LMFDB - Embedded newform 9025.2.a.ck.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:	$1$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 22x^{14} + 190x^{12} - 820x^{10} + 1862x^{8} - 2154x^{6} + 1163x^{4} - 256x^{2} + 16 $ x^16 - 22x^14 + 190x^12 - 820x^10 + 1862x^8 - 2154x^6 + 1163x^4 - 256*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1805)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.31447$ 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)

$f(q)$	$=$	$q-2.31447 q^{2} +2.90369 q^{3} +3.35679 q^{4} -6.72051 q^{6} +2.34671 q^{7} -3.14026 q^{8} +5.43140 q^{9} +O(q^{10})$	\(q-2.31447 q^{2} +2.90369 q^{3} +3.35679 q^{4} -6.72051 q^{6} +2.34671 q^{7} -3.14026 q^{8} +5.43140 q^{9} -3.45040 q^{11} +9.74707 q^{12} -3.29131 q^{13} -5.43140 q^{14} +0.554464 q^{16} +1.38276 q^{17} -12.5708 q^{18} +6.81412 q^{21} +7.98585 q^{22} -8.90479 q^{23} -9.11833 q^{24} +7.61765 q^{26} +7.06003 q^{27} +7.87742 q^{28} +6.28677 q^{29} -7.39730 q^{31} +4.99722 q^{32} -10.0189 q^{33} -3.20036 q^{34} +18.2321 q^{36} -0.650489 q^{37} -9.55694 q^{39} +2.01205 q^{41} -15.7711 q^{42} -1.72022 q^{43} -11.5823 q^{44} +20.6099 q^{46} -6.32206 q^{47} +1.60999 q^{48} -1.49295 q^{49} +4.01510 q^{51} -11.0482 q^{52} +2.12219 q^{53} -16.3403 q^{54} -7.36928 q^{56} -14.5506 q^{58} -6.06907 q^{59} +2.96638 q^{61} +17.1209 q^{62} +12.7459 q^{63} -12.6749 q^{64} +23.1884 q^{66} +2.46462 q^{67} +4.64164 q^{68} -25.8567 q^{69} -5.69927 q^{71} -17.0560 q^{72} +6.88086 q^{73} +1.50554 q^{74} -8.09708 q^{77} +22.1193 q^{78} +13.8708 q^{79} +4.20592 q^{81} -4.65683 q^{82} -1.84546 q^{83} +22.8736 q^{84} +3.98141 q^{86} +18.2548 q^{87} +10.8351 q^{88} +6.47352 q^{89} -7.72375 q^{91} -29.8915 q^{92} -21.4795 q^{93} +14.6322 q^{94} +14.5104 q^{96} +7.61563 q^{97} +3.45539 q^{98} -18.7405 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 12 q^{4} - 10 q^{6} + 6 q^{9}+O(q^{10}) $ 16 * q + 12 * q^4 - 10 * q^6 + 6 * q^9	$ 16 q + 12 q^{4} - 10 q^{6} + 6 q^{9} - 22 q^{11} - 6 q^{14} + 8 q^{16} + 20 q^{21} - 14 q^{24} - 16 q^{26} - 2 q^{29} - 16 q^{31} + 8 q^{34} + 18 q^{36} - 36 q^{39} - 26 q^{41} - 64 q^{44} - 2 q^{46} - 20 q^{49} + 38 q^{51} - 12 q^{54} - 6 q^{56} - 10 q^{59} - 30 q^{61} - 16 q^{64} + 4 q^{66} - 68 q^{69} + 20 q^{71} - 40 q^{74} - 12 q^{79} - 48 q^{81} + 2 q^{84} + 20 q^{86} + 86 q^{91} + 38 q^{94} + 22 q^{96} - 32 q^{99}+O(q^{100}) $ 16 * q + 12 * q^4 - 10 * q^6 + 6 * q^9 - 22 * q^11 - 6 * q^14 + 8 * q^16 + 20 * q^21 - 14 * q^24 - 16 * q^26 - 2 * q^29 - 16 * q^31 + 8 * q^34 + 18 * q^36 - 36 * q^39 - 26 * q^41 - 64 * q^44 - 2 * q^46 - 20 * q^49 + 38 * q^51 - 12 * q^54 - 6 * q^56 - 10 * q^59 - 30 * q^61 - 16 * q^64 + 4 * q^66 - 68 * q^69 + 20 * q^71 - 40 * q^74 - 12 * q^79 - 48 * q^81 + 2 * q^84 + 20 * q^86 + 86 * q^91 + 38 * q^94 + 22 * q^96 - 32 * 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 $

$2$	−2.31447	−1.63658	−0.818290	−	0.574805i	$-0.805078\pi$
$2$	−2.31447	−1.63658	−0.818290	+	0.574805i	$0.805078\pi$
$3$	2.90369	1.67644	0.838222	−	0.545328i	$-0.183595\pi$
$3$	2.90369	1.67644	0.838222	+	0.545328i	$0.183595\pi$
$4$	3.35679	1.67840
$5$	0	0
$5$	0	0
$6$	−6.72051	−2.74364
$7$	2.34671	0.886973	0.443487	−	0.896281i	$-0.353741\pi$
$7$	2.34671	0.886973	0.443487	+	0.896281i	$0.353741\pi$
$8$	−3.14026	−1.11025
$9$	5.43140	1.81047
$10$	0	0
$11$	−3.45040	−1.04033	−0.520167	−	0.854065i	$-0.674130\pi$
$11$	−3.45040	−1.04033	−0.520167	+	0.854065i	$0.674130\pi$
$12$	9.74707	2.81374
$13$	−3.29131	−0.912845	−0.456423	−	0.889763i	$-0.650869\pi$
$13$	−3.29131	−0.912845	−0.456423	+	0.889763i	$0.650869\pi$
$14$	−5.43140	−1.45160
$15$	0	0
$16$	0.554464	0.138616
$17$	1.38276	0.335369	0.167684	−	0.985841i	$-0.446371\pi$
$17$	1.38276	0.335369	0.167684	+	0.985841i	$0.446371\pi$
$18$	−12.5708	−2.96298
$19$	0	0
$19$	0	0
$20$	0	0
$21$	6.81412	1.48696
$22$	7.98585	1.70259
$23$	−8.90479	−1.85678	−0.928389	−	0.371610i	$-0.878806\pi$
$23$	−8.90479	−1.85678	−0.928389	+	0.371610i	$0.878806\pi$
$24$	−9.11833	−1.86127
$25$	0	0
$26$	7.61765	1.49394
$27$	7.06003	1.35870
$28$	7.87742	1.48869
$29$	6.28677	1.16742	0.583712	−	0.811961i	$-0.301600\pi$
$29$	6.28677	1.16742	0.583712	+	0.811961i	$0.301600\pi$
$30$	0	0
$31$	−7.39730	−1.32859	−0.664297	−	0.747468i	$-0.731269\pi$
$31$	−7.39730	−1.32859	−0.664297	+	0.747468i	$0.731269\pi$
$32$	4.99722	0.883393
$33$	−10.0189	−1.74406
$34$	−3.20036	−0.548858
$35$	0	0
$36$	18.2321	3.03868
$37$	−0.650489	−0.106940	−0.0534698	−	0.998569i	$-0.517028\pi$
$37$	−0.650489	−0.106940	−0.0534698	+	0.998569i	$0.517028\pi$
$38$	0	0
$39$	−9.55694	−1.53033
$40$	0	0
$41$	2.01205	0.314229	0.157115	−	0.987580i	$-0.449781\pi$
$41$	2.01205	0.314229	0.157115	+	0.987580i	$0.449781\pi$
$42$	−15.7711	−2.43353
$43$	−1.72022	−0.262332	−0.131166	−	0.991360i	$-0.541872\pi$
$43$	−1.72022	−0.262332	−0.131166	+	0.991360i	$0.541872\pi$
$44$	−11.5823	−1.74609
$45$	0	0
$46$	20.6099	3.03877
$47$	−6.32206	−0.922167	−0.461083	−	0.887357i	$-0.652539\pi$
$47$	−6.32206	−0.922167	−0.461083	+	0.887357i	$0.652539\pi$
$48$	1.60999	0.232382
$49$	−1.49295	−0.213278
$50$	0	0
$51$	4.01510	0.562227
$52$	−11.0482	−1.53212
$53$	2.12219	0.291505	0.145752	−	0.989321i	$-0.453440\pi$
$53$	2.12219	0.291505	0.145752	+	0.989321i	$0.453440\pi$
$54$	−16.3403	−2.22363
$55$	0	0
$56$	−7.36928	−0.984761
