LMFDB - Embedded newform 414.8.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [414,8,Mod(1,414)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(414, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("414.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 414 = 2 \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	414.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:	$129.327400550$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 62310x^{2} - 465075x + 877928625 $ x^4 - x^3 - 62310x^2 - 465075x + 877928625
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	no (minimal twist has level 138)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-181.010$ of defining polynomial
Character	$\chi$	$=$	414.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.00000 q^{2} +64.0000 q^{4} -322.021 q^{5} -1215.22 q^{7} +512.000 q^{8} +O(q^{10})$	\(q+8.00000 q^{2} +64.0000 q^{4} -322.021 q^{5} -1215.22 q^{7} +512.000 q^{8} -2576.17 q^{10} -2427.99 q^{11} -9169.50 q^{13} -9721.74 q^{14} +4096.00 q^{16} -34187.7 q^{17} +20831.4 q^{19} -20609.3 q^{20} -19424.0 q^{22} -12167.0 q^{23} +25572.3 q^{25} -73356.0 q^{26} -77773.9 q^{28} -62949.8 q^{29} +943.720 q^{31} +32768.0 q^{32} -273502. q^{34} +391325. q^{35} +83972.1 q^{37} +166651. q^{38} -164875. q^{40} +212907. q^{41} +22412.1 q^{43} -155392. q^{44} -97336.0 q^{46} -1.06462e6 q^{47} +653210. q^{49} +204578. q^{50} -586848. q^{52} +173560. q^{53} +781864. q^{55} -622191. q^{56} -503599. q^{58} -1.39877e6 q^{59} +1.45872e6 q^{61} +7549.76 q^{62} +262144. q^{64} +2.95277e6 q^{65} +3.53235e6 q^{67} -2.18801e6 q^{68} +3.13060e6 q^{70} +567484. q^{71} -684969. q^{73} +671777. q^{74} +1.33321e6 q^{76} +2.95054e6 q^{77} +1.07546e6 q^{79} -1.31900e6 q^{80} +1.70326e6 q^{82} -5.12255e6 q^{83} +1.10091e7 q^{85} +179297. q^{86} -1.24313e6 q^{88} +1.77386e6 q^{89} +1.11429e7 q^{91} -778688. q^{92} -8.51696e6 q^{94} -6.70815e6 q^{95} +1.74617e7 q^{97} +5.22568e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 32 q^{2} + 256 q^{4} + 162 q^{5} + 1218 q^{7} + 2048 q^{8}+O(q^{10}) $ 4 * q + 32 * q^2 + 256 * q^4 + 162 * q^5 + 1218 * q^7 + 2048 * q^8	\( 4 q + 32 q^{2} + 256 q^{4} + 162 q^{5} + 1218 q^{7} + 2048 q^{8} + 1296 q^{10} - 1820 q^{11} - 6508 q^{13} + 9744 q^{14} + 16384 q^{16} - 12870 q^{17} + 74474 q^{19} + 10368 q^{20} - 14560 q^{22} - 48668 q^{23} + 192544 q^{25} - 52064 q^{26} + 77952 q^{28} + 94452 q^{29} - 39420 q^{31} + 131072 q^{32} - 102960 q^{34} + 788392 q^{35} + 75848 q^{37} + 595792 q^{38} + 82944 q^{40} + 519032 q^{41} + 716946 q^{43} - 116480 q^{44} - 389344 q^{46} + 12232 q^{47} + 1656176 q^{49} + 1540352 q^{50} - 416512 q^{52} - 1053394 q^{53} + 1155136 q^{55} + 623616 q^{56} + 755616 q^{58} - 4398344 q^{59} + 5328312 q^{61} - 315360 q^{62} + 1048576 q^{64} - 6366708 q^{65} + 11303966 q^{67} - 823680 q^{68} + 6307136 q^{70} - 8395320 q^{71} + 7107008 q^{73} + 606784 q^{74} + 4766336 q^{76} - 9789160 q^{77} + 9785086 q^{79} + 663552 q^{80} + 4152256 q^{82} - 2424316 q^{83} + 25790076 q^{85} + 5735568 q^{86} - 931840 q^{88} - 3019218 q^{89} + 1693028 q^{91} - 3114752 q^{92} + 97856 q^{94} + 12329136 q^{95} + 32226876 q^{97} + 13249408 q^{98}+O(q^{100}) \) 4 * q + 32 * q^2 + 256 * q^4 + 162 * q^5 + 1218 * q^7 + 2048 * q^8 + 1296 * q^10 - 1820 * q^11 - 6508 * q^13 + 9744 * q^14 + 16384 * q^16 - 12870 * q^17 + 74474 * q^19 + 10368 * q^20 - 14560 * q^22 - 48668 * q^23 + 192544 * q^25 - 52064 * q^26 + 77952 * q^28 + 94452 * q^29 - 39420 * q^31 + 131072 * q^32 - 102960 * q^34 + 788392 * q^35 + 75848 * q^37 + 595792 * q^38 + 82944 * q^40 + 519032 * q^41 + 716946 * q^43 - 116480 * q^44 - 389344 * q^46 + 12232 * q^47 + 1656176 * q^49 + 1540352 * q^50 - 416512 * q^52 - 1053394 * q^53 + 1155136 * q^55 + 623616 * q^56 + 755616 * q^58 - 4398344 * q^59 + 5328312 * q^61 - 315360 * q^62 + 1048576 * q^64 - 6366708 * q^65 + 11303966 * q^67 - 823680 * q^68 + 6307136 * q^70 - 8395320 * q^71 + 7107008 * q^73 + 606784 * q^74 + 4766336 * q^76 - 9789160 * q^77 + 9785086 * q^79 + 663552 * q^80 + 4152256 * q^82 - 2424316 * q^83 + 25790076 * q^85 + 5735568 * q^86 - 931840 * q^88 - 3019218 * q^89 + 1693028 * q^91 - 3114752 * q^92 + 97856 * q^94 + 12329136 * q^95 + 32226876 * q^97 + 13249408 * q^98

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 $

$2$	8.00000	0.707107
$2$	8.00000	0.707107
$3$	0	0
$3$	0	0
$4$	64.0000	0.500000
$5$	−322.021	−1.15210	−0.576048	−	0.817416i	$-0.695406\pi$
$5$	−322.021	−1.15210	−0.576048	+	0.817416i	$0.695406\pi$
$6$	0	0
$7$	−1215.22	−1.33909	−0.669546	−	0.742770i	$-0.733512\pi$
$7$	−1215.22	−1.33909	−0.669546	+	0.742770i	$0.733512\pi$
$8$	512.000	0.353553
$9$	0	0
$10$	−2576.17	−0.814655
$11$	−2427.99	−0.550013	−0.275007	−	0.961442i	$-0.588680\pi$
$11$	−2427.99	−0.550013	−0.275007	+	0.961442i	$0.588680\pi$
$12$	0	0
$13$	−9169.50	−1.15756	−0.578780	−	0.815483i	$-0.696471\pi$
$13$	−9169.50	−1.15756	−0.578780	+	0.815483i	$0.696471\pi$
$14$	−9721.74	−0.946882
$15$	0	0
$16$	4096.00	0.250000
$17$	−34187.7	−1.68771	−0.843857	−	0.536568i	$-0.819720\pi$
$17$	−34187.7	−1.68771	−0.843857	+	0.536568i	$0.819720\pi$
$18$	0	0
$19$	20831.4	0.696757	0.348379	−	0.937354i	$-0.386732\pi$
$19$	20831.4	0.696757	0.348379	+	0.937354i	$0.386732\pi$
$20$	−20609.3	−0.576048
$21$	0	0
$22$	−19424.0	−0.388918
$23$	−12167.0	−0.208514
$23$	−12167.0	−0.208514
$24$	0	0
$25$	25572.3	0.327326
$26$	−73356.0	−0.818519
$27$	0	0
$28$	−77773.9	−0.669546
$29$	−62949.8	−0.479294	−0.239647	−	0.970860i	$-0.577032\pi$
$29$	−62949.8	−0.479294	−0.239647	+	0.970860i	$0.577032\pi$
$30$	0	0
$31$	943.720	0.00568954	0.00284477	−	0.999996i	$-0.499094\pi$
