LMFDB - Embedded newform 5625.2.a.bc.1.3

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			1.3
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	$-0.602531\pi$
$2$	−0.895394	−0.633139	−0.316569	+	0.948569i	$0.602531\pi$
$3$	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	$-0.911855\pi$
$7$	−5.08992	−1.92381	−0.961904	+	0.273389i	$0.911855\pi$
$8$	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	$-0.595582\pi$
$13$	−2.13295	−0.591575	−0.295787	+	0.955254i	$0.595582\pi$
$14$	4.55748	1.21804
$15$	0	0
$16$	−0.167607	−0.0419018
$17$	7.75001	1.87965	0.939826	−	0.341653i	$-0.110987\pi$
$17$	7.75001	1.87965	0.939826	+	0.341653i	$0.110987\pi$
$18$	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	$-0.721170\pi$
$23$	−6.14107	−1.28050	−0.640251	+	0.768166i	$0.721170\pi$
$24$	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	$-0.499587\pi$
$37$	0.0157706	0.00259267	0.00129634	+	0.999999i	$0.499587\pi$
$38$	−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	$-0.406056\pi$
$43$	3.81468	0.581733	0.290866	+	0.956764i	$0.406056\pi$
$44$	3.16715	0.477466
$45$	0	0
$46$	5.49868	0.810736
$47$	−0.897385	−0.130897	−0.0654486	−	0.997856i	$-0.520848\pi$
$47$	−0.897385	−0.130897	−0.0654486	+	0.997856i	$0.520848\pi$
$48$	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	$-0.719609\pi$
$53$	−9.26724	−1.27295	−0.636477	+	0.771296i	$0.719609\pi$
$54$	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	$-0.390079\pi$
$67$	5.54154	0.677007	0.338504	+	0.940965i	$0.390079\pi$
$68$	−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	$-0.302842\pi$
$73$	9.92024	1.16108	0.580538	+	0.814233i	$0.302842\pi$
$74$	−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	$-0.487356\pi$
$83$	0.723557	0.0794207	0.0397104	+	0.999211i	$0.487356\pi$
$84$	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	$-0.463913\pi$
$97$	2.22836	0.226256	0.113128	+	0.993580i	$0.463913\pi$
$98$	−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	$-0.693023\pi$
$103$	−11.5680	−1.13983	−0.569913	+	0.821705i	$0.693023\pi$
$104$	−6.10816	−0.598954
$105$	0	0
$106$	8.29783	0.805956
$107$	1.10562	0.106884	0.0534419	−	0.998571i	$-0.482981\pi$
$107$	1.10562	0.106884	0.0534419	+	0.998571i	$0.482981\pi$
$108$	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	$-0.473918\pi$
$113$	1.74010	0.163695	0.0818474	+	0.996645i	$0.473918\pi$
$114$	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	$-0.357230\pi$
$127$	9.77368	0.867274	0.433637	+	0.901088i	$0.357230\pi$
$128$	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	$-0.754568\pi$
$137$	−16.7888	−1.43436	−0.717181	+	0.696887i	$0.754568\pi$
$138$	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	$-0.729641\pi$
$157$	−16.5512	−1.32093	−0.660465	+	0.750857i	$0.729641\pi$
$158$	−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	$-0.856743\pi$
$163$	−22.9917	−1.80085	−0.900424	+	0.435014i	$0.856743\pi$
$164$	4.46750	0.348853
$165$	0	0
$166$	−0.647868	−0.0502843
$167$	−17.1191	−1.32472	−0.662359	−	0.749187i	$-0.730445\pi$
$167$	−17.1191	−1.32472	−0.662359	+	0.749187i	$0.730445\pi$
$168$	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	$-0.459886\pi$
$173$	3.30634	0.251376	0.125688	+	0.992070i	$0.459886\pi$
$174$	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	$-0.278839\pi$
$193$	17.7887	1.28046	0.640230	+	0.768183i	$0.278839\pi$
$194$	−1.99526	−0.143251
$195$	0	0
$196$	−22.6560	−1.61828
$197$	−22.2222	−1.58327	−0.791633	−	0.610997i	$-0.790769\pi$
$197$	−22.2222	−1.58327	−0.791633	+	0.610997i	$0.790769\pi$
$198$	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	$-0.759368\pi$
$223$	−21.7310	−1.45521	−0.727607	+	0.685994i	$0.759368\pi$
$224$	28.3882	1.89677
$225$	0	0
$226$	−1.55807	−0.103642
$227$	27.4127	1.81944	0.909722	−	0.415218i	$-0.136295\pi$
$227$	27.4127	1.81944	0.909722	+	0.415218i	$0.136295\pi$
$228$	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	$-0.608436\pi$
$233$	−10.2000	−0.668224	−0.334112	+	0.942533i	$0.608436\pi$
$234$	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	$-0.706415\pi$
$257$	−19.3647	−1.20794	−0.603969	+	0.797008i	$0.706415\pi$
$258$	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	$-0.639549\pi$
$263$	−13.7684	−0.848996	−0.424498	+	0.905429i	$0.639549\pi$
$264$	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	$-0.525250\pi$
$277$	−2.63773	−0.158486	−0.0792428	+	0.996855i	$0.525250\pi$
