LMFDB - Embedded newform 825.6.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	825.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$132.316651346$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.37228$ 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)

$f(q)$	$=$	$q-9.37228 q^{2} -9.00000 q^{3} +55.8397 q^{4} +84.3505 q^{6} +105.081 q^{7} -223.432 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-9.37228 q^{2} -9.00000 q^{3} +55.8397 q^{4} +84.3505 q^{6} +105.081 q^{7} -223.432 q^{8} +81.0000 q^{9} -121.000 q^{11} -502.557 q^{12} -147.549 q^{13} -984.853 q^{14} +307.198 q^{16} +1432.49 q^{17} -759.155 q^{18} +2033.18 q^{19} -945.733 q^{21} +1134.05 q^{22} -828.853 q^{23} +2010.89 q^{24} +1382.87 q^{26} -729.000 q^{27} +5867.71 q^{28} +4633.44 q^{29} -9835.65 q^{31} +4270.67 q^{32} +1089.00 q^{33} -13425.7 q^{34} +4523.01 q^{36} -7134.34 q^{37} -19055.6 q^{38} +1327.94 q^{39} +18265.0 q^{41} +8863.68 q^{42} -13822.5 q^{43} -6756.60 q^{44} +7768.24 q^{46} -22991.2 q^{47} -2764.78 q^{48} -5764.89 q^{49} -12892.4 q^{51} -8239.08 q^{52} -14311.9 q^{53} +6832.39 q^{54} -23478.6 q^{56} -18298.7 q^{57} -43425.9 q^{58} -7081.12 q^{59} -18470.2 q^{61} +92182.5 q^{62} +8511.60 q^{63} -49856.3 q^{64} -10206.4 q^{66} -16229.5 q^{67} +79989.7 q^{68} +7459.68 q^{69} +28198.9 q^{71} -18098.0 q^{72} +39382.9 q^{73} +66865.1 q^{74} +113532. q^{76} -12714.9 q^{77} -12445.8 q^{78} -41243.7 q^{79} +6561.00 q^{81} -171185. q^{82} +23355.3 q^{83} -52809.4 q^{84} +129549. q^{86} -41701.0 q^{87} +27035.3 q^{88} -103803. q^{89} -15504.6 q^{91} -46282.9 q^{92} +88520.9 q^{93} +215480. q^{94} -38436.1 q^{96} +149289. q^{97} +54030.2 q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 13 q^{2} - 18 q^{3} + 37 q^{4} + 117 q^{6} - 146 q^{7} - 39 q^{8} + 162 q^{9}+O(q^{10}) $ 2 * q - 13 * q^2 - 18 * q^3 + 37 * q^4 + 117 * q^6 - 146 * q^7 - 39 * q^8 + 162 * q^9	\( 2 q - 13 q^{2} - 18 q^{3} + 37 q^{4} + 117 q^{6} - 146 q^{7} - 39 q^{8} + 162 q^{9} - 242 q^{11} - 333 q^{12} + 130 q^{13} - 74 q^{14} + 241 q^{16} + 728 q^{17} - 1053 q^{18} - 828 q^{19} + 1314 q^{21} + 1573 q^{22} + 238 q^{23} + 351 q^{24} + 376 q^{26} - 1458 q^{27} + 10598 q^{28} + 696 q^{29} - 10480 q^{31} - 1391 q^{32} + 2178 q^{33} - 10870 q^{34} + 2997 q^{36} + 1908 q^{37} - 8676 q^{38} - 1170 q^{39} + 36484 q^{41} + 666 q^{42} - 9768 q^{43} - 4477 q^{44} + 3898 q^{46} - 43742 q^{47} - 2169 q^{48} + 40470 q^{49} - 6552 q^{51} - 13468 q^{52} + 12174 q^{53} + 9477 q^{54} - 69786 q^{56} + 7452 q^{57} - 29142 q^{58} - 2788 q^{59} - 25302 q^{61} + 94520 q^{62} - 11826 q^{63} - 27199 q^{64} - 14157 q^{66} + 40520 q^{67} + 93262 q^{68} - 2142 q^{69} + 31386 q^{71} - 3159 q^{72} + 46780 q^{73} + 34062 q^{74} + 167436 q^{76} + 17666 q^{77} - 3384 q^{78} - 16850 q^{79} + 13122 q^{81} - 237278 q^{82} - 79440 q^{83} - 95382 q^{84} + 114840 q^{86} - 6264 q^{87} + 4719 q^{88} - 54204 q^{89} - 85192 q^{91} - 66382 q^{92} + 94320 q^{93} + 290758 q^{94} + 12519 q^{96} + 241568 q^{97} - 113697 q^{98} - 19602 q^{99}+O(q^{100}) \) 2 * q - 13 * q^2 - 18 * q^3 + 37 * q^4 + 117 * q^6 - 146 * q^7 - 39 * q^8 + 162 * q^9 - 242 * q^11 - 333 * q^12 + 130 * q^13 - 74 * q^14 + 241 * q^16 + 728 * q^17 - 1053 * q^18 - 828 * q^19 + 1314 * q^21 + 1573 * q^22 + 238 * q^23 + 351 * q^24 + 376 * q^26 - 1458 * q^27 + 10598 * q^28 + 696 * q^29 - 10480 * q^31 - 1391 * q^32 + 2178 * q^33 - 10870 * q^34 + 2997 * q^36 + 1908 * q^37 - 8676 * q^38 - 1170 * q^39 + 36484 * q^41 + 666 * q^42 - 9768 * q^43 - 4477 * q^44 + 3898 * q^46 - 43742 * q^47 - 2169 * q^48 + 40470 * q^49 - 6552 * q^51 - 13468 * q^52 + 12174 * q^53 + 9477 * q^54 - 69786 * q^56 + 7452 * q^57 - 29142 * q^58 - 2788 * q^59 - 25302 * q^61 + 94520 * q^62 - 11826 * q^63 - 27199 * q^64 - 14157 * q^66 + 40520 * q^67 + 93262 * q^68 - 2142 * q^69 + 31386 * q^71 - 3159 * q^72 + 46780 * q^73 + 34062 * q^74 + 167436 * q^76 + 17666 * q^77 - 3384 * q^78 - 16850 * q^79 + 13122 * q^81 - 237278 * q^82 - 79440 * q^83 - 95382 * q^84 + 114840 * q^86 - 6264 * q^87 + 4719 * q^88 - 54204 * q^89 - 85192 * q^91 - 66382 * q^92 + 94320 * q^93 + 290758 * q^94 + 12519 * q^96 + 241568 * q^97 - 113697 * q^98 - 19602 * 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 $

$2$	−9.37228	−1.65680	−0.828400	−	0.560136i	$-0.810749\pi$
$2$	−9.37228	−1.65680	−0.828400	+	0.560136i	$0.810749\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	55.8397	1.74499
$5$	0	0
$5$	0	0
$6$	84.3505	0.956554
$7$	105.081	0.810552	0.405276	−	0.914194i	$-0.367175\pi$
$7$	105.081	0.810552	0.405276	+	0.914194i	$0.367175\pi$
$8$	−223.432	−1.23430
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	−502.557	−1.00747
$13$	−147.549	−0.242146	−0.121073	−	0.992644i	$-0.538634\pi$
$13$	−147.549	−0.242146	−0.121073	+	0.992644i	$0.538634\pi$
$14$	−984.853	−1.34292
$15$	0	0
$16$	307.198	0.299998
$17$	1432.49	1.20218	0.601089	−	0.799182i	$-0.294734\pi$
$17$	1432.49	1.20218	0.601089	+	0.799182i	$0.294734\pi$
$18$	−759.155	−0.552267
$19$	2033.18	1.29209	0.646045	−	0.763300i	$-0.276422\pi$
$19$	2033.18	1.29209	0.646045	+	0.763300i	$0.276422\pi$
$20$	0	0
$21$	−945.733	−0.467972
$22$	1134.05	0.499544
$23$	−828.853	−0.326707	−0.163353	−	0.986568i	$-0.552231\pi$
$23$	−828.853	−0.326707	−0.163353	+	0.986568i	$0.552231\pi$
$24$	2010.89	0.712623
$25$	0	0
$26$	1382.87	0.401188
$27$	−729.000	−0.192450
$28$	5867.71	1.41440
$29$	4633.44	1.02308	0.511539	−	0.859260i	$-0.329076\pi$
$29$	4633.44	1.02308	0.511539	+	0.859260i	$0.329076\pi$
$30$	0	0
$31$	−9835.65	−1.83823	−0.919113	−	0.393994i	$-0.871093\pi$
$31$	−9835.65	−1.83823	−0.919113	+	0.393994i	$0.871093\pi$
$32$	4270.67	0.737261
$33$	1089.00	0.174078
$34$	−13425.7	−1.99177
$35$	0	0
$36$	4523.01	0.581663
$37$	−7134.34	−0.856741	−0.428371	−	0.903603i	$-0.640912\pi$
$37$	−7134.34	−0.856741	−0.428371	+	0.903603i	$0.640912\pi$
$38$	−19055.6	−2.14074
$39$	1327.94	0.139803
