LMFDB - Embedded newform 3025.2.a.bn.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3025,2,Mod(1,3025)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3025, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3025.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3025 = 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3025.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,5,1,9,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	yes
Analytic conductor:	$24.1547466114$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 7x^{6} + 18x^{5} + 16x^{4} - 30x^{3} - 12x^{2} + 12x + 4 $ x^8 - 3x^7 - 7x^6 + 18x^5 + 16x^4 - 30x^3 - 12x^2 + 12*x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 275)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.321622$ of defining polynomial
Character	$\chi$	$=$	3025.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.32162 q^{2} -2.02642 q^{3} -0.253315 q^{4} -2.67816 q^{6} +0.348258 q^{7} -2.97803 q^{8} +1.10639 q^{9} +0.513323 q^{12} +1.85577 q^{13} +0.460265 q^{14} -3.42920 q^{16} +1.79974 q^{17} +1.46223 q^{18} -7.15813 q^{19} -0.705717 q^{21} -8.41145 q^{23} +6.03475 q^{24} +2.45262 q^{26} +3.83726 q^{27} -0.0882188 q^{28} +5.80070 q^{29} +5.55599 q^{31} +1.42395 q^{32} +2.37858 q^{34} -0.280264 q^{36} -1.84812 q^{37} -9.46035 q^{38} -3.76057 q^{39} -9.03346 q^{41} -0.932691 q^{42} +6.76370 q^{43} -11.1168 q^{46} +7.26198 q^{47} +6.94901 q^{48} -6.87872 q^{49} -3.64704 q^{51} -0.470093 q^{52} -0.342859 q^{53} +5.07141 q^{54} -1.03712 q^{56} +14.5054 q^{57} +7.66633 q^{58} +0.199317 q^{59} -7.79614 q^{61} +7.34292 q^{62} +0.385308 q^{63} +8.74033 q^{64} +12.2451 q^{67} -0.455901 q^{68} +17.0451 q^{69} +9.83305 q^{71} -3.29486 q^{72} -1.01317 q^{73} -2.44251 q^{74} +1.81326 q^{76} -4.97005 q^{78} +6.63314 q^{79} -11.0951 q^{81} -11.9388 q^{82} +11.1253 q^{83} +0.178769 q^{84} +8.93906 q^{86} -11.7547 q^{87} +8.84524 q^{89} +0.646285 q^{91} +2.13074 q^{92} -11.2588 q^{93} +9.59760 q^{94} -2.88553 q^{96} -6.20234 q^{97} -9.09106 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 5 q^{2} + q^{3} + 9 q^{4} + q^{6} + 8 q^{7} + 12 q^{8} + 9 q^{9} + 3 q^{12} + 9 q^{13} + 9 q^{14} + 23 q^{16} + 19 q^{17} + 22 q^{18} + q^{19} + 5 q^{21} + 2 q^{23} + q^{24} - 2 q^{26} - 2 q^{27}+ \cdots + 26 q^{98}+O(q^{100}) $ 8 * q + 5 * q^2 + q^3 + 9 * q^4 + q^6 + 8 * q^7 + 12 * q^8 + 9 * q^9 + 3 * q^12 + 9 * q^13 + 9 * q^14 + 23 * q^16 + 19 * q^17 + 22 * q^18 + q^19 + 5 * q^21 + 2 * q^23 + q^24 - 2 * q^26 - 2 * q^27 + 9 * q^28 + 7 * q^29 - 5 * q^31 + 29 * q^32 + 10 * q^34 - 16 * q^36 - 8 * q^37 - 37 * q^38 - q^39 + 41 * q^42 + 14 * q^43 - 20 * q^46 + 11 * q^47 - 27 * q^48 - 12 * q^49 - 25 * q^51 - 7 * q^52 + 11 * q^53 + 30 * q^54 - 10 * q^56 - 2 * q^57 + 27 * q^58 + 17 * q^59 - 2 * q^61 + 25 * q^62 + 41 * q^63 + 30 * q^64 + 7 * q^67 + 66 * q^68 + 17 * q^71 - 19 * q^72 + 34 * q^73 - 6 * q^74 - 31 * q^76 - 17 * q^78 + 23 * q^79 - 4 * q^81 - 17 * q^82 + 41 * q^83 + 83 * q^84 + q^86 + 25 * q^87 - 11 * q^89 - 7 * q^91 + 33 * q^92 - 59 * q^93 + 50 * q^94 - 61 * q^96 + 2 * q^97 + 26 * q^98

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 $

$2$	1.32162	0.934528	0.467264	−	0.884118i	$-0.345240\pi$
$2$	1.32162	0.934528	0.467264	+	0.884118i	$0.345240\pi$
$3$	−2.02642	−1.16996	−0.584978	−	0.811049i	$-0.698897\pi$
$3$	−2.02642	−1.16996	−0.584978	+	0.811049i	$0.698897\pi$
$4$	−0.253315	−0.126657
$5$	0	0
$5$	0	0
$6$	−2.67816	−1.09336
$7$	0.348258	0.131629	0.0658145	−	0.997832i	$-0.479035\pi$
$7$	0.348258	0.131629	0.0658145	+	0.997832i	$0.479035\pi$
$8$	−2.97803	−1.05289
$9$	1.10639	0.368796
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0.513323	0.148183
$13$	1.85577	0.514697	0.257349	−	0.966319i	$-0.417151\pi$
$13$	1.85577	0.514697	0.257349	+	0.966319i	$0.417151\pi$
$14$	0.460265	0.123011
$15$	0	0
$16$	−3.42920	−0.857301
$17$	1.79974	0.436502	0.218251	−	0.975893i	$-0.429965\pi$
$17$	1.79974	0.436502	0.218251	+	0.975893i	$0.429965\pi$
$18$	1.46223	0.344650
$19$	−7.15813	−1.64219	−0.821094	−	0.570793i	$-0.806636\pi$
$19$	−7.15813	−1.64219	−0.821094	+	0.570793i	$0.806636\pi$
$20$	0	0
$21$	−0.705717	−0.154000
$22$	0	0
$23$	−8.41145	−1.75391	−0.876954	−	0.480574i	$-0.840428\pi$
$23$	−8.41145	−1.75391	−0.876954	+	0.480574i	$0.840428\pi$
$24$	6.03475	1.23184
$25$	0	0
$26$	2.45262	0.480999
$27$	3.83726	0.738481
$28$	−0.0882188	−0.0166718
$29$	5.80070	1.07716	0.538581	−	0.842573i	$-0.318960\pi$
$29$	5.80070	1.07716	0.538581	+	0.842573i	$0.318960\pi$
$30$	0	0
$31$	5.55599	0.997886	0.498943	−	0.866635i	$-0.333722\pi$
$31$	5.55599	0.997886	0.498943	+	0.866635i	$0.333722\pi$
$32$	1.42395	0.251722
$33$	0	0
$34$	2.37858	0.407923
$35$	0	0
$36$	−0.280264	−0.0467107
$37$	−1.84812	−0.303828	−0.151914	−	0.988394i	$-0.548544\pi$
$37$	−1.84812	−0.303828	−0.151914	+	0.988394i	$0.548544\pi$
$38$	−9.46035	−1.53467
$39$	−3.76057	−0.602173
$40$	0	0
$41$	−9.03346	−1.41079	−0.705394	−	0.708815i	$-0.749230\pi$
$41$	−9.03346	−1.41079	−0.705394	+	0.708815i	$0.749230\pi$
$42$	−0.932691	−0.143917
$43$	6.76370	1.03145	0.515727	−	0.856753i	$-0.327522\pi$
$43$	6.76370	1.03145	0.515727	+	0.856753i	$0.327522\pi$
$44$	0	0
$45$	0	0
$46$	−11.1168	−1.63908
$47$	7.26198	1.05927	0.529634	−	0.848226i	$-0.322329\pi$
$47$	7.26198	1.05927	0.529634	+	0.848226i	$0.322329\pi$
$48$	6.94901	1.00300
$49$	−6.87872	−0.982674
