LMFDB - Embedded newform 825.6.a.p.1.7

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 176x^{6} + 272x^{5} + 9055x^{4} - 15851x^{3} - 118840x^{2} + 149572x - 33248 $ x^8 - x^7 - 176x^6 + 272x^5 + 9055x^4 - 15851x^3 - 118840x^2 + 149572x - 33248
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-8.09383$ 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)

$f(q)$	$=$	$q+7.09383 q^{2} +9.00000 q^{3} +18.3224 q^{4} +63.8444 q^{6} -4.62438 q^{7} -97.0268 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+7.09383 q^{2} +9.00000 q^{3} +18.3224 q^{4} +63.8444 q^{6} -4.62438 q^{7} -97.0268 q^{8} +81.0000 q^{9} -121.000 q^{11} +164.901 q^{12} +38.0995 q^{13} -32.8046 q^{14} -1274.61 q^{16} +610.281 q^{17} +574.600 q^{18} +2102.49 q^{19} -41.6194 q^{21} -858.353 q^{22} -3875.26 q^{23} -873.241 q^{24} +270.272 q^{26} +729.000 q^{27} -84.7296 q^{28} -8371.56 q^{29} -1417.68 q^{31} -5936.98 q^{32} -1089.00 q^{33} +4329.23 q^{34} +1484.11 q^{36} +3071.11 q^{37} +14914.7 q^{38} +342.896 q^{39} -591.131 q^{41} -295.241 q^{42} +19206.4 q^{43} -2217.01 q^{44} -27490.4 q^{46} -11447.9 q^{47} -11471.5 q^{48} -16785.6 q^{49} +5492.53 q^{51} +698.074 q^{52} -17463.0 q^{53} +5171.40 q^{54} +448.689 q^{56} +18922.5 q^{57} -59386.4 q^{58} -16917.0 q^{59} -2597.86 q^{61} -10056.8 q^{62} -374.575 q^{63} -1328.50 q^{64} -7725.18 q^{66} -59848.5 q^{67} +11181.8 q^{68} -34877.3 q^{69} -62258.0 q^{71} -7859.17 q^{72} -35314.1 q^{73} +21785.9 q^{74} +38522.7 q^{76} +559.550 q^{77} +2432.44 q^{78} +78372.2 q^{79} +6561.00 q^{81} -4193.38 q^{82} -39714.4 q^{83} -762.567 q^{84} +136247. q^{86} -75344.0 q^{87} +11740.2 q^{88} +54307.3 q^{89} -176.187 q^{91} -71003.9 q^{92} -12759.2 q^{93} -81209.4 q^{94} -53432.8 q^{96} -33574.1 q^{97} -119074. q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 9 q^{2} + 72 q^{3} + 107 q^{4} - 81 q^{6} + 66 q^{7} - 207 q^{8} + 648 q^{9}+O(q^{10}) $ 8 * q - 9 * q^2 + 72 * q^3 + 107 * q^4 - 81 * q^6 + 66 * q^7 - 207 * q^8 + 648 * q^9	\( 8 q - 9 q^{2} + 72 q^{3} + 107 q^{4} - 81 q^{6} + 66 q^{7} - 207 q^{8} + 648 q^{9} - 968 q^{11} + 963 q^{12} - 382 q^{13} + 1048 q^{14} - 325 q^{16} - 288 q^{17} - 729 q^{18} - 988 q^{19} + 594 q^{21} + 1089 q^{22} - 5972 q^{23} - 1863 q^{24} - 14579 q^{26} + 5832 q^{27} - 5942 q^{28} + 1032 q^{29} - 4682 q^{31} - 3863 q^{32} - 8712 q^{33} + 2206 q^{34} + 8667 q^{36} + 17200 q^{37} - 11011 q^{38} - 3438 q^{39} - 13220 q^{41} + 9432 q^{42} - 22872 q^{43} - 12947 q^{44} + 9101 q^{46} - 6700 q^{47} - 2925 q^{48} - 43466 q^{49} - 2592 q^{51} - 5009 q^{52} - 6224 q^{53} - 6561 q^{54} + 20992 q^{56} - 8892 q^{57} - 33015 q^{58} - 77556 q^{59} + 11554 q^{61} - 12135 q^{62} + 5346 q^{63} - 149917 q^{64} + 9801 q^{66} - 20894 q^{67} - 91776 q^{68} - 53748 q^{69} - 21648 q^{71} - 16767 q^{72} - 64660 q^{73} - 179522 q^{74} + 24401 q^{76} - 7986 q^{77} - 131211 q^{78} - 22660 q^{79} + 52488 q^{81} + 56080 q^{82} - 100390 q^{83} - 53478 q^{84} + 47271 q^{86} + 9288 q^{87} + 25047 q^{88} - 25578 q^{89} + 73250 q^{91} - 95311 q^{92} - 42138 q^{93} - 170120 q^{94} - 34767 q^{96} - 142828 q^{97} - 303397 q^{98} - 78408 q^{99}+O(q^{100}) \) 8 * q - 9 * q^2 + 72 * q^3 + 107 * q^4 - 81 * q^6 + 66 * q^7 - 207 * q^8 + 648 * q^9 - 968 * q^11 + 963 * q^12 - 382 * q^13 + 1048 * q^14 - 325 * q^16 - 288 * q^17 - 729 * q^18 - 988 * q^19 + 594 * q^21 + 1089 * q^22 - 5972 * q^23 - 1863 * q^24 - 14579 * q^26 + 5832 * q^27 - 5942 * q^28 + 1032 * q^29 - 4682 * q^31 - 3863 * q^32 - 8712 * q^33 + 2206 * q^34 + 8667 * q^36 + 17200 * q^37 - 11011 * q^38 - 3438 * q^39 - 13220 * q^41 + 9432 * q^42 - 22872 * q^43 - 12947 * q^44 + 9101 * q^46 - 6700 * q^47 - 2925 * q^48 - 43466 * q^49 - 2592 * q^51 - 5009 * q^52 - 6224 * q^53 - 6561 * q^54 + 20992 * q^56 - 8892 * q^57 - 33015 * q^58 - 77556 * q^59 + 11554 * q^61 - 12135 * q^62 + 5346 * q^63 - 149917 * q^64 + 9801 * q^66 - 20894 * q^67 - 91776 * q^68 - 53748 * q^69 - 21648 * q^71 - 16767 * q^72 - 64660 * q^73 - 179522 * q^74 + 24401 * q^76 - 7986 * q^77 - 131211 * q^78 - 22660 * q^79 + 52488 * q^81 + 56080 * q^82 - 100390 * q^83 - 53478 * q^84 + 47271 * q^86 + 9288 * q^87 + 25047 * q^88 - 25578 * q^89 + 73250 * q^91 - 95311 * q^92 - 42138 * q^93 - 170120 * q^94 - 34767 * q^96 - 142828 * q^97 - 303397 * q^98 - 78408 * 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 $

$2$	7.09383	1.25402	0.627012	−	0.779010i	$-0.284278\pi$
$2$	7.09383	1.25402	0.627012	+	0.779010i	$0.284278\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	18.3224	0.572574
$5$	0	0
$5$	0	0
$6$	63.8444	0.724011
$7$	−4.62438	−0.0356705	−0.0178352	−	0.999841i	$-0.505677\pi$
$7$	−4.62438	−0.0356705	−0.0178352	+	0.999841i	$0.505677\pi$
$8$	−97.0268	−0.536002
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	164.901	0.330576
$13$	38.0995	0.0625261	0.0312631	−	0.999511i	$-0.490047\pi$
$13$	38.0995	0.0625261	0.0312631	+	0.999511i	$0.490047\pi$
$14$	−32.8046	−0.0447316
$15$	0	0
$16$	−1274.61	−1.24473
$17$	610.281	0.512163	0.256081	−	0.966655i	$-0.417569\pi$
$17$	610.281	0.512163	0.256081	+	0.966655i	$0.417569\pi$
$18$	574.600	0.418008
$19$	2102.49	1.33614	0.668068	−	0.744100i	$-0.267121\pi$
$19$	2102.49	1.33614	0.668068	+	0.744100i	$0.267121\pi$
$20$	0	0
$21$	−41.6194	−0.0205944
$22$	−858.353	−0.378102
$23$	−3875.26	−1.52750	−0.763750	−	0.645513i	$-0.776644\pi$
$23$	−3875.26	−1.52750	−0.763750	+	0.645513i	$0.776644\pi$
$24$	−873.241	−0.309461
$25$	0	0
$26$	270.272	0.0784092
$27$	729.000	0.192450
$28$	−84.7296	−0.0204240
$29$	−8371.56	−1.84847	−0.924233	−	0.381829i	$-0.875294\pi$
$29$	−8371.56	−1.84847	−0.924233	+	0.381829i	$0.875294\pi$
$30$	0	0
$31$	−1417.68	−0.264957	−0.132479	−	0.991186i	$-0.542294\pi$
$31$	−1417.68	−0.264957	−0.132479	+	0.991186i	$0.542294\pi$
$32$	−5936.98	−1.02492
$33$	−1089.00	−0.174078
$34$	4329.23	0.642264
$35$	0	0
$36$	1484.11	0.190858