$57$	0	0
$58$	−14.5506	−1.91058
$59$	−6.06907	−0.790126	−0.395063	−	0.918654i	$-0.629277\pi$
$59$	−6.06907	−0.790126	−0.395063	+	0.918654i	$0.629277\pi$
$60$	0	0
$61$	2.96638	0.379805	0.189903	−	0.981803i	$-0.439183\pi$
$61$	2.96638	0.379805	0.189903	+	0.981803i	$0.439183\pi$
$62$	17.1209	2.17435
$63$	12.7459	1.60584
$64$	−12.6749	−1.58436
$65$	0	0
$66$	23.1884	2.85430
$67$	2.46462	0.301101	0.150551	−	0.988602i	$-0.451895\pi$
$67$	2.46462	0.301101	0.150551	+	0.988602i	$0.451895\pi$
$68$	4.64164	0.562881
$69$	−25.8567	−3.11279
$70$	0	0
$71$	−5.69927	−0.676379	−0.338189	−	0.941078i	$-0.609814\pi$
$71$	−5.69927	−0.676379	−0.338189	+	0.941078i	$0.609814\pi$
$72$	−17.0560	−2.01007
$73$	6.88086	0.805344	0.402672	−	0.915344i	$-0.368081\pi$
$73$	6.88086	0.805344	0.402672	+	0.915344i	$0.368081\pi$
$74$	1.50554	0.175015
$75$	0	0
$76$	0	0
$77$	−8.09708	−0.922748
$78$	22.1193	2.50452
$79$	13.8708	1.56058	0.780292	−	0.625415i	$-0.215070\pi$
$79$	13.8708	1.56058	0.780292	+	0.625415i	$0.215070\pi$
$80$	0	0
$81$	4.20592	0.467325
$82$	−4.65683	−0.514261
$83$	−1.84546	−0.202566	−0.101283	−	0.994858i	$-0.532295\pi$
$83$	−1.84546	−0.202566	−0.101283	+	0.994858i	$0.532295\pi$
$84$	22.8736	2.49571
$85$	0	0
$86$	3.98141	0.429327
$87$	18.2548	1.95712
$88$	10.8351	1.15503
$89$	6.47352	0.686192	0.343096	−	0.939300i	$-0.388524\pi$
$89$	6.47352	0.686192	0.343096	+	0.939300i	$0.388524\pi$
$90$	0	0
$91$	−7.72375	−0.809669
$92$	−29.8915	−3.11641
$93$	−21.4795	−2.22732
$94$	14.6322	1.50920
$95$	0	0
$96$	14.5104	1.48096
$97$	7.61563	0.773251	0.386625	−	0.922237i	$-0.373641\pi$
$97$	7.61563	0.773251	0.386625	+	0.922237i	$0.373641\pi$
$98$	3.45539	0.349047
$99$	−18.7405	−1.88349
$100$	0	0
$101$	−16.5949	−1.65125	−0.825627	−	0.564216i	$-0.809179\pi$
$101$	−16.5949	−1.65125	−0.825627	+	0.564216i	$0.809179\pi$
$102$	−9.29286	−0.920130
$103$	8.09429	0.797555	0.398777	−	0.917048i	$-0.369435\pi$
$103$	8.09429	0.797555	0.398777	+	0.917048i	$0.369435\pi$
$104$	10.3356	1.01349
$105$	0	0
$106$	−4.91175	−0.477071
$107$	−4.57981	−0.442747	−0.221373	−	0.975189i	$-0.571054\pi$
$107$	−4.57981	−0.442747	−0.221373	+	0.975189i	$0.571054\pi$
$108$	23.6991	2.28044
$109$	−10.0829	−0.965762	−0.482881	−	0.875686i	$-0.660410\pi$
$109$	−10.0829	−0.965762	−0.482881	+	0.875686i	$0.660410\pi$
$110$	0	0
$111$	−1.88882	−0.179278
$112$	1.30117	0.122949
$113$	−15.1432	−1.42455	−0.712276	−	0.701900i	$-0.752335\pi$
$113$	−15.1432	−1.42455	−0.712276	+	0.701900i	$0.752335\pi$
$114$	0	0
$115$	0	0
$116$	21.1034	1.95940
$117$	−17.8764	−1.65268
$118$	14.0467	1.29310
$119$	3.24494	0.297463
$120$	0	0
$121$	0.905235	0.0822941
$122$	−6.86560	−0.621582
$123$	5.84236	0.526788
$124$	−24.8312	−2.22991
$125$	0	0
$126$	−29.5001	−2.62808
$127$	−1.24195	−0.110205	−0.0551027	−	0.998481i	$-0.517549\pi$
$127$	−1.24195	−0.110205	−0.0551027	+	0.998481i	$0.517549\pi$
$128$	19.3412	1.70954
$129$	−4.99499	−0.439784
$130$	0	0
$131$	−11.7603	−1.02750	−0.513751	−	0.857939i	$-0.671744\pi$
$131$	−11.7603	−1.02750	−0.513751	+	0.857939i	$0.671744\pi$
$132$	−33.6313	−2.92723
$133$	0	0
$134$	−5.70430	−0.492776
$135$	0	0
$136$	−4.34222	−0.372343
$137$	17.4117	1.48758	0.743792	−	0.668412i	$-0.233025\pi$
$137$	17.4117	1.48758	0.743792	+	0.668412i	$0.233025\pi$
$138$	59.8448	5.09432
$139$	−10.5771	−0.897137	−0.448568	−	0.893748i	$-0.648066\pi$
$139$	−10.5771	−0.897137	−0.448568	+	0.893748i	$0.648066\pi$
$140$	0	0
$141$	−18.3573	−1.54596
$142$	13.1908	1.10695
$143$	11.3563	0.949663
$144$	3.01152	0.250960
$145$	0	0
$146$	−15.9256	−1.31801
$147$	−4.33505	−0.357549
$148$	−2.18355	−0.179487
$149$	3.63524	0.297810	0.148905	−	0.988852i	$-0.452425\pi$
$149$	3.63524	0.297810	0.148905	+	0.988852i	$0.452425\pi$
$150$	0	0
$151$	−8.01277	−0.652070	−0.326035	−	0.945358i	$-0.605713\pi$
$151$	−8.01277	−0.652070	−0.326035	+	0.945358i	$0.605713\pi$
$152$	0	0
$153$	7.51033	0.607174
$154$	18.7405	1.51015
$155$	0	0
$156$	−32.0806	−2.56851
$157$	2.19221	0.174958	0.0874789	−	0.996166i	$-0.472119\pi$
$157$	2.19221	0.174958	0.0874789	+	0.996166i	$0.472119\pi$
$158$	−32.1036	−2.55402
$159$	6.16217	0.488692
$160$	0	0
$161$	−20.8970	−1.64691
$162$	−9.73450	−0.764814
$163$	1.75826	0.137718	0.0688588	−	0.997626i	$-0.478064\pi$
$163$	1.75826	0.137718	0.0688588	+	0.997626i	$0.478064\pi$
$164$	6.75402	0.527401
$165$	0	0
$166$	4.27128	0.331515
$167$	−3.62416	−0.280446	−0.140223	−	0.990120i	$-0.544782\pi$
$167$	−3.62416	−0.280446	−0.140223	+	0.990120i	$0.544782\pi$
$168$	−21.3981	−1.65090
$169$	−2.16728	−0.166714
$170$	0	0
$171$	0	0
$172$	−5.77443	−0.440296
$173$	−1.49717	−0.113828	−0.0569140	−	0.998379i	$-0.518126\pi$
$173$	−1.49717	−0.113828	−0.0569140	+	0.998379i	$0.518126\pi$
$174$	−42.2503	−3.20299
$175$	0	0
$176$	−1.91312	−0.144207
$177$	−17.6227	−1.32460
$178$	−14.9828	−1.12301
$179$	−12.9371	−0.966963	−0.483481	−	0.875355i	$-0.660628\pi$
$179$	−12.9371	−0.966963	−0.483481	+	0.875355i	$0.660628\pi$
$180$	0	0
$181$	10.0013	0.743390	0.371695	−	0.928355i	$-0.378777\pi$
$181$	10.0013	0.743390	0.371695	+	0.928355i	$0.378777\pi$
$182$	17.8764	1.32509
$183$	8.61343	0.636723
$184$	27.9633	2.06149
$185$	0	0
$186$	49.7137	3.64518
$187$	−4.77107	−0.348895
$188$	−21.2218	−1.54776
$189$	16.5679	1.20513
$190$	0	0
$191$	−24.3534	−1.76215	−0.881077	−	0.472973i	$-0.843181\pi$
$191$	−24.3534	−1.76215	−0.881077	+	0.472973i	$0.843181\pi$
