LMFDB - Embedded newform 4925.2.a.s.1.29

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4925,2,Mod(1,4925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4925, 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("4925.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4925 = 5^{2} \cdot 197 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4925.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:	$39.3263229955$
Analytic rank:	$0$
Dimension:	$49$
Twist minimal:	no (minimal twist has level 985)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.29
Character	$\chi$	$=$	4925.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.828486 q^{2} +2.62539 q^{3} -1.31361 q^{4} +2.17510 q^{6} -1.25071 q^{7} -2.74528 q^{8} +3.89268 q^{9} -5.56666 q^{11} -3.44874 q^{12} +4.43513 q^{13} -1.03620 q^{14} +0.352794 q^{16} -3.67759 q^{17} +3.22503 q^{18} +2.46665 q^{19} -3.28361 q^{21} -4.61190 q^{22} +3.29841 q^{23} -7.20743 q^{24} +3.67445 q^{26} +2.34362 q^{27} +1.64295 q^{28} +1.93477 q^{29} +7.45953 q^{31} +5.78285 q^{32} -14.6147 q^{33} -3.04683 q^{34} -5.11346 q^{36} +6.44125 q^{37} +2.04359 q^{38} +11.6440 q^{39} +1.59137 q^{41} -2.72043 q^{42} +7.59936 q^{43} +7.31242 q^{44} +2.73269 q^{46} +6.52045 q^{47} +0.926223 q^{48} -5.43571 q^{49} -9.65511 q^{51} -5.82604 q^{52} +7.51214 q^{53} +1.94166 q^{54} +3.43356 q^{56} +6.47593 q^{57} +1.60293 q^{58} +6.90090 q^{59} +13.8403 q^{61} +6.18011 q^{62} -4.86863 q^{63} +4.08542 q^{64} -12.1080 q^{66} +7.77589 q^{67} +4.83092 q^{68} +8.65961 q^{69} +14.4768 q^{71} -10.6865 q^{72} -14.3996 q^{73} +5.33649 q^{74} -3.24022 q^{76} +6.96230 q^{77} +9.64686 q^{78} -9.44708 q^{79} -5.52510 q^{81} +1.31843 q^{82} -12.5209 q^{83} +4.31339 q^{84} +6.29596 q^{86} +5.07953 q^{87} +15.2820 q^{88} +8.40950 q^{89} -5.54709 q^{91} -4.33282 q^{92} +19.5842 q^{93} +5.40210 q^{94} +15.1822 q^{96} -6.31891 q^{97} -4.50341 q^{98} -21.6692 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 49 q + 5 q^{2} + 22 q^{3} + 49 q^{4} + 2 q^{6} + 32 q^{7} + 15 q^{8} + 51 q^{9} - 2 q^{11} + 44 q^{12} + 32 q^{13} - 8 q^{14} + 49 q^{16} + 14 q^{17} + 25 q^{18} + 4 q^{19} + 10 q^{21} + 38 q^{22} + 24 q^{23}+ \cdots + 22 q^{99}+O(q^{100}) $ 49 * q + 5 * q^2 + 22 * q^3 + 49 * q^4 + 2 * q^6 + 32 * q^7 + 15 * q^8 + 51 * q^9 - 2 * q^11 + 44 * q^12 + 32 * q^13 - 8 * q^14 + 49 * q^16 + 14 * q^17 + 25 * q^18 + 4 * q^19 + 10 * q^21 + 38 * q^22 + 24 * q^23 - 2 * q^26 + 88 * q^27 + 80 * q^28 + 6 * q^29 - 2 * q^31 + 35 * q^32 + 48 * q^33 - 12 * q^34 + 43 * q^36 + 60 * q^37 + 34 * q^38 - 2 * q^41 + 36 * q^42 + 80 * q^43 - 8 * q^44 - 22 * q^46 + 48 * q^47 + 88 * q^48 + 65 * q^49 + 16 * q^51 + 96 * q^52 + 18 * q^53 + 42 * q^54 - 22 * q^56 + 40 * q^57 + 52 * q^58 + 8 * q^59 + 16 * q^61 + 38 * q^62 + 82 * q^63 + 55 * q^64 + 42 * q^66 + 146 * q^67 - 68 * q^68 - 6 * q^69 + 75 * q^72 + 78 * q^73 - 24 * q^74 - 4 * q^76 + 32 * q^77 - 38 * q^78 + 10 * q^79 + 41 * q^81 + 96 * q^82 + 64 * q^83 + 42 * q^84 - 28 * q^86 + 56 * q^87 + 72 * q^88 + 4 * q^89 + 60 * q^92 - 28 * q^93 - 8 * q^94 - 34 * q^96 + 124 * q^97 - 19 * q^98 + 22 * q^99

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 $

$2$	0.828486	0.585828	0.292914	−	0.956139i	$-0.405375\pi$
$2$	0.828486	0.585828	0.292914	+	0.956139i	$0.405375\pi$
$3$	2.62539	1.51577	0.757885	−	0.652388i	$-0.226233\pi$
$3$	2.62539	1.51577	0.757885	+	0.652388i	$0.226233\pi$
$4$	−1.31361	−0.656805
$5$	0	0
$5$	0	0
$6$	2.17510	0.887981
$7$	−1.25071	−0.472726	−0.236363	−	0.971665i	$-0.575955\pi$
$7$	−1.25071	−0.472726	−0.236363	+	0.971665i	$0.575955\pi$
$8$	−2.74528	−0.970603
$9$	3.89268	1.29756
$10$	0	0
$11$	−5.56666	−1.67841	−0.839205	−	0.543815i	$-0.816979\pi$
$11$	−5.56666	−1.67841	−0.839205	+	0.543815i	$0.816979\pi$
$12$	−3.44874	−0.995566
$13$	4.43513	1.23008	0.615042	−	0.788494i	$-0.289139\pi$
$13$	4.43513	1.23008	0.615042	+	0.788494i	$0.289139\pi$
$14$	−1.03620	−0.276936
$15$	0	0
$16$	0.352794	0.0881986
$17$	−3.67759	−0.891947	−0.445973	−	0.895046i	$-0.647142\pi$
$17$	−3.67759	−0.891947	−0.445973	+	0.895046i	$0.647142\pi$
$18$	3.22503	0.760146
$19$	2.46665	0.565889	0.282944	−	0.959136i	$-0.408689\pi$
$19$	2.46665	0.565889	0.282944	+	0.959136i	$0.408689\pi$
$20$	0	0
$21$	−3.28361	−0.716543
$22$	−4.61190	−0.983260
$23$	3.29841	0.687766	0.343883	−	0.939013i	$-0.388258\pi$
$23$	3.29841	0.687766	0.343883	+	0.939013i	$0.388258\pi$
$24$	−7.20743	−1.47121
$25$	0	0
$26$	3.67445	0.720618
$27$	2.34362	0.451030
$28$	1.64295	0.310489
$29$	1.93477	0.359278	0.179639	−	0.983733i	$-0.442507\pi$
$29$	1.93477	0.359278	0.179639	+	0.983733i	$0.442507\pi$
$30$	0	0
$31$	7.45953	1.33977	0.669885	−	0.742465i	$-0.266343\pi$
$31$	7.45953	1.33977	0.669885	+	0.742465i	$0.266343\pi$
$32$	5.78285	1.02227
$33$	−14.6147	−2.54408
$34$	−3.04683	−0.522528
$35$	0	0
$36$	−5.11346	−0.852243
$37$	6.44125	1.05894	0.529468	−	0.848330i	$-0.322392\pi$
$37$	6.44125	1.05894	0.529468	+	0.848330i	$0.322392\pi$
$38$	2.04359	0.331514
$39$	11.6440	1.86453
$40$	0	0
$41$	1.59137	0.248530	0.124265	−	0.992249i	$-0.460343\pi$
$41$	1.59137	0.248530	0.124265	+	0.992249i	$0.460343\pi$
$42$	−2.72043	−0.419771
$43$	7.59936	1.15889	0.579446	−	0.815011i	$-0.303269\pi$
$43$	7.59936	1.15889	0.579446	+	0.815011i	$0.303269\pi$
$44$	7.31242	1.10239
$45$	0	0
$46$	2.73269	0.402913
$47$	6.52045	0.951105	0.475553	−	0.879687i	$-0.342248\pi$
$47$	6.52045	0.951105	0.475553	+	0.879687i	$0.342248\pi$
