LMFDB - Embedded newform 1224.2.bq.a.937.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1224,2,Mod(145,1224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1224, base_ring=CyclotomicField(8))

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

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

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

chi := DirichletCharacter("1224.145");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1224 = 2^{3} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1224.bq (of order $8$, degree $4$, 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:	$9.77368920740$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 136)
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			937.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1224.937
Dual form			1224.2.bq.a.145.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.292893 - 0.707107i) q^{5} +(-1.70711 + 4.12132i) q^{7} +O(q^{10})$	$q+(-0.292893 - 0.707107i) q^{5} +(-1.70711 + 4.12132i) q^{7} +(-1.70711 - 0.707107i) q^{11} +(-1.00000 - 4.00000i) q^{17} +(-4.41421 - 4.41421i) q^{19} +(-4.53553 - 1.87868i) q^{23} +(3.12132 - 3.12132i) q^{25} +(0.878680 + 2.12132i) q^{29} +(5.12132 - 2.12132i) q^{31} +3.41421 q^{35} +(1.70711 - 0.707107i) q^{37} +(3.70711 - 8.94975i) q^{41} +(-1.24264 + 1.24264i) q^{43} -7.17157i q^{47} +(-9.12132 - 9.12132i) q^{49} +(-7.82843 - 7.82843i) q^{53} +1.41421i q^{55} +(-8.41421 + 8.41421i) q^{59} +(-4.87868 + 11.7782i) q^{61} -1.65685 q^{67} +(-2.29289 + 0.949747i) q^{71} +(-5.36396 - 12.9497i) q^{73} +(5.82843 - 5.82843i) q^{77} +(12.5355 + 5.19239i) q^{79} +(-1.24264 - 1.24264i) q^{83} +(-2.53553 + 1.87868i) q^{85} +9.65685i q^{89} +(-1.82843 + 4.41421i) q^{95} +(0.778175 + 1.87868i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} - 4 q^{7}+O(q^{10}) $ 4 * q - 4 * q^5 - 4 * q^7	$ 4 q - 4 q^{5} - 4 q^{7} - 4 q^{11} - 4 q^{17} - 12 q^{19} - 4 q^{23} + 4 q^{25} + 12 q^{29} + 12 q^{31} + 8 q^{35} + 4 q^{37} + 12 q^{41} + 12 q^{43} - 28 q^{49} - 20 q^{53} - 28 q^{59} - 28 q^{61} + 16 q^{67} - 12 q^{71} + 4 q^{73} + 12 q^{77} + 36 q^{79} + 12 q^{83} + 4 q^{85} + 4 q^{95} - 28 q^{97}+O(q^{100}) $ 4 * q - 4 * q^5 - 4 * q^7 - 4 * q^11 - 4 * q^17 - 12 * q^19 - 4 * q^23 + 4 * q^25 + 12 * q^29 + 12 * q^31 + 8 * q^35 + 4 * q^37 + 12 * q^41 + 12 * q^43 - 28 * q^49 - 20 * q^53 - 28 * q^59 - 28 * q^61 + 16 * q^67 - 12 * q^71 + 4 * q^73 + 12 * q^77 + 36 * q^79 + 12 * q^83 + 4 * q^85 + 4 * q^95 - 28 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times$.

$n$	$137$	$613$	$649$	$919$
$\chi(n)$	$1$	$1$	$e\left(\frac{7}{8}\right)$	$1$

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
$5$	−0.292893	−	0.707107i	−0.130986	−	0.316228i	0.844756	−	0.535151i	$-0.179745\pi$
$5$	−0.292893	−	0.707107i	−0.130986	−	0.316228i	−0.975742	+	0.218924i	$0.929745\pi$
$6$		0			0
$7$	−1.70711	+	4.12132i	−0.645226	+	1.55771i	0.174314	+	0.984690i	$0.444229\pi$
$7$	−1.70711	+	4.12132i	−0.645226	+	1.55771i	−0.819540	+	0.573023i	$0.805771\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.70711	−	0.707107i	−0.514712	−	0.213201i	0.110180	−	0.993912i	$-0.464857\pi$
$11$	−1.70711	−	0.707107i	−0.514712	−	0.213201i	−0.624892	+	0.780711i	$0.714857\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.00000	−	4.00000i	−0.242536	−	0.970143i
$17$	−1.00000	−	4.00000i	−0.242536	−	0.970143i
$18$		0			0
$19$	−4.41421	−	4.41421i	−1.01269	−	1.01269i	−0.999918	−	0.0127716i	$-0.995935\pi$
$19$	−4.41421	−	4.41421i	−1.01269	−	1.01269i	−0.0127716	−	0.999918i	$-0.504065\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.53553	−	1.87868i	−0.945724	−	0.391732i	−0.144102	−	0.989563i	$-0.546029\pi$
$23$	−4.53553	−	1.87868i	−0.945724	−	0.391732i	−0.801622	+	0.597831i	$0.796029\pi$
$24$		0			0
$25$	3.12132	−	3.12132i	0.624264	−	0.624264i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.878680	+	2.12132i	0.163167	+	0.393919i	0.984224	−	0.176926i	$-0.0566155\pi$
$29$	0.878680	+	2.12132i	0.163167	+	0.393919i	−0.821057	+	0.570846i	$0.806615\pi$
$30$		0			0
$31$	5.12132	−	2.12132i	0.919816	−	0.381000i	0.128010	−	0.991773i	$-0.459141\pi$
$31$	5.12132	−	2.12132i	0.919816	−	0.381000i	0.791806	+	0.610772i	$0.209141\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.41421			0.577107
$36$		0			0
$37$	1.70711	−	0.707107i	0.280647	−	0.116248i	−0.237920	−	0.971285i	$-0.576466\pi$
$37$	1.70711	−	0.707107i	0.280647	−	0.116248i	0.518567	+	0.855037i	$0.326466\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.70711	−	8.94975i	0.578953	−	1.39772i	−0.314801	−	0.949158i	$-0.601938\pi$
$41$	3.70711	−	8.94975i	0.578953	−	1.39772i	0.893754	−	0.448558i	$-0.148062\pi$
$42$		0			0
$43$	−1.24264	+	1.24264i	−0.189501	+	0.189501i	−0.795480	−	0.605979i	$-0.792781\pi$
$43$	−1.24264	+	1.24264i	−0.189501	+	0.189501i	0.605979	+	0.795480i	$0.292781\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	7.17157i		−	1.04608i	−0.852308	−	0.523041i	$-0.824798\pi$
$47$		−	7.17157i		−	1.04608i	0.852308	−	0.523041i	$-0.175202\pi$
$48$		0			0
$49$	−9.12132	−	9.12132i	−1.30305	−	1.30305i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−7.82843	−	7.82843i	−1.07532	−	1.07532i	−0.996922	−	0.0783948i	$-0.975021\pi$
$53$	−7.82843	−	7.82843i	−1.07532	−	1.07532i	−0.0783948	−	0.996922i	$-0.524979\pi$
$54$		0			0
$55$			1.41421i			0.190693i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−8.41421	+	8.41421i	−1.09544	+	1.09544i	−0.100500	+	0.994937i	$0.532044\pi$
$59$	−8.41421	+	8.41421i	−1.09544	+	1.09544i	−0.994937	+	0.100500i	$0.967956\pi$
$60$		0			0
$61$	−4.87868	+	11.7782i	−0.624651	+	1.50804i	0.221535	+	0.975152i	$0.428893\pi$
$61$	−4.87868	+	11.7782i	−0.624651	+	1.50804i	−0.846186	+	0.532888i	$0.821107\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−1.65685			−0.202417			−0.101208	−	0.994865i	$-0.532271\pi$
