LMFDB - Embedded newform 5625.2.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5625,2,Mod(1,5625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.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:	$44.9158511370$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.44400625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 11x^{4} - x^{3} + 29x^{2} + 3x - 1 $ x^6 - 11x^4 - x^3 + 29x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.16056$ of defining polynomial
Character	$\chi$	$=$	5625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.16056 q^{2} +2.66802 q^{4} -3.16056 q^{7} -1.44329 q^{8} +O(q^{10})$	$q-2.16056 q^{2} +2.66802 q^{4} -3.16056 q^{7} -1.44329 q^{8} -1.53835 q^{11} -5.24144 q^{13} +6.82858 q^{14} -2.21771 q^{16} -1.29068 q^{17} +5.44587 q^{19} +3.32370 q^{22} +6.44244 q^{23} +11.3244 q^{26} -8.43243 q^{28} +2.36361 q^{29} -4.46542 q^{31} +7.67809 q^{32} +2.78860 q^{34} -5.95751 q^{37} -11.7661 q^{38} +8.53219 q^{41} -8.48426 q^{43} -4.10435 q^{44} -13.9193 q^{46} -0.753070 q^{47} +2.98914 q^{49} -13.9843 q^{52} -9.74991 q^{53} +4.56162 q^{56} -5.10672 q^{58} -4.11270 q^{59} -10.6939 q^{61} +9.64781 q^{62} -12.1535 q^{64} -1.89864 q^{67} -3.44357 q^{68} +0.0708774 q^{71} +4.01123 q^{73} +12.8716 q^{74} +14.5297 q^{76} +4.86205 q^{77} +1.61849 q^{79} -18.4343 q^{82} -13.1488 q^{83} +18.3308 q^{86} +2.22029 q^{88} -7.27597 q^{89} +16.5659 q^{91} +17.1885 q^{92} +1.62705 q^{94} +10.1939 q^{97} -6.45821 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 10 q^{4} - 6 q^{7} + 3 q^{8}+O(q^{10}) $ 6 * q + 10 * q^4 - 6 * q^7 + 3 * q^8	$ 6 q + 10 q^{4} - 6 q^{7} + 3 q^{8} - 3 q^{11} - 6 q^{13} + 22 q^{14} + 18 q^{16} + 13 q^{17} + 11 q^{19} - 16 q^{22} + 13 q^{23} + 28 q^{26} - 7 q^{28} + 3 q^{29} - 11 q^{31} + 16 q^{32} + 15 q^{34} - 21 q^{37} - 9 q^{38} + q^{41} - 2 q^{43} - 9 q^{44} + 19 q^{46} + 14 q^{47} - 14 q^{49} - 13 q^{52} + 23 q^{53} + 35 q^{56} - 22 q^{58} - 9 q^{59} + 11 q^{61} - 23 q^{62} - 23 q^{64} - 8 q^{67} + 50 q^{68} + 8 q^{71} - 13 q^{73} + 22 q^{74} - 26 q^{76} - 13 q^{77} - 5 q^{79} + 13 q^{82} - 20 q^{83} + 37 q^{86} - 28 q^{88} + 4 q^{89} + 34 q^{91} + 61 q^{92} + 41 q^{94} + 7 q^{97} - 41 q^{98}+O(q^{100}) $ 6 * q + 10 * q^4 - 6 * q^7 + 3 * q^8 - 3 * q^11 - 6 * q^13 + 22 * q^14 + 18 * q^16 + 13 * q^17 + 11 * q^19 - 16 * q^22 + 13 * q^23 + 28 * q^26 - 7 * q^28 + 3 * q^29 - 11 * q^31 + 16 * q^32 + 15 * q^34 - 21 * q^37 - 9 * q^38 + q^41 - 2 * q^43 - 9 * q^44 + 19 * q^46 + 14 * q^47 - 14 * q^49 - 13 * q^52 + 23 * q^53 + 35 * q^56 - 22 * q^58 - 9 * q^59 + 11 * q^61 - 23 * q^62 - 23 * q^64 - 8 * q^67 + 50 * q^68 + 8 * q^71 - 13 * q^73 + 22 * q^74 - 26 * q^76 - 13 * q^77 - 5 * q^79 + 13 * q^82 - 20 * q^83 + 37 * q^86 - 28 * q^88 + 4 * q^89 + 34 * q^91 + 61 * q^92 + 41 * q^94 + 7 * q^97 - 41 * q^98

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 $

$2$	−2.16056	−1.52775	−0.763873	−	0.645366i	$-0.776705\pi$
$2$	−2.16056	−1.52775	−0.763873	+	0.645366i	$0.776705\pi$
$3$	0	0
$3$	0	0
$4$	2.66802	1.33401
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.16056	−1.19458	−0.597290	−	0.802026i	$-0.703756\pi$
$7$	−3.16056	−1.19458	−0.597290	+	0.802026i	$0.703756\pi$
$8$	−1.44329	−0.510282
$9$	0	0
$10$	0	0
$11$	−1.53835	−0.463830	−0.231915	−	0.972736i	$-0.574499\pi$
$11$	−1.53835	−0.463830	−0.231915	+	0.972736i	$0.574499\pi$
$12$	0	0
$13$	−5.24144	−1.45371	−0.726857	−	0.686789i	$-0.759019\pi$
$13$	−5.24144	−1.45371	−0.726857	+	0.686789i	$0.759019\pi$
$14$	6.82858	1.82501
$15$	0	0
$16$	−2.21771	−0.554428
$17$	−1.29068	−0.313037	−0.156518	−	0.987675i	$-0.550027\pi$
$17$	−1.29068	−0.313037	−0.156518	+	0.987675i	$0.550027\pi$
$18$	0	0
$19$	5.44587	1.24937	0.624685	−	0.780877i	$-0.285228\pi$
$19$	5.44587	1.24937	0.624685	+	0.780877i	$0.285228\pi$
$20$	0	0
$21$	0	0
$22$	3.32370	0.708615
$23$	6.44244	1.34334	0.671671	−	0.740850i	$-0.265577\pi$
$23$	6.44244	1.34334	0.671671	+	0.740850i	$0.265577\pi$
$24$	0	0
$25$	0	0
$26$	11.3244	2.22090
$27$	0	0
$28$	−8.43243	−1.59358
$29$	2.36361	0.438912	0.219456	−	0.975622i	$-0.429572\pi$
$29$	2.36361	0.438912	0.219456	+	0.975622i	$0.429572\pi$
$30$	0	0
$31$	−4.46542	−0.802014	−0.401007	−	0.916075i	$-0.631340\pi$
$31$	−4.46542	−0.802014	−0.401007	+	0.916075i	$0.631340\pi$
$32$	7.67809	1.35731
$33$	0	0
$34$	2.78860	0.478241
$35$	0	0
$36$	0	0
$37$	−5.95751	−0.979408	−0.489704	−	0.871889i	$-0.662895\pi$
$37$	−5.95751	−0.979408	−0.489704	+	0.871889i	$0.662895\pi$
$38$	−11.7661	−1.90872
$39$	0	0
$40$	0	0
$41$	8.53219	1.33250	0.666252	−	0.745726i	$-0.267897\pi$
$41$	8.53219	1.33250	0.666252	+	0.745726i	$0.267897\pi$
$42$	0	0
$43$	−8.48426	−1.29384	−0.646919	−	0.762559i	$-0.723943\pi$
$43$	−8.48426	−1.29384	−0.646919	+	0.762559i	$0.723943\pi$
$44$	−4.10435	−0.618754
$45$	0	0
$46$	−13.9193	−2.05228
$47$	−0.753070	−0.109847	−0.0549233	−	0.998491i	$-0.517491\pi$
$47$	−0.753070	−0.109847	−0.0549233	+	0.998491i	$0.517491\pi$
$48$	0	0
$49$	2.98914	0.427020
$50$	0	0
