LMFDB - Embedded newform 9025.2.a.bp.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9025,2,Mod(1,9025)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

Level:	$ N $	$=$	$ 9025 = 5^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9025.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,1,3,5,0,2,4,12,1,0,-2,6,7,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$72.0649878242$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.7537.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 4x + 3 $ x^4 - x^3 - 5x^2 + 4x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.04717$ 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)

$f(q)$	$=$	\(q-1.19091 q^{2} +3.04717 q^{3} -0.581734 q^{4} -3.62891 q^{6} +0.609175 q^{7} +3.07461 q^{8} +6.28525 q^{9} +4.48517 q^{11} -1.77264 q^{12} +4.43800 q^{13} -0.725473 q^{14} -2.49812 q^{16} -2.90343 q^{17} -7.48517 q^{18} +1.85626 q^{21} -5.34143 q^{22} +2.84726 q^{23} +9.36887 q^{24} -5.28525 q^{26} +10.0107 q^{27} -0.354378 q^{28} -1.11630 q^{29} +6.22908 q^{31} -3.17419 q^{32} +13.6671 q^{33} +3.45773 q^{34} -3.65635 q^{36} -3.77264 q^{37} +13.5233 q^{39} +8.30369 q^{41} -2.21064 q^{42} +9.98877 q^{43} -2.60918 q^{44} -3.39082 q^{46} +5.88500 q^{47} -7.61219 q^{48} -6.62891 q^{49} -8.84726 q^{51} -2.58173 q^{52} +8.44872 q^{53} -11.9219 q^{54} +1.87298 q^{56} +1.32941 q^{58} -10.2359 q^{59} -4.98199 q^{61} -7.41827 q^{62} +3.82882 q^{63} +8.77641 q^{64} -16.2762 q^{66} +8.47616 q^{67} +1.68903 q^{68} +8.67608 q^{69} -11.6199 q^{71} +19.3247 q^{72} -3.72325 q^{73} +4.49288 q^{74} +2.73225 q^{77} -16.1051 q^{78} -9.03817 q^{79} +11.6486 q^{81} -9.88894 q^{82} +2.12178 q^{83} -1.07985 q^{84} -11.8957 q^{86} -3.40155 q^{87} +13.7901 q^{88} -7.93217 q^{89} +2.70352 q^{91} -1.65635 q^{92} +18.9811 q^{93} -7.00850 q^{94} -9.67231 q^{96} -9.67256 q^{97} +7.89443 q^{98} +28.1904 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + 3 q^{3} + 5 q^{4} + 2 q^{6} + 4 q^{7} + 12 q^{8} + q^{9} - 2 q^{11} + 6 q^{12} + 7 q^{13} + q^{14} + 7 q^{16} + q^{17} - 10 q^{18} + 4 q^{21} + 2 q^{22} - 2 q^{23} + 23 q^{24} + 3 q^{26}+ \cdots + 38 q^{99}+O(q^{100}) $ 4 * q + q^2 + 3 * q^3 + 5 * q^4 + 2 * q^6 + 4 * q^7 + 12 * q^8 + q^9 - 2 * q^11 + 6 * q^12 + 7 * q^13 + q^14 + 7 * q^16 + q^17 - 10 * q^18 + 4 * q^21 + 2 * q^22 - 2 * q^23 + 23 * q^24 + 3 * q^26 + 12 * q^27 + 19 * q^28 + q^29 + 30 * q^32 + 19 * q^33 - 15 * q^34 - 7 * q^36 - 2 * q^37 + 15 * q^39 + 8 * q^41 + 15 * q^42 - q^43 - 12 * q^44 - 12 * q^46 + 12 * q^47 + 23 * q^48 - 10 * q^49 - 22 * q^51 - 3 * q^52 - 5 * q^53 - 34 * q^54 + 41 * q^56 + 27 * q^58 + 5 * q^59 - 37 * q^62 + 3 * q^63 + 56 * q^64 - 31 * q^66 + 4 * q^67 - 16 * q^68 + 9 * q^69 - 20 * q^71 + 17 * q^72 + 20 * q^73 + 25 * q^74 - 14 * q^77 - 18 * q^78 - 17 * q^79 + 12 * q^81 - 21 * q^82 - q^83 + 20 * q^84 - 8 * q^86 + 16 * q^87 - 7 * q^88 - 11 * q^89 - 6 * q^91 + q^92 + 8 * q^93 + 31 * q^94 + 21 * q^96 + q^97 + 9 * q^98 + 38 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.19091	−0.842100	−0.421050	−	0.907037i	$-0.638338\pi$
$2$	−1.19091	−0.842100	−0.421050	+	0.907037i	$0.638338\pi$
$3$	3.04717	1.75929	0.879643	−	0.475635i	$-0.157782\pi$
$3$	3.04717	1.75929	0.879643	+	0.475635i	$0.157782\pi$
$4$	−0.581734	−0.290867
$5$	0	0
$5$	0	0
$6$	−3.62891	−1.48149
$7$	0.609175	0.230247	0.115123	−	0.993351i	$-0.463274\pi$
$7$	0.609175	0.230247	0.115123	+	0.993351i	$0.463274\pi$
$8$	3.07461	1.08704
$9$	6.28525	2.09508
$10$	0	0
$11$	4.48517	1.35233	0.676164	−	0.736751i	$-0.263641\pi$
$11$	4.48517	1.35233	0.676164	+	0.736751i	$0.263641\pi$
$12$	−1.77264	−0.511718
$13$	4.43800	1.23088	0.615439	−	0.788184i	$-0.288979\pi$
$13$	4.43800	1.23088	0.615439	+	0.788184i	$0.288979\pi$
$14$	−0.725473	−0.193891
$15$	0	0
$16$	−2.49812	−0.624529
$17$	−2.90343	−0.704186	−0.352093	−	0.935965i	$-0.614530\pi$
$17$	−2.90343	−0.704186	−0.352093	+	0.935965i	$0.614530\pi$
$18$	−7.48517	−1.76427
$19$	0	0
$19$	0	0
$20$	0	0
$21$	1.85626	0.405069
$22$	−5.34143	−1.13880
$23$	2.84726	0.593694	0.296847	−	0.954925i	$-0.404065\pi$
$23$	2.84726	0.593694	0.296847	+	0.954925i	$0.404065\pi$
$24$	9.36887	1.91241
$25$	0	0
$26$	−5.28525	−1.03652
$27$	10.0107	1.92656
$28$	−0.354378	−0.0669712
$29$	−1.11630	−0.207291	−0.103646	−	0.994614i	$-0.533051\pi$
$29$	−1.11630	−0.207291	−0.103646	+	0.994614i	$0.533051\pi$
$30$	0	0
$31$	6.22908	1.11877	0.559387	−	0.828906i	$-0.311036\pi$
$31$	6.22908	1.11877	0.559387	+	0.828906i	$0.311036\pi$
$32$	−3.17419	−0.561123
$33$	13.6671	2.37913
$34$	3.45773	0.592995
$35$	0	0
$36$	−3.65635	−0.609391
$37$	−3.77264	−0.620219	−0.310109	−	0.950701i	$-0.600366\pi$
$37$	−3.77264	−0.620219	−0.310109	+	0.950701i	$0.600366\pi$
$38$	0	0
$39$	13.5233	2.16547
$40$	0	0
$41$	8.30369	1.29682	0.648409	−	0.761292i	$-0.275435\pi$
$41$	8.30369	1.29682	0.648409	+	0.761292i	$0.275435\pi$
$42$	−2.21064	−0.341109
$43$	9.98877	1.52327	0.761637	−	0.648004i	$-0.224396\pi$
$43$	9.98877	1.52327	0.761637	+	0.648004i	$0.224396\pi$
$44$	−2.60918	−0.393348
$45$	0	0
$46$	−3.39082	−0.499950
$47$	5.88500	0.858415	0.429208	−	0.903206i	$-0.358793\pi$
$47$	5.88500	0.858415	0.429208	+	0.903206i	$0.358793\pi$
$48$	−7.61219	−1.09872
$49$	−6.62891	−0.946986
$50$	0	0
$51$	−8.84726	−1.23886
$52$	−2.58173	−0.358022
$53$	8.44872	1.16052	0.580261	−	0.814431i	$-0.302951\pi$
$53$	8.44872	1.16052	0.580261	+	0.814431i	$0.302951\pi$
$54$	−11.9219	−1.62236
