LMFDB - Embedded newform 7225.2.a.bw.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7225,2,Mod(1,7225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7225, 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("7225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7225 = 5^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7225.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:	$57.6919154604$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 21 x^{13} - 2 x^{12} + 171 x^{11} + 30 x^{10} - 678 x^{9} - 153 x^{8} + 1350 x^{7} + 301 x^{6} + \cdots + 3 $ x^15 - 21x^13 - 2x^12 + 171x^11 + 30x^10 - 678x^9 - 153x^8 + 1350x^7 + 301x^6 - 1284x^5 - 183x^4 + 555x^3 + 18x^2 - 81*x + 3
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			1.8
Root			$0.0377149$ of defining polynomial
Character	$\chi$	$=$	7225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0377149 q^{2} -2.71683 q^{3} -1.99858 q^{4} -0.102465 q^{6} -0.366049 q^{7} -0.150806 q^{8} +4.38117 q^{9} +O(q^{10})$	\(q+0.0377149 q^{2} -2.71683 q^{3} -1.99858 q^{4} -0.102465 q^{6} -0.366049 q^{7} -0.150806 q^{8} +4.38117 q^{9} +1.43996 q^{11} +5.42980 q^{12} +4.61482 q^{13} -0.0138055 q^{14} +3.99147 q^{16} +0.165236 q^{18} -3.66987 q^{19} +0.994492 q^{21} +0.0543079 q^{22} +8.56831 q^{23} +0.409715 q^{24} +0.174047 q^{26} -3.75241 q^{27} +0.731577 q^{28} +8.61745 q^{29} +9.41475 q^{31} +0.452150 q^{32} -3.91212 q^{33} -8.75611 q^{36} +7.58559 q^{37} -0.138409 q^{38} -12.5377 q^{39} -6.18104 q^{41} +0.0375072 q^{42} +7.04354 q^{43} -2.87787 q^{44} +0.323153 q^{46} -3.75423 q^{47} -10.8441 q^{48} -6.86601 q^{49} -9.22307 q^{52} +10.3687 q^{53} -0.141522 q^{54} +0.0552024 q^{56} +9.97043 q^{57} +0.325007 q^{58} +6.93936 q^{59} +2.29314 q^{61} +0.355077 q^{62} -1.60372 q^{63} -7.96588 q^{64} -0.147545 q^{66} -5.77178 q^{67} -23.2786 q^{69} +6.76686 q^{71} -0.660708 q^{72} +4.33935 q^{73} +0.286090 q^{74} +7.33453 q^{76} -0.527094 q^{77} -0.472858 q^{78} +4.06429 q^{79} -2.94884 q^{81} -0.233117 q^{82} +11.6274 q^{83} -1.98757 q^{84} +0.265647 q^{86} -23.4122 q^{87} -0.217154 q^{88} -10.6763 q^{89} -1.68925 q^{91} -17.1244 q^{92} -25.5783 q^{93} -0.141590 q^{94} -1.22842 q^{96} +8.52612 q^{97} -0.258951 q^{98} +6.30870 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 9 q^{3} + 12 q^{4} + 9 q^{6} + 12 q^{7} + 6 q^{8} + 12 q^{9}+O(q^{10}) $ 15 * q + 9 * q^3 + 12 * q^4 + 9 * q^6 + 12 * q^7 + 6 * q^8 + 12 * q^9	$ 15 q + 9 q^{3} + 12 q^{4} + 9 q^{6} + 12 q^{7} + 6 q^{8} + 12 q^{9} + 6 q^{11} + 24 q^{12} + 6 q^{16} + 12 q^{18} + 6 q^{19} + 30 q^{21} + 12 q^{22} + 36 q^{23} + 18 q^{24} + 36 q^{26} + 36 q^{27} + 24 q^{28} - 18 q^{29} + 12 q^{32} + 12 q^{33} - 9 q^{36} + 12 q^{37} - 6 q^{38} + 9 q^{39} - 18 q^{41} + 36 q^{42} - 3 q^{43} - 12 q^{44} + 21 q^{46} - 3 q^{47} - 12 q^{48} + 15 q^{49} - 27 q^{52} + 21 q^{54} - 6 q^{56} + 39 q^{57} + 18 q^{58} - 12 q^{59} - 15 q^{61} + 54 q^{62} + 60 q^{63} - 36 q^{64} + 18 q^{66} - 24 q^{67} + 42 q^{69} + 6 q^{71} + 66 q^{72} - 9 q^{73} - 36 q^{74} - 18 q^{76} + 30 q^{77} + 30 q^{78} - 9 q^{79} + 51 q^{81} - 36 q^{82} + 15 q^{83} + 9 q^{84} - 36 q^{86} - 51 q^{87} + 30 q^{88} - 24 q^{89} + 27 q^{91} + 15 q^{92} - 42 q^{93} - 57 q^{94} + 42 q^{96} + 48 q^{97} + 12 q^{99}+O(q^{100}) $ 15 * q + 9 * q^3 + 12 * q^4 + 9 * q^6 + 12 * q^7 + 6 * q^8 + 12 * q^9 + 6 * q^11 + 24 * q^12 + 6 * q^16 + 12 * q^18 + 6 * q^19 + 30 * q^21 + 12 * q^22 + 36 * q^23 + 18 * q^24 + 36 * q^26 + 36 * q^27 + 24 * q^28 - 18 * q^29 + 12 * q^32 + 12 * q^33 - 9 * q^36 + 12 * q^37 - 6 * q^38 + 9 * q^39 - 18 * q^41 + 36 * q^42 - 3 * q^43 - 12 * q^44 + 21 * q^46 - 3 * q^47 - 12 * q^48 + 15 * q^49 - 27 * q^52 + 21 * q^54 - 6 * q^56 + 39 * q^57 + 18 * q^58 - 12 * q^59 - 15 * q^61 + 54 * q^62 + 60 * q^63 - 36 * q^64 + 18 * q^66 - 24 * q^67 + 42 * q^69 + 6 * q^71 + 66 * q^72 - 9 * q^73 - 36 * q^74 - 18 * q^76 + 30 * q^77 + 30 * q^78 - 9 * q^79 + 51 * q^81 - 36 * q^82 + 15 * q^83 + 9 * q^84 - 36 * q^86 - 51 * q^87 + 30 * q^88 - 24 * q^89 + 27 * q^91 + 15 * q^92 - 42 * q^93 - 57 * q^94 + 42 * q^96 + 48 * q^97 + 12 * 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 $

$2$	0.0377149	0.0266685	0.0133342	−	0.999911i	$-0.495755\pi$
$2$	0.0377149	0.0266685	0.0133342	+	0.999911i	$0.495755\pi$
$3$	−2.71683	−1.56856	−0.784282	−	0.620405i	$-0.786968\pi$
$3$	−2.71683	−1.56856	−0.784282	+	0.620405i	$0.786968\pi$
$4$	−1.99858	−0.999289
$5$	0	0
$5$	0	0
$6$	−0.102465	−0.0418312
$7$	−0.366049	−0.138353	−0.0691767	−	0.997604i	$-0.522037\pi$
$7$	−0.366049	−0.138353	−0.0691767	+	0.997604i	$0.522037\pi$
$8$	−0.150806	−0.0533180
$9$	4.38117	1.46039
$10$	0	0
$11$	1.43996	0.434164	0.217082	−	0.976153i	$-0.430346\pi$
$11$	1.43996	0.434164	0.217082	+	0.976153i	$0.430346\pi$
$12$	5.42980	1.56745
$13$	4.61482	1.27992	0.639960	−	0.768408i	$-0.278951\pi$
$13$	4.61482	1.27992	0.639960	+	0.768408i	$0.278951\pi$
$14$	−0.0138055	−0.00368968
$15$	0	0
$16$	3.99147	0.997867
$17$	0	0
$17$	0	0
$18$	0.165236	0.0389464
$19$	−3.66987	−0.841927	−0.420963	−	0.907078i	$-0.638308\pi$
$19$	−3.66987	−0.841927	−0.420963	+	0.907078i	$0.638308\pi$
$20$	0	0
$21$	0.994492	0.217016
$22$	0.0543079	0.0115785
$23$	8.56831	1.78662	0.893308	−	0.449445i	$-0.148378\pi$
$23$	8.56831	1.78662	0.893308	+	0.449445i	$0.148378\pi$
$24$	0.409715	0.0836327
$25$	0	0
$26$	0.174047	0.0341335
$27$	−3.75241	−0.722152
$28$	0.731577	0.138255
$29$	8.61745	1.60022	0.800110	−	0.599853i	$-0.204774\pi$
$29$	8.61745	1.60022	0.800110	+	0.599853i	$0.204774\pi$
$30$	0	0
$31$	9.41475	1.69094	0.845470	−	0.534024i	$-0.179321\pi$
$31$	9.41475	1.69094	0.845470	+	0.534024i	$0.179321\pi$
