LMFDB - Embedded newform 5355.2.a.bd.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5355,2,Mod(1,5355)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5355.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5355, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5355 = 3^{2} \cdot 5 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5355.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,1,0,5,-3,0,3,3,0,-1,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$42.7598902824$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1785)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.51414$ of defining polynomial
Character	$\chi$	$=$	5355.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.51414 q^{2} +4.32088 q^{4} -1.00000 q^{5} +1.00000 q^{7} +5.83502 q^{8} -2.51414 q^{10} +1.70739 q^{11} -1.32088 q^{13} +2.51414 q^{14} +6.02827 q^{16} +1.00000 q^{17} +2.00000 q^{19} -4.32088 q^{20} +4.29261 q^{22} +6.64177 q^{23} +1.00000 q^{25} -3.32088 q^{26} +4.32088 q^{28} -2.34916 q^{29} +4.34916 q^{31} +3.48586 q^{32} +2.51414 q^{34} -1.00000 q^{35} +2.00000 q^{37} +5.02827 q^{38} -5.83502 q^{40} -6.00000 q^{41} +12.0565 q^{43} +7.37743 q^{44} +16.6983 q^{46} +0.641769 q^{47} +1.00000 q^{49} +2.51414 q^{50} -5.70739 q^{52} -2.67912 q^{53} -1.70739 q^{55} +5.83502 q^{56} -5.90611 q^{58} -7.61350 q^{59} +6.93438 q^{61} +10.9344 q^{62} -3.29261 q^{64} +1.32088 q^{65} +8.64177 q^{67} +4.32088 q^{68} -2.51414 q^{70} -8.34916 q^{71} +0.386505 q^{73} +5.02827 q^{74} +8.64177 q^{76} +1.70739 q^{77} +6.38650 q^{79} -6.02827 q^{80} -15.0848 q^{82} +6.00000 q^{83} -1.00000 q^{85} +30.3118 q^{86} +9.96265 q^{88} +6.97173 q^{89} -1.32088 q^{91} +28.6983 q^{92} +1.61350 q^{94} -2.00000 q^{95} -6.25526 q^{97} +2.51414 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 5 q^{4} - 3 q^{5} + 3 q^{7} + 3 q^{8} - q^{10} + 4 q^{13} + q^{14} + 5 q^{16} + 3 q^{17} + 6 q^{19} - 5 q^{20} + 18 q^{22} + 4 q^{23} + 3 q^{25} - 2 q^{26} + 5 q^{28} + 14 q^{29} - 8 q^{31}+ \cdots + q^{98}+O(q^{100}) $ 3 * q + q^2 + 5 * q^4 - 3 * q^5 + 3 * q^7 + 3 * q^8 - q^10 + 4 * q^13 + q^14 + 5 * q^16 + 3 * q^17 + 6 * q^19 - 5 * q^20 + 18 * q^22 + 4 * q^23 + 3 * q^25 - 2 * q^26 + 5 * q^28 + 14 * q^29 - 8 * q^31 + 17 * q^32 + q^34 - 3 * q^35 + 6 * q^37 + 2 * q^38 - 3 * q^40 - 18 * q^41 + 10 * q^43 - 12 * q^44 + 8 * q^46 - 14 * q^47 + 3 * q^49 + q^50 - 12 * q^52 - 16 * q^53 + 3 * q^56 - 20 * q^58 - 20 * q^59 + 10 * q^61 + 22 * q^62 - 15 * q^64 - 4 * q^65 + 10 * q^67 + 5 * q^68 - q^70 - 4 * q^71 + 4 * q^73 + 2 * q^74 + 10 * q^76 + 22 * q^79 - 5 * q^80 - 6 * q^82 + 18 * q^83 - 3 * q^85 + 46 * q^86 + 6 * q^88 + 34 * q^89 + 4 * q^91 + 44 * q^92 + 2 * q^94 - 6 * q^95 + q^98

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 $

$2$	2.51414	1.77776	0.888882	−	0.458137i	$-0.151483\pi$
$2$	2.51414	1.77776	0.888882	+	0.458137i	$0.151483\pi$
$3$	0	0
$3$	0	0
$4$	4.32088	2.16044
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	5.83502	2.06299
$9$	0	0
$10$	−2.51414	−0.795040
$11$	1.70739	0.514797	0.257399	−	0.966305i	$-0.417135\pi$
$11$	1.70739	0.514797	0.257399	+	0.966305i	$0.417135\pi$
$12$	0	0
$13$	−1.32088	−0.366347	−0.183174	−	0.983081i	$-0.558637\pi$
$13$	−1.32088	−0.366347	−0.183174	+	0.983081i	$0.558637\pi$
$14$	2.51414	0.671931
$15$	0	0
$16$	6.02827	1.50707
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	−4.32088	−0.966179
$21$	0	0
$22$	4.29261	0.915188
$23$	6.64177	1.38490	0.692452	−	0.721464i	$-0.256530\pi$
$23$	6.64177	1.38490	0.692452	+	0.721464i	$0.256530\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−3.32088	−0.651279
$27$	0	0
$28$	4.32088	0.816570
$29$	−2.34916	−0.436228	−0.218114	−	0.975923i	$-0.569990\pi$
$29$	−2.34916	−0.436228	−0.218114	+	0.975923i	$0.569990\pi$
$30$	0	0
$31$	4.34916	0.781132	0.390566	−	0.920575i	$-0.372279\pi$
$31$	4.34916	0.781132	0.390566	+	0.920575i	$0.372279\pi$
$32$	3.48586	0.616219
$33$	0	0
$34$	2.51414	0.431171
$35$	−1.00000	−0.169031
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	5.02827	0.815694
$39$	0	0
$40$	−5.83502	−0.922598
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	12.0565	1.83861	0.919303	−	0.393550i	$-0.128753\pi$
$43$	12.0565	1.83861	0.919303	+	0.393550i	$0.128753\pi$
$44$	7.37743	1.11219
$45$	0	0
$46$	16.6983	2.46203
$47$	0.641769	0.0936116	0.0468058	−	0.998904i	$-0.485096\pi$
$47$	0.641769	0.0936116	0.0468058	+	0.998904i	$0.485096\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	2.51414	0.355553
$51$	0	0
$52$	−5.70739	−0.791472
$53$	−2.67912	−0.368005	−0.184002	−	0.982926i	$-0.558905\pi$
$53$	−2.67912	−0.368005	−0.184002	+	0.982926i	$0.558905\pi$
$54$	0	0
$55$	−1.70739	−0.230224
$56$	5.83502	0.779738
$57$	0	0
$58$	−5.90611	−0.775510
$59$	−7.61350	−0.991193	−0.495596	−	0.868553i	$-0.665050\pi$
$59$	−7.61350	−0.991193	−0.495596	+	0.868553i	$0.665050\pi$
$60$	0	0
$61$	6.93438	0.887856	0.443928	−	0.896062i	$-0.353585\pi$
$61$	6.93438	0.887856	0.443928	+	0.896062i	$0.353585\pi$
$62$	10.9344	1.38867