$55$	0	0
$56$	1.87298	0.250287
$57$	0	0
$58$	1.32941	0.174560
$59$	−10.2359	−1.33259	−0.666297	−	0.745686i	$-0.732122\pi$
$59$	−10.2359	−1.33259	−0.666297	+	0.745686i	$0.732122\pi$
$60$	0	0
$61$	−4.98199	−0.637878	−0.318939	−	0.947775i	$-0.603327\pi$
$61$	−4.98199	−0.637878	−0.318939	+	0.947775i	$0.603327\pi$
$62$	−7.41827	−0.942121
$63$	3.82882	0.482386
$64$	8.77641	1.09705
$65$	0	0
$66$	−16.2762	−2.00347
$67$	8.47616	1.03553	0.517764	−	0.855524i	$-0.326765\pi$
$67$	8.47616	1.03553	0.517764	+	0.855524i	$0.326765\pi$
$68$	1.68903	0.204825
$69$	8.67608	1.04448
$70$	0	0
$71$	−11.6199	−1.37903	−0.689514	−	0.724272i	$-0.742176\pi$
$71$	−11.6199	−1.37903	−0.689514	+	0.724272i	$0.742176\pi$
$72$	19.3247	2.27744
$73$	−3.72325	−0.435773	−0.217887	−	0.975974i	$-0.569916\pi$
$73$	−3.72325	−0.435773	−0.217887	+	0.975974i	$0.569916\pi$
$74$	4.49288	0.522286
$75$	0	0
$76$	0	0
$77$	2.73225	0.311369
$78$	−16.1051	−1.82354
$79$	−9.03817	−1.01687	−0.508437	−	0.861099i	$-0.669776\pi$
$79$	−9.03817	−1.01687	−0.508437	+	0.861099i	$0.669776\pi$
$80$	0	0
$81$	11.6486	1.29429
$82$	−9.88894	−1.09205
$83$	2.12178	0.232896	0.116448	−	0.993197i	$-0.462849\pi$
$83$	2.12178	0.232896	0.116448	+	0.993197i	$0.462849\pi$
$84$	−1.07985	−0.117821
$85$	0	0
$86$	−11.8957	−1.28275
$87$	−3.40155	−0.364684
$88$	13.7901	1.47003
$89$	−7.93217	−0.840808	−0.420404	−	0.907337i	$-0.638112\pi$
$89$	−7.93217	−0.840808	−0.420404	+	0.907337i	$0.638112\pi$
$90$	0	0
$91$	2.70352	0.283406
$92$	−1.65635	−0.172686
$93$	18.9811	1.96824
$94$	−7.00850	−0.722872
$95$	0	0
$96$	−9.67231	−0.987176
$97$	−9.67256	−0.982099	−0.491050	−	0.871132i	$-0.663387\pi$
$97$	−9.67256	−0.982099	−0.491050	+	0.871132i	$0.663387\pi$
$98$	7.89443	0.797458
$99$	28.1904	2.83324
$100$	0	0
$101$	−0.971265	−0.0966444	−0.0483222	−	0.998832i	$-0.515387\pi$
$101$	−0.971265	−0.0966444	−0.0483222	+	0.998832i	$0.515387\pi$
$102$	10.5363	1.04325
$103$	−3.34143	−0.329241	−0.164620	−	0.986357i	$-0.552640\pi$
$103$	−3.34143	−0.329241	−0.164620	+	0.986357i	$0.552640\pi$
$104$	13.6451	1.33801
$105$	0	0
$106$	−10.0617	−0.977275
$107$	9.51655	0.920000	0.460000	−	0.887919i	$-0.347849\pi$
$107$	9.51655	0.920000	0.460000	+	0.887919i	$0.347849\pi$
$108$	−5.82358	−0.560374
$109$	−5.54357	−0.530978	−0.265489	−	0.964114i	$-0.585533\pi$
$109$	−5.54357	−0.530978	−0.265489	+	0.964114i	$0.585533\pi$
$110$	0	0
$111$	−11.4959	−1.09114
$112$	−1.52179	−0.143796
$113$	1.54134	0.144997	0.0724987	−	0.997369i	$-0.476903\pi$
$113$	1.54134	0.144997	0.0724987	+	0.997369i	$0.476903\pi$
$114$	0	0
$115$	0	0
$116$	0.649388	0.0602942
$117$	27.8939	2.57879
$118$	12.1900	1.12218
$119$	−1.76870	−0.162136
$120$	0	0
$121$	9.11672	0.828793
$122$	5.93310	0.537158
$123$	25.3028	2.28147
$124$	−3.62367	−0.325415
$125$	0	0
$126$	−4.55978	−0.406217
$127$	2.30549	0.204579	0.102289	−	0.994755i	$-0.467383\pi$
$127$	2.30549	0.204579	0.102289	+	0.994755i	$0.467383\pi$
$128$	−4.10353	−0.362704
$129$	30.4375	2.67987
$130$	0	0
$131$	−12.9181	−1.12866	−0.564330	−	0.825549i	$-0.690865\pi$
$131$	−12.9181	−1.12866	−0.564330	+	0.825549i	$0.690865\pi$
$132$	−7.95060	−0.692011
$133$	0	0
$134$	−10.0943	−0.872018
$135$	0	0
$136$	−8.92693	−0.765478
$137$	12.7335	1.08790	0.543950	−	0.839118i	$-0.316928\pi$
$137$	12.7335	1.08790	0.543950	+	0.839118i	$0.316928\pi$
$138$	−10.3324	−0.879554
$139$	10.6087	0.899816	0.449908	−	0.893075i	$-0.351457\pi$
$139$	10.6087	0.899816	0.449908	+	0.893075i	$0.351457\pi$
$140$	0	0
$141$	17.9326	1.51020
$142$	13.8383	1.16128
$143$	19.9052	1.66455
$144$	−15.7013	−1.30844
$145$	0	0
$146$	4.43405	0.366965
$147$	−20.1994	−1.66602
$148$	2.19468	0.180401
$149$	3.77307	0.309102	0.154551	−	0.987985i	$-0.450607\pi$
$149$	3.77307	0.309102	0.154551	+	0.987985i	$0.450607\pi$
$150$	0	0
$151$	9.51562	0.774370	0.387185	−	0.922002i	$-0.373447\pi$
$151$	9.51562	0.774370	0.387185	+	0.922002i	$0.373447\pi$
$152$	0	0
$153$	−18.2488	−1.47533
$154$	−3.25387	−0.262204
$155$	0	0
$156$	−7.86699	−0.629863
$157$	3.45643	0.275853	0.137927	−	0.990442i	$-0.455956\pi$
$157$	3.45643	0.275853	0.137927	+	0.990442i	$0.455956\pi$
$158$	10.7636	0.856309
$159$	25.7447	2.04169
$160$	0	0
$161$	1.73448	0.136696
$162$	−13.8725	−1.08992
$163$	−6.65283	−0.521090	−0.260545	−	0.965462i	$-0.583902\pi$
$163$	−6.65283	−0.521090	−0.260545	+	0.965462i	$0.583902\pi$
$164$	−4.83054	−0.377202
$165$	0	0
$166$	−2.52685	−0.196122
$167$	−16.4555	−1.27336	−0.636682	−	0.771126i	$-0.719694\pi$
$167$	−16.4555	−1.27336	−0.636682	+	0.771126i	$0.719694\pi$
$168$	5.70728	0.440327
$169$	6.69581	0.515062
$170$	0	0
$171$	0	0
$172$	−5.81081	−0.443070
$173$	−22.7824	−1.73212	−0.866058	−	0.499943i	$-0.833354\pi$
$173$	−22.7824	−1.73212	−0.866058	+	0.499943i	$0.833354\pi$
$174$	4.05094	0.307101
$175$	0	0
$176$	−11.2045	−0.844569
$177$	−31.1904	−2.34441
$178$	9.44650	0.708045
$179$	2.32916	0.174090	0.0870449	−	0.996204i	$-0.472258\pi$
$179$	2.32916	0.174090	0.0870449	+	0.996204i	$0.472258\pi$
$180$	0	0
$181$	22.3392	1.66046	0.830230	−	0.557421i	$-0.188209\pi$
$181$	22.3392	1.66046	0.830230	+	0.557421i	$0.188209\pi$
$182$	−3.21965	−0.238656
$183$	−15.1810	−1.12221
$184$	8.75421	0.645369
$185$	0	0
$186$	−22.6047	−1.65746
$187$	−13.0224	−0.952291
$188$	−3.42350	−0.249685
$189$	6.09829	0.443585
$190$	0	0
$191$	2.23766	0.161911	0.0809556	−	0.996718i	$-0.474203\pi$
$191$	2.23766	0.161911	0.0809556	+	0.996718i	$0.474203\pi$
$192$	26.7432	1.93003
