LMFDB - Embedded newform 6025.2.a.j.1.2

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

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q-2.68235 q^{2} -1.41400 q^{3} +5.19500 q^{4} +3.79283 q^{6} +3.05254 q^{7} -8.57011 q^{8} -1.00061 q^{9} +O(q^{10})$	\(q-2.68235 q^{2} -1.41400 q^{3} +5.19500 q^{4} +3.79283 q^{6} +3.05254 q^{7} -8.57011 q^{8} -1.00061 q^{9} +2.27833 q^{11} -7.34571 q^{12} +0.616505 q^{13} -8.18799 q^{14} +12.5980 q^{16} -4.32204 q^{17} +2.68400 q^{18} -6.06942 q^{19} -4.31629 q^{21} -6.11127 q^{22} +2.54169 q^{23} +12.1181 q^{24} -1.65368 q^{26} +5.65685 q^{27} +15.8580 q^{28} -4.14556 q^{29} +5.31530 q^{31} -16.6521 q^{32} -3.22155 q^{33} +11.5932 q^{34} -5.19819 q^{36} +2.80922 q^{37} +16.2803 q^{38} -0.871736 q^{39} -0.655977 q^{41} +11.5778 q^{42} -2.06759 q^{43} +11.8359 q^{44} -6.81771 q^{46} -1.80394 q^{47} -17.8136 q^{48} +2.31803 q^{49} +6.11136 q^{51} +3.20274 q^{52} +1.85617 q^{53} -15.1737 q^{54} -26.1606 q^{56} +8.58214 q^{57} +11.1198 q^{58} +9.52747 q^{59} -3.91122 q^{61} -14.2575 q^{62} -3.05442 q^{63} +19.4707 q^{64} +8.64132 q^{66} +3.72928 q^{67} -22.4530 q^{68} -3.59395 q^{69} -6.80163 q^{71} +8.57537 q^{72} +16.2227 q^{73} -7.53530 q^{74} -31.5306 q^{76} +6.95470 q^{77} +2.33830 q^{78} +6.36558 q^{79} -4.99693 q^{81} +1.75956 q^{82} -15.5877 q^{83} -22.4231 q^{84} +5.54599 q^{86} +5.86181 q^{87} -19.5255 q^{88} -8.49697 q^{89} +1.88191 q^{91} +13.2041 q^{92} -7.51582 q^{93} +4.83879 q^{94} +23.5460 q^{96} -6.92655 q^{97} -6.21776 q^{98} -2.27973 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 6 q^{2} - 15 q^{3} + 32 q^{4} - q^{6} - 19 q^{7} - 15 q^{8} + 32 q^{9}+O(q^{10}) $ 25 * q - 6 * q^2 - 15 * q^3 + 32 * q^4 - q^6 - 19 * q^7 - 15 * q^8 + 32 * q^9	$ 25 q - 6 q^{2} - 15 q^{3} + 32 q^{4} - q^{6} - 19 q^{7} - 15 q^{8} + 32 q^{9} + 2 q^{11} - 20 q^{12} - 14 q^{13} - 5 q^{14} + 38 q^{16} - 7 q^{17} - 9 q^{18} + 30 q^{19} + q^{21} - q^{22} - 43 q^{23} - 6 q^{24} - 22 q^{26} - 42 q^{27} - 32 q^{28} - 4 q^{29} + 14 q^{31} - 26 q^{32} - 4 q^{33} + 7 q^{34} + 15 q^{36} - 16 q^{37} - 14 q^{38} - 21 q^{39} - q^{41} + 25 q^{42} - 35 q^{43} - 52 q^{44} - 27 q^{46} - 50 q^{47} - 26 q^{48} + 46 q^{49} - 7 q^{51} - 3 q^{52} - 4 q^{53} - 31 q^{54} - 51 q^{56} - 2 q^{58} + 6 q^{59} + 19 q^{61} - 28 q^{63} + 49 q^{64} - 27 q^{66} - 65 q^{67} + 25 q^{68} + 2 q^{69} - 34 q^{71} + 10 q^{72} - 8 q^{73} - 42 q^{74} + 71 q^{76} - q^{77} + 59 q^{78} - 12 q^{79} + 29 q^{81} - 11 q^{82} - 41 q^{83} - 10 q^{84} - 13 q^{86} - 40 q^{87} + 52 q^{88} - 24 q^{89} + 46 q^{91} - 85 q^{92} + 30 q^{93} + 14 q^{94} - 30 q^{96} - 9 q^{97} + 64 q^{98} + 2 q^{99}+O(q^{100}) $ 25 * q - 6 * q^2 - 15 * q^3 + 32 * q^4 - q^6 - 19 * q^7 - 15 * q^8 + 32 * q^9 + 2 * q^11 - 20 * q^12 - 14 * q^13 - 5 * q^14 + 38 * q^16 - 7 * q^17 - 9 * q^18 + 30 * q^19 + q^21 - q^22 - 43 * q^23 - 6 * q^24 - 22 * q^26 - 42 * q^27 - 32 * q^28 - 4 * q^29 + 14 * q^31 - 26 * q^32 - 4 * q^33 + 7 * q^34 + 15 * q^36 - 16 * q^37 - 14 * q^38 - 21 * q^39 - q^41 + 25 * q^42 - 35 * q^43 - 52 * q^44 - 27 * q^46 - 50 * q^47 - 26 * q^48 + 46 * q^49 - 7 * q^51 - 3 * q^52 - 4 * q^53 - 31 * q^54 - 51 * q^56 - 2 * q^58 + 6 * q^59 + 19 * q^61 - 28 * q^63 + 49 * q^64 - 27 * q^66 - 65 * q^67 + 25 * q^68 + 2 * q^69 - 34 * q^71 + 10 * q^72 - 8 * q^73 - 42 * q^74 + 71 * q^76 - q^77 + 59 * q^78 - 12 * q^79 + 29 * q^81 - 11 * q^82 - 41 * q^83 - 10 * q^84 - 13 * q^86 - 40 * q^87 + 52 * q^88 - 24 * q^89 + 46 * q^91 - 85 * q^92 + 30 * q^93 + 14 * q^94 - 30 * q^96 - 9 * q^97 + 64 * q^98 + 2 * 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 $

$2$	−2.68235	−1.89671	−0.948354	−	0.317214i	$-0.897253\pi$
$2$	−2.68235	−1.89671	−0.948354	+	0.317214i	$0.897253\pi$
$3$	−1.41400	−0.816371	−0.408186	−	0.912899i	$-0.633838\pi$
$3$	−1.41400	−0.816371	−0.408186	+	0.912899i	$0.633838\pi$
$4$	5.19500	2.59750
$5$	0	0
$5$	0	0
$6$	3.79283	1.54842
$7$	3.05254	1.15375	0.576877	−	0.816831i	$-0.304271\pi$
$7$	3.05254	1.15375	0.576877	+	0.816831i	$0.304271\pi$
$8$	−8.57011	−3.02999
$9$	−1.00061	−0.333538
$10$	0	0
$11$	2.27833	0.686942	0.343471	−	0.939163i	$-0.388397\pi$
$11$	2.27833	0.686942	0.343471	+	0.939163i	$0.388397\pi$
$12$	−7.34571	−2.12052
$13$	0.616505	0.170988	0.0854938	−	0.996339i	$-0.472753\pi$
$13$	0.616505	0.170988	0.0854938	+	0.996339i	$0.472753\pi$
$14$	−8.18799	−2.18833
$15$	0	0
$16$	12.5980	3.14951
$17$	−4.32204	−1.04825	−0.524125	−	0.851641i	$-0.675608\pi$
$17$	−4.32204	−1.04825	−0.524125	+	0.851641i	$0.675608\pi$
$18$	2.68400	0.632624
$19$	−6.06942	−1.39242	−0.696210	−	0.717838i	$-0.745132\pi$
$19$	−6.06942	−1.39242	−0.696210	+	0.717838i	$0.745132\pi$
$20$	0	0
$21$	−4.31629	−0.941891
$22$	−6.11127	−1.30293
$23$	2.54169	0.529980	0.264990	−	0.964251i	$-0.414631\pi$
$23$	2.54169	0.529980	0.264990	+	0.964251i	$0.414631\pi$
$24$	12.1181	2.47360
$25$	0	0
$26$	−1.65368	−0.324314
$27$	5.65685	1.08866
$28$	15.8580	2.99687
$29$	−4.14556	−0.769811	−0.384906	−	0.922956i	$-0.625766\pi$
$29$	−4.14556	−0.769811	−0.384906	+	0.922956i	$0.625766\pi$
$30$	0	0
$31$	5.31530	0.954656	0.477328	−	0.878725i	$-0.341605\pi$
$31$	5.31530	0.954656	0.477328	+	0.878725i	$0.341605\pi$
$32$	−16.6521	−2.94371
$33$	−3.22155	−0.560799
$34$	11.5932	1.98822
$35$	0	0
$36$	−5.19819	−0.866365
$37$	2.80922	0.461832	0.230916	−	0.972974i	$-0.425828\pi$
$37$	2.80922	0.461832	0.230916	+	0.972974i	$0.425828\pi$
$38$	16.2803	2.64101
$39$	−0.871736	−0.139589
$40$	0	0
$41$	−0.655977	−0.102446	−0.0512232	−	0.998687i	$-0.516312\pi$
$41$	−0.655977	−0.102446	−0.0512232	+	0.998687i	$0.516312\pi$
