LMFDB - Embedded newform 825.4.c.f.199.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,4,Mod(199,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("825.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	825.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$48.6765757547$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	825.199
Dual form			825.4.c.f.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -3.00000i q^{3} +7.00000 q^{4} +3.00000 q^{6} +26.0000i q^{7} +15.0000i q^{8} -9.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -3.00000i q^{3} +7.00000 q^{4} +3.00000 q^{6} +26.0000i q^{7} +15.0000i q^{8} -9.00000 q^{9} +11.0000 q^{11} -21.0000i q^{12} -32.0000i q^{13} -26.0000 q^{14} +41.0000 q^{16} -74.0000i q^{17} -9.00000i q^{18} +60.0000 q^{19} +78.0000 q^{21} +11.0000i q^{22} -182.000i q^{23} +45.0000 q^{24} +32.0000 q^{26} +27.0000i q^{27} +182.000i q^{28} +90.0000 q^{29} -8.00000 q^{31} +161.000i q^{32} -33.0000i q^{33} +74.0000 q^{34} -63.0000 q^{36} +66.0000i q^{37} +60.0000i q^{38} -96.0000 q^{39} +422.000 q^{41} +78.0000i q^{42} +408.000i q^{43} +77.0000 q^{44} +182.000 q^{46} +506.000i q^{47} -123.000i q^{48} -333.000 q^{49} -222.000 q^{51} -224.000i q^{52} +348.000i q^{53} -27.0000 q^{54} -390.000 q^{56} -180.000i q^{57} +90.0000i q^{58} +200.000 q^{59} +132.000 q^{61} -8.00000i q^{62} -234.000i q^{63} +167.000 q^{64} +33.0000 q^{66} +1036.00i q^{67} -518.000i q^{68} -546.000 q^{69} +762.000 q^{71} -135.000i q^{72} -542.000i q^{73} -66.0000 q^{74} +420.000 q^{76} +286.000i q^{77} -96.0000i q^{78} +550.000 q^{79} +81.0000 q^{81} +422.000i q^{82} -132.000i q^{83} +546.000 q^{84} -408.000 q^{86} -270.000i q^{87} +165.000i q^{88} -570.000 q^{89} +832.000 q^{91} -1274.00i q^{92} +24.0000i q^{93} -506.000 q^{94} +483.000 q^{96} -14.0000i q^{97} -333.000i q^{98} -99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 14 q^{4} + 6 q^{6} - 18 q^{9}+O(q^{10}) $ 2 * q + 14 * q^4 + 6 * q^6 - 18 * q^9	$ 2 q + 14 q^{4} + 6 q^{6} - 18 q^{9} + 22 q^{11} - 52 q^{14} + 82 q^{16} + 120 q^{19} + 156 q^{21} + 90 q^{24} + 64 q^{26} + 180 q^{29} - 16 q^{31} + 148 q^{34} - 126 q^{36} - 192 q^{39} + 844 q^{41} + 154 q^{44} + 364 q^{46} - 666 q^{49} - 444 q^{51} - 54 q^{54} - 780 q^{56} + 400 q^{59} + 264 q^{61} + 334 q^{64} + 66 q^{66} - 1092 q^{69} + 1524 q^{71} - 132 q^{74} + 840 q^{76} + 1100 q^{79} + 162 q^{81} + 1092 q^{84} - 816 q^{86} - 1140 q^{89} + 1664 q^{91} - 1012 q^{94} + 966 q^{96} - 198 q^{99}+O(q^{100}) $ 2 * q + 14 * q^4 + 6 * q^6 - 18 * q^9 + 22 * q^11 - 52 * q^14 + 82 * q^16 + 120 * q^19 + 156 * q^21 + 90 * q^24 + 64 * q^26 + 180 * q^29 - 16 * q^31 + 148 * q^34 - 126 * q^36 - 192 * q^39 + 844 * q^41 + 154 * q^44 + 364 * q^46 - 666 * q^49 - 444 * q^51 - 54 * q^54 - 780 * q^56 + 400 * q^59 + 264 * q^61 + 334 * q^64 + 66 * q^66 - 1092 * q^69 + 1524 * q^71 - 132 * q^74 + 840 * q^76 + 1100 * q^79 + 162 * q^81 + 1092 * q^84 - 816 * q^86 - 1140 * q^89 + 1664 * q^91 - 1012 * q^94 + 966 * q^96 - 198 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$.

$n$	$376$	$551$	$727$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	1.00000i	0.353553i	0.984251	+	0.176777i	$0.0565670\pi$
$2$	1.00000i	0.353553i	−0.984251	+	0.176777i	$0.943433\pi$
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	7.00000	0.875000
$5$	0	0
$5$	0	0
$6$	3.00000	0.204124
$7$	26.0000i	1.40387i	0.712242	+	0.701934i	$0.247680\pi$
$7$	26.0000i	1.40387i	−0.712242	+	0.701934i	$0.752320\pi$
$8$	15.0000i	0.662913i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	− 21.0000i	− 0.505181i
$13$	− 32.0000i	− 0.682708i	−0.939935	−	0.341354i	$-0.889115\pi$
$13$	− 32.0000i	− 0.682708i	0.939935	−	0.341354i	$-0.110885\pi$
$14$	−26.0000	−0.496342
$15$	0	0
$16$	41.0000	0.640625
$17$	− 74.0000i	− 1.05574i	−0.849324	−	0.527872i	$-0.822990\pi$
$17$	− 74.0000i	− 1.05574i	0.849324	−	0.527872i	$-0.177010\pi$
$18$	− 9.00000i	− 0.117851i
$19$	60.0000	0.724471	0.362235	−	0.932087i	$-0.382014\pi$
$19$	60.0000	0.724471	0.362235	+	0.932087i	$0.382014\pi$
$20$	0	0
$21$	78.0000	0.810524
$22$	11.0000i	0.106600i
$23$	− 182.000i	− 1.64998i	−0.565145	−	0.824992i	$-0.691180\pi$
$23$	− 182.000i	− 1.64998i	0.565145	−	0.824992i	$-0.308820\pi$
$24$	45.0000	0.382733
$25$	0	0
$26$	32.0000	0.241374
$27$	27.0000i	0.192450i
$28$	182.000i	1.22838i
$29$	90.0000	0.576296	0.288148	−	0.957586i	$-0.406961\pi$
$29$	90.0000	0.576296	0.288148	+	0.957586i	$0.406961\pi$
$30$	0	0
$31$	−8.00000	−0.0463498	−0.0231749	−	0.999731i	$-0.507377\pi$
$31$	−8.00000	−0.0463498	−0.0231749	+	0.999731i	$0.507377\pi$
$32$	161.000i	0.889408i
$33$	− 33.0000i	− 0.174078i
$34$	74.0000	0.373262
$35$	0	0
$36$	−63.0000	−0.291667
$37$	66.0000i	0.293252i	0.989192	+	0.146626i	$0.0468414\pi$
$37$	66.0000i	0.293252i	−0.989192	+	0.146626i	$0.953159\pi$
$38$	60.0000i	0.256139i
$39$	−96.0000	−0.394162
$40$	0	0
$41$	422.000	1.60745	0.803724	−	0.595003i	$-0.202849\pi$
$41$	422.000	1.60745	0.803724	+	0.595003i	$0.202849\pi$
$42$	78.0000i	0.286563i
$43$	408.000i	1.44696i	0.690344	+	0.723482i	$0.257459\pi$
$43$	408.000i	1.44696i	−0.690344	+	0.723482i	$0.742541\pi$
$44$	77.0000	0.263822
$45$	0	0
$46$	182.000	0.583357
$47$	506.000i	1.57038i	0.619257	+	0.785188i	$0.287434\pi$
$47$	506.000i	1.57038i	−0.619257	+	0.785188i	$0.712566\pi$
$48$	− 123.000i	− 0.369865i
$49$	−333.000	−0.970845
$50$	0	0
$51$	−222.000	−0.609534
$52$	− 224.000i	− 0.597369i
$53$	348.000i	0.901915i	0.892546	+	0.450957i	$0.148917\pi$
$53$	348.000i	0.901915i	−0.892546	+	0.450957i	$0.851083\pi$
$54$	−27.0000	−0.0680414
$55$	0	0
$56$	−390.000	−0.930642
$57$	− 180.000i	− 0.418273i
$58$	90.0000i	0.203751i
$59$	200.000	0.441318	0.220659	−	0.975351i	$-0.429179\pi$
