LMFDB - Embedded newform 7225.2.a.bt.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:	$1$
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	$-0.213032\pi$
$3$	2.71683	1.56856	0.784282	+	0.620405i	$0.213032\pi$
$4$	−1.99858	−0.999289
$5$	0	0
$5$	0	0
$6$	0.102465	0.0418312
$7$	0.366049	0.138353	0.0691767	−	0.997604i	$-0.477963\pi$
$7$	0.366049	0.138353	0.0691767	+	0.997604i	$0.477963\pi$
$8$	−0.150806	−0.0533180
$9$	4.38117	1.46039
$10$	0	0
$11$	−1.43996	−0.434164	−0.217082	−	0.976153i	$-0.569654\pi$
$11$	−1.43996	−0.434164	−0.217082	+	0.976153i	$0.569654\pi$
$12$	−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	$-0.851622\pi$
$23$	−8.56831	−1.78662	−0.893308	+	0.449445i	$0.851622\pi$
$24$	−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	$-0.795226\pi$
$29$	−8.61745	−1.60022	−0.800110	+	0.599853i	$0.795226\pi$
$30$	0	0
$31$	−9.41475	−1.69094	−0.845470	−	0.534024i	$-0.820679\pi$
$31$	−9.41475	−1.69094	−0.845470	+	0.534024i	$0.820679\pi$
$32$	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	$-0.714303\pi$
$37$	−7.58559	−1.24706	−0.623532	+	0.781798i	$0.714303\pi$
$38$	−0.138409	−0.0224529
$39$	12.5377	2.00763
$40$	0	0
$41$	6.18104	0.965316	0.482658	−	0.875809i	$-0.339671\pi$
$41$	6.18104	0.965316	0.482658	+	0.875809i	$0.339671\pi$
$42$	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	$-0.546898\pi$
$61$	−2.29314	−0.293607	−0.146803	+	0.989166i	$0.546898\pi$
$62$	−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	$-0.631525\pi$
$71$	−6.76686	−0.803079	−0.401539	+	0.915842i	$0.631525\pi$
$72$	−0.660708	−0.0778651
$73$	−4.33935	−0.507883	−0.253941	−	0.967220i	$-0.581727\pi$
$73$	−4.33935	−0.507883	−0.253941	+	0.967220i	$0.581727\pi$
$74$	−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	$-0.573426\pi$
$79$	−4.06429	−0.457268	−0.228634	+	0.973512i	$0.573426\pi$
$80$	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	$-0.642491\pi$
$97$	−8.52612	−0.865696	−0.432848	+	0.901467i	$0.642491\pi$
$98$	−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	$-0.242824\pi$
$107$	14.9548	1.44573	0.722866	+	0.690988i	$0.242824\pi$
$108$	−7.49949	−0.721639
$109$	11.8358	1.13367	0.566834	−	0.823832i	$-0.308168\pi$
$109$	11.8358	1.13367	0.566834	+	0.823832i	$0.308168\pi$
$110$	0	0
$111$	−20.6088	−1.95610
$112$	1.46107	0.138058
$113$	6.67548	0.627976	0.313988	−	0.949427i	$-0.398335\pi$
$113$	6.67548	0.627976	0.313988	+	0.949427i	$0.398335\pi$
$114$	−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	$-0.830867\pi$
$131$	−19.7350	−1.72425	−0.862126	+	0.506694i	$0.830867\pi$
$132$	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	$-0.398960\pi$
$139$	7.35975	0.624246	0.312123	+	0.950042i	$0.398960\pi$
$140$	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	$-0.374876\pi$
$163$	9.78073	0.766086	0.383043	+	0.923731i	$0.374876\pi$
$164$	−12.3533	−0.964629
$165$	0	0
$166$	0.438528	0.0340364
$167$	−10.4388	−0.807776	−0.403888	−	0.914808i	$-0.632341\pi$
$167$	−10.4388	−0.807776	−0.403888	+	0.914808i	$0.632341\pi$
$168$	−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	$-0.615629\pi$
$173$	−9.34704	−0.710642	−0.355321	+	0.934744i	$0.615629\pi$
$174$	−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	$-0.650309\pi$
