LMFDB - Embedded newform 6160.2.a.bq.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6160,2,Mod(1,6160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6160, 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("6160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6160.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:	$49.1878476451$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.25488.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} - 6x + 6 $ x^4 - 9x^2 - 6x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3080)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.47535$ of defining polynomial
Character	$\chi$	$=$	6160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.732051 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.46410 q^{9} +O(q^{10})$	$q+0.732051 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.46410 q^{9} -1.00000 q^{11} -3.08003 q^{13} +0.732051 q^{15} -3.33477 q^{17} +6.76279 q^{19} +0.732051 q^{21} -2.34798 q^{23} +1.00000 q^{25} -4.00000 q^{27} +3.29869 q^{29} -8.03074 q^{31} -0.732051 q^{33} +1.00000 q^{35} +11.0175 q^{37} -2.25474 q^{39} -6.37872 q^{41} +4.16007 q^{43} -2.46410 q^{45} -10.7989 q^{47} +1.00000 q^{49} -2.44122 q^{51} +6.50805 q^{53} -1.00000 q^{55} +4.95071 q^{57} -5.61593 q^{59} +2.54413 q^{61} -2.46410 q^{63} -3.08003 q^{65} +0.414807 q^{67} -1.71884 q^{69} -5.46410 q^{71} -9.49484 q^{73} +0.732051 q^{75} -1.00000 q^{77} -3.11612 q^{79} +4.46410 q^{81} -11.6242 q^{83} -3.33477 q^{85} +2.41481 q^{87} -3.46410 q^{89} -3.08003 q^{91} -5.87891 q^{93} +6.76279 q^{95} +8.06682 q^{97} +2.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 4 * q^5 + 4 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} + 4 q^{5} + 4 q^{7} + 4 q^{9} - 4 q^{11} - 2 q^{13} - 4 q^{15} + 2 q^{17} - 2 q^{19} - 4 q^{21} - 6 q^{23} + 4 q^{25} - 16 q^{27} - 2 q^{29} - 10 q^{31} + 4 q^{33} + 4 q^{35} + 10 q^{37} - 4 q^{39} - 4 q^{43} + 4 q^{45} - 14 q^{47} + 4 q^{49} - 8 q^{51} + 2 q^{53} - 4 q^{55} + 8 q^{57} - 26 q^{59} - 14 q^{61} + 4 q^{63} - 2 q^{65} - 24 q^{67} + 12 q^{69} - 8 q^{71} - 2 q^{73} - 4 q^{75} - 4 q^{77} - 2 q^{79} + 4 q^{81} - 12 q^{83} + 2 q^{85} - 16 q^{87} - 2 q^{91} + 16 q^{93} - 2 q^{95} + 10 q^{97} - 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^5 + 4 * q^7 + 4 * q^9 - 4 * q^11 - 2 * q^13 - 4 * q^15 + 2 * q^17 - 2 * q^19 - 4 * q^21 - 6 * q^23 + 4 * q^25 - 16 * q^27 - 2 * q^29 - 10 * q^31 + 4 * q^33 + 4 * q^35 + 10 * q^37 - 4 * q^39 - 4 * q^43 + 4 * q^45 - 14 * q^47 + 4 * q^49 - 8 * q^51 + 2 * q^53 - 4 * q^55 + 8 * q^57 - 26 * q^59 - 14 * q^61 + 4 * q^63 - 2 * q^65 - 24 * q^67 + 12 * q^69 - 8 * q^71 - 2 * q^73 - 4 * q^75 - 4 * q^77 - 2 * q^79 + 4 * q^81 - 12 * q^83 + 2 * q^85 - 16 * q^87 - 2 * q^91 + 16 * q^93 - 2 * q^95 + 10 * q^97 - 4 * 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
$3$	0.732051	0.422650	0.211325	−	0.977416i	$-0.432222\pi$
$3$	0.732051	0.422650	0.211325	+	0.977416i	$0.432222\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.46410	−0.821367
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−3.08003	−0.854247	−0.427124	−	0.904193i	$-0.640473\pi$
$13$	−3.08003	−0.854247	−0.427124	+	0.904193i	$0.640473\pi$
$14$	0	0
$15$	0.732051	0.189015
$16$	0	0
$17$	−3.33477	−0.808801	−0.404401	−	0.914582i	$-0.632520\pi$
$17$	−3.33477	−0.808801	−0.404401	+	0.914582i	$0.632520\pi$
$18$	0	0
$19$	6.76279	1.55149	0.775745	−	0.631046i	$-0.217374\pi$
$19$	6.76279	1.55149	0.775745	+	0.631046i	$0.217374\pi$
$20$	0	0
$21$	0.732051	0.159747
$22$	0	0
$23$	−2.34798	−0.489588	−0.244794	−	0.969575i	$-0.578720\pi$
$23$	−2.34798	−0.489588	−0.244794	+	0.969575i	$0.578720\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	3.29869	0.612551	0.306275	−	0.951943i	$-0.400917\pi$
$29$	3.29869	0.612551	0.306275	+	0.951943i	$0.400917\pi$
$30$	0	0
$31$	−8.03074	−1.44236	−0.721182	−	0.692746i	$-0.756401\pi$
$31$	−8.03074	−1.44236	−0.721182	+	0.692746i	$0.756401\pi$
$32$	0	0
$33$	−0.732051	−0.127434
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	11.0175	1.81127	0.905635	−	0.424057i	$-0.139394\pi$
$37$	11.0175	1.81127	0.905635	+	0.424057i	$0.139394\pi$
$38$	0	0
$39$	−2.25474	−0.361047
$40$	0	0
$41$	−6.37872	−0.996189	−0.498094	−	0.867123i	$-0.665967\pi$
$41$	−6.37872	−0.996189	−0.498094	+	0.867123i	$0.665967\pi$
$42$	0	0
$43$	4.16007	0.634404	0.317202	−	0.948358i	$-0.397257\pi$
$43$	4.16007	0.634404	0.317202	+	0.948358i	$0.397257\pi$
$44$	0	0
$45$	−2.46410	−0.367327
$46$	0	0
$47$	−10.7989	−1.57518	−0.787589	−	0.616201i	$-0.788671\pi$
$47$	−10.7989	−1.57518	−0.787589	+	0.616201i	$0.788671\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.44122	−0.341840
$52$	0	0
$53$	6.50805	0.893949	0.446975	−	0.894547i	$-0.352501\pi$