$50$	0	0
$51$	−3.64704	−0.510688
$52$	−0.470093	−0.0651902
$53$	−0.342859	−0.0470953	−0.0235476	−	0.999723i	$-0.507496\pi$
$53$	−0.342859	−0.0470953	−0.0235476	+	0.999723i	$0.507496\pi$
$54$	5.07141	0.690131
$55$	0	0
$56$	−1.03712	−0.138591
$57$	14.5054	1.92129
$58$	7.66633	1.00664
$59$	0.199317	0.0259488	0.0129744	−	0.999916i	$-0.495870\pi$
$59$	0.199317	0.0259488	0.0129744	+	0.999916i	$0.495870\pi$
$60$	0	0
$61$	−7.79614	−0.998193	−0.499097	−	0.866546i	$-0.666335\pi$
$61$	−7.79614	−0.998193	−0.499097	+	0.866546i	$0.666335\pi$
$62$	7.34292	0.932552
$63$	0.385308	0.0485442
$64$	8.74033	1.09254
$65$	0	0
$66$	0	0
$67$	12.2451	1.49597	0.747986	−	0.663715i	$-0.231021\pi$
$67$	12.2451	1.49597	0.747986	+	0.663715i	$0.231021\pi$
$68$	−0.455901	−0.0552862
$69$	17.0451	2.05199
$70$	0	0
$71$	9.83305	1.16697	0.583484	−	0.812125i	$-0.301689\pi$
$71$	9.83305	1.16697	0.583484	+	0.812125i	$0.301689\pi$
$72$	−3.29486	−0.388302
$73$	−1.01317	−0.118582	−0.0592910	−	0.998241i	$-0.518884\pi$
$73$	−1.01317	−0.118582	−0.0592910	+	0.998241i	$0.518884\pi$
$74$	−2.44251	−0.283936
$75$	0	0
$76$	1.81326	0.207995
$77$	0	0
$78$	−4.97005	−0.562748
$79$	6.63314	0.746286	0.373143	−	0.927774i	$-0.378280\pi$
$79$	6.63314	0.746286	0.373143	+	0.927774i	$0.378280\pi$
$80$	0	0
$81$	−11.0951	−1.23279
$82$	−11.9388	−1.31842
$83$	11.1253	1.22116	0.610582	−	0.791953i	$-0.290936\pi$
$83$	11.1253	1.22116	0.610582	+	0.791953i	$0.290936\pi$
$84$	0.178769	0.0195052
$85$	0	0
$86$	8.93906	0.963923
$87$	−11.7547	−1.26023
$88$	0	0
$89$	8.84524	0.937594	0.468797	−	0.883306i	$-0.344688\pi$
$89$	8.84524	0.937594	0.468797	+	0.883306i	$0.344688\pi$
$90$	0	0
$91$	0.646285	0.0677491
$92$	2.13074	0.222145
$93$	−11.2588	−1.16748
$94$	9.59760	0.989917
$95$	0	0
$96$	−2.88553	−0.294503
$97$	−6.20234	−0.629752	−0.314876	−	0.949133i	$-0.601963\pi$
$97$	−6.20234	−0.629752	−0.314876	+	0.949133i	$0.601963\pi$
$98$	−9.09106	−0.918336
$99$	0	0
$100$	0	0
$101$	−3.66784	−0.364963	−0.182482	−	0.983209i	$-0.558413\pi$
$101$	−3.66784	−0.364963	−0.182482	+	0.983209i	$0.558413\pi$
$102$	−4.82001	−0.477252
$103$	19.6820	1.93932	0.969661	−	0.244454i	$-0.0786087\pi$
$103$	19.6820	1.93932	0.969661	+	0.244454i	$0.0786087\pi$
$104$	−5.52653	−0.541921
$105$	0	0
$106$	−0.453130	−0.0440118
$107$	0.301384	0.0291359	0.0145679	−	0.999894i	$-0.495363\pi$
$107$	0.301384	0.0291359	0.0145679	+	0.999894i	$0.495363\pi$
$108$	−0.972034	−0.0935341
$109$	−10.9371	−1.04759	−0.523794	−	0.851845i	$-0.675484\pi$
$109$	−10.9371	−1.04759	−0.523794	+	0.851845i	$0.675484\pi$
$110$	0	0
$111$	3.74506	0.355466
$112$	−1.19425	−0.112846
$113$	5.80228	0.545833	0.272916	−	0.962038i	$-0.412012\pi$
$113$	5.80228	0.545833	0.272916	+	0.962038i	$0.412012\pi$
$114$	19.1707	1.79550
$115$	0	0
$116$	−1.46940	−0.136431
$117$	2.05320	0.189818
$118$	0.263422	0.0242499
$119$	0.626774	0.0574563
$120$	0	0
$121$	0	0
$122$	−10.3035	−0.932839
$123$	18.3056	1.65056
$124$	−1.40742	−0.126390
$125$	0	0
$126$	0.509231	0.0453659
$127$	12.3505	1.09593	0.547964	−	0.836502i	$-0.315403\pi$
$127$	12.3505	1.09593	0.547964	+	0.836502i	$0.315403\pi$
$128$	8.70351	0.769289
$129$	−13.7061	−1.20676
$130$	0	0
$131$	7.81632	0.682915	0.341458	−	0.939897i	$-0.389079\pi$
$131$	7.81632	0.682915	0.341458	+	0.939897i	$0.389079\pi$
$132$	0	0
$133$	−2.49287	−0.216160
$134$	16.1833	1.39803
$135$	0	0
$136$	−5.35969	−0.459590
$137$	9.58073	0.818537	0.409269	−	0.912414i	$-0.365784\pi$
$137$	9.58073	0.818537	0.409269	+	0.912414i	$0.365784\pi$
$138$	22.5272	1.91765
$139$	19.1306	1.62264	0.811319	−	0.584604i	$-0.198750\pi$
$139$	19.1306	1.62264	0.811319	+	0.584604i	$0.198750\pi$
$140$	0	0
$141$	−14.7158	−1.23930
$142$	12.9956	1.09056
$143$	0	0
$144$	−3.79403	−0.316169
$145$	0	0
$146$	−1.33902	−0.110818
$147$	13.9392	1.14968
$148$	0.468155	0.0384821
$149$	5.77501	0.473107	0.236554	−	0.971618i	$-0.423982\pi$
$149$	5.77501	0.473107	0.236554	+	0.971618i	$0.423982\pi$
$150$	0	0
$151$	13.3681	1.08788	0.543942	−	0.839123i	$-0.316931\pi$
$151$	13.3681	1.08788	0.543942	+	0.839123i	$0.316931\pi$
$152$	21.3171	1.72905
$153$	1.99121	0.160980
$154$	0	0
$155$	0	0
$156$	0.952608	0.0762697
$157$	13.4451	1.07303	0.536516	−	0.843890i	$-0.319740\pi$
$157$	13.4451	1.07303	0.536516	+	0.843890i	$0.319740\pi$
$158$	8.76650	0.697425
$159$	0.694776	0.0550994
$160$	0	0
$161$	−2.92935	−0.230865
$162$	−14.6635	−1.15207
$163$	−3.13538	−0.245582	−0.122791	−	0.992433i	$-0.539184\pi$
$163$	−3.13538	−0.245582	−0.122791	+	0.992433i	$0.539184\pi$
$164$	2.28831	0.178687
$165$	0	0
$166$	14.7035	1.14121
$167$	−2.62877	−0.203420	−0.101710	−	0.994814i	$-0.532431\pi$
$167$	−2.62877	−0.203420	−0.101710	+	0.994814i	$0.532431\pi$
$168$	2.10165	0.162146
$169$	−9.55613	−0.735087
$170$	0	0
$171$	−7.91967	−0.605632
$172$	−1.71335	−0.130641
$173$	3.92271	0.298238	0.149119	−	0.988819i	$-0.452356\pi$
$173$	3.92271	0.298238	0.149119	+	0.988819i	$0.452356\pi$
$174$	−15.5352	−1.17772
$175$	0	0
$176$	0	0
$177$	−0.403900	−0.0303590
$178$	11.6901	0.876207
$179$	−3.31488	−0.247766	−0.123883	−	0.992297i	$-0.539535\pi$
$179$	−3.31488	−0.247766	−0.123883	+	0.992297i	$0.539535\pi$
$180$	0	0
$181$	16.9048	1.25652	0.628262	−	0.778002i	$-0.283767\pi$
$181$	16.9048	1.25652	0.628262	+	0.778002i	$0.283767\pi$
$182$	0.854145	0.0633134
$183$	15.7983	1.16784
$184$	25.0495	1.84668
$185$	0	0
$186$	−14.8799	−1.09104
$187$	0	0
$188$	−1.83957	−0.134164
