LMFDB - Embedded newform 5625.2.a.u.1.6

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			1.6
Root			$-0.895394$ 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)

$f(q)$	$=$	$q+0.895394 q^{2} -1.19827 q^{4} +5.08992 q^{7} -2.86371 q^{8} +O(q^{10})$	$q+0.895394 q^{2} -1.19827 q^{4} +5.08992 q^{7} -2.86371 q^{8} -2.64310 q^{11} +2.13295 q^{13} +4.55748 q^{14} -0.167607 q^{16} -7.75001 q^{17} +3.08652 q^{19} -2.36661 q^{22} +6.14107 q^{23} +1.90983 q^{26} -6.09909 q^{28} +4.13435 q^{29} -2.74277 q^{31} +5.57735 q^{32} -6.93931 q^{34} -0.0157706 q^{37} +2.76365 q^{38} -3.72829 q^{41} -3.81468 q^{43} +3.16715 q^{44} +5.49868 q^{46} +0.897385 q^{47} +18.9072 q^{49} -2.55585 q^{52} +9.26724 q^{53} -14.5760 q^{56} +3.70187 q^{58} +11.0693 q^{59} +6.38696 q^{61} -2.45585 q^{62} +5.32913 q^{64} -5.54154 q^{67} +9.28660 q^{68} +0.0828976 q^{71} -9.92024 q^{73} -0.0141209 q^{74} -3.69848 q^{76} -13.4532 q^{77} +5.30049 q^{79} -3.33829 q^{82} -0.723557 q^{83} -3.41564 q^{86} +7.56907 q^{88} +13.2548 q^{89} +10.8565 q^{91} -7.35867 q^{92} +0.803513 q^{94} -2.22836 q^{97} +16.9294 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.895394	0.633139	0.316569	−	0.948569i	$-0.397469\pi$
$2$	0.895394	0.633139	0.316569	+	0.948569i	$0.397469\pi$
$3$	0	0
$3$	0	0
$4$	−1.19827	−0.599135
$5$	0	0
$5$	0	0
$6$	0	0
$7$	5.08992	1.92381	0.961904	−	0.273389i	$-0.0881446\pi$
$7$	5.08992	1.92381	0.961904	+	0.273389i	$0.0881446\pi$
$8$	−2.86371	−1.01247
$9$	0	0
$10$	0	0
$11$	−2.64310	−0.796925	−0.398462	−	0.917185i	$-0.630456\pi$
$11$	−2.64310	−0.796925	−0.398462	+	0.917185i	$0.630456\pi$
$12$	0	0
$13$	2.13295	0.591575	0.295787	−	0.955254i	$-0.404418\pi$
$13$	2.13295	0.591575	0.295787	+	0.955254i	$0.404418\pi$
$14$	4.55748	1.21804
$15$	0	0
$16$	−0.167607	−0.0419018
$17$	−7.75001	−1.87965	−0.939826	−	0.341653i	$-0.889013\pi$
$17$	−7.75001	−1.87965	−0.939826	+	0.341653i	$0.889013\pi$
$18$	0	0
$19$	3.08652	0.708096	0.354048	−	0.935227i	$-0.384805\pi$
$19$	3.08652	0.708096	0.354048	+	0.935227i	$0.384805\pi$
$20$	0	0
$21$	0	0
$22$	−2.36661	−0.504564
$23$	6.14107	1.28050	0.640251	−	0.768166i	$-0.278830\pi$
$23$	6.14107	1.28050	0.640251	+	0.768166i	$0.278830\pi$
$24$	0	0
$25$	0	0
$26$	1.90983	0.374549
$27$	0	0
$28$	−6.09909	−1.15262
$29$	4.13435	0.767730	0.383865	−	0.923389i	$-0.374593\pi$
$29$	4.13435	0.767730	0.383865	+	0.923389i	$0.374593\pi$
$30$	0	0
$31$	−2.74277	−0.492615	−0.246308	−	0.969192i	$-0.579217\pi$
$31$	−2.74277	−0.492615	−0.246308	+	0.969192i	$0.579217\pi$
$32$	5.57735	0.985945
$33$	0	0
$34$	−6.93931	−1.19008
$35$	0	0
$36$	0	0
$37$	−0.0157706	−0.00259267	−0.00129634	−	0.999999i	$-0.500413\pi$
$37$	−0.0157706	−0.00259267	−0.00129634	+	0.999999i	$0.500413\pi$
$38$	2.76365	0.448323
$39$	0	0
$40$	0	0
$41$	−3.72829	−0.582262	−0.291131	−	0.956683i	$-0.594031\pi$
$41$	−3.72829	−0.582262	−0.291131	+	0.956683i	$0.594031\pi$
$42$	0	0
$43$	−3.81468	−0.581733	−0.290866	−	0.956764i	$-0.593944\pi$
$43$	−3.81468	−0.581733	−0.290866	+	0.956764i	$0.593944\pi$
$44$	3.16715	0.477466
$45$	0	0
$46$	5.49868	0.810736
$47$	0.897385	0.130897	0.0654486	−	0.997856i	$-0.479152\pi$
$47$	0.897385	0.130897	0.0654486	+	0.997856i	$0.479152\pi$
$48$	0	0
$49$	18.9072	2.70103
$50$	0	0
$51$	0	0
$52$	−2.55585	−0.354433
$53$	9.26724	1.27295	0.636477	−	0.771296i	$-0.280391\pi$
$53$	9.26724	1.27295	0.636477	+	0.771296i	$0.280391\pi$
$54$	0	0
$55$	0	0
$56$	−14.5760	−1.94781
$57$	0	0
$58$	3.70187	0.486080
$59$	11.0693	1.44111	0.720553	−	0.693400i	$-0.243888\pi$
$59$	11.0693	1.44111	0.720553	+	0.693400i	$0.243888\pi$
$60$	0	0
$61$	6.38696	0.817766	0.408883	−	0.912587i	$-0.365918\pi$
$61$	6.38696	0.817766	0.408883	+	0.912587i	$0.365918\pi$
$62$	−2.45585	−0.311894
$63$	0	0
$64$	5.32913	0.666142
$65$	0	0
$66$	0	0
$67$	−5.54154	−0.677007	−0.338504	−	0.940965i	$-0.609921\pi$
$67$	−5.54154	−0.677007	−0.338504	+	0.940965i	$0.609921\pi$
$68$	9.28660	1.12617
$69$	0	0
$70$	0	0
$71$	0.0828976	0.00983814	0.00491907	−	0.999988i	$-0.498434\pi$
$71$	0.0828976	0.00983814	0.00491907	+	0.999988i	$0.498434\pi$
$72$	0	0
$73$	−9.92024	−1.16108	−0.580538	−	0.814233i	$-0.697158\pi$
$73$	−9.92024	−1.16108	−0.580538	+	0.814233i	$0.697158\pi$
$74$	−0.0141209	−0.00164152
$75$	0	0
$76$	−3.69848	−0.424245
$77$	−13.4532	−1.53313
$78$	0	0
$79$	5.30049	0.596352	0.298176	−	0.954511i	$-0.403622\pi$
$79$	5.30049	0.596352	0.298176	+	0.954511i	$0.403622\pi$
$80$	0	0
$81$	0	0
$82$	−3.33829	−0.368652
$83$	−0.723557	−0.0794207	−0.0397104	−	0.999211i	$-0.512644\pi$
$83$	−0.723557	−0.0794207	−0.0397104	+	0.999211i	$0.512644\pi$
