LMFDB - Embedded newform 9025.2.a.z.1.1

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:	$1$
Dimension:	$3$
Coefficient field:	3.3.361.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x + 7 $ x^3 - x^2 - 6*x + 7
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			1.1
Root			$2.28514$ 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)

$f(q)$	$=$	\(q-2.28514 q^{2} +2.50702 q^{3} +3.22188 q^{4} -5.72889 q^{6} -3.50702 q^{7} -2.79216 q^{8} +3.28514 q^{9} -4.50702 q^{11} +8.07730 q^{12} +5.00000 q^{13} +8.01404 q^{14} -0.0632663 q^{16} -0.158610 q^{17} -7.50702 q^{18} -8.79216 q^{21} +10.2992 q^{22} -1.15861 q^{23} -7.00000 q^{24} -11.4257 q^{26} +0.714858 q^{27} -11.2992 q^{28} -3.50702 q^{29} -2.28514 q^{31} +5.72889 q^{32} -11.2992 q^{33} +0.362446 q^{34} +10.5843 q^{36} +10.9648 q^{37} +12.5351 q^{39} +6.07730 q^{41} +20.0913 q^{42} +3.34841 q^{43} -14.5211 q^{44} +2.64759 q^{46} -3.06327 q^{47} -0.158610 q^{48} +5.29918 q^{49} -0.397638 q^{51} +16.1094 q^{52} +5.74293 q^{53} -1.63355 q^{54} +9.79216 q^{56} +8.01404 q^{58} +3.06327 q^{59} -0.873467 q^{61} +5.22188 q^{62} -11.5211 q^{63} -12.9648 q^{64} +25.8202 q^{66} +8.44375 q^{67} -0.511021 q^{68} -2.90466 q^{69} -16.2359 q^{71} -9.17265 q^{72} +7.15861 q^{73} -25.0561 q^{74} +15.8062 q^{77} -28.6445 q^{78} -10.1265 q^{79} -8.06327 q^{81} -13.8875 q^{82} -4.85543 q^{83} -28.3273 q^{84} -7.65159 q^{86} -8.79216 q^{87} +12.5843 q^{88} -1.11250 q^{89} -17.5351 q^{91} -3.73290 q^{92} -5.72889 q^{93} +7.00000 q^{94} +14.3624 q^{96} -1.61951 q^{97} -12.1094 q^{98} -14.8062 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} - q^{3} + 7 q^{4} - 6 q^{6} - 2 q^{7} + 6 q^{8} + 4 q^{9} - 5 q^{11} + 4 q^{12} + 15 q^{13} + 7 q^{14} + 3 q^{16} - q^{17} - 14 q^{18} - 12 q^{21} + 8 q^{22} - 4 q^{23} - 21 q^{24} - 5 q^{26}+ \cdots - 13 q^{99}+O(q^{100}) $ 3 * q - q^2 - q^3 + 7 * q^4 - 6 * q^6 - 2 * q^7 + 6 * q^8 + 4 * q^9 - 5 * q^11 + 4 * q^12 + 15 * q^13 + 7 * q^14 + 3 * q^16 - q^17 - 14 * q^18 - 12 * q^21 + 8 * q^22 - 4 * q^23 - 21 * q^24 - 5 * q^26 + 8 * q^27 - 11 * q^28 - 2 * q^29 - q^31 + 6 * q^32 - 11 * q^33 - 25 * q^34 + 3 * q^36 + 2 * q^37 - 5 * q^39 - 2 * q^41 + 23 * q^42 + q^43 - 18 * q^44 - 24 * q^46 - 6 * q^47 - q^48 - 7 * q^49 - 6 * q^51 + 35 * q^52 - 11 * q^53 + 10 * q^54 + 15 * q^56 + 7 * q^58 + 6 * q^59 - 9 * q^61 + 13 * q^62 - 9 * q^63 - 8 * q^64 + 29 * q^66 + 20 * q^67 - 34 * q^68 - 5 * q^69 - 29 * q^71 - 11 * q^72 + 22 * q^73 - 7 * q^74 + 16 * q^77 - 30 * q^78 - 24 * q^79 - 21 * q^81 - 31 * q^82 + 3 * q^83 - 28 * q^84 - 32 * q^86 - 12 * q^87 + 9 * q^88 - 14 * q^89 - 10 * q^91 - 41 * q^92 - 6 * q^93 + 21 * q^94 + 17 * q^96 - 7 * q^97 - 23 * q^98 - 13 * 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 $

$2$	−2.28514	−1.61584	−0.807920	−	0.589292i	$-0.799407\pi$
$2$	−2.28514	−1.61584	−0.807920	+	0.589292i	$0.799407\pi$
$3$	2.50702	1.44743	0.723714	−	0.690100i	$-0.242434\pi$
$3$	2.50702	1.44743	0.723714	+	0.690100i	$0.242434\pi$
$4$	3.22188	1.61094
$5$	0	0
$5$	0	0
$6$	−5.72889	−2.33881
$7$	−3.50702	−1.32553	−0.662764	−	0.748828i	$-0.730617\pi$
$7$	−3.50702	−1.32553	−0.662764	+	0.748828i	$0.730617\pi$
$8$	−2.79216	−0.987178
$9$	3.28514	1.09505
$10$	0	0
$11$	−4.50702	−1.35892	−0.679459	−	0.733714i	$-0.737785\pi$
$11$	−4.50702	−1.35892	−0.679459	+	0.733714i	$0.737785\pi$
$12$	8.07730	2.33172
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	8.01404	2.14184
$15$	0	0
$16$	−0.0632663	−0.0158166
$17$	−0.158610	−0.0384685	−0.0192343	−	0.999815i	$-0.506123\pi$
$17$	−0.158610	−0.0384685	−0.0192343	+	0.999815i	$0.506123\pi$
$18$	−7.50702	−1.76942
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−8.79216	−1.91861
$22$	10.2992	2.19579
$23$	−1.15861	−0.241587	−0.120793	−	0.992678i	$-0.538544\pi$
$23$	−1.15861	−0.241587	−0.120793	+	0.992678i	$0.538544\pi$
$24$	−7.00000	−1.42887
$25$	0	0
$26$	−11.4257	−2.24077
$27$	0.714858	0.137574
$28$	−11.2992	−2.13534
$29$	−3.50702	−0.651237	−0.325619	−	0.945501i	$-0.605573\pi$
$29$	−3.50702	−0.651237	−0.325619	+	0.945501i	$0.605573\pi$
$30$	0	0
$31$	−2.28514	−0.410424	−0.205212	−	0.978718i	$-0.565788\pi$
$31$	−2.28514	−0.410424	−0.205212	+	0.978718i	$0.565788\pi$
$32$	5.72889	1.01274
$33$	−11.2992	−1.96693
$34$	0.362446	0.0621590
$35$	0	0
$36$	10.5843	1.76405
$37$	10.9648	1.80260	0.901302	−	0.433192i	$-0.142613\pi$
$37$	10.9648	1.80260	0.901302	+	0.433192i	$0.142613\pi$
$38$	0	0
$39$	12.5351	2.00722
$40$	0	0
$41$	6.07730	0.949115	0.474558	−	0.880224i	$-0.342608\pi$
$41$	6.07730	0.949115	0.474558	+	0.880224i	$0.342608\pi$
$42$	20.0913	3.10016
$43$	3.34841	0.510628	0.255314	−	0.966858i	$-0.417821\pi$
$43$	3.34841	0.510628	0.255314	+	0.966858i	$0.417821\pi$
$44$	−14.5211	−2.18913
$45$	0	0
$46$	2.64759	0.390366
$47$	−3.06327	−0.446823	−0.223412	−	0.974724i	$-0.571719\pi$
$47$	−3.06327	−0.446823	−0.223412	+	0.974724i	$0.571719\pi$
$48$	−0.158610	−0.0228934
$49$	5.29918	0.757026
$50$	0	0
$51$	−0.397638	−0.0556804
$52$	16.1094	2.23397
$53$	5.74293	0.788852	0.394426	−	0.918928i	$-0.370943\pi$
$53$	5.74293	0.788852	0.394426	+	0.918928i	$0.370943\pi$
$54$	−1.63355	−0.222298
$55$	0	0
$56$	9.79216	1.30853
$57$	0	0
$58$	8.01404	1.05229
$59$	3.06327	0.398803	0.199402	−	0.979918i	$-0.436100\pi$
$59$	3.06327	0.398803	0.199402	+	0.979918i	$0.436100\pi$
$60$	0	0
$61$	−0.873467	−0.111836	−0.0559180	−	0.998435i	$-0.517809\pi$
