LMFDB - Embedded newform 1925.2.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1925,2,Mod(1,1925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1925, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1925.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1925 = 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1925.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:	$15.3712023891$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 13x^{5} + 10x^{4} + 47x^{3} - 25x^{2} - 35x + 20 $ x^7 - x^6 - 13x^5 + 10x^4 + 47x^3 - 25x^2 - 35*x + 20
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.78276$ of defining polynomial
Character	$\chi$	$=$	1925.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.78276 q^{2} +3.17748 q^{3} +5.74378 q^{4} -8.84217 q^{6} -1.00000 q^{7} -10.4180 q^{8} +7.09637 q^{9} +O(q^{10})$	\(q-2.78276 q^{2} +3.17748 q^{3} +5.74378 q^{4} -8.84217 q^{6} -1.00000 q^{7} -10.4180 q^{8} +7.09637 q^{9} -1.00000 q^{11} +18.2507 q^{12} +0.540418 q^{13} +2.78276 q^{14} +17.5034 q^{16} +2.54598 q^{17} -19.7475 q^{18} +8.20511 q^{19} -3.17748 q^{21} +2.78276 q^{22} +4.34185 q^{23} -33.1031 q^{24} -1.50385 q^{26} +13.0161 q^{27} -5.74378 q^{28} -3.40469 q^{29} +1.01277 q^{31} -27.8718 q^{32} -3.17748 q^{33} -7.08485 q^{34} +40.7600 q^{36} -5.20486 q^{37} -22.8329 q^{38} +1.71717 q^{39} +1.80143 q^{41} +8.84217 q^{42} -7.06796 q^{43} -5.74378 q^{44} -12.0823 q^{46} -8.84217 q^{47} +55.6167 q^{48} +1.00000 q^{49} +8.08979 q^{51} +3.10404 q^{52} +2.77177 q^{53} -36.2208 q^{54} +10.4180 q^{56} +26.0716 q^{57} +9.47444 q^{58} -0.0440307 q^{59} +5.52934 q^{61} -2.81831 q^{62} -7.09637 q^{63} +42.5538 q^{64} +8.84217 q^{66} +3.03518 q^{67} +14.6235 q^{68} +13.7961 q^{69} -2.69419 q^{71} -73.9303 q^{72} +9.20158 q^{73} +14.4839 q^{74} +47.1283 q^{76} +1.00000 q^{77} -4.77847 q^{78} +16.3409 q^{79} +20.0693 q^{81} -5.01295 q^{82} -8.06822 q^{83} -18.2507 q^{84} +19.6685 q^{86} -10.8183 q^{87} +10.4180 q^{88} -9.88721 q^{89} -0.540418 q^{91} +24.9386 q^{92} +3.21806 q^{93} +24.6057 q^{94} -88.5620 q^{96} +8.21406 q^{97} -2.78276 q^{98} -7.09637 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{2} + 13 q^{4} - q^{6} - 7 q^{7} - 6 q^{8} + 13 q^{9}+O(q^{10}) $ 7 * q - q^2 + 13 * q^4 - q^6 - 7 * q^7 - 6 * q^8 + 13 * q^9	$ 7 q - q^{2} + 13 q^{4} - q^{6} - 7 q^{7} - 6 q^{8} + 13 q^{9} - 7 q^{11} + 21 q^{12} + 3 q^{13} + q^{14} + 29 q^{16} - 14 q^{18} + 18 q^{19} + q^{22} + 7 q^{23} - 22 q^{24} + 13 q^{26} - 6 q^{27} - 13 q^{28} - 2 q^{29} + 24 q^{31} - 33 q^{32} + 33 q^{34} + 44 q^{36} - 21 q^{37} - 9 q^{38} + 10 q^{39} - 10 q^{41} + q^{42} + 2 q^{43} - 13 q^{44} + 3 q^{46} - q^{47} + 77 q^{48} + 7 q^{49} + 29 q^{51} - 9 q^{52} + 11 q^{53} - 47 q^{54} + 6 q^{56} - 7 q^{57} + 33 q^{58} + q^{59} + 28 q^{61} - 16 q^{62} - 13 q^{63} + 48 q^{64} + q^{66} + 46 q^{68} + 33 q^{69} - 10 q^{71} - 36 q^{72} + 11 q^{73} - 6 q^{74} + 20 q^{76} + 7 q^{77} + 31 q^{78} + 19 q^{79} + 27 q^{81} - 8 q^{82} + 19 q^{83} - 21 q^{84} + 55 q^{86} + 12 q^{87} + 6 q^{88} - 2 q^{89} - 3 q^{91} - 40 q^{92} - 58 q^{93} + 21 q^{94} - 23 q^{96} - 8 q^{97} - q^{98} - 13 q^{99}+O(q^{100}) $ 7 * q - q^2 + 13 * q^4 - q^6 - 7 * q^7 - 6 * q^8 + 13 * q^9 - 7 * q^11 + 21 * q^12 + 3 * q^13 + q^14 + 29 * q^16 - 14 * q^18 + 18 * q^19 + q^22 + 7 * q^23 - 22 * q^24 + 13 * q^26 - 6 * q^27 - 13 * q^28 - 2 * q^29 + 24 * q^31 - 33 * q^32 + 33 * q^34 + 44 * q^36 - 21 * q^37 - 9 * q^38 + 10 * q^39 - 10 * q^41 + q^42 + 2 * q^43 - 13 * q^44 + 3 * q^46 - q^47 + 77 * q^48 + 7 * q^49 + 29 * q^51 - 9 * q^52 + 11 * q^53 - 47 * q^54 + 6 * q^56 - 7 * q^57 + 33 * q^58 + q^59 + 28 * q^61 - 16 * q^62 - 13 * q^63 + 48 * q^64 + q^66 + 46 * q^68 + 33 * q^69 - 10 * q^71 - 36 * q^72 + 11 * q^73 - 6 * q^74 + 20 * q^76 + 7 * q^77 + 31 * q^78 + 19 * q^79 + 27 * q^81 - 8 * q^82 + 19 * q^83 - 21 * q^84 + 55 * q^86 + 12 * q^87 + 6 * q^88 - 2 * q^89 - 3 * q^91 - 40 * q^92 - 58 * q^93 + 21 * q^94 - 23 * q^96 - 8 * q^97 - q^98 - 13 * 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 $

$2$	−2.78276	−1.96771	−0.983856	−	0.178963i	$-0.942726\pi$
$2$	−2.78276	−1.96771	−0.983856	+	0.178963i	$0.942726\pi$
$3$	3.17748	1.83452	0.917259	−	0.398291i	$-0.130397\pi$
$3$	3.17748	1.83452	0.917259	+	0.398291i	$0.130397\pi$
$4$	5.74378	2.87189
$5$	0	0
$5$	0	0
$6$	−8.84217	−3.60980
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−10.4180	−3.68334
$9$	7.09637	2.36546
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	18.2507	5.26853
$13$	0.540418	0.149885	0.0749424	−	0.997188i	$-0.476123\pi$
$13$	0.540418	0.149885	0.0749424	+	0.997188i	$0.476123\pi$
$14$	2.78276	0.743725
$15$	0	0
$16$	17.5034	4.37586
$17$	2.54598	0.617490	0.308745	−	0.951145i	$-0.400091\pi$
$17$	2.54598	0.617490	0.308745	+	0.951145i	$0.400091\pi$
$18$	−19.7475	−4.65454
$19$	8.20511	1.88238	0.941191	−	0.337875i	$-0.109708\pi$
$19$	8.20511	1.88238	0.941191	+	0.337875i	$0.109708\pi$
$20$	0	0
$21$	−3.17748	−0.693383
$22$	2.78276	0.593287
$23$	4.34185	0.905338	0.452669	−	0.891679i	$-0.350472\pi$
$23$	4.34185	0.905338	0.452669	+	0.891679i	$0.350472\pi$
$24$	−33.1031	−6.75715
$25$	0	0
$26$	−1.50385	−0.294930
$27$	13.0161	2.50495
$28$	−5.74378	−1.08547
$29$	−3.40469	−0.632235	−0.316117	−	0.948720i	$-0.602379\pi$
$29$	−3.40469	−0.632235	−0.316117	+	0.948720i	$0.602379\pi$
$30$	0	0
$31$	1.01277	0.181899	0.0909496	−	0.995855i	$-0.471010\pi$
$31$	1.01277	0.181899	0.0909496	+	0.995855i	$0.471010\pi$
$32$	−27.8718	−4.92708
$33$	−3.17748	−0.553128
$34$	−7.08485	−1.21504
$35$	0	0
$36$	40.7600	6.79333
$37$	−5.20486	−0.855674	−0.427837	−	0.903856i	$-0.640724\pi$