$192$	−36.8039	−2.65609
$193$	−6.49316	−0.467388	−0.233694	−	0.972310i	$-0.575081\pi$
$193$	−6.49316	−0.467388	−0.233694	+	0.972310i	$0.575081\pi$
$194$	−17.6262	−1.26549
$195$	0	0
$196$	−5.01151	−0.357965
$197$	−2.13662	−0.152228	−0.0761138	−	0.997099i	$-0.524251\pi$
$197$	−2.13662	−0.152228	−0.0761138	+	0.997099i	$0.524251\pi$
$198$	43.3744	3.08248
$199$	−17.6660	−1.25231	−0.626156	−	0.779697i	$-0.715373\pi$
$199$	−17.6660	−1.25231	−0.626156	+	0.779697i	$0.715373\pi$
$200$	0	0
$201$	7.15648	0.504779
$202$	38.4085	2.70241
$203$	14.7532	1.03547
$204$	13.4779	0.943639
$205$	0	0
$206$	−18.7340	−1.30526
$207$	−48.3655	−3.36164
$208$	−1.82491	−0.126535
$209$	0	0
$210$	0	0
$211$	−18.9939	−1.30759	−0.653796	−	0.756671i	$-0.726825\pi$
$211$	−18.9939	−1.30759	−0.653796	+	0.756671i	$0.726825\pi$
$212$	7.12374	0.489261
$213$	−16.5489	−1.13391
$214$	10.5999	0.724591
$215$	0	0
$216$	−22.1703	−1.50850
$217$	−17.3593	−1.17843
$218$	23.3365	1.58055
$219$	19.9799	1.35012
$220$	0	0
$221$	−4.55109	−0.306140
$222$	4.37161	0.293404
$223$	20.2503	1.35606	0.678029	−	0.735035i	$-0.262834\pi$
$223$	20.2503	1.35606	0.678029	+	0.735035i	$0.262834\pi$
$224$	11.7270	0.783546
$225$	0	0
$226$	35.0485	2.33139
$227$	−15.8466	−1.05177	−0.525887	−	0.850554i	$-0.676266\pi$
$227$	−15.8466	−1.05177	−0.525887	+	0.850554i	$0.676266\pi$
$228$	0	0
$229$	9.87340	0.652453	0.326226	−	0.945292i	$-0.394223\pi$
$229$	9.87340	0.652453	0.326226	+	0.945292i	$0.394223\pi$
$230$	0	0
$231$	−23.5114	−1.54694
$232$	−19.7421	−1.29613
$233$	−3.74263	−0.245188	−0.122594	−	0.992457i	$-0.539121\pi$
$233$	−3.74263	−0.245188	−0.122594	+	0.992457i	$0.539121\pi$
$234$	41.3745	2.70474
$235$	0	0
$236$	−20.3726	−1.32614
$237$	40.2764	2.61623
$238$	−7.51033	−0.486822
$239$	−21.2541	−1.37482	−0.687408	−	0.726272i	$-0.741251\pi$
$239$	−21.2541	−1.37482	−0.687408	+	0.726272i	$0.741251\pi$
$240$	0	0
$241$	27.7142	1.78523	0.892613	−	0.450824i	$-0.148870\pi$
$241$	27.7142	1.78523	0.892613	+	0.450824i	$0.148870\pi$
$242$	−2.09514	−0.134681
$243$	−8.96741	−0.575260
$244$	9.95750	0.637464
$245$	0	0
$246$	−13.5220	−0.862130
$247$	0	0
$248$	23.2294	1.47507
$249$	−5.35865	−0.339591
$250$	0	0
$251$	−17.0524	−1.07634	−0.538168	−	0.842838i	$-0.680883\pi$
$251$	−17.0524	−1.07634	−0.538168	+	0.842838i	$0.680883\pi$
$252$	42.7854	2.69523
$253$	30.7251	1.93167
$254$	2.87446	0.180360
$255$	0	0
$256$	−19.4150	−1.21344
$257$	8.87646	0.553699	0.276849	−	0.960913i	$-0.410710\pi$
$257$	8.87646	0.553699	0.276849	+	0.960913i	$0.410710\pi$
$258$	11.5608	0.719743
$259$	−1.52651	−0.0948526
$260$	0	0
$261$	34.1460	2.11358
$262$	27.2189	1.68159
$263$	−28.6398	−1.76600	−0.883002	−	0.469370i	$-0.844481\pi$
$263$	−28.6398	−1.76600	−0.883002	+	0.469370i	$0.844481\pi$
$264$	31.4618	1.93634
$265$	0	0
$266$	0	0
$267$	18.7971	1.15036
$268$	8.27321	0.505367
$269$	29.2060	1.78072	0.890362	−	0.455254i	$-0.150451\pi$
$269$	29.2060	1.78072	0.890362	+	0.455254i	$0.150451\pi$
$270$	0	0
$271$	−13.1166	−0.796779	−0.398390	−	0.917216i	$-0.630431\pi$
$271$	−13.1166	−0.796779	−0.398390	+	0.917216i	$0.630431\pi$
$272$	0.766691	0.0464875
$273$	−22.4274	−1.35737
$274$	−40.2990	−2.43455
$275$	0	0
$276$	−86.7957	−5.22449
$277$	6.23517	0.374635	0.187317	−	0.982299i	$-0.440021\pi$
$277$	6.23517	0.374635	0.187317	+	0.982299i	$0.440021\pi$
$278$	24.4804	1.46824
$279$	−40.1777	−2.40538
$280$	0	0
$281$	16.8127	1.00296	0.501480	−	0.865169i	$-0.332789\pi$
$281$	16.8127	1.00296	0.501480	+	0.865169i	$0.332789\pi$
$282$	42.4875	2.53009
$283$	15.7304	0.935076	0.467538	−	0.883973i	$-0.345141\pi$
$283$	15.7304	0.935076	0.467538	+	0.883973i	$0.345141\pi$
$284$	−19.1312	−1.13523
$285$	0	0
$286$	−26.2839	−1.55420
$287$	4.72169	0.278713
$288$	27.1419	1.59935
$289$	−15.0880	−0.887528
$290$	0	0
$291$	22.1134	1.29631
$292$	23.0976	1.35169
$293$	−18.3605	−1.07263	−0.536317	−	0.844016i	$-0.680185\pi$
$293$	−18.3605	−1.07263	−0.536317	+	0.844016i	$0.680185\pi$
$294$	10.0334	0.585158
$295$	0	0
$296$	2.04270	0.118730
$297$	−24.3599	−1.41351
$298$	−8.41366	−0.487390
$299$	29.3084	1.69495
$300$	0	0
$301$	−4.03687	−0.232681
$302$	18.5454	1.06717
$303$	−48.1864	−2.76824
$304$	0	0
$305$	0	0
$306$	−17.3825	−0.993689
$307$	21.4942	1.22674	0.613370	−	0.789796i	$-0.289814\pi$
$307$	21.4942	1.22674	0.613370	+	0.789796i	$0.289814\pi$
$308$	−27.1802	−1.54874
$309$	23.5033	1.33706
$310$	0	0
$311$	−31.3890	−1.77990	−0.889952	−	0.456054i	$-0.849262\pi$
$311$	−31.3890	−1.77990	−0.889952	+	0.456054i	$0.849262\pi$
$312$	30.0112	1.69905
$313$	21.6308	1.22264	0.611321	−	0.791382i	$-0.290638\pi$
$313$	21.6308	1.22264	0.611321	+	0.791382i	$0.290638\pi$
$314$	−5.07382	−0.286332
$315$	0	0
$316$	46.5613	2.61928
$317$	−26.0643	−1.46392	−0.731959	−	0.681348i	$-0.761394\pi$
$317$	−26.0643	−1.46392	−0.731959	+	0.681348i	$0.761394\pi$
$318$	−14.2622	−0.799784
$319$	−21.6918	−1.21451
$320$	0	0
$321$	−13.2983	−0.742241
$322$	48.3655	2.69530
$323$	0	0
$324$	14.1184	0.784356
$325$	0	0
$326$	−4.06945	−0.225386
$327$	−29.2775	−1.61905
$328$	−6.31835	−0.348872
$329$	−14.8360	−0.817937
$330$	0	0
$331$	−1.38704	−0.0762389	−0.0381194	−	0.999273i	$-0.512137\pi$
$331$	−1.38704	−0.0762389	−0.0381194	+	0.999273i	$0.512137\pi$
$332$	−6.19484	−0.339986
$333$	−3.53306	−0.193611
$334$	8.38802	0.458972
$335$	0	0
$336$	3.77818	0.206117
