LMFDB - Embedded newform 804.2.g.c.401.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [804,2,Mod(401,804)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(804, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("804.401");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 804 = 2^{2} \cdot 3 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	804.g (of order $2$, degree $1$, 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:	$6.41997232251$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{14} - x^{12} - 27x^{10} + 88x^{8} - 243x^{6} - 81x^{4} - 729x^{2} + 6561 $ x^16 - x^14 - x^12 - 27x^10 + 88x^8 - 243x^6 - 81x^4 - 729*x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			401.12
Root			$0.760738 + 1.55605i$ of defining polynomial
Character	$\chi$	$=$	804.401
Dual form			804.2.g.c.401.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.760738 + 1.55605i) q^{3} +3.48890 q^{5} -1.71955i q^{7} +(-1.84255 + 2.36749i) q^{9} +O(q^{10})$	\(q+(0.760738 + 1.55605i) q^{3} +3.48890 q^{5} -1.71955i q^{7} +(-1.84255 + 2.36749i) q^{9} -0.624556 q^{11} +2.24717i q^{13} +(2.65414 + 5.42888i) q^{15} +3.10515i q^{17} +5.86412 q^{19} +(2.67570 - 1.30813i) q^{21} -1.75326i q^{23} +7.17240 q^{25} +(-5.08562 - 1.06606i) q^{27} +0.0875179i q^{29} +0.222271i q^{31} +(-0.475124 - 0.971838i) q^{33} -5.99934i q^{35} +2.86412 q^{37} +(-3.49669 + 1.70950i) q^{39} -5.27770 q^{41} -7.30890i q^{43} +(-6.42848 + 8.25991i) q^{45} +1.62982i q^{47} +4.04313 q^{49} +(-4.83176 + 2.36221i) q^{51} -7.57926 q^{53} -2.17901 q^{55} +(4.46106 + 9.12484i) q^{57} -4.77090i q^{59} +11.6804i q^{61} +(4.07102 + 3.16837i) q^{63} +7.84013i q^{65} +(-3.68511 - 7.30890i) q^{67} +(2.72816 - 1.33377i) q^{69} +5.72141i q^{71} +5.99338 q^{73} +(5.45632 + 11.1606i) q^{75} +1.07396i q^{77} -8.28105i q^{79} +(-2.20999 - 8.72445i) q^{81} -8.79064i q^{83} +10.8336i q^{85} +(-0.136182 + 0.0665782i) q^{87} +13.2903i q^{89} +3.86412 q^{91} +(-0.345863 + 0.169090i) q^{93} +20.4593 q^{95} -18.0171i q^{97} +(1.15078 - 1.47863i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{9}+O(q^{10}) $ 16 * q + 2 * q^9	$ 16 q + 2 q^{9} + 18 q^{15} + 28 q^{19} - 16 q^{21} + 14 q^{33} - 20 q^{37} - 4 q^{49} - 32 q^{55} + 4 q^{67} - 16 q^{73} + 6 q^{81} - 4 q^{91} - 30 q^{93}+O(q^{100}) $ 16 * q + 2 * q^9 + 18 * q^15 + 28 * q^19 - 16 * q^21 + 14 * q^33 - 20 * q^37 - 4 * q^49 - 32 * q^55 + 4 * q^67 - 16 * q^73 + 6 * q^81 - 4 * q^91 - 30 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/804\mathbb{Z}\right)^\times$.

$n$	$269$	$337$	$403$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	0.760738	+	1.55605i	0.439212	+	0.898383i
$3$	0.760738	+	1.55605i	0.439212	+	0.898383i
$4$		0			0
$5$	3.48890			1.56028			0.780141	−	0.625604i	$-0.215147\pi$
$5$	3.48890			1.56028			0.780141	+	0.625604i	$0.215147\pi$
$6$		0			0
$7$		−	1.71955i		−	0.649930i	−0.945726	−	0.324965i	$-0.894647\pi$
$7$		−	1.71955i		−	0.649930i	0.945726	−	0.324965i	$-0.105353\pi$
$8$		0			0
$9$	−1.84255	+	2.36749i	−0.614185	+	0.789162i
$10$		0			0
$11$	−0.624556			−0.188311			−0.0941554	−	0.995558i	$-0.530015\pi$
$11$	−0.624556			−0.188311			−0.0941554	+	0.995558i	$0.530015\pi$
$12$		0			0
$13$			2.24717i			0.623252i	0.950205	+	0.311626i	$0.100874\pi$
$13$			2.24717i			0.623252i	−0.950205	+	0.311626i	$0.899126\pi$
$14$		0			0
$15$	2.65414	+	5.42888i	0.685295	+	1.40173i
$16$		0			0
$17$			3.10515i			0.753110i	0.926394	+	0.376555i	$0.122891\pi$
$17$			3.10515i			0.753110i	−0.926394	+	0.376555i	$0.877109\pi$
$18$		0			0
$19$	5.86412			1.34532			0.672661	−	0.739951i	$-0.265151\pi$
$19$	5.86412			1.34532			0.672661	+	0.739951i	$0.265151\pi$
$20$		0			0
$21$	2.67570	−	1.30813i	0.583886	−	0.285457i
$22$		0			0
$23$		−	1.75326i		−	0.365581i	−0.983152	−	0.182790i	$-0.941487\pi$
$23$		−	1.75326i		−	0.365581i	0.983152	−	0.182790i	$-0.0585130\pi$
$24$		0			0
$25$	7.17240			1.43448
$26$		0			0
$27$	−5.08562	−	1.06606i	−0.978728	−	0.205164i
$28$		0			0
$29$			0.0875179i			0.0162517i	0.999967	+	0.00812583i	$0.00258656\pi$
$29$			0.0875179i			0.0162517i	−0.999967	+	0.00812583i	$0.997413\pi$
$30$		0			0
$31$			0.222271i			0.0399210i	0.999801	+	0.0199605i	$0.00635405\pi$
$31$			0.222271i			0.0399210i	−0.999801	+	0.0199605i	$0.993646\pi$
$32$		0			0
$33$	−0.475124	−	0.971838i	−0.0827085	−	0.169175i
$34$		0			0
$35$		−	5.99934i		−	1.01407i
$36$		0			0
$37$	2.86412			0.470859			0.235429	−	0.971891i	$-0.424350\pi$
$37$	2.86412			0.470859			0.235429	+	0.971891i	$0.424350\pi$
$38$		0			0
$39$	−3.49669	+	1.70950i	−0.559919	+	0.273740i
$40$		0			0
$41$	−5.27770			−0.824239			−0.412119	−	0.911130i	$-0.635211\pi$
$41$	−5.27770			−0.824239			−0.412119	+	0.911130i	$0.635211\pi$
$42$		0			0
$43$		−	7.30890i		−	1.11460i	−0.830312	−	0.557298i	$-0.811838\pi$
$43$		−	7.30890i		−	1.11460i	0.830312	−	0.557298i	$-0.188162\pi$
$44$		0			0
$45$	−6.42848	+	8.25991i	−0.958301	+	1.23132i
$46$		0			0
$47$			1.62982i			0.237734i	0.992910	+	0.118867i	$0.0379262\pi$
$47$			1.62982i			0.237734i	−0.992910	+	0.118867i	$0.962074\pi$
$48$		0			0
$49$	4.04313			0.577591
$50$		0			0
$51$	−4.83176	+	2.36221i	−0.676582	+	0.330775i
$52$		0			0
$53$	−7.57926			−1.04109			−0.520546	−	0.853834i	$-0.674271\pi$
$53$	−7.57926			−1.04109			−0.520546	+	0.853834i	$0.674271\pi$
$54$		0			0
$55$	−2.17901			−0.293818
$56$		0			0
$57$	4.46106	+	9.12484i	0.590882	+	1.20861i
$58$		0			0
$59$		−	4.77090i		−	0.621118i	−0.950554	−	0.310559i	$-0.899484\pi$
$59$		−	4.77090i		−	0.621118i	0.950554	−	0.310559i	$-0.100516\pi$
$60$		0			0
$61$			11.6804i			1.49552i	0.663970	+	0.747759i	$0.268870\pi$
$61$			11.6804i			1.49552i	−0.663970	+	0.747759i	$0.731130\pi$
$62$		0			0
$63$	4.07102	+	3.16837i	0.512900	+	0.399177i
