LMFDB - Embedded newform 2401.2.a.g.1.18

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2401,2,Mod(1,2401)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2401, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2401.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2401 = 7^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2401.a (trivial)

Newform invariants

comment:select newform

sage:traces = [18,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$19.1720815253$
Analytic rank:	$1$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 26x^{16} + 249x^{14} - 1126x^{12} + 2746x^{10} - 3811x^{8} + 2997x^{6} - 1246x^{4} + 224x^{2} - 7 $ x^18 - 26x^16 + 249x^14 - 1126x^12 + 2746x^10 - 3811x^8 + 2997x^6 - 1246x^4 + 224x^2 - 7
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Root			$-1.18059$ of defining polynomial
Character	$\chi$	$=$	2401.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.09443 q^{2} +1.18059 q^{3} +2.38663 q^{4} -2.10761 q^{5} +2.47265 q^{6} +0.809772 q^{8} -1.60622 q^{9} -4.41423 q^{10} -5.55856 q^{11} +2.81762 q^{12} -3.52854 q^{13} -2.48821 q^{15} -3.07725 q^{16} +5.15551 q^{17} -3.36411 q^{18} +6.10565 q^{19} -5.03008 q^{20} -11.6420 q^{22} -1.67491 q^{23} +0.956005 q^{24} -0.557991 q^{25} -7.39029 q^{26} -5.43803 q^{27} -4.45437 q^{29} -5.21138 q^{30} +3.10516 q^{31} -8.06463 q^{32} -6.56236 q^{33} +10.7978 q^{34} -3.83345 q^{36} -0.856461 q^{37} +12.7879 q^{38} -4.16575 q^{39} -1.70668 q^{40} -12.5027 q^{41} +8.41637 q^{43} -13.2662 q^{44} +3.38528 q^{45} -3.50798 q^{46} -3.41721 q^{47} -3.63296 q^{48} -1.16867 q^{50} +6.08652 q^{51} -8.42134 q^{52} -11.3474 q^{53} -11.3896 q^{54} +11.7153 q^{55} +7.20825 q^{57} -9.32935 q^{58} -5.61291 q^{59} -5.93844 q^{60} +4.91468 q^{61} +6.50353 q^{62} -10.7363 q^{64} +7.43679 q^{65} -13.7444 q^{66} +2.99498 q^{67} +12.3043 q^{68} -1.97738 q^{69} -4.06181 q^{71} -1.30067 q^{72} -0.0322509 q^{73} -1.79380 q^{74} -0.658756 q^{75} +14.5719 q^{76} -8.72487 q^{78} +3.33384 q^{79} +6.48564 q^{80} -1.60142 q^{81} -26.1861 q^{82} +3.27947 q^{83} -10.8658 q^{85} +17.6275 q^{86} -5.25876 q^{87} -4.50117 q^{88} +2.41677 q^{89} +7.09022 q^{90} -3.99740 q^{92} +3.66590 q^{93} -7.15709 q^{94} -12.8683 q^{95} -9.52099 q^{96} -5.66202 q^{97} +8.92826 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 8 q^{4} - 6 q^{8} - 2 q^{9} - 28 q^{11} - 34 q^{15} - 12 q^{16} + 2 q^{18} - 20 q^{22} - 42 q^{23} + 34 q^{25} - 6 q^{29} - 46 q^{30} - 12 q^{32} - 10 q^{36} - 70 q^{39} - 48 q^{43} - 94 q^{44} - 40 q^{46}+ \cdots - 6 q^{99}+O(q^{100}) $ 18 * q + 8 * q^4 - 6 * q^8 - 2 * q^9 - 28 * q^11 - 34 * q^15 - 12 * q^16 + 2 * q^18 - 20 * q^22 - 42 * q^23 + 34 * q^25 - 6 * q^29 - 46 * q^30 - 12 * q^32 - 10 * q^36 - 70 * q^39 - 48 * q^43 - 94 * q^44 - 40 * q^46 - 48 * q^50 + 24 * q^51 - 70 * q^53 + 36 * q^57 - 42 * q^58 - 58 * q^64 - 14 * q^65 - 14 * q^67 - 104 * q^71 + 2 * q^74 + 70 * q^78 - 28 * q^79 + 50 * q^81 - 6 * q^85 + 70 * q^86 + 16 * q^88 - 90 * q^92 - 60 * q^93 - 142 * q^95 - 6 * q^99

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 $

$2$	2.09443	1.48098	0.740492	−	0.672065i	$-0.234592\pi$
$2$	2.09443	1.48098	0.740492	+	0.672065i	$0.234592\pi$
$3$	1.18059	0.681612	0.340806	−	0.940134i	$-0.389300\pi$
$3$	1.18059	0.681612	0.340806	+	0.940134i	$0.389300\pi$
$4$	2.38663	1.19332
$5$	−2.10761	−0.942551	−0.471275	−	0.881986i	$-0.656206\pi$
$5$	−2.10761	−0.942551	−0.471275	+	0.881986i	$0.656206\pi$
$6$	2.47265	1.00946
$7$	0	0
$7$	0	0
$8$	0.809772	0.286298
$9$	−1.60622	−0.535406
$10$	−4.41423	−1.39590
$11$	−5.55856	−1.67597	−0.837985	−	0.545694i	$-0.816266\pi$
$11$	−5.55856	−1.67597	−0.837985	+	0.545694i	$0.816266\pi$
$12$	2.81762	0.813378
$13$	−3.52854	−0.978642	−0.489321	−	0.872104i	$-0.662755\pi$
$13$	−3.52854	−0.978642	−0.489321	+	0.872104i	$0.662755\pi$
$14$	0	0
$15$	−2.48821	−0.642453
$16$	−3.07725	−0.769313
$17$	5.15551	1.25039	0.625197	−	0.780467i	$-0.285018\pi$
$17$	5.15551	1.25039	0.625197	+	0.780467i	$0.285018\pi$
$18$	−3.36411	−0.792928
$19$	6.10565	1.40073	0.700367	−	0.713783i	$-0.253020\pi$
$19$	6.10565	1.40073	0.700367	+	0.713783i	$0.253020\pi$
$20$	−5.03008	−1.12476
$21$	0	0
$22$	−11.6420	−2.48208
$23$	−1.67491	−0.349243	−0.174622	−	0.984636i	$-0.555870\pi$
$23$	−1.67491	−0.349243	−0.174622	+	0.984636i	$0.555870\pi$
$24$	0.956005	0.195144
$25$	−0.557991	−0.111598
$26$	−7.39029	−1.44935
$27$	−5.43803	−1.04655
$28$	0	0
$29$	−4.45437	−0.827155	−0.413578	−	0.910469i	$-0.635721\pi$
$29$	−4.45437	−0.827155	−0.413578	+	0.910469i	$0.635721\pi$
$30$	−5.21138	−0.951464
$31$	3.10516	0.557703	0.278851	−	0.960334i	$-0.410046\pi$
$31$	3.10516	0.557703	0.278851	+	0.960334i	$0.410046\pi$
$32$	−8.06463	−1.42564
$33$	−6.56236	−1.14236
$34$	10.7978	1.85182
$35$	0	0
$36$	−3.83345	−0.638908
$37$	−0.856461	−0.140801	−0.0704007	−	0.997519i	$-0.522428\pi$
$37$	−0.856461	−0.140801	−0.0704007	+	0.997519i	$0.522428\pi$
$38$	12.7879	2.07446
$39$	−4.16575	−0.667054
$40$	−1.70668	−0.269850
$41$	−12.5027	−1.95260	−0.976299	−	0.216428i	$-0.930559\pi$
$41$	−12.5027	−1.95260	−0.976299	+	0.216428i	$0.930559\pi$
$42$	0	0
$43$	8.41637	1.28349	0.641743	−	0.766920i	$-0.278212\pi$
$43$	8.41637	1.28349	0.641743	+	0.766920i	$0.278212\pi$
$44$	−13.2662	−1.99996
$45$	3.38528	0.504647
$46$	−3.50798	−0.517224
$47$	−3.41721	−0.498451	−0.249225	−	0.968446i	$-0.580176\pi$
$47$	−3.41721	−0.498451	−0.249225	+	0.968446i	$0.580176\pi$
$48$	−3.63296	−0.524373
$49$	0	0
$50$	−1.16867	−0.165275
