LMFDB - Embedded newform 650.6.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [650,6,Mod(1,650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(650, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("650.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 650 = 2 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	650.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:	$104.249482878$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1458804.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 361x - 1139 $ x^3 - x^2 - 361*x - 1139
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-16.6075$ of defining polynomial
Character	$\chi$	$=$	650.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000 q^{2} -9.60750 q^{3} +16.0000 q^{4} -38.4300 q^{6} -16.6832 q^{7} +64.0000 q^{8} -150.696 q^{9} +O(q^{10})$	\(q+4.00000 q^{2} -9.60750 q^{3} +16.0000 q^{4} -38.4300 q^{6} -16.6832 q^{7} +64.0000 q^{8} -150.696 q^{9} -693.425 q^{11} -153.720 q^{12} -169.000 q^{13} -66.7328 q^{14} +256.000 q^{16} -826.423 q^{17} -602.784 q^{18} -1787.39 q^{19} +160.284 q^{21} -2773.70 q^{22} +2375.24 q^{23} -614.880 q^{24} -676.000 q^{26} +3782.43 q^{27} -266.931 q^{28} +2020.54 q^{29} +6723.34 q^{31} +1024.00 q^{32} +6662.08 q^{33} -3305.69 q^{34} -2411.14 q^{36} +2855.18 q^{37} -7149.54 q^{38} +1623.67 q^{39} -19394.3 q^{41} +641.135 q^{42} +14028.7 q^{43} -11094.8 q^{44} +9500.97 q^{46} +19989.2 q^{47} -2459.52 q^{48} -16528.7 q^{49} +7939.86 q^{51} -2704.00 q^{52} -3915.90 q^{53} +15129.7 q^{54} -1067.72 q^{56} +17172.3 q^{57} +8082.17 q^{58} -24027.0 q^{59} +34515.9 q^{61} +26893.4 q^{62} +2514.09 q^{63} +4096.00 q^{64} +26648.3 q^{66} -20445.0 q^{67} -13222.8 q^{68} -22820.1 q^{69} +43931.2 q^{71} -9644.54 q^{72} +18271.5 q^{73} +11420.7 q^{74} -28598.2 q^{76} +11568.5 q^{77} +6494.67 q^{78} -41359.0 q^{79} +279.391 q^{81} -77577.4 q^{82} -58383.5 q^{83} +2564.54 q^{84} +56114.6 q^{86} -19412.3 q^{87} -44379.2 q^{88} +48741.2 q^{89} +2819.46 q^{91} +38003.9 q^{92} -64594.5 q^{93} +79956.9 q^{94} -9838.08 q^{96} +151991. q^{97} -66114.7 q^{98} +104496. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 12 q^{2} + 22 q^{3} + 48 q^{4} + 88 q^{6} + 234 q^{7} + 192 q^{8} + 155 q^{9}+O(q^{10}) $ 3 * q + 12 * q^2 + 22 * q^3 + 48 * q^4 + 88 * q^6 + 234 * q^7 + 192 * q^8 + 155 * q^9	\( 3 q + 12 q^{2} + 22 q^{3} + 48 q^{4} + 88 q^{6} + 234 q^{7} + 192 q^{8} + 155 q^{9} + 48 q^{11} + 352 q^{12} - 507 q^{13} + 936 q^{14} + 768 q^{16} - 1506 q^{17} + 620 q^{18} - 360 q^{19} + 6904 q^{21} + 192 q^{22} + 2370 q^{23} + 1408 q^{24} - 2028 q^{26} + 10168 q^{27} + 3744 q^{28} - 3078 q^{29} - 5388 q^{31} + 3072 q^{32} + 22572 q^{33} - 6024 q^{34} + 2480 q^{36} + 25362 q^{37} - 1440 q^{38} - 3718 q^{39} - 15906 q^{41} + 27616 q^{42} + 39306 q^{43} + 768 q^{44} + 9480 q^{46} + 17778 q^{47} + 5632 q^{48} + 7767 q^{49} - 33972 q^{51} - 8112 q^{52} - 9246 q^{53} + 40672 q^{54} + 14976 q^{56} + 2996 q^{57} - 12312 q^{58} - 77760 q^{59} + 17982 q^{61} - 21552 q^{62} + 128762 q^{63} + 12288 q^{64} + 90288 q^{66} + 2922 q^{67} - 24096 q^{68} - 31860 q^{69} - 4944 q^{71} + 9920 q^{72} - 43278 q^{73} + 101448 q^{74} - 5760 q^{76} + 144420 q^{77} - 14872 q^{78} + 42120 q^{79} + 146519 q^{81} - 63624 q^{82} - 58098 q^{83} + 110464 q^{84} + 157224 q^{86} - 2016 q^{87} + 3072 q^{88} + 19614 q^{89} - 39546 q^{91} + 37920 q^{92} - 162796 q^{93} + 71112 q^{94} + 22528 q^{96} + 87078 q^{97} + 31068 q^{98} + 350340 q^{99}+O(q^{100}) \) 3 * q + 12 * q^2 + 22 * q^3 + 48 * q^4 + 88 * q^6 + 234 * q^7 + 192 * q^8 + 155 * q^9 + 48 * q^11 + 352 * q^12 - 507 * q^13 + 936 * q^14 + 768 * q^16 - 1506 * q^17 + 620 * q^18 - 360 * q^19 + 6904 * q^21 + 192 * q^22 + 2370 * q^23 + 1408 * q^24 - 2028 * q^26 + 10168 * q^27 + 3744 * q^28 - 3078 * q^29 - 5388 * q^31 + 3072 * q^32 + 22572 * q^33 - 6024 * q^34 + 2480 * q^36 + 25362 * q^37 - 1440 * q^38 - 3718 * q^39 - 15906 * q^41 + 27616 * q^42 + 39306 * q^43 + 768 * q^44 + 9480 * q^46 + 17778 * q^47 + 5632 * q^48 + 7767 * q^49 - 33972 * q^51 - 8112 * q^52 - 9246 * q^53 + 40672 * q^54 + 14976 * q^56 + 2996 * q^57 - 12312 * q^58 - 77760 * q^59 + 17982 * q^61 - 21552 * q^62 + 128762 * q^63 + 12288 * q^64 + 90288 * q^66 + 2922 * q^67 - 24096 * q^68 - 31860 * q^69 - 4944 * q^71 + 9920 * q^72 - 43278 * q^73 + 101448 * q^74 - 5760 * q^76 + 144420 * q^77 - 14872 * q^78 + 42120 * q^79 + 146519 * q^81 - 63624 * q^82 - 58098 * q^83 + 110464 * q^84 + 157224 * q^86 - 2016 * q^87 + 3072 * q^88 + 19614 * q^89 - 39546 * q^91 + 37920 * q^92 - 162796 * q^93 + 71112 * q^94 + 22528 * q^96 + 87078 * q^97 + 31068 * q^98 + 350340 * 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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	−9.60750	−0.616321	−0.308161	−	0.951334i	$-0.599713\pi$
$3$	−9.60750	−0.616321	−0.308161	+	0.951334i	$0.599713\pi$
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	−38.4300	−0.435805
$7$	−16.6832	−0.128687	−0.0643434	−	0.997928i	$-0.520495\pi$
$7$	−16.6832	−0.128687	−0.0643434	+	0.997928i	$0.520495\pi$
$8$	64.0000	0.353553
$9$	−150.696	−0.620148
$10$	0	0
$11$	−693.425	−1.72790	−0.863948	−	0.503580i	$-0.832016\pi$
$11$	−693.425	−1.72790	−0.863948	+	0.503580i	$0.832016\pi$
$12$	−153.720	−0.308161
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	−66.7328	−0.0909953
$15$	0	0
$16$	256.000	0.250000
$17$	−826.423	−0.693553	−0.346777	−	0.937948i	$-0.612724\pi$
$17$	−826.423	−0.693553	−0.346777	+	0.937948i	$0.612724\pi$
$18$	−602.784	−0.438511
$19$	−1787.39	−1.13588	−0.567942	−	0.823068i	$-0.692260\pi$
$19$	−1787.39	−1.13588	−0.567942	+	0.823068i	$0.692260\pi$
$20$	0	0
$21$	160.284	0.0793124
$22$	−2773.70	−1.22181
$23$	2375.24	0.936242	0.468121	−	0.883664i	$-0.344931\pi$
$23$	2375.24	0.936242	0.468121	+	0.883664i	$0.344931\pi$
$24$	−614.880	−0.217903
$25$	0	0
$26$	−676.000	−0.196116
$27$	3782.43	0.998532
$28$	−266.931	−0.0643434
$29$	2020.54	0.446142	0.223071	−	0.974802i	$-0.428392\pi$
