LMFDB - Embedded newform 6025.2.a.n.1.1

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			1.1
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)

$f(q)$	$=$	$q-2.56980 q^{2} +0.961984 q^{3} +4.60385 q^{4} -2.47210 q^{6} +2.40958 q^{7} -6.69138 q^{8} -2.07459 q^{9} +O(q^{10})$	\(q-2.56980 q^{2} +0.961984 q^{3} +4.60385 q^{4} -2.47210 q^{6} +2.40958 q^{7} -6.69138 q^{8} -2.07459 q^{9} +2.76006 q^{11} +4.42884 q^{12} +1.28191 q^{13} -6.19212 q^{14} +7.98777 q^{16} +6.01132 q^{17} +5.33127 q^{18} +2.42658 q^{19} +2.31797 q^{21} -7.09278 q^{22} +4.77890 q^{23} -6.43700 q^{24} -3.29424 q^{26} -4.88167 q^{27} +11.0933 q^{28} +7.77112 q^{29} -4.82125 q^{31} -7.14419 q^{32} +2.65513 q^{33} -15.4479 q^{34} -9.55110 q^{36} +9.25158 q^{37} -6.23582 q^{38} +1.23317 q^{39} +3.19571 q^{41} -5.95672 q^{42} +7.89597 q^{43} +12.7069 q^{44} -12.2808 q^{46} +4.18466 q^{47} +7.68411 q^{48} -1.19394 q^{49} +5.78279 q^{51} +5.90171 q^{52} +2.01798 q^{53} +12.5449 q^{54} -16.1234 q^{56} +2.33433 q^{57} -19.9702 q^{58} +7.91117 q^{59} -6.64175 q^{61} +12.3896 q^{62} -4.99887 q^{63} +2.38357 q^{64} -6.82314 q^{66} +1.72857 q^{67} +27.6752 q^{68} +4.59723 q^{69} -13.9300 q^{71} +13.8818 q^{72} -8.98346 q^{73} -23.7747 q^{74} +11.1716 q^{76} +6.65056 q^{77} -3.16901 q^{78} -3.90169 q^{79} +1.52767 q^{81} -8.21233 q^{82} -3.72858 q^{83} +10.6716 q^{84} -20.2910 q^{86} +7.47570 q^{87} -18.4686 q^{88} -11.3155 q^{89} +3.08885 q^{91} +22.0014 q^{92} -4.63796 q^{93} -10.7537 q^{94} -6.87260 q^{96} +4.15783 q^{97} +3.06819 q^{98} -5.72597 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 9 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 20 q^{7} + 27 q^{8} + 38 q^{9}+O(q^{10}) $ 40 * q + 9 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 20 * q^7 + 27 * q^8 + 38 * q^9	$ 40 q + 9 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 20 q^{7} + 27 q^{8} + 38 q^{9} + q^{11} + 26 q^{12} + 11 q^{13} - q^{14} + 43 q^{16} + 20 q^{17} + 18 q^{18} + 2 q^{21} + 23 q^{22} + 79 q^{23} - 2 q^{24} + 2 q^{26} + 26 q^{27} + 30 q^{28} + 2 q^{29} + q^{31} + 68 q^{32} + 20 q^{33} + 5 q^{34} + 32 q^{36} + 16 q^{37} + 45 q^{38} - 2 q^{39} - 2 q^{41} + 19 q^{42} + 25 q^{43} + 3 q^{44} + 14 q^{46} + 88 q^{47} + 75 q^{48} + 40 q^{49} - 10 q^{51} + 18 q^{52} + 34 q^{53} + 4 q^{54} - 15 q^{56} + 51 q^{57} + 53 q^{58} + q^{59} + 9 q^{61} + 39 q^{62} + 110 q^{63} + 17 q^{64} + 26 q^{66} + 30 q^{67} + 44 q^{68} - 7 q^{69} + 5 q^{71} + 18 q^{72} + 23 q^{73} - 18 q^{74} + 43 q^{76} + 30 q^{77} + 46 q^{78} + 5 q^{79} + 44 q^{81} + 5 q^{82} + 65 q^{83} - 65 q^{84} + 40 q^{86} + 33 q^{87} + 71 q^{88} - 9 q^{89} + q^{91} + 117 q^{92} + 68 q^{93} - 72 q^{94} + 83 q^{96} - 8 q^{97} + 76 q^{98} - 40 q^{99}+O(q^{100}) $ 40 * q + 9 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 20 * q^7 + 27 * q^8 + 38 * q^9 + q^11 + 26 * q^12 + 11 * q^13 - q^14 + 43 * q^16 + 20 * q^17 + 18 * q^18 + 2 * q^21 + 23 * q^22 + 79 * q^23 - 2 * q^24 + 2 * q^26 + 26 * q^27 + 30 * q^28 + 2 * q^29 + q^31 + 68 * q^32 + 20 * q^33 + 5 * q^34 + 32 * q^36 + 16 * q^37 + 45 * q^38 - 2 * q^39 - 2 * q^41 + 19 * q^42 + 25 * q^43 + 3 * q^44 + 14 * q^46 + 88 * q^47 + 75 * q^48 + 40 * q^49 - 10 * q^51 + 18 * q^52 + 34 * q^53 + 4 * q^54 - 15 * q^56 + 51 * q^57 + 53 * q^58 + q^59 + 9 * q^61 + 39 * q^62 + 110 * q^63 + 17 * q^64 + 26 * q^66 + 30 * q^67 + 44 * q^68 - 7 * q^69 + 5 * q^71 + 18 * q^72 + 23 * q^73 - 18 * q^74 + 43 * q^76 + 30 * q^77 + 46 * q^78 + 5 * q^79 + 44 * q^81 + 5 * q^82 + 65 * q^83 - 65 * q^84 + 40 * q^86 + 33 * q^87 + 71 * q^88 - 9 * q^89 + q^91 + 117 * q^92 + 68 * q^93 - 72 * q^94 + 83 * q^96 - 8 * q^97 + 76 * q^98 - 40 * 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 $

$2$	−2.56980	−1.81712	−0.908560	−	0.417754i	$-0.862818\pi$
$2$	−2.56980	−1.81712	−0.908560	+	0.417754i	$0.862818\pi$
$3$	0.961984	0.555402	0.277701	−	0.960668i	$-0.410428\pi$
$3$	0.961984	0.555402	0.277701	+	0.960668i	$0.410428\pi$
$4$	4.60385	2.30193
$5$	0	0
$5$	0	0
$6$	−2.47210	−1.00923
$7$	2.40958	0.910734	0.455367	−	0.890304i	$-0.349508\pi$
$7$	2.40958	0.910734	0.455367	+	0.890304i	$0.349508\pi$
$8$	−6.69138	−2.36576
$9$	−2.07459	−0.691529
$10$	0	0
$11$	2.76006	0.832188	0.416094	−	0.909322i	$-0.363399\pi$
$11$	2.76006	0.832188	0.416094	+	0.909322i	$0.363399\pi$
$12$	4.42884	1.27849
$13$	1.28191	0.355537	0.177768	−	0.984072i	$-0.443112\pi$
$13$	1.28191	0.355537	0.177768	+	0.984072i	$0.443112\pi$
$14$	−6.19212	−1.65491
$15$	0	0
$16$	7.98777	1.99694
$17$	6.01132	1.45796	0.728980	−	0.684535i	$-0.239995\pi$
$17$	6.01132	1.45796	0.728980	+	0.684535i	$0.239995\pi$
$18$	5.33127	1.25659
$19$	2.42658	0.556696	0.278348	−	0.960480i	$-0.410213\pi$
$19$	2.42658	0.556696	0.278348	+	0.960480i	$0.410213\pi$
$20$	0	0
$21$	2.31797	0.505823
$22$	−7.09278	−1.51219
$23$	4.77890	0.996470	0.498235	−	0.867042i	$-0.333982\pi$
$23$	4.77890	0.996470	0.498235	+	0.867042i	$0.333982\pi$
$24$	−6.43700	−1.31395
$25$	0	0
$26$	−3.29424	−0.646053
$27$	−4.88167	−0.939478
$28$	11.0933	2.09644
$29$	7.77112	1.44306	0.721531	−	0.692383i	$-0.243439\pi$
$29$	7.77112	1.44306	0.721531	+	0.692383i	$0.243439\pi$
$30$	0	0
$31$	−4.82125	−0.865921	−0.432961	−	0.901413i	$-0.642531\pi$
$31$	−4.82125	−0.865921	−0.432961	+	0.901413i	$0.642531\pi$
$32$	−7.14419	−1.26293
$33$	2.65513	0.462199
$34$	−15.4479	−2.64929
$35$	0	0
$36$	−9.55110	−1.59185
$37$	9.25158	1.52095	0.760475	−	0.649367i	$-0.224966\pi$
$37$	9.25158	1.52095	0.760475	+	0.649367i	$0.224966\pi$
$38$	−6.23582	−1.01158
$39$	1.23317	0.197466
$40$	0	0
$41$	3.19571	0.499087	0.249543	−	0.968364i	$-0.419720\pi$
