LMFDB - Embedded newform 825.6.a.m.1.5

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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 152x^{5} + 358x^{4} + 5771x^{3} - 13444x^{2} - 51316x + 92576 $ x^7 - 2x^6 - 152x^5 + 358x^4 + 5771x^3 - 13444x^2 - 51316x + 92576
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-3.33276$ 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)

$f(q)$	$=$	$q+2.33276 q^{2} +9.00000 q^{3} -26.5582 q^{4} +20.9949 q^{6} -146.260 q^{7} -136.602 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+2.33276 q^{2} +9.00000 q^{3} -26.5582 q^{4} +20.9949 q^{6} -146.260 q^{7} -136.602 q^{8} +81.0000 q^{9} +121.000 q^{11} -239.024 q^{12} +158.241 q^{13} -341.189 q^{14} +531.202 q^{16} +1926.04 q^{17} +188.954 q^{18} -32.9364 q^{19} -1316.34 q^{21} +282.264 q^{22} -2077.06 q^{23} -1229.42 q^{24} +369.139 q^{26} +729.000 q^{27} +3884.40 q^{28} +1641.87 q^{29} -3563.71 q^{31} +5610.45 q^{32} +1089.00 q^{33} +4492.99 q^{34} -2151.22 q^{36} +5396.83 q^{37} -76.8329 q^{38} +1424.17 q^{39} +2803.48 q^{41} -3070.71 q^{42} +1140.23 q^{43} -3213.54 q^{44} -4845.29 q^{46} +15797.7 q^{47} +4780.82 q^{48} +4584.95 q^{49} +17334.4 q^{51} -4202.61 q^{52} -24043.7 q^{53} +1700.58 q^{54} +19979.4 q^{56} -296.428 q^{57} +3830.09 q^{58} +5002.23 q^{59} -4798.86 q^{61} -8313.28 q^{62} -11847.0 q^{63} -3910.63 q^{64} +2540.38 q^{66} -34126.4 q^{67} -51152.2 q^{68} -18693.6 q^{69} +10059.7 q^{71} -11064.8 q^{72} -71743.6 q^{73} +12589.5 q^{74} +874.733 q^{76} -17697.4 q^{77} +3322.25 q^{78} -14305.9 q^{79} +6561.00 q^{81} +6539.84 q^{82} -66464.5 q^{83} +34959.6 q^{84} +2659.88 q^{86} +14776.8 q^{87} -16528.9 q^{88} -11104.6 q^{89} -23144.4 q^{91} +55163.1 q^{92} -32073.4 q^{93} +36852.3 q^{94} +50494.0 q^{96} -56522.6 q^{97} +10695.6 q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 9 q^{2} + 63 q^{3} + 95 q^{4} - 81 q^{6} - 65 q^{7} - 207 q^{8} + 567 q^{9}+O(q^{10}) $ 7 * q - 9 * q^2 + 63 * q^3 + 95 * q^4 - 81 * q^6 - 65 * q^7 - 207 * q^8 + 567 * q^9	\( 7 q - 9 q^{2} + 63 q^{3} + 95 q^{4} - 81 q^{6} - 65 q^{7} - 207 q^{8} + 567 q^{9} + 847 q^{11} + 855 q^{12} + 113 q^{13} - 1212 q^{14} + 499 q^{16} - 1030 q^{17} - 729 q^{18} - 3803 q^{19} - 585 q^{21} - 1089 q^{22} - 514 q^{23} - 1863 q^{24} - 12111 q^{26} + 5103 q^{27} + 342 q^{28} - 2698 q^{29} - 17233 q^{31} - 9943 q^{32} + 7623 q^{33} + 4090 q^{34} + 7695 q^{36} - 23182 q^{37} + 11943 q^{38} + 1017 q^{39} - 16158 q^{41} - 10908 q^{42} + 4249 q^{43} + 11495 q^{44} - 28769 q^{46} - 7580 q^{47} + 4491 q^{48} + 37140 q^{49} - 9270 q^{51} + 23887 q^{52} - 20574 q^{53} - 6561 q^{54} - 73276 q^{56} - 34227 q^{57} - 8733 q^{58} - 364 q^{59} - 28127 q^{61} + 71917 q^{62} - 5265 q^{63} + 43379 q^{64} - 9801 q^{66} + 21493 q^{67} - 160660 q^{68} - 4626 q^{69} - 177084 q^{71} - 16767 q^{72} - 78670 q^{73} + 196750 q^{74} - 32701 q^{76} - 7865 q^{77} - 108999 q^{78} - 187432 q^{79} + 45927 q^{81} - 179552 q^{82} + 44592 q^{83} + 3078 q^{84} - 110433 q^{86} - 24282 q^{87} - 25047 q^{88} - 151168 q^{89} - 230153 q^{91} + 44767 q^{92} - 155097 q^{93} + 54040 q^{94} - 89487 q^{96} + 55589 q^{97} - 478341 q^{98} + 68607 q^{99}+O(q^{100}) \) 7 * q - 9 * q^2 + 63 * q^3 + 95 * q^4 - 81 * q^6 - 65 * q^7 - 207 * q^8 + 567 * q^9 + 847 * q^11 + 855 * q^12 + 113 * q^13 - 1212 * q^14 + 499 * q^16 - 1030 * q^17 - 729 * q^18 - 3803 * q^19 - 585 * q^21 - 1089 * q^22 - 514 * q^23 - 1863 * q^24 - 12111 * q^26 + 5103 * q^27 + 342 * q^28 - 2698 * q^29 - 17233 * q^31 - 9943 * q^32 + 7623 * q^33 + 4090 * q^34 + 7695 * q^36 - 23182 * q^37 + 11943 * q^38 + 1017 * q^39 - 16158 * q^41 - 10908 * q^42 + 4249 * q^43 + 11495 * q^44 - 28769 * q^46 - 7580 * q^47 + 4491 * q^48 + 37140 * q^49 - 9270 * q^51 + 23887 * q^52 - 20574 * q^53 - 6561 * q^54 - 73276 * q^56 - 34227 * q^57 - 8733 * q^58 - 364 * q^59 - 28127 * q^61 + 71917 * q^62 - 5265 * q^63 + 43379 * q^64 - 9801 * q^66 + 21493 * q^67 - 160660 * q^68 - 4626 * q^69 - 177084 * q^71 - 16767 * q^72 - 78670 * q^73 + 196750 * q^74 - 32701 * q^76 - 7865 * q^77 - 108999 * q^78 - 187432 * q^79 + 45927 * q^81 - 179552 * q^82 + 44592 * q^83 + 3078 * q^84 - 110433 * q^86 - 24282 * q^87 - 25047 * q^88 - 151168 * q^89 - 230153 * q^91 + 44767 * q^92 - 155097 * q^93 + 54040 * q^94 - 89487 * q^96 + 55589 * q^97 - 478341 * q^98 + 68607 * 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 $

$2$	2.33276	0.412378	0.206189	−	0.978512i	$-0.433894\pi$
$2$	2.33276	0.412378	0.206189	+	0.978512i	$0.433894\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−26.5582	−0.829944
$5$	0	0
$5$	0	0
$6$	20.9949	0.238087
$7$	−146.260	−1.12818	−0.564092	−	0.825712i	$-0.690774\pi$
$7$	−146.260	−1.12818	−0.564092	+	0.825712i	$0.690774\pi$
$8$	−136.602	−0.754629
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−239.024	−0.479169
$13$	158.241	0.259694	0.129847	−	0.991534i	$-0.458551\pi$
$13$	158.241	0.259694	0.129847	+	0.991534i	$0.458551\pi$
$14$	−341.189	−0.465238
$15$	0	0
$16$	531.202	0.518752
$17$	1926.04	1.61638	0.808189	−	0.588923i	$-0.200448\pi$
$17$	1926.04	1.61638	0.808189	+	0.588923i	$0.200448\pi$
$18$	188.954	0.137459
$19$	−32.9364	−0.0209311	−0.0104656	−	0.999945i	$-0.503331\pi$
$19$	−32.9364	−0.0209311	−0.0104656	+	0.999945i	$0.503331\pi$
$20$	0	0
$21$	−1316.34	−0.651358
$22$	282.264	0.124337
$23$	−2077.06	−0.818710	−0.409355	−	0.912375i	$-0.634246\pi$
$23$	−2077.06	−0.818710	−0.409355	+	0.912375i	$0.634246\pi$
$24$	−1229.42	−0.435685
$25$	0	0
$26$	369.139	0.107092
$27$	729.000	0.192450
$28$	3884.40	0.936330
$29$	1641.87	0.362530	0.181265	−	0.983434i	$-0.441981\pi$
$29$	1641.87	0.362530	0.181265	+	0.983434i	$0.441981\pi$
$30$	0	0
$31$	−3563.71	−0.666036	−0.333018	−	0.942920i	$-0.608067\pi$
$31$	−3563.71	−0.666036	−0.333018	+	0.942920i	$0.608067\pi$
$32$	5610.45	0.968551
$33$	1089.00	0.174078
$34$	4492.99	0.666559
$35$	0	0
$36$	−2151.22	−0.276648
