LMFDB - Embedded newform 3025.2.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3025,2,Mod(1,3025)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$24.1547466114$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 605)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.517638$ 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)

$f(q)$	$=$	$q-0.517638 q^{2} +1.93185 q^{3} -1.73205 q^{4} -1.00000 q^{6} +3.34607 q^{7} +1.93185 q^{8} +0.732051 q^{9} +O(q^{10})$	\(q-0.517638 q^{2} +1.93185 q^{3} -1.73205 q^{4} -1.00000 q^{6} +3.34607 q^{7} +1.93185 q^{8} +0.732051 q^{9} -3.34607 q^{12} -4.24264 q^{13} -1.73205 q^{14} +2.46410 q^{16} -3.86370 q^{17} -0.378937 q^{18} -4.19615 q^{19} +6.46410 q^{21} -3.48477 q^{23} +3.73205 q^{24} +2.19615 q^{26} -4.38134 q^{27} -5.79555 q^{28} +6.92820 q^{29} -8.73205 q^{31} -5.13922 q^{32} +2.00000 q^{34} -1.26795 q^{36} -1.79315 q^{37} +2.17209 q^{38} -8.19615 q^{39} +1.73205 q^{41} -3.34607 q^{42} +6.45189 q^{43} +1.80385 q^{46} -11.4524 q^{47} +4.76028 q^{48} +4.19615 q^{49} -7.46410 q^{51} +7.34847 q^{52} -2.17209 q^{53} +2.26795 q^{54} +6.46410 q^{56} -8.10634 q^{57} -3.58630 q^{58} -1.26795 q^{59} -11.7321 q^{61} +4.52004 q^{62} +2.44949 q^{63} -2.26795 q^{64} -2.20925 q^{67} +6.69213 q^{68} -6.73205 q^{69} -8.19615 q^{71} +1.41421 q^{72} +4.89898 q^{73} +0.928203 q^{74} +7.26795 q^{76} +4.24264 q^{78} +1.46410 q^{79} -10.6603 q^{81} -0.896575 q^{82} +9.89949 q^{83} -11.1962 q^{84} -3.33975 q^{86} +13.3843 q^{87} +0.464102 q^{89} -14.1962 q^{91} +6.03579 q^{92} -16.8690 q^{93} +5.92820 q^{94} -9.92820 q^{96} +9.14162 q^{97} -2.17209 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{6} - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^6 - 4 * q^9	$ 4 q - 4 q^{6} - 4 q^{9} - 4 q^{16} + 4 q^{19} + 12 q^{21} + 8 q^{24} - 12 q^{26} - 28 q^{31} + 8 q^{34} - 12 q^{36} - 12 q^{39} + 28 q^{46} - 4 q^{49} - 16 q^{51} + 16 q^{54} + 12 q^{56} - 12 q^{59} - 40 q^{61} - 16 q^{64} - 20 q^{69} - 12 q^{71} - 24 q^{74} + 36 q^{76} - 8 q^{79} - 8 q^{81} - 24 q^{84} - 48 q^{86} - 12 q^{89} - 36 q^{91} - 4 q^{94} - 12 q^{96}+O(q^{100}) $ 4 * q - 4 * q^6 - 4 * q^9 - 4 * q^16 + 4 * q^19 + 12 * q^21 + 8 * q^24 - 12 * q^26 - 28 * q^31 + 8 * q^34 - 12 * q^36 - 12 * q^39 + 28 * q^46 - 4 * q^49 - 16 * q^51 + 16 * q^54 + 12 * q^56 - 12 * q^59 - 40 * q^61 - 16 * q^64 - 20 * q^69 - 12 * q^71 - 24 * q^74 + 36 * q^76 - 8 * q^79 - 8 * q^81 - 24 * q^84 - 48 * q^86 - 12 * q^89 - 36 * q^91 - 4 * q^94 - 12 * q^96

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 $

$2$	−0.517638	−0.366025	−0.183013	−	0.983111i	$-0.558585\pi$
$2$	−0.517638	−0.366025	−0.183013	+	0.983111i	$0.558585\pi$
$3$	1.93185	1.11536	0.557678	−	0.830058i	$-0.311693\pi$
$3$	1.93185	1.11536	0.557678	+	0.830058i	$0.311693\pi$
$4$	−1.73205	−0.866025
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	3.34607	1.26469	0.632347	−	0.774685i	$-0.282092\pi$
$7$	3.34607	1.26469	0.632347	+	0.774685i	$0.282092\pi$
$8$	1.93185	0.683013
$9$	0.732051	0.244017
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−3.34607	−0.965926
$13$	−4.24264	−1.17670	−0.588348	−	0.808608i	$-0.700222\pi$
$13$	−4.24264	−1.17670	−0.588348	+	0.808608i	$0.700222\pi$
$14$	−1.73205	−0.462910
$15$	0	0
$16$	2.46410	0.616025
$17$	−3.86370	−0.937086	−0.468543	−	0.883441i	$-0.655221\pi$
$17$	−3.86370	−0.937086	−0.468543	+	0.883441i	$0.655221\pi$
$18$	−0.378937	−0.0893164
$19$	−4.19615	−0.962663	−0.481332	−	0.876539i	$-0.659847\pi$
$19$	−4.19615	−0.962663	−0.481332	+	0.876539i	$0.659847\pi$
$20$	0	0
$21$	6.46410	1.41058
$22$	0	0
$23$	−3.48477	−0.726624	−0.363312	−	0.931668i	$-0.618354\pi$
$23$	−3.48477	−0.726624	−0.363312	+	0.931668i	$0.618354\pi$
$24$	3.73205	0.761802
$25$	0	0
$26$	2.19615	0.430701
$27$	−4.38134	−0.843190
$28$	−5.79555	−1.09526
$29$	6.92820	1.28654	0.643268	−	0.765641i	$-0.277578\pi$
$29$	6.92820	1.28654	0.643268	+	0.765641i	$0.277578\pi$
$30$	0	0
$31$	−8.73205	−1.56832	−0.784161	−	0.620557i	$-0.786907\pi$
$31$	−8.73205	−1.56832	−0.784161	+	0.620557i	$0.786907\pi$
$32$	−5.13922	−0.908494
$33$	0	0
$34$	2.00000	0.342997
$35$	0	0
$36$	−1.26795	−0.211325
$37$	−1.79315	−0.294792	−0.147396	−	0.989078i	$-0.547089\pi$
$37$	−1.79315	−0.294792	−0.147396	+	0.989078i	$0.547089\pi$
$38$	2.17209	0.352359
$39$	−8.19615	−1.31243
$40$	0	0
$41$	1.73205	0.270501	0.135250	−	0.990811i	$-0.456816\pi$
$41$	1.73205	0.270501	0.135250	+	0.990811i	$0.456816\pi$
$42$	−3.34607	−0.516309
$43$	6.45189	0.983905	0.491952	−	0.870622i	$-0.336283\pi$
$43$	6.45189	0.983905	0.491952	+	0.870622i	$0.336283\pi$
$44$	0	0
$45$	0	0
$46$	1.80385	0.265963
$47$	−11.4524	−1.67051	−0.835253	−	0.549866i	$-0.814679\pi$
$47$	−11.4524	−1.67051	−0.835253	+	0.549866i	$0.814679\pi$
$48$	4.76028	0.687087
$49$	4.19615	0.599450
$50$	0	0
$51$	−7.46410	−1.04518
$52$	7.34847	1.01905
$53$	−2.17209	−0.298359	−0.149180	−	0.988810i	$-0.547663\pi$
$53$	−2.17209	−0.298359	−0.149180	+	0.988810i	$0.547663\pi$
$54$	2.26795	0.308629
$55$	0	0
$56$	6.46410	0.863802
$57$	−8.10634	−1.07371
$58$	−3.58630	−0.470905
$59$	−1.26795	−0.165073	−0.0825365	−	0.996588i	$-0.526302\pi$
$59$	−1.26795	−0.165073	−0.0825365	+	0.996588i	$0.526302\pi$
$60$	0	0
$61$	−11.7321	−1.50214	−0.751068	−	0.660225i	$-0.770461\pi$
$61$	−11.7321	−1.50214	−0.751068	+	0.660225i	$0.770461\pi$
$62$	4.52004	0.574046
$63$	2.44949	0.308607
