LMFDB - Embedded newform 3025.2.a.y.1.1

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			1.1
Root			$-1.93185$ 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)

$f(q)$	$=$	$q-1.93185 q^{2} +0.517638 q^{3} +1.73205 q^{4} -1.00000 q^{6} -0.896575 q^{7} +0.517638 q^{8} -2.73205 q^{9} +O(q^{10})$	\(q-1.93185 q^{2} +0.517638 q^{3} +1.73205 q^{4} -1.00000 q^{6} -0.896575 q^{7} +0.517638 q^{8} -2.73205 q^{9} +0.896575 q^{12} +4.24264 q^{13} +1.73205 q^{14} -4.46410 q^{16} -1.03528 q^{17} +5.27792 q^{18} +6.19615 q^{19} -0.464102 q^{21} -6.31319 q^{23} +0.267949 q^{24} -8.19615 q^{26} -2.96713 q^{27} -1.55291 q^{28} -6.92820 q^{29} -5.26795 q^{31} +7.58871 q^{32} +2.00000 q^{34} -4.73205 q^{36} +6.69213 q^{37} -11.9700 q^{38} +2.19615 q^{39} -1.73205 q^{41} +0.896575 q^{42} +10.6945 q^{43} +12.1962 q^{46} +4.10394 q^{47} -2.31079 q^{48} -6.19615 q^{49} -0.535898 q^{51} +7.34847 q^{52} +11.9700 q^{53} +5.73205 q^{54} -0.464102 q^{56} +3.20736 q^{57} +13.3843 q^{58} -4.73205 q^{59} -8.26795 q^{61} +10.1769 q^{62} +2.44949 q^{63} -5.73205 q^{64} -14.9372 q^{67} -1.79315 q^{68} -3.26795 q^{69} +2.19615 q^{71} -1.41421 q^{72} +4.89898 q^{73} -12.9282 q^{74} +10.7321 q^{76} -4.24264 q^{78} -5.46410 q^{79} +6.66025 q^{81} +3.34607 q^{82} -9.89949 q^{83} -0.803848 q^{84} -20.6603 q^{86} -3.58630 q^{87} -6.46410 q^{89} -3.80385 q^{91} -10.9348 q^{92} -2.72689 q^{93} -7.92820 q^{94} +3.92820 q^{96} +0.656339 q^{97} +11.9700 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.93185	−1.36603	−0.683013	−	0.730406i	$-0.739331\pi$
$2$	−1.93185	−1.36603	−0.683013	+	0.730406i	$0.739331\pi$
$3$	0.517638	0.298858	0.149429	−	0.988772i	$-0.452256\pi$
$3$	0.517638	0.298858	0.149429	+	0.988772i	$0.452256\pi$
$4$	1.73205	0.866025
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	−0.896575	−0.338874	−0.169437	−	0.985541i	$-0.554195\pi$
$7$	−0.896575	−0.338874	−0.169437	+	0.985541i	$0.554195\pi$
$8$	0.517638	0.183013
$9$	−2.73205	−0.910684
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0.896575	0.258819
$13$	4.24264	1.17670	0.588348	−	0.808608i	$-0.299778\pi$
$13$	4.24264	1.17670	0.588348	+	0.808608i	$0.299778\pi$
$14$	1.73205	0.462910
$15$	0	0
$16$	−4.46410	−1.11603
$17$	−1.03528	−0.251091	−0.125546	−	0.992088i	$-0.540068\pi$
$17$	−1.03528	−0.251091	−0.125546	+	0.992088i	$0.540068\pi$
$18$	5.27792	1.24402
$19$	6.19615	1.42149	0.710747	−	0.703447i	$-0.248357\pi$
$19$	6.19615	1.42149	0.710747	+	0.703447i	$0.248357\pi$
$20$	0	0
$21$	−0.464102	−0.101275
$22$	0	0
$23$	−6.31319	−1.31639	−0.658196	−	0.752847i	$-0.728680\pi$
$23$	−6.31319	−1.31639	−0.658196	+	0.752847i	$0.728680\pi$
$24$	0.267949	0.0546949
$25$	0	0
$26$	−8.19615	−1.60740
$27$	−2.96713	−0.571024
$28$	−1.55291	−0.293473
$29$	−6.92820	−1.28654	−0.643268	−	0.765641i	$-0.722422\pi$
$29$	−6.92820	−1.28654	−0.643268	+	0.765641i	$0.722422\pi$
$30$	0	0
$31$	−5.26795	−0.946152	−0.473076	−	0.881022i	$-0.656856\pi$
$31$	−5.26795	−0.946152	−0.473076	+	0.881022i	$0.656856\pi$
$32$	7.58871	1.34151
$33$	0	0
$34$	2.00000	0.342997
$35$	0	0
$36$	−4.73205	−0.788675
$37$	6.69213	1.10018	0.550090	−	0.835106i	$-0.314594\pi$
$37$	6.69213	1.10018	0.550090	+	0.835106i	$0.314594\pi$
$38$	−11.9700	−1.94180
$39$	2.19615	0.351666
$40$	0	0
$41$	−1.73205	−0.270501	−0.135250	−	0.990811i	$-0.543184\pi$
$41$	−1.73205	−0.270501	−0.135250	+	0.990811i	$0.543184\pi$
$42$	0.896575	0.138345
$43$	10.6945	1.63090	0.815451	−	0.578827i	$-0.196489\pi$
$43$	10.6945	1.63090	0.815451	+	0.578827i	$0.196489\pi$
$44$	0	0
$45$	0	0
$46$	12.1962	1.79822
$47$	4.10394	0.598621	0.299311	−	0.954156i	$-0.403243\pi$
$47$	4.10394	0.598621	0.299311	+	0.954156i	$0.403243\pi$
$48$	−2.31079	−0.333534
$49$	−6.19615	−0.885165
$50$	0	0
$51$	−0.535898	−0.0750408
$52$	7.34847	1.01905
$53$	11.9700	1.64421	0.822106	−	0.569334i	$-0.192799\pi$
$53$	11.9700	1.64421	0.822106	+	0.569334i	$0.192799\pi$
$54$	5.73205	0.780033
$55$	0	0
$56$	−0.464102	−0.0620182
$57$	3.20736	0.424826
$58$	13.3843	1.75744
$59$	−4.73205	−0.616061	−0.308030	−	0.951377i	$-0.599670\pi$
$59$	−4.73205	−0.616061	−0.308030	+	0.951377i	$0.599670\pi$
$60$	0	0
$61$	−8.26795	−1.05860	−0.529301	−	0.848434i	$-0.677546\pi$
$61$	−8.26795	−1.05860	−0.529301	+	0.848434i	$0.677546\pi$
$62$	10.1769	1.29247
$63$	2.44949	0.308607
$64$	−5.73205	−0.716506
$65$	0	0
$66$	0	0
$67$	−14.9372	−1.82487	−0.912433	−	0.409226i	$-0.865799\pi$
$67$	−14.9372	−1.82487	−0.912433	+	0.409226i	$0.865799\pi$
$68$	−1.79315	−0.217451
$69$	−3.26795	−0.393415
$70$	0	0
$71$	2.19615	0.260635	0.130318	−	0.991472i	$-0.458400\pi$
$71$	2.19615	0.260635	0.130318	+	0.991472i	$0.458400\pi$
$72$	−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	$0.407445\pi$
$74$	−12.9282	−1.50287
$75$	0	0
$76$	10.7321	1.23105
$77$	0	0
$78$	−4.24264	−0.480384
$79$	−5.46410	−0.614759	−0.307380	−	0.951587i	$-0.599452\pi$
$79$	−5.46410	−0.614759	−0.307380	+	0.951587i	$0.599452\pi$
$80$	0	0
$81$	6.66025	0.740028
$82$	3.34607	0.369511
$83$	−9.89949	−1.08661	−0.543305	−	0.839535i	$-0.682827\pi$
$83$	−9.89949	−1.08661	−0.543305	+	0.839535i	$0.682827\pi$
$84$	−0.803848	−0.0877070
