LMFDB - Embedded newform 9025.2.a.ch.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9025,2,Mod(1,9025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9025, 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("9025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9025 = 5^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9025.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:	$72.0649878242$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 12x^{8} + 23x^{7} + 47x^{6} - 86x^{5} - 69x^{4} + 115x^{3} + 34x^{2} - 45x + 5 $ x^10 - 2x^9 - 12x^8 + 23x^7 + 47x^6 - 86x^5 - 69x^4 + 115x^3 + 34x^2 - 45*x + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-1.06941$ of defining polynomial
Character	$\chi$	$=$	9025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.06941 q^{2} -3.33626 q^{3} -0.856371 q^{4} -3.56781 q^{6} +4.50384 q^{7} -3.05462 q^{8} +8.13062 q^{9} +O(q^{10})$	\(q+1.06941 q^{2} -3.33626 q^{3} -0.856371 q^{4} -3.56781 q^{6} +4.50384 q^{7} -3.05462 q^{8} +8.13062 q^{9} -1.71622 q^{11} +2.85707 q^{12} -2.17068 q^{13} +4.81643 q^{14} -1.55389 q^{16} +1.77579 q^{17} +8.69493 q^{18} -15.0260 q^{21} -1.83534 q^{22} +2.25047 q^{23} +10.1910 q^{24} -2.32134 q^{26} -17.1171 q^{27} -3.85695 q^{28} -3.69265 q^{29} +9.44856 q^{31} +4.44750 q^{32} +5.72575 q^{33} +1.89904 q^{34} -6.96283 q^{36} +6.04438 q^{37} +7.24196 q^{39} -6.06586 q^{41} -16.0688 q^{42} +1.08803 q^{43} +1.46972 q^{44} +2.40667 q^{46} +4.75787 q^{47} +5.18417 q^{48} +13.2845 q^{49} -5.92450 q^{51} +1.85891 q^{52} +9.88132 q^{53} -18.3051 q^{54} -13.7575 q^{56} -3.94894 q^{58} +5.91109 q^{59} +2.02928 q^{61} +10.1043 q^{62} +36.6190 q^{63} +7.86396 q^{64} +6.12316 q^{66} -12.8860 q^{67} -1.52074 q^{68} -7.50816 q^{69} -13.3300 q^{71} -24.8359 q^{72} -3.20010 q^{73} +6.46390 q^{74} -7.72957 q^{77} +7.74459 q^{78} -3.60102 q^{79} +32.7151 q^{81} -6.48687 q^{82} -8.09553 q^{83} +12.8678 q^{84} +1.16355 q^{86} +12.3196 q^{87} +5.24240 q^{88} -4.43455 q^{89} -9.77639 q^{91} -1.92724 q^{92} -31.5228 q^{93} +5.08809 q^{94} -14.8380 q^{96} -8.90483 q^{97} +14.2066 q^{98} -13.9539 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{2} + 8 q^{4} + 4 q^{6} + 4 q^{7} - 3 q^{8} + 12 q^{9}+O(q^{10}) $ 10 * q - 2 * q^2 + 8 * q^4 + 4 * q^6 + 4 * q^7 - 3 * q^8 + 12 * q^9	$ 10 q - 2 q^{2} + 8 q^{4} + 4 q^{6} + 4 q^{7} - 3 q^{8} + 12 q^{9} - 3 q^{11} + 14 q^{12} + 5 q^{13} + 14 q^{14} - 4 q^{16} + 11 q^{17} - 4 q^{18} - 17 q^{21} + 34 q^{22} + 32 q^{23} - 7 q^{24} - 13 q^{26} - 30 q^{27} - 12 q^{28} + 8 q^{29} + 11 q^{31} - 8 q^{32} + 19 q^{33} + 20 q^{34} - 4 q^{36} - 35 q^{37} + 20 q^{39} - 16 q^{41} + 32 q^{42} + 26 q^{43} - 44 q^{44} + 21 q^{46} + 19 q^{47} + 16 q^{48} + 18 q^{49} - 15 q^{51} + 25 q^{52} - 16 q^{53} - q^{54} + 20 q^{56} + 7 q^{58} - 2 q^{59} + 8 q^{61} + 13 q^{62} + 77 q^{63} + q^{64} + 20 q^{66} + 40 q^{68} - 12 q^{69} - 2 q^{71} - 31 q^{72} + q^{73} + 18 q^{74} + 18 q^{77} - 41 q^{78} - 18 q^{79} + 58 q^{81} + 25 q^{82} + 46 q^{83} - 24 q^{84} + 33 q^{86} + 57 q^{87} + 37 q^{88} + 13 q^{89} - 67 q^{91} + 33 q^{92} + 4 q^{93} - 25 q^{94} - 40 q^{96} - 8 q^{97} + 62 q^{98} - 62 q^{99}+O(q^{100}) $ 10 * q - 2 * q^2 + 8 * q^4 + 4 * q^6 + 4 * q^7 - 3 * q^8 + 12 * q^9 - 3 * q^11 + 14 * q^12 + 5 * q^13 + 14 * q^14 - 4 * q^16 + 11 * q^17 - 4 * q^18 - 17 * q^21 + 34 * q^22 + 32 * q^23 - 7 * q^24 - 13 * q^26 - 30 * q^27 - 12 * q^28 + 8 * q^29 + 11 * q^31 - 8 * q^32 + 19 * q^33 + 20 * q^34 - 4 * q^36 - 35 * q^37 + 20 * q^39 - 16 * q^41 + 32 * q^42 + 26 * q^43 - 44 * q^44 + 21 * q^46 + 19 * q^47 + 16 * q^48 + 18 * q^49 - 15 * q^51 + 25 * q^52 - 16 * q^53 - q^54 + 20 * q^56 + 7 * q^58 - 2 * q^59 + 8 * q^61 + 13 * q^62 + 77 * q^63 + q^64 + 20 * q^66 + 40 * q^68 - 12 * q^69 - 2 * q^71 - 31 * q^72 + q^73 + 18 * q^74 + 18 * q^77 - 41 * q^78 - 18 * q^79 + 58 * q^81 + 25 * q^82 + 46 * q^83 - 24 * q^84 + 33 * q^86 + 57 * q^87 + 37 * q^88 + 13 * q^89 - 67 * q^91 + 33 * q^92 + 4 * q^93 - 25 * q^94 - 40 * q^96 - 8 * q^97 + 62 * q^98 - 62 * 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 $

$2$	1.06941	0.756184	0.378092	−	0.925768i	$-0.376580\pi$
$2$	1.06941	0.756184	0.378092	+	0.925768i	$0.376580\pi$
$3$	−3.33626	−1.92619	−0.963095	−	0.269163i	$-0.913253\pi$
$3$	−3.33626	−1.92619	−0.963095	+	0.269163i	$0.913253\pi$
$4$	−0.856371	−0.428186
$5$	0	0
$5$	0	0
$6$	−3.56781	−1.45655
$7$	4.50384	1.70229	0.851145	−	0.524931i	$-0.175909\pi$
$7$	4.50384	1.70229	0.851145	+	0.524931i	$0.175909\pi$
$8$	−3.05462	−1.07997
$9$	8.13062	2.71021
$10$	0	0
$11$	−1.71622	−0.517460	−0.258730	−	0.965950i	$-0.583304\pi$
$11$	−1.71622	−0.517460	−0.258730	+	0.965950i	$0.583304\pi$
$12$	2.85707	0.824766
$13$	−2.17068	−0.602039	−0.301019	−	0.953618i	$-0.597327\pi$
$13$	−2.17068	−0.602039	−0.301019	+	0.953618i	$0.597327\pi$
$14$	4.81643	1.28724
$15$	0	0
$16$	−1.55389	−0.388472
$17$	1.77579	0.430693	0.215346	−	0.976538i	$-0.430912\pi$
$17$	1.77579	0.430693	0.215346	+	0.976538i	$0.430912\pi$
$18$	8.69493	2.04941
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−15.0260	−3.27893
$22$	−1.83534	−0.391295
$23$	2.25047	0.469256	0.234628	−	0.972085i	$-0.424613\pi$
$23$	2.25047	0.469256	0.234628	+	0.972085i	$0.424613\pi$
$24$	10.1910	2.08023
$25$	0	0
$26$	−2.32134	−0.455252
$27$	−17.1171	−3.29418
$28$	−3.85695	−0.728896
$29$	−3.69265	−0.685707	−0.342854	−	0.939389i	$-0.611394\pi$
$29$	−3.69265	−0.685707	−0.342854	+	0.939389i	$0.611394\pi$
$30$	0	0
$31$	9.44856	1.69701	0.848505	−	0.529187i	$-0.177503\pi$
$31$	9.44856	1.69701	0.848505	+	0.529187i	$0.177503\pi$
