LMFDB - Embedded newform 1200.3.bg.g.193.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,3,Mod(193,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1200.193");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1200.bg (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$32.6976317232$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			193.2
Root			$-1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1200.193
Dual form			1200.3.bg.g.1057.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 + 1.22474i) q^{3} +(-6.12372 + 6.12372i) q^{7} +3.00000i q^{9} +O(q^{10})$	$q+(1.22474 + 1.22474i) q^{3} +(-6.12372 + 6.12372i) q^{7} +3.00000i q^{9} -6.00000 q^{11} +(3.67423 + 3.67423i) q^{13} +(17.1464 - 17.1464i) q^{17} +23.0000i q^{19} -15.0000 q^{21} +(12.2474 + 12.2474i) q^{23} +(-3.67423 + 3.67423i) q^{27} -6.00000i q^{29} -25.0000 q^{31} +(-7.34847 - 7.34847i) q^{33} +(-24.4949 + 24.4949i) q^{37} +9.00000i q^{39} -60.0000 q^{41} +(-60.0125 - 60.0125i) q^{43} +(-7.34847 + 7.34847i) q^{47} -26.0000i q^{49} +42.0000 q^{51} +(-24.4949 - 24.4949i) q^{53} +(-28.1691 + 28.1691i) q^{57} -18.0000i q^{59} -37.0000 q^{61} +(-18.3712 - 18.3712i) q^{63} +(25.7196 - 25.7196i) q^{67} +30.0000i q^{69} -132.000 q^{71} +(-24.4949 - 24.4949i) q^{73} +(36.7423 - 36.7423i) q^{77} +10.0000i q^{79} -9.00000 q^{81} +(-2.44949 - 2.44949i) q^{83} +(7.34847 - 7.34847i) q^{87} +132.000i q^{89} -45.0000 q^{91} +(-30.6186 - 30.6186i) q^{93} +(23.2702 - 23.2702i) q^{97} -18.0000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 24 q^{11} - 60 q^{21} - 100 q^{31} - 240 q^{41} + 168 q^{51} - 148 q^{61} - 528 q^{71} - 36 q^{81} - 180 q^{91}+O(q^{100}) $ 4 * q - 24 * q^11 - 60 * q^21 - 100 * q^31 - 240 * q^41 + 168 * q^51 - 148 * q^61 - 528 * q^71 - 36 * q^81 - 180 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$1$	$1$

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
$3$	1.22474	+	1.22474i	0.408248	+	0.408248i
$3$	1.22474	+	1.22474i	0.408248	+	0.408248i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−6.12372	+	6.12372i	−0.874818	+	0.874818i	−0.992993	−	0.118175i	$-0.962296\pi$
$7$	−6.12372	+	6.12372i	−0.874818	+	0.874818i	0.118175	+	0.992993i	$0.462296\pi$
$8$		0			0
$9$			3.00000i			0.333333i
$10$		0			0
$11$	−6.00000			−0.545455			−0.272727	−	0.962091i	$-0.587926\pi$
$11$	−6.00000			−0.545455			−0.272727	+	0.962091i	$0.587926\pi$
$12$		0			0
$13$	3.67423	+	3.67423i	0.282633	+	0.282633i	0.834158	−	0.551525i	$-0.185954\pi$
$13$	3.67423	+	3.67423i	0.282633	+	0.282633i	−0.551525	+	0.834158i	$0.685954\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	17.1464	−	17.1464i	1.00861	−	1.00861i	0.00865084	−	0.999963i	$-0.497246\pi$
$17$	17.1464	−	17.1464i	1.00861	−	1.00861i	0.999963	−	0.00865084i	$-0.00275368\pi$
$18$		0			0
$19$			23.0000i			1.21053i	0.796025	+	0.605263i	$0.206932\pi$
$19$			23.0000i			1.21053i	−0.796025	+	0.605263i	$0.793068\pi$
$20$		0			0
$21$	−15.0000			−0.714286
$22$		0			0
$23$	12.2474	+	12.2474i	0.532498	+	0.532498i	0.921315	−	0.388817i	$-0.127116\pi$
$23$	12.2474	+	12.2474i	0.532498	+	0.532498i	−0.388817	+	0.921315i	$0.627116\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−3.67423	+	3.67423i	−0.136083	+	0.136083i
$28$		0			0
$29$		−	6.00000i		−	0.206897i	−0.994635	−	0.103448i	$-0.967012\pi$
$29$		−	6.00000i		−	0.206897i	0.994635	−	0.103448i	$-0.0329876\pi$
$30$		0			0
$31$	−25.0000			−0.806452			−0.403226	−	0.915101i	$-0.632111\pi$
$31$	−25.0000			−0.806452			−0.403226	+	0.915101i	$0.632111\pi$
$32$		0			0
$33$	−7.34847	−	7.34847i	−0.222681	−	0.222681i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−24.4949	+	24.4949i	−0.662024	+	0.662024i	−0.955857	−	0.293833i	$-0.905069\pi$
$37$	−24.4949	+	24.4949i	−0.662024	+	0.662024i	0.293833	+	0.955857i	$0.405069\pi$
$38$		0			0
$39$			9.00000i			0.230769i
$40$		0			0
$41$	−60.0000			−1.46341			−0.731707	−	0.681619i	$-0.761276\pi$
$41$	−60.0000			−1.46341			−0.731707	+	0.681619i	$0.761276\pi$
$42$		0			0
$43$	−60.0125	−	60.0125i	−1.39564	−	1.39564i	−0.812046	−	0.583594i	$-0.801646\pi$
$43$	−60.0125	−	60.0125i	−1.39564	−	1.39564i	−0.583594	−	0.812046i	$-0.698354\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−7.34847	+	7.34847i	−0.156350	+	0.156350i	−0.780947	−	0.624597i	$-0.785263\pi$
$47$	−7.34847	+	7.34847i	−0.156350	+	0.156350i	0.624597	+	0.780947i	$0.285263\pi$
$48$		0			0
$49$		−	26.0000i		−	0.530612i
$50$		0			0
$51$	42.0000			0.823529
$52$		0			0
$53$	−24.4949	−	24.4949i	−0.462168	−	0.462168i	0.437198	−	0.899365i	$-0.355971\pi$
$53$	−24.4949	−	24.4949i	−0.462168	−	0.462168i	−0.899365	+	0.437198i	$0.855971\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−28.1691	+	28.1691i	−0.494195	+	0.494195i
$58$		0			0
$59$		−	18.0000i		−	0.305085i	−0.988297	−	0.152542i	$-0.951254\pi$
$59$		−	18.0000i		−	0.305085i	0.988297	−	0.152542i	$-0.0487461\pi$
$60$		0			0
$61$	−37.0000			−0.606557			−0.303279	−	0.952902i	$-0.598081\pi$
$61$	−37.0000			−0.606557			−0.303279	+	0.952902i	$0.598081\pi$
$62$		0			0
$63$	−18.3712	−	18.3712i	−0.291606	−	0.291606i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	25.7196	−	25.7196i	0.383875	−	0.383875i	−0.488621	−	0.872496i	$-0.662500\pi$
$67$	25.7196	−	25.7196i	0.383875	−	0.383875i	0.872496	+	0.488621i	$0.162500\pi$
$68$		0			0
$69$			30.0000i			0.434783i
$70$		0			0
$71$	−132.000			−1.85915			−0.929577	−	0.368627i	$-0.879828\pi$
$71$	−132.000			−1.85915			−0.929577	+	0.368627i	$0.879828\pi$
$72$		0			0
$73$	−24.4949	−	24.4949i	−0.335547	−	0.335547i	0.519142	−	0.854688i	$-0.326252\pi$
$73$	−24.4949	−	24.4949i	−0.335547	−	0.335547i	−0.854688	+	0.519142i	$0.826252\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	36.7423	−	36.7423i	0.477173	−	0.477173i
$78$		0			0
$79$			10.0000i			0.126582i	0.997995	+	0.0632911i	$0.0201597\pi$
