LMFDB - Embedded newform 5625.2.a.r.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:	$6$
Coefficient field:	6.6.46840000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 11x^{4} + 8x^{3} + 31x^{2} - 15x - 9 $ x^6 - x^5 - 11x^4 + 8x^3 + 31x^2 - 15x - 9
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			$-2.02791$ 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)

$f(q)$	$=$	$q-2.02791 q^{2} +2.11242 q^{4} -0.505614 q^{7} -0.227977 q^{8} +O(q^{10})$	$q-2.02791 q^{2} +2.11242 q^{4} -0.505614 q^{7} -0.227977 q^{8} -0.687513 q^{11} +5.78627 q^{13} +1.02534 q^{14} -3.76252 q^{16} -4.74333 q^{17} +4.23709 q^{19} +1.39422 q^{22} +8.36239 q^{23} -11.7340 q^{26} -1.06807 q^{28} -4.88758 q^{29} +2.68853 q^{31} +8.08601 q^{32} +9.61905 q^{34} +11.3806 q^{37} -8.59244 q^{38} -0.144247 q^{41} -5.94442 q^{43} -1.45232 q^{44} -16.9582 q^{46} -6.10650 q^{47} -6.74435 q^{49} +12.2230 q^{52} +10.8398 q^{53} +0.115269 q^{56} +9.91157 q^{58} +6.96817 q^{59} -3.98042 q^{61} -5.45211 q^{62} -8.87266 q^{64} +1.31351 q^{67} -10.0199 q^{68} +3.79321 q^{71} +10.9252 q^{73} -23.0787 q^{74} +8.95051 q^{76} +0.347616 q^{77} -1.89869 q^{79} +0.292519 q^{82} +2.51849 q^{83} +12.0548 q^{86} +0.156737 q^{88} -15.0809 q^{89} -2.92562 q^{91} +17.6649 q^{92} +12.3834 q^{94} -1.17394 q^{97} +13.6769 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{2} + 11 q^{4} + 2 q^{7} + 6 q^{8}+O(q^{10}) $ 6 * q + q^2 + 11 * q^4 + 2 * q^7 + 6 * q^8	$ 6 q + q^{2} + 11 q^{4} + 2 q^{7} + 6 q^{8} + 4 q^{14} + 17 q^{16} + 2 q^{17} - 2 q^{19} - 9 q^{22} - q^{23} - 37 q^{26} + 44 q^{28} - 31 q^{29} - 2 q^{31} + 33 q^{32} + 37 q^{34} + 22 q^{37} + 27 q^{38} - 33 q^{41} + 3 q^{43} + 11 q^{44} - 12 q^{46} - 6 q^{47} + 4 q^{49} + 33 q^{52} + 14 q^{53} + 30 q^{56} + q^{58} + 8 q^{59} + 34 q^{61} + 31 q^{62} + 12 q^{64} - 2 q^{67} + 27 q^{68} + 3 q^{71} + 36 q^{73} - 36 q^{74} + 27 q^{76} + 16 q^{77} + 25 q^{79} - 36 q^{82} - 12 q^{83} + 30 q^{86} - 56 q^{88} - 18 q^{89} + 28 q^{91} + 3 q^{92} - 50 q^{94} - 7 q^{97} + 15 q^{98}+O(q^{100}) $ 6 * q + q^2 + 11 * q^4 + 2 * q^7 + 6 * q^8 + 4 * q^14 + 17 * q^16 + 2 * q^17 - 2 * q^19 - 9 * q^22 - q^23 - 37 * q^26 + 44 * q^28 - 31 * q^29 - 2 * q^31 + 33 * q^32 + 37 * q^34 + 22 * q^37 + 27 * q^38 - 33 * q^41 + 3 * q^43 + 11 * q^44 - 12 * q^46 - 6 * q^47 + 4 * q^49 + 33 * q^52 + 14 * q^53 + 30 * q^56 + q^58 + 8 * q^59 + 34 * q^61 + 31 * q^62 + 12 * q^64 - 2 * q^67 + 27 * q^68 + 3 * q^71 + 36 * q^73 - 36 * q^74 + 27 * q^76 + 16 * q^77 + 25 * q^79 - 36 * q^82 - 12 * q^83 + 30 * q^86 - 56 * q^88 - 18 * q^89 + 28 * q^91 + 3 * q^92 - 50 * q^94 - 7 * q^97 + 15 * 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 $

$2$	−2.02791	−1.43395	−0.716975	−	0.697099i	$-0.754474\pi$
$2$	−2.02791	−1.43395	−0.716975	+	0.697099i	$0.754474\pi$
$3$	0	0
$3$	0	0
$4$	2.11242	1.05621
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.505614	−0.191104	−0.0955521	−	0.995424i	$-0.530462\pi$
$7$	−0.505614	−0.191104	−0.0955521	+	0.995424i	$0.530462\pi$
$8$	−0.227977	−0.0806021
$9$	0	0
$10$	0	0
$11$	−0.687513	−0.207293	−0.103647	−	0.994614i	$-0.533051\pi$
$11$	−0.687513	−0.207293	−0.103647	+	0.994614i	$0.533051\pi$
$12$	0	0
$13$	5.78627	1.60482	0.802412	−	0.596771i	$-0.203550\pi$
$13$	5.78627	1.60482	0.802412	+	0.596771i	$0.203550\pi$
$14$	1.02534	0.274034
$15$	0	0
$16$	−3.76252	−0.940631
$17$	−4.74333	−1.15043	−0.575214	−	0.818003i	$-0.695081\pi$
$17$	−4.74333	−1.15043	−0.575214	+	0.818003i	$0.695081\pi$
$18$	0	0
$19$	4.23709	0.972055	0.486027	−	0.873944i	$-0.338446\pi$
$19$	4.23709	0.972055	0.486027	+	0.873944i	$0.338446\pi$
$20$	0	0
$21$	0	0
$22$	1.39422	0.297248
$23$	8.36239	1.74368	0.871839	−	0.489792i	$-0.162927\pi$
$23$	8.36239	1.74368	0.871839	+	0.489792i	$0.162927\pi$
$24$	0	0
$25$	0	0
$26$	−11.7340	−2.30124
$27$	0	0
$28$	−1.06807	−0.201846
$29$	−4.88758	−0.907601	−0.453800	−	0.891103i	$-0.649932\pi$
$29$	−4.88758	−0.907601	−0.453800	+	0.891103i	$0.649932\pi$
$30$	0	0
$31$	2.68853	0.482875	0.241437	−	0.970416i	$-0.422381\pi$
$31$	2.68853	0.482875	0.241437	+	0.970416i	$0.422381\pi$
$32$	8.08601	1.42942
$33$	0	0
$34$	9.61905	1.64965
$35$	0	0
$36$	0	0
$37$	11.3806	1.87095	0.935476	−	0.353390i	$-0.114971\pi$
$37$	11.3806	1.87095	0.935476	+	0.353390i	$0.114971\pi$
$38$	−8.59244	−1.39388
$39$	0	0
$40$	0	0
$41$	−0.144247	−0.0225275	−0.0112638	−	0.999937i	$-0.503585\pi$
$41$	−0.144247	−0.0225275	−0.0112638	+	0.999937i	$0.503585\pi$
$42$	0	0
$43$	−5.94442	−0.906516	−0.453258	−	0.891379i	$-0.649738\pi$
$43$	−5.94442	−0.906516	−0.453258	+	0.891379i	$0.649738\pi$
$44$	−1.45232	−0.218945
$45$	0	0
$46$	−16.9582	−2.50035
$47$	−6.10650	−0.890725	−0.445362	−	0.895350i	$-0.646925\pi$
$47$	−6.10650	−0.890725	−0.445362	+	0.895350i	$0.646925\pi$
$48$	0	0
$49$	−6.74435	−0.963479
$50$	0	0
$51$	0	0
$52$	12.2230	1.69503
$53$	10.8398	1.48896	0.744481	−	0.667643i	$-0.232697\pi$
