LMFDB - Embedded newform 896.2.j.b.223.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [896,2,Mod(223,896)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(896, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 2]))

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

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

chi := DirichletCharacter("896.223");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 896 = 2^{7} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	896.j (of order $4$, 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:	$7.15459602111$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			223.1
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	896.223
Dual form			896.2.j.b.671.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.44949 - 2.44949i) q^{5} +(-1.00000 + 2.44949i) q^{7} -3.00000i q^{9} +O(q^{10})$	$q+(-2.44949 - 2.44949i) q^{5} +(-1.00000 + 2.44949i) q^{7} -3.00000i q^{9} +(-1.00000 - 1.00000i) q^{11} +(-2.44949 + 2.44949i) q^{13} +4.89898i q^{17} +(4.89898 + 4.89898i) q^{19} +4.00000 q^{23} +7.00000i q^{25} +(3.00000 + 3.00000i) q^{29} -4.89898 q^{31} +(8.44949 - 3.55051i) q^{35} +(-5.00000 + 5.00000i) q^{37} -4.89898 q^{41} +(5.00000 + 5.00000i) q^{43} +(-7.34847 + 7.34847i) q^{45} +4.89898 q^{47} +(-5.00000 - 4.89898i) q^{49} +(1.00000 - 1.00000i) q^{53} +4.89898i q^{55} +(-4.89898 + 4.89898i) q^{59} +(2.44949 - 2.44949i) q^{61} +(7.34847 + 3.00000i) q^{63} +12.0000 q^{65} +(-5.00000 + 5.00000i) q^{67} +2.00000 q^{71} -9.79796 q^{73} +(3.44949 - 1.44949i) q^{77} +4.00000i q^{79} -9.00000 q^{81} +(12.0000 - 12.0000i) q^{85} +(-3.55051 - 8.44949i) q^{91} -24.0000i q^{95} +4.89898i q^{97} +(-3.00000 + 3.00000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7}+O(q^{10}) $ 4 * q - 4 * q^7	$ 4 q - 4 q^{7} - 4 q^{11} + 16 q^{23} + 12 q^{29} + 24 q^{35} - 20 q^{37} + 20 q^{43} - 20 q^{49} + 4 q^{53} + 48 q^{65} - 20 q^{67} + 8 q^{71} + 4 q^{77} - 36 q^{81} + 48 q^{85} - 24 q^{91} - 12 q^{99}+O(q^{100}) $ 4 * q - 4 * q^7 - 4 * q^11 + 16 * q^23 + 12 * q^29 + 24 * q^35 - 20 * q^37 + 20 * q^43 - 20 * q^49 + 4 * q^53 + 48 * q^65 - 20 * q^67 + 8 * q^71 + 4 * q^77 - 36 * q^81 + 48 * q^85 - 24 * q^91 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/896\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$645$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{3}{4}\right)$

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		0.707107	−	0.707107i	$-0.250000\pi$
$3$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$4$		0			0
$5$	−2.44949	−	2.44949i	−1.09545	−	1.09545i	−0.994936	−	0.100509i	$-0.967953\pi$
$5$	−2.44949	−	2.44949i	−1.09545	−	1.09545i	−0.100509	−	0.994936i	$-0.532047\pi$
$6$		0			0
$7$	−1.00000	+	2.44949i	−0.377964	+	0.925820i
$7$	−1.00000	+	2.44949i	−0.377964	+	0.925820i
$8$		0			0
$9$		−	3.00000i		−	1.00000i
$10$		0			0
$11$	−1.00000	−	1.00000i	−0.301511	−	0.301511i	0.540094	−	0.841605i	$-0.318389\pi$
$11$	−1.00000	−	1.00000i	−0.301511	−	0.301511i	−0.841605	+	0.540094i	$0.818389\pi$
$12$		0			0
$13$	−2.44949	+	2.44949i	−0.679366	+	0.679366i	−0.959857	−	0.280491i	$-0.909503\pi$
$13$	−2.44949	+	2.44949i	−0.679366	+	0.679366i	0.280491	+	0.959857i	$0.409503\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			4.89898i			1.18818i	0.804400	+	0.594089i	$0.202487\pi$
$17$			4.89898i			1.18818i	−0.804400	+	0.594089i	$0.797513\pi$
$18$		0			0
$19$	4.89898	+	4.89898i	1.12390	+	1.12390i	0.991149	+	0.132754i	$0.0423820\pi$
$19$	4.89898	+	4.89898i	1.12390	+	1.12390i	0.132754	+	0.991149i	$0.457618\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.00000			0.834058			0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000			0.834058			0.417029	+	0.908893i	$0.363071\pi$
$24$		0			0
$25$			7.00000i			1.40000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.00000	+	3.00000i	0.557086	+	0.557086i	0.928477	−	0.371391i	$-0.121119\pi$
$29$	3.00000	+	3.00000i	0.557086	+	0.557086i	−0.371391	+	0.928477i	$0.621119\pi$
$30$		0			0
$31$	−4.89898			−0.879883			−0.439941	−	0.898027i	$-0.645001\pi$
$31$	−4.89898			−0.879883			−0.439941	+	0.898027i	$0.645001\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	8.44949	−	3.55051i	1.42822	−	0.600146i
$36$		0			0
$37$	−5.00000	+	5.00000i	−0.821995	+	0.821995i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	−5.00000	+	5.00000i	−0.821995	+	0.821995i	0.164399	+	0.986394i	$0.447432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.89898			−0.765092			−0.382546	−	0.923936i	$-0.624953\pi$
$41$	−4.89898			−0.765092			−0.382546	+	0.923936i	$0.624953\pi$
$42$		0			0
$43$	5.00000	+	5.00000i	0.762493	+	0.762493i	0.976772	−	0.214280i	$-0.0687403\pi$
$43$	5.00000	+	5.00000i	0.762493	+	0.762493i	−0.214280	+	0.976772i	$0.568740\pi$
$44$		0			0
$45$	−7.34847	+	7.34847i	−1.09545	+	1.09545i
$46$		0			0
$47$	4.89898			0.714590			0.357295	−	0.933992i	$-0.383699\pi$
$47$	4.89898			0.714590			0.357295	+	0.933992i	$0.383699\pi$
$48$		0			0
$49$	−5.00000	−	4.89898i	−0.714286	−	0.699854i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.00000	−	1.00000i	0.137361	−	0.137361i	−0.635083	−	0.772444i	$-0.719034\pi$
$53$	1.00000	−	1.00000i	0.137361	−	0.137361i	0.772444	+	0.635083i	$0.219034\pi$
$54$		0			0
$55$			4.89898i			0.660578i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.89898	+	4.89898i	−0.637793	+	0.637793i	−0.950011	−	0.312218i	$-0.898928\pi$
$59$	−4.89898	+	4.89898i	−0.637793	+	0.637793i	0.312218	+	0.950011i	$0.398928\pi$
$60$		0			0
$61$	2.44949	−	2.44949i	0.313625	−	0.313625i	−0.532687	−	0.846312i	$-0.678818\pi$
$61$	2.44949	−	2.44949i	0.313625	−	0.313625i	0.846312	+	0.532687i	$0.178818\pi$
$62$		0			0
$63$	7.34847	+	3.00000i	0.925820	+	0.377964i
$64$		0			0
$65$	12.0000			1.48842
$66$		0			0
$67$	−5.00000	+	5.00000i	−0.610847	+	0.610847i	−0.943167	−	0.332320i	$-0.892169\pi$
