LMFDB - Embedded newform 5625.2.a.s.1.4

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:	$1$
Dimension:	$8$
Coefficient field:	8.8.6152203125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 20x^{5} + 26x^{4} - 35x^{3} - 27x^{2} + 16x + 1 $ x^8 - 3x^7 - 8x^6 + 20x^5 + 26x^4 - 35x^3 - 27x^2 + 16*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 625)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.0573749$ 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)

$f(q)$	$=$	$q-1.05737 q^{2} -0.881958 q^{4} +1.01199 q^{7} +3.04731 q^{8} +O(q^{10})$	$q-1.05737 q^{2} -0.881958 q^{4} +1.01199 q^{7} +3.04731 q^{8} -5.12074 q^{11} +6.08528 q^{13} -1.07006 q^{14} -1.45823 q^{16} -3.19320 q^{17} +3.42871 q^{19} +5.41454 q^{22} -2.91916 q^{23} -6.43442 q^{26} -0.892537 q^{28} +1.55246 q^{29} -7.99699 q^{31} -4.55272 q^{32} +3.37640 q^{34} +8.40726 q^{37} -3.62544 q^{38} +1.86355 q^{41} -5.22402 q^{43} +4.51628 q^{44} +3.08665 q^{46} +4.80081 q^{47} -5.97587 q^{49} -5.36696 q^{52} -10.0499 q^{53} +3.08386 q^{56} -1.64153 q^{58} -2.89450 q^{59} -2.30966 q^{61} +8.45582 q^{62} +7.73040 q^{64} +4.64895 q^{67} +2.81626 q^{68} +7.73711 q^{71} -0.595540 q^{73} -8.88962 q^{74} -3.02398 q^{76} -5.18216 q^{77} +11.0593 q^{79} -1.97047 q^{82} -14.2973 q^{83} +5.52374 q^{86} -15.6045 q^{88} +6.39323 q^{89} +6.15827 q^{91} +2.57458 q^{92} -5.07625 q^{94} -14.0833 q^{97} +6.31873 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{2} + 11 q^{4} + 10 q^{7} - 15 q^{8}+O(q^{10}) $ 8 * q - 5 * q^2 + 11 * q^4 + 10 * q^7 - 15 * q^8	$ 8 q - 5 q^{2} + 11 q^{4} + 10 q^{7} - 15 q^{8} - q^{11} + 10 q^{13} + 8 q^{14} + 13 q^{16} - 15 q^{17} - 10 q^{19} - 5 q^{22} - 30 q^{23} - 11 q^{26} - 5 q^{28} - 10 q^{29} - 9 q^{31} - 30 q^{32} + 7 q^{34} - 10 q^{37} - 20 q^{38} + 4 q^{41} + 18 q^{44} - 9 q^{46} - 30 q^{47} - 4 q^{49} + 5 q^{52} - 10 q^{53} - 30 q^{58} + 5 q^{59} + 6 q^{61} - 10 q^{62} - 9 q^{64} + 10 q^{67} - 40 q^{68} + 9 q^{71} + 18 q^{74} - 10 q^{76} - 5 q^{77} - 20 q^{79} - 45 q^{82} - 40 q^{83} + 24 q^{86} - 40 q^{88} + 5 q^{89} + 6 q^{91} - 15 q^{92} + 47 q^{94} + 30 q^{98}+O(q^{100}) $ 8 * q - 5 * q^2 + 11 * q^4 + 10 * q^7 - 15 * q^8 - q^11 + 10 * q^13 + 8 * q^14 + 13 * q^16 - 15 * q^17 - 10 * q^19 - 5 * q^22 - 30 * q^23 - 11 * q^26 - 5 * q^28 - 10 * q^29 - 9 * q^31 - 30 * q^32 + 7 * q^34 - 10 * q^37 - 20 * q^38 + 4 * q^41 + 18 * q^44 - 9 * q^46 - 30 * q^47 - 4 * q^49 + 5 * q^52 - 10 * q^53 - 30 * q^58 + 5 * q^59 + 6 * q^61 - 10 * q^62 - 9 * q^64 + 10 * q^67 - 40 * q^68 + 9 * q^71 + 18 * q^74 - 10 * q^76 - 5 * q^77 - 20 * q^79 - 45 * q^82 - 40 * q^83 + 24 * q^86 - 40 * q^88 + 5 * q^89 + 6 * q^91 - 15 * q^92 + 47 * q^94 + 30 * 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 $

$2$	−1.05737	−0.747677	−0.373838	−	0.927494i	$-0.621959\pi$
$2$	−1.05737	−0.747677	−0.373838	+	0.927494i	$0.621959\pi$
$3$	0	0
$3$	0	0
$4$	−0.881958	−0.440979
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.01199	0.382498	0.191249	−	0.981542i	$-0.438746\pi$
$7$	1.01199	0.382498	0.191249	+	0.981542i	$0.438746\pi$
$8$	3.04731	1.07739
$9$	0	0
$10$	0	0
$11$	−5.12074	−1.54396	−0.771980	−	0.635647i	$-0.780734\pi$
$11$	−5.12074	−1.54396	−0.771980	+	0.635647i	$0.780734\pi$
$12$	0	0
$13$	6.08528	1.68775	0.843876	−	0.536538i	$-0.180268\pi$
$13$	6.08528	1.68775	0.843876	+	0.536538i	$0.180268\pi$
$14$	−1.07006	−0.285985
$15$	0	0
$16$	−1.45823	−0.364558
$17$	−3.19320	−0.774464	−0.387232	−	0.921982i	$-0.626569\pi$
$17$	−3.19320	−0.774464	−0.387232	+	0.921982i	$0.626569\pi$
$18$	0	0
$19$	3.42871	0.786601	0.393300	−	0.919410i	$-0.371333\pi$
$19$	3.42871	0.786601	0.393300	+	0.919410i	$0.371333\pi$
$20$	0	0
$21$	0	0
$22$	5.41454	1.15438
$23$	−2.91916	−0.608687	−0.304343	−	0.952562i	$-0.598437\pi$
$23$	−2.91916	−0.608687	−0.304343	+	0.952562i	$0.598437\pi$
$24$	0	0
$25$	0	0
$26$	−6.43442	−1.26189
$27$	0	0
$28$	−0.892537	−0.168674
$29$	1.55246	0.288285	0.144142	−	0.989557i	$-0.453958\pi$
$29$	1.55246	0.288285	0.144142	+	0.989557i	$0.453958\pi$
$30$	0	0
$31$	−7.99699	−1.43630	−0.718151	−	0.695887i	$-0.755011\pi$
$31$	−7.99699	−1.43630	−0.718151	+	0.695887i	$0.755011\pi$
$32$	−4.55272	−0.804815
$33$	0	0
$34$	3.37640	0.579049
$35$	0	0
$36$	0	0
$37$	8.40726	1.38214	0.691072	−	0.722786i	$-0.257139\pi$
$37$	8.40726	1.38214	0.691072	+	0.722786i	$0.257139\pi$
$38$	−3.62544	−0.588123
$39$	0	0
$40$	0	0
$41$	1.86355	0.291037	0.145519	−	0.989356i	$-0.453515\pi$
$41$	1.86355	0.291037	0.145519	+	0.989356i	$0.453515\pi$
$42$	0	0
$43$	−5.22402	−0.796655	−0.398328	−	0.917243i	$-0.630409\pi$
$43$	−5.22402	−0.796655	−0.398328	+	0.917243i	$0.630409\pi$
$44$	4.51628	0.680854
$45$	0	0
$46$	3.08665	0.455101
$47$	4.80081	0.700269	0.350135	−	0.936699i	$-0.386136\pi$
$47$	4.80081	0.700269	0.350135	+	0.936699i	$0.386136\pi$
$48$	0	0
$49$	−5.97587	−0.853695
$50$	0	0
$51$	0	0
$52$	−5.36696	−0.744264
$53$	−10.0499	−1.38046	−0.690232	−	0.723588i	$-0.742492\pi$
$53$	−10.0499	−1.38046	−0.690232	+	0.723588i	$0.742492\pi$
$54$	0	0
$55$	0	0
$56$	3.08386	0.412098
$57$	0	0
