LMFDB - Embedded newform 1104.2.e.e.47.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1104,2,Mod(47,1104)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1104, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1104.47");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1104 = 2^{4} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1104.e (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:	$8.81548438315$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.12960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 $ x^8 - 3x^6 + 8x^4 - 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			47.3
Root			$-1.40126 - 0.809017i$ of defining polynomial
Character	$\chi$	$=$	1104.47
Dual form			1104.2.e.e.47.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.61803 + 0.618034i) q^{3} +1.54757i q^{5} +2.42055i q^{7} +(2.23607 - 2.00000i) q^{9} +O(q^{10})$	\(q+(-1.61803 + 0.618034i) q^{3} +1.54757i q^{5} +2.42055i q^{7} +(2.23607 - 2.00000i) q^{9} +2.73205 q^{11} +5.46410 q^{13} +(-0.956449 - 2.50402i) q^{15} -4.19252i q^{17} -1.54757i q^{19} +(-1.49598 - 3.91653i) q^{21} -1.00000 q^{23} +2.60503 q^{25} +(-2.38197 + 4.61803i) q^{27} +4.47214i q^{29} +5.60503i q^{31} +(-4.42055 + 1.68850i) q^{33} -3.74597 q^{35} -0.872983 q^{37} +(-8.84110 + 3.37700i) q^{39} -4.22803i q^{41} +4.82140i q^{43} +(3.09514 + 3.46047i) q^{45} -1.60503 q^{47} +1.14093 q^{49} +(2.59112 + 6.78364i) q^{51} +11.2298i q^{53} +4.22803i q^{55} +(0.956449 + 2.50402i) q^{57} +7.70820 q^{59} +11.8012 q^{61} +(4.84110 + 5.41252i) q^{63} +8.45607i q^{65} +0.539533i q^{67} +(1.61803 - 0.618034i) q^{69} -13.3132 q^{71} -12.6742 q^{73} +(-4.21503 + 1.61000i) q^{75} +6.61307i q^{77} -10.8766i q^{79} +(1.00000 - 8.94427i) q^{81} +5.66025 q^{83} +6.48820 q^{85} +(-2.76393 - 7.23607i) q^{87} +13.2078i q^{89} +13.2261i q^{91} +(-3.46410 - 9.06914i) q^{93} +2.39497 q^{95} -12.2414 q^{97} +(6.10905 - 5.46410i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3}+O(q^{10}) $ 8 * q - 4 * q^3	$ 8 q - 4 q^{3} + 8 q^{11} + 16 q^{13} - 4 q^{15} - 16 q^{21} - 8 q^{23} - 24 q^{25} - 28 q^{27} - 4 q^{33} + 32 q^{35} + 24 q^{37} - 8 q^{39} + 24 q^{45} + 32 q^{47} - 8 q^{49} + 24 q^{51} + 4 q^{57} + 8 q^{59} + 8 q^{61} - 24 q^{63} + 4 q^{69} - 8 q^{71} + 16 q^{73} + 12 q^{75} + 8 q^{81} - 24 q^{83} - 40 q^{87} + 64 q^{95} + 56 q^{97}+O(q^{100}) $ 8 * q - 4 * q^3 + 8 * q^11 + 16 * q^13 - 4 * q^15 - 16 * q^21 - 8 * q^23 - 24 * q^25 - 28 * q^27 - 4 * q^33 + 32 * q^35 + 24 * q^37 - 8 * q^39 + 24 * q^45 + 32 * q^47 - 8 * q^49 + 24 * q^51 + 4 * q^57 + 8 * q^59 + 8 * q^61 - 24 * q^63 + 4 * q^69 - 8 * q^71 + 16 * q^73 + 12 * q^75 + 8 * q^81 - 24 * q^83 - 40 * q^87 + 64 * q^95 + 56 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1104\mathbb{Z}\right)^\times$.

$n$	$97$	$277$	$415$	$737$
$\chi(n)$	$1$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.61803	+	0.618034i	−0.934172	+	0.356822i
$3$	−1.61803	+	0.618034i	−0.934172	+	0.356822i
$4$		0			0
$5$			1.54757i			0.692093i	0.938217	+	0.346047i	$0.112476\pi$
$5$			1.54757i			0.692093i	−0.938217	+	0.346047i	$0.887524\pi$
$6$		0			0
$7$			2.42055i			0.914882i	0.889240	+	0.457441i	$0.151234\pi$
$7$			2.42055i			0.914882i	−0.889240	+	0.457441i	$0.848766\pi$
$8$		0			0
$9$	2.23607	−	2.00000i	0.745356	−	0.666667i
$10$		0			0
$11$	2.73205			0.823744			0.411872	−	0.911242i	$-0.364875\pi$
$11$	2.73205			0.823744			0.411872	+	0.911242i	$0.364875\pi$
$12$		0			0
$13$	5.46410			1.51547			0.757735	−	0.652563i	$-0.226306\pi$
$13$	5.46410			1.51547			0.757735	+	0.652563i	$0.226306\pi$
$14$		0			0
$15$	−0.956449	−	2.50402i	−0.246954	−	0.646534i
$16$		0			0
$17$		−	4.19252i		−	1.01683i	−0.861111	−	0.508417i	$-0.830231\pi$
$17$		−	4.19252i		−	1.01683i	0.861111	−	0.508417i	$-0.169769\pi$
$18$		0			0
$19$		−	1.54757i		−	0.355036i	−0.984118	−	0.177518i	$-0.943193\pi$
$19$		−	1.54757i		−	0.355036i	0.984118	−	0.177518i	$-0.0568069\pi$
$20$		0			0
$21$	−1.49598	−	3.91653i	−0.326450	−	0.854658i
$22$		0			0
$23$	−1.00000			−0.208514
$23$	−1.00000			−0.208514
$24$		0			0
$25$	2.60503			0.521007
$26$		0			0
$27$	−2.38197	+	4.61803i	−0.458410	+	0.888741i
$28$		0			0
$29$			4.47214i			0.830455i	0.909718	+	0.415227i	$0.136298\pi$
$29$			4.47214i			0.830455i	−0.909718	+	0.415227i	$0.863702\pi$
$30$		0			0
$31$			5.60503i			1.00669i	0.864084	+	0.503347i	$0.167898\pi$
$31$			5.60503i			1.00669i	−0.864084	+	0.503347i	$0.832102\pi$
$32$		0			0
$33$	−4.42055	+	1.68850i	−0.769519	+	0.293930i
$34$		0			0
$35$	−3.74597			−0.633184
$36$		0			0
$37$	−0.872983			−0.143518			−0.0717588	−	0.997422i	$-0.522861\pi$
$37$	−0.872983			−0.143518			−0.0717588	+	0.997422i	$0.522861\pi$
$38$		0			0
$39$	−8.84110	+	3.37700i	−1.41571	+	0.540753i
$40$		0			0
$41$		−	4.22803i		−	0.660308i	−0.943927	−	0.330154i	$-0.892899\pi$
$41$		−	4.22803i		−	0.660308i	0.943927	−	0.330154i	$-0.107101\pi$
$42$		0			0
$43$			4.82140i			0.735256i	0.929973	+	0.367628i	$0.119830\pi$
$43$			4.82140i			0.735256i	−0.929973	+	0.367628i	$0.880170\pi$
$44$		0			0
$45$	3.09514	+	3.46047i	0.461396	+	0.515856i
$46$		0			0
$47$	−1.60503			−0.234118			−0.117059	−	0.993125i	$-0.537347\pi$
$47$	−1.60503			−0.234118			−0.117059	+	0.993125i	$0.537347\pi$
$48$		0			0
$49$	1.14093			0.162990
$50$		0			0
$51$	2.59112	+	6.78364i	0.362829	+	0.949899i
$52$		0			0
$53$			11.2298i			1.54253i	0.636516	+	0.771264i	$0.280375\pi$
$53$			11.2298i			1.54253i	−0.636516	+	0.771264i	$0.719625\pi$
$54$		0			0
$55$			4.22803i			0.570108i
$56$		0			0
$57$	0.956449	+	2.50402i	0.126685	+	0.331665i
$58$		0			0
$59$	7.70820			1.00352			0.501761	−	0.865006i	$-0.332686\pi$
$59$	7.70820			1.00352			0.501761	+	0.865006i	$0.332686\pi$
$60$		0			0
$61$	11.8012			1.51099			0.755494	−	0.655156i	$-0.227397\pi$