$63$	0	0
$64$	−3.29261	−0.411576
$65$	1.32088	0.163836
$66$	0	0
$67$	8.64177	1.05576	0.527880	−	0.849319i	$-0.322987\pi$
$67$	8.64177	1.05576	0.527880	+	0.849319i	$0.322987\pi$
$68$	4.32088	0.523984
$69$	0	0
$70$	−2.51414	−0.300497
$71$	−8.34916	−0.990863	−0.495431	−	0.868647i	$-0.664990\pi$
$71$	−8.34916	−0.990863	−0.495431	+	0.868647i	$0.664990\pi$
$72$	0	0
$73$	0.386505	0.0452370	0.0226185	−	0.999744i	$-0.492800\pi$
$73$	0.386505	0.0452370	0.0226185	+	0.999744i	$0.492800\pi$
$74$	5.02827	0.584525
$75$	0	0
$76$	8.64177	0.991279
$77$	1.70739	0.194575
$78$	0	0
$79$	6.38650	0.718538	0.359269	−	0.933234i	$-0.383026\pi$
$79$	6.38650	0.718538	0.359269	+	0.933234i	$0.383026\pi$
$80$	−6.02827	−0.673982
$81$	0	0
$82$	−15.0848	−1.66584
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	−1.00000	−0.108465
$86$	30.3118	3.26861
$87$	0	0
$88$	9.96265	1.06202
$89$	6.97173	0.739001	0.369501	−	0.929230i	$-0.379529\pi$
$89$	6.97173	0.739001	0.369501	+	0.929230i	$0.379529\pi$
$90$	0	0
$91$	−1.32088	−0.138466
$92$	28.6983	2.99201
$93$	0	0
$94$	1.61350	0.166419
$95$	−2.00000	−0.205196
$96$	0	0
$97$	−6.25526	−0.635126	−0.317563	−	0.948237i	$-0.602864\pi$
$97$	−6.25526	−0.635126	−0.317563	+	0.948237i	$0.602864\pi$
$98$	2.51414	0.253966
$99$	0	0
$100$	4.32088	0.432088
$101$	−8.25526	−0.821429	−0.410715	−	0.911764i	$-0.634721\pi$
$101$	−8.25526	−0.821429	−0.410715	+	0.911764i	$0.634721\pi$
$102$	0	0
$103$	−2.93438	−0.289133	−0.144567	−	0.989495i	$-0.546179\pi$
$103$	−2.93438	−0.289133	−0.144567	+	0.989495i	$0.546179\pi$
$104$	−7.70739	−0.755772
$105$	0	0
$106$	−6.73566	−0.654225
$107$	19.9253	1.92625	0.963126	−	0.269051i	$-0.0867099\pi$
$107$	19.9253	1.92625	0.963126	+	0.269051i	$0.0867099\pi$
$108$	0	0
$109$	7.35823	0.704791	0.352395	−	0.935851i	$-0.385367\pi$
$109$	7.35823	0.704791	0.352395	+	0.935851i	$0.385367\pi$
$110$	−4.29261	−0.409284
$111$	0	0
$112$	6.02827	0.569618
$113$	0.971726	0.0914123	0.0457062	−	0.998955i	$-0.485446\pi$
$113$	0.971726	0.0914123	0.0457062	+	0.998955i	$0.485446\pi$
$114$	0	0
$115$	−6.64177	−0.619348
$116$	−10.1504	−0.942445
$117$	0	0
$118$	−19.1414	−1.76211
$119$	1.00000	0.0916698
$120$	0	0
$121$	−8.08482	−0.734984
$122$	17.4340	1.57840
$123$	0	0
$124$	18.7922	1.68759
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	−15.2498	−1.34790
$129$	0	0
$130$	3.32088	0.291261
$131$	−8.44305	−0.737673	−0.368836	−	0.929494i	$-0.620244\pi$
$131$	−8.44305	−0.737673	−0.368836	+	0.929494i	$0.620244\pi$
$132$	0	0
$133$	2.00000	0.173422
$134$	21.7266	1.87689
$135$	0	0
$136$	5.83502	0.500349
$137$	−0.735663	−0.0628520	−0.0314260	−	0.999506i	$-0.510005\pi$
$137$	−0.735663	−0.0628520	−0.0314260	+	0.999506i	$0.510005\pi$
$138$	0	0
$139$	1.12217	0.0951811	0.0475905	−	0.998867i	$-0.484846\pi$
$139$	1.12217	0.0951811	0.0475905	+	0.998867i	$0.484846\pi$
$140$	−4.32088	−0.365181
$141$	0	0
$142$	−20.9909	−1.76152
$143$	−2.25526	−0.188595
$144$	0	0
$145$	2.34916	0.195087
$146$	0.971726	0.0804206
$147$	0	0
$148$	8.64177	0.710349
$149$	−0.641769	−0.0525758	−0.0262879	−	0.999654i	$-0.508369\pi$
$149$	−0.641769	−0.0525758	−0.0262879	+	0.999654i	$0.508369\pi$
$150$	0	0
$151$	16.1131	1.31127	0.655633	−	0.755080i	$-0.272402\pi$
$151$	16.1131	1.31127	0.655633	+	0.755080i	$0.272402\pi$
$152$	11.6700	0.946565
$153$	0	0
$154$	4.29261	0.345908
$155$	−4.34916	−0.349333
$156$	0	0
$157$	−23.3774	−1.86572	−0.932861	−	0.360236i	$-0.882696\pi$
$157$	−23.3774	−1.86572	−0.932861	+	0.360236i	$0.882696\pi$
$158$	16.0565	1.27739
$159$	0	0
$160$	−3.48586	−0.275582
$161$	6.64177	0.523445
$162$	0	0
$163$	6.05655	0.474385	0.237193	−	0.971463i	$-0.423773\pi$
$163$	6.05655	0.474385	0.237193	+	0.971463i	$0.423773\pi$
$164$	−25.9253	−2.02443
$165$	0	0
$166$	15.0848	1.17081
$167$	9.08482	0.703005	0.351502	−	0.936187i	$-0.385671\pi$
$167$	9.08482	0.703005	0.351502	+	0.936187i	$0.385671\pi$
$168$	0	0
$169$	−11.2553	−0.865790
$170$	−2.51414	−0.192826
$171$	0	0
$172$	52.0950	3.97220
$173$	−24.9536	−1.89719	−0.948593	−	0.316499i	$-0.897493\pi$
$173$	−24.9536	−1.89719	−0.948593	+	0.316499i	$0.897493\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	10.2926	0.775835
$177$	0	0
$178$	17.5279	1.31377
$179$	−22.8861	−1.71059	−0.855294	−	0.518143i	$-0.826623\pi$
$179$	−22.8861	−1.71059	−0.855294	+	0.518143i	$0.826623\pi$
$180$	0	0
$181$	13.5761	1.00911	0.504554	−	0.863380i	$-0.331657\pi$
$181$	13.5761	1.00911	0.504554	+	0.863380i	$0.331657\pi$
$182$	−3.32088	−0.246160
$183$	0	0
$184$	38.7549	2.85705
$185$	−2.00000	−0.147043
$186$	0	0
$187$	1.70739	0.124857
$188$	2.77301	0.202243
$189$	0	0
$190$	−5.02827	−0.364789
$191$	2.77301	0.200648	0.100324	−	0.994955i	$-0.468012\pi$
$191$	2.77301	0.200648	0.100324	+	0.994955i	$0.468012\pi$
$192$	0	0
$193$	−8.84049	−0.636352	−0.318176	−	0.948032i	$-0.603070\pi$
$193$	−8.84049	−0.636352	−0.318176	+	0.948032i	$0.603070\pi$
$194$	−15.7266	−1.12910
$195$	0	0
$196$	4.32088	0.308635
$197$	2.44305	0.174060	0.0870301	−	0.996206i	$-0.472262\pi$