$32$	4.44750	0.786215
$33$	5.72575	0.996726
$34$	1.89904	0.325683
$35$	0	0
$36$	−6.96283	−1.16047
$37$	6.04438	0.993690	0.496845	−	0.867839i	$-0.334492\pi$
$37$	6.04438	0.993690	0.496845	+	0.867839i	$0.334492\pi$
$38$	0	0
$39$	7.24196	1.15964
$40$	0	0
$41$	−6.06586	−0.947328	−0.473664	−	0.880706i	$-0.657069\pi$
$41$	−6.06586	−0.947328	−0.473664	+	0.880706i	$0.657069\pi$
$42$	−16.0688	−2.47948
$43$	1.08803	0.165923	0.0829615	−	0.996553i	$-0.473562\pi$
$43$	1.08803	0.165923	0.0829615	+	0.996553i	$0.473562\pi$
$44$	1.46972	0.221569
$45$	0	0
$46$	2.40667	0.354844
$47$	4.75787	0.694006	0.347003	−	0.937864i	$-0.387199\pi$
$47$	4.75787	0.694006	0.347003	+	0.937864i	$0.387199\pi$
$48$	5.18417	0.748270
$49$	13.2845	1.89779
$50$	0	0
$51$	−5.92450	−0.829596
$52$	1.85891	0.257784
$53$	9.88132	1.35730	0.678652	−	0.734460i	$-0.262565\pi$
$53$	9.88132	1.35730	0.678652	+	0.734460i	$0.262565\pi$
$54$	−18.3051	−2.49101
$55$	0	0
$56$	−13.7575	−1.83842
$57$	0	0
$58$	−3.94894	−0.518521
$59$	5.91109	0.769558	0.384779	−	0.923009i	$-0.374278\pi$
$59$	5.91109	0.769558	0.384779	+	0.923009i	$0.374278\pi$
$60$	0	0
$61$	2.02928	0.259823	0.129912	−	0.991526i	$-0.458531\pi$
$61$	2.02928	0.259823	0.129912	+	0.991526i	$0.458531\pi$
$62$	10.1043	1.28325
$63$	36.6190	4.61356
$64$	7.86396	0.982995
$65$	0	0
$66$	6.12316	0.753708
$67$	−12.8860	−1.57428	−0.787138	−	0.616777i	$-0.788438\pi$
$67$	−12.8860	−1.57428	−0.787138	+	0.616777i	$0.788438\pi$
$68$	−1.52074	−0.184416
$69$	−7.50816	−0.903876
$70$	0	0
$71$	−13.3300	−1.58198	−0.790988	−	0.611832i	$-0.790433\pi$
$71$	−13.3300	−1.58198	−0.790988	+	0.611832i	$0.790433\pi$
$72$	−24.8359	−2.92694
$73$	−3.20010	−0.374543	−0.187271	−	0.982308i	$-0.559964\pi$
$73$	−3.20010	−0.374543	−0.187271	+	0.982308i	$0.559964\pi$
$74$	6.46390	0.751413
$75$	0	0
$76$	0	0
$77$	−7.72957	−0.880867
$78$	7.74459	0.876902
$79$	−3.60102	−0.405147	−0.202573	−	0.979267i	$-0.564930\pi$
$79$	−3.60102	−0.405147	−0.202573	+	0.979267i	$0.564930\pi$
$80$	0	0
$81$	32.7151	3.63501
$82$	−6.48687	−0.716355
$83$	−8.09553	−0.888600	−0.444300	−	0.895878i	$-0.646547\pi$
$83$	−8.09553	−0.888600	−0.444300	+	0.895878i	$0.646547\pi$
$84$	12.8678	1.40399
$85$	0	0
$86$	1.16355	0.125468
$87$	12.3196	1.32080
$88$	5.24240	0.558842
$89$	−4.43455	−0.470061	−0.235030	−	0.971988i	$-0.575519\pi$
$89$	−4.43455	−0.470061	−0.235030	+	0.971988i	$0.575519\pi$
$90$	0	0
$91$	−9.77639	−1.02484
$92$	−1.92724	−0.200929
$93$	−31.5228	−3.26876
$94$	5.08809	0.524796
$95$	0	0
$96$	−14.8380	−1.51440
$97$	−8.90483	−0.904149	−0.452075	−	0.891980i	$-0.649316\pi$
$97$	−8.90483	−0.904149	−0.452075	+	0.891980i	$0.649316\pi$
$98$	14.2066	1.43508
$99$	−13.9539	−1.40242
$100$	0	0
$101$	0.613988	0.0610941	0.0305470	−	0.999533i	$-0.490275\pi$
$101$	0.613988	0.0610941	0.0305470	+	0.999533i	$0.490275\pi$
$102$	−6.33570	−0.627327
$103$	12.3814	1.21998	0.609989	−	0.792410i	$-0.291174\pi$
$103$	12.3814	1.21998	0.609989	+	0.792410i	$0.291174\pi$
$104$	6.63061	0.650185
$105$	0	0
$106$	10.5671	1.02637
$107$	12.8291	1.24024	0.620118	−	0.784508i	$-0.287085\pi$
$107$	12.8291	1.24024	0.620118	+	0.784508i	$0.287085\pi$
$108$	14.6586	1.41052
$109$	−9.15332	−0.876729	−0.438365	−	0.898797i	$-0.644442\pi$
$109$	−9.15332	−0.876729	−0.438365	+	0.898797i	$0.644442\pi$
$110$	0	0
$111$	−20.1656	−1.91404
$112$	−6.99845	−0.661291
$113$	10.0464	0.945085	0.472542	−	0.881308i	$-0.343336\pi$
$113$	10.0464	0.945085	0.472542	+	0.881308i	$0.343336\pi$
$114$	0	0
$115$	0	0
$116$	3.16228	0.293610
$117$	−17.6490	−1.63165
$118$	6.32135	0.581928
$119$	7.99788	0.733164
$120$	0	0
$121$	−8.05459	−0.732235
$122$	2.17013	0.196474
$123$	20.2373	1.82473
$124$	−8.09147	−0.726635
$125$	0	0
$126$	39.1605	3.48870
$127$	−1.25641	−0.111488	−0.0557440	−	0.998445i	$-0.517753\pi$
$127$	−1.25641	−0.111488	−0.0557440	+	0.998445i	$0.517753\pi$
$128$	−0.485244	−0.0428900
$129$	−3.62995	−0.319599
$130$	0	0
$131$	−2.47375	−0.216132	−0.108066	−	0.994144i	$-0.534466\pi$
$131$	−2.47375	−0.216132	−0.108066	+	0.994144i	$0.534466\pi$
$132$	−4.90337	−0.426784
$133$	0	0
$134$	−13.7804	−1.19044
$135$	0	0
$136$	−5.42437	−0.465136
$137$	14.0830	1.20319	0.601597	−	0.798800i	$-0.294532\pi$
$137$	14.0830	1.20319	0.601597	+	0.798800i	$0.294532\pi$
$138$	−8.02927	−0.683497
$139$	−10.4991	−0.890521	−0.445261	−	0.895401i	$-0.646889\pi$
$139$	−10.4991	−0.890521	−0.445261	+	0.895401i	$0.646889\pi$
$140$	0	0
$141$	−15.8735	−1.33679
$142$	−14.2551	−1.19627
$143$	3.72537	0.311531
$144$	−12.6341	−1.05284
$145$	0	0
$146$	−3.42220	−0.283223
$147$	−44.3206	−3.65550
$148$	−5.17623	−0.425484
$149$	4.66838	0.382449	0.191224	−	0.981546i	$-0.438754\pi$
$149$	4.66838	0.382449	0.191224	+	0.981546i	$0.438754\pi$
$150$	0	0
$151$	−3.63312	−0.295659	−0.147830	−	0.989013i	$-0.547229\pi$
$151$	−3.63312	−0.295659	−0.147830	+	0.989013i	$0.547229\pi$
$152$	0	0
$153$	14.4383	1.16727
$154$	−8.26605	−0.666097
$155$	0	0
$156$	−6.20180	−0.496542
$157$	6.54875	0.522647	0.261324	−	0.965251i	$-0.415841\pi$
$157$	6.54875	0.522647	0.261324	+	0.965251i	$0.415841\pi$
$158$	−3.85096	−0.306366
$159$	−32.9666	−2.61442
$160$	0	0
$161$	10.1358	0.798810
$162$	34.9857	2.74874
$163$	−20.6191	−1.61501	−0.807505	−	0.589860i	$-0.799183\pi$
$163$	−20.6191	−1.61501	−0.807505	+	0.589860i	$0.799183\pi$
$164$	5.19463	0.405632
$165$	0	0
$166$	−8.65740	−0.671945
$167$	21.3765	1.65417	0.827083	−	0.562080i	$-0.189999\pi$
$167$	21.3765	1.65417	0.827083	+	0.562080i	$0.189999\pi$
$168$	45.8986	3.54115
$169$	−8.28814	−0.637549
$170$	0	0
$171$	0	0
$172$	−0.931757	−0.0710458