$64$	−2.26795	−0.283494
$65$	0	0
$66$	0	0
$67$	−2.20925	−0.269903	−0.134952	−	0.990852i	$-0.543088\pi$
$67$	−2.20925	−0.269903	−0.134952	+	0.990852i	$0.543088\pi$
$68$	6.69213	0.811540
$69$	−6.73205	−0.810444
$70$	0	0
$71$	−8.19615	−0.972704	−0.486352	−	0.873763i	$-0.661673\pi$
$71$	−8.19615	−0.972704	−0.486352	+	0.873763i	$0.661673\pi$
$72$	1.41421	0.166667
$73$	4.89898	0.573382	0.286691	−	0.958023i	$-0.407445\pi$
$73$	4.89898	0.573382	0.286691	+	0.958023i	$0.407445\pi$
$74$	0.928203	0.107901
$75$	0	0
$76$	7.26795	0.833691
$77$	0	0
$78$	4.24264	0.480384
$79$	1.46410	0.164724	0.0823622	−	0.996602i	$-0.473754\pi$
$79$	1.46410	0.164724	0.0823622	+	0.996602i	$0.473754\pi$
$80$	0	0
$81$	−10.6603	−1.18447
$82$	−0.896575	−0.0990102
$83$	9.89949	1.08661	0.543305	−	0.839535i	$-0.317173\pi$
$83$	9.89949	1.08661	0.543305	+	0.839535i	$0.317173\pi$
$84$	−11.1962	−1.22160
$85$	0	0
$86$	−3.33975	−0.360134
$87$	13.3843	1.43494
$88$	0	0
$89$	0.464102	0.0491947	0.0245973	−	0.999697i	$-0.492170\pi$
$89$	0.464102	0.0491947	0.0245973	+	0.999697i	$0.492170\pi$
$90$	0	0
$91$	−14.1962	−1.48816
$92$	6.03579	0.629275
$93$	−16.8690	−1.74924
$94$	5.92820	0.611447
$95$	0	0
$96$	−9.92820	−1.01329
$97$	9.14162	0.928191	0.464095	−	0.885785i	$-0.346379\pi$
$97$	9.14162	0.928191	0.464095	+	0.885785i	$0.346379\pi$
$98$	−2.17209	−0.219414
$99$	0	0
$100$	0	0
$101$	19.3923	1.92961	0.964803	−	0.262973i	$-0.0847030\pi$
$101$	19.3923	1.92961	0.964803	+	0.262973i	$0.0847030\pi$
$102$	3.86370	0.382564
$103$	−4.24264	−0.418040	−0.209020	−	0.977911i	$-0.567027\pi$
$103$	−4.24264	−0.418040	−0.209020	+	0.977911i	$0.567027\pi$
$104$	−8.19615	−0.803699
$105$	0	0
$106$	1.12436	0.109207
$107$	−2.96713	−0.286843	−0.143422	−	0.989662i	$-0.545810\pi$
$107$	−2.96713	−0.286843	−0.143422	+	0.989662i	$0.545810\pi$
$108$	7.58871	0.730224
$109$	9.19615	0.880832	0.440416	−	0.897794i	$-0.354831\pi$
$109$	9.19615	0.880832	0.440416	+	0.897794i	$0.354831\pi$
$110$	0	0
$111$	−3.46410	−0.328798
$112$	8.24504	0.779083
$113$	−2.82843	−0.266076	−0.133038	−	0.991111i	$-0.542473\pi$
$113$	−2.82843	−0.266076	−0.133038	+	0.991111i	$0.542473\pi$
$114$	4.19615	0.393006
$115$	0	0
$116$	−12.0000	−1.11417
$117$	−3.10583	−0.287134
$118$	0.656339	0.0604209
$119$	−12.9282	−1.18513
$120$	0	0
$121$	0	0
$122$	6.07296	0.549820
$123$	3.34607	0.301705
$124$	15.1244	1.35821
$125$	0	0
$126$	−1.26795	−0.112958
$127$	0.896575	0.0795582	0.0397791	−	0.999208i	$-0.487335\pi$
$127$	0.896575	0.0795582	0.0397791	+	0.999208i	$0.487335\pi$
$128$	11.4524	1.01226
$129$	12.4641	1.09740
$130$	0	0
$131$	−3.12436	−0.272976	−0.136488	−	0.990642i	$-0.543582\pi$
$131$	−3.12436	−0.272976	−0.136488	+	0.990642i	$0.543582\pi$
$132$	0	0
$133$	−14.0406	−1.21747
$134$	1.14359	0.0987914
$135$	0	0
$136$	−7.46410	−0.640041
$137$	8.86422	0.757321	0.378661	−	0.925536i	$-0.376385\pi$
$137$	8.86422	0.757321	0.378661	+	0.925536i	$0.376385\pi$
$138$	3.48477	0.296643
$139$	−14.5885	−1.23738	−0.618688	−	0.785636i	$-0.712336\pi$
$139$	−14.5885	−1.23738	−0.618688	+	0.785636i	$0.712336\pi$
$140$	0	0
$141$	−22.1244	−1.86321
$142$	4.24264	0.356034
$143$	0	0
$144$	1.80385	0.150321
$145$	0	0
$146$	−2.53590	−0.209872
$147$	8.10634	0.668600
$148$	3.10583	0.255298
$149$	−10.2679	−0.841183	−0.420592	−	0.907250i	$-0.638177\pi$
$149$	−10.2679	−0.841183	−0.420592	+	0.907250i	$0.638177\pi$
$150$	0	0
$151$	4.19615	0.341478	0.170739	−	0.985316i	$-0.445384\pi$
$151$	4.19615	0.341478	0.170739	+	0.985316i	$0.445384\pi$
$152$	−8.10634	−0.657511
$153$	−2.82843	−0.228665
$154$	0	0
$155$	0	0
$156$	14.1962	1.13660
$157$	21.8695	1.74538	0.872690	−	0.488275i	$-0.162374\pi$
$157$	21.8695	1.74538	0.872690	+	0.488275i	$0.162374\pi$
$158$	−0.757875	−0.0602933
$159$	−4.19615	−0.332777
$160$	0	0
$161$	−11.6603	−0.918957
$162$	5.51815	0.433547
$163$	5.13922	0.402534	0.201267	−	0.979536i	$-0.435494\pi$
$163$	5.13922	0.402534	0.201267	+	0.979536i	$0.435494\pi$
$164$	−3.00000	−0.234261
$165$	0	0
$166$	−5.12436	−0.397727
$167$	−0.517638	−0.0400560	−0.0200280	−	0.999799i	$-0.506376\pi$
$167$	−0.517638	−0.0400560	−0.0200280	+	0.999799i	$0.506376\pi$
$168$	12.4877	0.963446
$169$	5.00000	0.384615
$170$	0	0
$171$	−3.07180	−0.234906
$172$	−11.1750	−0.852086
$173$	−21.7680	−1.65499	−0.827495	−	0.561472i	$-0.810235\pi$
$173$	−21.7680	−1.65499	−0.827495	+	0.561472i	$0.810235\pi$
$174$	−6.92820	−0.525226
$175$	0	0
$176$	0	0
$177$	−2.44949	−0.184115
$178$	−0.240237	−0.0180065
$179$	−15.4641	−1.15584	−0.577921	−	0.816093i	$-0.696136\pi$
$179$	−15.4641	−1.15584	−0.577921	+	0.816093i	$0.696136\pi$
$180$	0	0
$181$	−17.3923	−1.29276	−0.646380	−	0.763016i	$-0.723718\pi$
$181$	−17.3923	−1.29276	−0.646380	+	0.763016i	$0.723718\pi$
$182$	7.34847	0.544705
$183$	−22.6646	−1.67541
$184$	−6.73205	−0.496293
$185$	0	0
$186$	8.73205	0.640265
$187$	0	0
$188$	19.8362	1.44670
$189$	−14.6603	−1.06638
$190$	0	0
$191$	−7.26795	−0.525890	−0.262945	−	0.964811i	$-0.584694\pi$
$191$	−7.26795	−0.525890	−0.262945	+	0.964811i	$0.584694\pi$
$192$	−4.38134	−0.316196
$193$	−15.1774	−1.09249	−0.546247	−	0.837624i	$-0.683944\pi$
$193$	−15.1774	−1.09249	−0.546247	+	0.837624i	$0.683944\pi$
$194$	−4.73205	−0.339741
$195$	0	0
$196$	−7.26795	−0.519139
$197$	−17.2480	−1.22887	−0.614433	−	0.788969i	$-0.710615\pi$
$197$	−17.2480	−1.22887	−0.614433	+	0.788969i	$0.710615\pi$
$198$	0	0
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	0	0