$42$	11.5778	1.78649
$43$	−2.06759	−0.315304	−0.157652	−	0.987495i	$-0.550392\pi$
$43$	−2.06759	−0.315304	−0.157652	+	0.987495i	$0.550392\pi$
$44$	11.8359	1.78433
$45$	0	0
$46$	−6.81771	−1.00522
$47$	−1.80394	−0.263131	−0.131566	−	0.991307i	$-0.542000\pi$
$47$	−1.80394	−0.263131	−0.131566	+	0.991307i	$0.542000\pi$
$48$	−17.8136	−2.57117
$49$	2.31803	0.331147
$50$	0	0
$51$	6.11136	0.855761
$52$	3.20274	0.444141
$53$	1.85617	0.254964	0.127482	−	0.991841i	$-0.459311\pi$
$53$	1.85617	0.254964	0.127482	+	0.991841i	$0.459311\pi$
$54$	−15.1737	−2.06487
$55$	0	0
$56$	−26.1606	−3.49586
$57$	8.58214	1.13673
$58$	11.1198	1.46011
$59$	9.52747	1.24037	0.620185	−	0.784455i	$-0.287057\pi$
$59$	9.52747	1.24037	0.620185	+	0.784455i	$0.287057\pi$
$60$	0	0
$61$	−3.91122	−0.500780	−0.250390	−	0.968145i	$-0.580559\pi$
$61$	−3.91122	−0.500780	−0.250390	+	0.968145i	$0.580559\pi$
$62$	−14.2575	−1.81070
$63$	−3.05442	−0.384821
$64$	19.4707	2.43384
$65$	0	0
$66$	8.64132	1.06367
$67$	3.72928	0.455604	0.227802	−	0.973708i	$-0.426846\pi$
$67$	3.72928	0.455604	0.227802	+	0.973708i	$0.426846\pi$
$68$	−22.4530	−2.72283
$69$	−3.59395	−0.432660
$70$	0	0
$71$	−6.80163	−0.807205	−0.403602	−	0.914934i	$-0.632242\pi$
$71$	−6.80163	−0.807205	−0.403602	+	0.914934i	$0.632242\pi$
$72$	8.57537	1.01062
$73$	16.2227	1.89872	0.949360	−	0.314189i	$-0.101733\pi$
$73$	16.2227	1.89872	0.949360	+	0.314189i	$0.101733\pi$
$74$	−7.53530	−0.875961
$75$	0	0
$76$	−31.5306	−3.61681
$77$	6.95470	0.792561
$78$	2.33830	0.264760
$79$	6.36558	0.716183	0.358092	−	0.933686i	$-0.383428\pi$
$79$	6.36558	0.716183	0.358092	+	0.933686i	$0.383428\pi$
$80$	0	0
$81$	−4.99693	−0.555214
$82$	1.75956	0.194311
$83$	−15.5877	−1.71097	−0.855487	−	0.517824i	$-0.826742\pi$
$83$	−15.5877	−1.71097	−0.855487	+	0.517824i	$0.826742\pi$
$84$	−22.4231	−2.44656
$85$	0	0
$86$	5.54599	0.598039
$87$	5.86181	0.628452
$88$	−19.5255	−2.08143
$89$	−8.49697	−0.900677	−0.450338	−	0.892858i	$-0.648697\pi$
$89$	−8.49697	−0.900677	−0.450338	+	0.892858i	$0.648697\pi$
$90$	0	0
$91$	1.88191	0.197278
$92$	13.2041	1.37662
$93$	−7.51582	−0.779354
$94$	4.83879	0.499083
$95$	0	0
$96$	23.5460	2.40316
$97$	−6.92655	−0.703284	−0.351642	−	0.936135i	$-0.614377\pi$
$97$	−6.92655	−0.703284	−0.351642	+	0.936135i	$0.614377\pi$
$98$	−6.21776	−0.628088
$99$	−2.27973	−0.229121
$100$	0	0
$101$	3.47837	0.346111	0.173056	−	0.984912i	$-0.444636\pi$
$101$	3.47837	0.346111	0.173056	+	0.984912i	$0.444636\pi$
$102$	−16.3928	−1.62313
$103$	3.97081	0.391256	0.195628	−	0.980678i	$-0.437326\pi$
$103$	3.97081	0.391256	0.195628	+	0.980678i	$0.437326\pi$
$104$	−5.28352	−0.518091
$105$	0	0
$106$	−4.97889	−0.483592
$107$	−7.96286	−0.769799	−0.384900	−	0.922958i	$-0.625764\pi$
$107$	−7.96286	−0.769799	−0.384900	+	0.922958i	$0.625764\pi$
$108$	29.3874	2.82780
$109$	−15.6095	−1.49512	−0.747560	−	0.664194i	$-0.768775\pi$
$109$	−15.6095	−1.49512	−0.747560	+	0.664194i	$0.768775\pi$
$110$	0	0
$111$	−3.97222	−0.377027
$112$	38.4561	3.63376
$113$	−12.8931	−1.21288	−0.606439	−	0.795130i	$-0.707403\pi$
$113$	−12.8931	−1.21288	−0.606439	+	0.795130i	$0.707403\pi$
$114$	−23.0203	−2.15605
$115$	0	0
$116$	−21.5362	−1.99959
$117$	−0.616883	−0.0570309
$118$	−25.5560	−2.35262
$119$	−13.1932	−1.20942
$120$	0	0
$121$	−5.80922	−0.528111
$122$	10.4913	0.949833
$123$	0.927549	0.0836343
$124$	27.6130	2.47972
$125$	0	0
$126$	8.19302	0.729892
$127$	19.7819	1.75536	0.877678	−	0.479251i	$-0.159092\pi$
$127$	19.7819	1.75536	0.877678	+	0.479251i	$0.159092\pi$
$128$	−18.9231	−1.67258
$129$	2.92356	0.257405
$130$	0	0
$131$	18.4080	1.60832	0.804159	−	0.594414i	$-0.202616\pi$
$131$	18.4080	1.60832	0.804159	+	0.594414i	$0.202616\pi$
$132$	−16.7359	−1.45668
$133$	−18.5272	−1.60651
$134$	−10.0032	−0.864147
$135$	0	0
$136$	37.0404	3.17619
$137$	−21.1935	−1.81068	−0.905341	−	0.424686i	$-0.860384\pi$
$137$	−21.1935	−1.81068	−0.905341	+	0.424686i	$0.860384\pi$
$138$	9.64022	0.820630
$139$	−3.94086	−0.334260	−0.167130	−	0.985935i	$-0.553450\pi$
$139$	−3.94086	−0.334260	−0.167130	+	0.985935i	$0.553450\pi$
$140$	0	0
$141$	2.55076	0.214813
$142$	18.2443	1.53103
$143$	1.40460	0.117459
$144$	−12.6058	−1.05048
$145$	0	0
$146$	−43.5149	−3.60132
$147$	−3.27768	−0.270339
$148$	14.5939	1.19961
$149$	11.3657	0.931117	0.465559	−	0.885017i	$-0.345853\pi$
$149$	11.3657	0.931117	0.465559	+	0.885017i	$0.345853\pi$
$150$	0	0
$151$	5.49904	0.447505	0.223753	−	0.974646i	$-0.428169\pi$
$151$	5.49904	0.447505	0.223753	+	0.974646i	$0.428169\pi$
$152$	52.0156	4.21902
$153$	4.32470	0.349631
$154$	−18.6549	−1.50326
$155$	0	0
$156$	−4.52867	−0.362584
$157$	−4.33842	−0.346244	−0.173122	−	0.984900i	$-0.555385\pi$
$157$	−4.33842	−0.346244	−0.173122	+	0.984900i	$0.555385\pi$
$158$	−17.0747	−1.35839
$159$	−2.62461	−0.208145
$160$	0	0
$161$	7.75863	0.611466
$162$	13.4035	1.05308
$163$	−4.13936	−0.324220	−0.162110	−	0.986773i	$-0.551830\pi$
$163$	−4.13936	−0.324220	−0.162110	+	0.986773i	$0.551830\pi$
$164$	−3.40780	−0.266105
$165$	0	0
$166$	41.8117	3.24522
$167$	−18.7468	−1.45067	−0.725334	−	0.688398i	$-0.758314\pi$
$167$	−18.7468	−1.45067	−0.725334	+	0.688398i	$0.758314\pi$
$168$	36.9911	2.85392
$169$	−12.6199	−0.970763
$170$	0	0
$171$	6.07315	0.464425
$172$	−10.7411	−0.819002
$173$	−6.84794	−0.520639	−0.260319	−	0.965523i	$-0.583828\pi$
$173$	−6.84794	−0.520639	−0.260319	+	0.965523i	$0.583828\pi$
$174$	−15.7234	−1.19199
$175$	0	0
$176$	28.7025	2.16353
$177$	−13.4718	−1.01260
$178$	22.7918	1.70832
$179$	−9.94377	−0.743232	−0.371616	−	0.928387i	$-0.621196\pi$
$179$	−9.94377	−0.743232	−0.371616	+	0.928387i	$0.621196\pi$
