LMFDB - Embedded newform 800.2.o.g.207.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,2,Mod(143,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("800.143");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	800.o (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$6.38803216170$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{20})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			207.4
Root			$-0.587785 - 0.809017i$ of defining polynomial
Character	$\chi$	$=$	800.207
Dual form			800.2.o.g.143.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.618034 - 0.618034i) q^{3} +(1.90211 - 1.90211i) q^{7} +2.23607i q^{9} +3.23607 q^{11} +(0.726543 + 0.726543i) q^{13} +(1.00000 + 1.00000i) q^{17} +2.00000i q^{19} -2.35114i q^{21} +(-4.25325 - 4.25325i) q^{23} +(3.23607 + 3.23607i) q^{27} +6.15537 q^{29} -8.50651i q^{31} +(2.00000 - 2.00000i) q^{33} +(-0.726543 + 0.726543i) q^{37} +0.898056 q^{39} +5.70820 q^{41} +(4.61803 - 4.61803i) q^{43} +(-3.35520 + 3.35520i) q^{47} -0.236068i q^{49} +1.23607 q^{51} +(-3.07768 - 3.07768i) q^{53} +(1.23607 + 1.23607i) q^{57} +0.472136i q^{59} -0.898056i q^{61} +(4.25325 + 4.25325i) q^{63} +(-4.61803 - 4.61803i) q^{67} -5.25731 q^{69} +11.4127i q^{71} +(4.70820 - 4.70820i) q^{73} +(6.15537 - 6.15537i) q^{77} +2.90617 q^{79} -2.70820 q^{81} +(-6.61803 + 6.61803i) q^{83} +(3.80423 - 3.80423i) q^{87} +2.47214i q^{89} +2.76393 q^{91} +(-5.25731 - 5.25731i) q^{93} +(-4.23607 - 4.23607i) q^{97} +7.23607i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} + 8 q^{11} + 8 q^{17} + 8 q^{27} + 16 q^{33} - 8 q^{41} + 28 q^{43} - 8 q^{51} - 8 q^{57} - 28 q^{67} - 16 q^{73} + 32 q^{81} - 44 q^{83} + 40 q^{91} - 16 q^{97}+O(q^{100}) $ 8 * q - 4 * q^3 + 8 * q^11 + 8 * q^17 + 8 * q^27 + 16 * q^33 - 8 * q^41 + 28 * q^43 - 8 * q^51 - 8 * q^57 - 28 * q^67 - 16 * q^73 + 32 * q^81 - 44 * q^83 + 40 * q^91 - 16 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	0.618034	−	0.618034i	0.356822	−	0.356822i	−0.505818	−	0.862640i	$-0.668809\pi$
$3$	0.618034	−	0.618034i	0.356822	−	0.356822i	0.862640	+	0.505818i	$0.168809\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.90211	−	1.90211i	0.718931	−	0.718931i	−0.249455	−	0.968386i	$-0.580252\pi$
$7$	1.90211	−	1.90211i	0.718931	−	0.718931i	0.968386	+	0.249455i	$0.0802515\pi$
$8$		0			0
$9$			2.23607i			0.745356i
$10$		0			0
$11$	3.23607			0.975711			0.487856	−	0.872924i	$-0.337779\pi$
$11$	3.23607			0.975711			0.487856	+	0.872924i	$0.337779\pi$
$12$		0			0
$13$	0.726543	+	0.726543i	0.201507	+	0.201507i	0.800645	−	0.599139i	$-0.204490\pi$
$13$	0.726543	+	0.726543i	0.201507	+	0.201507i	−0.599139	+	0.800645i	$0.704490\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.00000	+	1.00000i	0.242536	+	0.242536i	0.817898	−	0.575363i	$-0.195139\pi$
$17$	1.00000	+	1.00000i	0.242536	+	0.242536i	−0.575363	+	0.817898i	$0.695139\pi$
$18$		0			0
$19$			2.00000i			0.458831i	0.973329	+	0.229416i	$0.0736815\pi$
$19$			2.00000i			0.458831i	−0.973329	+	0.229416i	$0.926318\pi$
$20$		0			0
$21$		−	2.35114i		−	0.513061i
$22$		0			0
$23$	−4.25325	−	4.25325i	−0.886865	−	0.886865i	0.107356	−	0.994221i	$-0.465762\pi$
$23$	−4.25325	−	4.25325i	−0.886865	−	0.886865i	−0.994221	+	0.107356i	$0.965762\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	3.23607	+	3.23607i	0.622782	+	0.622782i
$28$		0			0
$29$	6.15537			1.14302			0.571511	−	0.820594i	$-0.306357\pi$
$29$	6.15537			1.14302			0.571511	+	0.820594i	$0.306357\pi$
$30$		0			0
$31$		−	8.50651i		−	1.52781i	−0.645326	−	0.763907i	$-0.723279\pi$
$31$		−	8.50651i		−	1.52781i	0.645326	−	0.763907i	$-0.276721\pi$
$32$		0			0
$33$	2.00000	−	2.00000i	0.348155	−	0.348155i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−0.726543	+	0.726543i	−0.119443	+	0.119443i	−0.764302	−	0.644859i	$-0.776916\pi$
$37$	−0.726543	+	0.726543i	−0.119443	+	0.119443i	0.644859	+	0.764302i	$0.276916\pi$
$38$		0			0
$39$	0.898056			0.143804
$40$		0			0
$41$	5.70820			0.891472			0.445736	−	0.895165i	$-0.352942\pi$
$41$	5.70820			0.891472			0.445736	+	0.895165i	$0.352942\pi$
$42$		0			0
$43$	4.61803	−	4.61803i	0.704244	−	0.704244i	−0.261075	−	0.965319i	$-0.584077\pi$
$43$	4.61803	−	4.61803i	0.704244	−	0.704244i	0.965319	+	0.261075i	$0.0840770\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.35520	+	3.35520i	−0.489406	+	0.489406i	−0.908119	−	0.418713i	$-0.862481\pi$
$47$	−3.35520	+	3.35520i	−0.489406	+	0.489406i	0.418713	+	0.908119i	$0.362481\pi$
$48$		0			0
$49$		−	0.236068i		−	0.0337240i
$50$		0			0
$51$	1.23607			0.173084
$52$		0			0
$53$	−3.07768	−	3.07768i	−0.422752	−	0.422752i	0.463398	−	0.886150i	$-0.346630\pi$
$53$	−3.07768	−	3.07768i	−0.422752	−	0.422752i	−0.886150	+	0.463398i	$0.846630\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	1.23607	+	1.23607i	0.163721	+	0.163721i
$58$		0			0
$59$			0.472136i			0.0614669i	0.999528	+	0.0307334i	$0.00978430\pi$
$59$			0.472136i			0.0614669i	−0.999528	+	0.0307334i	$0.990216\pi$
$60$		0			0
$61$		−	0.898056i		−	0.114984i	−0.998346	−	0.0574921i	$-0.981690\pi$
$61$		−	0.898056i		−	0.114984i	0.998346	−	0.0574921i	$-0.0183104\pi$
$62$		0			0
$63$	4.25325	+	4.25325i	0.535860	+	0.535860i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−4.61803	−	4.61803i	−0.564183	−	0.564183i	0.366310	−	0.930493i	$-0.380621\pi$
$67$	−4.61803	−	4.61803i	−0.564183	−	0.564183i	−0.930493	+	0.366310i	$0.880621\pi$
$68$		0			0
$69$	−5.25731			−0.632906
$70$		0			0
$71$			11.4127i			1.35444i	0.735783	+	0.677218i	$0.236815\pi$
$71$			11.4127i			1.35444i	−0.735783	+	0.677218i	$0.763185\pi$
$72$		0			0
$73$	4.70820	−	4.70820i	0.551054	−	0.551054i	−0.375691	−	0.926745i	$-0.622595\pi$
$73$	4.70820	−	4.70820i	0.551054	−	0.551054i	0.926745	+	0.375691i	$0.122595\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	6.15537	−	6.15537i	0.701469	−	0.701469i
$78$		0			0
$79$	2.90617			0.326970			0.163485	−	0.986546i	$-0.447727\pi$
$79$	2.90617			0.326970			0.163485	+	0.986546i	$0.447727\pi$
$80$		0			0
$81$	−2.70820			−0.300912