$193$	−4.54306	−0.327017	−0.163508	−	0.986542i	$-0.552281\pi$
$193$	−4.54306	−0.327017	−0.163508	+	0.986542i	$0.552281\pi$
$194$	11.5191	0.827026
$195$	0	0
$196$	3.85626	0.275447
$197$	19.2236	1.36962	0.684812	−	0.728720i	$-0.259884\pi$
$197$	19.2236	1.36962	0.684812	+	0.728720i	$0.259884\pi$
$198$	−33.5722	−2.38587
$199$	−6.15094	−0.436029	−0.218014	−	0.975946i	$-0.569958\pi$
$199$	−6.15094	−0.436029	−0.218014	+	0.975946i	$0.569958\pi$
$200$	0	0
$201$	25.8283	1.82179
$202$	1.15669	0.0813843
$203$	−0.680021	−0.0477281
$204$	5.14675	0.360345
$205$	0	0
$206$	3.97934	0.277254
$207$	17.8957	1.24384
$208$	−11.0866	−0.768720
$209$	0	0
$210$	0	0
$211$	−12.6932	−0.873837	−0.436919	−	0.899501i	$-0.643930\pi$
$211$	−12.6932	−0.873837	−0.436919	+	0.899501i	$0.643930\pi$
$212$	−4.91491	−0.337557
$213$	−35.4078	−2.42610
$214$	−11.3334	−0.774732
$215$	0	0
$216$	30.7791	2.09425
$217$	3.79460	0.257594
$218$	6.60189	0.447136
$219$	−11.3454	−0.766649
$220$	0	0
$221$	−12.8854	−0.866767
$222$	13.6906	0.918851
$223$	22.5376	1.50923	0.754614	−	0.656169i	$-0.227824\pi$
$223$	22.5376	1.50923	0.754614	+	0.656169i	$0.227824\pi$
$224$	−1.93364	−0.129197
$225$	0	0
$226$	−1.83560	−0.122102
$227$	18.1124	1.20216	0.601080	−	0.799189i	$-0.294737\pi$
$227$	18.1124	1.20216	0.601080	+	0.799189i	$0.294737\pi$
$228$	0	0
$229$	−9.41604	−0.622229	−0.311115	−	0.950372i	$-0.600702\pi$
$229$	−9.41604	−0.622229	−0.311115	+	0.950372i	$0.600702\pi$
$230$	0	0
$231$	8.32564	0.547787
$232$	−3.43218	−0.225334
$233$	−15.7000	−1.02854	−0.514271	−	0.857628i	$-0.671937\pi$
$233$	−15.7000	−1.02854	−0.514271	+	0.857628i	$0.671937\pi$
$234$	−33.2191	−2.17160
$235$	0	0
$236$	5.95455	0.387608
$237$	−27.5408	−1.78897
$238$	2.10636	0.136535
$239$	−23.4610	−1.51757	−0.758783	−	0.651344i	$-0.774205\pi$
$239$	−23.4610	−1.51757	−0.758783	+	0.651344i	$0.774205\pi$
$240$	0	0
$241$	−13.1694	−0.848314	−0.424157	−	0.905589i	$-0.639430\pi$
$241$	−13.1694	−0.848314	−0.424157	+	0.905589i	$0.639430\pi$
$242$	−10.8572	−0.697927
$243$	5.46321	0.350465
$244$	2.89819	0.185538
$245$	0	0
$246$	−30.1333	−1.92123
$247$	0	0
$248$	19.1520	1.21615
$249$	6.46544	0.409730
$250$	0	0
$251$	−17.3251	−1.09355	−0.546776	−	0.837279i	$-0.684145\pi$
$251$	−17.3251	−1.09355	−0.546776	+	0.837279i	$0.684145\pi$
$252$	−2.22736	−0.140310
$253$	12.7704	0.802869
$254$	−2.74563	−0.172276
$255$	0	0
$256$	−12.6659	−0.791618
$257$	−5.67960	−0.354283	−0.177142	−	0.984185i	$-0.556685\pi$
$257$	−5.67960	−0.354283	−0.177142	+	0.984185i	$0.556685\pi$
$258$	−36.2483	−2.25672
$259$	−2.29820	−0.142803
$260$	0	0
$261$	−7.01621	−0.434293
$262$	15.3843	0.950445
$263$	−5.65764	−0.348865	−0.174433	−	0.984669i	$-0.555809\pi$
$263$	−5.65764	−0.348865	−0.174433	+	0.984669i	$0.555809\pi$
$264$	42.0209	2.58621
$265$	0	0
$266$	0	0
$267$	−24.1707	−1.47922
$268$	−4.93087	−0.301201
$269$	23.9918	1.46280	0.731402	−	0.681946i	$-0.238866\pi$
$269$	23.9918	1.46280	0.731402	+	0.681946i	$0.238866\pi$
$270$	0	0
$271$	−21.2995	−1.29385	−0.646926	−	0.762553i	$-0.723946\pi$
$271$	−21.2995	−1.29385	−0.646926	+	0.762553i	$0.723946\pi$
$272$	7.25311	0.439785
$273$	8.23808	0.498591
$274$	−15.1645	−0.916121
$275$	0	0
$276$	−5.04717	−0.303804
$277$	0.821109	0.0493357	0.0246678	−	0.999696i	$-0.492147\pi$
$277$	0.821109	0.0493357	0.0246678	+	0.999696i	$0.492147\pi$
$278$	−12.6340	−0.757735
$279$	39.1513	2.34393
$280$	0	0
$281$	0.587479	0.0350461	0.0175230	−	0.999846i	$-0.494422\pi$
$281$	0.587479	0.0350461	0.0175230	+	0.999846i	$0.494422\pi$
$282$	−21.3561	−1.27174
$283$	−30.9424	−1.83933	−0.919667	−	0.392699i	$-0.871541\pi$
$283$	−30.9424	−1.83933	−0.919667	+	0.392699i	$0.871541\pi$
$284$	6.75969	0.401114
$285$	0	0
$286$	−23.7052	−1.40172
$287$	5.05840	0.298588
$288$	−19.9506	−1.17560
$289$	−8.57008	−0.504122
$290$	0	0
$291$	−29.4739	−1.72779
$292$	2.16594	0.126752
$293$	3.76271	0.219820	0.109910	−	0.993942i	$-0.464944\pi$
$293$	3.76271	0.219820	0.109910	+	0.993942i	$0.464944\pi$
$294$	24.0557	1.40296
$295$	0	0
$296$	−11.5994	−0.674202
$297$	44.8998	2.60535
$298$	−4.49338	−0.260295
$299$	12.6361	0.730765
$300$	0	0
$301$	6.08491	0.350728
$302$	−11.3322	−0.652097
$303$	−2.95961	−0.170025
$304$	0	0
$305$	0	0
$306$	21.7327	1.24237
$307$	20.3419	1.16097	0.580485	−	0.814271i	$-0.302863\pi$
$307$	20.3419	1.16097	0.580485	+	0.814271i	$0.302863\pi$
$308$	−1.58945	−0.0905670
$309$	−10.1819	−0.579228
$310$	0	0
$311$	−7.67830	−0.435397	−0.217698	−	0.976016i	$-0.569855\pi$
$311$	−7.67830	−0.435397	−0.217698	+	0.976016i	$0.569855\pi$
$312$	41.5790	2.35395
$313$	−23.9927	−1.35615	−0.678074	−	0.734993i	$-0.737185\pi$
$313$	−23.9927	−1.35615	−0.678074	+	0.734993i	$0.737185\pi$
$314$	−4.11630	−0.232296
$315$	0	0
$316$	5.25781	0.295775
$317$	1.03904	0.0583580	0.0291790	−	0.999574i	$-0.490711\pi$
$317$	1.03904	0.0583580	0.0291790	+	0.999574i	$0.490711\pi$
$318$	−30.6596	−1.71931
$319$	−5.00678	−0.280326
$320$	0	0
$321$	28.9986	1.61854
$322$	−2.06561	−0.115112
$323$	0	0
$324$	−6.77641	−0.376467
$325$	0	0
$326$	7.92292	0.438810
$327$	−16.8922	−0.934141
$328$	25.5306	1.40969
$329$	3.58500	0.197647
$330$	0	0
$331$	−30.8316	−1.69466	−0.847328	−	0.531069i	$-0.821790\pi$
$331$	−30.8316	−1.69466	−0.847328	+	0.531069i	$0.821790\pi$
$332$	−1.23431	−0.0677418
$333$	−23.7120	−1.29941
$334$	19.5970	1.07230
$335$	0	0
$336$	−4.63716	−0.252978
$337$	20.7153	1.12843	0.564217	−	0.825627i	$-0.309178\pi$