$58$	−1.64153	−0.215544
$59$	−2.89450	−0.376832	−0.188416	−	0.982089i	$-0.560335\pi$
$59$	−2.89450	−0.376832	−0.188416	+	0.982089i	$0.560335\pi$
$60$	0	0
$61$	−2.30966	−0.295722	−0.147861	−	0.989008i	$-0.547239\pi$
$61$	−2.30966	−0.295722	−0.147861	+	0.989008i	$0.547239\pi$
$62$	8.45582	1.07389
$63$	0	0
$64$	7.73040	0.966300
$65$	0	0
$66$	0	0
$67$	4.64895	0.567960	0.283980	−	0.958830i	$-0.408345\pi$
$67$	4.64895	0.567960	0.283980	+	0.958830i	$0.408345\pi$
$68$	2.81626	0.341522
$69$	0	0
$70$	0	0
$71$	7.73711	0.918226	0.459113	−	0.888378i	$-0.348167\pi$
$71$	7.73711	0.918226	0.459113	+	0.888378i	$0.348167\pi$
$72$	0	0
$73$	−0.595540	−0.0697027	−0.0348513	−	0.999393i	$-0.511096\pi$
$73$	−0.595540	−0.0697027	−0.0348513	+	0.999393i	$0.511096\pi$
$74$	−8.88962	−1.03340
$75$	0	0
$76$	−3.02398	−0.346875
$77$	−5.18216	−0.590562
$78$	0	0
$79$	11.0593	1.24427	0.622134	−	0.782910i	$-0.286266\pi$
$79$	11.0593	1.24427	0.622134	+	0.782910i	$0.286266\pi$
$80$	0	0
$81$	0	0
$82$	−1.97047	−0.217602
$83$	−14.2973	−1.56933	−0.784665	−	0.619920i	$-0.787165\pi$
$83$	−14.2973	−1.56933	−0.784665	+	0.619920i	$0.787165\pi$
$84$	0	0
$85$	0	0
$86$	5.52374	0.595641
$87$	0	0
$88$	−15.6045	−1.66344
$89$	6.39323	0.677681	0.338841	−	0.940844i	$-0.389965\pi$
$89$	6.39323	0.677681	0.338841	+	0.940844i	$0.389965\pi$
$90$	0	0
$91$	6.15827	0.645562
$92$	2.57458	0.268418
$93$	0	0
$94$	−5.07625	−0.523575
$95$	0	0
$96$	0	0
$97$	−14.0833	−1.42994	−0.714970	−	0.699155i	$-0.753560\pi$
$97$	−14.0833	−1.42994	−0.714970	+	0.699155i	$0.753560\pi$
$98$	6.31873	0.638288
$99$	0	0
$100$	0	0
$101$	10.5130	1.04608	0.523040	−	0.852308i	$-0.324798\pi$
$101$	10.5130	1.04608	0.523040	+	0.852308i	$0.324798\pi$
$102$	0	0
$103$	−3.07770	−0.303254	−0.151627	−	0.988438i	$-0.548451\pi$
$103$	−3.07770	−0.303254	−0.151627	+	0.988438i	$0.548451\pi$
$104$	18.5437	1.81836
$105$	0	0
$106$	10.6265	1.03214
$107$	5.24731	0.507277	0.253638	−	0.967299i	$-0.418373\pi$
$107$	5.24731	0.507277	0.253638	+	0.967299i	$0.418373\pi$
$108$	0	0
$109$	−6.51906	−0.624413	−0.312206	−	0.950014i	$-0.601068\pi$
$109$	−6.51906	−0.624413	−0.312206	+	0.950014i	$0.601068\pi$
$110$	0	0
$111$	0	0
$112$	−1.47572	−0.139443
$113$	−0.385555	−0.0362699	−0.0181350	−	0.999836i	$-0.505773\pi$
$113$	−0.385555	−0.0362699	−0.0181350	+	0.999836i	$0.505773\pi$
$114$	0	0
$115$	0	0
$116$	−1.36921	−0.127128
$117$	0	0
$118$	3.06058	0.281749
$119$	−3.23150	−0.296231
$120$	0	0
$121$	15.2220	1.38381
$122$	2.44218	0.221104
$123$	0	0
$124$	7.05301	0.633379
$125$	0	0
$126$	0	0
$127$	−4.64324	−0.412021	−0.206010	−	0.978550i	$-0.566048\pi$
$127$	−4.64324	−0.412021	−0.206010	+	0.978550i	$0.566048\pi$
$128$	0.931513	0.0823349
$129$	0	0
$130$	0	0
$131$	−16.2213	−1.41726	−0.708630	−	0.705581i	$-0.750686\pi$
$131$	−16.2213	−1.41726	−0.708630	+	0.705581i	$0.750686\pi$
$132$	0	0
$133$	3.46984	0.300873
$134$	−4.91568	−0.424650
$135$	0	0
$136$	−9.73066	−0.834397
$137$	19.1801	1.63867	0.819335	−	0.573315i	$-0.194343\pi$
$137$	19.1801	1.63867	0.819335	+	0.573315i	$0.194343\pi$
$138$	0	0
$139$	3.58765	0.304300	0.152150	−	0.988357i	$-0.451380\pi$
$139$	3.58765	0.304300	0.152150	+	0.988357i	$0.451380\pi$
$140$	0	0
$141$	0	0
$142$	−8.18102	−0.686536
$143$	−31.1611	−2.60582
$144$	0	0
$145$	0	0
$146$	0.629709	0.0521151
$147$	0	0
$148$	−7.41485	−0.609497
$149$	−1.60192	−0.131234	−0.0656170	−	0.997845i	$-0.520902\pi$
$149$	−1.60192	−0.131234	−0.0656170	+	0.997845i	$0.520902\pi$
$150$	0	0
$151$	6.74218	0.548671	0.274336	−	0.961634i	$-0.411542\pi$
$151$	6.74218	0.548671	0.274336	+	0.961634i	$0.411542\pi$
$152$	10.4484	0.847474
$153$	0	0
$154$	5.47948	0.441549
$155$	0	0
$156$	0	0
$157$	−14.2612	−1.13817	−0.569083	−	0.822280i	$-0.692702\pi$
$157$	−14.2612	−1.13817	−0.569083	+	0.822280i	$0.692702\pi$
$158$	−11.6938	−0.930311
$159$	0	0
$160$	0	0
$161$	−2.95417	−0.232822
$162$	0	0
$163$	−10.1141	−0.792194	−0.396097	−	0.918209i	$-0.629636\pi$
$163$	−10.1141	−0.792194	−0.396097	+	0.918209i	$0.629636\pi$
$164$	−1.64357	−0.128341
$165$	0	0
$166$	15.1176	1.17335
$167$	−16.7161	−1.29353	−0.646765	−	0.762689i	$-0.723879\pi$
$167$	−16.7161	−1.29353	−0.646765	+	0.762689i	$0.723879\pi$
$168$	0	0
$169$	24.0306	1.84851
$170$	0	0
$171$	0	0
$172$	4.60736	0.351308
$173$	−6.44267	−0.489827	−0.244914	−	0.969545i	$-0.578760\pi$
$173$	−6.44267	−0.489827	−0.244914	+	0.969545i	$0.578760\pi$
$174$	0	0
$175$	0	0
$176$	7.46723	0.562864
$177$	0	0
$178$	−6.76004	−0.506687
$179$	15.4342	1.15361	0.576804	−	0.816882i	$-0.304300\pi$
$179$	15.4342	1.15361	0.576804	+	0.816882i	$0.304300\pi$
$180$	0	0
$181$	1.27794	0.0949885	0.0474943	−	0.998872i	$-0.484876\pi$
$181$	1.27794	0.0949885	0.0474943	+	0.998872i	$0.484876\pi$
$182$	−6.51160	−0.482672
$183$	0	0
$184$	−8.89559	−0.655791
$185$	0	0
$186$	0	0
$187$	16.3515	1.19574
$188$	−4.23411	−0.308804
$189$	0	0
$190$	0	0
$191$	−5.83613	−0.422287	−0.211144	−	0.977455i	$-0.567719\pi$
$191$	−5.83613	−0.422287	−0.211144	+	0.977455i	$0.567719\pi$
$192$	0	0
$193$	−4.17773	−0.300719	−0.150360	−	0.988631i	$-0.548043\pi$
$193$	−4.17773	−0.300719	−0.150360	+	0.988631i	$0.548043\pi$
