LMFDB - Embedded newform 5625.2.a.g.1.1

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:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
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			1.1
Root			$-0.618034$ 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)

$f(q)$	$=$	$q-0.618034 q^{2} -1.61803 q^{4} +2.00000 q^{7} +2.23607 q^{8} +O(q^{10})$	$q-0.618034 q^{2} -1.61803 q^{4} +2.00000 q^{7} +2.23607 q^{8} +3.00000 q^{11} +1.00000 q^{13} -1.23607 q^{14} +1.85410 q^{16} +0.236068 q^{17} +6.70820 q^{19} -1.85410 q^{22} +7.61803 q^{23} -0.618034 q^{26} -3.23607 q^{28} +1.38197 q^{29} -4.70820 q^{31} -5.61803 q^{32} -0.145898 q^{34} +2.00000 q^{37} -4.14590 q^{38} +11.6180 q^{41} +9.61803 q^{43} -4.85410 q^{44} -4.70820 q^{46} -9.23607 q^{47} -3.00000 q^{49} -1.61803 q^{52} +6.76393 q^{53} +4.47214 q^{56} -0.854102 q^{58} +13.9443 q^{59} -4.70820 q^{61} +2.90983 q^{62} -0.236068 q^{64} -9.18034 q^{67} -0.381966 q^{68} +1.09017 q^{71} -2.29180 q^{73} -1.23607 q^{74} -10.8541 q^{76} +6.00000 q^{77} -15.8541 q^{79} -7.18034 q^{82} +9.00000 q^{83} -5.94427 q^{86} +6.70820 q^{88} -11.1803 q^{89} +2.00000 q^{91} -12.3262 q^{92} +5.70820 q^{94} +2.85410 q^{97} +1.85410 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} + 4 q^{7}+O(q^{10}) $ 2 * q + q^2 - q^4 + 4 * q^7	$ 2 q + q^{2} - q^{4} + 4 q^{7} + 6 q^{11} + 2 q^{13} + 2 q^{14} - 3 q^{16} - 4 q^{17} + 3 q^{22} + 13 q^{23} + q^{26} - 2 q^{28} + 5 q^{29} + 4 q^{31} - 9 q^{32} - 7 q^{34} + 4 q^{37} - 15 q^{38} + 21 q^{41} + 17 q^{43} - 3 q^{44} + 4 q^{46} - 14 q^{47} - 6 q^{49} - q^{52} + 18 q^{53} + 5 q^{58} + 10 q^{59} + 4 q^{61} + 17 q^{62} + 4 q^{64} + 4 q^{67} - 3 q^{68} - 9 q^{71} - 18 q^{73} + 2 q^{74} - 15 q^{76} + 12 q^{77} - 25 q^{79} + 8 q^{82} + 18 q^{83} + 6 q^{86} + 4 q^{91} - 9 q^{92} - 2 q^{94} - q^{97} - 3 q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 + 4 * q^7 + 6 * q^11 + 2 * q^13 + 2 * q^14 - 3 * q^16 - 4 * q^17 + 3 * q^22 + 13 * q^23 + q^26 - 2 * q^28 + 5 * q^29 + 4 * q^31 - 9 * q^32 - 7 * q^34 + 4 * q^37 - 15 * q^38 + 21 * q^41 + 17 * q^43 - 3 * q^44 + 4 * q^46 - 14 * q^47 - 6 * q^49 - q^52 + 18 * q^53 + 5 * q^58 + 10 * q^59 + 4 * q^61 + 17 * q^62 + 4 * q^64 + 4 * q^67 - 3 * q^68 - 9 * q^71 - 18 * q^73 + 2 * q^74 - 15 * q^76 + 12 * q^77 - 25 * q^79 + 8 * q^82 + 18 * q^83 + 6 * q^86 + 4 * q^91 - 9 * q^92 - 2 * q^94 - q^97 - 3 * 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 $

$2$	−0.618034	−0.437016	−0.218508	−	0.975835i	$-0.570119\pi$
$2$	−0.618034	−0.437016	−0.218508	+	0.975835i	$0.570119\pi$
$3$	0	0
$3$	0	0
$4$	−1.61803	−0.809017
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	2.23607	0.790569
$9$	0	0
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	−1.23607	−0.330353
$15$	0	0
$16$	1.85410	0.463525
$17$	0.236068	0.0572549	0.0286274	−	0.999590i	$-0.490886\pi$
$17$	0.236068	0.0572549	0.0286274	+	0.999590i	$0.490886\pi$
$18$	0	0
$19$	6.70820	1.53897	0.769484	−	0.638666i	$-0.220514\pi$
$19$	6.70820	1.53897	0.769484	+	0.638666i	$0.220514\pi$
$20$	0	0
$21$	0	0
$22$	−1.85410	−0.395296
$23$	7.61803	1.58847	0.794235	−	0.607611i	$-0.207872\pi$
$23$	7.61803	1.58847	0.794235	+	0.607611i	$0.207872\pi$
$24$	0	0
$25$	0	0
$26$	−0.618034	−0.121206
$27$	0	0
$28$	−3.23607	−0.611559
$29$	1.38197	0.256625	0.128312	−	0.991734i	$-0.459044\pi$
$29$	1.38197	0.256625	0.128312	+	0.991734i	$0.459044\pi$
$30$	0	0
$31$	−4.70820	−0.845618	−0.422809	−	0.906219i	$-0.638956\pi$
$31$	−4.70820	−0.845618	−0.422809	+	0.906219i	$0.638956\pi$
$32$	−5.61803	−0.993137
$33$	0	0
$34$	−0.145898	−0.0250213
$35$	0	0
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−4.14590	−0.672553
$39$	0	0
$40$	0	0
$41$	11.6180	1.81443	0.907216	−	0.420665i	$-0.138203\pi$
$41$	11.6180	1.81443	0.907216	+	0.420665i	$0.138203\pi$
$42$	0	0
$43$	9.61803	1.46674	0.733368	−	0.679832i	$-0.237947\pi$
$43$	9.61803	1.46674	0.733368	+	0.679832i	$0.237947\pi$
$44$	−4.85410	−0.731783
$45$	0	0
$46$	−4.70820	−0.694187
$47$	−9.23607	−1.34722	−0.673609	−	0.739087i	$-0.735257\pi$
$47$	−9.23607	−1.34722	−0.673609	+	0.739087i	$0.735257\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0	0
$52$	−1.61803	−0.224381
$53$	6.76393	0.929098	0.464549	−	0.885548i	$-0.346217\pi$
$53$	6.76393	0.929098	0.464549	+	0.885548i	$0.346217\pi$
$54$	0	0
$55$	0	0
$56$	4.47214	0.597614
$57$	0	0
$58$	−0.854102	−0.112149
$59$	13.9443	1.81539	0.907695	−	0.419631i	$-0.137841\pi$
$59$	13.9443	1.81539	0.907695	+	0.419631i	$0.137841\pi$
$60$	0	0
$61$	−4.70820	−0.602824	−0.301412	−	0.953494i	$-0.597458\pi$
$61$	−4.70820	−0.602824	−0.301412	+	0.953494i	$0.597458\pi$
$62$	2.90983	0.369549
$63$	0	0
$64$	−0.236068	−0.0295085
$65$	0	0
$66$	0	0
$67$	−9.18034	−1.12156	−0.560779	−	0.827966i	$-0.689498\pi$
$67$	−9.18034	−1.12156	−0.560779	+	0.827966i	$0.689498\pi$
