LMFDB - Embedded newform 336.2.k.c.209.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,2,Mod(209,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("336.209");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	336.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$2.68297350792$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.342102016.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16 $ x^8 + x^6 + 4x^4 + 4x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.7
Root			$0.599676 - 1.28078i$ of defining polynomial
Character	$\chi$	$=$	336.209
Dual form			336.2.k.c.209.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.66757 - 0.468213i) q^{3} +0.936426 q^{5} +(-1.56155 - 2.13578i) q^{7} +(2.56155 - 1.56155i) q^{9} +O(q^{10})$	\(q+(1.66757 - 0.468213i) q^{3} +0.936426 q^{5} +(-1.56155 - 2.13578i) q^{7} +(2.56155 - 1.56155i) q^{9} -5.12311i q^{11} +3.33513i q^{13} +(1.56155 - 0.438447i) q^{15} +4.27156 q^{17} +5.73384i q^{19} +(-3.60399 - 2.83041i) q^{21} +2.00000i q^{23} -4.12311 q^{25} +(3.54042 - 3.80335i) q^{27} -0.876894i q^{29} +6.14441i q^{31} +(-2.39871 - 8.54312i) q^{33} +(-1.46228 - 2.00000i) q^{35} -2.00000 q^{37} +(1.56155 + 5.56155i) q^{39} +0.525853 q^{41} +4.00000 q^{43} +(2.39871 - 1.46228i) q^{45} -10.4160 q^{47} +(-2.12311 + 6.67026i) q^{49} +(7.12311 - 2.00000i) q^{51} +0.876894i q^{53} -4.79741i q^{55} +(2.68466 + 9.56155i) q^{57} +10.0054 q^{59} -10.0054i q^{61} +(-7.33513 - 3.03246i) q^{63} +3.12311i q^{65} -10.2462 q^{67} +(0.936426 + 3.33513i) q^{69} +13.1231i q^{71} +3.74571i q^{73} +(-6.87555 + 1.93049i) q^{75} +(-10.9418 + 8.00000i) q^{77} -11.1231 q^{79} +(4.12311 - 8.00000i) q^{81} +3.33513 q^{83} +4.00000 q^{85} +(-0.410574 - 1.46228i) q^{87} -15.7392 q^{89} +(7.12311 - 5.20798i) q^{91} +(2.87689 + 10.2462i) q^{93} +5.36932i q^{95} +10.4160i q^{97} +(-8.00000 - 13.1231i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{7} + 4 q^{9}+O(q^{10}) $ 8 * q + 4 * q^7 + 4 * q^9	$ 8 q + 4 q^{7} + 4 q^{9} - 4 q^{15} - 8 q^{21} - 16 q^{37} - 4 q^{39} + 32 q^{43} + 16 q^{49} + 24 q^{51} - 28 q^{57} - 32 q^{63} - 16 q^{67} - 56 q^{79} + 32 q^{85} + 24 q^{91} + 56 q^{93} - 64 q^{99}+O(q^{100}) $ 8 * q + 4 * q^7 + 4 * q^9 - 4 * q^15 - 8 * q^21 - 16 * q^37 - 4 * q^39 + 32 * q^43 + 16 * q^49 + 24 * q^51 - 28 * q^57 - 32 * q^63 - 16 * q^67 - 56 * q^79 + 32 * q^85 + 24 * q^91 + 56 * q^93 - 64 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	1.66757	−	0.468213i	0.962770	−	0.270323i
$3$	1.66757	−	0.468213i	0.962770	−	0.270323i
$4$		0			0
$5$	0.936426			0.418783			0.209391	−	0.977832i	$-0.432852\pi$
$5$	0.936426			0.418783			0.209391	+	0.977832i	$0.432852\pi$
$6$		0			0
$7$	−1.56155	−	2.13578i	−0.590211	−	0.807249i
$7$	−1.56155	−	2.13578i	−0.590211	−	0.807249i
$8$		0			0
$9$	2.56155	−	1.56155i	0.853851	−	0.520518i
$10$		0			0
$11$		−	5.12311i		−	1.54467i	−0.635213	−	0.772337i	$-0.719088\pi$
$11$		−	5.12311i		−	1.54467i	0.635213	−	0.772337i	$-0.280912\pi$
$12$		0			0
$13$			3.33513i			0.924999i	0.886619	+	0.462500i	$0.153047\pi$
$13$			3.33513i			0.924999i	−0.886619	+	0.462500i	$0.846953\pi$
$14$		0			0
$15$	1.56155	−	0.438447i	0.403191	−	0.113207i
$16$		0			0
$17$	4.27156			1.03601			0.518003	−	0.855379i	$-0.326676\pi$
$17$	4.27156			1.03601			0.518003	+	0.855379i	$0.326676\pi$
$18$		0			0
$19$			5.73384i			1.31543i	0.753266	+	0.657716i	$0.228477\pi$
$19$			5.73384i			1.31543i	−0.753266	+	0.657716i	$0.771523\pi$
$20$		0			0
$21$	−3.60399	−	2.83041i	−0.786456	−	0.617647i
$22$		0			0
$23$			2.00000i			0.417029i	0.978019	+	0.208514i	$0.0668628\pi$
$23$			2.00000i			0.417029i	−0.978019	+	0.208514i	$0.933137\pi$
$24$		0			0
$25$	−4.12311			−0.824621
$26$		0			0
$27$	3.54042	−	3.80335i	0.681354	−	0.731954i
$28$		0			0
$29$		−	0.876894i		−	0.162835i	−0.996680	−	0.0814176i	$-0.974055\pi$
$29$		−	0.876894i		−	0.162835i	0.996680	−	0.0814176i	$-0.0259447\pi$
$30$		0			0
$31$			6.14441i			1.10357i	0.833987	+	0.551784i	$0.186053\pi$
$31$			6.14441i			1.10357i	−0.833987	+	0.551784i	$0.813947\pi$
$32$		0			0
$33$	−2.39871	−	8.54312i	−0.417561	−	1.48717i
$34$		0			0
$35$	−1.46228	−	2.00000i	−0.247170	−	0.338062i
$36$		0			0
$37$	−2.00000			−0.328798			−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000			−0.328798			−0.164399	+	0.986394i	$0.552568\pi$
$38$		0			0
$39$	1.56155	+	5.56155i	0.250049	+	0.890561i
$40$		0			0
$41$	0.525853			0.0821244			0.0410622	−	0.999157i	$-0.486926\pi$
$41$	0.525853			0.0821244			0.0410622	+	0.999157i	$0.486926\pi$
$42$		0			0
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$		0			0
$45$	2.39871	−	1.46228i	0.357578	−	0.217984i
$46$		0			0
$47$	−10.4160			−1.51933			−0.759663	−	0.650317i	$-0.774636\pi$
$47$	−10.4160			−1.51933			−0.759663	+	0.650317i	$0.774636\pi$
$48$		0			0
$49$	−2.12311	+	6.67026i	−0.303301	+	0.952895i
$50$		0			0
$51$	7.12311	−	2.00000i	0.997434	−	0.280056i
$52$		0			0
$53$			0.876894i			0.120451i	0.998185	+	0.0602254i	$0.0191819\pi$
$53$			0.876894i			0.120451i	−0.998185	+	0.0602254i	$0.980818\pi$
$54$		0			0
$55$		−	4.79741i		−	0.646883i
$56$		0			0
$57$	2.68466	+	9.56155i	0.355592	+	1.26646i
$58$		0			0
$59$	10.0054			1.30259			0.651296	−	0.758824i	$-0.274226\pi$
$59$	10.0054			1.30259			0.651296	+	0.758824i	$0.274226\pi$
$60$		0			0
$61$		−	10.0054i		−	1.28106i	−0.767933	−	0.640530i	$-0.778715\pi$
$61$		−	10.0054i		−	1.28106i	0.767933	−	0.640530i	$-0.221285\pi$
$62$		0			0
$63$	−7.33513	−	3.03246i	−0.924140	−	0.382055i
$64$		0			0
$65$			3.12311i			0.387374i
$66$		0			0
$67$	−10.2462			−1.25177			−0.625887	−	0.779914i	$-0.715263\pi$
$67$	−10.2462			−1.25177			−0.625887	+	0.779914i	$0.715263\pi$
$68$		0			0
$69$	0.936426	+	3.33513i	0.112732	+	0.401503i
$70$		0			0
$71$			13.1231i			1.55743i	0.627380	+	0.778713i	$0.284127\pi$
$71$			13.1231i			1.55743i	−0.627380	+	0.778713i	$0.715873\pi$
$72$		0			0
$73$			3.74571i			0.438402i	0.975680	+	0.219201i	$0.0703450\pi$
$73$			3.74571i			0.438402i	−0.975680	+	0.219201i	$0.929655\pi$