$85$	0	0
$86$	−20.6603	−2.22785
$87$	−3.58630	−0.384492
$88$	0	0
$89$	−6.46410	−0.685193	−0.342597	−	0.939483i	$-0.611306\pi$
$89$	−6.46410	−0.685193	−0.342597	+	0.939483i	$0.611306\pi$
$90$	0	0
$91$	−3.80385	−0.398752
$92$	−10.9348	−1.14003
$93$	−2.72689	−0.282765
$94$	−7.92820	−0.817732
$95$	0	0
$96$	3.92820	0.400921
$97$	0.656339	0.0666411	0.0333206	−	0.999445i	$-0.489392\pi$
$97$	0.656339	0.0666411	0.0333206	+	0.999445i	$0.489392\pi$
$98$	11.9700	1.20916
$99$	0	0
$100$	0	0
$101$	−1.39230	−0.138540	−0.0692698	−	0.997598i	$-0.522067\pi$
$101$	−1.39230	−0.138540	−0.0692698	+	0.997598i	$0.522067\pi$
$102$	1.03528	0.102508
$103$	4.24264	0.418040	0.209020	−	0.977911i	$-0.432973\pi$
$103$	4.24264	0.418040	0.209020	+	0.977911i	$0.432973\pi$
$104$	2.19615	0.215350
$105$	0	0
$106$	−23.1244	−2.24604
$107$	−4.38134	−0.423560	−0.211780	−	0.977317i	$-0.567926\pi$
$107$	−4.38134	−0.423560	−0.211780	+	0.977317i	$0.567926\pi$
$108$	−5.13922	−0.494521
$109$	−1.19615	−0.114571	−0.0572853	−	0.998358i	$-0.518244\pi$
$109$	−1.19615	−0.114571	−0.0572853	+	0.998358i	$0.518244\pi$
$110$	0	0
$111$	3.46410	0.328798
$112$	4.00240	0.378192
$113$	2.82843	0.266076	0.133038	−	0.991111i	$-0.457527\pi$
$113$	2.82843	0.266076	0.133038	+	0.991111i	$0.457527\pi$
$114$	−6.19615	−0.580323
$115$	0	0
$116$	−12.0000	−1.11417
$117$	−11.5911	−1.07160
$118$	9.14162	0.841554
$119$	0.928203	0.0850883
$120$	0	0
$121$	0	0
$122$	15.9725	1.44608
$123$	−0.896575	−0.0808415
$124$	−9.12436	−0.819391
$125$	0	0
$126$	−4.73205	−0.421565
$127$	−3.34607	−0.296915	−0.148458	−	0.988919i	$-0.547431\pi$
$127$	−3.34607	−0.296915	−0.148458	+	0.988919i	$0.547431\pi$
$128$	−4.10394	−0.362740
$129$	5.53590	0.487409
$130$	0	0
$131$	21.1244	1.84564	0.922822	−	0.385227i	$-0.125877\pi$
$131$	21.1244	1.84564	0.922822	+	0.385227i	$0.125877\pi$
$132$	0	0
$133$	−5.55532	−0.481707
$134$	28.8564	2.49281
$135$	0	0
$136$	−0.535898	−0.0459529
$137$	−13.7632	−1.17587	−0.587935	−	0.808908i	$-0.700059\pi$
$137$	−13.7632	−1.17587	−0.587935	+	0.808908i	$0.700059\pi$
$138$	6.31319	0.537415
$139$	16.5885	1.40701	0.703507	−	0.710688i	$-0.251616\pi$
$139$	16.5885	1.40701	0.703507	+	0.710688i	$0.251616\pi$
$140$	0	0
$141$	2.12436	0.178903
$142$	−4.24264	−0.356034
$143$	0	0
$144$	12.1962	1.01635
$145$	0	0
$146$	−9.46410	−0.783255
$147$	−3.20736	−0.264539
$148$	11.5911	0.952783
$149$	−13.7321	−1.12497	−0.562487	−	0.826806i	$-0.690155\pi$
$149$	−13.7321	−1.12497	−0.562487	+	0.826806i	$0.690155\pi$
$150$	0	0
$151$	−6.19615	−0.504236	−0.252118	−	0.967697i	$-0.581127\pi$
$151$	−6.19615	−0.504236	−0.252118	+	0.967697i	$0.581127\pi$
$152$	3.20736	0.260152
$153$	2.82843	0.228665
$154$	0	0
$155$	0	0
$156$	3.80385	0.304552
$157$	−12.0716	−0.963417	−0.481709	−	0.876331i	$-0.659984\pi$
$157$	−12.0716	−0.963417	−0.481709	+	0.876331i	$0.659984\pi$
$158$	10.5558	0.839777
$159$	6.19615	0.491387
$160$	0	0
$161$	5.66025	0.446091
$162$	−12.8666	−1.01090
$163$	−7.58871	−0.594393	−0.297197	−	0.954816i	$-0.596052\pi$
$163$	−7.58871	−0.594393	−0.297197	+	0.954816i	$0.596052\pi$
$164$	−3.00000	−0.234261
$165$	0	0
$166$	19.1244	1.48434
$167$	−1.93185	−0.149491	−0.0747456	−	0.997203i	$-0.523814\pi$
$167$	−1.93185	−0.149491	−0.0747456	+	0.997203i	$0.523814\pi$
$168$	−0.240237	−0.0185347
$169$	5.00000	0.384615
$170$	0	0
$171$	−16.9282	−1.29453
$172$	18.5235	1.41240
$173$	−7.62587	−0.579784	−0.289892	−	0.957059i	$-0.593619\pi$
$173$	−7.62587	−0.579784	−0.289892	+	0.957059i	$0.593619\pi$
$174$	6.92820	0.525226
$175$	0	0
$176$	0	0
$177$	−2.44949	−0.184115
$178$	12.4877	0.935992
$179$	−8.53590	−0.638003	−0.319002	−	0.947754i	$-0.603348\pi$
$179$	−8.53590	−0.638003	−0.319002	+	0.947754i	$0.603348\pi$
$180$	0	0
$181$	3.39230	0.252148	0.126074	−	0.992021i	$-0.459762\pi$
$181$	3.39230	0.252148	0.126074	+	0.992021i	$0.459762\pi$
$182$	7.34847	0.544705
$183$	−4.27981	−0.316372
$184$	−3.26795	−0.240916
$185$	0	0
$186$	5.26795	0.386265
$187$	0	0
$188$	7.10823	0.518421
$189$	2.66025	0.193505
$190$	0	0
$191$	−10.7321	−0.776544	−0.388272	−	0.921545i	$-0.626928\pi$
$191$	−10.7321	−0.776544	−0.388272	+	0.921545i	$0.626928\pi$
$192$	−2.96713	−0.214134
$193$	10.2784	0.739858	0.369929	−	0.929060i	$-0.379382\pi$
$193$	10.2784	0.739858	0.369929	+	0.929060i	$0.379382\pi$
$194$	−1.26795	−0.0910334
$195$	0	0
$196$	−10.7321	−0.766575
$197$	2.55103	0.181753	0.0908765	−	0.995862i	$-0.471033\pi$
$197$	2.55103	0.181753	0.0908765	+	0.995862i	$0.471033\pi$
$198$	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
$201$	−7.73205	−0.545377
$202$	2.68973	0.189248
$203$	6.21166	0.435973
$204$	−0.928203	−0.0649872
$205$	0	0
$206$	−8.19615	−0.571053
$207$	17.2480	1.19882
$208$	−18.9396	−1.31322
$209$	0	0
$210$	0	0
$211$	11.1244	0.765832	0.382916	−	0.923783i	$-0.374920\pi$
$211$	11.1244	0.765832	0.382916	+	0.923783i	$0.374920\pi$
$212$	20.7327	1.42393
$213$	1.13681	0.0778931
$214$	8.46410	0.578594
$215$	0	0
$216$	−1.53590	−0.104505
$217$	4.72311	0.320626
$218$	2.31079	0.156506
$219$	2.53590	0.171360
$220$	0	0
$221$	−4.39230	−0.295458
$222$	−6.69213	−0.449146
$223$	−5.79555	−0.388099	−0.194050	−	0.980992i	$-0.562162\pi$