$189$	1.33635	0.0972055
$190$	0	0
$191$	−21.0157	−1.52064	−0.760320	−	0.649549i	$-0.774958\pi$
$191$	−21.0157	−1.52064	−0.760320	+	0.649549i	$0.774958\pi$
$192$	−17.7116	−1.27822
$193$	−3.31329	−0.238496	−0.119248	−	0.992864i	$-0.538048\pi$
$193$	−3.31329	−0.238496	−0.119248	+	0.992864i	$0.538048\pi$
$194$	−8.19715	−0.588521
$195$	0	0
$196$	1.74248	0.124463
$197$	6.97218	0.496747	0.248374	−	0.968664i	$-0.420104\pi$
$197$	6.97218	0.496747	0.248374	+	0.968664i	$0.420104\pi$
$198$	0	0
$199$	−19.2184	−1.36236	−0.681178	−	0.732118i	$-0.738532\pi$
$199$	−19.2184	−1.36236	−0.681178	+	0.732118i	$0.738532\pi$
$200$	0	0
$201$	−24.8136	−1.75022
$202$	−4.84749	−0.341068
$203$	2.02014	0.141786
$204$	0.923849	0.0646824
$205$	0	0
$206$	26.0121	1.81235
$207$	−9.30632	−0.646834
$208$	−6.36380	−0.441250
$209$	0	0
$210$	0	0
$211$	4.29419	0.295624	0.147812	−	0.989015i	$-0.452777\pi$
$211$	4.29419	0.295624	0.147812	+	0.989015i	$0.452777\pi$
$212$	0.0868512	0.00596496
$213$	−19.9259	−1.36530
$214$	0.398315	0.0272283
$215$	0	0
$216$	−11.4275	−0.777541
$217$	1.93492	0.131351
$218$	−14.4548	−0.979000
$219$	2.05310	0.138736
$220$	0	0
$221$	3.33991	0.224666
$222$	4.94956	0.332193
$223$	−10.5884	−0.709053	−0.354526	−	0.935046i	$-0.615358\pi$
$223$	−10.5884	−0.709053	−0.354526	+	0.935046i	$0.615358\pi$
$224$	0.495902	0.0331339
$225$	0	0
$226$	7.66843	0.510096
$227$	5.77505	0.383303	0.191652	−	0.981463i	$-0.438616\pi$
$227$	5.77505	0.383303	0.191652	+	0.981463i	$0.438616\pi$
$228$	−3.67443	−0.243345
$229$	−5.10650	−0.337447	−0.168723	−	0.985663i	$-0.553964\pi$
$229$	−5.10650	−0.337447	−0.168723	+	0.985663i	$0.553964\pi$
$230$	0	0
$231$	0	0
$232$	−17.2747	−1.13414
$233$	27.7861	1.82033	0.910164	−	0.414248i	$-0.135955\pi$
$233$	27.7861	1.82033	0.910164	+	0.414248i	$0.135955\pi$
$234$	2.71355	0.177390
$235$	0	0
$236$	−0.0504899	−0.00328661
$237$	−13.4415	−0.873122
$238$	0.828359	0.0536945
$239$	−6.74725	−0.436444	−0.218222	−	0.975899i	$-0.570026\pi$
$239$	−6.74725	−0.436444	−0.218222	+	0.975899i	$0.570026\pi$
$240$	0	0
$241$	−13.0610	−0.841331	−0.420666	−	0.907216i	$-0.638203\pi$
$241$	−13.0610	−0.841331	−0.420666	+	0.907216i	$0.638203\pi$
$242$	0	0
$243$	10.9715	0.703823
$244$	1.97488	0.126429
$245$	0	0
$246$	24.1931	1.54249
$247$	−13.2838	−0.845230
$248$	−16.5459	−1.05067
$249$	−22.5446	−1.42871
$250$	0	0
$251$	−30.1456	−1.90277	−0.951386	−	0.308000i	$-0.900340\pi$
$251$	−30.1456	−1.90277	−0.951386	+	0.308000i	$0.900340\pi$
$252$	−0.0976041	−0.00614848
$253$	0	0
$254$	16.3227	1.02418
$255$	0	0
$256$	−5.97791	−0.373619
$257$	−0.362964	−0.0226411	−0.0113205	−	0.999936i	$-0.503604\pi$
$257$	−0.362964	−0.0226411	−0.0113205	+	0.999936i	$0.503604\pi$
$258$	−18.1143	−1.12775
$259$	−0.643620	−0.0399926
$260$	0	0
$261$	6.41782	0.397253
$262$	10.3302	0.638203
$263$	22.5498	1.39048	0.695239	−	0.718779i	$-0.255299\pi$
$263$	22.5498	1.39048	0.695239	+	0.718779i	$0.255299\pi$
$264$	0	0
$265$	0	0
$266$	−3.29464	−0.202007
$267$	−17.9242	−1.09694
$268$	−3.10185	−0.189476
$269$	−6.14725	−0.374804	−0.187402	−	0.982283i	$-0.560007\pi$
$269$	−6.14725	−0.374804	−0.187402	+	0.982283i	$0.560007\pi$
$270$	0	0
$271$	16.6961	1.01422	0.507109	−	0.861882i	$-0.330714\pi$
$271$	16.6961	1.01422	0.507109	+	0.861882i	$0.330714\pi$
$272$	−6.17168	−0.374213
$273$	−1.30965	−0.0792634
$274$	12.6621	0.764946
$275$	0	0
$276$	−4.31779	−0.259900
$277$	−22.3299	−1.34167	−0.670835	−	0.741606i	$-0.734064\pi$
$277$	−22.3299	−1.34167	−0.670835	+	0.741606i	$0.734064\pi$
$278$	25.2835	1.51640
$279$	6.14708	0.368016
$280$	0	0
$281$	23.8551	1.42308	0.711538	−	0.702648i	$-0.247999\pi$
$281$	23.8551	1.42308	0.711538	+	0.702648i	$0.247999\pi$
$282$	−19.4488	−1.15816
$283$	0.145890	0.00867224	0.00433612	−	0.999991i	$-0.498620\pi$
$283$	0.145890	0.00867224	0.00433612	+	0.999991i	$0.498620\pi$
$284$	−2.49086	−0.147805
$285$	0	0
$286$	0	0
$287$	−3.14597	−0.185701
$288$	1.57544	0.0928338
$289$	−13.7609	−0.809466
$290$	0	0
$291$	12.5686	0.736782
$292$	0.256650	0.0150193
$293$	−7.67480	−0.448367	−0.224183	−	0.974547i	$-0.571971\pi$
$293$	−7.67480	−0.448367	−0.224183	+	0.974547i	$0.571971\pi$
$294$	18.4223	1.07441
$295$	0	0
$296$	5.50374	0.319899
$297$	0	0
$298$	7.63238	0.442132
$299$	−15.6097	−0.902732
$300$	0	0
$301$	2.35551	0.135769
$302$	17.6676	1.01666
$303$	7.43258	0.426991
$304$	24.5467	1.40785
$305$	0	0
$306$	2.63163	0.150440
$307$	−9.69537	−0.553344	−0.276672	−	0.960964i	$-0.589232\pi$
$307$	−9.69537	−0.553344	−0.276672	+	0.960964i	$0.589232\pi$
$308$	0	0
$309$	−39.8840	−2.26892
$310$	0	0
$311$	25.1584	1.42660	0.713300	−	0.700859i	$-0.247200\pi$
$311$	25.1584	1.42660	0.713300	+	0.700859i	$0.247200\pi$
$312$	11.1991	0.634024
$313$	10.8737	0.614616	0.307308	−	0.951610i	$-0.400572\pi$
$313$	10.8737	0.614616	0.307308	+	0.951610i	$0.400572\pi$
$314$	17.7693	1.00278
$315$	0	0
$316$	−1.68027	−0.0945227
$317$	−25.4344	−1.42854	−0.714268	−	0.699872i	$-0.753240\pi$
$317$	−25.4344	−1.42854	−0.714268	+	0.699872i	$0.753240\pi$
$318$	0.918232	0.0514919
$319$	0	0
$320$	0	0
$321$	−0.610730	−0.0340876
$322$	−3.87149	−0.215750
$323$	−12.8828	−0.716818
$324$	2.81054	0.156141
$325$	0	0
$326$	−4.14379	−0.229503
$327$	22.1632	1.22563
$328$	26.9019	1.48541
$329$	2.52904	0.139431
$330$	0	0
$331$	−10.3091	−0.566638	−0.283319	−	0.959026i	$-0.591435\pi$
$331$	−10.3091	−0.566638	−0.283319	+	0.959026i	$0.591435\pi$