$181$	−12.2389	−0.909712	−0.454856	+	0.890565i	$0.650309\pi$
$182$	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	$-0.393046\pi$
$193$	9.16123	0.659440	0.329720	+	0.944079i	$0.393046\pi$
$194$	−0.321562	−0.0230868
$195$	0	0
$196$	13.7223	0.980161
$197$	9.52822	0.678858	0.339429	−	0.940632i	$-0.389766\pi$
$197$	9.52822	0.678858	0.339429	+	0.940632i	$0.389766\pi$
$198$	−0.237932	−0.0169091
$199$	−5.17883	−0.367118	−0.183559	−	0.983009i	$-0.558762\pi$
$199$	−5.17883	−0.367118	−0.183559	+	0.983009i	$0.558762\pi$
$200$	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	$-0.345396\pi$
$211$	13.5622	0.933663	0.466831	+	0.884346i	$0.345396\pi$
$212$	−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	$-0.797304\pi$
$227$	−24.2273	−1.60802	−0.804010	+	0.594615i	$0.797304\pi$
$228$	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	$-0.563321\pi$
$233$	−6.03303	−0.395237	−0.197619	+	0.980279i	$0.563321\pi$
$234$	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	$-0.412133\pi$
$241$	8.46220	0.545098	0.272549	+	0.962142i	$0.412133\pi$
$242$	−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	$-0.630460\pi$
$269$	−13.0709	−0.796949	−0.398475	+	0.917179i	$0.630460\pi$
$270$	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	$-0.693318\pi$
$277$	−18.9958	−1.14135	−0.570675	+	0.821176i	$0.693318\pi$
$278$	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	$-0.562135\pi$
$283$	−6.52599	−0.387930	−0.193965	+	0.981008i	$0.562135\pi$
$284$	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	$-0.525057\pi$
$311$	−2.77354	−0.157273	−0.0786365	+	0.996903i	$0.525057\pi$
$312$	−1.89076	−0.107043
$313$	5.81322	0.328583	0.164291	−	0.986412i	$-0.447466\pi$
$313$	5.81322	0.328583	0.164291	+	0.986412i	$0.447466\pi$
$314$	0.472923	0.0266886
$315$	0	0
$316$	8.12279	0.456943
$317$	17.9295	1.00702	0.503510	−	0.863989i	$-0.332042\pi$
$317$	17.9295	1.00702	0.503510	+	0.863989i	$0.332042\pi$
$318$	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	$-0.327864\pi$
$337$	18.9012	1.02961	0.514807	+	0.857306i	$0.327864\pi$
$338$	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	$-0.536880\pi$
$347$	−4.30688	−0.231205	−0.115603	+	0.993296i	$0.536880\pi$
$348$	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	$-0.739313\pi$
$367$	−26.1677	−1.36594	−0.682972	+	0.730444i	$0.739313\pi$
$368$	−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	$-0.685985\pi$
$379$	−21.4773	−1.10321	−0.551607	+	0.834104i	$0.685985\pi$
$380$	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	$-0.868378\pi$
$397$	−36.4911	−1.83144	−0.915719	+	0.401819i	$0.868378\pi$
$398$	−0.195319	−0.00979047
$399$	−3.64966	−0.182712
$400$	0	0
$401$	16.3012	0.814042	0.407021	−	0.913419i	$-0.366568\pi$
$401$	16.3012	0.814042	0.407021	+	0.913419i	$0.366568\pi$
$402$	−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	$-0.612504\pi$
$419$	−14.1701	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$420$	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	$-0.432525\pi$
$431$	8.73582	0.420790	0.210395	+	0.977616i	$0.432525\pi$
$432$	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	$-0.366077\pi$
$439$	17.1151	0.816859	0.408429	+	0.912790i	$0.366077\pi$
$440$	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	$-0.645067\pi$