$67$	−1.65685			−0.202417			−0.101208	+	0.994865i	$0.532271\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−2.29289	+	0.949747i	−0.272116	+	0.112714i	−0.514569	−	0.857449i	$-0.672048\pi$
$71$	−2.29289	+	0.949747i	−0.272116	+	0.112714i	0.242453	+	0.970163i	$0.422048\pi$
$72$		0			0
$73$	−5.36396	−	12.9497i	−0.627804	−	1.51565i	−0.842345	−	0.538938i	$-0.818826\pi$
$73$	−5.36396	−	12.9497i	−0.627804	−	1.51565i	0.214541	−	0.976715i	$-0.431174\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	5.82843	−	5.82843i	0.664211	−	0.664211i
$78$		0			0
$79$	12.5355	+	5.19239i	1.41036	+	0.584189i	0.952418	−	0.304794i	$-0.0985876\pi$
$79$	12.5355	+	5.19239i	1.41036	+	0.584189i	0.457939	+	0.888983i	$0.348588\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.24264	−	1.24264i	−0.136398	−	0.136398i	0.635612	−	0.772009i	$-0.280748\pi$
$83$	−1.24264	−	1.24264i	−0.136398	−	0.136398i	−0.772009	+	0.635612i	$0.780748\pi$
$84$		0			0
$85$	−2.53553	+	1.87868i	−0.275017	+	0.203771i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			9.65685i			1.02362i	0.859097	+	0.511812i	$0.171026\pi$
$89$			9.65685i			1.02362i	−0.859097	+	0.511812i	$0.828974\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.82843	+	4.41421i	−0.187593	+	0.452889i
$96$		0			0
$97$	0.778175	+	1.87868i	0.0790117	+	0.190751i	0.958449	−	0.285263i	$-0.0920810\pi$
$97$	0.778175	+	1.87868i	0.0790117	+	0.190751i	−0.879438	+	0.476014i	$0.842081\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−10.0000			−0.995037			−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000			−0.995037			−0.497519	+	0.867453i	$0.665755\pi$
$102$		0			0
$103$	−13.6569			−1.34565			−0.672825	−	0.739802i	$-0.734919\pi$
$103$	−13.6569			−1.34565			−0.672825	+	0.739802i	$0.734919\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.636039	+	1.53553i	0.0614882	+	0.148446i	0.951637	−	0.307224i	$-0.0994000\pi$
$107$	0.636039	+	1.53553i	0.0614882	+	0.148446i	−0.890149	+	0.455669i	$0.849400\pi$
$108$		0			0
$109$	4.77817	−	11.5355i	0.457666	−	1.10490i	−0.511674	−	0.859180i	$-0.670974\pi$
$109$	4.77817	−	11.5355i	0.457666	−	1.10490i	0.969340	−	0.245724i	$-0.0790257\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.12132	+	0.464466i	0.105485	+	0.0436933i	0.434802	−	0.900526i	$-0.356818\pi$
$113$	1.12132	+	0.464466i	0.105485	+	0.0436933i	−0.329317	+	0.944219i	$0.606818\pi$
$114$		0			0
$115$			3.75736i			0.350376i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	18.1924	+	2.70711i	1.66769	+	0.248160i
$120$		0			0
$121$	−5.36396	−	5.36396i	−0.487633	−	0.487633i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−6.65685	−	2.75736i	−0.595407	−	0.246626i
$126$		0			0
$127$	7.24264	−	7.24264i	0.642680	−	0.642680i	−0.308533	−	0.951214i	$-0.599838\pi$
$127$	7.24264	−	7.24264i	0.642680	−	0.642680i	0.951214	+	0.308533i	$0.0998381\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	0.636039	+	1.53553i	0.0555710	+	0.134160i	0.949227	−	0.314593i	$-0.101868\pi$
$131$	0.636039	+	1.53553i	0.0555710	+	0.134160i	−0.893656	+	0.448753i	$0.851868\pi$
$132$		0			0
$133$	25.7279	−	10.6569i	2.23089	−	0.924066i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−14.0000			−1.19610			−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000			−1.19610			−0.598050	+	0.801459i	$0.704058\pi$
$138$		0			0
$139$	−18.1924	+	7.53553i	−1.54306	+	0.639156i	−0.982044	−	0.188651i	$-0.939589\pi$
$139$	−18.1924	+	7.53553i	−1.54306	+	0.639156i	−0.561014	+	0.827806i	$0.689589\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	1.24264	−	1.24264i	0.103196	−	0.103196i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			17.6569i			1.44651i	0.690583	+	0.723253i	$0.257354\pi$
$149$			17.6569i			1.44651i	−0.690583	+	0.723253i	$0.742646\pi$
$150$		0			0
$151$	0.0710678	+	0.0710678i	0.00578342	+	0.00578342i	0.709993	−	0.704209i	$-0.248698\pi$
$151$	0.0710678	+	0.0710678i	0.00578342	+	0.00578342i	−0.704209	+	0.709993i	$0.748698\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.00000	−	3.00000i	−0.240966	−	0.240966i
$156$		0			0
$157$		−	7.31371i		−	0.583697i	−0.956464	−	0.291849i	$-0.905730\pi$
$157$		−	7.31371i		−	0.583697i	0.956464	−	0.291849i	$-0.0942704\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	15.4853	−	15.4853i	1.22041	−	1.22041i
$162$		0			0
$163$	−5.70711	+	13.7782i	−0.447015	+	1.07919i	0.526420	+	0.850225i	$0.323534\pi$
$163$	−5.70711	+	13.7782i	−0.447015	+	1.07919i	−0.973435	+	0.228965i	$0.926466\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.70711	−	0.707107i	0.132100	−	0.0547176i	−0.315654	−	0.948874i	$-0.602224\pi$
$167$	1.70711	−	0.707107i	0.132100	−	0.0547176i	0.447754	+	0.894157i	$0.352224\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	22.7782	−	9.43503i	1.73179	−	0.717332i	0.732460	−	0.680810i	$-0.238372\pi$
$173$	22.7782	−	9.43503i	1.73179	−	0.717332i	0.999333	−	0.0365215i	$-0.0116278\pi$
$174$		0			0
$175$	7.53553	+	18.1924i	0.569633	+	1.37522i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	6.89949	−	6.89949i	0.515692	−	0.515692i	−0.400573	−	0.916265i	$-0.631189\pi$
$179$	6.89949	−	6.89949i	0.515692	−	0.515692i	0.916265	+	0.400573i	$0.131189\pi$
$180$		0			0
$181$	−1.12132	−	0.464466i	−0.0833471	−	0.0345235i	0.340620	−	0.940201i	$-0.389363\pi$
$181$	−1.12132	−	0.464466i	−0.0833471	−	0.0345235i	−0.423967	+	0.905677i	$0.639363\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.00000	−	1.00000i	−0.0735215	−	0.0735215i
$186$		0			0
$187$	−1.12132	+	7.53553i	−0.0819991	+	0.551053i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			3.17157i			0.229487i	0.993395	+	0.114743i	$0.0366046\pi$
$191$			3.17157i			0.229487i	−0.993395	+	0.114743i	$0.963395\pi$
$192$		0			0
$193$	1.70711	+	0.707107i	0.122880	+	0.0508987i	0.443276	−	0.896385i	$-0.353816\pi$
$193$	1.70711	+	0.707107i	0.122880	+	0.0508987i	−0.320396	+	0.947284i	$0.603816\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1.36396	−	3.29289i	0.0971782	−	0.234609i	−0.867813	−	0.496891i	$-0.834475\pi$
