LMFDB - Embedded newform 9025.2.a.m.1.2

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:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
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			1.2
Root			$-0.618034$ 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)

$f(q)$	$=$	$q-0.381966 q^{2} -2.23607 q^{3} -1.85410 q^{4} +0.854102 q^{6} -0.236068 q^{7} +1.47214 q^{8} +2.00000 q^{9} +O(q^{10})$	\(q-0.381966 q^{2} -2.23607 q^{3} -1.85410 q^{4} +0.854102 q^{6} -0.236068 q^{7} +1.47214 q^{8} +2.00000 q^{9} -3.47214 q^{11} +4.14590 q^{12} +2.00000 q^{13} +0.0901699 q^{14} +3.14590 q^{16} -1.85410 q^{17} -0.763932 q^{18} +0.527864 q^{21} +1.32624 q^{22} -3.23607 q^{23} -3.29180 q^{24} -0.763932 q^{26} +2.23607 q^{27} +0.437694 q^{28} +6.00000 q^{29} +4.85410 q^{31} -4.14590 q^{32} +7.76393 q^{33} +0.708204 q^{34} -3.70820 q^{36} -10.8541 q^{37} -4.47214 q^{39} -8.61803 q^{41} -0.201626 q^{42} +9.56231 q^{43} +6.43769 q^{44} +1.23607 q^{46} +5.38197 q^{47} -7.03444 q^{48} -6.94427 q^{49} +4.14590 q^{51} -3.70820 q^{52} +8.61803 q^{53} -0.854102 q^{54} -0.347524 q^{56} -2.29180 q^{58} -7.23607 q^{59} +14.5623 q^{61} -1.85410 q^{62} -0.472136 q^{63} -4.70820 q^{64} -2.96556 q^{66} -4.70820 q^{67} +3.43769 q^{68} +7.23607 q^{69} -13.4721 q^{71} +2.94427 q^{72} -2.70820 q^{73} +4.14590 q^{74} +0.819660 q^{77} +1.70820 q^{78} +1.00000 q^{79} -11.0000 q^{81} +3.29180 q^{82} +10.0902 q^{83} -0.978714 q^{84} -3.65248 q^{86} -13.4164 q^{87} -5.11146 q^{88} -9.70820 q^{89} -0.472136 q^{91} +6.00000 q^{92} -10.8541 q^{93} -2.05573 q^{94} +9.27051 q^{96} -9.61803 q^{97} +2.65248 q^{98} -6.94427 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} + 3 q^{4} - 5 q^{6} + 4 q^{7} - 6 q^{8} + 4 q^{9}+O(q^{10}) $ 2 * q - 3 * q^2 + 3 * q^4 - 5 * q^6 + 4 * q^7 - 6 * q^8 + 4 * q^9	$ 2 q - 3 q^{2} + 3 q^{4} - 5 q^{6} + 4 q^{7} - 6 q^{8} + 4 q^{9} + 2 q^{11} + 15 q^{12} + 4 q^{13} - 11 q^{14} + 13 q^{16} + 3 q^{17} - 6 q^{18} + 10 q^{21} - 13 q^{22} - 2 q^{23} - 20 q^{24} - 6 q^{26} + 21 q^{28} + 12 q^{29} + 3 q^{31} - 15 q^{32} + 20 q^{33} - 12 q^{34} + 6 q^{36} - 15 q^{37} - 15 q^{41} - 25 q^{42} - q^{43} + 33 q^{44} - 2 q^{46} + 13 q^{47} + 15 q^{48} + 4 q^{49} + 15 q^{51} + 6 q^{52} + 15 q^{53} + 5 q^{54} - 32 q^{56} - 18 q^{58} - 10 q^{59} + 9 q^{61} + 3 q^{62} + 8 q^{63} + 4 q^{64} - 35 q^{66} + 4 q^{67} + 27 q^{68} + 10 q^{69} - 18 q^{71} - 12 q^{72} + 8 q^{73} + 15 q^{74} + 24 q^{77} - 10 q^{78} + 2 q^{79} - 22 q^{81} + 20 q^{82} + 9 q^{83} + 45 q^{84} + 24 q^{86} - 46 q^{88} - 6 q^{89} + 8 q^{91} + 12 q^{92} - 15 q^{93} - 22 q^{94} - 15 q^{96} - 17 q^{97} - 26 q^{98} + 4 q^{99}+O(q^{100}) $ 2 * q - 3 * q^2 + 3 * q^4 - 5 * q^6 + 4 * q^7 - 6 * q^8 + 4 * q^9 + 2 * q^11 + 15 * q^12 + 4 * q^13 - 11 * q^14 + 13 * q^16 + 3 * q^17 - 6 * q^18 + 10 * q^21 - 13 * q^22 - 2 * q^23 - 20 * q^24 - 6 * q^26 + 21 * q^28 + 12 * q^29 + 3 * q^31 - 15 * q^32 + 20 * q^33 - 12 * q^34 + 6 * q^36 - 15 * q^37 - 15 * q^41 - 25 * q^42 - q^43 + 33 * q^44 - 2 * q^46 + 13 * q^47 + 15 * q^48 + 4 * q^49 + 15 * q^51 + 6 * q^52 + 15 * q^53 + 5 * q^54 - 32 * q^56 - 18 * q^58 - 10 * q^59 + 9 * q^61 + 3 * q^62 + 8 * q^63 + 4 * q^64 - 35 * q^66 + 4 * q^67 + 27 * q^68 + 10 * q^69 - 18 * q^71 - 12 * q^72 + 8 * q^73 + 15 * q^74 + 24 * q^77 - 10 * q^78 + 2 * q^79 - 22 * q^81 + 20 * q^82 + 9 * q^83 + 45 * q^84 + 24 * q^86 - 46 * q^88 - 6 * q^89 + 8 * q^91 + 12 * q^92 - 15 * q^93 - 22 * q^94 - 15 * q^96 - 17 * q^97 - 26 * q^98 + 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.381966	−0.270091	−0.135045	−	0.990839i	$-0.543118\pi$
$2$	−0.381966	−0.270091	−0.135045	+	0.990839i	$0.543118\pi$
$3$	−2.23607	−1.29099	−0.645497	−	0.763763i	$-0.723350\pi$
$3$	−2.23607	−1.29099	−0.645497	+	0.763763i	$0.723350\pi$
$4$	−1.85410	−0.927051
$5$	0	0
$5$	0	0
$6$	0.854102	0.348686
$7$	−0.236068	−0.0892253	−0.0446127	−	0.999004i	$-0.514205\pi$
$7$	−0.236068	−0.0892253	−0.0446127	+	0.999004i	$0.514205\pi$
$8$	1.47214	0.520479
$9$	2.00000	0.666667
$10$	0	0
$11$	−3.47214	−1.04689	−0.523444	−	0.852060i	$-0.675353\pi$
$11$	−3.47214	−1.04689	−0.523444	+	0.852060i	$0.675353\pi$
$12$	4.14590	1.19682
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0.0901699	0.0240989
$15$	0	0
$16$	3.14590	0.786475
$17$	−1.85410	−0.449686	−0.224843	−	0.974395i	$-0.572187\pi$
$17$	−1.85410	−0.449686	−0.224843	+	0.974395i	$0.572187\pi$
$18$	−0.763932	−0.180061
$19$	0	0
$19$	0	0
$20$	0	0
$21$	0.527864	0.115189
$22$	1.32624	0.282755
$23$	−3.23607	−0.674767	−0.337383	−	0.941367i	$-0.609542\pi$
$23$	−3.23607	−0.674767	−0.337383	+	0.941367i	$0.609542\pi$
$24$	−3.29180	−0.671935
$25$	0	0
$26$	−0.763932	−0.149819
$27$	2.23607	0.430331
$28$	0.437694	0.0827164
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	4.85410	0.871822	0.435911	−	0.899990i	$-0.356426\pi$
$31$	4.85410	0.871822	0.435911	+	0.899990i	$0.356426\pi$
$32$	−4.14590	−0.732898
$33$	7.76393	1.35153
$34$	0.708204	0.121456
$35$	0	0
$36$	−3.70820	−0.618034
$37$	−10.8541	−1.78440	−0.892202	−	0.451637i	$-0.850840\pi$
$37$	−10.8541	−1.78440	−0.892202	+	0.451637i	$0.850840\pi$
$38$	0	0
$39$	−4.47214	−0.716115
$40$	0	0
$41$	−8.61803	−1.34591	−0.672955	−	0.739683i	$-0.734975\pi$
$41$	−8.61803	−1.34591	−0.672955	+	0.739683i	$0.734975\pi$
$42$	−0.201626	−0.0311116
$43$	9.56231	1.45824	0.729119	−	0.684387i	$-0.239930\pi$
$43$	9.56231	1.45824	0.729119	+	0.684387i	$0.239930\pi$
$44$	6.43769	0.970519
$45$	0	0
$46$	1.23607	0.182248