$68$	−0.381966	−0.0463202
$69$	0	0
$70$	0	0
$71$	1.09017	0.129379	0.0646897	−	0.997905i	$-0.479394\pi$
$71$	1.09017	0.129379	0.0646897	+	0.997905i	$0.479394\pi$
$72$	0	0
$73$	−2.29180	−0.268234	−0.134117	−	0.990965i	$-0.542820\pi$
$73$	−2.29180	−0.268234	−0.134117	+	0.990965i	$0.542820\pi$
$74$	−1.23607	−0.143690
$75$	0	0
$76$	−10.8541	−1.24505
$77$	6.00000	0.683763
$78$	0	0
$79$	−15.8541	−1.78373	−0.891863	−	0.452306i	$-0.850602\pi$
$79$	−15.8541	−1.78373	−0.891863	+	0.452306i	$0.850602\pi$
$80$	0	0
$81$	0	0
$82$	−7.18034	−0.792936
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	0	0
$85$	0	0
$86$	−5.94427	−0.640987
$87$	0	0
$88$	6.70820	0.715097
$89$	−11.1803	−1.18511	−0.592557	−	0.805529i	$-0.701881\pi$
$89$	−11.1803	−1.18511	−0.592557	+	0.805529i	$0.701881\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	−12.3262	−1.28510
$93$	0	0
$94$	5.70820	0.588756
$95$	0	0
$96$	0	0
$97$	2.85410	0.289790	0.144895	−	0.989447i	$-0.453716\pi$
$97$	2.85410	0.289790	0.144895	+	0.989447i	$0.453716\pi$
$98$	1.85410	0.187293
$99$	0	0
$100$	0	0
$101$	11.6180	1.15604	0.578019	−	0.816023i	$-0.303826\pi$
$101$	11.6180	1.15604	0.578019	+	0.816023i	$0.303826\pi$
$102$	0	0
$103$	−12.4164	−1.22343	−0.611713	−	0.791080i	$-0.709519\pi$
$103$	−12.4164	−1.22343	−0.611713	+	0.791080i	$0.709519\pi$
$104$	2.23607	0.219265
$105$	0	0
$106$	−4.18034	−0.406031
$107$	−7.85410	−0.759285	−0.379642	−	0.925133i	$-0.623953\pi$
$107$	−7.85410	−0.759285	−0.379642	+	0.925133i	$0.623953\pi$
$108$	0	0
$109$	−10.8541	−1.03963	−0.519817	−	0.854278i	$-0.674000\pi$
$109$	−10.8541	−1.03963	−0.519817	+	0.854278i	$0.674000\pi$
$110$	0	0
$111$	0	0
$112$	3.70820	0.350392
$113$	−8.23607	−0.774784	−0.387392	−	0.921915i	$-0.626624\pi$
$113$	−8.23607	−0.774784	−0.387392	+	0.921915i	$0.626624\pi$
$114$	0	0
$115$	0	0
$116$	−2.23607	−0.207614
$117$	0	0
$118$	−8.61803	−0.793354
$119$	0.472136	0.0432806
$120$	0	0
$121$	−2.00000	−0.181818
$122$	2.90983	0.263444
$123$	0	0
$124$	7.61803	0.684120
$125$	0	0
$126$	0	0
$127$	17.6525	1.56640	0.783202	−	0.621767i	$-0.213585\pi$
$127$	17.6525	1.56640	0.783202	+	0.621767i	$0.213585\pi$
$128$	11.3820	1.00603
$129$	0	0
$130$	0	0
$131$	−8.18034	−0.714720	−0.357360	−	0.933967i	$-0.616323\pi$
$131$	−8.18034	−0.714720	−0.357360	+	0.933967i	$0.616323\pi$
$132$	0	0
$133$	13.4164	1.16335
$134$	5.67376	0.490138
$135$	0	0
$136$	0.527864	0.0452640
$137$	20.5623	1.75676	0.878378	−	0.477966i	$-0.158626\pi$
$137$	20.5623	1.75676	0.878378	+	0.477966i	$0.158626\pi$
$138$	0	0
$139$	−13.4164	−1.13796	−0.568982	−	0.822350i	$-0.692663\pi$
$139$	−13.4164	−1.13796	−0.568982	+	0.822350i	$0.692663\pi$
$140$	0	0
$141$	0	0
$142$	−0.673762	−0.0565409
$143$	3.00000	0.250873
$144$	0	0
$145$	0	0
$146$	1.41641	0.117223
$147$	0	0
$148$	−3.23607	−0.266003
$149$	1.90983	0.156459	0.0782297	−	0.996935i	$-0.475073\pi$
$149$	1.90983	0.156459	0.0782297	+	0.996935i	$0.475073\pi$
$150$	0	0
$151$	−4.38197	−0.356599	−0.178300	−	0.983976i	$-0.557060\pi$
$151$	−4.38197	−0.356599	−0.178300	+	0.983976i	$0.557060\pi$
$152$	15.0000	1.21666
$153$	0	0
$154$	−3.70820	−0.298816
$155$	0	0
$156$	0	0
$157$	−3.85410	−0.307591	−0.153795	−	0.988103i	$-0.549150\pi$
$157$	−3.85410	−0.307591	−0.153795	+	0.988103i	$0.549150\pi$
$158$	9.79837	0.779517
$159$	0	0
$160$	0	0
$161$	15.2361	1.20077
$162$	0	0
$163$	15.2705	1.19608	0.598039	−	0.801467i	$-0.295947\pi$
$163$	15.2705	1.19608	0.598039	+	0.801467i	$0.295947\pi$
$164$	−18.7984	−1.46791
$165$	0	0
$166$	−5.56231	−0.431719
$167$	−6.79837	−0.526074	−0.263037	−	0.964786i	$-0.584724\pi$
$167$	−6.79837	−0.526074	−0.263037	+	0.964786i	$0.584724\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	−15.5623	−1.18661
$173$	12.0902	0.919199	0.459599	−	0.888126i	$-0.347993\pi$
$173$	12.0902	0.919199	0.459599	+	0.888126i	$0.347993\pi$
$174$	0	0
$175$	0	0
$176$	5.56231	0.419275
$177$	0	0
$178$	6.90983	0.517914
$179$	−15.6525	−1.16992	−0.584960	−	0.811062i	$-0.698890\pi$
$179$	−15.6525	−1.16992	−0.584960	+	0.811062i	$0.698890\pi$
$180$	0	0
$181$	−3.52786	−0.262224	−0.131112	−	0.991368i	$-0.541855\pi$
$181$	−3.52786	−0.262224	−0.131112	+	0.991368i	$0.541855\pi$
$182$	−1.23607	−0.0916235
$183$	0	0
$184$	17.0344	1.25580
$185$	0	0
$186$	0	0
$187$	0.708204	0.0517890
$188$	14.9443	1.08992
$189$	0	0
$190$	0	0
$191$	−1.67376	−0.121109	−0.0605546	−	0.998165i	$-0.519287\pi$
$191$	−1.67376	−0.121109	−0.0605546	+	0.998165i	$0.519287\pi$
$192$	0	0
$193$	11.0000	0.791797	0.395899	−	0.918294i	$-0.370433\pi$
$193$	11.0000	0.791797	0.395899	+	0.918294i	$0.370433\pi$
$194$	−1.76393	−0.126643
$195$	0	0
$196$	4.85410	0.346722
$197$	11.0902	0.790142	0.395071	−	0.918651i	$-0.370720\pi$
$197$	11.0902	0.790142	0.395071	+	0.918651i	$0.370720\pi$
$198$	0	0