$67$	−5.00000	+	5.00000i	−0.610847	+	0.610847i	0.332320	+	0.943167i	$0.392169\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2.00000			0.237356			0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000			0.237356			0.118678	+	0.992933i	$0.462134\pi$
$72$		0			0
$73$	−9.79796			−1.14676			−0.573382	−	0.819288i	$-0.694369\pi$
$73$	−9.79796			−1.14676			−0.573382	+	0.819288i	$0.694369\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.44949	−	1.44949i	0.393106	−	0.165185i
$78$		0			0
$79$			4.00000i			0.450035i	0.974355	+	0.225018i	$0.0722440\pi$
$79$			4.00000i			0.450035i	−0.974355	+	0.225018i	$0.927756\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$83$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$84$		0			0
$85$	12.0000	−	12.0000i	1.30158	−	1.30158i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	−3.55051	−	8.44949i	−0.372195	−	0.885747i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	24.0000i		−	2.46235i
$96$		0			0
$97$			4.89898i			0.497416i	0.968579	+	0.248708i	$0.0800060\pi$
$97$			4.89898i			0.497416i	−0.968579	+	0.248708i	$0.919994\pi$
$98$		0			0
$99$	−3.00000	+	3.00000i	−0.301511	+	0.301511i
$100$		0			0
$101$	2.44949	+	2.44949i	0.243733	+	0.243733i	0.818393	−	0.574659i	$-0.194865\pi$
$101$	2.44949	+	2.44949i	0.243733	+	0.243733i	−0.574659	+	0.818393i	$0.694865\pi$
$102$		0			0
$103$			9.79796i			0.965422i	0.875780	+	0.482711i	$0.160348\pi$
$103$			9.79796i			0.965422i	−0.875780	+	0.482711i	$0.839652\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.00000	−	3.00000i	−0.290021	−	0.290021i	0.547068	−	0.837088i	$-0.315744\pi$
$107$	−3.00000	−	3.00000i	−0.290021	−	0.290021i	−0.837088	+	0.547068i	$0.815744\pi$
$108$		0			0
$109$	−7.00000	−	7.00000i	−0.670478	−	0.670478i	0.287348	−	0.957826i	$-0.407226\pi$
$109$	−7.00000	−	7.00000i	−0.670478	−	0.670478i	−0.957826	+	0.287348i	$0.907226\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.00000			0.376288			0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000			0.376288			0.188144	+	0.982141i	$0.439753\pi$
$114$		0			0
$115$	−9.79796	−	9.79796i	−0.913664	−	0.913664i
$116$		0			0
$117$	7.34847	+	7.34847i	0.679366	+	0.679366i
$118$		0			0
$119$	−12.0000	−	4.89898i	−1.10004	−	0.449089i
$120$		0			0
$121$		−	9.00000i		−	0.818182i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	4.89898	−	4.89898i	0.438178	−	0.438178i
$126$		0			0
$127$			18.0000i			1.59724i	0.601834	+	0.798621i	$0.294437\pi$
$127$			18.0000i			1.59724i	−0.601834	+	0.798621i	$0.705563\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−9.79796	−	9.79796i	−0.856052	−	0.856052i	0.134819	−	0.990870i	$-0.456955\pi$
$131$	−9.79796	−	9.79796i	−0.856052	−	0.856052i	−0.990870	+	0.134819i	$0.956955\pi$
$132$		0			0
$133$	−16.8990	+	7.10102i	−1.46533	+	0.615737i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			8.00000i			0.683486i	0.939793	+	0.341743i	$0.111017\pi$
$137$			8.00000i			0.683486i	−0.939793	+	0.341743i	$0.888983\pi$
$138$		0			0
$139$	−4.89898	+	4.89898i	−0.415526	+	0.415526i	−0.883658	−	0.468132i	$-0.844927\pi$
$139$	−4.89898	+	4.89898i	−0.415526	+	0.415526i	0.468132	+	0.883658i	$0.344927\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	4.89898			0.409673
$144$		0			0
$145$		−	14.6969i		−	1.22051i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.00000	+	3.00000i	−0.245770	+	0.245770i	−0.819232	−	0.573462i	$-0.805600\pi$
$149$	−3.00000	+	3.00000i	−0.245770	+	0.245770i	0.573462	+	0.819232i	$0.305600\pi$
$150$		0			0
$151$	12.0000			0.976546			0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000			0.976546			0.488273	+	0.872691i	$0.337627\pi$
$152$		0			0
$153$	14.6969			1.18818
$154$		0			0
$155$	12.0000	+	12.0000i	0.963863	+	0.963863i
$156$		0			0
$157$	12.2474	−	12.2474i	0.977453	−	0.977453i	−0.0222985	−	0.999751i	$-0.507098\pi$
$157$	12.2474	−	12.2474i	0.977453	−	0.977453i	0.999751	+	0.0222985i	$0.00709843\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−4.00000	+	9.79796i	−0.315244	+	0.772187i
$162$		0			0
$163$	1.00000	−	1.00000i	0.0783260	−	0.0783260i	−0.666858	−	0.745184i	$-0.732361\pi$
$163$	1.00000	−	1.00000i	0.0783260	−	0.0783260i	0.745184	+	0.666858i	$0.232361\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	19.5959i		−	1.51638i	−0.652035	−	0.758189i	$-0.726085\pi$
$167$		−	19.5959i		−	1.51638i	0.652035	−	0.758189i	$-0.273915\pi$
$168$		0			0
$169$			1.00000i			0.0769231i
$170$		0			0
$171$	14.6969	−	14.6969i	1.12390	−	1.12390i
$172$		0			0
$173$	−2.44949	+	2.44949i	−0.186231	+	0.186231i	−0.794065	−	0.607833i	$-0.792039\pi$
$173$	−2.44949	+	2.44949i	−0.186231	+	0.186231i	0.607833	+	0.794065i	$0.292039\pi$
$174$		0			0
$175$	−17.1464	−	7.00000i	−1.29615	−	0.529150i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−3.00000	+	3.00000i	−0.224231	+	0.224231i	−0.810277	−	0.586047i	$-0.800683\pi$
$179$	−3.00000	+	3.00000i	−0.224231	+	0.224231i	0.586047	+	0.810277i	$0.300683\pi$
$180$		0			0
$181$	2.44949	+	2.44949i	0.182069	+	0.182069i	0.792257	−	0.610188i	$-0.208906\pi$
$181$	2.44949	+	2.44949i	0.182069	+	0.182069i	−0.610188	+	0.792257i	$0.708906\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	24.4949			1.80090
$186$		0			0
$187$	4.89898	−	4.89898i	0.358249	−	0.358249i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			20.0000i			1.44715i	0.690246	+	0.723575i	$0.257502\pi$
$191$			20.0000i			1.44715i	−0.690246	+	0.723575i	$0.742498\pi$
$192$		0			0
$193$	−16.0000			−1.15171			−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000			−1.15171			−0.575853	+	0.817554i	$0.695330\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	15.0000	−	15.0000i	1.06871	−	1.06871i	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	15.0000	−	15.0000i	1.06871	−	1.06871i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$		0			0
$199$		−	9.79796i		−	0.694559i	−0.937762	−	0.347279i	$-0.887106\pi$