$31$	943.720	0.00568954	0.00284477	+	0.999996i	$0.499094\pi$
$32$	32768.0	0.176777
$33$	0	0
$34$	−273502.	−1.19339
$35$	391325.	1.54276
$36$	0	0
$37$	83972.1	0.272539	0.136270	−	0.990672i	$-0.456489\pi$
$37$	83972.1	0.272539	0.136270	+	0.990672i	$0.456489\pi$
$38$	166651.	0.492682
$39$	0	0
$40$	−164875.	−0.407328
$41$	212907.	0.482443	0.241222	−	0.970470i	$-0.422452\pi$
$41$	212907.	0.482443	0.241222	+	0.970470i	$0.422452\pi$
$42$	0	0
$43$	22412.1	0.0429876	0.0214938	−	0.999769i	$-0.493158\pi$
$43$	22412.1	0.0429876	0.0214938	+	0.999769i	$0.493158\pi$
$44$	−155392.	−0.275007
$45$	0	0
$46$	−97336.0	−0.147442
$47$	−1.06462e6	−1.49573	−0.747864	−	0.663853i	$-0.768920\pi$
$47$	−1.06462e6	−1.49573	−0.747864	+	0.663853i	$0.768920\pi$
$48$	0	0
$49$	653210.	0.793170
$50$	204578.	0.231454
$51$	0	0
$52$	−586848.	−0.578780
$53$	173560.	0.160135	0.0800673	−	0.996789i	$-0.474486\pi$
$53$	173560.	0.160135	0.0800673	+	0.996789i	$0.474486\pi$
$54$	0	0
$55$	781864.	0.633668
$56$	−622191.	−0.473441
$57$	0	0
$58$	−503599.	−0.338912
$59$	−1.39877e6	−0.886674	−0.443337	−	0.896355i	$-0.646206\pi$
$59$	−1.39877e6	−0.886674	−0.443337	+	0.896355i	$0.646206\pi$
$60$	0	0
$61$	1.45872e6	0.822847	0.411423	−	0.911444i	$-0.365032\pi$
$61$	1.45872e6	0.822847	0.411423	+	0.911444i	$0.365032\pi$
$62$	7549.76	0.00402311
$63$	0	0
$64$	262144.	0.125000
$65$	2.95277e6	1.33362
$66$	0	0
$67$	3.53235e6	1.43483	0.717417	−	0.696644i	$-0.245324\pi$
$67$	3.53235e6	1.43483	0.717417	+	0.696644i	$0.245324\pi$
$68$	−2.18801e6	−0.843857
$69$	0	0
$70$	3.13060e6	1.09090
$71$	567484.	0.188170	0.0940848	−	0.995564i	$-0.470008\pi$
$71$	567484.	0.188170	0.0940848	+	0.995564i	$0.470008\pi$
$72$	0	0
$73$	−684969.	−0.206082	−0.103041	−	0.994677i	$-0.532857\pi$
$73$	−684969.	−0.206082	−0.103041	+	0.994677i	$0.532857\pi$
$74$	671777.	0.192714
$75$	0	0
$76$	1.33321e6	0.348379
$77$	2.95054e6	0.736519
$78$	0	0
$79$	1.07546e6	0.245413	0.122707	−	0.992443i	$-0.460843\pi$
$79$	1.07546e6	0.245413	0.122707	+	0.992443i	$0.460843\pi$
$80$	−1.31900e6	−0.288024
$81$	0	0
$82$	1.70326e6	0.341139
$83$	−5.12255e6	−0.983360	−0.491680	−	0.870776i	$-0.663617\pi$
$83$	−5.12255e6	−0.983360	−0.491680	+	0.870776i	$0.663617\pi$
$84$	0	0
$85$	1.10091e7	1.94441
$86$	179297.	0.0303968
$87$	0	0
$88$	−1.24313e6	−0.194459
$89$	1.77386e6	0.266719	0.133360	−	0.991068i	$-0.457423\pi$
$89$	1.77386e6	0.266719	0.133360	+	0.991068i	$0.457423\pi$
$90$	0	0
$91$	1.11429e7	1.55008
$92$	−778688.	−0.104257
$93$	0	0
$94$	−8.51696e6	−1.05764
$95$	−6.70815e6	−0.802731
$96$	0	0
$97$	1.74617e7	1.94261	0.971306	−	0.237832i	$-0.0764369\pi$
$97$	1.74617e7	1.94261	0.971306	+	0.237832i	$0.0764369\pi$
$98$	5.22568e6	0.560856
$99$	0	0
$100$	1.63663e6	0.163663
$101$	9.01006e6	0.870167	0.435084	−	0.900390i	$-0.356719\pi$
$101$	9.01006e6	0.870167	0.435084	+	0.900390i	$0.356719\pi$
$102$	0	0
$103$	1.49001e7	1.34356	0.671782	−	0.740749i	$-0.265529\pi$
$103$	1.49001e7	1.34356	0.671782	+	0.740749i	$0.265529\pi$
$104$	−4.69478e6	−0.409260
$105$	0	0
$106$	1.38848e6	0.113232
$107$	−2.27434e7	−1.79478	−0.897392	−	0.441233i	$-0.854541\pi$
$107$	−2.27434e7	−1.79478	−0.897392	+	0.441233i	$0.854541\pi$
$108$	0	0
$109$	−1.55126e7	−1.14734	−0.573670	−	0.819087i	$-0.694481\pi$
$109$	−1.55126e7	−1.14734	−0.573670	+	0.819087i	$0.694481\pi$
$110$	6.25492e6	0.448071
$111$	0	0
$112$	−4.97753e6	−0.334773
$113$	5.37742e6	0.350590	0.175295	−	0.984516i	$-0.443912\pi$
$113$	5.37742e6	0.350590	0.175295	+	0.984516i	$0.443912\pi$
$114$	0	0
$115$	3.91803e6	0.240229
$116$	−4.02879e6	−0.239647
$117$	0	0
$118$	−1.11902e7	−0.626973
$119$	4.15455e7	2.26001
$120$	0	0
$121$	−1.35920e7	−0.697485
$122$	1.16698e7	0.581841
$123$	0	0
$124$	60398.1	0.00284477
$125$	1.69231e7	0.774986
$126$	0	0
$127$	−1.78204e7	−0.771979	−0.385989	−	0.922503i	$-0.626140\pi$
$127$	−1.78204e7	−0.771979	−0.385989	+	0.922503i	$0.626140\pi$
$128$	2.09715e6	0.0883883
$129$	0	0
$130$	2.36221e7	0.943013
$131$	−1.44603e7	−0.561987	−0.280994	−	0.959710i	$-0.590664\pi$
$131$	−1.44603e7	−0.561987	−0.280994	+	0.959710i	$0.590664\pi$
$132$	0	0
$133$	−2.53147e7	−0.933023
$134$	2.82588e7	1.01458
$135$	0	0
$136$	−1.75041e7	−0.596697
$137$	−2.64405e7	−0.878511	−0.439255	−	0.898362i	$-0.644758\pi$
$137$	−2.64405e7	−0.878511	−0.439255	+	0.898362i	$0.644758\pi$
$138$	0	0
$139$	−635959.	−0.0200853	−0.0100426	−	0.999950i	$-0.503197\pi$
$139$	−635959.	−0.0200853	−0.0100426	+	0.999950i	$0.503197\pi$
$140$	2.50448e7	0.771382
$141$	0	0
$142$	4.53987e6	0.133056
$143$	2.22635e7	0.636674
$144$	0	0
$145$	2.02712e7	0.552192
$146$	−5.47975e6	−0.145722
$147$	0	0
$148$	5.37421e6	0.136270
$149$	3.80259e7	0.941732	0.470866	−	0.882205i	$-0.343942\pi$
$149$	3.80259e7	0.941732	0.470866	+	0.882205i	$0.343942\pi$
$150$	0	0
$151$	−4.55892e7	−1.07756	−0.538781	−	0.842446i	$-0.681115\pi$
$151$	−4.55892e7	−1.07756	−0.538781	+	0.842446i	$0.681115\pi$
$152$	1.06657e7	0.246341
$153$	0	0
$154$	2.36043e7	0.520798
$155$	−303897.	−0.00655490
$156$	0	0
$157$	4.54874e7	0.938086	0.469043	−	0.883175i	$-0.344599\pi$
$157$	4.54874e7	0.938086	0.469043	+	0.883175i	$0.344599\pi$
$158$	8.60365e6	0.173533
$159$	0	0
$160$	−1.05520e7	−0.203664
$161$	1.47855e7	0.279220
$162$	0	0
$163$	−7.75863e6	−0.140323	−0.0701614	−	0.997536i	$-0.522351\pi$
$163$	−7.75863e6	−0.140323	−0.0701614	+	0.997536i	$0.522351\pi$
$164$	1.36260e7	0.241222
$165$	0	0
$166$	−4.09804e7	−0.695341
$167$	8.75023e7	1.45382	0.726912	−	0.686731i	$-0.240955\pi$
$167$	8.75023e7	1.45382	0.726912	+	0.686731i	$0.240955\pi$
$168$	0	0
$169$	2.13312e7	0.339947
$170$	8.80732e7	1.37490
$171$	0	0
$172$	1.43438e6	0.0214938
$173$	−3.52351e7	−0.517386	−0.258693	−	0.965960i	$-0.583292\pi$
