LMFDB - Embedded newform 2394.2.f.b.2015.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2394,2,Mod(2015,2394)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2394, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2394.2015");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2394.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:	$19.1161862439$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2015.4
Character	$\chi$	$=$	2394.2015
Dual form			2394.2.f.b.2015.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +4.23211 q^{5} +(2.64176 + 0.145265i) q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{4} +4.23211 q^{5} +(2.64176 + 0.145265i) q^{7} -1.00000i q^{8} +4.23211i q^{10} +0.875754i q^{11} -3.49874i q^{13} +(-0.145265 + 2.64176i) q^{14} +1.00000 q^{16} +0.180186 q^{17} +1.00000i q^{19} -4.23211 q^{20} -0.875754 q^{22} -1.13808i q^{23} +12.9107 q^{25} +3.49874 q^{26} +(-2.64176 - 0.145265i) q^{28} +9.84665i q^{29} +1.21499i q^{31} +1.00000i q^{32} +0.180186i q^{34} +(11.1802 + 0.614778i) q^{35} -6.29513 q^{37} -1.00000 q^{38} -4.23211i q^{40} +8.20621 q^{41} +0.254708 q^{43} -0.875754i q^{44} +1.13808 q^{46} -8.88217 q^{47} +(6.95780 + 0.767512i) q^{49} +12.9107i q^{50} +3.49874i q^{52} -10.5147i q^{53} +3.70629i q^{55} +(0.145265 - 2.64176i) q^{56} -9.84665 q^{58} +13.0770 q^{59} +6.24942i q^{61} -1.21499 q^{62} -1.00000 q^{64} -14.8070i q^{65} -8.07646 q^{67} -0.180186 q^{68} +(-0.614778 + 11.1802i) q^{70} -3.26508i q^{71} -16.2323i q^{73} -6.29513i q^{74} -1.00000i q^{76} +(-0.127217 + 2.31353i) q^{77} -12.6056 q^{79} +4.23211 q^{80} +8.20621i q^{82} +10.6139 q^{83} +0.762568 q^{85} +0.254708i q^{86} +0.875754 q^{88} +1.87911 q^{89} +(0.508245 - 9.24284i) q^{91} +1.13808i q^{92} -8.88217i q^{94} +4.23211i q^{95} +6.10878i q^{97} +(-0.767512 + 6.95780i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{4} + 4 q^{7}+O(q^{10}) $ 24 * q - 24 * q^4 + 4 * q^7	$ 24 q - 24 q^{4} + 4 q^{7} + 4 q^{14} + 24 q^{16} - 32 q^{17} - 8 q^{22} + 16 q^{25} - 4 q^{28} + 24 q^{35} - 24 q^{38} + 8 q^{41} + 16 q^{43} - 8 q^{46} - 16 q^{49} - 4 q^{56} + 16 q^{58} - 16 q^{59} + 16 q^{62} - 24 q^{64} - 24 q^{67} + 32 q^{68} - 16 q^{70} - 8 q^{77} - 40 q^{79} + 64 q^{83} + 40 q^{85} + 8 q^{88} + 64 q^{89} + 8 q^{91} + 16 q^{98}+O(q^{100}) $ 24 * q - 24 * q^4 + 4 * q^7 + 4 * q^14 + 24 * q^16 - 32 * q^17 - 8 * q^22 + 16 * q^25 - 4 * q^28 + 24 * q^35 - 24 * q^38 + 8 * q^41 + 16 * q^43 - 8 * q^46 - 16 * q^49 - 4 * q^56 + 16 * q^58 - 16 * q^59 + 16 * q^62 - 24 * q^64 - 24 * q^67 + 32 * q^68 - 16 * q^70 - 8 * q^77 - 40 * q^79 + 64 * q^83 + 40 * q^85 + 8 * q^88 + 64 * q^89 + 8 * q^91 + 16 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2394\mathbb{Z}\right)^\times$.