$41$	3.19571	0.499087	0.249543	+	0.968364i	$0.419720\pi$
$42$	−5.95672	−0.919142
$43$	7.89597	1.20412	0.602062	−	0.798449i	$-0.294346\pi$
$43$	7.89597	1.20412	0.602062	+	0.798449i	$0.294346\pi$
$44$	12.7069	1.91564
$45$	0	0
$46$	−12.2808	−1.81071
$47$	4.18466	0.610395	0.305197	−	0.952289i	$-0.401278\pi$
$47$	4.18466	0.610395	0.305197	+	0.952289i	$0.401278\pi$
$48$	7.68411	1.10911
$49$	−1.19394	−0.170563
$50$	0	0
$51$	5.78279	0.809753
$52$	5.90171	0.818420
$53$	2.01798	0.277190	0.138595	−	0.990349i	$-0.455741\pi$
$53$	2.01798	0.277190	0.138595	+	0.990349i	$0.455741\pi$
$54$	12.5449	1.70715
$55$	0	0
$56$	−16.1234	−2.15458
$57$	2.33433	0.309190
$58$	−19.9702	−2.62222
$59$	7.91117	1.02995	0.514973	−	0.857206i	$-0.327802\pi$
$59$	7.91117	1.02995	0.514973	+	0.857206i	$0.327802\pi$
$60$	0	0
$61$	−6.64175	−0.850389	−0.425195	−	0.905102i	$-0.639794\pi$
$61$	−6.64175	−0.850389	−0.425195	+	0.905102i	$0.639794\pi$
$62$	12.3896	1.57348
$63$	−4.99887	−0.629799
$64$	2.38357	0.297947
$65$	0	0
$66$	−6.82314	−0.839871
$67$	1.72857	0.211178	0.105589	−	0.994410i	$-0.466327\pi$
$67$	1.72857	0.211178	0.105589	+	0.994410i	$0.466327\pi$
$68$	27.6752	3.35612
$69$	4.59723	0.553441
$70$	0	0
$71$	−13.9300	−1.65319	−0.826593	−	0.562800i	$-0.809724\pi$
$71$	−13.9300	−1.65319	−0.826593	+	0.562800i	$0.809724\pi$
$72$	13.8818	1.63599
$73$	−8.98346	−1.05143	−0.525717	−	0.850659i	$-0.676203\pi$
$73$	−8.98346	−1.05143	−0.525717	+	0.850659i	$0.676203\pi$
$74$	−23.7747	−2.76375
$75$	0	0
$76$	11.1716	1.28147
$77$	6.65056	0.757902
$78$	−3.16901	−0.358819
$79$	−3.90169	−0.438975	−0.219487	−	0.975615i	$-0.570438\pi$
$79$	−3.90169	−0.438975	−0.219487	+	0.975615i	$0.570438\pi$
$80$	0	0
$81$	1.52767	0.169741
$82$	−8.21233	−0.906901
$83$	−3.72858	−0.409265	−0.204632	−	0.978839i	$-0.565600\pi$
$83$	−3.72858	−0.409265	−0.204632	+	0.978839i	$0.565600\pi$
$84$	10.6716	1.16437
$85$	0	0
$86$	−20.2910	−2.18804
$87$	7.47570	0.801479
$88$	−18.4686	−1.96876
$89$	−11.3155	−1.19944	−0.599722	−	0.800208i	$-0.704722\pi$
$89$	−11.3155	−1.19944	−0.599722	+	0.800208i	$0.704722\pi$
$90$	0	0
$91$	3.08885	0.323800
$92$	22.0014	2.29380
$93$	−4.63796	−0.480934
$94$	−10.7537	−1.10916
$95$	0	0
$96$	−6.87260	−0.701431
$97$	4.15783	0.422164	0.211082	−	0.977468i	$-0.432301\pi$
$97$	4.15783	0.422164	0.211082	+	0.977468i	$0.432301\pi$
$98$	3.06819	0.309934
$99$	−5.72597	−0.575482
$100$	0	0
$101$	−4.30829	−0.428690	−0.214345	−	0.976758i	$-0.568762\pi$
$101$	−4.30829	−0.428690	−0.214345	+	0.976758i	$0.568762\pi$
$102$	−14.8606	−1.47142
$103$	−4.95130	−0.487866	−0.243933	−	0.969792i	$-0.578438\pi$
$103$	−4.95130	−0.487866	−0.243933	+	0.969792i	$0.578438\pi$
$104$	−8.57772	−0.841115
$105$	0	0
$106$	−5.18579	−0.503688
$107$	4.59299	0.444022	0.222011	−	0.975044i	$-0.428738\pi$
$107$	4.59299	0.444022	0.222011	+	0.975044i	$0.428738\pi$
$108$	−22.4745	−2.16261
$109$	−6.90969	−0.661828	−0.330914	−	0.943661i	$-0.607357\pi$
$109$	−6.90969	−0.661828	−0.330914	+	0.943661i	$0.607357\pi$
$110$	0	0
$111$	8.89987	0.844738
$112$	19.2471	1.81868
$113$	−14.8042	−1.39266	−0.696331	−	0.717721i	$-0.745185\pi$
$113$	−14.8042	−1.39266	−0.696331	+	0.717721i	$0.745185\pi$
$114$	−5.99876	−0.561835
$115$	0	0
$116$	35.7771	3.32182
$117$	−2.65943	−0.245864
$118$	−20.3301	−1.87154
$119$	14.4847	1.32781
$120$	0	0
$121$	−3.38209	−0.307463
$122$	17.0679	1.54526
$123$	3.07423	0.277194
$124$	−22.1963	−1.99329
$125$	0	0
$126$	12.8461	1.14442
$127$	18.7773	1.66622	0.833109	−	0.553109i	$-0.186559\pi$
$127$	18.7773	1.66622	0.833109	+	0.553109i	$0.186559\pi$
$128$	8.16308	0.721521
$129$	7.59580	0.668773
$130$	0	0
$131$	10.9358	0.955466	0.477733	−	0.878505i	$-0.341459\pi$
$131$	10.9358	0.955466	0.477733	+	0.878505i	$0.341459\pi$
$132$	12.2238	1.06395
$133$	5.84703	0.507002
$134$	−4.44206	−0.383736
$135$	0	0
$136$	−40.2240	−3.44918
$137$	16.7262	1.42902	0.714509	−	0.699626i	$-0.246650\pi$
$137$	16.7262	1.42902	0.714509	+	0.699626i	$0.246650\pi$
$138$	−11.8139	−1.00567
$139$	2.44886	0.207709	0.103855	−	0.994592i	$-0.466882\pi$
$139$	2.44886	0.207709	0.103855	+	0.994592i	$0.466882\pi$
$140$	0	0
$141$	4.02557	0.339014
$142$	35.7973	3.00404
$143$	3.53813	0.295874
$144$	−16.5713	−1.38094
$145$	0	0
$146$	23.0857	1.91058
$147$	−1.14856	−0.0947313
$148$	42.5929	3.50112
$149$	23.2240	1.90258	0.951290	−	0.308297i	$-0.0997590\pi$
$149$	23.2240	1.90258	0.951290	+	0.308297i	$0.0997590\pi$
$150$	0	0
$151$	3.13685	0.255274	0.127637	−	0.991821i	$-0.459261\pi$
$151$	3.13685	0.255274	0.127637	+	0.991821i	$0.459261\pi$
$152$	−16.2372	−1.31701
$153$	−12.4710	−1.00822
$154$	−17.0906	−1.37720
$155$	0	0
$156$	5.67735	0.454552
$157$	−2.03612	−0.162500	−0.0812501	−	0.996694i	$-0.525891\pi$
$157$	−2.03612	−0.162500	−0.0812501	+	0.996694i	$0.525891\pi$
$158$	10.0266	0.797670
$159$	1.94126	0.153952
$160$	0	0
$161$	11.5151	0.907519
$162$	−3.92580	−0.308440
$163$	−2.15661	−0.168919	−0.0844593	−	0.996427i	$-0.526916\pi$
$163$	−2.15661	−0.168919	−0.0844593	+	0.996427i	$0.526916\pi$
$164$	14.7126	1.14886
$165$	0	0
$166$	9.58169	0.743683
$167$	−17.4203	−1.34803	−0.674013	−	0.738719i	$-0.735431\pi$
$167$	−17.4203	−1.34803	−0.674013	+	0.738719i	$0.735431\pi$
$168$	−15.5104	−1.19666
$169$	−11.3567	−0.873594
$170$	0	0
$171$	−5.03415	−0.384971
$172$	36.3519	2.77181
$173$	−7.88577	−0.599544	−0.299772	−	0.954011i	$-0.596911\pi$
$173$	−7.88577	−0.599544	−0.299772	+	0.954011i	$0.596911\pi$
$174$	−19.2110	−1.45638
$175$	0	0
$176$	22.0467	1.66183
$177$	7.61042	0.572034
$178$	29.0786	2.17953
$179$	−7.00366	−0.523478	−0.261739	−	0.965139i	$-0.584296\pi$