$84$	0	0
$85$	0	0
$86$	−3.41564	−0.368318
$87$	0	0
$88$	7.56907	0.806866
$89$	13.2548	1.40500	0.702500	−	0.711683i	$-0.252067\pi$
$89$	13.2548	1.40500	0.702500	+	0.711683i	$0.252067\pi$
$90$	0	0
$91$	10.8565	1.13808
$92$	−7.35867	−0.767194
$93$	0	0
$94$	0.803513	0.0828760
$95$	0	0
$96$	0	0
$97$	−2.22836	−0.226256	−0.113128	−	0.993580i	$-0.536087\pi$
$97$	−2.22836	−0.226256	−0.113128	+	0.993580i	$0.536087\pi$
$98$	16.9294	1.71013
$99$	0	0
$100$	0	0
$101$	−15.4908	−1.54140	−0.770698	−	0.637201i	$-0.780092\pi$
$101$	−15.4908	−1.54140	−0.770698	+	0.637201i	$0.780092\pi$
$102$	0	0
$103$	11.5680	1.13983	0.569913	−	0.821705i	$-0.306977\pi$
$103$	11.5680	1.13983	0.569913	+	0.821705i	$0.306977\pi$
$104$	−6.10816	−0.598954
$105$	0	0
$106$	8.29783	0.805956
$107$	−1.10562	−0.106884	−0.0534419	−	0.998571i	$-0.517019\pi$
$107$	−1.10562	−0.106884	−0.0534419	+	0.998571i	$0.517019\pi$
$108$	0	0
$109$	10.2626	0.982976	0.491488	−	0.870884i	$-0.336453\pi$
$109$	10.2626	0.982976	0.491488	+	0.870884i	$0.336453\pi$
$110$	0	0
$111$	0	0
$112$	−0.853106	−0.0806109
$113$	−1.74010	−0.163695	−0.0818474	−	0.996645i	$-0.526082\pi$
$113$	−1.74010	−0.163695	−0.0818474	+	0.996645i	$0.526082\pi$
$114$	0	0
$115$	0	0
$116$	−4.95407	−0.459974
$117$	0	0
$118$	9.91142	0.912421
$119$	−39.4469	−3.61609
$120$	0	0
$121$	−4.01402	−0.364911
$122$	5.71884	0.517760
$123$	0	0
$124$	3.28657	0.295143
$125$	0	0
$126$	0	0
$127$	−9.77368	−0.867274	−0.433637	−	0.901088i	$-0.642770\pi$
$127$	−9.77368	−0.867274	−0.433637	+	0.901088i	$0.642770\pi$
$128$	−6.38302	−0.564185
$129$	0	0
$130$	0	0
$131$	10.9407	0.955893	0.477947	−	0.878389i	$-0.341381\pi$
$131$	10.9407	0.955893	0.477947	+	0.878389i	$0.341381\pi$
$132$	0	0
$133$	15.7101	1.36224
$134$	−4.96186	−0.428640
$135$	0	0
$136$	22.1938	1.90310
$137$	16.7888	1.43436	0.717181	−	0.696887i	$-0.245432\pi$
$137$	16.7888	1.43436	0.717181	+	0.696887i	$0.245432\pi$
$138$	0	0
$139$	−5.48342	−0.465098	−0.232549	−	0.972585i	$-0.574707\pi$
$139$	−5.48342	−0.465098	−0.232549	+	0.972585i	$0.574707\pi$
$140$	0	0
$141$	0	0
$142$	0.0742260	0.00622891
$143$	−5.63761	−0.471440
$144$	0	0
$145$	0	0
$146$	−8.88251	−0.735122
$147$	0	0
$148$	0.0188974	0.00155336
$149$	0.273699	0.0224223	0.0112112	−	0.999937i	$-0.496431\pi$
$149$	0.273699	0.0224223	0.0112112	+	0.999937i	$0.496431\pi$
$150$	0	0
$151$	1.60377	0.130513	0.0652564	−	0.997869i	$-0.479213\pi$
$151$	1.60377	0.130513	0.0652564	+	0.997869i	$0.479213\pi$
$152$	−8.83890	−0.716929
$153$	0	0
$154$	−12.0459	−0.970684
$155$	0	0
$156$	0	0
$157$	16.5512	1.32093	0.660465	−	0.750857i	$-0.270359\pi$
$157$	16.5512	1.32093	0.660465	+	0.750857i	$0.270359\pi$
$158$	4.74603	0.377574
$159$	0	0
$160$	0	0
$161$	31.2575	2.46344
$162$	0	0
$163$	22.9917	1.80085	0.900424	−	0.435014i	$-0.143257\pi$
$163$	22.9917	1.80085	0.900424	+	0.435014i	$0.143257\pi$
$164$	4.46750	0.348853
$165$	0	0
$166$	−0.647868	−0.0502843
$167$	17.1191	1.32472	0.662359	−	0.749187i	$-0.269555\pi$
$167$	17.1191	1.32472	0.662359	+	0.749187i	$0.269555\pi$
$168$	0	0
$169$	−8.45051	−0.650039
$170$	0	0
$171$	0	0
$172$	4.57101	0.348537
$173$	−3.30634	−0.251376	−0.125688	−	0.992070i	$-0.540114\pi$
$173$	−3.30634	−0.251376	−0.125688	+	0.992070i	$0.540114\pi$
$174$	0	0
$175$	0	0
$176$	0.443002	0.0333926
$177$	0	0
$178$	11.8682	0.889561
$179$	−12.1351	−0.907020	−0.453510	−	0.891251i	$-0.649828\pi$
$179$	−12.1351	−0.907020	−0.453510	+	0.891251i	$0.649828\pi$
$180$	0	0
$181$	15.5920	1.15894	0.579472	−	0.814992i	$-0.303259\pi$
$181$	15.5920	1.15894	0.579472	+	0.814992i	$0.303259\pi$
$182$	9.72088	0.720560
$183$	0	0
$184$	−17.5863	−1.29648
$185$	0	0
$186$	0	0
$187$	20.4840	1.49794
$188$	−1.07531	−0.0784251
$189$	0	0
$190$	0	0
$191$	4.82025	0.348781	0.174391	−	0.984677i	$-0.444204\pi$
$191$	4.82025	0.348781	0.174391	+	0.984677i	$0.444204\pi$
$192$	0	0
$193$	−17.7887	−1.28046	−0.640230	−	0.768183i	$-0.721161\pi$
$193$	−17.7887	−1.28046	−0.640230	+	0.768183i	$0.721161\pi$
$194$	−1.99526	−0.143251
$195$	0	0
$196$	−22.6560	−1.61828
$197$	22.2222	1.58327	0.791633	−	0.610997i	$-0.209231\pi$
$197$	22.2222	1.58327	0.791633	+	0.610997i	$0.209231\pi$
$198$	0	0
$199$	−16.3687	−1.16035	−0.580175	−	0.814492i	$-0.697016\pi$
$199$	−16.3687	−1.16035	−0.580175	+	0.814492i	$0.697016\pi$
$200$	0	0
$201$	0	0
$202$	−13.8704	−0.975918
$203$	21.0435	1.47696
$204$	0	0
$205$	0	0
$206$	10.3579	0.721668
$207$	0	0
$208$	−0.357498	−0.0247880
$209$	−8.15798	−0.564299
$210$	0	0
$211$	−14.7224	−1.01353	−0.506765	−	0.862084i	$-0.669159\pi$
$211$	−14.7224	−1.01353	−0.506765	+	0.862084i	$0.669159\pi$
