LMFDB - Embedded newform 1875.2.a.m.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1875,2,Mod(1,1875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1875, 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("1875.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1875 = 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1875.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:	$14.9719503790$
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			1.5
Root			$1.08982$ of defining polynomial
Character	$\chi$	$=$	1875.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0898194 q^{2} +1.00000 q^{3} -1.99193 q^{4} +0.0898194 q^{6} -4.36070 q^{7} -0.358553 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.0898194 q^{2} +1.00000 q^{3} -1.99193 q^{4} +0.0898194 q^{6} -4.36070 q^{7} -0.358553 q^{8} +1.00000 q^{9} +4.39094 q^{11} -1.99193 q^{12} +1.98166 q^{13} -0.391676 q^{14} +3.95166 q^{16} -0.997022 q^{17} +0.0898194 q^{18} +1.35096 q^{19} -4.36070 q^{21} +0.394392 q^{22} -2.35651 q^{23} -0.358553 q^{24} +0.177991 q^{26} +1.00000 q^{27} +8.68622 q^{28} -7.97856 q^{29} -3.67761 q^{31} +1.07204 q^{32} +4.39094 q^{33} -0.0895519 q^{34} -1.99193 q^{36} +1.43706 q^{37} +0.121342 q^{38} +1.98166 q^{39} -5.98248 q^{41} -0.391676 q^{42} -2.68554 q^{43} -8.74646 q^{44} -0.211660 q^{46} -10.9393 q^{47} +3.95166 q^{48} +12.0157 q^{49} -0.997022 q^{51} -3.94732 q^{52} -11.0510 q^{53} +0.0898194 q^{54} +1.56354 q^{56} +1.35096 q^{57} -0.716629 q^{58} +6.68895 q^{59} -9.45570 q^{61} -0.330321 q^{62} -4.36070 q^{63} -7.80703 q^{64} +0.394392 q^{66} -12.9219 q^{67} +1.98600 q^{68} -2.35651 q^{69} +7.32257 q^{71} -0.358553 q^{72} +0.424804 q^{73} +0.129076 q^{74} -2.69101 q^{76} -19.1476 q^{77} +0.177991 q^{78} -6.35531 q^{79} +1.00000 q^{81} -0.537343 q^{82} -0.737011 q^{83} +8.68622 q^{84} -0.241213 q^{86} -7.97856 q^{87} -1.57439 q^{88} -9.78736 q^{89} -8.64141 q^{91} +4.69401 q^{92} -3.67761 q^{93} -0.982560 q^{94} +1.07204 q^{96} -0.0337081 q^{97} +1.07925 q^{98} +4.39094 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{2} + 8 q^{3} + 4 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q - 4 * q^2 + 8 * q^3 + 4 * q^4 - 4 * q^6 - 8 * q^7 - 12 * q^8 + 8 * q^9	$ 8 q - 4 q^{2} + 8 q^{3} + 4 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 8 q^{9} + 2 q^{11} + 4 q^{12} - 16 q^{13} + 6 q^{14} - 16 q^{17} - 4 q^{18} - 14 q^{19} - 8 q^{21} - 12 q^{22} - 14 q^{23} - 12 q^{24} + 6 q^{26} + 8 q^{27} - 16 q^{28} + 2 q^{29} - 22 q^{31} + 2 q^{32} + 2 q^{33} - 12 q^{34} + 4 q^{36} - 28 q^{37} + 16 q^{38} - 16 q^{39} + 8 q^{41} + 6 q^{42} - 20 q^{43} + 22 q^{44} - 2 q^{46} - 10 q^{47} - 16 q^{51} - 16 q^{52} - 44 q^{53} - 4 q^{54} + 30 q^{56} - 14 q^{57} - 8 q^{58} + 14 q^{59} - 20 q^{61} - 16 q^{62} - 8 q^{63} + 6 q^{64} - 12 q^{66} - 16 q^{67} + 2 q^{68} - 14 q^{69} + 16 q^{71} - 12 q^{72} - 24 q^{73} + 26 q^{74} - 16 q^{76} - 46 q^{77} + 6 q^{78} - 30 q^{79} + 8 q^{81} - 16 q^{82} - 12 q^{83} - 16 q^{84} + 32 q^{86} + 2 q^{87} - 32 q^{88} + 16 q^{89} - 12 q^{91} + 2 q^{92} - 22 q^{93} + 14 q^{94} + 2 q^{96} - 16 q^{97} - 4 q^{98} + 2 q^{99}+O(q^{100}) $ 8 * q - 4 * q^2 + 8 * q^3 + 4 * q^4 - 4 * q^6 - 8 * q^7 - 12 * q^8 + 8 * q^9 + 2 * q^11 + 4 * q^12 - 16 * q^13 + 6 * q^14 - 16 * q^17 - 4 * q^18 - 14 * q^19 - 8 * q^21 - 12 * q^22 - 14 * q^23 - 12 * q^24 + 6 * q^26 + 8 * q^27 - 16 * q^28 + 2 * q^29 - 22 * q^31 + 2 * q^32 + 2 * q^33 - 12 * q^34 + 4 * q^36 - 28 * q^37 + 16 * q^38 - 16 * q^39 + 8 * q^41 + 6 * q^42 - 20 * q^43 + 22 * q^44 - 2 * q^46 - 10 * q^47 - 16 * q^51 - 16 * q^52 - 44 * q^53 - 4 * q^54 + 30 * q^56 - 14 * q^57 - 8 * q^58 + 14 * q^59 - 20 * q^61 - 16 * q^62 - 8 * q^63 + 6 * q^64 - 12 * q^66 - 16 * q^67 + 2 * q^68 - 14 * q^69 + 16 * q^71 - 12 * q^72 - 24 * q^73 + 26 * q^74 - 16 * q^76 - 46 * q^77 + 6 * q^78 - 30 * q^79 + 8 * q^81 - 16 * q^82 - 12 * q^83 - 16 * q^84 + 32 * q^86 + 2 * q^87 - 32 * q^88 + 16 * q^89 - 12 * q^91 + 2 * q^92 - 22 * q^93 + 14 * q^94 + 2 * q^96 - 16 * q^97 - 4 * q^98 + 2 * q^99

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	$-0.489890\pi$
$2$	0.0898194	0.0635119	0.0317560	+	0.999496i	$0.489890\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.99193	−0.995966
$5$	0	0
$5$	0	0
$6$	0.0898194	0.0366686
$7$	−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
$9$	1.00000	0.333333
$10$	0	0
$11$	4.39094	1.32392	0.661959	−	0.749540i	$-0.269725\pi$
$11$	4.39094	1.32392	0.661959	+	0.749540i	$0.269725\pi$
$12$	−1.99193	−0.575021
$13$	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	$-0.538580\pi$
$17$	−0.997022	−0.241813	−0.120907	+	0.992664i	$0.538580\pi$
$18$	0.0898194	0.0211706
