LMFDB - Embedded newform 5625.2.a.u.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5625,2,Mod(1,5625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, 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("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.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:	$44.9158511370$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.13366265625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 12x^{6} + 10x^{5} + 41x^{4} - 20x^{3} - 48x^{2} + 8x + 16 $ x^8 - x^7 - 12x^6 + 10x^5 + 41x^4 - 20x^3 - 48x^2 + 8x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1875)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.23365$ of defining polynomial
Character	$\chi$	$=$	5625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23365 q^{2} +2.98921 q^{4} -1.03143 q^{7} -2.20956 q^{8} +O(q^{10})$	$q-2.23365 q^{2} +2.98921 q^{4} -1.03143 q^{7} -2.20956 q^{8} -6.17643 q^{11} +0.937763 q^{13} +2.30385 q^{14} -1.04303 q^{16} +6.56329 q^{17} +5.67453 q^{19} +13.7960 q^{22} -1.64660 q^{23} -2.09464 q^{26} -3.08316 q^{28} -8.35819 q^{29} +5.53371 q^{31} +6.74889 q^{32} -14.6601 q^{34} -1.29548 q^{37} -12.6749 q^{38} +4.98106 q^{41} +7.75619 q^{43} -18.4627 q^{44} +3.67793 q^{46} +7.67288 q^{47} -5.93616 q^{49} +2.80317 q^{52} -0.500546 q^{53} +2.27900 q^{56} +18.6693 q^{58} +1.19340 q^{59} -12.3637 q^{61} -12.3604 q^{62} -12.9886 q^{64} -7.58851 q^{67} +19.6191 q^{68} -10.6125 q^{71} -7.98638 q^{73} +2.89365 q^{74} +16.9624 q^{76} +6.37054 q^{77} -13.9213 q^{79} -11.1260 q^{82} -1.46223 q^{83} -17.3246 q^{86} +13.6472 q^{88} +8.51161 q^{89} -0.967234 q^{91} -4.92203 q^{92} -17.1386 q^{94} +3.75623 q^{97} +13.2593 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} + 9 q^{4} + 12 q^{7} - 3 q^{8}+O(q^{10}) $ 8 * q - q^2 + 9 * q^4 + 12 * q^7 - 3 * q^8	$ 8 q - q^{2} + 9 q^{4} + 12 q^{7} - 3 q^{8} - 12 q^{11} + 14 q^{13} - 16 q^{14} + 15 q^{16} + q^{17} + 16 q^{19} + 18 q^{22} + 4 q^{23} + 34 q^{26} - 21 q^{28} - 2 q^{29} + 13 q^{31} + 18 q^{32} - 37 q^{34} - 8 q^{37} + 24 q^{38} + 12 q^{41} + 20 q^{43} - 47 q^{44} + 33 q^{46} + 15 q^{47} + 30 q^{49} - q^{52} + 4 q^{53} - 60 q^{56} + 2 q^{58} - 14 q^{59} + 10 q^{61} - 4 q^{62} + 41 q^{64} + 19 q^{67} + 33 q^{68} - 21 q^{71} - 19 q^{73} + 9 q^{74} - q^{76} + 11 q^{77} + 10 q^{79} + 24 q^{82} + 27 q^{83} - 42 q^{86} + 53 q^{88} + 9 q^{89} - 12 q^{91} + 63 q^{92} + 14 q^{94} + 24 q^{97} + 24 q^{98}+O(q^{100}) $ 8 * q - q^2 + 9 * q^4 + 12 * q^7 - 3 * q^8 - 12 * q^11 + 14 * q^13 - 16 * q^14 + 15 * q^16 + q^17 + 16 * q^19 + 18 * q^22 + 4 * q^23 + 34 * q^26 - 21 * q^28 - 2 * q^29 + 13 * q^31 + 18 * q^32 - 37 * q^34 - 8 * q^37 + 24 * q^38 + 12 * q^41 + 20 * q^43 - 47 * q^44 + 33 * q^46 + 15 * q^47 + 30 * q^49 - q^52 + 4 * q^53 - 60 * q^56 + 2 * q^58 - 14 * q^59 + 10 * q^61 - 4 * q^62 + 41 * q^64 + 19 * q^67 + 33 * q^68 - 21 * q^71 - 19 * q^73 + 9 * q^74 - q^76 + 11 * q^77 + 10 * q^79 + 24 * q^82 + 27 * q^83 - 42 * q^86 + 53 * q^88 + 9 * q^89 - 12 * q^91 + 63 * q^92 + 14 * q^94 + 24 * q^97 + 24 * q^98

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 $

$2$	−2.23365	−1.57943	−0.789716	−	0.613472i	$-0.789772\pi$
$2$	−2.23365	−1.57943	−0.789716	+	0.613472i	$0.789772\pi$
$3$	0	0
$3$	0	0
$4$	2.98921	1.49461
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.03143	−0.389843	−0.194921	−	0.980819i	$-0.562445\pi$
$7$	−1.03143	−0.389843	−0.194921	+	0.980819i	$0.562445\pi$
$8$	−2.20956	−0.781198
$9$	0	0
$10$	0	0
$11$	−6.17643	−1.86227	−0.931133	−	0.364681i	$-0.881178\pi$
$11$	−6.17643	−1.86227	−0.931133	+	0.364681i	$0.881178\pi$
$12$	0	0
$13$	0.937763	0.260089	0.130044	−	0.991508i	$-0.458488\pi$
$13$	0.937763	0.260089	0.130044	+	0.991508i	$0.458488\pi$
$14$	2.30385	0.615730
$15$	0	0
$16$	−1.04303	−0.260757
$17$	6.56329	1.59183	0.795916	−	0.605407i	$-0.206990\pi$
$17$	6.56329	1.59183	0.795916	+	0.605407i	$0.206990\pi$
$18$	0	0
$19$	5.67453	1.30183	0.650913	−	0.759152i	$-0.274386\pi$
$19$	5.67453	1.30183	0.650913	+	0.759152i	$0.274386\pi$
$20$	0	0
$21$	0	0
$22$	13.7960	2.94132
$23$	−1.64660	−0.343339	−0.171669	−	0.985155i	$-0.554916\pi$
$23$	−1.64660	−0.343339	−0.171669	+	0.985155i	$0.554916\pi$
$24$	0	0
$25$	0	0
$26$	−2.09464	−0.410792
$27$	0	0
$28$	−3.08316	−0.582662
$29$	−8.35819	−1.55208	−0.776039	−	0.630685i	$-0.782774\pi$
$29$	−8.35819	−1.55208	−0.776039	+	0.630685i	$0.782774\pi$
$30$	0	0
$31$	5.53371	0.993883	0.496942	−	0.867784i	$-0.334456\pi$
$31$	5.53371	0.993883	0.496942	+	0.867784i	$0.334456\pi$
$32$	6.74889	1.19305
$33$	0	0
$34$	−14.6601	−2.51419
$35$	0	0
$36$	0	0
$37$	−1.29548	−0.212976	−0.106488	−	0.994314i	$-0.533960\pi$
$37$	−1.29548	−0.212976	−0.106488	+	0.994314i	$0.533960\pi$
$38$	−12.6749	−2.05615
$39$	0	0
$40$	0	0
$41$	4.98106	0.777910	0.388955	−	0.921257i	$-0.372836\pi$
$41$	4.98106	0.777910	0.388955	+	0.921257i	$0.372836\pi$
$42$	0	0
$43$	7.75619	1.18281	0.591404	−	0.806376i	$-0.298574\pi$