$59$	200.000	0.441318	0.220659	+	0.975351i	$0.429179\pi$
$60$	0	0
$61$	132.000	0.277063	0.138532	−	0.990358i	$-0.455762\pi$
$61$	132.000	0.277063	0.138532	+	0.990358i	$0.455762\pi$
$62$	− 8.00000i	− 0.0163871i
$63$	− 234.000i	− 0.467956i
$64$	167.000	0.326172
$65$	0	0
$66$	33.0000	0.0615457
$67$	1036.00i	1.88907i	0.328414	+	0.944534i	$0.393486\pi$
$67$	1036.00i	1.88907i	−0.328414	+	0.944534i	$0.606514\pi$
$68$	− 518.000i	− 0.923775i
$69$	−546.000	−0.952618
$70$	0	0
$71$	762.000	1.27370	0.636850	−	0.770987i	$-0.280237\pi$
$71$	762.000	1.27370	0.636850	+	0.770987i	$0.280237\pi$
$72$	− 135.000i	− 0.220971i
$73$	− 542.000i	− 0.868990i	−0.900674	−	0.434495i	$-0.856927\pi$
$73$	− 542.000i	− 0.868990i	0.900674	−	0.434495i	$-0.143073\pi$
$74$	−66.0000	−0.103680
$75$	0	0
$76$	420.000	0.633912
$77$	286.000i	0.423282i
$78$	− 96.0000i	− 0.139357i
$79$	550.000	0.783289	0.391645	−	0.920117i	$-0.371906\pi$
$79$	550.000	0.783289	0.391645	+	0.920117i	$0.371906\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	422.000i	0.568318i
$83$	− 132.000i	− 0.174565i	−0.996184	−	0.0872824i	$-0.972182\pi$
$83$	− 132.000i	− 0.174565i	0.996184	−	0.0872824i	$-0.0278183\pi$
$84$	546.000	0.709208
$85$	0	0
$86$	−408.000	−0.511579
$87$	− 270.000i	− 0.332725i
$88$	165.000i	0.199876i
$89$	−570.000	−0.678875	−0.339438	−	0.940629i	$-0.610237\pi$
$89$	−570.000	−0.678875	−0.339438	+	0.940629i	$0.610237\pi$
$90$	0	0
$91$	832.000	0.958432
$92$	− 1274.00i	− 1.44374i
$93$	24.0000i	0.0267600i
$94$	−506.000	−0.555212
$95$	0	0
$96$	483.000	0.513500
$97$	− 14.0000i	− 0.0146545i	−0.999973	−	0.00732724i	$-0.997668\pi$
$97$	− 14.0000i	− 0.0146545i	0.999973	−	0.00732724i	$-0.00233235\pi$
$98$	− 333.000i	− 0.343246i
$99$	−99.0000	−0.100504
$100$	0	0
$101$	1702.00	1.67679	0.838393	−	0.545067i	$-0.183496\pi$
$101$	1702.00	1.67679	0.838393	+	0.545067i	$0.183496\pi$
$102$	− 222.000i	− 0.215503i
$103$	− 1132.00i	− 1.08291i	−0.840731	−	0.541453i	$-0.817874\pi$
$103$	− 1132.00i	− 1.08291i	0.840731	−	0.541453i	$-0.182126\pi$
$104$	480.000	0.452576
$105$	0	0
$106$	−348.000	−0.318875
$107$	− 564.000i	− 0.509570i	−0.966998	−	0.254785i	$-0.917995\pi$
$107$	− 564.000i	− 0.509570i	0.966998	−	0.254785i	$-0.0820046\pi$
$108$	189.000i	0.168394i
$109$	320.000	0.281197	0.140598	−	0.990067i	$-0.455097\pi$
$109$	320.000	0.281197	0.140598	+	0.990067i	$0.455097\pi$
$110$	0	0
$111$	198.000	0.169309
$112$	1066.00i	0.899353i
$113$	− 2142.00i	− 1.78321i	−0.452817	−	0.891604i	$-0.649581\pi$
$113$	− 2142.00i	− 1.78321i	0.452817	−	0.891604i	$-0.350419\pi$
$114$	180.000	0.147882
$115$	0	0
$116$	630.000	0.504259
$117$	288.000i	0.227569i
$118$	200.000i	0.156030i
$119$	1924.00	1.48212
$120$	0	0
$121$	121.000	0.0909091
$122$	132.000i	0.0979567i
$123$	− 1266.00i	− 0.928060i
$124$	−56.0000	−0.0405560
$125$	0	0
$126$	234.000	0.165447
$127$	1606.00i	1.12212i	0.827775	+	0.561061i	$0.189607\pi$
$127$	1606.00i	1.12212i	−0.827775	+	0.561061i	$0.810393\pi$
$128$	1455.00i	1.00473i
$129$	1224.00	0.835405
$130$	0	0
$131$	−1908.00	−1.27254	−0.636270	−	0.771466i	$-0.719524\pi$
$131$	−1908.00	−1.27254	−0.636270	+	0.771466i	$0.719524\pi$
$132$	− 231.000i	− 0.152318i
$133$	1560.00i	1.01706i
$134$	−1036.00	−0.667886
$135$	0	0
$136$	1110.00	0.699866
$137$	2186.00i	1.36323i	0.731711	+	0.681615i	$0.238722\pi$
$137$	2186.00i	1.36323i	−0.731711	+	0.681615i	$0.761278\pi$
$138$	− 546.000i	− 0.336801i
$139$	−2740.00	−1.67197	−0.835985	−	0.548753i	$-0.815103\pi$
$139$	−2740.00	−1.67197	−0.835985	+	0.548753i	$0.815103\pi$
$140$	0	0
$141$	1518.00	0.906657
$142$	762.000i	0.450321i
$143$	− 352.000i	− 0.205844i
$144$	−369.000	−0.213542
$145$	0	0
$146$	542.000	0.307235
$147$	999.000i	0.560518i
$148$	462.000i	0.256596i
$149$	1310.00	0.720264	0.360132	−	0.932901i	$-0.382732\pi$
$149$	1310.00	0.720264	0.360132	+	0.932901i	$0.382732\pi$
$150$	0	0
$151$	−1198.00	−0.645641	−0.322821	−	0.946460i	$-0.604631\pi$
$151$	−1198.00	−0.645641	−0.322821	+	0.946460i	$0.604631\pi$
$152$	900.000i	0.480261i
$153$	666.000i	0.351914i
$154$	−286.000	−0.149653
$155$	0	0
$156$	−672.000	−0.344891
$157$	− 2114.00i	− 1.07462i	−0.843384	−	0.537311i	$-0.819440\pi$
$157$	− 2114.00i	− 1.07462i	0.843384	−	0.537311i	$-0.180560\pi$
$158$	550.000i	0.276934i
$159$	1044.00	0.520721
$160$	0	0
$161$	4732.00	2.31636
$162$	81.0000i	0.0392837i
$163$	3868.00i	1.85868i	0.369223	+	0.929341i	$0.379624\pi$
$163$	3868.00i	1.85868i	−0.369223	+	0.929341i	$0.620376\pi$
$164$	2954.00	1.40652
$165$	0	0
$166$	132.000	0.0617180
$167$	− 2004.00i	− 0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$	− 2004.00i	− 0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	1170.00i	0.537306i
$169$	1173.00	0.533910
$170$	0	0
$171$	−540.000	−0.241490
$172$	2856.00i	1.26609i
$173$	678.000i	0.297962i	0.988840	+	0.148981i	$0.0475993\pi$
$173$	678.000i	0.297962i	−0.988840	+	0.148981i	$0.952401\pi$
$174$	270.000	0.117636
$175$	0	0
$176$	451.000	0.193156
$177$	− 600.000i	− 0.254795i
$178$	− 570.000i	− 0.240019i
$179$	1680.00	0.701503	0.350752	−	0.936469i	$-0.385926\pi$
$179$	1680.00	0.701503	0.350752	+	0.936469i	$0.385926\pi$
$180$	0	0
$181$	−4358.00	−1.78966	−0.894828	−	0.446412i	$-0.852702\pi$
$181$	−4358.00	−1.78966	−0.894828	+	0.446412i	$0.852702\pi$
$182$	832.000i	0.338857i
$183$	− 396.000i	− 0.159963i
$184$	2730.00	1.09379
$185$	0	0
$186$	−24.0000	−0.00946110
$187$	− 814.000i	− 0.318319i
$188$	3542.00i	1.37408i
$189$	−702.000	−0.270175
$190$	0	0
$191$	−1778.00	−0.673568	−0.336784	−	0.941582i	$-0.609339\pi$
$191$	−1778.00	−0.673568	−0.336784	+	0.941582i	$0.609339\pi$
$192$	− 501.000i	− 0.188315i
$193$	− 3962.00i	− 1.47767i	−0.673884	−	0.738837i	$-0.735375\pi$
$193$	− 3962.00i	− 1.47767i	0.673884	−	0.738837i	$-0.264625\pi$