$19$	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
$21$	−4.36070	−0.951583
$22$	0.394392	0.0840846
$23$	−2.35651	−0.491366	−0.245683	−	0.969350i	$-0.579012\pi$
$23$	−2.35651	−0.491366	−0.245683	+	0.969350i	$0.579012\pi$
$24$	−0.358553	−0.0731893
$25$	0	0
$26$	0.177991	0.0349069
$27$	1.00000	0.192450
$28$	8.68622	1.64154
$29$	−7.97856	−1.48158	−0.740790	−	0.671736i	$-0.765549\pi$
$29$	−7.97856	−1.48158	−0.740790	+	0.671736i	$0.765549\pi$
$30$	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
$33$	4.39094	0.764365
$34$	−0.0895519	−0.0153580
$35$	0	0
$36$	−1.99193	−0.331989
$37$	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
$39$	1.98166	0.317319
$40$	0	0
$41$	−5.98248	−0.934306	−0.467153	−	0.884177i	$-0.654720\pi$
$41$	−5.98248	−0.934306	−0.467153	+	0.884177i	$0.654720\pi$
$42$	−0.391676	−0.0604369
$43$	−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	$-0.794018\pi$
$47$	−10.9393	−1.59566	−0.797829	+	0.602883i	$0.794018\pi$
$48$	3.95166	0.570373
$49$	12.0157	1.71653
$50$	0	0
$51$	−0.997022	−0.139611
$52$	−3.94732	−0.547395
$53$	−11.0510	−1.51798	−0.758989	−	0.651104i	$-0.774306\pi$
$53$	−11.0510	−1.51798	−0.758989	+	0.651104i	$0.774306\pi$
$54$	0.0898194	0.0122229
$55$	0	0
$56$	1.56354	0.208937
$57$	1.35096	0.178938
$58$	−0.716629	−0.0940980
$59$	6.68895	0.870827	0.435414	−	0.900231i	$-0.356602\pi$
$59$	6.68895	0.870827	0.435414	+	0.900231i	$0.356602\pi$
$60$	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
$63$	−4.36070	−0.549397
$64$	−7.80703	−0.975879
$65$	0	0
$66$	0.394392	0.0485463
$67$	−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
$69$	−2.35651	−0.283690
$70$	0	0
$71$	7.32257	0.869029	0.434515	−	0.900665i	$-0.356920\pi$
$71$	7.32257	0.869029	0.434515	+	0.900665i	$0.356920\pi$
$72$	−0.358553	−0.0422559
$73$	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
$78$	0.177991	0.0201535
$79$	−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
$81$	1.00000	0.111111
$82$	−0.537343	−0.0593396
$83$	−0.737011	−0.0808975	−0.0404487	−	0.999182i	$-0.512879\pi$
$83$	−0.737011	−0.0808975	−0.0404487	+	0.999182i	$0.512879\pi$
$84$	8.68622	0.947745
$85$	0	0
$86$	−0.241213	−0.0260107
$87$	−7.97856	−0.855391
$88$	−1.57439	−0.167830
$89$	−9.78736	−1.03746	−0.518729	−	0.854939i	$-0.673595\pi$
$89$	−9.78736	−1.03746	−0.518729	+	0.854939i	$0.673595\pi$
$90$	0	0
$91$	−8.64141	−0.905866
$92$	4.69401	0.489384
$93$	−3.67761	−0.381351
$94$	−0.982560	−0.101343
$95$	0	0
$96$	1.07204	0.109415
$97$	−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
$99$	4.39094	0.441306
$100$	0	0
$101$	3.19390	0.317805	0.158902	−	0.987294i	$-0.449204\pi$
$101$	3.19390	0.317805	0.158902	+	0.987294i	$0.449204\pi$
$102$	−0.0895519	−0.00886696
$103$	−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	$-0.534244\pi$
$107$	−2.22136	−0.214747	−0.107373	+	0.994219i	$0.534244\pi$
$108$	−1.99193	−0.191674
$109$	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
$111$	1.43706	0.136399
$112$	−17.2320	−1.62827
$113$	−1.71021	−0.160883	−0.0804415	−	0.996759i	$-0.525633\pi$
$113$	−1.71021	−0.160883	−0.0804415	+	0.996759i	$0.525633\pi$
$114$	0.121342	0.0113647
$115$	0	0
$116$	15.8927	1.47560
$117$	1.98166	0.183204
$118$	0.600798	0.0553079
$119$	4.34771	0.398554
$120$	0	0
$121$	8.28037	0.752761
$122$	−0.849306	−0.0768925
$123$	−5.98248	−0.539422
$124$	7.32556	0.657855
$125$	0	0
$126$	−0.391676	−0.0348933
$127$	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
$129$	−2.68554	−0.236448
$130$	0	0
$131$	16.4718	1.43915	0.719574	−	0.694416i	$-0.244337\pi$
$131$	16.4718	1.43915	0.719574	+	0.694416i	$0.244337\pi$
$132$	−8.74646	−0.761282
$133$	−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	$-0.364492\pi$
$137$	9.66732	0.825935	0.412967	+	0.910746i	$0.364492\pi$
$138$	−0.211660	−0.0180177
$139$	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
$141$	−10.9393	−0.921254
$142$	0.657709	0.0551937
$143$	8.70133	0.727642
$144$	3.95166	0.329305
$145$	0	0
$146$	0.0381557	0.00315778
$147$	12.0157	0.991040
$148$	−2.86252	−0.235298
$149$	13.6843	1.12106	0.560529	−	0.828134i	$-0.310598\pi$
$149$	13.6843	1.12106	0.560529	+	0.828134i	$0.310598\pi$
$150$	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
$153$	−0.997022	−0.0806044
$154$	−1.71983	−0.138587
$155$	0	0
$156$	−3.94732	−0.316039
$157$	−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
$159$	−11.0510	−0.876405
$160$	0	0
$161$	10.2760	0.809865
$162$	0.0898194	0.00705688
$163$	−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	$-0.590092\pi$
$167$	−7.21792	−0.558540	−0.279270	+	0.960213i	$0.590092\pi$
$168$	1.56354	0.120630
$169$	−9.07304	−0.697926
$170$	0	0
$171$	1.35096	0.103310
$172$	5.34941	0.407889
$173$	4.65009	0.353540	0.176770	−	0.984252i	$-0.443435\pi$