$201$	−4.26795	−0.301038
$202$	−10.0382	−0.706285
$203$	23.1822	1.62707
$204$	12.9282	0.905155
$205$	0	0
$206$	2.19615	0.153013
$207$	−2.55103	−0.177309
$208$	−10.4543	−0.724875
$209$	0	0
$210$	0	0
$211$	−13.1244	−0.903518	−0.451759	−	0.892140i	$-0.649203\pi$
$211$	−13.1244	−0.903518	−0.451759	+	0.892140i	$0.649203\pi$
$212$	3.76217	0.258387
$213$	−15.8338	−1.08491
$214$	1.53590	0.104992
$215$	0	0
$216$	−8.46410	−0.575909
$217$	−29.2180	−1.98345
$218$	−4.76028	−0.322407
$219$	9.46410	0.639525
$220$	0	0
$221$	16.3923	1.10267
$222$	1.79315	0.120348
$223$	−1.55291	−0.103991	−0.0519954	−	0.998647i	$-0.516558\pi$
$223$	−1.55291	−0.103991	−0.0519954	+	0.998647i	$0.516558\pi$
$224$	−17.1962	−1.14897
$225$	0	0
$226$	1.46410	0.0973906
$227$	−20.1136	−1.33498	−0.667492	−	0.744617i	$-0.732632\pi$
$227$	−20.1136	−1.33498	−0.667492	+	0.744617i	$0.732632\pi$
$228$	14.0406	0.929861
$229$	−4.07180	−0.269072	−0.134536	−	0.990909i	$-0.542954\pi$
$229$	−4.07180	−0.269072	−0.134536	+	0.990909i	$0.542954\pi$
$230$	0	0
$231$	0	0
$232$	13.3843	0.878720
$233$	−1.69161	−0.110821	−0.0554107	−	0.998464i	$-0.517647\pi$
$233$	−1.69161	−0.110821	−0.0554107	+	0.998464i	$0.517647\pi$
$234$	1.60770	0.105098
$235$	0	0
$236$	2.19615	0.142957
$237$	2.82843	0.183726
$238$	6.69213	0.433786
$239$	5.66025	0.366131	0.183066	−	0.983101i	$-0.441398\pi$
$239$	5.66025	0.366131	0.183066	+	0.983101i	$0.441398\pi$
$240$	0	0
$241$	14.1244	0.909830	0.454915	−	0.890535i	$-0.349670\pi$
$241$	14.1244	0.909830	0.454915	+	0.890535i	$0.349670\pi$
$242$	0	0
$243$	−7.45001	−0.477918
$244$	20.3205	1.30089
$245$	0	0
$246$	−1.73205	−0.110432
$247$	17.8028	1.13276
$248$	−16.8690	−1.07118
$249$	19.1244	1.21196
$250$	0	0
$251$	−3.80385	−0.240097	−0.120048	−	0.992768i	$-0.538305\pi$
$251$	−3.80385	−0.240097	−0.120048	+	0.992768i	$0.538305\pi$
$252$	−4.24264	−0.267261
$253$	0	0
$254$	−0.464102	−0.0291203
$255$	0	0
$256$	−1.39230	−0.0870191
$257$	−14.7985	−0.923103	−0.461552	−	0.887113i	$-0.652707\pi$
$257$	−14.7985	−0.923103	−0.461552	+	0.887113i	$0.652707\pi$
$258$	−6.45189	−0.401677
$259$	−6.00000	−0.372822
$260$	0	0
$261$	5.07180	0.313936
$262$	1.61729	0.0999162
$263$	6.79367	0.418915	0.209458	−	0.977818i	$-0.432830\pi$
$263$	6.79367	0.418915	0.209458	+	0.977818i	$0.432830\pi$
$264$	0	0
$265$	0	0
$266$	7.26795	0.445627
$267$	0.896575	0.0548695
$268$	3.82654	0.233743
$269$	−16.2679	−0.991874	−0.495937	−	0.868358i	$-0.665175\pi$
$269$	−16.2679	−0.991874	−0.495937	+	0.868358i	$0.665175\pi$
$270$	0	0
$271$	11.4641	0.696395	0.348197	−	0.937421i	$-0.386794\pi$
$271$	11.4641	0.696395	0.348197	+	0.937421i	$0.386794\pi$
$272$	−9.52056	−0.577269
$273$	−27.4249	−1.65983
$274$	−4.58846	−0.277199
$275$	0	0
$276$	11.6603	0.701865
$277$	31.1870	1.87385	0.936923	−	0.349535i	$-0.113660\pi$
$277$	31.1870	1.87385	0.936923	+	0.349535i	$0.113660\pi$
$278$	7.55154	0.452911
$279$	−6.39230	−0.382697
$280$	0	0
$281$	−17.3205	−1.03325	−0.516627	−	0.856210i	$-0.672813\pi$
$281$	−17.3205	−1.03325	−0.516627	+	0.856210i	$0.672813\pi$
$282$	11.4524	0.681981
$283$	8.06918	0.479663	0.239831	−	0.970815i	$-0.422908\pi$
$283$	8.06918	0.479663	0.239831	+	0.970815i	$0.422908\pi$
$284$	14.1962	0.842387
$285$	0	0
$286$	0	0
$287$	5.79555	0.342101
$288$	−3.76217	−0.221688
$289$	−2.07180	−0.121870
$290$	0	0
$291$	17.6603	1.03526
$292$	−8.48528	−0.496564
$293$	−15.0759	−0.880742	−0.440371	−	0.897816i	$-0.645153\pi$
$293$	−15.0759	−0.880742	−0.440371	+	0.897816i	$0.645153\pi$
$294$	−4.19615	−0.244725
$295$	0	0
$296$	−3.46410	−0.201347
$297$	0	0
$298$	5.31508	0.307894
$299$	14.7846	0.855016
$300$	0	0
$301$	21.5885	1.24434
$302$	−2.17209	−0.124990
$303$	37.4631	2.15220
$304$	−10.3397	−0.593025
$305$	0	0
$306$	1.46410	0.0836971
$307$	7.82894	0.446821	0.223411	−	0.974724i	$-0.428281\pi$
$307$	7.82894	0.446821	0.223411	+	0.974724i	$0.428281\pi$
$308$	0	0
$309$	−8.19615	−0.466263
$310$	0	0
$311$	10.0526	0.570028	0.285014	−	0.958523i	$-0.408002\pi$
$311$	10.0526	0.570028	0.285014	+	0.958523i	$0.408002\pi$
$312$	−15.8338	−0.896410
$313$	17.1464	0.969173	0.484587	−	0.874743i	$-0.338970\pi$
$313$	17.1464	0.969173	0.484587	+	0.874743i	$0.338970\pi$
$314$	−11.3205	−0.638853
$315$	0	0
$316$	−2.53590	−0.142655
$317$	12.6264	0.709168	0.354584	−	0.935024i	$-0.384622\pi$
$317$	12.6264	0.709168	0.354584	+	0.935024i	$0.384622\pi$
$318$	2.17209	0.121805
$319$	0	0
$320$	0	0
$321$	−5.73205	−0.319932
$322$	6.03579	0.336362
$323$	16.2127	0.902098
$324$	18.4641	1.02578
$325$	0	0
$326$	−2.66025	−0.147338
$327$	17.7656	0.982440
$328$	3.34607	0.184756
$329$	−38.3205	−2.11268
$330$	0	0
$331$	−18.1962	−1.00015	−0.500075	−	0.865982i	$-0.666694\pi$
$331$	−18.1962	−1.00015	−0.500075	+	0.865982i	$0.666694\pi$
$332$	−17.1464	−0.941033
$333$	−1.31268	−0.0719343
$334$	0.267949	0.0146615
$335$	0	0
$336$	15.9282	0.868955
$337$	17.6269	0.960199	0.480099	−	0.877214i	$-0.340601\pi$
$337$	17.6269	0.960199	0.480099	+	0.877214i	$0.340601\pi$
$338$	−2.58819	−0.140779
$339$	−5.46410	−0.296769
$340$	0	0
$341$	0	0
$342$	1.59008	0.0859816
$343$	−9.38186	−0.506573
$344$	12.4641	0.672019
$345$	0	0
$346$	11.2679	0.605769
$347$	4.38134	0.235203	0.117601	−	0.993061i	$-0.462479\pi$
$347$	4.38134	0.235203	0.117601	+	0.993061i	$0.462479\pi$
$348$	−23.1822	−1.24270
$349$	−7.32051	−0.391858	−0.195929	−	0.980618i	$-0.562772\pi$