$53$	6.50805	0.893949	0.446975	+	0.894547i	$0.352501\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	4.95071	0.655737
$58$	0	0
$59$	−5.61593	−0.731132	−0.365566	−	0.930785i	$-0.619125\pi$
$59$	−5.61593	−0.731132	−0.365566	+	0.930785i	$0.619125\pi$
$60$	0	0
$61$	2.54413	0.325743	0.162872	−	0.986647i	$-0.447924\pi$
$61$	2.54413	0.325743	0.162872	+	0.986647i	$0.447924\pi$
$62$	0	0
$63$	−2.46410	−0.310448
$64$	0	0
$65$	−3.08003	−0.382031
$66$	0	0
$67$	0.414807	0.0506767	0.0253384	−	0.999679i	$-0.491934\pi$
$67$	0.414807	0.0506767	0.0253384	+	0.999679i	$0.491934\pi$
$68$	0	0
$69$	−1.71884	−0.206924
$70$	0	0
$71$	−5.46410	−0.648470	−0.324235	−	0.945977i	$-0.605107\pi$
$71$	−5.46410	−0.648470	−0.324235	+	0.945977i	$0.605107\pi$
$72$	0	0
$73$	−9.49484	−1.11129	−0.555643	−	0.831421i	$-0.687528\pi$
$73$	−9.49484	−1.11129	−0.555643	+	0.831421i	$0.687528\pi$
$74$	0	0
$75$	0.732051	0.0845299
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−3.11612	−0.350591	−0.175295	−	0.984516i	$-0.556088\pi$
$79$	−3.11612	−0.350591	−0.175295	+	0.984516i	$0.556088\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	−11.6242	−1.27592	−0.637959	−	0.770070i	$-0.720221\pi$
$83$	−11.6242	−1.27592	−0.637959	+	0.770070i	$0.720221\pi$
$84$	0	0
$85$	−3.33477	−0.361707
$86$	0	0
$87$	2.41481	0.258894
$88$	0	0
$89$	−3.46410	−0.367194	−0.183597	−	0.983002i	$-0.558774\pi$
$89$	−3.46410	−0.367194	−0.183597	+	0.983002i	$0.558774\pi$
$90$	0	0
$91$	−3.08003	−0.322875
$92$	0	0
$93$	−5.87891	−0.609614
$94$	0	0
$95$	6.76279	0.693848
$96$	0	0
$97$	8.06682	0.819062	0.409531	−	0.912296i	$-0.365692\pi$
$97$	8.06682	0.819062	0.409531	+	0.912296i	$0.365692\pi$
$98$	0	0
$99$	2.46410	0.247652
$100$	0	0
$101$	−14.2855	−1.42146	−0.710729	−	0.703466i	$-0.751635\pi$
$101$	−14.2855	−1.42146	−0.710729	+	0.703466i	$0.751635\pi$
$102$	0	0
$103$	4.56664	0.449964	0.224982	−	0.974363i	$-0.427768\pi$
$103$	4.56664	0.449964	0.224982	+	0.974363i	$0.427768\pi$
$104$	0	0
$105$	0.732051	0.0714408
$106$	0	0
$107$	−14.6735	−1.41854	−0.709269	−	0.704938i	$-0.750975\pi$
$107$	−14.6735	−1.41854	−0.709269	+	0.704938i	$0.750975\pi$
$108$	0	0
$109$	−14.9229	−1.42935	−0.714675	−	0.699457i	$-0.753425\pi$
$109$	−14.9229	−1.42935	−0.714675	+	0.699457i	$0.753425\pi$
$110$	0	0
$111$	8.06539	0.765533
$112$	0	0
$113$	−0.928203	−0.0873180	−0.0436590	−	0.999046i	$-0.513902\pi$
$113$	−0.928203	−0.0873180	−0.0436590	+	0.999046i	$0.513902\pi$
$114$	0	0
$115$	−2.34798	−0.218950
$116$	0	0
$117$	7.58951	0.701651
$118$	0	0
$119$	−3.33477	−0.305698
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−4.66955	−0.421039
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−6.92820	−0.614779	−0.307389	−	0.951584i	$-0.599455\pi$
$127$	−6.92820	−0.614779	−0.307389	+	0.951584i	$0.599455\pi$
$128$	0	0
$129$	3.04538	0.268131
$130$	0	0
$131$	4.86138	0.424741	0.212370	−	0.977189i	$-0.431882\pi$
$131$	4.86138	0.424741	0.212370	+	0.977189i	$0.431882\pi$
$132$	0	0
$133$	6.76279	0.586408
$134$	0	0
$135$	−4.00000	−0.344265
$136$	0	0
$137$	3.74526	0.319979	0.159990	−	0.987119i	$-0.448854\pi$
$137$	3.74526	0.319979	0.159990	+	0.987119i	$0.448854\pi$
$138$	0	0
$139$	−12.2269	−1.03707	−0.518536	−	0.855056i	$-0.673523\pi$
$139$	−12.2269	−1.03707	−0.518536	+	0.855056i	$0.673523\pi$
$140$	0	0
$141$	−7.90533	−0.665748
$142$	0	0
$143$	3.08003	0.257565
$144$	0	0
$145$	3.29869	0.273941
$146$	0	0
$147$	0.732051	0.0603785
$148$	0	0
$149$	−4.06682	−0.333167	−0.166584	−	0.986027i	$-0.553274\pi$
$149$	−4.06682	−0.333167	−0.166584	+	0.986027i	$0.553274\pi$
$150$	0	0
$151$	−16.1322	−1.31282	−0.656411	−	0.754404i	$-0.727926\pi$
$151$	−16.1322	−1.31282	−0.656411	+	0.754404i	$0.727926\pi$
$152$	0	0
$153$	8.21722	0.664323
$154$	0	0
$155$	−8.03074	−0.645044
$156$	0	0
$157$	−23.2219	−1.85331	−0.926655	−	0.375912i	$-0.877330\pi$
$157$	−23.2219	−1.85331	−0.926655	+	0.375912i	$0.877330\pi$
$158$	0	0
$159$	4.76422	0.377827
$160$	0	0
$161$	−2.34798	−0.185047
$162$	0	0
$163$	−7.02288	−0.550074	−0.275037	−	0.961434i	$-0.588690\pi$
$163$	−7.02288	−0.550074	−0.275037	+	0.961434i	$0.588690\pi$
$164$	0	0
$165$	−0.732051	−0.0569901
$166$	0	0
$167$	−17.0883	−1.32233	−0.661165	−	0.750241i	$-0.729938\pi$
$167$	−17.0883	−1.32233	−0.661165	+	0.750241i	$0.729938\pi$
$168$	0	0
$169$	−3.51340	−0.270261
$170$	0	0
$171$	−16.6642	−1.27434
$172$	0	0
$173$	23.7886	1.80861	0.904305	−	0.426887i	$-0.140390\pi$
$173$	23.7886	1.80861	0.904305	+	0.426887i	$0.140390\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−4.11115	−0.309013
$178$	0	0
$179$	−18.4763	−1.38098	−0.690491	−	0.723341i	$-0.742606\pi$
$179$	−18.4763	−1.38098	−0.690491	+	0.723341i	$0.742606\pi$
$180$	0	0
$181$	24.6860	1.83490	0.917449	−	0.397854i	$-0.130245\pi$
$181$	24.6860	1.83490	0.917449	+	0.397854i	$0.130245\pi$
$182$	0	0
$183$	1.86244	0.137675
$184$	0	0
$185$	11.0175	0.810025
