LMFDB - Embedded newform 572.2.f.b.441.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [572,2,Mod(441,572)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(572, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("572.441");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 572 = 2^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	572.f (of order $2$, degree $1$, 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:	$4.56744299562$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{21})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 11x^{2} + 25 $ x^4 + 11*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			441.3
Root			$-2.79129i$ of defining polynomial
Character	$\chi$	$=$	572.441
Dual form			572.2.f.b.441.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.79129 q^{3} -3.79129i q^{7} +4.79129 q^{9} +O(q^{10})$	$q+2.79129 q^{3} -3.79129i q^{7} +4.79129 q^{9} -1.00000i q^{11} +(2.00000 - 3.00000i) q^{13} -7.58258 q^{17} +6.79129i q^{19} -10.5826i q^{21} +3.79129 q^{23} +5.00000 q^{25} +5.00000 q^{27} -1.58258 q^{29} +1.58258i q^{31} -2.79129i q^{33} +9.16515i q^{37} +(5.58258 - 8.37386i) q^{39} +5.37386i q^{41} +2.00000 q^{43} +6.00000i q^{47} -7.37386 q^{49} -21.1652 q^{51} -6.79129 q^{53} +18.9564i q^{57} -6.00000i q^{59} +10.0000 q^{61} -18.1652i q^{63} +15.1652i q^{67} +10.5826 q^{69} -10.7477i q^{71} -9.95644i q^{73} +13.9564 q^{75} -3.79129 q^{77} -8.00000 q^{79} -0.417424 q^{81} +11.3739i q^{83} -4.41742 q^{87} -6.00000i q^{89} +(-11.3739 - 7.58258i) q^{91} +4.41742i q^{93} -7.58258i q^{97} -4.79129i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 10 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 + 10 * q^9	$ 4 q + 2 q^{3} + 10 q^{9} + 8 q^{13} - 12 q^{17} + 6 q^{23} + 20 q^{25} + 20 q^{27} + 12 q^{29} + 4 q^{39} + 8 q^{43} - 2 q^{49} - 48 q^{51} - 18 q^{53} + 40 q^{61} + 24 q^{69} + 10 q^{75} - 6 q^{77} - 32 q^{79} - 20 q^{81} - 36 q^{87} - 18 q^{91}+O(q^{100}) $ 4 * q + 2 * q^3 + 10 * q^9 + 8 * q^13 - 12 * q^17 + 6 * q^23 + 20 * q^25 + 20 * q^27 + 12 * q^29 + 4 * q^39 + 8 * q^43 - 2 * q^49 - 48 * q^51 - 18 * q^53 + 40 * q^61 + 24 * q^69 + 10 * q^75 - 6 * q^77 - 32 * q^79 - 20 * q^81 - 36 * q^87 - 18 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/572\mathbb{Z}\right)^\times$.

$n$	$287$	$353$	$365$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	2.79129			1.61155			0.805775	−	0.592221i	$-0.201749\pi$
$3$	2.79129			1.61155			0.805775	+	0.592221i	$0.201749\pi$
$4$		0			0
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$		−	3.79129i		−	1.43297i	−0.697601	−	0.716486i	$-0.745749\pi$
$7$		−	3.79129i		−	1.43297i	0.697601	−	0.716486i	$-0.254251\pi$
$8$		0			0
$9$	4.79129			1.59710
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$	2.00000	−	3.00000i	0.554700	−	0.832050i
$13$	2.00000	−	3.00000i	0.554700	−	0.832050i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−7.58258			−1.83904			−0.919522	−	0.393038i	$-0.871424\pi$
$17$	−7.58258			−1.83904			−0.919522	+	0.393038i	$0.871424\pi$
$18$		0			0
$19$			6.79129i			1.55803i	0.627006	+	0.779014i	$0.284280\pi$
$19$			6.79129i			1.55803i	−0.627006	+	0.779014i	$0.715720\pi$
$20$		0			0
$21$		−	10.5826i		−	2.30931i
$22$		0			0
$23$	3.79129			0.790538			0.395269	−	0.918565i	$-0.370651\pi$
$23$	3.79129			0.790538			0.395269	+	0.918565i	$0.370651\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$		0			0
$27$	5.00000			0.962250
$28$		0			0
$29$	−1.58258			−0.293877			−0.146938	−	0.989146i	$-0.546942\pi$
$29$	−1.58258			−0.293877			−0.146938	+	0.989146i	$0.546942\pi$
$30$		0			0
$31$			1.58258i			0.284239i	0.989850	+	0.142119i	$0.0453917\pi$
$31$			1.58258i			0.284239i	−0.989850	+	0.142119i	$0.954608\pi$
$32$		0			0
$33$		−	2.79129i		−	0.485901i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			9.16515i			1.50674i	0.657596	+	0.753371i	$0.271573\pi$
$37$			9.16515i			1.50674i	−0.657596	+	0.753371i	$0.728427\pi$
$38$		0			0
$39$	5.58258	−	8.37386i	0.893928	−	1.34089i
$40$		0			0
$41$			5.37386i			0.839256i	0.907696	+	0.419628i	$0.137840\pi$
$41$			5.37386i			0.839256i	−0.907696	+	0.419628i	$0.862160\pi$
$42$		0			0
$43$	2.00000			0.304997			0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000			0.304997			0.152499	+	0.988304i	$0.451268\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			6.00000i			0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$			6.00000i			0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$		0			0
$49$	−7.37386			−1.05341
$50$		0			0
$51$	−21.1652			−2.96371
$52$		0			0
$53$	−6.79129			−0.932855			−0.466428	−	0.884559i	$-0.654459\pi$
$53$	−6.79129			−0.932855			−0.466428	+	0.884559i	$0.654459\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			18.9564i			2.51084i
$58$		0			0
$59$		−	6.00000i		−	0.781133i	−0.920575	−	0.390567i	$-0.872279\pi$
$59$		−	6.00000i		−	0.781133i	0.920575	−	0.390567i	$-0.127721\pi$
$60$		0			0
$61$	10.0000			1.28037			0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000			1.28037			0.640184	+	0.768221i	$0.278858\pi$
$62$		0			0
$63$		−	18.1652i		−	2.28859i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			15.1652i			1.85272i	0.376642	+	0.926359i	$0.377079\pi$
