LMFDB - Embedded newform 825.2.c.g.199.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,2,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, 2, names="a")

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

chi := DirichletCharacter("825.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
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:	$6.58765816676$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.6
Root			$-0.854638 - 0.854638i$ of defining polynomial
Character	$\chi$	$=$	825.199
Dual form			825.2.c.g.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.70928i q^{2} +1.00000i q^{3} -5.34017 q^{4} -2.70928 q^{6} -1.07838i q^{7} -9.04945i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+2.70928i q^{2} +1.00000i q^{3} -5.34017 q^{4} -2.70928 q^{6} -1.07838i q^{7} -9.04945i q^{8} -1.00000 q^{9} +1.00000 q^{11} -5.34017i q^{12} -4.34017i q^{13} +2.92162 q^{14} +13.8371 q^{16} -7.75872i q^{17} -2.70928i q^{18} -5.26180 q^{19} +1.07838 q^{21} +2.70928i q^{22} -2.15676i q^{23} +9.04945 q^{24} +11.7587 q^{26} -1.00000i q^{27} +5.75872i q^{28} -1.41855 q^{29} -4.68035 q^{31} +19.3896i q^{32} +1.00000i q^{33} +21.0205 q^{34} +5.34017 q^{36} +2.00000i q^{37} -14.2557i q^{38} +4.34017 q^{39} -9.41855 q^{41} +2.92162i q^{42} +7.60197i q^{43} -5.34017 q^{44} +5.84324 q^{46} -4.68035i q^{47} +13.8371i q^{48} +5.83710 q^{49} +7.75872 q^{51} +23.1773i q^{52} +0.156755i q^{53} +2.70928 q^{54} -9.75872 q^{56} -5.26180i q^{57} -3.84324i q^{58} -6.15676 q^{59} -4.15676 q^{61} -12.6803i q^{62} +1.07838i q^{63} -24.8576 q^{64} -2.70928 q^{66} +8.68035i q^{67} +41.4329i q^{68} +2.15676 q^{69} -4.68035 q^{71} +9.04945i q^{72} -10.4969i q^{73} -5.41855 q^{74} +28.0989 q^{76} -1.07838i q^{77} +11.7587i q^{78} +8.09890 q^{79} +1.00000 q^{81} -25.5174i q^{82} -11.0205i q^{83} -5.75872 q^{84} -20.5958 q^{86} -1.41855i q^{87} -9.04945i q^{88} +12.8371 q^{89} -4.68035 q^{91} +11.5174i q^{92} -4.68035i q^{93} +12.6803 q^{94} -19.3896 q^{96} -14.6803i q^{97} +15.8143i q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 10 q^{4} - 2 q^{6} - 6 q^{9}+O(q^{10}) $ 6 * q - 10 * q^4 - 2 * q^6 - 6 * q^9	$ 6 q - 10 q^{4} - 2 q^{6} - 6 q^{9} + 6 q^{11} + 24 q^{14} + 26 q^{16} - 16 q^{19} + 18 q^{24} + 20 q^{26} + 20 q^{29} + 16 q^{31} + 60 q^{34} + 10 q^{36} + 4 q^{39} - 28 q^{41} - 10 q^{44} + 48 q^{46} - 22 q^{49} - 4 q^{51} + 2 q^{54} - 8 q^{56} - 24 q^{59} - 12 q^{61} - 26 q^{64} - 2 q^{66} + 16 q^{71} - 4 q^{74} + 96 q^{76} - 24 q^{79} + 6 q^{81} + 16 q^{84} - 16 q^{86} + 20 q^{89} + 16 q^{91} + 32 q^{94} - 58 q^{96} - 6 q^{99}+O(q^{100}) $ 6 * q - 10 * q^4 - 2 * q^6 - 6 * q^9 + 6 * q^11 + 24 * q^14 + 26 * q^16 - 16 * q^19 + 18 * q^24 + 20 * q^26 + 20 * q^29 + 16 * q^31 + 60 * q^34 + 10 * q^36 + 4 * q^39 - 28 * q^41 - 10 * q^44 + 48 * q^46 - 22 * q^49 - 4 * q^51 + 2 * q^54 - 8 * q^56 - 24 * q^59 - 12 * q^61 - 26 * q^64 - 2 * q^66 + 16 * q^71 - 4 * q^74 + 96 * q^76 - 24 * q^79 + 6 * q^81 + 16 * q^84 - 16 * q^86 + 20 * q^89 + 16 * q^91 + 32 * q^94 - 58 * q^96 - 6 * 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 $

$2$	2.70928i	1.91575i	0.287190	+	0.957873i	$0.407279\pi$
$2$	2.70928i	1.91575i	−0.287190	+	0.957873i	$0.592721\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−5.34017	−2.67009
$5$	0	0
$5$	0	0
$6$	−2.70928	−1.10606
$7$	− 1.07838i	− 0.407588i	−0.979014	−	0.203794i	$-0.934673\pi$
$7$	− 1.07838i	− 0.407588i	0.979014	−	0.203794i	$-0.0653274\pi$
$8$	− 9.04945i	− 3.19946i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	− 5.34017i	− 1.54158i
$13$	− 4.34017i	− 1.20375i	−0.798591	−	0.601874i	$-0.794421\pi$
$13$	− 4.34017i	− 1.20375i	0.798591	−	0.601874i	$-0.205579\pi$
$14$	2.92162	0.780836
$15$	0	0
$16$	13.8371	3.45928
$17$	− 7.75872i	− 1.88177i	−0.338730	−	0.940883i	$-0.609997\pi$
$17$	− 7.75872i	− 1.88177i	0.338730	−	0.940883i	$-0.390003\pi$
$18$	− 2.70928i	− 0.638582i
$19$	−5.26180	−1.20714	−0.603569	−	0.797311i	$-0.706255\pi$
$19$	−5.26180	−1.20714	−0.603569	+	0.797311i	$0.706255\pi$
$20$	0	0
$21$	1.07838	0.235321
$22$	2.70928i	0.577619i
$23$	− 2.15676i	− 0.449715i	−0.974392	−	0.224857i	$-0.927808\pi$
$23$	− 2.15676i	− 0.449715i	0.974392	−	0.224857i	$-0.0721916\pi$
$24$	9.04945	1.84721
$25$	0	0
$26$	11.7587	2.30608
$27$	− 1.00000i	− 0.192450i
$28$	5.75872i	1.08830i
$29$	−1.41855	−0.263418	−0.131709	−	0.991288i	$-0.542046\pi$
$29$	−1.41855	−0.263418	−0.131709	+	0.991288i	$0.542046\pi$
$30$	0	0
$31$	−4.68035	−0.840615	−0.420307	−	0.907382i	$-0.638078\pi$
$31$	−4.68035	−0.840615	−0.420307	+	0.907382i	$0.638078\pi$
$32$	19.3896i	3.42763i
$33$	1.00000i	0.174078i
$34$	21.0205	3.60499
$35$	0	0
$36$	5.34017	0.890029
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	− 14.2557i	− 2.31257i
$39$	4.34017	0.694984
$40$	0	0
$41$	−9.41855	−1.47093	−0.735465	−	0.677562i	$-0.763036\pi$
$41$	−9.41855	−1.47093	−0.735465	+	0.677562i	$0.763036\pi$
$42$	2.92162i	0.450816i
$43$	7.60197i	1.15929i	0.814869	+	0.579645i	$0.196809\pi$
$43$	7.60197i	1.15929i	−0.814869	+	0.579645i	$0.803191\pi$
$44$	−5.34017	−0.805061
$45$	0	0
$46$	5.84324	0.861539
$47$	− 4.68035i	− 0.682699i	−0.939937	−	0.341349i	$-0.889116\pi$
$47$	− 4.68035i	− 0.682699i	0.939937	−	0.341349i	$-0.110884\pi$
$48$	13.8371i	1.99721i
$49$	5.83710	0.833872
$50$	0	0
$51$	7.75872	1.08644
$52$	23.1773i	3.21411i
$53$	0.156755i	0.0215320i	0.999942	+	0.0107660i	$0.00342699\pi$
$53$	0.156755i	0.0215320i	−0.999942	+	0.0107660i	$0.996573\pi$
$54$	2.70928	0.368686
$55$	0	0
$56$	−9.75872	−1.30406
$57$	− 5.26180i	− 0.696942i
$58$	− 3.84324i	− 0.504643i
$59$	−6.15676	−0.801541	−0.400771	−	0.916178i	$-0.631258\pi$
