LMFDB - Embedded newform 1104.2.e.g.47.2

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:	8.0.3814238552064.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{6} - 4x^{5} - 12x^{3} + 9x^{2} + 81 $ x^8 + x^6 - 4x^5 - 12x^3 + 9*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			47.2
Root			$1.66736 + 0.468935i$ of defining polynomial
Character	$\chi$	$=$	1104.47
Dual form			1104.2.e.g.47.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.66736 + 0.468935i) q^{3} +0.762065i q^{5} -0.762065i q^{7} +(2.56020 - 1.56377i) q^{9} +O(q^{10})$	\(q+(-1.66736 + 0.468935i) q^{3} +0.762065i q^{5} -0.762065i q^{7} +(2.56020 - 1.56377i) q^{9} -1.78567 q^{11} +0.799248 q^{13} +(-0.357359 - 1.27064i) q^{15} +4.92161i q^{17} -7.11128i q^{19} +(0.357359 + 1.27064i) q^{21} +1.00000 q^{23} +4.41926 q^{25} +(-3.53548 + 3.80794i) q^{27} -9.21928i q^{29} +7.02962i q^{31} +(2.97737 - 0.837365i) q^{33} +0.580743 q^{35} +3.78567 q^{37} +(-1.33264 + 0.374795i) q^{39} +5.00328i q^{41} -2.95173i q^{43} +(1.19169 + 1.95104i) q^{45} +4.58492 q^{47} +6.41926 q^{49} +(-2.30791 - 8.20612i) q^{51} +7.11128i q^{53} -1.36080i q^{55} +(3.33473 + 11.8571i) q^{57} +5.82076 q^{59} +5.57135 q^{61} +(-1.19169 - 1.95104i) q^{63} +0.609079i q^{65} +10.8251i q^{67} +(-1.66736 + 0.468935i) q^{69} +10.4057 q^{71} +9.78985 q^{73} +(-7.36851 + 2.07234i) q^{75} +1.36080i q^{77} +5.58715i q^{79} +(4.10925 - 8.00712i) q^{81} +7.38417 q^{83} -3.75059 q^{85} +(4.32324 + 15.3719i) q^{87} -4.47586i q^{89} -0.609079i q^{91} +(-3.29643 - 11.7209i) q^{93} +5.41926 q^{95} +5.38417 q^{97} +(-4.57168 + 2.79238i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{9}+O(q^{10}) $ 8 * q - 2 * q^9	$ 8 q - 2 q^{9} + 4 q^{11} - 4 q^{13} - 10 q^{15} + 10 q^{21} + 8 q^{23} - 16 q^{25} - 12 q^{27} - 10 q^{33} + 56 q^{35} + 12 q^{37} - 24 q^{39} - 6 q^{45} + 8 q^{47} - 38 q^{51} + 16 q^{59} + 8 q^{61} + 6 q^{63} + 24 q^{71} - 20 q^{73} - 52 q^{75} + 2 q^{81} + 20 q^{83} - 24 q^{85} - 40 q^{87} + 2 q^{93} - 8 q^{95} + 4 q^{97} - 62 q^{99}+O(q^{100}) $ 8 * q - 2 * q^9 + 4 * q^11 - 4 * q^13 - 10 * q^15 + 10 * q^21 + 8 * q^23 - 16 * q^25 - 12 * q^27 - 10 * q^33 + 56 * q^35 + 12 * q^37 - 24 * q^39 - 6 * q^45 + 8 * q^47 - 38 * q^51 + 16 * q^59 + 8 * q^61 + 6 * q^63 + 24 * q^71 - 20 * q^73 - 52 * q^75 + 2 * q^81 + 20 * q^83 - 24 * q^85 - 40 * q^87 + 2 * q^93 - 8 * q^95 + 4 * q^97 - 62 * q^99

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
$3$	−1.66736	+	0.468935i	−0.962653	+	0.270740i
$3$	−1.66736	+	0.468935i	−0.962653	+	0.270740i
$4$		0			0
$5$			0.762065i			0.340806i	0.985374	+	0.170403i	$0.0545069\pi$
$5$			0.762065i			0.340806i	−0.985374	+	0.170403i	$0.945493\pi$
$6$		0			0
$7$		−	0.762065i		−	0.288033i	−0.989575	−	0.144017i	$-0.953998\pi$
$7$		−	0.762065i		−	0.288033i	0.989575	−	0.144017i	$-0.0460019\pi$
$8$		0			0
$9$	2.56020	−	1.56377i	0.853400	−	0.521256i
$10$		0			0
$11$	−1.78567			−0.538401			−0.269201	−	0.963084i	$-0.586759\pi$
$11$	−1.78567			−0.538401			−0.269201	+	0.963084i	$0.586759\pi$
$12$		0			0
$13$	0.799248			0.221672			0.110836	−	0.993839i	$-0.464647\pi$
$13$	0.799248			0.221672			0.110836	+	0.993839i	$0.464647\pi$
$14$		0			0
$15$	−0.357359	−	1.27064i	−0.0922696	−	0.328078i
$16$		0			0
$17$			4.92161i			1.19367i	0.802365	+	0.596833i	$0.203575\pi$
$17$			4.92161i			1.19367i	−0.802365	+	0.596833i	$0.796425\pi$
$18$		0			0
$19$		−	7.11128i		−	1.63144i	−0.578447	−	0.815720i	$-0.696341\pi$
$19$		−	7.11128i		−	1.63144i	0.578447	−	0.815720i	$-0.303659\pi$
$20$		0			0
$21$	0.357359	+	1.27064i	0.0779821	+	0.277276i
$22$		0			0
$23$	1.00000			0.208514
$23$	1.00000			0.208514
$24$		0			0
$25$	4.41926			0.883851
$26$		0			0
$27$	−3.53548	+	3.80794i	−0.680403	+	0.732838i
$28$		0			0
$29$		−	9.21928i		−	1.71198i	−0.516994	−	0.855989i	$-0.672949\pi$
$29$		−	9.21928i		−	1.71198i	0.516994	−	0.855989i	$-0.327051\pi$
$30$		0			0
$31$			7.02962i			1.26256i	0.775557	+	0.631278i	$0.217469\pi$
$31$			7.02962i			1.26256i	−0.775557	+	0.631278i	$0.782531\pi$
$32$		0			0
$33$	2.97737	−	0.837365i	0.518293	−	0.145766i
$34$		0			0
$35$	0.580743			0.0981635
$36$		0			0
$37$	3.78567			0.622361			0.311181	−	0.950351i	$-0.399276\pi$
$37$	3.78567			0.622361			0.311181	+	0.950351i	$0.399276\pi$
$38$		0			0
$39$	−1.33264	+	0.374795i	−0.213393	+	0.0600153i
$40$		0			0
$41$			5.00328i			0.781381i	0.920522	+	0.390690i	$0.127764\pi$
$41$			5.00328i			0.781381i	−0.920522	+	0.390690i	$0.872236\pi$
$42$		0			0
$43$		−	2.95173i		−	0.450135i	−0.974343	−	0.225068i	$-0.927740\pi$
$43$		−	2.95173i		−	0.450135i	0.974343	−	0.225068i	$-0.0722603\pi$
$44$		0			0
$45$	1.19169	+	1.95104i	0.177647	+	0.290844i
$46$		0			0
$47$	4.58492			0.668780			0.334390	−	0.942435i	$-0.391470\pi$
$47$	4.58492			0.668780			0.334390	+	0.942435i	$0.391470\pi$
$48$		0			0
$49$	6.41926			0.917037
$50$		0			0
$51$	−2.30791	−	8.20612i	−0.323173	−	1.14909i
$52$		0			0
$53$			7.11128i			0.976810i	0.872617	+	0.488405i	$0.162421\pi$
$53$			7.11128i			0.976810i	−0.872617	+	0.488405i	$0.837579\pi$
$54$		0			0
$55$		−	1.36080i		−	0.183490i
$56$		0			0
$57$	3.33473	+	11.8571i	0.441695	+	1.57051i
$58$		0			0
$59$	5.82076			0.757799			0.378899	−	0.925438i	$-0.376303\pi$
$59$	5.82076			0.757799			0.378899	+	0.925438i	$0.376303\pi$
$60$		0			0
$61$	5.57135			0.713338			0.356669	−	0.934231i	$-0.383912\pi$
$61$	5.57135			0.713338			0.356669	+	0.934231i	$0.383912\pi$
$62$		0			0
$63$	−1.19169	−	1.95104i	−0.150139	−	0.245808i
$64$		0			0
$65$			0.609079i			0.0755469i
$66$		0			0
$67$			10.8251i			1.32249i	0.750168	+	0.661247i	$0.229972\pi$
$67$			10.8251i			1.32249i	−0.750168	+	0.661247i	$0.770028\pi$
$68$		0			0
$69$	−1.66736	+	0.468935i	−0.200727	+	0.0564531i
$70$		0			0