$197$	1.36396	−	3.29289i	0.0971782	−	0.234609i	0.964991	+	0.262282i	$0.0844750\pi$
$198$		0			0
$199$	5.22183	+	12.6066i	0.370165	+	0.893658i	0.993722	+	0.111880i	$0.0356872\pi$
$199$	5.22183	+	12.6066i	0.370165	+	0.893658i	−0.623556	+	0.781778i	$0.714313\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−10.2426			−0.718892
$204$		0			0
$205$	−7.41421			−0.517831
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.41421	+	10.6569i	0.305338	+	0.737150i
$210$		0			0
$211$	5.80761	−	14.0208i	0.399812	−	0.965233i	−0.587898	−	0.808935i	$-0.700044\pi$
$211$	5.80761	−	14.0208i	0.399812	−	0.965233i	0.987710	−	0.156297i	$-0.0499558\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.24264	+	0.514719i	0.0847474	+	0.0351035i
$216$		0			0
$217$			24.7279i			1.67864i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−9.58579	−	9.58579i	−0.641912	−	0.641912i	0.309113	−	0.951025i	$-0.399968\pi$
$223$	−9.58579	−	9.58579i	−0.641912	−	0.641912i	−0.951025	+	0.309113i	$0.899968\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−11.3640	−	4.70711i	−0.754253	−	0.312422i	−0.0277772	−	0.999614i	$-0.508843\pi$
$227$	−11.3640	−	4.70711i	−0.754253	−	0.312422i	−0.726475	+	0.687192i	$0.758843\pi$
$228$		0			0
$229$	−7.00000	+	7.00000i	−0.462573	+	0.462573i	−0.899498	−	0.436925i	$-0.856068\pi$
$229$	−7.00000	+	7.00000i	−0.462573	+	0.462573i	0.436925	+	0.899498i	$0.356068\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	6.53553	+	15.7782i	0.428157	+	1.03366i	0.979871	+	0.199629i	$0.0639738\pi$
$233$	6.53553	+	15.7782i	0.428157	+	1.03366i	−0.551714	+	0.834033i	$0.686026\pi$
$234$		0			0
$235$	−5.07107	+	2.10051i	−0.330800	+	0.137022i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	11.3137			0.731823			0.365911	−	0.930650i	$-0.380757\pi$
$239$	11.3137			0.731823			0.365911	+	0.930650i	$0.380757\pi$
$240$		0			0
$241$	7.36396	−	3.05025i	0.474354	−	0.196484i	−0.132681	−	0.991159i	$-0.542359\pi$
$241$	7.36396	−	3.05025i	0.474354	−	0.196484i	0.607035	+	0.794675i	$0.292359\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.77817	+	9.12132i	−0.241379	+	0.582740i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$			4.82843i			0.304768i	0.988321	+	0.152384i	$0.0486950\pi$
$251$			4.82843i			0.304768i	−0.988321	+	0.152384i	$0.951305\pi$
$252$		0			0
$253$	6.41421	+	6.41421i	0.403258	+	0.403258i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	0.171573	+	0.171573i	0.0107024	+	0.0107024i	0.712438	−	0.701735i	$-0.247591\pi$
$257$	0.171573	+	0.171573i	0.0107024	+	0.0107024i	−0.701735	+	0.712438i	$0.747591\pi$
$258$		0			0
$259$			8.24264i			0.512173i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−11.2426	+	11.2426i	−0.693251	+	0.693251i	−0.962946	−	0.269695i	$-0.913077\pi$
$263$	−11.2426	+	11.2426i	−0.693251	+	0.693251i	0.269695	+	0.962946i	$0.413077\pi$
$264$		0			0
$265$	−3.24264	+	7.82843i	−0.199194	+	0.480896i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−7.36396	+	3.05025i	−0.448989	+	0.185977i	−0.595708	−	0.803201i	$-0.703129\pi$
$269$	−7.36396	+	3.05025i	−0.448989	+	0.185977i	0.146720	+	0.989178i	$0.453129\pi$
$270$		0			0
$271$	11.3137			0.687259			0.343629	−	0.939105i	$-0.388344\pi$
$271$	11.3137			0.687259			0.343629	+	0.939105i	$0.388344\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−7.53553	+	3.12132i	−0.454410	+	0.188223i
$276$		0			0
$277$	1.46447	+	3.53553i	0.0879912	+	0.212430i	0.961749	−	0.273931i	$-0.0883241\pi$
$277$	1.46447	+	3.53553i	0.0879912	+	0.212430i	−0.873758	+	0.486361i	$0.838324\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−20.3137	+	20.3137i	−1.21181	+	1.21181i	−0.241385	+	0.970429i	$0.577602\pi$
$281$	−20.3137	+	20.3137i	−1.21181	+	1.21181i	−0.970429	+	0.241385i	$0.922398\pi$
$282$		0			0
$283$	−19.9497	−	8.26346i	−1.18589	−	0.491211i	−0.299475	−	0.954104i	$-0.596812\pi$
$283$	−19.9497	−	8.26346i	−1.18589	−	0.491211i	−0.886414	+	0.462893i	$0.846812\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	30.5563	+	30.5563i	1.80368	+	1.80368i
$288$		0			0
$289$	−15.0000	+	8.00000i	−0.882353	+	0.470588i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	15.3137i		−	0.894636i	−0.894375	−	0.447318i	$-0.852379\pi$
$293$		−	15.3137i		−	0.894636i	0.894375	−	0.447318i	$-0.147621\pi$
$294$		0			0
$295$	8.41421	+	3.48528i	0.489894	+	0.202921i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−3.00000	−	7.24264i	−0.172917	−	0.417459i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	9.75736			0.558705
$306$		0			0
$307$	18.6274			1.06312			0.531561	−	0.847020i	$-0.321605\pi$
$307$	18.6274			1.06312			0.531561	+	0.847020i	$0.321605\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	2.29289	+	5.53553i	0.130018	+	0.313891i	0.975460	−	0.220176i	$-0.0706630\pi$
$311$	2.29289	+	5.53553i	0.130018	+	0.313891i	−0.845442	+	0.534067i	$0.820663\pi$
$312$		0			0
$313$	5.94975	−	14.3640i	0.336300	−	0.811899i	−0.661765	−	0.749711i	$-0.730192\pi$
$313$	5.94975	−	14.3640i	0.336300	−	0.811899i	0.998064	−	0.0621876i	$-0.0198077\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	15.9497	+	6.60660i	0.895827	+	0.371064i	0.782614	−	0.622507i	$-0.213886\pi$
$317$	15.9497	+	6.60660i	0.895827	+	0.371064i	0.113213	+	0.993571i	$0.463886\pi$
$318$		0			0
$319$		−	4.24264i		−	0.237542i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−13.2426	+	22.0711i	−0.736840	+	1.22807i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	29.5563	+	12.2426i	1.62949	+	0.674959i
$330$		0			0
$331$	15.7279	−	15.7279i	0.864485	−	0.864485i	−0.127370	−	0.991855i	$-0.540654\pi$
$331$	15.7279	−	15.7279i	0.864485	−	0.864485i	0.991855	+	0.127370i	$0.0406537\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	0.485281	+	1.17157i	0.0265138	+	0.0640099i
$336$		0			0
$337$	−3.94975	+	1.63604i	−0.215156	+	0.0891207i	−0.487658	−	0.873034i	$-0.662149\pi$
$337$	−3.94975	+	1.63604i	−0.215156	+	0.0891207i	0.272502	+	0.962155i	$0.412149\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−10.2426			−0.554670