$186$	0	0
$187$	3.33477	0.243863
$188$	0	0
$189$	−4.00000	−0.290957
$190$	0	0
$191$	−5.53627	−0.400591	−0.200295	−	0.979736i	$-0.564190\pi$
$191$	−5.53627	−0.400591	−0.200295	+	0.979736i	$0.564190\pi$
$192$	0	0
$193$	2.23224	0.160680	0.0803400	−	0.996768i	$-0.474399\pi$
$193$	2.23224	0.160680	0.0803400	+	0.996768i	$0.474399\pi$
$194$	0	0
$195$	−2.25474	−0.161465
$196$	0	0
$197$	−10.8521	−0.773181	−0.386591	−	0.922251i	$-0.626347\pi$
$197$	−10.8521	−0.773181	−0.386591	+	0.922251i	$0.626347\pi$
$198$	0	0
$199$	20.3020	1.43917	0.719584	−	0.694406i	$-0.244333\pi$
$199$	20.3020	1.43917	0.719584	+	0.694406i	$0.244333\pi$
$200$	0	0
$201$	0.303660	0.0214185
$202$	0	0
$203$	3.29869	0.231522
$204$	0	0
$205$	−6.37872	−0.445509
$206$	0	0
$207$	5.78567	0.402132
$208$	0	0
$209$	−6.76279	−0.467792
$210$	0	0
$211$	28.6324	1.97114	0.985569	−	0.169274i	$-0.0541423\pi$
$211$	28.6324	1.97114	0.985569	+	0.169274i	$0.0541423\pi$
$212$	0	0
$213$	−4.00000	−0.274075
$214$	0	0
$215$	4.16007	0.283714
$216$	0	0
$217$	−8.03074	−0.545162
$218$	0	0
$219$	−6.95071	−0.469685
$220$	0	0
$221$	10.2712	0.690917
$222$	0	0
$223$	−15.1912	−1.01728	−0.508638	−	0.860980i	$-0.669851\pi$
$223$	−15.1912	−1.01728	−0.508638	+	0.860980i	$0.669851\pi$
$224$	0	0
$225$	−2.46410	−0.164273
$226$	0	0
$227$	10.2215	0.678428	0.339214	−	0.940709i	$-0.389839\pi$
$227$	10.2215	0.678428	0.339214	+	0.940709i	$0.389839\pi$
$228$	0	0
$229$	−28.9136	−1.91066	−0.955332	−	0.295535i	$-0.904502\pi$
$229$	−28.9136	−1.91066	−0.955332	+	0.295535i	$0.904502\pi$
$230$	0	0
$231$	−0.732051	−0.0481654
$232$	0	0
$233$	−4.00392	−0.262305	−0.131153	−	0.991362i	$-0.541868\pi$
$233$	−4.00392	−0.262305	−0.131153	+	0.991362i	$0.541868\pi$
$234$	0	0
$235$	−10.7989	−0.704441
$236$	0	0
$237$	−2.28116	−0.148177
$238$	0	0
$239$	−22.8964	−1.48105	−0.740524	−	0.672030i	$-0.765422\pi$
$239$	−22.8964	−1.48105	−0.740524	+	0.672030i	$0.765422\pi$
$240$	0	0
$241$	21.8779	1.40928	0.704639	−	0.709566i	$-0.251109\pi$
$241$	21.8779	1.40928	0.704639	+	0.709566i	$0.251109\pi$
$242$	0	0
$243$	15.2679	0.979439
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−20.8296	−1.32536
$248$	0	0
$249$	−8.50948	−0.539266
$250$	0	0
$251$	1.37375	0.0867102	0.0433551	−	0.999060i	$-0.486195\pi$
$251$	1.37375	0.0867102	0.0433551	+	0.999060i	$0.486195\pi$
$252$	0	0
$253$	2.34798	0.147616
$254$	0	0
$255$	−2.44122	−0.152875
$256$	0	0
$257$	3.62952	0.226403	0.113201	−	0.993572i	$-0.463889\pi$
$257$	3.62952	0.226403	0.113201	+	0.993572i	$0.463889\pi$
$258$	0	0
$259$	11.0175	0.684596
$260$	0	0
$261$	−8.12830	−0.503129
$262$	0	0
$263$	24.8296	1.53106	0.765530	−	0.643400i	$-0.222477\pi$
$263$	24.8296	1.53106	0.765530	+	0.643400i	$0.222477\pi$
$264$	0	0
$265$	6.50805	0.399786
$266$	0	0
$267$	−2.53590	−0.155194
$268$	0	0
$269$	20.1005	1.22555	0.612773	−	0.790259i	$-0.290054\pi$
$269$	20.1005	1.22555	0.612773	+	0.790259i	$0.290054\pi$
$270$	0	0
$271$	6.49089	0.394294	0.197147	−	0.980374i	$-0.436832\pi$
$271$	6.49089	0.394294	0.197147	+	0.980374i	$0.436832\pi$
$272$	0	0
$273$	−2.25474	−0.136463
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	32.3327	1.94268	0.971342	−	0.237688i	$-0.0763896\pi$
$277$	32.3327	1.94268	0.971342	+	0.237688i	$0.0763896\pi$
$278$	0	0
$279$	19.7886	1.18471
$280$	0	0
$281$	−3.50911	−0.209336	−0.104668	−	0.994507i	$-0.533378\pi$
$281$	−3.50911	−0.209336	−0.104668	+	0.994507i	$0.533378\pi$
$282$	0	0
$283$	−21.1497	−1.25722	−0.628611	−	0.777720i	$-0.716376\pi$
$283$	−21.1497	−1.25722	−0.628611	+	0.777720i	$0.716376\pi$
$284$	0	0
$285$	4.95071	0.293254
$286$	0	0
$287$	−6.37872	−0.376524
$288$	0	0
$289$	−5.87928	−0.345840
$290$	0	0
$291$	5.90533	0.346176
$292$	0	0
$293$	17.7046	1.03431	0.517156	−	0.855891i	$-0.326991\pi$
$293$	17.7046	1.03431	0.517156	+	0.855891i	$0.326991\pi$
$294$	0	0
$295$	−5.61593	−0.326972
$296$	0	0
$297$	4.00000	0.232104
$298$	0	0
$299$	7.23186	0.418229
$300$	0	0
$301$	4.16007	0.239782
$302$	0	0
$303$	−10.4577	−0.600779
$304$	0	0
$305$	2.54413	0.145677
$306$	0	0
$307$	−25.2669	−1.44206	−0.721030	−	0.692904i	$-0.756331\pi$
$307$	−25.2669	−1.44206	−0.721030	+	0.692904i	$0.756331\pi$
$308$	0	0
$309$	3.34301	0.190177
$310$	0	0
$311$	20.0472	1.13677	0.568387	−	0.822762i	$-0.307568\pi$
$311$	20.0472	1.13677	0.568387	+	0.822762i	$0.307568\pi$
$312$	0	0
$313$	−22.2884	−1.25981	−0.629906	−	0.776671i	$-0.716907\pi$
$313$	−22.2884	−1.25981	−0.629906	+	0.776671i	$0.716907\pi$
$314$	0	0
$315$	−2.46410	−0.138836
$316$	0	0
$317$	−4.06682	−0.228416	−0.114208	−	0.993457i	$-0.536433\pi$
$317$	−4.06682	−0.228416	−0.114208	+	0.993457i	$0.536433\pi$
$318$	0	0
$319$	−3.29869	−0.184691
$320$	0	0
$321$	−10.7417	−0.599544
$322$	0	0
$323$	−22.5524	−1.25485
$324$	0	0
$325$	−3.08003	−0.170849
$326$	0	0
$327$	−10.9243	−0.604115
$328$	0	0