$212$	−11.1047	−0.762671
$213$	0	0
$214$	−0.989961	−0.0676723
$215$	0	0
$216$	0	0
$217$	−13.9604	−0.947697
$218$	9.18904	0.622360
$219$	0	0
$220$	0	0
$221$	−16.5304	−1.11196
$222$	0	0
$223$	21.7310	1.45521	0.727607	−	0.685994i	$-0.240632\pi$
$223$	21.7310	1.45521	0.727607	+	0.685994i	$0.240632\pi$
$224$	28.3882	1.89677
$225$	0	0
$226$	−1.55807	−0.103642
$227$	−27.4127	−1.81944	−0.909722	−	0.415218i	$-0.863705\pi$
$227$	−27.4127	−1.81944	−0.909722	+	0.415218i	$0.863705\pi$
$228$	0	0
$229$	19.4643	1.28624	0.643118	−	0.765767i	$-0.277640\pi$
$229$	19.4643	1.28624	0.643118	+	0.765767i	$0.277640\pi$
$230$	0	0
$231$	0	0
$232$	−11.8396	−0.777307
$233$	10.2000	0.668224	0.334112	−	0.942533i	$-0.391564\pi$
$233$	10.2000	0.668224	0.334112	+	0.942533i	$0.391564\pi$
$234$	0	0
$235$	0	0
$236$	−13.2641	−0.863418
$237$	0	0
$238$	−35.3205	−2.28949
$239$	3.18653	0.206120	0.103060	−	0.994675i	$-0.467137\pi$
$239$	3.18653	0.206120	0.103060	+	0.994675i	$0.467137\pi$
$240$	0	0
$241$	12.4630	0.802811	0.401405	−	0.915901i	$-0.368522\pi$
$241$	12.4630	0.802811	0.401405	+	0.915901i	$0.368522\pi$
$242$	−3.59413	−0.231040
$243$	0	0
$244$	−7.65330	−0.489953
$245$	0	0
$246$	0	0
$247$	6.58340	0.418892
$248$	7.85449	0.498760
$249$	0	0
$250$	0	0
$251$	−1.49139	−0.0941357	−0.0470679	−	0.998892i	$-0.514988\pi$
$251$	−1.49139	−0.0941357	−0.0470679	+	0.998892i	$0.514988\pi$
$252$	0	0
$253$	−16.2315	−1.02046
$254$	−8.75129	−0.549105
$255$	0	0
$256$	−16.3736	−1.02335
$257$	19.3647	1.20794	0.603969	−	0.797008i	$-0.293585\pi$
$257$	19.3647	1.20794	0.603969	+	0.797008i	$0.293585\pi$
$258$	0	0
$259$	−0.0802710	−0.00498780
$260$	0	0
$261$	0	0
$262$	9.79623	0.605213
$263$	13.7684	0.848996	0.424498	−	0.905429i	$-0.360451\pi$
$263$	13.7684	0.848996	0.424498	+	0.905429i	$0.360451\pi$
$264$	0	0
$265$	0	0
$266$	14.0667	0.862487
$267$	0	0
$268$	6.64027	0.405619
$269$	−2.85430	−0.174030	−0.0870149	−	0.996207i	$-0.527733\pi$
$269$	−2.85430	−0.174030	−0.0870149	+	0.996207i	$0.527733\pi$
$270$	0	0
$271$	21.9647	1.33426	0.667131	−	0.744941i	$-0.267522\pi$
$271$	21.9647	1.33426	0.667131	+	0.744941i	$0.267522\pi$
$272$	1.29896	0.0787608
$273$	0	0
$274$	15.0326	0.908150
$275$	0	0
$276$	0	0
$277$	2.63773	0.158486	0.0792428	−	0.996855i	$-0.474750\pi$
$277$	2.63773	0.158486	0.0792428	+	0.996855i	$0.474750\pi$
$278$	−4.90982	−0.294472
$279$	0	0
$280$	0	0
$281$	8.16358	0.486998	0.243499	−	0.969901i	$-0.421705\pi$
$281$	8.16358	0.486998	0.243499	+	0.969901i	$0.421705\pi$
$282$	0	0
$283$	−6.06194	−0.360345	−0.180173	−	0.983635i	$-0.557666\pi$
$283$	−6.06194	−0.360345	−0.180173	+	0.983635i	$0.557666\pi$
$284$	−0.0993338	−0.00589437
$285$	0	0
$286$	−5.04788	−0.298487
$287$	−18.9767	−1.12016
$288$	0	0
$289$	43.0626	2.53309
$290$	0	0
$291$	0	0
$292$	11.8871	0.695641
$293$	16.5940	0.969432	0.484716	−	0.874672i	$-0.338923\pi$
$293$	16.5940	0.969432	0.484716	+	0.874672i	$0.338923\pi$
$294$	0	0
$295$	0	0
$296$	0.0451624	0.00262501
$297$	0	0
$298$	0.245069	0.0141964
$299$	13.0986	0.757513
$300$	0	0
$301$	−19.4164	−1.11914
$302$	1.43600	0.0826327
$303$	0	0
$304$	−0.517323	−0.0296705
$305$	0	0
$306$	0	0
$307$	14.8424	0.847098	0.423549	−	0.905873i	$-0.360784\pi$
$307$	14.8424	0.847098	0.423549	+	0.905873i	$0.360784\pi$
$308$	16.1205	0.918552
$309$	0	0
$310$	0	0
$311$	−13.6460	−0.773796	−0.386898	−	0.922123i	$-0.626453\pi$
$311$	−13.6460	−0.773796	−0.386898	+	0.922123i	$0.626453\pi$
$312$	0	0
$313$	3.18194	0.179854	0.0899270	−	0.995948i	$-0.471337\pi$
$313$	3.18194	0.179854	0.0899270	+	0.995948i	$0.471337\pi$
$314$	14.8198	0.836332
$315$	0	0
$316$	−6.35142	−0.357295
$317$	−3.70586	−0.208142	−0.104071	−	0.994570i	$-0.533187\pi$
$317$	−3.70586	−0.208142	−0.104071	+	0.994570i	$0.533187\pi$
$318$	0	0
$319$	−10.9275	−0.611823
$320$	0	0
$321$	0	0
$322$	27.9878	1.55970
$323$	−23.9205	−1.33097
$324$	0	0
$325$	0	0
$326$	20.5866	1.14019
$327$	0	0
$328$	10.6768	0.589525
$329$	4.56762	0.251821
$330$	0	0
$331$	−32.3878	−1.78020	−0.890098	−	0.455769i	$-0.849364\pi$
$331$	−32.3878	−1.78020	−0.890098	+	0.455769i	$0.849364\pi$
$332$	0.867017	0.0475838
$333$	0	0
$334$	15.3284	0.838730
$335$	0	0
$336$	0	0
$337$	−21.9294	−1.19457	−0.597285	−	0.802029i	$-0.703754\pi$
$337$	−21.9294	−1.19457	−0.597285	+	0.802029i	$0.703754\pi$
$338$	−7.56653	−0.411565
$339$	0	0
$340$	0	0
$341$	7.24940	0.392577
$342$	0	0
$343$	60.6068	3.27246
$344$	10.9241	0.588990
$345$	0	0
$346$	−2.96047	−0.159156
$347$	24.8312	1.33301	0.666503	−	0.745502i	$-0.267790\pi$
$347$	24.8312	1.33301	0.666503	+	0.745502i	$0.267790\pi$
$348$	0	0
