LMFDB - Embedded newform 5625.2.a.bc.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:	$1$
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			$-1.52260$ 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)

$f(q)$	$=$	$q-1.52260 q^{2} +0.318310 q^{4} +0.990985 q^{7} +2.56054 q^{8} +O(q^{10})$	$q-1.52260 q^{2} +0.318310 q^{4} +0.990985 q^{7} +2.56054 q^{8} -5.97349 q^{11} -4.02992 q^{13} -1.50887 q^{14} -4.53530 q^{16} +0.476176 q^{17} +2.82322 q^{19} +9.09523 q^{22} -1.74696 q^{23} +6.13596 q^{26} +0.315440 q^{28} +1.41641 q^{29} +8.76141 q^{31} +1.78436 q^{32} -0.725025 q^{34} +8.06031 q^{37} -4.29864 q^{38} +5.50296 q^{41} +6.82255 q^{43} -1.90142 q^{44} +2.65993 q^{46} -9.62845 q^{47} -6.01795 q^{49} -1.28276 q^{52} +6.57994 q^{53} +2.53746 q^{56} -2.15662 q^{58} -13.0760 q^{59} +12.2013 q^{61} -13.3401 q^{62} +6.35373 q^{64} -11.3195 q^{67} +0.151571 q^{68} +5.43047 q^{71} +4.08935 q^{73} -12.2726 q^{74} +0.898661 q^{76} -5.91964 q^{77} +15.0070 q^{79} -8.37881 q^{82} -2.29898 q^{83} -10.3880 q^{86} -15.2954 q^{88} -6.26920 q^{89} -3.99359 q^{91} -0.556076 q^{92} +14.6603 q^{94} -13.2541 q^{97} +9.16293 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.52260	−1.07664	−0.538320	−	0.842740i	$-0.680941\pi$
$2$	−1.52260	−1.07664	−0.538320	+	0.842740i	$0.680941\pi$
$3$	0	0
$3$	0	0
$4$	0.318310	0.159155
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.990985	0.374557	0.187279	−	0.982307i	$-0.440033\pi$
$7$	0.990985	0.374557	0.187279	+	0.982307i	$0.440033\pi$
$8$	2.56054	0.905288
$9$	0	0
$10$	0	0
$11$	−5.97349	−1.80107	−0.900537	−	0.434779i	$-0.856826\pi$
$11$	−5.97349	−1.80107	−0.900537	+	0.434779i	$0.856826\pi$
$12$	0	0
$13$	−4.02992	−1.11770	−0.558850	−	0.829269i	$-0.688757\pi$
$13$	−4.02992	−1.11770	−0.558850	+	0.829269i	$0.688757\pi$
$14$	−1.50887	−0.403263
$15$	0	0
$16$	−4.53530	−1.13382
$17$	0.476176	0.115490	0.0577448	−	0.998331i	$-0.481609\pi$
$17$	0.476176	0.115490	0.0577448	+	0.998331i	$0.481609\pi$
$18$	0	0
$19$	2.82322	0.647692	0.323846	−	0.946110i	$-0.395024\pi$
$19$	2.82322	0.647692	0.323846	+	0.946110i	$0.395024\pi$
$20$	0	0
$21$	0	0
$22$	9.09523	1.93911
$23$	−1.74696	−0.364267	−0.182134	−	0.983274i	$-0.558300\pi$
$23$	−1.74696	−0.364267	−0.182134	+	0.983274i	$0.558300\pi$
$24$	0	0
$25$	0	0
$26$	6.13596	1.20336
$27$	0	0
$28$	0.315440	0.0596127
$29$	1.41641	0.263020	0.131510	−	0.991315i	$-0.458017\pi$
$29$	1.41641	0.263020	0.131510	+	0.991315i	$0.458017\pi$
$30$	0	0
$31$	8.76141	1.57360	0.786798	−	0.617211i	$-0.211737\pi$
$31$	8.76141	1.57360	0.786798	+	0.617211i	$0.211737\pi$
$32$	1.78436	0.315434
$33$	0	0
$34$	−0.725025	−0.124341
$35$	0	0
$36$	0	0
$37$	8.06031	1.32511	0.662553	−	0.749015i	$-0.269473\pi$
$37$	8.06031	1.32511	0.662553	+	0.749015i	$0.269473\pi$
$38$	−4.29864	−0.697332
$39$	0	0
$40$	0	0
$41$	5.50296	0.859418	0.429709	−	0.902967i	$-0.358616\pi$
$41$	5.50296	0.859418	0.429709	+	0.902967i	$0.358616\pi$
$42$	0	0
$43$	6.82255	1.04043	0.520214	−	0.854036i	$-0.325852\pi$
$43$	6.82255	1.04043	0.520214	+	0.854036i	$0.325852\pi$
$44$	−1.90142	−0.286650
$45$	0	0
$46$	2.65993	0.392185
$47$	−9.62845	−1.40445	−0.702227	−	0.711953i	$-0.747811\pi$
$47$	−9.62845	−1.40445	−0.702227	+	0.711953i	$0.747811\pi$
$48$	0	0
$49$	−6.01795	−0.859707
$50$	0	0
$51$	0	0
$52$	−1.28276	−0.177887
$53$	6.57994	0.903824	0.451912	−	0.892063i	$-0.350742\pi$
$53$	6.57994	0.903824	0.451912	+	0.892063i	$0.350742\pi$
$54$	0	0
$55$	0	0
$56$	2.53746	0.339082
$57$	0	0
$58$	−2.15662	−0.283179
$59$	−13.0760	−1.70235	−0.851177	−	0.524879i	$-0.824111\pi$
$59$	−13.0760	−1.70235	−0.851177	+	0.524879i	$0.824111\pi$
$60$	0	0
$61$	12.2013	1.56222	0.781108	−	0.624397i	$-0.214655\pi$
$61$	12.2013	1.56222	0.781108	+	0.624397i	$0.214655\pi$
$62$	−13.3401	−1.69420
$63$	0	0
$64$	6.35373	0.794216
$65$	0	0
$66$	0	0
$67$	−11.3195	−1.38289	−0.691445	−	0.722429i	$-0.743026\pi$
$67$	−11.3195	−1.38289	−0.691445	+	0.722429i	$0.743026\pi$
$68$	0.151571	0.0183807
$69$	0	0
$70$	0	0
$71$	5.43047	0.644478	0.322239	−	0.946658i	$-0.395564\pi$
$71$	5.43047	0.644478	0.322239	+	0.946658i	$0.395564\pi$
$72$	0	0
$73$	4.08935	0.478622	0.239311	−	0.970943i	$-0.423078\pi$
$73$	4.08935	0.478622	0.239311	+	0.970943i	$0.423078\pi$
$74$	−12.2726	−1.42666
$75$	0	0
$76$	0.898661	0.103083
$77$	−5.91964	−0.674605
$78$	0	0
$79$	15.0070	1.68842	0.844212	−	0.536009i	$-0.180069\pi$
$79$	15.0070	1.68842	0.844212	+	0.536009i	$0.180069\pi$
$80$	0	0
$81$	0	0
$82$	−8.37881	−0.925284
$83$	−2.29898	−0.252346	−0.126173	−	0.992008i	$-0.540269\pi$
$83$	−2.29898	−0.252346	−0.126173	+	0.992008i	$0.540269\pi$
