LMFDB - Embedded newform 6025.2.a.o.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6025,2,Mod(1,6025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6025, 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("6025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6025 = 5^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6025.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:	$48.1098672178$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	6025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.25059 q^{2} -0.826129 q^{3} +3.06516 q^{4} +1.85928 q^{6} +0.537315 q^{7} -2.39723 q^{8} -2.31751 q^{9} +O(q^{10})$	\(q-2.25059 q^{2} -0.826129 q^{3} +3.06516 q^{4} +1.85928 q^{6} +0.537315 q^{7} -2.39723 q^{8} -2.31751 q^{9} -6.19250 q^{11} -2.53222 q^{12} -4.65993 q^{13} -1.20928 q^{14} -0.735125 q^{16} +3.65398 q^{17} +5.21577 q^{18} +3.14664 q^{19} -0.443892 q^{21} +13.9368 q^{22} +1.86456 q^{23} +1.98042 q^{24} +10.4876 q^{26} +4.39295 q^{27} +1.64696 q^{28} -4.37534 q^{29} -3.31595 q^{31} +6.44893 q^{32} +5.11580 q^{33} -8.22362 q^{34} -7.10354 q^{36} -7.82228 q^{37} -7.08179 q^{38} +3.84970 q^{39} -3.29955 q^{41} +0.999019 q^{42} +5.43831 q^{43} -18.9810 q^{44} -4.19636 q^{46} -4.23654 q^{47} +0.607308 q^{48} -6.71129 q^{49} -3.01866 q^{51} -14.2834 q^{52} +1.19398 q^{53} -9.88673 q^{54} -1.28807 q^{56} -2.59953 q^{57} +9.84710 q^{58} +1.33544 q^{59} -5.94095 q^{61} +7.46285 q^{62} -1.24523 q^{63} -13.0437 q^{64} -11.5136 q^{66} -15.2850 q^{67} +11.2000 q^{68} -1.54037 q^{69} -9.28201 q^{71} +5.55561 q^{72} +3.58802 q^{73} +17.6048 q^{74} +9.64494 q^{76} -3.32732 q^{77} -8.66410 q^{78} -14.1677 q^{79} +3.32339 q^{81} +7.42594 q^{82} +6.68037 q^{83} -1.36060 q^{84} -12.2394 q^{86} +3.61460 q^{87} +14.8449 q^{88} -7.81264 q^{89} -2.50385 q^{91} +5.71517 q^{92} +2.73940 q^{93} +9.53471 q^{94} -5.32765 q^{96} +11.6841 q^{97} +15.1044 q^{98} +14.3512 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9}+O(q^{10}) $ 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9	$ 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9} + q^{11} + 14 q^{12} + 9 q^{13} - q^{14} + 43 q^{16} + 12 q^{17} + 42 q^{18} + 2 q^{21} + 5 q^{22} + 77 q^{23} - 2 q^{24} + 2 q^{26} + 38 q^{27} + 42 q^{28} + 2 q^{29} + q^{31} + 72 q^{32} + 20 q^{33} + 5 q^{34} + 32 q^{36} + 28 q^{37} + 23 q^{38} - 2 q^{39} - 2 q^{41} + 37 q^{42} + 31 q^{43} + 3 q^{44} + 14 q^{46} + 96 q^{47} + 13 q^{48} + 40 q^{49} - 10 q^{51} + 42 q^{52} + 54 q^{53} + 4 q^{54} - 15 q^{56} + 37 q^{57} + 27 q^{58} + q^{59} + 5 q^{61} + 39 q^{62} + 70 q^{63} + 65 q^{64} - 52 q^{66} + 34 q^{67} + 52 q^{68} + 21 q^{69} - 9 q^{71} + 70 q^{72} + 25 q^{73} + 22 q^{74} - 47 q^{76} + 54 q^{77} + 58 q^{78} + 13 q^{79} + 12 q^{81} - 5 q^{82} + 63 q^{83} + 95 q^{84} - 18 q^{86} + 47 q^{87} + 13 q^{88} + 19 q^{89} - 31 q^{91} + 137 q^{92} + 52 q^{93} + 120 q^{94} - 49 q^{96} + 36 q^{97} + 64 q^{98} + 16 q^{99}+O(q^{100}) $ 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9 + q^11 + 14 * q^12 + 9 * q^13 - q^14 + 43 * q^16 + 12 * q^17 + 42 * q^18 + 2 * q^21 + 5 * q^22 + 77 * q^23 - 2 * q^24 + 2 * q^26 + 38 * q^27 + 42 * q^28 + 2 * q^29 + q^31 + 72 * q^32 + 20 * q^33 + 5 * q^34 + 32 * q^36 + 28 * q^37 + 23 * q^38 - 2 * q^39 - 2 * q^41 + 37 * q^42 + 31 * q^43 + 3 * q^44 + 14 * q^46 + 96 * q^47 + 13 * q^48 + 40 * q^49 - 10 * q^51 + 42 * q^52 + 54 * q^53 + 4 * q^54 - 15 * q^56 + 37 * q^57 + 27 * q^58 + q^59 + 5 * q^61 + 39 * q^62 + 70 * q^63 + 65 * q^64 - 52 * q^66 + 34 * q^67 + 52 * q^68 + 21 * q^69 - 9 * q^71 + 70 * q^72 + 25 * q^73 + 22 * q^74 - 47 * q^76 + 54 * q^77 + 58 * q^78 + 13 * q^79 + 12 * q^81 - 5 * q^82 + 63 * q^83 + 95 * q^84 - 18 * q^86 + 47 * q^87 + 13 * q^88 + 19 * q^89 - 31 * q^91 + 137 * q^92 + 52 * q^93 + 120 * q^94 - 49 * q^96 + 36 * q^97 + 64 * q^98 + 16 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.25059	−1.59141	−0.795704	−	0.605686i	$-0.792899\pi$
$2$	−2.25059	−1.59141	−0.795704	+	0.605686i	$0.792899\pi$
$3$	−0.826129	−0.476966	−0.238483	−	0.971147i	$-0.576650\pi$
$3$	−0.826129	−0.476966	−0.238483	+	0.971147i	$0.576650\pi$
$4$	3.06516	1.53258
$5$	0	0
$5$	0	0
$6$	1.85928	0.759047
$7$	0.537315	0.203086	0.101543	−	0.994831i	$-0.467622\pi$
$7$	0.537315	0.203086	0.101543	+	0.994831i	$0.467622\pi$
$8$	−2.39723	−0.847550
$9$	−2.31751	−0.772504
$10$	0	0
$11$	−6.19250	−1.86711	−0.933554	−	0.358437i	$-0.883310\pi$
$11$	−6.19250	−1.86711	−0.933554	+	0.358437i	$0.883310\pi$
$12$	−2.53222	−0.730988
$13$	−4.65993	−1.29243	−0.646216	−	0.763155i	$-0.723649\pi$
$13$	−4.65993	−1.29243	−0.646216	+	0.763155i	$0.723649\pi$
$14$	−1.20928	−0.323193
$15$	0	0
$16$	−0.735125	−0.183781
$17$	3.65398	0.886221	0.443110	−	0.896467i	$-0.353875\pi$
$17$	3.65398	0.886221	0.443110	+	0.896467i	$0.353875\pi$
$18$	5.21577	1.22937
$19$	3.14664	0.721888	0.360944	−	0.932587i	$-0.382455\pi$
$19$	3.14664	0.721888	0.360944	+	0.932587i	$0.382455\pi$
$20$	0	0
$21$	−0.443892	−0.0968651
$22$	13.9368	2.97133
$23$	1.86456	0.388788	0.194394	−	0.980924i	$-0.437726\pi$
$23$	1.86456	0.388788	0.194394	+	0.980924i	$0.437726\pi$
$24$	1.98042	0.404252
$25$	0	0
$26$	10.4876	2.05679
$27$	4.39295	0.845424
$28$	1.64696	0.311245
$29$	−4.37534	−0.812480	−0.406240	−	0.913766i	$-0.633160\pi$
$29$	−4.37534	−0.812480	−0.406240	+	0.913766i	$0.633160\pi$
$30$	0	0
$31$	−3.31595	−0.595562	−0.297781	−	0.954634i	$-0.596247\pi$
$31$	−3.31595	−0.595562	−0.297781	+	0.954634i	$0.596247\pi$
$32$	6.44893	1.14002
$33$	5.11580	0.890547
$34$	−8.22362	−1.41034
$35$	0	0
$36$	−7.10354	−1.18392
$37$	−7.82228	−1.28598	−0.642988	−	0.765877i	$-0.722305\pi$