$173$	−3.52351e7	−0.517386	−0.258693	+	0.965960i	$0.583292\pi$
$174$	0	0
$175$	−3.10759e7	−0.438319
$176$	−9.94507e6	−0.137503
$177$	0	0
$178$	1.41909e7	0.188599
$179$	−7.75379e7	−1.01048	−0.505241	−	0.862978i	$-0.668596\pi$
$179$	−7.75379e7	−1.01048	−0.505241	+	0.862978i	$0.668596\pi$
$180$	0	0
$181$	1.02205e8	1.28114	0.640570	−	0.767900i	$-0.278698\pi$
$181$	1.02205e8	1.28114	0.640570	+	0.767900i	$0.278698\pi$
$182$	8.91435e7	1.09607
$183$	0	0
$184$	−6.22950e6	−0.0737210
$185$	−2.70407e7	−0.313991
$186$	0	0
$187$	8.30076e7	0.928265
$188$	−6.81357e7	−0.747864
$189$	0	0
$190$	−5.36652e7	−0.567617
$191$	1.09122e8	1.13317	0.566587	−	0.824002i	$-0.308263\pi$
$191$	1.09122e8	1.13317	0.566587	+	0.824002i	$0.308263\pi$
$192$	0	0
$193$	1.55728e7	0.155925	0.0779623	−	0.996956i	$-0.475159\pi$
$193$	1.55728e7	0.155925	0.0779623	+	0.996956i	$0.475159\pi$
$194$	1.39694e8	1.37363
$195$	0	0
$196$	4.18054e7	0.396585
$197$	−1.24247e8	−1.15785	−0.578927	−	0.815379i	$-0.696529\pi$
$197$	−1.24247e8	−1.15785	−0.578927	+	0.815379i	$0.696529\pi$
$198$	0	0
$199$	1.11906e8	1.00662	0.503311	−	0.864106i	$-0.332115\pi$
$199$	1.11906e8	1.00662	0.503311	+	0.864106i	$0.332115\pi$
$200$	1.30930e7	0.115727
$201$	0	0
$202$	7.20805e7	0.615301
$203$	7.64977e7	0.641819
$204$	0	0
$205$	−6.85604e7	−0.555821
$206$	1.19201e8	0.950043
$207$	0	0
$208$	−3.75583e7	−0.289390
$209$	−5.05786e7	−0.383226
$210$	0	0
$211$	−1.61070e8	−1.18039	−0.590197	−	0.807259i	$-0.700950\pi$
$211$	−1.61070e8	−1.18039	−0.590197	+	0.807259i	$0.700950\pi$
$212$	1.11079e7	0.0800673
$213$	0	0
$214$	−1.81947e8	−1.26910
$215$	−7.21716e6	−0.0495258
$216$	0	0
$217$	−1.14682e6	−0.00761882
$218$	−1.24101e8	−0.811292
$219$	0	0
$220$	5.00393e7	0.316834
$221$	3.13484e8	1.95363
$222$	0	0
$223$	2.50898e8	1.51506	0.757530	−	0.652800i	$-0.226406\pi$
$223$	2.50898e8	1.51506	0.757530	+	0.652800i	$0.226406\pi$
$224$	−3.98202e7	−0.236720
$225$	0	0
$226$	4.30194e7	0.247905
$227$	6.57275e7	0.372955	0.186478	−	0.982459i	$-0.440293\pi$
$227$	6.57275e7	0.372955	0.186478	+	0.982459i	$0.440293\pi$
$228$	0	0
$229$	−1.33398e8	−0.734049	−0.367024	−	0.930211i	$-0.619623\pi$
$229$	−1.33398e8	−0.734049	−0.367024	+	0.930211i	$0.619623\pi$
$230$	3.13442e7	0.169867
$231$	0	0
$232$	−3.22303e7	−0.169456
$233$	−2.19012e8	−1.13429	−0.567144	−	0.823619i	$-0.691952\pi$
$233$	−2.19012e8	−1.13429	−0.567144	+	0.823619i	$0.691952\pi$
$234$	0	0
$235$	3.42830e8	1.72322
$236$	−8.95212e7	−0.443337
$237$	0	0
$238$	3.32364e8	1.59807
$239$	−1.05348e7	−0.0499153	−0.0249576	−	0.999689i	$-0.507945\pi$
$239$	−1.05348e7	−0.0499153	−0.0249576	+	0.999689i	$0.507945\pi$
$240$	0	0
$241$	−2.86751e8	−1.31961	−0.659805	−	0.751437i	$-0.729361\pi$
$241$	−2.86751e8	−1.31961	−0.659805	+	0.751437i	$0.729361\pi$
$242$	−1.08736e8	−0.493197
$243$	0	0
$244$	9.33584e7	0.411423
$245$	−2.10347e8	−0.913808
$246$	0	0
$247$	−1.91014e8	−0.806539
$248$	483185.	0.00201156
$249$	0	0
$250$	1.35384e8	0.547998
$251$	2.52569e8	1.00814	0.504071	−	0.863662i	$-0.331835\pi$
$251$	2.52569e8	1.00814	0.504071	+	0.863662i	$0.331835\pi$
$252$	0	0
$253$	2.95414e7	0.114686
$254$	−1.42564e8	−0.545872
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	3.14547e8	1.15590	0.577949	−	0.816073i	$-0.303853\pi$
$257$	3.14547e8	1.15590	0.577949	+	0.816073i	$0.303853\pi$
$258$	0	0
$259$	−1.02044e8	−0.364955
$260$	1.88977e8	0.666811
$261$	0	0
$262$	−1.15682e8	−0.397385
$263$	1.20573e8	0.408700	0.204350	−	0.978898i	$-0.434492\pi$
$263$	1.20573e8	0.408700	0.204350	+	0.978898i	$0.434492\pi$
$264$	0	0
$265$	−5.58900e7	−0.184490
$266$	−2.02518e8	−0.659747
$267$	0	0
$268$	2.26070e8	0.717417
$269$	3.83257e8	1.20049	0.600244	−	0.799817i	$-0.295070\pi$
$269$	3.83257e8	1.20049	0.600244	+	0.799817i	$0.295070\pi$
$270$	0	0
$271$	3.19711e8	0.975810	0.487905	−	0.872897i	$-0.337761\pi$
$271$	3.19711e8	0.975810	0.487905	+	0.872897i	$0.337761\pi$
$272$	−1.40033e8	−0.421928
$273$	0	0
$274$	−2.11524e8	−0.621201
$275$	−6.20894e7	−0.180033
$276$	0	0
$277$	−6.58748e8	−1.86226	−0.931130	−	0.364688i	$-0.881176\pi$
$277$	−6.58748e8	−1.86226	−0.931130	+	0.364688i	$0.881176\pi$
$278$	−5.08767e6	−0.0142024
$279$	0	0
$280$	2.00358e8	0.545449
$281$	−9.20353e7	−0.247447	−0.123724	−	0.992317i	$-0.539484\pi$
$281$	−9.20353e7	−0.247447	−0.123724	+	0.992317i	$0.539484\pi$
$282$	0	0
$283$	−3.52691e8	−0.925001	−0.462501	−	0.886619i	$-0.653048\pi$
$283$	−3.52691e8	−0.925001	−0.462501	+	0.886619i	$0.653048\pi$
$284$	3.63190e7	0.0940848
$285$	0	0
$286$	1.78108e8	0.450197
$287$	−2.58728e8	−0.646036
$288$	0	0
$289$	7.58461e8	1.84838
$290$	1.62169e8	0.390459
$291$	0	0
$292$	−4.38380e7	−0.103041
$293$	−4.27438e8	−0.992742	−0.496371	−	0.868110i	$-0.665335\pi$
$293$	−4.27438e8	−0.992742	−0.496371	+	0.868110i	$0.665335\pi$
$294$	0	0
$295$	4.50433e8	1.02153
$296$	4.29937e7	0.0963571
$297$	0	0
$298$	3.04207e8	0.665905
$299$	1.11565e8	0.241368
$300$	0	0
$301$	−2.72356e7	−0.0575644
$302$	−3.64713e8	−0.761952
$303$	0	0
$304$	8.53255e7	0.174189
$305$	−4.69740e8	−0.947999
$306$	0	0
$307$	−7.95556e8	−1.56923	−0.784615	−	0.619984i	$-0.787139\pi$
$307$	−7.95556e8	−1.56923	−0.784615	+	0.619984i	$0.787139\pi$
$308$	1.88835e8	0.368260
$309$	0	0
$310$	−2.43118e6	−0.00463501
$311$	4.89472e8	0.922712	0.461356	−	0.887215i	$-0.347363\pi$
$311$	4.89472e8	0.922712	0.461356	+	0.887215i	$0.347363\pi$
$312$	0	0
$313$	−6.36293e8	−1.17288	−0.586438	−	0.809994i	$-0.699470\pi$
$313$	−6.36293e8	−1.17288	−0.586438	+	0.809994i	$0.699470\pi$
$314$	3.63899e8	0.663327
$315$	0	0
$316$	6.88292e7	0.122707
$317$	1.12299e9	1.98001	0.990007	−	0.141017i	$-0.0450372\pi$
$317$	1.12299e9	1.98001	0.990007	+	0.141017i	$0.0450372\pi$