$84$	0	0
$85$	0	0
$86$	−10.3880	−1.12017
$87$	0	0
$88$	−15.2954	−1.63049
$89$	−6.26920	−0.664533	−0.332267	−	0.943185i	$-0.607813\pi$
$89$	−6.26920	−0.664533	−0.332267	+	0.943185i	$0.607813\pi$
$90$	0	0
$91$	−3.99359	−0.418642
$92$	−0.556076	−0.0579749
$93$	0	0
$94$	14.6603	1.51209
$95$	0	0
$96$	0	0
$97$	−13.2541	−1.34575	−0.672875	−	0.739756i	$-0.734941\pi$
$97$	−13.2541	−1.34575	−0.672875	+	0.739756i	$0.734941\pi$
$98$	9.16293	0.925595
$99$	0	0
$100$	0	0
$101$	6.41452	0.638269	0.319134	−	0.947709i	$-0.396608\pi$
$101$	6.41452	0.638269	0.319134	+	0.947709i	$0.396608\pi$
$102$	0	0
$103$	−9.38309	−0.924543	−0.462272	−	0.886738i	$-0.652966\pi$
$103$	−9.38309	−0.924543	−0.462272	+	0.886738i	$0.652966\pi$
$104$	−10.3188	−1.01184
$105$	0	0
$106$	−10.0186	−0.973094
$107$	−13.4108	−1.29647	−0.648234	−	0.761441i	$-0.724492\pi$
$107$	−13.4108	−1.29647	−0.648234	+	0.761441i	$0.724492\pi$
$108$	0	0
$109$	2.81573	0.269698	0.134849	−	0.990866i	$-0.456945\pi$
$109$	2.81573	0.269698	0.134849	+	0.990866i	$0.456945\pi$
$110$	0	0
$111$	0	0
$112$	−4.49441	−0.424682
$113$	−19.4252	−1.82737	−0.913685	−	0.406423i	$-0.866776\pi$
$113$	−19.4252	−1.82737	−0.913685	+	0.406423i	$0.866776\pi$
$114$	0	0
$115$	0	0
$116$	0.450857	0.0418610
$117$	0	0
$118$	19.9096	1.83282
$119$	0.471883	0.0432574
$120$	0	0
$121$	24.6825	2.24387
$122$	−18.5777	−1.68194
$123$	0	0
$124$	2.78884	0.250446
$125$	0	0
$126$	0	0
$127$	−6.42366	−0.570008	−0.285004	−	0.958526i	$-0.591995\pi$
$127$	−6.42366	−0.570008	−0.285004	+	0.958526i	$0.591995\pi$
$128$	−13.2429	−1.17052
$129$	0	0
$130$	0	0
$131$	7.30057	0.637853	0.318927	−	0.947779i	$-0.396678\pi$
$131$	7.30057	0.637853	0.318927	+	0.947779i	$0.396678\pi$
$132$	0	0
$133$	2.79777	0.242598
$134$	17.2350	1.48888
$135$	0	0
$136$	1.21927	0.104551
$137$	−7.84860	−0.670552	−0.335276	−	0.942120i	$-0.608829\pi$
$137$	−7.84860	−0.670552	−0.335276	+	0.942120i	$0.608829\pi$
$138$	0	0
$139$	−17.0514	−1.44628	−0.723138	−	0.690703i	$-0.757301\pi$
$139$	−17.0514	−1.44628	−0.723138	+	0.690703i	$0.757301\pi$
$140$	0	0
$141$	0	0
$142$	−8.26844	−0.693872
$143$	24.0727	2.01306
$144$	0	0
$145$	0	0
$146$	−6.22644	−0.515304
$147$	0	0
$148$	2.56568	0.210897
$149$	−2.84365	−0.232961	−0.116480	−	0.993193i	$-0.537161\pi$
$149$	−2.84365	−0.232961	−0.116480	+	0.993193i	$0.537161\pi$
$150$	0	0
$151$	11.5744	0.941915	0.470958	−	0.882156i	$-0.343908\pi$
$151$	11.5744	0.941915	0.470958	+	0.882156i	$0.343908\pi$
$152$	7.22898	0.586348
$153$	0	0
$154$	9.01324	0.726307
$155$	0	0
$156$	0	0
$157$	−4.77270	−0.380903	−0.190452	−	0.981697i	$-0.560995\pi$
$157$	−4.77270	−0.380903	−0.190452	+	0.981697i	$0.560995\pi$
$158$	−22.8497	−1.81783
$159$	0	0
$160$	0	0
$161$	−1.73122	−0.136439
$162$	0	0
$163$	−2.98657	−0.233926	−0.116963	−	0.993136i	$-0.537316\pi$
$163$	−2.98657	−0.233926	−0.116963	+	0.993136i	$0.537316\pi$
$164$	1.75165	0.136781
$165$	0	0
$166$	3.50043	0.271686
$167$	18.6105	1.44012	0.720060	−	0.693912i	$-0.244114\pi$
$167$	18.6105	1.44012	0.720060	+	0.693912i	$0.244114\pi$
$168$	0	0
$169$	3.24028	0.249252
$170$	0	0
$171$	0	0
$172$	2.17169	0.165589
$173$	16.3578	1.24366	0.621829	−	0.783153i	$-0.286390\pi$
$173$	16.3578	1.24366	0.621829	+	0.783153i	$0.286390\pi$
$174$	0	0
$175$	0	0
$176$	27.0915	2.04210
$177$	0	0
$178$	9.54548	0.715464
$179$	15.2266	1.13809	0.569045	−	0.822307i	$-0.307313\pi$
$179$	15.2266	1.13809	0.569045	+	0.822307i	$0.307313\pi$
$180$	0	0
$181$	−3.44321	−0.255932	−0.127966	−	0.991779i	$-0.540845\pi$
$181$	−3.44321	−0.255932	−0.127966	+	0.991779i	$0.540845\pi$
$182$	6.08064	0.450727
$183$	0	0
$184$	−4.47317	−0.329767
$185$	0	0
$186$	0	0
$187$	−2.84443	−0.208005
$188$	−3.06483	−0.223526
$189$	0	0
$190$	0	0
$191$	−15.1752	−1.09804	−0.549020	−	0.835809i	$-0.684999\pi$
$191$	−15.1752	−1.09804	−0.549020	+	0.835809i	$0.684999\pi$
$192$	0	0
$193$	−14.0304	−1.00993	−0.504965	−	0.863140i	$-0.668495\pi$
$193$	−14.0304	−1.00993	−0.504965	+	0.863140i	$0.668495\pi$
$194$	20.1807	1.44889
$195$	0	0
$196$	−1.91557	−0.136827
$197$	23.7871	1.69476	0.847382	−	0.530984i	$-0.178177\pi$
$197$	23.7871	1.69476	0.847382	+	0.530984i	$0.178177\pi$
$198$	0	0
$199$	−15.9445	−1.13028	−0.565139	−	0.824996i	$-0.691177\pi$
$199$	−15.9445	−1.13028	−0.565139	+	0.824996i	$0.691177\pi$
$200$	0	0
$201$	0	0
$202$	−9.76675	−0.687186
$203$	1.40364	0.0985162
$204$	0	0
$205$	0	0
$206$	14.2867	0.995401
$207$	0	0
$208$	18.2769	1.26728
$209$	−16.8645	−1.16654
$210$	0	0
$211$	−6.49016	−0.446801	−0.223401	−	0.974727i	$-0.571716\pi$
$211$	−6.49016	−0.446801	−0.223401	+	0.974727i	$0.571716\pi$
