LMFDB - Embedded newform 1200.3.c.l.449.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,3,Mod(449,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1200.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1200.c (of order $2$, degree $1$, 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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 18x^{10} + 75x^{8} + 1270x^{6} + 14397x^{4} - 7740x^{2} + 39204 $ x^12 + 18x^10 + 75x^8 + 1270x^6 + 14397x^4 - 7740*x^2 + 39204
Coefficient ring:	$\Z[a_1, \ldots, a_{37}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 600)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.3
Root			$0.139571 - 3.56627i$ of defining polynomial
Character	$\chi$	$=$	1200.449
Dual form			1200.3.c.l.449.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.55378 - 2.56627i) q^{3} -1.34301i q^{7} +(-4.17150 + 7.97487i) q^{9} +O(q^{10})$	$q+(-1.55378 - 2.56627i) q^{3} -1.34301i q^{7} +(-4.17150 + 7.97487i) q^{9} -3.66586i q^{11} +7.34301i q^{13} -6.31089 q^{17} +30.7406 q^{19} +(-3.44653 + 2.08675i) q^{21} -26.2426 q^{23} +(26.9473 - 1.68602i) q^{27} +40.9060i q^{29} +9.97097 q^{31} +(-9.40758 + 5.69595i) q^{33} -39.4232i q^{37} +(18.8442 - 11.4095i) q^{39} +68.1695i q^{41} -24.0802i q^{43} +79.7705 q^{47} +47.1963 q^{49} +(9.80576 + 16.1955i) q^{51} +38.9657 q^{53} +(-47.7643 - 78.8888i) q^{57} +39.9648i q^{59} -64.2475 q^{61} +(10.7103 + 5.60237i) q^{63} +53.9454i q^{67} +(40.7754 + 67.3457i) q^{69} -136.706i q^{71} -74.6196i q^{73} -4.92328 q^{77} +79.6417 q^{79} +(-46.1971 - 66.5344i) q^{81} -0.0506471 q^{83} +(104.976 - 63.5592i) q^{87} +22.6492i q^{89} +9.86173 q^{91} +(-15.4927 - 25.5882i) q^{93} -35.0428i q^{97} +(29.2347 + 15.2921i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 32 q^{9}+O(q^{10}) $ 12 * q - 32 * q^9	$ 12 q - 32 q^{9} + 100 q^{19} - 36 q^{21} + 228 q^{31} - 12 q^{39} - 152 q^{49} + 12 q^{51} + 124 q^{61} - 312 q^{69} - 152 q^{79} - 448 q^{81} + 620 q^{91} - 500 q^{99}+O(q^{100}) $ 12 * q - 32 * q^9 + 100 * q^19 - 36 * q^21 + 228 * q^31 - 12 * q^39 - 152 * q^49 + 12 * q^51 + 124 * q^61 - 312 * q^69 - 152 * q^79 - 448 * q^81 + 620 * q^91 - 500 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$-1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.55378	−	2.56627i	−0.517928	−	0.855424i
$3$	−1.55378	−	2.56627i	−0.517928	−	0.855424i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	1.34301i		−	0.191858i	−0.995388	−	0.0959292i	$-0.969418\pi$
$7$		−	1.34301i		−	0.191858i	0.995388	−	0.0959292i	$-0.0305822\pi$
$8$		0			0
$9$	−4.17150	+	7.97487i	−0.463500	+	0.886097i
$10$		0			0
$11$		−	3.66586i		−	0.333260i	−0.986020	−	0.166630i	$-0.946712\pi$
$11$		−	3.66586i		−	0.333260i	0.986020	−	0.166630i	$-0.0532885\pi$
$12$		0			0
$13$			7.34301i			0.564847i	0.959290	+	0.282423i	$0.0911383\pi$
$13$			7.34301i			0.564847i	−0.959290	+	0.282423i	$0.908862\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6.31089			−0.371229			−0.185614	−	0.982623i	$-0.559428\pi$
$17$	−6.31089			−0.371229			−0.185614	+	0.982623i	$0.559428\pi$
$18$		0			0
$19$	30.7406			1.61793			0.808964	−	0.587858i	$-0.200029\pi$
$19$	30.7406			1.61793			0.808964	+	0.587858i	$0.200029\pi$
$20$		0			0
$21$	−3.44653	+	2.08675i	−0.164120	+	0.0993689i
$22$		0			0
$23$	−26.2426			−1.14098			−0.570492	−	0.821303i	$-0.693247\pi$
$23$	−26.2426			−1.14098			−0.570492	+	0.821303i	$0.693247\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	26.9473	−	1.68602i	0.998048	−	0.0624451i
$28$		0			0
$29$			40.9060i			1.41055i	0.708932	+	0.705277i	$0.249177\pi$
$29$			40.9060i			1.41055i	−0.708932	+	0.705277i	$0.750823\pi$
$30$		0			0
$31$	9.97097			0.321644			0.160822	−	0.986983i	$-0.448585\pi$
$31$	9.97097			0.321644			0.160822	+	0.986983i	$0.448585\pi$
$32$		0			0
$33$	−9.40758	+	5.69595i	−0.285078	+	0.172605i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	39.4232i		−	1.06549i	−0.846275	−	0.532746i	$-0.821160\pi$
$37$		−	39.4232i		−	1.06549i	0.846275	−	0.532746i	$-0.178840\pi$
$38$		0			0
$39$	18.8442	−	11.4095i	0.483184	−	0.292550i
$40$		0			0
$41$			68.1695i			1.66267i	0.555771	+	0.831335i	$0.312423\pi$
$41$			68.1695i			1.66267i	−0.555771	+	0.831335i	$0.687577\pi$
$42$		0			0
$43$		−	24.0802i		−	0.560005i	−0.959999	−	0.280003i	$-0.909665\pi$
$43$		−	24.0802i		−	0.560005i	0.959999	−	0.280003i	$-0.0903353\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	79.7705			1.69724			0.848622	−	0.529000i	$-0.177433\pi$
$47$	79.7705			1.69724			0.848622	+	0.529000i	$0.177433\pi$
$48$		0			0
$49$	47.1963			0.963190
$50$		0			0
$51$	9.80576	+	16.1955i	0.192270	+	0.317558i
$52$		0			0
$53$	38.9657			0.735202			0.367601	−	0.929984i	$-0.380179\pi$
$53$	38.9657			0.735202			0.367601	+	0.929984i	$0.380179\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−47.7643	−	78.8888i	−0.837971	−	1.38401i
$58$		0			0
$59$			39.9648i			0.677369i	0.940900	+	0.338684i	$0.109982\pi$
$59$			39.9648i			0.677369i	−0.940900	+	0.338684i	$0.890018\pi$
$60$		0			0
$61$	−64.2475			−1.05324			−0.526619	−	0.850101i	$-0.676541\pi$
$61$	−64.2475			−1.05324			−0.526619	+	0.850101i	$0.676541\pi$
$62$		0			0
$63$	10.7103	+	5.60237i	0.170005	+	0.0889265i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			53.9454i			0.805155i	0.915386	+	0.402577i	$0.131886\pi$
$67$			53.9454i			0.805155i	−0.915386	+	0.402577i	$0.868114\pi$
$68$		0			0
$69$	40.7754	+	67.3457i	0.590948	+	0.976025i
$70$		0			0
$71$		−	136.706i		−	1.92543i	−0.270511	−	0.962717i	$-0.587193\pi$
$71$		−	136.706i		−	1.92543i	0.270511	−	0.962717i	$-0.412807\pi$