$37$	5396.83	0.648089	0.324045	−	0.946042i	$-0.394957\pi$
$37$	5396.83	0.648089	0.324045	+	0.946042i	$0.394957\pi$
$38$	−76.8329	−0.00863154
$39$	1424.17	0.149934
$40$	0	0
$41$	2803.48	0.260458	0.130229	−	0.991484i	$-0.458429\pi$
$41$	2803.48	0.260458	0.130229	+	0.991484i	$0.458429\pi$
$42$	−3070.71	−0.268606
$43$	1140.23	0.0940417	0.0470208	−	0.998894i	$-0.485027\pi$
$43$	1140.23	0.0940417	0.0470208	+	0.998894i	$0.485027\pi$
$44$	−3213.54	−0.250238
$45$	0	0
$46$	−4845.29	−0.337618
$47$	15797.7	1.04316	0.521578	−	0.853204i	$-0.325344\pi$
$47$	15797.7	1.04316	0.521578	+	0.853204i	$0.325344\pi$
$48$	4780.82	0.299502
$49$	4584.95	0.272800
$50$	0	0
$51$	17334.4	0.933216
$52$	−4202.61	−0.215531
$53$	−24043.7	−1.17574	−0.587871	−	0.808955i	$-0.700034\pi$
$53$	−24043.7	−1.17574	−0.587871	+	0.808955i	$0.700034\pi$
$54$	1700.58	0.0793622
$55$	0	0
$56$	19979.4	0.851360
$57$	−296.428	−0.0120846
$58$	3830.09	0.149499
$59$	5002.23	0.187083	0.0935414	−	0.995615i	$-0.470181\pi$
$59$	5002.23	0.187083	0.0935414	+	0.995615i	$0.470181\pi$
$60$	0	0
$61$	−4798.86	−0.165125	−0.0825626	−	0.996586i	$-0.526310\pi$
$61$	−4798.86	−0.165125	−0.0825626	+	0.996586i	$0.526310\pi$
$62$	−8313.28	−0.274659
$63$	−11847.0	−0.376061
$64$	−3910.63	−0.119343
$65$	0	0
$66$	2540.38	0.0717858
$67$	−34126.4	−0.928759	−0.464379	−	0.885636i	$-0.653723\pi$
$67$	−34126.4	−0.928759	−0.464379	+	0.885636i	$0.653723\pi$
$68$	−51152.2	−1.34150
$69$	−18693.6	−0.472683
$70$	0	0
$71$	10059.7	0.236832	0.118416	−	0.992964i	$-0.462218\pi$
$71$	10059.7	0.236832	0.118416	+	0.992964i	$0.462218\pi$
$72$	−11064.8	−0.251543
$73$	−71743.6	−1.57571	−0.787855	−	0.615861i	$-0.788808\pi$
$73$	−71743.6	−1.57571	−0.787855	+	0.615861i	$0.788808\pi$
$74$	12589.5	0.267258
$75$	0	0
$76$	874.733	0.0173717
$77$	−17697.4	−0.340160
$78$	3322.25	0.0618296
$79$	−14305.9	−0.257899	−0.128949	−	0.991651i	$-0.541160\pi$
$79$	−14305.9	−0.257899	−0.128949	+	0.991651i	$0.541160\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	6539.84	0.107407
$83$	−66464.5	−1.05900	−0.529498	−	0.848311i	$-0.677620\pi$
$83$	−66464.5	−1.05900	−0.529498	+	0.848311i	$0.677620\pi$
$84$	34959.6	0.540591
$85$	0	0
$86$	2659.88	0.0387807
$87$	14776.8	0.209307
$88$	−16528.9	−0.227529
$89$	−11104.6	−0.148603	−0.0743017	−	0.997236i	$-0.523673\pi$
$89$	−11104.6	−0.148603	−0.0743017	+	0.997236i	$0.523673\pi$
$90$	0	0
$91$	−23144.4	−0.292982
$92$	55163.1	0.679484
$93$	−32073.4	−0.384536
$94$	36852.3	0.430174
$95$	0	0
$96$	50494.0	0.559193
$97$	−56522.6	−0.609948	−0.304974	−	0.952361i	$-0.598648\pi$
$97$	−56522.6	−0.609948	−0.304974	+	0.952361i	$0.598648\pi$
$98$	10695.6	0.112497
$99$	9801.00	0.100504
$100$	0	0
$101$	−21709.1	−0.211758	−0.105879	−	0.994379i	$-0.533766\pi$
$101$	−21709.1	−0.211758	−0.105879	+	0.994379i	$0.533766\pi$
$102$	40436.9	0.384838
$103$	−15441.9	−0.143419	−0.0717096	−	0.997426i	$-0.522845\pi$
$103$	−15441.9	−0.143419	−0.0717096	+	0.997426i	$0.522845\pi$
$104$	−21616.1	−0.195972
$105$	0	0
$106$	−56088.3	−0.484850
$107$	−33358.3	−0.281673	−0.140836	−	0.990033i	$-0.544979\pi$
$107$	−33358.3	−0.281673	−0.140836	+	0.990033i	$0.544979\pi$
$108$	−19360.9	−0.159723
$109$	−94499.4	−0.761838	−0.380919	−	0.924608i	$-0.624392\pi$
$109$	−94499.4	−0.761838	−0.380919	+	0.924608i	$0.624392\pi$
$110$	0	0
$111$	48571.5	0.374174
$112$	−77693.6	−0.585248
$113$	150985.	1.11234	0.556170	−	0.831069i	$-0.312270\pi$
$113$	150985.	1.11234	0.556170	+	0.831069i	$0.312270\pi$
$114$	−691.496	−0.00498342
$115$	0	0
$116$	−43605.1	−0.300879
$117$	12817.5	0.0865646
$118$	11669.0	0.0771488
$119$	−281702.	−1.82357
$120$	0	0
$121$	14641.0	0.0909091
$122$	−11194.6	−0.0680940
$123$	25231.3	0.150375
$124$	94645.7	0.552773
$125$	0	0
$126$	−27636.3	−0.155079
$127$	−348511.	−1.91737	−0.958687	−	0.284462i	$-0.908185\pi$
$127$	−348511.	−1.91737	−0.958687	+	0.284462i	$0.908185\pi$
$128$	−188657.	−1.01777
$129$	10262.0	0.0542950
$130$	0	0
$131$	59482.3	0.302837	0.151419	−	0.988470i	$-0.451616\pi$
$131$	59482.3	0.302837	0.151419	+	0.988470i	$0.451616\pi$
$132$	−28921.9	−0.144475
$133$	4817.28	0.0236142
$134$	−79608.7	−0.383000
$135$	0	0
$136$	−263102.	−1.21977
$137$	148254.	0.674848	0.337424	−	0.941353i	$-0.390444\pi$
$137$	148254.	0.674848	0.337424	+	0.941353i	$0.390444\pi$
$138$	−43607.7	−0.194924
$139$	−196689.	−0.863462	−0.431731	−	0.902002i	$-0.642097\pi$
$139$	−196689.	−0.863462	−0.431731	+	0.902002i	$0.642097\pi$
$140$	0	0
$141$	142179.	0.602266
$142$	23467.0	0.0976645
$143$	19147.2	0.0783006
$144$	43027.4	0.172917
$145$	0	0
$146$	−167361.	−0.649788
$147$	41264.5	0.157501
$148$	−143330.	−0.537878
$149$	−305815.	−1.12848	−0.564239	−	0.825611i	$-0.690830\pi$
$149$	−305815.	−1.12848	−0.564239	+	0.825611i	$0.690830\pi$
$150$	0	0
$151$	−220420.	−0.786701	−0.393350	−	0.919389i	$-0.628684\pi$
$151$	−220420.	−0.786701	−0.393350	+	0.919389i	$0.628684\pi$
$152$	4499.20	0.0157952
$153$	156009.	0.538793
$154$	−41283.9	−0.140275
$155$	0	0
$156$	−37823.5	−0.124437
$157$	585267.	1.89498	0.947491	−	0.319782i	$-0.103610\pi$
$157$	585267.	1.89498	0.947491	+	0.319782i	$0.103610\pi$
$158$	−33372.4	−0.106352
$159$	−216393.	−0.678815
$160$	0	0
$161$	303791.	0.923656
$162$	15305.3	0.0458198
$163$	−109151.	−0.321780	−0.160890	−	0.986972i	$-0.551436\pi$
$163$	−109151.	−0.321780	−0.160890	+	0.986972i	$0.551436\pi$
$164$	−74455.3	−0.216165
$165$	0	0
$166$	−155046.	−0.436707
$167$	243334.	0.675168	0.337584	−	0.941295i	$-0.390390\pi$
$167$	243334.	0.675168	0.337584	+	0.941295i	$0.390390\pi$
$168$	179815.	0.491533
$169$	−346253.	−0.932559
$170$	0	0
$171$	−2667.85	−0.00697704
$172$	−30282.4	−0.0780494
$173$	398905.	1.01334	0.506669	−	0.862141i	$-0.330877\pi$
$173$	398905.	1.01334	0.506669	+	0.862141i	$0.330877\pi$
$174$	34470.8	0.0863134
$175$	0	0
$176$	64275.5	0.156410
$177$	45020.1	0.108012
$178$	−25904.5	−0.0612808
$179$	198924.	0.464039	0.232019	−	0.972711i	$-0.425467\pi$
