0.201104
$718$	0	0
$719$	−33.5662	−1.25181	−0.625904	−	0.779900i	$-0.715270\pi$
$719$	−33.5662	−1.25181	−0.625904	+	0.779900i	$0.715270\pi$
$720$	0	0
$721$	6.43383	0.239608
$722$	0	0
$723$	20.3228	0.755813
$724$	0	0
$725$	3.23014	0.119965
$726$	0	0
$727$	−30.3493	−1.12559	−0.562796	−	0.826596i	$-0.690274\pi$
$727$	−30.3493	−1.12559	−0.562796	+	0.826596i	$0.690274\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−5.40223	−0.199809
$732$	0	0
$733$	−25.3121	−0.934924	−0.467462	−	0.884013i	$-0.654832\pi$
$733$	−25.3121	−0.934924	−0.467462	+	0.884013i	$0.654832\pi$
$734$	0	0
$735$	−0.161400	−0.00595334
$736$	0	0
$737$	−34.0132	−1.25289
$738$	0	0
$739$	−28.8691	−1.06197	−0.530984	−	0.847382i	$-0.678178\pi$
$739$	−28.8691	−1.06197	−0.530984	+	0.847382i	$0.678178\pi$
$740$	0	0
$741$	7.77647	0.285676
$742$	0	0
$743$	51.7765	1.89949	0.949747	−	0.313018i	$-0.101340\pi$
$743$	51.7765	1.89949	0.949747	+	0.313018i	$0.101340\pi$
$744$	0	0
$745$	9.93126	0.363853
$746$	0	0
$747$	−13.0066	−0.475887
$748$	0	0
$749$	−48.0952	−1.75736
$750$	0	0
$751$	31.6217	1.15389	0.576946	−	0.816782i	$-0.304244\pi$
$751$	31.6217	1.15389	0.576946	+	0.816782i	$0.304244\pi$
$752$	0	0
$753$	−8.00000	−0.291536
$754$	0	0
$755$	13.0066	0.473359
$756$	0	0
$757$	−8.76986	−0.318746	−0.159373	−	0.987218i	$-0.550947\pi$
$757$	−8.76986	−0.318746	−0.159373	+	0.987218i	$0.550947\pi$
$758$	0	0
$759$	−16.7143	−0.606692
$760$	0	0
$761$	−39.1191	−1.41807	−0.709033	−	0.705175i	$-0.750869\pi$
$761$	−39.1191	−1.41807	−0.709033	+	0.705175i	$0.750869\pi$
$762$	0	0
$763$	−29.3676	−1.06318
$764$	0	0
$765$	4.39154	0.158777
$766$	0	0
$767$	9.00662	0.325210
$768$	0	0
$769$	43.5662	1.57104	0.785518	−	0.618839i	$-0.212396\pi$
$769$	43.5662	1.57104	0.785518	+	0.618839i	$0.212396\pi$
$770$	0	0
$771$	−28.0132	−1.00887
$772$	0	0
$773$	31.7037	1.14030	0.570151	−	0.821540i	$-0.306885\pi$
$773$	31.7037	1.14030	0.570151	+	0.821540i	$0.306885\pi$
$774$	0	0
$775$	−5.00662	−0.179843
$776$	0	0
$777$	11.4842	0.411993
$778$	0	0
$779$	−86.7963	−3.10980
$780$	0	0
$781$	−14.0820	−0.503893
$782$	0	0
$783$	3.23014	0.115436
$784$	0	0
$785$	4.46029	0.159194
$786$	0	0
$787$	34.0993	1.21551	0.607754	−	0.794125i	$-0.292071\pi$
$787$	34.0993	1.21551	0.607754	+	0.794125i	$0.292071\pi$
$788$	0	0
$789$	18.0132	0.641288
$790$	0	0
$791$	−1.20368	−0.0427980
$792$	0	0
$793$	0.391544	0.0139041
$794$	0	0
$795$	12.3915	0.439483
$796$	0	0
$797$	−54.4312	−1.92805	−0.964027	−	0.265806i	$-0.914362\pi$
$797$	−54.4312	−1.92805	−0.964027	+	0.265806i	$0.914362\pi$
$798$	0	0
$799$	39.5529	1.39928
$800$	0	0
$801$	−1.62169	−0.0572995
$802$	0	0
$803$	−5.23014	−0.184568
$804$	0	0
$805$	−16.7143	−0.589103
$806$	0	0
$807$	3.23014	0.113706
$808$	0	0
$809$	−25.5794	−0.899324	−0.449662	−	0.893199i	$-0.648456\pi$
$809$	−25.5794	−0.899324	−0.449662	+	0.893199i	$0.648456\pi$
$810$	0	0