$329$	−10.7989	−0.595361
$330$	0	0
$331$	12.5749	0.691178	0.345589	−	0.938386i	$-0.387679\pi$
$331$	12.5749	0.691178	0.345589	+	0.938386i	$0.387679\pi$
$332$	0	0
$333$	−27.1483	−1.48772
$334$	0	0
$335$	0.414807	0.0226633
$336$	0	0
$337$	28.5913	1.55747	0.778735	−	0.627353i	$-0.215862\pi$
$337$	28.5913	1.55747	0.778735	+	0.627353i	$0.215862\pi$
$338$	0	0
$339$	−0.679492	−0.0369049
$340$	0	0
$341$	8.03074	0.434889
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−1.71884	−0.0925394
$346$	0	0
$347$	19.9179	1.06925	0.534624	−	0.845090i	$-0.320453\pi$
$347$	19.9179	1.06925	0.534624	+	0.845090i	$0.320453\pi$
$348$	0	0
$349$	14.1683	0.758412	0.379206	−	0.925312i	$-0.376197\pi$
$349$	14.1683	0.758412	0.379206	+	0.925312i	$0.376197\pi$
$350$	0	0
$351$	12.3201	0.657600
$352$	0	0
$353$	−13.4588	−0.716337	−0.358169	−	0.933657i	$-0.616599\pi$
$353$	−13.4588	−0.716337	−0.358169	+	0.933657i	$0.616599\pi$
$354$	0	0
$355$	−5.46410	−0.290004
$356$	0	0
$357$	−2.44122	−0.129203
$358$	0	0
$359$	14.2308	0.751073	0.375537	−	0.926808i	$-0.377458\pi$
$359$	14.2308	0.751073	0.375537	+	0.926808i	$0.377458\pi$
$360$	0	0
$361$	26.7353	1.40712
$362$	0	0
$363$	0.732051	0.0384227
$364$	0	0
$365$	−9.49484	−0.496983
$366$	0	0
$367$	−13.3963	−0.699279	−0.349639	−	0.936884i	$-0.613696\pi$
$367$	−13.3963	−0.699279	−0.349639	+	0.936884i	$0.613696\pi$
$368$	0	0
$369$	15.7178	0.818237
$370$	0	0
$371$	6.50805	0.337881
$372$	0	0
$373$	15.8789	0.822179	0.411089	−	0.911595i	$-0.365148\pi$
$373$	15.8789	0.822179	0.411089	+	0.911595i	$0.365148\pi$
$374$	0	0
$375$	0.732051	0.0378029
$376$	0	0
$377$	−10.1601	−0.523270
$378$	0	0
$379$	−22.6245	−1.16214	−0.581072	−	0.813852i	$-0.697367\pi$
$379$	−22.6245	−1.16214	−0.581072	+	0.813852i	$0.697367\pi$
$380$	0	0
$381$	−5.07180	−0.259836
$382$	0	0
$383$	−10.9854	−0.561326	−0.280663	−	0.959806i	$-0.590554\pi$
$383$	−10.9854	−0.561326	−0.280663	+	0.959806i	$0.590554\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	−10.2508	−0.521079
$388$	0	0
$389$	−22.6960	−1.15073	−0.575366	−	0.817896i	$-0.695140\pi$
$389$	−22.6960	−1.15073	−0.575366	+	0.817896i	$0.695140\pi$
$390$	0	0
$391$	7.82999	0.395980
$392$	0	0
$393$	3.55878	0.179517
$394$	0	0
$395$	−3.11612	−0.156789
$396$	0	0
$397$	1.05321	0.0528591	0.0264295	−	0.999651i	$-0.491586\pi$
$397$	1.05321	0.0528591	0.0264295	+	0.999651i	$0.491586\pi$
$398$	0	0
$399$	4.95071	0.247845
$400$	0	0
$401$	19.7803	0.987782	0.493891	−	0.869524i	$-0.335574\pi$
$401$	19.7803	0.987782	0.493891	+	0.869524i	$0.335574\pi$
$402$	0	0
$403$	24.7349	1.23214
$404$	0	0
$405$	4.46410	0.221823
$406$	0	0
$407$	−11.0175	−0.546119
$408$	0	0
$409$	−8.47731	−0.419176	−0.209588	−	0.977790i	$-0.567212\pi$
$409$	−8.47731	−0.419176	−0.209588	+	0.977790i	$0.567212\pi$
$410$	0	0
$411$	2.74172	0.135239
$412$	0	0
$413$	−5.61593	−0.276342
$414$	0	0
$415$	−11.6242	−0.570608
$416$	0	0
$417$	−8.95071	−0.438318
$418$	0	0
$419$	−13.1679	−0.643295	−0.321648	−	0.946859i	$-0.604237\pi$
$419$	−13.1679	−0.643295	−0.321648	+	0.946859i	$0.604237\pi$
$420$	0	0
$421$	20.3162	0.990152	0.495076	−	0.868850i	$-0.335140\pi$
$421$	20.3162	0.990152	0.495076	+	0.868850i	$0.335140\pi$
$422$	0	0
$423$	26.6095	1.29380
$424$	0	0
$425$	−3.33477	−0.161760
$426$	0	0
$427$	2.54413	0.123119
$428$	0	0
$429$	2.25474	0.108860
$430$	0	0
$431$	−1.66849	−0.0803684	−0.0401842	−	0.999192i	$-0.512794\pi$
$431$	−1.66849	−0.0803684	−0.0401842	+	0.999192i	$0.512794\pi$
$432$	0	0
$433$	−6.09324	−0.292823	−0.146411	−	0.989224i	$-0.546772\pi$
$433$	−6.09324	−0.292823	−0.146411	+	0.989224i	$0.546772\pi$
$434$	0	0
$435$	2.41481	0.115781
$436$	0	0
$437$	−15.8789	−0.759591
$438$	0	0
$439$	4.02717	0.192206	0.0961031	−	0.995371i	$-0.469362\pi$
$439$	4.02717	0.192206	0.0961031	+	0.995371i	$0.469362\pi$
$440$	0	0
$441$	−2.46410	−0.117338
$442$	0	0
$443$	4.15615	0.197465	0.0987323	−	0.995114i	$-0.468521\pi$
$443$	4.15615	0.197465	0.0987323	+	0.995114i	$0.468521\pi$
$444$	0	0
$445$	−3.46410	−0.164214
$446$	0	0
$447$	−2.97712	−0.140813
$448$	0	0
$449$	−22.5359	−1.06353	−0.531767	−	0.846890i	$-0.678472\pi$
$449$	−22.5359	−1.06353	−0.531767	+	0.846890i	$0.678472\pi$
$450$	0	0
$451$	6.37872	0.300362
$452$	0	0
$453$	−11.8096	−0.554863
$454$	0	0
$455$	−3.08003	−0.144394
$456$	0	0
$457$	11.2279	0.525221	0.262611	−	0.964902i	$-0.415417\pi$
$457$	11.2279	0.525221	0.262611	+	0.964902i	$0.415417\pi$
$458$	0	0
$459$	13.3391	0.622616
$460$	0	0
$461$	−17.8482	−0.831272	−0.415636	−	0.909531i	$-0.636441\pi$
$461$	−17.8482	−0.831272	−0.415636	+	0.909531i	$0.636441\pi$
$462$	0	0
$463$	−23.4977	−1.09203	−0.546016	−	0.837775i	$-0.683856\pi$
$463$	−23.4977	−1.09203	−0.546016	+	0.837775i	$0.683856\pi$
$464$	0	0
$465$	−5.87891	−0.272628
$466$	0	0
$467$	−18.2576	−0.844862	−0.422431	−	0.906395i	$-0.638823\pi$
