LMFDB - Embedded newform 5625.2.a.bd.1.4

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:	8.8.5444000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 2x^{6} + 20x^{5} - 4x^{4} - 30x^{3} + 7x^{2} + 12x + 1 $ x^8 - 4x^7 - 2x^6 + 20x^5 - 4x^4 - 30x^3 + 7x^2 + 12*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.0898194$ 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)

$f(q)$	$=$	$q-0.0898194 q^{2} -1.99193 q^{4} -4.36070 q^{7} +0.358553 q^{8} +O(q^{10})$	$q-0.0898194 q^{2} -1.99193 q^{4} -4.36070 q^{7} +0.358553 q^{8} -4.39094 q^{11} +1.98166 q^{13} +0.391676 q^{14} +3.95166 q^{16} +0.997022 q^{17} +1.35096 q^{19} +0.394392 q^{22} +2.35651 q^{23} -0.177991 q^{26} +8.68622 q^{28} +7.97856 q^{29} -3.67761 q^{31} -1.07204 q^{32} -0.0895519 q^{34} +1.43706 q^{37} -0.121342 q^{38} +5.98248 q^{41} -2.68554 q^{43} +8.74646 q^{44} -0.211660 q^{46} +10.9393 q^{47} +12.0157 q^{49} -3.94732 q^{52} +11.0510 q^{53} -1.56354 q^{56} -0.716629 q^{58} -6.68895 q^{59} -9.45570 q^{61} +0.330321 q^{62} -7.80703 q^{64} -12.9219 q^{67} -1.98600 q^{68} -7.32257 q^{71} +0.424804 q^{73} -0.129076 q^{74} -2.69101 q^{76} +19.1476 q^{77} -6.35531 q^{79} -0.537343 q^{82} +0.737011 q^{83} +0.241213 q^{86} -1.57439 q^{88} +9.78736 q^{89} -8.64141 q^{91} -4.69401 q^{92} -0.982560 q^{94} -0.0337081 q^{97} -1.07925 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{2} + 4 q^{4} - 8 q^{7} + 12 q^{8}+O(q^{10}) $ 8 * q + 4 * q^2 + 4 * q^4 - 8 * q^7 + 12 * q^8	$ 8 q + 4 q^{2} + 4 q^{4} - 8 q^{7} + 12 q^{8} - 2 q^{11} - 16 q^{13} - 6 q^{14} + 16 q^{17} - 14 q^{19} - 12 q^{22} + 14 q^{23} - 6 q^{26} - 16 q^{28} - 2 q^{29} - 22 q^{31} - 2 q^{32} - 12 q^{34} - 28 q^{37} - 16 q^{38} - 8 q^{41} - 20 q^{43} - 22 q^{44} - 2 q^{46} + 10 q^{47} - 16 q^{52} + 44 q^{53} - 30 q^{56} - 8 q^{58} - 14 q^{59} - 20 q^{61} + 16 q^{62} + 6 q^{64} - 16 q^{67} - 2 q^{68} - 16 q^{71} - 24 q^{73} - 26 q^{74} - 16 q^{76} + 46 q^{77} - 30 q^{79} - 16 q^{82} + 12 q^{83} - 32 q^{86} - 32 q^{88} - 16 q^{89} - 12 q^{91} - 2 q^{92} + 14 q^{94} - 16 q^{97} + 4 q^{98}+O(q^{100}) $ 8 * q + 4 * q^2 + 4 * q^4 - 8 * q^7 + 12 * q^8 - 2 * q^11 - 16 * q^13 - 6 * q^14 + 16 * q^17 - 14 * q^19 - 12 * q^22 + 14 * q^23 - 6 * q^26 - 16 * q^28 - 2 * q^29 - 22 * q^31 - 2 * q^32 - 12 * q^34 - 28 * q^37 - 16 * q^38 - 8 * q^41 - 20 * q^43 - 22 * q^44 - 2 * q^46 + 10 * q^47 - 16 * q^52 + 44 * q^53 - 30 * q^56 - 8 * q^58 - 14 * q^59 - 20 * q^61 + 16 * q^62 + 6 * q^64 - 16 * q^67 - 2 * q^68 - 16 * q^71 - 24 * q^73 - 26 * q^74 - 16 * q^76 + 46 * q^77 - 30 * q^79 - 16 * q^82 + 12 * q^83 - 32 * q^86 - 32 * q^88 - 16 * q^89 - 12 * q^91 - 2 * q^92 + 14 * q^94 - 16 * q^97 + 4 * 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 $

$2$	−0.0898194	−0.0635119	−0.0317560	−	0.999496i	$-0.510110\pi$
$2$	−0.0898194	−0.0635119	−0.0317560	+	0.999496i	$0.510110\pi$
$3$	0	0
$3$	0	0
$4$	−1.99193	−0.995966
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.36070	−1.64819	−0.824095	−	0.566451i	$-0.808316\pi$
$7$	−4.36070	−1.64819	−0.824095	+	0.566451i	$0.808316\pi$
$8$	0.358553	0.126768
$9$	0	0
$10$	0	0
$11$	−4.39094	−1.32392	−0.661959	−	0.749540i	$-0.730275\pi$
$11$	−4.39094	−1.32392	−0.661959	+	0.749540i	$0.730275\pi$
$12$	0	0
$13$	1.98166	0.549612	0.274806	−	0.961500i	$-0.411386\pi$
$13$	1.98166	0.549612	0.274806	+	0.961500i	$0.411386\pi$
$14$	0.391676	0.104680
$15$	0	0
$16$	3.95166	0.987915
$17$	0.997022	0.241813	0.120907	−	0.992664i	$-0.461420\pi$
$17$	0.997022	0.241813	0.120907	+	0.992664i	$0.461420\pi$
$18$	0	0
$19$	1.35096	0.309931	0.154965	−	0.987920i	$-0.450473\pi$
$19$	1.35096	0.309931	0.154965	+	0.987920i	$0.450473\pi$
$20$	0	0
$21$	0	0
$22$	0.394392	0.0840846
$23$	2.35651	0.491366	0.245683	−	0.969350i	$-0.420988\pi$
$23$	2.35651	0.491366	0.245683	+	0.969350i	$0.420988\pi$
$24$	0	0
$25$	0	0
$26$	−0.177991	−0.0349069
$27$	0	0
$28$	8.68622	1.64154
$29$	7.97856	1.48158	0.740790	−	0.671736i	$-0.234451\pi$
$29$	7.97856	1.48158	0.740790	+	0.671736i	$0.234451\pi$
$30$	0	0
$31$	−3.67761	−0.660519	−0.330259	−	0.943890i	$-0.607136\pi$
$31$	−3.67761	−0.660519	−0.330259	+	0.943890i	$0.607136\pi$
$32$	−1.07204	−0.189512
$33$	0	0
$34$	−0.0895519	−0.0153580
$35$	0	0
$36$	0	0
$37$	1.43706	0.236251	0.118125	−	0.992999i	$-0.462312\pi$
$37$	1.43706	0.236251	0.118125	+	0.992999i	$0.462312\pi$
$38$	−0.121342	−0.0196843
$39$	0	0
$40$	0	0
$41$	5.98248	0.934306	0.467153	−	0.884177i	$-0.345280\pi$
$41$	5.98248	0.934306	0.467153	+	0.884177i	$0.345280\pi$
$42$	0	0
$43$	−2.68554	−0.409541	−0.204770	−	0.978810i	$-0.565645\pi$
$43$	−2.68554	−0.409541	−0.204770	+	0.978810i	$0.565645\pi$
$44$	8.74646	1.31858
$45$	0	0
$46$	−0.211660	−0.0312076
$47$	10.9393	1.59566	0.797829	−	0.602883i	$-0.205982\pi$
$47$	10.9393	1.59566	0.797829	+	0.602883i	$0.205982\pi$
$48$	0	0
$49$	12.0157	1.71653
$50$	0	0
$51$	0	0
$52$	−3.94732	−0.547395
$53$	11.0510	1.51798	0.758989	−	0.651104i	$-0.225694\pi$
$53$	11.0510	1.51798	0.758989	+	0.651104i	$0.225694\pi$
$54$	0	0
$55$	0	0
$56$	−1.56354	−0.208937
