LMFDB - Embedded newform 825.6.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,6,Mod(1,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	825.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:	$132.316651346$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.34253.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 52x + 48 $ x^3 - x^2 - 52*x + 48
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-7.17710$ of defining polynomial
Character	$\chi$	$=$	825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.17710 q^{2} -9.00000 q^{3} -5.19759 q^{4} +46.5939 q^{6} +123.437 q^{7} +192.576 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-5.17710 q^{2} -9.00000 q^{3} -5.19759 q^{4} +46.5939 q^{6} +123.437 q^{7} +192.576 q^{8} +81.0000 q^{9} -121.000 q^{11} +46.7783 q^{12} +500.053 q^{13} -639.048 q^{14} -830.662 q^{16} +422.631 q^{17} -419.345 q^{18} -932.948 q^{19} -1110.94 q^{21} +626.430 q^{22} +1225.18 q^{23} -1733.18 q^{24} -2588.83 q^{26} -729.000 q^{27} -641.576 q^{28} -2111.62 q^{29} -159.612 q^{31} -1862.00 q^{32} +1089.00 q^{33} -2188.00 q^{34} -421.005 q^{36} -5414.46 q^{37} +4829.97 q^{38} -4500.47 q^{39} -18066.7 q^{41} +5751.43 q^{42} -6815.47 q^{43} +628.908 q^{44} -6342.88 q^{46} +15098.9 q^{47} +7475.96 q^{48} -1570.23 q^{49} -3803.67 q^{51} -2599.07 q^{52} -15367.7 q^{53} +3774.11 q^{54} +23771.0 q^{56} +8396.53 q^{57} +10932.1 q^{58} +23400.5 q^{59} +10768.4 q^{61} +826.328 q^{62} +9998.42 q^{63} +36221.0 q^{64} -5637.87 q^{66} -14507.1 q^{67} -2196.66 q^{68} -11026.6 q^{69} -28114.0 q^{71} +15598.6 q^{72} +28836.7 q^{73} +28031.2 q^{74} +4849.08 q^{76} -14935.9 q^{77} +23299.4 q^{78} -8150.52 q^{79} +6561.00 q^{81} +93533.0 q^{82} +109864. q^{83} +5774.19 q^{84} +35284.4 q^{86} +19004.5 q^{87} -23301.7 q^{88} +69673.6 q^{89} +61725.2 q^{91} -6367.98 q^{92} +1436.51 q^{93} -78168.5 q^{94} +16758.0 q^{96} -91551.4 q^{97} +8129.23 q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 7 q^{2} - 27 q^{3} + 25 q^{4} - 63 q^{6} + 172 q^{7} + 231 q^{8} + 243 q^{9}+O(q^{10}) $ 3 * q + 7 * q^2 - 27 * q^3 + 25 * q^4 - 63 * q^6 + 172 * q^7 + 231 * q^8 + 243 * q^9	\( 3 q + 7 q^{2} - 27 q^{3} + 25 q^{4} - 63 q^{6} + 172 q^{7} + 231 q^{8} + 243 q^{9} - 363 q^{11} - 225 q^{12} + 654 q^{13} - 728 q^{14} - 415 q^{16} + 2366 q^{17} + 567 q^{18} - 2872 q^{19} - 1548 q^{21} - 847 q^{22} - 2272 q^{23} - 2079 q^{24} + 3422 q^{26} - 2187 q^{27} - 4592 q^{28} - 7738 q^{29} + 568 q^{31} - 1001 q^{32} + 3267 q^{33} + 2506 q^{34} + 2025 q^{36} + 9126 q^{37} - 13076 q^{38} - 5886 q^{39} - 8758 q^{41} + 6552 q^{42} + 14672 q^{43} - 3025 q^{44} - 28768 q^{46} + 19392 q^{47} + 3735 q^{48} - 26629 q^{49} - 21294 q^{51} + 61506 q^{52} + 4598 q^{53} - 5103 q^{54} + 2688 q^{56} + 25848 q^{57} - 8550 q^{58} - 9348 q^{59} - 60078 q^{61} + 14096 q^{62} + 13932 q^{63} - 7087 q^{64} + 7623 q^{66} + 38468 q^{67} - 59778 q^{68} + 20448 q^{69} - 74032 q^{71} + 18711 q^{72} + 44442 q^{73} + 82542 q^{74} - 98708 q^{76} - 20812 q^{77} - 30798 q^{78} - 108116 q^{79} + 19683 q^{81} + 92230 q^{82} + 81892 q^{83} + 41328 q^{84} + 126412 q^{86} + 69642 q^{87} - 27951 q^{88} + 167342 q^{89} - 31832 q^{91} - 72960 q^{92} - 5112 q^{93} + 12728 q^{94} + 9009 q^{96} - 159702 q^{97} - 163121 q^{98} - 29403 q^{99}+O(q^{100}) \) 3 * q + 7 * q^2 - 27 * q^3 + 25 * q^4 - 63 * q^6 + 172 * q^7 + 231 * q^8 + 243 * q^9 - 363 * q^11 - 225 * q^12 + 654 * q^13 - 728 * q^14 - 415 * q^16 + 2366 * q^17 + 567 * q^18 - 2872 * q^19 - 1548 * q^21 - 847 * q^22 - 2272 * q^23 - 2079 * q^24 + 3422 * q^26 - 2187 * q^27 - 4592 * q^28 - 7738 * q^29 + 568 * q^31 - 1001 * q^32 + 3267 * q^33 + 2506 * q^34 + 2025 * q^36 + 9126 * q^37 - 13076 * q^38 - 5886 * q^39 - 8758 * q^41 + 6552 * q^42 + 14672 * q^43 - 3025 * q^44 - 28768 * q^46 + 19392 * q^47 + 3735 * q^48 - 26629 * q^49 - 21294 * q^51 + 61506 * q^52 + 4598 * q^53 - 5103 * q^54 + 2688 * q^56 + 25848 * q^57 - 8550 * q^58 - 9348 * q^59 - 60078 * q^61 + 14096 * q^62 + 13932 * q^63 - 7087 * q^64 + 7623 * q^66 + 38468 * q^67 - 59778 * q^68 + 20448 * q^69 - 74032 * q^71 + 18711 * q^72 + 44442 * q^73 + 82542 * q^74 - 98708 * q^76 - 20812 * q^77 - 30798 * q^78 - 108116 * q^79 + 19683 * q^81 + 92230 * q^82 + 81892 * q^83 + 41328 * q^84 + 126412 * q^86 + 69642 * q^87 - 27951 * q^88 + 167342 * q^89 - 31832 * q^91 - 72960 * q^92 - 5112 * q^93 + 12728 * q^94 + 9009 * q^96 - 159702 * q^97 - 163121 * q^98 - 29403 * 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 $

$2$	−5.17710	−0.915191	−0.457596	−	0.889160i	$-0.651289\pi$
$2$	−5.17710	−0.915191	−0.457596	+	0.889160i	$0.651289\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	−5.19759	−0.162425
$5$	0	0
$5$	0	0
$6$	46.5939	0.528386
$7$	123.437	0.952141	0.476071	−	0.879407i	$-0.342061\pi$
$7$	123.437	0.952141	0.476071	+	0.879407i	$0.342061\pi$
$8$	192.576	1.06384
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	46.7783	0.0937759
$13$	500.053	0.820649	0.410324	−	0.911940i	$-0.365415\pi$
$13$	500.053	0.820649	0.410324	+	0.911940i	$0.365415\pi$
$14$	−639.048	−0.871392
$15$	0	0
$16$	−830.662	−0.811194
$17$	422.631	0.354682	0.177341	−	0.984150i	$-0.443251\pi$
$17$	422.631	0.354682	0.177341	+	0.984150i	$0.443251\pi$
$18$	−419.345	−0.305064
$19$	−932.948	−0.592889	−0.296445	−	0.955050i	$-0.595801\pi$
$19$	−932.948	−0.592889	−0.296445	+	0.955050i	$0.595801\pi$
$20$	0	0
$21$	−1110.94	−0.549719
$22$	626.430	0.275941
$23$	1225.18	0.482926	0.241463	−	0.970410i	$-0.422373\pi$
$23$	1225.18	0.482926	0.241463	+	0.970410i	$0.422373\pi$
$24$	−1733.18	−0.614209
$25$	0	0
$26$	−2588.83	−0.751051
$27$	−729.000	−0.192450
$28$	−641.576	−0.154651
$29$	−2111.62	−0.466251	−0.233126	−	0.972447i	$-0.574895\pi$
$29$	−2111.62	−0.466251	−0.233126	+	0.972447i	$0.574895\pi$
$30$	0	0
$31$	−159.612	−0.0298306	−0.0149153	−	0.999889i	$-0.504748\pi$
$31$	−159.612	−0.0298306	−0.0149153	+	0.999889i	$0.504748\pi$
$32$	−1862.00	−0.321444