$61$	11.8012			1.51099			0.755494	+	0.655156i	$0.227397\pi$
$62$		0			0
$63$	4.84110	+	5.41252i	0.609922	+	0.681913i
$64$		0			0
$65$			8.45607i			1.04885i
$66$		0			0
$67$			0.539533i			0.0659145i	0.999457	+	0.0329572i	$0.0104925\pi$
$67$			0.539533i			0.0659145i	−0.999457	+	0.0329572i	$0.989507\pi$
$68$		0			0
$69$	1.61803	−	0.618034i	0.194788	−	0.0744025i
$70$		0			0
$71$	−13.3132			−1.57999			−0.789995	−	0.613113i	$-0.789917\pi$
$71$	−13.3132			−1.57999			−0.789995	+	0.613113i	$0.789917\pi$
$72$		0			0
$73$	−12.6742			−1.48340			−0.741700	−	0.670732i	$-0.765980\pi$
$73$	−12.6742			−1.48340			−0.741700	+	0.670732i	$0.765980\pi$
$74$		0			0
$75$	−4.21503	+	1.61000i	−0.486710	+	0.185907i
$76$		0			0
$77$			6.61307i			0.753629i
$78$		0			0
$79$		−	10.8766i		−	1.22372i	−0.790968	−	0.611858i	$-0.790423\pi$
$79$		−	10.8766i		−	1.22372i	0.790968	−	0.611858i	$-0.209577\pi$
$80$		0			0
$81$	1.00000	−	8.94427i	0.111111	−	0.993808i
$82$		0			0
$83$	5.66025			0.621294			0.310647	−	0.950525i	$-0.399454\pi$
$83$	5.66025			0.621294			0.310647	+	0.950525i	$0.399454\pi$
$84$		0			0
$85$	6.48820			0.703745
$86$		0			0
$87$	−2.76393	−	7.23607i	−0.296325	−	0.775788i
$88$		0			0
$89$			13.2078i			1.40003i	0.714130	+	0.700013i	$0.246823\pi$
$89$			13.2078i			1.40003i	−0.714130	+	0.700013i	$0.753177\pi$
$90$		0			0
$91$			13.2261i			1.38648i
$92$		0			0
$93$	−3.46410	−	9.06914i	−0.359211	−	0.940426i
$94$		0			0
$95$	2.39497			0.245718
$96$		0			0
$97$	−12.2414			−1.24293			−0.621465	−	0.783442i	$-0.713462\pi$
$97$	−12.2414			−1.24293			−0.621465	+	0.783442i	$0.713462\pi$
$98$		0			0
$99$	6.10905	−	5.46410i	0.613983	−	0.549163i
$100$		0			0
$101$			0.164271i			0.0163456i	0.999967	+	0.00817280i	$0.00260151\pi$
$101$			0.164271i			0.0163456i	−0.999967	+	0.00817280i	$0.997398\pi$
$102$		0			0
$103$			5.51569i			0.543477i	0.962371	+	0.271738i	$0.0875985\pi$
$103$			5.51569i			0.543477i	−0.962371	+	0.271738i	$0.912401\pi$
$104$		0			0
$105$	6.06110	−	2.31513i	0.591503	−	0.225934i
$106$		0			0
$107$	10.0453			0.971115			0.485557	−	0.874205i	$-0.338617\pi$
$107$	10.0453			0.971115			0.485557	+	0.874205i	$0.338617\pi$
$108$		0			0
$109$	−0.487948			−0.0467370			−0.0233685	−	0.999727i	$-0.507439\pi$
$109$	−0.487948			−0.0467370			−0.0233685	+	0.999727i	$0.507439\pi$
$110$		0			0
$111$	1.41252	−	0.539533i	0.134070	−	0.0512102i
$112$		0			0
$113$		−	7.28765i		−	0.685565i	−0.939415	−	0.342782i	$-0.888631\pi$
$113$		−	7.28765i		−	0.685565i	0.939415	−	0.342782i	$-0.111369\pi$
$114$		0			0
$115$		−	1.54757i		−	0.144311i
$116$		0			0
$117$	12.2181	−	10.9282i	1.12956	−	1.01031i
$118$		0			0
$119$	10.1482			0.930284
$120$		0			0
$121$	−3.53590			−0.321445
$122$		0			0
$123$	2.61307	+	6.84110i	0.235612	+	0.616841i
$124$		0			0
$125$			11.7693i			1.05268i
$126$		0			0
$127$			7.56727i			0.671487i	0.941953	+	0.335743i	$0.108987\pi$
$127$			7.56727i			0.671487i	−0.941953	+	0.335743i	$0.891013\pi$
$128$		0			0
$129$	−2.97979	−	7.80119i	−0.262356	−	0.686856i
$130$		0			0
$131$	−10.8904			−0.951502			−0.475751	−	0.879580i	$-0.657824\pi$
$131$	−10.8904			−0.951502			−0.475751	+	0.879580i	$0.657824\pi$
$132$		0			0
$133$	3.74597			0.324817
$134$		0			0
$135$	−7.14672	−	3.68625i	−0.615092	−	0.317262i
$136$		0			0
$137$			11.0813i			0.946740i	0.880864	+	0.473370i	$0.156963\pi$
$137$			11.0813i			0.946740i	−0.880864	+	0.473370i	$0.843037\pi$
$138$		0			0
$139$			15.4641i			1.31165i	0.754914	+	0.655824i	$0.227679\pi$
$139$			15.4641i			1.31165i	−0.754914	+	0.655824i	$0.772321\pi$
$140$		0			0
$141$	2.59700	−	0.991966i	0.218707	−	0.0835386i
$142$		0			0
$143$	14.9282			1.24836
$144$		0			0
$145$	−6.92093			−0.574752
$146$		0			0
$147$	−1.84607	+	0.705135i	−0.152261	+	0.0581586i
$148$		0			0
$149$		−	9.37337i		−	0.767896i	−0.923355	−	0.383948i	$-0.874564\pi$
$149$		−	9.37337i		−	0.767896i	0.923355	−	0.383948i	$-0.125436\pi$
$150$		0			0
$151$		−	17.5340i		−	1.42690i	−0.700708	−	0.713448i	$-0.747132\pi$
$151$		−	17.5340i		−	1.42690i	0.700708	−	0.713448i	$-0.252868\pi$
$152$		0			0
$153$	−8.38503	−	9.37475i	−0.677890	−	0.757904i
$154$		0			0
$155$	−8.67417			−0.696726
$156$		0			0
$157$	−17.5472			−1.40042			−0.700208	−	0.713939i	$-0.746909\pi$
$157$	−17.5472			−1.40042			−0.700208	+	0.713939i	$0.746909\pi$
$158$		0			0
$159$	−6.94038	−	18.1702i	−0.550408	−	1.44099i
$160$		0			0
$161$		−	2.42055i		−	0.190766i
$162$		0			0
$163$		−	18.5592i		−	1.45367i	−0.686811	−	0.726836i	$-0.740990\pi$
$163$		−	18.5592i		−	1.45367i	0.686811	−	0.726836i	$-0.259010\pi$
$164$		0			0
$165$	−2.61307	−	6.84110i	−0.203427	−	0.532579i
$166$		0			0
$167$	18.6669			1.44449			0.722244	−	0.691638i	$-0.243111\pi$
$167$	18.6669			1.44449			0.722244	+	0.691638i	$0.243111\pi$
$168$		0			0
$169$	16.8564			1.29665
$170$		0			0
$171$	−3.09514	−	3.46047i	−0.236691	−	0.264628i
$172$		0			0
$173$			16.4721i			1.25235i	0.779681	+	0.626177i	$0.215381\pi$
$173$			16.4721i			1.25235i	−0.779681	+	0.626177i	$0.784619\pi$
$174$		0			0
$175$			6.30562i			0.476660i
$176$		0			0
$177$	−12.4721	+	4.76393i	−0.937463	+	0.358079i
$178$		0			0
$179$	−1.96224			−0.146664			−0.0733322	−	0.997308i	$-0.523363\pi$
$179$	−1.96224			−0.146664			−0.0733322	+	0.997308i	$0.523363\pi$
$180$		0			0
$181$	9.25802			0.688143			0.344072	−	0.938943i	$-0.388194\pi$
$181$	9.25802			0.688143			0.344072	+	0.938943i	$0.388194\pi$
$182$		0			0
$183$	−19.0947	+	7.29353i	−1.41152	+	0.539154i
$184$		0			0
$185$		−	1.35100i		−	0.0993276i
$186$		0			0
$187$		−	11.4542i		−	0.837612i
$188$		0			0
$189$	−11.1782	−	5.76567i	−0.813093	−	0.419391i
$190$		0			0
$191$	4.28187			0.309825			0.154912	−	0.987928i	$-0.450490\pi$
$191$	4.28187			0.309825			0.154912	+	0.987928i	$0.450490\pi$
$192$		0			0
$193$	3.35100			0.241210			0.120605	−	0.992701i	$-0.461517\pi$