$29$	2020.54	0.446142	0.223071	+	0.974802i	$0.428392\pi$
$30$	0	0
$31$	6723.34	1.25655	0.628277	−	0.777990i	$-0.283760\pi$
$31$	6723.34	1.25655	0.628277	+	0.777990i	$0.283760\pi$
$32$	1024.00	0.176777
$33$	6662.08	1.06494
$34$	−3305.69	−0.490416
$35$	0	0
$36$	−2411.14	−0.310074
$37$	2855.18	0.342870	0.171435	−	0.985195i	$-0.445160\pi$
$37$	2855.18	0.342870	0.171435	+	0.985195i	$0.445160\pi$
$38$	−7149.54	−0.803192
$39$	1623.67	0.170937
$40$	0	0
$41$	−19394.3	−1.80184	−0.900918	−	0.433989i	$-0.857106\pi$
$41$	−19394.3	−1.80184	−0.900918	+	0.433989i	$0.857106\pi$
$42$	641.135	0.0560823
$43$	14028.7	1.15703	0.578515	−	0.815672i	$-0.303632\pi$
$43$	14028.7	1.15703	0.578515	+	0.815672i	$0.303632\pi$
$44$	−11094.8	−0.863948
$45$	0	0
$46$	9500.97	0.662023
$47$	19989.2	1.31993	0.659966	−	0.751296i	$-0.270571\pi$
$47$	19989.2	1.31993	0.659966	+	0.751296i	$0.270571\pi$
$48$	−2459.52	−0.154080
$49$	−16528.7	−0.983440
$50$	0	0
$51$	7939.86	0.427452
$52$	−2704.00	−0.138675
$53$	−3915.90	−0.191488	−0.0957441	−	0.995406i	$-0.530523\pi$
$53$	−3915.90	−0.191488	−0.0957441	+	0.995406i	$0.530523\pi$
$54$	15129.7	0.706069
$55$	0	0
$56$	−1067.72	−0.0454977
$57$	17172.3	0.700070
$58$	8082.17	0.315470
$59$	−24027.0	−0.898605	−0.449303	−	0.893380i	$-0.648327\pi$
$59$	−24027.0	−0.898605	−0.449303	+	0.893380i	$0.648327\pi$
$60$	0	0
$61$	34515.9	1.18767	0.593833	−	0.804588i	$-0.297614\pi$
$61$	34515.9	1.18767	0.593833	+	0.804588i	$0.297614\pi$
$62$	26893.4	0.888518
$63$	2514.09	0.0798049
$64$	4096.00	0.125000
$65$	0	0
$66$	26648.3	0.753026
$67$	−20445.0	−0.556417	−0.278208	−	0.960521i	$-0.589741\pi$
$67$	−20445.0	−0.556417	−0.278208	+	0.960521i	$0.589741\pi$
$68$	−13222.8	−0.346777
$69$	−22820.1	−0.577026
$70$	0	0
$71$	43931.2	1.03426	0.517128	−	0.855908i	$-0.327001\pi$
$71$	43931.2	1.03426	0.517128	+	0.855908i	$0.327001\pi$
$72$	−9644.54	−0.219255
$73$	18271.5	0.401297	0.200649	−	0.979663i	$-0.435695\pi$
$73$	18271.5	0.401297	0.200649	+	0.979663i	$0.435695\pi$
$74$	11420.7	0.242445
$75$	0	0
$76$	−28598.2	−0.567942
$77$	11568.5	0.222358
$78$	6494.67	0.120871
$79$	−41359.0	−0.745593	−0.372797	−	0.927913i	$-0.621601\pi$
$79$	−41359.0	−0.745593	−0.372797	+	0.927913i	$0.621601\pi$
$80$	0	0
$81$	279.391	0.00473152
$82$	−77577.4	−1.27409
$83$	−58383.5	−0.930240	−0.465120	−	0.885248i	$-0.653989\pi$
$83$	−58383.5	−0.930240	−0.465120	+	0.885248i	$0.653989\pi$
$84$	2564.54	0.0396562
$85$	0	0
$86$	56114.6	0.818144
$87$	−19412.3	−0.274967
$88$	−44379.2	−0.610904
$89$	48741.2	0.652261	0.326130	−	0.945325i	$-0.394255\pi$
$89$	48741.2	0.652261	0.326130	+	0.945325i	$0.394255\pi$
$90$	0	0
$91$	2819.46	0.0356913
$92$	38003.9	0.468121
$93$	−64594.5	−0.774441
$94$	79956.9	0.933333
$95$	0	0
$96$	−9838.08	−0.108951
$97$	151991.	1.64017	0.820084	−	0.572244i	$-0.193927\pi$
$97$	151991.	1.64017	0.820084	+	0.572244i	$0.193927\pi$
$98$	−66114.7	−0.695397
$99$	104496.	1.07155
$100$	0	0
$101$	116976.	1.14102	0.570512	−	0.821290i	$-0.306745\pi$
$101$	116976.	1.14102	0.570512	+	0.821290i	$0.306745\pi$
$102$	31759.4	0.302254
$103$	−123435.	−1.14642	−0.573211	−	0.819408i	$-0.694302\pi$
$103$	−123435.	−1.14642	−0.573211	+	0.819408i	$0.694302\pi$
$104$	−10816.0	−0.0980581
$105$	0	0
$106$	−15663.6	−0.135403
$107$	158700.	1.34004	0.670021	−	0.742342i	$-0.266285\pi$
$107$	158700.	1.34004	0.670021	+	0.742342i	$0.266285\pi$
$108$	60518.9	0.499266
$109$	−43201.2	−0.348281	−0.174140	−	0.984721i	$-0.555715\pi$
$109$	−43201.2	−0.348281	−0.174140	+	0.984721i	$0.555715\pi$
$110$	0	0
$111$	−27431.1	−0.211318
$112$	−4270.90	−0.0321717
$113$	6402.49	0.0471685	0.0235843	−	0.999722i	$-0.492492\pi$
$113$	6402.49	0.0471685	0.0235843	+	0.999722i	$0.492492\pi$
$114$	68689.2	0.495024
$115$	0	0
$116$	32328.7	0.223071
$117$	25467.6	0.171998
$118$	−96107.8	−0.635410
$119$	13787.4	0.0892512
$120$	0	0
$121$	319787.	1.98563
$122$	138063.	0.839807
$123$	186331.	1.11051
$124$	107574.	0.628277
$125$	0	0
$126$	10056.4	0.0564306
$127$	−126776.	−0.697475	−0.348737	−	0.937221i	$-0.613389\pi$
$127$	−126776.	−0.697475	−0.348737	+	0.937221i	$0.613389\pi$
$128$	16384.0	0.0883883
$129$	−134780.	−0.713103
$130$	0	0
$131$	−374014.	−1.90419	−0.952093	−	0.305808i	$-0.901073\pi$
$131$	−374014.	−1.90419	−0.952093	+	0.305808i	$0.901073\pi$
$132$	106593.	0.532470
$133$	29819.3	0.146173
$134$	−81780.0	−0.393446
$135$	0	0
$136$	−52891.1	−0.245208
$137$	360787.	1.64229	0.821144	−	0.570721i	$-0.193336\pi$
$137$	360787.	1.64229	0.821144	+	0.570721i	$0.193336\pi$
$138$	−91280.5	−0.408019
$139$	406503.	1.78454	0.892271	−	0.451499i	$-0.149111\pi$
$139$	406503.	1.78454	0.892271	+	0.451499i	$0.149111\pi$
$140$	0	0
$141$	−192047.	−0.813502
$142$	175725.	0.731329
$143$	117189.	0.479232
$144$	−38578.2	−0.155037
$145$	0	0
$146$	73085.9	0.283760
$147$	158799.	0.606115
$148$	45682.9	0.171435
$149$	128834.	0.475407	0.237704	−	0.971338i	$-0.423605\pi$
$149$	128834.	0.475407	0.237704	+	0.971338i	$0.423605\pi$
$150$	0	0
$151$	180055.	0.642634	0.321317	−	0.946972i	$-0.395874\pi$
$151$	180055.	0.642634	0.321317	+	0.946972i	$0.395874\pi$
$152$	−114393.	−0.401596
$153$	124539.	0.430106
$154$	46274.2	0.157231
$155$	0	0
$156$	25978.7	0.0854684
$157$	−321316.	−1.04036	−0.520180	−	0.854057i	$-0.674135\pi$
$157$	−321316.	−1.04036	−0.520180	+	0.854057i	$0.674135\pi$
$158$	−165436.	−0.527214
$159$	37622.0	0.118018
$160$	0	0
$161$	−39626.6	−0.120482
$162$	1117.57	0.00334569
$163$	135115.	0.398322	0.199161	−	0.979967i	$-0.436178\pi$
$163$	135115.	0.398322	0.199161	+	0.979967i	$0.436178\pi$
$164$	−310309.	−0.900918
$165$	0	0
$166$	−233534.	−0.657779
$167$	−101528.	−0.281705	−0.140853	−	0.990031i	$-0.544984\pi$
$167$	−101528.	−0.281705	−0.140853	+	0.990031i	$0.544984\pi$
$168$	10258.2	0.0280412
$169$	28561.0	0.0769231
$170$	0	0
$171$	269352.	0.704417
$172$	224458.	0.578515
$173$	−309597.	−0.786470	−0.393235	−	0.919438i	$-0.628644\pi$
$173$	−309597.	−0.786470	−0.393235	+	0.919438i	$0.628644\pi$