$59$	−6.15676	−0.801541	−0.400771	+	0.916178i	$0.631258\pi$
$60$	0	0
$61$	−4.15676	−0.532218	−0.266109	−	0.963943i	$-0.585738\pi$
$61$	−4.15676	−0.532218	−0.266109	+	0.963943i	$0.585738\pi$
$62$	− 12.6803i	− 1.61041i
$63$	1.07838i	0.135863i
$64$	−24.8576	−3.10720
$65$	0	0
$66$	−2.70928	−0.333489
$67$	8.68035i	1.06047i	0.847850	+	0.530237i	$0.177897\pi$
$67$	8.68035i	1.06047i	−0.847850	+	0.530237i	$0.822103\pi$
$68$	41.4329i	5.02448i
$69$	2.15676	0.259643
$70$	0	0
$71$	−4.68035	−0.555455	−0.277727	−	0.960660i	$-0.589581\pi$
$71$	−4.68035	−0.555455	−0.277727	+	0.960660i	$0.589581\pi$
$72$	9.04945i	1.06649i
$73$	− 10.4969i	− 1.22857i	−0.789083	−	0.614286i	$-0.789444\pi$
$73$	− 10.4969i	− 1.22857i	0.789083	−	0.614286i	$-0.210556\pi$
$74$	−5.41855	−0.629894
$75$	0	0
$76$	28.0989	3.22316
$77$	− 1.07838i	− 0.122893i
$78$	11.7587i	1.33141i
$79$	8.09890	0.911197	0.455599	−	0.890185i	$-0.349425\pi$
$79$	8.09890	0.911197	0.455599	+	0.890185i	$0.349425\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 25.5174i	− 2.81793i
$83$	− 11.0205i	− 1.20966i	−0.796355	−	0.604830i	$-0.793241\pi$
$83$	− 11.0205i	− 1.20966i	0.796355	−	0.604830i	$-0.206759\pi$
$84$	−5.75872	−0.628328
$85$	0	0
$86$	−20.5958	−2.22090
$87$	− 1.41855i	− 0.152085i
$88$	− 9.04945i	− 0.964674i
$89$	12.8371	1.36073	0.680365	−	0.732873i	$-0.261821\pi$
$89$	12.8371	1.36073	0.680365	+	0.732873i	$0.261821\pi$
$90$	0	0
$91$	−4.68035	−0.490634
$92$	11.5174i	1.20078i
$93$	− 4.68035i	− 0.485329i
$94$	12.6803	1.30788
$95$	0	0
$96$	−19.3896	−1.97894
$97$	− 14.6803i	− 1.49056i	−0.666750	−	0.745282i	$-0.732315\pi$
$97$	− 14.6803i	− 1.49056i	0.666750	−	0.745282i	$-0.267685\pi$
$98$	15.8143i	1.59749i
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−15.5753	−1.54980	−0.774900	−	0.632083i	$-0.782200\pi$
$101$	−15.5753	−1.54980	−0.774900	+	0.632083i	$0.782200\pi$
$102$	21.0205i	2.08134i
$103$	6.83710i	0.673680i	0.941562	+	0.336840i	$0.109358\pi$
$103$	6.83710i	0.673680i	−0.941562	+	0.336840i	$0.890642\pi$
$104$	−39.2762	−3.85135
$105$	0	0
$106$	−0.424694	−0.0412499
$107$	− 6.34017i	− 0.612928i	−0.951882	−	0.306464i	$-0.900854\pi$
$107$	− 6.34017i	− 0.612928i	0.951882	−	0.306464i	$-0.0991458\pi$
$108$	5.34017i	0.513858i
$109$	−2.31351	−0.221594	−0.110797	−	0.993843i	$-0.535340\pi$
$109$	−2.31351	−0.221594	−0.110797	+	0.993843i	$0.535340\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	− 14.9216i	− 1.40996i
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	14.2557	1.33516
$115$	0	0
$116$	7.57531	0.703350
$117$	4.34017i	0.401249i
$118$	− 16.6803i	− 1.53555i
$119$	−8.36683	−0.766987
$120$	0	0
$121$	1.00000	0.0909091
$122$	− 11.2618i	− 1.01960i
$123$	− 9.41855i	− 0.849242i
$124$	24.9939	2.24451
$125$	0	0
$126$	−2.92162	−0.260279
$127$	2.24128i	0.198881i	0.995044	+	0.0994406i	$0.0317053\pi$
$127$	2.24128i	0.198881i	−0.995044	+	0.0994406i	$0.968295\pi$
$128$	− 28.5669i	− 2.52498i
$129$	−7.60197	−0.669316
$130$	0	0
$131$	8.68035	0.758405	0.379203	−	0.925314i	$-0.376198\pi$
$131$	8.68035	0.758405	0.379203	+	0.925314i	$0.376198\pi$
$132$	− 5.34017i	− 0.464802i
$133$	5.67420i	0.492016i
$134$	−23.5174	−2.03160
$135$	0	0
$136$	−70.2122	−6.02064
$137$	15.3607i	1.31235i	0.754608	+	0.656176i	$0.227827\pi$
$137$	15.3607i	1.31235i	−0.754608	+	0.656176i	$0.772173\pi$
$138$	5.84324i	0.497410i
$139$	−8.58145	−0.727869	−0.363935	−	0.931425i	$-0.618567\pi$
$139$	−8.58145	−0.727869	−0.363935	+	0.931425i	$0.618567\pi$
$140$	0	0
$141$	4.68035	0.394156
$142$	− 12.6803i	− 1.06411i
$143$	− 4.34017i	− 0.362943i
$144$	−13.8371	−1.15309
$145$	0	0
$146$	28.4391	2.35363
$147$	5.83710i	0.481436i
$148$	− 10.6803i	− 0.877919i
$149$	18.0989	1.48272	0.741360	−	0.671108i	$-0.234181\pi$
$149$	18.0989	1.48272	0.741360	+	0.671108i	$0.234181\pi$
$150$	0	0
$151$	22.9360	1.86651	0.933253	−	0.359221i	$-0.116958\pi$
$151$	22.9360	1.86651	0.933253	+	0.359221i	$0.116958\pi$
$152$	47.6163i	3.86220i
$153$	7.75872i	0.627256i
$154$	2.92162	0.235431
$155$	0	0
$156$	−23.1773	−1.85567
$157$	10.9939i	0.877405i	0.898632	+	0.438703i	$0.144562\pi$
$157$	10.9939i	0.877405i	−0.898632	+	0.438703i	$0.855438\pi$
$158$	21.9421i	1.74562i
$159$	−0.156755	−0.0124315
$160$	0	0
$161$	−2.32580	−0.183298
$162$	2.70928i	0.212861i
$163$	− 6.52359i	− 0.510967i	−0.966813	−	0.255484i	$-0.917765\pi$
$163$	− 6.52359i	− 0.510967i	0.966813	−	0.255484i	$-0.0822347\pi$
$164$	50.2967	3.92751
$165$	0	0
$166$	29.8576	2.31740
$167$	− 1.97334i	− 0.152701i	−0.997081	−	0.0763507i	$-0.975673\pi$
$167$	− 1.97334i	− 0.152701i	0.997081	−	0.0763507i	$-0.0243269\pi$
$168$	− 9.75872i	− 0.752902i
$169$	−5.83710	−0.449008
$170$	0	0
$171$	5.26180	0.402380
$172$	− 40.5958i	− 3.09540i
$173$	3.75872i	0.285770i	0.989739	+	0.142885i	$0.0456380\pi$
$173$	3.75872i	0.285770i	−0.989739	+	0.142885i	$0.954362\pi$
$174$	3.84324	0.291356
$175$	0	0
$176$	13.8371	1.04301
$177$	− 6.15676i	− 0.462770i
$178$	34.7792i	2.60681i
$179$	−15.1506	−1.13241	−0.566205	−	0.824264i	$-0.691589\pi$
$179$	−15.1506	−1.13241	−0.566205	+	0.824264i	$0.691589\pi$
$180$	0	0
$181$	4.83710	0.359539	0.179769	−	0.983709i	$-0.442465\pi$
$181$	4.83710	0.359539	0.179769	+	0.983709i	$0.442465\pi$
$182$	− 12.6803i	− 0.939930i
$183$	− 4.15676i	− 0.307276i
$184$	−19.5174	−1.43885
$185$	0	0
$186$	12.6803	0.929768
$187$	− 7.75872i	− 0.567374i
$188$	24.9939i	1.82286i
$189$	−1.07838	−0.0784404
$190$	0	0
$191$	2.52359	0.182601	0.0913003	−	0.995823i	$-0.470898\pi$
$191$	2.52359	0.182601	0.0913003	+	0.995823i	$0.470898\pi$
$192$	− 24.8576i	− 1.79394i
$193$	0.0266620i	0.00191917i	1.00000		0.000959586i	$0.000305446\pi$
$193$	0.0266620i	0.00191917i	−1.00000		0.000959586i	$0.999695\pi$
