LMFDB - Embedded newform 7280.2.a.bp.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7280,2,Mod(1,7280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7280, 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("7280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7280 = 2^{4} \cdot 5 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7280.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:	$58.1310926715$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3640)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.86081$ of defining polynomial
Character	$\chi$	$=$	7280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.46260 q^{3} -1.00000 q^{5} +1.00000 q^{7} -0.860806 q^{9} +O(q^{10})$	$q+1.46260 q^{3} -1.00000 q^{5} +1.00000 q^{7} -0.860806 q^{9} +1.39821 q^{11} +1.00000 q^{13} -1.46260 q^{15} +1.25901 q^{17} +0.462598 q^{19} +1.46260 q^{21} -8.04502 q^{23} +1.00000 q^{25} -5.64681 q^{27} -5.65722 q^{29} -5.46260 q^{31} +2.04502 q^{33} -1.00000 q^{35} -0.815790 q^{37} +1.46260 q^{39} +7.58242 q^{41} +1.07480 q^{43} +0.860806 q^{45} +8.69182 q^{47} +1.00000 q^{49} +1.84143 q^{51} -4.79641 q^{53} -1.39821 q^{55} +0.676596 q^{57} -6.43281 q^{59} -4.60179 q^{61} -0.860806 q^{63} -1.00000 q^{65} -2.13919 q^{67} -11.7666 q^{69} -0.128782 q^{71} -13.4882 q^{73} +1.46260 q^{75} +1.39821 q^{77} -3.06439 q^{79} -5.67660 q^{81} -1.44322 q^{83} -1.25901 q^{85} -8.27424 q^{87} -11.5076 q^{89} +1.00000 q^{91} -7.98959 q^{93} -0.462598 q^{95} -8.17380 q^{97} -1.20359 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 2 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9	$ 3 q + 2 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9} + q^{11} + 3 q^{13} - 2 q^{15} - 5 q^{17} - q^{19} + 2 q^{21} - 5 q^{23} + 3 q^{25} - q^{27} - 5 q^{29} - 14 q^{31} - 13 q^{33} - 3 q^{35} - 16 q^{37} + 2 q^{39} + 6 q^{41} + 8 q^{43} - 3 q^{45} - 9 q^{47} + 3 q^{49} - 20 q^{51} - 8 q^{53} - q^{55} + 10 q^{57} + 7 q^{59} - 17 q^{61} + 3 q^{63} - 3 q^{65} - 12 q^{67} - 5 q^{69} - 2 q^{71} + q^{73} + 2 q^{75} + q^{77} - 10 q^{79} - 25 q^{81} + 18 q^{83} + 5 q^{85} + 27 q^{87} - 13 q^{89} + 3 q^{91} - 20 q^{93} + q^{95} - 7 q^{97} - 10 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9 + q^11 + 3 * q^13 - 2 * q^15 - 5 * q^17 - q^19 + 2 * q^21 - 5 * q^23 + 3 * q^25 - q^27 - 5 * q^29 - 14 * q^31 - 13 * q^33 - 3 * q^35 - 16 * q^37 + 2 * q^39 + 6 * q^41 + 8 * q^43 - 3 * q^45 - 9 * q^47 + 3 * q^49 - 20 * q^51 - 8 * q^53 - q^55 + 10 * q^57 + 7 * q^59 - 17 * q^61 + 3 * q^63 - 3 * q^65 - 12 * q^67 - 5 * q^69 - 2 * q^71 + q^73 + 2 * q^75 + q^77 - 10 * q^79 - 25 * q^81 + 18 * q^83 + 5 * q^85 + 27 * q^87 - 13 * q^89 + 3 * q^91 - 20 * q^93 + q^95 - 7 * q^97 - 10 * 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
$3$	1.46260	0.844432	0.422216	−	0.906495i	$-0.361252\pi$
$3$	1.46260	0.844432	0.422216	+	0.906495i	$0.361252\pi$
$4$	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
$9$	−0.860806	−0.286935
$10$	0	0
$11$	1.39821	0.421575	0.210788	−	0.977532i	$-0.432397\pi$
$11$	1.39821	0.421575	0.210788	+	0.977532i	$0.432397\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−1.46260	−0.377641
$16$	0	0
$17$	1.25901	0.305356	0.152678	−	0.988276i	$-0.451210\pi$
$17$	1.25901	0.305356	0.152678	+	0.988276i	$0.451210\pi$
$18$	0	0
$19$	0.462598	0.106127	0.0530637	−	0.998591i	$-0.483101\pi$
$19$	0.462598	0.106127	0.0530637	+	0.998591i	$0.483101\pi$
$20$	0	0
$21$	1.46260	0.319165
$22$	0	0
$23$	−8.04502	−1.67750	−0.838751	−	0.544515i	$-0.816714\pi$
$23$	−8.04502	−1.67750	−0.838751	+	0.544515i	$0.816714\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.64681	−1.08673
$28$	0	0
$29$	−5.65722	−1.05052	−0.525260	−	0.850942i	$-0.676032\pi$
$29$	−5.65722	−1.05052	−0.525260	+	0.850942i	$0.676032\pi$
$30$	0	0
$31$	−5.46260	−0.981112	−0.490556	−	0.871410i	$-0.663206\pi$
$31$	−5.46260	−0.981112	−0.490556	+	0.871410i	$0.663206\pi$
$32$	0	0
$33$	2.04502	0.355992
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−0.815790	−0.134115	−0.0670575	−	0.997749i	$-0.521361\pi$
$37$	−0.815790	−0.134115	−0.0670575	+	0.997749i	$0.521361\pi$
$38$	0	0
$39$	1.46260	0.234203
$40$	0	0
$41$	7.58242	1.18417	0.592087	−	0.805874i	$-0.298304\pi$
$41$	7.58242	1.18417	0.592087	+	0.805874i	$0.298304\pi$
$42$	0	0
$43$	1.07480	0.163906	0.0819530	−	0.996636i	$-0.473884\pi$
$43$	1.07480	0.163906	0.0819530	+	0.996636i	$0.473884\pi$
$44$	0	0
$45$	0.860806	0.128321
$46$	0	0
$47$	8.69182	1.26783	0.633916	−	0.773402i	$-0.281446\pi$
$47$	8.69182	1.26783	0.633916	+	0.773402i	$0.281446\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.84143	0.257852
$52$	0	0
$53$	−4.79641	−0.658838	−0.329419	−	0.944184i	$-0.606853\pi$
$53$	−4.79641	−0.658838	−0.329419	+	0.944184i	$0.606853\pi$
$54$	0	0
$55$	−1.39821	−0.188534
$56$	0	0