$51$	0	0
$52$	−13.9843	−1.93927
$53$	−9.74991	−1.33925	−0.669626	−	0.742698i	$-0.733546\pi$
$53$	−9.74991	−1.33925	−0.669626	+	0.742698i	$0.733546\pi$
$54$	0	0
$55$	0	0
$56$	4.56162	0.609572
$57$	0	0
$58$	−5.10672	−0.670546
$59$	−4.11270	−0.535428	−0.267714	−	0.963498i	$-0.586268\pi$
$59$	−4.11270	−0.535428	−0.267714	+	0.963498i	$0.586268\pi$
$60$	0	0
$61$	−10.6939	−1.36922	−0.684610	−	0.728910i	$-0.740027\pi$
$61$	−10.6939	−1.36922	−0.684610	+	0.728910i	$0.740027\pi$
$62$	9.64781	1.22527
$63$	0	0
$64$	−12.1535	−1.51919
$65$	0	0
$66$	0	0
$67$	−1.89864	−0.231956	−0.115978	−	0.993252i	$-0.537000\pi$
$67$	−1.89864	−0.231956	−0.115978	+	0.993252i	$0.537000\pi$
$68$	−3.44357	−0.417594
$69$	0	0
$70$	0	0
$71$	0.0708774	0.00841160	0.00420580	−	0.999991i	$-0.498661\pi$
$71$	0.0708774	0.00841160	0.00420580	+	0.999991i	$0.498661\pi$
$72$	0	0
$73$	4.01123	0.469478	0.234739	−	0.972058i	$-0.424576\pi$
$73$	4.01123	0.469478	0.234739	+	0.972058i	$0.424576\pi$
$74$	12.8716	1.49629
$75$	0	0
$76$	14.5297	1.66667
$77$	4.86205	0.554082
$78$	0	0
$79$	1.61849	0.182094	0.0910472	−	0.995847i	$-0.470979\pi$
$79$	1.61849	0.182094	0.0910472	+	0.995847i	$0.470979\pi$
$80$	0	0
$81$	0	0
$82$	−18.4343	−2.03573
$83$	−13.1488	−1.44327	−0.721636	−	0.692272i	$-0.756610\pi$
$83$	−13.1488	−1.44327	−0.721636	+	0.692272i	$0.756610\pi$
$84$	0	0
$85$	0	0
$86$	18.3308	1.97666
$87$	0	0
$88$	2.22029	0.236684
$89$	−7.27597	−0.771251	−0.385626	−	0.922655i	$-0.626014\pi$
$89$	−7.27597	−0.771251	−0.385626	+	0.922655i	$0.626014\pi$
$90$	0	0
$91$	16.5659	1.73658
$92$	17.1885	1.79203
$93$	0	0
$94$	1.62705	0.167818
$95$	0	0
$96$	0	0
$97$	10.1939	1.03504	0.517518	−	0.855673i	$-0.326856\pi$
$97$	10.1939	1.03504	0.517518	+	0.855673i	$0.326856\pi$
$98$	−6.45821	−0.652378
$99$	0	0
$100$	0	0
$101$	−1.08759	−0.108219	−0.0541096	−	0.998535i	$-0.517232\pi$
$101$	−1.08759	−0.108219	−0.0541096	+	0.998535i	$0.517232\pi$
$102$	0	0
$103$	−4.93756	−0.486512	−0.243256	−	0.969962i	$-0.578216\pi$
$103$	−4.93756	−0.486512	−0.243256	+	0.969962i	$0.578216\pi$
$104$	7.56494	0.741803
$105$	0	0
$106$	21.0653	2.04604
$107$	−15.3059	−1.47967	−0.739837	−	0.672786i	$-0.765098\pi$
$107$	−15.3059	−1.47967	−0.739837	+	0.672786i	$0.765098\pi$
$108$	0	0
$109$	6.65900	0.637816	0.318908	−	0.947786i	$-0.396684\pi$
$109$	6.65900	0.637816	0.318908	+	0.947786i	$0.396684\pi$
$110$	0	0
$111$	0	0
$112$	7.00922	0.662309
$113$	8.97143	0.843961	0.421981	−	0.906605i	$-0.361335\pi$
$113$	8.97143	0.843961	0.421981	+	0.906605i	$0.361335\pi$
$114$	0	0
$115$	0	0
$116$	6.30616	0.585512
$117$	0	0
$118$	8.88573	0.817998
$119$	4.07928	0.373947
$120$	0	0
$121$	−8.63348	−0.784861
$122$	23.1049	2.09182
$123$	0	0
$124$	−11.9138	−1.06989
$125$	0	0
$126$	0	0
$127$	10.0185	0.888995	0.444497	−	0.895780i	$-0.353382\pi$
$127$	10.0185	0.888995	0.444497	+	0.895780i	$0.353382\pi$
$128$	10.9023	0.963635
$129$	0	0
$130$	0	0
$131$	6.37865	0.557305	0.278653	−	0.960392i	$-0.410112\pi$
$131$	6.37865	0.557305	0.278653	+	0.960392i	$0.410112\pi$
$132$	0	0
$133$	−17.2120	−1.49247
$134$	4.10214	0.354371
$135$	0	0
$136$	1.86284	0.159737
$137$	9.17732	0.784071	0.392036	−	0.919950i	$-0.371771\pi$
$137$	9.17732	0.784071	0.392036	+	0.919950i	$0.371771\pi$
$138$	0	0
$139$	−9.26539	−0.785880	−0.392940	−	0.919564i	$-0.628542\pi$
$139$	−9.26539	−0.785880	−0.392940	+	0.919564i	$0.628542\pi$
$140$	0	0
$141$	0	0
$142$	−0.153135	−0.0128508
$143$	8.06317	0.674276
$144$	0	0
$145$	0	0
$146$	−8.66649	−0.717244
$147$	0	0
$148$	−15.8947	−1.30654
$149$	10.0817	0.825928	0.412964	−	0.910747i	$-0.364494\pi$
$149$	10.0817	0.825928	0.412964	+	0.910747i	$0.364494\pi$
$150$	0	0
$151$	10.7375	0.873804	0.436902	−	0.899509i	$-0.356076\pi$
$151$	10.7375	0.873804	0.436902	+	0.899509i	$0.356076\pi$
$152$	−7.86000	−0.637530
$153$	0	0
$154$	−10.5048	−0.846497
$155$	0	0
$156$	0	0
$157$	17.4417	1.39200	0.696001	−	0.718041i	$-0.254961\pi$
$157$	17.4417	1.39200	0.696001	+	0.718041i	$0.254961\pi$
$158$	−3.49684	−0.278194
$159$	0	0
$160$	0	0
$161$	−20.3617	−1.60473
$162$	0	0
$163$	−16.2574	−1.27338	−0.636689	−	0.771120i	$-0.719697\pi$
$163$	−16.2574	−1.27338	−0.636689	+	0.771120i	$0.719697\pi$
$164$	22.7641	1.77757
$165$	0	0
$166$	28.4089	2.20496
$167$	11.3906	0.881430	0.440715	−	0.897647i	$-0.354725\pi$
$167$	11.3906	0.881430	0.440715	+	0.897647i	$0.354725\pi$
$168$	0	0
$169$	14.4727	1.11328
$170$	0	0
$171$	0	0
$172$	−22.6362	−1.72599
$173$	−9.45845	−0.719113	−0.359556	−	0.933123i	$-0.617072\pi$
$173$	−9.45845	−0.719113	−0.359556	+	0.933123i	$0.617072\pi$
$174$	0	0
$175$	0	0
$176$	3.41162	0.257161
$177$	0	0
$178$	15.7202	1.17828
$179$	−4.07747	−0.304764	−0.152382	−	0.988322i	$-0.548694\pi$
$179$	−4.07747	−0.304764	−0.152382	+	0.988322i	$0.548694\pi$
$180$	0	0
$181$	−11.1202	−0.826558	−0.413279	−	0.910604i	$-0.635617\pi$
$181$	−11.1202	−0.826558	−0.413279	+	0.910604i	$0.635617\pi$
$182$	−35.7916	−2.65305
$183$	0	0
$184$	−9.29833	−0.685482
$185$	0	0
$186$	0	0
$187$	1.98552	0.145196
$188$	−2.00921	−0.146536