$194$	39.7731	2.85554
$195$	0	0
$196$	−31.1711	−2.22651
$197$	− 21.1194i	− 1.50470i	−0.658766	−	0.752348i	$-0.728921\pi$
$197$	− 21.1194i	− 1.50470i	0.658766	−	0.752348i	$-0.271079\pi$
$198$	− 2.70928i	− 0.192540i
$199$	−10.5236	−0.745998	−0.372999	−	0.927832i	$-0.621670\pi$
$199$	−10.5236	−0.745998	−0.372999	+	0.927832i	$0.621670\pi$
$200$	0	0
$201$	−8.68035	−0.612264
$202$	− 42.1978i	− 2.96903i
$203$	1.52973i	0.107366i
$204$	−41.4329	−2.90089
$205$	0	0
$206$	−18.5236	−1.29060
$207$	2.15676i	0.149905i
$208$	− 60.0554i	− 4.16409i
$209$	−5.26180	−0.363966
$210$	0	0
$211$	9.57531	0.659191	0.329596	−	0.944122i	$-0.393088\pi$
$211$	9.57531	0.659191	0.329596	+	0.944122i	$0.393088\pi$
$212$	− 0.837101i	− 0.0574924i
$213$	− 4.68035i	− 0.320692i
$214$	17.1773	1.17421
$215$	0	0
$216$	−9.04945	−0.615737
$217$	5.04718i	0.342625i
$218$	− 6.26794i	− 0.424518i
$219$	10.4969	0.709317
$220$	0	0
$221$	−33.6742	−2.26517
$222$	− 5.41855i	− 0.363669i
$223$	− 2.15676i	− 0.144427i	−0.997389	−	0.0722135i	$-0.976994\pi$
$223$	− 2.15676i	− 0.144427i	0.997389	−	0.0722135i	$-0.0230063\pi$
$224$	20.9093	1.39706
$225$	0	0
$226$	16.2557	1.08131
$227$	− 9.65983i	− 0.641145i	−0.947224	−	0.320573i	$-0.896125\pi$
$227$	− 9.65983i	− 0.641145i	0.947224	−	0.320573i	$-0.103875\pi$
$228$	28.0989i	1.86089i
$229$	3.36069	0.222081	0.111040	−	0.993816i	$-0.464582\pi$
$229$	3.36069	0.222081	0.111040	+	0.993816i	$0.464582\pi$
$230$	0	0
$231$	1.07838	0.0709520
$232$	12.8371i	0.842797i
$233$	− 2.39803i	− 0.157100i	−0.996910	−	0.0785501i	$-0.974971\pi$
$233$	− 2.39803i	− 0.157100i	0.996910	−	0.0785501i	$-0.0250291\pi$
$234$	−11.7587	−0.768692
$235$	0	0
$236$	32.8781	2.14018
$237$	8.09890i	0.526080i
$238$	− 22.6681i	− 1.46935i
$239$	7.20394	0.465984	0.232992	−	0.972479i	$-0.425148\pi$
$239$	7.20394	0.465984	0.232992	+	0.972479i	$0.425148\pi$
$240$	0	0
$241$	−5.20394	−0.335215	−0.167608	−	0.985854i	$-0.553604\pi$
$241$	−5.20394	−0.335215	−0.167608	+	0.985854i	$0.553604\pi$
$242$	2.70928i	0.174159i
$243$	1.00000i	0.0641500i
$244$	22.1978	1.42107
$245$	0	0
$246$	25.5174	1.62693
$247$	22.8371i	1.45309i
$248$	42.3545i	2.68952i
$249$	11.0205	0.698397
$250$	0	0
$251$	15.3197	0.966968	0.483484	−	0.875353i	$-0.339371\pi$
$251$	15.3197	0.966968	0.483484	+	0.875353i	$0.339371\pi$
$252$	− 5.75872i	− 0.362765i
$253$	− 2.15676i	− 0.135594i
$254$	−6.07223	−0.381006
$255$	0	0
$256$	27.6803	1.73002
$257$	− 4.15676i	− 0.259291i	−0.991560	−	0.129646i	$-0.958616\pi$
$257$	− 4.15676i	− 0.259291i	0.991560	−	0.129646i	$-0.0413840\pi$
$258$	− 20.5958i	− 1.28224i
$259$	2.15676	0.134014
$260$	0	0
$261$	1.41855	0.0878061
$262$	23.5174i	1.45291i
$263$	− 18.7070i	− 1.15352i	−0.816912	−	0.576762i	$-0.804316\pi$
$263$	− 18.7070i	− 1.15352i	0.816912	−	0.576762i	$-0.195684\pi$
$264$	9.04945	0.556955
$265$	0	0
$266$	−15.3730	−0.942578
$267$	12.8371i	0.785618i
$268$	− 46.3545i	− 2.83155i
$269$	−23.3607	−1.42433	−0.712163	−	0.702014i	$-0.752284\pi$
$269$	−23.3607	−1.42433	−0.712163	+	0.702014i	$0.752284\pi$
$270$	0	0
$271$	−5.57531	−0.338676	−0.169338	−	0.985558i	$-0.554163\pi$
$271$	−5.57531	−0.338676	−0.169338	+	0.985558i	$0.554163\pi$
$272$	− 107.358i	− 6.50955i
$273$	− 4.68035i	− 0.283267i
$274$	−41.6163	−2.51414
$275$	0	0
$276$	−11.5174	−0.693269
$277$	26.0144i	1.56305i	0.623872	+	0.781526i	$0.285558\pi$
$277$	26.0144i	1.56305i	−0.623872	+	0.781526i	$0.714442\pi$
$278$	− 23.2495i	− 1.39441i
$279$	4.68035	0.280205
$280$	0	0
$281$	−9.41855	−0.561864	−0.280932	−	0.959728i	$-0.590643\pi$
$281$	−9.41855	−0.561864	−0.280932	+	0.959728i	$0.590643\pi$
$282$	12.6803i	0.755104i
$283$	14.2413i	0.846556i	0.906000	+	0.423278i	$0.139121\pi$
$283$	14.2413i	0.846556i	−0.906000	+	0.423278i	$0.860879\pi$
$284$	24.9939	1.48311
$285$	0	0
$286$	11.7587	0.695308
$287$	10.1568i	0.599534i
$288$	− 19.3896i	− 1.14254i
$289$	−43.1978	−2.54105
$290$	0	0
$291$	14.6803	0.860577
$292$	56.0554i	3.28039i
$293$	− 15.7587i	− 0.920634i	−0.887754	−	0.460317i	$-0.847736\pi$
$293$	− 15.7587i	− 0.920634i	0.887754	−	0.460317i	$-0.152264\pi$
$294$	−15.8143	−0.922310
$295$	0	0
$296$	18.0989	1.05198
$297$	− 1.00000i	− 0.0580259i
$298$	49.0349i	2.84052i
$299$	−9.36069	−0.541343
$300$	0	0
$301$	8.19779	0.472513
$302$	62.1399i	3.57575i
$303$	− 15.5753i	− 0.894778i
$304$	−72.8080	−4.17582
$305$	0	0
$306$	−21.0205	−1.20166
$307$	18.9216i	1.07991i	0.841693	+	0.539957i	$0.181560\pi$
$307$	18.9216i	1.07991i	−0.841693	+	0.539957i	$0.818440\pi$
$308$	5.75872i	0.328134i
$309$	−6.83710	−0.388949
$310$	0	0
$311$	−20.8781	−1.18389	−0.591945	−	0.805978i	$-0.701640\pi$
$311$	−20.8781	−1.18389	−0.591945	+	0.805978i	$0.701640\pi$
$312$	− 39.2762i	− 2.22358i
$313$	6.31351i	0.356861i	0.983953	+	0.178430i	$0.0571019\pi$
$313$	6.31351i	0.356861i	−0.983953	+	0.178430i	$0.942898\pi$
$314$	−29.7854	−1.68089
$315$	0	0
$316$	−43.2495	−2.43297
$317$	− 31.3607i	− 1.76139i	−0.473682	−	0.880696i	$-0.657075\pi$
$317$	− 31.3607i	− 1.76139i	0.473682	−	0.880696i	$-0.342925\pi$
$318$	− 0.424694i	− 0.0238156i
$319$	−1.41855	−0.0794236
$320$	0	0
$321$	6.34017	0.353874
$322$	− 6.30122i	− 0.351154i
$323$	40.8248i	2.27155i
$324$	−5.34017	−0.296676
$325$	0	0
$326$	17.6742	0.978884
$327$	− 2.31351i	− 0.127937i
$328$	85.2327i	4.70619i
$329$	−5.04718	−0.278260
$330$	0	0
$331$	19.2039	1.05554	0.527772	−	0.849386i	$-0.323028\pi$
$331$	19.2039	1.05554	0.527772	+	0.849386i	$0.323028\pi$
$332$	58.8515i	3.22989i
$333$	− 2.00000i	− 0.109599i
$334$	5.34632	0.292537
$335$	0	0
$336$	14.9216	0.814041
$337$	− 13.5031i	− 0.735559i	−0.929913	−	0.367780i	$-0.880118\pi$
$337$	− 13.5031i	− 0.735559i	0.929913	−	0.367780i	$-0.119882\pi$