$194$	14.8913	1.06913
$195$	0	0
$196$	5.27047	0.376462
$197$	17.4854	1.24579	0.622893	−	0.782307i	$-0.285957\pi$
$197$	17.4854	1.24579	0.622893	+	0.782307i	$0.285957\pi$
$198$	0	0
$199$	5.89046	0.417564	0.208782	−	0.977962i	$-0.433050\pi$
$199$	5.89046	0.417564	0.208782	+	0.977962i	$0.433050\pi$
$200$	0	0
$201$	0	0
$202$	−11.1162	−0.782130
$203$	1.57108	0.110268
$204$	0	0
$205$	0	0
$206$	3.25428	0.226736
$207$	0	0
$208$	−8.87375	−0.615284
$209$	−17.5575	−1.21448
$210$	0	0
$211$	−11.9773	−0.824549	−0.412275	−	0.911060i	$-0.635266\pi$
$211$	−11.9773	−0.824549	−0.412275	+	0.911060i	$0.635266\pi$
$212$	8.86362	0.608756
$213$	0	0
$214$	−5.54838	−0.379279
$215$	0	0
$216$	0	0
$217$	−8.09291	−0.549383
$218$	6.89309	0.466859
$219$	0	0
$220$	0	0
$221$	−19.4315	−1.30710
$222$	0	0
$223$	−1.27716	−0.0855248	−0.0427624	−	0.999085i	$-0.513616\pi$
$223$	−1.27716	−0.0855248	−0.0427624	+	0.999085i	$0.513616\pi$
$224$	−4.60733	−0.307840
$225$	0	0
$226$	0.407676	0.0271182
$227$	−28.1382	−1.86760	−0.933798	−	0.357802i	$-0.883526\pi$
$227$	−28.1382	−1.86760	−0.933798	+	0.357802i	$0.883526\pi$
$228$	0	0
$229$	11.2821	0.745544	0.372772	−	0.927923i	$-0.378407\pi$
$229$	11.2821	0.745544	0.372772	+	0.927923i	$0.378407\pi$
$230$	0	0
$231$	0	0
$232$	4.73083	0.310594
$233$	13.4719	0.882576	0.441288	−	0.897365i	$-0.354522\pi$
$233$	13.4719	0.882576	0.441288	+	0.897365i	$0.354522\pi$
$234$	0	0
$235$	0	0
$236$	2.55283	0.166175
$237$	0	0
$238$	3.41690	0.221485
$239$	−1.90251	−0.123063	−0.0615315	−	0.998105i	$-0.519598\pi$
$239$	−1.90251	−0.123063	−0.0615315	+	0.998105i	$0.519598\pi$
$240$	0	0
$241$	−19.5685	−1.26052	−0.630258	−	0.776386i	$-0.717051\pi$
$241$	−19.5685	−1.26052	−0.630258	+	0.776386i	$0.717051\pi$
$242$	−16.0953	−1.03465
$243$	0	0
$244$	2.03702	0.130407
$245$	0	0
$246$	0	0
$247$	20.8647	1.32759
$248$	−24.3693	−1.54745
$249$	0	0
$250$	0	0
$251$	−20.7096	−1.30718	−0.653590	−	0.756849i	$-0.726738\pi$
$251$	−20.7096	−1.30718	−0.653590	+	0.756849i	$0.726738\pi$
$252$	0	0
$253$	14.9483	0.939789
$254$	4.90964	0.308058
$255$	0	0
$256$	−16.4458	−1.02786
$257$	−15.0730	−0.940225	−0.470113	−	0.882606i	$-0.655787\pi$
$257$	−15.0730	−0.940225	−0.470113	+	0.882606i	$0.655787\pi$
$258$	0	0
$259$	8.50810	0.528667
$260$	0	0
$261$	0	0
$262$	17.1520	1.05965
$263$	−5.00327	−0.308515	−0.154257	−	0.988031i	$-0.549298\pi$
$263$	−5.00327	−0.308515	−0.154257	+	0.988031i	$0.549298\pi$
$264$	0	0
$265$	0	0
$266$	−3.66892	−0.224956
$267$	0	0
$268$	−4.10018	−0.250458
$269$	27.6804	1.68770	0.843852	−	0.536576i	$-0.180283\pi$
$269$	27.6804	1.68770	0.843852	+	0.536576i	$0.180283\pi$
$270$	0	0
$271$	−26.3928	−1.60325	−0.801624	−	0.597829i	$-0.796030\pi$
$271$	−26.3928	−1.60325	−0.801624	+	0.597829i	$0.796030\pi$
$272$	4.65642	0.282337
$273$	0	0
$274$	−20.2806	−1.22520
$275$	0	0
$276$	0	0
$277$	−7.12277	−0.427966	−0.213983	−	0.976837i	$-0.568644\pi$
$277$	−7.12277	−0.427966	−0.213983	+	0.976837i	$0.568644\pi$
$278$	−3.79349	−0.227518
$279$	0	0
$280$	0	0
$281$	10.3267	0.616037	0.308019	−	0.951380i	$-0.400334\pi$
$281$	10.3267	0.616037	0.308019	+	0.951380i	$0.400334\pi$
$282$	0	0
$283$	4.55987	0.271056	0.135528	−	0.990773i	$-0.456727\pi$
$283$	4.55987	0.271056	0.135528	+	0.990773i	$0.456727\pi$
$284$	−6.82381	−0.404918
$285$	0	0
$286$	32.9490	1.94831
$287$	1.88590	0.111321
$288$	0	0
$289$	−6.80350	−0.400206
$290$	0	0
$291$	0	0
$292$	0.525241	0.0307374
$293$	20.8237	1.21653	0.608267	−	0.793733i	$-0.291865\pi$
$293$	20.8237	1.21653	0.608267	+	0.793733i	$0.291865\pi$
$294$	0	0
$295$	0	0
$296$	25.6195	1.48910
$297$	0	0
$298$	1.69383	0.0981207
$299$	−17.7639	−1.02731
$300$	0	0
$301$	−5.28668	−0.304719
$302$	−7.12902	−0.410229
$303$	0	0
$304$	−4.99986	−0.286762
$305$	0	0
$306$	0	0
$307$	9.44200	0.538884	0.269442	−	0.963017i	$-0.413161\pi$
$307$	9.44200	0.538884	0.269442	+	0.963017i	$0.413161\pi$
$308$	4.57045	0.260425
$309$	0	0
$310$	0	0
$311$	−13.3804	−0.758732	−0.379366	−	0.925247i	$-0.623858\pi$
$311$	−13.3804	−0.758732	−0.379366	+	0.925247i	$0.623858\pi$
$312$	0	0
$313$	23.9947	1.35626	0.678131	−	0.734941i	$-0.262790\pi$
$313$	23.9947	1.35626	0.678131	+	0.734941i	$0.262790\pi$
$314$	15.0794	0.850980
$315$	0	0
$316$	−9.75384	−0.548697
$317$	−14.5616	−0.817862	−0.408931	−	0.912565i	$-0.634098\pi$
$317$	−14.5616	−0.817862	−0.408931	+	0.912565i	$0.634098\pi$
$318$	0	0
$319$	−7.94975	−0.445100
$320$	0	0
$321$	0	0
$322$	3.12367	0.174075
$323$	−10.9486	−0.609194
$324$	0	0
$325$	0	0
$326$	10.6944	0.592305
$327$	0	0
$328$	5.67880	0.313560
$329$	4.85839	0.267852
$330$	0	0
$331$	−24.4248	−1.34251	−0.671255	−	0.741226i	$-0.734245\pi$
$331$	−24.4248	−1.34251	−0.671255	+	0.741226i	$0.734245\pi$
$332$	12.6096	0.692042
$333$	0	0
$334$	17.6752	0.967143
$335$	0	0
$336$	0	0
$337$	−8.57221	−0.466958	−0.233479	−	0.972362i	$-0.575011\pi$
$337$	−8.57221	−0.466958	−0.233479	+	0.972362i	$0.575011\pi$
$338$	−25.4094	−1.38209
$339$	0	0
$340$	0	0
$341$	40.9505	2.21759
$342$	0	0
$343$	−13.1315	−0.709035
$344$	−15.9192	−0.858306
$345$	0	0
$346$	6.81232	0.366233