$40$	0	0
$41$	18265.0	1.69691	0.848456	−	0.529265i	$-0.177532\pi$
$41$	18265.0	1.69691	0.848456	+	0.529265i	$0.177532\pi$
$42$	8863.68	0.775337
$43$	−13822.5	−1.14003	−0.570016	−	0.821634i	$-0.693063\pi$
$43$	−13822.5	−1.14003	−0.570016	+	0.821634i	$0.693063\pi$
$44$	−6756.60	−0.526134
$45$	0	0
$46$	7768.24	0.541288
$47$	−22991.2	−1.51816	−0.759079	−	0.650999i	$-0.774350\pi$
$47$	−22991.2	−1.51816	−0.759079	+	0.650999i	$0.774350\pi$
$48$	−2764.78	−0.173204
$49$	−5764.89	−0.343005
$50$	0	0
$51$	−12892.4	−0.694078
$52$	−8239.08	−0.422542
$53$	−14311.9	−0.699856	−0.349928	−	0.936777i	$-0.613794\pi$
$53$	−14311.9	−0.699856	−0.349928	+	0.936777i	$0.613794\pi$
$54$	6832.39	0.318851
$55$	0	0
$56$	−23478.6	−1.00046
$57$	−18298.7	−0.745988
$58$	−43425.9	−1.69504
$59$	−7081.12	−0.264833	−0.132416	−	0.991194i	$-0.542274\pi$
$59$	−7081.12	−0.264833	−0.132416	+	0.991194i	$0.542274\pi$
$60$	0	0
$61$	−18470.2	−0.635547	−0.317774	−	0.948167i	$-0.602935\pi$
$61$	−18470.2	−0.635547	−0.317774	+	0.948167i	$0.602935\pi$
$62$	92182.5	3.04557
$63$	8511.60	0.270184
$64$	−49856.3	−1.52149
$65$	0	0
$66$	−10206.4	−0.288412
$67$	−16229.5	−0.441690	−0.220845	−	0.975309i	$-0.570881\pi$
$67$	−16229.5	−0.441690	−0.220845	+	0.975309i	$0.570881\pi$
$68$	79989.7	2.09779
$69$	7459.68	0.188624
$70$	0	0
$71$	28198.9	0.663875	0.331938	−	0.943301i	$-0.392298\pi$
$71$	28198.9	0.663875	0.331938	+	0.943301i	$0.392298\pi$
$72$	−18098.0	−0.411433
$73$	39382.9	0.864968	0.432484	−	0.901642i	$-0.357637\pi$
$73$	39382.9	0.864968	0.432484	+	0.901642i	$0.357637\pi$
$74$	66865.1	1.41945
$75$	0	0
$76$	113532.	2.25468
$77$	−12714.9	−0.244391
$78$	−12445.8	−0.231626
$79$	−41243.7	−0.743515	−0.371758	−	0.928330i	$-0.621245\pi$
$79$	−41243.7	−0.743515	−0.371758	+	0.928330i	$0.621245\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−171185.	−2.81145
$83$	23355.3	0.372126	0.186063	−	0.982538i	$-0.440427\pi$
$83$	23355.3	0.372126	0.186063	+	0.982538i	$0.440427\pi$
$84$	−52809.4	−0.816607
$85$	0	0
$86$	129549.	1.88880
$87$	−41701.0	−0.590675
$88$	27035.3	0.372155
$89$	−103803.	−1.38911	−0.694555	−	0.719440i	$-0.744399\pi$
$89$	−103803.	−1.38911	−0.694555	+	0.719440i	$0.744399\pi$
$90$	0	0
$91$	−15504.6	−0.196272
$92$	−46282.9	−0.570099
$93$	88520.9	1.06130
$94$	215480.	2.51528
$95$	0	0
$96$	−38436.1	−0.425658
$97$	149289.	1.61101	0.805503	−	0.592592i	$-0.201895\pi$
$97$	149289.	1.61101	0.805503	+	0.592592i	$0.201895\pi$
$98$	54030.2	0.568292
$99$	−9801.00	−0.100504
$100$	0	0
$101$	62410.1	0.608768	0.304384	−	0.952549i	$-0.401549\pi$
$101$	62410.1	0.608768	0.304384	+	0.952549i	$0.401549\pi$
$102$	120831.	1.14995
$103$	90047.2	0.836329	0.418165	−	0.908371i	$-0.362674\pi$
$103$	90047.2	0.836329	0.418165	+	0.908371i	$0.362674\pi$
$104$	32967.1	0.298881
$105$	0	0
$106$	134136.	1.15952
$107$	−120458.	−1.01713	−0.508566	−	0.861023i	$-0.669824\pi$
$107$	−120458.	−1.01713	−0.508566	+	0.861023i	$0.669824\pi$
$108$	−40707.1	−0.335823
$109$	−91854.0	−0.740511	−0.370256	−	0.928930i	$-0.620730\pi$
$109$	−91854.0	−0.740511	−0.370256	+	0.928930i	$0.620730\pi$
$110$	0	0
$111$	64209.1	0.494640
$112$	32280.8	0.243164
$113$	29619.6	0.218214	0.109107	−	0.994030i	$-0.465201\pi$
$113$	29619.6	0.218214	0.109107	+	0.994030i	$0.465201\pi$
$114$	171500.	1.23595
$115$	0	0
$116$	258730.	1.78526
$117$	−11951.5	−0.0807153
$118$	66366.2	0.438775
$119$	150528.	0.974428
$120$	0	0
$121$	14641.0	0.0909091
$122$	173108.	1.05298
$123$	−164385.	−0.979713
$124$	−549219.	−3.20768
$125$	0	0
$126$	−79773.1	−0.447641
$127$	−271128.	−1.49165	−0.745823	−	0.666145i	$-0.767943\pi$
$127$	−271128.	−1.49165	−0.745823	+	0.666145i	$0.767943\pi$
$128$	330606.	1.78355
$129$	124403.	0.658197
$130$	0	0
$131$	−223990.	−1.14038	−0.570191	−	0.821512i	$-0.693131\pi$
$131$	−223990.	−1.14038	−0.570191	+	0.821512i	$0.693131\pi$
$132$	60809.4	0.303764
$133$	213650.	1.04731
$134$	152107.	0.731792
$135$	0	0
$136$	−320064.	−1.48385
$137$	348229.	1.58513	0.792563	−	0.609789i	$-0.208746\pi$
$137$	348229.	1.58513	0.792563	+	0.609789i	$0.208746\pi$
$138$	−69914.2	−0.312513
$139$	333802.	1.46538	0.732692	−	0.680560i	$-0.238264\pi$
$139$	333802.	1.46538	0.732692	+	0.680560i	$0.238264\pi$
$140$	0	0
$141$	206921.	0.876509
$142$	−264288.	−1.09991
$143$	17853.4	0.0730098
$144$	24883.1	0.0999994
$145$	0	0
$146$	−369107.	−1.43308
$147$	51884.0	0.198034
$148$	−398379.	−1.49500
$149$	310249.	1.14484	0.572419	−	0.819961i	$-0.306005\pi$
$149$	310249.	1.14484	0.572419	+	0.819961i	$0.306005\pi$
$150$	0	0
$151$	−54751.5	−0.195413	−0.0977066	−	0.995215i	$-0.531151\pi$
$151$	−54751.5	−0.195413	−0.0977066	+	0.995215i	$0.531151\pi$
$152$	−454278.	−1.59482
$153$	116032.	0.400726
$154$	119167.	0.404907
$155$	0	0
$156$	74151.7	0.243955
$157$	−221556.	−0.717354	−0.358677	−	0.933462i	$-0.616772\pi$
$157$	−221556.	−0.717354	−0.358677	+	0.933462i	$0.616772\pi$
$158$	386547.	1.23186
$159$	128807.	0.404062
$160$	0	0
$161$	−87097.1	−0.264813
$162$	−61491.5	−0.184089
$163$	225239.	0.664010	0.332005	−	0.943278i	$-0.392275\pi$
$163$	225239.	0.664010	0.332005	+	0.943278i	$0.392275\pi$
$164$	1.01991e6	2.96109
$165$	0	0
$166$	−218892.	−0.616539
$167$	−186796.	−0.518294	−0.259147	−	0.965838i	$-0.583441\pi$
$167$	−186796.	−0.518294	−0.259147	+	0.965838i	$0.583441\pi$
$168$	211307.	0.577618
$169$	−349522.	−0.941365
$170$	0	0
$171$	164688.	0.430697
$172$	−771846.	−1.98934
$173$	713662.	1.81292	0.906458	−	0.422296i	$-0.138776\pi$
$173$	713662.	1.81292	0.906458	+	0.422296i	$0.138776\pi$
$174$	390833.	0.978630
$175$	0	0
$176$	−37171.0	−0.0904529
$177$	63730.1	0.152901
$178$	972875.	2.30148
$179$	789450.	1.84159	0.920793	−	0.390051i	$-0.127543\pi$
$179$	789450.	1.84159	0.920793	+	0.390051i	$0.127543\pi$
$180$	0	0
$181$	40330.7	0.0915039	0.0457519	−	0.998953i	$-0.485432\pi$
$181$	40330.7	0.0915039	0.0457519	+	0.998953i	$0.485432\pi$
$182$	145314.	0.325184
$183$	166232.	0.366934