$342$		0			0
$343$	24.3137	−	10.0711i	1.31282	−	0.543787i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	7.36396	−	17.7782i	0.395318	−	0.954382i	−0.593443	−	0.804876i	$-0.702232\pi$
$347$	7.36396	−	17.7782i	0.395318	−	0.954382i	0.988761	−	0.149506i	$-0.0477684\pi$
$348$		0			0
$349$	1.48528	−	1.48528i	0.0795053	−	0.0795053i	−0.666236	−	0.745741i	$-0.732096\pi$
$349$	1.48528	−	1.48528i	0.0795053	−	0.0795053i	0.745741	+	0.666236i	$0.232096\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	8.97056i		−	0.477455i	−0.971087	−	0.238727i	$-0.923270\pi$
$353$		−	8.97056i		−	0.477455i	0.971087	−	0.238727i	$-0.0767302\pi$
$354$		0			0
$355$	1.34315	+	1.34315i	0.0712868	+	0.0712868i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−9.72792	−	9.72792i	−0.513420	−	0.513420i	0.402153	−	0.915573i	$-0.368262\pi$
$359$	−9.72792	−	9.72792i	−0.513420	−	0.513420i	−0.915573	+	0.402153i	$0.868262\pi$
$360$		0			0
$361$			19.9706i			1.05108i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−7.58579	+	7.58579i	−0.397058	+	0.397058i
$366$		0			0
$367$	7.94975	−	19.1924i	0.414973	−	1.00183i	−0.568809	−	0.822470i	$-0.692596\pi$
$367$	7.94975	−	19.1924i	0.414973	−	1.00183i	0.983783	−	0.179365i	$-0.0574043\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	45.6274	−	18.8995i	2.36886	−	0.981213i
$372$		0			0
$373$	−31.9411			−1.65385			−0.826924	−	0.562313i	$-0.809912\pi$
$373$	−31.9411			−1.65385			−0.826924	+	0.562313i	$0.809912\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	4.05025	+	9.77817i	0.208047	+	0.502271i	0.993116	−	0.117138i	$-0.0373721\pi$
$379$	4.05025	+	9.77817i	0.208047	+	0.502271i	−0.785068	+	0.619409i	$0.787372\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	10.4142	−	10.4142i	0.532141	−	0.532141i	−0.389068	−	0.921209i	$-0.627203\pi$
$383$	10.4142	−	10.4142i	0.532141	−	0.532141i	0.921209	+	0.389068i	$0.127203\pi$
$384$		0			0
$385$	−5.82843	−	2.41421i	−0.297044	−	0.123040i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2.31371	+	2.31371i	0.117310	+	0.117310i	0.763325	−	0.646015i	$-0.223566\pi$
$389$	2.31371	+	2.31371i	0.117310	+	0.117310i	−0.646015	+	0.763325i	$0.723566\pi$
$390$		0			0
$391$	−2.97918	+	20.0208i	−0.150664	+	1.01250i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	10.3848i		−	0.522515i
$396$		0			0
$397$	6.87868	+	2.84924i	0.345231	+	0.142999i	0.548560	−	0.836111i	$-0.315176\pi$
$397$	6.87868	+	2.84924i	0.345231	+	0.142999i	−0.203329	+	0.979110i	$0.565176\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−0.979185	+	2.36396i	−0.0488982	+	0.118051i	−0.946441	−	0.322876i	$-0.895350\pi$
$401$	−0.979185	+	2.36396i	−0.0488982	+	0.118051i	0.897543	+	0.440927i	$0.145350\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.41421			−0.169236
$408$		0			0
$409$	18.0000			0.890043			0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000			0.890043			0.445021	+	0.895520i	$0.353196\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−20.3137	−	49.0416i	−0.999572	−	2.41318i
$414$		0			0
$415$	−0.514719	+	1.24264i	−0.0252665	+	0.0609988i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−4.53553	−	1.87868i	−0.221575	−	0.0917795i	0.269134	−	0.963103i	$-0.413262\pi$
$419$	−4.53553	−	1.87868i	−0.221575	−	0.0917795i	−0.490710	+	0.871323i	$0.663262\pi$
$420$		0			0
$421$			11.3137i			0.551396i	0.961244	+	0.275698i	$0.0889090\pi$
$421$			11.3137i			0.551396i	−0.961244	+	0.275698i	$0.911091\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−15.6066	−	9.36396i	−0.757031	−	0.454219i
$426$		0			0
$427$	−40.2132	−	40.2132i	−1.94605	−	1.94605i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	5.12132	+	2.12132i	0.246685	+	0.102180i	0.502600	−	0.864519i	$-0.332377\pi$
$431$	5.12132	+	2.12132i	0.246685	+	0.102180i	−0.255915	+	0.966699i	$0.582377\pi$
$432$		0			0
$433$	−1.82843	+	1.82843i	−0.0878686	+	0.0878686i	−0.749675	−	0.661806i	$-0.769790\pi$
$433$	−1.82843	+	1.82843i	−0.0878686	+	0.0878686i	0.661806	+	0.749675i	$0.269790\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	11.7279	+	28.3137i	0.561023	+	1.35443i
$438$		0			0
$439$	17.6066	−	7.29289i	0.840317	−	0.348071i	0.0793386	−	0.996848i	$-0.474719\pi$
$439$	17.6066	−	7.29289i	0.840317	−	0.348071i	0.760979	+	0.648777i	$0.224719\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−16.2843			−0.773689			−0.386845	−	0.922145i	$-0.626435\pi$
$443$	−16.2843			−0.773689			−0.386845	+	0.922145i	$0.626435\pi$
$444$		0			0
$445$	6.82843	−	2.82843i	0.323698	−	0.134080i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−9.46447	+	22.8492i	−0.446656	+	1.07832i	0.526911	+	0.849920i	$0.323350\pi$
$449$	−9.46447	+	22.8492i	−0.446656	+	1.07832i	−0.973567	+	0.228402i	$0.926650\pi$
$450$		0			0
$451$	−12.6569	+	12.6569i	−0.595988	+	0.595988i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	13.4853	+	13.4853i	0.630815	+	0.630815i	0.948272	−	0.317458i	$-0.102829\pi$
$457$	13.4853	+	13.4853i	0.630815	+	0.630815i	−0.317458	+	0.948272i	$0.602829\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	19.4853	+	19.4853i	0.907520	+	0.907520i	0.996072	−	0.0885516i	$-0.0282238\pi$
$461$	19.4853	+	19.4853i	0.907520	+	0.907520i	−0.0885516	+	0.996072i	$0.528224\pi$
$462$		0			0
$463$		−	4.14214i		−	0.192501i	−0.995357	−	0.0962507i	$-0.969315\pi$
$463$		−	4.14214i		−	0.192501i	0.995357	−	0.0962507i	$-0.0306850\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−12.4142	+	12.4142i	−0.574461	+	0.574461i	−0.933372	−	0.358911i	$-0.883148\pi$
$467$	−12.4142	+	12.4142i	−0.574461	+	0.574461i	0.358911	+	0.933372i	$0.383148\pi$
$468$		0			0
$469$	2.82843	−	6.82843i	0.130605	−	0.315307i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3.00000	−	1.24264i	0.137940	−	0.0571367i
$474$		0			0
$475$	−27.5563			−1.26437
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−7.46447	+	3.09188i	−0.341060	+	0.141272i	−0.546638	−	0.837369i	$-0.684093\pi$
