LMFDB - Embedded newform 6025.2.a.k.1.19

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:	$25$
Twist minimal:	no (minimal twist has level 1205)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
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)

$f(q)$	$=$	$q+2.06740 q^{2} -3.16922 q^{3} +2.27415 q^{4} -6.55204 q^{6} -3.93807 q^{7} +0.566773 q^{8} +7.04394 q^{9} +O(q^{10})$	\(q+2.06740 q^{2} -3.16922 q^{3} +2.27415 q^{4} -6.55204 q^{6} -3.93807 q^{7} +0.566773 q^{8} +7.04394 q^{9} -1.10618 q^{11} -7.20727 q^{12} -6.41389 q^{13} -8.14157 q^{14} -3.37655 q^{16} -7.11454 q^{17} +14.5626 q^{18} +0.694578 q^{19} +12.4806 q^{21} -2.28692 q^{22} +7.97331 q^{23} -1.79623 q^{24} -13.2601 q^{26} -12.8161 q^{27} -8.95575 q^{28} -9.53328 q^{29} -7.06598 q^{31} -8.11423 q^{32} +3.50572 q^{33} -14.7086 q^{34} +16.0190 q^{36} +5.67490 q^{37} +1.43597 q^{38} +20.3270 q^{39} -3.17397 q^{41} +25.8024 q^{42} -3.80187 q^{43} -2.51562 q^{44} +16.4840 q^{46} -1.99330 q^{47} +10.7010 q^{48} +8.50839 q^{49} +22.5475 q^{51} -14.5861 q^{52} +6.08486 q^{53} -26.4961 q^{54} -2.23199 q^{56} -2.20127 q^{57} -19.7091 q^{58} +1.55555 q^{59} +6.40303 q^{61} -14.6082 q^{62} -27.7395 q^{63} -10.0223 q^{64} +7.24774 q^{66} -0.552548 q^{67} -16.1795 q^{68} -25.2692 q^{69} +0.706993 q^{71} +3.99231 q^{72} +8.09751 q^{73} +11.7323 q^{74} +1.57957 q^{76} +4.35621 q^{77} +42.0241 q^{78} -9.29883 q^{79} +19.4853 q^{81} -6.56188 q^{82} -5.06419 q^{83} +28.3827 q^{84} -7.85998 q^{86} +30.2130 q^{87} -0.626953 q^{88} -12.5416 q^{89} +25.2583 q^{91} +18.1325 q^{92} +22.3936 q^{93} -4.12094 q^{94} +25.7157 q^{96} -5.95406 q^{97} +17.5903 q^{98} -7.79186 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 4 q^{2} - 9 q^{3} + 36 q^{4} + 7 q^{6} - 7 q^{7} + 15 q^{8} + 36 q^{9}+O(q^{10}) $ 25 * q + 4 * q^2 - 9 * q^3 + 36 * q^4 + 7 * q^6 - 7 * q^7 + 15 * q^8 + 36 * q^9	$ 25 q + 4 q^{2} - 9 q^{3} + 36 q^{4} + 7 q^{6} - 7 q^{7} + 15 q^{8} + 36 q^{9} + 10 q^{11} - 22 q^{12} - 10 q^{13} + 13 q^{14} + 54 q^{16} - q^{17} + 13 q^{18} + 50 q^{19} + 9 q^{21} - 11 q^{22} + 31 q^{23} + 22 q^{24} + 8 q^{26} - 42 q^{27} - 14 q^{28} + 4 q^{29} + 34 q^{31} + 44 q^{32} - 28 q^{33} + 33 q^{34} + 83 q^{36} - 14 q^{37} + 10 q^{38} + 23 q^{39} + 11 q^{41} - 23 q^{42} - 49 q^{43} + 20 q^{44} + 27 q^{46} + 28 q^{47} - 30 q^{48} + 66 q^{49} + 49 q^{51} - 39 q^{52} + 16 q^{53} + 5 q^{54} + 51 q^{56} - 10 q^{57} + 8 q^{58} + 30 q^{59} + 35 q^{61} + 18 q^{62} + 73 q^{64} - 13 q^{66} - 37 q^{67} - 11 q^{68} - 4 q^{69} + 12 q^{71} + 90 q^{72} - 36 q^{73} - 12 q^{74} + 57 q^{76} + 31 q^{77} + 9 q^{78} + 16 q^{79} + 65 q^{81} + 11 q^{82} - 43 q^{83} - 62 q^{84} - 9 q^{86} + 22 q^{87} - 20 q^{88} + 38 q^{89} + 86 q^{91} + 119 q^{92} - 10 q^{93} - 18 q^{94} - 34 q^{96} - 17 q^{97} + 32 q^{98} + 26 q^{99}+O(q^{100}) $ 25 * q + 4 * q^2 - 9 * q^3 + 36 * q^4 + 7 * q^6 - 7 * q^7 + 15 * q^8 + 36 * q^9 + 10 * q^11 - 22 * q^12 - 10 * q^13 + 13 * q^14 + 54 * q^16 - q^17 + 13 * q^18 + 50 * q^19 + 9 * q^21 - 11 * q^22 + 31 * q^23 + 22 * q^24 + 8 * q^26 - 42 * q^27 - 14 * q^28 + 4 * q^29 + 34 * q^31 + 44 * q^32 - 28 * q^33 + 33 * q^34 + 83 * q^36 - 14 * q^37 + 10 * q^38 + 23 * q^39 + 11 * q^41 - 23 * q^42 - 49 * q^43 + 20 * q^44 + 27 * q^46 + 28 * q^47 - 30 * q^48 + 66 * q^49 + 49 * q^51 - 39 * q^52 + 16 * q^53 + 5 * q^54 + 51 * q^56 - 10 * q^57 + 8 * q^58 + 30 * q^59 + 35 * q^61 + 18 * q^62 + 73 * q^64 - 13 * q^66 - 37 * q^67 - 11 * q^68 - 4 * q^69 + 12 * q^71 + 90 * q^72 - 36 * q^73 - 12 * q^74 + 57 * q^76 + 31 * q^77 + 9 * q^78 + 16 * q^79 + 65 * q^81 + 11 * q^82 - 43 * q^83 - 62 * q^84 - 9 * q^86 + 22 * q^87 - 20 * q^88 + 38 * q^89 + 86 * q^91 + 119 * q^92 - 10 * q^93 - 18 * q^94 - 34 * q^96 - 17 * q^97 + 32 * q^98 + 26 * 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 $

$2$	2.06740	1.46187	0.730937	−	0.682445i	$-0.239083\pi$
$2$	2.06740	1.46187	0.730937	+	0.682445i	$0.239083\pi$
$3$	−3.16922	−1.82975	−0.914874	−	0.403739i	$-0.867710\pi$
$3$	−3.16922	−1.82975	−0.914874	+	0.403739i	$0.867710\pi$
$4$	2.27415	1.13707
$5$	0	0
$5$	0	0
$6$	−6.55204	−2.67486
$7$	−3.93807	−1.48845	−0.744225	−	0.667929i	$-0.767181\pi$
$7$	−3.93807	−1.48845	−0.744225	+	0.667929i	$0.767181\pi$
$8$	0.566773	0.200384
$9$	7.04394	2.34798
$10$	0	0
$11$	−1.10618	−0.333526	−0.166763	−	0.985997i	$-0.553331\pi$
$11$	−1.10618	−0.333526	−0.166763	+	0.985997i	$0.553331\pi$
$12$	−7.20727	−2.08056
$13$	−6.41389	−1.77889	−0.889446	−	0.457040i	$-0.848910\pi$
$13$	−6.41389	−1.77889	−0.889446	+	0.457040i	$0.848910\pi$
$14$	−8.14157	−2.17593
$15$	0	0
$16$	−3.37655	−0.844137
$17$	−7.11454	−1.72553	−0.862764	−	0.505606i	$-0.831269\pi$
$17$	−7.11454	−1.72553	−0.862764	+	0.505606i	$0.831269\pi$
$18$	14.5626	3.43245
$19$	0.694578	0.159347	0.0796736	−	0.996821i	$-0.474612\pi$
$19$	0.694578	0.159347	0.0796736	+	0.996821i	$0.474612\pi$
$20$	0	0
$21$	12.4806	2.72349
$22$	−2.28692	−0.487572
$23$	7.97331	1.66255	0.831275	−	0.555861i	$-0.187611\pi$
$23$	7.97331	1.66255	0.831275	+	0.555861i	$0.187611\pi$
$24$	−1.79623	−0.366653
$25$	0	0
$26$	−13.2601	−2.60052
$27$	−12.8161	−2.46646
$28$	−8.95575	−1.69248
$29$	−9.53328	−1.77029	−0.885143	−	0.465320i	$-0.845939\pi$
$29$	−9.53328	−1.77029	−0.885143	+	0.465320i	$0.845939\pi$
$30$	0	0
$31$	−7.06598	−1.26909	−0.634544	−	0.772887i	$-0.718812\pi$
$31$	−7.06598	−1.26909	−0.634544	+	0.772887i	$0.718812\pi$
$32$	−8.11423	−1.43441
$33$	3.50572	0.610268
$34$	−14.7086	−2.52250
$35$	0	0
$36$	16.0190	2.66983
$37$	5.67490	0.932948	0.466474	−	0.884535i	$-0.345524\pi$
$37$	5.67490	0.932948	0.466474	+	0.884535i	$0.345524\pi$