$184$	185192.	0.403254
$185$	0	0
$186$	−829642.	−1.75836
$187$	−173331.	−0.362470
$188$	−1.28382e6	−2.64917
$189$	−76604.4	−0.155991
$190$	0	0
$191$	133736.	0.265256	0.132628	−	0.991166i	$-0.457658\pi$
$191$	133736.	0.265256	0.132628	+	0.991166i	$0.457658\pi$
$192$	448707.	0.878435
$193$	462266.	0.893303	0.446652	−	0.894708i	$-0.352616\pi$
$193$	462266.	0.893303	0.446652	+	0.894708i	$0.352616\pi$
$194$	−1.39917e6	−2.66912
$195$	0	0
$196$	−321910.	−0.598541
$197$	−96432.5	−0.177035	−0.0885173	−	0.996075i	$-0.528213\pi$
$197$	−96432.5	−0.177035	−0.0885173	+	0.996075i	$0.528213\pi$
$198$	91857.7	0.166515
$199$	−1.08264e6	−1.93800	−0.968999	−	0.247066i	$-0.920533\pi$
$199$	−1.08264e6	−1.93800	−0.968999	+	0.247066i	$0.920533\pi$
$200$	0	0
$201$	146065.	0.255010
$202$	−584925.	−1.00861
$203$	486889.	0.829258
$204$	−719907.	−1.21116
$205$	0	0
$206$	−843948.	−1.38563
$207$	−67137.1	−0.108902
$208$	−45326.7	−0.0726434
$209$	−246015.	−0.389580
$210$	0	0
$211$	875588.	1.35392	0.676961	−	0.736019i	$-0.263297\pi$
$211$	875588.	1.35392	0.676961	+	0.736019i	$0.263297\pi$
$212$	−799174.	−1.22124
$213$	−253790.	−0.383289
$214$	1.12897e6	1.68518
$215$	0	0
$216$	162882.	0.237541
$217$	−1.03354e6	−1.48998
$218$	860881.	1.22688
$219$	−354446.	−0.499390
$220$	0	0
$221$	−211362.	−0.291103
$222$	−601786.	−0.819520
$223$	−988042.	−1.33049	−0.665247	−	0.746623i	$-0.731674\pi$
$223$	−988042.	−1.33049	−0.665247	+	0.746623i	$0.731674\pi$
$224$	448769.	0.597589
$225$	0	0
$226$	−277603.	−0.361537
$227$	−67066.0	−0.0863849	−0.0431925	−	0.999067i	$-0.513753\pi$
$227$	−67066.0	−0.0863849	−0.0431925	+	0.999067i	$0.513753\pi$
$228$	−1.02179e6	−1.30174
$229$	−662131.	−0.834363	−0.417182	−	0.908823i	$-0.636982\pi$
$229$	−662131.	−0.834363	−0.417182	+	0.908823i	$0.636982\pi$
$230$	0	0
$231$	114434.	0.141099
$232$	−1.03526e6	−1.26278
$233$	−1.37961e6	−1.66482	−0.832411	−	0.554159i	$-0.813040\pi$
$233$	−1.37961e6	−1.66482	−0.832411	+	0.554159i	$0.813040\pi$
$234$	112012.	0.133729
$235$	0	0
$236$	−395407.	−0.462130
$237$	371193.	0.429269
$238$	−1.41079e6	−1.61443
$239$	423791.	0.479907	0.239954	−	0.970784i	$-0.422868\pi$
$239$	423791.	0.479907	0.239954	+	0.970784i	$0.422868\pi$
$240$	0	0
$241$	−1.34802e6	−1.49504	−0.747521	−	0.664238i	$-0.768756\pi$
$241$	−1.34802e6	−1.49504	−0.747521	+	0.664238i	$0.768756\pi$
$242$	−137220.	−0.150618
$243$	−59049.0	−0.0641500
$244$	−1.03137e6	−1.10902
$245$	0	0
$246$	1.54066e6	1.62319
$247$	−299994.	−0.312874
$248$	2.19760e6	2.26892
$249$	−210198.	−0.214847
$250$	0	0
$251$	−1.45914e6	−1.46188	−0.730940	−	0.682442i	$-0.760918\pi$
$251$	−1.45914e6	−1.46188	−0.730940	+	0.682442i	$0.760918\pi$
$252$	475285.	0.471468
$253$	100291.	0.0985057
$254$	2.54109e6	2.47136
$255$	0	0
$256$	−1.50313e6	−1.43349
$257$	−548503.	−0.518020	−0.259010	−	0.965875i	$-0.583396\pi$
$257$	−548503.	−0.518020	−0.259010	+	0.965875i	$0.583396\pi$
$258$	−1.16594e6	−1.09050
$259$	−749687.	−0.694434
$260$	0	0
$261$	375309.	0.341026
$262$	2.09930e6	1.88939
$263$	−905299.	−0.807054	−0.403527	−	0.914968i	$-0.632216\pi$
$263$	−905299.	−0.807054	−0.403527	+	0.914968i	$0.632216\pi$
$264$	−243317.	−0.214864
$265$	0	0
$266$	−2.00239e6	−1.73518
$267$	934231.	0.802003
$268$	−906248.	−0.770744
$269$	−164041.	−0.138220	−0.0691100	−	0.997609i	$-0.522016\pi$
$269$	−164041.	−0.138220	−0.0691100	+	0.997609i	$0.522016\pi$
$270$	0	0
$271$	989636.	0.818563	0.409282	−	0.912408i	$-0.365779\pi$
$271$	989636.	0.818563	0.409282	+	0.912408i	$0.365779\pi$
$272$	440058.	0.360651
$273$	139542.	0.113318
$274$	−3.26370e6	−2.62624
$275$	0	0
$276$	416546.	0.329147
$277$	−727149.	−0.569409	−0.284704	−	0.958615i	$-0.591895\pi$
$277$	−727149.	−0.569409	−0.284704	+	0.958615i	$0.591895\pi$
$278$	−3.12848e6	−2.42785
$279$	−796688.	−0.612742
$280$	0	0
$281$	−379100.	−0.286410	−0.143205	−	0.989693i	$-0.545741\pi$
$281$	−379100.	−0.286410	−0.143205	+	0.989693i	$0.545741\pi$
$282$	−1.93932e6	−1.45220
$283$	695371.	0.516120	0.258060	−	0.966129i	$-0.416917\pi$
$283$	695371.	0.516120	0.258060	+	0.966129i	$0.416917\pi$
$284$	1.57462e6	1.15846
$285$	0	0
$286$	−167327.	−0.120963
$287$	1.91931e6	1.37544
$288$	345925.	0.245754
$289$	632167.	0.445233
$290$	0	0
$291$	−1.34360e6	−0.930115
$292$	2.19913e6	1.50936
$293$	−268373.	−0.182629	−0.0913146	−	0.995822i	$-0.529107\pi$
$293$	−268373.	−0.182629	−0.0913146	+	0.995822i	$0.529107\pi$
$294$	−486272.	−0.328103
$295$	0	0
$296$	1.59404e6	1.05747
$297$	88209.0	0.0580259
$298$	−2.90774e6	−1.89677
$299$	122296.	0.0791107
$300$	0	0
$301$	−1.45249e6	−0.924055
$302$	513147.	0.323761
$303$	−561691.	−0.351472
$304$	624591.	0.387625
$305$	0	0
$306$	−1.08748e6	−0.663923
$307$	−1.78841e6	−1.08298	−0.541491	−	0.840706i	$-0.682140\pi$
$307$	−1.78841e6	−1.08298	−0.541491	+	0.840706i	$0.682140\pi$
$308$	−709993.	−0.426459
$309$	−810425.	−0.482855
$310$	0	0
$311$	362727.	0.212657	0.106328	−	0.994331i	$-0.466091\pi$
$311$	362727.	0.212657	0.106328	+	0.994331i	$0.466091\pi$
$312$	−296704.	−0.172559
$313$	1.14382e6	0.659926	0.329963	−	0.943994i	$-0.392964\pi$
$313$	1.14382e6	0.659926	0.329963	+	0.943994i	$0.392964\pi$
$314$	2.07648e6	1.18851
$315$	0	0
$316$	−2.30303e6	−1.29743
$317$	−3.43750e6	−1.92130	−0.960649	−	0.277766i	$-0.910406\pi$
$317$	−3.43750e6	−1.92130	−0.960649	+	0.277766i	$0.910406\pi$
$318$	−1.20722e6	−0.669451
$319$	−560647.	−0.308470
$320$	0	0
$321$	1.08412e6	0.587241
$322$	816298.	0.438742
$323$	2.91251e6	1.55332
$324$	366364.	0.193888
$325$	0	0
$326$	−2.11100e6	−1.10013
$327$	826686.	0.427534
$328$	−4.08098e6	−2.09450
$329$	−2.41595e6	−1.23055
$330$	0	0
$331$	1.11428e6	0.559015	0.279507	−	0.960144i	$-0.409829\pi$
$331$	1.11428e6	0.559015	0.279507	+	0.960144i	$0.409829\pi$
$332$	1.30415e6	0.649356
$333$	−577882.	−0.285580
$334$	1.75070e6	0.858710
$335$	0	0
$336$	−290528.	−0.140391
$337$	2.25804e6	1.08307	0.541535	−	0.840678i	$-0.317843\pi$
$337$	2.25804e6	1.08307	0.541535	+	0.840678i	$0.317843\pi$