$199$		−	9.79796i		−	0.694559i	0.937762	−	0.347279i	$-0.112894\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−10.3485	+	4.34847i	−0.726320	+	0.305203i
$204$		0			0
$205$	12.0000	+	12.0000i	0.838116	+	0.838116i
$206$		0			0
$207$		−	12.0000i		−	0.834058i
$208$		0			0
$209$		−	9.79796i		−	0.677739i
$210$		0			0
$211$	−1.00000	+	1.00000i	−0.0688428	+	0.0688428i	−0.740690	−	0.671847i	$-0.765501\pi$
$211$	−1.00000	+	1.00000i	−0.0688428	+	0.0688428i	0.671847	+	0.740690i	$0.265501\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	24.4949i		−	1.67054i
$216$		0			0
$217$	4.89898	−	12.0000i	0.332564	−	0.814613i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−12.0000	−	12.0000i	−0.807207	−	0.807207i
$222$		0			0
$223$	−9.79796			−0.656120			−0.328060	−	0.944657i	$-0.606395\pi$
$223$	−9.79796			−0.656120			−0.328060	+	0.944657i	$0.606395\pi$
$224$		0			0
$225$	21.0000			1.40000
$226$		0			0
$227$	19.5959	+	19.5959i	1.30063	+	1.30063i	0.927969	+	0.372658i	$0.121554\pi$
$227$	19.5959	+	19.5959i	1.30063	+	1.30063i	0.372658	+	0.927969i	$0.378446\pi$
$228$		0			0
$229$	−7.34847	−	7.34847i	−0.485601	−	0.485601i	0.421314	−	0.906915i	$-0.361569\pi$
$229$	−7.34847	−	7.34847i	−0.485601	−	0.485601i	−0.906915	+	0.421314i	$0.861569\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	4.00000i		−	0.262049i	−0.991379	−	0.131024i	$-0.958173\pi$
$233$		−	4.00000i		−	0.262049i	0.991379	−	0.131024i	$-0.0418266\pi$
$234$		0			0
$235$	−12.0000	−	12.0000i	−0.782794	−	0.782794i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			14.0000i			0.905585i	0.891616	+	0.452792i	$0.149572\pi$
$239$			14.0000i			0.905585i	−0.891616	+	0.452792i	$0.850428\pi$
$240$		0			0
$241$		−	24.4949i		−	1.57786i	−0.614486	−	0.788928i	$-0.710637\pi$
$241$		−	24.4949i		−	1.57786i	0.614486	−	0.788928i	$-0.289363\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.247449	+	24.2474i	0.0158089	+	1.54911i
$246$		0			0
$247$	−24.0000			−1.52708
$248$		0			0
$249$		0			0
$250$		0			0
$251$	14.6969	−	14.6969i	0.927663	−	0.927663i	−0.0698920	−	0.997555i	$-0.522265\pi$
$251$	14.6969	−	14.6969i	0.927663	−	0.927663i	0.997555	+	0.0698920i	$0.0222655\pi$
$252$		0			0
$253$	−4.00000	−	4.00000i	−0.251478	−	0.251478i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			29.3939i			1.83354i	0.399416	+	0.916770i	$0.369213\pi$
$257$			29.3939i			1.83354i	−0.399416	+	0.916770i	$0.630787\pi$
$258$		0			0
$259$	−7.24745	−	17.2474i	−0.450335	−	1.07170i
$260$		0			0
$261$	9.00000	−	9.00000i	0.557086	−	0.557086i
$262$		0			0
$263$	−26.0000			−1.60323			−0.801614	−	0.597841i	$-0.796025\pi$
$263$	−26.0000			−1.60323			−0.801614	+	0.597841i	$0.796025\pi$
$264$		0			0
$265$	−4.89898			−0.300942
$266$		0			0
$267$		0			0
$268$		0			0
$269$	7.34847	−	7.34847i	0.448044	−	0.448044i	−0.446660	−	0.894704i	$-0.647387\pi$
$269$	7.34847	−	7.34847i	0.448044	−	0.448044i	0.894704	+	0.446660i	$0.147387\pi$
$270$		0			0
$271$	−29.3939			−1.78555			−0.892775	−	0.450502i	$-0.851245\pi$
$271$	−29.3939			−1.78555			−0.892775	+	0.450502i	$0.851245\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	7.00000	−	7.00000i	0.422116	−	0.422116i
$276$		0			0
$277$	−5.00000	+	5.00000i	−0.300421	+	0.300421i	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−5.00000	+	5.00000i	−0.300421	+	0.300421i	0.540758	+	0.841178i	$0.318138\pi$
$278$		0			0
$279$			14.6969i			0.879883i
$280$		0			0
$281$		−	20.0000i		−	1.19310i	−0.802576	−	0.596550i	$-0.796538\pi$
$281$		−	20.0000i		−	1.19310i	0.802576	−	0.596550i	$-0.203462\pi$
$282$		0			0
$283$	−14.6969	+	14.6969i	−0.873642	+	0.873642i	−0.992867	−	0.119225i	$-0.961959\pi$
$283$	−14.6969	+	14.6969i	−0.873642	+	0.873642i	0.119225	+	0.992867i	$0.461959\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4.89898	−	12.0000i	0.289178	−	0.708338i
$288$		0			0
$289$	−7.00000			−0.411765
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−12.2474	−	12.2474i	−0.715504	−	0.715504i	0.252177	−	0.967681i	$-0.418853\pi$
$293$	−12.2474	−	12.2474i	−0.715504	−	0.715504i	−0.967681	+	0.252177i	$0.918853\pi$
$294$		0			0
$295$	24.0000			1.39733
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−9.79796	+	9.79796i	−0.566631	+	0.566631i
$300$		0			0
$301$	−17.2474	+	7.24745i	−0.994126	+	0.417736i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−12.0000			−0.687118
$306$		0			0
$307$	−4.89898	−	4.89898i	−0.279600	−	0.279600i	0.553350	−	0.832949i	$-0.313349\pi$
$307$	−4.89898	−	4.89898i	−0.279600	−	0.279600i	−0.832949	+	0.553350i	$0.813349\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			24.4949i			1.38898i	0.719503	+	0.694489i	$0.244370\pi$
$311$			24.4949i			1.38898i	−0.719503	+	0.694489i	$0.755630\pi$
$312$		0			0
$313$	14.6969			0.830720			0.415360	−	0.909657i	$-0.363656\pi$
$313$	14.6969			0.830720			0.415360	+	0.909657i	$0.363656\pi$
$314$		0			0
$315$	−10.6515	−	25.3485i	−0.600146	−	1.42822i
$316$		0			0
$317$	−13.0000	−	13.0000i	−0.730153	−	0.730153i	0.240497	−	0.970650i	$-0.422690\pi$
$317$	−13.0000	−	13.0000i	−0.730153	−	0.730153i	−0.970650	+	0.240497i	$0.922690\pi$
$318$		0			0
$319$		−	6.00000i		−	0.335936i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−24.0000	+	24.0000i	−1.33540	+	1.33540i
$324$		0			0
$325$	−17.1464	−	17.1464i	−0.951113	−	0.951113i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.89898	+	12.0000i	−0.270089	+	0.661581i
$330$		0			0
$331$	19.0000	+	19.0000i	1.04433	+	1.04433i	0.998971	+	0.0453639i	$0.0144447\pi$
$331$	19.0000	+	19.0000i	1.04433	+	1.04433i	0.0453639	+	0.998971i	$0.485555\pi$
$332$		0			0
$333$	15.0000	+	15.0000i	0.821995	+	0.821995i
$334$		0			0
$335$	24.4949			1.33830
$336$		0			0
$337$	28.0000			1.52526			0.762629	−	0.646837i	$-0.223908\pi$
$337$	28.0000			1.52526			0.762629	+	0.646837i	$0.223908\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	4.89898	+	4.89898i	0.265295	+	0.265295i
$342$		0			0