$37$	−5.20486	−0.855674	−0.427837	+	0.903856i	$0.640724\pi$
$38$	−22.8329	−3.70398
$39$	1.71717	0.274966
$40$	0	0
$41$	1.80143	0.281336	0.140668	−	0.990057i	$-0.455075\pi$
$41$	1.80143	0.281336	0.140668	+	0.990057i	$0.455075\pi$
$42$	8.84217	1.36438
$43$	−7.06796	−1.07785	−0.538927	−	0.842352i	$-0.681170\pi$
$43$	−7.06796	−1.07785	−0.538927	+	0.842352i	$0.681170\pi$
$44$	−5.74378	−0.865907
$45$	0	0
$46$	−12.0823	−1.78144
$47$	−8.84217	−1.28976	−0.644882	−	0.764282i	$-0.723093\pi$
$47$	−8.84217	−1.28976	−0.644882	+	0.764282i	$0.723093\pi$
$48$	55.6167	8.02759
$49$	1.00000	0.142857
$50$	0	0
$51$	8.08979	1.13280
$52$	3.10404	0.430453
$53$	2.77177	0.380732	0.190366	−	0.981713i	$-0.439033\pi$
$53$	2.77177	0.380732	0.190366	+	0.981713i	$0.439033\pi$
$54$	−36.2208	−4.92903
$55$	0	0
$56$	10.4180	1.39217
$57$	26.0716	3.45326
$58$	9.47444	1.24406
$59$	−0.0440307	−0.00573230	−0.00286615	−	0.999996i	$-0.500912\pi$
$59$	−0.0440307	−0.00573230	−0.00286615	+	0.999996i	$0.500912\pi$
$60$	0	0
$61$	5.52934	0.707959	0.353980	−	0.935253i	$-0.384828\pi$
$61$	5.52934	0.707959	0.353980	+	0.935253i	$0.384828\pi$
$62$	−2.81831	−0.357925
$63$	−7.09637	−0.894058
$64$	42.5538	5.31923
$65$	0	0
$66$	8.84217	1.08840
$67$	3.03518	0.370807	0.185403	−	0.982662i	$-0.440641\pi$
$67$	3.03518	0.370807	0.185403	+	0.982662i	$0.440641\pi$
$68$	14.6235	1.77336
$69$	13.7961	1.66086
$70$	0	0
$71$	−2.69419	−0.319741	−0.159871	−	0.987138i	$-0.551108\pi$
$71$	−2.69419	−0.319741	−0.159871	+	0.987138i	$0.551108\pi$
$72$	−73.9303	−8.71277
$73$	9.20158	1.07696	0.538482	−	0.842637i	$-0.318998\pi$
$73$	9.20158	1.07696	0.538482	+	0.842637i	$0.318998\pi$
$74$	14.4839	1.68372
$75$	0	0
$76$	47.1283	5.40599
$77$	1.00000	0.113961
$78$	−4.77847	−0.541055
$79$	16.3409	1.83850	0.919249	−	0.393677i	$-0.128797\pi$
$79$	16.3409	1.83850	0.919249	+	0.393677i	$0.128797\pi$
$80$	0	0
$81$	20.0693	2.22993
$82$	−5.01295	−0.553588
$83$	−8.06822	−0.885602	−0.442801	−	0.896620i	$-0.646015\pi$
$83$	−8.06822	−0.885602	−0.442801	+	0.896620i	$0.646015\pi$
$84$	−18.2507	−1.99132
$85$	0	0
$86$	19.6685	2.12091
$87$	−10.8183	−1.15985
$88$	10.4180	1.11057
$89$	−9.88721	−1.04804	−0.524021	−	0.851705i	$-0.675569\pi$
$89$	−9.88721	−1.04804	−0.524021	+	0.851705i	$0.675569\pi$
$90$	0	0
$91$	−0.540418	−0.0566512
$92$	24.9386	2.60003
$93$	3.21806	0.333697
$94$	24.6057	2.53788
$95$	0	0
$96$	−88.5620	−9.03883
$97$	8.21406	0.834012	0.417006	−	0.908904i	$-0.363079\pi$
$97$	8.21406	0.834012	0.417006	+	0.908904i	$0.363079\pi$
$98$	−2.78276	−0.281102
$99$	−7.09637	−0.713212
$100$	0	0
$101$	−6.80193	−0.676817	−0.338409	−	0.940999i	$-0.609889\pi$
$101$	−6.80193	−0.676817	−0.338409	+	0.940999i	$0.609889\pi$
$102$	−22.5120	−2.22902
$103$	−13.9410	−1.37364	−0.686822	−	0.726826i	$-0.740995\pi$
$103$	−13.9410	−1.37364	−0.686822	+	0.726826i	$0.740995\pi$
$104$	−5.63010	−0.552076
$105$	0	0
$106$	−7.71318	−0.749170
$107$	14.7322	1.42421	0.712106	−	0.702072i	$-0.247741\pi$
$107$	14.7322	1.42421	0.712106	+	0.702072i	$0.247741\pi$
$108$	74.7617	7.19395
$109$	3.47850	0.333180	0.166590	−	0.986026i	$-0.446724\pi$
$109$	3.47850	0.333180	0.166590	+	0.986026i	$0.446724\pi$
$110$	0	0
$111$	−16.5383	−1.56975
$112$	−17.5034	−1.65392
$113$	4.21004	0.396048	0.198024	−	0.980197i	$-0.436548\pi$
$113$	4.21004	0.396048	0.198024	+	0.980197i	$0.436548\pi$
$114$	−72.5510	−6.79503
$115$	0	0
$116$	−19.5558	−1.81571
$117$	3.83500	0.354546
$118$	0.122527	0.0112795
$119$	−2.54598	−0.233389
$120$	0	0
$121$	1.00000	0.0909091
$122$	−15.3868	−1.39306
$123$	5.72400	0.516116
$124$	5.81714	0.522394
$125$	0	0
$126$	19.7475	1.75925
$127$	−0.917075	−0.0813772	−0.0406886	−	0.999172i	$-0.512955\pi$
$127$	−0.917075	−0.0813772	−0.0406886	+	0.999172i	$0.512955\pi$
$128$	−62.6736	−5.53962
$129$	−22.4583	−1.97734
$130$	0	0
$131$	2.08992	0.182597	0.0912987	−	0.995824i	$-0.470898\pi$
$131$	2.08992	0.182597	0.0912987	+	0.995824i	$0.470898\pi$
$132$	−18.2507	−1.58852
$133$	−8.20511	−0.711473
$134$	−8.44620	−0.729641
$135$	0	0
$136$	−26.5241	−2.27442
$137$	22.5530	1.92683	0.963417	−	0.268007i	$-0.0863651\pi$
$137$	22.5530	1.92683	0.963417	+	0.268007i	$0.0863651\pi$
$138$	−38.3914	−3.26809
$139$	7.56074	0.641294	0.320647	−	0.947199i	$-0.396100\pi$
$139$	7.56074	0.641294	0.320647	+	0.947199i	$0.396100\pi$
$140$	0	0
$141$	−28.0958	−2.36609
$142$	7.49729	0.629158
$143$	−0.540418	−0.0451920
$144$	124.211	10.3509
$145$	0	0
$146$	−25.6058	−2.11915
$147$	3.17748	0.262074
$148$	−29.8956	−2.45740
$149$	−5.13544	−0.420711	−0.210356	−	0.977625i	$-0.567462\pi$
$149$	−5.13544	−0.420711	−0.210356	+	0.977625i	$0.567462\pi$
$150$	0	0
$151$	−12.7128	−1.03455	−0.517275	−	0.855820i	$-0.673053\pi$
$151$	−12.7128	−1.03455	−0.517275	+	0.855820i	$0.673053\pi$
$152$	−85.4813	−6.93345
$153$	18.0672	1.46065
$154$	−2.78276	−0.224242
$155$	0	0
$156$	9.86301	0.789673
$157$	8.01716	0.639839	0.319920	−	0.947445i	$-0.396344\pi$
$157$	8.01716	0.639839	0.319920	+	0.947445i	$0.396344\pi$
$158$	−45.4729	−3.61763
$159$	8.80723	0.698459
$160$	0	0
$161$	−4.34185	−0.342185
$162$	−55.8483	−4.38785
$163$	17.0406	1.33472	0.667362	−	0.744734i	$-0.267424\pi$
$163$	17.0406	1.33472	0.667362	+	0.744734i	$0.267424\pi$
$164$	10.3470	0.807966
$165$	0	0
$166$	22.4519	1.74261
$167$	19.5103	1.50975	0.754875	−	0.655868i	$-0.227697\pi$
$167$	19.5103	1.50975	0.754875	+	0.655868i	$0.227697\pi$
$168$	33.1031	2.55396
$169$	−12.7079	−0.977535
$170$	0	0
$171$	58.2265	4.45269
$172$	−40.5968	−3.09548
$173$	−1.67934	−0.127678	−0.0638391	−	0.997960i	$-0.520334\pi$
$173$	−1.67934	−0.127678	−0.0638391	+	0.997960i	$0.520334\pi$
$174$	30.1048	2.28224
$175$	0	0
$176$	−17.5034	−1.31937
