LMFDB - Embedded newform 4212.2.i.b.2809.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4212,2,Mod(1405,4212)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4212, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4212.1405");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4212 = 2^{2} \cdot 3^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4212.i (of order $3$, degree $2$, 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:	$33.6329893314$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 156)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			2809.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	4212.2809
Dual form			4212.2.i.b.1405.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 - 3.46410i) q^{5} +(1.00000 - 1.73205i) q^{7} +O(q^{10})$	$q+(-2.00000 - 3.46410i) q^{5} +(1.00000 - 1.73205i) q^{7} +(-2.00000 + 3.46410i) q^{11} +(-0.500000 - 0.866025i) q^{13} -2.00000 q^{17} -2.00000 q^{19} +(-5.50000 + 9.52628i) q^{25} +(-3.00000 + 5.19615i) q^{29} +(5.00000 + 8.66025i) q^{31} -8.00000 q^{35} +10.0000 q^{37} +(4.00000 + 6.92820i) q^{41} +(-2.00000 + 3.46410i) q^{43} +(-2.00000 + 3.46410i) q^{47} +(1.50000 + 2.59808i) q^{49} +10.0000 q^{53} +16.0000 q^{55} +(-4.00000 - 6.92820i) q^{59} +(7.00000 - 12.1244i) q^{61} +(-2.00000 + 3.46410i) q^{65} +(-1.00000 - 1.73205i) q^{67} -16.0000 q^{71} -10.0000 q^{73} +(4.00000 + 6.92820i) q^{77} +(8.00000 - 13.8564i) q^{79} +(4.00000 + 6.92820i) q^{85} +4.00000 q^{89} -2.00000 q^{91} +(4.00000 + 6.92820i) q^{95} +(1.00000 - 1.73205i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q - 4 * q^5 + 2 * q^7	$ 2 q - 4 q^{5} + 2 q^{7} - 4 q^{11} - q^{13} - 4 q^{17} - 4 q^{19} - 11 q^{25} - 6 q^{29} + 10 q^{31} - 16 q^{35} + 20 q^{37} + 8 q^{41} - 4 q^{43} - 4 q^{47} + 3 q^{49} + 20 q^{53} + 32 q^{55} - 8 q^{59} + 14 q^{61} - 4 q^{65} - 2 q^{67} - 32 q^{71} - 20 q^{73} + 8 q^{77} + 16 q^{79} + 8 q^{85} + 8 q^{89} - 4 q^{91} + 8 q^{95} + 2 q^{97}+O(q^{100}) $ 2 * q - 4 * q^5 + 2 * q^7 - 4 * q^11 - q^13 - 4 * q^17 - 4 * q^19 - 11 * q^25 - 6 * q^29 + 10 * q^31 - 16 * q^35 + 20 * q^37 + 8 * q^41 - 4 * q^43 - 4 * q^47 + 3 * q^49 + 20 * q^53 + 32 * q^55 - 8 * q^59 + 14 * q^61 - 4 * q^65 - 2 * q^67 - 32 * q^71 - 20 * q^73 + 8 * q^77 + 16 * q^79 + 8 * q^85 + 8 * q^89 - 4 * q^91 + 8 * q^95 + 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4212\mathbb{Z}\right)^\times$.

$n$	$2107$	$3485$	$3889$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.00000	−	3.46410i	−0.894427	−	1.54919i	−0.834512	−	0.550990i	$-0.814250\pi$
$5$	−2.00000	−	3.46410i	−0.894427	−	1.54919i	−0.0599153	−	0.998203i	$-0.519083\pi$
$6$		0			0
$7$	1.00000	−	1.73205i	0.377964	−	0.654654i	−0.612801	−	0.790237i	$-0.709957\pi$
$7$	1.00000	−	1.73205i	0.377964	−	0.654654i	0.990766	+	0.135583i	$0.0432908\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.00000	+	3.46410i	−0.603023	+	1.04447i	0.389338	+	0.921095i	$0.372704\pi$
$11$	−2.00000	+	3.46410i	−0.603023	+	1.04447i	−0.992361	+	0.123371i	$0.960630\pi$
$12$		0			0
$13$	−0.500000	−	0.866025i	−0.138675	−	0.240192i
$13$	−0.500000	−	0.866025i	−0.138675	−	0.240192i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$		0			0
$19$	−2.00000			−0.458831			−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000			−0.458831			−0.229416	+	0.973329i	$0.573682\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	−5.50000	+	9.52628i	−1.10000	+	1.90526i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.00000	+	5.19615i	−0.557086	+	0.964901i	0.440652	+	0.897678i	$0.354747\pi$
$29$	−3.00000	+	5.19615i	−0.557086	+	0.964901i	−0.997738	+	0.0672232i	$0.978586\pi$
$30$		0			0
$31$	5.00000	+	8.66025i	0.898027	+	1.55543i	0.830014	+	0.557743i	$0.188333\pi$
$31$	5.00000	+	8.66025i	0.898027	+	1.55543i	0.0680129	+	0.997684i	$0.478334\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−8.00000			−1.35225
$36$		0			0
$37$	10.0000			1.64399			0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000			1.64399			0.821995	+	0.569495i	$0.192861\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.00000	+	6.92820i	0.624695	+	1.08200i	0.988600	+	0.150567i	$0.0481100\pi$
$41$	4.00000	+	6.92820i	0.624695	+	1.08200i	−0.363905	+	0.931436i	$0.618557\pi$
$42$		0			0
$43$	−2.00000	+	3.46410i	−0.304997	+	0.528271i	−0.977261	−	0.212041i	$-0.931989\pi$
$43$	−2.00000	+	3.46410i	−0.304997	+	0.528271i	0.672264	+	0.740312i	$0.265322\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.00000	+	3.46410i	−0.291730	+	0.505291i	−0.974219	−	0.225605i	$-0.927564\pi$
$47$	−2.00000	+	3.46410i	−0.291730	+	0.505291i	0.682489	+	0.730896i	$0.260898\pi$
$48$		0			0
$49$	1.50000	+	2.59808i	0.214286	+	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	10.0000			1.37361			0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000			1.37361			0.686803	+	0.726844i	$0.259014\pi$
$54$		0			0
$55$	16.0000			2.15744
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.00000	−	6.92820i	−0.520756	−	0.901975i	−0.999709	−	0.0241347i	$-0.992317\pi$
$59$	−4.00000	−	6.92820i	−0.520756	−	0.901975i	0.478953	−	0.877841i	$-0.341016\pi$
$60$		0			0
$61$	7.00000	−	12.1244i	0.896258	−	1.55236i	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	7.00000	−	12.1244i	0.896258	−	1.55236i	0.832240	−	0.554416i	$-0.187058\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.00000	+	3.46410i	−0.248069	+	0.429669i
$66$		0			0
$67$	−1.00000	−	1.73205i	−0.122169	−	0.211604i	0.798454	−	0.602056i	$-0.205652\pi$
$67$	−1.00000	−	1.73205i	−0.122169	−	0.211604i	−0.920623	+	0.390453i	$0.872318\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−16.0000			−1.89885			−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000			−1.89885			−0.949425	+	0.313993i	$0.898333\pi$
$72$		0			0