$67$			15.1652i			1.85272i	−0.376642	+	0.926359i	$0.622921\pi$
$68$		0			0
$69$	10.5826			1.27399
$70$		0			0
$71$		−	10.7477i		−	1.27552i	−0.770235	−	0.637760i	$-0.779861\pi$
$71$		−	10.7477i		−	1.27552i	0.770235	−	0.637760i	$-0.220139\pi$
$72$		0			0
$73$		−	9.95644i		−	1.16531i	−0.812719	−	0.582657i	$-0.802013\pi$
$73$		−	9.95644i		−	1.16531i	0.812719	−	0.582657i	$-0.197987\pi$
$74$		0			0
$75$	13.9564			1.61155
$76$		0			0
$77$	−3.79129			−0.432057
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$	−0.417424			−0.0463805
$82$		0			0
$83$			11.3739i			1.24844i	0.781247	+	0.624222i	$0.214584\pi$
$83$			11.3739i			1.24844i	−0.781247	+	0.624222i	$0.785416\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−4.41742			−0.473598
$88$		0			0
$89$		−	6.00000i		−	0.635999i	−0.948091	−	0.317999i	$-0.896989\pi$
$89$		−	6.00000i		−	0.635999i	0.948091	−	0.317999i	$-0.103011\pi$
$90$		0			0
$91$	−11.3739	−	7.58258i	−1.19230	−	0.794870i
$92$		0			0
$93$			4.41742i			0.458066i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	7.58258i		−	0.769894i	−0.922939	−	0.384947i	$-0.874220\pi$
$97$		−	7.58258i		−	0.769894i	0.922939	−	0.384947i	$-0.125780\pi$
$98$		0			0
$99$		−	4.79129i		−	0.481543i
$100$		0			0
$101$	15.1652			1.50899			0.754494	−	0.656306i	$-0.227882\pi$
$101$	15.1652			1.50899			0.754494	+	0.656306i	$0.227882\pi$
$102$		0			0
$103$	7.37386			0.726568			0.363284	−	0.931678i	$-0.381655\pi$
$103$	7.37386			0.726568			0.363284	+	0.931678i	$0.381655\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−13.5826			−1.31308			−0.656539	−	0.754292i	$-0.727980\pi$
$107$	−13.5826			−1.31308			−0.656539	+	0.754292i	$0.727980\pi$
$108$		0			0
$109$		−	18.7913i		−	1.79988i	−0.436015	−	0.899939i	$-0.643610\pi$
$109$		−	18.7913i		−	1.79988i	0.436015	−	0.899939i	$-0.356390\pi$
$110$		0			0
$111$			25.5826i			2.42819i
$112$		0			0
$113$	−3.62614			−0.341118			−0.170559	−	0.985347i	$-0.554557\pi$
$113$	−3.62614			−0.341118			−0.170559	+	0.985347i	$0.554557\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	9.58258	−	14.3739i	0.885909	−	1.32886i
$118$		0			0
$119$			28.7477i			2.63530i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$			15.0000i			1.35250i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−2.00000			−0.177471			−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000			−0.177471			−0.0887357	+	0.996055i	$0.528283\pi$
$128$		0			0
$129$	5.58258			0.491518
$130$		0			0
$131$	−13.5826			−1.18672			−0.593358	−	0.804939i	$-0.702198\pi$
$131$	−13.5826			−1.18672			−0.593358	+	0.804939i	$0.702198\pi$
$132$		0			0
$133$	25.7477			2.23261
$134$		0			0
$135$		0			0
$136$		0			0
$137$			12.0000i			1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$			12.0000i			1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$		0			0
$139$	−14.7477			−1.25089			−0.625443	−	0.780270i	$-0.715082\pi$
$139$	−14.7477			−1.25089			−0.625443	+	0.780270i	$0.715082\pi$
$140$		0			0
$141$			16.7477i			1.41041i
$142$		0			0
$143$	−3.00000	−	2.00000i	−0.250873	−	0.167248i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−20.5826			−1.69762
$148$		0			0
$149$			6.62614i			0.542834i	0.962462	+	0.271417i	$0.0874923\pi$
$149$			6.62614i			0.542834i	−0.962462	+	0.271417i	$0.912508\pi$
$150$		0			0
$151$		−	12.0000i		−	0.976546i	−0.872691	−	0.488273i	$-0.837627\pi$
$151$		−	12.0000i		−	0.976546i	0.872691	−	0.488273i	$-0.162373\pi$
$152$		0			0
$153$	−36.3303			−2.93713
$154$		0			0
$155$		0			0
$156$		0			0
$157$	2.62614			0.209589			0.104794	−	0.994494i	$-0.466582\pi$
$157$	2.62614			0.209589			0.104794	+	0.994494i	$0.466582\pi$
$158$		0			0
$159$	−18.9564			−1.50334
$160$		0			0
$161$		−	14.3739i		−	1.13282i
$162$		0			0
$163$			15.1652i			1.18783i	0.804529	+	0.593913i	$0.202418\pi$
$163$			15.1652i			1.18783i	−0.804529	+	0.593913i	$0.797582\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	3.62614i		−	0.280599i	−0.990109	−	0.140299i	$-0.955193\pi$
$167$		−	3.62614i		−	0.280599i	0.990109	−	0.140299i	$-0.0448065\pi$
$168$		0			0
$169$	−5.00000	−	12.0000i	−0.384615	−	0.923077i
$170$		0			0
$171$			32.5390i			2.48832i
$172$		0			0
$173$	−4.41742			−0.335851			−0.167925	−	0.985800i	$-0.553707\pi$
$173$	−4.41742			−0.335851			−0.167925	+	0.985800i	$0.553707\pi$
$174$		0			0
$175$		−	18.9564i		−	1.43297i
$176$		0			0
$177$		−	16.7477i		−	1.25884i
$178$		0			0
$179$	−18.3303			−1.37007			−0.685036	−	0.728510i	$-0.740213\pi$
$179$	−18.3303			−1.37007			−0.685036	+	0.728510i	$0.740213\pi$
$180$		0			0
$181$	−0.373864			−0.0277891			−0.0138945	−	0.999903i	$-0.504423\pi$
$181$	−0.373864			−0.0277891			−0.0138945	+	0.999903i	$0.504423\pi$
$182$		0			0
$183$	27.9129			2.06338
$184$		0			0
$185$		0			0
$186$		0			0
$187$			7.58258i			0.554493i
$188$		0			0
$189$		−	18.9564i		−	1.37888i
$190$		0			0
$191$	−8.37386			−0.605912			−0.302956	−	0.953005i	$-0.597973\pi$
$191$	−8.37386			−0.605912			−0.302956	+	0.953005i	$0.597973\pi$
$192$		0			0
$193$			2.20871i			0.158987i	0.996835	+	0.0794933i	$0.0253302\pi$
$193$			2.20871i			0.158987i	−0.996835	+	0.0794933i	$0.974670\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			8.37386i			0.596613i	0.954470	+	0.298307i	$0.0964218\pi$
$197$			8.37386i			0.596613i	−0.954470	+	0.298307i	$0.903578\pi$