$43$	7.75619	1.18281	0.591404	+	0.806376i	$0.298574\pi$
$44$	−18.4627	−2.78335
$45$	0	0
$46$	3.67793	0.542281
$47$	7.67288	1.11920	0.559602	−	0.828761i	$-0.310954\pi$
$47$	7.67288	1.11920	0.559602	+	0.828761i	$0.310954\pi$
$48$	0	0
$49$	−5.93616	−0.848023
$50$	0	0
$51$	0	0
$52$	2.80317	0.388730
$53$	−0.500546	−0.0687553	−0.0343777	−	0.999409i	$-0.510945\pi$
$53$	−0.500546	−0.0687553	−0.0343777	+	0.999409i	$0.510945\pi$
$54$	0	0
$55$	0	0
$56$	2.27900	0.304544
$57$	0	0
$58$	18.6693	2.45140
$59$	1.19340	0.155367	0.0776837	−	0.996978i	$-0.475248\pi$
$59$	1.19340	0.155367	0.0776837	+	0.996978i	$0.475248\pi$
$60$	0	0
$61$	−12.3637	−1.58301	−0.791504	−	0.611164i	$-0.790701\pi$
$61$	−12.3637	−1.58301	−0.791504	+	0.611164i	$0.790701\pi$
$62$	−12.3604	−1.56977
$63$	0	0
$64$	−12.9886	−1.62358
$65$	0	0
$66$	0	0
$67$	−7.58851	−0.927084	−0.463542	−	0.886075i	$-0.653422\pi$
$67$	−7.58851	−0.927084	−0.463542	+	0.886075i	$0.653422\pi$
$68$	19.6191	2.37916
$69$	0	0
$70$	0	0
$71$	−10.6125	−1.25948	−0.629739	−	0.776807i	$-0.716838\pi$
$71$	−10.6125	−1.25948	−0.629739	+	0.776807i	$0.716838\pi$
$72$	0	0
$73$	−7.98638	−0.934735	−0.467367	−	0.884063i	$-0.654797\pi$
$73$	−7.98638	−0.934735	−0.467367	+	0.884063i	$0.654797\pi$
$74$	2.89365	0.336380
$75$	0	0
$76$	16.9624	1.94572
$77$	6.37054	0.725990
$78$	0	0
$79$	−13.9213	−1.56627	−0.783134	−	0.621854i	$-0.786380\pi$
$79$	−13.9213	−1.56627	−0.783134	+	0.621854i	$0.786380\pi$
$80$	0	0
$81$	0	0
$82$	−11.1260	−1.22866
$83$	−1.46223	−0.160500	−0.0802500	−	0.996775i	$-0.525572\pi$
$83$	−1.46223	−0.160500	−0.0802500	+	0.996775i	$0.525572\pi$
$84$	0	0
$85$	0	0
$86$	−17.3246	−1.86816
$87$	0	0
$88$	13.6472	1.45480
$89$	8.51161	0.902229	0.451115	−	0.892466i	$-0.351027\pi$
$89$	8.51161	0.902229	0.451115	+	0.892466i	$0.351027\pi$
$90$	0	0
$91$	−0.967234	−0.101394
$92$	−4.92203	−0.513157
$93$	0	0
$94$	−17.1386	−1.76771
$95$	0	0
$96$	0	0
$97$	3.75623	0.381387	0.190693	−	0.981650i	$-0.438926\pi$
$97$	3.75623	0.381387	0.190693	+	0.981650i	$0.438926\pi$
$98$	13.2593	1.33939
$99$	0	0
$100$	0	0
$101$	2.60566	0.259273	0.129636	−	0.991562i	$-0.458619\pi$
$101$	2.60566	0.259273	0.129636	+	0.991562i	$0.458619\pi$
$102$	0	0
$103$	17.2182	1.69656	0.848279	−	0.529550i	$-0.177639\pi$
$103$	17.2182	1.69656	0.848279	+	0.529550i	$0.177639\pi$
$104$	−2.07204	−0.203181
$105$	0	0
$106$	1.11805	0.108594
$107$	−5.54296	−0.535858	−0.267929	−	0.963439i	$-0.586339\pi$
$107$	−5.54296	−0.535858	−0.267929	+	0.963439i	$0.586339\pi$
$108$	0	0
$109$	−5.98458	−0.573219	−0.286610	−	0.958047i	$-0.592528\pi$
$109$	−5.98458	−0.573219	−0.286610	+	0.958047i	$0.592528\pi$
$110$	0	0
$111$	0	0
$112$	1.07581	0.101654
$113$	−0.323636	−0.0304451	−0.0152226	−	0.999884i	$-0.504846\pi$
$113$	−0.323636	−0.0304451	−0.0152226	+	0.999884i	$0.504846\pi$
$114$	0	0
$115$	0	0
$116$	−24.9844	−2.31975
$117$	0	0
$118$	−2.66564	−0.245392
$119$	−6.76955	−0.620564
$120$	0	0
$121$	27.1483	2.46803
$122$	27.6162	2.50025
$123$	0	0
$124$	16.5414	1.48546
$125$	0	0
$126$	0	0
$127$	−0.0904114	−0.00802271	−0.00401136	−	0.999992i	$-0.501277\pi$
$127$	−0.0904114	−0.00802271	−0.00401136	+	0.999992i	$0.501277\pi$
$128$	15.5143	1.37129
$129$	0	0
$130$	0	0
$131$	13.5700	1.18562	0.592809	−	0.805343i	$-0.298019\pi$
$131$	13.5700	1.18562	0.592809	+	0.805343i	$0.298019\pi$
$132$	0	0
$133$	−5.85286	−0.507507
$134$	16.9501	1.46427
$135$	0	0
$136$	−14.5020	−1.24354
$137$	13.9795	1.19435	0.597174	−	0.802111i	$-0.296290\pi$
$137$	13.9795	1.19435	0.597174	+	0.802111i	$0.296290\pi$
$138$	0	0
$139$	−2.21762	−0.188096	−0.0940481	−	0.995568i	$-0.529981\pi$
$139$	−2.21762	−0.188096	−0.0940481	+	0.995568i	$0.529981\pi$
$140$	0	0
$141$	0	0
$142$	23.7048	1.98926
$143$	−5.79203	−0.484354
$144$	0	0
$145$	0	0
$146$	17.8388	1.47635
$147$	0	0
$148$	−3.87247	−0.318315
$149$	−3.59402	−0.294433	−0.147217	−	0.989104i	$-0.547031\pi$
$149$	−3.59402	−0.294433	−0.147217	+	0.989104i	$0.547031\pi$
$150$	0	0
$151$	16.8624	1.37224	0.686121	−	0.727487i	$-0.259312\pi$
$151$	16.8624	1.37224	0.686121	+	0.727487i	$0.259312\pi$
$152$	−12.5382	−1.01698
$153$	0	0
$154$	−14.2296	−1.14665
$155$	0	0
$156$	0	0
$157$	−8.42895	−0.672703	−0.336352	−	0.941736i	$-0.609193\pi$
$157$	−8.42895	−0.672703	−0.336352	+	0.941736i	$0.609193\pi$
$158$	31.0953	2.47381
$159$	0	0
$160$	0	0
$161$	1.69834	0.133848
$162$	0	0
$163$	−11.3806	−0.891394	−0.445697	−	0.895184i	$-0.647044\pi$
$163$	−11.3806	−0.891394	−0.445697	+	0.895184i	$0.647044\pi$
$164$	14.8894	1.16267
$165$	0	0
$166$	3.26611	0.253499
$167$	−0.886875	−0.0686285	−0.0343142	−	0.999411i	$-0.510925\pi$
$167$	−0.886875	−0.0686285	−0.0343142	+	0.999411i	$0.510925\pi$
$168$	0	0
$169$	−12.1206	−0.932354
$170$	0	0
$171$	0	0
$172$	23.1849	1.76783
$173$	20.0184	1.52197	0.760986	−	0.648769i	$-0.224716\pi$
$173$	20.0184	1.52197	0.760986	+	0.648769i	$0.224716\pi$
$174$	0	0
$175$	0	0
$176$	6.44220	0.485599
$177$	0	0
$178$	−19.0120	−1.42501
$179$	3.38799	0.253231	0.126615	−	0.991952i	$-0.459589\pi$