$212$	2.09446	0.143848
$213$	0	0
$214$	20.4192	1.39583
$215$	0	0
$216$	0	0
$217$	8.68243	0.589402
$218$	−4.28723	−0.290368
$219$	0	0
$220$	0	0
$221$	−1.91895	−0.129083
$222$	0	0
$223$	−10.7454	−0.719562	−0.359781	−	0.933037i	$-0.617149\pi$
$223$	−10.7454	−0.719562	−0.359781	+	0.933037i	$0.617149\pi$
$224$	1.76828	0.118148
$225$	0	0
$226$	29.5768	1.96742
$227$	−6.84922	−0.454599	−0.227299	−	0.973825i	$-0.572990\pi$
$227$	−6.84922	−0.454599	−0.227299	+	0.973825i	$0.572990\pi$
$228$	0	0
$229$	2.21379	0.146291	0.0731457	−	0.997321i	$-0.476696\pi$
$229$	2.21379	0.146291	0.0731457	+	0.997321i	$0.476696\pi$
$230$	0	0
$231$	0	0
$232$	3.62677	0.238109
$233$	−9.39682	−0.615606	−0.307803	−	0.951450i	$-0.599594\pi$
$233$	−9.39682	−0.615606	−0.307803	+	0.951450i	$0.599594\pi$
$234$	0	0
$235$	0	0
$236$	−4.16223	−0.270938
$237$	0	0
$238$	−0.718489	−0.0465727
$239$	0.737315	0.0476930	0.0238465	−	0.999716i	$-0.492409\pi$
$239$	0.737315	0.0476930	0.0238465	+	0.999716i	$0.492409\pi$
$240$	0	0
$241$	−4.21895	−0.271767	−0.135883	−	0.990725i	$-0.543387\pi$
$241$	−4.21895	−0.271767	−0.135883	+	0.990725i	$0.543387\pi$
$242$	−37.5816	−2.41584
$243$	0	0
$244$	3.88379	0.248634
$245$	0	0
$246$	0	0
$247$	−11.3774	−0.723925
$248$	22.4339	1.42456
$249$	0	0
$250$	0	0
$251$	9.82121	0.619909	0.309955	−	0.950751i	$-0.399686\pi$
$251$	9.82121	0.619909	0.309955	+	0.950751i	$0.399686\pi$
$252$	0	0
$253$	10.4355	0.656072
$254$	9.78067	0.613694
$255$	0	0
$256$	7.45620	0.466012
$257$	0.0383881	0.00239458	0.00119729	−	0.999999i	$-0.499619\pi$
$257$	0.0383881	0.00239458	0.00119729	+	0.999999i	$0.499619\pi$
$258$	0	0
$259$	7.98764	0.496328
$260$	0	0
$261$	0	0
$262$	−11.1158	−0.686739
$263$	−14.2706	−0.879965	−0.439982	−	0.898006i	$-0.645015\pi$
$263$	−14.2706	−0.879965	−0.439982	+	0.898006i	$0.645015\pi$
$264$	0	0
$265$	0	0
$266$	−4.25989	−0.261191
$267$	0	0
$268$	−3.60309	−0.220094
$269$	−21.4090	−1.30533	−0.652666	−	0.757646i	$-0.726349\pi$
$269$	−21.4090	−1.30533	−0.652666	+	0.757646i	$0.726349\pi$
$270$	0	0
$271$	6.22804	0.378327	0.189163	−	0.981946i	$-0.439422\pi$
$271$	6.22804	0.378327	0.189163	+	0.981946i	$0.439422\pi$
$272$	−2.15960	−0.130945
$273$	0	0
$274$	11.9503	0.721943
$275$	0	0
$276$	0	0
$277$	14.0431	0.843768	0.421884	−	0.906650i	$-0.361369\pi$
$277$	14.0431	0.843768	0.421884	+	0.906650i	$0.361369\pi$
$278$	25.9624	1.55712
$279$	0	0
$280$	0	0
$281$	0.516993	0.0308412	0.0154206	−	0.999881i	$-0.495091\pi$
$281$	0.516993	0.0308412	0.0154206	+	0.999881i	$0.495091\pi$
$282$	0	0
$283$	−0.977154	−0.0580858	−0.0290429	−	0.999578i	$-0.509246\pi$
$283$	−0.977154	−0.0580858	−0.0290429	+	0.999578i	$0.509246\pi$
$284$	1.72857	0.102572
$285$	0	0
$286$	−36.6531	−2.16734
$287$	5.45335	0.321901
$288$	0	0
$289$	−16.7733	−0.986662
$290$	0	0
$291$	0	0
$292$	1.30168	0.0761751
$293$	2.75561	0.160984	0.0804922	−	0.996755i	$-0.474351\pi$
$293$	2.75561	0.160984	0.0804922	+	0.996755i	$0.474351\pi$
$294$	0	0
$295$	0	0
$296$	20.6387	1.19960
$297$	0	0
$298$	4.32974	0.250815
$299$	7.04013	0.407141
$300$	0	0
$301$	6.76104	0.389700
$302$	−17.6233	−1.01410
$303$	0	0
$304$	−12.8042	−0.734369
$305$	0	0
$306$	0	0
$307$	−16.1940	−0.924241	−0.462120	−	0.886817i	$-0.652911\pi$
$307$	−16.1940	−0.924241	−0.462120	+	0.886817i	$0.652911\pi$
$308$	−1.88428	−0.107367
$309$	0	0
$310$	0	0
$311$	−8.20094	−0.465033	−0.232516	−	0.972593i	$-0.574696\pi$
$311$	−8.20094	−0.465033	−0.232516	+	0.972593i	$0.574696\pi$
$312$	0	0
$313$	−20.0065	−1.13083	−0.565416	−	0.824806i	$-0.691284\pi$
$313$	−20.0065	−1.13083	−0.565416	+	0.824806i	$0.691284\pi$
$314$	7.26691	0.410096
$315$	0	0
$316$	4.77689	0.268721
$317$	−6.52708	−0.366597	−0.183299	−	0.983057i	$-0.558678\pi$
$317$	−6.52708	−0.366597	−0.183299	+	0.983057i	$0.558678\pi$
$318$	0	0
$319$	−8.46090	−0.473719
$320$	0	0
$321$	0	0
$322$	2.63595	0.146896
$323$	1.34435	0.0748017
$324$	0	0
$325$	0	0
$326$	4.54735	0.251854
$327$	0	0
$328$	14.0906	0.778021
$329$	−9.54165	−0.526048
$330$	0	0
$331$	33.2523	1.82771	0.913857	−	0.406037i	$-0.133090\pi$
$331$	33.2523	1.82771	0.913857	+	0.406037i	$0.133090\pi$
$332$	−0.731789	−0.0401621
$333$	0	0
$334$	−28.3363	−1.55049
$335$	0	0
$336$	0	0
$337$	−16.9653	−0.924160	−0.462080	−	0.886838i	$-0.652897\pi$
$337$	−16.9653	−0.924160	−0.462080	+	0.886838i	$0.652897\pi$
$338$	−4.93365	−0.268355
$339$	0	0
$340$	0	0
$341$	−52.3362	−2.83416
$342$	0	0
$343$	−12.9006	−0.696567
$344$	17.4694	0.941888
$345$	0	0
$346$	−24.9063	−1.33897
$347$	−20.1449	−1.08144	−0.540719	−	0.841203i	$-0.681848\pi$