$198$		0			0
$199$	−4.37386			−0.310055			−0.155027	−	0.987910i	$-0.549547\pi$
$199$	−4.37386			−0.310055			−0.155027	+	0.987910i	$0.549547\pi$
$200$		0			0
$201$			42.3303i			2.98575i
$202$		0			0
$203$			6.00000i			0.421117i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	18.1652			1.26257
$208$		0			0
$209$	6.79129			0.469763
$210$		0			0
$211$	16.0000			1.10149			0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000			1.10149			0.550743	+	0.834675i	$0.314345\pi$
$212$		0			0
$213$		−	30.0000i		−	2.05557i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	6.00000			0.407307
$218$		0			0
$219$		−	27.7913i		−	1.87796i
$220$		0			0
$221$	−15.1652	+	22.7477i	−1.02012	+	1.53018i
$222$		0			0
$223$		−	3.16515i		−	0.211954i	−0.994369	−	0.105977i	$-0.966203\pi$
$223$		−	3.16515i		−	0.211954i	0.994369	−	0.105977i	$-0.0337970\pi$
$224$		0			0
$225$	23.9564			1.59710
$226$		0			0
$227$		−	8.37386i		−	0.555793i	−0.960611	−	0.277896i	$-0.910363\pi$
$227$		−	8.37386i		−	0.555793i	0.960611	−	0.277896i	$-0.0896372\pi$
$228$		0			0
$229$		−	22.7477i		−	1.50321i	−0.659612	−	0.751606i	$-0.729279\pi$
$229$		−	22.7477i		−	1.50321i	0.659612	−	0.751606i	$-0.270721\pi$
$230$		0			0
$231$	−10.5826			−0.696282
$232$		0			0
$233$	10.7477			0.704107			0.352054	−	0.935980i	$-0.385484\pi$
$233$	10.7477			0.704107			0.352054	+	0.935980i	$0.385484\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−22.3303			−1.45051
$238$		0			0
$239$			8.37386i			0.541660i	0.962627	+	0.270830i	$0.0872982\pi$
$239$			8.37386i			0.541660i	−0.962627	+	0.270830i	$0.912702\pi$
$240$		0			0
$241$			5.37386i			0.346161i	0.984908	+	0.173080i	$0.0553721\pi$
$241$			5.37386i			0.346161i	−0.984908	+	0.173080i	$0.944628\pi$
$242$		0			0
$243$	−16.1652			−1.03699
$244$		0			0
$245$		0			0
$246$		0			0
$247$	20.3739	+	13.5826i	1.29636	+	0.864239i
$248$		0			0
$249$			31.7477i			2.01193i
$250$		0			0
$251$	3.79129			0.239304			0.119652	−	0.992816i	$-0.461822\pi$
$251$	3.79129			0.239304			0.119652	+	0.992816i	$0.461822\pi$
$252$		0			0
$253$		−	3.79129i		−	0.238356i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	28.1216			1.75418			0.877088	−	0.480329i	$-0.159483\pi$
$257$	28.1216			1.75418			0.877088	+	0.480329i	$0.159483\pi$
$258$		0			0
$259$	34.7477			2.15912
$260$		0			0
$261$	−7.58258			−0.469350
$262$		0			0
$263$	−10.4174			−0.642366			−0.321183	−	0.947017i	$-0.604080\pi$
$263$	−10.4174			−0.642366			−0.321183	+	0.947017i	$0.604080\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		−	16.7477i		−	1.02494i
$268$		0			0
$269$	29.3739			1.79096			0.895478	−	0.445106i	$-0.146834\pi$
$269$	29.3739			1.79096			0.895478	+	0.445106i	$0.146834\pi$
$270$		0			0
$271$		−	3.62614i		−	0.220272i	−0.993917	−	0.110136i	$-0.964871\pi$
$271$		−	3.62614i		−	0.220272i	0.993917	−	0.110136i	$-0.0351287\pi$
$272$		0			0
$273$	−31.7477	−	21.1652i	−1.92146	−	1.28097i
$274$		0			0
$275$		−	5.00000i		−	0.301511i
$276$		0			0
$277$	−8.74773			−0.525600			−0.262800	−	0.964850i	$-0.584646\pi$
$277$	−8.74773			−0.525600			−0.262800	+	0.964850i	$0.584646\pi$
$278$		0			0
$279$			7.58258i			0.453957i
$280$		0			0
$281$		−	25.1216i		−	1.49863i	−0.662215	−	0.749314i	$-0.730383\pi$
$281$		−	25.1216i		−	1.49863i	0.662215	−	0.749314i	$-0.269617\pi$
$282$		0			0
$283$	20.7477			1.23332			0.616662	−	0.787228i	$-0.288484\pi$
$283$	20.7477			1.23332			0.616662	+	0.787228i	$0.288484\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	20.3739			1.20263
$288$		0			0
$289$	40.4955			2.38209
$290$		0			0
$291$		−	21.1652i		−	1.24072i
$292$		0			0
$293$			6.00000i			0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$			6.00000i			0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		−	5.00000i		−	0.290129i
$298$		0			0
$299$	7.58258	−	11.3739i	0.438512	−	0.657768i
$300$		0			0
$301$		−	7.58258i		−	0.437052i
$302$		0			0
$303$	42.3303			2.43181
$304$		0			0
$305$		0			0
$306$		0			0
$307$			6.33030i			0.361289i	0.983548	+	0.180645i	$0.0578184\pi$
$307$			6.33030i			0.361289i	−0.983548	+	0.180645i	$0.942182\pi$
$308$		0			0
$309$	20.5826			1.17090
$310$		0			0
$311$	−23.5390			−1.33478			−0.667388	−	0.744711i	$-0.732588\pi$
$311$	−23.5390			−1.33478			−0.667388	+	0.744711i	$0.732588\pi$
$312$		0			0
$313$	9.37386			0.529842			0.264921	−	0.964270i	$-0.414654\pi$
$313$	9.37386			0.529842			0.264921	+	0.964270i	$0.414654\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	28.7477i		−	1.61463i	−0.590119	−	0.807317i	$-0.700919\pi$
$317$		−	28.7477i		−	1.61463i	0.590119	−	0.807317i	$-0.299081\pi$
$318$		0			0
$319$			1.58258i			0.0886072i
$320$		0			0
$321$	−37.9129			−2.11609
$322$		0			0
$323$		−	51.4955i		−	2.86528i
$324$		0			0
$325$	10.0000	−	15.0000i	0.554700	−	0.832050i
$326$		0			0
$327$		−	52.4519i		−	2.90060i
$328$		0			0
$329$	22.7477			1.25412
$330$		0			0
$331$			2.83485i			0.155817i	0.996960	+	0.0779087i	$0.0248243\pi$
$331$			2.83485i			0.155817i	−0.996960	+	0.0779087i	$0.975176\pi$
$332$		0			0
$333$			43.9129i			2.40641i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−2.00000			−0.108947			−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000			−0.108947			−0.0544735	+	0.998515i	$0.517348\pi$