$194$	14.0000	0.00518114
$195$	0	0
$196$	−2331.00	−0.849490
$197$	− 374.000i	− 0.135261i	−0.997710	−	0.0676304i	$-0.978456\pi$
$197$	− 374.000i	− 0.135261i	0.997710	−	0.0676304i	$-0.0215439\pi$
$198$	− 99.0000i	− 0.0355335i
$199$	−2100.00	−0.748066	−0.374033	−	0.927415i	$-0.622025\pi$
$199$	−2100.00	−0.748066	−0.374033	+	0.927415i	$0.622025\pi$
$200$	0	0
$201$	3108.00	1.09065
$202$	1702.00i	0.592833i
$203$	2340.00i	0.809043i
$204$	−1554.00	−0.533342
$205$	0	0
$206$	1132.00	0.382865
$207$	1638.00i	0.549995i
$208$	− 1312.00i	− 0.437360i
$209$	660.000	0.218436
$210$	0	0
$211$	2232.00	0.728233	0.364117	−	0.931353i	$-0.381371\pi$
$211$	2232.00	0.728233	0.364117	+	0.931353i	$0.381371\pi$
$212$	2436.00i	0.789175i
$213$	− 2286.00i	− 0.735372i
$214$	564.000	0.180160
$215$	0	0
$216$	−405.000	−0.127578
$217$	− 208.000i	− 0.0650689i
$218$	320.000i	0.0994180i
$219$	−1626.00	−0.501712
$220$	0	0
$221$	−2368.00	−0.720764
$222$	198.000i	0.0598599i
$223$	2128.00i	0.639020i	0.947583	+	0.319510i	$0.103518\pi$
$223$	2128.00i	0.639020i	−0.947583	+	0.319510i	$0.896482\pi$
$224$	−4186.00	−1.24861
$225$	0	0
$226$	2142.00	0.630459
$227$	− 2964.00i	− 0.866641i	−0.901240	−	0.433321i	$-0.857342\pi$
$227$	− 2964.00i	− 0.866641i	0.901240	−	0.433321i	$-0.142658\pi$
$228$	− 1260.00i	− 0.365989i
$229$	2550.00	0.735846	0.367923	−	0.929856i	$-0.380069\pi$
$229$	2550.00	0.735846	0.367923	+	0.929856i	$0.380069\pi$
$230$	0	0
$231$	858.000	0.244382
$232$	1350.00i	0.382034i
$233$	− 3042.00i	− 0.855314i	−0.903941	−	0.427657i	$-0.859339\pi$
$233$	− 3042.00i	− 0.855314i	0.903941	−	0.427657i	$-0.140661\pi$
$234$	−288.000	−0.0804579
$235$	0	0
$236$	1400.00	0.386154
$237$	− 1650.00i	− 0.452232i
$238$	1924.00i	0.524010i
$239$	−2700.00	−0.730747	−0.365373	−	0.930861i	$-0.619059\pi$
$239$	−2700.00	−0.730747	−0.365373	+	0.930861i	$0.619059\pi$
$240$	0	0
$241$	−578.000	−0.154491	−0.0772453	−	0.997012i	$-0.524612\pi$
$241$	−578.000	−0.154491	−0.0772453	+	0.997012i	$0.524612\pi$
$242$	121.000i	0.0321412i
$243$	− 243.000i	− 0.0641500i
$244$	924.000	0.242430
$245$	0	0
$246$	1266.00	0.328119
$247$	− 1920.00i	− 0.494602i
$248$	− 120.000i	− 0.0307258i
$249$	−396.000	−0.100785
$250$	0	0
$251$	3752.00	0.943522	0.471761	−	0.881726i	$-0.343618\pi$
$251$	3752.00	0.943522	0.471761	+	0.881726i	$0.343618\pi$
$252$	− 1638.00i	− 0.409462i
$253$	− 2002.00i	− 0.497489i
$254$	−1606.00	−0.396730
$255$	0	0
$256$	−119.000	−0.0290527
$257$	− 674.000i	− 0.163591i	−0.996649	−	0.0817957i	$-0.973935\pi$
$257$	− 674.000i	− 0.163591i	0.996649	−	0.0817957i	$-0.0260655\pi$
$258$	1224.00i	0.295360i
$259$	−1716.00	−0.411687
$260$	0	0
$261$	−810.000	−0.192099
$262$	− 1908.00i	− 0.449911i
$263$	− 4352.00i	− 1.02036i	−0.860066	−	0.510182i	$-0.829578\pi$
$263$	− 4352.00i	− 1.02036i	0.860066	−	0.510182i	$-0.170422\pi$
$264$	495.000	0.115398
$265$	0	0
$266$	−1560.00	−0.359585
$267$	1710.00i	0.391949i
$268$	7252.00i	1.65293i
$269$	−500.000	−0.113329	−0.0566646	−	0.998393i	$-0.518047\pi$
$269$	−500.000	−0.113329	−0.0566646	+	0.998393i	$0.518047\pi$
$270$	0	0
$271$	−6538.00	−1.46552	−0.732759	−	0.680489i	$-0.761768\pi$
$271$	−6538.00	−1.46552	−0.732759	+	0.680489i	$0.761768\pi$
$272$	− 3034.00i	− 0.676336i
$273$	− 2496.00i	− 0.553351i
$274$	−2186.00	−0.481975
$275$	0	0
$276$	−3822.00	−0.833541
$277$	− 124.000i	− 0.0268969i	−0.999910	−	0.0134484i	$-0.995719\pi$
$277$	− 124.000i	− 0.0268969i	0.999910	−	0.0134484i	$-0.00428090\pi$
$278$	− 2740.00i	− 0.591131i
$279$	72.0000	0.0154499
$280$	0	0
$281$	3642.00	0.773180	0.386590	−	0.922252i	$-0.373653\pi$
$281$	3642.00	0.773180	0.386590	+	0.922252i	$0.373653\pi$
$282$	1518.00i	0.320552i
$283$	4648.00i	0.976307i	0.872758	+	0.488154i	$0.162329\pi$
$283$	4648.00i	0.976307i	−0.872758	+	0.488154i	$0.837671\pi$
$284$	5334.00	1.11449
$285$	0	0
$286$	352.000	0.0727769
$287$	10972.0i	2.25664i
$288$	− 1449.00i	− 0.296469i
$289$	−563.000	−0.114594
$290$	0	0
$291$	−42.0000	−0.00846077
$292$	− 3794.00i	− 0.760367i
$293$	− 3102.00i	− 0.618501i	−0.950981	−	0.309250i	$-0.899922\pi$
$293$	− 3102.00i	− 0.618501i	0.950981	−	0.309250i	$-0.100078\pi$
$294$	−999.000	−0.198173
$295$	0	0
$296$	−990.000	−0.194401
$297$	297.000i	0.0580259i
$298$	1310.00i	0.254652i
$299$	−5824.00	−1.12646
$300$	0	0
$301$	−10608.0	−2.03135
$302$	− 1198.00i	− 0.228269i
$303$	− 5106.00i	− 0.968093i
$304$	2460.00	0.464114
$305$	0	0
$306$	−666.000	−0.124421
$307$	− 1244.00i	− 0.231267i	−0.993292	−	0.115633i	$-0.963110\pi$
$307$	− 1244.00i	− 0.231267i	0.993292	−	0.115633i	$-0.0368897\pi$
$308$	2002.00i	0.370372i
$309$	−3396.00	−0.625216
$310$	0	0
$311$	2082.00	0.379612	0.189806	−	0.981822i	$-0.439214\pi$
$311$	2082.00	0.379612	0.189806	+	0.981822i	$0.439214\pi$
$312$	− 1440.00i	− 0.261295i
$313$	2378.00i	0.429433i	0.976676	+	0.214716i	$0.0688827\pi$
$313$	2378.00i	0.429433i	−0.976676	+	0.214716i	$0.931117\pi$
$314$	2114.00	0.379936
$315$	0	0
$316$	3850.00	0.685378
$317$	496.000i	0.0878806i	0.999034	+	0.0439403i	$0.0139911\pi$
$317$	496.000i	0.0878806i	−0.999034	+	0.0439403i	$0.986009\pi$
$318$	1044.00i	0.184103i
$319$	990.000	0.173760
$320$	0	0
$321$	−1692.00	−0.294200
$322$	4732.00i	0.818957i
$323$	− 4440.00i	− 0.764855i
$324$	567.000	0.0972222
$325$	0	0
$326$	−3868.00	−0.657143
$327$	− 960.000i	− 0.162349i
$328$	6330.00i	1.06560i
$329$	−13156.0	−2.20460
$330$	0	0
$331$	−2708.00	−0.449683	−0.224842	−	0.974395i	$-0.572186\pi$
$331$	−2708.00	−0.449683	−0.224842	+	0.974395i	$0.572186\pi$
$332$	− 924.000i	− 0.152744i
$333$	− 594.000i	− 0.0977507i
$334$	2004.00	0.328305
$335$	0	0
$336$	3198.00	0.519242
$337$	− 4034.00i	− 0.652065i	−0.945359	−	0.326033i	$-0.894288\pi$
$337$	− 4034.00i	− 0.652065i	0.945359	−	0.326033i	$-0.105712\pi$
$338$	1173.00i	0.188766i
