LMFDB - Embedded newform 1200.3.bg.h.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 300)
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.h.1057.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 + 1.22474i) q^{3} +(-2.44949 + 2.44949i) q^{7} +3.00000i q^{9} +O(q^{10})$	$q+(1.22474 + 1.22474i) q^{3} +(-2.44949 + 2.44949i) q^{7} +3.00000i q^{9} -6.00000 q^{11} +(-12.2474 - 12.2474i) q^{13} +(14.6969 - 14.6969i) q^{17} -10.0000i q^{19} -6.00000 q^{21} +(29.3939 + 29.3939i) q^{23} +(-3.67423 + 3.67423i) q^{27} -48.0000i q^{29} +26.0000 q^{31} +(-7.34847 - 7.34847i) q^{33} +(31.8434 - 31.8434i) q^{37} -30.0000i q^{39} +30.0000 q^{41} +(29.3939 + 29.3939i) q^{43} +(14.6969 - 14.6969i) q^{47} +37.0000i q^{49} +36.0000 q^{51} +(14.6969 + 14.6969i) q^{53} +(12.2474 - 12.2474i) q^{57} +78.0000i q^{59} +2.00000 q^{61} +(-7.34847 - 7.34847i) q^{63} +(63.6867 - 63.6867i) q^{67} +72.0000i q^{69} -120.000 q^{71} +(83.2827 + 83.2827i) q^{73} +(14.6969 - 14.6969i) q^{77} -74.0000i q^{79} -9.00000 q^{81} +(-44.0908 - 44.0908i) q^{83} +(58.7878 - 58.7878i) q^{87} -150.000i q^{89} +60.0000 q^{91} +(31.8434 + 31.8434i) q^{93} +(4.89898 - 4.89898i) q^{97} -18.0000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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	−0.860082	−	0.510155i	$-0.829588\pi$
$7$	−2.44949	+	2.44949i	−0.349927	+	0.349927i	0.510155	+	0.860082i	$0.329588\pi$
$8$		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	0.0563023	−	0.998414i	$-0.482069\pi$
$13$	−12.2474	−	12.2474i	−0.942111	−	0.942111i	−0.998414	+	0.0563023i	$0.982069\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	14.6969	−	14.6969i	0.864526	−	0.864526i	−0.127334	−	0.991860i	$-0.540642\pi$
$17$	14.6969	−	14.6969i	0.864526	−	0.864526i	0.991860	+	0.127334i	$0.0406420\pi$
$18$		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	$0.109144\pi$
$23$	29.3939	+	29.3939i	1.27799	+	1.27799i	0.336206	+	0.941788i	$0.390856\pi$
$24$		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	−0.130780	−	0.991411i	$-0.541748\pi$
$37$	31.8434	−	31.8434i	0.860632	−	0.860632i	0.991411	+	0.130780i	$0.0417481\pi$
$38$		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	0.960805	−	0.277226i	$-0.0894151\pi$
$43$	29.3939	+	29.3939i	0.683579	+	0.683579i	−0.277226	+	0.960805i	$0.589415\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	14.6969	−	14.6969i	0.312701	−	0.312701i	−0.533254	−	0.845955i	$-0.679031\pi$
$47$	14.6969	−	14.6969i	0.312701	−	0.312701i	0.845955	+	0.533254i	$0.179031\pi$
$48$		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	0.832031	−	0.554730i	$-0.187178\pi$
$53$	14.6969	+	14.6969i	0.277301	+	0.277301i	−0.554730	+	0.832031i	$0.687178\pi$
$54$		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	−0.0482853	−	0.998834i	$-0.515376\pi$
$67$	63.6867	−	63.6867i	0.950548	−	0.950548i	0.998834	+	0.0482853i	$0.0153757\pi$
$68$		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	$0.0487533\pi$
$73$	83.2827	+	83.2827i	1.14086	+	1.14086i	0.152565	+	0.988293i	$0.451247\pi$
$74$		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	0.389719	−	0.920934i	$-0.372572\pi$
$83$	−44.0908	−	44.0908i	−0.531215	−	0.531215i	−0.920934	+	0.389719i	$0.872572\pi$
$84$		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	−0.681403	−	0.731908i	$-0.738630\pi$
$97$	4.89898	−	4.89898i	0.0505049	−	0.0505049i	0.731908	+	0.681403i	$0.238630\pi$
$98$		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	0.475455	−	0.879740i	$-0.342283\pi$