$51$	6.08652	0.852283
$52$	−8.42134	−1.16783
$53$	−11.3474	−1.55869	−0.779345	−	0.626595i	$-0.784448\pi$
$53$	−11.3474	−1.55869	−0.779345	+	0.626595i	$0.784448\pi$
$54$	−11.3896	−1.54992
$55$	11.7153	1.57969
$56$	0	0
$57$	7.20825	0.954756
$58$	−9.32935	−1.22500
$59$	−5.61291	−0.730739	−0.365369	−	0.930863i	$-0.619057\pi$
$59$	−5.61291	−0.730739	−0.365369	+	0.930863i	$0.619057\pi$
$60$	−5.93844	−0.766650
$61$	4.91468	0.629260	0.314630	−	0.949214i	$-0.398120\pi$
$61$	4.91468	0.629260	0.314630	+	0.949214i	$0.398120\pi$
$62$	6.50353	0.825949
$63$	0	0
$64$	−10.7363	−1.34204
$65$	7.43679	0.922420
$66$	−13.7444	−1.69182
$67$	2.99498	0.365896	0.182948	−	0.983123i	$-0.441436\pi$
$67$	2.99498	0.365896	0.182948	+	0.983123i	$0.441436\pi$
$68$	12.3043	1.49212
$69$	−1.97738	−0.238048
$70$	0	0
$71$	−4.06181	−0.482048	−0.241024	−	0.970519i	$-0.577483\pi$
$71$	−4.06181	−0.482048	−0.241024	+	0.970519i	$0.577483\pi$
$72$	−1.30067	−0.153285
$73$	−0.0322509	−0.00377468	−0.00188734	−	0.999998i	$-0.500601\pi$
$73$	−0.0322509	−0.00377468	−0.00188734	+	0.999998i	$0.500601\pi$
$74$	−1.79380	−0.208525
$75$	−0.658756	−0.0760666
$76$	14.5719	1.67152
$77$	0	0
$78$	−8.72487	−0.987897
$79$	3.33384	0.375087	0.187543	−	0.982256i	$-0.439948\pi$
$79$	3.33384	0.375087	0.187543	+	0.982256i	$0.439948\pi$
$80$	6.48564	0.725117
$81$	−1.60142	−0.177935
$82$	−26.1861	−2.89177
$83$	3.27947	0.359969	0.179984	−	0.983669i	$-0.442395\pi$
$83$	3.27947	0.359969	0.179984	+	0.983669i	$0.442395\pi$
$84$	0	0
$85$	−10.8658	−1.17856
$86$	17.6275	1.90082
$87$	−5.25876	−0.563798
$88$	−4.50117	−0.479826
$89$	2.41677	0.256177	0.128089	−	0.991763i	$-0.459116\pi$
$89$	2.41677	0.256177	0.128089	+	0.991763i	$0.459116\pi$
$90$	7.09022	0.747375
$91$	0	0
$92$	−3.99740	−0.416758
$93$	3.66590	0.380136
$94$	−7.15709	−0.738198
$95$	−12.8683	−1.32026
$96$	−9.52099	−0.971732
$97$	−5.66202	−0.574891	−0.287446	−	0.957797i	$-0.592806\pi$
$97$	−5.66202	−0.574891	−0.287446	+	0.957797i	$0.592806\pi$
$98$	0	0
$99$	8.92826	0.897324
$100$	−1.33172	−0.133172
$101$	−10.4928	−1.04407	−0.522037	−	0.852923i	$-0.674828\pi$
$101$	−10.4928	−1.04407	−0.522037	+	0.852923i	$0.674828\pi$
$102$	12.7478	1.26222
$103$	14.2767	1.40672	0.703361	−	0.710833i	$-0.251682\pi$
$103$	14.2767	1.40672	0.703361	+	0.710833i	$0.251682\pi$
$104$	−2.85732	−0.280183
$105$	0	0
$106$	−23.7664	−2.30840
$107$	0.389725	0.0376761	0.0188381	−	0.999823i	$-0.494003\pi$
$107$	0.389725	0.0376761	0.0188381	+	0.999823i	$0.494003\pi$
$108$	−12.9786	−1.24886
$109$	13.2936	1.27330	0.636649	−	0.771154i	$-0.280320\pi$
$109$	13.2936	1.27330	0.636649	+	0.771154i	$0.280320\pi$
$110$	24.5368	2.33949
$111$	−1.01113	−0.0959718
$112$	0	0
$113$	2.06284	0.194056	0.0970280	−	0.995282i	$-0.469066\pi$
$113$	2.06284	0.194056	0.0970280	+	0.995282i	$0.469066\pi$
$114$	15.0972	1.41398
$115$	3.53006	0.329180
$116$	−10.6309	−0.987057
$117$	5.66761	0.523971
$118$	−11.7558	−1.08221
$119$	0	0
$120$	−2.01488	−0.183933
$121$	19.8976	1.80887
$122$	10.2934	0.931925
$123$	−14.7605	−1.33091
$124$	7.41086	0.665515
$125$	11.7141	1.04774
$126$	0	0
$127$	−16.5136	−1.46535	−0.732674	−	0.680580i	$-0.761728\pi$
$127$	−16.5136	−1.46535	−0.732674	+	0.680580i	$0.761728\pi$
$128$	−6.35713	−0.561896
$129$	9.93625	0.874838
$130$	15.5758	1.36609
$131$	20.7437	1.81239	0.906194	−	0.422862i	$-0.138975\pi$
$131$	20.7437	1.81239	0.906194	+	0.422862i	$0.138975\pi$
$132$	−15.6619	−1.36320
$133$	0	0
$134$	6.27278	0.541886
$135$	11.4612	0.986427
$136$	4.17479	0.357985
$137$	6.92736	0.591844	0.295922	−	0.955212i	$-0.404373\pi$
$137$	6.92736	0.591844	0.295922	+	0.955212i	$0.404373\pi$
$138$	−4.14148	−0.352546
$139$	8.01600	0.679908	0.339954	−	0.940442i	$-0.389588\pi$
$139$	8.01600	0.679908	0.339954	+	0.940442i	$0.389588\pi$
$140$	0	0
$141$	−4.03431	−0.339750
$142$	−8.50717	−0.713906
$143$	19.6136	1.64017
$144$	4.94274	0.411895
$145$	9.38806	0.779636
$146$	−0.0675471	−0.00559024
$147$	0	0
$148$	−2.04406	−0.168020
$149$	17.2234	1.41099	0.705497	−	0.708713i	$-0.250724\pi$
$149$	17.2234	1.41099	0.705497	+	0.708713i	$0.250724\pi$
$150$	−1.37972	−0.112653
$151$	−15.7877	−1.28478	−0.642391	−	0.766377i	$-0.722057\pi$
$151$	−15.7877	−1.28478	−0.642391	+	0.766377i	$0.722057\pi$
$152$	4.94419	0.401027
$153$	−8.28087	−0.669468
$154$	0	0
$155$	−6.54445	−0.525663
$156$	−9.94211	−0.796006
$157$	−0.996829	−0.0795556	−0.0397778	−	0.999209i	$-0.512665\pi$
$157$	−0.996829	−0.0795556	−0.0397778	+	0.999209i	$0.512665\pi$
$158$	6.98249	0.555497
$159$	−13.3966	−1.06242
$160$	16.9971	1.34374
$161$	0	0
$162$	−3.35405	−0.263519
$163$	3.00571	0.235425	0.117713	−	0.993048i	$-0.462444\pi$
$163$	3.00571	0.235425	0.117713	+	0.993048i	$0.462444\pi$
$164$	−29.8394	−2.33006
$165$	13.8309	1.07673
$166$	6.86862	0.533108
$167$	19.7807	1.53068	0.765340	−	0.643627i	$-0.222571\pi$
$167$	19.7807	1.53068	0.765340	+	0.643627i	$0.222571\pi$
$168$	0	0
$169$	−0.549373	−0.0422594
$170$	−22.7576	−1.74543
$171$	−9.80701	−0.749961
$172$	20.0868	1.53160
$173$	4.92633	0.374542	0.187271	−	0.982308i	$-0.440036\pi$
$173$	4.92633	0.374542	0.187271	+	0.982308i	$0.440036\pi$
$174$	−11.0141	−0.834977
$175$	0	0
$176$	17.1051	1.28935
$177$	−6.62652	−0.498080
$178$	5.06175	0.379395
$179$	−4.19995	−0.313919	−0.156960	−	0.987605i	$-0.550169\pi$
$179$	−4.19995	−0.313919	−0.156960	+	0.987605i	$0.550169\pi$
$180$	8.07940	0.602203
$181$	−13.7539	−1.02232	−0.511158	−	0.859487i	$-0.670783\pi$
$181$	−13.7539	−1.02232	−0.511158	+	0.859487i	$0.670783\pi$
$182$	0	0