$179$	3.38799	0.253231	0.126615	+	0.991952i	$0.459589\pi$
$180$	0	0
$181$	5.41991	0.402859	0.201429	−	0.979503i	$-0.435441\pi$
$181$	5.41991	0.402859	0.201429	+	0.979503i	$0.435441\pi$
$182$	2.16047	0.160144
$183$	0	0
$184$	3.63825	0.268216
$185$	0	0
$186$	0	0
$187$	−40.5377	−2.96441
$188$	22.9359	1.67277
$189$	0	0
$190$	0	0
$191$	11.2760	0.815901	0.407950	−	0.913004i	$-0.366244\pi$
$191$	11.2760	0.815901	0.407950	+	0.913004i	$0.366244\pi$
$192$	0	0
$193$	−6.22383	−0.448001	−0.224000	−	0.974589i	$-0.571912\pi$
$193$	−6.22383	−0.448001	−0.224000	+	0.974589i	$0.571912\pi$
$194$	−8.39011	−0.602375
$195$	0	0
$196$	−17.7444	−1.26746
$197$	19.7296	1.40568	0.702839	−	0.711349i	$-0.251916\pi$
$197$	19.7296	1.40568	0.702839	+	0.711349i	$0.251916\pi$
$198$	0	0
$199$	18.5731	1.31661	0.658307	−	0.752750i	$-0.271273\pi$
$199$	18.5731	1.31661	0.658307	+	0.752750i	$0.271273\pi$
$200$	0	0
$201$	0	0
$202$	−5.82014	−0.409504
$203$	8.62087	0.605066
$204$	0	0
$205$	0	0
$206$	−38.4595	−2.67960
$207$	0	0
$208$	−0.978114	−0.0678200
$209$	−35.0483	−2.42434
$210$	0	0
$211$	2.71998	0.187251	0.0936255	−	0.995607i	$-0.470154\pi$
$211$	2.71998	0.187251	0.0936255	+	0.995607i	$0.470154\pi$
$212$	−1.49624	−0.102762
$213$	0	0
$214$	12.3811	0.846352
$215$	0	0
$216$	0	0
$217$	−5.70762	−0.387458
$218$	13.3675	0.905361
$219$	0	0
$220$	0	0
$221$	6.15481	0.414017
$222$	0	0
$223$	7.04569	0.471814	0.235907	−	0.971776i	$-0.424194\pi$
$223$	7.04569	0.471814	0.235907	+	0.971776i	$0.424194\pi$
$224$	−6.96099	−0.465101
$225$	0	0
$226$	0.722892	0.0480860
$227$	−16.9671	−1.12615	−0.563073	−	0.826407i	$-0.690381\pi$
$227$	−16.9671	−1.12615	−0.563073	+	0.826407i	$0.690381\pi$
$228$	0	0
$229$	5.14606	0.340061	0.170030	−	0.985439i	$-0.445613\pi$
$229$	5.14606	0.340061	0.170030	+	0.985439i	$0.445613\pi$
$230$	0	0
$231$	0	0
$232$	18.4679	1.21248
$233$	−27.3476	−1.79160	−0.895799	−	0.444460i	$-0.853396\pi$
$233$	−27.3476	−1.79160	−0.895799	+	0.444460i	$0.853396\pi$
$234$	0	0
$235$	0	0
$236$	3.56733	0.232213
$237$	0	0
$238$	15.1208	0.980139
$239$	−19.6994	−1.27425	−0.637123	−	0.770762i	$-0.719876\pi$
$239$	−19.6994	−1.27425	−0.637123	+	0.770762i	$0.719876\pi$
$240$	0	0
$241$	−9.57114	−0.616532	−0.308266	−	0.951300i	$-0.599749\pi$
$241$	−9.57114	−0.616532	−0.308266	+	0.951300i	$0.599749\pi$
$242$	−60.6400	−3.89809
$243$	0	0
$244$	−36.9577	−2.36597
$245$	0	0
$246$	0	0
$247$	5.32136	0.338590
$248$	−12.2271	−0.776420
$249$	0	0
$250$	0	0
$251$	5.82514	0.367680	0.183840	−	0.982956i	$-0.441147\pi$
$251$	5.82514	0.367680	0.183840	+	0.982956i	$0.441147\pi$
$252$	0	0
$253$	10.1701	0.639388
$254$	0.201948	0.0126713
$255$	0	0
$256$	−8.67641	−0.542276
$257$	3.47558	0.216801	0.108400	−	0.994107i	$-0.465427\pi$
$257$	3.47558	0.216801	0.108400	+	0.994107i	$0.465427\pi$
$258$	0	0
$259$	1.33619	0.0830270
$260$	0	0
$261$	0	0
$262$	−30.3107	−1.87260
$263$	22.7227	1.40114	0.700570	−	0.713584i	$-0.252929\pi$
$263$	22.7227	1.40114	0.700570	+	0.713584i	$0.252929\pi$
$264$	0	0
$265$	0	0
$266$	13.0733	0.801573
$267$	0	0
$268$	−22.6837	−1.38563
$269$	−1.63546	−0.0997160	−0.0498580	−	0.998756i	$-0.515877\pi$
$269$	−1.63546	−0.0997160	−0.0498580	+	0.998756i	$0.515877\pi$
$270$	0	0
$271$	12.4498	0.756275	0.378137	−	0.925750i	$-0.376565\pi$
$271$	12.4498	0.756275	0.378137	+	0.925750i	$0.376565\pi$
$272$	−6.84571	−0.415082
$273$	0	0
$274$	−31.2254	−1.88639
$275$	0	0
$276$	0	0
$277$	2.16353	0.129994	0.0649971	−	0.997885i	$-0.479296\pi$
$277$	2.16353	0.129994	0.0649971	+	0.997885i	$0.479296\pi$
$278$	4.95340	0.297085
$279$	0	0
$280$	0	0
$281$	−4.97817	−0.296973	−0.148486	−	0.988914i	$-0.547440\pi$
$281$	−4.97817	−0.296973	−0.148486	+	0.988914i	$0.547440\pi$
$282$	0	0
$283$	−17.3357	−1.03050	−0.515249	−	0.857041i	$-0.672301\pi$
$283$	−17.3357	−1.03050	−0.515249	+	0.857041i	$0.672301\pi$
$284$	−31.7232	−1.88242
$285$	0	0
$286$	12.9374	0.765004
$287$	−5.13759	−0.303263
$288$	0	0
$289$	26.0768	1.53393
$290$	0	0
$291$	0	0
$292$	−23.8730	−1.39706
$293$	−20.1863	−1.17930	−0.589648	−	0.807661i	$-0.700733\pi$
$293$	−20.1863	−1.17930	−0.589648	+	0.807661i	$0.700733\pi$
$294$	0	0
$295$	0	0
$296$	2.86244	0.166376
$297$	0	0
$298$	8.02779	0.465038
$299$	−1.54412	−0.0892986
$300$	0	0
$301$	−7.99994	−0.461109
$302$	−37.6648	−2.16736
$303$	0	0
$304$	−5.91870	−0.339461
$305$	0	0
$306$	0	0
$307$	24.7882	1.41474	0.707369	−	0.706844i	$-0.249882\pi$
$307$	24.7882	1.41474	0.707369	+	0.706844i	$0.249882\pi$
$308$	19.0429	1.08507
$309$	0	0
$310$	0	0
$311$	28.1046	1.59367	0.796834	−	0.604198i	$-0.206507\pi$
$311$	28.1046	1.59367	0.796834	+	0.604198i	$0.206507\pi$
$312$	0	0
$313$	−27.1858	−1.53663	−0.768317	−	0.640070i	$-0.778905\pi$
$313$	−27.1858	−1.53663	−0.768317	+	0.640070i	$0.778905\pi$
$314$	18.8274	1.06249
$315$	0	0
$316$	−41.6137	−2.34095
$317$	14.9749	0.841073	0.420536	−	0.907276i	$-0.361842\pi$
$317$	14.9749	0.841073	0.420536	+	0.907276i	$0.361842\pi$
$318$	0	0
$319$	51.6238	2.89038
$320$	0	0