$82$		0			0
$83$	−6.61803	+	6.61803i	−0.726424	+	0.726424i	−0.969905	−	0.243482i	$-0.921710\pi$
$83$	−6.61803	+	6.61803i	−0.726424	+	0.726424i	0.243482	+	0.969905i	$0.421710\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	3.80423	−	3.80423i	0.407856	−	0.407856i
$88$		0			0
$89$			2.47214i			0.262046i	0.991379	+	0.131023i	$0.0418262\pi$
$89$			2.47214i			0.262046i	−0.991379	+	0.131023i	$0.958174\pi$
$90$		0			0
$91$	2.76393			0.289739
$92$		0			0
$93$	−5.25731	−	5.25731i	−0.545158	−	0.545158i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−4.23607	−	4.23607i	−0.430108	−	0.430108i	0.458557	−	0.888665i	$-0.348366\pi$
$97$	−4.23607	−	4.23607i	−0.430108	−	0.430108i	−0.888665	+	0.458557i	$0.848366\pi$
$98$		0			0
$99$			7.23607i			0.727252i
$100$		0			0
$101$		−	2.90617i		−	0.289175i	−0.989492	−	0.144587i	$-0.953815\pi$
$101$		−	2.90617i		−	0.289175i	0.989492	−	0.144587i	$-0.0461855\pi$
$102$		0			0
$103$	3.35520	+	3.35520i	0.330597	+	0.330597i	0.852813	−	0.522216i	$-0.174894\pi$
$103$	3.35520	+	3.35520i	0.330597	+	0.330597i	−0.522216	+	0.852813i	$0.674894\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.909830	+	0.909830i	0.0879566	+	0.0879566i	0.749716	−	0.661760i	$-0.230190\pi$
$107$	0.909830	+	0.909830i	0.0879566	+	0.0879566i	−0.661760	+	0.749716i	$0.730190\pi$
$108$		0			0
$109$	−14.6619			−1.40435			−0.702176	−	0.712003i	$-0.747788\pi$
$109$	−14.6619			−1.40435			−0.702176	+	0.712003i	$0.747788\pi$
$110$		0			0
$111$			0.898056i			0.0852397i
$112$		0			0
$113$	−8.70820	+	8.70820i	−0.819199	+	0.819199i	−0.985992	−	0.166793i	$-0.946659\pi$
$113$	−8.70820	+	8.70820i	−0.819199	+	0.819199i	0.166793	+	0.985992i	$0.446659\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1.62460	+	1.62460i	−0.150194	+	0.150194i
$118$		0			0
$119$	3.80423			0.348733
$120$		0			0
$121$	−0.527864			−0.0479876
$122$		0			0
$123$	3.52786	−	3.52786i	0.318097	−	0.318097i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−2.80017	+	2.80017i	−0.248475	+	0.248475i	−0.820344	−	0.571870i	$-0.806218\pi$
$127$	−2.80017	+	2.80017i	−0.248475	+	0.248475i	0.571870	+	0.820344i	$0.306218\pi$
$128$		0			0
$129$		−	5.70820i		−	0.502579i
$130$		0			0
$131$	−13.7082			−1.19769			−0.598846	−	0.800864i	$-0.704374\pi$
$131$	−13.7082			−1.19769			−0.598846	+	0.800864i	$0.704374\pi$
$132$		0			0
$133$	3.80423	+	3.80423i	0.329868	+	0.329868i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	5.47214	+	5.47214i	0.467516	+	0.467516i	0.901109	−	0.433593i	$-0.142754\pi$
$137$	5.47214	+	5.47214i	0.467516	+	0.467516i	−0.433593	+	0.901109i	$0.642754\pi$
$138$		0			0
$139$		−	21.4164i		−	1.81652i	−0.418411	−	0.908258i	$-0.637413\pi$
$139$		−	21.4164i		−	1.81652i	0.418411	−	0.908258i	$-0.362587\pi$
$140$		0			0
$141$			4.14725i			0.349262i
$142$		0			0
$143$	2.35114	+	2.35114i	0.196612	+	0.196612i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−0.145898	−	0.145898i	−0.0120335	−	0.0120335i
$148$		0			0
$149$	12.8658			1.05400			0.527002	−	0.849864i	$-0.323316\pi$
$149$	12.8658			1.05400			0.527002	+	0.849864i	$0.323316\pi$
$150$		0			0
$151$		−	6.71040i		−	0.546084i	−0.962002	−	0.273042i	$-0.911970\pi$
$151$		−	6.71040i		−	0.546084i	0.962002	−	0.273042i	$-0.0880298\pi$
$152$		0			0
$153$	−2.23607	+	2.23607i	−0.180775	+	0.180775i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−13.9353	+	13.9353i	−1.11216	+	1.11216i	−0.119303	+	0.992858i	$0.538066\pi$
$157$	−13.9353	+	13.9353i	−1.11216	+	1.11216i	−0.992858	+	0.119303i	$0.961934\pi$
$158$		0			0
$159$	−3.80423			−0.301695
$160$		0			0
$161$	−16.1803			−1.27519
$162$		0			0
$163$	−13.8541	+	13.8541i	−1.08514	+	1.08514i	−0.0891157	+	0.996021i	$0.528404\pi$
$163$	−13.8541	+	13.8541i	−1.08514	+	1.08514i	−0.996021	+	0.0891157i	$0.971596\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−8.05748	+	8.05748i	−0.623507	+	0.623507i	−0.946426	−	0.322920i	$-0.895336\pi$
$167$	−8.05748	+	8.05748i	−0.623507	+	0.623507i	0.322920	+	0.946426i	$0.395336\pi$
$168$		0			0
$169$		−	11.9443i		−	0.918790i
$170$		0			0
$171$	−4.47214			−0.341993
$172$		0			0
$173$	14.4904	+	14.4904i	1.10168	+	1.10168i	0.994208	+	0.107474i	$0.0342762\pi$
$173$	14.4904	+	14.4904i	1.10168	+	1.10168i	0.107474	+	0.994208i	$0.465724\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	0.291796	+	0.291796i	0.0219327	+	0.0219327i
$178$		0			0
$179$			7.52786i			0.562659i	0.959611	+	0.281329i	$0.0907754\pi$
$179$			7.52786i			0.562659i	−0.959611	+	0.281329i	$0.909225\pi$
$180$		0			0
$181$			15.2169i			1.13106i	0.824726	+	0.565532i	$0.191329\pi$
$181$			15.2169i			1.13106i	−0.824726	+	0.565532i	$0.808671\pi$
$182$		0			0
$183$	−0.555029	−	0.555029i	−0.0410289	−	0.0410289i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	3.23607	+	3.23607i	0.236645	+	0.236645i
$188$		0			0
$189$	12.3107			0.895474
$190$		0			0
$191$			13.2088i			0.955755i	0.878427	+	0.477877i	$0.158594\pi$
$191$			13.2088i			0.955755i	−0.878427	+	0.477877i	$0.841406\pi$
$192$		0			0
$193$	−1.47214	+	1.47214i	−0.105967	+	0.105967i	−0.758102	−	0.652136i	$-0.773873\pi$
$193$	−1.47214	+	1.47214i	−0.105967	+	0.105967i	0.652136	+	0.758102i	$0.273873\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	9.23305	−	9.23305i	0.657828	−	0.657828i	−0.297038	−	0.954866i	$-0.595999\pi$
$197$	9.23305	−	9.23305i	0.657828	−	0.657828i	0.954866	+	0.297038i	$0.0959988\pi$
$198$		0			0
$199$	−21.7153			−1.53936			−0.769678	−	0.638432i	$-0.779583\pi$
$199$	−21.7153			−1.53936			−0.769678	+	0.638432i	$0.779583\pi$
$200$		0			0
$201$	−5.70820			−0.402626
$202$		0			0
$203$	11.7082	−	11.7082i	0.821755	−	0.821755i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	9.51057	−	9.51057i	0.661030	−	0.661030i
$208$		0			0
$209$			6.47214i			0.447687i
$210$		0			0
$211$	−2.29180			−0.157774			−0.0788869	−	0.996884i	$-0.525137\pi$
$211$	−2.29180			−0.157774			−0.0788869	+	0.996884i	$0.525137\pi$
$212$		0			0
$213$	7.05342	+	7.05342i	0.483293	+	0.483293i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−16.1803	−	16.1803i	−1.09839	−	1.09839i