$183$	5.80220	0.428911
$184$	−1.35630	−0.0999876
$185$	1.80508	0.132712
$186$	7.67797	0.562976
$187$	−28.6572	−2.09562
$188$	−8.15561	−0.594809
$189$	0	0
$190$	−26.9518	−1.95529
$191$	−9.70361	−0.702129	−0.351064	−	0.936351i	$-0.614180\pi$
$191$	−9.70361	−0.702129	−0.351064	+	0.936351i	$0.614180\pi$
$192$	−12.6751	−0.914747
$193$	12.2965	0.885121	0.442561	−	0.896739i	$-0.354070\pi$
$193$	12.2965	0.885121	0.442561	+	0.896739i	$0.354070\pi$
$194$	−11.8587	−0.851405
$195$	8.77977	0.628732
$196$	0	0
$197$	−22.4084	−1.59654	−0.798268	−	0.602303i	$-0.794250\pi$
$197$	−22.4084	−1.59654	−0.798268	+	0.602303i	$0.794250\pi$
$198$	18.6996	1.32892
$199$	−2.37614	−0.168440	−0.0842200	−	0.996447i	$-0.526840\pi$
$199$	−2.37614	−0.168440	−0.0842200	+	0.996447i	$0.526840\pi$
$200$	−0.451845	−0.0319503
$201$	3.53584	0.249399
$202$	−21.9765	−1.54626
$203$	0	0
$204$	14.5263	1.01704
$205$	26.3508	1.84042
$206$	29.9015	2.08333
$207$	2.69027	0.186987
$208$	10.8582	0.752882
$209$	−33.9387	−2.34759
$210$	0	0
$211$	−3.08061	−0.212078	−0.106039	−	0.994362i	$-0.533817\pi$
$211$	−3.08061	−0.212078	−0.106039	+	0.994362i	$0.533817\pi$
$212$	−27.0821	−1.86001
$213$	−4.79531	−0.328570
$214$	0.816251	0.0557978
$215$	−17.7384	−1.20975
$216$	−4.40357	−0.299625
$217$	0	0
$218$	27.8425	1.88573
$219$	−0.0380749	−0.00257286
$220$	27.9600	1.88506
$221$	−18.1914	−1.22369
$222$	−2.11773	−0.142133
$223$	−22.1618	−1.48406	−0.742032	−	0.670364i	$-0.766138\pi$
$223$	−22.1618	−1.48406	−0.742032	+	0.670364i	$0.766138\pi$
$224$	0	0
$225$	0.896254	0.0597503
$226$	4.32048	0.287394
$227$	11.2660	0.747751	0.373875	−	0.927479i	$-0.378029\pi$
$227$	11.2660	0.747751	0.373875	+	0.927479i	$0.378029\pi$
$228$	17.2034	1.13933
$229$	−11.6378	−0.769048	−0.384524	−	0.923115i	$-0.625634\pi$
$229$	−11.6378	−0.769048	−0.384524	+	0.923115i	$0.625634\pi$
$230$	7.39345	0.487510
$231$	0	0
$232$	−3.60702	−0.236813
$233$	−16.0658	−1.05251	−0.526253	−	0.850328i	$-0.676403\pi$
$233$	−16.0658	−1.05251	−0.526253	+	0.850328i	$0.676403\pi$
$234$	11.8704	0.775992
$235$	7.20213	0.469815
$236$	−13.3959	−0.872002
$237$	3.93589	0.255663
$238$	0	0
$239$	−15.2125	−0.984014	−0.492007	−	0.870591i	$-0.663737\pi$
$239$	−15.2125	−0.984014	−0.492007	+	0.870591i	$0.663737\pi$
$240$	7.65686	0.494248
$241$	−19.8427	−1.27818	−0.639091	−	0.769131i	$-0.720689\pi$
$241$	−19.8427	−1.27818	−0.639091	+	0.769131i	$0.720689\pi$
$242$	41.6741	2.67891
$243$	14.4235	0.925268
$244$	11.7295	0.750906
$245$	0	0
$246$	−30.9149	−1.97106
$247$	−21.5441	−1.37082
$248$	2.51447	0.159669
$249$	3.87170	0.245359
$250$	24.5343	1.55168
$251$	1.73626	0.109592	0.0547958	−	0.998498i	$-0.482549\pi$
$251$	1.73626	0.109592	0.0547958	+	0.998498i	$0.482549\pi$
$252$	0	0
$253$	9.31010	0.585321
$254$	−34.5866	−2.17016
$255$	−12.8280	−0.803320
$256$	8.15802	0.509877
$257$	−13.7942	−0.860457	−0.430228	−	0.902720i	$-0.641567\pi$
$257$	−13.7942	−0.860457	−0.430228	+	0.902720i	$0.641567\pi$
$258$	20.8108	1.29562
$259$	0	0
$260$	17.7489	1.10074
$261$	7.15468	0.442864
$262$	43.4463	2.68412
$263$	−6.13125	−0.378069	−0.189035	−	0.981970i	$-0.560536\pi$
$263$	−6.13125	−0.378069	−0.189035	+	0.981970i	$0.560536\pi$
$264$	−5.31401	−0.327055
$265$	23.9159	1.46914
$266$	0	0
$267$	2.85321	0.174613
$268$	7.14792	0.436629
$269$	−22.3069	−1.36008	−0.680038	−	0.733177i	$-0.738037\pi$
$269$	−22.3069	−1.36008	−0.680038	+	0.733177i	$0.738037\pi$
$270$	24.0048	1.46088
$271$	9.92693	0.603018	0.301509	−	0.953463i	$-0.402510\pi$
$271$	9.92693	0.603018	0.301509	+	0.953463i	$0.402510\pi$
$272$	−15.8648	−0.961945
$273$	0	0
$274$	14.5089	0.876512
$275$	3.10162	0.187035
$276$	−4.71927	−0.284067
$277$	−15.8522	−0.952464	−0.476232	−	0.879320i	$-0.657998\pi$
$277$	−15.8522	−0.952464	−0.476232	+	0.879320i	$0.657998\pi$
$278$	16.7889	1.00693
$279$	−4.98756	−0.298597
$280$	0	0
$281$	−3.52496	−0.210281	−0.105141	−	0.994457i	$-0.533529\pi$
$281$	−3.52496	−0.210281	−0.105141	+	0.994457i	$0.533529\pi$
$282$	−8.44956	−0.503164
$283$	−6.02827	−0.358344	−0.179172	−	0.983818i	$-0.557342\pi$
$283$	−6.02827	−0.358344	−0.179172	+	0.983818i	$0.557342\pi$
$284$	−9.69404	−0.575236
$285$	−15.1922	−0.899906
$286$	41.0794	2.42907
$287$	0	0
$288$	12.9535	0.763295
$289$	9.57928	0.563487
$290$	19.6626	1.15463
$291$	−6.68451	−0.391853
$292$	−0.0769709	−0.00450438
$293$	24.1583	1.41134	0.705672	−	0.708539i	$-0.250645\pi$
$293$	24.1583	1.41134	0.705672	+	0.708539i	$0.250645\pi$
$294$	0	0
$295$	11.8298	0.688758
$296$	−0.693538	−0.0403111
$297$	30.2276	1.75399
$298$	36.0732	2.08966
$299$	5.91000	0.341784
$300$	−1.57221	−0.0907714
$301$	0	0
$302$	−33.0661	−1.90274
$303$	−12.3877	−0.711653
$304$	−18.7886	−1.07760
$305$	−10.3582	−0.593110
$306$	−17.3437	−0.991473
$307$	−26.0363	−1.48597	−0.742985	−	0.669308i	$-0.766591\pi$
$307$	−26.0363	−1.48597	−0.742985	+	0.669308i	$0.766591\pi$
$308$	0	0
$309$	16.8548	0.958838
$310$	−13.7069	−0.778499
$311$	0.282119	0.0159975	0.00799875	−	0.999968i	$-0.497454\pi$
$311$	0.282119	0.0159975	0.00799875	+	0.999968i	$0.497454\pi$
$312$	−3.37331	−0.190976
$313$	−21.8398	−1.23446	−0.617229	−	0.786783i	$-0.711745\pi$
$313$	−21.8398	−1.23446	−0.617229	+	0.786783i	$0.711745\pi$
$314$	−2.08779	−0.117821
$315$	0	0
$316$	7.95665	0.447597
$317$	−6.20916	−0.348741	−0.174370	−	0.984680i	$-0.555789\pi$
$317$	−6.20916	−0.348741	−0.174370	+	0.984680i	$0.555789\pi$
$318$	−28.0583	−1.57343
$319$	24.7599	1.38629
$320$	22.6279	1.26494
$321$	0.460104	0.0256805
$322$	0	0
$323$	31.4778	1.75147