$318$	0	0
$319$	1.52842e8	0.263618
$320$	−8.44158e7	−0.144012
$321$	0	0
$322$	1.18284e8	0.197438
$323$	−7.12179e8	−1.17593
$324$	0	0
$325$	−2.34485e8	−0.378899
$326$	−6.20690e7	−0.0992233
$327$	0	0
$328$	1.09008e8	0.170569
$329$	1.29375e9	2.00292
$330$	0	0
$331$	−5.11366e8	−0.775057	−0.387529	−	0.921858i	$-0.626671\pi$
$331$	−5.11366e8	−0.775057	−0.387529	+	0.921858i	$0.626671\pi$
$332$	−3.27843e8	−0.491680
$333$	0	0
$334$	7.00018e8	1.02801
$335$	−1.13749e9	−1.65307
$336$	0	0
$337$	8.33425e8	1.18621	0.593105	−	0.805125i	$-0.297902\pi$
$337$	8.33425e8	1.18621	0.593105	+	0.805125i	$0.297902\pi$
$338$	1.70650e8	0.240379
$339$	0	0
$340$	7.04585e8	0.972204
$341$	−2.29135e6	−0.00312932
$342$	0	0
$343$	2.06992e8	0.276965
$344$	1.14750e7	0.0151984
$345$	0	0
$346$	−2.81881e8	−0.365847
$347$	3.39842e8	0.436640	0.218320	−	0.975877i	$-0.429942\pi$
$347$	3.39842e8	0.436640	0.218320	+	0.975877i	$0.429942\pi$
$348$	0	0
$349$	−1.26924e9	−1.59829	−0.799143	−	0.601141i	$-0.794713\pi$
$349$	−1.26924e9	−1.59829	−0.799143	+	0.601141i	$0.794713\pi$
$350$	−2.48607e8	−0.309939
$351$	0	0
$352$	−7.95605e7	−0.0972296
$353$	6.62670e8	0.801836	0.400918	−	0.916114i	$-0.368691\pi$
$353$	6.62670e8	0.801836	0.400918	+	0.916114i	$0.368691\pi$
$354$	0	0
$355$	−1.82742e8	−0.216790
$356$	1.13527e8	0.133360
$357$	0	0
$358$	−6.20303e8	−0.714518
$359$	−1.36679e9	−1.55909	−0.779543	−	0.626349i	$-0.784548\pi$
$359$	−1.36679e9	−1.55909	−0.779543	+	0.626349i	$0.784548\pi$
$360$	0	0
$361$	−4.59923e8	−0.514529
$362$	8.17639e8	0.905903
$363$	0	0
$364$	7.13148e8	0.775041
$365$	2.20574e8	0.237427
$366$	0	0
$367$	−1.16268e9	−1.22780	−0.613901	−	0.789383i	$-0.710400\pi$
$367$	−1.16268e9	−1.22780	−0.613901	+	0.789383i	$0.710400\pi$
$368$	−4.98360e7	−0.0521286
$369$	0	0
$370$	−2.16326e8	−0.222025
$371$	−2.10914e8	−0.214435
$372$	0	0
$373$	−9.98010e8	−0.995758	−0.497879	−	0.867246i	$-0.665888\pi$
$373$	−9.98010e8	−0.995758	−0.497879	+	0.867246i	$0.665888\pi$
$374$	6.64061e8	0.656383
$375$	0	0
$376$	−5.45086e8	−0.528819
$377$	5.77219e8	0.554812
$378$	0	0
$379$	9.80397e8	0.925049	0.462524	−	0.886607i	$-0.346944\pi$
$379$	9.80397e8	0.925049	0.462524	+	0.886607i	$0.346944\pi$
$380$	−4.29322e8	−0.401366
$381$	0	0
$382$	8.72978e8	0.801275
$383$	−1.91672e9	−1.74327	−0.871633	−	0.490159i	$-0.836939\pi$
$383$	−1.91672e9	−1.74327	−0.871633	+	0.490159i	$0.836939\pi$
$384$	0	0
$385$	−9.50135e8	−0.848541
$386$	1.24582e8	0.110255
$387$	0	0
$388$	1.11755e9	0.971306
$389$	2.29204e9	1.97424	0.987118	−	0.159992i	$-0.0511469\pi$
$389$	2.29204e9	1.97424	0.987118	+	0.159992i	$0.0511469\pi$
$390$	0	0
$391$	4.15962e8	0.351913
$392$	3.34443e8	0.280428
$393$	0	0
$394$	−9.93976e8	−0.818727
$395$	−3.46319e8	−0.282740
$396$	0	0
$397$	3.39501e8	0.272317	0.136158	−	0.990687i	$-0.456524\pi$
$397$	3.39501e8	0.272317	0.136158	+	0.990687i	$0.456524\pi$
$398$	8.95245e8	0.711789
$399$	0	0
$400$	1.04744e8	0.0818314
$401$	7.09395e8	0.549393	0.274696	−	0.961531i	$-0.411423\pi$
$401$	7.09395e8	0.549393	0.274696	+	0.961531i	$0.411423\pi$
$402$	0	0
$403$	−8.65344e6	−0.00658599
$404$	5.76644e8	0.435084
$405$	0	0
$406$	6.11982e8	0.453834
$407$	−2.03884e8	−0.149900
$408$	0	0
$409$	1.78592e9	1.29072	0.645359	−	0.763879i	$-0.276708\pi$
$409$	1.78592e9	1.29072	0.645359	+	0.763879i	$0.276708\pi$
$410$	−5.48483e8	−0.393025
$411$	0	0
$412$	9.53605e8	0.671782
$413$	1.69981e9	1.18734
$414$	0	0
$415$	1.64957e9	1.13293
$416$	−3.00466e8	−0.204630
$417$	0	0
$418$	−4.04629e8	−0.270982
$419$	2.90324e9	1.92812	0.964058	−	0.265691i	$-0.0856001\pi$
$419$	2.90324e9	1.92812	0.964058	+	0.265691i	$0.0856001\pi$
$420$	0	0
$421$	1.86394e9	1.21743	0.608717	−	0.793387i	$-0.291684\pi$
$421$	1.86394e9	1.21743	0.608717	+	0.793387i	$0.291684\pi$
$422$	−1.28856e9	−0.834665
$423$	0	0
$424$	8.88629e7	0.0566161
$425$	−8.74259e8	−0.552432
$426$	0	0
$427$	−1.77267e9	−1.10187
$428$	−1.45558e9	−0.897392
$429$	0	0
$430$	−5.77373e7	−0.0350201
$431$	−1.50064e9	−0.902830	−0.451415	−	0.892314i	$-0.649081\pi$
$431$	−1.50064e9	−0.902830	−0.451415	+	0.892314i	$0.649081\pi$
$432$	0	0
$433$	−1.96980e9	−1.16605	−0.583023	−	0.812456i	$-0.698130\pi$
$433$	−1.96980e9	−1.16605	−0.583023	+	0.812456i	$0.698130\pi$
$434$	−9.17460e6	−0.00538732
$435$	0	0
$436$	−9.92807e8	−0.573670
$437$	−2.53456e8	−0.145284
$438$	0	0
$439$	9.62749e8	0.543109	0.271555	−	0.962423i	$-0.412462\pi$
$439$	9.62749e8	0.543109	0.271555	+	0.962423i	$0.412462\pi$
$440$	4.00315e8	0.224036
$441$	0	0
$442$	2.50787e9	1.38143
$443$	−1.24690e9	−0.681425	−0.340713	−	0.940167i	$-0.610668\pi$
$443$	−1.24690e9	−0.681425	−0.340713	+	0.940167i	$0.610668\pi$
$444$	0	0
$445$	−5.71220e8	−0.307286
$446$	2.00718e9	1.07131
$447$	0	0
$448$	−3.18562e8	−0.167387
$449$	−3.50551e9	−1.82763	−0.913816	−	0.406129i	$-0.866878\pi$
$449$	−3.50551e9	−1.82763	−0.913816	+	0.406129i	$0.866878\pi$
$450$	0	0
$451$	−5.16937e8	−0.265350
$452$	3.44155e8	0.175295
$453$	0	0
$454$	5.25820e8	0.263719
$455$	−3.58825e9	−1.78584
$456$	0	0
$457$	−1.19776e9	−0.587032	−0.293516	−	0.955954i	$-0.594825\pi$
$457$	−1.19776e9	−0.587032	−0.293516	+	0.955954i	$0.594825\pi$
$458$	−1.06718e9	−0.519051
$459$	0	0
$460$	2.50754e8	0.120114
$461$	−2.41420e9	−1.14768	−0.573840	−	0.818967i	$-0.694547\pi$
$461$	−2.41420e9	−1.14768	−0.573840	+	0.818967i	$0.694547\pi$
$462$	0	0
$463$	−5.47261e8	−0.256248	−0.128124	−	0.991758i	$-0.540896\pi$
$463$	−5.47261e8	−0.256248	−0.128124	+	0.991758i	$0.540896\pi$
$464$	−2.57843e8	−0.119823
$465$	0	0
$466$	−1.75210e9	−0.802062
$467$	−3.89790e9	−1.77101	−0.885505	−	0.464630i	$-0.846187\pi$
$467$	−3.89790e9	−1.77101	−0.885505	+	0.464630i	$0.846187\pi$
$468$	0	0
$469$	−4.29257e9	−1.92138