$64$		0			0
$65$			7.84013i			0.972448i
$66$		0			0
$67$	−3.68511	−	7.30890i	−0.450208	−	0.892924i
$67$	−3.68511	−	7.30890i	−0.450208	−	0.892924i
$68$		0			0
$69$	2.72816	−	1.33377i	0.328432	−	0.160568i
$70$		0			0
$71$			5.72141i			0.679007i	0.940605	+	0.339503i	$0.110259\pi$
$71$			5.72141i			0.679007i	−0.940605	+	0.339503i	$0.889741\pi$
$72$		0			0
$73$	5.99338			0.701472			0.350736	−	0.936474i	$-0.385931\pi$
$73$	5.99338			0.701472			0.350736	+	0.936474i	$0.385931\pi$
$74$		0			0
$75$	5.45632	+	11.1606i	0.630041	+	1.28871i
$76$		0			0
$77$			1.07396i			0.122389i
$78$		0			0
$79$		−	8.28105i		−	0.931691i	−0.884866	−	0.465845i	$-0.845750\pi$
$79$		−	8.28105i		−	0.931691i	0.884866	−	0.465845i	$-0.154250\pi$
$80$		0			0
$81$	−2.20999	−	8.72445i	−0.245554	−	0.969383i
$82$		0			0
$83$		−	8.79064i		−	0.964898i	−0.875924	−	0.482449i	$-0.839747\pi$
$83$		−	8.79064i		−	0.964898i	0.875924	−	0.482449i	$-0.160253\pi$
$84$		0			0
$85$			10.8336i			1.17506i
$86$		0			0
$87$	−0.136182	+	0.0665782i	−0.0146002	+	0.00713793i
$88$		0			0
$89$			13.2903i			1.40877i	0.709817	+	0.704386i	$0.248778\pi$
$89$			13.2903i			1.40877i	−0.709817	+	0.704386i	$0.751222\pi$
$90$		0			0
$91$	3.86412			0.405070
$92$		0			0
$93$	−0.345863	+	0.169090i	−0.0358644	+	0.0175338i
$94$		0			0
$95$	20.4593			2.09908
$96$		0			0
$97$		−	18.0171i		−	1.82936i	−0.404178	−	0.914680i	$-0.632442\pi$
$97$		−	18.0171i		−	1.82936i	0.404178	−	0.914680i	$-0.367558\pi$
$98$		0			0
$99$	1.15078	−	1.47863i	0.115658	−	0.148608i
$100$		0			0
$101$	−19.2102			−1.91149			−0.955743	−	0.294203i	$-0.904946\pi$
$101$	−19.2102			−1.91149			−0.955743	+	0.294203i	$0.904946\pi$
$102$		0			0
$103$	−13.2155			−1.30216			−0.651082	−	0.759007i	$-0.725685\pi$
$103$	−13.2155			−1.30216			−0.651082	+	0.759007i	$0.725685\pi$
$104$		0			0
$105$	9.33525	−	4.56393i	0.911027	−	0.445394i
$106$		0			0
$107$			18.5229i			1.79067i	0.445392	+	0.895336i	$0.353064\pi$
$107$			18.5229i			1.79067i	−0.445392	+	0.895336i	$0.646936\pi$
$108$		0			0
$109$		−	2.99705i		−	0.287065i	−0.989646	−	0.143533i	$-0.954154\pi$
$109$		−	2.99705i		−	0.287065i	0.989646	−	0.143533i	$-0.0458462\pi$
$110$		0			0
$111$	2.17885	+	4.45670i	0.206807	+	0.423012i
$112$		0			0
$113$	0.539693			0.0507700			0.0253850	−	0.999678i	$-0.491919\pi$
$113$	0.539693			0.0507700			0.0253850	+	0.999678i	$0.491919\pi$
$114$		0			0
$115$		−	6.11695i		−	0.570409i
$116$		0			0
$117$	−5.32013	−	4.14052i	−0.491847	−	0.382792i
$118$		0			0
$119$	5.33948			0.489469
$120$		0			0
$121$	−10.6099			−0.964539
$122$		0			0
$123$	−4.01495	−	8.21234i	−0.362016	−	0.740482i
$124$		0			0
$125$	7.57926			0.677910
$126$		0			0
$127$	6.77124			0.600850			0.300425	−	0.953805i	$-0.402871\pi$
$127$	6.77124			0.600850			0.300425	+	0.953805i	$0.402871\pi$
$128$		0			0
$129$	11.3730	−	5.56016i	1.00133	−	0.489545i
$130$		0			0
$131$		−	13.7520i		−	1.20151i	−0.799432	−	0.600757i	$-0.794866\pi$
$131$		−	13.7520i		−	1.20151i	0.799432	−	0.600757i	$-0.205134\pi$
$132$		0			0
$133$		−	10.0837i		−	0.874365i
$134$		0			0
$135$	−17.7432	−	3.71938i	−1.52709	−	0.320113i
$136$		0			0
$137$	−1.85058			−0.158106			−0.0790529	−	0.996870i	$-0.525190\pi$
$137$	−1.85058			−0.158106			−0.0790529	+	0.996870i	$0.525190\pi$
$138$		0			0
$139$		−	18.9893i		−	1.61065i	−0.592835	−	0.805324i	$-0.701991\pi$
$139$		−	18.9893i		−	1.61065i	0.592835	−	0.805324i	$-0.298009\pi$
$140$		0			0
$141$	−2.53607	+	1.23987i	−0.213576	+	0.104416i
$142$		0			0
$143$		−	1.40348i		−	0.117365i
$144$		0			0
$145$			0.305341i			0.0253572i
$146$		0			0
$147$	3.07577	+	6.29130i	0.253685	+	0.518898i
$148$		0			0
$149$		−	17.0681i		−	1.39827i	−0.714989	−	0.699135i	$-0.753569\pi$
$149$		−	17.0681i		−	1.39827i	0.714989	−	0.699135i	$-0.246431\pi$
$150$		0			0
$151$	3.19121			0.259697			0.129848	−	0.991534i	$-0.458551\pi$
$151$	3.19121			0.259697			0.129848	+	0.991534i	$0.458551\pi$
$152$		0			0
$153$	−7.35141	−	5.72141i	−0.594326	−	0.462549i
$154$		0			0
$155$			0.775479i			0.0622880i
$156$		0			0
$157$	12.0365			0.960619			0.480309	−	0.877099i	$-0.340524\pi$
$157$	12.0365			0.960619			0.480309	+	0.877099i	$0.340524\pi$
$158$		0			0
$159$	−5.76583	−	11.7937i	−0.457260	−	0.935299i
$160$		0			0
$161$	−3.01483			−0.237602
$162$		0			0
$163$	−6.02432			−0.471861			−0.235931	−	0.971770i	$-0.575814\pi$
$163$	−6.02432			−0.471861			−0.235931	+	0.971770i	$0.575814\pi$
$164$		0			0
$165$	−1.65766	−	3.39064i	−0.129049	−	0.263961i
$166$		0			0
$167$		−	8.05109i		−	0.623012i	−0.950244	−	0.311506i	$-0.899167\pi$
$167$		−	8.05109i		−	0.623012i	0.950244	−	0.311506i	$-0.100833\pi$
$168$		0			0
$169$	7.95025			0.611558
$170$		0			0
$171$	−10.8050	+	13.8832i	−0.826276	+	1.06168i
$172$		0			0
$173$			11.0122i			0.837246i	0.908160	+	0.418623i	$0.137487\pi$
$173$			11.0122i			0.837246i	−0.908160	+	0.418623i	$0.862513\pi$
$174$		0			0
$175$		−	12.3333i		−	0.932311i
$176$		0			0
$177$	7.42373	−	3.62941i	0.558002	−	0.272803i
$178$		0			0
$179$	−18.0459			−1.34882			−0.674409	−	0.738358i	$-0.735601\pi$
$179$	−18.0459			−1.34882			−0.674409	+	0.738358i	$0.735601\pi$
$180$		0			0
$181$	−8.74581			−0.650071			−0.325035	−	0.945702i	$-0.605376\pi$
$181$	−8.74581			−0.650071			−0.325035	+	0.945702i	$0.605376\pi$
$182$		0			0
$183$	−18.1752	+	8.88570i	−1.34355	+	0.656850i
$184$		0			0
$185$	9.99262			0.734672
$186$		0			0
$187$		−	1.93934i		−	0.141819i
$188$		0			0
$189$	−1.83315	+	8.74499i	−0.133342	+	0.636105i
$190$		0			0
$191$	9.54165			0.690409			0.345205	−	0.938527i	$-0.387810\pi$
$191$	9.54165			0.690409			0.345205	+	0.938527i	$0.387810\pi$
$192$		0			0
$193$	12.9436			0.931703			0.465851	−	0.884863i	$-0.345748\pi$
$193$	12.9436			0.931703			0.465851	+	0.884863i	$0.345748\pi$