$278$	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	$-0.442334\pi$
$283$	6.06194	0.360345	0.180173	+	0.983635i	$0.442334\pi$
$284$	−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	$-0.661077\pi$
$293$	−16.5940	−0.969432	−0.484716	+	0.874672i	$0.661077\pi$
$294$	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	$-0.639216\pi$
$307$	−14.8424	−0.847098	−0.423549	+	0.905873i	$0.639216\pi$
$308$	−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	$-0.528663\pi$
$313$	−3.18194	−0.179854	−0.0899270	+	0.995948i	$0.528663\pi$
$314$	14.8198	0.836332
$315$	0	0
$316$	−6.35142	−0.357295
$317$	3.70586	0.208142	0.104071	−	0.994570i	$-0.466813\pi$
$317$	3.70586	0.208142	0.104071	+	0.994570i	$0.466813\pi$
$318$	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	$-0.296246\pi$
$337$	21.9294	1.19457	0.597285	+	0.802029i	$0.296246\pi$
$338$	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	$-0.732210\pi$
$347$	−24.8312	−1.33301	−0.666503	+	0.745502i	$0.732210\pi$
$348$	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	$-0.482965\pi$
$353$	2.00997	0.106980	0.0534900	+	0.998568i	$0.482965\pi$
$354$	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	$-0.495075\pi$
$367$	0.592824	0.0309452	0.0154726	+	0.999880i	$0.495075\pi$
$368$	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	$-0.569992\pi$
$373$	−8.42516	−0.436238	−0.218119	+	0.975922i	$0.569992\pi$
$374$	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	$-0.581755\pi$
$383$	−9.94283	−0.508055	−0.254027	+	0.967197i	$0.581755\pi$
$384$	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	$-0.475399\pi$
$397$	3.07681	0.154421	0.0772105	+	0.997015i	$0.475399\pi$
$398$	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	$-0.390618\pi$
$433$	14.0213	0.673821	0.336911	+	0.941537i	$0.390618\pi$
$434$	−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	$-0.564973\pi$
$443$	−8.53290	−0.405410	−0.202705	+	0.979240i	$0.564973\pi$
$444$	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	$-0.621049\pi$
$457$	−15.8701	−0.742372	−0.371186	+	0.928559i	$0.621049\pi$
$458$	−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	$-0.405002\pi$
$463$	12.6538	0.588070	0.294035	+	0.955795i	$0.405002\pi$
$464$	−0.692947	−0.0321693
$465$	0	0
$466$	9.13301	0.423078
$467$	11.1891	0.517769	0.258884	−	0.965908i	$-0.416645\pi$
$467$	11.1891	0.517769	0.258884	+	0.965908i	$0.416645\pi$
$468$	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	$-0.305109\pi$
$487$	25.3662	1.14945	0.574726	+	0.818346i	$0.305109\pi$
$488$	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	$-0.465081\pi$
$503$	4.91088	0.218966	0.109483	+	0.993989i	$0.465081\pi$
$504$	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	$-0.367112\pi$
$523$	18.5451	0.810919	0.405459	+	0.914113i	$0.367112\pi$
$524$	−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	$-0.734003\pi$
$547$	−31.3724	−1.34139	−0.670693	+	0.741735i	$0.734003\pi$
$548$	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	$-0.635258\pi$
$557$	−19.4590	−0.824506	−0.412253	+	0.911069i	$0.635258\pi$
$558$	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	$-0.595477\pi$
$563$	−14.0217	−0.590944	−0.295472	+	0.955351i	$0.595477\pi$
$564$	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	$-0.622524\pi$
$577$	−18.0389	−0.750968	−0.375484	+	0.926829i	$0.622524\pi$
$578$	−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	$-0.433000\pi$
$587$	10.1243	0.417874	0.208937	+	0.977929i	$0.433000\pi$
$588$	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	$-0.445830\pi$
$593$	8.24833	0.338718	0.169359	+	0.985554i	$0.445830\pi$
$594$	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	$-0.287098\pi$
$607$	30.5545	1.24017	0.620084	+	0.784535i	$0.287098\pi$
$608$	−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	$-0.438614\pi$
$613$	9.49041	0.383314	0.191657	+	0.981462i	$0.438614\pi$
$614$	13.2898	0.536331
$615$	0	0
$616$	38.5259	1.55225
$617$	−28.4765	−1.14642	−0.573210	−	0.819408i	$-0.694302\pi$
$617$	−28.4765	−1.14642	−0.573210	+	0.819408i	$0.694302\pi$
$618$	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	$-0.400259\pi$
$643$	15.6325	0.616487	0.308244	+	0.951307i	$0.400259\pi$
$644$	−37.4550	−1.47593
$645$	0	0
$646$	−21.4183	−0.842692
$647$	−34.4383	−1.35391	−0.676955	−	0.736025i	$-0.736701\pi$
$647$	−34.4383	−1.35391	−0.676955	+	0.736025i	$0.736701\pi$