$339$	−6426.00	−1.02954
$340$	0	0
$341$	−88.0000	−0.0139750
$342$	− 540.000i	− 0.0853797i
$343$	260.000i	0.0409291i
$344$	−6120.00	−0.959210
$345$	0	0
$346$	−678.000	−0.105345
$347$	− 11084.0i	− 1.71476i	−0.514687	−	0.857378i	$-0.672092\pi$
$347$	− 11084.0i	− 1.71476i	0.514687	−	0.857378i	$-0.327908\pi$
$348$	− 1890.00i	− 0.291134i
$349$	3120.00	0.478538	0.239269	−	0.970953i	$-0.423092\pi$
$349$	3120.00	0.478538	0.239269	+	0.970953i	$0.423092\pi$
$350$	0	0
$351$	864.000	0.131387
$352$	1771.00i	0.268167i
$353$	− 5622.00i	− 0.847674i	−0.905739	−	0.423837i	$-0.860683\pi$
$353$	− 5622.00i	− 0.847674i	0.905739	−	0.423837i	$-0.139317\pi$
$354$	600.000	0.0900837
$355$	0	0
$356$	−3990.00	−0.594016
$357$	− 5772.00i	− 0.855705i
$358$	1680.00i	0.248019i
$359$	8500.00	1.24962	0.624809	−	0.780778i	$-0.285177\pi$
$359$	8500.00	1.24962	0.624809	+	0.780778i	$0.285177\pi$
$360$	0	0
$361$	−3259.00	−0.475142
$362$	− 4358.00i	− 0.632739i
$363$	− 363.000i	− 0.0524864i
$364$	5824.00	0.838628
$365$	0	0
$366$	396.000	0.0565553
$367$	− 7144.00i	− 1.01611i	−0.861324	−	0.508057i	$-0.830364\pi$
$367$	− 7144.00i	− 1.01611i	0.861324	−	0.508057i	$-0.169636\pi$
$368$	− 7462.00i	− 1.05702i
$369$	−3798.00	−0.535816
$370$	0	0
$371$	−9048.00	−1.26617
$372$	168.000i	0.0234150i
$373$	− 632.000i	− 0.0877312i	−0.999037	−	0.0438656i	$-0.986033\pi$
$373$	− 632.000i	− 0.0877312i	0.999037	−	0.0438656i	$-0.0139673\pi$
$374$	814.000	0.112543
$375$	0	0
$376$	−7590.00	−1.04102
$377$	− 2880.00i	− 0.393442i
$378$	− 702.000i	− 0.0955211i
$379$	4220.00	0.571944	0.285972	−	0.958238i	$-0.407684\pi$
$379$	4220.00	0.571944	0.285972	+	0.958238i	$0.407684\pi$
$380$	0	0
$381$	4818.00	0.647857
$382$	− 1778.00i	− 0.238142i
$383$	8458.00i	1.12842i	0.825632	+	0.564208i	$0.190819\pi$
$383$	8458.00i	1.12842i	−0.825632	+	0.564208i	$0.809181\pi$
$384$	4365.00	0.580079
$385$	0	0
$386$	3962.00	0.522437
$387$	− 3672.00i	− 0.482321i
$388$	− 98.0000i	− 0.0128227i
$389$	−1740.00	−0.226790	−0.113395	−	0.993550i	$-0.536173\pi$
$389$	−1740.00	−0.226790	−0.113395	+	0.993550i	$0.536173\pi$
$390$	0	0
$391$	−13468.0	−1.74196
$392$	− 4995.00i	− 0.643586i
$393$	5724.00i	0.734701i
$394$	374.000	0.0478219
$395$	0	0
$396$	−693.000	−0.0879408
$397$	5126.00i	0.648027i	0.946053	+	0.324013i	$0.105032\pi$
$397$	5126.00i	0.648027i	−0.946053	+	0.324013i	$0.894968\pi$
$398$	− 2100.00i	− 0.264481i
$399$	4680.00	0.587201
$400$	0	0
$401$	−3098.00	−0.385802	−0.192901	−	0.981218i	$-0.561790\pi$
$401$	−3098.00	−0.385802	−0.192901	+	0.981218i	$0.561790\pi$
$402$	3108.00i	0.385604i
$403$	256.000i	0.0316433i
$404$	11914.0	1.46719
$405$	0	0
$406$	−2340.00	−0.286040
$407$	726.000i	0.0884189i
$408$	− 3330.00i	− 0.404068i
$409$	−6390.00	−0.772531	−0.386265	−	0.922388i	$-0.626235\pi$
$409$	−6390.00	−0.772531	−0.386265	+	0.922388i	$0.626235\pi$
$410$	0	0
$411$	6558.00	0.787062
$412$	− 7924.00i	− 0.947542i
$413$	5200.00i	0.619553i
$414$	−1638.00	−0.194452
$415$	0	0
$416$	5152.00	0.607206
$417$	8220.00i	0.965312i
$418$	660.000i	0.0772288i
$419$	−9760.00	−1.13796	−0.568982	−	0.822350i	$-0.692663\pi$
$419$	−9760.00	−1.13796	−0.568982	+	0.822350i	$0.692663\pi$
$420$	0	0
$421$	−5138.00	−0.594800	−0.297400	−	0.954753i	$-0.596119\pi$
$421$	−5138.00	−0.594800	−0.297400	+	0.954753i	$0.596119\pi$
$422$	2232.00i	0.257469i
$423$	− 4554.00i	− 0.523459i
$424$	−5220.00	−0.597891
$425$	0	0
$426$	2286.00	0.259993
$427$	3432.00i	0.388960i
$428$	− 3948.00i	− 0.445873i
$429$	−1056.00	−0.118844
$430$	0	0
$431$	−7008.00	−0.783210	−0.391605	−	0.920133i	$-0.628080\pi$
$431$	−7008.00	−0.783210	−0.391605	+	0.920133i	$0.628080\pi$
$432$	1107.00i	0.123288i
$433$	5578.00i	0.619080i	0.950886	+	0.309540i	$0.100175\pi$
$433$	5578.00i	0.619080i	−0.950886	+	0.309540i	$0.899825\pi$
$434$	208.000	0.0230053
$435$	0	0
$436$	2240.00	0.246047
$437$	− 10920.0i	− 1.19536i
$438$	− 1626.00i	− 0.177382i
$439$	10430.0	1.13393	0.566967	−	0.823741i	$-0.308117\pi$
$439$	10430.0	1.13393	0.566967	+	0.823741i	$0.308117\pi$
$440$	0	0
$441$	2997.00	0.323615
$442$	− 2368.00i	− 0.254829i
$443$	− 4432.00i	− 0.475329i	−0.971347	−	0.237664i	$-0.923618\pi$
$443$	− 4432.00i	− 0.475329i	0.971347	−	0.237664i	$-0.0763819\pi$
$444$	1386.00	0.148146
$445$	0	0
$446$	−2128.00	−0.225928
$447$	− 3930.00i	− 0.415845i
$448$	4342.00i	0.457902i
$449$	6290.00	0.661121	0.330561	−	0.943785i	$-0.392762\pi$
$449$	6290.00	0.661121	0.330561	+	0.943785i	$0.392762\pi$
$450$	0	0
$451$	4642.00	0.484664
$452$	− 14994.0i	− 1.56031i
$453$	3594.00i	0.372761i
$454$	2964.00	0.306404
$455$	0	0
$456$	2700.00	0.277279
$457$	− 3054.00i	− 0.312604i	−0.987709	−	0.156302i	$-0.950043\pi$
$457$	− 3054.00i	− 0.312604i	0.987709	−	0.156302i	$-0.0499573\pi$
$458$	2550.00i	0.260161i
$459$	1998.00	0.203178
$460$	0	0
$461$	12882.0	1.30146	0.650732	−	0.759308i	$-0.274462\pi$
$461$	12882.0	1.30146	0.650732	+	0.759308i	$0.274462\pi$
$462$	858.000i	0.0864021i
$463$	6148.00i	0.617110i	0.951207	+	0.308555i	$0.0998453\pi$
$463$	6148.00i	0.617110i	−0.951207	+	0.308555i	$0.900155\pi$
$464$	3690.00	0.369190
$465$	0	0
$466$	3042.00	0.302399
$467$	− 5124.00i	− 0.507731i	−0.967240	−	0.253866i	$-0.918298\pi$
$467$	− 5124.00i	− 0.507731i	0.967240	−	0.253866i	$-0.0817021\pi$
$468$	2016.00i	0.199123i
$469$	−26936.0	−2.65200
$470$	0	0
$471$	−6342.00	−0.620433
$472$	3000.00i	0.292555i
$473$	4488.00i	0.436276i
$474$	1650.00	0.159888
$475$	0	0
$476$	13468.0	1.29686
$477$	− 3132.00i	− 0.300638i
$478$	− 2700.00i	− 0.258358i
$479$	16520.0	1.57582	0.787910	−	0.615790i	$-0.211163\pi$
$479$	16520.0	1.57582	0.787910	+	0.615790i	$0.211163\pi$
$480$	0	0
$481$	2112.00	0.200206
$482$	− 578.000i	− 0.0546207i
$483$	− 14196.0i	− 1.33735i
$484$	847.000	0.0795455
$485$	0	0