$32$	0.452150	0.0799296
$33$	−3.91212	−0.681013
$34$	0	0
$35$	0	0
$36$	−8.75611	−1.45935
$37$	7.58559	1.24706	0.623532	−	0.781798i	$-0.285697\pi$
$37$	7.58559	1.24706	0.623532	+	0.781798i	$0.285697\pi$
$38$	−0.138409	−0.0224529
$39$	−12.5377	−2.00763
$40$	0	0
$41$	−6.18104	−0.965316	−0.482658	−	0.875809i	$-0.660329\pi$
$41$	−6.18104	−0.965316	−0.482658	+	0.875809i	$0.660329\pi$
$42$	0.0375072	0.00578749
$43$	7.04354	1.07413	0.537065	−	0.843541i	$-0.319533\pi$
$43$	7.04354	1.07413	0.537065	+	0.843541i	$0.319533\pi$
$44$	−2.87787	−0.433855
$45$	0	0
$46$	0.323153	0.0476463
$47$	−3.75423	−0.547610	−0.273805	−	0.961785i	$-0.588282\pi$
$47$	−3.75423	−0.547610	−0.273805	+	0.961785i	$0.588282\pi$
$48$	−10.8441	−1.56522
$49$	−6.86601	−0.980858
$50$	0	0
$51$	0	0
$52$	−9.22307	−1.27901
$53$	10.3687	1.42425	0.712124	−	0.702054i	$-0.247734\pi$
$53$	10.3687	1.42425	0.712124	+	0.702054i	$0.247734\pi$
$54$	−0.141522	−0.0192587
$55$	0	0
$56$	0.0552024	0.00737673
$57$	9.97043	1.32062
$58$	0.325007	0.0426754
$59$	6.93936	0.903428	0.451714	−	0.892163i	$-0.350813\pi$
$59$	6.93936	0.903428	0.451714	+	0.892163i	$0.350813\pi$
$60$	0	0
$61$	2.29314	0.293607	0.146803	−	0.989166i	$-0.453102\pi$
$61$	2.29314	0.293607	0.146803	+	0.989166i	$0.453102\pi$
$62$	0.355077	0.0450948
$63$	−1.60372	−0.202050
$64$	−7.96588	−0.995735
$65$	0	0
$66$	−0.147545	−0.0181616
$67$	−5.77178	−0.705135	−0.352567	−	0.935786i	$-0.614691\pi$
$67$	−5.77178	−0.705135	−0.352567	+	0.935786i	$0.614691\pi$
$68$	0	0
$69$	−23.2786	−2.80242
$70$	0	0
$71$	6.76686	0.803079	0.401539	−	0.915842i	$-0.368475\pi$
$71$	6.76686	0.803079	0.401539	+	0.915842i	$0.368475\pi$
$72$	−0.660708	−0.0778651
$73$	4.33935	0.507883	0.253941	−	0.967220i	$-0.418273\pi$
$73$	4.33935	0.507883	0.253941	+	0.967220i	$0.418273\pi$
$74$	0.286090	0.0332573
$75$	0	0
$76$	7.33453	0.841328
$77$	−0.527094	−0.0600680
$78$	−0.472858	−0.0535406
$79$	4.06429	0.457268	0.228634	−	0.973512i	$-0.426574\pi$
$79$	4.06429	0.457268	0.228634	+	0.973512i	$0.426574\pi$
$80$	0	0
$81$	−2.94884	−0.327649
$82$	−0.233117	−0.0257435
$83$	11.6274	1.27628	0.638139	−	0.769921i	$-0.279705\pi$
$83$	11.6274	1.27628	0.638139	+	0.769921i	$0.279705\pi$
$84$	−1.98757	−0.216862
$85$	0	0
$86$	0.265647	0.0286454
$87$	−23.4122	−2.51005
$88$	−0.217154	−0.0231487
$89$	−10.6763	−1.13169	−0.565843	−	0.824513i	$-0.691449\pi$
$89$	−10.6763	−1.13169	−0.565843	+	0.824513i	$0.691449\pi$
$90$	0	0
$91$	−1.68925	−0.177081
$92$	−17.1244	−1.78535
$93$	−25.5783	−2.65235
$94$	−0.141590	−0.0146039
$95$	0	0
$96$	−1.22842	−0.125375
$97$	8.52612	0.865696	0.432848	−	0.901467i	$-0.357509\pi$
$97$	8.52612	0.865696	0.432848	+	0.901467i	$0.357509\pi$
$98$	−0.258951	−0.0261580
$99$	6.30870	0.634048
$100$	0	0
$101$	−10.2422	−1.01914	−0.509569	−	0.860430i	$-0.670195\pi$
$101$	−10.2422	−1.01914	−0.509569	+	0.860430i	$0.670195\pi$
$102$	0	0
$103$	6.77080	0.667146	0.333573	−	0.942724i	$-0.391746\pi$
$103$	6.77080	0.667146	0.333573	+	0.942724i	$0.391746\pi$
$104$	−0.695942	−0.0682428
$105$	0	0
$106$	0.391054	0.0379825
$107$	−14.9548	−1.44573	−0.722866	−	0.690988i	$-0.757176\pi$
$107$	−14.9548	−1.44573	−0.722866	+	0.690988i	$0.757176\pi$
$108$	7.49949	0.721639
$109$	−11.8358	−1.13367	−0.566834	−	0.823832i	$-0.691832\pi$
$109$	−11.8358	−1.13367	−0.566834	+	0.823832i	$0.691832\pi$
$110$	0	0
$111$	−20.6088	−1.95610
$112$	−1.46107	−0.138058
$113$	−6.67548	−0.627976	−0.313988	−	0.949427i	$-0.601665\pi$
$113$	−6.67548	−0.627976	−0.313988	+	0.949427i	$0.601665\pi$
$114$	0.376034	0.0352188
$115$	0	0
$116$	−17.2226	−1.59908
$117$	20.2183	1.86918
$118$	0.261718	0.0240931
$119$	0	0
$120$	0	0
$121$	−8.92652	−0.811502
$122$	0.0864857	0.00783005
$123$	16.7928	1.51416
$124$	−18.8161	−1.68974
$125$	0	0
$126$	−0.0604843	−0.00538837
$127$	4.84668	0.430073	0.215037	−	0.976606i	$-0.431013\pi$
$127$	4.84668	0.430073	0.215037	+	0.976606i	$0.431013\pi$
$128$	−1.20473	−0.106484
$129$	−19.1361	−1.68484
$130$	0	0
$131$	19.7350	1.72425	0.862126	−	0.506694i	$-0.169133\pi$
$131$	19.7350	1.72425	0.862126	+	0.506694i	$0.169133\pi$
$132$	7.81868	0.680529
$133$	1.34335	0.116483
$134$	−0.217682	−0.0188049
$135$	0	0
$136$	0	0
$137$	4.66953	0.398945	0.199473	−	0.979903i	$-0.436077\pi$
$137$	4.66953	0.398945	0.199473	+	0.979903i	$0.436077\pi$
$138$	−0.877953	−0.0747363
$139$	−7.35975	−0.624246	−0.312123	−	0.950042i	$-0.601040\pi$
$139$	−7.35975	−0.624246	−0.312123	+	0.950042i	$0.601040\pi$
$140$	0	0
$141$	10.1996	0.858961
$142$	0.255212	0.0214169
$143$	6.64514	0.555694
$144$	17.4873	1.45728
$145$	0	0
$146$	0.163658	0.0135445
$147$	18.6538	1.53854
$148$	−15.1604	−1.24618
$149$	−22.3595	−1.83176	−0.915879	−	0.401455i	$-0.868505\pi$
$149$	−22.3595	−1.83176	−0.915879	+	0.401455i	$0.868505\pi$
$150$	0	0
$151$	15.0735	1.22667	0.613333	−	0.789824i	$-0.289828\pi$
$151$	15.0735	1.22667	0.613333	+	0.789824i	$0.289828\pi$
$152$	0.553439	0.0448898
$153$	0	0
$154$	−0.0198793	−0.00160192
$155$	0	0
$156$	25.0575	2.00621
$157$	12.5394	1.00075	0.500377	−	0.865808i	$-0.333195\pi$
$157$	12.5394	1.00075	0.500377	+	0.865808i	$0.333195\pi$
$158$	0.153284	0.0121946
$159$	−28.1699	−2.23402
$160$	0	0
$161$	−3.13642	−0.247184
$162$	−0.111215	−0.00873791
$163$	−9.78073	−0.766086	−0.383043	−	0.923731i	$-0.625124\pi$
$163$	−9.78073	−0.766086	−0.383043	+	0.923731i	$0.625124\pi$
$164$	12.3533	0.964629
$165$	0	0
$166$	0.438528	0.0340364
$167$	10.4388	0.807776	0.403888	−	0.914808i	$-0.367659\pi$
$167$	10.4388	0.807776	0.403888	+	0.914808i	$0.367659\pi$
$168$	−0.149976	−0.0115709
$169$	8.29652	0.638194
$170$	0	0
$171$	−16.0783	−1.22954
$172$	−14.0771	−1.07337
$173$	9.34704	0.710642	0.355321	−	0.934744i	$-0.384371\pi$