$74$		0			0
$75$	−6.87555	+	1.93049i	−0.793920	+	0.222914i
$76$		0			0
$77$	−10.9418	+	8.00000i	−1.24694	+	0.911685i
$78$		0			0
$79$	−11.1231			−1.25145			−0.625724	−	0.780045i	$-0.715196\pi$
$79$	−11.1231			−1.25145			−0.625724	+	0.780045i	$0.715196\pi$
$80$		0			0
$81$	4.12311	−	8.00000i	0.458123	−	0.888889i
$82$		0			0
$83$	3.33513			0.366078			0.183039	−	0.983106i	$-0.441406\pi$
$83$	3.33513			0.366078			0.183039	+	0.983106i	$0.441406\pi$
$84$		0			0
$85$	4.00000			0.433861
$86$		0			0
$87$	−0.410574	−	1.46228i	−0.0440181	−	0.156773i
$88$		0			0
$89$	−15.7392			−1.66836			−0.834178	−	0.551496i	$-0.814057\pi$
$89$	−15.7392			−1.66836			−0.834178	+	0.551496i	$0.814057\pi$
$90$		0			0
$91$	7.12311	−	5.20798i	0.746704	−	0.545945i
$92$		0			0
$93$	2.87689	+	10.2462i	0.298320	+	1.06248i
$94$		0			0
$95$			5.36932i			0.550880i
$96$		0			0
$97$			10.4160i			1.05758i	0.848752	+	0.528791i	$0.177354\pi$
$97$			10.4160i			1.05758i	−0.848752	+	0.528791i	$0.822646\pi$
$98$		0			0
$99$	−8.00000	−	13.1231i	−0.804030	−	1.31892i
$100$		0			0
$101$	14.2770			1.42061			0.710305	−	0.703894i	$-0.248557\pi$
$101$	14.2770			1.42061			0.710305	+	0.703894i	$0.248557\pi$
$102$		0			0
$103$			2.39871i			0.236351i	0.992993	+	0.118176i	$0.0377046\pi$
$103$			2.39871i			0.236351i	−0.992993	+	0.118176i	$0.962295\pi$
$104$		0			0
$105$	−3.37487	−	2.65047i	−0.329354	−	0.258660i
$106$		0			0
$107$			1.12311i			0.108575i	0.998525	+	0.0542874i	$0.0172887\pi$
$107$			1.12311i			0.108575i	−0.998525	+	0.0542874i	$0.982711\pi$
$108$		0			0
$109$	4.24621			0.406713			0.203357	−	0.979105i	$-0.434815\pi$
$109$	4.24621			0.406713			0.203357	+	0.979105i	$0.434815\pi$
$110$		0			0
$111$	−3.33513	+	0.936426i	−0.316557	+	0.0888817i
$112$		0			0
$113$		−	17.3693i		−	1.63397i	−0.576660	−	0.816984i	$-0.695644\pi$
$113$		−	17.3693i		−	1.63397i	0.576660	−	0.816984i	$-0.304356\pi$
$114$		0			0
$115$			1.87285i			0.174644i
$116$		0			0
$117$	5.20798	+	8.54312i	0.481478	+	0.789811i
$118$		0			0
$119$	−6.67026	−	9.12311i	−0.611462	−	0.836314i
$120$		0			0
$121$	−15.2462			−1.38602
$122$		0			0
$123$	0.876894	−	0.246211i	0.0790669	−	0.0222001i
$124$		0			0
$125$	−8.54312			−0.764120
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		0			0
$129$	6.67026	−	1.87285i	0.587284	−	0.164895i
$130$		0			0
$131$	0.410574			0.0358720			0.0179360	−	0.999839i	$-0.494290\pi$
$131$	0.410574			0.0358720			0.0179360	+	0.999839i	$0.494290\pi$
$132$		0			0
$133$	12.2462	−	8.95369i	1.06188	−	0.776383i
$134$		0			0
$135$	3.31534	−	3.56155i	0.285339	−	0.306530i
$136$		0			0
$137$			14.2462i			1.21714i	0.793502	+	0.608568i	$0.208256\pi$
$137$			14.2462i			1.21714i	−0.793502	+	0.608568i	$0.791744\pi$
$138$		0			0
$139$			5.73384i			0.486338i	0.969984	+	0.243169i	$0.0781869\pi$
$139$			5.73384i			0.486338i	−0.969984	+	0.243169i	$0.921813\pi$
$140$		0			0
$141$	−17.3693	+	4.87689i	−1.46276	+	0.410709i
$142$		0			0
$143$	17.0862			1.42882
$144$		0			0
$145$		−	0.821147i		−	0.0681925i
$146$		0			0
$147$	−0.417313	+	12.1172i	−0.0344194	+	0.999407i
$148$		0			0
$149$		−	21.3693i		−	1.75064i	−0.483542	−	0.875321i	$-0.660650\pi$
$149$		−	21.3693i		−	1.75064i	0.483542	−	0.875321i	$-0.339350\pi$
$150$		0			0
$151$	20.4924			1.66765			0.833825	−	0.552029i	$-0.186146\pi$
$151$	20.4924			1.66765			0.833825	+	0.552029i	$0.186146\pi$
$152$		0			0
$153$	10.9418	−	6.67026i	0.884594	−	0.539259i
$154$		0			0
$155$			5.75379i			0.462155i
$156$		0			0
$157$			0.410574i			0.0327673i	0.999866	+	0.0163837i	$0.00521532\pi$
$157$			0.410574i			0.0327673i	−0.999866	+	0.0163837i	$0.994785\pi$
$158$		0			0
$159$	0.410574	+	1.46228i	0.0325606	+	0.115966i
$160$		0			0
$161$	4.27156	−	3.12311i	0.336646	−	0.246135i
$162$		0			0
$163$	2.24621			0.175937			0.0879684	−	0.996123i	$-0.471963\pi$
$163$	2.24621			0.175937			0.0879684	+	0.996123i	$0.471963\pi$
$164$		0			0
$165$	−2.24621	−	8.00000i	−0.174867	−	0.622799i
$166$		0			0
$167$	6.67026			0.516161			0.258080	−	0.966123i	$-0.416910\pi$
$167$	6.67026			0.516161			0.258080	+	0.966123i	$0.416910\pi$
$168$		0			0
$169$	1.87689			0.144376
$170$		0			0
$171$	8.95369	+	14.6875i	0.684706	+	1.12318i
$172$		0			0
$173$	10.5312			0.800676			0.400338	−	0.916368i	$-0.368893\pi$
$173$	10.5312			0.800676			0.400338	+	0.916368i	$0.368893\pi$
$174$		0			0
$175$	6.43845	+	8.80604i	0.486701	+	0.665674i
$176$		0			0
$177$	16.6847	−	4.68466i	1.25410	−	0.352120i
$178$		0			0
$179$		−	13.1231i		−	0.980867i	−0.871479	−	0.490433i	$-0.836838\pi$
$179$		−	13.1231i		−	0.980867i	0.871479	−	0.490433i	$-0.163162\pi$
$180$		0			0
$181$		−	24.1671i		−	1.79632i	−0.439664	−	0.898162i	$-0.644903\pi$
$181$		−	24.1671i		−	1.79632i	0.439664	−	0.898162i	$-0.355097\pi$
$182$		0			0
$183$	−4.68466	−	16.6847i	−0.346300	−	1.23337i
$184$		0			0
$185$	−1.87285			−0.137695
$186$		0			0
$187$		−	21.8836i		−	1.60029i
$188$		0			0
$189$	−13.6517	−	1.62243i	−0.993012	−	0.118014i
$190$		0			0
$191$		−	7.36932i		−	0.533225i	−0.963804	−	0.266613i	$-0.914096\pi$
$191$		−	7.36932i		−	0.533225i	0.963804	−	0.266613i	$-0.0859044\pi$
$192$		0			0
$193$	−1.12311			−0.0808429			−0.0404215	−	0.999183i	$-0.512870\pi$
$193$	−1.12311			−0.0808429			−0.0404215	+	0.999183i	$0.512870\pi$
$194$		0			0
$195$	1.46228	+	5.20798i	0.104716	+	0.372952i
$196$		0			0
$197$		−	8.87689i		−	0.632453i	−0.948684	−	0.316226i	$-0.897584\pi$
$197$		−	8.87689i		−	0.632453i	0.948684	−	0.316226i	$-0.102416\pi$
$198$		0			0
$199$		−	14.6875i		−	1.04117i	−0.853809	−	0.520586i	$-0.825714\pi$
$199$		−	14.6875i		−	1.04117i	0.853809	−	0.520586i	$-0.174286\pi$
$200$		0			0
$201$	−17.0862	+	4.79741i	−1.20517	+	0.338383i
$202$		0			0
$203$	−1.87285	+	1.36932i	−0.131448	+	0.0961072i
$204$		0			0
$205$	0.492423			0.0343923
$206$		0			0
$207$	3.12311	+	5.12311i	0.217071	+	0.356080i
$208$		0			0
$209$	29.3751			2.03192
$210$		0			0
$211$	5.75379			0.396107			0.198054	−	0.980191i	$-0.436538\pi$