$338$	3.27582e6	1.55965
$339$	−266576.	−0.125986
$340$	0	0
$341$	1.19011e6	0.554246
$342$	−1.54350e6	−0.713578
$343$	−2.37189e6	−1.08858
$344$	3.08840e6	1.40714
$345$	0	0
$346$	−6.68865e6	−3.00364
$347$	1.17055e6	0.521873	0.260937	−	0.965356i	$-0.415969\pi$
$347$	1.17055e6	0.521873	0.260937	+	0.965356i	$0.415969\pi$
$348$	−2.32857e6	−1.03072
$349$	1.21704e6	0.534863	0.267432	−	0.963577i	$-0.413825\pi$
$349$	1.21704e6	0.534863	0.267432	+	0.963577i	$0.413825\pi$
$350$	0	0
$351$	107563.	0.0466010
$352$	−516752.	−0.222293
$353$	−286188.	−0.122240	−0.0611201	−	0.998130i	$-0.519467\pi$
$353$	−286188.	−0.122240	−0.0611201	+	0.998130i	$0.519467\pi$
$354$	−597296.	−0.253327
$355$	0	0
$356$	−5.79635e6	−2.42398
$357$	−1.35475e6	−0.562586
$358$	−7.39895e6	−3.05114
$359$	1.19066e6	0.487588	0.243794	−	0.969827i	$-0.421608\pi$
$359$	1.19066e6	0.487588	0.243794	+	0.969827i	$0.421608\pi$
$360$	0	0
$361$	1.65774e6	0.669495
$362$	−377991.	−0.151604
$363$	−131769.	−0.0524864
$364$	−865774.	−0.342492
$365$	0	0
$366$	−1.55797e6	−0.607936
$367$	−3.11120e6	−1.20577	−0.602884	−	0.797829i	$-0.705982\pi$
$367$	−3.11120e6	−1.20577	−0.602884	+	0.797829i	$0.705982\pi$
$368$	−254622.	−0.0980114
$369$	1.47946e6	0.565638
$370$	0	0
$371$	−1.50392e6	−0.567270
$372$	4.94297e6	1.85196
$373$	−3.58536e6	−1.33432	−0.667160	−	0.744914i	$-0.732490\pi$
$373$	−3.58536e6	−1.33432	−0.667160	+	0.744914i	$0.732490\pi$
$374$	1.62451e6	0.600541
$375$	0	0
$376$	5.13697e6	1.87386
$377$	−683659.	−0.247734
$378$	717958.	0.258446
$379$	1.87822e6	0.671660	0.335830	−	0.941923i	$-0.390983\pi$
$379$	1.87822e6	0.671660	0.335830	+	0.941923i	$0.390983\pi$
$380$	0	0
$381$	2.44015e6	0.861202
$382$	−1.25341e6	−0.439476
$383$	662654.	0.230829	0.115414	−	0.993317i	$-0.463180\pi$
$383$	662654.	0.230829	0.115414	+	0.993317i	$0.463180\pi$
$384$	−2.97545e6	−1.02973
$385$	0	0
$386$	−4.33249e6	−1.48003
$387$	−1.11963e6	−0.380010
$388$	8.33622e6	2.81119
$389$	−5.08370e6	−1.70336	−0.851678	−	0.524065i	$-0.824415\pi$
$389$	−5.08370e6	−1.70336	−0.851678	+	0.524065i	$0.824415\pi$
$390$	0	0
$391$	−1.18732e6	−0.392760
$392$	1.28806e6	0.423371
$393$	2.01591e6	0.658400
$394$	903793.	0.293311
$395$	0	0
$396$	−547284.	−0.175378
$397$	518385.	0.165073	0.0825366	−	0.996588i	$-0.473698\pi$
$397$	518385.	0.165073	0.0825366	+	0.996588i	$0.473698\pi$
$398$	1.01468e7	3.21088
$399$	−1.92285e6	−0.604662
$400$	0	0
$401$	1.12538e6	0.349493	0.174747	−	0.984613i	$-0.444089\pi$
$401$	1.12538e6	0.349493	0.174747	+	0.984613i	$0.444089\pi$
$402$	−1.36896e6	−0.422500
$403$	1.45124e6	0.445119
$404$	3.48496e6	1.06229
$405$	0	0
$406$	−4.56326e6	−1.37392
$407$	863256.	0.258317
$408$	2.88057e6	0.856700
$409$	3.15394e6	0.932279	0.466139	−	0.884711i	$-0.345645\pi$
$409$	3.15394e6	0.932279	0.466139	+	0.884711i	$0.345645\pi$
$410$	0	0
$411$	−3.13406e6	−0.915173
$412$	5.02821e6	1.45939
$413$	−744094.	−0.214661
$414$	629228.	0.180429
$415$	0	0
$416$	−630133.	−0.178525
$417$	−3.00421e6	−0.846040
$418$	2.30572e6	0.645456
$419$	801315.	0.222981	0.111491	−	0.993765i	$-0.464438\pi$
$419$	801315.	0.222981	0.111491	+	0.993765i	$0.464438\pi$
$420$	0	0
$421$	−1.85449e6	−0.509940	−0.254970	−	0.966949i	$-0.582066\pi$
$421$	−1.85449e6	−0.509940	−0.254970	+	0.966949i	$0.582066\pi$
$422$	−8.20626e6	−2.24318
$423$	−1.86229e6	−0.506052
$424$	3.19775e6	0.863832
$425$	0	0
$426$	2.37859e6	0.635033
$427$	−1.94088e6	−0.515144
$428$	−6.72635e6	−1.77488
$429$	−160681.	−0.0421522
$430$	0	0
$431$	2.46004e6	0.637895	0.318947	−	0.947772i	$-0.396671\pi$
$431$	2.46004e6	0.637895	0.318947	+	0.947772i	$0.396671\pi$
$432$	−223948.	−0.0577347
$433$	−3.12223e6	−0.800287	−0.400144	−	0.916452i	$-0.631040\pi$
$433$	−3.12223e6	−0.800287	−0.400144	+	0.916452i	$0.631040\pi$
$434$	9.68667e6	2.46860
$435$	0	0
$436$	−5.12910e6	−1.29218
$437$	−1.68521e6	−0.422134
$438$	3.32197e6	0.827389
$439$	−6.09275e6	−1.50887	−0.754436	−	0.656374i	$-0.772089\pi$
$439$	−6.09275e6	−1.50887	−0.754436	+	0.656374i	$0.772089\pi$
$440$	0	0
$441$	−466956.	−0.114335
$442$	1.98094e6	0.482299
$443$	7.93457e6	1.92094	0.960470	−	0.278382i	$-0.0897982\pi$
$443$	7.93457e6	1.92094	0.960470	+	0.278382i	$0.0897982\pi$
$444$	3.58541e6	0.863141
$445$	0	0
$446$	9.26021e6	2.20436
$447$	−2.79224e6	−0.660973
$448$	−5.23897e6	−1.23325
$449$	4.61011e6	1.07918	0.539592	−	0.841926i	$-0.318578\pi$
$449$	4.61011e6	1.07918	0.539592	+	0.841926i	$0.318578\pi$
$450$	0	0
$451$	−2.21006e6	−0.511638
$452$	1.65395e6	0.380781
$453$	492764.	0.112822
$454$	628562.	0.143123
$455$	0	0
$456$	4.08850e6	0.920773
$457$	8.05458e6	1.80407	0.902033	−	0.431667i	$-0.142074\pi$
$457$	8.05458e6	1.80407	0.902033	+	0.431667i	$0.142074\pi$
$458$	6.20568e6	1.38237
$459$	−1.04428e6	−0.231359
$460$	0	0
$461$	3.22431e6	0.706617	0.353309	−	0.935507i	$-0.385057\pi$
$461$	3.22431e6	0.706617	0.353309	+	0.935507i	$0.385057\pi$
$462$	−1.07250e6	−0.233773
$463$	−1.29607e6	−0.280979	−0.140490	−	0.990082i	$-0.544868\pi$
$463$	−1.29607e6	−0.280979	−0.140490	+	0.990082i	$0.544868\pi$
$464$	1.42339e6	0.306922
$465$	0	0
$466$	1.29301e7	2.75828
$467$	2.13902e6	0.453860	0.226930	−	0.973911i	$-0.427131\pi$
$467$	2.13902e6	0.453860	0.226930	+	0.973911i	$0.427131\pi$
$468$	−667365.	−0.140847
$469$	−1.70542e6	−0.358012
$470$	0	0
$471$	1.99400e6	0.414165
$472$	1.58215e6	0.326883
$473$	1.67253e6	0.343732
$474$	−3.47893e6	−0.711213
$475$	0	0
$476$	8.40543e6	1.70037
$477$	−1.15927e6	−0.233285
$478$	−3.97189e6	−0.795111
$479$	−4.77343e6	−0.950586	−0.475293	−	0.879828i	$-0.657658\pi$
$479$	−4.77343e6	−0.950586	−0.475293	+	0.879828i	$0.657658\pi$
$480$	0	0
$481$	1.05266e6	0.207457
$482$	1.26340e7	2.47699
$483$	783873.	0.152890
$484$	817548.	0.158635
$485$	0	0
$486$	553424.	0.106284
$487$	−7.98376e6	−1.52541	−0.762703	−	0.646749i	$-0.776128\pi$
$487$	−7.98376e6	−1.52541	−0.762703	+	0.646749i	$0.776128\pi$
$488$	4.12684e6	0.784456
$489$	−2.02715e6	−0.383366