$47$	5.38197	0.785040	0.392520	−	0.919743i	$-0.371603\pi$
$47$	5.38197	0.785040	0.392520	+	0.919743i	$0.371603\pi$
$48$	−7.03444	−1.01533
$49$	−6.94427	−0.992039
$50$	0	0
$51$	4.14590	0.580542
$52$	−3.70820	−0.514235
$53$	8.61803	1.18378	0.591889	−	0.806019i	$-0.298382\pi$
$53$	8.61803	1.18378	0.591889	+	0.806019i	$0.298382\pi$
$54$	−0.854102	−0.116229
$55$	0	0
$56$	−0.347524	−0.0464399
$57$	0	0
$58$	−2.29180	−0.300928
$59$	−7.23607	−0.942056	−0.471028	−	0.882118i	$-0.656117\pi$
$59$	−7.23607	−0.942056	−0.471028	+	0.882118i	$0.656117\pi$
$60$	0	0
$61$	14.5623	1.86451	0.932256	−	0.361799i	$-0.117837\pi$
$61$	14.5623	1.86451	0.932256	+	0.361799i	$0.117837\pi$
$62$	−1.85410	−0.235471
$63$	−0.472136	−0.0594835
$64$	−4.70820	−0.588525
$65$	0	0
$66$	−2.96556	−0.365035
$67$	−4.70820	−0.575199	−0.287599	−	0.957751i	$-0.592857\pi$
$67$	−4.70820	−0.575199	−0.287599	+	0.957751i	$0.592857\pi$
$68$	3.43769	0.416882
$69$	7.23607	0.871120
$70$	0	0
$71$	−13.4721	−1.59885	−0.799424	−	0.600767i	$-0.794862\pi$
$71$	−13.4721	−1.59885	−0.799424	+	0.600767i	$0.794862\pi$
$72$	2.94427	0.346986
$73$	−2.70820	−0.316971	−0.158486	−	0.987361i	$-0.550661\pi$
$73$	−2.70820	−0.316971	−0.158486	+	0.987361i	$0.550661\pi$
$74$	4.14590	0.481951
$75$	0	0
$76$	0	0
$77$	0.819660	0.0934089
$78$	1.70820	0.193416
$79$	1.00000	0.112509	0.0562544	−	0.998416i	$-0.482084\pi$
$79$	1.00000	0.112509	0.0562544	+	0.998416i	$0.482084\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	3.29180	0.363518
$83$	10.0902	1.10754	0.553770	−	0.832670i	$-0.313189\pi$
$83$	10.0902	1.10754	0.553770	+	0.832670i	$0.313189\pi$
$84$	−0.978714	−0.106786
$85$	0	0
$86$	−3.65248	−0.393857
$87$	−13.4164	−1.43839
$88$	−5.11146	−0.544883
$89$	−9.70820	−1.02907	−0.514534	−	0.857470i	$-0.672035\pi$
$89$	−9.70820	−1.02907	−0.514534	+	0.857470i	$0.672035\pi$
$90$	0	0
$91$	−0.472136	−0.0494933
$92$	6.00000	0.625543
$93$	−10.8541	−1.12552
$94$	−2.05573	−0.212032
$95$	0	0
$96$	9.27051	0.946167
$97$	−9.61803	−0.976563	−0.488282	−	0.872686i	$-0.662376\pi$
$97$	−9.61803	−0.976563	−0.488282	+	0.872686i	$0.662376\pi$
$98$	2.65248	0.267941
$99$	−6.94427	−0.697926
$100$	0	0
$101$	16.0344	1.59549	0.797743	−	0.602997i	$-0.206027\pi$
$101$	16.0344	1.59549	0.797743	+	0.602997i	$0.206027\pi$
$102$	−1.58359	−0.156799
$103$	−3.23607	−0.318859	−0.159430	−	0.987209i	$-0.550966\pi$
$103$	−3.23607	−0.318859	−0.159430	+	0.987209i	$0.550966\pi$
$104$	2.94427	0.288710
$105$	0	0
$106$	−3.29180	−0.319727
$107$	13.1459	1.27086	0.635431	−	0.772158i	$-0.280822\pi$
$107$	13.1459	1.27086	0.635431	+	0.772158i	$0.280822\pi$
$108$	−4.14590	−0.398939
$109$	−9.94427	−0.952489	−0.476244	−	0.879313i	$-0.658002\pi$
$109$	−9.94427	−0.952489	−0.476244	+	0.879313i	$0.658002\pi$
$110$	0	0
$111$	24.2705	2.30365
$112$	−0.742646	−0.0701734
$113$	−6.32624	−0.595122	−0.297561	−	0.954703i	$-0.596173\pi$
$113$	−6.32624	−0.595122	−0.297561	+	0.954703i	$0.596173\pi$
$114$	0	0
$115$	0	0
$116$	−11.1246	−1.03289
$117$	4.00000	0.369800
$118$	2.76393	0.254441
$119$	0.437694	0.0401234
$120$	0	0
$121$	1.05573	0.0959753
$122$	−5.56231	−0.503588
$123$	19.2705	1.73756
$124$	−9.00000	−0.808224
$125$	0	0
$126$	0.180340	0.0160660
$127$	5.85410	0.519468	0.259734	−	0.965680i	$-0.416365\pi$
$127$	5.85410	0.519468	0.259734	+	0.965680i	$0.416365\pi$
$128$	10.0902	0.891853
$129$	−21.3820	−1.88258
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	−14.3951	−1.25293
$133$	0	0
$134$	1.79837	0.155356
$135$	0	0
$136$	−2.72949	−0.234052
$137$	4.76393	0.407010	0.203505	−	0.979074i	$-0.434767\pi$
$137$	4.76393	0.407010	0.203505	+	0.979074i	$0.434767\pi$
$138$	−2.76393	−0.235282
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	−12.0344	−1.01348
$142$	5.14590	0.431834
$143$	−6.94427	−0.580709
$144$	6.29180	0.524316
$145$	0	0
$146$	1.03444	0.0856110
$147$	15.5279	1.28072
$148$	20.1246	1.65423
$149$	−0.527864	−0.0432443	−0.0216222	−	0.999766i	$-0.506883\pi$
$149$	−0.527864	−0.0432443	−0.0216222	+	0.999766i	$0.506883\pi$
$150$	0	0
$151$	−18.7984	−1.52979	−0.764895	−	0.644155i	$-0.777209\pi$
$151$	−18.7984	−1.52979	−0.764895	+	0.644155i	$0.777209\pi$
$152$	0	0
$153$	−3.70820	−0.299791
$154$	−0.313082	−0.0252289
$155$	0	0
$156$	8.29180	0.663875
$157$	−24.1803	−1.92980	−0.964901	−	0.262615i	$-0.915415\pi$
$157$	−24.1803	−1.92980	−0.964901	+	0.262615i	$0.915415\pi$
$158$	−0.381966	−0.0303876
$159$	−19.2705	−1.52825
$160$	0	0
$161$	0.763932	0.0602063
$162$	4.20163	0.330111
$163$	7.23607	0.566773	0.283386	−	0.959006i	$-0.408542\pi$
$163$	7.23607	0.566773	0.283386	+	0.959006i	$0.408542\pi$
$164$	15.9787	1.24773
$165$	0	0
$166$	−3.85410	−0.299136
$167$	0.763932	0.0591148	0.0295574	−	0.999563i	$-0.490590\pi$
$167$	0.763932	0.0591148	0.0295574	+	0.999563i	$0.490590\pi$
$168$	0.777088	0.0599536
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	−17.7295	−1.35186
$173$	11.6525	0.885921	0.442961	−	0.896541i	$-0.353928\pi$
$173$	11.6525	0.885921	0.442961	+	0.896541i	$0.353928\pi$
$174$	5.12461	0.388496
$175$	0	0
$176$	−10.9230	−0.823351
$177$	16.1803	1.21619
$178$	3.70820	0.277942
$179$	−11.1803	−0.835658	−0.417829	−	0.908526i	$-0.637209\pi$
$179$	−11.1803	−0.835658	−0.417829	+	0.908526i	$0.637209\pi$
$180$	0	0
$181$	−5.29180	−0.393336	−0.196668	−	0.980470i	$-0.563012\pi$
$181$	−5.29180	−0.393336	−0.196668	+	0.980470i	$0.563012\pi$
$182$	0.180340	0.0133677
$183$	−32.5623	−2.40707
$184$	−4.76393	−0.351202
$185$	0	0
$186$	4.14590	0.303992
$187$	6.43769	0.470771
$188$	−9.97871	−0.727772
