LMFDB - Embedded newform 804.2.g.c.401.16

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			401.16
Root			$1.72824 + 0.114788i$ of defining polynomial
Character	$\chi$	$=$	804.401
Dual form			804.2.g.c.401.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.72824 + 0.114788i) q^{3} +2.40477 q^{5} +3.48008i q^{7} +(2.97365 + 0.396761i) q^{9} +O(q^{10})$	\(q+(1.72824 + 0.114788i) q^{3} +2.40477 q^{5} +3.48008i q^{7} +(2.97365 + 0.396761i) q^{9} -1.00557 q^{11} +1.58879i q^{13} +(4.15602 + 0.276037i) q^{15} +3.02716i q^{17} -3.52914 q^{19} +(-0.399470 + 6.01443i) q^{21} -5.89371i q^{23} +0.782904 q^{25} +(5.09364 + 1.02704i) q^{27} +6.29577i q^{29} -10.0710i q^{31} +(-1.73787 - 0.115427i) q^{33} +8.36879i q^{35} -6.52914 q^{37} +(-0.182374 + 2.74582i) q^{39} +4.16073 q^{41} -5.62403i q^{43} +(7.15093 + 0.954119i) q^{45} -2.23364i q^{47} -5.11098 q^{49} +(-0.347480 + 5.23167i) q^{51} +10.1411 q^{53} -2.41816 q^{55} +(-6.09921 - 0.405101i) q^{57} -2.62510i q^{59} -12.1980i q^{61} +(-1.38076 + 10.3485i) q^{63} +3.82068i q^{65} +(5.94730 - 5.62403i) q^{67} +(0.676524 - 10.1858i) q^{69} -9.00170i q^{71} -0.635252 q^{73} +(1.35305 + 0.0898676i) q^{75} -3.49946i q^{77} +9.44903i q^{79} +(8.68516 + 2.35966i) q^{81} +7.80612i q^{83} +7.27961i q^{85} +(-0.722675 + 10.8806i) q^{87} +15.8389i q^{89} -5.52914 q^{91} +(1.15602 - 17.4051i) q^{93} -8.48676 q^{95} -8.49912i q^{97} +(-2.99020 - 0.398970i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.72824	+	0.114788i	0.997802	+	0.0662726i
$3$	1.72824	+	0.114788i	0.997802	+	0.0662726i
$4$		0			0
$5$	2.40477			1.07544			0.537722	−	0.843122i	$-0.319285\pi$
$5$	2.40477			1.07544			0.537722	+	0.843122i	$0.319285\pi$
$6$		0			0
$7$			3.48008i			1.31535i	0.753303	+	0.657674i	$0.228460\pi$
$7$			3.48008i			1.31535i	−0.753303	+	0.657674i	$0.771540\pi$
$8$		0			0
$9$	2.97365	+	0.396761i	0.991216	+	0.132254i
$10$		0			0
$11$	−1.00557			−0.303190			−0.151595	−	0.988443i	$-0.548441\pi$
$11$	−1.00557			−0.303190			−0.151595	+	0.988443i	$0.548441\pi$
$12$		0			0
$13$			1.58879i			0.440652i	0.975426	+	0.220326i	$0.0707122\pi$
$13$			1.58879i			0.440652i	−0.975426	+	0.220326i	$0.929288\pi$
$14$		0			0
$15$	4.15602	+	0.276037i	1.07308	+	0.0712725i
$16$		0			0
$17$			3.02716i			0.734194i	0.930183	+	0.367097i	$0.119648\pi$
$17$			3.02716i			0.734194i	−0.930183	+	0.367097i	$0.880352\pi$
$18$		0			0
$19$	−3.52914			−0.809640			−0.404820	−	0.914396i	$-0.632666\pi$
$19$	−3.52914			−0.809640			−0.404820	+	0.914396i	$0.632666\pi$
$20$		0			0
$21$	−0.399470	+	6.01443i	−0.0871715	+	1.31246i
$22$		0			0
$23$		−	5.89371i		−	1.22892i	−0.788947	−	0.614461i	$-0.789373\pi$
$23$		−	5.89371i		−	1.22892i	0.788947	−	0.614461i	$-0.210627\pi$
$24$		0			0
$25$	0.782904			0.156581
$26$		0			0
$27$	5.09364	+	1.02704i	0.980272	+	0.197654i
$28$		0			0
$29$			6.29577i			1.16909i	0.811360	+	0.584547i	$0.198728\pi$
$29$			6.29577i			1.16909i	−0.811360	+	0.584547i	$0.801272\pi$
$30$		0			0
$31$		−	10.0710i		−	1.80880i	−0.426686	−	0.904400i	$-0.640319\pi$
$31$		−	10.0710i		−	1.80880i	0.426686	−	0.904400i	$-0.359681\pi$
$32$		0			0
$33$	−1.73787	−	0.115427i	−0.302524	−	0.0200932i
$34$		0			0
$35$			8.36879i			1.41458i
$36$		0			0
$37$	−6.52914			−1.07338			−0.536692	−	0.843778i	$-0.680326\pi$
$37$	−6.52914			−1.07338			−0.536692	+	0.843778i	$0.680326\pi$
$38$		0			0
$39$	−0.182374	+	2.74582i	−0.0292032	+	0.439684i
$40$		0			0
$41$	4.16073			0.649796			0.324898	−	0.945749i	$-0.394670\pi$
$41$	4.16073			0.649796			0.324898	+	0.945749i	$0.394670\pi$
$42$		0			0
$43$		−	5.62403i		−	0.857656i	−0.903386	−	0.428828i	$-0.858927\pi$
$43$		−	5.62403i		−	0.857656i	0.903386	−	0.428828i	$-0.141073\pi$
$44$		0			0
$45$	7.15093	+	0.954119i	1.06600	+	0.142232i
$46$		0			0
$47$		−	2.23364i		−	0.325809i	−0.986642	−	0.162905i	$-0.947914\pi$
$47$		−	2.23364i		−	0.325809i	0.986642	−	0.162905i	$-0.0520863\pi$
$48$		0			0
$49$	−5.11098			−0.730140
$50$		0			0
$51$	−0.347480	+	5.23167i	−0.0486569	+	0.732580i
$52$		0			0
$53$	10.1411			1.39299			0.696496	−	0.717561i	$-0.254741\pi$
$53$	10.1411			1.39299			0.696496	+	0.717561i	$0.254741\pi$
$54$		0			0
$55$	−2.41816			−0.326064
$56$		0			0
$57$	−6.09921	−	0.405101i	−0.807860	−	0.0536570i
$58$		0			0
$59$		−	2.62510i		−	0.341759i	−0.985292	−	0.170879i	$-0.945339\pi$
$59$		−	2.62510i		−	0.341759i	0.985292	−	0.170879i	$-0.0546609\pi$
$60$		0			0
$61$		−	12.1980i		−	1.56179i	−0.624663	−	0.780895i	$-0.714764\pi$
$61$		−	12.1980i		−	1.56179i	0.624663	−	0.780895i	$-0.285236\pi$
$62$		0			0
$63$	−1.38076	+	10.3485i	−0.173960	+	1.30379i
$64$		0			0
$65$			3.82068i			0.473897i
$66$		0			0
$67$	5.94730	−	5.62403i	0.726578	−	0.687084i
$67$	5.94730	−	5.62403i	0.726578	−	0.687084i
$68$		0			0
$69$	0.676524	−	10.1858i	0.0814439	−	1.22622i
$70$		0			0
$71$		−	9.00170i		−	1.06831i	−0.845388	−	0.534153i	$-0.820631\pi$
$71$		−	9.00170i		−	1.06831i	0.845388	−	0.534153i	$-0.179369\pi$
$72$		0			0
$73$	−0.635252			−0.0743507			−0.0371753	−	0.999309i	$-0.511836\pi$
$73$	−0.635252			−0.0743507			−0.0371753	+	0.999309i	$0.511836\pi$
$74$		0			0
$75$	1.35305	+	0.0898676i	0.156236	+	0.0103770i
$76$		0			0
$77$		−	3.49946i		−	0.398800i
$78$		0			0
$79$			9.44903i			1.06310i	0.847027	+	0.531550i	$0.178390\pi$
$79$			9.44903i			1.06310i	−0.847027	+	0.531550i	$0.821610\pi$
$80$		0			0
$81$	8.68516	+	2.35966i	0.965018	+	0.262184i
$82$		0			0
$83$			7.80612i			0.856833i	0.903581	+	0.428417i	$0.140928\pi$
$83$			7.80612i			0.856833i	−0.903581	+	0.428417i	$0.859072\pi$
$84$		0			0