$174$	−77649.4	−0.194431
$175$	0	0
$176$	−177517.	−0.431974
$177$	230839.	0.553829
$178$	194965.	0.461218
$179$	−191098.	−0.445782	−0.222891	−	0.974843i	$-0.571549\pi$
$179$	−191098.	−0.445782	−0.222891	+	0.974843i	$0.571549\pi$
$180$	0	0
$181$	694442.	1.57558	0.787789	−	0.615945i	$-0.211226\pi$
$181$	694442.	1.57558	0.787789	+	0.615945i	$0.211226\pi$
$182$	11277.8	0.0252376
$183$	−331611.	−0.731984
$184$	152015.	0.331012
$185$	0	0
$186$	−258378.	−0.547612
$187$	573062.	1.19839
$188$	319828.	0.659966
$189$	−63103.1	−0.128498
$190$	0	0
$191$	−581263.	−1.15289	−0.576447	−	0.817135i	$-0.695561\pi$
$191$	−581263.	−1.15289	−0.576447	+	0.817135i	$0.695561\pi$
$192$	−39352.3	−0.0770402
$193$	−53201.0	−0.102808	−0.0514039	−	0.998678i	$-0.516370\pi$
$193$	−53201.0	−0.102808	−0.0514039	+	0.998678i	$0.516370\pi$
$194$	607963.	1.15977
$195$	0	0
$196$	−264459.	−0.491720
$197$	877908.	1.61170	0.805848	−	0.592122i	$-0.201710\pi$
$197$	877908.	1.61170	0.805848	+	0.592122i	$0.201710\pi$
$198$	417985.	0.757702
$199$	1.03412e6	1.85114	0.925571	−	0.378574i	$-0.123585\pi$
$199$	1.03412e6	1.85114	0.925571	+	0.378574i	$0.123585\pi$
$200$	0	0
$201$	196425.	0.342931
$202$	467905.	0.806825
$203$	−33709.1	−0.0574125
$204$	127038.	0.213726
$205$	0	0
$206$	−493739.	−0.810642
$207$	−357939.	−0.580609
$208$	−43264.0	−0.0693375
$209$	1.23942e6	1.96269
$210$	0	0
$211$	−46774.3	−0.0723271	−0.0361635	−	0.999346i	$-0.511514\pi$
$211$	−46774.3	−0.0723271	−0.0361635	+	0.999346i	$0.511514\pi$
$212$	−62654.4	−0.0957441
$213$	−422069.	−0.637434
$214$	634801.	0.947552
$215$	0	0
$216$	242076.	0.353034
$217$	−112167.	−0.161702
$218$	−172805.	−0.246272
$219$	−175543.	−0.247328
$220$	0	0
$221$	139665.	0.192357
$222$	−109725.	−0.149424
$223$	302978.	0.407990	0.203995	−	0.978972i	$-0.434607\pi$
$223$	302978.	0.407990	0.203995	+	0.978972i	$0.434607\pi$
$224$	−17083.6	−0.0227488
$225$	0	0
$226$	25609.9	0.0333532
$227$	581763.	0.749345	0.374672	−	0.927157i	$-0.377755\pi$
$227$	581763.	0.749345	0.374672	+	0.927157i	$0.377755\pi$
$228$	274757.	0.350035
$229$	53955.7	0.0679905	0.0339953	−	0.999422i	$-0.489177\pi$
$229$	53955.7	0.0679905	0.0339953	+	0.999422i	$0.489177\pi$
$230$	0	0
$231$	−111145.	−0.137044
$232$	129315.	0.157735
$233$	−897805.	−1.08341	−0.541704	−	0.840569i	$-0.682221\pi$
$233$	−897805.	−1.08341	−0.541704	+	0.840569i	$0.682221\pi$
$234$	101870.	0.121621
$235$	0	0
$236$	−384431.	−0.449303
$237$	397356.	0.459525
$238$	55149.5	0.0631101
$239$	−1.02657e6	−1.16250	−0.581251	−	0.813724i	$-0.697437\pi$
$239$	−1.02657e6	−1.16250	−0.581251	+	0.813724i	$0.697437\pi$
$240$	0	0
$241$	1.13459e6	1.25833	0.629167	−	0.777270i	$-0.283396\pi$
$241$	1.13459e6	1.25833	0.629167	+	0.777270i	$0.283396\pi$
$242$	1.27915e6	1.40405
$243$	−921816.	−1.00145
$244$	552254.	0.593833
$245$	0	0
$246$	745324.	0.785249
$247$	302068.	0.315038
$248$	430294.	0.444259
$249$	560919.	0.573327
$250$	0	0
$251$	−697428.	−0.698740	−0.349370	−	0.936985i	$-0.613604\pi$
$251$	−697428.	−0.698740	−0.349370	+	0.936985i	$0.613604\pi$
$252$	40225.4	0.0399024
$253$	−1.64705e6	−1.61773
$254$	−507105.	−0.493189
$255$	0	0
$256$	65536.0	0.0625000
$257$	−1.56660e6	−1.47954	−0.739770	−	0.672860i	$-0.765066\pi$
$257$	−1.56660e6	−1.47954	−0.739770	+	0.672860i	$0.765066\pi$
$258$	−539121.	−0.504240
$259$	−47633.5	−0.0441228
$260$	0	0
$261$	−304487.	−0.276674
$262$	−1.49606e6	−1.34646
$263$	1.03876e6	0.926028	0.463014	−	0.886351i	$-0.346768\pi$
$263$	1.03876e6	0.926028	0.463014	+	0.886351i	$0.346768\pi$
$264$	426373.	0.376513
$265$	0	0
$266$	119277.	0.103360
$267$	−468281.	−0.402002
$268$	−327120.	−0.278208
$269$	107944.	0.0909532	0.0454766	−	0.998965i	$-0.485519\pi$
$269$	107944.	0.0909532	0.0454766	+	0.998965i	$0.485519\pi$
$270$	0	0
$271$	−1.06092e6	−0.877528	−0.438764	−	0.898602i	$-0.644584\pi$
$271$	−1.06092e6	−0.877528	−0.438764	+	0.898602i	$0.644584\pi$
$272$	−211564.	−0.173388
$273$	−27087.9	−0.0219973
$274$	1.44315e6	1.16127
$275$	0	0
$276$	−365122.	−0.288513
$277$	−523617.	−0.410028	−0.205014	−	0.978759i	$-0.565724\pi$
$277$	−523617.	−0.410028	−0.205014	+	0.978759i	$0.565724\pi$
$278$	1.62601e6	1.26186
$279$	−1.01318e6	−0.779249
$280$	0	0
$281$	1.24579e6	0.941192	0.470596	−	0.882349i	$-0.344039\pi$
$281$	1.24579e6	0.941192	0.470596	+	0.882349i	$0.344039\pi$
$282$	−768186.	−0.575233
$283$	−772266.	−0.573193	−0.286597	−	0.958051i	$-0.592524\pi$
$283$	−772266.	−0.573193	−0.286597	+	0.958051i	$0.592524\pi$
$284$	702900.	0.517128
$285$	0	0
$286$	468755.	0.338868
$287$	323559.	0.231873
$288$	−154313.	−0.109628
$289$	−736882.	−0.518984
$290$	0	0
$291$	−1.46025e6	−1.01087
$292$	292343.	0.200649
$293$	−1.55416e6	−1.05761	−0.528806	−	0.848743i	$-0.677360\pi$
$293$	−1.55416e6	−1.05761	−0.528806	+	0.848743i	$0.677360\pi$
$294$	635197.	0.428588
$295$	0	0
$296$	182731.	0.121223
$297$	−2.62283e6	−1.72536
$298$	515337.	0.336164
$299$	−401416.	−0.259667
$300$	0	0
$301$	−234043.	−0.148895
$302$	720222.	0.454411
$303$	−1.12385e6	−0.703237
$304$	−457571.	−0.283971
$305$	0	0
$306$	498154.	0.304131
$307$	836762.	0.506706	0.253353	−	0.967374i	$-0.418467\pi$
$307$	836762.	0.506706	0.253353	+	0.967374i	$0.418467\pi$
$308$	185097.	0.111179
$309$	1.18590e6	0.706564
$310$	0	0
$311$	2.18576e6	1.28145	0.640724	−	0.767772i	$-0.278634\pi$
$311$	2.18576e6	1.28145	0.640724	+	0.767772i	$0.278634\pi$
$312$	103915.	0.0604353
$313$	−1.05261e6	−0.607307	−0.303654	−	0.952782i	$-0.598207\pi$
$313$	−1.05261e6	−0.607307	−0.303654	+	0.952782i	$0.598207\pi$
$314$	−1.28526e6	−0.735645
$315$	0	0
$316$	−661744.	−0.372797
$317$	1.09831e6	0.613872	0.306936	−	0.951730i	$-0.400696\pi$
$317$	1.09831e6	0.613872	0.306936	+	0.951730i	$0.400696\pi$
$318$	150488.	0.0834515
$319$	−1.40109e6	−0.770887
$320$	0	0
$321$	−1.52471e6	−0.825896
$322$	−158506.	−0.0851936
$323$	1.47714e6	0.787797
$324$	4470.26	0.00236576
$325$	0	0
$326$	540460.	0.281656
$327$	415055.	0.214653
$328$	−1.24124e6	−0.637045
$329$	−333484.	−0.169858
$330$	0	0