$173$	4.65009	0.353540	0.176770	+	0.984252i	$0.443435\pi$
$174$	−0.716629	−0.0543275
$175$	0	0
$176$	17.3515	1.30792
$177$	6.68895	0.502772
$178$	−0.879095	−0.0658909
$179$	−7.20338	−0.538406	−0.269203	−	0.963083i	$-0.586760\pi$
$179$	−7.20338	−0.538406	−0.269203	+	0.963083i	$0.586760\pi$
$180$	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
$183$	−9.45570	−0.698986
$184$	0.844934	0.0622893
$185$	0	0
$186$	−0.330321	−0.0242203
$187$	−4.37786	−0.320141
$188$	21.7903	1.58922
$189$	−4.36070	−0.317194
$190$	0	0
$191$	21.5541	1.55960	0.779801	−	0.626027i	$-0.215320\pi$
$191$	21.5541	1.55960	0.779801	+	0.626027i	$0.215320\pi$
$192$	−7.80703	−0.563424
$193$	−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	$-0.879498\pi$
$197$	−26.0837	−1.85839	−0.929195	+	0.369590i	$0.879498\pi$
$198$	0.394392	0.0280282
$199$	−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
$201$	−12.9219	−0.911437
$202$	0.286874	0.0201844
$203$	34.7921	2.44193
$204$	1.98600	0.139048
$205$	0	0
$206$	−0.768264	−0.0535275
$207$	−2.35651	−0.163789
$208$	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
$213$	7.32257	0.501734
$214$	−0.199521	−0.0136390
$215$	0	0
$216$	−0.358553	−0.0243964
$217$	16.0370	1.08866
$218$	0.988224	0.0669310
$219$	0.424804	0.0287056
$220$	0	0
$221$	−1.97575	−0.132904
$222$	0.129076	0.00866299
$223$	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	$-0.538585\pi$
$227$	−3.64374	−0.241844	−0.120922	+	0.992662i	$0.538585\pi$
$228$	−2.69101	−0.178217
$229$	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
$231$	−19.1476	−1.25982
$232$	2.86074	0.187817
$233$	−7.85899	−0.514860	−0.257430	−	0.966297i	$-0.582876\pi$
$233$	−7.85899	−0.514860	−0.257430	+	0.966297i	$0.582876\pi$
$234$	0.177991	0.0116356
$235$	0	0
$236$	−13.3239	−0.867314
$237$	−6.35531	−0.412821
$238$	0.390509	0.0253130
$239$	−0.567301	−0.0366956	−0.0183478	−	0.999832i	$-0.505841\pi$
$239$	−0.567301	−0.0366956	−0.0183478	+	0.999832i	$0.505841\pi$
$240$	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
$243$	1.00000	0.0641500
$244$	18.8351	1.20580
$245$	0	0
$246$	−0.537343	−0.0342597
$247$	2.67713	0.170342
$248$	1.31862	0.0837324
$249$	−0.737011	−0.0467062
$250$	0	0
$251$	3.02533	0.190957	0.0954787	−	0.995431i	$-0.469562\pi$
$251$	3.02533	0.190957	0.0954787	+	0.995431i	$0.469562\pi$
$252$	8.68622	0.547181
$253$	−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	$-0.287353\pi$
$257$	19.8613	1.23891	0.619456	+	0.785032i	$0.287353\pi$
$258$	−0.241213	−0.0150173
$259$	−6.26658	−0.389386
$260$	0	0
$261$	−7.97856	−0.493860
$262$	1.47949	0.0914030
$263$	−22.8299	−1.40775	−0.703876	−	0.710323i	$-0.748549\pi$
$263$	−22.8299	−1.40775	−0.703876	+	0.710323i	$0.748549\pi$
$264$	−1.57439	−0.0968968
$265$	0	0
$266$	−0.529137	−0.0324435
$267$	−9.78736	−0.598977
$268$	25.7395	1.57229
$269$	14.8324	0.904347	0.452174	−	0.891930i	$-0.350649\pi$
$269$	14.8324	0.904347	0.452174	+	0.891930i	$0.350649\pi$
$270$	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
$273$	−8.64141	−0.523002
$274$	0.868313	0.0524567
$275$	0	0
$276$	4.69401	0.282546
$277$	−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
$279$	−3.67761	−0.220173
$280$	0	0
$281$	−20.4867	−1.22214	−0.611068	−	0.791578i	$-0.709260\pi$
$281$	−20.4867	−1.22214	−0.611068	+	0.791578i	$0.709260\pi$
$282$	−0.982560	−0.0585106
$283$	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
$288$	1.07204	0.0631707
$289$	−16.0059	−0.941526
$290$	0	0
$291$	−0.0337081	−0.00197600
$292$	−0.846181	−0.0495190
$293$	−28.5505	−1.66794	−0.833968	−	0.551812i	$-0.813937\pi$
$293$	−28.5505	−1.66794	−0.833968	+	0.551812i	$0.813937\pi$
$294$	1.07925	0.0629429
$295$	0	0
$296$	−0.515261	−0.0299489
$297$	4.39094	0.254788
$298$	1.22911	0.0712006
$299$	−4.66979	−0.270061
$300$	0	0
$301$	11.7108	0.675001
$302$	−1.01679	−0.0585095
$303$	3.19390	0.183485
$304$	5.33852	0.306185
$305$	0	0
$306$	−0.0895519	−0.00511934
$307$	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
$309$	−8.55342	−0.486587
$310$	0	0
$311$	17.5496	0.995146	0.497573	−	0.867422i	$-0.334225\pi$
$311$	17.5496	0.995146	0.497573	+	0.867422i	$0.334225\pi$
$312$	−0.710529	−0.0402258
$313$	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	$-0.349953\pi$
$317$	16.1708	0.908244	0.454122	+	0.890940i	$0.349953\pi$
$318$	−0.992598	−0.0556621
$319$	−35.0334	−1.96149
$320$	0	0
$321$	−2.22136	−0.123984
$322$	0.922988	0.0514361
$323$	−1.34693	−0.0749453
$324$	−1.99193	−0.110663
$325$	0	0
$326$	−0.411830	−0.0228092
$327$	11.0023	0.608431
$328$	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
$333$	1.43706	0.0787502
$334$	−0.648310	−0.0354739
$335$	0	0
$336$	−17.2320	−0.940083