$490$	0	0
$491$	−3.70031e6	−0.692682	−0.346341	−	0.938109i	$-0.612576\pi$
$491$	−3.70031e6	−0.692682	−0.346341	+	0.938109i	$0.612576\pi$
$492$	−9.17919e6	−1.70959
$493$	6.63736e6	1.22992
$494$	2.81163e6	0.518371
$495$	0	0
$496$	−3.02149e6	−0.551465
$497$	2.96318e6	0.538105
$498$	1.97003e6	0.355959
$499$	−7.05137e6	−1.26772	−0.633858	−	0.773449i	$-0.718530\pi$
$499$	−7.05137e6	−1.26772	−0.633858	+	0.773449i	$0.718530\pi$
$500$	0	0
$501$	1.68116e6	0.299237
$502$	1.36754e7	2.42204
$503$	7.77325e6	1.36988	0.684940	−	0.728599i	$-0.259828\pi$
$503$	7.77325e6	1.36988	0.684940	+	0.728599i	$0.259828\pi$
$504$	−1.90176e6	−0.333488
$505$	0	0
$506$	−939957.	−0.163204
$507$	3.14570e6	0.543498
$508$	−1.51397e7	−2.60290
$509$	4.11598e6	0.704173	0.352086	−	0.935968i	$-0.385472\pi$
$509$	4.11598e6	0.704173	0.352086	+	0.935968i	$0.385472\pi$
$510$	0	0
$511$	4.13841e6	0.701102
$512$	3.50836e6	0.591465
$513$	−1.48219e6	−0.248663
$514$	5.14073e6	0.858256
$515$	0	0
$516$	6.94661e6	1.14855
$517$	2.78193e6	0.457742
$518$	7.02628e6	1.15054
$519$	−6.42296e6	−1.04669
$520$	0	0
$521$	−2.56352e6	−0.413754	−0.206877	−	0.978367i	$-0.566330\pi$
$521$	−2.56352e6	−0.413754	−0.206877	+	0.978367i	$0.566330\pi$
$522$	−3.51750e6	−0.565012
$523$	3.69240e6	0.590274	0.295137	−	0.955455i	$-0.404635\pi$
$523$	3.69240e6	0.590274	0.295137	+	0.955455i	$0.404635\pi$
$524$	−1.25075e7	−1.98996
$525$	0	0
$526$	8.48471e6	1.33713
$527$	−1.40895e7	−2.20988
$528$	334539.	0.0522230
$529$	−5.74935e6	−0.893263
$530$	0	0
$531$	−573570.	−0.0882776
$532$	1.19301e7	1.82754
$533$	−2.69498e6	−0.410901
$534$	−8.75587e6	−1.32876
$535$	0	0
$536$	3.62618e6	0.545177
$537$	−7.10505e6	−1.06324
$538$	1.53743e6	0.229003
$539$	697552.	0.103420
$540$	0	0
$541$	−9.33636e6	−1.37146	−0.685732	−	0.727854i	$-0.740518\pi$
$541$	−9.33636e6	−1.37146	−0.685732	+	0.727854i	$0.740518\pi$
$542$	−9.27515e6	−1.35620
$543$	−362977.	−0.0528298
$544$	6.11769e6	0.886320
$545$	0	0
$546$	−1.30782e6	−0.187745
$547$	−1.30814e6	−0.186933	−0.0934664	−	0.995622i	$-0.529795\pi$
$547$	−1.30814e6	−0.186933	−0.0934664	+	0.995622i	$0.529795\pi$
$548$	1.94450e7	2.76603
$549$	−1.49609e6	−0.211849
$550$	0	0
$551$	9.42064e6	1.32191
$552$	−1.66673e6	−0.232819
$553$	−4.33395e6	−0.602658
$554$	6.81505e6	0.943397
$555$	0	0
$556$	1.86394e7	2.55708
$557$	−6.95945e6	−0.950467	−0.475234	−	0.879860i	$-0.657636\pi$
$557$	−6.95945e6	−0.950467	−0.475234	+	0.879860i	$0.657636\pi$
$558$	7.46678e6	1.01519
$559$	2.03950e6	0.276054
$560$	0	0
$561$	1.55998e6	0.209272
$562$	3.55303e6	0.474524
$563$	−6.90198e6	−0.917704	−0.458852	−	0.888513i	$-0.651739\pi$
$563$	−6.90198e6	−0.917704	−0.458852	+	0.888513i	$0.651739\pi$
$564$	1.15544e7	1.52950
$565$	0	0
$566$	−6.51721e6	−0.855107
$567$	689439.	0.0900613
$568$	−6.30054e6	−0.819421
$569$	3.31261e6	0.428933	0.214466	−	0.976731i	$-0.431199\pi$
$569$	3.31261e6	0.428933	0.214466	+	0.976731i	$0.431199\pi$
$570$	0	0
$571$	1.43930e7	1.84741	0.923703	−	0.383109i	$-0.125147\pi$
$571$	1.43930e7	1.84741	0.923703	+	0.383109i	$0.125147\pi$
$572$	996928.	0.127401
$573$	−1.20362e6	−0.153146
$574$	−1.79883e7	−2.27882
$575$	0	0
$576$	−4.03836e6	−0.507165
$577$	−3.31761e6	−0.414845	−0.207423	−	0.978251i	$-0.566508\pi$
$577$	−3.31761e6	−0.414845	−0.207423	+	0.978251i	$0.566508\pi$
$578$	−5.92484e6	−0.737662
$579$	−4.16040e6	−0.515749
$580$	0	0
$581$	2.45421e6	0.301628
$582$	1.25926e7	1.54101
$583$	1.73174e6	0.211015
$584$	−8.79939e6	−1.06763
$585$	0	0
$586$	2.51527e6	0.302580
$587$	−2.10735e6	−0.252431	−0.126215	−	0.992003i	$-0.540283\pi$
$587$	−2.10735e6	−0.252431	−0.126215	+	0.992003i	$0.540283\pi$
$588$	2.89719e6	0.345568
$589$	−1.99977e7	−2.37515
$590$	0	0
$591$	867893.	0.102211
$592$	−2.19166e6	−0.257021
$593$	−525039.	−0.0613133	−0.0306566	−	0.999530i	$-0.509760\pi$
$593$	−525039.	−0.0613133	−0.0306566	+	0.999530i	$0.509760\pi$
$594$	−826720.	−0.0961373
$595$	0	0
$596$	1.73242e7	1.99773
$597$	9.74380e6	1.11890
$598$	−1.14619e6	−0.131071
$599$	−7.02964e6	−0.800509	−0.400254	−	0.916404i	$-0.631078\pi$
$599$	−7.02964e6	−0.800509	−0.400254	+	0.916404i	$0.631078\pi$
$600$	0	0
$601$	−1.11158e6	−0.125532	−0.0627660	−	0.998028i	$-0.519992\pi$
$601$	−1.11158e6	−0.125532	−0.0627660	+	0.998028i	$0.519992\pi$
$602$	1.36132e7	1.53097
$603$	−1.31459e6	−0.147230
$604$	−3.05731e6	−0.340994
$605$	0	0
$606$	5.26433e6	0.582320
$607$	−1.53428e7	−1.69018	−0.845088	−	0.534627i	$-0.820452\pi$
$607$	−1.53428e7	−1.69018	−0.845088	+	0.534627i	$0.820452\pi$
$608$	8.68307e6	0.952608
$609$	−4.38200e6	−0.478772
$610$	0	0
$611$	3.39232e6	0.367616
$612$	6.47916e6	0.699263
$613$	−4.78522e6	−0.514341	−0.257170	−	0.966366i	$-0.582790\pi$
$613$	−4.78522e6	−0.514341	−0.257170	+	0.966366i	$0.582790\pi$
$614$	1.67615e7	1.79429
$615$	0	0
$616$	2.84091e6	0.301651
$617$	−1.69116e7	−1.78843	−0.894215	−	0.447639i	$-0.852265\pi$
$617$	−1.69116e7	−1.78843	−0.894215	+	0.447639i	$0.852265\pi$
$618$	7.59553e6	0.799994
$619$	−8.65985e6	−0.908414	−0.454207	−	0.890896i	$-0.650077\pi$
$619$	−8.65985e6	−0.908414	−0.454207	+	0.890896i	$0.650077\pi$
$620$	0	0
$621$	604234.	0.0628747
$622$	−3.39958e6	−0.352330
$623$	−1.09078e7	−1.12595
$624$	407941.	0.0419407
$625$	0	0
$626$	−1.07202e7	−1.09337
$627$	2.21414e6	0.224924
$628$	−1.23716e7	−1.25178
$629$	−1.02199e7	−1.02996
$630$	0	0
$631$	9.04083e6	0.903930	0.451965	−	0.892036i	$-0.350723\pi$
$631$	9.04083e6	0.903930	0.451965	+	0.892036i	$0.350723\pi$
$632$	9.21516e6	0.917720
$633$	−7.88029e6	−0.781687
$634$	3.22172e7	3.18321
$635$	0	0
$636$	7.19257e6	0.705084
$637$	850603.	0.0830574
$638$	5.25454e6	0.511073
$639$	2.28411e6	0.221292
$640$	0	0
$641$	−1.21113e7	−1.16424	−0.582122	−	0.813101i	$-0.697778\pi$
$641$	−1.21113e7	−1.16424	−0.582122	+	0.813101i	$0.697778\pi$
$642$	−1.01607e7	−0.972942
$643$	−2.24955e6	−0.214570	−0.107285	−	0.994228i	$-0.534216\pi$
$643$	−2.24955e6	−0.214570	−0.107285	+	0.994228i	$0.534216\pi$