$57$	0.676596	0.0896173
$58$	0	0
$59$	−6.43281	−0.837481	−0.418740	−	0.908106i	$-0.637528\pi$
$59$	−6.43281	−0.837481	−0.418740	+	0.908106i	$0.637528\pi$
$60$	0	0
$61$	−4.60179	−0.589199	−0.294600	−	0.955621i	$-0.595186\pi$
$61$	−4.60179	−0.589199	−0.294600	+	0.955621i	$0.595186\pi$
$62$	0	0
$63$	−0.860806	−0.108451
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−2.13919	−0.261344	−0.130672	−	0.991426i	$-0.541714\pi$
$67$	−2.13919	−0.261344	−0.130672	+	0.991426i	$0.541714\pi$
$68$	0	0
$69$	−11.7666	−1.41654
$70$	0	0
$71$	−0.128782	−0.0152836	−0.00764180	−	0.999971i	$-0.502432\pi$
$71$	−0.128782	−0.0152836	−0.00764180	+	0.999971i	$0.502432\pi$
$72$	0	0
$73$	−13.4882	−1.57868	−0.789340	−	0.613957i	$-0.789577\pi$
$73$	−13.4882	−1.57868	−0.789340	+	0.613957i	$0.789577\pi$
$74$	0	0
$75$	1.46260	0.168886
$76$	0	0
$77$	1.39821	0.159341
$78$	0	0
$79$	−3.06439	−0.344771	−0.172385	−	0.985030i	$-0.555147\pi$
$79$	−3.06439	−0.344771	−0.172385	+	0.985030i	$0.555147\pi$
$80$	0	0
$81$	−5.67660	−0.630733
$82$	0	0
$83$	−1.44322	−0.158414	−0.0792072	−	0.996858i	$-0.525239\pi$
$83$	−1.44322	−0.158414	−0.0792072	+	0.996858i	$0.525239\pi$
$84$	0	0
$85$	−1.25901	−0.136559
$86$	0	0
$87$	−8.27424	−0.887092
$88$	0	0
$89$	−11.5076	−1.21980	−0.609902	−	0.792477i	$-0.708791\pi$
$89$	−11.5076	−1.21980	−0.609902	+	0.792477i	$0.708791\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−7.98959	−0.828482
$94$	0	0
$95$	−0.462598	−0.0474616
$96$	0	0
$97$	−8.17380	−0.829923	−0.414962	−	0.909839i	$-0.636205\pi$
$97$	−8.17380	−0.829923	−0.414962	+	0.909839i	$0.636205\pi$
$98$	0	0
$99$	−1.20359	−0.120965
$100$	0	0
$101$	−13.4882	−1.34213	−0.671065	−	0.741399i	$-0.734163\pi$
$101$	−13.4882	−1.34213	−0.671065	+	0.741399i	$0.734163\pi$
$102$	0	0
$103$	−9.75622	−0.961308	−0.480654	−	0.876910i	$-0.659601\pi$
$103$	−9.75622	−0.961308	−0.480654	+	0.876910i	$0.659601\pi$
$104$	0	0
$105$	−1.46260	−0.142735
$106$	0	0
$107$	13.5720	1.31206	0.656028	−	0.754737i	$-0.272235\pi$
$107$	13.5720	1.31206	0.656028	+	0.754737i	$0.272235\pi$
$108$	0	0
$109$	14.7368	1.41153	0.705767	−	0.708444i	$-0.250603\pi$
$109$	14.7368	1.41153	0.705767	+	0.708444i	$0.250603\pi$
$110$	0	0
$111$	−1.19317	−0.113251
$112$	0	0
$113$	16.2638	1.52997	0.764986	−	0.644047i	$-0.222746\pi$
$113$	16.2638	1.52997	0.764986	+	0.644047i	$0.222746\pi$
$114$	0	0
$115$	8.04502	0.750202
$116$	0	0
$117$	−0.860806	−0.0795815
$118$	0	0
$119$	1.25901	0.115414
$120$	0	0
$121$	−9.04502	−0.822274
$122$	0	0
$123$	11.0900	0.999955
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	4.99104	0.442883	0.221441	−	0.975174i	$-0.428924\pi$
$127$	4.99104	0.442883	0.221441	+	0.975174i	$0.428924\pi$
$128$	0	0
$129$	1.57201	0.138407
$130$	0	0
$131$	9.35319	0.817192	0.408596	−	0.912715i	$-0.366018\pi$
$131$	9.35319	0.817192	0.408596	+	0.912715i	$0.366018\pi$
$132$	0	0
$133$	0.462598	0.0401124
$134$	0	0
$135$	5.64681	0.486000
$136$	0	0
$137$	−3.63640	−0.310678	−0.155339	−	0.987861i	$-0.549647\pi$
$137$	−3.63640	−0.310678	−0.155339	+	0.987861i	$0.549647\pi$
$138$	0	0
$139$	6.21881	0.527473	0.263736	−	0.964595i	$-0.415045\pi$
$139$	6.21881	0.527473	0.263736	+	0.964595i	$0.415045\pi$
$140$	0	0
$141$	12.7126	1.07060
$142$	0	0
$143$	1.39821	0.116924
$144$	0	0
$145$	5.65722	0.469807
$146$	0	0
$147$	1.46260	0.120633
$148$	0	0
$149$	16.7819	1.37482	0.687412	−	0.726268i	$-0.258747\pi$
$149$	16.7819	1.37482	0.687412	+	0.726268i	$0.258747\pi$
$150$	0	0
$151$	−21.8116	−1.77501	−0.887503	−	0.460802i	$-0.847562\pi$
$151$	−21.8116	−1.77501	−0.887503	+	0.460802i	$0.847562\pi$
$152$	0	0
$153$	−1.08377	−0.0876173
$154$	0	0
$155$	5.46260	0.438766
$156$	0	0
$157$	−22.7562	−1.81614	−0.908072	−	0.418814i	$-0.862446\pi$
$157$	−22.7562	−1.81614	−0.908072	+	0.418814i	$0.862446\pi$
$158$	0	0
$159$	−7.01523	−0.556344
$160$	0	0
$161$	−8.04502	−0.634036
$162$	0	0
$163$	9.87603	0.773551	0.386775	−	0.922174i	$-0.373589\pi$
$163$	9.87603	0.773551	0.386775	+	0.922174i	$0.373589\pi$
$164$	0	0
$165$	−2.04502	−0.159204
$166$	0	0
$167$	17.4224	1.34819	0.674093	−	0.738647i	$-0.264535\pi$
$167$	17.4224	1.34819	0.674093	+	0.738647i	$0.264535\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−0.398207	−0.0304517
$172$	0	0
$173$	−23.0796	−1.75471	−0.877356	−	0.479841i	$-0.840694\pi$
$173$	−23.0796	−1.75471	−0.877356	+	0.479841i	$0.840694\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−9.40862	−0.707195
$178$	0	0
$179$	−12.1842	−0.910691	−0.455345	−	0.890315i	$-0.650484\pi$
$179$	−12.1842	−0.910691	−0.455345	+	0.890315i	$0.650484\pi$
$180$	0	0
$181$	15.3595	1.14166	0.570830	−	0.821068i	$-0.306622\pi$
$181$	15.3595	1.14166	0.570830	+	0.821068i	$0.306622\pi$
$182$	0	0
$183$	−6.73057	−0.497538
$184$	0	0
$185$	0.815790	0.0599781
$186$	0	0
$187$	1.76036	0.128730
$188$	0	0
$189$	−5.64681	−0.410745