$72$		0			0
$73$		−	74.6196i		−	1.02219i	−0.859526	−	0.511093i	$-0.829241\pi$
$73$		−	74.6196i		−	1.02219i	0.859526	−	0.511093i	$-0.170759\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4.92328			−0.0639387
$78$		0			0
$79$	79.6417			1.00812			0.504062	−	0.863668i	$-0.331838\pi$
$79$	79.6417			1.00812			0.504062	+	0.863668i	$0.331838\pi$
$80$		0			0
$81$	−46.1971	−	66.5344i	−0.570335	−	0.821412i
$82$		0			0
$83$	−0.0506471			−0.000610206			−0.000305103	−	1.00000i	$-0.500097\pi$
$83$	−0.0506471			−0.000610206			−0.000305103		1.00000i	$0.500097\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	104.976	−	63.5592i	1.20662	−	0.730566i
$88$		0			0
$89$			22.6492i			0.254485i	0.991872	+	0.127243i	$0.0406126\pi$
$89$			22.6492i			0.254485i	−0.991872	+	0.127243i	$0.959387\pi$
$90$		0			0
$91$	9.86173			0.108371
$92$		0			0
$93$	−15.4927	−	25.5882i	−0.166589	−	0.275142i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	35.0428i		−	0.361266i	−0.983551	−	0.180633i	$-0.942185\pi$
$97$		−	35.0428i		−	0.361266i	0.983551	−	0.180633i	$-0.0578146\pi$
$98$		0			0
$99$	29.2347	+	15.2921i	0.295300	+	0.154466i
$100$		0			0
$101$			10.2711i			0.101694i	0.998706	+	0.0508472i	$0.0161921\pi$
$101$			10.2711i			0.101694i	−0.998706	+	0.0508472i	$0.983808\pi$
$102$		0			0
$103$		−	108.963i		−	1.05789i	−0.848656	−	0.528944i	$-0.822588\pi$
$103$		−	108.963i		−	1.05789i	0.848656	−	0.528944i	$-0.177412\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−113.251			−1.05842			−0.529209	−	0.848492i	$-0.677511\pi$
$107$	−113.251			−1.05842			−0.529209	+	0.848492i	$0.677511\pi$
$108$		0			0
$109$	105.722			0.969926			0.484963	−	0.874535i	$-0.338833\pi$
$109$	105.722			0.969926			0.484963	+	0.874535i	$0.338833\pi$
$110$		0			0
$111$	−101.171	+	61.2552i	−0.911448	+	0.551849i
$112$		0			0
$113$	168.318			1.48954			0.744770	−	0.667321i	$-0.232559\pi$
$113$	168.318			1.48954			0.744770	+	0.667321i	$0.232559\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−58.5595	−	30.6314i	−0.500509	−	0.261807i
$118$		0			0
$119$			8.47558i			0.0712234i
$120$		0			0
$121$	107.562			0.888938
$122$		0			0
$123$	174.941	−	105.921i	1.42229	−	0.861144i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	97.8159i		−	0.770204i	−0.922874	−	0.385102i	$-0.874166\pi$
$127$		−	97.8159i		−	0.770204i	0.922874	−	0.385102i	$-0.125834\pi$
$128$		0			0
$129$	−61.7964	+	37.4155i	−0.479042	+	0.290043i
$130$		0			0
$131$		−	88.9361i		−	0.678902i	−0.940624	−	0.339451i	$-0.889759\pi$
$131$		−	88.9361i		−	0.678902i	0.940624	−	0.339451i	$-0.110241\pi$
$132$		0			0
$133$		−	41.2850i		−	0.310413i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	241.635			1.76376			0.881880	−	0.471474i	$-0.156278\pi$
$137$	241.635			1.76376			0.881880	+	0.471474i	$0.156278\pi$
$138$		0			0
$139$	215.214			1.54830			0.774149	−	0.633003i	$-0.218178\pi$
$139$	215.214			1.54830			0.774149	+	0.633003i	$0.218178\pi$
$140$		0			0
$141$	−123.946	−	204.713i	−0.879051	−	1.45186i
$142$		0			0
$143$	26.9184			0.188241
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−73.3329	−	121.119i	−0.498864	−	0.823936i
$148$		0			0
$149$		−	104.338i		−	0.700257i	−0.936702	−	0.350128i	$-0.886138\pi$
$149$		−	104.338i		−	0.700257i	0.936702	−	0.350128i	$-0.113862\pi$
$150$		0			0
$151$	−92.1895			−0.610526			−0.305263	−	0.952268i	$-0.598744\pi$
$151$	−92.1895			−0.610526			−0.305263	+	0.952268i	$0.598744\pi$
$152$		0			0
$153$	26.3259	−	50.3285i	0.172065	−	0.328945i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			250.985i			1.59863i	0.600913	+	0.799314i	$0.294804\pi$
$157$			250.985i			1.59863i	−0.600913	+	0.799314i	$0.705196\pi$
$158$		0			0
$159$	−60.5443	−	99.9966i	−0.380782	−	0.628909i
$160$		0			0
$161$			35.2441i			0.218907i
$162$		0			0
$163$		−	87.6008i		−	0.537428i	−0.963220	−	0.268714i	$-0.913401\pi$
$163$		−	87.6008i		−	0.537428i	0.963220	−	0.268714i	$-0.0865987\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−116.164			−0.695590			−0.347795	−	0.937571i	$-0.613069\pi$
$167$	−116.164			−0.695590			−0.347795	+	0.937571i	$0.613069\pi$
$168$		0			0
$169$	115.080			0.680948
$170$		0			0
$171$	−128.235	+	245.153i	−0.749911	+	1.43364i
$172$		0			0
$173$	76.7440			0.443607			0.221804	−	0.975091i	$-0.428806\pi$
$173$	76.7440			0.443607			0.221804	+	0.975091i	$0.428806\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	102.560	−	62.0967i	0.579438	−	0.350829i
$178$		0			0
$179$			281.899i			1.57486i	0.616406	+	0.787428i	$0.288588\pi$
$179$			281.899i			1.57486i	−0.616406	+	0.787428i	$0.711412\pi$
$180$		0			0
$181$	−37.8892			−0.209333			−0.104666	−	0.994507i	$-0.533377\pi$
$181$	−37.8892			−0.209333			−0.104666	+	0.994507i	$0.533377\pi$
$182$		0			0
$183$	99.8268	+	164.877i	0.545502	+	0.900965i
$184$		0			0
$185$		0			0
$186$		0			0
$187$			23.1348i			0.123716i
$188$		0			0
$189$	−2.26434	−	36.1905i	−0.0119806	−	0.191484i
$190$		0			0
$191$		−	62.1819i		−	0.325560i	−0.986662	−	0.162780i	$-0.947954\pi$
$191$		−	62.1819i		−	0.325560i	0.986662	−	0.162780i	$-0.0520460\pi$
$192$		0			0
$193$		−	263.150i		−	1.36347i	−0.731597	−	0.681737i	$-0.761225\pi$
$193$		−	263.150i		−	1.36347i	0.731597	−	0.681737i	$-0.238775\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	180.658			0.917044			0.458522	−	0.888683i	$-0.348379\pi$
$197$	180.658			0.917044			0.458522	+	0.888683i	$0.348379\pi$
$198$		0			0
$199$	−136.985			−0.688365			−0.344183	−	0.938903i	$-0.611844\pi$
$199$	−136.985			−0.688365			−0.344183	+	0.938903i	$0.611844\pi$
$200$		0			0