$332$	−2.81821	−0.154669
$333$	−2.04473	−0.112051
$334$	−3.47424	−0.190102
$335$	0	0
$336$	2.42005	0.132024
$337$	−2.30404	−0.125509	−0.0627544	−	0.998029i	$-0.519988\pi$
$337$	−2.30404	−0.125509	−0.0627544	+	0.998029i	$0.519988\pi$
$338$	−12.6296	−0.686959
$339$	−11.7579	−0.638600
$340$	0	0
$341$	0	0
$342$	−10.4668	−0.565980
$343$	−4.83337	−0.260977
$344$	−20.1425	−1.08601
$345$	0	0
$346$	5.18434	0.278712
$347$	20.7387	1.11331	0.556655	−	0.830744i	$-0.312085\pi$
$347$	20.7387	1.11331	0.556655	+	0.830744i	$0.312085\pi$
$348$	2.97763	0.159618
$349$	3.71251	0.198726	0.0993630	−	0.995051i	$-0.468320\pi$
$349$	3.71251	0.198726	0.0993630	+	0.995051i	$0.468320\pi$
$350$	0	0
$351$	7.12106	0.380094
$352$	0	0
$353$	3.01441	0.160441	0.0802203	−	0.996777i	$-0.474438\pi$
$353$	3.01441	0.160441	0.0802203	+	0.996777i	$0.474438\pi$
$354$	−0.533803	−0.0283713
$355$	0	0
$356$	−2.24063	−0.118753
$357$	−1.27011	−0.0672213
$358$	−4.38102	−0.231544
$359$	−22.7391	−1.20013	−0.600063	−	0.799953i	$-0.704858\pi$
$359$	−22.7391	−1.20013	−0.600063	+	0.799953i	$0.704858\pi$
$360$	0	0
$361$	32.2389	1.69678
$362$	22.3418	1.17426
$363$	0	0
$364$	−0.163714	−0.00858092
$365$	0	0
$366$	20.8793	1.09138
$367$	−13.5435	−0.706964	−0.353482	−	0.935441i	$-0.615002\pi$
$367$	−13.5435	−0.706964	−0.353482	+	0.935441i	$0.615002\pi$
$368$	28.8446	1.50363
$369$	−9.99450	−0.520293
$370$	0	0
$371$	−0.119403	−0.00619910
$372$	2.85202	0.147870
$373$	−24.8281	−1.28555	−0.642774	−	0.766056i	$-0.722217\pi$
$373$	−24.8281	−1.28555	−0.642774	+	0.766056i	$0.722217\pi$
$374$	0	0
$375$	0	0
$376$	−21.6264	−1.11530
$377$	10.7648	0.554413
$378$	1.76616	0.0908413
$379$	6.16065	0.316451	0.158226	−	0.987403i	$-0.449423\pi$
$379$	6.16065	0.316451	0.158226	+	0.987403i	$0.449423\pi$
$380$	0	0
$381$	−25.0273	−1.28219
$382$	−27.7748	−1.42108
$383$	0.363669	0.0185826	0.00929131	−	0.999957i	$-0.497042\pi$
$383$	0.363669	0.0185826	0.00929131	+	0.999957i	$0.497042\pi$
$384$	−17.6370	−0.900034
$385$	0	0
$386$	−4.37892	−0.222881
$387$	7.48327	0.380396
$388$	1.57114	0.0797628
$389$	−1.86786	−0.0947044	−0.0473522	−	0.998878i	$-0.515078\pi$
$389$	−1.86786	−0.0947044	−0.0473522	+	0.998878i	$0.515078\pi$
$390$	0	0
$391$	−15.1384	−0.765584
$392$	20.4850	1.03465
$393$	−15.8392	−0.798980
$394$	9.21459	0.464224
$395$	0	0
$396$	0	0
$397$	0.384172	0.0192811	0.00964053	−	0.999954i	$-0.496931\pi$
$397$	0.384172	0.0192811	0.00964053	+	0.999954i	$0.496931\pi$
$398$	−25.3995	−1.27316
$399$	5.05162	0.252897
$400$	0	0
$401$	−22.6534	−1.13126	−0.565628	−	0.824660i	$-0.691366\pi$
$401$	−22.6534	−1.13126	−0.565628	+	0.824660i	$0.691366\pi$
$402$	−32.7943	−1.63563
$403$	10.3106	0.513609
$404$	0.929117	0.0462253
$405$	0	0
$406$	2.66986	0.132503
$407$	0	0
$408$	10.8610	0.537699
$409$	−29.2142	−1.44455	−0.722275	−	0.691606i	$-0.756903\pi$
$409$	−29.2142	−1.44455	−0.722275	+	0.691606i	$0.756903\pi$
$410$	0	0
$411$	−19.4146	−0.957652
$412$	−4.98573	−0.245629
$413$	0.0694136	0.00341562
$414$	−12.2994	−0.604484
$415$	0	0
$416$	2.64252	0.129560
$417$	−38.7667	−1.89841
$418$	0	0
$419$	−0.720765	−0.0352117	−0.0176058	−	0.999845i	$-0.505604\pi$
$419$	−0.720765	−0.0352117	−0.0176058	+	0.999845i	$0.505604\pi$
$420$	0	0
$421$	−23.4569	−1.14322	−0.571610	−	0.820525i	$-0.693681\pi$
$421$	−23.4569	−1.14322	−0.571610	+	0.820525i	$0.693681\pi$
$422$	5.67530	0.276269
$423$	8.03456	0.390654
$424$	1.02104	0.0495863
$425$	0	0
$426$	−26.3345	−1.27591
$427$	−2.71506	−0.131391
$428$	−0.0763449	−0.00369027
$429$	0	0
$430$	0	0
$431$	3.75868	0.181049	0.0905246	−	0.995894i	$-0.471146\pi$
$431$	3.75868	0.181049	0.0905246	+	0.995894i	$0.471146\pi$
$432$	−13.1587	−0.633100
$433$	13.6069	0.653906	0.326953	−	0.945041i	$-0.393978\pi$
$433$	13.6069	0.653906	0.326953	+	0.945041i	$0.393978\pi$
$434$	2.55723	0.122751
$435$	0	0
$436$	2.77054	0.132685
$437$	60.2103	2.88025
$438$	2.71342	0.129652
$439$	15.6217	0.745584	0.372792	−	0.927915i	$-0.378400\pi$
$439$	15.6217	0.745584	0.372792	+	0.927915i	$0.378400\pi$
$440$	0	0
$441$	−7.61052	−0.362406
$442$	4.41409	0.209957
$443$	1.41582	0.0672676	0.0336338	−	0.999434i	$-0.489292\pi$
$443$	1.41582	0.0672676	0.0336338	+	0.999434i	$0.489292\pi$
$444$	−0.948679	−0.0450223
$445$	0	0
$446$	−13.9939	−0.662630
$447$	−11.7026	−0.553514
$448$	3.04389	0.143810
$449$	6.90886	0.326049	0.163025	−	0.986622i	$-0.447875\pi$
$449$	6.90886	0.326049	0.163025	+	0.986622i	$0.447875\pi$
$450$	0	0
$451$	0	0
$452$	−1.46980	−0.0691338
$453$	−27.0895	−1.27278
$454$	7.63243	0.358208
$455$	0	0
$456$	−43.1975	−2.02291
$457$	42.3537	1.98122	0.990612	−	0.136706i	$-0.0436516\pi$
$457$	42.3537	1.98122	0.990612	+	0.136706i	$0.0436516\pi$
$458$	−6.74886	−0.315353
$459$	6.90608	0.322348
$460$	0	0
$461$	25.1563	1.17165	0.585823	−	0.810439i	$-0.300772\pi$
$461$	25.1563	1.17165	0.585823	+	0.810439i	$0.300772\pi$
$462$	0	0
$463$	−7.89251	−0.366796	−0.183398	−	0.983039i	$-0.558710\pi$
$463$	−7.89251	−0.366796	−0.183398	+	0.983039i	$0.558710\pi$
$464$	−19.8918	−0.923452
$465$	0	0
$466$	36.7227	1.70115
$467$	15.9711	0.739057	0.369528	−	0.929219i	$-0.379519\pi$
$467$	15.9711	0.739057	0.369528	+	0.929219i	$0.379519\pi$
$468$	−0.520105	−0.0240419
$469$	4.26443	0.196913
$470$	0	0
$471$	−27.2454	−1.25540
$472$	−0.593572	−0.0273214
$473$	0	0
$474$	−17.7646	−0.815957
$475$	0	0
$476$	−0.158771	−0.00727726
$477$	−0.379334	−0.0173685
$478$	−8.91732	−0.407869