$179$	−7.00366	−0.523478	−0.261739	+	0.965139i	$0.584296\pi$
$180$	0	0
$181$	−22.2733	−1.65556	−0.827782	−	0.561050i	$-0.810398\pi$
$181$	−22.2733	−1.65556	−0.827782	+	0.561050i	$0.810398\pi$
$182$	−7.93772	−0.588383
$183$	−6.38926	−0.472308
$184$	−31.9774	−2.35741
$185$	0	0
$186$	11.9186	0.873915
$187$	16.5916	1.21330
$188$	19.2656	1.40508
$189$	−11.7628	−0.855615
$190$	0	0
$191$	6.87996	0.497817	0.248908	−	0.968527i	$-0.419928\pi$
$191$	6.87996	0.497817	0.248908	+	0.968527i	$0.419928\pi$
$192$	2.29296	0.165480
$193$	−15.6852	−1.12904	−0.564522	−	0.825418i	$-0.690939\pi$
$193$	−15.6852	−1.12904	−0.564522	+	0.825418i	$0.690939\pi$
$194$	−10.6848	−0.767123
$195$	0	0
$196$	−5.49675	−0.392625
$197$	22.2648	1.58630	0.793152	−	0.609024i	$-0.208439\pi$
$197$	22.2648	1.58630	0.793152	+	0.609024i	$0.208439\pi$
$198$	14.7146	1.04572
$199$	0.724748	0.0513761	0.0256880	−	0.999670i	$-0.491822\pi$
$199$	0.724748	0.0513761	0.0256880	+	0.999670i	$0.491822\pi$
$200$	0	0
$201$	1.66285	0.117289
$202$	11.0714	0.778982
$203$	18.7251	1.31424
$204$	26.6231	1.86399
$205$	0	0
$206$	12.7238	0.886511
$207$	−9.91425	−0.689088
$208$	10.2396	0.709987
$209$	6.69749	0.463275
$210$	0	0
$211$	−8.31522	−0.572443	−0.286222	−	0.958163i	$-0.592399\pi$
$211$	−8.31522	−0.572443	−0.286222	+	0.958163i	$0.592399\pi$
$212$	9.29047	0.638072
$213$	−13.4004	−0.918183
$214$	−11.8031	−0.806841
$215$	0	0
$216$	32.6651	2.22258
$217$	−11.6172	−0.788624
$218$	17.7565	1.20262
$219$	−8.64195	−0.583969
$220$	0	0
$221$	7.70595	0.518358
$222$	−22.8709	−1.53499
$223$	−24.2506	−1.62394	−0.811970	−	0.583699i	$-0.801605\pi$
$223$	−24.2506	−1.62394	−0.811970	+	0.583699i	$0.801605\pi$
$224$	−17.2145	−1.15019
$225$	0	0
$226$	38.0438	2.53063
$227$	−19.4632	−1.29182	−0.645911	−	0.763413i	$-0.723522\pi$
$227$	−19.4632	−1.29182	−0.645911	+	0.763413i	$0.723522\pi$
$228$	10.7469	0.711732
$229$	13.8587	0.915810	0.457905	−	0.889001i	$-0.348600\pi$
$229$	13.8587	0.915810	0.457905	+	0.889001i	$0.348600\pi$
$230$	0	0
$231$	6.39774	0.420940
$232$	−51.9995	−3.41394
$233$	21.2772	1.39392	0.696958	−	0.717112i	$-0.254537\pi$
$233$	21.2772	1.39392	0.696958	+	0.717112i	$0.254537\pi$
$234$	6.83418	0.446765
$235$	0	0
$236$	36.4219	2.37086
$237$	−3.75337	−0.243807
$238$	−37.2228	−2.41280
$239$	18.7664	1.21390	0.606948	−	0.794741i	$-0.292394\pi$
$239$	18.7664	1.21390	0.606948	+	0.794741i	$0.292394\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	8.69129	0.558697
$243$	16.1146	1.03375
$244$	−30.5777	−1.95753
$245$	0	0
$246$	−7.90013	−0.503694
$247$	3.11065	0.197926
$248$	32.2608	2.04856
$249$	−3.58683	−0.227306
$250$	0	0
$251$	20.3673	1.28557	0.642786	−	0.766046i	$-0.277779\pi$
$251$	20.3673	1.28557	0.642786	+	0.766046i	$0.277779\pi$
$252$	−23.0141	−1.44975
$253$	13.1900	0.829251
$254$	−48.2539	−3.02772
$255$	0	0
$256$	−25.7446	−1.60904
$257$	−4.28573	−0.267336	−0.133668	−	0.991026i	$-0.542676\pi$
$257$	−4.28573	−0.267336	−0.133668	+	0.991026i	$0.542676\pi$
$258$	−19.5197	−1.21524
$259$	22.2924	1.38518
$260$	0	0
$261$	−16.1219	−0.997918
$262$	−28.1028	−1.73620
$263$	9.91891	0.611626	0.305813	−	0.952092i	$-0.401072\pi$
$263$	9.91891	0.611626	0.305813	+	0.952092i	$0.401072\pi$
$264$	−17.7665	−1.09345
$265$	0	0
$266$	−15.0257	−0.921283
$267$	−10.8854	−0.666173
$268$	7.95807	0.486116
$269$	−5.66235	−0.345240	−0.172620	−	0.984989i	$-0.555223\pi$
$269$	−5.66235	−0.345240	−0.172620	+	0.984989i	$0.555223\pi$
$270$	0	0
$271$	−1.59387	−0.0968208	−0.0484104	−	0.998828i	$-0.515416\pi$
$271$	−1.59387	−0.0968208	−0.0484104	+	0.998828i	$0.515416\pi$
$272$	48.0170	2.91146
$273$	2.97142	0.179839
$274$	−42.9830	−2.59670
$275$	0	0
$276$	21.1650	1.27398
$277$	−0.549080	−0.0329910	−0.0164955	−	0.999864i	$-0.505251\pi$
$277$	−0.549080	−0.0329910	−0.0164955	+	0.999864i	$0.505251\pi$
$278$	−6.29306	−0.377433
$279$	10.0021	0.598810
$280$	0	0
$281$	26.4741	1.57931	0.789656	−	0.613550i	$-0.210259\pi$
$281$	26.4741	1.57931	0.789656	+	0.613550i	$0.210259\pi$
$282$	−10.3449	−0.616030
$283$	−0.425122	−0.0252709	−0.0126354	−	0.999920i	$-0.504022\pi$
$283$	−0.425122	−0.0252709	−0.0126354	+	0.999920i	$0.504022\pi$
$284$	−64.1317	−3.80552
$285$	0	0
$286$	−9.09228	−0.537638
$287$	7.70031	0.454535
$288$	14.8212	0.873350
$289$	19.1360	1.12565
$290$	0	0
$291$	3.99977	0.234471
$292$	−41.3585	−2.42033
$293$	15.2147	0.888855	0.444427	−	0.895815i	$-0.353407\pi$
$293$	15.2147	0.888855	0.444427	+	0.895815i	$0.353407\pi$
$294$	2.95155	0.172138
$295$	0	0
$296$	−61.9058	−3.59820
$297$	−13.4737	−0.781822
$298$	−59.6808	−3.45722
$299$	6.12611	0.354282
$300$	0	0
$301$	19.0259	1.09664
$302$	−8.06108	−0.463863
$303$	−4.14450	−0.238095
$304$	19.3830	1.11169
$305$	0	0
$306$	32.0479	1.83206
$307$	16.1114	0.919527	0.459763	−	0.888041i	$-0.347934\pi$
$307$	16.1114	0.919527	0.459763	+	0.888041i	$0.347934\pi$
$308$	30.6182	1.74464
$309$	−4.76307	−0.270961
$310$	0	0
$311$	−33.9632	−1.92588	−0.962939	−	0.269719i	$-0.913069\pi$
$311$	−33.9632	−1.92588	−0.962939	+	0.269719i	$0.913069\pi$
$312$	−8.25163	−0.467157
$313$	−24.4072	−1.37958	−0.689790	−	0.724010i	$-0.742297\pi$
$313$	−24.4072	−1.37958	−0.689790	+	0.724010i	$0.742297\pi$
$314$	5.23242	0.295282
$315$	0	0
$316$	−17.9628	−1.01049
$317$	−0.545333	−0.0306289	−0.0153145	−	0.999883i	$-0.504875\pi$
$317$	−0.545333	−0.0306289	−0.0153145	+	0.999883i	$0.504875\pi$
$318$	−4.98864	−0.279749
$319$	21.4487	1.20090
$320$	0	0
$321$	4.41839	0.246610
$322$	−29.5915	−1.64907
$323$	14.5869	0.811639
$324$	7.03317	0.390732
$325$	0	0
$326$	5.54204	0.306945
$327$	−6.64701	−0.367581
$328$	−21.3837	−1.18072
$329$	10.0832	0.555907
$330$	0	0