$201$	138.439	−	83.8195i	0.688749	−	0.417013i
$202$		0			0
$203$	54.9372			0.270627
$204$		0			0
$205$		0			0
$206$		0			0
$207$	109.471	−	209.282i	0.528847	−	1.01102i
$208$		0			0
$209$		−	112.691i		−	0.539190i
$210$		0			0
$211$	139.434			0.660822			0.330411	−	0.943837i	$-0.392813\pi$
$211$	139.434			0.660822			0.330411	+	0.943837i	$0.392813\pi$
$212$		0			0
$213$	−350.824	+	212.411i	−1.64706	+	0.997237i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	13.3911i		−	0.0617102i
$218$		0			0
$219$	−191.494	+	115.943i	−0.874402	+	0.529419i
$220$		0			0
$221$		−	46.3409i		−	0.209687i
$222$		0			0
$223$			4.94040i			0.0221543i	0.999939	+	0.0110771i	$0.00352603\pi$
$223$			4.94040i			0.0221543i	−0.999939	+	0.0110771i	$0.996474\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−334.372			−1.47301			−0.736503	−	0.676434i	$-0.763524\pi$
$227$	−334.372			−1.47301			−0.736503	+	0.676434i	$0.763524\pi$
$228$		0			0
$229$	337.101			1.47206			0.736028	−	0.676951i	$-0.236699\pi$
$229$	337.101			1.47206			0.736028	+	0.676951i	$0.236699\pi$
$230$		0			0
$231$	7.64971	+	12.6345i	0.0331156	+	0.0546947i
$232$		0			0
$233$	−113.099			−0.485402			−0.242701	−	0.970101i	$-0.578033\pi$
$233$	−113.099			−0.485402			−0.242701	+	0.970101i	$0.578033\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−123.746	−	204.382i	−0.522135	−	0.862373i
$238$		0			0
$239$			130.740i			0.547029i	0.961868	+	0.273514i	$0.0881862\pi$
$239$			130.740i			0.547029i	−0.961868	+	0.273514i	$0.911814\pi$
$240$		0			0
$241$	336.727			1.39721			0.698604	−	0.715508i	$-0.253805\pi$
$241$	336.727			1.39721			0.698604	+	0.715508i	$0.253805\pi$
$242$		0			0
$243$	−98.9650	+	221.934i	−0.407264	+	0.913311i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			225.729i			0.913882i
$248$		0			0
$249$	0.0786948	+	0.129974i	0.000316043	+	0.000521985i
$250$		0			0
$251$			135.453i			0.539655i	0.962909	+	0.269828i	$0.0869667\pi$
$251$			135.453i			0.539655i	−0.962909	+	0.269828i	$0.913033\pi$
$252$		0			0
$253$			96.2017i			0.380244i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	104.440			0.406379			0.203190	−	0.979139i	$-0.434869\pi$
$257$	104.440			0.406379			0.203190	+	0.979139i	$0.434869\pi$
$258$		0			0
$259$	−52.9457			−0.204424
$260$		0			0
$261$	−326.220	−	170.640i	−1.24989	−	0.653792i
$262$		0			0
$263$	−326.365			−1.24093			−0.620466	−	0.784234i	$-0.713056\pi$
$263$	−326.365			−1.24093			−0.620466	+	0.784234i	$0.713056\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	58.1239	−	35.1919i	0.217693	−	0.131805i
$268$		0			0
$269$		−	56.9353i		−	0.211656i	−0.994384	−	0.105828i	$-0.966251\pi$
$269$		−	56.9353i		−	0.211656i	0.994384	−	0.105828i	$-0.0337492\pi$
$270$		0			0
$271$	110.853			0.409053			0.204526	−	0.978861i	$-0.434435\pi$
$271$	110.853			0.409053			0.204526	+	0.978861i	$0.434435\pi$
$272$		0			0
$273$	−15.3230	−	25.3079i	−0.0561282	−	0.0927028i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			492.842i			1.77921i	0.456728	+	0.889606i	$0.349021\pi$
$277$			492.842i			1.77921i	−0.456728	+	0.889606i	$0.650979\pi$
$278$		0			0
$279$	−41.5940	+	79.5172i	−0.149082	+	0.285008i
$280$		0			0
$281$			428.705i			1.52564i	0.646611	+	0.762820i	$0.276186\pi$
$281$			428.705i			1.52564i	−0.646611	+	0.762820i	$0.723814\pi$
$282$		0			0
$283$			289.871i			1.02428i	0.858903	+	0.512139i	$0.171147\pi$
$283$			289.871i			1.02428i	−0.858903	+	0.512139i	$0.828853\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	91.5522			0.318997
$288$		0			0
$289$	−249.173			−0.862189
$290$		0			0
$291$	−89.9293	+	54.4490i	−0.309035	+	0.187110i
$292$		0			0
$293$	−407.033			−1.38919			−0.694596	−	0.719400i	$-0.744417\pi$
$293$	−407.033			−1.38919			−0.694596	+	0.719400i	$0.744417\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−6.18070	−	98.7849i	−0.0208104	−	0.332609i
$298$		0			0
$299$		−	192.700i		−	0.644481i
$300$		0			0
$301$	−32.3400			−0.107442
$302$		0			0
$303$	26.3585	−	15.9591i	0.0869918	−	0.0526704i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			284.594i			0.927016i	0.886093	+	0.463508i	$0.153410\pi$
$307$			284.594i			0.927016i	−0.886093	+	0.463508i	$0.846590\pi$
$308$		0			0
$309$	−279.628	+	169.304i	−0.904944	+	0.547911i
$310$		0			0
$311$			32.0008i			0.102896i	0.998676	+	0.0514482i	$0.0163837\pi$
$311$			32.0008i			0.102896i	−0.998676	+	0.0514482i	$0.983616\pi$
$312$		0			0
$313$			142.954i			0.456722i	0.973576	+	0.228361i	$0.0733367\pi$
$313$			142.954i			0.456722i	−0.973576	+	0.228361i	$0.926663\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−248.052			−0.782500			−0.391250	−	0.920284i	$-0.627957\pi$
$317$	−248.052			−0.782500			−0.391250	+	0.920284i	$0.627957\pi$
$318$		0			0
$319$	149.956			0.470080
$320$		0			0
$321$	175.967	+	290.632i	0.548184	+	0.905395i
$322$		0			0
$323$	−194.001			−0.600622
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−164.269	−	271.311i	−0.502352	−	0.829698i
$328$		0			0
$329$		−	107.132i		−	0.325630i
$330$		0			0
$331$	377.660			1.14097			0.570484	−	0.821309i	$-0.306756\pi$
$331$	377.660			1.14097			0.570484	+	0.821309i	$0.306756\pi$
$332$		0			0
$333$	314.395	+	164.454i	0.944129	+	0.493856i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	30.2254i		−	0.0896896i	−0.998994	−	0.0448448i	$-0.985721\pi$
$337$		−	30.2254i		−	0.0896896i	0.998994	−	0.0448448i	$-0.0142793\pi$
$338$		0			0
$339$	−261.530	−	431.950i	−0.771475	−	1.27419i
$340$		0			0
$341$		−	36.5521i		−	0.107191i
$342$		0			0
$343$		−	129.193i		−	0.376655i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	424.938			1.22460			0.612302	−	0.790624i	$-0.290244\pi$