$48$	0.926223	0.133689
$49$	−5.43571	−0.776531
$50$	0	0
$51$	−9.65511	−1.35199
$52$	−5.82604	−0.807926
$53$	7.51214	1.03187	0.515936	−	0.856627i	$-0.327444\pi$
$53$	7.51214	1.03187	0.515936	+	0.856627i	$0.327444\pi$
$54$	1.94166	0.264226
$55$	0	0
$56$	3.43356	0.458829
$57$	6.47593	0.857758
$58$	1.60293	0.210475
$59$	6.90090	0.898421	0.449211	−	0.893426i	$-0.351705\pi$
$59$	6.90090	0.898421	0.449211	+	0.893426i	$0.351705\pi$
$60$	0	0
$61$	13.8403	1.77207	0.886033	−	0.463622i	$-0.153450\pi$
$61$	13.8403	1.77207	0.886033	+	0.463622i	$0.153450\pi$
$62$	6.18011	0.784875
$63$	−4.86863	−0.613389
$64$	4.08542	0.510677
$65$	0	0
$66$	−12.1080	−1.49040
$67$	7.77589	0.949976	0.474988	−	0.879992i	$-0.342452\pi$
$67$	7.77589	0.949976	0.474988	+	0.879992i	$0.342452\pi$
$68$	4.83092	0.585835
$69$	8.65961	1.04249
$70$	0	0
$71$	14.4768	1.71808	0.859041	−	0.511907i	$-0.171061\pi$
$71$	14.4768	1.71808	0.859041	+	0.511907i	$0.171061\pi$
$72$	−10.6865	−1.25941
$73$	−14.3996	−1.68534	−0.842671	−	0.538428i	$-0.819018\pi$
$73$	−14.3996	−1.68534	−0.842671	+	0.538428i	$0.819018\pi$
$74$	5.33649	0.620354
$75$	0	0
$76$	−3.24022	−0.371679
$77$	6.96230	0.793428
$78$	9.64686	1.09229
$79$	−9.44708	−1.06288	−0.531440	−	0.847096i	$-0.678349\pi$
$79$	−9.44708	−1.06288	−0.531440	+	0.847096i	$0.678349\pi$
$80$	0	0
$81$	−5.52510	−0.613900
$82$	1.31843	0.145596
$83$	−12.5209	−1.37435	−0.687176	−	0.726491i	$-0.741150\pi$
$83$	−12.5209	−1.37435	−0.687176	+	0.726491i	$0.741150\pi$
$84$	4.31339	0.470629
$85$	0	0
$86$	6.29596	0.678911
$87$	5.07953	0.544582
$88$	15.2820	1.62907
$89$	8.40950	0.891405	0.445703	−	0.895181i	$-0.352954\pi$
$89$	8.40950	0.891405	0.445703	+	0.895181i	$0.352954\pi$
$90$	0	0
$91$	−5.54709	−0.581493
$92$	−4.33282	−0.451728
$93$	19.5842	2.03078
$94$	5.40210	0.557184
$95$	0	0
$96$	15.1822	1.54953
$97$	−6.31891	−0.641588	−0.320794	−	0.947149i	$-0.603950\pi$
$97$	−6.31891	−0.641588	−0.320794	+	0.947149i	$0.603950\pi$
$98$	−4.50341	−0.454913
$99$	−21.6692	−2.17784
$100$	0	0
$101$	1.96974	0.195996	0.0979981	−	0.995187i	$-0.468756\pi$
$101$	1.96974	0.195996	0.0979981	+	0.995187i	$0.468756\pi$
$102$	−7.99913	−0.792032
$103$	4.14430	0.408350	0.204175	−	0.978934i	$-0.434549\pi$
$103$	4.14430	0.408350	0.204175	+	0.978934i	$0.434549\pi$
$104$	−12.1757	−1.19392
$105$	0	0
$106$	6.22370	0.604500
$107$	−11.7031	−1.13138	−0.565692	−	0.824617i	$-0.691391\pi$
$107$	−11.7031	−1.13138	−0.565692	+	0.824617i	$0.691391\pi$
$108$	−3.07861	−0.296239
$109$	5.83793	0.559172	0.279586	−	0.960121i	$-0.409803\pi$
$109$	5.83793	0.559172	0.279586	+	0.960121i	$0.409803\pi$
$110$	0	0
$111$	16.9108	1.60510
$112$	−0.441245	−0.0416937
$113$	−13.5004	−1.27001	−0.635007	−	0.772507i	$-0.719003\pi$
$113$	−13.5004	−1.27001	−0.635007	+	0.772507i	$0.719003\pi$
$114$	5.36522	0.502499
$115$	0	0
$116$	−2.54153	−0.235975
$117$	17.2645	1.59611
$118$	5.71730	0.526321
$119$	4.59962	0.421646
$120$	0	0
$121$	19.9877	1.81706
$122$	11.4665	1.03813
$123$	4.17797	0.376715
$124$	−9.79891	−0.879969
$125$	0	0
$126$	−4.03359	−0.359341
$127$	−0.808906	−0.0717788	−0.0358894	−	0.999356i	$-0.511426\pi$
$127$	−0.808906	−0.0717788	−0.0358894	+	0.999356i	$0.511426\pi$
$128$	−8.18098	−0.723103
$129$	19.9513	1.75661
$130$	0	0
$131$	−17.9546	−1.56870	−0.784348	−	0.620321i	$-0.787003\pi$
$131$	−17.9546	−1.56870	−0.784348	+	0.620321i	$0.787003\pi$
$132$	19.1980	1.67097
$133$	−3.08508	−0.267510
$134$	6.44222	0.556523
$135$	0	0
$136$	10.0960	0.865726
$137$	−0.404201	−0.0345332	−0.0172666	−	0.999851i	$-0.505496\pi$
$137$	−0.404201	−0.0345332	−0.0172666	+	0.999851i	$0.505496\pi$
$138$	7.17437	0.610723
$139$	17.1259	1.45260	0.726302	−	0.687376i	$-0.241237\pi$
$139$	17.1259	1.45260	0.726302	+	0.687376i	$0.241237\pi$
$140$	0	0
$141$	17.1187	1.44166
$142$	11.9938	1.00650
$143$	−24.6889	−2.06459
$144$	1.37331	0.114443
$145$	0	0
$146$	−11.9298	−0.987321
$147$	−14.2709	−1.17704
$148$	−8.46130	−0.695514
$149$	−8.78283	−0.719517	−0.359759	−	0.933045i	$-0.617141\pi$
$149$	−8.78283	−0.719517	−0.359759	+	0.933045i	$0.617141\pi$
$150$	0	0
$151$	−6.84679	−0.557184	−0.278592	−	0.960410i	$-0.589868\pi$
$151$	−6.84679	−0.557184	−0.278592	+	0.960410i	$0.589868\pi$
$152$	−6.77165	−0.549254
$153$	−14.3157	−1.15735
$154$	5.76817	0.464812
$155$	0	0
$156$	−15.2956	−1.22463
$157$	19.0201	1.51797	0.758985	−	0.651108i	$-0.225695\pi$
$157$	19.0201	1.51797	0.758985	+	0.651108i	$0.225695\pi$
$158$	−7.82678	−0.622665
$159$	19.7223	1.56408
$160$	0	0
$161$	−4.12537	−0.325124
$162$	−4.57747	−0.359640
$163$	−0.780206	−0.0611104	−0.0305552	−	0.999533i	$-0.509728\pi$
$163$	−0.780206	−0.0611104	−0.0305552	+	0.999533i	$0.509728\pi$
$164$	−2.09044	−0.163236
$165$	0	0
$166$	−10.3734	−0.805134
$167$	−9.22289	−0.713689	−0.356844	−	0.934164i	$-0.616147\pi$
$167$	−9.22289	−0.713689	−0.356844	+	0.934164i	$0.616147\pi$
$168$	9.01444	0.695479
$169$	6.67041	0.513109
$170$	0	0
$171$	9.60188	0.734274
$172$	−9.98260	−0.761166
$173$	−12.6520	−0.961916	−0.480958	−	0.876744i	$-0.659711\pi$
$173$	−12.6520	−0.961916	−0.480958	+	0.876744i	$0.659711\pi$
$174$	4.20832	0.319032
$175$	0	0
$176$	−1.96389	−0.148033
$177$	18.1176	1.36180
$178$	6.96715	0.522210
$179$	−14.2477	−1.06492	−0.532462	−	0.846454i	$-0.678733\pi$
$179$	−14.2477	−1.06492	−0.532462	+	0.846454i	$0.678733\pi$
$180$	0	0
$181$	24.4657	1.81852	0.909259	−	0.416230i	$-0.136649\pi$
$181$	24.4657	1.81852	0.909259	+	0.416230i	$0.136649\pi$
$182$	−4.59568	−0.340655
$183$	36.3361	2.68604