$37$	3071.11	0.368800	0.184400	−	0.982851i	$-0.440966\pi$
$37$	3071.11	0.368800	0.184400	+	0.982851i	$0.440966\pi$
$38$	14914.7	1.67555
$39$	342.896	0.0360995
$40$	0	0
$41$	−591.131	−0.0549192	−0.0274596	−	0.999623i	$-0.508742\pi$
$41$	−591.131	−0.0549192	−0.0274596	+	0.999623i	$0.508742\pi$
$42$	−295.241	−0.0258258
$43$	19206.4	1.58407	0.792037	−	0.610474i	$-0.209021\pi$
$43$	19206.4	1.58407	0.792037	+	0.610474i	$0.209021\pi$
$44$	−2217.01	−0.172638
$45$	0	0
$46$	−27490.4	−1.91552
$47$	−11447.9	−0.755929	−0.377965	−	0.925820i	$-0.623376\pi$
$47$	−11447.9	−0.755929	−0.377965	+	0.925820i	$0.623376\pi$
$48$	−11471.5	−0.718647
$49$	−16785.6	−0.998728
$50$	0	0
$51$	5492.53	0.295697
$52$	698.074	0.0358008
$53$	−17463.0	−0.853944	−0.426972	−	0.904265i	$-0.640420\pi$
$53$	−17463.0	−0.853944	−0.426972	+	0.904265i	$0.640420\pi$
$54$	5171.40	0.241337
$55$	0	0
$56$	448.689	0.0191194
$57$	18922.5	0.771419
$58$	−59386.4	−2.31802
$59$	−16917.0	−0.632692	−0.316346	−	0.948644i	$-0.602456\pi$
$59$	−16917.0	−0.632692	−0.316346	+	0.948644i	$0.602456\pi$
$60$	0	0
$61$	−2597.86	−0.0893904	−0.0446952	−	0.999001i	$-0.514232\pi$
$61$	−2597.86	−0.0893904	−0.0446952	+	0.999001i	$0.514232\pi$
$62$	−10056.8	−0.332262
$63$	−374.575	−0.0118902
$64$	−1328.50	−0.0405426
$65$	0	0
$66$	−7725.18	−0.218297
$67$	−59848.5	−1.62879	−0.814397	−	0.580308i	$-0.802932\pi$
$67$	−59848.5	−1.62879	−0.814397	+	0.580308i	$0.802932\pi$
$68$	11181.8	0.293251
$69$	−34877.3	−0.881902
$70$	0	0
$71$	−62258.0	−1.46572	−0.732858	−	0.680382i	$-0.761814\pi$
$71$	−62258.0	−1.46572	−0.732858	+	0.680382i	$0.761814\pi$
$72$	−7859.17	−0.178667
$73$	−35314.1	−0.775605	−0.387803	−	0.921742i	$-0.626766\pi$
$73$	−35314.1	−0.775605	−0.387803	+	0.921742i	$0.626766\pi$
$74$	21785.9	0.462484
$75$	0	0
$76$	38522.7	0.765037
$77$	559.550	0.0107550
$78$	2432.44	0.0452696
$79$	78372.2	1.41284	0.706422	−	0.707791i	$-0.250308\pi$
$79$	78372.2	1.41284	0.706422	+	0.707791i	$0.250308\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−4193.38	−0.0688700
$83$	−39714.4	−0.632781	−0.316390	−	0.948629i	$-0.602471\pi$
$83$	−39714.4	−0.632781	−0.316390	+	0.948629i	$0.602471\pi$
$84$	−762.567	−0.0117918
$85$	0	0
$86$	136247.	1.98646
$87$	−75344.0	−1.06721
$88$	11740.2	0.161611
$89$	54307.3	0.726747	0.363374	−	0.931643i	$-0.381625\pi$
$89$	54307.3	0.726747	0.363374	+	0.931643i	$0.381625\pi$
$90$	0	0
$91$	−176.187	−0.00223034
$92$	−71003.9	−0.874606
$93$	−12759.2	−0.152973
$94$	−81209.4	−0.947953
$95$	0	0
$96$	−53432.8	−0.591739
$97$	−33574.1	−0.362305	−0.181153	−	0.983455i	$-0.557983\pi$
$97$	−33574.1	−0.362305	−0.181153	+	0.983455i	$0.557983\pi$
$98$	−119074.	−1.25243
$99$	−9801.00	−0.100504
$100$	0	0
$101$	−203384.	−1.98387	−0.991935	−	0.126750i	$-0.959545\pi$
$101$	−203384.	−1.98387	−0.991935	+	0.126750i	$0.959545\pi$
$102$	38963.1	0.370811
$103$	95955.0	0.891199	0.445599	−	0.895233i	$-0.352991\pi$
$103$	95955.0	0.891199	0.445599	+	0.895233i	$0.352991\pi$
$104$	−3696.68	−0.0335141
$105$	0	0
$106$	−123880.	−1.07087
$107$	−170934.	−1.44334	−0.721669	−	0.692239i	$-0.756625\pi$
$107$	−170934.	−1.44334	−0.721669	+	0.692239i	$0.756625\pi$
$108$	13357.0	0.110192
$109$	−62167.4	−0.501183	−0.250592	−	0.968093i	$-0.580625\pi$
$109$	−62167.4	−0.501183	−0.250592	+	0.968093i	$0.580625\pi$
$110$	0	0
$111$	27640.0	0.212927
$112$	5894.27	0.0444002
$113$	−75966.1	−0.559660	−0.279830	−	0.960050i	$-0.590278\pi$
$113$	−75966.1	−0.559660	−0.279830	+	0.960050i	$0.590278\pi$
$114$	134233.	0.967377
$115$	0	0
$116$	−153387.	−1.05838
$117$	3086.06	0.0208420
$118$	−120006.	−0.793410
$119$	−2822.17	−0.0182691
$120$	0	0
$121$	14641.0	0.0909091
$122$	−18428.8	−0.112098
$123$	−5320.18	−0.0317076
$124$	−25975.3	−0.151707
$125$	0	0
$126$	−2657.17	−0.0149105
$127$	100022.	0.550285	0.275143	−	0.961403i	$-0.411275\pi$
$127$	100022.	0.550285	0.275143	+	0.961403i	$0.411275\pi$
$128$	180559.	0.974080
$129$	172858.	0.914565
$130$	0	0
$131$	322642.	1.64264	0.821319	−	0.570469i	$-0.193238\pi$
$131$	322642.	1.64264	0.821319	+	0.570469i	$0.193238\pi$
$132$	−19953.1	−0.0996723
$133$	−9722.74	−0.0476606
$134$	−424555.	−2.04255
$135$	0	0
$136$	−59213.6	−0.274520
$137$	−325855.	−1.48328	−0.741639	−	0.670799i	$-0.765951\pi$
$137$	−325855.	−1.48328	−0.741639	+	0.670799i	$0.765951\pi$
$138$	−247414.	−1.10593
$139$	415444.	1.82379	0.911895	−	0.410422i	$-0.134619\pi$
$139$	415444.	1.82379	0.911895	+	0.410422i	$0.134619\pi$
$140$	0	0
$141$	−103031.	−0.436436
$142$	−441648.	−1.83804
$143$	−4610.04	−0.0188523
$144$	−103243.	−0.414911
$145$	0	0
$146$	−250512.	−0.972627
$147$	−151071.	−0.576616
$148$	56270.0	0.211165
$149$	306627.	1.13147	0.565737	−	0.824585i	$-0.308592\pi$
$149$	306627.	1.13147	0.565737	+	0.824585i	$0.308592\pi$
$150$	0	0
$151$	−90347.8	−0.322460	−0.161230	−	0.986917i	$-0.551546\pi$
$151$	−90347.8	−0.322460	−0.161230	+	0.986917i	$0.551546\pi$
$152$	−203998.	−0.716172
$153$	49432.8	0.170721
$154$	3969.35	0.0134871
$155$	0	0
$156$	6282.66	0.0206696
$157$	−81270.2	−0.263137	−0.131569	−	0.991307i	$-0.542001\pi$
$157$	−81270.2	−0.263137	−0.131569	+	0.991307i	$0.542001\pi$
$158$	555959.	1.77174
$159$	−157167.	−0.493025
$160$	0	0
$161$	17920.7	0.0544866
$162$	46542.6	0.139336
$163$	178404.	0.525940	0.262970	−	0.964804i	$-0.415298\pi$
$163$	178404.	0.525940	0.262970	+	0.964804i	$0.415298\pi$
$164$	−10830.9	−0.0314453
$165$	0	0
$166$	−281727.	−0.793522
$167$	−626513.	−1.73836	−0.869178	−	0.494500i	$-0.835351\pi$
$167$	−626513.	−1.73836	−0.869178	+	0.494500i	$0.835351\pi$
$168$	4038.20	0.0110386
$169$	−369841.	−0.996090
$170$	0	0
$171$	170302.	0.445379
$172$	351907.	0.906999
$173$	325238.	0.826202	0.413101	−	0.910685i	$-0.364446\pi$
$173$	325238.	0.826202	0.413101	+	0.910685i	$0.364446\pi$
$174$	−534478.	−1.33831
$175$	0	0
$176$	154227.	0.375301
$177$	−152253.	−0.365285
$178$	385247.	0.911358
$179$	183002.	0.426897	0.213449	−	0.976954i	$-0.431530\pi$