$218$		0			0
$219$		−	5.81966i		−	0.393256i
$220$		0			0
$221$			1.45309i			0.0977451i
$222$		0			0
$223$	−14.2128	−	14.2128i	−0.951763	−	0.951763i	0.0471263	−	0.998889i	$-0.484994\pi$
$223$	−14.2128	−	14.2128i	−0.951763	−	0.951763i	−0.998889	+	0.0471263i	$0.984994\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−9.38197	−	9.38197i	−0.622703	−	0.622703i	0.323519	−	0.946222i	$-0.395134\pi$
$227$	−9.38197	−	9.38197i	−0.622703	−	0.622703i	−0.946222	+	0.323519i	$0.895134\pi$
$228$		0			0
$229$	−7.95148			−0.525449			−0.262724	−	0.964871i	$-0.584621\pi$
$229$	−7.95148			−0.525449			−0.262724	+	0.964871i	$0.584621\pi$
$230$		0			0
$231$		−	7.60845i		−	0.500599i
$232$		0			0
$233$	−5.47214	+	5.47214i	−0.358492	+	0.358492i	−0.863257	−	0.504765i	$-0.831579\pi$
$233$	−5.47214	+	5.47214i	−0.358492	+	0.358492i	0.504765	+	0.863257i	$0.331579\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	1.79611	−	1.79611i	0.116670	−	0.116670i
$238$		0			0
$239$	−13.4208			−0.868119			−0.434059	−	0.900884i	$-0.642919\pi$
$239$	−13.4208			−0.868119			−0.434059	+	0.900884i	$0.642919\pi$
$240$		0			0
$241$	11.2361			0.723779			0.361889	−	0.932221i	$-0.382132\pi$
$241$	11.2361			0.723779			0.361889	+	0.932221i	$0.382132\pi$
$242$		0			0
$243$	−11.3820	+	11.3820i	−0.730153	+	0.730153i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−1.45309	+	1.45309i	−0.0924576	+	0.0924576i
$248$		0			0
$249$			8.18034i			0.518408i
$250$		0			0
$251$	−0.180340			−0.0113830			−0.00569148	−	0.999984i	$-0.501812\pi$
$251$	−0.180340			−0.0113830			−0.00569148	+	0.999984i	$0.501812\pi$
$252$		0			0
$253$	−13.7638	−	13.7638i	−0.865324	−	0.865324i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−5.29180	−	5.29180i	−0.330093	−	0.330093i	0.522529	−	0.852622i	$-0.324989\pi$
$257$	−5.29180	−	5.29180i	−0.330093	−	0.330093i	−0.852622	+	0.522529i	$0.824989\pi$
$258$		0			0
$259$			2.76393i			0.171742i
$260$		0			0
$261$			13.7638i			0.851959i
$262$		0			0
$263$	7.50245	+	7.50245i	0.462621	+	0.462621i	0.899514	−	0.436893i	$-0.143921\pi$
$263$	7.50245	+	7.50245i	0.462621	+	0.462621i	−0.436893	+	0.899514i	$0.643921\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	1.52786	+	1.52786i	0.0935038	+	0.0935038i
$268$		0			0
$269$	−20.4742			−1.24833			−0.624167	−	0.781291i	$-0.714562\pi$
$269$	−20.4742			−1.24833			−0.624167	+	0.781291i	$0.714562\pi$
$270$		0			0
$271$			17.2250i			1.04635i	0.852227	+	0.523173i	$0.175252\pi$
$271$			17.2250i			1.04635i	−0.852227	+	0.523173i	$0.824748\pi$
$272$		0			0
$273$	1.70820	−	1.70820i	0.103385	−	0.103385i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−9.23305	+	9.23305i	−0.554760	+	0.554760i	−0.927811	−	0.373051i	$-0.878312\pi$
$277$	−9.23305	+	9.23305i	−0.554760	+	0.554760i	0.373051	+	0.927811i	$0.378312\pi$
$278$		0			0
$279$	19.0211			1.13877
$280$		0			0
$281$	−9.70820			−0.579143			−0.289571	−	0.957156i	$-0.593513\pi$
$281$	−9.70820			−0.579143			−0.289571	+	0.957156i	$0.593513\pi$
$282$		0			0
$283$	13.3820	−	13.3820i	0.795475	−	0.795475i	−0.186903	−	0.982378i	$-0.559845\pi$
$283$	13.3820	−	13.3820i	0.795475	−	0.795475i	0.982378	+	0.186903i	$0.0598450\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	10.8576	−	10.8576i	0.640907	−	0.640907i
$288$		0			0
$289$		−	15.0000i		−	0.882353i
$290$		0			0
$291$	−5.23607			−0.306944
$292$		0			0
$293$	3.07768	+	3.07768i	0.179800	+	0.179800i	0.791269	−	0.611469i	$-0.209421\pi$
$293$	3.07768	+	3.07768i	0.179800	+	0.179800i	−0.611469	+	0.791269i	$0.709421\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	10.4721	+	10.4721i	0.607655	+	0.607655i
$298$		0			0
$299$		−	6.18034i		−	0.357418i
$300$		0			0
$301$		−	17.5680i		−	1.01261i
$302$		0			0
$303$	−1.79611	−	1.79611i	−0.103184	−	0.103184i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−13.5623	−	13.5623i	−0.774042	−	0.774042i	0.204769	−	0.978810i	$-0.434356\pi$
$307$	−13.5623	−	13.5623i	−0.774042	−	0.774042i	−0.978810	+	0.204769i	$0.934356\pi$
$308$		0			0
$309$	4.14725			0.235929
$310$		0			0
$311$		−	20.8172i		−	1.18044i	−0.807243	−	0.590219i	$-0.799041\pi$
$311$		−	20.8172i		−	1.18044i	0.807243	−	0.590219i	$-0.200959\pi$
$312$		0			0
$313$	1.76393	−	1.76393i	0.0997033	−	0.0997033i	−0.655496	−	0.755199i	$-0.727540\pi$
$313$	1.76393	−	1.76393i	0.0997033	−	0.0997033i	0.755199	+	0.655496i	$0.227540\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3.97574	+	3.97574i	−0.223300	+	0.223300i	−0.809886	−	0.586587i	$-0.800471\pi$
$317$	−3.97574	+	3.97574i	−0.223300	+	0.223300i	0.586587	+	0.809886i	$0.300471\pi$
$318$		0			0
$319$	19.9192			1.11526
$320$		0			0
$321$	1.12461			0.0627697
$322$		0			0
$323$	−2.00000	+	2.00000i	−0.111283	+	0.111283i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−9.06154	+	9.06154i	−0.501104	+	0.501104i
$328$		0			0
$329$			12.7639i			0.703698i
$330$		0			0
$331$	30.0689			1.65274			0.826368	−	0.563131i	$-0.190403\pi$
$331$	30.0689			1.65274			0.826368	+	0.563131i	$0.190403\pi$
$332$		0			0
$333$	−1.62460	−	1.62460i	−0.0890274	−	0.0890274i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	19.9443	+	19.9443i	1.08643	+	1.08643i	0.995893	+	0.0905410i	$0.0288596\pi$
$337$	19.9443	+	19.9443i	1.08643	+	1.08643i	0.0905410	+	0.995893i	$0.471140\pi$
$338$		0			0
$339$			10.7639i			0.584617i
$340$		0			0
$341$		−	27.5276i		−	1.49071i
$342$		0			0
$343$	12.8658	+	12.8658i	0.694686	+	0.694686i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	26.0344	+	26.0344i	1.39760	+	1.39760i	0.806862	+	0.590740i	$0.201164\pi$
$347$	26.0344	+	26.0344i	1.39760	+	1.39760i	0.590740	+	0.806862i	$0.298836\pi$
$348$		0			0
$349$	16.6700			0.892324			0.446162	−	0.894952i	$-0.352791\pi$
$349$	16.6700			0.892324			0.446162	+	0.894952i	$0.352791\pi$
$350$		0			0
$351$			4.70228i			0.250989i
$352$		0			0
$353$	22.4164	−	22.4164i	1.19311	−	1.19311i	0.216914	−	0.976191i	$-0.430401\pi$