$184$	−9.05506	−0.667548
$185$	0	0
$186$	16.2252	1.18969
$187$	20.4719	1.49705
$188$	−8.56533	−0.624691
$189$	−2.93120	−0.213214
$190$	0	0
$191$	−3.60292	−0.260698	−0.130349	−	0.991468i	$-0.541610\pi$
$191$	−3.60292	−0.260698	−0.130349	+	0.991468i	$0.541610\pi$
$192$	10.7258	0.774070
$193$	−0.674279	−0.0485357	−0.0242678	−	0.999705i	$-0.507725\pi$
$193$	−0.674279	−0.0485357	−0.0242678	+	0.999705i	$0.507725\pi$
$194$	−5.23513	−0.375860
$195$	0	0
$196$	7.14041	0.510029
$197$	−1.00000	−0.0712470
$197$	−1.00000	−0.0712470
$198$	−17.9526	−1.27584
$199$	4.12802	0.292627	0.146314	−	0.989238i	$-0.453259\pi$
$199$	4.12802	0.292627	0.146314	+	0.989238i	$0.453259\pi$
$200$	0	0
$201$	20.4147	1.43995
$202$	1.63190	0.114820
$203$	−2.41984	−0.169840
$204$	12.6831	0.887992
$205$	0	0
$206$	3.43350	0.239223
$207$	12.8396	0.892416
$208$	1.56469	0.108492
$209$	−13.7310	−0.949794
$210$	0	0
$211$	−24.9117	−1.71499	−0.857495	−	0.514492i	$-0.827981\pi$
$211$	−24.9117	−1.71499	−0.857495	+	0.514492i	$0.827981\pi$
$212$	−9.86803	−0.677739
$213$	38.0073	2.60422
$214$	−9.69588	−0.662796
$215$	0	0
$216$	−6.43390	−0.437772
$217$	−9.32974	−0.633344
$218$	4.83664	0.327579
$219$	−37.8045	−2.55459
$220$	0	0
$221$	−16.3106	−1.09717
$222$	14.0104	0.940314
$223$	4.36939	0.292596	0.146298	−	0.989241i	$-0.453264\pi$
$223$	4.36939	0.292596	0.146298	+	0.989241i	$0.453264\pi$
$224$	−7.23269	−0.483254
$225$	0	0
$226$	−11.1849	−0.744010
$227$	13.6263	0.904410	0.452205	−	0.891914i	$-0.350638\pi$
$227$	13.6263	0.904410	0.452205	+	0.891914i	$0.350638\pi$
$228$	−8.50685	−0.563380
$229$	−5.29112	−0.349647	−0.174823	−	0.984600i	$-0.555935\pi$
$229$	−5.29112	−0.349647	−0.174823	+	0.984600i	$0.555935\pi$
$230$	0	0
$231$	18.2788	1.20265
$232$	−5.31148	−0.348716
$233$	6.68539	0.437974	0.218987	−	0.975728i	$-0.429725\pi$
$233$	6.68539	0.437974	0.218987	+	0.975728i	$0.429725\pi$
$234$	14.3034	0.935045
$235$	0	0
$236$	−9.06510	−0.590088
$237$	−24.8023	−1.61108
$238$	3.81072	0.247012
$239$	17.1044	1.10639	0.553196	−	0.833051i	$-0.313408\pi$
$239$	17.1044	1.10639	0.553196	+	0.833051i	$0.313408\pi$
$240$	0	0
$241$	5.50017	0.354297	0.177149	−	0.984184i	$-0.443313\pi$
$241$	5.50017	0.354297	0.177149	+	0.984184i	$0.443313\pi$
$242$	16.5595	1.06449
$243$	−21.5364	−1.38156
$244$	−18.1807	−1.16390
$245$	0	0
$246$	3.46139	0.220690
$247$	10.9399	0.696091
$248$	−20.4785	−1.30039
$249$	−32.8724	−2.08320
$250$	0	0
$251$	−12.3140	−0.777253	−0.388626	−	0.921395i	$-0.627050\pi$
$251$	−12.3140	−0.777253	−0.388626	+	0.921395i	$0.627050\pi$
$252$	6.39548	0.402877
$253$	−18.3611	−1.15435
$254$	−0.670168	−0.0420501
$255$	0	0
$256$	−14.9487	−0.934292
$257$	22.8292	1.42404	0.712022	−	0.702157i	$-0.247780\pi$
$257$	22.8292	1.42404	0.712022	+	0.702157i	$0.247780\pi$
$258$	16.5294	1.02907
$259$	−8.05616	−0.500586
$260$	0	0
$261$	7.53143	0.466184
$262$	−14.8751	−0.918987
$263$	25.3186	1.56121	0.780606	−	0.625023i	$-0.214910\pi$
$263$	25.3186	1.56121	0.780606	+	0.625023i	$0.214910\pi$
$264$	40.1213	2.46930
$265$	0	0
$266$	−2.55594	−0.156715
$267$	22.0782	1.35117
$268$	−10.2145	−0.623949
$269$	0.555341	0.0338598	0.0169299	−	0.999857i	$-0.494611\pi$
$269$	0.555341	0.0338598	0.0169299	+	0.999857i	$0.494611\pi$
$270$	0	0
$271$	15.4061	0.935852	0.467926	−	0.883768i	$-0.345001\pi$
$271$	15.4061	0.935852	0.467926	+	0.883768i	$0.345001\pi$
$272$	−1.29743	−0.0786685
$273$	−14.5633	−0.881409
$274$	−0.334875	−0.0202305
$275$	0	0
$276$	−11.3754	−0.684716
$277$	−10.3809	−0.623725	−0.311863	−	0.950127i	$-0.600953\pi$
$277$	−10.3809	−0.623725	−0.311863	+	0.950127i	$0.600953\pi$
$278$	14.1886	0.850976
$279$	29.0375	1.73843
$280$	0	0
$281$	12.4022	0.739855	0.369927	−	0.929061i	$-0.379383\pi$
$281$	12.4022	0.739855	0.369927	+	0.929061i	$0.379383\pi$
$282$	14.1826	0.844563
$283$	16.4352	0.976971	0.488485	−	0.872572i	$-0.337550\pi$
$283$	16.4352	0.976971	0.488485	+	0.872572i	$0.337550\pi$
$284$	−19.0169	−1.12845
$285$	0	0
$286$	−20.4544	−1.20949
$287$	−1.99035	−0.117487
$288$	22.5107	1.32646
$289$	−3.47533	−0.204431
$290$	0	0
$291$	−16.5896	−0.972500
$292$	18.9154	1.10694
$293$	−15.4094	−0.900228	−0.450114	−	0.892971i	$-0.648617\pi$
$293$	−15.4094	−0.900228	−0.450114	+	0.892971i	$0.648617\pi$
$294$	−11.8232	−0.689544
$295$	0	0
$296$	−17.6830	−1.02781
$297$	−13.0461	−0.757014
$298$	−7.27645	−0.421513
$299$	14.6289	0.846010
$300$	0	0
$301$	−9.50463	−0.547838
$302$	−5.67247	−0.326414
$303$	5.17133	0.297085
$304$	0.870221	0.0499106
$305$	0	0
$306$	−11.8603	−0.678010
$307$	−14.1683	−0.808626	−0.404313	−	0.914621i	$-0.632489\pi$
$307$	−14.1683	−0.808626	−0.404313	+	0.914621i	$0.632489\pi$
$308$	−9.14575	−0.521127
$309$	10.8804	0.618965
$310$	0	0
$311$	14.9512	0.847805	0.423903	−	0.905708i	$-0.360660\pi$
$311$	14.9512	0.847805	0.423903	+	0.905708i	$0.360660\pi$
$312$	−31.9659	−1.80971
$313$	2.76214	0.156125	0.0780627	−	0.996948i	$-0.475127\pi$
$313$	2.76214	0.156125	0.0780627	+	0.996948i	$0.475127\pi$
$314$	15.7579	0.889270
$315$	0	0
$316$	12.4098	0.698105
$317$	−5.61587	−0.315419	−0.157709	−	0.987486i	$-0.550411\pi$
$317$	−5.61587	−0.315419	−0.157709	+	0.987486i	$0.550411\pi$
$318$	16.3397	0.916282
$319$	−10.7702	−0.603015
$320$	0	0
$321$	−30.7253	−1.71492
$322$	−3.41781	−0.190467
$323$	−9.07134	−0.504743
$324$	7.25783	0.403213
$325$	0	0
$326$	−0.646390	−0.0358002
$327$	15.3268	0.847576
$328$	−4.36876	−0.241224
$329$	−8.15522	−0.449612
$330$	0	0
$331$	−12.0210	−0.660732	−0.330366	−	0.943853i	$-0.607172\pi$