$179$	183002.	0.426897	0.213449	+	0.976954i	$0.431530\pi$
$180$	0	0
$181$	−623259.	−1.41407	−0.707037	−	0.707177i	$-0.749968\pi$
$181$	−623259.	−1.41407	−0.707037	+	0.707177i	$0.749968\pi$
$182$	−1249.84	−0.00279689
$183$	−23380.7	−0.0516096
$184$	376004.	0.818743
$185$	0	0
$186$	−90511.3	−0.191832
$187$	−73844.0	−0.154423
$188$	−209753.	−0.432826
$189$	−3371.18	−0.00686478
$190$	0	0
$191$	179426.	0.355879	0.177939	−	0.984041i	$-0.443057\pi$
$191$	179426.	0.355879	0.177939	+	0.984041i	$0.443057\pi$
$192$	−11956.5	−0.0234073
$193$	904409.	1.74772	0.873859	−	0.486179i	$-0.161610\pi$
$193$	904409.	1.74772	0.873859	+	0.486179i	$0.161610\pi$
$194$	−238169.	−0.454339
$195$	0	0
$196$	−307552.	−0.571845
$197$	494917.	0.908588	0.454294	−	0.890852i	$-0.349892\pi$
$197$	494917.	0.908588	0.454294	+	0.890852i	$0.349892\pi$
$198$	−69526.6	−0.126034
$199$	−824334.	−1.47561	−0.737804	−	0.675015i	$-0.764137\pi$
$199$	−824334.	−1.47561	−0.737804	+	0.675015i	$0.764137\pi$
$200$	0	0
$201$	−538636.	−0.940385
$202$	−1.44277e6	−2.48782
$203$	38713.3	0.0659356
$204$	100636.	0.169308
$205$	0	0
$206$	680688.	1.11758
$207$	−313896.	−0.509166
$208$	−48561.9	−0.0778283
$209$	−254402.	−0.402860
$210$	0	0
$211$	4819.05	0.00745170	0.00372585	−	0.999993i	$-0.498814\pi$
$211$	4819.05	0.00745170	0.00372585	+	0.999993i	$0.498814\pi$
$212$	−319964.	−0.488946
$213$	−560322.	−0.846231
$214$	−1.21257e6	−1.80998
$215$	0	0
$216$	−70732.5	−0.103154
$217$	6555.92	0.00945114
$218$	−441005.	−0.628495
$219$	−317827.	−0.447796
$220$	0	0
$221$	23251.4	0.0320235
$222$	196073.	0.267015
$223$	225057.	0.303062	0.151531	−	0.988453i	$-0.451580\pi$
$223$	225057.	0.303062	0.151531	+	0.988453i	$0.451580\pi$
$224$	27454.9	0.0365594
$225$	0	0
$226$	−538891.	−0.701826
$227$	−847198.	−1.09124	−0.545620	−	0.838033i	$-0.683706\pi$
$227$	−847198.	−1.09124	−0.545620	+	0.838033i	$0.683706\pi$
$228$	346704.	0.441694
$229$	901257.	1.13569	0.567845	−	0.823135i	$-0.307777\pi$
$229$	901257.	1.13569	0.567845	+	0.823135i	$0.307777\pi$
$230$	0	0
$231$	5035.95	0.00620943
$232$	812265.	0.990782
$233$	439120.	0.529900	0.264950	−	0.964262i	$-0.414645\pi$
$233$	439120.	0.529900	0.264950	+	0.964262i	$0.414645\pi$
$234$	21892.0	0.0261364
$235$	0	0
$236$	−309959.	−0.362263
$237$	705350.	0.815706
$238$	−20020.0	−0.0229098
$239$	−922328.	−1.04446	−0.522229	−	0.852805i	$-0.674899\pi$
$239$	−922328.	−1.04446	−0.522229	+	0.852805i	$0.674899\pi$
$240$	0	0
$241$	−891658.	−0.988907	−0.494454	−	0.869204i	$-0.664632\pi$
$241$	−891658.	−0.988907	−0.494454	+	0.869204i	$0.664632\pi$
$242$	103861.	0.114002
$243$	59049.0	0.0641500
$244$	−47598.9	−0.0511826
$245$	0	0
$246$	−37740.4	−0.0397621
$247$	80104.1	0.0835434
$248$	137553.	0.142018
$249$	−357430.	−0.365336
$250$	0	0
$251$	−125904.	−0.126141	−0.0630704	−	0.998009i	$-0.520089\pi$
$251$	−125904.	−0.126141	−0.0630704	+	0.998009i	$0.520089\pi$
$252$	−6863.10	−0.00680799
$253$	468906.	0.460558
$254$	709541.	0.690070
$255$	0	0
$256$	1.32337e6	1.26206
$257$	864526.	0.816479	0.408240	−	0.912875i	$-0.366143\pi$
$257$	864526.	0.816479	0.408240	+	0.912875i	$0.366143\pi$
$258$	1.22622e6	1.14689
$259$	−14202.0	−0.0131553
$260$	0	0
$261$	−678096.	−0.616155
$262$	2.28876e6	2.05991
$263$	860845.	0.767424	0.383712	−	0.923453i	$-0.374645\pi$
$263$	860845.	0.767424	0.383712	+	0.923453i	$0.374645\pi$
$264$	105662.	0.0933060
$265$	0	0
$266$	−68971.4	−0.0597675
$267$	488766.	0.419588
$268$	−1.09657e6	−0.932605
$269$	−169703.	−0.142991	−0.0714956	−	0.997441i	$-0.522777\pi$
$269$	−169703.	−0.142991	−0.0714956	+	0.997441i	$0.522777\pi$
$270$	0	0
$271$	−368847.	−0.305087	−0.152543	−	0.988297i	$-0.548746\pi$
$271$	−368847.	−0.305087	−0.152543	+	0.988297i	$0.548746\pi$
$272$	−777869.	−0.637506
$273$	−1585.68	−0.00128768
$274$	−2.31156e6	−1.86006
$275$	0	0
$276$	−639035.	−0.504954
$277$	423587.	0.331698	0.165849	−	0.986151i	$-0.446964\pi$
$277$	423587.	0.331698	0.165849	+	0.986151i	$0.446964\pi$
$278$	2.94708e6	2.28708
$279$	−114832.	−0.0883190
$280$	0	0
$281$	2.39124e6	1.80658	0.903290	−	0.429031i	$-0.141145\pi$
$281$	2.39124e6	1.80658	0.903290	+	0.429031i	$0.141145\pi$
$282$	−730885.	−0.547301
$283$	−238848.	−0.177279	−0.0886393	−	0.996064i	$-0.528252\pi$
$283$	−238848.	−0.177279	−0.0886393	+	0.996064i	$0.528252\pi$
$284$	−1.14071e6	−0.839230
$285$	0	0
$286$	−32702.9	−0.0236413
$287$	2733.62	0.00195899
$288$	−480895.	−0.341641
$289$	−1.04741e6	−0.737690
$290$	0	0
$291$	−302167.	−0.209177
$292$	−647037.	−0.444091
$293$	−289624.	−0.197091	−0.0985454	−	0.995133i	$-0.531419\pi$
$293$	−289624.	−0.197091	−0.0985454	+	0.995133i	$0.531419\pi$
$294$	−1.07167e6	−0.723089
$295$	0	0
$296$	−297980.	−0.197678
$297$	−88209.0	−0.0580259
$298$	2.17516e6	1.41890
$299$	−147646.	−0.0955086
$300$	0	0
$301$	−88817.8	−0.0565046
$302$	−640912.	−0.404372
$303$	−1.83045e6	−1.14539
$304$	−2.67985e6	−1.66313
$305$	0	0
$306$	350668.	0.214088
$307$	2.74983e6	1.66517	0.832586	−	0.553895i	$-0.186859\pi$
$307$	2.74983e6	1.66517	0.832586	+	0.553895i	$0.186859\pi$
$308$	10252.3	0.00615806
$309$	863595.	0.514534
$310$	0	0
$311$	−669544.	−0.392534	−0.196267	−	0.980550i	$-0.562882\pi$
$311$	−669544.	−0.392534	−0.196267	+	0.980550i	$0.562882\pi$
$312$	−33270.1	−0.0193494
$313$	458501.	0.264533	0.132266	−	0.991214i	$-0.457775\pi$
$313$	458501.	0.264533	0.132266	+	0.991214i	$0.457775\pi$
$314$	−576516.	−0.329980
$315$	0	0
$316$	1.43596e6	0.808958
$317$	−1.69327e6	−0.946406	−0.473203	−	0.880953i	$-0.656902\pi$
$317$	−1.69327e6	−0.946406	−0.473203	+	0.880953i	$0.656902\pi$
$318$	−1.11492e6	−0.618264
$319$	1.01296e6	0.557333
$320$	0	0
$321$	−1.53840e6	−0.833311
$322$	127126.	0.0683275
$323$	1.28311e6	0.684319
$324$	120213.	0.0636193
$325$	0	0
$326$	1.26557e6	0.659540
$327$	−559507.	−0.289358
$328$	57355.6	0.0294368
$329$	52939.5	0.0269644
$330$	0	0
$331$	−2.96369e6	−1.48684	−0.743418	−	0.668827i	$-0.766797\pi$
$331$	−2.96369e6	−1.48684	−0.743418	+	0.668827i	$0.766797\pi$