$343$	17.0000	−	7.34847i	0.917914	−	0.396780i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3.00000	−	3.00000i	−0.161048	−	0.161048i	0.621983	−	0.783031i	$-0.286327\pi$
$347$	−3.00000	−	3.00000i	−0.161048	−	0.161048i	−0.783031	+	0.621983i	$0.786327\pi$
$348$		0			0
$349$	7.34847	−	7.34847i	0.393355	−	0.393355i	−0.482527	−	0.875881i	$-0.660281\pi$
$349$	7.34847	−	7.34847i	0.393355	−	0.393355i	0.875881	+	0.482527i	$0.160281\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			9.79796i			0.521493i	0.965407	+	0.260746i	$0.0839686\pi$
$353$			9.79796i			0.521493i	−0.965407	+	0.260746i	$0.916031\pi$
$354$		0			0
$355$	−4.89898	−	4.89898i	−0.260011	−	0.260011i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	20.0000			1.05556			0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000			1.05556			0.527780	+	0.849381i	$0.323025\pi$
$360$		0			0
$361$			29.0000i			1.52632i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	24.0000	+	24.0000i	1.25622	+	1.25622i
$366$		0			0
$367$	29.3939			1.53435			0.767174	−	0.641439i	$-0.221662\pi$
$367$	29.3939			1.53435			0.767174	+	0.641439i	$0.221662\pi$
$368$		0			0
$369$			14.6969i			0.765092i
$370$		0			0
$371$	1.44949	+	3.44949i	0.0752538	+	0.179089i
$372$		0			0
$373$	11.0000	−	11.0000i	0.569558	−	0.569558i	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	11.0000	−	11.0000i	0.569558	−	0.569558i	0.932005	+	0.362446i	$0.118058\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−14.6969			−0.756931
$378$		0			0
$379$	23.0000	+	23.0000i	1.18143	+	1.18143i	0.979374	+	0.202057i	$0.0647626\pi$
$379$	23.0000	+	23.0000i	1.18143	+	1.18143i	0.202057	+	0.979374i	$0.435237\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	14.6969			0.750978			0.375489	−	0.926827i	$-0.377475\pi$
$383$	14.6969			0.750978			0.375489	+	0.926827i	$0.377475\pi$
$384$		0			0
$385$	−12.0000	−	4.89898i	−0.611577	−	0.249675i
$386$		0			0
$387$	15.0000	−	15.0000i	0.762493	−	0.762493i
$388$		0			0
$389$	−13.0000	+	13.0000i	−0.659126	+	0.659126i	−0.955173	−	0.296047i	$-0.904331\pi$
$389$	−13.0000	+	13.0000i	−0.659126	+	0.659126i	0.296047	+	0.955173i	$0.404331\pi$
$390$		0			0
$391$			19.5959i			0.991008i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	9.79796	−	9.79796i	0.492989	−	0.492989i
$396$		0			0
$397$	−12.2474	+	12.2474i	−0.614682	+	0.614682i	−0.944162	−	0.329481i	$-0.893126\pi$
$397$	−12.2474	+	12.2474i	−0.614682	+	0.614682i	0.329481	+	0.944162i	$0.393126\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−8.00000			−0.399501			−0.199750	−	0.979847i	$-0.564013\pi$
$401$	−8.00000			−0.399501			−0.199750	+	0.979847i	$0.564013\pi$
$402$		0			0
$403$	12.0000	−	12.0000i	0.597763	−	0.597763i
$404$		0			0
$405$	22.0454	+	22.0454i	1.09545	+	1.09545i
$406$		0			0
$407$	10.0000			0.495682
$408$		0			0
$409$	−24.4949			−1.21119			−0.605597	−	0.795771i	$-0.707066\pi$
$409$	−24.4949			−1.21119			−0.605597	+	0.795771i	$0.707066\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−7.10102	−	16.8990i	−0.349418	−	0.831544i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4.89898	+	4.89898i	0.239331	+	0.239331i	0.816573	−	0.577242i	$-0.195871\pi$
$419$	4.89898	+	4.89898i	0.239331	+	0.239331i	−0.577242	+	0.816573i	$0.695871\pi$
$420$		0			0
$421$	−11.0000	+	11.0000i	−0.536107	+	0.536107i	−0.922383	−	0.386276i	$-0.873761\pi$
$421$	−11.0000	+	11.0000i	−0.536107	+	0.536107i	0.386276	+	0.922383i	$0.373761\pi$
$422$		0			0
$423$		−	14.6969i		−	0.714590i
$424$		0			0
$425$	−34.2929			−1.66345
$426$		0			0
$427$	3.55051	+	8.44949i	0.171821	+	0.408899i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			10.0000i			0.481683i	0.970564	+	0.240842i	$0.0774234\pi$
$431$			10.0000i			0.481683i	−0.970564	+	0.240842i	$0.922577\pi$
$432$		0			0
$433$		−	14.6969i		−	0.706290i	−0.935569	−	0.353145i	$-0.885112\pi$
$433$		−	14.6969i		−	0.706290i	0.935569	−	0.353145i	$-0.114888\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	19.5959	+	19.5959i	0.937400	+	0.937400i
$438$		0			0
$439$			14.6969i			0.701447i	0.936479	+	0.350723i	$0.114064\pi$
$439$			14.6969i			0.701447i	−0.936479	+	0.350723i	$0.885936\pi$
$440$		0			0
$441$	−14.6969	+	15.0000i	−0.699854	+	0.714286i
$442$		0			0
$443$	−5.00000	−	5.00000i	−0.237557	−	0.237557i	0.578281	−	0.815838i	$-0.303724\pi$
$443$	−5.00000	−	5.00000i	−0.237557	−	0.237557i	−0.815838	+	0.578281i	$0.803724\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	10.0000			0.471929			0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000			0.471929			0.235965	+	0.971762i	$0.424175\pi$
$450$		0			0
$451$	4.89898	+	4.89898i	0.230684	+	0.230684i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−12.0000	+	29.3939i	−0.562569	+	1.37801i
$456$		0			0
$457$			18.0000i			0.842004i	0.907060	+	0.421002i	$0.138322\pi$
$457$			18.0000i			0.842004i	−0.907060	+	0.421002i	$0.861678\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−22.0454	+	22.0454i	−1.02676	+	1.02676i	−0.0271249	+	0.999632i	$0.508635\pi$
$461$	−22.0454	+	22.0454i	−1.02676	+	1.02676i	−0.999632	+	0.0271249i	$0.991365\pi$
$462$		0			0
$463$		−	4.00000i		−	0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$		−	4.00000i		−	0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	19.5959	+	19.5959i	0.906791	+	0.906791i	0.996012	−	0.0892209i	$-0.0284377\pi$
$467$	19.5959	+	19.5959i	0.906791	+	0.906791i	−0.0892209	+	0.996012i	$0.528438\pi$
$468$		0			0
$469$	−7.24745	−	17.2474i	−0.334656	−	0.796413i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	10.0000i		−	0.459800i
$474$		0			0
$475$	−34.2929	+	34.2929i	−1.57346	+	1.57346i
$476$		0			0
$477$	−3.00000	−	3.00000i	−0.137361	−	0.137361i
$478$		0			0
$479$	−24.4949			−1.11920			−0.559600	−	0.828763i	$-0.689045\pi$
$479$	−24.4949			−1.11920			−0.559600	+	0.828763i	$0.689045\pi$
$480$		0			0
$481$		−	24.4949i		−	1.11687i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	12.0000	−	12.0000i	0.544892	−	0.544892i
$486$		0			0
