LMFDB - Embedded newform 1200.3.bg.h.193.1

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 300)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			193.1
Root			$1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1200.193
Dual form			1200.3.bg.h.1057.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 - 1.22474i) q^{3} +(2.44949 - 2.44949i) q^{7} +3.00000i q^{9} +O(q^{10})$	$q+(-1.22474 - 1.22474i) q^{3} +(2.44949 - 2.44949i) q^{7} +3.00000i q^{9} -6.00000 q^{11} +(12.2474 + 12.2474i) q^{13} +(-14.6969 + 14.6969i) q^{17} -10.0000i q^{19} -6.00000 q^{21} +(-29.3939 - 29.3939i) q^{23} +(3.67423 - 3.67423i) q^{27} -48.0000i q^{29} +26.0000 q^{31} +(7.34847 + 7.34847i) q^{33} +(-31.8434 + 31.8434i) q^{37} -30.0000i q^{39} +30.0000 q^{41} +(-29.3939 - 29.3939i) q^{43} +(-14.6969 + 14.6969i) q^{47} +37.0000i q^{49} +36.0000 q^{51} +(-14.6969 - 14.6969i) q^{53} +(-12.2474 + 12.2474i) q^{57} +78.0000i q^{59} +2.00000 q^{61} +(7.34847 + 7.34847i) q^{63} +(-63.6867 + 63.6867i) q^{67} +72.0000i q^{69} -120.000 q^{71} +(-83.2827 - 83.2827i) q^{73} +(-14.6969 + 14.6969i) q^{77} -74.0000i q^{79} -9.00000 q^{81} +(44.0908 + 44.0908i) q^{83} +(-58.7878 + 58.7878i) q^{87} -150.000i q^{89} +60.0000 q^{91} +(-31.8434 - 31.8434i) q^{93} +(-4.89898 + 4.89898i) q^{97} -18.0000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 24 q^{11} - 24 q^{21} + 104 q^{31} + 120 q^{41} + 144 q^{51} + 8 q^{61} - 480 q^{71} - 36 q^{81} + 240 q^{91}+O(q^{100}) $ 4 * q - 24 * q^11 - 24 * q^21 + 104 * q^31 + 120 * q^41 + 144 * q^51 + 8 * q^61 - 480 * q^71 - 36 * q^81 + 240 * 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
$7$	2.44949	−	2.44949i	0.349927	−	0.349927i	−0.510155	−	0.860082i	$-0.670412\pi$
$7$	2.44949	−	2.44949i	0.349927	−	0.349927i	0.860082	+	0.510155i	$0.170412\pi$
$8$		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
$13$	12.2474	+	12.2474i	0.942111	+	0.942111i	0.998414	−	0.0563023i	$-0.0179311\pi$
$13$	12.2474	+	12.2474i	0.942111	+	0.942111i	−0.0563023	+	0.998414i	$0.517931\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−14.6969	+	14.6969i	−0.864526	+	0.864526i	−0.991860	−	0.127334i	$-0.959358\pi$
$17$	−14.6969	+	14.6969i	−0.864526	+	0.864526i	0.127334	+	0.991860i	$0.459358\pi$
$18$		0			0
$19$		−	10.0000i		−	0.526316i	−0.964753	−	0.263158i	$-0.915236\pi$
$19$		−	10.0000i		−	0.526316i	0.964753	−	0.263158i	$-0.0847640\pi$
$20$		0			0
$21$	−6.00000			−0.285714
$22$		0			0
$23$	−29.3939	−	29.3939i	−1.27799	−	1.27799i	−0.941788	−	0.336206i	$-0.890856\pi$
$23$	−29.3939	−	29.3939i	−1.27799	−	1.27799i	−0.336206	−	0.941788i	$-0.609144\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	3.67423	−	3.67423i	0.136083	−	0.136083i
$28$		0			0
$29$		−	48.0000i		−	1.65517i	−0.561339	−	0.827586i	$-0.689713\pi$
$29$		−	48.0000i		−	1.65517i	0.561339	−	0.827586i	$-0.310287\pi$
$30$		0			0
$31$	26.0000			0.838710			0.419355	−	0.907822i	$-0.362256\pi$
$31$	26.0000			0.838710			0.419355	+	0.907822i	$0.362256\pi$
$32$		0			0
$33$	7.34847	+	7.34847i	0.222681	+	0.222681i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−31.8434	+	31.8434i	−0.860632	+	0.860632i	−0.991411	−	0.130780i	$-0.958252\pi$
$37$	−31.8434	+	31.8434i	−0.860632	+	0.860632i	0.130780	+	0.991411i	$0.458252\pi$
$38$		0			0
$39$		−	30.0000i		−	0.769231i
$40$		0			0
$41$	30.0000			0.731707			0.365854	−	0.930672i	$-0.380777\pi$
$41$	30.0000			0.731707			0.365854	+	0.930672i	$0.380777\pi$
$42$		0			0
$43$	−29.3939	−	29.3939i	−0.683579	−	0.683579i	0.277226	−	0.960805i	$-0.410585\pi$
$43$	−29.3939	−	29.3939i	−0.683579	−	0.683579i	−0.960805	+	0.277226i	$0.910585\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−14.6969	+	14.6969i	−0.312701	+	0.312701i	−0.845955	−	0.533254i	$-0.820969\pi$
$47$	−14.6969	+	14.6969i	−0.312701	+	0.312701i	0.533254	+	0.845955i	$0.320969\pi$
$48$		0			0
$49$			37.0000i			0.755102i
$50$		0			0
$51$	36.0000			0.705882
$52$		0			0
$53$	−14.6969	−	14.6969i	−0.277301	−	0.277301i	0.554730	−	0.832031i	$-0.312822\pi$
$53$	−14.6969	−	14.6969i	−0.277301	−	0.277301i	−0.832031	+	0.554730i	$0.812822\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−12.2474	+	12.2474i	−0.214868	+	0.214868i
$58$		0			0
$59$			78.0000i			1.32203i	0.750371	+	0.661017i	$0.229875\pi$
$59$			78.0000i			1.32203i	−0.750371	+	0.661017i	$0.770125\pi$
$60$		0			0
$61$	2.00000			0.0327869			0.0163934	−	0.999866i	$-0.494782\pi$
$61$	2.00000			0.0327869			0.0163934	+	0.999866i	$0.494782\pi$
$62$		0			0
$63$	7.34847	+	7.34847i	0.116642	+	0.116642i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−63.6867	+	63.6867i	−0.950548	+	0.950548i	−0.998834	−	0.0482853i	$-0.984624\pi$
$67$	−63.6867	+	63.6867i	−0.950548	+	0.950548i	0.0482853	+	0.998834i	$0.484624\pi$
$68$		0			0
$69$			72.0000i			1.04348i
$70$		0			0
$71$	−120.000			−1.69014			−0.845070	−	0.534655i	$-0.820442\pi$
$71$	−120.000			−1.69014			−0.845070	+	0.534655i	$0.820442\pi$
$72$		0			0
$73$	−83.2827	−	83.2827i	−1.14086	−	1.14086i	−0.988293	−	0.152565i	$-0.951247\pi$
$73$	−83.2827	−	83.2827i	−1.14086	−	1.14086i	−0.152565	−	0.988293i	$-0.548753\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−14.6969	+	14.6969i	−0.190869	+	0.190869i
$78$		0			0
$79$		−	74.0000i		−	0.936709i	−0.883541	−	0.468354i	$-0.844847\pi$
$79$		−	74.0000i		−	0.936709i	0.883541	−	0.468354i	$-0.155153\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$		0			0
$83$	44.0908	+	44.0908i	0.531215	+	0.531215i	0.920934	−	0.389719i	$-0.127428\pi$
$83$	44.0908	+	44.0908i	0.531215	+	0.531215i	−0.389719	+	0.920934i	$0.627428\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−58.7878	+	58.7878i	−0.675721	+	0.675721i
$88$		0			0
$89$		−	150.000i		−	1.68539i	−0.538389	−	0.842697i	$-0.680967\pi$