$57$	0	0
$58$	−0.716629	−0.0940980
$59$	−6.68895	−0.870827	−0.435414	−	0.900231i	$-0.643398\pi$
$59$	−6.68895	−0.870827	−0.435414	+	0.900231i	$0.643398\pi$
$60$	0	0
$61$	−9.45570	−1.21068	−0.605339	−	0.795967i	$-0.706963\pi$
$61$	−9.45570	−1.21068	−0.605339	+	0.795967i	$0.706963\pi$
$62$	0.330321	0.0419508
$63$	0	0
$64$	−7.80703	−0.975879
$65$	0	0
$66$	0	0
$67$	−12.9219	−1.57866	−0.789328	−	0.613972i	$-0.789571\pi$
$67$	−12.9219	−1.57866	−0.789328	+	0.613972i	$0.789571\pi$
$68$	−1.98600	−0.240838
$69$	0	0
$70$	0	0
$71$	−7.32257	−0.869029	−0.434515	−	0.900665i	$-0.643080\pi$
$71$	−7.32257	−0.869029	−0.434515	+	0.900665i	$0.643080\pi$
$72$	0	0
$73$	0.424804	0.0497195	0.0248598	−	0.999691i	$-0.492086\pi$
$73$	0.424804	0.0497195	0.0248598	+	0.999691i	$0.492086\pi$
$74$	−0.129076	−0.0150047
$75$	0	0
$76$	−2.69101	−0.308680
$77$	19.1476	2.18207
$78$	0	0
$79$	−6.35531	−0.715028	−0.357514	−	0.933908i	$-0.616376\pi$
$79$	−6.35531	−0.715028	−0.357514	+	0.933908i	$0.616376\pi$
$80$	0	0
$81$	0	0
$82$	−0.537343	−0.0593396
$83$	0.737011	0.0808975	0.0404487	−	0.999182i	$-0.487121\pi$
$83$	0.737011	0.0808975	0.0404487	+	0.999182i	$0.487121\pi$
$84$	0	0
$85$	0	0
$86$	0.241213	0.0260107
$87$	0	0
$88$	−1.57439	−0.167830
$89$	9.78736	1.03746	0.518729	−	0.854939i	$-0.326405\pi$
$89$	9.78736	1.03746	0.518729	+	0.854939i	$0.326405\pi$
$90$	0	0
$91$	−8.64141	−0.905866
$92$	−4.69401	−0.489384
$93$	0	0
$94$	−0.982560	−0.101343
$95$	0	0
$96$	0	0
$97$	−0.0337081	−0.00342254	−0.00171127	−	0.999999i	$-0.500545\pi$
$97$	−0.0337081	−0.00342254	−0.00171127	+	0.999999i	$0.500545\pi$
$98$	−1.07925	−0.109020
$99$	0	0
$100$	0	0
$101$	−3.19390	−0.317805	−0.158902	−	0.987294i	$-0.550796\pi$
$101$	−3.19390	−0.317805	−0.158902	+	0.987294i	$0.550796\pi$
$102$	0	0
$103$	−8.55342	−0.842794	−0.421397	−	0.906876i	$-0.638460\pi$
$103$	−8.55342	−0.842794	−0.421397	+	0.906876i	$0.638460\pi$
$104$	0.710529	0.0696731
$105$	0	0
$106$	−0.992598	−0.0964097
$107$	2.22136	0.214747	0.107373	−	0.994219i	$-0.465756\pi$
$107$	2.22136	0.214747	0.107373	+	0.994219i	$0.465756\pi$
$108$	0	0
$109$	11.0023	1.05383	0.526916	−	0.849917i	$-0.323348\pi$
$109$	11.0023	1.05383	0.526916	+	0.849917i	$0.323348\pi$
$110$	0	0
$111$	0	0
$112$	−17.2320	−1.62827
$113$	1.71021	0.160883	0.0804415	−	0.996759i	$-0.474367\pi$
$113$	1.71021	0.160883	0.0804415	+	0.996759i	$0.474367\pi$
$114$	0	0
$115$	0	0
$116$	−15.8927	−1.47560
$117$	0	0
$118$	0.600798	0.0553079
$119$	−4.34771	−0.398554
$120$	0	0
$121$	8.28037	0.752761
$122$	0.849306	0.0768925
$123$	0	0
$124$	7.32556	0.657855
$125$	0	0
$126$	0	0
$127$	12.5570	1.11425	0.557125	−	0.830429i	$-0.311905\pi$
$127$	12.5570	1.11425	0.557125	+	0.830429i	$0.311905\pi$
$128$	2.84531	0.251492
$129$	0	0
$130$	0	0
$131$	−16.4718	−1.43915	−0.719574	−	0.694416i	$-0.755663\pi$
$131$	−16.4718	−1.43915	−0.719574	+	0.694416i	$0.755663\pi$
$132$	0	0
$133$	−5.89112	−0.510825
$134$	1.16063	0.100263
$135$	0	0
$136$	0.357485	0.0306541
$137$	−9.66732	−0.825935	−0.412967	−	0.910746i	$-0.635508\pi$
$137$	−9.66732	−0.825935	−0.412967	+	0.910746i	$0.635508\pi$
$138$	0	0
$139$	13.5327	1.14783	0.573913	−	0.818916i	$-0.305425\pi$
$139$	13.5327	1.14783	0.573913	+	0.818916i	$0.305425\pi$
$140$	0	0
$141$	0	0
$142$	0.657709	0.0551937
$143$	−8.70133	−0.727642
$144$	0	0
$145$	0	0
$146$	−0.0381557	−0.00315778
$147$	0	0
$148$	−2.86252	−0.235298
$149$	−13.6843	−1.12106	−0.560529	−	0.828134i	$-0.689402\pi$
$149$	−13.6843	−1.12106	−0.560529	+	0.828134i	$0.689402\pi$
$150$	0	0
$151$	−11.3204	−0.921237	−0.460619	−	0.887598i	$-0.652372\pi$
$151$	−11.3204	−0.921237	−0.460619	+	0.887598i	$0.652372\pi$
$152$	0.484389	0.0392892
$153$	0	0
$154$	−1.71983	−0.138587
$155$	0	0
$156$	0	0
$157$	−8.56070	−0.683219	−0.341609	−	0.939842i	$-0.610972\pi$
$157$	−8.56070	−0.683219	−0.341609	+	0.939842i	$0.610972\pi$
$158$	0.570830	0.0454128
$159$	0	0
$160$	0	0
$161$	−10.2760	−0.809865
$162$	0	0
$163$	−4.58509	−0.359132	−0.179566	−	0.983746i	$-0.557469\pi$
$163$	−4.58509	−0.359132	−0.179566	+	0.983746i	$0.557469\pi$
$164$	−11.9167	−0.930537
$165$	0	0
$166$	−0.0661979	−0.00513795
$167$	7.21792	0.558540	0.279270	−	0.960213i	$-0.409908\pi$
$167$	7.21792	0.558540	0.279270	+	0.960213i	$0.409908\pi$
$168$	0	0
$169$	−9.07304	−0.697926
$170$	0	0
$171$	0	0
$172$	5.34941	0.407889
$173$	−4.65009	−0.353540	−0.176770	−	0.984252i	$-0.556565\pi$
$173$	−4.65009	−0.353540	−0.176770	+	0.984252i	$0.556565\pi$
$174$	0	0
$175$	0	0
$176$	−17.3515	−1.30792
$177$	0	0
$178$	−0.879095	−0.0658909
$179$	7.20338	0.538406	0.269203	−	0.963083i	$-0.413240\pi$
$179$	7.20338	0.538406	0.269203	+	0.963083i	$0.413240\pi$
$180$	0	0
$181$	−3.80424	−0.282767	−0.141383	−	0.989955i	$-0.545155\pi$
$181$	−3.80424	−0.282767	−0.141383	+	0.989955i	$0.545155\pi$
$182$	0.776167	0.0575333
$183$	0	0
$184$	0.844934	0.0622893
$185$	0	0
$186$	0	0
$187$	−4.37786	−0.320141
$188$	−21.7903	−1.58922
$189$	0	0
$190$	0	0
$191$	−21.5541	−1.55960	−0.779801	−	0.626027i	$-0.784680\pi$
$191$	−21.5541	−1.55960	−0.779801	+	0.626027i	$0.784680\pi$