$347$	−10.8076	−0.580182	−0.290091	−	0.956999i	$-0.593686\pi$
$347$	−10.8076	−0.580182	−0.290091	+	0.956999i	$0.593686\pi$
$348$	0	0
$349$	−8.13956	−0.435701	−0.217850	−	0.975982i	$-0.569904\pi$
$349$	−8.13956	−0.435701	−0.217850	+	0.975982i	$0.569904\pi$
$350$	0	0
$351$	0	0
$352$	23.3133	1.24260
$353$	−21.1661	−1.12656	−0.563279	−	0.826267i	$-0.690460\pi$
$353$	−21.1661	−1.12656	−0.563279	+	0.826267i	$0.690460\pi$
$354$	0	0
$355$	0	0
$356$	−5.63856	−0.298843
$357$	0	0
$358$	−16.3198	−0.862527
$359$	5.32585	0.281087	0.140544	−	0.990074i	$-0.455115\pi$
$359$	5.32585	0.281087	0.140544	+	0.990074i	$0.455115\pi$
$360$	0	0
$361$	−7.24392	−0.381259
$362$	−1.35126	−0.0710207
$363$	0	0
$364$	−5.43134	−0.284679
$365$	0	0
$366$	0	0
$367$	−17.0272	−0.888812	−0.444406	−	0.895826i	$-0.646585\pi$
$367$	−17.0272	−0.888812	−0.444406	+	0.895826i	$0.646585\pi$
$368$	4.25682	0.221902
$369$	0	0
$370$	0	0
$371$	−10.1705	−0.528025
$372$	0	0
$373$	−6.85131	−0.354748	−0.177374	−	0.984144i	$-0.556760\pi$
$373$	−6.85131	−0.354748	−0.177374	+	0.984144i	$0.556760\pi$
$374$	−17.2897	−0.894028
$375$	0	0
$376$	14.6295	0.754461
$377$	9.44716	0.486554
$378$	0	0
$379$	−7.38816	−0.379504	−0.189752	−	0.981832i	$-0.560768\pi$
$379$	−7.38816	−0.379504	−0.189752	+	0.981832i	$0.560768\pi$
$380$	0	0
$381$	0	0
$382$	6.17097	0.315734
$383$	−20.2032	−1.03233	−0.516167	−	0.856488i	$-0.672641\pi$
$383$	−20.2032	−1.03233	−0.516167	+	0.856488i	$0.672641\pi$
$384$	0	0
$385$	0	0
$386$	4.41742	0.224841
$387$	0	0
$388$	12.4209	0.630574
$389$	−8.80029	−0.446192	−0.223096	−	0.974796i	$-0.571616\pi$
$389$	−8.80029	−0.446192	−0.223096	+	0.974796i	$0.571616\pi$
$390$	0	0
$391$	9.32145	0.471406
$392$	−18.2103	−0.919760
$393$	0	0
$394$	−18.4887	−0.931446
$395$	0	0
$396$	0	0
$397$	6.09306	0.305802	0.152901	−	0.988242i	$-0.451138\pi$
$397$	6.09306	0.305802	0.152901	+	0.988242i	$0.451138\pi$
$398$	−6.22842	−0.312203
$399$	0	0
$400$	0	0
$401$	1.71924	0.0858547	0.0429274	−	0.999078i	$-0.486332\pi$
$401$	1.71924	0.0858547	0.0429274	+	0.999078i	$0.486332\pi$
$402$	0	0
$403$	−48.6639	−2.42412
$404$	−9.27201	−0.461300
$405$	0	0
$406$	−1.66122	−0.0824451
$407$	−43.0514	−2.13398
$408$	0	0
$409$	−27.7414	−1.37173	−0.685863	−	0.727731i	$-0.740575\pi$
$409$	−27.7414	−1.37173	−0.685863	+	0.727731i	$0.740575\pi$
$410$	0	0
$411$	0	0
$412$	2.71440	0.133729
$413$	−2.92922	−0.144138
$414$	0	0
$415$	0	0
$416$	−27.7046	−1.35833
$417$	0	0
$418$	18.5649	0.908039
$419$	−16.3540	−0.798944	−0.399472	−	0.916745i	$-0.630807\pi$
$419$	−16.3540	−0.798944	−0.399472	+	0.916745i	$0.630807\pi$
$420$	0	0
$421$	−9.94561	−0.484719	−0.242360	−	0.970186i	$-0.577921\pi$
$421$	−9.94561	−0.484719	−0.242360	+	0.970186i	$0.577921\pi$
$422$	12.6645	0.616496
$423$	0	0
$424$	−30.6253	−1.48729
$425$	0	0
$426$	0	0
$427$	−2.33736	−0.113113
$428$	−4.62791	−0.223699
$429$	0	0
$430$	0	0
$431$	20.5302	0.988903	0.494452	−	0.869205i	$-0.335369\pi$
$431$	20.5302	0.988903	0.494452	+	0.869205i	$0.335369\pi$
$432$	0	0
$433$	−25.9469	−1.24693	−0.623463	−	0.781853i	$-0.714275\pi$
$433$	−25.9469	−1.24693	−0.623463	+	0.781853i	$0.714275\pi$
$434$	8.55724	0.410761
$435$	0	0
$436$	5.74954	0.275353
$437$	−10.0090	−0.478794
$438$	0	0
$439$	25.1284	1.19931	0.599656	−	0.800258i	$-0.295304\pi$
$439$	25.1284	1.19931	0.599656	+	0.800258i	$0.295304\pi$
$440$	0	0
$441$	0	0
$442$	20.5464	0.977291
$443$	−27.0262	−1.28405	−0.642027	−	0.766682i	$-0.721906\pi$
$443$	−27.0262	−1.28405	−0.642027	+	0.766682i	$0.721906\pi$
$444$	0	0
$445$	0	0
$446$	1.35043	0.0639449
$447$	0	0
$448$	7.82312	0.369608
$449$	−37.9871	−1.79272	−0.896361	−	0.443325i	$-0.853799\pi$
$449$	−37.9871	−1.79272	−0.896361	+	0.443325i	$0.853799\pi$
$450$	0	0
$451$	−9.54273	−0.449350
$452$	0.340043	0.0159943
$453$	0	0
$454$	29.7526	1.39636
$455$	0	0
$456$	0	0
$457$	−29.9832	−1.40256	−0.701278	−	0.712888i	$-0.747387\pi$
$457$	−29.9832	−1.40256	−0.701278	+	0.712888i	$0.747387\pi$
$458$	−11.9294	−0.557426
$459$	0	0
$460$	0	0
$461$	2.13364	0.0993734	0.0496867	−	0.998765i	$-0.484178\pi$
$461$	2.13364	0.0993734	0.0496867	+	0.998765i	$0.484178\pi$
$462$	0	0
$463$	−21.9804	−1.02151	−0.510757	−	0.859725i	$-0.670635\pi$
$463$	−21.9804	−1.02151	−0.510757	+	0.859725i	$0.670635\pi$
$464$	−2.26385	−0.105097
$465$	0	0
$466$	−14.2449	−0.659882
$467$	−5.31093	−0.245761	−0.122880	−	0.992421i	$-0.539213\pi$
$467$	−5.31093	−0.245761	−0.122880	+	0.992421i	$0.539213\pi$
$468$	0	0
$469$	4.70471	0.217243
$470$	0	0
$471$	0	0
$472$	−8.82045	−0.405994
$473$	26.7508	1.23000
$474$	0	0
$475$	0	0
$476$	2.85005	0.130632
$477$	0	0
$478$	2.01166	0.0920113
$479$	21.0877	0.963522	0.481761	−	0.876303i	$-0.339997\pi$
$479$	21.0877	0.963522	0.481761	+	0.876303i	$0.339997\pi$
$480$	0	0
$481$	51.1605	2.33272
$482$	20.6912	0.942459
$483$	0	0
$484$	−13.4251	−0.610233
$485$	0	0
$486$	0	0
$487$	−26.6121	−1.20591	−0.602954	−	0.797776i	$-0.706010\pi$
$487$	−26.6121	−1.20591	−0.602954	+	0.797776i	$0.706010\pi$
$488$	−7.03825	−0.318607
$489$	0	0
$490$	0	0
$491$	−1.80069	−0.0812638	−0.0406319	−	0.999174i	$-0.512937\pi$