$79$			10.0000i			0.126582i	−0.997995	+	0.0632911i	$0.979840\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$		0			0
$83$	−2.44949	−	2.44949i	−0.0295119	−	0.0295119i	0.692197	−	0.721709i	$-0.256643\pi$
$83$	−2.44949	−	2.44949i	−0.0295119	−	0.0295119i	−0.721709	+	0.692197i	$0.756643\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	7.34847	−	7.34847i	0.0844652	−	0.0844652i
$88$		0			0
$89$			132.000i			1.48315i	0.670872	+	0.741573i	$0.265920\pi$
$89$			132.000i			1.48315i	−0.670872	+	0.741573i	$0.734080\pi$
$90$		0			0
$91$	−45.0000			−0.494505
$92$		0			0
$93$	−30.6186	−	30.6186i	−0.329232	−	0.329232i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	23.2702	−	23.2702i	0.239898	−	0.239898i	−0.576910	−	0.816808i	$-0.695741\pi$
$97$	23.2702	−	23.2702i	0.239898	−	0.239898i	0.816808	+	0.576910i	$0.195741\pi$
$98$		0			0
$99$		−	18.0000i		−	0.181818i
$100$		0			0
$101$	96.0000			0.950495			0.475248	−	0.879852i	$-0.342359\pi$
$101$	96.0000			0.950495			0.475248	+	0.879852i	$0.342359\pi$
$102$		0			0
$103$	19.5959	+	19.5959i	0.190252	+	0.190252i	0.795805	−	0.605553i	$-0.207048\pi$
$103$	19.5959	+	19.5959i	0.190252	+	0.190252i	−0.605553	+	0.795805i	$0.707048\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	29.3939	−	29.3939i	0.274709	−	0.274709i	−0.556283	−	0.830993i	$-0.687773\pi$
$107$	29.3939	−	29.3939i	0.274709	−	0.274709i	0.830993	+	0.556283i	$0.187773\pi$
$108$		0			0
$109$			167.000i			1.53211i	0.642775	+	0.766055i	$0.277783\pi$
$109$			167.000i			1.53211i	−0.642775	+	0.766055i	$0.722217\pi$
$110$		0			0
$111$	−60.0000			−0.540541
$112$		0			0
$113$	−58.7878	−	58.7878i	−0.520246	−	0.520246i	0.397400	−	0.917646i	$-0.369913\pi$
$113$	−58.7878	−	58.7878i	−0.520246	−	0.520246i	−0.917646	+	0.397400i	$0.869913\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−11.0227	+	11.0227i	−0.0942111	+	0.0942111i
$118$		0			0
$119$			210.000i			1.76471i
$120$		0			0
$121$	−85.0000			−0.702479
$122$		0			0
$123$	−73.4847	−	73.4847i	−0.597437	−	0.597437i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	44.0908	−	44.0908i	0.347172	−	0.347172i	−0.511883	−	0.859055i	$-0.671052\pi$
$127$	44.0908	−	44.0908i	0.347172	−	0.347172i	0.859055	+	0.511883i	$0.171052\pi$
$128$		0			0
$129$		−	147.000i		−	1.13953i
$130$		0			0
$131$	−108.000			−0.824427			−0.412214	−	0.911087i	$-0.635244\pi$
$131$	−108.000			−0.824427			−0.412214	+	0.911087i	$0.635244\pi$
$132$		0			0
$133$	−140.846	−	140.846i	−1.05899	−	1.05899i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	120.025	−	120.025i	0.876095	−	0.876095i	−0.117033	−	0.993128i	$-0.537338\pi$
$137$	120.025	−	120.025i	0.876095	−	0.876095i	0.993128	+	0.117033i	$0.0373384\pi$
$138$		0			0
$139$		−	58.0000i		−	0.417266i	−0.977994	−	0.208633i	$-0.933099\pi$
$139$		−	58.0000i		−	0.417266i	0.977994	−	0.208633i	$-0.0669014\pi$
$140$		0			0
$141$	−18.0000			−0.127660
$142$		0			0
$143$	−22.0454	−	22.0454i	−0.154164	−	0.154164i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	31.8434	−	31.8434i	0.216622	−	0.216622i
$148$		0			0
$149$			186.000i			1.24832i	0.781296	+	0.624161i	$0.214559\pi$
$149$			186.000i			1.24832i	−0.781296	+	0.624161i	$0.785441\pi$
$150$		0			0
$151$	83.0000			0.549669			0.274834	−	0.961492i	$-0.411377\pi$
$151$	83.0000			0.549669			0.274834	+	0.961492i	$0.411377\pi$
$152$		0			0
$153$	51.4393	+	51.4393i	0.336204	+	0.336204i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−45.3156	+	45.3156i	−0.288634	+	0.288634i	−0.836540	−	0.547906i	$-0.815425\pi$
$157$	−45.3156	+	45.3156i	−0.288634	+	0.288634i	0.547906	+	0.836540i	$0.315425\pi$
$158$		0			0
$159$		−	60.0000i		−	0.377358i
$160$		0			0
$161$	−150.000			−0.931677
$162$		0			0
$163$	−99.2043	−	99.2043i	−0.608616	−	0.608616i	0.333969	−	0.942584i	$-0.391612\pi$
$163$	−99.2043	−	99.2043i	−0.608616	−	0.608616i	−0.942584	+	0.333969i	$0.891612\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	97.9796	−	97.9796i	0.586704	−	0.586704i	−0.350033	−	0.936737i	$-0.613830\pi$
$167$	97.9796	−	97.9796i	0.586704	−	0.586704i	0.936737	+	0.350033i	$0.113830\pi$
$168$		0			0
$169$		−	142.000i		−	0.840237i
$170$		0			0
$171$	−69.0000			−0.403509
$172$		0			0
$173$	173.914	+	173.914i	1.00528	+	1.00528i	0.999986	+	0.00529594i	$0.00168576\pi$
$173$	173.914	+	173.914i	1.00528	+	1.00528i	0.00529594	+	0.999986i	$0.498314\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	22.0454	−	22.0454i	0.124550	−	0.124550i
$178$		0			0
$179$			150.000i			0.837989i	0.907989	+	0.418994i	$0.137617\pi$
$179$			150.000i			0.837989i	−0.907989	+	0.418994i	$0.862383\pi$
$180$		0			0
$181$	−215.000			−1.18785			−0.593923	−	0.804522i	$-0.702421\pi$
$181$	−215.000			−1.18785			−0.593923	+	0.804522i	$0.702421\pi$
$182$		0			0
$183$	−45.3156	−	45.3156i	−0.247626	−	0.247626i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−102.879	+	102.879i	−0.550153	+	0.550153i
$188$		0			0
$189$		−	45.0000i		−	0.238095i
$190$		0			0
$191$	234.000			1.22513			0.612565	−	0.790420i	$-0.290138\pi$
$191$	234.000			1.22513			0.612565	+	0.790420i	$0.290138\pi$
$192$		0			0
$193$	−64.9115	−	64.9115i	−0.336329	−	0.336329i	0.518655	−	0.854984i	$-0.326433\pi$
$193$	−64.9115	−	64.9115i	−0.336329	−	0.336329i	−0.854984	+	0.518655i	$0.826433\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−193.510	+	193.510i	−0.982283	+	0.982283i	−0.999846	−	0.0175631i	$-0.994409\pi$
$197$	−193.510	+	193.510i	−0.982283	+	0.982283i	0.0175631	+	0.999846i	$0.494409\pi$
$198$		0			0
$199$		−	11.0000i		−	0.0552764i	−0.999618	−	0.0276382i	$-0.991201\pi$
$199$		−	11.0000i		−	0.0552764i	0.999618	−	0.0276382i	$-0.00879863\pi$
$200$		0			0
$201$	63.0000			0.313433
$202$		0			0
$203$	36.7423	+	36.7423i	0.180997	+	0.180997i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−36.7423	+	36.7423i	−0.177499	+	0.177499i
$208$		0			0