$197$	2.44305	0.174060	0.0870301	+	0.996206i	$0.472262\pi$
$198$	0	0
$199$	11.6508	0.825906	0.412953	−	0.910752i	$-0.364497\pi$
$199$	11.6508	0.825906	0.412953	+	0.910752i	$0.364497\pi$
$200$	5.83502	0.412598
$201$	0	0
$202$	−20.7549	−1.46031
$203$	−2.34916	−0.164879
$204$	0	0
$205$	6.00000	0.419058
$206$	−7.37743	−0.514010
$207$	0	0
$208$	−7.96265	−0.552111
$209$	3.41478	0.236205
$210$	0	0
$211$	−5.61350	−0.386449	−0.193224	−	0.981155i	$-0.561895\pi$
$211$	−5.61350	−0.386449	−0.193224	+	0.981155i	$0.561895\pi$
$212$	−11.5761	−0.795053
$213$	0	0
$214$	50.0950	3.42442
$215$	−12.0565	−0.822250
$216$	0	0
$217$	4.34916	0.295240
$218$	18.4996	1.25295
$219$	0	0
$220$	−7.37743	−0.497386
$221$	−1.32088	−0.0888523
$222$	0	0
$223$	−12.9909	−0.869937	−0.434968	−	0.900446i	$-0.643240\pi$
$223$	−12.9909	−0.869937	−0.434968	+	0.900446i	$0.643240\pi$
$224$	3.48586	0.232909
$225$	0	0
$226$	2.44305	0.162509
$227$	10.0565	0.667477	0.333738	−	0.942666i	$-0.391690\pi$
$227$	10.0565	0.667477	0.333738	+	0.942666i	$0.391690\pi$
$228$	0	0
$229$	−4.82956	−0.319146	−0.159573	−	0.987186i	$-0.551012\pi$
$229$	−4.82956	−0.319146	−0.159573	+	0.987186i	$0.551012\pi$
$230$	−16.6983	−1.10105
$231$	0	0
$232$	−13.7074	−0.899934
$233$	−24.4996	−1.60502	−0.802511	−	0.596637i	$-0.796503\pi$
$233$	−24.4996	−1.60502	−0.802511	+	0.596637i	$0.796503\pi$
$234$	0	0
$235$	−0.641769	−0.0418644
$236$	−32.8970	−2.14141
$237$	0	0
$238$	2.51414	0.162967
$239$	13.9253	0.900753	0.450377	−	0.892839i	$-0.351290\pi$
$239$	13.9253	0.900753	0.450377	+	0.892839i	$0.351290\pi$
$240$	0	0
$241$	−4.40571	−0.283796	−0.141898	−	0.989881i	$-0.545321\pi$
$241$	−4.40571	−0.283796	−0.141898	+	0.989881i	$0.545321\pi$
$242$	−20.3263	−1.30663
$243$	0	0
$244$	29.9627	1.91816
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−2.64177	−0.168092
$248$	25.3774	1.61147
$249$	0	0
$250$	−2.51414	−0.159008
$251$	−3.22699	−0.203686	−0.101843	−	0.994800i	$-0.532474\pi$
$251$	−3.22699	−0.203686	−0.101843	+	0.994800i	$0.532474\pi$
$252$	0	0
$253$	11.3401	0.712945
$254$	5.02827	0.315502
$255$	0	0
$256$	−31.7549	−1.98468
$257$	−17.3401	−1.08164	−0.540822	−	0.841137i	$-0.681887\pi$
$257$	−17.3401	−1.08164	−0.540822	+	0.841137i	$0.681887\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	5.70739	0.353957
$261$	0	0
$262$	−21.2270	−1.31141
$263$	−22.2745	−1.37350	−0.686751	−	0.726893i	$-0.740964\pi$
$263$	−22.2745	−1.37350	−0.686751	+	0.726893i	$0.740964\pi$
$264$	0	0
$265$	2.67912	0.164577
$266$	5.02827	0.308303
$267$	0	0
$268$	37.3401	2.28091
$269$	−20.7549	−1.26545	−0.632723	−	0.774378i	$-0.718063\pi$
$269$	−20.7549	−1.26545	−0.632723	+	0.774378i	$0.718063\pi$
$270$	0	0
$271$	5.22699	0.317517	0.158759	−	0.987317i	$-0.449251\pi$
$271$	5.22699	0.317517	0.158759	+	0.987317i	$0.449251\pi$
$272$	6.02827	0.365518
$273$	0	0
$274$	−1.84956	−0.111736
$275$	1.70739	0.102959
$276$	0	0
$277$	12.0565	0.724408	0.362204	−	0.932099i	$-0.382024\pi$
$277$	12.0565	0.724408	0.362204	+	0.932099i	$0.382024\pi$
$278$	2.82128	0.169209
$279$	0	0
$280$	−5.83502	−0.348709
$281$	11.3582	0.677575	0.338788	−	0.940863i	$-0.389983\pi$
$281$	11.3582	0.677575	0.338788	+	0.940863i	$0.389983\pi$
$282$	0	0
$283$	4.58522	0.272563	0.136282	−	0.990670i	$-0.456485\pi$
$283$	4.58522	0.272563	0.136282	+	0.990670i	$0.456485\pi$
$284$	−36.0757	−2.14070
$285$	0	0
$286$	−5.67004	−0.335277
$287$	−6.00000	−0.354169
$288$	0	0
$289$	1.00000	0.0588235
$290$	5.90611	0.346818
$291$	0	0
$292$	1.67004	0.0977319
$293$	−17.3401	−1.01302	−0.506509	−	0.862234i	$-0.669064\pi$
$293$	−17.3401	−1.01302	−0.506509	+	0.862234i	$0.669064\pi$
$294$	0	0
$295$	7.61350	0.443275
$296$	11.6700	0.678307
$297$	0	0
$298$	−1.61350	−0.0934673
$299$	−8.77301	−0.507356
$300$	0	0
$301$	12.0565	0.694928
$302$	40.5105	2.33112
$303$	0	0
$304$	12.0565	0.691490
$305$	−6.93438	−0.397061
$306$	0	0
$307$	−8.48040	−0.484002	−0.242001	−	0.970276i	$-0.577804\pi$
$307$	−8.48040	−0.484002	−0.242001	+	0.970276i	$0.577804\pi$
$308$	7.37743	0.420368
$309$	0	0
$310$	−10.9344	−0.621031
$311$	24.9536	1.41499	0.707494	−	0.706719i	$-0.249826\pi$
$311$	24.9536	1.41499	0.707494	+	0.706719i	$0.249826\pi$
$312$	0	0
$313$	−21.8578	−1.23548	−0.617739	−	0.786383i	$-0.711951\pi$
$313$	−21.8578	−1.23548	−0.617739	+	0.786383i	$0.711951\pi$
$314$	−58.7741	−3.31681
$315$	0	0
$316$	27.5953	1.55236
$317$	34.5561	1.94087	0.970433	−	0.241369i	$-0.0775965\pi$
$317$	34.5561	1.94087	0.970433	+	0.241369i	$0.0775965\pi$
$318$	0	0
$319$	−4.01093	−0.224569
$320$	3.29261	0.184063
$321$	0	0
$322$	16.6983	0.930561
$323$	2.00000	0.111283
$324$	0	0
$325$	−1.32088	−0.0732695
$326$	15.2270	0.843345
$327$	0	0
$328$	−35.0101	−1.93311
$329$	0.641769	0.0353819
$330$	0	0
$331$	−10.8296	−0.595246	−0.297623	−	0.954683i	$-0.596194\pi$
$331$	−10.8296	−0.595246	−0.297623	+	0.954683i	$0.596194\pi$
$332$	25.9253	1.42284
$333$	0	0
$334$	22.8405	1.24978
$335$	−8.64177	−0.472150
$336$	0	0
$337$	−4.82956	−0.263083	−0.131541	−	0.991311i	$-0.541993\pi$
$337$	−4.82956	−0.263083	−0.131541	+	0.991311i	$0.541993\pi$