$192$	0	0
$193$	−3.15029	−0.226763	−0.113381	−	0.993552i	$-0.536168\pi$
$193$	−3.15029	−0.226763	−0.113381	+	0.993552i	$0.536168\pi$
$194$	0.00302764	0.000217372 0
$195$	0	0
$196$	−23.9345	−1.70961
$197$	26.0837	1.85839	0.929195	−	0.369590i	$-0.120502\pi$
$197$	26.0837	1.85839	0.929195	+	0.369590i	$0.120502\pi$
$198$	0	0
$199$	−24.2662	−1.72018	−0.860092	−	0.510139i	$-0.829594\pi$
$199$	−24.2662	−1.72018	−0.860092	+	0.510139i	$0.829594\pi$
$200$	0	0
$201$	0	0
$202$	0.286874	0.0201844
$203$	−34.7921	−2.44193
$204$	0	0
$205$	0	0
$206$	0.768264	0.0535275
$207$	0	0
$208$	7.83083	0.542970
$209$	−5.93197	−0.410323
$210$	0	0
$211$	16.3783	1.12753	0.563765	−	0.825935i	$-0.309352\pi$
$211$	16.3783	1.12753	0.563765	+	0.825935i	$0.309352\pi$
$212$	−22.0129	−1.51185
$213$	0	0
$214$	−0.199521	−0.0136390
$215$	0	0
$216$	0	0
$217$	16.0370	1.08866
$218$	−0.988224	−0.0669310
$219$	0	0
$220$	0	0
$221$	1.97575	0.132904
$222$	0	0
$223$	5.68295	0.380558	0.190279	−	0.981730i	$-0.439061\pi$
$223$	5.68295	0.380558	0.190279	+	0.981730i	$0.439061\pi$
$224$	4.67486	0.312352
$225$	0	0
$226$	−0.153610	−0.0102180
$227$	3.64374	0.241844	0.120922	−	0.992662i	$-0.461415\pi$
$227$	3.64374	0.241844	0.120922	+	0.992662i	$0.461415\pi$
$228$	0	0
$229$	1.66125	0.109779	0.0548893	−	0.998492i	$-0.482519\pi$
$229$	1.66125	0.109779	0.0548893	+	0.998492i	$0.482519\pi$
$230$	0	0
$231$	0	0
$232$	2.86074	0.187817
$233$	7.85899	0.514860	0.257430	−	0.966297i	$-0.417124\pi$
$233$	7.85899	0.514860	0.257430	+	0.966297i	$0.417124\pi$
$234$	0	0
$235$	0	0
$236$	13.3239	0.867314
$237$	0	0
$238$	0.390509	0.0253130
$239$	0.567301	0.0366956	0.0183478	−	0.999832i	$-0.494159\pi$
$239$	0.567301	0.0366956	0.0183478	+	0.999832i	$0.494159\pi$
$240$	0	0
$241$	−19.0081	−1.22442	−0.612211	−	0.790695i	$-0.709720\pi$
$241$	−19.0081	−1.22442	−0.612211	+	0.790695i	$0.709720\pi$
$242$	−0.743738	−0.0478093
$243$	0	0
$244$	18.8351	1.20580
$245$	0	0
$246$	0	0
$247$	2.67713	0.170342
$248$	−1.31862	−0.0837324
$249$	0	0
$250$	0	0
$251$	−3.02533	−0.190957	−0.0954787	−	0.995431i	$-0.530438\pi$
$251$	−3.02533	−0.190957	−0.0954787	+	0.995431i	$0.530438\pi$
$252$	0	0
$253$	−10.3473	−0.650529
$254$	−1.12786	−0.0707681
$255$	0	0
$256$	15.3585	0.959906
$257$	−19.8613	−1.23891	−0.619456	−	0.785032i	$-0.712647\pi$
$257$	−19.8613	−1.23891	−0.619456	+	0.785032i	$0.712647\pi$
$258$	0	0
$259$	−6.26658	−0.389386
$260$	0	0
$261$	0	0
$262$	1.47949	0.0914030
$263$	22.8299	1.40775	0.703876	−	0.710323i	$-0.251451\pi$
$263$	22.8299	1.40775	0.703876	+	0.710323i	$0.251451\pi$
$264$	0	0
$265$	0	0
$266$	0.529137	0.0324435
$267$	0	0
$268$	25.7395	1.57229
$269$	−14.8324	−0.904347	−0.452174	−	0.891930i	$-0.649351\pi$
$269$	−14.8324	−0.904347	−0.452174	+	0.891930i	$0.649351\pi$
$270$	0	0
$271$	−6.43720	−0.391032	−0.195516	−	0.980701i	$-0.562638\pi$
$271$	−6.43720	−0.391032	−0.195516	+	0.980701i	$0.562638\pi$
$272$	3.93989	0.238891
$273$	0	0
$274$	0.868313	0.0524567
$275$	0	0
$276$	0	0
$277$	−6.70976	−0.403150	−0.201575	−	0.979473i	$-0.564606\pi$
$277$	−6.70976	−0.403150	−0.201575	+	0.979473i	$0.564606\pi$
$278$	−1.21550	−0.0729006
$279$	0	0
$280$	0	0
$281$	20.4867	1.22214	0.611068	−	0.791578i	$-0.290740\pi$
$281$	20.4867	1.22214	0.611068	+	0.791578i	$0.290740\pi$
$282$	0	0
$283$	11.4177	0.678715	0.339357	−	0.940658i	$-0.389790\pi$
$283$	11.4177	0.678715	0.339357	+	0.940658i	$0.389790\pi$
$284$	14.5861	0.865524
$285$	0	0
$286$	0.781549	0.0462140
$287$	−26.0878	−1.53991
$288$	0	0
$289$	−16.0059	−0.941526
$290$	0	0
$291$	0	0
$292$	−0.846181	−0.0495190
$293$	28.5505	1.66794	0.833968	−	0.551812i	$-0.186063\pi$
$293$	28.5505	1.66794	0.833968	+	0.551812i	$0.186063\pi$
$294$	0	0
$295$	0	0
$296$	0.515261	0.0299489
$297$	0	0
$298$	1.22911	0.0712006
$299$	4.66979	0.270061
$300$	0	0
$301$	11.7108	0.675001
$302$	1.01679	0.0585095
$303$	0	0
$304$	5.33852	0.306185
$305$	0	0
$306$	0	0
$307$	20.5417	1.17238	0.586188	−	0.810175i	$-0.300628\pi$
$307$	20.5417	1.17238	0.586188	+	0.810175i	$0.300628\pi$
$308$	−38.1407	−2.17327
$309$	0	0
$310$	0	0
$311$	−17.5496	−0.995146	−0.497573	−	0.867422i	$-0.665775\pi$
$311$	−17.5496	−0.995146	−0.497573	+	0.867422i	$0.665775\pi$
$312$	0	0
$313$	2.98564	0.168758	0.0843790	−	0.996434i	$-0.473109\pi$
$313$	2.98564	0.168758	0.0843790	+	0.996434i	$0.473109\pi$
$314$	0.768918	0.0433925
$315$	0	0
$316$	12.6593	0.712143
$317$	−16.1708	−0.908244	−0.454122	−	0.890940i	$-0.650047\pi$
$317$	−16.1708	−0.908244	−0.454122	+	0.890940i	$0.650047\pi$
$318$	0	0
$319$	−35.0334	−1.96149
$320$	0	0
$321$	0	0
$322$	0.922988	0.0514361
$323$	1.34693	0.0749453
$324$	0	0
$325$	0	0
$326$	0.411830	0.0228092
$327$	0	0
$328$	2.14504	0.118440
$329$	−47.7030	−2.62995
$330$	0	0
$331$	−13.0705	−0.718418	−0.359209	−	0.933257i	$-0.616953\pi$
$331$	−13.0705	−0.718418	−0.359209	+	0.933257i	$0.616953\pi$
$332$	−1.46808	−0.0805711
$333$	0	0
$334$	−0.648310	−0.0354739
$335$	0	0
$336$	0	0
$337$	−26.8049	−1.46015	−0.730077	−	0.683365i	$-0.760516\pi$
