LMFDB - Embedded newform 1200.3.bg.p.1057.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,3,Mod(193,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))

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

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

chi := DirichletCharacter("1200.193");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1200.bg (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$32.6976317232$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 150)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1057.2
Root			$-1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1200.1057
Dual form			1200.3.bg.p.193.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 - 1.22474i) q^{3} +(4.77526 + 4.77526i) q^{7} -3.00000i q^{9} +O(q^{10})$	$q+(1.22474 - 1.22474i) q^{3} +(4.77526 + 4.77526i) q^{7} -3.00000i q^{9} -20.6969 q^{11} +(8.32577 - 8.32577i) q^{13} +(1.34847 + 1.34847i) q^{17} +33.6969i q^{19} +11.6969 q^{21} +(-16.0454 + 16.0454i) q^{23} +(-3.67423 - 3.67423i) q^{27} +8.69694i q^{29} +49.0908 q^{31} +(-25.3485 + 25.3485i) q^{33} +(36.4949 + 36.4949i) q^{37} -20.3939i q^{39} +33.3031 q^{41} +(-24.1237 + 24.1237i) q^{43} +(-13.9546 - 13.9546i) q^{47} -3.39388i q^{49} +3.30306 q^{51} +(-59.3939 + 59.3939i) q^{53} +(41.2702 + 41.2702i) q^{57} +30.0000i q^{59} -47.7878 q^{61} +(14.3258 - 14.3258i) q^{63} +(17.0227 + 17.0227i) q^{67} +39.3031i q^{69} +9.30306 q^{71} +(-22.2929 + 22.2929i) q^{73} +(-98.8332 - 98.8332i) q^{77} -27.3939i q^{79} -9.00000 q^{81} +(-63.4393 + 63.4393i) q^{83} +(10.6515 + 10.6515i) q^{87} +17.3939i q^{89} +79.5153 q^{91} +(60.1237 - 60.1237i) q^{93} +(-41.8763 - 41.8763i) q^{97} +62.0908i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 24 q^{7}+O(q^{10}) $ 4 * q + 24 * q^7	$ 4 q + 24 q^{7} - 24 q^{11} + 48 q^{13} - 24 q^{17} - 12 q^{21} + 24 q^{23} + 20 q^{31} - 72 q^{33} + 48 q^{37} + 192 q^{41} - 72 q^{43} - 144 q^{47} + 72 q^{51} - 120 q^{53} + 72 q^{57} + 44 q^{61} + 72 q^{63} + 24 q^{67} + 96 q^{71} + 48 q^{73} - 72 q^{77} - 36 q^{81} - 48 q^{83} + 72 q^{87} + 612 q^{91} + 216 q^{93} - 192 q^{97}+O(q^{100}) $ 4 * q + 24 * q^7 - 24 * q^11 + 48 * q^13 - 24 * q^17 - 12 * q^21 + 24 * q^23 + 20 * q^31 - 72 * q^33 + 48 * q^37 + 192 * q^41 - 72 * q^43 - 144 * q^47 + 72 * q^51 - 120 * q^53 + 72 * q^57 + 44 * q^61 + 72 * q^63 + 24 * q^67 + 96 * q^71 + 48 * q^73 - 72 * q^77 - 36 * q^81 - 48 * q^83 + 72 * q^87 + 612 * q^91 + 216 * q^93 - 192 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	1.22474	−	1.22474i	0.408248	−	0.408248i
$3$	1.22474	−	1.22474i	0.408248	−	0.408248i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	4.77526	+	4.77526i	0.682179	+	0.682179i	0.960491	−	0.278312i	$-0.0897748\pi$
$7$	4.77526	+	4.77526i	0.682179	+	0.682179i	−0.278312	+	0.960491i	$0.589775\pi$
$8$		0			0
$9$		−	3.00000i		−	0.333333i
$10$		0			0
$11$	−20.6969			−1.88154			−0.940770	−	0.339046i	$-0.889896\pi$
$11$	−20.6969			−1.88154			−0.940770	+	0.339046i	$0.889896\pi$
$12$		0			0
$13$	8.32577	−	8.32577i	0.640443	−	0.640443i	−0.310221	−	0.950664i	$-0.600403\pi$
$13$	8.32577	−	8.32577i	0.640443	−	0.640443i	0.950664	+	0.310221i	$0.100403\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.34847	+	1.34847i	0.0793217	+	0.0793217i	0.745654	−	0.666333i	$-0.232137\pi$
$17$	1.34847	+	1.34847i	0.0793217	+	0.0793217i	−0.666333	+	0.745654i	$0.732137\pi$
$18$		0			0
$19$			33.6969i			1.77352i	0.462227	+	0.886762i	$0.347050\pi$
$19$			33.6969i			1.77352i	−0.462227	+	0.886762i	$0.652950\pi$
$20$		0			0
$21$	11.6969			0.556997
$22$		0			0
$23$	−16.0454	+	16.0454i	−0.697626	+	0.697626i	−0.963898	−	0.266272i	$-0.914208\pi$
$23$	−16.0454	+	16.0454i	−0.697626	+	0.697626i	0.266272	+	0.963898i	$0.414208\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−3.67423	−	3.67423i	−0.136083	−	0.136083i
$28$		0			0
$29$			8.69694i			0.299894i	0.988694	+	0.149947i	$0.0479104\pi$
$29$			8.69694i			0.299894i	−0.988694	+	0.149947i	$0.952090\pi$
$30$		0			0
$31$	49.0908			1.58357			0.791787	−	0.610797i	$-0.209151\pi$
$31$	49.0908			1.58357			0.791787	+	0.610797i	$0.209151\pi$
$32$		0			0
$33$	−25.3485	+	25.3485i	−0.768135	+	0.768135i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	36.4949	+	36.4949i	0.986349	+	0.986349i	0.999908	−	0.0135595i	$-0.00431625\pi$
$37$	36.4949	+	36.4949i	0.986349	+	0.986349i	−0.0135595	+	0.999908i	$0.504316\pi$
$38$		0			0
$39$		−	20.3939i		−	0.522920i
$40$		0			0
$41$	33.3031			0.812270			0.406135	−	0.913813i	$-0.366876\pi$
$41$	33.3031			0.812270			0.406135	+	0.913813i	$0.366876\pi$
$42$		0			0
$43$	−24.1237	+	24.1237i	−0.561017	+	0.561017i	−0.929596	−	0.368579i	$-0.879844\pi$
$43$	−24.1237	+	24.1237i	−0.561017	+	0.561017i	0.368579	+	0.929596i	$0.379844\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−13.9546	−	13.9546i	−0.296906	−	0.296906i	0.542895	−	0.839801i	$-0.317328\pi$
$47$	−13.9546	−	13.9546i	−0.296906	−	0.296906i	−0.839801	+	0.542895i	$0.817328\pi$
$48$		0			0
$49$		−	3.39388i		−	0.0692628i
$50$		0			0
$51$	3.30306			0.0647659
$52$		0			0
$53$	−59.3939	+	59.3939i	−1.12064	+	1.12064i	−0.128994	+	0.991645i	$0.541175\pi$
$53$	−59.3939	+	59.3939i	−1.12064	+	1.12064i	−0.991645	+	0.128994i	$0.958825\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	41.2702	+	41.2702i	0.724038	+	0.724038i
$58$		0			0
$59$			30.0000i			0.508475i	0.967142	+	0.254237i	$0.0818244\pi$
$59$			30.0000i			0.508475i	−0.967142	+	0.254237i	$0.918176\pi$
$60$		0			0
$61$	−47.7878			−0.783406			−0.391703	−	0.920092i	$-0.628114\pi$
$61$	−47.7878			−0.783406			−0.391703	+	0.920092i	$0.628114\pi$
$62$		0			0
$63$	14.3258	−	14.3258i	0.227393	−	0.227393i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	17.0227	+	17.0227i	0.254070	+	0.254070i	0.822637	−	0.568567i	$-0.192502\pi$
$67$	17.0227	+	17.0227i	0.254070	+	0.254070i	−0.568567	+	0.822637i	$0.692502\pi$
$68$		0			0
$69$			39.3031i			0.569610i
$70$		0			0
