LMFDB - Embedded newform 5625.2.a.u.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

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

Embedding invariants

Embedding label			1.7
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	$-0.319059\pi$
$2$	1.52260	1.07664	0.538320	+	0.842740i	$0.319059\pi$
$3$	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	$-0.559967\pi$
$7$	−0.990985	−0.374557	−0.187279	+	0.982307i	$0.559967\pi$
$8$	−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	$-0.311243\pi$
$13$	4.02992	1.11770	0.558850	+	0.829269i	$0.311243\pi$
$14$	−1.50887	−0.403263
$15$	0	0
$16$	−4.53530	−1.13382
$17$	−0.476176	−0.115490	−0.0577448	−	0.998331i	$-0.518391\pi$
$17$	−0.476176	−0.115490	−0.0577448	+	0.998331i	$0.518391\pi$
$18$	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	$-0.441700\pi$
$23$	1.74696	0.364267	0.182134	+	0.983274i	$0.441700\pi$
$24$	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	$-0.730527\pi$
$37$	−8.06031	−1.32511	−0.662553	+	0.749015i	$0.730527\pi$
$38$	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	$-0.674148\pi$
$43$	−6.82255	−1.04043	−0.520214	+	0.854036i	$0.674148\pi$
$44$	−1.90142	−0.286650
$45$	0	0
$46$	2.65993	0.392185
$47$	9.62845	1.40445	0.702227	−	0.711953i	$-0.252189\pi$
$47$	9.62845	1.40445	0.702227	+	0.711953i	$0.252189\pi$
$48$	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	$-0.649258\pi$
$53$	−6.57994	−0.903824	−0.451912	+	0.892063i	$0.649258\pi$
$54$	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	$-0.256974\pi$
$67$	11.3195	1.38289	0.691445	+	0.722429i	$0.256974\pi$
$68$	−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	$-0.576922\pi$
$73$	−4.08935	−0.478622	−0.239311	+	0.970943i	$0.576922\pi$
$74$	−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	$-0.459731\pi$
$83$	2.29898	0.252346	0.126173	+	0.992008i	$0.459731\pi$
$84$	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	$-0.265059\pi$
$97$	13.2541	1.34575	0.672875	+	0.739756i	$0.265059\pi$
$98$	−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	$-0.347034\pi$
$103$	9.38309	0.924543	0.462272	+	0.886738i	$0.347034\pi$
$104$	−10.3188	−1.01184
$105$	0	0
$106$	−10.0186	−0.973094
$107$	13.4108	1.29647	0.648234	−	0.761441i	$-0.275508\pi$
$107$	13.4108	1.29647	0.648234	+	0.761441i	$0.275508\pi$
$108$	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	$-0.133224\pi$
$113$	19.4252	1.82737	0.913685	+	0.406423i	$0.133224\pi$
$114$	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	$-0.408005\pi$
$127$	6.42366	0.570008	0.285004	+	0.958526i	$0.408005\pi$
$128$	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	$-0.391171\pi$
$137$	7.84860	0.670552	0.335276	+	0.942120i	$0.391171\pi$
$138$	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	$-0.439005\pi$
$157$	4.77270	0.380903	0.190452	+	0.981697i	$0.439005\pi$
$158$	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	$-0.462684\pi$
$163$	2.98657	0.233926	0.116963	+	0.993136i	$0.462684\pi$
$164$	1.75165	0.136781
$165$	0	0
$166$	3.50043	0.271686
$167$	−18.6105	−1.44012	−0.720060	−	0.693912i	$-0.755886\pi$
$167$	−18.6105	−1.44012	−0.720060	+	0.693912i	$0.755886\pi$
$168$	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	$-0.713610\pi$
$173$	−16.3578	−1.24366	−0.621829	+	0.783153i	$0.713610\pi$
$174$	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	$-0.331505\pi$
$193$	14.0304	1.00993	0.504965	+	0.863140i	$0.331505\pi$
$194$	20.1807	1.44889
$195$	0	0
$196$	−1.91557	−0.136827
$197$	−23.7871	−1.69476	−0.847382	−	0.530984i	$-0.821823\pi$
$197$	−23.7871	−1.69476	−0.847382	+	0.530984i	$0.821823\pi$
$198$	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	$-0.382851\pi$
$223$	10.7454	0.719562	0.359781	+	0.933037i	$0.382851\pi$
$224$	1.76828	0.118148
$225$	0	0
$226$	29.5768	1.96742
$227$	6.84922	0.454599	0.227299	−	0.973825i	$-0.427010\pi$
$227$	6.84922	0.454599	0.227299	+	0.973825i	$0.427010\pi$
$228$	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	$-0.400406\pi$
$233$	9.39682	0.615606	0.307803	+	0.951450i	$0.400406\pi$
$234$	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	$-0.500381\pi$
$257$	−0.0383881	−0.00239458	−0.00119729	+	0.999999i	$0.500381\pi$
$258$	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	$-0.354985\pi$
$263$	14.2706	0.879965	0.439982	+	0.898006i	$0.354985\pi$