$338$	−28.2973	−1.53917
$339$	0	0
$340$	−4.32088	−0.234333
$341$	7.42571	0.402125
$342$	0	0
$343$	1.00000	0.0539949
$344$	70.3502	3.79303
$345$	0	0
$346$	−62.7367	−3.37275
$347$	−16.5105	−0.886332	−0.443166	−	0.896440i	$-0.646145\pi$
$347$	−16.5105	−0.886332	−0.443166	+	0.896440i	$0.646145\pi$
$348$	0	0
$349$	−21.3401	−1.14231	−0.571154	−	0.820843i	$-0.693504\pi$
$349$	−21.3401	−1.14231	−0.571154	+	0.820843i	$0.693504\pi$
$350$	2.51414	0.134386
$351$	0	0
$352$	5.95173	0.317228
$353$	−19.2835	−1.02636	−0.513180	−	0.858281i	$-0.671533\pi$
$353$	−19.2835	−1.02636	−0.513180	+	0.858281i	$0.671533\pi$
$354$	0	0
$355$	8.34916	0.443127
$356$	30.1240	1.59657
$357$	0	0
$358$	−57.5388	−3.04102
$359$	−10.5105	−0.554724	−0.277362	−	0.960765i	$-0.589460\pi$
$359$	−10.5105	−0.554724	−0.277362	+	0.960765i	$0.589460\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	34.1323	1.79395
$363$	0	0
$364$	−5.70739	−0.299148
$365$	−0.386505	−0.0202306
$366$	0	0
$367$	22.5671	1.17799	0.588996	−	0.808136i	$-0.299523\pi$
$367$	22.5671	1.17799	0.588996	+	0.808136i	$0.299523\pi$
$368$	40.0384	2.08715
$369$	0	0
$370$	−5.02827	−0.261408
$371$	−2.67912	−0.139093
$372$	0	0
$373$	−31.2088	−1.61593	−0.807966	−	0.589229i	$-0.799432\pi$
$373$	−31.2088	−1.61593	−0.807966	+	0.589229i	$0.799432\pi$
$374$	4.29261	0.221966
$375$	0	0
$376$	3.74474	0.193120
$377$	3.10297	0.159811
$378$	0	0
$379$	4.91518	0.252476	0.126238	−	0.992000i	$-0.459710\pi$
$379$	4.91518	0.252476	0.126238	+	0.992000i	$0.459710\pi$
$380$	−8.64177	−0.443313
$381$	0	0
$382$	6.97173	0.356705
$383$	19.2835	0.985343	0.492671	−	0.870215i	$-0.336020\pi$
$383$	19.2835	0.985343	0.492671	+	0.870215i	$0.336020\pi$
$384$	0	0
$385$	−1.70739	−0.0870166
$386$	−22.2262	−1.13128
$387$	0	0
$388$	−27.0283	−1.37215
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	6.64177	0.335889
$392$	5.83502	0.294713
$393$	0	0
$394$	6.14217	0.309438
$395$	−6.38650	−0.321340
$396$	0	0
$397$	0.386505	0.0193981	0.00969906	−	0.999953i	$-0.496913\pi$
$397$	0.386505	0.0193981	0.00969906	+	0.999953i	$0.496913\pi$
$398$	29.2918	1.46827
$399$	0	0
$400$	6.02827	0.301414
$401$	8.80314	0.439608	0.219804	−	0.975544i	$-0.429458\pi$
$401$	8.80314	0.439608	0.219804	+	0.975544i	$0.429458\pi$
$402$	0	0
$403$	−5.74474	−0.286166
$404$	−35.6700	−1.77465
$405$	0	0
$406$	−5.90611	−0.293115
$407$	3.41478	0.169264
$408$	0	0
$409$	−2.07469	−0.102587	−0.0512935	−	0.998684i	$-0.516334\pi$
$409$	−2.07469	−0.102587	−0.0512935	+	0.998684i	$0.516334\pi$
$410$	15.0848	0.744986
$411$	0	0
$412$	−12.6791	−0.624655
$413$	−7.61350	−0.374636
$414$	0	0
$415$	−6.00000	−0.294528
$416$	−4.60442	−0.225750
$417$	0	0
$418$	8.58522	0.419917
$419$	29.8397	1.45776	0.728882	−	0.684639i	$-0.240040\pi$
$419$	29.8397	1.45776	0.728882	+	0.684639i	$0.240040\pi$
$420$	0	0
$421$	−29.1414	−1.42026	−0.710132	−	0.704069i	$-0.751365\pi$
$421$	−29.1414	−1.42026	−0.710132	+	0.704069i	$0.751365\pi$
$422$	−14.1131	−0.687015
$423$	0	0
$424$	−15.6327	−0.759191
$425$	1.00000	0.0485071
$426$	0	0
$427$	6.93438	0.335578
$428$	86.0950	4.16156
$429$	0	0
$430$	−30.3118	−1.46177
$431$	−6.21792	−0.299507	−0.149753	−	0.988723i	$-0.547848\pi$
$431$	−6.21792	−0.299507	−0.149753	+	0.988723i	$0.547848\pi$
$432$	0	0
$433$	−2.60442	−0.125161	−0.0625803	−	0.998040i	$-0.519933\pi$
$433$	−2.60442	−0.125161	−0.0625803	+	0.998040i	$0.519933\pi$
$434$	10.9344	0.524867
$435$	0	0
$436$	31.7941	1.52266
$437$	13.2835	0.635438
$438$	0	0
$439$	−13.8205	−0.659616	−0.329808	−	0.944048i	$-0.606984\pi$
$439$	−13.8205	−0.659616	−0.329808	+	0.944048i	$0.606984\pi$
$440$	−9.96265	−0.474951
$441$	0	0
$442$	−3.32088	−0.157958
$443$	−17.9517	−0.852912	−0.426456	−	0.904508i	$-0.640238\pi$
$443$	−17.9517	−0.852912	−0.426456	+	0.904508i	$0.640238\pi$
$444$	0	0
$445$	−6.97173	−0.330492
$446$	−32.6610	−1.54654
$447$	0	0
$448$	−3.29261	−0.155561
$449$	39.8205	1.87924	0.939622	−	0.342213i	$-0.111176\pi$
$449$	39.8205	1.87924	0.939622	+	0.342213i	$0.111176\pi$
$450$	0	0
$451$	−10.2443	−0.482387
$452$	4.19872	0.197491
$453$	0	0
$454$	25.2835	1.18662
$455$	1.32088	0.0619240
$456$	0	0
$457$	26.0000	1.21623	0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000	1.21623	0.608114	+	0.793849i	$0.291926\pi$
$458$	−12.1422	−0.567366
$459$	0	0
$460$	−28.6983	−1.33807
$461$	−1.80128	−0.0838941	−0.0419471	−	0.999120i	$-0.513356\pi$
$461$	−1.80128	−0.0838941	−0.0419471	+	0.999120i	$0.513356\pi$
$462$	0	0
$463$	11.8688	0.551588	0.275794	−	0.961217i	$-0.411059\pi$
$463$	11.8688	0.551588	0.275794	+	0.961217i	$0.411059\pi$
$464$	−14.1614	−0.657425
$465$	0	0
$466$	−61.5953	−2.85335
$467$	18.0000	0.832941	0.416470	−	0.909149i	$-0.363267\pi$
$467$	18.0000	0.832941	0.416470	+	0.909149i	$0.363267\pi$
$468$	0	0
$469$	8.64177	0.399040
$470$	−1.61350	−0.0744250
$471$	0	0
$472$	−44.4249	−2.04482
$473$	20.5852	0.946509
$474$	0	0
$475$	2.00000	0.0917663
$476$	4.32088	0.198047
$477$	0	0
$478$	35.0101	1.60133
$479$	−2.89703	−0.132369	−0.0661844	−	0.997807i	$-0.521083\pi$