$479$	−7.46447	+	3.09188i	−0.341060	+	0.141272i	0.205578	+	0.978641i	$0.434093\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.10051	−	1.10051i	0.0499714	−	0.0499714i
$486$		0			0
$487$	−39.2635	−	16.2635i	−1.77920	−	0.736968i	−0.992876	−	0.119149i	$-0.961983\pi$
$487$	−39.2635	−	16.2635i	−1.77920	−	0.736968i	−0.786321	−	0.617819i	$-0.788017\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−11.5858	−	11.5858i	−0.522859	−	0.522859i	0.395575	−	0.918434i	$-0.370545\pi$
$491$	−11.5858	−	11.5858i	−0.522859	−	0.522859i	−0.918434	+	0.395575i	$0.870545\pi$
$492$		0			0
$493$	7.60660	−	5.63604i	0.342584	−	0.253834i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	11.0711i		−	0.496605i
$498$		0			0
$499$	−20.4350	−	8.46447i	−0.914798	−	0.378922i	−0.124906	−	0.992169i	$-0.539863\pi$
$499$	−20.4350	−	8.46447i	−0.914798	−	0.378922i	−0.789891	+	0.613247i	$0.789863\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	1.70711	−	4.12132i	0.0761161	−	0.183761i	−0.881241	−	0.472667i	$-0.843292\pi$
$503$	1.70711	−	4.12132i	0.0761161	−	0.183761i	0.957357	+	0.288906i	$0.0932915\pi$
$504$		0			0
$505$	2.92893	+	7.07107i	0.130336	+	0.314658i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−18.2843			−0.810436			−0.405218	−	0.914220i	$-0.632804\pi$
$509$	−18.2843			−0.810436			−0.405218	+	0.914220i	$0.632804\pi$
$510$		0			0
$511$	62.5269			2.76603
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4.00000	+	9.65685i	0.176261	+	0.425532i
$516$		0			0
$517$	−5.07107	+	12.2426i	−0.223025	+	0.538431i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−9.02082	−	3.73654i	−0.395209	−	0.163701i	0.176223	−	0.984350i	$-0.443612\pi$
$521$	−9.02082	−	3.73654i	−0.395209	−	0.163701i	−0.571432	+	0.820649i	$0.693612\pi$
$522$		0			0
$523$		−	24.1421i		−	1.05566i	−0.849349	−	0.527831i	$-0.823005\pi$
$523$		−	24.1421i		−	1.05566i	0.849349	−	0.527831i	$-0.176995\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−13.6066	−	18.3640i	−0.592713	−	0.799947i
$528$		0			0
$529$	0.778175	+	0.778175i	0.0338337	+	0.0338337i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	0.899495	−	0.899495i	0.0388886	−	0.0388886i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	9.12132	+	22.0208i	0.392883	+	0.948504i
$540$		0			0
$541$	0.0502525	−	0.0208153i	0.00216053	−	0.000894919i	−0.381603	−	0.924326i	$-0.624628\pi$
$541$	0.0502525	−	0.0208153i	0.00216053	−	0.000894919i	0.383763	+	0.923431i	$0.374628\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−9.55635			−0.409349
$546$		0			0
$547$	15.2635	−	6.32233i	0.652618	−	0.270323i	−0.0317103	−	0.999497i	$-0.510095\pi$
$547$	15.2635	−	6.32233i	0.652618	−	0.270323i	0.684329	+	0.729174i	$0.260095\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	5.48528	−	13.2426i	0.233681	−	0.564155i
$552$		0			0
$553$	−42.7990	+	42.7990i	−1.82000	+	1.82000i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			16.0000i			0.677942i	0.940797	+	0.338971i	$0.110079\pi$
$557$			16.0000i			0.677942i	−0.940797	+	0.338971i	$0.889921\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−15.5858	−	15.5858i	−0.656863	−	0.656863i	0.297774	−	0.954636i	$-0.403756\pi$
$563$	−15.5858	−	15.5858i	−0.656863	−	0.656863i	−0.954636	+	0.297774i	$0.903756\pi$
$564$		0			0
$565$		−	0.928932i		−	0.0390805i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	32.6569	−	32.6569i	1.36905	−	1.36905i	0.507244	−	0.861802i	$-0.330664\pi$
$569$	32.6569	−	32.6569i	1.36905	−	1.36905i	0.861802	−	0.507244i	$-0.169336\pi$
$570$		0			0
$571$	0.150758	−	0.363961i	0.00630901	−	0.0152313i	−0.920694	−	0.390286i	$-0.872376\pi$
$571$	0.150758	−	0.363961i	0.00630901	−	0.0152313i	0.927003	+	0.375055i	$0.122376\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−20.0208	+	8.29289i	−0.834926	+	0.345838i
$576$		0			0
$577$	14.0000			0.582828			0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000			0.582828			0.291414	+	0.956597i	$0.405874\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	7.24264	−	3.00000i	0.300475	−	0.124461i
$582$		0			0
$583$	7.82843	+	18.8995i	0.324220	+	0.782737i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−31.7279	+	31.7279i	−1.30955	+	1.30955i	−0.387814	+	0.921738i	$0.626770\pi$
$587$	−31.7279	+	31.7279i	−1.30955	+	1.30955i	−0.921738	+	0.387814i	$0.873230\pi$
$588$		0			0
$589$	−31.9706	−	13.2426i	−1.31732	−	0.545654i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−26.6569	−	26.6569i	−1.09467	−	1.09467i	−0.995023	−	0.0996425i	$-0.968230\pi$
$593$	−26.6569	−	26.6569i	−1.09467	−	1.09467i	−0.0996425	−	0.995023i	$-0.531770\pi$
$594$		0			0
$595$	−3.41421	−	13.6569i	−0.139969	−	0.559876i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	39.1716i		−	1.60051i	−0.599662	−	0.800254i	$-0.704698\pi$
$599$		−	39.1716i		−	1.60051i	0.599662	−	0.800254i	$-0.295302\pi$
$600$		0			0
$601$	−9.60660	−	3.97918i	−0.391861	−	0.162314i	0.178048	−	0.984022i	$-0.443022\pi$
$601$	−9.60660	−	3.97918i	−0.391861	−	0.162314i	−0.569910	+	0.821707i	$0.693022\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2.22183	+	5.36396i	−0.0903300	+	0.218076i
$606$		0			0
$607$	−2.09188	−	5.05025i	−0.0849069	−	0.204983i	0.875723	−	0.482813i	$-0.160385\pi$
$607$	−2.09188	−	5.05025i	−0.0849069	−	0.204983i	−0.960630	+	0.277830i	$0.910385\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−6.68629			−0.270057			−0.135028	−	0.990842i	$-0.543113\pi$
$613$	−6.68629			−0.270057			−0.135028	+	0.990842i	$0.543113\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	7.22183	+	17.4350i	0.290740	+	0.701908i	0.999995	−	0.00304356i	$-0.000968795\pi$
$617$	7.22183	+	17.4350i	0.290740	+	0.701908i	−0.709256	+	0.704951i	$0.750969\pi$
$618$		0			0
$619$	7.94975	−	19.1924i	0.319527	−	0.771407i	−0.679752	−	0.733442i	$-0.737912\pi$
$619$	7.94975	−	19.1924i	0.319527	−	0.771407i	0.999279	−	0.0379649i	$-0.0120875\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−39.7990	−	16.4853i	−1.59451	−	0.660469i