$33$	1089.00	0.174078
$34$	−2188.00	−0.324601
$35$	0	0
$36$	−421.005	−0.0541415
$37$	−5414.46	−0.650205	−0.325103	−	0.945679i	$-0.605399\pi$
$37$	−5414.46	−0.650205	−0.325103	+	0.945679i	$0.605399\pi$
$38$	4829.97	0.542607
$39$	−4500.47	−0.473802
$40$	0	0
$41$	−18066.7	−1.67849	−0.839244	−	0.543756i	$-0.817002\pi$
$41$	−18066.7	−1.67849	−0.839244	+	0.543756i	$0.817002\pi$
$42$	5751.43	0.503098
$43$	−6815.47	−0.562114	−0.281057	−	0.959691i	$-0.590685\pi$
$43$	−6815.47	−0.562114	−0.281057	+	0.959691i	$0.590685\pi$
$44$	628.908	0.0489729
$45$	0	0
$46$	−6342.88	−0.441969
$47$	15098.9	0.997012	0.498506	−	0.866886i	$-0.333882\pi$
$47$	15098.9	0.997012	0.498506	+	0.866886i	$0.333882\pi$
$48$	7475.96	0.468343
$49$	−1570.23	−0.0934270
$50$	0	0
$51$	−3803.67	−0.204775
$52$	−2599.07	−0.133294
$53$	−15367.7	−0.751484	−0.375742	−	0.926724i	$-0.622612\pi$
$53$	−15367.7	−0.751484	−0.375742	+	0.926724i	$0.622612\pi$
$54$	3774.11	0.176129
$55$	0	0
$56$	23771.0	1.01293
$57$	8396.53	0.342305
$58$	10932.1	0.426709
$59$	23400.5	0.875177	0.437588	−	0.899175i	$-0.355833\pi$
$59$	23400.5	0.875177	0.437588	+	0.899175i	$0.355833\pi$
$60$	0	0
$61$	10768.4	0.370532	0.185266	−	0.982688i	$-0.440685\pi$
$61$	10768.4	0.370532	0.185266	+	0.982688i	$0.440685\pi$
$62$	826.328	0.0273007
$63$	9998.42	0.317380
$64$	36221.0	1.10538
$65$	0	0
$66$	−5637.87	−0.159314
$67$	−14507.1	−0.394814	−0.197407	−	0.980322i	$-0.563252\pi$
$67$	−14507.1	−0.394814	−0.197407	+	0.980322i	$0.563252\pi$
$68$	−2196.66	−0.0576090
$69$	−11026.6	−0.278817
$70$	0	0
$71$	−28114.0	−0.661876	−0.330938	−	0.943653i	$-0.607365\pi$
$71$	−28114.0	−0.661876	−0.330938	+	0.943653i	$0.607365\pi$
$72$	15598.6	0.354614
$73$	28836.7	0.633342	0.316671	−	0.948536i	$-0.397435\pi$
$73$	28836.7	0.633342	0.316671	+	0.948536i	$0.397435\pi$
$74$	28031.2	0.595062
$75$	0	0
$76$	4849.08	0.0962998
$77$	−14935.9	−0.287081
$78$	23299.4	0.433619
$79$	−8150.52	−0.146932	−0.0734662	−	0.997298i	$-0.523406\pi$
$79$	−8150.52	−0.146932	−0.0734662	+	0.997298i	$0.523406\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	93533.0	1.53614
$83$	109864.	1.75049	0.875246	−	0.483679i	$-0.160700\pi$
$83$	109864.	1.75049	0.875246	+	0.483679i	$0.160700\pi$
$84$	5774.19	0.0892879
$85$	0	0
$86$	35284.4	0.514442
$87$	19004.5	0.269190
$88$	−23301.7	−0.320760
$89$	69673.6	0.932380	0.466190	−	0.884685i	$-0.345626\pi$
$89$	69673.6	0.932380	0.466190	+	0.884685i	$0.345626\pi$
$90$	0	0
$91$	61725.2	0.781374
$92$	−6367.98	−0.0784390
$93$	1436.51	0.0172227
$94$	−78168.5	−0.912457
$95$	0	0
$96$	16758.0	0.185586
$97$	−91551.4	−0.987952	−0.493976	−	0.869476i	$-0.664457\pi$
$97$	−91551.4	−0.987952	−0.493976	+	0.869476i	$0.664457\pi$
$98$	8129.23	0.0855036
$99$	−9801.00	−0.100504
$100$	0	0
$101$	21299.3	0.207760	0.103880	−	0.994590i	$-0.466874\pi$
$101$	21299.3	0.207760	0.103880	+	0.994590i	$0.466874\pi$
$102$	19692.0	0.187409
$103$	6548.06	0.0608163	0.0304081	−	0.999538i	$-0.490319\pi$
$103$	6548.06	0.0608163	0.0304081	+	0.999538i	$0.490319\pi$
$104$	96298.1	0.873040
$105$	0	0
$106$	79560.3	0.687752
$107$	−127171.	−1.07381	−0.536905	−	0.843643i	$-0.680407\pi$
$107$	−127171.	−1.07381	−0.536905	+	0.843643i	$0.680407\pi$
$108$	3789.04	0.0312586
$109$	57285.2	0.461823	0.230912	−	0.972975i	$-0.425829\pi$
$109$	57285.2	0.461823	0.230912	+	0.972975i	$0.425829\pi$
$110$	0	0
$111$	48730.1	0.375396
$112$	−102535.	−0.772371
$113$	67774.2	0.499308	0.249654	−	0.968335i	$-0.419683\pi$
$113$	67774.2	0.499308	0.249654	+	0.968335i	$0.419683\pi$
$114$	−43469.7	−0.313274
$115$	0	0
$116$	10975.3	0.0757307
$117$	40504.3	0.273550
$118$	−121147.	−0.800954
$119$	52168.4	0.337707
$120$	0	0
$121$	14641.0	0.0909091
$122$	−55749.1	−0.339108
$123$	162600.	0.969075
$124$	829.598	0.00484522
$125$	0	0
$126$	−51762.9	−0.290464
$127$	115352.	0.634622	0.317311	−	0.948322i	$-0.397220\pi$
$127$	115352.	0.634622	0.317311	+	0.948322i	$0.397220\pi$
$128$	−127936.	−0.690187
$129$	61339.2	0.324537
$130$	0	0
$131$	43276.4	0.220329	0.110165	−	0.993913i	$-0.464862\pi$
$131$	43276.4	0.220329	0.110165	+	0.993913i	$0.464862\pi$
$132$	−5660.17	−0.0282745
$133$	−115161.	−0.564514
$134$	75104.6	0.361331
$135$	0	0
$136$	81388.4	0.377325
$137$	−401439.	−1.82734	−0.913668	−	0.406461i	$-0.866763\pi$
$137$	−401439.	−1.82734	−0.913668	+	0.406461i	$0.866763\pi$
$138$	57085.9	0.255171
$139$	−302018.	−1.32586	−0.662928	−	0.748683i	$-0.730686\pi$
$139$	−302018.	−1.32586	−0.662928	+	0.748683i	$0.730686\pi$
$140$	0	0
$141$	−135890.	−0.575625
$142$	145549.	0.605743
$143$	−60506.4	−0.247435
$144$	−67283.6	−0.270398
$145$	0	0
$146$	−149290.	−0.579629
$147$	14132.1	0.0539401
$148$	28142.1	0.105609
$149$	326942.	1.20644	0.603218	−	0.797576i	$-0.293885\pi$
$149$	326942.	1.20644	0.603218	+	0.797576i	$0.293885\pi$
$150$	0	0
$151$	164681.	0.587761	0.293881	−	0.955842i	$-0.405053\pi$
$151$	164681.	0.587761	0.293881	+	0.955842i	$0.405053\pi$
$152$	−179663.	−0.630740
$153$	34233.1	0.118227
$154$	77324.8	0.262734
$155$	0	0
$156$	23391.6	0.0769571
$157$	−248620.	−0.804982	−0.402491	−	0.915424i	$-0.631856\pi$
$157$	−248620.	−0.804982	−0.402491	+	0.915424i	$0.631856\pi$
$158$	42196.1	0.134471
$159$	138309.	0.433870
$160$	0	0
$161$	151233.	0.459813
$162$	−33967.0	−0.101688
$163$	417656.	1.23126	0.615629	−	0.788036i	$-0.288902\pi$
$163$	417656.	1.23126	0.615629	+	0.788036i	$0.288902\pi$
$164$	93903.0	0.272628
$165$	0	0
$166$	−568777.	−1.60203
$167$	−704955.	−1.95601	−0.978004	−	0.208588i	$-0.933113\pi$
$167$	−704955.	−1.95601	−0.978004	+	0.208588i	$0.933113\pi$
$168$	−213939.	−0.584814
$169$	−121240.	−0.326535
$170$	0	0
$171$	−75568.8	−0.197630
$172$	35424.0	0.0913012
$173$	171060.	0.434545	0.217272	−	0.976111i	$-0.430284\pi$
$173$	171060.	0.434545	0.217272	+	0.976111i	$0.430284\pi$
$174$	−98388.5	−0.246361
$175$	0	0
$176$	100510.	0.244584
$177$	−210605.	−0.505284
$178$	−360707.	−0.853306
$179$	−166327.	−0.387999	−0.193999	−	0.981002i	$-0.562146\pi$