$338$	− 15.8143i	− 0.860185i
$339$	6.00000	0.325875
$340$	0	0
$341$	−4.68035	−0.253455
$342$	14.2557i	0.770857i
$343$	− 13.8432i	− 0.747465i
$344$	68.7936	3.70910
$345$	0	0
$346$	−10.1834	−0.547464
$347$	− 6.34017i	− 0.340358i	−0.985413	−	0.170179i	$-0.945565\pi$
$347$	− 6.34017i	− 0.340358i	0.985413	−	0.170179i	$-0.0544346\pi$
$348$	7.57531i	0.406079i
$349$	−16.1568	−0.864851	−0.432426	−	0.901670i	$-0.642342\pi$
$349$	−16.1568	−0.864851	−0.432426	+	0.901670i	$0.642342\pi$
$350$	0	0
$351$	−4.34017	−0.231661
$352$	19.3896i	1.03347i
$353$	− 13.2039i	− 0.702775i	−0.936230	−	0.351387i	$-0.885710\pi$
$353$	− 13.2039i	− 0.702775i	0.936230	−	0.351387i	$-0.114290\pi$
$354$	16.6803	0.886550
$355$	0	0
$356$	−68.5523	−3.63327
$357$	− 8.36683i	− 0.442820i
$358$	− 41.0472i	− 2.16941i
$359$	−3.31965	−0.175205	−0.0876023	−	0.996156i	$-0.527920\pi$
$359$	−3.31965	−0.175205	−0.0876023	+	0.996156i	$0.527920\pi$
$360$	0	0
$361$	8.68649	0.457184
$362$	13.1050i	0.688786i
$363$	1.00000i	0.0524864i
$364$	24.9939	1.31003
$365$	0	0
$366$	11.2618	0.588663
$367$	36.1445i	1.88673i	0.331762	+	0.943363i	$0.392357\pi$
$367$	36.1445i	1.88673i	−0.331762	+	0.943363i	$0.607643\pi$
$368$	− 29.8432i	− 1.55569i
$369$	9.41855	0.490310
$370$	0	0
$371$	0.169042	0.00877620
$372$	24.9939i	1.29587i
$373$	− 2.81044i	− 0.145519i	−0.997350	−	0.0727595i	$-0.976819\pi$
$373$	− 2.81044i	− 0.145519i	0.997350	−	0.0727595i	$-0.0231806\pi$
$374$	21.0205	1.08695
$375$	0	0
$376$	−42.3545	−2.18427
$377$	6.15676i	0.317089i
$378$	− 2.92162i	− 0.150272i
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	−2.24128	−0.114824
$382$	6.83710i	0.349817i
$383$	33.5585i	1.71476i	0.514685	+	0.857379i	$0.327909\pi$
$383$	33.5585i	1.71476i	−0.514685	+	0.857379i	$0.672091\pi$
$384$	28.5669	1.45780
$385$	0	0
$386$	−0.0722347	−0.00367665
$387$	− 7.60197i	− 0.386430i
$388$	78.3956i	3.97993i
$389$	−12.8371	−0.650867	−0.325433	−	0.945565i	$-0.605510\pi$
$389$	−12.8371	−0.650867	−0.325433	+	0.945565i	$0.605510\pi$
$390$	0	0
$391$	−16.7337	−0.846258
$392$	− 52.8225i	− 2.66794i
$393$	8.68035i	0.437866i
$394$	57.2183	2.88262
$395$	0	0
$396$	5.34017	0.268354
$397$	5.31965i	0.266986i	0.991050	+	0.133493i	$0.0426193\pi$
$397$	5.31965i	0.266986i	−0.991050	+	0.133493i	$0.957381\pi$
$398$	− 28.5113i	− 1.42914i
$399$	−5.67420	−0.284065
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	− 23.5174i	− 1.17294i
$403$	20.3135i	1.01189i
$404$	83.1748	4.13810
$405$	0	0
$406$	−4.14447	−0.205687
$407$	2.00000i	0.0991363i
$408$	− 70.2122i	− 3.47602i
$409$	−26.1978	−1.29540	−0.647699	−	0.761897i	$-0.724268\pi$
$409$	−26.1978	−1.29540	−0.647699	+	0.761897i	$0.724268\pi$
$410$	0	0
$411$	−15.3607	−0.757687
$412$	− 36.5113i	− 1.79878i
$413$	6.63931i	0.326699i
$414$	−5.84324	−0.287180
$415$	0	0
$416$	84.1543	4.12600
$417$	− 8.58145i	− 0.420235i
$418$	− 14.2557i	− 0.697267i
$419$	2.83710	0.138601	0.0693007	−	0.997596i	$-0.477923\pi$
$419$	2.83710	0.138601	0.0693007	+	0.997596i	$0.477923\pi$
$420$	0	0
$421$	11.4764	0.559326	0.279663	−	0.960098i	$-0.409777\pi$
$421$	11.4764	0.559326	0.279663	+	0.960098i	$0.409777\pi$
$422$	25.9421i	1.26284i
$423$	4.68035i	0.227566i
$424$	1.41855	0.0688909
$425$	0	0
$426$	12.6803	0.614365
$427$	4.48255i	0.216926i
$428$	33.8576i	1.63657i
$429$	4.34017	0.209546
$430$	0	0
$431$	23.5708	1.13536	0.567682	−	0.823248i	$-0.307840\pi$
$431$	23.5708	1.13536	0.567682	+	0.823248i	$0.307840\pi$
$432$	− 13.8371i	− 0.665738i
$433$	− 14.9939i	− 0.720559i	−0.932844	−	0.360279i	$-0.882681\pi$
$433$	− 14.9939i	− 0.720559i	0.932844	−	0.360279i	$-0.117319\pi$
$434$	−13.6742	−0.656383
$435$	0	0
$436$	12.3545	0.591676
$437$	11.3484i	0.542868i
$438$	28.4391i	1.35887i
$439$	−4.77924	−0.228101	−0.114050	−	0.993475i	$-0.536383\pi$
$439$	−4.77924	−0.228101	−0.114050	+	0.993475i	$0.536383\pi$
$440$	0	0
$441$	−5.83710	−0.277957
$442$	− 91.2327i	− 4.33950i
$443$	20.1978i	0.959626i	0.877371	+	0.479813i	$0.159296\pi$
$443$	20.1978i	0.959626i	−0.877371	+	0.479813i	$0.840704\pi$
$444$	10.6803	0.506867
$445$	0	0
$446$	5.84324	0.276686
$447$	18.0989i	0.856048i
$448$	26.8059i	1.26646i
$449$	21.5708	1.01799	0.508994	−	0.860770i	$-0.330018\pi$
$449$	21.5708	1.01799	0.508994	+	0.860770i	$0.330018\pi$
$450$	0	0
$451$	−9.41855	−0.443502
$452$	32.0410i	1.50708i
$453$	22.9360i	1.07763i
$454$	26.1711	1.22827
$455$	0	0
$456$	−47.6163	−2.22984
$457$	− 28.1711i	− 1.31779i	−0.752235	−	0.658895i	$-0.771024\pi$
$457$	− 28.1711i	− 1.31779i	0.752235	−	0.658895i	$-0.228976\pi$
$458$	9.10504i	0.425451i
$459$	−7.75872	−0.362146
$460$	0	0
$461$	−1.47187	−0.0685520	−0.0342760	−	0.999412i	$-0.510913\pi$
$461$	−1.47187	−0.0685520	−0.0342760	+	0.999412i	$0.510913\pi$
$462$	2.92162i	0.135926i
$463$	− 23.2039i	− 1.07838i	−0.842185	−	0.539189i	$-0.818731\pi$
$463$	− 23.2039i	− 1.07838i	0.842185	−	0.539189i	$-0.181269\pi$
$464$	−19.6286	−0.911236
$465$	0	0
$466$	6.49693	0.300964
$467$	− 14.1568i	− 0.655097i	−0.944834	−	0.327548i	$-0.893778\pi$
$467$	− 14.1568i	− 0.655097i	0.944834	−	0.327548i	$-0.106222\pi$
$468$	− 23.1773i	− 1.07137i
$469$	9.36069	0.432237
$470$	0	0
$471$	−10.9939	−0.506570
$472$	55.7152i	2.56450i
$473$	7.60197i	0.349539i
$474$	−21.9421	−1.00784
$475$	0	0
$476$	44.6803	2.04792
$477$	− 0.156755i	− 0.00717734i
$478$	19.5174i	0.892707i
$479$	13.8432	0.632514	0.316257	−	0.948674i	$-0.397574\pi$
$479$	13.8432	0.632514	0.316257	+	0.948674i	$0.397574\pi$
$480$	0	0
$481$	8.68035	0.395790
$482$	− 14.0989i	− 0.642187i
$483$	− 2.32580i	− 0.105827i
$484$	−5.34017	−0.242735
$485$	0	0
$486$	−2.70928	−0.122895
$487$	40.9939i	1.85761i	0.370570	+	0.928804i	$0.379162\pi$
$487$	40.9939i	1.85761i	−0.370570	+	0.928804i	$0.620838\pi$
