LMFDB - Embedded newform 7225.2.a.bu.1.5

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			1.5
Root			$0.788473$ 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)

$f(q)$	$=$	$q-0.788473 q^{2} -1.07579 q^{3} -1.37831 q^{4} +0.848234 q^{6} +2.07026 q^{7} +2.66371 q^{8} -1.84267 q^{9} +O(q^{10})$	\(q-0.788473 q^{2} -1.07579 q^{3} -1.37831 q^{4} +0.848234 q^{6} +2.07026 q^{7} +2.66371 q^{8} -1.84267 q^{9} +0.902211 q^{11} +1.48278 q^{12} +5.02210 q^{13} -1.63234 q^{14} +0.656358 q^{16} +1.45290 q^{18} +4.51882 q^{19} -2.22717 q^{21} -0.711369 q^{22} -2.03589 q^{23} -2.86560 q^{24} -3.95979 q^{26} +5.20971 q^{27} -2.85346 q^{28} -8.44232 q^{29} -2.80267 q^{31} -5.84493 q^{32} -0.970591 q^{33} +2.53977 q^{36} -1.78625 q^{37} -3.56297 q^{38} -5.40274 q^{39} -5.78401 q^{41} +1.75606 q^{42} -6.03159 q^{43} -1.24353 q^{44} +1.60524 q^{46} -8.94852 q^{47} -0.706105 q^{48} -2.71404 q^{49} -6.92201 q^{52} +2.77409 q^{53} -4.10772 q^{54} +5.51456 q^{56} -4.86131 q^{57} +6.65655 q^{58} -8.44590 q^{59} +14.0841 q^{61} +2.20983 q^{62} -3.81480 q^{63} +3.29586 q^{64} +0.765285 q^{66} +4.39303 q^{67} +2.19019 q^{69} +12.5424 q^{71} -4.90833 q^{72} +5.60061 q^{73} +1.40841 q^{74} -6.22833 q^{76} +1.86781 q^{77} +4.25991 q^{78} -8.55981 q^{79} -0.0765534 q^{81} +4.56054 q^{82} -7.40790 q^{83} +3.06973 q^{84} +4.75575 q^{86} +9.08219 q^{87} +2.40322 q^{88} +3.23212 q^{89} +10.3970 q^{91} +2.80608 q^{92} +3.01509 q^{93} +7.05567 q^{94} +6.28794 q^{96} -9.98548 q^{97} +2.13994 q^{98} -1.66248 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.788473	−0.557535	−0.278767	−	0.960359i	$-0.589926\pi$
$2$	−0.788473	−0.557535	−0.278767	+	0.960359i	$0.589926\pi$
$3$	−1.07579	−0.621109	−0.310555	−	0.950556i	$-0.600515\pi$
$3$	−1.07579	−0.621109	−0.310555	+	0.950556i	$0.600515\pi$
$4$	−1.37831	−0.689155
$5$	0	0
$5$	0	0
$6$	0.848234	0.346290
$7$	2.07026	0.782484	0.391242	−	0.920288i	$-0.372046\pi$
$7$	2.07026	0.782484	0.391242	+	0.920288i	$0.372046\pi$
$8$	2.66371	0.941763
$9$	−1.84267	−0.614224
$10$	0	0
$11$	0.902211	0.272027	0.136013	−	0.990707i	$-0.456571\pi$
$11$	0.902211	0.272027	0.136013	+	0.990707i	$0.456571\pi$
$12$	1.48278	0.428040
$13$	5.02210	1.39288	0.696440	−	0.717615i	$-0.254766\pi$
$13$	5.02210	1.39288	0.696440	+	0.717615i	$0.254766\pi$
$14$	−1.63234	−0.436262
$15$	0	0
$16$	0.656358	0.164089
$17$	0	0
$17$	0	0
$18$	1.45290	0.342451
$19$	4.51882	1.03669	0.518344	−	0.855172i	$-0.326549\pi$
$19$	4.51882	1.03669	0.518344	+	0.855172i	$0.326549\pi$
$20$	0	0
$21$	−2.22717	−0.486008
$22$	−0.711369	−0.151664
$23$	−2.03589	−0.424512	−0.212256	−	0.977214i	$-0.568081\pi$
$23$	−2.03589	−0.424512	−0.212256	+	0.977214i	$0.568081\pi$
$24$	−2.86560	−0.584937
$25$	0	0
$26$	−3.95979	−0.776579
$27$	5.20971	1.00261
$28$	−2.85346	−0.539252
$29$	−8.44232	−1.56770	−0.783850	−	0.620950i	$-0.786747\pi$
$29$	−8.44232	−1.56770	−0.783850	+	0.620950i	$0.786747\pi$
$30$	0	0
$31$	−2.80267	−0.503375	−0.251687	−	0.967809i	$-0.580985\pi$
$31$	−2.80267	−0.503375	−0.251687	+	0.967809i	$0.580985\pi$
$32$	−5.84493	−1.03325
$33$	−0.970591	−0.168958
$34$	0	0
$35$	0	0
$36$	2.53977	0.423295
$37$	−1.78625	−0.293658	−0.146829	−	0.989162i	$-0.546907\pi$
$37$	−1.78625	−0.293658	−0.146829	+	0.989162i	$0.546907\pi$
$38$	−3.56297	−0.577990
$39$	−5.40274	−0.865130
$40$	0	0
$41$	−5.78401	−0.903310	−0.451655	−	0.892193i	$-0.649166\pi$
$41$	−5.78401	−0.903310	−0.451655	+	0.892193i	$0.649166\pi$
$42$	1.75606	0.270966
$43$	−6.03159	−0.919809	−0.459905	−	0.887968i	$-0.652116\pi$
$43$	−6.03159	−0.919809	−0.459905	+	0.887968i	$0.652116\pi$
$44$	−1.24353	−0.187469
$45$	0	0
$46$	1.60524	0.236680
$47$	−8.94852	−1.30528	−0.652638	−	0.757670i	$-0.726338\pi$
$47$	−8.94852	−1.30528	−0.652638	+	0.757670i	$0.726338\pi$
$48$	−0.706105	−0.101917
$49$	−2.71404	−0.387719
$50$	0	0
$51$	0	0
$52$	−6.92201	−0.959910
$53$	2.77409	0.381050	0.190525	−	0.981682i	$-0.438981\pi$
$53$	2.77409	0.381050	0.190525	+	0.981682i	$0.438981\pi$
$54$	−4.10772	−0.558989
$55$	0	0
$56$	5.51456	0.736914
$57$	−4.86131	−0.643896
$58$	6.65655	0.874047
$59$	−8.44590	−1.09956	−0.549781	−	0.835309i	$-0.685289\pi$
$59$	−8.44590	−1.09956	−0.549781	+	0.835309i	$0.685289\pi$
$60$	0	0
$61$	14.0841	1.80328	0.901641	−	0.432485i	$-0.142363\pi$
$61$	14.0841	1.80328	0.901641	+	0.432485i	$0.142363\pi$
$62$	2.20983	0.280649
$63$	−3.81480	−0.480620
$64$	3.29586	0.411982
$65$	0	0
$66$	0.765285	0.0942001
$67$	4.39303	0.536695	0.268347	−	0.963322i	$-0.413523\pi$
$67$	4.39303	0.536695	0.268347	+	0.963322i	$0.413523\pi$
$68$	0	0
$69$	2.19019	0.263668
$70$	0	0
$71$	12.5424	1.48851	0.744256	−	0.667895i	$-0.232804\pi$
$71$	12.5424	1.48851	0.744256	+	0.667895i	$0.232804\pi$
$72$	−4.90833	−0.578453
$73$	5.60061	0.655502	0.327751	−	0.944764i	$-0.393709\pi$
$73$	5.60061	0.655502	0.327751	+	0.944764i	$0.393709\pi$
$74$	1.40841	0.163724
$75$	0	0
$76$	−6.22833	−0.714439
$77$	1.86781	0.212856
$78$	4.25991	0.482340
$79$	−8.55981	−0.963054	−0.481527	−	0.876431i	$-0.659918\pi$
$79$	−8.55981	−0.963054	−0.481527	+	0.876431i	$0.659918\pi$