$173$	−7.47782	−0.568528	−0.284264	−	0.958746i	$-0.591749\pi$
$173$	−7.47782	−0.568528	−0.284264	+	0.958746i	$0.591749\pi$
$174$	13.1747	0.998770
$175$	0	0
$176$	2.66681	0.201018
$177$	−19.7209	−1.48231
$178$	−4.74233	−0.355453
$179$	5.16304	0.385904	0.192952	−	0.981208i	$-0.438194\pi$
$179$	5.16304	0.385904	0.192952	+	0.981208i	$0.438194\pi$
$180$	0	0
$181$	3.66449	0.272379	0.136190	−	0.990683i	$-0.456514\pi$
$181$	3.66449	0.272379	0.136190	+	0.990683i	$0.456514\pi$
$182$	−10.4549	−0.774971
$183$	−6.77022	−0.500469
$184$	−6.87434	−0.506783
$185$	0	0
$186$	−33.7107	−2.47179
$187$	−3.04765	−0.222866
$188$	−4.07450	−0.297163
$189$	−77.0924	−5.60765
$190$	0	0
$191$	13.5459	0.980149	0.490074	−	0.871681i	$-0.336970\pi$
$191$	13.5459	0.980149	0.490074	+	0.871681i	$0.336970\pi$
$192$	−26.2362	−1.89343
$193$	−8.61483	−0.620109	−0.310054	−	0.950719i	$-0.600347\pi$
$193$	−8.61483	−0.620109	−0.310054	+	0.950719i	$0.600347\pi$
$194$	−9.52288	−0.683703
$195$	0	0
$196$	−11.3765	−0.812606
$197$	19.4257	1.38403	0.692013	−	0.721885i	$-0.256724\pi$
$197$	19.4257	1.38403	0.692013	+	0.721885i	$0.256724\pi$
$198$	−14.9224	−1.06049
$199$	7.96268	0.564459	0.282230	−	0.959347i	$-0.408926\pi$
$199$	7.96268	0.564459	0.282230	+	0.959347i	$0.408926\pi$
$200$	0	0
$201$	42.9910	3.03235
$202$	0.656602	0.0461984
$203$	−16.6311	−1.16727
$204$	5.07357	0.355221
$205$	0	0
$206$	13.2408	0.922528
$207$	18.2977	1.27178
$208$	3.37299	0.233875
$209$	0	0
$210$	0	0
$211$	−10.7955	−0.743194	−0.371597	−	0.928394i	$-0.621190\pi$
$211$	−10.7955	−0.743194	−0.371597	+	0.928394i	$0.621190\pi$
$212$	−8.46207	−0.581178
$213$	44.4722	3.04719
$214$	13.7195	0.937847
$215$	0	0
$216$	52.2861	3.55762
$217$	42.5547	2.88880
$218$	−9.78862	−0.662969
$219$	10.6763	0.721440
$220$	0	0
$221$	−3.85468	−0.259294
$222$	−21.5652	−1.44736
$223$	−3.77689	−0.252919	−0.126460	−	0.991972i	$-0.540361\pi$
$223$	−3.77689	−0.252919	−0.126460	+	0.991972i	$0.540361\pi$
$224$	20.0308	1.33837
$225$	0	0
$226$	10.7437	0.714658
$227$	−1.58551	−0.105234	−0.0526171	−	0.998615i	$-0.516756\pi$
$227$	−1.58551	−0.105234	−0.0526171	+	0.998615i	$0.516756\pi$
$228$	0	0
$229$	−15.5665	−1.02866	−0.514331	−	0.857592i	$-0.671960\pi$
$229$	−15.5665	−1.02866	−0.514331	+	0.857592i	$0.671960\pi$
$230$	0	0
$231$	25.7879	1.69672
$232$	11.2796	0.740544
$233$	−19.8875	−1.30287	−0.651435	−	0.758704i	$-0.725833\pi$
$233$	−19.8875	−1.30287	−0.651435	+	0.758704i	$0.725833\pi$
$234$	−18.8739	−1.23383
$235$	0	0
$236$	−5.06209	−0.329514
$237$	12.0139	0.780389
$238$	8.55298	0.554407
$239$	−8.58202	−0.555125	−0.277562	−	0.960708i	$-0.589526\pi$
$239$	−8.58202	−0.555125	−0.277562	+	0.960708i	$0.589526\pi$
$240$	0	0
$241$	12.2854	0.791373	0.395687	−	0.918386i	$-0.370507\pi$
$241$	12.2854	0.791373	0.395687	+	0.918386i	$0.370507\pi$
$242$	−8.61362	−0.553705
$243$	−57.7948	−3.70754
$244$	−1.73782	−0.111253
$245$	0	0
$246$	21.6419	1.37983
$247$	0	0
$248$	−28.8617	−1.83272
$249$	27.0088	1.71161
$250$	0	0
$251$	−25.1910	−1.59004	−0.795021	−	0.606581i	$-0.792540\pi$
$251$	−25.1910	−1.59004	−0.795021	+	0.606581i	$0.792540\pi$
$252$	−31.3594	−1.97546
$253$	−3.86231	−0.242821
$254$	−1.34361	−0.0843054
$255$	0	0
$256$	−16.2468	−1.01543
$257$	5.49152	0.342552	0.171276	−	0.985223i	$-0.445211\pi$
$257$	5.49152	0.342552	0.171276	+	0.985223i	$0.445211\pi$
$258$	−3.88189	−0.241676
$259$	27.2229	1.69155
$260$	0	0
$261$	−30.0235	−1.85841
$262$	−2.64544	−0.163436
$263$	2.70056	0.166524	0.0832619	−	0.996528i	$-0.473466\pi$
$263$	2.70056	0.166524	0.0832619	+	0.996528i	$0.473466\pi$
$264$	−17.4900	−1.07644
$265$	0	0
$266$	0	0
$267$	14.7948	0.905426
$268$	11.0352	0.674082
$269$	0.413875	0.0252344	0.0126172	−	0.999920i	$-0.495984\pi$
$269$	0.413875	0.0252344	0.0126172	+	0.999920i	$0.495984\pi$
$270$	0	0
$271$	7.04097	0.427709	0.213854	−	0.976866i	$-0.431398\pi$
$271$	7.04097	0.427709	0.213854	+	0.976866i	$0.431398\pi$
$272$	−2.75938	−0.167312
$273$	32.6166	1.97404
$274$	15.0605	0.909835
$275$	0	0
$276$	6.42977	0.387027
$277$	−17.3499	−1.04246	−0.521228	−	0.853418i	$-0.674526\pi$
$277$	−17.3499	−1.04246	−0.521228	+	0.853418i	$0.674526\pi$
$278$	−11.2278	−0.673398
$279$	76.8226	4.59925
$280$	0	0
$281$	7.81843	0.466408	0.233204	−	0.972428i	$-0.425079\pi$
$281$	7.81843	0.466408	0.233204	+	0.972428i	$0.425079\pi$
$282$	−16.9752	−1.01086
$283$	15.0611	0.895288	0.447644	−	0.894212i	$-0.352263\pi$
$283$	15.0611	0.895288	0.447644	+	0.894212i	$0.352263\pi$
$284$	11.4154	0.677379
$285$	0	0
$286$	3.98393	0.235575
$287$	−27.3196	−1.61263
$288$	36.1610	2.13080
$289$	−13.8466	−0.814504
$290$	0	0
$291$	29.7088	1.74156
$292$	2.74047	0.160374
$293$	22.6596	1.32379	0.661894	−	0.749597i	$-0.269753\pi$
$293$	22.6596	1.32379	0.661894	+	0.749597i	$0.269753\pi$
$294$	−47.3967	−2.76423
$295$	0	0
$296$	−18.4633	−1.07316
$297$	29.3767	1.70461
$298$	4.99240	0.289202
$299$	−4.88506	−0.282510
$300$	0	0
$301$	4.90031	0.282449
$302$	−3.88528	−0.223573
$303$	−2.04842	−0.117679
$304$	0	0
$305$	0	0
$306$	15.4404	0.882668
$307$	26.6086	1.51863	0.759315	−	0.650723i	$-0.225534\pi$
$307$	26.6086	1.51863	0.759315	+	0.650723i	$0.225534\pi$
$308$	6.61938	0.377174
$309$	−41.3076	−2.34991
$310$	0	0
$311$	16.8555	0.955789	0.477895	−	0.878417i	$-0.341400\pi$
$311$	16.8555	0.955789	0.477895	+	0.878417i	$0.341400\pi$
$312$	−22.1214	−1.25238
$313$	26.0803	1.47415	0.737073	−	0.675813i	$-0.236208\pi$
$313$	26.0803	1.47415	0.737073	+	0.675813i	$0.236208\pi$
$314$	7.00327	0.395217
$315$	0	0
$316$	3.08381	0.173478
$317$	−8.51947	−0.478501	−0.239251	−	0.970958i	$-0.576902\pi$