$177$	−0.139906	−0.0105160
$178$	27.5138	2.06224
$179$	8.31286	0.621332	0.310666	−	0.950519i	$-0.399448\pi$
$179$	8.31286	0.621332	0.310666	+	0.950519i	$0.399448\pi$
$180$	0	0
$181$	10.5117	0.781332	0.390666	−	0.920533i	$-0.372245\pi$
$181$	10.5117	0.781332	0.390666	+	0.920533i	$0.372245\pi$
$182$	1.50385	0.111473
$183$	17.5694	1.29876
$184$	−45.2336	−3.33466
$185$	0	0
$186$	−8.95510	−0.656620
$187$	−2.54598	−0.186180
$188$	−50.7875	−3.70406
$189$	−13.0161	−0.946784
$190$	0	0
$191$	−10.0954	−0.730481	−0.365240	−	0.930913i	$-0.619013\pi$
$191$	−10.0954	−0.730481	−0.365240	+	0.930913i	$0.619013\pi$
$192$	135.214	9.75821
$193$	−7.47850	−0.538314	−0.269157	−	0.963096i	$-0.586745\pi$
$193$	−7.47850	−0.538314	−0.269157	+	0.963096i	$0.586745\pi$
$194$	−22.8578	−1.64109
$195$	0	0
$196$	5.74378	0.410270
$197$	−2.79075	−0.198833	−0.0994164	−	0.995046i	$-0.531698\pi$
$197$	−2.79075	−0.198833	−0.0994164	+	0.995046i	$0.531698\pi$
$198$	19.7475	1.40340
$199$	−7.59441	−0.538353	−0.269177	−	0.963091i	$-0.586752\pi$
$199$	−7.59441	−0.538353	−0.269177	+	0.963091i	$0.586752\pi$
$200$	0	0
$201$	9.64423	0.680252
$202$	18.9282	1.33178
$203$	3.40469	0.238962
$204$	46.4659	3.25327
$205$	0	0
$206$	38.7944	2.70294
$207$	30.8113	2.14154
$208$	9.45916	0.655874
$209$	−8.20511	−0.567560
$210$	0	0
$211$	−4.55400	−0.313511	−0.156755	−	0.987637i	$-0.550103\pi$
$211$	−4.55400	−0.313511	−0.156755	+	0.987637i	$0.550103\pi$
$212$	15.9204	1.09342
$213$	−8.56072	−0.586571
$214$	−40.9962	−2.80244
$215$	0	0
$216$	−135.603	−9.22659
$217$	−1.01277	−0.0687514
$218$	−9.67984	−0.655602
$219$	29.2378	1.97571
$220$	0	0
$221$	1.37589	0.0925524
$222$	46.0223	3.08881
$223$	13.1563	0.881011	0.440506	−	0.897750i	$-0.354799\pi$
$223$	13.1563	0.881011	0.440506	+	0.897750i	$0.354799\pi$
$224$	27.8718	1.86226
$225$	0	0
$226$	−11.7156	−0.779307
$227$	−1.00982	−0.0670243	−0.0335121	−	0.999438i	$-0.510669\pi$
$227$	−1.00982	−0.0670243	−0.0335121	+	0.999438i	$0.510669\pi$
$228$	149.749	9.91739
$229$	−17.1062	−1.13041	−0.565204	−	0.824951i	$-0.691203\pi$
$229$	−17.1062	−1.13041	−0.565204	+	0.824951i	$0.691203\pi$
$230$	0	0
$231$	3.17748	0.209063
$232$	35.4702	2.32873
$233$	19.3085	1.26494	0.632471	−	0.774584i	$-0.282041\pi$
$233$	19.3085	1.26494	0.632471	+	0.774584i	$0.282041\pi$
$234$	−10.6719	−0.697644
$235$	0	0
$236$	−0.252902	−0.0164625
$237$	51.9229	3.37276
$238$	7.08485	0.459243
$239$	26.9717	1.74465	0.872326	−	0.488925i	$-0.162611\pi$
$239$	26.9717	1.74465	0.872326	+	0.488925i	$0.162611\pi$
$240$	0	0
$241$	−2.20800	−0.142230	−0.0711148	−	0.997468i	$-0.522656\pi$
$241$	−2.20800	−0.142230	−0.0711148	+	0.997468i	$0.522656\pi$
$242$	−2.78276	−0.178883
$243$	24.7215	1.58589
$244$	31.7593	2.03318
$245$	0	0
$246$	−15.9286	−1.01557
$247$	4.43419	0.282141
$248$	−10.5511	−0.669996
$249$	−25.6366	−1.62465
$250$	0	0
$251$	−23.1188	−1.45924	−0.729622	−	0.683851i	$-0.760304\pi$
$251$	−23.1188	−1.45924	−0.729622	+	0.683851i	$0.760304\pi$
$252$	−40.7600	−2.56764
$253$	−4.34185	−0.272970
$254$	2.55200	0.160127
$255$	0	0
$256$	89.2983	5.58114
$257$	−22.7273	−1.41769	−0.708846	−	0.705364i	$-0.750784\pi$
$257$	−22.7273	−1.41769	−0.708846	+	0.705364i	$0.750784\pi$
$258$	62.4962	3.89084
$259$	5.20486	0.323414
$260$	0	0
$261$	−24.1609	−1.49552
$262$	−5.81576	−0.359299
$263$	1.54428	0.0952246	0.0476123	−	0.998866i	$-0.484839\pi$
$263$	1.54428	0.0952246	0.0476123	+	0.998866i	$0.484839\pi$
$264$	33.1031	2.03736
$265$	0	0
$266$	22.8329	1.39997
$267$	−31.4164	−1.92265
$268$	17.4334	1.06492
$269$	29.4218	1.79388	0.896938	−	0.442156i	$-0.145786\pi$
$269$	29.4218	1.79388	0.896938	+	0.442156i	$0.145786\pi$
$270$	0	0
$271$	−25.2782	−1.53554	−0.767771	−	0.640725i	$-0.778634\pi$
$271$	−25.2782	−1.53554	−0.767771	+	0.640725i	$0.778634\pi$
$272$	44.5633	2.70205
$273$	−1.71717	−0.103928
$274$	−62.7597	−3.79145
$275$	0	0
$276$	79.2419	4.76980
$277$	−25.6054	−1.53848	−0.769239	−	0.638962i	$-0.779364\pi$
$277$	−25.6054	−1.53848	−0.769239	+	0.638962i	$0.779364\pi$
$278$	−21.0398	−1.26188
$279$	7.18700	0.430275
$280$	0	0
$281$	−26.2808	−1.56778	−0.783891	−	0.620898i	$-0.786768\pi$
$281$	−26.2808	−1.56778	−0.783891	+	0.620898i	$0.786768\pi$
$282$	78.1840	4.65579
$283$	−22.0282	−1.30944	−0.654721	−	0.755870i	$-0.727214\pi$
$283$	−22.0282	−1.30944	−0.654721	+	0.755870i	$0.727214\pi$
$284$	−15.4748	−0.918261
$285$	0	0
$286$	1.50385	0.0889248
$287$	−1.80143	−0.106335
$288$	−197.789	−11.6548
$289$	−10.5180	−0.618706
$290$	0	0
$291$	26.1000	1.53001
$292$	52.8518	3.09292
$293$	12.2364	0.714858	0.357429	−	0.933940i	$-0.383653\pi$
$293$	12.2364	0.714858	0.357429	+	0.933940i	$0.383653\pi$
$294$	−8.84217	−0.515686
$295$	0	0
$296$	54.2245	3.15173
$297$	−13.0161	−0.755272
$298$	14.2907	0.827839
$299$	2.34641	0.135696
$300$	0	0
$301$	7.06796	0.407391
$302$	35.3766	2.03569
$303$	−21.6130	−1.24163
$304$	143.618	8.23703
$305$	0	0
$306$	−50.2767	−2.87413
$307$	16.2577	0.927877	0.463939	−	0.885867i	$-0.346436\pi$
$307$	16.2577	0.927877	0.463939	+	0.885867i	$0.346436\pi$
$308$	5.74378	0.327282
$309$	−44.2971	−2.51998
$310$	0	0
$311$	−3.07307	−0.174258	−0.0871288	−	0.996197i	$-0.527769\pi$
$311$	−3.07307	−0.174258	−0.0871288	+	0.996197i	$0.527769\pi$
$312$	−17.8895	−1.01279
$313$	12.2521	0.692528	0.346264	−	0.938137i	$-0.387450\pi$
$313$	12.2521	0.692528	0.346264	+	0.938137i	$0.387450\pi$
$314$	−22.3099	−1.25902
$315$	0	0
$316$	93.8586	5.27996
$317$	−12.2444	−0.687715	−0.343858	−	0.939022i	$-0.611734\pi$
$317$	−12.2444	−0.687715	−0.343858	+	0.939022i	$0.611734\pi$
$318$	−24.5085	−1.37437
$319$	3.40469	0.190626
$320$	0	0
$321$	46.8112	2.61274
$322$	12.0823	0.673322