$488$	37.6163i	1.70281i
$489$	6.52359	0.295007
$490$	0	0
$491$	34.8371	1.57218	0.786088	−	0.618114i	$-0.212103\pi$
$491$	34.8371	1.57218	0.786088	+	0.618114i	$0.212103\pi$
$492$	50.2967i	2.26755i
$493$	11.0061i	0.495692i
$494$	−61.8720	−2.78375
$495$	0	0
$496$	−64.7624	−2.90792
$497$	5.04718i	0.226397i
$498$	29.8576i	1.33795i
$499$	−15.1506	−0.678235	−0.339117	−	0.940744i	$-0.610128\pi$
$499$	−15.1506	−0.678235	−0.339117	+	0.940744i	$0.610128\pi$
$500$	0	0
$501$	1.97334	0.0881622
$502$	41.5052i	1.85247i
$503$	− 6.65368i	− 0.296673i	−0.988937	−	0.148337i	$-0.952608\pi$
$503$	− 6.65368i	− 0.296673i	0.988937	−	0.148337i	$-0.0473919\pi$
$504$	9.75872	0.434688
$505$	0	0
$506$	5.84324	0.259764
$507$	− 5.83710i	− 0.259235i
$508$	− 11.9688i	− 0.531030i
$509$	−41.3484	−1.83274	−0.916368	−	0.400337i	$-0.868893\pi$
$509$	−41.3484	−1.83274	−0.916368	+	0.400337i	$0.868893\pi$
$510$	0	0
$511$	−11.3197	−0.500752
$512$	17.8599i	0.789303i
$513$	5.26180i	0.232314i
$514$	11.2618	0.496736
$515$	0	0
$516$	40.5958	1.78713
$517$	− 4.68035i	− 0.205841i
$518$	5.84324i	0.256737i
$519$	−3.75872	−0.164990
$520$	0	0
$521$	7.67420	0.336213	0.168106	−	0.985769i	$-0.446235\pi$
$521$	7.67420	0.336213	0.168106	+	0.985769i	$0.446235\pi$
$522$	3.84324i	0.168214i
$523$	23.2351i	1.01600i	0.861357	+	0.508001i	$0.169615\pi$
$523$	23.2351i	1.01600i	−0.861357	+	0.508001i	$0.830385\pi$
$524$	−46.3545	−2.02501
$525$	0	0
$526$	50.6824	2.20986
$527$	36.3135i	1.58184i
$528$	13.8371i	0.602183i
$529$	18.3484	0.797757
$530$	0	0
$531$	6.15676	0.267180
$532$	− 30.3012i	− 1.31372i
$533$	40.8781i	1.77063i
$534$	−34.7792	−1.50505
$535$	0	0
$536$	78.5523	3.39294
$537$	− 15.1506i	− 0.653797i
$538$	− 63.2905i	− 2.72865i
$539$	5.83710	0.251422
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	− 15.1050i	− 0.648817i
$543$	4.83710i	0.207580i
$544$	150.439	6.45001
$545$	0	0
$546$	12.6803	0.542669
$547$	23.0661i	0.986235i	0.869963	+	0.493117i	$0.164143\pi$
$547$	23.0661i	0.986235i	−0.869963	+	0.493117i	$0.835857\pi$
$548$	− 82.0288i	− 3.50409i
$549$	4.15676	0.177406
$550$	0	0
$551$	7.46412	0.317982
$552$	− 19.5174i	− 0.830718i
$553$	− 8.73367i	− 0.371393i
$554$	−70.4801	−2.99441
$555$	0	0
$556$	45.8264	1.94347
$557$	− 10.5958i	− 0.448960i	−0.974479	−	0.224480i	$-0.927932\pi$
$557$	− 10.5958i	− 0.448960i	0.974479	−	0.224480i	$-0.0720683\pi$
$558$	12.6803i	0.536802i
$559$	32.9939	1.39549
$560$	0	0
$561$	7.75872	0.327574
$562$	− 25.5174i	− 1.07639i
$563$	− 36.2122i	− 1.52616i	−0.646303	−	0.763080i	$-0.723686\pi$
$563$	− 36.2122i	− 1.52616i	0.646303	−	0.763080i	$-0.276314\pi$
$564$	−24.9939	−1.05243
$565$	0	0
$566$	−38.5835	−1.62179
$567$	− 1.07838i	− 0.0452876i
$568$	42.3545i	1.77716i
$569$	27.5753	1.15602	0.578008	−	0.816031i	$-0.303830\pi$
$569$	27.5753	1.15602	0.578008	+	0.816031i	$0.303830\pi$
$570$	0	0
$571$	−27.9299	−1.16883	−0.584414	−	0.811456i	$-0.698676\pi$
$571$	−27.9299	−1.16883	−0.584414	+	0.811456i	$0.698676\pi$
$572$	23.1773i	0.969091i
$573$	2.52359i	0.105425i
$574$	−27.5174	−1.14856
$575$	0	0
$576$	24.8576	1.03573
$577$	− 41.4017i	− 1.72358i	−0.507268	−	0.861788i	$-0.669345\pi$
$577$	− 41.4017i	− 1.72358i	0.507268	−	0.861788i	$-0.330655\pi$
$578$	− 117.035i	− 4.86800i
$579$	−0.0266620	−0.00110803
$580$	0	0
$581$	−11.8843	−0.493043
$582$	39.7731i	1.64865i
$583$	0.156755i	0.00649215i
$584$	−94.9914	−3.93077
$585$	0	0
$586$	42.6947	1.76370
$587$	− 8.48255i	− 0.350112i	−0.984558	−	0.175056i	$-0.943989\pi$
$587$	− 8.48255i	− 0.350112i	0.984558	−	0.175056i	$-0.0560107\pi$
$588$	− 31.1711i	− 1.28548i
$589$	24.6270	1.01474
$590$	0	0
$591$	21.1194	0.868737
$592$	27.6742i	1.13740i
$593$	7.56093i	0.310490i	0.987876	+	0.155245i	$0.0496167\pi$
$593$	7.56093i	0.310490i	−0.987876	+	0.155245i	$0.950383\pi$
$594$	2.70928	0.111163
$595$	0	0
$596$	−96.6512	−3.95899
$597$	− 10.5236i	− 0.430702i
$598$	− 25.3607i	− 1.03708i
$599$	−5.67420	−0.231842	−0.115921	−	0.993258i	$-0.536982\pi$
$599$	−5.67420	−0.231842	−0.115921	+	0.993258i	$0.536982\pi$
$600$	0	0
$601$	−1.31965	−0.0538298	−0.0269149	−	0.999638i	$-0.508568\pi$
$601$	−1.31965	−0.0538298	−0.0269149	+	0.999638i	$0.508568\pi$
$602$	22.2101i	0.905215i
$603$	− 8.68035i	− 0.353491i
$604$	−122.482	−4.98373
$605$	0	0
$606$	42.1978	1.71417
$607$	2.24128i	0.0909706i	0.998965	+	0.0454853i	$0.0144834\pi$
$607$	2.24128i	0.0909706i	−0.998965	+	0.0454853i	$0.985517\pi$
$608$	− 102.024i	− 4.13763i
$609$	−1.52973	−0.0619879
$610$	0	0
$611$	−20.3135	−0.821797
$612$	− 41.4329i	− 1.67483i
$613$	42.8638i	1.73125i	0.500692	+	0.865626i	$0.333079\pi$
$613$	42.8638i	1.73125i	−0.500692	+	0.865626i	$0.666921\pi$
$614$	−51.2639	−2.06884
$615$	0	0
$616$	−9.75872	−0.393190
$617$	− 11.3607i	− 0.457364i	−0.973501	−	0.228682i	$-0.926558\pi$
$617$	− 11.3607i	− 0.457364i	0.973501	−	0.228682i	$-0.0734416\pi$
$618$	− 18.5236i	− 0.745128i
$619$	45.1917	1.81641	0.908203	−	0.418530i	$-0.137455\pi$
$619$	45.1917	1.81641	0.908203	+	0.418530i	$0.137455\pi$
$620$	0	0
$621$	−2.15676	−0.0865476
$622$	− 56.5646i	− 2.26803i
$623$	− 13.8432i	− 0.554618i
$624$	60.0554	2.40414
$625$	0	0
$626$	−17.1050	−0.683655
$627$	− 5.26180i	− 0.210136i
$628$	− 58.7091i	− 2.34275i
$629$	15.5174	0.618721
$630$	0	0
$631$	−9.78992	−0.389731	−0.194865	−	0.980830i	$-0.562427\pi$
$631$	−9.78992	−0.389731	−0.194865	+	0.980830i	$0.562427\pi$
$632$	− 73.2905i	− 2.91534i
$633$	9.57531i	0.380584i
$634$	84.9647	3.37438
$635$	0	0
$636$	0.837101	0.0331932
$637$	− 25.3340i	− 1.00377i
$638$	− 3.84324i	− 0.152156i
$639$	4.68035	0.185152
$640$	0	0
$641$	0.210079	0.00829764	0.00414882	−	0.999991i	$-0.498679\pi$