$317$	−8.51947	−0.478501	−0.239251	+	0.970958i	$0.576902\pi$
$318$	−35.2547	−1.97699
$319$	6.33740	0.354826
$320$	0	0
$321$	−42.8012	−2.38893
$322$	10.8392	0.604047
$323$	0	0
$324$	−28.0162	−1.55646
$325$	0	0
$326$	−22.0502	−1.22125
$327$	30.5378	1.68875
$328$	18.5289	1.02309
$329$	21.4286	1.18140
$330$	0	0
$331$	6.58391	0.361884	0.180942	−	0.983494i	$-0.442085\pi$
$331$	6.58391	0.361884	0.180942	+	0.983494i	$0.442085\pi$
$332$	6.93278	0.380486
$333$	49.1446	2.69311
$334$	22.8602	1.25085
$335$	0	0
$336$	23.3486	1.27377
$337$	12.7609	0.695129	0.347565	−	0.937656i	$-0.387009\pi$
$337$	12.7609	0.695129	0.347565	+	0.937656i	$0.387009\pi$
$338$	−8.86338	−0.482105
$339$	−33.5173	−1.82041
$340$	0	0
$341$	−16.2158	−0.878135
$342$	0	0
$343$	28.3045	1.52830
$344$	−3.32352	−0.179192
$345$	0	0
$346$	−7.99683	−0.429912
$347$	33.7867	1.81377	0.906884	−	0.421381i	$-0.138454\pi$
$347$	33.7867	1.81377	0.906884	+	0.421381i	$0.138454\pi$
$348$	−10.5502	−0.565549
$349$	−20.2044	−1.08152	−0.540759	−	0.841178i	$-0.681863\pi$
$349$	−20.2044	−1.08152	−0.540759	+	0.841178i	$0.681863\pi$
$350$	0	0
$351$	37.1557	1.98322
$352$	−7.63290	−0.406835
$353$	29.4688	1.56847	0.784234	−	0.620465i	$-0.213056\pi$
$353$	29.4688	1.56847	0.784234	+	0.620465i	$0.213056\pi$
$354$	−21.0897	−1.12090
$355$	0	0
$356$	3.79762	0.201273
$357$	−26.6830	−1.41221
$358$	5.52139	0.291814
$359$	32.0768	1.69295	0.846473	−	0.532431i	$-0.178721\pi$
$359$	32.0768	1.69295	0.846473	+	0.532431i	$0.178721\pi$
$360$	0	0
$361$	0	0
$362$	3.91882	0.205969
$363$	26.8722	1.41042
$364$	8.37222	0.438824
$365$	0	0
$366$	−7.24011	−0.378447
$367$	25.8563	1.34969	0.674843	−	0.737961i	$-0.264211\pi$
$367$	25.8563	1.34969	0.674843	+	0.737961i	$0.264211\pi$
$368$	−3.49698	−0.182293
$369$	−49.3192	−2.56745
$370$	0	0
$371$	44.5038	2.31052
$372$	26.9952	1.39964
$373$	33.5019	1.73466	0.867332	−	0.497729i	$-0.165833\pi$
$373$	33.5019	1.73466	0.867332	+	0.497729i	$0.165833\pi$
$374$	−3.25918	−0.168528
$375$	0	0
$376$	−14.5335	−0.749507
$377$	8.01556	0.412823
$378$	−82.4431	−4.24041
$379$	0.157134	0.00807141	0.00403571	−	0.999992i	$-0.498715\pi$
$379$	0.157134	0.00807141	0.00403571	+	0.999992i	$0.498715\pi$
$380$	0	0
$381$	4.19169	0.214747
$382$	14.4861	0.741173
$383$	28.6838	1.46568	0.732838	−	0.680403i	$-0.238195\pi$
$383$	28.6838	1.46568	0.732838	+	0.680403i	$0.238195\pi$
$384$	1.61890	0.0826142
$385$	0	0
$386$	−9.21275	−0.468916
$387$	8.84635	0.449685
$388$	7.62584	0.387144
$389$	−2.18272	−0.110668	−0.0553342	−	0.998468i	$-0.517622\pi$
$389$	−2.18272	−0.110668	−0.0553342	+	0.998468i	$0.517622\pi$
$390$	0	0
$391$	3.99637	0.202105
$392$	−40.5792	−2.04956
$393$	8.25306	0.416312
$394$	20.7740	1.04658
$395$	0	0
$396$	11.9497	0.600497
$397$	−7.56061	−0.379456	−0.189728	−	0.981837i	$-0.560761\pi$
$397$	−7.56061	−0.379456	−0.189728	+	0.981837i	$0.560761\pi$
$398$	8.51533	0.426835
$399$	0	0
$400$	0	0
$401$	39.2000	1.95755	0.978776	−	0.204931i	$-0.0656970\pi$
$401$	39.2000	1.95755	0.978776	+	0.204931i	$0.0656970\pi$
$402$	45.9749	2.29302
$403$	−20.5098	−1.02167
$404$	−0.525801	−0.0261596
$405$	0	0
$406$	−17.7854	−0.882673
$407$	−10.3735	−0.514195
$408$	18.0971	0.895940
$409$	−3.31998	−0.164163	−0.0820813	−	0.996626i	$-0.526157\pi$
$409$	−3.31998	−0.164163	−0.0820813	+	0.996626i	$0.526157\pi$
$410$	0	0
$411$	−46.9846	−2.31758
$412$	−10.6031	−0.522377
$413$	26.6226	1.31001
$414$	19.5677	0.961700
$415$	0	0
$416$	−9.65412	−0.473332
$417$	35.0277	1.71531
$418$	0	0
$419$	30.8762	1.50840	0.754201	−	0.656644i	$-0.228024\pi$
$419$	30.8762	1.50840	0.754201	+	0.656644i	$0.228024\pi$
$420$	0	0
$421$	35.9752	1.75332	0.876662	−	0.481107i	$-0.159765\pi$
$421$	35.9752	1.75332	0.876662	+	0.481107i	$0.159765\pi$
$422$	−11.5448	−0.561991
$423$	38.6844	1.88090
$424$	−30.1837	−1.46585
$425$	0	0
$426$	47.5588	2.30423
$427$	9.13956	0.442294
$428$	−10.9865	−0.531051
$429$	−12.4288	−0.600068
$430$	0	0
$431$	−30.4458	−1.46652	−0.733262	−	0.679946i	$-0.762003\pi$
$431$	−30.4458	−1.46652	−0.733262	+	0.679946i	$0.762003\pi$
$432$	26.5980	1.27970
$433$	−21.4889	−1.03269	−0.516344	−	0.856381i	$-0.672707\pi$
$433$	−21.4889	−1.03269	−0.516344	+	0.856381i	$0.672707\pi$
$434$	45.5083	2.18447
$435$	0	0
$436$	7.83864	0.375403
$437$	0	0
$438$	11.4173	0.545542
$439$	8.96597	0.427923	0.213961	−	0.976842i	$-0.431363\pi$
$439$	8.96597	0.427923	0.213961	+	0.976842i	$0.431363\pi$
$440$	0	0
$441$	108.011	5.14340
$442$	−4.12222	−0.196074
$443$	10.7114	0.508915	0.254457	−	0.967084i	$-0.418103\pi$
$443$	10.7114	0.508915	0.254457	+	0.967084i	$0.418103\pi$
$444$	17.2693	0.819562
$445$	0	0
$446$	−4.03903	−0.191253
$447$	−15.5749	−0.736669
$448$	35.4180	1.67334
$449$	−5.89653	−0.278274	−0.139137	−	0.990273i	$-0.544433\pi$
$449$	−5.89653	−0.278274	−0.139137	+	0.990273i	$0.544433\pi$
$450$	0	0
$451$	10.4104	0.490205
$452$	−8.60344	−0.404672
$453$	12.1210	0.569495
$454$	−1.69556	−0.0795764
$455$	0	0
$456$	0	0
$457$	−7.82739	−0.366150	−0.183075	−	0.983099i	$-0.558605\pi$
$457$	−7.82739	−0.366150	−0.183075	+	0.983099i	$0.558605\pi$
$458$	−16.6469	−0.777859
$459$	−30.3963	−1.41878
$460$	0	0
$461$	−15.9326	−0.742055	−0.371028	−	0.928622i	$-0.620994\pi$
$461$	−15.9326	−0.742055	−0.371028	+	0.928622i	$0.620994\pi$
$462$	27.5777	1.28303
$463$	−0.872460	−0.0405466	−0.0202733	−	0.999794i	$-0.506454\pi$
$463$	−0.872460	−0.0405466	−0.0202733	+	0.999794i	$0.506454\pi$
$464$	5.73795	0.266378
$465$	0	0
$466$	−21.2678	−0.985210
$467$	7.78126	0.360074	0.180037	−	0.983660i	$-0.442378\pi$
$467$	7.78126	0.360074	0.180037	+	0.983660i	$0.442378\pi$
$468$	15.1141	0.698649