$323$	20.8900	1.16235
$324$	115.274	6.40410
$325$	0	0
$326$	−47.4200	−2.62635
$327$	11.0529	0.611224
$328$	−18.7674	−1.03626
$329$	8.84217	0.487485
$330$	0	0
$331$	−17.4468	−0.958961	−0.479481	−	0.877552i	$-0.659175\pi$
$331$	−17.4468	−0.958961	−0.479481	+	0.877552i	$0.659175\pi$
$332$	−46.3420	−2.54335
$333$	−36.9356	−2.02406
$334$	−54.2925	−2.97075
$335$	0	0
$336$	−55.6167	−3.03414
$337$	−1.92770	−0.105009	−0.0525044	−	0.998621i	$-0.516720\pi$
$337$	−1.92770	−0.105009	−0.0525044	+	0.998621i	$0.516720\pi$
$338$	35.3632	1.92351
$339$	13.3773	0.726556
$340$	0	0
$341$	−1.01277	−0.0548447
$342$	−162.031	−8.76161
$343$	−1.00000	−0.0539949
$344$	73.6344	3.97010
$345$	0	0
$346$	4.67322	0.251234
$347$	−20.9367	−1.12394	−0.561970	−	0.827158i	$-0.689956\pi$
$347$	−20.9367	−1.12394	−0.561970	+	0.827158i	$0.689956\pi$
$348$	−62.1380	−3.33095
$349$	−19.9680	−1.06886	−0.534430	−	0.845213i	$-0.679474\pi$
$349$	−19.9680	−1.06886	−0.534430	+	0.845213i	$0.679474\pi$
$350$	0	0
$351$	7.03414	0.375455
$352$	27.8718	1.48557
$353$	−17.5184	−0.932412	−0.466206	−	0.884676i	$-0.654380\pi$
$353$	−17.5184	−0.932412	−0.466206	+	0.884676i	$0.654380\pi$
$354$	0.389327	0.0206925
$355$	0	0
$356$	−56.7899	−3.00986
$357$	−8.08979	−0.428157
$358$	−23.1327	−1.22260
$359$	−12.4827	−0.658812	−0.329406	−	0.944188i	$-0.606848\pi$
$359$	−12.4827	−0.658812	−0.329406	+	0.944188i	$0.606848\pi$
$360$	0	0
$361$	48.3239	2.54336
$362$	−29.2517	−1.53744
$363$	3.17748	0.166774
$364$	−3.10404	−0.162696
$365$	0	0
$366$	−48.8914	−2.55559
$367$	−11.4673	−0.598586	−0.299293	−	0.954161i	$-0.596751\pi$
$367$	−11.4673	−0.598586	−0.299293	+	0.954161i	$0.596751\pi$
$368$	75.9972	3.96163
$369$	12.7836	0.665488
$370$	0	0
$371$	−2.77177	−0.143903
$372$	18.4838	0.958342
$373$	20.0536	1.03834	0.519168	−	0.854672i	$-0.326242\pi$
$373$	20.0536	1.03834	0.519168	+	0.854672i	$0.326242\pi$
$374$	7.08485	0.366349
$375$	0	0
$376$	92.1182	4.75063
$377$	−1.83995	−0.0947624
$378$	36.2208	1.86300
$379$	−10.1539	−0.521569	−0.260785	−	0.965397i	$-0.583981\pi$
$379$	−10.1539	−0.521569	−0.260785	+	0.965397i	$0.583981\pi$
$380$	0	0
$381$	−2.91399	−0.149288
$382$	28.0932	1.43738
$383$	−5.11528	−0.261378	−0.130689	−	0.991423i	$-0.541719\pi$
$383$	−5.11528	−0.261378	−0.130689	+	0.991423i	$0.541719\pi$
$384$	−199.144	−10.1625
$385$	0	0
$386$	20.8109	1.05925
$387$	−50.1569	−2.54962
$388$	47.1798	2.39519
$389$	−37.5193	−1.90230	−0.951152	−	0.308724i	$-0.900098\pi$
$389$	−37.5193	−1.90230	−0.951152	+	0.308724i	$0.900098\pi$
$390$	0	0
$391$	11.0542	0.559037
$392$	−10.4180	−0.526191
$393$	6.64069	0.334978
$394$	7.76600	0.391245
$395$	0	0
$396$	−40.7600	−2.04827
$397$	12.6630	0.635539	0.317769	−	0.948168i	$-0.397066\pi$
$397$	12.6630	0.635539	0.317769	+	0.948168i	$0.397066\pi$
$398$	21.1335	1.05932
$399$	−26.0716	−1.30521
$400$	0	0
$401$	−19.8625	−0.991886	−0.495943	−	0.868355i	$-0.665177\pi$
$401$	−19.8625	−0.991886	−0.495943	+	0.868355i	$0.665177\pi$
$402$	−26.8376	−1.33854
$403$	0.547320	0.0272639
$404$	−39.0688	−1.94374
$405$	0	0
$406$	−9.47444	−0.470209
$407$	5.20486	0.257995
$408$	−84.2798	−4.17247
$409$	−17.9670	−0.888409	−0.444204	−	0.895925i	$-0.646514\pi$
$409$	−17.9670	−0.888409	−0.444204	+	0.895925i	$0.646514\pi$
$410$	0	0
$411$	71.6617	3.53481
$412$	−80.0738	−3.94495
$413$	0.0440307	0.00216661
$414$	−85.7407	−4.21393
$415$	0	0
$416$	−15.0624	−0.738495
$417$	24.0241	1.17646
$418$	22.8329	1.11679
$419$	21.5040	1.05054	0.525270	−	0.850936i	$-0.323964\pi$
$419$	21.5040	1.05054	0.525270	+	0.850936i	$0.323964\pi$
$420$	0	0
$421$	11.2646	0.549001	0.274501	−	0.961587i	$-0.411487\pi$
$421$	11.2646	0.549001	0.274501	+	0.961587i	$0.411487\pi$
$422$	12.6727	0.616898
$423$	−62.7473	−3.05088
$424$	−28.8764	−1.40236
$425$	0	0
$426$	23.8225	1.15420
$427$	−5.52934	−0.267583
$428$	84.6183	4.09018
$429$	−1.71717	−0.0829055
$430$	0	0
$431$	−13.5880	−0.654509	−0.327254	−	0.944936i	$-0.606123\pi$
$431$	−13.5880	−0.654509	−0.327254	+	0.944936i	$0.606123\pi$
$432$	227.827	10.9613
$433$	−12.0833	−0.580688	−0.290344	−	0.956922i	$-0.593770\pi$
$433$	−12.0833	−0.580688	−0.290344	+	0.956922i	$0.593770\pi$
$434$	2.81831	0.135283
$435$	0	0
$436$	19.9797	0.956855
$437$	35.6253	1.70419
$438$	−81.3620	−3.88763
$439$	3.21020	0.153215	0.0766073	−	0.997061i	$-0.475591\pi$
$439$	3.21020	0.153215	0.0766073	+	0.997061i	$0.475591\pi$
$440$	0	0
$441$	7.09637	0.337922
$442$	−3.82878	−0.182116
$443$	−15.9952	−0.759957	−0.379979	−	0.924995i	$-0.624069\pi$
$443$	−15.9952	−0.759957	−0.379979	+	0.924995i	$0.624069\pi$
$444$	−94.9925	−4.50814
$445$	0	0
$446$	−36.6109	−1.73358
$447$	−16.3177	−0.771803
$448$	−42.5538	−2.01048
$449$	26.0838	1.23097	0.615485	−	0.788148i	$-0.288960\pi$
$449$	26.0838	1.23097	0.615485	+	0.788148i	$0.288960\pi$
$450$	0	0
$451$	−1.80143	−0.0848260
$452$	24.1816	1.13740
$453$	−40.3945	−1.89790
$454$	2.81010	0.131884
$455$	0	0
$456$	−271.615	−12.7195
$457$	4.44015	0.207701	0.103851	−	0.994593i	$-0.466884\pi$
$457$	4.44015	0.207701	0.103851	+	0.994593i	$0.466884\pi$
$458$	47.6025	2.22432
$459$	33.1388	1.54678
$460$	0	0
$461$	−35.5414	−1.65533	−0.827665	−	0.561223i	$-0.810331\pi$
$461$	−35.5414	−1.65533	−0.827665	+	0.561223i	$0.810331\pi$
$462$	−8.84217	−0.411375
$463$	21.3451	0.991991	0.495995	−	0.868325i	$-0.334803\pi$
$463$	21.3451	0.991991	0.495995	+	0.868325i	$0.334803\pi$
$464$	−59.5937	−2.76657
$465$	0	0
$466$	−53.7310	−2.48904
$467$	−13.6285	−0.630653	−0.315326	−	0.948983i	$-0.602114\pi$
$467$	−13.6285	−0.630653	−0.315326	+	0.948983i	$0.602114\pi$
$468$	22.0274	1.01822
$469$	−3.03518	−0.140152
$470$	0	0
$471$	25.4744	1.17380
$472$	0.458714	0.0211140