$470$	2.74264e9	1.21850
$471$	0	0
$472$	−7.16170e8	−0.313487
$473$	−5.44165e7	−0.0236438
$474$	0	0
$475$	5.32708e8	0.228066
$476$	2.65891e9	1.13000
$477$	0	0
$478$	−8.42784e7	−0.0352954
$479$	1.53884e9	0.639764	0.319882	−	0.947457i	$-0.396357\pi$
$479$	1.53884e9	0.639764	0.319882	+	0.947457i	$0.396357\pi$
$480$	0	0
$481$	−7.69982e8	−0.315481
$482$	−2.29401e9	−0.933106
$483$	0	0
$484$	−8.69889e8	−0.348743
$485$	−5.62304e9	−2.23808
$486$	0	0
$487$	−4.83701e9	−1.89769	−0.948846	−	0.315740i	$-0.897747\pi$
$487$	−4.83701e9	−1.89769	−0.948846	+	0.315740i	$0.897747\pi$
$488$	7.46867e8	0.290920
$489$	0	0
$490$	−1.68278e9	−0.646160
$491$	−4.03954e9	−1.54009	−0.770047	−	0.637987i	$-0.779767\pi$
$491$	−4.03954e9	−1.54009	−0.770047	+	0.637987i	$0.779767\pi$
$492$	0	0
$493$	2.15211e9	0.808910
$494$	−1.52811e9	−0.570309
$495$	0	0
$496$	3.86548e6	0.00142239
$497$	−6.89617e8	−0.251977
$498$	0	0
$499$	−2.60047e9	−0.936915	−0.468457	−	0.883486i	$-0.655190\pi$
$499$	−2.60047e9	−0.936915	−0.468457	+	0.883486i	$0.655190\pi$
$500$	1.08308e9	0.387493
$501$	0	0
$502$	2.02055e9	0.712865
$503$	−4.89901e9	−1.71641	−0.858203	−	0.513310i	$-0.828419\pi$
$503$	−4.89901e9	−1.71641	−0.858203	+	0.513310i	$0.828419\pi$
$504$	0	0
$505$	−2.90142e9	−1.00252
$506$	2.36331e8	0.0810951
$507$	0	0
$508$	−1.14051e9	−0.385989
$509$	−3.85047e9	−1.29420	−0.647101	−	0.762405i	$-0.724019\pi$
$509$	−3.85047e9	−1.29420	−0.647101	+	0.762405i	$0.724019\pi$
$510$	0	0
$511$	8.32385e8	0.275963
$512$	1.34218e8	0.0441942
$513$	0	0
$514$	2.51637e9	0.817343
$515$	−4.79813e9	−1.54791
$516$	0	0
$517$	2.58489e9	0.822670
$518$	−8.16355e8	−0.258062
$519$	0	0
$520$	1.51182e9	0.471506
$521$	−3.26852e8	−0.101256	−0.0506278	−	0.998718i	$-0.516122\pi$
$521$	−3.26852e8	−0.101256	−0.0506278	+	0.998718i	$0.516122\pi$
$522$	0	0
$523$	2.37138e9	0.724845	0.362423	−	0.932014i	$-0.381950\pi$
$523$	2.37138e9	0.724845	0.362423	+	0.932014i	$0.381950\pi$
$524$	−9.25457e8	−0.280994
$525$	0	0
$526$	9.64584e8	0.288995
$527$	−3.22636e7	−0.00960232
$528$	0	0
$529$	1.48036e8	0.0434783
$530$	−4.47120e8	−0.130454
$531$	0	0
$532$	−1.62014e9	−0.466511
$533$	−1.95225e9	−0.558457
$534$	0	0
$535$	7.32385e9	2.06776
$536$	1.80856e9	0.507290
$537$	0	0
$538$	3.06606e9	0.848873
$539$	−1.58599e9	−0.436254
$540$	0	0
$541$	−3.14800e8	−0.0854761	−0.0427381	−	0.999086i	$-0.513608\pi$
$541$	−3.14800e8	−0.0854761	−0.0427381	+	0.999086i	$0.513608\pi$
$542$	2.55769e9	0.690002
$543$	0	0
$544$	−1.12026e9	−0.298348
$545$	4.99538e9	1.32185
$546$	0	0
$547$	3.60555e9	0.941924	0.470962	−	0.882154i	$-0.343907\pi$
$547$	3.60555e9	0.941924	0.470962	+	0.882154i	$0.343907\pi$
$548$	−1.69219e9	−0.439255
$549$	0	0
$550$	−4.96715e8	−0.127303
$551$	−1.31134e9	−0.333951
$552$	0	0
$553$	−1.30691e9	−0.328631
$554$	−5.26999e9	−1.31682
$555$	0	0
$556$	−4.07014e7	−0.0100426
$557$	6.67431e9	1.63649	0.818244	−	0.574871i	$-0.194948\pi$
$557$	6.67431e9	1.63649	0.818244	+	0.574871i	$0.194948\pi$
$558$	0	0
$559$	−2.05508e8	−0.0497608
$560$	1.60287e9	0.385691
$561$	0	0
$562$	−7.36282e8	−0.174971
$563$	2.43029e9	0.573956	0.286978	−	0.957937i	$-0.407349\pi$
$563$	2.43029e9	0.573956	0.286978	+	0.957937i	$0.407349\pi$
$564$	0	0
$565$	−1.73164e9	−0.403914
$566$	−2.82153e9	−0.654075
$567$	0	0
$568$	2.90552e8	0.0665280
$569$	−3.25823e8	−0.0741461	−0.0370731	−	0.999313i	$-0.511803\pi$
$569$	−3.25823e8	−0.0741461	−0.0370731	+	0.999313i	$0.511803\pi$
$570$	0	0
$571$	−7.03454e9	−1.58128	−0.790641	−	0.612280i	$-0.790252\pi$
$571$	−7.03454e9	−1.58128	−0.790641	+	0.612280i	$0.790252\pi$
$572$	1.42486e9	0.318337
$573$	0	0
$574$	−2.06982e9	−0.456817
$575$	−3.11138e8	−0.0682521
$576$	0	0
$577$	1.26179e9	0.273447	0.136723	−	0.990609i	$-0.456343\pi$
$577$	1.26179e9	0.273447	0.136723	+	0.990609i	$0.456343\pi$
$578$	6.06769e9	1.30700
$579$	0	0
$580$	1.29735e9	0.276096
$581$	6.22501e9	1.31681
$582$	0	0
$583$	−4.21404e8	−0.0880762
$584$	−3.50704e8	−0.0728611
$585$	0	0
$586$	−3.41951e9	−0.701975
$587$	2.43647e9	0.497197	0.248598	−	0.968607i	$-0.420030\pi$
$587$	2.43647e9	0.497197	0.248598	+	0.968607i	$0.420030\pi$
$588$	0	0
$589$	1.96590e7	0.00396423
$590$	3.60346e9	0.722334
$591$	0	0
$592$	3.43950e8	0.0681348
$593$	−1.26627e9	−0.249365	−0.124682	−	0.992197i	$-0.539791\pi$
$593$	−1.26627e9	−0.249365	−0.124682	+	0.992197i	$0.539791\pi$
$594$	0	0
$595$	−1.33785e10	−2.60374
$596$	2.43366e9	0.470866
$597$	0	0
$598$	8.92522e8	0.170673
$599$	6.67077e9	1.26818	0.634092	−	0.773258i	$-0.281374\pi$
$599$	6.67077e9	1.26818	0.634092	+	0.773258i	$0.281374\pi$
$600$	0	0
$601$	−8.35545e8	−0.157003	−0.0785017	−	0.996914i	$-0.525014\pi$
$601$	−8.35545e8	−0.157003	−0.0785017	+	0.996914i	$0.525014\pi$
$602$	−2.17885e8	−0.0407042
$603$	0	0
$604$	−2.91771e9	−0.538781
$605$	4.37691e9	0.803570
$606$	0	0
$607$	1.32918e9	0.241226	0.120613	−	0.992700i	$-0.461514\pi$
$607$	1.32918e9	0.241226	0.120613	+	0.992700i	$0.461514\pi$
$608$	6.82604e8	0.123170
$609$	0	0
$610$	−3.75792e9	−0.670336
$611$	9.76204e9	1.73140
$612$	0	0
$613$	−5.66892e9	−0.994004	−0.497002	−	0.867749i	$-0.665566\pi$
$613$	−5.66892e9	−0.994004	−0.497002	+	0.867749i	$0.665566\pi$
$614$	−6.36445e9	−1.10961
$615$	0	0
$616$	1.51068e9	0.260399
$617$	8.23525e9	1.41149	0.705746	−	0.708465i	$-0.250612\pi$
$617$	8.23525e9	1.41149	0.705746	+	0.708465i	$0.250612\pi$
$618$	0	0
$619$	3.06591e9	0.519568	0.259784	−	0.965667i	$-0.416349\pi$
$619$	3.06591e9	0.519568	0.259784	+	0.965667i	$0.416349\pi$
$620$	−1.94494e7	−0.00327745
$621$	0	0
$622$	3.91577e9	0.652456
$623$	−2.15563e9	−0.357162
$624$	0	0
$625$	−7.44741e9	−1.22018
$626$	−5.09035e9	−0.829348
$627$	0	0
$628$	2.91119e9	0.469043
$629$	−2.87081e9	−0.459968