$349$	−1.99222	−0.106641	−0.0533204	−	0.998577i	$-0.516980\pi$
$349$	−1.99222	−0.106641	−0.0533204	+	0.998577i	$0.516980\pi$
$350$	0	0
$351$	0	0
$352$	−14.7415	−0.785724
$353$	−2.00997	−0.106980	−0.0534900	−	0.998568i	$-0.517035\pi$
$353$	−2.00997	−0.106980	−0.0534900	+	0.998568i	$0.517035\pi$
$354$	0	0
$355$	0	0
$356$	−15.8828	−0.841785
$357$	0	0
$358$	−10.8657	−0.574270
$359$	−24.2078	−1.27764	−0.638818	−	0.769358i	$-0.720576\pi$
$359$	−24.2078	−1.27764	−0.638818	+	0.769358i	$0.720576\pi$
$360$	0	0
$361$	−9.47340	−0.498600
$362$	13.9610	0.733772
$363$	0	0
$364$	−13.0091	−0.681861
$365$	0	0
$366$	0	0
$367$	−0.592824	−0.0309452	−0.0154726	−	0.999880i	$-0.504925\pi$
$367$	−0.592824	−0.0309452	−0.0154726	+	0.999880i	$0.504925\pi$
$368$	−1.02929	−0.0536553
$369$	0	0
$370$	0	0
$371$	47.1695	2.44892
$372$	0	0
$373$	8.42516	0.436238	0.218119	−	0.975922i	$-0.430008\pi$
$373$	8.42516	0.436238	0.218119	+	0.975922i	$0.430008\pi$
$374$	18.3413	0.948405
$375$	0	0
$376$	−2.56985	−0.132530
$377$	8.81838	0.454170
$378$	0	0
$379$	−18.6130	−0.956087	−0.478043	−	0.878336i	$-0.658654\pi$
$379$	−18.6130	−0.956087	−0.478043	+	0.878336i	$0.658654\pi$
$380$	0	0
$381$	0	0
$382$	4.31602	0.220827
$383$	9.94283	0.508055	0.254027	−	0.967197i	$-0.418245\pi$
$383$	9.94283	0.508055	0.254027	+	0.967197i	$0.418245\pi$
$384$	0	0
$385$	0	0
$386$	−15.9279	−0.810709
$387$	0	0
$388$	2.67018	0.135558
$389$	5.93571	0.300952	0.150476	−	0.988614i	$-0.451919\pi$
$389$	5.93571	0.300952	0.150476	+	0.988614i	$0.451919\pi$
$390$	0	0
$391$	−47.5934	−2.40690
$392$	−54.1449	−2.73473
$393$	0	0
$394$	19.8976	1.00243
$395$	0	0
$396$	0	0
$397$	−3.07681	−0.154421	−0.0772105	−	0.997015i	$-0.524601\pi$
$397$	−3.07681	−0.154421	−0.0772105	+	0.997015i	$0.524601\pi$
$398$	−14.6565	−0.734662
$399$	0	0
$400$	0	0
$401$	−20.6370	−1.03056	−0.515281	−	0.857021i	$-0.672313\pi$
$401$	−20.6370	−1.03056	−0.515281	+	0.857021i	$0.672313\pi$
$402$	0	0
$403$	−5.85019	−0.291419
$404$	18.5622	0.923505
$405$	0	0
$406$	18.8422	0.935124
$407$	0.0416833	0.00206616
$408$	0	0
$409$	20.8240	1.02968	0.514840	−	0.857286i	$-0.327851\pi$
$409$	20.8240	1.02968	0.514840	+	0.857286i	$0.327851\pi$
$410$	0	0
$411$	0	0
$412$	−13.8616	−0.682910
$413$	56.3421	2.77241
$414$	0	0
$415$	0	0
$416$	11.8962	0.583260
$417$	0	0
$418$	−7.30460	−0.357280
$419$	−11.3349	−0.553748	−0.276874	−	0.960906i	$-0.589298\pi$
$419$	−11.3349	−0.553748	−0.276874	+	0.960906i	$0.589298\pi$
$420$	0	0
$421$	−2.03813	−0.0993323	−0.0496662	−	0.998766i	$-0.515816\pi$
$421$	−2.03813	−0.0993323	−0.0496662	+	0.998766i	$0.515816\pi$
$422$	−13.1823	−0.641705
$423$	0	0
$424$	−26.5387	−1.28883
$425$	0	0
$426$	0	0
$427$	32.5091	1.57322
$428$	1.32483	0.0640379
$429$	0	0
$430$	0	0
$431$	17.9230	0.863319	0.431660	−	0.902037i	$-0.357928\pi$
$431$	17.9230	0.863319	0.431660	+	0.902037i	$0.357928\pi$
$432$	0	0
$433$	−14.0213	−0.673821	−0.336911	−	0.941537i	$-0.609382\pi$
$433$	−14.0213	−0.673821	−0.336911	+	0.941537i	$0.609382\pi$
$434$	−12.5001	−0.600024
$435$	0	0
$436$	−12.2973	−0.588936
$437$	18.9545	0.906718
$438$	0	0
$439$	−1.35014	−0.0644385	−0.0322192	−	0.999481i	$-0.510257\pi$
$439$	−1.35014	−0.0644385	−0.0322192	+	0.999481i	$0.510257\pi$
$440$	0	0
$441$	0	0
$442$	−14.8012	−0.704022
$443$	8.53290	0.405410	0.202705	−	0.979240i	$-0.435027\pi$
$443$	8.53290	0.405410	0.202705	+	0.979240i	$0.435027\pi$
$444$	0	0
$445$	0	0
$446$	19.4578	0.921353
$447$	0	0
$448$	27.1248	1.28153
$449$	33.6080	1.58606	0.793030	−	0.609183i	$-0.208503\pi$
$449$	33.6080	1.58606	0.793030	+	0.609183i	$0.208503\pi$
$450$	0	0
$451$	9.85425	0.464019
$452$	2.08511	0.0980753
$453$	0	0
$454$	−24.5451	−1.15196
$455$	0	0
$456$	0	0
$457$	15.8701	0.742372	0.371186	−	0.928559i	$-0.378951\pi$
$457$	15.8701	0.742372	0.371186	+	0.928559i	$0.378951\pi$
$458$	17.4282	0.814367
$459$	0	0
$460$	0	0
$461$	−12.1978	−0.568110	−0.284055	−	0.958808i	$-0.591680\pi$
$461$	−12.1978	−0.568110	−0.284055	+	0.958808i	$0.591680\pi$
$462$	0	0
$463$	−12.6538	−0.588070	−0.294035	−	0.955795i	$-0.594998\pi$
$463$	−12.6538	−0.588070	−0.294035	+	0.955795i	$0.594998\pi$
$464$	−0.692947	−0.0321693
$465$	0	0
$466$	9.13301	0.423078
$467$	−11.1891	−0.517769	−0.258884	−	0.965908i	$-0.583355\pi$
$467$	−11.1891	−0.517769	−0.258884	+	0.965908i	$0.583355\pi$
$468$	0	0
$469$	−28.2060	−1.30243
$470$	0	0
$471$	0	0
$472$	−31.6994	−1.45908
$473$	10.0826	0.463597
$474$	0	0
$475$	0	0
$476$	47.2680	2.16653
$477$	0	0
$478$	2.85320	0.130502
$479$	−12.5593	−0.573848	−0.286924	−	0.957953i	$-0.592633\pi$
$479$	−12.5593	−0.573848	−0.286924	+	0.957953i	$0.592633\pi$
$480$	0	0