$85$			7.27961i			0.789585i
$86$		0			0
$87$	−0.722675	+	10.8806i	−0.0774789	+	1.16652i
$88$		0			0
$89$			15.8389i			1.67893i	0.543418	+	0.839463i	$0.317130\pi$
$89$			15.8389i			1.67893i	−0.543418	+	0.839463i	$0.682870\pi$
$90$		0			0
$91$	−5.52914			−0.579611
$92$		0			0
$93$	1.15602	−	17.4051i	0.119874	−	1.80482i
$94$		0			0
$95$	−8.48676			−0.870723
$96$		0			0
$97$		−	8.49912i		−	0.862955i	−0.902124	−	0.431478i	$-0.857992\pi$
$97$		−	8.49912i		−	0.862955i	0.902124	−	0.431478i	$-0.142008\pi$
$98$		0			0
$99$	−2.99020	−	0.398970i	−0.300527	−	0.0400980i
$100$		0			0
$101$	10.4979			1.04458			0.522290	−	0.852768i	$-0.325078\pi$
$101$	10.4979			1.04458			0.522290	+	0.852768i	$0.325078\pi$
$102$		0			0
$103$	2.32808			0.229392			0.114696	−	0.993401i	$-0.463411\pi$
$103$	2.32808			0.229392			0.114696	+	0.993401i	$0.463411\pi$
$104$		0			0
$105$	−0.960633	+	14.4633i	−0.0937481	+	1.41147i
$106$		0			0
$107$		−	8.21877i		−	0.794539i	−0.917702	−	0.397269i	$-0.869958\pi$
$107$		−	8.21877i		−	0.794539i	0.917702	−	0.397269i	$-0.130042\pi$
$108$		0			0
$109$			3.41330i			0.326934i	0.986549	+	0.163467i	$0.0522678\pi$
$109$			3.41330i			0.326934i	−0.986549	+	0.163467i	$0.947732\pi$
$110$		0			0
$111$	−11.2839	−	0.749464i	−1.07102	−	0.0711359i
$112$		0			0
$113$	−8.57663			−0.806821			−0.403411	−	0.915019i	$-0.632175\pi$
$113$	−8.57663			−0.806821			−0.403411	+	0.915019i	$0.632175\pi$
$114$		0			0
$115$		−	14.1730i		−	1.32164i
$116$		0			0
$117$	−0.630373	+	4.72452i	−0.0582780	+	0.436782i
$118$		0			0
$119$	−10.5348			−0.965720
$120$		0			0
$121$	−9.98883			−0.908076
$122$		0			0
$123$	7.19074	+	0.477599i	0.648368	+	0.0430637i
$124$		0			0
$125$	−10.1411			−0.907051
$126$		0			0
$127$	9.05757			0.803730			0.401865	−	0.915699i	$-0.368362\pi$
$127$	9.05757			0.803730			0.401865	+	0.915699i	$0.368362\pi$
$128$		0			0
$129$	0.645568	−	9.71969i	0.0568391	−	0.855771i
$130$		0			0
$131$			10.8439i			0.947434i	0.880677	+	0.473717i	$0.157088\pi$
$131$			10.8439i			0.947434i	−0.880677	+	0.473717i	$0.842912\pi$
$132$		0			0
$133$		−	12.2817i		−	1.06496i
$134$		0			0
$135$	12.2490	+	2.46979i	1.05423	+	0.212565i
$136$		0			0
$137$	12.9395			1.10550			0.552749	−	0.833348i	$-0.313579\pi$
$137$	12.9395			1.10550			0.552749	+	0.833348i	$0.313579\pi$
$138$		0			0
$139$			6.57394i			0.557594i	0.960350	+	0.278797i	$0.0899357\pi$
$139$			6.57394i			0.557594i	−0.960350	+	0.278797i	$0.910064\pi$
$140$		0			0
$141$	0.256393	−	3.86026i	0.0215922	−	0.325093i
$142$		0			0
$143$		−	1.59764i		−	0.133601i
$144$		0			0
$145$			15.1398i			1.25730i
$146$		0			0
$147$	−8.83302	−	0.586677i	−0.728535	−	0.0483883i
$148$		0			0
$149$		−	6.60639i		−	0.541217i	−0.962690	−	0.270608i	$-0.912775\pi$
$149$		−	6.60639i		−	0.541217i	0.962690	−	0.270608i	$-0.0872248\pi$
$150$		0			0
$151$	−16.3127			−1.32751			−0.663756	−	0.747949i	$-0.731039\pi$
$151$	−16.3127			−1.32751			−0.663756	+	0.747949i	$0.731039\pi$
$152$		0			0
$153$	−1.20106	+	9.00170i	−0.0970999	+	0.727745i
$154$		0			0
$155$		−	24.2183i		−	1.94526i
$156$		0			0
$157$	−3.74624			−0.298982			−0.149491	−	0.988763i	$-0.547764\pi$
$157$	−3.74624			−0.298982			−0.149491	+	0.988763i	$0.547764\pi$
$158$		0			0
$159$	17.5263	+	1.16408i	1.38993	+	0.0923172i
$160$		0			0
$161$	20.5106			1.61646
$162$		0			0
$163$	−9.98467			−0.782059			−0.391030	−	0.920378i	$-0.627881\pi$
$163$	−9.98467			−0.782059			−0.391030	+	0.920378i	$0.627881\pi$
$164$		0			0
$165$	−4.17916	−	0.277574i	−0.325347	−	0.0216091i
$166$		0			0
$167$		−	18.2438i		−	1.41175i	−0.708338	−	0.705873i	$-0.750555\pi$
$167$		−	18.2438i		−	1.41175i	0.708338	−	0.705873i	$-0.249445\pi$
$168$		0			0
$169$	10.4757			0.805825
$170$		0			0
$171$	−10.4944	−	1.40023i	−0.802528	−	0.107078i
$172$		0			0
$173$		−	6.94236i		−	0.527817i	−0.964548	−	0.263909i	$-0.914988\pi$
$173$		−	6.94236i		−	0.527817i	0.964548	−	0.263909i	$-0.0850118\pi$
$174$		0			0
$175$			2.72457i			0.205958i
$176$		0			0
$177$	0.301329	−	4.53681i	0.0226492	−	0.341007i
$178$		0			0
$179$	2.92683			0.218762			0.109381	−	0.994000i	$-0.465113\pi$
$179$	2.92683			0.218762			0.109381	+	0.994000i	$0.465113\pi$
$180$		0			0
$181$	−17.5180			−1.30210			−0.651050	−	0.759035i	$-0.725671\pi$
$181$	−17.5180			−1.30210			−0.651050	+	0.759035i	$0.725671\pi$
$182$		0			0
$183$	1.40017	−	21.0810i	0.103504	−	1.55836i
$184$		0			0
$185$	−15.7011			−1.15436
$186$		0			0
$187$		−	3.04401i		−	0.222600i
$188$		0			0
$189$	−3.57418	+	17.7263i	−0.259983	+	1.28940i
$190$		0			0
$191$	−22.6602			−1.63963			−0.819817	−	0.572626i	$-0.805925\pi$
$191$	−22.6602			−1.63963			−0.819817	+	0.572626i	$0.805925\pi$
$192$		0			0
$193$	8.84048			0.636352			0.318176	−	0.948032i	$-0.396930\pi$
$193$	8.84048			0.636352			0.318176	+	0.948032i	$0.396930\pi$
$194$		0			0
$195$	−0.438566	+	6.60307i	−0.0314064	+	0.472855i
$196$		0			0
$197$	−10.1411			−0.722526			−0.361263	−	0.932464i	$-0.617654\pi$
$197$	−10.1411			−0.722526			−0.361263	+	0.932464i	$0.617654\pi$
$198$		0			0
$199$	−10.7142			−0.759507			−0.379753	−	0.925088i	$-0.623991\pi$
$199$	−10.7142			−0.759507			−0.379753	+	0.925088i	$0.623991\pi$
$200$		0			0
$201$	10.9239	−	9.03701i	0.770515	−	0.637422i
$202$		0			0
$203$	−21.9098			−1.53777
$204$		0			0
$205$	10.0056			0.698820
$206$		0			0
$207$	2.33840	−	17.5258i	0.162530	−	1.21813i
$208$		0			0
$209$	3.54879			0.245475
$210$		0			0
$211$	−2.48690			−0.171205			−0.0856025	−	0.996329i	$-0.527282\pi$
$211$	−2.48690			−0.171205			−0.0856025	+	0.996329i	$0.527282\pi$
$212$		0			0
$213$	1.03328	−	15.5571i	0.0707994	−	1.06596i
