LMFDB - Embedded newform 342.8.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [342,8,Mod(1,342)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(342, 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("342.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 342 = 2 \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	342.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:	$106.835678716$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{633}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 158 $ x^2 - x - 158
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-12.0797$ of defining polynomial
Character	$\chi$	$=$	342.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.00000 q^{2} +64.0000 q^{4} +312.472 q^{5} -766.767 q^{7} -512.000 q^{8} +O(q^{10})$	\(q-8.00000 q^{2} +64.0000 q^{4} +312.472 q^{5} -766.767 q^{7} -512.000 q^{8} -2499.78 q^{10} -252.042 q^{11} -1065.19 q^{13} +6134.14 q^{14} +4096.00 q^{16} +18769.7 q^{17} -6859.00 q^{19} +19998.2 q^{20} +2016.33 q^{22} -11559.9 q^{23} +19513.8 q^{25} +8521.55 q^{26} -49073.1 q^{28} +46290.0 q^{29} +46848.2 q^{31} -32768.0 q^{32} -150158. q^{34} -239593. q^{35} -182916. q^{37} +54872.0 q^{38} -159986. q^{40} -819661. q^{41} +477471. q^{43} -16130.7 q^{44} +92478.8 q^{46} -992580. q^{47} -235611. q^{49} -156111. q^{50} -68172.4 q^{52} +852516. q^{53} -78756.0 q^{55} +392585. q^{56} -370320. q^{58} +1.92404e6 q^{59} +209564. q^{61} -374786. q^{62} +262144. q^{64} -332843. q^{65} -2.32447e6 q^{67} +1.20126e6 q^{68} +1.91675e6 q^{70} +5.37237e6 q^{71} -3.71614e6 q^{73} +1.46333e6 q^{74} -438976. q^{76} +193257. q^{77} -1.30851e6 q^{79} +1.27989e6 q^{80} +6.55729e6 q^{82} +6.51683e6 q^{83} +5.86502e6 q^{85} -3.81976e6 q^{86} +129045. q^{88} -3.99586e6 q^{89} +816756. q^{91} -739831. q^{92} +7.94064e6 q^{94} -2.14325e6 q^{95} -6.90391e6 q^{97} +1.88489e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 16 q^{2} + 128 q^{4} - 155 q^{5} - 2238 q^{7} - 1024 q^{8}+O(q^{10}) $ 2 * q - 16 * q^2 + 128 * q^4 - 155 * q^5 - 2238 * q^7 - 1024 * q^8	\( 2 q - 16 q^{2} + 128 q^{4} - 155 q^{5} - 2238 q^{7} - 1024 q^{8} + 1240 q^{10} + 3295 q^{11} - 13427 q^{13} + 17904 q^{14} + 8192 q^{16} + 32256 q^{17} - 13718 q^{19} - 9920 q^{20} - 26360 q^{22} + 82525 q^{23} + 159919 q^{25} + 107416 q^{26} - 143232 q^{28} + 12749 q^{29} + 258944 q^{31} - 65536 q^{32} - 258048 q^{34} + 448167 q^{35} - 149260 q^{37} + 109744 q^{38} + 79360 q^{40} - 339130 q^{41} - 83869 q^{43} + 210880 q^{44} - 660200 q^{46} - 1471025 q^{47} + 1105372 q^{49} - 1279352 q^{50} - 859328 q^{52} + 945643 q^{53} - 1736899 q^{55} + 1145856 q^{56} - 101992 q^{58} + 969009 q^{59} - 1506755 q^{61} - 2071552 q^{62} + 524288 q^{64} + 5445956 q^{65} - 1848219 q^{67} + 2064384 q^{68} - 3585336 q^{70} + 3417184 q^{71} - 2499822 q^{73} + 1194080 q^{74} - 877952 q^{76} - 5025267 q^{77} + 2636926 q^{79} - 634880 q^{80} + 2713040 q^{82} + 10059354 q^{83} - 439425 q^{85} + 670952 q^{86} - 1687040 q^{88} + 3506160 q^{89} + 19003851 q^{91} + 5281600 q^{92} + 11768200 q^{94} + 1063145 q^{95} + 5893526 q^{97} - 8842976 q^{98}+O(q^{100}) \) 2 * q - 16 * q^2 + 128 * q^4 - 155 * q^5 - 2238 * q^7 - 1024 * q^8 + 1240 * q^10 + 3295 * q^11 - 13427 * q^13 + 17904 * q^14 + 8192 * q^16 + 32256 * q^17 - 13718 * q^19 - 9920 * q^20 - 26360 * q^22 + 82525 * q^23 + 159919 * q^25 + 107416 * q^26 - 143232 * q^28 + 12749 * q^29 + 258944 * q^31 - 65536 * q^32 - 258048 * q^34 + 448167 * q^35 - 149260 * q^37 + 109744 * q^38 + 79360 * q^40 - 339130 * q^41 - 83869 * q^43 + 210880 * q^44 - 660200 * q^46 - 1471025 * q^47 + 1105372 * q^49 - 1279352 * q^50 - 859328 * q^52 + 945643 * q^53 - 1736899 * q^55 + 1145856 * q^56 - 101992 * q^58 + 969009 * q^59 - 1506755 * q^61 - 2071552 * q^62 + 524288 * q^64 + 5445956 * q^65 - 1848219 * q^67 + 2064384 * q^68 - 3585336 * q^70 + 3417184 * q^71 - 2499822 * q^73 + 1194080 * q^74 - 877952 * q^76 - 5025267 * q^77 + 2636926 * q^79 - 634880 * q^80 + 2713040 * q^82 + 10059354 * q^83 - 439425 * q^85 + 670952 * q^86 - 1687040 * q^88 + 3506160 * q^89 + 19003851 * q^91 + 5281600 * q^92 + 11768200 * q^94 + 1063145 * q^95 + 5893526 * q^97 - 8842976 * 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
$5$	312.472	1.11793	0.558967	−	0.829190i	$-0.311198\pi$
$5$	312.472	1.11793	0.558967	+	0.829190i	$0.311198\pi$
$6$	0	0
$7$	−766.767	−0.844929	−0.422465	−	0.906379i	$-0.638835\pi$
$7$	−766.767	−0.844929	−0.422465	+	0.906379i	$0.638835\pi$
$8$	−512.000	−0.353553
$9$	0	0
$10$	−2499.78	−0.790499
$11$	−252.042	−0.0570950	−0.0285475	−	0.999592i	$-0.509088\pi$
$11$	−252.042	−0.0570950	−0.0285475	+	0.999592i	$0.509088\pi$
$12$	0	0
$13$	−1065.19	−0.134471	−0.0672353	−	0.997737i	$-0.521418\pi$
$13$	−1065.19	−0.134471	−0.0672353	+	0.997737i	$0.521418\pi$
$14$	6134.14	0.597455
$15$	0	0
$16$	4096.00	0.250000
$17$	18769.7	0.926589	0.463295	−	0.886204i	$-0.346667\pi$
$17$	18769.7	0.926589	0.463295	+	0.886204i	$0.346667\pi$
$18$	0	0
$19$	−6859.00	−0.229416
$19$	−6859.00	−0.229416
$20$	19998.2	0.558967
$21$	0	0
$22$	2016.33	0.0403722
$23$	−11559.9	−0.198109	−0.0990546	−	0.995082i	$-0.531582\pi$
$23$	−11559.9	−0.198109	−0.0990546	+	0.995082i	$0.531582\pi$
$24$	0	0
$25$	19513.8	0.249777
$26$	8521.55	0.0950850
$27$	0	0
$28$	−49073.1	−0.422465
$29$	46290.0	0.352448	0.176224	−	0.984350i	$-0.443612\pi$
$29$	46290.0	0.352448	0.176224	+	0.984350i	$0.443612\pi$
$30$	0	0
$31$	46848.2	0.282441	0.141220	−	0.989978i	$-0.454897\pi$
$31$	46848.2	0.282441	0.141220	+	0.989978i	$0.454897\pi$
$32$	−32768.0	−0.176777
$33$	0	0
$34$	−150158.	−0.655197
$35$	−239593.	−0.944575
$36$	0	0