$324$	−3.82199	−0.212333
$325$	1.96889	0.109215
$326$	6.29525	0.348662
$327$	15.6942	0.867894
$328$	−10.1244	−0.559024
$329$	0	0
$330$	28.9678	1.59462
$331$	5.59408	0.307479	0.153739	−	0.988111i	$-0.450868\pi$
$331$	5.59408	0.307479	0.153739	+	0.988111i	$0.450868\pi$
$332$	7.82689	0.429556
$333$	1.37566	0.0753859
$334$	41.4293	2.26691
$335$	−6.31225	−0.344875
$336$	0	0
$337$	−18.4785	−1.00659	−0.503295	−	0.864115i	$-0.667879\pi$
$337$	−18.4785	−1.00659	−0.503295	+	0.864115i	$0.667879\pi$
$338$	−1.15062	−0.0625856
$339$	2.43536	0.132271
$340$	−25.9326	−1.40639
$341$	−17.2602	−0.934692
$342$	−20.5401	−1.11068
$343$	0	0
$344$	6.81535	0.367459
$345$	4.16754	0.224373
$346$	10.3178	0.554691
$347$	−5.01410	−0.269171	−0.134585	−	0.990902i	$-0.542970\pi$
$347$	−5.01410	−0.269171	−0.134585	+	0.990902i	$0.542970\pi$
$348$	−12.5507	−0.672790
$349$	−0.785199	−0.0420307	−0.0210154	−	0.999779i	$-0.506690\pi$
$349$	−0.785199	−0.0420307	−0.0210154	+	0.999779i	$0.506690\pi$
$350$	0	0
$351$	19.1883	1.02420
$352$	44.8277	2.38933
$353$	−21.9676	−1.16922	−0.584610	−	0.811315i	$-0.698752\pi$
$353$	−21.9676	−1.16922	−0.584610	+	0.811315i	$0.698752\pi$
$354$	−13.8788	−0.737649
$355$	8.56070	0.454355
$356$	5.76794	0.305700
$357$	0	0
$358$	−8.79650	−0.464910
$359$	9.30124	0.490901	0.245450	−	0.969409i	$-0.421064\pi$
$359$	9.30124	0.490901	0.245450	+	0.969409i	$0.421064\pi$
$360$	2.74130	0.144479
$361$	18.2790	0.962053
$362$	−28.8065	−1.51403
$363$	23.4908	1.23295
$364$	0	0
$365$	0.0679721	0.00355782
$366$	12.1523	0.635211
$367$	7.20590	0.376145	0.188073	−	0.982155i	$-0.439776\pi$
$367$	7.20590	0.376145	0.188073	+	0.982155i	$0.439776\pi$
$368$	5.15413	0.268678
$369$	20.0821	1.04543
$370$	3.78062	0.196545
$371$	0	0
$372$	8.74916	0.453623
$373$	20.7892	1.07642	0.538211	−	0.842810i	$-0.319100\pi$
$373$	20.7892	1.07642	0.538211	+	0.842810i	$0.319100\pi$
$374$	−60.0205	−3.10359
$375$	13.8295	0.714150
$376$	−2.76716	−0.142705
$377$	15.7174	0.809489
$378$	0	0
$379$	−32.4958	−1.66920	−0.834598	−	0.550859i	$-0.814300\pi$
$379$	−32.4958	−1.66920	−0.834598	+	0.550859i	$0.814300\pi$
$380$	−30.7119	−1.57549
$381$	−19.4958	−0.998798
$382$	−20.3235	−1.03984
$383$	−0.609747	−0.0311566	−0.0155783	−	0.999879i	$-0.504959\pi$
$383$	−0.609747	−0.0311566	−0.0155783	+	0.999879i	$0.504959\pi$
$384$	−7.50514	−0.382995
$385$	0	0
$386$	25.7541	1.31085
$387$	−13.5185	−0.687185
$388$	−13.5132	−0.686027
$389$	−12.3414	−0.625734	−0.312867	−	0.949797i	$-0.601289\pi$
$389$	−12.3414	−0.625734	−0.312867	+	0.949797i	$0.601289\pi$
$390$	18.3886	0.931143
$391$	−8.63503	−0.436692
$392$	0	0
$393$	24.4897	1.23534
$394$	−46.9329	−2.36444
$395$	−7.02643	−0.353538
$396$	21.3085	1.07079
$397$	25.7656	1.29314	0.646569	−	0.762855i	$-0.276203\pi$
$397$	25.7656	1.29314	0.646569	+	0.762855i	$0.276203\pi$
$398$	−4.97665	−0.249457
$399$	0	0
$400$	1.71708	0.0858539
$401$	5.37781	0.268555	0.134277	−	0.990944i	$-0.457129\pi$
$401$	5.37781	0.268555	0.134277	+	0.990944i	$0.457129\pi$
$402$	7.40556	0.369356
$403$	−10.9567	−0.545791
$404$	−25.0425	−1.24591
$405$	3.37515	0.167713
$406$	0	0
$407$	4.76069	0.235979
$408$	4.92870	0.244007
$409$	33.2252	1.64288	0.821439	−	0.570296i	$-0.193172\pi$
$409$	33.2252	1.64288	0.821439	+	0.570296i	$0.193172\pi$
$410$	55.1899	2.72564
$411$	8.17834	0.403408
$412$	34.0732	1.67866
$413$	0	0
$414$	5.63458	0.276925
$415$	−6.91184	−0.339289
$416$	28.4564	1.39519
$417$	9.46357	0.463433
$418$	−71.0821	−3.47674
$419$	12.5948	0.615295	0.307647	−	0.951500i	$-0.400458\pi$
$419$	12.5948	0.615295	0.307647	+	0.951500i	$0.400458\pi$
$420$	0	0
$421$	−36.7360	−1.79040	−0.895202	−	0.445660i	$-0.852969\pi$
$421$	−36.7360	−1.79040	−0.895202	+	0.445660i	$0.852969\pi$
$422$	−6.45211	−0.314084
$423$	5.48878	0.266873
$424$	−9.18884	−0.446249
$425$	−2.87673	−0.139542
$426$	−10.0434	−0.486606
$427$	0	0
$428$	0.930130	0.0449595
$429$	23.1556	1.11796
$430$	−37.1518	−1.79162
$431$	26.0571	1.25513	0.627563	−	0.778565i	$-0.284052\pi$
$431$	26.0571	1.25513	0.627563	+	0.778565i	$0.284052\pi$
$432$	16.7342	0.805125
$433$	−16.1429	−0.775780	−0.387890	−	0.921706i	$-0.626796\pi$
$433$	−16.1429	−0.775780	−0.387890	+	0.921706i	$0.626796\pi$
$434$	0	0
$435$	11.0834	0.531409
$436$	31.7270	1.51945
$437$	−10.2264	−0.489197
$438$	−0.0797452	−0.00381037
$439$	−11.7198	−0.559353	−0.279677	−	0.960094i	$-0.590227\pi$
$439$	−11.7198	−0.559353	−0.279677	+	0.960094i	$0.590227\pi$
$440$	9.48670	0.452260
$441$	0	0
$442$	−38.1007	−1.81226
$443$	−22.3206	−1.06049	−0.530243	−	0.847846i	$-0.677899\pi$
$443$	−22.3206	−1.06049	−0.530243	+	0.847846i	$0.677899\pi$
$444$	−2.41319	−0.114525
$445$	−5.09361	−0.241460
$446$	−46.4163	−2.19788
$447$	20.3337	0.961750
$448$	0	0
$449$	6.54006	0.308644	0.154322	−	0.988021i	$-0.450681\pi$
$449$	6.54006	0.308644	0.154322	+	0.988021i	$0.450681\pi$
$450$	1.87714	0.0884892
$451$	69.4971	3.27249
$452$	4.92325	0.231570
$453$	−18.6387	−0.875722
$454$	23.5958	1.10741
$455$	0	0
$456$	5.83704	0.273344
$457$	−14.2532	−0.666735	−0.333368	−	0.942797i	$-0.608185\pi$
$457$	−14.2532	−0.666735	−0.333368	+	0.942797i	$0.608185\pi$
$458$	−24.3746	−1.13895
$459$	−28.0358	−1.30860
$460$	8.42495	0.392815
$461$	−38.4005	−1.78849	−0.894245	−	0.447577i	$-0.852287\pi$
$461$	−38.4005	−1.78849	−0.894245	+	0.447577i	$0.852287\pi$
$462$	0	0
$463$	−10.9006	−0.506593	−0.253296	−	0.967389i	$-0.581515\pi$
$463$	−10.9006	−0.506593	−0.253296	+	0.967389i	$0.581515\pi$
$464$	13.7072	0.636341
$465$	−7.72629	−0.358298
$466$	−33.6487	−1.55874