$331$	−12.0210	−0.660732	−0.330366	+	0.943853i	$0.607172\pi$
$332$	16.4476	0.902682
$333$	25.0737	1.37403
$334$	−7.64104	−0.418099
$335$	0	0
$336$	−1.15844	−0.0631981
$337$	18.5547	1.01074	0.505370	−	0.862903i	$-0.331356\pi$
$337$	18.5547	1.01074	0.505370	+	0.862903i	$0.331356\pi$
$338$	5.52635	0.300594
$339$	−35.4439	−1.92505
$340$	0	0
$341$	−41.5246	−2.24869
$342$	7.95502	0.430158
$343$	15.5535	0.839811
$344$	−20.8624	−1.12482
$345$	0	0
$346$	−10.4820	−0.563517
$347$	34.3835	1.84580	0.922902	−	0.385035i	$-0.125811\pi$
$347$	34.3835	1.84580	0.922902	+	0.385035i	$0.125811\pi$
$348$	−6.67252	−0.357685
$349$	−34.7319	−1.85916	−0.929579	−	0.368624i	$-0.879829\pi$
$349$	−34.7319	−1.85916	−0.929579	+	0.368624i	$0.879829\pi$
$350$	0	0
$351$	10.3943	0.554806
$352$	−32.1911	−1.71579
$353$	−19.4978	−1.03776	−0.518881	−	0.854846i	$-0.673651\pi$
$353$	−19.4978	−1.03776	−0.518881	+	0.854846i	$0.673651\pi$
$354$	15.0102	0.797781
$355$	0	0
$356$	−11.0468	−0.585480
$357$	12.0758	0.639118
$358$	−11.8040	−0.623863
$359$	−25.1653	−1.32817	−0.664087	−	0.747656i	$-0.731179\pi$
$359$	−25.1653	−1.32817	−0.664087	+	0.747656i	$0.731179\pi$
$360$	0	0
$361$	−12.9156	−0.679770
$362$	20.2695	1.06534
$363$	52.4755	2.75425
$364$	7.28671	0.381927
$365$	0	0
$366$	30.1040	1.57356
$367$	6.17921	0.322552	0.161276	−	0.986909i	$-0.448439\pi$
$367$	6.17921	0.322552	0.161276	+	0.986909i	$0.448439\pi$
$368$	1.16366	0.0606600
$369$	6.19469	0.322483
$370$	0	0
$371$	−9.39554	−0.487792
$372$	−25.7260	−1.33383
$373$	27.6858	1.43351	0.716757	−	0.697323i	$-0.245626\pi$
$373$	27.6858	1.43351	0.716757	+	0.697323i	$0.245626\pi$
$374$	16.9607	0.877016
$375$	0	0
$376$	−17.9005	−0.923146
$377$	8.58096	0.441942
$378$	−2.42846	−0.124907
$379$	21.8264	1.12115	0.560574	−	0.828104i	$-0.310580\pi$
$379$	21.8264	1.12115	0.560574	+	0.828104i	$0.310580\pi$
$380$	0	0
$381$	−2.12370	−0.108800
$382$	−2.98497	−0.152724
$383$	0.993450	0.0507629	0.0253815	−	0.999678i	$-0.491920\pi$
$383$	0.993450	0.0507629	0.0253815	+	0.999678i	$0.491920\pi$
$384$	−21.4783	−1.09606
$385$	0	0
$386$	−0.558631	−0.0284336
$387$	29.5818	1.50373
$388$	8.30059	0.421398
$389$	−17.9885	−0.912051	−0.456025	−	0.889967i	$-0.650727\pi$
$389$	−17.9885	−0.912051	−0.456025	+	0.889967i	$0.650727\pi$
$390$	0	0
$391$	−12.1302	−0.613450
$392$	14.9226	0.753703
$393$	−47.1377	−2.37778
$394$	−0.828486	−0.0417385
$395$	0	0
$396$	28.4649	1.43041
$397$	−4.06775	−0.204155	−0.102077	−	0.994776i	$-0.532549\pi$
$397$	−4.06775	−0.204155	−0.102077	+	0.994776i	$0.532549\pi$
$398$	3.42001	0.171429
$399$	−8.09953	−0.405484
$400$	0	0
$401$	30.9146	1.54380	0.771900	−	0.635743i	$-0.219306\pi$
$401$	30.9146	1.54380	0.771900	+	0.635743i	$0.219306\pi$
$402$	16.9133	0.843560
$403$	33.0840	1.64803
$404$	−2.58747	−0.128731
$405$	0	0
$406$	−2.00481	−0.0994969
$407$	−35.8562	−1.77733
$408$	26.5060	1.31224
$409$	23.9199	1.18276	0.591382	−	0.806391i	$-0.298582\pi$
$409$	23.9199	1.18276	0.591382	+	0.806391i	$0.298582\pi$
$410$	0	0
$411$	−1.06118	−0.0523444
$412$	−5.44400	−0.268206
$413$	−8.63106	−0.424707
$414$	10.6375	0.522803
$415$	0	0
$416$	25.6477	1.25748
$417$	44.9623	2.20181
$418$	−11.3760	−0.556416
$419$	9.22370	0.450607	0.225304	−	0.974289i	$-0.427663\pi$
$419$	9.22370	0.450607	0.225304	+	0.974289i	$0.427663\pi$
$420$	0	0
$421$	13.2461	0.645573	0.322787	−	0.946472i	$-0.395380\pi$
$421$	13.2461	0.645573	0.322787	+	0.946472i	$0.395380\pi$
$422$	−20.6390	−1.00469
$423$	25.3820	1.23411
$424$	−20.6229	−1.00154
$425$	0	0
$426$	31.4885	1.52562
$427$	−17.3102	−0.837701
$428$	15.3733	0.743099
$429$	−64.8179	−3.12944
$430$	0	0
$431$	−26.9184	−1.29662	−0.648308	−	0.761378i	$-0.724523\pi$
$431$	−26.9184	−1.29662	−0.648308	+	0.761378i	$0.724523\pi$
$432$	0.826817	0.0397803
$433$	28.9834	1.39285	0.696427	−	0.717627i	$-0.254772\pi$
$433$	28.9834	1.39285	0.696427	+	0.717627i	$0.254772\pi$
$434$	−7.72956	−0.371031
$435$	0	0
$436$	−7.66877	−0.367267
$437$	8.13603	0.389199
$438$	−31.3205	−1.49655
$439$	−34.1704	−1.63086	−0.815432	−	0.578853i	$-0.803501\pi$
$439$	−34.1704	−1.63086	−0.815432	+	0.578853i	$0.803501\pi$
$440$	0	0
$441$	−21.1595	−1.00759
$442$	−13.5131	−0.642753
$443$	7.51967	0.357270	0.178635	−	0.983915i	$-0.442832\pi$
$443$	7.51967	0.357270	0.178635	+	0.983915i	$0.442832\pi$
$444$	−22.2142	−1.05424
$445$	0	0
$446$	3.61998	0.171411
$447$	−23.0584	−1.09062
$448$	−5.10969	−0.241410
$449$	−3.72879	−0.175972	−0.0879862	−	0.996122i	$-0.528043\pi$
$449$	−3.72879	−0.175972	−0.0879862	+	0.996122i	$0.528043\pi$
$450$	0	0
$451$	−8.85861	−0.417136
$452$	17.7343	0.834151
$453$	−17.9755	−0.844563
$454$	11.2892	0.529829
$455$	0	0
$456$	−17.7782	−0.832542
$457$	−16.5767	−0.775424	−0.387712	−	0.921781i	$-0.626734\pi$
$457$	−16.5767	−0.775424	−0.387712	+	0.921781i	$0.626734\pi$
$458$	−4.38362	−0.204833
$459$	−8.61889	−0.402295
$460$	0	0
$461$	−2.48534	−0.115754	−0.0578769	−	0.998324i	$-0.518433\pi$
$461$	−2.48534	−0.115754	−0.0578769	+	0.998324i	$0.518433\pi$
$462$	15.1437	0.704548
$463$	29.0032	1.34789	0.673946	−	0.738781i	$-0.264598\pi$
$463$	29.0032	1.34789	0.673946	+	0.738781i	$0.264598\pi$
$464$	0.682576	0.0316878
$465$	0	0
$466$	5.53875	0.256578
$467$	34.5919	1.60072	0.800361	−	0.599518i	$-0.204641\pi$
$467$	34.5919	1.60072	0.800361	+	0.599518i	$0.204641\pi$
$468$	−22.6789	−1.04833
$469$	−9.72541	−0.449078
$470$	0	0
$471$	49.9352	2.30089
$472$	−18.9449	−0.872011
$473$	−42.3030	−1.94510
$474$	−20.5483	−0.943817
