LMFDB - Embedded newform 1200.3.l.r.401.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,3,Mod(401,1200)]

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, 0]))

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

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

chi := DirichletCharacter("1200.401");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1200.l (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:	$2$
Coefficient field:	$\Q(\sqrt{-5}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 5 $ x^2 + 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			401.2
Root			$2.23607i$ of defining polynomial
Character	$\chi$	$=$	1200.401
Dual form			1200.3.l.r.401.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.00000 + 2.23607i) q^{3} +2.00000 q^{7} +(-1.00000 + 8.94427i) q^{9} +O(q^{10})$	$q+(2.00000 + 2.23607i) q^{3} +2.00000 q^{7} +(-1.00000 + 8.94427i) q^{9} +13.4164i q^{11} -8.00000 q^{13} +13.4164i q^{17} +34.0000 q^{19} +(4.00000 + 4.47214i) q^{21} -40.2492i q^{23} +(-22.0000 + 15.6525i) q^{27} +40.2492i q^{29} -14.0000 q^{31} +(-30.0000 + 26.8328i) q^{33} -56.0000 q^{37} +(-16.0000 - 17.8885i) q^{39} +26.8328i q^{41} +8.00000 q^{43} +40.2492i q^{47} -45.0000 q^{49} +(-30.0000 + 26.8328i) q^{51} -40.2492i q^{53} +(68.0000 + 76.0263i) q^{57} +13.4164i q^{59} -46.0000 q^{61} +(-2.00000 + 17.8885i) q^{63} +32.0000 q^{67} +(90.0000 - 80.4984i) q^{69} +53.6656i q^{71} +106.000 q^{73} +26.8328i q^{77} +22.0000 q^{79} +(-79.0000 - 17.8885i) q^{81} +120.748i q^{83} +(-90.0000 + 80.4984i) q^{87} -107.331i q^{89} -16.0000 q^{91} +(-28.0000 - 31.3050i) q^{93} -122.000 q^{97} +(-120.000 - 13.4164i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} + 4 q^{7} - 2 q^{9}+O(q^{10}) $ 2 * q + 4 * q^3 + 4 * q^7 - 2 * q^9	$ 2 q + 4 q^{3} + 4 q^{7} - 2 q^{9} - 16 q^{13} + 68 q^{19} + 8 q^{21} - 44 q^{27} - 28 q^{31} - 60 q^{33} - 112 q^{37} - 32 q^{39} + 16 q^{43} - 90 q^{49} - 60 q^{51} + 136 q^{57} - 92 q^{61} - 4 q^{63} + 64 q^{67} + 180 q^{69} + 212 q^{73} + 44 q^{79} - 158 q^{81} - 180 q^{87} - 32 q^{91} - 56 q^{93} - 244 q^{97} - 240 q^{99}+O(q^{100}) $ 2 * q + 4 * q^3 + 4 * q^7 - 2 * q^9 - 16 * q^13 + 68 * q^19 + 8 * q^21 - 44 * q^27 - 28 * q^31 - 60 * q^33 - 112 * q^37 - 32 * q^39 + 16 * q^43 - 90 * q^49 - 60 * q^51 + 136 * q^57 - 92 * q^61 - 4 * q^63 + 64 * q^67 + 180 * q^69 + 212 * q^73 + 44 * q^79 - 158 * q^81 - 180 * q^87 - 32 * q^91 - 56 * q^93 - 244 * q^97 - 240 * 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
$3$	2.00000	+	2.23607i	0.666667	+	0.745356i
$3$	2.00000	+	2.23607i	0.666667	+	0.745356i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	2.00000			0.285714			0.142857	−	0.989743i	$-0.454371\pi$
$7$	2.00000			0.285714			0.142857	+	0.989743i	$0.454371\pi$
$8$		0			0
$9$	−1.00000	+	8.94427i	−0.111111	+	0.993808i
$10$		0			0
$11$			13.4164i			1.21967i	0.792527	+	0.609837i	$0.208765\pi$
$11$			13.4164i			1.21967i	−0.792527	+	0.609837i	$0.791235\pi$
$12$		0			0
$13$	−8.00000			−0.615385			−0.307692	−	0.951486i	$-0.599557\pi$
$13$	−8.00000			−0.615385			−0.307692	+	0.951486i	$0.599557\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			13.4164i			0.789200i	0.918853	+	0.394600i	$0.129117\pi$
$17$			13.4164i			0.789200i	−0.918853	+	0.394600i	$0.870883\pi$
$18$		0			0
$19$	34.0000			1.78947			0.894737	−	0.446594i	$-0.147363\pi$
$19$	34.0000			1.78947			0.894737	+	0.446594i	$0.147363\pi$
$20$		0			0
$21$	4.00000	+	4.47214i	0.190476	+	0.212959i
$22$		0			0
$23$		−	40.2492i		−	1.74997i	−0.484153	−	0.874983i	$-0.660872\pi$
$23$		−	40.2492i		−	1.74997i	0.484153	−	0.874983i	$-0.339128\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−22.0000	+	15.6525i	−0.814815	+	0.579721i
$28$		0			0
$29$			40.2492i			1.38790i	0.720021	+	0.693952i	$0.244132\pi$
$29$			40.2492i			1.38790i	−0.720021	+	0.693952i	$0.755868\pi$
$30$		0			0
$31$	−14.0000			−0.451613			−0.225806	−	0.974172i	$-0.572502\pi$
$31$	−14.0000			−0.451613			−0.225806	+	0.974172i	$0.572502\pi$
$32$		0			0
$33$	−30.0000	+	26.8328i	−0.909091	+	0.813116i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−56.0000			−1.51351			−0.756757	−	0.653697i	$-0.773217\pi$
$37$	−56.0000			−1.51351			−0.756757	+	0.653697i	$0.773217\pi$
$38$		0			0
$39$	−16.0000	−	17.8885i	−0.410256	−	0.458681i
$40$		0			0
$41$			26.8328i			0.654459i	0.944945	+	0.327229i	$0.106115\pi$
$41$			26.8328i			0.654459i	−0.944945	+	0.327229i	$0.893885\pi$
$42$		0			0
$43$	8.00000			0.186047			0.0930233	−	0.995664i	$-0.470347\pi$
$43$	8.00000			0.186047			0.0930233	+	0.995664i	$0.470347\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			40.2492i			0.856366i	0.903692	+	0.428183i	$0.140846\pi$
$47$			40.2492i			0.856366i	−0.903692	+	0.428183i	$0.859154\pi$
$48$		0			0
$49$	−45.0000			−0.918367
$50$		0			0
$51$	−30.0000	+	26.8328i	−0.588235	+	0.526134i
$52$		0			0
$53$		−	40.2492i		−	0.759419i	−0.925106	−	0.379710i	$-0.876024\pi$
$53$		−	40.2492i		−	0.759419i	0.925106	−	0.379710i	$-0.123976\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	68.0000	+	76.0263i	1.19298	+	1.33379i
$58$		0			0
$59$			13.4164i			0.227397i	0.993515	+	0.113698i	$0.0362697\pi$
$59$			13.4164i			0.227397i	−0.993515	+	0.113698i	$0.963730\pi$
$60$		0			0
$61$	−46.0000			−0.754098			−0.377049	−	0.926193i	$-0.623061\pi$
$61$	−46.0000			−0.754098			−0.377049	+	0.926193i	$0.623061\pi$
$62$		0			0
$63$	−2.00000	+	17.8885i	−0.0317460	+	0.283945i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	32.0000			0.477612			0.238806	−	0.971067i	$-0.423244\pi$
$67$	32.0000			0.477612			0.238806	+	0.971067i	$0.423244\pi$
$68$		0			0