$473$	7.06796	0.324985
$474$	−144.489	−6.63661
$475$	0	0
$476$	−14.6235	−0.670268
$477$	19.6695	0.900604
$478$	−75.0558	−3.43297
$479$	−33.3770	−1.52504	−0.762518	−	0.646967i	$-0.776037\pi$
$479$	−33.3770	−1.52504	−0.762518	+	0.646967i	$0.776037\pi$
$480$	0	0
$481$	−2.81280	−0.128253
$482$	6.14433	0.279867
$483$	−13.7961	−0.627745
$484$	5.74378	0.261081
$485$	0	0
$486$	−68.7942	−3.12057
$487$	−40.4197	−1.83159	−0.915796	−	0.401645i	$-0.868439\pi$
$487$	−40.4197	−1.83159	−0.915796	+	0.401645i	$0.868439\pi$
$488$	−57.6049	−2.60765
$489$	54.1462	2.44857
$490$	0	0
$491$	42.4781	1.91701	0.958506	−	0.285073i	$-0.0920179\pi$
$491$	42.4781	1.91701	0.958506	+	0.285073i	$0.0920179\pi$
$492$	32.8774	1.48223
$493$	−8.66826	−0.390399
$494$	−12.3393	−0.555171
$495$	0	0
$496$	17.7270	0.795965
$497$	2.69419	0.120851
$498$	71.3406	3.19685
$499$	18.8393	0.843361	0.421681	−	0.906744i	$-0.361440\pi$
$499$	18.8393	0.843361	0.421681	+	0.906744i	$0.361440\pi$
$500$	0	0
$501$	61.9935	2.76967
$502$	64.3341	2.87137
$503$	−27.4828	−1.22540	−0.612699	−	0.790316i	$-0.709916\pi$
$503$	−27.4828	−1.22540	−0.612699	+	0.790316i	$0.709916\pi$
$504$	73.9303	3.29312
$505$	0	0
$506$	12.0823	0.537125
$507$	−40.3792	−1.79330
$508$	−5.26747	−0.233706
$509$	28.5668	1.26620	0.633100	−	0.774070i	$-0.281782\pi$
$509$	28.5668	1.26620	0.633100	+	0.774070i	$0.281782\pi$
$510$	0	0
$511$	−9.20158	−0.407054
$512$	−123.149	−5.44246
$513$	106.799	4.71528
$514$	63.2448	2.78961
$515$	0	0
$516$	−128.995	−5.67871
$517$	8.84217	0.388878
$518$	−14.4839	−0.636386
$519$	−5.33608	−0.234228
$520$	0	0
$521$	−25.6041	−1.12174	−0.560868	−	0.827905i	$-0.689533\pi$
$521$	−25.6041	−1.12174	−0.560868	+	0.827905i	$0.689533\pi$
$522$	67.2342	2.94276
$523$	−37.6223	−1.64511	−0.822555	−	0.568686i	$-0.807452\pi$
$523$	−37.6223	−1.64511	−0.822555	+	0.568686i	$0.807452\pi$
$524$	12.0041	0.524399
$525$	0	0
$526$	−4.29738	−0.187375
$527$	2.57849	0.112321
$528$	−55.6167	−2.42041
$529$	−4.14836	−0.180364
$530$	0	0
$531$	−0.312458	−0.0135595
$532$	−47.1283	−2.04327
$533$	0.973524	0.0421680
$534$	87.4244	3.78322
$535$	0	0
$536$	−31.6207	−1.36581
$537$	26.4139	1.13984
$538$	−81.8738	−3.52983
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−5.86219	−0.252035	−0.126018	−	0.992028i	$-0.540220\pi$
$541$	−5.86219	−0.252035	−0.126018	+	0.992028i	$0.540220\pi$
$542$	70.3433	3.02150
$543$	33.4008	1.43337
$544$	−70.9610	−3.04243
$545$	0	0
$546$	4.77847	0.204499
$547$	3.42447	0.146420	0.0732099	−	0.997317i	$-0.476676\pi$
$547$	3.42447	0.146420	0.0732099	+	0.997317i	$0.476676\pi$
$548$	129.539	5.53365
$549$	39.2382	1.67465
$550$	0	0
$551$	−27.9359	−1.19011
$552$	−143.729	−6.11750
$553$	−16.3409	−0.694887
$554$	71.2537	3.02728
$555$	0	0
$556$	43.4272	1.84172
$557$	−12.1666	−0.515515	−0.257757	−	0.966210i	$-0.582984\pi$
$557$	−12.1666	−0.515515	−0.257757	+	0.966210i	$0.582984\pi$
$558$	−19.9997	−0.846656
$559$	−3.81965	−0.161554
$560$	0	0
$561$	−8.08979	−0.341551
$562$	73.1333	3.08494
$563$	−26.4409	−1.11435	−0.557175	−	0.830395i	$-0.688115\pi$
$563$	−26.4409	−1.11435	−0.557175	+	0.830395i	$0.688115\pi$
$564$	−161.376	−6.79516
$565$	0	0
$566$	61.2994	2.57661
$567$	−20.0693	−0.842833
$568$	28.0682	1.17771
$569$	9.50490	0.398466	0.199233	−	0.979952i	$-0.436155\pi$
$569$	9.50490	0.398466	0.199233	+	0.979952i	$0.436155\pi$
$570$	0	0
$571$	12.8174	0.536392	0.268196	−	0.963364i	$-0.413572\pi$
$571$	12.8174	0.536392	0.268196	+	0.963364i	$0.413572\pi$
$572$	−3.10404	−0.129786
$573$	−32.0781	−1.34008
$574$	5.01295	0.209237
$575$	0	0
$576$	301.977	12.5824
$577$	33.4748	1.39357	0.696786	−	0.717279i	$-0.254613\pi$
$577$	33.4748	1.39357	0.696786	+	0.717279i	$0.254613\pi$
$578$	29.2691	1.21743
$579$	−23.7628	−0.987547
$580$	0	0
$581$	8.06822	0.334726
$582$	−72.6302	−3.01062
$583$	−2.77177	−0.114795
$584$	−95.8626	−3.96682
$585$	0	0
$586$	−34.0510	−1.40663
$587$	12.3890	0.511350	0.255675	−	0.966763i	$-0.417702\pi$
$587$	12.3890	0.511350	0.255675	+	0.966763i	$0.417702\pi$
$588$	18.2507	0.752647
$589$	8.30991	0.342404
$590$	0	0
$591$	−8.86755	−0.364762
$592$	−91.1029	−3.74431
$593$	4.18654	0.171921	0.0859603	−	0.996299i	$-0.472604\pi$
$593$	4.18654	0.171921	0.0859603	+	0.996299i	$0.472604\pi$
$594$	36.2208	1.48616
$595$	0	0
$596$	−29.4968	−1.20824
$597$	−24.1311	−0.987619
$598$	−6.52951	−0.267011
$599$	23.0533	0.941933	0.470966	−	0.882151i	$-0.343905\pi$
$599$	23.0533	0.941933	0.470966	+	0.882151i	$0.343905\pi$
$600$	0	0
$601$	−10.0237	−0.408875	−0.204437	−	0.978880i	$-0.565536\pi$
$601$	−10.0237	−0.408875	−0.204437	+	0.978880i	$0.565536\pi$
$602$	−19.6685	−0.801627
$603$	21.5388	0.877127
$604$	−73.0192	−2.97111
$605$	0	0
$606$	60.1438	2.44318
$607$	−20.5333	−0.833420	−0.416710	−	0.909039i	$-0.636817\pi$
$607$	−20.5333	−0.833420	−0.416710	+	0.909039i	$0.636817\pi$
$608$	−228.691	−9.27465
$609$	10.8183	0.438381
$610$	0	0
$611$	−4.77847	−0.193316
$612$	103.774	4.19481
$613$	−26.2832	−1.06157	−0.530784	−	0.847507i	$-0.678102\pi$
$613$	−26.2832	−1.06157	−0.530784	+	0.847507i	$0.678102\pi$
$614$	−45.2414	−1.82580
$615$	0	0
$616$	−10.4180	−0.419755
$617$	2.30398	0.0927547	0.0463774	−	0.998924i	$-0.485232\pi$
$617$	2.30398	0.0927547	0.0463774	+	0.998924i	$0.485232\pi$
$618$	123.268	4.95858
$619$	−4.06644	−0.163444	−0.0817220	−	0.996655i	$-0.526042\pi$
$619$	−4.06644	−0.163444	−0.0817220	+	0.996655i	$0.526042\pi$
$620$	0	0
$621$	56.5140	2.26783
$622$	8.55162	0.342889
$623$	9.88721	0.396123
$624$	30.0563	1.20321
$625$	0	0
$626$	−34.0946	−1.36270
$627$	−26.0716	−1.04120
$628$	46.0488	1.83755
$629$	−13.2515	−0.528370
$630$	0	0
$631$	−16.0131	−0.637473	−0.318736	−	0.947843i	$-0.603258\pi$