$630$	0	0
$631$	1.10525e10	1.75129	0.875647	−	0.482952i	$-0.160435\pi$
$631$	1.10525e10	1.75129	0.875647	+	0.482952i	$0.160435\pi$
$632$	5.50634e8	0.0867667
$633$	0	0
$634$	8.98392e9	1.40008
$635$	5.73855e9	0.889394
$636$	0	0
$637$	−5.98961e9	−0.918143
$638$	1.22273e9	0.186406
$639$	0	0
$640$	−6.75326e8	−0.101832
$641$	−1.16396e10	−1.74556	−0.872778	−	0.488117i	$-0.837684\pi$
$641$	−1.16396e10	−1.74556	−0.872778	+	0.488117i	$0.837684\pi$
$642$	0	0
$643$	5.82160e9	0.863583	0.431792	−	0.901973i	$-0.357882\pi$
$643$	5.82160e9	0.863583	0.431792	+	0.901973i	$0.357882\pi$
$644$	9.46275e8	0.139610
$645$	0	0
$646$	−5.69743e9	−0.831506
$647$	5.74961e9	0.834591	0.417296	−	0.908771i	$-0.362978\pi$
$647$	5.74961e9	0.834591	0.417296	+	0.908771i	$0.362978\pi$
$648$	0	0
$649$	3.39620e9	0.487683
$650$	−1.87588e9	−0.267922
$651$	0	0
$652$	−4.96552e8	−0.0701614
$653$	−2.81258e9	−0.395284	−0.197642	−	0.980274i	$-0.563328\pi$
$653$	−2.81258e9	−0.395284	−0.197642	+	0.980274i	$0.563328\pi$
$654$	0	0
$655$	4.65650e9	0.647464
$656$	8.72067e8	0.120611
$657$	0	0
$658$	1.03500e10	1.41628
$659$	9.18794e9	1.25060	0.625301	−	0.780384i	$-0.284976\pi$
$659$	9.18794e9	1.25060	0.625301	+	0.780384i	$0.284976\pi$
$660$	0	0
$661$	2.48632e9	0.334851	0.167425	−	0.985885i	$-0.446455\pi$
$661$	2.48632e9	0.334851	0.167425	+	0.985885i	$0.446455\pi$
$662$	−4.09093e9	−0.548048
$663$	0	0
$664$	−2.62274e9	−0.347670
$665$	8.15186e9	1.07493
$666$	0	0
$667$	7.65911e8	0.0999396
$668$	5.60015e9	0.726912
$669$	0	0
$670$	−9.09991e9	−1.16889
$671$	−3.54178e9	−0.452577
$672$	0	0
$673$	8.94038e9	1.13058	0.565292	−	0.824891i	$-0.308763\pi$
$673$	8.94038e9	1.13058	0.565292	+	0.824891i	$0.308763\pi$
$674$	6.66740e9	0.838777
$675$	0	0
$676$	1.36520e9	0.169974
$677$	−5.70757e9	−0.706953	−0.353476	−	0.935443i	$-0.615001\pi$
$677$	−5.70757e9	−0.706953	−0.353476	+	0.935443i	$0.615001\pi$
$678$	0	0
$679$	−2.12198e10	−2.60134
$680$	5.63668e9	0.687452
$681$	0	0
$682$	−1.83308e7	−0.00221277
$683$	−1.24773e10	−1.49847	−0.749234	−	0.662305i	$-0.769578\pi$
$683$	−1.24773e10	−1.49847	−0.749234	+	0.662305i	$0.769578\pi$
$684$	0	0
$685$	8.51437e9	1.01213
$686$	1.65594e9	0.195843
$687$	0	0
$688$	9.18000e7	0.0107469
$689$	−1.59146e9	−0.185366
$690$	0	0
$691$	4.59750e9	0.530088	0.265044	−	0.964236i	$-0.414613\pi$
$691$	4.59750e9	0.530088	0.265044	+	0.964236i	$0.414613\pi$
$692$	−2.25505e9	−0.258693
$693$	0	0
$694$	2.71873e9	0.308751
$695$	2.04792e8	0.0231401
$696$	0	0
$697$	−7.27880e9	−0.814226
$698$	−1.01539e10	−1.13016
$699$	0	0
$700$	−1.98886e9	−0.219160
$701$	1.71002e10	1.87494	0.937471	−	0.348064i	$-0.113161\pi$
$701$	1.71002e10	1.87494	0.937471	+	0.348064i	$0.113161\pi$
$702$	0	0
$703$	1.74926e9	0.189894
$704$	−6.36484e8	−0.0687517
$705$	0	0
$706$	5.30136e9	0.566984
$707$	−1.09492e10	−1.16523
$708$	0	0
$709$	5.42381e9	0.571535	0.285768	−	0.958299i	$-0.407751\pi$
$709$	5.42381e9	0.571535	0.285768	+	0.958299i	$0.407751\pi$
$710$	−1.46193e9	−0.153293
$711$	0	0
$712$	9.08217e8	0.0942996
$713$	−1.14822e7	−0.00118635
$714$	0	0
$715$	−7.16930e9	−0.733510
$716$	−4.96243e9	−0.505241
$717$	0	0
$718$	−1.09343e10	−1.10244
$719$	3.73902e9	0.375151	0.187576	−	0.982250i	$-0.439937\pi$
$719$	3.73902e9	0.375151	0.187576	+	0.982250i	$0.439937\pi$
$720$	0	0
$721$	−1.81068e10	−1.79916
$722$	−3.67939e9	−0.363827
$723$	0	0
$724$	6.54111e9	0.640570
$725$	−1.60977e9	−0.156885
$726$	0	0
$727$	1.51333e10	1.46071	0.730354	−	0.683069i	$-0.239355\pi$
$727$	1.51333e10	1.46071	0.730354	+	0.683069i	$0.239355\pi$
$728$	5.70518e9	0.548037
$729$	0	0
$730$	1.76459e9	0.167886
$731$	−7.66219e8	−0.0725507
$732$	0	0
$733$	9.58708e9	0.899130	0.449565	−	0.893248i	$-0.351579\pi$
$733$	9.58708e9	0.899130	0.449565	+	0.893248i	$0.351579\pi$
$734$	−9.30142e9	−0.868187
$735$	0	0
$736$	−3.98688e8	−0.0368605
$737$	−8.57652e9	−0.789178
$738$	0	0
$739$	1.52443e10	1.38948	0.694740	−	0.719261i	$-0.255519\pi$
$739$	1.52443e10	1.38948	0.694740	+	0.719261i	$0.255519\pi$
$740$	−1.73061e9	−0.156996
$741$	0	0
$742$	−1.68731e9	−0.151629
$743$	2.88935e9	0.258428	0.129214	−	0.991617i	$-0.458755\pi$
$743$	2.88935e9	0.258428	0.129214	+	0.991617i	$0.458755\pi$
$744$	0	0
$745$	−1.22451e10	−1.08497
$746$	−7.98408e9	−0.704108
$747$	0	0
$748$	5.31248e9	0.464133
$749$	2.76382e10	2.40338
$750$	0	0
$751$	−2.42239e8	−0.0208691	−0.0104346	−	0.999946i	$-0.503321\pi$
$751$	−2.42239e8	−0.0208691	−0.0104346	+	0.999946i	$0.503321\pi$
$752$	−4.36069e9	−0.373932
$753$	0	0
$754$	4.61775e9	0.392311
$755$	1.46807e10	1.24146
$756$	0	0
$757$	−1.32362e10	−1.10899	−0.554495	−	0.832187i	$-0.687089\pi$
$757$	−1.32362e10	−1.10899	−0.554495	+	0.832187i	$0.687089\pi$
$758$	7.84317e9	0.654108
$759$	0	0
$760$	−3.43457e9	−0.283808
$761$	3.53947e9	0.291134	0.145567	−	0.989348i	$-0.453499\pi$
$761$	3.53947e9	0.291134	0.145567	+	0.989348i	$0.453499\pi$
$762$	0	0
$763$	1.88512e10	1.53639
$764$	6.98383e9	0.566587
$765$	0	0
$766$	−1.53338e10	−1.23268
$767$	1.28260e10	1.02638
$768$	0	0
$769$	1.57158e10	1.24622	0.623109	−	0.782135i	$-0.285869\pi$
$769$	1.57158e10	1.24622	0.623109	+	0.782135i	$0.285869\pi$
$770$	−7.60108e9	−0.600009
$771$	0	0
$772$	9.96656e8	0.0779623
$773$	7.29029e9	0.567697	0.283848	−	0.958869i	$-0.408389\pi$
$773$	7.29029e9	0.567697	0.283848	+	0.958869i	$0.408389\pi$
$774$	0	0
$775$	2.41331e7	0.00186233
$776$	8.94040e9	0.686817
$777$	0	0
$778$	1.83363e10	1.39600
$779$	4.43515e9	0.336146
$780$	0	0
$781$	−1.37785e9	−0.103496
$782$	3.32769e9	0.248840
$783$	0	0
$784$	2.67555e9	0.198293
$785$	−1.46479e10	−1.08076
$786$	0	0
$787$	4.79088e9	0.350351	0.175176	−	0.984537i	$-0.443951\pi$
$787$	4.79088e9	0.350351	0.175176	+	0.984537i	$0.443951\pi$