$321$	0	0
$322$	−3.79351	−0.211404
$323$	37.2436	2.07229
$324$	0	0
$325$	0	0
$326$	25.4202	1.40790
$327$	0	0
$328$	−11.0059	−0.607702
$329$	−7.91401	−0.436314
$330$	0	0
$331$	−13.9393	−0.766173	−0.383086	−	0.923713i	$-0.625139\pi$
$331$	−13.9393	−0.766173	−0.383086	+	0.923713i	$0.625139\pi$
$332$	−4.37090	−0.239885
$333$	0	0
$334$	1.98097	0.108394
$335$	0	0
$336$	0	0
$337$	35.3670	1.92656	0.963282	−	0.268492i	$-0.0865252\pi$
$337$	35.3670	1.92656	0.963282	+	0.268492i	$0.0865252\pi$
$338$	27.0732	1.47259
$339$	0	0
$340$	0	0
$341$	−34.1786	−1.85087
$342$	0	0
$343$	13.3427	0.720438
$344$	−17.1378	−0.924007
$345$	0	0
$346$	−44.7142	−2.40385
$347$	−9.34346	−0.501583	−0.250792	−	0.968041i	$-0.580691\pi$
$347$	−9.34346	−0.501583	−0.250792	+	0.968041i	$0.580691\pi$
$348$	0	0
$349$	6.92379	0.370622	0.185311	−	0.982680i	$-0.440671\pi$
$349$	6.92379	0.370622	0.185311	+	0.982680i	$0.440671\pi$
$350$	0	0
$351$	0	0
$352$	−41.6841	−2.22177
$353$	−5.13902	−0.273522	−0.136761	−	0.990604i	$-0.543669\pi$
$353$	−5.13902	−0.273522	−0.136761	+	0.990604i	$0.543669\pi$
$354$	0	0
$355$	0	0
$356$	25.4430	1.34848
$357$	0	0
$358$	−7.56761	−0.399961
$359$	19.0429	1.00504	0.502522	−	0.864564i	$-0.332406\pi$
$359$	19.0429	1.00504	0.502522	+	0.864564i	$0.332406\pi$
$360$	0	0
$361$	13.2003	0.694750
$362$	−12.1062	−0.636288
$363$	0	0
$364$	−2.89127	−0.151544
$365$	0	0
$366$	0	0
$367$	14.1916	0.740796	0.370398	−	0.928873i	$-0.379221\pi$
$367$	14.1916	0.740796	0.370398	+	0.928873i	$0.379221\pi$
$368$	1.71745	0.0895282
$369$	0	0
$370$	0	0
$371$	0.516277	0.0268038
$372$	0	0
$373$	35.4663	1.83637	0.918187	−	0.396148i	$-0.129653\pi$
$373$	35.4663	1.83637	0.918187	+	0.396148i	$0.129653\pi$
$374$	90.5473	4.68209
$375$	0	0
$376$	−16.9537	−0.874320
$377$	−7.83800	−0.403678
$378$	0	0
$379$	−1.99692	−0.102575	−0.0512874	−	0.998684i	$-0.516332\pi$
$379$	−1.99692	−0.102575	−0.0512874	+	0.998684i	$0.516332\pi$
$380$	0	0
$381$	0	0
$382$	−25.1866	−1.28866
$383$	9.43941	0.482331	0.241166	−	0.970484i	$-0.422470\pi$
$383$	9.43941	0.482331	0.241166	+	0.970484i	$0.422470\pi$
$384$	0	0
$385$	0	0
$386$	13.9019	0.707587
$387$	0	0
$388$	11.2282	0.570024
$389$	1.31535	0.0666908	0.0333454	−	0.999444i	$-0.489384\pi$
$389$	1.31535	0.0666908	0.0333454	+	0.999444i	$0.489384\pi$
$390$	0	0
$391$	−10.8071	−0.546538
$392$	13.1163	0.662474
$393$	0	0
$394$	−44.0692	−2.22017
$395$	0	0
$396$	0	0
$397$	−25.3086	−1.27020	−0.635100	−	0.772430i	$-0.719041\pi$
$397$	−25.3086	−1.27020	−0.635100	+	0.772430i	$0.719041\pi$
$398$	−41.4860	−2.07950
$399$	0	0
$400$	0	0
$401$	−13.4276	−0.670543	−0.335271	−	0.942122i	$-0.608828\pi$
$401$	−13.4276	−0.670543	−0.335271	+	0.942122i	$0.608828\pi$
$402$	0	0
$403$	5.18931	0.258498
$404$	7.78887	0.387511
$405$	0	0
$406$	−19.2560	−0.955661
$407$	8.00144	0.396617
$408$	0	0
$409$	−18.8088	−0.930035	−0.465018	−	0.885301i	$-0.653952\pi$
$409$	−18.8088	−0.930035	−0.465018	+	0.885301i	$0.653952\pi$
$410$	0	0
$411$	0	0
$412$	51.4688	2.53569
$413$	−1.23090	−0.0605688
$414$	0	0
$415$	0	0
$416$	6.32886	0.310298
$417$	0	0
$418$	78.2859	3.82909
$419$	21.7812	1.06408	0.532042	−	0.846718i	$-0.321425\pi$
$419$	21.7812	1.06408	0.532042	+	0.846718i	$0.321425\pi$
$420$	0	0
$421$	28.3679	1.38257	0.691283	−	0.722584i	$-0.257046\pi$
$421$	28.3679	1.38257	0.691283	+	0.722584i	$0.257046\pi$
$422$	−6.07549	−0.295750
$423$	0	0
$424$	1.10599	0.0537115
$425$	0	0
$426$	0	0
$427$	12.7522	0.617124
$428$	−16.5691	−0.800898
$429$	0	0
$430$	0	0
$431$	23.7069	1.14192	0.570960	−	0.820978i	$-0.306571\pi$
$431$	23.7069	1.14192	0.570960	+	0.820978i	$0.306571\pi$
$432$	0	0
$433$	7.11846	0.342091	0.171046	−	0.985263i	$-0.445285\pi$
$433$	7.11846	0.342091	0.171046	+	0.985263i	$0.445285\pi$
$434$	12.7488	0.611964
$435$	0	0
$436$	−17.8892	−0.856737
$437$	−9.34365	−0.446967
$438$	0	0
$439$	6.68142	0.318887	0.159443	−	0.987207i	$-0.449030\pi$
$439$	6.68142	0.318887	0.159443	+	0.987207i	$0.449030\pi$
$440$	0	0
$441$	0	0
$442$	−13.7477	−0.653912
$443$	−20.7065	−0.983797	−0.491898	−	0.870653i	$-0.663697\pi$
$443$	−20.7065	−0.983797	−0.491898	+	0.870653i	$0.663697\pi$
$444$	0	0
$445$	0	0
$446$	−15.7376	−0.745199
$447$	0	0
$448$	13.3968	0.632940
$449$	0.218310	0.0103027	0.00515134	−	0.999987i	$-0.498360\pi$
$449$	0.218310	0.0103027	0.00515134	+	0.999987i	$0.498360\pi$
$450$	0	0
$451$	−30.7652	−1.44867
$452$	−0.967418	−0.0455035
$453$	0	0
$454$	37.8987	1.77867
$455$	0	0
$456$	0	0
$457$	5.96061	0.278826	0.139413	−	0.990234i	$-0.455479\pi$
$457$	5.96061	0.278826	0.139413	+	0.990234i	$0.455479\pi$
$458$	−11.4945	−0.537103
$459$	0	0
$460$	0	0
$461$	12.8249	0.597314	0.298657	−	0.954360i	$-0.403461\pi$
$461$	12.8249	0.597314	0.298657	+	0.954360i	$0.403461\pi$
$462$	0	0
$463$	26.9181	1.25099	0.625496	−	0.780227i	$-0.284897\pi$
$463$	26.9181	1.25099	0.625496	+	0.780227i	$0.284897\pi$
$464$	8.71784	0.404716
$465$	0	0
$466$	61.0850	2.82971
$467$	26.8600	1.24293	0.621466	−	0.783441i	$-0.286537\pi$