$189$	0	0
$190$	0	0
$191$	−3.61135	−0.261308	−0.130654	−	0.991428i	$-0.541708\pi$
$191$	−3.61135	−0.261308	−0.130654	+	0.991428i	$0.541708\pi$
$192$	0	0
$193$	−17.0562	−1.22774	−0.613868	−	0.789409i	$-0.710387\pi$
$193$	−17.0562	−1.22774	−0.613868	+	0.789409i	$0.710387\pi$
$194$	−22.0246	−1.58127
$195$	0	0
$196$	7.97508	0.569648
$197$	4.40523	0.313860	0.156930	−	0.987610i	$-0.449840\pi$
$197$	4.40523	0.313860	0.156930	+	0.987610i	$0.449840\pi$
$198$	0	0
$199$	16.5956	1.17643	0.588214	−	0.808705i	$-0.299831\pi$
$199$	16.5956	1.17643	0.588214	+	0.808705i	$0.299831\pi$
$200$	0	0
$201$	0	0
$202$	2.34980	0.165332
$203$	−7.47034	−0.524315
$204$	0	0
$205$	0	0
$206$	10.6679	0.743267
$207$	0	0
$208$	11.6240	0.805980
$209$	−8.37767	−0.579495
$210$	0	0
$211$	18.5454	1.27671	0.638357	−	0.769740i	$-0.279614\pi$
$211$	18.5454	1.27671	0.638357	+	0.769740i	$0.279614\pi$
$212$	−26.0129	−1.78658
$213$	0	0
$214$	33.0693	2.26057
$215$	0	0
$216$	0	0
$217$	14.1132	0.958069
$218$	−14.3872	−0.974422
$219$	0	0
$220$	0	0
$221$	6.76504	0.455066
$222$	0	0
$223$	−0.771557	−0.0516673	−0.0258336	−	0.999666i	$-0.508224\pi$
$223$	−0.771557	−0.0516673	−0.0258336	+	0.999666i	$0.508224\pi$
$224$	−24.2671	−1.62141
$225$	0	0
$226$	−19.3833	−1.28936
$227$	−11.4314	−0.758727	−0.379363	−	0.925248i	$-0.623857\pi$
$227$	−11.4314	−0.758727	−0.379363	+	0.925248i	$0.623857\pi$
$228$	0	0
$229$	10.2248	0.675671	0.337835	−	0.941205i	$-0.390305\pi$
$229$	10.2248	0.675671	0.337835	+	0.941205i	$0.390305\pi$
$230$	0	0
$231$	0	0
$232$	−3.41139	−0.223969
$233$	22.5229	1.47553	0.737763	−	0.675059i	$-0.235882\pi$
$233$	22.5229	1.47553	0.737763	+	0.675059i	$0.235882\pi$
$234$	0	0
$235$	0	0
$236$	−10.9728	−0.714266
$237$	0	0
$238$	−8.81353	−0.571297
$239$	0.589074	0.0381040	0.0190520	−	0.999818i	$-0.493935\pi$
$239$	0.589074	0.0381040	0.0190520	+	0.999818i	$0.493935\pi$
$240$	0	0
$241$	1.50937	0.0972270	0.0486135	−	0.998818i	$-0.484520\pi$
$241$	1.50937	0.0972270	0.0486135	+	0.998818i	$0.484520\pi$
$242$	18.6531	1.19907
$243$	0	0
$244$	−28.5317	−1.82655
$245$	0	0
$246$	0	0
$247$	−28.5442	−1.81622
$248$	6.44492	0.409253
$249$	0	0
$250$	0	0
$251$	−11.8953	−0.750823	−0.375412	−	0.926858i	$-0.622499\pi$
$251$	−11.8953	−0.750823	−0.375412	+	0.926858i	$0.622499\pi$
$252$	0	0
$253$	−9.91073	−0.623082
$254$	−21.6455	−1.35816
$255$	0	0
$256$	0.752056	0.0470035
$257$	18.0492	1.12588	0.562939	−	0.826498i	$-0.309671\pi$
$257$	18.0492	1.12588	0.562939	+	0.826498i	$0.309671\pi$
$258$	0	0
$259$	18.8291	1.16998
$260$	0	0
$261$	0	0
$262$	−13.7815	−0.851421
$263$	19.1066	1.17817	0.589083	−	0.808072i	$-0.299489\pi$
$263$	19.1066	1.17817	0.589083	+	0.808072i	$0.299489\pi$
$264$	0	0
$265$	0	0
$266$	37.1876	2.28012
$267$	0	0
$268$	−5.06562	−0.309432
$269$	22.2250	1.35508	0.677542	−	0.735484i	$-0.263045\pi$
$269$	22.2250	1.35508	0.677542	+	0.735484i	$0.263045\pi$
$270$	0	0
$271$	20.2107	1.22772	0.613858	−	0.789417i	$-0.289617\pi$
$271$	20.2107	1.22772	0.613858	+	0.789417i	$0.289617\pi$
$272$	2.86237	0.173556
$273$	0	0
$274$	−19.8281	−1.19786
$275$	0	0
$276$	0	0
$277$	−16.0413	−0.963826	−0.481913	−	0.876219i	$-0.660058\pi$
$277$	−16.0413	−0.963826	−0.481913	+	0.876219i	$0.660058\pi$
$278$	20.0184	1.20063
$279$	0	0
$280$	0	0
$281$	−8.37308	−0.499496	−0.249748	−	0.968311i	$-0.580348\pi$
$281$	−8.37308	−0.499496	−0.249748	+	0.968311i	$0.580348\pi$
$282$	0	0
$283$	−1.12995	−0.0671685	−0.0335843	−	0.999436i	$-0.510692\pi$
$283$	−1.12995	−0.0671685	−0.0335843	+	0.999436i	$0.510692\pi$
$284$	0.189102	0.0112211
$285$	0	0
$286$	−17.4210	−1.03012
$287$	−26.9665	−1.59178
$288$	0	0
$289$	−15.3341	−0.902008
$290$	0	0
$291$	0	0
$292$	10.7020	0.626289
$293$	2.37857	0.138958	0.0694789	−	0.997583i	$-0.477866\pi$
$293$	2.37857	0.138958	0.0694789	+	0.997583i	$0.477866\pi$
$294$	0	0
$295$	0	0
$296$	8.59844	0.499774
$297$	0	0
$298$	−21.7822	−1.26181
$299$	−33.7676	−1.95283
$300$	0	0
$301$	26.8150	1.54559
$302$	−23.1990	−1.33495
$303$	0	0
$304$	−12.0774	−0.692686
$305$	0	0
$306$	0	0
$307$	2.89366	0.165150	0.0825748	−	0.996585i	$-0.473686\pi$
$307$	2.89366	0.165150	0.0825748	+	0.996585i	$0.473686\pi$
$308$	12.9720	0.739151
$309$	0	0
$310$	0	0
$311$	7.14545	0.405181	0.202591	−	0.979264i	$-0.435064\pi$
$311$	7.14545	0.405181	0.202591	+	0.979264i	$0.435064\pi$
$312$	0	0
$313$	23.0318	1.30184	0.650918	−	0.759148i	$-0.274384\pi$
$313$	23.0318	1.30184	0.650918	+	0.759148i	$0.274384\pi$
$314$	−37.6839	−2.12663
$315$	0	0
$316$	4.31816	0.242916
$317$	20.7750	1.16684	0.583419	−	0.812171i	$-0.301715\pi$
$317$	20.7750	1.16684	0.583419	+	0.812171i	$0.301715\pi$
$318$	0	0
$319$	−3.63606	−0.203581
$320$	0	0
$321$	0	0
$322$	43.9927	2.45162
$323$	−7.02890	−0.391098
$324$	0	0
$325$	0	0
$326$	35.1251	1.94540
$327$	0	0
$328$	−12.3145	−0.679953
$329$	2.38012	0.131220
$330$	0	0
$331$	−19.2504	−1.05810	−0.529048	−	0.848592i	$-0.677451\pi$
$331$	−19.2504	−1.05810	−0.529048	+	0.848592i	$0.677451\pi$
$332$	−35.0814	−1.92534
$333$	0	0
$334$	−24.6100	−1.34660