$644$	−4.86347e6	−0.462095
$645$	0	0
$646$	−2.72969e7	−2.57355
$647$	−8.21206e6	−0.771244	−0.385622	−	0.922657i	$-0.626013\pi$
$647$	−8.21206e6	−0.771244	−0.385622	+	0.922657i	$0.626013\pi$
$648$	−1.46594e6	−0.137144
$649$	856815.	0.0798501
$650$	0	0
$651$	9.30190e6	0.860239
$652$	1.25773e7	1.15869
$653$	1.15110e7	1.05641	0.528203	−	0.849118i	$-0.322866\pi$
$653$	1.15110e7	1.05641	0.528203	+	0.849118i	$0.322866\pi$
$654$	−7.74793e6	−0.708340
$655$	0	0
$656$	5.61097e6	0.509071
$657$	3.19001e6	0.288323
$658$	2.26429e7	2.03877
$659$	−1.79339e7	−1.60865	−0.804323	−	0.594192i	$-0.797472\pi$
$659$	−1.79339e7	−1.60865	−0.804323	+	0.594192i	$0.797472\pi$
$660$	0	0
$661$	−8.63046e6	−0.768299	−0.384150	−	0.923271i	$-0.625505\pi$
$661$	−8.63046e6	−0.768299	−0.384150	+	0.923271i	$0.625505\pi$
$662$	−1.04433e7	−0.926176
$663$	1.90226e6	0.168068
$664$	−5.21832e6	−0.459315
$665$	0	0
$666$	5.41607e6	0.473150
$667$	−3.84044e6	−0.334246
$668$	−1.04306e7	−0.904418
$669$	8.89238e6	0.768162
$670$	0	0
$671$	2.23490e6	0.191625
$672$	−4.03892e6	−0.345018
$673$	1.92878e7	1.64152	0.820758	−	0.571276i	$-0.193551\pi$
$673$	1.92878e7	1.64152	0.820758	+	0.571276i	$0.193551\pi$
$674$	−2.11630e7	−1.79443
$675$	0	0
$676$	−1.95172e7	−1.64267
$677$	−6.67400e6	−0.559647	−0.279824	−	0.960051i	$-0.590276\pi$
$677$	−6.67400e6	−0.559647	−0.279824	+	0.960051i	$0.590276\pi$
$678$	2.49843e6	0.208734
$679$	1.56875e7	1.30580
$680$	0	0
$681$	603594.	0.0498744
$682$	−1.11541e7	−0.918275
$683$	7.83722e6	0.642851	0.321426	−	0.946935i	$-0.395838\pi$
$683$	7.83722e6	0.642851	0.321426	+	0.946935i	$0.395838\pi$
$684$	9.19611e6	0.751561
$685$	0	0
$686$	2.22300e7	1.80355
$687$	5.95918e6	0.481720
$688$	−4.24626e6	−0.342007
$689$	2.11171e6	0.169467
$690$	0	0
$691$	−2.05838e7	−1.63995	−0.819974	−	0.572400i	$-0.806012\pi$
$691$	−2.05838e7	−1.63995	−0.819974	+	0.572400i	$0.806012\pi$
$692$	3.98507e7	3.16352
$693$	−1.02990e6	−0.0814635
$694$	−1.09707e7	−0.864640
$695$	0	0
$696$	9.31734e6	0.729069
$697$	2.61644e7	2.03999
$698$	−1.14065e7	−0.886162
$699$	1.24165e7	0.961185
$700$	0	0
$701$	−8.88356e6	−0.682797	−0.341399	−	0.939919i	$-0.610901\pi$
$701$	−8.88356e6	−0.682797	−0.341399	+	0.939919i	$0.610901\pi$
$702$	−1.00811e6	−0.0772086
$703$	−1.45054e7	−1.10699
$704$	6.03261e6	0.458748
$705$	0	0
$706$	2.68223e6	0.202528
$707$	6.55815e6	0.493438
$708$	3.55866e6	0.266811
$709$	6.33167e6	0.473045	0.236523	−	0.971626i	$-0.423992\pi$
$709$	6.33167e6	0.473045	0.236523	+	0.971626i	$0.423992\pi$
$710$	0	0
$711$	−3.34074e6	−0.247838
$712$	2.31930e7	1.71458
$713$	8.15231e6	0.600560
$714$	1.26971e7	0.932094
$715$	0	0
$716$	4.40826e7	3.21355
$717$	−3.81412e6	−0.277075
$718$	−1.11592e7	−0.807836
$719$	3.88936e6	0.280579	0.140290	−	0.990111i	$-0.455197\pi$
$719$	3.88936e6	0.280579	0.140290	+	0.990111i	$0.455197\pi$
$720$	0	0
$721$	9.46229e6	0.677888
$722$	−1.55368e7	−1.10922
$723$	1.21322e7	0.863163
$724$	2.25205e6	0.159673
$725$	0	0
$726$	1.23498e6	0.0869595
$727$	−2.34319e7	−1.64426	−0.822131	−	0.569299i	$-0.807215\pi$
$727$	−2.34319e7	−1.64426	−0.822131	+	0.569299i	$0.807215\pi$
$728$	3.46423e6	0.242258
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.98006e7	−1.37052
$732$	9.28235e6	0.640295
$733$	−1.77978e7	−1.22351	−0.611753	−	0.791049i	$-0.709535\pi$
$733$	−1.77978e7	−1.22351	−0.611753	+	0.791049i	$0.709535\pi$
$734$	2.91591e7	1.99772
$735$	0	0
$736$	−3.53976e6	−0.240868
$737$	1.96376e6	0.133174
$738$	−1.38659e7	−0.937149
$739$	3.15950e6	0.212817	0.106409	−	0.994322i	$-0.466065\pi$
$739$	3.15950e6	0.212817	0.106409	+	0.994322i	$0.466065\pi$
$740$	0	0
$741$	2.69994e6	0.180638
$742$	1.40952e7	0.939853
$743$	4.37086e6	0.290465	0.145233	−	0.989398i	$-0.453607\pi$
$743$	4.37086e6	0.290465	0.145233	+	0.989398i	$0.453607\pi$
$744$	−1.97784e7	−1.30996
$745$	0	0
$746$	3.36030e7	2.21070
$747$	1.89178e6	0.124042
$748$	−9.67875e6	−0.632507
$749$	−1.26579e7	−0.824438
$750$	0	0
$751$	1.72544e7	1.11635	0.558173	−	0.829724i	$-0.311502\pi$
$751$	1.72544e7	1.11635	0.558173	+	0.829724i	$0.311502\pi$
$752$	−7.06285e6	−0.455445
$753$	1.31322e7	0.844017
$754$	6.40745e6	0.410447
$755$	0	0
$756$	−4.27756e6	−0.272202
$757$	1.57084e7	0.996307	0.498153	−	0.867089i	$-0.334012\pi$
$757$	1.57084e7	0.996307	0.498153	+	0.867089i	$0.334012\pi$
$758$	−1.76033e7	−1.11281
$759$	−902621.	−0.0568723
$760$	0	0
$761$	5.01006e6	0.313604	0.156802	−	0.987630i	$-0.449882\pi$
$761$	5.01006e6	0.313604	0.156802	+	0.987630i	$0.449882\pi$
$762$	−2.28698e7	−1.42684
$763$	−9.65215e6	−0.600223
$764$	7.46778e6	0.462869
$765$	0	0
$766$	−6.21058e6	−0.382437
$767$	1.04481e6	0.0641282
$768$	1.35282e7	0.827629
$769$	3.47489e6	0.211897	0.105949	−	0.994372i	$-0.466212\pi$
$769$	3.47489e6	0.211897	0.105949	+	0.994372i	$0.466212\pi$
$770$	0	0
$771$	4.93653e6	0.299079
$772$	2.58128e7	1.55880
$773$	−2.29786e7	−1.38317	−0.691584	−	0.722296i	$-0.743087\pi$
$773$	−2.29786e7	−1.38317	−0.691584	+	0.722296i	$0.743087\pi$
$774$	1.04934e7	0.629602
$775$	0	0
$776$	−3.33558e7	−1.98846
$777$	6.74718e6	0.400931
$778$	4.76458e7	2.82212
$779$	3.71361e7	2.19256
$780$	0	0
$781$	−3.41207e6	−0.200166
$782$	1.11279e7	0.650724
$783$	−3.37778e6	−0.196892
$784$	−1.77096e6	−0.102901
$785$	0	0
$786$	−1.88937e7	−1.09084
$787$	1.40930e7	0.811087	0.405544	−	0.914076i	$-0.367082\pi$
$787$	1.40930e7	0.811087	0.405544	+	0.914076i	$0.367082\pi$
$788$	−5.38476e6	−0.308923
$789$	8.14769e6	0.465953
$790$	0	0
$791$	3.11247e6	0.176874
$792$	2.18986e6	0.124052
$793$	2.72526e6	0.153895
$794$	−4.85845e6	−0.273493
$795$	0	0
$796$	−6.04545e7	−3.38178
$797$	1.08281e7	0.603819	0.301909	−	0.953337i	$-0.402376\pi$
$797$	1.08281e7	0.603819	0.301909	+	0.953337i	$0.402376\pi$
$798$	1.80215e7	1.00181
$799$	−3.29346e7	−1.82510
$800$	0	0
$801$	−8.40808e6	−0.463037
$802$	−1.05474e7	−0.579041
$803$	−4.76533e6	−0.260798
$804$	8.15623e6	0.444989
$805$	0	0
$806$	−1.36014e7	−0.737474
$807$	1.47637e6	0.0798013