$199$	−1.70820	−0.121091	−0.0605457	−	0.998165i	$-0.519284\pi$
$199$	−1.70820	−0.121091	−0.0605457	+	0.998165i	$0.519284\pi$
$200$	0	0
$201$	0	0
$202$	−7.18034	−0.505207
$203$	2.76393	0.193990
$204$	0	0
$205$	0	0
$206$	7.67376	0.534656
$207$	0	0
$208$	1.85410	0.128559
$209$	20.1246	1.39205
$210$	0	0
$211$	−3.00000	−0.206529	−0.103264	−	0.994654i	$-0.532929\pi$
$211$	−3.00000	−0.206529	−0.103264	+	0.994654i	$0.532929\pi$
$212$	−10.9443	−0.751656
$213$	0	0
$214$	4.85410	0.331820
$215$	0	0
$216$	0	0
$217$	−9.41641	−0.639227
$218$	6.70820	0.454337
$219$	0	0
$220$	0	0
$221$	0.236068	0.0158797
$222$	0	0
$223$	16.8541	1.12863	0.564317	−	0.825558i	$-0.309140\pi$
$223$	16.8541	1.12863	0.564317	+	0.825558i	$0.309140\pi$
$224$	−11.2361	−0.750741
$225$	0	0
$226$	5.09017	0.338593
$227$	10.2361	0.679392	0.339696	−	0.940535i	$-0.389676\pi$
$227$	10.2361	0.679392	0.339696	+	0.940535i	$0.389676\pi$
$228$	0	0
$229$	6.18034	0.408408	0.204204	−	0.978928i	$-0.434539\pi$
$229$	6.18034	0.408408	0.204204	+	0.978928i	$0.434539\pi$
$230$	0	0
$231$	0	0
$232$	3.09017	0.202880
$233$	−12.1803	−0.797961	−0.398980	−	0.916959i	$-0.630636\pi$
$233$	−12.1803	−0.797961	−0.398980	+	0.916959i	$0.630636\pi$
$234$	0	0
$235$	0	0
$236$	−22.5623	−1.46868
$237$	0	0
$238$	−0.291796	−0.0189143
$239$	−23.6180	−1.52772	−0.763862	−	0.645380i	$-0.776699\pi$
$239$	−23.6180	−1.52772	−0.763862	+	0.645380i	$0.776699\pi$
$240$	0	0
$241$	−8.32624	−0.536340	−0.268170	−	0.963372i	$-0.586419\pi$
$241$	−8.32624	−0.536340	−0.268170	+	0.963372i	$0.586419\pi$
$242$	1.23607	0.0794575
$243$	0	0
$244$	7.61803	0.487695
$245$	0	0
$246$	0	0
$247$	6.70820	0.426833
$248$	−10.5279	−0.668520
$249$	0	0
$250$	0	0
$251$	−27.9787	−1.76600	−0.883000	−	0.469372i	$-0.844480\pi$
$251$	−27.9787	−1.76600	−0.883000	+	0.469372i	$0.844480\pi$
$252$	0	0
$253$	22.8541	1.43683
$254$	−10.9098	−0.684544
$255$	0	0
$256$	−6.56231	−0.410144
$257$	−20.2148	−1.26096	−0.630482	−	0.776204i	$-0.717143\pi$
$257$	−20.2148	−1.26096	−0.630482	+	0.776204i	$0.717143\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	0	0
$261$	0	0
$262$	5.05573	0.312344
$263$	25.5066	1.57280	0.786401	−	0.617716i	$-0.211942\pi$
$263$	25.5066	1.57280	0.786401	+	0.617716i	$0.211942\pi$
$264$	0	0
$265$	0	0
$266$	−8.29180	−0.508403
$267$	0	0
$268$	14.8541	0.907359
$269$	29.4721	1.79695	0.898474	−	0.439027i	$-0.144677\pi$
$269$	29.4721	1.79695	0.898474	+	0.439027i	$0.144677\pi$
$270$	0	0
$271$	15.4164	0.936480	0.468240	−	0.883601i	$-0.344888\pi$
$271$	15.4164	0.936480	0.468240	+	0.883601i	$0.344888\pi$
$272$	0.437694	0.0265391
$273$	0	0
$274$	−12.7082	−0.767731
$275$	0	0
$276$	0	0
$277$	30.9443	1.85926	0.929631	−	0.368493i	$-0.120126\pi$
$277$	30.9443	1.85926	0.929631	+	0.368493i	$0.120126\pi$
$278$	8.29180	0.497309
$279$	0	0
$280$	0	0
$281$	−8.18034	−0.487998	−0.243999	−	0.969775i	$-0.578459\pi$
$281$	−8.18034	−0.487998	−0.243999	+	0.969775i	$0.578459\pi$
$282$	0	0
$283$	−15.7082	−0.933756	−0.466878	−	0.884322i	$-0.654621\pi$
$283$	−15.7082	−0.933756	−0.466878	+	0.884322i	$0.654621\pi$
$284$	−1.76393	−0.104670
$285$	0	0
$286$	−1.85410	−0.109635
$287$	23.2361	1.37158
$288$	0	0
$289$	−16.9443	−0.996722
$290$	0	0
$291$	0	0
$292$	3.70820	0.217006
$293$	9.32624	0.544845	0.272422	−	0.962178i	$-0.412175\pi$
$293$	9.32624	0.544845	0.272422	+	0.962178i	$0.412175\pi$
$294$	0	0
$295$	0	0
$296$	4.47214	0.259938
$297$	0	0
$298$	−1.18034	−0.0683753
$299$	7.61803	0.440562
$300$	0	0
$301$	19.2361	1.10875
$302$	2.70820	0.155840
$303$	0	0
$304$	12.4377	0.713351
$305$	0	0
$306$	0	0
$307$	−2.14590	−0.122473	−0.0612364	−	0.998123i	$-0.519504\pi$
$307$	−2.14590	−0.122473	−0.0612364	+	0.998123i	$0.519504\pi$
$308$	−9.70820	−0.553176
$309$	0	0
$310$	0	0
$311$	22.4721	1.27428	0.637139	−	0.770749i	$-0.280118\pi$
$311$	22.4721	1.27428	0.637139	+	0.770749i	$0.280118\pi$
$312$	0	0
$313$	−15.7082	−0.887880	−0.443940	−	0.896056i	$-0.646420\pi$
$313$	−15.7082	−0.887880	−0.443940	+	0.896056i	$0.646420\pi$
$314$	2.38197	0.134422
$315$	0	0
$316$	25.6525	1.44306
$317$	0.437694	0.0245833	0.0122917	−	0.999924i	$-0.496087\pi$
$317$	0.437694	0.0245833	0.0122917	+	0.999924i	$0.496087\pi$
$318$	0	0
$319$	4.14590	0.232126
$320$	0	0
$321$	0	0
$322$	−9.41641	−0.524756
$323$	1.58359	0.0881134
$324$	0	0
$325$	0	0
$326$	−9.43769	−0.522706
$327$	0	0
$328$	25.9787	1.43443
$329$	−18.4721	−1.01840
$330$	0	0
$331$	29.6869	1.63174	0.815870	−	0.578235i	$-0.196258\pi$
$331$	29.6869	1.63174	0.815870	+	0.578235i	$0.196258\pi$
$332$	−14.5623	−0.799210
$333$	0	0
$334$	4.20163	0.229903
$335$	0	0
$336$	0	0
$337$	10.8197	0.589384	0.294692	−	0.955592i	$-0.404783\pi$
$337$	10.8197	0.589384	0.294692	+	0.955592i	$0.404783\pi$
$338$	7.41641	0.403399
$339$	0	0
$340$	0	0
$341$	−14.1246	−0.764891
$342$	0	0
$343$	−20.0000	−1.07990
$344$	21.5066	1.15956
$345$	0	0