$337$	18.9728	1.03352	0.516758	−	0.856131i	$-0.327139\pi$
$337$	18.9728	1.03352	0.516758	+	0.856131i	$0.327139\pi$
$338$	5.01611	0.272840
$339$	−43.9711	−2.38818
$340$	0	0
$341$	25.5236	1.38218
$342$	0	0
$343$	−19.9305	−1.07615
$344$	5.40195	0.291253
$345$	0	0
$346$	3.46517	0.186289
$347$	−26.6474	−1.43051	−0.715254	−	0.698865i	$-0.753689\pi$
$347$	−26.6474	−1.43051	−0.715254	+	0.698865i	$0.753689\pi$
$348$	61.2776	3.28482
$349$	−10.7692	−0.576460	−0.288230	−	0.957561i	$-0.593067\pi$
$349$	−10.7692	−0.576460	−0.288230	+	0.957561i	$0.593067\pi$
$350$	0	0
$351$	−23.2368	−1.24029
$352$	−17.2424	−0.919023
$353$	3.45433	0.183856	0.0919278	−	0.995766i	$-0.470697\pi$
$353$	3.45433	0.183856	0.0919278	+	0.995766i	$0.470697\pi$
$354$	40.7873	2.16782
$355$	0	0
$356$	21.7302	1.15170
$357$	9.42229	0.498680
$358$	29.9425	1.58251
$359$	−28.0316	−1.47945	−0.739725	−	0.672909i	$-0.765044\pi$
$359$	−28.0316	−1.47945	−0.739725	+	0.672909i	$0.765044\pi$
$360$	0	0
$361$	0	0
$362$	−23.1477	−1.21662
$363$	2.62852	0.137961
$364$	−25.9270	−1.35895
$365$	0	0
$366$	−19.9356	−1.04205
$367$	−0.122068	−0.00637192	−0.00318596	−	0.999995i	$-0.501014\pi$
$367$	−0.122068	−0.00637192	−0.00318596	+	0.999995i	$0.501014\pi$
$368$	−4.93739	−0.257379
$369$	10.9282	0.568901
$370$	0	0
$371$	4.98016	0.258557
$372$	−72.1021	−3.73832
$373$	−32.4779	−1.68164	−0.840820	−	0.541315i	$-0.817927\pi$
$373$	−32.4779	−1.68164	−0.840820	+	0.541315i	$0.817927\pi$
$374$	11.0425	0.570995
$375$	0	0
$376$	19.8529	1.02383
$377$	−20.6917	−1.06568
$378$	−38.3459	−1.97230
$379$	−32.8185	−1.68578	−0.842888	−	0.538090i	$-0.819146\pi$
$379$	−32.8185	−1.68578	−0.842888	+	0.538090i	$0.819146\pi$
$380$	0	0
$381$	−3.60624	−0.184753
$382$	56.3654	2.88391
$383$	23.5078	1.20119	0.600596	−	0.799553i	$-0.294930\pi$
$383$	23.5078	1.20119	0.600596	+	0.799553i	$0.294930\pi$
$384$	56.1609	2.86595
$385$	0	0
$386$	15.0282	0.764918
$387$	−9.34322	−0.474943
$388$	25.5641	1.29782
$389$	−32.8332	−1.66471	−0.832354	−	0.554244i	$-0.813007\pi$
$389$	−32.8332	−1.66471	−0.832354	+	0.554244i	$0.813007\pi$
$390$	0	0
$391$	−12.3132	−0.622705
$392$	4.68824	0.236792
$393$	−34.1482	−1.72255
$394$	4.94515	0.249133
$395$	0	0
$396$	−62.9079	−3.16124
$397$	35.3737	1.77535	0.887677	−	0.460467i	$-0.152318\pi$
$397$	35.3737	1.77535	0.887677	+	0.460467i	$0.152318\pi$
$398$	40.8876	2.04951
$399$	0	0
$400$	0	0
$401$	−3.00886	−0.150255	−0.0751276	−	0.997174i	$-0.523936\pi$
$401$	−3.00886	−0.150255	−0.0751276	+	0.997174i	$0.523936\pi$
$402$	−16.5635	−0.826112
$403$	24.3468	1.21280
$404$	−55.7056	−2.77146
$405$	0	0
$406$	−34.1460	−1.69464
$407$	2.24444	0.111253
$408$	−12.6085	−0.624212
$409$	−12.0147	−0.594087	−0.297043	−	0.954864i	$-0.596001\pi$
$409$	−12.0147	−0.594087	−0.297043	+	0.954864i	$0.596001\pi$
$410$	0	0
$411$	50.5582	2.49385
$412$	27.1709	1.33861
$413$	−14.2424	−0.700821
$414$	111.941	5.50159
$415$	0	0
$416$	−16.4474	−0.806401
$417$	−30.7126	−1.50400
$418$	0	0
$419$	−10.0485	−0.490900	−0.245450	−	0.969409i	$-0.578936\pi$
$419$	−10.0485	−0.490900	−0.245450	+	0.969409i	$0.578936\pi$
$420$	0	0
$421$	24.4034	1.18935	0.594675	−	0.803966i	$-0.297281\pi$
$421$	24.4034	1.18935	0.594675	+	0.803966i	$0.297281\pi$
$422$	43.9608	2.13998
$423$	−34.3376	−1.66955
$424$	−6.66422	−0.323643
$425$	0	0
$426$	38.3020	1.85574
$427$	6.96123	0.336877
$428$	−15.3735	−0.743104
$429$	32.9752	1.59206
$430$	0	0
$431$	−7.90872	−0.380950	−0.190475	−	0.981692i	$-0.561003\pi$
$431$	−7.90872	−0.380950	−0.190475	+	0.981692i	$0.561003\pi$
$432$	3.91453	0.188338
$433$	−1.34464	−0.0646191	−0.0323095	−	0.999478i	$-0.510286\pi$
$433$	−1.34464	−0.0646191	−0.0323095	+	0.999478i	$0.510286\pi$
$434$	40.1777	1.92859
$435$	0	0
$436$	−33.8460	−1.62093
$437$	0	0
$438$	−46.2429	−2.20957
$439$	11.3408	0.541268	0.270634	−	0.962682i	$-0.412767\pi$
$439$	11.3408	0.541268	0.270634	+	0.962682i	$0.412767\pi$
$440$	0	0
$441$	−8.10880	−0.386133
$442$	10.5334	0.501022
$443$	14.0383	0.666978	0.333489	−	0.942754i	$-0.391774\pi$
$443$	14.0383	0.666978	0.333489	+	0.942754i	$0.391774\pi$
$444$	−6.34036	−0.300900
$445$	0	0
$446$	−46.8687	−2.21930
$447$	10.5556	0.499262
$448$	−29.7443	−1.40528
$449$	3.88724	0.183450	0.0917252	−	0.995784i	$-0.470762\pi$
$449$	3.88724	0.183450	0.0917252	+	0.995784i	$0.470762\pi$
$450$	0	0
$451$	−6.94236	−0.326903
$452$	−50.8325	−2.39096
$453$	−23.2666	−1.09316
$454$	36.6765	1.72131
$455$	0	0
$456$	0	0
$457$	−31.0487	−1.45240	−0.726198	−	0.687486i	$-0.758714\pi$
$457$	−31.0487	−1.45240	−0.726198	+	0.687486i	$0.758714\pi$
$458$	−22.8517	−1.06779
$459$	9.76233	0.455667
$460$	0	0
$461$	31.2148	1.45382	0.726909	−	0.686734i	$-0.240956\pi$
$461$	31.2148	1.45382	0.726909	+	0.686734i	$0.240956\pi$
$462$	54.4165	2.53169
$463$	−29.2480	−1.35927	−0.679634	−	0.733551i	$-0.737861\pi$
$463$	−29.2480	−1.35927	−0.679634	+	0.733551i	$0.737861\pi$
$464$	3.48579	0.161824
$465$	0	0
$466$	8.66223	0.401270
$467$	12.5589	0.581155	0.290578	−	0.956851i	$-0.406153\pi$
$467$	12.5589	0.581155	0.290578	+	0.956851i	$0.406153\pi$
$468$	−60.0074	−2.77384
$469$	5.78375	0.267069
$470$	0	0
$471$	6.36551	0.293307
$472$	19.0585	0.877237
$473$	5.93545	0.272912
$474$	−93.2187	−4.28168
$475$	0	0
$476$	10.8926	0.499261
$477$	11.5265	0.527760
$478$	49.1921	2.25000
$479$	8.84075	0.403944	0.201972	−	0.979391i	$-0.435265\pi$