$37$	−7.82228	−1.28598	−0.642988	+	0.765877i	$0.722305\pi$
$38$	−7.08179	−1.14882
$39$	3.84970	0.616446
$40$	0	0
$41$	−3.29955	−0.515303	−0.257652	−	0.966238i	$-0.582949\pi$
$41$	−3.29955	−0.515303	−0.257652	+	0.966238i	$0.582949\pi$
$42$	0.999019	0.154152
$43$	5.43831	0.829334	0.414667	−	0.909973i	$-0.363898\pi$
$43$	5.43831	0.829334	0.414667	+	0.909973i	$0.363898\pi$
$44$	−18.9810	−2.86149
$45$	0	0
$46$	−4.19636	−0.618720
$47$	−4.23654	−0.617962	−0.308981	−	0.951068i	$-0.599988\pi$
$47$	−4.23654	−0.617962	−0.308981	+	0.951068i	$0.599988\pi$
$48$	0.607308	0.0876573
$49$	−6.71129	−0.958756
$50$	0	0
$51$	−3.01866	−0.422697
$52$	−14.2834	−1.98075
$53$	1.19398	0.164006	0.0820030	−	0.996632i	$-0.473868\pi$
$53$	1.19398	0.164006	0.0820030	+	0.996632i	$0.473868\pi$
$54$	−9.88673	−1.34541
$55$	0	0
$56$	−1.28807	−0.172126
$57$	−2.59953	−0.344316
$58$	9.84710	1.29299
$59$	1.33544	0.173860	0.0869298	−	0.996214i	$-0.472294\pi$
$59$	1.33544	0.173860	0.0869298	+	0.996214i	$0.472294\pi$
$60$	0	0
$61$	−5.94095	−0.760661	−0.380330	−	0.924851i	$-0.624190\pi$
$61$	−5.94095	−0.760661	−0.380330	+	0.924851i	$0.624190\pi$
$62$	7.46285	0.947783
$63$	−1.24523	−0.156885
$64$	−13.0437	−1.63046
$65$	0	0
$66$	−11.5136	−1.41722
$67$	−15.2850	−1.86737	−0.933683	−	0.358102i	$-0.883424\pi$
$67$	−15.2850	−1.86737	−0.933683	+	0.358102i	$0.883424\pi$
$68$	11.2000	1.35820
$69$	−1.54037	−0.185439
$70$	0	0
$71$	−9.28201	−1.10157	−0.550786	−	0.834646i	$-0.685672\pi$
$71$	−9.28201	−1.10157	−0.550786	+	0.834646i	$0.685672\pi$
$72$	5.55561	0.654735
$73$	3.58802	0.419947	0.209973	−	0.977707i	$-0.432662\pi$
$73$	3.58802	0.419947	0.209973	+	0.977707i	$0.432662\pi$
$74$	17.6048	2.04651
$75$	0	0
$76$	9.64494	1.10635
$77$	−3.32732	−0.379184
$78$	−8.66410	−0.981016
$79$	−14.1677	−1.59399	−0.796994	−	0.603987i	$-0.793578\pi$
$79$	−14.1677	−1.59399	−0.796994	+	0.603987i	$0.793578\pi$
$80$	0	0
$81$	3.32339	0.369265
$82$	7.42594	0.820057
$83$	6.68037	0.733266	0.366633	−	0.930366i	$-0.380510\pi$
$83$	6.68037	0.733266	0.366633	+	0.930366i	$0.380510\pi$
$84$	−1.36060	−0.148453
$85$	0	0
$86$	−12.2394	−1.31981
$87$	3.61460	0.387525
$88$	14.8449	1.58247
$89$	−7.81264	−0.828138	−0.414069	−	0.910245i	$-0.635893\pi$
$89$	−7.81264	−0.828138	−0.414069	+	0.910245i	$0.635893\pi$
$90$	0	0
$91$	−2.50385	−0.262475
$92$	5.71517	0.595848
$93$	2.73940	0.284063
$94$	9.53471	0.983430
$95$	0	0
$96$	−5.32765	−0.543751
$97$	11.6841	1.18634	0.593169	−	0.805078i	$-0.297877\pi$
$97$	11.6841	1.18634	0.593169	+	0.805078i	$0.297877\pi$
$98$	15.1044	1.52577
$99$	14.3512	1.44235
$100$	0	0
$101$	14.3776	1.43063	0.715314	−	0.698803i	$-0.246284\pi$
$101$	14.3776	1.43063	0.715314	+	0.698803i	$0.246284\pi$
$102$	6.79377	0.672683
$103$	−7.03032	−0.692718	−0.346359	−	0.938102i	$-0.612582\pi$
$103$	−7.03032	−0.692718	−0.346359	+	0.938102i	$0.612582\pi$
$104$	11.1709	1.09540
$105$	0	0
$106$	−2.68716	−0.261000
$107$	−11.0043	−1.06382	−0.531911	−	0.846800i	$-0.678526\pi$
$107$	−11.0043	−1.06382	−0.531911	+	0.846800i	$0.678526\pi$
$108$	13.4651	1.29568
$109$	−9.60580	−0.920068	−0.460034	−	0.887901i	$-0.652163\pi$
$109$	−9.60580	−0.920068	−0.460034	+	0.887901i	$0.652163\pi$
$110$	0	0
$111$	6.46221	0.613366
$112$	−0.394994	−0.0373234
$113$	8.31312	0.782032	0.391016	−	0.920384i	$-0.372124\pi$
$113$	8.31312	0.782032	0.391016	+	0.920384i	$0.372124\pi$
$114$	5.85048	0.547947
$115$	0	0
$116$	−13.4111	−1.24519
$117$	10.7994	0.998408
$118$	−3.00553	−0.276681
$119$	1.96334	0.179979
$120$	0	0
$121$	27.3470	2.48609
$122$	13.3706	1.21052
$123$	2.72585	0.245782
$124$	−10.1639	−0.912746
$125$	0	0
$126$	2.80251	0.249668
$127$	0.496281	0.0440378	0.0220189	−	0.999758i	$-0.492991\pi$
$127$	0.496281	0.0440378	0.0220189	+	0.999758i	$0.492991\pi$
$128$	16.4581	1.45470
$129$	−4.49274	−0.395564
$130$	0	0
$131$	−0.377470	−0.0329797	−0.0164898	−	0.999864i	$-0.505249\pi$
$131$	−0.377470	−0.0329797	−0.0164898	+	0.999864i	$0.505249\pi$
$132$	15.6807	1.36483
$133$	1.69074	0.146605
$134$	34.4004	2.97174
$135$	0	0
$136$	−8.75945	−0.751116
$137$	−7.89528	−0.674539	−0.337269	−	0.941408i	$-0.609503\pi$
$137$	−7.89528	−0.674539	−0.337269	+	0.941408i	$0.609503\pi$
$138$	3.46674	0.295108
$139$	−12.5173	−1.06171	−0.530854	−	0.847464i	$-0.678129\pi$
$139$	−12.5173	−1.06171	−0.530854	+	0.847464i	$0.678129\pi$
$140$	0	0
$141$	3.49993	0.294747
$142$	20.8900	1.75305
$143$	28.8566	2.41311
$144$	1.70366	0.141972
$145$	0	0
$146$	−8.07517	−0.668306
$147$	5.54439	0.457294
$148$	−23.9765	−1.97086
$149$	6.93533	0.568164	0.284082	−	0.958800i	$-0.408311\pi$
$149$	6.93533	0.568164	0.284082	+	0.958800i	$0.408311\pi$
$150$	0	0
$151$	5.49286	0.447003	0.223501	−	0.974704i	$-0.428251\pi$
$151$	5.49286	0.447003	0.223501	+	0.974704i	$0.428251\pi$
$152$	−7.54323	−0.611836
$153$	−8.46814	−0.684609
$154$	7.48844	0.603436
$155$	0	0
$156$	11.7999	0.944751
$157$	−18.5072	−1.47704	−0.738519	−	0.674232i	$-0.764475\pi$
$157$	−18.5072	−1.47704	−0.738519	+	0.674232i	$0.764475\pi$
$158$	31.8857	2.53669
$159$	−0.986383	−0.0782252
$160$	0	0
$161$	1.00186	0.0789574
$162$	−7.47959	−0.587652
$163$	3.50081	0.274205	0.137102	−	0.990557i	$-0.456221\pi$
$163$	3.50081	0.274205	0.137102	+	0.990557i	$0.456221\pi$
$164$	−10.1136	−0.789743
$165$	0	0
$166$	−15.0348	−1.16693
$167$	−18.0780	−1.39892	−0.699459	−	0.714673i	$-0.746575\pi$
$167$	−18.0780	−1.39892	−0.699459	+	0.714673i	$0.746575\pi$
$168$	1.06411	0.0820980
$169$	8.71492	0.670379
$170$	0	0
$171$	−7.29237	−0.557661
$172$	16.6693	1.27102
$173$	9.18528	0.698344	0.349172	−	0.937059i	$-0.386463\pi$
$173$	9.18528	0.698344	0.349172	+	0.937059i	$0.386463\pi$