$641$	0.210079	0.00829764	0.00414882	+	0.999991i	$0.498679\pi$
$642$	17.1773i	0.677933i
$643$	14.5236i	0.572754i	0.958117	+	0.286377i	$0.0924511\pi$
$643$	14.5236i	0.572754i	−0.958117	+	0.286377i	$0.907549\pi$
$644$	12.4202	0.489423
$645$	0	0
$646$	−110.606	−4.35172
$647$	− 15.4641i	− 0.607957i	−0.952679	−	0.303979i	$-0.901685\pi$
$647$	− 15.4641i	− 0.607957i	0.952679	−	0.303979i	$-0.0983152\pi$
$648$	− 9.04945i	− 0.355496i
$649$	−6.15676	−0.241674
$650$	0	0
$651$	−5.04718	−0.197815
$652$	34.8371i	1.36433i
$653$	− 17.8310i	− 0.697779i	−0.937164	−	0.348890i	$-0.886559\pi$
$653$	− 17.8310i	− 0.697779i	0.937164	−	0.348890i	$-0.113441\pi$
$654$	6.26794	0.245096
$655$	0	0
$656$	−130.325	−5.08835
$657$	10.4969i	0.409524i
$658$	− 13.6742i	− 0.533076i
$659$	32.3135	1.25876	0.629378	−	0.777099i	$-0.283310\pi$
$659$	32.3135	1.25876	0.629378	+	0.777099i	$0.283310\pi$
$660$	0	0
$661$	−5.68649	−0.221179	−0.110589	−	0.993866i	$-0.535274\pi$
$661$	−5.68649	−0.221179	−0.110589	+	0.993866i	$0.535274\pi$
$662$	52.0288i	2.02215i
$663$	− 33.6742i	− 1.30780i
$664$	−99.7296	−3.87026
$665$	0	0
$666$	5.41855	0.209965
$667$	3.05947i	0.118463i
$668$	10.5380i	0.407726i
$669$	2.15676	0.0833850
$670$	0	0
$671$	−4.15676	−0.160470
$672$	20.9093i	0.806595i
$673$	− 21.0205i	− 0.810281i	−0.914254	−	0.405141i	$-0.867223\pi$
$673$	− 21.0205i	− 0.810281i	0.914254	−	0.405141i	$-0.132777\pi$
$674$	36.5835	1.40915
$675$	0	0
$676$	31.1711	1.19889
$677$	− 36.7526i	− 1.41252i	−0.707954	−	0.706258i	$-0.750382\pi$
$677$	− 36.7526i	− 1.41252i	0.707954	−	0.706258i	$-0.249618\pi$
$678$	16.2557i	0.624295i
$679$	−15.8310	−0.607536
$680$	0	0
$681$	9.65983	0.370165
$682$	− 12.6803i	− 0.485556i
$683$	− 17.3074i	− 0.662248i	−0.943587	−	0.331124i	$-0.892572\pi$
$683$	− 17.3074i	− 0.662248i	0.943587	−	0.331124i	$-0.107428\pi$
$684$	−28.0989	−1.07439
$685$	0	0
$686$	37.5052	1.43195
$687$	3.36069i	0.128218i
$688$	105.189i	4.01030i
$689$	0.680346	0.0259191
$690$	0	0
$691$	17.6742	0.672358	0.336179	−	0.941798i	$-0.390865\pi$
$691$	17.6742	0.672358	0.336179	+	0.941798i	$0.390865\pi$
$692$	− 20.0722i	− 0.763032i
$693$	1.07838i	0.0409642i
$694$	17.1773	0.652040
$695$	0	0
$696$	−12.8371	−0.486589
$697$	73.0759i	2.76795i
$698$	− 43.7731i	− 1.65684i
$699$	2.39803	0.0907019
$700$	0	0
$701$	−17.1050	−0.646048	−0.323024	−	0.946391i	$-0.604700\pi$
$701$	−17.1050	−0.646048	−0.323024	+	0.946391i	$0.604700\pi$
$702$	− 11.7587i	− 0.443804i
$703$	− 10.5236i	− 0.396905i
$704$	−24.8576	−0.936857
$705$	0	0
$706$	35.7731	1.34634
$707$	16.7961i	0.631681i
$708$	32.8781i	1.23564i
$709$	−25.1506	−0.944551	−0.472276	−	0.881451i	$-0.656567\pi$
$709$	−25.1506	−0.944551	−0.472276	+	0.881451i	$0.656567\pi$
$710$	0	0
$711$	−8.09890	−0.303732
$712$	− 116.169i	− 4.35361i
$713$	10.0944i	0.378037i
$714$	22.6681	0.848331
$715$	0	0
$716$	80.9069	3.02363
$717$	7.20394i	0.269036i
$718$	− 8.99386i	− 0.335648i
$719$	1.78992	0.0667528	0.0333764	−	0.999443i	$-0.489374\pi$
$719$	1.78992	0.0667528	0.0333764	+	0.999443i	$0.489374\pi$
$720$	0	0
$721$	7.37298	0.274584
$722$	23.5341i	0.875848i
$723$	− 5.20394i	− 0.193536i
$724$	−25.8310	−0.960000
$725$	0	0
$726$	−2.70928	−0.100551
$727$	− 25.9877i	− 0.963831i	−0.876218	−	0.481915i	$-0.839941\pi$
$727$	− 25.9877i	− 0.963831i	0.876218	−	0.481915i	$-0.160059\pi$
$728$	42.3545i	1.56976i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	58.9816	2.18151
$732$	22.1978i	0.820454i
$733$	− 41.0205i	− 1.51513i	−0.652761	−	0.757564i	$-0.726390\pi$
$733$	− 41.0205i	− 1.51513i	0.652761	−	0.757564i	$-0.273610\pi$
$734$	−97.9253	−3.61449
$735$	0	0
$736$	41.8187	1.54146
$737$	8.68035i	0.319745i
$738$	25.5174i	0.939310i
$739$	47.6163	1.75160	0.875798	−	0.482678i	$-0.160336\pi$
$739$	47.6163	1.75160	0.875798	+	0.482678i	$0.160336\pi$
$740$	0	0
$741$	−22.8371	−0.838942
$742$	0.457980i	0.0168130i
$743$	0.550252i	0.0201868i	0.999949	+	0.0100934i	$0.00321288\pi$
$743$	0.550252i	0.0201868i	−0.999949	+	0.0100934i	$0.996787\pi$
$744$	−42.3545	−1.55279
$745$	0	0
$746$	7.61425	0.278778
$747$	11.0205i	0.403220i
$748$	41.4329i	1.51494i
$749$	−6.83710	−0.249822
$750$	0	0
$751$	41.5585	1.51649	0.758245	−	0.651969i	$-0.226057\pi$
$751$	41.5585	1.51649	0.758245	+	0.651969i	$0.226057\pi$
$752$	− 64.7624i	− 2.36164i
$753$	15.3197i	0.558279i
$754$	−16.6803	−0.607462
$755$	0	0
$756$	5.75872	0.209443
$757$	− 1.31965i	− 0.0479636i	−0.999712	−	0.0239818i	$-0.992366\pi$
$757$	− 1.31965i	− 0.0479636i	0.999712	−	0.0239818i	$-0.00763438\pi$
$758$	54.1855i	1.96811i
$759$	2.15676	0.0782853
$760$	0	0
$761$	−2.21461	−0.0802797	−0.0401399	−	0.999194i	$-0.512780\pi$
$761$	−2.21461	−0.0802797	−0.0401399	+	0.999194i	$0.512780\pi$
$762$	− 6.07223i	− 0.219974i
$763$	2.49484i	0.0903192i
$764$	−13.4764	−0.487559
$765$	0	0
$766$	−90.9192	−3.28504
$767$	26.7214i	0.964853i
$768$	27.6803i	0.998828i
$769$	14.3668	0.518081	0.259041	−	0.965866i	$-0.416594\pi$
$769$	14.3668	0.518081	0.259041	+	0.965866i	$0.416594\pi$
$770$	0	0
$771$	4.15676	0.149702
$772$	− 0.142380i	− 0.00512436i
$773$	40.1568i	1.44434i	0.691717	+	0.722169i	$0.256855\pi$
$773$	40.1568i	1.44434i	−0.691717	+	0.722169i	$0.743145\pi$
$774$	20.5958	0.740302
$775$	0	0
$776$	−132.849	−4.76900
$777$	2.15676i	0.0773732i
$778$	− 34.7792i	− 1.24690i
$779$	49.5585	1.77562
$780$	0	0
$781$	−4.68035	−0.167476
$782$	− 45.3361i	− 1.62122i
$783$	1.41855i	0.0506949i
$784$	80.7686	2.88459
$785$	0	0
$786$	−23.5174	−0.838840
$787$	− 49.5897i	− 1.76768i	−0.467788	−	0.883841i	$-0.654949\pi$
$787$	− 49.5897i	− 1.76768i	0.467788	−	0.883841i	$-0.345051\pi$
$788$	112.781i	4.01767i
$789$	18.7070	0.665987
$790$	0	0
$791$	−6.47027	−0.230056
$792$	9.04945i	0.321558i