$89$		−	150.000i		−	1.68539i	0.538389	−	0.842697i	$-0.319033\pi$
$90$		0			0
$91$	60.0000			0.659341
$92$		0			0
$93$	−31.8434	−	31.8434i	−0.342402	−	0.342402i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−4.89898	+	4.89898i	−0.0505049	+	0.0505049i	−0.731908	−	0.681403i	$-0.761370\pi$
$97$	−4.89898	+	4.89898i	−0.0505049	+	0.0505049i	0.681403	+	0.731908i	$0.261370\pi$
$98$		0			0
$99$		−	18.0000i		−	0.181818i
$100$		0			0
$101$	−12.0000			−0.118812			−0.0594059	−	0.998234i	$-0.518921\pi$
$101$	−12.0000			−0.118812			−0.0594059	+	0.998234i	$0.518921\pi$
$102$		0			0
$103$	41.6413	+	41.6413i	0.404285	+	0.404285i	0.879740	−	0.475455i	$-0.157717\pi$
$103$	41.6413	+	41.6413i	0.404285	+	0.404285i	−0.475455	+	0.879740i	$0.657717\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−102.879	+	102.879i	−0.961482	+	0.961482i	−0.999285	−	0.0378032i	$-0.987964\pi$
$107$	−102.879	+	102.879i	−0.961482	+	0.961482i	0.0378032	+	0.999285i	$0.487964\pi$
$108$		0			0
$109$			74.0000i			0.678899i	0.940624	+	0.339450i	$0.110241\pi$
$109$			74.0000i			0.678899i	−0.940624	+	0.339450i	$0.889759\pi$
$110$		0			0
$111$	78.0000			0.702703
$112$		0			0
$113$	−132.272	−	132.272i	−1.17055	−	1.17055i	−0.982078	−	0.188475i	$-0.939646\pi$
$113$	−132.272	−	132.272i	−1.17055	−	1.17055i	−0.188475	−	0.982078i	$-0.560354\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−36.7423	+	36.7423i	−0.314037	+	0.314037i
$118$		0			0
$119$			72.0000i			0.605042i
$120$		0			0
$121$	−85.0000			−0.702479
$122$		0			0
$123$	−36.7423	−	36.7423i	−0.298718	−	0.298718i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−95.5301	+	95.5301i	−0.752206	+	0.752206i	−0.974890	−	0.222685i	$-0.928518\pi$
$127$	−95.5301	+	95.5301i	−0.752206	+	0.752206i	0.222685	+	0.974890i	$0.428518\pi$
$128$		0			0
$129$			72.0000i			0.558140i
$130$		0			0
$131$	−102.000			−0.778626			−0.389313	−	0.921106i	$-0.627288\pi$
$131$	−102.000			−0.778626			−0.389313	+	0.921106i	$0.627288\pi$
$132$		0			0
$133$	−24.4949	−	24.4949i	−0.184172	−	0.184172i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−44.0908	+	44.0908i	−0.321831	+	0.321831i	−0.849469	−	0.527638i	$-0.823078\pi$
$137$	−44.0908	+	44.0908i	−0.321831	+	0.321831i	0.527638	+	0.849469i	$0.323078\pi$
$138$		0			0
$139$			122.000i			0.877698i	0.898561	+	0.438849i	$0.144614\pi$
$139$			122.000i			0.877698i	−0.898561	+	0.438849i	$0.855386\pi$
$140$		0			0
$141$	36.0000			0.255319
$142$		0			0
$143$	−73.4847	−	73.4847i	−0.513879	−	0.513879i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	45.3156	−	45.3156i	0.308269	−	0.308269i
$148$		0			0
$149$		−	36.0000i		−	0.241611i	−0.992676	−	0.120805i	$-0.961452\pi$
$149$		−	36.0000i		−	0.241611i	0.992676	−	0.120805i	$-0.0385477\pi$
$150$		0			0
$151$	−70.0000			−0.463576			−0.231788	−	0.972766i	$-0.574458\pi$
$151$	−70.0000			−0.463576			−0.231788	+	0.972766i	$0.574458\pi$
$152$		0			0
$153$	−44.0908	−	44.0908i	−0.288175	−	0.288175i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−139.621	+	139.621i	−0.889305	+	0.889305i	−0.994456	−	0.105151i	$-0.966467\pi$
$157$	−139.621	+	139.621i	−0.889305	+	0.889305i	0.105151	+	0.994456i	$0.466467\pi$
$158$		0			0
$159$			36.0000i			0.226415i
$160$		0			0
$161$	−144.000			−0.894410
$162$		0			0
$163$	−97.9796	−	97.9796i	−0.601102	−	0.601102i	0.339503	−	0.940605i	$-0.389741\pi$
$163$	−97.9796	−	97.9796i	−0.601102	−	0.601102i	−0.940605	+	0.339503i	$0.889741\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−29.3939	+	29.3939i	−0.176011	+	0.176011i	−0.789614	−	0.613603i	$-0.789719\pi$
$167$	−29.3939	+	29.3939i	−0.176011	+	0.176011i	0.613603	+	0.789614i	$0.289719\pi$
$168$		0			0
$169$			131.000i			0.775148i
$170$		0			0
$171$	30.0000			0.175439
$172$		0			0
$173$	191.060	+	191.060i	1.10439	+	1.10439i	0.993874	+	0.110520i	$0.0352517\pi$
$173$	191.060	+	191.060i	1.10439	+	1.10439i	0.110520	+	0.993874i	$0.464748\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	95.5301	−	95.5301i	0.539718	−	0.539718i
$178$		0			0
$179$		−	222.000i		−	1.24022i	−0.784514	−	0.620112i	$-0.787087\pi$
$179$		−	222.000i		−	1.24022i	0.784514	−	0.620112i	$-0.212913\pi$
$180$		0			0
$181$	190.000			1.04972			0.524862	−	0.851187i	$-0.324117\pi$
$181$	190.000			1.04972			0.524862	+	0.851187i	$0.324117\pi$
$182$		0			0
$183$	−2.44949	−	2.44949i	−0.0133852	−	0.0133852i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	88.1816	−	88.1816i	0.471560	−	0.471560i
$188$		0			0
$189$		−	18.0000i		−	0.0952381i
$190$		0			0
$191$	−204.000			−1.06806			−0.534031	−	0.845465i	$-0.679324\pi$
$191$	−204.000			−1.06806			−0.534031	+	0.845465i	$0.679324\pi$
$192$		0			0
$193$	68.5857	+	68.5857i	0.355366	+	0.355366i	0.862102	−	0.506735i	$-0.169148\pi$
$193$	68.5857	+	68.5857i	0.355366	+	0.355366i	−0.506735	+	0.862102i	$0.669148\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−205.757	+	205.757i	−1.04445	+	1.04445i	−0.0454876	+	0.998965i	$0.514484\pi$
$197$	−205.757	+	205.757i	−1.04445	+	1.04445i	−0.998965	+	0.0454876i	$0.985516\pi$
$198$		0			0
$199$			46.0000i			0.231156i	0.993298	+	0.115578i	$0.0368720\pi$
$199$			46.0000i			0.231156i	−0.993298	+	0.115578i	$0.963128\pi$
$200$		0			0
$201$	156.000			0.776119
$202$		0			0
$203$	−117.576	−	117.576i	−0.579190	−	0.579190i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	88.1816	−	88.1816i	0.425998	−	0.425998i
$208$		0			0
$209$			60.0000i			0.287081i
$210$		0			0
$211$	310.000			1.46919			0.734597	−	0.678504i	$-0.237371\pi$
$211$	310.000			1.46919			0.734597	+	0.678504i	$0.237371\pi$
$212$		0			0
$213$	146.969	+	146.969i	0.689997	+	0.689997i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	63.6867	−	63.6867i	0.293487	−	0.293487i
$218$		0			0
$219$			204.000i			0.931507i