$624$		0			0
$625$		−	16.5563i		−	0.662254i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−4.53553	−	6.12132i	−0.180844	−	0.244073i
$630$		0			0
$631$	19.3848	+	19.3848i	0.771696	+	0.771696i	0.978403	−	0.206707i	$-0.0662747\pi$
$631$	19.3848	+	19.3848i	0.771696	+	0.771696i	−0.206707	+	0.978403i	$0.566275\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−7.24264	−	3.00000i	−0.287415	−	0.119051i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−15.1213	−	36.5061i	−0.597256	−	1.44190i	−0.876366	−	0.481645i	$-0.840039\pi$
$641$	−15.1213	−	36.5061i	−0.597256	−	1.44190i	0.279110	−	0.960259i	$-0.409961\pi$
$642$		0			0
$643$	−9.02082	+	3.73654i	−0.355746	+	0.147355i	−0.553397	−	0.832918i	$-0.686669\pi$
$643$	−9.02082	+	3.73654i	−0.355746	+	0.147355i	0.197651	+	0.980272i	$0.436669\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	45.2548			1.77915			0.889576	−	0.456788i	$-0.151000\pi$
$647$	45.2548			1.77915			0.889576	+	0.456788i	$0.151000\pi$
$648$		0			0
$649$	20.3137	−	8.41421i	0.797383	−	0.330287i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−10.6360	+	25.6777i	−0.416220	+	1.00484i	0.567212	+	0.823572i	$0.308022\pi$
$653$	−10.6360	+	25.6777i	−0.416220	+	1.00484i	−0.983433	+	0.181273i	$0.941978\pi$
$654$		0			0
$655$	0.899495	−	0.899495i	0.0351462	−	0.0351462i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	50.7696i		−	1.97770i	−0.148912	−	0.988850i	$-0.547577\pi$
$659$		−	50.7696i		−	1.97770i	0.148912	−	0.988850i	$-0.452423\pi$
$660$		0			0
$661$	20.7990	+	20.7990i	0.808987	+	0.808987i	0.984481	−	0.175494i	$-0.0561521\pi$
$661$	20.7990	+	20.7990i	0.808987	+	0.808987i	−0.175494	+	0.984481i	$0.556152\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−15.0711	−	15.0711i	−0.584431	−	0.584431i
$666$		0			0
$667$		−	11.2721i		−	0.436457i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	16.6569	−	16.6569i	0.643031	−	0.643031i
$672$		0			0
$673$	−4.19239	+	10.1213i	−0.161605	+	0.390148i	−0.983852	−	0.178981i	$-0.942720\pi$
$673$	−4.19239	+	10.1213i	−0.161605	+	0.390148i	0.822248	+	0.569130i	$0.192720\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	42.0919	−	17.4350i	1.61772	−	0.670083i	0.623945	−	0.781468i	$-0.285529\pi$
$677$	42.0919	−	17.4350i	1.61772	−	0.670083i	0.993777	+	0.111385i	$0.0355288\pi$
$678$		0			0
$679$	−9.07107			−0.348116
$680$		0			0
$681$		0			0
$682$		0			0
$683$	37.0208	−	15.3345i	1.41656	−	0.586759i	0.462568	−	0.886584i	$-0.346928\pi$
$683$	37.0208	−	15.3345i	1.41656	−	0.586759i	0.953994	+	0.299825i	$0.0969282\pi$
$684$		0			0
$685$	4.10051	+	9.89949i	0.156672	+	0.378240i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−33.6066	−	13.9203i	−1.27846	−	0.529554i	−0.362931	−	0.931816i	$-0.618224\pi$
$691$	−33.6066	−	13.9203i	−1.27846	−	0.529554i	−0.915524	+	0.402262i	$0.868224\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	10.6569	+	10.6569i	0.404238	+	0.404238i
$696$		0			0
$697$	−39.5061	−	5.87868i	−1.49640	−	0.222671i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	51.3137i		−	1.93809i	−0.246880	−	0.969046i	$-0.579405\pi$
$701$		−	51.3137i		−	1.93809i	0.246880	−	0.969046i	$-0.420595\pi$
$702$		0			0
$703$	−10.6569	−	4.41421i	−0.401931	−	0.166485i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	17.0711	−	41.2132i	0.642024	−	1.54998i
$708$		0			0
$709$	15.8076	+	38.1630i	0.593667	+	1.43324i	0.879937	+	0.475091i	$0.157585\pi$
$709$	15.8076	+	38.1630i	0.593667	+	1.43324i	−0.286270	+	0.958149i	$0.592415\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−27.2132			−1.01914
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−8.05025	−	19.4350i	−0.300224	−	0.724804i	−0.999946	−	0.0103893i	$-0.996693\pi$
$719$	−8.05025	−	19.4350i	−0.300224	−	0.724804i	0.699722	−	0.714415i	$-0.253307\pi$
$720$		0			0
$721$	23.3137	−	56.2843i	0.868248	−	2.09614i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	9.36396	+	3.87868i	0.347769	+	0.144051i
$726$		0			0
$727$			21.7990i			0.808480i	0.914653	+	0.404240i	$0.132464\pi$
$727$			21.7990i			0.808480i	−0.914653	+	0.404240i	$0.867536\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	6.21320	+	3.72792i	0.229804	+	0.137882i
$732$		0			0
$733$	0.514719	+	0.514719i	0.0190116	+	0.0190116i	0.716549	−	0.697537i	$-0.245721\pi$
$733$	0.514719	+	0.514719i	0.0190116	+	0.0190116i	−0.697537	+	0.716549i	$0.745721\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2.82843	+	1.17157i	0.104186	+	0.0431554i
$738$		0			0
$739$	−3.58579	+	3.58579i	−0.131905	+	0.131905i	−0.769977	−	0.638072i	$-0.779732\pi$
$739$	−3.58579	+	3.58579i	−0.131905	+	0.131905i	0.638072	+	0.769977i	$0.279732\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	0.150758	+	0.363961i	0.00553076	+	0.0133524i	0.926621	−	0.375997i	$-0.122700\pi$
$743$	0.150758	+	0.363961i	0.00553076	+	0.0133524i	−0.921090	+	0.389350i	$0.872700\pi$
$744$		0			0
$745$	12.4853	−	5.17157i	0.457425	−	0.189472i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−7.41421			−0.270909
$750$		0			0
$751$	23.2635	−	9.63604i	0.848896	−	0.351624i	0.0845408	−	0.996420i	$-0.473058\pi$
$751$	23.2635	−	9.63604i	0.848896	−	0.351624i	0.764355	+	0.644796i	$0.223058\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	0.0294373	−	0.0710678i	0.00107133	−	0.00258642i
$756$		0			0
$757$	−23.9706	+	23.9706i	−0.871225	+	0.871225i	−0.992606	−	0.121381i	$-0.961268\pi$
$757$	−23.9706	+	23.9706i	−0.871225	+	0.871225i	0.121381	+	0.992606i	$0.461268\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			36.9706i			1.34018i	0.742279	+	0.670091i	$0.233745\pi$
$761$			36.9706i			1.34018i	−0.742279	+	0.670091i	$0.766255\pi$
$762$		0			0
$763$	39.3848	+	39.3848i	1.42582	+	1.42582i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	35.5980i		−	1.28370i	−0.766832	−	0.641848i	$-0.778168\pi$