$479$	−2.89703	−0.132369	−0.0661844	+	0.997807i	$0.521083\pi$
$480$	0	0
$481$	−2.64177	−0.120454
$482$	−11.0765	−0.504523
$483$	0	0
$484$	−34.9336	−1.58789
$485$	6.25526	0.284037
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	40.4623	1.83164
$489$	0	0
$490$	−2.51414	−0.113577
$491$	−30.8114	−1.39050	−0.695250	−	0.718768i	$-0.744706\pi$
$491$	−30.8114	−1.39050	−0.695250	+	0.718768i	$0.744706\pi$
$492$	0	0
$493$	−2.34916	−0.105801
$494$	−6.64177	−0.298827
$495$	0	0
$496$	26.2179	1.17722
$497$	−8.34916	−0.374511
$498$	0	0
$499$	−33.6519	−1.50647	−0.753233	−	0.657754i	$-0.771507\pi$
$499$	−33.6519	−1.50647	−0.753233	+	0.657754i	$0.771507\pi$
$500$	−4.32088	−0.193236
$501$	0	0
$502$	−8.11310	−0.362105
$503$	−36.6236	−1.63297	−0.816483	−	0.577369i	$-0.804079\pi$
$503$	−36.6236	−1.63297	−0.816483	+	0.577369i	$0.804079\pi$
$504$	0	0
$505$	8.25526	0.367354
$506$	28.5105	1.26745
$507$	0	0
$508$	8.64177	0.383416
$509$	41.6519	1.84619	0.923094	−	0.384575i	$-0.125652\pi$
$509$	41.6519	1.84619	0.923094	+	0.384575i	$0.125652\pi$
$510$	0	0
$511$	0.386505	0.0170980
$512$	−49.3365	−2.18038
$513$	0	0
$514$	−43.5953	−1.92291
$515$	2.93438	0.129304
$516$	0	0
$517$	1.09575	0.0481910
$518$	5.02827	0.220930
$519$	0	0
$520$	7.70739	0.337991
$521$	−9.41478	−0.412469	−0.206234	−	0.978503i	$-0.566121\pi$
$521$	−9.41478	−0.412469	−0.206234	+	0.978503i	$0.566121\pi$
$522$	0	0
$523$	−0.367304	−0.0160611	−0.00803053	−	0.999968i	$-0.502556\pi$
$523$	−0.367304	−0.0160611	−0.00803053	+	0.999968i	$0.502556\pi$
$524$	−36.4815	−1.59370
$525$	0	0
$526$	−56.0011	−2.44176
$527$	4.34916	0.189452
$528$	0	0
$529$	21.1131	0.917961
$530$	6.73566	0.292579
$531$	0	0
$532$	8.64177	0.374668
$533$	7.92531	0.343283
$534$	0	0
$535$	−19.9253	−0.861446
$536$	50.4249	2.17802
$537$	0	0
$538$	−52.1806	−2.24966
$539$	1.70739	0.0735425
$540$	0	0
$541$	25.1523	1.08138	0.540691	−	0.841221i	$-0.318163\pi$
$541$	25.1523	1.08138	0.540691	+	0.841221i	$0.318163\pi$
$542$	13.1414	0.564470
$543$	0	0
$544$	3.48586	0.149455
$545$	−7.35823	−0.315192
$546$	0	0
$547$	14.4540	0.618008	0.309004	−	0.951061i	$-0.400004\pi$
$547$	14.4540	0.618008	0.309004	+	0.951061i	$0.400004\pi$
$548$	−3.17872	−0.135788
$549$	0	0
$550$	4.29261	0.183038
$551$	−4.69832	−0.200155
$552$	0	0
$553$	6.38650	0.271582
$554$	30.3118	1.28783
$555$	0	0
$556$	4.84876	0.205633
$557$	9.69646	0.410852	0.205426	−	0.978673i	$-0.434142\pi$
$557$	9.69646	0.410852	0.205426	+	0.978673i	$0.434142\pi$
$558$	0	0
$559$	−15.9253	−0.673569
$560$	−6.02827	−0.254741
$561$	0	0
$562$	28.5561	1.20457
$563$	13.9253	0.586882	0.293441	−	0.955977i	$-0.405200\pi$
$563$	13.9253	0.586882	0.293441	+	0.955977i	$0.405200\pi$
$564$	0	0
$565$	−0.971726	−0.0408808
$566$	11.5279	0.484553
$567$	0	0
$568$	−48.7175	−2.04414
$569$	14.1131	0.591652	0.295826	−	0.955242i	$-0.404405\pi$
$569$	14.1131	0.591652	0.295826	+	0.955242i	$0.404405\pi$
$570$	0	0
$571$	13.6882	0.572833	0.286416	−	0.958105i	$-0.407536\pi$
$571$	13.6882	0.572833	0.286416	+	0.958105i	$0.407536\pi$
$572$	−9.74474	−0.407448
$573$	0	0
$574$	−15.0848	−0.629628
$575$	6.64177	0.276981
$576$	0	0
$577$	3.37743	0.140604	0.0703022	−	0.997526i	$-0.477604\pi$
$577$	3.37743	0.140604	0.0703022	+	0.997526i	$0.477604\pi$
$578$	2.51414	0.104574
$579$	0	0
$580$	10.1504	0.421474
$581$	6.00000	0.248922
$582$	0	0
$583$	−4.57429	−0.189448
$584$	2.25526	0.0933235
$585$	0	0
$586$	−43.5953	−1.80091
$587$	2.77301	0.114454	0.0572272	−	0.998361i	$-0.481774\pi$
$587$	2.77301	0.114454	0.0572272	+	0.998361i	$0.481774\pi$
$588$	0	0
$589$	8.69832	0.358408
$590$	19.1414	0.788038
$591$	0	0
$592$	12.0565	0.495521
$593$	−39.5844	−1.62554	−0.812769	−	0.582587i	$-0.802041\pi$
$593$	−39.5844	−1.62554	−0.812769	+	0.582587i	$0.802041\pi$
$594$	0	0
$595$	−1.00000	−0.0409960
$596$	−2.77301	−0.113587
$597$	0	0
$598$	−22.0565	−0.901959
$599$	13.3017	0.543492	0.271746	−	0.962369i	$-0.412399\pi$
$599$	13.3017	0.543492	0.271746	+	0.962369i	$0.412399\pi$
$600$	0	0
$601$	−15.1222	−0.616846	−0.308423	−	0.951249i	$-0.599801\pi$
$601$	−15.1222	−0.616846	−0.308423	+	0.951249i	$0.599801\pi$
$602$	30.3118	1.23542
$603$	0	0
$604$	69.6228	2.83291
$605$	8.08482	0.328695
$606$	0	0
$607$	−35.2654	−1.43138	−0.715689	−	0.698419i	$-0.753887\pi$
$607$	−35.2654	−1.43138	−0.715689	+	0.698419i	$0.753887\pi$
$608$	6.97173	0.282741
$609$	0	0
$610$	−17.4340	−0.705881
$611$	−0.847703	−0.0342944
$612$	0	0
$613$	−24.9427	−1.00742	−0.503712	−	0.863872i	$-0.668033\pi$
$613$	−24.9427	−1.00742	−0.503712	+	0.863872i	$0.668033\pi$
$614$	−21.3209	−0.860441
$615$	0	0
$616$	9.96265	0.401407
$617$	−32.4249	−1.30538	−0.652689	−	0.757626i	$-0.726359\pi$
$617$	−32.4249	−1.30538	−0.652689	+	0.757626i	$0.726359\pi$
$618$	0	0
$619$	17.4449	0.701170	0.350585	−	0.936531i	$-0.385983\pi$
$619$	17.4449	0.701170	0.350585	+	0.936531i	$0.385983\pi$
$620$	−18.7922	−0.754713
$621$	0	0
$622$	62.7367	2.51551
$623$	6.97173	0.279316
$624$	0	0
$625$	1.00000	0.0400000
$626$	−54.9536	−2.19639