$69$	90.0000	−	80.4984i	1.30435	−	1.16664i
$70$		0			0
$71$			53.6656i			0.755854i	0.925835	+	0.377927i	$0.123363\pi$
$71$			53.6656i			0.755854i	−0.925835	+	0.377927i	$0.876637\pi$
$72$		0			0
$73$	106.000			1.45205			0.726027	−	0.687666i	$-0.241365\pi$
$73$	106.000			1.45205			0.726027	+	0.687666i	$0.241365\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			26.8328i			0.348478i
$78$		0			0
$79$	22.0000			0.278481			0.139241	−	0.990259i	$-0.455534\pi$
$79$	22.0000			0.278481			0.139241	+	0.990259i	$0.455534\pi$
$80$		0			0
$81$	−79.0000	−	17.8885i	−0.975309	−	0.220846i
$82$		0			0
$83$			120.748i			1.45479i	0.686218	+	0.727396i	$0.259269\pi$
$83$			120.748i			1.45479i	−0.686218	+	0.727396i	$0.740731\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−90.0000	+	80.4984i	−1.03448	+	0.925270i
$88$		0			0
$89$		−	107.331i		−	1.20597i	−0.797753	−	0.602985i	$-0.793978\pi$
$89$		−	107.331i		−	1.20597i	0.797753	−	0.602985i	$-0.206022\pi$
$90$		0			0
$91$	−16.0000			−0.175824
$92$		0			0
$93$	−28.0000	−	31.3050i	−0.301075	−	0.336612i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−122.000			−1.25773			−0.628866	−	0.777514i	$-0.716481\pi$
$97$	−122.000			−1.25773			−0.628866	+	0.777514i	$0.716481\pi$
$98$		0			0
$99$	−120.000	−	13.4164i	−1.21212	−	0.135519i
$100$		0			0
$101$			174.413i			1.72686i	0.504465	+	0.863432i	$0.331690\pi$
$101$			174.413i			1.72686i	−0.504465	+	0.863432i	$0.668310\pi$
$102$		0			0
$103$	−46.0000			−0.446602			−0.223301	−	0.974750i	$-0.571683\pi$
$103$	−46.0000			−0.446602			−0.223301	+	0.974750i	$0.571683\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	13.4164i		−	0.125387i	−0.998033	−	0.0626935i	$-0.980031\pi$
$107$		−	13.4164i		−	0.125387i	0.998033	−	0.0626935i	$-0.0199691\pi$
$108$		0			0
$109$	86.0000			0.788991			0.394495	−	0.918898i	$-0.370919\pi$
$109$	86.0000			0.788991			0.394495	+	0.918898i	$0.370919\pi$
$110$		0			0
$111$	−112.000	−	125.220i	−1.00901	−	1.12811i
$112$		0			0
$113$			93.9149i			0.831105i	0.909569	+	0.415552i	$0.136412\pi$
$113$			93.9149i			0.831105i	−0.909569	+	0.415552i	$0.863588\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	8.00000	−	71.5542i	0.0683761	−	0.611574i
$118$		0			0
$119$			26.8328i			0.225486i
$120$		0			0
$121$	−59.0000			−0.487603
$122$		0			0
$123$	−60.0000	+	53.6656i	−0.487805	+	0.436306i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−34.0000			−0.267717			−0.133858	−	0.991000i	$-0.542737\pi$
$127$	−34.0000			−0.267717			−0.133858	+	0.991000i	$0.542737\pi$
$128$		0			0
$129$	16.0000	+	17.8885i	0.124031	+	0.138671i
$130$		0			0
$131$			147.580i			1.12657i	0.826263	+	0.563284i	$0.190462\pi$
$131$			147.580i			1.12657i	−0.826263	+	0.563284i	$0.809538\pi$
$132$		0			0
$133$	68.0000			0.511278
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	40.2492i		−	0.293790i	−0.989152	−	0.146895i	$-0.953072\pi$
$137$		−	40.2492i		−	0.293790i	0.989152	−	0.146895i	$-0.0469279\pi$
$138$		0			0
$139$	82.0000			0.589928			0.294964	−	0.955508i	$-0.404692\pi$
$139$	82.0000			0.589928			0.294964	+	0.955508i	$0.404692\pi$
$140$		0			0
$141$	−90.0000	+	80.4984i	−0.638298	+	0.570911i
$142$		0			0
$143$		−	107.331i		−	0.750568i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−90.0000	−	100.623i	−0.612245	−	0.684511i
$148$		0			0
$149$		−	147.580i		−	0.990473i	−0.868758	−	0.495237i	$-0.835081\pi$
$149$		−	147.580i		−	0.990473i	0.868758	−	0.495237i	$-0.164919\pi$
$150$		0			0
$151$	46.0000			0.304636			0.152318	−	0.988332i	$-0.451326\pi$
$151$	46.0000			0.304636			0.152318	+	0.988332i	$0.451326\pi$
$152$		0			0
$153$	−120.000	−	13.4164i	−0.784314	−	0.0876889i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−92.0000			−0.585987			−0.292994	−	0.956114i	$-0.594651\pi$
$157$	−92.0000			−0.585987			−0.292994	+	0.956114i	$0.594651\pi$
$158$		0			0
$159$	90.0000	−	80.4984i	0.566038	−	0.506280i
$160$		0			0
$161$		−	80.4984i		−	0.499990i
$162$		0			0
$163$	68.0000			0.417178			0.208589	−	0.978003i	$-0.433113\pi$
$163$	68.0000			0.417178			0.208589	+	0.978003i	$0.433113\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	67.0820i		−	0.401689i	−0.979623	−	0.200844i	$-0.935631\pi$
$167$		−	67.0820i		−	0.401689i	0.979623	−	0.200844i	$-0.0643686\pi$
$168$		0			0
$169$	−105.000			−0.621302
$170$		0			0
$171$	−34.0000	+	304.105i	−0.198830	+	1.77839i
$172$		0			0
$173$		−	120.748i		−	0.697963i	−0.937130	−	0.348982i	$-0.886528\pi$
$173$		−	120.748i		−	0.697963i	0.937130	−	0.348982i	$-0.113472\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−30.0000	+	26.8328i	−0.169492	+	0.151598i
$178$		0			0
$179$		−	281.745i		−	1.57399i	−0.616958	−	0.786996i	$-0.711635\pi$
$179$		−	281.745i		−	1.57399i	0.616958	−	0.786996i	$-0.288365\pi$
$180$		0			0
$181$	194.000			1.07182			0.535912	−	0.844274i	$-0.319968\pi$
$181$	194.000			1.07182			0.535912	+	0.844274i	$0.319968\pi$
$182$		0			0
$183$	−92.0000	−	102.859i	−0.502732	−	0.562072i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−180.000			−0.962567
$188$		0			0
$189$	−44.0000	+	31.3050i	−0.232804	+	0.165635i
$190$		0			0
$191$		−	80.4984i		−	0.421458i	−0.977545	−	0.210729i	$-0.932416\pi$
$191$		−	80.4984i		−	0.421458i	0.977545	−	0.210729i	$-0.0675837\pi$
$192$		0			0
$193$	−218.000			−1.12953			−0.564767	−	0.825251i	$-0.691034\pi$
$193$	−218.000			−1.12953			−0.564767	+	0.825251i	$0.691034\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			93.9149i			0.476725i	0.971176	+	0.238363i	$0.0766107\pi$
$197$			93.9149i			0.476725i	−0.971176	+	0.238363i	$0.923389\pi$
$198$		0			0
$199$	34.0000			0.170854			0.0854271	−	0.996344i	$-0.472775\pi$