$73$	−10.0000			−1.17041			−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000			−1.17041			−0.585206	+	0.810885i	$0.698986\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4.00000	+	6.92820i	0.455842	+	0.789542i
$78$		0			0
$79$	8.00000	−	13.8564i	0.900070	−	1.55897i	0.0726692	−	0.997356i	$-0.476848\pi$
$79$	8.00000	−	13.8564i	0.900070	−	1.55897i	0.827401	−	0.561611i	$-0.189818\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$83$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$84$		0			0
$85$	4.00000	+	6.92820i	0.433861	+	0.751469i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	4.00000			0.423999			0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000			0.423999			0.212000	+	0.977270i	$0.432002\pi$
$90$		0			0
$91$	−2.00000			−0.209657
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.00000	+	6.92820i	0.410391	+	0.710819i
$96$		0			0
$97$	1.00000	−	1.73205i	0.101535	−	0.175863i	−0.810782	−	0.585348i	$-0.800958\pi$
$97$	1.00000	−	1.73205i	0.101535	−	0.175863i	0.912317	+	0.409484i	$0.134291\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	5.00000	−	8.66025i	0.497519	−	0.861727i	−0.502477	−	0.864590i	$-0.667578\pi$
$101$	5.00000	−	8.66025i	0.497519	−	0.861727i	0.999996	+	0.00286291i	$0.000911295\pi$
$102$		0			0
$103$	4.00000	+	6.92820i	0.394132	+	0.682656i	0.992990	−	0.118199i	$-0.0377120\pi$
$103$	4.00000	+	6.92820i	0.394132	+	0.682656i	−0.598858	+	0.800855i	$0.704379\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$		0			0
$109$	−2.00000			−0.191565			−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000			−0.191565			−0.0957826	+	0.995402i	$0.530535\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.00000	+	5.19615i	0.282216	+	0.488813i	0.971930	−	0.235269i	$-0.0755971\pi$
$113$	3.00000	+	5.19615i	0.282216	+	0.488813i	−0.689714	+	0.724082i	$0.742264\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.00000	+	3.46410i	−0.183340	+	0.317554i
$120$		0			0
$121$	−2.50000	−	4.33013i	−0.227273	−	0.393648i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	24.0000			2.14663
$126$		0			0
$127$	12.0000			1.06483			0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000			1.06483			0.532414	+	0.846484i	$0.321285\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2.00000	+	3.46410i	0.174741	+	0.302660i	0.940072	−	0.340977i	$-0.110758\pi$
$131$	2.00000	+	3.46410i	0.174741	+	0.302660i	−0.765331	+	0.643637i	$0.777425\pi$
$132$		0			0
$133$	−2.00000	+	3.46410i	−0.173422	+	0.300376i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.00000	−	6.92820i	0.341743	−	0.591916i	−0.643013	−	0.765855i	$-0.722316\pi$
$137$	4.00000	−	6.92820i	0.341743	−	0.591916i	0.984757	+	0.173939i	$0.0556494\pi$
$138$		0			0
$139$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$139$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	4.00000			0.334497
$144$		0			0
$145$	24.0000			1.99309
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$149$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$150$		0			0
$151$	−9.00000	+	15.5885i	−0.732410	+	1.26857i	0.223441	+	0.974717i	$0.428271\pi$
$151$	−9.00000	+	15.5885i	−0.732410	+	1.26857i	−0.955851	+	0.293853i	$0.905062\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	20.0000	−	34.6410i	1.60644	−	2.78243i
$156$		0			0
$157$	−1.00000	−	1.73205i	−0.0798087	−	0.138233i	0.823359	−	0.567521i	$-0.192098\pi$
$157$	−1.00000	−	1.73205i	−0.0798087	−	0.138233i	−0.903167	+	0.429289i	$0.858764\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	2.00000			0.156652			0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000			0.156652			0.0783260	+	0.996928i	$0.475042\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.00000	−	10.3923i	−0.464294	−	0.804181i	0.534875	−	0.844931i	$-0.320359\pi$
$167$	−6.00000	−	10.3923i	−0.464294	−	0.804181i	−0.999169	+	0.0407502i	$0.987025\pi$
$168$		0			0
$169$	−0.500000	+	0.866025i	−0.0384615	+	0.0666173i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.00000	−	1.73205i	0.0760286	−	0.131685i	−0.825505	−	0.564396i	$-0.809109\pi$
$173$	1.00000	−	1.73205i	0.0760286	−	0.131685i	0.901533	+	0.432710i	$0.142443\pi$
$174$		0			0
$175$	11.0000	+	19.0526i	0.831522	+	1.44024i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−20.0000	−	34.6410i	−1.47043	−	2.54686i
$186$		0			0
$187$	4.00000	−	6.92820i	0.292509	−	0.506640i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4.00000	+	6.92820i	−0.289430	+	0.501307i	−0.973674	−	0.227946i	$-0.926799\pi$
$191$	−4.00000	+	6.92820i	−0.289430	+	0.501307i	0.684244	+	0.729253i	$0.260132\pi$
$192$		0			0
$193$	7.00000	+	12.1244i	0.503871	+	0.872730i	0.999990	+	0.00447566i	$0.00142465\pi$
$193$	7.00000	+	12.1244i	0.503871	+	0.872730i	−0.496119	+	0.868255i	$0.665242\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	12.0000			0.854965			0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000			0.854965			0.427482	+	0.904024i	$0.359401\pi$
$198$		0			0
$199$	4.00000			0.283552			0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000			0.283552			0.141776	+	0.989899i	$0.454719\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	6.00000	+	10.3923i	0.421117	+	0.729397i
$204$		0			0
$205$	16.0000	−	27.7128i	1.11749	−	1.93555i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.00000	−	6.92820i	0.276686	−	0.479234i
$210$		0			0
$211$	6.00000	+	10.3923i	0.413057	+	0.715436i	0.995222	−	0.0976347i	$-0.0311277\pi$
$211$	6.00000	+	10.3923i	0.413057	+	0.715436i	−0.582165	+	0.813070i	$0.697794\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	16.0000			1.09119
$216$		0			0
$217$	20.0000			1.35769
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.00000	+	1.73205i	0.0672673	+	0.116510i
$222$		0			0
$223$	−7.00000	+	12.1244i	−0.468755	+	0.811907i	−0.999362	−	0.0357107i	$-0.988630\pi$
