LMFDB - Embedded newform 1200.3.c.l.449.9

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			449.9
Root			$-0.139571 - 3.56627i$ of defining polynomial
Character	$\chi$	$=$	1200.449
Dual form			1200.3.c.l.449.10

$q$-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	$0.0532885\pi$
$11$			3.66586i			0.333260i	−0.986020	+	0.166630i	$0.946712\pi$
$12$		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	$-0.440572\pi$
$17$	6.31089			0.371229			0.185614	+	0.982623i	$0.440572\pi$
$18$		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	$-0.306753\pi$
$23$	26.2426			1.14098			0.570492	+	0.821303i	$0.306753\pi$
$24$		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	$-0.750823\pi$
$29$		−	40.9060i		−	1.41055i	0.708932	−	0.705277i	$-0.249177\pi$
$30$		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	$-0.687577\pi$
$41$		−	68.1695i		−	1.66267i	0.555771	−	0.831335i	$-0.312423\pi$
$42$		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	$-0.822567\pi$
$47$	−79.7705			−1.69724			−0.848622	+	0.529000i	$0.822567\pi$
$48$		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	$-0.619821\pi$
$53$	−38.9657			−0.735202			−0.367601	+	0.929984i	$0.619821\pi$
$54$		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	$-0.890018\pi$
$59$		−	39.9648i		−	0.677369i	0.940900	−	0.338684i	$-0.109982\pi$
$60$		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	$0.412807\pi$
$71$			136.706i			1.92543i	−0.270511	+	0.962717i	$0.587193\pi$
$72$		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	$-0.499903\pi$
$83$	0.0506471			0.000610206			0.000305103		1.00000i	$0.499903\pi$
$84$		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	$-0.959387\pi$
$89$		−	22.6492i		−	0.254485i	0.991872	−	0.127243i	$-0.0406126\pi$
$90$		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	$-0.983808\pi$
$101$		−	10.2711i		−	0.101694i	0.998706	−	0.0508472i	$-0.0161921\pi$
$102$		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	$-0.322489\pi$
$107$	113.251			1.05842			0.529209	+	0.848492i	$0.322489\pi$
$108$		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	$-0.767441\pi$
$113$	−168.318			−1.48954			−0.744770	+	0.667321i	$0.767441\pi$
$114$		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	$0.110241\pi$
$131$			88.9361i			0.678902i	−0.940624	+	0.339451i	$0.889759\pi$
$132$		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	$-0.843722\pi$
$137$	−241.635			−1.76376			−0.881880	+	0.471474i	$0.843722\pi$
$138$		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	$0.113862\pi$
$149$			104.338i			0.700257i	−0.936702	+	0.350128i	$0.886138\pi$
$150$		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	$-0.386931\pi$
$167$	116.164			0.695590			0.347795	+	0.937571i	$0.386931\pi$
$168$		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	$-0.571194\pi$
$173$	−76.7440			−0.443607			−0.221804	+	0.975091i	$0.571194\pi$
$174$		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	$-0.711412\pi$
$179$		−	281.899i		−	1.57486i	0.616406	−	0.787428i	$-0.288588\pi$
$180$		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	$0.0520460\pi$
$191$			62.1819i			0.325560i	−0.986662	+	0.162780i	$0.947954\pi$
$192$		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	$-0.651621\pi$
$197$	−180.658			−0.917044			−0.458522	+	0.888683i	$0.651621\pi$
$198$		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	$-0.236476\pi$
$227$	334.372			1.47301			0.736503	+	0.676434i	$0.236476\pi$
$228$		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	$-0.421967\pi$
$233$	113.099			0.485402			0.242701	+	0.970101i	$0.421967\pi$
$234$		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	$-0.911814\pi$
$239$		−	130.740i		−	0.547029i	0.961868	−	0.273514i	$-0.0881862\pi$
$240$		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	$-0.913033\pi$
$251$		−	135.453i		−	0.539655i	0.962909	−	0.269828i	$-0.0869667\pi$
$252$		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	$-0.565131\pi$
$257$	−104.440			−0.406379			−0.203190	+	0.979139i	$0.565131\pi$
$258$		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	$-0.286944\pi$
$263$	326.365			1.24093			0.620466	+	0.784234i	$0.286944\pi$
$264$		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	$0.0337492\pi$
$269$			56.9353i			0.211656i	−0.994384	+	0.105828i	$0.966251\pi$