$190$	0	0
$191$	3.41758	0.247288	0.123644	−	0.992327i	$-0.460542\pi$
$191$	3.41758	0.247288	0.123644	+	0.992327i	$0.460542\pi$
$192$	0	0
$193$	−14.5526	−1.04752	−0.523761	−	0.851865i	$-0.675471\pi$
$193$	−14.5526	−1.04752	−0.523761	+	0.851865i	$0.675471\pi$
$194$	0	0
$195$	−1.46260	−0.104739
$196$	0	0
$197$	−3.09418	−0.220451	−0.110226	−	0.993907i	$-0.535157\pi$
$197$	−3.09418	−0.220451	−0.110226	+	0.993907i	$0.535157\pi$
$198$	0	0
$199$	18.3684	1.30210	0.651051	−	0.759034i	$-0.274328\pi$
$199$	18.3684	1.30210	0.651051	+	0.759034i	$0.274328\pi$
$200$	0	0
$201$	−3.12878	−0.220687
$202$	0	0
$203$	−5.65722	−0.397059
$204$	0	0
$205$	−7.58242	−0.529579
$206$	0	0
$207$	6.92520	0.481334
$208$	0	0
$209$	0.646809	0.0447407
$210$	0	0
$211$	−22.0554	−1.51836	−0.759179	−	0.650882i	$-0.774399\pi$
$211$	−22.0554	−1.51836	−0.759179	+	0.650882i	$0.774399\pi$
$212$	0	0
$213$	−0.188356	−0.0129060
$214$	0	0
$215$	−1.07480	−0.0733010
$216$	0	0
$217$	−5.46260	−0.370825
$218$	0	0
$219$	−19.7279	−1.33309
$220$	0	0
$221$	1.25901	0.0846904
$222$	0	0
$223$	−9.11982	−0.610708	−0.305354	−	0.952239i	$-0.598775\pi$
$223$	−9.11982	−0.610708	−0.305354	+	0.952239i	$0.598775\pi$
$224$	0	0
$225$	−0.860806	−0.0573871
$226$	0	0
$227$	6.00000	0.398234	0.199117	−	0.979976i	$-0.436193\pi$
$227$	6.00000	0.398234	0.199117	+	0.979976i	$0.436193\pi$
$228$	0	0
$229$	−11.6662	−0.770923	−0.385462	−	0.922724i	$-0.625958\pi$
$229$	−11.6662	−0.770923	−0.385462	+	0.922724i	$0.625958\pi$
$230$	0	0
$231$	2.04502	0.134552
$232$	0	0
$233$	−18.7819	−1.23044	−0.615220	−	0.788355i	$-0.710933\pi$
$233$	−18.7819	−1.23044	−0.615220	+	0.788355i	$0.710933\pi$
$234$	0	0
$235$	−8.69182	−0.566992
$236$	0	0
$237$	−4.48197	−0.291135
$238$	0	0
$239$	0.407170	0.0263377	0.0131688	−	0.999913i	$-0.495808\pi$
$239$	0.407170	0.0263377	0.0131688	+	0.999913i	$0.495808\pi$
$240$	0	0
$241$	−7.77704	−0.500963	−0.250482	−	0.968121i	$-0.580589\pi$
$241$	−7.77704	−0.500963	−0.250482	+	0.968121i	$0.580589\pi$
$242$	0	0
$243$	8.63785	0.554118
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	0.462598	0.0294344
$248$	0	0
$249$	−2.11086	−0.133770
$250$	0	0
$251$	14.8206	0.935468	0.467734	−	0.883869i	$-0.345070\pi$
$251$	14.8206	0.935468	0.467734	+	0.883869i	$0.345070\pi$
$252$	0	0
$253$	−11.2486	−0.707193
$254$	0	0
$255$	−1.84143	−0.115315
$256$	0	0
$257$	2.09003	0.130373	0.0651863	−	0.997873i	$-0.479236\pi$
$257$	2.09003	0.130373	0.0651863	+	0.997873i	$0.479236\pi$
$258$	0	0
$259$	−0.815790	−0.0506907
$260$	0	0
$261$	4.86977	0.301431
$262$	0	0
$263$	−0.497202	−0.0306588	−0.0153294	−	0.999882i	$-0.504880\pi$
$263$	−0.497202	−0.0306588	−0.0153294	+	0.999882i	$0.504880\pi$
$264$	0	0
$265$	4.79641	0.294641
$266$	0	0
$267$	−16.8310	−1.03004
$268$	0	0
$269$	−23.7279	−1.44671	−0.723357	−	0.690474i	$-0.757402\pi$
$269$	−23.7279	−1.44671	−0.723357	+	0.690474i	$0.757402\pi$
$270$	0	0
$271$	−11.5478	−0.701480	−0.350740	−	0.936473i	$-0.614070\pi$
$271$	−11.5478	−0.701480	−0.350740	+	0.936473i	$0.614070\pi$
$272$	0	0
$273$	1.46260	0.0885205
$274$	0	0
$275$	1.39821	0.0843151
$276$	0	0
$277$	−25.8504	−1.55320	−0.776600	−	0.629994i	$-0.783057\pi$
$277$	−25.8504	−1.55320	−0.776600	+	0.629994i	$0.783057\pi$
$278$	0	0
$279$	4.70224	0.281516
$280$	0	0
$281$	14.3684	0.857148	0.428574	−	0.903507i	$-0.359016\pi$
$281$	14.3684	0.857148	0.428574	+	0.903507i	$0.359016\pi$
$282$	0	0
$283$	−15.1336	−0.899599	−0.449800	−	0.893130i	$-0.648505\pi$
$283$	−15.1336	−0.899599	−0.449800	+	0.893130i	$0.648505\pi$
$284$	0	0
$285$	−0.676596	−0.0400781
$286$	0	0
$287$	7.58242	0.447576
$288$	0	0
$289$	−15.4149	−0.906758
$290$	0	0
$291$	−11.9550	−0.700814
$292$	0	0
$293$	−3.31444	−0.193632	−0.0968159	−	0.995302i	$-0.530866\pi$
$293$	−3.31444	−0.193632	−0.0968159	+	0.995302i	$0.530866\pi$
$294$	0	0
$295$	6.43281	0.374533
$296$	0	0
$297$	−7.89541	−0.458138
$298$	0	0
$299$	−8.04502	−0.465255
$300$	0	0
$301$	1.07480	0.0619506
$302$	0	0
$303$	−19.7279	−1.13334
$304$	0	0
$305$	4.60179	0.263498
$306$	0	0
$307$	8.49720	0.484961	0.242480	−	0.970156i	$-0.422039\pi$
$307$	8.49720	0.484961	0.242480	+	0.970156i	$0.422039\pi$
$308$	0	0
$309$	−14.2694	−0.811759
$310$	0	0
$311$	−15.0748	−0.854814	−0.427407	−	0.904059i	$-0.640573\pi$
$311$	−15.0748	−0.854814	−0.427407	+	0.904059i	$0.640573\pi$
$312$	0	0
$313$	−1.91478	−0.108230	−0.0541150	−	0.998535i	$-0.517234\pi$
$313$	−1.91478	−0.108230	−0.0541150	+	0.998535i	$0.517234\pi$
$314$	0	0
$315$	0.860806	0.0485009
$316$	0	0
$317$	−27.1890	−1.52709	−0.763544	−	0.645756i	$-0.776542\pi$
$317$	−27.1890	−1.52709	−0.763544	+	0.645756i	$0.776542\pi$
$318$	0	0
$319$	−7.90997	−0.442873
$320$	0	0
$321$	19.8504	1.10794
$322$	0	0
$323$	0.582418	0.0324066
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	21.5541	1.19194
$328$	0	0
$329$	8.69182	0.479196