$199$	34.0000			0.170854			0.0854271	+	0.996344i	$0.472775\pi$
$200$		0			0
$201$	64.0000	+	71.5542i	0.318408	+	0.355991i
$202$		0			0
$203$			80.4984i			0.396544i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	360.000	+	40.2492i	1.73913	+	0.194441i
$208$		0			0
$209$			456.158i			2.18257i
$210$		0			0
$211$	46.0000			0.218009			0.109005	−	0.994041i	$-0.465234\pi$
$211$	46.0000			0.218009			0.109005	+	0.994041i	$0.465234\pi$
$212$		0			0
$213$	−120.000	+	107.331i	−0.563380	+	0.503903i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−28.0000			−0.129032
$218$		0			0
$219$	212.000	+	237.023i	0.968037	+	1.08230i
$220$		0			0
$221$		−	107.331i		−	0.485662i
$222$		0			0
$223$	398.000			1.78475			0.892377	−	0.451291i	$-0.149036\pi$
$223$	398.000			1.78475			0.892377	+	0.451291i	$0.149036\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			335.410i			1.47758i	0.673937	+	0.738789i	$0.264602\pi$
$227$			335.410i			1.47758i	−0.673937	+	0.738789i	$0.735398\pi$
$228$		0			0
$229$	86.0000			0.375546			0.187773	−	0.982212i	$-0.439873\pi$
$229$	86.0000			0.375546			0.187773	+	0.982212i	$0.439873\pi$
$230$		0			0
$231$	−60.0000	+	53.6656i	−0.259740	+	0.232319i
$232$		0			0
$233$			308.577i			1.32437i	0.749342	+	0.662183i	$0.230370\pi$
$233$			308.577i			1.32437i	−0.749342	+	0.662183i	$0.769630\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	44.0000	+	49.1935i	0.185654	+	0.207567i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	134.000			0.556017			0.278008	−	0.960579i	$-0.410326\pi$
$241$	134.000			0.556017			0.278008	+	0.960579i	$0.410326\pi$
$242$		0			0
$243$	−118.000	−	212.426i	−0.485597	−	0.874183i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−272.000			−1.10121
$248$		0			0
$249$	−270.000	+	241.495i	−1.08434	+	0.969861i
$250$		0			0
$251$		−	308.577i		−	1.22939i	−0.788764	−	0.614696i	$-0.789279\pi$
$251$		−	308.577i		−	1.22939i	0.788764	−	0.614696i	$-0.210721\pi$
$252$		0			0
$253$	540.000			2.13439
$254$		0			0
$255$		0			0
$256$		0			0
$257$			174.413i			0.678651i	0.940669	+	0.339325i	$0.110199\pi$
$257$			174.413i			0.678651i	−0.940669	+	0.339325i	$0.889801\pi$
$258$		0			0
$259$	−112.000			−0.432432
$260$		0			0
$261$	−360.000	−	40.2492i	−1.37931	−	0.154212i
$262$		0			0
$263$		−	254.912i		−	0.969246i	−0.874723	−	0.484623i	$-0.838957\pi$
$263$		−	254.912i		−	0.969246i	0.874723	−	0.484623i	$-0.161043\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	240.000	−	214.663i	0.898876	−	0.803979i
$268$		0			0
$269$		−	335.410i		−	1.24688i	−0.781872	−	0.623439i	$-0.785735\pi$
$269$		−	335.410i		−	1.24688i	0.781872	−	0.623439i	$-0.214265\pi$
$270$		0			0
$271$	−2.00000			−0.00738007			−0.00369004	−	0.999993i	$-0.501175\pi$
$271$	−2.00000			−0.00738007			−0.00369004	+	0.999993i	$0.501175\pi$
$272$		0			0
$273$	−32.0000	−	35.7771i	−0.117216	−	0.131052i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	448.000			1.61733			0.808664	−	0.588270i	$-0.200191\pi$
$277$	448.000			1.61733			0.808664	+	0.588270i	$0.200191\pi$
$278$		0			0
$279$	14.0000	−	125.220i	0.0501792	−	0.448817i
$280$		0			0
$281$		−	187.830i		−	0.668433i	−0.942496	−	0.334217i	$-0.891528\pi$
$281$		−	187.830i		−	0.668433i	0.942496	−	0.334217i	$-0.108472\pi$
$282$		0			0
$283$	248.000			0.876325			0.438163	−	0.898896i	$-0.355629\pi$
$283$	248.000			0.876325			0.438163	+	0.898896i	$0.355629\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			53.6656i			0.186988i
$288$		0			0
$289$	109.000			0.377163
$290$		0			0
$291$	−244.000	−	272.800i	−0.838488	−	0.937458i
$292$		0			0
$293$		−	147.580i		−	0.503688i	−0.967768	−	0.251844i	$-0.918963\pi$
$293$		−	147.580i		−	0.503688i	0.967768	−	0.251844i	$-0.0810369\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−210.000	−	295.161i	−0.707071	−	0.993808i
$298$		0			0
$299$			321.994i			1.07690i
$300$		0			0
$301$	16.0000			0.0531561
$302$		0			0
$303$	−390.000	+	348.827i	−1.28713	+	1.15124i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	272.000			0.885993			0.442997	−	0.896523i	$-0.353915\pi$
$307$	272.000			0.885993			0.442997	+	0.896523i	$0.353915\pi$
$308$		0			0
$309$	−92.0000	−	102.859i	−0.297735	−	0.332877i
$310$		0			0
$311$			53.6656i			0.172558i	0.996271	+	0.0862792i	$0.0274977\pi$
$311$			53.6656i			0.172558i	−0.996271	+	0.0862792i	$0.972502\pi$
$312$		0			0
$313$	−278.000			−0.888179			−0.444089	−	0.895982i	$-0.646473\pi$
$313$	−278.000			−0.888179			−0.444089	+	0.895982i	$0.646473\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			120.748i			0.380907i	0.981696	+	0.190454i	$0.0609959\pi$
$317$			120.748i			0.380907i	−0.981696	+	0.190454i	$0.939004\pi$
$318$		0			0
$319$	−540.000			−1.69279
$320$		0			0
$321$	30.0000	−	26.8328i	0.0934579	−	0.0835913i
$322$		0			0
$323$			456.158i			1.41225i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	172.000	+	192.302i	0.525994	+	0.588079i
$328$		0			0
$329$			80.4984i			0.244676i
$330$		0			0
$331$	598.000			1.80665			0.903323	−	0.428960i	$-0.141120\pi$
$331$	598.000			1.80665			0.903323	+	0.428960i	$0.141120\pi$
$332$		0			0
$333$	56.0000	−	500.879i	0.168168	−	1.50414i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	358.000			1.06231			0.531157	−	0.847273i	$-0.321757\pi$
$337$	358.000			1.06231			0.531157	+	0.847273i	$0.321757\pi$
$338$		0			0
$339$	−210.000	+	187.830i	−0.619469	+	0.554070i
$340$		0			0
$341$		−	187.830i		−	0.550820i
$342$		0			0
$343$	−188.000			−0.548105
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	254.912i		−	0.734616i	−0.930099	−	0.367308i	$-0.880280\pi$
$347$		−	254.912i		−	0.734616i	0.930099	−	0.367308i	$-0.119720\pi$
$348$		0			0