$209$		−	138.000i		−	0.660287i
$210$		0			0
$211$	85.0000			0.402844			0.201422	−	0.979505i	$-0.435444\pi$
$211$	85.0000			0.402844			0.201422	+	0.979505i	$0.435444\pi$
$212$		0			0
$213$	−161.666	−	161.666i	−0.758997	−	0.758997i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	153.093	−	153.093i	0.705498	−	0.705498i
$218$		0			0
$219$		−	60.0000i		−	0.273973i
$220$		0			0
$221$	126.000			0.570136
$222$		0			0
$223$	−101.654	−	101.654i	−0.455847	−	0.455847i	0.441443	−	0.897289i	$-0.354467\pi$
$223$	−101.654	−	101.654i	−0.455847	−	0.455847i	−0.897289	+	0.441443i	$0.854467\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−195.959	+	195.959i	−0.863256	+	0.863256i	−0.991715	−	0.128459i	$-0.958997\pi$
$227$	−195.959	+	195.959i	−0.863256	+	0.863256i	0.128459	+	0.991715i	$0.458997\pi$
$228$		0			0
$229$		−	227.000i		−	0.991266i	−0.868532	−	0.495633i	$-0.834936\pi$
$229$		−	227.000i		−	0.991266i	0.868532	−	0.495633i	$-0.165064\pi$
$230$		0			0
$231$	90.0000			0.389610
$232$		0			0
$233$	102.879	+	102.879i	0.441539	+	0.441539i	0.892529	−	0.450990i	$-0.148929\pi$
$233$	102.879	+	102.879i	0.441539	+	0.441539i	−0.450990	+	0.892529i	$0.648929\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−12.2474	+	12.2474i	−0.0516770	+	0.0516770i
$238$		0			0
$239$			228.000i			0.953975i	0.878910	+	0.476987i	$0.158271\pi$
$239$			228.000i			0.953975i	−0.878910	+	0.476987i	$0.841729\pi$
$240$		0			0
$241$	191.000			0.792531			0.396266	−	0.918136i	$-0.370306\pi$
$241$	191.000			0.792531			0.396266	+	0.918136i	$0.370306\pi$
$242$		0			0
$243$	−11.0227	−	11.0227i	−0.0453609	−	0.0453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−84.5074	+	84.5074i	−0.342135	+	0.342135i
$248$		0			0
$249$		−	6.00000i		−	0.0240964i
$250$		0			0
$251$	−192.000			−0.764940			−0.382470	−	0.923968i	$-0.624927\pi$
$251$	−192.000			−0.764940			−0.382470	+	0.923968i	$0.624927\pi$
$252$		0			0
$253$	−73.4847	−	73.4847i	−0.290453	−	0.290453i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	142.070	−	142.070i	0.552803	−	0.552803i	−0.374446	−	0.927249i	$-0.622167\pi$
$257$	142.070	−	142.070i	0.552803	−	0.552803i	0.927249	+	0.374446i	$0.122167\pi$
$258$		0			0
$259$		−	300.000i		−	1.15830i
$260$		0			0
$261$	18.0000			0.0689655
$262$		0			0
$263$	249.848	+	249.848i	0.949992	+	0.949992i	0.998808	−	0.0488156i	$-0.0155447\pi$
$263$	249.848	+	249.848i	0.949992	+	0.949992i	−0.0488156	+	0.998808i	$0.515545\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−161.666	+	161.666i	−0.605492	+	0.605492i
$268$		0			0
$269$			462.000i			1.71747i	0.512418	+	0.858736i	$0.328750\pi$
$269$			462.000i			1.71747i	−0.512418	+	0.858736i	$0.671250\pi$
$270$		0			0
$271$	−446.000			−1.64576			−0.822878	−	0.568218i	$-0.807633\pi$
$271$	−446.000			−1.64576			−0.822878	+	0.568218i	$0.807633\pi$
$272$		0			0
$273$	−55.1135	−	55.1135i	−0.201881	−	0.201881i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−175.139	+	175.139i	−0.632269	+	0.632269i	−0.948637	−	0.316368i	$-0.897537\pi$
$277$	−175.139	+	175.139i	−0.632269	+	0.632269i	0.316368	+	0.948637i	$0.397537\pi$
$278$		0			0
$279$		−	75.0000i		−	0.268817i
$280$		0			0
$281$	−546.000			−1.94306			−0.971530	−	0.236916i	$-0.923864\pi$
$281$	−546.000			−1.94306			−0.971530	+	0.236916i	$0.923864\pi$
$282$		0			0
$283$	192.285	+	192.285i	0.679452	+	0.679452i	0.959876	−	0.280424i	$-0.0904751\pi$
$283$	192.285	+	192.285i	0.679452	+	0.679452i	−0.280424	+	0.959876i	$0.590475\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	367.423	−	367.423i	1.28022	−	1.28022i
$288$		0			0
$289$		−	299.000i		−	1.03460i
$290$		0			0
$291$	57.0000			0.195876
$292$		0			0
$293$	276.792	+	276.792i	0.944684	+	0.944684i	0.998548	−	0.0538645i	$-0.0171539\pi$
$293$	276.792	+	276.792i	0.944684	+	0.944684i	−0.0538645	+	0.998548i	$0.517154\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	22.0454	−	22.0454i	0.0742270	−	0.0742270i
$298$		0			0
$299$			90.0000i			0.301003i
$300$		0			0
$301$	735.000			2.44186
$302$		0			0
$303$	117.576	+	117.576i	0.388038	+	0.388038i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−86.9569	+	86.9569i	−0.283247	+	0.283247i	−0.834403	−	0.551155i	$-0.814187\pi$
$307$	−86.9569	+	86.9569i	−0.283247	+	0.283247i	0.551155	+	0.834403i	$0.314187\pi$
$308$		0			0
$309$			48.0000i			0.155340i
$310$		0			0
$311$	294.000			0.945338			0.472669	−	0.881240i	$-0.343291\pi$
$311$	294.000			0.945338			0.472669	+	0.881240i	$0.343291\pi$
$312$		0			0
$313$	393.143	+	393.143i	1.25605	+	1.25605i	0.952964	+	0.303085i	$0.0980165\pi$
$313$	393.143	+	393.143i	1.25605	+	1.25605i	0.303085	+	0.952964i	$0.401984\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	93.0806	−	93.0806i	0.293630	−	0.293630i	−0.544883	−	0.838512i	$-0.683426\pi$
$317$	93.0806	−	93.0806i	0.293630	−	0.293630i	0.838512	+	0.544883i	$0.183426\pi$
$318$		0			0
$319$			36.0000i			0.112853i
$320$		0			0
$321$	72.0000			0.224299
$322$		0			0
$323$	394.368	+	394.368i	1.22095	+	1.22095i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−204.532	+	204.532i	−0.625481	+	0.625481i
$328$		0			0
$329$		−	90.0000i		−	0.273556i
$330$		0			0
$331$	178.000			0.537764			0.268882	−	0.963173i	$-0.413346\pi$
$331$	178.000			0.537764			0.268882	+	0.963173i	$0.413346\pi$
$332$		0			0
$333$	−73.4847	−	73.4847i	−0.220675	−	0.220675i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−150.644	+	150.644i	−0.447014	+	0.447014i	−0.894361	−	0.447347i	$-0.852369\pi$
$337$	−150.644	+	150.644i	−0.447014	+	0.447014i	0.447347	+	0.894361i	$0.352369\pi$
$338$		0			0
$339$		−	144.000i		−	0.424779i
$340$		0			0
$341$	150.000			0.439883
$342$		0			0
$343$	−140.846	−	140.846i	−0.410629	−	0.410629i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	374.772	−	374.772i	1.08003	−	1.08003i	0.0835290	−	0.996505i	$-0.473381\pi$