$80$	0	0
$81$	−0.0765534	−0.00850593
$82$	4.56054	0.503627
$83$	−7.40790	−0.813122	−0.406561	−	0.913624i	$-0.633272\pi$
$83$	−7.40790	−0.813122	−0.406561	+	0.913624i	$0.633272\pi$
$84$	3.06973	0.334935
$85$	0	0
$86$	4.75575	0.512826
$87$	9.08219	0.973713
$88$	2.40322	0.256185
$89$	3.23212	0.342604	0.171302	−	0.985219i	$-0.445203\pi$
$89$	3.23212	0.342604	0.171302	+	0.985219i	$0.445203\pi$
$90$	0	0
$91$	10.3970	1.08991
$92$	2.80608	0.292554
$93$	3.01509	0.312651
$94$	7.05567	0.727737
$95$	0	0
$96$	6.28794	0.641760
$97$	−9.98548	−1.01387	−0.506936	−	0.861984i	$-0.669222\pi$
$97$	−9.98548	−1.01387	−0.506936	+	0.861984i	$0.669222\pi$
$98$	2.13994	0.216167
$99$	−1.66248	−0.167085
$100$	0	0
$101$	8.51374	0.847149	0.423574	−	0.905861i	$-0.360775\pi$
$101$	8.51374	0.847149	0.423574	+	0.905861i	$0.360775\pi$
$102$	0	0
$103$	7.45176	0.734244	0.367122	−	0.930173i	$-0.380343\pi$
$103$	7.45176	0.734244	0.367122	+	0.930173i	$0.380343\pi$
$104$	13.3774	1.31176
$105$	0	0
$106$	−2.18729	−0.212449
$107$	−13.3482	−1.29042	−0.645208	−	0.764007i	$-0.723229\pi$
$107$	−13.3482	−1.29042	−0.645208	+	0.764007i	$0.723229\pi$
$108$	−7.18059	−0.690953
$109$	19.4651	1.86442	0.932210	−	0.361919i	$-0.117878\pi$
$109$	19.4651	1.86442	0.932210	+	0.361919i	$0.117878\pi$
$110$	0	0
$111$	1.92163	0.182394
$112$	1.35883	0.128397
$113$	−7.85690	−0.739115	−0.369557	−	0.929208i	$-0.620491\pi$
$113$	−7.85690	−0.739115	−0.369557	+	0.929208i	$0.620491\pi$
$114$	3.83301	0.358995
$115$	0	0
$116$	11.6361	1.08039
$117$	−9.25407	−0.855539
$118$	6.65936	0.613044
$119$	0	0
$120$	0	0
$121$	−10.1860	−0.926001
$122$	−11.1049	−1.00539
$123$	6.22239	0.561054
$124$	3.86295	0.346903
$125$	0	0
$126$	3.00787	0.267962
$127$	18.3302	1.62655	0.813273	−	0.581883i	$-0.197684\pi$
$127$	18.3302	1.62655	0.813273	+	0.581883i	$0.197684\pi$
$128$	9.09117	0.803554
$129$	6.48874	0.571302
$130$	0	0
$131$	−14.5594	−1.27206	−0.636032	−	0.771663i	$-0.719425\pi$
$131$	−14.5594	−1.27206	−0.636032	+	0.771663i	$0.719425\pi$
$132$	1.33778	0.116438
$133$	9.35511	0.811191
$134$	−3.46379	−0.299226
$135$	0	0
$136$	0	0
$137$	−9.50780	−0.812306	−0.406153	−	0.913805i	$-0.633130\pi$
$137$	−9.50780	−0.812306	−0.406153	+	0.913805i	$0.633130\pi$
$138$	−1.72691	−0.147004
$139$	−7.55424	−0.640742	−0.320371	−	0.947292i	$-0.603808\pi$
$139$	−7.55424	−0.640742	−0.320371	+	0.947292i	$0.603808\pi$
$140$	0	0
$141$	9.62675	0.810719
$142$	−9.88936	−0.829897
$143$	4.53099	0.378900
$144$	−1.20945	−0.100788
$145$	0	0
$146$	−4.41593	−0.365465
$147$	2.91974	0.240816
$148$	2.46201	0.202376
$149$	−13.4039	−1.09809	−0.549044	−	0.835793i	$-0.685008\pi$
$149$	−13.4039	−1.09809	−0.549044	+	0.835793i	$0.685008\pi$
$150$	0	0
$151$	18.4749	1.50347	0.751733	−	0.659467i	$-0.229218\pi$
$151$	18.4749	1.50347	0.751733	+	0.659467i	$0.229218\pi$
$152$	12.0368	0.976314
$153$	0	0
$154$	−1.47272	−0.118675
$155$	0	0
$156$	7.44664	0.596209
$157$	−23.6328	−1.88611	−0.943053	−	0.332643i	$-0.892060\pi$
$157$	−23.6328	−1.88611	−0.943053	+	0.332643i	$0.892060\pi$
$158$	6.74919	0.536936
$159$	−2.98434	−0.236674
$160$	0	0
$161$	−4.21481	−0.332173
$162$	0.0603603	0.00474235
$163$	−3.34966	−0.262366	−0.131183	−	0.991358i	$-0.541878\pi$
$163$	−3.34966	−0.262366	−0.131183	+	0.991358i	$0.541878\pi$
$164$	7.97216	0.622521
$165$	0	0
$166$	5.84093	0.453344
$167$	2.50935	0.194179	0.0970897	−	0.995276i	$-0.469047\pi$
$167$	2.50935	0.194179	0.0970897	+	0.995276i	$0.469047\pi$
$168$	−5.93252	−0.457704
$169$	12.2215	0.940113
$170$	0	0
$171$	−8.32669	−0.636758
$172$	8.31340	0.633891
$173$	−25.7668	−1.95901	−0.979506	−	0.201415i	$-0.935446\pi$
$173$	−25.7668	−1.95901	−0.979506	+	0.201415i	$0.935446\pi$
$174$	−7.16106	−0.542879
$175$	0	0
$176$	0.592173	0.0446367
$177$	9.08603	0.682948
$178$	−2.54844	−0.191014
$179$	17.9590	1.34232	0.671161	−	0.741311i	$-0.265796\pi$
$179$	17.9590	1.34232	0.671161	+	0.741311i	$0.265796\pi$
$180$	0	0
$181$	4.83467	0.359358	0.179679	−	0.983725i	$-0.442494\pi$
$181$	4.83467	0.359358	0.179679	+	0.983725i	$0.442494\pi$
$182$	−8.19778	−0.607660
$183$	−15.1515	−1.12003
$184$	−5.42301	−0.399789
$185$	0	0
$186$	−2.37732	−0.174314
$187$	0	0
$188$	12.3338	0.899537
$189$	10.7854	0.784525
$190$	0	0
$191$	22.1295	1.60123	0.800616	−	0.599178i	$-0.204506\pi$
$191$	22.1295	1.60123	0.800616	+	0.599178i	$0.204506\pi$
$192$	−3.54566	−0.255886
$193$	−17.2102	−1.23882	−0.619408	−	0.785069i	$-0.712627\pi$
$193$	−17.2102	−1.23882	−0.619408	+	0.785069i	$0.712627\pi$
$194$	7.87328	0.565269
$195$	0	0
$196$	3.74078	0.267199
$197$	−3.97386	−0.283126	−0.141563	−	0.989929i	$-0.545213\pi$
$197$	−3.97386	−0.283126	−0.141563	+	0.989929i	$0.545213\pi$
$198$	1.31082	0.0931558
$199$	−14.0746	−0.997725	−0.498863	−	0.866681i	$-0.666249\pi$
$199$	−14.0746	−0.997725	−0.498863	+	0.866681i	$0.666249\pi$
$200$	0	0
$201$	−4.72599	−0.333346
$202$	−6.71286	−0.472315
$203$	−17.4778	−1.22670
$204$	0	0
$205$	0	0
$206$	−5.87551	−0.409366
$207$	3.75147	0.260745
$208$	3.29629	0.228557
$209$	4.07693	0.282007
$210$	0	0