$337$	−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
$339$	−1.71021	−0.0928858
$340$	0	0
$341$	−16.1482	−0.874473
$342$	0.121342	0.00656143
$343$	−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	$-0.260205\pi$
$347$	25.4859	1.36815	0.684077	+	0.729410i	$0.260205\pi$
$348$	15.8927	0.851941
$349$	−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
$351$	1.98166	0.105773
$352$	4.70727	0.250899
$353$	−14.6667	−0.780630	−0.390315	−	0.920681i	$-0.627634\pi$
$353$	−14.6667	−0.780630	−0.390315	+	0.920681i	$0.627634\pi$
$354$	0.600798	0.0319320
$355$	0	0
$356$	19.4958	1.03327
$357$	4.34771	0.230105
$358$	−0.647004	−0.0341952
$359$	−14.6205	−0.771642	−0.385821	−	0.922574i	$-0.626082\pi$
$359$	−14.6205	−0.771642	−0.385821	+	0.922574i	$0.626082\pi$
$360$	0	0
$361$	−17.1749	−0.903943
$362$	−0.341694	−0.0179591
$363$	8.28037	0.434607
$364$	17.2131	0.902212
$365$	0	0
$366$	−0.849306	−0.0443939
$367$	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
$369$	−5.98248	−0.311435
$370$	0	0
$371$	48.1903	2.50192
$372$	7.32556	0.379813
$373$	−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
$378$	−0.391676	−0.0201456
$379$	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
$381$	12.5570	0.643312
$382$	1.93598	0.0990533
$383$	−34.4334	−1.75947	−0.879733	−	0.475468i	$-0.842279\pi$
$383$	−34.4334	−1.75947	−0.879733	+	0.475468i	$0.842279\pi$
$384$	−2.84531	−0.145199
$385$	0	0
$386$	−0.282957	−0.0144021
$387$	−2.68554	−0.136514
$388$	0.0671442	0.00340873
$389$	13.5225	0.685619	0.342809	−	0.939405i	$-0.388622\pi$
$389$	13.5225	0.685619	0.342809	+	0.939405i	$0.388622\pi$
$390$	0	0
$391$	2.34949	0.118819
$392$	−4.30827	−0.217601
$393$	16.4718	0.830892
$394$	−2.34283	−0.118030
$395$	0	0
$396$	−8.74646	−0.439526
$397$	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
$399$	−5.89112	−0.294925
$400$	0	0
$401$	4.35977	0.217717	0.108858	−	0.994057i	$-0.465281\pi$
$401$	4.35977	0.217717	0.108858	+	0.994057i	$0.465281\pi$
$402$	−1.16063	−0.0578871
$403$	−7.28776	−0.363029
$404$	−6.36203	−0.316523
$405$	0	0
$406$	3.12501	0.155092
$407$	6.31003	0.312777
$408$	0.357485	0.0176982
$409$	−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
$411$	9.66732	0.476854
$412$	17.0378	0.839394
$413$	−29.1685	−1.43529
$414$	−0.211660	−0.0104025
$415$	0	0
$416$	2.12442	0.104158
$417$	13.5327	0.662698
$418$	0.532806	0.0260604
$419$	0.527867	0.0257880	0.0128940	−	0.999917i	$-0.495896\pi$
$419$	0.527867	0.0257880	0.0128940	+	0.999917i	$0.495896\pi$
$420$	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
$423$	−10.9393	−0.531886
$424$	3.96239	0.192430
$425$	0	0
$426$	0.657709	0.0318661
$427$	41.2335	1.99543
$428$	4.42480	0.213881
$429$	8.70133	0.420104
$430$	0	0
$431$	20.9913	1.01112	0.505559	−	0.862792i	$-0.331286\pi$
$431$	20.9913	1.01112	0.505559	+	0.862792i	$0.331286\pi$
$432$	3.95166	0.190124
$433$	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
$438$	0.0381557	0.00182315
$439$	−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
$441$	12.0157	0.572177
$442$	−0.177461	−0.00844096
$443$	1.60742	0.0763707	0.0381854	−	0.999271i	$-0.487842\pi$
$443$	1.60742	0.0763707	0.0381854	+	0.999271i	$0.487842\pi$
$444$	−2.86252	−0.135849
$445$	0	0
$446$	0.510439	0.0241700
$447$	13.6843	0.647244
$448$	34.0441	1.60843
$449$	−13.8291	−0.652634	−0.326317	−	0.945260i	$-0.605808\pi$
$449$	−13.8291	−0.652634	−0.326317	+	0.945260i	$0.605808\pi$
$450$	0	0
$451$	−26.2687	−1.23694
$452$	3.40662	0.160234
$453$	−11.3204	−0.531876
$454$	−0.327279	−0.0153600
$455$	0	0
$456$	−0.484389	−0.0226836
$457$	−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
$459$	−0.997022	−0.0465370
$460$	0	0
$461$	31.5264	1.46833	0.734166	−	0.678970i	$-0.237573\pi$
$461$	31.5264	1.46833	0.734166	+	0.678970i	$0.237573\pi$
$462$	−1.71983	−0.0800135
$463$	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	$-0.225342\pi$
$467$	32.8349	1.51942	0.759708	+	0.650265i	$0.225342\pi$
$468$	−3.94732	−0.182465
$469$	56.3484	2.60193
$470$	0	0
$471$	−8.56070	−0.394456
$472$	−2.39834	−0.110393
$473$	−11.7920	−0.542198
$474$	−0.570830	−0.0262191
$475$	0	0
$476$	−8.66035	−0.396947
$477$	−11.0510	−0.505992
$478$	−0.0509546	−0.00233061
$479$	7.48576	0.342033	0.171017	−	0.985268i	$-0.445295\pi$
$479$	7.48576	0.342033	0.171017	+	0.985268i	$0.445295\pi$
$480$	0	0
$481$	2.84775	0.129846
$482$	−1.70730	−0.0777654
$483$	10.2760	0.467576
$484$	−16.4939	−0.749724
$485$	0	0
$486$	0.0898194	0.00407429
$487$	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
$489$	−4.58509	−0.207345
$490$	0	0
$491$	−11.0668	−0.499438	−0.249719	−	0.968318i	$-0.580338\pi$
$491$	−11.0668	−0.499438	−0.249719	+	0.968318i	$0.580338\pi$
$492$	11.9167	0.537246
$493$	7.95479	0.358266