$173$	9.34704	0.710642	0.355321	+	0.934744i	$0.384371\pi$
$174$	−0.882988	−0.0669391
$175$	0	0
$176$	5.74754	0.433237
$177$	−18.8531	−1.41708
$178$	−0.402656	−0.0301804
$179$	4.81684	0.360028	0.180014	−	0.983664i	$-0.442386\pi$
$179$	4.81684	0.360028	0.180014	+	0.983664i	$0.442386\pi$
$180$	0	0
$181$	12.2389	0.909712	0.454856	−	0.890565i	$-0.349691\pi$
$181$	12.2389	0.909712	0.454856	+	0.890565i	$0.349691\pi$
$182$	−0.0637098	−0.00472249
$183$	−6.23008	−0.460541
$184$	−1.29215	−0.0952588
$185$	0	0
$186$	−0.964684	−0.0707340
$187$	0	0
$188$	7.50311	0.547221
$189$	1.37357	0.0999122
$190$	0	0
$191$	−17.8459	−1.29128	−0.645641	−	0.763641i	$-0.723410\pi$
$191$	−17.8459	−1.29128	−0.645641	+	0.763641i	$0.723410\pi$
$192$	21.6420	1.56187
$193$	−9.16123	−0.659440	−0.329720	−	0.944079i	$-0.606954\pi$
$193$	−9.16123	−0.659440	−0.329720	+	0.944079i	$0.606954\pi$
$194$	0.321562	0.0230868
$195$	0	0
$196$	13.7223	0.980161
$197$	−9.52822	−0.678858	−0.339429	−	0.940632i	$-0.610234\pi$
$197$	−9.52822	−0.678858	−0.339429	+	0.940632i	$0.610234\pi$
$198$	0.237932	0.0169091
$199$	5.17883	0.367118	0.183559	−	0.983009i	$-0.441238\pi$
$199$	5.17883	0.367118	0.183559	+	0.983009i	$0.441238\pi$
$200$	0	0
$201$	15.6809	1.10605
$202$	−0.386284	−0.0271789
$203$	−3.15440	−0.221396
$204$	0	0
$205$	0	0
$206$	0.255360	0.0177918
$207$	37.5392	2.60916
$208$	18.4199	1.27719
$209$	−5.28446	−0.365534
$210$	0	0
$211$	−13.5622	−0.933663	−0.466831	−	0.884346i	$-0.654604\pi$
$211$	−13.5622	−0.933663	−0.466831	+	0.884346i	$0.654604\pi$
$212$	−20.7226	−1.42323
$213$	−18.3844	−1.25968
$214$	−0.564018	−0.0385555
$215$	0	0
$216$	0.565887	0.0385037
$217$	−3.44626	−0.233947
$218$	−0.446388	−0.0302332
$219$	−11.7893	−0.796647
$220$	0	0
$221$	0	0
$222$	−0.777259	−0.0521662
$223$	−9.81594	−0.657324	−0.328662	−	0.944448i	$-0.606598\pi$
$223$	−9.81594	−0.657324	−0.328662	+	0.944448i	$0.606598\pi$
$224$	−0.165509	−0.0110585
$225$	0	0
$226$	−0.251765	−0.0167472
$227$	24.2273	1.60802	0.804010	−	0.594615i	$-0.202696\pi$
$227$	24.2273	1.60802	0.804010	+	0.594615i	$0.202696\pi$
$228$	−19.9267	−1.31968
$229$	−5.02401	−0.331996	−0.165998	−	0.986126i	$-0.553085\pi$
$229$	−5.02401	−0.331996	−0.165998	+	0.986126i	$0.553085\pi$
$230$	0	0
$231$	1.43203	0.0942204
$232$	−1.29956	−0.0853205
$233$	6.03303	0.395237	0.197619	−	0.980279i	$-0.436679\pi$
$233$	6.03303	0.395237	0.197619	+	0.980279i	$0.436679\pi$
$234$	0.762532	0.0498483
$235$	0	0
$236$	−13.8688	−0.902785
$237$	−11.0420	−0.717254
$238$	0	0
$239$	−3.44423	−0.222789	−0.111394	−	0.993776i	$-0.535532\pi$
$239$	−3.44423	−0.222789	−0.111394	+	0.993776i	$0.535532\pi$
$240$	0	0
$241$	−8.46220	−0.545098	−0.272549	−	0.962142i	$-0.587867\pi$
$241$	−8.46220	−0.545098	−0.272549	+	0.962142i	$0.587867\pi$
$242$	−0.336663	−0.0216415
$243$	19.2688	1.23609
$244$	−4.58302	−0.293398
$245$	0	0
$246$	0.633341	0.0403803
$247$	−16.9358	−1.07760
$248$	−1.41980	−0.0901575
$249$	−31.5898	−2.00192
$250$	0	0
$251$	22.2331	1.40334	0.701670	−	0.712502i	$-0.252438\pi$
$251$	22.2331	1.40334	0.701670	+	0.712502i	$0.252438\pi$
$252$	3.20516	0.201906
$253$	12.3380	0.775683
$254$	0.182792	0.0114694
$255$	0	0
$256$	15.8863	0.992896
$257$	−12.4364	−0.775762	−0.387881	−	0.921709i	$-0.626793\pi$
$257$	−12.4364	−0.775762	−0.387881	+	0.921709i	$0.626793\pi$
$258$	−0.721718	−0.0449322
$259$	−2.77670	−0.172535
$260$	0	0
$261$	37.7545	2.33695
$262$	0.744303	0.0459832
$263$	−2.44229	−0.150598	−0.0752992	−	0.997161i	$-0.523991\pi$
$263$	−2.44229	−0.150598	−0.0752992	+	0.997161i	$0.523991\pi$
$264$	0.589972	0.0363103
$265$	0	0
$266$	0.0506644	0.00310644
$267$	29.0057	1.77512
$268$	11.5353	0.704633
$269$	13.0709	0.796949	0.398475	−	0.917179i	$-0.369540\pi$
$269$	13.0709	0.796949	0.398475	+	0.917179i	$0.369540\pi$
$270$	0	0
$271$	−4.61868	−0.280565	−0.140282	−	0.990112i	$-0.544801\pi$
$271$	−4.61868	−0.280565	−0.140282	+	0.990112i	$0.544801\pi$
$272$	0	0
$273$	4.58940	0.277763
$274$	0.176111	0.0106393
$275$	0	0
$276$	46.5242	2.80043
$277$	18.9958	1.14135	0.570675	−	0.821176i	$-0.306682\pi$
$277$	18.9958	1.14135	0.570675	+	0.821176i	$0.306682\pi$
$278$	−0.277573	−0.0166477
$279$	41.2476	2.46943
$280$	0	0
$281$	8.88753	0.530186	0.265093	−	0.964223i	$-0.414597\pi$
$281$	8.88753	0.530186	0.265093	+	0.964223i	$0.414597\pi$
$282$	0.384677	0.0229072
$283$	6.52599	0.387930	0.193965	−	0.981008i	$-0.437865\pi$
$283$	6.52599	0.387930	0.193965	+	0.981008i	$0.437865\pi$
$284$	−13.5241	−0.802507
$285$	0	0
$286$	0.250621	0.0148195
$287$	2.26256	0.133555
$288$	1.98095	0.116728
$289$	0	0
$290$	0	0
$291$	−23.1640	−1.35790
$292$	−8.67254	−0.507522
$293$	−33.2490	−1.94243	−0.971213	−	0.238214i	$-0.923438\pi$
$293$	−33.2490	−1.94243	−0.971213	+	0.238214i	$0.923438\pi$
$294$	0.703526	0.0410305
$295$	0	0
$296$	−1.14395	−0.0664910
$297$	−5.40332	−0.313532
$298$	−0.843285	−0.0488502
$299$	39.5412	2.28672
$300$	0	0
$301$	−2.57828	−0.148610
$302$	0.568497	0.0327133
$303$	27.8263	1.59858
$304$	−14.6482	−0.840131
$305$	0	0
$306$	0	0
$307$	16.4964	0.941500	0.470750	−	0.882267i	$-0.343983\pi$
$307$	16.4964	0.941500	0.470750	+	0.882267i	$0.343983\pi$
$308$	1.05344	0.0600253
$309$	−18.3951	−1.04646
$310$	0	0
$311$	2.77354	0.157273	0.0786365	−	0.996903i	$-0.474943\pi$
$311$	2.77354	0.157273	0.0786365	+	0.996903i	$0.474943\pi$
$312$	1.89076	0.107043
$313$	−5.81322	−0.328583	−0.164291	−	0.986412i	$-0.552534\pi$
$313$	−5.81322	−0.328583	−0.164291	+	0.986412i	$0.552534\pi$
$314$	0.472923	0.0266886
$315$	0	0
$316$	−8.12279	−0.456943
$317$	−17.9295	−1.00702	−0.503510	−	0.863989i	$-0.667958\pi$
$317$	−17.9295	−1.00702	−0.503510	+	0.863989i	$0.667958\pi$