$211$	−5.15984	−0.355218	−0.177609	−	0.984101i	$-0.556836\pi$
$211$	−5.15984	−0.355218	−0.177609	+	0.984101i	$0.556836\pi$
$212$	−3.82355	−0.262603
$213$	−13.4930	−0.924528
$214$	10.5247	0.719451
$215$	0	0
$216$	13.8771	0.944220
$217$	−5.80225	−0.393882
$218$	−15.3477	−1.03948
$219$	−6.02509	−0.407138
$220$	0	0
$221$	0	0
$222$	−1.51516	−0.101691
$223$	12.5122	0.837880	0.418940	−	0.908014i	$-0.362402\pi$
$223$	12.5122	0.837880	0.418940	+	0.908014i	$0.362402\pi$
$224$	−12.1005	−0.808500
$225$	0	0
$226$	6.19496	0.412082
$227$	−28.0069	−1.85889	−0.929443	−	0.368967i	$-0.879712\pi$
$227$	−28.0069	−1.85889	−0.929443	+	0.368967i	$0.879712\pi$
$228$	6.70039	0.443744
$229$	20.0829	1.32711	0.663557	−	0.748125i	$-0.269046\pi$
$229$	20.0829	1.32711	0.663557	+	0.748125i	$0.269046\pi$
$230$	0	0
$231$	−2.00937	−0.132207
$232$	−22.4879	−1.47640
$233$	−3.54472	−0.232223	−0.116111	−	0.993236i	$-0.537043\pi$
$233$	−3.54472	−0.232223	−0.116111	+	0.993236i	$0.537043\pi$
$234$	7.29659	0.476993
$235$	0	0
$236$	11.6411	0.757769
$237$	9.20858	0.598162
$238$	0	0
$239$	−17.6098	−1.13908	−0.569542	−	0.821962i	$-0.692880\pi$
$239$	−17.6098	−1.13908	−0.569542	+	0.821962i	$0.692880\pi$
$240$	0	0
$241$	20.2549	1.30473	0.652366	−	0.757904i	$-0.273777\pi$
$241$	20.2549	1.30473	0.652366	+	0.757904i	$0.273777\pi$
$242$	8.03140	0.516278
$243$	−15.5468	−0.997326
$244$	−19.4122	−1.24274
$245$	0	0
$246$	−4.90619	−0.312807
$247$	22.6939	1.44398
$248$	−7.46550	−0.474059
$249$	7.96936	0.505038
$250$	0	0
$251$	4.69991	0.296656	0.148328	−	0.988938i	$-0.452611\pi$
$251$	4.69991	0.296656	0.148328	+	0.988938i	$0.452611\pi$
$252$	5.25798	0.331222
$253$	−1.83680	−0.115479
$254$	−14.4529	−0.906856
$255$	0	0
$256$	−13.7599	−0.859992
$257$	30.1169	1.87864	0.939320	−	0.343041i	$-0.111457\pi$
$257$	30.1169	1.87864	0.939320	+	0.343041i	$0.111457\pi$
$258$	−5.11620	−0.318521
$259$	−3.69800	−0.229782
$260$	0	0
$261$	15.5564	0.962918
$262$	11.4797	0.709220
$263$	9.50761	0.586264	0.293132	−	0.956072i	$-0.405302\pi$
$263$	9.50761	0.586264	0.293132	+	0.956072i	$0.405302\pi$
$264$	−2.58537	−0.159119
$265$	0	0
$266$	−7.37626	−0.452267
$267$	−3.47709	−0.212794
$268$	−6.05496	−0.369866
$269$	11.1736	0.681267	0.340634	−	0.940196i	$-0.389358\pi$
$269$	11.1736	0.681267	0.340634	+	0.940196i	$0.389358\pi$
$270$	0	0
$271$	−6.92805	−0.420849	−0.210425	−	0.977610i	$-0.567485\pi$
$271$	−6.92805	−0.420849	−0.210425	+	0.977610i	$0.567485\pi$
$272$	0	0
$273$	−11.1851	−0.676950
$274$	7.49665	0.452889
$275$	0	0
$276$	−3.01876	−0.181708
$277$	−23.5320	−1.41390	−0.706950	−	0.707263i	$-0.749930\pi$
$277$	−23.5320	−1.41390	−0.706950	+	0.707263i	$0.749930\pi$
$278$	5.95632	0.357236
$279$	5.16440	0.309185
$280$	0	0
$281$	−19.0200	−1.13464	−0.567320	−	0.823498i	$-0.692020\pi$
$281$	−19.0200	−1.13464	−0.567320	+	0.823498i	$0.692020\pi$
$282$	−7.59044	−0.452004
$283$	13.8428	0.822869	0.411435	−	0.911439i	$-0.365028\pi$
$283$	13.8428	0.822869	0.411435	+	0.911439i	$0.365028\pi$
$284$	−17.2873	−1.02582
$285$	0	0
$286$	−3.57256	−0.211250
$287$	−11.9744	−0.706826
$288$	10.7703	0.634645
$289$	0	0
$290$	0	0
$291$	10.7423	0.629725
$292$	−7.71938	−0.451742
$293$	−5.50682	−0.321712	−0.160856	−	0.986978i	$-0.551425\pi$
$293$	−5.50682	−0.321712	−0.160856	+	0.986978i	$0.551425\pi$
$294$	−2.30214	−0.134263
$295$	0	0
$296$	−4.75805	−0.276556
$297$	4.70025	0.272736
$298$	10.5686	0.612222
$299$	−10.2244	−0.591294
$300$	0	0
$301$	−12.4869	−0.719736
$302$	−14.5670	−0.838235
$303$	−9.15902	−0.526172
$304$	2.96596	0.170110
$305$	0	0
$306$	0	0
$307$	−2.81999	−0.160946	−0.0804728	−	0.996757i	$-0.525643\pi$
$307$	−2.81999	−0.160946	−0.0804728	+	0.996757i	$0.525643\pi$
$308$	−2.57442	−0.146691
$309$	−8.01655	−0.456045
$310$	0	0
$311$	−2.26079	−0.128198	−0.0640989	−	0.997944i	$-0.520417\pi$
$311$	−2.26079	−0.128198	−0.0640989	+	0.997944i	$0.520417\pi$
$312$	−14.3913	−0.814747
$313$	−19.4941	−1.10187	−0.550936	−	0.834548i	$-0.685729\pi$
$313$	−19.4941	−1.10187	−0.550936	+	0.834548i	$0.685729\pi$
$314$	18.6339	1.05157
$315$	0	0
$316$	11.7981	0.663694
$317$	15.1955	0.853462	0.426731	−	0.904378i	$-0.359665\pi$
$317$	15.1955	0.853462	0.426731	+	0.904378i	$0.359665\pi$
$318$	2.35307	0.131954
$319$	−7.61675	−0.426456
$320$	0	0
$321$	14.3598	0.801489
$322$	3.32326	0.185198
$323$	0	0
$324$	0.105514	0.00586191
$325$	0	0
$326$	2.64112	0.146278
$327$	−20.9404	−1.15801
$328$	−15.4069	−0.850704
$329$	−18.5257	−1.02136
$330$	0	0
$331$	7.96570	0.437834	0.218917	−	0.975743i	$-0.429748\pi$
$331$	7.96570	0.437834	0.218917	+	0.975743i	$0.429748\pi$
$332$	10.2104	0.560367
$333$	3.29147	0.180372
$334$	−1.97856	−0.108262
$335$	0	0
$336$	−1.46182	−0.0797487
$337$	−17.6321	−0.960482	−0.480241	−	0.877137i	$-0.659451\pi$
$337$	−17.6321	−0.960482	−0.480241	+	0.877137i	$0.659451\pi$
$338$	−9.63630	−0.524146
$339$	8.45239	0.459071
$340$	0	0
$341$	−2.52860	−0.136931
$342$	6.56538	0.355015
$343$	−20.1106	−1.08587
$344$	−16.0664	−0.866242
$345$	0	0
$346$	20.3164	1.09222
$347$	32.2774	1.73274	0.866371	−	0.499401i	$-0.166447\pi$