$38$	1.43597	0.232945
$39$	20.3270	3.25493
$40$	0	0
$41$	−3.17397	−0.495691	−0.247846	−	0.968800i	$-0.579723\pi$
$41$	−3.17397	−0.495691	−0.247846	+	0.968800i	$0.579723\pi$
$42$	25.8024	3.98140
$43$	−3.80187	−0.579779	−0.289890	−	0.957060i	$-0.593619\pi$
$43$	−3.80187	−0.579779	−0.289890	+	0.957060i	$0.593619\pi$
$44$	−2.51562	−0.379243
$45$	0	0
$46$	16.4840	2.43044
$47$	−1.99330	−0.290752	−0.145376	−	0.989376i	$-0.546439\pi$
$47$	−1.99330	−0.290752	−0.145376	+	0.989376i	$0.546439\pi$
$48$	10.7010	1.54456
$49$	8.50839	1.21548
$50$	0	0
$51$	22.5475	3.15728
$52$	−14.5861	−2.02273
$53$	6.08486	0.835820	0.417910	−	0.908488i	$-0.362763\pi$
$53$	6.08486	0.835820	0.417910	+	0.908488i	$0.362763\pi$
$54$	−26.4961	−3.60566
$55$	0	0
$56$	−2.23199	−0.298262
$57$	−2.20127	−0.291565
$58$	−19.7091	−2.58793
$59$	1.55555	0.202515	0.101258	−	0.994860i	$-0.467713\pi$
$59$	1.55555	0.202515	0.101258	+	0.994860i	$0.467713\pi$
$60$	0	0
$61$	6.40303	0.819824	0.409912	−	0.912125i	$-0.365559\pi$
$61$	6.40303	0.819824	0.409912	+	0.912125i	$0.365559\pi$
$62$	−14.6082	−1.85525
$63$	−27.7395	−3.49485
$64$	−10.0223	−1.25278
$65$	0	0
$66$	7.24774	0.892135
$67$	−0.552548	−0.0675045	−0.0337523	−	0.999430i	$-0.510746\pi$
$67$	−0.552548	−0.0675045	−0.0337523	+	0.999430i	$0.510746\pi$
$68$	−16.1795	−1.96205
$69$	−25.2692	−3.04205
$70$	0	0
$71$	0.706993	0.0839046	0.0419523	−	0.999120i	$-0.486642\pi$
$71$	0.706993	0.0839046	0.0419523	+	0.999120i	$0.486642\pi$
$72$	3.99231	0.470499
$73$	8.09751	0.947742	0.473871	−	0.880594i	$-0.342856\pi$
$73$	8.09751	0.947742	0.473871	+	0.880594i	$0.342856\pi$
$74$	11.7323	1.36385
$75$	0	0
$76$	1.57957	0.181189
$77$	4.35621	0.496437
$78$	42.0241	4.75829
$79$	−9.29883	−1.04620	−0.523100	−	0.852271i	$-0.675225\pi$
$79$	−9.29883	−1.04620	−0.523100	+	0.852271i	$0.675225\pi$
$80$	0	0
$81$	19.4853	2.16503
$82$	−6.56188	−0.724638
$83$	−5.06419	−0.555867	−0.277934	−	0.960600i	$-0.589650\pi$
$83$	−5.06419	−0.555867	−0.277934	+	0.960600i	$0.589650\pi$
$84$	28.3827	3.09681
$85$	0	0
$86$	−7.85998	−0.847564
$87$	30.2130	3.23918
$88$	−0.626953	−0.0668334
$89$	−12.5416	−1.32941	−0.664703	−	0.747108i	$-0.731442\pi$
$89$	−12.5416	−1.32941	−0.664703	+	0.747108i	$0.731442\pi$
$90$	0	0
$91$	25.2583	2.64779
$92$	18.1325	1.89044
$93$	22.3936	2.32211
$94$	−4.12094	−0.425043
$95$	0	0
$96$	25.7157	2.62460
$97$	−5.95406	−0.604543	−0.302271	−	0.953222i	$-0.597745\pi$
$97$	−5.95406	−0.604543	−0.302271	+	0.953222i	$0.597745\pi$
$98$	17.5903	1.77689
$99$	−7.79186	−0.783112
$100$	0	0
$101$	9.92477	0.987552	0.493776	−	0.869589i	$-0.335616\pi$
$101$	9.92477	0.987552	0.493776	+	0.869589i	$0.335616\pi$
$102$	46.6148	4.61555
$103$	−8.48846	−0.836393	−0.418196	−	0.908357i	$-0.637338\pi$
$103$	−8.48846	−0.836393	−0.418196	+	0.908357i	$0.637338\pi$
$104$	−3.63522	−0.356462
$105$	0	0
$106$	12.5798	1.22186
$107$	8.49550	0.821291	0.410646	−	0.911795i	$-0.365303\pi$
$107$	8.49550	0.821291	0.410646	+	0.911795i	$0.365303\pi$
$108$	−29.1458	−2.80455
$109$	−13.4601	−1.28925	−0.644623	−	0.764500i	$-0.722986\pi$
$109$	−13.4601	−1.28925	−0.644623	+	0.764500i	$0.722986\pi$
$110$	0	0
$111$	−17.9850	−1.70706
$112$	13.2971	1.25646
$113$	1.12644	0.105967	0.0529835	−	0.998595i	$-0.483127\pi$
$113$	1.12644	0.105967	0.0529835	+	0.998595i	$0.483127\pi$
$114$	−4.55091	−0.426231
$115$	0	0
$116$	−21.6801	−2.01294
$117$	−45.1790	−4.17680
$118$	3.21595	0.296052
$119$	28.0175	2.56836
$120$	0	0
$121$	−9.77637	−0.888761
$122$	13.2376	1.19848
$123$	10.0590	0.906991
$124$	−16.0691	−1.44305
$125$	0	0
$126$	−57.3487	−5.10903
$127$	−6.66279	−0.591227	−0.295613	−	0.955308i	$-0.595524\pi$
$127$	−6.66279	−0.591227	−0.295613	+	0.955308i	$0.595524\pi$
$128$	−4.49159	−0.397004
$129$	12.0489	1.06085
$130$	0	0
$131$	−0.802568	−0.0701207	−0.0350603	−	0.999385i	$-0.511162\pi$
$131$	−0.802568	−0.0701207	−0.0350603	+	0.999385i	$0.511162\pi$
$132$	7.97253	0.693920
$133$	−2.73530	−0.237180
$134$	−1.14234	−0.0986831
$135$	0	0
$136$	−4.03233	−0.345769
$137$	−7.99763	−0.683284	−0.341642	−	0.939830i	$-0.610983\pi$
$137$	−7.99763	−0.683284	−0.341642	+	0.939830i	$0.610983\pi$
$138$	−52.2415	−4.44709
$139$	11.8768	1.00738	0.503688	−	0.863886i	$-0.331976\pi$
$139$	11.8768	1.00738	0.503688	+	0.863886i	$0.331976\pi$
$140$	0	0
$141$	6.31719	0.532004
$142$	1.46164	0.122658
$143$	7.09491	0.593306
$144$	−23.7842	−1.98202
$145$	0	0
$146$	16.7408	1.38548
$147$	−26.9650	−2.22403
$148$	12.9056	1.06083
$149$	4.14163	0.339295	0.169648	−	0.985505i	$-0.445737\pi$
$149$	4.14163	0.339295	0.169648	+	0.985505i	$0.445737\pi$
$150$	0	0
$151$	18.4231	1.49925	0.749626	−	0.661861i	$-0.230233\pi$
$151$	18.4231	1.49925	0.749626	+	0.661861i	$0.230233\pi$
$152$	0.393668	0.0319307
$153$	−50.1144	−4.05151
$154$	9.00604	0.725727
$155$	0	0
$156$	46.2266	3.70109
$157$	−6.63249	−0.529330	−0.264665	−	0.964340i	$-0.585261\pi$
$157$	−6.63249	−0.529330	−0.264665	+	0.964340i	$0.585261\pi$
$158$	−19.2244	−1.52941
$159$	−19.2842	−1.52934
$160$	0	0
$161$	−31.3995	−2.47462
$162$	40.2839	3.16500
$163$	20.3092	1.59074	0.795369	−	0.606126i	$-0.207277\pi$
$163$	20.3092	1.59074	0.795369	+	0.606126i	$0.207277\pi$
$164$	−7.21808	−0.563638
$165$	0	0
$166$	−10.4697	−0.812608
$167$	20.5549	1.59058	0.795291	−	0.606228i	$-0.207318\pi$
$167$	20.5549	1.59058	0.795291	+	0.606228i	$0.207318\pi$
$168$	7.07366	0.545745
$169$	28.1379	2.16446
$170$	0	0
$171$	4.89257	0.374144
$172$	−8.64601	−0.659252
$173$	−5.98909	−0.455342	−0.227671	−	0.973738i	$-0.573111\pi$
$173$	−5.98909	−0.455342	−0.227671	+	0.973738i	$0.573111\pi$
$174$	62.4624	4.73527
$175$	0	0
$176$	3.73507	0.281541