$71$	10.4057			1.23493			0.617464	−	0.786599i	$-0.288160\pi$
$71$	10.4057			1.23493			0.617464	+	0.786599i	$0.288160\pi$
$72$		0			0
$73$	9.78985			1.14582			0.572908	−	0.819620i	$-0.305815\pi$
$73$	9.78985			1.14582			0.572908	+	0.819620i	$0.305815\pi$
$74$		0			0
$75$	−7.36851	+	2.07234i	−0.850842	+	0.239294i
$76$		0			0
$77$			1.36080i			0.155078i
$78$		0			0
$79$			5.58715i			0.628604i	0.949323	+	0.314302i	$0.101770\pi$
$79$			5.58715i			0.628604i	−0.949323	+	0.314302i	$0.898230\pi$
$80$		0			0
$81$	4.10925	−	8.00712i	0.456584	−	0.889681i
$82$		0			0
$83$	7.38417			0.810518			0.405259	−	0.914202i	$-0.367181\pi$
$83$	7.38417			0.810518			0.405259	+	0.914202i	$0.367181\pi$
$84$		0			0
$85$	−3.75059			−0.406808
$86$		0			0
$87$	4.32324	+	15.3719i	0.463500	+	1.64804i
$88$		0			0
$89$		−	4.47586i		−	0.474441i	−0.971456	−	0.237220i	$-0.923764\pi$
$89$		−	4.47586i		−	0.474441i	0.971456	−	0.237220i	$-0.0762363\pi$
$90$		0			0
$91$		−	0.609079i		−	0.0638488i
$92$		0			0
$93$	−3.29643	−	11.7209i	−0.341824	−	1.21540i
$94$		0			0
$95$	5.41926			0.556004
$96$		0			0
$97$	5.38417			0.546680			0.273340	−	0.961918i	$-0.411872\pi$
$97$	5.38417			0.546680			0.273340	+	0.961918i	$0.411872\pi$
$98$		0			0
$99$	−4.57168	+	2.79238i	−0.459472	+	0.280645i
$100$		0			0
$101$			5.74014i			0.571165i	0.958354	+	0.285582i	$0.0921871\pi$
$101$			5.74014i			0.571165i	−0.958354	+	0.285582i	$0.907813\pi$
$102$		0			0
$103$		−	9.13762i		−	0.900356i	−0.892939	−	0.450178i	$-0.851360\pi$
$103$		−	9.13762i		−	0.900356i	0.892939	−	0.450178i	$-0.148640\pi$
$104$		0			0
$105$	−0.968309	+	0.272330i	−0.0944973	+	0.0265767i
$106$		0			0
$107$	6.40150			0.618857			0.309428	−	0.950923i	$-0.399862\pi$
$107$	6.40150			0.618857			0.309428	+	0.950923i	$0.399862\pi$
$108$		0			0
$109$	−7.24002			−0.693468			−0.346734	−	0.937964i	$-0.612709\pi$
$109$	−7.24002			−0.693468			−0.346734	+	0.937964i	$0.612709\pi$
$110$		0			0
$111$	−6.31209	+	1.77523i	−0.599117	+	0.168498i
$112$		0			0
$113$		−	3.45394i		−	0.324920i	−0.986715	−	0.162460i	$-0.948057\pi$
$113$		−	3.45394i		−	0.324920i	0.986715	−	0.162460i	$-0.0519428\pi$
$114$		0			0
$115$			0.762065i			0.0710629i
$116$		0			0
$117$	2.04624	−	1.24984i	0.189175	−	0.115548i
$118$		0			0
$119$	3.75059			0.343816
$120$		0			0
$121$	−7.81137			−0.710124
$122$		0			0
$123$	−2.34621	−	8.34228i	−0.211551	−	0.752198i
$124$		0			0
$125$			7.17808i			0.642027i
$126$		0			0
$127$		−	13.9879i		−	1.24123i	−0.784117	−	0.620613i	$-0.786884\pi$
$127$		−	13.9879i		−	1.24123i	0.784117	−	0.620613i	$-0.213116\pi$
$128$		0			0
$129$	1.38417	+	4.92161i	0.121869	+	0.433324i
$130$		0			0
$131$	1.59432			0.139296			0.0696480	−	0.997572i	$-0.477812\pi$
$131$	1.59432			0.139296			0.0696480	+	0.997572i	$0.477812\pi$
$132$		0			0
$133$	−5.41926			−0.469909
$134$		0			0
$135$	−2.90190	−	2.69426i	−0.249755	−	0.231885i
$136$		0			0
$137$		−	15.5937i		−	1.33226i	−0.745836	−	0.666130i	$-0.767950\pi$
$137$		−	15.5937i		−	1.33226i	0.745836	−	0.666130i	$-0.232050\pi$
$138$		0			0
$139$			2.81361i			0.238647i	0.992855	+	0.119324i	$0.0380726\pi$
$139$			2.81361i			0.238647i	−0.992855	+	0.119324i	$0.961927\pi$
$140$		0			0
$141$	−7.64473	+	2.15003i	−0.643802	+	0.181065i
$142$		0			0
$143$	−1.42720			−0.119348
$144$		0			0
$145$	7.02569			0.583452
$146$		0			0
$147$	−10.7032	+	3.01021i	−0.882788	+	0.248278i
$148$		0			0
$149$		−	16.9545i		−	1.38897i	−0.719509	−	0.694484i	$-0.755633\pi$
$149$		−	16.9545i		−	1.38897i	0.719509	−	0.694484i	$-0.244367\pi$
$150$		0			0
$151$			5.50549i			0.448030i	0.974586	+	0.224015i	$0.0719165\pi$
$151$			5.50549i			0.448030i	−0.974586	+	0.224015i	$0.928084\pi$
$152$		0			0
$153$	7.69626	+	12.6003i	0.622206	+	1.01867i
$154$		0			0
$155$	−5.35702			−0.430286
$156$		0			0
$157$	−7.16984			−0.572216			−0.286108	−	0.958197i	$-0.592362\pi$
$157$	−7.16984			−0.572216			−0.286108	+	0.958197i	$0.592362\pi$
$158$		0			0
$159$	−3.33473	−	11.8571i	−0.264461	−	0.940328i
$160$		0			0
$161$		−	0.762065i		−	0.0600591i
$162$		0			0
$163$			20.0844i			1.57313i	0.617505	+	0.786567i	$0.288144\pi$
$163$			20.0844i			1.57313i	−0.617505	+	0.786567i	$0.711856\pi$
$164$		0			0
$165$	0.638126	+	2.26895i	0.0496781	+	0.176637i
$166$		0			0
$167$	3.97285			0.307429			0.153714	−	0.988115i	$-0.450876\pi$
$167$	3.97285			0.307429			0.153714	+	0.988115i	$0.450876\pi$
$168$		0			0
$169$	−12.3612			−0.950862
$170$		0			0
$171$	−11.1204	−	18.2063i	−0.850398	−	1.39227i
$172$		0			0
$173$			12.5351i			0.953026i	0.879167	+	0.476513i	$0.158100\pi$
$173$			12.5351i			0.953026i	−0.879167	+	0.476513i	$0.841900\pi$
$174$		0			0
$175$		−	3.36776i		−	0.254579i
$176$		0			0
$177$	−9.70532	+	2.72956i	−0.729497	+	0.205166i
$178$		0			0
$179$	18.2536			1.36434			0.682169	−	0.731195i	$-0.261037\pi$
$179$	18.2536			1.36434			0.682169	+	0.731195i	$0.261037\pi$
$180$		0			0
$181$	−18.5540			−1.37911			−0.689555	−	0.724234i	$-0.742194\pi$
$181$	−18.5540			−1.37911			−0.689555	+	0.724234i	$0.742194\pi$
$182$		0			0
$183$	−9.28946	+	2.61260i	−0.686697	+	0.193129i
$184$		0			0
$185$			2.88493i			0.212104i
$186$		0			0
$187$		−	8.78840i		−	0.642671i
$188$		0			0
$189$	2.90190	+	2.69426i	0.211082	+	0.195979i
$190$		0			0
$191$	−20.1254			−1.45622			−0.728110	−	0.685460i	$-0.759601\pi$
$191$	−20.1254			−1.45622			−0.728110	+	0.685460i	$0.759601\pi$
$192$		0			0
$193$	8.37060			0.602529			0.301264	−	0.953541i	$-0.402591\pi$
$193$	8.37060			0.602529			0.301264	+	0.953541i	$0.402591\pi$
$194$		0			0
$195$	−0.285618	−	1.01556i	−0.0204535	−	0.0727254i
$196$		0			0
$197$		−	14.8465i		−	1.05777i	−0.848694	−	0.528885i	$-0.822610\pi$
$197$		−	14.8465i		−	1.05777i	0.848694	−	0.528885i	$-0.177390\pi$
$198$		0			0
$199$			16.1524i			1.14501i	0.819901	+	0.572506i	$0.194029\pi$