$214$		0			0
$215$		−	13.5245i		−	0.922361i
$216$		0			0
$217$	35.0478			2.37920
$218$		0			0
$219$	−1.09787	−	0.0729190i	−0.0741872	−	0.00492741i
$220$		0			0
$221$	−4.80953			−0.323524
$222$		0			0
$223$	−24.0845			−1.61282			−0.806408	−	0.591359i	$-0.798591\pi$
$223$	−24.0845			−1.61282			−0.806408	+	0.591359i	$0.798591\pi$
$224$		0			0
$225$	2.32808	+	0.310626i	0.155205	+	0.0207084i
$226$		0			0
$227$			2.95692i			0.196257i	0.995174	+	0.0981287i	$0.0312857\pi$
$227$			2.95692i			0.196257i	−0.995174	+	0.0981287i	$0.968714\pi$
$228$		0			0
$229$			13.7369i			0.907761i	0.891063	+	0.453881i	$0.149961\pi$
$229$			13.7369i			0.907761i	−0.891063	+	0.453881i	$0.850039\pi$
$230$		0			0
$231$	0.401694	−	6.04792i	0.0264295	−	0.397924i
$232$		0			0
$233$	−7.12443			−0.466737			−0.233368	−	0.972388i	$-0.574975\pi$
$233$	−7.12443			−0.466737			−0.233368	+	0.972388i	$0.574975\pi$
$234$		0			0
$235$		−	5.37137i		−	0.350390i
$236$		0			0
$237$	−1.08463	+	16.3302i	−0.0704543	+	1.06076i
$238$		0			0
$239$	−14.3019			−0.925110			−0.462555	−	0.886591i	$-0.653067\pi$
$239$	−14.3019			−0.925110			−0.462555	+	0.886591i	$0.653067\pi$
$240$		0			0
$241$	−24.0216			−1.54737			−0.773684	−	0.633572i	$-0.781588\pi$
$241$	−24.0216			−1.54737			−0.773684	+	0.633572i	$0.781588\pi$
$242$		0			0
$243$	14.7392	+	5.07501i	0.945521	+	0.325562i
$244$		0			0
$245$	−12.2907			−0.785225
$246$		0			0
$247$		−	5.60708i		−	0.356770i
$248$		0			0
$249$	−0.896045	+	13.4909i	−0.0567846	+	0.854949i
$250$		0			0
$251$	20.2823			1.28021			0.640103	−	0.768289i	$-0.278892\pi$
$251$	20.2823			1.28021			0.640103	+	0.768289i	$0.278892\pi$
$252$		0			0
$253$			5.92652i			0.372597i
$254$		0			0
$255$	−0.835608	+	12.5809i	−0.0523278	+	0.787849i
$256$		0			0
$257$			22.7121i			1.41674i	0.705840	+	0.708372i	$0.250570\pi$
$257$			22.7121i			1.41674i	−0.705840	+	0.708372i	$0.749430\pi$
$258$		0			0
$259$		−	22.7220i		−	1.41187i
$260$		0			0
$261$	−2.49792	+	18.7214i	−0.154617	+	1.15882i
$262$		0			0
$263$			7.80612i			0.481346i	0.970606	+	0.240673i	$0.0773681\pi$
$263$			7.80612i			0.481346i	−0.970606	+	0.240673i	$0.922632\pi$
$264$		0			0
$265$	24.3871			1.49809
$266$		0			0
$267$	−1.81811	+	27.3735i	−0.111267	+	1.67523i
$268$		0			0
$269$			13.7210i			0.836585i	0.908312	+	0.418293i	$0.137371\pi$
$269$			13.7210i			0.836585i	−0.908312	+	0.418293i	$0.862629\pi$
$270$		0			0
$271$		−	2.56307i		−	0.155695i	−0.996965	−	0.0778476i	$-0.975195\pi$
$271$		−	2.56307i		−	0.155695i	0.996965	−	0.0778476i	$-0.0248047\pi$
$272$		0			0
$273$	−9.55570	−	0.634676i	−0.578337	−	0.0384123i
$274$		0			0
$275$	−0.787263			−0.0474737
$276$		0			0
$277$	−13.1525			−0.790258			−0.395129	−	0.918626i	$-0.629300\pi$
$277$	−13.1525			−0.790258			−0.395129	+	0.918626i	$0.629300\pi$
$278$		0			0
$279$	3.99577	−	29.9475i	0.239221	−	1.79291i
$280$		0			0
$281$	23.0052			1.37238			0.686188	−	0.727424i	$-0.259283\pi$
$281$	23.0052			1.37238			0.686188	+	0.727424i	$0.259283\pi$
$282$		0			0
$283$	−0.852349			−0.0506669			−0.0253334	−	0.999679i	$-0.508065\pi$
$283$	−0.852349			−0.0506669			−0.0253334	+	0.999679i	$0.508065\pi$
$284$		0			0
$285$	−14.6672	−	0.974174i	−0.868809	−	0.0577051i
$286$		0			0
$287$			14.4797i			0.854708i
$288$		0			0
$289$	7.83631			0.460960
$290$		0			0
$291$	0.975593	−	14.6885i	0.0571903	−	0.861058i
$292$		0			0
$293$		−	32.7235i		−	1.91172i	−0.293814	−	0.955862i	$-0.594925\pi$
$293$		−	32.7235i		−	1.91172i	0.293814	−	0.955862i	$-0.405075\pi$
$294$		0			0
$295$		−	6.31275i		−	0.367543i
$296$		0			0
$297$	−5.12200	−	1.03276i	−0.297209	−	0.0599266i
$298$		0			0
$299$	9.36389			0.541528
$300$		0			0
$301$	19.5721			1.12812
$302$		0			0
$303$	18.1429	+	1.20503i	1.04228	+	0.0692270i
$304$		0			0
$305$		−	29.3333i		−	1.67962i
$306$		0			0
$307$	−21.2802			−1.21453			−0.607264	−	0.794500i	$-0.707733\pi$
$307$	−21.2802			−1.21453			−0.607264	+	0.794500i	$0.707733\pi$
$308$		0			0
$309$	4.02349	+	0.267234i	0.228888	+	0.0152024i
$310$		0			0
$311$	−8.70337			−0.493523			−0.246761	−	0.969076i	$-0.579366\pi$
$311$	−8.70337			−0.493523			−0.246761	+	0.969076i	$0.579366\pi$
$312$		0			0
$313$			1.72237i			0.0973542i	0.998815	+	0.0486771i	$0.0155005\pi$
$313$			1.72237i			0.0973542i	−0.998815	+	0.0486771i	$0.984499\pi$
$314$		0			0
$315$	−3.32041	+	24.8858i	−0.187084	+	1.40216i
$316$		0			0
$317$			2.30281i			0.129339i	0.997907	+	0.0646694i	$0.0205993\pi$
$317$			2.30281i			0.129339i	−0.997907	+	0.0646694i	$0.979401\pi$
$318$		0			0
$319$		−	6.33082i		−	0.354458i
$320$		0			0
$321$	0.943413	−	14.2040i	0.0526562	−	0.792792i
$322$		0			0
$323$		−	10.6833i		−	0.594433i
$324$		0			0
$325$			1.24387i			0.0689977i
$326$		0			0
$327$	−0.391804	+	5.89900i	−0.0216668	+	0.326216i
$328$		0			0
$329$	7.77324			0.428553
$330$		0			0
$331$		−	1.75771i		−	0.0966126i	−0.998833	−	0.0483063i	$-0.984618\pi$
$331$		−	1.75771i		−	0.0966126i	0.998833	−	0.0483063i	$-0.0153824\pi$
$332$		0			0
$333$	−19.4154	−	2.59051i	−1.06396	−	0.141959i
$334$		0			0
$335$	14.3019	−	13.5245i	0.781394	−	0.738921i
$336$		0			0
$337$			32.4978i			1.77027i	0.465334	+	0.885135i	$0.345934\pi$
$337$			32.4978i			1.77027i	−0.465334	+	0.885135i	$0.654066\pi$
$338$		0			0
$339$	−14.8225	−	0.984490i	−0.805047	−	0.0534701i
$340$		0			0
$341$			10.1270i			0.548410i
$342$		0			0
$343$			6.57394i			0.354959i
$344$		0			0
$345$	1.62688	−	24.4944i	0.0875884	−	1.31873i
$346$		0			0
$347$	33.0665			1.77510			0.887552	−	0.460708i	$-0.152405\pi$
$347$	33.0665			1.77510			0.887552	+	0.460708i	$0.152405\pi$