$330$	0	0
$331$	18.9494	1.04155	0.520776	−	0.853693i	$-0.325643\pi$
$331$	18.9494	1.04155	0.520776	+	0.853693i	$0.325643\pi$
$332$	0	0
$333$	0.702237	0.0384823
$334$	0	0
$335$	2.13919	0.116877
$336$	0	0
$337$	25.2486	1.37538	0.687689	−	0.726005i	$-0.258625\pi$
$337$	25.2486	1.37538	0.687689	+	0.726005i	$0.258625\pi$
$338$	0	0
$339$	23.7875	1.29196
$340$	0	0
$341$	−7.63785	−0.413613
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	11.7666	0.633494
$346$	0	0
$347$	18.3476	0.984951	0.492475	−	0.870326i	$-0.336092\pi$
$347$	18.3476	0.984951	0.492475	+	0.870326i	$0.336092\pi$
$348$	0	0
$349$	17.2292	0.922259	0.461130	−	0.887333i	$-0.347444\pi$
$349$	17.2292	0.922259	0.461130	+	0.887333i	$0.347444\pi$
$350$	0	0
$351$	−5.64681	−0.301404
$352$	0	0
$353$	2.19462	0.116808	0.0584040	−	0.998293i	$-0.481399\pi$
$353$	2.19462	0.116808	0.0584040	+	0.998293i	$0.481399\pi$
$354$	0	0
$355$	0.128782	0.00683504
$356$	0	0
$357$	1.84143	0.0974589
$358$	0	0
$359$	30.5664	1.61323	0.806617	−	0.591075i	$-0.201296\pi$
$359$	30.5664	1.61323	0.806617	+	0.591075i	$0.201296\pi$
$360$	0	0
$361$	−18.7860	−0.988737
$362$	0	0
$363$	−13.2292	−0.694354
$364$	0	0
$365$	13.4882	0.706007
$366$	0	0
$367$	10.4072	0.543250	0.271625	−	0.962403i	$-0.412439\pi$
$367$	10.4072	0.543250	0.271625	+	0.962403i	$0.412439\pi$
$368$	0	0
$369$	−6.52699	−0.339781
$370$	0	0
$371$	−4.79641	−0.249017
$372$	0	0
$373$	16.4585	0.852186	0.426093	−	0.904679i	$-0.359890\pi$
$373$	16.4585	0.852186	0.426093	+	0.904679i	$0.359890\pi$
$374$	0	0
$375$	−1.46260	−0.0755283
$376$	0	0
$377$	−5.65722	−0.291362
$378$	0	0
$379$	−31.3836	−1.61207	−0.806035	−	0.591868i	$-0.798391\pi$
$379$	−31.3836	−1.61207	−0.806035	+	0.591868i	$0.798391\pi$
$380$	0	0
$381$	7.29988	0.373984
$382$	0	0
$383$	−27.5270	−1.40656	−0.703282	−	0.710911i	$-0.748283\pi$
$383$	−27.5270	−1.40656	−0.703282	+	0.710911i	$0.748283\pi$
$384$	0	0
$385$	−1.39821	−0.0712592
$386$	0	0
$387$	−0.925197	−0.0470304
$388$	0	0
$389$	−8.07625	−0.409482	−0.204741	−	0.978816i	$-0.565635\pi$
$389$	−8.07625	−0.409482	−0.204741	+	0.978816i	$0.565635\pi$
$390$	0	0
$391$	−10.1288	−0.512235
$392$	0	0
$393$	13.6800	0.690063
$394$	0	0
$395$	3.06439	0.154186
$396$	0	0
$397$	−9.33237	−0.468378	−0.234189	−	0.972191i	$-0.575243\pi$
$397$	−9.33237	−0.468378	−0.234189	+	0.972191i	$0.575243\pi$
$398$	0	0
$399$	0.676596	0.0338722
$400$	0	0
$401$	20.9073	1.04406	0.522030	−	0.852927i	$-0.325175\pi$
$401$	20.9073	1.04406	0.522030	+	0.852927i	$0.325175\pi$
$402$	0	0
$403$	−5.46260	−0.272111
$404$	0	0
$405$	5.67660	0.282072
$406$	0	0
$407$	−1.14064	−0.0565396
$408$	0	0
$409$	34.9169	1.72653	0.863265	−	0.504751i	$-0.168416\pi$
$409$	34.9169	1.72653	0.863265	+	0.504751i	$0.168416\pi$
$410$	0	0
$411$	−5.31859	−0.262347
$412$	0	0
$413$	−6.43281	−0.316538
$414$	0	0
$415$	1.44322	0.0708451
$416$	0	0
$417$	9.09563	0.445415
$418$	0	0
$419$	29.4674	1.43958	0.719789	−	0.694193i	$-0.244239\pi$
$419$	29.4674	1.43958	0.719789	+	0.694193i	$0.244239\pi$
$420$	0	0
$421$	−19.4882	−0.949799	−0.474899	−	0.880040i	$-0.657516\pi$
$421$	−19.4882	−0.949799	−0.474899	+	0.880040i	$0.657516\pi$
$422$	0	0
$423$	−7.48197	−0.363786
$424$	0	0
$425$	1.25901	0.0610711
$426$	0	0
$427$	−4.60179	−0.222696
$428$	0	0
$429$	2.04502	0.0987343
$430$	0	0
$431$	10.2605	0.494229	0.247115	−	0.968986i	$-0.420518\pi$
$431$	10.2605	0.494229	0.247115	+	0.968986i	$0.420518\pi$
$432$	0	0
$433$	−10.9211	−0.524832	−0.262416	−	0.964955i	$-0.584519\pi$
$433$	−10.9211	−0.524832	−0.262416	+	0.964955i	$0.584519\pi$
$434$	0	0
$435$	8.27424	0.396720
$436$	0	0
$437$	−3.72161	−0.178029
$438$	0	0
$439$	9.38365	0.447857	0.223929	−	0.974606i	$-0.428112\pi$
$439$	9.38365	0.447857	0.223929	+	0.974606i	$0.428112\pi$
$440$	0	0
$441$	−0.860806	−0.0409908
$442$	0	0
$443$	20.5872	0.978129	0.489065	−	0.872248i	$-0.337338\pi$
$443$	20.5872	0.978129	0.489065	+	0.872248i	$0.337338\pi$
$444$	0	0
$445$	11.5076	0.545513
$446$	0	0
$447$	24.5451	1.16094
$448$	0	0
$449$	−25.9612	−1.22519	−0.612594	−	0.790398i	$-0.709874\pi$
$449$	−25.9612	−1.22519	−0.612594	+	0.790398i	$0.709874\pi$
$450$	0	0
$451$	10.6018	0.499219
$452$	0	0
$453$	−31.9017	−1.49887
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	−13.5284	−0.632834	−0.316417	−	0.948620i	$-0.602480\pi$
$457$	−13.5284	−0.632834	−0.316417	+	0.948620i	$0.602480\pi$
$458$	0	0
$459$	−7.10941	−0.331839
$460$	0	0
$461$	−34.6129	−1.61208	−0.806041	−	0.591859i	$-0.798394\pi$
$461$	−34.6129	−1.61208	−0.806041	+	0.591859i	$0.798394\pi$
$462$	0	0
$463$	8.55263	0.397474	0.198737	−	0.980053i	$-0.436316\pi$
$463$	8.55263	0.397474	0.198737	+	0.980053i	$0.436316\pi$
$464$	0	0
$465$	7.98959	0.370508
$466$	0	0
$467$	−15.6274	−0.723151	−0.361576	−	0.932343i	$-0.617761\pi$
$467$	−15.6274	−0.723151	−0.361576	+	0.932343i	$0.617761\pi$