$347$	424.938			1.22460			0.612302	+	0.790624i	$0.290244\pi$
$348$		0			0
$349$	−263.069			−0.753779			−0.376889	−	0.926258i	$-0.623006\pi$
$349$	−263.069			−0.753779			−0.376889	+	0.926258i	$0.623006\pi$
$350$		0			0
$351$	12.3804	+	197.874i	0.0352719	+	0.563744i
$352$		0			0
$353$	352.772			0.999353			0.499677	−	0.866212i	$-0.333452\pi$
$353$	352.772			0.999353			0.499677	+	0.866212i	$0.333452\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	21.7506	−	13.1692i	0.0609262	−	0.0368886i
$358$		0			0
$359$		−	421.619i		−	1.17443i	−0.809432	−	0.587214i	$-0.800225\pi$
$359$		−	421.619i		−	1.17443i	0.809432	−	0.587214i	$-0.199775\pi$
$360$		0			0
$361$	583.987			1.61769
$362$		0			0
$363$	−167.127	−	276.032i	−0.460406	−	0.760419i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			605.435i			1.64969i	0.565361	+	0.824844i	$0.308737\pi$
$367$			605.435i			1.64969i	−0.565361	+	0.824844i	$0.691263\pi$
$368$		0			0
$369$	−543.643	−	284.369i	−1.47329	−	0.770649i
$370$		0			0
$371$		−	52.3313i		−	0.141055i
$372$		0			0
$373$			691.549i			1.85402i	0.375040	+	0.927009i	$0.377629\pi$
$373$			691.549i			1.85402i	−0.375040	+	0.927009i	$0.622371\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−300.373			−0.796747
$378$		0			0
$379$	179.894			0.474655			0.237327	−	0.971430i	$-0.423729\pi$
$379$	179.894			0.474655			0.237327	+	0.971430i	$0.423729\pi$
$380$		0			0
$381$	−251.022	+	151.985i	−0.658851	+	0.398910i
$382$		0			0
$383$	137.087			0.357929			0.178965	−	0.983856i	$-0.442725\pi$
$383$	137.087			0.357929			0.178965	+	0.983856i	$0.442725\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	192.037	+	100.451i	0.496219	+	0.259563i
$388$		0			0
$389$			682.795i			1.75526i	0.479341	+	0.877629i	$0.340876\pi$
$389$			682.795i			1.75526i	−0.479341	+	0.877629i	$0.659124\pi$
$390$		0			0
$391$	165.614			0.423566
$392$		0			0
$393$	−228.234	+	138.188i	−0.580749	+	0.351622i
$394$		0			0
$395$		0			0
$396$		0			0
$397$			538.381i			1.35612i	0.735005	+	0.678062i	$0.237180\pi$
$397$			538.381i			1.35612i	−0.735005	+	0.678062i	$0.762820\pi$
$398$		0			0
$399$	−105.948	+	64.1479i	−0.265535	+	0.160772i
$400$		0			0
$401$			113.772i			0.283720i	0.989887	+	0.141860i	$0.0453083\pi$
$401$			113.772i			0.283720i	−0.989887	+	0.141860i	$0.954692\pi$
$402$		0			0
$403$			73.2169i			0.181680i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−144.520			−0.355086
$408$		0			0
$409$	−438.806			−1.07288			−0.536438	−	0.843940i	$-0.680230\pi$
$409$	−438.806			−1.07288			−0.536438	+	0.843940i	$0.680230\pi$
$410$		0			0
$411$	−375.449	−	620.101i	−0.913501	−	1.50876i
$412$		0			0
$413$	53.6730			0.129959
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−334.396	−	552.296i	−0.801908	−	1.32445i
$418$		0			0
$419$		−	544.982i		−	1.30067i	−0.759646	−	0.650337i	$-0.774628\pi$
$419$		−	544.982i		−	1.30067i	0.759646	−	0.650337i	$-0.225372\pi$
$420$		0			0
$421$	−557.806			−1.32495			−0.662477	−	0.749082i	$-0.730495\pi$
$421$	−557.806			−1.32495			−0.662477	+	0.749082i	$0.730495\pi$
$422$		0			0
$423$	−332.763	+	636.159i	−0.786673	+	1.50392i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			86.2850i			0.202073i
$428$		0			0
$429$	−41.8254	−	69.0800i	−0.0974952	−	0.161026i
$430$		0			0
$431$		−	41.0366i		−	0.0952124i	−0.998866	−	0.0476062i	$-0.984841\pi$
$431$		−	41.0366i		−	0.0952124i	0.998866	−	0.0476062i	$-0.0151593\pi$
$432$		0			0
$433$		−	589.338i		−	1.36106i	−0.732721	−	0.680529i	$-0.761750\pi$
$433$		−	589.338i		−	1.36106i	0.732721	−	0.680529i	$-0.238250\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−806.715			−1.84603
$438$		0			0
$439$	−363.125			−0.827165			−0.413582	−	0.910467i	$-0.635723\pi$
$439$	−363.125			−0.827165			−0.413582	+	0.910467i	$0.635723\pi$
$440$		0			0
$441$	−196.880	+	376.385i	−0.446439	+	0.853480i
$442$		0			0
$443$	−142.457			−0.321573			−0.160786	−	0.986989i	$-0.551403\pi$
$443$	−142.457			−0.321573			−0.160786	+	0.986989i	$0.551403\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−267.760	+	162.119i	−0.599016	+	0.362683i
$448$		0			0
$449$		−	613.747i		−	1.36692i	−0.729988	−	0.683460i	$-0.760474\pi$
$449$		−	613.747i		−	1.36692i	0.729988	−	0.683460i	$-0.239526\pi$
$450$		0			0
$451$	249.900			0.554101
$452$		0			0
$453$	143.243	+	236.583i	0.316209	+	0.522259i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	66.7594i		−	0.146082i	−0.997329	−	0.0730409i	$-0.976730\pi$
$457$		−	66.7594i		−	0.146082i	0.997329	−	0.0730409i	$-0.0232704\pi$
$458$		0			0
$459$	−170.061	+	10.6403i	−0.370504	+	0.0231814i
$460$		0			0
$461$		−	891.544i		−	1.93393i	−0.254900	−	0.966967i	$-0.582043\pi$
$461$		−	891.544i		−	1.93393i	0.254900	−	0.966967i	$-0.417957\pi$
$462$		0			0
$463$		−	381.895i		−	0.824826i	−0.910997	−	0.412413i	$-0.864686\pi$
$463$		−	381.895i		−	0.824826i	0.910997	−	0.412413i	$-0.135314\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	546.640			1.17054			0.585268	−	0.810840i	$-0.300989\pi$
$467$	546.640			1.17054			0.585268	+	0.810840i	$0.300989\pi$
$468$		0			0
$469$	72.4491			0.154476
$470$		0			0
$471$	644.095	−	389.976i	1.36751	−	0.827975i
$472$		0			0
$473$	−88.2746			−0.186627
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−162.546	+	310.746i	−0.340766	+	0.651460i
$478$		0			0
$479$		−	227.312i		−	0.474555i	−0.971442	−	0.237277i	$-0.923745\pi$
$479$		−	227.312i		−	0.474555i	0.971442	−	0.237277i	$-0.0762550\pi$
$480$		0			0
$481$	289.485			0.601840
$482$		0			0