$347$	374.772	−	374.772i	1.08003	−	1.08003i	0.996505	−	0.0835290i	$-0.0266191\pi$
$348$		0			0
$349$			514.000i			1.47278i	0.676558	+	0.736390i	$0.263471\pi$
$349$			514.000i			1.47278i	−0.676558	+	0.736390i	$0.736529\pi$
$350$		0			0
$351$	−27.0000			−0.0769231
$352$		0			0
$353$	242.499	+	242.499i	0.686967	+	0.686967i	0.961561	−	0.274593i	$-0.0885432\pi$
$353$	242.499	+	242.499i	0.686967	+	0.686967i	−0.274593	+	0.961561i	$0.588543\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−257.196	+	257.196i	−0.720438	+	0.720438i
$358$		0			0
$359$		−	648.000i		−	1.80501i	−0.430675	−	0.902507i	$-0.641725\pi$
$359$		−	648.000i		−	1.80501i	0.430675	−	0.902507i	$-0.358275\pi$
$360$		0			0
$361$	−168.000			−0.465374
$362$		0			0
$363$	−104.103	−	104.103i	−0.286786	−	0.286786i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−143.295	+	143.295i	−0.390450	+	0.390450i	−0.874848	−	0.484398i	$-0.839039\pi$
$367$	−143.295	+	143.295i	−0.390450	+	0.390450i	0.484398	+	0.874848i	$0.339039\pi$
$368$		0			0
$369$		−	180.000i		−	0.487805i
$370$		0			0
$371$	300.000			0.808625
$372$		0			0
$373$	35.5176	+	35.5176i	0.0952215	+	0.0952215i	0.753113	−	0.657891i	$-0.228551\pi$
$373$	35.5176	+	35.5176i	0.0952215	+	0.0952215i	−0.657891	+	0.753113i	$0.728551\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	22.0454	−	22.0454i	0.0584759	−	0.0584759i
$378$		0			0
$379$			215.000i			0.567282i	0.958930	+	0.283641i	$0.0915425\pi$
$379$			215.000i			0.567282i	−0.958930	+	0.283641i	$0.908458\pi$
$380$		0			0
$381$	108.000			0.283465
$382$		0			0
$383$	−19.5959	−	19.5959i	−0.0511643	−	0.0511643i	0.681062	−	0.732226i	$-0.261519\pi$
$383$	−19.5959	−	19.5959i	−0.0511643	−	0.0511643i	−0.732226	+	0.681062i	$0.761519\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	180.037	−	180.037i	0.465213	−	0.465213i
$388$		0			0
$389$		−	192.000i		−	0.493573i	−0.969070	−	0.246787i	$-0.920625\pi$
$389$		−	192.000i		−	0.493573i	0.969070	−	0.246787i	$-0.0793747\pi$
$390$		0			0
$391$	420.000			1.07417
$392$		0			0
$393$	−132.272	−	132.272i	−0.336571	−	0.336571i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−366.199	+	366.199i	−0.922415	+	0.922415i	−0.997200	−	0.0747848i	$-0.976173\pi$
$397$	−366.199	+	366.199i	−0.922415	+	0.922415i	0.0747848	+	0.997200i	$0.476173\pi$
$398$		0			0
$399$		−	345.000i		−	0.864662i
$400$		0			0
$401$	228.000			0.568579			0.284289	−	0.958739i	$-0.408242\pi$
$401$	228.000			0.568579			0.284289	+	0.958739i	$0.408242\pi$
$402$		0			0
$403$	−91.8559	−	91.8559i	−0.227930	−	0.227930i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	146.969	−	146.969i	0.361104	−	0.361104i
$408$		0			0
$409$		−	707.000i		−	1.72861i	−0.502971	−	0.864303i	$-0.667760\pi$
$409$		−	707.000i		−	1.72861i	0.502971	−	0.864303i	$-0.332240\pi$
$410$		0			0
$411$	294.000			0.715328
$412$		0			0
$413$	110.227	+	110.227i	0.266894	+	0.266894i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	71.0352	−	71.0352i	0.170348	−	0.170348i
$418$		0			0
$419$			48.0000i			0.114558i	0.998358	+	0.0572792i	$0.0182425\pi$
$419$			48.0000i			0.114558i	−0.998358	+	0.0572792i	$0.981757\pi$
$420$		0			0
$421$	514.000			1.22090			0.610451	−	0.792054i	$-0.290988\pi$
$421$	514.000			1.22090			0.610451	+	0.792054i	$0.290988\pi$
$422$		0			0
$423$	−22.0454	−	22.0454i	−0.0521168	−	0.0521168i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	226.578	−	226.578i	0.530627	−	0.530627i
$428$		0			0
$429$		−	54.0000i		−	0.125874i
$430$		0			0
$431$	−30.0000			−0.0696056			−0.0348028	−	0.999394i	$-0.511080\pi$
$431$	−30.0000			−0.0696056			−0.0348028	+	0.999394i	$0.511080\pi$
$432$		0			0
$433$	−353.951	−	353.951i	−0.817439	−	0.817439i	0.168297	−	0.985736i	$-0.446173\pi$
$433$	−353.951	−	353.951i	−0.817439	−	0.817439i	−0.985736	+	0.168297i	$0.946173\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−281.691	+	281.691i	−0.644603	+	0.644603i
$438$		0			0
$439$		−	575.000i		−	1.30979i	−0.755718	−	0.654897i	$-0.772712\pi$
$439$		−	575.000i		−	1.30979i	0.755718	−	0.654897i	$-0.227288\pi$
$440$		0			0
$441$	78.0000			0.176871
$442$		0			0
$443$	112.677	+	112.677i	0.254349	+	0.254349i	0.822751	−	0.568402i	$-0.192438\pi$
$443$	112.677	+	112.677i	0.254349	+	0.254349i	−0.568402	+	0.822751i	$0.692438\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−227.803	+	227.803i	−0.509625	+	0.509625i
$448$		0			0
$449$		−	204.000i		−	0.454343i	−0.973855	−	0.227171i	$-0.927052\pi$
$449$		−	204.000i		−	0.454343i	0.973855	−	0.227171i	$-0.0729478\pi$
$450$		0			0
$451$	360.000			0.798226
$452$		0			0
$453$	101.654	+	101.654i	0.224401	+	0.224401i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	440.908	−	440.908i	0.964788	−	0.964788i	−0.0346127	−	0.999401i	$-0.511020\pi$
$457$	440.908	−	440.908i	0.964788	−	0.964788i	0.999401	+	0.0346127i	$0.0110198\pi$
$458$		0			0
$459$			126.000i			0.274510i
$460$		0			0
$461$	132.000			0.286334			0.143167	−	0.989699i	$-0.454271\pi$
$461$	132.000			0.286334			0.143167	+	0.989699i	$0.454271\pi$
$462$		0			0
$463$	436.009	+	436.009i	0.941704	+	0.941704i	0.998392	−	0.0566875i	$-0.0180539\pi$
$463$	436.009	+	436.009i	0.941704	+	0.941704i	−0.0566875	+	0.998392i	$0.518054\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−276.792	+	276.792i	−0.592703	+	0.592703i	−0.938361	−	0.345658i	$-0.887656\pi$
$467$	−276.792	+	276.792i	−0.592703	+	0.592703i	0.345658	+	0.938361i	$0.387656\pi$
$468$		0			0
$469$			315.000i			0.671642i
$470$		0			0
$471$	−111.000			−0.235669
$472$		0			0
$473$	360.075	+	360.075i	0.761258	+	0.761258i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	73.4847	−	73.4847i	0.154056	−	0.154056i
$478$		0			0
$479$			810.000i			1.69102i	0.533957	+	0.845511i	$0.320704\pi$
$479$			810.000i			1.69102i	−0.533957	+	0.845511i	$0.679296\pi$
$480$		0			0
$481$	−180.000			−0.374220
$482$		0			0