$486$	243.000	0.0226805
$487$	− 524.000i	− 0.0487571i	−0.999703	−	0.0243785i	$-0.992239\pi$
$487$	− 524.000i	− 0.0487571i	0.999703	−	0.0243785i	$-0.00776070\pi$
$488$	1980.00i	0.183669i
$489$	11604.0	1.07311
$490$	0	0
$491$	−15028.0	−1.38127	−0.690636	−	0.723203i	$-0.742669\pi$
$491$	−15028.0	−1.38127	−0.690636	+	0.723203i	$0.742669\pi$
$492$	− 8862.00i	− 0.812052i
$493$	− 6660.00i	− 0.608421i
$494$	1920.00	0.174868
$495$	0	0
$496$	−328.000	−0.0296928
$497$	19812.0i	1.78811i
$498$	− 396.000i	− 0.0356329i
$499$	−9020.00	−0.809200	−0.404600	−	0.914494i	$-0.632589\pi$
$499$	−9020.00	−0.809200	−0.404600	+	0.914494i	$0.632589\pi$
$500$	0	0
$501$	−6012.00	−0.536120
$502$	3752.00i	0.333586i
$503$	− 14812.0i	− 1.31299i	−0.754330	−	0.656495i	$-0.772038\pi$
$503$	− 14812.0i	− 1.31299i	0.754330	−	0.656495i	$-0.227962\pi$
$504$	3510.00	0.310214
$505$	0	0
$506$	2002.00	0.175889
$507$	− 3519.00i	− 0.308253i
$508$	11242.0i	0.981856i
$509$	−12660.0	−1.10245	−0.551223	−	0.834358i	$-0.685839\pi$
$509$	−12660.0	−1.10245	−0.551223	+	0.834358i	$0.685839\pi$
$510$	0	0
$511$	14092.0	1.21995
$512$	11521.0i	0.994455i
$513$	1620.00i	0.139424i
$514$	674.000	0.0578383
$515$	0	0
$516$	8568.00	0.730979
$517$	5566.00i	0.473486i
$518$	− 1716.00i	− 0.145553i
$519$	2034.00	0.172028
$520$	0	0
$521$	−3738.00	−0.314328	−0.157164	−	0.987573i	$-0.550235\pi$
$521$	−3738.00	−0.314328	−0.157164	+	0.987573i	$0.550235\pi$
$522$	− 810.000i	− 0.0679171i
$523$	− 6352.00i	− 0.531078i	−0.964100	−	0.265539i	$-0.914450\pi$
$523$	− 6352.00i	− 0.531078i	0.964100	−	0.265539i	$-0.0855498\pi$
$524$	−13356.0	−1.11347
$525$	0	0
$526$	4352.00	0.360753
$527$	592.000i	0.0489334i
$528$	− 1353.00i	− 0.111518i
$529$	−20957.0	−1.72245
$530$	0	0
$531$	−1800.00	−0.147106
$532$	10920.0i	0.889929i
$533$	− 13504.0i	− 1.09742i
$534$	−1710.00	−0.138575
$535$	0	0
$536$	−15540.0	−1.25229
$537$	− 5040.00i	− 0.405013i
$538$	− 500.000i	− 0.0400679i
$539$	−3663.00	−0.292721
$540$	0	0
$541$	−24728.0	−1.96514	−0.982569	−	0.185898i	$-0.940481\pi$
$541$	−24728.0	−1.96514	−0.982569	+	0.185898i	$0.940481\pi$
$542$	− 6538.00i	− 0.518139i
$543$	13074.0i	1.03326i
$544$	11914.0	0.938986
$545$	0	0
$546$	2496.00	0.195639
$547$	22756.0i	1.77875i	0.457178	+	0.889375i	$0.348860\pi$
$547$	22756.0i	1.77875i	−0.457178	+	0.889375i	$0.651140\pi$
$548$	15302.0i	1.19283i
$549$	−1188.00	−0.0923545
$550$	0	0
$551$	5400.00	0.417509
$552$	− 8190.00i	− 0.631503i
$553$	14300.0i	1.09963i
$554$	124.000	0.00950949
$555$	0	0
$556$	−19180.0	−1.46297
$557$	9526.00i	0.724649i	0.932052	+	0.362325i	$0.118017\pi$
$557$	9526.00i	0.724649i	−0.932052	+	0.362325i	$0.881983\pi$
$558$	72.0000i	0.00546237i
$559$	13056.0	0.987853
$560$	0	0
$561$	−2442.00	−0.183781
$562$	3642.00i	0.273360i
$563$	12068.0i	0.903385i	0.892174	+	0.451692i	$0.149180\pi$
$563$	12068.0i	0.903385i	−0.892174	+	0.451692i	$0.850820\pi$
$564$	10626.0	0.793325
$565$	0	0
$566$	−4648.00	−0.345177
$567$	2106.00i	0.155985i
$568$	11430.0i	0.844352i
$569$	−15090.0	−1.11179	−0.555893	−	0.831254i	$-0.687623\pi$
$569$	−15090.0	−1.11179	−0.555893	+	0.831254i	$0.687623\pi$
$570$	0	0
$571$	4412.00	0.323356	0.161678	−	0.986844i	$-0.448309\pi$
$571$	4412.00	0.323356	0.161678	+	0.986844i	$0.448309\pi$
$572$	− 2464.00i	− 0.180114i
$573$	5334.00i	0.388885i
$574$	−10972.0	−0.797844
$575$	0	0
$576$	−1503.00	−0.108724
$577$	3906.00i	0.281818i	0.990023	+	0.140909i	$0.0450025\pi$
$577$	3906.00i	0.281818i	−0.990023	+	0.140909i	$0.954998\pi$
$578$	− 563.000i	− 0.0405151i
$579$	−11886.0	−0.853135
$580$	0	0
$581$	3432.00	0.245066
$582$	− 42.0000i	− 0.00299133i
$583$	3828.00i	0.271937i
$584$	8130.00	0.576065
$585$	0	0
$586$	3102.00	0.218673
$587$	12016.0i	0.844895i	0.906387	+	0.422448i	$0.138829\pi$
$587$	12016.0i	0.844895i	−0.906387	+	0.422448i	$0.861171\pi$
$588$	6993.00i	0.490453i
$589$	−480.000	−0.0335790
$590$	0	0
$591$	−1122.00	−0.0780929
$592$	2706.00i	0.187865i
$593$	− 11342.0i	− 0.785430i	−0.919660	−	0.392715i	$-0.871536\pi$
$593$	− 11342.0i	− 0.785430i	0.919660	−	0.392715i	$-0.128464\pi$
$594$	−297.000	−0.0205152
$595$	0	0
$596$	9170.00	0.630231
$597$	6300.00i	0.431896i
$598$	− 5824.00i	− 0.398263i
$599$	−20690.0	−1.41130	−0.705651	−	0.708559i	$-0.749346\pi$
$599$	−20690.0	−1.41130	−0.705651	+	0.708559i	$0.749346\pi$
$600$	0	0
$601$	−598.000	−0.0405872	−0.0202936	−	0.999794i	$-0.506460\pi$
$601$	−598.000	−0.0405872	−0.0202936	+	0.999794i	$0.506460\pi$
$602$	− 10608.0i	− 0.718189i
$603$	− 9324.00i	− 0.629689i
$604$	−8386.00	−0.564936
$605$	0	0
$606$	5106.00	0.342272
$607$	166.000i	0.0111001i	0.999985	+	0.00555003i	$0.00176664\pi$
$607$	166.000i	0.0111001i	−0.999985	+	0.00555003i	$0.998233\pi$
$608$	9660.00i	0.644350i
$609$	7020.00	0.467101
$610$	0	0
$611$	16192.0	1.07211
$612$	4662.00i	0.307925i
$613$	20108.0i	1.32488i	0.749113	+	0.662442i	$0.230480\pi$
$613$	20108.0i	1.32488i	−0.749113	+	0.662442i	$0.769520\pi$
$614$	1244.00	0.0817651
$615$	0	0
$616$	−4290.00	−0.280599
$617$	2286.00i	0.149159i	0.997215	+	0.0745793i	$0.0237614\pi$
$617$	2286.00i	0.149159i	−0.997215	+	0.0745793i	$0.976239\pi$
$618$	− 3396.00i	− 0.221047i
$619$	25660.0	1.66618	0.833088	−	0.553141i	$-0.186571\pi$
$619$	25660.0	1.66618	0.833088	+	0.553141i	$0.186571\pi$
$620$	0	0
$621$	4914.00	0.317539
$622$	2082.00i	0.134213i
$623$	− 14820.0i	− 0.953051i
$624$	−3936.00	−0.252510
$625$	0	0
$626$	−2378.00	−0.151827
$627$	− 1980.00i	− 0.126114i
$628$	− 14798.0i	− 0.940294i
$629$	4884.00	0.309599
$630$	0	0
$631$	−11408.0	−0.719723	−0.359862	−	0.933006i	$-0.617176\pi$
$631$	−11408.0	−0.719723	−0.359862	+	0.933006i	$0.617176\pi$
$632$	8250.00i	0.519252i
$633$	− 6696.00i	− 0.420446i
$634$	−496.000	−0.0310705
$635$	0	0
$636$	7308.00	0.455631
$637$	10656.0i	0.662804i