$349$	−7.32051	−0.391858	−0.195929	+	0.980618i	$0.562772\pi$
$350$	0	0
$351$	18.5885	0.992178
$352$	0	0
$353$	13.0053	0.692204	0.346102	−	0.938197i	$-0.387505\pi$
$353$	13.0053	0.692204	0.346102	+	0.938197i	$0.387505\pi$
$354$	1.26795	0.0673907
$355$	0	0
$356$	−0.803848	−0.0426038
$357$	−24.9754	−1.32184
$358$	8.00481	0.423067
$359$	35.6603	1.88208	0.941038	−	0.338301i	$-0.109852\pi$
$359$	35.6603	1.88208	0.941038	+	0.338301i	$0.109852\pi$
$360$	0	0
$361$	−1.39230	−0.0732792
$362$	9.00292	0.473183
$363$	0	0
$364$	24.5885	1.28879
$365$	0	0
$366$	11.7321	0.613244
$367$	16.9062	0.882496	0.441248	−	0.897385i	$-0.354536\pi$
$367$	16.9062	0.882496	0.441248	+	0.897385i	$0.354536\pi$
$368$	−8.58682	−0.447619
$369$	1.26795	0.0660068
$370$	0	0
$371$	−7.26795	−0.377333
$372$	29.2180	1.51488
$373$	−0.175865	−0.00910597	−0.00455298	−	0.999990i	$-0.501449\pi$
$373$	−0.175865	−0.00910597	−0.00455298	+	0.999990i	$0.501449\pi$
$374$	0	0
$375$	0	0
$376$	−22.1244	−1.14098
$377$	−29.3939	−1.51386
$378$	7.58871	0.390321
$379$	1.46410	0.0752058	0.0376029	−	0.999293i	$-0.488028\pi$
$379$	1.46410	0.0752058	0.0376029	+	0.999293i	$0.488028\pi$
$380$	0	0
$381$	1.73205	0.0887357
$382$	3.76217	0.192489
$383$	38.4612	1.96527	0.982637	−	0.185539i	$-0.0594031\pi$
$383$	38.4612	1.96527	0.982637	+	0.185539i	$0.0594031\pi$
$384$	22.1244	1.12903
$385$	0	0
$386$	7.85641	0.399881
$387$	4.72311	0.240089
$388$	−15.8338	−0.803837
$389$	6.12436	0.310517	0.155259	−	0.987874i	$-0.450379\pi$
$389$	6.12436	0.310517	0.155259	+	0.987874i	$0.450379\pi$
$390$	0	0
$391$	13.4641	0.680909
$392$	8.10634	0.409432
$393$	−6.03579	−0.304465
$394$	8.92820	0.449796
$395$	0	0
$396$	0	0
$397$	−32.9802	−1.65523	−0.827614	−	0.561298i	$-0.810302\pi$
$397$	−32.9802	−1.65523	−0.827614	+	0.561298i	$0.810302\pi$
$398$	1.03528	0.0518937
$399$	−27.1244	−1.35792
$400$	0	0
$401$	2.32051	0.115881	0.0579403	−	0.998320i	$-0.481547\pi$
$401$	2.32051	0.115881	0.0579403	+	0.998320i	$0.481547\pi$
$402$	2.20925	0.110188
$403$	37.0470	1.84544
$404$	−33.5885	−1.67109
$405$	0	0
$406$	−12.0000	−0.595550
$407$	0	0
$408$	−14.4195	−0.713873
$409$	−26.1244	−1.29177	−0.645883	−	0.763436i	$-0.723511\pi$
$409$	−26.1244	−1.29177	−0.645883	+	0.763436i	$0.723511\pi$
$410$	0	0
$411$	17.1244	0.844682
$412$	7.34847	0.362033
$413$	−4.24264	−0.208767
$414$	1.32051	0.0648994
$415$	0	0
$416$	21.8038	1.06902
$417$	−28.1827	−1.38011
$418$	0	0
$419$	−12.3397	−0.602836	−0.301418	−	0.953492i	$-0.597460\pi$
$419$	−12.3397	−0.602836	−0.301418	+	0.953492i	$0.597460\pi$
$420$	0	0
$421$	−17.0526	−0.831091	−0.415545	−	0.909572i	$-0.636409\pi$
$421$	−17.0526	−0.831091	−0.415545	+	0.909572i	$0.636409\pi$
$422$	6.79367	0.330711
$423$	−8.38375	−0.407632
$424$	−4.19615	−0.203783
$425$	0	0
$426$	8.19615	0.397105
$427$	−39.2562	−1.89974
$428$	5.13922	0.248413
$429$	0	0
$430$	0	0
$431$	−5.07180	−0.244300	−0.122150	−	0.992512i	$-0.538979\pi$
$431$	−5.07180	−0.244300	−0.122150	+	0.992512i	$0.538979\pi$
$432$	−10.7961	−0.519426
$433$	28.7375	1.38104	0.690519	−	0.723314i	$-0.257382\pi$
$433$	28.7375	1.38104	0.690519	+	0.723314i	$0.257382\pi$
$434$	15.1244	0.725992
$435$	0	0
$436$	−15.9282	−0.762823
$437$	14.6226	0.699494
$438$	−4.89898	−0.234082
$439$	28.2487	1.34824	0.674119	−	0.738623i	$-0.264524\pi$
$439$	28.2487	1.34824	0.674119	+	0.738623i	$0.264524\pi$
$440$	0	0
$441$	3.07180	0.146276
$442$	−8.48528	−0.403604
$443$	−27.5636	−1.30958	−0.654792	−	0.755809i	$-0.727244\pi$
$443$	−27.5636	−1.30958	−0.654792	+	0.755809i	$0.727244\pi$
$444$	6.00000	0.284747
$445$	0	0
$446$	0.803848	0.0380633
$447$	−19.8362	−0.938218
$448$	−7.58871	−0.358533
$449$	34.5167	1.62894	0.814471	−	0.580204i	$-0.197027\pi$
$449$	34.5167	1.62894	0.814471	+	0.580204i	$0.197027\pi$
$450$	0	0
$451$	0	0
$452$	4.89898	0.230429
$453$	8.10634	0.380869
$454$	10.4115	0.488638
$455$	0	0
$456$	−15.6603	−0.733359
$457$	21.2132	0.992312	0.496156	−	0.868233i	$-0.334744\pi$
$457$	21.2132	0.992312	0.496156	+	0.868233i	$0.334744\pi$
$458$	2.10772	0.0984872
$459$	16.9282	0.790141
$460$	0	0
$461$	−33.0000	−1.53696	−0.768482	−	0.639872i	$-0.778987\pi$
$461$	−33.0000	−1.53696	−0.768482	+	0.639872i	$0.778987\pi$
$462$	0	0
$463$	20.3166	0.944194	0.472097	−	0.881547i	$-0.343497\pi$
$463$	20.3166	0.944194	0.472097	+	0.881547i	$0.343497\pi$
$464$	17.0718	0.792538
$465$	0	0
$466$	0.875644	0.0405634
$467$	0.619174	0.0286520	0.0143260	−	0.999897i	$-0.495440\pi$
$467$	0.619174	0.0286520	0.0143260	+	0.999897i	$0.495440\pi$
$468$	5.37945	0.248665
$469$	−7.39230	−0.341345
$470$	0	0
$471$	42.2487	1.94672
$472$	−2.44949	−0.112747
$473$	0	0
$474$	−1.46410	−0.0672484
$475$	0	0
$476$	22.3923	1.02635
$477$	−1.59008	−0.0728047
$478$	−2.92996	−0.134013
$479$	2.53590	0.115868	0.0579341	−	0.998320i	$-0.481549\pi$
$479$	2.53590	0.115868	0.0579341	+	0.998320i	$0.481549\pi$
$480$	0	0
$481$	7.60770	0.346881
$482$	−7.31130	−0.333021
$483$	−22.5259	−1.02496
$484$	0	0
$485$	0	0
$486$	3.85641	0.174930
$487$	−12.7279	−0.576757	−0.288379	−	0.957516i	$-0.593116\pi$
$487$	−12.7279	−0.576757	−0.288379	+	0.957516i	$0.593116\pi$
$488$	−22.6646	−1.02598
$489$	9.92820	0.448969
$490$	0	0
$491$	39.7128	1.79221	0.896107	−	0.443838i	$-0.146383\pi$
$491$	39.7128	1.79221	0.896107	+	0.443838i	$0.146383\pi$
$492$	−5.79555	−0.261284
$493$	−26.7685	−1.20559
$494$	−9.21539	−0.414620
$495$	0	0
$496$	−21.5167	−0.966127
$497$	−27.4249	−1.23017