$177$	−4.92988	−0.370552
$178$	−25.9285	−1.94342
$179$	13.3249	0.995947	0.497974	−	0.867192i	$-0.334078\pi$
$179$	13.3249	0.995947	0.497974	+	0.867192i	$0.334078\pi$
$180$	0	0
$181$	−7.24204	−0.538297	−0.269148	−	0.963099i	$-0.586742\pi$
$181$	−7.24204	−0.538297	−0.269148	+	0.963099i	$0.586742\pi$
$182$	52.2191	3.87074
$183$	−20.2926	−1.50007
$184$	4.51906	0.333149
$185$	0	0
$186$	46.2966	3.39463
$187$	7.86996	0.575508
$188$	−4.53305	−0.330607
$189$	50.4708	3.67121
$190$	0	0
$191$	20.2185	1.46296	0.731481	−	0.681862i	$-0.238830\pi$
$191$	20.2185	1.46296	0.731481	+	0.681862i	$0.238830\pi$
$192$	31.7627	2.29228
$193$	1.14758	0.0826047	0.0413023	−	0.999147i	$-0.486849\pi$
$193$	1.14758	0.0826047	0.0413023	+	0.999147i	$0.486849\pi$
$194$	−12.3094	−0.883765
$195$	0	0
$196$	19.3493	1.38210
$197$	−11.1443	−0.794002	−0.397001	−	0.917818i	$-0.629949\pi$
$197$	−11.1443	−0.794002	−0.397001	+	0.917818i	$0.629949\pi$
$198$	−16.1089	−1.14481
$199$	2.69927	0.191346	0.0956731	−	0.995413i	$-0.469500\pi$
$199$	2.69927	0.191346	0.0956731	+	0.995413i	$0.469500\pi$
$200$	0	0
$201$	1.75115	0.123516
$202$	20.5185	1.44368
$203$	37.5427	2.63498
$204$	51.2764	3.59006
$205$	0	0
$206$	−17.5491	−1.22270
$207$	56.1635	3.90364
$208$	21.6568	1.50163
$209$	−0.768328	−0.0531464
$210$	0	0
$211$	0.880031	0.0605839	0.0302919	−	0.999541i	$-0.490356\pi$
$211$	0.880031	0.0605839	0.0302919	+	0.999541i	$0.490356\pi$
$212$	13.8379	0.950389
$213$	−2.24061	−0.153524
$214$	17.5636	1.20062
$215$	0	0
$216$	−7.26383	−0.494241
$217$	27.8263	1.88897
$218$	−27.8275	−1.88472
$219$	−25.6628	−1.73413
$220$	0	0
$221$	45.6318	3.06953
$222$	−37.1822	−2.49551
$223$	20.1365	1.34844	0.674220	−	0.738531i	$-0.264480\pi$
$223$	20.1365	1.34844	0.674220	+	0.738531i	$0.264480\pi$
$224$	31.9544	2.13504
$225$	0	0
$226$	2.32881	0.154910
$227$	15.5755	1.03379	0.516893	−	0.856050i	$-0.327089\pi$
$227$	15.5755	1.03379	0.516893	+	0.856050i	$0.327089\pi$
$228$	−5.00601	−0.331531
$229$	−16.4467	−1.08683	−0.543413	−	0.839466i	$-0.682868\pi$
$229$	−16.4467	−1.08683	−0.543413	+	0.839466i	$0.682868\pi$
$230$	0	0
$231$	−13.8058	−0.908354
$232$	−5.40320	−0.354738
$233$	13.4312	0.879908	0.439954	−	0.898020i	$-0.354995\pi$
$233$	13.4312	0.879908	0.439954	+	0.898020i	$0.354995\pi$
$234$	−93.4032	−6.10596
$235$	0	0
$236$	3.53755	0.230275
$237$	29.4700	1.91428
$238$	57.9235	3.75462
$239$	9.35049	0.604833	0.302416	−	0.953176i	$-0.402207\pi$
$239$	9.35049	0.604833	0.302416	+	0.953176i	$0.402207\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	−20.2117	−1.29926
$243$	−23.3047	−1.49500
$244$	14.5614	0.932200
$245$	0	0
$246$	20.7960	1.32591
$247$	−4.45495	−0.283461
$248$	−4.00481	−0.254305
$249$	16.0495	1.01710
$250$	0	0
$251$	−15.4629	−0.976012	−0.488006	−	0.872840i	$-0.662276\pi$
$251$	−15.4629	−0.976012	−0.488006	+	0.872840i	$0.662276\pi$
$252$	−63.0838	−3.97390
$253$	−8.81992	−0.554504
$254$	−13.7747	−0.864298
$255$	0	0
$256$	10.7586	0.672413
$257$	−29.6348	−1.84857	−0.924284	−	0.381706i	$-0.875337\pi$
$257$	−29.6348	−1.84857	−0.924284	+	0.381706i	$0.875337\pi$
$258$	24.9100	1.55083
$259$	−22.3482	−1.38865
$260$	0	0
$261$	−67.1518	−4.15659
$262$	−1.65923	−0.102508
$263$	−21.0285	−1.29667	−0.648336	−	0.761354i	$-0.724535\pi$
$263$	−21.0285	−1.29667	−0.648336	+	0.761354i	$0.724535\pi$
$264$	1.98695	0.122288
$265$	0	0
$266$	−5.65496	−0.346728
$267$	39.7470	2.43248
$268$	−1.25658	−0.0767576
$269$	−24.2346	−1.47761	−0.738806	−	0.673918i	$-0.764610\pi$
$269$	−24.2346	−1.47761	−0.738806	+	0.673918i	$0.764610\pi$
$270$	0	0
$271$	6.81359	0.413896	0.206948	−	0.978352i	$-0.433647\pi$
$271$	6.81359	0.413896	0.206948	+	0.978352i	$0.433647\pi$
$272$	24.0226	1.45658
$273$	−80.0492	−4.84480
$274$	−16.5343	−0.998874
$275$	0	0
$276$	−57.4658	−3.45904
$277$	−2.26887	−0.136323	−0.0681615	−	0.997674i	$-0.521713\pi$
$277$	−2.26887	−0.136323	−0.0681615	+	0.997674i	$0.521713\pi$
$278$	24.5541	1.47266
$279$	−49.7724	−2.97979
$280$	0	0
$281$	−16.0818	−0.959359	−0.479679	−	0.877444i	$-0.659247\pi$
$281$	−16.0818	−0.959359	−0.479679	+	0.877444i	$0.659247\pi$
$282$	13.0602	0.777722
$283$	−26.1504	−1.55448	−0.777239	−	0.629205i	$-0.783381\pi$
$283$	−26.1504	−1.55448	−0.777239	+	0.629205i	$0.783381\pi$
$284$	1.60781	0.0954058
$285$	0	0
$286$	14.6680	0.867339
$287$	12.4993	0.737812
$288$	−57.1561	−3.36796
$289$	33.6166	1.97745
$290$	0	0
$291$	18.8697	1.10616
$292$	18.4149	1.07765
$293$	13.5755	0.793091	0.396545	−	0.918015i	$-0.370209\pi$
$293$	13.5755	0.793091	0.396545	+	0.918015i	$0.370209\pi$
$294$	−55.7474	−3.25125
$295$	0	0
$296$	3.21638	0.186948
$297$	14.1769	0.822629
$298$	8.56241	0.496007
$299$	−51.1399	−2.95750
$300$	0	0
$301$	14.9720	0.862973
$302$	38.0880	2.19172
$303$	−31.4538	−1.80697
$304$	−2.34528	−0.134511
$305$	0	0
$306$	−103.607	−5.92279
$307$	−17.0217	−0.971478	−0.485739	−	0.874104i	$-0.661449\pi$
$307$	−17.0217	−0.971478	−0.485739	+	0.874104i	$0.661449\pi$
$308$	9.90667	0.564485
$309$	26.9018	1.53039
$310$	0	0
$311$	10.0596	0.570425	0.285213	−	0.958464i	$-0.407936\pi$
$311$	10.0596	0.570425	0.285213	+	0.958464i	$0.407936\pi$
$312$	11.5208	0.652236
$313$	11.8099	0.667533	0.333767	−	0.942656i	$-0.391680\pi$
$313$	11.8099	0.667533	0.333767	+	0.942656i	$0.391680\pi$
$314$	−13.7120	−0.773814
$315$	0	0
$316$	−21.1469	−1.18961
$317$	−10.4070	−0.584518	−0.292259	−	0.956339i	$-0.594407\pi$
$317$	−10.4070	−0.584518	−0.292259	+	0.956339i	$0.594407\pi$
$318$	−39.8683	−2.23570
$319$	10.5455	0.590436
$320$	0	0
$321$	−26.9241	−1.50276
$322$	−64.9153	−3.61759
$323$	−4.94160	−0.274958
$324$	44.3124	2.46180
$325$	0	0
$326$	41.9872	2.32546