$469$	−58.0364	−2.67987
$470$	0	0
$471$	−21.8483	−1.00672
$472$	−18.0561	−0.831101
$473$	−1.86730	−0.0858585
$474$	12.8478	0.590118
$475$	0	0
$476$	−6.84915	−0.313930
$477$	80.3412	3.67857
$478$	−9.17766	−0.419776
$479$	7.29738	0.333426	0.166713	−	0.986005i	$-0.446685\pi$
$479$	7.29738	0.333426	0.166713	+	0.986005i	$0.446685\pi$
$480$	0	0
$481$	−13.1204	−0.598240
$482$	13.1381	0.598424
$483$	−33.8155	−1.53866
$484$	6.89772	0.313533
$485$	0	0
$486$	−61.8061	−2.80358
$487$	−25.0603	−1.13559	−0.567796	−	0.823169i	$-0.692204\pi$
$487$	−25.0603	−1.13559	−0.567796	+	0.823169i	$0.692204\pi$
$488$	−6.19869	−0.280602
$489$	68.7906	3.11082
$490$	0	0
$491$	32.9821	1.48846	0.744230	−	0.667924i	$-0.232817\pi$
$491$	32.9821	1.48846	0.744230	+	0.667924i	$0.232817\pi$
$492$	−17.3306	−0.781325
$493$	−6.55738	−0.295329
$494$	0	0
$495$	0	0
$496$	−14.6820	−0.659240
$497$	−60.0360	−2.69298
$498$	28.8833	1.29429
$499$	10.4688	0.468646	0.234323	−	0.972159i	$-0.424713\pi$
$499$	10.4688	0.468646	0.234323	+	0.972159i	$0.424713\pi$
$500$	0	0
$501$	−71.3176	−3.18624
$502$	−26.9394	−1.20237
$503$	14.6999	0.655437	0.327719	−	0.944775i	$-0.393720\pi$
$503$	14.6999	0.655437	0.327719	+	0.944775i	$0.393720\pi$
$504$	−111.857	−4.98251
$505$	0	0
$506$	−4.13037	−0.183618
$507$	27.6514	1.22804
$508$	1.07595	0.0477375
$509$	−1.07521	−0.0476577	−0.0238289	−	0.999716i	$-0.507586\pi$
$509$	−1.07521	−0.0476577	−0.0238289	+	0.999716i	$0.507586\pi$
$510$	0	0
$511$	−14.4127	−0.637580
$512$	−16.4040	−0.724960
$513$	0	0
$514$	5.87266	0.259032
$515$	0	0
$516$	3.10858	0.136848
$517$	−8.16555	−0.359120
$518$	29.1123	1.27912
$519$	24.9479	1.09509
$520$	0	0
$521$	−31.1865	−1.36631	−0.683153	−	0.730276i	$-0.739392\pi$
$521$	−31.1865	−1.36631	−0.683153	+	0.730276i	$0.739392\pi$
$522$	−32.1073	−1.40530
$523$	27.1727	1.18818	0.594090	−	0.804398i	$-0.297512\pi$
$523$	27.1727	1.18818	0.594090	+	0.804398i	$0.297512\pi$
$524$	2.11845	0.0925448
$525$	0	0
$526$	2.88800	0.125923
$527$	16.7787	0.730890
$528$	−8.89717	−0.387200
$529$	−17.9354	−0.779799
$530$	0	0
$531$	48.0608	2.08566
$532$	0	0
$533$	13.1671	0.570329
$534$	15.8216	0.684669
$535$	0	0
$536$	39.3618	1.70017
$537$	−17.2252	−0.743324
$538$	0.442600	0.0190819
$539$	−22.7992	−0.982030
$540$	0	0
$541$	−24.7037	−1.06210	−0.531048	−	0.847342i	$-0.678202\pi$
$541$	−24.7037	−1.06210	−0.531048	+	0.847342i	$0.678202\pi$
$542$	7.52966	0.323426
$543$	−12.2257	−0.524654
$544$	7.89785	0.338617
$545$	0	0
$546$	34.8804	1.49274
$547$	28.7905	1.23099	0.615497	−	0.788139i	$-0.288955\pi$
$547$	28.7905	1.23099	0.615497	+	0.788139i	$0.288955\pi$
$548$	−12.0603	−0.515190
$549$	16.4993	0.704174
$550$	0	0
$551$	0	0
$552$	22.9346	0.976160
$553$	−16.2184	−0.689677
$554$	−18.5541	−0.788288
$555$	0	0
$556$	8.99112	0.381308
$557$	35.8048	1.51710	0.758549	−	0.651616i	$-0.225909\pi$
$557$	35.8048	1.51710	0.758549	+	0.651616i	$0.225909\pi$
$558$	82.1545	3.47788
$559$	−2.36177	−0.0998921
$560$	0	0
$561$	10.1678	0.429283
$562$	8.36108	0.352691
$563$	12.6602	0.533565	0.266782	−	0.963757i	$-0.414039\pi$
$563$	12.6602	0.533565	0.266782	+	0.963757i	$0.414039\pi$
$564$	13.5936	0.572393
$565$	0	0
$566$	16.1064	0.677002
$567$	147.343	6.18784
$568$	40.7180	1.70849
$569$	0.805287	0.0337594	0.0168797	−	0.999858i	$-0.494627\pi$
$569$	0.805287	0.0337594	0.0168797	+	0.999858i	$0.494627\pi$
$570$	0	0
$571$	−31.1807	−1.30487	−0.652436	−	0.757844i	$-0.726253\pi$
$571$	−31.1807	−1.30487	−0.652436	+	0.757844i	$0.726253\pi$
$572$	−3.19030	−0.133393
$573$	−45.1927	−1.88795
$574$	−29.2158	−1.21944
$575$	0	0
$576$	63.9388	2.66412
$577$	29.9177	1.24549	0.622745	−	0.782425i	$-0.286017\pi$
$577$	29.9177	1.24549	0.622745	+	0.782425i	$0.286017\pi$
$578$	−14.8076	−0.615915
$579$	28.7413	1.19445
$580$	0	0
$581$	−36.4609	−1.51265
$582$	31.7708	1.31694
$583$	−16.9585	−0.702350
$584$	9.77507	0.404495
$585$	0	0
$586$	24.2323	1.00103
$587$	11.9462	0.493074	0.246537	−	0.969133i	$-0.420707\pi$
$587$	11.9462	0.493074	0.246537	+	0.969133i	$0.420707\pi$
$588$	37.9549	1.56523
$589$	0	0
$590$	0	0
$591$	−64.8092	−2.66590
$592$	−9.39228	−0.386020
$593$	−42.1737	−1.73187	−0.865933	−	0.500160i	$-0.833274\pi$
$593$	−42.1737	−1.73187	−0.865933	+	0.500160i	$0.833274\pi$
$594$	31.4156	1.28900
$595$	0	0
$596$	−3.99787	−0.163759
$597$	−26.5655	−1.08726
$598$	−5.22411	−0.213630
$599$	37.3867	1.52758	0.763789	−	0.645466i	$-0.223337\pi$
$599$	37.3867	1.52758	0.763789	+	0.645466i	$0.223337\pi$
$600$	0	0
$601$	−0.438043	−0.0178681	−0.00893406	−	0.999960i	$-0.502844\pi$
$601$	−0.438043	−0.0178681	−0.00893406	+	0.999960i	$0.502844\pi$
$602$	5.24042	0.213583
$603$	−104.771	−4.26661
$604$	3.11130	0.126597
$605$	0	0
$606$	−2.19059	−0.0889868
$607$	22.1376	0.898538	0.449269	−	0.893397i	$-0.351685\pi$
$607$	22.1376	0.898538	0.449269	+	0.893397i	$0.351685\pi$
$608$	0	0
$609$	55.4856	2.24839
$610$	0	0
$611$	−10.3278	−0.417819
$612$	−12.3645	−0.499807
$613$	−8.26710	−0.333905	−0.166953	−	0.985965i	$-0.553393\pi$
$613$	−8.26710	−0.333905	−0.166953	+	0.985965i	$0.553393\pi$
$614$	28.4553	1.14836
$615$	0	0
$616$	23.6109	0.951311
$617$	−2.73666	−0.110174	−0.0550869	−	0.998482i	$-0.517544\pi$
$617$	−2.73666	−0.110174	−0.0550869	+	0.998482i	$0.517544\pi$
$618$	−44.1746	−1.77696
$619$	9.91012	0.398322	0.199161	−	0.979967i	$-0.436178\pi$
$619$	9.91012	0.398322	0.199161	+	0.979967i	$0.436178\pi$
$620$	0	0
$621$	−38.5215	−1.54581
$622$	18.0254	0.722753
$623$	−19.9725	−0.800180
$624$	−11.2532	−0.450488
$625$	0	0
$626$	27.8904	1.11473
$627$	0	0
$628$	−5.60816	−0.223790
$629$	10.7336	0.427975