$353$	22.4164	−	22.4164i	1.19311	−	1.19311i	0.976191	−	0.216914i	$-0.0695992\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	2.35114	−	2.35114i	0.124436	−	0.124436i
$358$		0			0
$359$	19.9192			1.05129			0.525647	−	0.850703i	$-0.323823\pi$
$359$	19.9192			1.05129			0.525647	+	0.850703i	$0.323823\pi$
$360$		0			0
$361$	15.0000			0.789474
$362$		0			0
$363$	−0.326238	+	0.326238i	−0.0171231	+	0.0171231i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−12.2047	+	12.2047i	−0.637082	+	0.637082i	−0.949835	−	0.312753i	$-0.898749\pi$
$367$	−12.2047	+	12.2047i	−0.637082	+	0.637082i	0.312753	+	0.949835i	$0.398749\pi$
$368$		0			0
$369$			12.7639i			0.664464i
$370$		0			0
$371$	−11.7082			−0.607860
$372$		0			0
$373$	−22.4418	−	22.4418i	−1.16199	−	1.16199i	−0.984038	−	0.177956i	$-0.943052\pi$
$373$	−22.4418	−	22.4418i	−1.16199	−	1.16199i	−0.177956	−	0.984038i	$-0.556948\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4.47214	+	4.47214i	0.230327	+	0.230327i
$378$		0			0
$379$		−	0.111456i		−	0.00572512i	−0.999996	−	0.00286256i	$-0.999089\pi$
$379$		−	0.111456i		−	0.00572512i	0.999996	−	0.00286256i	$-0.000911182\pi$
$380$		0			0
$381$			3.46120i			0.177323i
$382$		0			0
$383$	1.00406	+	1.00406i	0.0513049	+	0.0513049i	0.732294	−	0.680989i	$-0.238450\pi$
$383$	1.00406	+	1.00406i	0.0513049	+	0.0513049i	−0.680989	+	0.732294i	$0.738450\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	10.3262	+	10.3262i	0.524912	+	0.524912i
$388$		0			0
$389$	4.14725			0.210274			0.105137	−	0.994458i	$-0.466472\pi$
$389$	4.14725			0.210274			0.105137	+	0.994458i	$0.466472\pi$
$390$		0			0
$391$		−	8.50651i		−	0.430193i
$392$		0			0
$393$	−8.47214	+	8.47214i	−0.427363	+	0.427363i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	24.4500	−	24.4500i	1.22711	−	1.22711i	0.262055	−	0.965053i	$-0.415600\pi$
$397$	24.4500	−	24.4500i	1.22711	−	1.22711i	0.965053	−	0.262055i	$-0.0844002\pi$
$398$		0			0
$399$	4.70228			0.235409
$400$		0			0
$401$	31.8885			1.59244			0.796219	−	0.605009i	$-0.206830\pi$
$401$	31.8885			1.59244			0.796219	+	0.605009i	$0.206830\pi$
$402$		0			0
$403$	6.18034	−	6.18034i	0.307865	−	0.307865i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−2.35114	+	2.35114i	−0.116542	+	0.116542i
$408$		0			0
$409$		−	21.5967i		−	1.06789i	−0.845519	−	0.533945i	$-0.820709\pi$
$409$		−	21.5967i		−	1.06789i	0.845519	−	0.533945i	$-0.179291\pi$
$410$		0			0
$411$	6.76393			0.333640
$412$		0			0
$413$	0.898056	+	0.898056i	0.0441904	+	0.0441904i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−13.2361	−	13.2361i	−0.648173	−	0.648173i
$418$		0			0
$419$			28.8328i			1.40858i	0.709915	+	0.704288i	$0.248733\pi$
$419$			28.8328i			1.40858i	−0.709915	+	0.704288i	$0.751267\pi$
$420$		0			0
$421$		−	28.4257i		−	1.38538i	−0.721234	−	0.692692i	$-0.756425\pi$
$421$		−	28.4257i		−	1.38538i	0.721234	−	0.692692i	$-0.243575\pi$
$422$		0			0
$423$	−7.50245	−	7.50245i	−0.364782	−	0.364782i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−1.70820	−	1.70820i	−0.0826658	−	0.0826658i
$428$		0			0
$429$	2.90617			0.140311
$430$		0			0
$431$		−	19.0211i		−	0.916216i	−0.888897	−	0.458108i	$-0.848527\pi$
$431$		−	19.0211i		−	0.916216i	0.888897	−	0.458108i	$-0.151473\pi$
$432$		0			0
$433$	−0.819660	+	0.819660i	−0.0393904	+	0.0393904i	−0.726528	−	0.687137i	$-0.758867\pi$
$433$	−0.819660	+	0.819660i	−0.0393904	+	0.0393904i	0.687137	+	0.726528i	$0.258867\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	8.50651	−	8.50651i	0.406921	−	0.406921i
$438$		0			0
$439$	−35.1361			−1.67695			−0.838477	−	0.544937i	$-0.816554\pi$
$439$	−35.1361			−1.67695			−0.838477	+	0.544937i	$0.816554\pi$
$440$		0			0
$441$	0.527864			0.0251364
$442$		0			0
$443$	−1.09017	+	1.09017i	−0.0517955	+	0.0517955i	−0.732530	−	0.680735i	$-0.761661\pi$
$443$	−1.09017	+	1.09017i	−0.0517955	+	0.0517955i	0.680735	+	0.732530i	$0.261661\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	7.95148	−	7.95148i	0.376092	−	0.376092i
$448$		0			0
$449$			17.5967i			0.830442i	0.909721	+	0.415221i	$0.136296\pi$
$449$			17.5967i			0.830442i	−0.909721	+	0.415221i	$0.863704\pi$
$450$		0			0
$451$	18.4721			0.869819
$452$		0			0
$453$	−4.14725	−	4.14725i	−0.194855	−	0.194855i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−9.65248	−	9.65248i	−0.451524	−	0.451524i	0.444336	−	0.895860i	$-0.353440\pi$
$457$	−9.65248	−	9.65248i	−0.451524	−	0.451524i	−0.895860	+	0.444336i	$0.853440\pi$
$458$		0			0
$459$			6.47214i			0.302093i
$460$		0			0
$461$			27.5276i			1.28209i	0.767503	+	0.641045i	$0.221499\pi$
$461$			27.5276i			1.28209i	−0.767503	+	0.641045i	$0.778501\pi$
$462$		0			0
$463$	−2.45714	−	2.45714i	−0.114193	−	0.114193i	0.647701	−	0.761894i	$-0.275730\pi$
$463$	−2.45714	−	2.45714i	−0.114193	−	0.114193i	−0.761894	+	0.647701i	$0.775730\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−18.3262	−	18.3262i	−0.848037	−	0.848037i	0.141851	−	0.989888i	$-0.454695\pi$
$467$	−18.3262	−	18.3262i	−0.848037	−	0.848037i	−0.989888	+	0.141851i	$0.954695\pi$
$468$		0			0
$469$	−17.5680			−0.811217
$470$		0			0
$471$			17.2250i			0.793687i
$472$		0			0
$473$	14.9443	−	14.9443i	0.687138	−	0.687138i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	6.88191	−	6.88191i	0.315101	−	0.315101i
$478$		0			0
$479$	−4.70228			−0.214853			−0.107426	−	0.994213i	$-0.534261\pi$
$479$	−4.70228			−0.214853			−0.107426	+	0.994213i	$0.534261\pi$
$480$		0			0
$481$	−1.05573			−0.0481371
$482$		0			0
$483$	−10.0000	+	10.0000i	−0.455016	+	0.455016i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	18.9151	−	18.9151i	0.857126	−	0.857126i	−0.133872	−	0.990999i	$-0.542741\pi$
$487$	18.9151	−	18.9151i	0.857126	−	0.857126i	0.990999	+	0.133872i	$0.0427412\pi$
$488$		0			0
$489$			17.1246i			0.774402i
$490$		0			0
$491$	15.2361			0.687594			0.343797	−	0.939044i	$-0.388287\pi$
$491$	15.2361			0.687594			0.343797	+	0.939044i	$0.388287\pi$
$492$		0			0
$493$	6.15537	+	6.15537i	0.277224	+	0.277224i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	21.7082	+	21.7082i	0.973746	+	0.973746i
$498$		0			0