$193$	3.35100			0.241210			0.120605	+	0.992701i	$0.461517\pi$
$194$		0			0
$195$	−5.22614	−	13.6822i	−0.374251	−	0.979803i
$196$		0			0
$197$		−	3.66880i		−	0.261391i	−0.991423	−	0.130695i	$-0.958279\pi$
$197$		−	3.66880i		−	0.261391i	0.991423	−	0.130695i	$-0.0417210\pi$
$198$		0			0
$199$		−	17.9125i		−	1.26978i	−0.772602	−	0.634891i	$-0.781045\pi$
$199$		−	17.9125i		−	1.26978i	0.772602	−	0.634891i	$-0.218955\pi$
$200$		0			0
$201$	−0.333450	−	0.872983i	−0.0235197	−	0.0615755i
$202$		0			0
$203$	−10.8250			−0.759768
$204$		0			0
$205$	6.54317			0.456995
$206$		0			0
$207$	−2.23607	+	2.00000i	−0.155417	+	0.139010i
$208$		0			0
$209$		−	4.22803i		−	0.292459i
$210$		0			0
$211$		−	21.8071i		−	1.50126i	−0.660723	−	0.750630i	$-0.729750\pi$
$211$		−	21.8071i		−	1.50126i	0.660723	−	0.750630i	$-0.270250\pi$
$212$		0			0
$213$	21.5413	−	8.22803i	1.47598	−	0.563776i
$214$		0			0
$215$	−7.46144			−0.508866
$216$		0			0
$217$	−13.5673			−0.921006
$218$		0			0
$219$	20.5072	−	7.83307i	1.38575	−	0.529310i
$220$		0			0
$221$		−	22.9083i		−	1.54098i
$222$		0			0
$223$			9.32766i			0.624626i	0.949979	+	0.312313i	$0.101104\pi$
$223$			9.32766i			0.624626i	−0.949979	+	0.312313i	$0.898896\pi$
$224$		0			0
$225$	5.82503	−	5.21007i	0.388336	−	0.347338i
$226$		0			0
$227$	−26.1557			−1.73602			−0.868008	−	0.496550i	$-0.834600\pi$
$227$	−26.1557			−1.73602			−0.868008	+	0.496550i	$0.834600\pi$
$228$		0			0
$229$	−1.05168			−0.0694969			−0.0347484	−	0.999396i	$-0.511063\pi$
$229$	−1.05168			−0.0694969			−0.0347484	+	0.999396i	$0.511063\pi$
$230$		0			0
$231$	−4.08710	−	10.7002i	−0.268912	−	0.704019i
$232$		0			0
$233$		−	5.80973i		−	0.380608i	−0.981725	−	0.190304i	$-0.939053\pi$
$233$		−	5.80973i		−	0.380608i	0.981725	−	0.190304i	$-0.0609474\pi$
$234$		0			0
$235$		−	2.48390i		−	0.162032i
$236$		0			0
$237$	6.72212	+	17.5987i	0.436649	+	1.14316i
$238$		0			0
$239$	23.2101			1.50133			0.750667	−	0.660680i	$-0.229732\pi$
$239$	23.2101			1.50133			0.750667	+	0.660680i	$0.229732\pi$
$240$		0			0
$241$	−11.9523			−0.769916			−0.384958	−	0.922934i	$-0.625784\pi$
$241$	−11.9523			−0.769916			−0.384958	+	0.922934i	$0.625784\pi$
$242$		0			0
$243$	3.90983	+	15.0902i	0.250816	+	0.968035i
$244$		0			0
$245$			1.76567i			0.112805i
$246$		0			0
$247$		−	8.45607i		−	0.538047i
$248$		0			0
$249$	−9.15848	+	3.49823i	−0.580395	+	0.221691i
$250$		0			0
$251$	17.5571			1.10819			0.554097	−	0.832452i	$-0.313064\pi$
$251$	17.5571			1.10819			0.554097	+	0.832452i	$0.313064\pi$
$252$		0			0
$253$	−2.73205			−0.171763
$254$		0			0
$255$	−10.4981	+	4.00993i	−0.657419	+	0.251112i
$256$		0			0
$257$			8.02169i			0.500380i	0.968197	+	0.250190i	$0.0804930\pi$
$257$			8.02169i			0.500380i	−0.968197	+	0.250190i	$0.919507\pi$
$258$		0			0
$259$		−	2.11310i		−	0.131302i
$260$		0			0
$261$	8.94427	+	10.0000i	0.553637	+	0.618984i
$262$		0			0
$263$	21.0665			1.29901			0.649507	−	0.760355i	$-0.274975\pi$
$263$	21.0665			1.29901			0.649507	+	0.760355i	$0.274975\pi$
$264$		0			0
$265$	−17.3788			−1.06757
$266$		0			0
$267$	−8.16288	−	21.3707i	−0.499560	−	1.30787i
$268$		0			0
$269$			8.58093i			0.523189i	0.965178	+	0.261594i	$0.0842482\pi$
$269$			8.58093i			0.523189i	−0.965178	+	0.261594i	$0.915752\pi$
$270$		0			0
$271$		−	21.5313i		−	1.30794i	−0.756522	−	0.653968i	$-0.773103\pi$
$271$		−	21.5313i		−	1.30794i	0.756522	−	0.653968i	$-0.226897\pi$
$272$		0			0
$273$	−8.17420	−	21.4003i	−0.494725	−	1.29521i
$274$		0			0
$275$	7.11709			0.429176
$276$		0			0
$277$	24.0278			1.44369			0.721846	−	0.692053i	$-0.243294\pi$
$277$	24.0278			1.44369			0.721846	+	0.692053i	$0.243294\pi$
$278$		0			0
$279$	11.2101	+	12.5332i	0.671129	+	0.750345i
$280$		0			0
$281$		−	28.1799i		−	1.68107i	−0.541755	−	0.840537i	$-0.682240\pi$
$281$		−	28.1799i		−	1.68107i	0.541755	−	0.840537i	$-0.317760\pi$
$282$		0			0
$283$		−	4.53226i		−	0.269415i	−0.990885	−	0.134708i	$-0.956991\pi$
$283$		−	4.53226i		−	0.269415i	0.990885	−	0.134708i	$-0.0430095\pi$
$284$		0			0
$285$	−3.87514	+	1.48017i	−0.229543	+	0.0876777i
$286$		0			0
$287$	10.2342			0.604104
$288$		0			0
$289$	−0.577202			−0.0339531
$290$		0			0
$291$	19.8071	−	7.56563i	1.16111	−	0.443505i
$292$		0			0
$293$			29.5277i			1.72503i	0.506034	+	0.862513i	$0.331111\pi$
$293$			29.5277i			1.72503i	−0.506034	+	0.862513i	$0.668889\pi$
$294$		0			0
$295$			11.9290i			0.694531i
$296$		0			0
$297$	−6.50765	+	12.6167i	−0.377612	+	0.732095i
$298$		0			0
$299$	−5.46410			−0.315997
$300$		0			0
$301$	−11.6704			−0.672673
$302$		0			0
$303$	−0.101525	−	0.265796i	−0.00583247	−	0.0152696i
$304$		0			0
$305$			18.2631i			1.04574i
$306$		0			0
$307$		−	2.27737i		−	0.129976i	−0.997886	−	0.0649882i	$-0.979299\pi$
$307$		−	2.27737i		−	0.129976i	0.997886	−	0.0649882i	$-0.0207010\pi$
$308$		0			0
$309$	−3.40888	−	8.92457i	−0.193925	−	0.507701i
$310$		0			0
$311$	12.1964			0.691595			0.345797	−	0.938309i	$-0.387608\pi$
$311$	12.1964			0.691595			0.345797	+	0.938309i	$0.387608\pi$
$312$		0			0
$313$	−28.4478			−1.60796			−0.803982	−	0.594654i	$-0.797289\pi$
$313$	−28.4478			−1.60796			−0.803982	+	0.594654i	$0.797289\pi$
$314$		0			0
$315$	−8.37624	+	7.49193i	−0.471947	+	0.422123i
$316$		0			0
$317$		−	23.4003i		−	1.31429i	−0.753762	−	0.657147i	$-0.771763\pi$
$317$		−	23.4003i		−	1.31429i	0.753762	−	0.657147i	$-0.228237\pi$
$318$		0			0
$319$			12.2181i			0.684082i
$320$		0			0
$321$	−16.2536	+	6.20833i	−0.907189	+	0.346515i
$322$		0			0
$323$	−6.48820			−0.361013
$324$		0			0
$325$	14.2342			0.789570
$326$		0			0
$327$	0.789517	−	0.301569i	0.0436604	−	0.0166768i
$328$		0			0
$329$		−	3.88507i		−	0.214191i
$330$		0			0
$331$		−	32.2792i		−	1.77423i	−0.461553	−	0.887113i	$-0.652708\pi$
$331$		−	32.2792i		−	1.77423i	0.461553	−	0.887113i	$-0.347292\pi$