$491$	−1.80069	−0.0812638	−0.0406319	+	0.999174i	$0.512937\pi$
$492$	0	0
$493$	−4.95731	−0.223266
$494$	−22.0618	−0.992607
$495$	0	0
$496$	11.6615	0.523616
$497$	7.82991	0.351220
$498$	0	0
$499$	28.6962	1.28462	0.642309	−	0.766446i	$-0.277977\pi$
$499$	28.6962	1.28462	0.642309	+	0.766446i	$0.277977\pi$
$500$	0	0
$501$	0	0
$502$	21.8978	0.977349
$503$	23.6827	1.05596	0.527980	−	0.849257i	$-0.322949\pi$
$503$	23.6827	1.05596	0.527980	+	0.849257i	$0.322949\pi$
$504$	0	0
$505$	0	0
$506$	−15.8059	−0.702658
$507$	0	0
$508$	4.09514	0.181693
$509$	−17.6957	−0.784346	−0.392173	−	0.919891i	$-0.628277\pi$
$509$	−17.6957	−0.784346	−0.392173	+	0.919891i	$0.628277\pi$
$510$	0	0
$511$	−0.602683	−0.0266611
$512$	15.5263	0.686172
$513$	0	0
$514$	15.9378	0.702985
$515$	0	0
$516$	0	0
$517$	−24.5837	−1.08119
$518$	−8.99625	−0.395273
$519$	0	0
$520$	0	0
$521$	−22.9756	−1.00658	−0.503289	−	0.864118i	$-0.667877\pi$
$521$	−22.9756	−1.00658	−0.503289	+	0.864118i	$0.667877\pi$
$522$	0	0
$523$	−28.9773	−1.26709	−0.633544	−	0.773707i	$-0.718400\pi$
$523$	−28.9773	−1.26709	−0.633544	+	0.773707i	$0.718400\pi$
$524$	14.3065	0.624982
$525$	0	0
$526$	5.29033	0.230669
$527$	25.5359	1.11236
$528$	0	0
$529$	−14.4785	−0.629500
$530$	0	0
$531$	0	0
$532$	−3.06025	−0.132679
$533$	11.3402	0.491199
$534$	0	0
$535$	0	0
$536$	14.1668	0.611912
$537$	0	0
$538$	−29.2686	−1.26186
$539$	30.6008	1.31807
$540$	0	0
$541$	24.4505	1.05121	0.525605	−	0.850729i	$-0.323839\pi$
$541$	24.4505	1.05121	0.525605	+	0.850729i	$0.323839\pi$
$542$	27.9071	1.19871
$543$	0	0
$544$	14.5377	0.623300
$545$	0	0
$546$	0	0
$547$	−6.50334	−0.278063	−0.139031	−	0.990288i	$-0.544399\pi$
$547$	−6.50334	−0.278063	−0.139031	+	0.990288i	$0.544399\pi$
$548$	−16.9161	−0.722619
$549$	0	0
$550$	0	0
$551$	5.32295	0.226765
$552$	0	0
$553$	11.1920	0.475930
$554$	7.53144	0.319980
$555$	0	0
$556$	−3.16416	−0.134190
$557$	−3.12305	−0.132328	−0.0661640	−	0.997809i	$-0.521076\pi$
$557$	−3.12305	−0.132328	−0.0661640	+	0.997809i	$0.521076\pi$
$558$	0	0
$559$	−31.7896	−1.34456
$560$	0	0
$561$	0	0
$562$	−10.9192	−0.460597
$563$	36.5252	1.53935	0.769677	−	0.638433i	$-0.220417\pi$
$563$	36.5252	1.53935	0.769677	+	0.638433i	$0.220417\pi$
$564$	0	0
$565$	0	0
$566$	−4.82149	−0.202663
$567$	0	0
$568$	23.5774	0.989285
$569$	37.7676	1.58330	0.791650	−	0.610975i	$-0.209222\pi$
$569$	37.7676	1.58330	0.791650	+	0.610975i	$0.209222\pi$
$570$	0	0
$571$	−41.2370	−1.72571	−0.862857	−	0.505448i	$-0.831327\pi$
$571$	−41.2370	−1.72571	−0.862857	+	0.505448i	$0.831327\pi$
$572$	27.4828	1.14911
$573$	0	0
$574$	−1.99410	−0.0832322
$575$	0	0
$576$	0	0
$577$	−4.26929	−0.177733	−0.0888663	−	0.996044i	$-0.528324\pi$
$577$	−4.26929	−0.177733	−0.0888663	+	0.996044i	$0.528324\pi$
$578$	7.19385	0.299225
$579$	0	0
$580$	0	0
$581$	−14.4688	−0.600266
$582$	0	0
$583$	51.4631	2.13138
$584$	−1.81480	−0.0750968
$585$	0	0
$586$	−22.0185	−0.909574
$587$	−11.4331	−0.471893	−0.235947	−	0.971766i	$-0.575819\pi$
$587$	−11.4331	−0.471893	−0.235947	+	0.971766i	$0.575819\pi$
$588$	0	0
$589$	−27.4194	−1.12980
$590$	0	0
$591$	0	0
$592$	−12.2597	−0.503872
$593$	−30.9375	−1.27045	−0.635225	−	0.772327i	$-0.719092\pi$
$593$	−30.9375	−1.27045	−0.635225	+	0.772327i	$0.719092\pi$
$594$	0	0
$595$	0	0
$596$	1.41282	0.0578715
$597$	0	0
$598$	18.7831	0.768098
$599$	46.1912	1.88732	0.943660	−	0.330916i	$-0.107358\pi$
$599$	46.1912	1.88732	0.943660	+	0.330916i	$0.107358\pi$
$600$	0	0
$601$	−38.0963	−1.55398	−0.776990	−	0.629513i	$-0.783254\pi$
$601$	−38.0963	−1.55398	−0.776990	+	0.629513i	$0.783254\pi$
$602$	5.59000	0.227831
$603$	0	0
$604$	−5.94632	−0.241953
$605$	0	0
$606$	0	0
$607$	38.6361	1.56819	0.784095	−	0.620641i	$-0.213127\pi$
$607$	38.6361	1.56819	0.784095	+	0.620641i	$0.213127\pi$
$608$	−15.6100	−0.633068
$609$	0	0
$610$	0	0
$611$	29.2142	1.18188
$612$	0	0
$613$	−10.7561	−0.434434	−0.217217	−	0.976123i	$-0.569698\pi$
$613$	−10.7561	−0.434434	−0.217217	+	0.976123i	$0.569698\pi$
$614$	−9.98374	−0.402911
$615$	0	0
$616$	−15.7916	−0.636264
$617$	10.8072	0.435082	0.217541	−	0.976051i	$-0.430196\pi$
$617$	10.8072	0.435082	0.217541	+	0.976051i	$0.430196\pi$
$618$	0	0
$619$	−36.8938	−1.48289	−0.741443	−	0.671016i	$-0.765858\pi$
$619$	−36.8938	−1.48289	−0.741443	+	0.671016i	$0.765858\pi$
$620$	0	0
$621$	0	0
$622$	14.1481	0.567286
$623$	6.46992	0.259212
$624$	0	0
$625$	0	0
$626$	−25.3714	−1.01405
$627$	0	0
$628$	12.5778	0.501907
$629$	−26.8460	−1.07042
$630$	0	0
$631$	39.2593	1.56289	0.781445	−	0.623974i	$-0.214483\pi$
$631$	39.2593	1.56289	0.781445	+	0.623974i	$0.214483\pi$
$632$	33.7011	1.34056
$633$	0	0
$634$	15.3971	0.611497
$635$	0	0
$636$	0	0
$637$	−36.3648	−1.44083
$638$	8.40586	0.332791
$639$	0	0
$640$	0	0
$641$	−8.34833	−0.329739	−0.164870	−	0.986315i	$-0.552720\pi$
$641$	−8.34833	−0.329739	−0.164870	+	0.986315i	$0.552720\pi$
$642$	0	0
$643$	−13.1408	−0.518223	−0.259112	−	0.965847i	$-0.583430\pi$
$643$	−13.1408	−0.518223	−0.259112	+	0.965847i	$0.583430\pi$
$644$	2.60546	0.102669
$645$	0	0
$646$	11.5767	0.455480
$647$	−26.0801	−1.02531	−0.512656	−	0.858594i	$-0.671339\pi$