$332$	−727663.	−0.362314
$333$	248760.	0.122933
$334$	−4.44437e6	−2.17994
$335$	0	0
$336$	53048.4	0.0256345
$337$	2.28507e6	1.09604	0.548018	−	0.836466i	$-0.315382\pi$
$337$	2.28507e6	1.09604	0.548018	+	0.836466i	$0.315382\pi$
$338$	−2.62359e6	−1.24912
$339$	−683695.	−0.323120
$340$	0	0
$341$	171540.	0.0798875
$342$	1.20809e6	0.558515
$343$	155345.	0.0712955
$344$	−1.86354e6	−0.849067
$345$	0	0
$346$	2.30718e6	1.03608
$347$	427019.	0.190381	0.0951905	−	0.995459i	$-0.469654\pi$
$347$	427019.	0.190381	0.0951905	+	0.995459i	$0.469654\pi$
$348$	−1.38048e6	−0.611058
$349$	−2.34367e6	−1.02999	−0.514995	−	0.857193i	$-0.672206\pi$
$349$	−2.34367e6	−1.02999	−0.514995	+	0.857193i	$0.672206\pi$
$350$	0	0
$351$	27774.6	0.0120332
$352$	718375.	0.309026
$353$	1.63784e6	0.699574	0.349787	−	0.936829i	$-0.386254\pi$
$353$	1.63784e6	0.699574	0.349787	+	0.936829i	$0.386254\pi$
$354$	−1.08005e6	−0.458076
$355$	0	0
$356$	995039.	0.416117
$357$	−25399.6	−0.0105477
$358$	1.29819e6	0.535339
$359$	2.80870e6	1.15019	0.575095	−	0.818087i	$-0.304965\pi$
$359$	2.80870e6	1.15019	0.575095	+	0.818087i	$0.304965\pi$
$360$	0	0
$361$	1.94439e6	0.785261
$362$	−4.42129e6	−1.77328
$363$	131769.	0.0524864
$364$	−3228.16	−0.00127703
$365$	0	0
$366$	−165859.	−0.0647196
$367$	−40056.0	−0.0155239	−0.00776197	−	0.999970i	$-0.502471\pi$
$367$	−40056.0	−0.0155239	−0.00776197	+	0.999970i	$0.502471\pi$
$368$	4.93943e6	1.90133
$369$	−47881.6	−0.0183064
$370$	0	0
$371$	80755.6	0.0304606
$372$	−233778.	−0.0875884
$373$	−1.60363e6	−0.596806	−0.298403	−	0.954440i	$-0.596454\pi$
$373$	−1.60363e6	−0.596806	−0.298403	+	0.954440i	$0.596454\pi$
$374$	−523837.	−0.193650
$375$	0	0
$376$	1.11075e6	0.405180
$377$	−318953.	−0.115577
$378$	−23914.5	−0.00860860
$379$	2.12659e6	0.760476	0.380238	−	0.924889i	$-0.375842\pi$
$379$	2.12659e6	0.760476	0.380238	+	0.924889i	$0.375842\pi$
$380$	0	0
$381$	900201.	0.317707
$382$	1.27282e6	0.446280
$383$	3.69333e6	1.28653	0.643267	−	0.765642i	$-0.277579\pi$
$383$	3.69333e6	1.28653	0.643267	+	0.765642i	$0.277579\pi$
$384$	1.62503e6	0.562386
$385$	0	0
$386$	6.41572e6	2.19168
$387$	1.55572e6	0.528024
$388$	−615157.	−0.207447
$389$	−215626.	−0.0722482	−0.0361241	−	0.999347i	$-0.511501\pi$
$389$	−215626.	−0.0722482	−0.0361241	+	0.999347i	$0.511501\pi$
$390$	0	0
$391$	−2.36500e6	−0.782328
$392$	1.62865e6	0.535320
$393$	2.90377e6	0.948378
$394$	3.51086e6	1.13939
$395$	0	0
$396$	−179578.	−0.0575458
$397$	−4.45417e6	−1.41837	−0.709186	−	0.705021i	$-0.750938\pi$
$397$	−4.45417e6	−1.41837	−0.709186	+	0.705021i	$0.750938\pi$
$398$	−5.84769e6	−1.85045
$399$	−87504.7	−0.0275169
$400$	0	0
$401$	5.31987e6	1.65211	0.826057	−	0.563586i	$-0.190579\pi$
$401$	5.31987e6	1.65211	0.826057	+	0.563586i	$0.190579\pi$
$402$	−3.82099e6	−1.17926
$403$	−54013.1	−0.0165667
$404$	−3.72647e6	−1.13591
$405$	0	0
$406$	274625.	0.0826848
$407$	−371604.	−0.111197
$408$	−532923.	−0.158494
$409$	−3.81493e6	−1.12766	−0.563830	−	0.825891i	$-0.690673\pi$
$409$	−3.81493e6	−1.12766	−0.563830	+	0.825891i	$0.690673\pi$
$410$	0	0
$411$	−2.93269e6	−0.856371
$412$	1.75812e6	0.510277
$413$	78230.5	0.0225684
$414$	−2.22672e6	−0.638506
$415$	0	0
$416$	−226196.	−0.0640844
$417$	3.73899e6	1.05297
$418$	−1.80468e6	−0.505196
$419$	−3.59908e6	−1.00151	−0.500756	−	0.865588i	$-0.666945\pi$
$419$	−3.59908e6	−1.00151	−0.500756	+	0.865588i	$0.666945\pi$
$420$	0	0
$421$	−2.15462e6	−0.592467	−0.296234	−	0.955115i	$-0.595731\pi$
$421$	−2.15462e6	−0.592467	−0.296234	+	0.955115i	$0.595731\pi$
$422$	34185.5	0.00934460
$423$	−927280.	−0.251976
$424$	1.69438e6	0.457716
$425$	0	0
$426$	−3.97483e6	−1.06119
$427$	12013.5	0.00318860
$428$	−3.13191e6	−0.826417
$429$	−41490.4	−0.0108844
$430$	0	0
$431$	−2.10237e6	−0.545149	−0.272575	−	0.962135i	$-0.587875\pi$
$431$	−2.10237e6	−0.545149	−0.272575	+	0.962135i	$0.587875\pi$
$432$	−929188.	−0.239549
$433$	−41848.5	−0.0107265	−0.00536327	−	0.999986i	$-0.501707\pi$
$433$	−41848.5	−0.0107265	−0.00536327	+	0.999986i	$0.501707\pi$
$434$	46506.5	0.0118519
$435$	0	0
$436$	−1.13905e6	−0.286964
$437$	−8.14771e6	−2.04095
$438$	−2.25461e6	−0.561546
$439$	−5.35761e6	−1.32681	−0.663407	−	0.748259i	$-0.730890\pi$
$439$	−5.35761e6	−1.32681	−0.663407	+	0.748259i	$0.730890\pi$
$440$	0	0
$441$	−1.35963e6	−0.332909
$442$	164942.	0.0401582
$443$	4.69738e6	1.13722	0.568612	−	0.822606i	$-0.307480\pi$
$443$	4.69738e6	1.13722	0.568612	+	0.822606i	$0.307480\pi$
$444$	506430.	0.121916
$445$	0	0
$446$	1.59652e6	0.380046
$447$	2.75964e6	0.653257
$448$	6143.50	0.00144617
$449$	7.10537e6	1.66330	0.831650	−	0.555300i	$-0.187397\pi$
$449$	7.10537e6	1.66330	0.831650	+	0.555300i	$0.187397\pi$
$450$	0	0
$451$	71526.9	0.0165588
$452$	−1.39188e6	−0.320447
$453$	−813131.	−0.186172
$454$	−6.00987e6	−1.36844
$455$	0	0
$456$	−1.83598e6	−0.413482
$457$	4.82566e6	1.08085	0.540426	−	0.841392i	$-0.318263\pi$
$457$	4.82566e6	1.08085	0.540426	+	0.841392i	$0.318263\pi$
$458$	6.39336e6	1.42418
$459$	444895.	0.0985657
$460$	0	0
$461$	−1.03634e6	−0.227117	−0.113558	−	0.993531i	$-0.536225\pi$
$461$	−1.03634e6	−0.227117	−0.113558	+	0.993531i	$0.536225\pi$
$462$	35724.2	0.00778677
$463$	457904.	0.0992709	0.0496355	−	0.998767i	$-0.484194\pi$
$463$	457904.	0.0992709	0.0496355	+	0.998767i	$0.484194\pi$
$464$	1.06704e7	2.30085
$465$	0	0
$466$	3.11504e6	0.664507
$467$	2.02783e6	0.430268	0.215134	−	0.976584i	$-0.430981\pi$
$467$	2.02783e6	0.430268	0.215134	+	0.976584i	$0.430981\pi$
$468$	56544.0	0.0119336
$469$	276762.	0.0580998
$470$	0	0
$471$	−731431.	−0.151922
$472$	1.64140e6	0.339124
$473$	−2.32398e6	−0.477616
$474$	5.00363e6	1.02291
$475$	0	0
$476$	−51708.9	−0.0104604
$477$	−1.41450e6	−0.284648
$478$	−6.54284e6	−1.30977
$479$	3.41441e6	0.679949	0.339975	−	0.940435i	$-0.389582\pi$
$479$	3.41441e6	0.679949	0.339975	+	0.940435i	$0.389582\pi$
$480$	0	0
$481$	117008.	0.0230596
$482$	−6.32526e6	−1.24011