$337$	−26.8049	−1.46015	−0.730077	+	0.683365i	$0.760516\pi$
$338$	0.814935	0.0443266
$339$	0	0
$340$	0	0
$341$	16.1482	0.874473
$342$	0	0
$343$	−21.8721	−1.18098
$344$	−0.962908	−0.0519165
$345$	0	0
$346$	0.417669	0.0224540
$347$	−25.4859	−1.36815	−0.684077	−	0.729410i	$-0.739795\pi$
$347$	−25.4859	−1.36815	−0.684077	+	0.729410i	$0.739795\pi$
$348$	0	0
$349$	−28.0435	−1.50113	−0.750566	−	0.660795i	$-0.770219\pi$
$349$	−28.0435	−1.50113	−0.750566	+	0.660795i	$0.770219\pi$
$350$	0	0
$351$	0	0
$352$	4.70727	0.250899
$353$	14.6667	0.780630	0.390315	−	0.920681i	$-0.372366\pi$
$353$	14.6667	0.780630	0.390315	+	0.920681i	$0.372366\pi$
$354$	0	0
$355$	0	0
$356$	−19.4958	−1.03327
$357$	0	0
$358$	−0.647004	−0.0341952
$359$	14.6205	0.771642	0.385821	−	0.922574i	$-0.373918\pi$
$359$	14.6205	0.771642	0.385821	+	0.922574i	$0.373918\pi$
$360$	0	0
$361$	−17.1749	−0.903943
$362$	0.341694	0.0179591
$363$	0	0
$364$	17.2131	0.902212
$365$	0	0
$366$	0	0
$367$	17.6940	0.923617	0.461808	−	0.886980i	$-0.347201\pi$
$367$	17.6940	0.923617	0.461808	+	0.886980i	$0.347201\pi$
$368$	9.31212	0.485428
$369$	0	0
$370$	0	0
$371$	−48.1903	−2.50192
$372$	0	0
$373$	−12.2293	−0.633210	−0.316605	−	0.948557i	$-0.602543\pi$
$373$	−12.2293	−0.633210	−0.316605	+	0.948557i	$0.602543\pi$
$374$	0.393217	0.0203328
$375$	0	0
$376$	3.92231	0.202278
$377$	15.8107	0.814295
$378$	0	0
$379$	28.0951	1.44315	0.721574	−	0.692338i	$-0.243419\pi$
$379$	28.0951	1.44315	0.721574	+	0.692338i	$0.243419\pi$
$380$	0	0
$381$	0	0
$382$	1.93598	0.0990533
$383$	34.4334	1.75947	0.879733	−	0.475468i	$-0.157721\pi$
$383$	34.4334	1.75947	0.879733	+	0.475468i	$0.157721\pi$
$384$	0	0
$385$	0	0
$386$	0.282957	0.0144021
$387$	0	0
$388$	0.0671442	0.00340873
$389$	−13.5225	−0.685619	−0.342809	−	0.939405i	$-0.611378\pi$
$389$	−13.5225	−0.685619	−0.342809	+	0.939405i	$0.611378\pi$
$390$	0	0
$391$	2.34949	0.118819
$392$	4.30827	0.217601
$393$	0	0
$394$	−2.34283	−0.118030
$395$	0	0
$396$	0	0
$397$	35.9744	1.80550	0.902751	−	0.430164i	$-0.141544\pi$
$397$	35.9744	1.80550	0.902751	+	0.430164i	$0.141544\pi$
$398$	2.17958	0.109252
$399$	0	0
$400$	0	0
$401$	−4.35977	−0.217717	−0.108858	−	0.994057i	$-0.534719\pi$
$401$	−4.35977	−0.217717	−0.108858	+	0.994057i	$0.534719\pi$
$402$	0	0
$403$	−7.28776	−0.363029
$404$	6.36203	0.316523
$405$	0	0
$406$	3.12501	0.155092
$407$	−6.31003	−0.312777
$408$	0	0
$409$	−18.1138	−0.895668	−0.447834	−	0.894117i	$-0.647804\pi$
$409$	−18.1138	−0.895668	−0.447834	+	0.894117i	$0.647804\pi$
$410$	0	0
$411$	0	0
$412$	17.0378	0.839394
$413$	29.1685	1.43529
$414$	0	0
$415$	0	0
$416$	−2.12442	−0.104158
$417$	0	0
$418$	0.532806	0.0260604
$419$	−0.527867	−0.0257880	−0.0128940	−	0.999917i	$-0.504104\pi$
$419$	−0.527867	−0.0257880	−0.0128940	+	0.999917i	$0.504104\pi$
$420$	0	0
$421$	18.6586	0.909365	0.454682	−	0.890654i	$-0.349753\pi$
$421$	18.6586	0.909365	0.454682	+	0.890654i	$0.349753\pi$
$422$	−1.47109	−0.0716117
$423$	0	0
$424$	3.96239	0.192430
$425$	0	0
$426$	0	0
$427$	41.2335	1.99543
$428$	−4.42480	−0.213881
$429$	0	0
$430$	0	0
$431$	−20.9913	−1.01112	−0.505559	−	0.862792i	$-0.668714\pi$
$431$	−20.9913	−1.01112	−0.505559	+	0.862792i	$0.668714\pi$
$432$	0	0
$433$	13.6639	0.656647	0.328324	−	0.944565i	$-0.393516\pi$
$433$	13.6639	0.656647	0.328324	+	0.944565i	$0.393516\pi$
$434$	−1.44043	−0.0691430
$435$	0	0
$436$	−21.9159	−1.04958
$437$	3.18354	0.152289
$438$	0	0
$439$	−4.33339	−0.206821	−0.103411	−	0.994639i	$-0.532976\pi$
$439$	−4.33339	−0.206821	−0.103411	+	0.994639i	$0.532976\pi$
$440$	0	0
$441$	0	0
$442$	−0.177461	−0.00844096
$443$	−1.60742	−0.0763707	−0.0381854	−	0.999271i	$-0.512158\pi$
$443$	−1.60742	−0.0763707	−0.0381854	+	0.999271i	$0.512158\pi$
$444$	0	0
$445$	0	0
$446$	−0.510439	−0.0241700
$447$	0	0
$448$	34.0441	1.60843
$449$	13.8291	0.652634	0.326317	−	0.945260i	$-0.394192\pi$
$449$	13.8291	0.652634	0.326317	+	0.945260i	$0.394192\pi$
$450$	0	0
$451$	−26.2687	−1.23694
$452$	−3.40662	−0.160234
$453$	0	0
$454$	−0.327279	−0.0153600
$455$	0	0
$456$	0	0
$457$	−20.1345	−0.941850	−0.470925	−	0.882173i	$-0.656080\pi$
$457$	−20.1345	−0.941850	−0.470925	+	0.882173i	$0.656080\pi$
$458$	−0.149213	−0.00697225
$459$	0	0
$460$	0	0
$461$	−31.5264	−1.46833	−0.734166	−	0.678970i	$-0.762427\pi$
$461$	−31.5264	−1.46833	−0.734166	+	0.678970i	$0.762427\pi$
$462$	0	0
$463$	0.0451697	0.00209921	0.00104961	−	0.999999i	$-0.499666\pi$
$463$	0.0451697	0.00209921	0.00104961	+	0.999999i	$0.499666\pi$
$464$	31.5285	1.46368
$465$	0	0
$466$	−0.705890	−0.0326997
$467$	−32.8349	−1.51942	−0.759708	−	0.650265i	$-0.774658\pi$
$467$	−32.8349	−1.51942	−0.759708	+	0.650265i	$0.774658\pi$
$468$	0	0
$469$	56.3484	2.60193
$470$	0	0
$471$	0	0
$472$	−2.39834	−0.110393
$473$	11.7920	0.542198
$474$	0	0
$475$	0	0
$476$	8.66035	0.396947
$477$	0	0
$478$	−0.0509546	−0.00233061
$479$	−7.48576	−0.342033	−0.171017	−	0.985268i	$-0.554705\pi$
$479$	−7.48576	−0.342033	−0.171017	+	0.985268i	$0.554705\pi$
$480$	0	0
$481$	2.84775	0.129846
$482$	1.70730	0.0777654
$483$	0	0
$484$	−16.4939	−0.749724