$631$	−16.0131	−0.637473	−0.318736	+	0.947843i	$0.603258\pi$
$632$	−170.241	−6.77181
$633$	−14.4702	−0.575141
$634$	34.0733	1.35323
$635$	0	0
$636$	50.5868	2.00590
$637$	0.540418	0.0214121
$638$	−9.47444	−0.375097
$639$	−19.1189	−0.756334
$640$	0	0
$641$	−5.14215	−0.203103	−0.101551	−	0.994830i	$-0.532381\pi$
$641$	−5.14215	−0.203103	−0.101551	+	0.994830i	$0.532381\pi$
$642$	−130.264	−5.14113
$643$	9.66481	0.381143	0.190571	−	0.981673i	$-0.438966\pi$
$643$	9.66481	0.381143	0.190571	+	0.981673i	$0.438966\pi$
$644$	−24.9386	−0.982719
$645$	0	0
$646$	−58.1320	−2.28717
$647$	−18.5979	−0.731160	−0.365580	−	0.930780i	$-0.619129\pi$
$647$	−18.5979	−0.731160	−0.365580	+	0.930780i	$0.619129\pi$
$648$	−209.083	−8.21357
$649$	0.0440307	0.00172835
$650$	0	0
$651$	−3.21806	−0.126126
$652$	97.8774	3.83318
$653$	−21.3407	−0.835125	−0.417563	−	0.908648i	$-0.637116\pi$
$653$	−21.3407	−0.835125	−0.417563	+	0.908648i	$0.637116\pi$
$654$	−30.7575	−1.20271
$655$	0	0
$656$	31.5312	1.23109
$657$	65.2978	2.54751
$658$	−24.6057	−0.959229
$659$	42.6387	1.66097	0.830484	−	0.557042i	$-0.188064\pi$
$659$	42.6387	1.66097	0.830484	+	0.557042i	$0.188064\pi$
$660$	0	0
$661$	−37.1188	−1.44375	−0.721876	−	0.692022i	$-0.756720\pi$
$661$	−37.1188	−1.44375	−0.721876	+	0.692022i	$0.756720\pi$
$662$	48.5503	1.88696
$663$	4.37186	0.169789
$664$	84.0551	3.26197
$665$	0	0
$666$	102.783	3.98276
$667$	−14.7826	−0.572386
$668$	112.063	4.33584
$669$	41.8039	1.61623
$670$	0	0
$671$	−5.52934	−0.213458
$672$	88.5620	3.41635
$673$	−15.5046	−0.597660	−0.298830	−	0.954306i	$-0.596596\pi$
$673$	−15.5046	−0.597660	−0.298830	+	0.954306i	$0.596596\pi$
$674$	5.36435	0.206627
$675$	0	0
$676$	−72.9916	−2.80737
$677$	24.3970	0.937653	0.468827	−	0.883290i	$-0.344677\pi$
$677$	24.3970	0.937653	0.468827	+	0.883290i	$0.344677\pi$
$678$	−37.2259	−1.42965
$679$	−8.21406	−0.315227
$680$	0	0
$681$	−3.20869	−0.122957
$682$	2.81831	0.107918
$683$	30.7671	1.17727	0.588634	−	0.808399i	$-0.299666\pi$
$683$	30.7671	1.17727	0.588634	+	0.808399i	$0.299666\pi$
$684$	334.440	12.7876
$685$	0	0
$686$	2.78276	0.106246
$687$	−54.3545	−2.07375
$688$	−123.714	−4.71654
$689$	1.49791	0.0570659
$690$	0	0
$691$	−27.0821	−1.03025	−0.515125	−	0.857115i	$-0.672255\pi$
$691$	−27.0821	−1.03025	−0.515125	+	0.857115i	$0.672255\pi$
$692$	−9.64578	−0.366678
$693$	7.09637	0.269569
$694$	58.2619	2.21159
$695$	0	0
$696$	112.706	4.27210
$697$	4.58640	0.173722
$698$	55.5661	2.10321
$699$	61.3524	2.32056
$700$	0	0
$701$	36.8593	1.39216	0.696078	−	0.717966i	$-0.254927\pi$
$701$	36.8593	1.39216	0.696078	+	0.717966i	$0.254927\pi$
$702$	−19.5744	−0.738787
$703$	−42.7065	−1.61070
$704$	−42.5538	−1.60381
$705$	0	0
$706$	48.7497	1.83472
$707$	6.80193	0.255813
$708$	−0.803591	−0.0302008
$709$	34.6636	1.30182	0.650909	−	0.759155i	$-0.274388\pi$
$709$	34.6636	1.30182	0.650909	+	0.759155i	$0.274388\pi$
$710$	0	0
$711$	115.961	4.34889
$712$	103.005	3.86029
$713$	4.39730	0.164680
$714$	22.5120	0.842489
$715$	0	0
$716$	47.7472	1.78440
$717$	85.7019	3.20059
$718$	34.7364	1.29635
$719$	11.9889	0.447111	0.223556	−	0.974691i	$-0.428234\pi$
$719$	11.9889	0.447111	0.223556	+	0.974691i	$0.428234\pi$
$720$	0	0
$721$	13.9410	0.519189
$722$	−134.474	−5.00460
$723$	−7.01586	−0.260923
$724$	60.3771	2.24390
$725$	0	0
$726$	−8.84217	−0.328164
$727$	18.0487	0.669389	0.334695	−	0.942327i	$-0.391367\pi$
$727$	18.0487	0.669389	0.334695	+	0.942327i	$0.391367\pi$
$728$	5.63010	0.208665
$729$	18.3441	0.679412
$730$	0	0
$731$	−17.9949	−0.665565
$732$	100.914	3.72991
$733$	25.4611	0.940429	0.470215	−	0.882552i	$-0.344177\pi$
$733$	25.4611	0.940429	0.470215	+	0.882552i	$0.344177\pi$
$734$	31.9107	1.17784
$735$	0	0
$736$	−121.015	−4.46068
$737$	−3.03518	−0.111802
$738$	−35.5738	−1.30949
$739$	−9.24728	−0.340167	−0.170083	−	0.985430i	$-0.554404\pi$
$739$	−9.24728	−0.340167	−0.170083	+	0.985430i	$0.554404\pi$
$740$	0	0
$741$	14.0895	0.517592
$742$	7.71318	0.283160
$743$	8.64209	0.317048	0.158524	−	0.987355i	$-0.449327\pi$
$743$	8.64209	0.317048	0.158524	+	0.987355i	$0.449327\pi$
$744$	−33.5259	−1.22912
$745$	0	0
$746$	−55.8044	−2.04314
$747$	−57.2550	−2.09485
$748$	−14.6235	−0.534689
$749$	−14.7322	−0.538302
$750$	0	0
$751$	−33.2981	−1.21507	−0.607533	−	0.794295i	$-0.707841\pi$
$751$	−33.2981	−1.21507	−0.607533	+	0.794295i	$0.707841\pi$
$752$	−154.768	−5.64382
$753$	−73.4594	−2.67701
$754$	5.12016	0.186465
$755$	0	0
$756$	−74.7617	−2.71906
$757$	−38.3806	−1.39497	−0.697484	−	0.716601i	$-0.745697\pi$
$757$	−38.3806	−1.39497	−0.697484	+	0.716601i	$0.745697\pi$
$758$	28.2558	1.02630
$759$	−13.7961	−0.500768
$760$	0	0
$761$	−26.9083	−0.975425	−0.487712	−	0.873004i	$-0.662169\pi$
$761$	−26.9083	−0.975425	−0.487712	+	0.873004i	$0.662169\pi$
$762$	8.10893	0.293756
$763$	−3.47850	−0.125930
$764$	−57.9860	−2.09786
$765$	0	0
$766$	14.2346	0.514317
$767$	−0.0237949	−0.000859185 0
$768$	283.743	10.2387
$769$	19.2624	0.694622	0.347311	−	0.937750i	$-0.387095\pi$
$769$	19.2624	0.694622	0.347311	+	0.937750i	$0.387095\pi$
$770$	0	0
$771$	−72.2156	−2.60078
$772$	−42.9548	−1.54598
$773$	−23.1008	−0.830879	−0.415439	−	0.909621i	$-0.636372\pi$
$773$	−23.1008	−0.830879	−0.415439	+	0.909621i	$0.636372\pi$
$774$	139.575	5.01691
$775$	0	0
$776$	−85.5745	−3.07195
$777$	16.5383	0.593309
$778$	104.407	3.74318
$779$	14.7809	0.529582
$780$	0	0
$781$	2.69419	0.0964056
$782$	−30.7614	−1.10002
$783$	−44.3158	−1.58372
$784$	17.5034	0.625122
$785$	0	0
$786$	−18.4795	−0.659141
$787$	−0.210698	−0.00751056	−0.00375528	−	0.999993i	$-0.501195\pi$
$787$	−0.210698	−0.00751056	−0.00375528	+	0.999993i	$0.501195\pi$