$347$	−20.1449	−1.08144	−0.540719	+	0.841203i	$0.681848\pi$
$348$	0	0
$349$	−29.8420	−1.59741	−0.798703	−	0.601725i	$-0.794480\pi$
$349$	−29.8420	−1.59741	−0.798703	+	0.601725i	$0.794480\pi$
$350$	0	0
$351$	0	0
$352$	−10.6589	−0.568120
$353$	0.982924	0.0523157	0.0261579	−	0.999658i	$-0.491673\pi$
$353$	0.982924	0.0523157	0.0261579	+	0.999658i	$0.491673\pi$
$354$	0	0
$355$	0	0
$356$	−1.99555	−0.105764
$357$	0	0
$358$	−23.1840	−1.22531
$359$	33.4999	1.76806	0.884030	−	0.467431i	$-0.154820\pi$
$359$	33.4999	1.76806	0.884030	+	0.467431i	$0.154820\pi$
$360$	0	0
$361$	−11.0294	−0.580495
$362$	5.24263	0.275546
$363$	0	0
$364$	−1.27120	−0.0666290
$365$	0	0
$366$	0	0
$367$	−0.352799	−0.0184160	−0.00920798	−	0.999958i	$-0.502931\pi$
$367$	−0.352799	−0.0184160	−0.00920798	+	0.999958i	$0.502931\pi$
$368$	7.92300	0.413015
$369$	0	0
$370$	0	0
$371$	6.52062	0.338534
$372$	0	0
$373$	24.2811	1.25723	0.628614	−	0.777718i	$-0.283623\pi$
$373$	24.2811	1.25723	0.628614	+	0.777718i	$0.283623\pi$
$374$	4.33093	0.223947
$375$	0	0
$376$	−24.6540	−1.27144
$377$	−5.70802	−0.293978
$378$	0	0
$379$	−36.4446	−1.87203	−0.936017	−	0.351956i	$-0.885517\pi$
$379$	−36.4446	−1.87203	−0.936017	+	0.351956i	$0.885517\pi$
$380$	0	0
$381$	0	0
$382$	23.1058	1.18219
$383$	6.63250	0.338905	0.169453	−	0.985538i	$-0.445800\pi$
$383$	6.63250	0.338905	0.169453	+	0.985538i	$0.445800\pi$
$384$	0	0
$385$	0	0
$386$	21.3627	1.08733
$387$	0	0
$388$	−4.21891	−0.214183
$389$	−2.94097	−0.149113	−0.0745566	−	0.997217i	$-0.523754\pi$
$389$	−2.94097	−0.149113	−0.0745566	+	0.997217i	$0.523754\pi$
$390$	0	0
$391$	−0.831862	−0.0420690
$392$	−15.4092	−0.778282
$393$	0	0
$394$	−36.2183	−1.82465
$395$	0	0
$396$	0	0
$397$	9.44435	0.473998	0.236999	−	0.971510i	$-0.423836\pi$
$397$	9.44435	0.473998	0.236999	+	0.971510i	$0.423836\pi$
$398$	24.2771	1.21690
$399$	0	0
$400$	0	0
$401$	−24.3968	−1.21832	−0.609159	−	0.793048i	$-0.708493\pi$
$401$	−24.3968	−1.21832	−0.609159	+	0.793048i	$0.708493\pi$
$402$	0	0
$403$	−35.3078	−1.75881
$404$	2.04181	0.101584
$405$	0	0
$406$	−2.13718	−0.106067
$407$	−48.1481	−2.38661
$408$	0	0
$409$	13.6073	0.672836	0.336418	−	0.941713i	$-0.390784\pi$
$409$	13.6073	0.672836	0.336418	+	0.941713i	$0.390784\pi$
$410$	0	0
$411$	0	0
$412$	−2.98673	−0.147146
$413$	−12.9582	−0.637629
$414$	0	0
$415$	0	0
$416$	−7.19085	−0.352560
$417$	0	0
$418$	25.6779	1.25595
$419$	−25.0654	−1.22452	−0.612262	−	0.790655i	$-0.709740\pi$
$419$	−25.0654	−1.22452	−0.612262	+	0.790655i	$0.709740\pi$
$420$	0	0
$421$	−5.56078	−0.271016	−0.135508	−	0.990776i	$-0.543267\pi$
$421$	−5.56078	−0.271016	−0.135508	+	0.990776i	$0.543267\pi$
$422$	9.88192	0.481044
$423$	0	0
$424$	16.8482	0.818221
$425$	0	0
$426$	0	0
$427$	12.0913	0.585139
$428$	−4.26878	−0.206339
$429$	0	0
$430$	0	0
$431$	−24.2599	−1.16856	−0.584279	−	0.811553i	$-0.698623\pi$
$431$	−24.2599	−1.16856	−0.584279	+	0.811553i	$0.698623\pi$
$432$	0	0
$433$	−38.8459	−1.86681	−0.933407	−	0.358821i	$-0.883179\pi$
$433$	−38.8459	−1.86681	−0.933407	+	0.358821i	$0.883179\pi$
$434$	−13.2199	−0.634574
$435$	0	0
$436$	0.896274	0.0429238
$437$	−4.93207	−0.235933
$438$	0	0
$439$	−9.22879	−0.440466	−0.220233	−	0.975447i	$-0.570682\pi$
$439$	−9.22879	−0.440466	−0.220233	+	0.975447i	$0.570682\pi$
$440$	0	0
$441$	0	0
$442$	2.92179	0.138976
$443$	−5.30591	−0.252091	−0.126046	−	0.992024i	$-0.540229\pi$
$443$	−5.30591	−0.252091	−0.126046	+	0.992024i	$0.540229\pi$
$444$	0	0
$445$	0	0
$446$	16.3609	0.774710
$447$	0	0
$448$	6.29645	0.297479
$449$	22.2199	1.04862	0.524311	−	0.851527i	$-0.324323\pi$
$449$	22.2199	1.04862	0.524311	+	0.851527i	$0.324323\pi$
$450$	0	0
$451$	−32.8719	−1.54788
$452$	−6.18324	−0.290835
$453$	0	0
$454$	10.4286	0.489440
$455$	0	0
$456$	0	0
$457$	6.77157	0.316761	0.158380	−	0.987378i	$-0.449373\pi$
$457$	6.77157	0.316761	0.158380	+	0.987378i	$0.449373\pi$
$458$	−3.37072	−0.157503
$459$	0	0
$460$	0	0
$461$	−33.9340	−1.58046	−0.790231	−	0.612809i	$-0.790040\pi$
$461$	−33.9340	−1.58046	−0.790231	+	0.612809i	$0.790040\pi$
$462$	0	0
$463$	28.8896	1.34262	0.671308	−	0.741179i	$-0.265733\pi$
$463$	28.8896	1.34262	0.671308	+	0.741179i	$0.265733\pi$
$464$	−6.42384	−0.298219
$465$	0	0
$466$	14.3076	0.662786
$467$	−31.2100	−1.44423	−0.722113	−	0.691776i	$-0.756829\pi$
$467$	−31.2100	−1.44423	−0.722113	+	0.691776i	$0.756829\pi$
$468$	0	0
$469$	−11.2174	−0.517972
$470$	0	0
$471$	0	0
$472$	−33.4817	−1.54112
$473$	−40.7544	−1.87389
$474$	0	0
$475$	0	0
$476$	0.150205	0.00688464
$477$	0	0
$478$	−1.12264	−0.0513482
$479$	14.6580	0.669741	0.334870	−	0.942264i	$-0.391307\pi$