$485$	0	0
$486$	0	0
$487$	25.4295	1.15232	0.576160	−	0.817337i	$-0.304551\pi$
$487$	25.4295	1.15232	0.576160	+	0.817337i	$0.304551\pi$
$488$	−3.39037	−0.153475
$489$	0	0
$490$	0	0
$491$	11.0668	0.499438	0.249719	−	0.968318i	$-0.419662\pi$
$491$	11.0668	0.499438	0.249719	+	0.968318i	$0.419662\pi$
$492$	0	0
$493$	7.95479	0.358266
$494$	−0.240458	−0.0108187
$495$	0	0
$496$	−14.5327	−0.652537
$497$	31.9315	1.43233
$498$	0	0
$499$	4.68157	0.209576	0.104788	−	0.994495i	$-0.466584\pi$
$499$	4.68157	0.209576	0.104788	+	0.994495i	$0.466584\pi$
$500$	0	0
$501$	0	0
$502$	0.271734	0.0121281
$503$	10.8285	0.482818	0.241409	−	0.970423i	$-0.422391\pi$
$503$	10.8285	0.482818	0.241409	+	0.970423i	$0.422391\pi$
$504$	0	0
$505$	0	0
$506$	0.929388	0.0413163
$507$	0	0
$508$	−25.0126	−1.10975
$509$	−11.5007	−0.509757	−0.254879	−	0.966973i	$-0.582036\pi$
$509$	−11.5007	−0.509757	−0.254879	+	0.966973i	$0.582036\pi$
$510$	0	0
$511$	−1.85244	−0.0819473
$512$	−7.07011	−0.312457
$513$	0	0
$514$	1.78393	0.0786856
$515$	0	0
$516$	0	0
$517$	−48.0338	−2.11252
$518$	0.562860	0.0247307
$519$	0	0
$520$	0	0
$521$	−0.797807	−0.0349525	−0.0174763	−	0.999847i	$-0.505563\pi$
$521$	−0.797807	−0.0349525	−0.0174763	+	0.999847i	$0.505563\pi$
$522$	0	0
$523$	−40.1429	−1.75533	−0.877663	−	0.479279i	$-0.840898\pi$
$523$	−40.1429	−1.75533	−0.877663	+	0.479279i	$0.840898\pi$
$524$	32.8107	1.43334
$525$	0	0
$526$	−2.05057	−0.0894090
$527$	−3.66666	−0.159722
$528$	0	0
$529$	−17.4469	−0.758559
$530$	0	0
$531$	0	0
$532$	11.7347	0.508764
$533$	11.8552	0.513506
$534$	0	0
$535$	0	0
$536$	−4.63317	−0.200123
$537$	0	0
$538$	1.33224	0.0574369
$539$	−52.7603	−2.27255
$540$	0	0
$541$	1.41016	0.0606277	0.0303138	−	0.999540i	$-0.490349\pi$
$541$	1.41016	0.0606277	0.0303138	+	0.999540i	$0.490349\pi$
$542$	0.578185	0.0248352
$543$	0	0
$544$	−1.06885	−0.0458265
$545$	0	0
$546$	0	0
$547$	−15.4621	−0.661110	−0.330555	−	0.943787i	$-0.607236\pi$
$547$	−15.4621	−0.661110	−0.330555	+	0.943787i	$0.607236\pi$
$548$	19.2566	0.822603
$549$	0	0
$550$	0	0
$551$	10.7787	0.459187
$552$	0	0
$553$	27.7136	1.17850
$554$	0.602667	0.0256049
$555$	0	0
$556$	−26.9562	−1.14320
$557$	−18.0445	−0.764568	−0.382284	−	0.924045i	$-0.624862\pi$
$557$	−18.0445	−0.764568	−0.382284	+	0.924045i	$0.624862\pi$
$558$	0	0
$559$	−5.32181	−0.225089
$560$	0	0
$561$	0	0
$562$	−1.84011	−0.0776202
$563$	−28.5327	−1.20251	−0.601254	−	0.799058i	$-0.705332\pi$
$563$	−28.5327	−1.20251	−0.601254	+	0.799058i	$0.705332\pi$
$564$	0	0
$565$	0	0
$566$	−1.02554	−0.0431065
$567$	0	0
$568$	−2.62553	−0.110165
$569$	−16.4072	−0.687826	−0.343913	−	0.939001i	$-0.611753\pi$
$569$	−16.4072	−0.687826	−0.343913	+	0.939001i	$0.611753\pi$
$570$	0	0
$571$	−8.34705	−0.349313	−0.174657	−	0.984629i	$-0.555882\pi$
$571$	−8.34705	−0.349313	−0.174657	+	0.984629i	$0.555882\pi$
$572$	17.3325	0.724707
$573$	0	0
$574$	2.34319	0.0978029
$575$	0	0
$576$	0	0
$577$	−8.43531	−0.351167	−0.175583	−	0.984465i	$-0.556181\pi$
$577$	−8.43531	−0.351167	−0.175583	+	0.984465i	$0.556181\pi$
$578$	1.43765	0.0597982
$579$	0	0
$580$	0	0
$581$	−3.21389	−0.133334
$582$	0	0
$583$	−48.5245	−2.00968
$584$	0.152315	0.00630283
$585$	0	0
$586$	−2.56439	−0.105934
$587$	−24.2852	−1.00236	−0.501179	−	0.865344i	$-0.667100\pi$
$587$	−24.2852	−1.00236	−0.501179	+	0.865344i	$0.667100\pi$
$588$	0	0
$589$	−4.96829	−0.204715
$590$	0	0
$591$	0	0
$592$	5.67876	0.233396
$593$	−28.4653	−1.16893	−0.584466	−	0.811418i	$-0.698696\pi$
$593$	−28.4653	−1.16893	−0.584466	+	0.811418i	$0.698696\pi$
$594$	0	0
$595$	0	0
$596$	27.2581	1.11654
$597$	0	0
$598$	−0.419438	−0.0171521
$599$	−16.0387	−0.655323	−0.327662	−	0.944795i	$-0.606261\pi$
$599$	−16.0387	−0.655323	−0.327662	+	0.944795i	$0.606261\pi$
$600$	0	0
$601$	−8.09005	−0.330000	−0.165000	−	0.986294i	$-0.552762\pi$
$601$	−8.09005	−0.330000	−0.165000	+	0.986294i	$0.552762\pi$
$602$	−1.05186	−0.0428706
$603$	0	0
$604$	22.5494	0.917521
$605$	0	0
$606$	0	0
$607$	0.434608	0.0176402	0.00882010	−	0.999961i	$-0.497192\pi$
$607$	0.434608	0.0176402	0.00882010	+	0.999961i	$0.497192\pi$
$608$	−1.44828	−0.0587356
$609$	0	0
$610$	0	0
$611$	21.6779	0.876994
$612$	0	0
$613$	39.3962	1.59120	0.795599	−	0.605824i	$-0.207156\pi$
$613$	39.3962	1.59120	0.795599	+	0.605824i	$0.207156\pi$
$614$	−1.84504	−0.0744599
$615$	0	0
$616$	6.86543	0.276616
$617$	15.4146	0.620568	0.310284	−	0.950644i	$-0.399576\pi$
$617$	15.4146	0.620568	0.310284	+	0.950644i	$0.399576\pi$
$618$	0	0
$619$	−10.6718	−0.428935	−0.214468	−	0.976731i	$-0.568802\pi$
$619$	−10.6718	−0.428935	−0.214468	+	0.976731i	$0.568802\pi$
$620$	0	0
$621$	0	0
$622$	1.57629	0.0632036
$623$	−42.6797	−1.70993
$624$	0	0
$625$	0	0
$626$	−0.268168	−0.0107182
$627$	0	0
$628$	17.0523	0.680463
$629$	1.43278	0.0571285
$630$	0	0
$631$	−18.4347	−0.733874	−0.366937	−	0.930246i	$-0.619594\pi$
$631$	−18.4347	−0.733874	−0.366937	+	0.930246i	$0.619594\pi$
$632$	−2.27871	−0.0906424
$633$	0	0
$634$	1.45245	0.0576843
$635$	0	0
$636$	0	0
$637$	23.8110	0.943427
$638$	3.14668	0.124578
$639$	0	0