$174$	−8.13498	−0.616711
$175$	0	0
$176$	4.55226	0.343139
$177$	−1.10325	−0.0829251
$178$	17.5831	1.31791
$179$	−9.65815	−0.721884	−0.360942	−	0.932588i	$-0.617545\pi$
$179$	−9.65815	−0.721884	−0.360942	+	0.932588i	$0.617545\pi$
$180$	0	0
$181$	−2.31978	−0.172428	−0.0862140	−	0.996277i	$-0.527477\pi$
$181$	−2.31978	−0.172428	−0.0862140	+	0.996277i	$0.527477\pi$
$182$	5.63514	0.417704
$183$	4.90799	0.362809
$184$	−4.46979	−0.329517
$185$	0	0
$186$	−6.16528	−0.452060
$187$	−22.6273	−1.65467
$188$	−12.9857	−0.947076
$189$	2.36040	0.171694
$190$	0	0
$191$	21.2836	1.54003	0.770015	−	0.638026i	$-0.220249\pi$
$191$	21.2836	1.54003	0.770015	+	0.638026i	$0.220249\pi$
$192$	10.7757	0.777672
$193$	0.929601	0.0669142	0.0334571	−	0.999440i	$-0.489348\pi$
$193$	0.929601	0.0669142	0.0334571	+	0.999440i	$0.489348\pi$
$194$	−26.2961	−1.88795
$195$	0	0
$196$	−20.5712	−1.46937
$197$	2.93516	0.209122	0.104561	−	0.994518i	$-0.466656\pi$
$197$	2.93516	0.209122	0.104561	+	0.994518i	$0.466656\pi$
$198$	−32.2986	−2.29536
$199$	2.86689	0.203228	0.101614	−	0.994824i	$-0.467599\pi$
$199$	2.86689	0.203228	0.101614	+	0.994824i	$0.467599\pi$
$200$	0	0
$201$	12.6274	0.890669
$202$	−32.3582	−2.27671
$203$	−2.35094	−0.165003
$204$	−9.25267	−0.647817
$205$	0	0
$206$	15.8224	1.10240
$207$	−4.32114	−0.300340
$208$	3.42563	0.237524
$209$	−19.4855	−1.34784
$210$	0	0
$211$	−19.7270	−1.35806	−0.679031	−	0.734110i	$-0.737600\pi$
$211$	−19.7270	−1.35806	−0.679031	+	0.734110i	$0.737600\pi$
$212$	3.65974	0.251352
$213$	7.66814	0.525412
$214$	24.7661	1.69298
$215$	0	0
$216$	−10.5309	−0.716539
$217$	−1.78171	−0.120950
$218$	21.6187	1.46420
$219$	−2.96417	−0.200300
$220$	0	0
$221$	−17.0273	−1.14538
$222$	−14.5438	−0.976116
$223$	−13.6199	−0.912055	−0.456027	−	0.889966i	$-0.650728\pi$
$223$	−13.6199	−0.912055	−0.456027	+	0.889966i	$0.650728\pi$
$224$	3.46511	0.231522
$225$	0	0
$226$	−18.7094	−1.24453
$227$	13.8400	0.918592	0.459296	−	0.888283i	$-0.348102\pi$
$227$	13.8400	0.918592	0.459296	+	0.888283i	$0.348102\pi$
$228$	−7.96797	−0.527691
$229$	5.32716	0.352028	0.176014	−	0.984388i	$-0.443680\pi$
$229$	5.32716	0.352028	0.176014	+	0.984388i	$0.443680\pi$
$230$	0	0
$231$	2.74880	0.180858
$232$	10.4887	0.688618
$233$	−11.6880	−0.765705	−0.382852	−	0.923810i	$-0.625058\pi$
$233$	−11.6880	−0.765705	−0.382852	+	0.923810i	$0.625058\pi$
$234$	−24.3051	−1.58887
$235$	0	0
$236$	4.09334	0.266453
$237$	11.7043	0.760278
$238$	−4.41868	−0.286420
$239$	5.22349	0.337879	0.168940	−	0.985626i	$-0.445966\pi$
$239$	5.22349	0.337879	0.168940	+	0.985626i	$0.445966\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	−61.5469	−3.95638
$243$	−15.9244	−1.02155
$244$	−18.2100	−1.16577
$245$	0	0
$246$	−6.13478	−0.391139
$247$	−14.6631	−0.932991
$248$	7.94911	0.504769
$249$	−5.51885	−0.349743
$250$	0	0
$251$	−27.8251	−1.75631	−0.878154	−	0.478378i	$-0.841225\pi$
$251$	−27.8251	−1.75631	−0.878154	+	0.478378i	$0.841225\pi$
$252$	−3.81684	−0.240438
$253$	−11.5463	−0.725909
$254$	−1.11693	−0.0700822
$255$	0	0
$256$	−10.9530	−0.684565
$257$	−5.15339	−0.321460	−0.160730	−	0.986998i	$-0.551385\pi$
$257$	−5.15339	−0.321460	−0.160730	+	0.986998i	$0.551385\pi$
$258$	10.1113	0.629503
$259$	−4.20303	−0.261164
$260$	0	0
$261$	10.1399	0.627644
$262$	0.849530	0.0524842
$263$	17.5989	1.08519	0.542596	−	0.839994i	$-0.317441\pi$
$263$	17.5989	1.08519	0.542596	+	0.839994i	$0.317441\pi$
$264$	−12.2638	−0.754783
$265$	0	0
$266$	−3.80516	−0.233309
$267$	6.45425	0.394994
$268$	−46.8511	−2.86188
$269$	0.504343	0.0307503	0.0153752	−	0.999882i	$-0.495106\pi$
$269$	0.504343	0.0307503	0.0153752	+	0.999882i	$0.495106\pi$
$270$	0	0
$271$	16.2581	0.987607	0.493803	−	0.869574i	$-0.335606\pi$
$271$	16.2581	0.987607	0.493803	+	0.869574i	$0.335606\pi$
$272$	−2.68613	−0.162871
$273$	2.06850	0.125192
$274$	17.7690	1.07347
$275$	0	0
$276$	−4.72147	−0.284199
$277$	−5.07050	−0.304657	−0.152329	−	0.988330i	$-0.548677\pi$
$277$	−5.07050	−0.304657	−0.152329	+	0.988330i	$0.548677\pi$
$278$	28.1714	1.68961
$279$	7.68475	0.460074
$280$	0	0
$281$	32.0241	1.91040	0.955200	−	0.295962i	$-0.0956401\pi$
$281$	32.0241	1.91040	0.955200	+	0.295962i	$0.0956401\pi$
$282$	−7.87690	−0.469063
$283$	23.3119	1.38575	0.692874	−	0.721059i	$-0.256344\pi$
$283$	23.3119	1.38575	0.692874	+	0.721059i	$0.256344\pi$
$284$	−28.4508	−1.68825
$285$	0	0
$286$	−64.9443	−3.84024
$287$	−1.77290	−0.104651
$288$	−14.9455	−0.880670
$289$	−3.64842	−0.214613
$290$	0	0
$291$	−9.65255	−0.565842
$292$	10.9979	0.643601
$293$	17.8206	1.04109	0.520544	−	0.853835i	$-0.325729\pi$
$293$	17.8206	1.04109	0.520544	+	0.853835i	$0.325729\pi$
$294$	−12.4782	−0.727741
$295$	0	0
$296$	18.7518	1.08993
$297$	−27.2033	−1.57850
$298$	−15.6086	−0.904181
$299$	−8.68872	−0.502482
$300$	0	0
$301$	2.92208	0.168426
$302$	−12.3622	−0.711364
$303$	−11.8778	−0.682361
$304$	−2.31317	−0.132669
$305$	0	0
$306$	19.0583	1.08949
$307$	12.2791	0.700807	0.350403	−	0.936599i	$-0.386045\pi$
$307$	12.2791	0.700807	0.350403	+	0.936599i	$0.386045\pi$
$308$	−10.1988	−0.581129
$309$	5.80795	0.330403
$310$	0	0
$311$	−4.51188	−0.255845	−0.127923	−	0.991784i	$-0.540831\pi$
$311$	−4.51188	−0.255845	−0.127923	+	0.991784i	$0.540831\pi$
$312$	−9.22863	−0.522468
$313$	−11.5065	−0.650388	−0.325194	−	0.945647i	$-0.605430\pi$
$313$	−11.5065	−0.650388	−0.325194	+	0.945647i	$0.605430\pi$
$314$	41.6522	2.35057
$315$	0	0
$316$	−43.4262	−2.44291
$317$	3.29598	0.185121	0.0925603	−	0.995707i	$-0.470495\pi$
$317$	3.29598	0.185121	0.0925603	+	0.995707i	$0.470495\pi$
$318$	2.21994	0.124488
$319$	27.0943	1.51699
$320$	0	0
$321$	9.09095	0.507407