$318$	−1.06243	−0.0595780
$319$	12.4088	0.694757
$320$	0	0
$321$	40.6296	2.26772
$322$	−0.118290	−0.00659203
$323$	0	0
$324$	5.89349	0.327416
$325$	0	0
$326$	−0.368880	−0.0204304
$327$	32.1560	1.77823
$328$	0.932138	0.0514687
$329$	1.37423	0.0757637
$330$	0	0
$331$	−33.9446	−1.86577	−0.932883	−	0.360179i	$-0.882716\pi$
$331$	−33.9446	−1.86577	−0.932883	+	0.360179i	$0.882716\pi$
$332$	−23.2383	−1.27537
$333$	33.2338	1.82120
$334$	0.393697	0.0215422
$335$	0	0
$336$	3.96948	0.216553
$337$	−18.9012	−1.02961	−0.514807	−	0.857306i	$-0.672136\pi$
$337$	−18.9012	−1.02961	−0.514807	+	0.857306i	$0.672136\pi$
$338$	0.312903	0.0170197
$339$	18.1361	0.985020
$340$	0	0
$341$	13.5568	0.734144
$342$	−0.606394	−0.0327900
$343$	5.07563	0.274058
$344$	−1.06221	−0.0572705
$345$	0	0
$346$	0.352523	0.0189518
$347$	4.30688	0.231205	0.115603	−	0.993296i	$-0.463120\pi$
$347$	4.30688	0.231205	0.115603	+	0.993296i	$0.463120\pi$
$348$	46.7910	2.50826
$349$	10.9822	0.587863	0.293932	−	0.955826i	$-0.405036\pi$
$349$	10.9822	0.587863	0.293932	+	0.955826i	$0.405036\pi$
$350$	0	0
$351$	−17.3167	−0.924297
$352$	0.651077	0.0347025
$353$	23.3983	1.24537	0.622683	−	0.782474i	$-0.286043\pi$
$353$	23.3983	1.24537	0.622683	+	0.782474i	$0.286043\pi$
$354$	−0.711042	−0.0377915
$355$	0	0
$356$	21.3374	1.13088
$357$	0	0
$358$	0.181667	0.00960140
$359$	19.3663	1.02212	0.511058	−	0.859546i	$-0.329254\pi$
$359$	19.3663	1.02212	0.511058	+	0.859546i	$0.329254\pi$
$360$	0	0
$361$	−5.53203	−0.291160
$362$	0.461590	0.0242606
$363$	24.2519	1.27289
$364$	3.37609	0.176955
$365$	0	0
$366$	−0.234967	−0.0122819
$367$	26.1677	1.36594	0.682972	−	0.730444i	$-0.260687\pi$
$367$	26.1677	1.36594	0.682972	+	0.730444i	$0.260687\pi$
$368$	34.2001	1.78280
$369$	−27.0802	−1.40974
$370$	0	0
$371$	−3.79544	−0.197049
$372$	51.1202	2.65046
$373$	13.1266	0.679670	0.339835	−	0.940485i	$-0.389629\pi$
$373$	13.1266	0.679670	0.339835	+	0.940485i	$0.389629\pi$
$374$	0	0
$375$	0	0
$376$	0.566160	0.0291975
$377$	39.7679	2.04815
$378$	0.0518040	0.00266451
$379$	21.4773	1.10321	0.551607	−	0.834104i	$-0.314015\pi$
$379$	21.4773	1.10321	0.551607	+	0.834104i	$0.314015\pi$
$380$	0	0
$381$	−13.1676	−0.674597
$382$	−0.673056	−0.0344365
$383$	17.3300	0.885522	0.442761	−	0.896640i	$-0.353999\pi$
$383$	17.3300	0.885522	0.442761	+	0.896640i	$0.353999\pi$
$384$	3.27306	0.167027
$385$	0	0
$386$	−0.345515	−0.0175863
$387$	30.8590	1.56865
$388$	−17.0401	−0.865080
$389$	−18.1062	−0.918020	−0.459010	−	0.888431i	$-0.651796\pi$
$389$	−18.1062	−0.918020	−0.459010	+	0.888431i	$0.651796\pi$
$390$	0	0
$391$	0	0
$392$	1.03544	0.0522974
$393$	−53.6166	−2.70460
$394$	−0.359356	−0.0181041
$395$	0	0
$396$	−12.6084	−0.633598
$397$	36.4911	1.83144	0.915719	−	0.401819i	$-0.131622\pi$
$397$	36.4911	1.83144	0.915719	+	0.401819i	$0.131622\pi$
$398$	0.195319	0.00979047
$399$	−3.64966	−0.182712
$400$	0	0
$401$	−16.3012	−0.814042	−0.407021	−	0.913419i	$-0.633432\pi$
$401$	−16.3012	−0.814042	−0.407021	+	0.913419i	$0.633432\pi$
$402$	0.591406	0.0294966
$403$	43.4473	2.16427
$404$	20.4698	1.01841
$405$	0	0
$406$	−0.118968	−0.00590429
$407$	10.9229	0.541430
$408$	0	0
$409$	37.1630	1.83759	0.918796	−	0.394733i	$-0.129163\pi$
$409$	37.1630	1.83759	0.918796	+	0.394733i	$0.129163\pi$
$410$	0	0
$411$	−12.6863	−0.625771
$412$	−13.5320	−0.666672
$413$	−2.54014	−0.124992
$414$	1.41579	0.0695823
$415$	0	0
$416$	2.08659	0.102303
$417$	19.9952	0.979170
$418$	−0.199303	−0.00974823
$419$	14.1701	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$419$	14.1701	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$420$	0	0
$421$	−22.8940	−1.11578	−0.557892	−	0.829914i	$-0.688390\pi$
$421$	−22.8940	−1.11578	−0.557892	+	0.829914i	$0.688390\pi$
$422$	−0.511499	−0.0248994
$423$	−16.4479	−0.799725
$424$	−1.56366	−0.0759380
$425$	0	0
$426$	−0.693367	−0.0335938
$427$	−0.839402	−0.0406215
$428$	29.8883	1.44470
$429$	−18.0537	−0.871642
$430$	0	0
$431$	−8.73582	−0.420790	−0.210395	−	0.977616i	$-0.567475\pi$
$431$	−8.73582	−0.420790	−0.210395	+	0.977616i	$0.567475\pi$
$432$	−14.9776	−0.720612
$433$	12.4809	0.599793	0.299897	−	0.953972i	$-0.403048\pi$
$433$	12.4809	0.599793	0.299897	+	0.953972i	$0.403048\pi$
$434$	−0.129975	−0.00623902
$435$	0	0
$436$	23.6549	1.13286
$437$	−31.4446	−1.50420
$438$	−0.444632	−0.0212454
$439$	−17.1151	−0.816859	−0.408429	−	0.912790i	$-0.633923\pi$
$439$	−17.1151	−0.816859	−0.408429	+	0.912790i	$0.633923\pi$
$440$	0	0
$441$	−30.0812	−1.43244
$442$	0	0
$443$	−24.5327	−1.16558	−0.582792	−	0.812621i	$-0.698040\pi$
$443$	−24.5327	−1.16558	−0.582792	+	0.812621i	$0.698040\pi$
$444$	41.1882	1.95471
$445$	0	0
$446$	−0.370208	−0.0175298
$447$	60.7469	2.87323
$448$	2.91590	0.137763
$449$	18.6522	0.880254	0.440127	−	0.897936i	$-0.354933\pi$
$449$	18.6522	0.880254	0.440127	+	0.897936i	$0.354933\pi$
$450$	0	0
$451$	−8.90043	−0.419105
$452$	13.3415	0.627530
$453$	−40.9522	−1.92410
$454$	0.913730	0.0428835
$455$	0	0
$456$	−1.50360	−0.0704126
$457$	−6.44295	−0.301389	−0.150694	−	0.988580i	$-0.548151\pi$
$457$	−6.44295	−0.301389	−0.150694	+	0.988580i	$0.548151\pi$
$458$	−0.189480	−0.00885383
$459$	0	0
$460$	0	0
$461$	11.2992	0.526256	0.263128	−	0.964761i	$-0.415246\pi$
$461$	11.2992	0.526256	0.263128	+	0.964761i	$0.415246\pi$
$462$	0.0540088	0.00251272
$463$	−18.3573	−0.853135	−0.426568	−	0.904456i	$-0.640277\pi$
$463$	−18.3573	−0.853135	−0.426568	+	0.904456i	$0.640277\pi$
$464$	34.3963	1.59681
$465$	0	0
$466$	0.227535	0.0105404
$467$	−11.8648	−0.549039	−0.274520	−	0.961581i	$-0.588519\pi$
$467$	−11.8648	−0.549039	−0.274520	+	0.961581i	$0.588519\pi$
$468$	−40.4078	−1.86785
$469$	2.11275	0.0975578
$470$	0	0