$37$	−182916.	−0.593672	−0.296836	−	0.954928i	$-0.595932\pi$
$37$	−182916.	−0.593672	−0.296836	+	0.954928i	$0.595932\pi$
$38$	54872.0	0.162221
$39$	0	0
$40$	−159986.	−0.395249
$41$	−819661.	−1.85734	−0.928669	−	0.370910i	$-0.879046\pi$
$41$	−819661.	−1.85734	−0.928669	+	0.370910i	$0.879046\pi$
$42$	0	0
$43$	477471.	0.915814	0.457907	−	0.889000i	$-0.348599\pi$
$43$	477471.	0.915814	0.457907	+	0.889000i	$0.348599\pi$
$44$	−16130.7	−0.0285475
$45$	0	0
$46$	92478.8	0.140084
$47$	−992580.	−1.39451	−0.697257	−	0.716821i	$-0.745596\pi$
$47$	−992580.	−1.39451	−0.697257	+	0.716821i	$0.745596\pi$
$48$	0	0
$49$	−235611.	−0.286095
$50$	−156111.	−0.176619
$51$	0	0
$52$	−68172.4	−0.0672353
$53$	852516.	0.786569	0.393285	−	0.919417i	$-0.371339\pi$
$53$	852516.	0.786569	0.393285	+	0.919417i	$0.371339\pi$
$54$	0	0
$55$	−78756.0	−0.0638284
$56$	392585.	0.298728
$57$	0	0
$58$	−370320.	−0.249218
$59$	1.92404e6	1.21964	0.609821	−	0.792539i	$-0.291241\pi$
$59$	1.92404e6	1.21964	0.609821	+	0.792539i	$0.291241\pi$
$60$	0	0
$61$	209564.	0.118212	0.0591062	−	0.998252i	$-0.481175\pi$
$61$	209564.	0.118212	0.0591062	+	0.998252i	$0.481175\pi$
$62$	−374786.	−0.199716
$63$	0	0
$64$	262144.	0.125000
$65$	−332843.	−0.150329
$66$	0	0
$67$	−2.32447e6	−0.944198	−0.472099	−	0.881546i	$-0.656503\pi$
$67$	−2.32447e6	−0.944198	−0.472099	+	0.881546i	$0.656503\pi$
$68$	1.20126e6	0.463295
$69$	0	0
$70$	1.91675e6	0.667916
$71$	5.37237e6	1.78140	0.890700	−	0.454591i	$-0.150215\pi$
$71$	5.37237e6	1.78140	0.890700	+	0.454591i	$0.150215\pi$
$72$	0	0
$73$	−3.71614e6	−1.11805	−0.559027	−	0.829150i	$-0.688825\pi$
$73$	−3.71614e6	−1.11805	−0.559027	+	0.829150i	$0.688825\pi$
$74$	1.46333e6	0.419790
$75$	0	0
$76$	−438976.	−0.114708
$77$	193257.	0.0482412
$78$	0	0
$79$	−1.30851e6	−0.298596	−0.149298	−	0.988792i	$-0.547701\pi$
$79$	−1.30851e6	−0.298596	−0.149298	+	0.988792i	$0.547701\pi$
$80$	1.27989e6	0.279484
$81$	0	0
$82$	6.55729e6	1.31334
$83$	6.51683e6	1.25102	0.625508	−	0.780218i	$-0.284892\pi$
$83$	6.51683e6	1.25102	0.625508	+	0.780218i	$0.284892\pi$
$84$	0	0
$85$	5.86502e6	1.03587
$86$	−3.81976e6	−0.647578
$87$	0	0
$88$	129045.	0.0201861
$89$	−3.99586e6	−0.600822	−0.300411	−	0.953810i	$-0.597124\pi$
$89$	−3.99586e6	−0.600822	−0.300411	+	0.953810i	$0.597124\pi$
$90$	0	0
$91$	816756.	0.113618
$92$	−739831.	−0.0990546
$93$	0	0
$94$	7.94064e6	0.986070
$95$	−2.14325e6	−0.256472
$96$	0	0
$97$	−6.90391e6	−0.768058	−0.384029	−	0.923321i	$-0.625464\pi$
$97$	−6.90391e6	−0.768058	−0.384029	+	0.923321i	$0.625464\pi$
$98$	1.88489e6	0.202299
$99$	0	0
$100$	1.24888e6	0.124888
$101$	−2.81610e6	−0.271971	−0.135986	−	0.990711i	$-0.543420\pi$
$101$	−2.81610e6	−0.271971	−0.135986	+	0.990711i	$0.543420\pi$
$102$	0	0
$103$	7.86273e6	0.708995	0.354497	−	0.935057i	$-0.384652\pi$
$103$	7.86273e6	0.708995	0.354497	+	0.935057i	$0.384652\pi$
$104$	545379.	0.0475425
$105$	0	0
$106$	−6.82013e6	−0.556188
$107$	−611626.	−0.0482662	−0.0241331	−	0.999709i	$-0.507683\pi$
$107$	−611626.	−0.0482662	−0.0241331	+	0.999709i	$0.507683\pi$
$108$	0	0
$109$	5.23210e6	0.386976	0.193488	−	0.981103i	$-0.438020\pi$
$109$	5.23210e6	0.386976	0.193488	+	0.981103i	$0.438020\pi$
$110$	630048.	0.0451335
$111$	0	0
$112$	−3.14068e6	−0.211232
$113$	3.58527e6	0.233747	0.116874	−	0.993147i	$-0.462713\pi$
$113$	3.58527e6	0.233747	0.116874	+	0.993147i	$0.462713\pi$
$114$	0	0
$115$	−3.61213e6	−0.221473
$116$	2.96256e6	0.176224
$117$	0	0
$118$	−1.53923e7	−0.862418
$119$	−1.43920e7	−0.782902
$120$	0	0
$121$	−1.94236e7	−0.996740
$122$	−1.67651e6	−0.0835888
$123$	0	0
$124$	2.99829e6	0.141220
$125$	−1.83144e7	−0.838700
$126$	0	0
$127$	−2.99465e7	−1.29728	−0.648639	−	0.761096i	$-0.724661\pi$
$127$	−2.99465e7	−1.29728	−0.648639	+	0.761096i	$0.724661\pi$
$128$	−2.09715e6	−0.0883883
$129$	0	0
$130$	2.66275e6	0.106299
$131$	8.11292e6	0.315303	0.157651	−	0.987495i	$-0.449608\pi$
$131$	8.11292e6	0.315303	0.157651	+	0.987495i	$0.449608\pi$
$132$	0	0
$133$	5.25926e6	0.193840
$134$	1.85958e7	0.667649
$135$	0	0
$136$	−9.61011e6	−0.327599
$137$	−1.44157e7	−0.478977	−0.239488	−	0.970899i	$-0.576980\pi$
$137$	−1.44157e7	−0.478977	−0.239488	+	0.970899i	$0.576980\pi$
$138$	0	0
$139$	−3.77446e7	−1.19207	−0.596036	−	0.802957i	$-0.703259\pi$
$139$	−3.77446e7	−1.19207	−0.596036	+	0.802957i	$0.703259\pi$
$140$	−1.53340e7	−0.472288
$141$	0	0
$142$	−4.29789e7	−1.25964
$143$	268473.	0.00767759
$144$	0	0
$145$	1.44643e7	0.394013
$146$	2.97292e7	0.790583
$147$	0	0
$148$	−1.17067e7	−0.296836
$149$	−7.30679e7	−1.80957	−0.904784	−	0.425872i	$-0.859967\pi$
$149$	−7.30679e7	−1.80957	−0.904784	+	0.425872i	$0.859967\pi$
$150$	0	0
$151$	−4.14056e7	−0.978678	−0.489339	−	0.872094i	$-0.662762\pi$
$151$	−4.14056e7	−0.978678	−0.489339	+	0.872094i	$0.662762\pi$
$152$	3.51181e6	0.0811107
$153$	0	0
$154$	−1.54606e6	−0.0341117
$155$	1.46388e7	0.315750
$156$	0	0
$157$	2.93945e7	0.606203	0.303101	−	0.952958i	$-0.401978\pi$
$157$	2.93945e7	0.606203	0.303101	+	0.952958i	$0.401978\pi$
$158$	1.04681e7	0.211139
$159$	0	0
$160$	−1.02391e7	−0.197625
$161$	8.86371e6	0.167388
$162$	0	0
$163$	9.03924e6	0.163484	0.0817420	−	0.996654i	$-0.473952\pi$
$163$	9.03924e6	0.163484	0.0817420	+	0.996654i	$0.473952\pi$
$164$	−5.24583e7	−0.928669
$165$	0	0
$166$	−5.21346e7	−0.884602
$167$	8.16728e7	1.35697	0.678484	−	0.734615i	$-0.262637\pi$
$167$	8.16728e7	1.35697	0.678484	+	0.734615i	$0.262637\pi$
$168$	0	0
$169$	−6.16139e7	−0.981918
$170$	−4.69202e7	−0.732468
$171$	0	0
$172$	3.05581e7	0.457907
$173$	−1.33191e8	−1.95575	−0.977874	−	0.209197i	$-0.932915\pi$
$173$	−1.33191e8	−1.95575	−0.977874	+	0.209197i	$0.932915\pi$
$174$	0	0
$175$	−1.49626e7	−0.211044
$176$	−1.03236e6	−0.0142737