$322$	−2.25477	−0.125653
$323$	11.4978	0.639752
$324$	10.1867	0.565928
$325$	0	0
$326$	−7.87889	−0.436372
$327$	7.93563	0.438841
$328$	7.90979	0.436745
$329$	−2.27636	−0.125500
$330$	0	0
$331$	11.9156	0.654941	0.327470	−	0.944861i	$-0.393804\pi$
$331$	11.9156	0.654941	0.327470	+	0.944861i	$0.393804\pi$
$332$	20.4764	1.12379
$333$	18.1282	0.993420
$334$	40.6862	2.22625
$335$	0	0
$336$	0.326316	0.0178020
$337$	−19.0018	−1.03509	−0.517546	−	0.855656i	$-0.673154\pi$
$337$	−19.0018	−1.03509	−0.517546	+	0.855656i	$0.673154\pi$
$338$	−19.6137	−1.06685
$339$	−6.86771	−0.373003
$340$	0	0
$341$	20.5340	1.11198
$342$	16.4121	0.887466
$343$	−7.36729	−0.397796
$344$	−13.0369	−0.702902
$345$	0	0
$346$	−20.6723	−1.11135
$347$	−3.64261	−0.195545	−0.0977727	−	0.995209i	$-0.531172\pi$
$347$	−3.64261	−0.195545	−0.0977727	+	0.995209i	$0.531172\pi$
$348$	11.0793	0.593913
$349$	26.4353	1.41505	0.707523	−	0.706690i	$-0.249812\pi$
$349$	26.4353	1.41505	0.707523	+	0.706690i	$0.249812\pi$
$350$	0	0
$351$	−20.4708	−1.09265
$352$	−39.9350	−2.12854
$353$	14.3105	0.761670	0.380835	−	0.924643i	$-0.375637\pi$
$353$	14.3105	0.761670	0.380835	+	0.924643i	$0.375637\pi$
$354$	2.48296	0.131968
$355$	0	0
$356$	−23.9470	−1.26919
$357$	−1.62197	−0.0858439
$358$	21.7365	1.14881
$359$	27.0008	1.42505	0.712523	−	0.701649i	$-0.247553\pi$
$359$	27.0008	1.42505	0.712523	+	0.701649i	$0.247553\pi$
$360$	0	0
$361$	−9.09867	−0.478877
$362$	5.22088	0.274403
$363$	−22.5922	−1.18578
$364$	−7.67470	−0.402263
$365$	0	0
$366$	−11.0459	−0.577377
$367$	18.2226	0.951209	0.475605	−	0.879659i	$-0.342229\pi$
$367$	18.2226	0.951209	0.475605	+	0.879659i	$0.342229\pi$
$368$	−1.37068	−0.0714519
$369$	7.64674	0.398074
$370$	0	0
$371$	0.641544	0.0333073
$372$	8.39670	0.435349
$373$	−32.0449	−1.65922	−0.829612	−	0.558340i	$-0.811438\pi$
$373$	−32.0449	−1.65922	−0.829612	+	0.558340i	$0.811438\pi$
$374$	50.9247	2.63325
$375$	0	0
$376$	10.1560	0.523754
$377$	20.3888	1.05008
$378$	−5.31229	−0.273235
$379$	11.0912	0.569719	0.284860	−	0.958569i	$-0.408053\pi$
$379$	11.0912	0.569719	0.284860	+	0.958569i	$0.408053\pi$
$380$	0	0
$381$	−0.409992	−0.0210045
$382$	−47.9007	−2.45082
$383$	17.4076	0.889489	0.444744	−	0.895658i	$-0.353294\pi$
$383$	17.4076	0.889489	0.444744	+	0.895658i	$0.353294\pi$
$384$	−13.5965	−0.693843
$385$	0	0
$386$	−2.09215	−0.106488
$387$	−12.6033	−0.640663
$388$	35.8135	1.81816
$389$	10.5174	0.533253	0.266626	−	0.963800i	$-0.414091\pi$
$389$	10.5174	0.533253	0.266626	+	0.963800i	$0.414091\pi$
$390$	0	0
$391$	6.81307	0.344552
$392$	16.0885	0.812594
$393$	0.311839	0.0157302
$394$	−6.60585	−0.332798
$395$	0	0
$396$	43.9886	2.21051
$397$	27.8491	1.39771	0.698854	−	0.715264i	$-0.253694\pi$
$397$	27.8491	1.39771	0.698854	+	0.715264i	$0.253694\pi$
$398$	−6.45219	−0.323419
$399$	−1.39677	−0.0699258
$400$	0	0
$401$	−29.9750	−1.49688	−0.748441	−	0.663202i	$-0.769197\pi$
$401$	−29.9750	−1.49688	−0.748441	+	0.663202i	$0.769197\pi$
$402$	−28.4191	−1.41742
$403$	15.4521	0.769724
$404$	44.0697	2.19255
$405$	0	0
$406$	5.29100	0.262588
$407$	48.4394	2.40105
$408$	7.23643	0.358257
$409$	−4.72474	−0.233624	−0.116812	−	0.993154i	$-0.537267\pi$
$409$	−4.72474	−0.233624	−0.116812	+	0.993154i	$0.537267\pi$
$410$	0	0
$411$	6.52252	0.321732
$412$	−21.5490	−1.06164
$413$	0.717553	0.0353085
$414$	9.72512	0.477963
$415$	0	0
$416$	−30.0515	−1.47340
$417$	10.3409	0.506398
$418$	43.8540	2.14497
$419$	−23.0045	−1.12384	−0.561921	−	0.827191i	$-0.689937\pi$
$419$	−23.0045	−1.12384	−0.561921	+	0.827191i	$0.689937\pi$
$420$	0	0
$421$	5.50816	0.268451	0.134226	−	0.990951i	$-0.457145\pi$
$421$	5.50816	0.268451	0.134226	+	0.990951i	$0.457145\pi$
$422$	44.3974	2.16123
$423$	9.81822	0.477378
$424$	−2.86225	−0.139003
$425$	0	0
$426$	−17.2578	−0.836145
$427$	−3.19216	−0.154480
$428$	−33.7298	−1.63039
$429$	−23.8393	−1.15097
$430$	0	0
$431$	−12.1577	−0.585615	−0.292807	−	0.956171i	$-0.594589\pi$
$431$	−12.1577	−0.585615	−0.292807	+	0.956171i	$0.594589\pi$
$432$	−3.22937	−0.155373
$433$	−25.4442	−1.22277	−0.611386	−	0.791333i	$-0.709388\pi$
$433$	−25.4442	−1.22277	−0.611386	+	0.791333i	$0.709388\pi$
$434$	4.00990	0.192481
$435$	0	0
$436$	−29.4433	−1.41008
$437$	5.86710	0.280661
$438$	6.67114	0.318759
$439$	32.0488	1.52961	0.764803	−	0.644264i	$-0.222836\pi$
$439$	32.0488	1.52961	0.764803	+	0.644264i	$0.222836\pi$
$440$	0	0
$441$	15.5535	0.740642
$442$	38.3215	1.82277
$443$	8.30472	0.394569	0.197285	−	0.980346i	$-0.436788\pi$
$443$	8.30472	0.394569	0.197285	+	0.980346i	$0.436788\pi$
$444$	19.8077	0.940032
$445$	0	0
$446$	30.6528	1.45145
$447$	−5.72948	−0.270995
$448$	−7.00855	−0.331123
$449$	26.7239	1.26118	0.630589	−	0.776117i	$-0.282813\pi$
$449$	26.7239	1.26118	0.630589	+	0.776117i	$0.282813\pi$
$450$	0	0
$451$	20.4324	0.962126
$452$	25.4810	1.19853
$453$	−4.53781	−0.213205
$454$	−31.1481	−1.46185
$455$	0	0
$456$	6.23168	0.291825
$457$	−32.3755	−1.51446	−0.757232	−	0.653146i	$-0.773449\pi$
$457$	−32.3755	−1.51446	−0.757232	+	0.653146i	$0.773449\pi$
$458$	−11.9892	−0.560221
$459$	16.0518	0.749232
$460$	0	0
$461$	31.4613	1.46530	0.732649	−	0.680607i	$-0.238284\pi$
$461$	31.4613	1.46530	0.732649	+	0.680607i	$0.238284\pi$
$462$	−6.18642	−0.287818
$463$	−20.2858	−0.942759	−0.471380	−	0.881930i	$-0.656244\pi$
$463$	−20.2858	−0.942759	−0.471380	+	0.881930i	$0.656244\pi$
$464$	3.21642	0.149319
$465$	0	0
$466$	26.3048	1.21855
$467$	16.7835	0.776649	0.388325	−	0.921523i	$-0.373054\pi$
$467$	16.7835	0.776649	0.388325	+	0.921523i	$0.373054\pi$
$468$	33.1020	1.53014
$469$	−8.21289	−0.379236
$470$	0	0
$471$	15.2894	0.704497
$472$	−3.20136	−0.147355