$467$	26.8600	1.24293	0.621466	+	0.783441i	$0.286537\pi$
$468$	0	0
$469$	7.82699	0.361417
$470$	0	0
$471$	0	0
$472$	−2.63689	−0.121373
$473$	−47.9056	−2.20270
$474$	0	0
$475$	0	0
$476$	−20.2356	−0.927499
$477$	0	0
$478$	44.0016	2.01258
$479$	−39.2952	−1.79544	−0.897722	−	0.440562i	$-0.854779\pi$
$479$	−39.2952	−1.79544	−0.897722	+	0.440562i	$0.854779\pi$
$480$	0	0
$481$	−1.21485	−0.0553925
$482$	21.3786	0.973770
$483$	0	0
$484$	81.1522	3.68874
$485$	0	0
$486$	0	0
$487$	37.3269	1.69144	0.845721	−	0.533626i	$-0.179171\pi$
$487$	37.3269	1.69144	0.845721	+	0.533626i	$0.179171\pi$
$488$	27.3183	1.23664
$489$	0	0
$490$	0	0
$491$	16.8123	0.758729	0.379364	−	0.925247i	$-0.376143\pi$
$491$	16.8123	0.758729	0.379364	+	0.925247i	$0.376143\pi$
$492$	0	0
$493$	−54.8573	−2.47065
$494$	−11.8861	−0.534780
$495$	0	0
$496$	−5.77182	−0.259162
$497$	10.9461	0.490998
$498$	0	0
$499$	−20.6970	−0.926526	−0.463263	−	0.886221i	$-0.653322\pi$
$499$	−20.6970	−0.926526	−0.463263	+	0.886221i	$0.653322\pi$
$500$	0	0
$501$	0	0
$502$	−13.0114	−0.580725
$503$	38.8631	1.73282	0.866410	−	0.499333i	$-0.166421\pi$
$503$	38.8631	1.73282	0.866410	+	0.499333i	$0.166421\pi$
$504$	0	0
$505$	0	0
$506$	−22.7165	−1.00987
$507$	0	0
$508$	−0.270259	−0.0119908
$509$	−20.8260	−0.923096	−0.461548	−	0.887115i	$-0.652706\pi$
$509$	−20.8260	−0.923096	−0.461548	+	0.887115i	$0.652706\pi$
$510$	0	0
$511$	8.23736	0.364400
$512$	−11.6486	−0.514799
$513$	0	0
$514$	−7.76325	−0.342422
$515$	0	0
$516$	0	0
$517$	−47.3910	−2.08425
$518$	−2.98459	−0.131135
$519$	0	0
$520$	0	0
$521$	−10.0302	−0.439430	−0.219715	−	0.975564i	$-0.570513\pi$
$521$	−10.0302	−0.439430	−0.219715	+	0.975564i	$0.570513\pi$
$522$	0	0
$523$	21.8640	0.956046	0.478023	−	0.878347i	$-0.341354\pi$
$523$	21.8640	0.956046	0.478023	+	0.878347i	$0.341354\pi$
$524$	40.5637	1.77203
$525$	0	0
$526$	−50.7546	−2.21300
$527$	36.3193	1.58210
$528$	0	0
$529$	−20.2887	−0.882118
$530$	0	0
$531$	0	0
$532$	−17.4954	−0.758524
$533$	4.67105	0.202326
$534$	0	0
$535$	0	0
$536$	16.7673	0.724236
$537$	0	0
$538$	3.65306	0.157495
$539$	36.6643	1.57924
$540$	0	0
$541$	39.2383	1.68699	0.843493	−	0.537140i	$-0.180495\pi$
$541$	39.2383	1.68699	0.843493	+	0.537140i	$0.180495\pi$
$542$	−27.8087	−1.19448
$543$	0	0
$544$	44.2949	1.89913
$545$	0	0
$546$	0	0
$547$	2.82535	0.120803	0.0604016	−	0.998174i	$-0.480762\pi$
$547$	2.82535	0.120803	0.0604016	+	0.998174i	$0.480762\pi$
$548$	41.7877	1.78508
$549$	0	0
$550$	0	0
$551$	−47.4288	−2.02053
$552$	0	0
$553$	14.3588	0.610598
$554$	−4.83259	−0.205317
$555$	0	0
$556$	−6.62895	−0.281130
$557$	10.4495	0.442759	0.221379	−	0.975188i	$-0.428944\pi$
$557$	10.4495	0.442759	0.221379	+	0.975188i	$0.428944\pi$
$558$	0	0
$559$	7.27347	0.307635
$560$	0	0
$561$	0	0
$562$	11.1195	0.469048
$563$	−5.74113	−0.241960	−0.120980	−	0.992655i	$-0.538604\pi$
$563$	−5.74113	−0.241960	−0.120980	+	0.992655i	$0.538604\pi$
$564$	0	0
$565$	0	0
$566$	38.7219	1.62760
$567$	0	0
$568$	23.4491	0.983901
$569$	−5.50079	−0.230605	−0.115302	−	0.993330i	$-0.536784\pi$
$569$	−5.50079	−0.230605	−0.115302	+	0.993330i	$0.536784\pi$
$570$	0	0
$571$	−9.36436	−0.391886	−0.195943	−	0.980615i	$-0.562777\pi$
$571$	−9.36436	−0.391886	−0.195943	+	0.980615i	$0.562777\pi$
$572$	−17.3136	−0.723919
$573$	0	0
$574$	11.4756	0.478983
$575$	0	0
$576$	0	0
$577$	23.9595	0.997448	0.498724	−	0.866761i	$-0.333802\pi$
$577$	23.9595	0.997448	0.498724	+	0.866761i	$0.333802\pi$
$578$	−58.2465	−2.42274
$579$	0	0
$580$	0	0
$581$	1.50818	0.0625698
$582$	0	0
$583$	3.09159	0.128041
$584$	17.6464	0.730213
$585$	0	0
$586$	45.0892	1.86262
$587$	22.4831	0.927978	0.463989	−	0.885841i	$-0.346418\pi$
$587$	22.4831	0.927978	0.463989	+	0.885841i	$0.346418\pi$
$588$	0	0
$589$	31.4012	1.29386
$590$	0	0
$591$	0	0
$592$	1.35122	0.0555349
$593$	31.0854	1.27652	0.638262	−	0.769819i	$-0.279654\pi$
$593$	31.0854	1.27652	0.638262	+	0.769819i	$0.279654\pi$
$594$	0	0
$595$	0	0
$596$	−10.7433	−0.440062
$597$	0	0
$598$	3.44902	0.141041
$599$	9.30580	0.380225	0.190112	−	0.981762i	$-0.439115\pi$
$599$	9.30580	0.380225	0.190112	+	0.981762i	$0.439115\pi$
$600$	0	0
$601$	39.0074	1.59114	0.795572	−	0.605859i	$-0.207170\pi$
$601$	39.0074	1.59114	0.795572	+	0.605859i	$0.207170\pi$
$602$	17.8691	0.728290
$603$	0	0
$604$	50.4053	2.05096
$605$	0	0
$606$	0	0
$607$	18.6229	0.755880	0.377940	−	0.925830i	$-0.376632\pi$
$607$	18.6229	0.755880	0.377940	+	0.925830i	$0.376632\pi$
$608$	38.2968	1.55314
$609$	0	0
$610$	0	0
$611$	7.19534	0.291092
$612$	0	0
$613$	−36.3935	−1.46992	−0.734960	−	0.678111i	$-0.762799\pi$
$613$	−36.3935	−1.46992	−0.734960	+	0.678111i	$0.762799\pi$
$614$	−55.3683	−2.23448
$615$	0	0
$616$	−14.0761	−0.567142
$617$	−0.139173	−0.00560289	−0.00280145	−	0.999996i	$-0.500892\pi$
$617$	−0.139173	−0.00560289	−0.00280145	+	0.999996i	$0.500892\pi$
$618$	0	0
$619$	1.04686	0.0420769	0.0210384	−	0.999779i	$-0.493303\pi$
$619$	1.04686	0.0420769	0.0210384	+	0.999779i	$0.493303\pi$
$620$	0	0