$189$	−0.527864	−0.0383965
$190$	0	0
$191$	−0.0557281	−0.00403234	−0.00201617	−	0.999998i	$-0.500642\pi$
$191$	−0.0557281	−0.00403234	−0.00201617	+	0.999998i	$0.500642\pi$
$192$	10.5279	0.759783
$193$	10.2705	0.739287	0.369644	−	0.929174i	$-0.379480\pi$
$193$	10.2705	0.739287	0.369644	+	0.929174i	$0.379480\pi$
$194$	3.67376	0.263761
$195$	0	0
$196$	12.8754	0.919671
$197$	−1.47214	−0.104885	−0.0524427	−	0.998624i	$-0.516701\pi$
$197$	−1.47214	−0.104885	−0.0524427	+	0.998624i	$0.516701\pi$
$198$	2.65248	0.188503
$199$	−7.41641	−0.525735	−0.262868	−	0.964832i	$-0.584668\pi$
$199$	−7.41641	−0.525735	−0.262868	+	0.964832i	$0.584668\pi$
$200$	0	0
$201$	10.5279	0.742578
$202$	−6.12461	−0.430926
$203$	−1.41641	−0.0994123
$204$	−7.68692	−0.538192
$205$	0	0
$206$	1.23607	0.0861209
$207$	−6.47214	−0.449845
$208$	6.29180	0.436258
$209$	0	0
$210$	0	0
$211$	−7.76393	−0.534491	−0.267246	−	0.963628i	$-0.586113\pi$
$211$	−7.76393	−0.534491	−0.267246	+	0.963628i	$0.586113\pi$
$212$	−15.9787	−1.09742
$213$	30.1246	2.06410
$214$	−5.02129	−0.343248
$215$	0	0
$216$	3.29180	0.223978
$217$	−1.14590	−0.0777886
$218$	3.79837	0.257258
$219$	6.05573	0.409208
$220$	0	0
$221$	−3.70820	−0.249441
$222$	−9.27051	−0.622196
$223$	16.8885	1.13094	0.565470	−	0.824769i	$-0.308695\pi$
$223$	16.8885	1.13094	0.565470	+	0.824769i	$0.308695\pi$
$224$	0.978714	0.0653931
$225$	0	0
$226$	2.41641	0.160737
$227$	−5.52786	−0.366897	−0.183449	−	0.983029i	$-0.558726\pi$
$227$	−5.52786	−0.366897	−0.183449	+	0.983029i	$0.558726\pi$
$228$	0	0
$229$	−10.6525	−0.703935	−0.351968	−	0.936012i	$-0.614487\pi$
$229$	−10.6525	−0.703935	−0.351968	+	0.936012i	$0.614487\pi$
$230$	0	0
$231$	−1.83282	−0.120590
$232$	8.83282	0.579903
$233$	25.4721	1.66874	0.834368	−	0.551208i	$-0.185833\pi$
$233$	25.4721	1.66874	0.834368	+	0.551208i	$0.185833\pi$
$234$	−1.52786	−0.0998796
$235$	0	0
$236$	13.4164	0.873334
$237$	−2.23607	−0.145248
$238$	−0.167184	−0.0108369
$239$	−12.7082	−0.822025	−0.411013	−	0.911630i	$-0.634825\pi$
$239$	−12.7082	−0.822025	−0.411013	+	0.911630i	$0.634825\pi$
$240$	0	0
$241$	−13.0000	−0.837404	−0.418702	−	0.908124i	$-0.637515\pi$
$241$	−13.0000	−0.837404	−0.418702	+	0.908124i	$0.637515\pi$
$242$	−0.403252	−0.0259220
$243$	17.8885	1.14755
$244$	−27.0000	−1.72850
$245$	0	0
$246$	−7.36068	−0.469300
$247$	0	0
$248$	7.14590	0.453765
$249$	−22.5623	−1.42983
$250$	0	0
$251$	17.0902	1.07872	0.539361	−	0.842075i	$-0.318666\pi$
$251$	17.0902	1.07872	0.539361	+	0.842075i	$0.318666\pi$
$252$	0.875388	0.0551443
$253$	11.2361	0.706406
$254$	−2.23607	−0.140303
$255$	0	0
$256$	5.56231	0.347644
$257$	−28.7984	−1.79639	−0.898197	−	0.439594i	$-0.855122\pi$
$257$	−28.7984	−1.79639	−0.898197	+	0.439594i	$0.855122\pi$
$258$	8.16718	0.508467
$259$	2.56231	0.159214
$260$	0	0
$261$	12.0000	0.742781
$262$	4.58359	0.283175
$263$	20.3820	1.25681	0.628403	−	0.777888i	$-0.283709\pi$
$263$	20.3820	1.25681	0.628403	+	0.777888i	$0.283709\pi$
$264$	11.4296	0.703441
$265$	0	0
$266$	0	0
$267$	21.7082	1.32852
$268$	8.72949	0.533238
$269$	−13.1803	−0.803620	−0.401810	−	0.915723i	$-0.631619\pi$
$269$	−13.1803	−0.803620	−0.401810	+	0.915723i	$0.631619\pi$
$270$	0	0
$271$	4.32624	0.262800	0.131400	−	0.991329i	$-0.458053\pi$
$271$	4.32624	0.262800	0.131400	+	0.991329i	$0.458053\pi$
$272$	−5.83282	−0.353666
$273$	1.05573	0.0638956
$274$	−1.81966	−0.109930
$275$	0	0
$276$	−13.4164	−0.807573
$277$	−12.0000	−0.721010	−0.360505	−	0.932757i	$-0.617396\pi$
$277$	−12.0000	−0.721010	−0.360505	+	0.932757i	$0.617396\pi$
$278$	5.34752	0.320723
$279$	9.70820	0.581215
$280$	0	0
$281$	2.38197	0.142096	0.0710481	−	0.997473i	$-0.477366\pi$
$281$	2.38197	0.142096	0.0710481	+	0.997473i	$0.477366\pi$
$282$	4.59675	0.273732
$283$	−28.2361	−1.67846	−0.839230	−	0.543777i	$-0.816994\pi$
$283$	−28.2361	−1.67846	−0.839230	+	0.543777i	$0.816994\pi$
$284$	24.9787	1.48221
$285$	0	0
$286$	2.65248	0.156844
$287$	2.03444	0.120089
$288$	−8.29180	−0.488599
$289$	−13.5623	−0.797783
$290$	0	0
$291$	21.5066	1.26074
$292$	5.02129	0.293849
$293$	4.52786	0.264521	0.132260	−	0.991215i	$-0.457777\pi$
$293$	4.52786	0.264521	0.132260	+	0.991215i	$0.457777\pi$
$294$	−5.93112	−0.345910
$295$	0	0
$296$	−15.9787	−0.928744
$297$	−7.76393	−0.450509
$298$	0.201626	0.0116799
$299$	−6.47214	−0.374293
$300$	0	0
$301$	−2.25735	−0.130112
$302$	7.18034	0.413182
$303$	−35.8541	−2.05976
$304$	0	0
$305$	0	0
$306$	1.41641	0.0809706
$307$	12.4377	0.709857	0.354928	−	0.934894i	$-0.384505\pi$
$307$	12.4377	0.709857	0.354928	+	0.934894i	$0.384505\pi$
$308$	−1.51973	−0.0865948
$309$	7.23607	0.411646
$310$	0	0
$311$	11.7639	0.667071	0.333536	−	0.942737i	$-0.391758\pi$
$311$	11.7639	0.667071	0.333536	+	0.942737i	$0.391758\pi$
$312$	−6.58359	−0.372723
$313$	2.05573	0.116197	0.0580983	−	0.998311i	$-0.481496\pi$
$313$	2.05573	0.116197	0.0580983	+	0.998311i	$0.481496\pi$
$314$	9.23607	0.521221
$315$	0	0
$316$	−1.85410	−0.104301
$317$	6.94427	0.390029	0.195015	−	0.980800i	$-0.437525\pi$
$317$	6.94427	0.390029	0.195015	+	0.980800i	$0.437525\pi$
$318$	7.36068	0.412766
$319$	−20.8328	−1.16641
$320$	0	0
$321$	−29.3951	−1.64068
$322$	−0.291796	−0.0162612
$323$	0	0
$324$	20.3951	1.13306
$325$	0	0
$326$	−2.76393	−0.153080
$327$	22.2361	1.22966
$328$	−12.6869	−0.700518
$329$	−1.27051	−0.0700455
$330$	0	0
$331$	6.56231	0.360697	0.180348	−	0.983603i	$-0.442277\pi$
$331$	6.56231	0.360697	0.180348	+	0.983603i	$0.442277\pi$
$332$	−18.7082	−1.02675
$333$	−21.7082	−1.18960