$338$		0			0
$339$	−10.1216			−0.549729
$340$		0			0
$341$	1.58258			0.0857013
$342$		0			0
$343$			1.41742i			0.0765337i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−12.0000			−0.644194			−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000			−0.644194			−0.322097	+	0.946707i	$0.604388\pi$
$348$		0			0
$349$		−	11.0436i		−	0.591148i	−0.955320	−	0.295574i	$-0.904489\pi$
$349$		−	11.0436i		−	0.591148i	0.955320	−	0.295574i	$-0.0955109\pi$
$350$		0			0
$351$	10.0000	−	15.0000i	0.533761	−	0.800641i
$352$		0			0
$353$			7.25227i			0.386000i	0.981199	+	0.193000i	$0.0618217\pi$
$353$			7.25227i			0.386000i	−0.981199	+	0.193000i	$0.938178\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			80.2432i			4.24692i
$358$		0			0
$359$			10.1216i			0.534197i	0.963669	+	0.267099i	$0.0860649\pi$
$359$			10.1216i			0.534197i	−0.963669	+	0.267099i	$0.913935\pi$
$360$		0			0
$361$	−27.1216			−1.42745
$362$		0			0
$363$	−2.79129			−0.146505
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−17.1216			−0.893740			−0.446870	−	0.894599i	$-0.647461\pi$
$367$	−17.1216			−0.893740			−0.446870	+	0.894599i	$0.647461\pi$
$368$		0			0
$369$			25.7477i			1.34037i
$370$		0			0
$371$			25.7477i			1.33676i
$372$		0			0
$373$	25.4955			1.32010			0.660052	−	0.751220i	$-0.270534\pi$
$373$	25.4955			1.32010			0.660052	+	0.751220i	$0.270534\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−3.16515	+	4.74773i	−0.163014	+	0.244520i
$378$		0			0
$379$			6.00000i			0.308199i	0.988055	+	0.154100i	$0.0492477\pi$
$379$			6.00000i			0.308199i	−0.988055	+	0.154100i	$0.950752\pi$
$380$		0			0
$381$	−5.58258			−0.286004
$382$		0			0
$383$			1.25227i			0.0639882i	0.999488	+	0.0319941i	$0.0101858\pi$
$383$			1.25227i			0.0639882i	−0.999488	+	0.0319941i	$0.989814\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	9.58258			0.487110
$388$		0			0
$389$	−33.9564			−1.72166			−0.860830	−	0.508893i	$-0.830055\pi$
$389$	−33.9564			−1.72166			−0.860830	+	0.508893i	$0.830055\pi$
$390$		0			0
$391$	−28.7477			−1.45384
$392$		0			0
$393$	−37.9129			−1.91245
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	19.5826i		−	0.982821i	−0.870928	−	0.491411i	$-0.836481\pi$
$397$		−	19.5826i		−	0.982821i	0.870928	−	0.491411i	$-0.163519\pi$
$398$		0			0
$399$	71.8693			3.59797
$400$		0			0
$401$			6.00000i			0.299626i	0.988714	+	0.149813i	$0.0478671\pi$
$401$			6.00000i			0.299626i	−0.988714	+	0.149813i	$0.952133\pi$
$402$		0			0
$403$	4.74773	+	3.16515i	0.236501	+	0.157667i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	9.16515			0.454300
$408$		0			0
$409$			9.16515i			0.453188i	0.973989	+	0.226594i	$0.0727590\pi$
$409$			9.16515i			0.453188i	−0.973989	+	0.226594i	$0.927241\pi$
$410$		0			0
$411$			33.4955i			1.65221i
$412$		0			0
$413$	−22.7477			−1.11934
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−41.1652			−2.01587
$418$		0			0
$419$	35.3739			1.72813			0.864063	−	0.503383i	$-0.167912\pi$
$419$	35.3739			1.72813			0.864063	+	0.503383i	$0.167912\pi$
$420$		0			0
$421$		−	16.7477i		−	0.816234i	−0.912930	−	0.408117i	$-0.866185\pi$
$421$		−	16.7477i		−	0.816234i	0.912930	−	0.408117i	$-0.133815\pi$
$422$		0			0
$423$			28.7477i			1.39776i
$424$		0			0
$425$	−37.9129			−1.83904
$426$		0			0
$427$		−	37.9129i		−	1.83473i
$428$		0			0
$429$	−8.37386	−	5.58258i	−0.404294	−	0.269529i
$430$		0			0
$431$			0.626136i			0.0301599i	0.999886	+	0.0150800i	$0.00480028\pi$
$431$			0.626136i			0.0301599i	−0.999886	+	0.0150800i	$0.995200\pi$
$432$		0			0
$433$	−4.37386			−0.210194			−0.105097	−	0.994462i	$-0.533515\pi$
$433$	−4.37386			−0.210194			−0.105097	+	0.994462i	$0.533515\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			25.7477i			1.23168i
$438$		0			0
$439$	−14.7477			−0.703871			−0.351935	−	0.936024i	$-0.614476\pi$
$439$	−14.7477			−0.703871			−0.351935	+	0.936024i	$0.614476\pi$
$440$		0			0
$441$	−35.3303			−1.68240
$442$		0			0
$443$	−3.79129			−0.180130			−0.0900648	−	0.995936i	$-0.528707\pi$
$443$	−3.79129			−0.180130			−0.0900648	+	0.995936i	$0.528707\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			18.4955i			0.874805i
$448$		0			0
$449$		−	33.4955i		−	1.58075i	−0.612624	−	0.790374i	$-0.709886\pi$
$449$		−	33.4955i		−	1.58075i	0.612624	−	0.790374i	$-0.290114\pi$
$450$		0			0
$451$	5.37386			0.253045
$452$		0			0
$453$		−	33.4955i		−	1.57375i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			15.6261i			0.730960i	0.930819	+	0.365480i	$0.119095\pi$
$457$			15.6261i			0.730960i	−0.930819	+	0.365480i	$0.880905\pi$
$458$		0			0
$459$	−37.9129			−1.76962
$460$		0			0
$461$			25.1216i			1.17003i	0.811023	+	0.585015i	$0.198911\pi$
$461$			25.1216i			1.17003i	−0.811023	+	0.585015i	$0.801089\pi$
$462$		0			0
$463$			10.7477i			0.499489i	0.968312	+	0.249745i	$0.0803467\pi$
$463$			10.7477i			0.499489i	−0.968312	+	0.249745i	$0.919653\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	30.3303			1.40352			0.701760	−	0.712414i	$-0.252398\pi$
$467$	30.3303			1.40352			0.701760	+	0.712414i	$0.252398\pi$
$468$		0			0
$469$	57.4955			2.65489
$470$		0			0
$471$	7.33030			0.337763
$472$		0			0
$473$		−	2.00000i		−	0.0919601i
$474$		0			0
$475$			33.9564i			1.55803i
$476$		0			0
$477$	−32.5390			−1.48986