$621$	0	0
$622$	−62.7761	−2.51709
$623$	−8.77910	−0.351727
$624$	0	0
$625$	0	0
$626$	60.7237	2.42701
$627$	0	0
$628$	−25.1959	−1.00543
$629$	−8.50261	−0.339021
$630$	0	0
$631$	25.6813	1.02236	0.511178	−	0.859475i	$-0.329209\pi$
$631$	25.6813	1.02236	0.511178	+	0.859475i	$0.329209\pi$
$632$	30.7599	1.22356
$633$	0	0
$634$	−33.4487	−1.32842
$635$	0	0
$636$	0	0
$637$	−5.56671	−0.220561
$638$	−115.310	−4.56516
$639$	0	0
$640$	0	0
$641$	12.7784	0.504718	0.252359	−	0.967634i	$-0.418794\pi$
$641$	12.7784	0.504718	0.252359	+	0.967634i	$0.418794\pi$
$642$	0	0
$643$	14.8374	0.585131	0.292565	−	0.956246i	$-0.405491\pi$
$643$	14.8374	0.585131	0.292565	+	0.956246i	$0.405491\pi$
$644$	5.07671	0.200050
$645$	0	0
$646$	−83.1893	−3.27304
$647$	−17.9733	−0.706602	−0.353301	−	0.935510i	$-0.614941\pi$
$647$	−17.9733	−0.706602	−0.353301	+	0.935510i	$0.614941\pi$
$648$	0	0
$649$	−7.37095	−0.289335
$650$	0	0
$651$	0	0
$652$	−34.0189	−1.33228
$653$	−3.36997	−0.131877	−0.0659385	−	0.997824i	$-0.521004\pi$
$653$	−3.36997	−0.131877	−0.0659385	+	0.997824i	$0.521004\pi$
$654$	0	0
$655$	0	0
$656$	−5.19539	−0.202846
$657$	0	0
$658$	17.6772	0.689128
$659$	28.9767	1.12877	0.564386	−	0.825511i	$-0.309113\pi$
$659$	28.9767	1.12877	0.564386	+	0.825511i	$0.309113\pi$
$660$	0	0
$661$	18.3608	0.714153	0.357076	−	0.934075i	$-0.383774\pi$
$661$	18.3608	0.714153	0.357076	+	0.934075i	$0.383774\pi$
$662$	31.1356	1.21012
$663$	0	0
$664$	3.23088	0.125382
$665$	0	0
$666$	0	0
$667$	13.7626	0.532889
$668$	−2.65106	−0.102573
$669$	0	0
$670$	0	0
$671$	76.3635	2.94798
$672$	0	0
$673$	−15.6585	−0.603591	−0.301796	−	0.953373i	$-0.597586\pi$
$673$	−15.6585	−0.603591	−0.301796	+	0.953373i	$0.597586\pi$
$674$	−78.9977	−3.04288
$675$	0	0
$676$	−36.2311	−1.39350
$677$	−32.5070	−1.24934	−0.624672	−	0.780887i	$-0.714767\pi$
$677$	−32.5070	−1.24934	−0.624672	+	0.780887i	$0.714767\pi$
$678$	0	0
$679$	−3.87427	−0.148681
$680$	0	0
$681$	0	0
$682$	76.3432	2.92333
$683$	0.00991954	0.000379561 0	0.000189780	−	1.00000i	$-0.499940\pi$
$683$	0.00991954	0.000379561 0	0.000189780		1.00000i	$0.499940\pi$
$684$	0	0
$685$	0	0
$686$	−29.8030	−1.13788
$687$	0	0
$688$	−8.08993	−0.308426
$689$	−0.469394	−0.0178825
$690$	0	0
$691$	15.1856	0.577688	0.288844	−	0.957376i	$-0.406729\pi$
$691$	15.1856	0.577688	0.288844	+	0.957376i	$0.406729\pi$
$692$	59.8393	2.27475
$693$	0	0
$694$	20.8701	0.792217
$695$	0	0
$696$	0	0
$697$	32.6921	1.23830
$698$	−15.4654	−0.585373
$699$	0	0
$700$	0	0
$701$	−38.0017	−1.43530	−0.717652	−	0.696402i	$-0.754783\pi$
$701$	−38.0017	−1.43530	−0.717652	+	0.696402i	$0.754783\pi$
$702$	0	0
$703$	−7.35123	−0.277257
$704$	80.2234	3.02353
$705$	0	0
$706$	11.4788	0.432010
$707$	−2.68754	−0.101076
$708$	0	0
$709$	23.2204	0.872062	0.436031	−	0.899932i	$-0.356384\pi$
$709$	23.2204	0.872062	0.436031	+	0.899932i	$0.356384\pi$
$710$	0	0
$711$	0	0
$712$	−18.8069	−0.704819
$713$	−9.11178	−0.341239
$714$	0	0
$715$	0	0
$716$	10.1274	0.378480
$717$	0	0
$718$	−42.5352	−1.58740
$719$	−23.3002	−0.868951	−0.434476	−	0.900684i	$-0.643066\pi$
$719$	−23.3002	−0.868951	−0.434476	+	0.900684i	$0.643066\pi$
$720$	0	0
$721$	−17.7593	−0.661390
$722$	−29.4848	−1.09731
$723$	0	0
$724$	16.2013	0.602116
$725$	0	0
$726$	0	0
$727$	23.3640	0.866523	0.433261	−	0.901268i	$-0.357363\pi$
$727$	23.3640	0.866523	0.433261	+	0.901268i	$0.357363\pi$
$728$	2.13716	0.0792085
$729$	0	0
$730$	0	0
$731$	50.9061	1.88283
$732$	0	0
$733$	10.8277	0.399931	0.199966	−	0.979803i	$-0.435917\pi$
$733$	10.8277	0.399931	0.199966	+	0.979803i	$0.435917\pi$
$734$	−31.6992	−1.17004
$735$	0	0
$736$	−11.1127	−0.409619
$737$	46.8699	1.72648
$738$	0	0
$739$	24.5505	0.903105	0.451553	−	0.892244i	$-0.350870\pi$
$739$	24.5505	0.903105	0.451553	+	0.892244i	$0.350870\pi$
$740$	0	0
$741$	0	0
$742$	−1.15318	−0.0423347
$743$	−0.745387	−0.0273456	−0.0136728	−	0.999907i	$-0.504352\pi$
$743$	−0.745387	−0.0273456	−0.0136728	+	0.999907i	$0.504352\pi$
$744$	0	0
$745$	0	0
$746$	−79.2194	−2.90043
$747$	0	0
$748$	−121.176	−4.43063
$749$	5.71716	0.208900
$750$	0	0
$751$	1.84004	0.0671440	0.0335720	−	0.999436i	$-0.489312\pi$
$751$	1.84004	0.0671440	0.0335720	+	0.999436i	$0.489312\pi$
$752$	−8.00304	−0.291841
$753$	0	0
$754$	17.5074	0.637582
$755$	0	0
$756$	0	0
$757$	24.4525	0.888742	0.444371	−	0.895843i	$-0.353427\pi$
$757$	24.4525	0.888742	0.444371	+	0.895843i	$0.353427\pi$
$758$	4.46043	0.162010
$759$	0	0
$760$	0	0
$761$	−15.6021	−0.565574	−0.282787	−	0.959183i	$-0.591259\pi$
$761$	−15.6021	−0.565574	−0.282787	+	0.959183i	$0.591259\pi$
$762$	0	0
$763$	6.17266	0.223465
$764$	33.7063	1.21945
$765$	0	0
$766$	−21.0844	−0.761810
$767$	1.11913	0.0404093
$768$	0	0
$769$	−9.63438	−0.347425	−0.173712	−	0.984796i	$-0.555576\pi$
$769$	−9.63438	−0.347425	−0.173712	+	0.984796i	$0.555576\pi$
$770$	0	0
$771$	0	0
$772$	−18.6043	−0.669585
$773$	5.00666	0.180077	0.0900386	−	0.995938i	$-0.471301\pi$
$773$	5.00666	0.180077	0.0900386	+	0.995938i	$0.471301\pi$
$774$	0	0