$349$	518.000			1.48424			0.742120	−	0.670267i	$-0.233820\pi$
$349$	518.000			1.48424			0.742120	+	0.670267i	$0.233820\pi$
$350$		0			0
$351$	176.000	−	125.220i	0.501425	−	0.356752i
$352$		0			0
$353$		−	281.745i		−	0.798143i	−0.916920	−	0.399072i	$-0.869332\pi$
$353$		−	281.745i		−	0.798143i	0.916920	−	0.399072i	$-0.130668\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−60.0000	+	53.6656i	−0.168067	+	0.150324i
$358$		0			0
$359$			348.827i			0.971662i	0.874053	+	0.485831i	$0.161483\pi$
$359$			348.827i			0.971662i	−0.874053	+	0.485831i	$0.838517\pi$
$360$		0			0
$361$	795.000			2.20222
$362$		0			0
$363$	−118.000	−	131.928i	−0.325069	−	0.363438i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−178.000			−0.485014			−0.242507	−	0.970150i	$-0.577970\pi$
$367$	−178.000			−0.485014			−0.242507	+	0.970150i	$0.577970\pi$
$368$		0			0
$369$	−240.000	−	26.8328i	−0.650407	−	0.0727177i
$370$		0			0
$371$		−	80.4984i		−	0.216977i
$372$		0			0
$373$	532.000			1.42627			0.713137	−	0.701025i	$-0.247274\pi$
$373$	532.000			1.42627			0.713137	+	0.701025i	$0.247274\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	321.994i		−	0.854095i
$378$		0			0
$379$	−86.0000			−0.226913			−0.113456	−	0.993543i	$-0.536192\pi$
$379$	−86.0000			−0.226913			−0.113456	+	0.993543i	$0.536192\pi$
$380$		0			0
$381$	−68.0000	−	76.0263i	−0.178478	−	0.199544i
$382$		0			0
$383$			120.748i			0.315268i	0.987498	+	0.157634i	$0.0503866\pi$
$383$			120.748i			0.315268i	−0.987498	+	0.157634i	$0.949613\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−8.00000	+	71.5542i	−0.0206718	+	0.184895i
$388$		0			0
$389$		−	415.909i		−	1.06917i	−0.845113	−	0.534587i	$-0.820467\pi$
$389$		−	415.909i		−	1.06917i	0.845113	−	0.534587i	$-0.179533\pi$
$390$		0			0
$391$	540.000			1.38107
$392$		0			0
$393$	−330.000	+	295.161i	−0.839695	+	0.751046i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	28.0000			0.0705290			0.0352645	−	0.999378i	$-0.488773\pi$
$397$	28.0000			0.0705290			0.0352645	+	0.999378i	$0.488773\pi$
$398$		0			0
$399$	136.000	+	152.053i	0.340852	+	0.381084i
$400$		0			0
$401$			268.328i			0.669148i	0.942370	+	0.334574i	$0.108592\pi$
$401$			268.328i			0.669148i	−0.942370	+	0.334574i	$0.891408\pi$
$402$		0			0
$403$	112.000			0.277916
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	751.319i		−	1.84599i
$408$		0			0
$409$	−214.000			−0.523227			−0.261614	−	0.965173i	$-0.584255\pi$
$409$	−214.000			−0.523227			−0.261614	+	0.965173i	$0.584255\pi$
$410$		0			0
$411$	90.0000	−	80.4984i	0.218978	−	0.195860i
$412$		0			0
$413$			26.8328i			0.0649705i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	164.000	+	183.358i	0.393285	+	0.439706i
$418$		0			0
$419$		−	174.413i		−	0.416261i	−0.978101	−	0.208130i	$-0.933262\pi$
$419$		−	174.413i		−	0.416261i	0.978101	−	0.208130i	$-0.0667379\pi$
$420$		0			0
$421$	−238.000			−0.565321			−0.282660	−	0.959220i	$-0.591217\pi$
$421$	−238.000			−0.565321			−0.282660	+	0.959220i	$0.591217\pi$
$422$		0			0
$423$	−360.000	−	40.2492i	−0.851064	−	0.0951518i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−92.0000			−0.215457
$428$		0			0
$429$	240.000	−	214.663i	0.559441	−	0.500379i
$430$		0			0
$431$		−	241.495i		−	0.560314i	−0.959954	−	0.280157i	$-0.909613\pi$
$431$		−	241.495i		−	0.560314i	0.959954	−	0.280157i	$-0.0903865\pi$
$432$		0			0
$433$	382.000			0.882217			0.441109	−	0.897454i	$-0.354585\pi$
$433$	382.000			0.882217			0.441109	+	0.897454i	$0.354585\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	1368.47i		−	3.13152i
$438$		0			0
$439$	274.000			0.624146			0.312073	−	0.950058i	$-0.398977\pi$
$439$	274.000			0.624146			0.312073	+	0.950058i	$0.398977\pi$
$440$		0			0
$441$	45.0000	−	402.492i	0.102041	−	0.912681i
$442$		0			0
$443$		−	764.735i		−	1.72626i	−0.504978	−	0.863132i	$-0.668499\pi$
$443$		−	764.735i		−	1.72626i	0.504978	−	0.863132i	$-0.331501\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	330.000	−	295.161i	0.738255	−	0.660315i
$448$		0			0
$449$			778.152i			1.73308i	0.499110	+	0.866539i	$0.333660\pi$
$449$			778.152i			1.73308i	−0.499110	+	0.866539i	$0.666340\pi$
$450$		0			0
$451$	−360.000			−0.798226
$452$		0			0
$453$	92.0000	+	102.859i	0.203091	+	0.227062i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−446.000			−0.975930			−0.487965	−	0.872863i	$-0.662261\pi$
$457$	−446.000			−0.975930			−0.487965	+	0.872863i	$0.662261\pi$
$458$		0			0
$459$	−210.000	−	295.161i	−0.457516	−	0.643052i
$460$		0			0
$461$			93.9149i			0.203720i	0.994799	+	0.101860i	$0.0324794\pi$
$461$			93.9149i			0.203720i	−0.994799	+	0.101860i	$0.967521\pi$
$462$		0			0
$463$	854.000			1.84449			0.922246	−	0.386603i	$-0.126352\pi$
$463$	854.000			1.84449			0.922246	+	0.386603i	$0.126352\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	13.4164i		−	0.0287289i	−0.999897	−	0.0143645i	$-0.995427\pi$
$467$		−	13.4164i		−	0.0287289i	0.999897	−	0.0143645i	$-0.00457251\pi$
$468$		0			0
$469$	64.0000			0.136461
$470$		0			0
$471$	−184.000	−	205.718i	−0.390658	−	0.436769i
$472$		0			0
$473$			107.331i			0.226916i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	360.000	+	40.2492i	0.754717	+	0.0843799i
$478$		0			0
$479$			53.6656i			0.112037i	0.998430	+	0.0560184i	$0.0178406\pi$
$479$			53.6656i			0.112037i	−0.998430	+	0.0560184i	$0.982159\pi$
$480$		0			0
$481$	448.000			0.931393
$482$		0			0
$483$	180.000	−	160.997i	0.372671	−	0.333327i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	2.00000			0.00410678			0.00205339	−	0.999998i	$-0.499346\pi$
$487$	2.00000			0.00410678			0.00205339	+	0.999998i	$0.499346\pi$
$488$		0			0