$479$	8.84075	0.403944	0.201972	+	0.979391i	$0.435265\pi$
$480$	0	0
$481$	2.14096	0.0976193
$482$	−64.1437	−2.92167
$483$	−60.6783	−2.76096
$484$	3.03868	0.138122
$485$	0	0
$486$	20.7548	0.941459
$487$	−7.47254	−0.338613	−0.169307	−	0.985563i	$-0.554153\pi$
$487$	−7.47254	−0.338613	−0.169307	+	0.985563i	$0.554153\pi$
$488$	−9.31518	−0.421679
$489$	5.10544	0.230876
$490$	0	0
$491$	22.1507	0.999647	0.499823	−	0.866127i	$-0.333398\pi$
$491$	22.1507	0.999647	0.499823	+	0.866127i	$0.333398\pi$
$492$	19.6116	0.884158
$493$	8.69310	0.391517
$494$	0	0
$495$	0	0
$496$	−4.10154	−0.184165
$497$	−13.3745	−0.599930
$498$	12.4025	0.555767
$499$	−3.30154	−0.147797	−0.0738985	−	0.997266i	$-0.523544\pi$
$499$	−3.30154	−0.147797	−0.0738985	+	0.997266i	$0.523544\pi$
$500$	0	0
$501$	−10.5234	−0.470152
$502$	39.4673	1.76151
$503$	21.3453	0.951739	0.475870	−	0.879516i	$-0.342133\pi$
$503$	21.3453	0.951739	0.475870	+	0.879516i	$0.342133\pi$
$504$	−40.0255	−1.78288
$505$	0	0
$506$	−71.1124	−3.16133
$507$	−6.29310	−0.279486
$508$	−4.16897	−0.184968
$509$	9.88579	0.438180	0.219090	−	0.975705i	$-0.429691\pi$
$509$	9.88579	0.438180	0.219090	+	0.975705i	$0.429691\pi$
$510$	0	0
$511$	16.1474	0.714319
$512$	6.25310	0.276351
$513$	0	0
$514$	−20.5443	−0.906172
$515$	0	0
$516$	−16.7671	−0.738132
$517$	21.8136	0.959361
$518$	3.53306	0.155234
$519$	−4.34732	−0.190826
$520$	0	0
$521$	−26.6149	−1.16602	−0.583009	−	0.812465i	$-0.698125\pi$
$521$	−26.6149	−1.16602	−0.583009	+	0.812465i	$0.698125\pi$
$522$	−79.0300	−3.45905
$523$	11.4971	0.502732	0.251366	−	0.967892i	$-0.419120\pi$
$523$	11.4971	0.502732	0.251366	+	0.967892i	$0.419120\pi$
$524$	−39.4769	−1.72455
$525$	0	0
$526$	66.2860	2.89021
$527$	−10.2287	−0.445569
$528$	−5.55511	−0.241755
$529$	56.2953	2.44762
$530$	0	0
$531$	−32.9636	−1.43050
$532$	0	0
$533$	−6.62227	−0.286842
$534$	−43.5053	−1.88266
$535$	0	0
$536$	−7.73954	−0.334297
$537$	−37.5652	−1.62106
$538$	−67.5966	−2.91430
$539$	5.15126	0.221881
$540$	0	0
$541$	11.5115	0.494919	0.247460	−	0.968898i	$-0.420404\pi$
$541$	11.5115	0.494919	0.247460	+	0.968898i	$0.420404\pi$
$542$	30.3581	1.30399
$543$	29.0406	1.24625
$544$	6.90996	0.296262
$545$	0	0
$546$	51.9076	2.22144
$547$	37.0866	1.58571	0.792854	−	0.609412i	$-0.208594\pi$
$547$	37.0866	1.58571	0.792854	+	0.609412i	$0.208594\pi$
$548$	58.4475	2.49675
$549$	16.1116	0.687625
$550$	0	0
$551$	0	0
$552$	81.1968	3.45597
$553$	32.5507	1.38420
$554$	−14.4311	−0.613120
$555$	0	0
$556$	−35.5051	−1.50575
$557$	32.6114	1.38179	0.690895	−	0.722955i	$-0.257217\pi$
$557$	32.6114	1.38179	0.690895	+	0.722955i	$0.257217\pi$
$558$	92.9903	3.93659
$559$	5.66179	0.239468
$560$	0	0
$561$	−13.8537	−0.584904
$562$	−38.9125	−1.64143
$563$	39.3465	1.65826	0.829128	−	0.559059i	$-0.188838\pi$
$563$	39.3465	1.65826	0.829128	+	0.559059i	$0.188838\pi$
$564$	−61.6216	−2.59474
$565$	0	0
$566$	−36.4076	−1.53033
$567$	9.87008	0.414505
$568$	17.8972	0.750949
$569$	−2.56181	−0.107397	−0.0536983	−	0.998557i	$-0.517101\pi$
$569$	−2.56181	−0.107397	−0.0536983	+	0.998557i	$0.517101\pi$
$570$	0	0
$571$	39.2894	1.64421	0.822105	−	0.569337i	$-0.192800\pi$
$571$	39.2894	1.64421	0.822105	+	0.569337i	$0.192800\pi$
$572$	38.1208	1.59391
$573$	−70.7148	−2.95415
$574$	−10.9282	−0.456136
$575$	0	0
$576$	−68.8423	−2.86843
$577$	41.0561	1.70919	0.854594	−	0.519297i	$-0.173806\pi$
$577$	41.0561	1.70919	0.854594	+	0.519297i	$0.173806\pi$
$578$	34.9207	1.45251
$579$	−18.8541	−0.783550
$580$	0	0
$581$	−4.33077	−0.179671
$582$	−51.1809	−2.12152
$583$	−7.32239	−0.303262
$584$	−21.6077	−0.894133
$585$	0	0
$586$	42.4950	1.75545
$587$	−10.7634	−0.444252	−0.222126	−	0.975018i	$-0.571300\pi$
$587$	−10.7634	−0.444252	−0.222126	+	0.975018i	$0.571300\pi$
$588$	−14.5519	−0.600109
$589$	0	0
$590$	0	0
$591$	−6.20407	−0.255201
$592$	−0.360673	−0.0148235
$593$	7.83424	0.321714	0.160857	−	0.986978i	$-0.448574\pi$
$593$	7.83424	0.321714	0.160857	+	0.986978i	$0.448574\pi$
$594$	56.3804	2.31332
$595$	0	0
$596$	12.2027	0.499843
$597$	−51.2967	−2.09943
$598$	−67.8336	−2.77392
$599$	29.7590	1.21592	0.607960	−	0.793968i	$-0.291988\pi$
$599$	29.7590	1.21592	0.607960	+	0.793968i	$0.291988\pi$
$600$	0	0
$601$	−5.85799	−0.238952	−0.119476	−	0.992837i	$-0.538122\pi$
$601$	−5.85799	−0.238952	−0.119476	+	0.992837i	$0.538122\pi$
$602$	9.34322	0.380801
$603$	13.3863	0.545134
$604$	−26.8972	−1.09443
$605$	0	0
$606$	111.526	4.53044
$607$	4.74137	0.192446	0.0962231	−	0.995360i	$-0.469324\pi$
$607$	4.74137	0.192446	0.0962231	+	0.995360i	$0.469324\pi$
$608$	0	0
$609$	42.8388	1.73591
$610$	0	0
$611$	20.8079	0.841796
$612$	25.2106	1.01908
$613$	−0.275200	−0.0111152	−0.00555761	−	0.999985i	$-0.501769\pi$
$613$	−0.275200	−0.0111152	−0.00555761	+	0.999985i	$0.501769\pi$
$614$	−49.7478	−2.00766
$615$	0	0
$616$	25.4269	1.02448
$617$	−14.4721	−0.582626	−0.291313	−	0.956628i	$-0.594092\pi$
$617$	−14.4721	−0.582626	−0.291313	+	0.956628i	$0.594092\pi$
$618$	−54.3978	−2.18820
$619$	22.3695	0.899108	0.449554	−	0.893253i	$-0.351583\pi$
$619$	22.3695	0.899108	0.449554	+	0.893253i	$0.351583\pi$
$620$	0	0
$621$	−62.8681	−2.52281
$622$	72.6489	2.91296
$623$	15.1915	0.608634
$624$	−5.29898	−0.212129
$625$	0	0
$626$	−50.0638	−2.00095
$627$	0	0
$628$	7.35881	0.293648
$629$	−0.899470	−0.0358642
$630$	0	0
$631$	−0.112343	−0.00447232	−0.00223616	−	0.999997i	$-0.500712\pi$