$487$	−12.0000			−0.543772			−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000			−0.543772			−0.271886	+	0.962329i	$0.587647\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−11.0000	−	11.0000i	−0.496423	−	0.496423i	0.413900	−	0.910323i	$-0.364166\pi$
$491$	−11.0000	−	11.0000i	−0.496423	−	0.496423i	−0.910323	+	0.413900i	$0.864166\pi$
$492$		0			0
$493$	−14.6969	+	14.6969i	−0.661917	+	0.661917i
$494$		0			0
$495$	14.6969			0.660578
$496$		0			0
$497$	−2.00000	+	4.89898i	−0.0897123	+	0.219749i
$498$		0			0
$499$	−23.0000	+	23.0000i	−1.02962	+	1.02962i	−0.0300737	+	0.999548i	$0.509574\pi$
$499$	−23.0000	+	23.0000i	−1.02962	+	1.02962i	−0.999548	+	0.0300737i	$0.990426\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	14.6969i		−	0.655304i	−0.944798	−	0.327652i	$-0.893743\pi$
$503$		−	14.6969i		−	0.655304i	0.944798	−	0.327652i	$-0.106257\pi$
$504$		0			0
$505$		−	12.0000i		−	0.533993i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7.34847	−	7.34847i	0.325715	−	0.325715i	−0.525239	−	0.850955i	$-0.676024\pi$
$509$	7.34847	−	7.34847i	0.325715	−	0.325715i	0.850955	+	0.525239i	$0.176024\pi$
$510$		0			0
$511$	9.79796	−	24.0000i	0.433436	−	1.06170i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	24.0000	−	24.0000i	1.05757	−	1.05757i
$516$		0			0
$517$	−4.89898	−	4.89898i	−0.215457	−	0.215457i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−4.89898			−0.214628			−0.107314	−	0.994225i	$-0.534225\pi$
$521$	−4.89898			−0.214628			−0.107314	+	0.994225i	$0.534225\pi$
$522$		0			0
$523$	−14.6969	+	14.6969i	−0.642652	+	0.642652i	−0.951207	−	0.308554i	$-0.900155\pi$
$523$	−14.6969	+	14.6969i	−0.642652	+	0.642652i	0.308554	+	0.951207i	$0.400155\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	24.0000i		−	1.04546i
$528$		0			0
$529$	−7.00000			−0.304348
$530$		0			0
$531$	14.6969	+	14.6969i	0.637793	+	0.637793i
$532$		0			0
$533$	12.0000	−	12.0000i	0.519778	−	0.519778i
$534$		0			0
$535$			14.6969i			0.635404i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.101021	+	9.89898i	0.00435126	+	0.426379i
$540$		0			0
$541$	29.0000	+	29.0000i	1.24681	+	1.24681i	0.957122	+	0.289685i	$0.0935507\pi$
$541$	29.0000	+	29.0000i	1.24681	+	1.24681i	0.289685	+	0.957122i	$0.406449\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			34.2929i			1.46894i
$546$		0			0
$547$	5.00000	−	5.00000i	0.213785	−	0.213785i	−0.592088	−	0.805873i	$-0.701696\pi$
$547$	5.00000	−	5.00000i	0.213785	−	0.213785i	0.805873	+	0.592088i	$0.201696\pi$
$548$		0			0
$549$	−7.34847	−	7.34847i	−0.313625	−	0.313625i
$550$		0			0
$551$			29.3939i			1.25222i
$552$		0			0
$553$	−9.79796	−	4.00000i	−0.416652	−	0.170097i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3.00000	−	3.00000i	−0.127114	−	0.127114i	0.640688	−	0.767802i	$-0.278649\pi$
$557$	−3.00000	−	3.00000i	−0.127114	−	0.127114i	−0.767802	+	0.640688i	$0.778649\pi$
$558$		0			0
$559$	−24.4949			−1.03602
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−24.4949	−	24.4949i	−1.03234	−	1.03234i	−0.999459	−	0.0328775i	$-0.989533\pi$
$563$	−24.4949	−	24.4949i	−1.03234	−	1.03234i	−0.0328775	−	0.999459i	$-0.510467\pi$
$564$		0			0
$565$	−9.79796	−	9.79796i	−0.412203	−	0.412203i
$566$		0			0
$567$	9.00000	−	22.0454i	0.377964	−	0.925820i
$568$		0			0
$569$			14.0000i			0.586911i	0.955973	+	0.293455i	$0.0948052\pi$
$569$			14.0000i			0.586911i	−0.955973	+	0.293455i	$0.905195\pi$
$570$		0			0
$571$	−31.0000	−	31.0000i	−1.29731	−	1.29731i	−0.930163	−	0.367146i	$-0.880335\pi$
$571$	−31.0000	−	31.0000i	−1.29731	−	1.29731i	−0.367146	−	0.930163i	$-0.619665\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			28.0000i			1.16768i
$576$		0			0
$577$			29.3939i			1.22368i	0.790980	+	0.611842i	$0.209571\pi$
$577$			29.3939i			1.22368i	−0.790980	+	0.611842i	$0.790429\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−2.00000			−0.0828315
$584$		0			0
$585$		−	36.0000i		−	1.48842i
$586$		0			0
$587$	24.4949	−	24.4949i	1.01101	−	1.01101i	0.0110739	−	0.999939i	$-0.496475\pi$
$587$	24.4949	−	24.4949i	1.01101	−	1.01101i	0.999939	−	0.0110739i	$-0.00352501\pi$
$588$		0			0
$589$	−24.0000	−	24.0000i	−0.988903	−	0.988903i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	39.1918i		−	1.60942i	−0.593671	−	0.804708i	$-0.702322\pi$
$593$		−	39.1918i		−	1.60942i	0.593671	−	0.804708i	$-0.297678\pi$
$594$		0			0
$595$	17.3939	+	41.3939i	0.713079	+	1.69698i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	10.0000			0.408589			0.204294	−	0.978909i	$-0.434510\pi$
$599$	10.0000			0.408589			0.204294	+	0.978909i	$0.434510\pi$
$600$		0			0
$601$	19.5959			0.799334			0.399667	−	0.916660i	$-0.369126\pi$
$601$	19.5959			0.799334			0.399667	+	0.916660i	$0.369126\pi$
$602$		0			0
$603$	15.0000	+	15.0000i	0.610847	+	0.610847i
$604$		0			0
$605$	−22.0454	+	22.0454i	−0.896273	+	0.896273i
$606$		0			0
$607$	29.3939			1.19306			0.596530	−	0.802591i	$-0.296546\pi$
$607$	29.3939			1.19306			0.596530	+	0.802591i	$0.296546\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−12.0000	+	12.0000i	−0.485468	+	0.485468i
$612$		0			0
$613$	−19.0000	+	19.0000i	−0.767403	+	0.767403i	−0.977649	−	0.210246i	$-0.932574\pi$
$613$	−19.0000	+	19.0000i	−0.767403	+	0.767403i	0.210246	+	0.977649i	$0.432574\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	2.00000i		−	0.0805170i	−0.999189	−	0.0402585i	$-0.987182\pi$
$617$		−	2.00000i		−	0.0805170i	0.999189	−	0.0402585i	$-0.0128181\pi$
$618$		0			0
$619$	−4.89898	+	4.89898i	−0.196907	+	0.196907i	−0.798673	−	0.601766i	$-0.794464\pi$
$619$	−4.89898	+	4.89898i	−0.196907	+	0.196907i	0.601766	+	0.798673i	$0.294464\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	11.0000			0.440000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−24.4949	−	24.4949i	−0.976676	−	0.976676i
$630$		0			0
$631$	−18.0000			−0.716569			−0.358284	−	0.933613i	$-0.616638\pi$
$631$	−18.0000			−0.716569			−0.358284	+	0.933613i	$0.616638\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	44.0908	−	44.0908i	1.74969	−	1.74969i