$211$	5.75379			0.396107			0.198054	+	0.980191i	$0.436538\pi$
$212$		0			0
$213$	6.14441	+	21.8836i	0.421008	+	1.49944i
$214$		0			0
$215$	3.74571			0.255455
$216$		0			0
$217$	13.1231	−	9.59482i	0.890854	−	0.651339i
$218$		0			0
$219$	1.75379	+	6.24621i	0.118510	+	0.422080i
$220$		0			0
$221$			14.2462i			0.958304i
$222$		0			0
$223$			5.32326i			0.356472i	0.983988	+	0.178236i	$0.0570391\pi$
$223$			5.32326i			0.356472i	−0.983988	+	0.178236i	$0.942961\pi$
$224$		0			0
$225$	−10.5616	+	6.43845i	−0.704104	+	0.429230i
$226$		0			0
$227$	−17.4968			−1.16130			−0.580652	−	0.814152i	$-0.697202\pi$
$227$	−17.4968			−1.16130			−0.580652	+	0.814152i	$0.697202\pi$
$228$		0			0
$229$			20.4214i			1.34948i	0.738055	+	0.674741i	$0.235745\pi$
$229$			20.4214i			1.34948i	−0.738055	+	0.674741i	$0.764255\pi$
$230$		0			0
$231$	−14.5005	+	18.4636i	−0.954063	+	1.21482i
$232$		0			0
$233$			8.00000i			0.524097i	0.965055	+	0.262049i	$0.0843981\pi$
$233$			8.00000i			0.524097i	−0.965055	+	0.262049i	$0.915602\pi$
$234$		0			0
$235$	−9.75379			−0.636267
$236$		0			0
$237$	−18.5485	+	5.20798i	−1.20486	+	0.338295i
$238$		0			0
$239$		−	12.2462i		−	0.792142i	−0.918220	−	0.396071i	$-0.870373\pi$
$239$		−	12.2462i		−	0.792142i	0.918220	−	0.396071i	$-0.129627\pi$
$240$		0			0
$241$			2.92456i			0.188387i	0.995554	+	0.0941937i	$0.0300273\pi$
$241$			2.92456i			0.188387i	−0.995554	+	0.0941937i	$0.969973\pi$
$242$		0			0
$243$	3.12985	−	15.2710i	0.200780	−	0.979636i
$244$		0			0
$245$	−1.98813	+	6.24621i	−0.127017	+	0.399056i
$246$		0			0
$247$	−19.1231			−1.21677
$248$		0			0
$249$	5.56155	−	1.56155i	0.352449	−	0.0989594i
$250$		0			0
$251$	−13.7511			−0.867962			−0.433981	−	0.900922i	$-0.642891\pi$
$251$	−13.7511			−0.867962			−0.433981	+	0.900922i	$0.642891\pi$
$252$		0			0
$253$	10.2462			0.644174
$254$		0			0
$255$	6.67026	−	1.87285i	0.417708	−	0.117283i
$256$		0			0
$257$	−19.4849			−1.21544			−0.607719	−	0.794152i	$-0.707915\pi$
$257$	−19.4849			−1.21544			−0.607719	+	0.794152i	$0.707915\pi$
$258$		0			0
$259$	3.12311	+	4.27156i	0.194060	+	0.265422i
$260$		0			0
$261$	−1.36932	−	2.24621i	−0.0847586	−	0.139037i
$262$		0			0
$263$		−	15.3693i		−	0.947713i	−0.880602	−	0.473856i	$-0.842862\pi$
$263$		−	15.3693i		−	0.947713i	0.880602	−	0.473856i	$-0.157138\pi$
$264$		0			0
$265$			0.821147i			0.0504427i
$266$		0			0
$267$	−26.2462	+	7.36932i	−1.60624	+	0.450995i
$268$		0			0
$269$	−2.80928			−0.171285			−0.0856424	−	0.996326i	$-0.527294\pi$
$269$	−2.80928			−0.171285			−0.0856424	+	0.996326i	$0.527294\pi$
$270$		0			0
$271$		−	17.6121i		−	1.06986i	−0.844897	−	0.534929i	$-0.820338\pi$
$271$		−	17.6121i		−	1.06986i	0.844897	−	0.534929i	$-0.179662\pi$
$272$		0			0
$273$	9.43980	−	12.0198i	0.571323	−	0.727471i
$274$		0			0
$275$			21.1231i			1.27377i
$276$		0			0
$277$	4.24621			0.255130			0.127565	−	0.991830i	$-0.459284\pi$
$277$	4.24621			0.255130			0.127565	+	0.991830i	$0.459284\pi$
$278$		0			0
$279$	9.59482	+	15.7392i	0.574427	+	0.942283i
$280$		0			0
$281$			4.49242i			0.267995i	0.990982	+	0.133998i	$0.0427815\pi$
$281$			4.49242i			0.267995i	−0.990982	+	0.133998i	$0.957219\pi$
$282$		0			0
$283$		−	3.86098i		−	0.229512i	−0.993394	−	0.114756i	$-0.963391\pi$
$283$		−	3.86098i		−	0.229512i	0.993394	−	0.114756i	$-0.0366086\pi$
$284$		0			0
$285$	2.51398	+	8.95369i	0.148916	+	0.530371i
$286$		0			0
$287$	−0.821147	−	1.12311i	−0.0484708	−	0.0662948i
$288$		0			0
$289$	1.24621			0.0733065
$290$		0			0
$291$	4.87689	+	17.3693i	0.285889	+	1.01821i
$292$		0			0
$293$	−15.3287			−0.895510			−0.447755	−	0.894156i	$-0.647776\pi$
$293$	−15.3287			−0.895510			−0.447755	+	0.894156i	$0.647776\pi$
$294$		0			0
$295$	9.36932			0.545503
$296$		0			0
$297$	−19.4849	−	18.1379i	−1.13063	−	1.05247i
$298$		0			0
$299$	−6.67026			−0.385751
$300$		0			0
$301$	−6.24621	−	8.54312i	−0.360026	−	0.492417i
$302$		0			0
$303$	23.8078	−	6.68466i	1.36772	−	0.384024i
$304$		0			0
$305$		−	9.36932i		−	0.536486i
$306$		0			0
$307$		−	0.115279i		−	0.00657934i	−0.999995	−	0.00328967i	$-0.998953\pi$
$307$		−	0.115279i		−	0.00657934i	0.999995	−	0.00328967i	$-0.00104714\pi$
$308$		0			0
$309$	1.12311	+	4.00000i	0.0638912	+	0.227552i
$310$		0			0
$311$	−9.59482			−0.544072			−0.272036	−	0.962287i	$-0.587697\pi$
$311$	−9.59482			−0.544072			−0.272036	+	0.962287i	$0.587697\pi$
$312$		0			0
$313$			6.67026i			0.377026i	0.982071	+	0.188513i	$0.0603667\pi$
$313$			6.67026i			0.377026i	−0.982071	+	0.188513i	$0.939633\pi$
$314$		0			0
$315$	−6.86881	−	2.83968i	−0.387014	−	0.159998i
$316$		0			0
$317$			27.6155i			1.55104i	0.631321	+	0.775521i	$0.282513\pi$
$317$			27.6155i			1.55104i	−0.631321	+	0.775521i	$0.717487\pi$
$318$		0			0
$319$	−4.49242			−0.251527
$320$		0			0
$321$	0.525853	+	1.87285i	0.0293502	+	0.104532i
$322$		0			0
$323$			24.4924i			1.36279i
$324$		0			0
$325$		−	13.7511i		−	0.762774i
$326$		0			0
$327$	7.08084	−	1.98813i	0.391571	−	0.109944i
$328$		0			0
$329$	16.2651	+	22.2462i	0.896723	+	1.22647i
$330$		0			0
$331$	−5.75379			−0.316257			−0.158128	−	0.987419i	$-0.550546\pi$
$331$	−5.75379			−0.316257			−0.158128	+	0.987419i	$0.550546\pi$
$332$		0			0
$333$	−5.12311	+	3.12311i	−0.280744	+	0.171145i
$334$		0			0
$335$	−9.59482			−0.524221
$336$		0			0
$337$	0.630683			0.0343555			0.0171777	−	0.999852i	$-0.494532\pi$
$337$	0.630683			0.0343555			0.0171777	+	0.999852i	$0.494532\pi$
$338$		0			0
$339$	−8.13254	−	28.9645i	−0.441699	−	1.57313i
$340$		0			0
$341$	31.4785			1.70465
$342$		0			0
$343$	17.5616	−	5.88148i	0.948235	−	0.317570i
$344$		0			0
$345$	0.876894	+	3.12311i	0.0472104	+	0.168142i
$346$		0			0
$347$			17.1231i			0.919216i	0.888122	+	0.459608i	$0.152010\pi$
$347$			17.1231i			0.919216i	−0.888122	+	0.459608i	$0.847990\pi$
$348$		0			0
$349$		−	20.4214i		−	1.09313i	−0.837416	−	0.546565i	$-0.815935\pi$
$349$		−	20.4214i		−	1.09313i	0.837416	−	0.546565i	$-0.184065\pi$
$350$		0			0
$351$	12.6847	+	11.8078i	0.677057	+	0.630252i
$352$		0			0