$494$	0.240458	0.0108187
$495$	0	0
$496$	−14.5327	−0.652537
$497$	−31.9315	−1.43233
$498$	−0.0661979	−0.00296640
$499$	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
$501$	−7.21792	−0.322473
$502$	0.271734	0.0121281
$503$	−10.8285	−0.482818	−0.241409	−	0.970423i	$-0.577609\pi$
$503$	−10.8285	−0.482818	−0.241409	+	0.970423i	$0.577609\pi$
$504$	1.56354	0.0696458
$505$	0	0
$506$	−0.929388	−0.0413163
$507$	−9.07304	−0.402948
$508$	−25.0126	−1.10975
$509$	11.5007	0.509757	0.254879	−	0.966973i	$-0.417964\pi$
$509$	11.5007	0.509757	0.254879	+	0.966973i	$0.417964\pi$
$510$	0	0
$511$	−1.85244	−0.0819473
$512$	7.07011	0.312457
$513$	1.35096	0.0596462
$514$	1.78393	0.0786856
$515$	0	0
$516$	5.34941	0.235495
$517$	−48.0338	−2.11252
$518$	−0.562860	−0.0247307
$519$	4.65009	0.204116
$520$	0	0
$521$	0.797807	0.0349525	0.0174763	−	0.999847i	$-0.494437\pi$
$521$	0.797807	0.0349525	0.0174763	+	0.999847i	$0.494437\pi$
$522$	−0.716629	−0.0313660
$523$	−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
$528$	17.3515	0.755127
$529$	−17.4469	−0.758559
$530$	0	0
$531$	6.68895	0.290276
$532$	11.7347	0.508764
$533$	−11.8552	−0.513506
$534$	−0.879095	−0.0380422
$535$	0	0
$536$	4.63317	0.200123
$537$	−7.20338	−0.310849
$538$	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
$543$	−3.80424	−0.163255
$544$	−1.06885	−0.0458265
$545$	0	0
$546$	−0.776167	−0.0332169
$547$	−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
$549$	−9.45570	−0.403560
$550$	0	0
$551$	−10.7787	−0.459187
$552$	0.844934	0.0359628
$553$	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	$-0.375138\pi$
$557$	18.0445	0.764568	0.382284	+	0.924045i	$0.375138\pi$
$558$	−0.330321	−0.0139836
$559$	−5.32181	−0.225089
$560$	0	0
$561$	−4.37786	−0.184834
$562$	−1.84011	−0.0776202
$563$	28.5327	1.20251	0.601254	−	0.799058i	$-0.294668\pi$
$563$	28.5327	1.20251	0.601254	+	0.799058i	$0.294668\pi$
$564$	21.7903	0.917538
$565$	0	0
$566$	1.02554	0.0431065
$567$	−4.36070	−0.183132
$568$	−2.62553	−0.110165
$569$	16.4072	0.687826	0.343913	−	0.939001i	$-0.388247\pi$
$569$	16.4072	0.687826	0.343913	+	0.939001i	$0.388247\pi$
$570$	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
$573$	21.5541	0.900437
$574$	2.34319	0.0978029
$575$	0	0
$576$	−7.80703	−0.325293
$577$	−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
$579$	−3.15029	−0.130921
$580$	0	0
$581$	3.21389	0.133334
$582$	−0.00302764	−0.000125500 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	$-0.332900\pi$
$587$	24.2852	1.00236	0.501179	+	0.865344i	$0.332900\pi$
$588$	−23.9345	−0.987042
$589$	−4.96829	−0.204715
$590$	0	0
$591$	−26.0837	−1.07294
$592$	5.67876	0.233396
$593$	28.4653	1.16893	0.584466	−	0.811418i	$-0.301304\pi$
$593$	28.4653	1.16893	0.584466	+	0.811418i	$0.301304\pi$
$594$	0.394392	0.0161821
$595$	0	0
$596$	−27.2581	−1.11654
$597$	−24.2662	−0.993149
$598$	−0.419438	−0.0171521
$599$	16.0387	0.655323	0.327662	−	0.944795i	$-0.393739\pi$
$599$	16.0387	0.655323	0.327662	+	0.944795i	$0.393739\pi$
$600$	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
$603$	−12.9219	−0.526219
$604$	22.5494	0.917521
$605$	0	0
$606$	0.286874	0.0116535
$607$	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
$609$	34.7921	1.40985
$610$	0	0
$611$	−21.6779	−0.876994
$612$	1.98600	0.0802793
$613$	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	$-0.600424\pi$
$617$	−15.4146	−0.620568	−0.310284	+	0.950644i	$0.600424\pi$
$618$	−0.768264	−0.0309041
$619$	−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
$621$	−2.35651	−0.0945635
$622$	1.57629	0.0632036
$623$	42.6797	1.70993
$624$	7.83083	0.313484
$625$	0	0
$626$	0.268168	0.0107182
$627$	5.93197	0.236900
$628$	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
$633$	16.3783	0.650980
$634$	1.45245	0.0576843
$635$	0	0
$636$	22.0129	0.872869
$637$	23.8110	0.943427
$638$	−3.14668	−0.124578
$639$	7.32257	0.289676
$640$	0	0
$641$	12.4281	0.490882	0.245441	−	0.969412i	$-0.421067\pi$
$641$	12.4281	0.490882	0.245441	+	0.969412i	$0.421067\pi$
$642$	−0.199521	−0.00787447
$643$	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	$-0.222169\pi$
$647$	38.9760	1.53230	0.766152	+	0.642660i	$0.222169\pi$
$648$	−0.358553	−0.0140853
$649$	29.3708	1.15290
$650$	0	0
$651$	16.0370	0.628539
$652$	9.13319	0.357683
$653$	−28.3389	−1.10899	−0.554493	−	0.832189i	$-0.687088\pi$
$653$	−28.3389	−1.10899	−0.554493	+	0.832189i	$0.687088\pi$
$654$	0.988224	0.0386426
$655$	0	0
$656$	−23.6407	−0.923015
$657$	0.424804	0.0165732
$658$	4.28465	0.167033
$659$	−22.3561	−0.870868	−0.435434	−	0.900221i	$-0.643405\pi$
$659$	−22.3561	−0.870868	−0.435434	+	0.900221i	$0.643405\pi$
$660$	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