$489$	136.000	+	152.053i	0.278119	+	0.310946i
$490$		0			0
$491$		−	469.574i		−	0.956363i	−0.878261	−	0.478182i	$-0.841296\pi$
$491$		−	469.574i		−	0.956363i	0.878261	−	0.478182i	$-0.158704\pi$
$492$		0			0
$493$	−540.000			−1.09533
$494$		0			0
$495$		0			0
$496$		0			0
$497$			107.331i			0.215958i
$498$		0			0
$499$	514.000			1.03006			0.515030	−	0.857172i	$-0.327781\pi$
$499$	514.000			1.03006			0.515030	+	0.857172i	$0.327781\pi$
$500$		0			0
$501$	150.000	−	134.164i	0.299401	−	0.267793i
$502$		0			0
$503$			657.404i			1.30697i	0.756941	+	0.653483i	$0.226693\pi$
$503$			657.404i			1.30697i	−0.756941	+	0.653483i	$0.773307\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−210.000	−	234.787i	−0.414201	−	0.463091i
$508$		0			0
$509$			308.577i			0.606242i	0.952952	+	0.303121i	$0.0980287\pi$
$509$			308.577i			0.606242i	−0.952952	+	0.303121i	$0.901971\pi$
$510$		0			0
$511$	212.000			0.414873
$512$		0			0
$513$	−748.000	+	532.184i	−1.45809	+	1.03740i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−540.000			−1.04449
$518$		0			0
$519$	270.000	−	241.495i	0.520231	−	0.465309i
$520$		0			0
$521$		−	670.820i		−	1.28756i	−0.765209	−	0.643782i	$-0.777365\pi$
$521$		−	670.820i		−	1.28756i	0.765209	−	0.643782i	$-0.222635\pi$
$522$		0			0
$523$	−832.000			−1.59082			−0.795411	−	0.606070i	$-0.792745\pi$
$523$	−832.000			−1.59082			−0.795411	+	0.606070i	$0.792745\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	187.830i		−	0.356413i
$528$		0			0
$529$	−1091.00			−2.06238
$530$		0			0
$531$	−120.000	−	13.4164i	−0.225989	−	0.0252663i
$532$		0			0
$533$		−	214.663i		−	0.402744i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	630.000	−	563.489i	1.17318	−	1.04933i
$538$		0			0
$539$		−	603.738i		−	1.12011i
$540$		0			0
$541$	314.000			0.580407			0.290203	−	0.956965i	$-0.406277\pi$
$541$	314.000			0.580407			0.290203	+	0.956965i	$0.406277\pi$
$542$		0			0
$543$	388.000	+	433.797i	0.714549	+	0.798890i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	536.000			0.979890			0.489945	−	0.871753i	$-0.337017\pi$
$547$	536.000			0.979890			0.489945	+	0.871753i	$0.337017\pi$
$548$		0			0
$549$	46.0000	−	411.437i	0.0837887	−	0.749429i
$550$		0			0
$551$			1368.47i			2.48362i
$552$		0			0
$553$	44.0000			0.0795660
$554$		0			0
$555$		0			0
$556$		0			0
$557$			898.899i			1.61382i	0.590672	+	0.806911i	$0.298863\pi$
$557$			898.899i			1.61382i	−0.590672	+	0.806911i	$0.701137\pi$
$558$		0			0
$559$	−64.0000			−0.114490
$560$		0			0
$561$	−360.000	−	402.492i	−0.641711	−	0.717455i
$562$		0			0
$563$		−	818.401i		−	1.45364i	−0.686827	−	0.726821i	$-0.740997\pi$
$563$		−	818.401i		−	1.45364i	0.686827	−	0.726821i	$-0.259003\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−158.000	−	35.7771i	−0.278660	−	0.0630989i
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	526.000			0.921191			0.460595	−	0.887610i	$-0.347636\pi$
$571$	526.000			0.921191			0.460595	+	0.887610i	$0.347636\pi$
$572$		0			0
$573$	180.000	−	160.997i	0.314136	−	0.280972i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−122.000			−0.211438			−0.105719	−	0.994396i	$-0.533714\pi$
$577$	−122.000			−0.211438			−0.105719	+	0.994396i	$0.533714\pi$
$578$		0			0
$579$	−436.000	−	487.463i	−0.753022	−	0.841905i
$580$		0			0
$581$			241.495i			0.415655i
$582$		0			0
$583$	540.000			0.926244
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	228.079i		−	0.388550i	−0.980947	−	0.194275i	$-0.937765\pi$
$587$		−	228.079i		−	0.388550i	0.980947	−	0.194275i	$-0.0622354\pi$
$588$		0			0
$589$	−476.000			−0.808149
$590$		0			0
$591$	−210.000	+	187.830i	−0.355330	+	0.317817i
$592$		0			0
$593$			737.902i			1.24435i	0.782876	+	0.622177i	$0.213752\pi$
$593$			737.902i			1.24435i	−0.782876	+	0.622177i	$0.786248\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	68.0000	+	76.0263i	0.113903	+	0.127347i
$598$		0			0
$599$		−	80.4984i		−	0.134388i	−0.997740	−	0.0671940i	$-0.978595\pi$
$599$		−	80.4984i		−	0.134388i	0.997740	−	0.0671940i	$-0.0214047\pi$
$600$		0			0
$601$	−766.000			−1.27454			−0.637271	−	0.770640i	$-0.719937\pi$
$601$	−766.000			−1.27454			−0.637271	+	0.770640i	$0.719937\pi$
$602$		0			0
$603$	−32.0000	+	286.217i	−0.0530680	+	0.474655i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	206.000			0.339374			0.169687	−	0.985498i	$-0.445724\pi$
$607$	206.000			0.339374			0.169687	+	0.985498i	$0.445724\pi$
$608$		0			0
$609$	−180.000	+	160.997i	−0.295567	+	0.264363i
$610$		0			0
$611$		−	321.994i		−	0.526995i
$612$		0			0
$613$	556.000			0.907015			0.453507	−	0.891253i	$-0.350173\pi$
$613$	556.000			0.907015			0.453507	+	0.891253i	$0.350173\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			389.076i			0.630593i	0.948993	+	0.315296i	$0.102104\pi$
$617$			389.076i			0.630593i	−0.948993	+	0.315296i	$0.897896\pi$
$618$		0			0
$619$	514.000			0.830372			0.415186	−	0.909737i	$-0.363717\pi$
$619$	514.000			0.830372			0.415186	+	0.909737i	$0.363717\pi$
$620$		0			0
$621$	630.000	+	885.483i	1.01449	+	1.42590i
$622$		0			0
$623$		−	214.663i		−	0.344563i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−1020.00	+	912.316i	−1.62679	+	1.45505i
$628$		0			0
$629$		−	751.319i		−	1.19447i
$630$		0			0
$631$	−1094.00			−1.73376			−0.866878	−	0.498520i	$-0.833877\pi$
$631$	−1094.00			−1.73376			−0.866878	+	0.498520i	$0.833877\pi$
$632$		0			0
$633$	92.0000	+	102.859i	0.145340	+	0.162495i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	360.000			0.565149
$638$		0			0
$639$	−480.000	−	53.6656i	−0.751174	−	0.0839838i
$640$		0			0
$641$		−	160.997i		−	0.251165i	−0.992083	−	0.125583i	$-0.959920\pi$