$337$	20.7153	1.12843	0.564217	+	0.825627i	$0.309178\pi$
$338$	−7.97410	−0.433734
$339$	4.69674	0.255092
$340$	0	0
$341$	27.9384	1.51295
$342$	0	0
$343$	−8.30239	−0.448287
$344$	30.7116	1.65586
$345$	0	0
$346$	27.1318	1.45862
$347$	−8.22136	−0.441346	−0.220673	−	0.975348i	$-0.570825\pi$
$347$	−8.22136	−0.441346	−0.220673	+	0.975348i	$0.570825\pi$
$348$	1.97880	0.106075
$349$	11.9216	0.638150	0.319075	−	0.947730i	$-0.396628\pi$
$349$	11.9216	0.638150	0.319075	+	0.947730i	$0.396628\pi$
$350$	0	0
$351$	44.4276	2.37137
$352$	−14.2368	−0.758823
$353$	11.7983	0.627959	0.313980	−	0.949430i	$-0.398338\pi$
$353$	11.7983	0.627959	0.313980	+	0.949430i	$0.398338\pi$
$354$	37.1450	1.97423
$355$	0	0
$356$	4.61441	0.244563
$357$	−5.38953	−0.285244
$358$	−2.77382	−0.146601
$359$	0.110812	0.00584841	0.00292420	−	0.999996i	$-0.499069\pi$
$359$	0.110812	0.00584841	0.00292420	+	0.999996i	$0.499069\pi$
$360$	0	0
$361$	0	0
$362$	−26.6040	−1.39827
$363$	27.7802	1.45808
$364$	−1.57273	−0.0824334
$365$	0	0
$366$	18.0792	0.945013
$367$	−11.7397	−0.612808	−0.306404	−	0.951902i	$-0.599126\pi$
$367$	−11.7397	−0.612808	−0.306404	+	0.951902i	$0.599126\pi$
$368$	−7.11278	−0.370779
$369$	52.1908	2.71694
$370$	0	0
$371$	5.14675	0.267206
$372$	−11.0419	−0.572498
$373$	14.5190	0.751763	0.375882	−	0.926668i	$-0.377340\pi$
$373$	14.5190	0.751763	0.375882	+	0.926668i	$0.377340\pi$
$374$	15.5085	0.801924
$375$	0	0
$376$	18.0941	0.933131
$377$	−4.95412	−0.255150
$378$	−7.26251	−0.373543
$379$	6.59023	0.338518	0.169259	−	0.985572i	$-0.445863\pi$
$379$	6.59023	0.338518	0.169259	+	0.985572i	$0.445863\pi$
$380$	0	0
$381$	7.02522	0.359913
$382$	−2.66485	−0.136345
$383$	−2.86921	−0.146610	−0.0733049	−	0.997310i	$-0.523355\pi$
$383$	−2.86921	−0.146610	−0.0733049	+	0.997310i	$0.523355\pi$
$384$	−12.5041	−0.638099
$385$	0	0
$386$	5.41038	0.275381
$387$	62.7819	3.19138
$388$	5.62686	0.285660
$389$	−6.33149	−0.321019	−0.160510	−	0.987034i	$-0.551314\pi$
$389$	−6.33149	−0.321019	−0.160510	+	0.987034i	$0.551314\pi$
$390$	0	0
$391$	−8.26682	−0.418071
$392$	−20.3813	−1.02941
$393$	−39.3637	−1.98563
$394$	−22.8936	−1.15336
$395$	0	0
$396$	−16.3993	−0.824097
$397$	30.5498	1.53325	0.766626	−	0.642094i	$-0.221934\pi$
$397$	30.5498	1.53325	0.766626	+	0.642094i	$0.221934\pi$
$398$	7.32522	0.367180
$399$	0	0
$400$	0	0
$401$	−30.3422	−1.51522	−0.757609	−	0.652709i	$-0.773633\pi$
$401$	−30.3422	−1.51522	−0.757609	+	0.652709i	$0.773633\pi$
$402$	−30.7592	−1.53413
$403$	27.6446	1.37708
$404$	0.565018	0.0281107
$405$	0	0
$406$	0.809843	0.0401919
$407$	−16.9209	−0.838740
$408$	−27.2019	−1.34669
$409$	14.9726	0.740345	0.370173	−	0.928963i	$-0.379299\pi$
$409$	14.9726	0.740345	0.370173	+	0.928963i	$0.379299\pi$
$410$	0	0
$411$	38.8013	1.91393
$412$	1.94382	0.0957653
$413$	−6.23543	−0.306825
$414$	−21.3122	−1.04744
$415$	0	0
$416$	−14.0871	−0.690675
$417$	32.3264	1.58303
$418$	0	0
$419$	−6.17419	−0.301629	−0.150815	−	0.988562i	$-0.548190\pi$
$419$	−6.17419	−0.301629	−0.150815	+	0.988562i	$0.548190\pi$
$420$	0	0
$421$	27.5428	1.34235	0.671177	−	0.741297i	$-0.265789\pi$
$421$	27.5428	1.34235	0.671177	+	0.741297i	$0.265789\pi$
$422$	15.1165	0.735858
$423$	36.9887	1.79845
$424$	25.9765	1.26153
$425$	0	0
$426$	42.1675	2.04302
$427$	−3.03490	−0.146869
$428$	−5.53610	−0.267598
$429$	60.6544	2.92842
$430$	0	0
$431$	15.0588	0.725358	0.362679	−	0.931914i	$-0.381862\pi$
$431$	15.0588	0.725358	0.362679	+	0.931914i	$0.381862\pi$
$432$	−25.0080	−1.20320
$433$	−0.970840	−0.0466556	−0.0233278	−	0.999728i	$-0.507426\pi$
$433$	−0.970840	−0.0466556	−0.0233278	+	0.999728i	$0.507426\pi$
$434$	−4.51902	−0.216920
$435$	0	0
$436$	3.22488	0.154444
$437$	0	0
$438$	13.5113	0.645596
$439$	27.4375	1.30952	0.654760	−	0.755837i	$-0.272770\pi$
$439$	27.4375	1.30952	0.654760	+	0.755837i	$0.272770\pi$
$440$	0	0
$441$	−41.6643	−1.98402
$442$	15.3454	0.729905
$443$	8.76544	0.416459	0.208229	−	0.978080i	$-0.433230\pi$
$443$	8.76544	0.416459	0.208229	+	0.978080i	$0.433230\pi$
$444$	6.68755	0.317377
$445$	0	0
$446$	−26.8402	−1.27092
$447$	11.4972	0.543798
$448$	5.34637	0.252592
$449$	9.63397	0.454655	0.227327	−	0.973818i	$-0.427001\pi$
$449$	9.63397	0.454655	0.227327	+	0.973818i	$0.427001\pi$
$450$	0	0
$451$	37.2434	1.75372
$452$	−0.896652	−0.0421750
$453$	28.9957	1.36234
$454$	−21.5702	−1.01234
$455$	0	0
$456$	0	0
$457$	10.6708	0.499161	0.249580	−	0.968354i	$-0.419707\pi$
$457$	10.6708	0.499161	0.249580	+	0.968354i	$0.419707\pi$
$458$	11.2137	0.523980
$459$	−29.0655	−1.35666
$460$	0	0
$461$	5.68680	0.264861	0.132430	−	0.991192i	$-0.457722\pi$
$461$	5.68680	0.264861	0.132430	+	0.991192i	$0.457722\pi$
$462$	−9.91509	−0.461292
$463$	−35.3550	−1.64309	−0.821543	−	0.570147i	$-0.806886\pi$
$463$	−35.3550	−1.64309	−0.821543	+	0.570147i	$0.806886\pi$
$464$	2.78864	0.129459
$465$	0	0
$466$	18.6973	0.866135
$467$	32.9071	1.52276	0.761380	−	0.648306i	$-0.224522\pi$
$467$	32.9071	1.52276	0.761380	+	0.648306i	$0.224522\pi$
$468$	−16.2269	−0.750086
$469$	5.16347	0.238427
$470$	0	0
$471$	10.5323	0.485305
$472$	−31.4713	−1.44858
$473$	44.8013	2.05997
$474$	32.7986	1.50649
$475$	0	0
$476$	1.02891	0.0471602
$477$	53.1023	2.43139
$478$	27.9399	1.27794
$479$	−9.05721	−0.413835	−0.206917	−	0.978358i	$-0.566343\pi$
$479$	−9.05721	−0.413835	−0.206917	+	0.978358i	$0.566343\pi$
$480$	0	0
$481$	−16.7430	−0.763414
$482$	15.6835	0.714366
$483$	5.28525	0.240487
$484$	−5.30351	−0.241069