$194$		0			0
$195$	−12.1996	+	5.96428i	−0.873631	+	0.427111i
$196$		0			0
$197$	7.57926			0.540000			0.270000	−	0.962860i	$-0.412976\pi$
$197$	7.57926			0.540000			0.270000	+	0.962860i	$0.412976\pi$
$198$		0			0
$199$	−20.0111			−1.41855			−0.709274	−	0.704933i	$-0.750977\pi$
$199$	−20.0111			−1.41855			−0.709274	+	0.704933i	$0.750977\pi$
$200$		0			0
$201$	8.56957	−	11.2944i	0.604451	−	0.796642i
$202$		0			0
$203$	0.150492			0.0105624
$204$		0			0
$205$	−18.4134			−1.28604
$206$		0			0
$207$	4.15083	+	3.23048i	0.288502	+	0.224534i
$208$		0			0
$209$	−3.66248			−0.253339
$210$		0			0
$211$	0.659681			0.0454143			0.0227072	−	0.999742i	$-0.492771\pi$
$211$	0.659681			0.0454143			0.0227072	+	0.999742i	$0.492771\pi$
$212$		0			0
$213$	−8.90278	+	4.35250i	−0.610008	+	0.298228i
$214$		0			0
$215$		−	25.5000i		−	1.73908i
$216$		0			0
$217$	0.382206			0.0259459
$218$		0			0
$219$	4.55940	+	9.32598i	0.308095	+	0.630191i
$220$		0			0
$221$	−6.97779			−0.469377
$222$		0			0
$223$	−11.5911			−0.776199			−0.388099	−	0.921618i	$-0.626868\pi$
$223$	−11.5911			−0.776199			−0.388099	+	0.921618i	$0.626868\pi$
$224$		0			0
$225$	−13.2155	+	16.9805i	−0.881035	+	1.13204i
$226$		0			0
$227$			5.61852i			0.372914i	0.982463	+	0.186457i	$0.0597005\pi$
$227$			5.61852i			0.372914i	−0.982463	+	0.186457i	$0.940299\pi$
$228$		0			0
$229$			9.77585i			0.646006i	0.946398	+	0.323003i	$0.104692\pi$
$229$			9.77585i			0.646006i	−0.946398	+	0.323003i	$0.895308\pi$
$230$		0			0
$231$	−1.67113	+	0.817001i	−0.109952	+	0.0537547i
$232$		0			0
$233$	9.45293			0.619282			0.309641	−	0.950854i	$-0.399791\pi$
$233$	9.45293			0.619282			0.309641	+	0.950854i	$0.399791\pi$
$234$		0			0
$235$			5.68627i			0.370932i
$236$		0			0
$237$	12.8857	−	6.29971i	0.837015	−	0.409210i
$238$		0			0
$239$	12.8570			0.831648			0.415824	−	0.909445i	$-0.363493\pi$
$239$	12.8570			0.831648			0.415824	+	0.909445i	$0.363493\pi$
$240$		0			0
$241$	16.9372			1.09102			0.545509	−	0.838105i	$-0.316336\pi$
$241$	16.9372			1.09102			0.545509	+	0.838105i	$0.316336\pi$
$242$		0			0
$243$	11.8944	−	10.0759i	0.763027	−	0.646367i
$244$		0			0
$245$	14.1061			0.901204
$246$		0			0
$247$			13.1777i			0.838474i
$248$		0			0
$249$	13.6786	−	6.68738i	0.866848	−	0.423795i
$250$		0			0
$251$	−15.1585			−0.956797			−0.478399	−	0.878143i	$-0.658783\pi$
$251$	−15.1585			−0.956797			−0.478399	+	0.878143i	$0.658783\pi$
$252$		0			0
$253$			1.09501i			0.0688428i
$254$		0			0
$255$	−16.8575	+	8.24150i	−1.05566	+	0.516103i
$256$		0			0
$257$		−	20.8253i		−	1.29904i	−0.760343	−	0.649522i	$-0.774969\pi$
$257$		−	20.8253i		−	1.29904i	0.760343	−	0.649522i	$-0.225031\pi$
$258$		0			0
$259$		−	4.92501i		−	0.306025i
$260$		0			0
$261$	−0.207197	−	0.161256i	−0.0128252	−	0.00998152i
$262$		0			0
$263$		−	8.79064i		−	0.542054i	−0.962572	−	0.271027i	$-0.912637\pi$
$263$		−	8.79064i		−	0.542054i	0.962572	−	0.271027i	$-0.0873633\pi$
$264$		0			0
$265$	−26.4433			−1.62440
$266$		0			0
$267$	−20.6804	+	10.1105i	−1.26562	+	0.618750i
$268$		0			0
$269$		−	23.1703i		−	1.41272i	−0.707854	−	0.706359i	$-0.750336\pi$
$269$		−	23.1703i		−	1.41272i	0.707854	−	0.706359i	$-0.249664\pi$
$270$		0			0
$271$			15.1170i			0.918292i	0.888361	+	0.459146i	$0.151845\pi$
$271$			15.1170i			0.918292i	−0.888361	+	0.459146i	$0.848155\pi$
$272$		0			0
$273$	2.93959	+	6.01275i	0.177912	+	0.363908i
$274$		0			0
$275$	−4.47957			−0.270128
$276$		0			0
$277$	−14.2519			−0.856314			−0.428157	−	0.903704i	$-0.640837\pi$
$277$	−14.2519			−0.856314			−0.428157	+	0.903704i	$0.640837\pi$
$278$		0			0
$279$	−0.526223	−	0.409546i	−0.0315041	−	0.0245189i
$280$		0			0
$281$	20.3937			1.21659			0.608293	−	0.793713i	$-0.291855\pi$
$281$	20.3937			1.21659			0.608293	+	0.793713i	$0.291855\pi$
$282$		0			0
$283$	12.1658			0.723180			0.361590	−	0.932337i	$-0.382234\pi$
$283$	12.1658			0.723180			0.361590	+	0.932337i	$0.382234\pi$
$284$		0			0
$285$	15.5642	+	31.8356i	0.921943	+	1.88578i
$286$		0			0
$287$			9.07529i			0.535698i
$288$		0			0
$289$	7.35803			0.432825
$290$		0			0
$291$	28.0354	−	13.7063i	1.64347	−	0.803478i
$292$		0			0
$293$		−	17.1264i		−	1.00053i	−0.865871	−	0.500267i	$-0.833235\pi$
$293$		−	17.1264i		−	1.00053i	0.865871	−	0.500267i	$-0.166765\pi$
$294$		0			0
$295$		−	16.6452i		−	0.969119i
$296$		0			0
$297$	3.17626	+	0.665815i	0.184305	+	0.0386345i
$298$		0			0
$299$	3.93987			0.227849
$300$		0			0
$301$	−12.5680			−0.724410
$302$		0			0
$303$	−14.6139	−	29.8919i	−0.839548	−	1.71725i
$304$		0			0
$305$			40.7516i			2.33343i
$306$		0			0
$307$	15.8145			0.902582			0.451291	−	0.892377i	$-0.350964\pi$
$307$	15.8145			0.902582			0.451291	+	0.892377i	$0.350964\pi$
$308$		0			0
$309$	−10.0536	−	20.5640i	−0.571927	−	1.16984i
$310$		0			0
$311$	−33.2506			−1.88547			−0.942736	−	0.333541i	$-0.891756\pi$
$311$	−33.2506			−1.88547			−0.942736	+	0.333541i	$0.891756\pi$
$312$		0			0
$313$			4.80215i			0.271434i	0.990748	+	0.135717i	$0.0433337\pi$
$313$			4.80215i			0.271434i	−0.990748	+	0.135717i	$0.956666\pi$
$314$		0			0
$315$	14.2034	+	11.0541i	0.800269	+	0.622829i
$316$		0			0
$317$			21.4735i			1.20607i	0.797714	+	0.603036i	$0.206043\pi$
$317$			21.4735i			1.20607i	−0.797714	+	0.603036i	$0.793957\pi$
$318$		0			0
$319$		−	0.0546598i		−	0.00306036i
$320$		0			0
$321$	−28.8224	+	14.0910i	−1.60871	+	0.786485i
$322$		0			0
$323$			18.2090i			1.01318i
$324$		0			0
$325$			16.1176i			0.894041i
$326$		0			0
$327$	4.66354	−	2.27997i	0.257894	−	0.126083i
$328$		0			0
$329$	2.80256			0.154510
$330$		0			0
$331$			6.52171i			0.358465i	0.983807	+	0.179233i	$0.0573615\pi$
$331$			6.52171i			0.358465i	−0.983807	+	0.179233i	$0.942638\pi$
$332$		0			0
$333$	−5.27730	+	6.78077i	−0.289194	+	0.371584i