$468$	0	0
$469$	−2.13919	−0.0987788
$470$	0	0
$471$	−33.2832	−1.53361
$472$	0	0
$473$	1.50280	0.0690987
$474$	0	0
$475$	0.462598	0.0212255
$476$	0	0
$477$	4.12878	0.189044
$478$	0	0
$479$	12.5824	0.574905	0.287453	−	0.957795i	$-0.407192\pi$
$479$	12.5824	0.574905	0.287453	+	0.957795i	$0.407192\pi$
$480$	0	0
$481$	−0.815790	−0.0371968
$482$	0	0
$483$	−11.7666	−0.535400
$484$	0	0
$485$	8.17380	0.371153
$486$	0	0
$487$	−34.1849	−1.54906	−0.774532	−	0.632534i	$-0.782015\pi$
$487$	−34.1849	−1.54906	−0.774532	+	0.632534i	$0.782015\pi$
$488$	0	0
$489$	14.4447	0.653211
$490$	0	0
$491$	−10.1496	−0.458045	−0.229023	−	0.973421i	$-0.573553\pi$
$491$	−10.1496	−0.458045	−0.229023	+	0.973421i	$0.573553\pi$
$492$	0	0
$493$	−7.12252	−0.320782
$494$	0	0
$495$	1.20359	0.0540971
$496$	0	0
$497$	−0.128782	−0.00577666
$498$	0	0
$499$	−18.9910	−0.850156	−0.425078	−	0.905157i	$-0.639753\pi$
$499$	−18.9910	−0.850156	−0.425078	+	0.905157i	$0.639753\pi$
$500$	0	0
$501$	25.4820	1.13845
$502$	0	0
$503$	−21.7729	−0.970805	−0.485403	−	0.874291i	$-0.661327\pi$
$503$	−21.7729	−0.970805	−0.485403	+	0.874291i	$0.661327\pi$
$504$	0	0
$505$	13.4882	0.600219
$506$	0	0
$507$	1.46260	0.0649563
$508$	0	0
$509$	−16.0554	−0.711644	−0.355822	−	0.934554i	$-0.615799\pi$
$509$	−16.0554	−0.711644	−0.355822	+	0.934554i	$0.615799\pi$
$510$	0	0
$511$	−13.4882	−0.596685
$512$	0	0
$513$	−2.61220	−0.115332
$514$	0	0
$515$	9.75622	0.429910
$516$	0	0
$517$	12.1530	0.534487
$518$	0	0
$519$	−33.7562	−1.48173
$520$	0	0
$521$	27.4916	1.20443	0.602215	−	0.798334i	$-0.294285\pi$
$521$	27.4916	1.20443	0.602215	+	0.798334i	$0.294285\pi$
$522$	0	0
$523$	2.34423	0.102506	0.0512530	−	0.998686i	$-0.483679\pi$
$523$	2.34423	0.102506	0.0512530	+	0.998686i	$0.483679\pi$
$524$	0	0
$525$	1.46260	0.0638330
$526$	0	0
$527$	−6.87748	−0.299588
$528$	0	0
$529$	41.7223	1.81401
$530$	0	0
$531$	5.53740	0.240303
$532$	0	0
$533$	7.58242	0.328431
$534$	0	0
$535$	−13.5720	−0.586769
$536$	0	0
$537$	−17.8206	−0.769016
$538$	0	0
$539$	1.39821	0.0602251
$540$	0	0
$541$	−21.8116	−0.937756	−0.468878	−	0.883263i	$-0.655342\pi$
$541$	−21.8116	−0.937756	−0.468878	+	0.883263i	$0.655342\pi$
$542$	0	0
$543$	22.4647	0.964053
$544$	0	0
$545$	−14.7368	−0.631257
$546$	0	0
$547$	−0.796415	−0.0340522	−0.0170261	−	0.999855i	$-0.505420\pi$
$547$	−0.796415	−0.0340522	−0.0170261	+	0.999855i	$0.505420\pi$
$548$	0	0
$549$	3.96125	0.169062
$550$	0	0
$551$	−2.61702	−0.111489
$552$	0	0
$553$	−3.06439	−0.130311
$554$	0	0
$555$	1.19317	0.0506474
$556$	0	0
$557$	15.7625	0.667878	0.333939	−	0.942595i	$-0.391622\pi$
$557$	15.7625	0.667878	0.333939	+	0.942595i	$0.391622\pi$
$558$	0	0
$559$	1.07480	0.0454593
$560$	0	0
$561$	2.57470	0.108704
$562$	0	0
$563$	9.49239	0.400056	0.200028	−	0.979790i	$-0.435897\pi$
$563$	9.49239	0.400056	0.200028	+	0.979790i	$0.435897\pi$
$564$	0	0
$565$	−16.2638	−0.684224
$566$	0	0
$567$	−5.67660	−0.238395
$568$	0	0
$569$	29.7327	1.24646	0.623230	−	0.782039i	$-0.285820\pi$
$569$	29.7327	1.24646	0.623230	+	0.782039i	$0.285820\pi$
$570$	0	0
$571$	−26.6032	−1.11331	−0.556656	−	0.830743i	$-0.687916\pi$
$571$	−26.6032	−1.11331	−0.556656	+	0.830743i	$0.687916\pi$
$572$	0	0
$573$	4.99855	0.208817
$574$	0	0
$575$	−8.04502	−0.335500
$576$	0	0
$577$	3.05398	0.127139	0.0635694	−	0.997977i	$-0.479752\pi$
$577$	3.05398	0.127139	0.0635694	+	0.997977i	$0.479752\pi$
$578$	0	0
$579$	−21.2847	−0.884560
$580$	0	0
$581$	−1.44322	−0.0598750
$582$	0	0
$583$	−6.70638	−0.277750
$584$	0	0
$585$	0.860806	0.0355899
$586$	0	0
$587$	21.3144	0.879741	0.439871	−	0.898061i	$-0.355024\pi$
$587$	21.3144	0.879741	0.439871	+	0.898061i	$0.355024\pi$
$588$	0	0
$589$	−2.52699	−0.104123
$590$	0	0
$591$	−4.52554	−0.186156
$592$	0	0
$593$	4.64681	0.190822	0.0954108	−	0.995438i	$-0.469584\pi$
$593$	4.64681	0.190822	0.0954108	+	0.995438i	$0.469584\pi$
$594$	0	0
$595$	−1.25901	−0.0516145
$596$	0	0
$597$	26.8656	1.09954
$598$	0	0
$599$	−21.0215	−0.858915	−0.429457	−	0.903087i	$-0.641295\pi$
$599$	−21.0215	−0.858915	−0.429457	+	0.903087i	$0.641295\pi$
$600$	0	0
$601$	34.2188	1.39581	0.697907	−	0.716188i	$-0.254115\pi$
$601$	34.2188	1.39581	0.697907	+	0.716188i	$0.254115\pi$
$602$	0	0
$603$	1.84143	0.0749889
$604$	0	0
$605$	9.04502	0.367732
$606$	0	0
$607$	42.4841	1.72438	0.862188	−	0.506588i	$-0.169094\pi$
$607$	42.4841	1.72438	0.862188	+	0.506588i	$0.169094\pi$
$608$	0	0
$609$	−8.27424	−0.335289
$610$	0	0
$611$	8.69182	0.351634
$612$	0	0
$613$	12.6018	0.508982	0.254491	−	0.967075i	$-0.418092\pi$
$613$	12.6018	0.508982	0.254491	+	0.967075i	$0.418092\pi$
$614$	0	0
$615$	−11.0900	−0.447193
$616$	0	0
$617$	−9.71747	−0.391210	−0.195605	−	0.980683i	$-0.562667\pi$
$617$	−9.71747	−0.391210	−0.195605	+	0.980683i	$0.562667\pi$
$618$	0	0
$619$	13.7473	0.552549	0.276274	−	0.961079i	$-0.410900\pi$