$199$			16.1524i			1.14501i	−0.819901	+	0.572506i	$0.805971\pi$
$200$		0			0
$201$	−5.07626	−	18.0493i	−0.358051	−	1.27310i
$202$		0			0
$203$	−7.02569			−0.493107
$204$		0			0
$205$	−3.81282			−0.266299
$206$		0			0
$207$	2.56020	−	1.56377i	0.177946	−	0.108689i
$208$		0			0
$209$			12.6984i			0.878369i
$210$		0			0
$211$		−	21.1809i		−	1.45815i	−0.684434	−	0.729075i	$-0.739950\pi$
$211$		−	21.1809i		−	1.45815i	0.684434	−	0.729075i	$-0.260050\pi$
$212$		0			0
$213$	−17.3501	+	4.87959i	−1.18881	+	0.334344i
$214$		0			0
$215$	2.24941			0.153409
$216$		0			0
$217$	5.35702			0.363658
$218$		0			0
$219$	−16.3232	+	4.59080i	−1.10302	+	0.310218i
$220$		0			0
$221$			3.93359i			0.264602i
$222$		0			0
$223$		−	3.04826i		−	0.204127i	−0.994778	−	0.102063i	$-0.967456\pi$
$223$		−	3.04826i		−	0.204127i	0.994778	−	0.102063i	$-0.0325444\pi$
$224$		0			0
$225$	11.3142	−	6.91070i	0.754279	−	0.460713i
$226$		0			0
$227$	25.3654			1.68356			0.841780	−	0.539821i	$-0.181508\pi$
$227$	25.3654			1.68356			0.841780	+	0.539821i	$0.181508\pi$
$228$		0			0
$229$	−29.1511			−1.92636			−0.963178	−	0.268864i	$-0.913352\pi$
$229$	−29.1511			−1.92636			−0.963178	+	0.268864i	$0.913352\pi$
$230$		0			0
$231$	−0.638126	−	2.26895i	−0.0419856	−	0.149286i
$232$		0			0
$233$		−	5.00328i		−	0.327776i	−0.986479	−	0.163888i	$-0.947597\pi$
$233$		−	5.00328i		−	0.327776i	0.986479	−	0.163888i	$-0.0524035\pi$
$234$		0			0
$235$			3.49401i			0.227924i
$236$		0			0
$237$	−2.62001	−	9.31581i	−0.170188	−	0.605127i
$238$		0			0
$239$	−4.76416			−0.308168			−0.154084	−	0.988058i	$-0.549243\pi$
$239$	−4.76416			−0.308168			−0.154084	+	0.988058i	$0.549243\pi$
$240$		0			0
$241$	3.35702			0.216245			0.108122	−	0.994138i	$-0.465516\pi$
$241$	3.35702			0.216245			0.108122	+	0.994138i	$0.465516\pi$
$242$		0			0
$243$	−3.09680	+	15.2778i	−0.198660	+	0.980069i
$244$		0			0
$245$			4.89189i			0.312531i
$246$		0			0
$247$		−	5.68368i		−	0.361644i
$248$		0			0
$249$	−12.3121	+	3.46269i	−0.780247	+	0.219439i
$250$		0			0
$251$	8.18718			0.516770			0.258385	−	0.966042i	$-0.416810\pi$
$251$	8.18718			0.516770			0.258385	+	0.966042i	$0.416810\pi$
$252$		0			0
$253$	−1.78567			−0.112264
$254$		0			0
$255$	6.25359	−	1.75878i	0.391615	−	0.110139i
$256$		0			0
$257$			28.1776i			1.75767i	0.477127	+	0.878834i	$0.341678\pi$
$257$			28.1776i			1.75767i	−0.477127	+	0.878834i	$0.658322\pi$
$258$		0			0
$259$		−	2.88493i		−	0.179261i
$260$		0			0
$261$	−14.4168	−	23.6032i	−0.892379	−	1.46100i
$262$		0			0
$263$	−5.59850			−0.345218			−0.172609	−	0.984990i	$-0.555220\pi$
$263$	−5.59850			−0.345218			−0.172609	+	0.984990i	$0.555220\pi$
$264$		0			0
$265$	−5.41926			−0.332902
$266$		0			0
$267$	2.09889	+	7.46289i	0.128450	+	0.456721i
$268$		0			0
$269$		−	19.2258i		−	1.17222i	−0.810231	−	0.586110i	$-0.800659\pi$
$269$		−	19.2258i		−	1.17222i	0.810231	−	0.586110i	$-0.199341\pi$
$270$		0			0
$271$		−	11.6436i		−	0.707299i	−0.935378	−	0.353649i	$-0.884941\pi$
$271$		−	11.6436i		−	0.707299i	0.935378	−	0.353649i	$-0.115059\pi$
$272$		0			0
$273$	0.285618	+	1.01556i	0.0172864	+	0.0614642i
$274$		0			0
$275$	−7.89135			−0.475867
$276$		0			0
$277$	−27.7982			−1.67023			−0.835116	−	0.550073i	$-0.814600\pi$
$277$	−27.7982			−1.67023			−0.835116	+	0.550073i	$0.814600\pi$
$278$		0			0
$279$	10.9927	+	17.9972i	0.658115	+	1.07747i
$280$		0			0
$281$		−	4.92161i		−	0.293599i	−0.989166	−	0.146799i	$-0.953103\pi$
$281$		−	4.92161i		−	0.293599i	0.989166	−	0.146799i	$-0.0468972\pi$
$282$		0			0
$283$			0.981855i			0.0583652i	0.999574	+	0.0291826i	$0.00929043\pi$
$283$			0.981855i			0.0583652i	−0.999574	+	0.0291826i	$0.990710\pi$
$284$		0			0
$285$	−9.03587	+	2.54128i	−0.535239	+	0.150532i
$286$		0			0
$287$	3.81282			0.225064
$288$		0			0
$289$	−7.22227			−0.424839
$290$		0			0
$291$	−8.97737	+	2.52482i	−0.526263	+	0.148008i
$292$		0			0
$293$		−	15.6502i		−	0.914292i	−0.889392	−	0.457146i	$-0.848872\pi$
$293$		−	15.6502i		−	0.914292i	0.889392	−	0.457146i	$-0.151128\pi$
$294$		0			0
$295$			4.43580i			0.258262i
$296$		0			0
$297$	6.31321	−	6.79974i	0.366330	−	0.394561i
$298$		0			0
$299$	0.799248			0.0462217
$300$		0			0
$301$	−2.24941			−0.129654
$302$		0			0
$303$	−2.69175	−	9.57089i	−0.154637	−	0.549833i
$304$		0			0
$305$			4.24573i			0.243110i
$306$		0			0
$307$			4.54267i			0.259264i	0.991562	+	0.129632i	$0.0413796\pi$
$307$			4.54267i			0.259264i	−0.991562	+	0.129632i	$0.958620\pi$
$308$		0			0
$309$	4.28495	+	15.2357i	0.243762	+	0.866730i
$310$		0			0
$311$	−21.8249			−1.23758			−0.618789	−	0.785557i	$-0.712377\pi$
$311$	−21.8249			−1.23758			−0.618789	+	0.785557i	$0.712377\pi$
$312$		0			0
$313$	11.9729			0.676746			0.338373	−	0.941012i	$-0.390124\pi$
$313$	11.9729			0.676746			0.338373	+	0.941012i	$0.390124\pi$
$314$		0			0
$315$	1.48682	−	0.908148i	0.0837727	−	0.0511683i
$316$		0			0
$317$		−	5.43416i		−	0.305213i	−0.988287	−	0.152607i	$-0.951233\pi$
$317$		−	5.43416i		−	0.305213i	0.988287	−	0.152607i	$-0.0487667\pi$
$318$		0			0
$319$			16.4626i			0.921731i
$320$		0			0
$321$	−10.6736	+	3.00189i	−0.595744	+	0.167549i
$322$		0			0
$323$	34.9990			1.94739
$324$		0			0
$325$	3.53208			0.195925
$326$		0			0
$327$	12.0717	−	3.39510i	0.667569	−	0.187749i
$328$		0			0
$329$		−	3.49401i		−	0.192631i
$330$		0			0
$331$		−	3.44624i		−	0.189423i	−0.995505	−	0.0947113i	$-0.969807\pi$
$331$		−	3.44624i		−	0.189423i	0.995505	−	0.0947113i	$-0.0301928\pi$
$332$		0			0
$333$	9.69209	−	5.91992i	0.531123	−	0.324410i
$334$		0			0
$335$	−8.24941			−0.450714
$336$		0			0
$337$	22.3827			1.21926			0.609632	−	0.792684i	$-0.291317\pi$
$337$	22.3827			1.21926			0.609632	+	0.792684i	$0.291317\pi$
$338$		0			0
$339$	1.61967	+	5.75898i	0.0879686	+	0.312785i