$348$		0			0
$349$	28.6663			1.53447			0.767237	−	0.641364i	$-0.221631\pi$
$349$	28.6663			1.53447			0.767237	+	0.641364i	$0.221631\pi$
$350$		0			0
$351$	−1.63175	+	8.09275i	−0.0870965	+	0.431959i
$352$		0			0
$353$	−12.8641			−0.684687			−0.342343	−	0.939575i	$-0.611221\pi$
$353$	−12.8641			−0.684687			−0.342343	+	0.939575i	$0.611221\pi$
$354$		0			0
$355$		−	21.6470i		−	1.14890i
$356$		0			0
$357$	−18.2066	−	1.20926i	−0.963597	−	0.0640008i
$358$		0			0
$359$			17.1184i			0.903477i	0.892150	+	0.451739i	$0.149196\pi$
$359$			17.1184i			0.903477i	−0.892150	+	0.451739i	$0.850804\pi$
$360$		0			0
$361$	−6.54518			−0.344483
$362$		0			0
$363$	−17.2631	−	1.14659i	−0.906079	−	0.0601805i
$364$		0			0
$365$	−1.52763			−0.0799600
$366$		0			0
$367$			18.8981i			0.986471i	0.869896	+	0.493235i	$0.164186\pi$
$367$			18.8981i			0.986471i	−0.869896	+	0.493235i	$0.835814\pi$
$368$		0			0
$369$	12.3725	+	1.65082i	0.644088	+	0.0859380i
$370$		0			0
$371$			35.2920i			1.83227i
$372$		0			0
$373$		−	27.0000i		−	1.39801i	−0.715119	−	0.699003i	$-0.753627\pi$
$373$		−	27.0000i		−	1.39801i	0.715119	−	0.699003i	$-0.246373\pi$
$374$		0			0
$375$	−17.5263	−	1.16408i	−0.905056	−	0.0601126i
$376$		0			0
$377$	−10.0027			−0.515164
$378$		0			0
$379$			13.7259i			0.705054i	0.935802	+	0.352527i	$0.114678\pi$
$379$			13.7259i			0.705054i	−0.935802	+	0.352527i	$0.885322\pi$
$380$		0			0
$381$	15.6537	+	1.03970i	0.801963	+	0.0532653i
$382$		0			0
$383$	22.6233			1.15600			0.577999	−	0.816038i	$-0.303834\pi$
$383$	22.6233			1.15600			0.577999	+	0.816038i	$0.303834\pi$
$384$		0			0
$385$		−	8.41539i		−	0.428888i
$386$		0			0
$387$	2.23140	−	16.7239i	0.113428	−	0.850122i
$388$		0			0
$389$			21.3306i			1.08150i	0.841182	+	0.540752i	$0.181860\pi$
$389$			21.3306i			1.08150i	−0.841182	+	0.540752i	$0.818140\pi$
$390$		0			0
$391$	17.8412			0.902267
$392$		0			0
$393$	−1.24474	+	18.7408i	−0.0627889	+	0.945351i
$394$		0			0
$395$			22.7227i			1.14330i
$396$		0			0
$397$	35.3023			1.77177			0.885885	−	0.463904i	$-0.153552\pi$
$397$	35.3023			1.77177			0.885885	+	0.463904i	$0.153552\pi$
$398$		0			0
$399$	1.40979	−	21.2258i	0.0705776	−	1.06262i
$400$		0			0
$401$	11.4019			0.569383			0.284691	−	0.958619i	$-0.408109\pi$
$401$	11.4019			0.569383			0.284691	+	0.958619i	$0.408109\pi$
$402$		0			0
$403$	16.0007			0.797052
$404$		0			0
$405$	20.8858	+	5.67443i	1.03782	+	0.281964i
$406$		0			0
$407$	6.56549			0.325439
$408$		0			0
$409$		−	13.1479i		−	0.650120i	−0.945693	−	0.325060i	$-0.894615\pi$
$409$		−	13.1479i		−	0.650120i	0.945693	−	0.325060i	$-0.105385\pi$
$410$		0			0
$411$	22.3627	+	1.48530i	1.10307	+	0.0732643i
$412$		0			0
$413$	9.13557			0.449532
$414$		0			0
$415$			18.7719i			0.921476i
$416$		0			0
$417$	−0.754606	+	11.3614i	−0.0369532	+	0.556368i
$418$		0			0
$419$		−	13.1614i		−	0.642978i	−0.946913	−	0.321489i	$-0.895817\pi$
$419$		−	13.1614i		−	0.642978i	0.946913	−	0.321489i	$-0.104183\pi$
$420$		0			0
$421$	4.94589			0.241048			0.120524	−	0.992710i	$-0.461543\pi$
$421$	4.94589			0.241048			0.120524	+	0.992710i	$0.461543\pi$
$422$		0			0
$423$	0.886220	−	6.64205i	0.0430895	−	0.322947i
$424$		0			0
$425$			2.36997i			0.114961i
$426$		0			0
$427$	42.4499			2.05430
$428$		0			0
$429$	0.183389	−	2.76111i	0.00885412	−	0.133308i
$430$		0			0
$431$			35.6698i			1.71815i	0.511847	+	0.859076i	$0.328961\pi$
$431$			35.6698i			1.71815i	−0.511847	+	0.859076i	$0.671039\pi$
$432$		0			0
$433$			25.6992i			1.23502i	0.786562	+	0.617512i	$0.211859\pi$
$433$			25.6992i			1.23502i	−0.786562	+	0.617512i	$0.788141\pi$
$434$		0			0
$435$	−1.73787	+	26.1653i	−0.0833243	+	1.25453i
$436$		0			0
$437$			20.7997i			0.994985i
$438$		0			0
$439$	−38.6879			−1.84647			−0.923237	−	0.384230i	$-0.874467\pi$
$439$	−38.6879			−1.84647			−0.923237	+	0.384230i	$0.874467\pi$
$440$		0			0
$441$	−15.1983	−	2.02784i	−0.723727	−	0.0965639i
$442$		0			0
$443$	21.0310			0.999211			0.499606	−	0.866253i	$-0.333478\pi$
$443$	21.0310			0.999211			0.499606	+	0.866253i	$0.333478\pi$
$444$		0			0
$445$			38.0890i			1.80559i
$446$		0			0
$447$	0.758331	−	11.4174i	0.0358678	−	0.540027i
$448$		0			0
$449$		−	17.1427i		−	0.809015i	−0.914535	−	0.404508i	$-0.867443\pi$
$449$		−	17.1427i		−	0.809015i	0.914535	−	0.404508i	$-0.132557\pi$
$450$		0			0
$451$	−4.18389			−0.197012
$452$		0			0
$453$	−28.1924	−	1.87250i	−1.32459	−	0.0879777i
$454$		0			0
$455$	−13.2963			−0.623340
$456$		0			0
$457$	−8.76687			−0.410097			−0.205048	−	0.978752i	$-0.565735\pi$
$457$	−8.76687			−0.410097			−0.205048	+	0.978752i	$0.565735\pi$
$458$		0			0
$459$	−3.10901	+	15.4193i	−0.145116	+	0.719710i
$460$		0			0
$461$			18.4969i			0.861486i	0.902475	+	0.430743i	$0.141749\pi$
$461$			18.4969i			0.861486i	−0.902475	+	0.430743i	$0.858251\pi$
$462$		0			0
$463$			27.2710i			1.26739i	0.773582	+	0.633696i	$0.218463\pi$
$463$			27.2710i			1.26739i	−0.773582	+	0.633696i	$0.781537\pi$
$464$		0			0
$465$	2.77996	−	41.8552i	0.128918	−	1.94099i
$466$		0			0
$467$			30.3387i			1.40391i	0.712221	+	0.701955i	$0.247689\pi$
$467$			30.3387i			1.40391i	−0.712221	+	0.701955i	$0.752311\pi$
$468$		0			0
$469$	19.5721	+	20.6971i	0.903755	+	0.955703i
$470$		0			0
$471$	−6.47441	−	0.430021i	−0.298325	−	0.0198143i
$472$		0			0
$473$			5.65534i			0.260033i
$474$		0			0
$475$	−2.76298			−0.126774
$476$		0			0
$477$	30.1562	+	4.02361i	1.38076	+	0.184228i
$478$		0			0
$479$		−	32.1159i		−	1.46741i	−0.679467	−	0.733706i	$-0.737789\pi$
$479$		−	32.1159i		−	1.46741i	0.679467	−	0.733706i	$-0.262211\pi$
$480$		0			0