$334$	−0.291796	−0.0159664
$335$	0	0
$336$	1.66061	0.0905935
$337$	−8.70820	−0.474366	−0.237183	−	0.971465i	$-0.576224\pi$
$337$	−8.70820	−0.474366	−0.237183	+	0.971465i	$0.576224\pi$
$338$	3.43769	0.186986
$339$	14.1459	0.768300
$340$	0	0
$341$	−16.8541	−0.912701
$342$	0	0
$343$	3.29180	0.177740
$344$	14.0770	0.758982
$345$	0	0
$346$	−4.45085	−0.239279
$347$	24.5279	1.31672	0.658362	−	0.752701i	$-0.271249\pi$
$347$	24.5279	1.31672	0.658362	+	0.752701i	$0.271249\pi$
$348$	24.8754	1.33346
$349$	31.0689	1.66308	0.831540	−	0.555465i	$-0.187460\pi$
$349$	31.0689	1.66308	0.831540	+	0.555465i	$0.187460\pi$
$350$	0	0
$351$	4.47214	0.238705
$352$	14.3951	0.767263
$353$	−25.4721	−1.35574	−0.677872	−	0.735179i	$-0.737098\pi$
$353$	−25.4721	−1.35574	−0.677872	+	0.735179i	$0.737098\pi$
$354$	−6.18034	−0.328481
$355$	0	0
$356$	18.0000	0.953998
$357$	−0.978714	−0.0517990
$358$	4.27051	0.225703
$359$	26.5967	1.40372	0.701861	−	0.712314i	$-0.252353\pi$
$359$	26.5967	1.40372	0.701861	+	0.712314i	$0.252353\pi$
$360$	0	0
$361$	0	0
$362$	2.02129	0.106236
$363$	−2.36068	−0.123904
$364$	0.875388	0.0458828
$365$	0	0
$366$	12.4377	0.650129
$367$	3.52786	0.184153	0.0920765	−	0.995752i	$-0.470650\pi$
$367$	3.52786	0.184153	0.0920765	+	0.995752i	$0.470650\pi$
$368$	−10.1803	−0.530687
$369$	−17.2361	−0.897274
$370$	0	0
$371$	−2.03444	−0.105623
$372$	20.1246	1.04341
$373$	−19.8541	−1.02801	−0.514003	−	0.857788i	$-0.671838\pi$
$373$	−19.8541	−1.02801	−0.514003	+	0.857788i	$0.671838\pi$
$374$	−2.45898	−0.127151
$375$	0	0
$376$	7.92299	0.408597
$377$	12.0000	0.618031
$378$	0.201626	0.0103705
$379$	30.5410	1.56879	0.784393	−	0.620264i	$-0.212974\pi$
$379$	30.5410	1.56879	0.784393	+	0.620264i	$0.212974\pi$
$380$	0	0
$381$	−13.0902	−0.670630
$382$	0.0212862	0.00108910
$383$	−36.0689	−1.84303	−0.921517	−	0.388338i	$-0.873049\pi$
$383$	−36.0689	−1.84303	−0.921517	+	0.388338i	$0.873049\pi$
$384$	−22.5623	−1.15138
$385$	0	0
$386$	−3.92299	−0.199675
$387$	19.1246	0.972159
$388$	17.8328	0.905324
$389$	4.47214	0.226746	0.113373	−	0.993552i	$-0.463834\pi$
$389$	4.47214	0.226746	0.113373	+	0.993552i	$0.463834\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	−10.2229	−0.516335
$393$	26.8328	1.35354
$394$	0.562306	0.0283286
$395$	0	0
$396$	12.8754	0.647013
$397$	−20.0344	−1.00550	−0.502750	−	0.864432i	$-0.667678\pi$
$397$	−20.0344	−1.00550	−0.502750	+	0.864432i	$0.667678\pi$
$398$	2.83282	0.141996
$399$	0	0
$400$	0	0
$401$	29.8885	1.49256	0.746281	−	0.665631i	$-0.231837\pi$
$401$	29.8885	1.49256	0.746281	+	0.665631i	$0.231837\pi$
$402$	−4.02129	−0.200564
$403$	9.70820	0.483600
$404$	−29.7295	−1.47910
$405$	0	0
$406$	0.541020	0.0268504
$407$	37.6869	1.86807
$408$	6.10333	0.302160
$409$	36.5623	1.80789	0.903945	−	0.427649i	$-0.140658\pi$
$409$	36.5623	1.80789	0.903945	+	0.427649i	$0.140658\pi$
$410$	0	0
$411$	−10.6525	−0.525448
$412$	6.00000	0.295599
$413$	1.70820	0.0840552
$414$	2.47214	0.121499
$415$	0	0
$416$	−8.29180	−0.406539
$417$	31.3050	1.53301
$418$	0	0
$419$	−18.4721	−0.902423	−0.451211	−	0.892417i	$-0.649008\pi$
$419$	−18.4721	−0.902423	−0.451211	+	0.892417i	$0.649008\pi$
$420$	0	0
$421$	−15.0000	−0.731055	−0.365528	−	0.930800i	$-0.619111\pi$
$421$	−15.0000	−0.731055	−0.365528	+	0.930800i	$0.619111\pi$
$422$	2.96556	0.144361
$423$	10.7639	0.523360
$424$	12.6869	0.616131
$425$	0	0
$426$	−11.5066	−0.557496
$427$	−3.43769	−0.166362
$428$	−24.3738	−1.17815
$429$	15.5279	0.749692
$430$	0	0
$431$	20.6525	0.994795	0.497397	−	0.867523i	$-0.334289\pi$
$431$	20.6525	0.994795	0.497397	+	0.867523i	$0.334289\pi$
$432$	7.03444	0.338445
$433$	11.3607	0.545959	0.272980	−	0.962020i	$-0.411991\pi$
$433$	11.3607	0.545959	0.272980	+	0.962020i	$0.411991\pi$
$434$	0.437694	0.0210100
$435$	0	0
$436$	18.4377	0.883005
$437$	0	0
$438$	−2.31308	−0.110523
$439$	−1.50658	−0.0719050	−0.0359525	−	0.999353i	$-0.511447\pi$
$439$	−1.50658	−0.0719050	−0.0359525	+	0.999353i	$0.511447\pi$
$440$	0	0
$441$	−13.8885	−0.661359
$442$	1.41641	0.0673717
$443$	28.3262	1.34582	0.672910	−	0.739724i	$-0.265044\pi$
$443$	28.3262	1.34582	0.672910	+	0.739724i	$0.265044\pi$
$444$	−45.0000	−2.13561
$445$	0	0
$446$	−6.45085	−0.305457
$447$	1.18034	0.0558282
$448$	1.11146	0.0525114
$449$	0.201626	0.00951533	0.00475766	−	0.999989i	$-0.498486\pi$
$449$	0.201626	0.00951533	0.00475766	+	0.999989i	$0.498486\pi$
$450$	0	0
$451$	29.9230	1.40902
$452$	11.7295	0.551709
$453$	42.0344	1.97495
$454$	2.11146	0.0990955
$455$	0	0
$456$	0	0
$457$	14.5623	0.681196	0.340598	−	0.940209i	$-0.389371\pi$
$457$	14.5623	0.681196	0.340598	+	0.940209i	$0.389371\pi$
$458$	4.06888	0.190126
$459$	−4.14590	−0.193514
$460$	0	0
$461$	21.0344	0.979672	0.489836	−	0.871815i	$-0.337057\pi$
$461$	21.0344	0.979672	0.489836	+	0.871815i	$0.337057\pi$
$462$	0.700073	0.0325704
$463$	40.8673	1.89926	0.949631	−	0.313370i	$-0.101458\pi$
$463$	40.8673	1.89926	0.949631	+	0.313370i	$0.101458\pi$
$464$	18.8754	0.876268
$465$	0	0
$466$	−9.72949	−0.450710
$467$	36.2705	1.67840	0.839200	−	0.543824i	$-0.183024\pi$
$467$	36.2705	1.67840	0.839200	+	0.543824i	$0.183024\pi$
$468$	−7.41641	−0.342824
$469$	1.11146	0.0513223
$470$	0	0
$471$	54.0689	2.49136
$472$	−10.6525	−0.490320
$473$	−33.2016	−1.52661
$474$	0.854102	0.0392302
$475$	0	0
$476$	−0.811529	−0.0371964
$477$	17.2361	0.789185
$478$	4.85410	0.222021
$479$	14.7426	0.673609	0.336804	−	0.941575i	$-0.390654\pi$
$479$	14.7426	0.673609	0.336804	+	0.941575i	$0.390654\pi$
$480$	0	0
$481$	−21.7082	−0.989809