$483$	90.4459	−	54.7617i	0.187259	−	0.113378i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			896.798i			1.84147i	0.390185	+	0.920737i	$0.372411\pi$
$487$			896.798i			1.84147i	−0.390185	+	0.920737i	$0.627589\pi$
$488$		0			0
$489$	−224.808	+	136.113i	−0.459729	+	0.278349i
$490$		0			0
$491$		−	368.073i		−	0.749639i	−0.927098	−	0.374819i	$-0.877705\pi$
$491$		−	368.073i		−	0.749639i	0.927098	−	0.374819i	$-0.122295\pi$
$492$		0			0
$493$		−	258.154i		−	0.523638i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−183.597			−0.369411
$498$		0			0
$499$	−471.446			−0.944782			−0.472391	−	0.881389i	$-0.656609\pi$
$499$	−471.446			−0.944782			−0.472391	+	0.881389i	$0.656609\pi$
$500$		0			0
$501$	180.493	+	298.107i	0.360266	+	0.595024i
$502$		0			0
$503$	601.673			1.19617			0.598085	−	0.801433i	$-0.295929\pi$
$503$	601.673			1.19617			0.598085	+	0.801433i	$0.295929\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−178.810	−	295.327i	−0.352682	−	0.582499i
$508$		0			0
$509$		−	471.175i		−	0.925687i	−0.886440	−	0.462844i	$-0.846829\pi$
$509$		−	471.175i		−	0.925687i	0.886440	−	0.462844i	$-0.153171\pi$
$510$		0			0
$511$	−100.215			−0.196115
$512$		0			0
$513$	828.377	−	51.8293i	1.61477	−	0.101032i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	292.427i		−	0.565623i
$518$		0			0
$519$	−119.244	−	196.946i	−0.229757	−	0.379472i
$520$		0			0
$521$			324.027i			0.621932i	0.950421	+	0.310966i	$0.100653\pi$
$521$			324.027i			0.621932i	−0.950421	+	0.310966i	$0.899347\pi$
$522$		0			0
$523$			88.5290i			0.169272i	0.996412	+	0.0846358i	$0.0269727\pi$
$523$			88.5290i			0.169272i	−0.996412	+	0.0846358i	$0.973027\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−62.9257			−0.119404
$528$		0			0
$529$	159.675			0.301844
$530$		0			0
$531$	−318.714	−	166.713i	−0.600214	−	0.313961i
$532$		0			0
$533$	−500.569			−0.939154
$534$		0			0
$535$		0			0
$536$		0			0
$537$	723.431	−	438.011i	1.34717	−	0.815663i
$538$		0			0
$539$		−	173.015i		−	0.320992i
$540$		0			0
$541$	−566.555			−1.04724			−0.523619	−	0.851953i	$-0.675418\pi$
$541$	−566.555			−1.04724			−0.523619	+	0.851953i	$0.675418\pi$
$542$		0			0
$543$	58.8717	+	97.2341i	0.108419	+	0.179068i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			8.15550i			0.0149095i	0.999972	+	0.00745475i	$0.00237294\pi$
$547$			8.15550i			0.0149095i	−0.999972	+	0.00745475i	$0.997627\pi$
$548$		0			0
$549$	268.009	−	512.366i	0.488176	−	0.933271i
$550$		0			0
$551$			1257.48i			2.28217i
$552$		0			0
$553$		−	106.960i		−	0.193417i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−403.842			−0.725031			−0.362516	−	0.931978i	$-0.618082\pi$
$557$	−403.842			−0.725031			−0.362516	+	0.931978i	$0.618082\pi$
$558$		0			0
$559$	176.821			0.316317
$560$		0			0
$561$	59.3702	−	35.9465i	0.105829	−	0.0640758i
$562$		0			0
$563$	319.709			0.567867			0.283933	−	0.958844i	$-0.408361\pi$
$563$	319.709			0.567867			0.283933	+	0.958844i	$0.408361\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−89.3563	+	62.0431i	−0.157595	+	0.109423i
$568$		0			0
$569$			705.574i			1.24003i	0.784592	+	0.620013i	$0.212873\pi$
$569$			705.574i			1.24003i	−0.784592	+	0.620013i	$0.787127\pi$
$570$		0			0
$571$	13.6020			0.0238214			0.0119107	−	0.999929i	$-0.496209\pi$
$571$	13.6020			0.0238214			0.0119107	+	0.999929i	$0.496209\pi$
$572$		0			0
$573$	−159.576	+	96.6173i	−0.278492	+	0.168617i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			426.416i			0.739023i	0.929226	+	0.369512i	$0.120475\pi$
$577$			426.416i			0.739023i	−0.929226	+	0.369512i	$0.879525\pi$
$578$		0			0
$579$	−675.316	+	408.879i	−1.16635	+	0.706182i
$580$		0			0
$581$			0.0680196i			0.000117073i
$582$		0			0
$583$		−	142.843i		−	0.245013i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	178.246			0.303656			0.151828	−	0.988407i	$-0.451484\pi$
$587$	178.246			0.303656			0.151828	+	0.988407i	$0.451484\pi$
$588$		0			0
$589$	306.514			0.520397
$590$		0			0
$591$	−280.703	−	463.617i	−0.474963	−	0.784461i
$592$		0			0
$593$	122.209			0.206086			0.103043	−	0.994677i	$-0.467142\pi$
$593$	122.209			0.206086			0.103043	+	0.994677i	$0.467142\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	212.845	+	351.540i	0.356524	+	0.588844i
$598$		0			0
$599$			874.960i			1.46070i	0.683072	+	0.730351i	$0.260643\pi$
$599$			874.960i			1.46070i	−0.683072	+	0.730351i	$0.739357\pi$
$600$		0			0
$601$	976.356			1.62455			0.812276	−	0.583273i	$-0.198228\pi$
$601$	976.356			1.62455			0.812276	+	0.583273i	$0.198228\pi$
$602$		0			0
$603$	−430.207	−	225.033i	−0.713445	−	0.373190i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	728.093i		−	1.19949i	−0.800189	−	0.599747i	$-0.795268\pi$
$607$		−	728.093i		−	1.19949i	0.800189	−	0.599747i	$-0.204732\pi$
$608$		0			0
$609$	−85.3606	−	140.984i	−0.140165	−	0.231500i
$610$		0			0
$611$			585.755i			0.958683i
$612$		0			0
$613$			368.041i			0.600394i	0.953877	+	0.300197i	$0.0970523\pi$
$613$			368.041i			0.600394i	−0.953877	+	0.300197i	$0.902948\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	550.637			0.892442			0.446221	−	0.894923i	$-0.352769\pi$
$617$	550.637			0.892442			0.446221	+	0.894923i	$0.352769\pi$
$618$		0			0
$619$	757.081			1.22307			0.611536	−	0.791217i	$-0.290552\pi$
$619$	757.081			1.22307			0.611536	+	0.791217i	$0.290552\pi$
$620$		0			0
$621$	−707.168	+	44.2455i	−1.13876	+	0.0712488i
$622$		0			0
$623$	30.4180			0.0488251
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−289.195	+	175.097i	−0.461236	+	0.279262i
$628$		0			0
$629$			248.796i			0.395541i
$630$		0			0