$223$	−7.00000	+	12.1244i	−0.468755	+	0.811907i	0.530607	+	0.847618i	$0.321964\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2.00000	+	3.46410i	−0.132745	+	0.229920i	−0.924734	−	0.380615i	$-0.875712\pi$
$227$	−2.00000	+	3.46410i	−0.132745	+	0.229920i	0.791989	+	0.610535i	$0.209046\pi$
$228$		0			0
$229$	1.00000	+	1.73205i	0.0660819	+	0.114457i	0.897173	−	0.441679i	$-0.145617\pi$
$229$	1.00000	+	1.73205i	0.0660819	+	0.114457i	−0.831092	+	0.556136i	$0.812283\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	18.0000			1.17922			0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000			1.17922			0.589610	+	0.807688i	$0.299282\pi$
$234$		0			0
$235$	16.0000			1.04372
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−12.0000	−	20.7846i	−0.776215	−	1.34444i	−0.934109	−	0.356988i	$-0.883804\pi$
$239$	−12.0000	−	20.7846i	−0.776215	−	1.34444i	0.157893	−	0.987456i	$-0.449530\pi$
$240$		0			0
$241$	1.00000	−	1.73205i	0.0644157	−	0.111571i	−0.832019	−	0.554747i	$-0.812815\pi$
$241$	1.00000	−	1.73205i	0.0644157	−	0.111571i	0.896435	+	0.443176i	$0.146148\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	6.00000	−	10.3923i	0.383326	−	0.663940i
$246$		0			0
$247$	1.00000	+	1.73205i	0.0636285	+	0.110208i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	20.0000			1.26239			0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000			1.26239			0.631194	+	0.775625i	$0.282565\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	15.0000	+	25.9808i	0.935674	+	1.62064i	0.773427	+	0.633885i	$0.218541\pi$
$257$	15.0000	+	25.9808i	0.935674	+	1.62064i	0.162247	+	0.986750i	$0.448126\pi$
$258$		0			0
$259$	10.0000	−	17.3205i	0.621370	−	1.07624i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−8.00000	+	13.8564i	−0.493301	+	0.854423i	−0.999970	−	0.00771799i	$-0.997543\pi$
$263$	−8.00000	+	13.8564i	−0.493301	+	0.854423i	0.506669	+	0.862141i	$0.330877\pi$
$264$		0			0
$265$	−20.0000	−	34.6410i	−1.22859	−	2.12798i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	14.0000			0.853595			0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000			0.853595			0.426798	+	0.904347i	$0.359642\pi$
$270$		0			0
$271$	−10.0000			−0.607457			−0.303728	−	0.952759i	$-0.598232\pi$
$271$	−10.0000			−0.607457			−0.303728	+	0.952759i	$0.598232\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−22.0000	−	38.1051i	−1.32665	−	2.29783i
$276$		0			0
$277$	−11.0000	+	19.0526i	−0.660926	+	1.14476i	0.319447	+	0.947604i	$0.396503\pi$
$277$	−11.0000	+	19.0526i	−0.660926	+	1.14476i	−0.980373	+	0.197153i	$0.936830\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−4.00000	+	6.92820i	−0.238620	+	0.413302i	−0.960319	−	0.278906i	$-0.910028\pi$
$281$	−4.00000	+	6.92820i	−0.238620	+	0.413302i	0.721699	+	0.692207i	$0.243362\pi$
$282$		0			0
$283$	8.00000	+	13.8564i	0.475551	+	0.823678i	0.999608	−	0.0280052i	$-0.00891551\pi$
$283$	8.00000	+	13.8564i	0.475551	+	0.823678i	−0.524057	+	0.851683i	$0.675582\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	16.0000			0.944450
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$		0			0
$292$		0			0
$293$	4.00000	+	6.92820i	0.233682	+	0.404750i	0.958889	−	0.283782i	$-0.0915890\pi$
$293$	4.00000	+	6.92820i	0.233682	+	0.404750i	−0.725206	+	0.688531i	$0.758256\pi$
$294$		0			0
$295$	−16.0000	+	27.7128i	−0.931556	+	1.61350i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	4.00000	+	6.92820i	0.230556	+	0.399335i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−56.0000			−3.20655
$306$		0			0
$307$	22.0000			1.25561			0.627803	−	0.778372i	$-0.283954\pi$
$307$	22.0000			1.25561			0.627803	+	0.778372i	$0.283954\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	12.0000	+	20.7846i	0.680458	+	1.17859i	0.974841	+	0.222900i	$0.0715523\pi$
$311$	12.0000	+	20.7846i	0.680458	+	1.17859i	−0.294384	+	0.955687i	$0.595114\pi$
$312$		0			0
$313$	3.00000	−	5.19615i	0.169570	−	0.293704i	−0.768699	−	0.639611i	$-0.779095\pi$
$313$	3.00000	−	5.19615i	0.169570	−	0.293704i	0.938269	+	0.345907i	$0.112429\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−6.00000	+	10.3923i	−0.336994	+	0.583690i	−0.983866	−	0.178908i	$-0.942743\pi$
$317$	−6.00000	+	10.3923i	−0.336994	+	0.583690i	0.646872	+	0.762598i	$0.276077\pi$
$318$		0			0
$319$	−12.0000	−	20.7846i	−0.671871	−	1.16371i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	4.00000			0.222566
$324$		0			0
$325$	11.0000			0.610170
$326$		0			0
$327$		0			0
$328$		0			0
$329$	4.00000	+	6.92820i	0.220527	+	0.381964i
$330$		0			0
$331$	−1.00000	+	1.73205i	−0.0549650	+	0.0952021i	−0.892199	−	0.451643i	$-0.850838\pi$
$331$	−1.00000	+	1.73205i	−0.0549650	+	0.0952021i	0.837234	+	0.546845i	$0.184171\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−4.00000	+	6.92820i	−0.218543	+	0.378528i
$336$		0			0
$337$	15.0000	+	25.9808i	0.817102	+	1.41526i	0.907809	+	0.419385i	$0.137754\pi$
$337$	15.0000	+	25.9808i	0.817102	+	1.41526i	−0.0907066	+	0.995878i	$0.528913\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−40.0000			−2.16612
$342$		0			0
$343$	20.0000			1.07990
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.00000	+	10.3923i	0.322097	+	0.557888i	0.980921	−	0.194409i	$-0.0622790\pi$
$347$	6.00000	+	10.3923i	0.322097	+	0.557888i	−0.658824	+	0.752297i	$0.728946\pi$
$348$		0			0
$349$	−9.00000	+	15.5885i	−0.481759	+	0.834431i	−0.999781	−	0.0209364i	$-0.993335\pi$
$349$	−9.00000	+	15.5885i	−0.481759	+	0.834431i	0.518022	+	0.855367i	$0.326669\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−8.00000	+	13.8564i	−0.425797	+	0.737502i	−0.996495	−	0.0836583i	$-0.973340\pi$
$353$	−8.00000	+	13.8564i	−0.425797	+	0.737502i	0.570697	+	0.821160i	$0.306673\pi$
$354$		0			0
$355$	32.0000	+	55.4256i	1.69838	+	2.94169i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	12.0000			0.633336			0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000			0.633336			0.316668	+	0.948536i	$0.397436\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		0			0