$641$		−	160.997i		−	0.251165i	0.992083	−	0.125583i	$-0.0400800\pi$
$642$		0			0
$643$	404.000			0.628305			0.314152	−	0.949373i	$-0.398280\pi$
$643$	404.000			0.628305			0.314152	+	0.949373i	$0.398280\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			1167.23i			1.80406i	0.431672	+	0.902031i	$0.357924\pi$
$647$			1167.23i			1.80406i	−0.431672	+	0.902031i	$0.642076\pi$
$648$		0			0
$649$	−180.000			−0.277350
$650$		0			0
$651$	−56.0000	−	62.6099i	−0.0860215	−	0.0961750i
$652$		0			0
$653$		−	764.735i		−	1.17111i	−0.810632	−	0.585555i	$-0.800876\pi$
$653$		−	764.735i		−	1.17111i	0.810632	−	0.585555i	$-0.199124\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−106.000	+	948.093i	−0.161339	+	1.44306i
$658$		0			0
$659$			791.568i			1.20117i	0.799563	+	0.600583i	$0.205065\pi$
$659$			791.568i			1.20117i	−0.799563	+	0.600583i	$0.794935\pi$
$660$		0			0
$661$	−118.000			−0.178517			−0.0892587	−	0.996008i	$-0.528450\pi$
$661$	−118.000			−0.178517			−0.0892587	+	0.996008i	$0.528450\pi$
$662$		0			0
$663$	240.000	−	214.663i	0.361991	−	0.323775i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	1620.00			2.42879
$668$		0			0
$669$	796.000	+	889.955i	1.18984	+	1.33028i
$670$		0			0
$671$		−	617.155i		−	0.919754i
$672$		0			0
$673$	−194.000			−0.288262			−0.144131	−	0.989559i	$-0.546039\pi$
$673$	−194.000			−0.288262			−0.144131	+	0.989559i	$0.546039\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	415.909i		−	0.614341i	−0.951655	−	0.307170i	$-0.900618\pi$
$677$		−	415.909i		−	0.614341i	0.951655	−	0.307170i	$-0.0993821\pi$
$678$		0			0
$679$	−244.000			−0.359352
$680$		0			0
$681$	−750.000	+	670.820i	−1.10132	+	0.985052i
$682$		0			0
$683$		−	93.9149i		−	0.137503i	−0.997634	−	0.0687517i	$-0.978098\pi$
$683$		−	93.9149i		−	0.137503i	0.997634	−	0.0687517i	$-0.0219016\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	172.000	+	192.302i	0.250364	+	0.279915i
$688$		0			0
$689$			321.994i			0.467335i
$690$		0			0
$691$	−122.000			−0.176556			−0.0882779	−	0.996096i	$-0.528136\pi$
$691$	−122.000			−0.176556			−0.0882779	+	0.996096i	$0.528136\pi$
$692$		0			0
$693$	−240.000	−	26.8328i	−0.346320	−	0.0387198i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−360.000			−0.516499
$698$		0			0
$699$	−690.000	+	617.155i	−0.987124	+	0.882911i
$700$		0			0
$701$			576.906i			0.822975i	0.911415	+	0.411488i	$0.134991\pi$
$701$			576.906i			0.822975i	−0.911415	+	0.411488i	$0.865009\pi$
$702$		0			0
$703$	−1904.00			−2.70839
$704$		0			0
$705$		0			0
$706$		0			0
$707$			348.827i			0.493390i
$708$		0			0
$709$	1166.00			1.64457			0.822285	−	0.569076i	$-0.192699\pi$
$709$	1166.00			1.64457			0.822285	+	0.569076i	$0.192699\pi$
$710$		0			0
$711$	−22.0000	+	196.774i	−0.0309423	+	0.276757i
$712$		0			0
$713$			563.489i			0.790307i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	214.663i		−	0.298557i	−0.988795	−	0.149279i	$-0.952305\pi$
$719$		−	214.663i		−	0.298557i	0.988795	−	0.149279i	$-0.0476951\pi$
$720$		0			0
$721$	−92.0000			−0.127601
$722$		0			0
$723$	268.000	+	299.633i	0.370678	+	0.414430i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−1318.00			−1.81293			−0.906465	−	0.422281i	$-0.861230\pi$
$727$	−1318.00			−1.81293			−0.906465	+	0.422281i	$0.861230\pi$
$728$		0			0
$729$	239.000	−	688.709i	0.327846	−	0.944731i
$730$		0			0
$731$			107.331i			0.146828i
$732$		0			0
$733$	136.000			0.185539			0.0927694	−	0.995688i	$-0.470428\pi$
$733$	136.000			0.185539			0.0927694	+	0.995688i	$0.470428\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			429.325i			0.582531i
$738$		0			0
$739$	682.000			0.922869			0.461434	−	0.887174i	$-0.347335\pi$
$739$	682.000			0.922869			0.461434	+	0.887174i	$0.347335\pi$
$740$		0			0
$741$	−544.000	−	608.210i	−0.734143	−	0.820797i
$742$		0			0
$743$			872.067i			1.17371i	0.809692	+	0.586855i	$0.199634\pi$
$743$			872.067i			1.17371i	−0.809692	+	0.586855i	$0.800366\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−1080.00	−	120.748i	−1.44578	−	0.161643i
$748$		0			0
$749$		−	26.8328i		−	0.0358249i
$750$		0			0
$751$	658.000			0.876165			0.438083	−	0.898935i	$-0.355658\pi$
$751$	658.000			0.876165			0.438083	+	0.898935i	$0.355658\pi$
$752$		0			0
$753$	690.000	−	617.155i	0.916335	−	0.819595i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1112.00			−1.46896			−0.734478	−	0.678632i	$-0.762573\pi$
$757$	−1112.00			−1.46896			−0.734478	+	0.678632i	$0.762573\pi$
$758$		0			0
$759$	1080.00	+	1207.48i	1.42292	+	1.59088i
$760$		0			0
$761$			80.4984i			0.105780i	0.998600	+	0.0528899i	$0.0168432\pi$
$761$			80.4984i			0.105780i	−0.998600	+	0.0528899i	$0.983157\pi$
$762$		0			0
$763$	172.000			0.225426
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	107.331i		−	0.139936i
$768$		0			0
$769$	206.000			0.267880			0.133940	−	0.990989i	$-0.457237\pi$
$769$	206.000			0.267880			0.133940	+	0.990989i	$0.457237\pi$
$770$		0			0
$771$	−390.000	+	348.827i	−0.505837	+	0.452434i
$772$		0			0
$773$		−	1086.73i		−	1.40586i	−0.711260	−	0.702930i	$-0.751875\pi$
$773$		−	1086.73i		−	1.40586i	0.711260	−	0.702930i	$-0.248125\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−224.000	−	250.440i	−0.288288	−	0.322316i
$778$		0			0
$779$			912.316i			1.17114i
$780$		0			0
$781$	−720.000			−0.921895
$782$		0			0
$783$	−630.000	−	885.483i	−0.804598	−	1.13088i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−1168.00			−1.48412			−0.742058	−	0.670335i	$-0.766150\pi$
$787$	−1168.00			−1.48412			−0.742058	+	0.670335i	$0.766150\pi$
$788$		0			0
$789$	570.000	−	509.823i	0.722433	−	0.646164i
$790$		0			0
$791$			187.830i			0.237459i
$792$		0			0
$793$	368.000			0.464061