$332$		0			0
$333$	−1.95205	+	1.74597i	−0.106972	+	0.0956784i
$334$		0			0
$335$	−0.834964			−0.0456190
$336$		0			0
$337$	−20.2414			−1.10262			−0.551311	−	0.834300i	$-0.685872\pi$
$337$	−20.2414			−1.10262			−0.551311	+	0.834300i	$0.685872\pi$
$338$		0			0
$339$	4.50402	+	11.7917i	0.244625	+	0.640436i
$340$		0			0
$341$			15.3132i			0.829258i
$342$		0			0
$343$			19.7055i			1.06400i
$344$		0			0
$345$	0.956449	+	2.50402i	0.0514935	+	0.134812i
$346$		0			0
$347$	20.6642			1.10931			0.554657	−	0.832079i	$-0.312849\pi$
$347$	20.6642			1.10931			0.554657	+	0.832079i	$0.312849\pi$
$348$		0			0
$349$	−14.3446			−0.767849			−0.383925	−	0.923364i	$-0.625428\pi$
$349$	−14.3446			−0.767849			−0.383925	+	0.923364i	$0.625428\pi$
$350$		0			0
$351$	−13.0153	+	25.2334i	−0.694706	+	1.34686i
$352$		0			0
$353$		−	0.559237i		−	0.0297652i	−0.999889	−	0.0148826i	$-0.995263\pi$
$353$		−	0.559237i		−	0.0297652i	0.999889	−	0.0148826i	$-0.00473745\pi$
$354$		0			0
$355$		−	20.6031i		−	1.09350i
$356$		0			0
$357$	−16.4201	+	6.27193i	−0.869046	+	0.331946i
$358$		0			0
$359$	4.97590			0.262618			0.131309	−	0.991342i	$-0.458082\pi$
$359$	4.97590			0.262618			0.131309	+	0.991342i	$0.458082\pi$
$360$		0			0
$361$	16.6050			0.873949
$362$		0			0
$363$	5.72120	−	2.18531i	0.300285	−	0.114699i
$364$		0			0
$365$		−	19.6141i		−	1.02665i
$366$		0			0
$367$			13.6306i			0.711513i	0.934579	+	0.355756i	$0.115777\pi$
$367$			13.6306i			0.711513i	−0.934579	+	0.355756i	$0.884223\pi$
$368$		0			0
$369$	−8.45607	−	9.45417i	−0.440205	−	0.492164i
$370$		0			0
$371$	−27.1822			−1.41123
$372$		0			0
$373$	9.25802			0.479362			0.239681	−	0.970852i	$-0.422957\pi$
$373$	9.25802			0.479362			0.239681	+	0.970852i	$0.422957\pi$
$374$		0			0
$375$	−7.27383	−	19.0431i	−0.375619	−	0.983383i
$376$		0			0
$377$			24.4362i			1.25853i
$378$		0			0
$379$			8.99560i			0.462073i	0.972945	+	0.231036i	$0.0742117\pi$
$379$			8.99560i			0.462073i	−0.972945	+	0.231036i	$0.925788\pi$
$380$		0			0
$381$	−4.67683	−	12.2441i	−0.239601	−	0.627284i
$382$		0			0
$383$	−1.92466			−0.0983456			−0.0491728	−	0.998790i	$-0.515659\pi$
$383$	−1.92466			−0.0983456			−0.0491728	+	0.998790i	$0.515659\pi$
$384$		0			0
$385$	−10.2342			−0.521582
$386$		0			0
$387$	9.64280	+	10.7810i	0.490171	+	0.548028i
$388$		0			0
$389$			28.5197i			1.44600i	0.690846	+	0.723002i	$0.257238\pi$
$389$			28.5197i			1.44600i	−0.690846	+	0.723002i	$0.742762\pi$
$390$		0			0
$391$			4.19252i			0.212025i
$392$		0			0
$393$	17.6211	−	6.73066i	0.888867	−	0.339517i
$394$		0			0
$395$	16.8323			0.846925
$396$		0			0
$397$	3.85641			0.193547			0.0967737	−	0.995306i	$-0.469148\pi$
$397$	3.85641			0.193547			0.0967737	+	0.995306i	$0.469148\pi$
$398$		0			0
$399$	−6.06110	+	2.31513i	−0.303435	+	0.115902i
$400$		0			0
$401$		−	13.9267i		−	0.695467i	−0.937593	−	0.347734i	$-0.886951\pi$
$401$		−	13.9267i		−	0.695467i	0.937593	−	0.347734i	$-0.113049\pi$
$402$		0			0
$403$			30.6265i			1.52561i
$404$		0			0
$405$	13.8419	+	1.54757i	0.687808	+	0.0768993i
$406$		0			0
$407$	−2.38503			−0.118222
$408$		0			0
$409$	36.1211			1.78607			0.893036	−	0.449985i	$-0.148571\pi$
$409$	36.1211			1.78607			0.893036	+	0.449985i	$0.148571\pi$
$410$		0			0
$411$	−6.84863	−	17.9299i	−0.337818	−	0.884419i
$412$		0			0
$413$			18.6581i			0.918105i
$414$		0			0
$415$			8.75963i			0.429993i
$416$		0			0
$417$	−9.55734	−	25.0214i	−0.468025	−	1.22531i
$418$		0			0
$419$	−27.6808			−1.35230			−0.676148	−	0.736766i	$-0.736352\pi$
$419$	−27.6808			−1.35230			−0.676148	+	0.736766i	$0.736352\pi$
$420$		0			0
$421$	2.16566			0.105548			0.0527739	−	0.998606i	$-0.483194\pi$
$421$	2.16566			0.105548			0.0527739	+	0.998606i	$0.483194\pi$
$422$		0			0
$423$	−3.58897	+	3.21007i	−0.174501	+	0.156079i
$424$		0			0
$425$		−	10.9217i		−	0.529778i
$426$		0			0
$427$			28.5654i			1.38238i
$428$		0			0
$429$	−24.1543	+	9.22614i	−1.16618	+	0.445442i
$430$		0			0
$431$	1.35366			0.0652036			0.0326018	−	0.999468i	$-0.489621\pi$
$431$	1.35366			0.0652036			0.0326018	+	0.999468i	$0.489621\pi$
$432$		0			0
$433$	25.9119			1.24525			0.622623	−	0.782522i	$-0.286067\pi$
$433$	25.9119			1.24525			0.622623	+	0.782522i	$0.286067\pi$
$434$		0			0
$435$	11.1983	−	4.27737i	0.536918	−	0.205084i
$436$		0			0
$437$			1.54757i			0.0740302i
$438$		0			0
$439$		−	3.95231i		−	0.188633i	−0.995542	−	0.0943166i	$-0.969933\pi$
$439$		−	3.95231i		−	0.188633i	0.995542	−	0.0943166i	$-0.0300666\pi$
$440$		0			0
$441$	2.55120	−	2.28187i	0.121486	−	0.108660i
$442$		0			0
$443$	−25.2001			−1.19729			−0.598647	−	0.801013i	$-0.704295\pi$
$443$	−25.2001			−1.19729			−0.598647	+	0.801013i	$0.704295\pi$
$444$		0			0
$445$	−20.4400			−0.968949
$446$		0			0
$447$	5.79306	+	15.1664i	0.274002	+	0.717347i
$448$		0			0
$449$			33.5764i			1.58457i	0.610153	+	0.792284i	$0.291108\pi$
$449$			33.5764i			1.58457i	−0.610153	+	0.792284i	$0.708892\pi$
$450$		0			0
$451$		−	11.5512i		−	0.543925i
$452$		0			0
$453$	10.8366	+	28.3706i	0.509148	+	1.33297i
$454$		0			0
$455$	−20.4683			−0.959571
$456$		0			0
$457$	31.7261			1.48408			0.742042	−	0.670353i	$-0.233857\pi$
$457$	31.7261			1.48408			0.742042	+	0.670353i	$0.233857\pi$
$458$		0			0
$459$	19.3612	+	9.98643i	0.903703	+	0.466127i
$460$		0			0
$461$		−	37.6795i		−	1.75491i	−0.479658	−	0.877456i	$-0.659239\pi$
$461$		−	37.6795i		−	1.75491i	0.479658	−	0.877456i	$-0.340761\pi$
$462$		0			0
$463$		−	17.8087i		−	0.827641i	−0.910358	−	0.413821i	$-0.864194\pi$
$463$		−	17.8087i		−	0.827641i	0.910358	−	0.413821i	$-0.135806\pi$
$464$		0			0
$465$	14.0351	−	5.36093i	0.650862	−	0.248607i
$466$		0			0
$467$	18.9384			0.876364			0.438182	−	0.898886i	$-0.355623\pi$
$467$	18.9384			0.876364			0.438182	+	0.898886i	$0.355623\pi$
$468$		0			0
$469$	−1.30597			−0.0603040
$470$		0			0