$177$	0	0
$178$	3.19669e7	0.424845
$179$	4.90910e7	0.639759	0.319879	−	0.947458i	$-0.396358\pi$
$179$	4.90910e7	0.639759	0.319879	+	0.947458i	$0.396358\pi$
$180$	0	0
$181$	8.11979e7	1.01782	0.508908	−	0.860821i	$-0.330049\pi$
$181$	8.11979e7	1.01782	0.508908	+	0.860821i	$0.330049\pi$
$182$	−6.53405e6	−0.0803401
$183$	0	0
$184$	5.91864e6	0.0700422
$185$	−5.71563e7	−0.663686
$186$	0	0
$187$	−4.73076e6	−0.0529036
$188$	−6.35251e7	−0.697257
$189$	0	0
$190$	1.71460e7	0.181353
$191$	−7.40713e7	−0.769190	−0.384595	−	0.923086i	$-0.625659\pi$
$191$	−7.40713e7	−0.769190	−0.384595	+	0.923086i	$0.625659\pi$
$192$	0	0
$193$	−9.70125e7	−0.971353	−0.485676	−	0.874139i	$-0.661427\pi$
$193$	−9.70125e7	−0.971353	−0.485676	+	0.874139i	$0.661427\pi$
$194$	5.52313e7	0.543099
$195$	0	0
$196$	−1.50791e7	−0.143047
$197$	4.34770e7	0.405161	0.202581	−	0.979266i	$-0.435067\pi$
$197$	4.34770e7	0.405161	0.202581	+	0.979266i	$0.435067\pi$
$198$	0	0
$199$	3.89651e7	0.350502	0.175251	−	0.984524i	$-0.443926\pi$
$199$	3.89651e7	0.350502	0.175251	+	0.984524i	$0.443926\pi$
$200$	−9.99108e6	−0.0883095
$201$	0	0
$202$	2.25288e7	0.192313
$203$	−3.54937e7	−0.297793
$204$	0	0
$205$	−2.56121e8	−2.07638
$206$	−6.29018e7	−0.501335
$207$	0	0
$208$	−4.36304e6	−0.0336176
$209$	1.72875e6	0.0130985
$210$	0	0
$211$	−3.85264e7	−0.282339	−0.141169	−	0.989985i	$-0.545086\pi$
$211$	−3.85264e7	−0.282339	−0.141169	+	0.989985i	$0.545086\pi$
$212$	5.45610e7	0.393285
$213$	0	0
$214$	4.89301e6	0.0341293
$215$	1.49196e8	1.02382
$216$	0	0
$217$	−3.59217e7	−0.238642
$218$	−4.18568e7	−0.273633
$219$	0	0
$220$	−5.04038e6	−0.0319142
$221$	−1.99934e7	−0.124599
$222$	0	0
$223$	−2.03401e8	−1.22825	−0.614124	−	0.789209i	$-0.710491\pi$
$223$	−2.03401e8	−1.22825	−0.614124	+	0.789209i	$0.710491\pi$
$224$	2.51254e7	0.149364
$225$	0	0
$226$	−2.86821e7	−0.165284
$227$	−2.24944e8	−1.27639	−0.638195	−	0.769875i	$-0.720319\pi$
$227$	−2.24944e8	−1.27639	−0.638195	+	0.769875i	$0.720319\pi$
$228$	0	0
$229$	−2.38882e8	−1.31450	−0.657248	−	0.753674i	$-0.728280\pi$
$229$	−2.38882e8	−1.31450	−0.657248	+	0.753674i	$0.728280\pi$
$230$	2.88971e7	0.156605
$231$	0	0
$232$	−2.37005e7	−0.124609
$233$	−3.23019e8	−1.67295	−0.836475	−	0.548005i	$-0.815387\pi$
$233$	−3.23019e8	−1.67295	−0.836475	+	0.548005i	$0.815387\pi$
$234$	0	0
$235$	−3.10153e8	−1.55897
$236$	1.23139e8	0.609821
$237$	0	0
$238$	1.15136e8	0.553595
$239$	5.18160e7	0.245511	0.122756	−	0.992437i	$-0.460827\pi$
$239$	5.18160e7	0.245511	0.122756	+	0.992437i	$0.460827\pi$
$240$	0	0
$241$	1.15926e8	0.533484	0.266742	−	0.963768i	$-0.414053\pi$
$241$	1.15926e8	0.533484	0.266742	+	0.963768i	$0.414053\pi$
$242$	1.55389e8	0.704802
$243$	0	0
$244$	1.34121e7	0.0591062
$245$	−7.36219e7	−0.319835
$246$	0	0
$247$	7.30617e6	0.0308497
$248$	−2.39863e7	−0.0998579
$249$	0	0
$250$	1.46515e8	0.593050
$251$	4.03761e7	0.161163	0.0805817	−	0.996748i	$-0.474322\pi$
$251$	4.03761e7	0.161163	0.0805817	+	0.996748i	$0.474322\pi$
$252$	0	0
$253$	2.91356e6	0.0113110
$254$	2.39572e8	0.917314
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	−2.67661e8	−0.983602	−0.491801	−	0.870708i	$-0.663661\pi$
$257$	−2.67661e8	−0.983602	−0.491801	+	0.870708i	$0.663661\pi$
$258$	0	0
$259$	1.40254e8	0.501611
$260$	−2.13020e7	−0.0751646
$261$	0	0
$262$	−6.49034e7	−0.222953
$263$	−5.12778e8	−1.73814	−0.869068	−	0.494692i	$-0.835281\pi$
$263$	−5.12778e8	−1.73814	−0.869068	+	0.494692i	$0.835281\pi$
$264$	0	0
$265$	2.66387e8	0.879333
$266$	−4.20740e7	−0.137066
$267$	0	0
$268$	−1.48766e8	−0.472099
$269$	−5.75792e8	−1.80357	−0.901784	−	0.432187i	$-0.857742\pi$
$269$	−5.75792e8	−1.80357	−0.901784	+	0.432187i	$0.857742\pi$
$270$	0	0
$271$	5.24839e8	1.60189	0.800946	−	0.598737i	$-0.204330\pi$
$271$	5.24839e8	1.60189	0.800946	+	0.598737i	$0.204330\pi$
$272$	7.68809e7	0.231647
$273$	0	0
$274$	1.15326e8	0.338688
$275$	−4.91829e6	−0.0142610
$276$	0	0
$277$	−4.64620e8	−1.31347	−0.656733	−	0.754123i	$-0.728062\pi$
$277$	−4.64620e8	−1.31347	−0.656733	+	0.754123i	$0.728062\pi$
$278$	3.01957e8	0.842923
$279$	0	0
$280$	1.22672e8	0.333958
$281$	1.75429e8	0.471660	0.235830	−	0.971794i	$-0.424219\pi$
$281$	1.75429e8	0.471660	0.235830	+	0.971794i	$0.424219\pi$
$282$	0	0
$283$	1.44948e8	0.380154	0.190077	−	0.981769i	$-0.439126\pi$
$283$	1.44948e8	0.380154	0.190077	+	0.981769i	$0.439126\pi$
$284$	3.43832e8	0.890700
$285$	0	0
$286$	−2.14779e6	−0.00542888
$287$	6.28489e8	1.56932
$288$	0	0
$289$	−5.80353e7	−0.141433
$290$	−1.15715e8	−0.278609
$291$	0	0
$292$	−2.37833e8	−0.559027
$293$	4.24522e8	0.985970	0.492985	−	0.870038i	$-0.335906\pi$
$293$	4.24522e8	0.985970	0.492985	+	0.870038i	$0.335906\pi$
$294$	0	0
$295$	6.01210e8	1.36348
$296$	9.36532e7	0.209895
$297$	0	0
$298$	5.84543e8	1.27956
$299$	1.23135e7	0.0266399
$300$	0	0
$301$	−3.66109e8	−0.773798
$302$	3.31245e8	0.692030
$303$	0	0
$304$	−2.80945e7	−0.0573539
$305$	6.54830e7	0.132154
$306$	0	0
$307$	4.27528e8	0.843297	0.421649	−	0.906759i	$-0.361452\pi$
$307$	4.27528e8	0.843297	0.421649	+	0.906759i	$0.361452\pi$
$308$	1.23685e7	0.0241206
$309$	0	0
$310$	−1.17110e8	−0.223269
$311$	1.70275e8	0.320989	0.160495	−	0.987037i	$-0.448691\pi$
$311$	1.70275e8	0.320989	0.160495	+	0.987037i	$0.448691\pi$
$312$	0	0
$313$	−4.02143e8	−0.741269	−0.370634	−	0.928779i	$-0.620860\pi$
$313$	−4.02143e8	−0.741269	−0.370634	+	0.928779i	$0.620860\pi$
$314$	−2.35156e8	−0.428650
$315$	0	0
$316$	−8.37450e7	−0.149298
$317$	1.90630e8	0.336112	0.168056	−	0.985777i	$-0.446251\pi$
$317$	1.90630e8	0.336112	0.168056	+	0.985777i	$0.446251\pi$
$318$	0	0
$319$	−1.16670e7	−0.0201230
$320$	8.19127e7	0.139742
$321$	0	0
$322$	−7.09097e7	−0.118361
$323$	−1.28742e8	−0.212574