$498$	−9.89949	−0.443607
$499$	17.6077	0.788229	0.394114	−	0.919061i	$-0.371051\pi$
$499$	17.6077	0.788229	0.394114	+	0.919061i	$0.371051\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	1.96902	0.0878815
$503$	7.38563	0.329309	0.164655	−	0.986351i	$-0.447349\pi$
$503$	7.38563	0.329309	0.164655	+	0.986351i	$0.447349\pi$
$504$	4.73205	0.210782
$505$	0	0
$506$	0	0
$507$	9.65926	0.428983
$508$	−1.55291	−0.0688994
$509$	−8.07180	−0.357776	−0.178888	−	0.983869i	$-0.557250\pi$
$509$	−8.07180	−0.357776	−0.178888	+	0.983869i	$0.557250\pi$
$510$	0	0
$511$	16.3923	0.725153
$512$	−22.1841	−0.980408
$513$	18.3848	0.811708
$514$	7.66025	0.337879
$515$	0	0
$516$	−21.5885	−0.950379
$517$	0	0
$518$	3.10583	0.136462
$519$	−42.0526	−1.84590
$520$	0	0
$521$	−9.24871	−0.405193	−0.202597	−	0.979262i	$-0.564938\pi$
$521$	−9.24871	−0.405193	−0.202597	+	0.979262i	$0.564938\pi$
$522$	−2.62536	−0.114909
$523$	−25.6317	−1.12080	−0.560398	−	0.828223i	$-0.689352\pi$
$523$	−25.6317	−1.12080	−0.560398	+	0.828223i	$0.689352\pi$
$524$	5.41154	0.236404
$525$	0	0
$526$	−3.51666	−0.153334
$527$	33.7381	1.46965
$528$	0	0
$529$	−10.8564	−0.472018
$530$	0	0
$531$	−0.928203	−0.0402806
$532$	24.3190	1.05436
$533$	−7.34847	−0.318298
$534$	−0.464102	−0.0200836
$535$	0	0
$536$	−4.26795	−0.184347
$537$	−29.8744	−1.28917
$538$	8.42091	0.363051
$539$	0	0
$540$	0	0
$541$	−2.60770	−0.112114	−0.0560568	−	0.998428i	$-0.517853\pi$
$541$	−2.60770	−0.112114	−0.0560568	+	0.998428i	$0.517853\pi$
$542$	−5.93426	−0.254898
$543$	−33.5994	−1.44189
$544$	19.8564	0.851336
$545$	0	0
$546$	14.1962	0.607539
$547$	28.7375	1.22873	0.614364	−	0.789023i	$-0.289413\pi$
$547$	28.7375	1.22873	0.614364	+	0.789023i	$0.289413\pi$
$548$	−15.3533	−0.655859
$549$	−8.58846	−0.366546
$550$	0	0
$551$	−29.0718	−1.23850
$552$	−13.0053	−0.553543
$553$	4.89898	0.208326
$554$	−16.1436	−0.685876
$555$	0	0
$556$	25.2679	1.07160
$557$	37.6018	1.59324	0.796619	−	0.604482i	$-0.206620\pi$
$557$	37.6018	1.59324	0.796619	+	0.604482i	$0.206620\pi$
$558$	3.30890	0.140077
$559$	−27.3731	−1.15776
$560$	0	0
$561$	0	0
$562$	8.96575	0.378198
$563$	−15.9725	−0.673159	−0.336579	−	0.941655i	$-0.609270\pi$
$563$	−15.9725	−0.673159	−0.336579	+	0.941655i	$0.609270\pi$
$564$	38.3205	1.61358
$565$	0	0
$566$	−4.17691	−0.175569
$567$	−35.6699	−1.49800
$568$	−15.8338	−0.664369
$569$	30.1244	1.26288	0.631439	−	0.775425i	$-0.282464\pi$
$569$	30.1244	1.26288	0.631439	+	0.775425i	$0.282464\pi$
$570$	0	0
$571$	−19.7128	−0.824956	−0.412478	−	0.910968i	$-0.635337\pi$
$571$	−19.7128	−0.824956	−0.412478	+	0.910968i	$0.635337\pi$
$572$	0	0
$573$	−14.0406	−0.586554
$574$	−3.00000	−0.125218
$575$	0	0
$576$	−1.66025	−0.0691773
$577$	3.76217	0.156621	0.0783105	−	0.996929i	$-0.475047\pi$
$577$	3.76217	0.156621	0.0783105	+	0.996929i	$0.475047\pi$
$578$	1.07244	0.0446077
$579$	−29.3205	−1.21852
$580$	0	0
$581$	33.1244	1.37423
$582$	−9.14162	−0.378932
$583$	0	0
$584$	9.46410	0.391627
$585$	0	0
$586$	7.80385	0.322374
$587$	20.3910	0.841625	0.420812	−	0.907148i	$-0.361745\pi$
$587$	20.3910	0.841625	0.420812	+	0.907148i	$0.361745\pi$
$588$	−14.0406	−0.579025
$589$	36.6410	1.50977
$590$	0	0
$591$	−33.3205	−1.37062
$592$	−4.41851	−0.181599
$593$	−0.859411	−0.0352918	−0.0176459	−	0.999844i	$-0.505617\pi$
$593$	−0.859411	−0.0352918	−0.0176459	+	0.999844i	$0.505617\pi$
$594$	0	0
$595$	0	0
$596$	17.7846	0.728486
$597$	−3.86370	−0.158131
$598$	−7.65308	−0.312958
$599$	−38.4449	−1.57081	−0.785407	−	0.618979i	$-0.787546\pi$
$599$	−38.4449	−1.57081	−0.785407	+	0.618979i	$0.787546\pi$
$600$	0	0
$601$	20.0000	0.815817	0.407909	−	0.913023i	$-0.366258\pi$
$601$	20.0000	0.815817	0.407909	+	0.913023i	$0.366258\pi$
$602$	−11.1750	−0.455459
$603$	−1.61729	−0.0658610
$604$	−7.26795	−0.295729
$605$	0	0
$606$	−19.3923	−0.787759
$607$	−36.3906	−1.47705	−0.738525	−	0.674226i	$-0.764477\pi$
$607$	−36.3906	−1.47705	−0.738525	+	0.674226i	$0.764477\pi$
$608$	21.5649	0.874574
$609$	44.7846	1.81476
$610$	0	0
$611$	48.5885	1.96568
$612$	4.89898	0.198030
$613$	−1.79315	−0.0724247	−0.0362123	−	0.999344i	$-0.511529\pi$
$613$	−1.79315	−0.0724247	−0.0362123	+	0.999344i	$0.511529\pi$
$614$	−4.05256	−0.163548
$615$	0	0
$616$	0	0
$617$	−13.0053	−0.523575	−0.261787	−	0.965126i	$-0.584312\pi$
$617$	−13.0053	−0.523575	−0.261787	+	0.965126i	$0.584312\pi$
$618$	4.24264	0.170664
$619$	12.7846	0.513857	0.256928	−	0.966430i	$-0.417290\pi$
$619$	12.7846	0.513857	0.256928	+	0.966430i	$0.417290\pi$
$620$	0	0
$621$	15.2679	0.612682
$622$	−5.20359	−0.208645
$623$	1.55291	0.0622162
$624$	−20.1962	−0.808493
$625$	0	0
$626$	−8.87564	−0.354742
$627$	0	0
$628$	−37.8792	−1.51154
$629$	6.92820	0.276246
$630$	0	0
$631$	−15.3205	−0.609900	−0.304950	−	0.952368i	$-0.598640\pi$
$631$	−15.3205	−0.609900	−0.304950	+	0.952368i	$0.598640\pi$
$632$	2.82843	0.112509
$633$	−25.3543	−1.00774
$634$	−6.53590	−0.259574
$635$	0	0
$636$	7.26795	0.288193
$637$	−17.8028	−0.705371
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	−7.85641	−0.310309	−0.155155	−	0.987890i	$-0.549588\pi$
$641$	−7.85641	−0.310309	−0.155155	+	0.987890i	$0.549588\pi$
$642$	2.96713	0.117103
$643$	10.5187	0.414816	0.207408	−	0.978255i	$-0.433497\pi$
$643$	10.5187	0.414816	0.207408	+	0.978255i	$0.433497\pi$
$644$	20.1962	0.795840
$645$	0	0
$646$	−8.39230	−0.330191
$647$	37.3615	1.46883	0.734416	−	0.678699i	$-0.237456\pi$