$479$	14.3625	0.656241	0.328121	−	0.944636i	$-0.393585\pi$
$479$	14.3625	0.656241	0.328121	+	0.944636i	$0.393585\pi$
$480$	0	0
$481$	−3.42967	−0.156380
$482$	−17.2617	−0.786248
$483$	5.93610	0.270102
$484$	0	0
$485$	0	0
$486$	14.5002	0.657742
$487$	−35.9315	−1.62821	−0.814105	−	0.580718i	$-0.802772\pi$
$487$	−35.9315	−1.62821	−0.814105	+	0.580718i	$0.802772\pi$
$488$	23.2171	1.05099
$489$	6.35360	0.287320
$490$	0	0
$491$	35.2176	1.58935	0.794673	−	0.607038i	$-0.207642\pi$
$491$	35.2176	1.58935	0.794673	+	0.607038i	$0.207642\pi$
$492$	−4.63708	−0.209056
$493$	10.4398	0.470183
$494$	−17.5562	−0.789891
$495$	0	0
$496$	−19.0526	−0.855488
$497$	3.42443	0.153607
$498$	−29.7955	−1.33517
$499$	26.4922	1.18595	0.592977	−	0.805219i	$-0.297952\pi$
$499$	26.4922	1.18595	0.592977	+	0.805219i	$0.297952\pi$
$500$	0	0
$501$	5.32699	0.237992
$502$	−39.8411	−1.77819
$503$	17.7436	0.791146	0.395573	−	0.918435i	$-0.370546\pi$
$503$	17.7436	0.791146	0.395573	+	0.918435i	$0.370546\pi$
$504$	−1.14746	−0.0511119
$505$	0	0
$506$	0	0
$507$	19.3647	0.860019
$508$	−3.12856	−0.138807
$509$	−13.7539	−0.609630	−0.304815	−	0.952412i	$-0.598595\pi$
$509$	−13.7539	−0.609630	−0.304815	+	0.952412i	$0.598595\pi$
$510$	0	0
$511$	−0.352843	−0.0156088
$512$	−25.3076	−1.11845
$513$	−27.4676	−1.21272
$514$	−0.479701	−0.0211587
$515$	0	0
$516$	3.47196	0.152845
$517$	0	0
$518$	−0.850623	−0.0373742
$519$	−7.94907	−0.348925
$520$	0	0
$521$	12.5792	0.551106	0.275553	−	0.961286i	$-0.411139\pi$
$521$	12.5792	0.551106	0.275553	+	0.961286i	$0.411139\pi$
$522$	8.48193	0.371244
$523$	−40.7967	−1.78392	−0.891958	−	0.452118i	$-0.850669\pi$
$523$	−40.7967	−1.78392	−0.891958	+	0.452118i	$0.850669\pi$
$524$	−1.97999	−0.0864962
$525$	0	0
$526$	29.8023	1.29944
$527$	9.99936	0.435579
$528$	0	0
$529$	47.7524	2.07619
$530$	0	0
$531$	0.220522	0.00956982
$532$	0.631482	0.0273782
$533$	−16.7640	−0.726129
$534$	−23.6890	−1.02512
$535$	0	0
$536$	−36.4661	−1.57510
$537$	6.71735	0.289875
$538$	−8.12434	−0.350265
$539$	0	0
$540$	0	0
$541$	32.6165	1.40229	0.701146	−	0.713018i	$-0.252672\pi$
$541$	32.6165	1.40229	0.701146	+	0.713018i	$0.252672\pi$
$542$	22.0660	0.947815
$543$	−34.2563	−1.47008
$544$	2.56275	0.109877
$545$	0	0
$546$	−1.73086	−0.0740739
$547$	5.72986	0.244991	0.122496	−	0.992469i	$-0.460910\pi$
$547$	5.72986	0.244991	0.122496	+	0.992469i	$0.460910\pi$
$548$	−2.42694	−0.103674
$549$	−8.62555	−0.368129
$550$	0	0
$551$	−41.5222	−1.76890
$552$	−50.7610	−2.16053
$553$	2.31004	0.0982329
$554$	−29.5116	−1.25383
$555$	0	0
$556$	−4.84607	−0.205519
$557$	40.6304	1.72157	0.860783	−	0.508971i	$-0.169974\pi$
$557$	40.6304	1.72157	0.860783	+	0.508971i	$0.169974\pi$
$558$	8.12412	0.343921
$559$	12.5519	0.530887
$560$	0	0
$561$	0	0
$562$	31.5274	1.32990
$563$	−14.6876	−0.619009	−0.309504	−	0.950898i	$-0.600163\pi$
$563$	−14.6876	−0.619009	−0.309504	+	0.950898i	$0.600163\pi$
$564$	3.72774	0.156966
$565$	0	0
$566$	0.192811	0.00810446
$567$	−3.86394	−0.162270
$568$	−29.2831	−1.22869
$569$	−24.5949	−1.03107	−0.515536	−	0.856868i	$-0.672407\pi$
$569$	−24.5949	−1.03107	−0.515536	+	0.856868i	$0.672407\pi$
$570$	0	0
$571$	−22.4923	−0.941273	−0.470637	−	0.882327i	$-0.655976\pi$
$571$	−22.4923	−0.941273	−0.470637	+	0.882327i	$0.655976\pi$
$572$	0	0
$573$	42.5866	1.77908
$574$	−4.15778	−0.173543
$575$	0	0
$576$	9.67019	0.402925
$577$	37.5334	1.56254	0.781268	−	0.624196i	$-0.214573\pi$
$577$	37.5334	1.56254	0.781268	+	0.624196i	$0.214573\pi$
$578$	−18.1867	−0.756469
$579$	6.71413	0.279030
$580$	0	0
$581$	3.87448	0.160741
$582$	16.6109	0.688543
$583$	0	0
$584$	3.01724	0.124854
$585$	0	0
$586$	−10.1432	−0.419011
$587$	−28.0102	−1.15611	−0.578053	−	0.816000i	$-0.696187\pi$
$587$	−28.0102	−1.15611	−0.578053	+	0.816000i	$0.696187\pi$
$588$	−3.53100	−0.145616
$589$	−39.7705	−1.63872
$590$	0	0
$591$	−14.1286	−0.581172
$592$	6.33756	0.260472
$593$	−17.7522	−0.728996	−0.364498	−	0.931204i	$-0.618759\pi$
$593$	−17.7522	−0.728996	−0.364498	+	0.931204i	$0.618759\pi$
$594$	0	0
$595$	0	0
$596$	−1.46289	−0.0599225
$597$	38.9446	1.59390
$598$	−20.6301	−0.843628
$599$	−24.4240	−0.997937	−0.498968	−	0.866620i	$-0.666288\pi$
$599$	−24.4240	−0.997937	−0.498968	+	0.866620i	$0.666288\pi$
$600$	0	0
$601$	−4.35978	−0.177839	−0.0889195	−	0.996039i	$-0.528341\pi$
$601$	−4.35978	−0.177839	−0.0889195	+	0.996039i	$0.528341\pi$
$602$	3.11309	0.126880
$603$	13.5478	0.551708
$604$	−3.38635	−0.137789
$605$	0	0
$606$	9.82307	0.399035
$607$	39.6180	1.60804	0.804022	−	0.594599i	$-0.202689\pi$
$607$	39.6180	1.60804	0.804022	+	0.594599i	$0.202689\pi$
$608$	−10.1928	−0.413374
$609$	−4.09365	−0.165883
$610$	0	0
$611$	13.4766	0.545203
$612$	−0.504404	−0.0203893
$613$	−2.60921	−0.105385	−0.0526926	−	0.998611i	$-0.516780\pi$
$613$	−2.60921	−0.105385	−0.0526926	+	0.998611i	$0.516780\pi$
$614$	−12.8136	−0.517115
$615$	0	0
$616$	0	0
$617$	−3.53110	−0.142157	−0.0710783	−	0.997471i	$-0.522644\pi$
$617$	−3.53110	−0.142157	−0.0710783	+	0.997471i	$0.522644\pi$
$618$	−52.7115	−2.12037
$619$	−6.03555	−0.242589	−0.121295	−	0.992617i	$-0.538705\pi$
$619$	−6.03555	−0.242589	−0.121295	+	0.992617i	$0.538705\pi$
$620$	0	0
$621$	−32.2769	−1.29523
$622$	33.2498	1.33320
$623$	3.08042	0.123415
$624$	12.8958	0.516243
$625$	0	0
$626$	14.3709	0.574376
$627$	0	0
$628$	−3.40583	−0.135907
$629$	−3.32613	−0.132622
$630$	0	0
$631$	1.08000	0.0429943	0.0214971	−	0.999769i	$-0.493157\pi$