$353$	−19.4849			−1.03708			−0.518539	−	0.855054i	$-0.673524\pi$
$353$	−19.4849			−1.03708			−0.518539	+	0.855054i	$0.673524\pi$
$354$		0			0
$355$			12.2888i			0.652223i
$356$		0			0
$357$	−15.3947	−	12.0903i	−0.814772	−	0.639885i
$358$		0			0
$359$		−	7.75379i		−	0.409229i	−0.978843	−	0.204615i	$-0.934406\pi$
$359$		−	7.75379i		−	0.409229i	0.978843	−	0.204615i	$-0.0655941\pi$
$360$		0			0
$361$	−13.8769			−0.730363
$362$		0			0
$363$	−25.4241	+	7.13848i	−1.33442	+	0.374673i
$364$		0			0
$365$			3.50758i			0.183595i
$366$		0			0
$367$			12.8147i			0.668921i	0.942410	+	0.334460i	$0.108554\pi$
$367$			12.8147i			0.668921i	−0.942410	+	0.334460i	$0.891446\pi$
$368$		0			0
$369$	1.34700	−	0.821147i	0.0701220	−	0.0427472i
$370$		0			0
$371$	1.87285	−	1.36932i	0.0972337	−	0.0710914i
$372$		0			0
$373$	−22.4924			−1.16461			−0.582307	−	0.812969i	$-0.697850\pi$
$373$	−22.4924			−1.16461			−0.582307	+	0.812969i	$0.697850\pi$
$374$		0			0
$375$	−14.2462	+	4.00000i	−0.735671	+	0.206559i
$376$		0			0
$377$	2.92456			0.150622
$378$		0			0
$379$	32.4924			1.66902			0.834512	−	0.550990i	$-0.185750\pi$
$379$	32.4924			1.66902			0.834512	+	0.550990i	$0.185750\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	23.7565			1.21390			0.606950	−	0.794740i	$-0.292393\pi$
$383$	23.7565			1.21390			0.606950	+	0.794740i	$0.292393\pi$
$384$		0			0
$385$	−10.2462	+	7.49141i	−0.522195	+	0.381798i
$386$		0			0
$387$	10.2462	−	6.24621i	0.520844	−	0.317513i
$388$		0			0
$389$		−	21.3693i		−	1.08347i	−0.840550	−	0.541734i	$-0.817768\pi$
$389$		−	21.3693i		−	1.08347i	0.840550	−	0.541734i	$-0.182232\pi$
$390$		0			0
$391$			8.54312i			0.432044i
$392$		0			0
$393$	0.684658	−	0.192236i	0.0345364	−	0.00969702i
$394$		0			0
$395$	−10.4160			−0.524084
$396$		0			0
$397$			7.90198i			0.396589i	0.980142	+	0.198295i	$0.0635403\pi$
$397$			7.90198i			0.396589i	−0.980142	+	0.198295i	$0.936460\pi$
$398$		0			0
$399$	16.2291	−	20.6647i	0.812473	−	1.03453i
$400$		0			0
$401$		−	20.8769i		−	1.04254i	−0.853391	−	0.521271i	$-0.825458\pi$
$401$		−	20.8769i		−	1.04254i	0.853391	−	0.521271i	$-0.174542\pi$
$402$		0			0
$403$	−20.4924			−1.02080
$404$		0			0
$405$	3.86098	−	7.49141i	0.191854	−	0.372251i
$406$		0			0
$407$			10.2462i			0.507886i
$408$		0			0
$409$			12.5194i			0.619044i	0.950892	+	0.309522i	$0.100169\pi$
$409$			12.5194i			0.619044i	−0.950892	+	0.309522i	$0.899831\pi$
$410$		0			0
$411$	6.67026	+	23.7565i	0.329020	+	1.17182i
$412$		0			0
$413$	−15.6240	−	21.3693i	−0.768805	−	1.05152i
$414$		0			0
$415$	3.12311			0.153307
$416$		0			0
$417$	2.68466	+	9.56155i	0.131468	+	0.468231i
$418$		0			0
$419$	16.6757			0.814659			0.407330	−	0.913281i	$-0.366460\pi$
$419$	16.6757			0.814659			0.407330	+	0.913281i	$0.366460\pi$
$420$		0			0
$421$	32.7386			1.59558			0.797792	−	0.602933i	$-0.206001\pi$
$421$	32.7386			1.59558			0.797792	+	0.602933i	$0.206001\pi$
$422$		0			0
$423$	−26.6811	+	16.2651i	−1.29728	+	0.790836i
$424$		0			0
$425$	−17.6121			−0.854312
$426$		0			0
$427$	−21.3693	+	15.6240i	−1.03413	+	0.756096i
$428$		0			0
$429$	28.4924	−	8.00000i	1.37563	−	0.386244i
$430$		0			0
$431$		−	6.00000i		−	0.289010i	−0.989504	−	0.144505i	$-0.953841\pi$
$431$		−	6.00000i		−	0.289010i	0.989504	−	0.144505i	$-0.0461589\pi$
$432$		0			0
$433$			2.92456i			0.140545i	0.997528	+	0.0702727i	$0.0223869\pi$
$433$			2.92456i			0.140545i	−0.997528	+	0.0702727i	$0.977613\pi$
$434$		0			0
$435$	−0.384472	−	1.36932i	−0.0184340	−	0.0656537i
$436$		0			0
$437$	−11.4677			−0.548573
$438$		0			0
$439$			9.06897i			0.432838i	0.976301	+	0.216419i	$0.0694378\pi$
$439$			9.06897i			0.432838i	−0.976301	+	0.216419i	$0.930562\pi$
$440$		0			0
$441$	4.97752	+	20.4016i	0.237025	+	0.971504i
$442$		0			0
$443$		−	14.8769i		−	0.706823i	−0.935468	−	0.353411i	$-0.885022\pi$
$443$		−	14.8769i		−	0.706823i	0.935468	−	0.353411i	$-0.114978\pi$
$444$		0			0
$445$	−14.7386			−0.698678
$446$		0			0
$447$	−10.0054	−	35.6347i	−0.473239	−	1.68547i
$448$		0			0
$449$			6.24621i			0.294777i	0.989079	+	0.147388i	$0.0470867\pi$
$449$			6.24621i			0.294777i	−0.989079	+	0.147388i	$0.952913\pi$
$450$		0			0
$451$		−	2.69400i		−	0.126855i
$452$		0			0
$453$	34.1725	−	9.59482i	1.60556	−	0.450804i
$454$		0			0
$455$	6.67026	−	4.87689i	0.312707	−	0.228632i
$456$		0			0
$457$	−29.6155			−1.38536			−0.692678	−	0.721247i	$-0.743569\pi$
$457$	−29.6155			−1.38536			−0.692678	+	0.721247i	$0.743569\pi$
$458$		0			0
$459$	15.1231	−	16.2462i	0.705886	−	0.758308i
$460$		0			0
$461$	−19.0744			−0.888382			−0.444191	−	0.895932i	$-0.646509\pi$
$461$	−19.0744			−0.888382			−0.444191	+	0.895932i	$0.646509\pi$
$462$		0			0
$463$	−23.6155			−1.09751			−0.548753	−	0.835984i	$-0.684897\pi$
$463$	−23.6155			−1.09751			−0.548753	+	0.835984i	$0.684897\pi$
$464$		0			0
$465$	2.69400	+	9.59482i	0.124931	+	0.444949i
$466$		0			0
$467$	16.6757			0.771658			0.385829	−	0.922570i	$-0.373916\pi$
$467$	16.6757			0.771658			0.385829	+	0.922570i	$0.373916\pi$
$468$		0			0
$469$	16.0000	+	21.8836i	0.738811	+	1.01049i
$470$		0			0
$471$	0.192236	+	0.684658i	0.00885776	+	0.0315474i
$472$		0			0
$473$		−	20.4924i		−	0.942243i
$474$		0			0
$475$		−	23.6412i		−	1.08473i
$476$		0			0
$477$	1.36932	+	2.24621i	0.0626967	+	0.102847i
$478$		0			0
$479$	−2.92456			−0.133626			−0.0668132	−	0.997765i	$-0.521283\pi$
$479$	−2.92456			−0.133626			−0.0668132	+	0.997765i	$0.521283\pi$
$480$		0			0
$481$		−	6.67026i		−	0.304138i
$482$		0			0
$483$	5.66083	−	7.20798i	0.257577	−	0.327975i
$484$		0			0
$485$			9.75379i			0.442897i
$486$		0			0
$487$	−8.00000			−0.362515			−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000			−0.362515			−0.181257	+	0.983436i	$0.558017\pi$
$488$		0			0
$489$	3.74571	−	1.05171i	0.169387	−	0.0475598i
$490$		0			0
$491$		−	37.1231i		−	1.67534i	−0.546175	−	0.837671i	$-0.683917\pi$
$491$		−	37.1231i		−	1.67534i	0.546175	−	0.837671i	$-0.316083\pi$
$492$		0			0
$493$		−	3.74571i		−	0.168698i
$494$		0			0