$346$	−7.47214	−0.401705
$347$	−21.2705	−1.14186	−0.570930	−	0.820998i	$-0.693417\pi$
$347$	−21.2705	−1.14186	−0.570930	+	0.820998i	$0.693417\pi$
$348$	0	0
$349$	2.76393	0.147950	0.0739749	−	0.997260i	$-0.476432\pi$
$349$	2.76393	0.147950	0.0739749	+	0.997260i	$0.476432\pi$
$350$	0	0
$351$	0	0
$352$	−16.8541	−0.898327
$353$	−14.6180	−0.778039	−0.389020	−	0.921229i	$-0.627186\pi$
$353$	−14.6180	−0.778039	−0.389020	+	0.921229i	$0.627186\pi$
$354$	0	0
$355$	0	0
$356$	18.0902	0.958777
$357$	0	0
$358$	9.67376	0.511274
$359$	−6.05573	−0.319609	−0.159805	−	0.987149i	$-0.551086\pi$
$359$	−6.05573	−0.319609	−0.159805	+	0.987149i	$0.551086\pi$
$360$	0	0
$361$	26.0000	1.36842
$362$	2.18034	0.114596
$363$	0	0
$364$	−3.23607	−0.169616
$365$	0	0
$366$	0	0
$367$	21.4721	1.12084	0.560418	−	0.828210i	$-0.310640\pi$
$367$	21.4721	1.12084	0.560418	+	0.828210i	$0.310640\pi$
$368$	14.1246	0.736296
$369$	0	0
$370$	0	0
$371$	13.5279	0.702332
$372$	0	0
$373$	9.41641	0.487563	0.243782	−	0.969830i	$-0.421612\pi$
$373$	9.41641	0.487563	0.243782	+	0.969830i	$0.421612\pi$
$374$	−0.437694	−0.0226326
$375$	0	0
$376$	−20.6525	−1.06507
$377$	1.38197	0.0711749
$378$	0	0
$379$	−11.3820	−0.584652	−0.292326	−	0.956319i	$-0.594429\pi$
$379$	−11.3820	−0.584652	−0.292326	+	0.956319i	$0.594429\pi$
$380$	0	0
$381$	0	0
$382$	1.03444	0.0529266
$383$	22.9443	1.17240	0.586199	−	0.810167i	$-0.300624\pi$
$383$	22.9443	1.17240	0.586199	+	0.810167i	$0.300624\pi$
$384$	0	0
$385$	0	0
$386$	−6.79837	−0.346028
$387$	0	0
$388$	−4.61803	−0.234445
$389$	30.6525	1.55414	0.777071	−	0.629413i	$-0.216705\pi$
$389$	30.6525	1.55414	0.777071	+	0.629413i	$0.216705\pi$
$390$	0	0
$391$	1.79837	0.0909477
$392$	−6.70820	−0.338815
$393$	0	0
$394$	−6.85410	−0.345305
$395$	0	0
$396$	0	0
$397$	11.4721	0.575770	0.287885	−	0.957665i	$-0.407048\pi$
$397$	11.4721	0.575770	0.287885	+	0.957665i	$0.407048\pi$
$398$	1.05573	0.0529189
$399$	0	0
$400$	0	0
$401$	−2.72949	−0.136304	−0.0681521	−	0.997675i	$-0.521710\pi$
$401$	−2.72949	−0.136304	−0.0681521	+	0.997675i	$0.521710\pi$
$402$	0	0
$403$	−4.70820	−0.234532
$404$	−18.7984	−0.935254
$405$	0	0
$406$	−1.70820	−0.0847767
$407$	6.00000	0.297409
$408$	0	0
$409$	35.1246	1.73680	0.868400	−	0.495864i	$-0.165149\pi$
$409$	35.1246	1.73680	0.868400	+	0.495864i	$0.165149\pi$
$410$	0	0
$411$	0	0
$412$	20.0902	0.989772
$413$	27.8885	1.37231
$414$	0	0
$415$	0	0
$416$	−5.61803	−0.275447
$417$	0	0
$418$	−12.4377	−0.608348
$419$	15.3262	0.748736	0.374368	−	0.927280i	$-0.377860\pi$
$419$	15.3262	0.748736	0.374368	+	0.927280i	$0.377860\pi$
$420$	0	0
$421$	14.3607	0.699897	0.349948	−	0.936769i	$-0.386199\pi$
$421$	14.3607	0.699897	0.349948	+	0.936769i	$0.386199\pi$
$422$	1.85410	0.0902563
$423$	0	0
$424$	15.1246	0.734516
$425$	0	0
$426$	0	0
$427$	−9.41641	−0.455692
$428$	12.7082	0.614274
$429$	0	0
$430$	0	0
$431$	−34.2361	−1.64909	−0.824547	−	0.565794i	$-0.808570\pi$
$431$	−34.2361	−1.64909	−0.824547	+	0.565794i	$0.808570\pi$
$432$	0	0
$433$	5.47214	0.262974	0.131487	−	0.991318i	$-0.458025\pi$
$433$	5.47214	0.262974	0.131487	+	0.991318i	$0.458025\pi$
$434$	5.81966	0.279353
$435$	0	0
$436$	17.5623	0.841082
$437$	51.1033	2.44460
$438$	0	0
$439$	−2.96556	−0.141538	−0.0707692	−	0.997493i	$-0.522545\pi$
$439$	−2.96556	−0.141538	−0.0707692	+	0.997493i	$0.522545\pi$
$440$	0	0
$441$	0	0
$442$	−0.145898	−0.00693966
$443$	7.41641	0.352364	0.176182	−	0.984358i	$-0.443625\pi$
$443$	7.41641	0.352364	0.176182	+	0.984358i	$0.443625\pi$
$444$	0	0
$445$	0	0
$446$	−10.4164	−0.493231
$447$	0	0
$448$	−0.472136	−0.0223063
$449$	21.5066	1.01496	0.507479	−	0.861664i	$-0.330577\pi$
$449$	21.5066	1.01496	0.507479	+	0.861664i	$0.330577\pi$
$450$	0	0
$451$	34.8541	1.64122
$452$	13.3262	0.626814
$453$	0	0
$454$	−6.32624	−0.296905
$455$	0	0
$456$	0	0
$457$	−25.8885	−1.21102	−0.605508	−	0.795840i	$-0.707030\pi$
$457$	−25.8885	−1.21102	−0.605508	+	0.795840i	$0.707030\pi$
$458$	−3.81966	−0.178481
$459$	0	0
$460$	0	0
$461$	−3.18034	−0.148123	−0.0740616	−	0.997254i	$-0.523596\pi$
$461$	−3.18034	−0.148123	−0.0740616	+	0.997254i	$0.523596\pi$
$462$	0	0
$463$	−26.6869	−1.24025	−0.620123	−	0.784505i	$-0.712917\pi$
$463$	−26.6869	−1.24025	−0.620123	+	0.784505i	$0.712917\pi$
$464$	2.56231	0.118952
$465$	0	0
$466$	7.52786	0.348722
$467$	16.4164	0.759661	0.379830	−	0.925056i	$-0.375982\pi$
$467$	16.4164	0.759661	0.379830	+	0.925056i	$0.375982\pi$
$468$	0	0
$469$	−18.3607	−0.847817
$470$	0	0
$471$	0	0
$472$	31.1803	1.43519
$473$	28.8541	1.32671
$474$	0	0
$475$	0	0
$476$	−0.763932	−0.0350148
$477$	0	0
$478$	14.5967	0.667640
$479$	19.7984	0.904611	0.452305	−	0.891863i	$-0.350602\pi$
$479$	19.7984	0.904611	0.452305	+	0.891863i	$0.350602\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	5.14590	0.234389
$483$	0	0
$484$	3.23607	0.147094