$363$		0			0
$364$		0			0
$365$	20.0000	+	34.6410i	1.04685	+	1.81319i
$366$		0			0
$367$	−2.00000	+	3.46410i	−0.104399	+	0.180825i	−0.913493	−	0.406855i	$-0.866625\pi$
$367$	−2.00000	+	3.46410i	−0.104399	+	0.180825i	0.809093	+	0.587680i	$0.199959\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	10.0000	−	17.3205i	0.519174	−	0.899236i
$372$		0			0
$373$	9.00000	+	15.5885i	0.466002	+	0.807140i	0.999246	−	0.0388219i	$-0.0123605\pi$
$373$	9.00000	+	15.5885i	0.466002	+	0.807140i	−0.533244	+	0.845962i	$0.679027\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	6.00000			0.309016
$378$		0			0
$379$	6.00000			0.308199			0.154100	−	0.988055i	$-0.450752\pi$
$379$	6.00000			0.308199			0.154100	+	0.988055i	$0.450752\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	6.00000	+	10.3923i	0.306586	+	0.531022i	0.977613	−	0.210411i	$-0.0674801\pi$
$383$	6.00000	+	10.3923i	0.306586	+	0.531022i	−0.671027	+	0.741433i	$0.734147\pi$
$384$		0			0
$385$	16.0000	−	27.7128i	0.815436	−	1.41238i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−13.0000	+	22.5167i	−0.659126	+	1.14164i	0.321716	+	0.946836i	$0.395740\pi$
$389$	−13.0000	+	22.5167i	−0.659126	+	1.14164i	−0.980842	+	0.194804i	$0.937593\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−64.0000			−3.22019
$396$		0			0
$397$	18.0000			0.903394			0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000			0.903394			0.451697	+	0.892171i	$0.350819\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−6.00000	−	10.3923i	−0.299626	−	0.518967i	0.676425	−	0.736512i	$-0.263528\pi$
$401$	−6.00000	−	10.3923i	−0.299626	−	0.518967i	−0.976050	+	0.217545i	$0.930195\pi$
$402$		0			0
$403$	5.00000	−	8.66025i	0.249068	−	0.431398i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−20.0000	+	34.6410i	−0.991363	+	1.71709i
$408$		0			0
$409$	−7.00000	−	12.1244i	−0.346128	−	0.599511i	0.639430	−	0.768849i	$-0.279170\pi$
$409$	−7.00000	−	12.1244i	−0.346128	−	0.599511i	−0.985558	+	0.169338i	$0.945837\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−16.0000			−0.787309
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−10.0000	−	17.3205i	−0.488532	−	0.846162i	0.511381	−	0.859354i	$-0.329134\pi$
$419$	−10.0000	−	17.3205i	−0.488532	−	0.846162i	−0.999913	+	0.0131919i	$0.995801\pi$
$420$		0			0
$421$	1.00000	−	1.73205i	0.0487370	−	0.0844150i	−0.840628	−	0.541613i	$-0.817814\pi$
$421$	1.00000	−	1.73205i	0.0487370	−	0.0844150i	0.889365	+	0.457198i	$0.151147\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	11.0000	−	19.0526i	0.533578	−	0.924185i
$426$		0			0
$427$	−14.0000	−	24.2487i	−0.677507	−	1.17348i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	20.0000			0.963366			0.481683	−	0.876346i	$-0.340026\pi$
$431$	20.0000			0.963366			0.481683	+	0.876346i	$0.340026\pi$
$432$		0			0
$433$	−18.0000			−0.865025			−0.432512	−	0.901628i	$-0.642373\pi$
$433$	−18.0000			−0.865025			−0.432512	+	0.901628i	$0.642373\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−8.00000	+	13.8564i	−0.381819	+	0.661330i	−0.991322	−	0.131453i	$-0.958036\pi$
$439$	−8.00000	+	13.8564i	−0.381819	+	0.661330i	0.609503	+	0.792784i	$0.291369\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.00000	+	3.46410i	−0.0950229	+	0.164584i	−0.909618	−	0.415445i	$-0.863626\pi$
$443$	−2.00000	+	3.46410i	−0.0950229	+	0.164584i	0.814595	+	0.580030i	$0.196959\pi$
$444$		0			0
$445$	−8.00000	−	13.8564i	−0.379236	−	0.656857i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−12.0000			−0.566315			−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000			−0.566315			−0.283158	+	0.959073i	$0.591382\pi$
$450$		0			0
$451$	−32.0000			−1.50682
$452$		0			0
$453$		0			0
$454$		0			0
$455$	4.00000	+	6.92820i	0.187523	+	0.324799i
$456$		0			0
$457$	9.00000	−	15.5885i	0.421002	−	0.729197i	−0.575036	−	0.818128i	$-0.695012\pi$
$457$	9.00000	−	15.5885i	0.421002	−	0.729197i	0.996038	+	0.0889312i	$0.0283451\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	6.00000	−	10.3923i	0.279448	−	0.484018i	−0.691800	−	0.722089i	$-0.743182\pi$
$461$	6.00000	−	10.3923i	0.279448	−	0.484018i	0.971248	+	0.238071i	$0.0765153\pi$
$462$		0			0
$463$	−11.0000	−	19.0526i	−0.511213	−	0.885448i	−0.999916	−	0.0129968i	$-0.995863\pi$
$463$	−11.0000	−	19.0526i	−0.511213	−	0.885448i	0.488702	−	0.872451i	$-0.337470\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	28.0000			1.29569			0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000			1.29569			0.647843	+	0.761774i	$0.275671\pi$
$468$		0			0
$469$	−4.00000			−0.184703
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−8.00000	−	13.8564i	−0.367840	−	0.637118i
$474$		0			0
$475$	11.0000	−	19.0526i	0.504715	−	0.874191i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−8.00000	+	13.8564i	−0.365529	+	0.633115i	−0.988861	−	0.148842i	$-0.952445\pi$
$479$	−8.00000	+	13.8564i	−0.365529	+	0.633115i	0.623332	+	0.781958i	$0.285779\pi$
$480$		0			0
$481$	−5.00000	−	8.66025i	−0.227980	−	0.394874i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−8.00000			−0.363261
$486$		0			0
$487$	−26.0000			−1.17817			−0.589086	−	0.808070i	$-0.700512\pi$
$487$	−26.0000			−1.17817			−0.589086	+	0.808070i	$0.700512\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	6.00000	+	10.3923i	0.270776	+	0.468998i	0.969061	−	0.246822i	$-0.0793863\pi$
$491$	6.00000	+	10.3923i	0.270776	+	0.468998i	−0.698285	+	0.715820i	$0.746053\pi$
$492$		0			0
$493$	6.00000	−	10.3923i	0.270226	−	0.468046i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−16.0000	+	27.7128i	−0.717698	+	1.24309i
$498$		0			0
$499$	3.00000	+	5.19615i	0.134298	+	0.232612i	0.925329	−	0.379165i	$-0.123789\pi$