$53$	10.8398	1.48896	0.744481	+	0.667643i	$0.232697\pi$
$54$	0	0
$55$	0	0
$56$	0.115269	0.0154034
$57$	0	0
$58$	9.91157	1.30145
$59$	6.96817	0.907179	0.453589	−	0.891211i	$-0.350143\pi$
$59$	6.96817	0.907179	0.453589	+	0.891211i	$0.350143\pi$
$60$	0	0
$61$	−3.98042	−0.509641	−0.254820	−	0.966988i	$-0.582016\pi$
$61$	−3.98042	−0.509641	−0.254820	+	0.966988i	$0.582016\pi$
$62$	−5.45211	−0.692418
$63$	0	0
$64$	−8.87266	−1.10908
$65$	0	0
$66$	0	0
$67$	1.31351	0.160470	0.0802352	−	0.996776i	$-0.474433\pi$
$67$	1.31351	0.160470	0.0802352	+	0.996776i	$0.474433\pi$
$68$	−10.0199	−1.21509
$69$	0	0
$70$	0	0
$71$	3.79321	0.450172	0.225086	−	0.974339i	$-0.427734\pi$
$71$	3.79321	0.450172	0.225086	+	0.974339i	$0.427734\pi$
$72$	0	0
$73$	10.9252	1.27870	0.639351	−	0.768915i	$-0.279203\pi$
$73$	10.9252	1.27870	0.639351	+	0.768915i	$0.279203\pi$
$74$	−23.0787	−2.68285
$75$	0	0
$76$	8.95051	1.02669
$77$	0.347616	0.0396146
$78$	0	0
$79$	−1.89869	−0.213620	−0.106810	−	0.994279i	$-0.534064\pi$
$79$	−1.89869	−0.213620	−0.106810	+	0.994279i	$0.534064\pi$
$80$	0	0
$81$	0	0
$82$	0.292519	0.0323033
$83$	2.51849	0.276441	0.138220	−	0.990402i	$-0.455862\pi$
$83$	2.51849	0.276441	0.138220	+	0.990402i	$0.455862\pi$
$84$	0	0
$85$	0	0
$86$	12.0548	1.29990
$87$	0	0
$88$	0.156737	0.0167083
$89$	−15.0809	−1.59858	−0.799289	−	0.600947i	$-0.794790\pi$
$89$	−15.0809	−1.59858	−0.799289	+	0.600947i	$0.794790\pi$
$90$	0	0
$91$	−2.92562	−0.306689
$92$	17.6649	1.84169
$93$	0	0
$94$	12.3834	1.27725
$95$	0	0
$96$	0	0
$97$	−1.17394	−0.119195	−0.0595977	−	0.998222i	$-0.518982\pi$
$97$	−1.17394	−0.119195	−0.0595977	+	0.998222i	$0.518982\pi$
$98$	13.6769	1.38158
$99$	0	0
$100$	0	0
$101$	−7.23262	−0.719672	−0.359836	−	0.933016i	$-0.617167\pi$
$101$	−7.23262	−0.719672	−0.359836	+	0.933016i	$0.617167\pi$
$102$	0	0
$103$	−18.4207	−1.81505	−0.907524	−	0.420000i	$-0.862030\pi$
$103$	−18.4207	−1.81505	−0.907524	+	0.420000i	$0.862030\pi$
$104$	−1.31914	−0.129352
$105$	0	0
$106$	−21.9822	−2.13510
$107$	17.5572	1.69732	0.848661	−	0.528937i	$-0.177409\pi$
$107$	17.5572	1.69732	0.848661	+	0.528937i	$0.177409\pi$
$108$	0	0
$109$	11.4098	1.09286	0.546429	−	0.837506i	$-0.315987\pi$
$109$	11.4098	1.09286	0.546429	+	0.837506i	$0.315987\pi$
$110$	0	0
$111$	0	0
$112$	1.90238	0.179758
$113$	9.64827	0.907632	0.453816	−	0.891095i	$-0.350062\pi$
$113$	9.64827	0.907632	0.453816	+	0.891095i	$0.350062\pi$
$114$	0	0
$115$	0	0
$116$	−10.3246	−0.958617
$117$	0	0
$118$	−14.1308	−1.30085
$119$	2.39830	0.219852
$120$	0	0
$121$	−10.5273	−0.957030
$122$	8.07194	0.730799
$123$	0	0
$124$	5.67931	0.510017
$125$	0	0
$126$	0	0
$127$	−4.32804	−0.384051	−0.192026	−	0.981390i	$-0.561506\pi$
$127$	−4.32804	−0.384051	−0.192026	+	0.981390i	$0.561506\pi$
$128$	1.82094	0.160950
$129$	0	0
$130$	0	0
$131$	10.1642	0.888047	0.444023	−	0.896015i	$-0.353551\pi$
$131$	10.1642	0.888047	0.444023	+	0.896015i	$0.353551\pi$
$132$	0	0
$133$	−2.14233	−0.185764
$134$	−2.66368	−0.230106
$135$	0	0
$136$	1.08137	0.0927269
$137$	5.57509	0.476312	0.238156	−	0.971227i	$-0.423457\pi$
$137$	5.57509	0.476312	0.238156	+	0.971227i	$0.423457\pi$
$138$	0	0
$139$	9.24097	0.783809	0.391904	−	0.920006i	$-0.371816\pi$
$139$	9.24097	0.783809	0.391904	+	0.920006i	$0.371816\pi$
$140$	0	0
$141$	0	0
$142$	−7.69230	−0.645523
$143$	−3.97814	−0.332669
$144$	0	0
$145$	0	0
$146$	−22.1554	−1.83359
$147$	0	0
$148$	24.0405	1.97612
$149$	9.91157	0.811988	0.405994	−	0.913876i	$-0.366925\pi$
$149$	9.91157	0.811988	0.405994	+	0.913876i	$0.366925\pi$
$150$	0	0
$151$	−21.8846	−1.78094	−0.890471	−	0.455041i	$-0.849625\pi$
$151$	−21.8846	−1.78094	−0.890471	+	0.455041i	$0.849625\pi$
$152$	−0.965960	−0.0783497
$153$	0	0
$154$	−0.704935	−0.0568053
$155$	0	0
$156$	0	0
$157$	−4.47331	−0.357009	−0.178504	−	0.983939i	$-0.557126\pi$
$157$	−4.47331	−0.357009	−0.178504	+	0.983939i	$0.557126\pi$
$158$	3.85038	0.306320
$159$	0	0
$160$	0	0
$161$	−4.22814	−0.333224
$162$	0	0
$163$	−14.2898	−1.11927	−0.559634	−	0.828740i	$-0.689058\pi$
$163$	−14.2898	−1.11927	−0.559634	+	0.828740i	$0.689058\pi$
$164$	−0.304709	−0.0237938
$165$	0	0
$166$	−5.10728	−0.396402
$167$	−0.473306	−0.0366255	−0.0183128	−	0.999832i	$-0.505829\pi$
$167$	−0.473306	−0.0366255	−0.0183128	+	0.999832i	$0.505829\pi$
$168$	0	0
$169$	20.4810	1.57546
$170$	0	0
$171$	0	0
$172$	−12.5571	−0.957471
$173$	2.14939	0.163415	0.0817074	−	0.996656i	$-0.473963\pi$
$173$	2.14939	0.163415	0.0817074	+	0.996656i	$0.473963\pi$
$174$	0	0
$175$	0	0
$176$	2.58678	0.194986
$177$	0	0
$178$	30.5828	2.29228
$179$	1.35831	0.101525	0.0507623	−	0.998711i	$-0.483835\pi$
$179$	1.35831	0.101525	0.0507623	+	0.998711i	$0.483835\pi$
$180$	0	0
$181$	3.37725	0.251029	0.125515	−	0.992092i	$-0.459942\pi$
$181$	3.37725	0.251029	0.125515	+	0.992092i	$0.459942\pi$
$182$	5.93290	0.439776
$183$	0	0
$184$	−1.90643	−0.140544
$185$	0	0
$186$	0	0
$187$	3.26110	0.238476