$335$	0	0
$336$	0	0
$337$	21.7064	1.18243	0.591213	−	0.806516i	$-0.298649\pi$
$337$	21.7064	1.18243	0.591213	+	0.806516i	$0.298649\pi$
$338$	−31.2690	−1.70081
$339$	0	0
$340$	0	0
$341$	6.86939	0.371998
$342$	0	0
$343$	12.6766	0.684470
$344$	12.2453	0.660221
$345$	0	0
$346$	20.4356	1.09862
$347$	1.71494	0.0920626	0.0460313	−	0.998940i	$-0.485343\pi$
$347$	1.71494	0.0920626	0.0460313	+	0.998940i	$0.485343\pi$
$348$	0	0
$349$	−15.5553	−0.832654	−0.416327	−	0.909215i	$-0.636683\pi$
$349$	−15.5553	−0.832654	−0.416327	+	0.909215i	$0.636683\pi$
$350$	0	0
$351$	0	0
$352$	−11.8116	−0.629560
$353$	−15.5536	−0.827833	−0.413916	−	0.910315i	$-0.635839\pi$
$353$	−15.5536	−0.827833	−0.413916	+	0.910315i	$0.635839\pi$
$354$	0	0
$355$	0	0
$356$	−19.4124	−1.02886
$357$	0	0
$358$	8.80961	0.465602
$359$	24.5371	1.29502	0.647508	−	0.762059i	$-0.275811\pi$
$359$	24.5371	1.29502	0.647508	+	0.762059i	$0.275811\pi$
$360$	0	0
$361$	10.6575	0.560924
$362$	24.0259	1.26277
$363$	0	0
$364$	44.1981	2.31661
$365$	0	0
$366$	0	0
$367$	28.1068	1.46716	0.733581	−	0.679602i	$-0.237847\pi$
$367$	28.1068	1.46716	0.733581	+	0.679602i	$0.237847\pi$
$368$	−14.2875	−0.744786
$369$	0	0
$370$	0	0
$371$	30.8152	1.59984
$372$	0	0
$373$	−9.70825	−0.502674	−0.251337	−	0.967900i	$-0.580870\pi$
$373$	−9.70825	−0.502674	−0.251337	+	0.967900i	$0.580870\pi$
$374$	−4.28984	−0.221823
$375$	0	0
$376$	1.08690	0.0560527
$377$	−12.3887	−0.638052
$378$	0	0
$379$	−22.2665	−1.14375	−0.571875	−	0.820340i	$-0.693784\pi$
$379$	−22.2665	−1.14375	−0.571875	+	0.820340i	$0.693784\pi$
$380$	0	0
$381$	0	0
$382$	7.80253	0.399212
$383$	13.6299	0.696458	0.348229	−	0.937410i	$-0.386783\pi$
$383$	13.6299	0.696458	0.348229	+	0.937410i	$0.386783\pi$
$384$	0	0
$385$	0	0
$386$	36.8510	1.87567
$387$	0	0
$388$	27.1976	1.38075
$389$	−19.2304	−0.975021	−0.487510	−	0.873117i	$-0.662095\pi$
$389$	−19.2304	−0.975021	−0.487510	+	0.873117i	$0.662095\pi$
$390$	0	0
$391$	−8.31515	−0.420515
$392$	−4.31421	−0.217900
$393$	0	0
$394$	−9.51777	−0.479498
$395$	0	0
$396$	0	0
$397$	22.2281	1.11560	0.557799	−	0.829976i	$-0.311646\pi$
$397$	22.2281	1.11560	0.557799	+	0.829976i	$0.311646\pi$
$398$	−35.8557	−1.79729
$399$	0	0
$400$	0	0
$401$	−4.71728	−0.235570	−0.117785	−	0.993039i	$-0.537579\pi$
$401$	−4.71728	−0.235570	−0.117785	+	0.993039i	$0.537579\pi$
$402$	0	0
$403$	23.4052	1.16590
$404$	−2.90171	−0.144365
$405$	0	0
$406$	16.1401	0.801020
$407$	9.16474	0.454279
$408$	0	0
$409$	−12.4807	−0.617130	−0.308565	−	0.951203i	$-0.599849\pi$
$409$	−12.4807	−0.617130	−0.308565	+	0.951203i	$0.599849\pi$
$410$	0	0
$411$	0	0
$412$	−13.1735	−0.649012
$413$	12.9984	0.639611
$414$	0	0
$415$	0	0
$416$	−40.2442	−1.97314
$417$	0	0
$418$	18.1005	0.885322
$419$	−34.9484	−1.70734	−0.853669	−	0.520815i	$-0.825628\pi$
$419$	−34.9484	−1.70734	−0.853669	+	0.520815i	$0.825628\pi$
$420$	0	0
$421$	−39.5601	−1.92804	−0.964020	−	0.265829i	$-0.914354\pi$
$421$	−39.5601	−1.92804	−0.964020	+	0.265829i	$0.914354\pi$
$422$	−40.0683	−1.95050
$423$	0	0
$424$	14.0720	0.683396
$425$	0	0
$426$	0	0
$427$	33.7989	1.63564
$428$	−40.8364	−1.97390
$429$	0	0
$430$	0	0
$431$	14.0584	0.677170	0.338585	−	0.940936i	$-0.390052\pi$
$431$	14.0584	0.677170	0.338585	+	0.940936i	$0.390052\pi$
$432$	0	0
$433$	−0.549678	−0.0264158	−0.0132079	−	0.999913i	$-0.504204\pi$
$433$	−0.549678	−0.0264158	−0.0132079	+	0.999913i	$0.504204\pi$
$434$	−30.4925	−1.46369
$435$	0	0
$436$	17.7663	0.850853
$437$	35.0847	1.67833
$438$	0	0
$439$	−2.97377	−0.141930	−0.0709651	−	0.997479i	$-0.522608\pi$
$439$	−2.97377	−0.141930	−0.0709651	+	0.997479i	$0.522608\pi$
$440$	0	0
$441$	0	0
$442$	−14.6163	−0.695225
$443$	36.6893	1.74316	0.871580	−	0.490253i	$-0.163095\pi$
$443$	36.6893	1.74316	0.871580	+	0.490253i	$0.163095\pi$
$444$	0	0
$445$	0	0
$446$	1.66700	0.0789345
$447$	0	0
$448$	38.4120	1.81480
$449$	17.8432	0.842071	0.421036	−	0.907044i	$-0.361667\pi$
$449$	17.8432	0.842071	0.421036	+	0.907044i	$0.361667\pi$
$450$	0	0
$451$	−13.1255	−0.618056
$452$	23.9359	1.12585
$453$	0	0
$454$	24.6982	1.15914
$455$	0	0
$456$	0	0
$457$	−24.1784	−1.13102	−0.565509	−	0.824742i	$-0.691320\pi$
$457$	−24.1784	−1.13102	−0.565509	+	0.824742i	$0.691320\pi$
$458$	−22.0912	−1.03225
$459$	0	0
$460$	0	0
$461$	37.3874	1.74130	0.870651	−	0.491901i	$-0.163698\pi$
$461$	37.3874	1.74130	0.870651	+	0.491901i	$0.163698\pi$
$462$	0	0
$463$	20.4314	0.949530	0.474765	−	0.880113i	$-0.342533\pi$
$463$	20.4314	0.949530	0.474765	+	0.880113i	$0.342533\pi$
$464$	−5.24181	−0.243345
$465$	0	0
$466$	−48.6622	−2.25423
$467$	−31.2124	−1.44434	−0.722169	−	0.691716i	$-0.756855\pi$
$467$	−31.2124	−1.44434	−0.722169	+	0.691716i	$0.756855\pi$
$468$	0	0
$469$	6.00078	0.277090
$470$	0	0
$471$	0	0
$472$	5.93584	0.273219
$473$	13.0518	0.600121
$474$	0	0
$475$	0	0
$476$	10.8836	0.498849
$477$	0	0
$478$	−1.27273	−0.0582133
$479$	−11.0970	−0.507033	−0.253516	−	0.967331i	$-0.581587\pi$
$479$	−11.0970	−0.507033	−0.253516	+	0.967331i	$0.581587\pi$
$480$	0	0