$179$	−166327.	−0.387999	−0.193999	+	0.981002i	$0.562146\pi$
$180$	0	0
$181$	−584391.	−1.32589	−0.662945	−	0.748668i	$-0.730694\pi$
$181$	−584391.	−1.32589	−0.662945	+	0.748668i	$0.730694\pi$
$182$	−319558.	−0.715107
$183$	−96915.5	−0.213927
$184$	235940.	0.513756
$185$	0	0
$186$	−7436.96	−0.0157621
$187$	−51138.3	−0.106941
$188$	−78477.8	−0.161939
$189$	−89985.8	−0.183240
$190$	0	0
$191$	−715510.	−1.41916	−0.709581	−	0.704624i	$-0.751116\pi$
$191$	−715510.	−1.41916	−0.709581	+	0.704624i	$0.751116\pi$
$192$	−325989.	−0.638189
$193$	922088.	1.78188	0.890941	−	0.454118i	$-0.150046\pi$
$193$	922088.	1.78188	0.890941	+	0.454118i	$0.150046\pi$
$194$	473971.	0.904165
$195$	0	0
$196$	8161.40	0.0151749
$197$	613632.	1.12653	0.563265	−	0.826276i	$-0.309545\pi$
$197$	613632.	1.12653	0.563265	+	0.826276i	$0.309545\pi$
$198$	50740.8	0.0919802
$199$	−378985.	−0.678406	−0.339203	−	0.940713i	$-0.610157\pi$
$199$	−378985.	−0.678406	−0.339203	+	0.940713i	$0.610157\pi$
$200$	0	0
$201$	130564.	0.227946
$202$	−110269.	−0.190140
$203$	−260652.	−0.443937
$204$	19769.9	0.0332606
$205$	0	0
$206$	−33900.0	−0.0556585
$207$	99239.5	0.160975
$208$	−415375.	−0.665705
$209$	112887.	0.178763
$210$	0	0
$211$	−473721.	−0.732515	−0.366257	−	0.930514i	$-0.619361\pi$
$211$	−473721.	−0.732515	−0.366257	+	0.930514i	$0.619361\pi$
$212$	79875.1	0.122060
$213$	253026.	0.382134
$214$	658376.	0.982741
$215$	0	0
$216$	−140388.	−0.204736
$217$	−19702.1	−0.0284029
$218$	−296571.	−0.422657
$219$	−259530.	−0.365660
$220$	0	0
$221$	211338.	0.291069
$222$	−252281.	−0.343559
$223$	822747.	1.10791	0.553954	−	0.832547i	$-0.313118\pi$
$223$	822747.	1.10791	0.553954	+	0.832547i	$0.313118\pi$
$224$	−229840.	−0.306060
$225$	0	0
$226$	−350874.	−0.456962
$227$	556097.	0.716285	0.358143	−	0.933667i	$-0.383410\pi$
$227$	556097.	0.716285	0.358143	+	0.933667i	$0.383410\pi$
$228$	−43641.7	−0.0555987
$229$	−634919.	−0.800073	−0.400036	−	0.916499i	$-0.631003\pi$
$229$	−634919.	−0.800073	−0.400036	+	0.916499i	$0.631003\pi$
$230$	0	0
$231$	134423.	0.165747
$232$	−406646.	−0.496017
$233$	906561.	1.09397	0.546987	−	0.837141i	$-0.315775\pi$
$233$	906561.	1.09397	0.546987	+	0.837141i	$0.315775\pi$
$234$	−209695.	−0.250350
$235$	0	0
$236$	−121626.	−0.142150
$237$	73354.7	0.0848314
$238$	−270081.	−0.309066
$239$	662586.	0.750322	0.375161	−	0.926960i	$-0.377587\pi$
$239$	662586.	0.750322	0.375161	+	0.926960i	$0.377587\pi$
$240$	0	0
$241$	−1.31823e6	−1.46200	−0.731001	−	0.682376i	$-0.760947\pi$
$241$	−1.31823e6	−1.46200	−0.731001	+	0.682376i	$0.760947\pi$
$242$	−75798.0	−0.0831992
$243$	−59049.0	−0.0641500
$244$	−55969.6	−0.0601836
$245$	0	0
$246$	−841797.	−0.886889
$247$	−466523.	−0.486554
$248$	−30737.4	−0.0317350
$249$	−988775.	−1.01065
$250$	0	0
$251$	−11073.9	−0.0110947	−0.00554737	−	0.999985i	$-0.501766\pi$
$251$	−11073.9	−0.0110947	−0.00554737	+	0.999985i	$0.501766\pi$
$252$	−51967.7	−0.0515504
$253$	−148247.	−0.145608
$254$	−597188.	−0.580800
$255$	0	0
$256$	−496734.	−0.473723
$257$	788596.	0.744769	0.372384	−	0.928079i	$-0.378540\pi$
$257$	788596.	0.744769	0.372384	+	0.928079i	$0.378540\pi$
$258$	−317560.	−0.297013
$259$	−668346.	−0.619087
$260$	0	0
$261$	−171041.	−0.155417
$262$	−224046.	−0.201644
$263$	−211325.	−0.188391	−0.0941956	−	0.995554i	$-0.530028\pi$
$263$	−211325.	−0.188391	−0.0941956	+	0.995554i	$0.530028\pi$
$264$	209715.	0.185191
$265$	0	0
$266$	596198.	0.516639
$267$	−627062.	−0.538310
$268$	75401.8	0.0641276
$269$	552342.	0.465401	0.232701	−	0.972548i	$-0.425244\pi$
$269$	552342.	0.465401	0.232701	+	0.972548i	$0.425244\pi$
$270$	0	0
$271$	−1.49255e6	−1.23454	−0.617271	−	0.786751i	$-0.711762\pi$
$271$	−1.49255e6	−1.23454	−0.617271	+	0.786751i	$0.711762\pi$
$272$	−351063.	−0.287715
$273$	−555527.	−0.451126
$274$	2.07829e6	1.67236
$275$	0	0
$276$	57311.8	0.0452868
$277$	−1.01431e6	−0.794275	−0.397138	−	0.917759i	$-0.629996\pi$
$277$	−1.01431e6	−0.794275	−0.397138	+	0.917759i	$0.629996\pi$
$278$	1.56358e6	1.21341
$279$	−12928.6	−0.00994352
$280$	0	0
$281$	−658216.	−0.497282	−0.248641	−	0.968596i	$-0.579984\pi$
$281$	−658216.	−0.497282	−0.248641	+	0.968596i	$0.579984\pi$
$282$	703517.	0.526807
$283$	−1.65981e6	−1.23195	−0.615974	−	0.787767i	$-0.711237\pi$
$283$	−1.65981e6	−1.23195	−0.615974	+	0.787767i	$0.711237\pi$
$284$	146125.	0.107505
$285$	0	0
$286$	313248.	0.226450
$287$	−2.23010e6	−1.59816
$288$	−150822.	−0.107148
$289$	−1.24124e6	−0.874201
$290$	0	0
$291$	823963.	0.570394
$292$	−149881.	−0.102870
$293$	410864.	0.279594	0.139797	−	0.990180i	$-0.455355\pi$
$293$	410864.	0.279594	0.139797	+	0.990180i	$0.455355\pi$
$294$	−73163.1	−0.0493655
$295$	0	0
$296$	−1.04269e6	−0.691715
$297$	88209.0	0.0580259
$298$	−1.69261e6	−1.10412
$299$	612654.	0.396312
$300$	0	0
$301$	−841283.	−0.535212
$302$	−852570.	−0.537914
$303$	−191694.	−0.119950
$304$	774965.	0.480948
$305$	0	0
$306$	−177228.	−0.108200
$307$	−1.34831e6	−0.816477	−0.408238	−	0.912875i	$-0.633857\pi$
$307$	−1.34831e6	−0.816477	−0.408238	+	0.912875i	$0.633857\pi$
$308$	77630.7	0.0466291
$309$	−58932.6	−0.0351123
$310$	0	0
$311$	−2.52585e6	−1.48083	−0.740417	−	0.672148i	$-0.765372\pi$
$311$	−2.52585e6	−1.48083	−0.740417	+	0.672148i	$0.765372\pi$
$312$	−866682.	−0.504050
$313$	−2.23061e6	−1.28695	−0.643476	−	0.765466i	$-0.722509\pi$
$313$	−2.23061e6	−1.28695	−0.643476	+	0.765466i	$0.722509\pi$
$314$	1.28713e6	0.736713
$315$	0	0
$316$	42363.0	0.0238654
$317$	120908.	0.0675780	0.0337890	−	0.999429i	$-0.489243\pi$
$317$	120908.	0.0675780	0.0337890	+	0.999429i	$0.489243\pi$
$318$	−716043.	−0.397074
$319$	255506.	0.140580
$320$	0	0
$321$	1.14454e6	0.619964
$322$	−782948.	−0.420817
$323$	−394292.	−0.210287
$324$	−34101.4	−0.0180472
$325$	0	0
$326$	−2.16225e6	−1.12684
$327$	−515566.	−0.266634
$328$	−3.47920e6	−1.78564
$329$	1.86377e6	0.949296
$330$	0	0
$331$	578480.	0.290214	0.145107	−	0.989416i	$-0.453647\pi$
$331$	578480.	0.290214	0.145107	+	0.989416i	$0.453647\pi$