$n$	$533$	$1009$	$1711$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	4.23211			1.89266			0.946328	−	0.323208i	$-0.104761\pi$
$5$	4.23211			1.89266			0.946328	+	0.323208i	$0.104761\pi$
$6$		0			0
$7$	2.64176	+	0.145265i	0.998492	+	0.0549051i
$7$	2.64176	+	0.145265i	0.998492	+	0.0549051i
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$			4.23211i			1.33831i
$11$			0.875754i			0.264050i	0.991246	+	0.132025i	$0.0421479\pi$
$11$			0.875754i			0.264050i	−0.991246	+	0.132025i	$0.957852\pi$
$12$		0			0
$13$		−	3.49874i		−	0.970376i	−0.874410	−	0.485188i	$-0.838751\pi$
$13$		−	3.49874i		−	0.970376i	0.874410	−	0.485188i	$-0.161249\pi$
$14$	−0.145265	+	2.64176i	−0.0388238	+	0.706040i
$15$		0			0
$16$	1.00000			0.250000
$17$	0.180186			0.0437016			0.0218508	−	0.999761i	$-0.493044\pi$
$17$	0.180186			0.0437016			0.0218508	+	0.999761i	$0.493044\pi$
$18$		0			0
$19$			1.00000i			0.229416i
$19$			1.00000i			0.229416i
$20$	−4.23211			−0.946328
$21$		0			0
$22$	−0.875754			−0.186711
$23$		−	1.13808i		−	0.237306i	−0.992936	−	0.118653i	$-0.962142\pi$
$23$		−	1.13808i		−	0.237306i	0.992936	−	0.118653i	$-0.0378576\pi$
$24$		0			0
$25$	12.9107			2.58215
$26$	3.49874			0.686160
$27$		0			0
$28$	−2.64176	−	0.145265i	−0.499246	−	0.0274525i
$29$			9.84665i			1.82848i	0.405177	+	0.914238i	$0.367210\pi$
$29$			9.84665i			1.82848i	−0.405177	+	0.914238i	$0.632790\pi$
$30$		0			0
$31$			1.21499i			0.218218i	0.994030	+	0.109109i	$0.0347998\pi$
$31$			1.21499i			0.218218i	−0.994030	+	0.109109i	$0.965200\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$			0.180186i			0.0309017i
$35$	11.1802	+	0.614778i	1.88980	+	0.103916i
$36$		0			0
$37$	−6.29513			−1.03491			−0.517456	−	0.855710i	$-0.673121\pi$
$37$	−6.29513			−1.03491			−0.517456	+	0.855710i	$0.673121\pi$
$38$	−1.00000			−0.162221
$39$		0			0
$40$		−	4.23211i		−	0.669155i
$41$	8.20621			1.28160			0.640798	−	0.767710i	$-0.278604\pi$
$41$	8.20621			1.28160			0.640798	+	0.767710i	$0.278604\pi$
$42$		0			0
$43$	0.254708			0.0388425			0.0194213	−	0.999811i	$-0.493818\pi$
$43$	0.254708			0.0388425			0.0194213	+	0.999811i	$0.493818\pi$
$44$		−	0.875754i		−	0.132025i
$45$		0			0
$46$	1.13808			0.167800
$47$	−8.88217			−1.29560			−0.647799	−	0.761811i	$-0.724310\pi$
$47$	−8.88217			−1.29560			−0.647799	+	0.761811i	$0.724310\pi$
$48$		0			0
$49$	6.95780	+	0.767512i	0.993971	+	0.109645i
$50$			12.9107i			1.82585i
$51$		0			0
$52$			3.49874i			0.485188i
$53$		−	10.5147i		−	1.44431i	−0.691732	−	0.722154i	$-0.743152\pi$
$53$		−	10.5147i		−	1.44431i	0.691732	−	0.722154i	$-0.256848\pi$
$54$		0			0
$55$			3.70629i			0.499756i
$56$	0.145265	−	2.64176i	0.0194119	−	0.353020i
$57$		0			0
$58$	−9.84665			−1.29293
$59$	13.0770			1.70248			0.851242	−	0.524774i	$-0.175850\pi$