$180$	0	0
$181$	17.1721	1.27639	0.638195	−	0.769875i	$-0.279681\pi$
$181$	17.1721	1.27639	0.638195	+	0.769875i	$0.279681\pi$
$182$	−5.04794	−0.374178
$183$	5.53045	0.408822
$184$	−21.7826	−1.60583
$185$	0	0
$186$	20.1601	1.47821
$187$	−9.84704	−0.720087
$188$	−9.37145	−0.683483
$189$	17.2678	1.25605
$190$	0	0
$191$	9.71037	0.702618	0.351309	−	0.936260i	$-0.385737\pi$
$191$	9.71037	0.702618	0.351309	+	0.936260i	$0.385737\pi$
$192$	−27.5315	−1.98692
$193$	13.2416	0.953152	0.476576	−	0.879133i	$-0.341878\pi$
$193$	13.2416	0.953152	0.476576	+	0.879133i	$0.341878\pi$
$194$	18.5794	1.33392
$195$	0	0
$196$	12.0421	0.860153
$197$	−19.2007	−1.36799	−0.683995	−	0.729486i	$-0.739759\pi$
$197$	−19.2007	−1.36799	−0.683995	+	0.729486i	$0.739759\pi$
$198$	6.11503	0.434576
$199$	3.11313	0.220684	0.110342	−	0.993894i	$-0.464805\pi$
$199$	3.11313	0.220684	0.110342	+	0.993894i	$0.464805\pi$
$200$	0	0
$201$	−5.27318	−0.371942
$202$	−9.33021	−0.656472
$203$	−12.6545	−0.888172
$204$	31.7485	2.22284
$205$	0	0
$206$	−10.6511	−0.742098
$207$	−2.54325	−0.176768
$208$	7.76675	0.538527
$209$	−13.8281	−0.956512
$210$	0	0
$211$	1.59969	0.110127	0.0550637	−	0.998483i	$-0.482464\pi$
$211$	1.59969	0.110127	0.0550637	+	0.998483i	$0.482464\pi$
$212$	9.64278	0.662269
$213$	9.61748	0.658979
$214$	21.3592	1.46008
$215$	0	0
$216$	−48.4799	−3.29864
$217$	16.2252	1.10144
$218$	41.8702	2.83581
$219$	−22.9388	−1.55006
$220$	0	0
$221$	−2.66456	−0.179238
$222$	10.6549	0.715110
$223$	−14.6757	−0.982757	−0.491378	−	0.870946i	$-0.663507\pi$
$223$	−14.6757	−0.982757	−0.491378	+	0.870946i	$0.663507\pi$
$224$	−50.8313	−3.39631
$225$	0	0
$226$	34.5837	2.30048
$227$	26.3910	1.75163	0.875816	−	0.482646i	$-0.160324\pi$
$227$	26.3910	1.75163	0.875816	+	0.482646i	$0.160324\pi$
$228$	44.5842	2.95266
$229$	−10.5611	−0.697896	−0.348948	−	0.937142i	$-0.613461\pi$
$229$	−10.5611	−0.697896	−0.348948	+	0.937142i	$0.613461\pi$
$230$	0	0
$231$	−9.83392	−0.647024
$232$	35.5279	2.33252
$233$	7.55972	0.495254	0.247627	−	0.968855i	$-0.420349\pi$
$233$	7.55972	0.495254	0.247627	+	0.968855i	$0.420349\pi$
$234$	1.65470	0.108171
$235$	0	0
$236$	49.4952	3.22186
$237$	−9.00090	−0.584671
$238$	35.3889	2.29392
$239$	4.45269	0.288020	0.144010	−	0.989576i	$-0.454000\pi$
$239$	4.45269	0.288020	0.144010	+	0.989576i	$0.454000\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	15.5824	1.00167
$243$	−9.90492	−0.635401
$244$	−20.3188	−1.30078
$245$	0	0
$246$	−2.48801	−0.158630
$247$	−3.74183	−0.238087
$248$	−45.5527	−2.89260
$249$	22.0410	1.39679
$250$	0	0
$251$	15.6096	0.985270	0.492635	−	0.870236i	$-0.336034\pi$
$251$	15.6096	0.985270	0.492635	+	0.870236i	$0.336034\pi$
$252$	−15.8677	−0.999572
$253$	5.79081	0.364065
$254$	−53.0618	−3.32940
$255$	0	0
$256$	11.8169	0.738555
$257$	−28.0229	−1.74802	−0.874012	−	0.485905i	$-0.838490\pi$
$257$	−28.0229	−1.74802	−0.874012	+	0.485905i	$0.838490\pi$
$258$	−7.84201	−0.488222
$259$	8.57526	0.532841
$260$	0	0
$261$	4.14811	0.256761
$262$	−49.3768	−3.05051
$263$	3.10219	0.191289	0.0956445	−	0.995416i	$-0.469509\pi$
$263$	3.10219	0.191289	0.0956445	+	0.995416i	$0.469509\pi$
$264$	27.6090	1.69922
$265$	0	0
$266$	49.6964	3.04708
$267$	12.0147	0.735286
$268$	19.3736	1.18343
$269$	−30.1751	−1.83981	−0.919904	−	0.392143i	$-0.871734\pi$
$269$	−30.1751	−1.83981	−0.919904	+	0.392143i	$0.871734\pi$
$270$	0	0
$271$	13.0541	0.792982	0.396491	−	0.918039i	$-0.370228\pi$
$271$	13.0541	0.792982	0.396491	+	0.918039i	$0.370228\pi$
$272$	−54.4493	−3.30147
$273$	−2.66101	−0.161052
$274$	56.8483	3.43433
$275$	0	0
$276$	−18.6706	−1.12384
$277$	25.2635	1.51794	0.758970	−	0.651126i	$-0.225703\pi$
$277$	25.2635	1.51794	0.758970	+	0.651126i	$0.225703\pi$
$278$	10.5708	0.633993
$279$	−5.31857	−0.318414
$280$	0	0
$281$	0.422214	0.0251872	0.0125936	−	0.999921i	$-0.495991\pi$
$281$	0.422214	0.0251872	0.0125936	+	0.999921i	$0.495991\pi$
$282$	−6.84203	−0.407437
$283$	−23.0533	−1.37038	−0.685188	−	0.728366i	$-0.740280\pi$
$283$	−23.0533	−1.37038	−0.685188	+	0.728366i	$0.740280\pi$
$284$	−35.3345	−2.09672
$285$	0	0
$286$	−3.76763	−0.222785
$287$	−2.00240	−0.118198
$288$	16.6623	0.981838
$289$	1.68007	0.0988277
$290$	0	0
$291$	9.79411	0.574141
$292$	84.2768	4.93193
$293$	28.0787	1.64037	0.820187	−	0.572095i	$-0.193869\pi$
$293$	28.0787	1.64037	0.820187	+	0.572095i	$0.193869\pi$
$294$	8.79189	0.512753
$295$	0	0
$296$	−24.0753	−1.39935
$297$	12.8882	0.747847
$298$	−30.4869	−1.76606
$299$	1.56697	0.0906200
$300$	0	0
$301$	−6.31140	−0.363783
$302$	−14.7503	−0.848787
$303$	−4.91841	−0.282555
$304$	−76.4628	−4.38544
$305$	0	0
$306$	−11.6004	−0.663148
$307$	31.3571	1.78965	0.894823	−	0.446422i	$-0.147302\pi$
$307$	31.3571	1.78965	0.894823	+	0.446422i	$0.147302\pi$
$308$	36.1297	2.05868
$309$	−5.61472	−0.319410
$310$	0	0
$311$	−26.6179	−1.50936	−0.754681	−	0.656091i	$-0.772209\pi$
$311$	−26.6179	−1.50936	−0.754681	+	0.656091i	$0.772209\pi$
$312$	7.47087	0.422955
$313$	−16.7693	−0.947859	−0.473929	−	0.880563i	$-0.657165\pi$
$313$	−16.7693	−0.947859	−0.473929	+	0.880563i	$0.657165\pi$
$314$	11.6372	0.656723
$315$	0	0
$316$	33.0692	1.86029
$317$	34.2148	1.92170	0.960848	−	0.277077i	$-0.0893657\pi$
$317$	34.2148	1.92170	0.960848	+	0.277077i	$0.0893657\pi$
$318$	7.04013	0.394791
$319$	−9.44495	−0.528816
$320$	0	0
$321$	11.2595	0.628442
$322$	−20.8114	−1.15977
$323$	26.2323	1.45960
$324$	−25.9591	−1.44217
$325$	0	0
$326$	11.1032	0.614951
$327$	22.0718	1.22057
$328$	5.62180	0.310412
$329$	−5.50659	−0.303588
$330$	0	0
$331$	20.8025	1.14341	0.571703	−	0.820460i	$-0.306283\pi$