$485$	0	0
$486$	0	0
$487$	−14.3820	−0.651709	−0.325855	−	0.945420i	$-0.605652\pi$
$487$	−14.3820	−0.651709	−0.325855	+	0.945420i	$0.605652\pi$
$488$	−10.5279	−0.476574
$489$	0	0
$490$	0	0
$491$	−6.67376	−0.301183	−0.150591	−	0.988596i	$-0.548118\pi$
$491$	−6.67376	−0.301183	−0.150591	+	0.988596i	$0.548118\pi$
$492$	0	0
$493$	0.326238	0.0146930
$494$	−4.14590	−0.186533
$495$	0	0
$496$	−8.72949	−0.391966
$497$	2.18034	0.0978016
$498$	0	0
$499$	−15.0000	−0.671492	−0.335746	−	0.941953i	$-0.608988\pi$
$499$	−15.0000	−0.671492	−0.335746	+	0.941953i	$0.608988\pi$
$500$	0	0
$501$	0	0
$502$	17.2918	0.771771
$503$	−33.0344	−1.47293	−0.736466	−	0.676474i	$-0.763507\pi$
$503$	−33.0344	−1.47293	−0.736466	+	0.676474i	$0.763507\pi$
$504$	0	0
$505$	0	0
$506$	−14.1246	−0.627916
$507$	0	0
$508$	−28.5623	−1.26725
$509$	2.88854	0.128032	0.0640162	−	0.997949i	$-0.479609\pi$
$509$	2.88854	0.128032	0.0640162	+	0.997949i	$0.479609\pi$
$510$	0	0
$511$	−4.58359	−0.202766
$512$	−18.7082	−0.826794
$513$	0	0
$514$	12.4934	0.551061
$515$	0	0
$516$	0	0
$517$	−27.7082	−1.21861
$518$	−2.47214	−0.108619
$519$	0	0
$520$	0	0
$521$	−28.9098	−1.26656	−0.633281	−	0.773922i	$-0.718292\pi$
$521$	−28.9098	−1.26656	−0.633281	+	0.773922i	$0.718292\pi$
$522$	0	0
$523$	18.5623	0.811673	0.405836	−	0.913946i	$-0.366980\pi$
$523$	18.5623	0.811673	0.405836	+	0.913946i	$0.366980\pi$
$524$	13.2361	0.578220
$525$	0	0
$526$	−15.7639	−0.687340
$527$	−1.11146	−0.0484158
$528$	0	0
$529$	35.0344	1.52324
$530$	0	0
$531$	0	0
$532$	−21.7082	−0.941170
$533$	11.6180	0.503233
$534$	0	0
$535$	0	0
$536$	−20.5279	−0.886669
$537$	0	0
$538$	−18.2148	−0.785295
$539$	−9.00000	−0.387657
$540$	0	0
$541$	−39.7082	−1.70719	−0.853595	−	0.520938i	$-0.825582\pi$
$541$	−39.7082	−1.70719	−0.853595	+	0.520938i	$0.825582\pi$
$542$	−9.52786	−0.409257
$543$	0	0
$544$	−1.32624	−0.0568620
$545$	0	0
$546$	0	0
$547$	−11.2918	−0.482802	−0.241401	−	0.970425i	$-0.577607\pi$
$547$	−11.2918	−0.482802	−0.241401	+	0.970425i	$0.577607\pi$
$548$	−33.2705	−1.42125
$549$	0	0
$550$	0	0
$551$	9.27051	0.394937
$552$	0	0
$553$	−31.7082	−1.34837
$554$	−19.1246	−0.812527
$555$	0	0
$556$	21.7082	0.920633
$557$	−6.34752	−0.268953	−0.134477	−	0.990917i	$-0.542935\pi$
$557$	−6.34752	−0.268953	−0.134477	+	0.990917i	$0.542935\pi$
$558$	0	0
$559$	9.61803	0.406799
$560$	0	0
$561$	0	0
$562$	5.05573	0.213263
$563$	9.00000	0.379305	0.189652	−	0.981851i	$-0.439264\pi$
$563$	9.00000	0.379305	0.189652	+	0.981851i	$0.439264\pi$
$564$	0	0
$565$	0	0
$566$	9.70820	0.408066
$567$	0	0
$568$	2.43769	0.102283
$569$	4.14590	0.173805	0.0869025	−	0.996217i	$-0.472303\pi$
$569$	4.14590	0.173805	0.0869025	+	0.996217i	$0.472303\pi$
$570$	0	0
$571$	2.12461	0.0889122	0.0444561	−	0.999011i	$-0.485845\pi$
$571$	2.12461	0.0889122	0.0444561	+	0.999011i	$0.485845\pi$
$572$	−4.85410	−0.202960
$573$	0	0
$574$	−14.3607	−0.599403
$575$	0	0
$576$	0	0
$577$	−37.2705	−1.55159	−0.775796	−	0.630984i	$-0.782651\pi$
$577$	−37.2705	−1.55159	−0.775796	+	0.630984i	$0.782651\pi$
$578$	10.4721	0.435583
$579$	0	0
$580$	0	0
$581$	18.0000	0.746766
$582$	0	0
$583$	20.2918	0.840400
$584$	−5.12461	−0.212058
$585$	0	0
$586$	−5.76393	−0.238106
$587$	−23.3050	−0.961898	−0.480949	−	0.876748i	$-0.659708\pi$
$587$	−23.3050	−0.961898	−0.480949	+	0.876748i	$0.659708\pi$
$588$	0	0
$589$	−31.5836	−1.30138
$590$	0	0
$591$	0	0
$592$	3.70820	0.152406
$593$	15.3820	0.631662	0.315831	−	0.948816i	$-0.397717\pi$
$593$	15.3820	0.631662	0.315831	+	0.948816i	$0.397717\pi$
$594$	0	0
$595$	0	0
$596$	−3.09017	−0.126578
$597$	0	0
$598$	−4.70820	−0.192533
$599$	−5.72949	−0.234101	−0.117050	−	0.993126i	$-0.537344\pi$
$599$	−5.72949	−0.234101	−0.117050	+	0.993126i	$0.537344\pi$
$600$	0	0
$601$	−11.2918	−0.460602	−0.230301	−	0.973119i	$-0.573971\pi$
$601$	−11.2918	−0.460602	−0.230301	+	0.973119i	$0.573971\pi$
$602$	−11.8885	−0.484541
$603$	0	0
$604$	7.09017	0.288495
$605$	0	0
$606$	0	0
$607$	16.1459	0.655342	0.327671	−	0.944792i	$-0.393736\pi$
$607$	16.1459	0.655342	0.327671	+	0.944792i	$0.393736\pi$
$608$	−37.6869	−1.52841
$609$	0	0
$610$	0	0
$611$	−9.23607	−0.373651
$612$	0	0
$613$	46.1246	1.86296	0.931478	−	0.363798i	$-0.118520\pi$
$613$	46.1246	1.86296	0.931478	+	0.363798i	$0.118520\pi$
$614$	1.32624	0.0535226
$615$	0	0
$616$	13.4164	0.540562
$617$	20.7639	0.835924	0.417962	−	0.908464i	$-0.362744\pi$
$617$	20.7639	0.835924	0.417962	+	0.908464i	$0.362744\pi$
$618$	0	0
$619$	0.729490	0.0293207	0.0146603	−	0.999893i	$-0.495333\pi$
$619$	0.729490	0.0293207	0.0146603	+	0.999893i	$0.495333\pi$
$620$	0	0
$621$	0	0
$622$	−13.8885	−0.556880
$623$	−22.3607	−0.895862
$624$	0	0
$625$	0	0
$626$	9.70820	0.388018
$627$	0	0
$628$	6.23607	0.248846
$629$	0.472136	0.0188253
$630$	0	0
$631$	−15.2361	−0.606538	−0.303269	−	0.952905i	$-0.598078\pi$