$59$	13.0770			1.70248			0.851242	+	0.524774i	$0.175850\pi$
$60$		0			0
$61$			6.24942i			0.800156i	0.916481	+	0.400078i	$0.131017\pi$
$61$			6.24942i			0.800156i	−0.916481	+	0.400078i	$0.868983\pi$
$62$	−1.21499			−0.154303
$63$		0			0
$64$	−1.00000			−0.125000
$65$		−	14.8070i		−	1.83659i
$66$		0			0
$67$	−8.07646			−0.986696			−0.493348	−	0.869832i	$-0.664227\pi$
$67$	−8.07646			−0.986696			−0.493348	+	0.869832i	$0.664227\pi$
$68$	−0.180186			−0.0218508
$69$		0			0
$70$	−0.614778	+	11.1802i	−0.0734800	+	1.33629i
$71$		−	3.26508i		−	0.387494i	−0.981052	−	0.193747i	$-0.937936\pi$
$71$		−	3.26508i		−	0.387494i	0.981052	−	0.193747i	$-0.0620641\pi$
$72$		0			0
$73$		−	16.2323i		−	1.89985i	−0.312485	−	0.949923i	$-0.601161\pi$
$73$		−	16.2323i		−	1.89985i	0.312485	−	0.949923i	$-0.398839\pi$
$74$		−	6.29513i		−	0.731794i
$75$		0			0
$76$		−	1.00000i		−	0.114708i
$77$	−0.127217	+	2.31353i	−0.0144977	+	0.263652i
$78$		0			0
$79$	−12.6056			−1.41824			−0.709122	−	0.705085i	$-0.750909\pi$
$79$	−12.6056			−1.41824			−0.709122	+	0.705085i	$0.750909\pi$
$80$	4.23211			0.473164
$81$		0			0
$82$			8.20621i			0.906225i
$83$	10.6139			1.16503			0.582513	−	0.812821i	$-0.302069\pi$
$83$	10.6139			1.16503			0.582513	+	0.812821i	$0.302069\pi$
$84$		0			0
$85$	0.762568			0.0827122
$86$			0.254708i			0.0274658i
$87$		0			0
$88$	0.875754			0.0933557
$89$	1.87911			0.199185			0.0995925	−	0.995028i	$-0.468246\pi$
$89$	1.87911			0.199185			0.0995925	+	0.995028i	$0.468246\pi$
$90$		0			0
$91$	0.508245	−	9.24284i	0.0532786	−	0.968912i
$92$			1.13808i			0.118653i
$93$		0			0
$94$		−	8.88217i		−	0.916126i
$95$			4.23211i			0.434205i
$96$		0			0
$97$			6.10878i			0.620252i	0.950695	+	0.310126i	$0.100371\pi$
$97$			6.10878i			0.620252i	−0.950695	+	0.310126i	$0.899629\pi$
$98$	−0.767512	+	6.95780i	−0.0775304	+	0.702844i
$99$		0			0
$100$	−12.9107			−1.29107
$101$	−0.696515			−0.0693058			−0.0346529	−	0.999399i	$-0.511033\pi$
$101$	−0.696515			−0.0693058			−0.0346529	+	0.999399i	$0.511033\pi$
$102$		0			0
$103$			9.23113i			0.909570i	0.890601	+	0.454785i	$0.150284\pi$
$103$			9.23113i			0.909570i	−0.890601	+	0.454785i	$0.849716\pi$
$104$	−3.49874			−0.343080
$105$		0			0
$106$	10.5147			1.02128
$107$		−	14.2578i		−	1.37836i	−0.724592	−	0.689178i	$-0.757972\pi$
$107$		−	14.2578i		−	1.37836i	0.724592	−	0.689178i	$-0.242028\pi$
$108$		0			0
$109$	−0.289836			−0.0277612			−0.0138806	−	0.999904i	$-0.504418\pi$
$109$	−0.289836			−0.0277612			−0.0138806	+	0.999904i	$0.504418\pi$
$110$	−3.70629			−0.353381
$111$		0			0
$112$	2.64176	+	0.145265i	0.249623	+	0.0137263i
$113$		−	3.45428i		−	0.324952i	−0.986713	−	0.162476i	$-0.948052\pi$
$113$		−	3.45428i		−	0.324952i	0.986713	−	0.162476i	$-0.0519479\pi$
$114$		0			0
$115$		−	4.81647i		−	0.449138i
$116$		−	9.84665i		−	0.914238i