$332$	−571028.	−0.284323
$333$	−438571.	−0.216735
$334$	3.64963e6	1.79012
$335$	0	0
$336$	922812.	0.445929
$337$	1.78710e6	0.857186	0.428593	−	0.903498i	$-0.359009\pi$
$337$	1.78710e6	0.857186	0.428593	+	0.903498i	$0.359009\pi$
$338$	627674.	0.298842
$339$	−609968.	−0.288276
$340$	0	0
$341$	19313.1	0.00899425
$342$	391228.	0.180869
$343$	−2.26844e6	−1.04110
$344$	−1.31249e6	−0.598000
$345$	0	0
$346$	−885598.	−0.397692
$347$	1.46282e6	0.652179	0.326089	−	0.945339i	$-0.394269\pi$
$347$	1.46282e6	0.652179	0.326089	+	0.945339i	$0.394269\pi$
$348$	−98777.8	−0.0437231
$349$	1.48501e6	0.652627	0.326314	−	0.945262i	$-0.394193\pi$
$349$	1.48501e6	0.652627	0.326314	+	0.945262i	$0.394193\pi$
$350$	0	0
$351$	−364538.	−0.157934
$352$	225302.	0.0969189
$353$	−1.34528e6	−0.574616	−0.287308	−	0.957838i	$-0.592760\pi$
$353$	−1.34528e6	−0.574616	−0.287308	+	0.957838i	$0.592760\pi$
$354$	1.09032e6	0.462431
$355$	0	0
$356$	−362135.	−0.151442
$357$	−469515.	−0.194975
$358$	861092.	0.355093
$359$	−1.83956e6	−0.753316	−0.376658	−	0.926352i	$-0.622927\pi$
$359$	−1.83956e6	−0.753316	−0.376658	+	0.926352i	$0.622927\pi$
$360$	0	0
$361$	−1.60571e6	−0.648483
$362$	3.02546e6	1.21344
$363$	−131769.	−0.0524864
$364$	−320822.	−0.126914
$365$	0	0
$366$	501742.	0.195784
$367$	312079.	0.120948	0.0604741	−	0.998170i	$-0.480739\pi$
$367$	312079.	0.120948	0.0604741	+	0.998170i	$0.480739\pi$
$368$	−1.01771e6	−0.391746
$369$	−1.46340e6	−0.559496
$370$	0	0
$371$	−1.89695e6	−0.715519
$372$	−7466.38	−0.00279739
$373$	−3.38658e6	−1.26034	−0.630172	−	0.776455i	$-0.717016\pi$
$373$	−3.38658e6	−1.26034	−0.630172	+	0.776455i	$0.717016\pi$
$374$	264748.	0.0978710
$375$	0	0
$376$	2.90768e6	1.06066
$377$	−1.05592e6	−0.382629
$378$	465866.	0.167699
$379$	−1.62370e6	−0.580641	−0.290321	−	0.956929i	$-0.593762\pi$
$379$	−1.62370e6	−0.580641	−0.290321	+	0.956929i	$0.593762\pi$
$380$	0	0
$381$	−1.03817e6	−0.366399
$382$	3.70427e6	1.29881
$383$	3.06187e6	1.06657	0.533285	−	0.845935i	$-0.320957\pi$
$383$	3.06187e6	1.06657	0.533285	+	0.845935i	$0.320957\pi$
$384$	1.15142e6	0.398480
$385$	0	0
$386$	−4.77375e6	−1.63076
$387$	−552053.	−0.187371
$388$	475847.	0.160468
$389$	4.21840e6	1.41343	0.706715	−	0.707499i	$-0.250176\pi$
$389$	4.21840e6	1.41343	0.706715	+	0.707499i	$0.250176\pi$
$390$	0	0
$391$	517798.	0.171285
$392$	−302388.	−0.0993915
$393$	−389487.	−0.127207
$394$	−3.17684e6	−1.03099
$395$	0	0
$396$	50941.6	0.0163243
$397$	−513185.	−0.163417	−0.0817085	−	0.996656i	$-0.526038\pi$
$397$	−513185.	−0.163417	−0.0817085	+	0.996656i	$0.526038\pi$
$398$	1.96205e6	0.620871
$399$	1.03645e6	0.325922
$400$	0	0
$401$	−2.18458e6	−0.678432	−0.339216	−	0.940709i	$-0.610162\pi$
$401$	−2.18458e6	−0.678432	−0.339216	+	0.940709i	$0.610162\pi$
$402$	−675942.	−0.208614
$403$	−79814.5	−0.0244804
$404$	−110705.	−0.0337453
$405$	0	0
$406$	1.34942e6	0.406287
$407$	655149.	0.196044
$408$	−732496.	−0.217849
$409$	1.22307e6	0.361529	0.180764	−	0.983526i	$-0.442143\pi$
$409$	1.22307e6	0.361529	0.180764	+	0.983526i	$0.442143\pi$
$410$	0	0
$411$	3.61295e6	1.05501
$412$	−34034.1	−0.00987806
$413$	2.88850e6	0.833292
$414$	−513774.	−0.147323
$415$	0	0
$416$	−931098.	−0.263792
$417$	2.71816e6	0.765483
$418$	−584426.	−0.163602
$419$	2.42879e6	0.675858	0.337929	−	0.941172i	$-0.390274\pi$
$419$	2.42879e6	0.675858	0.337929	+	0.941172i	$0.390274\pi$
$420$	0	0
$421$	−727467.	−0.200036	−0.100018	−	0.994986i	$-0.531890\pi$
$421$	−727467.	−0.200036	−0.100018	+	0.994986i	$0.531890\pi$
$422$	2.45250e6	0.670391
$423$	1.22301e6	0.332337
$424$	−2.95945e6	−0.799460
$425$	0	0
$426$	−1.30994e6	−0.349726
$427$	1.32922e6	0.352799
$428$	660981.	0.174413
$429$	544557.	0.142857
$430$	0	0
$431$	6.18223e6	1.60307	0.801534	−	0.597949i	$-0.204018\pi$
$431$	6.18223e6	1.60307	0.801534	+	0.597949i	$0.204018\pi$
$432$	605553.	0.156114
$433$	−2.42337e6	−0.621154	−0.310577	−	0.950548i	$-0.600522\pi$
$433$	−2.42337e6	−0.621154	−0.310577	+	0.950548i	$0.600522\pi$
$434$	102000.	0.0259941
$435$	0	0
$436$	−297745.	−0.0750115
$437$	−1.14303e6	−0.286321
$438$	1.34361e6	0.334649
$439$	−3.79993e6	−0.941054	−0.470527	−	0.882385i	$-0.655936\pi$
$439$	−3.79993e6	−0.941054	−0.470527	+	0.882385i	$0.655936\pi$
$440$	0	0
$441$	−127188.	−0.0311423
$442$	−1.09412e6	−0.266384
$443$	5.58044e6	1.35101	0.675506	−	0.737354i	$-0.263925\pi$
$443$	5.58044e6	1.35101	0.675506	+	0.737354i	$0.263925\pi$
$444$	−253279.	−0.0609736
$445$	0	0
$446$	−4.25945e6	−1.01395
$447$	−2.94247e6	−0.696537
$448$	4.47102e6	1.05247
$449$	2.69609e6	0.631130	0.315565	−	0.948904i	$-0.397806\pi$
$449$	2.69609e6	0.631130	0.315565	+	0.948904i	$0.397806\pi$
$450$	0	0
$451$	2.18607e6	0.506083
$452$	−352263.	−0.0810999
$453$	−1.48213e6	−0.339344
$454$	−2.87897e6	−0.655538
$455$	0	0
$456$	1.61697e6	0.364158
$457$	−1.14257e6	−0.255912	−0.127956	−	0.991780i	$-0.540842\pi$
$457$	−1.14257e6	−0.255912	−0.127956	+	0.991780i	$0.540842\pi$
$458$	3.28704e6	0.732220
$459$	−308098.	−0.0682585
$460$	0	0
$461$	5.37091e6	1.17705	0.588526	−	0.808478i	$-0.299708\pi$
$461$	5.37091e6	1.17705	0.588526	+	0.808478i	$0.299708\pi$
$462$	−695923.	−0.151690
$463$	3.80227e6	0.824311	0.412155	−	0.911114i	$-0.364776\pi$
$463$	3.80227e6	0.824311	0.412155	+	0.911114i	$0.364776\pi$
$464$	1.75404e6	0.378220
$465$	0	0
$466$	−4.69336e6	−1.00120
$467$	−9.41694e6	−1.99810	−0.999051	−	0.0435609i	$-0.986130\pi$
$467$	−9.41694e6	−1.99810	−0.999051	+	0.0435609i	$0.986130\pi$
$468$	−210525.	−0.0444312
$469$	−1.79071e6	−0.375919
$470$	0	0
$471$	2.23758e6	0.464757
$472$	4.50638e6	0.931049
$473$	824672.	0.169484
$474$	−379765.	−0.0776370
$475$	0	0
$476$	−271150.	−0.0548519
$477$	−1.24479e6	−0.250495
$478$	−3.43028e6	−0.686688
$479$	−4.69047e6	−0.934065	−0.467033	−	0.884240i	$-0.654677\pi$
$479$	−4.69047e6	−0.934065	−0.467033	+	0.884240i	$0.654677\pi$
$480$	0	0
$481$	−2.70751e6	−0.533590
$482$	6.82461e6	1.33801
$483$	−1.36110e6	−0.265473
$484$	−76097.9	−0.0147659
$485$	0	0