$223$	−5.79555	−0.388099	−0.194050	+	0.980992i	$0.562162\pi$
$224$	−6.80385	−0.454601
$225$	0	0
$226$	−5.46410	−0.363467
$227$	−21.5278	−1.42885	−0.714424	−	0.699713i	$-0.753311\pi$
$227$	−21.5278	−1.42885	−0.714424	+	0.699713i	$0.753311\pi$
$228$	5.55532	0.367910
$229$	−17.9282	−1.18473	−0.592365	−	0.805670i	$-0.701805\pi$
$229$	−17.9282	−1.18473	−0.592365	+	0.805670i	$0.701805\pi$
$230$	0	0
$231$	0	0
$232$	−3.58630	−0.235452
$233$	−13.0053	−0.852007	−0.426004	−	0.904721i	$-0.640079\pi$
$233$	−13.0053	−0.852007	−0.426004	+	0.904721i	$0.640079\pi$
$234$	22.3923	1.46383
$235$	0	0
$236$	−8.19615	−0.533524
$237$	−2.82843	−0.183726
$238$	−1.79315	−0.116233
$239$	−11.6603	−0.754239	−0.377120	−	0.926165i	$-0.623085\pi$
$239$	−11.6603	−0.754239	−0.377120	+	0.926165i	$0.623085\pi$
$240$	0	0
$241$	−10.1244	−0.652167	−0.326084	−	0.945341i	$-0.605729\pi$
$241$	−10.1244	−0.652167	−0.326084	+	0.945341i	$0.605729\pi$
$242$	0	0
$243$	12.3490	0.792188
$244$	−14.3205	−0.916777
$245$	0	0
$246$	1.73205	0.110432
$247$	26.2880	1.67267
$248$	−2.72689	−0.173158
$249$	−5.12436	−0.324743
$250$	0	0
$251$	−14.1962	−0.896053	−0.448027	−	0.894020i	$-0.647873\pi$
$251$	−14.1962	−0.896053	−0.448027	+	0.894020i	$0.647873\pi$
$252$	4.24264	0.267261
$253$	0	0
$254$	6.46410	0.405594
$255$	0	0
$256$	19.3923	1.21202
$257$	5.00052	0.311924	0.155962	−	0.987763i	$-0.450152\pi$
$257$	5.00052	0.311924	0.155962	+	0.987763i	$0.450152\pi$
$258$	−10.6945	−0.665813
$259$	−6.00000	−0.372822
$260$	0	0
$261$	18.9282	1.17163
$262$	−40.8091	−2.52120
$263$	−21.4906	−1.32517	−0.662584	−	0.748988i	$-0.730540\pi$
$263$	−21.4906	−1.32517	−0.662584	+	0.748988i	$0.730540\pi$
$264$	0	0
$265$	0	0
$266$	10.7321	0.658024
$267$	−3.34607	−0.204776
$268$	−25.8719	−1.58038
$269$	−19.7321	−1.20308	−0.601542	−	0.798841i	$-0.705447\pi$
$269$	−19.7321	−1.20308	−0.601542	+	0.798841i	$0.705447\pi$
$270$	0	0
$271$	4.53590	0.275536	0.137768	−	0.990465i	$-0.456007\pi$
$271$	4.53590	0.275536	0.137768	+	0.990465i	$0.456007\pi$
$272$	4.62158	0.280224
$273$	−1.96902	−0.119170
$274$	26.5885	1.60627
$275$	0	0
$276$	−5.66025	−0.340707
$277$	22.7017	1.36402	0.682008	−	0.731345i	$-0.261107\pi$
$277$	22.7017	1.36402	0.682008	+	0.731345i	$0.261107\pi$
$278$	−32.0464	−1.92202
$279$	14.3923	0.861645
$280$	0	0
$281$	17.3205	1.03325	0.516627	−	0.856210i	$-0.327187\pi$
$281$	17.3205	1.03325	0.516627	+	0.856210i	$0.327187\pi$
$282$	−4.10394	−0.244386
$283$	−30.1146	−1.79013	−0.895063	−	0.445939i	$-0.852870\pi$
$283$	−30.1146	−1.79013	−0.895063	+	0.445939i	$0.852870\pi$
$284$	3.80385	0.225717
$285$	0	0
$286$	0	0
$287$	1.55291	0.0916656
$288$	−20.7327	−1.22169
$289$	−15.9282	−0.936953
$290$	0	0
$291$	0.339746	0.0199163
$292$	8.48528	0.496564
$293$	−9.41902	−0.550265	−0.275133	−	0.961406i	$-0.588722\pi$
$293$	−9.41902	−0.550265	−0.275133	+	0.961406i	$0.588722\pi$
$294$	6.19615	0.361367
$295$	0	0
$296$	3.46410	0.201347
$297$	0	0
$298$	26.5283	1.53674
$299$	−26.7846	−1.54899
$300$	0	0
$301$	−9.58846	−0.552669
$302$	11.9700	0.688799
$303$	−0.720710	−0.0414037
$304$	−27.6603	−1.58642
$305$	0	0
$306$	−5.46410	−0.312362
$307$	−17.6269	−1.00602	−0.503010	−	0.864280i	$-0.667774\pi$
$307$	−17.6269	−1.00602	−0.503010	+	0.864280i	$0.667774\pi$
$308$	0	0
$309$	2.19615	0.124935
$310$	0	0
$311$	−28.0526	−1.59071	−0.795357	−	0.606141i	$-0.792717\pi$
$311$	−28.0526	−1.59071	−0.795357	+	0.606141i	$0.792717\pi$
$312$	1.13681	0.0643593
$313$	17.1464	0.969173	0.484587	−	0.874743i	$-0.338970\pi$
$313$	17.1464	0.969173	0.484587	+	0.874743i	$0.338970\pi$
$314$	23.3205	1.31605
$315$	0	0
$316$	−9.46410	−0.532397
$317$	6.96953	0.391448	0.195724	−	0.980659i	$-0.437294\pi$
$317$	6.96953	0.391448	0.195724	+	0.980659i	$0.437294\pi$
$318$	−11.9700	−0.671247
$319$	0	0
$320$	0	0
$321$	−2.26795	−0.126585
$322$	−10.9348	−0.609371
$323$	−6.41473	−0.356925
$324$	11.5359	0.640883
$325$	0	0
$326$	14.6603	0.811956
$327$	−0.619174	−0.0342404
$328$	−0.896575	−0.0495051
$329$	−3.67949	−0.202857
$330$	0	0
$331$	−7.80385	−0.428938	−0.214469	−	0.976731i	$-0.568802\pi$
$331$	−7.80385	−0.428938	−0.214469	+	0.976731i	$0.568802\pi$
$332$	−17.1464	−0.941033
$333$	−18.2832	−1.00192
$334$	3.73205	0.204209
$335$	0	0
$336$	2.07180	0.113026
$337$	−7.82894	−0.426470	−0.213235	−	0.977001i	$-0.568400\pi$
$337$	−7.82894	−0.426470	−0.213235	+	0.977001i	$0.568400\pi$
$338$	−9.65926	−0.525394
$339$	1.46410	0.0795191
$340$	0	0
$341$	0	0
$342$	32.7028	1.76836
$343$	11.8313	0.638833
$344$	5.53590	0.298476
$345$	0	0
$346$	14.7321	0.792000
$347$	2.96713	0.159284	0.0796419	−	0.996824i	$-0.474622\pi$
$347$	2.96713	0.159284	0.0796419	+	0.996824i	$0.474622\pi$
$348$	−6.21166	−0.332980
$349$	27.3205	1.46243	0.731217	−	0.682145i	$-0.238953\pi$
$349$	27.3205	1.46243	0.731217	+	0.682145i	$0.238953\pi$
$350$	0	0
$351$	−12.5885	−0.671922
$352$	0	0
$353$	1.69161	0.0900356	0.0450178	−	0.998986i	$-0.485666\pi$
$353$	1.69161	0.0900356	0.0450178	+	0.998986i	$0.485666\pi$
$354$	4.73205	0.251506
$355$	0	0
$356$	−11.1962	−0.593395
$357$	0.480473	0.0254293
$358$	16.4901	0.871528
$359$	18.3397	0.967935	0.483967	−	0.875086i	$-0.339195\pi$