$103$	−41.6413	−	41.6413i	−0.404285	−	0.404285i	−0.879740	+	0.475455i	$0.842283\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	102.879	−	102.879i	0.961482	−	0.961482i	−0.0378032	−	0.999285i	$-0.512036\pi$
$107$	102.879	−	102.879i	0.961482	−	0.961482i	0.999285	+	0.0378032i	$0.0120360\pi$
$108$		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	$0.0603543\pi$
$113$	132.272	+	132.272i	1.17055	+	1.17055i	0.188475	+	0.982078i	$0.439646\pi$
$114$		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	−0.222685	−	0.974890i	$-0.571482\pi$
$127$	95.5301	−	95.5301i	0.752206	−	0.752206i	0.974890	+	0.222685i	$0.0714821\pi$
$128$		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	−0.527638	−	0.849469i	$-0.676922\pi$
$137$	44.0908	−	44.0908i	0.321831	−	0.321831i	0.849469	+	0.527638i	$0.176922\pi$
$138$		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	−0.105151	−	0.994456i	$-0.533533\pi$
$157$	139.621	−	139.621i	0.889305	−	0.889305i	0.994456	+	0.105151i	$0.0335326\pi$
$158$		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	0.940605	−	0.339503i	$-0.110259\pi$
$163$	97.9796	+	97.9796i	0.601102	+	0.601102i	−0.339503	+	0.940605i	$0.610259\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	29.3939	−	29.3939i	0.176011	−	0.176011i	−0.613603	−	0.789614i	$-0.710281\pi$
$167$	29.3939	−	29.3939i	0.176011	−	0.176011i	0.789614	+	0.613603i	$0.210281\pi$
$168$		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	$-0.964748\pi$
$173$	−191.060	−	191.060i	−1.10439	−	1.10439i	−0.110520	−	0.993874i	$-0.535252\pi$
$174$		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	0.506735	−	0.862102i	$-0.330852\pi$
$193$	−68.5857	−	68.5857i	−0.355366	−	0.355366i	−0.862102	+	0.506735i	$0.830852\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	205.757	−	205.757i	1.04445	−	1.04445i	0.0454876	−	0.998965i	$-0.485516\pi$
$197$	205.757	−	205.757i	1.04445	−	1.04445i	0.998965	−	0.0454876i	$-0.0144841\pi$
$198$		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	0.986653	−	0.162834i	$-0.0520635\pi$
$223$	183.712	+	183.712i	0.823819	+	0.823819i	−0.162834	+	0.986653i	$0.552064\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−73.4847	+	73.4847i	−0.323721	+	0.323721i	−0.850193	−	0.526472i	$-0.823515\pi$
$227$	−73.4847	+	73.4847i	−0.323721	+	0.323721i	0.526472	+	0.850193i	$0.323515\pi$
$228$		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	0.846991	−	0.531607i	$-0.178411\pi$
$233$	73.4847	+	73.4847i	0.315385	+	0.315385i	−0.531607	+	0.846991i	$0.678411\pi$
$234$		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	−0.878341	−	0.478035i	$-0.841349\pi$
$257$	−102.879	+	102.879i	−0.400306	+	0.400306i	0.478035	+	0.878341i	$0.341349\pi$
$258$		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	0.447319	−	0.894374i	$-0.352379\pi$
$263$	−117.576	−	117.576i	−0.447055	−	0.447055i	−0.894374	+	0.447319i	$0.852379\pi$
$264$		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	−0.837798	−	0.545981i	$-0.816157\pi$
$277$	−80.8332	+	80.8332i	−0.291816	+	0.291816i	0.545981	+	0.837798i	$0.316157\pi$
$278$		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	0.0131927	−	0.999913i	$-0.495801\pi$
$283$	−279.242	−	279.242i	−0.986720	−	0.986720i	−0.999913	+	0.0131927i	$0.995801\pi$
$284$		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	0.180932	−	0.983496i	$-0.442088\pi$
$293$	−235.151	−	235.151i	−0.802563	−	0.802563i	−0.983496	+	0.180932i	$0.942088\pi$
$294$		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	−0.775260	−	0.631642i	$-0.782381\pi$