$482$	4.96556	0.226175
$483$	−1.70820	−0.0777260
$484$	−1.95743	−0.0889740
$485$	0	0
$486$	−6.83282	−0.309943
$487$	−30.1246	−1.36508	−0.682538	−	0.730850i	$-0.739124\pi$
$487$	−30.1246	−1.36508	−0.682538	+	0.730850i	$0.739124\pi$
$488$	21.4377	0.970439
$489$	−16.1803	−0.731700
$490$	0	0
$491$	−32.2361	−1.45479	−0.727397	−	0.686217i	$-0.759270\pi$
$491$	−32.2361	−1.45479	−0.727397	+	0.686217i	$0.759270\pi$
$492$	−35.7295	−1.61081
$493$	−11.1246	−0.501027
$494$	0	0
$495$	0	0
$496$	15.2705	0.685666
$497$	3.18034	0.142658
$498$	8.61803	0.386183
$499$	38.5967	1.72783	0.863914	−	0.503640i	$-0.168006\pi$
$499$	38.5967	1.72783	0.863914	+	0.503640i	$0.168006\pi$
$500$	0	0
$501$	−1.70820	−0.0763169
$502$	−6.52786	−0.291353
$503$	−23.7426	−1.05863	−0.529316	−	0.848425i	$-0.677551\pi$
$503$	−23.7426	−1.05863	−0.529316	+	0.848425i	$0.677551\pi$
$504$	−0.695048	−0.0309599
$505$	0	0
$506$	−4.29180	−0.190794
$507$	20.1246	0.893765
$508$	−10.8541	−0.481573
$509$	−16.0344	−0.710714	−0.355357	−	0.934731i	$-0.615641\pi$
$509$	−16.0344	−0.710714	−0.355357	+	0.934731i	$0.615641\pi$
$510$	0	0
$511$	0.639320	0.0282819
$512$	−22.3050	−0.985749
$513$	0	0
$514$	11.0000	0.485189
$515$	0	0
$516$	39.6443	1.74524
$517$	−18.6869	−0.821850
$518$	−0.978714	−0.0430022
$519$	−26.0557	−1.14372
$520$	0	0
$521$	15.7639	0.690630	0.345315	−	0.938487i	$-0.387772\pi$
$521$	15.7639	0.690630	0.345315	+	0.938487i	$0.387772\pi$
$522$	−4.58359	−0.200618
$523$	−4.61803	−0.201933	−0.100966	−	0.994890i	$-0.532193\pi$
$523$	−4.61803	−0.201933	−0.100966	+	0.994890i	$0.532193\pi$
$524$	22.2492	0.971962
$525$	0	0
$526$	−7.78522	−0.339452
$527$	−9.00000	−0.392046
$528$	24.4245	1.06294
$529$	−12.5279	−0.544690
$530$	0	0
$531$	−14.4721	−0.628037
$532$	0	0
$533$	−17.2361	−0.746577
$534$	−8.29180	−0.358821
$535$	0	0
$536$	−6.93112	−0.299379
$537$	25.0000	1.07883
$538$	5.03444	0.217050
$539$	24.1115	1.03855
$540$	0	0
$541$	38.6525	1.66180	0.830900	−	0.556422i	$-0.187826\pi$
$541$	38.6525	1.66180	0.830900	+	0.556422i	$0.187826\pi$
$542$	−1.65248	−0.0709799
$543$	11.8328	0.507795
$544$	7.68692	0.329574
$545$	0	0
$546$	−0.403252	−0.0172576
$547$	20.4164	0.872943	0.436471	−	0.899718i	$-0.356228\pi$
$547$	20.4164	0.872943	0.436471	+	0.899718i	$0.356228\pi$
$548$	−8.83282	−0.377319
$549$	29.1246	1.24301
$550$	0	0
$551$	0	0
$552$	10.6525	0.453399
$553$	−0.236068	−0.0100386
$554$	4.58359	0.194738
$555$	0	0
$556$	25.9574	1.10084
$557$	−4.49342	−0.190392	−0.0951962	−	0.995459i	$-0.530348\pi$
$557$	−4.49342	−0.190392	−0.0951962	+	0.995459i	$0.530348\pi$
$558$	−3.70820	−0.156981
$559$	19.1246	0.808885
$560$	0	0
$561$	−14.3951	−0.607763
$562$	−0.909830	−0.0383789
$563$	32.5967	1.37379	0.686895	−	0.726757i	$-0.258973\pi$
$563$	32.5967	1.37379	0.686895	+	0.726757i	$0.258973\pi$
$564$	22.3131	0.939550
$565$	0	0
$566$	10.7852	0.453337
$567$	2.59675	0.109053
$568$	−19.8328	−0.832166
$569$	11.2918	0.473377	0.236688	−	0.971586i	$-0.423938\pi$
$569$	11.2918	0.473377	0.236688	+	0.971586i	$0.423938\pi$
$570$	0	0
$571$	−36.2492	−1.51698	−0.758491	−	0.651683i	$-0.774063\pi$
$571$	−36.2492	−1.51698	−0.758491	+	0.651683i	$0.774063\pi$
$572$	12.8754	0.538347
$573$	0.124612	0.00520573
$574$	−0.777088	−0.0324350
$575$	0	0
$576$	−9.41641	−0.392350
$577$	15.3050	0.637153	0.318577	−	0.947897i	$-0.396795\pi$
$577$	15.3050	0.637153	0.318577	+	0.947897i	$0.396795\pi$
$578$	5.18034	0.215474
$579$	−22.9656	−0.954416
$580$	0	0
$581$	−2.38197	−0.0988206
$582$	−8.21478	−0.340514
$583$	−29.9230	−1.23928
$584$	−3.98684	−0.164977
$585$	0	0
$586$	−1.72949	−0.0714446
$587$	43.3262	1.78827	0.894133	−	0.447802i	$-0.147793\pi$
$587$	43.3262	1.78827	0.894133	+	0.447802i	$0.147793\pi$
$588$	−28.7902	−1.18729
$589$	0	0
$590$	0	0
$591$	3.29180	0.135406
$592$	−34.1459	−1.40339
$593$	36.5967	1.50285	0.751424	−	0.659819i	$-0.229367\pi$
$593$	36.5967	1.50285	0.751424	+	0.659819i	$0.229367\pi$
$594$	2.96556	0.121678
$595$	0	0
$596$	0.978714	0.0400897
$597$	16.5836	0.678721
$598$	2.47214	0.101093
$599$	17.0689	0.697416	0.348708	−	0.937231i	$-0.386621\pi$
$599$	17.0689	0.697416	0.348708	+	0.937231i	$0.386621\pi$
$600$	0	0
$601$	−41.6869	−1.70044	−0.850222	−	0.526424i	$-0.823533\pi$
$601$	−41.6869	−1.70044	−0.850222	+	0.526424i	$0.823533\pi$
$602$	0.862233	0.0351420
$603$	−9.41641	−0.383466
$604$	34.8541	1.41819
$605$	0	0
$606$	13.6950	0.556323
$607$	19.9787	0.810911	0.405455	−	0.914115i	$-0.367113\pi$
$607$	19.9787	0.810911	0.405455	+	0.914115i	$0.367113\pi$
$608$	0	0
$609$	3.16718	0.128341
$610$	0	0
$611$	10.7639	0.435462
$612$	6.87539	0.277921
$613$	−30.5066	−1.23215	−0.616075	−	0.787688i	$-0.711278\pi$
$613$	−30.5066	−1.23215	−0.616075	+	0.787688i	$0.711278\pi$
$614$	−4.75078	−0.191726
$615$	0	0
$616$	1.20665	0.0486174
$617$	28.5066	1.14763	0.573816	−	0.818984i	$-0.305463\pi$
$617$	28.5066	1.14763	0.573816	+	0.818984i	$0.305463\pi$
$618$	−2.76393	−0.111182
$619$	−1.50658	−0.0605545	−0.0302772	−	0.999542i	$-0.509639\pi$
$619$	−1.50658	−0.0605545	−0.0302772	+	0.999542i	$0.509639\pi$
$620$	0	0
$621$	−7.23607	−0.290373
$622$	−4.49342	−0.180170
$623$	2.29180	0.0918189
$624$	−14.0689	−0.563206
$625$	0	0
$626$	−0.785218	−0.0313836
$627$	0	0
$628$	44.8328	1.78902
$629$	20.1246	0.802421
$630$	0	0
$631$	12.9787	0.516674	0.258337	−	0.966055i	$-0.416825\pi$
$631$	12.9787	0.516674	0.258337	+	0.966055i	$0.416825\pi$
$632$	1.47214	0.0585584
$633$	17.3607	0.690025
$634$	−2.65248	−0.105343
$635$	0	0
$636$	35.7295	1.41677