$481$		−	10.3735i		−	0.472989i
$482$		0			0
$483$	35.4473	+	2.35436i	1.61291	+	0.107127i
$484$		0			0
$485$		−	20.4384i		−	0.928060i
$486$		0			0
$487$		−	16.2926i		−	0.738287i	−0.929372	−	0.369143i	$-0.879651\pi$
$487$		−	16.2926i		−	0.738287i	0.929372	−	0.369143i	$-0.120349\pi$
$488$		0			0
$489$	−17.2559	−	1.14612i	−0.780340	−	0.0518291i
$490$		0			0
$491$			18.4705i			0.833562i	0.909007	+	0.416781i	$0.136842\pi$
$491$			18.4705i			0.833562i	−0.909007	+	0.416781i	$0.863158\pi$
$492$		0			0
$493$	−19.0583			−0.858342
$494$		0			0
$495$	−7.19074	−	0.959431i	−0.323200	−	0.0431232i
$496$		0			0
$497$	31.3267			1.40519
$498$		0			0
$499$			42.7473i			1.91363i	0.290697	+	0.956815i	$0.406113\pi$
$499$			42.7473i			1.91363i	−0.290697	+	0.956815i	$0.593887\pi$
$500$		0			0
$501$	2.09416	−	31.5297i	0.0935601	−	1.40864i
$502$		0			0
$503$	−32.4077			−1.44499			−0.722494	−	0.691378i	$-0.757004\pi$
$503$	−32.4077			−1.44499			−0.722494	+	0.691378i	$0.757004\pi$
$504$		0			0
$505$	25.2450			1.12339
$506$		0			0
$507$	18.1046	+	1.20248i	0.804054	+	0.0534041i
$508$		0			0
$509$		−	13.0680i		−	0.579228i	−0.957144	−	0.289614i	$-0.906473\pi$
$509$		−	13.0680i		−	0.579228i	0.957144	−	0.289614i	$-0.0935269\pi$
$510$		0			0
$511$		−	2.21073i		−	0.0977970i
$512$		0			0
$513$	−17.9762	−	3.62456i	−0.793667	−	0.160028i
$514$		0			0
$515$	5.59849			0.246699
$516$		0			0
$517$			2.24607i			0.0987821i
$518$		0			0
$519$	0.796896	−	11.9981i	0.0349798	−	0.526657i
$520$		0			0
$521$	9.74750			0.427046			0.213523	−	0.976938i	$-0.431506\pi$
$521$	9.74750			0.427046			0.213523	+	0.976938i	$0.431506\pi$
$522$		0			0
$523$	7.59858			0.332263			0.166131	−	0.986104i	$-0.446872\pi$
$523$	7.59858			0.332263			0.166131	+	0.986104i	$0.446872\pi$
$524$		0			0
$525$	−0.312747	+	4.70872i	−0.0136494	+	0.205505i
$526$		0			0
$527$	30.4864			1.32801
$528$		0			0
$529$	−11.7358			−0.510251
$530$		0			0
$531$	1.04154	−	7.80612i	0.0451989	−	0.338757i
$532$		0			0
$533$			6.61054i			0.286334i
$534$		0			0
$535$		−	19.7642i		−	0.854482i
$536$		0			0
$537$	5.05828	+	0.335964i	0.218281	+	0.0144979i
$538$		0			0
$539$	5.13944			0.221371
$540$		0			0
$541$		−	16.4590i		−	0.707629i	−0.935316	−	0.353815i	$-0.884884\pi$
$541$		−	16.4590i		−	0.707629i	0.935316	−	0.353815i	$-0.115116\pi$
$542$		0			0
$543$	−30.2753	−	2.01084i	−1.29924	−	0.0862936i
$544$		0			0
$545$			8.20818i			0.351600i
$546$		0			0
$547$			11.9269i			0.509958i	0.966947	+	0.254979i	$0.0820686\pi$
$547$			11.9269i			0.509958i	−0.966947	+	0.254979i	$0.917931\pi$
$548$		0			0
$549$	4.83968	−	36.2724i	0.206553	−	1.54807i
$550$		0			0
$551$		−	22.2186i		−	0.946546i
$552$		0			0
$553$	−32.8834			−1.39835
$554$		0			0
$555$	−27.1352	−	1.80229i	−1.15183	−	0.0765028i
$556$		0			0
$557$		−	8.04650i		−	0.340941i	−0.985363	−	0.170471i	$-0.945471\pi$
$557$		−	8.04650i		−	0.340941i	0.985363	−	0.170471i	$-0.0545288\pi$
$558$		0			0
$559$	8.93543			0.377928
$560$		0			0
$561$	0.349415	−	5.26079i	0.0147523	−	0.222111i
$562$		0			0
$563$	17.0248			0.717511			0.358755	−	0.933432i	$-0.383201\pi$
$563$	17.0248			0.717511			0.358755	+	0.933432i	$0.383201\pi$
$564$		0			0
$565$	−20.6248			−0.867691
$566$		0			0
$567$	−8.21181	+	30.2251i	−0.344863	+	1.26933i
$568$		0			0
$569$		−	18.1503i		−	0.760902i	−0.924801	−	0.380451i	$-0.875769\pi$
$569$		−	18.1503i		−	0.760902i	0.924801	−	0.380451i	$-0.124231\pi$
$570$		0			0
$571$	33.8258			1.41557			0.707784	−	0.706429i	$-0.249695\pi$
$571$	33.8258			1.41557			0.707784	+	0.706429i	$0.249695\pi$
$572$		0			0
$573$	−39.1623	−	2.60111i	−1.63603	−	0.108663i
$574$		0			0
$575$		−	4.61420i		−	0.192426i
$576$		0			0
$577$		−	39.1979i		−	1.63183i	−0.578171	−	0.815916i	$-0.696233\pi$
$577$		−	39.1979i		−	1.63183i	0.578171	−	0.815916i	$-0.303767\pi$
$578$		0			0
$579$	15.2785	+	1.01478i	0.634953	+	0.0421727i
$580$		0			0
$581$	−27.1660			−1.12703
$582$		0			0
$583$	−10.1976			−0.422341
$584$		0			0
$585$	−1.51590	+	11.3614i	−0.0626747	+	0.469734i
$586$		0			0
$587$	−2.34104			−0.0966251			−0.0483126	−	0.998832i	$-0.515384\pi$
$587$	−2.34104			−0.0966251			−0.0483126	+	0.998832i	$0.515384\pi$
$588$		0			0
$589$			35.5419i			1.46448i
$590$		0			0
$591$	−17.5263	−	1.16408i	−0.720937	−	0.0478837i
$592$		0			0
$593$	−14.6039			−0.599711			−0.299856	−	0.953985i	$-0.596939\pi$
$593$	−14.6039			−0.599711			−0.299856	+	0.953985i	$0.596939\pi$
$594$		0			0
$595$	−25.3337			−1.03858
$596$		0			0
$597$	−18.5167	−	1.22985i	−0.757837	−	0.0503345i
$598$		0			0
$599$	22.4218			0.916131			0.458066	−	0.888918i	$-0.348542\pi$
$599$	22.4218			0.916131			0.458066	+	0.888918i	$0.348542\pi$
$600$		0			0
$601$	−19.6824			−0.802860			−0.401430	−	0.915890i	$-0.631487\pi$
$601$	−19.6824			−0.802860			−0.401430	+	0.915890i	$0.631487\pi$
$602$		0			0
$603$	19.9166	−	14.3642i	0.811065	−	0.584956i
$604$		0			0
$605$	−24.0208			−0.976585
$606$		0			0
$607$	4.47227			0.181524			0.0907619	−	0.995873i	$-0.471070\pi$
$607$	4.47227			0.181524			0.0907619	+	0.995873i	$0.471070\pi$
$608$		0			0
$609$	−37.8654	−	2.51497i	−1.53439	−	0.101912i
$610$		0			0
$611$	3.54879			0.143569
$612$		0			0
$613$	24.2817			0.980727			0.490363	−	0.871518i	$-0.336864\pi$
$613$	24.2817			0.980727			0.490363	+	0.871518i	$0.336864\pi$
$614$		0			0
$615$	17.2921	+	1.14851i	0.697283	+	0.0463126i
$616$		0			0
$617$		−	7.63721i		−	0.307463i	−0.988113	−	0.153731i	$-0.950871\pi$
$617$		−	7.63721i		−	0.307463i	0.988113	−	0.153731i	$-0.0491290\pi$
$618$		0			0
$619$	20.1700			0.810699			0.405350	−	0.914162i	$-0.367150\pi$