$71$	9.30306			0.131029			0.0655145	−	0.997852i	$-0.479131\pi$
$71$	9.30306			0.131029			0.0655145	+	0.997852i	$0.479131\pi$
$72$		0			0
$73$	−22.2929	+	22.2929i	−0.305382	+	0.305382i	−0.843115	−	0.537733i	$-0.819281\pi$
$73$	−22.2929	+	22.2929i	−0.305382	+	0.305382i	0.537733	+	0.843115i	$0.319281\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−98.8332	−	98.8332i	−1.28355	−	1.28355i
$78$		0			0
$79$		−	27.3939i		−	0.346758i	−0.984855	−	0.173379i	$-0.944531\pi$
$79$		−	27.3939i		−	0.346758i	0.984855	−	0.173379i	$-0.0554685\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$		0			0
$83$	−63.4393	+	63.4393i	−0.764329	+	0.764329i	−0.977102	−	0.212773i	$-0.931751\pi$
$83$	−63.4393	+	63.4393i	−0.764329	+	0.764329i	0.212773	+	0.977102i	$0.431751\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	10.6515	+	10.6515i	0.122431	+	0.122431i
$88$		0			0
$89$			17.3939i			0.195437i	0.995214	+	0.0977184i	$0.0311545\pi$
$89$			17.3939i			0.195437i	−0.995214	+	0.0977184i	$0.968846\pi$
$90$		0			0
$91$	79.5153			0.873795
$92$		0			0
$93$	60.1237	−	60.1237i	0.646492	−	0.646492i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−41.8763	−	41.8763i	−0.431714	−	0.431714i	0.457497	−	0.889211i	$-0.348746\pi$
$97$	−41.8763	−	41.8763i	−0.431714	−	0.431714i	−0.889211	+	0.457497i	$0.848746\pi$
$98$		0			0
$99$			62.0908i			0.627180i
$100$		0			0
$101$	162.879			1.61266			0.806330	−	0.591467i	$-0.201451\pi$
$101$	162.879			1.61266			0.806330	+	0.591467i	$0.201451\pi$
$102$		0			0
$103$	−36.9898	+	36.9898i	−0.359124	+	0.359124i	−0.863490	−	0.504366i	$-0.831726\pi$
$103$	−36.9898	+	36.9898i	−0.359124	+	0.359124i	0.504366	+	0.863490i	$0.331726\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	134.091	+	134.091i	1.25319	+	1.25319i	0.954284	+	0.298901i	$0.0966201\pi$
$107$	134.091	+	134.091i	1.25319	+	1.25319i	0.298901	+	0.954284i	$0.403380\pi$
$108$		0			0
$109$			145.000i			1.33028i	0.746721	+	0.665138i	$0.231627\pi$
$109$			145.000i			1.33028i	−0.746721	+	0.665138i	$0.768373\pi$
$110$		0			0
$111$	89.3939			0.805350
$112$		0			0
$113$	13.2122	−	13.2122i	0.116923	−	0.116923i	−0.646225	−	0.763147i	$-0.723653\pi$
$113$	13.2122	−	13.2122i	0.116923	−	0.116923i	0.763147	+	0.646225i	$0.223653\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−24.9773	−	24.9773i	−0.213481	−	0.213481i
$118$		0			0
$119$			12.8786i			0.108223i
$120$		0			0
$121$	307.363			2.54019
$122$		0			0
$123$	40.7878	−	40.7878i	0.331608	−	0.331608i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−50.6969	−	50.6969i	−0.399188	−	0.399188i	0.478758	−	0.877947i	$-0.341087\pi$
$127$	−50.6969	−	50.6969i	−0.399188	−	0.399188i	−0.877947	+	0.478758i	$0.841087\pi$
$128$		0			0
$129$			59.0908i			0.458068i
$130$		0			0
$131$	156.272			1.19292			0.596460	−	0.802643i	$-0.296574\pi$
$131$	156.272			1.19292			0.596460	+	0.802643i	$0.296574\pi$
$132$		0			0
$133$	−160.911	+	160.911i	−1.20986	+	1.20986i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	91.3485	+	91.3485i	0.666777	+	0.666777i	0.956969	−	0.290191i	$-0.0937190\pi$
$137$	91.3485	+	91.3485i	0.666777	+	0.666777i	−0.290191	+	0.956969i	$0.593719\pi$
$138$		0			0
$139$			35.2122i			0.253326i	0.991946	+	0.126663i	$0.0404266\pi$
$139$			35.2122i			0.253326i	−0.991946	+	0.126663i	$0.959573\pi$
$140$		0			0
$141$	−34.1816			−0.242423
$142$		0			0
$143$	−172.318	+	172.318i	−1.20502	+	1.20502i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−4.15663	−	4.15663i	−0.0282764	−	0.0282764i
$148$		0			0
$149$		−	4.78775i		−	0.0321326i	−0.999871	−	0.0160663i	$-0.994886\pi$
$149$		−	4.78775i		−	0.0321326i	0.999871	−	0.0160663i	$-0.00511428\pi$
$150$		0			0
$151$	−37.8786			−0.250851			−0.125426	−	0.992103i	$-0.540030\pi$
$151$	−37.8786			−0.250851			−0.125426	+	0.992103i	$0.540030\pi$
$152$		0			0
$153$	4.04541	−	4.04541i	0.0264406	−	0.0264406i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	29.1339	+	29.1339i	0.185566	+	0.185566i	0.793776	−	0.608210i	$-0.208112\pi$
$157$	29.1339	+	29.1339i	0.185566	+	0.185566i	−0.608210	+	0.793776i	$0.708112\pi$
$158$		0			0
$159$			145.485i			0.914998i
$160$		0			0
$161$	−153.242			−0.951813
$162$		0			0
$163$	193.856	−	193.856i	1.18930	−	1.18930i	0.212038	−	0.977261i	$-0.431990\pi$
$163$	193.856	−	193.856i	1.18930	−	1.18930i	0.977261	−	0.212038i	$-0.0680102\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−167.666	−	167.666i	−1.00399	−	1.00399i	−0.999992	−	0.00399795i	$-0.998727\pi$
$167$	−167.666	−	167.666i	−1.00399	−	1.00399i	−0.00399795	−	0.999992i	$-0.501273\pi$
$168$		0			0
$169$			30.3633i			0.179664i
$170$		0			0
$171$	101.091			0.591174
$172$		0			0
$173$	−74.8332	+	74.8332i	−0.432562	+	0.432562i	−0.889499	−	0.456937i	$-0.848946\pi$
$173$	−74.8332	+	74.8332i	−0.432562	+	0.432562i	0.456937	+	0.889499i	$0.348946\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	36.7423	+	36.7423i	0.207584	+	0.207584i
$178$		0			0
$179$		−	229.151i		−	1.28017i	−0.768303	−	0.640087i	$-0.778898\pi$
$179$		−	229.151i		−	1.28017i	0.768303	−	0.640087i	$-0.221102\pi$
$180$		0			0
$181$	−135.182			−0.746860			−0.373430	−	0.927658i	$-0.621818\pi$
$181$	−135.182			−0.746860			−0.373430	+	0.927658i	$0.621818\pi$
$182$		0			0
$183$	−58.5278	+	58.5278i	−0.319824	+	0.319824i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−27.9092	−	27.9092i	−0.149247	−	0.149247i
$188$		0			0
$189$		−	35.0908i		−	0.185666i
$190$		0			0
$191$	21.9092			0.114708			0.0573539	−	0.998354i	$-0.481734\pi$
$191$	21.9092			0.114708			0.0573539	+	0.998354i	$0.481734\pi$
$192$		0			0
$193$	98.4870	−	98.4870i	0.510295	−	0.510295i	−0.404322	−	0.914617i	$-0.632492\pi$
$193$	98.4870	−	98.4870i	0.510295	−	0.510295i	0.914617	+	0.404322i	$0.132492\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−63.4393	−	63.4393i	−0.322027	−	0.322027i	0.527517	−	0.849544i	$-0.323123\pi$
$197$	−63.4393	−	63.4393i	−0.322027	−	0.322027i	−0.849544	+	0.527517i	$0.823123\pi$
$198$		0			0