$220$		0			0
$221$	−360.000			−1.62896
$222$		0			0
$223$	−183.712	−	183.712i	−0.823819	−	0.823819i	0.162834	−	0.986653i	$-0.447936\pi$
$223$	−183.712	−	183.712i	−0.823819	−	0.823819i	−0.986653	+	0.162834i	$0.947936\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	73.4847	−	73.4847i	0.323721	−	0.323721i	−0.526472	−	0.850193i	$-0.676485\pi$
$227$	73.4847	−	73.4847i	0.323721	−	0.323721i	0.850193	+	0.526472i	$0.176485\pi$
$228$		0			0
$229$		−	242.000i		−	1.05677i	−0.849005	−	0.528384i	$-0.822798\pi$
$229$		−	242.000i		−	1.05677i	0.849005	−	0.528384i	$-0.177202\pi$
$230$		0			0
$231$	36.0000			0.155844
$232$		0			0
$233$	−73.4847	−	73.4847i	−0.315385	−	0.315385i	0.531607	−	0.846991i	$-0.321589\pi$
$233$	−73.4847	−	73.4847i	−0.315385	−	0.315385i	−0.846991	+	0.531607i	$0.821589\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−90.6311	+	90.6311i	−0.382410	+	0.382410i
$238$		0			0
$239$		−	324.000i		−	1.35565i	−0.735224	−	0.677824i	$-0.762923\pi$
$239$		−	324.000i		−	1.35565i	0.735224	−	0.677824i	$-0.237077\pi$
$240$		0			0
$241$	398.000			1.65145			0.825726	−	0.564071i	$-0.190766\pi$
$241$	398.000			1.65145			0.825726	+	0.564071i	$0.190766\pi$
$242$		0			0
$243$	11.0227	+	11.0227i	0.0453609	+	0.0453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	122.474	−	122.474i	0.495848	−	0.495848i
$248$		0			0
$249$		−	108.000i		−	0.433735i
$250$		0			0
$251$	162.000			0.645418			0.322709	−	0.946498i	$-0.395406\pi$
$251$	162.000			0.645418			0.322709	+	0.946498i	$0.395406\pi$
$252$		0			0
$253$	176.363	+	176.363i	0.697088	+	0.697088i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	102.879	−	102.879i	0.400306	−	0.400306i	−0.478035	−	0.878341i	$-0.658651\pi$
$257$	102.879	−	102.879i	0.400306	−	0.400306i	0.878341	+	0.478035i	$0.158651\pi$
$258$		0			0
$259$			156.000i			0.602317i
$260$		0			0
$261$	144.000			0.551724
$262$		0			0
$263$	117.576	+	117.576i	0.447055	+	0.447055i	0.894374	−	0.447319i	$-0.147621\pi$
$263$	117.576	+	117.576i	0.447055	+	0.447055i	−0.447319	+	0.894374i	$0.647621\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−183.712	+	183.712i	−0.688059	+	0.688059i
$268$		0			0
$269$			48.0000i			0.178439i	0.996012	+	0.0892193i	$0.0284372\pi$
$269$			48.0000i			0.178439i	−0.996012	+	0.0892193i	$0.971563\pi$
$270$		0			0
$271$	46.0000			0.169742			0.0848708	−	0.996392i	$-0.472952\pi$
$271$	46.0000			0.169742			0.0848708	+	0.996392i	$0.472952\pi$
$272$		0			0
$273$	−73.4847	−	73.4847i	−0.269175	−	0.269175i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	80.8332	−	80.8332i	0.291816	−	0.291816i	−0.545981	−	0.837798i	$-0.683843\pi$
$277$	80.8332	−	80.8332i	0.291816	−	0.291816i	0.837798	+	0.545981i	$0.183843\pi$
$278$		0			0
$279$			78.0000i			0.279570i
$280$		0			0
$281$	−414.000			−1.47331			−0.736655	−	0.676269i	$-0.763596\pi$
$281$	−414.000			−1.47331			−0.736655	+	0.676269i	$0.763596\pi$
$282$		0			0
$283$	279.242	+	279.242i	0.986720	+	0.986720i	0.999913	−	0.0131927i	$-0.00419950\pi$
$283$	279.242	+	279.242i	0.986720	+	0.986720i	−0.0131927	+	0.999913i	$0.504199\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	73.4847	−	73.4847i	0.256044	−	0.256044i
$288$		0			0
$289$		−	143.000i		−	0.494810i
$290$		0			0
$291$	12.0000			0.0412371
$292$		0			0
$293$	235.151	+	235.151i	0.802563	+	0.802563i	0.983496	−	0.180932i	$-0.0579115\pi$
$293$	235.151	+	235.151i	0.802563	+	0.802563i	−0.180932	+	0.983496i	$0.557912\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−22.0454	+	22.0454i	−0.0742270	+	0.0742270i
$298$		0			0
$299$		−	720.000i		−	2.40803i
$300$		0			0
$301$	−144.000			−0.478405
$302$		0			0
$303$	14.6969	+	14.6969i	0.0485047	+	0.0485047i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	44.0908	−	44.0908i	0.143618	−	0.143618i	−0.631642	−	0.775260i	$-0.717619\pi$
$307$	44.0908	−	44.0908i	0.143618	−	0.143618i	0.775260	+	0.631642i	$0.217619\pi$
$308$		0			0
$309$		−	102.000i		−	0.330097i
$310$		0			0
$311$	−204.000			−0.655949			−0.327974	−	0.944687i	$-0.606366\pi$
$311$	−204.000			−0.655949			−0.327974	+	0.944687i	$0.606366\pi$
$312$		0			0
$313$	−372.322	−	372.322i	−1.18953	−	1.18953i	−0.977198	−	0.212331i	$-0.931895\pi$
$313$	−372.322	−	372.322i	−1.18953	−	1.18953i	−0.212331	−	0.977198i	$-0.568105\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	352.727	−	352.727i	1.11270	−	1.11270i	0.119918	−	0.992784i	$-0.461737\pi$
$317$	352.727	−	352.727i	1.11270	−	1.11270i	0.992784	−	0.119918i	$-0.0382632\pi$
$318$		0			0
$319$			288.000i			0.902821i
$320$		0			0
$321$	252.000			0.785047
$322$		0			0
$323$	146.969	+	146.969i	0.455014	+	0.455014i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	90.6311	−	90.6311i	0.277159	−	0.277159i
$328$		0			0
$329$			72.0000i			0.218845i
$330$		0			0
$331$	−542.000			−1.63746			−0.818731	−	0.574177i	$-0.805322\pi$
$331$	−542.000			−1.63746			−0.818731	+	0.574177i	$0.805322\pi$
$332$		0			0
$333$	−95.5301	−	95.5301i	−0.286877	−	0.286877i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	347.828	−	347.828i	1.03213	−	1.03213i	0.0326628	−	0.999466i	$-0.489601\pi$
$337$	347.828	−	347.828i	1.03213	−	1.03213i	0.999466	−	0.0326628i	$-0.0103987\pi$
$338$		0			0
$339$			324.000i			0.955752i
$340$		0			0
$341$	−156.000			−0.457478
$342$		0			0
$343$	210.656	+	210.656i	0.614158	+	0.614158i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	191.060	−	191.060i	0.550606	−	0.550606i	−0.376010	−	0.926616i	$-0.622704\pi$
$347$	191.060	−	191.060i	0.550606	−	0.550606i	0.926616	+	0.376010i	$0.122704\pi$
$348$		0			0
$349$			358.000i			1.02579i	0.858452	+	0.512894i	$0.171427\pi$
$349$			358.000i			1.02579i	−0.858452	+	0.512894i	$0.828573\pi$
$350$		0			0
$351$	90.0000			0.256410
$352$		0			0