$495$	−7.49141	−	12.2888i	−0.336714	−	0.552341i
$496$		0			0
$497$	28.0281	−	20.4924i	1.25723	−	0.919211i
$498$		0			0
$499$	−28.9848			−1.29754			−0.648770	−	0.760985i	$-0.724716\pi$
$499$	−28.9848			−1.29754			−0.648770	+	0.760985i	$0.724716\pi$
$500$		0			0
$501$	11.1231	−	3.12311i	0.496944	−	0.139530i
$502$		0			0
$503$	17.0862			0.761838			0.380919	−	0.924609i	$-0.375608\pi$
$503$	17.0862			0.761838			0.380919	+	0.924609i	$0.375608\pi$
$504$		0			0
$505$	13.3693			0.594927
$506$		0			0
$507$	3.12985	−	0.878787i	0.139001	−	0.0390283i
$508$		0			0
$509$	38.8546			1.72220			0.861100	−	0.508436i	$-0.169776\pi$
$509$	38.8546			1.72220			0.861100	+	0.508436i	$0.169776\pi$
$510$		0			0
$511$	8.00000	−	5.84912i	0.353899	−	0.258750i
$512$		0			0
$513$	21.8078	+	20.3002i	0.962836	+	0.896275i
$514$		0			0
$515$			2.24621i			0.0989799i
$516$		0			0
$517$			53.3621i			2.34686i
$518$		0			0
$519$	17.5616	−	4.93087i	0.770867	−	0.216441i
$520$		0			0
$521$	−18.6638			−0.817676			−0.408838	−	0.912607i	$-0.634066\pi$
$521$	−18.6638			−0.817676			−0.408838	+	0.912607i	$0.634066\pi$
$522$		0			0
$523$			16.1498i			0.706182i	0.935589	+	0.353091i	$0.114869\pi$
$523$			16.1498i			0.706182i	−0.935589	+	0.353091i	$0.885131\pi$
$524$		0			0
$525$	14.8596	+	11.6701i	0.648528	+	0.509325i
$526$		0			0
$527$			26.2462i			1.14330i
$528$		0			0
$529$	19.0000			0.826087
$530$		0			0
$531$	25.6294	−	15.6240i	1.11222	−	0.678022i
$532$		0			0
$533$			1.75379i			0.0759650i
$534$		0			0
$535$			1.05171i			0.0454692i
$536$		0			0
$537$	−6.14441	−	21.8836i	−0.265151	−	0.944349i
$538$		0			0
$539$	34.1725	+	10.8769i	1.47191	+	0.468501i
$540$		0			0
$541$	12.2462			0.526506			0.263253	−	0.964727i	$-0.415205\pi$
$541$	12.2462			0.526506			0.263253	+	0.964727i	$0.415205\pi$
$542$		0			0
$543$	−11.3153	−	40.3002i	−0.485588	−	1.72945i
$544$		0			0
$545$	3.97626			0.170324
$546$		0			0
$547$	21.7538			0.930125			0.465062	−	0.885278i	$-0.346032\pi$
$547$	21.7538			0.930125			0.465062	+	0.885278i	$0.346032\pi$
$548$		0			0
$549$	−15.6240	−	25.6294i	−0.666814	−	1.09383i
$550$		0			0
$551$	5.02797			0.214199
$552$		0			0
$553$	17.3693	+	23.7565i	0.738618	+	1.01023i
$554$		0			0
$555$	−3.12311	+	0.876894i	−0.132568	+	0.0372221i
$556$		0			0
$557$			11.6155i			0.492166i	0.969249	+	0.246083i	$0.0791435\pi$
$557$			11.6155i			0.492166i	−0.969249	+	0.246083i	$0.920856\pi$
$558$		0			0
$559$			13.3405i			0.564244i
$560$		0			0
$561$	−10.2462	−	36.4924i	−0.432595	−	1.54071i
$562$		0			0
$563$	−12.9300			−0.544933			−0.272466	−	0.962165i	$-0.587839\pi$
$563$	−12.9300			−0.544933			−0.272466	+	0.962165i	$0.587839\pi$
$564$		0			0
$565$		−	16.2651i		−	0.684277i
$566$		0			0
$567$	−23.5247	+	3.68638i	−0.987944	+	0.154813i
$568$		0			0
$569$		−	28.1080i		−	1.17835i	−0.808007	−	0.589173i	$-0.799454\pi$
$569$		−	28.1080i		−	1.17835i	0.808007	−	0.589173i	$-0.200546\pi$
$570$		0			0
$571$	22.7386			0.951582			0.475791	−	0.879558i	$-0.342162\pi$
$571$	22.7386			0.951582			0.475791	+	0.879558i	$0.342162\pi$
$572$		0			0
$573$	−3.45041	−	12.2888i	−0.144143	−	0.513373i
$574$		0			0
$575$		−	8.24621i		−	0.343891i
$576$		0			0
$577$		−	26.6811i		−	1.11075i	−0.831601	−	0.555373i	$-0.812575\pi$
$577$		−	26.6811i		−	1.11075i	0.831601	−	0.555373i	$-0.187425\pi$
$578$		0			0
$579$	−1.87285	+	0.525853i	−0.0778331	+	0.0218537i
$580$		0			0
$581$	−5.20798	−	7.12311i	−0.216064	−	0.295516i
$582$		0			0
$583$	4.49242			0.186057
$584$		0			0
$585$	4.87689	+	8.00000i	0.201635	+	0.330759i
$586$		0			0
$587$	−12.1088			−0.499784			−0.249892	−	0.968274i	$-0.580395\pi$
$587$	−12.1088			−0.499784			−0.249892	+	0.968274i	$0.580395\pi$
$588$		0			0
$589$	−35.2311			−1.45167
$590$		0			0
$591$	−4.15628	−	14.8028i	−0.170966	−	0.608906i
$592$		0			0
$593$	1.34700			0.0553147			0.0276573	−	0.999617i	$-0.491195\pi$
$593$	1.34700			0.0553147			0.0276573	+	0.999617i	$0.491195\pi$
$594$		0			0
$595$	−6.24621	−	8.54312i	−0.256070	−	0.350234i
$596$		0			0
$597$	−6.87689	−	24.4924i	−0.281453	−	1.00241i
$598$		0			0
$599$			16.6307i			0.679511i	0.940514	+	0.339756i	$0.110344\pi$
$599$			16.6307i			0.679511i	−0.940514	+	0.339756i	$0.889656\pi$
$600$		0			0
$601$			17.0862i			0.696962i	0.937316	+	0.348481i	$0.113302\pi$
$601$			17.0862i			0.696962i	−0.937316	+	0.348481i	$0.886698\pi$
$602$		0			0
$603$	−26.2462	+	16.0000i	−1.06883	+	0.651570i
$604$		0			0
$605$	−14.2770			−0.580441
$606$		0			0
$607$			46.9871i			1.90715i	0.301157	+	0.953575i	$0.402627\pi$
$607$			46.9871i			1.90715i	−0.301157	+	0.953575i	$0.597373\pi$
$608$		0			0
$609$	−2.48197	+	3.16032i	−0.100575	+	0.128063i
$610$		0			0
$611$		−	34.7386i		−	1.40537i
$612$		0			0
$613$	−38.4924			−1.55469			−0.777347	−	0.629072i	$-0.783435\pi$
$613$	−38.4924			−1.55469			−0.777347	+	0.629072i	$0.783435\pi$
$614$		0			0
$615$	0.821147	−	0.230559i	0.0331118	−	0.00929702i
$616$		0			0
$617$			41.3693i			1.66547i	0.553675	+	0.832733i	$0.313225\pi$
$617$			41.3693i			1.66547i	−0.553675	+	0.832733i	$0.686775\pi$
$618$		0			0
$619$		−	4.68213i		−	0.188191i	−0.995563	−	0.0940954i	$-0.970004\pi$
$619$		−	4.68213i		−	0.188191i	0.995563	−	0.0940954i	$-0.0299959\pi$
$620$		0			0
$621$	7.60669	+	7.08084i	0.305246	+	0.284144i
$622$		0			0
$623$	24.5776	+	33.6155i	0.984683	+	1.34678i
$624$		0			0
$625$	12.6155			0.504621
$626$		0			0
$627$	48.9848	−	13.7538i	1.95627	−	0.549273i
$628$		0			0
$629$	−8.54312			−0.340636
$630$		0			0
$631$	−28.8769			−1.14957			−0.574786	−	0.818304i	$-0.694915\pi$
$631$	−28.8769			−1.14957			−0.574786	+	0.818304i	$0.694915\pi$
$632$		0			0
$633$	9.59482	−	2.69400i	0.381360	−	0.107077i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−22.2462	−	7.08084i	−0.881427	−	0.280553i
$638$		0			0
$639$	20.4924	+	33.6155i	0.810668	+	1.32981i
$640$		0			0
$641$			29.8617i			1.17947i	0.807598	+	0.589734i	$0.200767\pi$
$641$			29.8617i			1.17947i	−0.807598	+	0.589734i	$0.799233\pi$
$642$		0			0
$643$			13.2252i			0.521553i	0.965399	+	0.260776i	$0.0839785\pi$