$199$			181.091i			0.910004i	0.890490	+	0.455002i	$0.150361\pi$
$199$			181.091i			0.910004i	−0.890490	+	0.455002i	$0.849639\pi$
$200$		0			0
$201$	41.6969			0.207447
$202$		0			0
$203$	−41.5301	+	41.5301i	−0.204582	+	0.204582i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	48.1362	+	48.1362i	0.232542	+	0.232542i
$208$		0			0
$209$		−	697.423i		−	3.33695i
$210$		0			0
$211$	−28.7276			−0.136150			−0.0680748	−	0.997680i	$-0.521686\pi$
$211$	−28.7276			−0.136150			−0.0680748	+	0.997680i	$0.521686\pi$
$212$		0			0
$213$	11.3939	−	11.3939i	0.0534924	−	0.0534924i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	234.421	+	234.421i	1.08028	+	1.08028i
$218$		0			0
$219$			54.6061i			0.249343i
$220$		0			0
$221$	22.4541			0.101602
$222$		0			0
$223$	−33.1793	+	33.1793i	−0.148786	+	0.148786i	−0.777576	−	0.628789i	$-0.783551\pi$
$223$	−33.1793	+	33.1793i	−0.148786	+	0.148786i	0.628789	+	0.777576i	$0.283551\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−252.879	−	252.879i	−1.11400	−	1.11400i	−0.992604	−	0.121399i	$-0.961262\pi$
$227$	−252.879	−	252.879i	−1.11400	−	1.11400i	−0.121399	−	0.992604i	$-0.538738\pi$
$228$		0			0
$229$			275.000i			1.20087i	0.799672	+	0.600437i	$0.205007\pi$
$229$			275.000i			1.20087i	−0.799672	+	0.600437i	$0.794993\pi$
$230$		0			0
$231$	−242.091			−1.04801
$232$		0			0
$233$	−223.757	+	223.757i	−0.960331	+	0.960331i	−0.999243	−	0.0389116i	$-0.987611\pi$
$233$	−223.757	+	223.757i	−0.960331	+	0.960331i	0.0389116	+	0.999243i	$0.487611\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−33.5505	−	33.5505i	−0.141563	−	0.141563i
$238$		0			0
$239$		−	305.666i		−	1.27894i	−0.768817	−	0.639469i	$-0.779154\pi$
$239$		−	305.666i		−	1.27894i	0.768817	−	0.639469i	$-0.220846\pi$
$240$		0			0
$241$	167.000			0.692946			0.346473	−	0.938060i	$-0.387379\pi$
$241$	167.000			0.692946			0.346473	+	0.938060i	$0.387379\pi$
$242$		0			0
$243$	−11.0227	+	11.0227i	−0.0453609	+	0.0453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	280.553	+	280.553i	1.13584	+	1.13584i
$248$		0			0
$249$			155.394i			0.624072i
$250$		0			0
$251$	−261.576			−1.04213			−0.521067	−	0.853516i	$-0.674466\pi$
$251$	−261.576			−1.04213			−0.521067	+	0.853516i	$0.674466\pi$
$252$		0			0
$253$	332.091	−	332.091i	1.31261	−	1.31261i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−94.5153	−	94.5153i	−0.367764	−	0.367764i	0.498897	−	0.866661i	$-0.333738\pi$
$257$	−94.5153	−	94.5153i	−0.367764	−	0.367764i	−0.866661	+	0.498897i	$0.833738\pi$
$258$		0			0
$259$			348.545i			1.34573i
$260$		0			0
$261$	26.0908			0.0999648
$262$		0			0
$263$	34.1816	−	34.1816i	0.129968	−	0.129968i	−0.639130	−	0.769098i	$-0.720706\pi$
$263$	34.1816	−	34.1816i	0.129968	−	0.129968i	0.769098	+	0.639130i	$0.220706\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	21.3031	+	21.3031i	0.0797867	+	0.0797867i
$268$		0			0
$269$			389.605i			1.44835i	0.689618	+	0.724173i	$0.257778\pi$
$269$			389.605i			1.44835i	−0.689618	+	0.724173i	$0.742222\pi$
$270$		0			0
$271$	−156.788			−0.578553			−0.289276	−	0.957246i	$-0.593415\pi$
$271$	−156.788			−0.578553			−0.289276	+	0.957246i	$0.593415\pi$
$272$		0			0
$273$	97.3860	−	97.3860i	0.356725	−	0.356725i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−285.588	−	285.588i	−1.03100	−	1.03100i	−0.999504	−	0.0314999i	$-0.989972\pi$
$277$	−285.588	−	285.588i	−1.03100	−	1.03100i	−0.0314999	−	0.999504i	$-0.510028\pi$
$278$		0			0
$279$		−	147.272i		−	0.527858i
$280$		0			0
$281$	−20.1520			−0.0717155			−0.0358577	−	0.999357i	$-0.511416\pi$
$281$	−20.1520			−0.0717155			−0.0358577	+	0.999357i	$0.511416\pi$
$282$		0			0
$283$	−170.573	+	170.573i	−0.602732	+	0.602732i	−0.941037	−	0.338304i	$-0.890147\pi$
$283$	−170.573	+	170.573i	−0.602732	+	0.602732i	0.338304	+	0.941037i	$0.390147\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	159.031	+	159.031i	0.554114	+	0.554114i
$288$		0			0
$289$		−	285.363i		−	0.987416i
$290$		0			0
$291$	−102.576			−0.352493
$292$		0			0
$293$	−259.621	+	259.621i	−0.886078	+	0.886078i	−0.994144	−	0.108066i	$-0.965534\pi$
$293$	−259.621	+	259.621i	−0.886078	+	0.886078i	0.108066	+	0.994144i	$0.465534\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	76.0454	+	76.0454i	0.256045	+	0.256045i
$298$		0			0
$299$			267.181i			0.893581i
$300$		0			0
$301$	−230.394			−0.765428
$302$		0			0
$303$	199.485	−	199.485i	0.658365	−	0.658365i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	118.911	+	118.911i	0.387334	+	0.387334i	0.873735	−	0.486402i	$-0.161691\pi$
$307$	118.911	+	118.911i	0.387334	+	0.387334i	−0.486402	+	0.873735i	$0.661691\pi$
$308$		0			0
$309$			90.6061i			0.293224i
$310$		0			0
$311$	52.7878			0.169736			0.0848678	−	0.996392i	$-0.472953\pi$
$311$	52.7878			0.169736			0.0848678	+	0.996392i	$0.472953\pi$
$312$		0			0
$313$	238.553	−	238.553i	0.762150	−	0.762150i	−0.214561	−	0.976711i	$-0.568832\pi$
$313$	238.553	−	238.553i	0.762150	−	0.762150i	0.976711	+	0.214561i	$0.0688321\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	48.9398	+	48.9398i	0.154384	+	0.154384i	0.780073	−	0.625689i	$-0.215182\pi$
$317$	48.9398	+	48.9398i	0.154384	+	0.154384i	−0.625689	+	0.780073i	$0.715182\pi$
$318$		0			0
$319$		−	180.000i		−	0.564263i
$320$		0			0
$321$	328.454			1.02322
$322$		0			0
$323$	−45.4393	+	45.4393i	−0.140679	+	0.140679i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	177.588	+	177.588i	0.543083	+	0.543083i
$328$		0			0
$329$		−	133.273i		−	0.405087i
$330$		0			0
$331$	291.939			0.881990			0.440995	−	0.897510i	$-0.354626\pi$
$331$	291.939			0.881990			0.440995	+	0.897510i	$0.354626\pi$
$332$		0			0
$333$	109.485	−	109.485i	0.328783	−	0.328783i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	266.462	+	266.462i	0.790688	+	0.790688i	0.981606	−	0.190918i	$-0.0611463\pi$
$337$	266.462	+	266.462i	0.790688	+	0.790688i	−0.190918	+	0.981606i	$0.561146\pi$
$338$		0			0
$339$		−	32.3633i		−	0.0954668i
$340$		0			0