$637$	−13.8885	−0.550284
$638$	7.95743	0.315038
$639$	−26.9443	−1.06590
$640$	0	0
$641$	30.7082	1.21290	0.606451	−	0.795121i	$-0.292593\pi$
$641$	30.7082	1.21290	0.606451	+	0.795121i	$0.292593\pi$
$642$	11.2279	0.443131
$643$	24.3050	0.958494	0.479247	−	0.877680i	$-0.340910\pi$
$643$	24.3050	0.958494	0.479247	+	0.877680i	$0.340910\pi$
$644$	−1.41641	−0.0558143
$645$	0	0
$646$	0	0
$647$	−16.5066	−0.648941	−0.324470	−	0.945896i	$-0.605186\pi$
$647$	−16.5066	−0.648941	−0.324470	+	0.945896i	$0.605186\pi$
$648$	−16.1935	−0.636141
$649$	25.1246	0.986227
$650$	0	0
$651$	2.56231	0.100425
$652$	−13.4164	−0.525427
$653$	26.1246	1.02234	0.511168	−	0.859481i	$-0.329213\pi$
$653$	26.1246	1.02234	0.511168	+	0.859481i	$0.329213\pi$
$654$	−8.49342	−0.332119
$655$	0	0
$656$	−27.1115	−1.05852
$657$	−5.41641	−0.211314
$658$	0.485292	0.0189186
$659$	−43.0689	−1.67773	−0.838863	−	0.544343i	$-0.816779\pi$
$659$	−43.0689	−1.67773	−0.838863	+	0.544343i	$0.816779\pi$
$660$	0	0
$661$	−1.58359	−0.0615946	−0.0307973	−	0.999526i	$-0.509805\pi$
$661$	−1.58359	−0.0615946	−0.0307973	+	0.999526i	$0.509805\pi$
$662$	−2.50658	−0.0974209
$663$	8.29180	0.322027
$664$	14.8541	0.576451
$665$	0	0
$666$	8.29180	0.321301
$667$	−19.4164	−0.751806
$668$	−1.41641	−0.0548025
$669$	−37.7639	−1.46004
$670$	0	0
$671$	−50.5623	−1.95194
$672$	−2.18847	−0.0844221
$673$	−25.1803	−0.970631	−0.485315	−	0.874339i	$-0.661295\pi$
$673$	−25.1803	−0.970631	−0.485315	+	0.874339i	$0.661295\pi$
$674$	3.32624	0.128122
$675$	0	0
$676$	16.6869	0.641805
$677$	−24.9098	−0.957363	−0.478681	−	0.877989i	$-0.658885\pi$
$677$	−24.9098	−0.957363	−0.478681	+	0.877989i	$0.658885\pi$
$678$	−5.40325	−0.207511
$679$	2.27051	0.0871342
$680$	0	0
$681$	12.3607	0.473662
$682$	6.43769	0.246512
$683$	−12.2016	−0.466882	−0.233441	−	0.972371i	$-0.574999\pi$
$683$	−12.2016	−0.466882	−0.233441	+	0.972371i	$0.574999\pi$
$684$	0	0
$685$	0	0
$686$	−1.25735	−0.0480060
$687$	23.8197	0.908777
$688$	30.0820	1.14687
$689$	17.2361	0.656642
$690$	0	0
$691$	23.6525	0.899783	0.449891	−	0.893083i	$-0.351463\pi$
$691$	23.6525	0.899783	0.449891	+	0.893083i	$0.351463\pi$
$692$	−21.6049	−0.821294
$693$	1.63932	0.0622726
$694$	−9.36881	−0.355635
$695$	0	0
$696$	−19.7508	−0.748651
$697$	15.9787	0.605237
$698$	−11.8673	−0.449182
$699$	−56.9574	−2.15433
$700$	0	0
$701$	−31.0557	−1.17296	−0.586479	−	0.809964i	$-0.699486\pi$
$701$	−31.0557	−1.17296	−0.586479	+	0.809964i	$0.699486\pi$
$702$	−1.70820	−0.0644720
$703$	0	0
$704$	16.3475	0.616120
$705$	0	0
$706$	9.72949	0.366174
$707$	−3.78522	−0.142358
$708$	−30.0000	−1.12747
$709$	−44.8328	−1.68373	−0.841866	−	0.539687i	$-0.818543\pi$
$709$	−44.8328	−1.68373	−0.841866	+	0.539687i	$0.818543\pi$
$710$	0	0
$711$	2.00000	0.0750059
$712$	−14.2918	−0.535608
$713$	−15.7082	−0.588277
$714$	0.373835	0.0139904
$715$	0	0
$716$	20.7295	0.774697
$717$	28.4164	1.06123
$718$	−10.1591	−0.379133
$719$	−7.76393	−0.289546	−0.144773	−	0.989465i	$-0.546245\pi$
$719$	−7.76393	−0.289546	−0.144773	+	0.989465i	$0.546245\pi$
$720$	0	0
$721$	0.763932	0.0284503
$722$	0	0
$723$	29.0689	1.08108
$724$	9.81153	0.364643
$725$	0	0
$726$	0.901699	0.0334652
$727$	19.6869	0.730147	0.365074	−	0.930979i	$-0.381044\pi$
$727$	19.6869	0.730147	0.365074	+	0.930979i	$0.381044\pi$
$728$	−0.695048	−0.0257602
$729$	−7.00000	−0.259259
$730$	0	0
$731$	−17.7295	−0.655749
$732$	60.3738	2.23148
$733$	−19.1591	−0.707656	−0.353828	−	0.935311i	$-0.615120\pi$
$733$	−19.1591	−0.707656	−0.353828	+	0.935311i	$0.615120\pi$
$734$	−1.34752	−0.0497380
$735$	0	0
$736$	13.4164	0.494535
$737$	16.3475	0.602169
$738$	6.58359	0.242345
$739$	−18.5623	−0.682825	−0.341413	−	0.939913i	$-0.610905\pi$
$739$	−18.5623	−0.682825	−0.341413	+	0.939913i	$0.610905\pi$
$740$	0	0
$741$	0	0
$742$	0.777088	0.0285278
$743$	−13.2361	−0.485584	−0.242792	−	0.970078i	$-0.578063\pi$
$743$	−13.2361	−0.485584	−0.242792	+	0.970078i	$0.578063\pi$
$744$	−15.9787	−0.585808
$745$	0	0
$746$	7.58359	0.277655
$747$	20.1803	0.738360
$748$	−11.9361	−0.436429
$749$	−3.10333	−0.113393
$750$	0	0
$751$	47.5623	1.73557	0.867787	−	0.496937i	$-0.165542\pi$
$751$	47.5623	1.73557	0.867787	+	0.496937i	$0.165542\pi$
$752$	16.9311	0.617414
$753$	−38.2148	−1.39262
$754$	−4.58359	−0.166925
$755$	0	0
$756$	0.978714	0.0355955
$757$	28.8885	1.04997	0.524986	−	0.851111i	$-0.324071\pi$
$757$	28.8885	1.04997	0.524986	+	0.851111i	$0.324071\pi$
$758$	−11.6656	−0.423715
$759$	−25.1246	−0.911966
$760$	0	0
$761$	29.6180	1.07365	0.536826	−	0.843693i	$-0.319623\pi$
$761$	29.6180	1.07365	0.536826	+	0.843693i	$0.319623\pi$
$762$	5.00000	0.181131
$763$	2.34752	0.0849861
$764$	0.103326	0.00373819
$765$	0	0
$766$	13.7771	0.497786
$767$	−14.4721	−0.522559
$768$	−12.4377	−0.448807
$769$	−18.2705	−0.658851	−0.329426	−	0.944181i	$-0.606855\pi$
$769$	−18.2705	−0.658851	−0.329426	+	0.944181i	$0.606855\pi$
$770$	0	0
$771$	64.3951	2.31913
$772$	−19.0426	−0.685357
$773$	−18.9656	−0.682144	−0.341072	−	0.940037i	$-0.610790\pi$
$773$	−18.9656	−0.682144	−0.341072	+	0.940037i	$0.610790\pi$
$774$	−7.30495	−0.262571
$775$	0	0
$776$	−14.1591	−0.508280
$777$	−5.72949	−0.205544
$778$	−1.70820	−0.0612421
$779$	0	0
$780$	0	0
$781$	46.7771	1.67382
$782$	−2.29180	−0.0819545
$783$	13.4164	0.479463
$784$	−21.8460	−0.780213
$785$	0	0
$786$	−10.2492	−0.365578
$787$	−5.59675	−0.199503	−0.0997513	−	0.995012i	$-0.531805\pi$
$787$	−5.59675	−0.199503	−0.0997513	+	0.995012i	$0.531805\pi$
$788$	2.72949	0.0972341
$789$	−45.5755	−1.62253
$790$	0	0