$353$	−426.211	−	426.211i	−1.20740	−	1.20740i	−0.971867	−	0.235530i	$-0.924317\pi$
$353$	−426.211	−	426.211i	−1.20740	−	1.20740i	−0.235530	−	0.971867i	$-0.575683\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	88.1816	−	88.1816i	0.247007	−	0.247007i
$358$		0			0
$359$		−	132.000i		−	0.367688i	−0.982955	−	0.183844i	$-0.941146\pi$
$359$		−	132.000i		−	0.367688i	0.982955	−	0.183844i	$-0.0588541\pi$
$360$		0			0
$361$	261.000			0.722992
$362$		0			0
$363$	104.103	+	104.103i	0.286786	+	0.286786i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−120.025	+	120.025i	−0.327044	+	0.327044i	−0.851461	−	0.524418i	$-0.824283\pi$
$367$	−120.025	+	120.025i	−0.327044	+	0.327044i	0.524418	+	0.851461i	$0.324283\pi$
$368$		0			0
$369$			90.0000i			0.243902i
$370$		0			0
$371$	−72.0000			−0.194070
$372$		0			0
$373$	409.065	+	409.065i	1.09669	+	1.09669i	0.994795	+	0.101893i	$0.0324900\pi$
$373$	409.065	+	409.065i	1.09669	+	1.09669i	0.101893	+	0.994795i	$0.467510\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	587.878	−	587.878i	1.55936	−	1.55936i
$378$		0			0
$379$			26.0000i			0.0686016i	0.999412	+	0.0343008i	$0.0109204\pi$
$379$			26.0000i			0.0686016i	−0.999412	+	0.0343008i	$0.989080\pi$
$380$		0			0
$381$	234.000			0.614173
$382$		0			0
$383$	426.211	+	426.211i	1.11282	+	1.11282i	0.992767	+	0.120056i	$0.0383074\pi$
$383$	426.211	+	426.211i	1.11282	+	1.11282i	0.120056	+	0.992767i	$0.461693\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	88.1816	−	88.1816i	0.227860	−	0.227860i
$388$		0			0
$389$		−	72.0000i		−	0.185090i	−0.995709	−	0.0925450i	$-0.970500\pi$
$389$		−	72.0000i		−	0.185090i	0.995709	−	0.0925450i	$-0.0295002\pi$
$390$		0			0
$391$	864.000			2.20972
$392$		0			0
$393$	124.924	+	124.924i	0.317873	+	0.317873i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−227.803	+	227.803i	−0.573810	+	0.573810i	−0.933191	−	0.359381i	$-0.882988\pi$
$397$	−227.803	+	227.803i	−0.573810	+	0.573810i	0.359381	+	0.933191i	$0.382988\pi$
$398$		0			0
$399$			60.0000i			0.150376i
$400$		0			0
$401$	−414.000			−1.03242			−0.516209	−	0.856462i	$-0.672657\pi$
$401$	−414.000			−1.03242			−0.516209	+	0.856462i	$0.672657\pi$
$402$		0			0
$403$	318.434	+	318.434i	0.790158	+	0.790158i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	191.060	−	191.060i	0.469435	−	0.469435i
$408$		0			0
$409$		−	482.000i		−	1.17848i	−0.807957	−	0.589242i	$-0.799426\pi$
$409$		−	482.000i		−	1.17848i	0.807957	−	0.589242i	$-0.200574\pi$
$410$		0			0
$411$	108.000			0.262774
$412$		0			0
$413$	191.060	+	191.060i	0.462615	+	0.462615i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	149.419	−	149.419i	0.358319	−	0.358319i
$418$		0			0
$419$		−	126.000i		−	0.300716i	−0.988632	−	0.150358i	$-0.951957\pi$
$419$		−	126.000i		−	0.300716i	0.988632	−	0.150358i	$-0.0480426\pi$
$420$		0			0
$421$	430.000			1.02138			0.510689	−	0.859766i	$-0.329390\pi$
$421$	430.000			1.02138			0.510689	+	0.859766i	$0.329390\pi$
$422$		0			0
$423$	−44.0908	−	44.0908i	−0.104234	−	0.104234i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	4.89898	−	4.89898i	0.0114730	−	0.0114730i
$428$		0			0
$429$			180.000i			0.419580i
$430$		0			0
$431$	−228.000			−0.529002			−0.264501	−	0.964385i	$-0.585207\pi$
$431$	−228.000			−0.529002			−0.264501	+	0.964385i	$0.585207\pi$
$432$		0			0
$433$	−249.848	−	249.848i	−0.577016	−	0.577016i	0.357064	−	0.934080i	$-0.383778\pi$
$433$	−249.848	−	249.848i	−0.577016	−	0.577016i	−0.934080	+	0.357064i	$0.883778\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−293.939	+	293.939i	−0.672629	+	0.672629i
$438$		0			0
$439$		−	194.000i		−	0.441913i	−0.975284	−	0.220957i	$-0.929082\pi$
$439$		−	194.000i		−	0.441913i	0.975284	−	0.220957i	$-0.0709180\pi$
$440$		0			0
$441$	−111.000			−0.251701
$442$		0			0
$443$	−338.030	−	338.030i	−0.763046	−	0.763046i	0.213825	−	0.976872i	$-0.431408\pi$
$443$	−338.030	−	338.030i	−0.763046	−	0.763046i	−0.976872	+	0.213825i	$0.931408\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−44.0908	+	44.0908i	−0.0986372	+	0.0986372i
$448$		0			0
$449$		−	6.00000i		−	0.0133630i	−0.999978	−	0.00668151i	$-0.997873\pi$
$449$		−	6.00000i		−	0.0133630i	0.999978	−	0.00668151i	$-0.00212681\pi$
$450$		0			0
$451$	−180.000			−0.399113
$452$		0			0
$453$	85.7321	+	85.7321i	0.189254	+	0.189254i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	367.423	−	367.423i	0.803990	−	0.803990i	−0.179727	−	0.983717i	$-0.557521\pi$
$457$	367.423	−	367.423i	0.803990	−	0.803990i	0.983717	+	0.179727i	$0.0575213\pi$
$458$		0			0
$459$			108.000i			0.235294i
$460$		0			0
$461$	−204.000			−0.442516			−0.221258	−	0.975215i	$-0.571016\pi$
$461$	−204.000			−0.442516			−0.221258	+	0.975215i	$0.571016\pi$
$462$		0			0
$463$	−100.429	−	100.429i	−0.216909	−	0.216909i	0.590285	−	0.807195i	$-0.299015\pi$
$463$	−100.429	−	100.429i	−0.216909	−	0.216909i	−0.807195	+	0.590285i	$0.799015\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	191.060	−	191.060i	0.409122	−	0.409122i	−0.472310	−	0.881432i	$-0.656580\pi$
$467$	191.060	−	191.060i	0.409122	−	0.409122i	0.881432	+	0.472310i	$0.156580\pi$
$468$		0			0
$469$			312.000i			0.665245i
$470$		0			0
$471$	342.000			0.726115
$472$		0			0
$473$	176.363	+	176.363i	0.372861	+	0.372861i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	44.0908	−	44.0908i	0.0924336	−	0.0924336i
$478$		0			0
$479$			888.000i			1.85386i	0.375232	+	0.926931i	$0.377563\pi$
$479$			888.000i			1.85386i	−0.375232	+	0.926931i	$0.622437\pi$
$480$		0			0
$481$	−780.000			−1.62162
$482$		0			0
$483$	176.363	+	176.363i	0.365141	+	0.365141i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	531.539	−	531.539i	1.09146	−	1.09146i	0.0960831	−	0.995373i	$-0.469369\pi$