$471$	28.3919	−	10.8447i	1.30823	−	0.499699i
$472$		0			0
$473$			13.1723i			0.605663i
$474$		0			0
$475$		−	4.03147i		−	0.184976i
$476$		0			0
$477$	22.4595	+	25.1105i	1.02835	+	1.14973i
$478$		0			0
$479$	−0.178695			−0.00816480			−0.00408240	−	0.999992i	$-0.501299\pi$
$479$	−0.178695			−0.00816480			−0.00408240	+	0.999992i	$0.501299\pi$
$480$		0			0
$481$	−4.77007			−0.217496
$482$		0			0
$483$	1.49598	+	3.91653i	0.0680696	+	0.178208i
$484$		0			0
$485$		−	18.9445i		−	0.860224i
$486$		0			0
$487$		−	10.0118i		−	0.453676i	−0.973932	−	0.226838i	$-0.927161\pi$
$487$		−	10.0118i		−	0.453676i	0.973932	−	0.226838i	$-0.0728388\pi$
$488$		0			0
$489$	11.4702	+	30.0295i	0.518702	+	1.35798i
$490$		0			0
$491$	−32.1005			−1.44868			−0.724338	−	0.689445i	$-0.757854\pi$
$491$	−32.1005			−1.44868			−0.724338	+	0.689445i	$0.757854\pi$
$492$		0			0
$493$	18.7495			0.844435
$494$		0			0
$495$	8.45607	+	9.45417i	0.380072	+	0.424933i
$496$		0			0
$497$		−	32.2254i		−	1.44551i
$498$		0			0
$499$			8.82503i			0.395063i	0.980297	+	0.197531i	$0.0632924\pi$
$499$			8.82503i			0.395063i	−0.980297	+	0.197531i	$0.936708\pi$
$500$		0			0
$501$	−30.2037	+	11.5368i	−1.34940	+	0.515425i
$502$		0			0
$503$	−36.0629			−1.60797			−0.803983	−	0.594652i	$-0.797290\pi$
$503$	−36.0629			−1.60797			−0.803983	+	0.594652i	$0.797290\pi$
$504$		0			0
$505$	−0.254221			−0.0113127
$506$		0			0
$507$	−27.2742	+	10.4178i	−1.21129	+	0.462672i
$508$		0			0
$509$		−	13.6661i		−	0.605741i	−0.953032	−	0.302870i	$-0.902055\pi$
$509$		−	13.6661i		−	0.605741i	0.953032	−	0.302870i	$-0.0979449\pi$
$510$		0			0
$511$		−	30.6785i		−	1.35714i
$512$		0			0
$513$	7.14672	+	3.68625i	0.315535	+	0.162752i
$514$		0			0
$515$	−8.53590			−0.376137
$516$		0			0
$517$	−4.38503			−0.192854
$518$		0			0
$519$	−10.1803	−	26.6525i	−0.446867	−	1.16991i
$520$		0			0
$521$			24.7984i			1.08644i	0.839591	+	0.543220i	$0.182795\pi$
$521$			24.7984i			1.08644i	−0.839591	+	0.543220i	$0.817205\pi$
$522$		0			0
$523$			19.3447i			0.845885i	0.906157	+	0.422942i	$0.139003\pi$
$523$			19.3447i			0.845885i	−0.906157	+	0.422942i	$0.860997\pi$
$524$		0			0
$525$	−3.89709	−	10.2027i	−0.170083	−	0.445283i
$526$		0			0
$527$	23.4992			1.02364
$528$		0			0
$529$	1.00000			0.0434783
$530$		0			0
$531$	17.2361	−	15.4164i	0.747982	−	0.669015i
$532$		0			0
$533$		−	23.1024i		−	1.00068i
$534$		0			0
$535$			15.5458i			0.672102i
$536$		0			0
$537$	3.17497	−	1.21273i	0.137010	−	0.0523331i
$538$		0			0
$539$	3.11709			0.134262
$540$		0			0
$541$	−17.0519			−0.733120			−0.366560	−	0.930394i	$-0.619465\pi$
$541$	−17.0519			−0.733120			−0.366560	+	0.930394i	$0.619465\pi$
$542$		0			0
$543$	−14.9798	+	5.72177i	−0.642844	+	0.245545i
$544$		0			0
$545$		−	0.755133i		−	0.0323464i
$546$		0			0
$547$			27.9973i			1.19708i	0.801093	+	0.598540i	$0.204252\pi$
$547$			27.9973i			1.19708i	−0.801093	+	0.598540i	$0.795748\pi$
$548$		0			0
$549$	26.3883	−	23.6024i	1.12622	−	1.00732i
$550$		0			0
$551$	6.92093			0.294842
$552$		0			0
$553$	26.3274			1.11956
$554$		0			0
$555$	0.834964	+	2.18597i	0.0354423	+	0.0927891i
$556$		0			0
$557$		−	18.9479i		−	0.802849i	−0.915892	−	0.401424i	$-0.868515\pi$
$557$		−	18.9479i		−	0.802849i	0.915892	−	0.401424i	$-0.131485\pi$
$558$		0			0
$559$			26.3446i			1.11426i
$560$		0			0
$561$	7.07907	+	18.5332i	0.298878	+	0.782474i
$562$		0			0
$563$	−35.3990			−1.49189			−0.745944	−	0.666009i	$-0.768001\pi$
$563$	−35.3990			−1.49189			−0.745944	+	0.666009i	$0.768001\pi$
$564$		0			0
$565$	11.2781			0.474475
$566$		0			0
$567$	21.6501	+	2.42055i	0.909217	+	0.101654i
$568$		0			0
$569$		−	36.4179i		−	1.52672i	−0.645975	−	0.763359i	$-0.723549\pi$
$569$		−	36.4179i		−	1.52672i	0.645975	−	0.763359i	$-0.276451\pi$
$570$		0			0
$571$		−	8.88516i		−	0.371832i	−0.982566	−	0.185916i	$-0.940475\pi$
$571$		−	8.88516i		−	0.371832i	0.982566	−	0.185916i	$-0.0595253\pi$
$572$		0			0
$573$	−6.92820	+	2.64634i	−0.289430	+	0.110552i
$574$		0			0
$575$	−2.60503			−0.108637
$576$		0			0
$577$	−25.3859			−1.05683			−0.528415	−	0.848986i	$-0.677213\pi$
$577$	−25.3859			−1.05683			−0.528415	+	0.848986i	$0.677213\pi$
$578$		0			0
$579$	−5.42203	+	2.07103i	−0.225332	+	0.0860692i
$580$		0			0
$581$			13.7009i			0.568411i
$582$		0			0
$583$			30.6803i			1.27065i
$584$		0			0
$585$	16.9121	+	18.9083i	0.699231	+	0.781764i
$586$		0			0
$587$	−7.83142			−0.323237			−0.161619	−	0.986853i	$-0.551671\pi$
$587$	−7.83142			−0.323237			−0.161619	+	0.986853i	$0.551671\pi$
$588$		0			0
$589$	8.67417			0.357413
$590$		0			0
$591$	2.26744	+	5.93624i	0.0932701	+	0.244184i
$592$		0			0
$593$		−	44.1562i		−	1.81328i	−0.421909	−	0.906638i	$-0.638640\pi$
$593$		−	44.1562i		−	1.81328i	0.421909	−	0.906638i	$-0.361360\pi$
$594$		0			0
$595$			15.7050i			0.643843i
$596$		0			0
$597$	11.0705	+	28.9830i	0.453086	+	1.18620i
$598$		0			0
$599$	−32.7091			−1.33646			−0.668228	−	0.743956i	$-0.732947\pi$
$599$	−32.7091			−1.33646			−0.668228	+	0.743956i	$0.732947\pi$
$600$		0			0
$601$	4.16876			0.170047			0.0850237	−	0.996379i	$-0.472903\pi$
$601$	4.16876			0.170047			0.0850237	+	0.996379i	$0.472903\pi$
$602$		0			0
$603$	1.07907	+	1.20643i	0.0439430	+	0.0491298i
$604$		0			0
$605$		−	5.47204i		−	0.222470i
$606$		0			0
$607$		−	14.6203i		−	0.593421i	−0.954967	−	0.296711i	$-0.904110\pi$
$607$		−	14.6203i		−	0.593421i	0.954967	−	0.296711i	$-0.0958897\pi$
$608$		0			0
$609$	17.5153	−	6.69024i	0.709755	−	0.271102i
$610$		0			0
$611$	−8.77007			−0.354799
$612$		0			0
$613$	−2.23392			−0.0902270			−0.0451135	−	0.998982i	$-0.514365\pi$
$613$	−2.23392			−0.0902270			−0.0451135	+	0.998982i	$0.514365\pi$
$614$		0			0
$615$	−10.5871	+	4.04390i	−0.426912	+	0.163066i
$616$		0			0
$617$			23.0497i			0.927946i	0.885849	+	0.463973i	$0.153576\pi$