$619$	20.1700			0.810699			0.405350	+	0.914162i	$0.367150\pi$
$620$		0			0
$621$	6.05306	−	30.0204i	0.242901	−	1.20468i
$622$		0			0
$623$	−55.1209			−2.20837
$624$		0			0
$625$	−28.3016			−1.13206
$626$		0			0
$627$	6.13317	+	0.407357i	0.244935	+	0.0162683i
$628$		0			0
$629$		−	19.7647i		−	0.788072i
$630$		0			0
$631$		−	40.9970i		−	1.63206i	−0.578007	−	0.816032i	$-0.696169\pi$
$631$		−	40.9970i		−	1.63206i	0.578007	−	0.816032i	$-0.303831\pi$
$632$		0			0
$633$	−4.29796	−	0.285465i	−0.170829	−	0.0113462i
$634$		0			0
$635$	21.7814			0.864367
$636$		0			0
$637$		−	8.12030i		−	0.321738i
$638$		0			0
$639$	3.57153	−	26.7679i	0.141287	−	1.05892i
$640$		0			0
$641$	−2.20160			−0.0869581			−0.0434790	−	0.999054i	$-0.513844\pi$
$641$	−2.20160			−0.0869581			−0.0434790	+	0.999054i	$0.513844\pi$
$642$		0			0
$643$	−30.5553			−1.20499			−0.602493	−	0.798124i	$-0.705826\pi$
$643$	−30.5553			−1.20499			−0.602493	+	0.798124i	$0.705826\pi$
$644$		0			0
$645$	1.55244	−	23.3736i	0.0611273	−	0.920334i
$646$		0			0
$647$	11.9608			0.470228			0.235114	−	0.971968i	$-0.424454\pi$
$647$	11.9608			0.470228			0.235114	+	0.971968i	$0.424454\pi$
$648$		0			0
$649$			2.63972i			0.103618i
$650$		0			0
$651$	60.5711	+	4.02305i	2.37397	+	0.157676i
$652$		0			0
$653$	−34.5043			−1.35026			−0.675128	−	0.737700i	$-0.735912\pi$
$653$	−34.5043			−1.35026			−0.675128	+	0.737700i	$0.735912\pi$
$654$		0			0
$655$			26.0770i			1.01891i
$656$		0			0
$657$	−1.88902	−	0.252044i	−0.0736976	−	0.00983316i
$658$		0			0
$659$			6.85950i			0.267208i	0.991035	+	0.133604i	$0.0426550\pi$
$659$			6.85950i			0.267208i	−0.991035	+	0.133604i	$0.957345\pi$
$660$		0			0
$661$			38.1922i			1.48551i	0.669566	+	0.742753i	$0.266480\pi$
$661$			38.1922i			1.48551i	−0.669566	+	0.742753i	$0.733520\pi$
$662$		0			0
$663$	−8.31204	−	0.552074i	−0.322813	−	0.0214408i
$664$		0			0
$665$		−	29.5346i		−	1.14530i
$666$		0			0
$667$	37.1054			1.43673
$668$		0			0
$669$	−41.6238	−	2.76460i	−1.60927	−	0.106886i
$670$		0			0
$671$			12.2659i			0.473519i
$672$		0			0
$673$		−	31.3366i		−	1.20794i	−0.797008	−	0.603968i	$-0.793585\pi$
$673$		−	31.3366i		−	1.20794i	0.797008	−	0.603968i	$-0.206415\pi$
$674$		0			0
$675$	3.98783	+	0.804072i	0.153492	+	0.0309487i
$676$		0			0
$677$	−36.2747			−1.39415			−0.697074	−	0.716999i	$-0.745515\pi$
$677$	−36.2747			−1.39415			−0.697074	+	0.716999i	$0.745515\pi$
$678$		0			0
$679$	29.5777			1.13509
$680$		0			0
$681$	−0.339417	+	5.11027i	−0.0130065	+	0.195826i
$682$		0			0
$683$	2.72296			0.104191			0.0520956	−	0.998642i	$-0.483410\pi$
$683$	2.72296			0.104191			0.0520956	+	0.998642i	$0.483410\pi$
$684$		0			0
$685$	31.1166			1.18890
$686$		0			0
$687$	−1.57683	+	23.7407i	−0.0601597	+	0.905765i
$688$		0			0
$689$			16.1122i			0.613825i
$690$		0			0
$691$	−3.92666			−0.149377			−0.0746887	−	0.997207i	$-0.523796\pi$
$691$	−3.92666			−0.149377			−0.0746887	+	0.997207i	$0.523796\pi$
$692$		0			0
$693$	1.38845	−	10.4062i	0.0527429	−	0.395297i
$694$		0			0
$695$			15.8088i			0.599661i
$696$		0			0
$697$			12.5952i			0.477076i
$698$		0			0
$699$	−12.3127	−	0.817796i	−0.465711	−	0.0309319i
$700$		0			0
$701$	−9.06182			−0.342260			−0.171130	−	0.985248i	$-0.554742\pi$
$701$	−9.06182			−0.342260			−0.171130	+	0.985248i	$0.554742\pi$
$702$		0			0
$703$	23.0422			0.869055
$704$		0			0
$705$	0.616567	−	9.28304i	0.0232212	−	0.349619i
$706$		0			0
$707$			36.5335i			1.37399i
$708$		0			0
$709$	−14.7100			−0.552445			−0.276223	−	0.961094i	$-0.589083\pi$
$709$	−14.7100			−0.552445			−0.276223	+	0.961094i	$0.589083\pi$
$710$		0			0
$711$	−3.74901	+	28.0981i	−0.140599	+	1.05376i
$712$		0			0
$713$	−59.3553			−2.22287
$714$		0			0
$715$		−	3.84195i		−	0.143681i
$716$		0			0
$717$	−24.7171	−	1.64167i	−0.923077	−	0.0613095i
$718$		0			0
$719$		−	35.0485i		−	1.30709i	−0.756888	−	0.653545i	$-0.773281\pi$
$719$		−	35.0485i		−	1.30709i	0.756888	−	0.653545i	$-0.226719\pi$
$720$		0			0
$721$			8.10191i			0.301731i
$722$		0			0
$723$	−41.5152	−	2.75738i	−1.54397	−	0.102548i
$724$		0			0
$725$			4.92898i			0.183058i
$726$		0			0
$727$		−	33.9711i		−	1.25992i	−0.776628	−	0.629960i	$-0.783071\pi$
$727$		−	33.9711i		−	1.25992i	0.776628	−	0.629960i	$-0.216929\pi$
$728$		0			0
$729$	24.8904	+	10.4627i	0.921866	+	0.387508i
$730$		0			0
$731$	17.0248			0.629686
$732$		0			0
$733$			31.9611i			1.18051i	0.807217	+	0.590255i	$0.200973\pi$
$733$			31.9611i			1.18051i	−0.807217	+	0.590255i	$0.799027\pi$
$734$		0			0
$735$	−21.2414	−	1.41082i	−0.783499	−	0.0520389i
$736$		0			0
$737$	−5.98041	+	5.65534i	−0.220291	+	0.208317i
$738$		0			0
$739$			1.52798i			0.0562075i	0.999605	+	0.0281037i	$0.00894688\pi$
$739$			1.52798i			0.0562075i	−0.999605	+	0.0281037i	$0.991053\pi$
$740$		0			0
$741$	0.643623	−	9.69039i	0.0236441	−	0.355986i
$742$		0			0
$743$		−	40.0113i		−	1.46787i	−0.679218	−	0.733936i	$-0.737681\pi$
$743$		−	40.0113i		−	1.46787i	0.679218	−	0.733936i	$-0.262319\pi$
$744$		0			0
$745$		−	15.8868i		−	0.582048i
$746$		0			0
$747$	−3.09717	+	23.2127i	−0.113319	+	0.849307i
$748$		0			0
$749$	28.6020			1.04510
$750$		0			0
$751$	15.8586			0.578689			0.289345	−	0.957225i	$-0.406563\pi$
$751$	15.8586			0.578689			0.289345	+	0.957225i	$0.406563\pi$
$752$		0			0
$753$	35.0527	+	2.32815i	1.27739	+	0.0848425i
$754$		0			0
$755$	−39.2284			−1.42767
$756$		0			0
$757$			0.529669i			0.0192512i	0.999954	+	0.00962558i	$0.00306396\pi$
$757$			0.529669i			0.0192512i	−0.999954	+	0.00962558i	$0.996936\pi$
$758$		0			0
$759$	−0.680291	+	10.2425i	−0.0246930	+	0.371778i
$760$		0			0