$467$	−18.2576	−0.844862	−0.422431	+	0.906395i	$0.638823\pi$
$468$	0	0
$469$	0.414807	0.0191540
$470$	0	0
$471$	−16.9996	−0.783301
$472$	0	0
$473$	−4.16007	−0.191280
$474$	0	0
$475$	6.76279	0.310298
$476$	0	0
$477$	−16.0365	−0.734261
$478$	0	0
$479$	35.0062	1.59947	0.799736	−	0.600352i	$-0.204973\pi$
$479$	35.0062	1.59947	0.799736	+	0.600352i	$0.204973\pi$
$480$	0	0
$481$	−33.9344	−1.54727
$482$	0	0
$483$	−1.71884	−0.0782100
$484$	0	0
$485$	8.06682	0.366296
$486$	0	0
$487$	4.20439	0.190519	0.0952595	−	0.995452i	$-0.469632\pi$
$487$	4.20439	0.190519	0.0952595	+	0.995452i	$0.469632\pi$
$488$	0	0
$489$	−5.14110	−0.232489
$490$	0	0
$491$	−34.1355	−1.54051	−0.770257	−	0.637734i	$-0.779872\pi$
$491$	−34.1355	−1.54051	−0.770257	+	0.637734i	$0.779872\pi$
$492$	0	0
$493$	−11.0004	−0.495432
$494$	0	0
$495$	2.46410	0.110753
$496$	0	0
$497$	−5.46410	−0.245098
$498$	0	0
$499$	−25.9053	−1.15968	−0.579841	−	0.814730i	$-0.696885\pi$
$499$	−25.9053	−1.15968	−0.579841	+	0.814730i	$0.696885\pi$
$500$	0	0
$501$	−12.5095	−0.558882
$502$	0	0
$503$	22.7310	1.01353	0.506763	−	0.862085i	$-0.330842\pi$
$503$	22.7310	1.01353	0.506763	+	0.862085i	$0.330842\pi$
$504$	0	0
$505$	−14.2855	−0.635695
$506$	0	0
$507$	−2.57198	−0.114226
$508$	0	0
$509$	−12.6774	−0.561915	−0.280957	−	0.959720i	$-0.590652\pi$
$509$	−12.6774	−0.561915	−0.280957	+	0.959720i	$0.590652\pi$
$510$	0	0
$511$	−9.49484	−0.420027
$512$	0	0
$513$	−27.0512	−1.19434
$514$	0	0
$515$	4.56664	0.201230
$516$	0	0
$517$	10.7989	0.474934
$518$	0	0
$519$	17.4144	0.764409
$520$	0	0
$521$	−41.4270	−1.81495	−0.907475	−	0.420107i	$-0.861993\pi$
$521$	−41.4270	−1.81495	−0.907475	+	0.420107i	$0.861993\pi$
$522$	0	0
$523$	−14.8032	−0.647299	−0.323649	−	0.946177i	$-0.604910\pi$
$523$	−14.8032	−0.647299	−0.323649	+	0.946177i	$0.604910\pi$
$524$	0	0
$525$	0.732051	0.0319493
$526$	0	0
$527$	26.7807	1.16659
$528$	0	0
$529$	−17.4870	−0.760303
$530$	0	0
$531$	13.8382	0.600528
$532$	0	0
$533$	19.6467	0.850992
$534$	0	0
$535$	−14.6735	−0.634389
$536$	0	0
$537$	−13.5256	−0.583672
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	13.9232	0.598606	0.299303	−	0.954158i	$-0.403246\pi$
$541$	13.9232	0.598606	0.299303	+	0.954158i	$0.403246\pi$
$542$	0	0
$543$	18.0714	0.775519
$544$	0	0
$545$	−14.9229	−0.639225
$546$	0	0
$547$	2.73102	0.116770	0.0583851	−	0.998294i	$-0.481405\pi$
$547$	2.73102	0.116770	0.0583851	+	0.998294i	$0.481405\pi$
$548$	0	0
$549$	−6.26901	−0.267555
$550$	0	0
$551$	22.3083	0.950367
$552$	0	0
$553$	−3.11612	−0.132511
$554$	0	0
$555$	8.06539	0.342357
$556$	0	0
$557$	15.6817	0.664456	0.332228	−	0.943199i	$-0.392200\pi$
$557$	15.6817	0.664456	0.332228	+	0.943199i	$0.392200\pi$
$558$	0	0
$559$	−12.8131	−0.541938
$560$	0	0
$561$	2.44122	0.103069
$562$	0	0
$563$	−3.59700	−0.151595	−0.0757977	−	0.997123i	$-0.524150\pi$
$563$	−3.59700	−0.151595	−0.0757977	+	0.997123i	$0.524150\pi$
$564$	0	0
$565$	−0.928203	−0.0390498
$566$	0	0
$567$	4.46410	0.187475
$568$	0	0
$569$	16.8453	0.706193	0.353097	−	0.935587i	$-0.385129\pi$
$569$	16.8453	0.706193	0.353097	+	0.935587i	$0.385129\pi$
$570$	0	0
$571$	3.81352	0.159591	0.0797954	−	0.996811i	$-0.474573\pi$
$571$	3.81352	0.159591	0.0797954	+	0.996811i	$0.474573\pi$
$572$	0	0
$573$	−4.05283	−0.169310
$574$	0	0
$575$	−2.34798	−0.0979176
$576$	0	0
$577$	−4.06682	−0.169304	−0.0846521	−	0.996411i	$-0.526978\pi$
$577$	−4.06682	−0.169304	−0.0846521	+	0.996411i	$0.526978\pi$
$578$	0	0
$579$	1.63411	0.0679114
$580$	0	0
$581$	−11.6242	−0.482252
$582$	0	0
$583$	−6.50805	−0.269536
$584$	0	0
$585$	7.58951	0.313788
$586$	0	0
$587$	−20.5456	−0.848006	−0.424003	−	0.905661i	$-0.639375\pi$
$587$	−20.5456	−0.848006	−0.424003	+	0.905661i	$0.639375\pi$
$588$	0	0
$589$	−54.3102	−2.23781
$590$	0	0
$591$	−7.94430	−0.326785
$592$	0	0
$593$	−42.5381	−1.74683	−0.873415	−	0.486976i	$-0.838100\pi$
$593$	−42.5381	−1.74683	−0.873415	+	0.486976i	$0.838100\pi$
$594$	0	0
$595$	−3.33477	−0.136712
$596$	0	0
$597$	14.8621	0.608264
$598$	0	0
$599$	45.5421	1.86080	0.930399	−	0.366549i	$-0.119461\pi$
$599$	45.5421	1.86080	0.930399	+	0.366549i	$0.119461\pi$
$600$	0	0
$601$	16.6831	0.680519	0.340260	−	0.940332i	$-0.389485\pi$
$601$	16.6831	0.680519	0.340260	+	0.940332i	$0.389485\pi$
$602$	0	0
$603$	−1.02213	−0.0416242
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	37.7081	1.53053	0.765263	−	0.643717i	$-0.222609\pi$
$607$	37.7081	1.53053	0.765263	+	0.643717i	$0.222609\pi$
$608$	0	0
$609$	2.41481	0.0978529
$610$	0	0
$611$	33.2609	1.34559
$612$	0	0
$613$	40.4577	1.63407	0.817035	−	0.576587i	$-0.195616\pi$
$613$	40.4577	1.63407	0.817035	+	0.576587i	$0.195616\pi$
$614$	0	0
$615$	−4.66955	−0.188294
$616$	0	0
$617$	11.1936	0.450639	0.225319	−	0.974285i	$-0.427657\pi$
$617$	11.1936	0.450639	0.225319	+	0.974285i	$0.427657\pi$
$618$	0	0