$347$	32.2774	1.73274	0.866371	+	0.499401i	$0.166447\pi$
$348$	−12.5181	−0.671039
$349$	−26.1383	−1.39915	−0.699575	−	0.714559i	$-0.746627\pi$
$349$	−26.1383	−1.39915	−0.699575	+	0.714559i	$0.746627\pi$
$350$	0	0
$351$	26.1637	1.39651
$352$	−5.27336	−0.281071
$353$	12.1939	0.649018	0.324509	−	0.945883i	$-0.394801\pi$
$353$	12.1939	0.649018	0.324509	+	0.945883i	$0.394801\pi$
$354$	−7.16409	−0.380767
$355$	0	0
$356$	−4.45486	−0.236107
$357$	0	0
$358$	−14.1602	−0.748391
$359$	1.09010	0.0575330	0.0287665	−	0.999586i	$-0.490842\pi$
$359$	1.09010	0.0575330	0.0287665	+	0.999586i	$0.490842\pi$
$360$	0	0
$361$	1.41972	0.0747219
$362$	−3.81201	−0.200355
$363$	10.9580	0.575148
$364$	−14.3303	−0.751114
$365$	0	0
$366$	11.9466	0.624458
$367$	13.6855	0.714376	0.357188	−	0.934033i	$-0.383736\pi$
$367$	13.6855	0.714376	0.357188	+	0.934033i	$0.383736\pi$
$368$	−1.33627	−0.0696579
$369$	10.6580	0.554835
$370$	0	0
$371$	5.74308	0.298166
$372$	−4.15573	−0.215465
$373$	7.67264	0.397274	0.198637	−	0.980073i	$-0.436348\pi$
$373$	7.67264	0.397274	0.198637	+	0.980073i	$0.436348\pi$
$374$	0	0
$375$	0	0
$376$	−23.8362	−1.22926
$377$	−42.3982	−2.18362
$378$	−8.50403	−0.437400
$379$	−26.3147	−1.35170	−0.675848	−	0.737041i	$-0.736223\pi$
$379$	−26.3147	−1.35170	−0.675848	+	0.737041i	$0.736223\pi$
$380$	0	0
$381$	−19.7195	−1.01026
$382$	−17.4485	−0.892743
$383$	26.1213	1.33474	0.667368	−	0.744728i	$-0.267421\pi$
$383$	26.1213	1.33474	0.667368	+	0.744728i	$0.267421\pi$
$384$	−9.78021	−0.499094
$385$	0	0
$386$	13.5698	0.690683
$387$	11.1142	0.564968
$388$	13.7631	0.698714
$389$	−28.5958	−1.44986	−0.724932	−	0.688821i	$-0.758129\pi$
$389$	−28.5958	−1.44986	−0.724932	+	0.688821i	$0.758129\pi$
$390$	0	0
$391$	0	0
$392$	−7.22940	−0.365140
$393$	15.6629	0.790090
$394$	3.13328	0.157852
$395$	0	0
$396$	2.29141	0.115148
$397$	8.70559	0.436921	0.218460	−	0.975846i	$-0.429897\pi$
$397$	8.70559	0.436921	0.218460	+	0.975846i	$0.429897\pi$
$398$	11.0975	0.556267
$399$	−10.0642	−0.503838
$400$	0	0
$401$	−31.7417	−1.58511	−0.792553	−	0.609803i	$-0.791249\pi$
$401$	−31.7417	−1.58511	−0.792553	+	0.609803i	$0.791249\pi$
$402$	3.72632	0.185852
$403$	−14.0753	−0.701140
$404$	−11.7346	−0.583817
$405$	0	0
$406$	13.7808	0.683928
$407$	−1.61157	−0.0798828
$408$	0	0
$409$	15.9369	0.788028	0.394014	−	0.919104i	$-0.371086\pi$
$409$	15.9369	0.788028	0.394014	+	0.919104i	$0.371086\pi$
$410$	0	0
$411$	10.2284	0.504531
$412$	−10.2708	−0.506008
$413$	−17.4852	−0.860389
$414$	−2.95793	−0.145374
$415$	0	0
$416$	−29.3538	−1.43919
$417$	8.12679	0.397971
$418$	−3.21455	−0.157229
$419$	−13.1828	−0.644022	−0.322011	−	0.946736i	$-0.604359\pi$
$419$	−13.1828	−0.644022	−0.322011	+	0.946736i	$0.604359\pi$
$420$	0	0
$421$	0.0723712	0.00352716	0.00176358	−	0.999998i	$-0.499439\pi$
$421$	0.0723712	0.00352716	0.00176358	+	0.999998i	$0.499439\pi$
$422$	4.06840	0.198047
$423$	16.4892	0.801731
$424$	7.38936	0.358859
$425$	0	0
$426$	10.6389	0.515457
$427$	29.1577	1.41104
$428$	18.3979	0.889296
$429$	−4.87441	−0.235339
$430$	0	0
$431$	19.1010	0.920063	0.460032	−	0.887902i	$-0.347838\pi$
$431$	19.1010	0.920063	0.460032	+	0.887902i	$0.347838\pi$
$432$	3.41943	0.164518
$433$	9.76517	0.469284	0.234642	−	0.972082i	$-0.424608\pi$
$433$	9.76517	0.469284	0.234642	+	0.972082i	$0.424608\pi$
$434$	4.57492	0.219603
$435$	0	0
$436$	−26.8290	−1.28487
$437$	−9.19980	−0.440086
$438$	4.75063	0.226994
$439$	9.92378	0.473636	0.236818	−	0.971554i	$-0.423895\pi$
$439$	9.92378	0.473636	0.236818	+	0.971554i	$0.423895\pi$
$440$	0	0
$441$	5.00107	0.238146
$442$	0	0
$443$	−16.7955	−0.797976	−0.398988	−	0.916956i	$-0.630639\pi$
$443$	−16.7955	−0.797976	−0.398988	+	0.916956i	$0.630639\pi$
$444$	−2.64861	−0.125697
$445$	0	0
$446$	−9.86555	−0.467148
$447$	14.4198	0.682033
$448$	6.82328	0.322369
$449$	10.4651	0.493880	0.246940	−	0.969031i	$-0.420575\pi$
$449$	10.4651	0.493880	0.246940	+	0.969031i	$0.420575\pi$
$450$	0	0
$451$	−5.21839	−0.245725
$452$	10.8292	0.509365
$453$	−19.8752	−0.933817
$454$	22.0827	1.03639
$455$	0	0
$456$	−12.9491	−0.606397
$457$	−37.0481	−1.73304	−0.866518	−	0.499146i	$-0.833647\pi$
$457$	−37.0481	−1.73304	−0.866518	+	0.499146i	$0.833647\pi$
$458$	−15.8348	−0.739913
$459$	0	0
$460$	0	0
$461$	−17.7022	−0.824472	−0.412236	−	0.911077i	$-0.635252\pi$
$461$	−17.7022	−0.824472	−0.412236	+	0.911077i	$0.635252\pi$
$462$	1.58434	0.0737100
$463$	−28.0982	−1.30584	−0.652918	−	0.757429i	$-0.726455\pi$
$463$	−28.0982	−1.30584	−0.652918	+	0.757429i	$0.726455\pi$
$464$	−5.54118	−0.257243
$465$	0	0
$466$	2.79492	0.129472
$467$	32.4081	1.49967	0.749835	−	0.661625i	$-0.230133\pi$
$467$	32.4081	1.49967	0.749835	+	0.661625i	$0.230133\pi$
$468$	12.7550	0.589599
$469$	9.09471	0.419955
$470$	0	0
$471$	25.4240	1.17148
$472$	−22.4974	−1.03553
$473$	−5.44177	−0.250213
$474$	−7.26072	−0.333496
$475$	0	0
$476$	0	0
$477$	−5.11173	−0.234050
$478$	13.8849	0.635079
$479$	18.7989	0.858943	0.429472	−	0.903080i	$-0.358700\pi$
$479$	18.7989	0.858943	0.429472	+	0.903080i	$0.358700\pi$
$480$	0	0
$481$	−8.97073	−0.409030
$482$	−15.9704	−0.727433