$485$	0	0
$486$	−6.50619	−0.295127
$487$	−16.5206	−0.748620	−0.374310	−	0.927304i	$-0.622120\pi$
$487$	−16.5206	−0.748620	−0.374310	+	0.927304i	$0.622120\pi$
$488$	−15.3177	−0.693399
$489$	−20.2723	−0.916745
$490$	0	0
$491$	−1.39125	−0.0627862	−0.0313931	−	0.999507i	$-0.509994\pi$
$491$	−1.39125	−0.0627862	−0.0313931	+	0.999507i	$0.509994\pi$
$492$	−14.7195	−0.663605
$493$	3.24109	0.145972
$494$	0	0
$495$	0	0
$496$	−15.5610	−0.698708
$497$	−7.07856	−0.317517
$498$	−7.69975	−0.345034
$499$	−16.6651	−0.746033	−0.373016	−	0.927825i	$-0.621676\pi$
$499$	−16.6651	−0.746033	−0.373016	+	0.927825i	$0.621676\pi$
$500$	0	0
$501$	−50.1427	−2.24021
$502$	20.6327	0.920881
$503$	−15.6391	−0.697314	−0.348657	−	0.937250i	$-0.613362\pi$
$503$	−15.6391	−0.697314	−0.348657	+	0.937250i	$0.613362\pi$
$504$	11.7721	0.524373
$505$	0	0
$506$	−15.2084	−0.676096
$507$	20.4033	0.906141
$508$	−1.34118	−0.0595053
$509$	19.1540	0.848988	0.424494	−	0.905431i	$-0.360452\pi$
$509$	19.1540	0.848988	0.424494	+	0.905431i	$0.360452\pi$
$510$	0	0
$511$	−2.26811	−0.100335
$512$	23.2910	1.02933
$513$	0	0
$514$	6.76389	0.298342
$515$	0	0
$516$	−17.7065	−0.779487
$517$	26.3952	1.16086
$518$	2.73695	0.120255
$519$	−69.4220	−3.04729
$520$	0	0
$521$	19.2394	0.842892	0.421446	−	0.906853i	$-0.361523\pi$
$521$	19.2394	0.842892	0.421446	+	0.906853i	$0.361523\pi$
$522$	8.35567	0.365718
$523$	6.63945	0.290323	0.145161	−	0.989408i	$-0.453630\pi$
$523$	6.63945	0.290323	0.145161	+	0.989408i	$0.453630\pi$
$524$	7.51490	0.328290
$525$	0	0
$526$	6.73774	0.293779
$527$	−18.0857	−0.787825
$528$	−34.1419	−1.48584
$529$	−14.8931	−0.647528
$530$	0	0
$531$	−64.3349	−2.79190
$532$	0	0
$533$	36.8517	1.59623
$534$	28.7851	1.24565
$535$	0	0
$536$	26.0609	1.12566
$537$	7.09736	0.306274
$538$	−28.5720	−1.23183
$539$	−29.7317	−1.28064
$540$	0	0
$541$	41.7511	1.79502	0.897510	−	0.440994i	$-0.145374\pi$
$541$	41.7511	1.79502	0.897510	+	0.440994i	$0.145374\pi$
$542$	25.3658	1.08955
$543$	68.0714	2.92122
$544$	9.21606	0.395135
$545$	0	0
$546$	−9.81081	−0.419864
$547$	12.2052	0.521855	0.260927	−	0.965358i	$-0.415972\pi$
$547$	12.2052	0.521855	0.260927	+	0.965358i	$0.415972\pi$
$548$	−7.40754	−0.316434
$549$	−31.3131	−1.33641
$550$	0	0
$551$	0	0
$552$	26.6756	1.13539
$553$	−5.50583	−0.234132
$554$	−0.977867	−0.0415456
$555$	0	0
$556$	−6.17143	−0.261727
$557$	−35.1548	−1.48956	−0.744779	−	0.667311i	$-0.767445\pi$
$557$	−35.1548	−1.48956	−0.744779	+	0.667311i	$0.767445\pi$
$558$	−46.6257	−1.97382
$559$	44.3301	1.87496
$560$	0	0
$561$	−39.6814	−1.67535
$562$	−0.699634	−0.0295123
$563$	17.8406	0.751891	0.375945	−	0.926642i	$-0.377318\pi$
$563$	17.8406	0.751891	0.375945	+	0.926642i	$0.377318\pi$
$564$	−10.4320	−0.439267
$565$	0	0
$566$	36.8496	1.54890
$567$	7.09606	0.298007
$568$	−35.7267	−1.49906
$569$	−31.6042	−1.32492	−0.662459	−	0.749098i	$-0.730487\pi$
$569$	−31.6042	−1.32492	−0.662459	+	0.749098i	$0.730487\pi$
$570$	0	0
$571$	−4.73053	−0.197967	−0.0989833	−	0.995089i	$-0.531559\pi$
$571$	−4.73053	−0.197967	−0.0989833	+	0.995089i	$0.531559\pi$
$572$	−11.5795	−0.484164
$573$	6.81852	0.284848
$574$	−6.02410	−0.251441
$575$	0	0
$576$	55.1620	2.29841
$577$	−24.4074	−1.01609	−0.508047	−	0.861330i	$-0.669632\pi$
$577$	−24.4074	−1.01609	−0.508047	+	0.861330i	$0.669632\pi$
$578$	10.2062	0.424522
$579$	−13.8435	−0.575316
$580$	0	0
$581$	1.29254	0.0536235
$582$	35.1008	1.45497
$583$	37.8939	1.56941
$584$	−11.4475	−0.473703
$585$	0	0
$586$	−4.48104	−0.185110
$587$	28.6154	1.18109	0.590543	−	0.807006i	$-0.298914\pi$
$587$	28.6154	1.18109	0.590543	+	0.807006i	$0.298914\pi$
$588$	11.7507	0.484590
$589$	0	0
$590$	0	0
$591$	58.5776	2.40956
$592$	9.42450	0.387345
$593$	−3.71511	−0.152561	−0.0762807	−	0.997086i	$-0.524305\pi$
$593$	−3.71511	−0.152561	−0.0762807	+	0.997086i	$0.524305\pi$
$594$	−53.4716	−2.19397
$595$	0	0
$596$	−2.19492	−0.0899076
$597$	−18.7430	−0.767099
$598$	−15.0485	−0.615378
$599$	−7.09940	−0.290074	−0.145037	−	0.989426i	$-0.546330\pi$
$599$	−7.09940	−0.290074	−0.145037	+	0.989426i	$0.546330\pi$
$600$	0	0
$601$	11.0596	0.451131	0.225566	−	0.974228i	$-0.427577\pi$
$601$	11.0596	0.451131	0.225566	+	0.974228i	$0.427577\pi$
$602$	−7.24658	−0.295349
$603$	53.2748	2.16952
$604$	−5.53556	−0.225239
$605$	0	0
$606$	3.52463	0.143178
$607$	27.1193	1.10074	0.550369	−	0.834921i	$-0.314487\pi$
$607$	27.1193	1.10074	0.550369	+	0.834921i	$0.314487\pi$
$608$	0	0
$609$	−2.07214	−0.0839673
$610$	0	0
$611$	26.1176	1.05660
$612$	10.6160	0.429125
$613$	−41.8312	−1.68954	−0.844772	−	0.535126i	$-0.820264\pi$
$613$	−41.8312	−1.68954	−0.844772	+	0.535126i	$0.820264\pi$
$614$	−24.2253	−0.977654
$615$	0	0
$616$	8.40062	0.338471
$617$	17.7052	0.712786	0.356393	−	0.934336i	$-0.384006\pi$
$617$	17.7052	0.712786	0.356393	+	0.934336i	$0.384006\pi$
$618$	12.1257	0.487768
$619$	−39.8064	−1.59995	−0.799976	−	0.600032i	$-0.795155\pi$
$619$	−39.8064	−1.59995	−0.799976	+	0.600032i	$0.795155\pi$
$620$	0	0
$621$	28.5031	1.14379
$622$	9.14416	0.366648
$623$	−4.83208	−0.193593
$624$	−33.7829	−1.35240
$625$	0	0
$626$	28.5732	1.14201
$627$	0	0
$628$	−2.01072	−0.0802366
$629$	10.9536	0.436749
$630$	0	0
$631$	46.1980	1.83911	0.919557	−	0.392956i	$-0.128547\pi$
$631$	46.1980	1.83911	0.919557	+	0.392956i	$0.128547\pi$
$632$	−27.7889	−1.10538
$633$	−38.6784	−1.53733
$634$	−1.23740	−0.0491433
$635$	0	0
$636$	−14.9766	−0.593860
$637$	−29.4191	−1.16563