$264$	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	$-0.638631\pi$
$277$	−14.0431	−0.843768	−0.421884	+	0.906650i	$0.638631\pi$
$278$	−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	$-0.490754\pi$
$283$	0.977154	0.0580858	0.0290429	+	0.999578i	$0.490754\pi$
$284$	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	$-0.525649\pi$
$293$	−2.75561	−0.160984	−0.0804922	+	0.996755i	$0.525649\pi$
$294$	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	$-0.347089\pi$
$307$	16.1940	0.924241	0.462120	+	0.886817i	$0.347089\pi$
$308$	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	$-0.308716\pi$
$313$	20.0065	1.13083	0.565416	+	0.824806i	$0.308716\pi$
$314$	7.26691	0.410096
$315$	0	0
$316$	4.77689	0.268721
$317$	6.52708	0.366597	0.183299	−	0.983057i	$-0.441322\pi$
$317$	6.52708	0.366597	0.183299	+	0.983057i	$0.441322\pi$
$318$	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	$-0.347103\pi$
$337$	16.9653	0.924160	0.462080	+	0.886838i	$0.347103\pi$
$338$	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	$-0.318152\pi$
$347$	20.1449	1.08144	0.540719	+	0.841203i	$0.318152\pi$
$348$	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	$-0.508327\pi$
$353$	−0.982924	−0.0523157	−0.0261579	+	0.999658i	$0.508327\pi$
$354$	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	$-0.497069\pi$
$367$	0.352799	0.0184160	0.00920798	+	0.999958i	$0.497069\pi$
$368$	−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	$-0.716377\pi$
$373$	−24.2811	−1.25723	−0.628614	+	0.777718i	$0.716377\pi$
$374$	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	$-0.554200\pi$
$383$	−6.63250	−0.338905	−0.169453	+	0.985538i	$0.554200\pi$
$384$	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	$-0.576164\pi$
$397$	−9.44435	−0.473998	−0.236999	+	0.971510i	$0.576164\pi$
$398$	−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	$-0.116821\pi$
$433$	38.8459	1.86681	0.933407	+	0.358821i	$0.116821\pi$
$434$	−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	$-0.459771\pi$
$443$	5.30591	0.252091	0.126046	+	0.992024i	$0.459771\pi$
$444$	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	$-0.550627\pi$
$457$	−6.77157	−0.316761	−0.158380	+	0.987378i	$0.550627\pi$
$458$	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	$-0.734267\pi$
$463$	−28.8896	−1.34262	−0.671308	+	0.741179i	$0.734267\pi$
$464$	−6.42384	−0.298219
$465$	0	0
$466$	14.3076	0.662786
$467$	31.2100	1.44423	0.722113	−	0.691776i	$-0.243171\pi$
$467$	31.2100	1.44423	0.722113	+	0.691776i	$0.243171\pi$
$468$	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	$-0.459804\pi$
$487$	5.55868	0.251888	0.125944	+	0.992037i	$0.459804\pi$
$488$	−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	$-0.378098\pi$
$503$	16.7612	0.747345	0.373672	+	0.927561i	$0.378098\pi$
$504$	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	$-0.698161\pi$
$523$	−26.6701	−1.16620	−0.583102	+	0.812399i	$0.698161\pi$
$524$	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	$-0.441248\pi$
$547$	8.58474	0.367057	0.183529	+	0.983014i	$0.441248\pi$
$548$	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	$-0.355693\pi$
$557$	20.6736	0.875968	0.437984	+	0.898983i	$0.355693\pi$
$558$	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	$-0.585654\pi$
$563$	−12.6162	−0.531708	−0.265854	+	0.964013i	$0.585654\pi$
$564$	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	$-0.606293\pi$
$577$	−15.7460	−0.655515	−0.327757	+	0.944762i	$0.606293\pi$
$578$	−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	$-0.424573\pi$
$587$	11.3751	0.469501	0.234751	+	0.972056i	$0.424573\pi$
$588$	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	$-0.284913\pi$
$593$	30.4617	1.25091	0.625457	+	0.780259i	$0.284913\pi$
$594$	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	$-0.219727\pi$
$607$	37.9938	1.54212	0.771060	+	0.636762i	$0.219727\pi$
$608$	−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	$-0.666369\pi$
$613$	−24.7188	−0.998381	−0.499191	+	0.866492i	$0.666369\pi$
$614$	24.6570	0.995075
$615$	0	0
$616$	−15.1575	−0.610712
$617$	−29.8302	−1.20092	−0.600460	−	0.799655i	$-0.705016\pi$
$617$	−29.8302	−1.20092	−0.600460	+	0.799655i	$0.705016\pi$
$618$	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	$-0.349447\pi$
$643$	23.1026	0.911077	0.455538	+	0.890216i	$0.349447\pi$
$644$	−0.551063	−0.0217149
$645$	0	0
$646$	−2.04691	−0.0805345