$483$	4.53426	0.206316
$484$	14.0395	0.638158
$485$	0	0
$486$	12.2582	0.556044
$487$	−9.16522	−0.415316	−0.207658	−	0.978202i	$-0.566584\pi$
$487$	−9.16522	−0.415316	−0.207658	+	0.978202i	$0.566584\pi$
$488$	37.5159	1.69826
$489$	3.60354	0.162958
$490$	0	0
$491$	−31.9128	−1.44021	−0.720103	−	0.693867i	$-0.755905\pi$
$491$	−31.9128	−1.44021	−0.720103	+	0.693867i	$0.755905\pi$
$492$	−8.57639	−0.386653
$493$	0	0
$494$	−17.8936	−0.805070
$495$	0	0
$496$	−1.83956	−0.0825985
$497$	25.9660	1.16474
$498$	−6.28363	−0.281576
$499$	−27.0644	−1.21157	−0.605785	−	0.795628i	$-0.707141\pi$
$499$	−27.0644	−1.21157	−0.605785	+	0.795628i	$0.707141\pi$
$500$	0	0
$501$	−2.69954	−0.120607
$502$	−3.70575	−0.165396
$503$	−31.3690	−1.39867	−0.699337	−	0.714793i	$-0.746521\pi$
$503$	−31.3690	−1.39867	−0.699337	+	0.714793i	$0.746521\pi$
$504$	−10.1615	−0.452630
$505$	0	0
$506$	1.44827	0.0643833
$507$	−13.1478	−0.583913
$508$	−25.2647	−1.12094
$509$	31.7249	1.40618	0.703090	−	0.711101i	$-0.251803\pi$
$509$	31.7249	1.40618	0.703090	+	0.711101i	$0.251803\pi$
$510$	0	0
$511$	11.5947	0.512919
$512$	−7.33306	−0.324078
$513$	23.5417	1.03939
$514$	−23.7464	−1.04741
$515$	0	0
$516$	−8.94349	−0.393715
$517$	−8.07345	−0.355070
$518$	2.91577	0.128112
$519$	27.7197	1.21676
$520$	0	0
$521$	2.43658	0.106749	0.0533743	−	0.998575i	$-0.483002\pi$
$521$	2.43658	0.106749	0.0533743	+	0.998575i	$0.483002\pi$
$522$	−12.2658	−0.536860
$523$	−4.81223	−0.210424	−0.105212	−	0.994450i	$-0.533552\pi$
$523$	−4.81223	−0.210424	−0.105212	+	0.994450i	$0.533552\pi$
$524$	20.0674	0.876649
$525$	0	0
$526$	−7.49649	−0.326863
$527$	0	0
$528$	−0.637055	−0.0277243
$529$	−18.8552	−0.819790
$530$	0	0
$531$	15.5630	0.675377
$532$	−12.8942	−0.559037
$533$	−29.0479	−1.25820
$534$	2.74159	0.118640
$535$	0	0
$536$	11.7018	0.505439
$537$	−19.3202	−0.833729
$538$	−8.81009	−0.379830
$539$	−2.44863	−0.105470
$540$	0	0
$541$	−16.2448	−0.698418	−0.349209	−	0.937045i	$-0.613550\pi$
$541$	−16.2448	−0.698418	−0.349209	+	0.937045i	$0.613550\pi$
$542$	5.46258	0.234638
$543$	−5.20110	−0.223200
$544$	0	0
$545$	0	0
$546$	8.81911	0.377423
$547$	−41.1006	−1.75733	−0.878667	−	0.477436i	$-0.841566\pi$
$547$	−41.1006	−1.75733	−0.878667	+	0.477436i	$0.841566\pi$
$548$	13.1047	0.559805
$549$	−25.9523	−1.10762
$550$	0	0
$551$	−38.1493	−1.62522
$552$	5.83403	0.248313
$553$	−17.7210	−0.753574
$554$	18.5544	0.788299
$555$	0	0
$556$	10.4121	0.441571
$557$	10.2936	0.436153	0.218077	−	0.975932i	$-0.430022\pi$
$557$	10.2936	0.436153	0.218077	+	0.975932i	$0.430022\pi$
$558$	−4.07199	−0.172381
$559$	−30.2912	−1.28118
$560$	0	0
$561$	0	0
$562$	14.9968	0.632601
$563$	−25.6216	−1.07982	−0.539911	−	0.841722i	$-0.681542\pi$
$563$	−25.6216	−1.07982	−0.539911	+	0.841722i	$0.681542\pi$
$564$	−13.2686	−0.558711
$565$	0	0
$566$	−10.9147	−0.458778
$567$	−0.158485	−0.00665575
$568$	33.4093	1.40182
$569$	−14.6943	−0.616018	−0.308009	−	0.951384i	$-0.599663\pi$
$569$	−14.6943	−0.616018	−0.308009	+	0.951384i	$0.599663\pi$
$570$	0	0
$571$	−0.748113	−0.0313076	−0.0156538	−	0.999877i	$-0.504983\pi$
$571$	−0.748113	−0.0313076	−0.0156538	+	0.999877i	$0.504983\pi$
$572$	−6.24511	−0.261121
$573$	−23.8067	−0.994540
$574$	9.44148	0.394080
$575$	0	0
$576$	−6.07318	−0.253049
$577$	−4.98871	−0.207683	−0.103841	−	0.994594i	$-0.533113\pi$
$577$	−4.98871	−0.207683	−0.103841	+	0.994594i	$0.533113\pi$
$578$	0	0
$579$	18.5146	0.769440
$580$	0	0
$581$	−15.3363	−0.636255
$582$	−8.47002	−0.351093
$583$	2.50281	0.103656
$584$	14.9184	0.617327
$585$	0	0
$586$	4.34198	0.179366
$587$	−25.1019	−1.03607	−0.518033	−	0.855361i	$-0.673335\pi$
$587$	−25.1019	−1.03607	−0.518033	+	0.855361i	$0.673335\pi$
$588$	−4.02431	−0.165960
$589$	−12.6648	−0.521842
$590$	0	0
$591$	4.27505	0.175852
$592$	−1.17242	−0.0481861
$593$	0.924289	0.0379560	0.0189780	−	0.999820i	$-0.493959\pi$
$593$	0.924289	0.0379560	0.0189780	+	0.999820i	$0.493959\pi$
$594$	−3.70602	−0.152060
$595$	0	0
$596$	18.4747	0.756753
$597$	15.1414	0.619696
$598$	8.06169	0.329667
$599$	6.56172	0.268104	0.134052	−	0.990974i	$-0.457201\pi$
$599$	6.56172	0.268104	0.134052	+	0.990974i	$0.457201\pi$
$600$	0	0
$601$	−11.4662	−0.467718	−0.233859	−	0.972271i	$-0.575135\pi$
$601$	−11.4662	−0.467718	−0.233859	+	0.972271i	$0.575135\pi$
$602$	9.84562	0.401278
$603$	−8.09492	−0.329650
$604$	−25.4642	−1.03612
$605$	0	0
$606$	7.22164	0.293359
$607$	−21.1625	−0.858961	−0.429480	−	0.903076i	$-0.641303\pi$
$607$	−21.1625	−0.858961	−0.429480	+	0.903076i	$0.641303\pi$
$608$	−26.4122	−1.07116
$609$	18.8025	0.761914
$610$	0	0
$611$	−44.9403	−1.81809
$612$	0	0
$613$	−15.9265	−0.643264	−0.321632	−	0.946865i	$-0.604231\pi$
$613$	−15.9265	−0.643264	−0.321632	+	0.946865i	$0.604231\pi$
$614$	2.22349	0.0897328
$615$	0	0
$616$	4.97529	0.200460
$617$	18.0614	0.727126	0.363563	−	0.931570i	$-0.381560\pi$
$617$	18.0614	0.727126	0.363563	+	0.931570i	$0.381560\pi$
$618$	6.32083	0.254261
$619$	−1.31449	−0.0528340	−0.0264170	−	0.999651i	$-0.508410\pi$
$619$	−1.31449	−0.0528340	−0.0264170	+	0.999651i	$0.508410\pi$