$499$	3.00000	+	5.19615i	0.134298	+	0.232612i	−0.791031	+	0.611776i	$0.790455\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−16.0000			−0.713405			−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000			−0.713405			−0.356702	+	0.934218i	$0.616099\pi$
$504$		0			0
$505$	−40.0000			−1.77998
$506$		0			0
$507$		0			0
$508$		0			0
$509$	6.00000	+	10.3923i	0.265945	+	0.460631i	0.967811	−	0.251679i	$-0.0809826\pi$
$509$	6.00000	+	10.3923i	0.265945	+	0.460631i	−0.701866	+	0.712309i	$0.747649\pi$
$510$		0			0
$511$	−10.0000	+	17.3205i	−0.442374	+	0.766214i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	16.0000	−	27.7128i	0.705044	−	1.22117i
$516$		0			0
$517$	−8.00000	−	13.8564i	−0.351840	−	0.609404i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	2.00000			0.0876216			0.0438108	−	0.999040i	$-0.486050\pi$
$521$	2.00000			0.0876216			0.0438108	+	0.999040i	$0.486050\pi$
$522$		0			0
$523$	16.0000			0.699631			0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000			0.699631			0.349816	+	0.936819i	$0.386244\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−10.0000	−	17.3205i	−0.435607	−	0.754493i
$528$		0			0
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	4.00000	−	6.92820i	0.173259	−	0.300094i
$534$		0			0
$535$	24.0000	+	41.5692i	1.03761	+	1.79719i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−12.0000			−0.516877
$540$		0			0
$541$	38.0000			1.63375			0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000			1.63375			0.816874	+	0.576816i	$0.195705\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4.00000	+	6.92820i	0.171341	+	0.296772i
$546$		0			0
$547$	−16.0000	+	27.7128i	−0.684111	+	1.18491i	0.289605	+	0.957146i	$0.406476\pi$
$547$	−16.0000	+	27.7128i	−0.684111	+	1.18491i	−0.973715	+	0.227768i	$0.926857\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	6.00000	−	10.3923i	0.255609	−	0.442727i
$552$		0			0
$553$	−16.0000	−	27.7128i	−0.680389	−	1.17847i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−24.0000			−1.01691			−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000			−1.01691			−0.508456	+	0.861088i	$0.669784\pi$
$558$		0			0
$559$	4.00000			0.169182
$560$		0			0
$561$		0			0
$562$		0			0
$563$	14.0000	+	24.2487i	0.590030	+	1.02196i	0.994228	+	0.107290i	$0.0342173\pi$
$563$	14.0000	+	24.2487i	0.590030	+	1.02196i	−0.404198	+	0.914671i	$0.632449\pi$
$564$		0			0
$565$	12.0000	−	20.7846i	0.504844	−	0.874415i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3.00000	−	5.19615i	0.125767	−	0.217834i	−0.796266	−	0.604947i	$-0.793194\pi$
$569$	3.00000	−	5.19615i	0.125767	−	0.217834i	0.922032	+	0.387113i	$0.126528\pi$
$570$		0			0
$571$	4.00000	+	6.92820i	0.167395	+	0.289936i	0.937503	−	0.347977i	$-0.113131\pi$
$571$	4.00000	+	6.92820i	0.167395	+	0.289936i	−0.770108	+	0.637913i	$0.779798\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−30.0000			−1.24892			−0.624458	−	0.781058i	$-0.714680\pi$
$577$	−30.0000			−1.24892			−0.624458	+	0.781058i	$0.714680\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−20.0000	+	34.6410i	−0.828315	+	1.43468i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−4.00000	+	6.92820i	−0.165098	+	0.285958i	−0.936690	−	0.350160i	$-0.886127\pi$
$587$	−4.00000	+	6.92820i	−0.165098	+	0.285958i	0.771592	+	0.636117i	$0.219461\pi$
$588$		0			0
$589$	−10.0000	−	17.3205i	−0.412043	−	0.713679i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	36.0000			1.47834			0.739171	−	0.673517i	$-0.235217\pi$
$593$	36.0000			1.47834			0.739171	+	0.673517i	$0.235217\pi$
$594$		0			0
$595$	16.0000			0.655936
$596$		0			0
$597$		0			0
$598$		0			0
$599$	12.0000	+	20.7846i	0.490307	+	0.849236i	0.999938	−	0.0111569i	$-0.00355143\pi$
$599$	12.0000	+	20.7846i	0.490307	+	0.849236i	−0.509631	+	0.860393i	$0.670218\pi$
$600$		0			0
$601$	11.0000	−	19.0526i	0.448699	−	0.777170i	−0.549602	−	0.835426i	$-0.685221\pi$
$601$	11.0000	−	19.0526i	0.448699	−	0.777170i	0.998302	+	0.0582563i	$0.0185541\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−10.0000	+	17.3205i	−0.406558	+	0.704179i
$606$		0			0
$607$	2.00000	+	3.46410i	0.0811775	+	0.140604i	0.903756	−	0.428048i	$-0.140799\pi$
$607$	2.00000	+	3.46410i	0.0811775	+	0.140604i	−0.822578	+	0.568652i	$0.807465\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	4.00000			0.161823
$612$		0			0
$613$	−14.0000			−0.565455			−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000			−0.565455			−0.282727	+	0.959200i	$0.591239\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−24.0000	−	41.5692i	−0.966204	−	1.67351i	−0.706346	−	0.707867i	$-0.749658\pi$
$617$	−24.0000	−	41.5692i	−0.966204	−	1.67351i	−0.259858	−	0.965647i	$-0.583676\pi$
$618$		0			0
$619$	−7.00000	+	12.1244i	−0.281354	+	0.487319i	−0.971718	−	0.236143i	$-0.924117\pi$
$619$	−7.00000	+	12.1244i	−0.281354	+	0.487319i	0.690365	+	0.723462i	$0.257450\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	4.00000	−	6.92820i	0.160257	−	0.277573i
$624$		0			0
$625$	−20.5000	−	35.5070i	−0.820000	−	1.42028i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−20.0000			−0.797452
$630$		0			0
$631$	−30.0000			−1.19428			−0.597141	−	0.802137i	$-0.703697\pi$
$631$	−30.0000			−1.19428			−0.597141	+	0.802137i	$0.703697\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−24.0000	−	41.5692i	−0.952411	−	1.64962i
$636$		0			0
$637$	1.50000	−	2.59808i	0.0594322	−	0.102940i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	13.0000	−	22.5167i	0.513469	−	0.889355i	−0.486409	−	0.873731i	$-0.661693\pi$
$641$	13.0000	−	22.5167i	0.513469	−	0.889355i	0.999878	−	0.0156233i	$-0.00497325\pi$
$642$		0			0
$643$	−13.0000	−	22.5167i	−0.512670	−	0.887970i	−0.999892	−	0.0146923i	$-0.995323\pi$
$643$	−13.0000	−	22.5167i	−0.512670	−	0.887970i	0.487222	−	0.873278i	$-0.338010\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	24.0000			0.943537			0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000			0.943537			0.471769	+	0.881722i	$0.343616\pi$