$479$	14.6580	0.669741	0.334870	+	0.942264i	$0.391307\pi$
$480$	0	0
$481$	−32.4824	−1.48107
$482$	6.42377	0.292595
$483$	0	0
$484$	7.85670	0.357123
$485$	0	0
$486$	0	0
$487$	−5.55868	−0.251888	−0.125944	−	0.992037i	$-0.540196\pi$
$487$	−5.55868	−0.251888	−0.125944	+	0.992037i	$0.540196\pi$
$488$	31.2419	1.41425
$489$	0	0
$490$	0	0
$491$	17.4593	0.787925	0.393962	−	0.919127i	$-0.371104\pi$
$491$	17.4593	0.787925	0.393962	+	0.919127i	$0.371104\pi$
$492$	0	0
$493$	0.674459	0.0303761
$494$	17.3232	0.779407
$495$	0	0
$496$	−39.7356	−1.78418
$497$	5.38152	0.241394
$498$	0	0
$499$	16.3254	0.730825	0.365412	−	0.930846i	$-0.380928\pi$
$499$	16.3254	0.730825	0.365412	+	0.930846i	$0.380928\pi$
$500$	0	0
$501$	0	0
$502$	−14.9538	−0.667420
$503$	−16.7612	−0.747345	−0.373672	−	0.927561i	$-0.621902\pi$
$503$	−16.7612	−0.747345	−0.373672	+	0.927561i	$0.621902\pi$
$504$	0	0
$505$	0	0
$506$	−15.8890	−0.706354
$507$	0	0
$508$	−2.04472	−0.0907196
$509$	−8.35851	−0.370485	−0.185242	−	0.982693i	$-0.559307\pi$
$509$	−8.35851	−0.370485	−0.185242	+	0.982693i	$0.559307\pi$
$510$	0	0
$511$	4.05248	0.179271
$512$	15.1330	0.668791
$513$	0	0
$514$	−0.0584497	−0.00257811
$515$	0	0
$516$	0	0
$517$	57.5154	2.52953
$518$	−12.1620	−0.534367
$519$	0	0
$520$	0	0
$521$	45.6262	1.99892	0.999460	−	0.0328599i	$-0.0104615\pi$
$521$	45.6262	1.99892	0.999460	+	0.0328599i	$0.0104615\pi$
$522$	0	0
$523$	26.6701	1.16620	0.583102	−	0.812399i	$-0.301839\pi$
$523$	26.6701	1.16620	0.583102	+	0.812399i	$0.301839\pi$
$524$	2.32384	0.101518
$525$	0	0
$526$	21.7285	0.947406
$527$	4.17197	0.181734
$528$	0	0
$529$	−19.9481	−0.867309
$530$	0	0
$531$	0	0
$532$	0.890559	0.0386106
$533$	−22.1765	−0.960571
$534$	0	0
$535$	0	0
$536$	−28.9839	−1.25191
$537$	0	0
$538$	32.5974	1.40537
$539$	35.9481	1.54840
$540$	0	0
$541$	−0.225005	−0.00967370	−0.00483685	−	0.999988i	$-0.501540\pi$
$541$	−0.225005	−0.00967370	−0.00483685	+	0.999988i	$0.501540\pi$
$542$	−9.48282	−0.407322
$543$	0	0
$544$	0.849670	0.0364293
$545$	0	0
$546$	0	0
$547$	−8.58474	−0.367057	−0.183529	−	0.983014i	$-0.558752\pi$
$547$	−8.58474	−0.367057	−0.183529	+	0.983014i	$0.558752\pi$
$548$	−2.49829	−0.106722
$549$	0	0
$550$	0	0
$551$	3.99884	0.170356
$552$	0	0
$553$	14.8718	0.632411
$554$	−21.3820	−0.908435
$555$	0	0
$556$	−5.42762	−0.230182
$557$	−20.6736	−0.875968	−0.437984	−	0.898983i	$-0.644307\pi$
$557$	−20.6736	−0.875968	−0.437984	+	0.898983i	$0.644307\pi$
$558$	0	0
$559$	−27.4943	−1.16289
$560$	0	0
$561$	0	0
$562$	−0.787174	−0.0332049
$563$	12.6162	0.531708	0.265854	−	0.964013i	$-0.414346\pi$
$563$	12.6162	0.531708	0.265854	+	0.964013i	$0.414346\pi$
$564$	0	0
$565$	0	0
$566$	1.48781	0.0625375
$567$	0	0
$568$	13.9049	0.583439
$569$	−10.9897	−0.460714	−0.230357	−	0.973106i	$-0.573989\pi$
$569$	−10.9897	−0.460714	−0.230357	+	0.973106i	$0.573989\pi$
$570$	0	0
$571$	0.276517	0.0115719	0.00578593	−	0.999983i	$-0.498158\pi$
$571$	0.276517	0.0115719	0.00578593	+	0.999983i	$0.498158\pi$
$572$	7.66258	0.320389
$573$	0	0
$574$	−8.30327	−0.346572
$575$	0	0
$576$	0	0
$577$	15.7460	0.655515	0.327757	−	0.944762i	$-0.393707\pi$
$577$	15.7460	0.655515	0.327757	+	0.944762i	$0.393707\pi$
$578$	25.5390	1.06228
$579$	0	0
$580$	0	0
$581$	−2.27826	−0.0945180
$582$	0	0
$583$	−39.3052	−1.62785
$584$	10.4709	0.433291
$585$	0	0
$586$	−4.19569	−0.173322
$587$	−11.3751	−0.469501	−0.234751	−	0.972056i	$-0.575427\pi$
$587$	−11.3751	−0.469501	−0.234751	+	0.972056i	$0.575427\pi$
$588$	0	0
$589$	24.7354	1.01921
$590$	0	0
$591$	0	0
$592$	−36.5559	−1.50244
$593$	−30.4617	−1.25091	−0.625457	−	0.780259i	$-0.715087\pi$
$593$	−30.4617	−1.25091	−0.625457	+	0.780259i	$0.715087\pi$
$594$	0	0
$595$	0	0
$596$	−0.905161	−0.0370769
$597$	0	0
$598$	−10.7193	−0.438345
$599$	−3.09493	−0.126455	−0.0632277	−	0.997999i	$-0.520139\pi$
$599$	−3.09493	−0.126455	−0.0632277	+	0.997999i	$0.520139\pi$
$600$	0	0
$601$	9.19481	0.375064	0.187532	−	0.982258i	$-0.439951\pi$
$601$	9.19481	0.375064	0.187532	+	0.982258i	$0.439951\pi$
$602$	−10.2944	−0.419567
$603$	0	0
$604$	3.68426	0.149911
$605$	0	0
$606$	0	0
$607$	−37.9938	−1.54212	−0.771060	−	0.636762i	$-0.780273\pi$
$607$	−37.9938	−1.54212	−0.771060	+	0.636762i	$0.780273\pi$
$608$	5.03766	0.204304
$609$	0	0
$610$	0	0
$611$	38.8019	1.56976
$612$	0	0
$613$	24.7188	0.998381	0.499191	−	0.866492i	$-0.333631\pi$
$613$	24.7188	0.998381	0.499191	+	0.866492i	$0.333631\pi$
$614$	24.6570	0.995075
$615$	0	0
$616$	−15.1575	−0.610712
$617$	29.8302	1.20092	0.600460	−	0.799655i	$-0.294984\pi$
$617$	29.8302	1.20092	0.600460	+	0.799655i	$0.294984\pi$
$618$	0	0