$483$	161286.	0.0314579
$484$	268258.	0.0520522
$485$	0	0
$486$	418883.	0.0804456
$487$	−1.21543e6	−0.232224	−0.116112	−	0.993236i	$-0.537043\pi$
$487$	−1.21543e6	−0.232224	−0.116112	+	0.993236i	$0.537043\pi$
$488$	252062.	0.0479134
$489$	1.60564e6	0.303651
$490$	0	0
$491$	−4.85480e6	−0.908798	−0.454399	−	0.890798i	$-0.650146\pi$
$491$	−4.85480e6	−0.908798	−0.454399	+	0.890798i	$0.650146\pi$
$492$	−97478.3	−0.0181550
$493$	−5.10901e6	−0.946715
$494$	568244.	0.104765
$495$	0	0
$496$	1.80699e6	0.329801
$497$	287905.	0.0522827
$498$	−2.53555e6	−0.458140
$499$	7.01256e6	1.26074	0.630370	−	0.776295i	$-0.282903\pi$
$499$	7.01256e6	1.26074	0.630370	+	0.776295i	$0.282903\pi$
$500$	0	0
$501$	−5.63861e6	−1.00364
$502$	−893142.	−0.158184
$503$	7.65551e6	1.34913	0.674565	−	0.738215i	$-0.264331\pi$
$503$	7.65551e6	1.34913	0.674565	+	0.738215i	$0.264331\pi$
$504$	36343.8	0.00637315
$505$	0	0
$506$	3.32634e6	0.577551
$507$	−3.32857e6	−0.575093
$508$	1.83265e6	0.315079
$509$	7.70001e6	1.31734	0.658669	−	0.752433i	$-0.271120\pi$
$509$	7.70001e6	1.31734	0.658669	+	0.752433i	$0.271120\pi$
$510$	0	0
$511$	163306.	0.0276662
$512$	3.60985e6	0.608574
$513$	1.53272e6	0.257140
$514$	6.13279e6	1.02388
$515$	0	0
$516$	3.16716e6	0.523656
$517$	1.38520e6	0.227921
$518$	−100746.	−0.0164970
$519$	2.92714e6	0.477008
$520$	0	0
$521$	−3.03644e6	−0.490083	−0.245042	−	0.969513i	$-0.578802\pi$
$521$	−3.03644e6	−0.490083	−0.245042	+	0.969513i	$0.578802\pi$
$522$	−4.81030e6	−0.772673
$523$	−7.75936e6	−1.24043	−0.620214	−	0.784432i	$-0.712954\pi$
$523$	−7.75936e6	−1.24043	−0.620214	+	0.784432i	$0.712954\pi$
$524$	5.91156e6	0.940532
$525$	0	0
$526$	6.10669e6	0.962368
$527$	−865187.	−0.135701
$528$	1.38805e6	0.216680
$529$	8.58128e6	1.33325
$530$	0	0
$531$	−1.37027e6	−0.210897
$532$	−178144.	−0.0272892
$533$	−22521.8	−0.00343389
$534$	3.46722e6	0.526173
$535$	0	0
$536$	5.80691e6	0.873037
$537$	1.64702e6	0.246469
$538$	−1.20385e6	−0.179314
$539$	2.03106e6	0.301128
$540$	0	0
$541$	−8.80224e6	−1.29300	−0.646502	−	0.762912i	$-0.723769\pi$
$541$	−8.80224e6	−1.29300	−0.646502	+	0.762912i	$0.723769\pi$
$542$	−2.61654e6	−0.382586
$543$	−5.60933e6	−0.816416
$544$	−3.62323e6	−0.524927
$545$	0	0
$546$	−11248.6	−0.00161479
$547$	3.58640e6	0.512496	0.256248	−	0.966611i	$-0.417514\pi$
$547$	3.58640e6	0.512496	0.256248	+	0.966611i	$0.417514\pi$
$548$	−5.97043e6	−0.849286
$549$	−210426.	−0.0297968
$550$	0	0
$551$	−1.76012e7	−2.46980
$552$	3.38403e6	0.472701
$553$	−362423.	−0.0503968
$554$	3.00485e6	0.415957
$555$	0	0
$556$	7.61191e6	1.04426
$557$	−4.61032e6	−0.629641	−0.314821	−	0.949151i	$-0.601944\pi$
$557$	−4.61032e6	−0.629641	−0.314821	+	0.949151i	$0.601944\pi$
$558$	−814602.	−0.110754
$559$	731756.	0.0990459
$560$	0	0
$561$	−664596.	−0.0891561
$562$	1.69630e7	2.26549
$563$	−2.16200e6	−0.287464	−0.143732	−	0.989617i	$-0.545910\pi$
$563$	−2.16200e6	−0.287464	−0.143732	+	0.989617i	$0.545910\pi$
$564$	−1.88777e6	−0.249892
$565$	0	0
$566$	−1.69435e6	−0.222311
$567$	−30340.6	−0.00396338
$568$	6.04070e6	0.785627
$569$	1.71372e6	0.221901	0.110951	−	0.993826i	$-0.464610\pi$
$569$	1.71372e6	0.221901	0.110951	+	0.993826i	$0.464610\pi$
$570$	0	0
$571$	−3.80446e6	−0.488318	−0.244159	−	0.969735i	$-0.578512\pi$
$571$	−3.80446e6	−0.488318	−0.244159	+	0.969735i	$0.578512\pi$
$572$	−84466.9	−0.0107944
$573$	1.61484e6	0.205467
$574$	19391.8	0.00245662
$575$	0	0
$576$	−107609.	−0.0135142
$577$	9.84742e6	1.23135	0.615677	−	0.787998i	$-0.288883\pi$
$577$	9.84742e6	1.23135	0.615677	+	0.787998i	$0.288883\pi$
$578$	−7.43017e6	−0.925080
$579$	8.13968e6	1.00905
$580$	0	0
$581$	183655.	0.0225716
$582$	−2.14352e6	−0.262313
$583$	2.11302e6	0.257474
$584$	3.42641e6	0.415726
$585$	0	0
$586$	−2.05455e6	−0.247156
$587$	−4.48517e6	−0.537258	−0.268629	−	0.963244i	$-0.586571\pi$
$587$	−4.48517e6	−0.537258	−0.268629	+	0.963244i	$0.586571\pi$
$588$	−2.76797e6	−0.330155
$589$	−2.98067e6	−0.354019
$590$	0	0
$591$	4.45426e6	0.524574
$592$	−3.91446e6	−0.459058
$593$	−1.57719e7	−1.84182	−0.920909	−	0.389777i	$-0.872552\pi$
$593$	−1.57719e7	−1.84182	−0.920909	+	0.389777i	$0.872552\pi$
$594$	−625739.	−0.0727658
$595$	0	0
$596$	5.61813e6	0.647853
$597$	−7.41901e6	−0.851942
$598$	−1.04737e6	−0.119770
$599$	−8.13287e6	−0.926140	−0.463070	−	0.886322i	$-0.653252\pi$
$599$	−8.13287e6	−0.926140	−0.463070	+	0.886322i	$0.653252\pi$
$600$	0	0
$601$	6.56975e6	0.741930	0.370965	−	0.928647i	$-0.379027\pi$
$601$	6.56975e6	0.741930	0.370965	+	0.928647i	$0.379027\pi$
$602$	−630058.	−0.0708581
$603$	−4.84773e6	−0.542931
$604$	−1.65539e6	−0.184632
$605$	0	0
$606$	−1.29849e7	−1.43634
$607$	5.13693e6	0.565890	0.282945	−	0.959136i	$-0.408689\pi$
$607$	5.13693e6	0.565890	0.282945	+	0.959136i	$0.408689\pi$
$608$	−1.24825e7	−1.36944
$609$	348420.	0.0380680
$610$	0	0
$611$	−436160.	−0.0472653
$612$	905726.	0.0977503
$613$	4.41922e6	0.475001	0.237501	−	0.971387i	$-0.423672\pi$
$613$	4.41922e6	0.475001	0.237501	+	0.971387i	$0.423672\pi$
$614$	1.95068e7	2.08817
$615$	0	0
$616$	−54291.4	−0.00576473
$617$	5.70350e6	0.603154	0.301577	−	0.953442i	$-0.402487\pi$
$617$	5.70350e6	0.603154	0.301577	+	0.953442i	$0.402487\pi$
$618$	6.12619e6	0.645237
$619$	381437.	0.0400125	0.0200063	−	0.999800i	$-0.493631\pi$
$619$	381437.	0.0400125	0.0200063	+	0.999800i	$0.493631\pi$
$620$	0	0
$621$	−2.82506e6	−0.293967
$622$	−4.74963e6	−0.492247
$623$	−251138.	−0.0259234
$624$	−437057.	−0.0449342
$625$	0	0
$626$	3.25253e6	0.331730
$627$	−2.28962e6	−0.232592
$628$	−1.48906e6	−0.150665
$629$	1.87424e6	0.188886
$630$	0	0
$631$	−5.26994e6	−0.526905	−0.263453	−	0.964672i	$-0.584861\pi$
$631$	−5.26994e6	−0.526905	−0.263453	+	0.964672i	$0.584861\pi$
$632$	−7.60420e6	−0.757288
$633$	43371.5	0.00430224
$634$	−1.20117e7	−1.18682
$635$	0	0
$636$	−2.87967e6	−0.282293
$637$	−639524.	−0.0624466
$638$	7.18575e6	0.698909
$639$	−5.04290e6	−0.488572
$640$	0	0