$647$	37.3615	1.46883	0.734416	+	0.678699i	$0.237456\pi$
$648$	−20.5940	−0.809010
$649$	0	0
$650$	0	0
$651$	−56.4449	−2.21225
$652$	−8.90138	−0.348605
$653$	−12.1459	−0.475306	−0.237653	−	0.971350i	$-0.576378\pi$
$653$	−12.1459	−0.475306	−0.237653	+	0.971350i	$0.576378\pi$
$654$	−9.19615	−0.359598
$655$	0	0
$656$	4.26795	0.166635
$657$	3.58630	0.139915
$658$	19.8362	0.773294
$659$	−23.3205	−0.908438	−0.454219	−	0.890890i	$-0.650082\pi$
$659$	−23.3205	−0.908438	−0.454219	+	0.890890i	$0.650082\pi$
$660$	0	0
$661$	29.5885	1.15086	0.575429	−	0.817852i	$-0.304835\pi$
$661$	29.5885	1.15086	0.575429	+	0.817852i	$0.304835\pi$
$662$	9.41902	0.366081
$663$	31.6675	1.22986
$664$	19.1244	0.742169
$665$	0	0
$666$	0.679492	0.0263298
$667$	−24.1432	−0.934827
$668$	0.896575	0.0346895
$669$	−3.00000	−0.115987
$670$	0	0
$671$	0	0
$672$	−33.2204	−1.28151
$673$	40.8091	1.57308	0.786538	−	0.617542i	$-0.211871\pi$
$673$	40.8091	1.57308	0.786538	+	0.617542i	$0.211871\pi$
$674$	−9.12436	−0.351457
$675$	0	0
$676$	−8.66025	−0.333087
$677$	−20.3538	−0.782260	−0.391130	−	0.920335i	$-0.627916\pi$
$677$	−20.3538	−0.782260	−0.391130	+	0.920335i	$0.627916\pi$
$678$	2.82843	0.108625
$679$	30.5885	1.17388
$680$	0	0
$681$	−38.8564	−1.48898
$682$	0	0
$683$	−29.8372	−1.14169	−0.570844	−	0.821058i	$-0.693384\pi$
$683$	−29.8372	−1.14169	−0.570844	+	0.821058i	$0.693384\pi$
$684$	5.32051	0.203435
$685$	0	0
$686$	4.85641	0.185418
$687$	−7.86611	−0.300111
$688$	15.8981	0.606110
$689$	9.21539	0.351078
$690$	0	0
$691$	37.9090	1.44213	0.721063	−	0.692870i	$-0.243654\pi$
$691$	37.9090	1.44213	0.721063	+	0.692870i	$0.243654\pi$
$692$	37.7033	1.43326
$693$	0	0
$694$	−2.26795	−0.0860902
$695$	0	0
$696$	25.8564	0.980085
$697$	−6.69213	−0.253483
$698$	3.78937	0.143430
$699$	−3.26795	−0.123605
$700$	0	0
$701$	37.8564	1.42982	0.714908	−	0.699218i	$-0.246468\pi$
$701$	37.8564	1.42982	0.714908	+	0.699218i	$0.246468\pi$
$702$	−9.62209	−0.363163
$703$	7.52433	0.283786
$704$	0	0
$705$	0	0
$706$	−6.73205	−0.253364
$707$	64.8879	2.44036
$708$	4.24264	0.159448
$709$	2.60770	0.0979340	0.0489670	−	0.998800i	$-0.484407\pi$
$709$	2.60770	0.0979340	0.0489670	+	0.998800i	$0.484407\pi$
$710$	0	0
$711$	1.07180	0.0401955
$712$	0.896575	0.0336006
$713$	30.4292	1.13958
$714$	12.9282	0.483826
$715$	0	0
$716$	26.7846	1.00099
$717$	10.9348	0.408367
$718$	−18.4591	−0.688888
$719$	−50.1962	−1.87200	−0.936000	−	0.351999i	$-0.885502\pi$
$719$	−50.1962	−1.87200	−0.936000	+	0.351999i	$0.885502\pi$
$720$	0	0
$721$	−14.1962	−0.528692
$722$	0.720710	0.0268220
$723$	27.2862	1.01478
$724$	30.1244	1.11956
$725$	0	0
$726$	0	0
$727$	8.24504	0.305792	0.152896	−	0.988242i	$-0.451140\pi$
$727$	8.24504	0.305792	0.152896	+	0.988242i	$0.451140\pi$
$728$	−27.4249	−1.01643
$729$	17.5885	0.651424
$730$	0	0
$731$	−24.9282	−0.922003
$732$	39.2562	1.45095
$733$	9.31749	0.344149	0.172075	−	0.985084i	$-0.444953\pi$
$733$	9.31749	0.344149	0.172075	+	0.985084i	$0.444953\pi$
$734$	−8.75129	−0.323016
$735$	0	0
$736$	17.9090	0.660133
$737$	0	0
$738$	−0.656339	−0.0241602
$739$	−38.9282	−1.43200	−0.715999	−	0.698102i	$-0.754028\pi$
$739$	−38.9282	−1.43200	−0.715999	+	0.698102i	$0.754028\pi$
$740$	0	0
$741$	34.3923	1.26343
$742$	3.76217	0.138114
$743$	−43.3973	−1.59209	−0.796046	−	0.605235i	$-0.793079\pi$
$743$	−43.3973	−1.59209	−0.796046	+	0.605235i	$0.793079\pi$
$744$	−32.5885	−1.19475
$745$	0	0
$746$	0.0910347	0.00333302
$747$	7.24693	0.265151
$748$	0	0
$749$	−9.92820	−0.362769
$750$	0	0
$751$	48.6410	1.77494	0.887468	−	0.460869i	$-0.152462\pi$
$751$	48.6410	1.77494	0.887468	+	0.460869i	$0.152462\pi$
$752$	−28.2199	−1.02907
$753$	−7.34847	−0.267793
$754$	15.2154	0.554112
$755$	0	0
$756$	25.3923	0.923509
$757$	49.7749	1.80910	0.904549	−	0.426369i	$-0.140207\pi$
$757$	49.7749	1.80910	0.904549	+	0.426369i	$0.140207\pi$
$758$	−0.757875	−0.0275273
$759$	0	0
$760$	0	0
$761$	−19.8564	−0.719794	−0.359897	−	0.932992i	$-0.617188\pi$
$761$	−19.8564	−0.719794	−0.359897	+	0.932992i	$0.617188\pi$
$762$	−0.896575	−0.0324795
$763$	30.7709	1.11398
$764$	12.5885	0.455434
$765$	0	0
$766$	−19.9090	−0.719340
$767$	5.37945	0.194241
$768$	−2.68973	−0.0970571
$769$	−9.85641	−0.355431	−0.177716	−	0.984082i	$-0.556871\pi$
$769$	−9.85641	−0.355431	−0.177716	+	0.984082i	$0.556871\pi$
$770$	0	0
$771$	−28.5885	−1.02959
$772$	26.2880	0.946128
$773$	12.0444	0.433206	0.216603	−	0.976260i	$-0.430502\pi$
$773$	12.0444	0.433206	0.216603	+	0.976260i	$0.430502\pi$
$774$	−2.44486	−0.0878788
$775$	0	0
$776$	17.6603	0.633966
$777$	−11.5911	−0.415829
$778$	−3.17020	−0.113657
$779$	−7.26795	−0.260401
$780$	0	0
$781$	0	0
$782$	−6.96953	−0.249230
$783$	−30.3548	−1.08479
$784$	10.3397	0.369277
$785$	0	0
$786$	3.12436	0.111442
$787$	17.3867	0.619768	0.309884	−	0.950774i	$-0.399710\pi$
$787$	17.3867	0.619768	0.309884	+	0.950774i	$0.399710\pi$
$788$	29.8744	1.06423
$789$	13.1244	0.467239
$790$	0	0
$791$	−9.46410	−0.336505
$792$	0	0
$793$	49.7749	1.76756
$794$	17.0718	0.605855
$795$	0	0
$796$	3.46410	0.122782
$797$	−36.1875	−1.28183	−0.640914	−	0.767612i	$-0.721445\pi$
$797$	−36.1875	−1.28183	−0.640914	+	0.767612i	$0.721445\pi$
$798$	14.0406	0.497032
$799$	44.2487	1.56541
$800$	0	0
$801$	0.339746	0.0120043
$802$	−1.20118	−0.0424153
$803$	0	0
$804$	7.39230	0.260706
$805$	0	0
$806$	−19.1769	−0.675478
$807$	−31.4273	−1.10629
$808$	37.4631	1.31795
$809$	−14.5359	−0.511055	−0.255527	−	0.966802i	$-0.582249\pi$