$631$	−15.2361	−0.606538	−0.303269	+	0.952905i	$0.598078\pi$
$632$	−35.4508	−1.41016
$633$	0	0
$634$	−0.270510	−0.0107433
$635$	0	0
$636$	0	0
$637$	−3.00000	−0.118864
$638$	−2.56231	−0.101443
$639$	0	0
$640$	0	0
$641$	7.67376	0.303095	0.151548	−	0.988450i	$-0.451574\pi$
$641$	7.67376	0.303095	0.151548	+	0.988450i	$0.451574\pi$
$642$	0	0
$643$	−17.0902	−0.673971	−0.336985	−	0.941510i	$-0.609407\pi$
$643$	−17.0902	−0.673971	−0.336985	+	0.941510i	$0.609407\pi$
$644$	−24.6525	−0.971444
$645$	0	0
$646$	−0.978714	−0.0385070
$647$	10.0344	0.394495	0.197247	−	0.980354i	$-0.436800\pi$
$647$	10.0344	0.394495	0.197247	+	0.980354i	$0.436800\pi$
$648$	0	0
$649$	41.8328	1.64208
$650$	0	0
$651$	0	0
$652$	−24.7082	−0.967648
$653$	−1.65248	−0.0646664	−0.0323332	−	0.999477i	$-0.510294\pi$
$653$	−1.65248	−0.0646664	−0.0323332	+	0.999477i	$0.510294\pi$
$654$	0	0
$655$	0	0
$656$	21.5410	0.841036
$657$	0	0
$658$	11.4164	0.445058
$659$	−2.23607	−0.0871048	−0.0435524	−	0.999051i	$-0.513868\pi$
$659$	−2.23607	−0.0871048	−0.0435524	+	0.999051i	$0.513868\pi$
$660$	0	0
$661$	−30.8885	−1.20143	−0.600713	−	0.799465i	$-0.705116\pi$
$661$	−30.8885	−1.20143	−0.600713	+	0.799465i	$0.705116\pi$
$662$	−18.3475	−0.713097
$663$	0	0
$664$	20.1246	0.780986
$665$	0	0
$666$	0	0
$667$	10.5279	0.407641
$668$	11.0000	0.425603
$669$	0	0
$670$	0	0
$671$	−14.1246	−0.545275
$672$	0	0
$673$	−24.7771	−0.955087	−0.477543	−	0.878608i	$-0.658473\pi$
$673$	−24.7771	−0.955087	−0.477543	+	0.878608i	$0.658473\pi$
$674$	−6.68692	−0.257570
$675$	0	0
$676$	19.4164	0.746785
$677$	−9.11146	−0.350182	−0.175091	−	0.984552i	$-0.556022\pi$
$677$	−9.11146	−0.350182	−0.175091	+	0.984552i	$0.556022\pi$
$678$	0	0
$679$	5.70820	0.219061
$680$	0	0
$681$	0	0
$682$	8.72949	0.334269
$683$	48.5967	1.85950	0.929751	−	0.368188i	$-0.120022\pi$
$683$	48.5967	1.85950	0.929751	+	0.368188i	$0.120022\pi$
$684$	0	0
$685$	0	0
$686$	12.3607	0.471933
$687$	0	0
$688$	17.8328	0.679870
$689$	6.76393	0.257685
$690$	0	0
$691$	3.90983	0.148737	0.0743685	−	0.997231i	$-0.476306\pi$
$691$	3.90983	0.148737	0.0743685	+	0.997231i	$0.476306\pi$
$692$	−19.5623	−0.743647
$693$	0	0
$694$	13.1459	0.499011
$695$	0	0
$696$	0	0
$697$	2.74265	0.103885
$698$	−1.70820	−0.0646565
$699$	0	0
$700$	0	0
$701$	17.3475	0.655207	0.327603	−	0.944815i	$-0.393759\pi$
$701$	17.3475	0.655207	0.327603	+	0.944815i	$0.393759\pi$
$702$	0	0
$703$	13.4164	0.506009
$704$	−0.708204	−0.0266914
$705$	0	0
$706$	9.03444	0.340016
$707$	23.2361	0.873882
$708$	0	0
$709$	−29.7984	−1.11910	−0.559551	−	0.828796i	$-0.689026\pi$
$709$	−29.7984	−1.11910	−0.559551	+	0.828796i	$0.689026\pi$
$710$	0	0
$711$	0	0
$712$	−25.0000	−0.936915
$713$	−35.8673	−1.34324
$714$	0	0
$715$	0	0
$716$	25.3262	0.946486
$717$	0	0
$718$	3.74265	0.139674
$719$	−5.12461	−0.191116	−0.0955579	−	0.995424i	$-0.530464\pi$
$719$	−5.12461	−0.191116	−0.0955579	+	0.995424i	$0.530464\pi$
$720$	0	0
$721$	−24.8328	−0.924822
$722$	−16.0689	−0.598022
$723$	0	0
$724$	5.70820	0.212144
$725$	0	0
$726$	0	0
$727$	34.5623	1.28184	0.640922	−	0.767606i	$-0.278552\pi$
$727$	34.5623	1.28184	0.640922	+	0.767606i	$0.278552\pi$
$728$	4.47214	0.165748
$729$	0	0
$730$	0	0
$731$	2.27051	0.0839778
$732$	0	0
$733$	16.8541	0.622520	0.311260	−	0.950325i	$-0.399249\pi$
$733$	16.8541	0.622520	0.311260	+	0.950325i	$0.399249\pi$
$734$	−13.2705	−0.489823
$735$	0	0
$736$	−42.7984	−1.57757
$737$	−27.5410	−1.01449
$738$	0	0
$739$	11.7082	0.430693	0.215347	−	0.976538i	$-0.430912\pi$
$739$	11.7082	0.430693	0.215347	+	0.976538i	$0.430912\pi$
$740$	0	0
$741$	0	0
$742$	−8.36068	−0.306930
$743$	−25.4721	−0.934482	−0.467241	−	0.884130i	$-0.654752\pi$
$743$	−25.4721	−0.934482	−0.467241	+	0.884130i	$0.654752\pi$
$744$	0	0
$745$	0	0
$746$	−5.81966	−0.213073
$747$	0	0
$748$	−1.14590	−0.0418982
$749$	−15.7082	−0.573965
$750$	0	0
$751$	28.7082	1.04758	0.523789	−	0.851848i	$-0.324518\pi$
$751$	28.7082	1.04758	0.523789	+	0.851848i	$0.324518\pi$
$752$	−17.1246	−0.624470
$753$	0	0
$754$	−0.854102	−0.0311046
$755$	0	0
$756$	0	0
$757$	1.27051	0.0461775	0.0230887	−	0.999733i	$-0.492650\pi$
$757$	1.27051	0.0461775	0.0230887	+	0.999733i	$0.492650\pi$
$758$	7.03444	0.255502
$759$	0	0
$760$	0	0
$761$	19.1803	0.695287	0.347643	−	0.937627i	$-0.386982\pi$
$761$	19.1803	0.695287	0.347643	+	0.937627i	$0.386982\pi$
$762$	0	0
$763$	−21.7082	−0.785890
$764$	2.70820	0.0979794
$765$	0	0
$766$	−14.1803	−0.512357
$767$	13.9443	0.503498
$768$	0	0
$769$	26.3050	0.948581	0.474290	−	0.880368i	$-0.342705\pi$
$769$	26.3050	0.948581	0.474290	+	0.880368i	$0.342705\pi$
$770$	0	0
$771$	0	0
$772$	−17.7984	−0.640577
$773$	29.7771	1.07101	0.535504	−	0.844533i	$-0.320122\pi$
$773$	29.7771	1.07101	0.535504	+	0.844533i	$0.320122\pi$
$774$	0	0
$775$	0	0
$776$	6.38197	0.229099
$777$	0	0