$811$	18.4088	0.646422	0.323211	−	0.946327i	$-0.395238\pi$
$811$	18.4088	0.646422	0.323211	+	0.946327i	$0.395238\pi$
$812$	0	0
$813$	−10.2368	−0.359019
$814$	0	0
$815$	−4.15479	−0.145536
$816$	0	0
$817$	9.56617	0.334678
$818$	0	0
$819$	2.61507	0.0913780
$820$	0	0
$821$	23.6085	0.823941	0.411970	−	0.911197i	$-0.364841\pi$
$821$	23.6085	0.823941	0.411970	+	0.911197i	$0.364841\pi$
$822$	0	0
$823$	28.0265	0.976941	0.488471	−	0.872580i	$-0.337555\pi$
$823$	28.0265	0.976941	0.488471	+	0.872580i	$0.337555\pi$
$824$	0	0
$825$	−2.61507	−0.0910451
$826$	0	0
$827$	23.4669	0.816024	0.408012	−	0.912977i	$-0.366222\pi$
$827$	23.4669	0.816024	0.408012	+	0.912977i	$0.366222\pi$
$828$	0	0
$829$	5.38086	0.186885	0.0934425	−	0.995625i	$-0.470213\pi$
$829$	5.38086	0.186885	0.0934425	+	0.995625i	$0.470213\pi$
$830$	0	0
$831$	−3.23014	−0.112052
$832$	0	0
$833$	−0.708797	−0.0245584
$834$	0	0
$835$	6.54633	0.226545
$836$	0	0
$837$	−5.00662	−0.173054
$838$	0	0
$839$	0.0172994	0.000597242 0	0.000298621	−	1.00000i	$-0.499905\pi$
$839$	0.0172994	0.000597242 0	0.000298621		1.00000i	$0.499905\pi$
$840$	0	0
$841$	−18.5662	−0.640213
$842$	0	0
$843$	20.0132	0.689292
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	10.8824	0.373922
$848$	0	0
$849$	−14.0132	−0.480933
$850$	0	0
$851$	−28.0687	−0.962184
$852$	0	0
$853$	−13.1482	−0.450185	−0.225092	−	0.974337i	$-0.572268\pi$
$853$	−13.1482	−0.450185	−0.225092	+	0.974337i	$0.572268\pi$
$854$	0	0
$855$	−7.77647	−0.265950
$856$	0	0
$857$	22.5423	0.770029	0.385014	−	0.922911i	$-0.374196\pi$
$857$	22.5423	0.770029	0.385014	+	0.922911i	$0.374196\pi$
$858$	0	0
$859$	−46.4180	−1.58376	−0.791881	−	0.610676i	$-0.790898\pi$
$859$	−46.4180	−1.58376	−0.791881	+	0.610676i	$0.790898\pi$
$860$	0	0
$861$	−29.1879	−0.994720
$862$	0	0
$863$	24.6970	0.840697	0.420349	−	0.907363i	$-0.361908\pi$
$863$	24.6970	0.840697	0.420349	+	0.907363i	$0.361908\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	0	0
$867$	2.28566	0.0776249
$868$	0	0
$869$	−27.1746	−0.921836
$870$	0	0
$871$	−13.0066	−0.440712
$872$	0	0
$873$	5.93126	0.200743
$874$	0	0
$875$	−2.61507	−0.0884056
$876$	0	0
$877$	−0.460287	−0.0155428	−0.00777139	−	0.999970i	$-0.502474\pi$
$877$	−0.460287	−0.0155428	−0.00777139	+	0.999970i	$0.502474\pi$
$878$	0	0
$879$	−4.32280	−0.145804
$880$	0	0
$881$	−9.21691	−0.310526	−0.155263	−	0.987873i	$-0.549622\pi$
$881$	−9.21691	−0.310526	−0.155263	+	0.987873i	$0.549622\pi$
$882$	0	0
$883$	6.90734	0.232451	0.116225	−	0.993223i	$-0.462921\pi$
$883$	6.90734	0.232451	0.116225	+	0.993223i	$0.462921\pi$
$884$	0	0
$885$	−9.00662	−0.302754
$886$	0	0
$887$	9.16140	0.307610	0.153805	−	0.988101i	$-0.450847\pi$
$887$	9.16140	0.307610	0.153805	+	0.988101i	$0.450847\pi$
$888$	0	0
$889$	−27.3544	−0.917437
$890$	0	0
$891$	−2.61507	−0.0876082
$892$	0	0
$893$	−70.0397	−2.34379
$894$	0	0
$895$	12.7831	0.427291
$896$	0	0