$619$	22.1869	0.891766	0.445883	−	0.895091i	$-0.352890\pi$
$619$	22.1869	0.891766	0.445883	+	0.895091i	$0.352890\pi$
$620$	0	0
$621$	9.39193	0.376885
$622$	0	0
$623$	−3.46410	−0.138786
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.95071	−0.197712
$628$	0	0
$629$	−36.7410	−1.46496
$630$	0	0
$631$	−29.6142	−1.17892	−0.589462	−	0.807796i	$-0.700660\pi$
$631$	−29.6142	−1.17892	−0.589462	+	0.807796i	$0.700660\pi$
$632$	0	0
$633$	20.9604	0.833101
$634$	0	0
$635$	−6.92820	−0.274937
$636$	0	0
$637$	−3.08003	−0.122035
$638$	0	0
$639$	13.4641	0.532632
$640$	0	0
$641$	−9.53236	−0.376506	−0.188253	−	0.982121i	$-0.560282\pi$
$641$	−9.53236	−0.376506	−0.188253	+	0.982121i	$0.560282\pi$
$642$	0	0
$643$	21.3752	0.842955	0.421477	−	0.906839i	$-0.361512\pi$
$643$	21.3752	0.842955	0.421477	+	0.906839i	$0.361512\pi$
$644$	0	0
$645$	3.04538	0.119912
$646$	0	0
$647$	5.03111	0.197794	0.0988968	−	0.995098i	$-0.468469\pi$
$647$	5.03111	0.197794	0.0988968	+	0.995098i	$0.468469\pi$
$648$	0	0
$649$	5.61593	0.220445
$650$	0	0
$651$	−5.87891	−0.230413
$652$	0	0
$653$	30.9843	1.21251	0.606255	−	0.795270i	$-0.292671\pi$
$653$	30.9843	1.21251	0.606255	+	0.795270i	$0.292671\pi$
$654$	0	0
$655$	4.86138	0.189950
$656$	0	0
$657$	23.3963	0.912775
$658$	0	0
$659$	−5.87891	−0.229010	−0.114505	−	0.993423i	$-0.536528\pi$
$659$	−5.87891	−0.229010	−0.114505	+	0.993423i	$0.536528\pi$
$660$	0	0
$661$	−37.3548	−1.45293	−0.726467	−	0.687201i	$-0.758839\pi$
$661$	−37.3548	−1.45293	−0.726467	+	0.687201i	$0.758839\pi$
$662$	0	0
$663$	7.51905	0.292016
$664$	0	0
$665$	6.76279	0.262250
$666$	0	0
$667$	−7.74526	−0.299898
$668$	0	0
$669$	−11.1207	−0.429952
$670$	0	0
$671$	−2.54413	−0.0982152
$672$	0	0
$673$	−3.07646	−0.118589	−0.0592945	−	0.998241i	$-0.518885\pi$
$673$	−3.07646	−0.118589	−0.0592945	+	0.998241i	$0.518885\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	12.4123	0.477045	0.238523	−	0.971137i	$-0.423337\pi$
$677$	12.4123	0.477045	0.238523	+	0.971137i	$0.423337\pi$
$678$	0	0
$679$	8.06682	0.309576
$680$	0	0
$681$	7.48269	0.286737
$682$	0	0
$683$	0.729163	0.0279006	0.0139503	−	0.999903i	$-0.495559\pi$
$683$	0.729163	0.0279006	0.0139503	+	0.999903i	$0.495559\pi$
$684$	0	0
$685$	3.74526	0.143099
$686$	0	0
$687$	−21.1662	−0.807542
$688$	0	0
$689$	−20.0450	−0.763654
$690$	0	0
$691$	2.63203	0.100127	0.0500635	−	0.998746i	$-0.484058\pi$
$691$	2.63203	0.100127	0.0500635	+	0.998746i	$0.484058\pi$
$692$	0	0
$693$	2.46410	0.0936035
$694$	0	0
$695$	−12.2269	−0.463792
$696$	0	0
$697$	21.2716	0.805719
$698$	0	0
$699$	−2.93107	−0.110863
$700$	0	0
$701$	34.0297	1.28528	0.642642	−	0.766166i	$-0.277838\pi$
$701$	34.0297	1.28528	0.642642	+	0.766166i	$0.277838\pi$
$702$	0	0
$703$	74.5092	2.81017
$704$	0	0
$705$	−7.90533	−0.297732
$706$	0	0
$707$	−14.2855	−0.537261
$708$	0	0
$709$	−26.5767	−0.998110	−0.499055	−	0.866570i	$-0.666319\pi$
$709$	−26.5767	−0.998110	−0.499055	+	0.866570i	$0.666319\pi$
$710$	0	0
$711$	7.67843	0.287964
$712$	0	0
$713$	18.8560	0.706164
$714$	0	0
$715$	3.08003	0.115187
$716$	0	0
$717$	−16.7614	−0.625964
$718$	0	0
$719$	8.99214	0.335350	0.167675	−	0.985842i	$-0.446374\pi$
$719$	8.99214	0.335350	0.167675	+	0.985842i	$0.446374\pi$
$720$	0	0
$721$	4.56664	0.170070
$722$	0	0
$723$	16.0157	0.595631
$724$	0	0
$725$	3.29869	0.122510
$726$	0	0
$727$	42.1702	1.56400	0.782002	−	0.623275i	$-0.214198\pi$
$727$	42.1702	1.56400	0.782002	+	0.623275i	$0.214198\pi$
$728$	0	0
$729$	−2.21539	−0.0820515
$730$	0	0
$731$	−13.8729	−0.513107
$732$	0	0
$733$	4.79888	0.177251	0.0886253	−	0.996065i	$-0.471753\pi$
$733$	4.79888	0.177251	0.0886253	+	0.996065i	$0.471753\pi$
$734$	0	0
$735$	0.732051	0.0270021
$736$	0	0
$737$	−0.414807	−0.0152796
$738$	0	0
$739$	−51.1291	−1.88081	−0.940407	−	0.340050i	$-0.889556\pi$
$739$	−51.1291	−1.88081	−0.940407	+	0.340050i	$0.889556\pi$
$740$	0	0
$741$	−15.2483	−0.560162
$742$	0	0
$743$	30.5417	1.12047	0.560233	−	0.828335i	$-0.310712\pi$
$743$	30.5417	1.12047	0.560233	+	0.828335i	$0.310712\pi$
$744$	0	0
$745$	−4.06682	−0.148997
$746$	0	0
$747$	28.6431	1.04800
$748$	0	0
$749$	−14.6735	−0.536157
$750$	0	0
$751$	29.3977	1.07274	0.536369	−	0.843984i	$-0.319796\pi$
$751$	29.3977	1.07274	0.536369	+	0.843984i	$0.319796\pi$
$752$	0	0
$753$	1.00565	0.0366480
$754$	0	0
$755$	−16.1322	−0.587111
$756$	0	0
$757$	43.8246	1.59283	0.796417	−	0.604748i	$-0.206726\pi$
$757$	43.8246	1.59283	0.796417	+	0.604748i	$0.206726\pi$
$758$	0	0
$759$	1.71884	0.0623900
$760$	0	0
$761$	−37.6721	−1.36561	−0.682806	−	0.730600i	$-0.739240\pi$
$761$	−37.6721	−1.36561	−0.682806	+	0.730600i	$0.739240\pi$
$762$	0	0
$763$	−14.9229	−0.540244
$764$	0	0
$765$	8.21722	0.297094
$766$	0	0
$767$	17.2973	0.624568
$768$	0	0
$769$	−44.8061	−1.61575	−0.807874	−	0.589355i	$-0.799382\pi$
$769$	−44.8061	−1.61575	−0.807874	+	0.589355i	$0.799382\pi$