$775$	0	0
$776$	−8.29961	−0.297939
$777$	0	0
$778$	−2.93803	−0.105334
$779$	28.2651	1.01270
$780$	0	0
$781$	65.5477	2.34548
$782$	24.1393	0.863220
$783$	0	0
$784$	6.19159	0.221128
$785$	0	0
$786$	0	0
$787$	45.0282	1.60508	0.802541	−	0.596597i	$-0.203481\pi$
$787$	45.0282	1.60508	0.802541	+	0.596597i	$0.203481\pi$
$788$	58.9761	2.10093
$789$	0	0
$790$	0	0
$791$	0.333807	0.0118688
$792$	0	0
$793$	−11.5942	−0.411722
$794$	56.5306	2.00620
$795$	0	0
$796$	55.5190	1.96782
$797$	3.26600	0.115688	0.0578438	−	0.998326i	$-0.481577\pi$
$797$	3.26600	0.115688	0.0578438	+	0.998326i	$0.481577\pi$
$798$	0	0
$799$	50.3593	1.78158
$800$	0	0
$801$	0	0
$802$	29.9926	1.05908
$803$	49.3273	1.74072
$804$	0	0
$805$	0	0
$806$	−11.5911	−0.408280
$807$	0	0
$808$	−5.75736	−0.202543
$809$	24.5785	0.864135	0.432067	−	0.901841i	$-0.357784\pi$
$809$	24.5785	0.864135	0.432067	+	0.901841i	$0.357784\pi$
$810$	0	0
$811$	−17.0396	−0.598340	−0.299170	−	0.954200i	$-0.596710\pi$
$811$	−17.0396	−0.598340	−0.299170	+	0.954200i	$0.596710\pi$
$812$	25.7696	0.904336
$813$	0	0
$814$	−17.8725	−0.626430
$815$	0	0
$816$	0	0
$817$	44.0127	1.53981
$818$	42.0124	1.46893
$819$	0	0
$820$	0	0
$821$	3.30808	0.115453	0.0577264	−	0.998332i	$-0.481615\pi$
$821$	3.30808	0.115453	0.0577264	+	0.998332i	$0.481615\pi$
$822$	0	0
$823$	14.5583	0.507469	0.253735	−	0.967274i	$-0.418341\pi$
$823$	14.5583	0.507469	0.253735	+	0.967274i	$0.418341\pi$
$824$	−38.0446	−1.32535
$825$	0	0
$826$	2.74941	0.0956644
$827$	−8.34632	−0.290230	−0.145115	−	0.989415i	$-0.546355\pi$
$827$	−8.34632	−0.290230	−0.145115	+	0.989415i	$0.546355\pi$
$828$	0	0
$829$	11.1514	0.387303	0.193652	−	0.981070i	$-0.437967\pi$
$829$	11.1514	0.387303	0.193652	+	0.981070i	$0.437967\pi$
$830$	0	0
$831$	0	0
$832$	−12.1803	−0.422274
$833$	−38.9607	−1.34991
$834$	0	0
$835$	0	0
$836$	−104.767	−3.62344
$837$	0	0
$838$	−48.6518	−1.68065
$839$	−10.3600	−0.357665	−0.178833	−	0.983879i	$-0.557232\pi$
$839$	−10.3600	−0.357665	−0.178833	+	0.983879i	$0.557232\pi$
$840$	0	0
$841$	40.8594	1.40894
$842$	−63.3641	−2.18367
$843$	0	0
$844$	8.13060	0.279867
$845$	0	0
$846$	0	0
$847$	−28.0015	−0.962144
$848$	0.522085	0.0179285
$849$	0	0
$850$	0	0
$851$	2.13313	0.0731228
$852$	0	0
$853$	−16.3865	−0.561064	−0.280532	−	0.959845i	$-0.590511\pi$
$853$	−16.3865	−0.561064	−0.280532	+	0.959845i	$0.590511\pi$
$854$	−28.4841	−0.974706
$855$	0	0
$856$	12.2475	0.418611
$857$	−0.794707	−0.0271467	−0.0135733	−	0.999908i	$-0.504321\pi$
$857$	−0.794707	−0.0271467	−0.0135733	+	0.999908i	$0.504321\pi$
$858$	0	0
$859$	−1.06774	−0.0364307	−0.0182154	−	0.999834i	$-0.505798\pi$
$859$	−1.06774	−0.0364307	−0.0182154	+	0.999834i	$0.505798\pi$
$860$	0	0
$861$	0	0
$862$	−52.9530	−1.80359
$863$	−49.3832	−1.68102	−0.840511	−	0.541794i	$-0.817745\pi$
$863$	−49.3832	−1.68102	−0.840511	+	0.541794i	$0.817745\pi$
$864$	0	0
$865$	0	0
$866$	−15.9002	−0.540310
$867$	0	0
$868$	−17.0613	−0.579098
$869$	85.9839	2.91680
$870$	0	0
$871$	−7.11622	−0.241124
$872$	13.2233	0.447798
$873$	0	0
$874$	20.8705	0.705955
$875$	0	0
$876$	0	0
$877$	−46.1096	−1.55701	−0.778506	−	0.627638i	$-0.784022\pi$
$877$	−46.1096	−1.55701	−0.778506	+	0.627638i	$0.784022\pi$
$878$	−14.9240	−0.503660
$879$	0	0
$880$	0	0
$881$	14.0984	0.474988	0.237494	−	0.971389i	$-0.423674\pi$
$881$	14.0984	0.474988	0.237494	+	0.971389i	$0.423674\pi$
$882$	0	0
$883$	−54.9493	−1.84919	−0.924595	−	0.380951i	$-0.875597\pi$
$883$	−54.9493	−1.84919	−0.924595	+	0.380951i	$0.875597\pi$
$884$	18.3980	0.618793
$885$	0	0
$886$	46.2512	1.55384
$887$	−0.338258	−0.0113576	−0.00567879	−	0.999984i	$-0.501808\pi$
$887$	−0.338258	−0.0113576	−0.00567879	+	0.999984i	$0.501808\pi$
$888$	0	0
$889$	0.0932527	0.00312759
$890$	0	0
$891$	0	0
$892$	21.0611	0.705177
$893$	43.5399	1.45701
$894$	0	0
$895$	0	0
$896$	−16.0019	−0.534586
$897$	0	0
$898$	−0.487629	−0.0162724
$899$	−46.2518	−1.54258
$900$	0	0
$901$	−3.28523	−0.109447
$902$	68.7188	2.28808
$903$	0	0
$904$	0.715094	0.0237837
$905$	0	0
$906$	0	0
$907$	25.1534	0.835204	0.417602	−	0.908630i	$-0.362871\pi$
$907$	25.1534	0.835204	0.417602	+	0.908630i	$0.362871\pi$
$908$	−50.7183	−1.68315
$909$	0	0
$910$	0	0
$911$	16.2238	0.537519	0.268759	−	0.963207i	$-0.413386\pi$
$911$	16.2238	0.537519	0.268759	+	0.963207i	$0.413386\pi$
$912$	0	0
$913$	9.03134	0.298894
$914$	−13.3139	−0.440386
$915$	0	0
$916$	15.3827	0.508257
$917$	−13.9965	−0.462205
$918$	0	0
$919$	21.5696	0.711515	0.355758	−	0.934578i	$-0.384223\pi$
$919$	21.5696	0.711515	0.355758	+	0.934578i	$0.384223\pi$
$920$	0	0
$921$	0	0
$922$	−28.6464	−0.943417
$923$	−9.95205	−0.327576
$924$	0	0
$925$	0	0
$926$	−60.1258	−1.97586
$927$	0	0
$928$	−56.4085	−1.85170
$929$	37.6727	1.23600	0.618001	−	0.786177i	$-0.287943\pi$
$929$	37.6727	1.23600	0.618001	+	0.786177i	$0.287943\pi$
$930$	0	0
$931$	−33.6849	−1.10398
$932$	−81.7477	−2.67773
$933$	0	0
$934$	−59.9960	−1.96313
$935$	0	0
$936$	0	0
$937$	−42.7073	−1.39519	−0.697593	−	0.716495i	$-0.745745\pi$