$481$	−0.0336379	−0.00153376
$482$	11.1593	0.508291
$483$	0	0
$484$	4.80989	0.218631
$485$	0	0
$486$	0	0
$487$	−25.3662	−1.14945	−0.574726	−	0.818346i	$-0.694891\pi$
$487$	−25.3662	−1.14945	−0.574726	+	0.818346i	$0.694891\pi$
$488$	−18.2904	−0.827968
$489$	0	0
$490$	0	0
$491$	−20.6927	−0.933847	−0.466924	−	0.884298i	$-0.654638\pi$
$491$	−20.6927	−0.933847	−0.466924	+	0.884298i	$0.654638\pi$
$492$	0	0
$493$	−32.0413	−1.44307
$494$	5.89473	0.265217
$495$	0	0
$496$	0.459707	0.0206415
$497$	0.421942	0.0189267
$498$	0	0
$499$	40.3649	1.80698	0.903490	−	0.428608i	$-0.140996\pi$
$499$	40.3649	1.80698	0.903490	+	0.428608i	$0.140996\pi$
$500$	0	0
$501$	0	0
$502$	−1.33538	−0.0596010
$503$	−4.91088	−0.218966	−0.109483	−	0.993989i	$-0.534919\pi$
$503$	−4.91088	−0.218966	−0.109483	+	0.993989i	$0.534919\pi$
$504$	0	0
$505$	0	0
$506$	−14.5335	−0.646095
$507$	0	0
$508$	11.7115	0.519614
$509$	−16.5254	−0.732476	−0.366238	−	0.930521i	$-0.619354\pi$
$509$	−16.5254	−0.732476	−0.366238	+	0.930521i	$0.619354\pi$
$510$	0	0
$511$	−50.4932	−2.23369
$512$	−1.89476	−0.0837373
$513$	0	0
$514$	17.3390	0.764792
$515$	0	0
$516$	0	0
$517$	−2.37188	−0.104315
$518$	−0.0718741	−0.00315797
$519$	0	0
$520$	0	0
$521$	−10.2405	−0.448646	−0.224323	−	0.974515i	$-0.572017\pi$
$521$	−10.2405	−0.448646	−0.224323	+	0.974515i	$0.572017\pi$
$522$	0	0
$523$	−18.5451	−0.810919	−0.405459	−	0.914113i	$-0.632888\pi$
$523$	−18.5451	−0.810919	−0.405459	+	0.914113i	$0.632888\pi$
$524$	−13.1099	−0.572709
$525$	0	0
$526$	12.3281	0.537532
$527$	21.2564	0.925945
$528$	0	0
$529$	14.7128	0.639686
$530$	0	0
$531$	0	0
$532$	−18.8250	−0.816166
$533$	−7.95228	−0.344451
$534$	0	0
$535$	0	0
$536$	15.8694	0.685453
$537$	0	0
$538$	−2.55572	−0.110185
$539$	−49.9737	−2.15252
$540$	0	0
$541$	−22.9520	−0.986782	−0.493391	−	0.869808i	$-0.664243\pi$
$541$	−22.9520	−0.986782	−0.493391	+	0.869808i	$0.664243\pi$
$542$	19.6671	0.844773
$543$	0	0
$544$	−43.2245	−1.85323
$545$	0	0
$546$	0	0
$547$	31.3724	1.34139	0.670693	−	0.741735i	$-0.265997\pi$
$547$	31.3724	1.34139	0.670693	+	0.741735i	$0.265997\pi$
$548$	−20.1175	−0.859377
$549$	0	0
$550$	0	0
$551$	12.7608	0.543627
$552$	0	0
$553$	26.9791	1.14727
$554$	2.36180	0.100343
$555$	0	0
$556$	6.57063	0.278657
$557$	19.4590	0.824506	0.412253	−	0.911069i	$-0.364742\pi$
$557$	19.4590	0.824506	0.412253	+	0.911069i	$0.364742\pi$
$558$	0	0
$559$	−8.13653	−0.344138
$560$	0	0
$561$	0	0
$562$	7.30962	0.308338
$563$	14.0217	0.590944	0.295472	−	0.955351i	$-0.404523\pi$
$563$	14.0217	0.590944	0.295472	+	0.955351i	$0.404523\pi$
$564$	0	0
$565$	0	0
$566$	−5.42783	−0.228149
$567$	0	0
$568$	−0.237395	−0.00996086
$569$	40.4550	1.69596	0.847982	−	0.530025i	$-0.177818\pi$
$569$	40.4550	1.69596	0.847982	+	0.530025i	$0.177818\pi$
$570$	0	0
$571$	−10.9176	−0.456886	−0.228443	−	0.973557i	$-0.573363\pi$
$571$	−10.9176	−0.456886	−0.228443	+	0.973557i	$0.573363\pi$
$572$	6.75538	0.282457
$573$	0	0
$574$	−16.9916	−0.709216
$575$	0	0
$576$	0	0
$577$	18.0389	0.750968	0.375484	−	0.926829i	$-0.377476\pi$
$577$	18.0389	0.750968	0.375484	+	0.926829i	$0.377476\pi$
$578$	38.5580	1.60380
$579$	0	0
$580$	0	0
$581$	−3.68285	−0.152790
$582$	0	0
$583$	−24.4942	−1.01445
$584$	28.4087	1.17556
$585$	0	0
$586$	14.8582	0.613785
$587$	−10.1243	−0.417874	−0.208937	−	0.977929i	$-0.567000\pi$
$587$	−10.1243	−0.417874	−0.208937	+	0.977929i	$0.567000\pi$
$588$	0	0
$589$	−8.46560	−0.348819
$590$	0	0
$591$	0	0
$592$	0.00264326	0.000108637 0
$593$	−8.24833	−0.338718	−0.169359	−	0.985554i	$-0.554170\pi$
$593$	−8.24833	−0.338718	−0.169359	+	0.985554i	$0.554170\pi$
$594$	0	0
$595$	0	0
$596$	−0.327966	−0.0134340
$597$	0	0
$598$	11.7284	0.479611
$599$	−33.3982	−1.36461	−0.682307	−	0.731066i	$-0.739023\pi$
$599$	−33.3982	−1.36461	−0.682307	+	0.731066i	$0.739023\pi$
$600$	0	0
$601$	23.7251	0.967767	0.483884	−	0.875132i	$-0.339226\pi$
$601$	23.7251	0.967767	0.483884	+	0.875132i	$0.339226\pi$
$602$	−17.3853	−0.708572
$603$	0	0
$604$	−1.92175	−0.0781948
$605$	0	0
$606$	0	0
$607$	−30.5545	−1.24017	−0.620084	−	0.784535i	$-0.712902\pi$
$607$	−30.5545	−1.24017	−0.620084	+	0.784535i	$0.712902\pi$
$608$	17.2146	0.698144
$609$	0	0
$610$	0	0
$611$	1.91408	0.0774354
$612$	0	0
$613$	−9.49041	−0.383314	−0.191657	−	0.981462i	$-0.561386\pi$
$613$	−9.49041	−0.383314	−0.191657	+	0.981462i	$0.561386\pi$
$614$	13.2898	0.536331
$615$	0	0
$616$	38.5259	1.55225
$617$	28.4765	1.14642	0.573210	−	0.819408i	$-0.305698\pi$
$617$	28.4765	1.14642	0.573210	+	0.819408i	$0.305698\pi$
$618$	0	0
$619$	−6.76342	−0.271845	−0.135922	−	0.990719i	$-0.543400\pi$