$331$	20.8025	1.14341	0.571703	+	0.820460i	$0.306283\pi$
$332$	−80.9782	−4.44426
$333$	−2.81094	−0.154039
$334$	50.2853	2.75149
$335$	0	0
$336$	−54.3767	−2.96649
$337$	−14.4232	−0.785683	−0.392842	−	0.919606i	$-0.628508\pi$
$337$	−14.4232	−0.785683	−0.392842	+	0.919606i	$0.628508\pi$
$338$	33.8510	1.84125
$339$	18.2308	0.990159
$340$	0	0
$341$	12.1100	0.655793
$342$	−16.2903	−0.880879
$343$	−14.2919	−0.771692
$344$	17.7194	0.955368
$345$	0	0
$346$	18.3686	0.987500
$347$	12.7955	0.686897	0.343449	−	0.939171i	$-0.388405\pi$
$347$	12.7955	0.686897	0.343449	+	0.939171i	$0.388405\pi$
$348$	30.4521	1.63240
$349$	27.8480	1.49067	0.745334	−	0.666692i	$-0.232290\pi$
$349$	27.8480	1.49067	0.745334	+	0.666692i	$0.232290\pi$
$350$	0	0
$351$	3.48748	0.186148
$352$	−37.9390	−2.02215
$353$	−15.3466	−0.816818	−0.408409	−	0.912799i	$-0.633916\pi$
$353$	−15.3466	−0.816818	−0.408409	+	0.912799i	$0.633916\pi$
$354$	36.1361	1.92061
$355$	0	0
$356$	−44.1417	−2.33951
$357$	18.6552	0.987337
$358$	26.6727	1.40969
$359$	−0.500103	−0.0263944	−0.0131972	−	0.999913i	$-0.504201\pi$
$359$	−0.500103	−0.0263944	−0.0131972	+	0.999913i	$0.504201\pi$
$360$	0	0
$361$	17.8379	0.938834
$362$	−46.0615	−2.42094
$363$	8.21422	0.431135
$364$	9.77652	0.512429
$365$	0	0
$366$	−14.8346	−0.775417
$367$	30.9406	1.61508	0.807542	−	0.589810i	$-0.200797\pi$
$367$	30.9406	1.61508	0.807542	+	0.589810i	$0.200797\pi$
$368$	32.0203	1.66918
$369$	0.656380	0.0341698
$370$	0	0
$371$	5.66603	0.294165
$372$	−39.0447	−2.02437
$373$	−19.7798	−1.02416	−0.512080	−	0.858938i	$-0.671125\pi$
$373$	−19.7798	−1.02416	−0.512080	+	0.858938i	$0.671125\pi$
$374$	26.4132	1.36579
$375$	0	0
$376$	15.4599	0.797285
$377$	−2.55576	−0.131628
$378$	−46.3183	−2.38235
$379$	27.0713	1.39056	0.695279	−	0.718740i	$-0.255281\pi$
$379$	27.0713	1.39056	0.695279	+	0.718740i	$0.255281\pi$
$380$	0	0
$381$	−27.9715	−1.43302
$382$	−26.0466	−1.33266
$383$	4.35176	0.222365	0.111182	−	0.993800i	$-0.464536\pi$
$383$	4.35176	0.222365	0.111182	+	0.993800i	$0.464536\pi$
$384$	26.7572	1.36545
$385$	0	0
$386$	−35.5186	−1.80785
$387$	2.06886	0.105166
$388$	−35.9834	−1.82678
$389$	−14.2705	−0.723542	−0.361771	−	0.932267i	$-0.617828\pi$
$389$	−14.2705	−0.723542	−0.361771	+	0.932267i	$0.617828\pi$
$390$	0	0
$391$	−10.9853	−0.555551
$392$	−19.8657	−1.00337
$393$	−26.0289	−1.31298
$394$	51.5029	2.59468
$395$	0	0
$396$	−11.8432	−0.595142
$397$	−21.4639	−1.07724	−0.538620	−	0.842549i	$-0.681054\pi$
$397$	−21.4639	−1.07724	−0.538620	+	0.842549i	$0.681054\pi$
$398$	−8.35050	−0.418573
$399$	26.1974	1.31151
$400$	0	0
$401$	−27.7914	−1.38784	−0.693919	−	0.720053i	$-0.744117\pi$
$401$	−27.7914	−1.38784	−0.693919	+	0.720053i	$0.744117\pi$
$402$	14.1445	0.705465
$403$	3.27691	0.163234
$404$	18.0702	0.899024
$405$	0	0
$406$	33.9438	1.68460
$407$	6.40032	0.317252
$408$	−52.3750	−2.59295
$409$	−7.86615	−0.388956	−0.194478	−	0.980907i	$-0.562301\pi$
$409$	−7.86615	−0.388956	−0.194478	+	0.980907i	$0.562301\pi$
$410$	0	0
$411$	29.9675	1.47819
$412$	20.6284	1.01629
$413$	29.0830	1.43108
$414$	6.82190	0.335278
$415$	0	0
$416$	−10.2661	−0.503337
$417$	5.57237	0.272880
$418$	37.0919	1.81422
$419$	25.5622	1.24879	0.624397	−	0.781107i	$-0.285345\pi$
$419$	25.5622	1.24879	0.624397	+	0.781107i	$0.285345\pi$
$420$	0	0
$421$	−25.2911	−1.23261	−0.616307	−	0.787506i	$-0.711372\pi$
$421$	−25.2911	−1.23261	−0.616307	+	0.787506i	$0.711372\pi$
$422$	−4.29093	−0.208879
$423$	1.80504	0.0877642
$424$	−15.9075	−0.772539
$425$	0	0
$426$	−25.7974	−1.24989
$427$	−11.9392	−0.577776
$428$	−41.3671	−1.99955
$429$	−1.98610	−0.0958898
$430$	0	0
$431$	23.0672	1.11111	0.555554	−	0.831480i	$-0.312506\pi$
$431$	23.0672	1.11111	0.555554	+	0.831480i	$0.312506\pi$
$432$	71.2652	3.42875
$433$	−21.7182	−1.04371	−0.521856	−	0.853034i	$-0.674760\pi$
$433$	−21.7182	−1.04371	−0.521856	+	0.853034i	$0.674760\pi$
$434$	−43.5216	−2.08911
$435$	0	0
$436$	−81.0914	−3.88357
$437$	−15.4266	−0.737955
$438$	61.5299	2.94001
$439$	−6.33178	−0.302200	−0.151100	−	0.988519i	$-0.548281\pi$
$439$	−6.33178	−0.302200	−0.151100	+	0.988519i	$0.548281\pi$
$440$	0	0
$441$	−2.31945	−0.110450
$442$	7.14729	0.339962
$443$	−4.98082	−0.236646	−0.118323	−	0.992975i	$-0.537752\pi$
$443$	−4.98082	−0.236646	−0.118323	+	0.992975i	$0.537752\pi$
$444$	−20.6357	−0.979327
$445$	0	0
$446$	39.3653	1.86400
$447$	−16.0711	−0.760137
$448$	59.4353	2.80805
$449$	26.1530	1.23424	0.617118	−	0.786870i	$-0.288300\pi$
$449$	26.1530	1.23424	0.617118	+	0.786870i	$0.288300\pi$
$450$	0	0
$451$	−1.49453	−0.0703747
$452$	−66.9795	−3.15045
$453$	−7.77562	−0.365331
$454$	−70.7899	−3.32233
$455$	0	0
$456$	−73.5499	−3.44429
$457$	0.828266	0.0387446	0.0193723	−	0.999812i	$-0.493833\pi$
$457$	0.828266	0.0387446	0.0193723	+	0.999812i	$0.493833\pi$
$458$	28.3285	1.32371
$459$	−24.4492	−1.14119
$460$	0	0
$461$	9.13286	0.425360	0.212680	−	0.977122i	$-0.431781\pi$
$461$	9.13286	0.425360	0.212680	+	0.977122i	$0.431781\pi$
$462$	26.3780	1.22722
$463$	−23.3743	−1.08630	−0.543148	−	0.839637i	$-0.682768\pi$
$463$	−23.3743	−1.08630	−0.543148	+	0.839637i	$0.682768\pi$
$464$	−52.2259	−2.42453
$465$	0	0
$466$	−20.2778	−0.939352
$467$	−41.6087	−1.92542	−0.962711	−	0.270532i	$-0.912800\pi$
$467$	−41.6087	−1.92542	−0.962711	+	0.270532i	$0.912800\pi$
$468$	−3.20471	−0.148138
$469$	11.3838	0.525654
$470$	0	0
$471$	6.13451	0.282663
$472$	−81.6514	−3.75831
$473$	−4.71064	−0.216595
$474$	24.1436	1.10895
$475$	0	0
$476$	−68.5389	−3.14147
$477$	−1.85731	−0.0850402
$478$	−11.9437	−0.546291
$479$	−31.0885	−1.42047	−0.710234	−	0.703966i	$-0.751411\pi$
$479$	−31.0885	−1.42047	−0.710234	+	0.703966i	$0.751411\pi$