$630$	0	0
$631$	−33.0013	−1.31376	−0.656880	−	0.753995i	$-0.728124\pi$
$631$	−33.0013	−1.31376	−0.656880	+	0.753995i	$0.728124\pi$
$632$	10.9998	0.437547
$633$	36.0166	1.43153
$634$	−9.11078	−0.361835
$635$	0	0
$636$	28.2317	1.11946
$637$	−28.8365	−1.14254
$638$	6.77725	0.268314
$639$	−108.381	−4.28748
$640$	0	0
$641$	0.723239	0.0285662	0.0142831	−	0.999898i	$-0.495453\pi$
$641$	0.723239	0.0285662	0.0142831	+	0.999898i	$0.495453\pi$
$642$	−45.7719	−1.80647
$643$	−19.6825	−0.776201	−0.388100	−	0.921617i	$-0.626869\pi$
$643$	−19.6825	−0.776201	−0.388100	+	0.921617i	$0.626869\pi$
$644$	−8.67997	−0.342039
$645$	0	0
$646$	0	0
$647$	28.9654	1.13875	0.569373	−	0.822079i	$-0.307186\pi$
$647$	28.9654	1.13875	0.569373	+	0.822079i	$0.307186\pi$
$648$	−99.9321	−3.92571
$649$	−10.1447	−0.398216
$650$	0	0
$651$	−141.974	−5.56438
$652$	17.6576	0.691524
$653$	−12.2167	−0.478076	−0.239038	−	0.971010i	$-0.576832\pi$
$653$	−12.2167	−0.478076	−0.239038	+	0.971010i	$0.576832\pi$
$654$	32.6573	1.27700
$655$	0	0
$656$	9.42566	0.368010
$657$	−26.0187	−1.01509
$658$	22.9159	0.893355
$659$	−34.4530	−1.34210	−0.671050	−	0.741412i	$-0.734156\pi$
$659$	−34.4530	−1.34210	−0.671050	+	0.741412i	$0.734156\pi$
$660$	0	0
$661$	14.1098	0.548807	0.274404	−	0.961615i	$-0.411520\pi$
$661$	14.1098	0.548807	0.274404	+	0.961615i	$0.411520\pi$
$662$	7.04087	0.273651
$663$	12.8602	0.499449
$664$	24.7288	0.959662
$665$	0	0
$666$	52.5555	2.03648
$667$	−8.31020	−0.321772
$668$	−18.3062	−0.708290
$669$	12.6007	0.487170
$670$	0	0
$671$	−3.48270	−0.134448
$672$	−66.8280	−2.57795
$673$	−39.7565	−1.53250	−0.766250	−	0.642542i	$-0.777880\pi$
$673$	−39.7565	−1.53250	−0.766250	+	0.642542i	$0.777880\pi$
$674$	13.6466	0.525646
$675$	0	0
$676$	7.09772	0.272989
$677$	−33.8060	−1.29927	−0.649635	−	0.760246i	$-0.725078\pi$
$677$	−33.8060	−1.29927	−0.649635	+	0.760246i	$0.725078\pi$
$678$	−35.8436	−1.37657
$679$	−40.1059	−1.53912
$680$	0	0
$681$	5.28968	0.202701
$682$	−17.3413	−0.664032
$683$	32.2063	1.23234	0.616169	−	0.787614i	$-0.288684\pi$
$683$	32.2063	1.23234	0.616169	+	0.787614i	$0.288684\pi$
$684$	0	0
$685$	0	0
$686$	30.2690	1.15568
$687$	51.9338	1.98140
$688$	−1.69067	−0.0644564
$689$	−21.4492	−0.817149
$690$	0	0
$691$	36.6544	1.39440	0.697200	−	0.716876i	$-0.254429\pi$
$691$	36.6544	1.39440	0.697200	+	0.716876i	$0.254429\pi$
$692$	6.40379	0.243436
$693$	−62.8462	−2.38733
$694$	36.1317	1.37154
$695$	0	0
$696$	−37.6318	−1.42643
$697$	−10.7717	−0.408008
$698$	−21.6067	−0.817826
$699$	66.3497	2.50958
$700$	0	0
$701$	33.1730	1.25293	0.626463	−	0.779451i	$-0.284502\pi$
$701$	33.1730	1.25293	0.626463	+	0.779451i	$0.284502\pi$
$702$	39.7345	1.49968
$703$	0	0
$704$	−13.4963	−0.508661
$705$	0	0
$706$	31.5142	1.18605
$707$	2.76530	0.104000
$708$	16.8884	0.634706
$709$	30.8209	1.15750	0.578752	−	0.815503i	$-0.303540\pi$
$709$	30.8209	1.15750	0.578752	+	0.815503i	$0.303540\pi$
$710$	0	0
$711$	−29.2785	−1.09803
$712$	13.5459	0.507652
$713$	21.2637	0.796332
$714$	−28.5349	−1.06789
$715$	0	0
$716$	−4.42148	−0.165238
$717$	28.6318	1.06928
$718$	34.3031	1.28018
$719$	0.922672	0.0344099	0.0172049	−	0.999852i	$-0.494523\pi$
$719$	0.922672	0.0344099	0.0172049	+	0.999852i	$0.494523\pi$
$720$	0	0
$721$	55.7639	2.07676
$722$	0	0
$723$	−40.9873	−1.52434
$724$	−3.13816	−0.116629
$725$	0	0
$726$	28.7373	1.06654
$727$	−41.1504	−1.52618	−0.763092	−	0.646290i	$-0.776320\pi$
$727$	−41.1504	−1.52618	−0.763092	+	0.646290i	$0.776320\pi$
$728$	29.8632	1.10680
$729$	94.6730	3.50641
$730$	0	0
$731$	1.93211	0.0714618
$732$	5.79782	0.214293
$733$	7.32416	0.270524	0.135262	−	0.990810i	$-0.456812\pi$
$733$	7.32416	0.270524	0.135262	+	0.990810i	$0.456812\pi$
$734$	27.6509	1.02061
$735$	0	0
$736$	10.0090	0.368936
$737$	22.1152	0.814625
$738$	−52.7422	−1.94147
$739$	38.0009	1.39789	0.698943	−	0.715177i	$-0.253654\pi$
$739$	38.0009	1.39789	0.698943	+	0.715177i	$0.253654\pi$
$740$	0	0
$741$	0	0
$742$	47.5926	1.74718
$743$	−0.547906	−0.0201007	−0.0100504	−	0.999949i	$-0.503199\pi$
$743$	−0.547906	−0.0201007	−0.0100504	+	0.999949i	$0.503199\pi$
$744$	96.2902	3.53017
$745$	0	0
$746$	35.8272	1.31173
$747$	−65.8216	−2.40829
$748$	2.60992	0.0954282
$749$	57.7802	2.11124
$750$	0	0
$751$	−29.1917	−1.06522	−0.532611	−	0.846360i	$-0.678789\pi$
$751$	−29.1917	−1.06522	−0.532611	+	0.846360i	$0.678789\pi$
$752$	−7.39318	−0.269602
$753$	84.0437	3.06272
$754$	8.57189	0.312170
$755$	0	0
$756$	66.0197	2.40111
$757$	−15.1017	−0.548880	−0.274440	−	0.961604i	$-0.588492\pi$
$757$	−15.1017	−0.548880	−0.274440	+	0.961604i	$0.588492\pi$
$758$	0.168040	0.00610347
$759$	12.8857	0.467720
$760$	0	0
$761$	19.5940	0.710281	0.355141	−	0.934813i	$-0.384433\pi$
$761$	19.5940	0.710281	0.355141	+	0.934813i	$0.384433\pi$
$762$	4.48262	0.162388
$763$	−41.2251	−1.49245
$764$	−11.6003	−0.419686
$765$	0	0
$766$	30.6747	1.10832
$767$	−12.8311	−0.463304
$768$	54.2037	1.95591
$769$	−23.8527	−0.860151	−0.430076	−	0.902793i	$-0.641513\pi$
$769$	−23.8527	−0.860151	−0.430076	+	0.902793i	$0.641513\pi$
$770$	0	0
$771$	−18.3211	−0.659819
$772$	7.37749	0.265522
$773$	2.51567	0.0904823	0.0452412	−	0.998976i	$-0.485594\pi$
$773$	2.51567	0.0904823	0.0452412	+	0.998976i	$0.485594\pi$
$774$	9.46034	0.340045
$775$	0	0
$776$	27.2009	0.976455
$777$	−90.8226	−3.25824
$778$	−2.33421	−0.0836856
$779$	0	0
$780$	0	0
$781$	22.8772	0.818609
$782$	4.27374	0.152829
$783$	63.2073	2.25884
$784$	−20.6427	−0.737238
$785$	0	0
$786$	8.82587	0.314809
$787$	−13.5792	−0.484046	−0.242023	−	0.970271i	$-0.577811\pi$
$787$	−13.5792	−0.484046	−0.242023	+	0.970271i	$0.577811\pi$