$647$	−26.0801	−1.02531	−0.512656	+	0.858594i	$0.671339\pi$
$648$	0	0
$649$	14.8220	0.581814
$650$	0	0
$651$	0	0
$652$	8.92018	0.349341
$653$	−23.0295	−0.901213	−0.450606	−	0.892723i	$-0.648792\pi$
$653$	−23.0295	−0.901213	−0.450606	+	0.892723i	$0.648792\pi$
$654$	0	0
$655$	0	0
$656$	−2.71749	−0.106100
$657$	0	0
$658$	−5.13714	−0.200267
$659$	−34.7580	−1.35398	−0.676990	−	0.735992i	$-0.736716\pi$
$659$	−34.7580	−1.35398	−0.676990	+	0.735992i	$0.736716\pi$
$660$	0	0
$661$	18.0033	0.700249	0.350124	−	0.936703i	$-0.386139\pi$
$661$	18.0033	0.700249	0.350124	+	0.936703i	$0.386139\pi$
$662$	25.8262	1.00376
$663$	0	0
$664$	−43.5682	−1.69078
$665$	0	0
$666$	0	0
$667$	−4.53188	−0.175475
$668$	14.7429	0.570420
$669$	0	0
$670$	0	0
$671$	11.8272	0.456583
$672$	0	0
$673$	14.8622	0.572896	0.286448	−	0.958096i	$-0.407526\pi$
$673$	14.8622	0.572896	0.286448	+	0.958096i	$0.407526\pi$
$674$	9.06404	0.349134
$675$	0	0
$676$	−21.1940	−0.815153
$677$	23.1238	0.888720	0.444360	−	0.895848i	$-0.353431\pi$
$677$	23.1238	0.888720	0.444360	+	0.895848i	$0.353431\pi$
$678$	0	0
$679$	−14.2522	−0.546949
$680$	0	0
$681$	0	0
$682$	−43.3000	−1.65804
$683$	19.6347	0.751302	0.375651	−	0.926761i	$-0.377419\pi$
$683$	19.6347	0.751302	0.375651	+	0.926761i	$0.377419\pi$
$684$	0	0
$685$	0	0
$686$	13.8849	0.530129
$687$	0	0
$688$	7.61783	0.290427
$689$	−61.1566	−2.32988
$690$	0	0
$691$	−5.87782	−0.223603	−0.111801	−	0.993731i	$-0.535662\pi$
$691$	−5.87782	−0.223603	−0.111801	+	0.993731i	$0.535662\pi$
$692$	5.68217	0.216004
$693$	0	0
$694$	11.4277	0.433788
$695$	0	0
$696$	0	0
$697$	−5.95067	−0.225398
$698$	8.60657	0.325763
$699$	0	0
$700$	0	0
$701$	50.0581	1.89067	0.945334	−	0.326103i	$-0.105736\pi$
$701$	50.0581	1.89067	0.945334	+	0.326103i	$0.105736\pi$
$702$	0	0
$703$	28.8261	1.08720
$704$	−39.5853	−1.49193
$705$	0	0
$706$	22.3805	0.842301
$707$	10.6391	0.400124
$708$	0	0
$709$	13.7506	0.516414	0.258207	−	0.966090i	$-0.416868\pi$
$709$	13.7506	0.516414	0.258207	+	0.966090i	$0.416868\pi$
$710$	0	0
$711$	0	0
$712$	19.4822	0.730125
$713$	23.3445	0.874258
$714$	0	0
$715$	0	0
$716$	−13.6123	−0.508717
$717$	0	0
$718$	−5.63142	−0.210163
$719$	10.5000	0.391582	0.195791	−	0.980646i	$-0.437273\pi$
$719$	10.5000	0.391582	0.195791	+	0.980646i	$0.437273\pi$
$720$	0	0
$721$	−3.11461	−0.115994
$722$	7.65954	0.285059
$723$	0	0
$724$	−1.12709	−0.0418880
$725$	0	0
$726$	0	0
$727$	−6.06366	−0.224889	−0.112444	−	0.993658i	$-0.535868\pi$
$727$	−6.06366	−0.224889	−0.112444	+	0.993658i	$0.535868\pi$
$728$	18.7662	0.695520
$729$	0	0
$730$	0	0
$731$	16.6813	0.616980
$732$	0	0
$733$	2.34070	0.0864558	0.0432279	−	0.999065i	$-0.486236\pi$
$733$	2.34070	0.0864558	0.0432279	+	0.999065i	$0.486236\pi$
$734$	18.0041	0.664544
$735$	0	0
$736$	13.2901	0.489880
$737$	−23.8061	−0.876907
$738$	0	0
$739$	−41.5352	−1.52790	−0.763949	−	0.645276i	$-0.776742\pi$
$739$	−41.5352	−1.52790	−0.763949	+	0.645276i	$0.776742\pi$
$740$	0	0
$741$	0	0
$742$	10.7540	0.394792
$743$	0.813821	0.0298562	0.0149281	−	0.999889i	$-0.495248\pi$
$743$	0.813821	0.0298562	0.0149281	+	0.999889i	$0.495248\pi$
$744$	0	0
$745$	0	0
$746$	7.24441	0.265237
$747$	0	0
$748$	−14.4214	−0.527297
$749$	5.31025	0.194032
$750$	0	0
$751$	31.8919	1.16375	0.581877	−	0.813277i	$-0.302319\pi$
$751$	31.8919	1.16375	0.581877	+	0.813277i	$0.302319\pi$
$752$	−7.00069	−0.255289
$753$	0	0
$754$	−9.98919	−0.363785
$755$	0	0
$756$	0	0
$757$	26.7040	0.970572	0.485286	−	0.874355i	$-0.338715\pi$
$757$	26.7040	0.970572	0.485286	+	0.874355i	$0.338715\pi$
$758$	7.81205	0.283746
$759$	0	0
$760$	0	0
$761$	−43.0644	−1.56108	−0.780542	−	0.625103i	$-0.785057\pi$
$761$	−43.0644	−1.56108	−0.780542	+	0.625103i	$0.785057\pi$
$762$	0	0
$763$	−6.59725	−0.238837
$764$	5.14722	0.186220
$765$	0	0
$766$	21.3623	0.771852
$767$	−17.6139	−0.636000
$768$	0	0
$769$	−1.05358	−0.0379930	−0.0189965	−	0.999820i	$-0.506047\pi$
$769$	−1.05358	−0.0379930	−0.0189965	+	0.999820i	$0.506047\pi$
$770$	0	0
$771$	0	0
$772$	3.68458	0.132611
$773$	13.2299	0.475845	0.237923	−	0.971284i	$-0.423534\pi$
$773$	13.2299	0.475845	0.237923	+	0.971284i	$0.423534\pi$
$774$	0	0
$775$	0	0
$776$	−42.9161	−1.54060
$777$	0	0
$778$	9.30521	0.333608
$779$	6.38957	0.228930
$780$	0	0
$781$	−39.6197	−1.41770
$782$	−9.85627	−0.352459
$783$	0	0
$784$	8.71421	0.311222
$785$	0	0
$786$	0	0
$787$	33.6378	1.19906	0.599528	−	0.800354i	$-0.295355\pi$
$787$	33.6378	1.19906	0.599528	+	0.800354i	$0.295355\pi$
$788$	−15.4214	−0.549366
$789$	0	0
$790$	0	0
$791$	−0.390179	−0.0138732
$792$	0	0
$793$	−14.0549	−0.499105
$794$	−6.44264	−0.228641
$795$	0	0
$796$	−5.19514	−0.184137
$797$	−6.89741	−0.244319	−0.122159	−	0.992511i	$-0.538982\pi$
$797$	−6.89741	−0.244319	−0.122159	+	0.992511i	$0.538982\pi$
$798$	0	0
$799$	−15.3299	−0.542333
$800$	0	0
$801$	0	0
$802$	−1.81788	−0.0641916
$803$	3.04960	0.107618
$804$	0	0
$805$	0	0
$806$	51.4560	1.81246
$807$	0	0
$808$	32.0363	1.12703
$809$	35.3330	1.24224	0.621121	−	0.783715i	$-0.286678\pi$
$809$	35.3330	1.24224	0.621121	+	0.783715i	$0.286678\pi$
$810$	0	0