$619$	13.7473	0.552549	0.276274	+	0.961079i	$0.410900\pi$
$620$	0	0
$621$	45.4287	1.82299
$622$	0	0
$623$	−11.5076	−0.461043
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0.946021	0.0377804
$628$	0	0
$629$	−1.02709	−0.0409528
$630$	0	0
$631$	−42.5664	−1.69454	−0.847271	−	0.531161i	$-0.821756\pi$
$631$	−42.5664	−1.69454	−0.847271	+	0.531161i	$0.821756\pi$
$632$	0	0
$633$	−32.2582	−1.28215
$634$	0	0
$635$	−4.99104	−0.198063
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	0.110856	0.00438541
$640$	0	0
$641$	−2.49575	−0.0985763	−0.0492882	−	0.998785i	$-0.515695\pi$
$641$	−2.49575	−0.0985763	−0.0492882	+	0.998785i	$0.515695\pi$
$642$	0	0
$643$	0.963947	0.0380144	0.0190072	−	0.999819i	$-0.493949\pi$
$643$	0.963947	0.0380144	0.0190072	+	0.999819i	$0.493949\pi$
$644$	0	0
$645$	−1.57201	−0.0618977
$646$	0	0
$647$	7.32755	0.288076	0.144038	−	0.989572i	$-0.453991\pi$
$647$	7.32755	0.288076	0.144038	+	0.989572i	$0.453991\pi$
$648$	0	0
$649$	−8.99440	−0.353061
$650$	0	0
$651$	−7.98959	−0.313137
$652$	0	0
$653$	−11.7812	−0.461033	−0.230517	−	0.973068i	$-0.574042\pi$
$653$	−11.7812	−0.461033	−0.230517	+	0.973068i	$0.574042\pi$
$654$	0	0
$655$	−9.35319	−0.365459
$656$	0	0
$657$	11.6108	0.452979
$658$	0	0
$659$	38.2147	1.48863	0.744316	−	0.667828i	$-0.232776\pi$
$659$	38.2147	1.48863	0.744316	+	0.667828i	$0.232776\pi$
$660$	0	0
$661$	−23.5679	−0.916683	−0.458342	−	0.888776i	$-0.651556\pi$
$661$	−23.5679	−0.916683	−0.458342	+	0.888776i	$0.651556\pi$
$662$	0	0
$663$	1.84143	0.0715152
$664$	0	0
$665$	−0.462598	−0.0179388
$666$	0	0
$667$	45.5124	1.76225
$668$	0	0
$669$	−13.3386	−0.515701
$670$	0	0
$671$	−6.43426	−0.248392
$672$	0	0
$673$	43.3870	1.67245	0.836223	−	0.548389i	$-0.184759\pi$
$673$	43.3870	1.67245	0.836223	+	0.548389i	$0.184759\pi$
$674$	0	0
$675$	−5.64681	−0.217346
$676$	0	0
$677$	0.691825	0.0265890	0.0132945	−	0.999912i	$-0.495768\pi$
$677$	0.691825	0.0265890	0.0132945	+	0.999912i	$0.495768\pi$
$678$	0	0
$679$	−8.17380	−0.313682
$680$	0	0
$681$	8.77559	0.336281
$682$	0	0
$683$	−35.8704	−1.37254	−0.686272	−	0.727345i	$-0.740754\pi$
$683$	−35.8704	−1.37254	−0.686272	+	0.727345i	$0.740754\pi$
$684$	0	0
$685$	3.63640	0.138940
$686$	0	0
$687$	−17.0629	−0.650992
$688$	0	0
$689$	−4.79641	−0.182729
$690$	0	0
$691$	−32.6337	−1.24144	−0.620722	−	0.784031i	$-0.713161\pi$
$691$	−32.6337	−1.24144	−0.620722	+	0.784031i	$0.713161\pi$
$692$	0	0
$693$	−1.20359	−0.0457204
$694$	0	0
$695$	−6.21881	−0.235893
$696$	0	0
$697$	9.54636	0.361594
$698$	0	0
$699$	−27.4703	−1.03902
$700$	0	0
$701$	6.83102	0.258004	0.129002	−	0.991644i	$-0.458823\pi$
$701$	6.83102	0.258004	0.129002	+	0.991644i	$0.458823\pi$
$702$	0	0
$703$	−0.377383	−0.0142333
$704$	0	0
$705$	−12.7126	−0.478786
$706$	0	0
$707$	−13.4882	−0.507277
$708$	0	0
$709$	−22.1350	−0.831299	−0.415650	−	0.909525i	$-0.636446\pi$
$709$	−22.1350	−0.831299	−0.415650	+	0.909525i	$0.636446\pi$
$710$	0	0
$711$	2.63785	0.0989269
$712$	0	0
$713$	43.9467	1.64582
$714$	0	0
$715$	−1.39821	−0.0522900
$716$	0	0
$717$	0.595527	0.0222403
$718$	0	0
$719$	43.2549	1.61313	0.806567	−	0.591142i	$-0.201323\pi$
$719$	43.2549	1.61313	0.806567	+	0.591142i	$0.201323\pi$
$720$	0	0
$721$	−9.75622	−0.363340
$722$	0	0
$723$	−11.3747	−0.423029
$724$	0	0
$725$	−5.65722	−0.210104
$726$	0	0
$727$	48.8109	1.81029	0.905147	−	0.425098i	$-0.139760\pi$
$727$	48.8109	1.81029	0.905147	+	0.425098i	$0.139760\pi$
$728$	0	0
$729$	29.6635	1.09865
$730$	0	0
$731$	1.35319	0.0500496
$732$	0	0
$733$	3.20359	0.118327	0.0591636	−	0.998248i	$-0.481157\pi$
$733$	3.20359	0.118327	0.0591636	+	0.998248i	$0.481157\pi$
$734$	0	0
$735$	−1.46260	−0.0539488
$736$	0	0
$737$	−2.99104	−0.110176
$738$	0	0
$739$	−8.18836	−0.301214	−0.150607	−	0.988594i	$-0.548123\pi$
$739$	−8.18836	−0.301214	−0.150607	+	0.988594i	$0.548123\pi$
$740$	0	0
$741$	0.676596	0.0248554
$742$	0	0
$743$	−35.7175	−1.31035	−0.655173	−	0.755479i	$-0.727404\pi$
$743$	−35.7175	−1.31035	−0.655173	+	0.755479i	$0.727404\pi$
$744$	0	0
$745$	−16.7819	−0.614840
$746$	0	0
$747$	1.24234	0.0454547
$748$	0	0
$749$	13.5720	0.495910
$750$	0	0
$751$	14.5374	0.530477	0.265239	−	0.964183i	$-0.414549\pi$
$751$	14.5374	0.530477	0.265239	+	0.964183i	$0.414549\pi$
$752$	0	0
$753$	21.6766	0.789939
$754$	0	0
$755$	21.8116	0.793807
$756$	0	0
$757$	19.2036	0.697966	0.348983	−	0.937129i	$-0.386527\pi$
$757$	19.2036	0.697966	0.348983	+	0.937129i	$0.386527\pi$
$758$	0	0
$759$	−16.4522	−0.597177
$760$	0	0
$761$	0.0401993	0.00145722	0.000728612	−	1.00000i	$-0.499768\pi$
$761$	0.0401993	0.00145722	0.000728612		1.00000i	$0.499768\pi$
$762$	0	0
$763$	14.7368	0.533509
$764$	0	0
$765$	1.08377	0.0391836
$766$	0	0
$767$	−6.43281	−0.232275
$768$	0	0
$769$	39.5783	1.42723	0.713614	−	0.700539i	$-0.247057\pi$
$769$	39.5783	1.42723	0.713614	+	0.700539i	$0.247057\pi$