$359$	18.3397	0.967935	0.483967	+	0.875086i	$0.339195\pi$
$360$	0	0
$361$	19.3923	1.02065
$362$	−6.55343	−0.344441
$363$	0	0
$364$	−6.58846	−0.345329
$365$	0	0
$366$	8.26795	0.432173
$367$	29.6341	1.54689	0.773444	−	0.633865i	$-0.218532\pi$
$367$	29.6341	1.54689	0.773444	+	0.633865i	$0.218532\pi$
$368$	28.1827	1.46913
$369$	4.73205	0.246341
$370$	0	0
$371$	−10.7321	−0.557180
$372$	−4.72311	−0.244882
$373$	−34.1170	−1.76651	−0.883255	−	0.468892i	$-0.844653\pi$
$373$	−34.1170	−1.76651	−0.883255	+	0.468892i	$0.844653\pi$
$374$	0	0
$375$	0	0
$376$	2.12436	0.109555
$377$	−29.3939	−1.51386
$378$	−5.13922	−0.264333
$379$	−5.46410	−0.280672	−0.140336	−	0.990104i	$-0.544818\pi$
$379$	−5.46410	−0.280672	−0.140336	+	0.990104i	$0.544818\pi$
$380$	0	0
$381$	−1.73205	−0.0887357
$382$	20.7327	1.06078
$383$	−23.7642	−1.21430	−0.607148	−	0.794589i	$-0.707686\pi$
$383$	−23.7642	−1.21430	−0.607148	+	0.794589i	$0.707686\pi$
$384$	−2.12436	−0.108408
$385$	0	0
$386$	−19.8564	−1.01066
$387$	−29.2180	−1.48523
$388$	1.13681	0.0577129
$389$	−18.1244	−0.918941	−0.459471	−	0.888193i	$-0.651961\pi$
$389$	−18.1244	−0.918941	−0.459471	+	0.888193i	$0.651961\pi$
$390$	0	0
$391$	6.53590	0.330535
$392$	−3.20736	−0.161996
$393$	10.9348	0.551586
$394$	−4.92820	−0.248279
$395$	0	0
$396$	0	0
$397$	−16.0096	−0.803500	−0.401750	−	0.915749i	$-0.631598\pi$
$397$	−16.0096	−0.803500	−0.401750	+	0.915749i	$0.631598\pi$
$398$	3.86370	0.193670
$399$	−2.87564	−0.143962
$400$	0	0
$401$	−32.3205	−1.61401	−0.807005	−	0.590545i	$-0.798913\pi$
$401$	−32.3205	−1.61401	−0.807005	+	0.590545i	$0.798913\pi$
$402$	14.9372	0.744999
$403$	−22.3500	−1.11333
$404$	−2.41154	−0.119979
$405$	0	0
$406$	−12.0000	−0.595550
$407$	0	0
$408$	−0.277401	−0.0137334
$409$	−1.87564	−0.0927446	−0.0463723	−	0.998924i	$-0.514766\pi$
$409$	−1.87564	−0.0927446	−0.0463723	+	0.998924i	$0.514766\pi$
$410$	0	0
$411$	−7.12436	−0.351419
$412$	7.34847	0.362033
$413$	4.24264	0.208767
$414$	−33.3205	−1.63761
$415$	0	0
$416$	32.1962	1.57855
$417$	8.58682	0.420498
$418$	0	0
$419$	−29.6603	−1.44900	−0.724499	−	0.689276i	$-0.757929\pi$
$419$	−29.6603	−1.44900	−0.724499	+	0.689276i	$0.757929\pi$
$420$	0	0
$421$	21.0526	1.02604	0.513019	−	0.858377i	$-0.328527\pi$
$421$	21.0526	1.02604	0.513019	+	0.858377i	$0.328527\pi$
$422$	−21.4906	−1.04615
$423$	−11.2122	−0.545154
$424$	6.19615	0.300912
$425$	0	0
$426$	−2.19615	−0.106404
$427$	7.41284	0.358732
$428$	−7.58871	−0.366814
$429$	0	0
$430$	0	0
$431$	−18.9282	−0.911739	−0.455870	−	0.890047i	$-0.650672\pi$
$431$	−18.9282	−0.911739	−0.455870	+	0.890047i	$0.650672\pi$
$432$	13.2456	0.637277
$433$	20.2523	0.973261	0.486631	−	0.873608i	$-0.338226\pi$
$433$	20.2523	0.973261	0.486631	+	0.873608i	$0.338226\pi$
$434$	−9.12436	−0.437983
$435$	0	0
$436$	−2.07180	−0.0992211
$437$	−39.1175	−1.87124
$438$	−4.89898	−0.234082
$439$	−20.2487	−0.966418	−0.483209	−	0.875505i	$-0.660529\pi$
$439$	−20.2487	−0.966418	−0.483209	+	0.875505i	$0.660529\pi$
$440$	0	0
$441$	16.9282	0.806105
$442$	8.48528	0.403604
$443$	−9.17878	−0.436097	−0.218049	−	0.975938i	$-0.569969\pi$
$443$	−9.17878	−0.436097	−0.218049	+	0.975938i	$0.569969\pi$
$444$	6.00000	0.284747
$445$	0	0
$446$	11.1962	0.530153
$447$	−7.10823	−0.336208
$448$	5.13922	0.242805
$449$	−10.5167	−0.496312	−0.248156	−	0.968720i	$-0.579825\pi$
$449$	−10.5167	−0.496312	−0.248156	+	0.968720i	$0.579825\pi$
$450$	0	0
$451$	0	0
$452$	4.89898	0.230429
$453$	−3.20736	−0.150695
$454$	41.5885	1.95184
$455$	0	0
$456$	1.66025	0.0777485
$457$	−21.2132	−0.992312	−0.496156	−	0.868233i	$-0.665256\pi$
$457$	−21.2132	−0.992312	−0.496156	+	0.868233i	$0.665256\pi$
$458$	34.6346	1.61837
$459$	3.07180	0.143379
$460$	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
$463$	−17.8671	−0.830356	−0.415178	−	0.909740i	$-0.636281\pi$
$463$	−17.8671	−0.830356	−0.415178	+	0.909740i	$0.636281\pi$
$464$	30.9282	1.43581
$465$	0	0
$466$	25.1244	1.16386
$467$	−17.7656	−0.822094	−0.411047	−	0.911614i	$-0.634837\pi$
$467$	−17.7656	−0.822094	−0.411047	+	0.911614i	$0.634837\pi$
$468$	−20.0764	−0.928032
$469$	13.3923	0.618399
$470$	0	0
$471$	−6.24871	−0.287925
$472$	−2.44949	−0.112747
$473$	0	0
$474$	5.46410	0.250974
$475$	0	0
$476$	1.60770	0.0736886
$477$	−32.7028	−1.49736
$478$	22.5259	1.03031
$479$	9.46410	0.432426	0.216213	−	0.976346i	$-0.430629\pi$
$479$	9.46410	0.432426	0.216213	+	0.976346i	$0.430629\pi$
$480$	0	0
$481$	28.3923	1.29458
$482$	19.5588	0.890877
$483$	2.92996	0.133318
$484$	0	0
$485$	0	0
$486$	−23.8564	−1.08215
$487$	12.7279	0.576757	0.288379	−	0.957516i	$-0.406884\pi$
$487$	12.7279	0.576757	0.288379	+	0.957516i	$0.406884\pi$
$488$	−4.27981	−0.193738
$489$	−3.92820	−0.177639
$490$	0	0
$491$	−15.7128	−0.709109	−0.354555	−	0.935035i	$-0.615368\pi$
$491$	−15.7128	−0.709109	−0.354555	+	0.935035i	$0.615368\pi$
$492$	−1.55291	−0.0700108
$493$	7.17260	0.323038
$494$	−50.7846	−2.28491
$495$	0	0
$496$	23.5167	1.05593
$497$	−1.96902	−0.0883225
$498$	9.89949	0.443607
$499$	38.3923	1.71868	0.859338	−	0.511408i	$-0.170876\pi$
$499$	38.3923	1.71868	0.859338	+	0.511408i	$0.170876\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	27.4249	1.22403