$340$		0			0
$341$		−	12.5526i		−	0.679762i
$342$		0			0
$343$		−	10.2263i		−	0.552171i
$344$		0			0
$345$	−0.357359	−	1.27064i	−0.0192395	−	0.0684089i
$346$		0			0
$347$	28.5975			1.53519			0.767596	−	0.640934i	$-0.221453\pi$
$347$	28.5975			1.53519			0.767596	+	0.640934i	$0.221453\pi$
$348$		0			0
$349$	12.8076			0.685575			0.342788	−	0.939413i	$-0.388629\pi$
$349$	12.8076			0.685575			0.342788	+	0.939413i	$0.388629\pi$
$350$		0			0
$351$	−2.82572	+	3.04349i	−0.150826	+	0.162449i
$352$		0			0
$353$		−	19.3388i		−	1.02930i	−0.857401	−	0.514649i	$-0.827922\pi$
$353$		−	19.3388i		−	1.02930i	0.857401	−	0.514649i	$-0.172078\pi$
$354$		0			0
$355$			7.92981i			0.420870i
$356$		0			0
$357$	−6.25359	+	1.75878i	−0.330975	+	0.0930845i
$358$		0			0
$359$	21.1698			1.11730			0.558651	−	0.829403i	$-0.311319\pi$
$359$	21.1698			1.11730			0.558651	+	0.829403i	$0.311319\pi$
$360$		0			0
$361$	−31.5703			−1.66160
$362$		0			0
$363$	13.0244	−	3.66302i	0.683603	−	0.192259i
$364$		0			0
$365$			7.46050i			0.390501i
$366$		0			0
$367$		−	24.7714i		−	1.29306i	−0.762890	−	0.646528i	$-0.776220\pi$
$367$		−	24.7714i		−	1.29306i	0.762890	−	0.646528i	$-0.223780\pi$
$368$		0			0
$369$	7.82397	+	12.8094i	0.407300	+	0.666830i
$370$		0			0
$371$	5.41926			0.281354
$372$		0			0
$373$	6.55402			0.339354			0.169677	−	0.985500i	$-0.445728\pi$
$373$	6.55402			0.339354			0.169677	+	0.985500i	$0.445728\pi$
$374$		0			0
$375$	−3.36605	−	11.9685i	−0.173822	−	0.618049i
$376$		0			0
$377$		−	7.36849i		−	0.379497i
$378$		0			0
$379$			34.1185i			1.75255i	0.481814	+	0.876274i	$0.339978\pi$
$379$			34.1185i			1.75255i	−0.481814	+	0.876274i	$0.660022\pi$
$380$		0			0
$381$	6.55942	+	23.3229i	0.336049	+	1.19487i
$382$		0			0
$383$	−24.0000			−1.22634			−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000			−1.22634			−0.613171	+	0.789950i	$0.710106\pi$
$384$		0			0
$385$	−1.03702			−0.0528513
$386$		0			0
$387$	−4.61583	−	7.55703i	−0.234636	−	0.384145i
$388$		0			0
$389$			31.7028i			1.60740i	0.595037	+	0.803699i	$0.297138\pi$
$389$			31.7028i			1.60740i	−0.595037	+	0.803699i	$0.702862\pi$
$390$		0			0
$391$			4.92161i			0.248897i
$392$		0			0
$393$	−2.65830	+	0.747630i	−0.134094	+	0.0377130i
$394$		0			0
$395$	−4.25777			−0.214232
$396$		0			0
$397$	−2.12118			−0.106459			−0.0532296	−	0.998582i	$-0.516952\pi$
$397$	−2.12118			−0.106459			−0.0532296	+	0.998582i	$0.516952\pi$
$398$		0			0
$399$	9.03587	−	2.54128i	0.452359	−	0.127223i
$400$		0			0
$401$			12.1859i			0.608534i	0.952587	+	0.304267i	$0.0984115\pi$
$401$			12.1859i			0.608534i	−0.952587	+	0.304267i	$0.901589\pi$
$402$		0			0
$403$			5.61841i			0.279873i
$404$		0			0
$405$	6.10195	+	3.13152i	0.303208	+	0.155606i
$406$		0			0
$407$	−6.75998			−0.335080
$408$		0			0
$409$	13.1820			0.651806			0.325903	−	0.945403i	$-0.394332\pi$
$409$	13.1820			0.651806			0.325903	+	0.945403i	$0.394332\pi$
$410$		0			0
$411$	7.31243	+	26.0004i	0.360696	+	1.28250i
$412$		0			0
$413$		−	4.43580i		−	0.218271i
$414$		0			0
$415$			5.62722i			0.276229i
$416$		0			0
$417$	−1.31940	−	4.69131i	−0.0646112	−	0.229734i
$418$		0			0
$419$	25.2484			1.23346			0.616732	−	0.787173i	$-0.288456\pi$
$419$	25.2484			1.23346			0.616732	+	0.787173i	$0.288456\pi$
$420$		0			0
$421$	−5.45434			−0.265828			−0.132914	−	0.991128i	$-0.542433\pi$
$421$	−5.45434			−0.265828			−0.132914	+	0.991128i	$0.542433\pi$
$422$		0			0
$423$	11.7383	−	7.16976i	0.570737	−	0.348606i
$424$		0			0
$425$			21.7499i			1.05502i
$426$		0			0
$427$		−	4.24573i		−	0.205465i
$428$		0			0
$429$	2.37966	−	0.669262i	0.114891	−	0.0323123i
$430$		0			0
$431$	−14.4099			−0.694099			−0.347049	−	0.937847i	$-0.612816\pi$
$431$	−14.4099			−0.694099			−0.347049	+	0.937847i	$0.612816\pi$
$432$		0			0
$433$	30.1955			1.45110			0.725552	−	0.688167i	$-0.241584\pi$
$433$	30.1955			1.45110			0.725552	+	0.688167i	$0.241584\pi$
$434$		0			0
$435$	−11.7144	+	3.29459i	−0.561662	+	0.157964i
$436$		0			0
$437$		−	7.11128i		−	0.340179i
$438$		0			0
$439$		−	7.30587i		−	0.348690i	−0.984685	−	0.174345i	$-0.944219\pi$
$439$		−	7.30587i		−	0.348690i	0.984685	−	0.174345i	$-0.0557808\pi$
$440$		0			0
$441$	16.4346	−	10.0382i	0.782599	−	0.478011i
$442$		0			0
$443$	−16.7642			−0.796489			−0.398245	−	0.917279i	$-0.630380\pi$
$443$	−16.7642			−0.796489			−0.398245	+	0.917279i	$0.630380\pi$
$444$		0			0
$445$	3.41090			0.161692
$446$		0			0
$447$	7.95056	+	28.2693i	0.376048	+	1.33709i
$448$		0			0
$449$		−	14.5285i		−	0.685644i	−0.939400	−	0.342822i	$-0.888617\pi$
$449$		−	14.5285i		−	0.685644i	0.939400	−	0.342822i	$-0.111383\pi$
$450$		0			0
$451$		−	8.93422i		−	0.420696i
$452$		0			0
$453$	−2.58171	−	9.17965i	−0.121299	−	0.431297i
$454$		0			0
$455$	0.464158			0.0217600
$456$		0			0
$457$	−28.1684			−1.31766			−0.658831	−	0.752291i	$-0.728949\pi$
$457$	−28.1684			−1.31766			−0.658831	+	0.752291i	$0.728949\pi$
$458$		0			0
$459$	−18.7412	−	17.4003i	−0.874764	−	0.812174i
$460$		0			0
$461$		−	16.6469i		−	0.775323i	−0.921802	−	0.387661i	$-0.873283\pi$
$461$		−	16.6469i		−	0.775323i	0.921802	−	0.387661i	$-0.126717\pi$
$462$		0			0
$463$			19.6864i			0.914907i	0.889234	+	0.457453i	$0.151238\pi$
$463$			19.6864i			0.914907i	−0.889234	+	0.457453i	$0.848762\pi$
$464$		0			0
$465$	8.93210	−	2.51209i	0.414216	−	0.116496i
$466$		0			0
$467$	−8.00836			−0.370583			−0.185291	−	0.982684i	$-0.559323\pi$
$467$	−8.00836			−0.370583			−0.185291	+	0.982684i	$0.559323\pi$
$468$		0			0
$469$	8.24941			0.380922
$470$		0			0
$471$	11.9547	−	3.36219i	0.550845	−	0.154922i
$472$		0			0
$473$			5.27083i			0.242353i
$474$		0			0
$475$		−	31.4266i		−	1.44195i
$476$		0			0
$477$	11.1204	+	18.2063i	0.509168	+	0.833609i
$478$		0			0