$486$	305703.	0.0587096
$487$	−7.07177e6	−1.35116	−0.675579	−	0.737288i	$-0.736106\pi$
$487$	−7.07177e6	−1.35116	−0.675579	+	0.737288i	$0.736106\pi$
$488$	2.07373e6	0.394187
$489$	−3.75890e6	−0.710867
$490$	0	0
$491$	906051.	0.169609	0.0848045	−	0.996398i	$-0.472973\pi$
$491$	906051.	0.169609	0.0848045	+	0.996398i	$0.472973\pi$
$492$	−845127.	−0.157402
$493$	−892434.	−0.165371
$494$	2.41524e6	0.445290
$495$	0	0
$496$	132584.	0.0241984
$497$	−3.47031e6	−0.630199
$498$	5.11899e6	0.924935
$499$	−7.13323e6	−1.28243	−0.641217	−	0.767360i	$-0.721570\pi$
$499$	−7.13323e6	−1.28243	−0.641217	+	0.767360i	$0.721570\pi$
$500$	0	0
$501$	6.34460e6	1.12930
$502$	57330.9	0.0101538
$503$	2.34927e6	0.414013	0.207006	−	0.978340i	$-0.433628\pi$
$503$	2.34927e6	0.414013	0.207006	+	0.978340i	$0.433628\pi$
$504$	1.92545e6	0.337642
$505$	0	0
$506$	767489.	0.133259
$507$	1.09116e6	0.188525
$508$	−599551.	−0.103078
$509$	1.97148e6	0.337286	0.168643	−	0.985677i	$-0.446062\pi$
$509$	1.97148e6	0.337286	0.168643	+	0.985677i	$0.446062\pi$
$510$	0	0
$511$	3.55952e6	0.603031
$512$	6.66559e6	1.12373
$513$	680119.	0.114102
$514$	−4.08264e6	−0.681606
$515$	0	0
$516$	−318816.	−0.0527128
$517$	−1.82697e6	−0.300610
$518$	3.46010e6	0.566583
$519$	−1.53954e6	−0.250884
$520$	0	0
$521$	1.20074e7	1.93801	0.969004	−	0.247046i	$-0.0794599\pi$
$521$	1.20074e7	1.93801	0.969004	+	0.247046i	$0.0794599\pi$
$522$	885497.	0.142236
$523$	−4.16772e6	−0.666260	−0.333130	−	0.942881i	$-0.608105\pi$
$523$	−4.16772e6	−0.666260	−0.333130	+	0.942881i	$0.608105\pi$
$524$	−224933.	−0.0357869
$525$	0	0
$526$	1.09405e6	0.172414
$527$	−67456.9	−0.0105804
$528$	−904591.	−0.141211
$529$	−4.93528e6	−0.766783
$530$	0	0
$531$	1.89544e6	0.291726
$532$	598557.	0.0916910
$533$	−9.03428e6	−1.37745
$534$	3.24637e6	0.492657
$535$	0	0
$536$	−2.79371e6	−0.420020
$537$	1.49694e6	0.224011
$538$	−2.85953e6	−0.425931
$539$	189998.	0.0281693
$540$	0	0
$541$	1.01106e7	1.48519	0.742594	−	0.669741i	$-0.233595\pi$
$541$	1.01106e7	1.48519	0.742594	+	0.669741i	$0.233595\pi$
$542$	7.72709e6	1.12984
$543$	5.25952e6	0.765503
$544$	−786938.	−0.114010
$545$	0	0
$546$	2.87602e6	0.412867
$547$	7.39087e6	1.05615	0.528077	−	0.849196i	$-0.322913\pi$
$547$	7.39087e6	1.05615	0.528077	+	0.849196i	$0.322913\pi$
$548$	2.08652e6	0.296805
$549$	872239.	0.123511
$550$	0	0
$551$	1.97003e6	0.276435
$552$	−2.12346e6	−0.296617
$553$	−1.00608e6	−0.139900
$554$	5.25119e6	0.726914
$555$	0	0
$556$	1.56977e6	0.215352
$557$	−1.52459e6	−0.208217	−0.104108	−	0.994566i	$-0.533199\pi$
$557$	−1.52459e6	−0.208217	−0.104108	+	0.994566i	$0.533199\pi$
$558$	66932.6	0.00910023
$559$	−3.40809e6	−0.461299
$560$	0	0
$561$	460245.	0.0617421
$562$	3.40765e6	0.455108
$563$	1.40845e7	1.87271	0.936353	−	0.351059i	$-0.114178\pi$
$563$	1.40845e7	1.87271	0.936353	+	0.351059i	$0.114178\pi$
$564$	706300.	0.0934957
$565$	0	0
$566$	8.59301e6	1.12747
$567$	809872.	0.105793
$568$	−5.41407e6	−0.704131
$569$	−1.30473e7	−1.68942	−0.844712	−	0.535221i	$-0.820228\pi$
$569$	−1.30473e7	−1.68942	−0.844712	+	0.535221i	$0.820228\pi$
$570$	0	0
$571$	−1.19873e7	−1.53862	−0.769312	−	0.638873i	$-0.779401\pi$
$571$	−1.19873e7	−1.53862	−0.769312	+	0.638873i	$0.779401\pi$
$572$	314487.	0.0401895
$573$	6.43959e6	0.819354
$574$	1.15455e7	1.46262
$575$	0	0
$576$	2.93390e6	0.368459
$577$	−568904.	−0.0711376	−0.0355688	−	0.999367i	$-0.511324\pi$
$577$	−568904.	−0.0711376	−0.0355688	+	0.999367i	$0.511324\pi$
$578$	6.42603e6	0.800061
$579$	−8.29879e6	−1.02877
$580$	0	0
$581$	1.35613e7	1.66671
$582$	−4.26574e6	−0.522020
$583$	1.85949e6	0.226581
$584$	5.55325e6	0.673775
$585$	0	0
$586$	−2.12708e6	−0.255882
$587$	−2.93323e6	−0.351358	−0.175679	−	0.984447i	$-0.556212\pi$
$587$	−2.93323e6	−0.351358	−0.175679	+	0.984447i	$0.556212\pi$
$588$	−73452.6	−0.00876120
$589$	148910.	0.0176862
$590$	0	0
$591$	−5.52269e6	−0.650402
$592$	4.49758e6	0.527442
$593$	−5.90557e6	−0.689644	−0.344822	−	0.938668i	$-0.612061\pi$
$593$	−5.90557e6	−0.689644	−0.344822	+	0.938668i	$0.612061\pi$
$594$	−456667.	−0.0531048
$595$	0	0
$596$	−1.69931e6	−0.195955
$597$	3.41087e6	0.391678
$598$	−3.17178e6	−0.362702
$599$	1.07665e7	1.22605	0.613025	−	0.790064i	$-0.289953\pi$
$599$	1.07665e7	1.22605	0.613025	+	0.790064i	$0.289953\pi$
$600$	0	0
$601$	−1.19746e7	−1.35231	−0.676154	−	0.736760i	$-0.736355\pi$
$601$	−1.19746e7	−1.35231	−0.676154	+	0.736760i	$0.736355\pi$
$602$	4.35541e6	0.489822
$603$	−1.17507e6	−0.131605
$604$	−855944.	−0.0954669
$605$	0	0
$606$	992418.	0.109777
$607$	3.45506e6	0.380613	0.190307	−	0.981725i	$-0.439052\pi$
$607$	3.45506e6	0.380613	0.190307	+	0.981725i	$0.439052\pi$
$608$	1.73715e6	0.190580
$609$	2.34587e6	0.256307
$610$	0	0
$611$	7.55024e6	0.818197
$612$	−177929.	−0.0192030
$613$	8.22419e6	0.883979	0.441990	−	0.897020i	$-0.354273\pi$
$613$	8.22419e6	0.883979	0.441990	+	0.897020i	$0.354273\pi$
$614$	6.98034e6	0.747233
$615$	0	0
$616$	−2.87630e6	−0.305409
$617$	−9.83567e6	−1.04014	−0.520069	−	0.854124i	$-0.674094\pi$
$617$	−9.83567e6	−1.04014	−0.520069	+	0.854124i	$0.674094\pi$
$618$	305100.	0.0321345
$619$	267648.	0.0280762	0.0140381	−	0.999901i	$-0.495531\pi$
$619$	267648.	0.0280762	0.0140381	+	0.999901i	$0.495531\pi$
$620$	0	0
$621$	−893156.	−0.0929391
$622$	1.30766e7	1.35525
$623$	8.60032e6	0.887758
$624$	3.73837e6	0.384345
$625$	0	0
$626$	1.15481e7	1.17781
$627$	−1.01598e6	−0.103209
$628$	1.29222e6	0.130749
$629$	−2.28831e6	−0.230616
$630$	0	0
$631$	−4.17135e6	−0.417065	−0.208532	−	0.978015i	$-0.566869\pi$
$631$	−4.17135e6	−0.417065	−0.208532	+	0.978015i	$0.566869\pi$
$632$	−1.56959e6	−0.156313
$633$	4.26349e6	0.422918
$634$	−625951.	−0.0618468
$635$	0	0
$636$	−718876.	−0.0704711
$637$	−785197.	−0.0766708
$638$	−1.32278e6	−0.128658
$639$	−2.27723e6	−0.220625
$640$	0	0
$641$	8.24673e6	0.792751	0.396376	−	0.918088i	$-0.370268\pi$
$641$	8.24673e6	0.792751	0.396376	+	0.918088i	$0.370268\pi$
$642$	−5.92538e6	−0.567386