$327$	42.6581	2.35900
$328$	−1.79892	−0.0993289
$329$	7.84974	0.432770
$330$	0	0
$331$	−25.0178	−1.37510	−0.687551	−	0.726136i	$-0.741314\pi$
$331$	−25.0178	−1.37510	−0.687551	+	0.726136i	$0.741314\pi$
$332$	−11.5167	−0.632062
$333$	39.9737	2.19054
$334$	42.4951	2.32523
$335$	0	0
$336$	−42.1413	−2.29900
$337$	13.9771	0.761383	0.380691	−	0.924702i	$-0.375686\pi$
$337$	13.9771	0.761383	0.380691	+	0.924702i	$0.375686\pi$
$338$	58.1724	3.16416
$339$	−3.56995	−0.193893
$340$	0	0
$341$	7.81625	0.423274
$342$	10.1149	0.546951
$343$	−5.94016	−0.320739
$344$	−2.15479	−0.116179
$345$	0	0
$346$	−12.3819	−0.665653
$347$	−24.3336	−1.30630	−0.653148	−	0.757231i	$-0.726552\pi$
$347$	−24.3336	−1.30630	−0.653148	+	0.757231i	$0.726552\pi$
$348$	68.7089	3.68318
$349$	−7.04015	−0.376850	−0.188425	−	0.982088i	$-0.560338\pi$
$349$	−7.04015	−0.376850	−0.188425	+	0.982088i	$0.560338\pi$
$350$	0	0
$351$	82.2012	4.38757
$352$	8.97579	0.478411
$353$	6.77087	0.360377	0.180189	−	0.983632i	$-0.442329\pi$
$353$	6.77087	0.360377	0.180189	+	0.983632i	$0.442329\pi$
$354$	−10.1920	−0.541700
$355$	0	0
$356$	−28.5214	−1.51163
$357$	−88.7937	−4.69946
$358$	27.5478	1.45595
$359$	−31.1776	−1.64549	−0.822745	−	0.568411i	$-0.807558\pi$
$359$	−31.1776	−1.64549	−0.822745	+	0.568411i	$0.807558\pi$
$360$	0	0
$361$	−18.5176	−0.974608
$362$	−14.9722	−0.786922
$363$	30.9834	1.62621
$364$	57.4412	3.01074
$365$	0	0
$366$	−41.9529	−2.19291
$367$	−1.10879	−0.0578785	−0.0289393	−	0.999581i	$-0.509213\pi$
$367$	−1.10879	−0.0578785	−0.0289393	+	0.999581i	$0.509213\pi$
$368$	−26.9223	−1.40342
$369$	−22.3573	−1.16387
$370$	0	0
$371$	−23.9626	−1.24408
$372$	50.9264	2.64041
$373$	30.2471	1.56614	0.783069	−	0.621935i	$-0.213653\pi$
$373$	30.2471	1.56614	0.783069	+	0.621935i	$0.213653\pi$
$374$	16.2704	0.841320
$375$	0	0
$376$	−1.12975	−0.0582622
$377$	61.1454	3.14915
$378$	104.343	5.36684
$379$	16.0849	0.826223	0.413112	−	0.910680i	$-0.364442\pi$
$379$	16.0849	0.826223	0.413112	+	0.910680i	$0.364442\pi$
$380$	0	0
$381$	21.1158	1.08180
$382$	41.7998	2.13866
$383$	22.5204	1.15074	0.575369	−	0.817894i	$-0.304858\pi$
$383$	22.5204	1.15074	0.575369	+	0.817894i	$0.304858\pi$
$384$	14.2348	0.726417
$385$	0	0
$386$	2.37251	0.120758
$387$	−26.7801	−1.36131
$388$	−13.5404	−0.687410
$389$	−17.9740	−0.911319	−0.455660	−	0.890154i	$-0.650597\pi$
$389$	−17.9740	−0.911319	−0.455660	+	0.890154i	$0.650597\pi$
$390$	0	0
$391$	−56.7264	−2.86878
$392$	4.82233	0.243564
$393$	2.54351	0.128303
$394$	−23.0398	−1.16073
$395$	0	0
$396$	−17.7198	−0.890456
$397$	−15.8528	−0.795631	−0.397815	−	0.917465i	$-0.630232\pi$
$397$	−15.8528	−0.795631	−0.397815	+	0.917465i	$0.630232\pi$
$398$	5.58047	0.279724
$399$	8.66875	0.433980
$400$	0	0
$401$	16.5966	0.828792	0.414396	−	0.910097i	$-0.363993\pi$
$401$	16.5966	0.828792	0.414396	+	0.910097i	$0.363993\pi$
$402$	3.62032	0.180565
$403$	45.3204	2.25757
$404$	22.5704	1.12292
$405$	0	0
$406$	77.6158	3.85201
$407$	−6.27746	−0.311162
$408$	12.7793	0.632671
$409$	18.4612	0.912848	0.456424	−	0.889762i	$-0.349130\pi$
$409$	18.4612	0.912848	0.456424	+	0.889762i	$0.349130\pi$
$410$	0	0
$411$	25.3462	1.25024
$412$	−19.3040	−0.951040
$413$	−6.12586	−0.301434
$414$	116.113	5.70662
$415$	0	0
$416$	52.0437	2.55165
$417$	−37.6401	−1.84324
$418$	−1.58844	−0.0776933
$419$	31.7240	1.54982	0.774910	−	0.632072i	$-0.217795\pi$
$419$	31.7240	1.54982	0.774910	+	0.632072i	$0.217795\pi$
$420$	0	0
$421$	−0.862198	−0.0420210	−0.0210105	−	0.999779i	$-0.506688\pi$
$421$	−0.862198	−0.0420210	−0.0210105	+	0.999779i	$0.506688\pi$
$422$	1.81938	0.0885659
$423$	−14.0407	−0.682680
$424$	3.44873	0.167485
$425$	0	0
$426$	−4.63225	−0.224433
$427$	−25.2156	−1.22027
$428$	19.3200	0.933869
$429$	−22.4853	−1.08560
$430$	0	0
$431$	7.25982	0.349693	0.174847	−	0.984596i	$-0.444057\pi$
$431$	7.25982	0.349693	0.174847	+	0.984596i	$0.444057\pi$
$432$	43.2743	2.08203
$433$	5.12918	0.246493	0.123246	−	0.992376i	$-0.460669\pi$
$433$	5.12918	0.246493	0.123246	+	0.992376i	$0.460669\pi$
$434$	57.5282	2.76144
$435$	0	0
$436$	−30.6103	−1.46597
$437$	5.53809	0.264923
$438$	−53.0553	−2.53508
$439$	−21.9407	−1.04717	−0.523585	−	0.851973i	$-0.675406\pi$
$439$	−21.9407	−1.04717	−0.523585	+	0.851973i	$0.675406\pi$
$440$	0	0
$441$	59.9326	2.85393
$442$	94.3393	4.48726
$443$	−32.9507	−1.56554	−0.782768	−	0.622313i	$-0.786193\pi$
$443$	−32.9507	−1.56554	−0.782768	+	0.622313i	$0.786193\pi$
$444$	−40.9005	−1.94105
$445$	0	0
$446$	41.6302	1.97125
$447$	−13.1257	−0.620825
$448$	39.4684	1.86471
$449$	−29.8137	−1.40699	−0.703497	−	0.710698i	$-0.748379\pi$
$449$	−29.8137	−1.40699	−0.703497	+	0.710698i	$0.748379\pi$
$450$	0	0
$451$	3.51099	0.165326
$452$	2.56170	0.120492
$453$	−58.3869	−2.74326
$454$	32.2009	1.51126
$455$	0	0
$456$	−1.24762	−0.0584251
$457$	−28.6232	−1.33893	−0.669467	−	0.742841i	$-0.733478\pi$
$457$	−28.6232	−1.33893	−0.669467	+	0.742841i	$0.733478\pi$
$458$	−34.0018	−1.58880
$459$	91.1808	4.25595
$460$	0	0
$461$	−16.9797	−0.790822	−0.395411	−	0.918504i	$-0.629398\pi$
$461$	−16.9797	−0.790822	−0.395411	+	0.918504i	$0.629398\pi$
$462$	−28.5421	−1.32790
$463$	23.7172	1.10223	0.551116	−	0.834429i	$-0.314202\pi$
$463$	23.7172	1.10223	0.551116	+	0.834429i	$0.314202\pi$
$464$	32.1896	1.49436
$465$	0	0
$466$	27.7677	1.28631
$467$	−32.5642	−1.50689	−0.753446	−	0.657510i	$-0.771610\pi$
$467$	−32.5642	−1.50689	−0.753446	+	0.657510i	$0.771610\pi$
$468$	−102.744	−4.74933
$469$	2.17597	0.100477
$470$	0	0
$471$	21.0198	0.968541
$472$	0.881643	0.0405809
$473$	4.20555	0.193371
$474$	60.9264	2.79844
$475$	0	0
$476$	63.7160	2.92042