$61$	−0.873467	−0.111836	−0.0559180	+	0.998435i	$0.517809\pi$
$62$	5.22188	0.663179
$63$	−11.5211	−1.45152
$64$	−12.9648	−1.62060
$65$	0	0
$66$	25.8202	3.17825
$67$	8.44375	1.03157	0.515784	−	0.856719i	$-0.327501\pi$
$67$	8.44375	1.03157	0.515784	+	0.856719i	$0.327501\pi$
$68$	−0.511021	−0.0619704
$69$	−2.90466	−0.349680
$70$	0	0
$71$	−16.2359	−1.92685	−0.963424	−	0.267981i	$-0.913644\pi$
$71$	−16.2359	−1.92685	−0.963424	+	0.267981i	$0.913644\pi$
$72$	−9.17265	−1.08101
$73$	7.15861	0.837852	0.418926	−	0.908020i	$-0.362407\pi$
$73$	7.15861	0.837852	0.418926	+	0.908020i	$0.362407\pi$
$74$	−25.0561	−2.91272
$75$	0	0
$76$	0	0
$77$	15.8062	1.80128
$78$	−28.6445	−3.24335
$79$	−10.1265	−1.13932	−0.569662	−	0.821879i	$-0.692926\pi$
$79$	−10.1265	−1.13932	−0.569662	+	0.821879i	$0.692926\pi$
$80$	0	0
$81$	−8.06327	−0.895918
$82$	−13.8875	−1.53362
$83$	−4.85543	−0.532952	−0.266476	−	0.963841i	$-0.585859\pi$
$83$	−4.85543	−0.532952	−0.266476	+	0.963841i	$0.585859\pi$
$84$	−28.3273	−3.09076
$85$	0	0
$86$	−7.65159	−0.825092
$87$	−8.79216	−0.942619
$88$	12.5843	1.34149
$89$	−1.11250	−0.117924	−0.0589621	−	0.998260i	$-0.518779\pi$
$89$	−1.11250	−0.117924	−0.0589621	+	0.998260i	$0.518779\pi$
$90$	0	0
$91$	−17.5351	−1.83818
$92$	−3.73290	−0.389181
$93$	−5.72889	−0.594059
$94$	7.00000	0.721995
$95$	0	0
$96$	14.3624	1.46586
$97$	−1.61951	−0.164437	−0.0822184	−	0.996614i	$-0.526200\pi$
$97$	−1.61951	−0.164437	−0.0822184	+	0.996614i	$0.526200\pi$
$98$	−12.1094	−1.22323
$99$	−14.8062	−1.48808
$100$	0	0
$101$	12.3132	1.22521	0.612605	−	0.790389i	$-0.290121\pi$
$101$	12.3132	1.22521	0.612605	+	0.790389i	$0.290121\pi$
$102$	0.908659	0.0899707
$103$	−10.6164	−1.04606	−0.523032	−	0.852313i	$-0.675199\pi$
$103$	−10.6164	−1.04606	−0.523032	+	0.852313i	$0.675199\pi$
$104$	−13.9608	−1.36897
$105$	0	0
$106$	−13.1234	−1.27466
$107$	2.17265	0.210038	0.105019	−	0.994470i	$-0.466510\pi$
$107$	2.17265	0.210038	0.105019	+	0.994470i	$0.466510\pi$
$108$	2.30318	0.221624
$109$	15.8202	1.51530	0.757652	−	0.652659i	$-0.226347\pi$
$109$	15.8202	1.51530	0.757652	+	0.652659i	$0.226347\pi$
$110$	0	0
$111$	27.4890	2.60914
$112$	0.221876	0.0209653
$113$	−9.83828	−0.925507	−0.462754	−	0.886487i	$-0.653139\pi$
$113$	−9.83828	−0.925507	−0.462754	+	0.886487i	$0.653139\pi$
$114$	0	0
$115$	0	0
$116$	−11.2992	−1.04910
$117$	16.4257	1.51856
$118$	−7.00000	−0.644402
$119$	0.556248	0.0509911
$120$	0	0
$121$	9.31322	0.846656
$122$	1.99600	0.180709
$123$	15.2359	1.37378
$124$	−7.36245	−0.661167
$125$	0	0
$126$	26.3273	2.34542
$127$	−15.7109	−1.39411	−0.697056	−	0.717016i	$-0.745507\pi$
$127$	−15.7109	−1.39411	−0.697056	+	0.717016i	$0.745507\pi$
$128$	18.1686	1.60590
$129$	8.39452	0.739097
$130$	0	0
$131$	−11.5351	−1.00783	−0.503913	−	0.863754i	$-0.668107\pi$
$131$	−11.5351	−1.00783	−0.503913	+	0.863754i	$0.668107\pi$
$132$	−36.4046	−3.16861
$133$	0	0
$134$	−19.2952	−1.66685
$135$	0	0
$136$	0.442864	0.0379753
$137$	−0.109381	−0.00934503	−0.00467252	−	0.999989i	$-0.501487\pi$
$137$	−0.109381	−0.00934503	−0.00467252	+	0.999989i	$0.501487\pi$
$138$	6.63755	0.565026
$139$	1.44375	0.122457	0.0612287	−	0.998124i	$-0.480498\pi$
$139$	1.44375	0.122457	0.0612287	+	0.998124i	$0.480498\pi$
$140$	0	0
$141$	−7.67967	−0.646745
$142$	37.1014	3.11348
$143$	−22.5351	−1.88448
$144$	−0.207839	−0.0173199
$145$	0	0
$146$	−16.3584	−1.35383
$147$	13.2851	1.09574
$148$	35.3273	2.90388
$149$	1.72889	0.141637	0.0708183	−	0.997489i	$-0.477439\pi$
$149$	1.72889	0.141637	0.0708183	+	0.997489i	$0.477439\pi$
$150$	0	0
$151$	−20.1406	−1.63902	−0.819508	−	0.573068i	$-0.805753\pi$
$151$	−20.1406	−1.63902	−0.819508	+	0.573068i	$0.805753\pi$
$152$	0	0
$153$	−0.521056	−0.0421249
$154$	−36.1194	−2.91059
$155$	0	0
$156$	40.3865	3.23351
$157$	−3.77812	−0.301527	−0.150764	−	0.988570i	$-0.548173\pi$
$157$	−3.77812	−0.301527	−0.150764	+	0.988570i	$0.548173\pi$
$158$	23.1406	1.84096
$159$	14.3976	1.14181
$160$	0	0
$161$	4.06327	0.320230
$162$	18.4257	1.44766
$163$	1.61640	0.126606	0.0633031	−	0.997994i	$-0.479837\pi$
$163$	1.61640	0.126606	0.0633031	+	0.997994i	$0.479837\pi$
$164$	19.5803	1.52897
$165$	0	0
$166$	11.0953	0.861166
$167$	−6.49298	−0.502442	−0.251221	−	0.967930i	$-0.580832\pi$
$167$	−6.49298	−0.502442	−0.251221	+	0.967930i	$0.580832\pi$
$168$	24.5491	1.89401
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	10.7882	0.822589
$173$	−8.52106	−0.647844	−0.323922	−	0.946084i	$-0.605002\pi$
$173$	−8.52106	−0.647844	−0.323922	+	0.946084i	$0.605002\pi$
$174$	20.0913	1.52312
$175$	0	0
$176$	0.285142	0.0214934
$177$	7.67967	0.577239
$178$	2.54221	0.190547
$179$	−10.2711	−0.767698	−0.383849	−	0.923396i	$-0.625402\pi$
$179$	−10.2711	−0.767698	−0.383849	+	0.923396i	$0.625402\pi$
$180$	0	0
$181$	12.2671	0.911807	0.455903	−	0.890029i	$-0.349316\pi$
$181$	12.2671	0.911807	0.455903	+	0.890029i	$0.349316\pi$
$182$	40.0702	2.97020
$183$	−2.18980	−0.161875
$184$	3.23503	0.238489
$185$	0	0
$186$	13.0913	0.959904
$187$	0.714858	0.0522756
$188$	−9.86946	−0.719805
$189$	−2.50702	−0.182359
$190$	0	0
$191$	5.71085	0.413223	0.206611	−	0.978423i	$-0.433756\pi$
$191$	5.71085	0.413223	0.206611	+	0.978423i	$0.433756\pi$
$192$	−32.5030	−2.34570
$193$	10.1586	0.731233	0.365616	−	0.930766i	$-0.380858\pi$
$193$	10.1586	0.731233	0.365616	+	0.930766i	$0.380858\pi$
$194$	3.70082	0.265703
$195$	0	0