$643$	−1.07773e7	−1.02797	−0.513986	−	0.857799i	$-0.671832\pi$
$643$	−1.07773e7	−1.02797	−0.513986	+	0.857799i	$0.671832\pi$
$644$	−786046.	−0.0746850
$645$	0	0
$646$	2.04129e6	0.192453
$647$	−7.89194e6	−0.741179	−0.370590	−	0.928797i	$-0.620844\pi$
$647$	−7.89194e6	−0.741179	−0.370590	+	0.928797i	$0.620844\pi$
$648$	1.26349e6	0.118205
$649$	−2.83147e6	−0.263876
$650$	0	0
$651$	177319.	0.0163984
$652$	−2.17080e6	−0.199987
$653$	−1.47858e7	−1.35695	−0.678473	−	0.734625i	$-0.737358\pi$
$653$	−1.47858e7	−1.35695	−0.678473	+	0.734625i	$0.737358\pi$
$654$	2.66914e6	0.244021
$655$	0	0
$656$	1.50073e7	1.36158
$657$	2.33577e6	0.211114
$658$	−9.64891e6	−0.868788
$659$	−475088.	−0.0426148	−0.0213074	−	0.999773i	$-0.506783\pi$
$659$	−475088.	−0.0426148	−0.0213074	+	0.999773i	$0.506783\pi$
$660$	0	0
$661$	2.73604e6	0.243567	0.121784	−	0.992557i	$-0.461139\pi$
$661$	2.73604e6	0.243567	0.121784	+	0.992557i	$0.461139\pi$
$662$	−2.99485e6	−0.265601
$663$	−1.90204e6	−0.168049
$664$	2.11571e7	1.86224
$665$	0	0
$666$	2.27053e6	0.198354
$667$	−2.58711e6	−0.225165
$668$	3.66407e6	0.317704
$669$	−7.40472e6	−0.639651
$670$	0	0
$671$	−1.30297e6	−0.111720
$672$	2.06856e6	0.176704
$673$	−1.52861e7	−1.30094	−0.650471	−	0.759531i	$-0.725428\pi$
$673$	−1.52861e7	−1.30094	−0.650471	+	0.759531i	$0.725428\pi$
$674$	−9.25202e6	−0.784489
$675$	0	0
$676$	630157.	0.0530374
$677$	3.93225e6	0.329739	0.164869	−	0.986315i	$-0.447280\pi$
$677$	3.93225e6	0.329739	0.164869	+	0.986315i	$0.447280\pi$
$678$	3.15787e6	0.263827
$679$	−1.13009e7	−0.940670
$680$	0	0
$681$	−5.00487e6	−0.413547
$682$	−99985.7	−0.00823146
$683$	−3.42591e6	−0.281011	−0.140506	−	0.990080i	$-0.544873\pi$
$683$	−3.42591e6	−0.281011	−0.140506	+	0.990080i	$0.544873\pi$
$684$	392776.	0.0320999
$685$	0	0
$686$	1.17439e7	0.952803
$687$	5.71427e6	0.461922
$688$	5.66135e6	0.455984
$689$	−7.68467e6	−0.616705
$690$	0	0
$691$	1.81896e7	1.44920	0.724599	−	0.689171i	$-0.242025\pi$
$691$	1.81896e7	1.44920	0.724599	+	0.689171i	$0.242025\pi$
$692$	−889102.	−0.0705808
$693$	−1.20981e6	−0.0956938
$694$	−7.57316e6	−0.596868
$695$	0	0
$696$	3.65982e6	0.286376
$697$	−7.63552e6	−0.595328
$698$	−7.68804e6	−0.597279
$699$	−8.15905e6	−0.631607
$700$	0	0
$701$	−1.15598e6	−0.0888494	−0.0444247	−	0.999013i	$-0.514145\pi$
$701$	−1.15598e6	−0.0888494	−0.0444247	+	0.999013i	$0.514145\pi$
$702$	1.88725e6	0.144540
$703$	5.05141e6	0.385500
$704$	−4.38274e6	−0.333283
$705$	0	0
$706$	6.96468e6	0.525883
$707$	2.62913e6	0.197817
$708$	1.09464e6	0.0820705
$709$	2.42664e7	1.81296	0.906481	−	0.422246i	$-0.138758\pi$
$709$	2.42664e7	1.81296	0.906481	+	0.422246i	$0.138758\pi$
$710$	0	0
$711$	−660192.	−0.0489775
$712$	1.34174e7	0.991904
$713$	−195553.	−0.0144059
$714$	2.43073e6	0.178440
$715$	0	0
$716$	864499.	0.0630205
$717$	−5.96328e6	−0.433199
$718$	9.52359e6	0.689429
$719$	−2.25503e7	−1.62679	−0.813394	−	0.581714i	$-0.802382\pi$
$719$	−2.25503e7	−1.62679	−0.813394	+	0.581714i	$0.802382\pi$
$720$	0	0
$721$	808276.	0.0579057
$722$	8.31291e6	0.593486
$723$	1.18641e7	0.844088
$724$	3.03743e6	0.215357
$725$	0	0
$726$	682182.	0.0480351
$727$	2.17941e7	1.52933	0.764667	−	0.644425i	$-0.222903\pi$
$727$	2.17941e7	1.52933	0.764667	+	0.644425i	$0.222903\pi$
$728$	1.18868e7	0.831257
$729$	531441.	0.0370370
$730$	0	0
$731$	−2.88043e6	−0.199372
$732$	503727.	0.0347470
$733$	−1.73565e7	−1.19317	−0.596583	−	0.802551i	$-0.703476\pi$
$733$	−1.73565e7	−1.19317	−0.596583	+	0.802551i	$0.703476\pi$
$734$	−1.61567e6	−0.110691
$735$	0	0
$736$	−2.28129e6	−0.155233
$737$	1.75536e6	0.119041
$738$	7.57617e6	0.512046
$739$	1.16961e6	0.0787825	0.0393912	−	0.999224i	$-0.487458\pi$
$739$	1.16961e6	0.0787825	0.0393912	+	0.999224i	$0.487458\pi$
$740$	0	0
$741$	4.19871e6	0.280912
$742$	9.82071e6	0.654837
$743$	−2.98047e6	−0.198067	−0.0990336	−	0.995084i	$-0.531575\pi$
$743$	−2.98047e6	−0.198067	−0.0990336	+	0.995084i	$0.531575\pi$
$744$	276637.	0.0183222
$745$	0	0
$746$	1.75327e7	1.15346
$747$	8.89898e6	0.583497
$748$	265796.	0.0173698
$749$	−1.56976e7	−1.02242
$750$	0	0
$751$	−1.83578e7	−1.18774	−0.593868	−	0.804563i	$-0.702400\pi$
$751$	−1.83578e7	−1.18774	−0.593868	+	0.804563i	$0.702400\pi$
$752$	−1.25421e7	−0.808770
$753$	99665.3	0.00640555
$754$	5.46661e6	0.350178
$755$	0	0
$756$	467709.	0.0297626
$757$	−1.60050e7	−1.01511	−0.507557	−	0.861618i	$-0.669451\pi$
$757$	−1.60050e7	−1.01511	−0.507557	+	0.861618i	$0.669451\pi$
$758$	8.40607e6	0.531398
$759$	1.33422e6	0.0840665
$760$	0	0
$761$	−1.67436e7	−1.04807	−0.524033	−	0.851698i	$-0.675573\pi$
$761$	−1.67436e7	−1.04807	−0.524033	+	0.851698i	$0.675573\pi$
$762$	5.37469e6	0.335325
$763$	7.07113e6	0.439721
$764$	3.71892e6	0.230507
$765$	0	0
$766$	−1.58516e7	−0.976116
$767$	1.17015e7	0.718213
$768$	4.47061e6	0.273504
$769$	−1.73863e7	−1.06021	−0.530104	−	0.847933i	$-0.677847\pi$
$769$	−1.73863e7	−1.06021	−0.530104	+	0.847933i	$0.677847\pi$
$770$	0	0
$771$	−7.09736e6	−0.429993
$772$	−4.79263e6	−0.289422
$773$	−1.79095e7	−1.07804	−0.539021	−	0.842292i	$-0.681206\pi$
$773$	−1.79095e7	−1.07804	−0.539021	+	0.842292i	$0.681206\pi$
$774$	2.85804e6	0.171481
$775$	0	0
$776$	−1.76306e7	−1.05102
$777$	6.01511e6	0.357430
$778$	−2.18391e7	−1.29356
$779$	1.68552e7	0.995157
$780$	0	0
$781$	3.40179e6	0.199563
$782$	−2.68070e6	−0.156758
$783$	1.53937e6	0.0897301
$784$	1.30433e6	0.0757874
$785$	0	0
$786$	2.01642e6	0.116419
$787$	3.01291e7	1.73400	0.867000	−	0.498308i	$-0.166045\pi$
$787$	3.01291e7	1.73400	0.867000	+	0.498308i	$0.166045\pi$
$788$	−3.18941e6	−0.182976
$789$	1.90192e6	0.108768
$790$	0	0
$791$	8.36587e6	0.475412
$792$	−1.88744e6	−0.106920
$793$	5.38476e6	0.304077
$794$	2.65681e6	0.149558
$795$	0	0
$796$	1.96981e6	0.110190
$797$	5.51251e6	0.307400	0.153700	−	0.988118i	$-0.450881\pi$
$797$	5.51251e6	0.307400	0.153700	+	0.988118i	$0.450881\pi$
$798$	−5.36579e6	−0.298281
$799$	6.38125e6	0.353622
$800$	0	0
$801$	5.64356e6	0.310793
$802$	1.13098e7	0.620895