$471$	−34.0674	−1.56974
$472$	−1.04650	−0.0481690
$473$	10.1424	0.466348
$474$	−0.416448	−0.0191281
$475$	0	0
$476$	0	0
$477$	45.4270	2.07996
$478$	−0.129899	−0.00594144
$479$	−24.4474	−1.11703	−0.558516	−	0.829494i	$-0.688629\pi$
$479$	−24.4474	−1.11703	−0.558516	+	0.829494i	$0.688629\pi$
$480$	0	0
$481$	35.0061	1.59614
$482$	−0.319151	−0.0145369
$483$	8.52112	0.387724
$484$	17.8403	0.810925
$485$	0	0
$486$	0.726720	0.0329647
$487$	18.4617	0.836580	0.418290	−	0.908313i	$-0.362630\pi$
$487$	18.4617	0.836580	0.418290	+	0.908313i	$0.362630\pi$
$488$	−0.345820	−0.0156545
$489$	26.5726	1.20165
$490$	0	0
$491$	30.8780	1.39351	0.696753	−	0.717312i	$-0.254628\pi$
$491$	30.8780	1.39351	0.696753	+	0.717312i	$0.254628\pi$
$492$	−33.5618	−1.51308
$493$	0	0
$494$	−0.638732	−0.0287379
$495$	0	0
$496$	37.5787	1.68733
$497$	−2.47700	−0.111109
$498$	−1.19141	−0.0533882
$499$	−23.2095	−1.03900	−0.519500	−	0.854470i	$-0.673882\pi$
$499$	−23.2095	−1.03900	−0.519500	+	0.854470i	$0.673882\pi$
$500$	0	0
$501$	−28.3604	−1.26705
$502$	0.838519	0.0374250
$503$	22.9270	1.02226	0.511132	−	0.859502i	$-0.329226\pi$
$503$	22.9270	1.02226	0.511132	+	0.859502i	$0.329226\pi$
$504$	0.241851	0.0107729
$505$	0	0
$506$	0.465327	0.0206863
$507$	−22.5402	−1.00105
$508$	−9.68647	−0.429767
$509$	−16.6396	−0.737536	−0.368768	−	0.929521i	$-0.620220\pi$
$509$	−16.6396	−0.737536	−0.368768	+	0.929521i	$0.620220\pi$
$510$	0	0
$511$	−1.58841	−0.0702673
$512$	3.00862	0.132963
$513$	13.7709	0.607999
$514$	−0.469038	−0.0206884
$515$	0	0
$516$	38.2450	1.68364
$517$	−5.40593	−0.237752
$518$	−0.104723	−0.00460126
$519$	−25.3943	−1.11469
$520$	0	0
$521$	−15.8719	−0.695360	−0.347680	−	0.937613i	$-0.613030\pi$
$521$	−15.8719	−0.695360	−0.347680	+	0.937613i	$0.613030\pi$
$522$	1.42391	0.0623228
$523$	−41.6746	−1.82230	−0.911151	−	0.412072i	$-0.864805\pi$
$523$	−41.6746	−1.82230	−0.911151	+	0.412072i	$0.864805\pi$
$524$	−39.4419	−1.72303
$525$	0	0
$526$	−0.0921110	−0.00401623
$527$	0	0
$528$	−15.6151	−0.679560
$529$	50.4159	2.19200
$530$	0	0
$531$	30.4025	1.31936
$532$	−2.68479	−0.116401
$533$	−28.5243	−1.23553
$534$	1.09395	0.0473398
$535$	0	0
$536$	0.870419	0.0375964
$537$	−13.0866	−0.564727
$538$	0.492970	0.0212534
$539$	−9.88676	−0.425853
$540$	0	0
$541$	−38.7515	−1.66606	−0.833028	−	0.553231i	$-0.813395\pi$
$541$	−38.7515	−1.66606	−0.833028	+	0.553231i	$0.813395\pi$
$542$	−0.174193	−0.00748223
$543$	−33.2511	−1.42694
$544$	0	0
$545$	0	0
$546$	0.173089	0.00740752
$547$	−29.8009	−1.27420	−0.637098	−	0.770783i	$-0.719865\pi$
$547$	−29.8009	−1.27420	−0.637098	+	0.770783i	$0.719865\pi$
$548$	−9.33243	−0.398661
$549$	10.0467	0.428781
$550$	0	0
$551$	−31.6249	−1.34727
$552$	3.51056	0.149419
$553$	−1.48773	−0.0632646
$554$	0.716427	0.0304381
$555$	0	0
$556$	14.7090	0.623802
$557$	44.6719	1.89281	0.946405	−	0.322982i	$-0.104685\pi$
$557$	44.6719	1.89281	0.946405	+	0.322982i	$0.104685\pi$
$558$	1.55565	0.0658560
$559$	32.5047	1.37480
$560$	0	0
$561$	0	0
$562$	0.335193	0.0141392
$563$	−7.58402	−0.319628	−0.159814	−	0.987147i	$-0.551090\pi$
$563$	−7.58402	−0.319628	−0.159814	+	0.987147i	$0.551090\pi$
$564$	−20.3847	−0.858351
$565$	0	0
$566$	0.246127	0.0103455
$567$	1.07942	0.0453314
$568$	−1.02048	−0.0428186
$569$	−33.1675	−1.39045	−0.695227	−	0.718790i	$-0.744696\pi$
$569$	−33.1675	−1.39045	−0.695227	+	0.718790i	$0.744696\pi$
$570$	0	0
$571$	22.2140	0.929626	0.464813	−	0.885409i	$-0.346122\pi$
$571$	22.2140	0.929626	0.464813	+	0.885409i	$0.346122\pi$
$572$	−13.2808	−0.555299
$573$	48.4842	2.02546
$574$	0.0853323	0.00356170
$575$	0	0
$576$	−34.8999	−1.45416
$577$	3.97246	0.165376	0.0826879	−	0.996575i	$-0.473650\pi$
$577$	3.97246	0.165376	0.0826879	+	0.996575i	$0.473650\pi$
$578$	0	0
$579$	24.8895	1.03437
$580$	0	0
$581$	−4.25621	−0.176577
$582$	−0.873630	−0.0362131
$583$	14.9305	0.618356
$584$	−0.654401	−0.0270793
$585$	0	0
$586$	−1.25398	−0.0518015
$587$	−32.9000	−1.35793	−0.678965	−	0.734171i	$-0.737571\pi$
$587$	−32.9000	−1.35793	−0.678965	+	0.734171i	$0.737571\pi$
$588$	−37.2810	−1.53744
$589$	−34.5509	−1.42365
$590$	0	0
$591$	25.8866	1.06483
$592$	30.2776	1.24440
$593$	8.35698	0.343180	0.171590	−	0.985168i	$-0.445110\pi$
$593$	8.35698	0.343180	0.171590	+	0.985168i	$0.445110\pi$
$594$	−0.203786	−0.00836143
$595$	0	0
$596$	44.6871	1.83046
$597$	−14.0700	−0.575847
$598$	1.49129	0.0609835
$599$	19.3005	0.788597	0.394299	−	0.918982i	$-0.370988\pi$
$599$	19.3005	0.788597	0.394299	+	0.918982i	$0.370988\pi$
$600$	0	0
$601$	−30.5134	−1.24467	−0.622334	−	0.782752i	$-0.713816\pi$
$601$	−30.5134	−1.24467	−0.622334	+	0.782752i	$0.713816\pi$
$602$	−0.0972397	−0.00396319
$603$	−25.2872	−1.02977
$604$	−30.1256	−1.22579
$605$	0	0
$606$	1.04947	0.0426318
$607$	−9.56833	−0.388366	−0.194183	−	0.980965i	$-0.562206\pi$
$607$	−9.56833	−0.388366	−0.194183	+	0.980965i	$0.562206\pi$
$608$	−1.65933	−0.0672949
$609$	8.56999	0.347273
$610$	0	0
$611$	−17.3251	−0.700897
$612$	0	0
$613$	21.2504	0.858295	0.429148	−	0.903234i	$-0.358814\pi$
$613$	21.2504	0.858295	0.429148	+	0.903234i	$0.358814\pi$
$614$	0.622162	0.0251084
$615$	0	0
$616$	0.0794891	0.00320271
$617$	−5.82850	−0.234647	−0.117323	−	0.993094i	$-0.537431\pi$
$617$	−5.82850	−0.234647	−0.117323	+	0.993094i	$0.537431\pi$
$618$	−0.693770	−0.0279075
$619$	17.6026	0.707510	0.353755	−	0.935338i	$-0.384905\pi$
$619$	17.6026	0.707510	0.353755	+	0.935338i	$0.384905\pi$
$620$	0	0
$621$	−32.1518	−1.29021
$622$	0.104604	0.00419423
$623$	3.90805	0.156573
$624$	−50.0437	−2.00335
$625$	0	0
$626$	−0.219245	−0.00876281
$627$	14.3570	0.573363
$628$	−25.0610	−1.00004
$629$	0	0
$630$	0	0
$631$	44.1395	1.75717	0.878584	−	0.477588i	$-0.158489\pi$