$636$		0			0
$637$	24.2474	−	0.247449i	0.960719	−	0.00980428i
$638$		0			0
$639$		−	6.00000i		−	0.237356i
$640$		0			0
$641$	32.0000			1.26392			0.631962	−	0.774999i	$-0.282250\pi$
$641$	32.0000			1.26392			0.631962	+	0.774999i	$0.282250\pi$
$642$		0			0
$643$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$643$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	19.5959i		−	0.770395i	−0.922834	−	0.385198i	$-0.874133\pi$
$647$		−	19.5959i		−	0.770395i	0.922834	−	0.385198i	$-0.125867\pi$
$648$		0			0
$649$	9.79796			0.384604
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−15.0000	−	15.0000i	−0.586995	−	0.586995i	0.349821	−	0.936817i	$-0.386242\pi$
$653$	−15.0000	−	15.0000i	−0.586995	−	0.586995i	−0.936817	+	0.349821i	$0.886242\pi$
$654$		0			0
$655$			48.0000i			1.87552i
$656$		0			0
$657$			29.3939i			1.14676i
$658$		0			0
$659$	27.0000	−	27.0000i	1.05177	−	1.05177i	0.0531861	−	0.998585i	$-0.483062\pi$
$659$	27.0000	−	27.0000i	1.05177	−	1.05177i	0.998585	−	0.0531861i	$-0.0169376\pi$
$660$		0			0
$661$	2.44949	+	2.44949i	0.0952741	+	0.0952741i	0.753137	−	0.657863i	$-0.228540\pi$
$661$	2.44949	+	2.44949i	0.0952741	+	0.0952741i	−0.657863	+	0.753137i	$0.728540\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	58.7878	+	24.0000i	2.27969	+	0.930680i
$666$		0			0
$667$	12.0000	+	12.0000i	0.464642	+	0.464642i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−4.89898			−0.189123
$672$		0			0
$673$	−6.00000			−0.231283			−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000			−0.231283			−0.115642	+	0.993291i	$0.536892\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	7.34847	+	7.34847i	0.282425	+	0.282425i	0.834075	−	0.551651i	$-0.186002\pi$
$677$	7.34847	+	7.34847i	0.282425	+	0.282425i	−0.551651	+	0.834075i	$0.686002\pi$
$678$		0			0
$679$	−12.0000	−	4.89898i	−0.460518	−	0.188006i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	25.0000	+	25.0000i	0.956598	+	0.956598i	0.999097	−	0.0424981i	$-0.0135317\pi$
$683$	25.0000	+	25.0000i	0.956598	+	0.956598i	−0.0424981	+	0.999097i	$0.513532\pi$
$684$		0			0
$685$	19.5959	−	19.5959i	0.748722	−	0.748722i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			4.89898i			0.186636i
$690$		0			0
$691$	−9.79796	−	9.79796i	−0.372732	−	0.372732i	0.495739	−	0.868471i	$-0.334897\pi$
$691$	−9.79796	−	9.79796i	−0.372732	−	0.372732i	−0.868471	+	0.495739i	$0.834897\pi$
$692$		0			0
$693$	−4.34847	−	10.3485i	−0.165185	−	0.393106i
$694$		0			0
$695$	24.0000			0.910372
$696$		0			0
$697$		−	24.0000i		−	0.909065i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	29.0000	+	29.0000i	1.09531	+	1.09531i	0.994951	+	0.100364i	$0.0320008\pi$
$701$	29.0000	+	29.0000i	1.09531	+	1.09531i	0.100364	+	0.994951i	$0.467999\pi$
$702$		0			0
$703$	−48.9898			−1.84769
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−8.44949	+	3.55051i	−0.317776	+	0.133531i
$708$		0			0
$709$	−23.0000	+	23.0000i	−0.863783	+	0.863783i	−0.991775	−	0.127992i	$-0.959147\pi$
$709$	−23.0000	+	23.0000i	−0.863783	+	0.863783i	0.127992	+	0.991775i	$0.459147\pi$
$710$		0			0
$711$	12.0000			0.450035
$712$		0			0
$713$	−19.5959			−0.733873
$714$		0			0
$715$	−12.0000	−	12.0000i	−0.448775	−	0.448775i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−24.4949			−0.913506			−0.456753	−	0.889594i	$-0.650988\pi$
$719$	−24.4949			−0.913506			−0.456753	+	0.889594i	$0.650988\pi$
$720$		0			0
$721$	−24.0000	−	9.79796i	−0.893807	−	0.364895i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−21.0000	+	21.0000i	−0.779920	+	0.779920i
$726$		0			0
$727$		−	19.5959i		−	0.726772i	−0.931639	−	0.363386i	$-0.881621\pi$
$727$		−	19.5959i		−	0.726772i	0.931639	−	0.363386i	$-0.118379\pi$
$728$		0			0
$729$			27.0000i			1.00000i
$730$		0			0
$731$	−24.4949	+	24.4949i	−0.905977	+	0.905977i
$732$		0			0
$733$	22.0454	−	22.0454i	0.814266	−	0.814266i	−0.171005	−	0.985270i	$-0.554701\pi$
$733$	22.0454	−	22.0454i	0.814266	−	0.814266i	0.985270	+	0.171005i	$0.0547013\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	10.0000			0.368355
$738$		0			0
$739$	17.0000	−	17.0000i	0.625355	−	0.625355i	−0.321541	−	0.946896i	$-0.604201\pi$
$739$	17.0000	−	17.0000i	0.625355	−	0.625355i	0.946896	+	0.321541i	$0.104201\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	4.00000			0.146746			0.0733729	−	0.997305i	$-0.476624\pi$
$743$	4.00000			0.146746			0.0733729	+	0.997305i	$0.476624\pi$
$744$		0			0
$745$	14.6969			0.538454
$746$		0			0
$747$		0			0
$748$		0			0
$749$	10.3485	−	4.34847i	0.378125	−	0.158890i
$750$		0			0
$751$		−	10.0000i		−	0.364905i	−0.983215	−	0.182453i	$-0.941596\pi$
$751$		−	10.0000i		−	0.364905i	0.983215	−	0.182453i	$-0.0584036\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−29.3939	−	29.3939i	−1.06975	−	1.06975i
$756$		0			0
$757$	5.00000	−	5.00000i	0.181728	−	0.181728i	−0.610380	−	0.792108i	$-0.708983\pi$
$757$	5.00000	−	5.00000i	0.181728	−	0.181728i	0.792108	+	0.610380i	$0.208983\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	44.0908			1.59829			0.799145	−	0.601138i	$-0.205286\pi$
$761$	44.0908			1.59829			0.799145	+	0.601138i	$0.205286\pi$
$762$		0			0
$763$	24.1464	−	10.1464i	0.874159	−	0.367325i
$764$		0			0
$765$	−36.0000	−	36.0000i	−1.30158	−	1.30158i
$766$		0			0
$767$		−	24.0000i		−	0.866590i
$768$		0			0
$769$		−	34.2929i		−	1.23663i	−0.785930	−	0.618316i	$-0.787815\pi$
$769$		−	34.2929i		−	1.23663i	0.785930	−	0.618316i	$-0.212185\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	12.2474	+	12.2474i	0.440510	+	0.440510i	0.892183	−	0.451673i	$-0.149173\pi$
$773$	12.2474	+	12.2474i	0.440510	+	0.440510i	−0.451673	+	0.892183i	$0.649173\pi$
$774$		0			0
$775$		−	34.2929i		−	1.23184i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−24.0000	−	24.0000i	−0.859889	−	0.859889i
$780$		0			0