$499$		−	11.8885i		−	0.532204i	−0.963945	−	0.266102i	$-0.914264\pi$
$499$		−	11.8885i		−	0.532204i	0.963945	−	0.266102i	$-0.0857359\pi$
$500$		0			0
$501$			9.95959i			0.444962i
$502$		0			0
$503$	−16.5640	−	16.5640i	−0.738552	−	0.738552i	0.233746	−	0.972298i	$-0.424902\pi$
$503$	−16.5640	−	16.5640i	−0.738552	−	0.738552i	−0.972298	+	0.233746i	$0.924902\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−7.38197	−	7.38197i	−0.327845	−	0.327845i
$508$		0			0
$509$	10.8576			0.481257			0.240628	−	0.970617i	$-0.422646\pi$
$509$	10.8576			0.481257			0.240628	+	0.970617i	$0.422646\pi$
$510$		0			0
$511$		−	17.9111i		−	0.792339i
$512$		0			0
$513$	−6.47214	+	6.47214i	−0.285752	+	0.285752i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−10.8576	+	10.8576i	−0.477519	+	0.477519i
$518$		0			0
$519$	17.9111			0.786209
$520$		0			0
$521$	−0.472136			−0.0206847			−0.0103423	−	0.999947i	$-0.503292\pi$
$521$	−0.472136			−0.0206847			−0.0103423	+	0.999947i	$0.503292\pi$
$522$		0			0
$523$	25.7426	−	25.7426i	1.12565	−	1.12565i	0.134770	−	0.990877i	$-0.456970\pi$
$523$	25.7426	−	25.7426i	1.12565	−	1.12565i	0.990877	−	0.134770i	$-0.0430297\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	8.50651	−	8.50651i	0.370549	−	0.370549i
$528$		0			0
$529$			13.1803i			0.573058i
$530$		0			0
$531$	−1.05573			−0.0458147
$532$		0			0
$533$	4.14725	+	4.14725i	0.179637	+	0.179637i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	4.65248	+	4.65248i	0.200769	+	0.200769i
$538$		0			0
$539$		−	0.763932i		−	0.0329049i
$540$		0			0
$541$		−	12.3107i		−	0.529280i	−0.964347	−	0.264640i	$-0.914747\pi$
$541$		−	12.3107i		−	0.529280i	0.964347	−	0.264640i	$-0.0852531\pi$
$542$		0			0
$543$	9.40456	+	9.40456i	0.403588	+	0.403588i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	11.5623	+	11.5623i	0.494369	+	0.494369i	0.909679	−	0.415311i	$-0.136327\pi$
$547$	11.5623	+	11.5623i	0.494369	+	0.494369i	−0.415311	+	0.909679i	$0.636327\pi$
$548$		0			0
$549$	2.00811			0.0857042
$550$		0			0
$551$			12.3107i			0.524455i
$552$		0			0
$553$	5.52786	−	5.52786i	0.235069	−	0.235069i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−23.5519	+	23.5519i	−0.997926	+	0.997926i	−0.999998	−	0.00207187i	$-0.999341\pi$
$557$	−23.5519	+	23.5519i	−0.997926	+	0.997926i	0.00207187	+	0.999998i	$0.499341\pi$
$558$		0			0
$559$	6.71040			0.283820
$560$		0			0
$561$	4.00000			0.168880
$562$		0			0
$563$	−4.32624	+	4.32624i	−0.182329	+	0.182329i	−0.792370	−	0.610041i	$-0.791153\pi$
$563$	−4.32624	+	4.32624i	−0.182329	+	0.182329i	0.610041	+	0.792370i	$0.291153\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−5.15131	+	5.15131i	−0.216335	+	0.216335i
$568$		0			0
$569$		−	27.1246i		−	1.13712i	−0.822641	−	0.568561i	$-0.807500\pi$
$569$		−	27.1246i		−	1.13712i	0.822641	−	0.568561i	$-0.192500\pi$
$570$		0			0
$571$	−22.6525			−0.947977			−0.473988	−	0.880531i	$-0.657186\pi$
$571$	−22.6525			−0.947977			−0.473988	+	0.880531i	$0.657186\pi$
$572$		0			0
$573$	8.16348	+	8.16348i	0.341034	+	0.341034i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−26.2361	−	26.2361i	−1.09222	−	1.09222i	−0.995291	−	0.0969307i	$-0.969097\pi$
$577$	−26.2361	−	26.2361i	−1.09222	−	1.09222i	−0.0969307	−	0.995291i	$-0.530903\pi$
$578$		0			0
$579$			1.81966i			0.0756225i
$580$		0			0
$581$			25.1765i			1.04450i
$582$		0			0
$583$	−9.95959	−	9.95959i	−0.412484	−	0.412484i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	17.0902	+	17.0902i	0.705387	+	0.705387i	0.965562	−	0.260175i	$-0.0837802\pi$
$587$	17.0902	+	17.0902i	0.705387	+	0.705387i	−0.260175	+	0.965562i	$0.583780\pi$
$588$		0			0
$589$	17.0130			0.701009
$590$		0			0
$591$		−	11.4127i		−	0.469455i
$592$		0			0
$593$	−20.4164	+	20.4164i	−0.838401	+	0.838401i	−0.988648	−	0.150247i	$-0.951993\pi$
$593$	−20.4164	+	20.4164i	−0.838401	+	0.838401i	0.150247	+	0.988648i	$0.451993\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−13.4208	+	13.4208i	−0.549276	+	0.549276i
$598$		0			0
$599$	6.49839			0.265517			0.132759	−	0.991148i	$-0.457617\pi$
$599$	6.49839			0.265517			0.132759	+	0.991148i	$0.457617\pi$
$600$		0			0
$601$	−17.7082			−0.722333			−0.361166	−	0.932501i	$-0.617621\pi$
$601$	−17.7082			−0.722333			−0.361166	+	0.932501i	$0.617621\pi$
$602$		0			0
$603$	10.3262	−	10.3262i	0.420517	−	0.420517i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−32.6789	+	32.6789i	−1.32640	+	1.32640i	−0.417908	+	0.908489i	$0.637237\pi$
$607$	−32.6789	+	32.6789i	−1.32640	+	1.32640i	−0.908489	+	0.417908i	$0.862763\pi$
$608$		0			0
$609$		−	14.4721i		−	0.586441i
$610$		0			0
$611$	−4.87539			−0.197237
$612$		0			0
$613$	19.5357	+	19.5357i	0.789038	+	0.789038i	0.981337	−	0.192298i	$-0.0615941\pi$
$613$	19.5357	+	19.5357i	0.789038	+	0.789038i	−0.192298	+	0.981337i	$0.561594\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−4.88854	−	4.88854i	−0.196805	−	0.196805i	0.601824	−	0.798629i	$-0.294441\pi$
$617$	−4.88854	−	4.88854i	−0.196805	−	0.196805i	−0.798629	+	0.601824i	$0.794441\pi$
$618$		0			0
$619$		−	35.3050i		−	1.41903i	−0.704692	−	0.709513i	$-0.748915\pi$
$619$		−	35.3050i		−	1.41903i	0.704692	−	0.709513i	$-0.251085\pi$
$620$		0			0
$621$		−	27.5276i		−	1.10465i
$622$		0			0
$623$	4.70228	+	4.70228i	0.188393	+	0.188393i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	4.00000	+	4.00000i	0.159745	+	0.159745i
$628$		0			0
$629$	−1.45309			−0.0579383
$630$		0			0
$631$			22.6134i			0.900223i	0.892972	+	0.450112i	$0.148616\pi$
$631$			22.6134i			0.900223i	−0.892972	+	0.450112i	$0.851384\pi$
$632$		0			0
$633$	−1.41641	+	1.41641i	−0.0562972	+	0.0562972i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	0.171513	−	0.171513i	0.00679561	−	0.00679561i
$638$		0			0
$639$	−25.5195			−1.00954
$640$		0			0
$641$	−38.6525			−1.52668			−0.763341	−	0.645996i	$-0.776442\pi$
$641$	−38.6525			−1.52668			−0.763341	+	0.645996i	$0.776442\pi$
$642$		0			0
$643$	−11.5623	+	11.5623i	−0.455973	+	0.455973i	−0.897331	−	0.441358i	$-0.854497\pi$