$479$	−29.4197			−1.34422			−0.672110	−	0.740452i	$-0.734612\pi$
$479$	−29.4197			−1.34422			−0.672110	+	0.740452i	$0.734612\pi$
$480$		0			0
$481$	3.02569			0.137960
$482$		0			0
$483$	0.357359	+	1.27064i	0.0162604	+	0.0578161i
$484$		0			0
$485$			4.10309i			0.186312i
$486$		0			0
$487$			21.2522i			0.963028i	0.876438	+	0.481514i	$0.159913\pi$
$487$			21.2522i			0.963028i	−0.876438	+	0.481514i	$0.840087\pi$
$488$		0			0
$489$	−9.41829	−	33.4880i	−0.425910	−	1.51438i
$490$		0			0
$491$	−36.9760			−1.66870			−0.834352	−	0.551232i	$-0.814158\pi$
$491$	−36.9760			−1.66870			−0.834352	+	0.551232i	$0.814158\pi$
$492$		0			0
$493$	45.3737			2.04353
$494$		0			0
$495$	−2.12798	−	3.48392i	−0.0956454	−	0.156591i
$496$		0			0
$497$		−	7.92981i		−	0.355700i
$498$		0			0
$499$		−	24.4638i		−	1.09515i	−0.836757	−	0.547574i	$-0.815551\pi$
$499$		−	24.4638i		−	1.09515i	0.836757	−	0.547574i	$-0.184449\pi$
$500$		0			0
$501$	−6.62419	+	1.86301i	−0.295947	+	0.0832331i
$502$		0			0
$503$	34.4182			1.53463			0.767316	−	0.641269i	$-0.221592\pi$
$503$	34.4182			1.53463			0.767316	+	0.641269i	$0.221592\pi$
$504$		0			0
$505$	−4.37436			−0.194656
$506$		0			0
$507$	20.6106	−	5.79660i	0.915350	−	0.257436i
$508$		0			0
$509$			18.7565i			0.831369i	0.909509	+	0.415684i	$0.136458\pi$
$509$			18.7565i			0.831369i	−0.909509	+	0.415684i	$0.863542\pi$
$510$		0			0
$511$		−	7.46050i		−	0.330033i
$512$		0			0
$513$	27.0793	+	25.1418i	1.19558	+	1.11004i
$514$		0			0
$515$	6.96346			0.306847
$516$		0			0
$517$	−8.18718			−0.360072
$518$		0			0
$519$	−5.87814	−	20.9006i	−0.258022	−	0.917433i
$520$		0			0
$521$			4.28281i			0.187633i	0.995589	+	0.0938167i	$0.0299068\pi$
$521$			4.28281i			0.187633i	−0.995589	+	0.0938167i	$0.970093\pi$
$522$		0			0
$523$		−	6.60907i		−	0.288995i	−0.989505	−	0.144497i	$-0.953844\pi$
$523$		−	6.60907i		−	0.288995i	0.989505	−	0.144497i	$-0.0461565\pi$
$524$		0			0
$525$	1.57926	+	5.61528i	0.0689246	+	0.245071i
$526$		0			0
$527$	−34.5970			−1.50707
$528$		0			0
$529$	1.00000			0.0434783
$530$		0			0
$531$	14.9023	−	9.10233i	0.646705	−	0.395007i
$532$		0			0
$533$			3.99886i			0.173210i
$534$		0			0
$535$			4.87836i			0.210910i
$536$		0			0
$537$	−30.4354	+	8.55974i	−1.31338	+	0.369380i
$538$		0			0
$539$	−11.4627			−0.493734
$540$		0			0
$541$	9.61062			0.413193			0.206596	−	0.978426i	$-0.433761\pi$
$541$	9.61062			0.413193			0.206596	+	0.978426i	$0.433761\pi$
$542$		0			0
$543$	30.9363	−	8.70062i	1.32760	−	0.373379i
$544$		0			0
$545$		−	5.51736i		−	0.236338i
$546$		0			0
$547$			19.6151i			0.838682i	0.907829	+	0.419341i	$0.137739\pi$
$547$			19.6151i			0.838682i	−0.907829	+	0.419341i	$0.862261\pi$
$548$		0			0
$549$	14.2638	−	8.71230i	0.608763	−	0.371832i
$550$		0			0
$551$	−65.5609			−2.79299
$552$		0			0
$553$	4.25777			0.181059
$554$		0			0
$555$	−1.35284	−	4.81023i	−0.0574250	−	0.204183i
$556$		0			0
$557$			17.1743i			0.727698i	0.931458	+	0.363849i	$0.118538\pi$
$557$			17.1743i			0.727698i	−0.931458	+	0.363849i	$0.881462\pi$
$558$		0			0
$559$		−	2.35917i		−	0.0997821i
$560$		0			0
$561$	4.12118	+	14.6534i	0.173997	+	0.618669i
$562$		0			0
$563$	−7.50118			−0.316137			−0.158068	−	0.987428i	$-0.550527\pi$
$563$	−7.50118			−0.316137			−0.158068	+	0.987428i	$0.550527\pi$
$564$		0			0
$565$	2.63213			0.110734
$566$		0			0
$567$	−6.10195	−	3.13152i	−0.256258	−	0.131511i
$568$		0			0
$569$			17.5132i			0.734190i	0.930183	+	0.367095i	$0.119648\pi$
$569$			17.5132i			0.734190i	−0.930183	+	0.367095i	$0.880352\pi$
$570$		0			0
$571$		−	9.40782i		−	0.393705i	−0.980433	−	0.196852i	$-0.936928\pi$
$571$		−	9.40782i		−	0.393705i	0.980433	−	0.196852i	$-0.0630720\pi$
$572$		0			0
$573$	33.5563	−	9.43748i	1.40183	−	0.394256i
$574$		0			0
$575$	4.41926			0.184296
$576$		0			0
$577$	8.72907			0.363396			0.181698	−	0.983354i	$-0.441841\pi$
$577$	8.72907			0.363396			0.181698	+	0.983354i	$0.441841\pi$
$578$		0			0
$579$	−13.9568	+	3.92526i	−0.580026	+	0.163128i
$580$		0			0
$581$		−	5.62722i		−	0.233456i
$582$		0			0
$583$		−	12.6984i		−	0.525915i
$584$		0			0
$585$	0.952459	+	1.55936i	0.0393793	+	0.0644718i
$586$		0			0
$587$	−38.0397			−1.57007			−0.785033	−	0.619454i	$-0.787354\pi$
$587$	−38.0397			−1.57007			−0.785033	+	0.619454i	$0.787354\pi$
$588$		0			0
$589$	49.9896			2.05978
$590$		0			0
$591$	6.96204	+	24.7545i	0.286380	+	1.01826i
$592$		0			0
$593$			33.1808i			1.36257i	0.732016	+	0.681287i	$0.238579\pi$
$593$			33.1808i			1.36257i	−0.732016	+	0.681287i	$0.761421\pi$
$594$		0			0
$595$			2.85819i			0.117174i
$596$		0			0
$597$	−7.57441	−	26.9319i	−0.310000	−	1.10225i
$598$		0			0
$599$	25.0809			1.02478			0.512389	−	0.858754i	$-0.328761\pi$
$599$	25.0809			1.02478			0.512389	+	0.858754i	$0.328761\pi$
$600$		0			0
$601$	7.62940			0.311210			0.155605	−	0.987819i	$-0.450267\pi$
$601$	7.62940			0.311210			0.155605	+	0.987819i	$0.450267\pi$
$602$		0			0
$603$	16.9279	+	27.7144i	0.689358	+	1.12862i
$604$		0			0
$605$		−	5.95277i		−	0.242014i
$606$		0			0
$607$			41.2862i			1.67576i	0.545858	+	0.837878i	$0.316204\pi$
$607$			41.2862i			1.67576i	−0.545858	+	0.837878i	$0.683796\pi$
$608$		0			0
$609$	11.7144	−	3.29459i	0.474691	−	0.133504i
$610$		0			0
$611$	3.66449			0.148249
$612$		0			0
$613$	22.9910			0.928599			0.464299	−	0.885678i	$-0.346306\pi$
$613$	22.9910			0.928599			0.464299	+	0.885678i	$0.346306\pi$
$614$		0			0
$615$	6.35736	−	1.78796i	0.256353	−	0.0720977i
$616$		0			0
$617$			15.4868i			0.623477i	0.950168	+	0.311738i	$0.100911\pi$
$617$			15.4868i			0.623477i	−0.950168	+	0.311738i	$0.899089\pi$
$618$		0			0
$619$		−	18.0658i		−	0.726126i	−0.931765	−	0.363063i	$-0.881731\pi$
$619$		−	18.0658i		−	0.726126i	0.931765	−	0.363063i	$-0.118269\pi$
$620$		0			0
$621$	−3.53548	+	3.80794i	−0.141874	+	0.152807i