$631$	−0.112343	−0.00447232	−0.00223616	+	0.999997i	$0.500712\pi$
$632$	−43.5578	−1.73264
$633$	−55.1523	−2.19211
$634$	60.3252	2.39582
$635$	0	0
$636$	20.6851	0.820218
$637$	4.91375	0.194690
$638$	50.2052	1.98764
$639$	−30.9550	−1.22456
$640$	0	0
$641$	−26.8374	−1.06001	−0.530006	−	0.847994i	$-0.677810\pi$
$641$	−26.8374	−1.06001	−0.530006	+	0.847994i	$0.677810\pi$
$642$	30.7787	1.21474
$643$	3.66175	0.144405	0.0722027	−	0.997390i	$-0.476997\pi$
$643$	3.66175	0.144405	0.0722027	+	0.997390i	$0.476997\pi$
$644$	−70.1468	−2.76417
$645$	0	0
$646$	0	0
$647$	29.3983	1.15577	0.577883	−	0.816120i	$-0.303879\pi$
$647$	29.3983	1.15577	0.577883	+	0.816120i	$0.303879\pi$
$648$	−13.2077	−0.518847
$649$	20.9407	0.821995
$650$	0	0
$651$	−50.4061	−1.97557
$652$	5.90211	0.231144
$653$	−20.3618	−0.796819	−0.398409	−	0.917208i	$-0.630438\pi$
$653$	−20.3618	−0.796819	−0.398409	+	0.917208i	$0.630438\pi$
$654$	67.7619	2.64970
$655$	0	0
$656$	1.11561	0.0435572
$657$	37.3727	1.45805
$658$	34.3376	1.33862
$659$	−25.7612	−1.00352	−0.501758	−	0.865008i	$-0.667313\pi$
$659$	−25.7612	−1.00352	−0.501758	+	0.865008i	$0.667313\pi$
$660$	0	0
$661$	9.09527	0.353765	0.176883	−	0.984232i	$-0.443399\pi$
$661$	9.09527	0.353765	0.176883	+	0.984232i	$0.443399\pi$
$662$	3.21028	0.124771
$663$	−13.2150	−0.513226
$664$	5.79523	0.224899
$665$	0	0
$666$	8.17719	0.316860
$667$	−55.9824	−2.16765
$668$	−12.1655	−0.470699
$669$	58.8004	2.27336
$670$	0	0
$671$	−10.2352	−0.395124
$672$	34.0517	1.31357
$673$	8.00415	0.308537	0.154269	−	0.988029i	$-0.450698\pi$
$673$	8.00415	0.308537	0.154269	+	0.988029i	$0.450698\pi$
$674$	−43.9121	−1.69143
$675$	0	0
$676$	−7.27510	−0.279812
$677$	1.04002	0.0399712	0.0199856	−	0.999800i	$-0.493638\pi$
$677$	1.04002	0.0399712	0.0199856	+	0.999800i	$0.493638\pi$
$678$	101.770	3.90845
$679$	17.8717	0.685853
$680$	0	0
$681$	−46.0135	−1.76324
$682$	−59.0738	−2.26205
$683$	31.1559	1.19215	0.596073	−	0.802930i	$-0.296727\pi$
$683$	31.1559	1.19215	0.596073	+	0.802930i	$0.296727\pi$
$684$	0	0
$685$	0	0
$686$	46.1286	1.76120
$687$	28.6693	1.09380
$688$	−0.953802	−0.0363634
$689$	−6.98478	−0.266099
$690$	0	0
$691$	23.1659	0.881272	0.440636	−	0.897686i	$-0.354753\pi$
$691$	23.1659	0.881272	0.440636	+	0.897686i	$0.354753\pi$
$692$	−5.02570	−0.191048
$693$	−43.9785	−1.67061
$694$	61.6747	2.34114
$695$	0	0
$696$	−57.3248	−2.17289
$697$	2.78218	0.105383
$698$	24.9250	0.943423
$699$	−10.8674	−0.411044
$700$	0	0
$701$	10.2358	0.386601	0.193300	−	0.981140i	$-0.438081\pi$
$701$	10.2358	0.386601	0.193300	+	0.981140i	$0.438081\pi$
$702$	53.7809	2.02983
$703$	0	0
$704$	43.7333	1.64826
$705$	0	0
$706$	−7.99497	−0.300895
$707$	−38.9434	−1.46462
$708$	−59.1557	−2.22321
$709$	24.3417	0.914171	0.457085	−	0.889423i	$-0.348893\pi$
$709$	24.3417	0.914171	0.457085	+	0.889423i	$0.348893\pi$
$710$	0	0
$711$	75.3378	2.82539
$712$	−20.3285	−0.761843
$713$	65.8715	2.46691
$714$	−21.8076	−0.816131
$715$	0	0
$716$	−43.4271	−1.62295
$717$	−61.7154	−2.30480
$718$	64.8784	2.42124
$719$	−22.3900	−0.835007	−0.417504	−	0.908675i	$-0.637095\pi$
$719$	−22.3900	−0.835007	−0.417504	+	0.908675i	$0.637095\pi$
$720$	0	0
$721$	18.9950	0.707410
$722$	0	0
$723$	80.4732	2.99283
$724$	33.5723	1.24770
$725$	0	0
$726$	−6.08364	−0.225785
$727$	−6.53528	−0.242380	−0.121190	−	0.992629i	$-0.538671\pi$
$727$	−6.53528	−0.242380	−0.121190	+	0.992629i	$0.538671\pi$
$728$	24.2546	0.898934
$729$	−38.6563	−1.43172
$730$	0	0
$731$	−2.37866	−0.0879778
$732$	28.9135	1.06867
$733$	−10.5576	−0.389954	−0.194977	−	0.980808i	$-0.562463\pi$
$733$	−10.5576	−0.389954	−0.194977	+	0.980808i	$0.562463\pi$
$734$	0.282524	0.0104282
$735$	0	0
$736$	−44.4992	−1.64026
$737$	−8.50391	−0.313246
$738$	−25.2931	−0.931053
$739$	9.05935	0.333254	0.166627	−	0.986020i	$-0.446712\pi$
$739$	9.05935	0.333254	0.166627	+	0.986020i	$0.446712\pi$
$740$	0	0
$741$	0	0
$742$	−11.5265	−0.423150
$743$	−10.6892	−0.392147	−0.196074	−	0.980589i	$-0.562819\pi$
$743$	−10.6892	−0.392147	−0.196074	+	0.980589i	$0.562819\pi$
$744$	67.4510	2.47288
$745$	0	0
$746$	75.1692	2.75214
$747$	−10.0235	−0.366739
$748$	−16.0155	−0.585584
$749$	−10.7475	−0.392705
$750$	0	0
$751$	−50.9596	−1.85954	−0.929772	−	0.368136i	$-0.879996\pi$
$751$	−50.9596	−1.85954	−0.929772	+	0.368136i	$0.879996\pi$
$752$	−3.50535	−0.127827
$753$	−49.5147	−1.80442
$754$	47.8904	1.74407
$755$	0	0
$756$	55.6148	2.02269
$757$	48.7783	1.77288	0.886438	−	0.462847i	$-0.153172\pi$
$757$	48.7783	1.77288	0.886438	+	0.462847i	$0.153172\pi$
$758$	75.9577	2.75891
$759$	89.2160	3.23834
$760$	0	0
$761$	30.3424	1.09991	0.549956	−	0.835193i	$-0.314644\pi$
$761$	30.3424	1.09991	0.549956	+	0.835193i	$0.314644\pi$
$762$	8.34654	0.302363
$763$	−23.6615	−0.856605
$764$	−81.7494	−2.95759
$765$	0	0
$766$	−54.4082	−1.96585
$767$	19.9752	0.721263
$768$	−56.3751	−2.03426
$769$	13.4786	0.486052	0.243026	−	0.970020i	$-0.421860\pi$
$769$	13.4786	0.486052	0.243026	+	0.970020i	$0.421860\pi$
$770$	0	0
$771$	25.7745	0.928245
$772$	−21.7962	−0.784461
$773$	16.3898	0.589500	0.294750	−	0.955574i	$-0.404764\pi$
$773$	16.3898	0.589500	0.294750	+	0.955574i	$0.404764\pi$
$774$	21.6247	0.777282
$775$	0	0
$776$	−23.9151	−0.858501
$777$	−4.43250	−0.159015
$778$	75.9915	2.72443
$779$	0	0
$780$	0	0
$781$	19.6647	0.703659
$782$	28.4986	1.01911