$324$	0	0
$325$	−2.07860e7	−0.0335876
$326$	−7.23139e7	−0.115601
$327$	0	0
$328$	4.19666e8	0.656668
$329$	7.61077e8	1.17827
$330$	0	0
$331$	−4.12008e8	−0.624464	−0.312232	−	0.950006i	$-0.601077\pi$
$331$	−4.12008e8	−0.624464	−0.312232	+	0.950006i	$0.601077\pi$
$332$	4.17077e8	0.625508
$333$	0	0
$334$	−6.53382e8	−0.959522
$335$	−7.26333e8	−1.05555
$336$	0	0
$337$	−6.99272e8	−0.995271	−0.497636	−	0.867386i	$-0.665798\pi$
$337$	−6.99272e8	−0.995271	−0.497636	+	0.867386i	$0.665798\pi$
$338$	4.92911e8	0.694321
$339$	0	0
$340$	3.75361e8	0.517933
$341$	−1.18077e7	−0.0161259
$342$	0	0
$343$	8.12125e8	1.08666
$344$	−2.44465e8	−0.323789
$345$	0	0
$346$	1.06553e9	1.38292
$347$	9.29885e8	1.19475	0.597374	−	0.801963i	$-0.296211\pi$
$347$	9.29885e8	1.19475	0.597374	+	0.801963i	$0.296211\pi$
$348$	0	0
$349$	−4.43776e8	−0.558823	−0.279412	−	0.960171i	$-0.590139\pi$
$349$	−4.43776e8	−0.558823	−0.279412	+	0.960171i	$0.590139\pi$
$350$	1.19700e8	0.149231
$351$	0	0
$352$	8.25890e6	0.0100931
$353$	1.49388e9	1.80761	0.903804	−	0.427946i	$-0.140763\pi$
$353$	1.49388e9	1.80761	0.903804	+	0.427946i	$0.140763\pi$
$354$	0	0
$355$	1.67872e9	1.99149
$356$	−2.55735e8	−0.300411
$357$	0	0
$358$	−3.92728e8	−0.452378
$359$	1.77937e8	0.202972	0.101486	−	0.994837i	$-0.467640\pi$
$359$	1.77937e8	0.202972	0.101486	+	0.994837i	$0.467640\pi$
$360$	0	0
$361$	4.70459e7	0.0526316
$362$	−6.49583e8	−0.719705
$363$	0	0
$364$	5.22724e7	0.0568090
$365$	−1.16119e9	−1.24991
$366$	0	0
$367$	−1.37515e9	−1.45218	−0.726088	−	0.687601i	$-0.758664\pi$
$367$	−1.37515e9	−1.45218	−0.726088	+	0.687601i	$0.758664\pi$
$368$	−4.73492e7	−0.0495273
$369$	0	0
$370$	4.57250e8	0.469297
$371$	−6.53681e8	−0.664595
$372$	0	0
$373$	1.55845e9	1.55493	0.777466	−	0.628925i	$-0.216505\pi$
$373$	1.55845e9	1.55493	0.777466	+	0.628925i	$0.216505\pi$
$374$	3.78461e7	0.0374085
$375$	0	0
$376$	5.08201e8	0.493035
$377$	−4.93079e7	−0.0473938
$378$	0	0
$379$	8.84549e8	0.834613	0.417306	−	0.908766i	$-0.362974\pi$
$379$	8.84549e8	0.834613	0.417306	+	0.908766i	$0.362974\pi$
$380$	−1.37168e8	−0.128236
$381$	0	0
$382$	5.92571e8	0.543899
$383$	−1.44573e9	−1.31490	−0.657449	−	0.753499i	$-0.728365\pi$
$383$	−1.44573e9	−1.31490	−0.657449	+	0.753499i	$0.728365\pi$
$384$	0	0
$385$	6.03875e7	0.0539305
$386$	7.76100e8	0.686850
$387$	0	0
$388$	−4.41850e8	−0.384029
$389$	1.02164e9	0.879986	0.439993	−	0.898001i	$-0.354981\pi$
$389$	1.02164e9	0.879986	0.439993	+	0.898001i	$0.354981\pi$
$390$	0	0
$391$	−2.16975e8	−0.183566
$392$	1.20633e8	0.101150
$393$	0	0
$394$	−3.47816e8	−0.286492
$395$	−4.08874e8	−0.333811
$396$	0	0
$397$	1.80050e8	0.144419	0.0722097	−	0.997389i	$-0.476995\pi$
$397$	1.80050e8	0.144419	0.0722097	+	0.997389i	$0.476995\pi$
$398$	−3.11721e8	−0.247842
$399$	0	0
$400$	7.99286e7	0.0624442
$401$	−5.06670e8	−0.392392	−0.196196	−	0.980565i	$-0.562859\pi$
$401$	−5.06670e8	−0.392392	−0.196196	+	0.980565i	$0.562859\pi$
$402$	0	0
$403$	−4.99025e7	−0.0379799
$404$	−1.80230e8	−0.135986
$405$	0	0
$406$	2.83949e8	0.210572
$407$	4.61026e7	0.0338957
$408$	0	0
$409$	1.14898e9	0.830388	0.415194	−	0.909733i	$-0.363714\pi$
$409$	1.14898e9	0.830388	0.415194	+	0.909733i	$0.363714\pi$
$410$	2.04897e9	1.46822
$411$	0	0
$412$	5.03215e8	0.354497
$413$	−1.47529e9	−1.03051
$414$	0	0
$415$	2.03633e9	1.39855
$416$	3.49043e7	0.0237713
$417$	0	0
$418$	−1.38300e7	−0.00926203
$419$	−2.02289e9	−1.34346	−0.671729	−	0.740797i	$-0.734448\pi$
$419$	−2.02289e9	−1.34346	−0.671729	+	0.740797i	$0.734448\pi$
$420$	0	0
$421$	2.55503e9	1.66882	0.834408	−	0.551147i	$-0.185810\pi$
$421$	2.55503e9	1.66882	0.834408	+	0.551147i	$0.185810\pi$
$422$	3.08211e8	0.199644
$423$	0	0
$424$	−4.36488e8	−0.278094
$425$	3.66269e8	0.231441
$426$	0	0
$427$	−1.60687e8	−0.0998811
$428$	−3.91441e7	−0.0241331
$429$	0	0
$430$	−1.19357e9	−0.723950
$431$	−1.47316e9	−0.886295	−0.443147	−	0.896449i	$-0.646138\pi$
$431$	−1.47316e9	−0.886295	−0.443147	+	0.896449i	$0.646138\pi$
$432$	0	0
$433$	−9.33243e8	−0.552443	−0.276221	−	0.961094i	$-0.589082\pi$
$433$	−9.33243e8	−0.552443	−0.276221	+	0.961094i	$0.589082\pi$
$434$	2.87373e8	0.168746
$435$	0	0
$436$	3.34855e8	0.193488
$437$	7.92890e7	0.0454494
$438$	0	0
$439$	−1.94411e9	−1.09672	−0.548358	−	0.836244i	$-0.684747\pi$
$439$	−1.94411e9	−1.09672	−0.548358	+	0.836244i	$0.684747\pi$
$440$	4.03231e7	0.0225668
$441$	0	0
$442$	1.59947e8	0.0881047
$443$	2.35592e9	1.28750	0.643751	−	0.765235i	$-0.277377\pi$
$443$	2.35592e9	1.28750	0.643751	+	0.765235i	$0.277377\pi$
$444$	0	0
$445$	−1.24860e9	−0.671679
$446$	1.62721e9	0.868503
$447$	0	0
$448$	−2.01003e8	−0.105616
$449$	1.83374e9	0.956042	0.478021	−	0.878349i	$-0.341354\pi$
$449$	1.83374e9	0.956042	0.478021	+	0.878349i	$0.341354\pi$
$450$	0	0
$451$	2.06589e8	0.106045
$452$	2.29457e8	0.116874
$453$	0	0
$454$	1.79955e9	0.902544
$455$	2.55213e8	0.127018
$456$	0	0
$457$	−3.76683e9	−1.84616	−0.923079	−	0.384611i	$-0.874336\pi$
$457$	−3.76683e9	−1.84616	−0.923079	+	0.384611i	$0.874336\pi$
$458$	1.91106e9	0.929489
$459$	0	0
$460$	−2.31176e8	−0.110737
$461$	−3.30560e9	−1.57144	−0.785720	−	0.618583i	$-0.787707\pi$
$461$	−3.30560e9	−1.57144	−0.785720	+	0.618583i	$0.787707\pi$
$462$	0	0
$463$	−3.66326e9	−1.71528	−0.857638	−	0.514253i	$-0.828069\pi$
$463$	−3.66326e9	−1.71528	−0.857638	+	0.514253i	$0.828069\pi$
$464$	1.89604e8	0.0881119
$465$	0	0
$466$	2.58416e9	1.18295
$467$	2.22674e9	1.01172	0.505860	−	0.862616i	$-0.331175\pi$
$467$	2.22674e9	1.01172	0.505860	+	0.862616i	$0.331175\pi$
$468$	0	0
$469$	1.78233e9	0.797780
$470$	2.48123e9	1.10236
$471$	0	0
$472$	−9.85110e8	−0.431209
$473$	−1.20342e8	−0.0522883
$474$	0	0
$475$	−1.33845e8	−0.0573028
$476$	−9.21090e8	−0.391451