$331$	−8.69918	−0.478150	−0.239075	−	0.971001i	$-0.576844\pi$
$331$	−8.69918	−0.478150	−0.239075	+	0.971001i	$0.576844\pi$
$332$	−17.1658	−0.942098
$333$	−19.1932	−1.05178
$334$	44.7667	2.44953
$335$	0	0
$336$	18.5154	1.01010
$337$	−17.7435	−0.966552	−0.483276	−	0.875468i	$-0.660553\pi$
$337$	−17.7435	−0.966552	−0.483276	+	0.875468i	$0.660553\pi$
$338$	29.1845	1.58742
$339$	−14.2414	−0.773487
$340$	0	0
$341$	−13.3069	−0.720609
$342$	12.9367	0.699539
$343$	−19.7439	−1.06607
$344$	−52.8349	−2.84867
$345$	0	0
$346$	20.2648	1.08944
$347$	−3.47050	−0.186306	−0.0931532	−	0.995652i	$-0.529695\pi$
$347$	−3.47050	−0.186306	−0.0931532	+	0.995652i	$0.529695\pi$
$348$	34.4170	1.84495
$349$	−14.9330	−0.799347	−0.399673	−	0.916658i	$-0.630876\pi$
$349$	−14.9330	−0.799347	−0.399673	+	0.916658i	$0.630876\pi$
$350$	0	0
$351$	−6.25785	−0.334019
$352$	−19.7184	−1.05099
$353$	−35.2607	−1.87674	−0.938368	−	0.345637i	$-0.887663\pi$
$353$	−35.2607	−1.87674	−0.938368	+	0.345637i	$0.887663\pi$
$354$	−19.5572	−1.03945
$355$	0	0
$356$	−52.0951	−2.76103
$357$	13.9341	0.737470
$358$	17.9980	0.951223
$359$	4.06661	0.214627	0.107314	−	0.994225i	$-0.465775\pi$
$359$	4.06661	0.214627	0.107314	+	0.994225i	$0.465775\pi$
$360$	0	0
$361$	−13.1117	−0.690090
$362$	57.2380	3.00836
$363$	−3.25352	−0.170765
$364$	14.2206	0.745363
$365$	0	0
$366$	16.4191	0.858240
$367$	−12.1950	−0.636573	−0.318286	−	0.947995i	$-0.603107\pi$
$367$	−12.1950	−0.636573	−0.318286	+	0.947995i	$0.603107\pi$
$368$	38.1728	1.98989
$369$	−6.62978	−0.345133
$370$	0	0
$371$	4.86247	0.252447
$372$	−21.3525	−1.10708
$373$	20.0717	1.03928	0.519638	−	0.854387i	$-0.326067\pi$
$373$	20.0717	1.03928	0.519638	+	0.854387i	$0.326067\pi$
$374$	−42.6370	−2.20471
$375$	0	0
$376$	−28.0011	−1.44405
$377$	9.96185	0.513061
$378$	30.2279	1.55476
$379$	−3.33465	−0.171289	−0.0856446	−	0.996326i	$-0.527295\pi$
$379$	−3.33465	−0.171289	−0.0856446	+	0.996326i	$0.527295\pi$
$380$	0	0
$381$	18.0635	0.925421
$382$	−17.6801	−0.904593
$383$	−14.8263	−0.757589	−0.378794	−	0.925481i	$-0.623661\pi$
$383$	−14.8263	−0.757589	−0.378794	+	0.925481i	$0.623661\pi$
$384$	7.85275	0.400734
$385$	0	0
$386$	40.3077	2.05161
$387$	−16.3809	−0.832687
$388$	19.1421	0.971791
$389$	1.08650	0.0550879	0.0275439	−	0.999621i	$-0.491231\pi$
$389$	1.08650	0.0550879	0.0275439	+	0.999621i	$0.491231\pi$
$390$	0	0
$391$	28.7275	1.45281
$392$	7.98913	0.403512
$393$	10.5201	0.530667
$394$	−57.2161	−2.88250
$395$	0	0
$396$	−26.3616	−1.32472
$397$	22.6258	1.13556	0.567778	−	0.823182i	$-0.307803\pi$
$397$	22.6258	1.13556	0.567778	+	0.823182i	$0.307803\pi$
$398$	−1.86246	−0.0933565
$399$	5.62475	0.281590
$400$	0	0
$401$	1.35455	0.0676428	0.0338214	−	0.999428i	$-0.489232\pi$
$401$	1.35455	0.0676428	0.0338214	+	0.999428i	$0.489232\pi$
$402$	−4.27320	−0.213128
$403$	−6.18039	−0.307867
$404$	−19.8347	−0.986814
$405$	0	0
$406$	−48.1197	−2.38814
$407$	25.5349	1.26572
$408$	−38.6949	−1.91568
$409$	33.8119	1.67189	0.835945	−	0.548813i	$-0.184920\pi$
$409$	33.8119	1.67189	0.835945	+	0.548813i	$0.184920\pi$
$410$	0	0
$411$	16.0904	0.793679
$412$	−22.7950	−1.12303
$413$	19.0626	0.938007
$414$	25.4776	1.25216
$415$	0	0
$416$	−9.15818	−0.449017
$417$	2.35576	0.115362
$418$	−17.2112	−0.841827
$419$	6.43048	0.314150	0.157075	−	0.987587i	$-0.449794\pi$
$419$	6.43048	0.314150	0.157075	+	0.987587i	$0.449794\pi$
$420$	0	0
$421$	12.8887	0.628157	0.314079	−	0.949397i	$-0.398304\pi$
$421$	12.8887	0.628157	0.314079	+	0.949397i	$0.398304\pi$
$422$	21.3684	1.04020
$423$	−8.68143	−0.422106
$424$	−13.5030	−0.655765
$425$	0	0
$426$	34.4364	1.66845
$427$	−16.0038	−0.774478
$428$	21.1455	1.02211
$429$	3.40363	0.164329
$430$	0	0
$431$	−14.3106	−0.689316	−0.344658	−	0.938728i	$-0.612005\pi$
$431$	−14.3106	−0.689316	−0.344658	+	0.938728i	$0.612005\pi$
$432$	−38.9937	−1.87608
$433$	−5.31418	−0.255383	−0.127692	−	0.991814i	$-0.540757\pi$
$433$	−5.31418	−0.255383	−0.127692	+	0.991814i	$0.540757\pi$
$434$	29.8537	1.43302
$435$	0	0
$436$	−31.8112	−1.52348
$437$	11.5964	0.554730
$438$	22.2080	1.06114
$439$	−37.7150	−1.80004	−0.900020	−	0.435849i	$-0.856448\pi$
$439$	−37.7150	−1.80004	−0.900020	+	0.435849i	$0.856448\pi$
$440$	0	0
$441$	2.47694	0.117950
$442$	−19.8027	−0.941919
$443$	−10.2200	−0.485567	−0.242783	−	0.970081i	$-0.578060\pi$
$443$	−10.2200	−0.485567	−0.242783	+	0.970081i	$0.578060\pi$
$444$	40.9737	1.94453
$445$	0	0
$446$	62.3191	2.95089
$447$	22.3411	1.05670
$448$	5.74340	0.271350
$449$	8.72689	0.411847	0.205924	−	0.978568i	$-0.433980\pi$
$449$	8.72689	0.411847	0.205924	+	0.978568i	$0.433980\pi$
$450$	0	0
$451$	8.82035	0.415334
$452$	−68.1564	−3.20581
$453$	3.01760	0.141779
$454$	50.0166	2.34740
$455$	0	0
$456$	−15.6199	−0.731468
$457$	8.00249	0.374341	0.187170	−	0.982327i	$-0.440068\pi$
$457$	8.00249	0.374341	0.187170	+	0.982327i	$0.440068\pi$
$458$	−35.6141	−1.66414
$459$	−29.3453	−1.36972
$460$	0	0
$461$	−16.7451	−0.779898	−0.389949	−	0.920836i	$-0.627507\pi$
$461$	−16.7451	−0.779898	−0.389949	+	0.920836i	$0.627507\pi$
$462$	−16.4409	−0.764899
$463$	15.5116	0.720883	0.360442	−	0.932782i	$-0.382626\pi$
$463$	15.5116	0.720883	0.360442	+	0.932782i	$0.382626\pi$
$464$	62.0739	2.88171
$465$	0	0
$466$	−54.6781	−2.53291
$467$	24.4421	1.13105	0.565523	−	0.824732i	$-0.308674\pi$
$467$	24.4421	1.13105	0.565523	+	0.824732i	$0.308674\pi$
$468$	−12.2436	−0.565961
$469$	4.16511	0.192327
$470$	0	0
$471$	−1.95872	−0.0902529
$472$	−52.9366	−2.43660
$473$	21.7933	1.00206
$474$	9.64539	0.443027
$475$	0	0
$476$	66.6856	3.05653
$477$	−4.18647	−0.191685
$478$	−48.2258	−2.20580