$481$	31.2259	1.42378
$482$	−3.26108	−0.148538
$483$	0	0
$484$	−23.0343	−1.04701
$485$	0	0
$486$	0	0
$487$	7.13702	0.323409	0.161705	−	0.986839i	$-0.448301\pi$
$487$	7.13702	0.323409	0.161705	+	0.986839i	$0.448301\pi$
$488$	15.4345	0.698688
$489$	0	0
$490$	0	0
$491$	−7.23348	−0.326442	−0.163221	−	0.986590i	$-0.552188\pi$
$491$	−7.23348	−0.326442	−0.163221	+	0.986590i	$0.552188\pi$
$492$	0	0
$493$	−3.05067	−0.137395
$494$	61.6715	2.77473
$495$	0	0
$496$	9.90303	0.444659
$497$	−0.224012	−0.0100483
$498$	0	0
$499$	16.0169	0.717014	0.358507	−	0.933527i	$-0.383286\pi$
$499$	16.0169	0.717014	0.358507	+	0.933527i	$0.383286\pi$
$500$	0	0
$501$	0	0
$502$	25.7005	1.14707
$503$	33.7963	1.50690	0.753452	−	0.657503i	$-0.228387\pi$
$503$	33.7963	1.50690	0.753452	+	0.657503i	$0.228387\pi$
$504$	0	0
$505$	0	0
$506$	21.4127	0.951912
$507$	0	0
$508$	26.7294	1.18593
$509$	36.4270	1.61460	0.807300	−	0.590142i	$-0.200928\pi$
$509$	36.4270	1.61460	0.807300	+	0.590142i	$0.200928\pi$
$510$	0	0
$511$	−12.6777	−0.560829
$512$	−23.4294	−1.03544
$513$	0	0
$514$	−38.9964	−1.72006
$515$	0	0
$516$	0	0
$517$	1.15849	0.0509502
$518$	−40.6813	−1.78743
$519$	0	0
$520$	0	0
$521$	−25.4993	−1.11714	−0.558572	−	0.829456i	$-0.688651\pi$
$521$	−25.4993	−1.11714	−0.558572	+	0.829456i	$0.688651\pi$
$522$	0	0
$523$	−7.03174	−0.307477	−0.153738	−	0.988112i	$-0.549131\pi$
$523$	−7.03174	−0.307477	−0.153738	+	0.988112i	$0.549131\pi$
$524$	17.0184	0.743450
$525$	0	0
$526$	−41.2811	−1.79994
$527$	5.76345	0.251060
$528$	0	0
$529$	18.5050	0.804565
$530$	0	0
$531$	0	0
$532$	−45.9220	−1.99097
$533$	−44.7210	−1.93708
$534$	0	0
$535$	0	0
$536$	2.74030	0.118363
$537$	0	0
$538$	−48.0185	−2.07022
$539$	−4.59834	−0.198065
$540$	0	0
$541$	−27.1476	−1.16717	−0.583584	−	0.812053i	$-0.698350\pi$
$541$	−27.1476	−1.16717	−0.583584	+	0.812053i	$0.698350\pi$
$542$	−43.6665	−1.87564
$543$	0	0
$544$	−9.90999	−0.424887
$545$	0	0
$546$	0	0
$547$	44.0769	1.88459	0.942296	−	0.334782i	$-0.108663\pi$
$547$	44.0769	1.88459	0.942296	+	0.334782i	$0.108663\pi$
$548$	24.4853	1.04596
$549$	0	0
$550$	0	0
$551$	12.8719	0.548363
$552$	0	0
$553$	−5.11533	−0.217526
$554$	34.6581	1.47248
$555$	0	0
$556$	−24.7202	−1.04837
$557$	9.85667	0.417641	0.208820	−	0.977954i	$-0.433038\pi$
$557$	9.85667	0.417641	0.208820	+	0.977954i	$0.433038\pi$
$558$	0	0
$559$	44.4697	1.88087
$560$	0	0
$561$	0	0
$562$	18.0905	0.763103
$563$	12.3917	0.522249	0.261125	−	0.965305i	$-0.415907\pi$
$563$	12.3917	0.522249	0.261125	+	0.965305i	$0.415907\pi$
$564$	0	0
$565$	0	0
$566$	2.44132	0.102616
$567$	0	0
$568$	−0.102297	−0.00429228
$569$	11.7158	0.491150	0.245575	−	0.969378i	$-0.421023\pi$
$569$	11.7158	0.491150	0.245575	+	0.969378i	$0.421023\pi$
$570$	0	0
$571$	28.9797	1.21276	0.606382	−	0.795173i	$-0.292620\pi$
$571$	28.9797	1.21276	0.606382	+	0.795173i	$0.292620\pi$
$572$	21.5127	0.899491
$573$	0	0
$574$	58.2628	2.43184
$575$	0	0
$576$	0	0
$577$	−32.8024	−1.36558	−0.682790	−	0.730614i	$-0.739234\pi$
$577$	−32.8024	−1.36558	−0.682790	+	0.730614i	$0.739234\pi$
$578$	33.1303	1.37804
$579$	0	0
$580$	0	0
$581$	41.5577	1.72410
$582$	0	0
$583$	14.9988	0.621186
$584$	−5.78938	−0.239566
$585$	0	0
$586$	−5.13905	−0.212292
$587$	−18.6395	−0.769334	−0.384667	−	0.923055i	$-0.625684\pi$
$587$	−18.6395	−0.769334	−0.384667	+	0.923055i	$0.625684\pi$
$588$	0	0
$589$	−24.3181	−1.00201
$590$	0	0
$591$	0	0
$592$	13.2120	0.543012
$593$	−1.55882	−0.0640129	−0.0320064	−	0.999488i	$-0.510190\pi$
$593$	−1.55882	−0.0640129	−0.0320064	+	0.999488i	$0.510190\pi$
$594$	0	0
$595$	0	0
$596$	26.8983	1.10180
$597$	0	0
$598$	72.9570	2.98343
$599$	31.5470	1.28898	0.644489	−	0.764614i	$-0.277070\pi$
$599$	31.5470	1.28898	0.644489	+	0.764614i	$0.277070\pi$
$600$	0	0
$601$	−32.3999	−1.32162	−0.660809	−	0.750554i	$-0.729787\pi$
$601$	−32.3999	−1.32162	−0.660809	+	0.750554i	$0.729787\pi$
$602$	−57.9354	−2.36127
$603$	0	0
$604$	28.6478	1.16566
$605$	0	0
$606$	0	0
$607$	24.8410	1.00827	0.504133	−	0.863626i	$-0.331812\pi$
$607$	24.8410	1.00827	0.504133	+	0.863626i	$0.331812\pi$
$608$	41.8139	1.69578
$609$	0	0
$610$	0	0
$611$	3.94717	0.159685
$612$	0	0
$613$	−3.94558	−0.159360	−0.0796802	−	0.996820i	$-0.525390\pi$
$613$	−3.94558	−0.159360	−0.0796802	+	0.996820i	$0.525390\pi$
$614$	−6.25192	−0.252307
$615$	0	0
$616$	−7.01737	−0.282738
$617$	17.4821	0.703801	0.351900	−	0.936037i	$-0.385536\pi$
$617$	17.4821	0.703801	0.351900	+	0.936037i	$0.385536\pi$
$618$	0	0
$619$	−14.3869	−0.578260	−0.289130	−	0.957290i	$-0.593366\pi$
$619$	−14.3869	−0.578260	−0.289130	+	0.957290i	$0.593366\pi$
$620$	0	0
$621$	0	0
$622$	−15.4382	−0.619014
$623$	22.9961	0.921321
$624$	0	0
$625$	0	0
$626$	−49.7617	−1.98888
$627$	0	0
$628$	46.5349	1.85694
$629$	7.68926	0.306591
$630$	0	0
$631$	−17.6758	−0.703664	−0.351832	−	0.936063i	$-0.614441\pi$
$631$	−17.6758	−0.703664	−0.351832	+	0.936063i	$0.614441\pi$
$632$	−2.33596	−0.0929194
$633$	0	0
$634$	−44.8855	−1.78263
$635$	0	0
$636$	0	0
$637$	−15.6674	−0.620764
$638$	7.85594	0.311019