$788$	−16.0294	−0.571025
$789$	4.90693	0.174691
$790$	0	0
$791$	−4.21004	−0.149692
$792$	73.9303	2.62700
$793$	2.98815	0.106112
$794$	−35.2382	−1.25056
$795$	0	0
$796$	−43.6206	−1.54609
$797$	6.09559	0.215917	0.107958	−	0.994155i	$-0.465569\pi$
$797$	6.09559	0.215917	0.107958	+	0.994155i	$0.465569\pi$
$798$	72.5510	2.56828
$799$	−22.5120	−0.796416
$800$	0	0
$801$	−70.1633	−2.47910
$802$	55.2726	1.95174
$803$	−9.20158	−0.324717
$804$	55.3943	1.95361
$805$	0	0
$806$	−1.52306	−0.0536476
$807$	93.4870	3.29090
$808$	70.8628	2.49295
$809$	35.1318	1.23517	0.617584	−	0.786505i	$-0.288111\pi$
$809$	35.1318	1.23517	0.617584	+	0.786505i	$0.288111\pi$
$810$	0	0
$811$	47.9560	1.68396	0.841981	−	0.539508i	$-0.181390\pi$
$811$	47.9560	1.68396	0.841981	+	0.539508i	$0.181390\pi$
$812$	19.5558	0.686273
$813$	−80.3209	−2.81698
$814$	−14.4839	−0.507660
$815$	0	0
$816$	141.599	4.95696
$817$	−57.9934	−2.02893
$818$	49.9978	1.74813
$819$	−3.83500	−0.134006
$820$	0	0
$821$	13.9583	0.487147	0.243574	−	0.969882i	$-0.421680\pi$
$821$	13.9583	0.487147	0.243574	+	0.969882i	$0.421680\pi$
$822$	−199.418	−6.95549
$823$	14.1607	0.493611	0.246806	−	0.969065i	$-0.420619\pi$
$823$	14.1607	0.493611	0.246806	+	0.969065i	$0.420619\pi$
$824$	145.238	5.05959
$825$	0	0
$826$	−0.122527	−0.00426326
$827$	45.7887	1.59223	0.796114	−	0.605147i	$-0.206886\pi$
$827$	45.7887	1.59223	0.796114	+	0.605147i	$0.206886\pi$
$828$	176.974	6.15026
$829$	−29.3291	−1.01864	−0.509320	−	0.860577i	$-0.670103\pi$
$829$	−29.3291	−1.01864	−0.509320	+	0.860577i	$0.670103\pi$
$830$	0	0
$831$	−81.3605	−2.82236
$832$	22.9968	0.797271
$833$	2.54598	0.0882129
$834$	−66.8534	−2.31494
$835$	0	0
$836$	−47.1283	−1.62997
$837$	13.1824	0.455649
$838$	−59.8406	−2.06716
$839$	8.96518	0.309512	0.154756	−	0.987953i	$-0.450541\pi$
$839$	8.96518	0.309512	0.154756	+	0.987953i	$0.450541\pi$
$840$	0	0
$841$	−17.4081	−0.600279
$842$	−31.3466	−1.08028
$843$	−83.5067	−2.87612
$844$	−26.1572	−0.900367
$845$	0	0
$846$	174.611	6.00325
$847$	−1.00000	−0.0343604
$848$	48.5154	1.66603
$849$	−69.9943	−2.40220
$850$	0	0
$851$	−22.5987	−0.774674
$852$	−49.1709	−1.68457
$853$	−33.3108	−1.14054	−0.570270	−	0.821457i	$-0.693161\pi$
$853$	−33.3108	−1.14054	−0.570270	+	0.821457i	$0.693161\pi$
$854$	15.3868	0.526527
$855$	0	0
$856$	−153.480	−5.24585
$857$	28.5649	0.975759	0.487880	−	0.872911i	$-0.337770\pi$
$857$	28.5649	0.975759	0.487880	+	0.872911i	$0.337770\pi$
$858$	4.77847	0.163134
$859$	−51.5003	−1.75717	−0.878583	−	0.477589i	$-0.841511\pi$
$859$	−51.5003	−1.75717	−0.878583	+	0.477589i	$0.841511\pi$
$860$	0	0
$861$	−5.72400	−0.195074
$862$	37.8121	1.28788
$863$	22.9081	0.779801	0.389901	−	0.920857i	$-0.372509\pi$
$863$	22.9081	0.779801	0.389901	+	0.920857i	$0.372509\pi$
$864$	−362.783	−12.3421
$865$	0	0
$866$	33.6251	1.14263
$867$	−33.4207	−1.13503
$868$	−5.81714	−0.197446
$869$	−16.3409	−0.554328
$870$	0	0
$871$	1.64027	0.0555783
$872$	−36.2392	−1.22721
$873$	58.2900	1.97282
$874$	−99.1369	−3.35336
$875$	0	0
$876$	167.936	5.67402
$877$	−0.188122	−0.00635244	−0.00317622	−	0.999995i	$-0.501011\pi$
$877$	−0.188122	−0.00635244	−0.00317622	+	0.999995i	$0.501011\pi$
$878$	−8.93324	−0.301482
$879$	38.8809	1.31142
$880$	0	0
$881$	−31.3936	−1.05768	−0.528839	−	0.848722i	$-0.677372\pi$
$881$	−31.3936	−1.05768	−0.528839	+	0.848722i	$0.677372\pi$
$882$	−19.7475	−0.664934
$883$	3.04846	0.102589	0.0512943	−	0.998684i	$-0.483665\pi$
$883$	3.04846	0.102589	0.0512943	+	0.998684i	$0.483665\pi$
$884$	7.90281	0.265800
$885$	0	0
$886$	44.5110	1.49538
$887$	14.5683	0.489157	0.244579	−	0.969629i	$-0.421350\pi$
$887$	14.5683	0.489157	0.244579	+	0.969629i	$0.421350\pi$
$888$	172.297	5.78191
$889$	0.917075	0.0307577
$890$	0	0
$891$	−20.0693	−0.672348
$892$	75.5669	2.53017
$893$	−72.5510	−2.42783
$894$	45.4084	1.51869
$895$	0	0
$896$	62.6736	2.09378
$897$	7.45567	0.248938
$898$	−72.5851	−2.42220
$899$	−3.44817	−0.115003
$900$	0	0
$901$	7.05686	0.235098
$902$	5.01295	0.166913
$903$	22.4583	0.747366
$904$	−43.8604	−1.45878
$905$	0	0
$906$	112.408	3.73452
$907$	−21.2523	−0.705669	−0.352835	−	0.935686i	$-0.614782\pi$
$907$	−21.2523	−0.705669	−0.352835	+	0.935686i	$0.614782\pi$
$908$	−5.80020	−0.192486
$909$	−48.2690	−1.60098
$910$	0	0
$911$	−41.5842	−1.37774	−0.688872	−	0.724883i	$-0.741894\pi$
$911$	−41.5842	−1.37774	−0.688872	+	0.724883i	$0.741894\pi$
$912$	456.342	15.1110
$913$	8.06822	0.267019
$914$	−12.3559	−0.408697
$915$	0	0
$916$	−98.2541	−3.24641
$917$	−2.08992	−0.0690153
$918$	−92.2173	−3.04363
$919$	11.1271	0.367050	0.183525	−	0.983015i	$-0.441249\pi$
$919$	11.1271	0.367050	0.183525	+	0.983015i	$0.441249\pi$
$920$	0	0
$921$	51.6586	1.70221
$922$	98.9034	3.25721
$923$	−1.45599	−0.0479244
$924$	18.2507	0.600405
$925$	0	0
$926$	−59.3984	−1.95195
$927$	−98.9302	−3.24930
$928$	94.8948	3.11507
$929$	36.5610	1.19953	0.599763	−	0.800178i	$-0.295261\pi$
$929$	36.5610	1.19953	0.599763	+	0.800178i	$0.295261\pi$
$930$	0	0
$931$	8.20511	0.268912
$932$	110.904	3.63277
$933$	−9.76460	−0.319679
$934$	37.9250	1.24094
$935$	0	0
$936$	−39.9532	−1.30591
$937$	−33.3482	−1.08944	−0.544719	−	0.838618i	$-0.683364\pi$
$937$	−33.3482	−1.08944	−0.544719	+	0.838618i	$0.683364\pi$
$938$	8.44620	0.275778
$939$	38.9307	1.27046
$940$	0	0
$941$	21.7498	0.709022	0.354511	−	0.935052i	$-0.384647\pi$
$941$	21.7498	0.709022	0.354511	+	0.935052i	$0.384647\pi$
$942$	−70.8892	−2.30969
$943$	7.82153	0.254704
$944$	−0.770687	−0.0250837
$945$	0	0
$946$	−19.6685	−0.639477
$947$	−26.9557	−0.875942	−0.437971	−	0.898989i	$-0.644303\pi$
$947$	−26.9557	−0.875942	−0.437971	+	0.898989i	$0.644303\pi$
$948$	298.234	9.68618