$196$	17.0733	1.21952
$197$	−16.2038	−1.15448	−0.577238	−	0.816576i	$-0.695869\pi$
$197$	−16.2038	−1.15448	−0.577238	+	0.816576i	$0.695869\pi$
$198$	33.8343	2.40450
$199$	0.334372	0.0237030	0.0118515	−	0.999930i	$-0.496227\pi$
$199$	0.334372	0.0237030	0.0118515	+	0.999930i	$0.496227\pi$
$200$	0	0
$201$	21.1686	1.49312
$202$	−28.1375	−1.97974
$203$	12.2992	0.863233
$204$	−1.28114	−0.0896977
$205$	0	0
$206$	24.2600	1.69027
$207$	−3.80620	−0.264549
$208$	−0.316332	−0.0219336
$209$	0	0
$210$	0	0
$211$	2.02807	0.139618	0.0698092	−	0.997560i	$-0.477761\pi$
$211$	2.02807	0.139618	0.0698092	+	0.997560i	$0.477761\pi$
$212$	18.5030	1.27079
$213$	−40.7037	−2.78897
$214$	−4.96481	−0.339387
$215$	0	0
$216$	−1.99600	−0.135810
$217$	8.01404	0.544028
$218$	−36.1515	−2.44849
$219$	17.9468	1.21273
$220$	0	0
$221$	−0.793049	−0.0533463
$222$	−62.8162	−4.21595
$223$	19.2319	1.28786	0.643932	−	0.765083i	$-0.277302\pi$
$223$	19.2319	1.28786	0.643932	+	0.765083i	$0.277302\pi$
$224$	−20.0913	−1.34241
$225$	0	0
$226$	22.4819	1.49547
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	−12.9788	−0.857666	−0.428833	−	0.903384i	$-0.641075\pi$
$229$	−12.9788	−0.857666	−0.428833	+	0.903384i	$0.641075\pi$
$230$	0	0
$231$	39.6264	2.60723
$232$	9.79216	0.642887
$233$	−27.0733	−1.77363	−0.886815	−	0.462124i	$-0.847088\pi$
$233$	−27.0733	−1.77363	−0.886815	+	0.462124i	$0.847088\pi$
$234$	−37.5351	−2.45375
$235$	0	0
$236$	9.86946	0.642447
$237$	−25.3874	−1.64909
$238$	−1.27111	−0.0823935
$239$	20.0602	1.29758	0.648792	−	0.760966i	$-0.275275\pi$
$239$	20.0602	1.29758	0.648792	+	0.760966i	$0.275275\pi$
$240$	0	0
$241$	−21.5843	−1.39037	−0.695184	−	0.718832i	$-0.744677\pi$
$241$	−21.5843	−1.39037	−0.695184	+	0.718832i	$0.744677\pi$
$242$	−21.2820	−1.36806
$243$	−22.3593	−1.43435
$244$	−2.81420	−0.180161
$245$	0	0
$246$	−34.8162	−2.21980
$247$	0	0
$248$	6.38049	0.405161
$249$	−12.1726	−0.771410
$250$	0	0
$251$	−7.26799	−0.458752	−0.229376	−	0.973338i	$-0.573668\pi$
$251$	−7.26799	−0.458752	−0.229376	+	0.973338i	$0.573668\pi$
$252$	−37.1194	−2.33830
$253$	5.22188	0.328297
$254$	35.9015	2.25266
$255$	0	0
$256$	−15.5883	−0.974270
$257$	−15.0773	−0.940496	−0.470248	−	0.882534i	$-0.655836\pi$
$257$	−15.0773	−0.940496	−0.470248	+	0.882534i	$0.655836\pi$
$258$	−19.1827	−1.19426
$259$	−38.4538	−2.38940
$260$	0	0
$261$	−11.5211	−0.713135
$262$	26.3593	1.62848
$263$	−20.1546	−1.24279	−0.621393	−	0.783499i	$-0.713433\pi$
$263$	−20.1546	−1.24279	−0.621393	+	0.783499i	$0.713433\pi$
$264$	31.5491	1.94171
$265$	0	0
$266$	0	0
$267$	−2.78905	−0.170687
$268$	27.2047	1.66179
$269$	−7.72489	−0.470995	−0.235497	−	0.971875i	$-0.575672\pi$
$269$	−7.72489	−0.470995	−0.235497	+	0.971875i	$0.575672\pi$
$270$	0	0
$271$	−5.28514	−0.321050	−0.160525	−	0.987032i	$-0.551319\pi$
$271$	−5.28514	−0.321050	−0.160525	+	0.987032i	$0.551319\pi$
$272$	0.0100347	0.000608441 0
$273$	−43.9608	−2.66063
$274$	0.249951	0.0151001
$275$	0	0
$276$	−9.35844	−0.563312
$277$	−30.6264	−1.84016	−0.920082	−	0.391726i	$-0.871878\pi$
$277$	−30.6264	−1.84016	−0.920082	+	0.391726i	$0.871878\pi$
$278$	−3.29918	−0.197872
$279$	−7.50702	−0.449433
$280$	0	0
$281$	13.3624	0.797137	0.398568	−	0.917139i	$-0.369507\pi$
$281$	13.3624	0.797137	0.398568	+	0.917139i	$0.369507\pi$
$282$	17.5491	1.04504
$283$	−4.09534	−0.243443	−0.121721	−	0.992564i	$-0.538841\pi$
$283$	−4.09534	−0.243443	−0.121721	+	0.992564i	$0.538841\pi$
$284$	−52.3101	−3.10403
$285$	0	0
$286$	51.4959	3.04502
$287$	−21.3132	−1.25808
$288$	18.8202	1.10899
$289$	−16.9748	−0.998520
$290$	0	0
$291$	−4.06015	−0.238010
$292$	23.0642	1.34973
$293$	−12.1726	−0.711134	−0.355567	−	0.934651i	$-0.615712\pi$
$293$	−12.1726	−0.711134	−0.355567	+	0.934651i	$0.615712\pi$
$294$	−30.3584	−1.77054
$295$	0	0
$296$	−30.6155	−1.77949
$297$	−3.22188	−0.186952
$298$	−3.95077	−0.228862
$299$	−5.79305	−0.335021
$300$	0	0
$301$	−11.7429	−0.676851
$302$	46.0241	2.64839
$303$	30.8695	1.77340
$304$	0	0
$305$	0	0
$306$	1.19069	0.0680670
$307$	−24.5351	−1.40029	−0.700146	−	0.714000i	$-0.746882\pi$
$307$	−24.5351	−1.40029	−0.700146	+	0.714000i	$0.746882\pi$
$308$	50.9256	2.90176
$309$	−26.6155	−1.51410
$310$	0	0
$311$	−10.2038	−0.578606	−0.289303	−	0.957238i	$-0.593424\pi$
$311$	−10.2038	−0.578606	−0.289303	+	0.957238i	$0.593424\pi$
$312$	−35.0000	−1.98148
$313$	31.9820	1.80773	0.903864	−	0.427821i	$-0.140719\pi$
$313$	31.9820	1.80773	0.903864	+	0.427821i	$0.140719\pi$
$314$	8.63355	0.487219
$315$	0	0
$316$	−32.6264	−1.83538
$317$	−2.23591	−0.125581	−0.0627907	−	0.998027i	$-0.520000\pi$
$317$	−2.23591	−0.125581	−0.0627907	+	0.998027i	$0.520000\pi$
$318$	−32.9007	−1.84498
$319$	15.8062	0.884977
$320$	0	0
$321$	5.44687	0.304014
$322$	−9.28514	−0.517441
$323$	0	0
$324$	−25.9788	−1.44327
$325$	0	0
$326$	−3.69370	−0.204575
$327$	39.6616	2.19329
$328$	−16.9688	−0.936946
$329$	10.7429	0.592277
$330$	0	0
$331$	−10.0913	−0.554670	−0.277335	−	0.960773i	$-0.589451\pi$
$331$	−10.0913	−0.554670	−0.277335	+	0.960773i	$0.589451\pi$
$332$	−15.6436	−0.858553
$333$	36.0210	1.97394
$334$	14.8374	0.811866
$335$	0	0
$336$	0.556248	0.0303458
$337$	−5.61240	−0.305727	−0.152863	−	0.988247i	$-0.548849\pi$
$337$	−5.61240	−0.305727	−0.152863	+	0.988247i	$0.548849\pi$
$338$	−27.4217	−1.49154
$339$	−24.6647	−1.33960
$340$	0	0
$341$	10.2992	0.557732
$342$	0	0
$343$	5.96481	0.322069
$344$	−9.34930	−0.504080