$487$	531.539	−	531.539i	1.09146	−	1.09146i	0.995373	−	0.0960831i	$-0.0306315\pi$
$488$		0			0
$489$			240.000i			0.490798i
$490$		0			0
$491$	−534.000			−1.08758			−0.543788	−	0.839223i	$-0.683011\pi$
$491$	−534.000			−1.08758			−0.543788	+	0.839223i	$0.683011\pi$
$492$		0			0
$493$	705.453	+	705.453i	1.43094	+	1.43094i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−293.939	+	293.939i	−0.591426	+	0.591426i
$498$		0			0
$499$			658.000i			1.31864i	0.751864	+	0.659319i	$0.229155\pi$
$499$			658.000i			1.31864i	−0.751864	+	0.659319i	$0.770845\pi$
$500$		0			0
$501$	72.0000			0.143713
$502$		0			0
$503$	−191.060	−	191.060i	−0.379841	−	0.379841i	0.491203	−	0.871045i	$-0.336557\pi$
$503$	−191.060	−	191.060i	−0.379841	−	0.379841i	−0.871045	+	0.491203i	$0.836557\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	160.442	−	160.442i	0.316453	−	0.316453i
$508$		0			0
$509$		−	324.000i		−	0.636542i	−0.948000	−	0.318271i	$-0.896898\pi$
$509$		−	324.000i		−	0.636542i	0.948000	−	0.318271i	$-0.103102\pi$
$510$		0			0
$511$	−408.000			−0.798434
$512$		0			0
$513$	−36.7423	−	36.7423i	−0.0716225	−	0.0716225i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	88.1816	−	88.1816i	0.170564	−	0.170564i
$518$		0			0
$519$		−	468.000i		−	0.901734i
$520$		0			0
$521$	342.000			0.656430			0.328215	−	0.944603i	$-0.393553\pi$
$521$	342.000			0.656430			0.328215	+	0.944603i	$0.393553\pi$
$522$		0			0
$523$	−578.080	−	578.080i	−1.10531	−	1.10531i	−0.993758	−	0.111557i	$-0.964416\pi$
$523$	−578.080	−	578.080i	−1.10531	−	1.10531i	−0.111557	−	0.993758i	$-0.535584\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−382.120	+	382.120i	−0.725086	+	0.725086i
$528$		0			0
$529$			1199.00i			2.26654i
$530$		0			0
$531$	−234.000			−0.440678
$532$		0			0
$533$	367.423	+	367.423i	0.689350	+	0.689350i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−271.893	+	271.893i	−0.506319	+	0.506319i
$538$		0			0
$539$		−	222.000i		−	0.411874i
$540$		0			0
$541$	98.0000			0.181146			0.0905730	−	0.995890i	$-0.471130\pi$
$541$	98.0000			0.181146			0.0905730	+	0.995890i	$0.471130\pi$
$542$		0			0
$543$	−232.702	−	232.702i	−0.428548	−	0.428548i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	519.292	−	519.292i	0.949345	−	0.949345i	−0.0494323	−	0.998777i	$-0.515741\pi$
$547$	519.292	−	519.292i	0.949345	−	0.949345i	0.998777	+	0.0494323i	$0.0157412\pi$
$548$		0			0
$549$			6.00000i			0.0109290i
$550$		0			0
$551$	−480.000			−0.871143
$552$		0			0
$553$	−181.262	−	181.262i	−0.327780	−	0.327780i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−396.817	+	396.817i	−0.712419	+	0.712419i	−0.967041	−	0.254622i	$-0.918049\pi$
$557$	−396.817	+	396.817i	−0.712419	+	0.712419i	0.254622	+	0.967041i	$0.418049\pi$
$558$		0			0
$559$		−	720.000i		−	1.28801i
$560$		0			0
$561$	−216.000			−0.385027
$562$		0			0
$563$	73.4847	+	73.4847i	0.130523	+	0.130523i	0.769350	−	0.638827i	$-0.220580\pi$
$563$	73.4847	+	73.4847i	0.130523	+	0.130523i	−0.638827	+	0.769350i	$0.720580\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−22.0454	+	22.0454i	−0.0388808	+	0.0388808i
$568$		0			0
$569$		−	762.000i		−	1.33919i	−0.742726	−	0.669596i	$-0.766467\pi$
$569$		−	762.000i		−	1.33919i	0.742726	−	0.669596i	$-0.233533\pi$
$570$		0			0
$571$	850.000			1.48862			0.744308	−	0.667836i	$-0.232779\pi$
$571$	850.000			1.48862			0.744308	+	0.667836i	$0.232779\pi$
$572$		0			0
$573$	249.848	+	249.848i	0.436035	+	0.436035i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−4.89898	+	4.89898i	−0.00849043	+	0.00849043i	−0.711339	−	0.702849i	$-0.751911\pi$
$577$	−4.89898	+	4.89898i	−0.00849043	+	0.00849043i	0.702849	+	0.711339i	$0.251911\pi$
$578$		0			0
$579$		−	168.000i		−	0.290155i
$580$		0			0
$581$	216.000			0.371773
$582$		0			0
$583$	88.1816	+	88.1816i	0.151255	+	0.151255i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−132.272	+	132.272i	−0.225336	+	0.225336i	−0.810741	−	0.585405i	$-0.800936\pi$
$587$	−132.272	+	132.272i	−0.225336	+	0.225336i	0.585405	+	0.810741i	$0.300936\pi$
$588$		0			0
$589$		−	260.000i		−	0.441426i
$590$		0			0
$591$	504.000			0.852792
$592$		0			0
$593$	102.879	+	102.879i	0.173488	+	0.173488i	0.788510	−	0.615022i	$-0.210853\pi$
$593$	102.879	+	102.879i	0.173488	+	0.173488i	−0.615022	+	0.788510i	$0.710853\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	56.3383	−	56.3383i	0.0943690	−	0.0943690i
$598$		0			0
$599$			732.000i			1.22204i	0.791616	+	0.611018i	$0.209240\pi$
$599$			732.000i			1.22204i	−0.791616	+	0.611018i	$0.790760\pi$
$600$		0			0
$601$	778.000			1.29451			0.647255	−	0.762274i	$-0.275917\pi$
$601$	778.000			1.29451			0.647255	+	0.762274i	$0.275917\pi$
$602$		0			0
$603$	−191.060	−	191.060i	−0.316849	−	0.316849i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	105.328	−	105.328i	0.173522	−	0.173522i	−0.615003	−	0.788525i	$-0.710845\pi$
$607$	105.328	−	105.328i	0.173522	−	0.173522i	0.788525	+	0.615003i	$0.210845\pi$
$608$		0			0
$609$			288.000i			0.472906i
$610$		0			0
$611$	−360.000			−0.589198
$612$		0			0
$613$	−71.0352	−	71.0352i	−0.115881	−	0.115881i	0.646788	−	0.762670i	$-0.276112\pi$
$613$	−71.0352	−	71.0352i	−0.115881	−	0.115881i	−0.762670	+	0.646788i	$0.776112\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	455.605	−	455.605i	0.738420	−	0.738420i	−0.233852	−	0.972272i	$-0.575133\pi$
$617$	455.605	−	455.605i	0.738420	−	0.738420i	0.972272	+	0.233852i	$0.0751332\pi$
$618$		0			0
$619$		−	362.000i		−	0.584814i	−0.956294	−	0.292407i	$-0.905544\pi$
$619$		−	362.000i		−	0.584814i	0.956294	−	0.292407i	$-0.0944562\pi$
$620$		0			0
$621$	−216.000			−0.347826
$622$		0			0
$623$	−367.423	−	367.423i	−0.589765	−	0.589765i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	73.4847	−	73.4847i	0.117200	−	0.117200i