$643$			13.2252i			0.521553i	−0.965399	+	0.260776i	$0.916021\pi$
$644$		0			0
$645$	6.24621	−	1.75379i	0.245944	−	0.0690554i
$646$		0			0
$647$	−40.8427			−1.60569			−0.802847	−	0.596185i	$-0.796682\pi$
$647$	−40.8427			−1.60569			−0.802847	+	0.596185i	$0.796682\pi$
$648$		0			0
$649$		−	51.2587i		−	2.01208i
$650$		0			0
$651$	17.3912	−	22.1444i	0.681616	−	0.867908i
$652$		0			0
$653$		−	10.6307i		−	0.416011i	−0.978128	−	0.208005i	$-0.933303\pi$
$653$		−	10.6307i		−	0.416011i	0.978128	−	0.208005i	$-0.0666972\pi$
$654$		0			0
$655$	0.384472			0.0150226
$656$		0			0
$657$	5.84912	+	9.59482i	0.228196	+	0.374330i
$658$		0			0
$659$		−	6.87689i		−	0.267886i	−0.990989	−	0.133943i	$-0.957236\pi$
$659$		−	6.87689i		−	0.267886i	0.990989	−	0.133943i	$-0.0427639\pi$
$660$		0			0
$661$			17.4968i			0.680547i	0.940327	+	0.340273i	$0.110520\pi$
$661$			17.4968i			0.680547i	−0.940327	+	0.340273i	$0.889480\pi$
$662$		0			0
$663$	6.67026	+	23.7565i	0.259052	+	0.922626i
$664$		0			0
$665$	11.4677	−	8.38447i	0.444697	−	0.325136i
$666$		0			0
$667$	1.75379			0.0679070
$668$		0			0
$669$	2.49242	+	8.87689i	0.0963626	+	0.343201i
$670$		0			0
$671$	−51.2587			−1.97882
$672$		0			0
$673$	−18.4924			−0.712831			−0.356415	−	0.934328i	$-0.616001\pi$
$673$	−18.4924			−0.712831			−0.356415	+	0.934328i	$0.616001\pi$
$674$		0			0
$675$	−14.5975	+	15.6816i	−0.561859	+	0.603585i
$676$		0			0
$677$	9.71010			0.373190			0.186595	−	0.982437i	$-0.440255\pi$
$677$	9.71010			0.373190			0.186595	+	0.982437i	$0.440255\pi$
$678$		0			0
$679$	22.2462	−	16.2651i	0.853731	−	0.624197i
$680$		0			0
$681$	−29.1771	+	8.19224i	−1.11807	+	0.313927i
$682$		0			0
$683$			23.3693i			0.894202i	0.894483	+	0.447101i	$0.147544\pi$
$683$			23.3693i			0.894202i	−0.894483	+	0.447101i	$0.852456\pi$
$684$		0			0
$685$			13.3405i			0.509715i
$686$		0			0
$687$	9.56155	+	34.0540i	0.364796	+	1.29924i
$688$		0			0
$689$	−2.92456			−0.111417
$690$		0			0
$691$			22.8201i			0.868116i	0.900885	+	0.434058i	$0.142919\pi$
$691$			22.8201i			0.868116i	−0.900885	+	0.434058i	$0.857081\pi$
$692$		0			0
$693$	−15.5356	+	37.5787i	−0.590150	+	1.42750i
$694$		0			0
$695$			5.36932i			0.203670i
$696$		0			0
$697$	2.24621			0.0850813
$698$		0			0
$699$	3.74571	+	13.3405i	0.141676	+	0.504585i
$700$		0			0
$701$		−	7.12311i		−	0.269036i	−0.990911	−	0.134518i	$-0.957051\pi$
$701$		−	7.12311i		−	0.269036i	0.990911	−	0.134518i	$-0.0429486\pi$
$702$		0			0
$703$		−	11.4677i		−	0.432512i
$704$		0			0
$705$	−16.2651	+	4.56685i	−0.612579	+	0.171998i
$706$		0			0
$707$	−22.2942	−	30.4924i	−0.838460	−	1.14679i
$708$		0			0
$709$	−16.2462			−0.610139			−0.305070	−	0.952330i	$-0.598680\pi$
$709$	−16.2462			−0.610139			−0.305070	+	0.952330i	$0.598680\pi$
$710$		0			0
$711$	−28.4924	+	17.3693i	−1.06855	+	0.651400i
$712$		0			0
$713$	−12.2888			−0.460220
$714$		0			0
$715$	16.0000			0.598366
$716$		0			0
$717$	−5.73384	−	20.4214i	−0.214134	−	0.762650i
$718$		0			0
$719$	−37.0970			−1.38349			−0.691743	−	0.722144i	$-0.743157\pi$
$719$	−37.0970			−1.38349			−0.691743	+	0.722144i	$0.743157\pi$
$720$		0			0
$721$	5.12311	−	3.74571i	0.190794	−	0.139497i
$722$		0			0
$723$	1.36932	+	4.87689i	0.0509254	+	0.181374i
$724$		0			0
$725$			3.61553i			0.134277i
$726$		0			0
$727$		−	43.4720i		−	1.61229i	−0.591720	−	0.806144i	$-0.701551\pi$
$727$		−	43.4720i		−	1.61229i	0.591720	−	0.806144i	$-0.298449\pi$
$728$		0			0
$729$	−1.93087	−	26.9309i	−0.0715137	−	0.997440i
$730$		0			0
$731$	17.0862			0.631957
$732$		0			0
$733$			21.2425i			0.784610i	0.919835	+	0.392305i	$0.128322\pi$
$733$			21.2425i			0.784610i	−0.919835	+	0.392305i	$0.871678\pi$
$734$		0			0
$735$	−0.390783	+	11.3468i	−0.0144142	+	0.418534i
$736$		0			0
$737$			52.4924i			1.93358i
$738$		0			0
$739$	−7.50758			−0.276171			−0.138085	−	0.990420i	$-0.544095\pi$
$739$	−7.50758			−0.276171			−0.138085	+	0.990420i	$0.544095\pi$
$740$		0			0
$741$	−31.8890	+	8.95369i	−1.17147	+	0.328922i
$742$		0			0
$743$		−	30.0000i		−	1.10059i	−0.834969	−	0.550297i	$-0.814515\pi$
$743$		−	30.0000i		−	1.10059i	0.834969	−	0.550297i	$-0.185485\pi$
$744$		0			0
$745$		−	20.0108i		−	0.733139i
$746$		0			0
$747$	8.54312	−	5.20798i	0.312576	−	0.190550i
$748$		0			0
$749$	2.39871	−	1.75379i	0.0876468	−	0.0640820i
$750$		0			0
$751$	−16.0000			−0.583848			−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000			−0.583848			−0.291924	+	0.956441i	$0.594295\pi$
$752$		0			0
$753$	−22.9309	+	6.43845i	−0.835647	+	0.234630i
$754$		0			0
$755$	19.1896			0.698383
$756$		0			0
$757$	−22.4924			−0.817501			−0.408751	−	0.912646i	$-0.634035\pi$
$757$	−22.4924			−0.817501			−0.408751	+	0.912646i	$0.634035\pi$
$758$		0			0
$759$	17.0862	−	4.79741i	0.620191	−	0.174135i
$760$		0			0
$761$	15.5087			0.562189			0.281095	−	0.959680i	$-0.409303\pi$
$761$	15.5087			0.562189			0.281095	+	0.959680i	$0.409303\pi$
$762$		0			0
$763$	−6.63068	−	9.06897i	−0.240047	−	0.328319i
$764$		0			0
$765$	10.2462	−	6.24621i	0.370453	−	0.225832i
$766$		0			0
$767$			33.3693i			1.20490i
$768$		0			0
$769$		−	42.9461i		−	1.54868i	−0.632771	−	0.774339i	$-0.718083\pi$
$769$		−	42.9461i		−	1.54868i	0.632771	−	0.774339i	$-0.281917\pi$
$770$		0			0
$771$	−32.4924	+	9.12311i	−1.17019	+	0.328561i
$772$		0			0
$773$	14.2770			0.513506			0.256753	−	0.966477i	$-0.417347\pi$
$773$	14.2770			0.513506			0.256753	+	0.966477i	$0.417347\pi$
$774$		0			0
$775$		−	25.3341i		−	0.910026i
$776$		0			0
$777$	7.20798	+	5.66083i	0.258585	+	0.203081i
$778$		0			0
$779$			3.01515i			0.108029i
$780$		0			0
$781$	67.2311			2.40572
$782$		0			0
$783$	−3.33513	−	3.10457i	−0.119188	−	0.110948i
$784$		0			0
$785$			0.384472i			0.0137224i
$786$		0			0
$787$			39.0852i			1.39324i	0.717443	+	0.696618i	$0.245313\pi$
$787$			39.0852i			1.39324i	−0.717443	+	0.696618i	$0.754687\pi$
$788$		0			0
$789$	−7.19612	−	25.6294i	−0.256189	−	0.912429i
$790$		0			0
$791$	−37.0970	+	27.1231i	−1.31902	+	0.964387i
$792$		0			0