$631$	1.08000	0.0429943	0.0214971	+	0.999769i	$0.493157\pi$
$632$	−19.7537	−0.785760
$633$	−8.70184	−0.345867
$634$	−33.6146	−1.33501
$635$	0	0
$636$	−0.175997	−0.00697874
$637$	−12.7653	−0.505780
$638$	0	0
$639$	10.8792	0.430373
$640$	0	0
$641$	9.88535	0.390448	0.195224	−	0.980759i	$-0.437457\pi$
$641$	9.88535	0.390448	0.195224	+	0.980759i	$0.437457\pi$
$642$	−0.807155	−0.0318559
$643$	28.8003	1.13577	0.567886	−	0.823108i	$-0.307762\pi$
$643$	28.8003	1.13577	0.567886	+	0.823108i	$0.307762\pi$
$644$	0.742048	0.0292408
$645$	0	0
$646$	−17.0262	−0.669887
$647$	18.3084	0.719776	0.359888	−	0.932995i	$-0.382815\pi$
$647$	18.3084	0.719776	0.359888	+	0.932995i	$0.382815\pi$
$648$	33.0415	1.29799
$649$	0	0
$650$	0	0
$651$	−3.92096	−0.153674
$652$	0.794238	0.0311048
$653$	−31.9408	−1.24994	−0.624970	−	0.780649i	$-0.714889\pi$
$653$	−31.9408	−1.24994	−0.624970	+	0.780649i	$0.714889\pi$
$654$	29.2914	1.14539
$655$	0	0
$656$	30.9775	1.20947
$657$	−1.12095	−0.0437326
$658$	3.34244	0.130302
$659$	27.0408	1.05336	0.526679	−	0.850064i	$-0.323437\pi$
$659$	27.0408	1.05336	0.526679	+	0.850064i	$0.323437\pi$
$660$	0	0
$661$	47.2504	1.83783	0.918915	−	0.394456i	$-0.129067\pi$
$661$	47.2504	1.83783	0.918915	+	0.394456i	$0.129067\pi$
$662$	−13.6247	−0.529539
$663$	−6.76806	−0.262850
$664$	−33.1316	−1.28575
$665$	0	0
$666$	−2.70236	−0.104714
$667$	−48.7923	−1.88924
$668$	0.665906	0.0257647
$669$	21.4566	0.829560
$670$	0	0
$671$	0	0
$672$	−1.00491	−0.0387651
$673$	−37.8762	−1.46002	−0.730011	−	0.683435i	$-0.760485\pi$
$673$	−37.8762	−1.46002	−0.730011	+	0.683435i	$0.760485\pi$
$674$	−3.04507	−0.117292
$675$	0	0
$676$	2.42071	0.0931041
$677$	−33.9038	−1.30303	−0.651514	−	0.758637i	$-0.725866\pi$
$677$	−33.9038	−1.30303	−0.651514	+	0.758637i	$0.725866\pi$
$678$	−15.5395	−0.596790
$679$	−2.16001	−0.0828937
$680$	0	0
$681$	−11.7027	−0.448448
$682$	0	0
$683$	−38.6515	−1.47896	−0.739480	−	0.673178i	$-0.764929\pi$
$683$	−38.6515	−1.47896	−0.739480	+	0.673178i	$0.764929\pi$
$684$	2.00617	0.0767078
$685$	0	0
$686$	−6.38789	−0.243891
$687$	10.3479	0.394798
$688$	−23.1941	−0.884267
$689$	−0.636266	−0.0242398
$690$	0	0
$691$	26.3748	1.00334	0.501672	−	0.865058i	$-0.332718\pi$
$691$	26.3748	1.00334	0.501672	+	0.865058i	$0.332718\pi$
$692$	−0.993680	−0.0377741
$693$	0	0
$694$	27.4087	1.04042
$695$	0	0
$696$	35.0058	1.32689
$697$	−16.2579	−0.615812
$698$	4.90653	0.185715
$699$	−56.3064	−2.12970
$700$	0	0
$701$	18.5841	0.701910	0.350955	−	0.936392i	$-0.385857\pi$
$701$	18.5841	0.701910	0.350955	+	0.936392i	$0.385857\pi$
$702$	9.41135	0.355209
$703$	13.2291	0.498943
$704$	0	0
$705$	0	0
$706$	3.98391	0.149936
$707$	−1.27735	−0.0480398
$708$	0.102314	0.00384519
$709$	23.7065	0.890315	0.445158	−	0.895452i	$-0.353148\pi$
$709$	23.7065	0.890315	0.445158	+	0.895452i	$0.353148\pi$
$710$	0	0
$711$	7.33882	0.275227
$712$	−26.3414	−0.987186
$713$	−46.7339	−1.75020
$714$	−1.67860	−0.0628202
$715$	0	0
$716$	0.839708	0.0313814
$717$	13.6728	0.510619
$718$	−30.0525	−1.12155
$719$	−30.8533	−1.15063	−0.575317	−	0.817930i	$-0.695121\pi$
$719$	−30.8533	−1.15063	−0.575317	+	0.817930i	$0.695121\pi$
$720$	0	0
$721$	6.85439	0.255271
$722$	42.6076	1.58569
$723$	26.4671	0.984320
$724$	−4.28224	−0.159148
$725$	0	0
$726$	0	0
$727$	24.1222	0.894642	0.447321	−	0.894373i	$-0.352378\pi$
$727$	24.1222	0.894642	0.447321	+	0.894373i	$0.352378\pi$
$728$	−1.92466	−0.0713326
$729$	11.0523	0.409344
$730$	0	0
$731$	12.1729	0.450232
$732$	−4.00193	−0.147916
$733$	13.8477	0.511475	0.255738	−	0.966746i	$-0.417682\pi$
$733$	13.8477	0.511475	0.255738	+	0.966746i	$0.417682\pi$
$734$	−17.8994	−0.660677
$735$	0	0
$736$	−11.9775	−0.441496
$737$	0	0
$738$	−13.2090	−0.486228
$739$	27.5086	1.01192	0.505961	−	0.862557i	$-0.331138\pi$
$739$	27.5086	1.01192	0.505961	+	0.862557i	$0.331138\pi$
$740$	0	0
$741$	26.9187	0.988881
$742$	−0.157806	−0.00579323
$743$	2.69535	0.0988828	0.0494414	−	0.998777i	$-0.484256\pi$
$743$	2.69535	0.0988828	0.0494414	+	0.998777i	$0.484256\pi$
$744$	33.5290	1.22923
$745$	0	0
$746$	−32.8133	−1.20138
$747$	12.3089	0.450360
$748$	0	0
$749$	0.104959	0.00383512
$750$	0	0
$751$	22.9844	0.838713	0.419357	−	0.907822i	$-0.362256\pi$
$751$	22.9844	0.838713	0.419357	+	0.907822i	$0.362256\pi$
$752$	−24.9028	−0.908112
$753$	61.0877	2.22616
$754$	14.2269	0.518114
$755$	0	0
$756$	−0.338518	−0.0123118
$757$	−34.9560	−1.27050	−0.635249	−	0.772307i	$-0.719103\pi$
$757$	−34.9560	−1.27050	−0.635249	+	0.772307i	$0.719103\pi$
$758$	8.14205	0.295733
$759$	0	0
$760$	0	0
$761$	37.2243	1.34938	0.674690	−	0.738102i	$-0.264277\pi$
$761$	37.2243	1.34938	0.674690	+	0.738102i	$0.264277\pi$
$762$	−33.0766	−1.19824
$763$	−3.80894	−0.137893
$764$	5.32358	0.192600
$765$	0	0
$766$	0.480633	0.0173660
$767$	0.369886	0.0133558
$768$	12.1138	0.437118
$769$	30.1272	1.08642	0.543208	−	0.839598i	$-0.317210\pi$
$769$	30.1272	1.08642	0.543208	+	0.839598i	$0.317210\pi$
$770$	0	0
$771$	0.735518	0.0264890
$772$	0.839306	0.0302073
$773$	−30.9081	−1.11169	−0.555844	−	0.831287i	$-0.687605\pi$
$773$	−30.9081	−1.11169	−0.555844	+	0.831287i	$0.687605\pi$
$774$	9.89006	0.355491
$775$	0	0
$776$	18.4708	0.663062
$777$	1.30425	0.0467896
$778$	−2.46861	−0.0885039
$779$	64.6627	2.31678
$780$	0	0
$781$	0	0
$782$	−20.0073	−0.715460
$783$	22.2588	0.795464
$784$	23.5885	0.842447
$785$	0	0
$786$	−20.9334	−0.746669