$619$	−20.9427	−0.841757	−0.420878	−	0.907117i	$-0.638278\pi$
$619$	−20.9427	−0.841757	−0.420878	+	0.907117i	$0.638278\pi$
$620$	0	0
$621$	0	0
$622$	12.4867	0.500673
$623$	−6.21268	−0.248906
$624$	0	0
$625$	0	0
$626$	30.4618	1.21750
$627$	0	0
$628$	−1.51920	−0.0606226
$629$	3.83812	0.153036
$630$	0	0
$631$	18.1871	0.724017	0.362009	−	0.932175i	$-0.382091\pi$
$631$	18.1871	0.724017	0.362009	+	0.932175i	$0.382091\pi$
$632$	38.4261	1.52851
$633$	0	0
$634$	9.93813	0.394693
$635$	0	0
$636$	0	0
$637$	24.2519	0.960894
$638$	12.8826	0.510025
$639$	0	0
$640$	0	0
$641$	−3.94506	−0.155821	−0.0779103	−	0.996960i	$-0.524825\pi$
$641$	−3.94506	−0.155821	−0.0779103	+	0.996960i	$0.524825\pi$
$642$	0	0
$643$	−23.1026	−0.911077	−0.455538	−	0.890216i	$-0.650553\pi$
$643$	−23.1026	−0.911077	−0.455538	+	0.890216i	$0.650553\pi$
$644$	−0.551063	−0.0217149
$645$	0	0
$646$	−2.04691	−0.0805345
$647$	−9.35622	−0.367831	−0.183916	−	0.982942i	$-0.558877\pi$
$647$	−9.35622	−0.367831	−0.183916	+	0.982942i	$0.558877\pi$
$648$	0	0
$649$	78.1095	3.06607
$650$	0	0
$651$	0	0
$652$	−0.950655	−0.0372305
$653$	2.29702	0.0898892	0.0449446	−	0.998989i	$-0.485689\pi$
$653$	2.29702	0.0898892	0.0449446	+	0.998989i	$0.485689\pi$
$654$	0	0
$655$	0	0
$656$	−24.9576	−0.974429
$657$	0	0
$658$	14.5281	0.566365
$659$	17.5619	0.684116	0.342058	−	0.939679i	$-0.388876\pi$
$659$	17.5619	0.684116	0.342058	+	0.939679i	$0.388876\pi$
$660$	0	0
$661$	−6.88170	−0.267667	−0.133834	−	0.991004i	$-0.542729\pi$
$661$	−6.88170	−0.267667	−0.133834	+	0.991004i	$0.542729\pi$
$662$	−50.6300	−1.96779
$663$	0	0
$664$	−5.88664	−0.228446
$665$	0	0
$666$	0	0
$667$	−2.47441	−0.0958097
$668$	5.92390	0.229202
$669$	0	0
$670$	0	0
$671$	−72.8842	−2.81367
$672$	0	0
$673$	12.4307	0.479168	0.239584	−	0.970876i	$-0.422989\pi$
$673$	12.4307	0.479168	0.239584	+	0.970876i	$0.422989\pi$
$674$	25.8314	0.994989
$675$	0	0
$676$	1.03141	0.0396697
$677$	−6.61696	−0.254310	−0.127155	−	0.991883i	$-0.540585\pi$
$677$	−6.61696	−0.254310	−0.127155	+	0.991883i	$0.540585\pi$
$678$	0	0
$679$	−13.1346	−0.504060
$680$	0	0
$681$	0	0
$682$	79.6870	3.05137
$683$	−18.6885	−0.715097	−0.357549	−	0.933895i	$-0.616387\pi$
$683$	−18.6885	−0.715097	−0.357549	+	0.933895i	$0.616387\pi$
$684$	0	0
$685$	0	0
$686$	19.6424	0.749952
$687$	0	0
$688$	−30.9423	−1.17966
$689$	−26.5166	−1.01020
$690$	0	0
$691$	−18.4352	−0.701306	−0.350653	−	0.936505i	$-0.614040\pi$
$691$	−18.4352	−0.701306	−0.350653	+	0.936505i	$0.614040\pi$
$692$	5.20684	0.197934
$693$	0	0
$694$	30.6727	1.16432
$695$	0	0
$696$	0	0
$697$	2.62038	0.0992538
$698$	45.4375	1.71983
$699$	0	0
$700$	0	0
$701$	−14.9443	−0.564437	−0.282219	−	0.959350i	$-0.591070\pi$
$701$	−14.9443	−0.564437	−0.282219	+	0.959350i	$0.591070\pi$
$702$	0	0
$703$	22.7561	0.858261
$704$	−37.9539	−1.43044
$705$	0	0
$706$	−1.49660	−0.0563253
$707$	6.35670	0.239068
$708$	0	0
$709$	44.4198	1.66822	0.834110	−	0.551598i	$-0.185982\pi$
$709$	44.4198	1.66822	0.834110	+	0.551598i	$0.185982\pi$
$710$	0	0
$711$	0	0
$712$	−16.0525	−0.601594
$713$	−15.3059	−0.573209
$714$	0	0
$715$	0	0
$716$	4.84678	0.181133
$717$	0	0
$718$	−51.0070	−1.90356
$719$	38.9409	1.45225	0.726125	−	0.687563i	$-0.241319\pi$
$719$	38.9409	1.45225	0.726125	+	0.687563i	$0.241319\pi$
$720$	0	0
$721$	−9.29850	−0.346294
$722$	16.7934	0.624984
$723$	0	0
$724$	−1.09601	−0.0407328
$725$	0	0
$726$	0	0
$727$	−2.48415	−0.0921320	−0.0460660	−	0.998938i	$-0.514668\pi$
$727$	−2.48415	−0.0921320	−0.0460660	+	0.998938i	$0.514668\pi$
$728$	−10.2258	−0.378992
$729$	0	0
$730$	0	0
$731$	3.24873	0.120159
$732$	0	0
$733$	31.7905	1.17421	0.587105	−	0.809511i	$-0.300267\pi$
$733$	31.7905	1.17421	0.587105	+	0.809511i	$0.300267\pi$
$734$	0.537172	0.0198274
$735$	0	0
$736$	−3.11722	−0.114902
$737$	67.6166	2.49069
$738$	0	0
$739$	−32.4392	−1.19329	−0.596647	−	0.802504i	$-0.703501\pi$
$739$	−32.4392	−1.19329	−0.596647	+	0.802504i	$0.703501\pi$
$740$	0	0
$741$	0	0
$742$	−9.92829	−0.364479
$743$	−2.02921	−0.0744447	−0.0372223	−	0.999307i	$-0.511851\pi$
$743$	−2.02921	−0.0744447	−0.0372223	+	0.999307i	$0.511851\pi$
$744$	0	0
$745$	0	0
$746$	−36.9704	−1.35358
$747$	0	0
$748$	−0.905410	−0.0331051
$749$	−13.2899	−0.485602
$750$	0	0
$751$	−31.8934	−1.16381	−0.581904	−	0.813257i	$-0.697692\pi$
$751$	−31.8934	−1.16381	−0.581904	+	0.813257i	$0.697692\pi$
$752$	43.6679	1.59241
$753$	0	0
$754$	8.69102	0.316508
$755$	0	0
$756$	0	0
$757$	−39.2724	−1.42738	−0.713689	−	0.700463i	$-0.752977\pi$
$757$	−39.2724	−1.42738	−0.713689	+	0.700463i	$0.752977\pi$
$758$	55.4905	2.01551
$759$	0	0
$760$	0	0