$331$	2.32534e6	1.16659	0.583293	−	0.812262i	$-0.301764\pi$
$331$	2.32534e6	1.16659	0.583293	+	0.812262i	$0.301764\pi$
$332$	−934136.	−0.465120
$333$	−430264.	−0.212630
$334$	−406112.	−0.199196
$335$	0	0
$336$	41032.6	0.0198281
$337$	2.86939e6	1.37630	0.688152	−	0.725567i	$-0.258422\pi$
$337$	2.86939e6	1.37630	0.688152	+	0.725567i	$0.258422\pi$
$338$	114244.	0.0543928
$339$	−61511.9	−0.0290710
$340$	0	0
$341$	−4.66214e6	−2.17120
$342$	1.07741e6	0.498098
$343$	556145.	0.255242
$344$	897834.	0.409072
$345$	0	0
$346$	−1.23839e6	−0.556118
$347$	3.31346e6	1.47726	0.738632	−	0.674109i	$-0.235472\pi$
$347$	3.31346e6	1.47726	0.738632	+	0.674109i	$0.235472\pi$
$348$	−310598.	−0.137483
$349$	−3.47506e6	−1.52721	−0.763605	−	0.645684i	$-0.776572\pi$
$349$	−3.47506e6	−1.52721	−0.763605	+	0.645684i	$0.776572\pi$
$350$	0	0
$351$	−639231.	−0.276943
$352$	−710067.	−0.305452
$353$	2.82052e6	1.20474	0.602368	−	0.798218i	$-0.294224\pi$
$353$	2.82052e6	1.20474	0.602368	+	0.798218i	$0.294224\pi$
$354$	923356.	0.391617
$355$	0	0
$356$	779859.	0.326130
$357$	−132462.	−0.0550074
$358$	−764390.	−0.315215
$359$	−2.32351e6	−0.951500	−0.475750	−	0.879580i	$-0.657823\pi$
$359$	−2.32351e6	−0.951500	−0.475750	+	0.879580i	$0.657823\pi$
$360$	0	0
$361$	718647.	0.290234
$362$	2.77777e6	1.11410
$363$	−3.07236e6	−1.22378
$364$	45111.3	0.0178456
$365$	0	0
$366$	−1.32644e6	−0.517591
$367$	2.12636e6	0.824084	0.412042	−	0.911165i	$-0.364816\pi$
$367$	2.12636e6	0.824084	0.412042	+	0.911165i	$0.364816\pi$
$368$	608062.	0.234061
$369$	2.92265e6	1.11741
$370$	0	0
$371$	65329.7	0.0246420
$372$	−1.03351e6	−0.387221
$373$	−3.90096e6	−1.45178	−0.725888	−	0.687813i	$-0.758571\pi$
$373$	−3.90096e6	−1.45178	−0.725888	+	0.687813i	$0.758571\pi$
$374$	2.29225e6	0.847389
$375$	0	0
$376$	1.27931e6	0.466666
$377$	−341471.	−0.123737
$378$	−252412.	−0.0908617
$379$	−3.71050e6	−1.32689	−0.663444	−	0.748226i	$-0.730906\pi$
$379$	−3.71050e6	−1.32689	−0.663444	+	0.748226i	$0.730906\pi$
$380$	0	0
$381$	1.21800e6	0.429868
$382$	−2.32505e6	−0.815219
$383$	3.60895e6	1.25714	0.628571	−	0.777752i	$-0.283640\pi$
$383$	3.60895e6	1.25714	0.628571	+	0.777752i	$0.283640\pi$
$384$	−157409.	−0.0544756
$385$	0	0
$386$	−212804.	−0.0726961
$387$	−2.11406e6	−0.717530
$388$	2.43185e6	0.820084
$389$	−1.65714e6	−0.555245	−0.277622	−	0.960690i	$-0.589546\pi$
$389$	−1.65714e6	−0.555245	−0.277622	+	0.960690i	$0.589546\pi$
$390$	0	0
$391$	−1.96295e6	−0.649334
$392$	−1.05783e6	−0.347698
$393$	3.59334e6	1.17359
$394$	3.51163e6	1.13964
$395$	0	0
$396$	1.67194e6	0.535776
$397$	2.10413e6	0.670035	0.335017	−	0.942212i	$-0.391258\pi$
$397$	2.10413e6	0.670035	0.335017	+	0.942212i	$0.391258\pi$
$398$	4.13649e6	1.30896
$399$	−286489.	−0.0900897
$400$	0	0
$401$	2.98174e6	0.925994	0.462997	−	0.886360i	$-0.346774\pi$
$401$	2.98174e6	0.925994	0.462997	+	0.886360i	$0.346774\pi$
$402$	785701.	0.242489
$403$	−1.13625e6	−0.348505
$404$	1.87162e6	0.570512
$405$	0	0
$406$	−134836.	−0.0405968
$407$	−1.97985e6	−0.592443
$408$	508151.	0.151127
$409$	2.98041e6	0.880983	0.440491	−	0.897757i	$-0.354804\pi$
$409$	2.98041e6	0.880983	0.440491	+	0.897757i	$0.354804\pi$
$410$	0	0
$411$	−3.46626e6	−1.01218
$412$	−1.97496e6	−0.573211
$413$	400846.	0.115639
$414$	−1.43176e6	−0.410552
$415$	0	0
$416$	−173056.	−0.0490290
$417$	−3.90548e6	−1.09985
$418$	4.95767e6	1.38783
$419$	−7.06233e6	−1.96523	−0.982614	−	0.185663i	$-0.940557\pi$
$419$	−7.06233e6	−1.96523	−0.982614	+	0.185663i	$0.940557\pi$
$420$	0	0
$421$	299952.	0.0824795	0.0412398	−	0.999149i	$-0.486869\pi$
$421$	299952.	0.0824795	0.0412398	+	0.999149i	$0.486869\pi$
$422$	−187097.	−0.0511430
$423$	−3.01230e6	−0.818553
$424$	−250618.	−0.0677013
$425$	0	0
$426$	−1.68828e6	−0.450734
$427$	−575835.	−0.152837
$428$	2.53920e6	0.670021
$429$	−1.12589e6	−0.295361
$430$	0	0
$431$	−1.29958e6	−0.336983	−0.168492	−	0.985703i	$-0.553890\pi$
$431$	−1.29958e6	−0.336983	−0.168492	+	0.985703i	$0.553890\pi$
$432$	968303.	0.249633
$433$	5.44298e6	1.39514	0.697569	−	0.716518i	$-0.254265\pi$
$433$	5.44298e6	1.39514	0.697569	+	0.716518i	$0.254265\pi$
$434$	−448667.	−0.114341
$435$	0	0
$436$	−691219.	−0.174140
$437$	−4.24547e6	−1.06346
$438$	−702172.	−0.174887
$439$	−4.35230e6	−1.07785	−0.538924	−	0.842355i	$-0.681169\pi$
$439$	−4.35230e6	−1.07785	−0.538924	+	0.842355i	$0.681169\pi$
$440$	0	0
$441$	2.49080e6	0.609878
$442$	558662.	0.136017
$443$	2.75827e6	0.667771	0.333886	−	0.942614i	$-0.391640\pi$
$443$	2.75827e6	0.667771	0.333886	+	0.942614i	$0.391640\pi$
$444$	−438898.	−0.105659
$445$	0	0
$446$	1.21191e6	0.288493
$447$	−1.23778e6	−0.293004
$448$	−68334.3	−0.0160858
$449$	720400.	0.168639	0.0843195	−	0.996439i	$-0.473128\pi$
$449$	720400.	0.168639	0.0843195	+	0.996439i	$0.473128\pi$
$450$	0	0
$451$	1.34485e7	3.11339
$452$	102440.	0.0235843
$453$	−1.72988e6	−0.396069
$454$	2.32705e6	0.529867
$455$	0	0
$456$	1.09903e6	0.247512
$457$	6.87411e6	1.53966	0.769832	−	0.638247i	$-0.220340\pi$
$457$	6.87411e6	1.53966	0.769832	+	0.638247i	$0.220340\pi$
$458$	215823.	0.0480766
$459$	−3.12589e6	−0.692535
$460$	0	0
$461$	3.25115e6	0.712501	0.356250	−	0.934391i	$-0.384055\pi$
$461$	3.25115e6	0.712501	0.356250	+	0.934391i	$0.384055\pi$
$462$	−444579.	−0.0969045
$463$	1.41176e6	0.306061	0.153031	−	0.988221i	$-0.451097\pi$
$463$	1.41176e6	0.306061	0.153031	+	0.988221i	$0.451097\pi$
$464$	517259.	0.111535
$465$	0	0
$466$	−3.59122e6	−0.766085
$467$	−9.07019e6	−1.92453	−0.962264	−	0.272117i	$-0.912276\pi$
$467$	−9.07019e6	−1.92453	−0.962264	+	0.272117i	$0.912276\pi$
$468$	407482.	0.0859991
$469$	341088.	0.0716035
$470$	0	0
$471$	3.08704e6	0.641196
$472$	−1.53773e6	−0.317705
$473$	−9.72782e6	−1.99923
$474$	1.58943e6	0.324933
$475$	0	0
$476$	220598.	0.0446256
$477$	590110.	0.118751
$478$	−4.10628e6	−0.822013
$479$	−3.57727e6	−0.712382	−0.356191	−	0.934413i	$-0.615925\pi$
$479$	−3.57727e6	−0.712382	−0.356191	+	0.934413i	$0.615925\pi$
$480$	0	0
$481$	−482525.	−0.0950949
$482$	4.53836e6	0.889777