$467$	5.09348	0.235698	0.117849	−	0.993032i	$-0.462400\pi$
$467$	5.09348	0.235698	0.117849	+	0.993032i	$0.462400\pi$
$468$	13.5265	0.625262
$469$	0	0
$470$	15.0843	0.695789
$471$	−1.17684	−0.0542260
$472$	−4.54518	−0.209209
$473$	−46.7829	−2.15108
$474$	8.24343	0.378633
$475$	−3.40690	−0.156319
$476$	0	0
$477$	18.2264	0.834532
$478$	−31.8615	−1.45731
$479$	−4.50141	−0.205675	−0.102837	−	0.994698i	$-0.532792\pi$
$479$	−4.50141	−0.205675	−0.102837	+	0.994698i	$0.532792\pi$
$480$	20.0665	0.915907
$481$	3.02206	0.137794
$482$	−41.5591	−1.89297
$483$	0	0
$484$	47.4882	2.15856
$485$	11.9333	0.541864
$486$	30.2090	1.37031
$487$	−5.32549	−0.241321	−0.120660	−	0.992694i	$-0.538501\pi$
$487$	−5.32549	−0.241321	−0.120660	+	0.992694i	$0.538501\pi$
$488$	3.97977	0.180156
$489$	3.54850	0.160469
$490$	0	0
$491$	33.3244	1.50391	0.751954	−	0.659215i	$-0.229111\pi$
$491$	33.3244	1.50391	0.751954	+	0.659215i	$0.229111\pi$
$492$	−35.2280	−1.58820
$493$	−22.9645	−1.03427
$494$	−45.1225	−2.03016
$495$	−18.8173	−0.845773
$496$	−9.55535	−0.429048
$497$	0	0
$498$	8.10900	0.363373
$499$	2.67398	0.119704	0.0598520	−	0.998207i	$-0.480937\pi$
$499$	2.67398	0.119704	0.0598520	+	0.998207i	$0.480937\pi$
$500$	27.9572	1.25028
$501$	23.3529	1.04333
$502$	3.63646	0.162303
$503$	−10.9759	−0.489391	−0.244696	−	0.969600i	$-0.578688\pi$
$503$	−10.9759	−0.489391	−0.244696	+	0.969600i	$0.578688\pi$
$504$	0	0
$505$	22.1147	0.984093
$506$	19.4993	0.866852
$507$	−0.648581	−0.0288045
$508$	−39.4119	−1.74862
$509$	−6.30294	−0.279373	−0.139686	−	0.990196i	$-0.544609\pi$
$509$	−6.30294	−0.279373	−0.139686	+	0.990196i	$0.544609\pi$
$510$	−26.8673	−1.18971
$511$	0	0
$512$	29.8007	1.31702
$513$	−33.2028	−1.46594
$514$	−28.8909	−1.27432
$515$	−30.0896	−1.32591
$516$	23.7142	1.04396
$517$	18.9948	0.835388
$518$	0	0
$519$	5.81595	0.255292
$520$	6.02210	0.264087
$521$	−32.4862	−1.42324	−0.711622	−	0.702562i	$-0.752039\pi$
$521$	−32.4862	−1.42324	−0.711622	+	0.702562i	$0.752039\pi$
$522$	14.9850	0.655874
$523$	−31.2901	−1.36822	−0.684110	−	0.729379i	$-0.739809\pi$
$523$	−31.2901	−1.36822	−0.684110	+	0.729379i	$0.739809\pi$
$524$	49.5076	2.16275
$525$	0	0
$526$	−12.8415	−0.559915
$527$	16.0087	0.697348
$528$	20.1940	0.878833
$529$	−20.1947	−0.878029
$530$	50.0902	2.17578
$531$	9.01555	0.391242
$532$	0	0
$533$	44.1164	1.91089
$534$	5.97584	0.258600
$535$	−0.821388	−0.0355117
$536$	2.42525	0.104755
$537$	−4.95841	−0.213971
$538$	−46.7202	−2.01425
$539$	0	0
$540$	27.3538	1.17712
$541$	36.3847	1.56430	0.782150	−	0.623090i	$-0.214123\pi$
$541$	36.3847	1.56430	0.782150	+	0.623090i	$0.214123\pi$
$542$	20.7912	0.893061
$543$	−16.2376	−0.696822
$544$	−41.5773	−1.78261
$545$	−28.0177	−1.20015
$546$	0	0
$547$	16.9976	0.726767	0.363383	−	0.931640i	$-0.381622\pi$
$547$	16.9976	0.726767	0.363383	+	0.931640i	$0.381622\pi$
$548$	16.5331	0.706257
$549$	−7.89404	−0.336909
$550$	6.49613	0.276996
$551$	−27.1968	−1.15862
$552$	−1.60123	−0.0681527
$553$	0	0
$554$	−33.2012	−1.41058
$555$	2.13106	0.0904583
$556$	19.1312	0.811345
$557$	−13.7439	−0.582347	−0.291173	−	0.956670i	$-0.594046\pi$
$557$	−13.7439	−0.582347	−0.291173	+	0.956670i	$0.594046\pi$
$558$	−10.4461	−0.442218
$559$	−29.6976	−1.25607
$560$	0	0
$561$	−33.8323	−1.42840
$562$	−7.38277	−0.311423
$563$	−2.23872	−0.0943508	−0.0471754	−	0.998887i	$-0.515022\pi$
$563$	−2.23872	−0.0943508	−0.0471754	+	0.998887i	$0.515022\pi$
$564$	−9.62840	−0.405429
$565$	−4.34766	−0.182908
$566$	−12.6258	−0.530701
$567$	0	0
$568$	−3.28914	−0.138009
$569$	−5.57202	−0.233591	−0.116796	−	0.993156i	$-0.537262\pi$
$569$	−5.57202	−0.233591	−0.116796	+	0.993156i	$0.537262\pi$
$570$	−31.8189	−1.33275
$571$	30.5173	1.27711	0.638555	−	0.769576i	$-0.279532\pi$
$571$	30.5173	1.27711	0.638555	+	0.769576i	$0.279532\pi$
$572$	46.8105	1.95725
$573$	−11.4559	−0.478579
$574$	0	0
$575$	0.934585	0.0389749
$576$	17.2448	0.718534
$577$	−8.00757	−0.333359	−0.166680	−	0.986011i	$-0.553305\pi$
$577$	−8.00757	−0.333359	−0.166680	+	0.986011i	$0.553305\pi$
$578$	20.0631	0.834516
$579$	14.5171	0.603309
$580$	22.4058	0.930351
$581$	0	0
$582$	−14.0002	−0.580328
$583$	63.0754	2.61232
$584$	−0.0261158	−0.00108068
$585$	−11.9451	−0.493869
$586$	50.5979	2.09018
$587$	17.5467	0.724229	0.362114	−	0.932134i	$-0.382055\pi$
$587$	17.5467	0.724229	0.362114	+	0.932134i	$0.382055\pi$
$588$	0	0
$589$	18.9590	0.781192
$590$	24.7767	1.02004
$591$	−26.4551	−1.08822
$592$	2.63555	0.108320
$593$	−9.49850	−0.390057	−0.195028	−	0.980798i	$-0.562480\pi$
$593$	−9.49850	−0.390057	−0.195028	+	0.980798i	$0.562480\pi$
$594$	63.3097	2.59763
$595$	0	0
$596$	41.1059	1.68376
$597$	−2.80524	−0.114811
$598$	12.3781	0.506177
$599$	−17.8187	−0.728052	−0.364026	−	0.931389i	$-0.618598\pi$
$599$	−17.8187	−0.728052	−0.364026	+	0.931389i	$0.618598\pi$
$600$	−0.533442	−0.0217777
$601$	33.3698	1.36118	0.680592	−	0.732663i	$-0.261723\pi$
$601$	33.3698	1.36118	0.680592	+	0.732663i	$0.261723\pi$
$602$	0	0
$603$	−4.81059	−0.195903
$604$	−37.6793	−1.53315
$605$	−41.9363	−1.70495
$606$	−25.9451	−1.05395
$607$	−1.13437	−0.0460428	−0.0230214	−	0.999735i	$-0.507329\pi$
$607$	−1.13437	−0.0460428	−0.0230214	+	0.999735i	$0.507329\pi$
$608$	−49.2399	−1.99694
$609$	0	0
$610$	−21.6945	−0.878386
$611$	12.0578	0.487805
$612$	−19.7634	−0.798887
$613$	−29.6517	−1.19762	−0.598811	−	0.800891i	$-0.704360\pi$
$613$	−29.6517	−1.19762	−0.598811	+	0.800891i	$0.704360\pi$
$614$	−54.5312	−2.20070
$615$	31.1094	1.25445
$616$	0	0