$640$	0	0
$641$	−12.4281	−0.490882	−0.245441	−	0.969412i	$-0.578933\pi$
$641$	−12.4281	−0.490882	−0.245441	+	0.969412i	$0.578933\pi$
$642$	0	0
$643$	1.84657	0.0728218	0.0364109	−	0.999337i	$-0.488407\pi$
$643$	1.84657	0.0728218	0.0364109	+	0.999337i	$0.488407\pi$
$644$	20.4692	0.806598
$645$	0	0
$646$	−0.120981	−0.00475992
$647$	−38.9760	−1.53230	−0.766152	−	0.642660i	$-0.777831\pi$
$647$	−38.9760	−1.53230	−0.766152	+	0.642660i	$0.777831\pi$
$648$	0	0
$649$	29.3708	1.15290
$650$	0	0
$651$	0	0
$652$	9.13319	0.357683
$653$	28.3389	1.10899	0.554493	−	0.832189i	$-0.312912\pi$
$653$	28.3389	1.10899	0.554493	+	0.832189i	$0.312912\pi$
$654$	0	0
$655$	0	0
$656$	23.6407	0.923015
$657$	0	0
$658$	4.28465	0.167033
$659$	22.3561	0.870868	0.435434	−	0.900221i	$-0.356595\pi$
$659$	22.3561	0.870868	0.435434	+	0.900221i	$0.356595\pi$
$660$	0	0
$661$	29.2143	1.13631	0.568153	−	0.822923i	$-0.307658\pi$
$661$	29.2143	1.13631	0.568153	+	0.822923i	$0.307658\pi$
$662$	1.17398	0.0456281
$663$	0	0
$664$	0.264258	0.0102552
$665$	0	0
$666$	0	0
$667$	18.8015	0.727998
$668$	−14.3776	−0.556287
$669$	0	0
$670$	0	0
$671$	41.5194	1.60284
$672$	0	0
$673$	−3.21546	−0.123947	−0.0619735	−	0.998078i	$-0.519739\pi$
$673$	−3.21546	−0.123947	−0.0619735	+	0.998078i	$0.519739\pi$
$674$	2.40760	0.0927372
$675$	0	0
$676$	18.0729	0.695111
$677$	−47.2470	−1.81585	−0.907925	−	0.419133i	$-0.862334\pi$
$677$	−47.2470	−1.81585	−0.907925	+	0.419133i	$0.862334\pi$
$678$	0	0
$679$	0.146991	0.00564099
$680$	0	0
$681$	0	0
$682$	−1.45042	−0.0555395
$683$	−37.2527	−1.42543	−0.712717	−	0.701452i	$-0.752536\pi$
$683$	−37.2527	−1.42543	−0.712717	+	0.701452i	$0.752536\pi$
$684$	0	0
$685$	0	0
$686$	1.96454	0.0750064
$687$	0	0
$688$	−10.6123	−0.404591
$689$	21.8994	0.834299
$690$	0	0
$691$	−10.9827	−0.417803	−0.208901	−	0.977937i	$-0.566989\pi$
$691$	−10.9827	−0.417803	−0.208901	+	0.977937i	$0.566989\pi$
$692$	9.26267	0.352114
$693$	0	0
$694$	2.28913	0.0868941
$695$	0	0
$696$	0	0
$697$	5.96466	0.225928
$698$	2.51885	0.0953399
$699$	0	0
$700$	0	0
$701$	−22.4086	−0.846361	−0.423180	−	0.906046i	$-0.639086\pi$
$701$	−22.4086	−0.846361	−0.423180	+	0.906046i	$0.639086\pi$
$702$	0	0
$703$	1.94140	0.0732213
$704$	34.2802	1.29198
$705$	0	0
$706$	−1.31736	−0.0495793
$707$	13.9276	0.523803
$708$	0	0
$709$	−14.6297	−0.549431	−0.274716	−	0.961526i	$-0.588584\pi$
$709$	−14.6297	−0.549431	−0.274716	+	0.961526i	$0.588584\pi$
$710$	0	0
$711$	0	0
$712$	3.50929	0.131516
$713$	−8.66633	−0.324557
$714$	0	0
$715$	0	0
$716$	−14.3486	−0.536234
$717$	0	0
$718$	−1.31321	−0.0490085
$719$	28.4977	1.06278	0.531392	−	0.847126i	$-0.321669\pi$
$719$	28.4977	1.06278	0.531392	+	0.847126i	$0.321669\pi$
$720$	0	0
$721$	37.2989	1.38908
$722$	1.54264	0.0574112
$723$	0	0
$724$	7.57778	0.281626
$725$	0	0
$726$	0	0
$727$	−44.0064	−1.63211	−0.816054	−	0.577976i	$-0.803843\pi$
$727$	−44.0064	−1.63211	−0.816054	+	0.577976i	$0.803843\pi$
$728$	−3.09840	−0.114835
$729$	0	0
$730$	0	0
$731$	−2.67754	−0.0990324
$732$	0	0
$733$	27.3227	1.00919	0.504593	−	0.863357i	$-0.331643\pi$
$733$	27.3227	1.00919	0.504593	+	0.863357i	$0.331643\pi$
$734$	−1.58926	−0.0586607
$735$	0	0
$736$	−2.52628	−0.0931198
$737$	56.7391	2.09001
$738$	0	0
$739$	16.4678	0.605776	0.302888	−	0.953026i	$-0.402049\pi$
$739$	16.4678	0.605776	0.302888	+	0.953026i	$0.402049\pi$
$740$	0	0
$741$	0	0
$742$	4.32843	0.158901
$743$	−35.6012	−1.30608	−0.653041	−	0.757322i	$-0.726507\pi$
$743$	−35.6012	−1.30608	−0.653041	+	0.757322i	$0.726507\pi$
$744$	0	0
$745$	0	0
$746$	1.09843	0.0402164
$747$	0	0
$748$	8.72041	0.318850
$749$	−9.68668	−0.353944
$750$	0	0
$751$	46.0748	1.68129	0.840647	−	0.541583i	$-0.182175\pi$
$751$	46.0748	1.68129	0.840647	+	0.541583i	$0.182175\pi$
$752$	43.2283	1.57638
$753$	0	0
$754$	−1.42011	−0.0517174
$755$	0	0
$756$	0	0
$757$	−36.6482	−1.33200	−0.666000	−	0.745951i	$-0.731995\pi$
$757$	−36.6482	−1.33200	−0.666000	+	0.745951i	$0.731995\pi$
$758$	−2.52348	−0.0916571
$759$	0	0
$760$	0	0
$761$	27.2078	0.986282	0.493141	−	0.869949i	$-0.335849\pi$
$761$	27.2078	0.986282	0.493141	+	0.869949i	$0.335849\pi$
$762$	0	0
$763$	−47.9779	−1.73692
$764$	42.9344	1.55331
$765$	0	0
$766$	−3.09279	−0.111747
$767$	−13.2552	−0.478617
$768$	0	0
$769$	26.5327	0.956794	0.478397	−	0.878144i	$-0.341218\pi$
$769$	26.5327	0.956794	0.478397	+	0.878144i	$0.341218\pi$
$770$	0	0
$771$	0	0
$772$	6.27516	0.225848
$773$	−28.7677	−1.03470	−0.517351	−	0.855773i	$-0.673082\pi$
$773$	−28.7677	−1.03470	−0.517351	+	0.855773i	$0.673082\pi$
$774$	0	0
$775$	0	0
$776$	−0.0120861	−0.000433867 0
$777$	0	0
$778$	1.21458	0.0435450
$779$	8.08206	0.289570
$780$	0	0
$781$	32.1530	1.15052
$782$	−0.211030	−0.00754641
$783$	0	0
$784$	47.4820	1.69579
$785$	0	0
$786$	0	0
$787$	−24.8663	−0.886387	−0.443193	−	0.896426i	$-0.646155\pi$
$787$	−24.8663	−0.886387	−0.443193	+	0.896426i	$0.646155\pi$
$788$	−51.9571	−1.85089
$789$	0	0
$790$	0	0
$791$	−7.45771	−0.265166
$792$	0	0
$793$	−18.7379	−0.665404
$794$	−3.23120	−0.114671
$795$	0	0
$796$	48.3366	1.71325