$897$	−6.39154	−0.213407
$898$	0	0
$899$	−16.1721	−0.539369
$900$	0	0
$901$	54.4180	1.81293
$902$	0	0
$903$	3.21691	0.107052
$904$	0	0
$905$	7.16140	0.238053
$906$	0	0
$907$	−1.23014	−0.0408462	−0.0204231	−	0.999791i	$-0.506501\pi$
$907$	−1.23014	−0.0408462	−0.0204231	+	0.999791i	$0.506501\pi$
$908$	0	0
$909$	−6.78309	−0.224981
$910$	0	0
$911$	−26.7699	−0.886925	−0.443462	−	0.896293i	$-0.646250\pi$
$911$	−26.7699	−0.886925	−0.443462	+	0.896293i	$0.646250\pi$
$912$	0	0
$913$	34.0132	1.12567
$914$	0	0
$915$	−0.391544	−0.0129440
$916$	0	0
$917$	−26.1853	−0.864715
$918$	0	0
$919$	23.9313	0.789419	0.394710	−	0.918806i	$-0.370845\pi$
$919$	23.9313	0.789419	0.394710	+	0.918806i	$0.370845\pi$
$920$	0	0
$921$	6.92464	0.228175
$922$	0	0
$923$	−5.38493	−0.177247
$924$	0	0
$925$	−4.39154	−0.144393
$926$	0	0
$927$	−2.46029	−0.0808064
$928$	0	0
$929$	−44.0820	−1.44628	−0.723141	−	0.690700i	$-0.757302\pi$
$929$	−44.0820	−1.44628	−0.723141	+	0.690700i	$0.757302\pi$
$930$	0	0
$931$	1.25513	0.0411351
$932$	0	0
$933$	7.24337	0.237137
$934$	0	0
$935$	−11.4842	−0.375574
$936$	0	0
$937$	−60.3493	−1.97152	−0.985762	−	0.168145i	$-0.946222\pi$
$937$	−60.3493	−1.97152	−0.985762	+	0.168145i	$0.946222\pi$
$938$	0	0
$939$	−32.0132	−1.04471
$940$	0	0
$941$	33.3121	1.08594	0.542972	−	0.839751i	$-0.317299\pi$
$941$	33.3121	1.08594	0.542972	+	0.839751i	$0.317299\pi$
$942$	0	0
$943$	71.3386	2.32310
$944$	0	0
$945$	−2.61507	−0.0850683
$946$	0	0
$947$	−30.1257	−0.978955	−0.489477	−	0.872016i	$-0.662812\pi$
$947$	−30.1257	−0.978955	−0.489477	+	0.872016i	$0.662812\pi$
$948$	0	0
$949$	−2.00000	−0.0649227
$950$	0	0
$951$	31.5662	1.02360
$952$	0	0
$953$	−4.05551	−0.131371	−0.0656855	−	0.997840i	$-0.520923\pi$
$953$	−4.05551	−0.131371	−0.0656855	+	0.997840i	$0.520923\pi$
$954$	0	0
$955$	−2.76986	−0.0896305
$956$	0	0
$957$	−8.44706	−0.273055
$958$	0	0
$959$	26.1507	0.844451
$960$	0	0
$961$	−5.93380	−0.191413
$962$	0	0
$963$	18.3915	0.592659
$964$	0	0
$965$	−14.8518	−0.478097
$966$	0	0
$967$	46.5463	1.49683	0.748415	−	0.663231i	$-0.230815\pi$
$967$	46.5463	1.49683	0.748415	+	0.663231i	$0.230815\pi$
$968$	0	0
$969$	−34.1507	−1.09708
$970$	0	0
$971$	−13.2301	−0.424576	−0.212288	−	0.977207i	$-0.568091\pi$
$971$	−13.2301	−0.424576	−0.212288	+	0.977207i	$0.568091\pi$
$972$	0	0
$973$	20.7408	0.664920
$974$	0	0
$975$	−1.00000	−0.0320256
$976$	0	0
$977$	31.5662	1.00989	0.504946	−	0.863151i	$-0.331513\pi$
$977$	31.5662	1.00989	0.504946	+	0.863151i	$0.331513\pi$
$978$	0	0
$979$	4.24083	0.135537
$980$	0	0
$981$	11.2301	0.358551
$982$	0	0
$983$	−48.5596	−1.54881	−0.774405	−	0.632691i	$-0.781951\pi$
$983$	−48.5596	−1.54881	−0.774405	+	0.632691i	$0.781951\pi$
$984$	0	0
$985$	8.46029	0.269567
$986$	0	0
$987$	−23.5529	−0.749698
$988$	0	0
$989$	−7.86251	−0.250013
$990$	0	0
$991$	0.0687429	0.00218369	0.00109184	−	0.999999i	$-0.499652\pi$