$478$		0			0
$479$			21.4955i			0.982152i	0.871117	+	0.491076i	$0.163396\pi$
$479$			21.4955i			0.982152i	−0.871117	+	0.491076i	$0.836604\pi$
$480$		0			0
$481$	27.4955	+	18.3303i	1.25368	+	0.835790i
$482$		0			0
$483$		−	40.1216i		−	1.82560i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			7.25227i			0.328632i	0.986408	+	0.164316i	$0.0525417\pi$
$487$			7.25227i			0.328632i	−0.986408	+	0.164316i	$0.947458\pi$
$488$		0			0
$489$			42.3303i			1.91424i
$490$		0			0
$491$	29.0780			1.31227			0.656137	−	0.754642i	$-0.272190\pi$
$491$	29.0780			1.31227			0.656137	+	0.754642i	$0.272190\pi$
$492$		0			0
$493$	12.0000			0.540453
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−40.7477			−1.82779
$498$		0			0
$499$		−	29.0780i		−	1.30171i	−0.759201	−	0.650856i	$-0.774410\pi$
$499$		−	29.0780i		−	1.30171i	0.759201	−	0.650856i	$-0.225590\pi$
$500$		0			0
$501$		−	10.1216i		−	0.452199i
$502$		0			0
$503$	−13.9129			−0.620345			−0.310172	−	0.950680i	$-0.600387\pi$
$503$	−13.9129			−0.620345			−0.310172	+	0.950680i	$0.600387\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−13.9564	−	33.4955i	−0.619827	−	1.48759i
$508$		0			0
$509$		−	39.4955i		−	1.75061i	−0.483576	−	0.875303i	$-0.660662\pi$
$509$		−	39.4955i		−	1.75061i	0.483576	−	0.875303i	$-0.339338\pi$
$510$		0			0
$511$	−37.7477			−1.66986
$512$		0			0
$513$			33.9564i			1.49921i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	6.00000			0.263880
$518$		0			0
$519$	−12.3303			−0.541240
$520$		0			0
$521$	−26.2087			−1.14822			−0.574112	−	0.818777i	$-0.694653\pi$
$521$	−26.2087			−1.14822			−0.574112	+	0.818777i	$0.694653\pi$
$522$		0			0
$523$	6.74773			0.295058			0.147529	−	0.989058i	$-0.452868\pi$
$523$	6.74773			0.295058			0.147529	+	0.989058i	$0.452868\pi$
$524$		0			0
$525$		−	52.9129i		−	2.30931i
$526$		0			0
$527$		−	12.0000i		−	0.522728i
$528$		0			0
$529$	−8.62614			−0.375049
$530$		0			0
$531$		−	28.7477i		−	1.24755i
$532$		0			0
$533$	16.1216	+	10.7477i	0.698304	+	0.465536i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−51.1652			−2.20794
$538$		0			0
$539$			7.37386i			0.317615i
$540$		0			0
$541$		−	5.37386i		−	0.231040i	−0.993305	−	0.115520i	$-0.963146\pi$
$541$		−	5.37386i		−	0.231040i	0.993305	−	0.115520i	$-0.0368535\pi$
$542$		0			0
$543$	−1.04356			−0.0447835
$544$		0			0
$545$		0			0
$546$		0			0
$547$	24.7477			1.05814			0.529068	−	0.848579i	$-0.322542\pi$
$547$	24.7477			1.05814			0.529068	+	0.848579i	$0.322542\pi$
$548$		0			0
$549$	47.9129			2.04487
$550$		0			0
$551$		−	10.7477i		−	0.457869i
$552$		0			0
$553$			30.3303i			1.28978i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			29.3739i			1.24461i	0.782775	+	0.622305i	$0.213804\pi$
$557$			29.3739i			1.24461i	−0.782775	+	0.622305i	$0.786196\pi$
$558$		0			0
$559$	4.00000	−	6.00000i	0.169182	−	0.253773i
$560$		0			0
$561$			21.1652i			0.893593i
$562$		0			0
$563$	−7.58258			−0.319567			−0.159784	−	0.987152i	$-0.551080\pi$
$563$	−7.58258			−0.319567			−0.159784	+	0.987152i	$0.551080\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$			1.58258i			0.0664619i
$568$		0			0
$569$	−13.2523			−0.555564			−0.277782	−	0.960644i	$-0.589599\pi$
$569$	−13.2523			−0.555564			−0.277782	+	0.960644i	$0.589599\pi$
$570$		0			0
$571$	−9.25227			−0.387196			−0.193598	−	0.981081i	$-0.562016\pi$
$571$	−9.25227			−0.387196			−0.193598	+	0.981081i	$0.562016\pi$
$572$		0			0
$573$	−23.3739			−0.976457
$574$		0			0
$575$	18.9564			0.790538
$576$		0			0
$577$		−	0.330303i		−	0.0137507i	−0.999976	−	0.00687534i	$-0.997811\pi$
$577$		−	0.330303i		−	0.0137507i	0.999976	−	0.00687534i	$-0.00218851\pi$
$578$		0			0
$579$			6.16515i			0.256215i
$580$		0			0
$581$	43.1216			1.78899
$582$		0			0
$583$			6.79129i			0.281266i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	22.7477i		−	0.938899i	−0.882959	−	0.469450i	$-0.844452\pi$
$587$		−	22.7477i		−	0.938899i	0.882959	−	0.469450i	$-0.155548\pi$
$588$		0			0
$589$	−10.7477			−0.442852
$590$		0			0
$591$			23.3739i			0.961472i
$592$		0			0
$593$		−	4.12159i		−	0.169253i	−0.996413	−	0.0846267i	$-0.973030\pi$
$593$		−	4.12159i		−	0.169253i	0.996413	−	0.0846267i	$-0.0269698\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−12.2087			−0.499669
$598$		0			0
$599$	3.62614			0.148160			0.0740800	−	0.997252i	$-0.476398\pi$
$599$	3.62614			0.148160			0.0740800	+	0.997252i	$0.476398\pi$
$600$		0			0
$601$	−12.7477			−0.519991			−0.259995	−	0.965610i	$-0.583721\pi$
$601$	−12.7477			−0.519991			−0.259995	+	0.965610i	$0.583721\pi$
$602$		0			0
$603$			72.6606i			2.95897i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	8.00000			0.324710			0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000			0.324710			0.162355	+	0.986732i	$0.448091\pi$
$608$		0			0
$609$			16.7477i			0.678652i
$610$		0			0
$611$	18.0000	+	12.0000i	0.728202	+	0.485468i
$612$		0			0
$613$			30.7913i			1.24365i	0.783157	+	0.621824i	$0.213608\pi$
$613$			30.7913i			1.24365i	−0.783157	+	0.621824i	$0.786392\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			40.7477i			1.64044i	0.572047	+	0.820221i	$0.306149\pi$
$617$			40.7477i			1.64044i	−0.572047	+	0.820221i	$0.693851\pi$
$618$		0			0
$619$		−	1.25227i		−	0.0503331i	−0.999683	−	0.0251665i	$-0.991988\pi$