$627$	0	0
$628$	−101.011	−4.03079
$629$	2.00000	0.0797452
$630$	0	0
$631$	36.3865	1.44852	0.724262	−	0.689525i	$-0.242181\pi$
$631$	36.3865	1.44852	0.724262	+	0.689525i	$0.242181\pi$
$632$	37.2654	1.48234
$633$	0	0
$634$	86.8789	3.45040
$635$	−2.00000	−0.0793676
$636$	0	0
$637$	−1.32088	−0.0523353
$638$	−10.0840	−0.399230
$639$	0	0
$640$	15.2498	0.602801
$641$	−30.3876	−1.20024	−0.600118	−	0.799911i	$-0.704880\pi$
$641$	−30.3876	−1.20024	−0.600118	+	0.799911i	$0.704880\pi$
$642$	0	0
$643$	−12.1131	−0.477694	−0.238847	−	0.971057i	$-0.576769\pi$
$643$	−12.1131	−0.477694	−0.238847	+	0.971057i	$0.576769\pi$
$644$	28.6983	1.13087
$645$	0	0
$646$	5.02827	0.197835
$647$	−34.2262	−1.34557	−0.672785	−	0.739838i	$-0.734902\pi$
$647$	−34.2262	−1.34557	−0.672785	+	0.739838i	$0.734902\pi$
$648$	0	0
$649$	−12.9992	−0.510263
$650$	−3.32088	−0.130256
$651$	0	0
$652$	26.1696	1.02488
$653$	0.971726	0.0380266	0.0190133	−	0.999819i	$-0.493948\pi$
$653$	0.971726	0.0380266	0.0190133	+	0.999819i	$0.493948\pi$
$654$	0	0
$655$	8.44305	0.329897
$656$	−36.1696	−1.41219
$657$	0	0
$658$	1.61350	0.0629006
$659$	−17.1523	−0.668159	−0.334079	−	0.942545i	$-0.608425\pi$
$659$	−17.1523	−0.668159	−0.334079	+	0.942545i	$0.608425\pi$
$660$	0	0
$661$	−23.0957	−0.898321	−0.449160	−	0.893451i	$-0.648277\pi$
$661$	−23.0957	−0.898321	−0.449160	+	0.893451i	$0.648277\pi$
$662$	−27.2270	−1.05821
$663$	0	0
$664$	35.0101	1.35866
$665$	−2.00000	−0.0775567
$666$	0	0
$667$	−15.6026	−0.604134
$668$	39.2545	1.51880
$669$	0	0
$670$	−21.7266	−0.839371
$671$	11.8397	0.457066
$672$	0	0
$673$	31.5097	1.21461	0.607305	−	0.794468i	$-0.292250\pi$
$673$	31.5097	1.21461	0.607305	+	0.794468i	$0.292250\pi$
$674$	−12.1422	−0.467699
$675$	0	0
$676$	−48.6327	−1.87049
$677$	34.3502	1.32019	0.660093	−	0.751184i	$-0.270517\pi$
$677$	34.3502	1.32019	0.660093	+	0.751184i	$0.270517\pi$
$678$	0	0
$679$	−6.25526	−0.240055
$680$	−5.83502	−0.223763
$681$	0	0
$682$	18.6692	0.714882
$683$	−8.11310	−0.310439	−0.155219	−	0.987880i	$-0.549608\pi$
$683$	−8.11310	−0.310439	−0.155219	+	0.987880i	$0.549608\pi$
$684$	0	0
$685$	0.735663	0.0281082
$686$	2.51414	0.0959902
$687$	0	0
$688$	72.6802	2.77091
$689$	3.53880	0.134818
$690$	0	0
$691$	5.00907	0.190554	0.0952771	−	0.995451i	$-0.469626\pi$
$691$	5.00907	0.190554	0.0952771	+	0.995451i	$0.469626\pi$
$692$	−107.822	−4.09876
$693$	0	0
$694$	−41.5097	−1.57569
$695$	−1.12217	−0.0425663
$696$	0	0
$697$	−6.00000	−0.227266
$698$	−53.6519	−2.03075
$699$	0	0
$700$	4.32088	0.163314
$701$	−37.4532	−1.41459	−0.707294	−	0.706920i	$-0.750084\pi$
$701$	−37.4532	−1.41459	−0.707294	+	0.706920i	$0.750084\pi$
$702$	0	0
$703$	4.00000	0.150863
$704$	−5.62177	−0.211878
$705$	0	0
$706$	−48.4815	−1.82462
$707$	−8.25526	−0.310471
$708$	0	0
$709$	−18.1131	−0.680252	−0.340126	−	0.940380i	$-0.610470\pi$
$709$	−18.1131	−0.680252	−0.340126	+	0.940380i	$0.610470\pi$
$710$	20.9909	0.787775
$711$	0	0
$712$	40.6802	1.52455
$713$	28.8861	1.08179
$714$	0	0
$715$	2.25526	0.0843421
$716$	−98.8882	−3.69563
$717$	0	0
$718$	−26.4249	−0.986169
$719$	−6.97173	−0.260002	−0.130001	−	0.991514i	$-0.541498\pi$
$719$	−6.97173	−0.260002	−0.130001	+	0.991514i	$0.541498\pi$
$720$	0	0
$721$	−2.93438	−0.109282
$722$	−37.7121	−1.40350
$723$	0	0
$724$	58.6610	2.18012
$725$	−2.34916	−0.0872456
$726$	0	0
$727$	−39.3702	−1.46016	−0.730080	−	0.683361i	$-0.760517\pi$
$727$	−39.3702	−1.46016	−0.730080	+	0.683361i	$0.760517\pi$
$728$	−7.70739	−0.285655
$729$	0	0
$730$	−0.971726	−0.0359652
$731$	12.0565	0.445928
$732$	0	0
$733$	−12.0373	−0.444610	−0.222305	−	0.974977i	$-0.571358\pi$
$733$	−12.0373	−0.444610	−0.222305	+	0.974977i	$0.571358\pi$
$734$	56.7367	2.09419
$735$	0	0
$736$	23.1523	0.853405
$737$	14.7549	0.543502
$738$	0	0
$739$	−21.6700	−0.797145	−0.398573	−	0.917137i	$-0.630494\pi$
$739$	−21.6700	−0.797145	−0.398573	+	0.917137i	$0.630494\pi$
$740$	−8.64177	−0.317678
$741$	0	0
$742$	−6.73566	−0.247274
$743$	−21.3966	−0.784966	−0.392483	−	0.919759i	$-0.628384\pi$
$743$	−21.3966	−0.784966	−0.392483	+	0.919759i	$0.628384\pi$
$744$	0	0
$745$	0.641769	0.0235126
$746$	−78.4633	−2.87275
$747$	0	0
$748$	7.37743	0.269746
$749$	19.9253	0.728055
$750$	0	0
$751$	−13.7266	−0.500890	−0.250445	−	0.968131i	$-0.580577\pi$
$751$	−13.7266	−0.500890	−0.250445	+	0.968131i	$0.580577\pi$
$752$	3.86876	0.141079
$753$	0	0
$754$	7.80128	0.284106
$755$	−16.1131	−0.586416
$756$	0	0
$757$	−33.5279	−1.21859	−0.609296	−	0.792943i	$-0.708548\pi$
$757$	−33.5279	−1.21859	−0.609296	+	0.792943i	$0.708548\pi$
$758$	12.3574	0.448842
$759$	0	0
$760$	−11.6700	−0.423317
$761$	−24.8296	−0.900071	−0.450035	−	0.893011i	$-0.648589\pi$
$761$	−24.8296	−0.900071	−0.450035	+	0.893011i	$0.648589\pi$
$762$	0	0
$763$	7.35823	0.266386
$764$	11.9819	0.433488
$765$	0	0
$766$	48.4815	1.75171
$767$	10.0565	0.363121
$768$	0	0
$769$	−17.2654	−0.622606	−0.311303	−	0.950311i	$-0.600765\pi$
$769$	−17.2654	−0.622606	−0.311303	+	0.950311i	$0.600765\pi$
$770$	−4.29261	−0.154695
$771$	0	0
$772$	−38.1987	−1.37480