$619$	−6.76342	−0.271845	−0.135922	+	0.990719i	$0.543400\pi$
$620$	0	0
$621$	0	0
$622$	−12.2186	−0.489920
$623$	67.4656	2.70295
$624$	0	0
$625$	0	0
$626$	2.84909	0.113873
$627$	0	0
$628$	−19.8328	−0.791415
$629$	0.122222	0.00487332
$630$	0	0
$631$	−11.2740	−0.448812	−0.224406	−	0.974496i	$-0.572044\pi$
$631$	−11.2740	−0.448812	−0.224406	+	0.974496i	$0.572044\pi$
$632$	−15.1791	−0.603791
$633$	0	0
$634$	−3.31821	−0.131783
$635$	0	0
$636$	0	0
$637$	40.3282	1.59786
$638$	−9.78442	−0.387369
$639$	0	0
$640$	0	0
$641$	−12.4989	−0.493676	−0.246838	−	0.969057i	$-0.579391\pi$
$641$	−12.4989	−0.493676	−0.246838	+	0.969057i	$0.579391\pi$
$642$	0	0
$643$	−15.6325	−0.616487	−0.308244	−	0.951307i	$-0.599741\pi$
$643$	−15.6325	−0.616487	−0.308244	+	0.951307i	$0.599741\pi$
$644$	−37.4550	−1.47593
$645$	0	0
$646$	−21.4183	−0.842692
$647$	34.4383	1.35391	0.676955	−	0.736025i	$-0.263299\pi$
$647$	34.4383	1.35391	0.676955	+	0.736025i	$0.263299\pi$
$648$	0	0
$649$	−29.2574	−1.14845
$650$	0	0
$651$	0	0
$652$	−27.5503	−1.07895
$653$	−34.6831	−1.35725	−0.678627	−	0.734483i	$-0.737425\pi$
$653$	−34.6831	−1.35725	−0.678627	+	0.734483i	$0.737425\pi$
$654$	0	0
$655$	0	0
$656$	0.624889	0.0243978
$657$	0	0
$658$	4.08981	0.159438
$659$	4.40271	0.171505	0.0857526	−	0.996316i	$-0.472671\pi$
$659$	4.40271	0.171505	0.0857526	+	0.996316i	$0.472671\pi$
$660$	0	0
$661$	14.4318	0.561333	0.280666	−	0.959805i	$-0.409445\pi$
$661$	14.4318	0.561333	0.280666	+	0.959805i	$0.409445\pi$
$662$	−28.9999	−1.12711
$663$	0	0
$664$	2.07206	0.0804115
$665$	0	0
$666$	0	0
$667$	25.3894	0.983080
$668$	−20.5133	−0.793685
$669$	0	0
$670$	0	0
$671$	−16.8814	−0.651698
$672$	0	0
$673$	2.75839	0.106328	0.0531640	−	0.998586i	$-0.483069\pi$
$673$	2.75839	0.106328	0.0531640	+	0.998586i	$0.483069\pi$
$674$	−19.6354	−0.756328
$675$	0	0
$676$	10.1260	0.389461
$677$	−25.9431	−0.997073	−0.498537	−	0.866869i	$-0.666129\pi$
$677$	−25.9431	−0.997073	−0.498537	+	0.866869i	$0.666129\pi$
$678$	0	0
$679$	−11.3422	−0.435273
$680$	0	0
$681$	0	0
$682$	6.49107	0.248556
$683$	16.5958	0.635019	0.317510	−	0.948255i	$-0.397153\pi$
$683$	16.5958	0.635019	0.317510	+	0.948255i	$0.397153\pi$
$684$	0	0
$685$	0	0
$686$	54.2670	2.07192
$687$	0	0
$688$	0.639367	0.0243756
$689$	19.7666	0.753047
$690$	0	0
$691$	12.4942	0.475300	0.237650	−	0.971351i	$-0.423623\pi$
$691$	12.4942	0.475300	0.237650	+	0.971351i	$0.423623\pi$
$692$	3.96189	0.150608
$693$	0	0
$694$	22.2337	0.843978
$695$	0	0
$696$	0	0
$697$	28.8943	1.09445
$698$	−1.78382	−0.0675185
$699$	0	0
$700$	0	0
$701$	−50.7235	−1.91580	−0.957900	−	0.287101i	$-0.907308\pi$
$701$	−50.7235	−1.91580	−0.957900	+	0.287101i	$0.907308\pi$
$702$	0	0
$703$	−0.0486762	−0.00183586
$704$	−14.0854	−0.530865
$705$	0	0
$706$	−1.79971	−0.0677331
$707$	−78.8470	−2.96535
$708$	0	0
$709$	−11.7870	−0.442671	−0.221335	−	0.975198i	$-0.571042\pi$
$709$	−11.7870	−0.442671	−0.221335	+	0.975198i	$0.571042\pi$
$710$	0	0
$711$	0	0
$712$	−37.9578	−1.42253
$713$	−16.8435	−0.630795
$714$	0	0
$715$	0	0
$716$	14.5411	0.543428
$717$	0	0
$718$	−21.6755	−0.808921
$719$	−7.52191	−0.280520	−0.140260	−	0.990115i	$-0.544794\pi$
$719$	−7.52191	−0.280520	−0.140260	+	0.990115i	$0.544794\pi$
$720$	0	0
$721$	58.8800	2.19281
$722$	−8.48242	−0.315683
$723$	0	0
$724$	−18.6834	−0.694364
$725$	0	0
$726$	0	0
$727$	−7.51316	−0.278648	−0.139324	−	0.990247i	$-0.544493\pi$
$727$	−7.51316	−0.278648	−0.139324	+	0.990247i	$0.544493\pi$
$728$	−31.0900	−1.15227
$729$	0	0
$730$	0	0
$731$	29.5638	1.09346
$732$	0	0
$733$	−4.39758	−0.162428	−0.0812141	−	0.996697i	$-0.525880\pi$
$733$	−4.39758	−0.162428	−0.0812141	+	0.996697i	$0.525880\pi$
$734$	−0.530811	−0.0195926
$735$	0	0
$736$	34.2509	1.26250
$737$	14.6469	0.539524
$738$	0	0
$739$	−45.4541	−1.67206	−0.836028	−	0.548686i	$-0.815128\pi$
$739$	−45.4541	−1.67206	−0.836028	+	0.548686i	$0.815128\pi$
$740$	0	0
$741$	0	0
$742$	42.2352	1.55050
$743$	−45.0800	−1.65382	−0.826912	−	0.562331i	$-0.809905\pi$
$743$	−45.0800	−1.65382	−0.826912	+	0.562331i	$0.809905\pi$
$744$	0	0
$745$	0	0
$746$	7.54384	0.276199
$747$	0	0
$748$	−24.5454	−0.897469
$749$	−5.62749	−0.205624
$750$	0	0
$751$	−14.7555	−0.538435	−0.269217	−	0.963079i	$-0.586765\pi$
$751$	−14.7555	−0.538435	−0.269217	+	0.963079i	$0.586765\pi$
$752$	−0.150408	−0.00548482
$753$	0	0
$754$	7.89592	0.287552
$755$	0	0
$756$	0	0
$757$	−41.8730	−1.52190	−0.760950	−	0.648811i	$-0.775267\pi$
$757$	−41.8730	−1.52190	−0.760950	+	0.648811i	$0.775267\pi$
$758$	−16.6660	−0.605336
$759$	0	0
$760$	0	0
$761$	1.58549	0.0574738	0.0287369	−	0.999587i	$-0.490851\pi$