$619$		−	1.25227i		−	0.0503331i	0.999683	−	0.0251665i	$-0.00801161\pi$
$620$		0			0
$621$	18.9564			0.760696
$622$		0			0
$623$	−22.7477			−0.911368
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$	18.9564			0.757047
$628$		0			0
$629$		−	69.4955i		−	2.77097i
$630$		0			0
$631$		−	16.7477i		−	0.666716i	−0.942800	−	0.333358i	$-0.891818\pi$
$631$		−	16.7477i		−	0.666716i	0.942800	−	0.333358i	$-0.108182\pi$
$632$		0			0
$633$	44.6606			1.77510
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−14.7477	+	22.1216i	−0.584326	+	0.876489i
$638$		0			0
$639$		−	51.4955i		−	2.03713i
$640$		0			0
$641$	−3.46099			−0.136701			−0.0683503	−	0.997661i	$-0.521774\pi$
$641$	−3.46099			−0.136701			−0.0683503	+	0.997661i	$0.521774\pi$
$642$		0			0
$643$		−	14.8348i		−	0.585029i	−0.956261	−	0.292515i	$-0.905508\pi$
$643$		−	14.8348i		−	0.585029i	0.956261	−	0.292515i	$-0.0944920\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−14.7042			−0.578080			−0.289040	−	0.957317i	$-0.593336\pi$
$647$	−14.7042			−0.578080			−0.289040	+	0.957317i	$0.593336\pi$
$648$		0			0
$649$	−6.00000			−0.235521
$650$		0			0
$651$	16.7477			0.656395
$652$		0			0
$653$	−24.3303			−0.952118			−0.476059	−	0.879413i	$-0.657935\pi$
$653$	−24.3303			−0.952118			−0.476059	+	0.879413i	$0.657935\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	47.7042i		−	1.86112i
$658$		0			0
$659$	48.6606			1.89555			0.947774	−	0.318943i	$-0.103328\pi$
$659$	48.6606			1.89555			0.947774	+	0.318943i	$0.103328\pi$
$660$		0			0
$661$		−	15.1652i		−	0.589856i	−0.955519	−	0.294928i	$-0.904704\pi$
$661$		−	15.1652i		−	0.589856i	0.955519	−	0.294928i	$-0.0952957\pi$
$662$		0			0
$663$	−42.3303	+	63.4955i	−1.64397	+	2.46596i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−6.00000			−0.232321
$668$		0			0
$669$		−	8.83485i		−	0.341575i
$670$		0			0
$671$		−	10.0000i		−	0.386046i
$672$		0			0
$673$	0.747727			0.0288228			0.0144114	−	0.999896i	$-0.495413\pi$
$673$	0.747727			0.0288228			0.0144114	+	0.999896i	$0.495413\pi$
$674$		0			0
$675$	25.0000			0.962250
$676$		0			0
$677$	−18.0000			−0.691796			−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000			−0.691796			−0.345898	+	0.938272i	$0.612426\pi$
$678$		0			0
$679$	−28.7477			−1.10324
$680$		0			0
$681$		−	23.3739i		−	0.895688i
$682$		0			0
$683$			45.4955i			1.74084i	0.492314	+	0.870418i	$0.336151\pi$
$683$			45.4955i			1.74084i	−0.492314	+	0.870418i	$0.663849\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		−	63.4955i		−	2.42250i
$688$		0			0
$689$	−13.5826	+	20.3739i	−0.517455	+	0.776182i
$690$		0			0
$691$			16.4174i			0.624548i	0.949992	+	0.312274i	$0.101091\pi$
$691$			16.4174i			0.624548i	−0.949992	+	0.312274i	$0.898909\pi$
$692$		0			0
$693$	−18.1652			−0.690037
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	40.7477i		−	1.54343i
$698$		0			0
$699$	30.0000			1.13470
$700$		0			0
$701$	46.7477			1.76564			0.882819	−	0.469714i	$-0.155643\pi$
$701$	46.7477			1.76564			0.882819	+	0.469714i	$0.155643\pi$
$702$		0			0
$703$	−62.2432			−2.34755
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	57.4955i		−	2.16234i
$708$		0			0
$709$		−	34.7477i		−	1.30498i	−0.757798	−	0.652489i	$-0.773725\pi$
$709$		−	34.7477i		−	1.30498i	0.757798	−	0.652489i	$-0.226275\pi$
$710$		0			0
$711$	−38.3303			−1.43750
$712$		0			0
$713$			6.00000i			0.224702i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			23.3739i			0.872913i
$718$		0			0
$719$	32.8348			1.22453			0.612267	−	0.790651i	$-0.290258\pi$
$719$	32.8348			1.22453			0.612267	+	0.790651i	$0.290258\pi$
$720$		0			0
$721$		−	27.9564i		−	1.04115i
$722$		0			0
$723$			15.0000i			0.557856i
$724$		0			0
$725$	−7.91288			−0.293877
$726$		0			0
$727$	−42.1216			−1.56220			−0.781102	−	0.624404i	$-0.785342\pi$
$727$	−42.1216			−1.56220			−0.781102	+	0.624404i	$0.785342\pi$
$728$		0			0
$729$	−43.8693			−1.62479
$730$		0			0
$731$	−15.1652			−0.560903
$732$		0			0
$733$			29.3739i			1.08495i	0.840072	+	0.542474i	$0.182512\pi$
$733$			29.3739i			1.08495i	−0.840072	+	0.542474i	$0.817488\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	15.1652			0.558616
$738$		0			0
$739$			30.9564i			1.13875i	0.822078	+	0.569376i	$0.192815\pi$
$739$			30.9564i			1.13875i	−0.822078	+	0.569376i	$0.807185\pi$
$740$		0			0
$741$	56.8693	+	37.9129i	2.08915	+	1.39276i
$742$		0			0
$743$			21.4955i			0.788592i	0.918984	+	0.394296i	$0.129011\pi$
$743$			21.4955i			0.788592i	−0.918984	+	0.394296i	$0.870989\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			54.4955i			1.99388i
$748$		0			0
$749$			51.4955i			1.88160i
$750$		0			0
$751$	−50.6170			−1.84704			−0.923521	−	0.383548i	$-0.874702\pi$
$751$	−50.6170			−1.84704			−0.923521	+	0.383548i	$0.874702\pi$
$752$		0			0
$753$	10.5826			0.385650
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−26.6261			−0.967743			−0.483872	−	0.875139i	$-0.660770\pi$
$757$	−26.6261			−0.967743			−0.483872	+	0.875139i	$0.660770\pi$
$758$		0			0
$759$		−	10.5826i		−	0.384123i
$760$		0			0
$761$			26.8693i			0.974012i	0.873399	+	0.487006i	$0.161911\pi$
$761$			26.8693i			0.974012i	−0.873399	+	0.487006i	$0.838089\pi$
$762$		0			0
$763$	−71.2432			−2.57918