$617$	−10.6733	−0.429691	−0.214846	−	0.976648i	$-0.568925\pi$
$617$	−10.6733	−0.429691	−0.214846	+	0.976648i	$0.568925\pi$
$618$	35.3013	1.42002
$619$	22.0552	0.886472	0.443236	−	0.896405i	$-0.353830\pi$
$619$	22.0552	0.886472	0.443236	+	0.896405i	$0.353830\pi$
$620$	−15.6192	−0.627282
$621$	9.10823	0.365501
$622$	0.590878	0.0236921
$623$	0	0
$624$	12.8191	0.513173
$625$	−21.8987	−0.875948
$626$	−45.7419	−1.82821
$627$	−40.0675	−1.60014
$628$	−2.37906	−0.0949349
$629$	−4.41549	−0.176057
$630$	0	0
$631$	−16.2924	−0.648591	−0.324295	−	0.945956i	$-0.605127\pi$
$631$	−16.2924	−0.648591	−0.324295	+	0.945956i	$0.605127\pi$
$632$	2.69965	0.107386
$633$	−3.63692	−0.144555
$634$	−13.0046	−0.516480
$635$	34.8042	1.38116
$636$	−31.9728	−1.26780
$637$	0	0
$638$	51.8578	2.05307
$639$	6.52415	0.258091
$640$	13.3983	0.529616
$641$	−11.1451	−0.440204	−0.220102	−	0.975477i	$-0.570639\pi$
$641$	−11.1451	−0.440204	−0.220102	+	0.975477i	$0.570639\pi$
$642$	0.963655	0.0380324
$643$	35.1654	1.38679	0.693395	−	0.720558i	$-0.256114\pi$
$643$	35.1654	1.38679	0.693395	+	0.720558i	$0.256114\pi$
$644$	0	0
$645$	−20.9417	−0.824579
$646$	65.9279	2.59390
$647$	17.7401	0.697436	0.348718	−	0.937228i	$-0.386617\pi$
$647$	17.7401	0.697436	0.348718	+	0.937228i	$0.386617\pi$
$648$	−1.29678	−0.0509424
$649$	31.1997	1.22470
$650$	4.12371	0.161745
$651$	0	0
$652$	7.17353	0.280937
$653$	−18.5516	−0.725980	−0.362990	−	0.931793i	$-0.618244\pi$
$653$	−18.5516	−0.725980	−0.362990	+	0.931793i	$0.618244\pi$
$654$	32.8705	1.28534
$655$	−43.7196	−1.70827
$656$	38.4740	1.50216
$657$	0.0518019	0.00202098
$658$	0	0
$659$	5.25401	0.204667	0.102334	−	0.994750i	$-0.467369\pi$
$659$	5.25401	0.204667	0.102334	+	0.994750i	$0.467369\pi$
$660$	33.0092	1.28488
$661$	33.7439	1.31249	0.656243	−	0.754549i	$-0.272144\pi$
$661$	33.7439	1.31249	0.656243	+	0.754549i	$0.272144\pi$
$662$	11.7164	0.455371
$663$	−21.4766	−0.834081
$664$	2.65563	0.103058
$665$	0	0
$666$	2.88123	0.111645
$667$	7.46067	0.288878
$668$	47.2093	1.82658
$669$	−26.1639	−1.01156
$670$	−13.2206	−0.510755
$671$	−27.3185	−1.05462
$672$	0	0
$673$	22.6013	0.871216	0.435608	−	0.900136i	$-0.356533\pi$
$673$	22.6013	0.871216	0.435608	+	0.900136i	$0.356533\pi$
$674$	−38.7020	−1.49074
$675$	3.03437	0.116793
$676$	−1.31115	−0.0504288
$677$	14.8936	0.572409	0.286205	−	0.958169i	$-0.407606\pi$
$677$	14.8936	0.572409	0.286205	+	0.958169i	$0.407606\pi$
$678$	5.10070	0.195891
$679$	0	0
$680$	−8.79881	−0.337419
$681$	13.3005	0.509676
$682$	−36.1503	−1.38427
$683$	−21.4490	−0.820725	−0.410363	−	0.911922i	$-0.634598\pi$
$683$	−21.4490	−0.820725	−0.410363	+	0.911922i	$0.634598\pi$
$684$	−23.4057	−0.894940
$685$	−14.6002	−0.557843
$686$	0	0
$687$	−13.7394	−0.524192
$688$	−25.8993	−0.987402
$689$	40.0399	1.52540
$690$	8.72861	0.332292
$691$	−7.51992	−0.286071	−0.143036	−	0.989718i	$-0.545686\pi$
$691$	−7.51992	−0.286071	−0.143036	+	0.989718i	$0.545686\pi$
$692$	11.7573	0.446947
$693$	0	0
$694$	−10.5017	−0.398638
$695$	−16.8946	−0.640848
$696$	−4.25840	−0.161414
$697$	−64.4579	−2.44152
$698$	−1.64454	−0.0622469
$699$	−18.9670	−0.717400
$700$	0	0
$701$	−19.0033	−0.717743	−0.358872	−	0.933387i	$-0.616838\pi$
$701$	−19.0033	−0.717743	−0.358872	+	0.933387i	$0.616838\pi$
$702$	40.1886	1.51682
$703$	−5.22926	−0.197225
$704$	59.6783	2.24921
$705$	8.50273	0.320231
$706$	−46.0097	−1.73160
$707$	0	0
$708$	−15.8151	−0.594367
$709$	45.9406	1.72534	0.862668	−	0.505771i	$-0.168792\pi$
$709$	45.9406	1.72534	0.862668	+	0.505771i	$0.168792\pi$
$710$	17.9298	0.672892
$711$	−5.35487	−0.200823
$712$	1.95703	0.0733429
$713$	−5.20086	−0.194774
$714$	0	0
$715$	−41.3378	−1.54595
$716$	−10.0237	−0.374605
$717$	−17.9597	−0.670715
$718$	19.4808	0.727016
$719$	−25.1688	−0.938637	−0.469319	−	0.883029i	$-0.655500\pi$
$719$	−25.1688	−0.938637	−0.469319	+	0.883029i	$0.655500\pi$
$720$	−10.4173	−0.388232
$721$	0	0
$722$	38.2841	1.42479
$723$	−23.4260	−0.871223
$724$	−32.8254	−1.21995
$725$	2.48549	0.0923089
$726$	49.1999	1.82598
$727$	−32.3491	−1.19976	−0.599882	−	0.800089i	$-0.704786\pi$
$727$	−32.3491	−1.19976	−0.599882	+	0.800089i	$0.704786\pi$
$728$	0	0
$729$	21.8324	0.808608
$730$	0.142363	0.00526908
$731$	43.3907	1.60486
$732$	13.8477	0.511826
$733$	17.4880	0.645935	0.322967	−	0.946410i	$-0.395320\pi$
$733$	17.4880	0.645935	0.322967	+	0.946410i	$0.395320\pi$
$734$	15.0923	0.557065
$735$	0	0
$736$	13.5075	0.497895
$737$	−16.6478	−0.613230
$738$	42.0605	1.54827
$739$	43.4102	1.59687	0.798436	−	0.602080i	$-0.205661\pi$
$739$	43.4102	1.59687	0.798436	+	0.602080i	$0.205661\pi$
$740$	4.30807	0.158368
$741$	−25.4346	−0.934364
$742$	0	0
$743$	34.6597	1.27154	0.635770	−	0.771879i	$-0.280683\pi$
$743$	34.6597	1.27154	0.635770	+	0.771879i	$0.280683\pi$
$744$	2.96855	0.108832
$745$	−36.3001	−1.32993
$746$	43.5414	1.59416
$747$	−5.26754	−0.192729
$748$	−68.3942	−2.50074
$749$	0	0
$750$	28.9648	1.05765
$751$	−22.4021	−0.817463	−0.408731	−	0.912655i	$-0.634029\pi$
$751$	−22.4021	−0.817463	−0.408731	+	0.912655i	$0.634029\pi$
$752$	10.5156	0.383465
$753$	2.04980	0.0746988
$754$	32.9190	1.19884
$755$	33.2742	1.21097
$756$	0	0
$757$	−7.70043	−0.279877	−0.139938	−	0.990160i	$-0.544690\pi$
$757$	−7.70043	−0.279877	−0.139938	+	0.990160i	$0.544690\pi$
$758$	−68.0601	−2.47205
$759$	10.9914	0.398962
$760$	−10.4204	−0.377988
$761$	32.5770	1.18092	0.590458	−	0.807068i	$-0.298947\pi$
$761$	32.5770	1.18092	0.590458	+	0.807068i	$0.298947\pi$
$762$	−40.8325	−1.47920
$763$	0	0
$764$	−23.1589	−0.837861
$765$	17.4528	0.631008