$631$	44.1395	1.75717	0.878584	+	0.477588i	$0.158489\pi$
$632$	−0.612919	−0.0243806
$633$	36.8463	1.46451
$634$	−0.676209	−0.0268557
$635$	0	0
$636$	56.2998	2.23243
$637$	−31.6854	−1.25542
$638$	0.467996	0.0185281
$639$	29.6468	1.17281
$640$	0	0
$641$	10.4786	0.413882	0.206941	−	0.978353i	$-0.433649\pi$
$641$	10.4786	0.413882	0.206941	+	0.978353i	$0.433649\pi$
$642$	1.53234	0.0604767
$643$	8.62068	0.339967	0.169983	−	0.985447i	$-0.445629\pi$
$643$	8.62068	0.339967	0.169983	+	0.985447i	$0.445629\pi$
$644$	6.26837	0.247009
$645$	0	0
$646$	0	0
$647$	−9.59804	−0.377338	−0.188669	−	0.982041i	$-0.560417\pi$
$647$	−9.59804	−0.377338	−0.188669	+	0.982041i	$0.560417\pi$
$648$	0.444703	0.0174696
$649$	9.99238	0.392235
$650$	0	0
$651$	9.36290	0.366961
$652$	19.5475	0.765541
$653$	−33.3525	−1.30518	−0.652592	−	0.757709i	$-0.726318\pi$
$653$	−33.3525	−1.30518	−0.652592	+	0.757709i	$0.726318\pi$
$654$	1.21276	0.0474227
$655$	0	0
$656$	−24.6714	−0.963256
$657$	19.0115	0.741708
$658$	0.0518290	0.00202050
$659$	−37.9334	−1.47768	−0.738838	−	0.673884i	$-0.764625\pi$
$659$	−37.9334	−1.47768	−0.738838	+	0.673884i	$0.764625\pi$
$660$	0	0
$661$	19.0909	0.742551	0.371275	−	0.928523i	$-0.378921\pi$
$661$	19.0909	0.742551	0.371275	+	0.928523i	$0.378921\pi$
$662$	−1.28022	−0.0497572
$663$	0	0
$664$	−1.75349	−0.0680486
$665$	0	0
$666$	1.25341	0.0485687
$667$	73.8369	2.85898
$668$	−20.8627	−0.807201
$669$	26.6683	1.03105
$670$	0	0
$671$	3.30203	0.127473
$672$	0.449660	0.0173460
$673$	−28.7021	−1.10638	−0.553191	−	0.833054i	$-0.686590\pi$
$673$	−28.7021	−1.10638	−0.553191	+	0.833054i	$0.686590\pi$
$674$	−0.712857	−0.0274582
$675$	0	0
$676$	−16.5812	−0.637740
$677$	−3.97611	−0.152814	−0.0764071	−	0.997077i	$-0.524345\pi$
$677$	−3.97611	−0.152814	−0.0764071	+	0.997077i	$0.524345\pi$
$678$	0.684004	0.0262690
$679$	−3.12097	−0.119772
$680$	0	0
$681$	−65.8214	−2.52228
$682$	0.511295	0.0195785
$683$	17.1195	0.655060	0.327530	−	0.944841i	$-0.393784\pi$
$683$	17.1195	0.655060	0.327530	+	0.944841i	$0.393784\pi$
$684$	32.1338	1.22867
$685$	0	0
$686$	0.191427	0.00730872
$687$	13.6494	0.520757
$688$	28.1141	1.07184
$689$	47.8495	1.82292
$690$	0	0
$691$	1.78983	0.0680883	0.0340441	−	0.999420i	$-0.489161\pi$
$691$	1.78983	0.0680883	0.0340441	+	0.999420i	$0.489161\pi$
$692$	−18.6808	−0.710137
$693$	−2.30929	−0.0877227
$694$	0.162434	0.00616589
$695$	0	0
$696$	3.53070	0.133831
$697$	0	0
$698$	0.414193	0.0156774
$699$	−16.3907	−0.619954
$700$	0	0
$701$	−23.8905	−0.902331	−0.451166	−	0.892440i	$-0.648992\pi$
$701$	−23.8905	−0.902331	−0.451166	+	0.892440i	$0.648992\pi$
$702$	−0.653098	−0.0246496
$703$	−27.8382	−1.04994
$704$	−11.4705	−0.432312
$705$	0	0
$706$	0.882466	0.0332120
$707$	3.74914	0.141001
$708$	37.6793	1.41608
$709$	−26.5187	−0.995930	−0.497965	−	0.867197i	$-0.665919\pi$
$709$	−26.5187	−0.995930	−0.497965	+	0.867197i	$0.665919\pi$
$710$	0	0
$711$	17.8063	0.667790
$712$	1.61005	0.0603393
$713$	80.6685	3.02106
$714$	0	0
$715$	0	0
$716$	−9.62684	−0.359772
$717$	9.35739	0.349458
$718$	0.730400	0.0272583
$719$	3.78411	0.141123	0.0705617	−	0.997507i	$-0.477521\pi$
$719$	3.78411	0.141123	0.0705617	+	0.997507i	$0.477521\pi$
$720$	0	0
$721$	−2.47844	−0.0923019
$722$	−0.208640	−0.00776479
$723$	22.9904	0.855021
$724$	−24.4604	−0.909065
$725$	0	0
$726$	0.914657	0.0339461
$727$	−8.11641	−0.301021	−0.150510	−	0.988608i	$-0.548092\pi$
$727$	−8.11641	−0.301021	−0.150510	+	0.988608i	$0.548092\pi$
$728$	0.254749	0.00944162
$729$	−43.5034	−1.61124
$730$	0	0
$731$	0	0
$732$	12.4513	0.460213
$733$	−4.71764	−0.174250	−0.0871250	−	0.996197i	$-0.527768\pi$
$733$	−4.71764	−0.174250	−0.0871250	+	0.996197i	$0.527768\pi$
$734$	0.986914	0.0364277
$735$	0	0
$736$	3.87416	0.142804
$737$	−8.31111	−0.306144
$738$	−1.02133	−0.0375956
$739$	−23.0955	−0.849582	−0.424791	−	0.905291i	$-0.639652\pi$
$739$	−23.0955	−0.849582	−0.424791	+	0.905291i	$0.639652\pi$
$740$	0	0
$741$	46.0117	1.69028
$742$	−0.143145	−0.00525501
$743$	−37.7461	−1.38477	−0.692384	−	0.721529i	$-0.743440\pi$
$743$	−37.7461	−1.38477	−0.692384	+	0.721529i	$0.743440\pi$
$744$	3.85736	0.141418
$745$	0	0
$746$	0.495070	0.0181258
$747$	50.9418	1.86386
$748$	0	0
$749$	5.47417	0.200022
$750$	0	0
$751$	−9.87924	−0.360499	−0.180249	−	0.983621i	$-0.557690\pi$
$751$	−9.87924	−0.360499	−0.180249	+	0.983621i	$0.557690\pi$
$752$	−14.9849	−0.546442
$753$	−60.4035	−2.20123
$754$	1.49985	0.0546211
$755$	0	0
$756$	−2.74518	−0.0998412
$757$	27.9633	1.01634	0.508172	−	0.861255i	$-0.330321\pi$
$757$	27.9633	1.01634	0.508172	+	0.861255i	$0.330321\pi$
$758$	0.810015	0.0294211
$759$	−33.5203	−1.21671
$760$	0	0
$761$	32.6749	1.18447	0.592233	−	0.805767i	$-0.298247\pi$
$761$	32.6749	1.18447	0.592233	+	0.805767i	$0.298247\pi$
$762$	−0.496616	−0.0179905
$763$	4.33249	0.156847
$764$	35.6663	1.29036
$765$	0	0
$766$	0.653600	0.0236155
$767$	32.0239	1.15631
$768$	−43.1605	−1.55742
$769$	−3.60934	−0.130156	−0.0650780	−	0.997880i	$-0.520730\pi$
$769$	−3.60934	−0.130156	−0.0650780	+	0.997880i	$0.520730\pi$
$770$	0	0
$771$	33.7876	1.21683
$772$	18.3094	0.658971
$773$	−34.5157	−1.24144	−0.620721	−	0.784031i	$-0.713160\pi$
$773$	−34.5157	−1.24144	−0.620721	+	0.784031i	$0.713160\pi$
$774$	1.16384	0.0418335
$775$	0	0
$776$	−1.28579	−0.0461572
$777$	7.54381	0.270633
$778$	−0.682874	−0.0244822
$779$	22.6836	0.812725
$780$	0	0
$781$	9.74399	0.348667
$782$	0	0
$783$	−32.3362	−1.15560
$784$	−27.4054	−0.978766
$785$	0	0
$786$	−2.02215	−0.0721275
$787$	−10.3491	−0.368904	−0.184452	−	0.982842i	$-0.559051\pi$
$787$	−10.3491	−0.368904	−0.184452	+	0.982842i	$0.559051\pi$
$788$	19.0429	0.678375
$789$	6.63530	0.236223