$479$	−37.1364	−1.69681	−0.848403	−	0.529351i	$-0.822435\pi$
$479$	−37.1364	−1.69681	−0.848403	+	0.529351i	$0.822435\pi$
$480$	0	0
$481$	11.8597	0.540754
$482$	−2.56980	−0.117051
$483$	11.0774	0.504038
$484$	−15.5707	−0.707758
$485$	0	0
$486$	−41.4113	−1.87845
$487$	−36.9350	−1.67369	−0.836843	−	0.547443i	$-0.815601\pi$
$487$	−36.9350	−1.67369	−0.836843	+	0.547443i	$0.815601\pi$
$488$	44.4425	2.01182
$489$	−2.07462	−0.0938176
$490$	0	0
$491$	−3.28780	−0.148376	−0.0741882	−	0.997244i	$-0.523637\pi$
$491$	−3.28780	−0.148376	−0.0741882	+	0.997244i	$0.523637\pi$
$492$	14.1533	0.638080
$493$	46.7147	2.10392
$494$	−7.99373	−0.359655
$495$	0	0
$496$	−38.5110	−1.72920
$497$	−33.5654	−1.50561
$498$	9.21743	0.413043
$499$	−4.08606	−0.182917	−0.0914586	−	0.995809i	$-0.529153\pi$
$499$	−4.08606	−0.182917	−0.0914586	+	0.995809i	$0.529153\pi$
$500$	0	0
$501$	−16.7581	−0.748696
$502$	−52.3398	−2.33604
$503$	22.0549	0.983381	0.491690	−	0.870770i	$-0.336379\pi$
$503$	22.0549	0.983381	0.491690	+	0.870770i	$0.336379\pi$
$504$	33.4494	1.48995
$505$	0	0
$506$	−33.8957	−1.50685
$507$	−10.9250	−0.485195
$508$	86.4481	3.83551
$509$	28.2247	1.25104	0.625519	−	0.780209i	$-0.284887\pi$
$509$	28.2247	1.25104	0.625519	+	0.780209i	$0.284887\pi$
$510$	0	0
$511$	−21.6463	−0.957577
$512$	49.8322	2.20229
$513$	−11.8458	−0.523003
$514$	11.0134	0.485782
$515$	0	0
$516$	34.9700	1.53947
$517$	11.5499	0.507963
$518$	−57.2869	−2.51704
$519$	−7.58599	−0.332988
$520$	0	0
$521$	37.1763	1.62873	0.814363	−	0.580356i	$-0.197087\pi$
$521$	37.1763	1.62873	0.814363	+	0.580356i	$0.197087\pi$
$522$	41.4299	1.81334
$523$	−1.04536	−0.0457105	−0.0228552	−	0.999739i	$-0.507276\pi$
$523$	−1.04536	−0.0457105	−0.0228552	+	0.999739i	$0.507276\pi$
$524$	50.3469	2.19941
$525$	0	0
$526$	−25.4896	−1.11140
$527$	−28.9821	−1.26248
$528$	21.2086	0.922984
$529$	−0.162090	−0.00704741
$530$	0	0
$531$	−16.4124	−0.712237
$532$	26.9189	1.16708
$533$	4.09661	0.177444
$534$	27.9732	1.21052
$535$	0	0
$536$	−11.5665	−0.499596
$537$	−6.73741	−0.290741
$538$	14.5511	0.627342
$539$	−3.29535	−0.141941
$540$	0	0
$541$	−23.9366	−1.02911	−0.514557	−	0.857456i	$-0.672044\pi$
$541$	−23.9366	−1.02911	−0.514557	+	0.857456i	$0.672044\pi$
$542$	4.09593	0.175935
$543$	−21.4266	−0.919503
$544$	−42.9460	−1.84130
$545$	0	0
$546$	−7.63596	−0.326789
$547$	42.6740	1.82461	0.912304	−	0.409514i	$-0.134302\pi$
$547$	42.6740	1.82461	0.912304	+	0.409514i	$0.134302\pi$
$548$	77.0051	3.28950
$549$	13.7789	0.588069
$550$	0	0
$551$	18.8572	0.803346
$552$	−30.7618	−1.30931
$553$	−9.40142	−0.399789
$554$	1.41102	0.0599487
$555$	0	0
$556$	11.2742	0.478131
$557$	−14.6507	−0.620770	−0.310385	−	0.950611i	$-0.600458\pi$
$557$	−14.6507	−0.620770	−0.310385	+	0.950611i	$0.600458\pi$
$558$	−25.7033	−1.08811
$559$	10.1219	0.428111
$560$	0	0
$561$	15.9608	0.673867
$562$	−68.0330	−2.86980
$563$	−41.1119	−1.73266	−0.866329	−	0.499474i	$-0.833527\pi$
$563$	−41.1119	−1.73266	−0.866329	+	0.499474i	$0.833527\pi$
$564$	18.5332	0.780387
$565$	0	0
$566$	1.09248	0.0459202
$567$	3.68104	0.154589
$568$	93.2109	3.91104
$569$	−25.8285	−1.08279	−0.541393	−	0.840770i	$-0.682103\pi$
$569$	−25.8285	−1.08279	−0.541393	+	0.840770i	$0.682103\pi$
$570$	0	0
$571$	42.4266	1.77550	0.887750	−	0.460326i	$-0.152268\pi$
$571$	42.4266	1.77550	0.887750	+	0.460326i	$0.152268\pi$
$572$	16.2890	0.681079
$573$	6.61842	0.276488
$574$	−19.7882	−0.825945
$575$	0	0
$576$	−4.94493	−0.206039
$577$	13.5474	0.563984	0.281992	−	0.959417i	$-0.409005\pi$
$577$	13.5474	0.563984	0.281992	+	0.959417i	$0.409005\pi$
$578$	−49.1755	−2.04543
$579$	−15.0889	−0.627073
$580$	0	0
$581$	−8.98429	−0.372731
$582$	−10.2786	−0.426061
$583$	5.56973	0.230674
$584$	60.1117	2.48744
$585$	0	0
$586$	−39.0988	−1.61516
$587$	−2.89175	−0.119355	−0.0596777	−	0.998218i	$-0.519007\pi$
$587$	−2.89175	−0.119355	−0.0596777	+	0.998218i	$0.519007\pi$
$588$	−5.28778	−0.218064
$589$	−11.6991	−0.482055
$590$	0	0
$591$	21.4184	0.881036
$592$	73.8995	3.03725
$593$	2.77354	0.113895	0.0569477	−	0.998377i	$-0.481863\pi$
$593$	2.77354	0.113895	0.0569477	+	0.998377i	$0.481863\pi$
$594$	34.6246	1.42067
$595$	0	0
$596$	106.920	4.37960
$597$	0.697196	0.0285344
$598$	−15.7428	−0.643773
$599$	−13.4449	−0.549345	−0.274673	−	0.961538i	$-0.588569\pi$
$599$	−13.4449	−0.549345	−0.274673	+	0.961538i	$0.588569\pi$
$600$	0	0
$601$	19.2296	0.784393	0.392197	−	0.919881i	$-0.371715\pi$
$601$	19.2296	0.784393	0.392197	+	0.919881i	$0.371715\pi$
$602$	−48.8928	−1.99272
$603$	−3.58606	−0.146036
$604$	14.4416	0.587621
$605$	0	0
$606$	10.6505	0.432648
$607$	26.9188	1.09260	0.546300	−	0.837590i	$-0.316036\pi$
$607$	26.9188	1.09260	0.546300	+	0.837590i	$0.316036\pi$
$608$	−17.3359	−0.703065
$609$	18.0133	0.729934
$610$	0	0
$611$	5.36434	0.217018
$612$	−57.4147	−2.32085
$613$	17.8828	0.722278	0.361139	−	0.932512i	$-0.382388\pi$
$613$	17.8828	0.722278	0.361139	+	0.932512i	$0.382388\pi$
$614$	−41.4031	−1.67089
$615$	0	0
$616$	−44.5014	−1.79301
$617$	−0.455631	−0.0183430	−0.00917152	−	0.999958i	$-0.502919\pi$
$617$	−0.455631	−0.0183430	−0.00917152	+	0.999958i	$0.502919\pi$
$618$	12.2401	0.492370
$619$	36.4206	1.46387	0.731934	−	0.681376i	$-0.238618\pi$
$619$	36.4206	1.46387	0.731934	+	0.681376i	$0.238618\pi$
$620$	0	0
$621$	−23.3290	−0.936162
$622$	87.2786	3.49955
$623$	−27.2656	−1.09237
$624$	9.85031	0.394328
$625$	0	0
$626$	62.7217	2.50686
$627$	6.44288	0.257304
$628$	−9.37401	−0.374064
$629$	55.6142	2.21748
$630$	0	0
$631$	33.3334	1.32698	0.663491	−	0.748184i	$-0.269074\pi$
$631$	33.3334	1.32698	0.663491	+	0.748184i	$0.269074\pi$
$632$	26.1077	1.03851