$797$	−31.4206	−1.11298	−0.556488	−	0.830856i	$-0.687851\pi$
$797$	−31.4206	−1.11298	−0.556488	+	0.830856i	$0.687851\pi$
$798$	0	0
$799$	10.9067	0.385851
$800$	0	0
$801$	0	0
$802$	0.391592	0.0138276
$803$	−1.86529	−0.0658246
$804$	0	0
$805$	0	0
$806$	0.654583	0.0230567
$807$	0	0
$808$	−1.14518	−0.0402874
$809$	−50.5027	−1.77558	−0.887790	−	0.460249i	$-0.847760\pi$
$809$	−50.5027	−1.77558	−0.887790	+	0.460249i	$0.847760\pi$
$810$	0	0
$811$	25.1612	0.883528	0.441764	−	0.897131i	$-0.354353\pi$
$811$	25.1612	0.883528	0.441764	+	0.897131i	$0.354353\pi$
$812$	69.3035	2.43208
$813$	0	0
$814$	0.566763	0.0198650
$815$	0	0
$816$	0	0
$817$	−3.62804	−0.126929
$818$	1.62697	0.0568856
$819$	0	0
$820$	0	0
$821$	20.6307	0.720017	0.360009	−	0.932949i	$-0.382774\pi$
$821$	20.6307	0.720017	0.360009	+	0.932949i	$0.382774\pi$
$822$	0	0
$823$	35.8618	1.25006	0.625032	−	0.780599i	$-0.285086\pi$
$823$	35.8618	1.25006	0.625032	+	0.780599i	$0.285086\pi$
$824$	−3.06686	−0.106839
$825$	0	0
$826$	−2.61990	−0.0911580
$827$	4.73642	0.164701	0.0823507	−	0.996603i	$-0.473757\pi$
$827$	4.73642	0.164701	0.0823507	+	0.996603i	$0.473757\pi$
$828$	0	0
$829$	−28.4768	−0.989039	−0.494520	−	0.869166i	$-0.664656\pi$
$829$	−28.4768	−0.989039	−0.494520	+	0.869166i	$0.664656\pi$
$830$	0	0
$831$	0	0
$832$	−15.4708	−0.536355
$833$	11.9799	0.415080
$834$	0	0
$835$	0	0
$836$	11.8161	0.408668
$837$	0	0
$838$	0.0474127	0.00163784
$839$	0.765715	0.0264354	0.0132177	−	0.999913i	$-0.495793\pi$
$839$	0.765715	0.0264354	0.0132177	+	0.999913i	$0.495793\pi$
$840$	0	0
$841$	34.6574	1.19508
$842$	−1.67591	−0.0577555
$843$	0	0
$844$	−32.6245	−1.12298
$845$	0	0
$846$	0	0
$847$	−36.1082	−1.24069
$848$	43.6700	1.49963
$849$	0	0
$850$	0	0
$851$	3.38644	0.116086
$852$	0	0
$853$	−34.7289	−1.18910	−0.594548	−	0.804060i	$-0.702669\pi$
$853$	−34.7289	−1.18910	−0.594548	+	0.804060i	$0.702669\pi$
$854$	−3.70357	−0.126734
$855$	0	0
$856$	0.796475	0.0272230
$857$	54.2561	1.85335	0.926676	−	0.375860i	$-0.122653\pi$
$857$	54.2561	1.85335	0.926676	+	0.375860i	$0.122653\pi$
$858$	0	0
$859$	15.9095	0.542824	0.271412	−	0.962463i	$-0.412509\pi$
$859$	15.9095	0.542824	0.271412	+	0.962463i	$0.412509\pi$
$860$	0	0
$861$	0	0
$862$	1.88543	0.0642180
$863$	22.4714	0.764935	0.382467	−	0.923969i	$-0.375074\pi$
$863$	22.4714	0.764935	0.382467	+	0.923969i	$0.375074\pi$
$864$	0	0
$865$	0	0
$866$	−1.22729	−0.0417049
$867$	0	0
$868$	−31.9446	−1.08427
$869$	27.9058	0.946639
$870$	0	0
$871$	−25.6067	−0.867649
$872$	3.94492	0.133592
$873$	0	0
$874$	−0.285944	−0.00967219
$875$	0	0
$876$	0	0
$877$	−27.0756	−0.914278	−0.457139	−	0.889395i	$-0.651126\pi$
$877$	−27.0756	−0.914278	−0.457139	+	0.889395i	$0.651126\pi$
$878$	0.389222	0.0131356
$879$	0	0
$880$	0	0
$881$	7.81317	0.263232	0.131616	−	0.991301i	$-0.457983\pi$
$881$	7.81317	0.263232	0.131616	+	0.991301i	$0.457983\pi$
$882$	0	0
$883$	58.3010	1.96199	0.980993	−	0.194042i	$-0.0621598\pi$
$883$	58.3010	1.96199	0.980993	+	0.194042i	$0.0621598\pi$
$884$	−3.93557	−0.132367
$885$	0	0
$886$	0.144377	0.00485045
$887$	−8.64610	−0.290307	−0.145154	−	0.989409i	$-0.546368\pi$
$887$	−8.64610	−0.290307	−0.145154	+	0.989409i	$0.546368\pi$
$888$	0	0
$889$	−54.7571	−1.83650
$890$	0	0
$891$	0	0
$892$	−11.3200	−0.379023
$893$	14.7785	0.494543
$894$	0	0
$895$	0	0
$896$	−12.4075	−0.414507
$897$	0	0
$898$	−1.24212	−0.0414500
$899$	−29.3420	−0.978612
$900$	0	0
$901$	11.0181	0.367067
$902$	2.35944	0.0785608
$903$	0	0
$904$	0.613201	0.0203948
$905$	0	0
$906$	0	0
$907$	40.4367	1.34268	0.671339	−	0.741151i	$-0.265720\pi$
$907$	40.4367	1.34268	0.671339	+	0.741151i	$0.265720\pi$
$908$	−7.25809	−0.240868
$909$	0	0
$910$	0	0
$911$	50.5643	1.67527	0.837635	−	0.546230i	$-0.183937\pi$
$911$	50.5643	1.67527	0.837635	+	0.546230i	$0.183937\pi$
$912$	0	0
$913$	−3.23617	−0.107102
$914$	1.80847	0.0598187
$915$	0	0
$916$	−3.30910	−0.109336
$917$	71.8286	2.37199
$918$	0	0
$919$	−43.3566	−1.43020	−0.715101	−	0.699021i	$-0.753619\pi$
$919$	−43.3566	−1.43020	−0.715101	+	0.699021i	$0.753619\pi$
$920$	0	0
$921$	0	0
$922$	2.83169	0.0932566
$923$	−14.5108	−0.477629
$924$	0	0
$925$	0	0
$926$	−0.00405712	−0.000133325 0
$927$	0	0
$928$	−8.55335	−0.280777
$929$	34.1746	1.12123	0.560617	−	0.828075i	$-0.310564\pi$
$929$	34.1746	1.12123	0.560617	+	0.828075i	$0.310564\pi$
$930$	0	0
$931$	16.2327	0.532006
$932$	−15.6546	−0.512783
$933$	0	0
$934$	2.94921	0.0965010
$935$	0	0
$936$	0	0
$937$	−27.8996	−0.911441	−0.455720	−	0.890123i	$-0.650618\pi$
$937$	−27.8996	−0.911441	−0.455720	+	0.890123i	$0.650618\pi$
$938$	−5.06118	−0.165253
$939$	0	0
$940$	0	0
$941$	54.3261	1.77098	0.885490	−	0.464659i	$-0.153823\pi$
$941$	54.3261	1.77098	0.885490	+	0.464659i	$0.153823\pi$
$942$	0	0
$943$	14.0978	0.459086
$944$	−26.4325	−0.860303
$945$	0	0
$946$	−1.05915	−0.0344361
$947$	4.23171	0.137512	0.0687561	−	0.997633i	$-0.478097\pi$
$947$	4.23171	0.137512	0.0687561	+	0.997633i	$0.478097\pi$
$948$	0	0
$949$	0.841815	0.0273265
$950$	0	0
$951$	0	0
$952$	−1.55889	−0.0505238