$477$	0	0
$478$	−4.14528e8	−0.173603
$479$	−1.43508e9	−0.596625	−0.298313	−	0.954468i	$-0.596424\pi$
$479$	−1.43508e9	−0.596625	−0.298313	+	0.954468i	$0.596424\pi$
$480$	0	0
$481$	1.94842e8	0.0798314
$482$	−9.27409e8	−0.377230
$483$	0	0
$484$	−1.24311e9	−0.498370
$485$	−2.15728e9	−0.858638
$486$	0	0
$487$	−2.11642e9	−0.830330	−0.415165	−	0.909746i	$-0.636276\pi$
$487$	−2.11642e9	−0.830330	−0.415165	+	0.909746i	$0.636276\pi$
$488$	−1.07297e8	−0.0417944
$489$	0	0
$490$	5.88975e8	0.226157
$491$	−4.07105e9	−1.55211	−0.776053	−	0.630668i	$-0.782781\pi$
$491$	−4.07105e9	−1.55211	−0.776053	+	0.630668i	$0.782781\pi$
$492$	0	0
$493$	8.68852e8	0.326574
$494$	−5.84493e7	−0.0218140
$495$	0	0
$496$	1.91890e8	0.0706102
$497$	−4.11935e9	−1.50516
$498$	0	0
$499$	3.93039e9	1.41607	0.708034	−	0.706178i	$-0.249582\pi$
$499$	3.93039e9	1.41607	0.708034	+	0.706178i	$0.249582\pi$
$500$	−1.17212e9	−0.419350
$501$	0	0
$502$	−3.23009e8	−0.113960
$503$	1.70030e9	0.595714	0.297857	−	0.954610i	$-0.403728\pi$
$503$	1.70030e9	0.595714	0.297857	+	0.954610i	$0.403728\pi$
$504$	0	0
$505$	−8.79952e8	−0.304046
$506$	−2.33085e7	−0.00799812
$507$	0	0
$508$	−1.91658e9	−0.648639
$509$	2.62955e8	0.0883830	0.0441915	−	0.999023i	$-0.485929\pi$
$509$	2.62955e8	0.0883830	0.0441915	+	0.999023i	$0.485929\pi$
$510$	0	0
$511$	2.84942e9	0.944676
$512$	−1.34218e8	−0.0441942
$513$	0	0
$514$	2.14129e9	0.695512
$515$	2.45688e9	0.792609
$516$	0	0
$517$	2.50171e8	0.0796197
$518$	−1.12203e9	−0.354693
$519$	0	0
$520$	1.70416e8	0.0531494
$521$	1.86990e9	0.579277	0.289639	−	0.957136i	$-0.406465\pi$
$521$	1.86990e9	0.579277	0.289639	+	0.957136i	$0.406465\pi$
$522$	0	0
$523$	−5.49460e9	−1.67950	−0.839750	−	0.542972i	$-0.817299\pi$
$523$	−5.49460e9	−1.67950	−0.839750	+	0.542972i	$0.817299\pi$
$524$	5.19227e8	0.157651
$525$	0	0
$526$	4.10222e9	1.22905
$527$	8.79329e8	0.261706
$528$	0	0
$529$	−3.27120e9	−0.960753
$530$	−2.13110e9	−0.621782
$531$	0	0
$532$	3.36592e8	0.0969200
$533$	8.73098e8	0.249757
$534$	0	0
$535$	−1.91116e8	−0.0539584
$536$	1.19013e9	0.333824
$537$	0	0
$538$	4.60634e9	1.27532
$539$	5.93838e7	0.0163346
$540$	0	0
$541$	6.97892e9	1.89495	0.947475	−	0.319830i	$-0.103626\pi$
$541$	6.97892e9	1.89495	0.947475	+	0.319830i	$0.103626\pi$
$542$	−4.19871e9	−1.13271
$543$	0	0
$544$	−6.15047e8	−0.163799
$545$	1.63489e9	0.432613
$546$	0	0
$547$	−5.87859e9	−1.53574	−0.767870	−	0.640606i	$-0.778683\pi$
$547$	−5.87859e9	−1.53574	−0.767870	+	0.640606i	$0.778683\pi$
$548$	−9.22606e8	−0.239488
$549$	0	0
$550$	3.93464e7	0.0100841
$551$	−3.17503e8	−0.0808570
$552$	0	0
$553$	1.00333e9	0.252292
$554$	3.71696e9	0.928761
$555$	0	0
$556$	−2.41565e9	−0.596036
$557$	2.03155e9	0.498119	0.249060	−	0.968488i	$-0.419878\pi$
$557$	2.03155e9	0.498119	0.249060	+	0.968488i	$0.419878\pi$
$558$	0	0
$559$	−5.08599e8	−0.123150
$560$	−9.81374e8	−0.236144
$561$	0	0
$562$	−1.40343e9	−0.333514
$563$	−1.68161e9	−0.397142	−0.198571	−	0.980087i	$-0.563630\pi$
$563$	−1.68161e9	−0.397142	−0.198571	+	0.980087i	$0.563630\pi$
$564$	0	0
$565$	1.12030e9	0.261314
$566$	−1.15958e9	−0.268810
$567$	0	0
$568$	−2.75065e9	−0.629820
$569$	4.57746e9	1.04167	0.520837	−	0.853656i	$-0.325620\pi$
$569$	4.57746e9	1.04167	0.520837	+	0.853656i	$0.325620\pi$
$570$	0	0
$571$	−5.79221e9	−1.30202	−0.651010	−	0.759069i	$-0.725655\pi$
$571$	−5.79221e9	−1.30202	−0.651010	+	0.759069i	$0.725655\pi$
$572$	1.71823e7	0.00383879
$573$	0	0
$574$	−5.02791e9	−1.10968
$575$	−2.25577e8	−0.0494831
$576$	0	0
$577$	−3.74692e9	−0.812005	−0.406003	−	0.913872i	$-0.633078\pi$
$577$	−3.74692e9	−0.812005	−0.406003	+	0.913872i	$0.633078\pi$
$578$	4.64282e8	0.100008
$579$	0	0
$580$	9.25718e8	0.197007
$581$	−4.99689e9	−1.05702
$582$	0	0
$583$	−2.14869e8	−0.0449091
$584$	1.90267e9	0.395292
$585$	0	0
$586$	−3.39618e9	−0.697186
$587$	5.69193e9	1.16152	0.580760	−	0.814075i	$-0.302756\pi$
$587$	5.69193e9	1.16152	0.580760	+	0.814075i	$0.302756\pi$
$588$	0	0
$589$	−3.21332e8	−0.0647963
$590$	−4.80968e9	−0.964126
$591$	0	0
$592$	−7.49226e8	−0.148418
$593$	−1.45596e9	−0.286720	−0.143360	−	0.989671i	$-0.545791\pi$
$593$	−1.45596e9	−0.286720	−0.143360	+	0.989671i	$0.545791\pi$
$594$	0	0
$595$	−4.49711e9	−0.875233
$596$	−4.67635e9	−0.904784
$597$	0	0
$598$	−9.85079e7	−0.0188372
$599$	6.21735e9	1.18198	0.590992	−	0.806677i	$-0.298736\pi$
$599$	6.21735e9	1.18198	0.590992	+	0.806677i	$0.298736\pi$
$600$	0	0
$601$	9.98776e9	1.87675	0.938377	−	0.345613i	$-0.112329\pi$
$601$	9.98776e9	1.87675	0.938377	+	0.345613i	$0.112329\pi$
$602$	2.92887e9	0.547158
$603$	0	0
$604$	−2.64996e9	−0.489339
$605$	−6.06935e9	−1.11429
$606$	0	0
$607$	3.17827e9	0.576807	0.288404	−	0.957509i	$-0.406876\pi$
$607$	3.17827e9	0.576807	0.288404	+	0.957509i	$0.406876\pi$
$608$	2.24756e8	0.0405554
$609$	0	0
$610$	−5.23864e8	−0.0934467
$611$	1.05729e9	0.187521
$612$	0	0
$613$	−6.34969e8	−0.111337	−0.0556687	−	0.998449i	$-0.517729\pi$
$613$	−6.34969e8	−0.111337	−0.0556687	+	0.998449i	$0.517729\pi$
$614$	−3.42023e9	−0.596301
$615$	0	0
$616$	−9.89477e7	−0.0170558
$617$	4.48835e7	0.00769287	0.00384643	−	0.999993i	$-0.498776\pi$
$617$	4.48835e7	0.00769287	0.00384643	+	0.999993i	$0.498776\pi$
$618$	0	0
$619$	−9.79132e8	−0.165930	−0.0829648	−	0.996552i	$-0.526439\pi$
$619$	−9.79132e8	−0.165930	−0.0829648	+	0.996552i	$0.526439\pi$
$620$	9.36881e8	0.157875
$621$	0	0
$622$	−1.36220e9	−0.226974
$623$	3.06390e9	0.507652
$624$	0	0
$625$	−7.24724e9	−1.18739
$626$	3.21715e9	0.524156
$627$	0	0
$628$	1.88125e9	0.303101
$629$	−3.43330e9	−0.550090
$630$	0	0
$631$	−2.42563e9	−0.384345	−0.192172	−	0.981361i	$-0.561553\pi$
$631$	−2.42563e9	−0.384345	−0.192172	+	0.981361i	$0.561553\pi$