$808$	−1.39444e7	−0.751402
$809$	−2.27494e7	−1.22208	−0.611040	−	0.791600i	$-0.709248\pi$
$809$	−2.27494e7	−1.22208	−0.611040	+	0.791600i	$0.709248\pi$
$810$	0	0
$811$	−1.33958e7	−0.715179	−0.357590	−	0.933879i	$-0.616401\pi$
$811$	−1.33958e7	−0.715179	−0.357590	+	0.933879i	$0.616401\pi$
$812$	2.71877e7	1.44705
$813$	−8.90673e6	−0.472598
$814$	−8.09067e6	−0.427980
$815$	0	0
$816$	−3.96052e6	−0.208222
$817$	−2.81038e7	−1.47302
$818$	−2.95596e7	−1.54460
$819$	−1.25588e6	−0.0654240
$820$	0	0
$821$	−1.77359e7	−0.918323	−0.459161	−	0.888353i	$-0.651850\pi$
$821$	−1.77359e7	−0.918323	−0.459161	+	0.888353i	$0.651850\pi$
$822$	2.93733e7	1.51626
$823$	2.24560e7	1.15567	0.577834	−	0.816154i	$-0.303898\pi$
$823$	2.24560e7	1.15567	0.577834	+	0.816154i	$0.303898\pi$
$824$	−2.01194e7	−1.03228
$825$	0	0
$826$	6.97386e6	0.355650
$827$	1.97457e7	1.00394	0.501972	−	0.864884i	$-0.332608\pi$
$827$	1.97457e7	1.00394	0.501972	+	0.864884i	$0.332608\pi$
$828$	−3.74891e6	−0.190033
$829$	2.61754e7	1.32284	0.661418	−	0.750017i	$-0.269955\pi$
$829$	2.61754e7	1.32284	0.661418	+	0.750017i	$0.269955\pi$
$830$	0	0
$831$	6.54434e6	0.328748
$832$	7.35624e6	0.368424
$833$	−8.25814e6	−0.412354
$834$	2.81563e7	1.40172
$835$	0	0
$836$	−1.37374e7	−0.679812
$837$	7.17019e6	0.353767
$838$	−7.51015e6	−0.369435
$839$	−3.69992e6	−0.181463	−0.0907313	−	0.995875i	$-0.528920\pi$
$839$	−3.69992e6	−0.181463	−0.0907313	+	0.995875i	$0.528920\pi$
$840$	0	0
$841$	957652.	0.0466893
$842$	1.73808e7	0.844869
$843$	3.41190e6	0.165359
$844$	4.88925e7	2.36258
$845$	0	0
$846$	1.74539e7	0.838428
$847$	1.53850e6	0.0736866
$848$	−4.39660e6	−0.209956
$849$	−6.25834e6	−0.297982
$850$	0	0
$851$	5.91332e6	0.279903
$852$	−1.41716e7	−0.668834
$853$	−2.90121e7	−1.36523	−0.682617	−	0.730776i	$-0.739158\pi$
$853$	−2.90121e7	−1.36523	−0.682617	+	0.730776i	$0.739158\pi$
$854$	1.81905e7	0.853492
$855$	0	0
$856$	2.69142e7	1.25544
$857$	3.60120e7	1.67493	0.837463	−	0.546494i	$-0.184038\pi$
$857$	3.60120e7	1.67493	0.837463	+	0.546494i	$0.184038\pi$
$858$	1.50594e6	0.0698378
$859$	−2.14715e7	−0.992841	−0.496421	−	0.868082i	$-0.665353\pi$
$859$	−2.14715e7	−0.992841	−0.496421	+	0.868082i	$0.665353\pi$
$860$	0	0
$861$	−1.72738e7	−0.794108
$862$	−2.30562e7	−1.05686
$863$	−1.43154e7	−0.654298	−0.327149	−	0.944973i	$-0.606088\pi$
$863$	−1.43154e7	−0.654298	−0.327149	+	0.944973i	$0.606088\pi$
$864$	−3.11332e6	−0.141886
$865$	0	0
$866$	2.92625e7	1.32592
$867$	−5.68950e6	−0.257055
$868$	−5.77128e7	−2.60000
$869$	4.99049e6	0.224178
$870$	0	0
$871$	2.39464e6	0.106953
$872$	2.05231e7	0.914013
$873$	1.20924e7	0.537002
$874$	1.57943e7	0.699392
$875$	0	0
$876$	−1.97921e7	−0.871430
$877$	−2.25088e7	−0.988221	−0.494110	−	0.869399i	$-0.664506\pi$
$877$	−2.25088e7	−0.988221	−0.494110	+	0.869399i	$0.664506\pi$
$878$	5.71030e7	2.49990
$879$	2.41536e6	0.105441
$880$	0	0
$881$	8.39629e6	0.364458	0.182229	−	0.983256i	$-0.441669\pi$
$881$	8.39629e6	0.364458	0.182229	+	0.983256i	$0.441669\pi$
$882$	4.37644e6	0.189431
$883$	−1.08666e7	−0.469021	−0.234510	−	0.972114i	$-0.575349\pi$
$883$	−1.08666e7	−0.469021	−0.234510	+	0.972114i	$0.575349\pi$
$884$	−1.18024e7	−0.507971
$885$	0	0
$886$	−7.43650e7	−3.18262
$887$	−1.53488e7	−0.655038	−0.327519	−	0.944845i	$-0.606212\pi$
$887$	−1.53488e7	−0.655038	−0.327519	+	0.944845i	$0.606212\pi$
$888$	−1.43464e7	−0.610533
$889$	−2.84905e7	−1.20906
$890$	0	0
$891$	−793881.	−0.0335013
$892$	−5.51719e7	−2.32170
$893$	−4.67453e7	−1.96160
$894$	2.61696e7	1.09510
$895$	0	0
$896$	3.47405e7	1.44566
$897$	−1.10067e6	−0.0456746
$898$	−4.32073e7	−1.78799
$899$	−4.55729e7	−1.88065
$900$	0	0
$901$	−2.05017e7	−0.841352
$902$	2.07133e7	0.847683
$903$	1.30724e7	0.533503
$904$	−6.61796e6	−0.269341
$905$	0	0
$906$	−4.61832e6	−0.186923
$907$	−7.93009e6	−0.320081	−0.160041	−	0.987110i	$-0.551162\pi$
$907$	−7.93009e6	−0.320081	−0.160041	+	0.987110i	$0.551162\pi$
$908$	−3.74494e6	−0.150741
$909$	5.05522e6	0.202923
$910$	0	0
$911$	−4.31907e7	−1.72423	−0.862114	−	0.506715i	$-0.830860\pi$
$911$	−4.31907e7	−1.72423	−0.862114	+	0.506715i	$0.830860\pi$
$912$	−5.62131e6	−0.223795
$913$	−2.82599e6	−0.112200
$914$	−7.54898e7	−2.98898
$915$	0	0
$916$	−3.69732e7	−1.45595
$917$	−2.35372e7	−0.924340
$918$	9.78733e6	0.383316
$919$	−2.17987e7	−0.851414	−0.425707	−	0.904861i	$-0.639975\pi$
$919$	−2.17987e7	−0.851414	−0.425707	+	0.904861i	$0.639975\pi$
$920$	0	0
$921$	1.60957e7	0.625260
$922$	−3.02191e7	−1.17072
$923$	−4.16072e6	−0.160755
$924$	6.38994e6	0.246216
$925$	0	0
$926$	1.21471e7	0.465527
$927$	7.29382e6	0.278776
$928$	1.97879e7	0.754276
$929$	−1.33390e7	−0.507089	−0.253545	−	0.967324i	$-0.581596\pi$
$929$	−1.33390e7	−0.507089	−0.253545	+	0.967324i	$0.581596\pi$
$930$	0	0
$931$	−1.17211e7	−0.443194
$932$	−7.70371e7	−2.90510
$933$	−3.26454e6	−0.122777
$934$	−2.00475e7	−0.751956
$935$	0	0
$936$	2.67034e6	0.0996269
$937$	−1.33697e7	−0.497476	−0.248738	−	0.968571i	$-0.580016\pi$
$937$	−1.33697e7	−0.497476	−0.248738	+	0.968571i	$0.580016\pi$
$938$	1.59836e7	0.593155
$939$	−1.02943e7	−0.381008
$940$	0	0
$941$	2.28926e7	0.842793	0.421397	−	0.906876i	$-0.361540\pi$
$941$	2.28926e7	0.842793	0.421397	+	0.906876i	$0.361540\pi$
$942$	−1.86883e7	−0.686189
$943$	−1.51390e7	−0.554393
$944$	−2.17531e6	−0.0794494
$945$	0	0
$946$	−1.56754e7	−0.569496
$947$	−3.33079e6	−0.120690	−0.0603451	−	0.998178i	$-0.519220\pi$
$947$	−3.33079e6	−0.120690	−0.0603451	+	0.998178i	$0.519220\pi$
$948$	2.07273e7	0.749069
$949$	−5.81089e6	−0.209449
$950$	0	0
$951$	3.09375e7	1.10926
$952$	−3.36328e7	−1.20274
$953$	−4.68461e7	−1.67087	−0.835433	−	0.549593i	$-0.814783\pi$
$953$	−4.68461e7	−1.67087	−0.835433	+	0.549593i	$0.814783\pi$
$954$	1.08650e7	0.386508
$955$	0	0
$956$	2.36644e7	0.837433
$957$	5.04582e6	0.178095
$958$	4.47379e7	1.57493
$959$	3.65924e7	1.28483
$960$	0	0
$961$	6.81109e7	2.37907
$962$	−9.86586e6	−0.343714
$963$	−9.75712e6	−0.339044