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−18.0000	−	12.0000i	−0.649942	−	0.433295i
$768$		0			0
$769$			21.7913i			0.785814i	0.919578	+	0.392907i	$0.128531\pi$
$769$			21.7913i			0.785814i	−0.919578	+	0.392907i	$0.871469\pi$
$770$		0			0
$771$	78.4955			2.82694
$772$		0			0
$773$		−	52.7477i		−	1.89720i	−0.316472	−	0.948602i	$-0.602498\pi$
$773$		−	52.7477i		−	1.89720i	0.316472	−	0.948602i	$-0.397502\pi$
$774$		0			0
$775$			7.91288i			0.284239i
$776$		0			0
$777$	96.9909			3.47953
$778$		0			0
$779$	−36.4955			−1.30759
$780$		0			0
$781$	−10.7477			−0.384584
$782$		0			0
$783$	−7.91288			−0.282783
$784$		0			0
$785$		0			0
$786$		0			0
$787$			46.1216i			1.64406i	0.569446	+	0.822029i	$0.307158\pi$
$787$			46.1216i			1.64406i	−0.569446	+	0.822029i	$0.692842\pi$
$788$		0			0
$789$	−29.0780			−1.03521
$790$		0			0
$791$			13.7477i			0.488813i
$792$		0			0
$793$	20.0000	−	30.0000i	0.710221	−	1.06533i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	18.0000			0.637593			0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000			0.637593			0.318796	+	0.947823i	$0.396721\pi$
$798$		0			0
$799$		−	45.4955i		−	1.60951i
$800$		0			0
$801$		−	28.7477i		−	1.01575i
$802$		0			0
$803$	−9.95644			−0.351355
$804$		0			0
$805$		0			0
$806$		0			0
$807$	81.9909			2.88622
$808$		0			0
$809$	2.83485			0.0996680			0.0498340	−	0.998758i	$-0.484131\pi$
$809$	2.83485			0.0996680			0.0498340	+	0.998758i	$0.484131\pi$
$810$		0			0
$811$		−	37.2867i		−	1.30931i	−0.755926	−	0.654657i	$-0.772813\pi$
$811$		−	37.2867i		−	1.30931i	0.755926	−	0.654657i	$-0.227187\pi$
$812$		0			0
$813$		−	10.1216i		−	0.354980i
$814$		0			0
$815$		0			0
$816$		0			0
$817$			13.5826i			0.475194i
$818$		0			0
$819$	−54.4955	−	36.3303i	−1.90423	−	1.26948i
$820$		0			0
$821$		−	39.4955i		−	1.37840i	−0.724571	−	0.689200i	$-0.757962\pi$
$821$		−	39.4955i		−	1.37840i	0.724571	−	0.689200i	$-0.242038\pi$
$822$		0			0
$823$	−5.87841			−0.204908			−0.102454	−	0.994738i	$-0.532670\pi$
$823$	−5.87841			−0.204908			−0.102454	+	0.994738i	$0.532670\pi$
$824$		0			0
$825$		−	13.9564i		−	0.485901i
$826$		0			0
$827$		−	32.3739i		−	1.12575i	−0.826542	−	0.562875i	$-0.809695\pi$
$827$		−	32.3739i		−	1.12575i	0.826542	−	0.562875i	$-0.190305\pi$
$828$		0			0
$829$	9.37386			0.325568			0.162784	−	0.986662i	$-0.447953\pi$
$829$	9.37386			0.325568			0.162784	+	0.986662i	$0.447953\pi$
$830$		0			0
$831$	−24.4174			−0.847031
$832$		0			0
$833$	55.9129			1.93727
$834$		0			0
$835$		0			0
$836$		0			0
$837$			7.91288i			0.273509i
$838$		0			0
$839$			57.4955i			1.98496i	0.122393	+	0.992482i	$0.460943\pi$
$839$			57.4955i			1.98496i	−0.122393	+	0.992482i	$0.539057\pi$
$840$		0			0
$841$	−26.4955			−0.913636
$842$		0			0
$843$		−	70.1216i		−	2.41512i
$844$		0			0
$845$		0			0
$846$		0			0
$847$			3.79129i			0.130270i
$848$		0			0
$849$	57.9129			1.98756
$850$		0			0
$851$			34.7477i			1.19114i
$852$		0			0
$853$		−	9.95644i		−	0.340902i	−0.985366	−	0.170451i	$-0.945478\pi$
$853$		−	9.95644i		−	0.340902i	0.985366	−	0.170451i	$-0.0545225\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−6.33030			−0.216239			−0.108120	−	0.994138i	$-0.534483\pi$
$857$	−6.33030			−0.216239			−0.108120	+	0.994138i	$0.534483\pi$
$858$		0			0
$859$	−6.87841			−0.234688			−0.117344	−	0.993091i	$-0.537438\pi$
$859$	−6.87841			−0.234688			−0.117344	+	0.993091i	$0.537438\pi$
$860$		0			0
$861$	56.8693			1.93810
$862$		0			0
$863$			10.7477i			0.365857i	0.983126	+	0.182928i	$0.0585577\pi$
$863$			10.7477i			0.365857i	−0.983126	+	0.182928i	$0.941442\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	113.034			3.83885
$868$		0			0
$869$			8.00000i			0.271381i
$870$		0			0
$871$	45.4955	+	30.3303i	1.54155	+	1.02770i
$872$		0			0
$873$		−	36.3303i		−	1.22959i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			11.5390i			0.389645i	0.980839	+	0.194822i	$0.0624131\pi$
$877$			11.5390i			0.389645i	−0.980839	+	0.194822i	$0.937587\pi$
$878$		0			0
$879$			16.7477i			0.564887i
$880$		0			0
$881$	−8.37386			−0.282123			−0.141061	−	0.990001i	$-0.545051\pi$
$881$	−8.37386			−0.282123			−0.141061	+	0.990001i	$0.545051\pi$
$882$		0			0
$883$	12.3739			0.416414			0.208207	−	0.978085i	$-0.433237\pi$
$883$	12.3739			0.416414			0.208207	+	0.978085i	$0.433237\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.25227			0.0420472			0.0210236	−	0.999779i	$-0.493307\pi$
$887$	1.25227			0.0420472			0.0210236	+	0.999779i	$0.493307\pi$
$888$		0			0
$889$			7.58258i			0.254311i
$890$		0			0
$891$			0.417424i			0.0139842i
$892$		0			0
$893$	−40.7477			−1.36357
$894$		0			0
$895$		0			0
$896$		0			0
$897$	21.1652	−	31.7477i	0.706684	−	1.06003i
$898$		0			0
$899$		−	2.50455i		−	0.0835313i
$900$		0			0
$901$	51.4955			1.71556
$902$		0			0
$903$		−	21.1652i		−	0.704332i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−9.87841			−0.328007			−0.164004	−	0.986460i	$-0.552441\pi$
$907$	−9.87841			−0.328007			−0.164004	+	0.986460i	$0.552441\pi$
$908$		0			0
$909$	72.6606			2.41000
$910$		0			0
$911$	17.0436			0.564678			0.282339	−	0.959315i	$-0.408890\pi$
$911$	17.0436			0.564678			0.282339	+	0.959315i	$0.408890\pi$
$912$		0			0
$913$	11.3739			0.376420
$914$		0			0
$915$		0			0
$916$		0			0
$917$			51.4955i			1.70053i