$663$	−1.97575	−0.0767319
$664$	0.264258	0.0102552
$665$	0	0
$666$	0.129076	0.00500158
$667$	18.8015	0.727998
$668$	14.3776	0.556287
$669$	5.68295	0.219715
$670$	0	0
$671$	−41.5194	−1.60284
$672$	−4.67486	−0.180336
$673$	−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	$-0.137666\pi$
$677$	47.2470	1.81585	0.907925	+	0.419133i	$0.137666\pi$
$678$	−0.153610	−0.00589936
$679$	0.146991	0.00564099
$680$	0	0
$681$	−3.64374	−0.139629
$682$	−1.45042	−0.0555395
$683$	37.2527	1.42543	0.712717	−	0.701452i	$-0.247464\pi$
$683$	37.2527	1.42543	0.712717	+	0.701452i	$0.247464\pi$
$684$	−2.69101	−0.102893
$685$	0	0
$686$	−1.96454	−0.0750064
$687$	1.66125	0.0633807
$688$	−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
$693$	−19.1476	−0.727357
$694$	2.28913	0.0868941
$695$	0	0
$696$	2.86074	0.108436
$697$	5.96466	0.225928
$698$	−2.51885	−0.0953399
$699$	−7.85899	−0.297254
$700$	0	0
$701$	22.4086	0.846361	0.423180	−	0.906046i	$-0.360914\pi$
$701$	22.4086	0.846361	0.423180	+	0.906046i	$0.360914\pi$
$702$	0.177991	0.00671784
$703$	1.94140	0.0732213
$704$	−34.2802	−1.29198
$705$	0	0
$706$	−1.31736	−0.0495793
$707$	−13.9276	−0.523803
$708$	−13.3239	−0.500744
$709$	−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
$711$	−6.35531	−0.238343
$712$	3.50929	0.131516
$713$	8.66633	0.324557
$714$	0.390509	0.0146144
$715$	0	0
$716$	14.3486	0.536234
$717$	−0.567301	−0.0211862
$718$	−1.31321	−0.0490085
$719$	−28.4977	−1.06278	−0.531392	−	0.847126i	$-0.678331\pi$
$719$	−28.4977	−1.06278	−0.531392	+	0.847126i	$0.678331\pi$
$720$	0	0
$721$	37.2989	1.38908
$722$	−1.54264	−0.0574112
$723$	−19.0081	−0.706920
$724$	7.57778	0.281626
$725$	0	0
$726$	0.743738	0.0276027
$727$	−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
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.67754	0.0990324
$732$	18.8351	0.696166
$733$	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
$738$	−0.537343	−0.0197799
$739$	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
$741$	2.67713	0.0983468
$742$	4.32843	0.158901
$743$	35.6012	1.30608	0.653041	−	0.757322i	$-0.273493\pi$
$743$	35.6012	1.30608	0.653041	+	0.757322i	$0.273493\pi$
$744$	1.31862	0.0483430
$745$	0	0
$746$	−1.09843	−0.0402164
$747$	−0.737011	−0.0269658
$748$	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
$753$	3.02533	0.110249
$754$	−1.42011	−0.0517174
$755$	0	0
$756$	8.68622	0.315915
$757$	−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
$759$	−10.3473	−0.375583
$760$	0	0
$761$	−27.2078	−0.986282	−0.493141	−	0.869949i	$-0.664151\pi$
$761$	−27.2078	−0.986282	−0.493141	+	0.869949i	$0.664151\pi$
$762$	1.12786	0.0408580
$763$	−47.9779	−1.73692
$764$	−42.9344	−1.55331
$765$	0	0
$766$	−3.09279	−0.111747
$767$	13.2552	0.478617
$768$	15.3585	0.554202
$769$	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
$771$	19.8613	0.715286
$772$	6.27516	0.225848
$773$	28.7677	1.03470	0.517351	−	0.855773i	$-0.326918\pi$
$773$	28.7677	1.03470	0.517351	+	0.855773i	$0.326918\pi$
$774$	−0.241213	−0.00867024
$775$	0	0
$776$	0.0120861	0.000433867 0
$777$	−6.26658	−0.224812
$778$	1.21458	0.0435450
$779$	−8.08206	−0.289570
$780$	0	0
$781$	32.1530	1.15052
$782$	0.211030	0.00754641
$783$	−7.97856	−0.285130
$784$	47.4820	1.69579
$785$	0	0
$786$	1.47949	0.0527716
$787$	−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
$789$	−22.8299	−0.812766
$790$	0	0
$791$	7.45771	0.265166
$792$	−1.57439	−0.0559434
$793$	−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	$-0.312149\pi$
$797$	31.4206	1.11298	0.556488	+	0.830856i	$0.312149\pi$
$798$	−0.529137	−0.0187312
$799$	10.9067	0.385851
$800$	0	0
$801$	−9.78736	−0.345819
$802$	0.391592	0.0138276
$803$	1.86529	0.0658246
$804$	25.7395	0.907761
$805$	0	0
$806$	−0.654583	−0.0230567
$807$	14.8324	0.522125
$808$	−1.14518	−0.0402874
$809$	50.5027	1.77558	0.887790	−	0.460249i	$-0.152240\pi$
$809$	50.5027	1.77558	0.887790	+	0.460249i	$0.152240\pi$
$810$	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
$813$	−6.43720	−0.225762
$814$	0.566763	0.0198650
$815$	0	0
$816$	−3.93989	−0.137924
$817$	−3.62804	−0.126929
$818$	−1.62697	−0.0568856
$819$	−8.64141	−0.301955
$820$	0	0
$821$	−20.6307	−0.720017	−0.360009	−	0.932949i	$-0.617226\pi$
$821$	−20.6307	−0.720017	−0.360009	+	0.932949i	$0.617226\pi$
$822$	0.868313	0.0302859
$823$	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	$-0.526243\pi$
$827$	−4.73642	−0.164701	−0.0823507	+	0.996603i	$0.526243\pi$
$828$	4.69401	0.163128
$829$	−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
$831$	−6.70976	−0.232759
$832$	−15.4708	−0.536355
$833$	−11.9799	−0.415080
$834$	1.21550	0.0420892
$835$	0	0
$836$	−11.8161	−0.408668
$837$	−3.67761	−0.127117