$761$	1.58549	0.0574738	0.0287369	+	0.999587i	$0.490851\pi$
$762$	0	0
$763$	52.2356	1.89106
$764$	−5.77596	−0.208967
$765$	0	0
$766$	8.90275	0.321669
$767$	23.6104	0.852522
$768$	0	0
$769$	17.0820	0.615993	0.307996	−	0.951388i	$-0.400342\pi$
$769$	17.0820	0.615993	0.307996	+	0.951388i	$0.400342\pi$
$770$	0	0
$771$	0	0
$772$	21.3157	0.767169
$773$	−51.0507	−1.83617	−0.918083	−	0.396388i	$-0.870263\pi$
$773$	−51.0507	−1.83617	−0.918083	+	0.396388i	$0.870263\pi$
$774$	0	0
$775$	0	0
$776$	6.38138	0.229078
$777$	0	0
$778$	5.31479	0.190545
$779$	−11.5075	−0.412297
$780$	0	0
$781$	−0.219107	−0.00784025
$782$	−42.6148	−1.52390
$783$	0	0
$784$	−3.16899	−0.113178
$785$	0	0
$786$	0	0
$787$	−41.2363	−1.46992	−0.734958	−	0.678113i	$-0.762798\pi$
$787$	−41.2363	−1.46992	−0.734958	+	0.678113i	$0.762798\pi$
$788$	−26.6282	−0.948590
$789$	0	0
$790$	0	0
$791$	−8.85696	−0.314917
$792$	0	0
$793$	13.6231	0.483770
$794$	−2.75496	−0.0977699
$795$	0	0
$796$	19.6142	0.695206
$797$	−38.0725	−1.34860	−0.674299	−	0.738458i	$-0.735554\pi$
$797$	−38.0725	−1.34860	−0.674299	+	0.738458i	$0.735554\pi$
$798$	0	0
$799$	−6.95474	−0.246041
$800$	0	0
$801$	0	0
$802$	−18.4782	−0.652489
$803$	26.2202	0.925290
$804$	0	0
$805$	0	0
$806$	−5.23822	−0.184508
$807$	0	0
$808$	44.3613	1.56062
$809$	−33.8845	−1.19131	−0.595657	−	0.803239i	$-0.703108\pi$
$809$	−33.8845	−1.19131	−0.595657	+	0.803239i	$0.703108\pi$
$810$	0	0
$811$	9.04529	0.317623	0.158812	−	0.987309i	$-0.449234\pi$
$811$	9.04529	0.317623	0.158812	+	0.987309i	$0.449234\pi$
$812$	−25.2158	−0.884902
$813$	0	0
$814$	0.0373229	0.00130817
$815$	0	0
$816$	0	0
$817$	−11.7741	−0.411923
$818$	18.6457	0.651930
$819$	0	0
$820$	0	0
$821$	−42.7195	−1.49092	−0.745461	−	0.666549i	$-0.767771\pi$
$821$	−42.7195	−1.49092	−0.745461	+	0.666549i	$0.767771\pi$
$822$	0	0
$823$	−47.3852	−1.65174	−0.825872	−	0.563858i	$-0.809317\pi$
$823$	−47.3852	−1.65174	−0.825872	+	0.563858i	$0.809317\pi$
$824$	−33.1273	−1.15404
$825$	0	0
$826$	50.4483	1.75532
$827$	39.0928	1.35939	0.679695	−	0.733495i	$-0.262112\pi$
$827$	39.0928	1.35939	0.679695	+	0.733495i	$0.262112\pi$
$828$	0	0
$829$	30.5779	1.06202	0.531008	−	0.847367i	$-0.321814\pi$
$829$	30.5779	1.06202	0.531008	+	0.847367i	$0.321814\pi$
$830$	0	0
$831$	0	0
$832$	11.3668	0.394073
$833$	−146.531	−5.07701
$834$	0	0
$835$	0	0
$836$	9.77546	0.338091
$837$	0	0
$838$	−10.1492	−0.350599
$839$	−30.3660	−1.04835	−0.524176	−	0.851610i	$-0.675627\pi$
$839$	−30.3660	−1.04835	−0.524176	+	0.851610i	$0.675627\pi$
$840$	0	0
$841$	−11.9071	−0.410590
$842$	−1.82493	−0.0628911
$843$	0	0
$844$	17.6414	0.607241
$845$	0	0
$846$	0	0
$847$	−20.4310	−0.702019
$848$	−1.55326	−0.0533390
$849$	0	0
$850$	0	0
$851$	−0.0968484	−0.00331992
$852$	0	0
$853$	−3.41673	−0.116987	−0.0584933	−	0.998288i	$-0.518630\pi$
$853$	−3.41673	−0.116987	−0.0584933	+	0.998288i	$0.518630\pi$
$854$	29.1084	0.996070
$855$	0	0
$856$	3.16616	0.108217
$857$	42.0948	1.43793	0.718966	−	0.695045i	$-0.244616\pi$
$857$	42.0948	1.43793	0.718966	+	0.695045i	$0.244616\pi$
$858$	0	0
$859$	−33.1680	−1.13168	−0.565840	−	0.824515i	$-0.691448\pi$
$859$	−33.1680	−1.13168	−0.565840	+	0.824515i	$0.691448\pi$
$860$	0	0
$861$	0	0
$862$	16.0481	0.546601
$863$	15.5446	0.529143	0.264572	−	0.964366i	$-0.414769\pi$
$863$	15.5446	0.529143	0.264572	+	0.964366i	$0.414769\pi$
$864$	0	0
$865$	0	0
$866$	−12.5546	−0.426622
$867$	0	0
$868$	16.7284	0.567798
$869$	−14.0097	−0.475248
$870$	0	0
$871$	−11.8199	−0.400500
$872$	−29.3890	−0.995238
$873$	0	0
$874$	16.9718	0.574079
$875$	0	0
$876$	0	0
$877$	−35.7948	−1.20871	−0.604353	−	0.796717i	$-0.706568\pi$
$877$	−35.7948	−1.20871	−0.604353	+	0.796717i	$0.706568\pi$
$878$	−1.20890	−0.0407985
$879$	0	0
$880$	0	0
$881$	−32.9640	−1.11059	−0.555293	−	0.831655i	$-0.687394\pi$
$881$	−32.9640	−1.11059	−0.555293	+	0.831655i	$0.687394\pi$
$882$	0	0
$883$	19.2788	0.648784	0.324392	−	0.945923i	$-0.394840\pi$
$883$	19.2788	0.648784	0.324392	+	0.945923i	$0.394840\pi$
$884$	19.8079	0.666211
$885$	0	0
$886$	7.64030	0.256681
$887$	26.4213	0.887141	0.443571	−	0.896239i	$-0.353712\pi$
$887$	26.4213	0.887141	0.443571	+	0.896239i	$0.353712\pi$
$888$	0	0
$889$	−49.7472	−1.66847
$890$	0	0
$891$	0	0
$892$	−26.0396	−0.871870
$893$	2.76980	0.0926877
$894$	0	0
$895$	0	0
$896$	−32.4890	−1.08538
$897$	0	0
$898$	30.0924	1.00420
$899$	−11.3396	−0.378196
$900$	0	0
$901$	−71.8212	−2.39271
$902$	8.82343	0.293788
$903$	0	0
$904$	4.98314	0.165737
$905$	0	0
$906$	0	0
$907$	4.32841	0.143722	0.0718612	−	0.997415i	$-0.477106\pi$
$907$	4.32841	0.143722	0.0718612	+	0.997415i	$0.477106\pi$