$639$	0	0
$640$	0	0
$641$	2.74127	0.108274	0.0541369	−	0.998534i	$-0.482759\pi$
$641$	2.74127	0.108274	0.0541369	+	0.998534i	$0.482759\pi$
$642$	0	0
$643$	20.9276	0.825303	0.412651	−	0.910889i	$-0.364603\pi$
$643$	20.9276	0.825303	0.412651	+	0.910889i	$0.364603\pi$
$644$	−54.3254	−2.14072
$645$	0	0
$646$	15.1864	0.597499
$647$	1.84667	0.0726002	0.0363001	−	0.999341i	$-0.488443\pi$
$647$	1.84667	0.0726002	0.0363001	+	0.999341i	$0.488443\pi$
$648$	0	0
$649$	6.32678	0.248348
$650$	0	0
$651$	0	0
$652$	−43.3751	−1.69870
$653$	18.4133	0.720567	0.360284	−	0.932843i	$-0.382680\pi$
$653$	18.4133	0.720567	0.360284	+	0.932843i	$0.382680\pi$
$654$	0	0
$655$	0	0
$656$	−18.9220	−0.738778
$657$	0	0
$658$	−5.14240	−0.200472
$659$	13.3800	0.521210	0.260605	−	0.965446i	$-0.416078\pi$
$659$	13.3800	0.521210	0.260605	+	0.965446i	$0.416078\pi$
$660$	0	0
$661$	39.1834	1.52406	0.762030	−	0.647542i	$-0.224203\pi$
$661$	39.1834	1.52406	0.762030	+	0.647542i	$0.224203\pi$
$662$	41.5915	1.61650
$663$	0	0
$664$	18.9776	0.736476
$665$	0	0
$666$	0	0
$667$	15.2274	0.589608
$668$	30.3903	1.17584
$669$	0	0
$670$	0	0
$671$	16.4510	0.635086
$672$	0	0
$673$	19.3911	0.747473	0.373736	−	0.927535i	$-0.378077\pi$
$673$	19.3911	0.747473	0.373736	+	0.927535i	$0.378077\pi$
$674$	−46.8981	−1.80645
$675$	0	0
$676$	38.6133	1.48513
$677$	45.6836	1.75576	0.877882	−	0.478876i	$-0.158956\pi$
$677$	45.6836	1.75576	0.877882	+	0.478876i	$0.158956\pi$
$678$	0	0
$679$	−32.2185	−1.23643
$680$	0	0
$681$	0	0
$682$	−14.8417	−0.568319
$683$	−11.5893	−0.443451	−0.221725	−	0.975109i	$-0.571169\pi$
$683$	−11.5893	−0.443451	−0.221725	+	0.975109i	$0.571169\pi$
$684$	0	0
$685$	0	0
$686$	−27.3885	−1.04570
$687$	0	0
$688$	18.8157	0.717340
$689$	51.1035	1.94689
$690$	0	0
$691$	−21.6424	−0.823317	−0.411658	−	0.911338i	$-0.635050\pi$
$691$	−21.6424	−0.823317	−0.411658	+	0.911338i	$0.635050\pi$
$692$	−25.2353	−0.959303
$693$	0	0
$694$	−3.70522	−0.140648
$695$	0	0
$696$	0	0
$697$	−11.0124	−0.417123
$698$	33.6081	1.27208
$699$	0	0
$700$	0	0
$701$	8.61904	0.325537	0.162768	−	0.986664i	$-0.447958\pi$
$701$	8.61904	0.325537	0.162768	+	0.986664i	$0.447958\pi$
$702$	0	0
$703$	−32.4438	−1.22364
$704$	18.6964	0.704648
$705$	0	0
$706$	33.6044	1.26472
$707$	3.43739	0.129276
$708$	0	0
$709$	16.0001	0.600895	0.300447	−	0.953798i	$-0.402864\pi$
$709$	16.0001	0.600895	0.300447	+	0.953798i	$0.402864\pi$
$710$	0	0
$711$	0	0
$712$	10.5014	0.393555
$713$	−28.7682	−1.07738
$714$	0	0
$715$	0	0
$716$	−10.8788	−0.406558
$717$	0	0
$718$	−53.0138	−1.97846
$719$	48.5838	1.81187	0.905935	−	0.423416i	$-0.139169\pi$
$719$	48.5838	1.81187	0.905935	+	0.423416i	$0.139169\pi$
$720$	0	0
$721$	15.6055	0.581177
$722$	−23.0263	−0.856949
$723$	0	0
$724$	−29.6689	−1.10264
$725$	0	0
$726$	0	0
$727$	25.4328	0.943251	0.471625	−	0.881799i	$-0.343667\pi$
$727$	25.4328	0.943251	0.471625	+	0.881799i	$0.343667\pi$
$728$	−23.9094	−0.886143
$729$	0	0
$730$	0	0
$731$	10.9505	0.405019
$732$	0	0
$733$	−18.6190	−0.687708	−0.343854	−	0.939023i	$-0.611733\pi$
$733$	−18.6190	−0.687708	−0.343854	+	0.939023i	$0.611733\pi$
$734$	−60.7264	−2.24145
$735$	0	0
$736$	49.4656	1.82333
$737$	2.92078	0.107588
$738$	0	0
$739$	46.8872	1.72478	0.862388	−	0.506249i	$-0.168968\pi$
$739$	46.8872	1.72478	0.862388	+	0.506249i	$0.168968\pi$
$740$	0	0
$741$	0	0
$742$	−66.5780	−2.44416
$743$	42.7061	1.56674	0.783368	−	0.621558i	$-0.213500\pi$
$743$	42.7061	1.56674	0.783368	+	0.621558i	$0.213500\pi$
$744$	0	0
$745$	0	0
$746$	20.9753	0.767959
$747$	0	0
$748$	5.29742	0.193693
$749$	48.3751	1.76759
$750$	0	0
$751$	−2.64893	−0.0966608	−0.0483304	−	0.998831i	$-0.515390\pi$
$751$	−2.64893	−0.0966608	−0.0483304	+	0.998831i	$0.515390\pi$
$752$	1.67009	0.0609021
$753$	0	0
$754$	26.7666	0.974781
$755$	0	0
$756$	0	0
$757$	52.9153	1.92324	0.961620	−	0.274383i	$-0.0884737\pi$
$757$	52.9153	1.92324	0.961620	+	0.274383i	$0.0884737\pi$
$758$	48.1080	1.74736
$759$	0	0
$760$	0	0
$761$	−50.7787	−1.84073	−0.920363	−	0.391064i	$-0.872107\pi$
$761$	−50.7787	−1.84073	−0.920363	+	0.391064i	$0.872107\pi$
$762$	0	0
$763$	−21.0462	−0.761922
$764$	−9.63514	−0.348587
$765$	0	0
$766$	−29.4483	−1.06401
$767$	21.5565	0.778358
$768$	0	0
$769$	24.5805	0.886396	0.443198	−	0.896424i	$-0.353844\pi$
$769$	24.5805	0.886396	0.443198	+	0.896424i	$0.353844\pi$
$770$	0	0
$771$	0	0
$772$	−45.5064	−1.63781
$773$	20.0112	0.719754	0.359877	−	0.933000i	$-0.382819\pi$
$773$	20.0112	0.719754	0.359877	+	0.933000i	$0.382819\pi$
$774$	0	0
$775$	0	0
$776$	−14.7128	−0.528159
$777$	0	0
$778$	41.5485	1.48958
$779$	46.4653	1.66479
$780$	0	0
$781$	−0.109034	−0.00390155
$782$	17.9654	0.642441
$783$	0	0
$784$	−6.62905	−0.236752
$785$	0	0
$786$	0	0
$787$	8.99272	0.320556	0.160278	−	0.987072i	$-0.448761\pi$
$787$	8.99272	0.320556	0.160278	+	0.987072i	$0.448761\pi$
$788$	11.7532	0.418692
$789$	0	0
$790$	0	0
$791$	−28.3547	−1.00818
$792$	0	0
$793$	56.0517	1.99045
$794$	−48.0252	−1.70435
$795$	0	0
$796$	44.2773	1.56937