$179$	198924.	0.464039	0.232019	+	0.972711i	$0.425467\pi$
$180$	0	0
$181$	−474113.	−1.07569	−0.537843	−	0.843045i	$-0.680761\pi$
$181$	−474113.	−1.07569	−0.537843	+	0.843045i	$0.680761\pi$
$182$	−53990.3	−0.120819
$183$	−43189.7	−0.0953350
$184$	283732.	0.617822
$185$	0	0
$186$	−74819.5	−0.158574
$187$	233051.	0.487356
$188$	−419559.	−0.865761
$189$	−106623.	−0.217119
$190$	0	0
$191$	201548.	0.399757	0.199878	−	0.979821i	$-0.435945\pi$
$191$	201548.	0.399757	0.199878	+	0.979821i	$0.435945\pi$
$192$	−35195.7	−0.0689028
$193$	−205191.	−0.396519	−0.198260	−	0.980150i	$-0.563529\pi$
$193$	−205191.	−0.396519	−0.198260	+	0.980150i	$0.563529\pi$
$194$	−131854.	−0.251529
$195$	0	0
$196$	−121768.	−0.226409
$197$	99895.6	0.183392	0.0916961	−	0.995787i	$-0.470771\pi$
$197$	99895.6	0.183392	0.0916961	+	0.995787i	$0.470771\pi$
$198$	22863.4	0.0414455
$199$	−371365.	−0.664765	−0.332382	−	0.943145i	$-0.607852\pi$
$199$	−371365.	−0.664765	−0.332382	+	0.943145i	$0.607852\pi$
$200$	0	0
$201$	−307137.	−0.536219
$202$	−50642.3	−0.0873242
$203$	−240139.	−0.409000
$204$	−460370.	−0.774518
$205$	0	0
$206$	−36022.3	−0.0591429
$207$	−168242.	−0.272903
$208$	84058.1	0.134717
$209$	−3985.31	−0.00631097
$210$	0	0
$211$	−285877.	−0.442052	−0.221026	−	0.975268i	$-0.570940\pi$
$211$	−285877.	−0.442052	−0.221026	+	0.975268i	$0.570940\pi$
$212$	638558.	0.975800
$213$	90537.7	0.136735
$214$	−77817.0	−0.116156
$215$	0	0
$216$	−99583.2	−0.145228
$217$	521227.	0.751412
$218$	−220445.	−0.314165
$219$	−645693.	−0.909737
$220$	0	0
$221$	304779.	0.419763
$222$	113306.	0.154301
$223$	−1.12395e6	−1.51351	−0.756756	−	0.653698i	$-0.773217\pi$
$223$	−1.12395e6	−1.51351	−0.756756	+	0.653698i	$0.773217\pi$
$224$	−820583.	−1.09270
$225$	0	0
$226$	352212.	0.458705
$227$	−998160.	−1.28569	−0.642844	−	0.765997i	$-0.722246\pi$
$227$	−998160.	−1.28569	−0.642844	+	0.765997i	$0.722246\pi$
$228$	7872.60	0.0100295
$229$	−964804.	−1.21577	−0.607884	−	0.794026i	$-0.707981\pi$
$229$	−964804.	−1.21577	−0.607884	+	0.794026i	$0.707981\pi$
$230$	0	0
$231$	−159277.	−0.196392
$232$	−224283.	−0.273575
$233$	−550785.	−0.664649	−0.332324	−	0.943165i	$-0.607833\pi$
$233$	−550785.	−0.664649	−0.332324	+	0.943165i	$0.607833\pi$
$234$	29900.3	0.0356973
$235$	0	0
$236$	−132850.	−0.155268
$237$	−128754.	−0.148898
$238$	−657144.	−0.752001
$239$	−413388.	−0.468127	−0.234063	−	0.972221i	$-0.575202\pi$
$239$	−413388.	−0.468127	−0.234063	+	0.972221i	$0.575202\pi$
$240$	0	0
$241$	213029.	0.236263	0.118132	−	0.992998i	$-0.462310\pi$
$241$	213029.	0.236263	0.118132	+	0.992998i	$0.462310\pi$
$242$	34154.0	0.0374889
$243$	59049.0	0.0641500
$244$	127449.	0.137045
$245$	0	0
$246$	58858.6	0.0620115
$247$	−5211.90	−0.00543568
$248$	486811.	0.502610
$249$	−598181.	−0.611412
$250$	0	0
$251$	1.62034e6	1.62339	0.811694	−	0.584082i	$-0.198546\pi$
$251$	1.62034e6	1.62339	0.811694	+	0.584082i	$0.198546\pi$
$252$	314637.	0.312110
$253$	−251325.	−0.246850
$254$	−812993.	−0.790683
$255$	0	0
$256$	−314951.	−0.300361
$257$	1.03963e6	0.981849	0.490924	−	0.871202i	$-0.336659\pi$
$257$	1.03963e6	0.981849	0.490924	+	0.871202i	$0.336659\pi$
$258$	23938.9	0.0223901
$259$	−789340.	−0.731164
$260$	0	0
$261$	132991.	0.120843
$262$	138758.	0.124883
$263$	207828.	0.185274	0.0926372	−	0.995700i	$-0.470470\pi$
$263$	207828.	0.185274	0.0926372	+	0.995700i	$0.470470\pi$
$264$	−148760.	−0.131364
$265$	0	0
$266$	11237.6	0.00973796
$267$	−99941.6	−0.0857963
$268$	906335.	0.770818
$269$	302188.	0.254622	0.127311	−	0.991863i	$-0.459365\pi$
$269$	302188.	0.254622	0.127311	+	0.991863i	$0.459365\pi$
$270$	0	0
$271$	1.17189e6	0.969310	0.484655	−	0.874705i	$-0.338945\pi$
$271$	1.17189e6	0.969310	0.484655	+	0.874705i	$0.338945\pi$
$272$	1.02312e6	0.838499
$273$	−208299.	−0.169153
$274$	345842.	0.278292
$275$	0	0
$276$	496468.	0.392300
$277$	2.15093e6	1.68433	0.842164	−	0.539221i	$-0.181281\pi$
$277$	2.15093e6	1.68433	0.842164	+	0.539221i	$0.181281\pi$
$278$	−458829.	−0.356073
$279$	−288660.	−0.222012
$280$	0	0
$281$	−2.31245e6	−1.74705	−0.873527	−	0.486775i	$-0.838173\pi$
$281$	−2.31245e6	−1.74705	−0.873527	+	0.486775i	$0.838173\pi$
$282$	331670.	0.248361
$283$	−655859.	−0.486793	−0.243397	−	0.969927i	$-0.578262\pi$
$283$	−655859.	−0.486793	−0.243397	+	0.969927i	$0.578262\pi$
$284$	−267169.	−0.196558
$285$	0	0
$286$	44665.9	0.0322894
$287$	−410036.	−0.293844
$288$	454446.	0.322850
$289$	2.28977e6	1.61268
$290$	0	0
$291$	−508703.	−0.352154
$292$	1.90538e6	1.30775
$293$	602815.	0.410218	0.205109	−	0.978739i	$-0.434245\pi$
$293$	602815.	0.410218	0.205109	+	0.978739i	$0.434245\pi$
$294$	96260.3	0.0649500
$295$	0	0
$296$	−737221.	−0.489067
$297$	88209.0	0.0580259
$298$	−713394.	−0.465360
$299$	−328677.	−0.212614
$300$	0	0
$301$	−166770.	−0.106096
$302$	−514189.	−0.324418
$303$	−195382.	−0.122258
$304$	−17495.9	−0.0108581
$305$	0	0
$306$	363932.	0.222186
$307$	62538.0	0.0378703	0.0189351	−	0.999821i	$-0.493972\pi$
$307$	62538.0	0.0378703	0.0189351	+	0.999821i	$0.493972\pi$
$308$	470013.	0.282314
$309$	−138977.	−0.0828031
$310$	0	0
$311$	−1.16927e6	−0.685510	−0.342755	−	0.939425i	$-0.611360\pi$
$311$	−1.16927e6	−0.685510	−0.342755	+	0.939425i	$0.611360\pi$
$312$	−194545.	−0.113145
$313$	−2.51021e6	−1.44827	−0.724135	−	0.689659i	$-0.757761\pi$
$313$	−2.51021e6	−1.44827	−0.724135	+	0.689659i	$0.757761\pi$
$314$	1.36529e6	0.781449
$315$	0	0
$316$	379940.	0.214041
$317$	1.58370e6	0.885167	0.442584	−	0.896727i	$-0.354062\pi$
$317$	1.58370e6	0.885167	0.442584	+	0.896727i	$0.354062\pi$
$318$	−504794.	−0.279928
$319$	198666.	0.109307
$320$	0	0
$321$	−300225.	−0.162624
$322$	708672.	0.380895
$323$	−63436.9	−0.0338326
$324$	−174248.	−0.0922160
$325$	0	0
$326$	−254623.	−0.132695
$327$	−850495.	−0.439848
$328$	−382961.	−0.196549
$329$	−2.31057e6	−1.17687
$330$	0	0
$331$	−2.73382e6	−1.37151	−0.685757	−	0.727831i	$-0.740529\pi$
$331$	−2.73382e6	−1.37151	−0.685757	+	0.727831i	$0.740529\pi$