$643$	−11.5623	+	11.5623i	−0.455973	+	0.455973i	0.441358	+	0.897331i	$0.354497\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	20.0252	−	20.0252i	0.787271	−	0.787271i	−0.193775	−	0.981046i	$-0.562073\pi$
$647$	20.0252	−	20.0252i	0.787271	−	0.787271i	0.981046	+	0.193775i	$0.0620731\pi$
$648$		0			0
$649$			1.52786i			0.0599739i
$650$		0			0
$651$	−20.0000			−0.783862
$652$		0			0
$653$	−20.0907	−	20.0907i	−0.786210	−	0.786210i	0.194661	−	0.980871i	$-0.437639\pi$
$653$	−20.0907	−	20.0907i	−0.786210	−	0.786210i	−0.980871	+	0.194661i	$0.937639\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	10.5279	+	10.5279i	0.410731	+	0.410731i
$658$		0			0
$659$		−	18.0000i		−	0.701180i	−0.936529	−	0.350590i	$-0.885981\pi$
$659$		−	18.0000i		−	0.701180i	0.936529	−	0.350590i	$-0.114019\pi$
$660$		0			0
$661$			3.80423i			0.147967i	0.997259	+	0.0739836i	$0.0235713\pi$
$661$			3.80423i			0.147967i	−0.997259	+	0.0739836i	$0.976429\pi$
$662$		0			0
$663$	0.898056	+	0.898056i	0.0348776	+	0.0348776i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−26.1803	−	26.1803i	−1.01371	−	1.01371i
$668$		0			0
$669$	−17.5680			−0.679220
$670$		0			0
$671$		−	2.90617i		−	0.112191i
$672$		0			0
$673$	17.2918	−	17.2918i	0.666550	−	0.666550i	−0.290366	−	0.956916i	$-0.593777\pi$
$673$	17.2918	−	17.2918i	0.666550	−	0.666550i	0.956916	+	0.290366i	$0.0937770\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−7.77997	+	7.77997i	−0.299008	+	0.299008i	−0.840625	−	0.541617i	$-0.817812\pi$
$677$	−7.77997	+	7.77997i	−0.299008	+	0.299008i	0.541617	+	0.840625i	$0.317812\pi$
$678$		0			0
$679$	−16.1150			−0.618435
$680$		0			0
$681$	−11.5967			−0.444388
$682$		0			0
$683$	−22.7984	+	22.7984i	−0.872356	+	0.872356i	−0.992729	−	0.120373i	$-0.961591\pi$
$683$	−22.7984	+	22.7984i	−0.872356	+	0.872356i	0.120373	+	0.992729i	$0.461591\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−4.91428	+	4.91428i	−0.187492	+	0.187492i
$688$		0			0
$689$		−	4.47214i		−	0.170375i
$690$		0			0
$691$	−9.12461			−0.347117			−0.173558	−	0.984824i	$-0.555527\pi$
$691$	−9.12461			−0.347117			−0.173558	+	0.984824i	$0.555527\pi$
$692$		0			0
$693$	13.7638	+	13.7638i	0.522844	+	0.522844i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	5.70820	+	5.70820i	0.216214	+	0.216214i
$698$		0			0
$699$			6.76393i			0.255835i
$700$		0			0
$701$			19.7072i			0.744330i	0.928167	+	0.372165i	$0.121384\pi$
$701$			19.7072i			0.744330i	−0.928167	+	0.372165i	$0.878616\pi$
$702$		0			0
$703$	−1.45309	−	1.45309i	−0.0548041	−	0.0548041i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−5.52786	−	5.52786i	−0.207897	−	0.207897i
$708$		0			0
$709$	−3.24920			−0.122026			−0.0610131	−	0.998137i	$-0.519433\pi$
$709$	−3.24920			−0.122026			−0.0610131	+	0.998137i	$0.519433\pi$
$710$		0			0
$711$			6.49839i			0.243709i
$712$		0			0
$713$	−36.1803	+	36.1803i	−1.35496	+	1.35496i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−8.29451	+	8.29451i	−0.309764	+	0.309764i
$718$		0			0
$719$	4.01623			0.149780			0.0748900	−	0.997192i	$-0.476139\pi$
$719$	4.01623			0.149780			0.0748900	+	0.997192i	$0.476139\pi$
$720$		0			0
$721$	12.7639			0.475354
$722$		0			0
$723$	6.94427	−	6.94427i	0.258260	−	0.258260i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	9.51057	−	9.51057i	0.352727	−	0.352727i	−0.508396	−	0.861123i	$-0.669761\pi$
$727$	9.51057	−	9.51057i	0.352727	−	0.352727i	0.861123	+	0.508396i	$0.169761\pi$
$728$		0			0
$729$			5.94427i			0.220158i
$730$		0			0
$731$	9.23607			0.341608
$732$		0			0
$733$	19.1926	+	19.1926i	0.708896	+	0.708896i	0.966303	−	0.257407i	$-0.0828680\pi$
$733$	19.1926	+	19.1926i	0.708896	+	0.708896i	−0.257407	+	0.966303i	$0.582868\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−14.9443	−	14.9443i	−0.550479	−	0.550479i
$738$		0			0
$739$			9.41641i			0.346388i	0.984888	+	0.173194i	$0.0554088\pi$
$739$			9.41641i			0.346388i	−0.984888	+	0.173194i	$0.944591\pi$
$740$		0			0
$741$			1.79611i			0.0659818i
$742$		0			0
$743$	−4.80828	−	4.80828i	−0.176399	−	0.176399i	0.613385	−	0.789784i	$-0.289807\pi$
$743$	−4.80828	−	4.80828i	−0.176399	−	0.176399i	−0.789784	+	0.613385i	$0.789807\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−14.7984	−	14.7984i	−0.541444	−	0.541444i
$748$		0			0
$749$	3.46120			0.126469
$750$		0			0
$751$		−	11.4127i		−	0.416455i	−0.978080	−	0.208227i	$-0.933231\pi$
$751$		−	11.4127i		−	0.416455i	0.978080	−	0.208227i	$-0.0667694\pi$
$752$		0			0
$753$	−0.111456	+	0.111456i	−0.00406169	+	0.00406169i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	31.7154	−	31.7154i	1.15272	−	1.15272i	0.166709	−	0.986006i	$-0.446686\pi$
$757$	31.7154	−	31.7154i	1.15272	−	1.15272i	0.986006	−	0.166709i	$-0.0533140\pi$
$758$		0			0
$759$	−17.0130			−0.617533
$760$		0			0
$761$	2.94427			0.106730			0.0533649	−	0.998575i	$-0.483005\pi$
$761$	2.94427			0.106730			0.0533649	+	0.998575i	$0.483005\pi$
$762$		0			0
$763$	−27.8885	+	27.8885i	−1.00963	+	1.00963i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−0.343027	+	0.343027i	−0.0123860	+	0.0123860i
$768$		0			0
$769$		−	6.47214i		−	0.233391i	−0.993168	−	0.116696i	$-0.962770\pi$
$769$		−	6.47214i		−	0.233391i	0.993168	−	0.116696i	$-0.0372302\pi$
$770$		0			0
$771$	−6.54102			−0.235569
$772$		0			0
$773$	−31.5034	−	31.5034i	−1.13310	−	1.13310i	−0.989659	−	0.143439i	$-0.954184\pi$
$773$	−31.5034	−	31.5034i	−1.13310	−	1.13310i	−0.143439	−	0.989659i	$-0.545816\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	1.70820	+	1.70820i	0.0612815	+	0.0612815i
$778$		0			0
$779$			11.4164i			0.409035i
$780$		0			0
$781$			36.9322i			1.32154i
$782$		0			0
$783$	19.9192	+	19.9192i	0.711854	+	0.711854i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	17.8541	+	17.8541i	0.636430	+	0.636430i	0.949673	−	0.313243i	$-0.101415\pi$
$787$	17.8541	+	17.8541i	0.636430	+	0.636430i	−0.313243	+	0.949673i	$0.601415\pi$
$788$		0			0
$789$	9.27354			0.330147
$790$		0			0
$791$			33.1280i			1.17790i
$792$		0			0