$638$	990.000i	0.0614333i
$639$	−6858.00	−0.424567
$640$	0	0
$641$	−3378.00	−0.208148	−0.104074	−	0.994570i	$-0.533188\pi$
$641$	−3378.00	−0.208148	−0.104074	+	0.994570i	$0.533188\pi$
$642$	− 1692.00i	− 0.104015i
$643$	− 11212.0i	− 0.687649i	−0.939034	−	0.343824i	$-0.888278\pi$
$643$	− 11212.0i	− 0.687649i	0.939034	−	0.343824i	$-0.111722\pi$
$644$	33124.0	2.02681
$645$	0	0
$646$	4440.00	0.270417
$647$	86.0000i	0.00522567i	0.999997	+	0.00261284i	$0.000831692\pi$
$647$	86.0000i	0.00522567i	−0.999997	+	0.00261284i	$0.999168\pi$
$648$	1215.00i	0.0736570i
$649$	2200.00	0.133062
$650$	0	0
$651$	−624.000	−0.0375676
$652$	27076.0i	1.62635i
$653$	− 4432.00i	− 0.265601i	−0.991143	−	0.132801i	$-0.957603\pi$
$653$	− 4432.00i	− 0.265601i	0.991143	−	0.132801i	$-0.0423970\pi$
$654$	960.000	0.0573990
$655$	0	0
$656$	17302.0	1.02977
$657$	4878.00i	0.289663i
$658$	− 13156.0i	− 0.779444i
$659$	−4580.00	−0.270731	−0.135365	−	0.990796i	$-0.543221\pi$
$659$	−4580.00	−0.270731	−0.135365	+	0.990796i	$0.543221\pi$
$660$	0	0
$661$	4282.00	0.251967	0.125984	−	0.992032i	$-0.459791\pi$
$661$	4282.00	0.251967	0.125984	+	0.992032i	$0.459791\pi$
$662$	− 2708.00i	− 0.158987i
$663$	7104.00i	0.416133i
$664$	1980.00	0.115721
$665$	0	0
$666$	594.000	0.0345601
$667$	− 16380.0i	− 0.950879i
$668$	− 14028.0i	− 0.812514i
$669$	6384.00	0.368938
$670$	0	0
$671$	1452.00	0.0835378
$672$	12558.0i	0.720886i
$673$	8438.00i	0.483300i	0.970363	+	0.241650i	$0.0776886\pi$
$673$	8438.00i	0.483300i	−0.970363	+	0.241650i	$0.922311\pi$
$674$	4034.00	0.230540
$675$	0	0
$676$	8211.00	0.467171
$677$	− 34494.0i	− 1.95822i	−0.203341	−	0.979108i	$-0.565180\pi$
$677$	− 34494.0i	− 1.95822i	0.203341	−	0.979108i	$-0.434820\pi$
$678$	− 6426.00i	− 0.363996i
$679$	364.000	0.0205730
$680$	0	0
$681$	−8892.00	−0.500356
$682$	− 88.0000i	− 0.00494090i
$683$	− 13712.0i	− 0.768192i	−0.923293	−	0.384096i	$-0.874513\pi$
$683$	− 13712.0i	− 0.768192i	0.923293	−	0.384096i	$-0.125487\pi$
$684$	−3780.00	−0.211304
$685$	0	0
$686$	−260.000	−0.0144706
$687$	− 7650.00i	− 0.424841i
$688$	16728.0i	0.926961i
$689$	11136.0	0.615744
$690$	0	0
$691$	11372.0	0.626066	0.313033	−	0.949742i	$-0.398655\pi$
$691$	11372.0	0.626066	0.313033	+	0.949742i	$0.398655\pi$
$692$	4746.00i	0.260717i
$693$	− 2574.00i	− 0.141094i
$694$	11084.0	0.606258
$695$	0	0
$696$	4050.00	0.220567
$697$	− 31228.0i	− 1.69705i
$698$	3120.00i	0.169189i
$699$	−9126.00	−0.493815
$700$	0	0
$701$	−6398.00	−0.344721	−0.172360	−	0.985034i	$-0.555139\pi$
$701$	−6398.00	−0.344721	−0.172360	+	0.985034i	$0.555139\pi$
$702$	864.000i	0.0464524i
$703$	3960.00i	0.212453i
$704$	1837.00	0.0983445
$705$	0	0
$706$	5622.00	0.299698
$707$	44252.0i	2.35399i
$708$	− 4200.00i	− 0.222946i
$709$	5830.00	0.308816	0.154408	−	0.988007i	$-0.450653\pi$
$709$	5830.00	0.308816	0.154408	+	0.988007i	$0.450653\pi$
$710$	0	0
$711$	−4950.00	−0.261096
$712$	− 8550.00i	− 0.450035i
$713$	1456.00i	0.0764763i
$714$	5772.00	0.302537
$715$	0	0
$716$	11760.0	0.613815
$717$	8100.00i	0.421897i
$718$	8500.00i	0.441807i
$719$	−34530.0	−1.79103	−0.895516	−	0.445030i	$-0.853193\pi$
$719$	−34530.0	−1.79103	−0.895516	+	0.445030i	$0.853193\pi$
$720$	0	0
$721$	29432.0	1.52026
$722$	− 3259.00i	− 0.167988i
$723$	1734.00i	0.0891952i
$724$	−30506.0	−1.56595
$725$	0	0
$726$	363.000	0.0185567
$727$	17316.0i	0.883377i	0.897169	+	0.441688i	$0.145620\pi$
$727$	17316.0i	0.883377i	−0.897169	+	0.441688i	$0.854380\pi$
$728$	12480.0i	0.635357i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	30192.0	1.52762
$732$	− 2772.00i	− 0.139967i
$733$	− 27072.0i	− 1.36416i	−0.731279	−	0.682079i	$-0.761076\pi$
$733$	− 27072.0i	− 1.36416i	0.731279	−	0.682079i	$-0.238924\pi$
$734$	7144.00	0.359250
$735$	0	0
$736$	29302.0	1.46751
$737$	11396.0i	0.569575i
$738$	− 3798.00i	− 0.189439i
$739$	17320.0	0.862147	0.431073	−	0.902317i	$-0.358135\pi$
$739$	17320.0	0.862147	0.431073	+	0.902317i	$0.358135\pi$
$740$	0	0
$741$	−5760.00	−0.285559
$742$	− 9048.00i	− 0.447658i
$743$	14588.0i	0.720299i	0.932895	+	0.360149i	$0.117274\pi$
$743$	14588.0i	0.720299i	−0.932895	+	0.360149i	$0.882726\pi$
$744$	−360.000	−0.0177396
$745$	0	0
$746$	632.000	0.0310176
$747$	1188.00i	0.0581883i
$748$	− 5698.00i	− 0.278529i
$749$	14664.0	0.715368
$750$	0	0
$751$	26152.0	1.27071	0.635353	−	0.772222i	$-0.280855\pi$
$751$	26152.0	1.27071	0.635353	+	0.772222i	$0.280855\pi$
$752$	20746.0i	1.00602i
$753$	− 11256.0i	− 0.544743i
$754$	2880.00	0.139103
$755$	0	0
$756$	−4914.00	−0.236403
$757$	1066.00i	0.0511815i	0.999673	+	0.0255908i	$0.00814669\pi$
$757$	1066.00i	0.0511815i	−0.999673	+	0.0255908i	$0.991853\pi$
$758$	4220.00i	0.202213i
$759$	−6006.00	−0.287225
$760$	0	0
$761$	−37518.0	−1.78716	−0.893578	−	0.448907i	$-0.851813\pi$
$761$	−37518.0	−1.78716	−0.893578	+	0.448907i	$0.851813\pi$
$762$	4818.00i	0.229052i
$763$	8320.00i	0.394763i
$764$	−12446.0	−0.589372
$765$	0	0
$766$	−8458.00	−0.398956
$767$	− 6400.00i	− 0.301292i
$768$	357.000i	0.0167736i
$769$	17290.0	0.810785	0.405392	−	0.914143i	$-0.367135\pi$
$769$	17290.0	0.810785	0.405392	+	0.914143i	$0.367135\pi$
$770$	0	0
$771$	−2022.00	−0.0944495
$772$	− 27734.0i	− 1.29296i
$773$	− 17172.0i	− 0.799009i	−0.916731	−	0.399504i	$-0.869182\pi$
$773$	− 17172.0i	− 0.799009i	0.916731	−	0.399504i	$-0.130818\pi$
$774$	3672.00	0.170526
$775$	0	0
$776$	210.000	0.00971464
$777$	5148.00i	0.237688i
$778$	− 1740.00i	− 0.0801825i
$779$	25320.0	1.16455
$780$	0	0
$781$	8382.00	0.384035
$782$	− 13468.0i	− 0.615876i
$783$	2430.00i	0.110908i
$784$	−13653.0	−0.621948
$785$	0	0
$786$	−5724.00	−0.259756
$787$	9536.00i	0.431921i	0.976402	+	0.215960i	$0.0692882\pi$
$787$	9536.00i	0.431921i	−0.976402	+	0.215960i	$0.930712\pi$
$788$	− 2618.00i	− 0.118353i