$647$	9.35622	0.367831	0.183916	−	0.982942i	$-0.441123\pi$
$647$	9.35622	0.367831	0.183916	+	0.982942i	$0.441123\pi$
$648$	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	$-0.514311\pi$
$653$	−2.29702	−0.0898892	−0.0449446	+	0.998989i	$0.514311\pi$
$654$	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	$-0.577011\pi$
$673$	−12.4307	−0.479168	−0.239584	+	0.970876i	$0.577011\pi$
$674$	25.8314	0.994989
$675$	0	0
$676$	1.03141	0.0396697
$677$	6.61696	0.254310	0.127155	−	0.991883i	$-0.459415\pi$
$677$	6.61696	0.254310	0.127155	+	0.991883i	$0.459415\pi$
$678$	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	$-0.383613\pi$
$683$	18.6885	0.715097	0.357549	+	0.933895i	$0.383613\pi$
$684$	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	$-0.485332\pi$
$727$	2.48415	0.0921320	0.0460660	+	0.998938i	$0.485332\pi$
$728$	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	$-0.699733\pi$
$733$	−31.7905	−1.17421	−0.587105	+	0.809511i	$0.699733\pi$
$734$	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	$-0.488149\pi$
$743$	2.02921	0.0744447	0.0372223	+	0.999307i	$0.488149\pi$
$744$	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	$-0.247023\pi$
$757$	39.2724	1.42738	0.713689	+	0.700463i	$0.247023\pi$
$758$	−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	$-0.451343\pi$
$773$	8.46681	0.304530	0.152265	+	0.988340i	$0.451343\pi$
$774$	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	$-0.586123\pi$
$787$	−14.9960	−0.534551	−0.267275	+	0.963620i	$0.586123\pi$
$788$	−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	$-0.438089\pi$
$797$	10.9127	0.386549	0.193274	+	0.981145i	$0.438089\pi$
$798$	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	$-0.211042\pi$
$823$	45.2205	1.57629	0.788144	+	0.615491i	$0.211042\pi$
$824$	−24.0258	−0.836978
$825$	0	0
$826$	19.7301	0.686497
$827$	25.6349	0.891411	0.445706	−	0.895180i	$-0.352953\pi$
$827$	25.6349	0.891411	0.445706	+	0.895180i	$0.352953\pi$
$828$	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	$-0.524309\pi$
$853$	−4.45661	−0.152591	−0.0762956	+	0.997085i	$0.524309\pi$
$854$	−18.4102	−0.629984
$855$	0	0
$856$	−34.3388	−1.17368
$857$	−48.0738	−1.64217	−0.821084	−	0.570807i	$-0.806630\pi$
$857$	−48.0738	−1.64217	−0.821084	+	0.570807i	$0.806630\pi$
$858$	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	$-0.543422\pi$
$863$	−7.99001	−0.271983	−0.135992	+	0.990710i	$0.543422\pi$
$864$	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	$-0.326281\pi$
$877$	30.7432	1.03812	0.519062	+	0.854737i	$0.326281\pi$
$878$	−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	$-0.705906\pi$
$883$	−35.8185	−1.20539	−0.602694	+	0.797972i	$0.705906\pi$
$884$	−0.610821	−0.0205441
$885$	0	0
$886$	8.07877	0.271412
$887$	30.3154	1.01789	0.508946	−	0.860798i	$-0.330035\pi$
$887$	30.3154	1.01789	0.508946	+	0.860798i	$0.330035\pi$
$888$	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	$-0.535999\pi$
$907$	−6.79755	−0.225709	−0.112854	+	0.993612i	$0.535999\pi$
$908$	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	$-0.492624\pi$
$937$	1.41847	0.0463395	0.0231698	+	0.999732i	$0.492624\pi$
$938$	−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	$-0.286938\pi$
$947$	38.1885	1.24096	0.620480	+	0.784222i	$0.286938\pi$
$948$	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	$-0.457304\pi$
$953$	8.25677	0.267463	0.133731	+	0.991018i	$0.457304\pi$
$954$	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	$-0.254842\pi$
$967$	43.3033	1.39254	0.696270	+	0.717780i	$0.254842\pi$
$968$	−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	$-0.723941\pi$
$977$	−40.4411	−1.29383	−0.646913	+	0.762563i	$0.723941\pi$
$978$	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	$-0.110599\pi$
$983$	58.9585	1.88048	0.940241	+	0.340509i	$0.110599\pi$
$984$	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	$-0.184029\pi$
$997$	52.8873	1.67496	0.837479	+	0.546469i	$0.184029\pi$
$998$	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.u.1.7		8
3.2	odd	2		1875.2.a.o.1.2	yes	8
5.4	even	2		5625.2.a.bc.1.2		8
15.2	even	4		1875.2.b.g.1249.4		16
15.8	even	4		1875.2.b.g.1249.13		16
15.14	odd	2		1875.2.a.n.1.7	✓	8

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

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

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