$332$	1.76518e6	0.878908
$333$	437144.	0.216030
$334$	567641.	0.278424
$335$	0	0
$336$	−699242.	−0.337893
$337$	3.08115e6	1.47787	0.738937	−	0.673774i	$-0.235328\pi$
$337$	3.08115e6	1.47787	0.738937	+	0.673774i	$0.235328\pi$
$338$	−807725.	−0.384567
$339$	1.35886e6	0.642210
$340$	0	0
$341$	−431209.	−0.200817
$342$	−6223.46	−0.00287718
$343$	1.78760e6	0.820416
$344$	−155758.	−0.0709666
$345$	0	0
$346$	930551.	0.417878
$347$	1.31963e6	0.588338	0.294169	−	0.955753i	$-0.404957\pi$
$347$	1.31963e6	0.588338	0.294169	+	0.955753i	$0.404957\pi$
$348$	−392446.	−0.173713
$349$	−4.16706e6	−1.83133	−0.915664	−	0.401944i	$-0.868335\pi$
$349$	−4.16706e6	−1.83133	−0.915664	+	0.401944i	$0.868335\pi$
$350$	0	0
$351$	115358.	0.0499781
$352$	678864.	0.292029
$353$	−3.56475e6	−1.52262	−0.761310	−	0.648388i	$-0.775444\pi$
$353$	−3.56475e6	−1.52262	−0.761310	+	0.648388i	$0.775444\pi$
$354$	105021.	0.0445419
$355$	0	0
$356$	294919.	0.123333
$357$	−2.53532e6	−1.05284
$358$	464042.	0.191359
$359$	−402049.	−0.164643	−0.0823215	−	0.996606i	$-0.526233\pi$
$359$	−402049.	−0.164643	−0.0823215	+	0.996606i	$0.526233\pi$
$360$	0	0
$361$	−2.47501e6	−0.999562
$362$	−1.10599e6	−0.443589
$363$	131769.	0.0524864
$364$	614673.	0.243159
$365$	0	0
$366$	−100751.	−0.0393141
$367$	−1.69899e6	−0.658455	−0.329227	−	0.944251i	$-0.606788\pi$
$367$	−1.69899e6	−0.658455	−0.329227	+	0.944251i	$0.606788\pi$
$368$	−1.10334e6	−0.424708
$369$	227081.	0.0868192
$370$	0	0
$371$	3.51663e6	1.32645
$372$	851812.	0.319144
$373$	1.78079e6	0.662738	0.331369	−	0.943501i	$-0.392490\pi$
$373$	1.78079e6	0.662738	0.331369	+	0.943501i	$0.392490\pi$
$374$	543652.	0.200975
$375$	0	0
$376$	−2.15800e6	−0.787195
$377$	259811.	0.0941466
$378$	−248727.	−0.0895352
$379$	−2.12968e6	−0.761580	−0.380790	−	0.924662i	$-0.624348\pi$
$379$	−2.12968e6	−0.761580	−0.380790	+	0.924662i	$0.624348\pi$
$380$	0	0
$381$	−3.13660e6	−1.10700
$382$	470165.	0.164851
$383$	−3.06780e6	−1.06864	−0.534318	−	0.845284i	$-0.679432\pi$
$383$	−3.06780e6	−1.06864	−0.534318	+	0.845284i	$0.679432\pi$
$384$	−1.69791e6	−0.587607
$385$	0	0
$386$	−478661.	−0.163516
$387$	92358.4	0.0313472
$388$	1.50114e6	0.506223
$389$	784173.	0.262747	0.131374	−	0.991333i	$-0.458061\pi$
$389$	784173.	0.262747	0.131374	+	0.991333i	$0.458061\pi$
$390$	0	0
$391$	−4.00051e6	−1.32335
$392$	−626315.	−0.205863
$393$	535340.	0.174843
$394$	233033.	0.0756269
$395$	0	0
$396$	−260297.	−0.0834125
$397$	−4.54374e6	−1.44690	−0.723448	−	0.690379i	$-0.757444\pi$
$397$	−4.54374e6	−1.44690	−0.723448	+	0.690379i	$0.757444\pi$
$398$	−866306.	−0.274134
$399$	43355.5	0.0136336
$400$	0	0
$401$	3.45110e6	1.07176	0.535879	−	0.844295i	$-0.319980\pi$
$401$	3.45110e6	1.07176	0.535879	+	0.844295i	$0.319980\pi$
$402$	−716478.	−0.221125
$403$	−563926.	−0.172965
$404$	576556.	0.175747
$405$	0	0
$406$	−560188.	−0.168663
$407$	653017.	0.195406
$408$	−2.36791e6	−0.704232
$409$	3.80281e6	1.12408	0.562039	−	0.827111i	$-0.310017\pi$
$409$	3.80281e6	1.12408	0.562039	+	0.827111i	$0.310017\pi$
$410$	0	0
$411$	1.33429e6	0.389624
$412$	410109.	0.119030
$413$	−731626.	−0.211064
$414$	−392469.	−0.112539
$415$	0	0
$416$	887804.	0.251527
$417$	−1.77020e6	−0.498520
$418$	−9296.78	−0.00260251
$419$	−2.60701e6	−0.725450	−0.362725	−	0.931896i	$-0.618153\pi$
$419$	−2.60701e6	−0.725450	−0.362725	+	0.931896i	$0.618153\pi$
$420$	0	0
$421$	−5.05826e6	−1.39090	−0.695449	−	0.718575i	$-0.744795\pi$
$421$	−5.05826e6	−1.39090	−0.695449	+	0.718575i	$0.744795\pi$
$422$	−666883.	−0.182292
$423$	1.27961e6	0.347718
$424$	3.28443e6	0.887248
$425$	0	0
$426$	211203.	0.0563866
$427$	701880.	0.186292
$428$	885937.	0.233773
$429$	172325.	0.0452069
$430$	0	0
$431$	−3.62455e6	−0.939856	−0.469928	−	0.882705i	$-0.655720\pi$
$431$	−3.62455e6	−0.939856	−0.469928	+	0.882705i	$0.655720\pi$
$432$	387246.	0.0998339
$433$	6.53089e6	1.67399	0.836995	−	0.547211i	$-0.184310\pi$
$433$	6.53089e6	1.67399	0.836995	+	0.547211i	$0.184310\pi$
$434$	1.21590e6	0.309866
$435$	0	0
$436$	2.50974e6	0.632283
$437$	68411.1	0.0171365
$438$	−1.50625e6	−0.375155
$439$	−4.25980e6	−1.05494	−0.527471	−	0.849573i	$-0.676860\pi$
$439$	−4.25980e6	−1.05494	−0.527471	+	0.849573i	$0.676860\pi$
$440$	0	0
$441$	371381.	0.0909333
$442$	710977.	0.173101
$443$	−1.87861e6	−0.454807	−0.227404	−	0.973801i	$-0.573024\pi$
$443$	−1.87861e6	−0.454807	−0.227404	+	0.973801i	$0.573024\pi$
$444$	−1.28997e6	−0.310544
$445$	0	0
$446$	−2.62191e6	−0.624139
$447$	−2.75234e6	−0.651527
$448$	571969.	0.134641
$449$	−296649.	−0.0694428	−0.0347214	−	0.999397i	$-0.511054\pi$
$449$	−296649.	−0.0694428	−0.0347214	+	0.999397i	$0.511054\pi$
$450$	0	0
$451$	339220.	0.0785309
$452$	−4.00989e6	−0.923180
$453$	−1.98378e6	−0.454202
$454$	−2.32847e6	−0.530189
$455$	0	0
$456$	40492.8	0.00911938
$457$	2.30010e6	0.515177	0.257588	−	0.966255i	$-0.417072\pi$
$457$	2.30010e6	0.515177	0.257588	+	0.966255i	$0.417072\pi$
$458$	−2.25066e6	−0.501356
$459$	1.40408e6	0.311072
$460$	0	0
$461$	7.72547e6	1.69306	0.846530	−	0.532340i	$-0.178687\pi$
$461$	7.72547e6	1.69306	0.846530	+	0.532340i	$0.178687\pi$
$462$	−371555.	−0.0809876
$463$	6.54474e6	1.41886	0.709430	−	0.704776i	$-0.248952\pi$
$463$	6.54474e6	1.41886	0.709430	+	0.704776i	$0.248952\pi$
$464$	872164.	0.188063
$465$	0	0
$466$	−1.28485e6	−0.274087
$467$	−6.94728e6	−1.47409	−0.737043	−	0.675846i	$-0.763778\pi$
$467$	−6.94728e6	−1.47409	−0.737043	+	0.675846i	$0.763778\pi$
$468$	−340411.	−0.0718438
$469$	4.99132e6	1.04781
$470$	0	0
$471$	5.26741e6	1.09407
$472$	−683317.	−0.141178
$473$	137968.	0.0283546
$474$	−300351.	−0.0614022
$475$	0	0
$476$	7.48151e6	1.51346
$477$	−1.94754e6	−0.391914
$478$	−964336.	−0.193045
$479$	−3.98195e6	−0.792971	−0.396485	−	0.918041i	$-0.629770\pi$
$479$	−3.98195e6	−0.792971	−0.396485	+	0.918041i	$0.629770\pi$
$480$	0	0
$481$	854002.	0.168305
$482$	496946.	0.0974298
$483$	2.73412e6	0.533273
$484$	−388839.	−0.0754495
$485$	0	0
$486$	137747.	0.0264541