$632$	6.69960e8	0.105570
$633$	0	0
$634$	−1.52504e9	−0.237667
$635$	−9.35744e9	−1.45027
$636$	0	0
$637$	2.50972e8	0.0384713
$638$	9.33361e7	0.0142291
$639$	0	0
$640$	−6.55302e8	−0.0988124
$641$	−9.23132e8	−0.138440	−0.0692199	−	0.997601i	$-0.522051\pi$
$641$	−9.23132e8	−0.138440	−0.0692199	+	0.997601i	$0.522051\pi$
$642$	0	0
$643$	6.86334e9	1.01812	0.509058	−	0.860732i	$-0.329994\pi$
$643$	6.86334e9	1.01812	0.509058	+	0.860732i	$0.329994\pi$
$644$	5.67278e8	0.0836942
$645$	0	0
$646$	1.02993e9	0.150313
$647$	−5.02399e8	−0.0729262	−0.0364631	−	0.999335i	$-0.511609\pi$
$647$	−5.02399e8	−0.0729262	−0.0364631	+	0.999335i	$0.511609\pi$
$648$	0	0
$649$	−4.84939e8	−0.0696355
$650$	1.66288e8	0.0237500
$651$	0	0
$652$	5.78512e8	0.0817420
$653$	−4.23523e9	−0.595225	−0.297612	−	0.954687i	$-0.596190\pi$
$653$	−4.23523e9	−0.595225	−0.297612	+	0.954687i	$0.596190\pi$
$654$	0	0
$655$	2.53506e9	0.352488
$656$	−3.35733e9	−0.464334
$657$	0	0
$658$	−6.08862e9	−0.833160
$659$	1.34554e10	1.83146	0.915731	−	0.401791i	$-0.131612\pi$
$659$	1.34554e10	1.83146	0.915731	+	0.401791i	$0.131612\pi$
$660$	0	0
$661$	1.23518e10	1.66351	0.831753	−	0.555146i	$-0.187338\pi$
$661$	1.23518e10	1.66351	0.831753	+	0.555146i	$0.187338\pi$
$662$	3.29606e9	0.441563
$663$	0	0
$664$	−3.33662e9	−0.442301
$665$	1.64337e9	0.216700
$666$	0	0
$667$	−5.35106e8	−0.0698231
$668$	5.22706e9	0.678484
$669$	0	0
$670$	5.81067e9	0.746387
$671$	−5.28189e7	−0.00674933
$672$	0	0
$673$	−1.52638e9	−0.193024	−0.0965119	−	0.995332i	$-0.530769\pi$
$673$	−1.52638e9	−0.193024	−0.0965119	+	0.995332i	$0.530769\pi$
$674$	5.59417e9	0.703763
$675$	0	0
$676$	−3.94329e9	−0.490959
$677$	1.55415e9	0.192500	0.0962502	−	0.995357i	$-0.469315\pi$
$677$	1.55415e9	0.192500	0.0962502	+	0.995357i	$0.469315\pi$
$678$	0	0
$679$	5.29369e9	0.648954
$680$	−3.00289e9	−0.366234
$681$	0	0
$682$	9.44616e7	0.0114028
$683$	3.21824e9	0.386497	0.193249	−	0.981150i	$-0.438098\pi$
$683$	3.21824e9	0.386497	0.193249	+	0.981150i	$0.438098\pi$
$684$	0	0
$685$	−4.50451e9	−0.535465
$686$	−6.49700e9	−0.768384
$687$	0	0
$688$	1.95572e9	0.228953
$689$	−9.08095e8	−0.105770
$690$	0	0
$691$	9.15999e9	1.05614	0.528071	−	0.849200i	$-0.322916\pi$
$691$	9.15999e9	1.05614	0.528071	+	0.849200i	$0.322916\pi$
$692$	−8.52421e9	−0.977874
$693$	0	0
$694$	−7.43908e9	−0.844814
$695$	−1.17941e10	−1.33266
$696$	0	0
$697$	−1.53848e10	−1.72099
$698$	3.55021e9	0.395148
$699$	0	0
$700$	−9.57604e8	−0.105522
$701$	7.97717e9	0.874653	0.437327	−	0.899303i	$-0.355925\pi$
$701$	7.97717e9	0.874653	0.437327	+	0.899303i	$0.355925\pi$
$702$	0	0
$703$	1.25462e9	0.136198
$704$	−6.60712e7	−0.00713687
$705$	0	0
$706$	−1.19510e10	−1.27817
$707$	2.15929e9	0.229796
$708$	0	0
$709$	7.97010e9	0.839851	0.419925	−	0.907559i	$-0.362056\pi$
$709$	7.97010e9	0.839851	0.419925	+	0.907559i	$0.362056\pi$
$710$	−1.34297e10	−1.40819
$711$	0	0
$712$	2.04588e9	0.212423
$713$	−5.41559e8	−0.0559541
$714$	0	0
$715$	8.38904e7	0.00858304
$716$	3.14182e9	0.319879
$717$	0	0
$718$	−1.42350e9	−0.143523
$719$	−1.00263e10	−1.00598	−0.502991	−	0.864292i	$-0.667767\pi$
$719$	−1.00263e10	−1.00598	−0.502991	+	0.864292i	$0.667767\pi$
$720$	0	0
$721$	−6.02888e9	−0.599050
$722$	−3.76367e8	−0.0372161
$723$	0	0
$724$	5.19666e9	0.508908
$725$	9.03295e8	0.0880333
$726$	0	0
$727$	−1.65401e10	−1.59650	−0.798248	−	0.602329i	$-0.794240\pi$
$727$	−1.65401e10	−1.59650	−0.798248	+	0.602329i	$0.794240\pi$
$728$	−4.18179e8	−0.0401701
$729$	0	0
$730$	9.28953e9	0.883820
$731$	8.96200e9	0.848583
$732$	0	0
$733$	9.81994e9	0.920969	0.460484	−	0.887668i	$-0.347676\pi$
$733$	9.81994e9	0.920969	0.460484	+	0.887668i	$0.347676\pi$
$734$	1.10012e10	1.02684
$735$	0	0
$736$	3.78793e8	0.0350211
$737$	5.85864e8	0.0539089
$738$	0	0
$739$	5.41304e9	0.493385	0.246692	−	0.969094i	$-0.420656\pi$
$739$	5.41304e9	0.493385	0.246692	+	0.969094i	$0.420656\pi$
$740$	−3.65800e9	−0.331843
$741$	0	0
$742$	5.22945e9	0.469940
$743$	8.26105e9	0.738881	0.369440	−	0.929254i	$-0.379549\pi$
$743$	8.26105e9	0.738881	0.369440	+	0.929254i	$0.379549\pi$
$744$	0	0
$745$	−2.28317e10	−2.02298
$746$	−1.24676e10	−1.09950
$747$	0	0
$748$	−3.02768e8	−0.0264518
$749$	4.68975e8	0.0407815
$750$	0	0
$751$	−1.44332e10	−1.24343	−0.621715	−	0.783243i	$-0.713564\pi$
$751$	−1.44332e10	−1.24343	−0.621715	+	0.783243i	$0.713564\pi$
$752$	−4.06561e9	−0.348628
$753$	0	0
$754$	3.94463e8	0.0335125
$755$	−1.29381e10	−1.09410
$756$	0	0
$757$	−8.31294e9	−0.696496	−0.348248	−	0.937402i	$-0.613223\pi$
$757$	−8.31294e9	−0.696496	−0.348248	+	0.937402i	$0.613223\pi$
$758$	−7.07640e9	−0.590160
$759$	0	0
$760$	1.09734e9	0.0906764
$761$	1.96636e10	1.61739	0.808697	−	0.588226i	$-0.200173\pi$
$761$	1.96636e10	1.61739	0.808697	+	0.588226i	$0.200173\pi$
$762$	0	0
$763$	−4.01180e9	−0.326967
$764$	−4.74057e9	−0.384595
$765$	0	0
$766$	1.15659e10	0.929774
$767$	−2.04948e9	−0.164006
$768$	0	0
$769$	1.27102e10	1.00788	0.503942	−	0.863738i	$-0.331883\pi$
$769$	1.27102e10	1.00788	0.503942	+	0.863738i	$0.331883\pi$
$770$	−4.83100e8	−0.0381346
$771$	0	0
$772$	−6.20880e9	−0.485676
$773$	2.87614e9	0.223966	0.111983	−	0.993710i	$-0.464280\pi$
$773$	2.87614e9	0.223966	0.111983	+	0.993710i	$0.464280\pi$
$774$	0	0
$775$	9.14188e8	0.0705472
$776$	3.53480e9	0.271549
$777$	0	0
$778$	−8.17315e9	−0.622244
$779$	5.62206e9	0.426102
$780$	0	0
$781$	−1.35406e9	−0.101709
$782$	1.73580e9	0.129801
$783$	0	0
$784$	−9.65063e8	−0.0715236
$785$	9.18498e9	0.677695
$786$	0	0
$787$	5.61813e9	0.410847	0.205424	−	0.978673i	$-0.434143\pi$
$787$	5.61813e9	0.410847	0.205424	+	0.978673i	$0.434143\pi$
$788$	2.78253e9	0.202581
$789$	0	0
$790$	3.27100e9	0.236040
$791$	−2.74906e9	−0.197500