$789$	−13056.0	−0.589108
$790$	0	0
$791$	55692.0	2.50339
$792$	− 1485.00i	− 0.0666252i
$793$	− 4224.00i	− 0.189153i
$794$	−5126.00	−0.229112
$795$	0	0
$796$	−14700.0	−0.654557
$797$	20516.0i	0.911812i	0.890028	+	0.455906i	$0.150685\pi$
$797$	20516.0i	0.911812i	−0.890028	+	0.455906i	$0.849315\pi$
$798$	4680.00i	0.207607i
$799$	37444.0	1.65791
$800$	0	0
$801$	5130.00	0.226292
$802$	− 3098.00i	− 0.136402i
$803$	− 5962.00i	− 0.262010i
$804$	21756.0	0.954322
$805$	0	0
$806$	−256.000	−0.0111876
$807$	1500.00i	0.0654306i
$808$	25530.0i	1.11156i
$809$	−22470.0	−0.976518	−0.488259	−	0.872699i	$-0.662368\pi$
$809$	−22470.0	−0.976518	−0.488259	+	0.872699i	$0.662368\pi$
$810$	0	0
$811$	−3368.00	−0.145828	−0.0729140	−	0.997338i	$-0.523230\pi$
$811$	−3368.00	−0.145828	−0.0729140	+	0.997338i	$0.523230\pi$
$812$	16380.0i	0.707913i
$813$	19614.0i	0.846117i
$814$	−726.000	−0.0312608
$815$	0	0
$816$	−9102.00	−0.390483
$817$	24480.0i	1.04828i
$818$	− 6390.00i	− 0.273131i
$819$	−7488.00	−0.319477
$820$	0	0
$821$	−10738.0	−0.456466	−0.228233	−	0.973607i	$-0.573295\pi$
$821$	−10738.0	−0.456466	−0.228233	+	0.973607i	$0.573295\pi$
$822$	6558.00i	0.278268i
$823$	− 15912.0i	− 0.673946i	−0.941514	−	0.336973i	$-0.890597\pi$
$823$	− 15912.0i	− 0.673946i	0.941514	−	0.336973i	$-0.109403\pi$
$824$	16980.0	0.717872
$825$	0	0
$826$	−5200.00	−0.219045
$827$	− 22924.0i	− 0.963900i	−0.876199	−	0.481950i	$-0.839929\pi$
$827$	− 22924.0i	− 0.963900i	0.876199	−	0.481950i	$-0.160071\pi$
$828$	11466.0i	0.481245i
$829$	41690.0	1.74663	0.873313	−	0.487159i	$-0.161967\pi$
$829$	41690.0	1.74663	0.873313	+	0.487159i	$0.161967\pi$
$830$	0	0
$831$	−372.000	−0.0155289
$832$	− 5344.00i	− 0.222680i
$833$	24642.0i	1.02496i
$834$	−8220.00	−0.341289
$835$	0	0
$836$	4620.00	0.191132
$837$	− 216.000i	− 0.00892001i
$838$	− 9760.00i	− 0.402331i
$839$	16450.0	0.676898	0.338449	−	0.940985i	$-0.390098\pi$
$839$	16450.0	0.676898	0.338449	+	0.940985i	$0.390098\pi$
$840$	0	0
$841$	−16289.0	−0.667883
$842$	− 5138.00i	− 0.210294i
$843$	− 10926.0i	− 0.446396i
$844$	15624.0	0.637204
$845$	0	0
$846$	4554.00	0.185071
$847$	3146.00i	0.127624i
$848$	14268.0i	0.577789i
$849$	13944.0	0.563671
$850$	0	0
$851$	12012.0	0.483861
$852$	− 16002.0i	− 0.643450i
$853$	− 30892.0i	− 1.24000i	−0.784601	−	0.620001i	$-0.787132\pi$
$853$	− 30892.0i	− 1.24000i	0.784601	−	0.620001i	$-0.212868\pi$
$854$	−3432.00	−0.137518
$855$	0	0
$856$	8460.00	0.337800
$857$	38906.0i	1.55076i	0.631493	+	0.775381i	$0.282442\pi$
$857$	38906.0i	1.55076i	−0.631493	+	0.775381i	$0.717558\pi$
$858$	− 1056.00i	− 0.0420178i
$859$	1020.00	0.0405145	0.0202572	−	0.999795i	$-0.493551\pi$
$859$	1020.00	0.0405145	0.0202572	+	0.999795i	$0.493551\pi$
$860$	0	0
$861$	32916.0	1.30287
$862$	− 7008.00i	− 0.276907i
$863$	15078.0i	0.594741i	0.954762	+	0.297370i	$0.0961096\pi$
$863$	15078.0i	0.594741i	−0.954762	+	0.297370i	$0.903890\pi$
$864$	−4347.00	−0.171167
$865$	0	0
$866$	−5578.00	−0.218878
$867$	1689.00i	0.0661608i
$868$	− 1456.00i	− 0.0569353i
$869$	6050.00	0.236171
$870$	0	0
$871$	33152.0	1.28968
$872$	4800.00i	0.186409i
$873$	126.000i	0.00488483i
$874$	10920.0	0.422625
$875$	0	0
$876$	−11382.0	−0.438998
$877$	− 22704.0i	− 0.874184i	−0.899417	−	0.437092i	$-0.856008\pi$
$877$	− 22704.0i	− 0.874184i	0.899417	−	0.437092i	$-0.143992\pi$
$878$	10430.0i	0.400906i
$879$	−9306.00	−0.357092
$880$	0	0
$881$	−19358.0	−0.740281	−0.370141	−	0.928976i	$-0.620690\pi$
$881$	−19358.0	−0.740281	−0.370141	+	0.928976i	$0.620690\pi$
$882$	2997.00i	0.114415i
$883$	− 11252.0i	− 0.428833i	−0.976742	−	0.214417i	$-0.931215\pi$
$883$	− 11252.0i	− 0.428833i	0.976742	−	0.214417i	$-0.0687851\pi$
$884$	−16576.0	−0.630669
$885$	0	0
$886$	4432.00	0.168054
$887$	− 43684.0i	− 1.65362i	−0.562478	−	0.826812i	$-0.690152\pi$
$887$	− 43684.0i	− 1.65362i	0.562478	−	0.826812i	$-0.309848\pi$
$888$	2970.00i	0.112237i
$889$	−41756.0	−1.57531
$890$	0	0
$891$	891.000	0.0335013
$892$	14896.0i	0.559142i
$893$	30360.0i	1.13769i
$894$	3930.00	0.147023
$895$	0	0
$896$	−37830.0	−1.41050
$897$	17472.0i	0.650360i
$898$	6290.00i	0.233742i
$899$	−720.000	−0.0267112
$900$	0	0
$901$	25752.0	0.952190
$902$	4642.00i	0.171354i
$903$	31824.0i	1.17280i
$904$	32130.0	1.18211
$905$	0	0
$906$	−3594.00	−0.131791
$907$	− 45804.0i	− 1.67684i	−0.545022	−	0.838422i	$-0.683479\pi$
$907$	− 45804.0i	− 1.67684i	0.545022	−	0.838422i	$-0.316521\pi$
$908$	− 20748.0i	− 0.758311i
$909$	−15318.0	−0.558928
$910$	0	0
$911$	−15318.0	−0.557089	−0.278544	−	0.960423i	$-0.589852\pi$
$911$	−15318.0	−0.557089	−0.278544	+	0.960423i	$0.589852\pi$
$912$	− 7380.00i	− 0.267956i
$913$	− 1452.00i	− 0.0526333i
$914$	3054.00	0.110522
$915$	0	0
$916$	17850.0	0.643865
$917$	− 49608.0i	− 1.78648i
$918$	1998.00i	0.0718342i
$919$	−11350.0	−0.407401	−0.203701	−	0.979033i	$-0.565297\pi$
$919$	−11350.0	−0.407401	−0.203701	+	0.979033i	$0.565297\pi$
$920$	0	0
$921$	−3732.00	−0.133522
$922$	12882.0i	0.460137i
$923$	− 24384.0i	− 0.869566i
$924$	6006.00	0.213834
$925$	0	0
$926$	−6148.00	−0.218181
$927$	10188.0i	0.360969i
$928$	14490.0i	0.512562i
$929$	−33030.0	−1.16650	−0.583250	−	0.812292i	$-0.698219\pi$
$929$	−33030.0	−1.16650	−0.583250	+	0.812292i	$0.698219\pi$
$930$	0	0
$931$	−19980.0	−0.703349
$932$	− 21294.0i	− 0.748399i
$933$	− 6246.00i	− 0.219169i
$934$	5124.00	0.179510
$935$	0	0
$936$	−4320.00	−0.150859
$937$	10006.0i	0.348860i	0.984670	+	0.174430i	$0.0558083\pi$
$937$	10006.0i	0.348860i	−0.984670	+	0.174430i	$0.944192\pi$
$938$	− 26936.0i	− 0.937624i
$939$	7134.00	0.247933
$940$	0	0
$941$	2622.00	0.0908340	0.0454170	−	0.998968i	$-0.485538\pi$
$941$	2622.00	0.0908340	0.0454170	+	0.998968i	$0.485538\pi$
$942$	− 6342.00i	− 0.219356i
$943$	− 76804.0i	− 2.65226i