$188$	−12.8995	−0.940792
$189$	0	0
$190$	0	0
$191$	−21.8942	−1.58421	−0.792103	−	0.610388i	$-0.791014\pi$
$191$	−21.8942	−1.58421	−0.792103	+	0.610388i	$0.791014\pi$
$192$	0	0
$193$	−20.3890	−1.46763	−0.733817	−	0.679347i	$-0.762263\pi$
$193$	−20.3890	−1.46763	−0.733817	+	0.679347i	$0.762263\pi$
$194$	2.38064	0.170920
$195$	0	0
$196$	−14.2469	−1.01764
$197$	24.9688	1.77895	0.889477	−	0.456980i	$-0.151069\pi$
$197$	24.9688	1.77895	0.889477	+	0.456980i	$0.151069\pi$
$198$	0	0
$199$	−10.2138	−0.724038	−0.362019	−	0.932171i	$-0.617913\pi$
$199$	−10.2138	−0.724038	−0.362019	+	0.932171i	$0.617913\pi$
$200$	0	0
$201$	0	0
$202$	14.6671	1.03197
$203$	2.47123	0.173446
$204$	0	0
$205$	0	0
$206$	37.3556	2.60269
$207$	0	0
$208$	−21.7710	−1.50955
$209$	−2.91305	−0.201500
$210$	0	0
$211$	5.46907	0.376507	0.188253	−	0.982121i	$-0.439717\pi$
$211$	5.46907	0.376507	0.188253	+	0.982121i	$0.439717\pi$
$212$	22.8982	1.57266
$213$	0	0
$214$	−35.6045	−2.43387
$215$	0	0
$216$	0	0
$217$	−1.35936	−0.0922794
$218$	−23.1380	−1.56710
$219$	0	0
$220$	0	0
$221$	−27.4462	−1.84623
$222$	0	0
$223$	22.8887	1.53274	0.766370	−	0.642399i	$-0.222061\pi$
$223$	22.8887	1.53274	0.766370	+	0.642399i	$0.222061\pi$
$224$	−4.08840	−0.273168
$225$	0	0
$226$	−19.5658	−1.30150
$227$	12.9219	0.857659	0.428829	−	0.903386i	$-0.358926\pi$
$227$	12.9219	0.857659	0.428829	+	0.903386i	$0.358926\pi$
$228$	0	0
$229$	14.5252	0.959850	0.479925	−	0.877309i	$-0.340664\pi$
$229$	14.5252	0.959850	0.479925	+	0.877309i	$0.340664\pi$
$230$	0	0
$231$	0	0
$232$	1.11426	0.0731545
$233$	9.94748	0.651681	0.325841	−	0.945425i	$-0.394353\pi$
$233$	9.94748	0.651681	0.325841	+	0.945425i	$0.394353\pi$
$234$	0	0
$235$	0	0
$236$	14.7197	0.958171
$237$	0	0
$238$	−4.86353	−0.315256
$239$	−1.58509	−0.102531	−0.0512656	−	0.998685i	$-0.516326\pi$
$239$	−1.58509	−0.102531	−0.0512656	+	0.998685i	$0.516326\pi$
$240$	0	0
$241$	23.7485	1.52978	0.764889	−	0.644163i	$-0.222794\pi$
$241$	23.7485	1.52978	0.764889	+	0.644163i	$0.222794\pi$
$242$	21.3485	1.37233
$243$	0	0
$244$	−8.40832	−0.538288
$245$	0	0
$246$	0	0
$247$	24.5170	1.55998
$248$	−0.612924	−0.0389207
$249$	0	0
$250$	0	0
$251$	−20.2887	−1.28061	−0.640307	−	0.768119i	$-0.721193\pi$
$251$	−20.2887	−1.28061	−0.640307	+	0.768119i	$0.721193\pi$
$252$	0	0
$253$	−5.74925	−0.361452
$254$	8.77687	0.550710
$255$	0	0
$256$	14.0526	0.878289
$257$	−21.8997	−1.36607	−0.683033	−	0.730387i	$-0.739340\pi$
$257$	−21.8997	−1.36607	−0.683033	+	0.730387i	$0.739340\pi$
$258$	0	0
$259$	−5.75417	−0.357547
$260$	0	0
$261$	0	0
$262$	−20.6120	−1.27341
$263$	−4.76922	−0.294083	−0.147041	−	0.989130i	$-0.546975\pi$
$263$	−4.76922	−0.294083	−0.147041	+	0.989130i	$0.546975\pi$
$264$	0	0
$265$	0	0
$266$	4.34446	0.266376
$267$	0	0
$268$	2.77468	0.169490
$269$	12.0532	0.734899	0.367450	−	0.930043i	$-0.380231\pi$
$269$	12.0532	0.734899	0.367450	+	0.930043i	$0.380231\pi$
$270$	0	0
$271$	27.5843	1.67562	0.837812	−	0.545959i	$-0.183834\pi$
$271$	27.5843	1.67562	0.837812	+	0.545959i	$0.183834\pi$
$272$	17.8469	1.08213
$273$	0	0
$274$	−11.3058	−0.683008
$275$	0	0
$276$	0	0
$277$	−28.9858	−1.74159	−0.870795	−	0.491646i	$-0.836395\pi$
$277$	−28.9858	−1.74159	−0.870795	+	0.491646i	$0.836395\pi$
$278$	−18.7399	−1.12394
$279$	0	0
$280$	0	0
$281$	−5.93418	−0.354003	−0.177002	−	0.984211i	$-0.556640\pi$
$281$	−5.93418	−0.354003	−0.177002	+	0.984211i	$0.556640\pi$
$282$	0	0
$283$	15.2381	0.905812	0.452906	−	0.891558i	$-0.350387\pi$
$283$	15.2381	0.905812	0.452906	+	0.891558i	$0.350387\pi$
$284$	8.01286	0.475476
$285$	0	0
$286$	8.06731	0.477030
$287$	0.0729331	0.00430511
$288$	0	0
$289$	5.49921	0.323483
$290$	0	0
$291$	0	0
$292$	23.0787	1.35058
$293$	1.73298	0.101242	0.0506208	−	0.998718i	$-0.483880\pi$
$293$	1.73298	0.101242	0.0506208	+	0.998718i	$0.483880\pi$
$294$	0	0
$295$	0	0
$296$	−2.59451	−0.150803
$297$	0	0
$298$	−20.0998	−1.16435
$299$	48.3871	2.79830
$300$	0	0
$301$	3.00558	0.173239
$302$	44.3800	2.55378
$303$	0	0
$304$	−15.9421	−0.914344
$305$	0	0
$306$	0	0
$307$	−11.6087	−0.662541	−0.331271	−	0.943536i	$-0.607477\pi$
$307$	−11.6087	−0.662541	−0.331271	+	0.943536i	$0.607477\pi$
$308$	0.734312	0.0418413
$309$	0	0
$310$	0	0
$311$	−6.60870	−0.374745	−0.187372	−	0.982289i	$-0.559997\pi$
$311$	−6.60870	−0.374745	−0.187372	+	0.982289i	$0.559997\pi$
$312$	0	0
$313$	−12.5410	−0.708860	−0.354430	−	0.935082i	$-0.615325\pi$
$313$	−12.5410	−0.708860	−0.354430	+	0.935082i	$0.615325\pi$
$314$	9.07146	0.511932
$315$	0	0
$316$	−4.01084	−0.225627
$317$	−11.7318	−0.658925	−0.329463	−	0.944169i	$-0.606868\pi$
$317$	−11.7318	−0.658925	−0.329463	+	0.944169i	$0.606868\pi$
$318$	0	0
$319$	3.36028	0.188139
$320$	0	0
$321$	0	0
$322$	8.57429	0.477827
$323$	−20.0979	−1.11828
$324$	0	0
$325$	0	0
$326$	28.9785	1.60497
$327$	0	0
$328$	0.0328849	0.00181577
$329$	3.08753	0.170221
$330$	0	0
$331$	8.16236	0.448644	0.224322	−	0.974515i	$-0.427983\pi$