$334$		0			0
$335$	−12.8570	−	25.5000i	−0.702451	−	1.39321i
$336$		0			0
$337$		−	0.635921i		−	0.0346408i	−0.999850	−	0.0173204i	$-0.994486\pi$
$337$		−	0.635921i		−	0.0346408i	0.999850	−	0.0173204i	$-0.00551353\pi$
$338$		0			0
$339$	0.410565	+	0.839787i	0.0222988	+	0.0456110i
$340$		0			0
$341$		−	0.138821i		−	0.00751756i
$342$		0			0
$343$		−	18.9893i		−	1.02532i
$344$		0			0
$345$	9.51826	−	4.65340i	0.512446	−	0.250531i
$346$		0			0
$347$	8.89350			0.477428			0.238714	−	0.971090i	$-0.423274\pi$
$347$	8.89350			0.477428			0.238714	+	0.971090i	$0.423274\pi$
$348$		0			0
$349$	16.4121			0.878520			0.439260	−	0.898360i	$-0.355241\pi$
$349$	16.4121			0.878520			0.439260	+	0.898360i	$0.355241\pi$
$350$		0			0
$351$	2.39562	−	11.4282i	0.127868	−	0.609994i
$352$		0			0
$353$	−27.9729			−1.48885			−0.744425	−	0.667706i	$-0.767276\pi$
$353$	−27.9729			−1.48885			−0.744425	+	0.667706i	$0.767276\pi$
$354$		0			0
$355$			19.9614i			1.05944i
$356$		0			0
$357$	4.06195	+	8.30847i	0.214981	+	0.439731i
$358$		0			0
$359$			2.46849i			0.130282i	0.997876	+	0.0651411i	$0.0207497\pi$
$359$			2.46849i			0.130282i	−0.997876	+	0.0651411i	$0.979250\pi$
$360$		0			0
$361$	15.3879			0.809891
$362$		0			0
$363$	−8.07138	−	16.5095i	−0.423638	−	0.866526i
$364$		0			0
$365$	20.9103			1.09449
$366$		0			0
$367$		−	16.5621i		−	0.864535i	−0.901746	−	0.432267i	$-0.857714\pi$
$367$		−	16.5621i		−	0.864535i	0.901746	−	0.432267i	$-0.142286\pi$
$368$		0			0
$369$	9.72445	−	12.4949i	0.506235	−	0.650458i
$370$		0			0
$371$			13.0329i			0.676637i
$372$		0			0
$373$		−	6.16272i		−	0.319093i	−0.987190	−	0.159547i	$-0.948997\pi$
$373$		−	6.16272i		−	0.319093i	0.987190	−	0.159547i	$-0.0510032\pi$
$374$		0			0
$375$	5.76583	+	11.7937i	0.297746	+	0.609023i
$376$		0			0
$377$	−0.196667			−0.0101289
$378$		0			0
$379$			30.0337i			1.54273i	0.636394	+	0.771364i	$0.280425\pi$
$379$			30.0337i			1.54273i	−0.636394	+	0.771364i	$0.719575\pi$
$380$		0			0
$381$	5.15114	+	10.5363i	0.263901	+	0.539794i
$382$		0			0
$383$	−23.4124			−1.19632			−0.598158	−	0.801378i	$-0.704100\pi$
$383$	−23.4124			−1.19632			−0.598158	+	0.801378i	$0.704100\pi$
$384$		0			0
$385$			3.74693i			0.190961i
$386$		0			0
$387$	17.3037	+	13.4670i	0.879597	+	0.684568i
$388$		0			0
$389$			16.7093i			0.847197i	0.905850	+	0.423599i	$0.139233\pi$
$389$			16.7093i			0.847197i	−0.905850	+	0.423599i	$0.860767\pi$
$390$		0			0
$391$	5.44415			0.275323
$392$		0			0
$393$	21.3987	−	10.4616i	1.07942	−	0.527720i
$394$		0			0
$395$		−	28.8917i		−	1.45370i
$396$		0			0
$397$	−0.0807556			−0.00405301			−0.00202650	−	0.999998i	$-0.500645\pi$
$397$	−0.0807556			−0.00405301			−0.00202650	+	0.999998i	$0.500645\pi$
$398$		0			0
$399$	15.6907	−	7.67104i	0.785515	−	0.384032i
$400$		0			0
$401$	3.06101			0.152859			0.0764297	−	0.997075i	$-0.475648\pi$
$401$	3.06101			0.152859			0.0764297	+	0.997075i	$0.475648\pi$
$402$		0			0
$403$	−0.499479			−0.0248808
$404$		0			0
$405$	−7.71041	−	30.4387i	−0.383133	−	1.51251i
$406$		0			0
$407$	−1.78881			−0.0886678
$408$		0			0
$409$			37.9785i			1.87792i	0.344030	+	0.938959i	$0.388208\pi$
$409$			37.9785i			1.87792i	−0.344030	+	0.938959i	$0.611792\pi$
$410$		0			0
$411$	−1.40781	−	2.87959i	−0.0694421	−	0.142040i
$412$		0			0
$413$	−8.20382			−0.403683
$414$		0			0
$415$		−	30.6696i		−	1.50551i
$416$		0			0
$417$	29.5482	−	14.4459i	1.44698	−	0.707417i
$418$		0			0
$419$			29.2728i			1.43007i	0.699088	+	0.715036i	$0.253590\pi$
$419$			29.2728i			1.43007i	−0.699088	+	0.715036i	$0.746410\pi$
$420$		0			0
$421$	28.3138			1.37993			0.689966	−	0.723842i	$-0.257625\pi$
$421$	28.3138			1.37993			0.689966	+	0.723842i	$0.257625\pi$
$422$		0			0
$423$	−3.85858	−	3.00303i	−0.187610	−	0.146012i
$424$		0			0
$425$			22.2714i			1.08032i
$426$		0			0
$427$	20.0850			0.971982
$428$		0			0
$429$	2.18388	−	1.06768i	0.105439	−	0.0515482i
$430$		0			0
$431$			30.8114i			1.48413i	0.670327	+	0.742066i	$0.266154\pi$
$431$			30.8114i			1.48413i	−0.670327	+	0.742066i	$0.733846\pi$
$432$		0			0
$433$			5.58686i			0.268487i	0.990948	+	0.134244i	$0.0428605\pi$
$433$			5.58686i			0.268487i	−0.990948	+	0.134244i	$0.957139\pi$
$434$		0			0
$435$	−0.475124	+	0.232284i	−0.0227805	+	0.0111372i
$436$		0			0
$437$		−	10.2814i		−	0.491824i
$438$		0			0
$439$	14.5250			0.693243			0.346621	−	0.938005i	$-0.387329\pi$
$439$	14.5250			0.693243			0.346621	+	0.938005i	$0.387329\pi$
$440$		0			0
$441$	−7.44970	+	9.57207i	−0.354747	+	0.455813i
$442$		0			0
$443$	33.0153			1.56860			0.784302	−	0.620379i	$-0.213021\pi$
$443$	33.0153			1.56860			0.784302	+	0.620379i	$0.213021\pi$
$444$		0			0
$445$			46.3686i			2.19808i
$446$		0			0
$447$	26.5587	−	12.9843i	1.25618	−	0.614138i
$448$		0			0
$449$			16.9757i			0.801131i	0.916268	+	0.400566i	$0.131186\pi$
$449$			16.9757i			0.801131i	−0.916268	+	0.400566i	$0.868814\pi$
$450$		0			0
$451$	3.29622			0.155213
$452$		0			0
$453$	2.42767	+	4.96566i	0.114062	+	0.233307i
$454$		0			0
$455$	13.4815			0.632023
$456$		0			0
$457$	−27.6962			−1.29557			−0.647787	−	0.761822i	$-0.724305\pi$
$457$	−27.6962			−1.29557			−0.647787	+	0.761822i	$0.724305\pi$
$458$		0			0
$459$	3.31028	−	15.7916i	0.154511	−	0.737090i
$460$		0			0
$461$		−	31.7549i		−	1.47897i	−0.673172	−	0.739486i	$-0.735069\pi$
$461$		−	31.7549i		−	1.47897i	0.673172	−	0.739486i	$-0.264931\pi$
$462$		0			0
$463$		−	12.6525i		−	0.588013i	−0.955803	−	0.294006i	$-0.905011\pi$
$463$		−	12.6525i		−	0.588013i	0.955803	−	0.294006i	$-0.0949887\pi$
$464$		0			0
$465$	−1.20668	+	0.589937i	−0.0559585	+	0.0273577i
$466$		0			0
$467$			31.8494i			1.47381i	0.675994	+	0.736907i	$0.263715\pi$
$467$			31.8494i			1.47381i	−0.675994	+	0.736907i	$0.736285\pi$
$468$		0			0