$794$		0			0
$795$		0			0
$796$		0			0
$797$			1194.06i			1.49819i	0.662461	+	0.749097i	$0.269512\pi$
$797$			1194.06i			1.49819i	−0.662461	+	0.749097i	$0.730488\pi$
$798$		0			0
$799$	−540.000			−0.675845
$800$		0			0
$801$	960.000	+	107.331i	1.19850	+	0.133997i
$802$		0			0
$803$			1422.14i			1.77103i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	750.000	−	670.820i	0.929368	−	0.831252i
$808$		0			0
$809$		−	912.316i		−	1.12771i	−0.825875	−	0.563854i	$-0.809318\pi$
$809$		−	912.316i		−	1.12771i	0.825875	−	0.563854i	$-0.190682\pi$
$810$		0			0
$811$	−674.000			−0.831073			−0.415536	−	0.909577i	$-0.636406\pi$
$811$	−674.000			−0.831073			−0.415536	+	0.909577i	$0.636406\pi$
$812$		0			0
$813$	−4.00000	−	4.47214i	−0.00492005	−	0.00550078i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	272.000			0.332925
$818$		0			0
$819$	16.0000	−	143.108i	0.0195360	−	0.174735i
$820$		0			0
$821$			1355.06i			1.65050i	0.564771	+	0.825248i	$0.308965\pi$
$821$			1355.06i			1.65050i	−0.564771	+	0.825248i	$0.691035\pi$
$822$		0			0
$823$	98.0000			0.119077			0.0595383	−	0.998226i	$-0.481037\pi$
$823$	98.0000			0.119077			0.0595383	+	0.998226i	$0.481037\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			818.401i			0.989602i	0.869006	+	0.494801i	$0.164759\pi$
$827$			818.401i			0.989602i	−0.869006	+	0.494801i	$0.835241\pi$
$828$		0			0
$829$	398.000			0.480097			0.240048	−	0.970761i	$-0.422837\pi$
$829$	398.000			0.480097			0.240048	+	0.970761i	$0.422837\pi$
$830$		0			0
$831$	896.000	+	1001.76i	1.07822	+	1.20549i
$832$		0			0
$833$		−	603.738i		−	0.724776i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	308.000	−	219.135i	0.367981	−	0.261810i
$838$		0			0
$839$		−	992.814i		−	1.18333i	−0.806184	−	0.591665i	$-0.798471\pi$
$839$		−	992.814i		−	1.18333i	0.806184	−	0.591665i	$-0.201529\pi$
$840$		0			0
$841$	−779.000			−0.926278
$842$		0			0
$843$	420.000	−	375.659i	0.498221	−	0.445622i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−118.000			−0.139315
$848$		0			0
$849$	496.000	+	554.545i	0.584217	+	0.653174i
$850$		0			0
$851$			2253.96i			2.64860i
$852$		0			0
$853$	−668.000			−0.783118			−0.391559	−	0.920153i	$-0.628064\pi$
$853$	−668.000			−0.783118			−0.391559	+	0.920153i	$0.628064\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	308.577i		−	0.360067i	−0.983660	−	0.180033i	$-0.942379\pi$
$857$		−	308.577i		−	0.360067i	0.983660	−	0.180033i	$-0.0576206\pi$
$858$		0			0
$859$	−278.000			−0.323632			−0.161816	−	0.986821i	$-0.551735\pi$
$859$	−278.000			−0.323632			−0.161816	+	0.986821i	$0.551735\pi$
$860$		0			0
$861$	−120.000	+	107.331i	−0.139373	+	0.124659i
$862$		0			0
$863$		−	362.243i		−	0.419749i	−0.977728	−	0.209874i	$-0.932695\pi$
$863$		−	362.243i		−	0.419749i	0.977728	−	0.209874i	$-0.0673055\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	218.000	+	243.731i	0.251442	+	0.281120i
$868$		0			0
$869$			295.161i			0.339656i
$870$		0			0
$871$	−256.000			−0.293915
$872$		0			0
$873$	122.000	−	1091.20i	0.139748	−	1.24994i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−572.000			−0.652223			−0.326112	−	0.945331i	$-0.605739\pi$
$877$	−572.000			−0.652223			−0.326112	+	0.945331i	$0.605739\pi$
$878$		0			0
$879$	330.000	−	295.161i	0.375427	−	0.335792i
$880$		0			0
$881$			804.984i			0.913717i	0.889539	+	0.456858i	$0.151025\pi$
$881$			804.984i			0.913717i	−0.889539	+	0.456858i	$0.848975\pi$
$882$		0			0
$883$	−1132.00			−1.28199			−0.640997	−	0.767544i	$-0.721479\pi$
$883$	−1132.00			−1.28199			−0.640997	+	0.767544i	$0.721479\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			1220.89i			1.37643i	0.725507	+	0.688215i	$0.241605\pi$
$887$			1220.89i			1.37643i	−0.725507	+	0.688215i	$0.758395\pi$
$888$		0			0
$889$	−68.0000			−0.0764904
$890$		0			0
$891$	240.000	−	1059.90i	0.269360	−	1.18956i
$892$		0			0
$893$			1368.47i			1.53245i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−720.000	+	643.988i	−0.802676	+	0.717935i
$898$		0			0
$899$		−	563.489i		−	0.626795i
$900$		0			0
$901$	540.000			0.599334
$902$		0			0
$903$	32.0000	+	35.7771i	0.0354374	+	0.0396203i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	716.000			0.789416			0.394708	−	0.918807i	$-0.370846\pi$
$907$	716.000			0.789416			0.394708	+	0.918807i	$0.370846\pi$
$908$		0			0
$909$	−1560.00	−	174.413i	−1.71617	−	0.191874i
$910$		0			0
$911$		−	1261.14i		−	1.38435i	−0.721730	−	0.692175i	$-0.756653\pi$
$911$		−	1261.14i		−	1.38435i	0.721730	−	0.692175i	$-0.243347\pi$
$912$		0			0
$913$	−1620.00			−1.77437
$914$		0			0
$915$		0			0
$916$		0			0
$917$			295.161i			0.321877i
$918$		0			0
$919$	−866.000			−0.942329			−0.471164	−	0.882045i	$-0.656166\pi$
$919$	−866.000			−0.942329			−0.471164	+	0.882045i	$0.656166\pi$
$920$		0			0
$921$	544.000	+	608.210i	0.590662	+	0.660381i
$922$		0			0
$923$		−	429.325i		−	0.465141i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	46.0000	−	411.437i	0.0496224	−	0.443837i
$928$		0			0
$929$		−	295.161i		−	0.317719i	−0.987301	−	0.158860i	$-0.949218\pi$
$929$		−	295.161i		−	0.317719i	0.987301	−	0.158860i	$-0.0507817\pi$
$930$		0			0
$931$	−1530.00			−1.64339
$932$		0			0
$933$	−120.000	+	107.331i	−0.128617	+	0.115039i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1018.00			1.08645			0.543223	−	0.839588i	$-0.317204\pi$
$937$	1018.00			1.08645			0.543223	+	0.839588i	$0.317204\pi$
$938$		0			0
$939$	−556.000	−	621.627i	−0.592119	−	0.662009i
$940$		0			0
$941$		−	1194.06i		−	1.26893i	−0.772953	−	0.634463i	$-0.781221\pi$
$941$		−	1194.06i		−	1.26893i	0.772953	−	0.634463i	$-0.218779\pi$
$942$		0			0
$943$	1080.00			1.14528