$631$	−101.694			−0.161164			−0.0805820	−	0.996748i	$-0.525678\pi$
$631$	−101.694			−0.161164			−0.0805820	+	0.996748i	$0.525678\pi$
$632$		0			0
$633$	−216.650	−	357.824i	−0.342259	−	0.565283i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			346.563i			0.544055i
$638$		0			0
$639$	1090.21	+	570.269i	1.70612	+	0.892439i
$640$		0			0
$641$		−	1008.28i		−	1.57298i	−0.617601	−	0.786491i	$-0.711895\pi$
$641$		−	1008.28i		−	1.57298i	0.617601	−	0.786491i	$-0.288105\pi$
$642$		0			0
$643$			1126.32i			1.75167i	0.482615	+	0.875833i	$0.339687\pi$
$643$			1126.32i			1.75167i	−0.482615	+	0.875833i	$0.660313\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−410.296			−0.634152			−0.317076	−	0.948400i	$-0.602701\pi$
$647$	−410.296			−0.634152			−0.317076	+	0.948400i	$0.602701\pi$
$648$		0			0
$649$	146.505			0.225740
$650$		0			0
$651$	−34.3652	+	20.8069i	−0.0527884	+	0.0319614i
$652$		0			0
$653$	−393.103			−0.601996			−0.300998	−	0.953625i	$-0.597320\pi$
$653$	−393.103			−0.601996			−0.300998	+	0.953625i	$0.597320\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	595.081	+	311.276i	0.905755	+	0.473784i
$658$		0			0
$659$		−	882.684i		−	1.33943i	−0.742618	−	0.669715i	$-0.766416\pi$
$659$		−	882.684i		−	1.33943i	0.742618	−	0.669715i	$-0.233584\pi$
$660$		0			0
$661$	260.491			0.394087			0.197043	−	0.980395i	$-0.436866\pi$
$661$	260.491			0.394087			0.197043	+	0.980395i	$0.436866\pi$
$662$		0			0
$663$	−118.923	+	72.0038i	−0.179372	+	0.108603i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	1073.48i		−	1.60942i
$668$		0			0
$669$	12.6784	−	7.67632i	0.0189513	−	0.0114743i
$670$		0			0
$671$			235.522i			0.351002i
$672$		0			0
$673$		−	69.6242i		−	0.103454i	−0.998661	−	0.0517268i	$-0.983528\pi$
$673$		−	69.6242i		−	0.103454i	0.998661	−	0.0517268i	$-0.0164725\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1148.22			1.69604			0.848019	−	0.529967i	$-0.177796\pi$
$677$	1148.22			1.69604			0.848019	+	0.529967i	$0.177796\pi$
$678$		0			0
$679$	−47.0628			−0.0693119
$680$		0			0
$681$	519.543	+	858.091i	0.762912	+	1.26004i
$682$		0			0
$683$	205.570			0.300981			0.150490	−	0.988611i	$-0.451915\pi$
$683$	205.570			0.300981			0.150490	+	0.988611i	$0.451915\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−523.782	−	865.092i	−0.762420	−	1.25923i
$688$		0			0
$689$			286.125i			0.415276i
$690$		0			0
$691$	45.2341			0.0654618			0.0327309	−	0.999464i	$-0.489580\pi$
$691$	45.2341			0.0654618			0.0327309	+	0.999464i	$0.489580\pi$
$692$		0			0
$693$	20.5375	−	39.2625i	0.0296356	−	0.0566558i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	430.210i		−	0.617231i
$698$		0			0
$699$	175.731	+	290.242i	0.251404	+	0.415225i
$700$		0			0
$701$		−	743.766i		−	1.06101i	−0.847683	−	0.530504i	$-0.822003\pi$
$701$		−	743.766i		−	1.06101i	0.847683	−	0.530504i	$-0.177997\pi$
$702$		0			0
$703$		−	1211.90i		−	1.72389i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	13.7942			0.0195109
$708$		0			0
$709$	283.784			0.400259			0.200130	−	0.979769i	$-0.435864\pi$
$709$	283.784			0.400259			0.200130	+	0.979769i	$0.435864\pi$
$710$		0			0
$711$	−332.226	+	635.132i	−0.467266	+	0.893294i
$712$		0			0
$713$	−261.665			−0.366991
$714$		0			0
$715$		0			0
$716$		0			0
$717$	335.514	−	203.142i	0.467942	−	0.283322i
$718$		0			0
$719$			1088.15i			1.51343i	0.653747	+	0.756713i	$0.273196\pi$
$719$			1088.15i			1.51343i	−0.653747	+	0.756713i	$0.726804\pi$
$720$		0			0
$721$	−146.338			−0.202965
$722$		0			0
$723$	−523.202	−	864.134i	−0.723654	−	1.19521i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	248.930i		−	0.342407i	−0.985236	−	0.171203i	$-0.945234\pi$
$727$		−	248.930i		−	0.342407i	0.985236	−	0.171203i	$-0.0547655\pi$
$728$		0			0
$729$	723.315	−	90.8673i	0.992201	−	0.124646i
$730$		0			0
$731$			151.968i			0.207890i
$732$		0			0
$733$			235.310i			0.321023i	0.987034	+	0.160512i	$0.0513144\pi$
$733$			235.310i			0.321023i	−0.987034	+	0.160512i	$0.948686\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	197.756			0.268326
$738$		0			0
$739$	356.976			0.483053			0.241527	−	0.970394i	$-0.422352\pi$
$739$	356.976			0.483053			0.241527	+	0.970394i	$0.422352\pi$
$740$		0			0
$741$	579.282	−	350.734i	0.781756	−	0.473325i
$742$		0			0
$743$	230.874			0.310732			0.155366	−	0.987857i	$-0.450344\pi$
$743$	230.874			0.310732			0.155366	+	0.987857i	$0.450344\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	0.211275	−	0.403904i	0.000282831	−	0.000540702i
$748$		0			0
$749$			152.097i			0.203066i
$750$		0			0
$751$	−620.963			−0.826848			−0.413424	−	0.910539i	$-0.635667\pi$
$751$	−620.963			−0.826848			−0.413424	+	0.910539i	$0.635667\pi$
$752$		0			0
$753$	347.610	−	210.466i	0.461634	−	0.279503i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	28.7800i		−	0.0380184i	−0.999819	−	0.0190092i	$-0.993949\pi$
$757$		−	28.7800i		−	0.0380184i	0.999819	−	0.0190092i	$-0.00605119\pi$
$758$		0			0
$759$	246.880	−	149.477i	0.325270	−	0.196939i
$760$		0			0
$761$		−	439.052i		−	0.576940i	−0.957489	−	0.288470i	$-0.906853\pi$
$761$		−	439.052i		−	0.576940i	0.957489	−	0.288470i	$-0.0931466\pi$
$762$		0			0
$763$		−	141.986i		−	0.186088i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−293.462			−0.382610
$768$		0			0
$769$	354.830			0.461417			0.230709	−	0.973023i	$-0.425896\pi$
$769$	354.830			0.461417			0.230709	+	0.973023i	$0.425896\pi$
$770$		0			0
$771$	−162.277	−	268.020i	−0.210475	−	0.347627i
$772$		0			0
$773$	−954.595			−1.23492			−0.617461	−	0.786601i	$-0.711839\pi$