$809$	−14.5359	−0.511055	−0.255527	+	0.966802i	$0.582249\pi$
$810$	0	0
$811$	18.9808	0.666505	0.333252	−	0.942838i	$-0.391854\pi$
$811$	18.9808	0.666505	0.333252	+	0.942838i	$0.391854\pi$
$812$	−40.1528	−1.40909
$813$	22.1469	0.776727
$814$	0	0
$815$	0	0
$816$	−18.3923	−0.643859
$817$	−27.0731	−0.947169
$818$	13.5230	0.472819
$819$	−10.3923	−0.363137
$820$	0	0
$821$	29.1962	1.01895	0.509476	−	0.860485i	$-0.329839\pi$
$821$	29.1962	1.01895	0.509476	+	0.860485i	$0.329839\pi$
$822$	−8.86422	−0.309175
$823$	6.75650	0.235517	0.117758	−	0.993042i	$-0.462429\pi$
$823$	6.75650	0.235517	0.117758	+	0.993042i	$0.462429\pi$
$824$	−8.19615	−0.285526
$825$	0	0
$826$	2.19615	0.0764139
$827$	34.0798	1.18507	0.592536	−	0.805544i	$-0.298127\pi$
$827$	34.0798	1.18507	0.592536	+	0.805544i	$0.298127\pi$
$828$	4.41851	0.153554
$829$	−9.98076	−0.346646	−0.173323	−	0.984865i	$-0.555450\pi$
$829$	−9.98076	−0.346646	−0.173323	+	0.984865i	$0.555450\pi$
$830$	0	0
$831$	60.2487	2.09000
$832$	9.62209	0.333586
$833$	−16.2127	−0.561736
$834$	14.5885	0.505157
$835$	0	0
$836$	0	0
$837$	38.2581	1.32239
$838$	6.38752	0.220653
$839$	−38.7846	−1.33899	−0.669497	−	0.742815i	$-0.733490\pi$
$839$	−38.7846	−1.33899	−0.669497	+	0.742815i	$0.733490\pi$
$840$	0	0
$841$	19.0000	0.655172
$842$	8.82705	0.304200
$843$	−33.4607	−1.15245
$844$	22.7321	0.782469
$845$	0	0
$846$	4.33975	0.149204
$847$	0	0
$848$	−5.35225	−0.183797
$849$	15.5885	0.534994
$850$	0	0
$851$	6.24871	0.214203
$852$	27.4249	0.939560
$853$	46.3644	1.58749	0.793744	−	0.608252i	$-0.208129\pi$
$853$	46.3644	1.58749	0.793744	+	0.608252i	$0.208129\pi$
$854$	20.3205	0.695353
$855$	0	0
$856$	−5.73205	−0.195917
$857$	−29.3195	−1.00154	−0.500768	−	0.865581i	$-0.666949\pi$
$857$	−29.3195	−1.00154	−0.500768	+	0.865581i	$0.666949\pi$
$858$	0	0
$859$	−22.1962	−0.757323	−0.378661	−	0.925535i	$-0.623616\pi$
$859$	−22.1962	−0.757323	−0.378661	+	0.925535i	$0.623616\pi$
$860$	0	0
$861$	11.1962	0.381564
$862$	2.62536	0.0894200
$863$	−17.1093	−0.582406	−0.291203	−	0.956661i	$-0.594056\pi$
$863$	−17.1093	−0.582406	−0.291203	+	0.956661i	$0.594056\pi$
$864$	22.5167	0.766032
$865$	0	0
$866$	−14.8756	−0.505495
$867$	−4.00240	−0.135929
$868$	50.6071	1.71772
$869$	0	0
$870$	0	0
$871$	9.37307	0.317594
$872$	17.7656	0.601619
$873$	6.69213	0.226494
$874$	−7.56922	−0.256033
$875$	0	0
$876$	−16.3923	−0.553845
$877$	−11.8957	−0.401690	−0.200845	−	0.979623i	$-0.564369\pi$
$877$	−11.8957	−0.401690	−0.200845	+	0.979623i	$0.564369\pi$
$878$	−14.6226	−0.493489
$879$	−29.1244	−0.982340
$880$	0	0
$881$	−26.9090	−0.906586	−0.453293	−	0.891362i	$-0.649751\pi$
$881$	−26.9090	−0.906586	−0.453293	+	0.891362i	$0.649751\pi$
$882$	−1.59008	−0.0535407
$883$	21.5649	0.725718	0.362859	−	0.931844i	$-0.381801\pi$
$883$	21.5649	0.725718	0.362859	+	0.931844i	$0.381801\pi$
$884$	−28.3923	−0.954937
$885$	0	0
$886$	14.2679	0.479341
$887$	−15.6950	−0.526988	−0.263494	−	0.964661i	$-0.584875\pi$
$887$	−15.6950	−0.526988	−0.263494	+	0.964661i	$0.584875\pi$
$888$	−6.69213	−0.224573
$889$	3.00000	0.100617
$890$	0	0
$891$	0	0
$892$	2.68973	0.0900587
$893$	48.0561	1.60813
$894$	10.2679	0.343412
$895$	0	0
$896$	38.3205	1.28020
$897$	28.5617	0.953646
$898$	−17.8671	−0.596234
$899$	−60.4974	−2.01770
$900$	0	0
$901$	8.39230	0.279588
$902$	0	0
$903$	41.7057	1.38788
$904$	−5.46410	−0.181733
$905$	0	0
$906$	−4.19615	−0.139408
$907$	30.1146	0.999938	0.499969	−	0.866043i	$-0.333345\pi$
$907$	30.1146	0.999938	0.499969	+	0.866043i	$0.333345\pi$
$908$	34.8377	1.15613
$909$	14.1962	0.470857
$910$	0	0
$911$	−30.2487	−1.00218	−0.501092	−	0.865394i	$-0.667068\pi$
$911$	−30.2487	−1.00218	−0.501092	+	0.865394i	$0.667068\pi$
$912$	−19.9749	−0.661434
$913$	0	0
$914$	−10.9808	−0.363211
$915$	0	0
$916$	7.05256	0.233023
$917$	−10.4543	−0.345231
$918$	−8.76268	−0.289212
$919$	−18.9808	−0.626118	−0.313059	−	0.949734i	$-0.601354\pi$
$919$	−18.9808	−0.626118	−0.313059	+	0.949734i	$0.601354\pi$
$920$	0	0
$921$	15.1244	0.498364
$922$	17.0821	0.562568
$923$	34.7733	1.14458
$924$	0	0
$925$	0	0
$926$	−10.5167	−0.345599
$927$	−3.10583	−0.102009
$928$	−35.6055	−1.16881
$929$	−43.8564	−1.43888	−0.719441	−	0.694554i	$-0.755602\pi$
$929$	−43.8564	−1.43888	−0.719441	+	0.694554i	$0.755602\pi$
$930$	0	0
$931$	−17.6077	−0.577069
$932$	2.92996	0.0959741
$933$	19.4201	0.635784
$934$	−0.320508	−0.0104873
$935$	0	0
$936$	−6.00000	−0.196116
$937$	31.1870	1.01884	0.509418	−	0.860519i	$-0.329861\pi$
$937$	31.1870	1.01884	0.509418	+	0.860519i	$0.329861\pi$
$938$	3.82654	0.124941
$939$	33.1244	1.08097
$940$	0	0
$941$	−27.0000	−0.880175	−0.440087	−	0.897955i	$-0.645053\pi$
$941$	−27.0000	−0.880175	−0.440087	+	0.897955i	$0.645053\pi$
$942$	−21.8695	−0.712548
$943$	−6.03579	−0.196552
$944$	−3.12436	−0.101689
$945$	0	0
$946$	0	0
$947$	19.9749	0.649096	0.324548	−	0.945869i	$-0.394788\pi$
$947$	19.9749	0.649096	0.324548	+	0.945869i	$0.394788\pi$
$948$	−4.89898	−0.159111
$949$	−20.7846	−0.674697
$950$	0	0
$951$	24.3923	0.790975
$952$	−24.9754	−0.809456
$953$	−16.6932	−0.540745	−0.270372	−	0.962756i	$-0.587147\pi$
$953$	−16.6932	−0.540745	−0.270372	+	0.962756i	$0.587147\pi$
$954$	0.823085	0.0266484
$955$	0	0
$956$	−9.80385	−0.317079
$957$	0	0
$958$	−1.31268	−0.0424107
$959$	29.6603	0.957780
$960$	0	0
$961$	45.2487	1.45964
$962$	−3.93803	−0.126967
$963$	−2.17209	−0.0699946