$483$	−183.712	−	183.712i	−0.380356	−	0.380356i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−488.673	+	488.673i	−1.00344	+	1.00344i	−0.00344166	+	0.999994i	$0.501096\pi$
$487$	−488.673	+	488.673i	−1.00344	+	1.00344i	−0.999994	+	0.00344166i	$0.998904\pi$
$488$		0			0
$489$		−	243.000i		−	0.496933i
$490$		0			0
$491$	−348.000			−0.708758			−0.354379	−	0.935102i	$-0.615308\pi$
$491$	−348.000			−0.708758			−0.354379	+	0.935102i	$0.615308\pi$
$492$		0			0
$493$	−102.879	−	102.879i	−0.208679	−	0.208679i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	808.332	−	808.332i	1.62642	−	1.62642i
$498$		0			0
$499$		−	227.000i		−	0.454910i	−0.973789	−	0.227455i	$-0.926960\pi$
$499$		−	227.000i		−	0.454910i	0.973789	−	0.227455i	$-0.0730404\pi$
$500$		0			0
$501$	240.000			0.479042
$502$		0			0
$503$	137.171	+	137.171i	0.272707	+	0.272707i	0.830189	−	0.557482i	$-0.188233\pi$
$503$	137.171	+	137.171i	0.272707	+	0.272707i	−0.557482	+	0.830189i	$0.688233\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	173.914	−	173.914i	0.343025	−	0.343025i
$508$		0			0
$509$			618.000i			1.21415i	0.794646	+	0.607073i	$0.207656\pi$
$509$			618.000i			1.21415i	−0.794646	+	0.607073i	$0.792344\pi$
$510$		0			0
$511$	300.000			0.587084
$512$		0			0
$513$	−84.5074	−	84.5074i	−0.164732	−	0.164732i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	44.0908	−	44.0908i	0.0852820	−	0.0852820i
$518$		0			0
$519$			426.000i			0.820809i
$520$		0			0
$521$	690.000			1.32438			0.662188	−	0.749338i	$-0.269628\pi$
$521$	690.000			1.32438			0.662188	+	0.749338i	$0.269628\pi$
$522$		0			0
$523$	−255.972	−	255.972i	−0.489430	−	0.489430i	0.418697	−	0.908126i	$-0.362487\pi$
$523$	−255.972	−	255.972i	−0.489430	−	0.489430i	−0.908126	+	0.418697i	$0.862487\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−428.661	+	428.661i	−0.813398	+	0.813398i
$528$		0			0
$529$		−	229.000i		−	0.432892i
$530$		0			0
$531$	54.0000			0.101695
$532$		0			0
$533$	−220.454	−	220.454i	−0.413610	−	0.413610i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−183.712	+	183.712i	−0.342108	+	0.342108i
$538$		0			0
$539$			156.000i			0.289425i
$540$		0			0
$541$	−325.000			−0.600739			−0.300370	−	0.953823i	$-0.597110\pi$
$541$	−325.000			−0.600739			−0.300370	+	0.953823i	$0.597110\pi$
$542$		0			0
$543$	−263.320	−	263.320i	−0.484936	−	0.484936i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	489.898	−	489.898i	0.895609	−	0.895609i	−0.0994353	−	0.995044i	$-0.531704\pi$
$547$	489.898	−	489.898i	0.895609	−	0.895609i	0.995044	+	0.0994353i	$0.0317036\pi$
$548$		0			0
$549$		−	111.000i		−	0.202186i
$550$		0			0
$551$	138.000			0.250454
$552$		0			0
$553$	−61.2372	−	61.2372i	−0.110736	−	0.110736i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−154.318	+	154.318i	−0.277052	+	0.277052i	−0.831931	−	0.554879i	$-0.812764\pi$
$557$	−154.318	+	154.318i	−0.277052	+	0.277052i	0.554879	+	0.831931i	$0.312764\pi$
$558$		0			0
$559$		−	441.000i		−	0.788909i
$560$		0			0
$561$	−252.000			−0.449198
$562$		0			0
$563$	609.923	+	609.923i	1.08334	+	1.08334i	0.996195	+	0.0871492i	$0.0277757\pi$
$563$	609.923	+	609.923i	1.08334	+	1.08334i	0.0871492	+	0.996195i	$0.472224\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	55.1135	−	55.1135i	0.0972020	−	0.0972020i
$568$		0			0
$569$		−	198.000i		−	0.347979i	−0.984748	−	0.173989i	$-0.944334\pi$
$569$		−	198.000i		−	0.347979i	0.984748	−	0.173989i	$-0.0556659\pi$
$570$		0			0
$571$	169.000			0.295972			0.147986	−	0.988989i	$-0.452721\pi$
$571$	169.000			0.295972			0.147986	+	0.988989i	$0.452721\pi$
$572$		0			0
$573$	286.590	+	286.590i	0.500158	+	0.500158i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	243.724	−	243.724i	0.422399	−	0.422399i	−0.463630	−	0.886029i	$-0.653453\pi$
$577$	243.724	−	243.724i	0.422399	−	0.422399i	0.886029	+	0.463630i	$0.153453\pi$
$578$		0			0
$579$		−	159.000i		−	0.274611i
$580$		0			0
$581$	30.0000			0.0516351
$582$		0			0
$583$	146.969	+	146.969i	0.252092	+	0.252092i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−289.040	+	289.040i	−0.492402	+	0.492402i	−0.909062	−	0.416660i	$-0.863200\pi$
$587$	−289.040	+	289.040i	−0.492402	+	0.492402i	0.416660	+	0.909062i	$0.363200\pi$
$588$		0			0
$589$		−	575.000i		−	0.976231i
$590$		0			0
$591$	−474.000			−0.802030
$592$		0			0
$593$	421.312	+	421.312i	0.710476	+	0.710476i	0.966635	−	0.256159i	$-0.0824570\pi$
$593$	421.312	+	421.312i	0.710476	+	0.710476i	−0.256159	+	0.966635i	$0.582457\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	13.4722	−	13.4722i	0.0225665	−	0.0225665i
$598$		0			0
$599$			144.000i			0.240401i	0.992750	+	0.120200i	$0.0383537\pi$
$599$			144.000i			0.240401i	−0.992750	+	0.120200i	$0.961646\pi$
$600$		0			0
$601$	−899.000			−1.49584			−0.747920	−	0.663789i	$-0.768947\pi$
$601$	−899.000			−1.49584			−0.747920	+	0.663789i	$0.768947\pi$
$602$		0			0
$603$	77.1589	+	77.1589i	0.127958	+	0.127958i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−4.89898	+	4.89898i	−0.00807081	+	0.00807081i	−0.711131	−	0.703060i	$-0.751817\pi$
$607$	−4.89898	+	4.89898i	−0.00807081	+	0.00807081i	0.703060	+	0.711131i	$0.251817\pi$
$608$		0			0
$609$			90.0000i			0.147783i
$610$		0			0
$611$	−54.0000			−0.0883797
$612$		0			0
$613$	259.646	+	259.646i	0.423566	+	0.423566i	0.886430	−	0.462864i	$-0.153178\pi$
$613$	259.646	+	259.646i	0.423566	+	0.423566i	−0.462864	+	0.886430i	$0.653178\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	9.79796	−	9.79796i	0.0158800	−	0.0158800i	−0.699122	−	0.715002i	$-0.746426\pi$
$617$	9.79796	−	9.79796i	0.0158800	−	0.0158800i	0.715002	+	0.699122i	$0.246426\pi$
$618$		0			0
$619$			637.000i			1.02908i	0.857467	+	0.514540i	$0.172037\pi$
$619$			637.000i			1.02908i	−0.857467	+	0.514540i	$0.827963\pi$
$620$		0			0
$621$	−90.0000			−0.144928
$622$		0			0