$341$	−1016.03			−2.97956
$342$		0			0
$343$	250.194	−	250.194i	0.729429	−	0.729429i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−141.773	−	141.773i	−0.408568	−	0.408568i	0.472671	−	0.881239i	$-0.343290\pi$
$347$	−141.773	−	141.773i	−0.408568	−	0.408568i	−0.881239	+	0.472671i	$0.843290\pi$
$348$		0			0
$349$		−	183.939i		−	0.527045i	−0.964653	−	0.263523i	$-0.915116\pi$
$349$		−	183.939i		−	0.527045i	0.964653	−	0.263523i	$-0.0848844\pi$
$350$		0			0
$351$	−61.1816			−0.174307
$352$		0			0
$353$	54.7423	−	54.7423i	0.155077	−	0.155077i	−0.625304	−	0.780381i	$-0.715025\pi$
$353$	54.7423	−	54.7423i	0.155077	−	0.155077i	0.780381	+	0.625304i	$0.215025\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	15.7730	+	15.7730i	0.0441820	+	0.0441820i
$358$		0			0
$359$		−	648.272i		−	1.80577i	−0.429879	−	0.902886i	$-0.641444\pi$
$359$		−	648.272i		−	1.80577i	0.429879	−	0.902886i	$-0.358556\pi$
$360$		0			0
$361$	−774.484			−2.14538
$362$		0			0
$363$	376.442	−	376.442i	1.03703	−	1.03703i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−56.6594	−	56.6594i	−0.154385	−	0.154385i	0.625688	−	0.780073i	$-0.284818\pi$
$367$	−56.6594	−	56.6594i	−0.154385	−	0.154385i	−0.780073	+	0.625688i	$0.784818\pi$
$368$		0			0
$369$		−	99.9092i		−	0.270757i
$370$		0			0
$371$	−567.242			−1.52895
$372$		0			0
$373$	155.815	−	155.815i	0.417735	−	0.417735i	−0.466688	−	0.884422i	$-0.654553\pi$
$373$	155.815	−	155.815i	0.417735	−	0.417735i	0.884422	+	0.466688i	$0.154553\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	72.4087	+	72.4087i	0.192065	+	0.192065i
$378$		0			0
$379$		−	674.181i		−	1.77884i	−0.457090	−	0.889420i	$-0.651108\pi$
$379$		−	674.181i		−	1.77884i	0.457090	−	0.889420i	$-0.348892\pi$
$380$		0			0
$381$	−124.182			−0.325936
$382$		0			0
$383$	244.182	−	244.182i	0.637550	−	0.637550i	−0.312401	−	0.949950i	$-0.601133\pi$
$383$	244.182	−	244.182i	0.637550	−	0.637550i	0.949950	+	0.312401i	$0.101133\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	72.3712	+	72.3712i	0.187006	+	0.187006i
$388$		0			0
$389$			44.7582i			0.115060i	0.998344	+	0.0575298i	$0.0183224\pi$
$389$			44.7582i			0.115060i	−0.998344	+	0.0575298i	$0.981678\pi$
$390$		0			0
$391$	−43.2735			−0.110674
$392$		0			0
$393$	191.394	−	191.394i	0.487007	−	0.487007i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−548.648	−	548.648i	−1.38199	−	1.38199i	−0.841090	−	0.540896i	$-0.818085\pi$
$397$	−548.648	−	548.648i	−1.38199	−	1.38199i	−0.540896	−	0.841090i	$-0.681915\pi$
$398$		0			0
$399$			394.151i			0.987847i
$400$		0			0
$401$	273.303			0.681554			0.340777	−	0.940144i	$-0.389310\pi$
$401$	273.303			0.681554			0.340777	+	0.940144i	$0.389310\pi$
$402$		0			0
$403$	408.719	−	408.719i	1.01419	−	1.01419i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−755.333	−	755.333i	−1.85585	−	1.85585i
$408$		0			0
$409$		−	197.182i		−	0.482107i	−0.970512	−	0.241053i	$-0.922507\pi$
$409$		−	197.182i		−	0.482107i	0.970512	−	0.241053i	$-0.0774929\pi$
$410$		0			0
$411$	223.757			0.544421
$412$		0			0
$413$	−143.258	+	143.258i	−0.346871	+	0.346871i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	43.1260	+	43.1260i	0.103420	+	0.103420i
$418$		0			0
$419$		−	613.485i		−	1.46416i	−0.681217	−	0.732082i	$-0.738549\pi$
$419$		−	613.485i		−	1.46416i	0.681217	−	0.732082i	$-0.261451\pi$
$420$		0			0
$421$	128.120			0.304324			0.152162	−	0.988356i	$-0.451376\pi$
$421$	128.120			0.304324			0.152162	+	0.988356i	$0.451376\pi$
$422$		0			0
$423$	−41.8638	+	41.8638i	−0.0989687	+	0.0989687i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−228.199	−	228.199i	−0.534423	−	0.534423i
$428$		0			0
$429$			422.091i			0.983895i
$430$		0			0
$431$	393.242			0.912394			0.456197	−	0.889879i	$-0.349211\pi$
$431$	393.242			0.912394			0.456197	+	0.889879i	$0.349211\pi$
$432$		0			0
$433$	−221.381	+	221.381i	−0.511273	+	0.511273i	−0.914917	−	0.403643i	$-0.867744\pi$
$433$	−221.381	+	221.381i	−0.511273	+	0.511273i	0.403643	+	0.914917i	$0.367744\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−540.681	−	540.681i	−1.23726	−	1.23726i
$438$		0			0
$439$			361.091i			0.822530i	0.911516	+	0.411265i	$0.134913\pi$
$439$			361.091i			0.822530i	−0.911516	+	0.411265i	$0.865087\pi$
$440$		0			0
$441$	−10.1816			−0.0230876
$442$		0			0
$443$	−440.636	+	440.636i	−0.994663	+	0.994663i	−0.999986	−	0.00532284i	$-0.998306\pi$
$443$	−440.636	+	440.636i	−0.994663	+	0.994663i	0.00532284	+	0.999986i	$0.498306\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−5.86378	−	5.86378i	−0.0131181	−	0.0131181i
$448$		0			0
$449$		−	280.849i		−	0.625499i	−0.949836	−	0.312749i	$-0.898750\pi$
$449$		−	280.849i		−	0.625499i	0.949836	−	0.312749i	$-0.101250\pi$
$450$		0			0
$451$	−689.271			−1.52832
$452$		0			0
$453$	−46.3916	+	46.3916i	−0.102410	+	0.102410i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	261.576	+	261.576i	0.572375	+	0.572375i	0.932792	−	0.360416i	$-0.117365\pi$
$457$	261.576	+	261.576i	0.572375	+	0.572375i	−0.360416	+	0.932792i	$0.617365\pi$
$458$		0			0
$459$		−	9.90918i		−	0.0215886i
$460$		0			0
$461$	−9.30306			−0.0201802			−0.0100901	−	0.999949i	$-0.503212\pi$
$461$	−9.30306			−0.0201802			−0.0100901	+	0.999949i	$0.503212\pi$
$462$		0			0
$463$	−252.495	+	252.495i	−0.545345	+	0.545345i	−0.925091	−	0.379746i	$-0.876012\pi$
$463$	−252.495	+	252.495i	−0.545345	+	0.545345i	0.379746	+	0.925091i	$0.376012\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	366.470	+	366.470i	0.784732	+	0.784732i	0.980625	−	0.195893i	$-0.0627606\pi$
$467$	366.470	+	366.470i	0.784732	+	0.784732i	−0.195893	+	0.980625i	$0.562761\pi$
$468$		0			0
$469$			162.576i			0.346643i
$470$		0			0
$471$	71.3633			0.151514
$472$		0			0
$473$	499.287	−	499.287i	1.05558	−	1.05558i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	178.182	+	178.182i	0.373546	+	0.373546i
$478$		0			0
$479$			442.182i			0.923135i	0.887105	+	0.461567i	$0.152713\pi$