$477$	42.8614	1.96249
$478$	19.3312	0.884189
$479$	10.3432	0.472591	0.236295	−	0.971681i	$-0.424067\pi$
$479$	10.3432	0.472591	0.236295	+	0.971681i	$0.424067\pi$
$480$	0	0
$481$	−36.3982	−1.65961
$482$	2.06740	0.0941675
$483$	99.5117	4.52794
$484$	−22.2329	−1.01059
$485$	0	0
$486$	−48.1801	−2.18549
$487$	−18.6602	−0.845572	−0.422786	−	0.906229i	$-0.638948\pi$
$487$	−18.6602	−0.845572	−0.422786	+	0.906229i	$0.638948\pi$
$488$	3.62906	0.164280
$489$	−64.3642	−2.91065
$490$	0	0
$491$	−5.71840	−0.258068	−0.129034	−	0.991640i	$-0.541188\pi$
$491$	−5.71840	−0.258068	−0.129034	+	0.991640i	$0.541188\pi$
$492$	22.8757	1.03132
$493$	67.8249	3.05468
$494$	−9.21016	−0.414385
$495$	0	0
$496$	23.8586	1.07128
$497$	−2.78419	−0.124888
$498$	33.1808	1.48687
$499$	17.8970	0.801181	0.400590	−	0.916257i	$-0.368805\pi$
$499$	17.8970	0.801181	0.400590	+	0.916257i	$0.368805\pi$
$500$	0	0
$501$	−65.1428	−2.91037
$502$	−31.9681	−1.42681
$503$	31.2884	1.39508	0.697540	−	0.716545i	$-0.254278\pi$
$503$	31.2884	1.39508	0.697540	+	0.716545i	$0.254278\pi$
$504$	−15.7220	−0.700314
$505$	0	0
$506$	−18.2343	−0.810614
$507$	−89.1753	−3.96041
$508$	−15.1522	−0.672268
$509$	−35.8630	−1.58960	−0.794801	−	0.606871i	$-0.792425\pi$
$509$	−35.8630	−1.58960	−0.794801	+	0.606871i	$0.792425\pi$
$510$	0	0
$511$	−31.8886	−1.41067
$512$	31.2255	1.37999
$513$	−8.90180	−0.393024
$514$	−61.2670	−2.70237
$515$	0	0
$516$	27.4011	1.20626
$517$	2.20495	0.0969734
$518$	−46.2026	−2.03003
$519$	18.9807	0.833162
$520$	0	0
$521$	16.3179	0.714899	0.357449	−	0.933933i	$-0.383646\pi$
$521$	16.3179	0.714899	0.357449	+	0.933933i	$0.383646\pi$
$522$	−138.830	−6.07641
$523$	−0.423244	−0.0185072	−0.00925358	−	0.999957i	$-0.502946\pi$
$523$	−0.423244	−0.0185072	−0.00925358	+	0.999957i	$0.502946\pi$
$524$	−1.82516	−0.0797324
$525$	0	0
$526$	−43.4743	−1.89557
$527$	50.2712	2.18985
$528$	−11.8372	−0.515150
$529$	40.5737	1.76408
$530$	0	0
$531$	10.9572	0.475502
$532$	−6.22047	−0.269692
$533$	20.3575	0.881782
$534$	82.1731	3.55598
$535$	0	0
$536$	−0.313169	−0.0135269
$537$	−42.2294	−1.82233
$538$	−50.1027	−2.16008
$539$	−9.41181	−0.405396
$540$	0	0
$541$	26.6456	1.14558	0.572791	−	0.819701i	$-0.305860\pi$
$541$	26.6456	1.14558	0.572791	+	0.819701i	$0.305860\pi$
$542$	14.0864	0.605064
$543$	22.9516	0.984948
$544$	57.7290	2.47511
$545$	0	0
$546$	−165.494	−7.08248
$547$	0.0900603	0.00385070	0.00192535	−	0.999998i	$-0.499387\pi$
$547$	0.0900603	0.00385070	0.00192535	+	0.999998i	$0.499387\pi$
$548$	−18.1878	−0.776944
$549$	45.1025	1.92493
$550$	0	0
$551$	−6.62161	−0.282090
$552$	−14.3219	−0.609580
$553$	36.6195	1.55722
$554$	−4.69066	−0.199287
$555$	0	0
$556$	27.0096	1.14546
$557$	−19.0201	−0.805906	−0.402953	−	0.915221i	$-0.632016\pi$
$557$	−19.0201	−0.805906	−0.402953	+	0.915221i	$0.632016\pi$
$558$	−102.899	−4.35608
$559$	24.3847	1.03136
$560$	0	0
$561$	−24.9416	−1.05304
$562$	−33.2475	−1.40246
$563$	−17.7714	−0.748975	−0.374488	−	0.927232i	$-0.622181\pi$
$563$	−17.7714	−0.748975	−0.374488	+	0.927232i	$0.622181\pi$
$564$	14.3662	0.604927
$565$	0	0
$566$	−54.0633	−2.27245
$567$	−76.7343	−3.22254
$568$	0.400704	0.0168132
$569$	−30.8642	−1.29389	−0.646947	−	0.762535i	$-0.723954\pi$
$569$	−30.8642	−1.29389	−0.646947	+	0.762535i	$0.723954\pi$
$570$	0	0
$571$	−17.1043	−0.715793	−0.357897	−	0.933761i	$-0.616506\pi$
$571$	−17.1043	−0.715793	−0.357897	+	0.933761i	$0.616506\pi$
$572$	16.1349	0.674633
$573$	−64.0769	−2.67685
$574$	25.8411	1.07859
$575$	0	0
$576$	−70.5962	−2.94151
$577$	42.6550	1.77575	0.887876	−	0.460083i	$-0.152181\pi$
$577$	42.6550	1.77575	0.887876	+	0.460083i	$0.152181\pi$
$578$	69.4991	2.89078
$579$	−3.63693	−0.151146
$580$	0	0
$581$	19.9431	0.827381
$582$	39.0112	1.61707
$583$	−6.73095	−0.278767
$584$	4.58945	0.189913
$585$	0	0
$586$	28.0661	1.15940
$587$	−10.0486	−0.414749	−0.207374	−	0.978262i	$-0.566492\pi$
$587$	−10.0486	−0.414749	−0.207374	+	0.978262i	$0.566492\pi$
$588$	−61.3223	−2.52889
$589$	−4.90788	−0.202226
$590$	0	0
$591$	35.3189	1.45282
$592$	−19.1616	−0.787536
$593$	20.2450	0.831362	0.415681	−	0.909510i	$-0.363543\pi$
$593$	20.2450	0.831362	0.415681	+	0.909510i	$0.363543\pi$
$594$	29.3094	1.20258
$595$	0	0
$596$	9.41867	0.385804
$597$	−8.55457	−0.350115
$598$	−105.727	−4.32349
$599$	−6.24694	−0.255243	−0.127621	−	0.991823i	$-0.540734\pi$
$599$	−6.24694	−0.255243	−0.127621	+	0.991823i	$0.540734\pi$
$600$	0	0
$601$	14.6109	0.595993	0.297996	−	0.954567i	$-0.403682\pi$
$601$	14.6109	0.595993	0.297996	+	0.954567i	$0.403682\pi$
$602$	30.9532	1.26156
$603$	−3.89212	−0.158499
$604$	41.8969	1.70476
$605$	0	0
$606$	−65.0275	−2.64156
$607$	41.2610	1.67473	0.837367	−	0.546641i	$-0.184094\pi$
$607$	41.2610	1.67473	0.837367	+	0.546641i	$0.184094\pi$
$608$	−5.63596	−0.228569
$609$	−118.981	−4.82135
$610$	0	0
$611$	12.7848	0.517217
$612$	−113.967	−4.60686
$613$	−26.8141	−1.08301	−0.541506	−	0.840697i	$-0.682146\pi$
$613$	−26.8141	−1.08301	−0.541506	+	0.840697i	$0.682146\pi$
$614$	−35.1906	−1.42018
$615$	0	0
$616$	2.46898	0.0994782
$617$	−24.9755	−1.00547	−0.502737	−	0.864439i	$-0.667674\pi$
$617$	−24.9755	−1.00547	−0.502737	+	0.864439i	$0.667674\pi$
$618$	55.6168	2.23723
$619$	−1.48566	−0.0597136	−0.0298568	−	0.999554i	$-0.509505\pi$
$619$	−1.48566	−0.0597136	−0.0298568	+	0.999554i	$0.509505\pi$
$620$	0	0
$621$	−102.187	−4.10062
$622$	20.7971	0.833889
$623$	49.3897	1.97876
$624$	−68.6351	−2.74760
$625$	0	0
$626$	24.4157	0.975849
$627$	2.43500	0.0972445
$628$	−15.0833	−0.601887
$629$	−40.3743	−1.60983
$630$	0	0
$631$	39.1945	1.56031	0.780155	−	0.625586i	$-0.215140\pi$