$949$	4.97270	0.161421
$950$	0	0
$951$	−38.9064	−1.26163
$952$	26.5241	0.859652
$953$	−45.4594	−1.47257	−0.736287	−	0.676669i	$-0.763423\pi$
$953$	−45.4594	−1.47257	−0.736287	+	0.676669i	$0.763423\pi$
$954$	−54.7356	−1.77213
$955$	0	0
$956$	154.919	5.01045
$957$	10.8183	0.349707
$958$	92.8804	3.00083
$959$	−22.5530	−0.728275
$960$	0	0
$961$	−29.9743	−0.966913
$962$	7.82735	0.252364
$963$	104.545	3.36891
$964$	−12.6822	−0.408467
$965$	0	0
$966$	38.3914	1.23522
$967$	9.72756	0.312817	0.156409	−	0.987692i	$-0.450008\pi$
$967$	9.72756	0.312817	0.156409	+	0.987692i	$0.450008\pi$
$968$	−10.4180	−0.334849
$969$	66.3776	2.13236
$970$	0	0
$971$	59.4901	1.90913	0.954565	−	0.298003i	$-0.0963207\pi$
$971$	59.4901	1.90913	0.954565	+	0.298003i	$0.0963207\pi$
$972$	141.995	4.55449
$973$	−7.56074	−0.242386
$974$	112.478	3.60404
$975$	0	0
$976$	96.7824	3.09793
$977$	−22.1361	−0.708198	−0.354099	−	0.935208i	$-0.615212\pi$
$977$	−22.1361	−0.708198	−0.354099	+	0.935208i	$0.615212\pi$
$978$	−150.676	−4.81809
$979$	9.88721	0.315997
$980$	0	0
$981$	24.6847	0.788122
$982$	−118.207	−3.77213
$983$	23.5832	0.752186	0.376093	−	0.926582i	$-0.377267\pi$
$983$	23.5832	0.752186	0.376093	+	0.926582i	$0.377267\pi$
$984$	−59.6330	−1.90103
$985$	0	0
$986$	24.1217	0.768192
$987$	28.0958	0.894300
$988$	25.4690	0.810276
$989$	−30.6880	−0.975822
$990$	0	0
$991$	−11.7425	−0.373014	−0.186507	−	0.982454i	$-0.559717\pi$
$991$	−11.7425	−0.373014	−0.186507	+	0.982454i	$0.559717\pi$
$992$	−28.2278	−0.896233
$993$	−55.4367	−1.75923
$994$	−7.49729	−0.237800
$995$	0	0
$996$	−147.251	−4.66582
$997$	−57.5250	−1.82183	−0.910917	−	0.412589i	$-0.864625\pi$
$997$	−57.5250	−1.82183	−0.910917	+	0.412589i	$0.864625\pi$
$998$	−52.4252	−1.65949
$999$	−67.7471	−2.14342

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1925.2.a.ba.1.1	✓	7
5.2	odd	4		1925.2.b.q.1849.1		14
5.3	odd	4		1925.2.b.q.1849.14		14
5.4	even	2		1925.2.a.bc.1.7	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1925.2.a.ba.1.1	✓	7	1.1	even	1	trivial
1925.2.a.bc.1.7	yes	7	5.4	even	2
1925.2.b.q.1849.1		14	5.2	odd	4
1925.2.b.q.1849.14		14	5.3	odd	4

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:	\( N \)	\(=\)	\( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1925.a (trivial)

\(f(q)\)	\(=\)	\(q-2.78276 q^{2} +3.17748 q^{3} +5.74378 q^{4} -8.84217 q^{6} -1.00000 q^{7} -10.4180 q^{8} +7.09637 q^{9} +O(q^{10})\)	\(q-2.78276 q^{2} +3.17748 q^{3} +5.74378 q^{4} -8.84217 q^{6} -1.00000 q^{7} -10.4180 q^{8} +7.09637 q^{9} -1.00000 q^{11} +18.2507 q^{12} +0.540418 q^{13} +2.78276 q^{14} +17.5034 q^{16} +2.54598 q^{17} -19.7475 q^{18} +8.20511 q^{19} -3.17748 q^{21} +2.78276 q^{22} +4.34185 q^{23} -33.1031 q^{24} -1.50385 q^{26} +13.0161 q^{27} -5.74378 q^{28} -3.40469 q^{29} +1.01277 q^{31} -27.8718 q^{32} -3.17748 q^{33} -7.08485 q^{34} +40.7600 q^{36} -5.20486 q^{37} -22.8329 q^{38} +1.71717 q^{39} +1.80143 q^{41} +8.84217 q^{42} -7.06796 q^{43} -5.74378 q^{44} -12.0823 q^{46} -8.84217 q^{47} +55.6167 q^{48} +1.00000 q^{49} +8.08979 q^{51} +3.10404 q^{52} +2.77177 q^{53} -36.2208 q^{54} +10.4180 q^{56} +26.0716 q^{57} +9.47444 q^{58} -0.0440307 q^{59} +5.52934 q^{61} -2.81831 q^{62} -7.09637 q^{63} +42.5538 q^{64} +8.84217 q^{66} +3.03518 q^{67} +14.6235 q^{68} +13.7961 q^{69} -2.69419 q^{71} -73.9303 q^{72} +9.20158 q^{73} +14.4839 q^{74} +47.1283 q^{76} +1.00000 q^{77} -4.77847 q^{78} +16.3409 q^{79} +20.0693 q^{81} -5.01295 q^{82} -8.06822 q^{83} -18.2507 q^{84} +19.6685 q^{86} -10.8183 q^{87} +10.4180 q^{88} -9.88721 q^{89} -0.540418 q^{91} +24.9386 q^{92} +3.21806 q^{93} +24.6057 q^{94} -88.5620 q^{96} +8.21406 q^{97} -2.78276 q^{98} -7.09637 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{2} + 13 q^{4} - q^{6} - 7 q^{7} - 6 q^{8} + 13 q^{9}+O(q^{10}) \) 7 * q - q^2 + 13 * q^4 - q^6 - 7 * q^7 - 6 * q^8 + 13 * q^9	\( 7 q - q^{2} + 13 q^{4} - q^{6} - 7 q^{7} - 6 q^{8} + 13 q^{9} - 7 q^{11} + 21 q^{12} + 3 q^{13} + q^{14} + 29 q^{16} - 14 q^{18} + 18 q^{19} + q^{22} + 7 q^{23} - 22 q^{24} + 13 q^{26} - 6 q^{27} - 13 q^{28} - 2 q^{29} + 24 q^{31} - 33 q^{32} + 33 q^{34} + 44 q^{36} - 21 q^{37} - 9 q^{38} + 10 q^{39} - 10 q^{41} + q^{42} + 2 q^{43} - 13 q^{44} + 3 q^{46} - q^{47} + 77 q^{48} + 7 q^{49} + 29 q^{51} - 9 q^{52} + 11 q^{53} - 47 q^{54} + 6 q^{56} - 7 q^{57} + 33 q^{58} + q^{59} + 28 q^{61} - 16 q^{62} - 13 q^{63} + 48 q^{64} + q^{66} + 46 q^{68} + 33 q^{69} - 10 q^{71} - 36 q^{72} + 11 q^{73} - 6 q^{74} + 20 q^{76} + 7 q^{77} + 31 q^{78} + 19 q^{79} + 27 q^{81} - 8 q^{82} + 19 q^{83} - 21 q^{84} + 55 q^{86} + 12 q^{87} + 6 q^{88} - 2 q^{89} - 3 q^{91} - 40 q^{92} - 58 q^{93} + 21 q^{94} - 23 q^{96} - 8 q^{97} - q^{98} - 13 q^{99}+O(q^{100}) \) 7 * q - q^2 + 13 * q^4 - q^6 - 7 * q^7 - 6 * q^8 + 13 * q^9 - 7 * q^11 + 21 * q^12 + 3 * q^13 + q^14 + 29 * q^16 - 14 * q^18 + 18 * q^19 + q^22 + 7 * q^23 - 22 * q^24 + 13 * q^26 - 6 * q^27 - 13 * q^28 - 2 * q^29 + 24 * q^31 - 33 * q^32 + 33 * q^34 + 44 * q^36 - 21 * q^37 - 9 * q^38 + 10 * q^39 - 10 * q^41 + q^42 + 2 * q^43 - 13 * q^44 + 3 * q^46 - q^47 + 77 * q^48 + 7 * q^49 + 29 * q^51 - 9 * q^52 + 11 * q^53 - 47 * q^54 + 6 * q^56 - 7 * q^57 + 33 * q^58 + q^59 + 28 * q^61 - 16 * q^62 - 13 * q^63 + 48 * q^64 + q^66 + 46 * q^68 + 33 * q^69 - 10 * q^71 - 36 * q^72 + 11 * q^73 - 6 * q^74 + 20 * q^76 + 7 * q^77 + 31 * q^78 + 19 * q^79 + 27 * q^81 - 8 * q^82 + 19 * q^83 - 21 * q^84 + 55 * q^86 + 12 * q^87 + 6 * q^88 - 2 * q^89 - 3 * q^91 - 40 * q^92 - 58 * q^93 + 21 * q^94 - 23 * q^96 - 8 * q^97 - q^98 - 13 * q^99

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