$769$		−	35.5980i		−	1.28370i	0.766832	−	0.641848i	$-0.221832\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−33.2843	+	33.2843i	−1.19715	+	1.19715i	−0.222136	+	0.975016i	$0.571303\pi$
$773$	−33.2843	+	33.2843i	−1.19715	+	1.19715i	−0.975016	+	0.222136i	$0.928697\pi$
$774$		0			0
$775$	9.36396	−	22.6066i	0.336363	−	0.812053i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−55.8701	+	23.1421i	−2.00175	+	0.829153i
$780$		0			0
$781$	4.58579			0.164092
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−5.17157	+	2.14214i	−0.184581	+	0.0764561i
$786$		0			0
$787$	−16.4350	−	39.6777i	−0.585846	−	1.41436i	−0.887442	−	0.460920i	$-0.847519\pi$
$787$	−16.4350	−	39.6777i	−0.585846	−	1.41436i	0.301596	−	0.953436i	$-0.402481\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−3.82843	+	3.82843i	−0.136123	+	0.136123i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	7.48528	+	7.48528i	0.265142	+	0.265142i	0.827139	−	0.561997i	$-0.189967\pi$
$797$	7.48528	+	7.48528i	0.265142	+	0.265142i	−0.561997	+	0.827139i	$0.689967\pi$
$798$		0			0
$799$	−28.6863	+	7.17157i	−1.01485	+	0.253712i
$800$		0			0
$801$		0			0
$802$		0			0
$803$			25.8995i			0.913973i
$804$		0			0
$805$	−15.4853	−	6.41421i	−0.545784	−	0.226071i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.778175	+	1.87868i	−0.0273592	+	0.0660509i	−0.936969	−	0.349413i	$-0.886381\pi$
$809$	−0.778175	+	1.87868i	−0.0273592	+	0.0660509i	0.909609	+	0.415464i	$0.136381\pi$
$810$		0			0
$811$	1.50610	+	3.63604i	0.0528862	+	0.127679i	0.948114	−	0.317929i	$-0.102988\pi$
$811$	1.50610	+	3.63604i	0.0528862	+	0.127679i	−0.895228	+	0.445608i	$0.852988\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	11.4142			0.399822
$816$		0			0
$817$	10.9706			0.383811
$818$		0			0
$819$		0			0
$820$		0			0
$821$	17.5650	+	42.4056i	0.613022	+	1.47997i	0.859665	+	0.510859i	$0.170673\pi$
$821$	17.5650	+	42.4056i	0.613022	+	1.47997i	−0.246643	+	0.969106i	$0.579327\pi$
$822$		0			0
$823$	6.77817	−	16.3640i	0.236272	−	0.570412i	−0.760619	−	0.649198i	$-0.775105\pi$
$823$	6.77817	−	16.3640i	0.236272	−	0.570412i	0.996892	+	0.0787865i	$0.0251045\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.12132	+	0.464466i	0.0389921	+	0.0161511i	0.402094	−	0.915598i	$-0.368282\pi$
$827$	1.12132	+	0.464466i	0.0389921	+	0.0161511i	−0.363102	+	0.931749i	$0.618282\pi$
$828$		0			0
$829$		−	13.9411i		−	0.484195i	−0.970252	−	0.242098i	$-0.922165\pi$
$829$		−	13.9411i		−	0.484195i	0.970252	−	0.242098i	$-0.0778354\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−27.3640	+	45.6066i	−0.948105	+	1.58018i
$834$		0			0
$835$	−1.00000	−	1.00000i	−0.0346064	−	0.0346064i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	0.636039	+	0.263456i	0.0219585	+	0.00909551i	0.393636	−	0.919267i	$-0.371217\pi$
$839$	0.636039	+	0.263456i	0.0219585	+	0.00909551i	−0.371677	+	0.928362i	$0.621217\pi$
$840$		0			0
$841$	16.7782	−	16.7782i	0.578558	−	0.578558i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.80761	−	9.19239i	−0.130986	−	0.316228i
$846$		0			0
$847$	31.2635	−	12.9497i	1.07423	−	0.444959i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−9.07107			−0.310952
$852$		0			0
$853$	−15.9497	+	6.60660i	−0.546109	+	0.226206i	−0.638642	−	0.769504i	$-0.720504\pi$
$853$	−15.9497	+	6.60660i	−0.546109	+	0.226206i	0.0925332	+	0.995710i	$0.470504\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	5.36396	−	12.9497i	0.183229	−	0.442355i	−0.805399	−	0.592733i	$-0.798049\pi$
$857$	5.36396	−	12.9497i	0.183229	−	0.442355i	0.988629	+	0.150378i	$0.0480490\pi$
$858$		0			0
$859$	−4.27208	+	4.27208i	−0.145761	+	0.145761i	−0.776222	−	0.630460i	$-0.782866\pi$
$859$	−4.27208	+	4.27208i	−0.145761	+	0.145761i	0.630460	+	0.776222i	$0.282866\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			44.1421i			1.50262i	0.659952	+	0.751308i	$0.270577\pi$
$863$			44.1421i			1.50262i	−0.659952	+	0.751308i	$0.729423\pi$
$864$		0			0
$865$	−13.3431	−	13.3431i	−0.453681	−	0.453681i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−17.7279	−	17.7279i	−0.601379	−	0.601379i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	22.7279	−	22.7279i	0.768344	−	0.768344i
$876$		0			0
$877$	8.77817	−	21.1924i	0.296418	−	0.715616i	−0.703570	−	0.710626i	$-0.748412\pi$
$877$	8.77817	−	21.1924i	0.296418	−	0.715616i	0.999988	−	0.00498980i	$-0.00158831\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	23.2635	−	9.63604i	0.783766	−	0.324646i	0.0453316	−	0.998972i	$-0.485566\pi$
$881$	23.2635	−	9.63604i	0.783766	−	0.324646i	0.738434	+	0.674326i	$0.235566\pi$
$882$		0			0
$883$	−49.2548			−1.65756			−0.828779	−	0.559577i	$-0.810964\pi$
$883$	−49.2548			−1.65756			−0.828779	+	0.559577i	$0.810964\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	23.8492	−	9.87868i	0.800779	−	0.331694i	0.0555107	−	0.998458i	$-0.482321\pi$
$887$	23.8492	−	9.87868i	0.800779	−	0.331694i	0.745268	+	0.666764i	$0.232321\pi$
$888$		0			0
$889$	17.4853	+	42.2132i	0.586438	+	1.41579i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−31.6569	+	31.6569i	−1.05936	+	1.05936i
$894$		0			0
$895$	−6.89949	−	2.85786i	−0.230625	−	0.0955279i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	9.00000	+	9.00000i	0.300167	+	0.300167i
$900$		0			0
$901$	−23.4853	+	39.1421i	−0.782408	+	1.30401i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			0.928932i			0.0308788i
$906$		0			0
$907$	33.0208	+	13.6777i	1.09644	+	0.454160i	0.856247	−	0.516566i	$-0.172790\pi$
$907$	33.0208	+	13.6777i	1.09644	+	0.454160i	0.240191	+	0.970726i	$0.422790\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−16.9203	+	40.8492i	−0.560595	+	1.35340i	0.348696	+	0.937236i	$0.386624\pi$
$911$	−16.9203	+	40.8492i	−0.560595	+	1.35340i	−0.909291	+	0.416160i	$0.863376\pi$
$912$		0			0
$913$	1.24264	+	3.00000i	0.0411254	+	0.0992855i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−7.41421			−0.244839
$918$		0			0
$919$	−12.2843			−0.405221			−0.202610	−	0.979259i	$-0.564942\pi$