$803$	−3.48924e6	−0.190960
$804$	−678616.	−0.0370241
$805$	0	0
$806$	413208.	0.0224043
$807$	−4.97108e6	−0.268700
$808$	4.10173e6	0.221024
$809$	3.11564e7	1.67369	0.836847	−	0.547436i	$-0.184396\pi$
$809$	3.11564e7	1.67369	0.836847	+	0.547436i	$0.184396\pi$
$810$	0	0
$811$	−1.11336e7	−0.594406	−0.297203	−	0.954814i	$-0.596054\pi$
$811$	−1.11336e7	−0.594406	−0.297203	+	0.954814i	$0.596054\pi$
$812$	1.35476e6	0.0721063
$813$	1.34330e7	0.712763
$814$	−3.39178e6	−0.179418
$815$	0	0
$816$	3.15957e6	0.166113
$817$	6.35848e6	0.333271
$818$	−6.33195e6	−0.330868
$819$	4.99974e6	0.260458
$820$	0	0
$821$	−3.44603e7	−1.78427	−0.892136	−	0.451767i	$-0.850794\pi$
$821$	−3.44603e7	−1.78427	−0.892136	+	0.451767i	$0.850794\pi$
$822$	−1.87046e7	−0.965539
$823$	9.74417e6	0.501470	0.250735	−	0.968056i	$-0.419328\pi$
$823$	9.74417e6	0.501470	0.250735	+	0.968056i	$0.419328\pi$
$824$	1.26100e6	0.0646989
$825$	0	0
$826$	−1.49541e7	−0.762622
$827$	9.63214e6	0.489733	0.244866	−	0.969557i	$-0.421256\pi$
$827$	9.63214e6	0.489733	0.244866	+	0.969557i	$0.421256\pi$
$828$	−515806.	−0.0261463
$829$	−2.79310e7	−1.41156	−0.705780	−	0.708431i	$-0.749403\pi$
$829$	−2.79310e7	−1.41156	−0.705780	+	0.708431i	$0.749403\pi$
$830$	0	0
$831$	9.12879e6	0.458575
$832$	1.81124e7	0.907126
$833$	−663626.	−0.0331368
$834$	−1.40722e7	−0.700564
$835$	0	0
$836$	−586739.	−0.0290355
$837$	116357.	0.00574090
$838$	−1.25741e7	−0.618540
$839$	−3.51475e7	−1.72381	−0.861906	−	0.507068i	$-0.830729\pi$
$839$	−3.51475e7	−1.72381	−0.861906	+	0.507068i	$0.830729\pi$
$840$	0	0
$841$	−1.60522e7	−0.782610
$842$	3.76617e6	0.183071
$843$	5.92394e6	0.287106
$844$	2.46221e6	0.118978
$845$	0	0
$846$	−6.33165e6	−0.304152
$847$	1.80725e6	0.0865583
$848$	1.27654e7	0.609599
$849$	1.49383e7	0.711265
$850$	0	0
$851$	−6.63368e6	−0.314001
$852$	−1.31512e6	−0.0620680
$853$	−2.50498e7	−1.17878	−0.589389	−	0.807849i	$-0.700631\pi$
$853$	−2.50498e7	−1.17878	−0.589389	+	0.807849i	$0.700631\pi$
$854$	−6.88152e6	−0.322879
$855$	0	0
$856$	−2.44900e7	−1.14236
$857$	1.89631e7	0.881975	0.440987	−	0.897513i	$-0.354628\pi$
$857$	1.89631e7	0.881975	0.440987	+	0.897513i	$0.354628\pi$
$858$	−2.81923e6	−0.130741
$859$	−3.04605e7	−1.40849	−0.704245	−	0.709957i	$-0.748714\pi$
$859$	−3.04605e7	−1.40849	−0.704245	+	0.709957i	$0.748714\pi$
$860$	0	0
$861$	2.00709e7	0.922696
$862$	−3.20060e7	−1.46711
$863$	−3.46176e7	−1.58223	−0.791115	−	0.611667i	$-0.790499\pi$
$863$	−3.46176e7	−1.58223	−0.791115	+	0.611667i	$0.790499\pi$
$864$	1.35740e6	0.0618619
$865$	0	0
$866$	1.25460e7	0.568475
$867$	1.11712e7	0.504720
$868$	102403.	0.00461333
$869$	986213.	0.0443018
$870$	0	0
$871$	−7.25430e6	−0.324004
$872$	1.10317e7	0.491307
$873$	−7.41566e6	−0.329317
$874$	5.91758e6	0.262039
$875$	0	0
$876$	1.34893e6	0.0593922
$877$	3.61979e7	1.58922	0.794611	−	0.607119i	$-0.207675\pi$
$877$	3.61979e7	1.58922	0.794611	+	0.607119i	$0.207675\pi$
$878$	1.96727e7	0.861245
$879$	−3.69777e6	−0.161424
$880$	0	0
$881$	2.40371e7	1.04338	0.521690	−	0.853135i	$-0.325302\pi$
$881$	2.40371e7	1.04338	0.521690	+	0.853135i	$0.325302\pi$
$882$	658468.	0.0285012
$883$	−3.64938e7	−1.57513	−0.787567	−	0.616229i	$-0.788659\pi$
$883$	−3.64938e7	−1.57513	−0.787567	+	0.616229i	$0.788659\pi$
$884$	−1.09845e6	−0.0472768
$885$	0	0
$886$	−2.88905e7	−1.23644
$887$	−2.29197e7	−0.978137	−0.489069	−	0.872245i	$-0.662663\pi$
$887$	−2.29197e7	−0.978137	−0.489069	+	0.872245i	$0.662663\pi$
$888$	9.38424e6	0.399362
$889$	1.42387e7	0.604250
$890$	0	0
$891$	−793881.	−0.0335013
$892$	−4.27630e6	−0.179952
$893$	−1.40865e7	−0.591117
$894$	1.52335e7	0.637464
$895$	0	0
$896$	−1.57920e7	−0.657156
$897$	−5.51389e6	−0.228811
$898$	−1.39579e7	−0.577605
$899$	337039.	0.0139085
$900$	0	0
$901$	−6.49487e6	−0.266538
$902$	−1.13175e7	−0.463163
$903$	7.57155e6	0.309005
$904$	1.30517e7	0.531184
$905$	0	0
$906$	7.67313e6	0.310565
$907$	2.66664e7	1.07633	0.538167	−	0.842838i	$-0.319117\pi$
$907$	2.66664e7	1.07633	0.538167	+	0.842838i	$0.319117\pi$
$908$	−2.89036e6	−0.116342
$909$	1.72524e6	0.0692533
$910$	0	0
$911$	1.73286e7	0.691778	0.345889	−	0.938276i	$-0.387577\pi$
$911$	1.73286e7	0.691778	0.345889	+	0.938276i	$0.387577\pi$
$912$	−6.97468e6	−0.277675
$913$	−1.32935e7	−0.527793
$914$	5.91518e6	0.234208
$915$	0	0
$916$	3.30005e6	0.129952
$917$	5.34192e6	0.209785
$918$	1.59505e6	0.0624696
$919$	−1.72198e7	−0.672573	−0.336286	−	0.941760i	$-0.609171\pi$
$919$	−1.72198e7	−0.672573	−0.336286	+	0.941760i	$0.609171\pi$
$920$	0	0
$921$	1.21348e7	0.471393
$922$	−2.78058e7	−1.07723
$923$	−1.40585e7	−0.543168
$924$	−698677.	−0.0269213
$925$	0	0
$926$	−1.96848e7	−0.754402
$927$	530393.	0.0202721
$928$	3.93183e6	0.149874
$929$	5.47137e6	0.207997	0.103998	−	0.994577i	$-0.466836\pi$
$929$	5.47137e6	0.207997	0.103998	+	0.994577i	$0.466836\pi$
$930$	0	0
$931$	1.46494e6	0.0553919
$932$	−4.71193e6	−0.177688
$933$	2.27326e7	0.854960
$934$	4.87525e7	1.82865
$935$	0	0
$936$	7.80014e6	0.291013
$937$	−8.04892e6	−0.299494	−0.149747	−	0.988724i	$-0.547846\pi$
$937$	−8.04892e6	−0.299494	−0.149747	+	0.988724i	$0.547846\pi$
$938$	9.27072e6	0.344038
$939$	2.00755e7	0.743022
$940$	0	0
$941$	−1.13721e7	−0.418665	−0.209333	−	0.977844i	$-0.567129\pi$
$941$	−1.13721e7	−0.418665	−0.209333	+	0.977844i	$0.567129\pi$
$942$	−1.15842e7	−0.425341
$943$	−2.21349e7	−0.810584
$944$	−1.94379e7	−0.709938
$945$	0	0
$946$	−4.26941e6	−0.155110
$947$	−4.97017e7	−1.80093	−0.900463	−	0.434932i	$-0.856772\pi$
$947$	−4.97017e7	−1.80093	−0.900463	+	0.434932i	$0.856772\pi$
$948$	−381267.	−0.0137787
$949$	1.44199e7	0.519751
$950$	0	0
$951$	−1.08817e6	−0.0390162
$952$	1.00464e7	0.359266
$953$	−2.95809e7	−1.05507	−0.527533	−	0.849534i	$-0.676883\pi$
$953$	−2.95809e7	−1.05507	−0.527533	+	0.849534i	$0.676883\pi$
$954$	6.44438e6	0.229251
$955$	0	0
$956$	−3.44385e6	−0.121871
$957$	−2.29955e6	−0.0811639
$958$	2.42830e7	0.854849
$959$	−4.95526e7	−1.73988
$960$	0	0
$961$	−2.86037e7	−0.999110
$962$	1.40171e7	0.488337