$631$	39.1945	1.56031	0.780155	+	0.625586i	$0.215140\pi$
$632$	−5.27033	−0.209642
$633$	−2.78901	−0.110853
$634$	−21.5155	−0.854491
$635$	0	0
$636$	−43.8552	−1.73897
$637$	−54.5719	−2.16222
$638$	21.8018	0.863142
$639$	4.98002	0.197006
$640$	0	0
$641$	35.6718	1.40895	0.704475	−	0.709729i	$-0.251183\pi$
$641$	35.6718	1.40895	0.704475	+	0.709729i	$0.251183\pi$
$642$	−55.6629	−2.19684
$643$	30.0030	1.18320	0.591602	−	0.806230i	$-0.298496\pi$
$643$	30.0030	1.18320	0.591602	+	0.806230i	$0.298496\pi$
$644$	−71.4070	−2.81383
$645$	0	0
$646$	−10.2163	−0.401954
$647$	28.6911	1.12797	0.563983	−	0.825787i	$-0.309269\pi$
$647$	28.6911	1.12797	0.563983	+	0.825787i	$0.309269\pi$
$648$	11.0437	0.433838
$649$	−1.72072	−0.0675441
$650$	0	0
$651$	−88.1877	−3.45635
$652$	46.1861	1.80879
$653$	−0.645712	−0.0252687	−0.0126343	−	0.999920i	$-0.504022\pi$
$653$	−0.645712	−0.0252687	−0.0126343	+	0.999920i	$0.504022\pi$
$654$	88.1914	3.44855
$655$	0	0
$656$	10.7171	0.418432
$657$	57.0384	2.22528
$658$	16.2286	0.632655
$659$	25.2001	0.981658	0.490829	−	0.871256i	$-0.336694\pi$
$659$	25.2001	0.981658	0.490829	+	0.871256i	$0.336694\pi$
$660$	0	0
$661$	−17.1752	−0.668037	−0.334019	−	0.942566i	$-0.608405\pi$
$661$	−17.1752	−0.668037	−0.334019	+	0.942566i	$0.608405\pi$
$662$	−51.7218	−2.01023
$663$	−144.617	−5.61647
$664$	−2.87025	−0.111387
$665$	0	0
$666$	82.6416	3.20230
$667$	−76.0118	−2.94319
$668$	46.7448	1.80861
$669$	−63.8169	−2.46731
$670$	0	0
$671$	−7.08290	−0.273432
$672$	−101.270	−3.90659
$673$	−33.6733	−1.29801	−0.649006	−	0.760783i	$-0.724815\pi$
$673$	−33.6733	−1.29801	−0.649006	+	0.760783i	$0.724815\pi$
$674$	28.8963	1.11305
$675$	0	0
$676$	63.9898	2.46115
$677$	−4.89563	−0.188154	−0.0940772	−	0.995565i	$-0.529990\pi$
$677$	−4.89563	−0.188154	−0.0940772	+	0.995565i	$0.529990\pi$
$678$	−7.38051	−0.283447
$679$	23.4475	0.899832
$680$	0	0
$681$	−49.3623	−1.89157
$682$	16.1593	0.618772
$683$	−21.6750	−0.829371	−0.414685	−	0.909965i	$-0.636108\pi$
$683$	−21.6750	−0.829371	−0.414685	+	0.909965i	$0.636108\pi$
$684$	11.1264	0.425429
$685$	0	0
$686$	−12.2807	−0.468879
$687$	52.1230	1.98862
$688$	12.8372	0.489413
$689$	−39.0276	−1.48683
$690$	0	0
$691$	23.9435	0.910852	0.455426	−	0.890274i	$-0.349487\pi$
$691$	23.9435	0.910852	0.455426	+	0.890274i	$0.349487\pi$
$692$	−13.6201	−0.517758
$693$	30.6849	1.16562
$694$	−50.3073	−1.90964
$695$	0	0
$696$	17.1239	0.649081
$697$	22.5814	0.855330
$698$	−14.5548	−0.550907
$699$	−42.5665	−1.61001
$700$	0	0
$701$	8.23018	0.310850	0.155425	−	0.987848i	$-0.450325\pi$
$701$	8.23018	0.310850	0.155425	+	0.987848i	$0.450325\pi$
$702$	169.943	6.41408
$703$	3.94166	0.148663
$704$	11.0864	0.417835
$705$	0	0
$706$	13.9981	0.526826
$707$	−39.0844	−1.46992
$708$	−11.2113	−0.421345
$709$	−12.9098	−0.484839	−0.242419	−	0.970172i	$-0.577941\pi$
$709$	−12.9098	−0.484839	−0.242419	+	0.970172i	$0.577941\pi$
$710$	0	0
$711$	−65.5004	−2.45646
$712$	−7.10824	−0.266392
$713$	−56.3393	−2.10992
$714$	−183.572	−6.87002
$715$	0	0
$716$	30.3027	1.13247
$717$	−29.6337	−1.10669
$718$	−64.4565	−2.40550
$719$	21.9808	0.819745	0.409873	−	0.912143i	$-0.365573\pi$
$719$	21.9808	0.819745	0.409873	+	0.912143i	$0.365573\pi$
$720$	0	0
$721$	33.4281	1.24493
$722$	−38.2832	−1.42475
$723$	−3.16922	−0.117864
$724$	−16.4695	−0.612083
$725$	0	0
$726$	64.0552	2.37731
$727$	−40.2120	−1.49138	−0.745690	−	0.666293i	$-0.767880\pi$
$727$	−40.2120	−1.49138	−0.745690	+	0.666293i	$0.767880\pi$
$728$	14.3157	0.530577
$729$	15.4018	0.570436
$730$	0	0
$731$	27.0485	1.00043
$732$	−46.1483	−1.70569
$733$	−12.2022	−0.450699	−0.225349	−	0.974278i	$-0.572352\pi$
$733$	−12.2022	−0.450699	−0.225349	+	0.974278i	$0.572352\pi$
$734$	−2.29232	−0.0846110
$735$	0	0
$736$	−64.6973	−2.38477
$737$	0.611218	0.0225145
$738$	−46.2215	−1.70144
$739$	−39.7538	−1.46237	−0.731183	−	0.682181i	$-0.761032\pi$
$739$	−39.7538	−1.46237	−0.731183	+	0.682181i	$0.761032\pi$
$740$	0	0
$741$	14.1187	0.518663
$742$	−49.5403	−1.81868
$743$	−30.3070	−1.11186	−0.555928	−	0.831230i	$-0.687637\pi$
$743$	−30.3070	−1.11186	−0.555928	+	0.831230i	$0.687637\pi$
$744$	12.6921	0.465315
$745$	0	0
$746$	62.5330	2.28950
$747$	−35.6719	−1.30517
$748$	17.8974	0.654395
$749$	−33.4559	−1.22245
$750$	0	0
$751$	−24.8267	−0.905941	−0.452970	−	0.891526i	$-0.649636\pi$
$751$	−24.8267	−0.905941	−0.452970	+	0.891526i	$0.649636\pi$
$752$	6.73046	0.245435
$753$	49.0054	1.78586
$754$	126.412	4.60365
$755$	0	0
$756$	114.778	4.17444
$757$	−6.63918	−0.241305	−0.120653	−	0.992695i	$-0.538499\pi$
$757$	−6.63918	−0.241305	−0.120653	+	0.992695i	$0.538499\pi$
$758$	33.2538	1.20783
$759$	27.9522	1.01460
$760$	0	0
$761$	−30.4841	−1.10505	−0.552523	−	0.833497i	$-0.686335\pi$
$761$	−30.4841	−1.10505	−0.552523	+	0.833497i	$0.686335\pi$
$762$	43.6549	1.58145
$763$	53.0069	1.91898
$764$	45.9799	1.66350
$765$	0	0
$766$	46.5587	1.68223
$767$	−9.97712	−0.360253
$768$	−34.0964	−1.23035
$769$	−54.3931	−1.96147	−0.980733	−	0.195355i	$-0.937414\pi$
$769$	−54.3931	−1.96147	−0.980733	+	0.195355i	$0.937414\pi$
$770$	0	0
$771$	93.9191	3.38241
$772$	2.60977	0.0939276
$773$	−40.1012	−1.44234	−0.721170	−	0.692758i	$-0.756395\pi$
$773$	−40.1012	−1.44234	−0.721170	+	0.692758i	$0.756395\pi$
$774$	−55.3653	−1.99006
$775$	0	0
$776$	−3.37460	−0.121141
$777$	70.8262	2.54088
$778$	−37.1595	−1.33223
$779$	−2.20457	−0.0789870
$780$	0	0
$781$	−0.782061	−0.0279844
$782$	−117.276	−4.19379
$783$	122.180	4.36634
$784$	−28.7290	−1.02604
$785$	0	0
$786$	5.25846	0.187563
$787$	−6.33041	−0.225655	−0.112827	−	0.993615i	$-0.535991\pi$
$787$	−6.33041	−0.225655	−0.112827	+	0.993615i	$0.535991\pi$