$622$		0			0
$623$	−3.41090			−0.136655
$624$		0			0
$625$	16.6261			0.665045
$626$		0			0
$627$	−5.95474	−	21.1729i	−0.237809	−	0.845564i
$628$		0			0
$629$			18.6316i			0.742891i
$630$		0			0
$631$		−	45.4858i		−	1.81076i	−0.424601	−	0.905381i	$-0.639585\pi$
$631$		−	45.4858i		−	1.81076i	0.424601	−	0.905381i	$-0.360415\pi$
$632$		0			0
$633$	9.93244	+	35.3162i	0.394779	+	1.40369i
$634$		0			0
$635$	10.6597			0.423017
$636$		0			0
$637$	5.13058			0.203281
$638$		0			0
$639$	26.6406	−	16.2721i	1.05389	−	0.643714i
$640$		0			0
$641$		−	48.4182i		−	1.91240i	−0.292710	−	0.956201i	$-0.594557\pi$
$641$		−	48.4182i		−	1.91240i	0.292710	−	0.956201i	$-0.405443\pi$
$642$		0			0
$643$			10.6885i			0.421513i	0.977539	+	0.210756i	$0.0675927\pi$
$643$			10.6885i			0.421513i	−0.977539	+	0.210756i	$0.932407\pi$
$644$		0			0
$645$	−3.75059	+	1.05483i	−0.147679	+	0.0415338i
$646$		0			0
$647$	36.5933			1.43863			0.719315	−	0.694684i	$-0.244456\pi$
$647$	36.5933			1.43863			0.719315	+	0.694684i	$0.244456\pi$
$648$		0			0
$649$	−10.3940			−0.408000
$650$		0			0
$651$	−8.93210	+	2.51209i	−0.350077	+	0.0984567i
$652$		0			0
$653$		−	9.49553i		−	0.371589i	−0.982589	−	0.185794i	$-0.940514\pi$
$653$		−	9.49553i		−	0.371589i	0.982589	−	0.185794i	$-0.0594859\pi$
$654$		0			0
$655$			1.21497i			0.0474729i
$656$		0			0
$657$	25.0640	−	15.3091i	0.977839	−	0.597264i
$658$		0			0
$659$	−40.5783			−1.58070			−0.790352	−	0.612653i	$-0.790102\pi$
$659$	−40.5783			−1.58070			−0.790352	+	0.612653i	$0.790102\pi$
$660$		0			0
$661$	−1.69582			−0.0659596			−0.0329798	−	0.999456i	$-0.510500\pi$
$661$	−1.69582			−0.0659596			−0.0329798	+	0.999456i	$0.510500\pi$
$662$		0			0
$663$	−1.84460	−	6.55872i	−0.0716382	−	0.254720i
$664$		0			0
$665$		−	4.12983i		−	0.160148i
$666$		0			0
$667$		−	9.21928i		−	0.356972i
$668$		0			0
$669$	1.42943	+	5.08256i	0.0552651	+	0.196503i
$670$		0			0
$671$	−9.94861			−0.384062
$672$		0			0
$673$	24.0664			0.927692			0.463846	−	0.885916i	$-0.346469\pi$
$673$	24.0664			0.927692			0.463846	+	0.885916i	$0.346469\pi$
$674$		0			0
$675$	−15.6242	+	16.8283i	−0.601375	+	0.647720i
$676$		0			0
$677$		−	25.0805i		−	0.963924i	−0.876192	−	0.481962i	$-0.839924\pi$
$677$		−	25.0805i		−	0.963924i	0.876192	−	0.481962i	$-0.160076\pi$
$678$		0			0
$679$		−	4.10309i		−	0.157462i
$680$		0			0
$681$	−42.2933	+	11.8947i	−1.62068	+	0.455806i
$682$		0			0
$683$	6.03133			0.230782			0.115391	−	0.993320i	$-0.463188\pi$
$683$	6.03133			0.230782			0.115391	+	0.993320i	$0.463188\pi$
$684$		0			0
$685$	11.8834			0.454042
$686$		0			0
$687$	48.6054	−	13.6699i	1.85441	−	0.521541i
$688$		0			0
$689$			5.68368i			0.216531i
$690$		0			0
$691$			36.4880i			1.38807i	0.719942	+	0.694034i	$0.244168\pi$
$691$			36.4880i			1.38807i	−0.719942	+	0.694034i	$0.755832\pi$
$692$		0			0
$693$	2.12798	+	3.48392i	0.0808351	+	0.132343i
$694$		0			0
$695$	−2.14415			−0.0813323
$696$		0			0
$697$	−24.6242			−0.932708
$698$		0			0
$699$	2.34621	+	8.34228i	0.0887418	+	0.315534i
$700$		0			0
$701$			36.0319i			1.36091i	0.732792	+	0.680453i	$0.238217\pi$
$701$			36.0319i			1.36091i	−0.732792	+	0.680453i	$0.761783\pi$
$702$		0			0
$703$		−	26.9210i		−	1.01534i
$704$		0			0
$705$	−1.63846	−	5.82578i	−0.0617080	−	0.219412i
$706$		0			0
$707$	4.37436			0.164515
$708$		0			0
$709$	−11.4356			−0.429471			−0.214736	−	0.976672i	$-0.568889\pi$
$709$	−11.4356			−0.429471			−0.214736	+	0.976672i	$0.568889\pi$
$710$		0			0
$711$	8.73701	+	14.3042i	0.327664	+	0.536450i
$712$		0			0
$713$			7.02962i			0.263261i
$714$		0			0
$715$		−	1.08762i		−	0.0406745i
$716$		0			0
$717$	7.94359	−	2.23408i	0.296659	−	0.0834333i
$718$		0			0
$719$	−2.24941			−0.0838889			−0.0419445	−	0.999120i	$-0.513355\pi$
$719$	−2.24941			−0.0838889			−0.0419445	+	0.999120i	$0.513355\pi$
$720$		0			0
$721$	−6.96346			−0.259333
$722$		0			0
$723$	−5.59738	+	1.57422i	−0.208169	+	0.0585460i
$724$		0			0
$725$		−	40.7424i		−	1.51313i
$726$		0			0
$727$			46.5849i			1.72774i	0.503717	+	0.863869i	$0.331965\pi$
$727$			46.5849i			1.72774i	−0.503717	+	0.863869i	$0.668035\pi$
$728$		0			0
$729$	−2.00078	−	26.9258i	−0.0741031	−	0.997251i
$730$		0			0
$731$	14.5273			0.537311
$732$		0			0
$733$	11.3654			0.419790			0.209895	−	0.977724i	$-0.432688\pi$
$733$	11.3654			0.419790			0.209895	+	0.977724i	$0.432688\pi$
$734$		0			0
$735$	−2.29398	−	8.15656i	−0.0846146	−	0.300859i
$736$		0			0
$737$		−	19.3301i		−	0.712032i
$738$		0			0
$739$		−	1.59545i		−	0.0586896i	−0.999569	−	0.0293448i	$-0.990658\pi$
$739$		−	1.59545i		−	0.0586896i	0.999569	−	0.0293448i	$-0.00934208\pi$
$740$		0			0
$741$	2.66527	+	9.47675i	0.0979113	+	0.348137i
$742$		0			0
$743$	−53.0612			−1.94663			−0.973313	−	0.229480i	$-0.926297\pi$
$743$	−53.0612			−1.94663			−0.973313	+	0.229480i	$0.926297\pi$
$744$		0			0
$745$	12.9204			0.473368
$746$		0			0
$747$	18.9050	−	11.5471i	0.691696	−	0.422488i
$748$		0			0
$749$		−	4.87836i		−	0.178251i
$750$		0			0
$751$		−	41.4331i		−	1.51192i	−0.654620	−	0.755958i	$-0.727171\pi$
$751$		−	41.4331i		−	1.51192i	0.654620	−	0.755958i	$-0.272829\pi$
$752$		0			0
$753$	−13.6510	+	3.83925i	−0.497470	+	0.139910i
$754$		0			0
$755$	−4.19554			−0.152691
$756$		0			0
$757$	−9.44598			−0.343320			−0.171660	−	0.985156i	$-0.554913\pi$
$757$	−9.44598			−0.343320			−0.171660	+	0.985156i	$0.554913\pi$
$758$		0			0
$759$	2.97737	−	0.837365i	0.108072	−	0.0303944i
$760$		0			0
$761$		−	48.5922i		−	1.76147i	−0.473614	−	0.880733i	$-0.657051\pi$
$761$		−	48.5922i		−	1.76147i	0.473614	−	0.880733i	$-0.342949\pi$
$762$		0			0
$763$			5.51736i			0.199742i
$764$		0			0
$765$	−9.60226	+	5.86505i	−0.347170	+	0.212051i
$766$		0			0
$767$	4.65223			0.167982
$768$		0			0