$811$	−3.97201	−0.139476	−0.0697380	−	0.997565i	$-0.522216\pi$
$811$	−3.97201	−0.139476	−0.0697380	+	0.997565i	$0.522216\pi$
$812$	−1.38563	−0.0486261
$813$	0	0
$814$	45.5214	1.59552
$815$	0	0
$816$	0	0
$817$	−17.9117	−0.626650
$818$	29.3331	1.02561
$819$	0	0
$820$	0	0
$821$	−24.3758	−0.850720	−0.425360	−	0.905024i	$-0.639853\pi$
$821$	−24.3758	−0.850720	−0.425360	+	0.905024i	$0.639853\pi$
$822$	0	0
$823$	22.2604	0.775947	0.387974	−	0.921670i	$-0.373175\pi$
$823$	22.2604	0.775947	0.387974	+	0.921670i	$0.373175\pi$
$824$	−9.37870	−0.326722
$825$	0	0
$826$	3.09729	0.107768
$827$	−26.6196	−0.925656	−0.462828	−	0.886448i	$-0.653165\pi$
$827$	−26.6196	−0.925656	−0.462828	+	0.886448i	$0.653165\pi$
$828$	0	0
$829$	−7.83334	−0.272063	−0.136032	−	0.990704i	$-0.543435\pi$
$829$	−7.83334	−0.272063	−0.136032	+	0.990704i	$0.543435\pi$
$830$	0	0
$831$	0	0
$832$	47.0416	1.63088
$833$	19.0821	0.661156
$834$	0	0
$835$	0	0
$836$	15.4850	0.535561
$837$	0	0
$838$	17.2923	0.597352
$839$	−5.87401	−0.202793	−0.101397	−	0.994846i	$-0.532331\pi$
$839$	−5.87401	−0.202793	−0.101397	+	0.994846i	$0.532331\pi$
$840$	0	0
$841$	−26.5899	−0.916892
$842$	10.5162	0.362413
$843$	0	0
$844$	10.5635	0.363609
$845$	0	0
$846$	0	0
$847$	15.4045	0.529306
$848$	14.6551	0.503260
$849$	0	0
$850$	0	0
$851$	−24.5421	−0.841293
$852$	0	0
$853$	−28.1446	−0.963654	−0.481827	−	0.876266i	$-0.660027\pi$
$853$	−28.1446	−0.963654	−0.481827	+	0.876266i	$0.660027\pi$
$854$	2.47147	0.0845719
$855$	0	0
$856$	15.9902	0.546534
$857$	36.2976	1.23990	0.619951	−	0.784640i	$-0.287152\pi$
$857$	36.2976	1.23990	0.619951	+	0.784640i	$0.287152\pi$
$858$	0	0
$859$	3.82400	0.130473	0.0652366	−	0.997870i	$-0.479220\pi$
$859$	3.82400	0.130473	0.0652366	+	0.997870i	$0.479220\pi$
$860$	0	0
$861$	0	0
$862$	−21.7081	−0.739380
$863$	−25.5955	−0.871280	−0.435640	−	0.900121i	$-0.643478\pi$
$863$	−25.5955	−0.871280	−0.435640	+	0.900121i	$0.643478\pi$
$864$	0	0
$865$	0	0
$866$	27.4356	0.932298
$867$	0	0
$868$	7.13761	0.242266
$869$	−56.6318	−1.92110
$870$	0	0
$871$	28.2902	0.958575
$872$	−19.8656	−0.672734
$873$	0	0
$874$	10.5832	0.357983
$875$	0	0
$876$	0	0
$877$	14.8579	0.501715	0.250857	−	0.968024i	$-0.419287\pi$
$877$	14.8579	0.501715	0.250857	+	0.968024i	$0.419287\pi$
$878$	−26.5701	−0.896697
$879$	0	0
$880$	0	0
$881$	−2.11362	−0.0712096	−0.0356048	−	0.999366i	$-0.511336\pi$
$881$	−2.11362	−0.0712096	−0.0356048	+	0.999366i	$0.511336\pi$
$882$	0	0
$883$	−51.0828	−1.71907	−0.859537	−	0.511074i	$-0.829248\pi$
$883$	−51.0828	−1.71907	−0.859537	+	0.511074i	$0.829248\pi$
$884$	17.1378	0.576405
$885$	0	0
$886$	28.5768	0.960058
$887$	−45.5799	−1.53042	−0.765212	−	0.643779i	$-0.777366\pi$
$887$	−45.5799	−1.53042	−0.765212	+	0.643779i	$0.777366\pi$
$888$	0	0
$889$	−4.69893	−0.157597
$890$	0	0
$891$	0	0
$892$	1.12640	0.0377147
$893$	16.4606	0.550833
$894$	0	0
$895$	0	0
$896$	0.942686	0.0314929
$897$	0	0
$898$	40.1666	1.34038
$899$	−12.4150	−0.414064
$900$	0	0
$901$	32.0914	1.06912
$902$	10.0902	0.335968
$903$	0	0
$904$	−1.17490	−0.0390767
$905$	0	0
$906$	0	0
$907$	−27.5215	−0.913837	−0.456919	−	0.889508i	$-0.651047\pi$
$907$	−27.5215	−0.913837	−0.456919	+	0.889508i	$0.651047\pi$
$908$	24.8167	0.823570
$909$	0	0
$910$	0	0
$911$	−50.4110	−1.67019	−0.835096	−	0.550104i	$-0.814588\pi$
$911$	−50.4110	−1.67019	−0.835096	+	0.550104i	$0.814588\pi$
$912$	0	0
$913$	73.2126	2.42298
$914$	31.7035	1.04866
$915$	0	0
$916$	−9.95037	−0.328770
$917$	−16.4158	−0.542099
$918$	0	0
$919$	12.6543	0.417427	0.208714	−	0.977977i	$-0.433072\pi$
$919$	12.6543	0.417427	0.208714	+	0.977977i	$0.433072\pi$
$920$	0	0
$921$	0	0
$922$	−2.25606	−0.0742992
$923$	47.0825	1.54974
$924$	0	0
$925$	0	0
$926$	23.2415	0.763763
$927$	0	0
$928$	−7.06793	−0.232016
$929$	3.03961	0.0997265	0.0498633	−	0.998756i	$-0.484121\pi$
$929$	3.03961	0.0997265	0.0498633	+	0.998756i	$0.484121\pi$
$930$	0	0
$931$	−20.4895	−0.671517
$932$	−11.8817	−0.389198
$933$	0	0
$934$	5.61565	0.183750
$935$	0	0
$936$	0	0
$937$	0.0480749	0.00157054	0.000785268	−	1.00000i	$-0.499750\pi$
$937$	0.0480749	0.00157054	0.000785268		1.00000i	$0.499750\pi$
$938$	−4.97465	−0.162428
$939$	0	0
$940$	0	0
$941$	1.25393	0.0408769	0.0204384	−	0.999791i	$-0.493494\pi$
$941$	1.25393	0.0408769	0.0204384	+	0.999791i	$0.493494\pi$
$942$	0	0
$943$	−5.43999	−0.177150
$944$	4.22086	0.137377
$945$	0	0
$946$	−28.2856	−0.919646
$947$	−44.5985	−1.44926	−0.724628	−	0.689140i	$-0.757989\pi$
$947$	−44.5985	−1.44926	−0.724628	+	0.689140i	$0.757989\pi$
$948$	0	0
$949$	−3.62403	−0.117641
$950$	0	0
$951$	0	0
$952$	−9.84737	−0.319155
$953$	41.8227	1.35477	0.677385	−	0.735629i	$-0.263113\pi$
$953$	41.8227	1.35477	0.677385	+	0.735629i	$0.263113\pi$
$954$	0	0
$955$	0	0
$956$	1.67793	0.0542682
$957$	0	0
$958$	−22.2976	−0.720404
$959$	19.4102	0.626788
$960$	0	0
$961$	32.9518	1.06296
$962$	−54.0958	−1.74412
$963$	0	0
$964$	17.2586	0.555861
$965$	0	0
$966$	0	0
$967$	44.4986	1.43098	0.715490	−	0.698623i	$-0.246204\pi$
$967$	44.4986	1.43098	0.715490	+	0.698623i	$0.246204\pi$
$968$	46.3860	1.49090
$969$	0	0