$783$	44.3848	1.58618
$784$	−0.827786	−0.0295638
$785$	0	0
$786$	79.0352	2.81909
$787$	−13.6646	−0.487090	−0.243545	−	0.969890i	$-0.578310\pi$
$787$	−13.6646	−0.487090	−0.243545	+	0.969890i	$0.578310\pi$
$788$	−7.17218	−0.255498
$789$	−83.1609	−2.96061
$790$	0	0
$791$	−35.5367	−1.26354
$792$	58.8500	2.09114
$793$	−9.76326	−0.346704
$794$	−81.8715	−2.90551
$795$	0	0
$796$	−59.3012	−2.10188
$797$	33.2973	1.17945	0.589725	−	0.807604i	$-0.299236\pi$
$797$	33.2973	1.17945	0.589725	+	0.807604i	$0.299236\pi$
$798$	0	0
$799$	−8.74189	−0.309266
$800$	0	0
$801$	35.1603	1.24233
$802$	6.96393	0.245905
$803$	−23.7417	−0.837827
$804$	24.0228	0.847219
$805$	0	0
$806$	−56.3501	−1.98485
$807$	84.8052	2.98528
$808$	52.1123	1.83330
$809$	49.3206	1.73402	0.867010	−	0.498291i	$-0.166039\pi$
$809$	49.3206	1.73402	0.867010	+	0.498291i	$0.166039\pi$
$810$	0	0
$811$	−40.7084	−1.42947	−0.714733	−	0.699398i	$-0.753452\pi$
$811$	−40.7084	−1.42947	−0.714733	+	0.699398i	$0.753452\pi$
$812$	49.5235	1.73793
$813$	−38.0866	−1.33576
$814$	−5.19471	−0.182074
$815$	0	0
$816$	2.22623	0.0779337
$817$	0	0
$818$	27.8076	0.972271
$819$	−41.9508	−1.46588
$820$	0	0
$821$	−32.8536	−1.14660	−0.573300	−	0.819346i	$-0.694337\pi$
$821$	−32.8536	−1.14660	−0.573300	+	0.819346i	$0.694337\pi$
$822$	−117.016	−4.08139
$823$	−19.7037	−0.686826	−0.343413	−	0.939184i	$-0.611583\pi$
$823$	−19.7037	−0.686826	−0.343413	+	0.939184i	$0.611583\pi$
$824$	−25.4182	−0.885484
$825$	0	0
$826$	32.9636	1.14695
$827$	−12.4937	−0.434447	−0.217224	−	0.976122i	$-0.569700\pi$
$827$	−12.4937	−0.434447	−0.217224	+	0.976122i	$0.569700\pi$
$828$	−162.353	−5.64215
$829$	1.71981	0.0597316	0.0298658	−	0.999554i	$-0.490492\pi$
$829$	1.71981	0.0597316	0.0298658	+	0.999554i	$0.490492\pi$
$830$	0	0
$831$	18.1050	0.628055
$832$	41.7169	1.44627
$833$	−2.06439	−0.0715268
$834$	71.0834	2.46142
$835$	0	0
$836$	0	0
$837$	−52.2252	−1.80517
$838$	23.2570	0.803398
$839$	14.8429	0.512432	0.256216	−	0.966620i	$-0.417524\pi$
$839$	14.8429	0.512432	0.256216	+	0.966620i	$0.417524\pi$
$840$	0	0
$841$	10.5235	0.362878
$842$	−56.4811	−1.94647
$843$	48.8188	1.68141
$844$	−63.7585	−2.19466
$845$	0	0
$846$	79.4736	2.73236
$847$	2.12432	0.0729926
$848$	1.17668	0.0404073
$849$	45.6762	1.56760
$850$	0	0
$851$	5.79247	0.198563
$852$	−55.5512	−1.90315
$853$	−37.7010	−1.29086	−0.645430	−	0.763820i	$-0.723322\pi$
$853$	−37.7010	−1.29086	−0.645430	+	0.763820i	$0.723322\pi$
$854$	−16.1116	−0.551327
$855$	0	0
$856$	14.3818	0.491559
$857$	8.91399	0.304496	0.152248	−	0.988342i	$-0.451349\pi$
$857$	8.91399	0.304496	0.152248	+	0.988342i	$0.451349\pi$
$858$	−76.3203	−2.60553
$859$	−7.01741	−0.239431	−0.119715	−	0.992808i	$-0.538198\pi$
$859$	−7.01741	−0.239431	−0.119715	+	0.992808i	$0.538198\pi$
$860$	0	0
$861$	13.7103	0.467247
$862$	18.3045	0.623455
$863$	−22.9207	−0.780231	−0.390115	−	0.920766i	$-0.627565\pi$
$863$	−22.9207	−0.780231	−0.390115	+	0.920766i	$0.627565\pi$
$864$	35.2806	1.20027
$865$	0	0
$866$	3.11212	0.105754
$867$	−43.8108	−1.48789
$868$	−58.2717	−1.97787
$869$	−47.8597	−1.62353
$870$	0	0
$871$	−8.11182	−0.274859
$872$	31.6628	1.07224
$873$	41.3636	1.39994
$874$	0	0
$875$	0	0
$876$	67.0683	2.26603
$877$	7.56012	0.255287	0.127644	−	0.991820i	$-0.459259\pi$
$877$	7.56012	0.255287	0.127644	+	0.991820i	$0.459259\pi$
$878$	−26.2481	−0.885829
$879$	−53.3133	−1.79821
$880$	0	0
$881$	6.69446	0.225542	0.112771	−	0.993621i	$-0.464027\pi$
$881$	6.69446	0.225542	0.112771	+	0.993621i	$0.464027\pi$
$882$	18.7676	0.631938
$883$	−14.9282	−0.502375	−0.251188	−	0.967938i	$-0.580821\pi$
$883$	−14.9282	−0.502375	−0.251188	+	0.967938i	$0.580821\pi$
$884$	−15.2771	−0.513823
$885$	0	0
$886$	−32.4912	−1.09156
$887$	−25.5872	−0.859134	−0.429567	−	0.903035i	$-0.641334\pi$
$887$	−25.5872	−0.859134	−0.429567	+	0.903035i	$0.641334\pi$
$888$	5.93137	0.199044
$889$	−2.91450	−0.0977492
$890$	0	0
$891$	−14.5121	−0.486174
$892$	67.9759	2.27600
$893$	0	0
$894$	−24.4306	−0.817083
$895$	0	0
$896$	45.3883	1.51632
$897$	85.1025	2.84149
$898$	−8.99692	−0.300231
$899$	−46.5051	−1.55103
$900$	0	0
$901$	2.93448	0.0977616
$902$	16.0679	0.535003
$903$	−11.7218	−0.390077
$904$	47.5535	1.58161
$905$	0	0
$906$	53.8499	1.78904
$907$	−11.0018	−0.365308	−0.182654	−	0.983177i	$-0.558469\pi$
$907$	−11.0018	−0.365308	−0.182654	+	0.983177i	$0.558469\pi$
$908$	−53.1936	−1.76529
$909$	−90.1336	−2.98954
$910$	0	0
$911$	38.0282	1.25993	0.629966	−	0.776623i	$-0.283069\pi$
$911$	38.0282	1.25993	0.629966	+	0.776623i	$0.283069\pi$
$912$	0	0
$913$	6.36758	0.210736
$914$	71.8614	2.37696
$915$	0	0
$916$	33.1429	1.09507
$917$	−27.5980	−0.911367
$918$	−22.5947	−0.745735
$919$	7.71405	0.254463	0.127231	−	0.991873i	$-0.459391\pi$
$919$	7.71405	0.254463	0.127231	+	0.991873i	$0.459391\pi$
$920$	0	0
$921$	62.4125	2.05656
$922$	−72.2459	−2.37929
$923$	18.7581	0.617429
$924$	−78.9229	−2.59637
$925$	0	0
$926$	67.6936	2.22455
$927$	43.9634	1.44395
$928$	31.4164	1.03129
$929$	22.8264	0.748909	0.374455	−	0.927245i	$-0.377830\pi$
$929$	22.8264	0.748909	0.374455	+	0.927245i	$0.377830\pi$
$930$	0	0
$931$	0	0
$932$	−12.5632	−0.411523
$933$	−91.1437	−2.98391
$934$	−29.0672	−0.951107
$935$	0	0
$936$	56.1366	1.83488
$937$	−14.8699	−0.485779	−0.242889	−	0.970054i	$-0.578095\pi$
$937$	−14.8699	−0.485779	−0.242889	+	0.970054i	$0.578095\pi$
$938$	−13.3863	−0.437079