$623$	−808.332	−	808.332i	−1.29748	−	1.29748i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	169.015	−	169.015i	0.269561	−	0.269561i
$628$		0			0
$629$			840.000i			1.33545i
$630$		0			0
$631$	−287.000			−0.454834			−0.227417	−	0.973798i	$-0.573028\pi$
$631$	−287.000			−0.454834			−0.227417	+	0.973798i	$0.573028\pi$
$632$		0			0
$633$	104.103	+	104.103i	0.164460	+	0.164460i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	95.5301	−	95.5301i	0.149969	−	0.149969i
$638$		0			0
$639$		−	396.000i		−	0.619718i
$640$		0			0
$641$	−60.0000			−0.0936037			−0.0468019	−	0.998904i	$-0.514903\pi$
$641$	−60.0000			−0.0936037			−0.0468019	+	0.998904i	$0.514903\pi$
$642$		0			0
$643$	191.060	+	191.060i	0.297139	+	0.297139i	0.839892	−	0.542753i	$-0.182618\pi$
$643$	191.060	+	191.060i	0.297139	+	0.297139i	−0.542753	+	0.839892i	$0.682618\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	656.463	−	656.463i	1.01463	−	1.01463i	0.0147349	−	0.999891i	$-0.495310\pi$
$647$	656.463	−	656.463i	1.01463	−	1.01463i	0.999891	−	0.0147349i	$-0.00469044\pi$
$648$		0			0
$649$			108.000i			0.166410i
$650$		0			0
$651$	375.000			0.576037
$652$		0			0
$653$	−751.993	−	751.993i	−1.15160	−	1.15160i	−0.986232	−	0.165365i	$-0.947120\pi$
$653$	−751.993	−	751.993i	−1.15160	−	1.15160i	−0.165365	−	0.986232i	$-0.552880\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	73.4847	−	73.4847i	0.111849	−	0.111849i
$658$		0			0
$659$			420.000i			0.637329i	0.947868	+	0.318665i	$0.103234\pi$
$659$			420.000i			0.637329i	−0.947868	+	0.318665i	$0.896766\pi$
$660$		0			0
$661$	−158.000			−0.239032			−0.119516	−	0.992832i	$-0.538134\pi$
$661$	−158.000			−0.239032			−0.119516	+	0.992832i	$0.538134\pi$
$662$		0			0
$663$	154.318	+	154.318i	0.232757	+	0.232757i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	73.4847	−	73.4847i	0.110172	−	0.110172i
$668$		0			0
$669$		−	249.000i		−	0.372197i
$670$		0			0
$671$	222.000			0.330849
$672$		0			0
$673$	122.474	+	122.474i	0.181983	+	0.181983i	0.792219	−	0.610236i	$-0.208926\pi$
$673$	122.474	+	122.474i	0.181983	+	0.181983i	−0.610236	+	0.792219i	$0.708926\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−242.499	+	242.499i	−0.358197	+	0.358197i	−0.863148	−	0.504951i	$-0.831511\pi$
$677$	−242.499	+	242.499i	−0.358197	+	0.358197i	0.504951	+	0.863148i	$0.331511\pi$
$678$		0			0
$679$			285.000i			0.419735i
$680$		0			0
$681$	−480.000			−0.704846
$682$		0			0
$683$	−19.5959	−	19.5959i	−0.0286909	−	0.0286909i	0.692616	−	0.721307i	$-0.256458\pi$
$683$	−19.5959	−	19.5959i	−0.0286909	−	0.0286909i	−0.721307	+	0.692616i	$0.756458\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	278.017	−	278.017i	0.404683	−	0.404683i
$688$		0			0
$689$		−	180.000i		−	0.261248i
$690$		0			0
$691$	74.0000			0.107091			0.0535456	−	0.998565i	$-0.482948\pi$
$691$	74.0000			0.107091			0.0535456	+	0.998565i	$0.482948\pi$
$692$		0			0
$693$	110.227	+	110.227i	0.159058	+	0.159058i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−1028.79	+	1028.79i	−1.47602	+	1.47602i
$698$		0			0
$699$			252.000i			0.360515i
$700$		0			0
$701$	−102.000			−0.145506			−0.0727532	−	0.997350i	$-0.523179\pi$
$701$	−102.000			−0.145506			−0.0727532	+	0.997350i	$0.523179\pi$
$702$		0			0
$703$	−563.383	−	563.383i	−0.801398	−	0.801398i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−587.878	+	587.878i	−0.831510	+	0.831510i
$708$		0			0
$709$			817.000i			1.15233i	0.817334	+	0.576164i	$0.195451\pi$
$709$			817.000i			1.15233i	−0.817334	+	0.576164i	$0.804549\pi$
$710$		0			0
$711$	−30.0000			−0.0421941
$712$		0			0
$713$	−306.186	−	306.186i	−0.429434	−	0.429434i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−279.242	+	279.242i	−0.389459	+	0.389459i
$718$		0			0
$719$		−	6.00000i		−	0.00834492i	−0.999991	−	0.00417246i	$-0.998672\pi$
$719$		−	6.00000i		−	0.00834492i	0.999991	−	0.00417246i	$-0.00132814\pi$
$720$		0			0
$721$	−240.000			−0.332871
$722$		0			0
$723$	233.926	+	233.926i	0.323549	+	0.323549i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	45.3156	−	45.3156i	0.0623323	−	0.0623323i	−0.675253	−	0.737586i	$-0.735966\pi$
$727$	45.3156	−	45.3156i	0.0623323	−	0.0623323i	0.737586	+	0.675253i	$0.235966\pi$
$728$		0			0
$729$		−	27.0000i		−	0.0370370i
$730$		0			0
$731$	−2058.00			−2.81532
$732$		0			0
$733$	−484.999	−	484.999i	−0.661663	−	0.661663i	0.294109	−	0.955772i	$-0.404977\pi$
$733$	−484.999	−	484.999i	−0.661663	−	0.661663i	−0.955772	+	0.294109i	$0.904977\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−154.318	+	154.318i	−0.209387	+	0.209387i
$738$		0			0
$739$		−	298.000i		−	0.403248i	−0.979463	−	0.201624i	$-0.935378\pi$
$739$		−	298.000i		−	0.403248i	0.979463	−	0.201624i	$-0.0646218\pi$
$740$		0			0
$741$	−207.000			−0.279352
$742$		0			0
$743$	−249.848	−	249.848i	−0.336269	−	0.336269i	0.518692	−	0.854961i	$-0.326419\pi$
$743$	−249.848	−	249.848i	−0.336269	−	0.336269i	−0.854961	+	0.518692i	$0.826419\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	7.34847	−	7.34847i	0.00983731	−	0.00983731i
$748$		0			0
$749$			360.000i			0.480641i
$750$		0			0
$751$	−1186.00			−1.57923			−0.789614	−	0.613604i	$-0.789719\pi$
$751$	−1186.00			−1.57923			−0.789614	+	0.613604i	$0.789719\pi$
$752$		0			0
$753$	−235.151	−	235.151i	−0.312286	−	0.312286i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−743.420	+	743.420i	−0.982061	+	0.982061i	−0.999842	−	0.0177810i	$-0.994340\pi$
$757$	−743.420	+	743.420i	−0.982061	+	0.982061i	0.0177810	+	0.999842i	$0.494340\pi$
$758$		0			0
$759$		−	180.000i		−	0.237154i
$760$		0			0
$761$	−576.000			−0.756899			−0.378449	−	0.925622i	$-0.623543\pi$
$761$	−576.000			−0.756899			−0.378449	+	0.925622i	$0.623543\pi$
$762$		0			0
$763$	−1022.66	−	1022.66i	−1.34032	−	1.34032i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	66.1362	−	66.1362i	0.0862271	−	0.0862271i
$768$		0			0