$617$			23.0497i			0.927946i	−0.885849	+	0.463973i	$0.846424\pi$
$618$		0			0
$619$			41.9082i			1.68443i	0.539138	+	0.842217i	$0.318750\pi$
$619$			41.9082i			1.68443i	−0.539138	+	0.842217i	$0.681250\pi$
$620$		0			0
$621$	2.38197	−	4.61803i	0.0955850	−	0.185315i
$622$		0			0
$623$	−31.9702			−1.28086
$624$		0			0
$625$	−5.18863			−0.207545
$626$		0			0
$627$	2.61307	+	6.84110i	0.104356	+	0.273207i
$628$		0			0
$629$			3.66000i			0.145934i
$630$		0			0
$631$			26.4994i			1.05492i	0.849579	+	0.527462i	$0.176856\pi$
$631$			26.4994i			1.05492i	−0.849579	+	0.527462i	$0.823144\pi$
$632$		0			0
$633$	13.4775	+	35.2846i	0.535683	+	1.40244i
$634$		0			0
$635$	−11.7109			−0.464731
$636$		0			0
$637$	6.23417			0.247007
$638$		0			0
$639$	−29.7693	+	26.6265i	−1.17766	+	1.05333i
$640$		0			0
$641$		−	43.2231i		−	1.70721i	−0.520922	−	0.853604i	$-0.674412\pi$
$641$		−	43.2231i		−	1.70721i	0.520922	−	0.853604i	$-0.325588\pi$
$642$		0			0
$643$			22.9991i			0.906995i	0.891258	+	0.453498i	$0.149824\pi$
$643$			22.9991i			0.906995i	−0.891258	+	0.453498i	$0.850176\pi$
$644$		0			0
$645$	12.0729	−	4.61142i	0.475369	−	0.181575i
$646$		0			0
$647$	26.6364			1.04719			0.523593	−	0.851969i	$-0.324591\pi$
$647$	26.6364			1.04719			0.523593	+	0.851969i	$0.324591\pi$
$648$		0			0
$649$	21.0592			0.826646
$650$		0			0
$651$	21.9523	−	8.38503i	0.860379	−	0.328635i
$652$		0			0
$653$		−	45.4371i		−	1.77809i	−0.457819	−	0.889045i	$-0.651369\pi$
$653$		−	45.4371i		−	1.77809i	0.457819	−	0.889045i	$-0.348631\pi$
$654$		0			0
$655$		−	16.8537i		−	0.658528i
$656$		0			0
$657$	−28.3403	+	25.3483i	−1.10566	+	0.988933i
$658$		0			0
$659$	11.8189			0.460399			0.230199	−	0.973143i	$-0.426062\pi$
$659$	11.8189			0.460399			0.230199	+	0.973143i	$0.426062\pi$
$660$		0			0
$661$	9.46436			0.368121			0.184060	−	0.982915i	$-0.441076\pi$
$661$	9.46436			0.368121			0.184060	+	0.982915i	$0.441076\pi$
$662$		0			0
$663$	14.1581	+	37.0665i	0.549856	+	1.43954i
$664$		0			0
$665$			5.79714i			0.224803i
$666$		0			0
$667$		−	4.47214i		−	0.173162i
$668$		0			0
$669$	−5.76481	−	15.0925i	−0.222881	−	0.583509i
$670$		0			0
$671$	32.2414			1.24467
$672$		0			0
$673$	−7.68056			−0.296064			−0.148032	−	0.988983i	$-0.547294\pi$
$673$	−7.68056			−0.296064			−0.148032	+	0.988983i	$0.547294\pi$
$674$		0			0
$675$	−6.20510	+	12.0301i	−0.238835	+	0.463040i
$676$		0			0
$677$		−	5.56211i		−	0.213769i	−0.994271	−	0.106885i	$-0.965912\pi$
$677$		−	5.56211i		−	0.213769i	0.994271	−	0.106885i	$-0.0340875\pi$
$678$		0			0
$679$		−	29.6310i		−	1.13713i
$680$		0			0
$681$	42.3209	−	16.1651i	1.62174	−	0.619449i
$682$		0			0
$683$	29.0142			1.11020			0.555098	−	0.831785i	$-0.312681\pi$
$683$	29.0142			1.11020			0.555098	+	0.831785i	$0.312681\pi$
$684$		0			0
$685$	−17.1491			−0.655233
$686$		0			0
$687$	1.70165	−	0.649973i	0.0649221	−	0.0247980i
$688$		0			0
$689$			61.3606i			2.33765i
$690$		0			0
$691$		−	10.9255i		−	0.415627i	−0.978168	−	0.207814i	$-0.933365\pi$
$691$		−	10.9255i		−	0.415627i	0.978168	−	0.207814i	$-0.0666347\pi$
$692$		0			0
$693$	13.2261	+	14.7873i	0.502419	+	0.561722i
$694$		0			0
$695$	−23.9317			−0.907783
$696$		0			0
$697$	−17.7261			−0.671424
$698$		0			0
$699$	3.59061	+	9.40034i	0.135809	+	0.355553i
$700$		0			0
$701$			15.8528i			0.598751i	0.954135	+	0.299375i	$0.0967783\pi$
$701$			15.8528i			0.598751i	−0.954135	+	0.299375i	$0.903222\pi$
$702$		0			0
$703$			1.35100i			0.0509540i
$704$		0			0
$705$	1.53513	+	4.01903i	0.0578165	+	0.151366i
$706$		0			0
$707$	−0.397627			−0.0149543
$708$		0			0
$709$	−14.5362			−0.545917			−0.272958	−	0.962026i	$-0.588002\pi$
$709$	−14.5362			−0.545917			−0.272958	+	0.962026i	$0.588002\pi$
$710$		0			0
$711$	−21.7532	−	24.3209i	−0.815810	−	0.912103i
$712$		0			0
$713$		−	5.60503i		−	0.209910i
$714$		0			0
$715$			23.1024i			0.863981i
$716$		0			0
$717$	−37.5547	+	14.3446i	−1.40251	+	0.535709i
$718$		0			0
$719$	−22.3347			−0.832943			−0.416472	−	0.909149i	$-0.636733\pi$
$719$	−22.3347			−0.832943			−0.416472	+	0.909149i	$0.636733\pi$
$720$		0			0
$721$	−13.3510			−0.497217
$722$		0			0
$723$	19.3392	−	7.38693i	0.719234	−	0.274723i
$724$		0			0
$725$			11.6501i			0.432673i
$726$		0			0
$727$		−	35.7328i		−	1.32526i	−0.748949	−	0.662628i	$-0.769441\pi$
$727$		−	35.7328i		−	1.32526i	0.748949	−	0.662628i	$-0.230559\pi$
$728$		0			0
$729$	−15.6525	−	22.0000i	−0.579721	−	0.814815i
$730$		0			0
$731$	20.2138			0.747634
$732$		0			0
$733$	27.7735			1.02584			0.512920	−	0.858437i	$-0.328564\pi$
$733$	27.7735			1.02584			0.512920	+	0.858437i	$0.328564\pi$
$734$		0			0
$735$	−1.09124	−	2.85691i	−0.0402512	−	0.105379i
$736$		0			0
$737$			1.47403i			0.0542967i
$738$		0			0
$739$		−	22.4767i		−	0.826820i	−0.910545	−	0.413410i	$-0.864338\pi$
$739$		−	22.4767i		−	0.826820i	0.910545	−	0.413410i	$-0.135662\pi$
$740$		0			0
$741$	5.22614	+	13.6822i	0.191987	+	0.502628i
$742$		0			0
$743$	17.1769			0.630160			0.315080	−	0.949065i	$-0.397969\pi$
$743$	17.1769			0.630160			0.315080	+	0.949065i	$0.397969\pi$
$744$		0			0
$745$	14.5059			0.531456
$746$		0			0
$747$	12.6567	−	11.3205i	0.463085	−	0.414196i
$748$		0			0
$749$			24.3151i			0.888456i
$750$		0			0
$751$			0.674584i			0.0246159i	0.999924	+	0.0123080i	$0.00391785\pi$
$751$			0.674584i			0.0246159i	−0.999924	+	0.0123080i	$0.996082\pi$
$752$		0			0
$753$	−28.4080	+	10.8509i	−1.03524	+	0.395428i
$754$		0			0
$755$	27.1351			0.987546
$756$		0			0
$757$	−22.5231			−0.818614			−0.409307	−	0.912397i	$-0.634230\pi$
$757$	−22.5231			−0.818614			−0.409307	+	0.912397i	$0.634230\pi$
$758$		0			0
$759$	4.42055	−	1.68850i	0.160456	−	0.0612887i
$760$		0			0
$761$		−	53.6058i		−	1.94321i	−0.236608	−	0.971605i	$-0.576036\pi$
$761$		−	53.6058i		−	1.94321i	0.236608	−	0.971605i	$-0.423964\pi$
$762$		0			0