$331$	8.16236	0.448644	0.224322	+	0.974515i	$0.427983\pi$
$332$	5.32012	0.291979
$333$	0	0
$334$	0.959822	0.0525191
$335$	0	0
$336$	0	0
$337$	11.1478	0.607259	0.303629	−	0.952790i	$-0.401802\pi$
$337$	11.1478	0.607259	0.303629	+	0.952790i	$0.401802\pi$
$338$	−41.5336	−2.25913
$339$	0	0
$340$	0	0
$341$	−1.84840	−0.100097
$342$	0	0
$343$	6.94934	0.375229
$344$	1.35519	0.0730671
$345$	0	0
$346$	−4.35876	−0.234329
$347$	−10.2954	−0.552687	−0.276343	−	0.961059i	$-0.589123\pi$
$347$	−10.2954	−0.552687	−0.276343	+	0.961059i	$0.589123\pi$
$348$	0	0
$349$	20.8288	1.11494	0.557471	−	0.830196i	$-0.311772\pi$
$349$	20.8288	1.11494	0.557471	+	0.830196i	$0.311772\pi$
$350$	0	0
$351$	0	0
$352$	−5.55924	−0.296308
$353$	27.6165	1.46988	0.734939	−	0.678133i	$-0.237211\pi$
$353$	27.6165	1.46988	0.734939	+	0.678133i	$0.237211\pi$
$354$	0	0
$355$	0	0
$356$	−31.8573	−1.68843
$357$	0	0
$358$	−2.75452	−0.145581
$359$	−19.2421	−1.01556	−0.507781	−	0.861486i	$-0.669534\pi$
$359$	−19.2421	−1.01556	−0.507781	+	0.861486i	$0.669534\pi$
$360$	0	0
$361$	−1.04708	−0.0551095
$362$	−6.84877	−0.359963
$363$	0	0
$364$	−6.18014	−0.323927
$365$	0	0
$366$	0	0
$367$	−8.23097	−0.429653	−0.214826	−	0.976652i	$-0.568919\pi$
$367$	−8.23097	−0.429653	−0.214826	+	0.976652i	$0.568919\pi$
$368$	−31.4637	−1.64016
$369$	0	0
$370$	0	0
$371$	−5.48076	−0.284547
$372$	0	0
$373$	−29.1863	−1.51121	−0.755606	−	0.655027i	$-0.772657\pi$
$373$	−29.1863	−1.51121	−0.755606	+	0.655027i	$0.772657\pi$
$374$	−6.61323	−0.341962
$375$	0	0
$376$	1.39214	0.0717943
$377$	−28.2809	−1.45654
$378$	0	0
$379$	−4.34734	−0.223308	−0.111654	−	0.993747i	$-0.535615\pi$
$379$	−4.34734	−0.223308	−0.111654	+	0.993747i	$0.535615\pi$
$380$	0	0
$381$	0	0
$382$	44.3994	2.27167
$383$	−22.6224	−1.15595	−0.577976	−	0.816054i	$-0.696157\pi$
$383$	−22.6224	−1.15595	−0.577976	+	0.816054i	$0.696157\pi$
$384$	0	0
$385$	0	0
$386$	41.3471	2.10451
$387$	0	0
$388$	−2.47985	−0.125895
$389$	−11.2860	−0.572221	−0.286111	−	0.958197i	$-0.592362\pi$
$389$	−11.2860	−0.572221	−0.286111	+	0.958197i	$0.592362\pi$
$390$	0	0
$391$	−39.6656	−2.00598
$392$	1.53756	0.0776585
$393$	0	0
$394$	−50.6345	−2.55093
$395$	0	0
$396$	0	0
$397$	9.06606	0.455013	0.227506	−	0.973777i	$-0.426943\pi$
$397$	9.06606	0.455013	0.227506	+	0.973777i	$0.426943\pi$
$398$	20.7127	1.03823
$399$	0	0
$400$	0	0
$401$	11.9700	0.597752	0.298876	−	0.954292i	$-0.403388\pi$
$401$	11.9700	0.597752	0.298876	+	0.954292i	$0.403388\pi$
$402$	0	0
$403$	15.5566	0.774929
$404$	−15.2783	−0.760125
$405$	0	0
$406$	−5.01143	−0.248713
$407$	−7.82428	−0.387835
$408$	0	0
$409$	−23.1228	−1.14335	−0.571673	−	0.820481i	$-0.693706\pi$
$409$	−23.1228	−1.14335	−0.571673	+	0.820481i	$0.693706\pi$
$410$	0	0
$411$	0	0
$412$	−38.9123	−1.91707
$413$	−3.52321	−0.173366
$414$	0	0
$415$	0	0
$416$	46.7879	2.29396
$417$	0	0
$418$	5.90741	0.288941
$419$	32.8579	1.60521	0.802606	−	0.596509i	$-0.203446\pi$
$419$	32.8579	1.60521	0.802606	+	0.596509i	$0.203446\pi$
$420$	0	0
$421$	8.70039	0.424031	0.212016	−	0.977266i	$-0.431997\pi$
$421$	8.70039	0.424031	0.212016	+	0.977266i	$0.431997\pi$
$422$	−11.0908	−0.539891
$423$	0	0
$424$	−2.47123	−0.120014
$425$	0	0
$426$	0	0
$427$	2.01256	0.0973945
$428$	37.0883	1.79273
$429$	0	0
$430$	0	0
$431$	−5.70635	−0.274865	−0.137432	−	0.990511i	$-0.543885\pi$
$431$	−5.70635	−0.274865	−0.137432	+	0.990511i	$0.543885\pi$
$432$	0	0
$433$	18.2221	0.875700	0.437850	−	0.899048i	$-0.355740\pi$
$433$	18.2221	0.875700	0.437850	+	0.899048i	$0.355740\pi$
$434$	2.75666	0.132324
$435$	0	0
$436$	24.1022	1.15429
$437$	35.4322	1.69495
$438$	0	0
$439$	13.2419	0.632002	0.316001	−	0.948759i	$-0.397660\pi$
$439$	13.2419	0.632002	0.316001	+	0.948759i	$0.397660\pi$
$440$	0	0
$441$	0	0
$442$	55.6585	2.64740
$443$	26.3276	1.25086	0.625431	−	0.780279i	$-0.284923\pi$
$443$	26.3276	1.25086	0.625431	+	0.780279i	$0.284923\pi$
$444$	0	0
$445$	0	0
$446$	−46.4162	−2.19787
$447$	0	0
$448$	4.48614	0.211950
$449$	−22.2932	−1.05208	−0.526040	−	0.850460i	$-0.676324\pi$
$449$	−22.2932	−1.05208	−0.526040	+	0.850460i	$0.676324\pi$
$450$	0	0
$451$	0.0991714	0.00466980
$452$	20.3812	0.958650
$453$	0	0
$454$	−26.2045	−1.22984
$455$	0	0
$456$	0	0
$457$	17.3159	0.810002	0.405001	−	0.914316i	$-0.367271\pi$
$457$	17.3159	0.810002	0.405001	+	0.914316i	$0.367271\pi$
$458$	−29.4557	−1.37638
$459$	0	0
$460$	0	0
$461$	16.2204	0.755460	0.377730	−	0.925916i	$-0.376705\pi$
$461$	16.2204	0.755460	0.377730	+	0.925916i	$0.376705\pi$
$462$	0	0
$463$	20.1061	0.934412	0.467206	−	0.884149i	$-0.345261\pi$
$463$	20.1061	0.934412	0.467206	+	0.884149i	$0.345261\pi$
$464$	18.3896	0.853717
$465$	0	0
$466$	−20.1726	−0.934478
$467$	32.4922	1.50356	0.751780	−	0.659414i	$-0.229195\pi$
$467$	32.4922	1.50356	0.751780	+	0.659414i	$0.229195\pi$
$468$	0	0
$469$	−0.664128	−0.0306666
$470$	0	0
$471$	0	0
$472$	−1.58858	−0.0731205
$473$	4.08687	0.187914
$474$	0	0