$793$	0.652476	−	0.652476i	0.0231701	−	0.0231701i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	14.8334	−	14.8334i	0.525426	−	0.525426i	−0.393779	−	0.919205i	$-0.628833\pi$
$797$	14.8334	−	14.8334i	0.525426	−	0.525426i	0.919205	+	0.393779i	$0.128833\pi$
$798$		0			0
$799$	−6.71040			−0.237397
$800$		0			0
$801$	−5.52786			−0.195317
$802$		0			0
$803$	15.2361	−	15.2361i	0.537669	−	0.537669i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−12.6538	+	12.6538i	−0.445433	+	0.445433i
$808$		0			0
$809$		−	4.94427i		−	0.173831i	−0.996216	−	0.0869157i	$-0.972299\pi$
$809$		−	4.94427i		−	0.173831i	0.996216	−	0.0869157i	$-0.0277011\pi$
$810$		0			0
$811$	−26.0689			−0.915402			−0.457701	−	0.889106i	$-0.651327\pi$
$811$	−26.0689			−0.915402			−0.457701	+	0.889106i	$0.651327\pi$
$812$		0			0
$813$	10.6456	+	10.6456i	0.373359	+	0.373359i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	9.23607	+	9.23607i	0.323129	+	0.323129i
$818$		0			0
$819$			6.18034i			0.215959i
$820$		0			0
$821$			17.9111i			0.625101i	0.949901	+	0.312550i	$0.101183\pi$
$821$			17.9111i			0.625101i	−0.949901	+	0.312550i	$0.898817\pi$
$822$		0			0
$823$	13.8698	+	13.8698i	0.483472	+	0.483472i	0.906238	−	0.422767i	$-0.138941\pi$
$823$	13.8698	+	13.8698i	0.483472	+	0.483472i	−0.422767	+	0.906238i	$0.638941\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	8.14590	+	8.14590i	0.283261	+	0.283261i	0.834408	−	0.551147i	$-0.185810\pi$
$827$	8.14590	+	8.14590i	0.283261	+	0.283261i	−0.551147	+	0.834408i	$0.685810\pi$
$828$		0			0
$829$	−54.5002			−1.89287			−0.946436	−	0.322892i	$-0.895345\pi$
$829$	−54.5002			−1.89287			−0.946436	+	0.322892i	$0.895345\pi$
$830$		0			0
$831$			11.4127i			0.395901i
$832$		0			0
$833$	0.236068	−	0.236068i	0.00817927	−	0.00817927i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	27.5276	−	27.5276i	0.951494	−	0.951494i
$838$		0			0
$839$	15.2169			0.525346			0.262673	−	0.964885i	$-0.415396\pi$
$839$	15.2169			0.525346			0.262673	+	0.964885i	$0.415396\pi$
$840$		0			0
$841$	8.88854			0.306502
$842$		0			0
$843$	−6.00000	+	6.00000i	−0.206651	+	0.206651i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−1.00406	+	1.00406i	−0.0344998	+	0.0344998i
$848$		0			0
$849$		−	16.5410i		−	0.567686i
$850$		0			0
$851$	6.18034			0.211859
$852$		0			0
$853$	18.8496	+	18.8496i	0.645399	+	0.645399i	0.951877	−	0.306479i	$-0.0991509\pi$
$853$	18.8496	+	18.8496i	0.645399	+	0.645399i	−0.306479	+	0.951877i	$0.599151\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−35.8328	−	35.8328i	−1.22403	−	1.22403i	−0.966188	−	0.257837i	$-0.916990\pi$
$857$	−35.8328	−	35.8328i	−1.22403	−	1.22403i	−0.257837	−	0.966188i	$-0.583010\pi$
$858$		0			0
$859$			16.4721i			0.562022i	0.959705	+	0.281011i	$0.0906697\pi$
$859$			16.4721i			0.562022i	−0.959705	+	0.281011i	$0.909330\pi$
$860$		0			0
$861$		−	13.4208i		−	0.457379i
$862$		0			0
$863$	35.5851	+	35.5851i	1.21133	+	1.21133i	0.970589	+	0.240743i	$0.0773909\pi$
$863$	35.5851	+	35.5851i	1.21133	+	1.21133i	0.240743	+	0.970589i	$0.422609\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−9.27051	−	9.27051i	−0.314843	−	0.314843i
$868$		0			0
$869$	9.40456			0.319028
$870$		0			0
$871$		−	6.71040i		−	0.227373i
$872$		0			0
$873$	9.47214	−	9.47214i	0.320583	−	0.320583i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−21.5438	+	21.5438i	−0.727482	+	0.727482i	−0.970118	−	0.242636i	$-0.921988\pi$
$877$	−21.5438	+	21.5438i	−0.727482	+	0.727482i	0.242636	+	0.970118i	$0.421988\pi$
$878$		0			0
$879$	3.80423			0.128313
$880$		0			0
$881$	6.87539			0.231638			0.115819	−	0.993270i	$-0.463051\pi$
$881$	6.87539			0.231638			0.115819	+	0.993270i	$0.463051\pi$
$882$		0			0
$883$	−2.79837	+	2.79837i	−0.0941728	+	0.0941728i	−0.752624	−	0.658451i	$-0.771212\pi$
$883$	−2.79837	+	2.79837i	−0.0941728	+	0.0941728i	0.658451	+	0.752624i	$0.271212\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−11.5187	+	11.5187i	−0.386759	+	0.386759i	−0.873530	−	0.486770i	$-0.838175\pi$
$887$	−11.5187	+	11.5187i	−0.386759	+	0.386759i	0.486770	+	0.873530i	$0.338175\pi$
$888$		0			0
$889$			10.6525i			0.357273i
$890$		0			0
$891$	−8.76393			−0.293603
$892$		0			0
$893$	−6.71040	−	6.71040i	−0.224555	−	0.224555i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−3.81966	−	3.81966i	−0.127535	−	0.127535i
$898$		0			0
$899$		−	52.3607i		−	1.74633i
$900$		0			0
$901$		−	6.15537i		−	0.205065i
$902$		0			0
$903$	−10.8576	−	10.8576i	−0.361320	−	0.361320i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−7.67376	−	7.67376i	−0.254803	−	0.254803i	0.568133	−	0.822936i	$-0.307666\pi$
$907$	−7.67376	−	7.67376i	−0.254803	−	0.254803i	−0.822936	+	0.568133i	$0.807666\pi$
$908$		0			0
$909$	6.49839			0.215538
$910$		0			0
$911$			10.3026i			0.341341i	0.985328	+	0.170671i	$0.0545934\pi$
$911$			10.3026i			0.341341i	−0.985328	+	0.170671i	$0.945407\pi$
$912$		0			0
$913$	−21.4164	+	21.4164i	−0.708780	+	0.708780i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−26.0746	+	26.0746i	−0.861058	+	0.861058i
$918$		0			0
$919$	18.1231			0.597825			0.298913	−	0.954281i	$-0.403376\pi$
$919$	18.1231			0.597825			0.298913	+	0.954281i	$0.403376\pi$
$920$		0			0
$921$	−16.7639			−0.552390
$922$		0			0
$923$	−8.29180	+	8.29180i	−0.272928	+	0.272928i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−7.50245	+	7.50245i	−0.246413	+	0.246413i
$928$		0			0
$929$		−	36.6525i		−	1.20253i	−0.799050	−	0.601264i	$-0.794664\pi$
$929$		−	36.6525i		−	1.20253i	0.799050	−	0.601264i	$-0.205336\pi$
$930$		0			0
$931$	0.472136			0.0154736
$932$		0			0
$933$	−12.8658	−	12.8658i	−0.421206	−	0.421206i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	42.3050	+	42.3050i	1.38204	+	1.38204i	0.840987	+	0.541056i	$0.181975\pi$
$937$	42.3050	+	42.3050i	1.38204	+	1.38204i	0.541056	+	0.840987i	$0.318025\pi$
$938$		0			0
$939$		−	2.18034i		−	0.0711527i
$940$		0			0
$941$		−	37.6183i		−	1.22632i	−0.789959	−	0.613160i	$-0.789898\pi$
$941$		−	37.6183i		−	1.22632i	0.789959	−	0.613160i	$-0.210102\pi$
$942$		0			0