$480$	0	0
$481$	1.73190	0.0789677
$482$	2.68235	0.122178
$483$	−10.9707	−0.499183
$484$	−30.1789	−1.37177
$485$	0	0
$486$	26.5685	1.20517
$487$	18.4140	0.834420	0.417210	−	0.908810i	$-0.363008\pi$
$487$	18.4140	0.834420	0.417210	+	0.908810i	$0.363008\pi$
$488$	33.5196	1.51736
$489$	5.85305	0.264684
$490$	0	0
$491$	−38.6958	−1.74632	−0.873159	−	0.487435i	$-0.837932\pi$
$491$	−38.6958	−1.74632	−0.873159	+	0.487435i	$0.837932\pi$
$492$	4.81862	0.217240
$493$	17.9173	0.806955
$494$	10.0369	0.451581
$495$	0	0
$496$	66.9624	3.00670
$497$	−20.7623	−0.931315
$498$	−59.1216	−2.64930
$499$	−32.9752	−1.47617	−0.738087	−	0.674705i	$-0.764271\pi$
$499$	−32.9752	−1.47617	−0.738087	+	0.674705i	$0.764271\pi$
$500$	0	0
$501$	26.5078	1.18428
$502$	−41.8705	−1.86877
$503$	−14.1605	−0.631383	−0.315692	−	0.948862i	$-0.602237\pi$
$503$	−14.1605	−0.631383	−0.315692	+	0.948862i	$0.602237\pi$
$504$	26.1767	1.16600
$505$	0	0
$506$	−15.5330	−0.690525
$507$	17.8445	0.792503
$508$	102.767	4.55954
$509$	−6.62549	−0.293670	−0.146835	−	0.989161i	$-0.546909\pi$
$509$	−6.62549	−0.293670	−0.146835	+	0.989161i	$0.546909\pi$
$510$	0	0
$511$	49.5204	2.19066
$512$	6.14918	0.271758
$513$	−34.3338	−1.51588
$514$	75.1673	3.31549
$515$	0	0
$516$	15.1879	0.668610
$517$	−4.10996	−0.180756
$518$	−23.0018	−1.01064
$519$	9.68296	0.425035
$520$	0	0
$521$	10.9649	0.480382	0.240191	−	0.970726i	$-0.422790\pi$
$521$	10.9649	0.480382	0.240191	+	0.970726i	$0.422790\pi$
$522$	−11.1267	−0.487001
$523$	−12.8844	−0.563395	−0.281698	−	0.959503i	$-0.590898\pi$
$523$	−12.8844	−0.563395	−0.281698	+	0.959503i	$0.590898\pi$
$524$	95.6298	4.17761
$525$	0	0
$526$	−8.32115	−0.362820
$527$	−22.9730	−1.00072
$528$	−40.5852	−1.76624
$529$	−16.5398	−0.719121
$530$	0	0
$531$	−9.53332	−0.413711
$532$	−96.2487	−4.17291
$533$	−0.404413	−0.0175171
$534$	−32.2276	−1.39462
$535$	0	0
$536$	−31.9603	−1.38047
$537$	14.0604	0.606753
$538$	80.9402	3.48958
$539$	5.28122	0.227478
$540$	0	0
$541$	−16.0082	−0.688246	−0.344123	−	0.938925i	$-0.611824\pi$
$541$	−16.0082	−0.688246	−0.344123	+	0.938925i	$0.611824\pi$
$542$	−35.0158	−1.50406
$543$	−24.2813	−1.04201
$544$	71.9712	3.08574
$545$	0	0
$546$	7.13776	0.305468
$547$	−29.6344	−1.26707	−0.633537	−	0.773712i	$-0.718398\pi$
$547$	−29.6344	−1.26707	−0.633537	+	0.773712i	$0.718398\pi$
$548$	−110.100	−4.70325
$549$	3.91362	0.167029
$550$	0	0
$551$	25.1611	1.07190
$552$	30.8005	1.31096
$553$	19.4312	0.826299
$554$	−67.7657	−2.87909
$555$	0	0
$556$	−20.4728	−0.868240
$557$	28.7044	1.21625	0.608123	−	0.793843i	$-0.291923\pi$
$557$	28.7044	1.21625	0.608123	+	0.793843i	$0.291923\pi$
$558$	14.2663	0.603939
$559$	−1.27468	−0.0539131
$560$	0	0
$561$	13.9237	0.587858
$562$	−1.13253	−0.0477727
$563$	24.9838	1.05294	0.526470	−	0.850193i	$-0.323515\pi$
$563$	24.9838	1.05294	0.526470	+	0.850193i	$0.323515\pi$
$564$	13.2512	0.557976
$565$	0	0
$566$	61.8370	2.59920
$567$	−15.2533	−0.640580
$568$	58.2907	2.44582
$569$	31.5450	1.32243	0.661217	−	0.750195i	$-0.270040\pi$
$569$	31.5450	1.32243	0.661217	+	0.750195i	$0.270040\pi$
$570$	0	0
$571$	−4.27560	−0.178928	−0.0894642	−	0.995990i	$-0.528515\pi$
$571$	−4.27560	−0.178928	−0.0894642	+	0.995990i	$0.528515\pi$
$572$	7.29690	0.305099
$573$	−13.7304	−0.573597
$574$	5.37114	0.224187
$575$	0	0
$576$	−19.4827	−0.811779
$577$	27.2529	1.13455	0.567277	−	0.823527i	$-0.307997\pi$
$577$	27.2529	1.13455	0.567277	+	0.823527i	$0.307997\pi$
$578$	−4.50654	−0.187447
$579$	−18.7236	−0.778126
$580$	0	0
$581$	−47.5822	−1.97404
$582$	−26.2712	−1.08898
$583$	4.22895	0.175145
$584$	−139.030	−5.75311
$585$	0	0
$586$	−75.3169	−3.11131
$587$	−34.2238	−1.41257	−0.706284	−	0.707929i	$-0.749630\pi$
$587$	−34.2238	−1.41257	−0.706284	+	0.707929i	$0.749630\pi$
$588$	−17.0276	−0.702204
$589$	−32.2608	−1.32928
$590$	0	0
$591$	27.1497	1.11679
$592$	35.3906	1.45455
$593$	18.4325	0.756931	0.378465	−	0.925615i	$-0.376452\pi$
$593$	18.4325	0.756931	0.378465	+	0.925615i	$0.376452\pi$
$594$	−34.5706	−1.41845
$595$	0	0
$596$	59.0450	2.41858
$597$	−4.40195	−0.180160
$598$	−4.20315	−0.171880
$599$	−40.9616	−1.67365	−0.836823	−	0.547473i	$-0.815590\pi$
$599$	−40.9616	−1.67365	−0.836823	+	0.547473i	$0.815590\pi$
$600$	0	0
$601$	−1.94664	−0.0794051	−0.0397025	−	0.999212i	$-0.512641\pi$
$601$	−1.94664	−0.0794051	−0.0397025	+	0.999212i	$0.512641\pi$
$602$	16.9294	0.689990
$603$	−3.73157	−0.151961
$604$	28.5675	1.16240
$605$	0	0
$606$	13.1929	0.535925
$607$	−11.8130	−0.479473	−0.239737	−	0.970838i	$-0.577061\pi$
$607$	−11.8130	−0.479473	−0.239737	+	0.970838i	$0.577061\pi$
$608$	101.069	4.09888
$609$	17.8934	0.725078
$610$	0	0
$611$	−1.11214	−0.0449922
$612$	22.4668	0.908167
$613$	−20.6232	−0.832963	−0.416481	−	0.909144i	$-0.636737\pi$
$613$	−20.6232	−0.832963	−0.416481	+	0.909144i	$0.636737\pi$
$614$	−84.1108	−3.39443
$615$	0	0
$616$	−59.6025	−2.40145
$617$	−47.8743	−1.92735	−0.963673	−	0.267085i	$-0.913940\pi$
$617$	−47.8743	−1.92735	−0.963673	+	0.267085i	$0.913940\pi$
$618$	15.0606	0.605827
$619$	−41.4316	−1.66528	−0.832638	−	0.553817i	$-0.813171\pi$
$619$	−41.4316	−1.66528	−0.832638	+	0.553817i	$0.813171\pi$
$620$	0	0
$621$	14.3780	0.576969
$622$	71.3985	2.86282
$623$	−25.9374	−1.03916
$624$	−10.9822	−0.439638
$625$	0	0
$626$	44.9812	1.79781
$627$	19.5529	0.780869
$628$	−22.5381	−0.899368
$629$	−12.1416	−0.484116
$630$	0	0
$631$	8.56918	0.341134	0.170567	−	0.985346i	$-0.445440\pi$
$631$	8.56918	0.341134	0.170567	+	0.985346i	$0.445440\pi$
$632$	−54.5537	−2.17003
$633$	−2.26196	−0.0899048
$634$	−91.7761	−3.64489
$635$	0	0