$475$	0	0
$476$	−6.04210	−0.276939
$477$	29.2423	1.33891
$478$	14.1708	0.648155
$479$	14.6393	0.668885	0.334443	−	0.942416i	$-0.391452\pi$
$479$	14.6393	0.668885	0.334443	+	0.942416i	$0.391452\pi$
$480$	0	0
$481$	28.5678	1.30258
$482$	4.55681	0.207557
$483$	−10.8307	−0.492814
$484$	−26.2560	−1.19346
$485$	0	0
$486$	−17.8426	−0.809358
$487$	28.2463	1.27996	0.639980	−	0.768391i	$-0.278942\pi$
$487$	28.2463	1.27996	0.639980	+	0.768391i	$0.278942\pi$
$488$	−37.9955	−1.71997
$489$	−2.04835	−0.0926294
$490$	0	0
$491$	−1.19790	−0.0540604	−0.0270302	−	0.999635i	$-0.508605\pi$
$491$	−1.19790	−0.0540604	−0.0270302	+	0.999635i	$0.508605\pi$
$492$	−5.48822	−0.247428
$493$	−7.11529	−0.320457
$494$	9.06359	0.407790
$495$	0	0
$496$	2.63168	0.118166
$497$	−18.1064	−0.812181
$498$	−27.2343	−1.22040
$499$	−28.5113	−1.27634	−0.638171	−	0.769894i	$-0.720309\pi$
$499$	−28.5113	−1.27634	−0.638171	+	0.769894i	$0.720309\pi$
$500$	0	0
$501$	−24.2137	−1.08179
$502$	−10.2020	−0.455337
$503$	14.1483	0.630841	0.315421	−	0.948952i	$-0.397854\pi$
$503$	14.1483	0.630841	0.315421	+	0.948952i	$0.397854\pi$
$504$	13.3657	0.595357
$505$	0	0
$506$	−15.2119	−0.676253
$507$	17.5124	0.777755
$508$	1.06259	0.0471447
$509$	36.7074	1.62703	0.813514	−	0.581545i	$-0.197552\pi$
$509$	36.7074	1.62703	0.813514	+	0.581545i	$0.197552\pi$
$510$	0	0
$511$	18.0098	0.796705
$512$	3.97720	0.175769
$513$	5.78090	0.255233
$514$	18.9136	0.834245
$515$	0	0
$516$	−26.2082	−1.15375
$517$	−36.2971	−1.59634
$518$	−6.67442	−0.293257
$519$	−33.2165	−1.45804
$520$	0	0
$521$	−1.84046	−0.0806318	−0.0403159	−	0.999187i	$-0.512836\pi$
$521$	−1.84046	−0.0806318	−0.0403159	+	0.999187i	$0.512836\pi$
$522$	6.23969	0.273104
$523$	10.7185	0.468689	0.234345	−	0.972154i	$-0.424706\pi$
$523$	10.7185	0.468689	0.234345	+	0.972154i	$0.424706\pi$
$524$	23.5853	1.03033
$525$	0	0
$526$	20.9761	0.914602
$527$	−27.4331	−1.19500
$528$	−5.15597	−0.224385
$529$	−12.1205	−0.526978
$530$	0	0
$531$	26.8630	1.16575
$532$	4.05259	0.175702
$533$	7.05794	0.305713
$534$	18.2915	0.791551
$535$	0	0
$536$	−21.3470	−0.922050
$537$	−37.4058	−1.61418
$538$	0.460093	0.0198360
$539$	30.2588	1.30334
$540$	0	0
$541$	−12.5366	−0.538990	−0.269495	−	0.963002i	$-0.586857\pi$
$541$	−12.5366	−0.538990	−0.269495	+	0.963002i	$0.586857\pi$
$542$	12.7637	0.548249
$543$	64.2319	2.75646
$544$	−21.2669	−0.911813
$545$	0	0
$546$	−12.0655	−0.516354
$547$	−17.5238	−0.749262	−0.374631	−	0.927174i	$-0.622231\pi$
$547$	−17.5238	−0.749262	−0.374631	+	0.927174i	$0.622231\pi$
$548$	0.530962	0.0226816
$549$	53.8757	2.29936
$550$	0	0
$551$	4.77240	0.203311
$552$	−23.7731	−1.01185
$553$	11.8156	0.502450
$554$	−8.60039	−0.365396
$555$	0	0
$556$	−22.4968	−0.954078
$557$	−28.9383	−1.22615	−0.613077	−	0.790023i	$-0.710068\pi$
$557$	−28.9383	−1.22615	−0.613077	+	0.790023i	$0.710068\pi$
$558$	24.0572	1.01842
$559$	33.7042	1.42553
$560$	0	0
$561$	53.7467	2.26919
$562$	10.2751	0.433428
$563$	1.69109	0.0712709	0.0356354	−	0.999365i	$-0.488654\pi$
$563$	1.69109	0.0712709	0.0356354	+	0.999365i	$0.488654\pi$
$564$	−22.4873	−0.946888
$565$	0	0
$566$	13.6163	0.572337
$567$	6.91032	0.290206
$568$	−39.7429	−1.66758
$569$	−8.00803	−0.335714	−0.167857	−	0.985811i	$-0.553685\pi$
$569$	−8.00803	−0.335714	−0.167857	+	0.985811i	$0.553685\pi$
$570$	0	0
$571$	−0.952819	−0.0398742	−0.0199371	−	0.999801i	$-0.506347\pi$
$571$	−0.952819	−0.0398742	−0.0199371	+	0.999801i	$0.506347\pi$
$572$	32.4316	1.35603
$573$	−9.45907	−0.395158
$574$	−1.64898	−0.0688270
$575$	0	0
$576$	15.9032	0.662634
$577$	−8.25605	−0.343704	−0.171852	−	0.985123i	$-0.554975\pi$
$577$	−8.25605	−0.343704	−0.171852	+	0.985123i	$0.554975\pi$
$578$	−2.87926	−0.119761
$579$	−1.77025	−0.0735689
$580$	0	0
$581$	15.6601	0.649691
$582$	−13.7443	−0.569718
$583$	−41.8175	−1.73190
$584$	39.5309	1.63580
$585$	0	0
$586$	−12.7665	−0.527379
$587$	−13.1511	−0.542805	−0.271403	−	0.962466i	$-0.587487\pi$
$587$	−13.1511	−0.542805	−0.271403	+	0.962466i	$0.587487\pi$
$588$	18.7464	0.773087
$589$	18.4001	0.758161
$590$	0	0
$591$	−2.62539	−0.107994
$592$	2.27244	0.0933966
$593$	−26.5644	−1.09087	−0.545434	−	0.838154i	$-0.683635\pi$
$593$	−26.5644	−1.09087	−0.545434	+	0.838154i	$0.683635\pi$
$594$	−10.8086	−0.443480
$595$	0	0
$596$	11.5372	0.472583
$597$	10.8377	0.443556
$598$	12.1198	0.495617
$599$	13.7241	0.560752	0.280376	−	0.959890i	$-0.409541\pi$
$599$	13.7241	0.560752	0.280376	+	0.959890i	$0.409541\pi$
$600$	0	0
$601$	5.38268	0.219564	0.109782	−	0.993956i	$-0.464985\pi$
$601$	5.38268	0.219564	0.109782	+	0.993956i	$0.464985\pi$
$602$	−7.87445	−0.320939
$603$	30.2690	1.23265
$604$	8.99402	0.365962
$605$	0	0
$606$	4.28438	0.174041
$607$	6.50833	0.264165	0.132082	−	0.991239i	$-0.457834\pi$
$607$	6.50833	0.264165	0.132082	+	0.991239i	$0.457834\pi$
$608$	14.2643	0.578493
$609$	−6.35303	−0.257438
$610$	0	0
$611$	28.9191	1.16994
$612$	18.8052	0.760156
$613$	10.2305	0.413206	0.206603	−	0.978425i	$-0.433759\pi$
$613$	10.2305	0.413206	0.206603	+	0.978425i	$0.433759\pi$
$614$	−11.7382	−0.473716
$615$	0	0
$616$	−19.1135	−0.770103
$617$	−16.4663	−0.662910	−0.331455	−	0.943471i	$-0.607540\pi$
$617$	−16.4663	−0.662910	−0.331455	+	0.943471i	$0.607540\pi$
$618$	9.01427	0.362607
$619$	21.9351	0.881647	0.440823	−	0.897594i	$-0.354686\pi$
$619$	21.9351	0.881647	0.440823	+	0.897594i	$0.354686\pi$
$620$	0	0
$621$	7.73023	0.310203
$622$	12.3869	0.496668
$623$	−10.5179	−0.421390
$624$	4.10792	0.164449