$770$	0	0
$771$	2.65699	0.0956892
$772$	0	0
$773$	−16.5631	−0.595732	−0.297866	−	0.954608i	$-0.596275\pi$
$773$	−16.5631	−0.595732	−0.297866	+	0.954608i	$0.596275\pi$
$774$	0	0
$775$	−8.03074	−0.288473
$776$	0	0
$777$	8.06539	0.289344
$778$	0	0
$779$	−43.1379	−1.54558
$780$	0	0
$781$	5.46410	0.195521
$782$	0	0
$783$	−13.1947	−0.471542
$784$	0	0
$785$	−23.2219	−0.828826
$786$	0	0
$787$	−32.8911	−1.17244	−0.586220	−	0.810152i	$-0.699385\pi$
$787$	−32.8911	−1.17244	−0.586220	+	0.810152i	$0.699385\pi$
$788$	0	0
$789$	18.1765	0.647102
$790$	0	0
$791$	−0.928203	−0.0330031
$792$	0	0
$793$	−7.83602	−0.278265
$794$	0	0
$795$	4.76422	0.168970
$796$	0	0
$797$	−21.6778	−0.767868	−0.383934	−	0.923361i	$-0.625431\pi$
$797$	−21.6778	−0.767868	−0.383934	+	0.923361i	$0.625431\pi$
$798$	0	0
$799$	36.0118	1.27401
$800$	0	0
$801$	8.53590	0.301601
$802$	0	0
$803$	9.49484	0.335066
$804$	0	0
$805$	−2.34798	−0.0827555
$806$	0	0
$807$	14.7146	0.517977
$808$	0	0
$809$	−17.9443	−0.630888	−0.315444	−	0.948944i	$-0.602153\pi$
$809$	−17.9443	−0.630888	−0.315444	+	0.948944i	$0.602153\pi$
$810$	0	0
$811$	6.51196	0.228666	0.114333	−	0.993442i	$-0.463527\pi$
$811$	6.51196	0.228666	0.114333	+	0.993442i	$0.463527\pi$
$812$	0	0
$813$	4.75166	0.166648
$814$	0	0
$815$	−7.02288	−0.246001
$816$	0	0
$817$	28.1336	0.984272
$818$	0	0
$819$	7.58951	0.265199
$820$	0	0
$821$	31.0029	1.08201	0.541004	−	0.841020i	$-0.318045\pi$
$821$	31.0029	1.08201	0.541004	+	0.841020i	$0.318045\pi$
$822$	0	0
$823$	−32.1244	−1.11979	−0.559893	−	0.828565i	$-0.689158\pi$
$823$	−32.1244	−1.11979	−0.559893	+	0.828565i	$0.689158\pi$
$824$	0	0
$825$	−0.732051	−0.0254867
$826$	0	0
$827$	6.31622	0.219636	0.109818	−	0.993952i	$-0.464973\pi$
$827$	6.31622	0.219636	0.109818	+	0.993952i	$0.464973\pi$
$828$	0	0
$829$	−12.5398	−0.435526	−0.217763	−	0.976002i	$-0.569876\pi$
$829$	−12.5398	−0.435526	−0.217763	+	0.976002i	$0.569876\pi$
$830$	0	0
$831$	23.6692	0.821074
$832$	0	0
$833$	−3.33477	−0.115543
$834$	0	0
$835$	−17.0883	−0.591364
$836$	0	0
$837$	32.1230	1.11033
$838$	0	0
$839$	−17.4391	−0.602066	−0.301033	−	0.953614i	$-0.597331\pi$
$839$	−17.4391	−0.602066	−0.301033	+	0.953614i	$0.597331\pi$
$840$	0	0
$841$	−18.1187	−0.624781
$842$	0	0
$843$	−2.56884	−0.0884757
$844$	0	0
$845$	−3.51340	−0.120865
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−15.4827	−0.531364
$850$	0	0
$851$	−25.8690	−0.886777
$852$	0	0
$853$	12.7539	0.436684	0.218342	−	0.975872i	$-0.429935\pi$
$853$	12.7539	0.436684	0.218342	+	0.975872i	$0.429935\pi$
$854$	0	0
$855$	−16.6642	−0.569904
$856$	0	0
$857$	39.5357	1.35051	0.675256	−	0.737583i	$-0.264033\pi$
$857$	39.5357	1.35051	0.675256	+	0.737583i	$0.264033\pi$
$858$	0	0
$859$	7.05286	0.240641	0.120320	−	0.992735i	$-0.461608\pi$
$859$	7.05286	0.240641	0.120320	+	0.992735i	$0.461608\pi$
$860$	0	0
$861$	−4.66955	−0.159138
$862$	0	0
$863$	−13.1962	−0.449203	−0.224602	−	0.974451i	$-0.572108\pi$
$863$	−13.1962	−0.449203	−0.224602	+	0.974451i	$0.572108\pi$
$864$	0	0
$865$	23.7886	0.808835
$866$	0	0
$867$	−4.30393	−0.146169
$868$	0	0
$869$	3.11612	0.105707
$870$	0	0
$871$	−1.27762	−0.0432905
$872$	0	0
$873$	−19.8775	−0.672751
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	25.9610	0.876642	0.438321	−	0.898819i	$-0.355573\pi$
$877$	25.9610	0.876642	0.438321	+	0.898819i	$0.355573\pi$
$878$	0	0
$879$	12.9606	0.437152
$880$	0	0
$881$	9.05116	0.304941	0.152471	−	0.988308i	$-0.451277\pi$
$881$	9.05116	0.304941	0.152471	+	0.988308i	$0.451277\pi$
$882$	0	0
$883$	3.22583	0.108558	0.0542790	−	0.998526i	$-0.482714\pi$
$883$	3.22583	0.108558	0.0542790	+	0.998526i	$0.482714\pi$
$884$	0	0
$885$	−4.11115	−0.138195
$886$	0	0
$887$	22.4274	0.753037	0.376519	−	0.926409i	$-0.377121\pi$
$887$	22.4274	0.753037	0.376519	+	0.926409i	$0.377121\pi$
$888$	0	0
$889$	−6.92820	−0.232364
$890$	0	0
$891$	−4.46410	−0.149553
$892$	0	0
$893$	−73.0305	−2.44387
$894$	0	0
$895$	−18.4763	−0.617594
$896$	0	0
$897$	5.29409	0.176765
$898$	0	0
$899$	−26.4909	−0.883521
$900$	0	0
$901$	−21.7029	−0.723027
$902$	0	0
$903$	3.04538	0.101344
$904$	0	0
$905$	24.6860	0.820591
$906$	0	0
$907$	39.7968	1.32143	0.660715	−	0.750637i	$-0.270253\pi$
$907$	39.7968	1.32143	0.660715	+	0.750637i	$0.270253\pi$
$908$	0	0
$909$	35.2009	1.16754
$910$	0	0
$911$	13.7314	0.454942	0.227471	−	0.973785i	$-0.426954\pi$
$911$	13.7314	0.454942	0.227471	+	0.973785i	$0.426954\pi$
$912$	0	0
$913$	11.6242	0.384704
$914$	0	0
$915$	1.86244	0.0615702
$916$	0	0
$917$	4.86138	0.160537
$918$	0	0
$919$	−37.7253	−1.24444	−0.622221	−	0.782841i	$-0.713770\pi$
$919$	−37.7253	−1.24444	−0.622221	+	0.782841i	$0.713770\pi$
$920$	0	0
$921$	−18.4967	−0.609486
$922$	0	0
$923$	16.8296	0.553953
$924$	0	0
$925$	11.0175	0.362254
$926$	0	0
$927$	−11.2527	−0.369586
$928$	0	0