$769$	−6.92837			−0.249843			−0.124922	−	0.992167i	$-0.539868\pi$
$769$	−6.92837			−0.249843			−0.124922	+	0.992167i	$0.539868\pi$
$770$		0			0
$771$	−13.2134	−	46.9822i	−0.475870	−	1.69202i
$772$		0			0
$773$			44.2319i			1.59091i	0.606013	+	0.795455i	$0.292768\pi$
$773$			44.2319i			1.59091i	−0.606013	+	0.795455i	$0.707232\pi$
$774$		0			0
$775$			31.0657i			1.11591i
$776$		0			0
$777$	1.35284	+	4.81023i	0.0485330	+	0.172566i
$778$		0			0
$779$	35.5797			1.27478
$780$		0			0
$781$	−18.5812			−0.664886
$782$		0			0
$783$	35.1065	+	32.5946i	1.25460	+	1.16484i
$784$		0			0
$785$		−	5.46389i		−	0.195014i
$786$		0			0
$787$			35.0306i			1.24871i	0.781142	+	0.624354i	$0.214638\pi$
$787$			35.0306i			1.24871i	−0.781142	+	0.624354i	$0.785362\pi$
$788$		0			0
$789$	9.33473	−	2.62533i	0.332325	−	0.0934642i
$790$		0			0
$791$	−2.63213			−0.0935877
$792$		0			0
$793$	4.45289			0.158127
$794$		0			0
$795$	9.03587	−	2.54128i	0.320469	−	0.0901298i
$796$		0			0
$797$			31.8755i			1.12909i	0.825403	+	0.564544i	$0.190948\pi$
$797$			31.8755i			1.12909i	−0.825403	+	0.564544i	$0.809052\pi$
$798$		0			0
$799$			22.5652i			0.798300i
$800$		0			0
$801$	−6.99922	−	11.4591i	−0.247305	−	0.404888i
$802$		0			0
$803$	−17.4815			−0.616908
$804$		0			0
$805$	0.580743			0.0204685
$806$		0			0
$807$	9.01566	+	32.0565i	0.317366	+	1.12844i
$808$		0			0
$809$		−	18.0197i		−	0.633538i	−0.948503	−	0.316769i	$-0.897402\pi$
$809$		−	18.0197i		−	0.633538i	0.948503	−	0.316769i	$-0.102598\pi$
$810$		0			0
$811$			25.4682i			0.894309i	0.894457	+	0.447154i	$0.147563\pi$
$811$			25.4682i			0.894309i	−0.894457	+	0.447154i	$0.852437\pi$
$812$		0			0
$813$	5.46009	+	19.4141i	0.191494	+	0.680883i
$814$		0			0
$815$	−15.3056			−0.536133
$816$		0			0
$817$	−20.9906			−0.734368
$818$		0			0
$819$	−0.952459	−	1.55936i	−0.0332816	−	0.0544886i
$820$		0			0
$821$			52.9802i			1.84902i	0.381155	+	0.924511i	$0.375526\pi$
$821$			52.9802i			1.84902i	−0.381155	+	0.924511i	$0.624474\pi$
$822$		0			0
$823$			34.0010i			1.18520i	0.805497	+	0.592601i	$0.201899\pi$
$823$			34.0010i			1.18520i	−0.805497	+	0.592601i	$0.798101\pi$
$824$		0			0
$825$	13.1578	−	3.70053i	0.458094	−	0.128836i
$826$		0			0
$827$	−25.9653			−0.902903			−0.451452	−	0.892296i	$-0.649094\pi$
$827$	−25.9653			−0.902903			−0.451452	+	0.892296i	$0.649094\pi$
$828$		0			0
$829$	1.91808			0.0666177			0.0333089	−	0.999445i	$-0.489395\pi$
$829$	1.91808			0.0666177			0.0333089	+	0.999445i	$0.489395\pi$
$830$		0			0
$831$	46.3497	−	13.0355i	1.60785	−	0.452198i
$832$		0			0
$833$			31.5931i			1.09464i
$834$		0			0
$835$			3.02757i			0.104773i
$836$		0			0
$837$	−26.7683	−	24.8531i	−0.925249	−	0.859047i
$838$		0			0
$839$	32.5081			1.12230			0.561152	−	0.827713i	$-0.310358\pi$
$839$	32.5081			1.12230			0.561152	+	0.827713i	$0.310358\pi$
$840$		0			0
$841$	−55.9952			−1.93087
$842$		0			0
$843$	2.30791	+	8.20612i	0.0794888	+	0.282634i
$844$		0			0
$845$		−	9.42004i		−	0.324059i
$846$		0			0
$847$			5.95277i			0.204540i
$848$		0			0
$849$	−0.460426	−	1.63711i	−0.0158018	−	0.0561854i
$850$		0			0
$851$	3.78567			0.129771
$852$		0			0
$853$	14.9087			0.510464			0.255232	−	0.966880i	$-0.417848\pi$
$853$	14.9087			0.510464			0.255232	+	0.966880i	$0.417848\pi$
$854$		0			0
$855$	13.8744	−	8.47447i	0.474494	−	0.289821i
$856$		0			0
$857$			10.2741i			0.350957i	0.984483	+	0.175478i	$0.0561472\pi$
$857$			10.2741i			0.350957i	−0.984483	+	0.175478i	$0.943853\pi$
$858$		0			0
$859$		−	55.6512i		−	1.89879i	−0.314079	−	0.949397i	$-0.601696\pi$
$859$		−	55.6512i		−	1.89879i	0.314079	−	0.949397i	$-0.398304\pi$
$860$		0			0
$861$	−6.35736	+	1.78796i	−0.216658	+	0.0609337i
$862$		0			0
$863$	−30.9330			−1.05297			−0.526485	−	0.850184i	$-0.676490\pi$
$863$	−30.9330			−1.05297			−0.526485	+	0.850184i	$0.676490\pi$
$864$		0			0
$865$	−9.55256			−0.324797
$866$		0			0
$867$	12.0421	−	3.38677i	0.408972	−	0.115021i
$868$		0			0
$869$		−	9.97683i		−	0.338441i
$870$		0			0
$871$			8.65192i			0.293159i
$872$		0			0
$873$	13.7846	−	8.41960i	0.466537	−	0.284960i
$874$		0			0
$875$	5.47017			0.184925
$876$		0			0
$877$	16.4287			0.554756			0.277378	−	0.960761i	$-0.410535\pi$
$877$	16.4287			0.554756			0.277378	+	0.960761i	$0.410535\pi$
$878$		0			0
$879$	7.33891	+	26.0945i	0.247535	+	0.880146i
$880$		0			0
$881$			10.3054i			0.347197i	0.984817	+	0.173598i	$0.0555394\pi$
$881$			10.3054i			0.347197i	−0.984817	+	0.173598i	$0.944461\pi$
$882$		0			0
$883$		−	38.8235i		−	1.30651i	−0.757136	−	0.653257i	$-0.773402\pi$
$883$		−	38.8235i		−	1.30651i	0.757136	−	0.653257i	$-0.226598\pi$
$884$		0			0
$885$	−2.08010	−	7.39609i	−0.0699218	−	0.248617i
$886$		0			0
$887$	−31.4393			−1.05563			−0.527814	−	0.849360i	$-0.676988\pi$
$887$	−31.4393			−1.05563			−0.527814	+	0.849360i	$0.676988\pi$
$888$		0			0
$889$	−10.6597			−0.357515
$890$		0			0
$891$	−7.33779	+	14.2981i	−0.245825	+	0.479005i
$892$		0			0
$893$		−	32.6047i		−	1.09107i
$894$		0			0
$895$			13.9104i			0.464974i
$896$		0			0
$897$	−1.33264	+	0.374795i	−0.0444954	+	0.0125140i
$898$		0			0
$899$	64.8080			2.16147
$900$		0			0
$901$	−34.9990			−1.16598
$902$		0			0
$903$	3.75059	−	1.05483i	0.124812	−	0.0351025i
$904$		0			0
$905$		−	14.1394i		−	0.470008i
$906$		0			0
$907$		−	2.75263i		−	0.0913996i	−0.998955	−	0.0456998i	$-0.985448\pi$
$907$		−	2.75263i		−	0.0913996i	0.998955	−	0.0456998i	$-0.0145518\pi$
$908$		0			0
$909$	8.97625	+	14.6959i	0.297723	+	0.487432i
$910$		0			0
$911$	11.2212			0.371776			0.185888	−	0.982571i	$-0.440484\pi$
$911$	11.2212			0.371776			0.185888	+	0.982571i	$0.440484\pi$
$912$		0			0
$913$	−13.1857			−0.436384
$914$		0			0
$915$	−1.99097	−	7.07917i	−0.0658194	−	0.234030i
$916$		0			0
$917$		−	1.21497i		−	0.0401219i
$918$		0			0