$944$	8200.00	0.282720
$945$	0	0
$946$	−4488.00	−0.154247
$947$	39876.0i	1.36832i	0.729334	+	0.684158i	$0.239830\pi$
$947$	39876.0i	1.36832i	−0.729334	+	0.684158i	$0.760170\pi$
$948$	− 11550.0i	− 0.395703i
$949$	−17344.0	−0.593267
$950$	0	0
$951$	1488.00	0.0507379
$952$	28860.0i	0.982519i
$953$	38918.0i	1.32285i	0.750011	+	0.661426i	$0.230048\pi$
$953$	38918.0i	1.32285i	−0.750011	+	0.661426i	$0.769952\pi$
$954$	3132.00	0.106292
$955$	0	0
$956$	−18900.0	−0.639403
$957$	− 2970.00i	− 0.100320i
$958$	16520.0i	0.557137i
$959$	−56836.0	−1.91380
$960$	0	0
$961$	−29727.0	−0.997852
$962$	2112.00i	0.0707834i
$963$	5076.00i	0.169857i
$964$	−4046.00	−0.135179
$965$	0	0
$966$	14196.0	0.472825
$967$	− 1114.00i	− 0.0370464i	−0.999828	−	0.0185232i	$-0.994104\pi$
$967$	− 1114.00i	− 0.0370464i	0.999828	−	0.0185232i	$-0.00589645\pi$
$968$	1815.00i	0.0602648i
$969$	−13320.0	−0.441589
$970$	0	0
$971$	−1688.00	−0.0557884	−0.0278942	−	0.999611i	$-0.508880\pi$
$971$	−1688.00	−0.0557884	−0.0278942	+	0.999611i	$0.508880\pi$
$972$	− 1701.00i	− 0.0561313i
$973$	− 71240.0i	− 2.34722i
$974$	524.000	0.0172382
$975$	0	0
$976$	5412.00	0.177494
$977$	41826.0i	1.36963i	0.728715	+	0.684817i	$0.240118\pi$
$977$	41826.0i	1.36963i	−0.728715	+	0.684817i	$0.759882\pi$
$978$	11604.0i	0.379402i
$979$	−6270.00	−0.204689
$980$	0	0
$981$	−2880.00	−0.0937322
$982$	− 15028.0i	− 0.488353i
$983$	978.000i	0.0317328i	0.999874	+	0.0158664i	$0.00505065\pi$
$983$	978.000i	0.0317328i	−0.999874	+	0.0158664i	$0.994949\pi$
$984$	18990.0	0.615223
$985$	0	0
$986$	6660.00	0.215109
$987$	39468.0i	1.27283i
$988$	− 13440.0i	− 0.432777i
$989$	74256.0	2.38747
$990$	0	0
$991$	47272.0	1.51528	0.757641	−	0.652671i	$-0.226352\pi$
$991$	47272.0	1.51528	0.757641	+	0.652671i	$0.226352\pi$
$992$	− 1288.00i	− 0.0412238i
$993$	8124.00i	0.259625i
$994$	−19812.0	−0.632192
$995$	0	0
$996$	−2772.00	−0.0881869
$997$	− 51104.0i	− 1.62335i	−0.584109	−	0.811675i	$-0.698556\pi$
$997$	− 51104.0i	− 1.62335i	0.584109	−	0.811675i	$-0.301444\pi$
$998$	− 9020.00i	− 0.286095i
$999$	−1782.00	−0.0564364

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.c.f.199.2		2
5.2	odd	4		33.4.a.b.1.1	✓	1
5.3	odd	4		825.4.a.f.1.1		1
5.4	even	2	inner	825.4.c.f.199.1		2
15.2	even	4		99.4.a.a.1.1		1
15.8	even	4		2475.4.a.e.1.1		1
20.7	even	4		528.4.a.h.1.1		1
35.27	even	4		1617.4.a.d.1.1		1
40.27	even	4		2112.4.a.h.1.1		1
40.37	odd	4		2112.4.a.u.1.1		1
55.32	even	4		363.4.a.d.1.1		1
60.47	odd	4		1584.4.a.l.1.1		1
165.32	odd	4		1089.4.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.a.b.1.1	✓	1	5.2	odd	4
99.4.a.a.1.1		1	15.2	even	4
363.4.a.d.1.1		1	55.32	even	4
528.4.a.h.1.1		1	20.7	even	4
825.4.a.f.1.1		1	5.3	odd	4
825.4.c.f.199.1		2	5.4	even	2	inner
825.4.c.f.199.2		2	1.1	even	1	trivial
1089.4.a.e.1.1		1	165.32	odd	4
1584.4.a.l.1.1		1	60.47	odd	4
1617.4.a.d.1.1		1	35.27	even	4
2112.4.a.h.1.1		1	40.27	even	4
2112.4.a.u.1.1		1	40.37	odd	4
2475.4.a.e.1.1		1	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	825.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -3.00000i q^{3} +7.00000 q^{4} +3.00000 q^{6} +26.0000i q^{7} +15.0000i q^{8} -9.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -3.00000i q^{3} +7.00000 q^{4} +3.00000 q^{6} +26.0000i q^{7} +15.0000i q^{8} -9.00000 q^{9} +11.0000 q^{11} -21.0000i q^{12} -32.0000i q^{13} -26.0000 q^{14} +41.0000 q^{16} -74.0000i q^{17} -9.00000i q^{18} +60.0000 q^{19} +78.0000 q^{21} +11.0000i q^{22} -182.000i q^{23} +45.0000 q^{24} +32.0000 q^{26} +27.0000i q^{27} +182.000i q^{28} +90.0000 q^{29} -8.00000 q^{31} +161.000i q^{32} -33.0000i q^{33} +74.0000 q^{34} -63.0000 q^{36} +66.0000i q^{37} +60.0000i q^{38} -96.0000 q^{39} +422.000 q^{41} +78.0000i q^{42} +408.000i q^{43} +77.0000 q^{44} +182.000 q^{46} +506.000i q^{47} -123.000i q^{48} -333.000 q^{49} -222.000 q^{51} -224.000i q^{52} +348.000i q^{53} -27.0000 q^{54} -390.000 q^{56} -180.000i q^{57} +90.0000i q^{58} +200.000 q^{59} +132.000 q^{61} -8.00000i q^{62} -234.000i q^{63} +167.000 q^{64} +33.0000 q^{66} +1036.00i q^{67} -518.000i q^{68} -546.000 q^{69} +762.000 q^{71} -135.000i q^{72} -542.000i q^{73} -66.0000 q^{74} +420.000 q^{76} +286.000i q^{77} -96.0000i q^{78} +550.000 q^{79} +81.0000 q^{81} +422.000i q^{82} -132.000i q^{83} +546.000 q^{84} -408.000 q^{86} -270.000i q^{87} +165.000i q^{88} -570.000 q^{89} +832.000 q^{91} -1274.00i q^{92} +24.0000i q^{93} -506.000 q^{94} +483.000 q^{96} -14.0000i q^{97} -333.000i q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 14 q^{4} + 6 q^{6} - 18 q^{9}+O(q^{10}) \) 2 * q + 14 * q^4 + 6 * q^6 - 18 * q^9	\( 2 q + 14 q^{4} + 6 q^{6} - 18 q^{9} + 22 q^{11} - 52 q^{14} + 82 q^{16} + 120 q^{19} + 156 q^{21} + 90 q^{24} + 64 q^{26} + 180 q^{29} - 16 q^{31} + 148 q^{34} - 126 q^{36} - 192 q^{39} + 844 q^{41} + 154 q^{44} + 364 q^{46} - 666 q^{49} - 444 q^{51} - 54 q^{54} - 780 q^{56} + 400 q^{59} + 264 q^{61} + 334 q^{64} + 66 q^{66} - 1092 q^{69} + 1524 q^{71} - 132 q^{74} + 840 q^{76} + 1100 q^{79} + 162 q^{81} + 1092 q^{84} - 816 q^{86} - 1140 q^{89} + 1664 q^{91} - 1012 q^{94} + 966 q^{96} - 198 q^{99}+O(q^{100}) \) 2 * q + 14 * q^4 + 6 * q^6 - 18 * q^9 + 22 * q^11 - 52 * q^14 + 82 * q^16 + 120 * q^19 + 156 * q^21 + 90 * q^24 + 64 * q^26 + 180 * q^29 - 16 * q^31 + 148 * q^34 - 126 * q^36 - 192 * q^39 + 844 * q^41 + 154 * q^44 + 364 * q^46 - 666 * q^49 - 444 * q^51 - 54 * q^54 - 780 * q^56 + 400 * q^59 + 264 * q^61 + 334 * q^64 + 66 * q^66 - 1092 * q^69 + 1524 * q^71 - 132 * q^74 + 840 * q^76 + 1100 * q^79 + 162 * q^81 + 1092 * q^84 - 816 * q^86 - 1140 * q^89 + 1664 * q^91 - 1012 * q^94 + 966 * q^96 - 198 * q^99

Introduction

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