$773$	−12.6418	−0.454693	−0.227346	−	0.973814i	$-0.573005\pi$
$773$	−12.6418	−0.454693	−0.227346	+	0.973814i	$0.573005\pi$
$774$	0	0
$775$	4.34916	0.156226
$776$	−36.4996	−1.31026
$777$	0	0
$778$	15.0848	0.540817
$779$	−12.0000	−0.429945
$780$	0	0
$781$	−14.2553	−0.510093
$782$	16.6983	0.597131
$783$	0	0
$784$	6.02827	0.215295
$785$	23.3774	0.834376
$786$	0	0
$787$	1.35823	0.0484157	0.0242079	−	0.999707i	$-0.492294\pi$
$787$	1.35823	0.0484157	0.0242079	+	0.999707i	$0.492294\pi$
$788$	10.5561	0.376047
$789$	0	0
$790$	−16.0565	−0.571266
$791$	0.971726	0.0345506
$792$	0	0
$793$	−9.15951	−0.325264
$794$	0.971726	0.0344853
$795$	0	0
$796$	50.3419	1.78432
$797$	11.1704	0.395677	0.197839	−	0.980235i	$-0.436608\pi$
$797$	11.1704	0.395677	0.197839	+	0.980235i	$0.436608\pi$
$798$	0	0
$799$	0.641769	0.0227042
$800$	3.48586	0.123244
$801$	0	0
$802$	22.1323	0.781519
$803$	0.659914	0.0232879
$804$	0	0
$805$	−6.64177	−0.234092
$806$	−14.4431	−0.508735
$807$	0	0
$808$	−48.1696	−1.69460
$809$	32.8031	1.15330	0.576648	−	0.816992i	$-0.304360\pi$
$809$	32.8031	1.15330	0.576648	+	0.816992i	$0.304360\pi$
$810$	0	0
$811$	28.3492	0.995474	0.497737	−	0.867328i	$-0.334165\pi$
$811$	28.3492	0.995474	0.497737	+	0.867328i	$0.334165\pi$
$812$	−10.1504	−0.356211
$813$	0	0
$814$	8.58522	0.300912
$815$	−6.05655	−0.212152
$816$	0	0
$817$	24.1131	0.843610
$818$	−5.21606	−0.182375
$819$	0	0
$820$	25.9253	0.905351
$821$	27.8205	0.970942	0.485471	−	0.874253i	$-0.338648\pi$
$821$	27.8205	0.970942	0.485471	+	0.874253i	$0.338648\pi$
$822$	0	0
$823$	19.4039	0.676376	0.338188	−	0.941079i	$-0.390186\pi$
$823$	19.4039	0.676376	0.338188	+	0.941079i	$0.390186\pi$
$824$	−17.1222	−0.596479
$825$	0	0
$826$	−19.1414	−0.666013
$827$	39.8506	1.38574	0.692871	−	0.721062i	$-0.256346\pi$
$827$	39.8506	1.38574	0.692871	+	0.721062i	$0.256346\pi$
$828$	0	0
$829$	55.3219	1.92141	0.960705	−	0.277571i	$-0.0895293\pi$
$829$	55.3219	1.92141	0.960705	+	0.277571i	$0.0895293\pi$
$830$	−15.0848	−0.523602
$831$	0	0
$832$	4.34916	0.150780
$833$	1.00000	0.0346479
$834$	0	0
$835$	−9.08482	−0.314393
$836$	14.7549	0.510308
$837$	0	0
$838$	75.0211	2.59156
$839$	36.4815	1.25948	0.629740	−	0.776806i	$-0.283161\pi$
$839$	36.4815	1.25948	0.629740	+	0.776806i	$0.283161\pi$
$840$	0	0
$841$	−23.4815	−0.809705
$842$	−73.2654	−2.52489
$843$	0	0
$844$	−24.2553	−0.834901
$845$	11.2553	0.387193
$846$	0	0
$847$	−8.08482	−0.277798
$848$	−16.1504	−0.554608
$849$	0	0
$850$	2.51414	0.0862342
$851$	13.2835	0.455354
$852$	0	0
$853$	52.8970	1.81116	0.905580	−	0.424176i	$-0.139436\pi$
$853$	52.8970	1.81116	0.905580	+	0.424176i	$0.139436\pi$
$854$	17.4340	0.596579
$855$	0	0
$856$	116.265	3.97384
$857$	10.0109	0.341967	0.170983	−	0.985274i	$-0.445306\pi$
$857$	10.0109	0.341967	0.170983	+	0.985274i	$0.445306\pi$
$858$	0	0
$859$	−0.567076	−0.0193484	−0.00967419	−	0.999953i	$-0.503079\pi$
$859$	−0.567076	−0.0193484	−0.00967419	+	0.999953i	$0.503079\pi$
$860$	−52.0950	−1.77642
$861$	0	0
$862$	−15.6327	−0.532452
$863$	−32.8031	−1.11663	−0.558316	−	0.829628i	$-0.688552\pi$
$863$	−32.8031	−1.11663	−0.558316	+	0.829628i	$0.688552\pi$
$864$	0	0
$865$	24.9536	0.848447
$866$	−6.54787	−0.222506
$867$	0	0
$868$	18.7922	0.637849
$869$	10.9043	0.369901
$870$	0	0
$871$	−11.4148	−0.386775
$872$	42.9354	1.45398
$873$	0	0
$874$	33.3966	1.12966
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−52.1696	−1.76164	−0.880822	−	0.473448i	$-0.843009\pi$
$877$	−52.1696	−1.76164	−0.880822	+	0.473448i	$0.843009\pi$
$878$	−34.7466	−1.17264
$879$	0	0
$880$	−10.2926	−0.346964
$881$	40.6802	1.37055	0.685275	−	0.728284i	$-0.259682\pi$
$881$	40.6802	1.37055	0.685275	+	0.728284i	$0.259682\pi$
$882$	0	0
$883$	−42.1131	−1.41722	−0.708609	−	0.705601i	$-0.750677\pi$
$883$	−42.1131	−1.41722	−0.708609	+	0.705601i	$0.750677\pi$
$884$	−5.70739	−0.191960
$885$	0	0
$886$	−45.1331	−1.51628
$887$	−54.1696	−1.81884	−0.909419	−	0.415880i	$-0.863473\pi$
$887$	−54.1696	−1.81884	−0.909419	+	0.415880i	$0.863473\pi$
$888$	0	0
$889$	2.00000	0.0670778
$890$	−17.5279	−0.587536
$891$	0	0
$892$	−56.1323	−1.87945
$893$	1.28354	0.0429520
$894$	0	0
$895$	22.8861	0.764998
$896$	−15.2498	−0.509460
$897$	0	0
$898$	100.114	3.34085
$899$	−10.2169	−0.340751
$900$	0	0
$901$	−2.67912	−0.0892543
$902$	−25.7557	−0.857570
$903$	0	0
$904$	5.67004	0.188583
$905$	−13.5761	−0.451286
$906$	0	0
$907$	36.8223	1.22267	0.611333	−	0.791374i	$-0.290634\pi$
$907$	36.8223	1.22267	0.611333	+	0.791374i	$0.290634\pi$
$908$	43.4532	1.44204
$909$	0	0
$910$	3.32088	0.110086
$911$	1.70739	0.0565683	0.0282842	−	0.999600i	$-0.490996\pi$
$911$	1.70739	0.0565683	0.0282842	+	0.999600i	$0.490996\pi$
$912$	0	0
$913$	10.2443	0.339038
$914$	65.3676	2.16217
$915$	0	0
$916$	−20.8680	−0.689497
$917$	−8.44305	−0.278814
$918$	0	0
$919$	23.8506	0.786759	0.393380	−	0.919376i	$-0.371306\pi$
$919$	23.8506	0.786759	0.393380	+	0.919376i	$0.371306\pi$
$920$	−38.7549	−1.27771
$921$	0	0
$922$	−4.52867	−0.149144
$923$	11.0283	0.363000
$924$	0	0