$475$	0	0
$476$	5.06621	0.232209
$477$	0	0
$478$	3.21443	0.147025
$479$	8.99966	0.411205	0.205603	−	0.978636i	$-0.434085\pi$
$479$	8.99966	0.411205	0.205603	+	0.978636i	$0.434085\pi$
$480$	0	0
$481$	65.8510	3.00255
$482$	−48.1599	−2.19362
$483$	0	0
$484$	−22.2381	−1.01082
$485$	0	0
$486$	0	0
$487$	40.1153	1.81780	0.908898	−	0.417018i	$-0.136925\pi$
$487$	40.1153	1.81780	0.908898	+	0.417018i	$0.136925\pi$
$488$	0.907446	0.0410781
$489$	0	0
$490$	0	0
$491$	35.7772	1.61460	0.807302	−	0.590138i	$-0.200927\pi$
$491$	35.7772	1.61460	0.807302	+	0.590138i	$0.200927\pi$
$492$	0	0
$493$	23.1834	1.04413
$494$	−49.7182	−2.23693
$495$	0	0
$496$	−10.1157	−0.454207
$497$	−1.91790	−0.0860297
$498$	0	0
$499$	14.9592	0.669665	0.334832	−	0.942278i	$-0.391320\pi$
$499$	14.9592	0.669665	0.334832	+	0.942278i	$0.391320\pi$
$500$	0	0
$501$	0	0
$502$	41.1437	1.83633
$503$	33.8165	1.50780	0.753901	−	0.656988i	$-0.228170\pi$
$503$	33.8165	1.50780	0.753901	+	0.656988i	$0.228170\pi$
$504$	0	0
$505$	0	0
$506$	11.6590	0.518304
$507$	0	0
$508$	−9.14263	−0.405639
$509$	−4.54546	−0.201474	−0.100737	−	0.994913i	$-0.532120\pi$
$509$	−4.54546	−0.201474	−0.100737	+	0.994913i	$0.532120\pi$
$510$	0	0
$511$	−5.52395	−0.244365
$512$	−32.1393	−1.42037
$513$	0	0
$514$	44.4107	1.95887
$515$	0	0
$516$	0	0
$517$	4.19830	0.184641
$518$	11.6689	0.512704
$519$	0	0
$520$	0	0
$521$	27.5808	1.20834	0.604169	−	0.796857i	$-0.293505\pi$
$521$	27.5808	1.20834	0.604169	+	0.796857i	$0.293505\pi$
$522$	0	0
$523$	−8.22959	−0.359855	−0.179927	−	0.983680i	$-0.557586\pi$
$523$	−8.22959	−0.359855	−0.179927	+	0.983680i	$0.557586\pi$
$524$	21.4710	0.937964
$525$	0	0
$526$	9.67155	0.421700
$527$	−12.7526	−0.555513
$528$	0	0
$529$	46.9295	2.04041
$530$	0	0
$531$	0	0
$532$	−4.52550	−0.196206
$533$	−0.834650	−0.0361527
$534$	0	0
$535$	0	0
$536$	−0.299450	−0.0129343
$537$	0	0
$538$	−24.4429	−1.05381
$539$	4.63683	0.199723
$540$	0	0
$541$	17.8165	0.765993	0.382996	−	0.923750i	$-0.374892\pi$
$541$	17.8165	0.765993	0.382996	+	0.923750i	$0.374892\pi$
$542$	−55.9384	−2.40276
$543$	0	0
$544$	−38.3547	−1.64444
$545$	0	0
$546$	0	0
$547$	−9.56762	−0.409082	−0.204541	−	0.978858i	$-0.565570\pi$
$547$	−9.56762	−0.409082	−0.204541	+	0.978858i	$0.565570\pi$
$548$	11.7769	0.503086
$549$	0	0
$550$	0	0
$551$	−20.7091	−0.882238
$552$	0	0
$553$	0.960007	0.0408236
$554$	58.7807	2.49735
$555$	0	0
$556$	19.5208	0.827866
$557$	20.5472	0.870613	0.435306	−	0.900282i	$-0.356640\pi$
$557$	20.5472	0.870613	0.435306	+	0.900282i	$0.356640\pi$
$558$	0	0
$559$	−34.3961	−1.45480
$560$	0	0
$561$	0	0
$562$	12.0340	0.507623
$563$	−18.7411	−0.789842	−0.394921	−	0.918715i	$-0.629228\pi$
$563$	−18.7411	−0.789842	−0.394921	+	0.918715i	$0.629228\pi$
$564$	0	0
$565$	0	0
$566$	−30.9015	−1.29889
$567$	0	0
$568$	−0.864766	−0.0362848
$569$	−4.66977	−0.195767	−0.0978834	−	0.995198i	$-0.531207\pi$
$569$	−4.66977	−0.195767	−0.0978834	+	0.995198i	$0.531207\pi$
$570$	0	0
$571$	14.9699	0.626471	0.313235	−	0.949676i	$-0.398587\pi$
$571$	14.9699	0.626471	0.313235	+	0.949676i	$0.398587\pi$
$572$	−8.40350	−0.351368
$573$	0	0
$574$	−0.147902	−0.00617330
$575$	0	0
$576$	0	0
$577$	−15.6712	−0.652400	−0.326200	−	0.945301i	$-0.605768\pi$
$577$	−15.6712	−0.652400	−0.326200	+	0.945301i	$0.605768\pi$
$578$	−11.1519	−0.463858
$579$	0	0
$580$	0	0
$581$	−1.27339	−0.0528290
$582$	0	0
$583$	−7.45251	−0.308652
$584$	−2.49070	−0.103066
$585$	0	0
$586$	−3.51432	−0.145175
$587$	30.0310	1.23951	0.619756	−	0.784794i	$-0.287231\pi$
$587$	30.0310	1.23951	0.619756	+	0.784794i	$0.287231\pi$
$588$	0	0
$589$	11.3916	0.469381
$590$	0	0
$591$	0	0
$592$	−42.8196	−1.75987
$593$	26.2392	1.07751	0.538757	−	0.842461i	$-0.318894\pi$
$593$	26.2392	1.07751	0.538757	+	0.842461i	$0.318894\pi$
$594$	0	0
$595$	0	0
$596$	20.9374	0.857630
$597$	0	0
$598$	−98.1246	−4.01261
$599$	15.5681	0.636095	0.318048	−	0.948075i	$-0.396973\pi$
$599$	15.5681	0.636095	0.318048	+	0.948075i	$0.396973\pi$
$600$	0	0
$601$	−6.27437	−0.255937	−0.127969	−	0.991778i	$-0.540846\pi$
$601$	−6.27437	−0.255937	−0.127969	+	0.991778i	$0.540846\pi$
$602$	−6.09505	−0.248416
$603$	0	0
$604$	−46.2294	−1.88105
$605$	0	0
$606$	0	0
$607$	8.47616	0.344037	0.172018	−	0.985094i	$-0.444971\pi$
$607$	8.47616	0.344037	0.172018	+	0.985094i	$0.444971\pi$
$608$	34.2611	1.38947
$609$	0	0
$610$	0	0
$611$	−35.3339	−1.42946
$612$	0	0
$613$	6.60744	0.266872	0.133436	−	0.991057i	$-0.457399\pi$
$613$	6.60744	0.266872	0.133436	+	0.991057i	$0.457399\pi$
$614$	23.5413	0.950051
$615$	0	0
$616$	−0.0792486	−0.00319302
$617$	−25.1919	−1.01419	−0.507094	−	0.861891i	$-0.669280\pi$
$617$	−25.1919	−1.01419	−0.507094	+	0.861891i	$0.669280\pi$
$618$	0	0
$619$	1.45782	0.0585949	0.0292975	−	0.999571i	$-0.490673\pi$
$619$	1.45782	0.0585949	0.0292975	+	0.999571i	$0.490673\pi$
$620$	0	0
$621$	0	0
$622$	13.4018	0.537365
$623$	7.62514	0.305495
$624$	0	0
$625$	0	0
$626$	25.4321	1.01647