$620$	0	0
$621$	−10.6064	−0.425619
$622$	1.78257	0.0714747
$623$	6.69132	0.268082
$624$	−3.54613	−0.141959
$625$	0	0
$626$	15.3706	0.614332
$627$	−4.38593	−0.175157
$628$	32.5734	1.29982
$629$	0	0
$630$	0	0
$631$	−12.1119	−0.482166	−0.241083	−	0.970505i	$-0.577503\pi$
$631$	−12.1119	−0.482166	−0.241083	+	0.970505i	$0.577503\pi$
$632$	−22.8008	−0.906969
$633$	5.55092	0.220629
$634$	−11.9812	−0.475835
$635$	0	0
$636$	4.11335	0.163105
$637$	−13.6302	−0.540046
$638$	6.00561	0.237764
$639$	−23.1116	−0.914279
$640$	0	0
$641$	17.3463	0.685139	0.342569	−	0.939493i	$-0.388703\pi$
$641$	17.3463	0.685139	0.342569	+	0.939493i	$0.388703\pi$
$642$	−11.3224	−0.446858
$643$	7.34129	0.289512	0.144756	−	0.989467i	$-0.453760\pi$
$643$	7.34129	0.289512	0.144756	+	0.989467i	$0.453760\pi$
$644$	5.80931	0.228919
$645$	0	0
$646$	0	0
$647$	18.2953	0.719261	0.359630	−	0.933095i	$-0.382903\pi$
$647$	18.2953	0.719261	0.359630	+	0.933095i	$0.382903\pi$
$648$	−0.203916	−0.00801057
$649$	−7.61998	−0.299110
$650$	0	0
$651$	6.24202	0.244644
$652$	4.61687	0.180811
$653$	−27.3978	−1.07216	−0.536079	−	0.844168i	$-0.680095\pi$
$653$	−27.3978	−1.07216	−0.536079	+	0.844168i	$0.680095\pi$
$654$	16.5110	0.645630
$655$	0	0
$656$	−3.79638	−0.148224
$657$	−10.3201	−0.402625
$658$	14.6070	0.569442
$659$	−39.0437	−1.52093	−0.760463	−	0.649381i	$-0.775028\pi$
$659$	−39.0437	−1.52093	−0.760463	+	0.649381i	$0.775028\pi$
$660$	0	0
$661$	8.56725	0.333228	0.166614	−	0.986022i	$-0.446717\pi$
$661$	8.56725	0.333228	0.166614	+	0.986022i	$0.446717\pi$
$662$	−6.28074	−0.244108
$663$	0	0
$664$	−19.7325	−0.765768
$665$	0	0
$666$	−2.59524	−0.100563
$667$	17.1876	0.665507
$668$	−3.45866	−0.133820
$669$	−13.4606	−0.520415
$670$	0	0
$671$	12.7068	0.490541
$672$	13.0176	0.502167
$673$	25.7678	0.993276	0.496638	−	0.867958i	$-0.334568\pi$
$673$	25.7678	0.993276	0.496638	+	0.867958i	$0.334568\pi$
$674$	13.9024	0.535502
$675$	0	0
$676$	−16.8450	−0.647884
$677$	8.31015	0.319385	0.159693	−	0.987167i	$-0.448950\pi$
$677$	8.31015	0.319385	0.159693	+	0.987167i	$0.448950\pi$
$678$	−6.66449	−0.255948
$679$	−20.6725	−0.793338
$680$	0	0
$681$	30.1296	1.15457
$682$	1.99373	0.0763440
$683$	−40.0297	−1.53169	−0.765846	−	0.643024i	$-0.777680\pi$
$683$	−40.0297	−1.53169	−0.765846	+	0.643024i	$0.777680\pi$
$684$	11.4768	0.438825
$685$	0	0
$686$	15.8566	0.605409
$687$	−21.6050	−0.824283
$688$	−3.95888	−0.150931
$689$	13.9317	0.530757
$690$	0	0
$691$	12.0028	0.456609	0.228305	−	0.973590i	$-0.426682\pi$
$691$	12.0028	0.456609	0.228305	+	0.973590i	$0.426682\pi$
$692$	35.5146	1.35006
$693$	−3.44175	−0.130741
$694$	−25.4499	−0.966064
$695$	0	0
$696$	24.1923	0.917006
$697$	0	0
$698$	20.6093	0.780075
$699$	3.81339	0.144236
$700$	0	0
$701$	2.25640	0.0852229	0.0426115	−	0.999092i	$-0.486432\pi$
$701$	2.25640	0.0852229	0.0426115	+	0.999092i	$0.486432\pi$
$702$	−20.6294	−0.778605
$703$	−8.07174	−0.304431
$704$	2.97356	0.112070
$705$	0	0
$706$	−9.61459	−0.361850
$707$	17.6256	0.662880
$708$	−12.5234	−0.470657
$709$	23.0989	0.867499	0.433749	−	0.901034i	$-0.357190\pi$
$709$	23.0989	0.867499	0.433749	+	0.901034i	$0.357190\pi$
$710$	0	0
$711$	15.7729	0.591531
$712$	8.60942	0.322652
$713$	5.70592	0.213688
$714$	0	0
$715$	0	0
$716$	−24.7531	−0.925068
$717$	18.9445	0.707496
$718$	−0.859511	−0.0320767
$719$	17.7725	0.662804	0.331402	−	0.943490i	$-0.392478\pi$
$719$	17.7725	0.662804	0.331402	+	0.943490i	$0.392478\pi$
$720$	0	0
$721$	15.4271	0.574534
$722$	−1.11941	−0.0416601
$723$	−21.7900	−0.810381
$724$	−6.66367	−0.247653
$725$	0	0
$726$	−8.64012	−0.320665
$727$	6.09357	0.225998	0.112999	−	0.993595i	$-0.463954\pi$
$727$	6.09357	0.225998	0.112999	+	0.993595i	$0.463954\pi$
$728$	27.6947	1.02643
$729$	16.9548	0.627954
$730$	0	0
$731$	0	0
$732$	20.8835	0.771877
$733$	12.8326	0.473983	0.236992	−	0.971512i	$-0.423839\pi$
$733$	12.8326	0.473983	0.236992	+	0.971512i	$0.423839\pi$
$734$	−10.7906	−0.398289
$735$	0	0
$736$	11.8996	0.438626
$737$	3.96344	0.145995
$738$	−8.40357	−0.309340
$739$	14.0525	0.516930	0.258465	−	0.966021i	$-0.416783\pi$
$739$	14.0525	0.516930	0.258465	+	0.966021i	$0.416783\pi$
$740$	0	0
$741$	−24.4140	−0.896870
$742$	−4.52826	−0.166238
$743$	−42.5949	−1.56266	−0.781328	−	0.624121i	$-0.785457\pi$
$743$	−42.5949	−1.56266	−0.781328	+	0.624121i	$0.785457\pi$
$744$	8.03132	0.294443
$745$	0	0
$746$	−6.04968	−0.221494
$747$	13.6503	0.499439
$748$	0	0
$749$	−27.6341	−1.00973
$750$	0	0
$751$	−23.1590	−0.845084	−0.422542	−	0.906343i	$-0.638862\pi$
$751$	−23.1590	−0.845084	−0.422542	+	0.906343i	$0.638862\pi$
$752$	−5.87343	−0.214182
$753$	−5.05613	−0.184255
$754$	33.4298	1.21744
$755$	0	0
$756$	−14.8657	−0.540659
$757$	−7.68037	−0.279148	−0.139574	−	0.990212i	$-0.544573\pi$
$757$	−7.68037	−0.279148	−0.139574	+	0.990212i	$0.544573\pi$
$758$	20.7485	0.753618
$759$	1.97601	0.0717248
$760$	0	0
$761$	27.4276	0.994250	0.497125	−	0.867679i	$-0.334389\pi$
$761$	27.4276	0.994250	0.497125	+	0.867679i	$0.334389\pi$
$762$	15.5483	0.563256
$763$	40.2978	1.45888
$764$	−30.5013	−1.10350
$765$	0	0