$648$		0			0
$649$	32.0000			1.25611
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−23.0000	−	39.8372i	−0.900060	−	1.55895i	−0.827415	−	0.561591i	$-0.810189\pi$
$653$	−23.0000	−	39.8372i	−0.900060	−	1.55895i	−0.0726446	−	0.997358i	$-0.523144\pi$
$654$		0			0
$655$	8.00000	−	13.8564i	0.312586	−	0.541415i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−6.00000	+	10.3923i	−0.233727	+	0.404827i	−0.958902	−	0.283738i	$-0.908425\pi$
$659$	−6.00000	+	10.3923i	−0.233727	+	0.404827i	0.725175	+	0.688565i	$0.241759\pi$
$660$		0			0
$661$	−5.00000	−	8.66025i	−0.194477	−	0.336845i	0.752252	−	0.658876i	$-0.228968\pi$
$661$	−5.00000	−	8.66025i	−0.194477	−	0.336845i	−0.946729	+	0.322031i	$0.895634\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	16.0000			0.620453
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	28.0000	+	48.4974i	1.08093	+	1.87222i
$672$		0			0
$673$	−11.0000	+	19.0526i	−0.424019	+	0.734422i	−0.996328	−	0.0856156i	$-0.972714\pi$
$673$	−11.0000	+	19.0526i	−0.424019	+	0.734422i	0.572309	+	0.820038i	$0.306048\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	19.0000	−	32.9090i	0.730229	−	1.26479i	−0.226556	−	0.973998i	$-0.572747\pi$
$677$	19.0000	−	32.9090i	0.730229	−	1.26479i	0.956785	−	0.290796i	$-0.0939201\pi$
$678$		0			0
$679$	−2.00000	−	3.46410i	−0.0767530	−	0.132940i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−20.0000			−0.765279			−0.382639	−	0.923898i	$-0.624985\pi$
$683$	−20.0000			−0.765279			−0.382639	+	0.923898i	$0.624985\pi$
$684$		0			0
$685$	−32.0000			−1.22266
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−5.00000	−	8.66025i	−0.190485	−	0.329929i
$690$		0			0
$691$	−5.00000	+	8.66025i	−0.190209	+	0.329452i	−0.945319	−	0.326146i	$-0.894250\pi$
$691$	−5.00000	+	8.66025i	−0.190209	+	0.329452i	0.755110	+	0.655598i	$0.227583\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−8.00000	−	13.8564i	−0.303022	−	0.524849i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	2.00000			0.0755390			0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000			0.0755390			0.0377695	+	0.999286i	$0.487975\pi$
$702$		0			0
$703$	−20.0000			−0.754314
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−10.0000	−	17.3205i	−0.376089	−	0.651405i
$708$		0			0
$709$	−3.00000	+	5.19615i	−0.112667	+	0.195146i	−0.916845	−	0.399244i	$-0.869273\pi$
$709$	−3.00000	+	5.19615i	−0.112667	+	0.195146i	0.804178	+	0.594389i	$0.202606\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−8.00000	−	13.8564i	−0.299183	−	0.518200i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	48.0000			1.79010			0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000			1.79010			0.895049	+	0.445968i	$0.147140\pi$
$720$		0			0
$721$	16.0000			0.595871
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−33.0000	−	57.1577i	−1.22559	−	2.12278i
$726$		0			0
$727$	−22.0000	+	38.1051i	−0.815935	+	1.41324i	0.0927199	+	0.995692i	$0.470444\pi$
$727$	−22.0000	+	38.1051i	−0.815935	+	1.41324i	−0.908655	+	0.417548i	$0.862889\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4.00000	−	6.92820i	0.147945	−	0.256249i
$732$		0			0
$733$	−11.0000	−	19.0526i	−0.406294	−	0.703722i	0.588177	−	0.808732i	$-0.299846\pi$
$733$	−11.0000	−	19.0526i	−0.406294	−	0.703722i	−0.994471	+	0.105010i	$0.966513\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	8.00000			0.294684
$738$		0			0
$739$	22.0000			0.809283			0.404642	−	0.914475i	$-0.367396\pi$
$739$	22.0000			0.809283			0.404642	+	0.914475i	$0.367396\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	24.0000	+	41.5692i	0.880475	+	1.52503i	0.850814	+	0.525467i	$0.176109\pi$
$743$	24.0000	+	41.5692i	0.880475	+	1.52503i	0.0296605	+	0.999560i	$0.490557\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−12.0000	+	20.7846i	−0.438470	+	0.759453i
$750$		0			0
$751$	−4.00000	−	6.92820i	−0.145962	−	0.252814i	0.783769	−	0.621052i	$-0.213294\pi$
$751$	−4.00000	−	6.92820i	−0.145962	−	0.252814i	−0.929731	+	0.368238i	$0.879961\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	72.0000			2.62035
$756$		0			0
$757$	18.0000			0.654221			0.327111	−	0.944986i	$-0.393925\pi$
$757$	18.0000			0.654221			0.327111	+	0.944986i	$0.393925\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	20.0000	+	34.6410i	0.724999	+	1.25574i	0.958974	+	0.283493i	$0.0914933\pi$
$761$	20.0000	+	34.6410i	0.724999	+	1.25574i	−0.233975	+	0.972243i	$0.575173\pi$
$762$		0			0
$763$	−2.00000	+	3.46410i	−0.0724049	+	0.125409i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−4.00000	+	6.92820i	−0.144432	+	0.250163i
$768$		0			0
$769$	11.0000	+	19.0526i	0.396670	+	0.687053i	0.993313	−	0.115454i	$-0.0368323\pi$
$769$	11.0000	+	19.0526i	0.396670	+	0.687053i	−0.596643	+	0.802507i	$0.703499\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	48.0000			1.72644			0.863220	−	0.504828i	$-0.168444\pi$
$773$	48.0000			1.72644			0.863220	+	0.504828i	$0.168444\pi$
$774$		0			0
$775$	−110.000			−3.95132
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−8.00000	−	13.8564i	−0.286630	−	0.496457i
$780$		0			0
$781$	32.0000	−	55.4256i	1.14505	−	1.98328i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−4.00000	+	6.92820i	−0.142766	+	0.247278i
$786$		0			0
$787$	25.0000	+	43.3013i	0.891154	+	1.54352i	0.838494	+	0.544911i	$0.183437\pi$
$787$	25.0000	+	43.3013i	0.891154	+	1.54352i	0.0526599	+	0.998613i	$0.483230\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	12.0000			0.426671
$792$		0			0
$793$	−14.0000			−0.497155
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−13.0000	−	22.5167i	−0.460484	−	0.797581i	0.538501	−	0.842625i	$-0.318991\pi$
$797$	−13.0000	−	22.5167i	−0.460484	−	0.797581i	−0.998985	+	0.0450436i	$0.985657\pi$
$798$		0			0