$628$		0			0
$629$		−	936.000i		−	1.48808i
$630$		0			0
$631$	478.000			0.757528			0.378764	−	0.925493i	$-0.376349\pi$
$631$	478.000			0.757528			0.378764	+	0.925493i	$0.376349\pi$
$632$		0			0
$633$	−379.671	−	379.671i	−0.599796	−	0.599796i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−453.156	+	453.156i	−0.711390	+	0.711390i
$638$		0			0
$639$		−	360.000i		−	0.563380i
$640$		0			0
$641$	354.000			0.552262			0.276131	−	0.961120i	$-0.410948\pi$
$641$	354.000			0.552262			0.276131	+	0.961120i	$0.410948\pi$
$642$		0			0
$643$	−264.545	−	264.545i	−0.411423	−	0.411423i	0.470811	−	0.882234i	$-0.343961\pi$
$643$	−264.545	−	264.545i	−0.411423	−	0.411423i	−0.882234	+	0.470811i	$0.843961\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	484.999	−	484.999i	0.749612	−	0.749612i	−0.224794	−	0.974406i	$-0.572171\pi$
$647$	484.999	−	484.999i	0.749612	−	0.749612i	0.974406	+	0.224794i	$0.0721710\pi$
$648$		0			0
$649$		−	468.000i		−	0.721109i
$650$		0			0
$651$	−156.000			−0.239631
$652$		0			0
$653$	29.3939	+	29.3939i	0.0450136	+	0.0450136i	0.729255	−	0.684242i	$-0.239867\pi$
$653$	29.3939	+	29.3939i	0.0450136	+	0.0450136i	−0.684242	+	0.729255i	$0.739867\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	249.848	−	249.848i	0.380286	−	0.380286i
$658$		0			0
$659$		−	414.000i		−	0.628225i	−0.949386	−	0.314112i	$-0.898293\pi$
$659$		−	414.000i		−	0.628225i	0.949386	−	0.314112i	$-0.101707\pi$
$660$		0			0
$661$	−1202.00			−1.81846			−0.909228	−	0.416298i	$-0.863327\pi$
$661$	−1202.00			−1.81846			−0.909228	+	0.416298i	$0.863327\pi$
$662$		0			0
$663$	440.908	+	440.908i	0.665020	+	0.665020i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−1410.91	+	1410.91i	−2.11530	+	2.11530i
$668$		0			0
$669$			450.000i			0.672646i
$670$		0			0
$671$	−12.0000			−0.0178838
$672$		0			0
$673$	−171.464	−	171.464i	−0.254776	−	0.254776i	0.568149	−	0.822925i	$-0.307660\pi$
$673$	−171.464	−	171.464i	−0.254776	−	0.254776i	−0.822925	+	0.568149i	$0.807660\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−235.151	+	235.151i	−0.347343	+	0.347343i	−0.859119	−	0.511776i	$-0.828988\pi$
$677$	−235.151	+	235.151i	−0.347343	+	0.347343i	0.511776	+	0.859119i	$0.328988\pi$
$678$		0			0
$679$			24.0000i			0.0353461i
$680$		0			0
$681$	−180.000			−0.264317
$682$		0			0
$683$	690.756	+	690.756i	1.01136	+	1.01136i	0.999935	+	0.0114212i	$0.00363555\pi$
$683$	690.756	+	690.756i	1.01136	+	1.01136i	0.0114212	+	0.999935i	$0.496364\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−296.388	+	296.388i	−0.431424	+	0.431424i
$688$		0			0
$689$		−	360.000i		−	0.522496i
$690$		0			0
$691$	−778.000			−1.12590			−0.562952	−	0.826489i	$-0.690335\pi$
$691$	−778.000			−1.12590			−0.562952	+	0.826489i	$0.690335\pi$
$692$		0			0
$693$	−44.0908	−	44.0908i	−0.0636231	−	0.0636231i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−440.908	+	440.908i	−0.632580	+	0.632580i
$698$		0			0
$699$			180.000i			0.257511i
$700$		0			0
$701$	−84.0000			−0.119829			−0.0599144	−	0.998204i	$-0.519083\pi$
$701$	−84.0000			−0.119829			−0.0599144	+	0.998204i	$0.519083\pi$
$702$		0			0
$703$	318.434	+	318.434i	0.452964	+	0.452964i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−29.3939	+	29.3939i	−0.0415755	+	0.0415755i
$708$		0			0
$709$			502.000i			0.708039i	0.935238	+	0.354020i	$0.115185\pi$
$709$			502.000i			0.708039i	−0.935238	+	0.354020i	$0.884815\pi$
$710$		0			0
$711$	222.000			0.312236
$712$		0			0
$713$	−764.241	−	764.241i	−1.07187	−	1.07187i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−396.817	+	396.817i	−0.553441	+	0.553441i
$718$		0			0
$719$			1356.00i			1.88595i	0.332860	+	0.942976i	$0.391986\pi$
$719$			1356.00i			1.88595i	−0.332860	+	0.942976i	$0.608014\pi$
$720$		0			0
$721$	204.000			0.282940
$722$		0			0
$723$	−487.448	−	487.448i	−0.674203	−	0.674203i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−815.680	+	815.680i	−1.12198	+	1.12198i	−0.130537	+	0.991443i	$0.541670\pi$
$727$	−815.680	+	815.680i	−1.12198	+	1.12198i	−0.991443	+	0.130537i	$0.958330\pi$
$728$		0			0
$729$		−	27.0000i		−	0.0370370i
$730$		0			0
$731$	864.000			1.18194
$732$		0			0
$733$	−962.649	−	962.649i	−1.31330	−	1.31330i	−0.918968	−	0.394333i	$-0.870976\pi$
$733$	−962.649	−	962.649i	−1.31330	−	1.31330i	−0.394333	−	0.918968i	$-0.629024\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	382.120	−	382.120i	0.518481	−	0.518481i
$738$		0			0
$739$		−	610.000i		−	0.825440i	−0.910858	−	0.412720i	$-0.864579\pi$
$739$		−	610.000i		−	0.825440i	0.910858	−	0.412720i	$-0.135421\pi$
$740$		0			0
$741$	−300.000			−0.404858
$742$		0			0
$743$	−44.0908	−	44.0908i	−0.0593416	−	0.0593416i	0.676813	−	0.736155i	$-0.263360\pi$
$743$	−44.0908	−	44.0908i	−0.0593416	−	0.0593416i	−0.736155	+	0.676813i	$0.763360\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−132.272	+	132.272i	−0.177072	+	0.177072i
$748$		0			0
$749$			504.000i			0.672897i
$750$		0			0
$751$	1058.00			1.40879			0.704394	−	0.709809i	$-0.251219\pi$
$751$	1058.00			1.40879			0.704394	+	0.709809i	$0.251219\pi$
$752$		0			0
$753$	−198.409	−	198.409i	−0.263491	−	0.263491i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	26.9444	−	26.9444i	0.0355936	−	0.0355936i	−0.689086	−	0.724680i	$-0.741988\pi$
$757$	26.9444	−	26.9444i	0.0355936	−	0.0355936i	0.724680	+	0.689086i	$0.241988\pi$
$758$		0			0
$759$		−	432.000i		−	0.569170i
$760$		0			0
$761$	1122.00			1.47438			0.737188	−	0.675688i	$-0.236153\pi$
$761$	1122.00			1.47438			0.737188	+	0.675688i	$0.236153\pi$
$762$		0			0
$763$	181.262	+	181.262i	0.237565	+	0.237565i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−955.301	+	955.301i	−1.24550	+	1.24550i
$768$		0			0
$769$		−	274.000i		−	0.356307i	−0.984003	−	0.178153i	$-0.942988\pi$