$761$		−	32.3498i		−	1.17268i	−0.810065	−	0.586340i	$-0.800568\pi$
$761$		−	32.3498i		−	1.17268i	0.810065	−	0.586340i	$-0.199432\pi$
$762$		0			0
$763$	−11.8786			−0.430033
$764$		0			0
$765$	−2.88827	+	21.6470i	−0.104426	+	0.782649i
$766$		0			0
$767$	4.17074			0.150597
$768$		0			0
$769$			13.9970i			0.504744i	0.967630	+	0.252372i	$0.0812107\pi$
$769$			13.9970i			0.504744i	−0.967630	+	0.252372i	$0.918789\pi$
$770$		0			0
$771$	−2.60707	+	39.2521i	−0.0938913	+	1.41363i
$772$		0			0
$773$		−	33.3659i		−	1.20009i	−0.799967	−	0.600044i	$-0.795150\pi$
$773$		−	33.3659i		−	1.20009i	0.799967	−	0.600044i	$-0.204850\pi$
$774$		0			0
$775$		−	7.88460i		−	0.283223i
$776$		0			0
$777$	2.60820	−	39.2691i	0.0935685	−	1.40877i
$778$		0			0
$779$	−14.6838			−0.526101
$780$		0			0
$781$			9.05182i			0.323900i
$782$		0			0
$783$	−6.46599	+	32.0684i	−0.231076	+	1.14603i
$784$		0			0
$785$	−9.00882			−0.321539
$786$		0			0
$787$			10.9770i			0.391288i	0.980675	+	0.195644i	$0.0626797\pi$
$787$			10.9770i			0.391288i	−0.980675	+	0.195644i	$0.937320\pi$
$788$		0			0
$789$	−0.896045	+	13.4909i	−0.0319001	+	0.480288i
$790$		0			0
$791$		−	29.8474i		−	1.06125i
$792$		0			0
$793$	19.3801			0.688206
$794$		0			0
$795$	42.1468	+	2.79933i	1.49479	+	0.0992820i
$796$		0			0
$797$		−	33.0974i		−	1.17237i	−0.810178	−	0.586185i	$-0.800629\pi$
$797$		−	33.0974i		−	1.17237i	0.810178	−	0.586185i	$-0.199371\pi$
$798$		0			0
$799$	6.76157			0.239207
$800$		0			0
$801$	−6.28428	+	47.0994i	−0.222044	+	1.66418i
$802$		0			0
$803$	0.638789			0.0225424
$804$		0			0
$805$	49.3232			1.73841
$806$		0			0
$807$	−1.57500	+	23.7133i	−0.0554427	+	0.834746i
$808$		0			0
$809$	−3.19372			−0.112285			−0.0561426	−	0.998423i	$-0.517880\pi$
$809$	−3.19372			−0.112285			−0.0561426	+	0.998423i	$0.517880\pi$
$810$		0			0
$811$			35.3729i			1.24211i	0.783766	+	0.621056i	$0.213296\pi$
$811$			35.3729i			1.24211i	−0.783766	+	0.621056i	$0.786704\pi$
$812$		0			0
$813$	0.294208	−	4.42960i	0.0103183	−	0.155353i
$814$		0			0
$815$	−24.0108			−0.841062
$816$		0			0
$817$			19.8480i			0.694393i
$818$		0			0
$819$	−16.4417	−	2.19375i	−0.574520	−	0.0766558i
$820$		0			0
$821$		−	19.0763i		−	0.665768i	−0.942968	−	0.332884i	$-0.891978\pi$
$821$		−	19.0763i		−	0.665768i	0.942968	−	0.332884i	$-0.108022\pi$
$822$		0			0
$823$	−11.8258			−0.412223			−0.206112	−	0.978528i	$-0.566081\pi$
$823$	−11.8258			−0.412223			−0.206112	+	0.978528i	$0.566081\pi$
$824$		0			0
$825$	−1.36058	−	0.0903679i	−0.0473694	−	0.00314621i
$826$		0			0
$827$		−	9.52199i		−	0.331112i	−0.986200	−	0.165556i	$-0.947058\pi$
$827$		−	9.52199i		−	0.331112i	0.986200	−	0.165556i	$-0.0529419\pi$
$828$		0			0
$829$	15.6367			0.543084			0.271542	−	0.962427i	$-0.412466\pi$
$829$	15.6367			0.543084			0.271542	+	0.962427i	$0.412466\pi$
$830$		0			0
$831$	−22.7308	−	1.50975i	−0.788521	−	0.0523725i
$832$		0			0
$833$		−	15.4718i		−	0.536065i
$834$		0			0
$835$		−	43.8721i		−	1.51826i
$836$		0			0
$837$	10.3433	−	51.2979i	0.357516	−	1.77312i
$838$		0			0
$839$		−	13.3597i		−	0.461229i	−0.973045	−	0.230614i	$-0.925926\pi$
$839$		−	13.3597i		−	0.461229i	0.973045	−	0.230614i	$-0.0740736\pi$
$840$		0			0
$841$	−10.6367			−0.366781
$842$		0			0
$843$	39.7586	+	2.64071i	1.36936	+	0.0909510i
$844$		0			0
$845$	25.1917			0.866620
$846$		0			0
$847$		−	34.7620i		−	1.19444i
$848$		0			0
$849$	−1.47307	−	0.0978390i	−0.0505555	−	0.00335783i
$850$		0			0
$851$			38.4808i			1.31911i
$852$		0			0
$853$	13.2956			0.455232			0.227616	−	0.973751i	$-0.426907\pi$
$853$	13.2956			0.455232			0.227616	+	0.973751i	$0.426907\pi$
$854$		0			0
$855$	−25.2366	−	3.36722i	−0.863074	−	0.115156i
$856$		0			0
$857$	−6.70395			−0.229002			−0.114501	−	0.993423i	$-0.536527\pi$
$857$	−6.70395			−0.229002			−0.114501	+	0.993423i	$0.536527\pi$
$858$		0			0
$859$	−11.4223			−0.389725			−0.194862	−	0.980831i	$-0.562426\pi$
$859$	−11.4223			−0.389725			−0.194862	+	0.980831i	$0.562426\pi$
$860$		0			0
$861$	−1.66209	+	25.0244i	−0.0566437	+	0.852829i
$862$		0			0
$863$		−	47.1153i		−	1.60382i	−0.597442	−	0.801912i	$-0.703816\pi$
$863$		−	47.1153i		−	1.60382i	0.597442	−	0.801912i	$-0.296184\pi$
$864$		0			0
$865$		−	16.6947i		−	0.567638i
$866$		0			0
$867$	13.5431	+	0.899511i	0.459946	+	0.0305490i
$868$		0			0
$869$		−	9.50164i		−	0.322321i
$870$		0			0
$871$	8.93543	+	9.44903i	0.302765	+	0.320168i
$872$		0			0
$873$	3.37212	−	25.2734i	0.114129	−	0.855375i
$874$		0			0
$875$		−	35.2920i		−	1.19309i
$876$		0			0
$877$	9.66733			0.326442			0.163221	−	0.986589i	$-0.447812\pi$
$877$	9.66733			0.326442			0.163221	+	0.986589i	$0.447812\pi$
$878$		0			0
$879$	3.75624	−	56.5541i	0.126695	−	1.90752i
$880$		0			0
$881$			56.1873i			1.89300i	0.322707	+	0.946499i	$0.395407\pi$
$881$			56.1873i			1.89300i	−0.322707	+	0.946499i	$0.604593\pi$
$882$		0			0
$883$			7.92106i			0.266565i	0.991078	+	0.133282i	$0.0425517\pi$
$883$			7.92106i			0.266565i	−0.991078	+	0.133282i	$0.957448\pi$
$884$		0			0
$885$	0.724625	−	10.9100i	0.0243580	−	0.366735i
$886$		0			0
$887$			17.3663i			0.583105i	0.956555	+	0.291552i	$0.0941718\pi$
$887$			17.3663i			0.583105i	−0.956555	+	0.291552i	$0.905828\pi$
$888$		0			0
$889$			31.5211i			1.05718i
$890$		0			0
$891$	−8.73352	−	2.37280i	−0.292584	−	0.0794916i
$892$		0			0
$893$			7.88281i			0.263788i
$894$		0			0
$895$	7.03835			0.235266
$896$		0			0
$897$	16.1831	+	1.07486i	0.540337	+	0.0358885i
$898$		0			0
$899$	63.4045			2.11466
$900$		0			0
$901$			30.6988i			1.02273i
$902$		0			0
$903$	33.8253	+	2.24663i	1.12564	+	0.0747632i