$770$	0	0
$771$	3.05688	0.110091
$772$	0	0
$773$	−39.8504	−1.43332	−0.716660	−	0.697423i	$-0.754330\pi$
$773$	−39.8504	−1.43332	−0.716660	+	0.697423i	$0.754330\pi$
$774$	0	0
$775$	−5.46260	−0.196222
$776$	0	0
$777$	−1.19317	−0.0428048
$778$	0	0
$779$	3.50761	0.125673
$780$	0	0
$781$	−0.180064	−0.00644319
$782$	0	0
$783$	31.9452	1.14163
$784$	0	0
$785$	22.7562	0.812204
$786$	0	0
$787$	39.4141	1.40496	0.702481	−	0.711703i	$-0.252076\pi$
$787$	39.4141	1.40496	0.702481	+	0.711703i	$0.252076\pi$
$788$	0	0
$789$	−0.727207	−0.0258893
$790$	0	0
$791$	16.2638	0.578275
$792$	0	0
$793$	−4.60179	−0.163414
$794$	0	0
$795$	7.01523	0.248805
$796$	0	0
$797$	−22.1094	−0.783155	−0.391578	−	0.920145i	$-0.628071\pi$
$797$	−22.1094	−0.783155	−0.391578	+	0.920145i	$0.628071\pi$
$798$	0	0
$799$	10.9431	0.387140
$800$	0	0
$801$	9.90582	0.350005
$802$	0	0
$803$	−18.8594	−0.665532
$804$	0	0
$805$	8.04502	0.283550
$806$	0	0
$807$	−34.7044	−1.22165
$808$	0	0
$809$	−26.2951	−0.924485	−0.462243	−	0.886753i	$-0.652955\pi$
$809$	−26.2951	−0.924485	−0.462243	+	0.886753i	$0.652955\pi$
$810$	0	0
$811$	−2.70638	−0.0950340	−0.0475170	−	0.998870i	$-0.515131\pi$
$811$	−2.70638	−0.0950340	−0.0475170	+	0.998870i	$0.515131\pi$
$812$	0	0
$813$	−16.8898	−0.592352
$814$	0	0
$815$	−9.87603	−0.345942
$816$	0	0
$817$	0.497202	0.0173949
$818$	0	0
$819$	−0.860806	−0.0300790
$820$	0	0
$821$	47.5845	1.66071	0.830356	−	0.557233i	$-0.188137\pi$
$821$	47.5845	1.66071	0.830356	+	0.557233i	$0.188137\pi$
$822$	0	0
$823$	24.6323	0.858626	0.429313	−	0.903156i	$-0.358756\pi$
$823$	24.6323	0.858626	0.429313	+	0.903156i	$0.358756\pi$
$824$	0	0
$825$	2.04502	0.0711983
$826$	0	0
$827$	16.7785	0.583445	0.291723	−	0.956503i	$-0.405772\pi$
$827$	16.7785	0.583445	0.291723	+	0.956503i	$0.405772\pi$
$828$	0	0
$829$	−21.9821	−0.763469	−0.381734	−	0.924272i	$-0.624673\pi$
$829$	−21.9821	−0.763469	−0.381734	+	0.924272i	$0.624673\pi$
$830$	0	0
$831$	−37.8087	−1.31157
$832$	0	0
$833$	1.25901	0.0436222
$834$	0	0
$835$	−17.4224	−0.602927
$836$	0	0
$837$	30.8462	1.06620
$838$	0	0
$839$	−50.6143	−1.74740	−0.873700	−	0.486465i	$-0.838286\pi$
$839$	−50.6143	−1.74740	−0.873700	+	0.486465i	$0.838286\pi$
$840$	0	0
$841$	3.00415	0.103591
$842$	0	0
$843$	21.0152	0.723803
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−9.04502	−0.310790
$848$	0	0
$849$	−22.1344	−0.759650
$850$	0	0
$851$	6.56304	0.224978
$852$	0	0
$853$	9.55118	0.327026	0.163513	−	0.986541i	$-0.447717\pi$
$853$	9.55118	0.327026	0.163513	+	0.986541i	$0.447717\pi$
$854$	0	0
$855$	0.398207	0.0136184
$856$	0	0
$857$	41.3193	1.41144	0.705719	−	0.708491i	$-0.250624\pi$
$857$	41.3193	1.41144	0.705719	+	0.708491i	$0.250624\pi$
$858$	0	0
$859$	40.7015	1.38872	0.694358	−	0.719630i	$-0.255688\pi$
$859$	40.7015	1.38872	0.694358	+	0.719630i	$0.255688\pi$
$860$	0	0
$861$	11.0900	0.377947
$862$	0	0
$863$	38.6039	1.31409	0.657046	−	0.753850i	$-0.271806\pi$
$863$	38.6039	1.31409	0.657046	+	0.753850i	$0.271806\pi$
$864$	0	0
$865$	23.0796	0.784731
$866$	0	0
$867$	−22.5458	−0.765695
$868$	0	0
$869$	−4.28465	−0.145347
$870$	0	0
$871$	−2.13919	−0.0724838
$872$	0	0
$873$	7.03605	0.238134
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−35.0256	−1.18273	−0.591366	−	0.806403i	$-0.701411\pi$
$877$	−35.0256	−1.18273	−0.591366	+	0.806403i	$0.701411\pi$
$878$	0	0
$879$	−4.84770	−0.163509
$880$	0	0
$881$	−7.60246	−0.256133	−0.128067	−	0.991766i	$-0.540877\pi$
$881$	−7.60246	−0.256133	−0.128067	+	0.991766i	$0.540877\pi$
$882$	0	0
$883$	−38.2396	−1.28687	−0.643433	−	0.765502i	$-0.722491\pi$
$883$	−38.2396	−1.28687	−0.643433	+	0.765502i	$0.722491\pi$
$884$	0	0
$885$	9.40862	0.316267
$886$	0	0
$887$	15.2084	0.510648	0.255324	−	0.966856i	$-0.417818\pi$
$887$	15.2084	0.510648	0.255324	+	0.966856i	$0.417818\pi$
$888$	0	0
$889$	4.99104	0.167394
$890$	0	0
$891$	−7.93706	−0.265901
$892$	0	0
$893$	4.02082	0.134552
$894$	0	0
$895$	12.1842	0.407273
$896$	0	0
$897$	−11.7666	−0.392876
$898$	0	0
$899$	30.9031	1.03068
$900$	0	0
$901$	−6.03875	−0.201180
$902$	0	0
$903$	1.57201	0.0523131
$904$	0	0
$905$	−15.3595	−0.510566
$906$	0	0
$907$	−43.1440	−1.43257	−0.716287	−	0.697806i	$-0.754160\pi$
$907$	−43.1440	−1.43257	−0.716287	+	0.697806i	$0.754160\pi$
$908$	0	0
$909$	11.6108	0.385104
$910$	0	0
$911$	−1.59698	−0.0529102	−0.0264551	−	0.999650i	$-0.508422\pi$
$911$	−1.59698	−0.0529102	−0.0264551	+	0.999650i	$0.508422\pi$
$912$	0	0
$913$	−2.01793	−0.0667836
$914$	0	0
$915$	6.73057	0.222506
$916$	0	0
$917$	9.35319	0.308870
$918$	0	0
$919$	−19.8310	−0.654165	−0.327082	−	0.944996i	$-0.606065\pi$
$919$	−19.8310	−0.654165	−0.327082	+	0.944996i	$0.606065\pi$
$920$	0	0
$921$	12.4280	0.409516
$922$	0	0
$923$	−0.128782	−0.00423891
$924$	0	0
$925$	−0.815790	−0.0268230
$926$	0	0
$927$	8.39821	0.275833
$928$	0	0