$793$	18.0410i	0.640656i
$794$	−14.4124	−0.511477
$795$	0	0
$796$	56.1978	1.99188
$797$	46.7091i	1.65452i	0.561818	+	0.827261i	$0.310102\pi$
$797$	46.7091i	1.65452i	−0.561818	+	0.827261i	$0.689898\pi$
$798$	− 15.3730i	− 0.544198i
$799$	−36.3135	−1.28468
$800$	0	0
$801$	−12.8371	−0.453577
$802$	5.41855i	0.191336i
$803$	− 10.4969i	− 0.370429i
$804$	46.3545	1.63480
$805$	0	0
$806$	−55.0349	−1.93852
$807$	− 23.3607i	− 0.822335i
$808$	140.948i	4.95853i
$809$	18.5814	0.653289	0.326644	−	0.945147i	$-0.394082\pi$
$809$	18.5814	0.653289	0.326644	+	0.945147i	$0.394082\pi$
$810$	0	0
$811$	27.3028	0.958732	0.479366	−	0.877615i	$-0.340867\pi$
$811$	27.3028	0.958732	0.479366	+	0.877615i	$0.340867\pi$
$812$	− 8.16904i	− 0.286677i
$813$	− 5.57531i	− 0.195535i
$814$	−5.41855	−0.189920
$815$	0	0
$816$	107.358	3.75829
$817$	− 40.0000i	− 1.39942i
$818$	− 70.9770i	− 2.48165i
$819$	4.68035	0.163545
$820$	0	0
$821$	−31.2085	−1.08918	−0.544592	−	0.838701i	$-0.683315\pi$
$821$	−31.2085	−1.08918	−0.544592	+	0.838701i	$0.683315\pi$
$822$	− 41.6163i	− 1.45154i
$823$	− 50.1855i	− 1.74936i	−0.484704	−	0.874678i	$-0.661073\pi$
$823$	− 50.1855i	− 1.74936i	0.484704	−	0.874678i	$-0.338927\pi$
$824$	61.8720	2.15541
$825$	0	0
$826$	−17.9877	−0.625873
$827$	− 27.3874i	− 0.952352i	−0.879350	−	0.476176i	$-0.842023\pi$
$827$	− 27.3874i	− 0.952352i	0.879350	−	0.476176i	$-0.157977\pi$
$828$	− 11.5174i	− 0.400259i
$829$	26.1978	0.909887	0.454943	−	0.890520i	$-0.349659\pi$
$829$	26.1978	0.909887	0.454943	+	0.890520i	$0.349659\pi$
$830$	0	0
$831$	−26.0144	−0.902429
$832$	107.886i	3.74029i
$833$	− 45.2885i	− 1.56915i
$834$	23.2495	0.805065
$835$	0	0
$836$	28.0989	0.971821
$837$	4.68035i	0.161776i
$838$	7.68649i	0.265525i
$839$	−7.20394	−0.248708	−0.124354	−	0.992238i	$-0.539686\pi$
$839$	−7.20394	−0.248708	−0.124354	+	0.992238i	$0.539686\pi$
$840$	0	0
$841$	−26.9877	−0.930611
$842$	31.0928i	1.07153i
$843$	− 9.41855i	− 0.324392i
$844$	−51.1338	−1.76010
$845$	0	0
$846$	−12.6803	−0.435959
$847$	− 1.07838i	− 0.0370535i
$848$	2.16904i	0.0744852i
$849$	−14.2413	−0.488759
$850$	0	0
$851$	4.31351	0.147865
$852$	24.9939i	0.856275i
$853$	39.8043i	1.36287i	0.731877	+	0.681437i	$0.238644\pi$
$853$	39.8043i	1.36287i	−0.731877	+	0.681437i	$0.761356\pi$
$854$	−12.1445	−0.415575
$855$	0	0
$856$	−57.3751	−1.96104
$857$	36.9504i	1.26220i	0.775701	+	0.631100i	$0.217396\pi$
$857$	36.9504i	1.26220i	−0.775701	+	0.631100i	$0.782604\pi$
$858$	11.7587i	0.401436i
$859$	−57.5052	−1.96205	−0.981025	−	0.193879i	$-0.937893\pi$
$859$	−57.5052	−1.96205	−0.981025	+	0.193879i	$0.937893\pi$
$860$	0	0
$861$	−10.1568	−0.346141
$862$	63.8597i	2.17507i
$863$	− 1.89657i	− 0.0645599i	−0.999479	−	0.0322800i	$-0.989723\pi$
$863$	− 1.89657i	− 0.0645599i	0.999479	−	0.0322800i	$-0.0102768\pi$
$864$	19.3896	0.659648
$865$	0	0
$866$	40.6225	1.38041
$867$	− 43.1978i	− 1.46707i
$868$	− 26.9528i	− 0.914838i
$869$	8.09890	0.274736
$870$	0	0
$871$	37.6742	1.27654
$872$	20.9360i	0.708982i
$873$	14.6803i	0.496854i
$874$	−30.7460	−1.04000
$875$	0	0
$876$	−56.0554	−1.89394
$877$	− 32.5380i	− 1.09873i	−0.835583	−	0.549365i	$-0.814870\pi$
$877$	− 32.5380i	− 1.09873i	0.835583	−	0.549365i	$-0.185130\pi$
$878$	− 12.9483i	− 0.436983i
$879$	15.7587	0.531529
$880$	0	0
$881$	18.1978	0.613099	0.306550	−	0.951855i	$-0.400825\pi$
$881$	18.1978	0.613099	0.306550	+	0.951855i	$0.400825\pi$
$882$	− 15.8143i	− 0.532496i
$883$	36.3956i	1.22481i	0.790545	+	0.612405i	$0.209798\pi$
$883$	36.3956i	1.22481i	−0.790545	+	0.612405i	$0.790202\pi$
$884$	179.826	6.04821
$885$	0	0
$886$	−54.7214	−1.83840
$887$	− 27.8699i	− 0.935780i	−0.883787	−	0.467890i	$-0.845014\pi$
$887$	− 27.8699i	− 0.935780i	0.883787	−	0.467890i	$-0.154986\pi$
$888$	18.0989i	0.607359i
$889$	2.41694	0.0810616
$890$	0	0
$891$	1.00000	0.0335013
$892$	11.5174i	0.385633i
$893$	24.6270i	0.824112i
$894$	−49.0349	−1.63997
$895$	0	0
$896$	−30.8059	−1.02915
$897$	− 9.36069i	− 0.312544i
$898$	58.4412i	1.95021i
$899$	6.63931	0.221433
$900$	0	0
$901$	1.21622	0.0405182
$902$	− 25.5174i	− 0.849638i
$903$	8.19779i	0.272805i
$904$	−54.2967	−1.80588
$905$	0	0
$906$	−62.1399	−2.06446
$907$	27.9376i	0.927653i	0.885926	+	0.463826i	$0.153524\pi$
$907$	27.9376i	0.927653i	−0.885926	+	0.463826i	$0.846476\pi$
$908$	51.5851i	1.71191i
$909$	15.5753	0.516600
$910$	0	0
$911$	−11.8843	−0.393744	−0.196872	−	0.980429i	$-0.563078\pi$
$911$	−11.8843	−0.393744	−0.196872	+	0.980429i	$0.563078\pi$
$912$	− 72.8080i	− 2.41091i
$913$	− 11.0205i	− 0.364726i
$914$	76.3234	2.52455
$915$	0	0
$916$	−17.9467	−0.592975
$917$	− 9.36069i	− 0.309117i
$918$	− 21.0205i	− 0.693781i
$919$	45.6041	1.50434	0.752170	−	0.658970i	$-0.229007\pi$
$919$	45.6041	1.50434	0.752170	+	0.658970i	$0.229007\pi$
$920$	0	0
$921$	−18.9216	−0.623489
$922$	− 3.98771i	− 0.131328i
$923$	20.3135i	0.668627i
$924$	−5.75872	−0.189448
$925$	0	0
$926$	62.8659	2.06590
$927$	− 6.83710i	− 0.224560i
$928$	− 27.5052i	− 0.902901i
$929$	25.1506	0.825165	0.412582	−	0.910920i	$-0.364627\pi$
$929$	25.1506	0.825165	0.412582	+	0.910920i	$0.364627\pi$
$930$	0	0
$931$	−30.7136	−1.00660
$932$	12.8059i	0.419471i
$933$	− 20.8781i	− 0.683520i
$934$	38.3545	1.25500
$935$	0	0
$936$	39.2762	1.28378
$937$	− 5.33403i	− 0.174255i	−0.996197	−	0.0871276i	$-0.972231\pi$
$937$	− 5.33403i	− 0.174255i	0.996197	−	0.0871276i	$-0.0277688\pi$
$938$	25.3607i	0.828056i
$939$	−6.31351	−0.206034
$940$	0	0
$941$	56.8203	1.85229	0.926144	−	0.377170i	$-0.123103\pi$
$941$	56.8203	1.85229	0.926144	+	0.377170i	$0.123103\pi$
$942$	− 29.7854i	− 0.970460i
$943$	20.3135i	0.661499i
$944$	−85.1917	−2.77275
$945$	0	0
$946$	−20.5958	−0.669628