$970$	0	0
$971$	−3.66499	−0.117615	−0.0588076	−	0.998269i	$-0.518730\pi$
$971$	−3.66499	−0.117615	−0.0588076	+	0.998269i	$0.518730\pi$
$972$	0	0
$973$	3.63068	0.116394
$974$	28.1389	0.901629
$975$	0	0
$976$	3.36802	0.107808
$977$	−15.7554	−0.504060	−0.252030	−	0.967719i	$-0.581098\pi$
$977$	−15.7554	−0.504060	−0.252030	+	0.967719i	$0.581098\pi$
$978$	0	0
$979$	−32.7381	−1.04631
$980$	0	0
$981$	0	0
$982$	1.90400	0.0607591
$983$	−26.9019	−0.858036	−0.429018	−	0.903296i	$-0.641140\pi$
$983$	−26.9019	−0.858036	−0.429018	+	0.903296i	$0.641140\pi$
$984$	0	0
$985$	0	0
$986$	5.24174	0.166931
$987$	0	0
$988$	−18.4018	−0.585438
$989$	15.2497	0.484914
$990$	0	0
$991$	14.0680	0.446885	0.223442	−	0.974717i	$-0.428271\pi$
$991$	14.0680	0.446885	0.223442	+	0.974717i	$0.428271\pi$
$992$	36.4081	1.15596
$993$	0	0
$994$	−8.27915	−0.262599
$995$	0	0
$996$	0	0
$997$	0.618644	0.0195926	0.00979632	−	0.999952i	$-0.496882\pi$
$997$	0.618644	0.0195926	0.00979632	+	0.999952i	$0.496882\pi$
$998$	−30.3426	−0.960479
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.s.1.4		8
3.2	odd	2		625.2.a.g.1.5	yes	8
5.4	even	2		5625.2.a.be.1.5		8
12.11	even	2		10000.2.a.be.1.6		8
15.2	even	4		625.2.b.d.624.11		16
15.8	even	4		625.2.b.d.624.6		16
15.14	odd	2		625.2.a.e.1.4	✓	8
60.59	even	2		10000.2.a.bn.1.3		8
75.2	even	20		625.2.e.j.124.6		32
75.8	even	20		625.2.e.k.374.6		32
75.11	odd	10		625.2.d.m.501.3		16
75.14	odd	10		625.2.d.q.501.2		16
75.17	even	20		625.2.e.k.374.3		32
75.23	even	20		625.2.e.j.124.3		32
75.29	odd	10		625.2.d.p.376.3		16
75.38	even	20		625.2.e.j.499.6		32
75.41	odd	10		625.2.d.m.126.3		16
75.44	odd	10		625.2.d.p.251.3		16
75.47	even	20		625.2.e.k.249.6		32
75.53	even	20		625.2.e.k.249.3		32
75.56	odd	10		625.2.d.n.251.2		16
75.59	odd	10		625.2.d.q.126.2		16
75.62	even	20		625.2.e.j.499.3		32
75.71	odd	10		625.2.d.n.376.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
625.2.a.e.1.4	✓	8	15.14	odd	2
625.2.a.g.1.5	yes	8	3.2	odd	2
625.2.b.d.624.6		16	15.8	even	4
625.2.b.d.624.11		16	15.2	even	4
625.2.d.m.126.3		16	75.41	odd	10
625.2.d.m.501.3		16	75.11	odd	10
625.2.d.n.251.2		16	75.56	odd	10
625.2.d.n.376.2		16	75.71	odd	10
625.2.d.p.251.3		16	75.44	odd	10
625.2.d.p.376.3		16	75.29	odd	10
625.2.d.q.126.2		16	75.59	odd	10
625.2.d.q.501.2		16	75.14	odd	10
625.2.e.j.124.3		32	75.23	even	20
625.2.e.j.124.6		32	75.2	even	20
625.2.e.j.499.3		32	75.62	even	20
625.2.e.j.499.6		32	75.38	even	20
625.2.e.k.249.3		32	75.53	even	20
625.2.e.k.249.6		32	75.47	even	20
625.2.e.k.374.3		32	75.17	even	20
625.2.e.k.374.6		32	75.8	even	20
5625.2.a.s.1.4		8	1.1	even	1	trivial
5625.2.a.be.1.5		8	5.4	even	2
10000.2.a.be.1.6		8	12.11	even	2
10000.2.a.bn.1.3		8	60.59	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)

\(f(q)\)	\(=\)	\(q-1.05737 q^{2} -0.881958 q^{4} +1.01199 q^{7} +3.04731 q^{8} +O(q^{10})\)	\(q-1.05737 q^{2} -0.881958 q^{4} +1.01199 q^{7} +3.04731 q^{8} -5.12074 q^{11} +6.08528 q^{13} -1.07006 q^{14} -1.45823 q^{16} -3.19320 q^{17} +3.42871 q^{19} +5.41454 q^{22} -2.91916 q^{23} -6.43442 q^{26} -0.892537 q^{28} +1.55246 q^{29} -7.99699 q^{31} -4.55272 q^{32} +3.37640 q^{34} +8.40726 q^{37} -3.62544 q^{38} +1.86355 q^{41} -5.22402 q^{43} +4.51628 q^{44} +3.08665 q^{46} +4.80081 q^{47} -5.97587 q^{49} -5.36696 q^{52} -10.0499 q^{53} +3.08386 q^{56} -1.64153 q^{58} -2.89450 q^{59} -2.30966 q^{61} +8.45582 q^{62} +7.73040 q^{64} +4.64895 q^{67} +2.81626 q^{68} +7.73711 q^{71} -0.595540 q^{73} -8.88962 q^{74} -3.02398 q^{76} -5.18216 q^{77} +11.0593 q^{79} -1.97047 q^{82} -14.2973 q^{83} +5.52374 q^{86} -15.6045 q^{88} +6.39323 q^{89} +6.15827 q^{91} +2.57458 q^{92} -5.07625 q^{94} -14.0833 q^{97} +6.31873 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{2} + 11 q^{4} + 10 q^{7} - 15 q^{8}+O(q^{10}) \) 8 * q - 5 * q^2 + 11 * q^4 + 10 * q^7 - 15 * q^8	\( 8 q - 5 q^{2} + 11 q^{4} + 10 q^{7} - 15 q^{8} - q^{11} + 10 q^{13} + 8 q^{14} + 13 q^{16} - 15 q^{17} - 10 q^{19} - 5 q^{22} - 30 q^{23} - 11 q^{26} - 5 q^{28} - 10 q^{29} - 9 q^{31} - 30 q^{32} + 7 q^{34} - 10 q^{37} - 20 q^{38} + 4 q^{41} + 18 q^{44} - 9 q^{46} - 30 q^{47} - 4 q^{49} + 5 q^{52} - 10 q^{53} - 30 q^{58} + 5 q^{59} + 6 q^{61} - 10 q^{62} - 9 q^{64} + 10 q^{67} - 40 q^{68} + 9 q^{71} + 18 q^{74} - 10 q^{76} - 5 q^{77} - 20 q^{79} - 45 q^{82} - 40 q^{83} + 24 q^{86} - 40 q^{88} + 5 q^{89} + 6 q^{91} - 15 q^{92} + 47 q^{94} + 30 q^{98}+O(q^{100}) \) 8 * q - 5 * q^2 + 11 * q^4 + 10 * q^7 - 15 * q^8 - q^11 + 10 * q^13 + 8 * q^14 + 13 * q^16 - 15 * q^17 - 10 * q^19 - 5 * q^22 - 30 * q^23 - 11 * q^26 - 5 * q^28 - 10 * q^29 - 9 * q^31 - 30 * q^32 + 7 * q^34 - 10 * q^37 - 20 * q^38 + 4 * q^41 + 18 * q^44 - 9 * q^46 - 30 * q^47 - 4 * q^49 + 5 * q^52 - 10 * q^53 - 30 * q^58 + 5 * q^59 + 6 * q^61 - 10 * q^62 - 9 * q^64 + 10 * q^67 - 40 * q^68 + 9 * q^71 + 18 * q^74 - 10 * q^76 - 5 * q^77 - 20 * q^79 - 45 * q^82 - 40 * q^83 + 24 * q^86 - 40 * q^88 + 5 * q^89 + 6 * q^91 - 15 * q^92 + 47 * q^94 + 30 * q^98

Introduction

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