$778$	−18.9443	−0.679185
$779$	77.9361	2.79235
$780$	0	0
$781$	3.27051	0.117028
$782$	−1.11146	−0.0397456
$783$	0	0
$784$	−5.56231	−0.198654
$785$	0	0
$786$	0	0
$787$	−23.8541	−0.850307	−0.425153	−	0.905121i	$-0.639780\pi$
$787$	−23.8541	−0.850307	−0.425153	+	0.905121i	$0.639780\pi$
$788$	−17.9443	−0.639238
$789$	0	0
$790$	0	0
$791$	−16.4721	−0.585682
$792$	0	0
$793$	−4.70820	−0.167193
$794$	−7.09017	−0.251621
$795$	0	0
$796$	2.76393	0.0979650
$797$	46.0132	1.62987	0.814935	−	0.579553i	$-0.196773\pi$
$797$	46.0132	1.62987	0.814935	+	0.579553i	$0.196773\pi$
$798$	0	0
$799$	−2.18034	−0.0771349
$800$	0	0
$801$	0	0
$802$	1.68692	0.0595671
$803$	−6.87539	−0.242627
$804$	0	0
$805$	0	0
$806$	2.90983	0.102494
$807$	0	0
$808$	25.9787	0.913928
$809$	−24.9230	−0.876246	−0.438123	−	0.898915i	$-0.644356\pi$
$809$	−24.9230	−0.876246	−0.438123	+	0.898915i	$0.644356\pi$
$810$	0	0
$811$	37.7771	1.32653	0.663266	−	0.748383i	$-0.269170\pi$
$811$	37.7771	1.32653	0.663266	+	0.748383i	$0.269170\pi$
$812$	−4.47214	−0.156941
$813$	0	0
$814$	−3.70820	−0.129972
$815$	0	0
$816$	0	0
$817$	64.5197	2.25726
$818$	−21.7082	−0.759010
$819$	0	0
$820$	0	0
$821$	11.9443	0.416858	0.208429	−	0.978038i	$-0.433165\pi$
$821$	11.9443	0.416858	0.208429	+	0.978038i	$0.433165\pi$
$822$	0	0
$823$	8.43769	0.294120	0.147060	−	0.989128i	$-0.453019\pi$
$823$	8.43769	0.294120	0.147060	+	0.989128i	$0.453019\pi$
$824$	−27.7639	−0.967202
$825$	0	0
$826$	−17.2361	−0.599720
$827$	2.02129	0.0702870	0.0351435	−	0.999382i	$-0.488811\pi$
$827$	2.02129	0.0702870	0.0351435	+	0.999382i	$0.488811\pi$
$828$	0	0
$829$	−9.87539	−0.342986	−0.171493	−	0.985185i	$-0.554859\pi$
$829$	−9.87539	−0.342986	−0.171493	+	0.985185i	$0.554859\pi$
$830$	0	0
$831$	0	0
$832$	−0.236068	−0.00818418
$833$	−0.708204	−0.0245378
$834$	0	0
$835$	0	0
$836$	−32.5623	−1.12619
$837$	0	0
$838$	−9.47214	−0.327210
$839$	−48.2148	−1.66456	−0.832280	−	0.554356i	$-0.812965\pi$
$839$	−48.2148	−1.66456	−0.832280	+	0.554356i	$0.812965\pi$
$840$	0	0
$841$	−27.0902	−0.934144
$842$	−8.87539	−0.305866
$843$	0	0
$844$	4.85410	0.167085
$845$	0	0
$846$	0	0
$847$	−4.00000	−0.137442
$848$	12.5410	0.430660
$849$	0	0
$850$	0	0
$851$	15.2361	0.522286
$852$	0	0
$853$	−53.3951	−1.82821	−0.914107	−	0.405473i	$-0.867107\pi$
$853$	−53.3951	−1.82821	−0.914107	+	0.405473i	$0.867107\pi$
$854$	5.81966	0.199145
$855$	0	0
$856$	−17.5623	−0.600267
$857$	26.9443	0.920399	0.460199	−	0.887816i	$-0.347778\pi$
$857$	26.9443	0.920399	0.460199	+	0.887816i	$0.347778\pi$
$858$	0	0
$859$	−25.1246	−0.857241	−0.428620	−	0.903485i	$-0.641000\pi$
$859$	−25.1246	−0.857241	−0.428620	+	0.903485i	$0.641000\pi$
$860$	0	0
$861$	0	0
$862$	21.1591	0.720680
$863$	−45.0689	−1.53416	−0.767081	−	0.641550i	$-0.778292\pi$
$863$	−45.0689	−1.53416	−0.767081	+	0.641550i	$0.778292\pi$
$864$	0	0
$865$	0	0
$866$	−3.38197	−0.114924
$867$	0	0
$868$	15.2361	0.517146
$869$	−47.5623	−1.61344
$870$	0	0
$871$	−9.18034	−0.311064
$872$	−24.2705	−0.821903
$873$	0	0
$874$	−31.5836	−1.06833
$875$	0	0
$876$	0	0
$877$	−2.87539	−0.0970950	−0.0485475	−	0.998821i	$-0.515459\pi$
$877$	−2.87539	−0.0970950	−0.0485475	+	0.998821i	$0.515459\pi$
$878$	1.83282	0.0618545
$879$	0	0
$880$	0	0
$881$	−3.90983	−0.131726	−0.0658628	−	0.997829i	$-0.520980\pi$
$881$	−3.90983	−0.131726	−0.0658628	+	0.997829i	$0.520980\pi$
$882$	0	0
$883$	−53.7984	−1.81046	−0.905230	−	0.424923i	$-0.860301\pi$
$883$	−53.7984	−1.81046	−0.905230	+	0.424923i	$0.860301\pi$
$884$	−0.381966	−0.0128469
$885$	0	0
$886$	−4.58359	−0.153989
$887$	−21.9230	−0.736102	−0.368051	−	0.929806i	$-0.619975\pi$
$887$	−21.9230	−0.736102	−0.368051	+	0.929806i	$0.619975\pi$
$888$	0	0
$889$	35.3050	1.18409
$890$	0	0
$891$	0	0
$892$	−27.2705	−0.913084
$893$	−61.9574	−2.07333
$894$	0	0
$895$	0	0
$896$	22.7639	0.760490
$897$	0	0
$898$	−13.2918	−0.443553
$899$	−6.50658	−0.217007
$900$	0	0
$901$	1.59675	0.0531954
$902$	−21.5410	−0.717238
$903$	0	0
$904$	−18.4164	−0.612521
$905$	0	0
$906$	0	0
$907$	17.0000	0.564476	0.282238	−	0.959344i	$-0.408923\pi$
$907$	17.0000	0.564476	0.282238	+	0.959344i	$0.408923\pi$
$908$	−16.5623	−0.549639
$909$	0	0
$910$	0	0
$911$	20.8885	0.692068	0.346034	−	0.938222i	$-0.387528\pi$
$911$	20.8885	0.692068	0.346034	+	0.938222i	$0.387528\pi$
$912$	0	0
$913$	27.0000	0.893570
$914$	16.0000	0.529233
$915$	0	0
$916$	−10.0000	−0.330409
$917$	−16.3607	−0.540277
$918$	0	0
$919$	−5.00000	−0.164935	−0.0824674	−	0.996594i	$-0.526280\pi$
$919$	−5.00000	−0.164935	−0.0824674	+	0.996594i	$0.526280\pi$
$920$	0	0
$921$	0	0
$922$	1.96556	0.0647322
$923$	1.09017	0.0358834
$924$	0	0
$925$	0	0
$926$	16.4934	0.542007
$927$	0	0
$928$	−7.76393	−0.254864
$929$	−19.5967	−0.642948	−0.321474	−	0.946918i	$-0.604178\pi$
$929$	−19.5967	−0.642948	−0.321474	+	0.946918i	$0.604178\pi$
$930$	0	0