$345$	0	0
$346$	19.4718	1.04681
$347$	−5.47494	−0.293910	−0.146955	−	0.989143i	$-0.546947\pi$
$347$	−5.47494	−0.293910	−0.146955	+	0.989143i	$0.546947\pi$
$348$	−28.3273	−1.51850
$349$	−18.8202	−1.00742	−0.503712	−	0.863872i	$-0.668033\pi$
$349$	−18.8202	−1.00742	−0.503712	+	0.863872i	$0.668033\pi$
$350$	0	0
$351$	3.57429	0.190781
$352$	−25.8202	−1.37622
$353$	12.7008	0.675996	0.337998	−	0.941147i	$-0.390250\pi$
$353$	12.7008	0.675996	0.337998	+	0.941147i	$0.390250\pi$
$354$	−17.5491	−0.932726
$355$	0	0
$356$	−3.58432	−0.189969
$357$	1.39452	0.0738060
$358$	23.4709	1.24048
$359$	−11.0804	−0.584802	−0.292401	−	0.956296i	$-0.594454\pi$
$359$	−11.0804	−0.584802	−0.292401	+	0.956296i	$0.594454\pi$
$360$	0	0
$361$	0	0
$362$	−28.0321	−1.47333
$363$	23.3484	1.22547
$364$	−56.4959	−2.96119
$365$	0	0
$366$	5.00400	0.261563
$367$	20.6405	1.07742	0.538712	−	0.842490i	$-0.318911\pi$
$367$	20.6405	1.07742	0.538712	+	0.842490i	$0.318911\pi$
$368$	0.0733010	0.00382108
$369$	19.9648	1.03933
$370$	0	0
$371$	−20.1406	−1.04565
$372$	−18.4578	−0.956992
$373$	−4.55313	−0.235752	−0.117876	−	0.993028i	$-0.537609\pi$
$373$	−4.55313	−0.235752	−0.117876	+	0.993028i	$0.537609\pi$
$374$	−1.63355	−0.0844689
$375$	0	0
$376$	8.55313	0.441094
$377$	−17.5351	−0.903103
$378$	5.72889	0.294663
$379$	−19.9187	−1.02315	−0.511577	−	0.859237i	$-0.670939\pi$
$379$	−19.9187	−1.02315	−0.511577	+	0.859237i	$0.670939\pi$
$380$	0	0
$381$	−39.3874	−2.01788
$382$	−13.0501	−0.667702
$383$	19.9748	1.02067	0.510333	−	0.859977i	$-0.329522\pi$
$383$	19.9748	1.02067	0.510333	+	0.859977i	$0.329522\pi$
$384$	45.5491	2.32442
$385$	0	0
$386$	−23.2139	−1.18155
$387$	11.0000	0.559161
$388$	−5.21787	−0.264897
$389$	13.8022	0.699799	0.349900	−	0.936787i	$-0.386216\pi$
$389$	13.8022	0.699799	0.349900	+	0.936787i	$0.386216\pi$
$390$	0	0
$391$	0.183767	0.00929349
$392$	−14.7962	−0.747319
$393$	−28.9187	−1.45876
$394$	37.0281	1.86545
$395$	0	0
$396$	−47.7037	−2.39720
$397$	16.5491	0.830577	0.415289	−	0.909690i	$-0.363681\pi$
$397$	16.5491	0.830577	0.415289	+	0.909690i	$0.363681\pi$
$398$	−0.764087	−0.0383002
$399$	0	0
$400$	0	0
$401$	8.52506	0.425721	0.212861	−	0.977083i	$-0.431722\pi$
$401$	8.52506	0.425721	0.212861	+	0.977083i	$0.431722\pi$
$402$	−48.3734	−2.41264
$403$	−11.4257	−0.569155
$404$	39.6717	1.97374
$405$	0	0
$406$	−28.1054	−1.39485
$407$	−49.4186	−2.44959
$408$	1.11027	0.0549665
$409$	−11.5663	−0.571916	−0.285958	−	0.958242i	$-0.592312\pi$
$409$	−11.5663	−0.571916	−0.285958	+	0.958242i	$0.592312\pi$
$410$	0	0
$411$	−0.274220	−0.0135263
$412$	−34.2047	−1.68515
$413$	−10.7429	−0.528625
$414$	8.69771	0.427469
$415$	0	0
$416$	28.6445	1.40441
$417$	3.61951	0.177248
$418$	0	0
$419$	−21.7149	−1.06084	−0.530420	−	0.847735i	$-0.677966\pi$
$419$	−21.7149	−1.06084	−0.530420	+	0.847735i	$0.677966\pi$
$420$	0	0
$421$	−6.92270	−0.337392	−0.168696	−	0.985668i	$-0.553956\pi$
$421$	−6.92270	−0.337392	−0.168696	+	0.985668i	$0.553956\pi$
$422$	−4.63444	−0.225601
$423$	−10.0633	−0.489293
$424$	−16.0352	−0.778738
$425$	0	0
$426$	93.0138	4.50654
$427$	3.06327	0.148242
$428$	7.00000	0.338358
$429$	−56.4959	−2.72765
$430$	0	0
$431$	−27.9788	−1.34769	−0.673847	−	0.738871i	$-0.735359\pi$
$431$	−27.9788	−1.34769	−0.673847	+	0.738871i	$0.735359\pi$
$432$	−0.0452264	−0.00217596
$433$	−6.25395	−0.300546	−0.150273	−	0.988645i	$-0.548015\pi$
$433$	−6.25395	−0.300546	−0.150273	+	0.988645i	$0.548015\pi$
$434$	−18.3132	−0.879063
$435$	0	0
$436$	50.9708	2.44106
$437$	0	0
$438$	−41.0109	−1.95958
$439$	31.1976	1.48898	0.744490	−	0.667633i	$-0.232693\pi$
$439$	31.1976	1.48898	0.744490	+	0.667633i	$0.232693\pi$
$440$	0	0
$441$	17.4086	0.828979
$442$	1.81223	0.0861990
$443$	32.7037	1.55380	0.776901	−	0.629623i	$-0.216791\pi$
$443$	32.7037	1.55380	0.776901	+	0.629623i	$0.216791\pi$
$444$	88.5661	4.20316
$445$	0	0
$446$	−43.9477	−2.08098
$447$	4.33437	0.205009
$448$	45.4678	2.14815
$449$	25.6304	1.20958	0.604788	−	0.796387i	$-0.293258\pi$
$449$	25.6304	1.20958	0.604788	+	0.796387i	$0.293258\pi$
$450$	0	0
$451$	−27.3905	−1.28977
$452$	−31.6977	−1.49093
$453$	−50.4928	−2.37236
$454$	9.14057	0.428988
$455$	0	0
$456$	0	0
$457$	−33.1646	−1.55138	−0.775688	−	0.631116i	$-0.782597\pi$
$457$	−33.1646	−1.55138	−0.775688	+	0.631116i	$0.782597\pi$
$458$	29.6585	1.38585
$459$	−0.113383	−0.00529229
$460$	0	0
$461$	2.17576	0.101335	0.0506677	−	0.998716i	$-0.483865\pi$
$461$	2.17576	0.101335	0.0506677	+	0.998716i	$0.483865\pi$
$462$	−90.5520	−4.21286
$463$	6.20072	0.288172	0.144086	−	0.989565i	$-0.453976\pi$
$463$	6.20072	0.288172	0.144086	+	0.989565i	$0.453976\pi$
$464$	0.221876	0.0103003
$465$	0	0
$466$	61.8664	2.86590
$467$	17.1546	0.793821	0.396910	−	0.917857i	$-0.370082\pi$
$467$	17.1546	0.793821	0.396910	+	0.917857i	$0.370082\pi$
$468$	52.9216	2.44630
$469$	−29.6124	−1.36737
$470$	0	0
$471$	−9.47183	−0.436439
$472$	−8.55313	−0.393690
$473$	−15.0913	−0.693901
$474$	58.0138	2.66466
$475$	0	0
$476$	1.79216	0.0821436
$477$	18.8664	0.863831
$478$	−45.8403	−2.09669
$479$	−35.7781	−1.63474	−0.817372	−	0.576110i	$-0.804570\pi$
$479$	−35.7781	−1.63474	−0.817372	+	0.576110i	$0.804570\pi$
$480$	0	0
$481$	54.8240	2.49976
$482$	49.3233	2.24661
$483$	10.1867	0.463510
$484$	30.0060	1.36391
$485$	0	0
$486$	51.0943	2.31768
$487$	23.7149	1.07462	0.537311	−	0.843384i	$-0.319440\pi$
$487$	23.7149	1.07462	0.537311	+	0.843384i	$0.319440\pi$