$791$	1.49342	0.0531000
$792$	−10.2229	−0.363255
$793$	29.1246	1.03425
$794$	7.65248	0.271576
$795$	0	0
$796$	13.7508	0.487383
$797$	54.8673	1.94350	0.971749	−	0.236017i	$-0.0758420\pi$
$797$	54.8673	1.94350	0.971749	+	0.236017i	$0.0758420\pi$
$798$	0	0
$799$	−9.97871	−0.353022
$800$	0	0
$801$	−19.4164	−0.686045
$802$	−11.4164	−0.403127
$803$	9.40325	0.331834
$804$	−19.5197	−0.688408
$805$	0	0
$806$	−3.70820	−0.130616
$807$	29.4721	1.03747
$808$	23.6049	0.830417
$809$	2.23607	0.0786160	0.0393080	−	0.999227i	$-0.487485\pi$
$809$	2.23607	0.0786160	0.0393080	+	0.999227i	$0.487485\pi$
$810$	0	0
$811$	−7.38197	−0.259216	−0.129608	−	0.991565i	$-0.541372\pi$
$811$	−7.38197	−0.259216	−0.129608	+	0.991565i	$0.541372\pi$
$812$	2.62616	0.0921603
$813$	−9.67376	−0.339274
$814$	−14.3951	−0.504549
$815$	0	0
$816$	13.0426	0.456581
$817$	0	0
$818$	−13.9656	−0.488294
$819$	−0.944272	−0.0329955
$820$	0	0
$821$	−36.9230	−1.28862	−0.644311	−	0.764764i	$-0.722856\pi$
$821$	−36.9230	−1.28862	−0.644311	+	0.764764i	$0.722856\pi$
$822$	4.06888	0.141919
$823$	−5.74265	−0.200176	−0.100088	−	0.994979i	$-0.531912\pi$
$823$	−5.74265	−0.200176	−0.100088	+	0.994979i	$0.531912\pi$
$824$	−4.76393	−0.165959
$825$	0	0
$826$	−0.652476	−0.0227025
$827$	−49.2492	−1.71256	−0.856282	−	0.516509i	$-0.827231\pi$
$827$	−49.2492	−1.71256	−0.856282	+	0.516509i	$0.827231\pi$
$828$	12.0000	0.417029
$829$	1.00000	0.0347314	0.0173657	−	0.999849i	$-0.494472\pi$
$829$	1.00000	0.0347314	0.0173657	+	0.999849i	$0.494472\pi$
$830$	0	0
$831$	26.8328	0.930820
$832$	−9.41641	−0.326455
$833$	12.8754	0.446106
$834$	−11.9574	−0.414052
$835$	0	0
$836$	0	0
$837$	10.8541	0.375173
$838$	7.05573	0.243736
$839$	49.7639	1.71804	0.859021	−	0.511941i	$-0.171073\pi$
$839$	49.7639	1.71804	0.859021	+	0.511941i	$0.171073\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	5.72949	0.197451
$843$	−5.32624	−0.183445
$844$	14.3951	0.495501
$845$	0	0
$846$	−4.11146	−0.141355
$847$	−0.249224	−0.00856342
$848$	27.1115	0.931011
$849$	63.1378	2.16688
$850$	0	0
$851$	35.1246	1.20406
$852$	−55.8541	−1.91353
$853$	40.1459	1.37457	0.687285	−	0.726388i	$-0.258802\pi$
$853$	40.1459	1.37457	0.687285	+	0.726388i	$0.258802\pi$
$854$	1.31308	0.0449328
$855$	0	0
$856$	19.3525	0.661457
$857$	47.1803	1.61165	0.805825	−	0.592154i	$-0.201722\pi$
$857$	47.1803	1.61165	0.805825	+	0.592154i	$0.201722\pi$
$858$	−5.93112	−0.202485
$859$	23.0902	0.787826	0.393913	−	0.919148i	$-0.371121\pi$
$859$	23.0902	0.787826	0.393913	+	0.919148i	$0.371121\pi$
$860$	0	0
$861$	−4.54915	−0.155035
$862$	−7.88854	−0.268685
$863$	32.9230	1.12071	0.560356	−	0.828252i	$-0.310664\pi$
$863$	32.9230	1.12071	0.560356	+	0.828252i	$0.310664\pi$
$864$	−9.27051	−0.315389
$865$	0	0
$866$	−4.33939	−0.147459
$867$	30.3262	1.02993
$868$	2.12461	0.0721140
$869$	−3.47214	−0.117784
$870$	0	0
$871$	−9.41641	−0.319063
$872$	−14.6393	−0.495750
$873$	−19.2361	−0.651042
$874$	0	0
$875$	0	0
$876$	−11.2279	−0.379357
$877$	15.9656	0.539119	0.269559	−	0.962984i	$-0.413122\pi$
$877$	15.9656	0.539119	0.269559	+	0.962984i	$0.413122\pi$
$878$	0.575462	0.0194209
$879$	−10.1246	−0.341495
$880$	0	0
$881$	−1.47214	−0.0495975	−0.0247988	−	0.999692i	$-0.507894\pi$
$881$	−1.47214	−0.0495975	−0.0247988	+	0.999692i	$0.507894\pi$
$882$	5.30495	0.178627
$883$	49.7771	1.67513	0.837566	−	0.546336i	$-0.183978\pi$
$883$	49.7771	1.67513	0.837566	+	0.546336i	$0.183978\pi$
$884$	6.87539	0.231244
$885$	0	0
$886$	−10.8197	−0.363494
$887$	−19.0557	−0.639829	−0.319914	−	0.947446i	$-0.603654\pi$
$887$	−19.0557	−0.639829	−0.319914	+	0.947446i	$0.603654\pi$
$888$	35.7295	1.19900
$889$	−1.38197	−0.0463497
$890$	0	0
$891$	38.1935	1.27953
$892$	−31.3131	−1.04844
$893$	0	0
$894$	−0.450850	−0.0150787
$895$	0	0
$896$	−2.38197	−0.0795759
$897$	14.4721	0.483211
$898$	−0.0770143	−0.00257000
$899$	29.1246	0.971360
$900$	0	0
$901$	−15.9787	−0.532328
$902$	−11.4296	−0.380563
$903$	5.04760	0.167974
$904$	−9.31308	−0.309749
$905$	0	0
$906$	−16.0557	−0.533416
$907$	54.9574	1.82483	0.912416	−	0.409265i	$-0.134215\pi$
$907$	54.9574	1.82483	0.912416	+	0.409265i	$0.134215\pi$
$908$	10.2492	0.340132
$909$	32.0689	1.06366
$910$	0	0
$911$	16.1459	0.534937	0.267469	−	0.963567i	$-0.413813\pi$
$911$	16.1459	0.534937	0.267469	+	0.963567i	$0.413813\pi$
$912$	0	0
$913$	−35.0344	−1.15947
$914$	−5.56231	−0.183985
$915$	0	0
$916$	19.7508	0.652584
$917$	2.83282	0.0935478
$918$	1.58359	0.0522663
$919$	7.50658	0.247619	0.123810	−	0.992306i	$-0.460489\pi$
$919$	7.50658	0.247619	0.123810	+	0.992306i	$0.460489\pi$
$920$	0	0
$921$	−27.8115	−0.916421
$922$	−8.03444	−0.264600
$923$	−26.9443	−0.886882
$924$	3.39823	0.111793
$925$	0	0
$926$	−15.6099	−0.512973
$927$	−6.47214	−0.212573
$928$	−24.8754	−0.816575
$929$	16.5279	0.542262	0.271131	−	0.962543i	$-0.412602\pi$
$929$	16.5279	0.542262	0.271131	+	0.962543i	$0.412602\pi$
$930$	0	0
$931$	0	0
$932$	−47.2279	−1.54700
$933$	−26.3050	−0.861185
$934$	−13.8541	−0.453320
$935$	0	0
$936$	5.88854	0.192473
$937$	−26.8328	−0.876590	−0.438295	−	0.898831i	$-0.644417\pi$
$937$	−26.8328	−0.876590	−0.438295	+	0.898831i	$0.644417\pi$
$938$	−0.424538	−0.0138617
$939$	−4.59675	−0.150009
$940$	0	0
$941$	17.2574	0.562574	0.281287	−	0.959624i	$-0.409239\pi$
$941$	17.2574	0.562574	0.281287	+	0.959624i	$0.409239\pi$
$942$	−20.6525	−0.672894
$943$	27.8885	0.908176
$944$	−22.7639	−0.740903
$945$	0	0
$946$	12.6819	0.412324
$947$	22.6869	0.737226	0.368613	−	0.929583i	$-0.379833\pi$