$641$	−735036.	−0.0706584	−0.0353292	−	0.999376i	$-0.511248\pi$
$641$	−735036.	−0.0706584	−0.0353292	+	0.999376i	$0.511248\pi$
$642$	−1.09132e7	−1.04499
$643$	1.45125e7	1.38425	0.692125	−	0.721778i	$-0.256675\pi$
$643$	1.45125e7	1.38425	0.692125	+	0.721778i	$0.256675\pi$
$644$	328349.	0.0311976
$645$	0	0
$646$	9.10218e6	0.858152
$647$	−1.87650e6	−0.176233	−0.0881167	−	0.996110i	$-0.528085\pi$
$647$	−1.87650e6	−0.176233	−0.0881167	+	0.996110i	$0.528085\pi$
$648$	−636593.	−0.0595558
$649$	2.04695e6	0.190764
$650$	0	0
$651$	59003.3	0.00545662
$652$	3.26879e6	0.301139
$653$	1.72496e7	1.58306	0.791529	−	0.611131i	$-0.209285\pi$
$653$	1.72496e7	1.58306	0.791529	+	0.611131i	$0.209285\pi$
$654$	−3.96904e6	−0.362862
$655$	0	0
$656$	753460.	0.0683598
$657$	−2.86044e6	−0.258535
$658$	375544.	0.0338139
$659$	7.60070e6	0.681773	0.340887	−	0.940104i	$-0.389273\pi$
$659$	7.60070e6	0.681773	0.340887	+	0.940104i	$0.389273\pi$
$660$	0	0
$661$	−74179.4	−0.00660358	−0.00330179	−	0.999995i	$-0.501051\pi$
$661$	−74179.4	−0.00660358	−0.00330179	+	0.999995i	$0.501051\pi$
$662$	−2.10239e7	−1.86453
$663$	209263.	0.0184888
$664$	3.85336e6	0.339172
$665$	0	0
$666$	1.76466e6	0.154161
$667$	3.24420e7	2.82353
$668$	−1.14792e7	−0.995337
$669$	2.02552e6	0.174973
$670$	0	0
$671$	314341.	0.0269522
$672$	247094.	0.0211076
$673$	5.80762e6	0.494265	0.247133	−	0.968982i	$-0.420512\pi$
$673$	5.80762e6	0.494265	0.247133	+	0.968982i	$0.420512\pi$
$674$	1.62099e7	1.37446
$675$	0	0
$676$	−6.77637e6	−0.570335
$677$	1.91692e7	1.60743	0.803716	−	0.595013i	$-0.202853\pi$
$677$	1.91692e7	1.60743	0.803716	+	0.595013i	$0.202853\pi$
$678$	−4.85002e6	−0.405199
$679$	155259.	0.0129236
$680$	0	0
$681$	−7.62478e6	−0.630028
$682$	1.21687e6	0.100181
$683$	−2.11907e7	−1.73817	−0.869087	−	0.494660i	$-0.835293\pi$
$683$	−2.11907e7	−1.73817	−0.869087	+	0.494660i	$0.835293\pi$
$684$	3.12034e6	0.255012
$685$	0	0
$686$	1.10199e6	0.0894062
$687$	8.11132e6	0.655691
$688$	−2.44806e7	−1.97175
$689$	−665332.	−0.0533938
$690$	0	0
$691$	1.24582e7	0.992565	0.496282	−	0.868161i	$-0.334698\pi$
$691$	1.24582e7	0.992565	0.496282	+	0.868161i	$0.334698\pi$
$692$	5.95913e6	0.473062
$693$	45323.6	0.00358502
$694$	3.02920e6	0.238742
$695$	0	0
$696$	7.31039e6	0.572028
$697$	−360756.	−0.0281276
$698$	−1.66256e7	−1.29163
$699$	3.95208e6	0.305938
$700$	0	0
$701$	582671.	0.0447845	0.0223923	−	0.999749i	$-0.492872\pi$
$701$	582671.	0.0447845	0.0223923	+	0.999749i	$0.492872\pi$
$702$	197028.	0.0150899
$703$	6.45699e6	0.492767
$704$	160749.	0.0122241
$705$	0	0
$706$	1.16185e7	0.877282
$707$	940525.	0.0707655
$708$	−2.78963e6	−0.209153
$709$	−3.31741e6	−0.247847	−0.123923	−	0.992292i	$-0.539548\pi$
$709$	−3.31741e6	−0.247847	−0.123923	+	0.992292i	$0.539548\pi$
$710$	0	0
$711$	6.34815e6	0.470948
$712$	−5.26926e6	−0.389538
$713$	5.49389e6	0.404722
$714$	−180180.	−0.0132270
$715$	0	0
$716$	3.35303e6	0.244430
$717$	−8.30096e6	−0.603018
$718$	1.99244e7	1.44236
$719$	−8.22478e6	−0.593338	−0.296669	−	0.954980i	$-0.595876\pi$
$719$	−8.22478e6	−0.593338	−0.296669	+	0.954980i	$0.595876\pi$
$720$	0	0
$721$	−443733.	−0.0317895
$722$	1.37931e7	0.984736
$723$	−8.02492e6	−0.570946
$724$	−1.14196e7	−0.809661
$725$	0	0
$726$	934746.	0.0658191
$727$	−5.99896e6	−0.420959	−0.210479	−	0.977598i	$-0.567503\pi$
$727$	−5.99896e6	−0.420959	−0.210479	+	0.977598i	$0.567503\pi$
$728$	17094.8	0.00119546
$729$	531441.	0.0370370
$730$	0	0
$731$	1.17213e7	0.811303
$732$	−428390.	−0.0295503
$733$	−1.02874e7	−0.707204	−0.353602	−	0.935396i	$-0.615043\pi$
$733$	−1.02874e7	−0.707204	−0.353602	+	0.935396i	$0.615043\pi$
$734$	−284150.	−0.0194674
$735$	0	0
$736$	2.30073e7	1.56557
$737$	7.24167e6	0.491100
$738$	−339664.	−0.0229567
$739$	4.35478e6	0.293329	0.146665	−	0.989186i	$-0.453146\pi$
$739$	4.35478e6	0.293329	0.146665	+	0.989186i	$0.453146\pi$
$740$	0	0
$741$	720937.	0.0482338
$742$	572866.	0.0381983
$743$	6.54892e6	0.435209	0.217604	−	0.976037i	$-0.430176\pi$
$743$	6.54892e6	0.435209	0.217604	+	0.976037i	$0.430176\pi$
$744$	1.23798e6	0.0819939
$745$	0	0
$746$	−1.13759e7	−0.748409
$747$	−3.21687e6	−0.210927
$748$	−1.35300e6	−0.0884185
$749$	790462.	0.0514845
$750$	0	0
$751$	1.42508e6	0.0922019	0.0461009	−	0.998937i	$-0.485320\pi$
$751$	1.42508e6	0.0922019	0.0461009	+	0.998937i	$0.485320\pi$
$752$	1.45916e7	0.940930
$753$	−1.13314e6	−0.0728275
$754$	−2.26259e6	−0.144937
$755$	0	0
$756$	−61767.9	−0.00393060
$757$	−1.59943e7	−1.01443	−0.507217	−	0.861818i	$-0.669326\pi$
$757$	−1.59943e7	−1.01443	−0.507217	+	0.861818i	$0.669326\pi$
$758$	1.50856e7	0.953654
$759$	4.22016e6	0.265904
$760$	0	0
$761$	−3.04747e7	−1.90756	−0.953778	−	0.300512i	$-0.902843\pi$
$761$	−3.04747e7	−1.90756	−0.953778	+	0.300512i	$0.902843\pi$
$762$	6.38587e6	0.398412
$763$	287486.	0.0178774
$764$	3.28751e6	0.203767
$765$	0	0
$766$	2.61998e7	1.61334
$767$	−644529.	−0.0395598
$768$	1.19103e7	0.728652
$769$	8.77682e6	0.535206	0.267603	−	0.963529i	$-0.413768\pi$
$769$	8.77682e6	0.535206	0.267603	+	0.963529i	$0.413768\pi$
$770$	0	0
$771$	7.78073e6	0.471394
$772$	1.65709e7	1.00070
$773$	867299.	0.0522060	0.0261030	−	0.999659i	$-0.491690\pi$
$773$	867299.	0.0522060	0.0261030	+	0.999659i	$0.491690\pi$
$774$	1.10360e7	0.662155
$775$	0	0
$776$	3.25758e6	0.194196
$777$	−127818.	−0.00759520
$778$	−1.52961e6	−0.0906009
$779$	−1.24285e6	−0.0733796
$780$	0	0
$781$	7.53322e6	0.441930
$782$	−1.67769e7	−0.981057
$783$	−6.10287e6	−0.355737
$784$	2.13951e7	1.24315
$785$	0	0
$786$	2.05989e7	1.18929
$787$	2.23177e7	1.28444	0.642218	−	0.766522i	$-0.278015\pi$
$787$	2.23177e7	1.28444	0.642218	+	0.766522i	$0.278015\pi$
$788$	9.06806e6	0.520234
$789$	7.74761e6	0.443073
$790$	0	0
$791$	351297.	0.0199633
$792$	950959.	0.0538702
$793$	−98977.2	−0.00558923
$794$	−3.15971e7	−1.77867
$795$	0	0
$796$	−1.51038e7	−0.844894
$797$	1.21353e7	0.676716	0.338358	−	0.941018i	$-0.390129\pi$
$797$	1.21353e7	0.676716	0.338358	+	0.941018i	$0.390129\pi$