$488$	2.43886	0.110402
$489$	4.05234	0.183253
$490$	0	0
$491$	−39.1867	−1.76847	−0.884235	−	0.467042i	$-0.845320\pi$
$491$	−39.1867	−1.76847	−0.884235	+	0.467042i	$0.845320\pi$
$492$	49.0882	2.21307
$493$	0.556248	0.0250521
$494$	0	0
$495$	0	0
$496$	0.144573	0.00649150
$497$	56.9397	2.55409
$498$	27.8162	1.24648
$499$	−9.37648	−0.419749	−0.209875	−	0.977728i	$-0.567306\pi$
$499$	−9.37648	−0.419749	−0.209875	+	0.977728i	$0.567306\pi$
$500$	0	0
$501$	−16.2780	−0.727249
$502$	16.6084	0.741269
$503$	−13.4930	−0.601622	−0.300811	−	0.953684i	$-0.597257\pi$
$503$	−13.4930	−0.601622	−0.300811	+	0.953684i	$0.597257\pi$
$504$	32.1686	1.43291
$505$	0	0
$506$	−11.9327	−0.530475
$507$	30.0842	1.33609
$508$	−50.6184	−2.24583
$509$	25.7069	1.13944	0.569718	−	0.821840i	$-0.307052\pi$
$509$	25.7069	1.13944	0.569718	+	0.821840i	$0.307052\pi$
$510$	0	0
$511$	−25.1054	−1.11060
$512$	−0.715746	−0.0316318
$513$	0	0
$514$	34.4538	1.51969
$515$	0	0
$516$	27.0461	1.19064
$517$	13.8062	0.607196
$518$	87.8724	3.86089
$519$	−21.3624	−0.937707
$520$	0	0
$521$	−37.7358	−1.65324	−0.826618	−	0.562763i	$-0.809738\pi$
$521$	−37.7358	−1.65324	−0.826618	+	0.562763i	$0.809738\pi$
$522$	26.3273	1.15231
$523$	−38.4006	−1.67914	−0.839570	−	0.543252i	$-0.817193\pi$
$523$	−38.4006	−1.67914	−0.839570	+	0.543252i	$0.817193\pi$
$524$	−37.1646	−1.62354
$525$	0	0
$526$	46.0561	2.00814
$527$	0.362446	0.0157884
$528$	0.714858	0.0311102
$529$	−21.6576	−0.941636
$530$	0	0
$531$	10.0633	0.436709
$532$	0	0
$533$	30.3865	1.31619
$534$	6.37337	0.275803
$535$	0	0
$536$	−23.5763	−1.01834
$537$	−25.7499	−1.11119
$538$	17.6525	0.761052
$539$	−23.8835	−1.02874
$540$	0	0
$541$	−12.8062	−0.550581	−0.275291	−	0.961361i	$-0.588774\pi$
$541$	−12.8062	−0.550581	−0.275291	+	0.961361i	$0.588774\pi$
$542$	12.0773	0.518765
$543$	30.7539	1.31977
$544$	−0.908659	−0.0389584
$545$	0	0
$546$	100.457	4.29915
$547$	18.2571	0.780616	0.390308	−	0.920684i	$-0.372369\pi$
$547$	18.2571	0.780616	0.390308	+	0.920684i	$0.372369\pi$
$548$	−0.352411	−0.0150543
$549$	−2.86946	−0.122466
$550$	0	0
$551$	0	0
$552$	8.11027	0.345196
$553$	35.5139	1.51021
$554$	69.9858	2.97341
$555$	0	0
$556$	4.65159	0.197271
$557$	−14.9047	−0.631531	−0.315765	−	0.948837i	$-0.602261\pi$
$557$	−14.9047	−0.631531	−0.315765	+	0.948837i	$0.602261\pi$
$558$	17.1546	0.726212
$559$	16.7420	0.708113
$560$	0	0
$561$	1.79216	0.0756651
$562$	−30.5351	−1.28805
$563$	−45.7810	−1.92944	−0.964720	−	0.263276i	$-0.915197\pi$
$563$	−45.7810	−1.92944	−0.964720	+	0.263276i	$0.915197\pi$
$564$	−24.7429	−1.04187
$565$	0	0
$566$	9.35844	0.393365
$567$	28.2780	1.18757
$568$	45.3333	1.90214
$569$	0.379598	0.0159136	0.00795679	−	0.999968i	$-0.497467\pi$
$569$	0.379598	0.0159136	0.00795679	+	0.999968i	$0.497467\pi$
$570$	0	0
$571$	−15.8514	−0.663361	−0.331681	−	0.943392i	$-0.607616\pi$
$571$	−15.8514	−0.663361	−0.331681	+	0.943392i	$0.607616\pi$
$572$	−72.6053	−3.03578
$573$	14.3172	0.598110
$574$	48.7037	2.03285
$575$	0	0
$576$	−42.5912	−1.77464
$577$	19.2350	0.800765	0.400382	−	0.916348i	$-0.368877\pi$
$577$	19.2350	0.800765	0.400382	+	0.916348i	$0.368877\pi$
$578$	38.7899	1.61345
$579$	25.4678	1.05841
$580$	0	0
$581$	17.0281	0.706444
$582$	9.27803	0.384587
$583$	−25.8835	−1.07199
$584$	−19.9880	−0.827109
$585$	0	0
$586$	27.8162	1.14908
$587$	−40.8483	−1.68599	−0.842995	−	0.537921i	$-0.819210\pi$
$587$	−40.8483	−1.68599	−0.842995	+	0.537921i	$0.819210\pi$
$588$	42.8031	1.76517
$589$	0	0
$590$	0	0
$591$	−40.6233	−1.67102
$592$	−0.693703	−0.0285110
$593$	14.2468	0.585047	0.292524	−	0.956258i	$-0.405505\pi$
$593$	14.2468	0.585047	0.292524	+	0.956258i	$0.405505\pi$
$594$	7.36245	0.302085
$595$	0	0
$596$	5.57028	0.228168
$597$	0.838276	0.0343083
$598$	13.2379	0.541340
$599$	−15.8514	−0.647672	−0.323836	−	0.946113i	$-0.604973\pi$
$599$	−15.8514	−0.647672	−0.323836	+	0.946113i	$0.604973\pi$
$600$	0	0
$601$	16.4718	0.671900	0.335950	−	0.941880i	$-0.390943\pi$
$601$	16.4718	0.671900	0.335950	+	0.941880i	$0.390943\pi$
$602$	26.8343	1.09368
$603$	27.7389	1.12962
$604$	−64.8904	−2.64035
$605$	0	0
$606$	−70.5411	−2.86554
$607$	10.3914	0.421774	0.210887	−	0.977510i	$-0.432365\pi$
$607$	10.3914	0.421774	0.210887	+	0.977510i	$0.432365\pi$
$608$	0	0
$609$	30.8343	1.24947
$610$	0	0
$611$	−15.3163	−0.619632
$612$	−1.67878	−0.0678606
$613$	21.7850	0.879890	0.439945	−	0.898025i	$-0.354998\pi$
$613$	21.7850	0.879890	0.439945	+	0.898025i	$0.354998\pi$
$614$	56.0662	2.26265
$615$	0	0
$616$	−44.1335	−1.77819
$617$	42.7710	1.72190	0.860948	−	0.508693i	$-0.169871\pi$
$617$	42.7710	1.72190	0.860948	+	0.508693i	$0.169871\pi$
$618$	60.8202	2.44655
$619$	28.7882	1.15709	0.578547	−	0.815649i	$-0.303620\pi$
$619$	28.7882	1.15709	0.578547	+	0.815649i	$0.303620\pi$
$620$	0	0
$621$	−0.828241	−0.0332362
$622$	23.3172	0.934935
$623$	3.90154	0.156312
$624$	−0.793049	−0.0317474
$625$	0	0
$626$	−73.0833	−2.92100
$627$	0	0
$628$	−12.1726	−0.485742
$629$	−1.73913	−0.0693435
$630$	0	0
$631$	−19.3874	−0.771800	−0.385900	−	0.922541i	$-0.626109\pi$
$631$	−19.3874	−0.771800	−0.385900	+	0.922541i	$0.626109\pi$
$632$	28.2749	1.12472
$633$	5.08442	0.202088
$634$	5.10938	0.202919
$635$	0	0
$636$	46.3874	1.83938
$637$	26.4959	1.04981
$638$	−36.1194	−1.42998
$639$	−53.3373	−2.10999
$640$	0	0
$641$	40.6866	1.60702	0.803512	−	0.595289i	$-0.202962\pi$