$919$	−12.2843			−0.405221			−0.202610	+	0.979259i	$0.564942\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	3.12132	−	7.53553i	0.102628	−	0.247767i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−23.3640	−	9.67767i	−0.766547	−	0.317514i	−0.0350740	−	0.999385i	$-0.511167\pi$
$929$	−23.3640	−	9.67767i	−0.766547	−	0.317514i	−0.731473	+	0.681871i	$0.761167\pi$
$930$		0			0
$931$			80.5269i			2.63916i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	5.65685	−	1.41421i	0.184999	−	0.0462497i
$936$		0			0
$937$	−22.3137	−	22.3137i	−0.728957	−	0.728957i	0.241455	−	0.970412i	$-0.422375\pi$
$937$	−22.3137	−	22.3137i	−0.728957	−	0.728957i	−0.970412	+	0.241455i	$0.922375\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	28.9203	+	11.9792i	0.942775	+	0.390510i	0.800511	−	0.599318i	$-0.204562\pi$
$941$	28.9203	+	11.9792i	0.942775	+	0.390510i	0.142264	+	0.989829i	$0.454562\pi$
$942$		0			0
$943$	−33.6274	+	33.6274i	−1.09506	+	1.09506i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	6.49390	+	15.6777i	0.211024	+	0.509456i	0.993581	−	0.113122i	$-0.0360852\pi$
$947$	6.49390	+	15.6777i	0.211024	+	0.509456i	−0.782558	+	0.622578i	$0.786085\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	50.0000			1.61966			0.809829	−	0.586665i	$-0.199560\pi$
$953$	50.0000			1.61966			0.809829	+	0.586665i	$0.199560\pi$
$954$		0			0
$955$	2.24264	−	0.928932i	0.0725701	−	0.0300595i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	23.8995	−	57.6985i	0.771755	−	1.86318i
$960$		0			0
$961$	−0.192388	+	0.192388i	−0.00620607	+	0.00620607i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	1.41421i		−	0.0455251i
$966$		0			0
$967$	−21.5858	−	21.5858i	−0.694152	−	0.694152i	0.268991	−	0.963143i	$-0.413310\pi$
$967$	−21.5858	−	21.5858i	−0.694152	−	0.694152i	−0.963143	+	0.268991i	$0.913310\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	20.6985	+	20.6985i	0.664246	+	0.664246i	0.956378	−	0.292132i	$-0.0943647\pi$
$971$	20.6985	+	20.6985i	0.664246	+	0.664246i	−0.292132	+	0.956378i	$0.594365\pi$
$972$		0			0
$973$		−	87.8406i		−	2.81604i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.68629	+	1.68629i	−0.0539492	+	0.0539492i	−0.733567	−	0.679617i	$-0.762146\pi$
$977$	−1.68629	+	1.68629i	−0.0539492	+	0.0539492i	0.679617	+	0.733567i	$0.262146\pi$
$978$		0			0
$979$	6.82843	−	16.4853i	0.218237	−	0.526872i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−28.9203	+	11.9792i	−0.922415	+	0.382077i	−0.792796	−	0.609487i	$-0.791375\pi$
$983$	−28.9203	+	11.9792i	−0.922415	+	0.382077i	−0.129619	+	0.991564i	$0.541375\pi$
$984$		0			0
$985$	−2.72792			−0.0869188
$986$		0			0
$987$		0			0
$988$		0			0
$989$	7.97056	−	3.30152i	0.253449	−	0.104982i
$990$		0			0
$991$	3.07969	+	7.43503i	0.0978296	+	0.236181i	0.965216	−	0.261454i	$-0.0842019\pi$
$991$	3.07969	+	7.43503i	0.0978296	+	0.236181i	−0.867386	+	0.497635i	$0.834202\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	7.38478	−	7.38478i	0.234113	−	0.234113i
$996$		0			0
$997$	38.6777	+	16.0208i	1.22493	+	0.507384i	0.898975	−	0.438000i	$-0.144313\pi$
$997$	38.6777	+	16.0208i	1.22493	+	0.507384i	0.325959	+	0.945384i	$0.394313\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1224.2.bq.a.937.1		4
3.2	odd	2		136.2.n.a.121.1	yes	4
12.11	even	2		272.2.v.e.257.1		4
17.9	even	8	inner	1224.2.bq.a.145.1		4
51.5	even	16		2312.2.b.j.577.1		4
51.14	even	16		2312.2.a.s.1.4		4
51.20	even	16		2312.2.a.s.1.1		4
51.26	odd	8		136.2.n.a.9.1	✓	4
51.29	even	16		2312.2.b.j.577.4		4
204.71	odd	16		4624.2.a.bm.1.4		4
204.167	odd	16		4624.2.a.bm.1.1		4
204.179	even	8		272.2.v.e.145.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.2.n.a.9.1	✓	4	51.26	odd	8
136.2.n.a.121.1	yes	4	3.2	odd	2
272.2.v.e.145.1		4	204.179	even	8
272.2.v.e.257.1		4	12.11	even	2
1224.2.bq.a.145.1		4	17.9	even	8	inner
1224.2.bq.a.937.1		4	1.1	even	1	trivial
2312.2.a.s.1.1		4	51.20	even	16
2312.2.a.s.1.4		4	51.14	even	16
2312.2.b.j.577.1		4	51.5	even	16
2312.2.b.j.577.4		4	51.29	even	16
4624.2.a.bm.1.1		4	204.167	odd	16
4624.2.a.bm.1.4		4	204.71	odd	16

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:	\( N \)	\(=\)	\( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1224.bq (of order \(8\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.292893 - 0.707107i) q^{5} +(-1.70711 + 4.12132i) q^{7} +O(q^{10})\)	\(q+(-0.292893 - 0.707107i) q^{5} +(-1.70711 + 4.12132i) q^{7} +(-1.70711 - 0.707107i) q^{11} +(-1.00000 - 4.00000i) q^{17} +(-4.41421 - 4.41421i) q^{19} +(-4.53553 - 1.87868i) q^{23} +(3.12132 - 3.12132i) q^{25} +(0.878680 + 2.12132i) q^{29} +(5.12132 - 2.12132i) q^{31} +3.41421 q^{35} +(1.70711 - 0.707107i) q^{37} +(3.70711 - 8.94975i) q^{41} +(-1.24264 + 1.24264i) q^{43} -7.17157i q^{47} +(-9.12132 - 9.12132i) q^{49} +(-7.82843 - 7.82843i) q^{53} +1.41421i q^{55} +(-8.41421 + 8.41421i) q^{59} +(-4.87868 + 11.7782i) q^{61} -1.65685 q^{67} +(-2.29289 + 0.949747i) q^{71} +(-5.36396 - 12.9497i) q^{73} +(5.82843 - 5.82843i) q^{77} +(12.5355 + 5.19239i) q^{79} +(-1.24264 - 1.24264i) q^{83} +(-2.53553 + 1.87868i) q^{85} +9.65685i q^{89} +(-1.82843 + 4.41421i) q^{95} +(0.778175 + 1.87868i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} - 4 q^{7}+O(q^{10}) \) 4 * q - 4 * q^5 - 4 * q^7	\( 4 q - 4 q^{5} - 4 q^{7} - 4 q^{11} - 4 q^{17} - 12 q^{19} - 4 q^{23} + 4 q^{25} + 12 q^{29} + 12 q^{31} + 8 q^{35} + 4 q^{37} + 12 q^{41} + 12 q^{43} - 28 q^{49} - 20 q^{53} - 28 q^{59} - 28 q^{61} + 16 q^{67} - 12 q^{71} + 4 q^{73} + 12 q^{77} + 36 q^{79} + 12 q^{83} + 4 q^{85} + 4 q^{95} - 28 q^{97}+O(q^{100}) \) 4 * q - 4 * q^5 - 4 * q^7 - 4 * q^11 - 4 * q^17 - 12 * q^19 - 4 * q^23 + 4 * q^25 + 12 * q^29 + 12 * q^31 + 8 * q^35 + 4 * q^37 + 12 * q^41 + 12 * q^43 - 28 * q^49 - 20 * q^53 - 28 * q^59 - 28 * q^61 + 16 * q^67 - 12 * q^71 + 4 * q^73 + 12 * q^77 + 36 * q^79 + 12 * q^83 + 4 * q^85 + 4 * q^95 - 28 * q^97

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