$963$	−1.03008e7	−0.357937
$964$	6.85161e6	0.237465
$965$	0	0
$966$	7.04654e6	0.242959
$967$	−3.10475e7	−1.06773	−0.533863	−	0.845571i	$-0.679260\pi$
$967$	−3.10475e7	−1.06773	−0.533863	+	0.845571i	$0.679260\pi$
$968$	2.81950e6	0.0967128
$969$	3.54863e6	0.121409
$970$	0	0
$971$	−1.89426e7	−0.644751	−0.322376	−	0.946612i	$-0.604481\pi$
$971$	−1.89426e7	−0.644751	−0.322376	+	0.946612i	$0.604481\pi$
$972$	306912.	0.0104195
$973$	−3.72803e7	−1.26240
$974$	3.66113e7	1.23657
$975$	0	0
$976$	−8.94489e6	−0.300573
$977$	−5.49466e7	−1.84164	−0.920819	−	0.389989i	$-0.872479\pi$
$977$	−5.49466e7	−1.84164	−0.920819	+	0.389989i	$0.872479\pi$
$978$	1.94602e7	0.650580
$979$	−8.43050e6	−0.281123
$980$	0	0
$981$	4.64010e6	0.153941
$982$	−4.69072e6	−0.155225
$983$	4.72203e7	1.55864	0.779318	−	0.626628i	$-0.215565\pi$
$983$	4.72203e7	1.55864	0.779318	+	0.626628i	$0.215565\pi$
$984$	3.13128e7	1.03094
$985$	0	0
$986$	4.62022e6	0.151346
$987$	−1.67739e7	−0.548076
$988$	2.42480e6	0.0790283
$989$	−8.35018e6	−0.271459
$990$	0	0
$991$	4.48844e7	1.45182	0.725908	−	0.687792i	$-0.241420\pi$
$991$	4.48844e7	1.45182	0.725908	+	0.687792i	$0.241420\pi$
$992$	297198.	0.00958885
$993$	−5.20632e6	−0.167555
$994$	1.79662e7	0.576753
$995$	0	0
$996$	5.13925e6	0.164154
$997$	−5.08797e7	−1.62109	−0.810543	−	0.585679i	$-0.800828\pi$
$997$	−5.08797e7	−1.62109	−0.810543	+	0.585679i	$0.800828\pi$
$998$	3.69295e7	1.17367
$999$	3.94714e6	0.125132

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.j.1.1		3
5.4	even	2		165.6.a.a.1.3	✓	3
15.14	odd	2		495.6.a.e.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.a.1.3	✓	3	5.4	even	2
495.6.a.e.1.1		3	15.14	odd	2
825.6.a.j.1.1		3	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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

\(f(q)\)	\(=\)	\(q-5.17710 q^{2} -9.00000 q^{3} -5.19759 q^{4} +46.5939 q^{6} +123.437 q^{7} +192.576 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-5.17710 q^{2} -9.00000 q^{3} -5.19759 q^{4} +46.5939 q^{6} +123.437 q^{7} +192.576 q^{8} +81.0000 q^{9} -121.000 q^{11} +46.7783 q^{12} +500.053 q^{13} -639.048 q^{14} -830.662 q^{16} +422.631 q^{17} -419.345 q^{18} -932.948 q^{19} -1110.94 q^{21} +626.430 q^{22} +1225.18 q^{23} -1733.18 q^{24} -2588.83 q^{26} -729.000 q^{27} -641.576 q^{28} -2111.62 q^{29} -159.612 q^{31} -1862.00 q^{32} +1089.00 q^{33} -2188.00 q^{34} -421.005 q^{36} -5414.46 q^{37} +4829.97 q^{38} -4500.47 q^{39} -18066.7 q^{41} +5751.43 q^{42} -6815.47 q^{43} +628.908 q^{44} -6342.88 q^{46} +15098.9 q^{47} +7475.96 q^{48} -1570.23 q^{49} -3803.67 q^{51} -2599.07 q^{52} -15367.7 q^{53} +3774.11 q^{54} +23771.0 q^{56} +8396.53 q^{57} +10932.1 q^{58} +23400.5 q^{59} +10768.4 q^{61} +826.328 q^{62} +9998.42 q^{63} +36221.0 q^{64} -5637.87 q^{66} -14507.1 q^{67} -2196.66 q^{68} -11026.6 q^{69} -28114.0 q^{71} +15598.6 q^{72} +28836.7 q^{73} +28031.2 q^{74} +4849.08 q^{76} -14935.9 q^{77} +23299.4 q^{78} -8150.52 q^{79} +6561.00 q^{81} +93533.0 q^{82} +109864. q^{83} +5774.19 q^{84} +35284.4 q^{86} +19004.5 q^{87} -23301.7 q^{88} +69673.6 q^{89} +61725.2 q^{91} -6367.98 q^{92} +1436.51 q^{93} -78168.5 q^{94} +16758.0 q^{96} -91551.4 q^{97} +8129.23 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 7 q^{2} - 27 q^{3} + 25 q^{4} - 63 q^{6} + 172 q^{7} + 231 q^{8} + 243 q^{9}+O(q^{10}) \) 3 * q + 7 * q^2 - 27 * q^3 + 25 * q^4 - 63 * q^6 + 172 * q^7 + 231 * q^8 + 243 * q^9	\( 3 q + 7 q^{2} - 27 q^{3} + 25 q^{4} - 63 q^{6} + 172 q^{7} + 231 q^{8} + 243 q^{9} - 363 q^{11} - 225 q^{12} + 654 q^{13} - 728 q^{14} - 415 q^{16} + 2366 q^{17} + 567 q^{18} - 2872 q^{19} - 1548 q^{21} - 847 q^{22} - 2272 q^{23} - 2079 q^{24} + 3422 q^{26} - 2187 q^{27} - 4592 q^{28} - 7738 q^{29} + 568 q^{31} - 1001 q^{32} + 3267 q^{33} + 2506 q^{34} + 2025 q^{36} + 9126 q^{37} - 13076 q^{38} - 5886 q^{39} - 8758 q^{41} + 6552 q^{42} + 14672 q^{43} - 3025 q^{44} - 28768 q^{46} + 19392 q^{47} + 3735 q^{48} - 26629 q^{49} - 21294 q^{51} + 61506 q^{52} + 4598 q^{53} - 5103 q^{54} + 2688 q^{56} + 25848 q^{57} - 8550 q^{58} - 9348 q^{59} - 60078 q^{61} + 14096 q^{62} + 13932 q^{63} - 7087 q^{64} + 7623 q^{66} + 38468 q^{67} - 59778 q^{68} + 20448 q^{69} - 74032 q^{71} + 18711 q^{72} + 44442 q^{73} + 82542 q^{74} - 98708 q^{76} - 20812 q^{77} - 30798 q^{78} - 108116 q^{79} + 19683 q^{81} + 92230 q^{82} + 81892 q^{83} + 41328 q^{84} + 126412 q^{86} + 69642 q^{87} - 27951 q^{88} + 167342 q^{89} - 31832 q^{91} - 72960 q^{92} - 5112 q^{93} + 12728 q^{94} + 9009 q^{96} - 159702 q^{97} - 163121 q^{98} - 29403 q^{99}+O(q^{100}) \) 3 * q + 7 * q^2 - 27 * q^3 + 25 * q^4 - 63 * q^6 + 172 * q^7 + 231 * q^8 + 243 * q^9 - 363 * q^11 - 225 * q^12 + 654 * q^13 - 728 * q^14 - 415 * q^16 + 2366 * q^17 + 567 * q^18 - 2872 * q^19 - 1548 * q^21 - 847 * q^22 - 2272 * q^23 - 2079 * q^24 + 3422 * q^26 - 2187 * q^27 - 4592 * q^28 - 7738 * q^29 + 568 * q^31 - 1001 * q^32 + 3267 * q^33 + 2506 * q^34 + 2025 * q^36 + 9126 * q^37 - 13076 * q^38 - 5886 * q^39 - 8758 * q^41 + 6552 * q^42 + 14672 * q^43 - 3025 * q^44 - 28768 * q^46 + 19392 * q^47 + 3735 * q^48 - 26629 * q^49 - 21294 * q^51 + 61506 * q^52 + 4598 * q^53 - 5103 * q^54 + 2688 * q^56 + 25848 * q^57 - 8550 * q^58 - 9348 * q^59 - 60078 * q^61 + 14096 * q^62 + 13932 * q^63 - 7087 * q^64 + 7623 * q^66 + 38468 * q^67 - 59778 * q^68 + 20448 * q^69 - 74032 * q^71 + 18711 * q^72 + 44442 * q^73 + 82542 * q^74 - 98708 * q^76 - 20812 * q^77 - 30798 * q^78 - 108116 * q^79 + 19683 * q^81 + 92230 * q^82 + 81892 * q^83 + 41328 * q^84 + 126412 * q^86 + 69642 * q^87 - 27951 * q^88 + 167342 * q^89 - 31832 * q^91 - 72960 * q^92 - 5112 * q^93 + 12728 * q^94 + 9009 * q^96 - 159702 * q^97 - 163121 * q^98 - 29403 * q^99

Introduction

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Modular forms

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Database

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