$641$	40.6866	1.60702	0.803512	+	0.595289i	$0.202962\pi$
$642$	−12.4469	−0.491239
$643$	34.1054	1.34498	0.672492	−	0.740104i	$-0.265224\pi$
$643$	34.1054	1.34498	0.672492	+	0.740104i	$0.265224\pi$
$644$	13.0913	0.515871
$645$	0	0
$646$	0	0
$647$	48.0029	1.88719	0.943595	−	0.331103i	$-0.107421\pi$
$647$	48.0029	1.88719	0.943595	+	0.331103i	$0.107421\pi$
$648$	22.5139	0.884431
$649$	−13.8062	−0.541941
$650$	0	0
$651$	20.0913	0.787442
$652$	5.20784	0.203955
$653$	−3.90866	−0.152958	−0.0764788	−	0.997071i	$-0.524368\pi$
$653$	−3.90866	−0.152958	−0.0764788	+	0.997071i	$0.524368\pi$
$654$	−90.6325	−3.54401
$655$	0	0
$656$	−0.384489	−0.0150118
$657$	23.5171	0.917488
$658$	−24.5491	−0.957025
$659$	13.0882	0.509845	0.254922	−	0.966961i	$-0.417950\pi$
$659$	13.0882	0.509845	0.254922	+	0.966961i	$0.417950\pi$
$660$	0	0
$661$	−23.9428	−0.931266	−0.465633	−	0.884978i	$-0.654173\pi$
$661$	−23.9428	−0.931266	−0.465633	+	0.884978i	$0.654173\pi$
$662$	23.0602	0.896258
$663$	−1.98819	−0.0772149
$664$	13.5571	0.526119
$665$	0	0
$666$	−82.3130	−3.18956
$667$	4.06327	0.157330
$668$	−20.9196	−0.809403
$669$	48.2148	1.86409
$670$	0	0
$671$	3.93673	0.151976
$672$	−50.3694	−1.94304
$673$	11.3304	0.436754	0.218377	−	0.975865i	$-0.429924\pi$
$673$	11.3304	0.436754	0.218377	+	0.975865i	$0.429924\pi$
$674$	12.8251	0.494005
$675$	0	0
$676$	38.6625	1.48702
$677$	−8.90466	−0.342234	−0.171117	−	0.985251i	$-0.554738\pi$
$677$	−8.90466	−0.342234	−0.171117	+	0.985251i	$0.554738\pi$
$678$	56.3624	2.16459
$679$	5.67967	0.217966
$680$	0	0
$681$	−10.0281	−0.384277
$682$	−23.5351	−0.901205
$683$	26.6977	1.02156	0.510780	−	0.859712i	$-0.329357\pi$
$683$	26.6977	1.02156	0.510780	+	0.859712i	$0.329357\pi$
$684$	0	0
$685$	0	0
$686$	−13.6304	−0.520412
$687$	−32.5382	−1.24141
$688$	−0.211842	−0.00807638
$689$	28.7147	1.09394
$690$	0	0
$691$	35.9708	1.36840	0.684198	−	0.729297i	$-0.260153\pi$
$691$	35.9708	1.36840	0.684198	+	0.729297i	$0.260153\pi$
$692$	−27.4538	−1.04364
$693$	51.9256	1.97249
$694$	12.5110	0.474912
$695$	0	0
$696$	24.5491	0.930532
$697$	−0.963920	−0.0365111
$698$	43.0069	1.62784
$699$	−67.8733	−2.56720
$700$	0	0
$701$	2.45087	0.0925681	0.0462840	−	0.998928i	$-0.485262\pi$
$701$	2.45087	0.0925681	0.0462840	+	0.998928i	$0.485262\pi$
$702$	−8.16776	−0.308272
$703$	0	0
$704$	58.4326	2.20226
$705$	0	0
$706$	−29.0232	−1.09230
$707$	−43.1827	−1.62405
$708$	24.7429	0.929896
$709$	−50.2881	−1.88861	−0.944304	−	0.329075i	$-0.893263\pi$
$709$	−50.2881	−1.88861	−0.944304	+	0.329075i	$0.893263\pi$
$710$	0	0
$711$	−33.2671	−1.24761
$712$	3.10627	0.116412
$713$	2.64759	0.0991530
$714$	−3.18668	−0.119259
$715$	0	0
$716$	−33.0922	−1.23671
$717$	50.2912	1.87816
$718$	25.3203	0.944946
$719$	−48.0983	−1.79376	−0.896881	−	0.442271i	$-0.854173\pi$
$719$	−48.0983	−1.79376	−0.896881	+	0.442271i	$0.854173\pi$
$720$	0	0
$721$	37.2319	1.38659
$722$	0	0
$723$	−54.1123	−2.01246
$724$	39.5231	1.46886
$725$	0	0
$726$	−53.3544	−1.98017
$727$	16.6797	0.618615	0.309307	−	0.950962i	$-0.399903\pi$
$727$	16.6797	0.618615	0.309307	+	0.950962i	$0.399903\pi$
$728$	48.9608	1.81461
$729$	−31.8655	−1.18020
$730$	0	0
$731$	−0.531091	−0.0196431
$732$	−7.05526	−0.260770
$733$	4.13365	0.152680	0.0763399	−	0.997082i	$-0.475677\pi$
$733$	4.13365	0.152680	0.0763399	+	0.997082i	$0.475677\pi$
$734$	−47.1664	−1.74094
$735$	0	0
$736$	−6.63755	−0.244663
$737$	−38.0561	−1.40182
$738$	−45.6224	−1.67938
$739$	46.9740	1.72796	0.863982	−	0.503522i	$-0.167963\pi$
$739$	46.9740	1.72796	0.863982	+	0.503522i	$0.167963\pi$
$740$	0	0
$741$	0	0
$742$	46.0241	1.68960
$743$	−41.2139	−1.51199	−0.755995	−	0.654577i	$-0.772847\pi$
$743$	−41.2139	−1.51199	−0.755995	+	0.654577i	$0.772847\pi$
$744$	15.9960	0.586442
$745$	0	0
$746$	10.4046	0.380938
$747$	−15.9508	−0.583608
$748$	2.30318	0.0842127
$749$	−7.61951	−0.278411
$750$	0	0
$751$	5.97885	0.218171	0.109086	−	0.994032i	$-0.465208\pi$
$751$	5.97885	0.218171	0.109086	+	0.994032i	$0.465208\pi$
$752$	0.193802	0.00706722
$753$	−18.2210	−0.664010
$754$	40.0702	1.45927
$755$	0	0
$756$	−8.07730	−0.293769
$757$	−36.3936	−1.32275	−0.661375	−	0.750056i	$-0.730027\pi$
$757$	−36.3936	−1.32275	−0.661375	+	0.750056i	$0.730027\pi$
$758$	45.5171	1.65325
$759$	13.0913	0.475186
$760$	0	0
$761$	−2.71397	−0.0983813	−0.0491907	−	0.998789i	$-0.515664\pi$
$761$	−2.71397	−0.0983813	−0.0491907	+	0.998789i	$0.515664\pi$
$762$	90.0058	3.26057
$763$	−55.4819	−2.00858
$764$	18.3997	0.665677
$765$	0	0
$766$	−45.6454	−1.64923
$767$	15.3163	0.553041
$768$	−39.0802	−1.41019
$769$	39.7991	1.43519	0.717596	−	0.696460i	$-0.245243\pi$
$769$	39.7991	1.43519	0.717596	+	0.696460i	$0.245243\pi$
$770$	0	0
$771$	−37.7991	−1.36130
$772$	32.7298	1.17797
$773$	−44.9748	−1.61763	−0.808816	−	0.588061i	$-0.799891\pi$
$773$	−44.9748	−1.61763	−0.808816	+	0.588061i	$0.799891\pi$
$774$	−25.1366	−0.903515
$775$	0	0
$776$	4.52194	0.162328
$777$	−96.4044	−3.45849
$778$	−31.5400	−1.13076
$779$	0	0
$780$	0	0
$781$	73.1756	2.61843
$782$	−0.419934	−0.0150168
$783$	−2.50702	−0.0895935
$784$	−0.335260	−0.0119736
$785$	0	0
$786$	66.0833	2.35711
$787$	17.7149	0.631466	0.315733	−	0.948848i	$-0.397750\pi$
$787$	17.7149	0.631466	0.315733	+	0.948848i	$0.397750\pi$
$788$	−52.2068	−1.85979
$789$	−50.5280	−1.79884
$790$	0	0
$791$	34.5030	1.22679
$792$	41.3413	1.46900
$793$	−4.36734	−0.155089
$794$	−37.8171	−1.34208
$795$	0	0
$796$	1.07730	0.0381840