$799$	4.00000	−	6.92820i	0.141510	−	0.245102i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	20.0000	−	34.6410i	0.705785	−	1.22245i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−38.0000			−1.33601			−0.668004	−	0.744157i	$-0.732851\pi$
$809$	−38.0000			−1.33601			−0.668004	+	0.744157i	$0.732851\pi$
$810$		0			0
$811$	6.00000			0.210688			0.105344	−	0.994436i	$-0.466406\pi$
$811$	6.00000			0.210688			0.105344	+	0.994436i	$0.466406\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4.00000	−	6.92820i	−0.140114	−	0.242684i
$816$		0			0
$817$	4.00000	−	6.92820i	0.139942	−	0.242387i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	12.0000	−	20.7846i	0.418803	−	0.725388i	−0.577016	−	0.816733i	$-0.695783\pi$
$821$	12.0000	−	20.7846i	0.418803	−	0.725388i	0.995819	+	0.0913446i	$0.0291165\pi$
$822$		0			0
$823$	−16.0000	−	27.7128i	−0.557725	−	0.966008i	−0.997686	−	0.0679910i	$-0.978341\pi$
$823$	−16.0000	−	27.7128i	−0.557725	−	0.966008i	0.439961	−	0.898017i	$-0.354992\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	20.0000			0.695468			0.347734	−	0.937593i	$-0.386951\pi$
$827$	20.0000			0.695468			0.347734	+	0.937593i	$0.386951\pi$
$828$		0			0
$829$	−14.0000			−0.486240			−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000			−0.486240			−0.243120	+	0.969996i	$0.578171\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−3.00000	−	5.19615i	−0.103944	−	0.180036i
$834$		0			0
$835$	−24.0000	+	41.5692i	−0.830554	+	1.43856i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−8.00000	+	13.8564i	−0.276191	+	0.478376i	−0.970435	−	0.241363i	$-0.922405\pi$
$839$	−8.00000	+	13.8564i	−0.276191	+	0.478376i	0.694244	+	0.719740i	$0.255739\pi$
$840$		0			0
$841$	−3.50000	−	6.06218i	−0.120690	−	0.209041i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4.00000			0.137604
$846$		0			0
$847$	−10.0000			−0.343604
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	7.00000	−	12.1244i	0.239675	−	0.415130i	−0.720946	−	0.692992i	$-0.756292\pi$
$853$	7.00000	−	12.1244i	0.239675	−	0.415130i	0.960621	+	0.277862i	$0.0896256\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−9.00000	+	15.5885i	−0.307434	+	0.532492i	−0.977800	−	0.209539i	$-0.932804\pi$
$857$	−9.00000	+	15.5885i	−0.307434	+	0.532492i	0.670366	+	0.742030i	$0.266137\pi$
$858$		0			0
$859$	−26.0000	−	45.0333i	−0.887109	−	1.53652i	−0.843278	−	0.537478i	$-0.819377\pi$
$859$	−26.0000	−	45.0333i	−0.887109	−	1.53652i	−0.0438309	−	0.999039i	$-0.513956\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−4.00000			−0.136162			−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000			−0.136162			−0.0680808	+	0.997680i	$0.521688\pi$
$864$		0			0
$865$	−8.00000			−0.272008
$866$		0			0
$867$		0			0
$868$		0			0
$869$	32.0000	+	55.4256i	1.08553	+	1.88019i
$870$		0			0
$871$	−1.00000	+	1.73205i	−0.0338837	+	0.0586883i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	24.0000	−	41.5692i	0.811348	−	1.40530i
$876$		0			0
$877$	−5.00000	−	8.66025i	−0.168838	−	0.292436i	0.769174	−	0.639040i	$-0.220668\pi$
$877$	−5.00000	−	8.66025i	−0.168838	−	0.292436i	−0.938012	+	0.346604i	$0.887335\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−14.0000			−0.471672			−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000			−0.471672			−0.235836	+	0.971793i	$0.575783\pi$
$882$		0			0
$883$	−32.0000			−1.07689			−0.538443	−	0.842662i	$-0.680987\pi$
$883$	−32.0000			−1.07689			−0.538443	+	0.842662i	$0.680987\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	24.0000	+	41.5692i	0.805841	+	1.39576i	0.915722	+	0.401813i	$0.131620\pi$
$887$	24.0000	+	41.5692i	0.805841	+	1.39576i	−0.109881	+	0.993945i	$0.535047\pi$
$888$		0			0
$889$	12.0000	−	20.7846i	0.402467	−	0.697093i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	4.00000	−	6.92820i	0.133855	−	0.231843i
$894$		0			0
$895$	24.0000	+	41.5692i	0.802232	+	1.38951i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−60.0000			−2.00111
$900$		0			0
$901$	−20.0000			−0.666297
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−4.00000	−	6.92820i	−0.132964	−	0.230301i
$906$		0			0
$907$	28.0000	−	48.4974i	0.929725	−	1.61033i	0.145944	−	0.989293i	$-0.453378\pi$
$907$	28.0000	−	48.4974i	0.929725	−	1.61033i	0.783781	−	0.621038i	$-0.213289\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−28.0000	+	48.4974i	−0.927681	+	1.60679i	−0.140490	+	0.990082i	$0.544868\pi$
$911$	−28.0000	+	48.4974i	−0.927681	+	1.60679i	−0.787191	+	0.616709i	$0.788465\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	8.00000			0.264183
$918$		0			0
$919$	56.0000			1.84727			0.923635	−	0.383274i	$-0.125203\pi$
$919$	56.0000			1.84727			0.923635	+	0.383274i	$0.125203\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	8.00000	+	13.8564i	0.263323	+	0.456089i
$924$		0			0
$925$	−55.0000	+	95.2628i	−1.80839	+	3.13222i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	8.00000	−	13.8564i	0.262471	−	0.454614i	−0.704427	−	0.709777i	$-0.748796\pi$
$929$	8.00000	−	13.8564i	0.262471	−	0.454614i	0.966898	+	0.255163i	$0.0821291\pi$
$930$		0			0
$931$	−3.00000	−	5.19615i	−0.0983210	−	0.170297i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−32.0000			−1.04651
$936$		0			0
$937$	−34.0000			−1.11073			−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000			−1.11073			−0.555366	+	0.831606i	$0.687422\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	4.00000	+	6.92820i	0.130396	+	0.225853i	0.923829	−	0.382804i	$-0.125042\pi$
$941$	4.00000	+	6.92820i	0.130396	+	0.225853i	−0.793433	+	0.608657i	$0.791708\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	4.00000	−	6.92820i	0.129983	−	0.225136i	−0.793687	−	0.608326i	$-0.791841\pi$
$947$	4.00000	−	6.92820i	0.129983	−	0.225136i	0.923670	+	0.383190i	$0.125175\pi$
$948$		0			0
$949$	5.00000	+	8.66025i	0.162307	+	0.281124i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	42.0000			1.36051			0.680257	−	0.732974i	$-0.261868\pi$