$503$	34.2557	1.52739	0.763693	−	0.645580i	$-0.223384\pi$
$503$	34.2557	1.52739	0.763693	+	0.645580i	$0.223384\pi$
$504$	1.26795	0.0564789
$505$	0	0
$506$	0	0
$507$	2.58819	0.114946
$508$	−5.79555	−0.257136
$509$	−21.9282	−0.971951	−0.485975	−	0.873973i	$-0.661535\pi$
$509$	−21.9282	−0.971951	−0.485975	+	0.873973i	$0.661535\pi$
$510$	0	0
$511$	−4.39230	−0.194304
$512$	−29.2552	−1.29291
$513$	−18.3848	−0.811708
$514$	−9.66025	−0.426096
$515$	0	0
$516$	9.58846	0.422108
$517$	0	0
$518$	11.5911	0.509284
$519$	−3.94744	−0.173273
$520$	0	0
$521$	39.2487	1.71952	0.859759	−	0.510701i	$-0.170614\pi$
$521$	39.2487	1.71952	0.859759	+	0.510701i	$0.170614\pi$
$522$	−36.5665	−1.60047
$523$	−8.66115	−0.378726	−0.189363	−	0.981907i	$-0.560642\pi$
$523$	−8.66115	−0.378726	−0.189363	+	0.981907i	$0.560642\pi$
$524$	36.5885	1.59837
$525$	0	0
$526$	41.5167	1.81021
$527$	5.45378	0.237570
$528$	0	0
$529$	16.8564	0.732887
$530$	0	0
$531$	12.9282	0.561036
$532$	−9.62209	−0.417171
$533$	−7.34847	−0.318298
$534$	6.46410	0.279729
$535$	0	0
$536$	−7.73205	−0.333974
$537$	−4.41851	−0.190673
$538$	38.1194	1.64344
$539$	0	0
$540$	0	0
$541$	−23.3923	−1.00571	−0.502857	−	0.864370i	$-0.667718\pi$
$541$	−23.3923	−1.00571	−0.502857	+	0.864370i	$0.667718\pi$
$542$	−8.76268	−0.376389
$543$	1.75599	0.0753566
$544$	−7.85641	−0.336841
$545$	0	0
$546$	3.80385	0.162790
$547$	20.2523	0.865924	0.432962	−	0.901412i	$-0.357468\pi$
$547$	20.2523	0.865924	0.432962	+	0.901412i	$0.357468\pi$
$548$	−23.8386	−1.01833
$549$	22.5885	0.964052
$550$	0	0
$551$	−42.9282	−1.82880
$552$	−1.69161	−0.0719999
$553$	4.89898	0.208326
$554$	−43.8564	−1.86328
$555$	0	0
$556$	28.7321	1.21851
$557$	6.48906	0.274950	0.137475	−	0.990505i	$-0.456101\pi$
$557$	6.48906	0.274950	0.137475	+	0.990505i	$0.456101\pi$
$558$	−27.8038	−1.17703
$559$	45.3731	1.91908
$560$	0	0
$561$	0	0
$562$	−33.4607	−1.41145
$563$	−6.07296	−0.255945	−0.127972	−	0.991778i	$-0.540847\pi$
$563$	−6.07296	−0.255945	−0.127972	+	0.991778i	$0.540847\pi$
$564$	3.67949	0.154935
$565$	0	0
$566$	58.1769	2.44536
$567$	−5.97142	−0.250776
$568$	1.13681	0.0476996
$569$	5.87564	0.246320	0.123160	−	0.992387i	$-0.460697\pi$
$569$	5.87564	0.246320	0.123160	+	0.992387i	$0.460697\pi$
$570$	0	0
$571$	35.7128	1.49453	0.747267	−	0.664524i	$-0.231365\pi$
$571$	35.7128	1.49453	0.747267	+	0.664524i	$0.231365\pi$
$572$	0	0
$573$	−5.55532	−0.232077
$574$	−3.00000	−0.125218
$575$	0	0
$576$	15.6603	0.652511
$577$	20.7327	0.863115	0.431557	−	0.902085i	$-0.357964\pi$
$577$	20.7327	0.863115	0.431557	+	0.902085i	$0.357964\pi$
$578$	30.7709	1.27990
$579$	5.32051	0.221113
$580$	0	0
$581$	8.87564	0.368224
$582$	−0.656339	−0.0272061
$583$	0	0
$584$	2.53590	0.104936
$585$	0	0
$586$	18.1962	0.751676
$587$	35.9473	1.48370	0.741852	−	0.670564i	$-0.233948\pi$
$587$	35.9473	1.48370	0.741852	+	0.670564i	$0.233948\pi$
$588$	−5.55532	−0.229097
$589$	−32.6410	−1.34495
$590$	0	0
$591$	1.32051	0.0543184
$592$	−29.8744	−1.22783
$593$	30.2533	1.24235	0.621177	−	0.783670i	$-0.286655\pi$
$593$	30.2533	1.24235	0.621177	+	0.783670i	$0.286655\pi$
$594$	0	0
$595$	0	0
$596$	−23.7846	−0.974256
$597$	−1.03528	−0.0423710
$598$	51.7439	2.11597
$599$	20.4449	0.835354	0.417677	−	0.908595i	$-0.362844\pi$
$599$	20.4449	0.835354	0.417677	+	0.908595i	$0.362844\pi$
$600$	0	0
$601$	20.0000	0.815817	0.407909	−	0.913023i	$-0.366258\pi$
$601$	20.0000	0.815817	0.407909	+	0.913023i	$0.366258\pi$
$602$	18.5235	0.754961
$603$	40.8091	1.66188
$604$	−10.7321	−0.436681
$605$	0	0
$606$	1.39230	0.0565585
$607$	31.4916	1.27821	0.639103	−	0.769121i	$-0.279306\pi$
$607$	31.4916	1.27821	0.639103	+	0.769121i	$0.279306\pi$
$608$	47.0208	1.90694
$609$	3.21539	0.130294
$610$	0	0
$611$	17.4115	0.704396
$612$	4.89898	0.198030
$613$	6.69213	0.270293	0.135146	−	0.990826i	$-0.456850\pi$
$613$	6.69213	0.270293	0.135146	+	0.990826i	$0.456850\pi$
$614$	34.0526	1.37425
$615$	0	0
$616$	0	0
$617$	−1.69161	−0.0681019	−0.0340509	−	0.999420i	$-0.510841\pi$
$617$	−1.69161	−0.0681019	−0.0340509	+	0.999420i	$0.510841\pi$
$618$	−4.24264	−0.170664
$619$	−28.7846	−1.15695	−0.578476	−	0.815700i	$-0.696352\pi$
$619$	−28.7846	−1.15695	−0.578476	+	0.815700i	$0.696352\pi$
$620$	0	0
$621$	18.7321	0.751691
$622$	54.1934	2.17296
$623$	5.79555	0.232194
$624$	−9.80385	−0.392468
$625$	0	0
$626$	−33.1244	−1.32392
$627$	0	0
$628$	−20.9086	−0.834344
$629$	−6.92820	−0.276246
$630$	0	0
$631$	19.3205	0.769137	0.384569	−	0.923096i	$-0.374350\pi$
$631$	19.3205	0.769137	0.384569	+	0.923096i	$0.374350\pi$
$632$	−2.82843	−0.112509
$633$	5.75839	0.228875
$634$	−13.4641	−0.534728
$635$	0	0
$636$	10.7321	0.425553
$637$	−26.2880	−1.04157
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	19.8564	0.784281	0.392140	−	0.919905i	$-0.371735\pi$
$641$	19.8564	0.784281	0.392140	+	0.919905i	$0.371735\pi$
$642$	4.38134	0.172918
$643$	−27.6651	−1.09100	−0.545502	−	0.838109i	$-0.683661\pi$
$643$	−27.6651	−1.09100	−0.545502	+	0.838109i	$0.683661\pi$
$644$	9.80385	0.386326
$645$	0	0
$646$	12.3923	0.487569
$647$	18.9767	0.746053	0.373026	−	0.927821i	$-0.378320\pi$