$797$	37.7564	1.33740	0.668700	−	0.743533i	$-0.266851\pi$
$797$	37.7564	1.33740	0.668700	+	0.743533i	$0.266851\pi$
$798$	0	0
$799$	0.971976	0.0343860
$800$	0	0
$801$	0	0
$802$	10.1920	0.359891
$803$	−6.17067	−0.217758
$804$	0	0
$805$	0	0
$806$	−50.5684	−1.78120
$807$	0	0
$808$	1.56971	0.0552223
$809$	48.9001	1.71924	0.859618	−	0.510937i	$-0.170702\pi$
$809$	48.9001	1.71924	0.859618	+	0.510937i	$0.170702\pi$
$810$	0	0
$811$	−18.0551	−0.633999	−0.317000	−	0.948426i	$-0.602675\pi$
$811$	−18.0551	−0.633999	−0.317000	+	0.948426i	$0.602675\pi$
$812$	−19.9310	−0.699441
$813$	0	0
$814$	−19.8010	−0.694024
$815$	0	0
$816$	0	0
$817$	−46.2042	−1.61648
$818$	26.9653	0.942819
$819$	0	0
$820$	0	0
$821$	−50.6097	−1.76629	−0.883146	−	0.469099i	$-0.844579\pi$
$821$	−50.6097	−1.76629	−0.883146	+	0.469099i	$0.844579\pi$
$822$	0	0
$823$	−26.8070	−0.934433	−0.467217	−	0.884143i	$-0.654743\pi$
$823$	−26.8070	−0.934433	−0.467217	+	0.884143i	$0.654743\pi$
$824$	7.12635	0.248258
$825$	0	0
$826$	−28.0839	−0.977163
$827$	21.3649	0.742929	0.371465	−	0.928447i	$-0.378856\pi$
$827$	21.3649	0.742929	0.371465	+	0.928447i	$0.378856\pi$
$828$	0	0
$829$	−48.1123	−1.67101	−0.835504	−	0.549484i	$-0.814824\pi$
$829$	−48.1123	−1.67101	−0.835504	+	0.549484i	$0.814824\pi$
$830$	0	0
$831$	0	0
$832$	63.7021	2.20847
$833$	−3.85803	−0.133673
$834$	0	0
$835$	0	0
$836$	−22.3518	−0.773052
$837$	0	0
$838$	75.5080	2.60838
$839$	41.8540	1.44496	0.722481	−	0.691391i	$-0.243002\pi$
$839$	41.8540	1.44496	0.722481	+	0.691391i	$0.243002\pi$
$840$	0	0
$841$	−23.4133	−0.807357
$842$	85.4719	2.94556
$843$	0	0
$844$	49.4794	1.70315
$845$	0	0
$846$	0	0
$847$	27.2866	0.937579
$848$	21.6225	0.742520
$849$	0	0
$850$	0	0
$851$	−38.3809	−1.31568
$852$	0	0
$853$	−58.2023	−1.99281	−0.996404	−	0.0847260i	$-0.972998\pi$
$853$	−58.2023	−1.99281	−0.996404	+	0.0847260i	$0.972998\pi$
$854$	−73.0245	−2.49885
$855$	0	0
$856$	22.0909	0.755051
$857$	3.22045	0.110008	0.0550042	−	0.998486i	$-0.482483\pi$
$857$	3.22045	0.110008	0.0550042	+	0.998486i	$0.482483\pi$
$858$	0	0
$859$	49.4431	1.68698	0.843488	−	0.537147i	$-0.180498\pi$
$859$	49.4431	1.68698	0.843488	+	0.537147i	$0.180498\pi$
$860$	0	0
$861$	0	0
$862$	−30.3740	−1.03454
$863$	−21.4924	−0.731611	−0.365806	−	0.930691i	$-0.619207\pi$
$863$	−21.4924	−0.731611	−0.365806	+	0.930691i	$0.619207\pi$
$864$	0	0
$865$	0	0
$866$	1.18761	0.0403567
$867$	0	0
$868$	37.6544	1.27807
$869$	−2.48981	−0.0844609
$870$	0	0
$871$	9.95163	0.337198
$872$	−9.61090	−0.325466
$873$	0	0
$874$	−75.8026	−2.56406
$875$	0	0
$876$	0	0
$877$	15.3793	0.519321	0.259661	−	0.965700i	$-0.416389\pi$
$877$	15.3793	0.519321	0.259661	+	0.965700i	$0.416389\pi$
$878$	6.42501	0.216833
$879$	0	0
$880$	0	0
$881$	−18.9275	−0.637682	−0.318841	−	0.947808i	$-0.603294\pi$
$881$	−18.9275	−0.637682	−0.318841	+	0.947808i	$0.603294\pi$
$882$	0	0
$883$	6.31708	0.212587	0.106293	−	0.994335i	$-0.466102\pi$
$883$	6.31708	0.212587	0.106293	+	0.994335i	$0.466102\pi$
$884$	18.0492	0.607062
$885$	0	0
$886$	−79.2694	−2.66311
$887$	−0.0721586	−0.00242285	−0.00121142	−	0.999999i	$-0.500386\pi$
$887$	−0.0721586	−0.00242285	−0.00121142	+	0.999999i	$0.500386\pi$
$888$	0	0
$889$	−31.6639	−1.06197
$890$	0	0
$891$	0	0
$892$	−2.05853	−0.0689246
$893$	−4.10113	−0.137239
$894$	0	0
$895$	0	0
$896$	−34.4573	−1.15114
$897$	0	0
$898$	−38.5512	−1.28647
$899$	−10.5545	−0.352013
$900$	0	0
$901$	12.5840	0.419235
$902$	28.3584	0.944233
$903$	0	0
$904$	−12.9484	−0.430658
$905$	0	0
$906$	0	0
$907$	−25.4951	−0.846550	−0.423275	−	0.906001i	$-0.639120\pi$
$907$	−25.4951	−0.846550	−0.423275	+	0.906001i	$0.639120\pi$
$908$	−30.4991	−1.01215
$909$	0	0
$910$	0	0
$911$	−3.55803	−0.117883	−0.0589414	−	0.998261i	$-0.518773\pi$
$911$	−3.55803	−0.117883	−0.0589414	+	0.998261i	$0.518773\pi$
$912$	0	0
$913$	20.2275	0.669434
$914$	52.2389	1.72791
$915$	0	0
$916$	27.2798	0.901351
$917$	−20.1601	−0.665745
$918$	0	0
$919$	7.96968	0.262896	0.131448	−	0.991323i	$-0.458037\pi$
$919$	7.96968	0.262896	0.131448	+	0.991323i	$0.458037\pi$
$920$	0	0
$921$	0	0
$922$	−80.7776	−2.66027
$923$	−0.371499	−0.0122280
$924$	0	0
$925$	0	0
$926$	−44.1434	−1.45064
$927$	0	0
$928$	18.1480	0.595738
$929$	−48.5424	−1.59263	−0.796313	−	0.604885i	$-0.793219\pi$
$929$	−48.5424	−1.59263	−0.796313	+	0.604885i	$0.793219\pi$
$930$	0	0
$931$	16.2785	0.533505
$932$	60.0916	1.96837
$933$	0	0
$934$	67.4363	2.20658
$935$	0	0
$936$	0	0
$937$	31.7296	1.03656	0.518280	−	0.855211i	$-0.326573\pi$
$937$	31.7296	1.03656	0.518280	+	0.855211i	$0.326573\pi$
$938$	−12.9650	−0.423324
$939$	0	0
$940$	0	0
$941$	−41.0068	−1.33678	−0.668392	−	0.743809i	$-0.733017\pi$
$941$	−41.0068	−1.33678	−0.668392	+	0.743809i	$0.733017\pi$
$942$	0	0
$943$	54.9681	1.79001
$944$	9.12079	0.296856
$945$	0	0
$946$	−28.1991	−0.916833
$947$	9.81789	0.319038	0.159519	−	0.987195i	$-0.449006\pi$
$947$	9.81789	0.319038	0.159519	+	0.987195i	$0.449006\pi$
$948$	0	0
$949$	−21.0246	−0.682487
$950$	0	0
$951$	0	0
$952$	−5.88761	−0.190818