$904$		0			0
$905$	−42.1266			−1.40034
$906$		0			0
$907$	−42.2484			−1.40284			−0.701418	−	0.712750i	$-0.747449\pi$
$907$	−42.2484			−1.40284			−0.701418	+	0.712750i	$0.747449\pi$
$908$		0			0
$909$	31.2170	+	4.16516i	1.03540	+	0.138150i
$910$		0			0
$911$			38.6141i			1.27934i	0.768649	+	0.639671i	$0.220929\pi$
$911$			38.6141i			1.27934i	−0.768649	+	0.639671i	$0.779071\pi$
$912$		0			0
$913$		−	7.84958i		−	0.259783i
$914$		0			0
$915$	3.36709	−	50.6950i	0.111313	−	1.67592i
$916$		0			0
$917$	−37.7376			−1.24621
$918$		0			0
$919$			6.07220i			0.200303i	0.994972	+	0.100152i	$0.0319328\pi$
$919$			6.07220i			0.200303i	−0.994972	+	0.100152i	$0.968067\pi$
$920$		0			0
$921$	−36.7774	−	2.44271i	−1.21186	−	0.0804899i
$922$		0			0
$923$	14.3019			0.470751
$924$		0			0
$925$	−5.11169			−0.168071
$926$		0			0
$927$	6.92289	+	0.923692i	0.227377	+	0.0303380i
$928$		0			0
$929$	31.1838			1.02311			0.511554	−	0.859251i	$-0.329070\pi$
$929$	31.1838			1.02311			0.511554	+	0.859251i	$0.329070\pi$
$930$		0			0
$931$	18.0374			0.591151
$932$		0			0
$933$	−15.0415	−	0.999038i	−0.492438	−	0.0327070i
$934$		0			0
$935$		−	7.32014i		−	0.239394i
$936$		0			0
$937$		−	1.03365i		−	0.0337678i	−0.999857	−	0.0168839i	$-0.994625\pi$
$937$		−	1.03365i		−	0.0337678i	0.999857	−	0.0168839i	$-0.00537456\pi$
$938$		0			0
$939$	−0.197707	+	2.97668i	−0.00645192	+	0.0971402i
$940$		0			0
$941$	55.6144			1.81298			0.906489	−	0.422230i	$-0.138753\pi$
$941$	55.6144			1.81298			0.906489	+	0.422230i	$0.138753\pi$
$942$		0			0
$943$		−	24.5221i		−	0.798549i
$944$		0			0
$945$	−8.59506	+	42.6276i	−0.279597	+	1.38668i
$946$		0			0
$947$		−	45.9267i		−	1.49242i	−0.665712	−	0.746209i	$-0.731872\pi$
$947$		−	45.9267i		−	1.49242i	0.665712	−	0.746209i	$-0.268128\pi$
$948$		0			0
$949$		−	1.00929i		−	0.0327628i
$950$		0			0
$951$	−0.264334	+	3.97982i	−0.00857162	+	0.129055i
$952$		0			0
$953$			23.1738i			0.750674i	0.926888	+	0.375337i	$0.122473\pi$
$953$			23.1738i			0.750674i	−0.926888	+	0.375337i	$0.877527\pi$
$954$		0			0
$955$	−54.4925			−1.76333
$956$		0			0
$957$	0.726699	−	10.9412i	0.0234908	−	0.353679i
$958$		0			0
$959$			45.0307i			1.45412i
$960$		0			0
$961$	−70.4244			−2.27176
$962$		0			0
$963$	3.26089	−	24.4397i	0.105081	−	0.787560i
$964$		0			0
$965$	21.2593			0.684361
$966$		0			0
$967$	−56.8732			−1.82892			−0.914460	−	0.404676i	$-0.867384\pi$
$967$	−56.8732			−1.82892			−0.914460	+	0.404676i	$0.867384\pi$
$968$		0			0
$969$	1.22631	−	18.4633i	0.0393946	−	0.593126i
$970$		0			0
$971$			55.5301i			1.78205i	0.453958	+	0.891023i	$0.350011\pi$
$971$			55.5301i			1.78205i	−0.453958	+	0.891023i	$0.649989\pi$
$972$		0			0
$973$	−22.8779			−0.733430
$974$		0			0
$975$	−0.142781	+	2.14972i	−0.00457266	+	0.0688460i
$976$		0			0
$977$			32.1001i			1.02697i	0.858098	+	0.513486i	$0.171646\pi$
$977$			32.1001i			1.02697i	−0.858098	+	0.513486i	$0.828354\pi$
$978$		0			0
$979$		−	15.9271i		−	0.509033i
$980$		0			0
$981$	−1.35426	+	10.1499i	−0.0432383	+	0.324063i
$982$		0			0
$983$	−45.0273			−1.43615			−0.718074	−	0.695966i	$-0.754976\pi$
$983$	−45.0273			−1.43615			−0.718074	+	0.695966i	$0.754976\pi$
$984$		0			0
$985$	−24.3871			−0.777036
$986$		0			0
$987$	13.4340	+	0.892271i	0.427610	+	0.0284013i
$988$		0			0
$989$	−33.1464			−1.05399
$990$		0			0
$991$			14.3808i			0.456820i	0.973565	+	0.228410i	$0.0733527\pi$
$991$			14.3808i			0.456820i	−0.973565	+	0.228410i	$0.926647\pi$
$992$		0			0
$993$	0.201763	−	3.03775i	0.00640277	−	0.0964002i
$994$		0			0
$995$	−25.7651			−0.816807
$996$		0			0
$997$	−22.8426			−0.723432			−0.361716	−	0.932288i	$-0.617809\pi$
$997$	−22.8426			−0.723432			−0.361716	+	0.932288i	$0.617809\pi$
$998$		0			0
$999$	−33.2571	−	6.70567i	−1.05221	−	0.212158i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.g.c.401.16	yes	16
3.2	odd	2	inner	804.2.g.c.401.2	yes	16
67.66	odd	2	inner	804.2.g.c.401.1	✓	16
201.200	even	2	inner	804.2.g.c.401.15	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.g.c.401.1	✓	16	67.66	odd	2	inner
804.2.g.c.401.2	yes	16	3.2	odd	2	inner
804.2.g.c.401.15	yes	16	201.200	even	2	inner
804.2.g.c.401.16	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)

\(f(q)\)	\(=\)	\(q+(1.72824 + 0.114788i) q^{3} +2.40477 q^{5} +3.48008i q^{7} +(2.97365 + 0.396761i) q^{9} +O(q^{10})\)	\(q+(1.72824 + 0.114788i) q^{3} +2.40477 q^{5} +3.48008i q^{7} +(2.97365 + 0.396761i) q^{9} -1.00557 q^{11} +1.58879i q^{13} +(4.15602 + 0.276037i) q^{15} +3.02716i q^{17} -3.52914 q^{19} +(-0.399470 + 6.01443i) q^{21} -5.89371i q^{23} +0.782904 q^{25} +(5.09364 + 1.02704i) q^{27} +6.29577i q^{29} -10.0710i q^{31} +(-1.73787 - 0.115427i) q^{33} +8.36879i q^{35} -6.52914 q^{37} +(-0.182374 + 2.74582i) q^{39} +4.16073 q^{41} -5.62403i q^{43} +(7.15093 + 0.954119i) q^{45} -2.23364i q^{47} -5.11098 q^{49} +(-0.347480 + 5.23167i) q^{51} +10.1411 q^{53} -2.41816 q^{55} +(-6.09921 - 0.405101i) q^{57} -2.62510i q^{59} -12.1980i q^{61} +(-1.38076 + 10.3485i) q^{63} +3.82068i q^{65} +(5.94730 - 5.62403i) q^{67} +(0.676524 - 10.1858i) q^{69} -9.00170i q^{71} -0.635252 q^{73} +(1.35305 + 0.0898676i) q^{75} -3.49946i q^{77} +9.44903i q^{79} +(8.68516 + 2.35966i) q^{81} +7.80612i q^{83} +7.27961i q^{85} +(-0.722675 + 10.8806i) q^{87} +15.8389i q^{89} -5.52914 q^{91} +(1.15602 - 17.4051i) q^{93} -8.48676 q^{95} -8.49912i q^{97} +(-2.99020 - 0.398970i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{9}+O(q^{10}) \) 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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