$636$	−13.6349	−0.540657
$637$	1.42907	0.0566220
$638$	25.3347	1.00301
$639$	6.80581	0.269233
$640$	0	0
$641$	−15.3155	−0.604926	−0.302463	−	0.953161i	$-0.597809\pi$
$641$	−15.3155	−0.604926	−0.302463	+	0.953161i	$0.597809\pi$
$642$	−30.2018	−1.19197
$643$	−23.5269	−0.927809	−0.463904	−	0.885885i	$-0.653552\pi$
$643$	−23.5269	−0.927809	−0.463904	+	0.885885i	$0.653552\pi$
$644$	40.3061	1.58828
$645$	0	0
$646$	−70.3642	−2.76844
$647$	−12.7010	−0.499328	−0.249664	−	0.968333i	$-0.580320\pi$
$647$	−12.7010	−0.499328	−0.249664	+	0.968333i	$0.580320\pi$
$648$	42.8242	1.68230
$649$	21.7067	0.852062
$650$	0	0
$651$	−22.9424	−0.899182
$652$	−21.5040	−0.842162
$653$	−24.5237	−0.959687	−0.479844	−	0.877354i	$-0.659307\pi$
$653$	−24.5237	−0.959687	−0.479844	+	0.877354i	$0.659307\pi$
$654$	−59.2043	−2.31507
$655$	0	0
$656$	−8.26402	−0.322656
$657$	−16.2326	−0.633296
$658$	14.7706	0.575818
$659$	−27.0064	−1.05202	−0.526010	−	0.850478i	$-0.676313\pi$
$659$	−27.0064	−1.05202	−0.526010	+	0.850478i	$0.676313\pi$
$660$	0	0
$661$	−44.2024	−1.71927	−0.859637	−	0.510905i	$-0.829311\pi$
$661$	−44.2024	−1.71927	−0.859637	+	0.510905i	$0.829311\pi$
$662$	−55.7995	−2.16871
$663$	3.76768	0.146325
$664$	133.588	5.18424
$665$	0	0
$666$	7.53993	0.292166
$667$	−10.5367	−0.407984
$668$	−97.3894	−3.76811
$669$	20.7514	0.802295
$670$	0	0
$671$	−8.91103	−0.344007
$672$	71.8753	2.77265
$673$	29.9966	1.15628	0.578142	−	0.815936i	$-0.303778\pi$
$673$	29.9966	1.15628	0.578142	+	0.815936i	$0.303778\pi$
$674$	38.6881	1.49021
$675$	0	0
$676$	−65.5605	−2.52156
$677$	48.3934	1.85991	0.929954	−	0.367676i	$-0.119846\pi$
$677$	48.3934	1.85991	0.929954	+	0.367676i	$0.119846\pi$
$678$	−48.9013	−1.87804
$679$	−21.1436	−0.811416
$680$	0	0
$681$	−37.3168	−1.42998
$682$	−32.4833	−1.24385
$683$	0.289497	0.0110773	0.00553864	−	0.999985i	$-0.498237\pi$
$683$	0.289497	0.0110773	0.00553864	+	0.999985i	$0.498237\pi$
$684$	31.5500	1.20634
$685$	0	0
$686$	38.3360	1.46367
$687$	14.9333	0.569743
$688$	−26.0475	−0.993052
$689$	1.14434	0.0435957
$690$	0	0
$691$	−9.05681	−0.344537	−0.172269	−	0.985050i	$-0.555110\pi$
$691$	−9.05681	−0.344537	−0.172269	+	0.985050i	$0.555110\pi$
$692$	−35.5750	−1.35236
$693$	−6.95897	−0.264349
$694$	−34.3219	−1.30284
$695$	0	0
$696$	−50.2363	−1.90420
$697$	2.83516	0.107389
$698$	−74.6980	−2.82736
$699$	−10.6894	−0.404311
$700$	0	0
$701$	−23.7083	−0.895451	−0.447725	−	0.894171i	$-0.647766\pi$
$701$	−23.7083	−0.895451	−0.447725	+	0.894171i	$0.647766\pi$
$702$	−9.35464	−0.353068
$703$	−17.0503	−0.643065
$704$	44.3607	1.67191
$705$	0	0
$706$	41.1650	1.54926
$707$	10.6179	0.399327
$708$	−69.9860	−2.63024
$709$	−3.27626	−0.123042	−0.0615212	−	0.998106i	$-0.519595\pi$
$709$	−3.27626	−0.123042	−0.0615212	+	0.998106i	$0.519595\pi$
$710$	0	0
$711$	−6.36948	−0.238874
$712$	72.8199	2.72904
$713$	13.5099	0.505949
$714$	−50.0397	−1.87269
$715$	0	0
$716$	−51.6579	−1.93055
$717$	−6.29608	−0.235132
$718$	1.34145	0.0500625
$719$	−16.3575	−0.610033	−0.305017	−	0.952347i	$-0.598662\pi$
$719$	−16.3575	−0.610033	−0.305017	+	0.952347i	$0.598662\pi$
$720$	0	0
$721$	12.1211	0.451413
$722$	−47.8474	−1.78069
$723$	1.41400	0.0525871
$724$	89.2090	3.31542
$725$	0	0
$726$	−22.0334	−0.817737
$727$	−52.4908	−1.94678	−0.973389	−	0.229159i	$-0.926402\pi$
$727$	−52.4908	−1.94678	−0.973389	+	0.229159i	$0.926402\pi$
$728$	−16.1282	−0.597749
$729$	28.9963	1.07394
$730$	0	0
$731$	8.93620	0.330517
$732$	28.7307	1.06192
$733$	27.7057	1.02334	0.511668	−	0.859184i	$-0.329028\pi$
$733$	27.7057	1.02334	0.511668	+	0.859184i	$0.329028\pi$
$734$	−82.9934	−3.06334
$735$	0	0
$736$	−42.3246	−1.56010
$737$	8.49651	0.312973
$738$	−1.76064	−0.0648101
$739$	−2.84613	−0.104697	−0.0523484	−	0.998629i	$-0.516671\pi$
$739$	−2.84613	−0.104697	−0.0523484	+	0.998629i	$0.516671\pi$
$740$	0	0
$741$	5.29093	0.194367
$742$	−15.1983	−0.557946
$743$	10.2791	0.377105	0.188553	−	0.982063i	$-0.439620\pi$
$743$	10.2791	0.377105	0.188553	+	0.982063i	$0.439620\pi$
$744$	64.4114	2.36144
$745$	0	0
$746$	53.0564	1.94253
$747$	15.5973	0.570675
$748$	−51.1554	−1.87043
$749$	−24.3070	−0.888158
$750$	0	0
$751$	−26.6112	−0.971058	−0.485529	−	0.874221i	$-0.661373\pi$
$751$	−26.6112	−0.971058	−0.485529	+	0.874221i	$0.661373\pi$
$752$	−22.7260	−0.828734
$753$	−22.0719	−0.804346
$754$	6.85544	0.249660
$755$	0	0
$756$	89.7062	3.26258
$757$	9.59259	0.348649	0.174324	−	0.984688i	$-0.444226\pi$
$757$	9.59259	0.348649	0.174324	+	0.984688i	$0.444226\pi$
$758$	−72.6146	−2.63748
$759$	−8.18819	−0.297212
$760$	0	0
$761$	18.1046	0.656291	0.328146	−	0.944627i	$-0.393576\pi$
$761$	18.1046	0.656291	0.328146	+	0.944627i	$0.393576\pi$
$762$	75.0293	2.71802
$763$	−47.6487	−1.72500
$764$	50.4454	1.82505
$765$	0	0
$766$	−11.6729	−0.421761
$767$	5.87373	0.212088
$768$	−16.7090	−0.602935
$769$	−25.5426	−0.921091	−0.460545	−	0.887636i	$-0.652346\pi$
$769$	−25.5426	−0.921091	−0.460545	+	0.887636i	$0.652346\pi$
$770$	0	0
$771$	39.6243	1.42704
$772$	68.7902	2.47581
$773$	−36.7086	−1.32032	−0.660159	−	0.751126i	$-0.729511\pi$
$773$	−36.7086	−1.32032	−0.660159	+	0.751126i	$0.729511\pi$
$774$	−5.54939	−0.199469
$775$	0	0
$776$	59.3613	2.13095
$777$	−12.1254	−0.434996
$778$	38.2784	1.37235
$779$	3.98140	0.142648
$780$	0	0
$781$	−15.4963	−0.554503
$782$	29.4665	1.05372
$783$	−23.4508	−0.838064
$784$	29.2026	1.04295
$785$	0	0
$786$	69.8186	2.49035
$787$	−39.2291	−1.39837	−0.699184	−	0.714942i	$-0.746453\pi$
$787$	−39.2291	−1.39837	−0.699184	+	0.714942i	$0.746453\pi$
$788$	−99.7475	−3.55336
$789$	−4.38648	−0.156163
$790$	0	0
$791$	−39.3567	−1.39936
$792$	19.5375	0.694235