$929$	−41.4335	−1.35939	−0.679694	−	0.733496i	$-0.737888\pi$
$929$	−41.4335	−1.35939	−0.679694	+	0.733496i	$0.737888\pi$
$930$	0	0
$931$	0.462598	0.0151611
$932$	0	0
$933$	−22.0484	−0.721832
$934$	0	0
$935$	−1.76036	−0.0575700
$936$	0	0
$937$	36.7923	1.20195	0.600976	−	0.799267i	$-0.294779\pi$
$937$	36.7923	1.20195	0.600976	+	0.799267i	$0.294779\pi$
$938$	0	0
$939$	−2.80056	−0.0913929
$940$	0	0
$941$	−13.2174	−0.430874	−0.215437	−	0.976518i	$-0.569118\pi$
$941$	−13.2174	−0.430874	−0.215437	+	0.976518i	$0.569118\pi$
$942$	0	0
$943$	−61.0007	−1.98646
$944$	0	0
$945$	5.64681	0.183691
$946$	0	0
$947$	32.4834	1.05557	0.527785	−	0.849378i	$-0.323023\pi$
$947$	32.4834	1.05557	0.527785	+	0.849378i	$0.323023\pi$
$948$	0	0
$949$	−13.4882	−0.437847
$950$	0	0
$951$	−39.7666	−1.28952
$952$	0	0
$953$	23.8325	0.772009	0.386005	−	0.922497i	$-0.373855\pi$
$953$	23.8325	0.772009	0.386005	+	0.922497i	$0.373855\pi$
$954$	0	0
$955$	−3.41758	−0.110590
$956$	0	0
$957$	−11.5691	−0.373976
$958$	0	0
$959$	−3.63640	−0.117425
$960$	0	0
$961$	−1.16002	−0.0374200
$962$	0	0
$963$	−11.6829	−0.376475
$964$	0	0
$965$	14.5526	0.468466
$966$	0	0
$967$	−12.0256	−0.386719	−0.193359	−	0.981128i	$-0.561938\pi$
$967$	−12.0256	−0.386719	−0.193359	+	0.981128i	$0.561938\pi$
$968$	0	0
$969$	0.851843	0.0273651
$970$	0	0
$971$	4.75140	0.152480	0.0762398	−	0.997090i	$-0.475709\pi$
$971$	4.75140	0.152480	0.0762398	+	0.997090i	$0.475709\pi$
$972$	0	0
$973$	6.21881	0.199366
$974$	0	0
$975$	1.46260	0.0468406
$976$	0	0
$977$	20.5139	0.656297	0.328149	−	0.944626i	$-0.393575\pi$
$977$	20.5139	0.656297	0.328149	+	0.944626i	$0.393575\pi$
$978$	0	0
$979$	−16.0900	−0.514240
$980$	0	0
$981$	−12.6856	−0.405019
$982$	0	0
$983$	5.35319	0.170740	0.0853701	−	0.996349i	$-0.472793\pi$
$983$	5.35319	0.170740	0.0853701	+	0.996349i	$0.472793\pi$
$984$	0	0
$985$	3.09418	0.0985887
$986$	0	0
$987$	12.7126	0.404648
$988$	0	0
$989$	−8.64681	−0.274953
$990$	0	0
$991$	−13.9508	−0.443163	−0.221581	−	0.975142i	$-0.571122\pi$
$991$	−13.9508	−0.443163	−0.221581	+	0.975142i	$0.571122\pi$
$992$	0	0
$993$	27.7153	0.879520
$994$	0	0
$995$	−18.3684	−0.582318
$996$	0	0
$997$	29.4120	0.931487	0.465743	−	0.884920i	$-0.345787\pi$
$997$	29.4120	0.931487	0.465743	+	0.884920i	$0.345787\pi$
$998$	0	0
$999$	4.60661	0.145747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7280.2.a.bp.1.2		3
4.3	odd	2		3640.2.a.o.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.o.1.2	✓	3	4.3	odd	2
7280.2.a.bp.1.2		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:	\( N \)	\(=\)	\( 7280 = 2^{4} \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7280.a (trivial)

\(f(q)\)	\(=\)	\(q+1.46260 q^{3} -1.00000 q^{5} +1.00000 q^{7} -0.860806 q^{9} +O(q^{10})\)	\(q+1.46260 q^{3} -1.00000 q^{5} +1.00000 q^{7} -0.860806 q^{9} +1.39821 q^{11} +1.00000 q^{13} -1.46260 q^{15} +1.25901 q^{17} +0.462598 q^{19} +1.46260 q^{21} -8.04502 q^{23} +1.00000 q^{25} -5.64681 q^{27} -5.65722 q^{29} -5.46260 q^{31} +2.04502 q^{33} -1.00000 q^{35} -0.815790 q^{37} +1.46260 q^{39} +7.58242 q^{41} +1.07480 q^{43} +0.860806 q^{45} +8.69182 q^{47} +1.00000 q^{49} +1.84143 q^{51} -4.79641 q^{53} -1.39821 q^{55} +0.676596 q^{57} -6.43281 q^{59} -4.60179 q^{61} -0.860806 q^{63} -1.00000 q^{65} -2.13919 q^{67} -11.7666 q^{69} -0.128782 q^{71} -13.4882 q^{73} +1.46260 q^{75} +1.39821 q^{77} -3.06439 q^{79} -5.67660 q^{81} -1.44322 q^{83} -1.25901 q^{85} -8.27424 q^{87} -11.5076 q^{89} +1.00000 q^{91} -7.98959 q^{93} -0.462598 q^{95} -8.17380 q^{97} -1.20359 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 2 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9	\( 3 q + 2 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9} + q^{11} + 3 q^{13} - 2 q^{15} - 5 q^{17} - q^{19} + 2 q^{21} - 5 q^{23} + 3 q^{25} - q^{27} - 5 q^{29} - 14 q^{31} - 13 q^{33} - 3 q^{35} - 16 q^{37} + 2 q^{39} + 6 q^{41} + 8 q^{43} - 3 q^{45} - 9 q^{47} + 3 q^{49} - 20 q^{51} - 8 q^{53} - q^{55} + 10 q^{57} + 7 q^{59} - 17 q^{61} + 3 q^{63} - 3 q^{65} - 12 q^{67} - 5 q^{69} - 2 q^{71} + q^{73} + 2 q^{75} + q^{77} - 10 q^{79} - 25 q^{81} + 18 q^{83} + 5 q^{85} + 27 q^{87} - 13 q^{89} + 3 q^{91} - 20 q^{93} + q^{95} - 7 q^{97} - 10 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9 + q^11 + 3 * q^13 - 2 * q^15 - 5 * q^17 - q^19 + 2 * q^21 - 5 * q^23 + 3 * q^25 - q^27 - 5 * q^29 - 14 * q^31 - 13 * q^33 - 3 * q^35 - 16 * q^37 + 2 * q^39 + 6 * q^41 + 8 * q^43 - 3 * q^45 - 9 * q^47 + 3 * q^49 - 20 * q^51 - 8 * q^53 - q^55 + 10 * q^57 + 7 * q^59 - 17 * q^61 + 3 * q^63 - 3 * q^65 - 12 * q^67 - 5 * q^69 - 2 * q^71 + q^73 + 2 * q^75 + q^77 - 10 * q^79 - 25 * q^81 + 18 * q^83 + 5 * q^85 + 27 * q^87 - 13 * q^89 + 3 * q^91 - 20 * q^93 + q^95 - 7 * q^97 - 10 * q^99

Introduction

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