$790$	0	0
$791$	2.44355	0.0868826
$792$	−0.951391	−0.0338062
$793$	10.5824	0.375793
$794$	1.37626	0.0488417
$795$	0	0
$796$	−10.3503	−0.366857
$797$	−27.7046	−0.981348	−0.490674	−	0.871343i	$-0.663249\pi$
$797$	−27.7046	−0.981348	−0.490674	+	0.871343i	$0.663249\pi$
$798$	−0.137647	−0.00487264
$799$	0	0
$800$	0	0
$801$	−46.7748	−1.65271
$802$	−0.614798	−0.0217093
$803$	6.24849	0.220504
$804$	−31.3396	−1.10526
$805$	0	0
$806$	1.63861	0.0577177
$807$	−35.5115	−1.25007
$808$	1.54459	0.0543384
$809$	−28.3530	−0.996837	−0.498419	−	0.866936i	$-0.666086\pi$
$809$	−28.3530	−0.996837	−0.498419	+	0.866936i	$0.666086\pi$
$810$	0	0
$811$	26.9233	0.945405	0.472702	−	0.881222i	$-0.343279\pi$
$811$	26.9233	0.945405	0.472702	+	0.881222i	$0.343279\pi$
$812$	6.30432	0.221238
$813$	12.5482	0.440083
$814$	0.411958	0.0144391
$815$	0	0
$816$	0	0
$817$	−25.8489	−0.904339
$818$	1.40160	0.0490058
$819$	−7.40088	−0.258608
$820$	0	0
$821$	4.11428	0.143589	0.0717946	−	0.997419i	$-0.477127\pi$
$821$	4.11428	0.143589	0.0717946	+	0.997419i	$0.477127\pi$
$822$	−0.478464	−0.0166884
$823$	−7.39842	−0.257893	−0.128946	−	0.991652i	$-0.541160\pi$
$823$	−7.39842	−0.257893	−0.128946	+	0.991652i	$0.541160\pi$
$824$	−1.02108	−0.0355709
$825$	0	0
$826$	−0.0958013	−0.00333335
$827$	−38.9992	−1.35614	−0.678068	−	0.734999i	$-0.737182\pi$
$827$	−38.9992	−1.35614	−0.678068	+	0.734999i	$0.737182\pi$
$828$	−75.0251	−2.60730
$829$	10.5054	0.364866	0.182433	−	0.983218i	$-0.441603\pi$
$829$	10.5054	0.364866	0.182433	+	0.983218i	$0.441603\pi$
$830$	0	0
$831$	−51.6085	−1.79028
$832$	−36.7611	−1.27446
$833$	0	0
$834$	0.754118	0.0261130
$835$	0	0
$836$	10.5614	0.365274
$837$	−35.3280	−1.22112
$838$	0.534425	0.0184614
$839$	−5.58244	−0.192727	−0.0963637	−	0.995346i	$-0.530721\pi$
$839$	−5.58244	−0.192727	−0.0963637	+	0.995346i	$0.530721\pi$
$840$	0	0
$841$	45.2604	1.56070
$842$	−0.863444	−0.0297563
$843$	−24.1459	−0.831630
$844$	27.1052	0.932999
$845$	0	0
$846$	−0.620332	−0.0213275
$847$	3.26754	0.112274
$848$	41.3862	1.42121
$849$	−17.7300	−0.608492
$850$	0	0
$851$	64.9957	2.22802
$852$	36.7427	1.25878
$853$	−14.3308	−0.490676	−0.245338	−	0.969438i	$-0.578899\pi$
$853$	−14.3308	−0.490676	−0.245338	+	0.969438i	$0.578899\pi$
$854$	−0.0316580	−0.00108331
$855$	0	0
$856$	2.25527	0.0770836
$857$	−17.4020	−0.594440	−0.297220	−	0.954809i	$-0.596059\pi$
$857$	−17.4020	−0.594440	−0.297220	+	0.954809i	$0.596059\pi$
$858$	−0.680895	−0.0232454
$859$	17.4630	0.595830	0.297915	−	0.954592i	$-0.403709\pi$
$859$	17.4630	0.595830	0.297915	+	0.954592i	$0.403709\pi$
$860$	0	0
$861$	−6.14699	−0.209489
$862$	−0.329471	−0.0112218
$863$	−35.3537	−1.20346	−0.601728	−	0.798701i	$-0.705521\pi$
$863$	−35.3537	−1.20346	−0.601728	+	0.798701i	$0.705521\pi$
$864$	−1.69665	−0.0577214
$865$	0	0
$866$	0.470716	0.0159956
$867$	0	0
$868$	6.88761	0.233781
$869$	5.85240	0.198529
$870$	0	0
$871$	−26.6357	−0.902516
$872$	1.78492	0.0604449
$873$	37.3544	1.26425
$874$	−1.18593	−0.0401147
$875$	0	0
$876$	23.5618	0.796080
$877$	−19.9641	−0.674141	−0.337071	−	0.941479i	$-0.609436\pi$
$877$	−19.9641	−0.674141	−0.337071	+	0.941479i	$0.609436\pi$
$878$	−0.645495	−0.0217844
$879$	90.3318	3.04682
$880$	0	0
$881$	36.5987	1.23304	0.616520	−	0.787339i	$-0.288542\pi$
$881$	36.5987	1.23304	0.616520	+	0.787339i	$0.288542\pi$
$882$	−1.13451	−0.0382009
$883$	−48.1401	−1.62004	−0.810022	−	0.586399i	$-0.800545\pi$
$883$	−48.1401	−1.62004	−0.810022	+	0.586399i	$0.800545\pi$
$884$	0	0
$885$	0	0
$886$	−0.925250	−0.0310844
$887$	24.8280	0.833644	0.416822	−	0.908988i	$-0.363144\pi$
$887$	24.8280	0.833644	0.416822	+	0.908988i	$0.363144\pi$
$888$	3.10793	0.104295
$889$	−1.77412	−0.0595021
$890$	0	0
$891$	−4.24621	−0.142253
$892$	19.6179	0.656857
$893$	13.7775	0.461048
$894$	2.29106	0.0766247
$895$	0	0
$896$	0.440991	0.0147325
$897$	−107.427	−3.58687
$898$	0.703468	0.0234750
$899$	81.1311	2.70587
$900$	0	0
$901$	0	0
$902$	−0.335679	−0.0111769
$903$	7.00475	0.233104
$904$	1.00670	0.0334824
$905$	0	0
$906$	−1.54451	−0.0513129
$907$	−41.0099	−1.36171	−0.680856	−	0.732417i	$-0.738392\pi$
$907$	−41.0099	−1.36171	−0.680856	+	0.732417i	$0.738392\pi$
$908$	−48.4201	−1.60688
$909$	−44.8729	−1.48834
$910$	0	0
$911$	−28.5011	−0.944284	−0.472142	−	0.881523i	$-0.656519\pi$
$911$	−28.5011	−0.944284	−0.472142	+	0.881523i	$0.656519\pi$
$912$	39.7966	1.31780
$913$	16.7430	0.554113
$914$	−0.242996	−0.00803758
$915$	0	0
$916$	10.0409	0.331760
$917$	−7.22396	−0.238556
$918$	0	0
$919$	−2.13827	−0.0705351	−0.0352675	−	0.999378i	$-0.511228\pi$
$919$	−2.13827	−0.0705351	−0.0352675	+	0.999378i	$0.511228\pi$
$920$	0	0
$921$	−44.8180	−1.47680
$922$	0.426149	0.0140345
$923$	31.2278	1.02788
$924$	−2.86202	−0.0941534
$925$	0	0
$926$	−0.692344	−0.0227518
$927$	29.6640	0.974294
$928$	3.89638	0.127905
$929$	57.8841	1.89912	0.949558	−	0.313591i	$-0.101532\pi$
$929$	57.8841	1.89912	0.949558	+	0.313591i	$0.101532\pi$
$930$	0	0
$931$	25.1974	0.825811
$932$	−12.0575	−0.394956
$933$	−7.53524	−0.246693
$934$	−0.447482	−0.0146420
$935$	0	0
$936$	−3.04904	−0.0996611
$937$	−42.3883	−1.38477	−0.692383	−	0.721530i	$-0.743439\pi$
$937$	−42.3883	−1.38477	−0.692383	+	0.721530i	$0.743439\pi$
$938$	0.0796823	0.00260172
$939$	15.7935	0.515403
$940$	0	0
$941$	−56.9695	−1.85715	−0.928577	−	0.371140i	$-0.878967\pi$
$941$	−56.9695	−1.85715	−0.928577	+	0.371140i	$0.878967\pi$
$942$	−1.28485	−0.0418627
$943$	−52.9610	−1.72465
$944$	27.6982	0.901501
$945$	0	0
$946$	0.382520	0.0124368
$947$	37.5106	1.21893	0.609466	−	0.792812i	$-0.291384\pi$
$947$	37.5106	1.21893	0.609466	+	0.792812i	$0.291384\pi$
$948$	22.0683	0.716744
$949$	20.0253	0.650049
$950$	0	0