$908$	32.8478	1.09009
$909$	0	0
$910$	0	0
$911$	−32.8983	−1.08997	−0.544985	−	0.838446i	$-0.683465\pi$
$911$	−32.8983	−1.08997	−0.544985	+	0.838446i	$0.683465\pi$
$912$	0	0
$913$	1.91243	0.0632923
$914$	14.2100	0.470024
$915$	0	0
$916$	−23.3235	−0.770630
$917$	55.6872	1.83895
$918$	0	0
$919$	49.8351	1.64391	0.821954	−	0.569554i	$-0.192884\pi$
$919$	49.8351	1.64391	0.821954	+	0.569554i	$0.192884\pi$
$920$	0	0
$921$	0	0
$922$	−10.9219	−0.359692
$923$	0.176817	0.00581999
$924$	0	0
$925$	0	0
$926$	−11.3301	−0.372330
$927$	0	0
$928$	23.0587	0.756940
$929$	−16.1832	−0.530955	−0.265477	−	0.964117i	$-0.585530\pi$
$929$	−16.1832	−0.530955	−0.265477	+	0.964117i	$0.585530\pi$
$930$	0	0
$931$	58.3575	1.91259
$932$	−12.2224	−0.400356
$933$	0	0
$934$	−10.0186	−0.327820
$935$	0	0
$936$	0	0
$937$	34.9825	1.14283	0.571413	−	0.820662i	$-0.306395\pi$
$937$	34.9825	1.14283	0.571413	+	0.820662i	$0.306395\pi$
$938$	−25.2555	−0.824620
$939$	0	0
$940$	0	0
$941$	−51.6450	−1.68358	−0.841790	−	0.539805i	$-0.818498\pi$
$941$	−51.6450	−1.68358	−0.841790	+	0.539805i	$0.818498\pi$
$942$	0	0
$943$	−22.8957	−0.745587
$944$	−1.85530	−0.0603849
$945$	0	0
$946$	9.02787	0.293521
$947$	1.85002	0.0601176	0.0300588	−	0.999548i	$-0.490431\pi$
$947$	1.85002	0.0601176	0.0300588	+	0.999548i	$0.490431\pi$
$948$	0	0
$949$	−21.1594	−0.686863
$950$	0	0
$951$	0	0
$952$	112.964	3.66120
$953$	2.18860	0.0708956	0.0354478	−	0.999372i	$-0.488714\pi$
$953$	2.18860	0.0708956	0.0354478	+	0.999372i	$0.488714\pi$
$954$	0	0
$955$	0	0
$956$	−3.81833	−0.123494
$957$	0	0
$958$	−11.2455	−0.363326
$959$	85.4534	2.75944
$960$	0	0
$961$	−23.4772	−0.757330
$962$	−0.0301192	−0.000971082 0
$963$	0	0
$964$	−14.9340	−0.480992
$965$	0	0
$966$	0	0
$967$	−13.2573	−0.426326	−0.213163	−	0.977017i	$-0.568377\pi$
$967$	−13.2573	−0.426326	−0.213163	+	0.977017i	$0.568377\pi$
$968$	11.4950	0.369463
$969$	0	0
$970$	0	0
$971$	35.2892	1.13248	0.566242	−	0.824239i	$-0.308397\pi$
$971$	35.2892	1.13248	0.566242	+	0.824239i	$0.308397\pi$
$972$	0	0
$973$	−27.9102	−0.894759
$974$	−22.7127	−0.727763
$975$	0	0
$976$	−1.07050	−0.0342659
$977$	−12.6542	−0.404844	−0.202422	−	0.979298i	$-0.564881\pi$
$977$	−12.6542	−0.404844	−0.202422	+	0.979298i	$0.564881\pi$
$978$	0	0
$979$	−35.0336	−1.11968
$980$	0	0
$981$	0	0
$982$	−18.5281	−0.591255
$983$	−22.3964	−0.714335	−0.357167	−	0.934040i	$-0.616257\pi$
$983$	−22.3964	−0.714335	−0.357167	+	0.934040i	$0.616257\pi$
$984$	0	0
$985$	0	0
$986$	−28.6895	−0.913661
$987$	0	0
$988$	−7.88869	−0.250973
$989$	−23.4262	−0.744910
$990$	0	0
$991$	−60.7912	−1.93110	−0.965548	−	0.260224i	$-0.916204\pi$
$991$	−60.7912	−1.93110	−0.965548	+	0.260224i	$0.916204\pi$
$992$	−15.2974	−0.485691
$993$	0	0
$994$	0.377804	0.0119832
$995$	0	0
$996$	0	0
$997$	−42.5995	−1.34914	−0.674570	−	0.738211i	$-0.735671\pi$
$997$	−42.5995	−1.34914	−0.674570	+	0.738211i	$0.735671\pi$
$998$	36.1425	1.14407
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.u.1.6		8
3.2	odd	2		1875.2.a.o.1.3	yes	8
5.4	even	2		5625.2.a.bc.1.3		8
15.2	even	4		1875.2.b.g.1249.6		16
15.8	even	4		1875.2.b.g.1249.11		16
15.14	odd	2		1875.2.a.n.1.6	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.n.1.6	✓	8	15.14	odd	2
1875.2.a.o.1.3	yes	8	3.2	odd	2
1875.2.b.g.1249.6		16	15.2	even	4
1875.2.b.g.1249.11		16	15.8	even	4
5625.2.a.u.1.6		8	1.1	even	1	trivial
5625.2.a.bc.1.3		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)

\(f(q)\)	\(=\)	\(q+0.895394 q^{2} -1.19827 q^{4} +5.08992 q^{7} -2.86371 q^{8} +O(q^{10})\)	\(q+0.895394 q^{2} -1.19827 q^{4} +5.08992 q^{7} -2.86371 q^{8} -2.64310 q^{11} +2.13295 q^{13} +4.55748 q^{14} -0.167607 q^{16} -7.75001 q^{17} +3.08652 q^{19} -2.36661 q^{22} +6.14107 q^{23} +1.90983 q^{26} -6.09909 q^{28} +4.13435 q^{29} -2.74277 q^{31} +5.57735 q^{32} -6.93931 q^{34} -0.0157706 q^{37} +2.76365 q^{38} -3.72829 q^{41} -3.81468 q^{43} +3.16715 q^{44} +5.49868 q^{46} +0.897385 q^{47} +18.9072 q^{49} -2.55585 q^{52} +9.26724 q^{53} -14.5760 q^{56} +3.70187 q^{58} +11.0693 q^{59} +6.38696 q^{61} -2.45585 q^{62} +5.32913 q^{64} -5.54154 q^{67} +9.28660 q^{68} +0.0828976 q^{71} -9.92024 q^{73} -0.0141209 q^{74} -3.69848 q^{76} -13.4532 q^{77} +5.30049 q^{79} -3.33829 q^{82} -0.723557 q^{83} -3.41564 q^{86} +7.56907 q^{88} +13.2548 q^{89} +10.8565 q^{91} -7.35867 q^{92} +0.803513 q^{94} -2.22836 q^{97} +16.9294 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 9 q^{4} + 12 q^{7} - 3 q^{8}+O(q^{10}) \) 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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