$766$	−20.5959	−0.744161
$767$	−42.4161	−1.53156
$768$	14.8028	0.534149
$769$	20.2469	0.730122	0.365061	−	0.930984i	$-0.381048\pi$
$769$	20.2469	0.730122	0.365061	+	0.930984i	$0.381048\pi$
$770$	0	0
$771$	−32.3995	−1.16684
$772$	23.7210	0.853736
$773$	30.6071	1.10086	0.550430	−	0.834881i	$-0.314464\pi$
$773$	30.6071	1.10086	0.550430	+	0.834881i	$0.314464\pi$
$774$	−8.76328	−0.314990
$775$	0	0
$776$	−26.5984	−0.954826
$777$	3.97828	0.142720
$778$	22.5470	0.808350
$779$	−26.1369	−0.936451
$780$	0	0
$781$	11.3159	0.404915
$782$	0	0
$783$	−43.9820	−1.57179
$784$	−1.78138	−0.0636207
$785$	0	0
$786$	−12.3498	−0.440503
$787$	−28.0170	−0.998699	−0.499349	−	0.866401i	$-0.666428\pi$
$787$	−28.0170	−0.998699	−0.499349	+	0.866401i	$0.666428\pi$
$788$	5.47721	0.195117
$789$	−10.2282	−0.364134
$790$	0	0
$791$	−16.2658	−0.578345
$792$	−4.42835	−0.157355
$793$	70.7316	2.51175
$794$	−6.86412	−0.243599
$795$	0	0
$796$	19.3992	0.687587
$797$	−15.3940	−0.545283	−0.272641	−	0.962116i	$-0.587897\pi$
$797$	−15.3940	−0.545283	−0.272641	+	0.962116i	$0.587897\pi$
$798$	7.93532	0.280907
$799$	0	0
$800$	0	0
$801$	−5.95573	−0.210435
$802$	25.0275	0.883752
$803$	5.05293	0.178314
$804$	6.51388	0.229727
$805$	0	0
$806$	11.0980	0.390910
$807$	−12.0205	−0.423141
$808$	22.6781	0.797813
$809$	−13.4231	−0.471932	−0.235966	−	0.971761i	$-0.575825\pi$
$809$	−13.4231	−0.471932	−0.235966	+	0.971761i	$0.575825\pi$
$810$	0	0
$811$	−16.5354	−0.580636	−0.290318	−	0.956930i	$-0.593761\pi$
$811$	−16.5354	−0.580636	−0.290318	+	0.956930i	$0.593761\pi$
$812$	24.0898	0.845386
$813$	7.45314	0.261393
$814$	1.27068	0.0445374
$815$	0	0
$816$	0	0
$817$	−27.2557	−0.953555
$818$	−12.5658	−0.439353
$819$	−19.1583	−0.669446
$820$	0	0
$821$	0.951598	0.0332110	0.0166055	−	0.999862i	$-0.494714\pi$
$821$	0.951598	0.0332110	0.0166055	+	0.999862i	$0.494714\pi$
$822$	−8.06484	−0.281294
$823$	−26.7507	−0.932469	−0.466235	−	0.884661i	$-0.654390\pi$
$823$	−26.7507	−0.932469	−0.466235	+	0.884661i	$0.654390\pi$
$824$	19.8493	0.691483
$825$	0	0
$826$	13.7866	0.479697
$827$	−25.0742	−0.871914	−0.435957	−	0.899968i	$-0.643590\pi$
$827$	−25.0742	−0.871914	−0.435957	+	0.899968i	$0.643590\pi$
$828$	−5.17069	−0.179694
$829$	6.49996	0.225753	0.112876	−	0.993609i	$-0.463994\pi$
$829$	6.49996	0.225753	0.112876	+	0.993609i	$0.463994\pi$
$830$	0	0
$831$	25.3155	0.878187
$832$	16.5521	0.573842
$833$	0	0
$834$	−6.40776	−0.221883
$835$	0	0
$836$	−5.61927	−0.194346
$837$	−14.6011	−0.504688
$838$	10.3943	0.359065
$839$	15.6114	0.538966	0.269483	−	0.963005i	$-0.413147\pi$
$839$	15.6114	0.538966	0.269483	+	0.963005i	$0.413147\pi$
$840$	0	0
$841$	42.2728	1.45768
$842$	−0.0570628	−0.00196651
$843$	20.4616	0.704735
$844$	7.11186	0.244800
$845$	0	0
$846$	−13.0013	−0.446993
$847$	−21.0877	−0.724581
$848$	1.82079	0.0625263
$849$	−14.8920	−0.511092
$850$	0	0
$851$	3.63660	0.124661
$852$	18.5976	0.637143
$853$	37.2357	1.27493	0.637463	−	0.770481i	$-0.279984\pi$
$853$	37.2357	1.27493	0.637463	+	0.770481i	$0.279984\pi$
$854$	−22.9900	−0.786703
$855$	0	0
$856$	−35.5556	−1.21526
$857$	22.3876	0.764747	0.382374	−	0.924008i	$-0.375107\pi$
$857$	22.3876	0.764747	0.382374	+	0.924008i	$0.375107\pi$
$858$	3.84334	0.131209
$859$	9.69288	0.330717	0.165358	−	0.986234i	$-0.447122\pi$
$859$	9.69288	0.330717	0.165358	+	0.986234i	$0.447122\pi$
$860$	0	0
$861$	12.8820	0.439016
$862$	−15.0606	−0.512967
$863$	−23.7777	−0.809403	−0.404702	−	0.914449i	$-0.632625\pi$
$863$	−23.7777	−0.809403	−0.404702	+	0.914449i	$0.632625\pi$
$864$	−30.4504	−1.03594
$865$	0	0
$866$	−7.69957	−0.261642
$867$	0	0
$868$	7.99730	0.271446
$869$	−7.72276	−0.261977
$870$	0	0
$871$	22.0623	0.747551
$872$	51.8493	1.75584
$873$	18.3999	0.622744
$874$	7.25380	0.245363
$875$	0	0
$876$	8.30445	0.280581
$877$	−54.8214	−1.85119	−0.925594	−	0.378519i	$-0.876434\pi$
$877$	−54.8214	−1.85119	−0.925594	+	0.378519i	$0.876434\pi$
$878$	−7.82463	−0.264069
$879$	5.92420	0.199818
$880$	0	0
$881$	−50.3600	−1.69667	−0.848336	−	0.529458i	$-0.822395\pi$
$881$	−50.3600	−1.69667	−0.848336	+	0.529458i	$0.822395\pi$
$882$	−3.94321	−0.132775
$883$	16.8491	0.567017	0.283508	−	0.958970i	$-0.408502\pi$
$883$	16.8491	0.567017	0.283508	+	0.958970i	$0.408502\pi$
$884$	0	0
$885$	0	0
$886$	13.2428	0.444900
$887$	17.1874	0.577096	0.288548	−	0.957465i	$-0.406827\pi$
$887$	17.1874	0.577096	0.288548	+	0.957465i	$0.406827\pi$
$888$	5.11867	0.171771
$889$	37.9483	1.27274
$890$	0	0
$891$	−0.0690673	−0.00231384
$892$	−17.2457	−0.577429
$893$	−40.4367	−1.35316
$894$	−11.3696	−0.380257
$895$	0	0
$896$	18.8211	0.628768
$897$	10.9994	0.367258
$898$	−8.25147	−0.275355
$899$	23.6611	0.789140
$900$	0	0
$901$	0	0
$902$	4.11456	0.137000
$903$	13.4334	0.447034
$904$	−20.9285	−0.696071
$905$	0	0
$906$	15.6710	0.520635
$907$	−16.3554	−0.543073	−0.271537	−	0.962428i	$-0.587532\pi$
$907$	−16.3554	−0.543073	−0.271537	+	0.962428i	$0.587532\pi$
$908$	38.6022	1.28106
$909$	−15.6880	−0.520339
$910$	0	0
$911$	−23.5675	−0.780826	−0.390413	−	0.920640i	$-0.627668\pi$