$627$	0	0
$628$	−9.44950	−0.377076
$629$	−53.9818	−2.15239
$630$	0	0
$631$	15.0828	0.600435	0.300218	−	0.953871i	$-0.402941\pi$
$631$	15.0828	0.600435	0.300218	+	0.953871i	$0.402941\pi$
$632$	0.432859	0.0172182
$633$	0	0
$634$	23.7911	0.944865
$635$	0	0
$636$	0	0
$637$	−39.0247	−1.54621
$638$	−6.81434	−0.269782
$639$	0	0
$640$	0	0
$641$	5.23264	0.206677	0.103338	−	0.994646i	$-0.467048\pi$
$641$	5.23264	0.206677	0.103338	+	0.994646i	$0.467048\pi$
$642$	0	0
$643$	36.4602	1.43785	0.718925	−	0.695088i	$-0.244634\pi$
$643$	36.4602	1.43785	0.718925	+	0.695088i	$0.244634\pi$
$644$	−8.93161	−0.351955
$645$	0	0
$646$	40.7568	1.60355
$647$	−5.06588	−0.199160	−0.0995801	−	0.995030i	$-0.531750\pi$
$647$	−5.06588	−0.199160	−0.0995801	+	0.995030i	$0.531750\pi$
$648$	0	0
$649$	−4.79071	−0.188052
$650$	0	0
$651$	0	0
$652$	−30.1862	−1.18218
$653$	−21.3639	−0.836033	−0.418017	−	0.908439i	$-0.637275\pi$
$653$	−21.3639	−0.836033	−0.418017	+	0.908439i	$0.637275\pi$
$654$	0	0
$655$	0	0
$656$	0.542731	0.0211901
$657$	0	0
$658$	−6.26124	−0.244089
$659$	47.2176	1.83934	0.919668	−	0.392696i	$-0.128458\pi$
$659$	47.2176	1.83934	0.919668	+	0.392696i	$0.128458\pi$
$660$	0	0
$661$	−23.7835	−0.925072	−0.462536	−	0.886601i	$-0.653060\pi$
$661$	−23.7835	−0.925072	−0.462536	+	0.886601i	$0.653060\pi$
$662$	−16.5525	−0.643332
$663$	0	0
$664$	−0.574159	−0.0222817
$665$	0	0
$666$	0	0
$667$	−40.8718	−1.58256
$668$	−0.999821	−0.0386842
$669$	0	0
$670$	0	0
$671$	2.73659	0.105645
$672$	0	0
$673$	26.1124	1.00656	0.503279	−	0.864124i	$-0.332127\pi$
$673$	26.1124	1.00656	0.503279	+	0.864124i	$0.332127\pi$
$674$	−22.6067	−0.870778
$675$	0	0
$676$	43.2644	1.66402
$677$	−11.8091	−0.453861	−0.226931	−	0.973911i	$-0.572869\pi$
$677$	−11.8091	−0.453861	−0.226931	+	0.973911i	$0.572869\pi$
$678$	0	0
$679$	0.593560	0.0227787
$680$	0	0
$681$	0	0
$682$	3.74839	0.143533
$683$	5.41899	0.207352	0.103676	−	0.994611i	$-0.466940\pi$
$683$	5.41899	0.207352	0.103676	+	0.994611i	$0.466940\pi$
$684$	0	0
$685$	0	0
$686$	−14.0926	−0.538059
$687$	0	0
$688$	22.3660	0.852696
$689$	62.7221	2.38952
$690$	0	0
$691$	48.1513	1.83176	0.915880	−	0.401451i	$-0.131494\pi$
$691$	48.1513	1.83176	0.915880	+	0.401451i	$0.131494\pi$
$692$	4.54041	0.172600
$693$	0	0
$694$	20.8782	0.792524
$695$	0	0
$696$	0	0
$697$	0.684210	0.0259163
$698$	−42.2390	−1.59877
$699$	0	0
$700$	0	0
$701$	−46.4747	−1.75532	−0.877662	−	0.479280i	$-0.840898\pi$
$701$	−46.4747	−1.75532	−0.877662	+	0.479280i	$0.840898\pi$
$702$	0	0
$703$	48.2204	1.81867
$704$	6.10007	0.229905
$705$	0	0
$706$	−56.0038	−2.10773
$707$	3.65691	0.137532
$708$	0	0
$709$	2.73598	0.102752	0.0513760	−	0.998679i	$-0.483639\pi$
$709$	2.73598	0.102752	0.0513760	+	0.998679i	$0.483639\pi$
$710$	0	0
$711$	0	0
$712$	3.43811	0.128849
$713$	22.4826	0.841979
$714$	0	0
$715$	0	0
$716$	2.86931	0.107231
$717$	0	0
$718$	39.0213	1.45626
$719$	−21.7083	−0.809584	−0.404792	−	0.914409i	$-0.632656\pi$
$719$	−21.7083	−0.809584	−0.404792	+	0.914409i	$0.632656\pi$
$720$	0	0
$721$	9.31378	0.346863
$722$	2.12338	0.0790242
$723$	0	0
$724$	7.13418	0.265140
$725$	0	0
$726$	0	0
$727$	25.8542	0.958879	0.479439	−	0.877575i	$-0.340840\pi$
$727$	25.8542	0.958879	0.479439	+	0.877575i	$0.340840\pi$
$728$	0.666975	0.0247197
$729$	0	0
$730$	0	0
$731$	28.1964	1.04288
$732$	0	0
$733$	−6.28263	−0.232054	−0.116027	−	0.993246i	$-0.537016\pi$
$733$	−6.28263	−0.232054	−0.116027	+	0.993246i	$0.537016\pi$
$734$	16.6917	0.616100
$735$	0	0
$736$	67.6184	2.49245
$737$	−0.903054	−0.0332644
$738$	0	0
$739$	23.7681	0.874325	0.437162	−	0.899383i	$-0.355983\pi$
$739$	23.7681	0.874325	0.437162	+	0.899383i	$0.355983\pi$
$740$	0	0
$741$	0	0
$742$	11.1145	0.408026
$743$	14.8532	0.544912	0.272456	−	0.962168i	$-0.412164\pi$
$743$	14.8532	0.544912	0.272456	+	0.962168i	$0.412164\pi$
$744$	0	0
$745$	0	0
$746$	59.1873	2.16700
$747$	0	0
$748$	6.88882	0.251880
$749$	−8.87719	−0.324365
$750$	0	0
$751$	26.1668	0.954841	0.477421	−	0.878675i	$-0.341572\pi$
$751$	26.1668	0.954841	0.477421	+	0.878675i	$0.341572\pi$
$752$	22.9758	0.837843
$753$	0	0
$754$	57.3511	2.08860
$755$	0	0
$756$	0	0
$757$	13.1012	0.476170	0.238085	−	0.971244i	$-0.423480\pi$
$757$	13.1012	0.476170	0.238085	+	0.971244i	$0.423480\pi$
$758$	8.81602	0.320212
$759$	0	0
$760$	0	0
$761$	36.3328	1.31706	0.658531	−	0.752554i	$-0.271178\pi$
$761$	36.3328	1.31706	0.658531	+	0.752554i	$0.271178\pi$
$762$	0	0
$763$	−5.76894	−0.208850
$764$	−46.2496	−1.67325
$765$	0	0
$766$	45.8762	1.65758
$767$	40.3198	1.45586
$768$	0	0
$769$	−18.6504	−0.672550	−0.336275	−	0.941764i	$-0.609167\pi$
$769$	−18.6504	−0.672550	−0.336275	+	0.941764i	$0.609167\pi$
$770$	0	0
$771$	0	0
$772$	−43.0702	−1.55013
$773$	−9.15449	−0.329264	−0.164632	−	0.986355i	$-0.552644\pi$
$773$	−9.15449	−0.329264	−0.164632	+	0.986355i	$0.552644\pi$
$774$	0	0
$775$	0	0
$776$	0.267631	0.00960740
$777$	0	0
$778$	22.8869	0.820536