$638$	5.96262	0.236062
$639$	−73.0340	−2.88918
$640$	0	0
$641$	−6.61348	−0.261217	−0.130608	−	0.991434i	$-0.541693\pi$
$641$	−6.61348	−0.261217	−0.130608	+	0.991434i	$0.541693\pi$
$642$	−34.5347	−1.36297
$643$	30.6152	1.20735	0.603673	−	0.797232i	$-0.293703\pi$
$643$	30.6152	1.20735	0.603673	+	0.797232i	$0.293703\pi$
$644$	−1.00901	−0.0397604
$645$	0	0
$646$	0	0
$647$	11.8979	0.467753	0.233877	−	0.972266i	$-0.424859\pi$
$647$	11.8979	0.467753	0.233877	+	0.972266i	$0.424859\pi$
$648$	35.8150	1.40695
$649$	−45.9095	−1.80211
$650$	0	0
$651$	11.5628	0.453182
$652$	3.87018	0.151568
$653$	1.42899	0.0559207	0.0279604	−	0.999609i	$-0.491099\pi$
$653$	1.42899	0.0559207	0.0279604	+	0.999609i	$0.491099\pi$
$654$	20.1171	0.786640
$655$	0	0
$656$	−20.7436	−0.809901
$657$	−23.4015	−0.912981
$658$	−4.26941	−0.166439
$659$	24.4970	0.954268	0.477134	−	0.878831i	$-0.341676\pi$
$659$	24.4970	0.954268	0.477134	+	0.878831i	$0.341676\pi$
$660$	0	0
$661$	3.22606	0.125479	0.0627396	−	0.998030i	$-0.480016\pi$
$661$	3.22606	0.125479	0.0627396	+	0.998030i	$0.480016\pi$
$662$	36.7176	1.42707
$663$	−39.2641	−1.52489
$664$	6.52366	0.253167
$665$	0	0
$666$	28.2389	1.09423
$667$	−3.17838	−0.123068
$668$	9.57273	0.370380
$669$	68.6759	2.65516
$670$	0	0
$671$	−22.3451	−0.862621
$672$	−5.89213	−0.227294
$673$	37.1424	1.43173	0.715866	−	0.698237i	$-0.246032\pi$
$673$	37.1424	1.43173	0.715866	+	0.698237i	$0.246032\pi$
$674$	−24.6700	−0.950254
$675$	0	0
$676$	−3.89518	−0.149815
$677$	−24.7550	−0.951412	−0.475706	−	0.879604i	$-0.657807\pi$
$677$	−24.7550	−0.951412	−0.475706	+	0.879604i	$0.657807\pi$
$678$	−5.59339	−0.214813
$679$	−5.89228	−0.226125
$680$	0	0
$681$	55.1914	2.11494
$682$	−33.2722	−1.27406
$683$	40.1153	1.53497	0.767484	−	0.641068i	$-0.221508\pi$
$683$	40.1153	1.53497	0.767484	+	0.641068i	$0.221508\pi$
$684$	0	0
$685$	0	0
$686$	9.88740	0.377503
$687$	−28.6923	−1.09468
$688$	−24.9531	−0.951328
$689$	37.4954	1.42846
$690$	0	0
$691$	39.4963	1.50251	0.751254	−	0.660013i	$-0.229449\pi$
$691$	39.4963	1.50251	0.751254	+	0.660013i	$0.229449\pi$
$692$	13.2533	0.503816
$693$	17.1729	0.652344
$694$	9.79090	0.371658
$695$	0	0
$696$	−10.4584	−0.396426
$697$	−24.1092	−0.913201
$698$	−14.1976	−0.537386
$699$	−47.8406	−1.80950
$700$	0	0
$701$	0.0219552	0.000829236 0	0.000414618	−	1.00000i	$-0.499868\pi$
$701$	0.0219552	0.000829236 0	0.000414618		1.00000i	$0.499868\pi$
$702$	−52.9092	−1.99693
$703$	0	0
$704$	39.3637	1.48357
$705$	0	0
$706$	−14.0507	−0.528805
$707$	−0.591670	−0.0222521
$708$	18.1445	0.681913
$709$	−17.8017	−0.668558	−0.334279	−	0.942474i	$-0.608493\pi$
$709$	−17.8017	−0.668558	−0.334279	+	0.942474i	$0.608493\pi$
$710$	0	0
$711$	−56.8071	−2.13043
$712$	−24.3883	−0.913992
$713$	17.7358	0.664210
$714$	6.41844	0.240204
$715$	0	0
$716$	−1.35495	−0.0506370
$717$	−71.4896	−2.66983
$718$	−0.131967	−0.00492495
$719$	18.8103	0.701506	0.350753	−	0.936468i	$-0.385926\pi$
$719$	18.8103	0.701506	0.350753	+	0.936468i	$0.385926\pi$
$720$	0	0
$721$	−2.03552	−0.0758066
$722$	0	0
$723$	−40.1294	−1.49243
$724$	−12.9955	−0.482973
$725$	0	0
$726$	−33.0837	−1.22785
$727$	−5.00301	−0.185552	−0.0927758	−	0.995687i	$-0.529574\pi$
$727$	−5.00301	−0.185552	−0.0927758	+	0.995687i	$0.529574\pi$
$728$	8.31227	0.308073
$729$	−18.2986	−0.677725
$730$	0	0
$731$	−29.0017	−1.07267
$732$	8.83129	0.326414
$733$	23.5259	0.868950	0.434475	−	0.900684i	$-0.356934\pi$
$733$	23.5259	0.868950	0.434475	+	0.900684i	$0.356934\pi$
$734$	13.9809	0.516046
$735$	0	0
$736$	−9.03774	−0.333136
$737$	38.0170	1.40037
$738$	−62.1545	−2.28794
$739$	−37.3836	−1.37518	−0.687590	−	0.726099i	$-0.741331\pi$
$739$	−37.3836	−1.37518	−0.687590	+	0.726099i	$0.741331\pi$
$740$	0	0
$741$	0	0
$742$	−6.12932	−0.225014
$743$	10.3886	0.381121	0.190560	−	0.981675i	$-0.438970\pi$
$743$	10.3886	0.381121	0.190560	+	0.981675i	$0.438970\pi$
$744$	58.3594	2.13956
$745$	0	0
$746$	−17.2908	−0.633060
$747$	13.3359	0.487937
$748$	7.57556	0.276990
$749$	5.79725	0.211827
$750$	0	0
$751$	31.2825	1.14152	0.570758	−	0.821119i	$-0.306650\pi$
$751$	31.2825	1.14152	0.570758	+	0.821119i	$0.306650\pi$
$752$	−14.7014	−0.536105
$753$	−52.7927	−1.92387
$754$	5.89991	0.214862
$755$	0	0
$756$	−3.54758	−0.129024
$757$	12.6241	0.458830	0.229415	−	0.973329i	$-0.426319\pi$
$757$	12.6241	0.458830	0.229415	+	0.973329i	$0.426319\pi$
$758$	−7.84837	−0.285066
$759$	38.9137	1.41248
$760$	0	0
$761$	11.5495	0.418668	0.209334	−	0.977844i	$-0.432870\pi$
$761$	11.5495	0.418668	0.209334	+	0.977844i	$0.432870\pi$
$762$	−8.36640	−0.303083
$763$	−3.37701	−0.122256
$764$	−1.30172	−0.0470946
$765$	0	0
$766$	3.41697	0.123460
$767$	−45.4267	−1.64026
$768$	−38.5951	−1.39268
$769$	−26.9207	−0.970784	−0.485392	−	0.874297i	$-0.661323\pi$
$769$	−26.9207	−0.970784	−0.485392	+	0.874297i	$0.661323\pi$
$770$	0	0
$771$	−17.3067	−0.623286
$772$	2.64286	0.0951184
$773$	21.3331	0.767299	0.383649	−	0.923479i	$-0.374667\pi$
$773$	21.3331	0.767299	0.383649	+	0.923479i	$0.374667\pi$
$774$	−74.7676	−2.68747
$775$	0	0
$776$	−29.7394	−1.06758
$777$	−7.00301	−0.251232
$778$	7.54024	0.270331
$779$	0	0
$780$	0	0
$781$	−52.1172	−1.86490
$782$	9.84503	0.352058
$783$	−11.1749	−0.399360
$784$	16.5598	0.591421
$785$	0	0
$786$	46.8786	1.67210
$787$	3.52489	0.125649	0.0628243	−	0.998025i	$-0.479989\pi$
$787$	3.52489	0.125649	0.0628243	+	0.998025i	$0.479989\pi$
$788$	−11.1830	−0.398379
$789$	−17.2398	−0.613753