$919$		−	15.0648i		−	0.496941i	−0.968640	−	0.248470i	$-0.920072\pi$
$919$		−	15.0648i		−	0.496941i	0.968640	−	0.248470i	$-0.0799278\pi$
$920$		0			0
$921$	−2.13021	−	7.57428i	−0.0701930	−	0.249581i
$922$		0			0
$923$	8.31672			0.273748
$924$		0			0
$925$	16.7299			0.550075
$926$		0			0
$927$	−14.2891	−	23.3941i	−0.469316	−	0.768364i
$928$		0			0
$929$			40.6623i			1.33409i	0.745019	+	0.667043i	$0.232440\pi$
$929$			40.6623i			1.33409i	−0.745019	+	0.667043i	$0.767560\pi$
$930$		0			0
$931$		−	45.6491i		−	1.49609i
$932$		0			0
$933$	36.3901	−	10.2345i	1.19136	−	0.335062i
$934$		0			0
$935$	6.69733			0.219026
$936$		0			0
$937$	−33.0182			−1.07866			−0.539328	−	0.842096i	$-0.681322\pi$
$937$	−33.0182			−1.07866			−0.539328	+	0.842096i	$0.681322\pi$
$938$		0			0
$939$	−19.9631	+	5.61449i	−0.651471	+	0.183222i
$940$		0			0
$941$		−	37.0363i		−	1.20735i	−0.797231	−	0.603674i	$-0.793703\pi$
$941$		−	37.0363i		−	1.20735i	0.797231	−	0.603674i	$-0.206297\pi$
$942$		0			0
$943$			5.00328i			0.162929i
$944$		0			0
$945$	−2.05320	+	2.21143i	−0.0667907	+	0.0719379i
$946$		0			0
$947$	−43.7360			−1.42123			−0.710614	−	0.703582i	$-0.751583\pi$
$947$	−43.7360			−1.42123			−0.710614	+	0.703582i	$0.751583\pi$
$948$		0			0
$949$	7.82452			0.253995
$950$		0			0
$951$	2.54827	+	9.06073i	0.0826333	+	0.293814i
$952$		0			0
$953$			1.15135i			0.0372960i	0.999826	+	0.0186480i	$0.00593619\pi$
$953$			1.15135i			0.0372960i	−0.999826	+	0.0186480i	$0.994064\pi$
$954$		0			0
$955$		−	15.3368i		−	0.496288i
$956$		0			0
$957$	−7.71990	−	27.4492i	−0.249549	−	0.887307i
$958$		0			0
$959$	−11.8834			−0.383735
$960$		0			0
$961$	−18.4155			−0.594048
$962$		0			0
$963$	16.3891	−	10.0105i	0.528132	−	0.322583i
$964$		0			0
$965$			6.37894i			0.205345i
$966$		0			0
$967$		−	54.3529i		−	1.74787i	−0.486042	−	0.873936i	$-0.661560\pi$
$967$		−	54.3529i		−	1.74787i	0.486042	−	0.873936i	$-0.338440\pi$
$968$		0			0
$969$	−58.3560	+	16.4122i	−1.87466	+	0.527237i
$970$		0			0
$971$	38.7926			1.24491			0.622457	−	0.782654i	$-0.286135\pi$
$971$	38.7926			1.24491			0.622457	+	0.782654i	$0.286135\pi$
$972$		0			0
$973$	2.14415			0.0687384
$974$		0			0
$975$	−5.88926	+	1.65632i	−0.188607	+	0.0530446i
$976$		0			0
$977$		−	40.7676i		−	1.30427i	−0.758102	−	0.652135i	$-0.773873\pi$
$977$		−	40.7676i		−	1.30427i	0.758102	−	0.652135i	$-0.226127\pi$
$978$		0			0
$979$			7.99243i			0.255439i
$980$		0			0
$981$	−18.5359	+	11.3217i	−0.591806	+	0.361475i
$982$		0			0
$983$	−36.7412			−1.17186			−0.585931	−	0.810361i	$-0.699271\pi$
$983$	−36.7412			−1.17186			−0.585931	+	0.810361i	$0.699271\pi$
$984$		0			0
$985$	11.3140			0.360494
$986$		0			0
$987$	1.63846	+	5.82578i	0.0521528	+	0.185437i
$988$		0			0
$989$		−	2.95173i		−	0.0938597i
$990$		0			0
$991$			11.5307i			0.366284i	0.983086	+	0.183142i	$0.0586269\pi$
$991$			11.5307i			0.366284i	−0.983086	+	0.183142i	$0.941373\pi$
$992$		0			0
$993$	1.61606	+	5.74613i	0.0512842	+	0.182348i
$994$		0			0
$995$	−12.3092			−0.390227
$996$		0			0
$997$	17.7936			0.563529			0.281765	−	0.959484i	$-0.409080\pi$
$997$	17.7936			0.563529			0.281765	+	0.959484i	$0.409080\pi$
$998$		0			0
$999$	−13.3842	+	14.4156i	−0.423456	+	0.456090i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.2.e.g.47.2	yes	8
3.2	odd	2		1104.2.e.f.47.8	yes	8
4.3	odd	2		1104.2.e.f.47.7	✓	8
12.11	even	2	inner	1104.2.e.g.47.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1104.2.e.f.47.7	✓	8	4.3	odd	2
1104.2.e.f.47.8	yes	8	3.2	odd	2
1104.2.e.g.47.1	yes	8	12.11	even	2	inner
1104.2.e.g.47.2	yes	8	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:	\( N \)	\(=\)	\( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1104.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.66736 + 0.468935i) q^{3} +0.762065i q^{5} -0.762065i q^{7} +(2.56020 - 1.56377i) q^{9} +O(q^{10})\)	\(q+(-1.66736 + 0.468935i) q^{3} +0.762065i q^{5} -0.762065i q^{7} +(2.56020 - 1.56377i) q^{9} -1.78567 q^{11} +0.799248 q^{13} +(-0.357359 - 1.27064i) q^{15} +4.92161i q^{17} -7.11128i q^{19} +(0.357359 + 1.27064i) q^{21} +1.00000 q^{23} +4.41926 q^{25} +(-3.53548 + 3.80794i) q^{27} -9.21928i q^{29} +7.02962i q^{31} +(2.97737 - 0.837365i) q^{33} +0.580743 q^{35} +3.78567 q^{37} +(-1.33264 + 0.374795i) q^{39} +5.00328i q^{41} -2.95173i q^{43} +(1.19169 + 1.95104i) q^{45} +4.58492 q^{47} +6.41926 q^{49} +(-2.30791 - 8.20612i) q^{51} +7.11128i q^{53} -1.36080i q^{55} +(3.33473 + 11.8571i) q^{57} +5.82076 q^{59} +5.57135 q^{61} +(-1.19169 - 1.95104i) q^{63} +0.609079i q^{65} +10.8251i q^{67} +(-1.66736 + 0.468935i) q^{69} +10.4057 q^{71} +9.78985 q^{73} +(-7.36851 + 2.07234i) q^{75} +1.36080i q^{77} +5.58715i q^{79} +(4.10925 - 8.00712i) q^{81} +7.38417 q^{83} -3.75059 q^{85} +(4.32324 + 15.3719i) q^{87} -4.47586i q^{89} -0.609079i q^{91} +(-3.29643 - 11.7209i) q^{93} +5.41926 q^{95} +5.38417 q^{97} +(-4.57168 + 2.79238i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{9}+O(q^{10}) \) 8 * q - 2 * q^9	\( 8 q - 2 q^{9} + 4 q^{11} - 4 q^{13} - 10 q^{15} + 10 q^{21} + 8 q^{23} - 16 q^{25} - 12 q^{27} - 10 q^{33} + 56 q^{35} + 12 q^{37} - 24 q^{39} - 6 q^{45} + 8 q^{47} - 38 q^{51} + 16 q^{59} + 8 q^{61} + 6 q^{63} + 24 q^{71} - 20 q^{73} - 52 q^{75} + 2 q^{81} + 20 q^{83} - 24 q^{85} - 40 q^{87} + 2 q^{93} - 8 q^{95} + 4 q^{97} - 62 q^{99}+O(q^{100}) \) 8 * q - 2 * q^9 + 4 * q^11 - 4 * q^13 - 10 * q^15 + 10 * q^21 + 8 * q^23 - 16 * q^25 - 12 * q^27 - 10 * q^33 + 56 * q^35 + 12 * q^37 - 24 * q^39 - 6 * q^45 + 8 * q^47 - 38 * q^51 + 16 * q^59 + 8 * q^61 + 6 * q^63 + 24 * q^71 - 20 * q^73 - 52 * q^75 + 2 * q^81 + 20 * q^83 - 24 * q^85 - 40 * q^87 + 2 * q^93 - 8 * q^95 + 4 * q^97 - 62 * q^99

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