$270$		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	$-0.723814\pi$
$281$		−	428.705i		−	1.52564i	0.646611	−	0.762820i	$-0.276186\pi$
$282$		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	$-0.255583\pi$
$293$	407.033			1.38919			0.694596	+	0.719400i	$0.255583\pi$
$294$		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	$-0.983616\pi$
$311$		−	32.0008i		−	0.102896i	0.998676	−	0.0514482i	$-0.0163837\pi$
$312$		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	$-0.372043\pi$
$317$	248.052			0.782500			0.391250	+	0.920284i	$0.372043\pi$
$318$		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	$-0.709756\pi$
$347$	−424.938			−1.22460			−0.612302	+	0.790624i	$0.709756\pi$
$348$		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	$-0.666548\pi$
$353$	−352.772			−0.999353			−0.499677	+	0.866212i	$0.666548\pi$
$354$		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	$0.199775\pi$
$359$			421.619i			1.17443i	−0.809432	+	0.587214i	$0.800225\pi$
$360$		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	$-0.557275\pi$
$383$	−137.087			−0.357929			−0.178965	+	0.983856i	$0.557275\pi$
$384$		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	$-0.659124\pi$
$389$		−	682.795i		−	1.75526i	0.479341	−	0.877629i	$-0.340876\pi$
$390$		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	$-0.954692\pi$
$401$		−	113.772i		−	0.283720i	0.989887	−	0.141860i	$-0.0453083\pi$
$402$		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	$0.225372\pi$
$419$			544.982i			1.30067i	−0.759646	+	0.650337i	$0.774628\pi$
$420$		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	$0.0151593\pi$
$431$			41.0366i			0.0952124i	−0.998866	+	0.0476062i	$0.984841\pi$
$432$		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	$-0.448597\pi$
$443$	142.457			0.321573			0.160786	+	0.986989i	$0.448597\pi$
$444$		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	$0.239526\pi$
$449$			613.747i			1.36692i	−0.729988	+	0.683460i	$0.760474\pi$
$450$		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	$0.417957\pi$
$461$			891.544i			1.93393i	−0.254900	+	0.966967i	$0.582043\pi$
$462$		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	$-0.699011\pi$
$467$	−546.640			−1.17054			−0.585268	+	0.810840i	$0.699011\pi$
$468$		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	$0.0762550\pi$
$479$			227.312i			0.474555i	−0.971442	+	0.237277i	$0.923745\pi$
$480$		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	$0.122295\pi$
$491$			368.073i			0.749639i	−0.927098	+	0.374819i	$0.877705\pi$
$492$		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	$-0.704071\pi$
$503$	−601.673			−1.19617			−0.598085	+	0.801433i	$0.704071\pi$
$504$		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	$0.153171\pi$
$509$			471.175i			0.925687i	−0.886440	+	0.462844i	$0.846829\pi$
$510$		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	$-0.899347\pi$
$521$		−	324.027i		−	0.621932i	0.950421	−	0.310966i	$-0.100653\pi$
$522$		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	$-0.381918\pi$
$557$	403.842			0.725031			0.362516	+	0.931978i	$0.381918\pi$
$558$		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	$-0.591639\pi$
$563$	−319.709			−0.567867			−0.283933	+	0.958844i	$0.591639\pi$
$564$		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	$-0.787127\pi$
$569$		−	705.574i		−	1.24003i	0.784592	−	0.620013i	$-0.212873\pi$
$570$		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	$-0.548516\pi$
$587$	−178.246			−0.303656			−0.151828	+	0.988407i	$0.548516\pi$
$588$		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	$-0.532858\pi$
$593$	−122.209			−0.206086			−0.103043	+	0.994677i	$0.532858\pi$
$594$		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	$-0.739357\pi$
$599$		−	874.960i		−	1.46070i	0.683072	−	0.730351i	$-0.260643\pi$
$600$		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	$-0.647231\pi$
$617$	−550.637			−0.892442			−0.446221	+	0.894923i	$0.647231\pi$
$618$		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	$0.288105\pi$
$641$			1008.28i			1.57298i	−0.617601	+	0.786491i	$0.711895\pi$
$642$		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	$-0.397299\pi$
$647$	410.296			0.634152			0.317076	+	0.948400i	$0.397299\pi$
$648$		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	$-0.402680\pi$