$937$	−42.7073	−1.39519	−0.697593	+	0.716495i	$0.745745\pi$
$938$	−17.4828	−0.570834
$939$	0	0
$940$	0	0
$941$	4.27023	0.139206	0.0696028	−	0.997575i	$-0.477827\pi$
$941$	4.27023	0.139206	0.0696028	+	0.997575i	$0.477827\pi$
$942$	0	0
$943$	−8.20179	−0.267087
$944$	−1.24475	−0.0405132
$945$	0	0
$946$	107.005	3.47902
$947$	26.4988	0.861097	0.430548	−	0.902567i	$-0.358320\pi$
$947$	26.4988	0.861097	0.430548	+	0.902567i	$0.358320\pi$
$948$	0	0
$949$	−7.48933	−0.243114
$950$	0	0
$951$	0	0
$952$	14.9577	0.484783
$953$	16.9907	0.550383	0.275191	−	0.961389i	$-0.411259\pi$
$953$	16.9907	0.550383	0.275191	+	0.961389i	$0.411259\pi$
$954$	0	0
$955$	0	0
$956$	−58.8856	−1.90450
$957$	0	0
$958$	87.7719	2.83578
$959$	−14.4188	−0.465608
$960$	0	0
$961$	−0.378072	−0.0121959
$962$	2.71356	0.0874887
$963$	0	0
$964$	−28.6102	−0.921472
$965$	0	0
$966$	0	0
$967$	2.79897	0.0900089	0.0450045	−	0.998987i	$-0.485670\pi$
$967$	2.79897	0.0900089	0.0450045	+	0.998987i	$0.485670\pi$
$968$	−59.9859	−1.92802
$969$	0	0
$970$	0	0
$971$	5.63046	0.180690	0.0903450	−	0.995911i	$-0.471203\pi$
$971$	5.63046	0.180690	0.0903450	+	0.995911i	$0.471203\pi$
$972$	0	0
$973$	2.28731	0.0733279
$974$	−83.3753	−2.67152
$975$	0	0
$976$	12.8957	0.412781
$977$	40.5445	1.29713	0.648567	−	0.761157i	$-0.275368\pi$
$977$	40.5445	1.29713	0.648567	+	0.761157i	$0.275368\pi$
$978$	0	0
$979$	−52.5714	−1.68019
$980$	0	0
$981$	0	0
$982$	−37.5529	−1.19836
$983$	16.6381	0.530674	0.265337	−	0.964156i	$-0.414517\pi$
$983$	16.6381	0.530674	0.265337	+	0.964156i	$0.414517\pi$
$984$	0	0
$985$	0	0
$986$	122.532	3.90222
$987$	0	0
$988$	15.9067	0.506059
$989$	−12.7713	−0.406104
$990$	0	0
$991$	1.05069	0.0333764	0.0166882	−	0.999861i	$-0.494688\pi$
$991$	1.05069	0.0333764	0.0166882	+	0.999861i	$0.494688\pi$
$992$	37.3464	1.18575
$993$	0	0
$994$	−24.4497	−0.775498
$995$	0	0
$996$	0	0
$997$	−0.318717	−0.0100939	−0.00504693	−	0.999987i	$-0.501606\pi$
$997$	−0.318717	−0.0100939	−0.00504693	+	0.999987i	$0.501606\pi$
$998$	46.2300	1.46339
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.u.1.2		8
3.2	odd	2		1875.2.a.o.1.7	yes	8
5.4	even	2		5625.2.a.bc.1.7		8
15.2	even	4		1875.2.b.g.1249.14		16
15.8	even	4		1875.2.b.g.1249.3		16
15.14	odd	2		1875.2.a.n.1.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.n.1.2	✓	8	15.14	odd	2
1875.2.a.o.1.7	yes	8	3.2	odd	2
1875.2.b.g.1249.3		16	15.8	even	4
1875.2.b.g.1249.14		16	15.2	even	4
5625.2.a.u.1.2		8	1.1	even	1	trivial
5625.2.a.bc.1.7		8	5.4	even	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:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q-2.23365 q^{2} +2.98921 q^{4} -1.03143 q^{7} -2.20956 q^{8} +O(q^{10})\)	\(q-2.23365 q^{2} +2.98921 q^{4} -1.03143 q^{7} -2.20956 q^{8} -6.17643 q^{11} +0.937763 q^{13} +2.30385 q^{14} -1.04303 q^{16} +6.56329 q^{17} +5.67453 q^{19} +13.7960 q^{22} -1.64660 q^{23} -2.09464 q^{26} -3.08316 q^{28} -8.35819 q^{29} +5.53371 q^{31} +6.74889 q^{32} -14.6601 q^{34} -1.29548 q^{37} -12.6749 q^{38} +4.98106 q^{41} +7.75619 q^{43} -18.4627 q^{44} +3.67793 q^{46} +7.67288 q^{47} -5.93616 q^{49} +2.80317 q^{52} -0.500546 q^{53} +2.27900 q^{56} +18.6693 q^{58} +1.19340 q^{59} -12.3637 q^{61} -12.3604 q^{62} -12.9886 q^{64} -7.58851 q^{67} +19.6191 q^{68} -10.6125 q^{71} -7.98638 q^{73} +2.89365 q^{74} +16.9624 q^{76} +6.37054 q^{77} -13.9213 q^{79} -11.1260 q^{82} -1.46223 q^{83} -17.3246 q^{86} +13.6472 q^{88} +8.51161 q^{89} -0.967234 q^{91} -4.92203 q^{92} -17.1386 q^{94} +3.75623 q^{97} +13.2593 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 9 q^{4} + 12 q^{7} - 3 q^{8}+O(q^{10}) \) 8 * q - q^2 + 9 * q^4 + 12 * q^7 - 3 * q^8	\( 8 q - q^{2} + 9 q^{4} + 12 q^{7} - 3 q^{8} - 12 q^{11} + 14 q^{13} - 16 q^{14} + 15 q^{16} + q^{17} + 16 q^{19} + 18 q^{22} + 4 q^{23} + 34 q^{26} - 21 q^{28} - 2 q^{29} + 13 q^{31} + 18 q^{32} - 37 q^{34} - 8 q^{37} + 24 q^{38} + 12 q^{41} + 20 q^{43} - 47 q^{44} + 33 q^{46} + 15 q^{47} + 30 q^{49} - q^{52} + 4 q^{53} - 60 q^{56} + 2 q^{58} - 14 q^{59} + 10 q^{61} - 4 q^{62} + 41 q^{64} + 19 q^{67} + 33 q^{68} - 21 q^{71} - 19 q^{73} + 9 q^{74} - q^{76} + 11 q^{77} + 10 q^{79} + 24 q^{82} + 27 q^{83} - 42 q^{86} + 53 q^{88} + 9 q^{89} - 12 q^{91} + 63 q^{92} + 14 q^{94} + 24 q^{97} + 24 q^{98}+O(q^{100}) \) 8 * q - q^2 + 9 * q^4 + 12 * q^7 - 3 * q^8 - 12 * q^11 + 14 * q^13 - 16 * q^14 + 15 * q^16 + q^17 + 16 * q^19 + 18 * q^22 + 4 * q^23 + 34 * q^26 - 21 * q^28 - 2 * q^29 + 13 * q^31 + 18 * q^32 - 37 * q^34 - 8 * q^37 + 24 * q^38 + 12 * q^41 + 20 * q^43 - 47 * q^44 + 33 * q^46 + 15 * q^47 + 30 * q^49 - q^52 + 4 * q^53 - 60 * q^56 + 2 * q^58 - 14 * q^59 + 10 * q^61 - 4 * q^62 + 41 * q^64 + 19 * q^67 + 33 * q^68 - 21 * q^71 - 19 * q^73 + 9 * q^74 - q^76 + 11 * q^77 + 10 * q^79 + 24 * q^82 + 27 * q^83 - 42 * q^86 + 53 * q^88 + 9 * q^89 - 12 * q^91 + 63 * q^92 + 14 * q^94 + 24 * q^97 + 24 * q^98

Introduction

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