$473$	−33.6767	−1.54846
$474$	−26.3417	−1.20991
$475$	0	0
$476$	6.01795	0.275832
$477$	−2.76706	−0.126695
$478$	−11.7559	−0.537704
$479$	30.8974	1.41174	0.705870	−	0.708342i	$-0.250556\pi$
$479$	30.8974	1.41174	0.705870	+	0.708342i	$0.250556\pi$
$480$	0	0
$481$	36.4513	1.66203
$482$	2.25059	0.102512
$483$	−0.827663	−0.0376600
$484$	83.8229	3.81013
$485$	0	0
$486$	35.8393	1.62570
$487$	−21.2896	−0.964726	−0.482363	−	0.875972i	$-0.660221\pi$
$487$	−21.2896	−0.964726	−0.482363	+	0.875972i	$0.660221\pi$
$488$	14.2418	0.644698
$489$	−2.89212	−0.130786
$490$	0	0
$491$	−29.6623	−1.33864	−0.669320	−	0.742974i	$-0.733415\pi$
$491$	−29.6623	−1.33864	−0.669320	+	0.742974i	$0.733415\pi$
$492$	8.35517	0.376680
$493$	−15.9874	−0.720037
$494$	33.0006	1.48477
$495$	0	0
$496$	2.43764	0.109453
$497$	−4.98737	−0.223714
$498$	12.4207	0.556584
$499$	−2.88071	−0.128958	−0.0644790	−	0.997919i	$-0.520539\pi$
$499$	−2.88071	−0.128958	−0.0644790	+	0.997919i	$0.520539\pi$
$500$	0	0
$501$	14.9348	0.667236
$502$	62.6230	2.79500
$503$	33.5275	1.49492	0.747458	−	0.664309i	$-0.231274\pi$
$503$	33.5275	1.49492	0.747458	+	0.664309i	$0.231274\pi$
$504$	2.98512	0.132968
$505$	0	0
$506$	25.9860	1.15522
$507$	−7.19965	−0.319748
$508$	1.52118	0.0674915
$509$	−34.9794	−1.55043	−0.775217	−	0.631695i	$-0.782360\pi$
$509$	−34.9794	−1.55043	−0.775217	+	0.631695i	$0.782360\pi$
$510$	0	0
$511$	1.92790	0.0852853
$512$	−8.26530	−0.365278
$513$	13.8230	0.610301
$514$	11.5982	0.511573
$515$	0	0
$516$	−13.7710	−0.606233
$517$	26.2347	1.15380
$518$	9.45930	0.415618
$519$	−7.58823	−0.333086
$520$	0	0
$521$	−25.8795	−1.13380	−0.566900	−	0.823786i	$-0.691858\pi$
$521$	−25.8795	−1.13380	−0.566900	+	0.823786i	$0.691858\pi$
$522$	−22.8208	−0.998838
$523$	−7.49693	−0.327818	−0.163909	−	0.986475i	$-0.552410\pi$
$523$	−7.49693	−0.327818	−0.163909	+	0.986475i	$0.552410\pi$
$524$	−1.15700	−0.0505440
$525$	0	0
$526$	−39.6078	−1.72698
$527$	−12.1164	−0.527800
$528$	−3.76075	−0.163666
$529$	−19.5234	−0.848844
$530$	0	0
$531$	−3.09490	−0.134307
$532$	5.18237	0.224684
$533$	15.3757	0.665994
$534$	−14.5259	−0.628596
$535$	0	0
$536$	36.6418	1.58269
$537$	7.97888	0.344314
$538$	−1.13507	−0.0489363
$539$	41.5596	1.79010
$540$	0	0
$541$	−11.7252	−0.504108	−0.252054	−	0.967713i	$-0.581106\pi$
$541$	−11.7252	−0.504108	−0.252054	+	0.967713i	$0.581106\pi$
$542$	−36.5902	−1.57168
$543$	1.91644	0.0822423
$544$	23.5643	1.01031
$545$	0	0
$546$	−4.65535	−0.199231
$547$	31.2836	1.33759	0.668796	−	0.743446i	$-0.266810\pi$
$547$	31.2836	1.33759	0.668796	+	0.743446i	$0.266810\pi$
$548$	−24.2003	−1.03378
$549$	13.7682	0.587613
$550$	0	0
$551$	−13.7676	−0.586520
$552$	3.69262	0.157168
$553$	−7.61251	−0.323717
$554$	11.4116	0.484834
$555$	0	0
$556$	−38.3676	−1.62715
$557$	−32.1850	−1.36372	−0.681860	−	0.731482i	$-0.738829\pi$
$557$	−32.1850	−1.36372	−0.681860	+	0.731482i	$0.738829\pi$
$558$	−17.2952	−0.732166
$559$	−25.3421	−1.07186
$560$	0	0
$561$	18.6930	0.789221
$562$	−72.0732	−3.04022
$563$	23.6954	0.998641	0.499320	−	0.866417i	$-0.333583\pi$
$563$	23.6954	0.998641	0.499320	+	0.866417i	$0.333583\pi$
$564$	10.7278	0.451723
$565$	0	0
$566$	−52.4655	−2.20529
$567$	1.78571	0.0749927
$568$	22.2512	0.933638
$569$	−33.7300	−1.41403	−0.707017	−	0.707197i	$-0.749959\pi$
$569$	−33.7300	−1.41403	−0.707017	+	0.707197i	$0.749959\pi$
$570$	0	0
$571$	−8.74903	−0.366135	−0.183068	−	0.983100i	$-0.558603\pi$
$571$	−8.74903	−0.366135	−0.183068	+	0.983100i	$0.558603\pi$
$572$	88.4500	3.69828
$573$	−17.5830	−0.734542
$574$	3.99007	0.166542
$575$	0	0
$576$	30.2288	1.25953
$577$	14.4483	0.601490	0.300745	−	0.953705i	$-0.402765\pi$
$577$	14.4483	0.601490	0.300745	+	0.953705i	$0.402765\pi$
$578$	8.21109	0.341536
$579$	−0.767971	−0.0319158
$580$	0	0
$581$	3.58947	0.148916
$582$	21.7239	0.900486
$583$	−7.39372	−0.306217
$584$	−8.60133	−0.355926
$585$	0	0
$586$	−40.1068	−1.65680
$587$	21.8146	0.900384	0.450192	−	0.892932i	$-0.351356\pi$
$587$	21.8146	0.900384	0.450192	+	0.892932i	$0.351356\pi$
$588$	16.9944	0.700839
$589$	−10.4341	−0.429929
$590$	0	0
$591$	−2.42482	−0.0997439
$592$	5.75035	0.236338
$593$	26.1209	1.07266	0.536329	−	0.844009i	$-0.319811\pi$
$593$	26.1209	1.07266	0.536329	+	0.844009i	$0.319811\pi$
$594$	61.2235	2.51203
$595$	0	0
$596$	21.2579	0.870757
$597$	−2.36842	−0.0969329
$598$	19.5547	0.799653
$599$	10.9913	0.449091	0.224545	−	0.974464i	$-0.427910\pi$
$599$	10.9913	0.449091	0.224545	+	0.974464i	$0.427910\pi$
$600$	0	0
$601$	36.4914	1.48852	0.744258	−	0.667892i	$-0.232803\pi$
$601$	36.4914	1.48852	0.744258	+	0.667892i	$0.232803\pi$
$602$	−6.57642	−0.268035
$603$	35.4233	1.44255
$604$	16.8365	0.685067
$605$	0	0
$606$	26.7320	1.08591
$607$	9.49692	0.385468	0.192734	−	0.981251i	$-0.438265\pi$
$607$	9.49692	0.385468	0.192734	+	0.981251i	$0.438265\pi$
$608$	20.2925	0.822968
$609$	1.94218	0.0787010
$610$	0	0
$611$	19.7420	0.798674
$612$	−25.9562	−1.04922
$613$	−8.81702	−0.356116	−0.178058	−	0.984020i	$-0.556981\pi$
$613$	−8.81702	−0.356116	−0.178058	+	0.984020i	$0.556981\pi$
$614$	−27.6353	−1.11527
$615$	0	0
$616$	7.97637	0.321377
$617$	7.58531	0.305373	0.152687	−	0.988275i	$-0.451208\pi$
$617$	7.58531	0.305373	0.152687	+	0.988275i	$0.451208\pi$
$618$	−13.0713	−0.525805
$619$	28.6537	1.15169	0.575845	−	0.817559i	$-0.304673\pi$
$619$	28.6537	1.15169	0.575845	+	0.817559i	$0.304673\pi$
$620$	0	0
$621$	8.19092	0.328690
$622$	10.1544	0.407154
$623$	−4.19785	−0.168183
$624$	−2.83001	−0.113291
$625$	0	0
$626$	25.8965	1.03503
$627$	16.0976	0.642875
$628$	−56.7276	−2.26368
$629$	−28.5825	−1.13966
$630$	0	0
$631$	20.8242	0.829000	0.414500	−	0.910049i	$-0.363957\pi$