$648$	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	$-0.262575\pi$
$653$	34.6831	1.35725	0.678627	+	0.734483i	$0.262575\pi$
$654$	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	$-0.516931\pi$
$673$	−2.75839	−0.106328	−0.0531640	+	0.998586i	$0.516931\pi$
$674$	−19.6354	−0.756328
$675$	0	0
$676$	10.1260	0.389461
$677$	25.9431	0.997073	0.498537	−	0.866869i	$-0.333871\pi$
$677$	25.9431	0.997073	0.498537	+	0.866869i	$0.333871\pi$
$678$	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	$-0.602847\pi$
$683$	−16.5958	−0.635019	−0.317510	+	0.948255i	$0.602847\pi$
$684$	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	$-0.455507\pi$
$727$	7.51316	0.278648	0.139324	+	0.990247i	$0.455507\pi$
$728$	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	$-0.474120\pi$
$733$	4.39758	0.162428	0.0812141	+	0.996697i	$0.474120\pi$
$734$	−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	$-0.190095\pi$
$743$	45.0800	1.65382	0.826912	+	0.562331i	$0.190095\pi$
$744$	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	$-0.224733\pi$
$757$	41.8730	1.52190	0.760950	+	0.648811i	$0.224733\pi$
$758$	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	$-0.129737\pi$
$773$	51.0507	1.83617	0.918083	+	0.396388i	$0.129737\pi$
$774$	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	$-0.237202\pi$
$787$	41.2363	1.46992	0.734958	+	0.678113i	$0.237202\pi$
$788$	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	$-0.264446\pi$
$797$	38.0725	1.34860	0.674299	+	0.738458i	$0.264446\pi$
$798$	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	$-0.190683\pi$
$823$	47.3852	1.65174	0.825872	+	0.563858i	$0.190683\pi$
$824$	−33.1273	−1.15404
$825$	0	0
$826$	50.4483	1.75532
$827$	−39.0928	−1.35939	−0.679695	−	0.733495i	$-0.737888\pi$
$827$	−39.0928	−1.35939	−0.679695	+	0.733495i	$0.737888\pi$
$828$	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	$-0.481370\pi$
$853$	3.41673	0.116987	0.0584933	+	0.998288i	$0.481370\pi$
$854$	29.1084	0.996070
$855$	0	0
$856$	3.16616	0.108217
$857$	−42.0948	−1.43793	−0.718966	−	0.695045i	$-0.755384\pi$
$857$	−42.0948	−1.43793	−0.718966	+	0.695045i	$0.755384\pi$
$858$	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	$-0.585231\pi$
$863$	−15.5446	−0.529143	−0.264572	+	0.964366i	$0.585231\pi$
$864$	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	$-0.293432\pi$
$877$	35.7948	1.20871	0.604353	+	0.796717i	$0.293432\pi$
$878$	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	$-0.605160\pi$
$883$	−19.2788	−0.648784	−0.324392	+	0.945923i	$0.605160\pi$
$884$	19.8079	0.666211
$885$	0	0
$886$	7.64030	0.256681
$887$	−26.4213	−0.887141	−0.443571	−	0.896239i	$-0.646288\pi$
$887$	−26.4213	−0.887141	−0.443571	+	0.896239i	$0.646288\pi$
$888$	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	$-0.522894\pi$
$907$	−4.32841	−0.143722	−0.0718612	+	0.997415i	$0.522894\pi$
$908$	−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	$-0.693605\pi$
$937$	−34.9825	−1.14283	−0.571413	+	0.820662i	$0.693605\pi$
$938$	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	$-0.509569\pi$
$947$	−1.85002	−0.0601176	−0.0300588	+	0.999548i	$0.509569\pi$
$948$	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	$-0.511286\pi$
$953$	−2.18860	−0.0708956	−0.0354478	+	0.999372i	$0.511286\pi$
$954$	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	$-0.431623\pi$
$967$	13.2573	0.426326	0.213163	+	0.977017i	$0.431623\pi$
$968$	−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	$-0.435119\pi$
$977$	12.6542	0.404844	0.202422	+	0.979298i	$0.435119\pi$
$978$	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	$-0.383743\pi$
$983$	22.3964	0.714335	0.357167	+	0.934040i	$0.383743\pi$
$984$	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	$-0.264329\pi$
$997$	42.5995	1.34914	0.674570	+	0.738211i	$0.264329\pi$
$998$	−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.bc.1.3		8
3.2	odd	2		1875.2.a.n.1.6	✓	8
5.4	even	2		5625.2.a.u.1.6		8
15.2	even	4		1875.2.b.g.1249.11		16
15.8	even	4		1875.2.b.g.1249.6		16
15.14	odd	2		1875.2.a.o.1.3	yes	8

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

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

Modular forms

Varieties

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Representations

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Database

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