$479$			442.182i			0.923135i	−0.887105	+	0.461567i	$0.847287\pi$
$480$		0			0
$481$	607.696			1.26340
$482$		0			0
$483$	−187.682	+	187.682i	−0.388576	+	0.388576i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	100.083	+	100.083i	0.205509	+	0.205509i	0.802355	−	0.596846i	$-0.203580\pi$
$487$	100.083	+	100.083i	0.205509	+	0.205509i	−0.596846	+	0.802355i	$0.703580\pi$
$488$		0			0
$489$		−	474.848i		−	0.971059i
$490$		0			0
$491$	−286.788			−0.584089			−0.292045	−	0.956405i	$-0.594336\pi$
$491$	−286.788			−0.584089			−0.292045	+	0.956405i	$0.594336\pi$
$492$		0			0
$493$	−11.7276	+	11.7276i	−0.0237881	+	0.0237881i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	44.4245	+	44.4245i	0.0893853	+	0.0893853i
$498$		0			0
$499$		−	82.3643i		−	0.165059i	−0.996589	−	0.0825294i	$-0.973700\pi$
$499$		−	82.3643i		−	0.165059i	0.996589	−	0.0825294i	$-0.0262998\pi$
$500$		0			0
$501$	−410.697			−0.819754
$502$		0			0
$503$	219.848	−	219.848i	0.437073	−	0.437073i	−0.453952	−	0.891026i	$-0.649986\pi$
$503$	219.848	−	219.848i	0.437073	−	0.437073i	0.891026	+	0.453952i	$0.149986\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	37.1872	+	37.1872i	0.0733476	+	0.0733476i
$508$		0			0
$509$		−	805.696i		−	1.58290i	−0.611234	−	0.791450i	$-0.709327\pi$
$509$		−	805.696i		−	1.58290i	0.611234	−	0.791450i	$-0.290673\pi$
$510$		0			0
$511$	−212.908			−0.416650
$512$		0			0
$513$	123.810	−	123.810i	0.241346	−	0.241346i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	288.817	+	288.817i	0.558641	+	0.558641i
$518$		0			0
$519$			183.303i			0.353185i
$520$		0			0
$521$	372.879			0.715698			0.357849	−	0.933779i	$-0.383510\pi$
$521$	372.879			0.715698			0.357849	+	0.933779i	$0.383510\pi$
$522$		0			0
$523$	56.1395	−	56.1395i	0.107341	−	0.107341i	−0.651396	−	0.758738i	$-0.725816\pi$
$523$	56.1395	−	56.1395i	0.107341	−	0.107341i	0.758738	+	0.651396i	$0.225816\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	66.1975	+	66.1975i	0.125612	+	0.125612i
$528$		0			0
$529$			14.0898i			0.0266348i
$530$		0			0
$531$	90.0000			0.169492
$532$		0			0
$533$	277.273	−	277.273i	0.520213	−	0.520213i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−280.652	−	280.652i	−0.522629	−	0.522629i
$538$		0			0
$539$			70.2429i			0.130321i
$540$		0			0
$541$	633.120			1.17028			0.585139	−	0.810933i	$-0.301040\pi$
$541$	633.120			1.17028			0.585139	+	0.810933i	$0.301040\pi$
$542$		0			0
$543$	−165.563	+	165.563i	−0.304904	+	0.304904i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	32.0408	+	32.0408i	0.0585755	+	0.0585755i	0.735788	−	0.677212i	$-0.236812\pi$
$547$	32.0408	+	32.0408i	0.0585755	+	0.0585755i	−0.677212	+	0.735788i	$0.736812\pi$
$548$		0			0
$549$			143.363i			0.261135i
$550$		0			0
$551$	−293.060			−0.531870
$552$		0			0
$553$	130.813	−	130.813i	0.236551	−	0.236551i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−701.741	−	701.741i	−1.25986	−	1.25986i	−0.951160	−	0.308699i	$-0.900106\pi$
$557$	−701.741	−	701.741i	−1.25986	−	1.25986i	−0.308699	−	0.951160i	$-0.599894\pi$
$558$		0			0
$559$			401.697i			0.718599i
$560$		0			0
$561$	−68.3633			−0.121860
$562$		0			0
$563$	−265.621	+	265.621i	−0.471796	+	0.471796i	−0.902495	−	0.430700i	$-0.858267\pi$
$563$	−265.621	+	265.621i	−0.471796	+	0.471796i	0.430700	+	0.902495i	$0.358267\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−42.9773	−	42.9773i	−0.0757977	−	0.0757977i
$568$		0			0
$569$		−	946.999i		−	1.66432i	−0.554534	−	0.832161i	$-0.687104\pi$
$569$		−	946.999i		−	1.66432i	0.554534	−	0.832161i	$-0.312896\pi$
$570$		0			0
$571$	337.334			0.590777			0.295389	−	0.955377i	$-0.404551\pi$
$571$	337.334			0.590777			0.295389	+	0.955377i	$0.404551\pi$
$572$		0			0
$573$	26.8332	−	26.8332i	0.0468293	−	0.0468293i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	325.547	+	325.547i	0.564207	+	0.564207i	0.930500	−	0.366293i	$-0.119373\pi$
$577$	325.547	+	325.547i	0.564207	+	0.564207i	−0.366293	+	0.930500i	$0.619373\pi$
$578$		0			0
$579$		−	241.243i		−	0.416654i
$580$		0			0
$581$	−605.878			−1.04282
$582$		0			0
$583$	1229.27	−	1229.27i	2.10853	−	2.10853i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	381.514	+	381.514i	0.649939	+	0.649939i	0.952978	−	0.303039i	$-0.0980012\pi$
$587$	381.514	+	381.514i	0.649939	+	0.649939i	−0.303039	+	0.952978i	$0.598001\pi$
$588$		0			0
$589$			1654.21i			2.80851i
$590$		0			0
$591$	−155.394			−0.262934
$592$		0			0
$593$	−200.697	+	200.697i	−0.338443	+	0.338443i	−0.855781	−	0.517338i	$-0.826923\pi$
$593$	−200.697	+	200.697i	−0.338443	+	0.338443i	0.517338	+	0.855781i	$0.326923\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	221.790	+	221.790i	0.371508	+	0.371508i
$598$		0			0
$599$			112.182i			0.187282i	0.995606	+	0.0936408i	$0.0298505\pi$
$599$			112.182i			0.187282i	−0.995606	+	0.0936408i	$0.970149\pi$
$600$		0			0
$601$	−59.5449			−0.0990764			−0.0495382	−	0.998772i	$-0.515775\pi$
$601$	−59.5449			−0.0990764			−0.0495382	+	0.998772i	$0.515775\pi$
$602$		0			0
$603$	51.0681	−	51.0681i	0.0846901	−	0.0846901i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−406.070	−	406.070i	−0.668979	−	0.668979i	0.288500	−	0.957480i	$-0.406843\pi$
$607$	−406.070	−	406.070i	−0.668979	−	0.668979i	−0.957480	+	0.288500i	$0.906843\pi$
$608$		0			0
$609$			101.728i			0.167040i
$610$		0			0
$611$	−232.365			−0.380303
$612$		0			0
$613$	81.7480	−	81.7480i	0.133357	−	0.133357i	−0.637277	−	0.770635i	$-0.719939\pi$
$613$	81.7480	−	81.7480i	0.133357	−	0.133357i	0.770635	+	0.637277i	$0.219939\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	734.636	+	734.636i	1.19066	+	1.19066i	0.976883	+	0.213775i	$0.0685758\pi$
$617$	734.636	+	734.636i	1.19066	+	1.19066i	0.213775	+	0.976883i	$0.431424\pi$
$618$		0			0
$619$			266.303i			0.430215i	0.976590	+	0.215107i	$0.0690102\pi$
$619$			266.303i			0.430215i	−0.976590	+	0.215107i	$0.930990\pi$
$620$		0			0
$621$	117.909			0.189870
$622$		0			0