$483$	380713.	0.0742556
$484$	5.11660e6	0.992814
$485$	0	0
$486$	−3.68726e6	−0.708131
$487$	−9.72962e6	−1.85898	−0.929488	−	0.368853i	$-0.879751\pi$
$487$	−9.72962e6	−1.85898	−0.929488	+	0.368853i	$0.879751\pi$
$488$	2.20902e6	0.419903
$489$	−1.29812e6	−0.245495
$490$	0	0
$491$	−292730.	−0.0547978	−0.0273989	−	0.999625i	$-0.508722\pi$
$491$	−292730.	−0.0547978	−0.0273989	+	0.999625i	$0.508722\pi$
$492$	2.98130e6	0.555255
$493$	−1.66982e6	−0.309423
$494$	1.20827e6	0.222765
$495$	0	0
$496$	1.72118e6	0.314138
$497$	−732913.	−0.133095
$498$	2.24368e6	0.405403
$499$	1.95009e6	0.350594	0.175297	−	0.984516i	$-0.443911\pi$
$499$	1.95009e6	0.350594	0.175297	+	0.984516i	$0.443911\pi$
$500$	0	0
$501$	975431.	0.173621
$502$	−2.78971e6	−0.494083
$503$	3.52987e6	0.622069	0.311034	−	0.950399i	$-0.399325\pi$
$503$	3.52987e6	0.622069	0.311034	+	0.950399i	$0.399325\pi$
$504$	160902.	0.0282153
$505$	0	0
$506$	−6.58821e6	−1.14391
$507$	−274400.	−0.0474093
$508$	−2.02842e6	−0.348737
$509$	80297.4	0.0137375	0.00686874	−	0.999976i	$-0.497814\pi$
$509$	80297.4	0.0137375	0.00686874	+	0.999976i	$0.497814\pi$
$510$	0	0
$511$	−304826.	−0.0516417
$512$	262144.	0.0441942
$513$	−6.76067e6	−1.13422
$514$	−6.26642e6	−1.04619
$515$	0	0
$516$	−2.15648e6	−0.356551
$517$	−1.38610e7	−2.28071
$518$	−190534.	−0.0311995
$519$	2.97446e6	0.484718
$520$	0	0
$521$	6.04030e6	0.974909	0.487455	−	0.873148i	$-0.337925\pi$
$521$	6.04030e6	0.974909	0.487455	+	0.873148i	$0.337925\pi$
$522$	−1.21795e6	−0.195638
$523$	2.72218e6	0.435174	0.217587	−	0.976041i	$-0.430181\pi$
$523$	2.72218e6	0.435174	0.217587	+	0.976041i	$0.430181\pi$
$524$	−5.98422e6	−0.952093
$525$	0	0
$526$	4.15502e6	0.654801
$527$	−5.55632e6	−0.871487
$528$	1.70549e6	0.266235
$529$	−794572.	−0.123451
$530$	0	0
$531$	3.62077e6	0.557268
$532$	477109.	0.0730867
$533$	3.27764e6	0.499740
$534$	−1.87312e6	−0.284258
$535$	0	0
$536$	−1.30848e6	−0.196723
$537$	1.83597e6	0.274745
$538$	431776.	0.0643136
$539$	1.14614e7	1.69928
$540$	0	0
$541$	−3.59202e6	−0.527650	−0.263825	−	0.964571i	$-0.584984\pi$
$541$	−3.59202e6	−0.527650	−0.263825	+	0.964571i	$0.584984\pi$
$542$	−4.24370e6	−0.620506
$543$	−6.67185e6	−0.971062
$544$	−846257.	−0.122604
$545$	0	0
$546$	−108352.	−0.0155544
$547$	5.06329e6	0.723543	0.361772	−	0.932267i	$-0.382172\pi$
$547$	5.06329e6	0.723543	0.361772	+	0.932267i	$0.382172\pi$
$548$	5.77259e6	0.821144
$549$	−5.20140e6	−0.736529
$550$	0	0
$551$	−3.61149e6	−0.506765
$552$	−1.46049e6	−0.204009
$553$	690000.	0.0959480
$554$	−2.09447e6	−0.289934
$555$	0	0
$556$	6.50405e6	0.892271
$557$	1.11450e7	1.52209	0.761046	−	0.648698i	$-0.224686\pi$
$557$	1.11450e7	1.52209	0.761046	+	0.648698i	$0.224686\pi$
$558$	−4.05272e6	−0.551013
$559$	−2.37084e6	−0.320903
$560$	0	0
$561$	−5.50570e6	−0.738593
$562$	4.98315e6	0.665524
$563$	−2.52470e6	−0.335691	−0.167845	−	0.985813i	$-0.553681\pi$
$563$	−2.52470e6	−0.335691	−0.167845	+	0.985813i	$0.553681\pi$
$564$	−3.07274e6	−0.406751
$565$	0	0
$566$	−3.08906e6	−0.405309
$567$	−4661.14	−0.000608884 0
$568$	2.81160e6	0.365664
$569$	−1.14153e7	−1.47811	−0.739057	−	0.673643i	$-0.764729\pi$
$569$	−1.14153e7	−1.47811	−0.739057	+	0.673643i	$0.764729\pi$
$570$	0	0
$571$	−1.30087e7	−1.66972	−0.834862	−	0.550460i	$-0.814452\pi$
$571$	−1.30087e7	−1.66972	−0.834862	+	0.550460i	$0.814452\pi$
$572$	1.87502e6	0.239616
$573$	5.58448e6	0.710553
$574$	1.29424e6	0.163959
$575$	0	0
$576$	−617251.	−0.0775185
$577$	3.66964e6	0.458864	0.229432	−	0.973325i	$-0.426313\pi$
$577$	3.66964e6	0.458864	0.229432	+	0.973325i	$0.426313\pi$
$578$	−2.94753e6	−0.366977
$579$	511128.	0.0633627
$580$	0	0
$581$	974023.	0.119710
$582$	−5.84101e6	−0.714793
$583$	2.71538e6	0.330872
$584$	1.16937e6	0.141880
$585$	0	0
$586$	−6.21663e6	−0.747844
$587$	−9.50908e6	−1.13905	−0.569526	−	0.821974i	$-0.692873\pi$
$587$	−9.50908e6	−1.13905	−0.569526	+	0.821974i	$0.692873\pi$
$588$	2.54079e6	0.303057
$589$	−1.20172e7	−1.42730
$590$	0	0
$591$	−8.43450e6	−0.993323
$592$	730926.	0.0857174
$593$	966817.	0.112904	0.0564518	−	0.998405i	$-0.482021\pi$
$593$	966817.	0.112904	0.0564518	+	0.998405i	$0.482021\pi$
$594$	−1.04913e7	−1.22001
$595$	0	0
$596$	2.06135e6	0.237704
$597$	−9.93534e6	−1.14090
$598$	−1.60566e6	−0.183612
$599$	272872.	0.0310737	0.0155368	−	0.999879i	$-0.495054\pi$
$599$	272872.	0.0310737	0.0155368	+	0.999879i	$0.495054\pi$
$600$	0	0
$601$	1.15123e6	0.130010	0.0650050	−	0.997885i	$-0.479294\pi$
$601$	1.15123e6	0.130010	0.0650050	+	0.997885i	$0.479294\pi$
$602$	−936171.	−0.105284
$603$	3.08098e6	0.345061
$604$	2.88089e6	0.321317
$605$	0	0
$606$	−4.49540e6	−0.497264
$607$	3.34458e6	0.368443	0.184221	−	0.982885i	$-0.441024\pi$
$607$	3.34458e6	0.368443	0.184221	+	0.982885i	$0.441024\pi$
$608$	−1.83028e6	−0.200798
$609$	323860.	0.0353846
$610$	0	0
$611$	−3.37818e6	−0.366083
$612$	1.99262e6	0.215053
$613$	9.25214e6	0.994469	0.497234	−	0.867616i	$-0.334349\pi$
$613$	9.25214e6	0.994469	0.497234	+	0.867616i	$0.334349\pi$
$614$	3.34705e6	0.358295
$615$	0	0
$616$	740387.	0.0786153
$617$	−2.23244e6	−0.236084	−0.118042	−	0.993009i	$-0.537662\pi$
$617$	−2.23244e6	−0.236084	−0.118042	+	0.993009i	$0.537662\pi$
$618$	4.74360e6	0.499616
$619$	−1.14222e7	−1.19818	−0.599092	−	0.800680i	$-0.704472\pi$
$619$	−1.14222e7	−1.19818	−0.599092	+	0.800680i	$0.704472\pi$
$620$	0	0
$621$	8.98419e6	0.934867
$622$	8.74302e6	0.906120
$623$	−813159.	−0.0839373
$624$	415659.	0.0427342
$625$	0	0
$626$	−4.21046e6	−0.429431
$627$	−1.19077e7	−1.20965
$628$	−5.14106e6	−0.520180
$629$	−2.35958e6	−0.237798
$630$	0	0
$631$	−1.66436e6	−0.166408	−0.0832040	−	0.996533i	$-0.526515\pi$
$631$	−1.66436e6	−0.166408	−0.0832040	+	0.996533i	$0.526515\pi$
$632$	−2.64697e6	−0.263607
$633$	449384.	0.0445767
$634$	4.39325e6	0.434073
$635$	0	0
$636$	601952.	0.0590091
$637$	2.79335e6	0.272757
$638$	−5.60438e6	−0.545099
$639$	−6.62026e6	−0.641391
$640$	0	0
$641$	−5.71944e6	−0.549804	−0.274902	−	0.961472i	$-0.588645\pi$