$793$	33.3693			1.18498
$794$		0			0
$795$	0.384472	+	1.36932i	0.0136358	+	0.0485647i
$796$		0			0
$797$	−7.37613			−0.261276			−0.130638	−	0.991430i	$-0.541703\pi$
$797$	−7.37613			−0.261276			−0.130638	+	0.991430i	$0.541703\pi$
$798$		0			0
$799$	−44.4924			−1.57403
$800$		0			0
$801$	−40.3169	+	24.5776i	−1.42453	+	0.868408i
$802$		0			0
$803$	19.1896			0.677188
$804$		0			0
$805$	4.00000	−	2.92456i	0.140981	−	0.103077i
$806$		0			0
$807$	−4.68466	+	1.31534i	−0.164908	+	0.0463022i
$808$		0			0
$809$		−	19.1231i		−	0.672333i	−0.941803	−	0.336166i	$-0.890870\pi$
$809$		−	19.1231i		−	0.672333i	0.941803	−	0.336166i	$-0.109130\pi$
$810$		0			0
$811$		−	35.9300i		−	1.26167i	−0.775915	−	0.630837i	$-0.782712\pi$
$811$		−	35.9300i		−	1.26167i	0.775915	−	0.630837i	$-0.217288\pi$
$812$		0			0
$813$	−8.24621	−	29.3693i	−0.289207	−	1.03003i
$814$		0			0
$815$	2.10341			0.0736793
$816$		0			0
$817$			22.9354i			0.802406i
$818$		0			0
$819$	10.1137	−	24.4636i	0.353400	−	0.854829i
$820$		0			0
$821$			23.1231i			0.807002i	0.914979	+	0.403501i	$0.132207\pi$
$821$			23.1231i			0.807002i	−0.914979	+	0.403501i	$0.867793\pi$
$822$		0			0
$823$	12.8769			0.448860			0.224430	−	0.974490i	$-0.427948\pi$
$823$	12.8769			0.448860			0.224430	+	0.974490i	$0.427948\pi$
$824$		0			0
$825$	9.89012	+	35.2242i	0.344330	+	1.22635i
$826$		0			0
$827$		−	23.8617i		−	0.829754i	−0.909878	−	0.414877i	$-0.863825\pi$
$827$		−	23.8617i		−	0.829754i	0.909878	−	0.414877i	$-0.136175\pi$
$828$		0			0
$829$			43.3567i			1.50584i	0.658111	+	0.752921i	$0.271356\pi$
$829$			43.3567i			1.50584i	−0.658111	+	0.752921i	$0.728644\pi$
$830$		0			0
$831$	7.08084	−	1.98813i	0.245632	−	0.0689675i
$832$		0			0
$833$	−9.06897	+	28.4924i	−0.314221	+	0.987204i
$834$		0			0
$835$	6.24621			0.216159
$836$		0			0
$837$	23.3693	+	21.7538i	0.807762	+	0.751921i
$838$		0			0
$839$	39.2004			1.35335			0.676675	−	0.736282i	$-0.263420\pi$
$839$	39.2004			1.35335			0.676675	+	0.736282i	$0.263420\pi$
$840$		0			0
$841$	28.2311			0.973485
$842$		0			0
$843$	2.10341	+	7.49141i	0.0724453	+	0.258018i
$844$		0			0
$845$	1.75757			0.0604624
$846$		0			0
$847$	23.8078	+	32.5625i	0.818044	+	1.11886i
$848$		0			0
$849$	−1.80776	−	6.43845i	−0.0620423	−	0.220967i
$850$		0			0
$851$		−	4.00000i		−	0.137118i
$852$		0			0
$853$		−	40.4322i		−	1.38437i	−0.721720	−	0.692185i	$-0.756648\pi$
$853$		−	40.4322i		−	1.38437i	0.721720	−	0.692185i	$-0.243352\pi$
$854$		0			0
$855$	8.38447	+	13.7538i	0.286743	+	0.470370i
$856$		0			0
$857$	21.3578			0.729568			0.364784	−	0.931092i	$-0.381143\pi$
$857$	21.3578			0.729568			0.364784	+	0.931092i	$0.381143\pi$
$858$		0			0
$859$		−	29.2598i		−	0.998331i	−0.866507	−	0.499165i	$-0.833640\pi$
$859$		−	29.2598i		−	0.998331i	0.866507	−	0.499165i	$-0.166360\pi$
$860$		0			0
$861$	−1.89517	−	1.48838i	−0.0645872	−	0.0507239i
$862$		0			0
$863$		−	26.8769i		−	0.914900i	−0.889235	−	0.457450i	$-0.848763\pi$
$863$		−	26.8769i		−	0.914900i	0.889235	−	0.457450i	$-0.151237\pi$
$864$		0			0
$865$	9.86174			0.335309
$866$		0			0
$867$	2.07814	−	0.583493i	0.0705773	−	0.0198164i
$868$		0			0
$869$			56.9848i			1.93308i
$870$		0			0
$871$		−	34.1725i		−	1.15789i
$872$		0			0
$873$	16.2651	+	26.6811i	0.550490	+	0.903017i
$874$		0			0
$875$	13.3405	+	18.2462i	0.450992	+	0.616835i
$876$		0			0
$877$	−20.7386			−0.700294			−0.350147	−	0.936695i	$-0.613868\pi$
$877$	−20.7386			−0.700294			−0.350147	+	0.936695i	$0.613868\pi$
$878$		0			0
$879$	−25.5616	+	7.17708i	−0.862170	+	0.242077i
$880$		0			0
$881$	22.1789			0.747227			0.373614	−	0.927584i	$-0.378119\pi$
$881$	22.1789			0.747227			0.373614	+	0.927584i	$0.378119\pi$
$882$		0			0
$883$	5.75379			0.193630			0.0968152	−	0.995302i	$-0.469134\pi$
$883$	5.75379			0.193630			0.0968152	+	0.995302i	$0.469134\pi$
$884$		0			0
$885$	15.6240	−	4.38684i	0.525193	−	0.147462i
$886$		0			0
$887$	−12.5194			−0.420360			−0.210180	−	0.977663i	$-0.567405\pi$
$887$	−12.5194			−0.420360			−0.210180	+	0.977663i	$0.567405\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−40.9848	−	21.1231i	−1.37304	−	0.707651i
$892$		0			0
$893$		−	59.7235i		−	1.99857i
$894$		0			0
$895$		−	12.2888i		−	0.410770i
$896$		0			0
$897$	−11.1231	+	3.12311i	−0.371390	+	0.104277i
$898$		0			0
$899$	5.38800			0.179700
$900$		0			0
$901$			3.74571i			0.124788i
$902$		0			0
$903$	−14.4160	−	11.3217i	−0.479733	−	0.376761i
$904$		0			0
$905$		−	22.6307i		−	0.752269i
$906$		0			0
$907$	16.4924			0.547622			0.273811	−	0.961784i	$-0.411716\pi$
$907$	16.4924			0.547622			0.273811	+	0.961784i	$0.411716\pi$
$908$		0			0
$909$	36.5712	−	22.2942i	1.21299	−	0.739453i
$910$		0			0
$911$		−	6.00000i		−	0.198789i	−0.995048	−	0.0993944i	$-0.968309\pi$
$911$		−	6.00000i		−	0.198789i	0.995048	−	0.0993944i	$-0.0316906\pi$
$912$		0			0
$913$		−	17.0862i		−	0.565472i
$914$		0			0
$915$	−4.38684	−	15.6240i	−0.145024	−	0.516512i
$916$		0			0
$917$	−0.641132	−	0.876894i	−0.0211721	−	0.0289576i
$918$		0			0
$919$	47.6155			1.57069			0.785346	−	0.619057i	$-0.212485\pi$
$919$	47.6155			1.57069			0.785346	+	0.619057i	$0.212485\pi$
$920$		0			0
$921$	−0.0539753	−	0.192236i	−0.00177855	−	0.00633439i
$922$		0			0
$923$	−43.7673			−1.44062
$924$		0			0
$925$	8.24621			0.271134
$926$		0			0
$927$	3.74571	+	6.14441i	0.123025	+	0.201809i
$928$		0			0
$929$	−58.2243			−1.91028			−0.955138	−	0.296161i	$-0.904293\pi$
$929$	−58.2243			−1.91028			−0.955138	+	0.296161i	$0.904293\pi$
$930$		0			0
$931$	−38.2462	−	12.1735i	−1.25347	−	0.398972i
$932$		0			0
$933$	−16.0000	+	4.49242i	−0.523816	+	0.147075i
$934$		0			0
$935$		−	20.4924i		−	0.670174i
$936$		0			0
$937$			9.59482i			0.313449i	0.987642	+	0.156725i	$0.0500935\pi$
$937$			9.59482i			0.313449i	−0.987642	+	0.156725i	$0.949906\pi$
$938$		0			0
$939$	3.12311	+	11.1231i	0.101919	+	0.362989i
$940$		0			0
$941$	41.7792			1.36196			0.680981	−	0.732301i	$-0.261554\pi$
$941$	41.7792			1.36196			0.680981	+	0.732301i	$0.261554\pi$
$942$		0			0