$947$	22.6869	0.737226	0.368613	+	0.929583i	$0.379833\pi$
$948$	4.14590	0.134653
$949$	−5.41641	−0.175824
$950$	0	0
$951$	−15.5279	−0.503525
$952$	0.644345	0.0208833
$953$	13.4164	0.434600	0.217300	−	0.976105i	$-0.430275\pi$
$953$	13.4164	0.434600	0.217300	+	0.976105i	$0.430275\pi$
$954$	−6.58359	−0.213152
$955$	0	0
$956$	23.5623	0.762059
$957$	46.5836	1.50583
$958$	−5.63119	−0.181935
$959$	−1.12461	−0.0363156
$960$	0	0
$961$	−7.43769	−0.239926
$962$	8.29180	0.267338
$963$	26.2918	0.847241
$964$	24.1033	0.776316
$965$	0	0
$966$	0.652476	0.0209931
$967$	−5.09017	−0.163689	−0.0818444	−	0.996645i	$-0.526081\pi$
$967$	−5.09017	−0.163689	−0.0818444	+	0.996645i	$0.526081\pi$
$968$	1.55418	0.0499531
$969$	0	0
$970$	0	0
$971$	−43.7984	−1.40556	−0.702778	−	0.711409i	$-0.748057\pi$
$971$	−43.7984	−1.40556	−0.702778	+	0.711409i	$0.748057\pi$
$972$	−33.1672	−1.06384
$973$	3.30495	0.105952
$974$	11.5066	0.368695
$975$	0	0
$976$	45.8115	1.46639
$977$	−19.2148	−0.614735	−0.307368	−	0.951591i	$-0.599448\pi$
$977$	−19.2148	−0.614735	−0.307368	+	0.951591i	$0.599448\pi$
$978$	6.18034	0.197625
$979$	33.7082	1.07732
$980$	0	0
$981$	−19.8885	−0.634992
$982$	12.3131	0.392926
$983$	−33.9787	−1.08375	−0.541876	−	0.840458i	$-0.682286\pi$
$983$	−33.9787	−1.08375	−0.541876	+	0.840458i	$0.682286\pi$
$984$	28.3688	0.904365
$985$	0	0
$986$	4.24922	0.135323
$987$	2.84095	0.0904283
$988$	0	0
$989$	−30.9443	−0.983971
$990$	0	0
$991$	11.4164	0.362654	0.181327	−	0.983423i	$-0.441961\pi$
$991$	11.4164	0.362654	0.181327	+	0.983423i	$0.441961\pi$
$992$	−20.1246	−0.638957
$993$	−14.6738	−0.465658
$994$	−1.21478	−0.0385305
$995$	0	0
$996$	41.8328	1.32552
$997$	35.7426	1.13198	0.565990	−	0.824412i	$-0.308494\pi$
$997$	35.7426	1.13198	0.565990	+	0.824412i	$0.308494\pi$
$998$	−14.7426	−0.466670
$999$	−24.2705	−0.767885

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.m.1.2	yes	2
5.4	even	2		9025.2.a.t.1.1	yes	2
19.18	odd	2		9025.2.a.u.1.1	yes	2
95.94	odd	2		9025.2.a.l.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9025.2.a.l.1.2	✓	2	95.94	odd	2
9025.2.a.m.1.2	yes	2	1.1	even	1	trivial
9025.2.a.t.1.1	yes	2	5.4	even	2
9025.2.a.u.1.1	yes	2	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)

\(f(q)\)	\(=\)	\(q-0.381966 q^{2} -2.23607 q^{3} -1.85410 q^{4} +0.854102 q^{6} -0.236068 q^{7} +1.47214 q^{8} +2.00000 q^{9} +O(q^{10})\)	\(q-0.381966 q^{2} -2.23607 q^{3} -1.85410 q^{4} +0.854102 q^{6} -0.236068 q^{7} +1.47214 q^{8} +2.00000 q^{9} -3.47214 q^{11} +4.14590 q^{12} +2.00000 q^{13} +0.0901699 q^{14} +3.14590 q^{16} -1.85410 q^{17} -0.763932 q^{18} +0.527864 q^{21} +1.32624 q^{22} -3.23607 q^{23} -3.29180 q^{24} -0.763932 q^{26} +2.23607 q^{27} +0.437694 q^{28} +6.00000 q^{29} +4.85410 q^{31} -4.14590 q^{32} +7.76393 q^{33} +0.708204 q^{34} -3.70820 q^{36} -10.8541 q^{37} -4.47214 q^{39} -8.61803 q^{41} -0.201626 q^{42} +9.56231 q^{43} +6.43769 q^{44} +1.23607 q^{46} +5.38197 q^{47} -7.03444 q^{48} -6.94427 q^{49} +4.14590 q^{51} -3.70820 q^{52} +8.61803 q^{53} -0.854102 q^{54} -0.347524 q^{56} -2.29180 q^{58} -7.23607 q^{59} +14.5623 q^{61} -1.85410 q^{62} -0.472136 q^{63} -4.70820 q^{64} -2.96556 q^{66} -4.70820 q^{67} +3.43769 q^{68} +7.23607 q^{69} -13.4721 q^{71} +2.94427 q^{72} -2.70820 q^{73} +4.14590 q^{74} +0.819660 q^{77} +1.70820 q^{78} +1.00000 q^{79} -11.0000 q^{81} +3.29180 q^{82} +10.0902 q^{83} -0.978714 q^{84} -3.65248 q^{86} -13.4164 q^{87} -5.11146 q^{88} -9.70820 q^{89} -0.472136 q^{91} +6.00000 q^{92} -10.8541 q^{93} -2.05573 q^{94} +9.27051 q^{96} -9.61803 q^{97} +2.65248 q^{98} -6.94427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} + 3 q^{4} - 5 q^{6} + 4 q^{7} - 6 q^{8} + 4 q^{9}+O(q^{10}) \) 2 * q - 3 * q^2 + 3 * q^4 - 5 * q^6 + 4 * q^7 - 6 * q^8 + 4 * q^9	\( 2 q - 3 q^{2} + 3 q^{4} - 5 q^{6} + 4 q^{7} - 6 q^{8} + 4 q^{9} + 2 q^{11} + 15 q^{12} + 4 q^{13} - 11 q^{14} + 13 q^{16} + 3 q^{17} - 6 q^{18} + 10 q^{21} - 13 q^{22} - 2 q^{23} - 20 q^{24} - 6 q^{26} + 21 q^{28} + 12 q^{29} + 3 q^{31} - 15 q^{32} + 20 q^{33} - 12 q^{34} + 6 q^{36} - 15 q^{37} - 15 q^{41} - 25 q^{42} - q^{43} + 33 q^{44} - 2 q^{46} + 13 q^{47} + 15 q^{48} + 4 q^{49} + 15 q^{51} + 6 q^{52} + 15 q^{53} + 5 q^{54} - 32 q^{56} - 18 q^{58} - 10 q^{59} + 9 q^{61} + 3 q^{62} + 8 q^{63} + 4 q^{64} - 35 q^{66} + 4 q^{67} + 27 q^{68} + 10 q^{69} - 18 q^{71} - 12 q^{72} + 8 q^{73} + 15 q^{74} + 24 q^{77} - 10 q^{78} + 2 q^{79} - 22 q^{81} + 20 q^{82} + 9 q^{83} + 45 q^{84} + 24 q^{86} - 46 q^{88} - 6 q^{89} + 8 q^{91} + 12 q^{92} - 15 q^{93} - 22 q^{94} - 15 q^{96} - 17 q^{97} - 26 q^{98} + 4 q^{99}+O(q^{100}) \) 2 * q - 3 * q^2 + 3 * q^4 - 5 * q^6 + 4 * q^7 - 6 * q^8 + 4 * q^9 + 2 * q^11 + 15 * q^12 + 4 * q^13 - 11 * q^14 + 13 * q^16 + 3 * q^17 - 6 * q^18 + 10 * q^21 - 13 * q^22 - 2 * q^23 - 20 * q^24 - 6 * q^26 + 21 * q^28 + 12 * q^29 + 3 * q^31 - 15 * q^32 + 20 * q^33 - 12 * q^34 + 6 * q^36 - 15 * q^37 - 15 * q^41 - 25 * q^42 - q^43 + 33 * q^44 - 2 * q^46 + 13 * q^47 + 15 * q^48 + 4 * q^49 + 15 * q^51 + 6 * q^52 + 15 * q^53 + 5 * q^54 - 32 * q^56 - 18 * q^58 - 10 * q^59 + 9 * q^61 + 3 * q^62 + 8 * q^63 + 4 * q^64 - 35 * q^66 + 4 * q^67 + 27 * q^68 + 10 * q^69 - 18 * q^71 - 12 * q^72 + 8 * q^73 + 15 * q^74 + 24 * q^77 - 10 * q^78 + 2 * q^79 - 22 * q^81 + 20 * q^82 + 9 * q^83 + 45 * q^84 + 24 * q^86 - 46 * q^88 - 6 * q^89 + 8 * q^91 + 12 * q^92 - 15 * q^93 - 22 * q^94 - 15 * q^96 - 17 * q^97 - 26 * q^98 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more