$469$	−12.5680	+	6.33674i	−0.580338	+	0.292604i
$470$		0			0
$471$	9.15664	+	18.7294i	0.421916	+	0.863004i
$472$		0			0
$473$			4.56482i			0.209891i
$474$		0			0
$475$	42.0598			1.92984
$476$		0			0
$477$	13.9652	−	17.9438i	0.639423	−	0.821590i
$478$		0			0
$479$			28.2124i			1.28906i	0.764580	+	0.644529i	$0.222947\pi$
$479$			28.2124i			1.28906i	−0.764580	+	0.644529i	$0.777053\pi$
$480$		0			0
$481$			6.43616i			0.293463i
$482$		0			0
$483$	−2.29350	−	4.69121i	−0.104358	−	0.213458i
$484$		0			0
$485$		−	62.8598i		−	2.85432i
$486$		0			0
$487$		−	6.21140i		−	0.281466i	−0.990048	−	0.140733i	$-0.955054\pi$
$487$		−	6.21140i		−	0.281466i	0.990048	−	0.140733i	$-0.0449458\pi$
$488$		0			0
$489$	−4.58293	−	9.37412i	−0.207247	−	0.423912i
$490$		0			0
$491$			38.9227i			1.75656i	0.478150	+	0.878278i	$0.341308\pi$
$491$			38.9227i			1.75656i	−0.478150	+	0.878278i	$0.658692\pi$
$492$		0			0
$493$	−0.271756			−0.0122393
$494$		0			0
$495$	4.01495	−	5.15878i	0.180459	−	0.231870i
$496$		0			0
$497$	9.83828			0.441307
$498$		0			0
$499$			22.4120i			1.00330i	0.865071	+	0.501650i	$0.167274\pi$
$499$			22.4120i			1.00330i	−0.865071	+	0.501650i	$0.832726\pi$
$500$		0			0
$501$	12.5279	−	6.12477i	0.559703	−	0.273635i
$502$		0			0
$503$	19.3607			0.863250			0.431625	−	0.902053i	$-0.357940\pi$
$503$	19.3607			0.863250			0.431625	+	0.902053i	$0.357940\pi$
$504$		0			0
$505$	−67.0224			−2.98246
$506$		0			0
$507$	6.04806	+	12.3709i	0.268604	+	0.549413i
$508$		0			0
$509$			32.9320i			1.45969i	0.683615	+	0.729843i	$0.260407\pi$
$509$			32.9320i			1.45969i	−0.683615	+	0.729843i	$0.739593\pi$
$510$		0			0
$511$		−	10.3059i		−	0.455908i
$512$		0			0
$513$	−29.8227	−	6.25151i	−1.31670	−	0.276011i
$514$		0			0
$515$	−46.1076			−2.03174
$516$		0			0
$517$		−	1.01791i		−	0.0447678i
$518$		0			0
$519$	−17.1356	+	8.37744i	−0.752168	+	0.367729i
$520$		0			0
$521$	−9.81904			−0.430180			−0.215090	−	0.976594i	$-0.569005\pi$
$521$	−9.81904			−0.430180			−0.215090	+	0.976594i	$0.569005\pi$
$522$		0			0
$523$	−21.2023			−0.927112			−0.463556	−	0.886068i	$-0.653427\pi$
$523$	−21.2023			−0.927112			−0.463556	+	0.886068i	$0.653427\pi$
$524$		0			0
$525$	19.1912	−	9.38243i	0.837573	−	0.409483i
$526$		0			0
$527$	−0.690185			−0.0300649
$528$		0			0
$529$	19.9261			0.866351
$530$		0			0
$531$	11.2950	+	8.79064i	0.490163	+	0.381481i
$532$		0			0
$533$		−	11.8599i		−	0.513708i
$534$		0			0
$535$			64.6243i			2.79395i
$536$		0			0
$537$	−13.7282	−	28.0803i	−0.592417	−	1.21176i
$538$		0			0
$539$	−2.52517			−0.108767
$540$		0			0
$541$			30.9292i			1.32975i	0.746954	+	0.664875i	$0.231515\pi$
$541$			30.9292i			1.32975i	−0.746954	+	0.664875i	$0.768485\pi$
$542$		0			0
$543$	−6.65327	−	13.6089i	−0.285519	−	0.584013i
$544$		0			0
$545$		−	10.4564i		−	0.447902i
$546$		0			0
$547$			7.13487i			0.305065i	0.988298	+	0.152532i	$0.0487429\pi$
$547$			7.13487i			0.305065i	−0.988298	+	0.152532i	$0.951257\pi$
$548$		0			0
$549$	−27.6531	−	21.5217i	−1.18021	−	0.918524i
$550$		0			0
$551$			0.513215i			0.0218637i
$552$		0			0
$553$	−14.2397			−0.605534
$554$		0			0
$555$	7.60177	+	15.5490i	0.322677	+	0.660017i
$556$		0			0
$557$		−	10.7033i		−	0.453512i	−0.973952	−	0.226756i	$-0.927188\pi$
$557$		−	10.7033i		−	0.453512i	0.973952	−	0.226756i	$-0.0728120\pi$
$558$		0			0
$559$	16.4243			0.694674
$560$		0			0
$561$	3.01771	−	1.47533i	0.127408	−	0.0622886i
$562$		0			0
$563$	22.6952			0.956490			0.478245	−	0.878226i	$-0.341273\pi$
$563$	22.6952			0.956490			0.478245	+	0.878226i	$0.341273\pi$
$564$		0			0
$565$	1.88293			0.0792156
$566$		0			0
$567$	−15.0022	+	3.80019i	−0.630031	+	0.159593i
$568$		0			0
$569$		−	4.39191i		−	0.184119i	−0.995754	−	0.0920593i	$-0.970655\pi$
$569$		−	4.39191i		−	0.184119i	0.995754	−	0.0920593i	$-0.0293449\pi$
$570$		0			0
$571$	17.4685			0.731033			0.365516	−	0.930805i	$-0.380892\pi$
$571$	17.4685			0.731033			0.365516	+	0.930805i	$0.380892\pi$
$572$		0			0
$573$	7.25870	+	14.8472i	0.303236	+	0.620252i
$574$		0			0
$575$		−	12.5751i		−	0.524418i
$576$		0			0
$577$			5.51765i			0.229703i	0.993383	+	0.114851i	$0.0366392\pi$
$577$			5.51765i			0.229703i	−0.993383	+	0.114851i	$0.963361\pi$
$578$		0			0
$579$	9.84672	+	20.1409i	0.409216	+	0.837026i
$580$		0			0
$581$	−15.1160			−0.627117
$582$		0			0
$583$	4.73368			0.196049
$584$		0			0
$585$	−18.5614	−	14.4459i	−0.767419	−	0.597263i
$586$		0			0
$587$	8.25384			0.340673			0.170336	−	0.985386i	$-0.445515\pi$
$587$	8.25384			0.340673			0.170336	+	0.985386i	$0.445515\pi$
$588$		0			0
$589$			1.30342i			0.0537066i
$590$		0			0
$591$	5.76583	+	11.7937i	0.237175	+	0.485127i
$592$		0			0
$593$	−27.0282			−1.10991			−0.554957	−	0.831879i	$-0.687265\pi$
$593$	−27.0282			−1.10991			−0.554957	+	0.831879i	$0.687265\pi$
$594$		0			0
$595$	18.6289			0.763710
$596$		0			0
$597$	−15.2232	−	31.1382i	−0.623044	−	1.27440i
$598$		0			0
$599$	−37.6840			−1.53973			−0.769864	−	0.638208i	$-0.779676\pi$
$599$	−37.6840			−1.53973			−0.769864	+	0.638208i	$0.779676\pi$
$600$		0			0
$601$	5.11170			0.208510			0.104255	−	0.994551i	$-0.466754\pi$
$601$	5.11170			0.208510			0.104255	+	0.994551i	$0.466754\pi$
$602$		0			0
$603$	24.0937	+	4.74259i	0.981173	+	0.193133i
$604$		0			0
$605$	−37.0169			−1.50495
$606$		0			0
$607$	−19.1348			−0.776659			−0.388330	−	0.921521i	$-0.626948\pi$
$607$	−19.1348			−0.776659			−0.388330	+	0.921521i	$0.626948\pi$
$608$		0			0
$609$	0.114485	+	0.234172i	0.00463916	+	0.00948912i
$610$		0			0
$611$	−3.66248			−0.148168
$612$		0			0
$613$	−45.8135			−1.85039			−0.925194	−	0.379494i	$-0.876098\pi$
$613$	−45.8135			−1.85039			−0.925194	+	0.379494i	$0.876098\pi$
$614$		0			0
$615$	−14.0077	−	28.6520i	−0.564847	−	1.15536i
$616$		0			0