$487$	−2.10449e6	−0.402090	−0.201045	−	0.979582i	$-0.564434\pi$
$487$	−2.10449e6	−0.402090	−0.201045	+	0.979582i	$0.564434\pi$
$488$	655535.	0.124608
$489$	−982358.	−0.185779
$490$	0	0
$491$	−4.60384e6	−0.861820	−0.430910	−	0.902395i	$-0.641807\pi$
$491$	−4.60384e6	−0.861820	−0.430910	+	0.902395i	$0.641807\pi$
$492$	−670098.	−0.124803
$493$	3.16230e6	0.585985
$494$	−12158.1	−0.00224156
$495$	0	0
$496$	−1.89305e6	−0.345508
$497$	−1.47134e6	−0.267191
$498$	−1.39541e6	−0.252133
$499$	−2.17027e6	−0.390177	−0.195089	−	0.980786i	$-0.562499\pi$
$499$	−2.17027e6	−0.390177	−0.195089	+	0.980786i	$0.562499\pi$
$500$	0	0
$501$	2.19001e6	0.389808
$502$	3.77987e6	0.669450
$503$	1.16557e6	0.205409	0.102705	−	0.994712i	$-0.467250\pi$
$503$	1.16557e6	0.205409	0.102705	+	0.994712i	$0.467250\pi$
$504$	1.61834e6	0.283787
$505$	0	0
$506$	−586281.	−0.101796
$507$	−3.11627e6	−0.538413
$508$	9.25583e6	1.59131
$509$	498433.	0.0852732	0.0426366	−	0.999091i	$-0.486424\pi$
$509$	498433.	0.0852732	0.0426366	+	0.999091i	$0.486424\pi$
$510$	0	0
$511$	1.04932e7	1.77769
$512$	5.30231e6	0.893903
$513$	−24010.7	−0.00402820
$514$	2.42520e6	0.404893
$515$	0	0
$516$	−272542.	−0.0450618
$517$	1.91152e6	0.314523
$518$	−1.84134e6	−0.301516
$519$	3.59015e6	0.585051
$520$	0	0
$521$	1.11019e6	0.179186	0.0895929	−	0.995978i	$-0.471443\pi$
$521$	1.11019e6	0.179186	0.0895929	+	0.995978i	$0.471443\pi$
$522$	310237.	0.0498331
$523$	5.52154e6	0.882686	0.441343	−	0.897338i	$-0.354502\pi$
$523$	5.52154e6	0.882686	0.441343	+	0.897338i	$0.354502\pi$
$524$	−1.57974e6	−0.251338
$525$	0	0
$526$	484814.	0.0764031
$527$	−6.86384e6	−1.07657
$528$	578479.	0.0903031
$529$	−2.12215e6	−0.329714
$530$	0	0
$531$	405181.	0.0623609
$532$	−127938.	−0.0195984
$533$	443626.	0.0676392
$534$	−233140.	−0.0353805
$535$	0	0
$536$	4.66174e6	0.700868
$537$	1.79032e6	0.267913
$538$	704933.	0.105001
$539$	554779.	0.0822523
$540$	0	0
$541$	−1.17913e7	−1.73208	−0.866039	−	0.499977i	$-0.833342\pi$
$541$	−1.17913e7	−1.73208	−0.866039	+	0.499977i	$0.833342\pi$
$542$	2.73373e6	0.399722
$543$	−4.26702e6	−0.621047
$544$	1.08059e7	1.56554
$545$	0	0
$546$	−485912.	−0.0697552
$547$	1.11424e7	1.59224	0.796121	−	0.605137i	$-0.206882\pi$
$547$	1.11424e7	1.59224	0.796121	+	0.605137i	$0.206882\pi$
$548$	−3.93737e6	−0.560086
$549$	−388707.	−0.0550417
$550$	0	0
$551$	−54077.3	−0.00758815
$552$	2.55359e6	0.356700
$553$	2.09239e6	0.290957
$554$	5.01761e6	0.694580
$555$	0	0
$556$	5.22371e6	0.716626
$557$	−8.68222e6	−1.18575	−0.592875	−	0.805295i	$-0.702007\pi$
$557$	−8.68222e6	−1.18575	−0.592875	+	0.805295i	$0.702007\pi$
$558$	−673376.	−0.0915529
$559$	180431.	0.0244220
$560$	0	0
$561$	2.09746e6	0.281375
$562$	−5.39439e6	−0.720447
$563$	2.59561e6	0.345118	0.172559	−	0.984999i	$-0.444796\pi$
$563$	2.59561e6	0.345118	0.172559	+	0.984999i	$0.444796\pi$
$564$	−3.77603e6	−0.499847
$565$	0	0
$566$	−1.52996e6	−0.200743
$567$	−959611.	−0.125354
$568$	−1.37419e6	−0.178721
$569$	1.43008e7	1.85174	0.925869	−	0.377846i	$-0.123335\pi$
$569$	1.43008e7	1.85174	0.925869	+	0.377846i	$0.123335\pi$
$570$	0	0
$571$	3.13885e6	0.402884	0.201442	−	0.979500i	$-0.435437\pi$
$571$	3.13885e6	0.402884	0.201442	+	0.979500i	$0.435437\pi$
$572$	−508515.	−0.0649851
$573$	1.81394e6	0.230800
$574$	−956516.	−0.121175
$575$	0	0
$576$	−316761.	−0.0397810
$577$	7.88699e6	0.986215	0.493108	−	0.869968i	$-0.335861\pi$
$577$	7.88699e6	0.986215	0.493108	+	0.869968i	$0.335861\pi$
$578$	5.34149e6	0.665033
$579$	−1.84671e6	−0.228930
$580$	0	0
$581$	9.72109e6	1.19474
$582$	−1.18668e6	−0.145220
$583$	−2.90929e6	−0.354499
$584$	9.80035e6	1.18908
$585$	0	0
$586$	1.40622e6	0.169165
$587$	1.91771e6	0.229714	0.114857	−	0.993382i	$-0.463359\pi$
$587$	1.91771e6	0.229714	0.114857	+	0.993382i	$0.463359\pi$
$588$	−1.09591e6	−0.130717
$589$	117376.	0.0139409
$590$	0	0
$591$	899061.	0.105882
$592$	2.86681e6	0.336198
$593$	1.20632e7	1.40872	0.704362	−	0.709841i	$-0.251233\pi$
$593$	1.20632e7	1.40872	0.704362	+	0.709841i	$0.251233\pi$
$594$	205771.	0.0239286
$595$	0	0
$596$	8.12190e6	0.936574
$597$	−3.34228e6	−0.383802
$598$	−766726.	−0.0876773
$599$	−5.17336e6	−0.589123	−0.294562	−	0.955633i	$-0.595174\pi$
$599$	−5.17336e6	−0.589123	−0.294562	+	0.955633i	$0.595174\pi$
$600$	0	0
$601$	9.40838e6	1.06250	0.531250	−	0.847215i	$-0.321723\pi$
$601$	9.40838e6	1.06250	0.531250	+	0.847215i	$0.321723\pi$
$602$	−389034.	−0.0437518
$603$	−2.76423e6	−0.309586
$604$	5.85398e6	0.652918
$605$	0	0
$606$	−455781.	−0.0504167
$607$	−368372.	−0.0405802	−0.0202901	−	0.999794i	$-0.506459\pi$
$607$	−368372.	−0.0405802	−0.0202901	+	0.999794i	$0.506459\pi$
$608$	−184788.	−0.0202729
$609$	−2.16126e6	−0.236136
$610$	0	0
$611$	2.49985e6	0.270901
$612$	−4.14333e6	−0.447168
$613$	6.57960e6	0.707210	0.353605	−	0.935395i	$-0.384956\pi$
$613$	6.57960e6	0.707210	0.353605	+	0.935395i	$0.384956\pi$
$614$	145886.	0.0156169
$615$	0	0
$616$	2.41751e6	0.256695
$617$	−140601.	−0.0148688	−0.00743441	−	0.999972i	$-0.502366\pi$
$617$	−140601.	−0.0148688	−0.00743441	+	0.999972i	$0.502366\pi$
$618$	−324200.	−0.0341462
$619$	1.69769e7	1.78087	0.890433	−	0.455115i	$-0.150402\pi$
$619$	1.69769e7	1.78087	0.890433	+	0.455115i	$0.150402\pi$
$620$	0	0
$621$	−1.51418e6	−0.157561
$622$	−2.72763e6	−0.282689
$623$	1.62416e6	0.167652
$624$	756523.	0.0777787
$625$	0	0
$626$	−5.85572e6	−0.597234
$627$	−35867.8	−0.00364364
$628$	−1.55437e7	−1.57273
$629$	1.03945e7	1.04756
$630$	0	0
$631$	−1.16982e7	−1.16962	−0.584811	−	0.811170i	$-0.698831\pi$
$631$	−1.16982e7	−1.16962	−0.584811	+	0.811170i	$0.698831\pi$
$632$	1.95423e6	0.194618
$633$	−2.57289e6	−0.255219
$634$	3.69440e6	0.365023
$635$	0	0
$636$	5.74702e6	0.563378
$637$	725528.	0.0708444
$638$	463441.	0.0450757
$639$	814839.	0.0789441
$640$	0	0
$641$	1.21774e6	0.117060	0.0585301	−	0.998286i	$-0.481359\pi$
$641$	1.21774e6	0.117060	0.0585301	+	0.998286i	$0.481359\pi$
$642$	−700353.	−0.0670625