$761$	−14.2422	−0.516278	−0.258139	−	0.966108i	$-0.583109\pi$
$761$	−14.2422	−0.516278	−0.258139	+	0.966108i	$0.583109\pi$
$762$	0	0
$763$	2.79034	0.101017
$764$	−4.83042	−0.174758
$765$	0	0
$766$	−10.0987	−0.364879
$767$	52.6954	1.90272
$768$	0	0
$769$	−6.86409	−0.247526	−0.123763	−	0.992312i	$-0.539496\pi$
$769$	−6.86409	−0.247526	−0.123763	+	0.992312i	$0.539496\pi$
$770$	0	0
$771$	0	0
$772$	−4.46601	−0.160735
$773$	−8.46681	−0.304530	−0.152265	−	0.988340i	$-0.548657\pi$
$773$	−8.46681	−0.304530	−0.152265	+	0.988340i	$0.548657\pi$
$774$	0	0
$775$	0	0
$776$	−33.9377	−1.21829
$777$	0	0
$778$	4.47792	0.160541
$779$	15.5361	0.556638
$780$	0	0
$781$	−32.4389	−1.16075
$782$	1.26659	0.0452932
$783$	0	0
$784$	27.2932	0.974757
$785$	0	0
$786$	0	0
$787$	14.9960	0.534551	0.267275	−	0.963620i	$-0.413877\pi$
$787$	14.9960	0.534551	0.267275	+	0.963620i	$0.413877\pi$
$788$	7.57169	0.269730
$789$	0	0
$790$	0	0
$791$	−19.2501	−0.684455
$792$	0	0
$793$	−49.1703	−1.74609
$794$	−14.3800	−0.510326
$795$	0	0
$796$	−5.07530	−0.179889
$797$	−10.9127	−0.386549	−0.193274	−	0.981145i	$-0.561911\pi$
$797$	−10.9127	−0.386549	−0.193274	+	0.981145i	$0.561911\pi$
$798$	0	0
$799$	−4.58483	−0.162200
$800$	0	0
$801$	0	0
$802$	37.1465	1.31169
$803$	−24.4277	−0.862033
$804$	0	0
$805$	0	0
$806$	53.7597	1.89360
$807$	0	0
$808$	16.4246	0.577817
$809$	−13.8729	−0.487746	−0.243873	−	0.969807i	$-0.578418\pi$
$809$	−13.8729	−0.487746	−0.243873	+	0.969807i	$0.578418\pi$
$810$	0	0
$811$	13.1036	0.460129	0.230065	−	0.973175i	$-0.426106\pi$
$811$	13.1036	0.460129	0.230065	+	0.973175i	$0.426106\pi$
$812$	0.446793	0.0156793
$813$	0	0
$814$	73.3103	2.56953
$815$	0	0
$816$	0	0
$817$	19.2616	0.673878
$818$	−20.7184	−0.724403
$819$	0	0
$820$	0	0
$821$	−15.5589	−0.543011	−0.271505	−	0.962437i	$-0.587521\pi$
$821$	−15.5589	−0.543011	−0.271505	+	0.962437i	$0.587521\pi$
$822$	0	0
$823$	−45.2205	−1.57629	−0.788144	−	0.615491i	$-0.788958\pi$
$823$	−45.2205	−1.57629	−0.788144	+	0.615491i	$0.788958\pi$
$824$	−24.0258	−0.836978
$825$	0	0
$826$	19.7301	0.686497
$827$	−25.6349	−0.891411	−0.445706	−	0.895180i	$-0.647047\pi$
$827$	−25.6349	−0.891411	−0.445706	+	0.895180i	$0.647047\pi$
$828$	0	0
$829$	14.6302	0.508129	0.254064	−	0.967187i	$-0.418232\pi$
$829$	14.6302	0.508129	0.254064	+	0.967187i	$0.418232\pi$
$830$	0	0
$831$	0	0
$832$	−25.6050	−0.887695
$833$	−2.86560	−0.0992872
$834$	0	0
$835$	0	0
$836$	−5.36814	−0.185661
$837$	0	0
$838$	38.1645	1.31837
$839$	−37.4349	−1.29240	−0.646199	−	0.763169i	$-0.723642\pi$
$839$	−37.4349	−1.29240	−0.646199	+	0.763169i	$0.723642\pi$
$840$	0	0
$841$	−26.9938	−0.930820
$842$	8.46684	0.291787
$843$	0	0
$844$	−2.06588	−0.0711106
$845$	0	0
$846$	0	0
$847$	24.4600	0.840457
$848$	−29.8420	−1.02478
$849$	0	0
$850$	0	0
$851$	−14.0811	−0.482693
$852$	0	0
$853$	4.45661	0.152591	0.0762956	−	0.997085i	$-0.475691\pi$
$853$	4.45661	0.152591	0.0762956	+	0.997085i	$0.475691\pi$
$854$	−18.4102	−0.629984
$855$	0	0
$856$	−34.3388	−1.17368
$857$	48.0738	1.64217	0.821084	−	0.570807i	$-0.193370\pi$
$857$	48.0738	1.64217	0.821084	+	0.570807i	$0.193370\pi$
$858$	0	0
$859$	−40.5317	−1.38292	−0.691461	−	0.722414i	$-0.743033\pi$
$859$	−40.5317	−1.38292	−0.691461	+	0.722414i	$0.743033\pi$
$860$	0	0
$861$	0	0
$862$	36.9381	1.25812
$863$	7.99001	0.271983	0.135992	−	0.990710i	$-0.456578\pi$
$863$	7.99001	0.271983	0.135992	+	0.990710i	$0.456578\pi$
$864$	0	0
$865$	0	0
$866$	59.1467	2.00989
$867$	0	0
$868$	2.76370	0.0938062
$869$	−89.6444	−3.04098
$870$	0	0
$871$	45.6165	1.54566
$872$	7.20979	0.244154
$873$	0	0
$874$	7.50957	0.254015
$875$	0	0
$876$	0	0
$877$	−30.7432	−1.03812	−0.519062	−	0.854737i	$-0.673719\pi$
$877$	−30.7432	−1.03812	−0.519062	+	0.854737i	$0.673719\pi$
$878$	14.0518	0.474224
$879$	0	0
$880$	0	0
$881$	35.1508	1.18426	0.592130	−	0.805843i	$-0.298287\pi$
$881$	35.1508	1.18426	0.592130	+	0.805843i	$0.298287\pi$
$882$	0	0
$883$	35.8185	1.20539	0.602694	−	0.797972i	$-0.294094\pi$
$883$	35.8185	1.20539	0.602694	+	0.797972i	$0.294094\pi$
$884$	−0.610821	−0.0205441
$885$	0	0
$886$	8.07877	0.271412
$887$	−30.3154	−1.01789	−0.508946	−	0.860798i	$-0.669965\pi$
$887$	−30.3154	−1.01789	−0.508946	+	0.860798i	$0.669965\pi$
$888$	0	0
$889$	−6.36575	−0.213501
$890$	0	0
$891$	0	0
$892$	−3.42035	−0.114522
$893$	−27.1833	−0.909654
$894$	0	0
$895$	0	0
$896$	−13.1235	−0.438426
$897$	0	0
$898$	−33.8320	−1.12899
$899$	12.4097	0.413888
$900$	0	0
$901$	3.13321	0.104382
$902$	50.0507	1.66651
$903$	0	0
$904$	−49.7390	−1.65430
$905$	0	0
$906$	0	0
$907$	6.79755	0.225709	0.112854	−	0.993612i	$-0.464001\pi$