$449$	−18.6522	−0.880254	−0.440127	+	0.897936i	$0.645067\pi$
$450$	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	$-0.311371\pi$
$479$	24.4474	1.11703	0.558516	+	0.829494i	$0.311371\pi$
$480$	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	$-0.637370\pi$
$487$	−18.4617	−0.836580	−0.418290	+	0.908313i	$0.637370\pi$
$488$	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	$-0.326118\pi$
$499$	23.2095	1.03900	0.519500	+	0.854470i	$0.326118\pi$
$500$	0	0
$501$	−28.3604	−1.26705
$502$	0.838519	0.0374250
$503$	−22.9270	−1.02226	−0.511132	−	0.859502i	$-0.670774\pi$
$503$	−22.9270	−1.02226	−0.511132	+	0.859502i	$0.670774\pi$
$504$	−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	$-0.386970\pi$
$521$	15.8719	0.695360	0.347680	+	0.937613i	$0.386970\pi$
$522$	−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	$-0.186605\pi$
$541$	38.7515	1.66606	0.833028	+	0.553231i	$0.186605\pi$
$542$	−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	$-0.280135\pi$
$547$	29.8009	1.27420	0.637098	+	0.770783i	$0.280135\pi$
$548$	−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	$-0.653878\pi$
$571$	−22.2140	−0.929626	−0.464813	+	0.885409i	$0.653878\pi$
$572$	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	$-0.286184\pi$
$601$	30.5134	1.24467	0.622334	+	0.782752i	$0.286184\pi$
$602$	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	$-0.437794\pi$
$607$	9.56833	0.388366	0.194183	+	0.980965i	$0.437794\pi$
$608$	−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	$-0.462569\pi$
$617$	5.82850	0.234647	0.117323	+	0.993094i	$0.462569\pi$
$618$	0.693770	0.0279075
$619$	−17.6026	−0.707510	−0.353755	−	0.935338i	$-0.615095\pi$
$619$	−17.6026	−0.707510	−0.353755	+	0.935338i	$0.615095\pi$
$620$	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	$-0.566351\pi$
$641$	−10.4786	−0.413882	−0.206941	+	0.978353i	$0.566351\pi$
$642$	1.53234	0.0604767
$643$	−8.62068	−0.339967	−0.169983	−	0.985447i	$-0.554371\pi$
$643$	−8.62068	−0.339967	−0.169983	+	0.985447i	$0.554371\pi$
$644$	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	$-0.273682\pi$
$653$	33.3525	1.30518	0.652592	+	0.757709i	$0.273682\pi$
$654$	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	$-0.313410\pi$
$673$	28.7021	1.10638	0.553191	+	0.833054i	$0.313410\pi$
$674$	0.712857	0.0274582
$675$	0	0
$676$	−16.5812	−0.637740
$677$	3.97611	0.152814	0.0764071	−	0.997077i	$-0.475655\pi$
$677$	3.97611	0.152814	0.0764071	+	0.997077i	$0.475655\pi$
$678$	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	$-0.606216\pi$
$683$	−17.1195	−0.655060	−0.327530	+	0.944841i	$0.606216\pi$
$684$	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	$-0.510839\pi$
$691$	−1.78983	−0.0680883	−0.0340441	+	0.999420i	$0.510839\pi$
$692$	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	$-0.334081\pi$
$709$	26.5187	0.995930	0.497965	+	0.867197i	$0.334081\pi$
$710$	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	$-0.522479\pi$
$719$	−3.78411	−0.141123	−0.0705617	+	0.997507i	$0.522479\pi$
$720$	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	$-0.256560\pi$
$743$	37.7461	1.38477	0.692384	+	0.721529i	$0.256560\pi$
$744$	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	$-0.442310\pi$
$751$	9.87924	0.360499	0.180249	+	0.983621i	$0.442310\pi$