$641$	−5.71944e6	−0.549804	−0.274902	+	0.961472i	$0.588645\pi$
$642$	−6.09885e6	−0.583997
$643$	−2.71308e6	−0.258783	−0.129391	−	0.991594i	$-0.541302\pi$
$643$	−2.71308e6	−0.258783	−0.129391	+	0.991594i	$0.541302\pi$
$644$	−634026.	−0.0602410
$645$	0	0
$646$	5.90854e6	0.557056
$647$	−5.29299e6	−0.497096	−0.248548	−	0.968620i	$-0.579953\pi$
$647$	−5.29299e6	−0.497096	−0.248548	+	0.968620i	$0.579953\pi$
$648$	17881.0	0.00167284
$649$	1.66609e7	1.55270
$650$	0	0
$651$	1.07764e6	0.0996603
$652$	2.16184e6	0.199161
$653$	−8.54020e6	−0.783764	−0.391882	−	0.920016i	$-0.628176\pi$
$653$	−8.54020e6	−0.783764	−0.391882	+	0.920016i	$0.628176\pi$
$654$	1.66022e6	0.151782
$655$	0	0
$656$	−4.96495e6	−0.450459
$657$	−2.75344e6	−0.248864
$658$	−1.33394e6	−0.120108
$659$	8.44994e6	0.757950	0.378975	−	0.925407i	$-0.376277\pi$
$659$	8.44994e6	0.757950	0.378975	+	0.925407i	$0.376277\pi$
$660$	0	0
$661$	1.35980e7	1.21052	0.605258	−	0.796029i	$-0.293070\pi$
$661$	1.35980e7	1.21052	0.605258	+	0.796029i	$0.293070\pi$
$662$	9.30137e6	0.824901
$663$	−1.34184e6	−0.118554
$664$	−3.73654e6	−0.328889
$665$	0	0
$666$	−1.72106e6	−0.150352
$667$	4.79927e6	0.417697
$668$	−1.62445e6	−0.140853
$669$	−2.91087e6	−0.251453
$670$	0	0
$671$	−2.39342e7	−2.05216
$672$	164131.	0.0140206
$673$	2.95663e6	0.251629	0.125814	−	0.992054i	$-0.459846\pi$
$673$	2.95663e6	0.251629	0.125814	+	0.992054i	$0.459846\pi$
$674$	1.14775e7	0.973194
$675$	0	0
$676$	456976.	0.0384615
$677$	1.47847e7	1.23977	0.619886	−	0.784692i	$-0.287179\pi$
$677$	1.47847e7	1.23977	0.619886	+	0.784692i	$0.287179\pi$
$678$	−246047.	−0.0205563
$679$	−2.53569e6	−0.211068
$680$	0	0
$681$	−5.58929e6	−0.461837
$682$	−1.86485e7	−1.53527
$683$	−7.10311e6	−0.582636	−0.291318	−	0.956626i	$-0.594094\pi$
$683$	−7.10311e6	−0.582636	−0.291318	+	0.956626i	$0.594094\pi$
$684$	4.30963e6	0.352208
$685$	0	0
$686$	2.22458e6	0.180484
$687$	−518379.	−0.0419040
$688$	3.59133e6	0.289258
$689$	661787.	0.0531093
$690$	0	0
$691$	1.42091e6	0.113206	0.0566032	−	0.998397i	$-0.481973\pi$
$691$	1.42091e6	0.113206	0.0566032	+	0.998397i	$0.481973\pi$
$692$	−4.95356e6	−0.393235
$693$	−1.74333e6	−0.137895
$694$	1.32538e7	1.04458
$695$	0	0
$696$	−1.24239e6	−0.0972154
$697$	1.60279e7	1.24967
$698$	−1.39002e7	−1.07990
$699$	8.62566e6	0.667728
$700$	0	0
$701$	2.53557e7	1.94886	0.974431	−	0.224685i	$-0.0721353\pi$
$701$	2.53557e7	1.94886	0.974431	+	0.224685i	$0.0721353\pi$
$702$	−2.55693e6	−0.195828
$703$	−5.10331e6	−0.389460
$704$	−2.84027e6	−0.215987
$705$	0	0
$706$	1.12821e7	0.851877
$707$	−1.95154e6	−0.146835
$708$	3.69342e6	0.276915
$709$	−1.46662e7	−1.09572	−0.547862	−	0.836569i	$-0.684558\pi$
$709$	−1.46662e7	−1.09572	−0.547862	+	0.836569i	$0.684558\pi$
$710$	0	0
$711$	6.23263e6	0.462378
$712$	3.11944e6	0.230609
$713$	1.59696e7	1.17644
$714$	−529848.	−0.0388961
$715$	0	0
$716$	−3.05756e6	−0.222891
$717$	9.86277e6	0.716475
$718$	−9.29405e6	−0.672812
$719$	4.43789e6	0.320150	0.160075	−	0.987105i	$-0.448826\pi$
$719$	4.43789e6	0.320150	0.160075	+	0.987105i	$0.448826\pi$
$720$	0	0
$721$	2.05928e6	0.147529
$722$	2.87459e6	0.205226
$723$	−1.09006e7	−0.775538
$724$	1.11111e7	0.787789
$725$	0	0
$726$	−1.22894e7	−0.865347
$727$	−1.07950e7	−0.757505	−0.378752	−	0.925498i	$-0.623647\pi$
$727$	−1.07950e7	−0.757505	−0.378752	+	0.925498i	$0.623647\pi$
$728$	180445.	0.0126188
$729$	8.78845e6	0.612482
$730$	0	0
$731$	−1.15936e7	−0.802463
$732$	−5.30578e6	−0.365992
$733$	−1.67123e7	−1.14889	−0.574444	−	0.818544i	$-0.694782\pi$
$733$	−1.67123e7	−1.14889	−0.574444	+	0.818544i	$0.694782\pi$
$734$	8.50544e6	0.582716
$735$	0	0
$736$	2.43225e6	0.165506
$737$	1.41771e7	0.961430
$738$	1.16906e7	0.790125
$739$	−8.52029e6	−0.573909	−0.286955	−	0.957944i	$-0.592643\pi$
$739$	−8.52029e6	−0.573909	−0.286955	+	0.957944i	$0.592643\pi$
$740$	0	0
$741$	−2.90212e6	−0.194164
$742$	261319.	0.0174245
$743$	1.57051e7	1.04369	0.521843	−	0.853042i	$-0.325245\pi$
$743$	1.57051e7	1.04369	0.521843	+	0.853042i	$0.325245\pi$
$744$	−4.13405e6	−0.273806
$745$	0	0
$746$	−1.56039e7	−1.02656
$747$	8.79816e6	0.576886
$748$	9.16900e6	0.599194
$749$	−2.64763e6	−0.172446
$750$	0	0
$751$	−3.79051e6	−0.245243	−0.122622	−	0.992453i	$-0.539130\pi$
$751$	−3.79051e6	−0.245243	−0.122622	+	0.992453i	$0.539130\pi$
$752$	5.11724e6	0.329983
$753$	6.70054e6	0.430648
$754$	−1.36589e6	−0.0874956
$755$	0	0
$756$	−1.00965e6	−0.0642489
$757$	5.62004e6	0.356451	0.178226	−	0.983990i	$-0.442964\pi$
$757$	5.62004e6	0.356451	0.178226	+	0.983990i	$0.442964\pi$
$758$	−1.48420e7	−0.938251
$759$	1.58240e7	0.997041
$760$	0	0
$761$	−1.97957e6	−0.123911	−0.0619554	−	0.998079i	$-0.519734\pi$
$761$	−1.97957e6	−0.123911	−0.0619554	+	0.998079i	$0.519734\pi$
$762$	4.87201e6	0.303963
$763$	720734.	0.0448191
$764$	−9.30021e6	−0.576447
$765$	0	0
$766$	1.44358e7	0.888934
$767$	4.06056e6	0.249228
$768$	−629637.	−0.0385201
$769$	3.06187e7	1.86711	0.933557	−	0.358429i	$-0.116688\pi$
$769$	3.06187e7	1.86711	0.933557	+	0.358429i	$0.116688\pi$
$770$	0	0
$771$	1.50512e7	0.911872
$772$	−851215.	−0.0514039
$773$	−7.23639e6	−0.435585	−0.217793	−	0.975995i	$-0.569886\pi$
$773$	−7.23639e6	−0.435585	−0.217793	+	0.975995i	$0.569886\pi$
$774$	−8.45624e6	−0.507370
$775$	0	0
$776$	9.72742e6	0.579887
$777$	457639.	0.0271938
$778$	−6.62855e6	−0.392617
$779$	3.46652e7	2.04668
$780$	0	0
$781$	−3.04630e7	−1.78709
$782$	−7.85181e6	−0.459148
$783$	7.64256e6	0.445487
$784$	−4.23134e6	−0.245860
$785$	0	0
$786$	1.43734e7	0.829854
$787$	−1.68586e7	−0.970255	−0.485127	−	0.874444i	$-0.661227\pi$
$787$	−1.68586e7	−0.970255	−0.485127	+	0.874444i	$0.661227\pi$
$788$	1.40465e7	0.805848
$789$	−9.97985e6	−0.570731
$790$	0	0
$791$	−106814.	−0.00606997
$792$	6.68777e6	0.378851
$793$	−5.83318e6	−0.329399
$794$	8.41654e6	0.473786
$795$	0	0
$796$	1.65460e7	0.925571
$797$	1.33831e7	0.746293	0.373147	−	0.927772i	$-0.378279\pi$
$797$	1.33831e7	0.746293	0.373147	+	0.927772i	$0.378279\pi$
$798$	−1.14596e6	−0.0637031
$799$	−1.65196e7	−0.915443