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	737.902i		−	0.779200i	−0.920984	−	0.389600i	$-0.872613\pi$
$947$		−	737.902i		−	0.779200i	0.920984	−	0.389600i	$-0.127387\pi$
$948$		0			0
$949$	−848.000			−0.893572
$950$		0			0
$951$	−270.000	+	241.495i	−0.283912	+	0.253938i
$952$		0			0
$953$			1542.89i			1.61898i	0.587134	+	0.809489i	$0.300256\pi$
$953$			1542.89i			1.61898i	−0.587134	+	0.809489i	$0.699744\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−1080.00	−	1207.48i	−1.12853	−	1.26173i
$958$		0			0
$959$		−	80.4984i		−	0.0839400i
$960$		0			0
$961$	−765.000			−0.796046
$962$		0			0
$963$	120.000	+	13.4164i	0.124611	+	0.0139319i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1414.00			−1.46225			−0.731127	−	0.682241i	$-0.761005\pi$
$967$	−1414.00			−1.46225			−0.731127	+	0.682241i	$0.761005\pi$
$968$		0			0
$969$	−1020.00	+	912.316i	−1.05263	+	0.941502i
$970$		0			0
$971$			335.410i			0.345428i	0.984972	+	0.172714i	$0.0552536\pi$
$971$			335.410i			0.345428i	−0.984972	+	0.172714i	$0.944746\pi$
$972$		0			0
$973$	164.000			0.168551
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	40.2492i		−	0.0411967i	−0.999788	−	0.0205984i	$-0.993443\pi$
$977$		−	40.2492i		−	0.0411967i	0.999788	−	0.0205984i	$-0.00655713\pi$
$978$		0			0
$979$	1440.00			1.47089
$980$		0			0
$981$	−86.0000	+	769.207i	−0.0876656	+	0.784105i
$982$		0			0
$983$		−	469.574i		−	0.477695i	−0.971057	−	0.238848i	$-0.923230\pi$
$983$		−	469.574i		−	0.477695i	0.971057	−	0.238848i	$-0.0767696\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−180.000	+	160.997i	−0.182371	+	0.163117i
$988$		0			0
$989$		−	321.994i		−	0.325575i
$990$		0			0
$991$	−914.000			−0.922301			−0.461150	−	0.887322i	$-0.652563\pi$
$991$	−914.000			−0.922301			−0.461150	+	0.887322i	$0.652563\pi$
$992$		0			0
$993$	1196.00	+	1337.17i	1.20443	+	1.34659i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1264.00			1.26780			0.633902	−	0.773414i	$-0.281452\pi$
$997$	1264.00			1.26780			0.633902	+	0.773414i	$0.281452\pi$
$998$		0			0
$999$	1232.00	−	876.539i	1.23323	−	0.877416i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.l.r.401.2		2
3.2	odd	2	inner	1200.3.l.r.401.1		2
4.3	odd	2		300.3.g.d.101.1		2
5.2	odd	4		1200.3.c.e.449.3		4
5.3	odd	4		1200.3.c.e.449.2		4
5.4	even	2		240.3.l.a.161.1		2
12.11	even	2		300.3.g.d.101.2		2
15.2	even	4		1200.3.c.e.449.1		4
15.8	even	4		1200.3.c.e.449.4		4
15.14	odd	2		240.3.l.a.161.2		2
20.3	even	4		300.3.b.c.149.3		4
20.7	even	4		300.3.b.c.149.2		4
20.19	odd	2		60.3.g.a.41.2	yes	2
40.19	odd	2		960.3.l.a.641.1		2
40.29	even	2		960.3.l.d.641.2		2
60.23	odd	4		300.3.b.c.149.1		4
60.47	odd	4		300.3.b.c.149.4		4
60.59	even	2		60.3.g.a.41.1	✓	2
120.29	odd	2		960.3.l.d.641.1		2
120.59	even	2		960.3.l.a.641.2		2
180.59	even	6		1620.3.o.b.1241.1		4
180.79	odd	6		1620.3.o.b.701.1		4
180.119	even	6		1620.3.o.b.701.2		4
180.139	odd	6		1620.3.o.b.1241.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.3.g.a.41.1	✓	2	60.59	even	2
60.3.g.a.41.2	yes	2	20.19	odd	2
240.3.l.a.161.1		2	5.4	even	2
240.3.l.a.161.2		2	15.14	odd	2
300.3.b.c.149.1		4	60.23	odd	4
300.3.b.c.149.2		4	20.7	even	4
300.3.b.c.149.3		4	20.3	even	4
300.3.b.c.149.4		4	60.47	odd	4
300.3.g.d.101.1		2	4.3	odd	2
300.3.g.d.101.2		2	12.11	even	2
960.3.l.a.641.1		2	40.19	odd	2
960.3.l.a.641.2		2	120.59	even	2
960.3.l.d.641.1		2	120.29	odd	2
960.3.l.d.641.2		2	40.29	even	2
1200.3.c.e.449.1		4	15.2	even	4
1200.3.c.e.449.2		4	5.3	odd	4
1200.3.c.e.449.3		4	5.2	odd	4
1200.3.c.e.449.4		4	15.8	even	4
1200.3.l.r.401.1		2	3.2	odd	2	inner
1200.3.l.r.401.2		2	1.1	even	1	trivial
1620.3.o.b.701.1		4	180.79	odd	6
1620.3.o.b.701.2		4	180.119	even	6
1620.3.o.b.1241.1		4	180.59	even	6
1620.3.o.b.1241.2		4	180.139	odd	6

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

\(f(q)\)	\(=\)	\(q+(2.00000 + 2.23607i) q^{3} +2.00000 q^{7} +(-1.00000 + 8.94427i) q^{9} +O(q^{10})\)	\(q+(2.00000 + 2.23607i) q^{3} +2.00000 q^{7} +(-1.00000 + 8.94427i) q^{9} +13.4164i q^{11} -8.00000 q^{13} +13.4164i q^{17} +34.0000 q^{19} +(4.00000 + 4.47214i) q^{21} -40.2492i q^{23} +(-22.0000 + 15.6525i) q^{27} +40.2492i q^{29} -14.0000 q^{31} +(-30.0000 + 26.8328i) q^{33} -56.0000 q^{37} +(-16.0000 - 17.8885i) q^{39} +26.8328i q^{41} +8.00000 q^{43} +40.2492i q^{47} -45.0000 q^{49} +(-30.0000 + 26.8328i) q^{51} -40.2492i q^{53} +(68.0000 + 76.0263i) q^{57} +13.4164i q^{59} -46.0000 q^{61} +(-2.00000 + 17.8885i) q^{63} +32.0000 q^{67} +(90.0000 - 80.4984i) q^{69} +53.6656i q^{71} +106.000 q^{73} +26.8328i q^{77} +22.0000 q^{79} +(-79.0000 - 17.8885i) q^{81} +120.748i q^{83} +(-90.0000 + 80.4984i) q^{87} -107.331i q^{89} -16.0000 q^{91} +(-28.0000 - 31.3050i) q^{93} -122.000 q^{97} +(-120.000 - 13.4164i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} + 4 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q + 4 * q^3 + 4 * q^7 - 2 * q^9	\( 2 q + 4 q^{3} + 4 q^{7} - 2 q^{9} - 16 q^{13} + 68 q^{19} + 8 q^{21} - 44 q^{27} - 28 q^{31} - 60 q^{33} - 112 q^{37} - 32 q^{39} + 16 q^{43} - 90 q^{49} - 60 q^{51} + 136 q^{57} - 92 q^{61} - 4 q^{63} + 64 q^{67} + 180 q^{69} + 212 q^{73} + 44 q^{79} - 158 q^{81} - 180 q^{87} - 32 q^{91} - 56 q^{93} - 244 q^{97} - 240 q^{99}+O(q^{100}) \) 2 * q + 4 * q^3 + 4 * q^7 - 2 * q^9 - 16 * q^13 + 68 * q^19 + 8 * q^21 - 44 * q^27 - 28 * q^31 - 60 * q^33 - 112 * q^37 - 32 * q^39 + 16 * q^43 - 90 * q^49 - 60 * q^51 + 136 * q^57 - 92 * q^61 - 4 * q^63 + 64 * q^67 + 180 * q^69 + 212 * q^73 + 44 * q^79 - 158 * q^81 - 180 * q^87 - 32 * q^91 - 56 * q^93 - 244 * q^97 - 240 * q^99

Introduction

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