$781$	−2.00000	−	2.00000i	−0.0715656	−	0.0715656i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−60.0000			−2.14149
$786$		0			0
$787$	−4.89898	−	4.89898i	−0.174630	−	0.174630i	0.614380	−	0.789010i	$-0.289406\pi$
$787$	−4.89898	−	4.89898i	−0.174630	−	0.174630i	−0.789010	+	0.614380i	$0.789406\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−4.00000	+	9.79796i	−0.142224	+	0.348375i
$792$		0			0
$793$			12.0000i			0.426132i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−12.2474	+	12.2474i	−0.433827	+	0.433827i	−0.889928	−	0.456101i	$-0.849246\pi$
$797$	−12.2474	+	12.2474i	−0.433827	+	0.433827i	0.456101	+	0.889928i	$0.349246\pi$
$798$		0			0
$799$			24.0000i			0.849059i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	9.79796	+	9.79796i	0.345762	+	0.345762i
$804$		0			0
$805$	33.7980	−	14.2020i	1.19122	−	0.500556i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	46.0000i		−	1.61727i	−0.588308	−	0.808637i	$-0.700206\pi$
$809$		−	46.0000i		−	1.61727i	0.588308	−	0.808637i	$-0.299794\pi$
$810$		0			0
$811$	14.6969	−	14.6969i	0.516079	−	0.516079i	−0.400303	−	0.916383i	$-0.631095\pi$
$811$	14.6969	−	14.6969i	0.516079	−	0.516079i	0.916383	+	0.400303i	$0.131095\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4.89898			−0.171604
$816$		0			0
$817$			48.9898i			1.71394i
$818$		0			0
$819$	−25.3485	+	10.6515i	−0.885747	+	0.372195i
$820$		0			0
$821$	19.0000	−	19.0000i	0.663105	−	0.663105i	−0.293006	−	0.956111i	$-0.594656\pi$
$821$	19.0000	−	19.0000i	0.663105	−	0.663105i	0.956111	+	0.293006i	$0.0946556\pi$
$822$		0			0
$823$	−6.00000			−0.209147			−0.104573	−	0.994517i	$-0.533348\pi$
$823$	−6.00000			−0.209147			−0.104573	+	0.994517i	$0.533348\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−33.0000	−	33.0000i	−1.14752	−	1.14752i	−0.987038	−	0.160484i	$-0.948695\pi$
$827$	−33.0000	−	33.0000i	−1.14752	−	1.14752i	−0.160484	−	0.987038i	$-0.551305\pi$
$828$		0			0
$829$	7.34847	−	7.34847i	0.255223	−	0.255223i	−0.567885	−	0.823108i	$-0.692238\pi$
$829$	7.34847	−	7.34847i	0.255223	−	0.255223i	0.823108	+	0.567885i	$0.192238\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	24.0000	−	24.4949i	0.831551	−	0.848698i
$834$		0			0
$835$	−48.0000	+	48.0000i	−1.66111	+	1.66111i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	9.79796i		−	0.338263i	−0.985593	−	0.169132i	$-0.945904\pi$
$839$		−	9.79796i		−	0.338263i	0.985593	−	0.169132i	$-0.0540963\pi$
$840$		0			0
$841$		−	11.0000i		−	0.379310i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2.44949	−	2.44949i	0.0842650	−	0.0842650i
$846$		0			0
$847$	22.0454	+	9.00000i	0.757489	+	0.309244i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−20.0000	+	20.0000i	−0.685591	+	0.685591i
$852$		0			0
$853$	12.2474	+	12.2474i	0.419345	+	0.419345i	0.884978	−	0.465633i	$-0.154173\pi$
$853$	12.2474	+	12.2474i	0.419345	+	0.419345i	−0.465633	+	0.884978i	$0.654173\pi$
$854$		0			0
$855$	−72.0000			−2.46235
$856$		0			0
$857$	4.89898			0.167346			0.0836730	−	0.996493i	$-0.473335\pi$
$857$	4.89898			0.167346			0.0836730	+	0.996493i	$0.473335\pi$
$858$		0			0
$859$	−29.3939	+	29.3939i	−1.00291	+	1.00291i	−0.00291037	+	0.999996i	$0.500926\pi$
$859$	−29.3939	+	29.3939i	−1.00291	+	1.00291i	−0.999996	+	0.00291037i	$0.999074\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	4.00000i		−	0.136162i	−0.997680	−	0.0680808i	$-0.978312\pi$
$863$		−	4.00000i		−	0.136162i	0.997680	−	0.0680808i	$-0.0216876\pi$
$864$		0			0
$865$	12.0000			0.408012
$866$		0			0
$867$		0			0
$868$		0			0
$869$	4.00000	−	4.00000i	0.135691	−	0.135691i
$870$		0			0
$871$		−	24.4949i		−	0.829978i
$872$		0			0
$873$	14.6969			0.497416
$874$		0			0
$875$	7.10102	+	16.8990i	0.240058	+	0.571290i
$876$		0			0
$877$	−13.0000	−	13.0000i	−0.438979	−	0.438979i	0.452689	−	0.891668i	$-0.350465\pi$
$877$	−13.0000	−	13.0000i	−0.438979	−	0.438979i	−0.891668	+	0.452689i	$0.850465\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	48.9898i		−	1.65051i	−0.564762	−	0.825254i	$-0.691032\pi$
$881$		−	48.9898i		−	1.65051i	0.564762	−	0.825254i	$-0.308968\pi$
$882$		0			0
$883$	31.0000	−	31.0000i	1.04323	−	1.04323i	0.0442108	−	0.999022i	$-0.485923\pi$
$883$	31.0000	−	31.0000i	1.04323	−	1.04323i	0.999022	−	0.0442108i	$-0.0140773\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	19.5959i		−	0.657967i	−0.944336	−	0.328983i	$-0.893294\pi$
$887$		−	19.5959i		−	0.657967i	0.944336	−	0.328983i	$-0.106706\pi$
$888$		0			0
$889$	−44.0908	−	18.0000i	−1.47876	−	0.603701i
$890$		0			0
$891$	9.00000	+	9.00000i	0.301511	+	0.301511i
$892$		0			0
$893$	24.0000	+	24.0000i	0.803129	+	0.803129i
$894$		0			0
$895$	14.6969			0.491264
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−14.6969	−	14.6969i	−0.490170	−	0.490170i
$900$		0			0
$901$	4.89898	+	4.89898i	0.163209	+	0.163209i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	12.0000i		−	0.398893i
$906$		0			0
$907$	7.00000	+	7.00000i	0.232431	+	0.232431i	0.813707	−	0.581276i	$-0.197446\pi$
$907$	7.00000	+	7.00000i	0.232431	+	0.232431i	−0.581276	+	0.813707i	$0.697446\pi$
$908$		0			0
$909$	7.34847	−	7.34847i	0.243733	−	0.243733i
$910$		0			0
$911$		−	50.0000i		−	1.65657i	−0.560304	−	0.828287i	$-0.689316\pi$
$911$		−	50.0000i		−	1.65657i	0.560304	−	0.828287i	$-0.310684\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	33.7980	−	14.2020i	1.11611	−	0.468993i
$918$		0			0
$919$	−10.0000			−0.329870			−0.164935	−	0.986304i	$-0.552741\pi$
$919$	−10.0000			−0.329870			−0.164935	+	0.986304i	$0.552741\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−4.89898	+	4.89898i	−0.161252	+	0.161252i
$924$		0			0
$925$	−35.0000	−	35.0000i	−1.15079	−	1.15079i
$926$		0			0
$927$	29.3939			0.965422
$928$		0			0
$929$		−	34.2929i		−	1.12511i	−0.826759	−	0.562556i	$-0.809818\pi$
$929$		−	34.2929i		−	1.12511i	0.826759	−	0.562556i	$-0.190182\pi$
$930$		0			0
$931$	−0.494897	−	48.4949i	−0.0162196	−	1.58936i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−24.0000			−0.784884