$787$	−34.5178	−1.23043	−0.615214	−	0.788360i	$-0.710930\pi$
$787$	−34.5178	−1.23043	−0.615214	+	0.788360i	$0.710930\pi$
$788$	−1.76616	−0.0629167
$789$	−45.6953	−1.62680
$790$	0	0
$791$	2.02069	0.0718474
$792$	0	0
$793$	−14.4678	−0.513767
$794$	0.507731	0.0180187
$795$	0	0
$796$	4.86831	0.172552
$797$	10.2353	0.362554	0.181277	−	0.983432i	$-0.441977\pi$
$797$	10.2353	0.362554	0.181277	+	0.983432i	$0.441977\pi$
$798$	6.67633	0.236339
$799$	13.0697	0.462373
$800$	0	0
$801$	9.78626	0.345780
$802$	−29.9392	−1.05719
$803$	0	0
$804$	6.28566	0.221678
$805$	0	0
$806$	13.6268	0.479982
$807$	12.4569	0.438504
$808$	10.9229	0.384267
$809$	35.8637	1.26090	0.630450	−	0.776230i	$-0.282870\pi$
$809$	35.8637	1.26090	0.630450	+	0.776230i	$0.282870\pi$
$810$	0	0
$811$	18.5001	0.649625	0.324813	−	0.945778i	$-0.394699\pi$
$811$	18.5001	0.649625	0.324813	+	0.945778i	$0.394699\pi$
$812$	−0.511731	−0.0179582
$813$	−33.8334	−1.18659
$814$	0	0
$815$	0	0
$816$	12.5064	0.437813
$817$	−48.4155	−1.69384
$818$	−38.6101	−1.34997
$819$	0.715042	0.0249856
$820$	0	0
$821$	−27.2802	−0.952086	−0.476043	−	0.879422i	$-0.657929\pi$
$821$	−27.2802	−0.952086	−0.476043	+	0.879422i	$0.657929\pi$
$822$	−25.6588	−0.894953
$823$	−1.98450	−0.0691752	−0.0345876	−	0.999402i	$-0.511012\pi$
$823$	−1.98450	−0.0691752	−0.0345876	+	0.999402i	$0.511012\pi$
$824$	−58.6135	−2.04190
$825$	0	0
$826$	0.0917386	0.00319199
$827$	−12.3307	−0.428782	−0.214391	−	0.976748i	$-0.568777\pi$
$827$	−12.3307	−0.428782	−0.214391	+	0.976748i	$0.568777\pi$
$828$	2.35743	0.0819263
$829$	0.309635	0.0107541	0.00537704	−	0.999986i	$-0.498288\pi$
$829$	0.309635	0.0107541	0.00537704	+	0.999986i	$0.498288\pi$
$830$	0	0
$831$	45.2497	1.56970
$832$	16.2200	0.562328
$833$	−12.3799	−0.428939
$834$	−51.2350	−1.77412
$835$	0	0
$836$	0	0
$837$	21.3198	0.736920
$838$	−0.952579	−0.0329063
$839$	−6.70348	−0.231430	−0.115715	−	0.993282i	$-0.536916\pi$
$839$	−6.70348	−0.231430	−0.115715	+	0.993282i	$0.536916\pi$
$840$	0	0
$841$	4.64811	0.160280
$842$	−31.0012	−1.06837
$843$	−48.3405	−1.66493
$844$	−1.08778	−0.0374430
$845$	0	0
$846$	10.6187	0.365077
$847$	0	0
$848$	1.17573	0.0403748
$849$	−0.295634	−0.0101461
$850$	0	0
$851$	15.5453	0.532887
$852$	5.04753	0.172925
$853$	44.5713	1.52609	0.763047	−	0.646343i	$-0.223703\pi$
$853$	44.5713	1.52609	0.763047	+	0.646343i	$0.223703\pi$
$854$	−3.58829	−0.122789
$855$	0	0
$856$	−0.897530	−0.0306769
$857$	32.2444	1.10145	0.550723	−	0.834688i	$-0.314352\pi$
$857$	32.2444	1.10145	0.550723	+	0.834688i	$0.314352\pi$
$858$	0	0
$859$	−18.1192	−0.618220	−0.309110	−	0.951026i	$-0.600031\pi$
$859$	−18.1192	−0.618220	−0.309110	+	0.951026i	$0.600031\pi$
$860$	0	0
$861$	6.37506	0.217262
$862$	4.96755	0.169196
$863$	8.21590	0.279672	0.139836	−	0.990175i	$-0.455342\pi$
$863$	8.21590	0.279672	0.139836	+	0.990175i	$0.455342\pi$
$864$	5.46407	0.185892
$865$	0	0
$866$	17.9832	0.611093
$867$	27.8854	0.947039
$868$	−0.490143	−0.0166365
$869$	0	0
$870$	0	0
$871$	22.7240	0.769973
$872$	32.5711	1.10300
$873$	−6.86219	−0.232250
$874$	79.5752	2.69167
$875$	0	0
$876$	−0.520081	−0.0175719
$877$	−33.0314	−1.11539	−0.557695	−	0.830046i	$-0.688314\pi$
$877$	−33.0314	−1.11539	−0.557695	+	0.830046i	$0.688314\pi$
$878$	20.6460	0.696769
$879$	15.5524	0.524569
$880$	0	0
$881$	30.1182	1.01471	0.507354	−	0.861738i	$-0.330624\pi$
$881$	30.1182	1.01471	0.507354	+	0.861738i	$0.330624\pi$
$882$	−10.0582	−0.338678
$883$	−19.0574	−0.641332	−0.320666	−	0.947192i	$-0.603907\pi$
$883$	−19.0574	−0.641332	−0.320666	+	0.947192i	$0.603907\pi$
$884$	−0.846047	−0.0284556
$885$	0	0
$886$	1.87118	0.0628634
$887$	22.3685	0.751059	0.375530	−	0.926810i	$-0.377461\pi$
$887$	22.3685	0.751059	0.375530	+	0.926810i	$0.377461\pi$
$888$	−11.1529	−0.374267
$889$	4.30115	0.144256
$890$	0	0
$891$	0	0
$892$	2.68220	0.0898068
$893$	−51.9822	−1.73952
$894$	−15.4664	−0.517275
$895$	0	0
$896$	3.03106	0.101261
$897$	31.6318	1.05616
$898$	9.13091	0.304702
$899$	32.2286	1.07489
$900$	0	0
$901$	−0.617057	−0.0205572
$902$	0	0
$903$	−4.77326	−0.158844
$904$	−17.2794	−0.574703
$905$	0	0
$906$	−35.8021	−1.18944
$907$	−42.6593	−1.41648	−0.708240	−	0.705972i	$-0.750511\pi$
$907$	−42.6593	−1.41648	−0.708240	+	0.705972i	$0.750511\pi$
$908$	−1.46290	−0.0485482
$909$	−4.05805	−0.134597
$910$	0	0
$911$	−37.1240	−1.22997	−0.614987	−	0.788537i	$-0.710839\pi$
$911$	−37.1240	−1.22997	−0.614987	+	0.788537i	$0.710839\pi$
$912$	−49.7419	−1.64712
$913$	0	0
$914$	55.9756	1.85151
$915$	0	0
$916$	1.29355	0.0427401
$917$	2.72209	0.0898914
$918$	9.12723	0.301243
$919$	3.35848	0.110786	0.0553930	−	0.998465i	$-0.482359\pi$
$919$	3.35848	0.110786	0.0553930	+	0.998465i	$0.482359\pi$
$920$	0	0
$921$	19.6469	0.647388
$922$	33.2471	1.09494
$923$	18.2479	0.600636
$924$	0	0
$925$	0	0
$926$	−10.4309	−0.342781
$927$	21.7759	0.715214
$928$	8.25992	0.271145
$929$	−20.1391	−0.660742	−0.330371	−	0.943851i	$-0.607174\pi$
$929$	−20.1391	−0.660742	−0.330371	+	0.943851i	$0.607174\pi$
$930$	0	0
$931$	49.2388	1.61374
$932$	−7.03863	−0.230558
$933$	−50.9815	−1.66906
$934$	21.1078	0.690669
$935$	0	0
$936$	−6.11449	−0.199858
$937$	10.5300	0.344000	0.172000	−	0.985097i	$-0.444977\pi$
$937$	10.5300	0.344000	0.172000	+	0.985097i	$0.444977\pi$
$938$	5.63597	0.184021
$939$	−22.0346	−0.719073
$940$	0	0
$941$	−19.3745	−0.631589	−0.315795	−	0.948828i	$-0.602271\pi$