$790$	0	0
$791$	0.938948	0.0333852
$792$	86.6746	3.07985
$793$	−22.1100	−0.785151
$794$	−36.3821	−1.29115
$795$	0	0
$796$	3.57821	0.126826
$797$	−39.0084	−1.38175	−0.690875	−	0.722974i	$-0.742774\pi$
$797$	−39.0084	−1.38175	−0.690875	+	0.722974i	$0.742774\pi$
$798$	0	0
$799$	−17.0867	−0.604484
$800$	0	0
$801$	−49.8557	−1.76156
$802$	36.1348	1.27597
$803$	−16.6994	−0.589309
$804$	−15.0252	−0.529899
$805$	0	0
$806$	−32.9222	−1.15964
$807$	73.1071	2.57349
$808$	−2.98626	−0.105056
$809$	50.7196	1.78321	0.891604	−	0.452816i	$-0.149581\pi$
$809$	50.7196	1.78321	0.891604	+	0.452816i	$0.149581\pi$
$810$	0	0
$811$	−29.0376	−1.01965	−0.509824	−	0.860279i	$-0.670289\pi$
$811$	−29.0376	−1.01965	−0.509824	+	0.860279i	$0.670289\pi$
$812$	0.395591	0.0138825
$813$	−64.9032	−2.27625
$814$	20.1513	0.706303
$815$	0	0
$816$	22.1015	0.773706
$817$	0	0
$818$	−17.8310	−0.623445
$819$	16.9923	0.593759
$820$	0	0
$821$	32.9878	1.15128	0.575640	−	0.817703i	$-0.304753\pi$
$821$	32.9878	1.15128	0.575640	+	0.817703i	$0.304753\pi$
$822$	−46.2088	−1.61172
$823$	26.5533	0.925591	0.462796	−	0.886465i	$-0.346846\pi$
$823$	26.5533	0.925591	0.462796	+	0.886465i	$0.346846\pi$
$824$	−10.2736	−0.357898
$825$	0	0
$826$	7.42583	0.258378
$827$	−32.7666	−1.13941	−0.569703	−	0.821851i	$-0.692942\pi$
$827$	−32.7666	−1.13941	−0.569703	+	0.821851i	$0.692942\pi$
$828$	−10.4106	−0.361792
$829$	−32.4548	−1.12720	−0.563601	−	0.826047i	$-0.690584\pi$
$829$	−32.4548	−1.12720	−0.563601	+	0.826047i	$0.690584\pi$
$830$	0	0
$831$	2.50206	0.0867955
$832$	38.9497	1.35034
$833$	19.2466	0.666854
$834$	−38.4979	−1.33307
$835$	0	0
$836$	0	0
$837$	62.3576	2.15539
$838$	7.35291	0.254002
$839$	35.2098	1.21558	0.607788	−	0.794099i	$-0.292057\pi$
$839$	35.2098	1.21558	0.607788	+	0.794099i	$0.292057\pi$
$840$	0	0
$841$	−27.7539	−0.957030
$842$	−32.8010	−1.13040
$843$	1.79015	0.0616560
$844$	7.38408	0.254171
$845$	0	0
$846$	−44.0502	−1.51448
$847$	5.55368	0.190827
$848$	−21.1059	−0.724779
$849$	−94.2867	−3.23591
$850$	0	0
$851$	−10.7417	−0.368220
$852$	20.5979	0.705674
$853$	1.78825	0.0612286	0.0306143	−	0.999531i	$-0.490254\pi$
$853$	1.78825	0.0612286	0.0306143	+	0.999531i	$0.490254\pi$
$854$	3.61430	0.123679
$855$	0	0
$856$	29.2597	1.00008
$857$	3.61379	0.123445	0.0617224	−	0.998093i	$-0.480341\pi$
$857$	3.61379	0.123445	0.0617224	+	0.998093i	$0.480341\pi$
$858$	−72.2339	−2.46603
$859$	−25.3648	−0.865437	−0.432719	−	0.901529i	$-0.642446\pi$
$859$	−25.3648	−0.865437	−0.432719	+	0.901529i	$0.642446\pi$
$860$	0	0
$861$	15.4138	0.525301
$862$	−17.9337	−0.610824
$863$	−29.9878	−1.02080	−0.510398	−	0.859939i	$-0.670502\pi$
$863$	−29.9878	−1.02080	−0.510398	+	0.859939i	$0.670502\pi$
$864$	−31.7760	−1.08104
$865$	0	0
$866$	1.15618	0.0392887
$867$	−26.1145	−0.886895
$868$	−2.20745	−0.0749257
$869$	−40.5377	−1.37515
$870$	0	0
$871$	37.6172	1.27461
$872$	−17.0443	−0.577194
$873$	−60.7945	−2.05758
$874$	0	0
$875$	0	0
$876$	6.59999	0.222993
$877$	10.6478	0.359550	0.179775	−	0.983708i	$-0.442463\pi$
$877$	10.6478	0.359550	0.179775	+	0.983708i	$0.442463\pi$
$878$	−32.6756	−1.10275
$879$	11.4656	0.386726
$880$	0	0
$881$	−31.5797	−1.06395	−0.531973	−	0.846761i	$-0.678549\pi$
$881$	−31.5797	−1.06395	−0.531973	+	0.846761i	$0.678549\pi$
$882$	49.6185	1.67074
$883$	−17.0174	−0.572682	−0.286341	−	0.958128i	$-0.592439\pi$
$883$	−17.0174	−0.572682	−0.286341	+	0.958128i	$0.592439\pi$
$884$	7.49589	0.252114
$885$	0	0
$886$	−10.4388	−0.350700
$887$	−13.2290	−0.444187	−0.222093	−	0.975025i	$-0.571289\pi$
$887$	−13.2290	−0.444187	−0.222093	+	0.975025i	$0.571289\pi$
$888$	−35.3454	−1.18611
$889$	1.40445	0.0471036
$890$	0	0
$891$	52.2461	1.75031
$892$	−13.1109	−0.438985
$893$	0	0
$894$	−13.6921	−0.457933
$895$	0	0
$896$	−2.49977	−0.0835113
$897$	38.5044	1.28562
$898$	−11.4732	−0.382865
$899$	−6.95350	−0.231912
$900$	0	0
$901$	−24.5303	−0.817222
$902$	−44.3536	−1.47681
$903$	18.5418	0.617031
$904$	4.73903	0.157618
$905$	0	0
$906$	−34.5313	−1.14723
$907$	−14.8904	−0.494428	−0.247214	−	0.968961i	$-0.579515\pi$
$907$	−14.8904	−0.494428	−0.247214	+	0.968961i	$0.579515\pi$
$908$	−10.5366	−0.349669
$909$	−6.10464	−0.202478
$910$	0	0
$911$	49.5480	1.64160	0.820800	−	0.571216i	$-0.193528\pi$
$911$	49.5480	1.64160	0.820800	+	0.571216i	$0.193528\pi$
$912$	0	0
$913$	9.51655	0.314952
$914$	−12.7080	−0.420343
$915$	0	0
$916$	5.47763	0.180986
$917$	−7.86939	−0.259870
$918$	34.6143	1.14244
$919$	27.3835	0.903300	0.451650	−	0.892195i	$-0.350836\pi$
$919$	27.3835	0.903300	0.451650	+	0.892195i	$0.350836\pi$
$920$	0	0
$921$	61.9851	2.04248
$922$	−6.77247	−0.223039
$923$	−51.5691	−1.69742
$924$	−4.84331	−0.159333
$925$	0	0
$926$	42.1046	1.38364
$927$	−21.0017	−0.689787
$928$	3.54334	0.116316
$929$	52.1894	1.71228	0.856139	−	0.516745i	$-0.172857\pi$
$929$	52.1894	1.71228	0.856139	+	0.516745i	$0.172857\pi$
$930$	0	0
$931$	0	0
$932$	9.13323	0.299169
$933$	−23.3971	−0.765987
$934$	−39.1894	−1.28232
$935$	0	0
$936$	85.7630	2.80325
$937$	−15.5798	−0.508968	−0.254484	−	0.967077i	$-0.581906\pi$
$937$	−15.5798	−0.508968	−0.254484	+	0.967077i	$0.581906\pi$
$938$	−6.14922	−0.200779
$939$	−73.1099	−2.38585
$940$	0	0
$941$	58.6553	1.91211	0.956055	−	0.293189i	$-0.0947164\pi$
$941$	58.6553	1.91211	0.956055	+	0.293189i	$0.0947164\pi$
$942$	−12.5431	−0.408675
$943$	23.6427	0.769913
$944$	25.5704	0.832244
$945$	0	0
$946$	−53.3543	−1.73470
$947$	−12.3065	−0.399908	−0.199954	−	0.979805i	$-0.564079\pi$