$631$	20.8242	0.829000	0.414500	+	0.910049i	$0.363957\pi$
$632$	33.9632	1.35099
$633$	16.2970	0.647749
$634$	−7.41790	−0.294602
$635$	0	0
$636$	−3.02342	−0.119886
$637$	31.2741	1.23913
$638$	−60.9781	−2.41415
$639$	21.5112	0.850969
$640$	0	0
$641$	−17.9010	−0.707047	−0.353524	−	0.935426i	$-0.615017\pi$
$641$	−17.9010	−0.707047	−0.353524	+	0.935426i	$0.615017\pi$
$642$	−20.4600	−0.807492
$643$	31.6182	1.24690	0.623451	−	0.781863i	$-0.285730\pi$
$643$	31.6182	1.24690	0.623451	+	0.781863i	$0.285730\pi$
$644$	3.07085	0.121008
$645$	0	0
$646$	−25.8767	−1.01811
$647$	24.0466	0.945369	0.472685	−	0.881232i	$-0.343285\pi$
$647$	24.0466	0.945369	0.472685	+	0.881232i	$0.343285\pi$
$648$	−7.96694	−0.312971
$649$	−8.26971	−0.324614
$650$	0	0
$651$	1.47192	0.0576892
$652$	10.7305	0.420240
$653$	39.8417	1.55913	0.779563	−	0.626324i	$-0.215441\pi$
$653$	39.8417	1.55913	0.779563	+	0.626324i	$0.215441\pi$
$654$	−17.8598	−0.698375
$655$	0	0
$656$	2.42558	0.0947030
$657$	−8.31529	−0.324410
$658$	5.12315	0.199721
$659$	−46.2497	−1.80163	−0.900817	−	0.434199i	$-0.857032\pi$
$659$	−46.2497	−1.80163	−0.900817	+	0.434199i	$0.857032\pi$
$660$	0	0
$661$	−19.6109	−0.762777	−0.381389	−	0.924415i	$-0.624554\pi$
$661$	−19.6109	−0.762777	−0.381389	+	0.924415i	$0.624554\pi$
$662$	−26.8171	−1.04228
$663$	14.0667	0.546307
$664$	−16.0144	−0.621480
$665$	0	0
$666$	−40.7992	−1.58094
$667$	−8.15809	−0.315883
$668$	−55.4119	−2.14395
$669$	11.2518	0.435019
$670$	0	0
$671$	36.7893	1.42024
$672$	−2.86263	−0.110428
$673$	40.1715	1.54850	0.774248	−	0.632882i	$-0.218128\pi$
$673$	40.1715	1.54850	0.774248	+	0.632882i	$0.218128\pi$
$674$	42.7652	1.64725
$675$	0	0
$676$	26.7126	1.02741
$677$	−23.8424	−0.916340	−0.458170	−	0.888865i	$-0.651495\pi$
$677$	−23.8424	−0.916340	−0.458170	+	0.888865i	$0.651495\pi$
$678$	15.4564	0.593600
$679$	6.27803	0.240929
$680$	0	0
$681$	−11.4336	−0.438137
$682$	−46.2137	−1.76961
$683$	32.9444	1.26058	0.630291	−	0.776359i	$-0.282935\pi$
$683$	32.9444	1.26058	0.630291	+	0.776359i	$0.282935\pi$
$684$	−22.3523	−0.854660
$685$	0	0
$686$	16.5807	0.633056
$687$	−4.40092	−0.167906
$688$	−3.99783	−0.152416
$689$	−5.56387	−0.211966
$690$	0	0
$691$	−11.7745	−0.447922	−0.223961	−	0.974598i	$-0.571899\pi$
$691$	−11.7745	−0.447922	−0.223961	+	0.974598i	$0.571899\pi$
$692$	28.1543	1.07027
$693$	7.71111	0.292921
$694$	8.19802	0.311193
$695$	0	0
$696$	−8.66503	−0.328447
$697$	−12.0565	−0.456672
$698$	−59.4949	−2.25192
$699$	9.65578	0.365215
$700$	0	0
$701$	−42.3059	−1.59787	−0.798936	−	0.601416i	$-0.794603\pi$
$701$	−42.3059	−1.59787	−0.798936	+	0.601416i	$0.794603\pi$
$702$	46.0715	1.73885
$703$	−24.6139	−0.928330
$704$	80.7728	3.04424
$705$	0	0
$706$	−32.2070	−1.21213
$707$	7.72532	0.290541
$708$	−3.38162	−0.127089
$709$	4.07331	0.152976	0.0764882	−	0.997070i	$-0.475629\pi$
$709$	4.07331	0.152976	0.0764882	+	0.997070i	$0.475629\pi$
$710$	0	0
$711$	32.8338	1.23136
$712$	18.7287	0.701888
$713$	−6.18279	−0.231547
$714$	3.65040	0.136613
$715$	0	0
$716$	−29.6037	−1.10634
$717$	−4.31528	−0.161157
$718$	−60.7677	−2.26783
$719$	11.0619	0.412540	0.206270	−	0.978495i	$-0.433867\pi$
$719$	11.0619	0.412540	0.206270	+	0.978495i	$0.433867\pi$
$720$	0	0
$721$	−3.77750	−0.140681
$722$	20.4774	0.762089
$723$	0.826129	0.0307241
$724$	−7.11050	−0.264259
$725$	0	0
$726$	50.8457	1.88706
$727$	17.3643	0.644005	0.322003	−	0.946739i	$-0.395644\pi$
$727$	17.3643	0.644005	0.322003	+	0.946739i	$0.395644\pi$
$728$	6.00231	0.222461
$729$	3.18544	0.117979
$730$	0	0
$731$	19.8715	0.734973
$732$	15.0438	0.556034
$733$	−0.102270	−0.00377743	−0.00188872	−	0.999998i	$-0.500601\pi$
$733$	−0.102270	−0.00377743	−0.00188872	+	0.999998i	$0.500601\pi$
$734$	−41.0115	−1.51376
$735$	0	0
$736$	12.0244	0.443226
$737$	94.6526	3.48657
$738$	−17.2097	−0.633497
$739$	−47.2261	−1.73724	−0.868620	−	0.495479i	$-0.834992\pi$
$739$	−47.2261	−1.73724	−0.868620	+	0.495479i	$0.834992\pi$
$740$	0	0
$741$	12.1136	0.445005
$742$	−1.44385	−0.0530055
$743$	17.4927	0.641745	0.320873	−	0.947122i	$-0.396024\pi$
$743$	17.4927	0.641745	0.320873	+	0.947122i	$0.396024\pi$
$744$	−6.56699	−0.240758
$745$	0	0
$746$	72.1200	2.64050
$747$	−15.4818	−0.566451
$748$	−69.3561	−2.53591
$749$	−5.91276	−0.216048
$750$	0	0
$751$	4.09784	0.149532	0.0747662	−	0.997201i	$-0.476179\pi$
$751$	4.09784	0.149532	0.0747662	+	0.997201i	$0.476179\pi$
$752$	3.11438	0.113570
$753$	22.9872	0.837699
$754$	−45.8868	−1.67110
$755$	0	0
$756$	7.23500	0.263134
$757$	−32.3636	−1.17627	−0.588137	−	0.808761i	$-0.700138\pi$
$757$	−32.3636	−1.17627	−0.588137	+	0.808761i	$0.700138\pi$
$758$	−24.9619	−0.906656
$759$	9.53872	0.346234
$760$	0	0
$761$	−19.3464	−0.701306	−0.350653	−	0.936505i	$-0.614040\pi$
$761$	−19.3464	−0.701306	−0.350653	+	0.936505i	$0.614040\pi$
$762$	0.922725	0.0334268
$763$	−5.16134	−0.186853
$764$	65.2377	2.36022
$765$	0	0
$766$	−39.1775	−1.41554
$767$	−6.22306	−0.224702
$768$	9.04863	0.326514
$769$	−39.1117	−1.41040	−0.705202	−	0.709006i	$-0.749144\pi$
$769$	−39.1117	−1.41040	−0.705202	+	0.709006i	$0.749144\pi$
$770$	0	0
$771$	4.25737	0.153325
$772$	2.84937	0.102551
$773$	24.5839	0.884222	0.442111	−	0.896960i	$-0.354230\pi$
$773$	24.5839	0.884222	0.442111	+	0.896960i	$0.354230\pi$
$774$	28.3649	1.01956
$775$	0	0
$776$	−28.0094	−1.00548
$777$	3.47225	0.124566
$778$	−23.6703	−0.848622
$779$	−10.3825	−0.371991
$780$	0	0
$781$	57.4788	2.05675
$782$	−15.3334	−0.548323
$783$	−19.2207	−0.686890
$784$	4.93364	0.176201
$785$	0	0
$786$	−0.701822	−0.0250331
$787$	39.5266	1.40897	0.704486	−	0.709718i	$-0.251178\pi$
$787$	39.5266	1.40897	0.704486	+	0.709718i	$0.251178\pi$