$788$	−16.6356	−0.592620
$789$	−9.00978	−0.320757
$790$	0	0
$791$	45.2473	1.60881
$792$	42.6240	1.51458
$793$	−4.40493	−0.156424
$794$	−8.08536	−0.286939
$795$	0	0
$796$	−6.81901	−0.241693
$797$	−45.2770	−1.60379	−0.801897	−	0.597462i	$-0.796176\pi$
$797$	−45.2770	−1.60379	−0.801897	+	0.597462i	$0.796176\pi$
$798$	0	0
$799$	8.44898	0.298903
$800$	0	0
$801$	−36.0556	−1.27396
$802$	41.9207	1.48027
$803$	5.49207	0.193811
$804$	−36.8163	−1.29841
$805$	0	0
$806$	−21.9333	−0.772568
$807$	−1.38079	−0.0486062
$808$	−1.87550	−0.0659798
$809$	34.7396	1.22138	0.610690	−	0.791870i	$-0.290892\pi$
$809$	34.7396	1.22138	0.610690	+	0.791870i	$0.290892\pi$
$810$	0	0
$811$	46.9824	1.64977	0.824887	−	0.565298i	$-0.191239\pi$
$811$	46.9824	1.64977	0.824887	+	0.565298i	$0.191239\pi$
$812$	14.2424	0.499809
$813$	−23.4905	−0.823848
$814$	−11.0935	−0.388826
$815$	0	0
$816$	9.20600	0.322275
$817$	0	0
$818$	−3.55041	−0.124137
$819$	−79.4881	−2.77754
$820$	0	0
$821$	−28.5526	−0.996492	−0.498246	−	0.867036i	$-0.666022\pi$
$821$	−28.5526	−0.996492	−0.498246	+	0.867036i	$0.666022\pi$
$822$	−50.2456	−1.75252
$823$	0.369620	0.0128841	0.00644207	−	0.999979i	$-0.497949\pi$
$823$	0.369620	0.0128841	0.00644207	+	0.999979i	$0.497949\pi$
$824$	−37.8205	−1.31754
$825$	0	0
$826$	28.4703	0.990610
$827$	47.4660	1.65055	0.825277	−	0.564728i	$-0.191019\pi$
$827$	47.4660	1.65055	0.825277	+	0.564728i	$0.191019\pi$
$828$	−15.6696	−0.544558
$829$	1.94291	0.0674802	0.0337401	−	0.999431i	$-0.489258\pi$
$829$	1.94291	0.0674802	0.0337401	+	0.999431i	$0.489258\pi$
$830$	0	0
$831$	57.8838	2.00797
$832$	−17.0702	−0.591801
$833$	23.5906	0.817365
$834$	37.4588	1.29709
$835$	0	0
$836$	0	0
$837$	−161.731	−5.59026
$838$	33.0192	1.14063
$839$	−24.1890	−0.835097	−0.417549	−	0.908655i	$-0.637111\pi$
$839$	−24.1890	−0.835097	−0.417549	+	0.908655i	$0.637111\pi$
$840$	0	0
$841$	−15.3644	−0.529805
$842$	38.4721	1.32584
$843$	−26.0843	−0.898391
$844$	9.24497	0.318225
$845$	0	0
$846$	41.3693	1.42231
$847$	−36.2765	−1.24648
$848$	−15.3544	−0.527274
$849$	−50.2476	−1.72449
$850$	0	0
$851$	13.6027	0.466295
$852$	−38.0847	−1.30476
$853$	−42.2108	−1.44527	−0.722634	−	0.691230i	$-0.757069\pi$
$853$	−42.2108	−1.44527	−0.722634	+	0.691230i	$0.757069\pi$
$854$	9.77390	0.334456
$855$	0	0
$856$	−39.1880	−1.33942
$857$	−15.0385	−0.513706	−0.256853	−	0.966451i	$-0.582686\pi$
$857$	−15.0385	−0.513706	−0.256853	+	0.966451i	$0.582686\pi$
$858$	−13.2914	−0.453762
$859$	18.8065	0.641670	0.320835	−	0.947135i	$-0.396036\pi$
$859$	18.8065	0.641670	0.320835	+	0.947135i	$0.396036\pi$
$860$	0	0
$861$	91.1454	3.10623
$862$	−32.5590	−1.10896
$863$	−9.97415	−0.339524	−0.169762	−	0.985485i	$-0.554300\pi$
$863$	−9.97415	−0.339524	−0.169762	+	0.985485i	$0.554300\pi$
$864$	−76.1282	−2.58993
$865$	0	0
$866$	−22.9803	−0.780903
$867$	46.1957	1.56889
$868$	−36.4426	−1.23694
$869$	6.18015	0.209647
$870$	0	0
$871$	27.9714	0.947775
$872$	27.9599	0.946842
$873$	−72.4018	−2.45043
$874$	0	0
$875$	0	0
$876$	−9.14291	−0.308910
$877$	−5.00412	−0.168977	−0.0844885	−	0.996424i	$-0.526926\pi$
$877$	−5.00412	−0.168977	−0.0844885	+	0.996424i	$0.526926\pi$
$878$	9.58826	0.323588
$879$	−75.5983	−2.54987
$880$	0	0
$881$	−21.6047	−0.727880	−0.363940	−	0.931422i	$-0.618569\pi$
$881$	−21.6047	−0.727880	−0.363940	+	0.931422i	$0.618569\pi$
$882$	115.508	3.88936
$883$	23.9026	0.804386	0.402193	−	0.915555i	$-0.368248\pi$
$883$	23.9026	0.804386	0.402193	+	0.915555i	$0.368248\pi$
$884$	3.30104	0.111026
$885$	0	0
$886$	11.4548	0.384833
$887$	23.3858	0.785220	0.392610	−	0.919705i	$-0.371572\pi$
$887$	23.3858	0.785220	0.392610	+	0.919705i	$0.371572\pi$
$888$	61.5983	2.06710
$889$	−5.65864	−0.189785
$890$	0	0
$891$	−56.1463	−1.88097
$892$	3.23442	0.108296
$893$	0	0
$894$	−16.6559	−0.557057
$895$	0	0
$896$	−2.18546	−0.0730111
$897$	16.2978	0.544168
$898$	−6.30578	−0.210427
$899$	−34.8902	−1.16365
$900$	0	0
$901$	17.5472	0.584581
$902$	11.1329	0.370685
$903$	−16.3487	−0.544050
$904$	−30.6879	−1.02066
$905$	0	0
$906$	12.9623	0.430643
$907$	−28.3876	−0.942594	−0.471297	−	0.881975i	$-0.656214\pi$
$907$	−28.3876	−0.942594	−0.471297	+	0.881975i	$0.656214\pi$
$908$	1.35779	0.0450598
$909$	4.99210	0.165578
$910$	0	0
$911$	−5.25801	−0.174206	−0.0871029	−	0.996199i	$-0.527761\pi$
$911$	−5.25801	−0.174206	−0.0871029	+	0.996199i	$0.527761\pi$
$912$	0	0
$913$	13.8937	0.459815
$914$	−8.37065	−0.276877
$915$	0	0
$916$	13.3307	0.440459
$917$	−11.1414	−0.367920
$918$	−32.5060	−1.07286
$919$	−37.3036	−1.23053	−0.615266	−	0.788320i	$-0.710951\pi$
$919$	−37.3036	−1.23053	−0.615266	+	0.788320i	$0.710951\pi$
$920$	0	0
$921$	−88.7730	−2.92517
$922$	−17.0384	−0.561130
$923$	28.9351	0.952411
$924$	−22.0840	−0.726509
$925$	0	0
$926$	−0.933014	−0.0306607
$927$	100.669	3.30639
$928$	−16.4231	−0.539114
$929$	46.4592	1.52428	0.762139	−	0.647414i	$-0.224149\pi$
$929$	46.4592	1.52428	0.762139	+	0.647414i	$0.224149\pi$
$930$	0	0
$931$	0	0
$932$	17.0310	0.557870
$933$	−56.2344	−1.84103
$934$	8.32132	0.272282
$935$	0	0
$936$	53.9109	1.76213
$937$	−0.863983	−0.0282251	−0.0141125	−	0.999900i	$-0.504492\pi$
$937$	−0.863983	−0.0282251	−0.0141125	+	0.999900i	$0.504492\pi$
$938$	−62.0645	−2.02648
$939$	−87.0106	−2.83948
$940$	0	0
$941$	−31.0972	−1.01374	−0.506869	−	0.862023i	$-0.669197\pi$
$941$	−31.0972	−1.01374	−0.506869	+	0.862023i	$0.669197\pi$
$942$	−23.3647	−0.761264
$943$	−13.6511	−0.444539
$944$	−9.18516	−0.298951
$945$	0	0
$946$	−1.99690	−0.0649248
$947$	−36.9328	−1.20016	−0.600078	−	0.799942i	$-0.704864\pi$