$617$		−	44.6082i		−	1.79586i	−0.440140	−	0.897929i	$-0.645071\pi$
$617$		−	44.6082i		−	1.79586i	0.440140	−	0.897929i	$-0.354929\pi$
$618$		0			0
$619$	−24.2709			−0.975528			−0.487764	−	0.872976i	$-0.662187\pi$
$619$	−24.2709			−0.975528			−0.487764	+	0.872976i	$0.662187\pi$
$620$		0			0
$621$	−1.86909	+	8.91643i	−0.0750038	+	0.357804i
$622$		0			0
$623$	22.8534			0.915604
$624$		0			0
$625$	−9.41872			−0.376749
$626$		0			0
$627$	−2.78619	−	5.69898i	−0.111270	−	0.227595i
$628$		0			0
$629$			8.89354i			0.354609i
$630$		0			0
$631$		−	17.3812i		−	0.691934i	−0.938247	−	0.345967i	$-0.887551\pi$
$631$		−	17.3812i		−	0.691934i	0.938247	−	0.345967i	$-0.112449\pi$
$632$		0			0
$633$	0.501845	+	1.02649i	0.0199465	+	0.0407995i
$634$		0			0
$635$	23.6241			0.937495
$636$		0			0
$637$			9.08559i			0.359984i
$638$		0			0
$639$	−13.5454	−	10.5420i	−0.535847	−	0.417036i
$640$		0			0
$641$	−46.7822			−1.84779			−0.923893	−	0.382652i	$-0.875011\pi$
$641$	−46.7822			−1.84779			−0.923893	+	0.382652i	$0.875011\pi$
$642$		0			0
$643$	−27.4552			−1.08273			−0.541364	−	0.840788i	$-0.682092\pi$
$643$	−27.4552			−1.08273			−0.541364	+	0.840788i	$0.682092\pi$
$644$		0			0
$645$	39.6791	−	19.3988i	1.56236	−	0.763827i
$646$		0			0
$647$	−4.60312			−0.180967			−0.0904836	−	0.995898i	$-0.528841\pi$
$647$	−4.60312			−0.180967			−0.0904836	+	0.995898i	$0.528841\pi$
$648$		0			0
$649$			2.97970i			0.116963i
$650$		0			0
$651$	0.290759	+	0.594731i	0.0113957	+	0.0233093i
$652$		0			0
$653$	31.9364			1.24977			0.624884	−	0.780718i	$-0.285146\pi$
$653$	31.9364			1.24977			0.624884	+	0.780718i	$0.285146\pi$
$654$		0			0
$655$		−	47.9791i		−	1.87470i
$656$		0			0
$657$	−11.0431	+	14.1893i	−0.430834	+	0.553575i
$658$		0			0
$659$		−	22.7379i		−	0.885742i	−0.896585	−	0.442871i	$-0.853960\pi$
$659$		−	22.7379i		−	0.885742i	0.896585	−	0.442871i	$-0.146040\pi$
$660$		0			0
$661$		−	0.754844i		−	0.0293600i	−0.999892	−	0.0146800i	$-0.995327\pi$
$661$		−	0.754844i		−	0.0293600i	0.999892	−	0.0146800i	$-0.00467296\pi$
$662$		0			0
$663$	−5.30827	−	10.8578i	−0.206156	−	0.421680i
$664$		0			0
$665$		−	35.1809i		−	1.36426i
$666$		0			0
$667$	0.153442			0.00594129
$668$		0			0
$669$	−8.81781	−	18.0363i	−0.340916	−	0.697324i
$670$		0			0
$671$		−	7.29505i		−	0.281622i
$672$		0			0
$673$		−	42.0876i		−	1.62236i	−0.584798	−	0.811179i	$-0.698826\pi$
$673$		−	42.0876i		−	1.62236i	0.584798	−	0.811179i	$-0.301174\pi$
$674$		0			0
$675$	−36.4761	−	7.64621i	−1.40396	−	0.294303i
$676$		0			0
$677$	−9.04014			−0.347441			−0.173720	−	0.984795i	$-0.555579\pi$
$677$	−9.04014			−0.347441			−0.173720	+	0.984795i	$0.555579\pi$
$678$		0			0
$679$	−30.9814			−1.18896
$680$		0			0
$681$	−8.74267	+	4.27422i	−0.335020	+	0.163789i
$682$		0			0
$683$	35.5522			1.36037			0.680184	−	0.733042i	$-0.261900\pi$
$683$	35.5522			1.36037			0.680184	+	0.733042i	$0.261900\pi$
$684$		0			0
$685$	−6.45649			−0.246690
$686$		0			0
$687$	−15.2117	+	7.43686i	−0.580361	+	0.283734i
$688$		0			0
$689$		−	17.0319i		−	0.648862i
$690$		0			0
$691$	40.4178			1.53757			0.768783	−	0.639509i	$-0.220862\pi$
$691$	40.4178			1.53757			0.768783	+	0.639509i	$0.220862\pi$
$692$		0			0
$693$	−2.54258	−	1.97883i	−0.0965847	−	0.0751694i
$694$		0			0
$695$		−	66.2516i		−	2.51307i
$696$		0			0
$697$		−	16.3881i		−	0.620742i
$698$		0			0
$699$	7.19121	+	14.7092i	0.271996	+	0.556353i
$700$		0			0
$701$	35.9453			1.35763			0.678817	−	0.734308i	$-0.262493\pi$
$701$	35.9453			1.35763			0.678817	+	0.734308i	$0.262493\pi$
$702$		0			0
$703$	16.7956			0.633457
$704$		0			0
$705$	−8.84810	+	4.32577i	−0.333239	+	0.162918i
$706$		0			0
$707$			33.0330i			1.24233i
$708$		0			0
$709$	−19.4255			−0.729539			−0.364770	−	0.931098i	$-0.618852\pi$
$709$	−19.4255			−0.729539			−0.364770	+	0.931098i	$0.618852\pi$
$710$		0			0
$711$	19.6053	+	15.2583i	0.735255	+	0.572230i
$712$		0			0
$713$	0.389699			0.0145943
$714$		0			0
$715$		−	4.89660i		−	0.183123i
$716$		0			0
$717$	9.78078	+	20.0060i	0.365270	+	0.747138i
$718$		0			0
$719$			3.14973i			0.117465i	0.998274	+	0.0587326i	$0.0187059\pi$
$719$			3.14973i			0.117465i	−0.998274	+	0.0587326i	$0.981294\pi$
$720$		0			0
$721$			22.7248i			0.846316i
$722$		0			0
$723$	12.8847	+	26.3550i	0.479189	+	0.980152i
$724$		0			0
$725$			0.627713i			0.0233127i
$726$		0			0
$727$			17.5343i			0.650309i	0.945661	+	0.325155i	$0.105416\pi$
$727$			17.5343i			0.650309i	−0.945661	+	0.325155i	$0.894584\pi$
$728$		0			0
$729$	24.7270	+	10.8432i	0.915816	+	0.401598i
$730$		0			0
$731$	22.6952			0.839414
$732$		0			0
$733$		−	39.2276i		−	1.44890i	−0.689325	−	0.724452i	$-0.742093\pi$
$733$		−	39.2276i		−	1.44890i	0.689325	−	0.724452i	$-0.257907\pi$
$734$		0			0
$735$	10.7310	+	21.9497i	0.395820	+	0.809627i
$736$		0			0
$737$	2.30156	+	4.56482i	0.0847790	+	0.168147i
$738$		0			0
$739$			41.7141i			1.53448i	0.641362	+	0.767239i	$0.278370\pi$
$739$			41.7141i			1.53448i	−0.641362	+	0.767239i	$0.721630\pi$
$740$		0			0
$741$	−20.5050	+	10.0247i	−0.753271	+	0.368268i
$742$		0			0
$743$			4.41595i			0.162005i	0.996714	+	0.0810027i	$0.0258122\pi$
$743$			4.41595i			0.162005i	−0.996714	+	0.0810027i	$0.974188\pi$
$744$		0			0
$745$		−	59.5487i		−	2.18170i
$746$		0			0
$747$	20.8117	+	16.1972i	0.761461	+	0.592626i
$748$		0			0
$749$	31.8510			1.16381
$750$		0			0
$751$	−42.0786			−1.53547			−0.767735	−	0.640767i	$-0.778616\pi$
$751$	−42.0786			−1.53547			−0.767735	+	0.640767i	$0.778616\pi$
$752$		0			0
$753$	−11.5317	−	23.5873i	−0.420237	−	0.859571i
$754$		0			0
$755$	11.1338			0.405200
$756$		0			0
$757$			34.9243i			1.26935i	0.772781	+	0.634673i	$0.218865\pi$
$757$			34.9243i			1.26935i	−0.772781	+	0.634673i	$0.781135\pi$
$758$		0			0