$307$	−44.0908	+	44.0908i	−0.143618	+	0.143618i	0.631642	+	0.775260i	$0.282381\pi$
$308$		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	$0.0681054\pi$
$313$	372.322	+	372.322i	1.18953	+	1.18953i	0.212331	+	0.977198i	$0.431895\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−352.727	+	352.727i	−1.11270	+	1.11270i	−0.119918	+	0.992784i	$0.538263\pi$
$317$	−352.727	+	352.727i	−1.11270	+	1.11270i	−0.992784	+	0.119918i	$0.961737\pi$
$318$		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	$0.510399\pi$
$337$	−347.828	+	347.828i	−1.03213	+	1.03213i	−0.999466	+	0.0326628i	$0.989601\pi$
$338$		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	−0.926616	−	0.376010i	$-0.877296\pi$
$347$	−191.060	+	191.060i	−0.550606	+	0.550606i	0.376010	+	0.926616i	$0.377296\pi$
$348$		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	$0.0756826\pi$
$353$	426.211	+	426.211i	1.20740	+	1.20740i	0.235530	+	0.971867i	$0.424317\pi$
$354$		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	−0.524418	−	0.851461i	$-0.675717\pi$
$367$	120.025	−	120.025i	0.327044	−	0.327044i	0.851461	+	0.524418i	$0.175717\pi$
$368$		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	$-0.967510\pi$
$373$	−409.065	−	409.065i	−1.09669	−	1.09669i	−0.101893	−	0.994795i	$-0.532490\pi$
$374$		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	$-0.961693\pi$
$383$	−426.211	−	426.211i	−1.11282	−	1.11282i	−0.120056	−	0.992767i	$-0.538307\pi$
$384$		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	−0.359381	−	0.933191i	$-0.617012\pi$
$397$	227.803	−	227.803i	0.573810	−	0.573810i	0.933191	+	0.359381i	$0.117012\pi$
$398$		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	0.934080	−	0.357064i	$-0.116222\pi$
$433$	249.848	+	249.848i	0.577016	+	0.577016i	−0.357064	+	0.934080i	$0.616222\pi$
$434$		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	0.976872	−	0.213825i	$-0.0685924\pi$
$443$	338.030	+	338.030i	0.763046	+	0.763046i	−0.213825	+	0.976872i	$0.568592\pi$
$444$		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	−0.983717	−	0.179727i	$-0.942479\pi$
$457$	−367.423	+	367.423i	−0.803990	+	0.803990i	0.179727	+	0.983717i	$0.442479\pi$
$458$		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	0.807195	−	0.590285i	$-0.200985\pi$
$463$	100.429	+	100.429i	0.216909	+	0.216909i	−0.590285	+	0.807195i	$0.700985\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−191.060	+	191.060i	−0.409122	+	0.409122i	−0.881432	−	0.472310i	$-0.843420\pi$
$467$	−191.060	+	191.060i	−0.409122	+	0.409122i	0.472310	+	0.881432i	$0.343420\pi$
$468$		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	$0.530631\pi$
$487$	−531.539	+	531.539i	−1.09146	+	1.09146i	−0.995373	+	0.0960831i	$0.969369\pi$
$488$		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	0.871045	−	0.491203i	$-0.163443\pi$
$503$	191.060	+	191.060i	0.379841	+	0.379841i	−0.491203	+	0.871045i	$0.663443\pi$
$504$		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	$0.0355836\pi$
$523$	578.080	+	578.080i	1.10531	+	1.10531i	0.111557	+	0.993758i	$0.464416\pi$
$524$		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	−0.998777	−	0.0494323i	$-0.984259\pi$
$547$	−519.292	+	519.292i	−0.949345	+	0.949345i	0.0494323	+	0.998777i	$0.484259\pi$
$548$		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	−0.254622	−	0.967041i	$-0.581951\pi$
$557$	396.817	−	396.817i	0.712419	−	0.712419i	0.967041	+	0.254622i	$0.0819510\pi$
$558$		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	0.638827	−	0.769350i	$-0.279420\pi$