$623$	−83.0602	+	83.0602i	−0.133323	+	0.133323i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−854.166	−	854.166i	−1.36231	−	1.36231i
$628$		0			0
$629$			98.4245i			0.156478i
$630$		0			0
$631$	−456.546			−0.723528			−0.361764	−	0.932270i	$-0.617825\pi$
$631$	−456.546			−0.723528			−0.361764	+	0.932270i	$0.617825\pi$
$632$		0			0
$633$	−35.1839	+	35.1839i	−0.0555828	+	0.0555828i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−28.2566	−	28.2566i	−0.0443589	−	0.0443589i
$638$		0			0
$639$		−	27.9092i		−	0.0436763i
$640$		0			0
$641$	538.120			0.839501			0.419751	−	0.907639i	$-0.362118\pi$
$641$	538.120			0.839501			0.419751	+	0.907639i	$0.362118\pi$
$642$		0			0
$643$	31.5459	−	31.5459i	0.0490605	−	0.0490605i	−0.682151	−	0.731211i	$-0.738955\pi$
$643$	31.5459	−	31.5459i	0.0490605	−	0.0490605i	0.731211	+	0.682151i	$0.238955\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−112.849	−	112.849i	−0.174419	−	0.174419i	0.614499	−	0.788918i	$-0.289358\pi$
$647$	−112.849	−	112.849i	−0.174419	−	0.174419i	−0.788918	+	0.614499i	$0.789358\pi$
$648$		0			0
$649$		−	620.908i		−	0.956715i
$650$		0			0
$651$	574.212			0.882046
$652$		0			0
$653$	−491.803	+	491.803i	−0.753143	+	0.753143i	−0.975065	−	0.221921i	$-0.928767\pi$
$653$	−491.803	+	491.803i	−0.753143	+	0.753143i	0.221921	+	0.975065i	$0.428767\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	66.8786	+	66.8786i	0.101794	+	0.101794i
$658$		0			0
$659$		−	471.787i		−	0.715913i	−0.933738	−	0.357957i	$-0.883474\pi$
$659$		−	471.787i		−	0.715913i	0.933738	−	0.357957i	$-0.116526\pi$
$660$		0			0
$661$	−897.151			−1.35726			−0.678632	−	0.734479i	$-0.737427\pi$
$661$	−897.151			−1.35726			−0.678632	+	0.734479i	$0.737427\pi$
$662$		0			0
$663$	27.5005	−	27.5005i	0.0414789	−	0.0414789i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−139.546	−	139.546i	−0.209214	−	0.209214i
$668$		0			0
$669$			81.2724i			0.121483i
$670$		0			0
$671$	989.060			1.47401
$672$		0			0
$673$	105.526	−	105.526i	0.156799	−	0.156799i	−0.624348	−	0.781146i	$-0.714635\pi$
$673$	105.526	−	105.526i	0.156799	−	0.156799i	0.781146	+	0.624348i	$0.214635\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	547.015	+	547.015i	0.807998	+	0.807998i	0.984331	−	0.176332i	$-0.0564234\pi$
$677$	547.015	+	547.015i	0.807998	+	0.807998i	−0.176332	+	0.984331i	$0.556423\pi$
$678$		0			0
$679$		−	399.940i		−	0.589013i
$680$		0			0
$681$	−619.423			−0.909579
$682$		0			0
$683$	−533.271	+	533.271i	−0.780778	+	0.780778i	−0.979962	−	0.199184i	$-0.936171\pi$
$683$	−533.271	+	533.271i	−0.780778	+	0.780778i	0.199184	+	0.979962i	$0.436171\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	336.805	+	336.805i	0.490254	+	0.490254i
$688$		0			0
$689$			988.999i			1.43541i
$690$		0			0
$691$	1191.09			1.72372			0.861859	−	0.507147i	$-0.169300\pi$
$691$	1191.09			1.72372			0.861859	+	0.507147i	$0.169300\pi$
$692$		0			0
$693$	−296.499	+	296.499i	−0.427849	+	0.427849i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	44.9082	+	44.9082i	0.0644306	+	0.0644306i
$698$		0			0
$699$			548.091i			0.784107i
$700$		0			0
$701$	717.242			1.02317			0.511585	−	0.859233i	$-0.329059\pi$
$701$	717.242			1.02317			0.511585	+	0.859233i	$0.329059\pi$
$702$		0			0
$703$	−1229.77	+	1229.77i	−1.74931	+	1.74931i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	777.787	+	777.787i	1.10012	+	1.10012i
$708$		0			0
$709$		−	148.514i		−	0.209470i	−0.994500	−	0.104735i	$-0.966601\pi$
$709$		−	148.514i		−	0.209470i	0.994500	−	0.104735i	$-0.0333994\pi$
$710$		0			0
$711$	−82.1816			−0.115586
$712$		0			0
$713$	−787.682	+	787.682i	−1.10474	+	1.10474i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−374.363	−	374.363i	−0.522124	−	0.522124i
$718$		0			0
$719$		−	43.4847i		−	0.0604794i	−0.999543	−	0.0302397i	$-0.990373\pi$
$719$		−	43.4847i		−	0.0604794i	0.999543	−	0.0302397i	$-0.00962706\pi$
$720$		0			0
$721$	−353.271			−0.489974
$722$		0			0
$723$	204.532	−	204.532i	0.282894	−	0.282894i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	817.608	+	817.608i	1.12463	+	1.12463i	0.991036	+	0.133598i	$0.0426530\pi$
$727$	817.608	+	817.608i	1.12463	+	1.12463i	0.133598	+	0.991036i	$0.457347\pi$
$728$		0			0
$729$			27.0000i			0.0370370i
$730$		0			0
$731$	−65.0602			−0.0890016
$732$		0			0
$733$	−842.030	+	842.030i	−1.14874	+	1.14874i	−0.161944	+	0.986800i	$0.551777\pi$
$733$	−842.030	+	842.030i	−1.14874	+	1.14874i	−0.986800	+	0.161944i	$0.948223\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−352.318	−	352.318i	−0.478043	−	0.478043i
$738$		0			0
$739$			406.545i			0.550128i	0.961426	+	0.275064i	$0.0886991\pi$
$739$			406.545i			0.550128i	−0.961426	+	0.275064i	$0.911301\pi$
$740$		0			0
$741$	687.211			0.927411
$742$		0			0
$743$	−394.515	+	394.515i	−0.530976	+	0.530976i	−0.920863	−	0.389887i	$-0.872514\pi$
$743$	−394.515	+	394.515i	−0.530976	+	0.530976i	0.389887	+	0.920863i	$0.372514\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	190.318	+	190.318i	0.254776	+	0.254776i
$748$		0			0
$749$			1280.64i			1.70979i
$750$		0			0
$751$	612.788			0.815962			0.407981	−	0.912990i	$-0.366233\pi$
$751$	612.788			0.815962			0.407981	+	0.912990i	$0.366233\pi$
$752$		0			0
$753$	−320.363	+	320.363i	−0.425449	+	0.425449i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−181.547	−	181.547i	−0.239825	−	0.239825i	0.576953	−	0.816777i	$-0.304242\pi$
$757$	−181.547	−	181.547i	−0.239825	−	0.239825i	−0.816777	+	0.576953i	$0.804242\pi$
$758$		0			0
$759$		−	813.453i		−	1.07174i
$760$		0			0
$761$	497.271			0.653445			0.326722	−	0.945120i	$-0.394056\pi$
$761$	497.271			0.653445			0.326722	+	0.945120i	$0.394056\pi$
$762$		0			0
$763$	−692.412	+	692.412i	−0.907486	+	0.907486i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	249.773	+	249.773i	0.325649	+	0.325649i
$768$		0			0
$769$		−	220.271i		−	0.286439i	−0.989691	−	0.143219i	$-0.954255\pi$
$769$		−	220.271i		−	0.286439i	0.989691	−	0.143219i	$-0.0457455\pi$
$770$		0			0