$800$	0	0
$801$	−7.34510e6	−0.404498
$802$	1.19269e7	0.654777
$803$	−1.26699e7	−0.693400
$804$	3.14280e6	0.171466
$805$	0	0
$806$	−4.54498e6	−0.246430
$807$	−1.03707e6	−0.0560564
$808$	7.48648e6	0.403413
$809$	6.55526e6	0.352142	0.176071	−	0.984377i	$-0.443661\pi$
$809$	6.55526e6	0.352142	0.176071	+	0.984377i	$0.443661\pi$
$810$	0	0
$811$	1.94125e7	1.03640	0.518202	−	0.855258i	$-0.326602\pi$
$811$	1.94125e7	1.03640	0.518202	+	0.855258i	$0.326602\pi$
$812$	−539345.	−0.0287063
$813$	1.01928e7	0.540839
$814$	−7.91941e6	−0.418921
$815$	0	0
$816$	2.03260e6	0.106863
$817$	−2.50746e7	−1.31425
$818$	1.19216e7	0.622949
$819$	−424881.	−0.0221339
$820$	0	0
$821$	7.46292e6	0.386412	0.193206	−	0.981158i	$-0.438111\pi$
$821$	7.46292e6	0.386412	0.193206	+	0.981158i	$0.438111\pi$
$822$	−1.38650e7	−0.715718
$823$	2.21232e7	1.13854	0.569271	−	0.822150i	$-0.307225\pi$
$823$	2.21232e7	1.13854	0.569271	+	0.822150i	$0.307225\pi$
$824$	−7.89982e6	−0.405321
$825$	0	0
$826$	1.60339e6	0.0817688
$827$	−8.40565e6	−0.427374	−0.213687	−	0.976902i	$-0.568547\pi$
$827$	−8.40565e6	−0.427374	−0.213687	+	0.976902i	$0.568547\pi$
$828$	−5.72703e6	−0.290304
$829$	7.27092e6	0.367454	0.183727	−	0.982977i	$-0.441184\pi$
$829$	7.27092e6	0.367454	0.183727	+	0.982977i	$0.441184\pi$
$830$	0	0
$831$	5.03065e6	0.252709
$832$	−692224.	−0.0346688
$833$	1.36597e7	0.682068
$834$	−1.56219e7	−0.777713
$835$	0	0
$836$	1.98307e7	0.981346
$837$	2.54306e7	1.25471
$838$	−2.82493e7	−1.38963
$839$	2.59125e7	1.27088	0.635441	−	0.772150i	$-0.280818\pi$
$839$	2.59125e7	1.27088	0.635441	+	0.772150i	$0.280818\pi$
$840$	0	0
$841$	−1.64286e7	−0.800958
$842$	1.19981e6	0.0583218
$843$	−1.19689e7	−0.580077
$844$	−748388.	−0.0361635
$845$	0	0
$846$	−1.20492e7	−0.578804
$847$	−5.33507e6	−0.255524
$848$	−1.00247e6	−0.0478720
$849$	7.41955e6	0.353271
$850$	0	0
$851$	6.78174e6	0.321009
$852$	−6.75311e6	−0.318717
$853$	3.64342e7	1.71450	0.857248	−	0.514903i	$-0.172172\pi$
$853$	3.64342e7	1.71450	0.857248	+	0.514903i	$0.172172\pi$
$854$	−2.30334e6	−0.108072
$855$	0	0
$856$	1.01568e7	0.473776
$857$	2.62890e7	1.22271	0.611353	−	0.791358i	$-0.290625\pi$
$857$	2.62890e7	1.22271	0.611353	+	0.791358i	$0.290625\pi$
$858$	−4.50357e6	−0.208852
$859$	1.81329e7	0.838465	0.419233	−	0.907879i	$-0.362299\pi$
$859$	1.81329e7	0.838465	0.419233	+	0.907879i	$0.362299\pi$
$860$	0	0
$861$	−3.10860e6	−0.142908
$862$	−5.19831e6	−0.238283
$863$	4.85495e6	0.221900	0.110950	−	0.993826i	$-0.464611\pi$
$863$	4.85495e6	0.221900	0.110950	+	0.993826i	$0.464611\pi$
$864$	3.87321e6	0.176517
$865$	0	0
$866$	2.17719e7	0.986512
$867$	7.07960e6	0.319861
$868$	−1.79467e6	−0.0808509
$869$	2.86794e7	1.28831
$870$	0	0
$871$	3.45520e6	0.154322
$872$	−2.76488e6	−0.123136
$873$	−2.29044e7	−1.01715
$874$	−1.69819e7	−0.751982
$875$	0	0
$876$	−2.80869e6	−0.123664
$877$	3.52512e7	1.54766	0.773828	−	0.633395i	$-0.218339\pi$
$877$	3.52512e7	1.54766	0.773828	+	0.633395i	$0.218339\pi$
$878$	−1.74092e7	−0.762153
$879$	1.49316e7	0.651829
$880$	0	0
$881$	−3.12586e6	−0.135684	−0.0678422	−	0.997696i	$-0.521611\pi$
$881$	−3.12586e6	−0.135684	−0.0678422	+	0.997696i	$0.521611\pi$
$882$	9.96322e6	0.431249
$883$	1.58054e7	0.682188	0.341094	−	0.940029i	$-0.389202\pi$
$883$	1.58054e7	0.682188	0.341094	+	0.940029i	$0.389202\pi$
$884$	2.23465e6	0.0961786
$885$	0	0
$886$	1.10331e7	0.472186
$887$	1.75515e7	0.749038	0.374519	−	0.927219i	$-0.377808\pi$
$887$	1.75515e7	0.749038	0.374519	+	0.927219i	$0.377808\pi$
$888$	−1.75559e6	−0.0747122
$889$	2.11503e6	0.0897558
$890$	0	0
$891$	−193737.	−0.00817557
$892$	4.84766e6	0.203995
$893$	−3.57285e7	−1.49929
$894$	−4.95110e6	−0.207185
$895$	0	0
$896$	−273337.	−0.0113744
$897$	3.85660e6	0.160038
$898$	2.88160e6	0.119246
$899$	1.35848e7	0.560601
$900$	0	0
$901$	3.23619e6	0.132807
$902$	5.37941e7	2.20150
$903$	2.24856e6	0.0917669
$904$	409759.	0.0166766
$905$	0	0
$906$	−6.91953e6	−0.280063
$907$	403972.	0.0163055	0.00815273	−	0.999967i	$-0.497405\pi$
$907$	403972.	0.0163055	0.00815273	+	0.999967i	$0.497405\pi$
$908$	9.30821e6	0.374672
$909$	−1.76279e7	−0.707603
$910$	0	0
$911$	1.17873e6	0.0470564	0.0235282	−	0.999723i	$-0.492510\pi$
$911$	1.17873e6	0.0470564	0.0235282	+	0.999723i	$0.492510\pi$
$912$	4.39611e6	0.175017
$913$	4.04846e7	1.60736
$914$	2.74964e7	1.08871
$915$	0	0
$916$	863291.	0.0339953
$917$	6.23974e6	0.245044
$918$	−1.25036e7	−0.489696
$919$	5.16739e6	0.201828	0.100914	−	0.994895i	$-0.467823\pi$
$919$	5.16739e6	0.201828	0.100914	+	0.994895i	$0.467823\pi$
$920$	0	0
$921$	−8.03919e6	−0.312294
$922$	1.30046e7	0.503814
$923$	−7.42438e6	−0.286851
$924$	−1.77832e6	−0.0685218
$925$	0	0
$926$	5.64704e6	0.216418
$927$	1.86011e7	0.710951
$928$	2.06903e6	0.0788674
$929$	1.58742e7	0.603467	0.301733	−	0.953392i	$-0.402435\pi$
$929$	1.58742e7	0.603467	0.301733	+	0.953392i	$0.402435\pi$
$930$	0	0
$931$	2.95431e7	1.11707
$932$	−1.43649e7	−0.541704
$933$	−2.09996e7	−0.789783
$934$	−3.62808e7	−1.36085
$935$	0	0
$936$	1.62993e6	0.0608105
$937$	3.42717e7	1.27522	0.637612	−	0.770358i	$-0.279922\pi$
$937$	3.42717e7	1.27522	0.637612	+	0.770358i	$0.279922\pi$
$938$	1.36435e6	0.0506313
$939$	1.01130e7	0.374297
$940$	0	0
$941$	−557810.	−0.0205358	−0.0102679	−	0.999947i	$-0.503268\pi$
$941$	−557810.	−0.0205358	−0.0102679	+	0.999947i	$0.503268\pi$
$942$	1.23482e7	0.453394
$943$	−4.60662e7	−1.68696
$944$	−6.15090e6	−0.224651
$945$	0	0
$946$	−3.89113e7	−1.41367
$947$	3.22519e7	1.16864	0.584319	−	0.811524i	$-0.301362\pi$
$947$	3.22519e7	1.16864	0.584319	+	0.811524i	$0.301362\pi$
$948$	6.35770e6	0.229763
$949$	−3.08788e6	−0.111300
$950$	0	0
$951$	−1.05520e7	−0.378342
$952$	882392.	0.0315551
$953$	2.42726e7	0.865732	0.432866	−	0.901458i	$-0.357502\pi$
$953$	2.42726e7	0.865732	0.432866	+	0.901458i	$0.357502\pi$
$954$	2.36044e6	0.0839696
$955$	0	0
$956$	−1.64251e7	−0.581251
$957$	1.34610e7	0.475114
$958$	−1.43091e7	−0.503730
$959$	−6.01908e6	−0.211341
$960$	0	0
$961$	1.65742e7	0.578928