$779$	−0.611186	−0.0218980
$780$	0	0
$781$	−2.60788	−0.0933174
$782$	80.4383	2.87647
$783$	0	0
$784$	25.3758	0.906278
$785$	0	0
$786$	0	0
$787$	36.1383	1.28819	0.644095	−	0.764945i	$-0.277234\pi$
$787$	36.1383	1.28819	0.644095	+	0.764945i	$0.277234\pi$
$788$	52.7446	1.87895
$789$	0	0
$790$	0	0
$791$	−4.87830	−0.173452
$792$	0	0
$793$	−23.0318	−0.817884
$794$	−18.3852	−0.652465
$795$	0	0
$796$	−21.5759	−0.764736
$797$	−15.1491	−0.536608	−0.268304	−	0.963334i	$-0.586463\pi$
$797$	−15.1491	−0.536608	−0.268304	+	0.963334i	$0.586463\pi$
$798$	0	0
$799$	28.9652	1.02471
$800$	0	0
$801$	0	0
$802$	−24.2740	−0.857146
$803$	−7.51124	−0.265066
$804$	0	0
$805$	0	0
$806$	−31.5474	−1.11121
$807$	0	0
$808$	1.64887	0.0580071
$809$	−33.2017	−1.16731	−0.583654	−	0.812002i	$-0.698378\pi$
$809$	−33.2017	−1.16731	−0.583654	+	0.812002i	$0.698378\pi$
$810$	0	0
$811$	26.6013	0.934098	0.467049	−	0.884231i	$-0.345317\pi$
$811$	26.6013	0.934098	0.467049	+	0.884231i	$0.345317\pi$
$812$	5.22028	0.183196
$813$	0	0
$814$	15.8669	0.556136
$815$	0	0
$816$	0	0
$817$	−25.1870	−0.881183
$818$	46.8909	1.63950
$819$	0	0
$820$	0	0
$821$	−2.75480	−0.0961432	−0.0480716	−	0.998844i	$-0.515308\pi$
$821$	−2.75480	−0.0961432	−0.0480716	+	0.998844i	$0.515308\pi$
$822$	0	0
$823$	4.73309	0.164985	0.0824926	−	0.996592i	$-0.473712\pi$
$823$	4.73309	0.164985	0.0824926	+	0.996592i	$0.473712\pi$
$824$	4.19951	0.146297
$825$	0	0
$826$	7.14475	0.248598
$827$	−39.6187	−1.37768	−0.688838	−	0.724916i	$-0.741879\pi$
$827$	−39.6187	−1.37768	−0.688838	+	0.724916i	$0.741879\pi$
$828$	0	0
$829$	−16.0834	−0.558600	−0.279300	−	0.960204i	$-0.590102\pi$
$829$	−16.0834	−0.558600	−0.279300	+	0.960204i	$0.590102\pi$
$830$	0	0
$831$	0	0
$832$	−51.3396	−1.77988
$833$	31.9907	1.10841
$834$	0	0
$835$	0	0
$836$	−6.15359	−0.212826
$837$	0	0
$838$	−66.6328	−2.30179
$839$	36.4617	1.25880	0.629400	−	0.777082i	$-0.283301\pi$
$839$	36.4617	1.25880	0.629400	+	0.777082i	$0.283301\pi$
$840$	0	0
$841$	−5.11156	−0.176261
$842$	−17.6436	−0.608039
$843$	0	0
$844$	11.5530	0.397670
$845$	0	0
$846$	0	0
$847$	5.32277	0.182892
$848$	−40.7850	−1.40056
$849$	0	0
$850$	0	0
$851$	95.1686	3.26234
$852$	0	0
$853$	25.0460	0.857560	0.428780	−	0.903409i	$-0.358944\pi$
$853$	25.0460	0.857560	0.428780	+	0.903409i	$0.358944\pi$
$854$	−4.08129	−0.139659
$855$	0	0
$856$	−4.00265	−0.136808
$857$	−15.1391	−0.517140	−0.258570	−	0.965993i	$-0.583251\pi$
$857$	−15.1391	−0.517140	−0.258570	+	0.965993i	$0.583251\pi$
$858$	0	0
$859$	24.3545	0.830966	0.415483	−	0.909601i	$-0.363613\pi$
$859$	24.3545	0.830966	0.415483	+	0.909601i	$0.363613\pi$
$860$	0	0
$861$	0	0
$862$	11.5720	0.394142
$863$	29.0380	0.988466	0.494233	−	0.869329i	$-0.335449\pi$
$863$	29.0380	0.988466	0.494233	+	0.869329i	$0.335449\pi$
$864$	0	0
$865$	0	0
$866$	−36.9529	−1.25571
$867$	0	0
$868$	−2.87154	−0.0974665
$869$	1.30538	0.0442819
$870$	0	0
$871$	7.60031	0.257527
$872$	−2.60117	−0.0880866
$873$	0	0
$874$	−71.8533	−2.43047
$875$	0	0
$876$	0	0
$877$	16.8629	0.569420	0.284710	−	0.958614i	$-0.408103\pi$
$877$	16.8629	0.569420	0.284710	+	0.958614i	$0.408103\pi$
$878$	−26.8534	−0.906259
$879$	0	0
$880$	0	0
$881$	50.2279	1.69222	0.846110	−	0.533008i	$-0.178938\pi$
$881$	50.2279	1.69222	0.846110	+	0.533008i	$0.178938\pi$
$882$	0	0
$883$	7.42047	0.249719	0.124859	−	0.992174i	$-0.460152\pi$
$883$	7.42047	0.249719	0.124859	+	0.992174i	$0.460152\pi$
$884$	−57.9780	−1.95001
$885$	0	0
$886$	−53.3900	−1.79367
$887$	39.7855	1.33587	0.667933	−	0.744221i	$-0.267179\pi$
$887$	39.7855	1.33587	0.667933	+	0.744221i	$0.267179\pi$
$888$	0	0
$889$	2.18832	0.0733938
$890$	0	0
$891$	0	0
$892$	48.3505	1.61890
$893$	−25.8738	−0.865833
$894$	0	0
$895$	0	0
$896$	−0.920691	−0.0307581
$897$	0	0
$898$	45.2086	1.50863
$899$	−13.1404	−0.438258
$900$	0	0
$901$	−51.4168	−1.71294
$902$	−0.201111	−0.00669626
$903$	0	0
$904$	−2.19958	−0.0731571
$905$	0	0
$906$	0	0
$907$	−46.5970	−1.54723	−0.773614	−	0.633657i	$-0.781553\pi$
$907$	−46.5970	−1.54723	−0.773614	+	0.633657i	$0.781553\pi$
$908$	27.2965	0.905868
$909$	0	0
$910$	0	0
$911$	−47.5250	−1.57457	−0.787287	−	0.616586i	$-0.788515\pi$
$911$	−47.5250	−1.57457	−0.787287	+	0.616586i	$0.788515\pi$
$912$	0	0
$913$	−1.73150	−0.0573042
$914$	−35.1150	−1.16150
$915$	0	0
$916$	30.6833	1.01380
$917$	−5.13914	−0.169709
$918$	0	0
$919$	−29.4578	−0.971723	−0.485862	−	0.874036i	$-0.661494\pi$
$919$	−29.4578	−0.971723	−0.485862	+	0.874036i	$0.661494\pi$
$920$	0	0
$921$	0	0
$922$	−32.8936	−1.08329
$923$	21.9486	0.722446
$924$	0	0
$925$	0	0
$926$	−40.7735	−1.33990
$927$	0	0
$928$	−39.5210	−1.29734
$929$	−33.9108	−1.11258	−0.556288	−	0.830989i	$-0.687775\pi$
$929$	−33.9108	−1.11258	−0.556288	+	0.830989i	$0.687775\pi$
$930$	0	0
$931$	−28.5764	−0.936555
$932$	21.0133	0.688312
$933$	0	0
$934$	−65.8913	−2.15603
$935$	0	0
$936$	0	0
$937$	−16.1534	−0.527708	−0.263854	−	0.964563i	$-0.584994\pi$