$633$	−7.99911	−0.317936
$634$	1.40139	0.0556565
$635$	0	0
$636$	8.93728	0.354386
$637$	−1.53052	−0.0606416
$638$	−55.1189	−2.18218
$639$	28.8990	1.14323
$640$	0	0
$641$	32.7954	1.29534	0.647670	−	0.761921i	$-0.275744\pi$
$641$	32.7954	1.29534	0.647670	+	0.761921i	$0.275744\pi$
$642$	−11.3544	−0.448121
$643$	−13.9659	−0.550763	−0.275381	−	0.961335i	$-0.588804\pi$
$643$	−13.9659	−0.550763	−0.275381	+	0.961335i	$0.588804\pi$
$644$	53.0140	2.08904
$645$	0	0
$646$	−37.4855	−1.47485
$647$	−3.29858	−0.129681	−0.0648403	−	0.997896i	$-0.520654\pi$
$647$	−3.29858	−0.129681	−0.0648403	+	0.997896i	$0.520654\pi$
$648$	−10.2222	−0.401567
$649$	21.8353	0.857109
$650$	0	0
$651$	−11.1755	−0.438003
$652$	−9.92871	−0.388838
$653$	15.4869	0.606048	0.303024	−	0.952983i	$-0.402004\pi$
$653$	15.4869	0.606048	0.303024	+	0.952983i	$0.402004\pi$
$654$	17.0815	0.667938
$655$	0	0
$656$	25.5266	0.996647
$657$	18.6370	0.727097
$658$	−25.9119	−1.01015
$659$	2.29471	0.0893894	0.0446947	−	0.999001i	$-0.485769\pi$
$659$	2.29471	0.0893894	0.0446947	+	0.999001i	$0.485769\pi$
$660$	0	0
$661$	22.8134	0.887340	0.443670	−	0.896190i	$-0.353676\pi$
$661$	22.8134	0.887340	0.443670	+	0.896190i	$0.353676\pi$
$662$	22.3551	0.868857
$663$	7.41300	0.287897
$664$	24.9493	0.968222
$665$	0	0
$666$	49.3226	1.91121
$667$	37.1374	1.43797
$668$	−80.2007	−3.10306
$669$	−23.3287	−0.901939
$670$	0	0
$671$	−18.3316	−0.707684
$672$	−16.5600	−0.638818
$673$	3.49826	0.134848	0.0674240	−	0.997724i	$-0.478522\pi$
$673$	3.49826	0.134848	0.0674240	+	0.997724i	$0.478522\pi$
$674$	45.5973	1.75634
$675$	0	0
$676$	−52.2847	−2.01095
$677$	25.4706	0.978914	0.489457	−	0.872027i	$-0.337195\pi$
$677$	25.4706	0.978914	0.489457	+	0.872027i	$0.337195\pi$
$678$	36.5975	1.40552
$679$	10.0186	0.384479
$680$	0	0
$681$	−18.7233	−0.717480
$682$	34.1960	1.30943
$683$	25.5350	0.977069	0.488534	−	0.872545i	$-0.337532\pi$
$683$	25.5350	0.977069	0.488534	+	0.872545i	$0.337532\pi$
$684$	−23.1765	−0.886175
$685$	0	0
$686$	50.7379	1.93718
$687$	13.3319	0.508643
$688$	63.0712	2.40457
$689$	2.58686	0.0985514
$690$	0	0
$691$	−45.8219	−1.74315	−0.871574	−	0.490263i	$-0.836901\pi$
$691$	−45.8219	−1.74315	−0.871574	+	0.490263i	$0.836901\pi$
$692$	−36.3050	−1.38011
$693$	−13.7972	−0.524111
$694$	8.91849	0.338541
$695$	0	0
$696$	−50.0227	−1.89611
$697$	19.2105	0.727648
$698$	38.3748	1.45251
$699$	20.4683	0.774183
$700$	0	0
$701$	20.2179	0.763619	0.381810	−	0.924241i	$-0.375301\pi$
$701$	20.2179	0.763619	0.381810	+	0.924241i	$0.375301\pi$
$702$	16.0814	0.606953
$703$	22.4497	0.846706
$704$	6.57880	0.247948
$705$	0	0
$706$	90.6128	3.41026
$707$	−10.3811	−0.390423
$708$	35.0372	1.31678
$709$	31.1300	1.16911	0.584555	−	0.811354i	$-0.301269\pi$
$709$	31.1300	1.16911	0.584555	+	0.811354i	$0.301269\pi$
$710$	0	0
$711$	8.09440	0.303564
$712$	75.7165	2.83760
$713$	−23.0403	−0.862865
$714$	−35.8078	−1.34007
$715$	0	0
$716$	−32.2438	−1.20501
$717$	18.0530	0.674200
$718$	−10.4504	−0.390004
$719$	−0.297632	−0.0110998	−0.00554991	−	0.999985i	$-0.501767\pi$
$719$	−0.297632	−0.0110998	−0.00554991	+	0.999985i	$0.501767\pi$
$720$	0	0
$721$	−11.9305	−0.444316
$722$	33.6944	1.25398
$723$	0.961984	0.0357766
$724$	−102.543	−3.81099
$725$	0	0
$726$	8.36088	0.310302
$727$	−20.2020	−0.749250	−0.374625	−	0.927176i	$-0.622229\pi$
$727$	−20.2020	−0.749250	−0.374625	+	0.927176i	$0.622229\pi$
$728$	−20.6687	−0.766032
$729$	10.9190	0.404407
$730$	0	0
$731$	47.4652	1.75556
$732$	−29.4152	−1.08722
$733$	−47.4355	−1.75207	−0.876034	−	0.482248i	$-0.839820\pi$
$733$	−47.4355	−1.75207	−0.876034	+	0.482248i	$0.839820\pi$
$734$	31.3386	1.15673
$735$	0	0
$736$	−34.1414	−1.25847
$737$	4.77094	0.175740
$738$	17.0372	0.627148
$739$	1.24854	0.0459283	0.0229642	−	0.999736i	$-0.492690\pi$
$739$	1.24854	0.0459283	0.0229642	+	0.999736i	$0.492690\pi$
$740$	0	0
$741$	2.99239	0.109928
$742$	−12.4955	−0.458726
$743$	−33.3293	−1.22273	−0.611367	−	0.791347i	$-0.709380\pi$
$743$	−33.3293	−1.22273	−0.611367	+	0.791347i	$0.709380\pi$
$744$	31.0344	1.13777
$745$	0	0
$746$	−51.5803	−1.88849
$747$	7.73526	0.283018
$748$	76.3852	2.79292
$749$	11.0672	0.404386
$750$	0	0
$751$	−5.76748	−0.210458	−0.105229	−	0.994448i	$-0.533558\pi$
$751$	−5.76748	−0.210458	−0.105229	+	0.994448i	$0.533558\pi$
$752$	33.4261	1.21892
$753$	19.5930	0.714009
$754$	−25.5999	−0.932295
$755$	0	0
$756$	−54.1540	−1.96956
$757$	−33.9884	−1.23533	−0.617664	−	0.786442i	$-0.711921\pi$
$757$	−33.9884	−1.23533	−0.617664	+	0.786442i	$0.711921\pi$
$758$	8.56936	0.311253
$759$	12.6886	0.460567
$760$	0	0
$761$	−50.3248	−1.82427	−0.912136	−	0.409888i	$-0.865568\pi$
$761$	−50.3248	−1.82427	−0.912136	+	0.409888i	$0.865568\pi$
$762$	−46.4195	−1.68160
$763$	−16.6494	−0.602749
$764$	31.6744	1.14594
$765$	0	0
$766$	38.1006	1.37663
$767$	10.1414	0.366184
$768$	−24.7659	−0.893662
$769$	15.1953	0.547956	0.273978	−	0.961736i	$-0.411661\pi$
$769$	15.1953	0.547956	0.273978	+	0.961736i	$0.411661\pi$
$770$	0	0
$771$	−4.12280	−0.148479
$772$	−72.2123	−2.59898
$773$	23.7630	0.854695	0.427347	−	0.904088i	$-0.359448\pi$
$773$	23.7630	0.854695	0.427347	+	0.904088i	$0.359448\pi$
$774$	42.0955	1.51309
$775$	0	0
$776$	−27.8216	−0.998738
$777$	21.4449	0.769332
$778$	−2.79209	−0.100101
$779$	7.75465	0.277839
$780$	0	0
$781$	−38.4476	−1.37576
$782$	−73.8239	−2.63994
$783$	−37.9361	−1.35572
$784$	−9.53695	−0.340605
$785$	0	0
$786$	−27.0344	−0.964286
$787$	−13.1357	−0.468238	−0.234119	−	0.972208i	$-0.575220\pi$
$787$	−13.1357	−0.468238	−0.234119	+	0.972208i	$0.575220\pi$
$788$	102.504	3.65156
$789$	9.54184	0.339698
$790$	0	0
$791$	−35.6718	−1.26834