$943$			1.05171i			0.0342483i
$944$		0			0
$945$	−12.7838	−	1.51928i	−0.415856	−	0.0494223i
$946$		0			0
$947$		−	10.3845i		−	0.337450i	−0.985663	−	0.168725i	$-0.946035\pi$
$947$		−	10.3845i		−	0.337450i	0.985663	−	0.168725i	$-0.0539650\pi$
$948$		0			0
$949$	−12.4924			−0.405521
$950$		0			0
$951$	12.9300	+	46.0507i	0.419283	+	1.49330i
$952$		0			0
$953$		−	26.7386i		−	0.866149i	−0.901358	−	0.433075i	$-0.857429\pi$
$953$		−	26.7386i		−	0.866149i	0.901358	−	0.433075i	$-0.142571\pi$
$954$		0			0
$955$		−	6.90082i		−	0.223305i
$956$		0			0
$957$	−7.49141	+	2.10341i	−0.242163	+	0.0679936i
$958$		0			0
$959$	30.4268	−	22.2462i	0.982531	−	0.718368i
$960$		0			0
$961$	−6.75379			−0.217864
$962$		0			0
$963$	1.75379	+	2.87689i	0.0565151	+	0.0927066i
$964$		0			0
$965$	−1.05171			−0.0338556
$966$		0			0
$967$	4.49242			0.144467			0.0722333	−	0.997388i	$-0.476987\pi$
$967$	4.49242			0.144467			0.0722333	+	0.997388i	$0.476987\pi$
$968$		0			0
$969$	11.4677	+	40.8427i	0.368395	+	1.31206i
$970$		0			0
$971$	4.15628			0.133381			0.0666907	−	0.997774i	$-0.478756\pi$
$971$	4.15628			0.133381			0.0666907	+	0.997774i	$0.478756\pi$
$972$		0			0
$973$	12.2462	−	8.95369i	0.392596	−	0.287042i
$974$		0			0
$975$	−6.43845	−	22.9309i	−0.206195	−	0.734376i
$976$		0			0
$977$		−	54.2462i		−	1.73549i	−0.497010	−	0.867745i	$-0.665569\pi$
$977$		−	54.2462i		−	1.73549i	0.497010	−	0.867745i	$-0.334431\pi$
$978$		0			0
$979$			80.6338i			2.57707i
$980$		0			0
$981$	10.8769	−	6.63068i	0.347273	−	0.211701i
$982$		0			0
$983$	40.8427			1.30268			0.651340	−	0.758786i	$-0.274207\pi$
$983$	40.8427			1.30268			0.651340	+	0.758786i	$0.274207\pi$
$984$		0			0
$985$		−	8.31256i		−	0.264860i
$986$		0			0
$987$	37.5391	+	29.4815i	1.19488	+	0.938406i
$988$		0			0
$989$			8.00000i			0.254385i
$990$		0			0
$991$	−4.87689			−0.154920			−0.0774598	−	0.996995i	$-0.524681\pi$
$991$	−4.87689			−0.154920			−0.0774598	+	0.996995i	$0.524681\pi$
$992$		0			0
$993$	−9.59482	+	2.69400i	−0.304482	+	0.0854915i
$994$		0			0
$995$		−	13.7538i		−	0.436024i
$996$		0			0
$997$		−	16.6757i		−	0.528123i	−0.964506	−	0.264062i	$-0.914938\pi$
$997$		−	16.6757i		−	0.528123i	0.964506	−	0.264062i	$-0.0850623\pi$
$998$		0			0
$999$	−7.08084	+	7.60669i	−0.224028	+	0.240665i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.2.k.c.209.7		8
3.2	odd	2	inner	336.2.k.c.209.1		8
4.3	odd	2		168.2.k.a.41.2	yes	8
7.6	odd	2	inner	336.2.k.c.209.2		8
8.3	odd	2		1344.2.k.f.1217.7		8
8.5	even	2		1344.2.k.i.1217.2		8
12.11	even	2		168.2.k.a.41.8	yes	8
21.20	even	2	inner	336.2.k.c.209.8		8
24.5	odd	2		1344.2.k.i.1217.8		8
24.11	even	2		1344.2.k.f.1217.1		8
28.3	even	6		1176.2.u.a.1097.2		16
28.11	odd	6		1176.2.u.a.1097.7		16
28.19	even	6		1176.2.u.a.521.4		16
28.23	odd	6		1176.2.u.a.521.5		16
28.27	even	2		168.2.k.a.41.7	yes	8
56.13	odd	2		1344.2.k.i.1217.7		8
56.27	even	2		1344.2.k.f.1217.2		8
84.11	even	6		1176.2.u.a.1097.4		16
84.23	even	6		1176.2.u.a.521.2		16
84.47	odd	6		1176.2.u.a.521.7		16
84.59	odd	6		1176.2.u.a.1097.5		16
84.83	odd	2		168.2.k.a.41.1	✓	8
168.83	odd	2		1344.2.k.f.1217.8		8
168.125	even	2		1344.2.k.i.1217.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.k.a.41.1	✓	8	84.83	odd	2
168.2.k.a.41.2	yes	8	4.3	odd	2
168.2.k.a.41.7	yes	8	28.27	even	2
168.2.k.a.41.8	yes	8	12.11	even	2
336.2.k.c.209.1		8	3.2	odd	2	inner
336.2.k.c.209.2		8	7.6	odd	2	inner
336.2.k.c.209.7		8	1.1	even	1	trivial
336.2.k.c.209.8		8	21.20	even	2	inner
1176.2.u.a.521.2		16	84.23	even	6
1176.2.u.a.521.4		16	28.19	even	6
1176.2.u.a.521.5		16	28.23	odd	6
1176.2.u.a.521.7		16	84.47	odd	6
1176.2.u.a.1097.2		16	28.3	even	6
1176.2.u.a.1097.4		16	84.11	even	6
1176.2.u.a.1097.5		16	84.59	odd	6
1176.2.u.a.1097.7		16	28.11	odd	6
1344.2.k.f.1217.1		8	24.11	even	2
1344.2.k.f.1217.2		8	56.27	even	2
1344.2.k.f.1217.7		8	8.3	odd	2
1344.2.k.f.1217.8		8	168.83	odd	2
1344.2.k.i.1217.1		8	168.125	even	2
1344.2.k.i.1217.2		8	8.5	even	2
1344.2.k.i.1217.7		8	56.13	odd	2
1344.2.k.i.1217.8		8	24.5	odd	2

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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	336.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.66757 - 0.468213i) q^{3} +0.936426 q^{5} +(-1.56155 - 2.13578i) q^{7} +(2.56155 - 1.56155i) q^{9} +O(q^{10})\)	\(q+(1.66757 - 0.468213i) q^{3} +0.936426 q^{5} +(-1.56155 - 2.13578i) q^{7} +(2.56155 - 1.56155i) q^{9} -5.12311i q^{11} +3.33513i q^{13} +(1.56155 - 0.438447i) q^{15} +4.27156 q^{17} +5.73384i q^{19} +(-3.60399 - 2.83041i) q^{21} +2.00000i q^{23} -4.12311 q^{25} +(3.54042 - 3.80335i) q^{27} -0.876894i q^{29} +6.14441i q^{31} +(-2.39871 - 8.54312i) q^{33} +(-1.46228 - 2.00000i) q^{35} -2.00000 q^{37} +(1.56155 + 5.56155i) q^{39} +0.525853 q^{41} +4.00000 q^{43} +(2.39871 - 1.46228i) q^{45} -10.4160 q^{47} +(-2.12311 + 6.67026i) q^{49} +(7.12311 - 2.00000i) q^{51} +0.876894i q^{53} -4.79741i q^{55} +(2.68466 + 9.56155i) q^{57} +10.0054 q^{59} -10.0054i q^{61} +(-7.33513 - 3.03246i) q^{63} +3.12311i q^{65} -10.2462 q^{67} +(0.936426 + 3.33513i) q^{69} +13.1231i q^{71} +3.74571i q^{73} +(-6.87555 + 1.93049i) q^{75} +(-10.9418 + 8.00000i) q^{77} -11.1231 q^{79} +(4.12311 - 8.00000i) q^{81} +3.33513 q^{83} +4.00000 q^{85} +(-0.410574 - 1.46228i) q^{87} -15.7392 q^{89} +(7.12311 - 5.20798i) q^{91} +(2.87689 + 10.2462i) q^{93} +5.36932i q^{95} +10.4160i q^{97} +(-8.00000 - 13.1231i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{7} + 4 q^{9}+O(q^{10}) \) 8 * q + 4 * q^7 + 4 * q^9	\( 8 q + 4 q^{7} + 4 q^{9} - 4 q^{15} - 8 q^{21} - 16 q^{37} - 4 q^{39} + 32 q^{43} + 16 q^{49} + 24 q^{51} - 28 q^{57} - 32 q^{63} - 16 q^{67} - 56 q^{79} + 32 q^{85} + 24 q^{91} + 56 q^{93} - 64 q^{99}+O(q^{100}) \) 8 * q + 4 * q^7 + 4 * q^9 - 4 * q^15 - 8 * q^21 - 16 * q^37 - 4 * q^39 + 32 * q^43 + 16 * q^49 + 24 * q^51 - 28 * q^57 - 32 * q^63 - 16 * q^67 - 56 * q^79 + 32 * q^85 + 24 * q^91 + 56 * q^93 - 64 * q^99

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