$797$	−25.4930	−0.903008	−0.451504	−	0.892269i	$-0.649112\pi$
$797$	−25.4930	−0.903008	−0.451504	+	0.892269i	$0.649112\pi$
$798$	0	0
$799$	0.485864	0.0171886
$800$	0	0
$801$	−3.65471	−0.129133
$802$	−19.4810	−0.687897
$803$	−32.2640	−1.13857
$804$	68.2028	2.40533
$805$	0	0
$806$	26.1094	0.919664
$807$	−19.3664	−0.681731
$808$	−34.3805	−1.20950
$809$	−20.4678	−0.719610	−0.359805	−	0.933027i	$-0.617157\pi$
$809$	−20.4678	−0.719610	−0.359805	+	0.933027i	$0.617157\pi$
$810$	0	0
$811$	−22.9015	−0.804182	−0.402091	−	0.915600i	$-0.631716\pi$
$811$	−22.9015	−0.804182	−0.402091	+	0.915600i	$0.631716\pi$
$812$	39.6264	1.39062
$813$	−13.2500	−0.464696
$814$	112.929	3.95814
$815$	0	0
$816$	0.0251571	0.000880674 0
$817$	0	0
$818$	26.4306	0.924124
$819$	−57.6053	−2.01289
$820$	0	0
$821$	30.7218	1.07220	0.536099	−	0.844155i	$-0.319897\pi$
$821$	30.7218	1.07220	0.536099	+	0.844155i	$0.319897\pi$
$822$	0.626631	0.0218563
$823$	−4.19692	−0.146295	−0.0731476	−	0.997321i	$-0.523304\pi$
$823$	−4.19692	−0.146295	−0.0731476	+	0.997321i	$0.523304\pi$
$824$	29.6427	1.03265
$825$	0	0
$826$	24.5491	0.854173
$827$	21.3304	0.741730	0.370865	−	0.928687i	$-0.379061\pi$
$827$	21.3304	0.741730	0.370865	+	0.928687i	$0.379061\pi$
$828$	−12.2631	−0.426172
$829$	34.0390	1.18222	0.591112	−	0.806590i	$-0.298689\pi$
$829$	34.0390	1.18222	0.591112	+	0.806590i	$0.298689\pi$
$830$	0	0
$831$	−76.7810	−2.66350
$832$	−64.8240	−2.24737
$833$	−0.840502	−0.0291217
$834$	−8.27111	−0.286405
$835$	0	0
$836$	0	0
$837$	−1.63355	−0.0564638
$838$	49.6215	1.71415
$839$	10.4117	0.359451	0.179725	−	0.983717i	$-0.442479\pi$
$839$	10.4117	0.359451	0.179725	+	0.983717i	$0.442479\pi$
$840$	0	0
$841$	−16.7008	−0.575890
$842$	15.8193	0.545171
$843$	33.4999	1.15380
$844$	6.53421	0.224917
$845$	0	0
$846$	22.9960	0.790619
$847$	−32.6616	−1.12227
$848$	−0.363334	−0.0124769
$849$	−10.2671	−0.352366
$850$	0	0
$851$	−12.7039	−0.435485
$852$	−131.142	−4.49286
$853$	13.1305	0.449581	0.224790	−	0.974407i	$-0.427830\pi$
$853$	13.1305	0.449581	0.224790	+	0.974407i	$0.427830\pi$
$854$	−7.00000	−0.239535
$855$	0	0
$856$	−6.06638	−0.207345
$857$	43.2459	1.47725	0.738627	−	0.674115i	$-0.235475\pi$
$857$	43.2459	1.47725	0.738627	+	0.674115i	$0.235475\pi$
$858$	129.101	4.40744
$859$	31.4687	1.07370	0.536849	−	0.843678i	$-0.319614\pi$
$859$	31.4687	1.07370	0.536849	+	0.843678i	$0.319614\pi$
$860$	0	0
$861$	−53.4326	−1.82098
$862$	63.9356	2.17766
$863$	23.7842	0.809622	0.404811	−	0.914400i	$-0.367337\pi$
$863$	23.7842	0.809622	0.404811	+	0.914400i	$0.367337\pi$
$864$	4.09534	0.139326
$865$	0	0
$866$	14.2912	0.485634
$867$	−42.5562	−1.44529
$868$	25.8202	0.876396
$869$	45.6405	1.54825
$870$	0	0
$871$	42.2188	1.43053
$872$	−44.1726	−1.49587
$873$	−5.32033	−0.180066
$874$	0	0
$875$	0	0
$876$	57.8223	1.95363
$877$	38.1695	1.28889	0.644447	−	0.764649i	$-0.277088\pi$
$877$	38.1695	1.28889	0.644447	+	0.764649i	$0.277088\pi$
$878$	−71.2910	−2.40595
$879$	−30.5171	−1.02931
$880$	0	0
$881$	5.49209	0.185033	0.0925167	−	0.995711i	$-0.470509\pi$
$881$	5.49209	0.185033	0.0925167	+	0.995711i	$0.470509\pi$
$882$	−39.7810	−1.33950
$883$	34.2983	1.15423	0.577115	−	0.816663i	$-0.304179\pi$
$883$	34.2983	1.15423	0.577115	+	0.816663i	$0.304179\pi$
$884$	−2.55511	−0.0859375
$885$	0	0
$886$	−74.7327	−2.51069
$887$	37.5171	1.25970	0.629850	−	0.776717i	$-0.283116\pi$
$887$	37.5171	1.25970	0.629850	+	0.776717i	$0.283116\pi$
$888$	−76.7537	−2.57568
$889$	55.0983	1.84794
$890$	0	0
$891$	36.3413	1.21748
$892$	61.9628	2.07467
$893$	0	0
$894$	−9.90466	−0.331261
$895$	0	0
$896$	−63.7178	−2.12866
$897$	−14.5233	−0.484918
$898$	−58.5692	−1.95448
$899$	8.01404	0.267283
$900$	0	0
$901$	−0.910886	−0.0303460
$902$	62.5912	2.08406
$903$	−29.4397	−0.979694
$904$	27.4701	0.913640
$905$	0	0
$906$	115.383	3.83335
$907$	22.2531	0.738901	0.369450	−	0.929250i	$-0.379546\pi$
$907$	22.2531	0.738901	0.369450	+	0.929250i	$0.379546\pi$
$908$	−12.8875	−0.427687
$909$	40.4507	1.34166
$910$	0	0
$911$	10.0421	0.332710	0.166355	−	0.986066i	$-0.446800\pi$
$911$	10.0421	0.332710	0.166355	+	0.986066i	$0.446800\pi$
$912$	0	0
$913$	21.8835	0.724238
$914$	75.7859	2.50678
$915$	0	0
$916$	−41.8162	−1.38165
$917$	40.4538	1.33590
$918$	0.259097	0.00855149
$919$	6.06104	0.199935	0.0999676	−	0.994991i	$-0.468126\pi$
$919$	6.06104	0.199935	0.0999676	+	0.994991i	$0.468126\pi$
$920$	0	0
$921$	−61.5099	−2.02682
$922$	−4.97193	−0.163742
$923$	−81.1796	−2.67206
$924$	127.671	4.20008
$925$	0	0
$926$	−14.1695	−0.465640
$927$	−34.8764	−1.14549
$928$	−20.0913	−0.659531
$929$	−51.8764	−1.70201	−0.851004	−	0.525158i	$-0.824006\pi$
$929$	−51.8764	−1.70201	−0.851004	+	0.525158i	$0.824006\pi$
$930$	0	0
$931$	0	0
$932$	−87.2268	−2.85721
$933$	−25.5812	−0.837491
$934$	−39.2007	−1.28269
$935$	0	0
$936$	−45.8632	−1.49909
$937$	−25.1998	−0.823243	−0.411621	−	0.911355i	$-0.635037\pi$
$937$	−25.1998	−0.823243	−0.411621	+	0.911355i	$0.635037\pi$
$938$	67.6685	2.20946
$939$	80.1794	2.61655
$940$	0	0
$941$	−1.93585	−0.0631068	−0.0315534	−	0.999502i	$-0.510045\pi$
$941$	−1.93585	−0.0631068	−0.0315534	+	0.999502i	$0.510045\pi$
$942$	21.6445	0.705215
$943$	−7.04122	−0.229294
$944$	−0.193802	−0.00630770
$945$	0	0
$946$	34.4859	1.12123
$947$	−31.3624	−1.01914	−0.509571	−	0.860428i	$-0.670196\pi$
$947$	−31.3624	−1.01914	−0.509571	+	0.860428i	$0.670196\pi$
$948$	−81.7951	−2.65658