$838$	0.0474127	0.00163784
$839$	−0.765715	−0.0264354	−0.0132177	−	0.999913i	$-0.504207\pi$
$839$	−0.765715	−0.0264354	−0.0132177	+	0.999913i	$0.504207\pi$
$840$	0	0
$841$	34.6574	1.19508
$842$	1.67591	0.0577555
$843$	−20.4867	−0.705600
$844$	−32.6245	−1.12298
$845$	0	0
$846$	−0.982560	−0.0337811
$847$	−36.1082	−1.24069
$848$	−43.6700	−1.49963
$849$	11.4177	0.391856
$850$	0	0
$851$	−3.38644	−0.116086
$852$	−14.5861	−0.499710
$853$	−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	$-0.877347\pi$
$857$	−54.2561	−1.85335	−0.926676	+	0.375860i	$0.877347\pi$
$858$	0.781549	0.0266816
$859$	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
$861$	26.0878	0.889070
$862$	1.88543	0.0642180
$863$	−22.4714	−0.764935	−0.382467	−	0.923969i	$-0.624926\pi$
$863$	−22.4714	−0.764935	−0.382467	+	0.923969i	$0.624926\pi$
$864$	1.07204	0.0364716
$865$	0	0
$866$	1.22729	0.0417049
$867$	−16.0059	−0.543590
$868$	−31.9446	−1.08427
$869$	−27.9058	−0.946639
$870$	0	0
$871$	−25.6067	−0.867649
$872$	−3.94492	−0.133592
$873$	−0.0337081	−0.00114085
$874$	−0.285944	−0.00967219
$875$	0	0
$876$	−0.846181	−0.0285898
$877$	−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
$879$	−28.5505	−0.962984
$880$	0	0
$881$	−7.81317	−0.263232	−0.131616	−	0.991301i	$-0.542017\pi$
$881$	−7.81317	−0.263232	−0.131616	+	0.991301i	$0.542017\pi$
$882$	1.07925	0.0363401
$883$	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	$-0.453632\pi$
$887$	8.64610	0.290307	0.145154	+	0.989409i	$0.453632\pi$
$888$	−0.515261	−0.0172910
$889$	−54.7571	−1.83650
$890$	0	0
$891$	4.39094	0.147102
$892$	−11.3200	−0.379023
$893$	−14.7785	−0.494543
$894$	1.22911	0.0411077
$895$	0	0
$896$	12.4075	0.414507
$897$	−4.66979	−0.155920
$898$	−1.24212	−0.0414500
$899$	29.3420	0.978612
$900$	0	0
$901$	11.0181	0.367067
$902$	−2.35944	−0.0785608
$903$	11.7108	0.389712
$904$	0.613201	0.0203948
$905$	0	0
$906$	−1.01679	−0.0337805
$907$	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
$909$	3.19390	0.105935
$910$	0	0
$911$	−50.5643	−1.67527	−0.837635	−	0.546230i	$-0.816063\pi$
$911$	−50.5643	−1.67527	−0.837635	+	0.546230i	$0.816063\pi$
$912$	5.33852	0.176776
$913$	−3.23617	−0.107102
$914$	−1.80847	−0.0598187
$915$	0	0
$916$	−3.30910	−0.109336
$917$	−71.8286	−2.37199
$918$	−0.0895519	−0.00295565
$919$	−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
$921$	20.5417	0.676871
$922$	2.83169	0.0932566
$923$	14.5108	0.477629
$924$	38.1407	1.25474
$925$	0	0
$926$	0.00405712	0.000133325 0
$927$	−8.55342	−0.280931
$928$	−8.55335	−0.280777
$929$	−34.1746	−1.12123	−0.560617	−	0.828075i	$-0.689436\pi$
$929$	−34.1746	−1.12123	−0.560617	+	0.828075i	$0.689436\pi$
$930$	0	0
$931$	16.2327	0.532006
$932$	15.6546	0.512783
$933$	17.5496	0.574548
$934$	2.94921	0.0965010
$935$	0	0
$936$	−0.710529	−0.0232244
$937$	−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
$939$	2.98564	0.0974325
$940$	0	0
$941$	−54.3261	−1.77098	−0.885490	−	0.464659i	$-0.846177\pi$
$941$	−54.3261	−1.77098	−0.885490	+	0.464659i	$0.846177\pi$
$942$	−0.768918	−0.0250527
$943$	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	$-0.521903\pi$
$947$	−4.23171	−0.137512	−0.0687561	+	0.997633i	$0.521903\pi$
$948$	12.6593	0.411156
$949$	0.841815	0.0273265
$950$	0	0
$951$	16.1708	0.524375
$952$	−1.55889	−0.0505238
$953$	47.9307	1.55263	0.776314	−	0.630346i	$-0.217087\pi$
$953$	47.9307	1.55263	0.776314	+	0.630346i	$0.217087\pi$
$954$	−0.992598	−0.0321366
$955$	0	0
$956$	1.13002	0.0365476
$957$	−35.0334	−1.13247
$958$	0.672367	0.0217232
$959$	−42.1563	−1.36130
$960$	0	0
$961$	−17.4752	−0.563715
$962$	0.255783	0.00824679
$963$	−2.22136	−0.0715823
$964$	37.8629	1.21948
$965$	0	0
$966$	0.922988	0.0296966
$967$	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
$969$	−1.34693	−0.0432697
$970$	0	0
$971$	16.0740	0.515838	0.257919	−	0.966167i	$-0.416963\pi$
$971$	16.0740	0.515838	0.257919	+	0.966167i	$0.416963\pi$
$972$	−1.99193	−0.0638913
$973$	−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	$-0.307355\pi$
$977$	35.5665	1.13787	0.568937	+	0.822381i	$0.307355\pi$
$978$	−0.411830	−0.0131689
$979$	−42.9757	−1.37351
$980$	0	0
$981$	11.0023	0.351278
$982$	−0.994014	−0.0317202
$983$	−18.4711	−0.589138	−0.294569	−	0.955630i	$-0.595176\pi$
$983$	−18.4711	−0.589138	−0.294569	+	0.955630i	$0.595176\pi$
$984$	2.14504	0.0683812
$985$	0	0
$986$	0.714495	0.0227542
$987$	47.7030	1.51840
$988$	−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
$993$	−13.0705	−0.414779
$994$	−2.86807	−0.0909698
$995$	0	0
$996$	1.46808	0.0465178
$997$	−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
$999$	1.43706	0.0454665

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.a.m.1.5		8
3.2	odd	2		5625.2.a.bd.1.4		8