$773$	−954.595			−1.23492			−0.617461	+	0.786601i	$0.711839\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	82.2663	+	135.873i	0.105877	+	0.174869i
$778$		0			0
$779$			2095.57i			2.69008i
$780$		0			0
$781$	−501.144			−0.641669
$782$		0			0
$783$	68.9683	+	1102.31i	0.0880822	+	1.40780i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	679.060i		−	0.862846i	−0.902150	−	0.431423i	$-0.858012\pi$
$787$		−	679.060i		−	0.862846i	0.902150	−	0.431423i	$-0.141988\pi$
$788$		0			0
$789$	507.101	+	837.541i	0.642713	+	1.06152i
$790$		0			0
$791$		−	226.053i		−	0.285781i
$792$		0			0
$793$		−	471.770i		−	0.594918i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−284.382			−0.356815			−0.178408	−	0.983957i	$-0.557095\pi$
$797$	−284.382			−0.356815			−0.178408	+	0.983957i	$0.557095\pi$
$798$		0			0
$799$	−503.422			−0.630066
$800$		0			0
$801$	−180.624	−	94.4811i	−0.225498	−	0.117954i
$802$		0			0
$803$	−273.545			−0.340653
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−146.112	+	88.4653i	−0.181055	+	0.109622i
$808$		0			0
$809$		−	33.2847i		−	0.0411431i	−0.999788	−	0.0205715i	$-0.993451\pi$
$809$		−	33.2847i		−	0.0411431i	0.999788	−	0.0205715i	$-0.00654859\pi$
$810$		0			0
$811$	−930.829			−1.14775			−0.573877	−	0.818941i	$-0.694561\pi$
$811$	−930.829			−1.14775			−0.573877	+	0.818941i	$0.694561\pi$
$812$		0			0
$813$	−172.242	−	284.480i	−0.211860	−	0.349914i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	740.241i		−	0.906048i
$818$		0			0
$819$	−41.1382	+	78.6460i	−0.0502298	+	0.0960268i
$820$		0			0
$821$			192.628i			0.234626i	0.993095	+	0.117313i	$0.0374281\pi$
$821$			192.628i			0.234626i	−0.993095	+	0.117313i	$0.962572\pi$
$822$		0			0
$823$			326.743i			0.397015i	0.980099	+	0.198507i	$0.0636094\pi$
$823$			326.743i			0.397015i	−0.980099	+	0.198507i	$0.936391\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	225.268			0.272392			0.136196	−	0.990682i	$-0.456512\pi$
$827$	225.268			0.272392			0.136196	+	0.990682i	$0.456512\pi$
$828$		0			0
$829$	875.339			1.05590			0.527949	−	0.849276i	$-0.322961\pi$
$829$	875.339			1.05590			0.527949	+	0.849276i	$0.322961\pi$
$830$		0			0
$831$	1264.77	−	765.770i	1.52198	−	0.921505i
$832$		0			0
$833$	−297.851			−0.357564
$834$		0			0
$835$		0			0
$836$		0			0
$837$	268.691	−	16.8112i	0.321017	−	0.0200851i
$838$		0			0
$839$		−	1614.70i		−	1.92456i	−0.272064	−	0.962279i	$-0.587706\pi$
$839$		−	1614.70i		−	1.92456i	0.272064	−	0.962279i	$-0.412294\pi$
$840$		0			0
$841$	−832.305			−0.989661
$842$		0			0
$843$	1100.17	−	666.115i	1.30507	−	0.790172i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	144.456i		−	0.170550i
$848$		0			0
$849$	743.887	−	450.396i	0.876191	−	0.530502i
$850$		0			0
$851$			1034.57i			1.21571i
$852$		0			0
$853$		−	1133.27i		−	1.32856i	−0.747482	−	0.664282i	$-0.768737\pi$
$853$		−	1133.27i		−	1.32856i	0.747482	−	0.664282i	$-0.231263\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	800.665			0.934264			0.467132	−	0.884187i	$-0.345287\pi$
$857$	800.665			0.934264			0.467132	+	0.884187i	$0.345287\pi$
$858$		0			0
$859$	1156.84			1.34672			0.673362	−	0.739313i	$-0.264850\pi$
$859$	1156.84			1.34672			0.673362	+	0.739313i	$0.264850\pi$
$860$		0			0
$861$	−142.252	−	234.948i	−0.165218	−	0.272878i
$862$		0			0
$863$	178.321			0.206629			0.103315	−	0.994649i	$-0.467055\pi$
$863$	178.321			0.206629			0.103315	+	0.994649i	$0.467055\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	387.161	+	639.445i	0.446552	+	0.737537i
$868$		0			0
$869$		−	291.955i		−	0.335967i
$870$		0			0
$871$	−396.121			−0.454789
$872$		0			0
$873$	279.462	+	146.181i	0.320116	+	0.167447i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	354.008i		−	0.403658i	−0.979421	−	0.201829i	$-0.935311\pi$
$877$		−	354.008i		−	0.403658i	0.979421	−	0.201829i	$-0.0646886\pi$
$878$		0			0
$879$	632.442	+	1044.56i	0.719502	+	1.18835i
$880$		0			0
$881$			828.055i			0.939903i	0.882692	+	0.469951i	$0.155729\pi$
$881$			828.055i			0.939903i	−0.882692	+	0.469951i	$0.844271\pi$
$882$		0			0
$883$			943.554i			1.06858i	0.845302	+	0.534289i	$0.179420\pi$
$883$			943.554i			1.06858i	−0.845302	+	0.534289i	$0.820580\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1503.35			1.69487			0.847437	−	0.530896i	$-0.178144\pi$
$887$	1503.35			1.69487			0.847437	+	0.530896i	$0.178144\pi$
$888$		0			0
$889$	−131.368			−0.147770
$890$		0			0
$891$	−243.906	+	169.352i	−0.273744	+	0.190069i
$892$		0			0
$893$	2452.19			2.74602
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−494.520	+	299.414i	−0.551305	+	0.333795i
$898$		0			0
$899$			407.873i			0.453696i
$900$		0			0
$901$	−245.908			−0.272928
$902$		0			0
$903$	50.2493	+	82.9931i	0.0556471	+	0.0919082i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	268.683i		−	0.296233i	−0.988970	−	0.148116i	$-0.952679\pi$
$907$		−	268.683i		−	0.296233i	0.988970	−	0.148116i	$-0.0473210\pi$
$908$		0			0
$909$	−81.9110	−	42.8461i	−0.0901111	−	0.0471354i
$910$		0			0
$911$		−	807.241i		−	0.886104i	−0.896496	−	0.443052i	$-0.853896\pi$
$911$		−	807.241i		−	0.886104i	0.896496	−	0.443052i	$-0.146104\pi$
$912$		0			0
$913$			0.185665i			0.000203357i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−119.442			−0.130253
$918$		0			0
$919$	−1359.86			−1.47972			−0.739860	−	0.672761i	$-0.765108\pi$
$919$	−1359.86			−1.47972			−0.739860	+	0.672761i	$0.765108\pi$
$920$		0			0
$921$	730.346	−	442.198i	0.792992	−	0.480128i
$922$		0			0
$923$	1003.83			1.08758
$924$		0			0
$925$		0			0
$926$		0			0
$927$	868.962	+	454.538i	0.937392	+	0.490332i