$643$	1.22181e7	1.16541	0.582703	−	0.812685i	$-0.301995\pi$
$643$	1.22181e7	1.16541	0.582703	+	0.812685i	$0.301995\pi$
$644$	−8.06815e6	−0.766583
$645$	0	0
$646$	−147983.	−0.0139518
$647$	−1.38354e7	−1.29936	−0.649682	−	0.760206i	$-0.725098\pi$
$647$	−1.38354e7	−1.29936	−0.649682	+	0.760206i	$0.725098\pi$
$648$	−896248.	−0.0838476
$649$	605270.	0.0564076
$650$	0	0
$651$	4.69105e6	0.433828
$652$	2.89885e6	0.267059
$653$	−7.07482e6	−0.649280	−0.324640	−	0.945838i	$-0.605243\pi$
$653$	−7.07482e6	−0.649280	−0.324640	+	0.945838i	$0.605243\pi$
$654$	−1.98400e6	−0.181383
$655$	0	0
$656$	1.48921e6	0.135113
$657$	−5.81124e6	−0.525237
$658$	−5.39001e6	−0.485316
$659$	−1.42464e7	−1.27788	−0.638942	−	0.769255i	$-0.720627\pi$
$659$	−1.42464e7	−1.27788	−0.638942	+	0.769255i	$0.720627\pi$
$660$	0	0
$661$	−1.44156e7	−1.28330	−0.641651	−	0.766997i	$-0.721750\pi$
$661$	−1.44156e7	−1.28330	−0.641651	+	0.766997i	$0.721750\pi$
$662$	−6.37735e6	−0.565582
$663$	2.74301e6	0.242350
$664$	9.07921e6	0.799149
$665$	0	0
$666$	1.01975e6	0.0890859
$667$	−3.41026e6	−0.296807
$668$	−6.46252e6	−0.560352
$669$	−1.01156e7	−0.873826
$670$	0	0
$671$	−580662.	−0.0497871
$672$	−7.38525e6	−0.630873
$673$	4.41234e6	0.375519	0.187759	−	0.982215i	$-0.439877\pi$
$673$	4.41234e6	0.375519	0.187759	+	0.982215i	$0.439877\pi$
$674$	7.18758e6	0.609443
$675$	0	0
$676$	9.19586e6	0.773972
$677$	−2.15482e7	−1.80692	−0.903461	−	0.428670i	$-0.858982\pi$
$677$	−2.15482e7	−1.80692	−0.903461	+	0.428670i	$0.858982\pi$
$678$	3.16991e6	0.264833
$679$	8.26699e6	0.688134
$680$	0	0
$681$	−8.98344e6	−0.742292
$682$	−1.00591e6	−0.0828127
$683$	1.93575e7	1.58781	0.793903	−	0.608045i	$-0.208046\pi$
$683$	1.93575e7	1.58781	0.793903	+	0.608045i	$0.208046\pi$
$684$	70853.4	0.00579056
$685$	0	0
$686$	4.17004e6	0.338321
$687$	−8.68324e6	−0.701923
$688$	605691.	0.0487843
$689$	−3.80471e6	−0.305333
$690$	0	0
$691$	7.29763e6	0.581416	0.290708	−	0.956812i	$-0.406109\pi$
$691$	7.29763e6	0.581416	0.290708	+	0.956812i	$0.406109\pi$
$692$	−1.05942e7	−0.841014
$693$	−1.43349e6	−0.113387
$694$	3.07837e6	0.242618
$695$	0	0
$696$	−2.01855e6	−0.157949
$697$	5.39960e6	0.420998
$698$	−9.72076e6	−0.755200
$699$	−4.95707e6	−0.383735
$700$	0	0
$701$	1.83574e7	1.41096	0.705481	−	0.708729i	$-0.250731\pi$
$701$	1.83574e7	1.41096	0.705481	+	0.708729i	$0.250731\pi$
$702$	269103.	0.0206099
$703$	−177753.	−0.0135652
$704$	−473187.	−0.0359833
$705$	0	0
$706$	−8.31570e6	−0.627895
$707$	3.17518e6	0.238902
$708$	−1.19565e6	−0.0896442
$709$	749166.	0.0559709	0.0279855	−	0.999608i	$-0.491091\pi$
$709$	749166.	0.0559709	0.0279855	+	0.999608i	$0.491091\pi$
$710$	0	0
$711$	−1.15878e6	−0.0859662
$712$	1.51692e6	0.112140
$713$	7.40205e6	0.545291
$714$	−5.91430e6	−0.434168
$715$	0	0
$716$	−5.28306e6	−0.385126
$717$	−3.72049e6	−0.270273
$718$	−937885.	−0.0678951
$719$	1.65136e7	1.19129	0.595647	−	0.803246i	$-0.296896\pi$
$719$	1.65136e7	1.19129	0.595647	+	0.803246i	$0.296896\pi$
$720$	0	0
$721$	2.25853e6	0.161803
$722$	−5.77362e6	−0.412197
$723$	1.91726e6	0.136407
$724$	1.25916e7	0.892759
$725$	0	0
$726$	307386.	0.0216442
$727$	1.79988e7	1.26301	0.631505	−	0.775372i	$-0.282438\pi$
$727$	1.79988e7	1.26301	0.631505	+	0.775372i	$0.282438\pi$
$728$	3.16157e6	0.221093
$729$	531441.	0.0370370
$730$	0	0
$731$	2.19612e6	0.152007
$732$	1.14704e6	0.0791228
$733$	127875.	0.00879075	0.00439537	−	0.999990i	$-0.498601\pi$
$733$	127875.	0.00879075	0.00439537	+	0.999990i	$0.498601\pi$
$734$	−3.96334e6	−0.271532
$735$	0	0
$736$	−1.16533e7	−0.792962
$737$	−4.12929e6	−0.280031
$738$	529727.	0.0358023
$739$	−7.32976e6	−0.493718	−0.246859	−	0.969051i	$-0.579398\pi$
$739$	−7.32976e6	−0.493718	−0.246859	+	0.969051i	$0.579398\pi$
$740$	0	0
$741$	−46907.1	−0.00313829
$742$	8.20346e6	0.547000
$743$	2.18733e7	1.45359	0.726797	−	0.686852i	$-0.241008\pi$
$743$	2.18733e7	1.45359	0.726797	+	0.686852i	$0.241008\pi$
$744$	4.38130e6	0.290182
$745$	0	0
$746$	4.15417e6	0.273298
$747$	−5.38363e6	−0.352999
$748$	−6.18941e6	−0.404479
$749$	4.87898e6	0.317779
$750$	0	0
$751$	−2.28695e7	−1.47964	−0.739820	−	0.672805i	$-0.765089\pi$
$751$	−2.28695e7	−1.47964	−0.739820	+	0.672805i	$0.765089\pi$
$752$	8.39177e6	0.541139
$753$	1.45831e7	0.937264
$754$	606078.	0.0388240
$755$	0	0
$756$	2.83173e6	0.180197
$757$	1.68271e7	1.06726	0.533629	−	0.845719i	$-0.320828\pi$
$757$	1.68271e7	1.06726	0.533629	+	0.845719i	$0.320828\pi$
$758$	−4.96803e6	−0.314059
$759$	−2.26192e6	−0.142519
$760$	0	0
$761$	−1.72838e7	−1.08188	−0.540938	−	0.841062i	$-0.681931\pi$
$761$	−1.72838e7	−1.08188	−0.540938	+	0.841062i	$0.681931\pi$
$762$	−7.31694e6	−0.456501
$763$	1.38215e7	0.859494
$764$	−5.35277e6	−0.331776
$765$	0	0
$766$	−7.15644e6	−0.440682
$767$	791560.	0.0485842
$768$	−2.83456e6	−0.173413
$769$	1.11705e7	0.681173	0.340587	−	0.940213i	$-0.389374\pi$
$769$	1.11705e7	0.681173	0.340587	+	0.940213i	$0.389374\pi$
$770$	0	0
$771$	9.35664e6	0.566871
$772$	5.44950e6	0.329089
$773$	−1.77002e7	−1.06544	−0.532719	−	0.846292i	$-0.678830\pi$
$773$	−1.77002e7	−1.06544	−0.532719	+	0.846292i	$0.678830\pi$
$774$	215450.	0.0129269
$775$	0	0
$776$	7.72112e6	0.460284
$777$	−7.10406e6	−0.422138
$778$	1.82929e6	0.108351
$779$	−92336.5	−0.00545167
$780$	0	0
$781$	1.21723e6	0.0714076
$782$	−9.33223e6	−0.545718
$783$	1.19692e6	0.0697688
$784$	2.43553e6	0.141515
$785$	0	0
$786$	1.24882e6	0.0721015
$787$	−3.55985e6	−0.204878	−0.102439	−	0.994739i	$-0.532665\pi$
$787$	−3.55985e6	−0.204878	−0.102439	+	0.994739i	$0.532665\pi$
$788$	−2.65305e6	−0.152205
$789$	1.87045e6	0.106968
$790$	0	0
$791$	−2.20830e7	−1.25492
$792$	−1.33884e6	−0.0758430
$793$	−759377.	−0.0428820
$794$	−1.05995e7	−0.596668
$795$	0	0
$796$	9.86279e6	0.551718
$797$	2.09403e7	1.16771	0.583857	−	0.811856i	$-0.301543\pi$
$797$	2.09403e7	1.16771	0.583857	+	0.811856i	$0.301543\pi$
$798$	101138.	0.00562222
$799$	3.04270e7	1.68613
$800$	0	0
$801$	−899475.	−0.0495345