5.2	odd	4		1875.2.b.h.1249.9		16
5.3	odd	4		1875.2.b.h.1249.8		16
5.4	even	2		1875.2.a.p.1.4		8
15.14	odd	2		5625.2.a.t.1.5		8
25.3	odd	20		375.2.i.c.49.2		16
25.4	even	10		375.2.g.d.76.3		16
25.6	even	5		375.2.g.e.301.2		16
25.8	odd	20		75.2.i.a.64.3	yes	16
25.17	odd	20		375.2.i.c.199.2		16
25.19	even	10		375.2.g.d.301.3		16
25.21	even	5		375.2.g.e.76.2		16
25.22	odd	20		75.2.i.a.34.3	✓	16
75.8	even	20		225.2.m.b.64.2		16
75.47	even	20		225.2.m.b.109.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.i.a.34.3	✓	16	25.22	odd	20
75.2.i.a.64.3	yes	16	25.8	odd	20
225.2.m.b.64.2		16	75.8	even	20
225.2.m.b.109.2		16	75.47	even	20
375.2.g.d.76.3		16	25.4	even	10
375.2.g.d.301.3		16	25.19	even	10
375.2.g.e.76.2		16	25.21	even	5
375.2.g.e.301.2		16	25.6	even	5
375.2.i.c.49.2		16	25.3	odd	20
375.2.i.c.199.2		16	25.17	odd	20
1875.2.a.m.1.5		8	1.1	even	1	trivial
1875.2.a.p.1.4		8	5.4	even	2
1875.2.b.h.1249.8		16	5.3	odd	4
1875.2.b.h.1249.9		16	5.2	odd	4
5625.2.a.t.1.5		8	15.14	odd	2
5625.2.a.bd.1.4		8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1875 = 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1875.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0898194 q^{2} +1.00000 q^{3} -1.99193 q^{4} +0.0898194 q^{6} -4.36070 q^{7} -0.358553 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.0898194 q^{2} +1.00000 q^{3} -1.99193 q^{4} +0.0898194 q^{6} -4.36070 q^{7} -0.358553 q^{8} +1.00000 q^{9} +4.39094 q^{11} -1.99193 q^{12} +1.98166 q^{13} -0.391676 q^{14} +3.95166 q^{16} -0.997022 q^{17} +0.0898194 q^{18} +1.35096 q^{19} -4.36070 q^{21} +0.394392 q^{22} -2.35651 q^{23} -0.358553 q^{24} +0.177991 q^{26} +1.00000 q^{27} +8.68622 q^{28} -7.97856 q^{29} -3.67761 q^{31} +1.07204 q^{32} +4.39094 q^{33} -0.0895519 q^{34} -1.99193 q^{36} +1.43706 q^{37} +0.121342 q^{38} +1.98166 q^{39} -5.98248 q^{41} -0.391676 q^{42} -2.68554 q^{43} -8.74646 q^{44} -0.211660 q^{46} -10.9393 q^{47} +3.95166 q^{48} +12.0157 q^{49} -0.997022 q^{51} -3.94732 q^{52} -11.0510 q^{53} +0.0898194 q^{54} +1.56354 q^{56} +1.35096 q^{57} -0.716629 q^{58} +6.68895 q^{59} -9.45570 q^{61} -0.330321 q^{62} -4.36070 q^{63} -7.80703 q^{64} +0.394392 q^{66} -12.9219 q^{67} +1.98600 q^{68} -2.35651 q^{69} +7.32257 q^{71} -0.358553 q^{72} +0.424804 q^{73} +0.129076 q^{74} -2.69101 q^{76} -19.1476 q^{77} +0.177991 q^{78} -6.35531 q^{79} +1.00000 q^{81} -0.537343 q^{82} -0.737011 q^{83} +8.68622 q^{84} -0.241213 q^{86} -7.97856 q^{87} -1.57439 q^{88} -9.78736 q^{89} -8.64141 q^{91} +4.69401 q^{92} -3.67761 q^{93} -0.982560 q^{94} +1.07204 q^{96} -0.0337081 q^{97} +1.07925 q^{98} +4.39094 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} + 8 q^{3} + 4 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q - 4 * q^2 + 8 * q^3 + 4 * q^4 - 4 * q^6 - 8 * q^7 - 12 * q^8 + 8 * q^9	\( 8 q - 4 q^{2} + 8 q^{3} + 4 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 8 q^{9} + 2 q^{11} + 4 q^{12} - 16 q^{13} + 6 q^{14} - 16 q^{17} - 4 q^{18} - 14 q^{19} - 8 q^{21} - 12 q^{22} - 14 q^{23} - 12 q^{24} + 6 q^{26} + 8 q^{27} - 16 q^{28} + 2 q^{29} - 22 q^{31} + 2 q^{32} + 2 q^{33} - 12 q^{34} + 4 q^{36} - 28 q^{37} + 16 q^{38} - 16 q^{39} + 8 q^{41} + 6 q^{42} - 20 q^{43} + 22 q^{44} - 2 q^{46} - 10 q^{47} - 16 q^{51} - 16 q^{52} - 44 q^{53} - 4 q^{54} + 30 q^{56} - 14 q^{57} - 8 q^{58} + 14 q^{59} - 20 q^{61} - 16 q^{62} - 8 q^{63} + 6 q^{64} - 12 q^{66} - 16 q^{67} + 2 q^{68} - 14 q^{69} + 16 q^{71} - 12 q^{72} - 24 q^{73} + 26 q^{74} - 16 q^{76} - 46 q^{77} + 6 q^{78} - 30 q^{79} + 8 q^{81} - 16 q^{82} - 12 q^{83} - 16 q^{84} + 32 q^{86} + 2 q^{87} - 32 q^{88} + 16 q^{89} - 12 q^{91} + 2 q^{92} - 22 q^{93} + 14 q^{94} + 2 q^{96} - 16 q^{97} - 4 q^{98} + 2 q^{99}+O(q^{100}) \) 8 * q - 4 * q^2 + 8 * q^3 + 4 * q^4 - 4 * q^6 - 8 * q^7 - 12 * q^8 + 8 * q^9 + 2 * q^11 + 4 * q^12 - 16 * q^13 + 6 * q^14 - 16 * q^17 - 4 * q^18 - 14 * q^19 - 8 * q^21 - 12 * q^22 - 14 * q^23 - 12 * q^24 + 6 * q^26 + 8 * q^27 - 16 * q^28 + 2 * q^29 - 22 * q^31 + 2 * q^32 + 2 * q^33 - 12 * q^34 + 4 * q^36 - 28 * q^37 + 16 * q^38 - 16 * q^39 + 8 * q^41 + 6 * q^42 - 20 * q^43 + 22 * q^44 - 2 * q^46 - 10 * q^47 - 16 * q^51 - 16 * q^52 - 44 * q^53 - 4 * q^54 + 30 * q^56 - 14 * q^57 - 8 * q^58 + 14 * q^59 - 20 * q^61 - 16 * q^62 - 8 * q^63 + 6 * q^64 - 12 * q^66 - 16 * q^67 + 2 * q^68 - 14 * q^69 + 16 * q^71 - 12 * q^72 - 24 * q^73 + 26 * q^74 - 16 * q^76 - 46 * q^77 + 6 * q^78 - 30 * q^79 + 8 * q^81 - 16 * q^82 - 12 * q^83 - 16 * q^84 + 32 * q^86 + 2 * q^87 - 32 * q^88 + 16 * q^89 - 12 * q^91 + 2 * q^92 - 22 * q^93 + 14 * q^94 + 2 * q^96 - 16 * q^97 - 4 * q^98 + 2 * q^99

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