$939$	62.8090	2.04969
$940$	0	0
$941$	56.0412	1.82689	0.913445	−	0.406962i	$-0.133412\pi$
$941$	56.0412	1.82689	0.913445	+	0.406962i	$0.133412\pi$
$942$	−14.7328	−0.480021
$943$	−17.9169	−0.583454
$944$	−3.36508	−0.109524
$945$	0	0
$946$	−13.7375	−0.446643
$947$	52.2948	1.69935	0.849676	−	0.527305i	$-0.176798\pi$
$947$	52.2948	1.69935	0.849676	+	0.527305i	$0.176798\pi$
$948$	135.200	4.39108
$949$	−22.6471	−0.735155
$950$	0	0
$951$	−75.6827	−2.45418
$952$	−10.1899	−0.330258
$953$	−0.366484	−0.0118716	−0.00593578	−	0.999982i	$-0.501889\pi$
$953$	−0.366484	−0.0118716	−0.00593578	+	0.999982i	$0.501889\pi$
$954$	−26.6777	−0.863722
$955$	0	0
$956$	−71.3457	−2.30748
$957$	−62.9863	−2.03606
$958$	−20.4617	−0.661087
$959$	40.8603	1.31945
$960$	0	0
$961$	23.7201	0.765164
$962$	−4.95520	−0.159762
$963$	−24.8748	−0.801579
$964$	93.0306	2.99631
$965$	0	0
$966$	140.438	4.51853
$967$	−20.1566	−0.648193	−0.324097	−	0.946024i	$-0.605060\pi$
$967$	−20.1566	−0.648193	−0.324097	+	0.946024i	$0.605060\pi$
$968$	−2.84267	−0.0913669
$969$	0	0
$970$	0	0
$971$	−24.7007	−0.792682	−0.396341	−	0.918103i	$-0.629720\pi$
$971$	−24.7007	−0.792682	−0.396341	+	0.918103i	$0.629720\pi$
$972$	−30.1017	−0.965513
$973$	−24.8214	−0.795736
$974$	17.2950	0.554168
$975$	0	0
$976$	1.64475	0.0526471
$977$	10.2194	0.326946	0.163473	−	0.986548i	$-0.447730\pi$
$977$	10.2194	0.326946	0.163473	+	0.986548i	$0.447730\pi$
$978$	−11.8164	−0.377847
$979$	−22.3362	−0.713868
$980$	0	0
$981$	−54.7640	−1.74848
$982$	−51.2672	−1.63600
$983$	−21.2356	−0.677311	−0.338656	−	0.940910i	$-0.609972\pi$
$983$	−21.2356	−0.677311	−0.338656	+	0.940910i	$0.609972\pi$
$984$	−18.3465	−0.584865
$985$	0	0
$986$	−20.1199	−0.640750
$987$	−43.0792	−1.37123
$988$	0	0
$989$	15.3182	0.487091
$990$	0	0
$991$	−20.7117	−0.657929	−0.328964	−	0.944342i	$-0.606700\pi$
$991$	−20.7117	−0.657929	−0.328964	+	0.944342i	$0.606700\pi$
$992$	−36.9660	−1.17367
$993$	−4.02754	−0.127810
$994$	30.9550	0.981833
$995$	0	0
$996$	−17.9879	−0.569967
$997$	−49.8656	−1.57926	−0.789630	−	0.613583i	$-0.789727\pi$
$997$	−49.8656	−1.57926	−0.789630	+	0.613583i	$0.789727\pi$
$998$	7.64132	0.241882
$999$	−4.59247	−0.145299

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.ck.1.2		16
5.2	odd	4		1805.2.b.j.1084.2	yes	16
5.3	odd	4		1805.2.b.j.1084.15	yes	16
5.4	even	2	inner	9025.2.a.ck.1.15		16
19.18	odd	2		9025.2.a.cl.1.15		16
95.18	even	4		1805.2.b.i.1084.2	✓	16
95.37	even	4		1805.2.b.i.1084.15	yes	16
95.94	odd	2		9025.2.a.cl.1.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1805.2.b.i.1084.2	✓	16	95.18	even	4
1805.2.b.i.1084.15	yes	16	95.37	even	4
1805.2.b.j.1084.2	yes	16	5.2	odd	4
1805.2.b.j.1084.15	yes	16	5.3	odd	4
9025.2.a.ck.1.2		16	1.1	even	1	trivial
9025.2.a.ck.1.15		16	5.4	even	2	inner
9025.2.a.cl.1.2		16	95.94	odd	2
9025.2.a.cl.1.15		16	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)

\(f(q)\)	\(=\)	\(q-2.31447 q^{2} +2.90369 q^{3} +3.35679 q^{4} -6.72051 q^{6} +2.34671 q^{7} -3.14026 q^{8} +5.43140 q^{9} +O(q^{10})\)	\(q-2.31447 q^{2} +2.90369 q^{3} +3.35679 q^{4} -6.72051 q^{6} +2.34671 q^{7} -3.14026 q^{8} +5.43140 q^{9} -3.45040 q^{11} +9.74707 q^{12} -3.29131 q^{13} -5.43140 q^{14} +0.554464 q^{16} +1.38276 q^{17} -12.5708 q^{18} +6.81412 q^{21} +7.98585 q^{22} -8.90479 q^{23} -9.11833 q^{24} +7.61765 q^{26} +7.06003 q^{27} +7.87742 q^{28} +6.28677 q^{29} -7.39730 q^{31} +4.99722 q^{32} -10.0189 q^{33} -3.20036 q^{34} +18.2321 q^{36} -0.650489 q^{37} -9.55694 q^{39} +2.01205 q^{41} -15.7711 q^{42} -1.72022 q^{43} -11.5823 q^{44} +20.6099 q^{46} -6.32206 q^{47} +1.60999 q^{48} -1.49295 q^{49} +4.01510 q^{51} -11.0482 q^{52} +2.12219 q^{53} -16.3403 q^{54} -7.36928 q^{56} -14.5506 q^{58} -6.06907 q^{59} +2.96638 q^{61} +17.1209 q^{62} +12.7459 q^{63} -12.6749 q^{64} +23.1884 q^{66} +2.46462 q^{67} +4.64164 q^{68} -25.8567 q^{69} -5.69927 q^{71} -17.0560 q^{72} +6.88086 q^{73} +1.50554 q^{74} -8.09708 q^{77} +22.1193 q^{78} +13.8708 q^{79} +4.20592 q^{81} -4.65683 q^{82} -1.84546 q^{83} +22.8736 q^{84} +3.98141 q^{86} +18.2548 q^{87} +10.8351 q^{88} +6.47352 q^{89} -7.72375 q^{91} -29.8915 q^{92} -21.4795 q^{93} +14.6322 q^{94} +14.5104 q^{96} +7.61563 q^{97} +3.45539 q^{98} -18.7405 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 12 q^{4} - 10 q^{6} + 6 q^{9}+O(q^{10}) \) 16 * q + 12 * q^4 - 10 * q^6 + 6 * q^9	\( 16 q + 12 q^{4} - 10 q^{6} + 6 q^{9} - 22 q^{11} - 6 q^{14} + 8 q^{16} + 20 q^{21} - 14 q^{24} - 16 q^{26} - 2 q^{29} - 16 q^{31} + 8 q^{34} + 18 q^{36} - 36 q^{39} - 26 q^{41} - 64 q^{44} - 2 q^{46} - 20 q^{49} + 38 q^{51} - 12 q^{54} - 6 q^{56} - 10 q^{59} - 30 q^{61} - 16 q^{64} + 4 q^{66} - 68 q^{69} + 20 q^{71} - 40 q^{74} - 12 q^{79} - 48 q^{81} + 2 q^{84} + 20 q^{86} + 86 q^{91} + 38 q^{94} + 22 q^{96} - 32 q^{99}+O(q^{100}) \) 16 * q + 12 * q^4 - 10 * q^6 + 6 * q^9 - 22 * q^11 - 6 * q^14 + 8 * q^16 + 20 * q^21 - 14 * q^24 - 16 * q^26 - 2 * q^29 - 16 * q^31 + 8 * q^34 + 18 * q^36 - 36 * q^39 - 26 * q^41 - 64 * q^44 - 2 * q^46 - 20 * q^49 + 38 * q^51 - 12 * q^54 - 6 * q^56 - 10 * q^59 - 30 * q^61 - 16 * q^64 + 4 * q^66 - 68 * q^69 + 20 * q^71 - 40 * q^74 - 12 * q^79 - 48 * q^81 + 2 * q^84 + 20 * q^86 + 86 * q^91 + 38 * q^94 + 22 * q^96 - 32 * q^99

Introduction

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