$759$	−1.70389	+	0.833018i	−0.0618472	+	0.0302366i
$760$		0			0
$761$			29.5800i			1.07227i	0.844131	+	0.536137i	$0.180117\pi$
$761$			29.5800i			1.07227i	−0.844131	+	0.536137i	$0.819883\pi$
$762$		0			0
$763$	−5.15358			−0.186572
$764$		0			0
$765$	−25.6483	−	19.9614i	−0.927316	−	0.721707i
$766$		0			0
$767$	10.7210			0.387113
$768$		0			0
$769$			11.2185i			0.404549i	0.979329	+	0.202274i	$0.0648332\pi$
$769$			11.2185i			0.404549i	−0.979329	+	0.202274i	$0.935167\pi$
$770$		0			0
$771$	32.4050	−	15.8426i	1.16704	−	0.570556i
$772$		0			0
$773$		−	34.9545i		−	1.25723i	−0.777718	−	0.628614i	$-0.783623\pi$
$773$		−	34.9545i		−	1.25723i	0.777718	−	0.628614i	$-0.216377\pi$
$774$		0			0
$775$			1.59421i			0.0572658i
$776$		0			0
$777$	7.66354	−	3.74664i	0.274928	−	0.134410i
$778$		0			0
$779$	−30.9491			−1.10887
$780$		0			0
$781$		−	3.57335i		−	0.127864i
$782$		0			0
$783$	0.0932994	−	0.445082i	0.00333425	−	0.0159059i
$784$		0			0
$785$	41.9942			1.49884
$786$		0			0
$787$			33.4330i			1.19176i	0.803074	+	0.595879i	$0.203196\pi$
$787$			33.4330i			1.19176i	−0.803074	+	0.595879i	$0.796804\pi$
$788$		0			0
$789$	13.6786	−	6.68738i	0.486972	−	0.238077i
$790$		0			0
$791$		−	0.928031i		−	0.0329970i
$792$		0			0
$793$	−26.2477			−0.932084
$794$		0			0
$795$	−20.1164	−	41.1469i	−0.713455	−	1.45933i
$796$		0			0
$797$			41.7758i			1.47977i	0.672731	+	0.739887i	$0.265121\pi$
$797$			41.7758i			1.47977i	−0.672731	+	0.739887i	$0.734879\pi$
$798$		0			0
$799$	−5.06084			−0.179040
$800$		0			0
$801$	−31.4647	−	24.4882i	−1.11175	−	0.865247i
$802$		0			0
$803$	−3.74321			−0.132095
$804$		0			0
$805$	−10.5184			−0.370726
$806$		0			0
$807$	36.0540	−	17.6265i	1.26916	−	0.620484i
$808$		0			0
$809$	49.5965			1.74372			0.871860	−	0.489755i	$-0.162914\pi$
$809$	49.5965			1.74372			0.871860	+	0.489755i	$0.162914\pi$
$810$		0			0
$811$			10.0723i			0.353686i	0.984239	+	0.176843i	$0.0565885\pi$
$811$			10.0723i			0.353686i	−0.984239	+	0.176843i	$0.943412\pi$
$812$		0			0
$813$	−23.5227	+	11.5001i	−0.824978	+	0.403325i
$814$		0			0
$815$	−21.0182			−0.736237
$816$		0			0
$817$		−	42.8603i		−	1.49949i
$818$		0			0
$819$	−7.11986	+	9.14826i	−0.248788	+	0.319666i
$820$		0			0
$821$			43.6525i			1.52348i	0.647881	+	0.761741i	$0.275655\pi$
$821$			43.6525i			1.52348i	−0.647881	+	0.761741i	$0.724345\pi$
$822$		0			0
$823$	4.53152			0.157959			0.0789795	−	0.996876i	$-0.474834\pi$
$823$	4.53152			0.157959			0.0789795	+	0.996876i	$0.474834\pi$
$824$		0			0
$825$	−3.40778	−	6.97041i	−0.118644	−	0.242678i
$826$		0			0
$827$		−	29.3357i		−	1.02010i	−0.860144	−	0.510052i	$-0.829626\pi$
$827$		−	29.3357i		−	1.02010i	0.860144	−	0.510052i	$-0.170374\pi$
$828$		0			0
$829$	−23.9923			−0.833288			−0.416644	−	0.909070i	$-0.636794\pi$
$829$	−23.9923			−0.833288			−0.416644	+	0.909070i	$0.636794\pi$
$830$		0			0
$831$	−10.8420	−	22.1766i	−0.376104	−	0.769298i
$832$		0			0
$833$			12.5546i			0.434989i
$834$		0			0
$835$		−	28.0894i		−	0.972074i
$836$		0			0
$837$	0.236954	−	1.13038i	0.00819033	−	0.0390718i
$838$		0			0
$839$		−	49.9709i		−	1.72519i	−0.505898	−	0.862593i	$-0.668839\pi$
$839$		−	49.9709i		−	1.72519i	0.505898	−	0.862593i	$-0.331161\pi$
$840$		0			0
$841$	28.9923			0.999736
$842$		0			0
$843$	15.5143	+	31.7335i	0.534339	+	1.09296i
$844$		0			0
$845$	27.7376			0.954202
$846$		0			0
$847$			18.2443i			0.626883i
$848$		0			0
$849$	9.25497	+	18.9305i	0.317630	+	0.649693i
$850$		0			0
$851$		−	5.02156i		−	0.172137i
$852$		0			0
$853$	−19.8388			−0.679269			−0.339634	−	0.940558i	$-0.610303\pi$
$853$	−19.8388			−0.679269			−0.339634	+	0.940558i	$0.610303\pi$
$854$		0			0
$855$	−37.6974	+	48.4371i	−1.28922	+	1.65652i
$856$		0			0
$857$	9.56473			0.326725			0.163363	−	0.986566i	$-0.447766\pi$
$857$	9.56473			0.326725			0.163363	+	0.986566i	$0.447766\pi$
$858$		0			0
$859$	−15.7646			−0.537882			−0.268941	−	0.963157i	$-0.586674\pi$
$859$	−15.7646			−0.537882			−0.268941	+	0.963157i	$0.586674\pi$
$860$		0			0
$861$	−14.1216	+	6.90392i	−0.481262	+	0.235285i
$862$		0			0
$863$			36.5879i			1.24547i	0.782435	+	0.622733i	$0.213978\pi$
$863$			36.5879i			1.24547i	−0.782435	+	0.622733i	$0.786022\pi$
$864$		0			0
$865$			38.4206i			1.30634i
$866$		0			0
$867$	5.59753	+	11.4494i	0.190102	+	0.388843i
$868$		0			0
$869$			5.17198i			0.175448i
$870$		0			0
$871$	16.4243	−	8.28105i	0.556516	−	0.280593i
$872$		0			0
$873$	42.6553	+	33.1975i	1.44366	+	1.12357i
$874$		0			0
$875$		−	13.0329i		−	0.440594i
$876$		0			0
$877$	−22.0410			−0.744271			−0.372136	−	0.928178i	$-0.621374\pi$
$877$	−22.0410			−0.744271			−0.372136	+	0.928178i	$0.621374\pi$
$878$		0			0
$879$	26.6494	−	13.0287i	0.898863	−	0.439447i
$880$		0			0
$881$		−	44.8226i		−	1.51011i	−0.655660	−	0.755056i	$-0.727609\pi$
$881$		−	44.8226i		−	1.51011i	0.655660	−	0.755056i	$-0.272391\pi$
$882$		0			0
$883$		−	49.9951i		−	1.68247i	−0.540670	−	0.841235i	$-0.681829\pi$
$883$		−	49.9951i		−	1.68247i	0.540670	−	0.841235i	$-0.318171\pi$
$884$		0			0
$885$	25.9006	−	12.6626i	0.870640	−	0.425649i
$886$		0			0
$887$			17.2072i			0.577760i	0.957365	+	0.288880i	$0.0932829\pi$
$887$			17.2072i			0.577760i	−0.957365	+	0.288880i	$0.906717\pi$
$888$		0			0
$889$		−	11.6435i		−	0.390511i
$890$		0			0
$891$	1.38026	+	5.44891i	0.0462405	+	0.182545i
$892$		0			0
$893$			9.55746i			0.319828i
$894$		0			0
$895$	−62.9604			−2.10454
$896$		0			0
$897$	2.99721	+	6.13062i	0.100074	+	0.204695i
$898$		0			0
$899$	−0.0194527			−0.000648782
$900$		0			0
$901$		−	23.5348i		−	0.784057i
$902$		0			0
$903$	−9.56099	−	19.5564i	−0.318170	−	0.650798i
$904$		0			0
$905$	−30.5132			−1.01429
$906$		0			0
$907$	28.9373			0.960847			0.480424	−	0.877037i	$-0.340483\pi$
$907$	28.9373			0.960847			0.480424	+	0.877037i	$0.340483\pi$