$793$	−2.41128	−0.0856272
$794$	57.5736	2.04321
$795$	0	0
$796$	16.1727	0.573227
$797$	40.9734	1.45135	0.725677	−	0.688036i	$-0.241527\pi$
$797$	40.9734	1.45135	0.725677	+	0.688036i	$0.241527\pi$
$798$	−70.2705	−2.48755
$799$	7.79669	0.275827
$800$	0	0
$801$	8.50218	0.300410
$802$	74.5464	2.63232
$803$	36.9606	1.30431
$804$	−27.3942	−0.966119
$805$	0	0
$806$	−8.78982	−0.309608
$807$	42.6675	1.50197
$808$	−29.8100	−1.04871
$809$	14.9376	0.525179	0.262590	−	0.964908i	$-0.415423\pi$
$809$	14.9376	0.525179	0.262590	+	0.964908i	$0.415423\pi$
$810$	0	0
$811$	16.9412	0.594886	0.297443	−	0.954740i	$-0.403866\pi$
$811$	16.9412	0.594886	0.297443	+	0.954740i	$0.403866\pi$
$812$	−65.7402	−2.30703
$813$	−18.4585	−0.647368
$814$	−17.1679	−0.601734
$815$	0	0
$816$	76.9911	2.69523
$817$	12.5490	0.439036
$818$	21.0998	0.737736
$819$	−1.88306	−0.0657996
$820$	0	0
$821$	19.5411	0.681989	0.340994	−	0.940065i	$-0.389236\pi$
$821$	19.5411	0.681989	0.340994	+	0.940065i	$0.389236\pi$
$822$	−80.3833	−2.80369
$823$	−1.26835	−0.0442119	−0.0221060	−	0.999756i	$-0.507037\pi$
$823$	−1.26835	−0.0442119	−0.0221060	+	0.999756i	$0.507037\pi$
$824$	−34.0303	−1.18550
$825$	0	0
$826$	−78.0108	−2.71434
$827$	15.4880	0.538570	0.269285	−	0.963061i	$-0.413213\pi$
$827$	15.4880	0.538570	0.269285	+	0.963061i	$0.413213\pi$
$828$	−13.2122	−0.459156
$829$	6.16681	0.214182	0.107091	−	0.994249i	$-0.465846\pi$
$829$	6.16681	0.214182	0.107091	+	0.994249i	$0.465846\pi$
$830$	0	0
$831$	−35.7226	−1.23920
$832$	12.0038	0.416157
$833$	−10.0186	−0.347124
$834$	−14.9470	−0.517574
$835$	0	0
$836$	−71.8371	−2.48454
$837$	30.0679	1.03930
$838$	−68.5667	−2.36860
$839$	45.5786	1.57355	0.786774	−	0.617242i	$-0.211750\pi$
$839$	45.5786	1.57355	0.786774	+	0.617242i	$0.211750\pi$
$840$	0	0
$841$	−11.8143	−0.407391
$842$	67.8396	2.33791
$843$	−0.597010	−0.0205621
$844$	8.31040	0.286056
$845$	0	0
$846$	−4.84176	−0.166463
$847$	−17.7329	−0.609310
$848$	23.3840	0.803011
$849$	32.5973	1.11874
$850$	0	0
$851$	7.14017	0.244762
$852$	49.9628	1.71170
$853$	−26.6751	−0.913339	−0.456670	−	0.889636i	$-0.650958\pi$
$853$	−26.6751	−0.913339	−0.456670	+	0.889636i	$0.650958\pi$
$854$	32.0250	1.09587
$855$	0	0
$856$	68.2426	2.33249
$857$	26.3184	0.899019	0.449510	−	0.893276i	$-0.351599\pi$
$857$	26.3184	0.899019	0.449510	+	0.893276i	$0.351599\pi$
$858$	5.32741	0.181875
$859$	34.2020	1.16696	0.583479	−	0.812129i	$-0.301691\pi$
$859$	34.2020	1.16696	0.583479	+	0.812129i	$0.301691\pi$
$860$	0	0
$861$	2.83139	0.0964934
$862$	−61.8743	−2.10745
$863$	−22.2872	−0.758664	−0.379332	−	0.925261i	$-0.623846\pi$
$863$	−22.2872	−0.758664	−0.379332	+	0.925261i	$0.623846\pi$
$864$	−94.1986	−3.20470
$865$	0	0
$866$	58.2559	1.97962
$867$	−2.37561	−0.0806801
$868$	84.2899	2.86099
$869$	14.5029	0.491976
$870$	0	0
$871$	2.29912	0.0779026
$872$	133.775	4.53020
$873$	6.93080	0.234572
$874$	41.3796	1.39968
$875$	0	0
$876$	−119.167	−4.02628
$877$	4.69860	0.158661	0.0793303	−	0.996848i	$-0.474722\pi$
$877$	4.69860	0.158661	0.0793303	+	0.996848i	$0.474722\pi$
$878$	16.9841	0.573184
$879$	−39.7032	−1.33915
$880$	0	0
$881$	−2.55822	−0.0861886	−0.0430943	−	0.999071i	$-0.513722\pi$
$881$	−2.55822	−0.0861886	−0.0430943	+	0.999071i	$0.513722\pi$
$882$	6.22157	0.209491
$883$	−10.4567	−0.351896	−0.175948	−	0.984399i	$-0.556299\pi$
$883$	−10.4567	−0.351896	−0.175948	+	0.984399i	$0.556299\pi$
$884$	−13.8424	−0.465570
$885$	0	0
$886$	13.3603	0.448848
$887$	3.96728	0.133208	0.0666042	−	0.997779i	$-0.478784\pi$
$887$	3.96728	0.133208	0.0666042	+	0.997779i	$0.478784\pi$
$888$	34.0424	1.14239
$889$	60.3850	2.02525
$890$	0	0
$891$	−11.3846	−0.381400
$892$	−76.2402	−2.55271
$893$	10.9488	0.366389
$894$	43.1083	1.44176
$895$	0	0
$896$	−57.7636	−1.92975
$897$	−2.21569	−0.0739796
$898$	−70.1515	−2.34099
$899$	−22.0349	−0.734905
$900$	0	0
$901$	−8.02243	−0.267266
$902$	4.00886	0.133480
$903$	8.92429	0.296982
$904$	110.495	3.67501
$905$	0	0
$906$	20.8569	0.692925
$907$	−18.7178	−0.621513	−0.310756	−	0.950490i	$-0.600582\pi$
$907$	−18.7178	−0.621513	−0.310756	+	0.950490i	$0.600582\pi$
$908$	137.101	4.54986
$909$	−3.48051	−0.115441
$910$	0	0
$911$	−15.6782	−0.519443	−0.259722	−	0.965684i	$-0.583631\pi$
$911$	−15.6782	−0.519443	−0.259722	+	0.965684i	$0.583631\pi$
$912$	108.118	3.58015
$913$	−35.5139	−1.17534
$914$	−2.22170	−0.0734872
$915$	0	0
$916$	−54.8649	−1.81279
$917$	56.1914	1.85560
$918$	65.5812	2.16450
$919$	−11.9951	−0.395683	−0.197842	−	0.980234i	$-0.563393\pi$
$919$	−11.9951	−0.395683	−0.197842	+	0.980234i	$0.563393\pi$
$920$	0	0
$921$	−44.3388	−1.46101
$922$	−24.4975	−0.806783
$923$	−4.19324	−0.138022
$924$	−51.0872	−1.68065
$925$	0	0
$926$	62.6981	2.06039
$927$	−3.97325	−0.130499
$928$	69.0324	2.26610
$929$	14.4591	0.474389	0.237194	−	0.971462i	$-0.423772\pi$
$929$	14.4591	0.474389	0.237194	+	0.971462i	$0.423772\pi$
$930$	0	0
$931$	−14.0691	−0.461095
$932$	39.2728	1.28642
$933$	37.6376	1.23220
$934$	111.609	3.65196
$935$	0	0
$936$	5.28676	0.172803
$937$	−24.6021	−0.803715	−0.401858	−	0.915702i	$-0.631635\pi$
$937$	−24.6021	−0.803715	−0.401858	+	0.915702i	$0.631635\pi$
$938$	−30.5353	−0.997012
$939$	23.7118	0.773804
$940$	0	0
$941$	32.2200	1.05034	0.525171	−	0.850997i	$-0.324001\pi$
$941$	32.2200	1.05034	0.525171	+	0.850997i	$0.324001\pi$
$942$	−16.4549	−0.536130
$943$	−1.66729	−0.0542945
$944$	120.027	3.90656
$945$	0	0
$946$	12.6356	0.410818
$947$	−43.3046	−1.40721	−0.703605	−	0.710591i	$-0.748427\pi$
$947$	−43.3046	−1.40721	−0.703605	+	0.710591i	$0.748427\pi$
$948$	−46.7597	−1.51868
$949$	10.0014	0.324658
$950$	0	0