$929$	−6.32300	−0.207451	−0.103725	−	0.994606i	$-0.533076\pi$
$929$	−6.32300	−0.207451	−0.103725	+	0.994606i	$0.533076\pi$
$930$	0	0
$931$	6.76279	0.221641
$932$	0	0
$933$	14.6756	0.480457
$934$	0	0
$935$	3.33477	0.109059
$936$	0	0
$937$	−4.90536	−0.160251	−0.0801255	−	0.996785i	$-0.525532\pi$
$937$	−4.90536	−0.160251	−0.0801255	+	0.996785i	$0.525532\pi$
$938$	0	0
$939$	−16.3162	−0.532460
$940$	0	0
$941$	−21.7760	−0.709877	−0.354939	−	0.934890i	$-0.615498\pi$
$941$	−21.7760	−0.709877	−0.354939	+	0.934890i	$0.615498\pi$
$942$	0	0
$943$	14.9771	0.487722
$944$	0	0
$945$	−4.00000	−0.130120
$946$	0	0
$947$	−57.9569	−1.88334	−0.941672	−	0.336531i	$-0.890746\pi$
$947$	−57.9569	−1.88334	−0.941672	+	0.336531i	$0.890746\pi$
$948$	0	0
$949$	29.2444	0.949314
$950$	0	0
$951$	−2.97712	−0.0965398
$952$	0	0
$953$	−5.48057	−0.177533	−0.0887666	−	0.996052i	$-0.528293\pi$
$953$	−5.48057	−0.177533	−0.0887666	+	0.996052i	$0.528293\pi$
$954$	0	0
$955$	−5.53627	−0.179150
$956$	0	0
$957$	−2.41481	−0.0780596
$958$	0	0
$959$	3.74526	0.120941
$960$	0	0
$961$	33.4928	1.08041
$962$	0	0
$963$	36.1569	1.16514
$964$	0	0
$965$	2.23224	0.0718583
$966$	0	0
$967$	6.46839	0.208009	0.104005	−	0.994577i	$-0.466834\pi$
$967$	6.46839	0.208009	0.104005	+	0.994577i	$0.466834\pi$
$968$	0	0
$969$	−16.5095	−0.530361
$970$	0	0
$971$	25.1679	0.807677	0.403839	−	0.914830i	$-0.367676\pi$
$971$	25.1679	0.807677	0.403839	+	0.914830i	$0.367676\pi$
$972$	0	0
$973$	−12.2269	−0.391976
$974$	0	0
$975$	−2.25474	−0.0722095
$976$	0	0
$977$	28.3209	0.906065	0.453033	−	0.891494i	$-0.350342\pi$
$977$	28.3209	0.906065	0.453033	+	0.891494i	$0.350342\pi$
$978$	0	0
$979$	3.46410	0.110713
$980$	0	0
$981$	36.7714	1.17402
$982$	0	0
$983$	41.3348	1.31837	0.659187	−	0.751979i	$-0.270900\pi$
$983$	41.3348	1.31837	0.659187	+	0.751979i	$0.270900\pi$
$984$	0	0
$985$	−10.8521	−0.345777
$986$	0	0
$987$	−7.90533	−0.251629
$988$	0	0
$989$	−9.76776	−0.310597
$990$	0	0
$991$	−2.11506	−0.0671872	−0.0335936	−	0.999436i	$-0.510695\pi$
$991$	−2.11506	−0.0671872	−0.0335936	+	0.999436i	$0.510695\pi$
$992$	0	0
$993$	9.20545	0.292126
$994$	0	0
$995$	20.3020	0.643615
$996$	0	0
$997$	23.5213	0.744926	0.372463	−	0.928047i	$-0.378513\pi$
$997$	23.5213	0.744926	0.372463	+	0.928047i	$0.378513\pi$
$998$	0	0
$999$	−44.0701	−1.39432

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.bq.1.3		4
4.3	odd	2		3080.2.a.q.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3080.2.a.q.1.1	✓	4	4.3	odd	2
6160.2.a.bq.1.3		4	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:	\( N \)	\(=\)	\( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6160.a (trivial)

\(f(q)\)	\(=\)	\(q+0.732051 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.46410 q^{9} +O(q^{10})\)	\(q+0.732051 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.46410 q^{9} -1.00000 q^{11} -3.08003 q^{13} +0.732051 q^{15} -3.33477 q^{17} +6.76279 q^{19} +0.732051 q^{21} -2.34798 q^{23} +1.00000 q^{25} -4.00000 q^{27} +3.29869 q^{29} -8.03074 q^{31} -0.732051 q^{33} +1.00000 q^{35} +11.0175 q^{37} -2.25474 q^{39} -6.37872 q^{41} +4.16007 q^{43} -2.46410 q^{45} -10.7989 q^{47} +1.00000 q^{49} -2.44122 q^{51} +6.50805 q^{53} -1.00000 q^{55} +4.95071 q^{57} -5.61593 q^{59} +2.54413 q^{61} -2.46410 q^{63} -3.08003 q^{65} +0.414807 q^{67} -1.71884 q^{69} -5.46410 q^{71} -9.49484 q^{73} +0.732051 q^{75} -1.00000 q^{77} -3.11612 q^{79} +4.46410 q^{81} -11.6242 q^{83} -3.33477 q^{85} +2.41481 q^{87} -3.46410 q^{89} -3.08003 q^{91} -5.87891 q^{93} +6.76279 q^{95} +8.06682 q^{97} +2.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 4 * q^5 + 4 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} + 4 q^{5} + 4 q^{7} + 4 q^{9} - 4 q^{11} - 2 q^{13} - 4 q^{15} + 2 q^{17} - 2 q^{19} - 4 q^{21} - 6 q^{23} + 4 q^{25} - 16 q^{27} - 2 q^{29} - 10 q^{31} + 4 q^{33} + 4 q^{35} + 10 q^{37} - 4 q^{39} - 4 q^{43} + 4 q^{45} - 14 q^{47} + 4 q^{49} - 8 q^{51} + 2 q^{53} - 4 q^{55} + 8 q^{57} - 26 q^{59} - 14 q^{61} + 4 q^{63} - 2 q^{65} - 24 q^{67} + 12 q^{69} - 8 q^{71} - 2 q^{73} - 4 q^{75} - 4 q^{77} - 2 q^{79} + 4 q^{81} - 12 q^{83} + 2 q^{85} - 16 q^{87} - 2 q^{91} + 16 q^{93} - 2 q^{95} + 10 q^{97} - 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^5 + 4 * q^7 + 4 * q^9 - 4 * q^11 - 2 * q^13 - 4 * q^15 + 2 * q^17 - 2 * q^19 - 4 * q^21 - 6 * q^23 + 4 * q^25 - 16 * q^27 - 2 * q^29 - 10 * q^31 + 4 * q^33 + 4 * q^35 + 10 * q^37 - 4 * q^39 - 4 * q^43 + 4 * q^45 - 14 * q^47 + 4 * q^49 - 8 * q^51 + 2 * q^53 - 4 * q^55 + 8 * q^57 - 26 * q^59 - 14 * q^61 + 4 * q^63 - 2 * q^65 - 24 * q^67 + 12 * q^69 - 8 * q^71 - 2 * q^73 - 4 * q^75 - 4 * q^77 - 2 * q^79 + 4 * q^81 - 12 * q^83 + 2 * q^85 - 16 * q^87 - 2 * q^91 + 16 * q^93 - 2 * q^95 + 10 * q^97 - 4 * q^99

Introduction

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