$925$	2.00000	0.0657596
$926$	29.8397	0.980593
$927$	0	0
$928$	−8.18884	−0.268812
$929$	8.75486	0.287238	0.143619	−	0.989633i	$-0.454126\pi$
$929$	8.75486	0.287238	0.143619	+	0.989633i	$0.454126\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	−105.860	−3.46756
$933$	0	0
$934$	45.2545	1.48077
$935$	−1.70739	−0.0558376
$936$	0	0
$937$	−16.5479	−0.540596	−0.270298	−	0.962777i	$-0.587122\pi$
$937$	−16.5479	−0.540596	−0.270298	+	0.962777i	$0.587122\pi$
$938$	21.7266	0.709398
$939$	0	0
$940$	−2.77301	−0.0904456
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	0	0
$943$	−39.8506	−1.29771
$944$	−45.8962	−1.49380
$945$	0	0
$946$	51.7541	1.68267
$947$	7.92531	0.257538	0.128769	−	0.991675i	$-0.458897\pi$
$947$	7.92531	0.257538	0.128769	+	0.991675i	$0.458897\pi$
$948$	0	0
$949$	−0.510528	−0.0165724
$950$	5.02827	0.163139
$951$	0	0
$952$	5.83502	0.189114
$953$	−2.20699	−0.0714914	−0.0357457	−	0.999361i	$-0.511381\pi$
$953$	−2.20699	−0.0714914	−0.0357457	+	0.999361i	$0.511381\pi$
$954$	0	0
$955$	−2.77301	−0.0897325
$956$	60.1696	1.94603
$957$	0	0
$958$	−7.28354	−0.235320
$959$	−0.735663	−0.0237558
$960$	0	0
$961$	−12.0848	−0.389833
$962$	−6.64177	−0.214139
$963$	0	0
$964$	−19.0365	−0.613126
$965$	8.84049	0.284585
$966$	0	0
$967$	−33.9637	−1.09220	−0.546100	−	0.837720i	$-0.683888\pi$
$967$	−33.9637	−1.09220	−0.546100	+	0.837720i	$0.683888\pi$
$968$	−47.1751	−1.51627
$969$	0	0
$970$	15.7266	0.504950
$971$	57.8962	1.85798	0.928989	−	0.370107i	$-0.120679\pi$
$971$	57.8962	1.85798	0.928989	+	0.370107i	$0.120679\pi$
$972$	0	0
$973$	1.12217	0.0359751
$974$	−40.2262	−1.28893
$975$	0	0
$976$	41.8023	1.33806
$977$	52.9619	1.69440	0.847200	−	0.531274i	$-0.178287\pi$
$977$	52.9619	1.69440	0.847200	+	0.531274i	$0.178287\pi$
$978$	0	0
$979$	11.9035	0.380436
$980$	−4.32088	−0.138026
$981$	0	0
$982$	−77.4641	−2.47198
$983$	38.8789	1.24004	0.620022	−	0.784584i	$-0.287124\pi$
$983$	38.8789	1.24004	0.620022	+	0.784584i	$0.287124\pi$
$984$	0	0
$985$	−2.44305	−0.0778421
$986$	−5.90611	−0.188089
$987$	0	0
$988$	−11.4148	−0.363152
$989$	80.0768	2.54629
$990$	0	0
$991$	−22.4996	−0.714723	−0.357362	−	0.933966i	$-0.616324\pi$
$991$	−22.4996	−0.714723	−0.357362	+	0.933966i	$0.616324\pi$
$992$	15.1606	0.481349
$993$	0	0
$994$	−20.9909	−0.665792
$995$	−11.6508	−0.369357
$996$	0	0
$997$	18.5561	0.587679	0.293840	−	0.955855i	$-0.405067\pi$
$997$	18.5561	0.587679	0.293840	+	0.955855i	$0.405067\pi$
$998$	−84.6055	−2.67814
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5355.2.a.bd.1.3		3
3.2	odd	2		1785.2.a.x.1.1	✓	3
15.14	odd	2		8925.2.a.bp.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1785.2.a.x.1.1	✓	3	3.2	odd	2
5355.2.a.bd.1.3		3	1.1	even	1	trivial
8925.2.a.bp.1.3		3	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5355 = 3^{2} \cdot 5 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5355.a (trivial)

\(f(q)\)	\(=\)	\(q+2.51414 q^{2} +4.32088 q^{4} -1.00000 q^{5} +1.00000 q^{7} +5.83502 q^{8} -2.51414 q^{10} +1.70739 q^{11} -1.32088 q^{13} +2.51414 q^{14} +6.02827 q^{16} +1.00000 q^{17} +2.00000 q^{19} -4.32088 q^{20} +4.29261 q^{22} +6.64177 q^{23} +1.00000 q^{25} -3.32088 q^{26} +4.32088 q^{28} -2.34916 q^{29} +4.34916 q^{31} +3.48586 q^{32} +2.51414 q^{34} -1.00000 q^{35} +2.00000 q^{37} +5.02827 q^{38} -5.83502 q^{40} -6.00000 q^{41} +12.0565 q^{43} +7.37743 q^{44} +16.6983 q^{46} +0.641769 q^{47} +1.00000 q^{49} +2.51414 q^{50} -5.70739 q^{52} -2.67912 q^{53} -1.70739 q^{55} +5.83502 q^{56} -5.90611 q^{58} -7.61350 q^{59} +6.93438 q^{61} +10.9344 q^{62} -3.29261 q^{64} +1.32088 q^{65} +8.64177 q^{67} +4.32088 q^{68} -2.51414 q^{70} -8.34916 q^{71} +0.386505 q^{73} +5.02827 q^{74} +8.64177 q^{76} +1.70739 q^{77} +6.38650 q^{79} -6.02827 q^{80} -15.0848 q^{82} +6.00000 q^{83} -1.00000 q^{85} +30.3118 q^{86} +9.96265 q^{88} +6.97173 q^{89} -1.32088 q^{91} +28.6983 q^{92} +1.61350 q^{94} -2.00000 q^{95} -6.25526 q^{97} +2.51414 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 5 q^{4} - 3 q^{5} + 3 q^{7} + 3 q^{8} - q^{10} + 4 q^{13} + q^{14} + 5 q^{16} + 3 q^{17} + 6 q^{19} - 5 q^{20} + 18 q^{22} + 4 q^{23} + 3 q^{25} - 2 q^{26} + 5 q^{28} + 14 q^{29} - 8 q^{31}+ \cdots + q^{98}+O(q^{100}) \) 3 * q + q^2 + 5 * q^4 - 3 * q^5 + 3 * q^7 + 3 * q^8 - q^10 + 4 * q^13 + q^14 + 5 * q^16 + 3 * q^17 + 6 * q^19 - 5 * q^20 + 18 * q^22 + 4 * q^23 + 3 * q^25 - 2 * q^26 + 5 * q^28 + 14 * q^29 - 8 * q^31 + 17 * q^32 + q^34 - 3 * q^35 + 6 * q^37 + 2 * q^38 - 3 * q^40 - 18 * q^41 + 10 * q^43 - 12 * q^44 + 8 * q^46 - 14 * q^47 + 3 * q^49 + q^50 - 12 * q^52 - 16 * q^53 + 3 * q^56 - 20 * q^58 - 20 * q^59 + 10 * q^61 + 22 * q^62 - 15 * q^64 - 4 * q^65 + 10 * q^67 + 5 * q^68 - q^70 - 4 * q^71 + 4 * q^73 + 2 * q^74 + 10 * q^76 + 22 * q^79 - 5 * q^80 - 6 * q^82 + 18 * q^83 - 3 * q^85 + 46 * q^86 + 6 * q^88 + 34 * q^89 + 4 * q^91 + 44 * q^92 + 2 * q^94 - 6 * q^95 + q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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