$769$			491.000i			0.638492i	0.947672	+	0.319246i	$0.103430\pi$
$769$			491.000i			0.638492i	−0.947672	+	0.319246i	$0.896570\pi$
$770$		0			0
$771$	348.000			0.451362
$772$		0			0
$773$	−519.292	−	519.292i	−0.671788	−	0.671788i	0.286340	−	0.958128i	$-0.407561\pi$
$773$	−519.292	−	519.292i	−0.671788	−	0.671788i	−0.958128	+	0.286340i	$0.907561\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	367.423	−	367.423i	0.472874	−	0.472874i
$778$		0			0
$779$		−	1380.00i		−	1.77150i
$780$		0			0
$781$	792.000			1.01408
$782$		0			0
$783$	22.0454	+	22.0454i	0.0281551	+	0.0281551i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−123.699	+	123.699i	−0.157178	+	0.157178i	−0.781315	−	0.624137i	$-0.785451\pi$
$787$	−123.699	+	123.699i	−0.157178	+	0.157178i	0.624137	+	0.781315i	$0.285451\pi$
$788$		0			0
$789$			612.000i			0.775665i
$790$		0			0
$791$	720.000			0.910240
$792$		0			0
$793$	−135.947	−	135.947i	−0.171433	−	0.171433i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	460.504	−	460.504i	0.577797	−	0.577797i	−0.356499	−	0.934296i	$-0.616030\pi$
$797$	460.504	−	460.504i	0.577797	−	0.577797i	0.934296	+	0.356499i	$0.116030\pi$
$798$		0			0
$799$			252.000i			0.315394i
$800$		0			0
$801$	−396.000			−0.494382
$802$		0			0
$803$	146.969	+	146.969i	0.183025	+	0.183025i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−565.832	+	565.832i	−0.701155	+	0.701155i
$808$		0			0
$809$		−	900.000i		−	1.11248i	−0.831020	−	0.556242i	$-0.812243\pi$
$809$		−	900.000i		−	1.11248i	0.831020	−	0.556242i	$-0.187757\pi$
$810$		0			0
$811$	−1477.00			−1.82121			−0.910604	−	0.413280i	$-0.864383\pi$
$811$	−1477.00			−1.82121			−0.910604	+	0.413280i	$0.864383\pi$
$812$		0			0
$813$	−546.236	−	546.236i	−0.671877	−	0.671877i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	1380.29	−	1380.29i	1.68946	−	1.68946i
$818$		0			0
$819$		−	135.000i		−	0.164835i
$820$		0			0
$821$	−786.000			−0.957369			−0.478685	−	0.877987i	$-0.658886\pi$
$821$	−786.000			−0.957369			−0.478685	+	0.877987i	$0.658886\pi$
$822$		0			0
$823$	371.098	+	371.098i	0.450909	+	0.450909i	0.895656	−	0.444747i	$-0.146707\pi$
$823$	371.098	+	371.098i	0.450909	+	0.450909i	−0.444747	+	0.895656i	$0.646707\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−788.736	+	788.736i	−0.953731	+	0.953731i	−0.998976	−	0.0452447i	$-0.985593\pi$
$827$	−788.736	+	788.736i	−0.953731	+	0.953731i	0.0452447	+	0.998976i	$0.485593\pi$
$828$		0			0
$829$		−	170.000i		−	0.205066i	−0.994730	−	0.102533i	$-0.967305\pi$
$829$		−	170.000i		−	0.205066i	0.994730	−	0.102533i	$-0.0326948\pi$
$830$		0			0
$831$	−429.000			−0.516245
$832$		0			0
$833$	−445.807	−	445.807i	−0.535183	−	0.535183i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	91.8559	−	91.8559i	0.109744	−	0.109744i
$838$		0			0
$839$		−	990.000i		−	1.17998i	−0.807412	−	0.589988i	$-0.799132\pi$
$839$		−	990.000i		−	1.17998i	0.807412	−	0.589988i	$-0.200868\pi$
$840$		0			0
$841$	805.000			0.957194
$842$		0			0
$843$	−668.711	−	668.711i	−0.793251	−	0.793251i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	520.517	−	520.517i	0.614541	−	0.614541i
$848$		0			0
$849$			471.000i			0.554770i
$850$		0			0
$851$	−600.000			−0.705053
$852$		0			0
$853$	723.824	+	723.824i	0.848563	+	0.848563i	0.989954	−	0.141391i	$-0.0451574\pi$
$853$	723.824	+	723.824i	0.848563	+	0.848563i	−0.141391	+	0.989954i	$0.545157\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−739.746	+	739.746i	−0.863181	+	0.863181i	−0.991706	−	0.128525i	$-0.958976\pi$
$857$	−739.746	+	739.746i	−0.863181	+	0.863181i	0.128525	+	0.991706i	$0.458976\pi$
$858$		0			0
$859$			1510.00i			1.75786i	0.476953	+	0.878929i	$0.341741\pi$
$859$			1510.00i			1.75786i	−0.476953	+	0.878929i	$0.658259\pi$
$860$		0			0
$861$	900.000			1.04530
$862$		0			0
$863$	−372.322	−	372.322i	−0.431428	−	0.431428i	0.457686	−	0.889114i	$-0.348678\pi$
$863$	−372.322	−	372.322i	−0.431428	−	0.431428i	−0.889114	+	0.457686i	$0.848678\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	366.199	−	366.199i	0.422375	−	0.422375i
$868$		0			0
$869$		−	60.0000i		−	0.0690449i
$870$		0			0
$871$	189.000			0.216992
$872$		0			0
$873$	69.8105	+	69.8105i	0.0799662	+	0.0799662i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	300.062	−	300.062i	0.342147	−	0.342147i	−0.515027	−	0.857174i	$-0.672218\pi$
$877$	300.062	−	300.062i	0.342147	−	0.342147i	0.857174	+	0.515027i	$0.172218\pi$
$878$		0			0
$879$			678.000i			0.771331i
$880$		0			0
$881$	−216.000			−0.245176			−0.122588	−	0.992458i	$-0.539119\pi$
$881$	−216.000			−0.245176			−0.122588	+	0.992458i	$0.539119\pi$
$882$		0			0
$883$	846.299	+	846.299i	0.958436	+	0.958436i	0.999170	−	0.0407343i	$-0.0129697\pi$
$883$	846.299	+	846.299i	0.958436	+	0.958436i	−0.0407343	+	0.999170i	$0.512970\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	996.942	−	996.942i	1.12395	−	1.12395i	0.132807	−	0.991142i	$-0.457601\pi$
$887$	996.942	−	996.942i	1.12395	−	1.12395i	0.991142	−	0.132807i	$-0.0423989\pi$
$888$		0			0
$889$			540.000i			0.607424i
$890$		0			0
$891$	54.0000			0.0606061
$892$		0			0
$893$	−169.015	−	169.015i	−0.189266	−	0.189266i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−110.227	+	110.227i	−0.122884	+	0.122884i
$898$		0			0
$899$			150.000i			0.166852i
$900$		0			0
$901$	−840.000			−0.932297
$902$		0			0
$903$	900.187	+	900.187i	0.996885	+	0.996885i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	53.8888	−	53.8888i	0.0594143	−	0.0594143i	−0.676775	−	0.736190i	$-0.736623\pi$
$907$	53.8888	−	53.8888i	0.0594143	−	0.0594143i	0.736190	+	0.676775i	$0.236623\pi$
$908$		0			0
$909$			288.000i			0.316832i
$910$		0			0
$911$	558.000			0.612514			0.306257	−	0.951949i	$-0.400923\pi$
$911$	558.000			0.612514			0.306257	+	0.951949i	$0.400923\pi$
$912$		0			0