$792$	38.3147	1.36145
$793$	−8.51410	−0.302345
$794$	−58.1437	−2.06344
$795$	0	0
$796$	3.33664	0.118264
$797$	−5.90153	−0.209043	−0.104521	−	0.994523i	$-0.533331\pi$
$797$	−5.90153	−0.209043	−0.104521	+	0.994523i	$0.533331\pi$
$798$	−14.4545	−0.511682
$799$	25.1553	0.889931
$800$	0	0
$801$	23.4750	0.829450
$802$	−3.48091	−0.122915
$803$	−24.7949	−0.874991
$804$	7.65554	0.269990
$805$	0	0
$806$	15.8823	0.559431
$807$	−5.44709	−0.191747
$808$	28.8284	1.01418
$809$	1.27468	0.0448153	0.0224076	−	0.999749i	$-0.492867\pi$
$809$	1.27468	0.0448153	0.0224076	+	0.999749i	$0.492867\pi$
$810$	0	0
$811$	−34.2824	−1.20382	−0.601910	−	0.798564i	$-0.705593\pi$
$811$	−34.2824	−1.20382	−0.601910	+	0.798564i	$0.705593\pi$
$812$	86.2077	3.02530
$813$	−1.53328	−0.0537745
$814$	−65.6194	−2.29996
$815$	0	0
$816$	46.1916	1.61703
$817$	19.1602	0.670331
$818$	−86.8897	−3.03803
$819$	−6.40809	−0.223917
$820$	0	0
$821$	38.5877	1.34672	0.673360	−	0.739315i	$-0.264850\pi$
$821$	38.5877	1.34672	0.673360	+	0.739315i	$0.264850\pi$
$822$	−41.3490	−1.44221
$823$	47.9481	1.67137	0.835683	−	0.549213i	$-0.185072\pi$
$823$	47.9481	1.67137	0.835683	+	0.549213i	$0.185072\pi$
$824$	33.1310	1.15417
$825$	0	0
$826$	−48.9869	−1.70447
$827$	36.4866	1.26876	0.634382	−	0.773019i	$-0.281255\pi$
$827$	36.4866	1.26876	0.634382	+	0.773019i	$0.281255\pi$
$828$	−45.6438	−1.58623
$829$	30.4656	1.05811	0.529056	−	0.848587i	$-0.322546\pi$
$829$	30.4656	1.05811	0.529056	+	0.848587i	$0.322546\pi$
$830$	0	0
$831$	−0.528206	−0.0183233
$832$	3.05552	0.105931
$833$	−7.17718	−0.248675
$834$	−6.05382	−0.209627
$835$	0	0
$836$	30.8343	1.06643
$837$	23.5357	0.813514
$838$	−16.5250	−0.570848
$839$	−16.0710	−0.554833	−0.277417	−	0.960750i	$-0.589478\pi$
$839$	−16.0710	−0.554833	−0.277417	+	0.960750i	$0.589478\pi$
$840$	0	0
$841$	31.3903	1.08243
$842$	−33.1214	−1.14144
$843$	25.4676	0.877152
$844$	−38.2821	−1.31772
$845$	0	0
$846$	22.3095	0.767017
$847$	−8.14941	−0.280017
$848$	16.1191	0.553533
$849$	−0.408961	−0.0140355
$850$	0	0
$851$	44.2124	1.51558
$852$	−61.6937	−2.11359
$853$	42.0171	1.43864	0.719319	−	0.694680i	$-0.244454\pi$
$853$	42.0171	1.43864	0.719319	+	0.694680i	$0.244454\pi$
$854$	41.1265	1.40732
$855$	0	0
$856$	−30.7335	−1.05045
$857$	−14.3149	−0.488988	−0.244494	−	0.969651i	$-0.578622\pi$
$857$	−14.3149	−0.488988	−0.244494	+	0.969651i	$0.578622\pi$
$858$	−8.74663	−0.298605
$859$	−35.1104	−1.19795	−0.598975	−	0.800767i	$-0.704425\pi$
$859$	−35.1104	−1.19795	−0.598975	+	0.800767i	$0.704425\pi$
$860$	0	0
$861$	7.40758	0.252450
$862$	36.7753	1.25257
$863$	−15.2640	−0.519592	−0.259796	−	0.965664i	$-0.583655\pi$
$863$	−15.2640	−0.519592	−0.259796	+	0.965664i	$0.583655\pi$
$864$	34.8756	1.18649
$865$	0	0
$866$	13.6564	0.464062
$867$	18.4085	0.625185
$868$	−53.4837	−1.81536
$869$	−10.7689	−0.365310
$870$	0	0
$871$	2.21586	0.0750816
$872$	46.2353	1.56573
$873$	−8.62579	−0.291939
$874$	−29.8004	−1.00801
$875$	0	0
$876$	−39.7863	−1.34425
$877$	13.6238	0.460043	0.230021	−	0.973186i	$-0.426120\pi$
$877$	13.6238	0.460043	0.230021	+	0.973186i	$0.426120\pi$
$878$	96.9199	3.27089
$879$	14.6363	0.493671
$880$	0	0
$881$	−20.2263	−0.681441	−0.340720	−	0.940165i	$-0.610671\pi$
$881$	−20.2263	−0.681441	−0.340720	+	0.940165i	$0.610671\pi$
$882$	−6.36524	−0.214329
$883$	25.4683	0.857075	0.428537	−	0.903524i	$-0.359029\pi$
$883$	25.4683	0.857075	0.428537	+	0.903524i	$0.359029\pi$
$884$	35.4771	1.19322
$885$	0	0
$886$	26.2633	0.882333
$887$	38.3721	1.28841	0.644205	−	0.764853i	$-0.277188\pi$
$887$	38.3721	1.28841	0.644205	+	0.764853i	$0.277188\pi$
$888$	−59.5524	−1.99845
$889$	45.2454	1.51748
$890$	0	0
$891$	4.21645	0.141256
$892$	−111.646	−3.73819
$893$	10.1544	0.339804
$894$	−57.4120	−1.92015
$895$	0	0
$896$	19.6696	0.657114
$897$	5.89322	0.196769
$898$	−22.4263	−0.748376
$899$	−37.4665	−1.24958
$900$	0	0
$901$	12.1307	0.404132
$902$	−22.6665	−0.754712
$903$	18.3027	0.609074
$904$	99.0605	3.29470
$905$	0	0
$906$	−7.75463	−0.257630
$907$	−9.43444	−0.313265	−0.156633	−	0.987657i	$-0.550064\pi$
$907$	−9.43444	−0.313265	−0.156633	+	0.987657i	$0.550064\pi$
$908$	−89.6060	−2.97368
$909$	8.93791	0.296452
$910$	0	0
$911$	45.5096	1.50780	0.753901	−	0.656988i	$-0.228170\pi$
$911$	45.5096	1.50780	0.753901	+	0.656988i	$0.228170\pi$
$912$	18.6461	0.617434
$913$	−10.2911	−0.340585
$914$	−20.5648	−0.680222
$915$	0	0
$916$	63.8036	2.10813
$917$	26.3506	0.870175
$918$	75.4114	2.48895
$919$	7.61377	0.251155	0.125578	−	0.992084i	$-0.459922\pi$
$919$	7.61377	0.251155	0.125578	+	0.992084i	$0.459922\pi$
$920$	0	0
$921$	15.4989	0.510707
$922$	43.0316	1.41717
$923$	−17.8570	−0.587769
$924$	29.4542	0.968974
$925$	0	0
$926$	−39.8616	−1.30993
$927$	10.2719	0.337373
$928$	−55.5184	−1.82248
$929$	−6.32853	−0.207632	−0.103816	−	0.994597i	$-0.533105\pi$
$929$	−6.32853	−0.207632	−0.103816	+	0.994597i	$0.533105\pi$
$930$	0	0
$931$	−2.89720	−0.0949519
$932$	97.9571	3.20869
$933$	−32.6721	−1.06964
$934$	−62.8113	−2.05525
$935$	0	0
$936$	17.7952	0.581655
$937$	−7.95709	−0.259947	−0.129973	−	0.991517i	$-0.541489\pi$
$937$	−7.95709	−0.259947	−0.129973	+	0.991517i	$0.541489\pi$
$938$	−10.7035	−0.349481
$939$	−23.4794	−0.766221
$940$	0	0
$941$	−4.65531	−0.151759	−0.0758793	−	0.997117i	$-0.524176\pi$
$941$	−4.65531	−0.151759	−0.0758793	+	0.997117i	$0.524176\pi$
$942$	5.03350	0.164000
$943$	15.2720	0.497325
$944$	63.1926	2.05674
$945$	0	0
$946$	−56.0044	−1.82086
$947$	3.24236	0.105363	0.0526813	−	0.998611i	$-0.483223\pi$
$947$	3.24236	0.105363	0.0526813	+	0.998611i	$0.483223\pi$
$948$	−17.2800	−0.561227
$949$	−11.5160	−0.373824