$647$	18.9767	0.746053	0.373026	+	0.927821i	$0.378320\pi$
$648$	3.44760	0.135435
$649$	0	0
$650$	0	0
$651$	2.44486	0.0958218
$652$	−13.1440	−0.514760
$653$	−31.9449	−1.25010	−0.625050	−	0.780584i	$-0.714922\pi$
$653$	−31.9449	−1.25010	−0.625050	+	0.780584i	$0.714922\pi$
$654$	1.19615	0.0467733
$655$	0	0
$656$	7.73205	0.301886
$657$	−13.3843	−0.522170
$658$	7.10823	0.277108
$659$	11.3205	0.440984	0.220492	−	0.975389i	$-0.429234\pi$
$659$	11.3205	0.440984	0.220492	+	0.975389i	$0.429234\pi$
$660$	0	0
$661$	−1.58846	−0.0617838	−0.0308919	−	0.999523i	$-0.509835\pi$
$661$	−1.58846	−0.0617838	−0.0308919	+	0.999523i	$0.509835\pi$
$662$	15.0759	0.585941
$663$	−2.27362	−0.0883003
$664$	−5.12436	−0.198864
$665$	0	0
$666$	35.3205	1.36864
$667$	43.7391	1.69358
$668$	−3.34607	−0.129463
$669$	−3.00000	−0.115987
$670$	0	0
$671$	0	0
$672$	−3.52193	−0.135861
$673$	−1.61729	−0.0623418	−0.0311709	−	0.999514i	$-0.509924\pi$
$673$	−1.61729	−0.0623418	−0.0311709	+	0.999514i	$0.509924\pi$
$674$	15.1244	0.582568
$675$	0	0
$676$	8.66025	0.333087
$677$	−9.04008	−0.347439	−0.173719	−	0.984795i	$-0.555579\pi$
$677$	−9.04008	−0.347439	−0.173719	+	0.984795i	$0.555579\pi$
$678$	−2.82843	−0.108625
$679$	−0.588457	−0.0225829
$680$	0	0
$681$	−11.1436	−0.427023
$682$	0	0
$683$	22.4887	0.860507	0.430253	−	0.902708i	$-0.358424\pi$
$683$	22.4887	0.860507	0.430253	+	0.902708i	$0.358424\pi$
$684$	−29.3205	−1.12110
$685$	0	0
$686$	−22.8564	−0.872662
$687$	−9.28032	−0.354066
$688$	−47.7415	−1.82013
$689$	50.7846	1.93474
$690$	0	0
$691$	−27.9090	−1.06171	−0.530854	−	0.847464i	$-0.678129\pi$
$691$	−27.9090	−1.06171	−0.530854	+	0.847464i	$0.678129\pi$
$692$	−13.2084	−0.502108
$693$	0	0
$694$	−5.73205	−0.217586
$695$	0	0
$696$	−1.85641	−0.0703669
$697$	1.79315	0.0679204
$698$	−52.7792	−1.99772
$699$	−6.73205	−0.254630
$700$	0	0
$701$	10.1436	0.383118	0.191559	−	0.981481i	$-0.438646\pi$
$701$	10.1436	0.383118	0.191559	+	0.981481i	$0.438646\pi$
$702$	24.3190	0.917863
$703$	41.4655	1.56390
$704$	0	0
$705$	0	0
$706$	−3.26795	−0.122991
$707$	1.24831	0.0469474
$708$	−4.24264	−0.159448
$709$	23.3923	0.878516	0.439258	−	0.898361i	$-0.355241\pi$
$709$	23.3923	0.878516	0.439258	+	0.898361i	$0.355241\pi$
$710$	0	0
$711$	14.9282	0.559851
$712$	−3.34607	−0.125399
$713$	33.2576	1.24551
$714$	−0.928203	−0.0347371
$715$	0	0
$716$	−14.7846	−0.552527
$717$	−6.03579	−0.225411
$718$	−35.4297	−1.32222
$719$	−39.8038	−1.48443	−0.742217	−	0.670160i	$-0.766225\pi$
$719$	−39.8038	−1.48443	−0.742217	+	0.670160i	$0.766225\pi$
$720$	0	0
$721$	−3.80385	−0.141663
$722$	−37.4631	−1.39423
$723$	−5.24075	−0.194906
$724$	5.87564	0.218367
$725$	0	0
$726$	0	0
$727$	4.00240	0.148441	0.0742205	−	0.997242i	$-0.476353\pi$
$727$	4.00240	0.148441	0.0742205	+	0.997242i	$0.476353\pi$
$728$	−1.96902	−0.0729766
$729$	−13.5885	−0.503276
$730$	0	0
$731$	−11.0718	−0.409505
$732$	−7.41284	−0.273986
$733$	34.7733	1.28438	0.642191	−	0.766545i	$-0.278026\pi$
$733$	34.7733	1.28438	0.642191	+	0.766545i	$0.278026\pi$
$734$	−57.2487	−2.11309
$735$	0	0
$736$	−47.9090	−1.76595
$737$	0	0
$738$	−9.14162	−0.336508
$739$	−25.0718	−0.922281	−0.461140	−	0.887327i	$-0.652560\pi$
$739$	−25.0718	−0.922281	−0.461140	+	0.887327i	$0.652560\pi$
$740$	0	0
$741$	13.6077	0.499891
$742$	20.7327	0.761122
$743$	−8.04197	−0.295031	−0.147516	−	0.989060i	$-0.547128\pi$
$743$	−8.04197	−0.295031	−0.147516	+	0.989060i	$0.547128\pi$
$744$	−1.41154	−0.0517497
$745$	0	0
$746$	65.9090	2.41310
$747$	27.0459	0.989559
$748$	0	0
$749$	3.92820	0.143533
$750$	0	0
$751$	−20.6410	−0.753201	−0.376601	−	0.926376i	$-0.622907\pi$
$751$	−20.6410	−0.753201	−0.376601	+	0.926376i	$0.622907\pi$
$752$	−18.3204	−0.668076
$753$	−7.34847	−0.267793
$754$	56.7846	2.06797
$755$	0	0
$756$	4.60770	0.167580
$757$	−35.0779	−1.27493	−0.637465	−	0.770480i	$-0.720017\pi$
$757$	−35.0779	−1.27493	−0.637465	+	0.770480i	$0.720017\pi$
$758$	10.5558	0.383405
$759$	0	0
$760$	0	0
$761$	7.85641	0.284795	0.142397	−	0.989810i	$-0.454519\pi$
$761$	7.85641	0.284795	0.142397	+	0.989810i	$0.454519\pi$
$762$	3.34607	0.121215
$763$	1.07244	0.0388250
$764$	−18.5885	−0.672507
$765$	0	0
$766$	45.9090	1.65876
$767$	−20.0764	−0.724916
$768$	10.0382	0.362222
$769$	17.8564	0.643918	0.321959	−	0.946754i	$-0.395659\pi$
$769$	17.8564	0.643918	0.321959	+	0.946754i	$0.395659\pi$
$770$	0	0
$771$	2.58846	0.0932210
$772$	17.8028	0.640736
$773$	51.6424	1.85745	0.928723	−	0.370774i	$-0.120907\pi$
$773$	51.6424	1.85745	0.928723	+	0.370774i	$0.120907\pi$
$774$	56.4449	2.02887
$775$	0	0
$776$	0.339746	0.0121962
$777$	−3.10583	−0.111421
$778$	35.0136	1.25530
$779$	−10.7321	−0.384516
$780$	0	0
$781$	0	0
$782$	−12.6264	−0.451519
$783$	20.5569	0.734642
$784$	27.6603	0.987866
$785$	0	0
$786$	−21.1244	−0.753481
$787$	4.65874	0.166066	0.0830331	−	0.996547i	$-0.473539\pi$
$787$	4.65874	0.166066	0.0830331	+	0.996547i	$0.473539\pi$
$788$	4.41851	0.157403
$789$	−11.1244	−0.396038
$790$	0	0
$791$	−2.53590	−0.0901662
$792$	0	0
$793$	−35.0779	−1.24565
$794$	30.9282	1.09760
$795$	0	0
$796$	−3.46410	−0.122782
$797$	−7.90327	−0.279948	−0.139974	−	0.990155i	$-0.544702\pi$