$964$	−24.4641	−0.787936
$965$	0	0
$966$	11.6603	0.375163
$967$	−4.24264	−0.136434	−0.0682171	−	0.997671i	$-0.521731\pi$
$967$	−4.24264	−0.136434	−0.0682171	+	0.997671i	$0.521731\pi$
$968$	0	0
$969$	31.3205	1.00616
$970$	0	0
$971$	−24.9282	−0.799984	−0.399992	−	0.916519i	$-0.630987\pi$
$971$	−24.9282	−0.799984	−0.399992	+	0.916519i	$0.630987\pi$
$972$	12.9038	0.413889
$973$	−48.8139	−1.56490
$974$	6.58846	0.211108
$975$	0	0
$976$	−28.9090	−0.925353
$977$	29.1165	0.931519	0.465759	−	0.884911i	$-0.345781\pi$
$977$	29.1165	0.931519	0.465759	+	0.884911i	$0.345781\pi$
$978$	−5.13922	−0.164334
$979$	0	0
$980$	0	0
$981$	6.73205	0.214938
$982$	−20.5569	−0.655996
$983$	56.0237	1.78688	0.893439	−	0.449184i	$-0.148285\pi$
$983$	56.0237	1.78688	0.893439	+	0.449184i	$0.148285\pi$
$984$	6.46410	0.206068
$985$	0	0
$986$	13.8564	0.441278
$987$	−74.0295	−2.35639
$988$	−30.8353	−0.981001
$989$	−22.4833	−0.714929
$990$	0	0
$991$	19.9090	0.632429	0.316215	−	0.948688i	$-0.397588\pi$
$991$	19.9090	0.632429	0.316215	+	0.948688i	$0.397588\pi$
$992$	44.8759	1.42481
$993$	−35.1523	−1.11552
$994$	14.1962	0.450275
$995$	0	0
$996$	−33.1244	−1.04959
$997$	34.1170	1.08050	0.540248	−	0.841506i	$-0.318330\pi$
$997$	34.1170	1.08050	0.540248	+	0.841506i	$0.318330\pi$
$998$	−9.11441	−0.288512
$999$	7.85641	0.248566

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3025.2.a.y.1.2		4
5.2	odd	4		605.2.b.d.364.2	✓	4
5.3	odd	4		605.2.b.d.364.3	yes	4
5.4	even	2	inner	3025.2.a.y.1.3		4
11.10	odd	2		3025.2.a.z.1.3		4
55.2	even	20		605.2.j.e.444.3		16
55.3	odd	20		605.2.j.f.9.3		16
55.7	even	20		605.2.j.e.269.2		16
55.8	even	20		605.2.j.e.9.2		16
55.13	even	20		605.2.j.e.444.2		16
55.17	even	20		605.2.j.e.124.2		16
55.18	even	20		605.2.j.e.269.3		16
55.27	odd	20		605.2.j.f.124.3		16
55.28	even	20		605.2.j.e.124.3		16
55.32	even	4		605.2.b.e.364.3	yes	4
55.37	odd	20		605.2.j.f.269.3		16
55.38	odd	20		605.2.j.f.124.2		16
55.42	odd	20		605.2.j.f.444.2		16
55.43	even	4		605.2.b.e.364.2	yes	4
55.47	odd	20		605.2.j.f.9.2		16
55.48	odd	20		605.2.j.f.269.2		16
55.52	even	20		605.2.j.e.9.3		16
55.53	odd	20		605.2.j.f.444.3		16
55.54	odd	2		3025.2.a.z.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.b.d.364.2	✓	4	5.2	odd	4
605.2.b.d.364.3	yes	4	5.3	odd	4
605.2.b.e.364.2	yes	4	55.43	even	4
605.2.b.e.364.3	yes	4	55.32	even	4
605.2.j.e.9.2		16	55.8	even	20
605.2.j.e.9.3		16	55.52	even	20
605.2.j.e.124.2		16	55.17	even	20
605.2.j.e.124.3		16	55.28	even	20
605.2.j.e.269.2		16	55.7	even	20
605.2.j.e.269.3		16	55.18	even	20
605.2.j.e.444.2		16	55.13	even	20
605.2.j.e.444.3		16	55.2	even	20
605.2.j.f.9.2		16	55.47	odd	20
605.2.j.f.9.3		16	55.3	odd	20
605.2.j.f.124.2		16	55.38	odd	20
605.2.j.f.124.3		16	55.27	odd	20
605.2.j.f.269.2		16	55.48	odd	20
605.2.j.f.269.3		16	55.37	odd	20
605.2.j.f.444.2		16	55.42	odd	20
605.2.j.f.444.3		16	55.53	odd	20
3025.2.a.y.1.2		4	1.1	even	1	trivial
3025.2.a.y.1.3		4	5.4	even	2	inner
3025.2.a.z.1.2		4	55.54	odd	2
3025.2.a.z.1.3		4	11.10	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:	\( N \)	\(=\)	\( 3025 = 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3025.a (trivial)

\(f(q)\)	\(=\)	\(q-0.517638 q^{2} +1.93185 q^{3} -1.73205 q^{4} -1.00000 q^{6} +3.34607 q^{7} +1.93185 q^{8} +0.732051 q^{9} +O(q^{10})\)	\(q-0.517638 q^{2} +1.93185 q^{3} -1.73205 q^{4} -1.00000 q^{6} +3.34607 q^{7} +1.93185 q^{8} +0.732051 q^{9} -3.34607 q^{12} -4.24264 q^{13} -1.73205 q^{14} +2.46410 q^{16} -3.86370 q^{17} -0.378937 q^{18} -4.19615 q^{19} +6.46410 q^{21} -3.48477 q^{23} +3.73205 q^{24} +2.19615 q^{26} -4.38134 q^{27} -5.79555 q^{28} +6.92820 q^{29} -8.73205 q^{31} -5.13922 q^{32} +2.00000 q^{34} -1.26795 q^{36} -1.79315 q^{37} +2.17209 q^{38} -8.19615 q^{39} +1.73205 q^{41} -3.34607 q^{42} +6.45189 q^{43} +1.80385 q^{46} -11.4524 q^{47} +4.76028 q^{48} +4.19615 q^{49} -7.46410 q^{51} +7.34847 q^{52} -2.17209 q^{53} +2.26795 q^{54} +6.46410 q^{56} -8.10634 q^{57} -3.58630 q^{58} -1.26795 q^{59} -11.7321 q^{61} +4.52004 q^{62} +2.44949 q^{63} -2.26795 q^{64} -2.20925 q^{67} +6.69213 q^{68} -6.73205 q^{69} -8.19615 q^{71} +1.41421 q^{72} +4.89898 q^{73} +0.928203 q^{74} +7.26795 q^{76} +4.24264 q^{78} +1.46410 q^{79} -10.6603 q^{81} -0.896575 q^{82} +9.89949 q^{83} -11.1962 q^{84} -3.33975 q^{86} +13.3843 q^{87} +0.464102 q^{89} -14.1962 q^{91} +6.03579 q^{92} -16.8690 q^{93} +5.92820 q^{94} -9.92820 q^{96} +9.14162 q^{97} -2.17209 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{6} - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^6 - 4 * q^9	\( 4 q - 4 q^{6} - 4 q^{9} - 4 q^{16} + 4 q^{19} + 12 q^{21} + 8 q^{24} - 12 q^{26} - 28 q^{31} + 8 q^{34} - 12 q^{36} - 12 q^{39} + 28 q^{46} - 4 q^{49} - 16 q^{51} + 16 q^{54} + 12 q^{56} - 12 q^{59} - 40 q^{61} - 16 q^{64} - 20 q^{69} - 12 q^{71} - 24 q^{74} + 36 q^{76} - 8 q^{79} - 8 q^{81} - 24 q^{84} - 48 q^{86} - 12 q^{89} - 36 q^{91} - 4 q^{94} - 12 q^{96}+O(q^{100}) \) 4 * q - 4 * q^6 - 4 * q^9 - 4 * q^16 + 4 * q^19 + 12 * q^21 + 8 * q^24 - 12 * q^26 - 28 * q^31 + 8 * q^34 - 12 * q^36 - 12 * q^39 + 28 * q^46 - 4 * q^49 - 16 * q^51 + 16 * q^54 + 12 * q^56 - 12 * q^59 - 40 * q^61 - 16 * q^64 - 20 * q^69 - 12 * q^71 - 24 * q^74 + 36 * q^76 - 8 * q^79 - 8 * q^81 - 24 * q^84 - 48 * q^86 - 12 * q^89 - 36 * q^91 - 4 * q^94 - 12 * q^96

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