$117$		0			0
$118$			13.0770i			1.20384i
$119$	0.476009	+	0.0261748i	0.0436357	+	0.00239944i
$120$		0			0
$121$	10.2331			0.930278
$122$	−6.24942			−0.565796
$123$		0			0
$124$		−	1.21499i		−	0.109109i
$125$	33.4791			2.99446
$126$		0			0
$127$	−11.9614			−1.06140			−0.530701	−	0.847559i	$-0.678071\pi$
$127$	−11.9614			−1.06140			−0.530701	+	0.847559i	$0.678071\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	14.8070			1.29866
$131$	15.0295			1.31314			0.656569	−	0.754266i	$-0.272007\pi$
$131$	15.0295			1.31314			0.656569	+	0.754266i	$0.272007\pi$
$132$		0			0
$133$	−0.145265	+	2.64176i	−0.0125961	+	0.229070i
$134$		−	8.07646i		−	0.697700i
$135$		0			0
$136$		−	0.180186i		−	0.0154509i
$137$			4.35267i			0.371873i	0.982562	+	0.185937i	$0.0595319\pi$
$137$			4.35267i			0.371873i	−0.982562	+	0.185937i	$0.940468\pi$
$138$		0			0
$139$			11.1373i			0.944652i	0.881424	+	0.472326i	$0.156585\pi$
$139$			11.1373i			0.944652i	−0.881424	+	0.472326i	$0.843415\pi$
$140$	−11.1802	−	0.614778i	−0.944901	−	0.0519582i
$141$		0			0
$142$	3.26508			0.274000
$143$	3.06404			0.256228
$144$		0			0
$145$			41.6721i			3.46068i
$146$	16.2323			1.34339
$147$		0			0
$148$	6.29513			0.517456
$149$		−	15.8320i		−	1.29700i	−0.761213	−	0.648502i	$-0.775396\pi$
$149$		−	15.8320i		−	1.29700i	0.761213	−	0.648502i	$-0.224604\pi$
$150$		0			0
$151$	−13.0465			−1.06171			−0.530853	−	0.847464i	$-0.678128\pi$
$151$	−13.0465			−1.06171			−0.530853	+	0.847464i	$0.678128\pi$
$152$	1.00000			0.0811107
$153$		0			0
$154$	−2.31353	−	0.127217i	−0.186430	−	0.0102514i
$155$			5.14196i			0.413012i
$156$		0			0
$157$			9.88029i			0.788533i	0.918996	+	0.394267i	$0.129001\pi$
$157$			9.88029i			0.788533i	−0.918996	+	0.394267i	$0.870999\pi$
$158$		−	12.6056i		−	1.00285i
$159$		0			0
$160$			4.23211i			0.334578i
$161$	0.165323	−	3.00653i	0.0130293	−	0.236948i
$162$		0			0
$163$	−16.6370			−1.30311			−0.651556	−	0.758600i	$-0.725884\pi$
$163$	−16.6370			−1.30311			−0.651556	+	0.758600i	$0.725884\pi$
$164$	−8.20621			−0.640798
$165$		0			0
$166$			10.6139i			0.823798i
$167$	−22.2000			−1.71789			−0.858943	−	0.512071i	$-0.828879\pi$
$167$	−22.2000			−1.71789			−0.858943	+	0.512071i	$0.828879\pi$
$168$		0			0
$169$	0.758811			0.0583701
$170$			0.762568i			0.0584863i
$171$		0			0
$172$	−0.254708			−0.0194213
$173$	−0.122870			−0.00934163			−0.00467081	−	0.999989i	$-0.501487\pi$
$173$	−0.122870			−0.00934163			−0.00467081	+	0.999989i	$0.501487\pi$
$174$		0			0
$175$	34.1071	+	1.87548i	2.57825	+	0.141773i
$176$			0.875754i			0.0660125i
$177$		0			0
$178$			1.87911i			0.140845i
$179$			6.05614i			0.452657i	0.974051	+	0.226329i	$0.0726723\pi$
$179$			6.05614i			0.452657i	−0.974051	+	0.226329i	$0.927328\pi$
$180$		0			0
$181$			8.02414i			0.596430i	0.954499	+	0.298215i	$0.0963912\pi$