$798$	−620743.	−0.0345068
$799$	−6.98644e6	−0.387159
$800$	0	0
$801$	4.39889e6	0.242249
$802$	3.77382e7	2.07179
$803$	4.27300e6	0.233854
$804$	−9.86909e6	−0.538440
$805$	0	0
$806$	−383160.	−0.0207751
$807$	−1.52733e6	−0.0825561
$808$	1.97337e7	1.06336
$809$	−1.25683e7	−0.675160	−0.337580	−	0.941297i	$-0.609608\pi$
$809$	−1.25683e7	−0.675160	−0.337580	+	0.941297i	$0.609608\pi$
$810$	0	0
$811$	6.29850e6	0.336268	0.168134	−	0.985764i	$-0.446226\pi$
$811$	6.29850e6	0.336268	0.168134	+	0.985764i	$0.446226\pi$
$812$	709319.	0.0377530
$813$	−3.31963e6	−0.176142
$814$	−2.63610e6	−0.139444
$815$	0	0
$816$	−7.00082e6	−0.368064
$817$	4.03814e7	2.11654
$818$	−2.70624e7	−1.41411
$819$	−14271.1	−0.000743445 0
$820$	0	0
$821$	1.89136e7	0.979303	0.489651	−	0.871918i	$-0.337124\pi$
$821$	1.89136e7	0.979303	0.489651	+	0.871918i	$0.337124\pi$
$822$	−2.08040e7	−1.07391
$823$	2.67549e7	1.37691	0.688453	−	0.725281i	$-0.258290\pi$
$823$	2.67549e7	1.37691	0.688453	+	0.725281i	$0.258290\pi$
$824$	−9.31020e6	−0.477684
$825$	0	0
$826$	554954.	0.0283013
$827$	−1.76676e7	−0.898283	−0.449142	−	0.893461i	$-0.648270\pi$
$827$	−1.76676e7	−0.898283	−0.449142	+	0.893461i	$0.648270\pi$
$828$	−5.75132e6	−0.291535
$829$	−2.37092e7	−1.19820	−0.599102	−	0.800673i	$-0.704476\pi$
$829$	−2.37092e7	−1.19820	−0.599102	+	0.800673i	$0.704476\pi$
$830$	0	0
$831$	3.81228e6	0.191506
$832$	−50615.3	−0.00253497
$833$	−1.02439e7	−0.511511
$834$	2.65238e7	1.32044
$835$	0	0
$836$	−4.66124e6	−0.230667
$837$	−1.03349e6	−0.0509910
$838$	−2.55312e7	−1.25592
$839$	−1.98735e7	−0.974696	−0.487348	−	0.873208i	$-0.662036\pi$
$839$	−1.98735e7	−0.974696	−0.487348	+	0.873208i	$0.662036\pi$
$840$	0	0
$841$	4.95719e7	2.41683
$842$	−1.52845e7	−0.742968
$843$	2.15211e7	1.04303
$844$	88296.4	0.00426665
$845$	0	0
$846$	−6.57796e6	−0.315984
$847$	−67705.6	−0.00324277
$848$	2.22585e7	1.06293
$849$	−2.14964e6	−0.102352
$850$	0	0
$851$	−1.19013e7	−0.563342
$852$	−1.02664e7	−0.484530
$853$	−5.52250e6	−0.259874	−0.129937	−	0.991522i	$-0.541478\pi$
$853$	−5.52250e6	−0.259874	−0.129937	+	0.991522i	$0.541478\pi$
$854$	85221.6	0.00399857
$855$	0	0
$856$	1.65851e7	0.773632
$857$	2.02109e7	0.940014	0.470007	−	0.882663i	$-0.344251\pi$
$857$	2.02109e7	0.940014	0.470007	+	0.882663i	$0.344251\pi$
$858$	−294326.	−0.0136493
$859$	2.05904e7	0.952097	0.476048	−	0.879419i	$-0.342069\pi$
$859$	2.05904e7	0.952097	0.476048	+	0.879419i	$0.342069\pi$
$860$	0	0
$861$	24602.6	0.00113103
$862$	−1.49138e7	−0.683630
$863$	−1.59735e7	−0.730086	−0.365043	−	0.930991i	$-0.618946\pi$
$863$	−1.59735e7	−0.730086	−0.365043	+	0.930991i	$0.618946\pi$
$864$	−4.32806e6	−0.197246
$865$	0	0
$866$	−296866.	−0.0134513
$867$	−9.42672e6	−0.425905
$868$	120120.	0.00541148
$869$	−9.48304e6	−0.425989
$870$	0	0
$871$	−2.28020e6	−0.101842
$872$	6.03190e6	0.268635
$873$	−2.71950e6	−0.120768
$874$	−5.77984e7	−2.55940
$875$	0	0
$876$	−5.82334e6	−0.256396
$877$	1.40128e7	0.615215	0.307607	−	0.951513i	$-0.400472\pi$
$877$	1.40128e7	0.615215	0.307607	+	0.951513i	$0.400472\pi$
$878$	−3.80060e7	−1.66386
$879$	−2.60662e6	−0.113790
$880$	0	0
$881$	−2.71339e7	−1.17780	−0.588901	−	0.808205i	$-0.700439\pi$
$881$	−2.71339e7	−1.17780	−0.588901	+	0.808205i	$0.700439\pi$
$882$	−9.64501e6	−0.417476
$883$	1.60023e7	0.690687	0.345344	−	0.938476i	$-0.387762\pi$
$883$	1.60023e7	0.690687	0.345344	+	0.938476i	$0.387762\pi$
$884$	426021.	0.0183358
$885$	0	0
$886$	3.33224e7	1.42611
$887$	2.23865e7	0.955381	0.477690	−	0.878528i	$-0.341474\pi$
$887$	2.23865e7	0.955381	0.477690	+	0.878528i	$0.341474\pi$
$888$	−2.68182e6	−0.114129
$889$	−462542.	−0.0196289
$890$	0	0
$891$	−793881.	−0.0335013
$892$	4.12358e6	0.173525
$893$	−2.40692e7	−1.01003
$894$	1.95764e7	0.819200
$895$	0	0
$896$	−834975.	−0.0347459
$897$	−1.32881e6	−0.0551419
$898$	5.04042e7	2.08582
$899$	1.18682e7	0.489764
$900$	0	0
$901$	−1.06573e7	−0.437358
$902$	507399.	0.0207651
$903$	−799361.	−0.0326230
$904$	7.37075e6	0.299979
$905$	0	0
$906$	−5.76821e6	−0.233464
$907$	−6.33552e6	−0.255719	−0.127860	−	0.991792i	$-0.540811\pi$
$907$	−6.33552e6	−0.255719	−0.127860	+	0.991792i	$0.540811\pi$
$908$	−1.55227e7	−0.624816
$909$	−1.64741e7	−0.661290
$910$	0	0
$911$	6.57264e6	0.262388	0.131194	−	0.991357i	$-0.458119\pi$
$911$	6.57264e6	0.262388	0.131194	+	0.991357i	$0.458119\pi$
$912$	−2.41187e7	−0.960211
$913$	4.80545e6	0.190791
$914$	3.42324e7	1.35541
$915$	0	0
$916$	1.65132e7	0.650267
$917$	−1.49202e6	−0.0585937
$918$	3.15601e6	0.123604
$919$	−4.55407e7	−1.77873	−0.889366	−	0.457196i	$-0.848854\pi$
$919$	−4.55407e7	−1.77873	−0.889366	+	0.457196i	$0.848854\pi$
$920$	0	0
$921$	2.47484e7	0.961388
$922$	−7.35160e6	−0.284809
$923$	−2.37200e6	−0.0916455
$924$	92270.6	0.00355536
$925$	0	0
$926$	3.24829e6	0.124488
$927$	7.77235e6	0.297066
$928$	4.97018e7	1.89453
$929$	−3.24053e7	−1.23190	−0.615952	−	0.787784i	$-0.711229\pi$
$929$	−3.24053e7	−1.23190	−0.615952	+	0.787784i	$0.711229\pi$
$930$	0	0
$931$	−3.52917e7	−1.33444
$932$	8.04572e6	0.303407
$933$	−6.02589e6	−0.226630
$934$	1.43851e7	0.539567
$935$	0	0
$936$	−299431.	−0.0111714
$937$	1.53794e7	0.572257	0.286129	−	0.958191i	$-0.407632\pi$
$937$	1.53794e7	0.572257	0.286129	+	0.958191i	$0.407632\pi$
$938$	1.96330e6	0.0728585
$939$	4.12651e6	0.152728
$940$	0	0
$941$	−2.92624e7	−1.07730	−0.538648	−	0.842531i	$-0.681065\pi$
$941$	−2.92624e7	−1.07730	−0.538648	+	0.842531i	$0.681065\pi$
$942$	−5.18865e6	−0.190514
$943$	2.29079e6	0.0838891
$944$	2.15625e7	0.787533
$945$	0	0
$946$	−1.64859e7	−0.598942
$947$	2.32692e7	0.843154	0.421577	−	0.906793i	$-0.361477\pi$
$947$	2.32692e7	0.843154	0.421577	+	0.906793i	$0.361477\pi$
$948$	1.29237e7	0.467052
$949$	−1.34545e6	−0.0484956
$950$	0	0
$951$	−1.52394e7	−0.546408
$952$	273826.	0.00979226
$953$	1.42689e7	0.508932	0.254466	−	0.967082i	$-0.418100\pi$
$953$	1.42689e7	0.508932	0.254466	+	0.967082i	$0.418100\pi$