$947$	−12.3065	−0.399908	−0.199954	+	0.979805i	$0.564079\pi$
$948$	16.0214	0.520352
$949$	−16.5238	−0.536384
$950$	0	0
$951$	3.16612	0.102668
$952$	−5.43807	−0.176249
$953$	8.26997	0.267891	0.133945	−	0.990989i	$-0.457235\pi$
$953$	8.26997	0.267891	0.133945	+	0.990989i	$0.457235\pi$
$954$	−63.2401	−2.04747
$955$	0	0
$956$	13.6481	0.441410
$957$	−15.2565	−0.493173
$958$	10.7863	0.348490
$959$	7.75696	0.250485
$960$	0	0
$961$	7.80138	0.251657
$962$	19.9394	0.642871
$963$	59.8139	1.92748
$964$	7.66108	0.246747
$965$	0	0
$966$	−6.29426	−0.202514
$967$	−14.2247	−0.457436	−0.228718	−	0.973493i	$-0.573453\pi$
$967$	−14.2247	−0.457436	−0.228718	+	0.973493i	$0.573453\pi$
$968$	28.0304	0.900931
$969$	0	0
$970$	0	0
$971$	−9.73861	−0.312527	−0.156263	−	0.987715i	$-0.549945\pi$
$971$	−9.73861	−0.312527	−0.156263	+	0.987715i	$0.549945\pi$
$972$	−3.17814	−0.101939
$973$	6.46254	0.207180
$974$	19.6745	0.630413
$975$	0	0
$976$	12.4456	0.398374
$977$	−6.35308	−0.203253	−0.101627	−	0.994823i	$-0.532405\pi$
$977$	−6.35308	−0.203253	−0.101627	+	0.994823i	$0.532405\pi$
$978$	24.1425	0.771991
$979$	−35.5771	−1.13705
$980$	0	0
$981$	−34.8427	−1.11244
$982$	1.65685	0.0528723
$983$	9.88087	0.315151	0.157575	−	0.987507i	$-0.449632\pi$
$983$	9.88087	0.315151	0.157575	+	0.987507i	$0.449632\pi$
$984$	77.7962	2.48005
$985$	0	0
$986$	−3.85985	−0.122923
$987$	10.9241	0.347718
$988$	0	0
$989$	28.4406	0.904358
$990$	0	0
$991$	−37.3350	−1.18599	−0.592993	−	0.805208i	$-0.702054\pi$
$991$	−37.3350	−1.18599	−0.592993	+	0.805208i	$0.702054\pi$
$992$	−19.7723	−0.627771
$993$	−93.9491	−2.98138
$994$	8.42992	0.267381
$995$	0	0
$996$	−3.76117	−0.119177
$997$	43.6948	1.38383	0.691914	−	0.721980i	$-0.256768\pi$
$997$	43.6948	1.38383	0.691914	+	0.721980i	$0.256768\pi$
$998$	19.8466	0.628234
$999$	−37.7669	−1.19489

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.bp.1.2		4
5.4	even	2		1805.2.a.i.1.3		4
19.8	odd	6		475.2.e.e.26.2		8
19.12	odd	6		475.2.e.e.201.2		8
19.18	odd	2		9025.2.a.bg.1.3		4
95.8	even	12		475.2.j.c.349.3		16
95.12	even	12		475.2.j.c.49.3		16
95.27	even	12		475.2.j.c.349.6		16
95.69	odd	6		95.2.e.c.11.3	✓	8
95.84	odd	6		95.2.e.c.26.3	yes	8
95.88	even	12		475.2.j.c.49.6		16
95.94	odd	2		1805.2.a.o.1.2		4
285.164	even	6		855.2.k.h.676.2		8
285.179	even	6		855.2.k.h.406.2		8
380.179	even	6		1520.2.q.o.881.4		8
380.259	even	6		1520.2.q.o.961.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.e.c.11.3	✓	8	95.69	odd	6
95.2.e.c.26.3	yes	8	95.84	odd	6
475.2.e.e.26.2		8	19.8	odd	6
475.2.e.e.201.2		8	19.12	odd	6
475.2.j.c.49.3		16	95.12	even	12
475.2.j.c.49.6		16	95.88	even	12
475.2.j.c.349.3		16	95.8	even	12
475.2.j.c.349.6		16	95.27	even	12
855.2.k.h.406.2		8	285.179	even	6
855.2.k.h.676.2		8	285.164	even	6
1520.2.q.o.881.4		8	380.179	even	6
1520.2.q.o.961.4		8	380.259	even	6
1805.2.a.i.1.3		4	5.4	even	2
1805.2.a.o.1.2		4	95.94	odd	2
9025.2.a.bg.1.3		4	19.18	odd	2
9025.2.a.bp.1.2		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q-1.19091 q^{2} +3.04717 q^{3} -0.581734 q^{4} -3.62891 q^{6} +0.609175 q^{7} +3.07461 q^{8} +6.28525 q^{9} +4.48517 q^{11} -1.77264 q^{12} +4.43800 q^{13} -0.725473 q^{14} -2.49812 q^{16} -2.90343 q^{17} -7.48517 q^{18} +1.85626 q^{21} -5.34143 q^{22} +2.84726 q^{23} +9.36887 q^{24} -5.28525 q^{26} +10.0107 q^{27} -0.354378 q^{28} -1.11630 q^{29} +6.22908 q^{31} -3.17419 q^{32} +13.6671 q^{33} +3.45773 q^{34} -3.65635 q^{36} -3.77264 q^{37} +13.5233 q^{39} +8.30369 q^{41} -2.21064 q^{42} +9.98877 q^{43} -2.60918 q^{44} -3.39082 q^{46} +5.88500 q^{47} -7.61219 q^{48} -6.62891 q^{49} -8.84726 q^{51} -2.58173 q^{52} +8.44872 q^{53} -11.9219 q^{54} +1.87298 q^{56} +1.32941 q^{58} -10.2359 q^{59} -4.98199 q^{61} -7.41827 q^{62} +3.82882 q^{63} +8.77641 q^{64} -16.2762 q^{66} +8.47616 q^{67} +1.68903 q^{68} +8.67608 q^{69} -11.6199 q^{71} +19.3247 q^{72} -3.72325 q^{73} +4.49288 q^{74} +2.73225 q^{77} -16.1051 q^{78} -9.03817 q^{79} +11.6486 q^{81} -9.88894 q^{82} +2.12178 q^{83} -1.07985 q^{84} -11.8957 q^{86} -3.40155 q^{87} +13.7901 q^{88} -7.93217 q^{89} +2.70352 q^{91} -1.65635 q^{92} +18.9811 q^{93} -7.00850 q^{94} -9.67231 q^{96} -9.67256 q^{97} +7.89443 q^{98} +28.1904 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + 3 q^{3} + 5 q^{4} + 2 q^{6} + 4 q^{7} + 12 q^{8} + q^{9} - 2 q^{11} + 6 q^{12} + 7 q^{13} + q^{14} + 7 q^{16} + q^{17} - 10 q^{18} + 4 q^{21} + 2 q^{22} - 2 q^{23} + 23 q^{24} + 3 q^{26}+ \cdots + 38 q^{99}+O(q^{100}) \) 4 * q + q^2 + 3 * q^3 + 5 * q^4 + 2 * q^6 + 4 * q^7 + 12 * q^8 + q^9 - 2 * q^11 + 6 * q^12 + 7 * q^13 + q^14 + 7 * q^16 + q^17 - 10 * q^18 + 4 * q^21 + 2 * q^22 - 2 * q^23 + 23 * q^24 + 3 * q^26 + 12 * q^27 + 19 * q^28 + q^29 + 30 * q^32 + 19 * q^33 - 15 * q^34 - 7 * q^36 - 2 * q^37 + 15 * q^39 + 8 * q^41 + 15 * q^42 - q^43 - 12 * q^44 - 12 * q^46 + 12 * q^47 + 23 * q^48 - 10 * q^49 - 22 * q^51 - 3 * q^52 - 5 * q^53 - 34 * q^54 + 41 * q^56 + 27 * q^58 + 5 * q^59 - 37 * q^62 + 3 * q^63 + 56 * q^64 - 31 * q^66 + 4 * q^67 - 16 * q^68 + 9 * q^69 - 20 * q^71 + 17 * q^72 + 20 * q^73 + 25 * q^74 - 14 * q^77 - 18 * q^78 - 17 * q^79 + 12 * q^81 - 21 * q^82 - q^83 + 20 * q^84 - 8 * q^86 + 16 * q^87 - 7 * q^88 - 11 * q^89 - 6 * q^91 + q^92 + 8 * q^93 + 31 * q^94 + 21 * q^96 + q^97 + 9 * q^98 + 38 * q^99

Introduction

L-functions

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