$951$	−48.3796	−1.56882
$952$	113.067	3.66454
$953$	0.153043	0.00495755	0.00247878	−	0.999997i	$-0.499211\pi$
$953$	0.153043	0.00495755	0.00247878	+	0.999997i	$0.499211\pi$
$954$	4.98194	0.161296
$955$	0	0
$956$	23.1317	0.748133
$957$	13.3551	0.431710
$958$	83.3901	2.69421
$959$	−64.6940	−2.08908
$960$	0	0
$961$	−2.74757	−0.0886314
$962$	−4.64555	−0.149779
$963$	7.96775	0.256757
$964$	−5.19500	−0.167320
$965$	0	0
$966$	29.4272	0.946805
$967$	20.4152	0.656508	0.328254	−	0.944589i	$-0.393540\pi$
$967$	20.4152	0.656508	0.328254	+	0.944589i	$0.393540\pi$
$968$	49.7857	1.60017
$969$	−37.0924	−1.19158
$970$	0	0
$971$	−7.17602	−0.230289	−0.115145	−	0.993349i	$-0.536733\pi$
$971$	−7.17602	−0.230289	−0.115145	+	0.993349i	$0.536733\pi$
$972$	−51.4561	−1.65045
$973$	−12.0297	−0.385653
$974$	−49.3929	−1.58265
$975$	0	0
$976$	−49.2736	−1.57721
$977$	−29.7338	−0.951270	−0.475635	−	0.879643i	$-0.657782\pi$
$977$	−29.7338	−0.951270	−0.475635	+	0.879643i	$0.657782\pi$
$978$	−15.6999	−0.502028
$979$	−19.3589	−0.618712
$980$	0	0
$981$	15.6191	0.498679
$982$	103.796	3.31226
$983$	35.0519	1.11798	0.558991	−	0.829174i	$-0.311189\pi$
$983$	35.0519	1.11798	0.558991	+	0.829174i	$0.311189\pi$
$984$	−7.94920	−0.253411
$985$	0	0
$986$	−48.0605	−1.53056
$987$	7.78630	0.247841
$988$	−19.4388	−0.618430
$989$	−5.25517	−0.167105
$990$	0	0
$991$	−0.413408	−0.0131323	−0.00656616	−	0.999978i	$-0.502090\pi$
$991$	−0.413408	−0.0131323	−0.00656616	+	0.999978i	$0.502090\pi$
$992$	−88.5110	−2.81023
$993$	−29.4146	−0.933444
$994$	55.6917	1.76643
$995$	0	0
$996$	114.503	3.62816
$997$	−15.4443	−0.489127	−0.244564	−	0.969633i	$-0.578645\pi$
$997$	−15.4443	−0.489127	−0.244564	+	0.969633i	$0.578645\pi$
$998$	88.4511	2.79987
$999$	15.8913	0.502779

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.j.1.2		25
5.4	even	2		1205.2.a.e.1.24	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.a.e.1.24	✓	25	5.4	even	2
6025.2.a.j.1.2		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)

\(f(q)\)	\(=\)	\(q-2.68235 q^{2} -1.41400 q^{3} +5.19500 q^{4} +3.79283 q^{6} +3.05254 q^{7} -8.57011 q^{8} -1.00061 q^{9} +O(q^{10})\)	\(q-2.68235 q^{2} -1.41400 q^{3} +5.19500 q^{4} +3.79283 q^{6} +3.05254 q^{7} -8.57011 q^{8} -1.00061 q^{9} +2.27833 q^{11} -7.34571 q^{12} +0.616505 q^{13} -8.18799 q^{14} +12.5980 q^{16} -4.32204 q^{17} +2.68400 q^{18} -6.06942 q^{19} -4.31629 q^{21} -6.11127 q^{22} +2.54169 q^{23} +12.1181 q^{24} -1.65368 q^{26} +5.65685 q^{27} +15.8580 q^{28} -4.14556 q^{29} +5.31530 q^{31} -16.6521 q^{32} -3.22155 q^{33} +11.5932 q^{34} -5.19819 q^{36} +2.80922 q^{37} +16.2803 q^{38} -0.871736 q^{39} -0.655977 q^{41} +11.5778 q^{42} -2.06759 q^{43} +11.8359 q^{44} -6.81771 q^{46} -1.80394 q^{47} -17.8136 q^{48} +2.31803 q^{49} +6.11136 q^{51} +3.20274 q^{52} +1.85617 q^{53} -15.1737 q^{54} -26.1606 q^{56} +8.58214 q^{57} +11.1198 q^{58} +9.52747 q^{59} -3.91122 q^{61} -14.2575 q^{62} -3.05442 q^{63} +19.4707 q^{64} +8.64132 q^{66} +3.72928 q^{67} -22.4530 q^{68} -3.59395 q^{69} -6.80163 q^{71} +8.57537 q^{72} +16.2227 q^{73} -7.53530 q^{74} -31.5306 q^{76} +6.95470 q^{77} +2.33830 q^{78} +6.36558 q^{79} -4.99693 q^{81} +1.75956 q^{82} -15.5877 q^{83} -22.4231 q^{84} +5.54599 q^{86} +5.86181 q^{87} -19.5255 q^{88} -8.49697 q^{89} +1.88191 q^{91} +13.2041 q^{92} -7.51582 q^{93} +4.83879 q^{94} +23.5460 q^{96} -6.92655 q^{97} -6.21776 q^{98} -2.27973 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 6 q^{2} - 15 q^{3} + 32 q^{4} - q^{6} - 19 q^{7} - 15 q^{8} + 32 q^{9}+O(q^{10}) \) 25 * q - 6 * q^2 - 15 * q^3 + 32 * q^4 - q^6 - 19 * q^7 - 15 * q^8 + 32 * q^9	\( 25 q - 6 q^{2} - 15 q^{3} + 32 q^{4} - q^{6} - 19 q^{7} - 15 q^{8} + 32 q^{9} + 2 q^{11} - 20 q^{12} - 14 q^{13} - 5 q^{14} + 38 q^{16} - 7 q^{17} - 9 q^{18} + 30 q^{19} + q^{21} - q^{22} - 43 q^{23} - 6 q^{24} - 22 q^{26} - 42 q^{27} - 32 q^{28} - 4 q^{29} + 14 q^{31} - 26 q^{32} - 4 q^{33} + 7 q^{34} + 15 q^{36} - 16 q^{37} - 14 q^{38} - 21 q^{39} - q^{41} + 25 q^{42} - 35 q^{43} - 52 q^{44} - 27 q^{46} - 50 q^{47} - 26 q^{48} + 46 q^{49} - 7 q^{51} - 3 q^{52} - 4 q^{53} - 31 q^{54} - 51 q^{56} - 2 q^{58} + 6 q^{59} + 19 q^{61} - 28 q^{63} + 49 q^{64} - 27 q^{66} - 65 q^{67} + 25 q^{68} + 2 q^{69} - 34 q^{71} + 10 q^{72} - 8 q^{73} - 42 q^{74} + 71 q^{76} - q^{77} + 59 q^{78} - 12 q^{79} + 29 q^{81} - 11 q^{82} - 41 q^{83} - 10 q^{84} - 13 q^{86} - 40 q^{87} + 52 q^{88} - 24 q^{89} + 46 q^{91} - 85 q^{92} + 30 q^{93} + 14 q^{94} - 30 q^{96} - 9 q^{97} + 64 q^{98} + 2 q^{99}+O(q^{100}) \) 25 * q - 6 * q^2 - 15 * q^3 + 32 * q^4 - q^6 - 19 * q^7 - 15 * q^8 + 32 * q^9 + 2 * q^11 - 20 * q^12 - 14 * q^13 - 5 * q^14 + 38 * q^16 - 7 * q^17 - 9 * q^18 + 30 * q^19 + q^21 - q^22 - 43 * q^23 - 6 * q^24 - 22 * q^26 - 42 * q^27 - 32 * q^28 - 4 * q^29 + 14 * q^31 - 26 * q^32 - 4 * q^33 + 7 * q^34 + 15 * q^36 - 16 * q^37 - 14 * q^38 - 21 * q^39 - q^41 + 25 * q^42 - 35 * q^43 - 52 * q^44 - 27 * q^46 - 50 * q^47 - 26 * q^48 + 46 * q^49 - 7 * q^51 - 3 * q^52 - 4 * q^53 - 31 * q^54 - 51 * q^56 - 2 * q^58 + 6 * q^59 + 19 * q^61 - 28 * q^63 + 49 * q^64 - 27 * q^66 - 65 * q^67 + 25 * q^68 + 2 * q^69 - 34 * q^71 + 10 * q^72 - 8 * q^73 - 42 * q^74 + 71 * q^76 - q^77 + 59 * q^78 - 12 * q^79 + 29 * q^81 - 11 * q^82 - 41 * q^83 - 10 * q^84 - 13 * q^86 - 40 * q^87 + 52 * q^88 - 24 * q^89 + 46 * q^91 - 85 * q^92 + 30 * q^93 + 14 * q^94 - 30 * q^96 - 9 * q^97 + 64 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

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Database

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