$788$	−25.3439	−0.902839
$789$	66.6439	2.37259
$790$	0	0
$791$	−4.43602	−0.157727
$792$	−4.41622	−0.156923
$793$	−41.0683	−1.45838
$794$	−32.7742	−1.16311
$795$	0	0
$796$	6.13854	0.217575
$797$	−30.5292	−1.08140	−0.540700	−	0.841216i	$-0.681841\pi$
$797$	−30.5292	−1.08140	−0.540700	+	0.841216i	$0.681841\pi$
$798$	17.9218	0.634424
$799$	14.1814	0.501701
$800$	0	0
$801$	−88.3422	−3.12142
$802$	34.3117	1.21159
$803$	−8.95731	−0.316096
$804$	3.98236	0.140447
$805$	0	0
$806$	93.6955	3.30028
$807$	76.8048	2.70366
$808$	5.62509	0.197890
$809$	7.45083	0.261957	0.130979	−	0.991385i	$-0.458188\pi$
$809$	7.45083	0.261957	0.130979	+	0.991385i	$0.458188\pi$
$810$	0	0
$811$	−11.1983	−0.393227	−0.196614	−	0.980481i	$-0.562994\pi$
$811$	−11.1983	−0.393227	−0.196614	+	0.980481i	$0.562994\pi$
$812$	85.3777	2.99617
$813$	−21.5938	−0.757326
$814$	−12.9780	−0.454880
$815$	0	0
$816$	−76.1328	−2.66518
$817$	−2.64069	−0.0923862
$818$	38.1667	1.33447
$819$	177.918	6.21696
$820$	0	0
$821$	−15.1459	−0.528595	−0.264298	−	0.964441i	$-0.585140\pi$
$821$	−15.1459	−0.528595	−0.264298	+	0.964441i	$0.585140\pi$
$822$	52.4008	1.82769
$823$	46.3367	1.61519	0.807597	−	0.589735i	$-0.200768\pi$
$823$	46.3367	1.61519	0.807597	+	0.589735i	$0.200768\pi$
$824$	−4.81103	−0.167600
$825$	0	0
$826$	−12.6646	−0.440658
$827$	23.1746	0.805859	0.402930	−	0.915231i	$-0.367992\pi$
$827$	23.1746	0.805859	0.402930	+	0.915231i	$0.367992\pi$
$828$	127.724	4.43872
$829$	32.8037	1.13932	0.569660	−	0.821881i	$-0.307075\pi$
$829$	32.8037	1.13932	0.569660	+	0.821881i	$0.307075\pi$
$830$	0	0
$831$	7.19053	0.249437
$832$	64.2817	2.22857
$833$	−60.5333	−2.09735
$834$	−77.8172	−2.69459
$835$	0	0
$836$	−1.74729	−0.0604314
$837$	90.5585	3.13016
$838$	65.5863	2.26564
$839$	−33.1220	−1.14350	−0.571749	−	0.820429i	$-0.693735\pi$
$839$	−33.1220	−1.14350	−0.571749	+	0.820429i	$0.693735\pi$
$840$	0	0
$841$	61.8834	2.13391
$842$	−1.78251	−0.0614293
$843$	50.9667	1.75539
$844$	2.00132	0.0688883
$845$	0	0
$846$	−29.0277	−0.997992
$847$	38.5000	1.32288
$848$	−20.5458	−0.705546
$849$	82.8763	2.84431
$850$	0	0
$851$	45.2478	1.55107
$852$	−5.09549	−0.174569
$853$	−10.9658	−0.375460	−0.187730	−	0.982221i	$-0.560113\pi$
$853$	−10.9658	−0.375460	−0.187730	+	0.982221i	$0.560113\pi$
$854$	−52.1307	−1.78388
$855$	0	0
$856$	4.81502	0.164574
$857$	−31.2649	−1.06799	−0.533994	−	0.845488i	$-0.679310\pi$
$857$	−31.2649	−1.06799	−0.533994	+	0.845488i	$0.679310\pi$
$858$	−46.4862	−1.58701
$859$	22.7878	0.777511	0.388755	−	0.921341i	$-0.372905\pi$
$859$	22.7878	0.777511	0.388755	+	0.921341i	$0.372905\pi$
$860$	0	0
$861$	−39.6131	−1.35001
$862$	15.0090	0.511207
$863$	21.0242	0.715673	0.357836	−	0.933784i	$-0.383515\pi$
$863$	21.0242	0.715673	0.357836	+	0.933784i	$0.383515\pi$
$864$	103.993	3.53791
$865$	0	0
$866$	10.6041	0.360341
$867$	−106.538	−3.61824
$868$	63.2812	2.14790
$869$	10.2862	0.348935
$870$	0	0
$871$	3.54398	0.120083
$872$	−7.62884	−0.258345
$873$	−41.9400	−1.41945
$874$	11.4495	0.387284
$875$	0	0
$876$	−58.3609	−1.97183
$877$	−48.2074	−1.62785	−0.813925	−	0.580970i	$-0.802673\pi$
$877$	−48.2074	−1.62785	−0.813925	+	0.580970i	$0.802673\pi$
$878$	−45.3602	−1.53083
$879$	−43.0238	−1.45116
$880$	0	0
$881$	−13.7705	−0.463939	−0.231969	−	0.972723i	$-0.574517\pi$
$881$	−13.7705	−0.463939	−0.231969	+	0.972723i	$0.574517\pi$
$882$	123.905	4.17209
$883$	0.177399	0.00596995	0.00298497	−	0.999996i	$-0.499050\pi$
$883$	0.177399	0.00596995	0.00298497	+	0.999996i	$0.499050\pi$
$884$	103.774	3.49028
$885$	0	0
$886$	−68.1224	−2.28862
$887$	−26.9823	−0.905976	−0.452988	−	0.891517i	$-0.649642\pi$
$887$	−26.9823	−0.905976	−0.452988	+	0.891517i	$0.649642\pi$
$888$	−10.1934	−0.342068
$889$	26.2385	0.880012
$890$	0	0
$891$	−21.5542	−0.722093
$892$	45.7934	1.53328
$893$	−1.38450	−0.0463305
$894$	−27.1361	−0.907568
$895$	0	0
$896$	17.6882	0.590921
$897$	162.074	5.41148
$898$	−61.6368	−2.05685
$899$	67.3620	2.24665
$900$	0	0
$901$	−43.2910	−1.44223
$902$	7.25862	0.241685
$903$	−47.4496	−1.57902
$904$	0.638438	0.0212341
$905$	0	0
$906$	−120.709	−4.01029
$907$	26.2952	0.873117	0.436558	−	0.899676i	$-0.356197\pi$
$907$	26.2952	0.873117	0.436558	+	0.899676i	$0.356197\pi$
$908$	35.4211	1.17549
$909$	69.9095	2.31875
$910$	0	0
$911$	−28.4511	−0.942626	−0.471313	−	0.881966i	$-0.656220\pi$
$911$	−28.4511	−0.942626	−0.471313	+	0.881966i	$0.656220\pi$
$912$	7.43269	0.246121
$913$	5.60191	0.185396
$914$	−59.1756	−1.95735
$915$	0	0
$916$	−37.4021	−1.23580
$917$	3.16057	0.104371
$918$	188.507	6.22167
$919$	−54.3129	−1.79162	−0.895808	−	0.444440i	$-0.853403\pi$
$919$	−54.3129	−1.79162	−0.895808	+	0.444440i	$0.853403\pi$
$920$	0	0
$921$	53.9454	1.77756
$922$	−35.1038	−1.15608
$923$	−4.53457	−0.149257
$924$	−31.3964	−1.03287
$925$	0	0
$926$	49.0330	1.61132
$927$	−59.7922	−1.96383
$928$	77.3552	2.53931
$929$	28.3938	0.931569	0.465785	−	0.884898i	$-0.345772\pi$
$929$	28.3938	0.931569	0.465785	+	0.884898i	$0.345772\pi$
$930$	0	0
$931$	5.90975	0.193684
$932$	30.5446	1.00052
$933$	−31.8809	−1.04373
$934$	−67.3232	−2.20288
$935$	0	0
$936$	−25.6062	−0.836966
$937$	12.9962	0.424568	0.212284	−	0.977208i	$-0.431910\pi$
$937$	12.9962	0.424568	0.212284	+	0.977208i	$0.431910\pi$
$938$	4.49861	0.146885
$939$	−37.4280	−1.22142
$940$	0	0
$941$	−33.6776	−1.09786	−0.548929	−	0.835869i	$-0.684964\pi$
$941$	−33.6776	−1.09786	−0.548929	+	0.835869i	$0.684964\pi$
$942$	43.4564	1.41588
$943$	−25.3071	−0.824112
$944$	−5.25239	−0.170951
$945$	0	0
$946$	8.69456	0.282684
$947$	14.4792	0.470510	0.235255	−	0.971934i	$-0.424407\pi$
$947$	14.4792	0.470510	0.235255	+	0.971934i	$0.424407\pi$