$947$	−36.9328	−1.20016	−0.600078	+	0.799942i	$0.704864\pi$
$948$	−10.2884	−0.334151
$949$	6.94639	0.225489
$950$	0	0
$951$	28.4232	0.921684
$952$	−24.4305	−0.791796
$953$	34.0793	1.10394	0.551968	−	0.833865i	$-0.313877\pi$
$953$	34.0793	1.10394	0.551968	+	0.833865i	$0.313877\pi$
$954$	85.9174	2.78168
$955$	0	0
$956$	7.34939	0.237696
$957$	−21.1432	−0.683462
$958$	7.80386	0.252131
$959$	63.4276	2.04818
$960$	0	0
$961$	58.2752	1.87984
$962$	−14.0311	−0.452380
$963$	104.309	3.36130
$964$	−10.5209	−0.338855
$965$	0	0
$966$	−36.1625	−1.16351
$967$	−11.7372	−0.377443	−0.188722	−	0.982031i	$-0.560434\pi$
$967$	−11.7372	−0.377443	−0.188722	+	0.982031i	$0.560434\pi$
$968$	24.6037	0.790793
$969$	0	0
$970$	0	0
$971$	57.0534	1.83093	0.915466	−	0.402395i	$-0.131822\pi$
$971$	57.0534	1.83093	0.915466	+	0.402395i	$0.131822\pi$
$972$	49.4938	1.58751
$973$	−47.2862	−1.51593
$974$	−26.7997	−0.858717
$975$	0	0
$976$	−3.15328	−0.100934
$977$	7.68771	0.245952	0.122976	−	0.992410i	$-0.460756\pi$
$977$	7.68771	0.245952	0.122976	+	0.992410i	$0.460756\pi$
$978$	73.5650	2.35235
$979$	7.61066	0.243238
$980$	0	0
$981$	−74.4222	−2.37612
$982$	35.2712	1.12555
$983$	26.1250	0.833258	0.416629	−	0.909077i	$-0.363211\pi$
$983$	26.1250	0.833258	0.416629	+	0.909077i	$0.363211\pi$
$984$	−61.8172	−1.97066
$985$	0	0
$986$	−7.01250	−0.223323
$987$	−71.4915	−2.27560
$988$	0	0
$989$	2.44858	0.0778603
$990$	0	0
$991$	−16.0875	−0.511037	−0.255519	−	0.966804i	$-0.582246\pi$
$991$	−16.0875	−0.511037	−0.255519	+	0.966804i	$0.582246\pi$
$992$	42.0225	1.33422
$993$	−21.9656	−0.697058
$994$	−64.2028	−2.03639
$995$	0	0
$996$	−23.1295	−0.732887
$997$	0.736253	0.0233174	0.0116587	−	0.999932i	$-0.496289\pi$
$997$	0.736253	0.0233174	0.0116587	+	0.999932i	$0.496289\pi$
$998$	11.1954	0.354383
$999$	−103.462	−3.27339

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.ch.1.7	yes	10
5.4	even	2		9025.2.a.ci.1.4	yes	10
19.18	odd	2		9025.2.a.cj.1.4	yes	10
95.94	odd	2		9025.2.a.cg.1.7	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9025.2.a.cg.1.7	✓	10	95.94	odd	2
9025.2.a.ch.1.7	yes	10	1.1	even	1	trivial
9025.2.a.ci.1.4	yes	10	5.4	even	2
9025.2.a.cj.1.4	yes	10	19.18	odd	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:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q+1.06941 q^{2} -3.33626 q^{3} -0.856371 q^{4} -3.56781 q^{6} +4.50384 q^{7} -3.05462 q^{8} +8.13062 q^{9} +O(q^{10})\)	\(q+1.06941 q^{2} -3.33626 q^{3} -0.856371 q^{4} -3.56781 q^{6} +4.50384 q^{7} -3.05462 q^{8} +8.13062 q^{9} -1.71622 q^{11} +2.85707 q^{12} -2.17068 q^{13} +4.81643 q^{14} -1.55389 q^{16} +1.77579 q^{17} +8.69493 q^{18} -15.0260 q^{21} -1.83534 q^{22} +2.25047 q^{23} +10.1910 q^{24} -2.32134 q^{26} -17.1171 q^{27} -3.85695 q^{28} -3.69265 q^{29} +9.44856 q^{31} +4.44750 q^{32} +5.72575 q^{33} +1.89904 q^{34} -6.96283 q^{36} +6.04438 q^{37} +7.24196 q^{39} -6.06586 q^{41} -16.0688 q^{42} +1.08803 q^{43} +1.46972 q^{44} +2.40667 q^{46} +4.75787 q^{47} +5.18417 q^{48} +13.2845 q^{49} -5.92450 q^{51} +1.85891 q^{52} +9.88132 q^{53} -18.3051 q^{54} -13.7575 q^{56} -3.94894 q^{58} +5.91109 q^{59} +2.02928 q^{61} +10.1043 q^{62} +36.6190 q^{63} +7.86396 q^{64} +6.12316 q^{66} -12.8860 q^{67} -1.52074 q^{68} -7.50816 q^{69} -13.3300 q^{71} -24.8359 q^{72} -3.20010 q^{73} +6.46390 q^{74} -7.72957 q^{77} +7.74459 q^{78} -3.60102 q^{79} +32.7151 q^{81} -6.48687 q^{82} -8.09553 q^{83} +12.8678 q^{84} +1.16355 q^{86} +12.3196 q^{87} +5.24240 q^{88} -4.43455 q^{89} -9.77639 q^{91} -1.92724 q^{92} -31.5228 q^{93} +5.08809 q^{94} -14.8380 q^{96} -8.90483 q^{97} +14.2066 q^{98} -13.9539 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{2} + 8 q^{4} + 4 q^{6} + 4 q^{7} - 3 q^{8} + 12 q^{9}+O(q^{10}) \) 10 * q - 2 * q^2 + 8 * q^4 + 4 * q^6 + 4 * q^7 - 3 * q^8 + 12 * q^9	\( 10 q - 2 q^{2} + 8 q^{4} + 4 q^{6} + 4 q^{7} - 3 q^{8} + 12 q^{9} - 3 q^{11} + 14 q^{12} + 5 q^{13} + 14 q^{14} - 4 q^{16} + 11 q^{17} - 4 q^{18} - 17 q^{21} + 34 q^{22} + 32 q^{23} - 7 q^{24} - 13 q^{26} - 30 q^{27} - 12 q^{28} + 8 q^{29} + 11 q^{31} - 8 q^{32} + 19 q^{33} + 20 q^{34} - 4 q^{36} - 35 q^{37} + 20 q^{39} - 16 q^{41} + 32 q^{42} + 26 q^{43} - 44 q^{44} + 21 q^{46} + 19 q^{47} + 16 q^{48} + 18 q^{49} - 15 q^{51} + 25 q^{52} - 16 q^{53} - q^{54} + 20 q^{56} + 7 q^{58} - 2 q^{59} + 8 q^{61} + 13 q^{62} + 77 q^{63} + q^{64} + 20 q^{66} + 40 q^{68} - 12 q^{69} - 2 q^{71} - 31 q^{72} + q^{73} + 18 q^{74} + 18 q^{77} - 41 q^{78} - 18 q^{79} + 58 q^{81} + 25 q^{82} + 46 q^{83} - 24 q^{84} + 33 q^{86} + 57 q^{87} + 37 q^{88} + 13 q^{89} - 67 q^{91} + 33 q^{92} + 4 q^{93} - 25 q^{94} - 40 q^{96} - 8 q^{97} + 62 q^{98} - 62 q^{99}+O(q^{100}) \) 10 * q - 2 * q^2 + 8 * q^4 + 4 * q^6 + 4 * q^7 - 3 * q^8 + 12 * q^9 - 3 * q^11 + 14 * q^12 + 5 * q^13 + 14 * q^14 - 4 * q^16 + 11 * q^17 - 4 * q^18 - 17 * q^21 + 34 * q^22 + 32 * q^23 - 7 * q^24 - 13 * q^26 - 30 * q^27 - 12 * q^28 + 8 * q^29 + 11 * q^31 - 8 * q^32 + 19 * q^33 + 20 * q^34 - 4 * q^36 - 35 * q^37 + 20 * q^39 - 16 * q^41 + 32 * q^42 + 26 * q^43 - 44 * q^44 + 21 * q^46 + 19 * q^47 + 16 * q^48 + 18 * q^49 - 15 * q^51 + 25 * q^52 - 16 * q^53 - q^54 + 20 * q^56 + 7 * q^58 - 2 * q^59 + 8 * q^61 + 13 * q^62 + 77 * q^63 + q^64 + 20 * q^66 + 40 * q^68 - 12 * q^69 - 2 * q^71 - 31 * q^72 + q^73 + 18 * q^74 + 18 * q^77 - 41 * q^78 - 18 * q^79 + 58 * q^81 + 25 * q^82 + 46 * q^83 - 24 * q^84 + 33 * q^86 + 57 * q^87 + 37 * q^88 + 13 * q^89 - 67 * q^91 + 33 * q^92 + 4 * q^93 - 25 * q^94 - 40 * q^96 - 8 * q^97 + 62 * q^98 - 62 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more