$954$	−1.00342e7	−0.356955
$955$	0	0
$956$	−1.68992e7	−0.598029
$957$	9.11663e6	0.321777
$958$	2.42212e7	0.852672
$959$	1.50688e6	0.0529092
$960$	0	0
$961$	−2.66193e7	−0.929798
$962$	830034.	0.0289173
$963$	−1.38456e7	−0.481112
$964$	−1.63373e7	−0.566223
$965$	0	0
$966$	1.14414e6	0.0394489
$967$	4.25962e7	1.46489	0.732444	−	0.680827i	$-0.238380\pi$
$967$	4.25962e7	1.46489	0.732444	+	0.680827i	$0.238380\pi$
$968$	−1.42057e6	−0.0487275
$969$	1.15480e7	0.395092
$970$	0	0
$971$	−3.93095e7	−1.33798	−0.668990	−	0.743271i	$-0.733273\pi$
$971$	−3.93095e7	−1.33798	−0.668990	+	0.743271i	$0.733273\pi$
$972$	1.08192e6	0.0367306
$973$	−1.92117e6	−0.0650555
$974$	−8.62205e6	−0.291215
$975$	0	0
$976$	3.31125e6	0.111267
$977$	−5.54613e7	−1.85889	−0.929444	−	0.368962i	$-0.879713\pi$
$977$	−5.54613e7	−1.85889	−0.929444	+	0.368962i	$0.879713\pi$
$978$	1.13901e7	0.380786
$979$	−6.57119e6	−0.219123
$980$	0	0
$981$	−5.03556e6	−0.167061
$982$	−3.44391e7	−1.13965
$983$	−5.37070e7	−1.77275	−0.886375	−	0.462968i	$-0.846784\pi$
$983$	−5.37070e7	−1.77275	−0.886375	+	0.462968i	$0.846784\pi$
$984$	516200.	0.0169954
$985$	0	0
$986$	−3.62424e7	−1.18720
$987$	476455.	0.0155679
$988$	1.46770e6	0.0478348
$989$	−7.44298e7	−2.41967
$990$	0	0
$991$	−803096.	−0.0259767	−0.0129883	−	0.999916i	$-0.504134\pi$
$991$	−803096.	−0.0259767	−0.0129883	+	0.999916i	$0.504134\pi$
$992$	8.41677e6	0.271560
$993$	−2.66732e7	−0.858425
$994$	2.04235e6	0.0655638
$995$	0	0
$996$	−6.54896e6	−0.209182
$997$	6.90977e6	0.220154	0.110077	−	0.993923i	$-0.464890\pi$
$997$	6.90977e6	0.220154	0.110077	+	0.993923i	$0.464890\pi$
$998$	4.97459e7	1.58100
$999$	2.23884e6	0.0709756

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.p.1.7	✓	8
5.4	even	2		825.6.a.q.1.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.6.a.p.1.7	✓	8	1.1	even	1	trivial
825.6.a.q.1.2	yes	8	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+7.09383 q^{2} +9.00000 q^{3} +18.3224 q^{4} +63.8444 q^{6} -4.62438 q^{7} -97.0268 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+7.09383 q^{2} +9.00000 q^{3} +18.3224 q^{4} +63.8444 q^{6} -4.62438 q^{7} -97.0268 q^{8} +81.0000 q^{9} -121.000 q^{11} +164.901 q^{12} +38.0995 q^{13} -32.8046 q^{14} -1274.61 q^{16} +610.281 q^{17} +574.600 q^{18} +2102.49 q^{19} -41.6194 q^{21} -858.353 q^{22} -3875.26 q^{23} -873.241 q^{24} +270.272 q^{26} +729.000 q^{27} -84.7296 q^{28} -8371.56 q^{29} -1417.68 q^{31} -5936.98 q^{32} -1089.00 q^{33} +4329.23 q^{34} +1484.11 q^{36} +3071.11 q^{37} +14914.7 q^{38} +342.896 q^{39} -591.131 q^{41} -295.241 q^{42} +19206.4 q^{43} -2217.01 q^{44} -27490.4 q^{46} -11447.9 q^{47} -11471.5 q^{48} -16785.6 q^{49} +5492.53 q^{51} +698.074 q^{52} -17463.0 q^{53} +5171.40 q^{54} +448.689 q^{56} +18922.5 q^{57} -59386.4 q^{58} -16917.0 q^{59} -2597.86 q^{61} -10056.8 q^{62} -374.575 q^{63} -1328.50 q^{64} -7725.18 q^{66} -59848.5 q^{67} +11181.8 q^{68} -34877.3 q^{69} -62258.0 q^{71} -7859.17 q^{72} -35314.1 q^{73} +21785.9 q^{74} +38522.7 q^{76} +559.550 q^{77} +2432.44 q^{78} +78372.2 q^{79} +6561.00 q^{81} -4193.38 q^{82} -39714.4 q^{83} -762.567 q^{84} +136247. q^{86} -75344.0 q^{87} +11740.2 q^{88} +54307.3 q^{89} -176.187 q^{91} -71003.9 q^{92} -12759.2 q^{93} -81209.4 q^{94} -53432.8 q^{96} -33574.1 q^{97} -119074. q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 9 q^{2} + 72 q^{3} + 107 q^{4} - 81 q^{6} + 66 q^{7} - 207 q^{8} + 648 q^{9}+O(q^{10}) \) 8 * q - 9 * q^2 + 72 * q^3 + 107 * q^4 - 81 * q^6 + 66 * q^7 - 207 * q^8 + 648 * q^9	\( 8 q - 9 q^{2} + 72 q^{3} + 107 q^{4} - 81 q^{6} + 66 q^{7} - 207 q^{8} + 648 q^{9} - 968 q^{11} + 963 q^{12} - 382 q^{13} + 1048 q^{14} - 325 q^{16} - 288 q^{17} - 729 q^{18} - 988 q^{19} + 594 q^{21} + 1089 q^{22} - 5972 q^{23} - 1863 q^{24} - 14579 q^{26} + 5832 q^{27} - 5942 q^{28} + 1032 q^{29} - 4682 q^{31} - 3863 q^{32} - 8712 q^{33} + 2206 q^{34} + 8667 q^{36} + 17200 q^{37} - 11011 q^{38} - 3438 q^{39} - 13220 q^{41} + 9432 q^{42} - 22872 q^{43} - 12947 q^{44} + 9101 q^{46} - 6700 q^{47} - 2925 q^{48} - 43466 q^{49} - 2592 q^{51} - 5009 q^{52} - 6224 q^{53} - 6561 q^{54} + 20992 q^{56} - 8892 q^{57} - 33015 q^{58} - 77556 q^{59} + 11554 q^{61} - 12135 q^{62} + 5346 q^{63} - 149917 q^{64} + 9801 q^{66} - 20894 q^{67} - 91776 q^{68} - 53748 q^{69} - 21648 q^{71} - 16767 q^{72} - 64660 q^{73} - 179522 q^{74} + 24401 q^{76} - 7986 q^{77} - 131211 q^{78} - 22660 q^{79} + 52488 q^{81} + 56080 q^{82} - 100390 q^{83} - 53478 q^{84} + 47271 q^{86} + 9288 q^{87} + 25047 q^{88} - 25578 q^{89} + 73250 q^{91} - 95311 q^{92} - 42138 q^{93} - 170120 q^{94} - 34767 q^{96} - 142828 q^{97} - 303397 q^{98} - 78408 q^{99}+O(q^{100}) \) 8 * q - 9 * q^2 + 72 * q^3 + 107 * q^4 - 81 * q^6 + 66 * q^7 - 207 * q^8 + 648 * q^9 - 968 * q^11 + 963 * q^12 - 382 * q^13 + 1048 * q^14 - 325 * q^16 - 288 * q^17 - 729 * q^18 - 988 * q^19 + 594 * q^21 + 1089 * q^22 - 5972 * q^23 - 1863 * q^24 - 14579 * q^26 + 5832 * q^27 - 5942 * q^28 + 1032 * q^29 - 4682 * q^31 - 3863 * q^32 - 8712 * q^33 + 2206 * q^34 + 8667 * q^36 + 17200 * q^37 - 11011 * q^38 - 3438 * q^39 - 13220 * q^41 + 9432 * q^42 - 22872 * q^43 - 12947 * q^44 + 9101 * q^46 - 6700 * q^47 - 2925 * q^48 - 43466 * q^49 - 2592 * q^51 - 5009 * q^52 - 6224 * q^53 - 6561 * q^54 + 20992 * q^56 - 8892 * q^57 - 33015 * q^58 - 77556 * q^59 + 11554 * q^61 - 12135 * q^62 + 5346 * q^63 - 149917 * q^64 + 9801 * q^66 - 20894 * q^67 - 91776 * q^68 - 53748 * q^69 - 21648 * q^71 - 16767 * q^72 - 64660 * q^73 - 179522 * q^74 + 24401 * q^76 - 7986 * q^77 - 131211 * q^78 - 22660 * q^79 + 52488 * q^81 + 56080 * q^82 - 100390 * q^83 - 53478 * q^84 + 47271 * q^86 + 9288 * q^87 + 25047 * q^88 - 25578 * q^89 + 73250 * q^91 - 95311 * q^92 - 42138 * q^93 - 170120 * q^94 - 34767 * q^96 - 142828 * q^97 - 303397 * q^98 - 78408 * q^99

Introduction

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