$948$	67.0192	2.17668
$949$	−51.9365	−1.68593
$950$	0	0
$951$	32.9822	1.06952
$952$	15.8796	0.514660
$953$	1.98683	0.0643597	0.0321798	−	0.999482i	$-0.489755\pi$
$953$	1.98683	0.0643597	0.0321798	+	0.999482i	$0.489755\pi$
$954$	88.6117	2.86891
$955$	0	0
$956$	21.2644	0.687739
$957$	−33.4210	−1.08035
$958$	21.3835	0.690868
$959$	31.4952	1.01703
$960$	0	0
$961$	18.9281	0.610584
$962$	−75.2497	−2.42615
$963$	59.8418	1.92838
$964$	2.27415	0.0732454
$965$	0	0
$966$	205.731	6.61928
$967$	−33.2320	−1.06867	−0.534335	−	0.845273i	$-0.679438\pi$
$967$	−33.2320	−1.06867	−0.534335	+	0.845273i	$0.679438\pi$
$968$	−5.54098	−0.178094
$969$	15.6610	0.503104
$970$	0	0
$971$	34.7575	1.11542	0.557710	−	0.830036i	$-0.311680\pi$
$971$	34.7575	1.11542	0.557710	+	0.830036i	$0.311680\pi$
$972$	−52.9983	−1.69992
$973$	−46.7716	−1.49943
$974$	−38.5780	−1.23612
$975$	0	0
$976$	−21.6201	−0.692044
$977$	15.6365	0.500255	0.250128	−	0.968213i	$-0.419527\pi$
$977$	15.6365	0.500255	0.250128	+	0.968213i	$0.419527\pi$
$978$	−133.067	−4.25500
$979$	13.8733	0.443391
$980$	0	0
$981$	−94.8123	−3.02712
$982$	−11.8222	−0.377262
$983$	−53.5264	−1.70723	−0.853613	−	0.520908i	$-0.825594\pi$
$983$	−53.5264	−1.70723	−0.853613	+	0.520908i	$0.825594\pi$
$984$	5.70118	0.181747
$985$	0	0
$986$	140.221	4.46555
$987$	−24.8775	−0.791861
$988$	−10.1312	−0.322317
$989$	−30.3135	−0.963913
$990$	0	0
$991$	−29.5818	−0.939698	−0.469849	−	0.882747i	$-0.655692\pi$
$991$	−29.5818	−0.939698	−0.469849	+	0.882747i	$0.655692\pi$
$992$	57.3350	1.82039
$993$	79.2868	2.51609
$994$	−5.75603	−0.182570
$995$	0	0
$996$	36.4990	1.15652
$997$	36.8830	1.16810	0.584048	−	0.811719i	$-0.301468\pi$
$997$	36.8830	1.16810	0.584048	+	0.811719i	$0.301468\pi$
$998$	37.0003	1.17122
$999$	−72.7303	−2.30108

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.k.1.19		25
5.4	even	2		1205.2.a.d.1.7	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.a.d.1.7	✓	25	5.4	even	2
6025.2.a.k.1.19		25	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)

\(f(q)\)	\(=\)	\(q+2.06740 q^{2} -3.16922 q^{3} +2.27415 q^{4} -6.55204 q^{6} -3.93807 q^{7} +0.566773 q^{8} +7.04394 q^{9} +O(q^{10})\)	\(q+2.06740 q^{2} -3.16922 q^{3} +2.27415 q^{4} -6.55204 q^{6} -3.93807 q^{7} +0.566773 q^{8} +7.04394 q^{9} -1.10618 q^{11} -7.20727 q^{12} -6.41389 q^{13} -8.14157 q^{14} -3.37655 q^{16} -7.11454 q^{17} +14.5626 q^{18} +0.694578 q^{19} +12.4806 q^{21} -2.28692 q^{22} +7.97331 q^{23} -1.79623 q^{24} -13.2601 q^{26} -12.8161 q^{27} -8.95575 q^{28} -9.53328 q^{29} -7.06598 q^{31} -8.11423 q^{32} +3.50572 q^{33} -14.7086 q^{34} +16.0190 q^{36} +5.67490 q^{37} +1.43597 q^{38} +20.3270 q^{39} -3.17397 q^{41} +25.8024 q^{42} -3.80187 q^{43} -2.51562 q^{44} +16.4840 q^{46} -1.99330 q^{47} +10.7010 q^{48} +8.50839 q^{49} +22.5475 q^{51} -14.5861 q^{52} +6.08486 q^{53} -26.4961 q^{54} -2.23199 q^{56} -2.20127 q^{57} -19.7091 q^{58} +1.55555 q^{59} +6.40303 q^{61} -14.6082 q^{62} -27.7395 q^{63} -10.0223 q^{64} +7.24774 q^{66} -0.552548 q^{67} -16.1795 q^{68} -25.2692 q^{69} +0.706993 q^{71} +3.99231 q^{72} +8.09751 q^{73} +11.7323 q^{74} +1.57957 q^{76} +4.35621 q^{77} +42.0241 q^{78} -9.29883 q^{79} +19.4853 q^{81} -6.56188 q^{82} -5.06419 q^{83} +28.3827 q^{84} -7.85998 q^{86} +30.2130 q^{87} -0.626953 q^{88} -12.5416 q^{89} +25.2583 q^{91} +18.1325 q^{92} +22.3936 q^{93} -4.12094 q^{94} +25.7157 q^{96} -5.95406 q^{97} +17.5903 q^{98} -7.79186 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 4 q^{2} - 9 q^{3} + 36 q^{4} + 7 q^{6} - 7 q^{7} + 15 q^{8} + 36 q^{9}+O(q^{10}) \) 25 * q + 4 * q^2 - 9 * q^3 + 36 * q^4 + 7 * q^6 - 7 * q^7 + 15 * q^8 + 36 * q^9	\( 25 q + 4 q^{2} - 9 q^{3} + 36 q^{4} + 7 q^{6} - 7 q^{7} + 15 q^{8} + 36 q^{9} + 10 q^{11} - 22 q^{12} - 10 q^{13} + 13 q^{14} + 54 q^{16} - q^{17} + 13 q^{18} + 50 q^{19} + 9 q^{21} - 11 q^{22} + 31 q^{23} + 22 q^{24} + 8 q^{26} - 42 q^{27} - 14 q^{28} + 4 q^{29} + 34 q^{31} + 44 q^{32} - 28 q^{33} + 33 q^{34} + 83 q^{36} - 14 q^{37} + 10 q^{38} + 23 q^{39} + 11 q^{41} - 23 q^{42} - 49 q^{43} + 20 q^{44} + 27 q^{46} + 28 q^{47} - 30 q^{48} + 66 q^{49} + 49 q^{51} - 39 q^{52} + 16 q^{53} + 5 q^{54} + 51 q^{56} - 10 q^{57} + 8 q^{58} + 30 q^{59} + 35 q^{61} + 18 q^{62} + 73 q^{64} - 13 q^{66} - 37 q^{67} - 11 q^{68} - 4 q^{69} + 12 q^{71} + 90 q^{72} - 36 q^{73} - 12 q^{74} + 57 q^{76} + 31 q^{77} + 9 q^{78} + 16 q^{79} + 65 q^{81} + 11 q^{82} - 43 q^{83} - 62 q^{84} - 9 q^{86} + 22 q^{87} - 20 q^{88} + 38 q^{89} + 86 q^{91} + 119 q^{92} - 10 q^{93} - 18 q^{94} - 34 q^{96} - 17 q^{97} + 32 q^{98} + 26 q^{99}+O(q^{100}) \) 25 * q + 4 * q^2 - 9 * q^3 + 36 * q^4 + 7 * q^6 - 7 * q^7 + 15 * q^8 + 36 * q^9 + 10 * q^11 - 22 * q^12 - 10 * q^13 + 13 * q^14 + 54 * q^16 - q^17 + 13 * q^18 + 50 * q^19 + 9 * q^21 - 11 * q^22 + 31 * q^23 + 22 * q^24 + 8 * q^26 - 42 * q^27 - 14 * q^28 + 4 * q^29 + 34 * q^31 + 44 * q^32 - 28 * q^33 + 33 * q^34 + 83 * q^36 - 14 * q^37 + 10 * q^38 + 23 * q^39 + 11 * q^41 - 23 * q^42 - 49 * q^43 + 20 * q^44 + 27 * q^46 + 28 * q^47 - 30 * q^48 + 66 * q^49 + 49 * q^51 - 39 * q^52 + 16 * q^53 + 5 * q^54 + 51 * q^56 - 10 * q^57 + 8 * q^58 + 30 * q^59 + 35 * q^61 + 18 * q^62 + 73 * q^64 - 13 * q^66 - 37 * q^67 - 11 * q^68 - 4 * q^69 + 12 * q^71 + 90 * q^72 - 36 * q^73 - 12 * q^74 + 57 * q^76 + 31 * q^77 + 9 * q^78 + 16 * q^79 + 65 * q^81 + 11 * q^82 - 43 * q^83 - 62 * q^84 - 9 * q^86 + 22 * q^87 - 20 * q^88 + 38 * q^89 + 86 * q^91 + 119 * q^92 - 10 * q^93 - 18 * q^94 - 34 * q^96 - 17 * q^97 + 32 * q^98 + 26 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more