$788$	8.99673	0.320495
$789$	−14.5389	−0.517600
$790$	0	0
$791$	4.46677	0.158820
$792$	−34.4031	−1.22246
$793$	27.6844	0.983102
$794$	−62.6770	−2.22432
$795$	0	0
$796$	8.78746	0.311463
$797$	−36.8227	−1.30433	−0.652163	−	0.758079i	$-0.726138\pi$
$797$	−36.8227	−1.30433	−0.652163	+	0.758079i	$0.726138\pi$
$798$	3.14355	0.111280
$799$	−15.4802	−0.547651
$800$	0	0
$801$	18.1059	0.639740
$802$	67.4615	2.38215
$803$	−22.2188	−0.784085
$804$	38.7050	1.36502
$805$	0	0
$806$	−34.7763	−1.22494
$807$	−0.416652	−0.0146668
$808$	−34.4665	−1.21253
$809$	29.5871	1.04023	0.520114	−	0.854097i	$-0.325890\pi$
$809$	29.5871	1.04023	0.520114	+	0.854097i	$0.325890\pi$
$810$	0	0
$811$	50.7120	1.78074	0.890370	−	0.455238i	$-0.150446\pi$
$811$	50.7120	1.78074	0.890370	+	0.455238i	$0.150446\pi$
$812$	−7.20599	−0.252881
$813$	−13.4313	−0.471055
$814$	−109.017	−3.82106
$815$	0	0
$816$	2.21909	0.0776837
$817$	17.1124	0.598686
$818$	10.6335	0.371790
$819$	5.80270	0.202763
$820$	0	0
$821$	33.7705	1.17860	0.589299	−	0.807915i	$-0.299404\pi$
$821$	33.7705	1.17860	0.589299	+	0.807915i	$0.299404\pi$
$822$	−14.6795	−0.512007
$823$	39.7659	1.38615	0.693076	−	0.720864i	$-0.256255\pi$
$823$	39.7659	1.38615	0.693076	+	0.720864i	$0.256255\pi$
$824$	16.8533	0.587113
$825$	0	0
$826$	−1.61492	−0.0561901
$827$	24.4271	0.849414	0.424707	−	0.905331i	$-0.360377\pi$
$827$	24.4271	0.849414	0.424707	+	0.905331i	$0.360377\pi$
$828$	−13.2450	−0.460295
$829$	4.25660	0.147838	0.0739189	−	0.997264i	$-0.476449\pi$
$829$	4.25660	0.147838	0.0739189	+	0.997264i	$0.476449\pi$
$830$	0	0
$831$	4.18889	0.145311
$832$	60.7825	2.10725
$833$	−24.5229	−0.849670
$834$	−23.2732	−0.805886
$835$	0	0
$836$	−59.7263	−2.06568
$837$	−14.5668	−0.503503
$838$	51.7736	1.78849
$839$	−22.5416	−0.778224	−0.389112	−	0.921190i	$-0.627218\pi$
$839$	−22.5416	−0.778224	−0.389112	+	0.921190i	$0.627218\pi$
$840$	0	0
$841$	−9.85639	−0.339875
$842$	−12.3966	−0.427216
$843$	−26.4561	−0.911195
$844$	−60.4663	−2.08134
$845$	0	0
$846$	−22.0968	−0.759703
$847$	14.6940	0.504890
$848$	−0.877725	−0.0301412
$849$	−19.2586	−0.660954
$850$	0	0
$851$	−14.5851	−0.499972
$852$	23.5041	0.805236
$853$	−46.0865	−1.57797	−0.788986	−	0.614412i	$-0.789393\pi$
$853$	−46.0865	−1.57797	−0.788986	+	0.614412i	$0.789393\pi$
$854$	7.18425	0.245840
$855$	0	0
$856$	26.3798	0.901643
$857$	29.2174	0.998046	0.499023	−	0.866589i	$-0.333692\pi$
$857$	29.2174	0.998046	0.499023	+	0.866589i	$0.333692\pi$
$858$	53.6524	1.83166
$859$	−27.8289	−0.949511	−0.474756	−	0.880118i	$-0.657464\pi$
$859$	−27.8289	−0.949511	−0.474756	+	0.880118i	$0.657464\pi$
$860$	0	0
$861$	1.46464	0.0499149
$862$	27.3619	0.931952
$863$	41.2024	1.40255	0.701274	−	0.712892i	$-0.252615\pi$
$863$	41.2024	1.40255	0.701274	+	0.712892i	$0.252615\pi$
$864$	28.3298	0.963800
$865$	0	0
$866$	57.2646	1.94593
$867$	3.01406	0.102363
$868$	−5.46123	−0.185366
$869$	87.7333	2.97615
$870$	0	0
$871$	71.2272	2.41344
$872$	23.0273	0.779804
$873$	−27.0780	−0.916450
$874$	−13.2044	−0.446647
$875$	0	0
$876$	−9.08565	−0.306976
$877$	−27.5073	−0.928855	−0.464428	−	0.885611i	$-0.653740\pi$
$877$	−27.5073	−0.928855	−0.464428	+	0.885611i	$0.653740\pi$
$878$	−72.1288	−2.43423
$879$	−14.7221	−0.496563
$880$	0	0
$881$	42.5035	1.43198	0.715990	−	0.698111i	$-0.245976\pi$
$881$	42.5035	1.43198	0.715990	+	0.698111i	$0.245976\pi$
$882$	−35.0045	−1.17866
$883$	−29.6322	−0.997204	−0.498602	−	0.866831i	$-0.666153\pi$
$883$	−29.6322	−0.997204	−0.498602	+	0.866831i	$0.666153\pi$
$884$	−52.1913	−1.75538
$885$	0	0
$886$	−18.6905	−0.627921
$887$	55.0299	1.84772	0.923862	−	0.382725i	$-0.125014\pi$
$887$	55.0299	1.84772	0.923862	+	0.382725i	$0.125014\pi$
$888$	−15.4914	−0.519859
$889$	0.266660	0.00894347
$890$	0	0
$891$	−20.5801	−0.689458
$892$	−41.7471	−1.39780
$893$	−13.3308	−0.446100
$894$	12.8947	0.431264
$895$	0	0
$896$	8.84317	0.295429
$897$	7.17800	0.239667
$898$	−60.1445	−2.00705
$899$	14.5084	0.483883
$900$	0	0
$901$	4.36279	0.145345
$902$	−45.9851	−1.53114
$903$	−2.41402	−0.0803335
$904$	−19.9285	−0.662812
$905$	0	0
$906$	10.2128	0.339296
$907$	−35.9423	−1.19344	−0.596722	−	0.802448i	$-0.703530\pi$
$907$	−35.9423	−1.19344	−0.596722	+	0.802448i	$0.703530\pi$
$908$	42.4217	1.40781
$909$	−33.3203	−1.10517
$910$	0	0
$911$	34.2507	1.13478	0.567388	−	0.823451i	$-0.307954\pi$
$911$	34.2507	1.13478	0.567388	+	0.823451i	$0.307954\pi$
$912$	1.91098	0.0632788
$913$	−41.3682	−1.36909
$914$	72.8641	2.41013
$915$	0	0
$916$	16.3286	0.539511
$917$	−0.202820	−0.00669772
$918$	−36.1259	−1.19233
$919$	−16.0328	−0.528875	−0.264437	−	0.964403i	$-0.585186\pi$
$919$	−16.0328	−0.528875	−0.264437	+	0.964403i	$0.585186\pi$
$920$	0	0
$921$	−10.1441	−0.334261
$922$	−70.8065	−2.33189
$923$	43.2535	1.42371
$924$	8.42550	0.277179
$925$	0	0
$926$	45.6549	1.50031
$927$	16.2928	0.535127
$928$	−28.2163	−0.926245
$929$	−4.12484	−0.135331	−0.0676657	−	0.997708i	$-0.521555\pi$
$929$	−4.12484	−0.135331	−0.0676657	+	0.997708i	$0.521555\pi$
$930$	0	0
$931$	−21.1180	−0.692115
$932$	−35.8255	−1.17350
$933$	3.72740	0.122029
$934$	−37.7728	−1.23597
$935$	0	0
$936$	−25.8888	−0.846201
$937$	59.3917	1.94024	0.970120	−	0.242624i	$-0.0780080\pi$
$937$	59.3917	1.94024	0.970120	+	0.242624i	$0.0780080\pi$
$938$	18.4838	0.603519
$939$	9.50588	0.310213
$940$	0	0
$941$	43.4437	1.41622	0.708112	−	0.706100i	$-0.249547\pi$
$941$	43.4437	1.41622	0.708112	+	0.706100i	$0.249547\pi$
$942$	−34.4101	−1.12114
$943$	−6.15221	−0.200344
$944$	−0.981715	−0.0319521
$945$	0	0
$946$	75.7924	2.46422
$947$	−58.8257	−1.91158	−0.955789	−	0.294053i	$-0.904996\pi$