$769$		−	274.000i		−	0.356307i	0.984003	−	0.178153i	$-0.0570123\pi$
$770$		0			0
$771$	−252.000			−0.326848
$772$		0			0
$773$	−382.120	−	382.120i	−0.494334	−	0.494334i	0.415334	−	0.909669i	$-0.363665\pi$
$773$	−382.120	−	382.120i	−0.494334	−	0.494334i	−0.909669	+	0.415334i	$0.863665\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	191.060	−	191.060i	0.245895	−	0.245895i
$778$		0			0
$779$		−	300.000i		−	0.385109i
$780$		0			0
$781$	720.000			0.921895
$782$		0			0
$783$	−176.363	−	176.363i	−0.225240	−	0.225240i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	421.312	−	421.312i	0.535340	−	0.535340i	−0.386817	−	0.922157i	$-0.626425\pi$
$787$	421.312	−	421.312i	0.535340	−	0.535340i	0.922157	+	0.386817i	$0.126425\pi$
$788$		0			0
$789$		−	288.000i		−	0.365019i
$790$		0			0
$791$	−648.000			−0.819216
$792$		0			0
$793$	24.4949	+	24.4949i	0.0308889	+	0.0308889i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−396.817	+	396.817i	−0.497889	+	0.497889i	−0.910780	−	0.412891i	$-0.864519\pi$
$797$	−396.817	+	396.817i	−0.497889	+	0.497889i	0.412891	+	0.910780i	$0.364519\pi$
$798$		0			0
$799$		−	432.000i		−	0.540676i
$800$		0			0
$801$	450.000			0.561798
$802$		0			0
$803$	499.696	+	499.696i	0.622286	+	0.622286i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	58.7878	−	58.7878i	0.0728473	−	0.0728473i
$808$		0			0
$809$			618.000i			0.763906i	0.924182	+	0.381953i	$0.124748\pi$
$809$			618.000i			0.763906i	−0.924182	+	0.381953i	$0.875252\pi$
$810$		0			0
$811$	−790.000			−0.974106			−0.487053	−	0.873372i	$-0.661928\pi$
$811$	−790.000			−0.974106			−0.487053	+	0.873372i	$0.661928\pi$
$812$		0			0
$813$	−56.3383	−	56.3383i	−0.0692968	−	0.0692968i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−293.939	+	293.939i	−0.359778	+	0.359778i
$818$		0			0
$819$			180.000i			0.219780i
$820$		0			0
$821$	1056.00			1.28624			0.643118	−	0.765767i	$-0.277640\pi$
$821$	1056.00			1.28624			0.643118	+	0.765767i	$0.277640\pi$
$822$		0			0
$823$	−75.9342	−	75.9342i	−0.0922651	−	0.0922651i	0.659468	−	0.751733i	$-0.270782\pi$
$823$	−75.9342	−	75.9342i	−0.0922651	−	0.0922651i	−0.751733	+	0.659468i	$0.770782\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1072.88	+	1072.88i	−1.29731	+	1.29731i	−0.367149	+	0.930162i	$0.619666\pi$
$827$	−1072.88	+	1072.88i	−1.29731	+	1.29731i	−0.930162	+	0.367149i	$0.880334\pi$
$828$		0			0
$829$		−	314.000i		−	0.378770i	−0.981903	−	0.189385i	$-0.939351\pi$
$829$		−	314.000i		−	0.378770i	0.981903	−	0.189385i	$-0.0606494\pi$
$830$		0			0
$831$	−198.000			−0.238267
$832$		0			0
$833$	−543.787	−	543.787i	−0.652805	−	0.652805i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	95.5301	−	95.5301i	0.114134	−	0.114134i
$838$		0			0
$839$			348.000i			0.414779i	0.978258	+	0.207390i	$0.0664968\pi$
$839$			348.000i			0.414779i	−0.978258	+	0.207390i	$0.933503\pi$
$840$		0			0
$841$	−1463.00			−1.73960
$842$		0			0
$843$	507.044	+	507.044i	0.601476	+	0.601476i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−208.207	+	208.207i	−0.245817	+	0.245817i
$848$		0			0
$849$		−	684.000i		−	0.805654i
$850$		0			0
$851$	1872.00			2.19976
$852$		0			0
$853$	144.520	+	144.520i	0.169425	+	0.169425i	0.786727	−	0.617301i	$-0.211774\pi$
$853$	144.520	+	144.520i	0.169425	+	0.169425i	−0.617301	+	0.786727i	$0.711774\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−543.787	+	543.787i	−0.634524	+	0.634524i	−0.949199	−	0.314676i	$-0.898104\pi$
$857$	−543.787	+	543.787i	−0.634524	+	0.634524i	0.314676	+	0.949199i	$0.398104\pi$
$858$		0			0
$859$			1030.00i			1.19907i	0.800349	+	0.599534i	$0.204648\pi$
$859$			1030.00i			1.19907i	−0.800349	+	0.599534i	$0.795352\pi$
$860$		0			0
$861$	−180.000			−0.209059
$862$		0			0
$863$	−940.604	−	940.604i	−1.08992	−	1.08992i	−0.995535	−	0.0943881i	$-0.969911\pi$
$863$	−940.604	−	940.604i	−1.08992	−	1.08992i	−0.0943881	−	0.995535i	$-0.530089\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−175.139	+	175.139i	−0.202005	+	0.202005i
$868$		0			0
$869$			444.000i			0.510932i
$870$		0			0
$871$	−1560.00			−1.79104
$872$		0			0
$873$	−14.6969	−	14.6969i	−0.0168350	−	0.0168350i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	688.307	−	688.307i	0.784842	−	0.784842i	−0.195801	−	0.980644i	$-0.562731\pi$
$877$	688.307	−	688.307i	0.784842	−	0.784842i	0.980644	+	0.195801i	$0.0627308\pi$
$878$		0			0
$879$		−	576.000i		−	0.655290i
$880$		0			0
$881$	−1290.00			−1.46425			−0.732123	−	0.681173i	$-0.761470\pi$
$881$	−1290.00			−1.46425			−0.732123	+	0.681173i	$0.761470\pi$
$882$		0			0
$883$	333.131	+	333.131i	0.377271	+	0.377271i	0.870117	−	0.492845i	$-0.164043\pi$
$883$	333.131	+	333.131i	0.377271	+	0.377271i	−0.492845	+	0.870117i	$0.664043\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	676.059	−	676.059i	0.762186	−	0.762186i	−0.214531	−	0.976717i	$-0.568822\pi$
$887$	676.059	−	676.059i	0.762186	−	0.762186i	0.976717	+	0.214531i	$0.0688223\pi$
$888$		0			0
$889$			468.000i			0.526434i
$890$		0			0
$891$	54.0000			0.0606061
$892$		0			0
$893$	146.969	+	146.969i	0.164579	+	0.164579i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−881.816	+	881.816i	−0.983073	+	0.983073i
$898$		0			0
$899$		−	1248.00i		−	1.38821i
$900$		0			0
$901$	432.000			0.479467
$902$		0			0
$903$	176.363	+	176.363i	0.195308	+	0.195308i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	215.555	−	215.555i	0.237657	−	0.237657i	−0.578222	−	0.815879i	$-0.696253\pi$
$907$	215.555	−	215.555i	0.237657	−	0.237657i	0.815879	+	0.578222i	$0.196253\pi$
$908$		0			0
$909$		−	36.0000i		−	0.0396040i
$910$		0			0
$911$	−408.000			−0.447859			−0.223930	−	0.974605i	$-0.571889\pi$
$911$	−408.000			−0.447859			−0.223930	+	0.974605i	$0.571889\pi$