$931$	−20.1246	−0.659558
$932$	19.7082	0.645564
$933$	0	0
$934$	−10.1459	−0.331984
$935$	0	0
$936$	0	0
$937$	−16.4164	−0.536301	−0.268150	−	0.963377i	$-0.586412\pi$
$937$	−16.4164	−0.536301	−0.268150	+	0.963377i	$0.586412\pi$
$938$	11.3475	0.370510
$939$	0	0
$940$	0	0
$941$	−11.0213	−0.359284	−0.179642	−	0.983732i	$-0.557494\pi$
$941$	−11.0213	−0.359284	−0.179642	+	0.983732i	$0.557494\pi$
$942$	0	0
$943$	88.5066	2.88217
$944$	25.8541	0.841479
$945$	0	0
$946$	−17.8328	−0.579795
$947$	29.8328	0.969436	0.484718	−	0.874670i	$-0.338922\pi$
$947$	29.8328	0.969436	0.484718	+	0.874670i	$0.338922\pi$
$948$	0	0
$949$	−2.29180	−0.0743948
$950$	0	0
$951$	0	0
$952$	1.05573	0.0342163
$953$	−59.9443	−1.94179	−0.970893	−	0.239515i	$-0.923012\pi$
$953$	−59.9443	−1.94179	−0.970893	+	0.239515i	$0.923012\pi$
$954$	0	0
$955$	0	0
$956$	38.2148	1.23595
$957$	0	0
$958$	−12.2361	−0.395329
$959$	41.1246	1.32798
$960$	0	0
$961$	−8.83282	−0.284930
$962$	−1.23607	−0.0398524
$963$	0	0
$964$	13.4721	0.433908
$965$	0	0
$966$	0	0
$967$	8.58359	0.276030	0.138015	−	0.990430i	$-0.455928\pi$
$967$	8.58359	0.276030	0.138015	+	0.990430i	$0.455928\pi$
$968$	−4.47214	−0.143740
$969$	0	0
$970$	0	0
$971$	5.88854	0.188972	0.0944862	−	0.995526i	$-0.469879\pi$
$971$	5.88854	0.188972	0.0944862	+	0.995526i	$0.469879\pi$
$972$	0	0
$973$	−26.8328	−0.860221
$974$	8.88854	0.284807
$975$	0	0
$976$	−8.72949	−0.279424
$977$	−6.34752	−0.203075	−0.101538	−	0.994832i	$-0.532376\pi$
$977$	−6.34752	−0.203075	−0.101538	+	0.994832i	$0.532376\pi$
$978$	0	0
$979$	−33.5410	−1.07198
$980$	0	0
$981$	0	0
$982$	4.12461	0.131622
$983$	−22.3820	−0.713874	−0.356937	−	0.934128i	$-0.616179\pi$
$983$	−22.3820	−0.713874	−0.356937	+	0.934128i	$0.616179\pi$
$984$	0	0
$985$	0	0
$986$	−0.201626	−0.00642108
$987$	0	0
$988$	−10.8541	−0.345315
$989$	73.2705	2.32987
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	26.4508	0.839815
$993$	0	0
$994$	−1.34752	−0.0427409
$995$	0	0
$996$	0	0
$997$	61.0689	1.93407	0.967035	−	0.254642i	$-0.0819575\pi$
$997$	61.0689	1.93407	0.967035	+	0.254642i	$0.0819575\pi$
$998$	9.27051	0.293453
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.g.1.1		2
3.2	odd	2		1875.2.a.b.1.2	✓	2
5.4	even	2		5625.2.a.b.1.2		2
15.2	even	4		1875.2.b.a.1249.3		4
15.8	even	4		1875.2.b.a.1249.2		4
15.14	odd	2		1875.2.a.c.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.b.1.2	✓	2	3.2	odd	2
1875.2.a.c.1.1	yes	2	15.14	odd	2
1875.2.b.a.1249.2		4	15.8	even	4
1875.2.b.a.1249.3		4	15.2	even	4
5625.2.a.b.1.2		2	5.4	even	2
5625.2.a.g.1.1		2	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)

\(f(q)\)	\(=\)	\(q-0.618034 q^{2} -1.61803 q^{4} +2.00000 q^{7} +2.23607 q^{8} +O(q^{10})\)	\(q-0.618034 q^{2} -1.61803 q^{4} +2.00000 q^{7} +2.23607 q^{8} +3.00000 q^{11} +1.00000 q^{13} -1.23607 q^{14} +1.85410 q^{16} +0.236068 q^{17} +6.70820 q^{19} -1.85410 q^{22} +7.61803 q^{23} -0.618034 q^{26} -3.23607 q^{28} +1.38197 q^{29} -4.70820 q^{31} -5.61803 q^{32} -0.145898 q^{34} +2.00000 q^{37} -4.14590 q^{38} +11.6180 q^{41} +9.61803 q^{43} -4.85410 q^{44} -4.70820 q^{46} -9.23607 q^{47} -3.00000 q^{49} -1.61803 q^{52} +6.76393 q^{53} +4.47214 q^{56} -0.854102 q^{58} +13.9443 q^{59} -4.70820 q^{61} +2.90983 q^{62} -0.236068 q^{64} -9.18034 q^{67} -0.381966 q^{68} +1.09017 q^{71} -2.29180 q^{73} -1.23607 q^{74} -10.8541 q^{76} +6.00000 q^{77} -15.8541 q^{79} -7.18034 q^{82} +9.00000 q^{83} -5.94427 q^{86} +6.70820 q^{88} -11.1803 q^{89} +2.00000 q^{91} -12.3262 q^{92} +5.70820 q^{94} +2.85410 q^{97} +1.85410 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} + 4 q^{7}+O(q^{10}) \) 2 * q + q^2 - q^4 + 4 * q^7	\( 2 q + q^{2} - q^{4} + 4 q^{7} + 6 q^{11} + 2 q^{13} + 2 q^{14} - 3 q^{16} - 4 q^{17} + 3 q^{22} + 13 q^{23} + q^{26} - 2 q^{28} + 5 q^{29} + 4 q^{31} - 9 q^{32} - 7 q^{34} + 4 q^{37} - 15 q^{38} + 21 q^{41} + 17 q^{43} - 3 q^{44} + 4 q^{46} - 14 q^{47} - 6 q^{49} - q^{52} + 18 q^{53} + 5 q^{58} + 10 q^{59} + 4 q^{61} + 17 q^{62} + 4 q^{64} + 4 q^{67} - 3 q^{68} - 9 q^{71} - 18 q^{73} + 2 q^{74} - 15 q^{76} + 12 q^{77} - 25 q^{79} + 8 q^{82} + 18 q^{83} + 6 q^{86} + 4 q^{91} - 9 q^{92} - 2 q^{94} - q^{97} - 3 q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 + 4 * q^7 + 6 * q^11 + 2 * q^13 + 2 * q^14 - 3 * q^16 - 4 * q^17 + 3 * q^22 + 13 * q^23 + q^26 - 2 * q^28 + 5 * q^29 + 4 * q^31 - 9 * q^32 - 7 * q^34 + 4 * q^37 - 15 * q^38 + 21 * q^41 + 17 * q^43 - 3 * q^44 + 4 * q^46 - 14 * q^47 - 6 * q^49 - q^52 + 18 * q^53 + 5 * q^58 + 10 * q^59 + 4 * q^61 + 17 * q^62 + 4 * q^64 + 4 * q^67 - 3 * q^68 - 9 * q^71 - 18 * q^73 + 2 * q^74 - 15 * q^76 + 12 * q^77 - 25 * q^79 + 8 * q^82 + 18 * q^83 + 6 * q^86 + 4 * q^91 - 9 * q^92 - 2 * q^94 - q^97 - 3 * q^98

Introduction

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