$625$	0	0
$626$	2.28840	0.0914627
$627$	−36.0493	−1.43967
$628$	−24.9850	−0.997011
$629$	−23.6883	−0.944514
$630$	0	0
$631$	14.4810	0.576481	0.288240	−	0.957558i	$-0.406930\pi$
$631$	14.4810	0.576481	0.288240	+	0.957558i	$0.406930\pi$
$632$	25.9349	1.03163
$633$	−65.4029	−2.59953
$634$	−4.65267	−0.184781
$635$	0	0
$636$	−25.9074	−1.02730
$637$	−24.1081	−0.955198
$638$	−8.92296	−0.353263
$639$	56.3536	2.22931
$640$	0	0
$641$	−14.6151	−0.577262	−0.288631	−	0.957440i	$-0.593200\pi$
$641$	−14.6151	−0.577262	−0.288631	+	0.957440i	$0.593200\pi$
$642$	−25.4555	−1.00465
$643$	−36.3205	−1.43234	−0.716170	−	0.697925i	$-0.754107\pi$
$643$	−36.3205	−1.43234	−0.716170	+	0.697925i	$0.754107\pi$
$644$	5.41913	0.213543
$645$	0	0
$646$	−7.51548	−0.295693
$647$	−20.2839	−0.797443	−0.398721	−	0.917072i	$-0.630546\pi$
$647$	−20.2839	−0.797443	−0.398721	+	0.917072i	$0.630546\pi$
$648$	15.1680	0.595854
$649$	−38.4150	−1.50792
$650$	0	0
$651$	−24.4942	−0.960004
$652$	1.02489	0.0401377
$653$	−9.57379	−0.374651	−0.187326	−	0.982298i	$-0.559982\pi$
$653$	−9.57379	−0.374651	−0.187326	+	0.982298i	$0.559982\pi$
$654$	12.6981	0.496534
$655$	0	0
$656$	0.561427	0.0219200
$657$	−56.0529	−2.18683
$658$	−6.75648	−0.263395
$659$	−27.0032	−1.05189	−0.525947	−	0.850517i	$-0.676289\pi$
$659$	−27.0032	−1.05189	−0.525947	+	0.850517i	$0.676289\pi$
$660$	0	0
$661$	−48.7702	−1.89694	−0.948470	−	0.316868i	$-0.897369\pi$
$661$	−48.7702	−1.89694	−0.948470	+	0.316868i	$0.897369\pi$
$662$	−9.95920	−0.387075
$663$	−42.8217	−1.66306
$664$	34.3735	1.33395
$665$	0	0
$666$	20.7732	0.804946
$667$	6.38166	0.247099
$668$	12.1153	0.468754
$669$	11.4713	0.443508
$670$	0	0
$671$	−77.0441	−2.97425
$672$	−18.9886	−0.732502
$673$	42.9571	1.65587	0.827937	−	0.560821i	$-0.189514\pi$
$673$	42.9571	1.65587	0.827937	+	0.560821i	$0.189514\pi$
$674$	15.3723	0.592120
$675$	0	0
$676$	−8.76233	−0.337013
$677$	−35.6547	−1.37032	−0.685160	−	0.728393i	$-0.740268\pi$
$677$	−35.6547	−1.37032	−0.685160	+	0.728393i	$0.740268\pi$
$678$	−29.3648	−1.12775
$679$	7.90315	0.303295
$680$	0	0
$681$	35.7744	1.37088
$682$	−34.4026	−1.31734
$683$	−0.933116	−0.0357047	−0.0178523	−	0.999841i	$-0.505683\pi$
$683$	−0.933116	−0.0357047	−0.0178523	+	0.999841i	$0.505683\pi$
$684$	−12.6131	−0.482275
$685$	0	0
$686$	12.8859	0.491985
$687$	−13.8913	−0.529984
$688$	2.68101	0.102213
$689$	33.3173	1.26929
$690$	0	0
$691$	−19.3165	−0.734836	−0.367418	−	0.930056i	$-0.619758\pi$
$691$	−19.3165	−0.734836	−0.367418	+	0.930056i	$0.619758\pi$
$692$	16.6198	0.631792
$693$	27.1020	1.02952
$694$	28.4863	1.08132
$695$	0	0
$696$	−13.9447	−0.528573
$697$	−5.85241	−0.221676
$698$	−28.7749	−1.08915
$699$	17.5518	0.663869
$700$	0	0
$701$	−41.4399	−1.56517	−0.782583	−	0.622547i	$-0.786098\pi$
$701$	−41.4399	−1.56517	−0.782583	+	0.622547i	$0.786098\pi$
$702$	8.61152	0.325021
$703$	15.8883	0.599240
$704$	−22.7421	−0.857126
$705$	0	0
$706$	−16.1536	−0.607950
$707$	−2.46358	−0.0926525
$708$	−23.7994	−0.894438
$709$	51.9958	1.95274	0.976371	−	0.216102i	$-0.0693343\pi$
$709$	51.9958	1.95274	0.976371	+	0.216102i	$0.0693343\pi$
$710$	0	0
$711$	−36.7744	−1.37915
$712$	−23.0864	−0.865201
$713$	24.6046	0.921448
$714$	10.0046	0.374414
$715$	0	0
$716$	18.7160	0.699448
$717$	44.9057	1.67704
$718$	−20.8491	−0.778081
$719$	−2.29676	−0.0856549	−0.0428274	−	0.999082i	$-0.513637\pi$
$719$	−2.29676	−0.0856549	−0.0428274	+	0.999082i	$0.513637\pi$
$720$	0	0
$721$	−5.18334	−0.193037
$722$	−10.7004	−0.398228
$723$	14.4401	0.537033
$724$	−32.1384	−1.19441
$725$	0	0
$726$	43.4752	1.61352
$727$	17.7008	0.656485	0.328242	−	0.944594i	$-0.393544\pi$
$727$	17.7008	0.656485	0.328242	+	0.944594i	$0.393544\pi$
$728$	15.2283	0.564399
$729$	−39.9662	−1.48023
$730$	0	0
$731$	−27.9473	−1.03367
$732$	−47.7315	−1.76421
$733$	−39.3110	−1.45198	−0.725992	−	0.687703i	$-0.758619\pi$
$733$	−39.3110	−1.45198	−0.725992	+	0.687703i	$0.758619\pi$
$734$	5.11939	0.188960
$735$	0	0
$736$	19.0742	0.703084
$737$	−43.2857	−1.59445
$738$	5.13221	0.188919
$739$	34.8854	1.28328	0.641640	−	0.767006i	$-0.278254\pi$
$739$	34.8854	1.28328	0.641640	+	0.767006i	$0.278254\pi$
$740$	0	0
$741$	28.7216	1.05511
$742$	−7.78408	−0.285762
$743$	18.7028	0.686138	0.343069	−	0.939310i	$-0.388534\pi$
$743$	18.7028	0.686138	0.343069	+	0.939310i	$0.388534\pi$
$744$	−53.7640	−1.97109
$745$	0	0
$746$	22.9373	0.839793
$747$	−48.7400	−1.78330
$748$	−26.8921	−0.983272
$749$	14.6373	0.534834
$750$	0	0
$751$	40.0760	1.46239	0.731197	−	0.682166i	$-0.238962\pi$
$751$	40.0760	1.46239	0.731197	+	0.682166i	$0.238962\pi$
$752$	2.30038	0.0838861
$753$	−32.3291	−1.17814
$754$	7.10921	0.258902
$755$	0	0
$756$	3.85046	0.140040
$757$	−22.7584	−0.827168	−0.413584	−	0.910466i	$-0.635723\pi$
$757$	−22.7584	−0.827168	−0.413584	+	0.910466i	$0.635723\pi$
$758$	18.0829	0.656801
$759$	−48.2051	−1.74973
$760$	0	0
$761$	−21.8639	−0.792566	−0.396283	−	0.918128i	$-0.629700\pi$
$761$	−21.8639	−0.792566	−0.396283	+	0.918128i	$0.629700\pi$
$762$	−1.75945	−0.0637382
$763$	−7.30158	−0.264335
$764$	4.73283	0.171228
$765$	0	0
$766$	0.823060	0.0297384
$767$	30.6064	1.10513
$768$	−39.2461	−1.41617
$769$	15.9462	0.575034	0.287517	−	0.957776i	$-0.407170\pi$
$769$	15.9462	0.575034	0.287517	+	0.957776i	$0.407170\pi$
$770$	0	0
$771$	59.9355	2.15852
$772$	0.885740	0.0318785
$773$	18.4449	0.663418	0.331709	−	0.943382i	$-0.392375\pi$
$773$	18.4449	0.663418	0.331709	+	0.943382i	$0.392375\pi$
$774$	24.5081	0.880927