$918$		0			0
$919$	−42.7477			−1.41012			−0.705059	−	0.709149i	$-0.749079\pi$
$919$	−42.7477			−1.41012			−0.705059	+	0.709149i	$0.749079\pi$
$920$		0			0
$921$			17.6697i			0.582236i
$922$		0			0
$923$	−32.2432	−	21.4955i	−1.06130	−	0.707531i
$924$		0			0
$925$			45.8258i			1.50674i
$926$		0			0
$927$	35.3303			1.16040
$928$		0			0
$929$		−	44.2432i		−	1.45157i	−0.687921	−	0.725786i	$-0.741476\pi$
$929$		−	44.2432i		−	1.45157i	0.687921	−	0.725786i	$-0.258524\pi$
$930$		0			0
$931$		−	50.0780i		−	1.64124i
$932$		0			0
$933$	−65.7042			−2.15106
$934$		0			0
$935$		0			0
$936$		0			0
$937$	25.4955			0.832900			0.416450	−	0.909159i	$-0.363274\pi$
$937$	25.4955			0.832900			0.416450	+	0.909159i	$0.363274\pi$
$938$		0			0
$939$	26.1652			0.853867
$940$		0			0
$941$		−	32.3739i		−	1.05536i	−0.849444	−	0.527679i	$-0.823062\pi$
$941$		−	32.3739i		−	1.05536i	0.849444	−	0.527679i	$-0.176938\pi$
$942$		0			0
$943$			20.3739i			0.663464i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			18.0000i			0.584921i	0.956278	+	0.292461i	$0.0944741\pi$
$947$			18.0000i			0.584921i	−0.956278	+	0.292461i	$0.905526\pi$
$948$		0			0
$949$	−29.8693	−	19.9129i	−0.969599	−	0.646399i
$950$		0			0
$951$		−	80.2432i		−	2.60206i
$952$		0			0
$953$	−53.4083			−1.73007			−0.865033	−	0.501715i	$-0.832703\pi$
$953$	−53.4083			−1.73007			−0.865033	+	0.501715i	$0.832703\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			4.41742i			0.142795i
$958$		0			0
$959$	45.4955			1.46912
$960$		0			0
$961$	28.4955			0.919208
$962$		0			0
$963$	−65.0780			−2.09711
$964$		0			0
$965$		0			0
$966$		0			0
$967$			42.7913i			1.37608i	0.725675	+	0.688038i	$0.241528\pi$
$967$			42.7913i			1.37608i	−0.725675	+	0.688038i	$0.758472\pi$
$968$		0			0
$969$		−	143.739i		−	4.61755i
$970$		0			0
$971$	33.0345			1.06013			0.530063	−	0.847958i	$-0.322168\pi$
$971$	33.0345			1.06013			0.530063	+	0.847958i	$0.322168\pi$
$972$		0			0
$973$			55.9129i			1.79248i
$974$		0			0
$975$	27.9129	−	41.8693i	0.893928	−	1.34089i
$976$		0			0
$977$			3.49545i			0.111829i	0.998436	+	0.0559147i	$0.0178075\pi$
$977$			3.49545i			0.111829i	−0.998436	+	0.0559147i	$0.982192\pi$
$978$		0			0
$979$	−6.00000			−0.191761
$980$		0			0
$981$		−	90.0345i		−	2.87458i
$982$		0			0
$983$			7.25227i			0.231312i	0.993289	+	0.115656i	$0.0368970\pi$
$983$			7.25227i			0.231312i	−0.993289	+	0.115656i	$0.963103\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	63.4955			2.02108
$988$		0			0
$989$	7.58258			0.241112
$990$		0			0
$991$	21.1216			0.670949			0.335475	−	0.942049i	$-0.391103\pi$
$991$	21.1216			0.670949			0.335475	+	0.942049i	$0.391103\pi$
$992$		0			0
$993$			7.91288i			0.251108i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−58.9909			−1.86826			−0.934130	−	0.356932i	$-0.883823\pi$
$997$	−58.9909			−1.86826			−0.934130	+	0.356932i	$0.883823\pi$
$998$		0			0
$999$			45.8258i			1.44986i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.f.b.441.3	✓	4
3.2	odd	2		5148.2.e.b.1585.1		4
4.3	odd	2		2288.2.j.f.1585.2		4
13.5	odd	4		7436.2.a.j.1.2		2
13.8	odd	4		7436.2.a.i.1.2		2
13.12	even	2	inner	572.2.f.b.441.4	yes	4
39.38	odd	2		5148.2.e.b.1585.4		4
52.51	odd	2		2288.2.j.f.1585.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.f.b.441.3	✓	4	1.1	even	1	trivial
572.2.f.b.441.4	yes	4	13.12	even	2	inner
2288.2.j.f.1585.1		4	52.51	odd	2
2288.2.j.f.1585.2		4	4.3	odd	2
5148.2.e.b.1585.1		4	3.2	odd	2
5148.2.e.b.1585.4		4	39.38	odd	2
7436.2.a.i.1.2		2	13.8	odd	4
7436.2.a.j.1.2		2	13.5	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:	\( N \)	\(=\)	\( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	572.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.79129 q^{3} -3.79129i q^{7} +4.79129 q^{9} +O(q^{10})\)	\(q+2.79129 q^{3} -3.79129i q^{7} +4.79129 q^{9} -1.00000i q^{11} +(2.00000 - 3.00000i) q^{13} -7.58258 q^{17} +6.79129i q^{19} -10.5826i q^{21} +3.79129 q^{23} +5.00000 q^{25} +5.00000 q^{27} -1.58258 q^{29} +1.58258i q^{31} -2.79129i q^{33} +9.16515i q^{37} +(5.58258 - 8.37386i) q^{39} +5.37386i q^{41} +2.00000 q^{43} +6.00000i q^{47} -7.37386 q^{49} -21.1652 q^{51} -6.79129 q^{53} +18.9564i q^{57} -6.00000i q^{59} +10.0000 q^{61} -18.1652i q^{63} +15.1652i q^{67} +10.5826 q^{69} -10.7477i q^{71} -9.95644i q^{73} +13.9564 q^{75} -3.79129 q^{77} -8.00000 q^{79} -0.417424 q^{81} +11.3739i q^{83} -4.41742 q^{87} -6.00000i q^{89} +(-11.3739 - 7.58258i) q^{91} +4.41742i q^{93} -7.58258i q^{97} -4.79129i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 10 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 + 10 * q^9	\( 4 q + 2 q^{3} + 10 q^{9} + 8 q^{13} - 12 q^{17} + 6 q^{23} + 20 q^{25} + 20 q^{27} + 12 q^{29} + 4 q^{39} + 8 q^{43} - 2 q^{49} - 48 q^{51} - 18 q^{53} + 40 q^{61} + 24 q^{69} + 10 q^{75} - 6 q^{77} - 32 q^{79} - 20 q^{81} - 36 q^{87} - 18 q^{91}+O(q^{100}) \) 4 * q + 2 * q^3 + 10 * q^9 + 8 * q^13 - 12 * q^17 + 6 * q^23 + 20 * q^25 + 20 * q^27 + 12 * q^29 + 4 * q^39 + 8 * q^43 - 2 * q^49 - 48 * q^51 - 18 * q^53 + 40 * q^61 + 24 * q^69 + 10 * q^75 - 6 * q^77 - 32 * q^79 - 20 * q^81 - 36 * q^87 - 18 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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