$908$		0			0
$909$	35.3958	−	45.4799i	1.17401	−	1.50847i
$910$		0			0
$911$			22.1909i			0.735217i	0.929981	+	0.367608i	$0.119823\pi$
$911$			22.1909i			0.735217i	−0.929981	+	0.367608i	$0.880177\pi$
$912$		0			0
$913$			5.49025i			0.181701i
$914$		0			0
$915$	−63.4113	+	31.0013i	−2.09631	+	1.02487i
$916$		0			0
$917$	−23.6472			−0.780900
$918$		0			0
$919$		−	53.6849i		−	1.77090i	−0.464733	−	0.885451i	$-0.653850\pi$
$919$		−	53.6849i		−	1.77090i	0.464733	−	0.885451i	$-0.346150\pi$
$920$		0			0
$921$	12.0307	+	24.6081i	0.396425	+	0.810864i
$922$		0			0
$923$	−12.8570			−0.423192
$924$		0			0
$925$	20.5426			0.675437
$926$		0			0
$927$	24.3503	−	31.2876i	0.799770	−	1.02762i
$928$		0			0
$929$	−16.1302			−0.529216			−0.264608	−	0.964356i	$-0.585243\pi$
$929$	−16.1302			−0.529216			−0.264608	+	0.964356i	$0.585243\pi$
$930$		0			0
$931$	23.7094			0.777045
$932$		0			0
$933$	−25.2950	−	51.7395i	−0.828122	−	1.69388i
$934$		0			0
$935$		−	6.76617i		−	0.221277i
$936$		0			0
$937$			4.53412i			0.148123i	0.997254	+	0.0740616i	$0.0235962\pi$
$937$			4.53412i			0.148123i	−0.997254	+	0.0740616i	$0.976404\pi$
$938$		0			0
$939$	−7.47237	+	3.65318i	−0.243851	+	0.119217i
$940$		0			0
$941$	15.3047			0.498918			0.249459	−	0.968385i	$-0.419747\pi$
$941$	15.3047			0.498918			0.249459	+	0.968385i	$0.419747\pi$
$942$		0			0
$943$			9.25320i			0.301326i
$944$		0			0
$945$	−6.39567	+	30.5104i	−0.208051	+	0.992503i
$946$		0			0
$947$		−	33.7123i		−	1.09550i	−0.836641	−	0.547751i	$-0.815484\pi$
$947$		−	33.7123i		−	1.09550i	0.836641	−	0.547751i	$-0.184516\pi$
$948$		0			0
$949$			13.4681i			0.437194i
$950$		0			0
$951$	−33.4138	+	16.3357i	−1.08352	+	0.529722i
$952$		0			0
$953$		−	16.9379i		−	0.548672i	−0.961634	−	0.274336i	$-0.911542\pi$
$953$		−	16.9379i		−	0.548672i	0.961634	−	0.274336i	$-0.0884581\pi$
$954$		0			0
$955$	33.2898			1.07723
$956$		0			0
$957$	0.0850532	−	0.0415818i	0.00274938	−	0.00134415i
$958$		0			0
$959$			3.18218i			0.102758i
$960$		0			0
$961$	30.9506			0.998406
$962$		0			0
$963$	−43.8526	−	34.1294i	−1.41313	−	1.09980i
$964$		0			0
$965$	45.1590			1.45372
$966$		0			0
$967$	36.8202			1.18406			0.592029	−	0.805916i	$-0.298327\pi$
$967$	36.8202			1.18406			0.592029	+	0.805916i	$0.298327\pi$
$968$		0			0
$969$	−28.3340	+	13.8523i	−0.910220	+	0.444999i
$970$		0			0
$971$		−	25.6564i		−	0.823353i	−0.911330	−	0.411677i	$-0.864943\pi$
$971$		−	25.6564i		−	0.823353i	0.911330	−	0.411677i	$-0.135057\pi$
$972$		0			0
$973$	−32.6531			−1.04681
$974$		0			0
$975$	−25.0797	+	12.2612i	−0.803192	+	0.392674i
$976$		0			0
$977$			34.3987i			1.10051i	0.834996	+	0.550256i	$0.185470\pi$
$977$			34.3987i			1.10051i	−0.834996	+	0.550256i	$0.814530\pi$
$978$		0			0
$979$		−	8.30056i		−	0.265287i
$980$		0			0
$981$	7.09547	+	5.52222i	0.226541	+	0.176311i
$982$		0			0
$983$	−4.29039			−0.136842			−0.0684210	−	0.997657i	$-0.521796\pi$
$983$	−4.29039			−0.136842			−0.0684210	+	0.997657i	$0.521796\pi$
$984$		0			0
$985$	26.4433			0.842552
$986$		0			0
$987$	2.13202	+	4.36092i	0.0678629	+	0.138809i
$988$		0			0
$989$	−12.8144			−0.407475
$990$		0			0
$991$		−	1.60311i		−	0.0509245i	−0.999676	−	0.0254623i	$-0.991894\pi$
$991$		−	1.60311i		−	0.0509245i	0.999676	−	0.0254623i	$-0.00810576\pi$
$992$		0			0
$993$	−10.1481	+	4.96131i	−0.322039	+	0.157442i
$994$		0			0
$995$	−69.8166			−2.21334
$996$		0			0
$997$	22.5548			0.714318			0.357159	−	0.934044i	$-0.383745\pi$
$997$	22.5548			0.714318			0.357159	+	0.934044i	$0.383745\pi$
$998$		0			0
$999$	−14.5658	−	3.05333i	−0.460842	−	0.0966030i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.g.c.401.12	yes	16
3.2	odd	2	inner	804.2.g.c.401.6	yes	16
67.66	odd	2	inner	804.2.g.c.401.5	✓	16
201.200	even	2	inner	804.2.g.c.401.11	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.g.c.401.5	✓	16	67.66	odd	2	inner
804.2.g.c.401.6	yes	16	3.2	odd	2	inner
804.2.g.c.401.11	yes	16	201.200	even	2	inner
804.2.g.c.401.12	yes	16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	804.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.760738 + 1.55605i) q^{3} +3.48890 q^{5} -1.71955i q^{7} +(-1.84255 + 2.36749i) q^{9} +O(q^{10})\)	\(q+(0.760738 + 1.55605i) q^{3} +3.48890 q^{5} -1.71955i q^{7} +(-1.84255 + 2.36749i) q^{9} -0.624556 q^{11} +2.24717i q^{13} +(2.65414 + 5.42888i) q^{15} +3.10515i q^{17} +5.86412 q^{19} +(2.67570 - 1.30813i) q^{21} -1.75326i q^{23} +7.17240 q^{25} +(-5.08562 - 1.06606i) q^{27} +0.0875179i q^{29} +0.222271i q^{31} +(-0.475124 - 0.971838i) q^{33} -5.99934i q^{35} +2.86412 q^{37} +(-3.49669 + 1.70950i) q^{39} -5.27770 q^{41} -7.30890i q^{43} +(-6.42848 + 8.25991i) q^{45} +1.62982i q^{47} +4.04313 q^{49} +(-4.83176 + 2.36221i) q^{51} -7.57926 q^{53} -2.17901 q^{55} +(4.46106 + 9.12484i) q^{57} -4.77090i q^{59} +11.6804i q^{61} +(4.07102 + 3.16837i) q^{63} +7.84013i q^{65} +(-3.68511 - 7.30890i) q^{67} +(2.72816 - 1.33377i) q^{69} +5.72141i q^{71} +5.99338 q^{73} +(5.45632 + 11.1606i) q^{75} +1.07396i q^{77} -8.28105i q^{79} +(-2.20999 - 8.72445i) q^{81} -8.79064i q^{83} +10.8336i q^{85} +(-0.136182 + 0.0665782i) q^{87} +13.2903i q^{89} +3.86412 q^{91} +(-0.345863 + 0.169090i) q^{93} +20.4593 q^{95} -18.0171i q^{97} +(1.15078 - 1.47863i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{9}+O(q^{10}) \) 16 * q + 2 * q^9	\( 16 q + 2 q^{9} + 18 q^{15} + 28 q^{19} - 16 q^{21} + 14 q^{33} - 20 q^{37} - 4 q^{49} - 32 q^{55} + 4 q^{67} - 16 q^{73} + 6 q^{81} - 4 q^{91} - 30 q^{93}+O(q^{100}) \) 16 * q + 2 * q^9 + 18 * q^15 + 28 * q^19 - 16 * q^21 + 14 * q^33 - 20 * q^37 - 4 * q^49 - 32 * q^55 + 4 * q^67 - 16 * q^73 + 6 * q^81 - 4 * q^91 - 30 * q^93

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