$788$	−7.95181e9	−0.578927
$789$	0	0
$790$	−2.77055e9	−0.199927
$791$	−6.53474e9	−0.469473
$792$	0	0
$793$	−1.33758e10	−0.952495
$794$	2.71601e9	0.192557
$795$	0	0
$796$	7.16196e9	0.503311
$797$	−1.42361e10	−0.996061	−0.498030	−	0.867160i	$-0.665943\pi$
$797$	−1.42361e10	−0.996061	−0.498030	+	0.867160i	$0.665943\pi$
$798$	0	0
$799$	3.63969e10	2.52436
$800$	8.37953e8	0.0578635
$801$	0	0
$802$	5.67516e9	0.388479
$803$	1.66310e9	0.113348
$804$	0	0
$805$	−4.76125e9	−0.321689
$806$	−6.92275e7	−0.00465700
$807$	0	0
$808$	4.61315e9	0.307651
$809$	5.21148e9	0.346052	0.173026	−	0.984917i	$-0.444646\pi$
$809$	5.21148e9	0.346052	0.173026	+	0.984917i	$0.444646\pi$
$810$	0	0
$811$	1.88724e10	1.24238	0.621190	−	0.783660i	$-0.286649\pi$
$811$	1.88724e10	1.24238	0.621190	+	0.783660i	$0.286649\pi$
$812$	4.89585e9	0.320909
$813$	0	0
$814$	−1.63107e9	−0.105995
$815$	2.49844e9	0.161665
$816$	0	0
$817$	4.66876e8	0.0299519
$818$	1.42874e10	0.912676
$819$	0	0
$820$	−4.38787e9	−0.277910
$821$	2.70475e10	1.70579	0.852895	−	0.522083i	$-0.174845\pi$
$821$	2.70475e10	1.70579	0.852895	+	0.522083i	$0.174845\pi$
$822$	0	0
$823$	1.91612e10	1.19818	0.599091	−	0.800681i	$-0.295529\pi$
$823$	1.91612e10	1.19818	0.599091	+	0.800681i	$0.295529\pi$
$824$	7.62884e9	0.475021
$825$	0	0
$826$	1.35985e10	0.839576
$827$	4.35287e9	0.267613	0.133806	−	0.991007i	$-0.457280\pi$
$827$	4.35287e9	0.267613	0.133806	+	0.991007i	$0.457280\pi$
$828$	0	0
$829$	2.88917e10	1.76129	0.880647	−	0.473773i	$-0.157108\pi$
$829$	2.88917e10	1.76129	0.880647	+	0.473773i	$0.157108\pi$
$830$	1.31965e10	0.801099
$831$	0	0
$832$	−2.40373e9	−0.144695
$833$	−2.23317e10	−1.33864
$834$	0	0
$835$	−2.81775e10	−1.67494
$836$	−3.23703e9	−0.191613
$837$	0	0
$838$	2.32259e10	1.36338
$839$	3.00184e10	1.75477	0.877385	−	0.479787i	$-0.159286\pi$
$839$	3.00184e10	1.75477	0.877385	+	0.479787i	$0.159286\pi$
$840$	0	0
$841$	−1.32872e10	−0.770278
$842$	1.49115e10	0.860856
$843$	0	0
$844$	−1.03085e10	−0.590197
$845$	−6.86908e9	−0.391652
$846$	0	0
$847$	1.65172e10	0.933998
$848$	7.10904e8	0.0400336
$849$	0	0
$850$	−6.99407e9	−0.390628
$851$	−1.02169e9	−0.0568283
$852$	0	0
$853$	−2.85225e10	−1.57350	−0.786748	−	0.617274i	$-0.788237\pi$
$853$	−2.85225e10	−1.57350	−0.786748	+	0.617274i	$0.788237\pi$
$854$	−1.41813e10	−0.779139
$855$	0	0
$856$	−1.16446e10	−0.634552
$857$	4.24891e9	0.230592	0.115296	−	0.993331i	$-0.463218\pi$
$857$	4.24891e9	0.230592	0.115296	+	0.993331i	$0.463218\pi$
$858$	0	0
$859$	8.53814e9	0.459608	0.229804	−	0.973237i	$-0.426192\pi$
$859$	8.53814e9	0.459608	0.229804	+	0.973237i	$0.426192\pi$
$860$	−4.61898e8	−0.0247629
$861$	0	0
$862$	−1.20051e10	−0.638397
$863$	−2.83058e10	−1.49912	−0.749562	−	0.661935i	$-0.769736\pi$
$863$	−2.83058e10	−1.49912	−0.749562	+	0.661935i	$0.769736\pi$
$864$	0	0
$865$	1.13464e10	0.596079
$866$	−1.57584e10	−0.824518
$867$	0	0
$868$	−7.33968e7	−0.00380941
$869$	−2.61120e9	−0.134981
$870$	0	0
$871$	−3.23899e10	−1.66091
$872$	−7.94245e9	−0.405646
$873$	0	0
$874$	−2.02765e9	−0.102731
$875$	−2.05652e10	−1.03778
$876$	0	0
$877$	−1.68914e10	−0.845603	−0.422802	−	0.906222i	$-0.638953\pi$
$877$	−1.68914e10	−0.845603	−0.422802	+	0.906222i	$0.638953\pi$
$878$	7.70199e9	0.384036
$879$	0	0
$880$	3.20252e9	0.158417
$881$	−5.68506e9	−0.280104	−0.140052	−	0.990144i	$-0.544727\pi$
$881$	−5.68506e9	−0.280104	−0.140052	+	0.990144i	$0.544727\pi$
$882$	0	0
$883$	1.99081e10	0.973120	0.486560	−	0.873647i	$-0.338251\pi$
$883$	1.99081e10	0.973120	0.486560	+	0.873647i	$0.338251\pi$
$884$	2.00630e10	0.976816
$885$	0	0
$886$	−9.97519e9	−0.481841
$887$	2.55857e10	1.23102	0.615511	−	0.788129i	$-0.288950\pi$
$887$	2.55857e10	1.23102	0.615511	+	0.788129i	$0.288950\pi$
$888$	0	0
$889$	2.16557e10	1.03375
$890$	−4.56976e9	−0.217284
$891$	0	0
$892$	1.60575e10	0.757530
$893$	−2.21776e10	−1.04216
$894$	0	0
$895$	2.49688e10	1.16417
$896$	−2.54850e9	−0.118360
$897$	0	0
$898$	−2.80441e10	−1.29233
$899$	−5.94070e7	−0.00272696
$900$	0	0
$901$	−5.93363e9	−0.270261
$902$	−4.13549e9	−0.187631
$903$	0	0
$904$	2.75324e9	0.123952
$905$	−3.29121e10	−1.47600
$906$	0	0
$907$	2.35476e10	1.04790	0.523952	−	0.851748i	$-0.324457\pi$
$907$	2.35476e10	1.04790	0.523952	+	0.851748i	$0.324457\pi$
$908$	4.20656e9	0.186478
$909$	0	0
$910$	−2.87060e10	−1.26278
$911$	2.61275e10	1.14494	0.572472	−	0.819924i	$-0.305985\pi$
$911$	2.61275e10	1.14494	0.572472	+	0.819924i	$0.305985\pi$
$912$	0	0
$913$	1.24375e10	0.540861
$914$	−9.58204e9	−0.415094
$915$	0	0
$916$	−8.53746e9	−0.367024
$917$	1.75724e10	0.752553
$918$	0	0
$919$	−3.01464e10	−1.28124	−0.640622	−	0.767857i	$-0.721323\pi$
$919$	−3.01464e10	−1.28124	−0.640622	+	0.767857i	$0.721323\pi$
$920$	2.00603e9	0.0849337
$921$	0	0
$922$	−1.93136e10	−0.811532
$923$	−5.20355e9	−0.217818
$924$	0	0
$925$	2.14736e9	0.0892090
$926$	−4.37808e9	−0.181195
$927$	0	0
$928$	−2.06274e9	−0.0847279
$929$	7.13952e9	0.292156	0.146078	−	0.989273i	$-0.453335\pi$
$929$	7.13952e9	0.292156	0.146078	+	0.989273i	$0.453335\pi$
$930$	0	0
$931$	1.36073e10	0.552647
$932$	−1.40168e10	−0.567144
$933$	0	0
$934$	−3.11832e10	−1.25229
$935$	−2.67302e10	−1.06945
$936$	0	0
$937$	−3.18765e10	−1.26585	−0.632926	−	0.774213i	$-0.718146\pi$
$937$	−3.18765e10	−1.26585	−0.632926	+	0.774213i	$0.718146\pi$
$938$	−3.43406e10	−1.35862
$939$	0	0
$940$	2.19411e10	0.861611
$941$	4.77865e10	1.86957	0.934784	−	0.355216i	$-0.115593\pi$
$941$	4.77865e10	1.86957	0.934784	+	0.355216i	$0.115593\pi$
$942$	0	0
$943$	−2.59044e9	−0.100596
$944$	−5.72936e9	−0.221669
$945$	0	0
$946$	−4.35332e8	−0.0167187
$947$	2.59008e10	0.991035	0.495518	−	0.868598i	$-0.334978\pi$
$947$	2.59008e10	0.991035	0.495518	+	0.868598i	$0.334978\pi$