$792$	0	0
$793$	−2.23227e8	−0.0158961
$794$	−1.44040e9	−0.102120
$795$	0	0
$796$	2.49377e9	0.175251
$797$	4.98926e9	0.349085	0.174543	−	0.984650i	$-0.444155\pi$
$797$	4.98926e9	0.349085	0.174543	+	0.984650i	$0.444155\pi$
$798$	0	0
$799$	−1.86305e10	−1.29214
$800$	−6.39429e8	−0.0441547
$801$	0	0
$802$	4.05336e9	0.277463
$803$	9.36623e8	0.0638352
$804$	0	0
$805$	2.76966e9	0.187129
$806$	3.99220e8	0.0268559
$807$	0	0
$808$	1.44184e9	0.0961563
$809$	−2.60176e10	−1.72762	−0.863809	−	0.503820i	$-0.831927\pi$
$809$	−2.60176e10	−1.72762	−0.863809	+	0.503820i	$0.831927\pi$
$810$	0	0
$811$	1.36033e10	0.895513	0.447756	−	0.894156i	$-0.352223\pi$
$811$	1.36033e10	0.895513	0.447756	+	0.894156i	$0.352223\pi$
$812$	−2.27160e9	−0.148897
$813$	0	0
$814$	−3.68820e8	−0.0239679
$815$	2.82451e9	0.182764
$816$	0	0
$817$	−3.27497e9	−0.210102
$818$	−9.19184e9	−0.587173
$819$	0	0
$820$	−1.63918e10	−1.03819
$821$	−5.59803e9	−0.353048	−0.176524	−	0.984296i	$-0.556485\pi$
$821$	−5.59803e9	−0.353048	−0.176524	+	0.984296i	$0.556485\pi$
$822$	0	0
$823$	2.83609e10	1.77346	0.886728	−	0.462292i	$-0.152973\pi$
$823$	2.83609e10	1.77346	0.886728	+	0.462292i	$0.152973\pi$
$824$	−4.02572e9	−0.250667
$825$	0	0
$826$	1.18023e10	0.728682
$827$	1.94087e10	1.19324	0.596619	−	0.802524i	$-0.296510\pi$
$827$	1.94087e10	1.19324	0.596619	+	0.802524i	$0.296510\pi$
$828$	0	0
$829$	1.25701e10	0.766301	0.383150	−	0.923686i	$-0.374839\pi$
$829$	1.25701e10	0.766301	0.383150	+	0.923686i	$0.374839\pi$
$830$	−1.62906e10	−0.988927
$831$	0	0
$832$	−2.79234e8	−0.0168088
$833$	−4.42236e9	−0.265092
$834$	0	0
$835$	2.55205e10	1.51700
$836$	1.10640e8	0.00654924
$837$	0	0
$838$	1.61831e10	0.949968
$839$	−1.95535e10	−1.14303	−0.571515	−	0.820592i	$-0.693644\pi$
$839$	−1.95535e10	−1.14303	−0.571515	+	0.820592i	$0.693644\pi$
$840$	0	0
$841$	−1.51071e10	−0.875781
$842$	−2.04402e10	−1.18003
$843$	0	0
$844$	−2.46569e9	−0.141169
$845$	−1.92526e10	−1.09772
$846$	0	0
$847$	1.48934e10	0.842175
$848$	3.49191e9	0.196642
$849$	0	0
$850$	−2.93016e9	−0.163653
$851$	2.11449e9	0.117612
$852$	0	0
$853$	1.14196e10	0.629986	0.314993	−	0.949094i	$-0.397998\pi$
$853$	1.14196e10	0.629986	0.314993	+	0.949094i	$0.397998\pi$
$854$	1.28550e9	0.0706266
$855$	0	0
$856$	3.13153e8	0.0170647
$857$	1.94401e10	1.05503	0.527516	−	0.849545i	$-0.323123\pi$
$857$	1.94401e10	1.05503	0.527516	+	0.849545i	$0.323123\pi$
$858$	0	0
$859$	1.07841e10	0.580509	0.290254	−	0.956950i	$-0.406260\pi$
$859$	1.07841e10	0.580509	0.290254	+	0.956950i	$0.406260\pi$
$860$	9.54856e9	0.511910
$861$	0	0
$862$	1.17852e10	0.626705
$863$	8.45236e8	0.0447651	0.0223826	−	0.999749i	$-0.492875\pi$
$863$	8.45236e8	0.0447651	0.0223826	+	0.999749i	$0.492875\pi$
$864$	0	0
$865$	−4.16184e10	−2.18640
$866$	7.46595e9	0.390636
$867$	0	0
$868$	−2.29899e9	−0.119321
$869$	3.29800e8	0.0170483
$870$	0	0
$871$	2.47602e9	0.126967
$872$	−2.67884e9	−0.136817
$873$	0	0
$874$	−6.34312e8	−0.0321376
$875$	1.40428e10	0.708642
$876$	0	0
$877$	2.17563e10	1.08915	0.544574	−	0.838713i	$-0.316691\pi$
$877$	2.17563e10	1.08915	0.544574	+	0.838713i	$0.316691\pi$
$878$	1.55529e10	0.775495
$879$	0	0
$880$	−3.22584e8	−0.0159571
$881$	−2.18056e10	−1.07436	−0.537182	−	0.843466i	$-0.680511\pi$
$881$	−2.18056e10	−1.07436	−0.537182	+	0.843466i	$0.680511\pi$
$882$	0	0
$883$	2.25662e9	0.110305	0.0551526	−	0.998478i	$-0.482435\pi$
$883$	2.25662e9	0.110305	0.0551526	+	0.998478i	$0.482435\pi$
$884$	−1.27958e9	−0.0622995
$885$	0	0
$886$	−1.88474e10	−0.910401
$887$	1.72271e10	0.828857	0.414428	−	0.910082i	$-0.363982\pi$
$887$	1.72271e10	0.828857	0.414428	+	0.910082i	$0.363982\pi$
$888$	0	0
$889$	2.29620e10	1.09611
$890$	9.98877e9	0.474949
$891$	0	0
$892$	−1.30177e10	−0.614124
$893$	6.80810e9	0.319923
$894$	0	0
$895$	1.53396e10	0.715208
$896$	1.60803e9	0.0746819
$897$	0	0
$898$	−1.46700e10	−0.676023
$899$	2.16861e9	0.0995455
$900$	0	0
$901$	1.60015e10	0.728827
$902$	−1.65271e9	−0.0749849
$903$	0	0
$904$	−1.83566e9	−0.0826422
$905$	2.53721e10	1.13785
$906$	0	0
$907$	2.32557e10	1.03491	0.517456	−	0.855710i	$-0.326879\pi$
$907$	2.32557e10	1.03491	0.517456	+	0.855710i	$0.326879\pi$
$908$	−1.43964e10	−0.638195
$909$	0	0
$910$	−2.04171e9	−0.0898150
$911$	4.22775e10	1.85266	0.926329	−	0.376715i	$-0.122946\pi$
$911$	4.22775e10	1.85266	0.926329	+	0.376715i	$0.122946\pi$
$912$	0	0
$913$	−1.64251e9	−0.0714267
$914$	3.01346e10	1.30543
$915$	0	0
$916$	−1.52885e10	−0.657248
$917$	−6.22072e9	−0.266408
$918$	0	0
$919$	2.74897e10	1.16833	0.584164	−	0.811635i	$-0.301422\pi$
$919$	2.74897e10	1.16833	0.584164	+	0.811635i	$0.301422\pi$
$920$	1.84941e9	0.0783026
$921$	0	0
$922$	2.64448e10	1.11118
$923$	−5.72261e9	−0.239546
$924$	0	0
$925$	−3.56940e9	−0.148286
$926$	2.93061e10	1.21288
$927$	0	0
$928$	−1.51683e9	−0.0623045
$929$	2.42814e10	0.993614	0.496807	−	0.867861i	$-0.334506\pi$
$929$	2.42814e10	0.993614	0.496807	+	0.867861i	$0.334506\pi$
$930$	0	0
$931$	1.61606e9	0.0656346
$932$	−2.06732e10	−0.836475
$933$	0	0
$934$	−1.78139e10	−0.715393
$935$	−1.47823e9	−0.0591427
$936$	0	0
$937$	−3.53708e10	−1.40461	−0.702305	−	0.711876i	$-0.747846\pi$
$937$	−3.53708e10	−1.40461	−0.702305	+	0.711876i	$0.747846\pi$
$938$	−1.42586e10	−0.564116
$939$	0	0
$940$	−1.98498e10	−0.779487
$941$	3.60255e10	1.40944	0.704719	−	0.709486i	$-0.251073\pi$
$941$	3.60255e10	1.40944	0.704719	+	0.709486i	$0.251073\pi$
$942$	0	0
$943$	9.47516e9	0.367956
$944$	7.88088e9	0.304911
$945$	0	0
$946$	9.62740e8	0.0369734
$947$	2.20930e10	0.845336	0.422668	−	0.906285i	$-0.361094\pi$
$947$	2.20930e10	0.845336	0.422668	+	0.906285i	$0.361094\pi$
$948$	0	0
$949$	3.95842e9	0.150345
$950$	1.07076e9	0.0405192
$951$	0	0