$991$	0.0687429	0.00218369	0.00109184	+	0.999999i	$0.499652\pi$
$992$	0	0
$993$	1.78970	0.0567945
$994$	0	0
$995$	11.5529	0.366253
$996$	0	0
$997$	−3.84928	−0.121908	−0.0609540	−	0.998141i	$-0.519414\pi$
$997$	−3.84928	−0.121908	−0.0609540	+	0.998141i	$0.519414\pi$
$998$	0	0
$999$	−4.39154	−0.138942

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6240.2.a.cc.1.2	yes	3
4.3	odd	2		6240.2.a.bz.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6240.2.a.bz.1.2	✓	3	4.3	odd	2
6240.2.a.cc.1.2	yes	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:	\( N \)	\(=\)	\( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6240.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} -2.61507 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} -2.61507 q^{7} +1.00000 q^{9} -2.61507 q^{11} -1.00000 q^{13} +1.00000 q^{15} +4.39154 q^{17} -7.77647 q^{19} -2.61507 q^{21} +6.39154 q^{23} +1.00000 q^{25} +1.00000 q^{27} +3.23014 q^{29} -5.00662 q^{31} -2.61507 q^{33} -2.61507 q^{35} -4.39154 q^{37} -1.00000 q^{39} +11.1614 q^{41} -1.23014 q^{43} +1.00000 q^{45} +9.00662 q^{47} -0.161400 q^{49} +4.39154 q^{51} +12.3915 q^{53} -2.61507 q^{55} -7.77647 q^{57} -9.00662 q^{59} -0.391544 q^{61} -2.61507 q^{63} -1.00000 q^{65} +13.0066 q^{67} +6.39154 q^{69} +5.38493 q^{71} +2.00000 q^{73} +1.00000 q^{75} +6.83860 q^{77} +10.3915 q^{79} +1.00000 q^{81} -13.0066 q^{83} +4.39154 q^{85} +3.23014 q^{87} -1.62169 q^{89} +2.61507 q^{91} -5.00662 q^{93} -7.77647 q^{95} +5.93126 q^{97} -2.61507 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{5} - 5 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 3 * q^5 - 5 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} + 3 q^{5} - 5 q^{7} + 3 q^{9} - 5 q^{11} - 3 q^{13} + 3 q^{15} - 9 q^{17} - 4 q^{19} - 5 q^{21} - 3 q^{23} + 3 q^{25} + 3 q^{27} + 4 q^{29} + 10 q^{31} - 5 q^{33} - 5 q^{35} + 9 q^{37} - 3 q^{39} + 17 q^{41} + 2 q^{43} + 3 q^{45} + 2 q^{47} + 16 q^{49} - 9 q^{51} + 15 q^{53} - 5 q^{55} - 4 q^{57} - 2 q^{59} + 21 q^{61} - 5 q^{63} - 3 q^{65} + 14 q^{67} - 3 q^{69} + 19 q^{71} + 6 q^{73} + 3 q^{75} + 37 q^{77} + 9 q^{79} + 3 q^{81} - 14 q^{83} - 9 q^{85} + 4 q^{87} + 23 q^{89} + 5 q^{91} + 10 q^{93} - 4 q^{95} + 7 q^{97} - 5 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^5 - 5 * q^7 + 3 * q^9 - 5 * q^11 - 3 * q^13 + 3 * q^15 - 9 * q^17 - 4 * q^19 - 5 * q^21 - 3 * q^23 + 3 * q^25 + 3 * q^27 + 4 * q^29 + 10 * q^31 - 5 * q^33 - 5 * q^35 + 9 * q^37 - 3 * q^39 + 17 * q^41 + 2 * q^43 + 3 * q^45 + 2 * q^47 + 16 * q^49 - 9 * q^51 + 15 * q^53 - 5 * q^55 - 4 * q^57 - 2 * q^59 + 21 * q^61 - 5 * q^63 - 3 * q^65 + 14 * q^67 - 3 * q^69 + 19 * q^71 + 6 * q^73 + 3 * q^75 + 37 * q^77 + 9 * q^79 + 3 * q^81 - 14 * q^83 - 9 * q^85 + 4 * q^87 + 23 * q^89 + 5 * q^91 + 10 * q^93 - 4 * q^95 + 7 * q^97 - 5 * q^99

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