$947$	20.9939i	0.682209i	0.940025	+	0.341104i	$0.110801\pi$
$947$	20.9939i	0.682209i	−0.940025	+	0.341104i	$0.889199\pi$
$948$	− 43.2495i	− 1.40468i
$949$	−45.5585	−1.47889
$950$	0	0
$951$	31.3607	1.01694
$952$	75.7152i	2.45395i
$953$	− 25.2351i	− 0.817446i	−0.912658	−	0.408723i	$-0.865974\pi$
$953$	− 25.2351i	− 0.817446i	0.912658	−	0.408723i	$-0.134026\pi$
$954$	0.424694	0.0137500
$955$	0	0
$956$	−38.4703	−1.24422
$957$	− 1.41855i	− 0.0458552i
$958$	37.5052i	1.21174i
$959$	16.5646	0.534900
$960$	0	0
$961$	−9.09436	−0.293367
$962$	23.5174i	0.758233i
$963$	6.34017i	0.204309i
$964$	27.7899	0.895053
$965$	0	0
$966$	6.30122	0.202739
$967$	− 13.1317i	− 0.422287i	−0.977455	−	0.211144i	$-0.932281\pi$
$967$	− 13.1317i	− 0.422287i	0.977455	−	0.211144i	$-0.0677187\pi$
$968$	− 9.04945i	− 0.290860i
$969$	−40.8248	−1.31148
$970$	0	0
$971$	8.94053	0.286915	0.143458	−	0.989656i	$-0.454178\pi$
$971$	8.94053	0.286915	0.143458	+	0.989656i	$0.454178\pi$
$972$	− 5.34017i	− 0.171286i
$973$	9.25404i	0.296671i
$974$	−111.064	−3.55871
$975$	0	0
$976$	−57.5174	−1.84109
$977$	− 50.3956i	− 1.61230i	−0.591713	−	0.806149i	$-0.701548\pi$
$977$	− 50.3956i	− 1.61230i	0.591713	−	0.806149i	$-0.298452\pi$
$978$	17.6742i	0.565159i
$979$	12.8371	0.410276
$980$	0	0
$981$	2.31351	0.0738647
$982$	94.3833i	3.01189i
$983$	− 32.1978i	− 1.02695i	−0.858105	−	0.513475i	$-0.828358\pi$
$983$	− 32.1978i	− 1.02695i	0.858105	−	0.513475i	$-0.171642\pi$
$984$	−85.2327	−2.71712
$985$	0	0
$986$	−29.8187	−0.949620
$987$	− 5.04718i	− 0.160654i
$988$	− 121.954i	− 3.87988i
$989$	16.3956	0.521349
$990$	0	0
$991$	−46.7747	−1.48585	−0.742924	−	0.669376i	$-0.766562\pi$
$991$	−46.7747	−1.48585	−0.742924	+	0.669376i	$0.766562\pi$
$992$	− 90.7501i	− 2.88132i
$993$	19.2039i	0.609419i
$994$	−13.6742	−0.433719
$995$	0	0
$996$	−58.8515	−1.86478
$997$	− 38.2122i	− 1.21019i	−0.796153	−	0.605096i	$-0.793135\pi$
$997$	− 38.2122i	− 1.21019i	0.796153	−	0.605096i	$-0.206865\pi$
$998$	− 41.0472i	− 1.29933i
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.2.c.g.199.6		6
3.2	odd	2		2475.2.c.r.199.1		6
5.2	odd	4		165.2.a.c.1.1	✓	3
5.3	odd	4		825.2.a.k.1.3		3
5.4	even	2	inner	825.2.c.g.199.1		6
15.2	even	4		495.2.a.e.1.3		3
15.8	even	4		2475.2.a.bb.1.1		3
15.14	odd	2		2475.2.c.r.199.6		6
20.7	even	4		2640.2.a.be.1.2		3
35.27	even	4		8085.2.a.bk.1.1		3
55.32	even	4		1815.2.a.m.1.3		3
55.43	even	4		9075.2.a.cf.1.1		3
60.47	odd	4		7920.2.a.cj.1.2		3
165.32	odd	4		5445.2.a.z.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.a.c.1.1	✓	3	5.2	odd	4
495.2.a.e.1.3		3	15.2	even	4
825.2.a.k.1.3		3	5.3	odd	4
825.2.c.g.199.1		6	5.4	even	2	inner
825.2.c.g.199.6		6	1.1	even	1	trivial
1815.2.a.m.1.3		3	55.32	even	4
2475.2.a.bb.1.1		3	15.8	even	4
2475.2.c.r.199.1		6	3.2	odd	2
2475.2.c.r.199.6		6	15.14	odd	2
2640.2.a.be.1.2		3	20.7	even	4
5445.2.a.z.1.1		3	165.32	odd	4
7920.2.a.cj.1.2		3	60.47	odd	4
8085.2.a.bk.1.1		3	35.27	even	4
9075.2.a.cf.1.1		3	55.43	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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	825.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.70928i q^{2} +1.00000i q^{3} -5.34017 q^{4} -2.70928 q^{6} -1.07838i q^{7} -9.04945i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+2.70928i q^{2} +1.00000i q^{3} -5.34017 q^{4} -2.70928 q^{6} -1.07838i q^{7} -9.04945i q^{8} -1.00000 q^{9} +1.00000 q^{11} -5.34017i q^{12} -4.34017i q^{13} +2.92162 q^{14} +13.8371 q^{16} -7.75872i q^{17} -2.70928i q^{18} -5.26180 q^{19} +1.07838 q^{21} +2.70928i q^{22} -2.15676i q^{23} +9.04945 q^{24} +11.7587 q^{26} -1.00000i q^{27} +5.75872i q^{28} -1.41855 q^{29} -4.68035 q^{31} +19.3896i q^{32} +1.00000i q^{33} +21.0205 q^{34} +5.34017 q^{36} +2.00000i q^{37} -14.2557i q^{38} +4.34017 q^{39} -9.41855 q^{41} +2.92162i q^{42} +7.60197i q^{43} -5.34017 q^{44} +5.84324 q^{46} -4.68035i q^{47} +13.8371i q^{48} +5.83710 q^{49} +7.75872 q^{51} +23.1773i q^{52} +0.156755i q^{53} +2.70928 q^{54} -9.75872 q^{56} -5.26180i q^{57} -3.84324i q^{58} -6.15676 q^{59} -4.15676 q^{61} -12.6803i q^{62} +1.07838i q^{63} -24.8576 q^{64} -2.70928 q^{66} +8.68035i q^{67} +41.4329i q^{68} +2.15676 q^{69} -4.68035 q^{71} +9.04945i q^{72} -10.4969i q^{73} -5.41855 q^{74} +28.0989 q^{76} -1.07838i q^{77} +11.7587i q^{78} +8.09890 q^{79} +1.00000 q^{81} -25.5174i q^{82} -11.0205i q^{83} -5.75872 q^{84} -20.5958 q^{86} -1.41855i q^{87} -9.04945i q^{88} +12.8371 q^{89} -4.68035 q^{91} +11.5174i q^{92} -4.68035i q^{93} +12.6803 q^{94} -19.3896 q^{96} -14.6803i q^{97} +15.8143i q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 10 q^{4} - 2 q^{6} - 6 q^{9}+O(q^{10}) \) 6 * q - 10 * q^4 - 2 * q^6 - 6 * q^9	\( 6 q - 10 q^{4} - 2 q^{6} - 6 q^{9} + 6 q^{11} + 24 q^{14} + 26 q^{16} - 16 q^{19} + 18 q^{24} + 20 q^{26} + 20 q^{29} + 16 q^{31} + 60 q^{34} + 10 q^{36} + 4 q^{39} - 28 q^{41} - 10 q^{44} + 48 q^{46} - 22 q^{49} - 4 q^{51} + 2 q^{54} - 8 q^{56} - 24 q^{59} - 12 q^{61} - 26 q^{64} - 2 q^{66} + 16 q^{71} - 4 q^{74} + 96 q^{76} - 24 q^{79} + 6 q^{81} + 16 q^{84} - 16 q^{86} + 20 q^{89} + 16 q^{91} + 32 q^{94} - 58 q^{96} - 6 q^{99}+O(q^{100}) \) 6 * q - 10 * q^4 - 2 * q^6 - 6 * q^9 + 6 * q^11 + 24 * q^14 + 26 * q^16 - 16 * q^19 + 18 * q^24 + 20 * q^26 + 20 * q^29 + 16 * q^31 + 60 * q^34 + 10 * q^36 + 4 * q^39 - 28 * q^41 - 10 * q^44 + 48 * q^46 - 22 * q^49 - 4 * q^51 + 2 * q^54 - 8 * q^56 - 24 * q^59 - 12 * q^61 - 26 * q^64 - 2 * q^66 + 16 * q^71 - 4 * q^74 + 96 * q^76 - 24 * q^79 + 6 * q^81 + 16 * q^84 - 16 * q^86 + 20 * q^89 + 16 * q^91 + 32 * q^94 - 58 * q^96 - 6 * q^99

Introduction

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