$949$	35.7930	1.16189
$950$	0	0
$951$	−5.60548	−0.181770
$952$	−1.55313	−0.0503373
$953$	−52.9085	−1.71387	−0.856937	−	0.515422i	$-0.827635\pi$
$953$	−52.9085	−1.71387	−0.856937	+	0.515422i	$0.827635\pi$
$954$	−43.1123	−1.39581
$955$	0	0
$956$	64.6313	2.09033
$957$	39.6264	1.28094
$958$	81.7581	2.64148
$959$	0.383601	0.0123871
$960$	0	0
$961$	−25.7781	−0.831552
$962$	−125.281	−4.03921
$963$	7.13746	0.230001
$964$	−69.5420	−2.23980
$965$	0	0
$966$	−23.2780	−0.748958
$967$	60.0281	1.93037	0.965186	−	0.261563i	$-0.0842380\pi$
$967$	60.0281	1.93037	0.965186	+	0.261563i	$0.0842380\pi$
$968$	−26.0040	−0.835800
$969$	0	0
$970$	0	0
$971$	1.75005	0.0561618	0.0280809	−	0.999606i	$-0.491060\pi$
$971$	1.75005	0.0561618	0.0280809	+	0.999606i	$0.491060\pi$
$972$	−72.0390	−2.31065
$973$	−5.06327	−0.162321
$974$	−54.1918	−1.73642
$975$	0	0
$976$	0.0552611	0.00176886
$977$	−5.47583	−0.175187	−0.0875937	−	0.996156i	$-0.527918\pi$
$977$	−5.47583	−0.175187	−0.0875937	+	0.996156i	$0.527918\pi$
$978$	−9.26018	−0.296108
$979$	5.01404	0.160249
$980$	0	0
$981$	51.9717	1.65933
$982$	89.5472	2.85756
$983$	−5.05926	−0.161365	−0.0806827	−	0.996740i	$-0.525710\pi$
$983$	−5.05926	−0.161365	−0.0806827	+	0.996740i	$0.525710\pi$
$984$	−42.5411	−1.35616
$985$	0	0
$986$	−1.27111	−0.0404802
$987$	26.9327	0.857278
$988$	0	0
$989$	−3.87950	−0.123361
$990$	0	0
$991$	0.0984581	0.00312762	0.00156381	−	0.999999i	$-0.499502\pi$
$991$	0.0984581	0.00312762	0.00156381	+	0.999999i	$0.499502\pi$
$992$	−13.0913	−0.415650
$993$	−25.2992	−0.802845
$994$	−130.115	−4.12700
$995$	0	0
$996$	−39.2188	−1.24269
$997$	−18.3593	−0.581446	−0.290723	−	0.956807i	$-0.593896\pi$
$997$	−18.3593	−0.581446	−0.290723	+	0.956807i	$0.593896\pi$
$998$	21.4266	0.678247
$999$	7.83828	0.247992

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.z.1.1		3
5.4	even	2		1805.2.a.h.1.3		3
19.7	even	3		475.2.e.d.201.3		6
19.11	even	3		475.2.e.d.26.3		6
19.18	odd	2		9025.2.a.ba.1.3		3
95.7	odd	12		475.2.j.b.49.5		12
95.49	even	6		95.2.e.b.26.1	yes	6
95.64	even	6		95.2.e.b.11.1	✓	6
95.68	odd	12		475.2.j.b.349.5		12
95.83	odd	12		475.2.j.b.49.2		12
95.87	odd	12		475.2.j.b.349.2		12
95.94	odd	2		1805.2.a.g.1.1		3
285.239	odd	6		855.2.k.g.406.3		6
285.254	odd	6		855.2.k.g.676.3		6
380.159	odd	6		1520.2.q.j.961.1		6
380.239	odd	6		1520.2.q.j.881.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.e.b.11.1	✓	6	95.64	even	6
95.2.e.b.26.1	yes	6	95.49	even	6
475.2.e.d.26.3		6	19.11	even	3
475.2.e.d.201.3		6	19.7	even	3
475.2.j.b.49.2		12	95.83	odd	12
475.2.j.b.49.5		12	95.7	odd	12
475.2.j.b.349.2		12	95.87	odd	12
475.2.j.b.349.5		12	95.68	odd	12
855.2.k.g.406.3		6	285.239	odd	6
855.2.k.g.676.3		6	285.254	odd	6
1520.2.q.j.881.1		6	380.239	odd	6
1520.2.q.j.961.1		6	380.159	odd	6
1805.2.a.g.1.1		3	95.94	odd	2
1805.2.a.h.1.3		3	5.4	even	2
9025.2.a.z.1.1		3	1.1	even	1	trivial
9025.2.a.ba.1.3		3	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)

\(f(q)\)	\(=\)	\(q-2.28514 q^{2} +2.50702 q^{3} +3.22188 q^{4} -5.72889 q^{6} -3.50702 q^{7} -2.79216 q^{8} +3.28514 q^{9} -4.50702 q^{11} +8.07730 q^{12} +5.00000 q^{13} +8.01404 q^{14} -0.0632663 q^{16} -0.158610 q^{17} -7.50702 q^{18} -8.79216 q^{21} +10.2992 q^{22} -1.15861 q^{23} -7.00000 q^{24} -11.4257 q^{26} +0.714858 q^{27} -11.2992 q^{28} -3.50702 q^{29} -2.28514 q^{31} +5.72889 q^{32} -11.2992 q^{33} +0.362446 q^{34} +10.5843 q^{36} +10.9648 q^{37} +12.5351 q^{39} +6.07730 q^{41} +20.0913 q^{42} +3.34841 q^{43} -14.5211 q^{44} +2.64759 q^{46} -3.06327 q^{47} -0.158610 q^{48} +5.29918 q^{49} -0.397638 q^{51} +16.1094 q^{52} +5.74293 q^{53} -1.63355 q^{54} +9.79216 q^{56} +8.01404 q^{58} +3.06327 q^{59} -0.873467 q^{61} +5.22188 q^{62} -11.5211 q^{63} -12.9648 q^{64} +25.8202 q^{66} +8.44375 q^{67} -0.511021 q^{68} -2.90466 q^{69} -16.2359 q^{71} -9.17265 q^{72} +7.15861 q^{73} -25.0561 q^{74} +15.8062 q^{77} -28.6445 q^{78} -10.1265 q^{79} -8.06327 q^{81} -13.8875 q^{82} -4.85543 q^{83} -28.3273 q^{84} -7.65159 q^{86} -8.79216 q^{87} +12.5843 q^{88} -1.11250 q^{89} -17.5351 q^{91} -3.73290 q^{92} -5.72889 q^{93} +7.00000 q^{94} +14.3624 q^{96} -1.61951 q^{97} -12.1094 q^{98} -14.8062 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} - q^{3} + 7 q^{4} - 6 q^{6} - 2 q^{7} + 6 q^{8} + 4 q^{9} - 5 q^{11} + 4 q^{12} + 15 q^{13} + 7 q^{14} + 3 q^{16} - q^{17} - 14 q^{18} - 12 q^{21} + 8 q^{22} - 4 q^{23} - 21 q^{24} - 5 q^{26}+ \cdots - 13 q^{99}+O(q^{100}) \) 3 * q - q^2 - q^3 + 7 * q^4 - 6 * q^6 - 2 * q^7 + 6 * q^8 + 4 * q^9 - 5 * q^11 + 4 * q^12 + 15 * q^13 + 7 * q^14 + 3 * q^16 - q^17 - 14 * q^18 - 12 * q^21 + 8 * q^22 - 4 * q^23 - 21 * q^24 - 5 * q^26 + 8 * q^27 - 11 * q^28 - 2 * q^29 - q^31 + 6 * q^32 - 11 * q^33 - 25 * q^34 + 3 * q^36 + 2 * q^37 - 5 * q^39 - 2 * q^41 + 23 * q^42 + q^43 - 18 * q^44 - 24 * q^46 - 6 * q^47 - q^48 - 7 * q^49 - 6 * q^51 + 35 * q^52 - 11 * q^53 + 10 * q^54 + 15 * q^56 + 7 * q^58 + 6 * q^59 - 9 * q^61 + 13 * q^62 - 9 * q^63 - 8 * q^64 + 29 * q^66 + 20 * q^67 - 34 * q^68 - 5 * q^69 - 29 * q^71 - 11 * q^72 + 22 * q^73 - 7 * q^74 + 16 * q^77 - 30 * q^78 - 24 * q^79 - 21 * q^81 - 31 * q^82 + 3 * q^83 - 28 * q^84 - 32 * q^86 - 12 * q^87 + 9 * q^88 - 14 * q^89 - 10 * q^91 - 41 * q^92 - 6 * q^93 + 21 * q^94 + 17 * q^96 - 7 * q^97 - 23 * q^98 - 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more