$763$		−	1.18110i		−	0.0427588i
$764$		0			0
$765$	14.5081	−	12.9764i	0.524540	−	0.469163i
$766$		0			0
$767$	42.1184			1.52081
$768$		0			0
$769$	−38.3724			−1.38375			−0.691873	−	0.722019i	$-0.743214\pi$
$769$	−38.3724			−1.38375			−0.691873	+	0.722019i	$0.743214\pi$
$770$		0			0
$771$	−4.95768	−	12.9794i	−0.178546	−	0.467441i
$772$		0			0
$773$			16.6111i			0.597459i	0.954338	+	0.298729i	$0.0965628\pi$
$773$			16.6111i			0.597459i	−0.954338	+	0.298729i	$0.903437\pi$
$774$		0			0
$775$			14.6013i			0.524494i
$776$		0			0
$777$	1.30597	+	3.41907i	0.0468513	+	0.122658i
$778$		0			0
$779$	−6.54317			−0.234433
$780$		0			0
$781$	−36.3724			−1.30151
$782$		0			0
$783$	−20.6525	−	10.6525i	−0.738059	−	0.380688i
$784$		0			0
$785$		−	27.1554i		−	0.969218i
$786$		0			0
$787$		−	4.37260i		−	0.155866i	−0.996959	−	0.0779332i	$-0.975168\pi$
$787$		−	4.37260i		−	0.155866i	0.996959	−	0.0779332i	$-0.0248321\pi$
$788$		0			0
$789$	−34.0863	+	13.0198i	−1.21350	+	0.463517i
$790$		0			0
$791$	17.6401			0.627211
$792$		0			0
$793$	64.4829			2.28985
$794$		0			0
$795$	28.1195	−	10.7407i	0.997297	−	0.380934i
$796$		0			0
$797$			12.2300i			0.433210i	0.976259	+	0.216605i	$0.0694983\pi$
$797$			12.2300i			0.433210i	−0.976259	+	0.216605i	$0.930502\pi$
$798$		0			0
$799$			6.72913i			0.238060i
$800$		0			0
$801$	26.4156	+	29.5336i	0.933351	+	1.04352i
$802$		0			0
$803$	−34.6265			−1.22194
$804$		0			0
$805$	3.74597			0.132028
$806$		0			0
$807$	−5.30331	−	13.8842i	−0.186685	−	0.488748i
$808$		0			0
$809$			11.4676i			0.403179i	0.979470	+	0.201589i	$0.0646106\pi$
$809$			11.4676i			0.403179i	−0.979470	+	0.201589i	$0.935389\pi$
$810$		0			0
$811$			2.78880i			0.0979280i	0.998801	+	0.0489640i	$0.0155920\pi$
$811$			2.78880i			0.0979280i	−0.998801	+	0.0489640i	$0.984408\pi$
$812$		0			0
$813$	13.3071	+	34.8384i	0.466700	+	1.22184i
$814$		0			0
$815$	28.7217			1.00608
$816$		0			0
$817$	7.46144			0.261043
$818$		0			0
$819$	26.4523	+	29.5745i	0.924317	+	1.03342i
$820$		0			0
$821$		−	30.6831i		−	1.07085i	−0.844584	−	0.535424i	$-0.820152\pi$
$821$		−	30.6831i		−	1.07085i	0.844584	−	0.535424i	$-0.179848\pi$
$822$		0			0
$823$		−	14.2476i		−	0.496639i	−0.968678	−	0.248320i	$-0.920122\pi$
$823$		−	14.2476i		−	0.496639i	0.968678	−	0.248320i	$-0.0798783\pi$
$824$		0			0
$825$	−11.5157	+	4.39860i	−0.400925	+	0.153140i
$826$		0			0
$827$	3.23285			0.112417			0.0562086	−	0.998419i	$-0.482099\pi$
$827$	3.23285			0.112417			0.0562086	+	0.998419i	$0.482099\pi$
$828$		0			0
$829$	−17.4990			−0.607766			−0.303883	−	0.952709i	$-0.598283\pi$
$829$	−17.4990			−0.607766			−0.303883	+	0.952709i	$0.598283\pi$
$830$		0			0
$831$	−38.8778	+	14.8500i	−1.34866	+	0.515141i
$832$		0			0
$833$		−	4.78338i		−	0.165734i
$834$		0			0
$835$			28.8883i			0.999721i
$836$		0			0
$837$	−25.8842	−	13.3510i	−0.894690	−	0.461478i
$838$		0			0
$839$	−24.0629			−0.830745			−0.415372	−	0.909651i	$-0.636349\pi$
$839$	−24.0629			−0.830745			−0.415372	+	0.909651i	$0.636349\pi$
$840$		0			0
$841$	9.00000			0.310345
$842$		0			0
$843$	17.4162	+	45.5961i	0.599844	+	1.57041i
$844$		0			0
$845$			26.0864i			0.897400i
$846$		0			0
$847$		−	8.55882i		−	0.294085i
$848$		0			0
$849$	2.80109	+	7.33336i	0.0961333	+	0.251680i
$850$		0			0
$851$	0.872983			0.0299255
$852$		0			0
$853$	22.2819			0.762917			0.381458	−	0.924386i	$-0.375422\pi$
$853$	22.2819			0.762917			0.381458	+	0.924386i	$0.375422\pi$
$854$		0			0
$855$	5.35531	−	4.78993i	0.183148	−	0.163812i
$856$		0			0
$857$		−	9.32481i		−	0.318530i	−0.987236	−	0.159265i	$-0.949088\pi$
$857$		−	9.32481i		−	0.318530i	0.987236	−	0.159265i	$-0.0509124\pi$
$858$		0			0
$859$		−	34.1750i		−	1.16604i	−0.812459	−	0.583018i	$-0.801872\pi$
$859$		−	34.1750i		−	1.16604i	0.812459	−	0.583018i	$-0.198128\pi$
$860$		0			0
$861$	−16.5592	+	6.32507i	−0.564337	+	0.215558i
$862$		0			0
$863$	28.7601			0.979007			0.489503	−	0.872001i	$-0.337178\pi$
$863$	28.7601			0.979007			0.489503	+	0.872001i	$0.337178\pi$
$864$		0			0
$865$	−25.4917			−0.866745
$866$		0			0
$867$	0.933933	−	0.356731i	0.0317180	−	0.0121152i
$868$		0			0
$869$		−	29.7155i		−	1.00803i
$870$		0			0
$871$			2.94807i			0.0998914i
$872$		0			0
$873$	−27.3727	+	24.4829i	−0.926425	+	0.828620i
$874$		0			0
$875$	−28.4882			−0.963077
$876$		0			0
$877$	−3.10671			−0.104906			−0.0524531	−	0.998623i	$-0.516704\pi$
$877$	−3.10671			−0.104906			−0.0524531	+	0.998623i	$0.516704\pi$
$878$		0			0
$879$	−18.2491	−	47.7768i	−0.615528	−	1.61147i
$880$		0			0
$881$			45.7005i			1.53969i	0.638231	+	0.769845i	$0.279666\pi$
$881$			45.7005i			1.53969i	−0.638231	+	0.769845i	$0.720334\pi$
$882$		0			0
$883$			28.4067i			0.955963i	0.878370	+	0.477981i	$0.158631\pi$
$883$			28.4067i			0.955963i	−0.878370	+	0.477981i	$0.841369\pi$
$884$		0			0
$885$	−7.37251	−	19.3015i	−0.247824	−	0.648812i
$886$		0			0
$887$	−56.5279			−1.89802			−0.949011	−	0.315243i	$-0.897914\pi$
$887$	−56.5279			−1.89802			−0.949011	+	0.315243i	$0.897914\pi$
$888$		0			0
$889$	−18.3170			−0.614331
$890$		0			0
$891$	2.73205	−	24.4362i	0.0915271	−	0.818644i
$892$		0			0
$893$			2.48390i			0.0831205i
$894$		0			0
$895$		−	3.03670i		−	0.101506i
$896$		0			0
$897$	8.84110	−	3.37700i	0.295196	−	0.112755i
$898$		0			0
$899$	−25.0665			−0.836014
$900$		0			0
$901$	47.0810			1.56850
$902$		0			0
$903$	18.8832	−	7.21273i	0.628393	−	0.240025i
$904$		0			0
$905$			14.3274i			0.476259i
$906$		0			0
$907$		−	33.0196i		−	1.09640i	−0.836348	−	0.548200i	$-0.815313\pi$
$907$		−	33.0196i		−	1.09640i	0.836348	−	0.548200i	$-0.184687\pi$
$908$		0			0
$909$	0.328542	+	0.367322i	0.0108971	+	0.0121833i
$910$		0			0
$911$	−32.3042			−1.07029			−0.535143	−	0.844762i	$-0.679742\pi$
$911$	−32.3042			−1.07029			−0.535143	+	0.844762i	$0.679742\pi$
$912$		0			0
$913$	15.4641			0.511787