$563$	−73.4847	−	73.4847i	−0.130523	−	0.130523i	−0.769350	+	0.638827i	$0.779420\pi$
$564$		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	−0.702849	−	0.711339i	$-0.748089\pi$
$577$	4.89898	−	4.89898i	0.00849043	−	0.00849043i	0.711339	+	0.702849i	$0.248089\pi$
$578$		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	−0.585405	−	0.810741i	$-0.699064\pi$
$587$	132.272	−	132.272i	0.225336	−	0.225336i	0.810741	+	0.585405i	$0.199064\pi$
$588$		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	0.615022	−	0.788510i	$-0.289147\pi$
$593$	−102.879	−	102.879i	−0.173488	−	0.173488i	−0.788510	+	0.615022i	$0.789147\pi$
$594$		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	−0.788525	−	0.615003i	$-0.789155\pi$
$607$	−105.328	+	105.328i	−0.173522	+	0.173522i	0.615003	+	0.788525i	$0.289155\pi$
$608$		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	0.762670	−	0.646788i	$-0.223888\pi$
$613$	71.0352	+	71.0352i	0.115881	+	0.115881i	−0.646788	+	0.762670i	$0.723888\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−455.605	+	455.605i	−0.738420	+	0.738420i	−0.972272	−	0.233852i	$-0.924867\pi$
$617$	−455.605	+	455.605i	−0.738420	+	0.738420i	0.233852	+	0.972272i	$0.424867\pi$
$618$		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	0.882234	−	0.470811i	$-0.156039\pi$
$643$	264.545	+	264.545i	0.411423	+	0.411423i	−0.470811	+	0.882234i	$0.656039\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−484.999	+	484.999i	−0.749612	+	0.749612i	−0.974406	−	0.224794i	$-0.927829\pi$
$647$	−484.999	+	484.999i	−0.749612	+	0.749612i	0.224794	+	0.974406i	$0.427829\pi$
$648$		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	0.684242	−	0.729255i	$-0.260133\pi$
$653$	−29.3939	−	29.3939i	−0.0450136	−	0.0450136i	−0.729255	+	0.684242i	$0.760133\pi$
$654$		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	0.822925	−	0.568149i	$-0.192340\pi$
$673$	171.464	+	171.464i	0.254776	+	0.254776i	−0.568149	+	0.822925i	$0.692340\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	235.151	−	235.151i	0.347343	−	0.347343i	−0.511776	−	0.859119i	$-0.671012\pi$
$677$	235.151	−	235.151i	0.347343	−	0.347343i	0.859119	+	0.511776i	$0.171012\pi$
$678$		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	$-0.996364\pi$
$683$	−690.756	−	690.756i	−1.01136	−	1.01136i	−0.0114212	−	0.999935i	$-0.503636\pi$
$684$		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	$-0.458330\pi$
$727$	815.680	−	815.680i	1.12198	−	1.12198i	0.991443	−	0.130537i	$-0.0416703\pi$
$728$		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	$0.129024\pi$
$733$	962.649	+	962.649i	1.31330	+	1.31330i	0.394333	+	0.918968i	$0.370976\pi$
$734$		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	0.736155	−	0.676813i	$-0.236640\pi$
$743$	44.0908	+	44.0908i	0.0593416	+	0.0593416i	−0.676813	+	0.736155i	$0.736640\pi$
$744$		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	−0.724680	−	0.689086i	$-0.758012\pi$
$757$	−26.9444	+	26.9444i	−0.0355936	+	0.0355936i	0.689086	+	0.724680i	$0.258012\pi$
$758$		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	0.909669	−	0.415334i	$-0.136335\pi$
$773$	382.120	+	382.120i	0.494334	+	0.494334i	−0.415334	+	0.909669i	$0.636335\pi$
$774$		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	−0.922157	−	0.386817i	$-0.873575\pi$
$787$	−421.312	+	421.312i	−0.535340	+	0.535340i	0.386817	+	0.922157i	$0.373575\pi$
$788$		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	−0.412891	−	0.910780i	$-0.635481\pi$
$797$	396.817	−	396.817i	0.497889	−	0.497889i	0.910780	+	0.412891i	$0.135481\pi$