$950$	0	0
$951$	−0.524601	−0.0170114
$952$	−96.9228	−3.14129
$953$	27.7286	0.898216	0.449108	−	0.893478i	$-0.351742\pi$
$953$	27.7286	0.898216	0.449108	+	0.893478i	$0.351742\pi$
$954$	10.7584	0.348315
$955$	0	0
$956$	86.3977	2.79430
$957$	20.6333	0.666981
$958$	95.4330	3.08330
$959$	40.3031	1.30146
$960$	0	0
$961$	−7.75559	−0.250180
$962$	−30.4769	−0.982615
$963$	−9.52857	−0.307054
$964$	4.60385	0.148280
$965$	0	0
$966$	−28.4666	−0.915897
$967$	38.0713	1.22429	0.612145	−	0.790746i	$-0.290307\pi$
$967$	38.0713	1.22429	0.612145	+	0.790746i	$0.290307\pi$
$968$	22.6309	0.727383
$969$	14.0324	0.450786
$970$	0	0
$971$	−18.0090	−0.577935	−0.288968	−	0.957339i	$-0.593312\pi$
$971$	−18.0090	−0.577935	−0.288968	+	0.957339i	$0.593312\pi$
$972$	74.1893	2.37962
$973$	5.90070	0.189168
$974$	94.9155	3.04129
$975$	0	0
$976$	−53.0528	−1.69818
$977$	−33.0043	−1.05590	−0.527951	−	0.849275i	$-0.677040\pi$
$977$	−33.0043	−1.05590	−0.527951	+	0.849275i	$0.677040\pi$
$978$	5.33136	0.170478
$979$	−31.2315	−0.998163
$980$	0	0
$981$	14.3347	0.457673
$982$	8.44898	0.269618
$983$	23.3592	0.745043	0.372522	−	0.928024i	$-0.378493\pi$
$983$	23.3592	0.745043	0.372522	+	0.928024i	$0.378493\pi$
$984$	−20.5708	−0.655773
$985$	0	0
$986$	−120.047	−3.82308
$987$	9.69992	0.308752
$988$	14.3210	0.455611
$989$	37.7341	1.19987
$990$	0	0
$991$	17.3157	0.550050	0.275025	−	0.961437i	$-0.411314\pi$
$991$	17.3157	0.550050	0.275025	+	0.961437i	$0.411314\pi$
$992$	34.4439	1.09359
$993$	−8.36848	−0.265566
$994$	86.2562	2.73588
$995$	0	0
$996$	−16.5133	−0.523243
$997$	−5.87493	−0.186061	−0.0930305	−	0.995663i	$-0.529655\pi$
$997$	−5.87493	−0.186061	−0.0930305	+	0.995663i	$0.529655\pi$
$998$	10.5003	0.332382
$999$	−45.1632	−1.42890

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.n.1.1	yes	40
5.4	even	2		6025.2.a.m.1.40	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6025.2.a.m.1.40	✓	40	5.4	even	2
6025.2.a.n.1.1	yes	40	1.1	even	1	trivial

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q-2.56980 q^{2} +0.961984 q^{3} +4.60385 q^{4} -2.47210 q^{6} +2.40958 q^{7} -6.69138 q^{8} -2.07459 q^{9} +O(q^{10})\)	\(q-2.56980 q^{2} +0.961984 q^{3} +4.60385 q^{4} -2.47210 q^{6} +2.40958 q^{7} -6.69138 q^{8} -2.07459 q^{9} +2.76006 q^{11} +4.42884 q^{12} +1.28191 q^{13} -6.19212 q^{14} +7.98777 q^{16} +6.01132 q^{17} +5.33127 q^{18} +2.42658 q^{19} +2.31797 q^{21} -7.09278 q^{22} +4.77890 q^{23} -6.43700 q^{24} -3.29424 q^{26} -4.88167 q^{27} +11.0933 q^{28} +7.77112 q^{29} -4.82125 q^{31} -7.14419 q^{32} +2.65513 q^{33} -15.4479 q^{34} -9.55110 q^{36} +9.25158 q^{37} -6.23582 q^{38} +1.23317 q^{39} +3.19571 q^{41} -5.95672 q^{42} +7.89597 q^{43} +12.7069 q^{44} -12.2808 q^{46} +4.18466 q^{47} +7.68411 q^{48} -1.19394 q^{49} +5.78279 q^{51} +5.90171 q^{52} +2.01798 q^{53} +12.5449 q^{54} -16.1234 q^{56} +2.33433 q^{57} -19.9702 q^{58} +7.91117 q^{59} -6.64175 q^{61} +12.3896 q^{62} -4.99887 q^{63} +2.38357 q^{64} -6.82314 q^{66} +1.72857 q^{67} +27.6752 q^{68} +4.59723 q^{69} -13.9300 q^{71} +13.8818 q^{72} -8.98346 q^{73} -23.7747 q^{74} +11.1716 q^{76} +6.65056 q^{77} -3.16901 q^{78} -3.90169 q^{79} +1.52767 q^{81} -8.21233 q^{82} -3.72858 q^{83} +10.6716 q^{84} -20.2910 q^{86} +7.47570 q^{87} -18.4686 q^{88} -11.3155 q^{89} +3.08885 q^{91} +22.0014 q^{92} -4.63796 q^{93} -10.7537 q^{94} -6.87260 q^{96} +4.15783 q^{97} +3.06819 q^{98} -5.72597 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 9 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 20 q^{7} + 27 q^{8} + 38 q^{9}+O(q^{10}) \) 40 * q + 9 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 20 * q^7 + 27 * q^8 + 38 * q^9	\( 40 q + 9 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 20 q^{7} + 27 q^{8} + 38 q^{9} + q^{11} + 26 q^{12} + 11 q^{13} - q^{14} + 43 q^{16} + 20 q^{17} + 18 q^{18} + 2 q^{21} + 23 q^{22} + 79 q^{23} - 2 q^{24} + 2 q^{26} + 26 q^{27} + 30 q^{28} + 2 q^{29} + q^{31} + 68 q^{32} + 20 q^{33} + 5 q^{34} + 32 q^{36} + 16 q^{37} + 45 q^{38} - 2 q^{39} - 2 q^{41} + 19 q^{42} + 25 q^{43} + 3 q^{44} + 14 q^{46} + 88 q^{47} + 75 q^{48} + 40 q^{49} - 10 q^{51} + 18 q^{52} + 34 q^{53} + 4 q^{54} - 15 q^{56} + 51 q^{57} + 53 q^{58} + q^{59} + 9 q^{61} + 39 q^{62} + 110 q^{63} + 17 q^{64} + 26 q^{66} + 30 q^{67} + 44 q^{68} - 7 q^{69} + 5 q^{71} + 18 q^{72} + 23 q^{73} - 18 q^{74} + 43 q^{76} + 30 q^{77} + 46 q^{78} + 5 q^{79} + 44 q^{81} + 5 q^{82} + 65 q^{83} - 65 q^{84} + 40 q^{86} + 33 q^{87} + 71 q^{88} - 9 q^{89} + q^{91} + 117 q^{92} + 68 q^{93} - 72 q^{94} + 83 q^{96} - 8 q^{97} + 76 q^{98} - 40 q^{99}+O(q^{100}) \) 40 * q + 9 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 20 * q^7 + 27 * q^8 + 38 * q^9 + q^11 + 26 * q^12 + 11 * q^13 - q^14 + 43 * q^16 + 20 * q^17 + 18 * q^18 + 2 * q^21 + 23 * q^22 + 79 * q^23 - 2 * q^24 + 2 * q^26 + 26 * q^27 + 30 * q^28 + 2 * q^29 + q^31 + 68 * q^32 + 20 * q^33 + 5 * q^34 + 32 * q^36 + 16 * q^37 + 45 * q^38 - 2 * q^39 - 2 * q^41 + 19 * q^42 + 25 * q^43 + 3 * q^44 + 14 * q^46 + 88 * q^47 + 75 * q^48 + 40 * q^49 - 10 * q^51 + 18 * q^52 + 34 * q^53 + 4 * q^54 - 15 * q^56 + 51 * q^57 + 53 * q^58 + q^59 + 9 * q^61 + 39 * q^62 + 110 * q^63 + 17 * q^64 + 26 * q^66 + 30 * q^67 + 44 * q^68 - 7 * q^69 + 5 * q^71 + 18 * q^72 + 23 * q^73 - 18 * q^74 + 43 * q^76 + 30 * q^77 + 46 * q^78 + 5 * q^79 + 44 * q^81 + 5 * q^82 + 65 * q^83 - 65 * q^84 + 40 * q^86 + 33 * q^87 + 71 * q^88 - 9 * q^89 + q^91 + 117 * q^92 + 68 * q^93 - 72 * q^94 + 83 * q^96 - 8 * q^97 + 76 * q^98 - 40 * q^99

Introduction

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