$802$	8.05060e6	0.441970
$803$	−8.68098e6	−0.475094
$804$	8.15702e6	0.445032
$805$	0	0
$806$	−1.31550e6	−0.0713271
$807$	2.71969e6	0.147006
$808$	2.96552e6	0.159799
$809$	−2.22981e7	−1.19784	−0.598918	−	0.800810i	$-0.704402\pi$
$809$	−2.22981e7	−1.19784	−0.598918	+	0.800810i	$0.704402\pi$
$810$	0	0
$811$	−7.95262e6	−0.424578	−0.212289	−	0.977207i	$-0.568092\pi$
$811$	−7.95262e6	−0.424578	−0.212289	+	0.977207i	$0.568092\pi$
$812$	6.37768e6	0.339447
$813$	1.05470e7	0.559631
$814$	1.52333e6	0.0805812
$815$	0	0
$816$	9.20805e6	0.484108
$817$	−37555.0	−0.00196840
$818$	8.87105e6	0.463545
$819$	−1.87469e6	−0.0976608
$820$	0	0
$821$	−1.23384e7	−0.638852	−0.319426	−	0.947611i	$-0.603490\pi$
$821$	−1.23384e7	−0.638852	−0.319426	+	0.947611i	$0.603490\pi$
$822$	3.11258e6	0.160672
$823$	−1.87024e6	−0.0962493	−0.0481247	−	0.998841i	$-0.515324\pi$
$823$	−1.87024e6	−0.0962493	−0.0481247	+	0.998841i	$0.515324\pi$
$824$	2.10940e6	0.108228
$825$	0	0
$826$	−1.70671e6	−0.0870381
$827$	−2.63791e7	−1.34121	−0.670604	−	0.741816i	$-0.733965\pi$
$827$	−2.63791e7	−1.34121	−0.670604	+	0.741816i	$0.733965\pi$
$828$	4.46821e6	0.226495
$829$	2.85869e7	1.44471	0.722355	−	0.691522i	$-0.243060\pi$
$829$	2.85869e7	1.44471	0.722355	+	0.691522i	$0.243060\pi$
$830$	0	0
$831$	1.93584e7	0.972447
$832$	−618824.	−0.0309926
$833$	8.83079e6	0.440948
$834$	−4.12946e6	−0.205579
$835$	0	0
$836$	105843.	0.00523776
$837$	−2.59794e6	−0.128179
$838$	−6.08153e6	−0.299159
$839$	−1.62633e7	−0.797634	−0.398817	−	0.917031i	$-0.630579\pi$
$839$	−1.62633e7	−0.797634	−0.398817	+	0.917031i	$0.630579\pi$
$840$	0	0
$841$	−1.78154e7	−0.868572
$842$	−1.17997e7	−0.573576
$843$	−2.08120e7	−1.00866
$844$	7.59238e6	0.366878
$845$	0	0
$846$	2.98503e6	0.143391
$847$	−2.14139e6	−0.102562
$848$	−1.27721e7	−0.609918
$849$	−5.90273e6	−0.281050
$850$	0	0
$851$	−1.12096e7	−0.530597
$852$	−2.40452e6	−0.113483
$853$	1.29208e6	0.0608021	0.0304010	−	0.999538i	$-0.490322\pi$
$853$	1.29208e6	0.0608021	0.0304010	+	0.999538i	$0.490322\pi$
$854$	1.63732e6	0.0768225
$855$	0	0
$856$	4.55683e6	0.212558
$857$	−8.96034e6	−0.416747	−0.208374	−	0.978049i	$-0.566817\pi$
$857$	−8.96034e6	−0.416747	−0.208374	+	0.978049i	$0.566817\pi$
$858$	401993.	0.0186423
$859$	−4.85453e6	−0.224473	−0.112236	−	0.993682i	$-0.535801\pi$
$859$	−4.85453e6	−0.224473	−0.112236	+	0.993682i	$0.535801\pi$
$860$	0	0
$861$	−3.69032e6	−0.169651
$862$	−8.45522e6	−0.387576
$863$	6.39862e6	0.292455	0.146228	−	0.989251i	$-0.453287\pi$
$863$	6.39862e6	0.292455	0.146228	+	0.989251i	$0.453287\pi$
$864$	4.09001e6	0.186398
$865$	0	0
$866$	1.52350e7	0.690316
$867$	2.06079e7	0.931080
$868$	−1.38429e7	−0.623630
$869$	−1.73102e6	−0.0777593
$870$	0	0
$871$	−5.40020e6	−0.241193
$872$	1.29088e7	0.574905
$873$	−4.57833e6	−0.203316
$874$	159587.	0.00706673
$875$	0	0
$876$	1.71485e7	0.755031
$877$	1.53595e7	0.674340	0.337170	−	0.941444i	$-0.390530\pi$
$877$	1.53595e7	0.674340	0.337170	+	0.941444i	$0.390530\pi$
$878$	−9.93711e6	−0.435035
$879$	5.42533e6	0.236840
$880$	0	0
$881$	−2.14534e7	−0.931228	−0.465614	−	0.884988i	$-0.654167\pi$
$881$	−2.14534e7	−0.931228	−0.465614	+	0.884988i	$0.654167\pi$
$882$	866343.	0.0374989
$883$	−3.30136e6	−0.142492	−0.0712460	−	0.997459i	$-0.522698\pi$
$883$	−3.30136e6	−0.142492	−0.0712460	+	0.997459i	$0.522698\pi$
$884$	−8.09439e6	−0.348380
$885$	0	0
$886$	−4.38235e6	−0.187553
$887$	−1.34325e7	−0.573256	−0.286628	−	0.958042i	$-0.592534\pi$
$887$	−1.34325e7	−0.573256	−0.286628	+	0.958042i	$0.592534\pi$
$888$	−6.63498e6	−0.282363
$889$	5.09732e7	2.16315
$890$	0	0
$891$	793881.	0.0335013
$892$	2.98502e7	1.25613
$893$	−520320.	−0.0218344
$894$	−6.42054e6	−0.268675
$895$	0	0
$896$	2.75929e7	1.14823
$897$	−2.95809e6	−0.122753
$898$	−692011.	−0.0286367
$899$	−5.85114e6	−0.241458
$900$	0	0
$901$	−4.63091e7	−1.90044
$902$	791321.	0.0323844
$903$	−1.50093e6	−0.0612548
$904$	−2.06249e7	−0.839404
$905$	0	0
$906$	−4.62770e6	−0.187303
$907$	2.40573e6	0.0971023	0.0485512	−	0.998821i	$-0.484540\pi$
$907$	2.40573e6	0.0971023	0.0485512	+	0.998821i	$0.484540\pi$
$908$	2.65093e7	1.06705
$909$	−1.75844e6	−0.0705859
$910$	0	0
$911$	−4.38090e7	−1.74891	−0.874455	−	0.485107i	$-0.838781\pi$
$911$	−4.38090e7	−1.74891	−0.874455	+	0.485107i	$0.838781\pi$
$912$	−157463.	−0.00626891
$913$	−8.04221e6	−0.319300
$914$	5.36559e6	0.212448
$915$	0	0
$916$	2.56235e7	1.00902
$917$	−8.69987e6	−0.341656
$918$	3.27539e6	0.128279
$919$	4.10650e6	0.160392	0.0801962	−	0.996779i	$-0.474445\pi$
$919$	4.10650e6	0.160392	0.0801962	+	0.996779i	$0.474445\pi$
$920$	0	0
$921$	562842.	0.0218644
$922$	1.80217e7	0.698181
$923$	1.59187e6	0.0615039
$924$	4.23011e6	0.162994
$925$	0	0
$926$	1.52673e7	0.585107
$927$	−1.25079e6	−0.0478064
$928$	9.21161e6	0.351128
$929$	−7.91355e6	−0.300838	−0.150419	−	0.988622i	$-0.548062\pi$
$929$	−7.91355e6	−0.300838	−0.150419	+	0.988622i	$0.548062\pi$
$930$	0	0
$931$	−151012.	−0.00571001
$932$	1.46279e7	0.551622
$933$	−1.05234e7	−0.395780
$934$	−1.62064e7	−0.607880
$935$	0	0
$936$	−1.75091e6	−0.0653241
$937$	5.82636e6	0.216795	0.108397	−	0.994108i	$-0.465428\pi$
$937$	5.82636e6	0.216795	0.108397	+	0.994108i	$0.465428\pi$
$938$	1.16436e7	0.432094
$939$	−2.25919e7	−0.836159
$940$	0	0
$941$	−4.94213e7	−1.81945	−0.909725	−	0.415210i	$-0.863708\pi$
$941$	−4.94213e7	−1.81945	−0.909725	+	0.415210i	$0.863708\pi$
$942$	1.22876e7	0.451170
$943$	−5.82300e6	−0.213239
$944$	2.65720e6	0.0970496
$945$	0	0
$946$	321845.	0.0116928
$947$	1.11808e7	0.405135	0.202567	−	0.979268i	$-0.435071\pi$
$947$	1.11808e7	0.405135	0.202567	+	0.979268i	$0.435071\pi$
$948$	3.41946e6	0.123577
$949$	−1.13528e7	−0.409202
$950$	0	0
$951$	1.42533e7	0.511051
$952$	3.84812e7	1.37612
$953$	−1.89338e7	−0.675315	−0.337658	−	0.941269i	$-0.609635\pi$
$953$	−1.89338e7	−0.675315	−0.337658	+	0.941269i	$0.609635\pi$
$954$	−4.54315e6	−0.161617
$955$	0	0
$956$	1.09789e7	0.388519
$957$	1.78799e6	0.0631083