$912$		0			0
$913$	−264.545	−	264.545i	−0.289753	−	0.289753i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−249.848	+	249.848i	−0.272462	+	0.272462i
$918$		0			0
$919$		−	574.000i		−	0.624592i	−0.949985	−	0.312296i	$-0.898902\pi$
$919$		−	574.000i		−	0.624592i	0.949985	−	0.312296i	$-0.101098\pi$
$920$		0			0
$921$	−108.000			−0.117264
$922$		0			0
$923$	−1469.69	−	1469.69i	−1.59230	−	1.59230i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−124.924	+	124.924i	−0.134762	+	0.134762i
$928$		0			0
$929$			474.000i			0.510226i	0.966911	+	0.255113i	$0.0821127\pi$
$929$			474.000i			0.510226i	−0.966911	+	0.255113i	$0.917887\pi$
$930$		0			0
$931$	370.000			0.397422
$932$		0			0
$933$	249.848	+	249.848i	0.267790	+	0.267790i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−362.524	+	362.524i	−0.386899	+	0.386899i	−0.873580	−	0.486681i	$-0.838207\pi$
$937$	−362.524	+	362.524i	−0.386899	+	0.386899i	0.486681	+	0.873580i	$0.338207\pi$
$938$		0			0
$939$			912.000i			0.971246i
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$	−881.816	−	881.816i	−0.935118	−	0.935118i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−102.879	+	102.879i	−0.108636	+	0.108636i	−0.759336	−	0.650699i	$-0.774476\pi$
$947$	−102.879	+	102.879i	−0.108636	+	0.108636i	0.650699	+	0.759336i	$0.274476\pi$
$948$		0			0
$949$		−	2040.00i		−	2.14963i
$950$		0			0
$951$	−864.000			−0.908517
$952$		0			0
$953$	338.030	+	338.030i	0.354701	+	0.354701i	0.861855	−	0.507155i	$-0.169303\pi$
$953$	338.030	+	338.030i	0.354701	+	0.354701i	−0.507155	+	0.861855i	$0.669303\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	352.727	−	352.727i	0.368575	−	0.368575i
$958$		0			0
$959$			216.000i			0.225235i
$960$		0			0
$961$	−285.000			−0.296566
$962$		0			0
$963$	−308.636	−	308.636i	−0.320494	−	0.320494i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−85.7321	+	85.7321i	−0.0886579	+	0.0886579i	−0.750045	−	0.661387i	$-0.769968\pi$
$967$	−85.7321	+	85.7321i	−0.0886579	+	0.0886579i	0.661387	+	0.750045i	$0.269968\pi$
$968$		0			0
$969$		−	360.000i		−	0.371517i
$970$		0			0
$971$	1542.00			1.58805			0.794027	−	0.607883i	$-0.207981\pi$
$971$	1542.00			1.58805			0.794027	+	0.607883i	$0.207981\pi$
$972$		0			0
$973$	298.838	+	298.838i	0.307130	+	0.307130i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−543.787	+	543.787i	−0.556588	+	0.556588i	−0.928334	−	0.371746i	$-0.878759\pi$
$977$	−543.787	+	543.787i	−0.556588	+	0.556588i	0.371746	+	0.928334i	$0.378759\pi$
$978$		0			0
$979$			900.000i			0.919305i
$980$		0			0
$981$	−222.000			−0.226300
$982$		0			0
$983$	426.211	+	426.211i	0.433582	+	0.433582i	0.889845	−	0.456263i	$-0.150812\pi$
$983$	426.211	+	426.211i	0.433582	+	0.433582i	−0.456263	+	0.889845i	$0.650812\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	88.1816	−	88.1816i	0.0893431	−	0.0893431i
$988$		0			0
$989$			1728.00i			1.74722i
$990$		0			0
$991$	262.000			0.264379			0.132190	−	0.991224i	$-0.457799\pi$
$991$	262.000			0.264379			0.132190	+	0.991224i	$0.457799\pi$
$992$		0			0
$993$	663.812	+	663.812i	0.668491	+	0.668491i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1124.32	+	1124.32i	−1.12770	+	1.12770i	−0.137148	+	0.990551i	$0.543794\pi$
$997$	−1124.32	+	1124.32i	−1.12770	+	1.12770i	−0.990551	+	0.137148i	$0.956206\pi$
$998$		0			0
$999$			234.000i			0.234234i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.bg.h.193.1		4
4.3	odd	2		300.3.k.b.193.2	yes	4
5.2	odd	4	inner	1200.3.bg.h.1057.1		4
5.3	odd	4	inner	1200.3.bg.h.1057.2		4
5.4	even	2	inner	1200.3.bg.h.193.2		4
12.11	even	2		900.3.l.c.793.1		4
20.3	even	4		300.3.k.b.157.1	✓	4
20.7	even	4		300.3.k.b.157.2	yes	4
20.19	odd	2		300.3.k.b.193.1	yes	4
60.23	odd	4		900.3.l.c.757.2		4
60.47	odd	4		900.3.l.c.757.1		4
60.59	even	2		900.3.l.c.793.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.3.k.b.157.1	✓	4	20.3	even	4
300.3.k.b.157.2	yes	4	20.7	even	4
300.3.k.b.193.1	yes	4	20.19	odd	2
300.3.k.b.193.2	yes	4	4.3	odd	2
900.3.l.c.757.1		4	60.47	odd	4
900.3.l.c.757.2		4	60.23	odd	4
900.3.l.c.793.1		4	12.11	even	2
900.3.l.c.793.2		4	60.59	even	2
1200.3.bg.h.193.1		4	1.1	even	1	trivial
1200.3.bg.h.193.2		4	5.4	even	2	inner
1200.3.bg.h.1057.1		4	5.2	odd	4	inner
1200.3.bg.h.1057.2		4	5.3	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)

\(f(q)\)	\(=\)	\(q+(-1.22474 - 1.22474i) q^{3} +(2.44949 - 2.44949i) q^{7} +3.00000i q^{9} +O(q^{10})\)	\(q+(-1.22474 - 1.22474i) q^{3} +(2.44949 - 2.44949i) q^{7} +3.00000i q^{9} -6.00000 q^{11} +(12.2474 + 12.2474i) q^{13} +(-14.6969 + 14.6969i) q^{17} -10.0000i q^{19} -6.00000 q^{21} +(-29.3939 - 29.3939i) q^{23} +(3.67423 - 3.67423i) q^{27} -48.0000i q^{29} +26.0000 q^{31} +(7.34847 + 7.34847i) q^{33} +(-31.8434 + 31.8434i) q^{37} -30.0000i q^{39} +30.0000 q^{41} +(-29.3939 - 29.3939i) q^{43} +(-14.6969 + 14.6969i) q^{47} +37.0000i q^{49} +36.0000 q^{51} +(-14.6969 - 14.6969i) q^{53} +(-12.2474 + 12.2474i) q^{57} +78.0000i q^{59} +2.00000 q^{61} +(7.34847 + 7.34847i) q^{63} +(-63.6867 + 63.6867i) q^{67} +72.0000i q^{69} -120.000 q^{71} +(-83.2827 - 83.2827i) q^{73} +(-14.6969 + 14.6969i) q^{77} -74.0000i q^{79} -9.00000 q^{81} +(44.0908 + 44.0908i) q^{83} +(-58.7878 + 58.7878i) q^{87} -150.000i q^{89} +60.0000 q^{91} +(-31.8434 - 31.8434i) q^{93} +(-4.89898 + 4.89898i) q^{97} -18.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 24 q^{11} - 24 q^{21} + 104 q^{31} + 120 q^{41} + 144 q^{51} + 8 q^{61} - 480 q^{71} - 36 q^{81} + 240 q^{91}+O(q^{100}) \) 4 * q - 24 * q^11 - 24 * q^21 + 104 * q^31 + 120 * q^41 + 144 * q^51 + 8 * q^61 - 480 * q^71 - 36 * q^81 + 240 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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