$907$	6.79755	0.225709	0.112854	+	0.993612i	$0.464001\pi$
$908$	−2.18018	−0.0723517
$909$	0	0
$910$	0	0
$911$	34.3310	1.13744	0.568719	−	0.822532i	$-0.307439\pi$
$911$	34.3310	1.13744	0.568719	+	0.822532i	$0.307439\pi$
$912$	0	0
$913$	13.7329	0.454494
$914$	−10.3104	−0.341037
$915$	0	0
$916$	0.704672	0.0232830
$917$	7.23475	0.238913
$918$	0	0
$919$	6.72902	0.221970	0.110985	−	0.993822i	$-0.464599\pi$
$919$	6.72902	0.221970	0.110985	+	0.993822i	$0.464599\pi$
$920$	0	0
$921$	0	0
$922$	51.6679	1.70159
$923$	−21.8844	−0.720333
$924$	0	0
$925$	0	0
$926$	−43.9874	−1.44551
$927$	0	0
$928$	2.52739	0.0829656
$929$	4.70020	0.154209	0.0771043	−	0.997023i	$-0.475433\pi$
$929$	4.70020	0.154209	0.0771043	+	0.997023i	$0.475433\pi$
$930$	0	0
$931$	−16.9900	−0.556825
$932$	−2.99110	−0.0979768
$933$	0	0
$934$	47.5203	1.55491
$935$	0	0
$936$	0	0
$937$	−1.41847	−0.0463395	−0.0231698	−	0.999732i	$-0.507376\pi$
$937$	−1.41847	−0.0463395	−0.0231698	+	0.999732i	$0.507376\pi$
$938$	17.0796	0.557669
$939$	0	0
$940$	0	0
$941$	48.7929	1.59060	0.795302	−	0.606213i	$-0.207312\pi$
$941$	48.7929	1.59060	0.795302	+	0.606213i	$0.207312\pi$
$942$	0	0
$943$	−9.61347	−0.313058
$944$	59.3037	1.93017
$945$	0	0
$946$	62.0526	2.01751
$947$	−38.1885	−1.24096	−0.620480	−	0.784222i	$-0.713062\pi$
$947$	−38.1885	−1.24096	−0.620480	+	0.784222i	$0.713062\pi$
$948$	0	0
$949$	−16.4798	−0.534955
$950$	0	0
$951$	0	0
$952$	1.20828	0.0391604
$953$	−8.25677	−0.267463	−0.133731	−	0.991018i	$-0.542696\pi$
$953$	−8.25677	−0.267463	−0.133731	+	0.991018i	$0.542696\pi$
$954$	0	0
$955$	0	0
$956$	0.234695	0.00759057
$957$	0	0
$958$	−22.3183	−0.721070
$959$	−7.77785	−0.251160
$960$	0	0
$961$	45.7623	1.47620
$962$	49.4577	1.59458
$963$	0	0
$964$	−1.34293	−0.0432530
$965$	0	0
$966$	0	0
$967$	−43.3033	−1.39254	−0.696270	−	0.717780i	$-0.745158\pi$
$967$	−43.3033	−1.39254	−0.696270	+	0.717780i	$0.745158\pi$
$968$	63.2007	2.03135
$969$	0	0
$970$	0	0
$971$	35.1452	1.12786	0.563932	−	0.825821i	$-0.309288\pi$
$971$	35.1452	1.12786	0.563932	+	0.825821i	$0.309288\pi$
$972$	0	0
$973$	−16.8976	−0.541713
$974$	8.46364	0.271192
$975$	0	0
$976$	−55.3365	−1.77128
$977$	40.4411	1.29383	0.646913	−	0.762563i	$-0.276059\pi$
$977$	40.4411	1.29383	0.646913	+	0.762563i	$0.276059\pi$
$978$	0	0
$979$	37.4490	1.19687
$980$	0	0
$981$	0	0
$982$	−26.5835	−0.848312
$983$	−58.9585	−1.88048	−0.940241	−	0.340509i	$-0.889401\pi$
$983$	−58.9585	−1.88048	−0.940241	+	0.340509i	$0.889401\pi$
$984$	0	0
$985$	0	0
$986$	−1.02693	−0.0327042
$987$	0	0
$988$	−3.62153	−0.115216
$989$	−11.9187	−0.378994
$990$	0	0
$991$	11.0145	0.349888	0.174944	−	0.984578i	$-0.444026\pi$
$991$	11.0145	0.349888	0.174944	+	0.984578i	$0.444026\pi$
$992$	15.6335	0.496365
$993$	0	0
$994$	−8.19390	−0.259895
$995$	0	0
$996$	0	0
$997$	−52.8873	−1.67496	−0.837479	−	0.546469i	$-0.815971\pi$
$997$	−52.8873	−1.67496	−0.837479	+	0.546469i	$0.815971\pi$
$998$	−24.8570	−0.786836
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.bc.1.2		8
3.2	odd	2		1875.2.a.n.1.7	✓	8
5.4	even	2		5625.2.a.u.1.7		8
15.2	even	4		1875.2.b.g.1249.13		16
15.8	even	4		1875.2.b.g.1249.4		16
15.14	odd	2		1875.2.a.o.1.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.n.1.7	✓	8	3.2	odd	2
1875.2.a.o.1.2	yes	8	15.14	odd	2
1875.2.b.g.1249.4		16	15.8	even	4
1875.2.b.g.1249.13		16	15.2	even	4
5625.2.a.u.1.7		8	5.4	even	2
5625.2.a.bc.1.2		8	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)

\(f(q)\)	\(=\)	\(q-1.52260 q^{2} +0.318310 q^{4} +0.990985 q^{7} +2.56054 q^{8} +O(q^{10})\)	\(q-1.52260 q^{2} +0.318310 q^{4} +0.990985 q^{7} +2.56054 q^{8} -5.97349 q^{11} -4.02992 q^{13} -1.50887 q^{14} -4.53530 q^{16} +0.476176 q^{17} +2.82322 q^{19} +9.09523 q^{22} -1.74696 q^{23} +6.13596 q^{26} +0.315440 q^{28} +1.41641 q^{29} +8.76141 q^{31} +1.78436 q^{32} -0.725025 q^{34} +8.06031 q^{37} -4.29864 q^{38} +5.50296 q^{41} +6.82255 q^{43} -1.90142 q^{44} +2.65993 q^{46} -9.62845 q^{47} -6.01795 q^{49} -1.28276 q^{52} +6.57994 q^{53} +2.53746 q^{56} -2.15662 q^{58} -13.0760 q^{59} +12.2013 q^{61} -13.3401 q^{62} +6.35373 q^{64} -11.3195 q^{67} +0.151571 q^{68} +5.43047 q^{71} +4.08935 q^{73} -12.2726 q^{74} +0.898661 q^{76} -5.91964 q^{77} +15.0070 q^{79} -8.37881 q^{82} -2.29898 q^{83} -10.3880 q^{86} -15.2954 q^{88} -6.26920 q^{89} -3.99359 q^{91} -0.556076 q^{92} +14.6603 q^{94} -13.2541 q^{97} +9.16293 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} + 9 q^{4} - 12 q^{7} + 3 q^{8}+O(q^{10}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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