$953$	−53.1118	−1.72046	−0.860230	−	0.509906i	$-0.829680\pi$
$953$	−53.1118	−1.72046	−0.860230	+	0.509906i	$0.829680\pi$
$954$	0	0
$955$	0	0
$956$	1.57166	0.0508311
$957$	0	0
$958$	23.9756	0.774618
$959$	−29.0055	−0.936635
$960$	0	0
$961$	−11.0600	−0.356774
$962$	−67.4654	−2.17517
$963$	0	0
$964$	4.02702	0.129702
$965$	0	0
$966$	0	0
$967$	−2.44732	−0.0787004	−0.0393502	−	0.999225i	$-0.512529\pi$
$967$	−2.44732	−0.0787004	−0.0393502	+	0.999225i	$0.512529\pi$
$968$	12.4606	0.400500
$969$	0	0
$970$	0	0
$971$	47.9683	1.53938	0.769688	−	0.638421i	$-0.220412\pi$
$971$	47.9683	1.53938	0.769688	+	0.638421i	$0.220412\pi$
$972$	0	0
$973$	29.2838	0.938796
$974$	−15.4200	−0.494088
$975$	0	0
$976$	23.7161	0.759134
$977$	−23.6950	−0.758070	−0.379035	−	0.925382i	$-0.623744\pi$
$977$	−23.6950	−0.758070	−0.379035	+	0.925382i	$0.623744\pi$
$978$	0	0
$979$	11.1930	0.357730
$980$	0	0
$981$	0	0
$982$	15.6284	0.498721
$983$	−12.9505	−0.413057	−0.206529	−	0.978441i	$-0.566217\pi$
$983$	−12.9505	−0.413057	−0.206529	+	0.978441i	$0.566217\pi$
$984$	0	0
$985$	0	0
$986$	6.59116	0.209905
$987$	0	0
$988$	−76.1565	−2.42286
$989$	−54.6593	−1.73807
$990$	0	0
$991$	44.4967	1.41348	0.706742	−	0.707471i	$-0.250164\pi$
$991$	44.4967	1.41348	0.706742	+	0.707471i	$0.250164\pi$
$992$	−34.2859	−1.08858
$993$	0	0
$994$	0.483992	0.0153513
$995$	0	0
$996$	0	0
$997$	−55.5513	−1.75933	−0.879663	−	0.475597i	$-0.842232\pi$
$997$	−55.5513	−1.75933	−0.879663	+	0.475597i	$0.842232\pi$
$998$	−34.6054	−1.09542
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.p.1.2		6
3.2	odd	2		1875.2.a.j.1.5		6
5.4	even	2		5625.2.a.q.1.5		6
15.2	even	4		1875.2.b.f.1249.10		12
15.8	even	4		1875.2.b.f.1249.3		12
15.14	odd	2		1875.2.a.k.1.2		6
25.6	even	5		225.2.h.d.136.3		12
25.21	even	5		225.2.h.d.91.3		12
75.8	even	20		375.2.i.d.199.6		24
75.17	even	20		375.2.i.d.199.1		24
75.29	odd	10		375.2.g.c.76.3		12
75.44	odd	10		375.2.g.c.301.3		12
75.47	even	20		375.2.i.d.49.6		24
75.53	even	20		375.2.i.d.49.1		24
75.56	odd	10		75.2.g.c.61.1	yes	12
75.71	odd	10		75.2.g.c.16.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.g.c.16.1	✓	12	75.71	odd	10
75.2.g.c.61.1	yes	12	75.56	odd	10
225.2.h.d.91.3		12	25.21	even	5
225.2.h.d.136.3		12	25.6	even	5
375.2.g.c.76.3		12	75.29	odd	10
375.2.g.c.301.3		12	75.44	odd	10
375.2.i.d.49.1		24	75.53	even	20
375.2.i.d.49.6		24	75.47	even	20
375.2.i.d.199.1		24	75.17	even	20
375.2.i.d.199.6		24	75.8	even	20
1875.2.a.j.1.5		6	3.2	odd	2
1875.2.a.k.1.2		6	15.14	odd	2
1875.2.b.f.1249.3		12	15.8	even	4
1875.2.b.f.1249.10		12	15.2	even	4
5625.2.a.p.1.2		6	1.1	even	1	trivial
5625.2.a.q.1.5		6	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q-2.16056 q^{2} +2.66802 q^{4} -3.16056 q^{7} -1.44329 q^{8} +O(q^{10})\)	\(q-2.16056 q^{2} +2.66802 q^{4} -3.16056 q^{7} -1.44329 q^{8} -1.53835 q^{11} -5.24144 q^{13} +6.82858 q^{14} -2.21771 q^{16} -1.29068 q^{17} +5.44587 q^{19} +3.32370 q^{22} +6.44244 q^{23} +11.3244 q^{26} -8.43243 q^{28} +2.36361 q^{29} -4.46542 q^{31} +7.67809 q^{32} +2.78860 q^{34} -5.95751 q^{37} -11.7661 q^{38} +8.53219 q^{41} -8.48426 q^{43} -4.10435 q^{44} -13.9193 q^{46} -0.753070 q^{47} +2.98914 q^{49} -13.9843 q^{52} -9.74991 q^{53} +4.56162 q^{56} -5.10672 q^{58} -4.11270 q^{59} -10.6939 q^{61} +9.64781 q^{62} -12.1535 q^{64} -1.89864 q^{67} -3.44357 q^{68} +0.0708774 q^{71} +4.01123 q^{73} +12.8716 q^{74} +14.5297 q^{76} +4.86205 q^{77} +1.61849 q^{79} -18.4343 q^{82} -13.1488 q^{83} +18.3308 q^{86} +2.22029 q^{88} -7.27597 q^{89} +16.5659 q^{91} +17.1885 q^{92} +1.62705 q^{94} +10.1939 q^{97} -6.45821 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 10 q^{4} - 6 q^{7} + 3 q^{8}+O(q^{10}) \) 6 * q + 10 * q^4 - 6 * q^7 + 3 * q^8	\( 6 q + 10 q^{4} - 6 q^{7} + 3 q^{8} - 3 q^{11} - 6 q^{13} + 22 q^{14} + 18 q^{16} + 13 q^{17} + 11 q^{19} - 16 q^{22} + 13 q^{23} + 28 q^{26} - 7 q^{28} + 3 q^{29} - 11 q^{31} + 16 q^{32} + 15 q^{34} - 21 q^{37} - 9 q^{38} + q^{41} - 2 q^{43} - 9 q^{44} + 19 q^{46} + 14 q^{47} - 14 q^{49} - 13 q^{52} + 23 q^{53} + 35 q^{56} - 22 q^{58} - 9 q^{59} + 11 q^{61} - 23 q^{62} - 23 q^{64} - 8 q^{67} + 50 q^{68} + 8 q^{71} - 13 q^{73} + 22 q^{74} - 26 q^{76} - 13 q^{77} - 5 q^{79} + 13 q^{82} - 20 q^{83} + 37 q^{86} - 28 q^{88} + 4 q^{89} + 34 q^{91} + 61 q^{92} + 41 q^{94} + 7 q^{97} - 41 q^{98}+O(q^{100}) \) 6 * q + 10 * q^4 - 6 * q^7 + 3 * q^8 - 3 * q^11 - 6 * q^13 + 22 * q^14 + 18 * q^16 + 13 * q^17 + 11 * q^19 - 16 * q^22 + 13 * q^23 + 28 * q^26 - 7 * q^28 + 3 * q^29 - 11 * q^31 + 16 * q^32 + 15 * q^34 - 21 * q^37 - 9 * q^38 + q^41 - 2 * q^43 - 9 * q^44 + 19 * q^46 + 14 * q^47 - 14 * q^49 - 13 * q^52 + 23 * q^53 + 35 * q^56 - 22 * q^58 - 9 * q^59 + 11 * q^61 - 23 * q^62 - 23 * q^64 - 8 * q^67 + 50 * q^68 + 8 * q^71 - 13 * q^73 + 22 * q^74 - 26 * q^76 - 13 * q^77 - 5 * q^79 + 13 * q^82 - 20 * q^83 + 37 * q^86 - 28 * q^88 + 4 * q^89 + 34 * q^91 + 61 * q^92 + 41 * q^94 + 7 * q^97 - 41 * q^98

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