$947$	−58.8257	−1.91158	−0.955789	+	0.294053i	$0.904996\pi$
$948$	35.8756	1.16519
$949$	−16.7199	−0.542752
$950$	0	0
$951$	−2.72290	−0.0882962
$952$	−4.70659	−0.152541
$953$	−29.2961	−0.948995	−0.474498	−	0.880257i	$-0.657370\pi$
$953$	−29.2961	−0.948995	−0.474498	+	0.880257i	$0.657370\pi$
$954$	6.22753	0.201624
$955$	0	0
$956$	16.0108	0.517827
$957$	−22.3834	−0.723552
$958$	−69.5375	−2.24665
$959$	−4.24225	−0.136989
$960$	0	0
$961$	−20.0045	−0.645305
$962$	−82.0369	−2.64497
$963$	25.5025	0.821807
$964$	−3.06516	−0.0987221
$965$	0	0
$966$	1.86273	0.0599324
$967$	−11.3823	−0.366031	−0.183015	−	0.983110i	$-0.558586\pi$
$967$	−11.3823	−0.366031	−0.183015	+	0.983110i	$0.558586\pi$
$968$	−65.5571	−2.10709
$969$	−9.49863	−0.305140
$970$	0	0
$971$	−23.8913	−0.766708	−0.383354	−	0.923602i	$-0.625231\pi$
$971$	−23.8913	−0.766708	−0.383354	+	0.923602i	$0.625231\pi$
$972$	−48.8108	−1.56561
$973$	−6.72576	−0.215618
$974$	47.9143	1.53527
$975$	0	0
$976$	4.36734	0.139795
$977$	20.2700	0.648495	0.324248	−	0.945972i	$-0.394889\pi$
$977$	20.2700	0.648495	0.324248	+	0.945972i	$0.394889\pi$
$978$	6.50898	0.208134
$979$	48.3797	1.54622
$980$	0	0
$981$	22.2615	0.710756
$982$	66.7577	2.13032
$983$	−52.0342	−1.65963	−0.829816	−	0.558037i	$-0.811555\pi$
$983$	−52.0342	−1.65963	−0.829816	+	0.558037i	$0.811555\pi$
$984$	−6.53451	−0.208313
$985$	0	0
$986$	35.9811	1.14587
$987$	1.88056	0.0598590
$988$	−44.9447	−1.42988
$989$	10.1401	0.322435
$990$	0	0
$991$	27.3954	0.870242	0.435121	−	0.900372i	$-0.356706\pi$
$991$	27.3954	0.870242	0.435121	+	0.900372i	$0.356706\pi$
$992$	−21.3843	−0.678953
$993$	−9.84383	−0.312384
$994$	11.2245	0.356020
$995$	0	0
$996$	−16.9161	−0.536008
$997$	7.64855	0.242232	0.121116	−	0.992638i	$-0.461353\pi$
$997$	7.64855	0.242232	0.121116	+	0.992638i	$0.461353\pi$
$998$	6.48329	0.205225
$999$	−34.3629	−1.08719

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.o.1.4	yes	40
5.4	even	2		6025.2.a.l.1.37	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6025.2.a.l.1.37	✓	40	5.4	even	2
6025.2.a.o.1.4	yes	40	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6025 = 5^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6025.a (trivial)

\(f(q)\)	\(=\)	\(q-2.25059 q^{2} -0.826129 q^{3} +3.06516 q^{4} +1.85928 q^{6} +0.537315 q^{7} -2.39723 q^{8} -2.31751 q^{9} +O(q^{10})\)	\(q-2.25059 q^{2} -0.826129 q^{3} +3.06516 q^{4} +1.85928 q^{6} +0.537315 q^{7} -2.39723 q^{8} -2.31751 q^{9} -6.19250 q^{11} -2.53222 q^{12} -4.65993 q^{13} -1.20928 q^{14} -0.735125 q^{16} +3.65398 q^{17} +5.21577 q^{18} +3.14664 q^{19} -0.443892 q^{21} +13.9368 q^{22} +1.86456 q^{23} +1.98042 q^{24} +10.4876 q^{26} +4.39295 q^{27} +1.64696 q^{28} -4.37534 q^{29} -3.31595 q^{31} +6.44893 q^{32} +5.11580 q^{33} -8.22362 q^{34} -7.10354 q^{36} -7.82228 q^{37} -7.08179 q^{38} +3.84970 q^{39} -3.29955 q^{41} +0.999019 q^{42} +5.43831 q^{43} -18.9810 q^{44} -4.19636 q^{46} -4.23654 q^{47} +0.607308 q^{48} -6.71129 q^{49} -3.01866 q^{51} -14.2834 q^{52} +1.19398 q^{53} -9.88673 q^{54} -1.28807 q^{56} -2.59953 q^{57} +9.84710 q^{58} +1.33544 q^{59} -5.94095 q^{61} +7.46285 q^{62} -1.24523 q^{63} -13.0437 q^{64} -11.5136 q^{66} -15.2850 q^{67} +11.2000 q^{68} -1.54037 q^{69} -9.28201 q^{71} +5.55561 q^{72} +3.58802 q^{73} +17.6048 q^{74} +9.64494 q^{76} -3.32732 q^{77} -8.66410 q^{78} -14.1677 q^{79} +3.32339 q^{81} +7.42594 q^{82} +6.68037 q^{83} -1.36060 q^{84} -12.2394 q^{86} +3.61460 q^{87} +14.8449 q^{88} -7.81264 q^{89} -2.50385 q^{91} +5.71517 q^{92} +2.73940 q^{93} +9.53471 q^{94} -5.32765 q^{96} +11.6841 q^{97} +15.1044 q^{98} +14.3512 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9}+O(q^{10}) \) 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9	\( 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9} + q^{11} + 14 q^{12} + 9 q^{13} - q^{14} + 43 q^{16} + 12 q^{17} + 42 q^{18} + 2 q^{21} + 5 q^{22} + 77 q^{23} - 2 q^{24} + 2 q^{26} + 38 q^{27} + 42 q^{28} + 2 q^{29} + q^{31} + 72 q^{32} + 20 q^{33} + 5 q^{34} + 32 q^{36} + 28 q^{37} + 23 q^{38} - 2 q^{39} - 2 q^{41} + 37 q^{42} + 31 q^{43} + 3 q^{44} + 14 q^{46} + 96 q^{47} + 13 q^{48} + 40 q^{49} - 10 q^{51} + 42 q^{52} + 54 q^{53} + 4 q^{54} - 15 q^{56} + 37 q^{57} + 27 q^{58} + q^{59} + 5 q^{61} + 39 q^{62} + 70 q^{63} + 65 q^{64} - 52 q^{66} + 34 q^{67} + 52 q^{68} + 21 q^{69} - 9 q^{71} + 70 q^{72} + 25 q^{73} + 22 q^{74} - 47 q^{76} + 54 q^{77} + 58 q^{78} + 13 q^{79} + 12 q^{81} - 5 q^{82} + 63 q^{83} + 95 q^{84} - 18 q^{86} + 47 q^{87} + 13 q^{88} + 19 q^{89} - 31 q^{91} + 137 q^{92} + 52 q^{93} + 120 q^{94} - 49 q^{96} + 36 q^{97} + 64 q^{98} + 16 q^{99}+O(q^{100}) \) 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9 + q^11 + 14 * q^12 + 9 * q^13 - q^14 + 43 * q^16 + 12 * q^17 + 42 * q^18 + 2 * q^21 + 5 * q^22 + 77 * q^23 - 2 * q^24 + 2 * q^26 + 38 * q^27 + 42 * q^28 + 2 * q^29 + q^31 + 72 * q^32 + 20 * q^33 + 5 * q^34 + 32 * q^36 + 28 * q^37 + 23 * q^38 - 2 * q^39 - 2 * q^41 + 37 * q^42 + 31 * q^43 + 3 * q^44 + 14 * q^46 + 96 * q^47 + 13 * q^48 + 40 * q^49 - 10 * q^51 + 42 * q^52 + 54 * q^53 + 4 * q^54 - 15 * q^56 + 37 * q^57 + 27 * q^58 + q^59 + 5 * q^61 + 39 * q^62 + 70 * q^63 + 65 * q^64 - 52 * q^66 + 34 * q^67 + 52 * q^68 + 21 * q^69 - 9 * q^71 + 70 * q^72 + 25 * q^73 + 22 * q^74 - 47 * q^76 + 54 * q^77 + 58 * q^78 + 13 * q^79 + 12 * q^81 - 5 * q^82 + 63 * q^83 + 95 * q^84 - 18 * q^86 + 47 * q^87 + 13 * q^88 + 19 * q^89 - 31 * q^91 + 137 * q^92 + 52 * q^93 + 120 * q^94 - 49 * q^96 + 36 * q^97 + 64 * q^98 + 16 * q^99

Introduction

L-functions

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