$797$	−7.90327	−0.279948	−0.139974	+	0.990155i	$0.544702\pi$
$798$	5.55532	0.196656
$799$	−4.24871	−0.150309
$800$	0	0
$801$	17.6603	0.623994
$802$	62.4384	2.20478
$803$	0	0
$804$	−13.3923	−0.472310
$805$	0	0
$806$	43.1769	1.52084
$807$	−10.2141	−0.359552
$808$	−0.720710	−0.0253545
$809$	−21.4641	−0.754638	−0.377319	−	0.926083i	$-0.623154\pi$
$809$	−21.4641	−0.754638	−0.377319	+	0.926083i	$0.623154\pi$
$810$	0	0
$811$	−32.9808	−1.15811	−0.579056	−	0.815288i	$-0.696579\pi$
$811$	−32.9808	−1.15811	−0.579056	+	0.815288i	$0.696579\pi$
$812$	10.7589	0.377564
$813$	2.34795	0.0823463
$814$	0	0
$815$	0	0
$816$	2.39230	0.0837474
$817$	66.2650	2.31832
$818$	3.62347	0.126692
$819$	10.3923	0.363137
$820$	0	0
$821$	18.8038	0.656259	0.328129	−	0.944633i	$-0.393582\pi$
$821$	18.8038	0.656259	0.328129	+	0.944633i	$0.393582\pi$
$822$	13.7632	0.480047
$823$	−48.3978	−1.68704	−0.843521	−	0.537096i	$-0.819521\pi$
$823$	−48.3978	−1.68704	−0.843521	+	0.537096i	$0.819521\pi$
$824$	2.19615	0.0765066
$825$	0	0
$826$	−8.19615	−0.285181
$827$	−26.7314	−0.929540	−0.464770	−	0.885431i	$-0.653863\pi$
$827$	−26.7314	−0.929540	−0.464770	+	0.885431i	$0.653863\pi$
$828$	29.8744	1.03821
$829$	41.9808	1.45805	0.729026	−	0.684486i	$-0.239973\pi$
$829$	41.9808	1.45805	0.729026	+	0.684486i	$0.239973\pi$
$830$	0	0
$831$	11.7513	0.407648
$832$	−24.3190	−0.843111
$833$	6.41473	0.222257
$834$	−16.5885	−0.574411
$835$	0	0
$836$	0	0
$837$	15.6307	0.540275
$838$	57.2992	1.97937
$839$	2.78461	0.0961354	0.0480677	−	0.998844i	$-0.484694\pi$
$839$	2.78461	0.0961354	0.0480677	+	0.998844i	$0.484694\pi$
$840$	0	0
$841$	19.0000	0.655172
$842$	−40.6704	−1.40160
$843$	8.96575	0.308797
$844$	19.2679	0.663230
$845$	0	0
$846$	21.6603	0.744695
$847$	0	0
$848$	−53.4355	−1.83498
$849$	−15.5885	−0.534994
$850$	0	0
$851$	−42.2487	−1.44827
$852$	1.96902	0.0674574
$853$	12.4233	0.425366	0.212683	−	0.977121i	$-0.431780\pi$
$853$	12.4233	0.425366	0.212683	+	0.977121i	$0.431780\pi$
$854$	−14.3205	−0.490038
$855$	0	0
$856$	−2.26795	−0.0775169
$857$	24.4206	0.834191	0.417095	−	0.908863i	$-0.363048\pi$
$857$	24.4206	0.834191	0.417095	+	0.908863i	$0.363048\pi$
$858$	0	0
$859$	−11.8038	−0.402742	−0.201371	−	0.979515i	$-0.564540\pi$
$859$	−11.8038	−0.402742	−0.201371	+	0.979515i	$0.564540\pi$
$860$	0	0
$861$	0.803848	0.0273951
$862$	36.5665	1.24546
$863$	9.76079	0.332261	0.166131	−	0.986104i	$-0.446873\pi$
$863$	9.76079	0.332261	0.166131	+	0.986104i	$0.446873\pi$
$864$	−22.5167	−0.766032
$865$	0	0
$866$	−39.1244	−1.32950
$867$	−8.24504	−0.280016
$868$	8.18067	0.277670
$869$	0	0
$870$	0	0
$871$	−63.3731	−2.14731
$872$	−0.619174	−0.0209679
$873$	−1.79315	−0.0606890
$874$	75.5692	2.55617
$875$	0	0
$876$	4.39230	0.148402
$877$	55.9865	1.89053	0.945265	−	0.326302i	$-0.105803\pi$
$877$	55.9865	1.89053	0.945265	+	0.326302i	$0.105803\pi$
$878$	39.1175	1.32015
$879$	−4.87564	−0.164451
$880$	0	0
$881$	38.9090	1.31088	0.655438	−	0.755249i	$-0.272484\pi$
$881$	38.9090	1.31088	0.655438	+	0.755249i	$0.272484\pi$
$882$	−32.7028	−1.10116
$883$	47.0208	1.58238	0.791188	−	0.611574i	$-0.209463\pi$
$883$	47.0208	1.58238	0.791188	+	0.611574i	$0.209463\pi$
$884$	−7.60770	−0.255874
$885$	0	0
$886$	17.7321	0.595720
$887$	8.34658	0.280251	0.140125	−	0.990134i	$-0.455249\pi$
$887$	8.34658	0.280251	0.140125	+	0.990134i	$0.455249\pi$
$888$	1.79315	0.0601742
$889$	3.00000	0.100617
$890$	0	0
$891$	0	0
$892$	−10.0382	−0.336104
$893$	25.4286	0.850937
$894$	13.7321	0.459268
$895$	0	0
$896$	3.67949	0.122923
$897$	−13.8647	−0.462930
$898$	20.3166	0.677975
$899$	36.4974	1.21726
$900$	0	0
$901$	−12.3923	−0.412848
$902$	0	0
$903$	−4.96335	−0.165170
$904$	1.46410	0.0486953
$905$	0	0
$906$	6.19615	0.205853
$907$	−8.06918	−0.267933	−0.133966	−	0.990986i	$-0.542771\pi$
$907$	−8.06918	−0.267933	−0.133966	+	0.990986i	$0.542771\pi$
$908$	−37.2872	−1.23742
$909$	3.80385	0.126166
$910$	0	0
$911$	18.2487	0.604607	0.302303	−	0.953212i	$-0.402244\pi$
$911$	18.2487	0.604607	0.302303	+	0.953212i	$0.402244\pi$
$912$	−14.3180	−0.474116
$913$	0	0
$914$	40.9808	1.35552
$915$	0	0
$916$	−31.0526	−1.02601
$917$	−18.9396	−0.625440
$918$	−5.93426	−0.195860
$919$	32.9808	1.08793	0.543967	−	0.839106i	$-0.316921\pi$
$919$	32.9808	1.08793	0.543967	+	0.839106i	$0.316921\pi$
$920$	0	0
$921$	−9.12436	−0.300658
$922$	63.7511	2.09953
$923$	9.31749	0.306689
$924$	0	0
$925$	0	0
$926$	34.5167	1.13429
$927$	−11.5911	−0.380702
$928$	−52.5761	−1.72589
$929$	−16.1436	−0.529654	−0.264827	−	0.964296i	$-0.585315\pi$
$929$	−16.1436	−0.529654	−0.264827	+	0.964296i	$0.585315\pi$
$930$	0	0
$931$	−38.3923	−1.25826
$932$	−22.5259	−0.737860
$933$	−14.5211	−0.475399
$934$	34.3205	1.12300
$935$	0	0
$936$	−6.00000	−0.196116
$937$	22.7017	0.741634	0.370817	−	0.928706i	$-0.379078\pi$
$937$	22.7017	0.741634	0.370817	+	0.928706i	$0.379078\pi$
$938$	−25.8719	−0.844749
$939$	8.87564	0.289646
$940$	0	0
$941$	−27.0000	−0.880175	−0.440087	−	0.897955i	$-0.645053\pi$
$941$	−27.0000	−0.880175	−0.440087	+	0.897955i	$0.645053\pi$
$942$	12.0716	0.393313
$943$	10.9348	0.356085
$944$	21.1244	0.687539
$945$	0	0