$964$	−7.52730e7	−2.60883
$965$	0	0
$966$	−7.34668e6	−0.253308
$967$	3.65887e7	1.25829	0.629146	−	0.777287i	$-0.283405\pi$
$967$	3.65887e7	1.25829	0.629146	+	0.777287i	$0.283405\pi$
$968$	−3.27127e6	−0.112209
$969$	−2.62126e7	−0.896811
$970$	0	0
$971$	1.62147e7	0.551901	0.275951	−	0.961172i	$-0.411007\pi$
$971$	1.62147e7	0.551901	0.275951	+	0.961172i	$0.411007\pi$
$972$	−3.29728e6	−0.111941
$973$	3.50764e7	1.18777
$974$	7.48261e7	2.52729
$975$	0	0
$976$	−5.67403e6	−0.190663
$977$	1.50181e6	0.0503360	0.0251680	−	0.999683i	$-0.491988\pi$
$977$	1.50181e6	0.0503360	0.0251680	+	0.999683i	$0.491988\pi$
$978$	1.89990e7	0.635161
$979$	1.25602e7	0.418832
$980$	0	0
$981$	−7.44017e6	−0.246837
$982$	3.46803e7	1.14764
$983$	−3.72690e7	−1.23017	−0.615083	−	0.788462i	$-0.710878\pi$
$983$	−3.72690e7	−1.23017	−0.615083	+	0.788462i	$0.710878\pi$
$984$	3.67288e7	1.20926
$985$	0	0
$986$	−6.22072e7	−2.03774
$987$	2.17435e7	0.710456
$988$	−1.67516e7	−0.545962
$989$	1.14569e7	0.372456
$990$	0	0
$991$	−3.20015e7	−1.03511	−0.517554	−	0.855651i	$-0.673157\pi$
$991$	−3.20015e7	−1.03511	−0.517554	+	0.855651i	$0.673157\pi$
$992$	−4.20049e7	−1.35525
$993$	−1.00285e7	−0.322747
$994$	−2.77718e7	−0.891534
$995$	0	0
$996$	−1.17374e7	−0.374906
$997$	5.59172e7	1.78159	0.890795	−	0.454406i	$-0.150148\pi$
$997$	5.59172e7	1.78159	0.890795	+	0.454406i	$0.150148\pi$
$998$	6.60874e7	2.10035
$999$	5.20094e6	0.164880

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.c.1.1		2
5.4	even	2		33.6.a.e.1.2	✓	2
15.14	odd	2		99.6.a.d.1.1		2
20.19	odd	2		528.6.a.o.1.1		2
55.54	odd	2		363.6.a.f.1.1		2
165.164	even	2		1089.6.a.p.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.a.e.1.2	✓	2	5.4	even	2
99.6.a.d.1.1		2	15.14	odd	2
363.6.a.f.1.1		2	55.54	odd	2
528.6.a.o.1.1		2	20.19	odd	2
825.6.a.c.1.1		2	1.1	even	1	trivial
1089.6.a.p.1.2		2	165.164	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)

\(f(q)\)	\(=\)	\(q-9.37228 q^{2} -9.00000 q^{3} +55.8397 q^{4} +84.3505 q^{6} +105.081 q^{7} -223.432 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-9.37228 q^{2} -9.00000 q^{3} +55.8397 q^{4} +84.3505 q^{6} +105.081 q^{7} -223.432 q^{8} +81.0000 q^{9} -121.000 q^{11} -502.557 q^{12} -147.549 q^{13} -984.853 q^{14} +307.198 q^{16} +1432.49 q^{17} -759.155 q^{18} +2033.18 q^{19} -945.733 q^{21} +1134.05 q^{22} -828.853 q^{23} +2010.89 q^{24} +1382.87 q^{26} -729.000 q^{27} +5867.71 q^{28} +4633.44 q^{29} -9835.65 q^{31} +4270.67 q^{32} +1089.00 q^{33} -13425.7 q^{34} +4523.01 q^{36} -7134.34 q^{37} -19055.6 q^{38} +1327.94 q^{39} +18265.0 q^{41} +8863.68 q^{42} -13822.5 q^{43} -6756.60 q^{44} +7768.24 q^{46} -22991.2 q^{47} -2764.78 q^{48} -5764.89 q^{49} -12892.4 q^{51} -8239.08 q^{52} -14311.9 q^{53} +6832.39 q^{54} -23478.6 q^{56} -18298.7 q^{57} -43425.9 q^{58} -7081.12 q^{59} -18470.2 q^{61} +92182.5 q^{62} +8511.60 q^{63} -49856.3 q^{64} -10206.4 q^{66} -16229.5 q^{67} +79989.7 q^{68} +7459.68 q^{69} +28198.9 q^{71} -18098.0 q^{72} +39382.9 q^{73} +66865.1 q^{74} +113532. q^{76} -12714.9 q^{77} -12445.8 q^{78} -41243.7 q^{79} +6561.00 q^{81} -171185. q^{82} +23355.3 q^{83} -52809.4 q^{84} +129549. q^{86} -41701.0 q^{87} +27035.3 q^{88} -103803. q^{89} -15504.6 q^{91} -46282.9 q^{92} +88520.9 q^{93} +215480. q^{94} -38436.1 q^{96} +149289. q^{97} +54030.2 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 13 q^{2} - 18 q^{3} + 37 q^{4} + 117 q^{6} - 146 q^{7} - 39 q^{8} + 162 q^{9}+O(q^{10}) \) 2 * q - 13 * q^2 - 18 * q^3 + 37 * q^4 + 117 * q^6 - 146 * q^7 - 39 * q^8 + 162 * q^9	\( 2 q - 13 q^{2} - 18 q^{3} + 37 q^{4} + 117 q^{6} - 146 q^{7} - 39 q^{8} + 162 q^{9} - 242 q^{11} - 333 q^{12} + 130 q^{13} - 74 q^{14} + 241 q^{16} + 728 q^{17} - 1053 q^{18} - 828 q^{19} + 1314 q^{21} + 1573 q^{22} + 238 q^{23} + 351 q^{24} + 376 q^{26} - 1458 q^{27} + 10598 q^{28} + 696 q^{29} - 10480 q^{31} - 1391 q^{32} + 2178 q^{33} - 10870 q^{34} + 2997 q^{36} + 1908 q^{37} - 8676 q^{38} - 1170 q^{39} + 36484 q^{41} + 666 q^{42} - 9768 q^{43} - 4477 q^{44} + 3898 q^{46} - 43742 q^{47} - 2169 q^{48} + 40470 q^{49} - 6552 q^{51} - 13468 q^{52} + 12174 q^{53} + 9477 q^{54} - 69786 q^{56} + 7452 q^{57} - 29142 q^{58} - 2788 q^{59} - 25302 q^{61} + 94520 q^{62} - 11826 q^{63} - 27199 q^{64} - 14157 q^{66} + 40520 q^{67} + 93262 q^{68} - 2142 q^{69} + 31386 q^{71} - 3159 q^{72} + 46780 q^{73} + 34062 q^{74} + 167436 q^{76} + 17666 q^{77} - 3384 q^{78} - 16850 q^{79} + 13122 q^{81} - 237278 q^{82} - 79440 q^{83} - 95382 q^{84} + 114840 q^{86} - 6264 q^{87} + 4719 q^{88} - 54204 q^{89} - 85192 q^{91} - 66382 q^{92} + 94320 q^{93} + 290758 q^{94} + 12519 q^{96} + 241568 q^{97} - 113697 q^{98} - 19602 q^{99}+O(q^{100}) \) 2 * q - 13 * q^2 - 18 * q^3 + 37 * q^4 + 117 * q^6 - 146 * q^7 - 39 * q^8 + 162 * q^9 - 242 * q^11 - 333 * q^12 + 130 * q^13 - 74 * q^14 + 241 * q^16 + 728 * q^17 - 1053 * q^18 - 828 * q^19 + 1314 * q^21 + 1573 * q^22 + 238 * q^23 + 351 * q^24 + 376 * q^26 - 1458 * q^27 + 10598 * q^28 + 696 * q^29 - 10480 * q^31 - 1391 * q^32 + 2178 * q^33 - 10870 * q^34 + 2997 * q^36 + 1908 * q^37 - 8676 * q^38 - 1170 * q^39 + 36484 * q^41 + 666 * q^42 - 9768 * q^43 - 4477 * q^44 + 3898 * q^46 - 43742 * q^47 - 2169 * q^48 + 40470 * q^49 - 6552 * q^51 - 13468 * q^52 + 12174 * q^53 + 9477 * q^54 - 69786 * q^56 + 7452 * q^57 - 29142 * q^58 - 2788 * q^59 - 25302 * q^61 + 94520 * q^62 - 11826 * q^63 - 27199 * q^64 - 14157 * q^66 + 40520 * q^67 + 93262 * q^68 - 2142 * q^69 + 31386 * q^71 - 3159 * q^72 + 46780 * q^73 + 34062 * q^74 + 167436 * q^76 + 17666 * q^77 - 3384 * q^78 - 16850 * q^79 + 13122 * q^81 - 237278 * q^82 - 79440 * q^83 - 95382 * q^84 + 114840 * q^86 - 6264 * q^87 + 4719 * q^88 - 54204 * q^89 - 85192 * q^91 - 66382 * q^92 + 94320 * q^93 + 290758 * q^94 + 12519 * q^96 + 241568 * q^97 - 113697 * q^98 - 19602 * q^99

Introduction

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