$936$		0			0
$937$	29.3939			0.960256			0.480128	−	0.877198i	$-0.340590\pi$
$937$	29.3939			0.960256			0.480128	+	0.877198i	$0.340590\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	26.9444	−	26.9444i	0.878362	−	0.878362i	−0.115003	−	0.993365i	$-0.536688\pi$
$941$	26.9444	−	26.9444i	0.878362	−	0.878362i	0.993365	+	0.115003i	$0.0366878\pi$
$942$		0			0
$943$	−19.5959			−0.638131
$944$		0			0
$945$		0			0
$946$		0			0
$947$	15.0000	−	15.0000i	0.487435	−	0.487435i	−0.420061	−	0.907496i	$-0.637991\pi$
$947$	15.0000	−	15.0000i	0.487435	−	0.487435i	0.907496	+	0.420061i	$0.137991\pi$
$948$		0			0
$949$	24.0000	−	24.0000i	0.779073	−	0.779073i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			16.0000i			0.518291i	0.965838	+	0.259145i	$0.0834409\pi$
$953$			16.0000i			0.518291i	−0.965838	+	0.259145i	$0.916559\pi$
$954$		0			0
$955$	48.9898	−	48.9898i	1.58527	−	1.58527i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−19.5959	−	8.00000i	−0.632785	−	0.258333i
$960$		0			0
$961$	−7.00000			−0.225806
$962$		0			0
$963$	−9.00000	+	9.00000i	−0.290021	+	0.290021i
$964$		0			0
$965$	39.1918	+	39.1918i	1.26163	+	1.26163i
$966$		0			0
$967$	28.0000			0.900419			0.450210	−	0.892923i	$-0.351349\pi$
$967$	28.0000			0.900419			0.450210	+	0.892923i	$0.351349\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−9.79796	+	9.79796i	−0.314431	+	0.314431i	−0.846624	−	0.532192i	$-0.821368\pi$
$971$	−9.79796	+	9.79796i	−0.314431	+	0.314431i	0.532192	+	0.846624i	$0.321368\pi$
$972$		0			0
$973$	−7.10102	−	16.8990i	−0.227648	−	0.541756i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	38.0000			1.21573			0.607864	−	0.794041i	$-0.292027\pi$
$977$	38.0000			1.21573			0.607864	+	0.794041i	$0.292027\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−21.0000	+	21.0000i	−0.670478	+	0.670478i
$982$		0			0
$983$			9.79796i			0.312506i	0.987717	+	0.156253i	$0.0499416\pi$
$983$			9.79796i			0.312506i	−0.987717	+	0.156253i	$0.950058\pi$
$984$		0			0
$985$	−73.4847			−2.34142
$986$		0			0
$987$		0			0
$988$		0			0
$989$	20.0000	+	20.0000i	0.635963	+	0.635963i
$990$		0			0
$991$			60.0000i			1.90596i	0.303029	+	0.952981i	$0.402002\pi$
$991$			60.0000i			1.90596i	−0.303029	+	0.952981i	$0.597998\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−24.0000	+	24.0000i	−0.760851	+	0.760851i
$996$		0			0
$997$	7.34847	+	7.34847i	0.232728	+	0.232728i	0.813831	−	0.581102i	$-0.197378\pi$
$997$	7.34847	+	7.34847i	0.232728	+	0.232728i	−0.581102	+	0.813831i	$0.697378\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.j.b.223.1		4
4.3	odd	2		896.2.j.e.223.1		4
7.6	odd	2	inner	896.2.j.b.223.2		4
8.3	odd	2		448.2.j.b.111.2		4
8.5	even	2		112.2.j.a.83.2	yes	4
16.3	odd	4		112.2.j.a.27.1	✓	4
16.5	even	4		896.2.j.e.671.2		4
16.11	odd	4	inner	896.2.j.b.671.2		4
16.13	even	4		448.2.j.b.335.1		4
28.27	even	2		896.2.j.e.223.2		4
56.5	odd	6		784.2.w.d.227.1		8
56.13	odd	2		112.2.j.a.83.1	yes	4
56.27	even	2		448.2.j.b.111.1		4
56.37	even	6		784.2.w.d.227.2		8
56.45	odd	6		784.2.w.d.19.2		8
56.53	even	6		784.2.w.d.19.1		8
112.3	even	12		784.2.w.d.411.2		8
112.13	odd	4		448.2.j.b.335.2		4
112.19	even	12		784.2.w.d.619.1		8
112.27	even	4	inner	896.2.j.b.671.1		4
112.51	odd	12		784.2.w.d.619.2		8
112.67	odd	12		784.2.w.d.411.1		8
112.69	odd	4		896.2.j.e.671.1		4
112.83	even	4		112.2.j.a.27.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.j.a.27.1	✓	4	16.3	odd	4
112.2.j.a.27.2	yes	4	112.83	even	4
112.2.j.a.83.1	yes	4	56.13	odd	2
112.2.j.a.83.2	yes	4	8.5	even	2
448.2.j.b.111.1		4	56.27	even	2
448.2.j.b.111.2		4	8.3	odd	2
448.2.j.b.335.1		4	16.13	even	4
448.2.j.b.335.2		4	112.13	odd	4
784.2.w.d.19.1		8	56.53	even	6
784.2.w.d.19.2		8	56.45	odd	6
784.2.w.d.227.1		8	56.5	odd	6
784.2.w.d.227.2		8	56.37	even	6
784.2.w.d.411.1		8	112.67	odd	12
784.2.w.d.411.2		8	112.3	even	12
784.2.w.d.619.1		8	112.19	even	12
784.2.w.d.619.2		8	112.51	odd	12
896.2.j.b.223.1		4	1.1	even	1	trivial
896.2.j.b.223.2		4	7.6	odd	2	inner
896.2.j.b.671.1		4	112.27	even	4	inner
896.2.j.b.671.2		4	16.11	odd	4	inner
896.2.j.e.223.1		4	4.3	odd	2
896.2.j.e.223.2		4	28.27	even	2
896.2.j.e.671.1		4	112.69	odd	4
896.2.j.e.671.2		4	16.5	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:	\( N \)	\(=\)	\( 896 = 2^{7} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	896.j (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.44949 - 2.44949i) q^{5} +(-1.00000 + 2.44949i) q^{7} -3.00000i q^{9} +O(q^{10})\)	\(q+(-2.44949 - 2.44949i) q^{5} +(-1.00000 + 2.44949i) q^{7} -3.00000i q^{9} +(-1.00000 - 1.00000i) q^{11} +(-2.44949 + 2.44949i) q^{13} +4.89898i q^{17} +(4.89898 + 4.89898i) q^{19} +4.00000 q^{23} +7.00000i q^{25} +(3.00000 + 3.00000i) q^{29} -4.89898 q^{31} +(8.44949 - 3.55051i) q^{35} +(-5.00000 + 5.00000i) q^{37} -4.89898 q^{41} +(5.00000 + 5.00000i) q^{43} +(-7.34847 + 7.34847i) q^{45} +4.89898 q^{47} +(-5.00000 - 4.89898i) q^{49} +(1.00000 - 1.00000i) q^{53} +4.89898i q^{55} +(-4.89898 + 4.89898i) q^{59} +(2.44949 - 2.44949i) q^{61} +(7.34847 + 3.00000i) q^{63} +12.0000 q^{65} +(-5.00000 + 5.00000i) q^{67} +2.00000 q^{71} -9.79796 q^{73} +(3.44949 - 1.44949i) q^{77} +4.00000i q^{79} -9.00000 q^{81} +(12.0000 - 12.0000i) q^{85} +(-3.55051 - 8.44949i) q^{91} -24.0000i q^{95} +4.89898i q^{97} +(-3.00000 + 3.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7}+O(q^{10}) \) 4 * q - 4 * q^7	\( 4 q - 4 q^{7} - 4 q^{11} + 16 q^{23} + 12 q^{29} + 24 q^{35} - 20 q^{37} + 20 q^{43} - 20 q^{49} + 4 q^{53} + 48 q^{65} - 20 q^{67} + 8 q^{71} + 4 q^{77} - 36 q^{81} + 48 q^{85} - 24 q^{91} - 12 q^{99}+O(q^{100}) \) 4 * q - 4 * q^7 - 4 * q^11 + 16 * q^23 + 12 * q^29 + 24 * q^35 - 20 * q^37 + 20 * q^43 - 20 * q^49 + 4 * q^53 + 48 * q^65 - 20 * q^67 + 8 * q^71 + 4 * q^77 - 36 * q^81 + 48 * q^85 - 24 * q^91 - 12 * q^99

Introduction

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