$941$	−19.3745	−0.631589	−0.315795	+	0.948828i	$0.602271\pi$
$942$	−36.0081	−1.17321
$943$	75.9844	2.47439
$944$	−0.683498	−0.0222460
$945$	0	0
$946$	0	0
$947$	−3.79793	−0.123416	−0.0617081	−	0.998094i	$-0.519655\pi$
$947$	−3.79793	−0.123416	−0.0617081	+	0.998094i	$0.519655\pi$
$948$	3.40494	0.110587
$949$	−1.88020	−0.0610339
$950$	0	0
$951$	51.5407	1.67132
$952$	−1.86655	−0.0604953
$953$	0.535767	0.0173552	0.00867760	−	0.999962i	$-0.497238\pi$
$953$	0.535767	0.0173552	0.00867760	+	0.999962i	$0.497238\pi$
$954$	−0.501337	−0.0162314
$955$	0	0
$956$	1.70918	0.0552788
$957$	0	0
$958$	18.9819	0.613276
$959$	3.33656	0.107743
$960$	0	0
$961$	−0.130938	−0.00422380
$962$	−4.53273	−0.146141
$963$	0.333447	0.0107452
$964$	3.30854	0.106561
$965$	0	0
$966$	7.84528	0.252418
$967$	36.5291	1.17470	0.587348	−	0.809335i	$-0.300172\pi$
$967$	36.5291	1.17470	0.587348	+	0.809335i	$0.300172\pi$
$968$	0	0
$969$	26.1060	0.838645
$970$	0	0
$971$	20.6281	0.661986	0.330993	−	0.943633i	$-0.392616\pi$
$971$	20.6281	0.661986	0.330993	+	0.943633i	$0.392616\pi$
$972$	−2.77925	−0.0891444
$973$	6.66238	0.213586
$974$	−47.4878	−1.52161
$975$	0	0
$976$	26.7345	0.855752
$977$	23.5349	0.752950	0.376475	−	0.926427i	$-0.377136\pi$
$977$	23.5349	0.752950	0.376475	+	0.926427i	$0.377136\pi$
$978$	8.39706	0.268508
$979$	0	0
$980$	0	0
$981$	−12.1007	−0.386346
$982$	46.5443	1.48529
$983$	40.6736	1.29729	0.648643	−	0.761093i	$-0.275337\pi$
$983$	40.6736	1.29729	0.648643	+	0.761093i	$0.275337\pi$
$984$	−54.5146	−1.73786
$985$	0	0
$986$	13.7974	0.439400
$987$	−5.12490	−0.163127
$988$	3.36499	0.107055
$989$	−56.8925	−1.80908
$990$	0	0
$991$	23.1498	0.735377	0.367688	−	0.929949i	$-0.380149\pi$
$991$	23.1498	0.735377	0.367688	+	0.929949i	$0.380149\pi$
$992$	7.91147	0.251189
$993$	20.8905	0.662941
$994$	4.52581	0.143550
$995$	0	0
$996$	5.71088	0.180956
$997$	−5.32844	−0.168753	−0.0843767	−	0.996434i	$-0.526890\pi$
$997$	−5.32844	−0.168753	−0.0843767	+	0.996434i	$0.526890\pi$
$998$	35.0127	1.10831
$999$	−7.09170	−0.224371

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3025.2.a.bn.1.5		8
5.4	even	2		3025.2.a.bi.1.4		8
11.5	even	5		275.2.h.e.201.2	yes	16
11.9	even	5		275.2.h.e.26.2	yes	16
11.10	odd	2		3025.2.a.bj.1.4		8
55.9	even	10		275.2.h.c.26.3	✓	16
55.27	odd	20		275.2.z.c.124.3		32
55.38	odd	20		275.2.z.c.124.6		32
55.42	odd	20		275.2.z.c.224.6		32
55.49	even	10		275.2.h.c.201.3	yes	16
55.53	odd	20		275.2.z.c.224.3		32
55.54	odd	2		3025.2.a.bm.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.2.h.c.26.3	✓	16	55.9	even	10
275.2.h.c.201.3	yes	16	55.49	even	10
275.2.h.e.26.2	yes	16	11.9	even	5
275.2.h.e.201.2	yes	16	11.5	even	5
275.2.z.c.124.3		32	55.27	odd	20
275.2.z.c.124.6		32	55.38	odd	20
275.2.z.c.224.3		32	55.53	odd	20
275.2.z.c.224.6		32	55.42	odd	20
3025.2.a.bi.1.4		8	5.4	even	2
3025.2.a.bj.1.4		8	11.10	odd	2
3025.2.a.bm.1.5		8	55.54	odd	2
3025.2.a.bn.1.5		8	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:	\( N \)	\(=\)	\( 3025 = 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3025.a (trivial)

\(f(q)\)	\(=\)	\(q+1.32162 q^{2} -2.02642 q^{3} -0.253315 q^{4} -2.67816 q^{6} +0.348258 q^{7} -2.97803 q^{8} +1.10639 q^{9} +0.513323 q^{12} +1.85577 q^{13} +0.460265 q^{14} -3.42920 q^{16} +1.79974 q^{17} +1.46223 q^{18} -7.15813 q^{19} -0.705717 q^{21} -8.41145 q^{23} +6.03475 q^{24} +2.45262 q^{26} +3.83726 q^{27} -0.0882188 q^{28} +5.80070 q^{29} +5.55599 q^{31} +1.42395 q^{32} +2.37858 q^{34} -0.280264 q^{36} -1.84812 q^{37} -9.46035 q^{38} -3.76057 q^{39} -9.03346 q^{41} -0.932691 q^{42} +6.76370 q^{43} -11.1168 q^{46} +7.26198 q^{47} +6.94901 q^{48} -6.87872 q^{49} -3.64704 q^{51} -0.470093 q^{52} -0.342859 q^{53} +5.07141 q^{54} -1.03712 q^{56} +14.5054 q^{57} +7.66633 q^{58} +0.199317 q^{59} -7.79614 q^{61} +7.34292 q^{62} +0.385308 q^{63} +8.74033 q^{64} +12.2451 q^{67} -0.455901 q^{68} +17.0451 q^{69} +9.83305 q^{71} -3.29486 q^{72} -1.01317 q^{73} -2.44251 q^{74} +1.81326 q^{76} -4.97005 q^{78} +6.63314 q^{79} -11.0951 q^{81} -11.9388 q^{82} +11.1253 q^{83} +0.178769 q^{84} +8.93906 q^{86} -11.7547 q^{87} +8.84524 q^{89} +0.646285 q^{91} +2.13074 q^{92} -11.2588 q^{93} +9.59760 q^{94} -2.88553 q^{96} -6.20234 q^{97} -9.09106 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 5 q^{2} + q^{3} + 9 q^{4} + q^{6} + 8 q^{7} + 12 q^{8} + 9 q^{9} + 3 q^{12} + 9 q^{13} + 9 q^{14} + 23 q^{16} + 19 q^{17} + 22 q^{18} + q^{19} + 5 q^{21} + 2 q^{23} + q^{24} - 2 q^{26} - 2 q^{27}+ \cdots + 26 q^{98}+O(q^{100}) \) 8 * q + 5 * q^2 + q^3 + 9 * q^4 + q^6 + 8 * q^7 + 12 * q^8 + 9 * q^9 + 3 * q^12 + 9 * q^13 + 9 * q^14 + 23 * q^16 + 19 * q^17 + 22 * q^18 + q^19 + 5 * q^21 + 2 * q^23 + q^24 - 2 * q^26 - 2 * q^27 + 9 * q^28 + 7 * q^29 - 5 * q^31 + 29 * q^32 + 10 * q^34 - 16 * q^36 - 8 * q^37 - 37 * q^38 - q^39 + 41 * q^42 + 14 * q^43 - 20 * q^46 + 11 * q^47 - 27 * q^48 - 12 * q^49 - 25 * q^51 - 7 * q^52 + 11 * q^53 + 30 * q^54 - 10 * q^56 - 2 * q^57 + 27 * q^58 + 17 * q^59 - 2 * q^61 + 25 * q^62 + 41 * q^63 + 30 * q^64 + 7 * q^67 + 66 * q^68 + 17 * q^71 - 19 * q^72 + 34 * q^73 - 6 * q^74 - 31 * q^76 - 17 * q^78 + 23 * q^79 - 4 * q^81 - 17 * q^82 + 41 * q^83 + 83 * q^84 + q^86 + 25 * q^87 - 11 * q^89 - 7 * q^91 + 33 * q^92 - 59 * q^93 + 50 * q^94 - 61 * q^96 + 2 * q^97 + 26 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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