$913$	14.6969	+	14.6969i	0.0160974	+	0.0160974i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	661.362	−	661.362i	0.721224	−	0.721224i
$918$		0			0
$919$			755.000i			0.821545i	0.911738	+	0.410773i	$0.134741\pi$
$919$			755.000i			0.821545i	−0.911738	+	0.410773i	$0.865259\pi$
$920$		0			0
$921$	−213.000			−0.231270
$922$		0			0
$923$	−484.999	−	484.999i	−0.525459	−	0.525459i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−58.7878	+	58.7878i	−0.0634172	+	0.0634172i
$928$		0			0
$929$			762.000i			0.820237i	0.912032	+	0.410118i	$0.134513\pi$
$929$			762.000i			0.820237i	−0.912032	+	0.410118i	$0.865487\pi$
$930$		0			0
$931$	598.000			0.642320
$932$		0			0
$933$	360.075	+	360.075i	0.385932	+	0.385932i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	388.244	−	388.244i	0.414348	−	0.414348i	−0.468902	−	0.883250i	$-0.655350\pi$
$937$	388.244	−	388.244i	0.414348	−	0.414348i	0.883250	+	0.468902i	$0.155350\pi$
$938$		0			0
$939$			963.000i			1.02556i
$940$		0			0
$941$	−690.000			−0.733262			−0.366631	−	0.930366i	$-0.619489\pi$
$941$	−690.000			−0.733262			−0.366631	+	0.930366i	$0.619489\pi$
$942$		0			0
$943$	−734.847	−	734.847i	−0.779265	−	0.779265i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	100.429	−	100.429i	0.106050	−	0.106050i	−0.652091	−	0.758141i	$-0.726108\pi$
$947$	100.429	−	100.429i	0.106050	−	0.106050i	0.758141	+	0.652091i	$0.226108\pi$
$948$		0			0
$949$		−	180.000i		−	0.189673i
$950$		0			0
$951$	228.000			0.239748
$952$		0			0
$953$	−749.544	−	749.544i	−0.786510	−	0.786510i	0.194410	−	0.980920i	$-0.437721\pi$
$953$	−749.544	−	749.544i	−0.786510	−	0.786510i	−0.980920	+	0.194410i	$0.937721\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−44.0908	+	44.0908i	−0.0460719	+	0.0460719i
$958$		0			0
$959$			1470.00i			1.53285i
$960$		0			0
$961$	−336.000			−0.349636
$962$		0			0
$963$	88.1816	+	88.1816i	0.0915697	+	0.0915697i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−4.89898	+	4.89898i	−0.00506616	+	0.00506616i	−0.709635	−	0.704569i	$-0.751140\pi$
$967$	−4.89898	+	4.89898i	−0.00506616	+	0.00506616i	0.704569	+	0.709635i	$0.251140\pi$
$968$		0			0
$969$			966.000i			0.996904i
$970$		0			0
$971$	−90.0000			−0.0926880			−0.0463440	−	0.998926i	$-0.514757\pi$
$971$	−90.0000			−0.0926880			−0.0463440	+	0.998926i	$0.514757\pi$
$972$		0			0
$973$	355.176	+	355.176i	0.365032	+	0.365032i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−301.287	+	301.287i	−0.308380	+	0.308380i	−0.844281	−	0.535901i	$-0.819972\pi$
$977$	−301.287	+	301.287i	−0.308380	+	0.308380i	0.535901	+	0.844281i	$0.319972\pi$
$978$		0			0
$979$		−	792.000i		−	0.808989i
$980$		0			0
$981$	−501.000			−0.510703
$982$		0			0
$983$	−465.403	−	465.403i	−0.473452	−	0.473452i	0.429578	−	0.903030i	$-0.358662\pi$
$983$	−465.403	−	465.403i	−0.473452	−	0.473452i	−0.903030	+	0.429578i	$0.858662\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	110.227	−	110.227i	0.111679	−	0.111679i
$988$		0			0
$989$		−	1470.00i		−	1.48635i
$990$		0			0
$991$	−1067.00			−1.07669			−0.538345	−	0.842724i	$-0.680950\pi$
$991$	−1067.00			−1.07669			−0.538345	+	0.842724i	$0.680950\pi$
$992$		0			0
$993$	218.005	+	218.005i	0.219541	+	0.219541i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	279.242	−	279.242i	0.280082	−	0.280082i	−0.553060	−	0.833142i	$-0.686540\pi$
$997$	279.242	−	279.242i	0.280082	−	0.280082i	0.833142	+	0.553060i	$0.186540\pi$
$998$		0			0
$999$		−	180.000i		−	0.180180i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.bg.g.193.2		4
4.3	odd	2		75.3.f.b.43.1	yes	4
5.2	odd	4	inner	1200.3.bg.g.1057.2		4
5.3	odd	4	inner	1200.3.bg.g.1057.1		4
5.4	even	2	inner	1200.3.bg.g.193.1		4
12.11	even	2		225.3.g.b.118.2		4
20.3	even	4		75.3.f.b.7.2	yes	4
20.7	even	4		75.3.f.b.7.1	✓	4
20.19	odd	2		75.3.f.b.43.2	yes	4
60.23	odd	4		225.3.g.b.82.1		4
60.47	odd	4		225.3.g.b.82.2		4
60.59	even	2		225.3.g.b.118.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.3.f.b.7.1	✓	4	20.7	even	4
75.3.f.b.7.2	yes	4	20.3	even	4
75.3.f.b.43.1	yes	4	4.3	odd	2
75.3.f.b.43.2	yes	4	20.19	odd	2
225.3.g.b.82.1		4	60.23	odd	4
225.3.g.b.82.2		4	60.47	odd	4
225.3.g.b.118.1		4	60.59	even	2
225.3.g.b.118.2		4	12.11	even	2
1200.3.bg.g.193.1		4	5.4	even	2	inner
1200.3.bg.g.193.2		4	1.1	even	1	trivial
1200.3.bg.g.1057.1		4	5.3	odd	4	inner
1200.3.bg.g.1057.2		4	5.2	odd	4	inner

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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1200.bg (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 + 1.22474i) q^{3} +(-6.12372 + 6.12372i) q^{7} +3.00000i q^{9} +O(q^{10})\)	\(q+(1.22474 + 1.22474i) q^{3} +(-6.12372 + 6.12372i) q^{7} +3.00000i q^{9} -6.00000 q^{11} +(3.67423 + 3.67423i) q^{13} +(17.1464 - 17.1464i) q^{17} +23.0000i q^{19} -15.0000 q^{21} +(12.2474 + 12.2474i) q^{23} +(-3.67423 + 3.67423i) q^{27} -6.00000i q^{29} -25.0000 q^{31} +(-7.34847 - 7.34847i) q^{33} +(-24.4949 + 24.4949i) q^{37} +9.00000i q^{39} -60.0000 q^{41} +(-60.0125 - 60.0125i) q^{43} +(-7.34847 + 7.34847i) q^{47} -26.0000i q^{49} +42.0000 q^{51} +(-24.4949 - 24.4949i) q^{53} +(-28.1691 + 28.1691i) q^{57} -18.0000i q^{59} -37.0000 q^{61} +(-18.3712 - 18.3712i) q^{63} +(25.7196 - 25.7196i) q^{67} +30.0000i q^{69} -132.000 q^{71} +(-24.4949 - 24.4949i) q^{73} +(36.7423 - 36.7423i) q^{77} +10.0000i q^{79} -9.00000 q^{81} +(-2.44949 - 2.44949i) q^{83} +(7.34847 - 7.34847i) q^{87} +132.000i q^{89} -45.0000 q^{91} +(-30.6186 - 30.6186i) q^{93} +(23.2702 - 23.2702i) q^{97} -18.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 24 q^{11} - 60 q^{21} - 100 q^{31} - 240 q^{41} + 168 q^{51} - 148 q^{61} - 528 q^{71} - 36 q^{81} - 180 q^{91}+O(q^{100}) \) 4 * q - 24 * q^11 - 60 * q^21 - 100 * q^31 - 240 * q^41 + 168 * q^51 - 148 * q^61 - 528 * q^71 - 36 * q^81 - 180 * q^91

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