$952$	7.36872e9	0.276798
$953$	2.36409e9	0.0884787	0.0442393	−	0.999021i	$-0.485914\pi$
$953$	2.36409e9	0.0884787	0.0442393	+	0.999021i	$0.485914\pi$
$954$	0	0
$955$	−2.31452e10	−0.859903
$956$	3.31623e9	0.122756
$957$	0	0
$958$	1.14806e10	0.421878
$959$	1.10535e10	0.404702
$960$	0	0
$961$	−2.53179e10	−0.920227
$962$	−1.55873e9	−0.0564493
$963$	0	0
$964$	7.41927e9	0.266742
$965$	−3.03137e10	−1.08591
$966$	0	0
$967$	2.69098e10	0.957014	0.478507	−	0.878084i	$-0.341178\pi$
$967$	2.69098e10	0.957014	0.478507	+	0.878084i	$0.341178\pi$
$968$	9.94491e9	0.352401
$969$	0	0
$970$	1.72582e10	0.607149
$971$	4.71594e9	0.165311	0.0826553	−	0.996578i	$-0.473660\pi$
$971$	4.71594e9	0.165311	0.0826553	+	0.996578i	$0.473660\pi$
$972$	0	0
$973$	2.89413e10	1.00722
$974$	1.69314e10	0.587132
$975$	0	0
$976$	8.58375e8	0.0295531
$977$	−1.79514e10	−0.615839	−0.307919	−	0.951412i	$-0.599633\pi$
$977$	−1.79514e10	−0.615839	−0.307919	+	0.951412i	$0.599633\pi$
$978$	0	0
$979$	1.00712e9	0.0343039
$980$	−4.71180e9	−0.159917
$981$	0	0
$982$	3.25684e10	1.09750
$983$	−2.39450e10	−0.804039	−0.402019	−	0.915631i	$-0.631692\pi$
$983$	−2.39450e10	−0.804039	−0.402019	+	0.915631i	$0.631692\pi$
$984$	0	0
$985$	1.35854e10	0.452944
$986$	−6.95082e9	−0.230923
$987$	0	0
$988$	4.67595e8	0.0154248
$989$	−5.51949e9	−0.181431
$990$	0	0
$991$	1.07956e10	0.352362	0.176181	−	0.984358i	$-0.443626\pi$
$991$	1.07956e10	0.352362	0.176181	+	0.984358i	$0.443626\pi$
$992$	−1.53512e9	−0.0499289
$993$	0	0
$994$	3.29548e10	1.06431
$995$	1.21755e10	0.391838
$996$	0	0
$997$	−3.54933e10	−1.13426	−0.567131	−	0.823627i	$-0.691947\pi$
$997$	−3.54933e10	−1.13426	−0.567131	+	0.823627i	$0.691947\pi$
$998$	−3.14431e10	−1.00131
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.8.a.g.1.2		2
3.2	odd	2		38.8.a.d.1.2	✓	2
12.11	even	2		304.8.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.a.d.1.2	✓	2	3.2	odd	2
304.8.a.d.1.1		2	12.11	even	2
342.8.a.g.1.2		2	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:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	342.a (trivial)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} +64.0000 q^{4} +312.472 q^{5} -766.767 q^{7} -512.000 q^{8} +O(q^{10})\)	\(q-8.00000 q^{2} +64.0000 q^{4} +312.472 q^{5} -766.767 q^{7} -512.000 q^{8} -2499.78 q^{10} -252.042 q^{11} -1065.19 q^{13} +6134.14 q^{14} +4096.00 q^{16} +18769.7 q^{17} -6859.00 q^{19} +19998.2 q^{20} +2016.33 q^{22} -11559.9 q^{23} +19513.8 q^{25} +8521.55 q^{26} -49073.1 q^{28} +46290.0 q^{29} +46848.2 q^{31} -32768.0 q^{32} -150158. q^{34} -239593. q^{35} -182916. q^{37} +54872.0 q^{38} -159986. q^{40} -819661. q^{41} +477471. q^{43} -16130.7 q^{44} +92478.8 q^{46} -992580. q^{47} -235611. q^{49} -156111. q^{50} -68172.4 q^{52} +852516. q^{53} -78756.0 q^{55} +392585. q^{56} -370320. q^{58} +1.92404e6 q^{59} +209564. q^{61} -374786. q^{62} +262144. q^{64} -332843. q^{65} -2.32447e6 q^{67} +1.20126e6 q^{68} +1.91675e6 q^{70} +5.37237e6 q^{71} -3.71614e6 q^{73} +1.46333e6 q^{74} -438976. q^{76} +193257. q^{77} -1.30851e6 q^{79} +1.27989e6 q^{80} +6.55729e6 q^{82} +6.51683e6 q^{83} +5.86502e6 q^{85} -3.81976e6 q^{86} +129045. q^{88} -3.99586e6 q^{89} +816756. q^{91} -739831. q^{92} +7.94064e6 q^{94} -2.14325e6 q^{95} -6.90391e6 q^{97} +1.88489e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 16 q^{2} + 128 q^{4} - 155 q^{5} - 2238 q^{7} - 1024 q^{8}+O(q^{10}) \) 2 * q - 16 * q^2 + 128 * q^4 - 155 * q^5 - 2238 * q^7 - 1024 * q^8	\( 2 q - 16 q^{2} + 128 q^{4} - 155 q^{5} - 2238 q^{7} - 1024 q^{8} + 1240 q^{10} + 3295 q^{11} - 13427 q^{13} + 17904 q^{14} + 8192 q^{16} + 32256 q^{17} - 13718 q^{19} - 9920 q^{20} - 26360 q^{22} + 82525 q^{23} + 159919 q^{25} + 107416 q^{26} - 143232 q^{28} + 12749 q^{29} + 258944 q^{31} - 65536 q^{32} - 258048 q^{34} + 448167 q^{35} - 149260 q^{37} + 109744 q^{38} + 79360 q^{40} - 339130 q^{41} - 83869 q^{43} + 210880 q^{44} - 660200 q^{46} - 1471025 q^{47} + 1105372 q^{49} - 1279352 q^{50} - 859328 q^{52} + 945643 q^{53} - 1736899 q^{55} + 1145856 q^{56} - 101992 q^{58} + 969009 q^{59} - 1506755 q^{61} - 2071552 q^{62} + 524288 q^{64} + 5445956 q^{65} - 1848219 q^{67} + 2064384 q^{68} - 3585336 q^{70} + 3417184 q^{71} - 2499822 q^{73} + 1194080 q^{74} - 877952 q^{76} - 5025267 q^{77} + 2636926 q^{79} - 634880 q^{80} + 2713040 q^{82} + 10059354 q^{83} - 439425 q^{85} + 670952 q^{86} - 1687040 q^{88} + 3506160 q^{89} + 19003851 q^{91} + 5281600 q^{92} + 11768200 q^{94} + 1063145 q^{95} + 5893526 q^{97} - 8842976 q^{98}+O(q^{100}) \) 2 * q - 16 * q^2 + 128 * q^4 - 155 * q^5 - 2238 * q^7 - 1024 * q^8 + 1240 * q^10 + 3295 * q^11 - 13427 * q^13 + 17904 * q^14 + 8192 * q^16 + 32256 * q^17 - 13718 * q^19 - 9920 * q^20 - 26360 * q^22 + 82525 * q^23 + 159919 * q^25 + 107416 * q^26 - 143232 * q^28 + 12749 * q^29 + 258944 * q^31 - 65536 * q^32 - 258048 * q^34 + 448167 * q^35 - 149260 * q^37 + 109744 * q^38 + 79360 * q^40 - 339130 * q^41 - 83869 * q^43 + 210880 * q^44 - 660200 * q^46 - 1471025 * q^47 + 1105372 * q^49 - 1279352 * q^50 - 859328 * q^52 + 945643 * q^53 - 1736899 * q^55 + 1145856 * q^56 - 101992 * q^58 + 969009 * q^59 - 1506755 * q^61 - 2071552 * q^62 + 524288 * q^64 + 5445956 * q^65 - 1848219 * q^67 + 2064384 * q^68 - 3585336 * q^70 + 3417184 * q^71 - 2499822 * q^73 + 1194080 * q^74 - 877952 * q^76 - 5025267 * q^77 + 2636926 * q^79 - 634880 * q^80 + 2713040 * q^82 + 10059354 * q^83 - 439425 * q^85 + 670952 * q^86 - 1687040 * q^88 + 3506160 * q^89 + 19003851 * q^91 + 5281600 * q^92 + 11768200 * q^94 + 1063145 * q^95 + 5893526 * q^97 - 8842976 * q^98

Introduction

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