$766$	−1.27707	−0.0461425
$767$	19.8054	0.715132
$768$	9.63125	0.347538
$769$	30.7801	1.10996	0.554980	−	0.831864i	$-0.312726\pi$
$769$	30.7801	1.10996	0.554980	+	0.831864i	$0.312726\pi$
$770$	0	0
$771$	−16.2852	−0.586497
$772$	29.3472	1.05623
$773$	51.0528	1.83624	0.918122	−	0.396299i	$-0.129706\pi$
$773$	51.0528	1.83624	0.918122	+	0.396299i	$0.129706\pi$
$774$	−28.3136	−1.01771
$775$	−1.73265	−0.0622385
$776$	−4.58495	−0.164590
$777$	0	0
$778$	−25.8482	−0.926703
$779$	−76.3373	−2.73507
$780$	20.9541	0.750276
$781$	22.5778	0.807898
$782$	−18.0854	−0.646734
$783$	24.2230	0.865659
$784$	0	0
$785$	2.10092	0.0749852
$786$	51.2920	1.82953
$787$	37.7076	1.34413	0.672065	−	0.740492i	$-0.265408\pi$
$787$	37.7076	1.34413	0.672065	+	0.740492i	$0.265408\pi$
$788$	−53.4807	−1.90517
$789$	−7.23847	−0.257696
$790$	−14.7164	−0.523585
$791$	0	0
$792$	7.22985	0.256902
$793$	−17.3417	−0.615821
$794$	53.9642	1.91512
$795$	28.2348	1.00139
$796$	−5.67097	−0.201002
$797$	33.6182	1.19082	0.595408	−	0.803423i	$-0.296990\pi$
$797$	33.6182	1.19082	0.595408	+	0.803423i	$0.296990\pi$
$798$	0	0
$799$	−17.6174	−0.623260
$800$	4.49999	0.159099
$801$	−3.88186	−0.137159
$802$	11.2634	0.397726
$803$	0.179268	0.00632624
$804$	8.43874	0.297611
$805$	0	0
$806$	−22.9480	−0.808308
$807$	−26.3352	−0.927044
$808$	−8.49679	−0.298916
$809$	14.5627	0.511997	0.255999	−	0.966677i	$-0.417596\pi$
$809$	14.5627	0.511997	0.255999	+	0.966677i	$0.417596\pi$
$810$	7.06902	0.248380
$811$	31.0325	1.08970	0.544849	−	0.838534i	$-0.316587\pi$
$811$	31.0325	1.08970	0.544849	+	0.838534i	$0.316587\pi$
$812$	0	0
$813$	11.7196	0.411024
$814$	9.97093	0.349481
$815$	−6.33486	−0.221900
$816$	−18.7298	−0.655673
$817$	51.3875	1.79782
$818$	69.5877	2.43308
$819$	0	0
$820$	62.8897	2.19620
$821$	2.10999	0.0736393	0.0368197	−	0.999322i	$-0.488277\pi$
$821$	2.10999	0.0736393	0.0368197	+	0.999322i	$0.488277\pi$
$822$	17.1290	0.597441
$823$	−5.62312	−0.196010	−0.0980048	−	0.995186i	$-0.531246\pi$
$823$	−5.62312	−0.196010	−0.0980048	+	0.995186i	$0.531246\pi$
$824$	11.5609	0.402741
$825$	3.66173	0.127485
$826$	0	0
$827$	−4.45427	−0.154890	−0.0774451	−	0.996997i	$-0.524676\pi$
$827$	−4.45427	−0.154890	−0.0774451	+	0.996997i	$0.524676\pi$
$828$	6.42069	0.223134
$829$	−36.9782	−1.28431	−0.642153	−	0.766577i	$-0.721959\pi$
$829$	−36.9782	−1.28431	−0.642153	+	0.766577i	$0.721959\pi$
$830$	−14.4764	−0.502482
$831$	−18.7148	−0.649210
$832$	37.8835	1.31337
$833$	0	0
$834$	19.8208	0.686337
$835$	−41.6900	−1.44274
$836$	−80.9991	−2.80141
$837$	−16.8859	−0.583664
$838$	26.3788	0.911242
$839$	30.8364	1.06459	0.532296	−	0.846559i	$-0.321329\pi$
$839$	30.8364	1.06459	0.532296	+	0.846559i	$0.321329\pi$
$840$	0	0
$841$	−9.15862	−0.315815
$842$	−76.9410	−2.65156
$843$	−4.16151	−0.143330
$844$	−7.35227	−0.253076
$845$	1.15786	0.0398317
$846$	11.4958	0.395235
$847$	0	0
$848$	34.9189	1.19912
$849$	−7.11689	−0.244251
$850$	−6.02510	−0.206659
$851$	1.43450	0.0491739
$852$	−11.4446	−0.392087
$853$	−21.3913	−0.732423	−0.366211	−	0.930532i	$-0.619345\pi$
$853$	−21.3913	−0.732423	−0.366211	+	0.930532i	$0.619345\pi$
$854$	0	0
$855$	20.6693	0.706876
$856$	0.315589	0.0107866
$857$	−16.8080	−0.574151	−0.287075	−	0.957908i	$-0.592683\pi$
$857$	−16.8080	−0.574151	−0.287075	+	0.957908i	$0.592683\pi$
$858$	48.4977	1.65568
$859$	3.75848	0.128238	0.0641188	−	0.997942i	$-0.479576\pi$
$859$	3.75848	0.128238	0.0641188	+	0.997942i	$0.479576\pi$
$860$	−42.3351	−1.44361
$861$	0	0
$862$	54.5748	1.85882
$863$	−40.6969	−1.38534	−0.692669	−	0.721256i	$-0.743565\pi$
$863$	−40.6969	−1.38534	−0.692669	+	0.721256i	$0.743565\pi$
$864$	43.8557	1.49200
$865$	−10.3828	−0.353025
$866$	−33.8102	−1.14892
$867$	11.3092	0.384079
$868$	0	0
$869$	−18.5314	−0.628634
$870$	23.2134	0.787008
$871$	−10.5679	−0.358081
$872$	10.7648	0.364542
$873$	9.09444	0.307800
$874$	−21.4185	−0.724493
$875$	0	0
$876$	−0.0908708	−0.00307024
$877$	−30.0005	−1.01304	−0.506522	−	0.862227i	$-0.669069\pi$
$877$	−30.0005	−1.01304	−0.506522	+	0.862227i	$0.669069\pi$
$878$	−24.5462	−0.828394
$879$	28.5210	0.961988
$880$	−36.0508	−1.21527
$881$	19.8479	0.668691	0.334346	−	0.942451i	$-0.391485\pi$
$881$	19.8479	0.668691	0.334346	+	0.942451i	$0.391485\pi$
$882$	0	0
$883$	−26.1083	−0.878616	−0.439308	−	0.898337i	$-0.644776\pi$
$883$	−26.1083	−0.878616	−0.439308	+	0.898337i	$0.644776\pi$
$884$	−43.4163	−1.46025
$885$	13.9661	0.469466
$886$	−46.7490	−1.57056
$887$	−8.39540	−0.281890	−0.140945	−	0.990017i	$-0.545014\pi$
$887$	−8.39540	−0.281890	−0.140945	+	0.990017i	$0.545014\pi$
$888$	−0.818782	−0.0274765
$889$	0	0
$890$	−10.6682	−0.357599
$891$	8.90156	0.298214
$892$	−52.8921	−1.77096
$893$	−20.8643	−0.698197
$894$	42.5875	1.42434
$895$	8.85185	0.295885
$896$	0	0
$897$	6.97727	0.232964
$898$	13.6977	0.457098
$899$	−13.8315	−0.461306
$900$	2.13903	0.0713009
$901$	−58.5018	−1.94898
$902$	145.557	4.84651
$903$	0	0
$904$	1.67043	0.0555578
$905$	28.9877	0.963584
$906$	−39.0374	−1.29693
$907$	49.5208	1.64431	0.822156	−	0.569263i	$-0.192771\pi$
$907$	49.5208	1.64431	0.822156	+	0.569263i	$0.192771\pi$
$908$	26.8878	0.892303
$909$	16.8537	0.559004
$910$	0	0
$911$	−42.4966	−1.40797	−0.703987	−	0.710213i	$-0.748599\pi$
$911$	−42.4966	−1.40797	−0.703987	+	0.710213i	$0.748599\pi$
$912$	−22.1816	−0.734506
$913$	−18.2291	−0.603297
$914$	−29.8523	−0.987425
$915$	−12.2288	−0.404270
$916$	−27.7752	−0.917717
$917$	0	0
$918$	−58.7191	−1.93802
$919$	−45.0944	−1.48753	−0.743764	−	0.668442i	$-0.766961\pi$