$653$	393.103			0.601996			0.300998	+	0.953625i	$0.402680\pi$
$654$		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	$0.233584\pi$
$659$			882.684i			1.33943i	−0.742618	+	0.669715i	$0.766416\pi$
$660$		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	$-0.822204\pi$
$677$	−1148.22			−1.69604			−0.848019	+	0.529967i	$0.822204\pi$
$678$		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	$-0.548085\pi$
$683$	−205.570			−0.300981			−0.150490	+	0.988611i	$0.548085\pi$
$684$		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	$0.177997\pi$
$701$			743.766i			1.06101i	−0.847683	+	0.530504i	$0.822003\pi$
$702$		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	$-0.726804\pi$
$719$		−	1088.15i		−	1.51343i	0.653747	−	0.756713i	$-0.273196\pi$
$720$		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	$-0.549656\pi$
$743$	−230.874			−0.310732			−0.155366	+	0.987857i	$0.549656\pi$
$744$		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	$0.0931466\pi$
$761$			439.052i			0.576940i	−0.957489	+	0.288470i	$0.906853\pi$
$762$		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	$-0.288161\pi$
$773$	954.595			1.23492			0.617461	+	0.786601i	$0.288161\pi$
$774$		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	$-0.442905\pi$
$797$	284.382			0.356815			0.178408	+	0.983957i	$0.442905\pi$
$798$		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	$0.00654859\pi$
$809$			33.2847i			0.0411431i	−0.999788	+	0.0205715i	$0.993451\pi$
$810$		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	$-0.962572\pi$
$821$		−	192.628i		−	0.234626i	0.993095	−	0.117313i	$-0.0374281\pi$
$822$		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	$-0.543488\pi$
$827$	−225.268			−0.272392			−0.136196	+	0.990682i	$0.543488\pi$
$828$		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	$0.412294\pi$
$839$			1614.70i			1.92456i	−0.272064	+	0.962279i	$0.587706\pi$
$840$		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	$-0.654713\pi$
$857$	−800.665			−0.934264			−0.467132	+	0.884187i	$0.654713\pi$
$858$		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	$-0.532945\pi$
$863$	−178.321			−0.206629			−0.103315	+	0.994649i	$0.532945\pi$
$864$		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	$-0.844271\pi$
$881$		−	828.055i		−	0.939903i	0.882692	−	0.469951i	$-0.155729\pi$
$882$		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	$-0.821856\pi$
$887$	−1503.35			−1.69487			−0.847437	+	0.530896i	$0.821856\pi$
$888$		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	$0.146104\pi$
$911$			807.241i			0.886104i	−0.896496	+	0.443052i	$0.853896\pi$
$912$		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	$-0.949346\pi$
$929$		−	294.426i		−	0.316928i	0.987365	−	0.158464i	$-0.0506542\pi$
$930$		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	$-0.934127\pi$
$941$		−	386.698i		−	0.410944i	0.978663	−	0.205472i	$-0.0658729\pi$
$942$		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	$-0.284236\pi$
$947$	1187.76			1.25423			0.627115	+	0.778927i	$0.284236\pi$
$948$		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	$-0.478682\pi$
$953$	127.552			0.133842			0.0669211	+	0.997758i	$0.478682\pi$
$954$		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	$0.282965\pi$
$971$			1507.80i			1.55283i	−0.630220	+	0.776417i	$0.717035\pi$
$972$		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	$-0.257361\pi$
$977$	1349.37			1.38113			0.690566	+	0.723269i	$0.257361\pi$
$978$		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	$-0.382359\pi$
$983$	710.166			0.722448			0.361224	+	0.932479i	$0.382359\pi$
$984$		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.449.9		12
3.2	odd	2	inner	1200.3.c.l.449.3		12
4.3	odd	2		600.3.c.c.449.4		12
5.2	odd	4		1200.3.l.w.401.1		6
5.3	odd	4		1200.3.l.v.401.6		6
5.4	even	2	inner	1200.3.c.l.449.4		12
12.11	even	2		600.3.c.c.449.10		12
15.2	even	4		1200.3.l.w.401.2		6
15.8	even	4		1200.3.l.v.401.5		6
15.14	odd	2	inner	1200.3.c.l.449.10		12
20.3	even	4		600.3.l.e.401.1	yes	6
20.7	even	4		600.3.l.d.401.6	yes	6
20.19	odd	2		600.3.c.c.449.9		12
60.23	odd	4		600.3.l.e.401.2	yes	6
60.47	odd	4		600.3.l.d.401.5	✓	6
60.59	even	2		600.3.c.c.449.3		12

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

Related objects

Downloads

Learn more