$962$	−1.93010e6	−0.0672423
$963$	−2.39155e7	−0.831024
$964$	1.81534e7	0.629167
$965$	0	0
$966$	1.52285e6	0.0525067
$967$	4.36346e7	1.50060	0.750300	−	0.661097i	$-0.229909\pi$
$967$	4.36346e7	1.50060	0.750300	+	0.661097i	$0.229909\pi$
$968$	2.04664e7	0.702026
$969$	−1.41916e7	−0.485536
$970$	0	0
$971$	−2.49001e7	−0.847526	−0.423763	−	0.905773i	$-0.639291\pi$
$971$	−2.49001e7	−0.847526	−0.423763	+	0.905773i	$0.639291\pi$
$972$	−1.47490e7	−0.500724
$973$	−6.78177e6	−0.229647
$974$	−3.89185e7	−1.31449
$975$	0	0
$976$	8.83606e6	0.296916
$977$	−4.31252e7	−1.44542	−0.722711	−	0.691150i	$-0.757104\pi$
$977$	−4.31252e7	−1.44542	−0.722711	+	0.691150i	$0.757104\pi$
$978$	−5.19247e6	−0.173591
$979$	−3.37984e7	−1.12704
$980$	0	0
$981$	6.51024e6	0.215986
$982$	−1.17092e6	−0.0387479
$983$	−2.51180e7	−0.829089	−0.414544	−	0.910029i	$-0.636059\pi$
$983$	−2.51180e7	−0.829089	−0.414544	+	0.910029i	$0.636059\pi$
$984$	1.19252e7	0.392625
$985$	0	0
$986$	−6.67929e6	−0.218795
$987$	3.20395e6	0.104687
$988$	4.83309e6	0.157519
$989$	3.33214e7	1.08326
$990$	0	0
$991$	−1.35160e7	−0.437182	−0.218591	−	0.975817i	$-0.570146\pi$
$991$	−1.35160e7	−0.437182	−0.218591	+	0.975817i	$0.570146\pi$
$992$	6.88470e6	0.222129
$993$	−2.23407e7	−0.718992
$994$	−2.93165e6	−0.0941124
$995$	0	0
$996$	8.97471e6	0.286663
$997$	−3.69725e7	−1.17799	−0.588994	−	0.808138i	$-0.700476\pi$
$997$	−3.69725e7	−1.17799	−0.588994	+	0.808138i	$0.700476\pi$
$998$	7.80038e6	0.247907
$999$	1.07995e7	0.342366

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	650.6.a.j.1.1		3
5.2	odd	4		650.6.b.j.599.6		6
5.3	odd	4		650.6.b.j.599.1		6
5.4	even	2		130.6.a.f.1.3	✓	3
20.19	odd	2		1040.6.a.l.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.6.a.f.1.3	✓	3	5.4	even	2
650.6.a.j.1.1		3	1.1	even	1	trivial
650.6.b.j.599.1		6	5.3	odd	4
650.6.b.j.599.6		6	5.2	odd	4
1040.6.a.l.1.1		3	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	650.a (trivial)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} -9.60750 q^{3} +16.0000 q^{4} -38.4300 q^{6} -16.6832 q^{7} +64.0000 q^{8} -150.696 q^{9} +O(q^{10})\)	\(q+4.00000 q^{2} -9.60750 q^{3} +16.0000 q^{4} -38.4300 q^{6} -16.6832 q^{7} +64.0000 q^{8} -150.696 q^{9} -693.425 q^{11} -153.720 q^{12} -169.000 q^{13} -66.7328 q^{14} +256.000 q^{16} -826.423 q^{17} -602.784 q^{18} -1787.39 q^{19} +160.284 q^{21} -2773.70 q^{22} +2375.24 q^{23} -614.880 q^{24} -676.000 q^{26} +3782.43 q^{27} -266.931 q^{28} +2020.54 q^{29} +6723.34 q^{31} +1024.00 q^{32} +6662.08 q^{33} -3305.69 q^{34} -2411.14 q^{36} +2855.18 q^{37} -7149.54 q^{38} +1623.67 q^{39} -19394.3 q^{41} +641.135 q^{42} +14028.7 q^{43} -11094.8 q^{44} +9500.97 q^{46} +19989.2 q^{47} -2459.52 q^{48} -16528.7 q^{49} +7939.86 q^{51} -2704.00 q^{52} -3915.90 q^{53} +15129.7 q^{54} -1067.72 q^{56} +17172.3 q^{57} +8082.17 q^{58} -24027.0 q^{59} +34515.9 q^{61} +26893.4 q^{62} +2514.09 q^{63} +4096.00 q^{64} +26648.3 q^{66} -20445.0 q^{67} -13222.8 q^{68} -22820.1 q^{69} +43931.2 q^{71} -9644.54 q^{72} +18271.5 q^{73} +11420.7 q^{74} -28598.2 q^{76} +11568.5 q^{77} +6494.67 q^{78} -41359.0 q^{79} +279.391 q^{81} -77577.4 q^{82} -58383.5 q^{83} +2564.54 q^{84} +56114.6 q^{86} -19412.3 q^{87} -44379.2 q^{88} +48741.2 q^{89} +2819.46 q^{91} +38003.9 q^{92} -64594.5 q^{93} +79956.9 q^{94} -9838.08 q^{96} +151991. q^{97} -66114.7 q^{98} +104496. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 12 q^{2} + 22 q^{3} + 48 q^{4} + 88 q^{6} + 234 q^{7} + 192 q^{8} + 155 q^{9}+O(q^{10}) \) 3 * q + 12 * q^2 + 22 * q^3 + 48 * q^4 + 88 * q^6 + 234 * q^7 + 192 * q^8 + 155 * q^9	\( 3 q + 12 q^{2} + 22 q^{3} + 48 q^{4} + 88 q^{6} + 234 q^{7} + 192 q^{8} + 155 q^{9} + 48 q^{11} + 352 q^{12} - 507 q^{13} + 936 q^{14} + 768 q^{16} - 1506 q^{17} + 620 q^{18} - 360 q^{19} + 6904 q^{21} + 192 q^{22} + 2370 q^{23} + 1408 q^{24} - 2028 q^{26} + 10168 q^{27} + 3744 q^{28} - 3078 q^{29} - 5388 q^{31} + 3072 q^{32} + 22572 q^{33} - 6024 q^{34} + 2480 q^{36} + 25362 q^{37} - 1440 q^{38} - 3718 q^{39} - 15906 q^{41} + 27616 q^{42} + 39306 q^{43} + 768 q^{44} + 9480 q^{46} + 17778 q^{47} + 5632 q^{48} + 7767 q^{49} - 33972 q^{51} - 8112 q^{52} - 9246 q^{53} + 40672 q^{54} + 14976 q^{56} + 2996 q^{57} - 12312 q^{58} - 77760 q^{59} + 17982 q^{61} - 21552 q^{62} + 128762 q^{63} + 12288 q^{64} + 90288 q^{66} + 2922 q^{67} - 24096 q^{68} - 31860 q^{69} - 4944 q^{71} + 9920 q^{72} - 43278 q^{73} + 101448 q^{74} - 5760 q^{76} + 144420 q^{77} - 14872 q^{78} + 42120 q^{79} + 146519 q^{81} - 63624 q^{82} - 58098 q^{83} + 110464 q^{84} + 157224 q^{86} - 2016 q^{87} + 3072 q^{88} + 19614 q^{89} - 39546 q^{91} + 37920 q^{92} - 162796 q^{93} + 71112 q^{94} + 22528 q^{96} + 87078 q^{97} + 31068 q^{98} + 350340 q^{99}+O(q^{100}) \) 3 * q + 12 * q^2 + 22 * q^3 + 48 * q^4 + 88 * q^6 + 234 * q^7 + 192 * q^8 + 155 * q^9 + 48 * q^11 + 352 * q^12 - 507 * q^13 + 936 * q^14 + 768 * q^16 - 1506 * q^17 + 620 * q^18 - 360 * q^19 + 6904 * q^21 + 192 * q^22 + 2370 * q^23 + 1408 * q^24 - 2028 * q^26 + 10168 * q^27 + 3744 * q^28 - 3078 * q^29 - 5388 * q^31 + 3072 * q^32 + 22572 * q^33 - 6024 * q^34 + 2480 * q^36 + 25362 * q^37 - 1440 * q^38 - 3718 * q^39 - 15906 * q^41 + 27616 * q^42 + 39306 * q^43 + 768 * q^44 + 9480 * q^46 + 17778 * q^47 + 5632 * q^48 + 7767 * q^49 - 33972 * q^51 - 8112 * q^52 - 9246 * q^53 + 40672 * q^54 + 14976 * q^56 + 2996 * q^57 - 12312 * q^58 - 77760 * q^59 + 17982 * q^61 - 21552 * q^62 + 128762 * q^63 + 12288 * q^64 + 90288 * q^66 + 2922 * q^67 - 24096 * q^68 - 31860 * q^69 - 4944 * q^71 + 9920 * q^72 - 43278 * q^73 + 101448 * q^74 - 5760 * q^76 + 144420 * q^77 - 14872 * q^78 + 42120 * q^79 + 146519 * q^81 - 63624 * q^82 - 58098 * q^83 + 110464 * q^84 + 157224 * q^86 - 2016 * q^87 + 3072 * q^88 + 19614 * q^89 - 39546 * q^91 + 37920 * q^92 - 162796 * q^93 + 71112 * q^94 + 22528 * q^96 + 87078 * q^97 + 31068 * q^98 + 350340 * q^99

Introduction

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