$958$	−9.28894e6	−0.327004
$959$	−2.16837e7	−0.761353
$960$	0	0
$961$	−1.59291e7	−0.556396
$962$	1.99218e6	0.0694052
$963$	−2.70202e6	−0.0938909
$964$	−5.65767e6	−0.196085
$965$	0	0
$966$	6.37805e6	0.219910
$967$	2.87893e7	0.990069	0.495034	−	0.868873i	$-0.335155\pi$
$967$	2.87893e7	0.990069	0.495034	+	0.868873i	$0.335155\pi$
$968$	−2.00000e6	−0.0686026
$969$	−570932.	−0.0195333
$970$	0	0
$971$	5.23023e7	1.78022	0.890108	−	0.455750i	$-0.150629\pi$
$971$	5.23023e7	1.78022	0.890108	+	0.455750i	$0.150629\pi$
$972$	−1.56824e6	−0.0532410
$973$	2.87677e7	0.974145
$974$	−4.90926e6	−0.165813
$975$	0	0
$976$	−2.54916e6	−0.0856590
$977$	5.27070e7	1.76657	0.883286	−	0.468834i	$-0.155326\pi$
$977$	5.27070e7	1.76657	0.883286	+	0.468834i	$0.155326\pi$
$978$	−2.29161e6	−0.0766114
$979$	−1.34366e6	−0.0448056
$980$	0	0
$981$	−7.65445e6	−0.253946
$982$	−1.07397e7	−0.355396
$983$	−5.51463e7	−1.82026	−0.910128	−	0.414328i	$-0.864017\pi$
$983$	−5.51463e7	−1.82026	−0.910128	+	0.414328i	$0.864017\pi$
$984$	−3.44665e6	−0.113478
$985$	0	0
$986$	7.37690e6	0.241647
$987$	−2.07951e7	−0.679467
$988$	138419.	0.00451131
$989$	−2.36833e6	−0.0769929
$990$	0	0
$991$	−4.11719e7	−1.33173	−0.665866	−	0.746071i	$-0.731938\pi$
$991$	−4.11719e7	−1.33173	−0.665866	+	0.746071i	$0.731938\pi$
$992$	−1.99940e7	−0.645090
$993$	−2.46044e7	−0.791844
$994$	−3.43228e6	−0.110184
$995$	0	0
$996$	1.58866e7	0.507438
$997$	−4.61912e7	−1.47171	−0.735853	−	0.677141i	$-0.763219\pi$
$997$	−4.61912e7	−1.47171	−0.735853	+	0.677141i	$0.763219\pi$
$998$	−5.06272e6	−0.160900
$999$	3.93429e6	0.124725

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.m.1.5	✓	7
5.4	even	2		825.6.a.o.1.3	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.6.a.m.1.5	✓	7	1.1	even	1	trivial
825.6.a.o.1.3	yes	7	5.4	even	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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

\(f(q)\)	\(=\)	\(q+2.33276 q^{2} +9.00000 q^{3} -26.5582 q^{4} +20.9949 q^{6} -146.260 q^{7} -136.602 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+2.33276 q^{2} +9.00000 q^{3} -26.5582 q^{4} +20.9949 q^{6} -146.260 q^{7} -136.602 q^{8} +81.0000 q^{9} +121.000 q^{11} -239.024 q^{12} +158.241 q^{13} -341.189 q^{14} +531.202 q^{16} +1926.04 q^{17} +188.954 q^{18} -32.9364 q^{19} -1316.34 q^{21} +282.264 q^{22} -2077.06 q^{23} -1229.42 q^{24} +369.139 q^{26} +729.000 q^{27} +3884.40 q^{28} +1641.87 q^{29} -3563.71 q^{31} +5610.45 q^{32} +1089.00 q^{33} +4492.99 q^{34} -2151.22 q^{36} +5396.83 q^{37} -76.8329 q^{38} +1424.17 q^{39} +2803.48 q^{41} -3070.71 q^{42} +1140.23 q^{43} -3213.54 q^{44} -4845.29 q^{46} +15797.7 q^{47} +4780.82 q^{48} +4584.95 q^{49} +17334.4 q^{51} -4202.61 q^{52} -24043.7 q^{53} +1700.58 q^{54} +19979.4 q^{56} -296.428 q^{57} +3830.09 q^{58} +5002.23 q^{59} -4798.86 q^{61} -8313.28 q^{62} -11847.0 q^{63} -3910.63 q^{64} +2540.38 q^{66} -34126.4 q^{67} -51152.2 q^{68} -18693.6 q^{69} +10059.7 q^{71} -11064.8 q^{72} -71743.6 q^{73} +12589.5 q^{74} +874.733 q^{76} -17697.4 q^{77} +3322.25 q^{78} -14305.9 q^{79} +6561.00 q^{81} +6539.84 q^{82} -66464.5 q^{83} +34959.6 q^{84} +2659.88 q^{86} +14776.8 q^{87} -16528.9 q^{88} -11104.6 q^{89} -23144.4 q^{91} +55163.1 q^{92} -32073.4 q^{93} +36852.3 q^{94} +50494.0 q^{96} -56522.6 q^{97} +10695.6 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 9 q^{2} + 63 q^{3} + 95 q^{4} - 81 q^{6} - 65 q^{7} - 207 q^{8} + 567 q^{9}+O(q^{10}) \) 7 * q - 9 * q^2 + 63 * q^3 + 95 * q^4 - 81 * q^6 - 65 * q^7 - 207 * q^8 + 567 * q^9	\( 7 q - 9 q^{2} + 63 q^{3} + 95 q^{4} - 81 q^{6} - 65 q^{7} - 207 q^{8} + 567 q^{9} + 847 q^{11} + 855 q^{12} + 113 q^{13} - 1212 q^{14} + 499 q^{16} - 1030 q^{17} - 729 q^{18} - 3803 q^{19} - 585 q^{21} - 1089 q^{22} - 514 q^{23} - 1863 q^{24} - 12111 q^{26} + 5103 q^{27} + 342 q^{28} - 2698 q^{29} - 17233 q^{31} - 9943 q^{32} + 7623 q^{33} + 4090 q^{34} + 7695 q^{36} - 23182 q^{37} + 11943 q^{38} + 1017 q^{39} - 16158 q^{41} - 10908 q^{42} + 4249 q^{43} + 11495 q^{44} - 28769 q^{46} - 7580 q^{47} + 4491 q^{48} + 37140 q^{49} - 9270 q^{51} + 23887 q^{52} - 20574 q^{53} - 6561 q^{54} - 73276 q^{56} - 34227 q^{57} - 8733 q^{58} - 364 q^{59} - 28127 q^{61} + 71917 q^{62} - 5265 q^{63} + 43379 q^{64} - 9801 q^{66} + 21493 q^{67} - 160660 q^{68} - 4626 q^{69} - 177084 q^{71} - 16767 q^{72} - 78670 q^{73} + 196750 q^{74} - 32701 q^{76} - 7865 q^{77} - 108999 q^{78} - 187432 q^{79} + 45927 q^{81} - 179552 q^{82} + 44592 q^{83} + 3078 q^{84} - 110433 q^{86} - 24282 q^{87} - 25047 q^{88} - 151168 q^{89} - 230153 q^{91} + 44767 q^{92} - 155097 q^{93} + 54040 q^{94} - 89487 q^{96} + 55589 q^{97} - 478341 q^{98} + 68607 q^{99}+O(q^{100}) \) 7 * q - 9 * q^2 + 63 * q^3 + 95 * q^4 - 81 * q^6 - 65 * q^7 - 207 * q^8 + 567 * q^9 + 847 * q^11 + 855 * q^12 + 113 * q^13 - 1212 * q^14 + 499 * q^16 - 1030 * q^17 - 729 * q^18 - 3803 * q^19 - 585 * q^21 - 1089 * q^22 - 514 * q^23 - 1863 * q^24 - 12111 * q^26 + 5103 * q^27 + 342 * q^28 - 2698 * q^29 - 17233 * q^31 - 9943 * q^32 + 7623 * q^33 + 4090 * q^34 + 7695 * q^36 - 23182 * q^37 + 11943 * q^38 + 1017 * q^39 - 16158 * q^41 - 10908 * q^42 + 4249 * q^43 + 11495 * q^44 - 28769 * q^46 - 7580 * q^47 + 4491 * q^48 + 37140 * q^49 - 9270 * q^51 + 23887 * q^52 - 20574 * q^53 - 6561 * q^54 - 73276 * q^56 - 34227 * q^57 - 8733 * q^58 - 364 * q^59 - 28127 * q^61 + 71917 * q^62 - 5265 * q^63 + 43379 * q^64 - 9801 * q^66 + 21493 * q^67 - 160660 * q^68 - 4626 * q^69 - 177084 * q^71 - 16767 * q^72 - 78670 * q^73 + 196750 * q^74 - 32701 * q^76 - 7865 * q^77 - 108999 * q^78 - 187432 * q^79 + 45927 * q^81 - 179552 * q^82 + 44592 * q^83 + 3078 * q^84 - 110433 * q^86 - 24282 * q^87 - 25047 * q^88 - 151168 * q^89 - 230153 * q^91 + 44767 * q^92 - 155097 * q^93 + 54040 * q^94 - 89487 * q^96 + 55589 * q^97 - 478341 * q^98 + 68607 * q^99

Introduction

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