$951$	48.7114	1.57957
$952$	0	0
$953$	−2.42534	−0.0785646	−0.0392823	−	0.999228i	$-0.512507\pi$
$953$	−2.42534	−0.0785646	−0.0392823	+	0.999228i	$0.512507\pi$
$954$	1.71328	0.0554693
$955$	0	0
$956$	6.88356	0.222630
$957$	−33.7125	−1.08977
$958$	−0.922033	−0.0297895
$959$	−1.70928	−0.0551954
$960$	0	0
$961$	57.6375	1.85927
$962$	1.32025	0.0425667
$963$	−65.5194	−2.11133
$964$	16.9124	0.544710
$965$	0	0
$966$	0.321373	0.0103400
$967$	4.85146	0.156012	0.0780062	−	0.996953i	$-0.475145\pi$
$967$	4.85146	0.156012	0.0780062	+	0.996953i	$0.475145\pi$
$968$	1.34617	0.0432677
$969$	0	0
$970$	0	0
$971$	−3.42903	−0.110043	−0.0550214	−	0.998485i	$-0.517523\pi$
$971$	−3.42903	−0.110043	−0.0550214	+	0.998485i	$0.517523\pi$
$972$	−38.5101	−1.23521
$973$	2.69403	0.0863666
$974$	0.696282	0.0223103
$975$	0	0
$976$	9.15300	0.292980
$977$	42.5661	1.36181	0.680905	−	0.732372i	$-0.261587\pi$
$977$	42.5661	1.36181	0.680905	+	0.732372i	$0.261587\pi$
$978$	1.00218	0.0320463
$979$	−15.3734	−0.491337
$980$	0	0
$981$	−51.8549	−1.65560
$982$	1.16456	0.0371627
$983$	−13.1827	−0.420462	−0.210231	−	0.977652i	$-0.567422\pi$
$983$	−13.1827	−0.420462	−0.210231	+	0.977652i	$0.567422\pi$
$984$	−2.53246	−0.0807319
$985$	0	0
$986$	0	0
$987$	−3.73355	−0.118840
$988$	33.8475	1.07683
$989$	60.3513	1.91906
$990$	0	0
$991$	−4.84861	−0.154021	−0.0770106	−	0.997030i	$-0.524538\pi$
$991$	−4.84861	−0.154021	−0.0770106	+	0.997030i	$0.524538\pi$
$992$	4.25688	0.135156
$993$	92.2219	2.92657
$994$	−0.0934199	−0.00296310
$995$	0	0
$996$	63.1347	2.00050
$997$	−58.4194	−1.85016	−0.925081	−	0.379770i	$-0.876003\pi$
$997$	−58.4194	−1.85016	−0.925081	+	0.379770i	$0.876003\pi$
$998$	−0.875345	−0.0277086
$999$	−28.4643	−0.900570

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.bw.1.8	yes	15
5.4	even	2		7225.2.a.bu.1.8	yes	15
17.16	even	2		7225.2.a.bt.1.8	✓	15
85.84	even	2		7225.2.a.bv.1.8	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7225.2.a.bt.1.8	✓	15	17.16	even	2
7225.2.a.bu.1.8	yes	15	5.4	even	2
7225.2.a.bv.1.8	yes	15	85.84	even	2
7225.2.a.bw.1.8	yes	15	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:	\( N \)	\(=\)	\( 7225 = 5^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7225.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0377149 q^{2} -2.71683 q^{3} -1.99858 q^{4} -0.102465 q^{6} -0.366049 q^{7} -0.150806 q^{8} +4.38117 q^{9} +O(q^{10})\)	\(q+0.0377149 q^{2} -2.71683 q^{3} -1.99858 q^{4} -0.102465 q^{6} -0.366049 q^{7} -0.150806 q^{8} +4.38117 q^{9} +1.43996 q^{11} +5.42980 q^{12} +4.61482 q^{13} -0.0138055 q^{14} +3.99147 q^{16} +0.165236 q^{18} -3.66987 q^{19} +0.994492 q^{21} +0.0543079 q^{22} +8.56831 q^{23} +0.409715 q^{24} +0.174047 q^{26} -3.75241 q^{27} +0.731577 q^{28} +8.61745 q^{29} +9.41475 q^{31} +0.452150 q^{32} -3.91212 q^{33} -8.75611 q^{36} +7.58559 q^{37} -0.138409 q^{38} -12.5377 q^{39} -6.18104 q^{41} +0.0375072 q^{42} +7.04354 q^{43} -2.87787 q^{44} +0.323153 q^{46} -3.75423 q^{47} -10.8441 q^{48} -6.86601 q^{49} -9.22307 q^{52} +10.3687 q^{53} -0.141522 q^{54} +0.0552024 q^{56} +9.97043 q^{57} +0.325007 q^{58} +6.93936 q^{59} +2.29314 q^{61} +0.355077 q^{62} -1.60372 q^{63} -7.96588 q^{64} -0.147545 q^{66} -5.77178 q^{67} -23.2786 q^{69} +6.76686 q^{71} -0.660708 q^{72} +4.33935 q^{73} +0.286090 q^{74} +7.33453 q^{76} -0.527094 q^{77} -0.472858 q^{78} +4.06429 q^{79} -2.94884 q^{81} -0.233117 q^{82} +11.6274 q^{83} -1.98757 q^{84} +0.265647 q^{86} -23.4122 q^{87} -0.217154 q^{88} -10.6763 q^{89} -1.68925 q^{91} -17.1244 q^{92} -25.5783 q^{93} -0.141590 q^{94} -1.22842 q^{96} +8.52612 q^{97} -0.258951 q^{98} +6.30870 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 9 q^{3} + 12 q^{4} + 9 q^{6} + 12 q^{7} + 6 q^{8} + 12 q^{9}+O(q^{10}) \) 15 * q + 9 * q^3 + 12 * q^4 + 9 * q^6 + 12 * q^7 + 6 * q^8 + 12 * q^9	\( 15 q + 9 q^{3} + 12 q^{4} + 9 q^{6} + 12 q^{7} + 6 q^{8} + 12 q^{9} + 6 q^{11} + 24 q^{12} + 6 q^{16} + 12 q^{18} + 6 q^{19} + 30 q^{21} + 12 q^{22} + 36 q^{23} + 18 q^{24} + 36 q^{26} + 36 q^{27} + 24 q^{28} - 18 q^{29} + 12 q^{32} + 12 q^{33} - 9 q^{36} + 12 q^{37} - 6 q^{38} + 9 q^{39} - 18 q^{41} + 36 q^{42} - 3 q^{43} - 12 q^{44} + 21 q^{46} - 3 q^{47} - 12 q^{48} + 15 q^{49} - 27 q^{52} + 21 q^{54} - 6 q^{56} + 39 q^{57} + 18 q^{58} - 12 q^{59} - 15 q^{61} + 54 q^{62} + 60 q^{63} - 36 q^{64} + 18 q^{66} - 24 q^{67} + 42 q^{69} + 6 q^{71} + 66 q^{72} - 9 q^{73} - 36 q^{74} - 18 q^{76} + 30 q^{77} + 30 q^{78} - 9 q^{79} + 51 q^{81} - 36 q^{82} + 15 q^{83} + 9 q^{84} - 36 q^{86} - 51 q^{87} + 30 q^{88} - 24 q^{89} + 27 q^{91} + 15 q^{92} - 42 q^{93} - 57 q^{94} + 42 q^{96} + 48 q^{97} + 12 q^{99}+O(q^{100}) \) 15 * q + 9 * q^3 + 12 * q^4 + 9 * q^6 + 12 * q^7 + 6 * q^8 + 12 * q^9 + 6 * q^11 + 24 * q^12 + 6 * q^16 + 12 * q^18 + 6 * q^19 + 30 * q^21 + 12 * q^22 + 36 * q^23 + 18 * q^24 + 36 * q^26 + 36 * q^27 + 24 * q^28 - 18 * q^29 + 12 * q^32 + 12 * q^33 - 9 * q^36 + 12 * q^37 - 6 * q^38 + 9 * q^39 - 18 * q^41 + 36 * q^42 - 3 * q^43 - 12 * q^44 + 21 * q^46 - 3 * q^47 - 12 * q^48 + 15 * q^49 - 27 * q^52 + 21 * q^54 - 6 * q^56 + 39 * q^57 + 18 * q^58 - 12 * q^59 - 15 * q^61 + 54 * q^62 + 60 * q^63 - 36 * q^64 + 18 * q^66 - 24 * q^67 + 42 * q^69 + 6 * q^71 + 66 * q^72 - 9 * q^73 - 36 * q^74 - 18 * q^76 + 30 * q^77 + 30 * q^78 - 9 * q^79 + 51 * q^81 - 36 * q^82 + 15 * q^83 + 9 * q^84 - 36 * q^86 - 51 * q^87 + 30 * q^88 - 24 * q^89 + 27 * q^91 + 15 * q^92 - 42 * q^93 - 57 * q^94 + 42 * q^96 + 48 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

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Database

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