$948$	0	0
$949$	6.28082e9	0.238553
$950$	4.26166e9	0.161267
$951$	0	0
$952$	2.12713e10	0.799033
$953$	2.34707e9	0.0878416	0.0439208	−	0.999035i	$-0.486015\pi$
$953$	2.34707e9	0.0878416	0.0439208	+	0.999035i	$0.486015\pi$
$954$	0	0
$955$	−3.51396e10	−1.30553
$956$	−6.74227e8	−0.0249576
$957$	0	0
$958$	1.23107e10	0.452381
$959$	3.21309e10	1.17641
$960$	0	0
$961$	−2.75117e10	−0.999968
$962$	−6.15986e9	−0.223079
$963$	0	0
$964$	−1.83521e10	−0.659805
$965$	−5.01475e9	−0.179640
$966$	0	0
$967$	−1.82845e10	−0.650264	−0.325132	−	0.945669i	$-0.605409\pi$
$967$	−1.82845e10	−0.650264	−0.325132	+	0.945669i	$0.605409\pi$
$968$	−6.95911e9	−0.246598
$969$	0	0
$970$	−4.49843e10	−1.58256
$971$	−1.59060e10	−0.557564	−0.278782	−	0.960354i	$-0.589931\pi$
$971$	−1.59060e10	−0.557564	−0.278782	+	0.960354i	$0.589931\pi$
$972$	0	0
$973$	7.72828e8	0.0268960
$974$	−3.86961e10	−1.34187
$975$	0	0
$976$	5.97494e9	0.205712
$977$	−1.68994e10	−0.579750	−0.289875	−	0.957065i	$-0.593614\pi$
$977$	−1.68994e10	−0.579750	−0.289875	+	0.957065i	$0.593614\pi$
$978$	0	0
$979$	−4.30693e9	−0.146699
$980$	−1.34622e10	−0.456904
$981$	0	0
$982$	−3.23163e10	−1.08901
$983$	−2.45605e10	−0.824706	−0.412353	−	0.911024i	$-0.635293\pi$
$983$	−2.45605e10	−0.824706	−0.412353	+	0.911024i	$0.635293\pi$
$984$	0	0
$985$	4.00101e10	1.33396
$986$	1.72169e10	0.571986
$987$	0	0
$988$	−1.22249e10	−0.403269
$989$	−2.72688e8	−0.00896353
$990$	0	0
$991$	1.19911e10	0.391383	0.195691	−	0.980666i	$-0.437305\pi$
$991$	1.19911e10	0.391383	0.195691	+	0.980666i	$0.437305\pi$
$992$	3.09238e7	0.00100578
$993$	0	0
$994$	−5.51693e9	−0.178174
$995$	−3.60359e10	−1.15972
$996$	0	0
$997$	−2.17158e10	−0.693972	−0.346986	−	0.937870i	$-0.612795\pi$
$997$	−2.17158e10	−0.693972	−0.346986	+	0.937870i	$0.612795\pi$
$998$	−2.08038e10	−0.662499
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.8.a.l.1.1		4
3.2	odd	2		138.8.a.f.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.8.a.f.1.4	✓	4	3.2	odd	2
414.8.a.l.1.1		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	414.a (trivial)

\(f(q)\)	\(=\)	\(q+8.00000 q^{2} +64.0000 q^{4} -322.021 q^{5} -1215.22 q^{7} +512.000 q^{8} +O(q^{10})\)	\(q+8.00000 q^{2} +64.0000 q^{4} -322.021 q^{5} -1215.22 q^{7} +512.000 q^{8} -2576.17 q^{10} -2427.99 q^{11} -9169.50 q^{13} -9721.74 q^{14} +4096.00 q^{16} -34187.7 q^{17} +20831.4 q^{19} -20609.3 q^{20} -19424.0 q^{22} -12167.0 q^{23} +25572.3 q^{25} -73356.0 q^{26} -77773.9 q^{28} -62949.8 q^{29} +943.720 q^{31} +32768.0 q^{32} -273502. q^{34} +391325. q^{35} +83972.1 q^{37} +166651. q^{38} -164875. q^{40} +212907. q^{41} +22412.1 q^{43} -155392. q^{44} -97336.0 q^{46} -1.06462e6 q^{47} +653210. q^{49} +204578. q^{50} -586848. q^{52} +173560. q^{53} +781864. q^{55} -622191. q^{56} -503599. q^{58} -1.39877e6 q^{59} +1.45872e6 q^{61} +7549.76 q^{62} +262144. q^{64} +2.95277e6 q^{65} +3.53235e6 q^{67} -2.18801e6 q^{68} +3.13060e6 q^{70} +567484. q^{71} -684969. q^{73} +671777. q^{74} +1.33321e6 q^{76} +2.95054e6 q^{77} +1.07546e6 q^{79} -1.31900e6 q^{80} +1.70326e6 q^{82} -5.12255e6 q^{83} +1.10091e7 q^{85} +179297. q^{86} -1.24313e6 q^{88} +1.77386e6 q^{89} +1.11429e7 q^{91} -778688. q^{92} -8.51696e6 q^{94} -6.70815e6 q^{95} +1.74617e7 q^{97} +5.22568e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 32 q^{2} + 256 q^{4} + 162 q^{5} + 1218 q^{7} + 2048 q^{8}+O(q^{10}) \) 4 * q + 32 * q^2 + 256 * q^4 + 162 * q^5 + 1218 * q^7 + 2048 * q^8	\( 4 q + 32 q^{2} + 256 q^{4} + 162 q^{5} + 1218 q^{7} + 2048 q^{8} + 1296 q^{10} - 1820 q^{11} - 6508 q^{13} + 9744 q^{14} + 16384 q^{16} - 12870 q^{17} + 74474 q^{19} + 10368 q^{20} - 14560 q^{22} - 48668 q^{23} + 192544 q^{25} - 52064 q^{26} + 77952 q^{28} + 94452 q^{29} - 39420 q^{31} + 131072 q^{32} - 102960 q^{34} + 788392 q^{35} + 75848 q^{37} + 595792 q^{38} + 82944 q^{40} + 519032 q^{41} + 716946 q^{43} - 116480 q^{44} - 389344 q^{46} + 12232 q^{47} + 1656176 q^{49} + 1540352 q^{50} - 416512 q^{52} - 1053394 q^{53} + 1155136 q^{55} + 623616 q^{56} + 755616 q^{58} - 4398344 q^{59} + 5328312 q^{61} - 315360 q^{62} + 1048576 q^{64} - 6366708 q^{65} + 11303966 q^{67} - 823680 q^{68} + 6307136 q^{70} - 8395320 q^{71} + 7107008 q^{73} + 606784 q^{74} + 4766336 q^{76} - 9789160 q^{77} + 9785086 q^{79} + 663552 q^{80} + 4152256 q^{82} - 2424316 q^{83} + 25790076 q^{85} + 5735568 q^{86} - 931840 q^{88} - 3019218 q^{89} + 1693028 q^{91} - 3114752 q^{92} + 97856 q^{94} + 12329136 q^{95} + 32226876 q^{97} + 13249408 q^{98}+O(q^{100}) \) 4 * q + 32 * q^2 + 256 * q^4 + 162 * q^5 + 1218 * q^7 + 2048 * q^8 + 1296 * q^10 - 1820 * q^11 - 6508 * q^13 + 9744 * q^14 + 16384 * q^16 - 12870 * q^17 + 74474 * q^19 + 10368 * q^20 - 14560 * q^22 - 48668 * q^23 + 192544 * q^25 - 52064 * q^26 + 77952 * q^28 + 94452 * q^29 - 39420 * q^31 + 131072 * q^32 - 102960 * q^34 + 788392 * q^35 + 75848 * q^37 + 595792 * q^38 + 82944 * q^40 + 519032 * q^41 + 716946 * q^43 - 116480 * q^44 - 389344 * q^46 + 12232 * q^47 + 1656176 * q^49 + 1540352 * q^50 - 416512 * q^52 - 1053394 * q^53 + 1155136 * q^55 + 623616 * q^56 + 755616 * q^58 - 4398344 * q^59 + 5328312 * q^61 - 315360 * q^62 + 1048576 * q^64 - 6366708 * q^65 + 11303966 * q^67 - 823680 * q^68 + 6307136 * q^70 - 8395320 * q^71 + 7107008 * q^73 + 606784 * q^74 + 4766336 * q^76 - 9789160 * q^77 + 9785086 * q^79 + 663552 * q^80 + 4152256 * q^82 - 2424316 * q^83 + 25790076 * q^85 + 5735568 * q^86 - 931840 * q^88 - 3019218 * q^89 + 1693028 * q^91 - 3114752 * q^92 + 97856 * q^94 + 12329136 * q^95 + 32226876 * q^97 + 13249408 * q^98

Introduction

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