$752$	−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	$-0.440949\pi$
$787$	10.3491	0.368904	0.184452	+	0.982842i	$0.440949\pi$
$788$	−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	$-0.333914\pi$
$809$	28.3530	0.996837	0.498419	+	0.866936i	$0.333914\pi$
$810$	0	0
$811$	−26.9233	−0.945405	−0.472702	−	0.881222i	$-0.656721\pi$
$811$	−26.9233	−0.945405	−0.472702	+	0.881222i	$0.656721\pi$
$812$	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	$-0.522873\pi$
$821$	−4.11428	−0.143589	−0.0717946	+	0.997419i	$0.522873\pi$
$822$	0.478464	0.0166884
$823$	7.39842	0.257893	0.128946	−	0.991652i	$-0.458840\pi$
$823$	7.39842	0.257893	0.128946	+	0.991652i	$0.458840\pi$
$824$	−1.02108	−0.0355709
$825$	0	0
$826$	0.0958013	0.00333335
$827$	38.9992	1.35614	0.678068	−	0.734999i	$-0.262818\pi$
$827$	38.9992	1.35614	0.678068	+	0.734999i	$0.262818\pi$
$828$	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	$-0.469279\pi$
$839$	5.58244	0.192727	0.0963637	+	0.995346i	$0.469279\pi$
$840$	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	$-0.421101\pi$
$853$	14.3308	0.490676	0.245338	+	0.969438i	$0.421101\pi$
$854$	−0.0316580	−0.00108331
$855$	0	0
$856$	−2.25527	−0.0770836
$857$	17.4020	0.594440	0.297220	−	0.954809i	$-0.403941\pi$
$857$	17.4020	0.594440	0.297220	+	0.954809i	$0.403941\pi$
$858$	−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	$-0.390564\pi$
$877$	19.9641	0.674141	0.337071	+	0.941479i	$0.390564\pi$
$878$	0.645495	0.0217844
$879$	−90.3318	−3.04682
$880$	0	0
$881$	−36.5987	−1.23304	−0.616520	−	0.787339i	$-0.711458\pi$
$881$	−36.5987	−1.23304	−0.616520	+	0.787339i	$0.711458\pi$
$882$	−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	$-0.636856\pi$
$887$	−24.8280	−0.833644	−0.416822	+	0.908988i	$0.636856\pi$
$888$	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	$-0.261608\pi$
$907$	41.0099	1.36171	0.680856	+	0.732417i	$0.261608\pi$
$908$	48.4201	1.60688
$909$	−44.8729	−1.48834
$910$	0	0
$911$	28.5011	0.944284	0.472142	−	0.881523i	$-0.343481\pi$
$911$	28.5011	0.944284	0.472142	+	0.881523i	$0.343481\pi$
$912$	−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	$-0.898468\pi$
$929$	−57.8841	−1.89912	−0.949558	+	0.313591i	$0.898468\pi$
$930$	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	$-0.121033\pi$
$941$	56.9695	1.85715	0.928577	+	0.371140i	$0.121033\pi$
$942$	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	$-0.708616\pi$
$947$	−37.5106	−1.21893	−0.609466	+	0.792812i	$0.708616\pi$
$948$	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	$-0.432578\pi$
$983$	13.1827	0.420462	0.210231	+	0.977652i	$0.432578\pi$
$984$	−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	$-0.475462\pi$
$991$	4.84861	0.154021	0.0770106	+	0.997030i	$0.475462\pi$
$992$	−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	$-0.123997\pi$
$997$	58.4194	1.85016	0.925081	+	0.379770i	$0.123997\pi$
$998$	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.bt.1.8	✓	15
5.4	even	2		7225.2.a.bv.1.8	yes	15
17.16	even	2		7225.2.a.bw.1.8	yes	15
85.84	even	2		7225.2.a.bu.1.8	yes	15

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

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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