$937$	−16.1534	−0.527708	−0.263854	+	0.964563i	$0.584994\pi$
$938$	1.34679	0.0439743
$939$	0	0
$940$	0	0
$941$	−32.2573	−1.05156	−0.525778	−	0.850622i	$-0.676226\pi$
$941$	−32.2573	−1.05156	−0.525778	+	0.850622i	$0.676226\pi$
$942$	0	0
$943$	−1.20625	−0.0392808
$944$	−26.2179	−0.853320
$945$	0	0
$946$	−8.28780	−0.269460
$947$	−20.9811	−0.681796	−0.340898	−	0.940100i	$-0.610731\pi$
$947$	−20.9811	−0.681796	−0.340898	+	0.940100i	$0.610731\pi$
$948$	0	0
$949$	63.2164	2.05209
$950$	0	0
$951$	0	0
$952$	−0.546757	−0.0177205
$953$	−19.0273	−0.616354	−0.308177	−	0.951329i	$-0.599719\pi$
$953$	−19.0273	−0.616354	−0.308177	+	0.951329i	$0.599719\pi$
$954$	0	0
$955$	0	0
$956$	−3.34838	−0.108294
$957$	0	0
$958$	−18.2505	−0.589647
$959$	−2.81885	−0.0910253
$960$	0	0
$961$	−23.7718	−0.766832
$962$	−133.540	−4.30550
$963$	0	0
$964$	50.1669	1.61577
$965$	0	0
$966$	0	0
$967$	−17.7570	−0.571027	−0.285513	−	0.958375i	$-0.592164\pi$
$967$	−17.7570	−0.571027	−0.285513	+	0.958375i	$0.592164\pi$
$968$	2.39999	0.0771386
$969$	0	0
$970$	0	0
$971$	19.0906	0.612648	0.306324	−	0.951927i	$-0.400901\pi$
$971$	19.0906	0.612648	0.306324	+	0.951927i	$0.400901\pi$
$972$	0	0
$973$	−4.67236	−0.149789
$974$	−81.3502	−2.60663
$975$	0	0
$976$	14.9764	0.479384
$977$	3.22285	0.103108	0.0515541	−	0.998670i	$-0.483583\pi$
$977$	3.22285	0.103108	0.0515541	+	0.998670i	$0.483583\pi$
$978$	0	0
$979$	10.3684	0.331374
$980$	0	0
$981$	0	0
$982$	−72.5530	−2.31526
$983$	−35.6008	−1.13549	−0.567744	−	0.823205i	$-0.692184\pi$
$983$	−35.6008	−1.13549	−0.567744	+	0.823205i	$0.692184\pi$
$984$	0	0
$985$	0	0
$986$	−47.0139	−1.49723
$987$	0	0
$988$	51.7901	1.64766
$989$	−49.7096	−1.58067
$990$	0	0
$991$	−17.0654	−0.542100	−0.271050	−	0.962565i	$-0.587371\pi$
$991$	−17.0654	−0.542100	−0.271050	+	0.962565i	$0.587371\pi$
$992$	21.7395	0.690230
$993$	0	0
$994$	3.88933	0.123362
$995$	0	0
$996$	0	0
$997$	−36.5010	−1.15600	−0.577998	−	0.816038i	$-0.696166\pi$
$997$	−36.5010	−1.15600	−0.577998	+	0.816038i	$0.696166\pi$
$998$	−30.3359	−0.960265
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.r.1.2		6
3.2	odd	2		1875.2.a.i.1.5	✓	6
5.4	even	2		5625.2.a.o.1.5		6
15.2	even	4		1875.2.b.e.1249.9		12
15.8	even	4		1875.2.b.e.1249.4		12
15.14	odd	2		1875.2.a.l.1.2	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.i.1.5	✓	6	3.2	odd	2
1875.2.a.l.1.2	yes	6	15.14	odd	2
1875.2.b.e.1249.4		12	15.8	even	4
1875.2.b.e.1249.9		12	15.2	even	4
5625.2.a.o.1.5		6	5.4	even	2
5625.2.a.r.1.2		6	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-2.02791 q^{2} +2.11242 q^{4} -0.505614 q^{7} -0.227977 q^{8} +O(q^{10})\)	\(q-2.02791 q^{2} +2.11242 q^{4} -0.505614 q^{7} -0.227977 q^{8} -0.687513 q^{11} +5.78627 q^{13} +1.02534 q^{14} -3.76252 q^{16} -4.74333 q^{17} +4.23709 q^{19} +1.39422 q^{22} +8.36239 q^{23} -11.7340 q^{26} -1.06807 q^{28} -4.88758 q^{29} +2.68853 q^{31} +8.08601 q^{32} +9.61905 q^{34} +11.3806 q^{37} -8.59244 q^{38} -0.144247 q^{41} -5.94442 q^{43} -1.45232 q^{44} -16.9582 q^{46} -6.10650 q^{47} -6.74435 q^{49} +12.2230 q^{52} +10.8398 q^{53} +0.115269 q^{56} +9.91157 q^{58} +6.96817 q^{59} -3.98042 q^{61} -5.45211 q^{62} -8.87266 q^{64} +1.31351 q^{67} -10.0199 q^{68} +3.79321 q^{71} +10.9252 q^{73} -23.0787 q^{74} +8.95051 q^{76} +0.347616 q^{77} -1.89869 q^{79} +0.292519 q^{82} +2.51849 q^{83} +12.0548 q^{86} +0.156737 q^{88} -15.0809 q^{89} -2.92562 q^{91} +17.6649 q^{92} +12.3834 q^{94} -1.17394 q^{97} +13.6769 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{2} + 11 q^{4} + 2 q^{7} + 6 q^{8}+O(q^{10}) \) 6 * q + q^2 + 11 * q^4 + 2 * q^7 + 6 * q^8	\( 6 q + q^{2} + 11 q^{4} + 2 q^{7} + 6 q^{8} + 4 q^{14} + 17 q^{16} + 2 q^{17} - 2 q^{19} - 9 q^{22} - q^{23} - 37 q^{26} + 44 q^{28} - 31 q^{29} - 2 q^{31} + 33 q^{32} + 37 q^{34} + 22 q^{37} + 27 q^{38} - 33 q^{41} + 3 q^{43} + 11 q^{44} - 12 q^{46} - 6 q^{47} + 4 q^{49} + 33 q^{52} + 14 q^{53} + 30 q^{56} + q^{58} + 8 q^{59} + 34 q^{61} + 31 q^{62} + 12 q^{64} - 2 q^{67} + 27 q^{68} + 3 q^{71} + 36 q^{73} - 36 q^{74} + 27 q^{76} + 16 q^{77} + 25 q^{79} - 36 q^{82} - 12 q^{83} + 30 q^{86} - 56 q^{88} - 18 q^{89} + 28 q^{91} + 3 q^{92} - 50 q^{94} - 7 q^{97} + 15 q^{98}+O(q^{100}) \) 6 * q + q^2 + 11 * q^4 + 2 * q^7 + 6 * q^8 + 4 * q^14 + 17 * q^16 + 2 * q^17 - 2 * q^19 - 9 * q^22 - q^23 - 37 * q^26 + 44 * q^28 - 31 * q^29 - 2 * q^31 + 33 * q^32 + 37 * q^34 + 22 * q^37 + 27 * q^38 - 33 * q^41 + 3 * q^43 + 11 * q^44 - 12 * q^46 - 6 * q^47 + 4 * q^49 + 33 * q^52 + 14 * q^53 + 30 * q^56 + q^58 + 8 * q^59 + 34 * q^61 + 31 * q^62 + 12 * q^64 - 2 * q^67 + 27 * q^68 + 3 * q^71 + 36 * q^73 - 36 * q^74 + 27 * q^76 + 16 * q^77 + 25 * q^79 - 36 * q^82 - 12 * q^83 + 30 * q^86 - 56 * q^88 - 18 * q^89 + 28 * q^91 + 3 * q^92 - 50 * q^94 - 7 * q^97 + 15 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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