$946$	0	0
$947$	14.3180	0.465273	0.232636	−	0.972564i	$-0.425265\pi$
$947$	14.3180	0.465273	0.232636	+	0.972564i	$0.425265\pi$
$948$	−4.89898	−0.159111
$949$	20.7846	0.674697
$950$	0	0
$951$	3.60770	0.116988
$952$	0.480473	0.0155722
$953$	31.3901	1.01683	0.508413	−	0.861114i	$-0.330233\pi$
$953$	31.3901	1.01683	0.508413	+	0.861114i	$0.330233\pi$
$954$	63.1769	2.04543
$955$	0	0
$956$	−20.1962	−0.653190
$957$	0	0
$958$	−18.2832	−0.590705
$959$	12.3397	0.398471
$960$	0	0
$961$	−3.24871	−0.104797
$962$	−54.8497	−1.76843
$963$	11.9700	0.385729
$964$	−17.5359	−0.564793
$965$	0	0
$966$	−5.66025	−0.182116
$967$	4.24264	0.136434	0.0682171	−	0.997671i	$-0.478269\pi$
$967$	4.24264	0.136434	0.0682171	+	0.997671i	$0.478269\pi$
$968$	0	0
$969$	−3.32051	−0.106670
$970$	0	0
$971$	−11.0718	−0.355311	−0.177655	−	0.984093i	$-0.556851\pi$
$971$	−11.0718	−0.355311	−0.177655	+	0.984093i	$0.556851\pi$
$972$	21.3891	0.686055
$973$	−14.8728	−0.476800
$974$	−24.5885	−0.787865
$975$	0	0
$976$	36.9090	1.18143
$977$	14.9743	0.479072	0.239536	−	0.970888i	$-0.423005\pi$
$977$	14.9743	0.479072	0.239536	+	0.970888i	$0.423005\pi$
$978$	7.58871	0.242660
$979$	0	0
$980$	0	0
$981$	3.26795	0.104338
$982$	30.3548	0.968661
$983$	15.0115	0.478793	0.239396	−	0.970922i	$-0.423050\pi$
$983$	15.0115	0.478793	0.239396	+	0.970922i	$0.423050\pi$
$984$	−0.464102	−0.0147950
$985$	0	0
$986$	−13.8564	−0.441278
$987$	−1.90465	−0.0606255
$988$	45.5322	1.44857
$989$	−67.5167	−2.14690
$990$	0	0
$991$	−45.9090	−1.45835	−0.729173	−	0.684329i	$-0.760095\pi$
$991$	−45.9090	−1.45835	−0.729173	+	0.684329i	$0.760095\pi$
$992$	−39.9769	−1.26927
$993$	−4.03957	−0.128192
$994$	3.80385	0.120651
$995$	0	0
$996$	−8.87564	−0.281236
$997$	0.175865	0.00556971	0.00278486	−	0.999996i	$-0.499114\pi$
$997$	0.175865	0.00556971	0.00278486	+	0.999996i	$0.499114\pi$
$998$	−74.1682	−2.34775
$999$	−19.8564	−0.628229

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3025.2.a.y.1.1		4
5.2	odd	4		605.2.b.d.364.1	✓	4
5.3	odd	4		605.2.b.d.364.4	yes	4
5.4	even	2	inner	3025.2.a.y.1.4		4
11.10	odd	2		3025.2.a.z.1.4		4
55.2	even	20		605.2.j.e.444.4		16
55.3	odd	20		605.2.j.f.9.4		16
55.7	even	20		605.2.j.e.269.1		16
55.8	even	20		605.2.j.e.9.1		16
55.13	even	20		605.2.j.e.444.1		16
55.17	even	20		605.2.j.e.124.1		16
55.18	even	20		605.2.j.e.269.4		16
55.27	odd	20		605.2.j.f.124.4		16
55.28	even	20		605.2.j.e.124.4		16
55.32	even	4		605.2.b.e.364.4	yes	4
55.37	odd	20		605.2.j.f.269.4		16
55.38	odd	20		605.2.j.f.124.1		16
55.42	odd	20		605.2.j.f.444.1		16
55.43	even	4		605.2.b.e.364.1	yes	4
55.47	odd	20		605.2.j.f.9.1		16
55.48	odd	20		605.2.j.f.269.1		16
55.52	even	20		605.2.j.e.9.4		16
55.53	odd	20		605.2.j.f.444.4		16
55.54	odd	2		3025.2.a.z.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.b.d.364.1	✓	4	5.2	odd	4
605.2.b.d.364.4	yes	4	5.3	odd	4
605.2.b.e.364.1	yes	4	55.43	even	4
605.2.b.e.364.4	yes	4	55.32	even	4
605.2.j.e.9.1		16	55.8	even	20
605.2.j.e.9.4		16	55.52	even	20
605.2.j.e.124.1		16	55.17	even	20
605.2.j.e.124.4		16	55.28	even	20
605.2.j.e.269.1		16	55.7	even	20
605.2.j.e.269.4		16	55.18	even	20
605.2.j.e.444.1		16	55.13	even	20
605.2.j.e.444.4		16	55.2	even	20
605.2.j.f.9.1		16	55.47	odd	20
605.2.j.f.9.4		16	55.3	odd	20
605.2.j.f.124.1		16	55.38	odd	20
605.2.j.f.124.4		16	55.27	odd	20
605.2.j.f.269.1		16	55.48	odd	20
605.2.j.f.269.4		16	55.37	odd	20
605.2.j.f.444.1		16	55.42	odd	20
605.2.j.f.444.4		16	55.53	odd	20
3025.2.a.y.1.1		4	1.1	even	1	trivial
3025.2.a.y.1.4		4	5.4	even	2	inner
3025.2.a.z.1.1		4	55.54	odd	2
3025.2.a.z.1.4		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)

\(f(q)\)	\(=\)	\(q-1.93185 q^{2} +0.517638 q^{3} +1.73205 q^{4} -1.00000 q^{6} -0.896575 q^{7} +0.517638 q^{8} -2.73205 q^{9} +O(q^{10})\)	\(q-1.93185 q^{2} +0.517638 q^{3} +1.73205 q^{4} -1.00000 q^{6} -0.896575 q^{7} +0.517638 q^{8} -2.73205 q^{9} +0.896575 q^{12} +4.24264 q^{13} +1.73205 q^{14} -4.46410 q^{16} -1.03528 q^{17} +5.27792 q^{18} +6.19615 q^{19} -0.464102 q^{21} -6.31319 q^{23} +0.267949 q^{24} -8.19615 q^{26} -2.96713 q^{27} -1.55291 q^{28} -6.92820 q^{29} -5.26795 q^{31} +7.58871 q^{32} +2.00000 q^{34} -4.73205 q^{36} +6.69213 q^{37} -11.9700 q^{38} +2.19615 q^{39} -1.73205 q^{41} +0.896575 q^{42} +10.6945 q^{43} +12.1962 q^{46} +4.10394 q^{47} -2.31079 q^{48} -6.19615 q^{49} -0.535898 q^{51} +7.34847 q^{52} +11.9700 q^{53} +5.73205 q^{54} -0.464102 q^{56} +3.20736 q^{57} +13.3843 q^{58} -4.73205 q^{59} -8.26795 q^{61} +10.1769 q^{62} +2.44949 q^{63} -5.73205 q^{64} -14.9372 q^{67} -1.79315 q^{68} -3.26795 q^{69} +2.19615 q^{71} -1.41421 q^{72} +4.89898 q^{73} -12.9282 q^{74} +10.7321 q^{76} -4.24264 q^{78} -5.46410 q^{79} +6.66025 q^{81} +3.34607 q^{82} -9.89949 q^{83} -0.803848 q^{84} -20.6603 q^{86} -3.58630 q^{87} -6.46410 q^{89} -3.80385 q^{91} -10.9348 q^{92} -2.72689 q^{93} -7.92820 q^{94} +3.92820 q^{96} +0.656339 q^{97} +11.9700 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{6} - 4 q^{9}+O(q^{10}) \) 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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