$928$		0			0
$929$			294.426i			0.316928i	0.987365	+	0.158464i	$0.0506542\pi$
$929$			294.426i			0.316928i	−0.987365	+	0.158464i	$0.949346\pi$
$930$		0			0
$931$	1450.85			1.55837
$932$		0			0
$933$	82.1227	−	49.7223i	0.0880200	−	0.0532929i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	1449.73i		−	1.54721i	−0.633671	−	0.773603i	$-0.718453\pi$
$937$		−	1449.73i		−	1.54721i	0.633671	−	0.773603i	$-0.281547\pi$
$938$		0			0
$939$	366.859	−	222.120i	0.390691	−	0.236549i
$940$		0			0
$941$			386.698i			0.410944i	0.978663	+	0.205472i	$0.0658729\pi$
$941$			386.698i			0.410944i	−0.978663	+	0.205472i	$0.934127\pi$
$942$		0			0
$943$		−	1788.95i		−	1.89708i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1187.76			−1.25423			−0.627115	−	0.778927i	$-0.715764\pi$
$947$	−1187.76			−1.25423			−0.627115	+	0.778927i	$0.715764\pi$
$948$		0			0
$949$	547.932			0.577378
$950$		0			0
$951$	385.420	+	636.570i	0.405279	+	0.669369i
$952$		0			0
$953$	−127.552			−0.133842			−0.0669211	−	0.997758i	$-0.521318\pi$
$953$	−127.552			−0.133842			−0.0669211	+	0.997758i	$0.521318\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−232.999	−	384.827i	−0.243468	−	0.402118i
$958$		0			0
$959$		−	324.518i		−	0.338392i
$960$		0			0
$961$	−861.580			−0.896545
$962$		0			0
$963$	472.426	−	903.159i	0.490577	−	0.937860i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	68.0305i		−	0.0703521i	−0.999381	−	0.0351760i	$-0.988801\pi$
$967$		−	68.0305i		−	0.0703521i	0.999381	−	0.0351760i	$-0.0111992\pi$
$968$		0			0
$969$	301.435	+	497.859i	0.311079	+	0.513786i
$970$		0			0
$971$		−	1507.80i		−	1.55283i	−0.630220	−	0.776417i	$-0.717035\pi$
$971$		−	1507.80i		−	1.55283i	0.630220	−	0.776417i	$-0.282965\pi$
$972$		0			0
$973$		−	289.034i		−	0.297054i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1349.37			−1.38113			−0.690566	−	0.723269i	$-0.742639\pi$
$977$	−1349.37			−1.38113			−0.690566	+	0.723269i	$0.742639\pi$
$978$		0			0
$979$	83.0286			0.0848096
$980$		0			0
$981$	−441.020	+	843.119i	−0.449561	+	0.859448i
$982$		0			0
$983$	−710.166			−0.722448			−0.361224	−	0.932479i	$-0.617641\pi$
$983$	−710.166			−0.722448			−0.361224	+	0.932479i	$0.617641\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−274.931	+	166.461i	−0.278552	+	0.168653i
$988$		0			0
$989$			631.928i			0.638957i
$990$		0			0
$991$	−1736.30			−1.75207			−0.876034	−	0.482249i	$-0.839820\pi$
$991$	−1736.30			−1.75207			−0.876034	+	0.482249i	$0.839820\pi$
$992$		0			0
$993$	−586.803	−	969.179i	−0.590940	−	0.976012i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	812.058i		−	0.814502i	−0.913316	−	0.407251i	$-0.866488\pi$
$997$		−	812.058i		−	0.814502i	0.913316	−	0.407251i	$-0.133512\pi$
$998$		0			0
$999$	−66.4683	−	1062.35i	−0.0665348	−	1.06341i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.c.l.449.3		12
3.2	odd	2	inner	1200.3.c.l.449.9		12
4.3	odd	2		600.3.c.c.449.10		12
5.2	odd	4		1200.3.l.w.401.2		6
5.3	odd	4		1200.3.l.v.401.5		6
5.4	even	2	inner	1200.3.c.l.449.10		12
12.11	even	2		600.3.c.c.449.4		12
15.2	even	4		1200.3.l.w.401.1		6
15.8	even	4		1200.3.l.v.401.6		6
15.14	odd	2	inner	1200.3.c.l.449.4		12
20.3	even	4		600.3.l.e.401.2	yes	6
20.7	even	4		600.3.l.d.401.5	✓	6
20.19	odd	2		600.3.c.c.449.3		12
60.23	odd	4		600.3.l.e.401.1	yes	6
60.47	odd	4		600.3.l.d.401.6	yes	6
60.59	even	2		600.3.c.c.449.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.3.c.c.449.3		12	20.19	odd	2
600.3.c.c.449.4		12	12.11	even	2
600.3.c.c.449.9		12	60.59	even	2
600.3.c.c.449.10		12	4.3	odd	2
600.3.l.d.401.5	✓	6	20.7	even	4
600.3.l.d.401.6	yes	6	60.47	odd	4
600.3.l.e.401.1	yes	6	60.23	odd	4
600.3.l.e.401.2	yes	6	20.3	even	4
1200.3.c.l.449.3		12	1.1	even	1	trivial
1200.3.c.l.449.4		12	15.14	odd	2	inner
1200.3.c.l.449.9		12	3.2	odd	2	inner
1200.3.c.l.449.10		12	5.4	even	2	inner
1200.3.l.v.401.5		6	5.3	odd	4
1200.3.l.v.401.6		6	15.8	even	4
1200.3.l.w.401.1		6	15.2	even	4
1200.3.l.w.401.2		6	5.2	odd	4

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.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.55378 - 2.56627i) q^{3} -1.34301i q^{7} +(-4.17150 + 7.97487i) q^{9} +O(q^{10})\)	\(q+(-1.55378 - 2.56627i) q^{3} -1.34301i q^{7} +(-4.17150 + 7.97487i) q^{9} -3.66586i q^{11} +7.34301i q^{13} -6.31089 q^{17} +30.7406 q^{19} +(-3.44653 + 2.08675i) q^{21} -26.2426 q^{23} +(26.9473 - 1.68602i) q^{27} +40.9060i q^{29} +9.97097 q^{31} +(-9.40758 + 5.69595i) q^{33} -39.4232i q^{37} +(18.8442 - 11.4095i) q^{39} +68.1695i q^{41} -24.0802i q^{43} +79.7705 q^{47} +47.1963 q^{49} +(9.80576 + 16.1955i) q^{51} +38.9657 q^{53} +(-47.7643 - 78.8888i) q^{57} +39.9648i q^{59} -64.2475 q^{61} +(10.7103 + 5.60237i) q^{63} +53.9454i q^{67} +(40.7754 + 67.3457i) q^{69} -136.706i q^{71} -74.6196i q^{73} -4.92328 q^{77} +79.6417 q^{79} +(-46.1971 - 66.5344i) q^{81} -0.0506471 q^{83} +(104.976 - 63.5592i) q^{87} +22.6492i q^{89} +9.86173 q^{91} +(-15.4927 - 25.5882i) q^{93} -35.0428i q^{97} +(29.2347 + 15.2921i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 32 q^{9}+O(q^{10}) \) 12 * q - 32 * q^9	\( 12 q - 32 q^{9} + 100 q^{19} - 36 q^{21} + 228 q^{31} - 12 q^{39} - 152 q^{49} + 12 q^{51} + 124 q^{61} - 312 q^{69} - 152 q^{79} - 448 q^{81} + 620 q^{91} - 500 q^{99}+O(q^{100}) \) 12 * q - 32 * q^9 + 100 * q^19 - 36 * q^21 + 228 * q^31 - 12 * q^39 - 152 * q^49 + 12 * q^51 + 124 * q^61 - 312 * q^69 - 152 * q^79 - 448 * q^81 + 620 * q^91 - 500 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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