$771$	−231.514			−0.300278
$772$		0			0
$773$	917.605	−	917.605i	1.18707	−	1.18707i	0.209196	−	0.977874i	$-0.432915\pi$
$773$	917.605	−	917.605i	1.18707	−	1.18707i	0.977874	−	0.209196i	$-0.0670848\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	426.879	+	426.879i	0.549393	+	0.549393i
$778$		0			0
$779$			1122.21i			1.44058i
$780$		0			0
$781$	−192.545			−0.246536
$782$		0			0
$783$	31.9546	−	31.9546i	0.0408105	−	0.0408105i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−862.194	−	862.194i	−1.09555	−	1.09555i	−0.994925	−	0.100620i	$-0.967917\pi$
$787$	−862.194	−	862.194i	−1.09555	−	1.09555i	−0.100620	−	0.994925i	$-0.532083\pi$
$788$		0			0
$789$		−	83.7276i		−	0.106119i
$790$		0			0
$791$	126.184			0.159524
$792$		0			0
$793$	−397.870	+	397.870i	−0.501727	+	0.501727i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−175.818	−	175.818i	−0.220600	−	0.220600i	0.588151	−	0.808751i	$-0.299856\pi$
$797$	−175.818	−	175.818i	−0.220600	−	0.220600i	−0.808751	+	0.588151i	$0.799856\pi$
$798$		0			0
$799$		−	37.6347i		−	0.0471022i
$800$		0			0
$801$	52.1816			0.0651456
$802$		0			0
$803$	461.394	−	461.394i	0.574588	−	0.574588i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	477.167	+	477.167i	0.591285	+	0.591285i
$808$		0			0
$809$			828.272i			1.02382i	0.859038	+	0.511911i	$0.171062\pi$
$809$			828.272i			1.02382i	−0.859038	+	0.511911i	$0.828938\pi$
$810$		0			0
$811$	116.909			0.144154			0.0720772	−	0.997399i	$-0.477037\pi$
$811$	116.909			0.144154			0.0720772	+	0.997399i	$0.477037\pi$
$812$		0			0
$813$	−192.025	+	192.025i	−0.236193	+	0.236193i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−812.896	−	812.896i	−0.994976	−	0.994976i
$818$		0			0
$819$		−	238.546i		−	0.291265i
$820$		0			0
$821$	632.424			0.770310			0.385155	−	0.922852i	$-0.374148\pi$
$821$	632.424			0.770310			0.385155	+	0.922852i	$0.374148\pi$
$822$		0			0
$823$	−1028.81	+	1028.81i	−1.25007	+	1.25007i	−0.294385	+	0.955687i	$0.595115\pi$
$823$	−1028.81	+	1028.81i	−1.25007	+	1.25007i	−0.955687	+	0.294385i	$0.904885\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−348.061	−	348.061i	−0.420872	−	0.420872i	0.464632	−	0.885504i	$-0.346187\pi$
$827$	−348.061	−	348.061i	−0.420872	−	0.420872i	−0.885504	+	0.464632i	$0.846187\pi$
$828$		0			0
$829$		−	901.392i		−	1.08732i	−0.839304	−	0.543662i	$-0.817037\pi$
$829$		−	901.392i		−	1.08732i	0.839304	−	0.543662i	$-0.182963\pi$
$830$		0			0
$831$	−699.545			−0.841811
$832$		0			0
$833$	4.57654	−	4.57654i	0.00549404	−	0.00549404i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−180.371	−	180.371i	−0.215497	−	0.215497i
$838$		0			0
$839$			99.1806i			0.118213i	0.998252	+	0.0591064i	$0.0188251\pi$
$839$			99.1806i			0.118213i	−0.998252	+	0.0591064i	$0.981175\pi$
$840$		0			0
$841$	765.363			0.910063
$842$		0			0
$843$	−24.6811	+	24.6811i	−0.0292777	+	0.0292777i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	1467.74	+	1467.74i	1.73287	+	1.73287i
$848$		0			0
$849$			417.817i			0.492129i
$850$		0			0
$851$	−1171.15			−1.37621
$852$		0			0
$853$	−662.612	+	662.612i	−0.776802	+	0.776802i	−0.979286	−	0.202484i	$-0.935099\pi$
$853$	−662.612	+	662.612i	−0.776802	+	0.776802i	0.202484	+	0.979286i	$0.435099\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−487.546	−	487.546i	−0.568898	−	0.568898i	0.362921	−	0.931820i	$-0.381779\pi$
$857$	−487.546	−	487.546i	−0.568898	−	0.568898i	−0.931820	+	0.362921i	$0.881779\pi$
$858$		0			0
$859$		−	110.849i		−	0.129044i	−0.997916	−	0.0645221i	$-0.979448\pi$
$859$		−	110.849i		−	0.129044i	0.997916	−	0.0645221i	$-0.0205523\pi$
$860$		0			0
$861$	389.544			0.452432
$862$		0			0
$863$	917.271	−	917.271i	1.06289	−	1.06289i	0.0650018	−	0.997885i	$-0.479295\pi$
$863$	917.271	−	917.271i	1.06289	−	1.06289i	0.997885	−	0.0650018i	$-0.0207053\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−349.497	−	349.497i	−0.403111	−	0.403111i
$868$		0			0
$869$			566.969i			0.652439i
$870$		0			0
$871$	283.454			0.325435
$872$		0			0
$873$	−125.629	+	125.629i	−0.143905	+	0.143905i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	53.7538	+	53.7538i	0.0612928	+	0.0612928i	0.737089	−	0.675796i	$-0.236200\pi$
$877$	53.7538	+	53.7538i	0.0612928	+	0.0612928i	−0.675796	+	0.737089i	$0.736200\pi$
$878$		0			0
$879$			635.939i			0.723480i
$880$		0			0
$881$	648.545			0.736146			0.368073	−	0.929797i	$-0.380018\pi$
$881$	648.545			0.736146			0.368073	+	0.929797i	$0.380018\pi$
$882$		0			0
$883$	191.381	−	191.381i	0.216740	−	0.216740i	−0.590383	−	0.807123i	$-0.701023\pi$
$883$	191.381	−	191.381i	0.216740	−	0.216740i	0.807123	+	0.590383i	$0.201023\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−468.347	−	468.347i	−0.528013	−	0.528013i	0.391967	−	0.919979i	$-0.371795\pi$
$887$	−468.347	−	468.347i	−0.528013	−	0.528013i	−0.919979	+	0.391967i	$0.871795\pi$
$888$		0			0
$889$		−	484.182i		−	0.544636i
$890$		0			0
$891$	186.272			0.209060
$892$		0			0
$893$	470.227	−	470.227i	0.526570	−	0.526570i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	327.228	+	327.228i	0.364803	+	0.364803i
$898$		0			0
$899$			426.940i			0.474905i
$900$		0			0
$901$	−160.182			−0.177782
$902$		0			0
$903$	−282.174	+	282.174i	−0.312485	+	0.312485i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−359.160	−	359.160i	−0.395987	−	0.395987i	0.480828	−	0.876815i	$-0.340336\pi$
$907$	−359.160	−	359.160i	−0.395987	−	0.395987i	−0.876815	+	0.480828i	$0.840336\pi$
$908$		0			0
$909$		−	488.636i		−	0.537553i
$910$		0			0
$911$	1698.00			1.86389			0.931943	−	0.362605i	$-0.118113\pi$
$911$	1698.00			1.86389			0.931943	+	0.362605i	$0.118113\pi$
$912$		0			0
$913$	1313.00	−	1313.00i	1.43811	−	1.43811i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	746.241	+	746.241i	0.813785	+	0.813785i
$918$		0			0
$919$		−	316.728i		−	0.344644i	−0.985041	−	0.172322i	$-0.944873\pi$
$919$		−	316.728i		−	0.344644i	0.985041	−	0.172322i	$-0.0551269\pi$
$920$		0			0