$943$	−24.2784	−	24.2784i	−0.790615	−	0.790615i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−2.14590	−	2.14590i	−0.0697323	−	0.0697323i	0.671381	−	0.741113i	$-0.265702\pi$
$947$	−2.14590	−	2.14590i	−0.0697323	−	0.0697323i	−0.741113	+	0.671381i	$0.765702\pi$
$948$		0			0
$949$	6.84142			0.222082
$950$		0			0
$951$			4.91428i			0.159357i
$952$		0			0
$953$	29.1803	−	29.1803i	0.945244	−	0.945244i	−0.0533329	−	0.998577i	$-0.516984\pi$
$953$	29.1803	−	29.1803i	0.945244	−	0.945244i	0.998577	+	0.0533329i	$0.0169844\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	12.3107	−	12.3107i	0.397950	−	0.397950i
$958$		0			0
$959$	20.8172			0.672224
$960$		0			0
$961$	−41.3607			−1.33422
$962$		0			0
$963$	−2.03444	+	2.03444i	−0.0655590	+	0.0655590i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−10.9637	+	10.9637i	−0.352567	+	0.352567i	−0.861064	−	0.508497i	$-0.830201\pi$
$967$	−10.9637	+	10.9637i	−0.352567	+	0.352567i	0.508497	+	0.861064i	$0.330201\pi$
$968$		0			0
$969$			2.47214i			0.0794164i
$970$		0			0
$971$	15.5967			0.500523			0.250262	−	0.968178i	$-0.419483\pi$
$971$	15.5967			0.500523			0.250262	+	0.968178i	$0.419483\pi$
$972$		0			0
$973$	−40.7364	−	40.7364i	−1.30595	−	1.30595i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	23.7639	+	23.7639i	0.760276	+	0.760276i	0.976372	−	0.216096i	$-0.0693326\pi$
$977$	23.7639	+	23.7639i	0.760276	+	0.760276i	−0.216096	+	0.976372i	$0.569333\pi$
$978$		0			0
$979$			8.00000i			0.255681i
$980$		0			0
$981$		−	32.7849i		−	1.04674i
$982$		0			0
$983$	18.0171	+	18.0171i	0.574655	+	0.574655i	0.933426	−	0.358770i	$-0.116804\pi$
$983$	18.0171	+	18.0171i	0.574655	+	0.574655i	−0.358770	+	0.933426i	$0.616804\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	7.88854	+	7.88854i	0.251095	+	0.251095i
$988$		0			0
$989$	−39.2833			−1.24914
$990$		0			0
$991$			14.3188i			0.454853i	0.973795	+	0.227427i	$0.0730312\pi$
$991$			14.3188i			0.454853i	−0.973795	+	0.227427i	$0.926969\pi$
$992$		0			0
$993$	18.5836	−	18.5836i	0.589732	−	0.589732i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−27.9112	+	27.9112i	−0.883955	+	0.883955i	−0.993934	−	0.109979i	$-0.964922\pi$
$997$	−27.9112	+	27.9112i	−0.883955	+	0.883955i	0.109979	+	0.993934i	$0.464922\pi$
$998$		0			0
$999$	−4.70228			−0.148774

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.2.o.g.207.4		8
4.3	odd	2		200.2.k.h.107.2		8
5.2	odd	4		160.2.o.a.143.2		8
5.3	odd	4	inner	800.2.o.g.143.3		8
5.4	even	2		160.2.o.a.47.1		8
8.3	odd	2	inner	800.2.o.g.207.3		8
8.5	even	2		200.2.k.h.107.4		8
15.2	even	4		1440.2.bi.c.1423.2		8
15.14	odd	2		1440.2.bi.c.847.3		8
20.3	even	4		200.2.k.h.43.4		8
20.7	even	4		40.2.k.a.3.1	✓	8
20.19	odd	2		40.2.k.a.27.3	yes	8
40.3	even	4	inner	800.2.o.g.143.4		8
40.13	odd	4		200.2.k.h.43.2		8
40.19	odd	2		160.2.o.a.47.2		8
40.27	even	4		160.2.o.a.143.1		8
40.29	even	2		40.2.k.a.27.1	yes	8
40.37	odd	4		40.2.k.a.3.3	yes	8
60.47	odd	4		360.2.w.c.163.4		8
60.59	even	2		360.2.w.c.307.2		8
80.19	odd	4		1280.2.n.m.767.4		8
80.27	even	4		1280.2.n.m.1023.3		8
80.29	even	4		1280.2.n.q.767.2		8
80.37	odd	4		1280.2.n.q.1023.1		8
80.59	odd	4		1280.2.n.q.767.1		8
80.67	even	4		1280.2.n.q.1023.2		8
80.69	even	4		1280.2.n.m.767.3		8
80.77	odd	4		1280.2.n.m.1023.4		8
120.29	odd	2		360.2.w.c.307.4		8
120.59	even	2		1440.2.bi.c.847.2		8
120.77	even	4		360.2.w.c.163.2		8
120.107	odd	4		1440.2.bi.c.1423.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.2.k.a.3.1	✓	8	20.7	even	4
40.2.k.a.3.3	yes	8	40.37	odd	4
40.2.k.a.27.1	yes	8	40.29	even	2
40.2.k.a.27.3	yes	8	20.19	odd	2
160.2.o.a.47.1		8	5.4	even	2
160.2.o.a.47.2		8	40.19	odd	2
160.2.o.a.143.1		8	40.27	even	4
160.2.o.a.143.2		8	5.2	odd	4
200.2.k.h.43.2		8	40.13	odd	4
200.2.k.h.43.4		8	20.3	even	4
200.2.k.h.107.2		8	4.3	odd	2
200.2.k.h.107.4		8	8.5	even	2
360.2.w.c.163.2		8	120.77	even	4
360.2.w.c.163.4		8	60.47	odd	4
360.2.w.c.307.2		8	60.59	even	2
360.2.w.c.307.4		8	120.29	odd	2
800.2.o.g.143.3		8	5.3	odd	4	inner
800.2.o.g.143.4		8	40.3	even	4	inner
800.2.o.g.207.3		8	8.3	odd	2	inner
800.2.o.g.207.4		8	1.1	even	1	trivial
1280.2.n.m.767.3		8	80.69	even	4
1280.2.n.m.767.4		8	80.19	odd	4
1280.2.n.m.1023.3		8	80.27	even	4
1280.2.n.m.1023.4		8	80.77	odd	4
1280.2.n.q.767.1		8	80.59	odd	4
1280.2.n.q.767.2		8	80.29	even	4
1280.2.n.q.1023.1		8	80.37	odd	4
1280.2.n.q.1023.2		8	80.67	even	4
1440.2.bi.c.847.2		8	120.59	even	2
1440.2.bi.c.847.3		8	15.14	odd	2
1440.2.bi.c.1423.2		8	15.2	even	4
1440.2.bi.c.1423.3		8	120.107	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	800.o (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.618034 - 0.618034i) q^{3} +(1.90211 - 1.90211i) q^{7} +2.23607i q^{9} +3.23607 q^{11} +(0.726543 + 0.726543i) q^{13} +(1.00000 + 1.00000i) q^{17} +2.00000i q^{19} -2.35114i q^{21} +(-4.25325 - 4.25325i) q^{23} +(3.23607 + 3.23607i) q^{27} +6.15537 q^{29} -8.50651i q^{31} +(2.00000 - 2.00000i) q^{33} +(-0.726543 + 0.726543i) q^{37} +0.898056 q^{39} +5.70820 q^{41} +(4.61803 - 4.61803i) q^{43} +(-3.35520 + 3.35520i) q^{47} -0.236068i q^{49} +1.23607 q^{51} +(-3.07768 - 3.07768i) q^{53} +(1.23607 + 1.23607i) q^{57} +0.472136i q^{59} -0.898056i q^{61} +(4.25325 + 4.25325i) q^{63} +(-4.61803 - 4.61803i) q^{67} -5.25731 q^{69} +11.4127i q^{71} +(4.70820 - 4.70820i) q^{73} +(6.15537 - 6.15537i) q^{77} +2.90617 q^{79} -2.70820 q^{81} +(-6.61803 + 6.61803i) q^{83} +(3.80423 - 3.80423i) q^{87} +2.47214i q^{89} +2.76393 q^{91} +(-5.25731 - 5.25731i) q^{93} +(-4.23607 - 4.23607i) q^{97} +7.23607i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} + 8 q^{11} + 8 q^{17} + 8 q^{27} + 16 q^{33} - 8 q^{41} + 28 q^{43} - 8 q^{51} - 8 q^{57} - 28 q^{67} - 16 q^{73} + 32 q^{81} - 44 q^{83} + 40 q^{91} - 16 q^{97}+O(q^{100}) \) 8 * q - 4 * q^3 + 8 * q^11 + 8 * q^17 + 8 * q^27 + 16 * q^33 - 8 * q^41 + 28 * q^43 - 8 * q^51 - 8 * q^57 - 28 * q^67 - 16 * q^73 + 32 * q^81 - 44 * q^83 + 40 * q^91 - 16 * q^97

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