$953$	−47.9307	−1.55263	−0.776314	−	0.630346i	$-0.782913\pi$
$953$	−47.9307	−1.55263	−0.776314	+	0.630346i	$0.782913\pi$
$954$	0	0
$955$	0	0
$956$	−1.13002	−0.0365476
$957$	0	0
$958$	0.672367	0.0217232
$959$	42.1563	1.36130
$960$	0	0
$961$	−17.4752	−0.563715
$962$	−0.255783	−0.00824679
$963$	0	0
$964$	37.8629	1.21948
$965$	0	0
$966$	0	0
$967$	20.8827	0.671544	0.335772	−	0.941943i	$-0.391003\pi$
$967$	20.8827	0.671544	0.335772	+	0.941943i	$0.391003\pi$
$968$	2.96895	0.0954257
$969$	0	0
$970$	0	0
$971$	−16.0740	−0.515838	−0.257919	−	0.966167i	$-0.583037\pi$
$971$	−16.0740	−0.515838	−0.257919	+	0.966167i	$0.583037\pi$
$972$	0	0
$973$	−59.0119	−1.89184
$974$	−2.28406	−0.0731860
$975$	0	0
$976$	−37.3657	−1.19605
$977$	−35.5665	−1.13787	−0.568937	−	0.822381i	$-0.692645\pi$
$977$	−35.5665	−1.13787	−0.568937	+	0.822381i	$0.692645\pi$
$978$	0	0
$979$	−42.9757	−1.37351
$980$	0	0
$981$	0	0
$982$	−0.994014	−0.0317202
$983$	18.4711	0.589138	0.294569	−	0.955630i	$-0.404824\pi$
$983$	18.4711	0.589138	0.294569	+	0.955630i	$0.404824\pi$
$984$	0	0
$985$	0	0
$986$	−0.714495	−0.0227542
$987$	0	0
$988$	−5.33266	−0.169655
$989$	−6.32849	−0.201234
$990$	0	0
$991$	−41.5907	−1.32117	−0.660586	−	0.750750i	$-0.729692\pi$
$991$	−41.5907	−1.32117	−0.660586	+	0.750750i	$0.729692\pi$
$992$	3.94256	0.125176
$993$	0	0
$994$	−2.86807	−0.0909698
$995$	0	0
$996$	0	0
$997$	−31.8374	−1.00830	−0.504151	−	0.863616i	$-0.668194\pi$
$997$	−31.8374	−1.00830	−0.504151	+	0.863616i	$0.668194\pi$
$998$	−0.420496	−0.0133106
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.bd.1.4		8
3.2	odd	2		1875.2.a.m.1.5		8
5.4	even	2		5625.2.a.t.1.5		8
15.2	even	4		1875.2.b.h.1249.9		16
15.8	even	4		1875.2.b.h.1249.8		16
15.14	odd	2		1875.2.a.p.1.4		8
25.8	odd	20		225.2.m.b.64.2		16
25.22	odd	20		225.2.m.b.109.2		16
75.8	even	20		75.2.i.a.64.3	yes	16
75.17	even	20		375.2.i.c.199.2		16
75.29	odd	10		375.2.g.d.76.3		16
75.44	odd	10		375.2.g.d.301.3		16
75.47	even	20		75.2.i.a.34.3	✓	16
75.53	even	20		375.2.i.c.49.2		16
75.56	odd	10		375.2.g.e.301.2		16
75.71	odd	10		375.2.g.e.76.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.i.a.34.3	✓	16	75.47	even	20
75.2.i.a.64.3	yes	16	75.8	even	20
225.2.m.b.64.2		16	25.8	odd	20
225.2.m.b.109.2		16	25.22	odd	20
375.2.g.d.76.3		16	75.29	odd	10
375.2.g.d.301.3		16	75.44	odd	10
375.2.g.e.76.2		16	75.71	odd	10
375.2.g.e.301.2		16	75.56	odd	10
375.2.i.c.49.2		16	75.53	even	20
375.2.i.c.199.2		16	75.17	even	20
1875.2.a.m.1.5		8	3.2	odd	2
1875.2.a.p.1.4		8	15.14	odd	2
1875.2.b.h.1249.8		16	15.8	even	4
1875.2.b.h.1249.9		16	15.2	even	4
5625.2.a.t.1.5		8	5.4	even	2
5625.2.a.bd.1.4		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)

\(f(q)\)	\(=\)	\(q-0.0898194 q^{2} -1.99193 q^{4} -4.36070 q^{7} +0.358553 q^{8} +O(q^{10})\)	\(q-0.0898194 q^{2} -1.99193 q^{4} -4.36070 q^{7} +0.358553 q^{8} -4.39094 q^{11} +1.98166 q^{13} +0.391676 q^{14} +3.95166 q^{16} +0.997022 q^{17} +1.35096 q^{19} +0.394392 q^{22} +2.35651 q^{23} -0.177991 q^{26} +8.68622 q^{28} +7.97856 q^{29} -3.67761 q^{31} -1.07204 q^{32} -0.0895519 q^{34} +1.43706 q^{37} -0.121342 q^{38} +5.98248 q^{41} -2.68554 q^{43} +8.74646 q^{44} -0.211660 q^{46} +10.9393 q^{47} +12.0157 q^{49} -3.94732 q^{52} +11.0510 q^{53} -1.56354 q^{56} -0.716629 q^{58} -6.68895 q^{59} -9.45570 q^{61} +0.330321 q^{62} -7.80703 q^{64} -12.9219 q^{67} -1.98600 q^{68} -7.32257 q^{71} +0.424804 q^{73} -0.129076 q^{74} -2.69101 q^{76} +19.1476 q^{77} -6.35531 q^{79} -0.537343 q^{82} +0.737011 q^{83} +0.241213 q^{86} -1.57439 q^{88} +9.78736 q^{89} -8.64141 q^{91} -4.69401 q^{92} -0.982560 q^{94} -0.0337081 q^{97} -1.07925 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} + 4 q^{4} - 8 q^{7} + 12 q^{8}+O(q^{10}) \) 8 * q + 4 * q^2 + 4 * q^4 - 8 * q^7 + 12 * q^8	\( 8 q + 4 q^{2} + 4 q^{4} - 8 q^{7} + 12 q^{8} - 2 q^{11} - 16 q^{13} - 6 q^{14} + 16 q^{17} - 14 q^{19} - 12 q^{22} + 14 q^{23} - 6 q^{26} - 16 q^{28} - 2 q^{29} - 22 q^{31} - 2 q^{32} - 12 q^{34} - 28 q^{37} - 16 q^{38} - 8 q^{41} - 20 q^{43} - 22 q^{44} - 2 q^{46} + 10 q^{47} - 16 q^{52} + 44 q^{53} - 30 q^{56} - 8 q^{58} - 14 q^{59} - 20 q^{61} + 16 q^{62} + 6 q^{64} - 16 q^{67} - 2 q^{68} - 16 q^{71} - 24 q^{73} - 26 q^{74} - 16 q^{76} + 46 q^{77} - 30 q^{79} - 16 q^{82} + 12 q^{83} - 32 q^{86} - 32 q^{88} - 16 q^{89} - 12 q^{91} - 2 q^{92} + 14 q^{94} - 16 q^{97} + 4 q^{98}+O(q^{100}) \) 8 * q + 4 * q^2 + 4 * q^4 - 8 * q^7 + 12 * q^8 - 2 * q^11 - 16 * q^13 - 6 * q^14 + 16 * q^17 - 14 * q^19 - 12 * q^22 + 14 * q^23 - 6 * q^26 - 16 * q^28 - 2 * q^29 - 22 * q^31 - 2 * q^32 - 12 * q^34 - 28 * q^37 - 16 * q^38 - 8 * q^41 - 20 * q^43 - 22 * q^44 - 2 * q^46 + 10 * q^47 - 16 * q^52 + 44 * q^53 - 30 * q^56 - 8 * q^58 - 14 * q^59 - 20 * q^61 + 16 * q^62 + 6 * q^64 - 16 * q^67 - 2 * q^68 - 16 * q^71 - 24 * q^73 - 26 * q^74 - 16 * q^76 + 46 * q^77 - 30 * q^79 - 16 * q^82 + 12 * q^83 - 32 * q^86 - 32 * q^88 - 16 * q^89 - 12 * q^91 - 2 * q^92 + 14 * q^94 - 16 * q^97 + 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more