$775$	0	0
$776$	17.3472	0.622727
$777$	−21.1506	−0.758773
$778$	−14.9032	−0.534305
$779$	3.92536	0.140641
$780$	0	0
$781$	−80.5875	−2.88365
$782$	−10.0497	−0.359377
$783$	4.53437	0.162045
$784$	−1.91769	−0.0684889
$785$	0	0
$786$	−39.0529	−1.39297
$787$	46.4201	1.65470	0.827349	−	0.561688i	$-0.189848\pi$
$787$	46.4201	1.65470	0.827349	+	0.561688i	$0.189848\pi$
$788$	1.31361	0.0467954
$789$	66.4713	2.36644
$790$	0	0
$791$	16.8852	0.600368
$792$	59.4880	2.11381
$793$	61.3835	2.17979
$794$	−3.37008	−0.119600
$795$	0	0
$796$	−5.42261	−0.192199
$797$	44.8723	1.58946	0.794729	−	0.606965i	$-0.207613\pi$
$797$	44.8723	1.58946	0.794729	+	0.606965i	$0.207613\pi$
$798$	−6.71035	−0.237544
$799$	−23.9795	−0.848335
$800$	0	0
$801$	32.7355	1.15665
$802$	25.6123	0.904402
$803$	80.1575	2.82870
$804$	−26.8170	−0.945764
$805$	0	0
$806$	27.4096	0.965463
$807$	1.45799	0.0513236
$808$	−5.40748	−0.190235
$809$	26.6725	0.937755	0.468877	−	0.883263i	$-0.344659\pi$
$809$	26.6725	0.937755	0.468877	+	0.883263i	$0.344659\pi$
$810$	0	0
$811$	−30.1724	−1.05950	−0.529748	−	0.848155i	$-0.677713\pi$
$811$	−30.1724	−1.05950	−0.529748	+	0.848155i	$0.677713\pi$
$812$	3.17873	0.111552
$813$	40.4470	1.41854
$814$	−29.7064	−1.04121
$815$	0	0
$816$	−3.40627	−0.119243
$817$	18.7450	0.655804
$818$	19.8173	0.692897
$819$	−21.5930	−0.754521
$820$	0	0
$821$	−26.7513	−0.933628	−0.466814	−	0.884355i	$-0.654598\pi$
$821$	−26.7513	−0.933628	−0.466814	+	0.884355i	$0.654598\pi$
$822$	−0.879177	−0.0306648
$823$	−21.0257	−0.732908	−0.366454	−	0.930436i	$-0.619428\pi$
$823$	−21.0257	−0.732908	−0.366454	+	0.930436i	$0.619428\pi$
$824$	−11.3773	−0.396346
$825$	0	0
$826$	−7.15071	−0.248805
$827$	4.29301	0.149283	0.0746413	−	0.997210i	$-0.476219\pi$
$827$	4.29301	0.149283	0.0746413	+	0.997210i	$0.476219\pi$
$828$	−16.8663	−0.586144
$829$	−5.07828	−0.176376	−0.0881879	−	0.996104i	$-0.528108\pi$
$829$	−5.07828	−0.176376	−0.0881879	+	0.996104i	$0.528108\pi$
$830$	0	0
$831$	−27.2538	−0.945424
$832$	18.1194	0.628177
$833$	19.9903	0.692624
$834$	37.2506	1.28988
$835$	0	0
$836$	18.0372	0.623830
$837$	17.4823	0.604277
$838$	7.64171	0.263979
$839$	−13.7575	−0.474962	−0.237481	−	0.971392i	$-0.576322\pi$
$839$	−13.7575	−0.474962	−0.237481	+	0.971392i	$0.576322\pi$
$840$	0	0
$841$	−25.2567	−0.870920
$842$	10.9742	0.378195
$843$	32.5607	1.12145
$844$	32.7243	1.12642
$845$	0	0
$846$	21.0286	0.722979
$847$	−24.9989	−0.858972
$848$	2.65024	0.0910097
$849$	43.1488	1.48086
$850$	0	0
$851$	21.2459	0.728299
$852$	−49.9268	−1.71046
$853$	−18.6226	−0.637626	−0.318813	−	0.947818i	$-0.603284\pi$
$853$	−18.6226	−0.637626	−0.318813	+	0.947818i	$0.603284\pi$
$854$	−14.3413	−0.490749
$855$	0	0
$856$	32.1284	1.09812
$857$	−3.11710	−0.106478	−0.0532391	−	0.998582i	$-0.516955\pi$
$857$	−3.11710	−0.106478	−0.0532391	+	0.998582i	$0.516955\pi$
$858$	−53.7008	−1.83331
$859$	−46.7701	−1.59577	−0.797887	−	0.602807i	$-0.794049\pi$
$859$	−46.7701	−1.59577	−0.797887	+	0.602807i	$0.794049\pi$
$860$	0	0
$861$	−5.22544	−0.178083
$862$	−22.3016	−0.759594
$863$	−48.9641	−1.66676	−0.833379	−	0.552702i	$-0.813597\pi$
$863$	−48.9641	−1.66676	−0.833379	+	0.552702i	$0.813597\pi$
$864$	13.5528	0.461076
$865$	0	0
$866$	24.0124	0.815974
$867$	−9.12409	−0.309870
$868$	12.2556	0.415984
$869$	52.5887	1.78395
$870$	0	0
$871$	34.4871	1.16855
$872$	−16.0268	−0.542734
$873$	−24.5975	−0.832498
$874$	6.74059	0.228004
$875$	0	0
$876$	49.6604	1.67787
$877$	43.4332	1.46663	0.733317	−	0.679887i	$-0.237971\pi$
$877$	43.4332	1.46663	0.733317	+	0.679887i	$0.237971\pi$
$878$	−28.3097	−0.955406
$879$	−40.4557	−1.36454
$880$	0	0
$881$	−43.3084	−1.45910	−0.729548	−	0.683930i	$-0.760269\pi$
$881$	−43.3084	−1.45910	−0.729548	+	0.683930i	$0.760269\pi$
$882$	−17.5303	−0.590277
$883$	4.15547	0.139843	0.0699214	−	0.997553i	$-0.477725\pi$
$883$	4.15547	0.139843	0.0699214	+	0.997553i	$0.477725\pi$
$884$	21.4258	0.720627
$885$	0	0
$886$	6.22994	0.209299
$887$	12.1615	0.408344	0.204172	−	0.978935i	$-0.434550\pi$
$887$	12.1615	0.408344	0.204172	+	0.978935i	$0.434550\pi$
$888$	−46.4249	−1.55792
$889$	1.01171	0.0339317
$890$	0	0
$891$	30.7564	1.03038
$892$	−5.73967	−0.192179
$893$	16.0837	0.538220
$894$	−19.1035	−0.638917
$895$	0	0
$896$	10.2321	0.341829
$897$	38.4065	1.28236
$898$	−3.08925	−0.103090
$899$	14.4325	0.481350
$900$	0	0
$901$	−27.6266	−0.920375
$902$	−7.33924	−0.244370
$903$	−24.9534	−0.830396
$904$	37.0625	1.23268
$905$	0	0
$906$	−14.8925	−0.494769
$907$	46.0244	1.52822	0.764108	−	0.645088i	$-0.223179\pi$
$907$	46.0244	1.52822	0.764108	+	0.645088i	$0.223179\pi$
$908$	−17.8997	−0.594021
$909$	7.66755	0.254317
$910$	0	0
$911$	40.1821	1.33129	0.665647	−	0.746267i	$-0.268156\pi$
$911$	40.1821	1.33129	0.665647	+	0.746267i	$0.268156\pi$
$912$	2.28467	0.0756530
$913$	69.6998	2.30673
$914$	−13.7335	−0.454265
$915$	0	0
$916$	6.95047	0.229650
$917$	22.4560	0.741563
$918$	−7.14063	−0.235676
$919$	9.95664	0.328439	0.164220	−	0.986424i	$-0.447489\pi$
$919$	9.95664	0.328439	0.164220	+	0.986424i	$0.447489\pi$
$920$	0	0
$921$	−37.1972	−1.22569
$922$	−2.05907	−0.0678119
$923$	64.2066	2.11339
$924$	−24.0112	−0.789909
$925$	0	0
$926$	24.0287	0.789633
$927$	16.1324	0.529858
$928$	11.1885	0.367280
$929$	8.56011	0.280848	0.140424	−	0.990091i	$-0.455153\pi$
$929$	8.56011	0.280848	0.140424	+	0.990091i	$0.455153\pi$
$930$	0	0
$931$	−13.4080	−0.439430
$932$	−8.78200	−0.287664
$933$	39.2528	1.28508
$934$	28.6589	0.937748
$935$	0	0