$914$		0			0
$915$	−11.2872	−	29.5504i	−0.373145	−	0.976905i
$916$		0			0
$917$		−	26.3609i		−	0.870513i
$918$		0			0
$919$		−	11.2860i		−	0.372291i	−0.982522	−	0.186146i	$-0.940400\pi$
$919$		−	11.2860i		−	0.372291i	0.982522	−	0.186146i	$-0.0595996\pi$
$920$		0			0
$921$	1.40749	+	3.68487i	0.0463785	+	0.121420i
$922$		0			0
$923$	−72.7449			−2.39443
$924$		0			0
$925$	−2.27415			−0.0747736
$926$		0			0
$927$	11.0314	+	12.3334i	0.362318	+	0.405084i
$928$		0			0
$929$			4.68942i			0.153855i	0.997037	+	0.0769275i	$0.0245110\pi$
$929$			4.68942i			0.153855i	−0.997037	+	0.0769275i	$0.975489\pi$
$930$		0			0
$931$		−	1.76567i		−	0.0578675i
$932$		0			0
$933$	−19.7342	+	7.53780i	−0.646069	+	0.246776i
$934$		0			0
$935$	17.7261			0.579706
$936$		0			0
$937$	25.2173			0.823815			0.411907	−	0.911226i	$-0.364863\pi$
$937$	25.2173			0.823815			0.411907	+	0.911226i	$0.364863\pi$
$938$		0			0
$939$	46.0295	−	17.5817i	1.50212	−	0.573757i
$940$		0			0
$941$		−	49.0546i		−	1.59913i	−0.600578	−	0.799566i	$-0.705063\pi$
$941$		−	49.0546i		−	1.59913i	0.600578	−	0.799566i	$-0.294937\pi$
$942$		0			0
$943$			4.22803i			0.137684i
$944$		0			0
$945$	8.92277	−	17.2990i	0.290258	−	0.562737i
$946$		0			0
$947$	10.0934			0.327992			0.163996	−	0.986461i	$-0.447562\pi$
$947$	10.0934			0.327992			0.163996	+	0.986461i	$0.447562\pi$
$948$		0			0
$949$	−69.2530			−2.24805
$950$		0			0
$951$	14.4622	+	37.8625i	0.468969	+	1.22778i
$952$		0			0
$953$			32.6242i			1.05680i	0.848995	+	0.528401i	$0.177208\pi$
$953$			32.6242i			1.05680i	−0.848995	+	0.528401i	$0.822792\pi$
$954$		0			0
$955$			6.62648i			0.214428i
$956$		0			0
$957$	−7.55120	−	19.7693i	−0.244096	−	0.639051i
$958$		0			0
$959$	−26.8229			−0.866156
$960$		0			0
$961$	−0.416408			−0.0134325
$962$		0			0
$963$	22.4619	−	20.0906i	0.723826	−	0.647410i
$964$		0			0
$965$			5.18590i			0.166940i
$966$		0			0
$967$		−	44.1093i		−	1.41846i	−0.704977	−	0.709230i	$-0.749043\pi$
$967$		−	44.1093i		−	1.41846i	0.704977	−	0.709230i	$-0.250957\pi$
$968$		0			0
$969$	10.4981	−	4.00993i	0.337249	−	0.128818i
$970$		0			0
$971$	18.4027			0.590570			0.295285	−	0.955409i	$-0.404585\pi$
$971$	18.4027			0.590570			0.295285	+	0.955409i	$0.404585\pi$
$972$		0			0
$973$	−37.4316			−1.20000
$974$		0			0
$975$	−23.0314	+	8.79720i	−0.737594	+	0.281736i
$976$		0			0
$977$		−	19.5452i		−	0.625305i	−0.949868	−	0.312653i	$-0.898782\pi$
$977$		−	19.5452i		−	0.625305i	0.949868	−	0.312653i	$-0.101218\pi$
$978$		0			0
$979$			36.0844i			1.15326i
$980$		0			0
$981$	−1.09109	+	0.975897i	−0.0348357	+	0.0311580i
$982$		0			0
$983$	−4.58359			−0.146194			−0.0730969	−	0.997325i	$-0.523288\pi$
$983$	−4.58359			−0.146194			−0.0730969	+	0.997325i	$0.523288\pi$
$984$		0			0
$985$	5.67771			0.180907
$986$		0			0
$987$	2.40110	+	6.28617i	0.0764280	+	0.200091i
$988$		0			0
$989$		−	4.82140i		−	0.153312i
$990$		0			0
$991$		−	14.3896i		−	0.457102i	−0.973532	−	0.228551i	$-0.926601\pi$
$991$		−	14.3896i		−	0.457102i	0.973532	−	0.228551i	$-0.0733988\pi$
$992$		0			0
$993$	19.9496	+	52.2288i	0.633083	+	1.65743i
$994$		0			0
$995$	27.7208			0.878808
$996$		0			0
$997$	−38.9839			−1.23463			−0.617316	−	0.786716i	$-0.711780\pi$
$997$	−38.9839			−1.23463			−0.617316	+	0.786716i	$0.711780\pi$
$998$		0			0
$999$	2.07942	−	4.03147i	0.0657898	−	0.127550i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.2.e.e.47.3	yes	8
3.2	odd	2		1104.2.e.h.47.8	yes	8
4.3	odd	2		1104.2.e.h.47.5	yes	8
12.11	even	2	inner	1104.2.e.e.47.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1104.2.e.e.47.2	✓	8	12.11	even	2	inner
1104.2.e.e.47.3	yes	8	1.1	even	1	trivial
1104.2.e.h.47.5	yes	8	4.3	odd	2
1104.2.e.h.47.8	yes	8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1104.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.61803 + 0.618034i) q^{3} +1.54757i q^{5} +2.42055i q^{7} +(2.23607 - 2.00000i) q^{9} +O(q^{10})\)	\(q+(-1.61803 + 0.618034i) q^{3} +1.54757i q^{5} +2.42055i q^{7} +(2.23607 - 2.00000i) q^{9} +2.73205 q^{11} +5.46410 q^{13} +(-0.956449 - 2.50402i) q^{15} -4.19252i q^{17} -1.54757i q^{19} +(-1.49598 - 3.91653i) q^{21} -1.00000 q^{23} +2.60503 q^{25} +(-2.38197 + 4.61803i) q^{27} +4.47214i q^{29} +5.60503i q^{31} +(-4.42055 + 1.68850i) q^{33} -3.74597 q^{35} -0.872983 q^{37} +(-8.84110 + 3.37700i) q^{39} -4.22803i q^{41} +4.82140i q^{43} +(3.09514 + 3.46047i) q^{45} -1.60503 q^{47} +1.14093 q^{49} +(2.59112 + 6.78364i) q^{51} +11.2298i q^{53} +4.22803i q^{55} +(0.956449 + 2.50402i) q^{57} +7.70820 q^{59} +11.8012 q^{61} +(4.84110 + 5.41252i) q^{63} +8.45607i q^{65} +0.539533i q^{67} +(1.61803 - 0.618034i) q^{69} -13.3132 q^{71} -12.6742 q^{73} +(-4.21503 + 1.61000i) q^{75} +6.61307i q^{77} -10.8766i q^{79} +(1.00000 - 8.94427i) q^{81} +5.66025 q^{83} +6.48820 q^{85} +(-2.76393 - 7.23607i) q^{87} +13.2078i q^{89} +13.2261i q^{91} +(-3.46410 - 9.06914i) q^{93} +2.39497 q^{95} -12.2414 q^{97} +(6.10905 - 5.46410i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3}+O(q^{10}) \) 8 * q - 4 * q^3	\( 8 q - 4 q^{3} + 8 q^{11} + 16 q^{13} - 4 q^{15} - 16 q^{21} - 8 q^{23} - 24 q^{25} - 28 q^{27} - 4 q^{33} + 32 q^{35} + 24 q^{37} - 8 q^{39} + 24 q^{45} + 32 q^{47} - 8 q^{49} + 24 q^{51} + 4 q^{57} + 8 q^{59} + 8 q^{61} - 24 q^{63} + 4 q^{69} - 8 q^{71} + 16 q^{73} + 12 q^{75} + 8 q^{81} - 24 q^{83} - 40 q^{87} + 64 q^{95} + 56 q^{97}+O(q^{100}) \) 8 * q - 4 * q^3 + 8 * q^11 + 16 * q^13 - 4 * q^15 - 16 * q^21 - 8 * q^23 - 24 * q^25 - 28 * q^27 - 4 * q^33 + 32 * q^35 + 24 * q^37 - 8 * q^39 + 24 * q^45 + 32 * q^47 - 8 * q^49 + 24 * q^51 + 4 * q^57 + 8 * q^59 + 8 * q^61 - 24 * q^63 + 4 * q^69 - 8 * q^71 + 16 * q^73 + 12 * q^75 + 8 * q^81 - 24 * q^83 - 40 * q^87 + 64 * q^95 + 56 * q^97

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