$919$	−45.0944	−1.48753	−0.743764	+	0.668442i	$0.766961\pi$
$920$	2.85854	0.0942433
$921$	−30.7381	−1.01285
$922$	−80.4272	−2.64873
$923$	14.3323	0.471753
$924$	0	0
$925$	0.477897	0.0157132
$926$	−22.8305	−0.750256
$927$	−22.9314	−0.753167
$928$	35.9228	1.17922
$929$	23.4623	0.769773	0.384886	−	0.922964i	$-0.374241\pi$
$929$	23.4623	0.769773	0.384886	+	0.922964i	$0.374241\pi$
$930$	−16.1822	−0.530634
$931$	0	0
$932$	−38.3431	−1.25597
$933$	0.333066	0.0109041
$934$	10.6679	0.349065
$935$	60.3982	1.97523
$936$	4.58947	0.150012
$937$	−4.85255	−0.158526	−0.0792630	−	0.996854i	$-0.525257\pi$
$937$	−4.85255	−0.158526	−0.0792630	+	0.996854i	$0.525257\pi$
$938$	0	0
$939$	−25.7838	−0.841421
$940$	17.1888	0.560638
$941$	23.5647	0.768186	0.384093	−	0.923294i	$-0.374514\pi$
$941$	23.5647	0.768186	0.384093	+	0.923294i	$0.374514\pi$
$942$	−2.46481	−0.0803079
$943$	20.9410	0.681932
$944$	17.2723	0.562167
$945$	0	0
$946$	−97.9835	−3.18572
$947$	−59.1290	−1.92143	−0.960717	−	0.277529i	$-0.910485\pi$
$947$	−59.1290	−1.92143	−0.960717	+	0.277529i	$0.910485\pi$
$948$	9.39351	0.305087
$949$	0.113799	0.00369406
$950$	−7.13550	−0.231506
$951$	−7.33044	−0.237706
$952$	0	0
$953$	8.08491	0.261896	0.130948	−	0.991389i	$-0.458198\pi$
$953$	8.08491	0.261896	0.130948	+	0.991389i	$0.458198\pi$
$954$	38.1740	1.23593
$955$	20.4514	0.661792
$956$	−36.3066	−1.17424
$957$	29.2311	0.944909
$958$	−9.42788	−0.304601
$959$	0	0
$960$	26.7142	0.862196
$961$	−21.3580	−0.688968
$962$	6.32949	0.204071
$963$	−0.625983	−0.0201720
$964$	−47.3572	−1.52527
$965$	−25.9162	−0.834272
$966$	0	0
$967$	−57.1542	−1.83795	−0.918977	−	0.394311i	$-0.870983\pi$
$967$	−57.1542	−1.83795	−0.918977	+	0.394311i	$0.870983\pi$
$968$	16.1125	0.517876
$969$	37.1622	1.19382
$970$	24.9935	0.802493
$971$	−10.6006	−0.340189	−0.170095	−	0.985428i	$-0.554407\pi$
$971$	−10.6006	−0.340189	−0.170095	+	0.985428i	$0.554407\pi$
$972$	34.4236	1.10414
$973$	0	0
$974$	−11.1539	−0.357392
$975$	2.32445	0.0744419
$976$	−15.1237	−0.484098
$977$	60.6099	1.93908	0.969542	−	0.244927i	$-0.0787638\pi$
$977$	60.6099	1.93908	0.969542	+	0.244927i	$0.0787638\pi$
$978$	7.43208	0.237652
$979$	−13.4338	−0.429345
$980$	0	0
$981$	−21.3524	−0.681731
$982$	69.7956	2.22727
$983$	−43.7727	−1.39613	−0.698066	−	0.716033i	$-0.745956\pi$
$983$	−43.7727	−1.39613	−0.698066	+	0.716033i	$0.745956\pi$
$984$	−11.9527	−0.381037
$985$	47.2282	1.50482
$986$	−48.0976	−1.53174
$987$	0	0
$988$	−51.4178	−1.63582
$989$	−14.0967	−0.448249
$990$	−39.4114	−1.25258
$991$	53.0254	1.68441	0.842204	−	0.539159i	$-0.181258\pi$
$991$	53.0254	1.68441	0.842204	+	0.539159i	$0.181258\pi$
$992$	−25.0419	−0.795082
$993$	6.60430	0.209581
$994$	0	0
$995$	5.00797	0.158763
$996$	9.24032	0.292791
$997$	37.9920	1.20322	0.601609	−	0.798791i	$-0.294527\pi$
$997$	37.9920	1.20322	0.601609	+	0.798791i	$0.294527\pi$
$998$	5.60047	0.177280
$999$	4.65747	0.147356

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2401.2.a.g.1.18	yes	18
7.6	odd	2	inner	2401.2.a.g.1.17	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2401.2.a.g.1.17	✓	18	7.6	odd	2	inner
2401.2.a.g.1.18	yes	18	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2401 = 7^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2401.a (trivial)

\(f(q)\)	\(=\)	\(q+2.09443 q^{2} +1.18059 q^{3} +2.38663 q^{4} -2.10761 q^{5} +2.47265 q^{6} +0.809772 q^{8} -1.60622 q^{9} -4.41423 q^{10} -5.55856 q^{11} +2.81762 q^{12} -3.52854 q^{13} -2.48821 q^{15} -3.07725 q^{16} +5.15551 q^{17} -3.36411 q^{18} +6.10565 q^{19} -5.03008 q^{20} -11.6420 q^{22} -1.67491 q^{23} +0.956005 q^{24} -0.557991 q^{25} -7.39029 q^{26} -5.43803 q^{27} -4.45437 q^{29} -5.21138 q^{30} +3.10516 q^{31} -8.06463 q^{32} -6.56236 q^{33} +10.7978 q^{34} -3.83345 q^{36} -0.856461 q^{37} +12.7879 q^{38} -4.16575 q^{39} -1.70668 q^{40} -12.5027 q^{41} +8.41637 q^{43} -13.2662 q^{44} +3.38528 q^{45} -3.50798 q^{46} -3.41721 q^{47} -3.63296 q^{48} -1.16867 q^{50} +6.08652 q^{51} -8.42134 q^{52} -11.3474 q^{53} -11.3896 q^{54} +11.7153 q^{55} +7.20825 q^{57} -9.32935 q^{58} -5.61291 q^{59} -5.93844 q^{60} +4.91468 q^{61} +6.50353 q^{62} -10.7363 q^{64} +7.43679 q^{65} -13.7444 q^{66} +2.99498 q^{67} +12.3043 q^{68} -1.97738 q^{69} -4.06181 q^{71} -1.30067 q^{72} -0.0322509 q^{73} -1.79380 q^{74} -0.658756 q^{75} +14.5719 q^{76} -8.72487 q^{78} +3.33384 q^{79} +6.48564 q^{80} -1.60142 q^{81} -26.1861 q^{82} +3.27947 q^{83} -10.8658 q^{85} +17.6275 q^{86} -5.25876 q^{87} -4.50117 q^{88} +2.41677 q^{89} +7.09022 q^{90} -3.99740 q^{92} +3.66590 q^{93} -7.15709 q^{94} -12.8683 q^{95} -9.52099 q^{96} -5.66202 q^{97} +8.92826 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 8 q^{4} - 6 q^{8} - 2 q^{9} - 28 q^{11} - 34 q^{15} - 12 q^{16} + 2 q^{18} - 20 q^{22} - 42 q^{23} + 34 q^{25} - 6 q^{29} - 46 q^{30} - 12 q^{32} - 10 q^{36} - 70 q^{39} - 48 q^{43} - 94 q^{44} - 40 q^{46}+ \cdots - 6 q^{99}+O(q^{100}) \) 18 * q + 8 * q^4 - 6 * q^8 - 2 * q^9 - 28 * q^11 - 34 * q^15 - 12 * q^16 + 2 * q^18 - 20 * q^22 - 42 * q^23 + 34 * q^25 - 6 * q^29 - 46 * q^30 - 12 * q^32 - 10 * q^36 - 70 * q^39 - 48 * q^43 - 94 * q^44 - 40 * q^46 - 48 * q^50 + 24 * q^51 - 70 * q^53 + 36 * q^57 - 42 * q^58 - 58 * q^64 - 14 * q^65 - 14 * q^67 - 104 * q^71 + 2 * q^74 + 70 * q^78 - 28 * q^79 + 50 * q^81 - 6 * q^85 + 70 * q^86 + 16 * q^88 - 90 * q^92 - 60 * q^93 - 142 * q^95 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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