$953$	42.0000			1.36051			0.680257	+	0.732974i	$0.261868\pi$
$954$		0			0
$955$	32.0000			1.03550
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−8.00000	−	13.8564i	−0.258333	−	0.447447i
$960$		0			0
$961$	−34.5000	+	59.7558i	−1.11290	+	1.92760i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	28.0000	−	48.4974i	0.901352	−	1.56119i
$966$		0			0
$967$	−11.0000	−	19.0526i	−0.353736	−	0.612689i	0.633165	−	0.774017i	$-0.281756\pi$
$967$	−11.0000	−	19.0526i	−0.353736	−	0.612689i	−0.986901	+	0.161328i	$0.948422\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	28.0000			0.898563			0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000			0.898563			0.449281	+	0.893390i	$0.351680\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−6.00000	−	10.3923i	−0.191957	−	0.332479i	0.753942	−	0.656941i	$-0.228150\pi$
$977$	−6.00000	−	10.3923i	−0.191957	−	0.332479i	−0.945899	+	0.324462i	$0.894817\pi$
$978$		0			0
$979$	−8.00000	+	13.8564i	−0.255681	+	0.442853i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	12.0000	−	20.7846i	0.382741	−	0.662926i	−0.608712	−	0.793391i	$-0.708314\pi$
$983$	12.0000	−	20.7846i	0.382741	−	0.662926i	0.991453	+	0.130465i	$0.0416470\pi$
$984$		0			0
$985$	−24.0000	−	41.5692i	−0.764704	−	1.32451i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−36.0000			−1.14358			−0.571789	−	0.820401i	$-0.693750\pi$
$991$	−36.0000			−1.14358			−0.571789	+	0.820401i	$0.693750\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−8.00000	−	13.8564i	−0.253617	−	0.439278i
$996$		0			0
$997$	−11.0000	+	19.0526i	−0.348373	+	0.603401i	−0.985961	−	0.166978i	$-0.946599\pi$
$997$	−11.0000	+	19.0526i	−0.348373	+	0.603401i	0.637587	+	0.770378i	$0.279933\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4212.2.i.b.2809.1		2
3.2	odd	2		4212.2.i.l.2809.1		2
9.2	odd	6		156.2.a.a.1.1	✓	1
9.4	even	3	inner	4212.2.i.b.1405.1		2
9.5	odd	6		4212.2.i.l.1405.1		2
9.7	even	3		468.2.a.d.1.1		1
36.7	odd	6		1872.2.a.s.1.1		1
36.11	even	6		624.2.a.e.1.1		1
45.2	even	12		3900.2.h.b.1249.2		2
45.29	odd	6		3900.2.a.m.1.1		1
45.38	even	12		3900.2.h.b.1249.1		2
63.20	even	6		7644.2.a.k.1.1		1
72.11	even	6		2496.2.a.o.1.1		1
72.29	odd	6		2496.2.a.bc.1.1		1
72.43	odd	6		7488.2.a.d.1.1		1
72.61	even	6		7488.2.a.c.1.1		1
117.2	even	12		2028.2.q.h.1837.2		4
117.11	even	12		2028.2.q.h.1837.1		4
117.20	even	12		2028.2.q.h.361.1		4
117.25	even	6		6084.2.a.b.1.1		1
117.29	odd	6		2028.2.i.e.529.1		2
117.34	odd	12		6084.2.b.j.4393.2		2
117.38	odd	6		2028.2.a.c.1.1		1
117.47	even	12		2028.2.b.a.337.1		2
117.56	odd	6		2028.2.i.g.2005.1		2
117.70	odd	12		6084.2.b.j.4393.1		2
117.74	odd	6		2028.2.i.e.2005.1		2
117.83	even	12		2028.2.b.a.337.2		2
117.101	odd	6		2028.2.i.g.529.1		2
117.110	even	12		2028.2.q.h.361.2		4
468.155	even	6		8112.2.a.bi.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.2.a.a.1.1	✓	1	9.2	odd	6
468.2.a.d.1.1		1	9.7	even	3
624.2.a.e.1.1		1	36.11	even	6
1872.2.a.s.1.1		1	36.7	odd	6
2028.2.a.c.1.1		1	117.38	odd	6
2028.2.b.a.337.1		2	117.47	even	12
2028.2.b.a.337.2		2	117.83	even	12
2028.2.i.e.529.1		2	117.29	odd	6
2028.2.i.e.2005.1		2	117.74	odd	6
2028.2.i.g.529.1		2	117.101	odd	6
2028.2.i.g.2005.1		2	117.56	odd	6
2028.2.q.h.361.1		4	117.20	even	12
2028.2.q.h.361.2		4	117.110	even	12
2028.2.q.h.1837.1		4	117.11	even	12
2028.2.q.h.1837.2		4	117.2	even	12
2496.2.a.o.1.1		1	72.11	even	6
2496.2.a.bc.1.1		1	72.29	odd	6
3900.2.a.m.1.1		1	45.29	odd	6
3900.2.h.b.1249.1		2	45.38	even	12
3900.2.h.b.1249.2		2	45.2	even	12
4212.2.i.b.1405.1		2	9.4	even	3	inner
4212.2.i.b.2809.1		2	1.1	even	1	trivial
4212.2.i.l.1405.1		2	9.5	odd	6
4212.2.i.l.2809.1		2	3.2	odd	2
6084.2.a.b.1.1		1	117.25	even	6
6084.2.b.j.4393.1		2	117.70	odd	12
6084.2.b.j.4393.2		2	117.34	odd	12
7488.2.a.c.1.1		1	72.61	even	6
7488.2.a.d.1.1		1	72.43	odd	6
7644.2.a.k.1.1		1	63.20	even	6
8112.2.a.bi.1.1		1	468.155	even	6

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:	\( N \)	\(=\)	\( 4212 = 2^{2} \cdot 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4212.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.00000 - 3.46410i) q^{5} +(1.00000 - 1.73205i) q^{7} +O(q^{10})\)	\(q+(-2.00000 - 3.46410i) q^{5} +(1.00000 - 1.73205i) q^{7} +(-2.00000 + 3.46410i) q^{11} +(-0.500000 - 0.866025i) q^{13} -2.00000 q^{17} -2.00000 q^{19} +(-5.50000 + 9.52628i) q^{25} +(-3.00000 + 5.19615i) q^{29} +(5.00000 + 8.66025i) q^{31} -8.00000 q^{35} +10.0000 q^{37} +(4.00000 + 6.92820i) q^{41} +(-2.00000 + 3.46410i) q^{43} +(-2.00000 + 3.46410i) q^{47} +(1.50000 + 2.59808i) q^{49} +10.0000 q^{53} +16.0000 q^{55} +(-4.00000 - 6.92820i) q^{59} +(7.00000 - 12.1244i) q^{61} +(-2.00000 + 3.46410i) q^{65} +(-1.00000 - 1.73205i) q^{67} -16.0000 q^{71} -10.0000 q^{73} +(4.00000 + 6.92820i) q^{77} +(8.00000 - 13.8564i) q^{79} +(4.00000 + 6.92820i) q^{85} +4.00000 q^{89} -2.00000 q^{91} +(4.00000 + 6.92820i) q^{95} +(1.00000 - 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q - 4 * q^5 + 2 * q^7	\( 2 q - 4 q^{5} + 2 q^{7} - 4 q^{11} - q^{13} - 4 q^{17} - 4 q^{19} - 11 q^{25} - 6 q^{29} + 10 q^{31} - 16 q^{35} + 20 q^{37} + 8 q^{41} - 4 q^{43} - 4 q^{47} + 3 q^{49} + 20 q^{53} + 32 q^{55} - 8 q^{59} + 14 q^{61} - 4 q^{65} - 2 q^{67} - 32 q^{71} - 20 q^{73} + 8 q^{77} + 16 q^{79} + 8 q^{85} + 8 q^{89} - 4 q^{91} + 8 q^{95} + 2 q^{97}+O(q^{100}) \) 2 * q - 4 * q^5 + 2 * q^7 - 4 * q^11 - q^13 - 4 * q^17 - 4 * q^19 - 11 * q^25 - 6 * q^29 + 10 * q^31 - 16 * q^35 + 20 * q^37 + 8 * q^41 - 4 * q^43 - 4 * q^47 + 3 * q^49 + 20 * q^53 + 32 * q^55 - 8 * q^59 + 14 * q^61 - 4 * q^65 - 2 * q^67 - 32 * q^71 - 20 * q^73 + 8 * q^77 + 16 * q^79 + 8 * q^85 + 8 * q^89 - 4 * q^91 + 8 * q^95 + 2 * q^97

Introduction

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