$798$		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	0.751733	−	0.659468i	$-0.229218\pi$
$823$	75.9342	+	75.9342i	0.0922651	+	0.0922651i	−0.659468	+	0.751733i	$0.729218\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1072.88	−	1072.88i	1.29731	−	1.29731i	0.367149	−	0.930162i	$-0.380334\pi$
$827$	1072.88	−	1072.88i	1.29731	−	1.29731i	0.930162	−	0.367149i	$-0.119666\pi$
$828$		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	0.617301	−	0.786727i	$-0.288226\pi$
$853$	−144.520	−	144.520i	−0.169425	−	0.169425i	−0.786727	+	0.617301i	$0.788226\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	543.787	−	543.787i	0.634524	−	0.634524i	−0.314676	−	0.949199i	$-0.601896\pi$
$857$	543.787	−	543.787i	0.634524	−	0.634524i	0.949199	+	0.314676i	$0.101896\pi$
$858$		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	$0.0300895\pi$
$863$	940.604	+	940.604i	1.08992	+	1.08992i	0.0943881	+	0.995535i	$0.469911\pi$
$864$		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	−0.980644	−	0.195801i	$-0.937269\pi$
$877$	−688.307	+	688.307i	−0.784842	+	0.784842i	0.195801	+	0.980644i	$0.437269\pi$
$878$		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	0.492845	−	0.870117i	$-0.335957\pi$
$883$	−333.131	−	333.131i	−0.377271	−	0.377271i	−0.870117	+	0.492845i	$0.835957\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−676.059	+	676.059i	−0.762186	+	0.762186i	−0.976717	−	0.214531i	$-0.931178\pi$
$887$	−676.059	+	676.059i	−0.762186	+	0.762186i	0.214531	+	0.976717i	$0.431178\pi$
$888$		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	−0.815879	−	0.578222i	$-0.803747\pi$
$907$	−215.555	+	215.555i	−0.237657	+	0.237657i	0.578222	+	0.815879i	$0.303747\pi$
$908$		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	−0.486681	−	0.873580i	$-0.661793\pi$
$937$	362.524	−	362.524i	0.386899	−	0.386899i	0.873580	+	0.486681i	$0.161793\pi$
$938$		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	−0.650699	−	0.759336i	$-0.725524\pi$
$947$	102.879	−	102.879i	0.108636	−	0.108636i	0.759336	+	0.650699i	$0.225524\pi$
$948$		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	0.507155	−	0.861855i	$-0.330697\pi$
$953$	−338.030	−	338.030i	−0.354701	−	0.354701i	−0.861855	+	0.507155i	$0.830697\pi$
$954$		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	−0.661387	−	0.750045i	$-0.730032\pi$
$967$	85.7321	−	85.7321i	0.0886579	−	0.0886579i	0.750045	+	0.661387i	$0.230032\pi$
$968$		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	−0.371746	−	0.928334i	$-0.621241\pi$
$977$	543.787	−	543.787i	0.556588	−	0.556588i	0.928334	+	0.371746i	$0.121241\pi$
$978$		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	0.456263	−	0.889845i	$-0.349188\pi$
$983$	−426.211	−	426.211i	−0.433582	−	0.433582i	−0.889845	+	0.456263i	$0.849188\pi$
$984$		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	$-0.456206\pi$
$997$	1124.32	−	1124.32i	1.12770	−	1.12770i	0.990551	−	0.137148i	$-0.0437937\pi$
$998$		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.193.2		4
4.3	odd	2		300.3.k.b.193.1	yes	4
5.2	odd	4	inner	1200.3.bg.h.1057.2		4
5.3	odd	4	inner	1200.3.bg.h.1057.1		4
5.4	even	2	inner	1200.3.bg.h.193.1		4
12.11	even	2		900.3.l.c.793.2		4
20.3	even	4		300.3.k.b.157.2	yes	4
20.7	even	4		300.3.k.b.157.1	✓	4
20.19	odd	2		300.3.k.b.193.2	yes	4
60.23	odd	4		900.3.l.c.757.1		4
60.47	odd	4		900.3.l.c.757.2		4
60.59	even	2		900.3.l.c.793.1		4

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

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