$911$	−23.5675	−0.780826	−0.390413	+	0.920640i	$0.627668\pi$
$912$	−3.19076	−0.105657
$913$	−6.68348	−0.221191
$914$	29.2114	0.966228
$915$	0	0
$916$	−27.6804	−0.914588
$917$	−30.1418	−0.995369
$918$	0	0
$919$	38.1020	1.25687	0.628435	−	0.777862i	$-0.283696\pi$
$919$	38.1020	1.25687	0.628435	+	0.777862i	$0.283696\pi$
$920$	0	0
$921$	3.03373	0.0999648
$922$	13.9577	0.459672
$923$	62.9893	2.07332
$924$	2.76954	0.0911112
$925$	0	0
$926$	22.1547	0.728049
$927$	−13.7311	−0.450990
$928$	49.3448	1.61982
$929$	37.6395	1.23491	0.617456	−	0.786606i	$-0.288163\pi$
$929$	37.6395	1.23491	0.617456	+	0.786606i	$0.288163\pi$
$930$	0	0
$931$	−12.2642	−0.401944
$932$	4.88573	0.160037
$933$	2.43214	0.0796248
$934$	−25.5529	−0.836118
$935$	0	0
$936$	−24.6501	−0.805715
$937$	−30.5201	−0.997049	−0.498525	−	0.866876i	$-0.666125\pi$
$937$	−30.5201	−0.997049	−0.498525	+	0.866876i	$0.666125\pi$
$938$	−7.17094	−0.234139
$939$	20.9716	0.684382
$940$	0	0
$941$	−7.73766	−0.252240	−0.126120	−	0.992015i	$-0.540253\pi$
$941$	−7.73766	−0.252240	−0.126120	+	0.992015i	$0.540253\pi$
$942$	−20.0462	−0.653139
$943$	11.7756	0.383466
$944$	−5.54353	−0.180427
$945$	0	0
$946$	4.29069	0.139502
$947$	32.9687	1.07134	0.535670	−	0.844427i	$-0.320059\pi$
$947$	32.9687	1.07134	0.535670	+	0.844427i	$0.320059\pi$
$948$	−12.6923	−0.412226
$949$	28.1268	0.913035
$950$	0	0
$951$	−16.3472	−0.530093
$952$	0	0
$953$	51.3121	1.66216	0.831080	−	0.556152i	$-0.187723\pi$
$953$	51.3121	1.66216	0.831080	+	0.556152i	$0.187723\pi$
$954$	4.03046	0.130491
$955$	0	0
$956$	24.2718	0.785006
$957$	8.19405	0.264876
$958$	−14.8224	−0.478891
$959$	−19.6836	−0.635616
$960$	0	0
$961$	−23.1450	−0.746614
$962$	7.07318	0.228048
$963$	24.5963	0.792603
$964$	−27.9175	−0.899162
$965$	0	0
$966$	−3.57514	−0.115028
$967$	48.9208	1.57319	0.786593	−	0.617471i	$-0.211843\pi$
$967$	48.9208	1.57319	0.786593	+	0.617471i	$0.211843\pi$
$968$	−27.1326	−0.872074
$969$	0	0
$970$	0	0
$971$	31.5368	1.01206	0.506031	−	0.862515i	$-0.331112\pi$
$971$	31.5368	1.01206	0.506031	+	0.862515i	$0.331112\pi$
$972$	21.4283	0.687312
$973$	−15.6392	−0.501370
$974$	7.22653	0.231553
$975$	0	0
$976$	9.24420	0.295900
$977$	45.0499	1.44127	0.720637	−	0.693312i	$-0.243849\pi$
$977$	45.0499	1.44127	0.720637	+	0.693312i	$0.243849\pi$
$978$	−2.84130	−0.0908546
$979$	2.91605	0.0931975
$980$	0	0
$981$	−35.8678	−1.14517
$982$	25.1624	0.802965
$983$	−10.6399	−0.339361	−0.169681	−	0.985499i	$-0.554274\pi$
$983$	−10.6399	−0.339361	−0.169681	+	0.985499i	$0.554274\pi$
$984$	16.5746	0.528380
$985$	0	0
$986$	0	0
$987$	19.9298	0.634374
$988$	−31.2793	−0.995127
$989$	12.2796	0.390470
$990$	0	0
$991$	26.5787	0.844299	0.422150	−	0.906526i	$-0.361276\pi$
$991$	26.5787	0.844299	0.422150	+	0.906526i	$0.361276\pi$
$992$	16.3814	0.520111
$993$	−8.56944	−0.271943
$994$	−20.4735	−0.649381
$995$	0	0
$996$	−10.9842	−0.348049
$997$	7.76783	0.246009	0.123005	−	0.992406i	$-0.460747\pi$
$997$	7.76783	0.246009	0.123005	+	0.992406i	$0.460747\pi$
$998$	21.3396	0.675493
$999$	−9.30584	−0.294424

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.bu.1.5	yes	15
5.4	even	2		7225.2.a.bw.1.11	yes	15
17.16	even	2		7225.2.a.bv.1.5	yes	15
85.84	even	2		7225.2.a.bt.1.11	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7225.2.a.bt.1.11	✓	15	85.84	even	2
7225.2.a.bu.1.5	yes	15	1.1	even	1	trivial
7225.2.a.bv.1.5	yes	15	17.16	even	2
7225.2.a.bw.1.11	yes	15	5.4	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)

\(f(q)\)	\(=\)	\(q-0.788473 q^{2} -1.07579 q^{3} -1.37831 q^{4} +0.848234 q^{6} +2.07026 q^{7} +2.66371 q^{8} -1.84267 q^{9} +O(q^{10})\)	\(q-0.788473 q^{2} -1.07579 q^{3} -1.37831 q^{4} +0.848234 q^{6} +2.07026 q^{7} +2.66371 q^{8} -1.84267 q^{9} +0.902211 q^{11} +1.48278 q^{12} +5.02210 q^{13} -1.63234 q^{14} +0.656358 q^{16} +1.45290 q^{18} +4.51882 q^{19} -2.22717 q^{21} -0.711369 q^{22} -2.03589 q^{23} -2.86560 q^{24} -3.95979 q^{26} +5.20971 q^{27} -2.85346 q^{28} -8.44232 q^{29} -2.80267 q^{31} -5.84493 q^{32} -0.970591 q^{33} +2.53977 q^{36} -1.78625 q^{37} -3.56297 q^{38} -5.40274 q^{39} -5.78401 q^{41} +1.75606 q^{42} -6.03159 q^{43} -1.24353 q^{44} +1.60524 q^{46} -8.94852 q^{47} -0.706105 q^{48} -2.71404 q^{49} -6.92201 q^{52} +2.77409 q^{53} -4.10772 q^{54} +5.51456 q^{56} -4.86131 q^{57} +6.65655 q^{58} -8.44590 q^{59} +14.0841 q^{61} +2.20983 q^{62} -3.81480 q^{63} +3.29586 q^{64} +0.765285 q^{66} +4.39303 q^{67} +2.19019 q^{69} +12.5424 q^{71} -4.90833 q^{72} +5.60061 q^{73} +1.40841 q^{74} -6.22833 q^{76} +1.86781 q^{77} +4.25991 q^{78} -8.55981 q^{79} -0.0765534 q^{81} +4.56054 q^{82} -7.40790 q^{83} +3.06973 q^{84} +4.75575 q^{86} +9.08219 q^{87} +2.40322 q^{88} +3.23212 q^{89} +10.3970 q^{91} +2.80608 q^{92} +3.01509 q^{93} +7.05567 q^{94} +6.28794 q^{96} -9.98548 q^{97} +2.13994 q^{98} -1.66248 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 9 q^{3} + 12 q^{4} + 9 q^{6} - 12 q^{7} - 6 q^{8} + 12 q^{9}+O(q^{10}) \) 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

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more