$936$	−47.3960	−1.54919
$937$	−36.6021	−1.19574	−0.597870	−	0.801593i	$-0.703986\pi$
$937$	−36.6021	−1.19574	−0.597870	+	0.801593i	$0.703986\pi$
$938$	−8.05737	−0.263082
$939$	7.25170	0.236650
$940$	0	0
$941$	53.9573	1.75896	0.879479	−	0.475939i	$-0.157892\pi$
$941$	53.9573	1.75896	0.879479	+	0.475939i	$0.157892\pi$
$942$	41.3707	1.34793
$943$	5.24899	0.170931
$944$	2.43460	0.0792395
$945$	0	0
$946$	−35.0475	−1.13949
$947$	29.5042	0.958757	0.479378	−	0.877608i	$-0.340862\pi$
$947$	29.5042	0.958757	0.479378	+	0.877608i	$0.340862\pi$
$948$	32.5805	1.05817
$949$	−63.8640	−2.07311
$950$	0	0
$951$	−14.7439	−0.478102
$952$	−12.6272	−0.409251
$953$	2.43741	0.0789553	0.0394777	−	0.999220i	$-0.487431\pi$
$953$	2.43741	0.0789553	0.0394777	+	0.999220i	$0.487431\pi$
$954$	24.2269	0.784374
$955$	0	0
$956$	−22.4685	−0.726684
$957$	−28.2760	−0.914033
$958$	12.1284	0.391852
$959$	0.505539	0.0163247
$960$	0	0
$961$	24.6445	0.794985
$962$	23.6680	0.763088
$963$	−45.5565	−1.46804
$964$	−7.22508	−0.232704
$965$	0	0
$966$	−8.97308	−0.288704
$967$	34.4353	1.10736	0.553682	−	0.832728i	$-0.313222\pi$
$967$	34.4353	1.10736	0.553682	+	0.832728i	$0.313222\pi$
$968$	−54.8718	−1.76365
$969$	−23.8158	−0.765074
$970$	0	0
$971$	7.50069	0.240709	0.120354	−	0.992731i	$-0.461597\pi$
$971$	7.50069	0.240709	0.120354	+	0.992731i	$0.461597\pi$
$972$	28.2905	0.907417
$973$	−21.4197	−0.686683
$974$	23.4016	0.749837
$975$	0	0
$976$	4.88277	0.156294
$977$	−18.9023	−0.604739	−0.302369	−	0.953191i	$-0.597778\pi$
$977$	−18.9023	−0.604739	−0.302369	+	0.953191i	$0.597778\pi$
$978$	−1.69703	−0.0542649
$979$	−46.8128	−1.49614
$980$	0	0
$981$	22.7252	0.725559
$982$	−0.992441	−0.0316701
$983$	51.0261	1.62748	0.813740	−	0.581229i	$-0.197428\pi$
$983$	51.0261	1.62748	0.813740	+	0.581229i	$0.197428\pi$
$984$	−11.4697	−0.365641
$985$	0	0
$986$	−5.89492	−0.187732
$987$	−21.4106	−0.681508
$988$	−14.3708	−0.457197
$989$	25.0658	0.797046
$990$	0	0
$991$	0.923477	0.0293352	0.0146676	−	0.999892i	$-0.495331\pi$
$991$	0.923477	0.0293352	0.0146676	+	0.999892i	$0.495331\pi$
$992$	43.1373	1.36961
$993$	−31.5597	−1.00152
$994$	−15.0009	−0.475799
$995$	0	0
$996$	43.1815	1.36826
$997$	−8.11305	−0.256943	−0.128471	−	0.991713i	$-0.541007\pi$
$997$	−8.11305	−0.256943	−0.128471	+	0.991713i	$0.541007\pi$
$998$	−23.6213	−0.747718
$999$	15.0959	0.477612

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4925.2.a.s.1.29		49
5.2	odd	4		985.2.b.a.789.60	yes	98
5.3	odd	4		985.2.b.a.789.39	✓	98
5.4	even	2		4925.2.a.r.1.21		49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
985.2.b.a.789.39	✓	98	5.3	odd	4
985.2.b.a.789.60	yes	98	5.2	odd	4
4925.2.a.r.1.21		49	5.4	even	2
4925.2.a.s.1.29		49	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:	\( N \)	\(=\)	\( 4925 = 5^{2} \cdot 197 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4925.a (trivial)

\(f(q)\)	\(=\)	\(q+0.828486 q^{2} +2.62539 q^{3} -1.31361 q^{4} +2.17510 q^{6} -1.25071 q^{7} -2.74528 q^{8} +3.89268 q^{9} -5.56666 q^{11} -3.44874 q^{12} +4.43513 q^{13} -1.03620 q^{14} +0.352794 q^{16} -3.67759 q^{17} +3.22503 q^{18} +2.46665 q^{19} -3.28361 q^{21} -4.61190 q^{22} +3.29841 q^{23} -7.20743 q^{24} +3.67445 q^{26} +2.34362 q^{27} +1.64295 q^{28} +1.93477 q^{29} +7.45953 q^{31} +5.78285 q^{32} -14.6147 q^{33} -3.04683 q^{34} -5.11346 q^{36} +6.44125 q^{37} +2.04359 q^{38} +11.6440 q^{39} +1.59137 q^{41} -2.72043 q^{42} +7.59936 q^{43} +7.31242 q^{44} +2.73269 q^{46} +6.52045 q^{47} +0.926223 q^{48} -5.43571 q^{49} -9.65511 q^{51} -5.82604 q^{52} +7.51214 q^{53} +1.94166 q^{54} +3.43356 q^{56} +6.47593 q^{57} +1.60293 q^{58} +6.90090 q^{59} +13.8403 q^{61} +6.18011 q^{62} -4.86863 q^{63} +4.08542 q^{64} -12.1080 q^{66} +7.77589 q^{67} +4.83092 q^{68} +8.65961 q^{69} +14.4768 q^{71} -10.6865 q^{72} -14.3996 q^{73} +5.33649 q^{74} -3.24022 q^{76} +6.96230 q^{77} +9.64686 q^{78} -9.44708 q^{79} -5.52510 q^{81} +1.31843 q^{82} -12.5209 q^{83} +4.31339 q^{84} +6.29596 q^{86} +5.07953 q^{87} +15.2820 q^{88} +8.40950 q^{89} -5.54709 q^{91} -4.33282 q^{92} +19.5842 q^{93} +5.40210 q^{94} +15.1822 q^{96} -6.31891 q^{97} -4.50341 q^{98} -21.6692 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q + 5 q^{2} + 22 q^{3} + 49 q^{4} + 2 q^{6} + 32 q^{7} + 15 q^{8} + 51 q^{9} - 2 q^{11} + 44 q^{12} + 32 q^{13} - 8 q^{14} + 49 q^{16} + 14 q^{17} + 25 q^{18} + 4 q^{19} + 10 q^{21} + 38 q^{22} + 24 q^{23}+ \cdots + 22 q^{99}+O(q^{100}) \) 49 * q + 5 * q^2 + 22 * q^3 + 49 * q^4 + 2 * q^6 + 32 * q^7 + 15 * q^8 + 51 * q^9 - 2 * q^11 + 44 * q^12 + 32 * q^13 - 8 * q^14 + 49 * q^16 + 14 * q^17 + 25 * q^18 + 4 * q^19 + 10 * q^21 + 38 * q^22 + 24 * q^23 - 2 * q^26 + 88 * q^27 + 80 * q^28 + 6 * q^29 - 2 * q^31 + 35 * q^32 + 48 * q^33 - 12 * q^34 + 43 * q^36 + 60 * q^37 + 34 * q^38 - 2 * q^41 + 36 * q^42 + 80 * q^43 - 8 * q^44 - 22 * q^46 + 48 * q^47 + 88 * q^48 + 65 * q^49 + 16 * q^51 + 96 * q^52 + 18 * q^53 + 42 * q^54 - 22 * q^56 + 40 * q^57 + 52 * q^58 + 8 * q^59 + 16 * q^61 + 38 * q^62 + 82 * q^63 + 55 * q^64 + 42 * q^66 + 146 * q^67 - 68 * q^68 - 6 * q^69 + 75 * q^72 + 78 * q^73 - 24 * q^74 - 4 * q^76 + 32 * q^77 - 38 * q^78 + 10 * q^79 + 41 * q^81 + 96 * q^82 + 64 * q^83 + 42 * q^84 - 28 * q^86 + 56 * q^87 + 72 * q^88 + 4 * q^89 + 60 * q^92 - 28 * q^93 - 8 * q^94 - 34 * q^96 + 124 * q^97 - 19 * q^98 + 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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