$921$	291.272			0.316257
$922$		0			0
$923$	77.4551	−	77.4551i	0.0839167	−	0.0839167i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	110.969	+	110.969i	0.119708	+	0.119708i
$928$		0			0
$929$		−	123.426i		−	0.132858i	−0.997791	−	0.0664292i	$-0.978839\pi$
$929$		−	123.426i		−	0.132858i	0.997791	−	0.0664292i	$-0.0211607\pi$
$930$		0			0
$931$	114.363			0.122839
$932$		0			0
$933$	64.6515	−	64.6515i	0.0692942	−	0.0692942i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	261.513	+	261.513i	0.279096	+	0.279096i	0.832748	−	0.553652i	$-0.186766\pi$
$937$	261.513	+	261.513i	0.279096	+	0.279096i	−0.553652	+	0.832748i	$0.686766\pi$
$938$		0			0
$939$		−	584.333i		−	0.622292i
$940$		0			0
$941$	−942.422			−1.00151			−0.500756	−	0.865589i	$-0.666945\pi$
$941$	−942.422			−1.00151			−0.500756	+	0.865589i	$0.666945\pi$
$942$		0			0
$943$	−534.361	+	534.361i	−0.566661	+	0.566661i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	986.983	+	986.983i	1.04222	+	1.04222i	0.999069	+	0.0431523i	$0.0137401\pi$
$947$	986.983	+	986.983i	1.04222	+	1.04222i	0.0431523	+	0.999069i	$0.486260\pi$
$948$		0			0
$949$			371.210i			0.391159i
$950$		0			0
$951$	119.878			0.126054
$952$		0			0
$953$	−112.454	+	112.454i	−0.118000	+	0.118000i	−0.763641	−	0.645641i	$-0.776590\pi$
$953$	−112.454	+	112.454i	−0.118000	+	0.118000i	0.645641	+	0.763641i	$0.276590\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−220.454	−	220.454i	−0.230360	−	0.230360i
$958$		0			0
$959$			872.424i			0.909723i
$960$		0			0
$961$	1448.91			1.50771
$962$		0			0
$963$	402.272	−	402.272i	0.417728	−	0.417728i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−700.009	−	700.009i	−0.723898	−	0.723898i	0.245499	−	0.969397i	$-0.421048\pi$
$967$	−700.009	−	700.009i	−0.723898	−	0.723898i	−0.969397	+	0.245499i	$0.921048\pi$
$968$		0			0
$969$			111.303i			0.114864i
$970$		0			0
$971$	1450.18			1.49349			0.746746	−	0.665109i	$-0.231615\pi$
$971$	1450.18			1.49349			0.746746	+	0.665109i	$0.231615\pi$
$972$		0			0
$973$	−168.147	+	168.147i	−0.172813	+	0.172813i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1146.14	+	1146.14i	1.17312	+	1.17312i	0.981462	+	0.191656i	$0.0613857\pi$
$977$	1146.14	+	1146.14i	1.17312	+	1.17312i	0.191656	+	0.981462i	$0.438614\pi$
$978$		0			0
$979$		−	360.000i		−	0.367722i
$980$		0			0
$981$	435.000			0.443425
$982$		0			0
$983$	760.332	−	760.332i	0.773481	−	0.773481i	−0.205232	−	0.978713i	$-0.565795\pi$
$983$	760.332	−	760.332i	0.773481	−	0.773481i	0.978713	+	0.205232i	$0.0657950\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−163.226	−	163.226i	−0.165376	−	0.165376i
$988$		0			0
$989$		−	774.150i		−	0.782760i
$990$		0			0
$991$	222.605			0.224627			0.112313	−	0.993673i	$-0.464174\pi$
$991$	222.605			0.224627			0.112313	+	0.993673i	$0.464174\pi$
$992$		0			0
$993$	357.551	−	357.551i	0.360071	−	0.360071i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	467.183	+	467.183i	0.468588	+	0.468588i	0.901457	−	0.432869i	$-0.142499\pi$
$997$	467.183	+	467.183i	0.468588	+	0.468588i	−0.432869	+	0.901457i	$0.642499\pi$
$998$		0			0
$999$		−	268.182i		−	0.268450i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.bg.p.1057.2		4
4.3	odd	2		150.3.f.a.7.1	✓	4
5.2	odd	4		1200.3.bg.a.193.1		4
5.3	odd	4	inner	1200.3.bg.p.193.2		4
5.4	even	2		1200.3.bg.a.1057.1		4
12.11	even	2		450.3.g.h.307.2		4
20.3	even	4		150.3.f.a.43.1	yes	4
20.7	even	4		150.3.f.c.43.2	yes	4
20.19	odd	2		150.3.f.c.7.2	yes	4
60.23	odd	4		450.3.g.h.343.2		4
60.47	odd	4		450.3.g.g.343.1		4
60.59	even	2		450.3.g.g.307.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.3.f.a.7.1	✓	4	4.3	odd	2
150.3.f.a.43.1	yes	4	20.3	even	4
150.3.f.c.7.2	yes	4	20.19	odd	2
150.3.f.c.43.2	yes	4	20.7	even	4
450.3.g.g.307.1		4	60.59	even	2
450.3.g.g.343.1		4	60.47	odd	4
450.3.g.h.307.2		4	12.11	even	2
450.3.g.h.343.2		4	60.23	odd	4
1200.3.bg.a.193.1		4	5.2	odd	4
1200.3.bg.a.1057.1		4	5.4	even	2
1200.3.bg.p.193.2		4	5.3	odd	4	inner
1200.3.bg.p.1057.2		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1200.bg (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 - 1.22474i) q^{3} +(4.77526 + 4.77526i) q^{7} -3.00000i q^{9} +O(q^{10})\)	\(q+(1.22474 - 1.22474i) q^{3} +(4.77526 + 4.77526i) q^{7} -3.00000i q^{9} -20.6969 q^{11} +(8.32577 - 8.32577i) q^{13} +(1.34847 + 1.34847i) q^{17} +33.6969i q^{19} +11.6969 q^{21} +(-16.0454 + 16.0454i) q^{23} +(-3.67423 - 3.67423i) q^{27} +8.69694i q^{29} +49.0908 q^{31} +(-25.3485 + 25.3485i) q^{33} +(36.4949 + 36.4949i) q^{37} -20.3939i q^{39} +33.3031 q^{41} +(-24.1237 + 24.1237i) q^{43} +(-13.9546 - 13.9546i) q^{47} -3.39388i q^{49} +3.30306 q^{51} +(-59.3939 + 59.3939i) q^{53} +(41.2702 + 41.2702i) q^{57} +30.0000i q^{59} -47.7878 q^{61} +(14.3258 - 14.3258i) q^{63} +(17.0227 + 17.0227i) q^{67} +39.3031i q^{69} +9.30306 q^{71} +(-22.2929 + 22.2929i) q^{73} +(-98.8332 - 98.8332i) q^{77} -27.3939i q^{79} -9.00000 q^{81} +(-63.4393 + 63.4393i) q^{83} +(10.6515 + 10.6515i) q^{87} +17.3939i q^{89} +79.5153 q^{91} +(60.1237 - 60.1237i) q^{93} +(-41.8763 - 41.8763i) q^{97} +62.0908i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 24 q^{7}+O(q^{10}) \) 4 * q + 24 * q^7	\( 4 q + 24 q^{7} - 24 q^{11} + 48 q^{13} - 24 q^{17} - 12 q^{21} + 24 q^{23} + 20 q^{31} - 72 q^{33} + 48 q^{37} + 192 q^{41} - 72 q^{43} - 144 q^{47} + 72 q^{51} - 120 q^{53} + 72 q^{57} + 44 q^{61} + 72 q^{63} + 24 q^{67} + 96 q^{71} + 48 q^{73} - 72 q^{77} - 36 q^{81} - 48 q^{83} + 72 q^{87} + 612 q^{91} + 216 q^{93} - 192 q^{97}+O(q^{100}) \) 4 * q + 24 * q^7 - 24 * q^11 + 48 * q^13 - 24 * q^17 - 12 * q^21 + 24 * q^23 + 20 * q^31 - 72 * q^33 + 48 * q^37 + 192 * q^41 - 72 * q^43 - 144 * q^47 + 72 * q^51 - 120 * q^53 + 72 * q^57 + 44 * q^61 + 72 * q^63 + 24 * q^67 + 96 * q^71 + 48 * q^73 - 72 * q^77 - 36 * q^81 - 48 * q^83 + 72 * q^87 + 612 * q^91 + 216 * q^93 - 192 * q^97

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