$181$			8.02414i			0.596430i	−0.954499	+	0.298215i	$0.903609\pi$
$182$	9.24284	+	0.508245i	0.685125	+	0.0376737i
$183$		0			0
$184$	−1.13808			−0.0839002
$185$	−26.6417			−1.95873
$186$		0			0
$187$			0.157799i			0.0115394i
$188$	8.88217			0.647799
$189$		0			0
$190$	−4.23211			−0.307029
$191$			17.3111i			1.25259i	0.779587	+	0.626294i	$0.215429\pi$
$191$			17.3111i			1.25259i	−0.779587	+	0.626294i	$0.784571\pi$
$192$		0			0
$193$	−10.0773			−0.725379			−0.362689	−	0.931910i	$-0.618141\pi$
$193$	−10.0773			−0.725379			−0.362689	+	0.931910i	$0.618141\pi$
$194$	−6.10878			−0.438585
$195$		0			0
$196$	−6.95780	−	0.767512i	−0.496985	−	0.0548223i
$197$			21.0687i			1.50108i	0.660825	+	0.750540i	$0.270206\pi$
$197$			21.0687i			1.50108i	−0.660825	+	0.750540i	$0.729794\pi$
$198$		0			0
$199$		−	9.30112i		−	0.659339i	−0.944096	−	0.329669i	$-0.893063\pi$
$199$		−	9.30112i		−	0.659339i	0.944096	−	0.329669i	$-0.106937\pi$
$200$		−	12.9107i		−	0.912927i
$201$		0			0
$202$		−	0.696515i		−	0.0490066i
$203$	−1.43038	+	26.0125i	−0.100393	+	1.82572i
$204$		0			0
$205$	34.7296			2.42562
$206$	−9.23113			−0.643163
$207$		0			0
$208$		−	3.49874i		−	0.242594i
$209$	−0.875754			−0.0605772
$210$		0			0
$211$	−6.46105			−0.444797			−0.222399	−	0.974956i	$-0.571389\pi$
$211$	−6.46105			−0.444797			−0.222399	+	0.974956i	$0.571389\pi$
$212$			10.5147i			0.722154i
$213$		0			0
$214$	14.2578			0.974645
$215$	1.07795			0.0735156
$216$		0			0
$217$	−0.176495	+	3.20970i	−0.0119813	+	0.217889i
$218$		−	0.289836i		−	0.0196302i
$219$		0			0
$220$		−	3.70629i		−	0.249878i
$221$		−	0.630426i		−	0.0424070i
$222$		0			0
$223$			1.18887i			0.0796128i	0.999207	+	0.0398064i	$0.0126741\pi$
$223$			1.18887i			0.0796128i	−0.999207	+	0.0398064i	$0.987326\pi$
$224$	−0.145265	+	2.64176i	−0.00970594	+	0.176510i
$225$		0			0
$226$	3.45428			0.229775
$227$	8.96755			0.595197			0.297599	−	0.954691i	$-0.403814\pi$
$227$	8.96755			0.595197			0.297599	+	0.954691i	$0.403814\pi$
$228$		0			0
$229$		−	12.1250i		−	0.801245i	−0.916243	−	0.400623i	$-0.868794\pi$
$229$		−	12.1250i		−	0.801245i	0.916243	−	0.400623i	$-0.131206\pi$
$230$	4.81647			0.317589
$231$		0			0
$232$	9.84665			0.646464
$233$		−	20.3722i		−	1.33463i	−0.744776	−	0.667315i	$-0.767444\pi$
$233$		−	20.3722i		−	1.33463i	0.744776	−	0.667315i	$-0.232556\pi$
$234$		0			0
$235$	−37.5903			−2.45212
$236$	−13.0770			−0.851242
$237$		0			0
$238$	−0.0261748	+	0.476009i	−0.00169666	+	0.0308551i
$239$			15.3450i			0.992588i	0.868154	+	0.496294i	$0.165306\pi$
$239$			15.3450i			0.992588i	−0.868154	+	0.496294i	$0.834694\pi$
$240$		0			0
$241$		−	6.49921i		−	0.418651i	−0.977846	−	0.209325i	$-0.932873\pi$
$241$		−	6.49921i		−	0.418651i	0.977846	−	0.209325i	$-0.0671268\pi$
$242$			10.2331i			0.657806i
$243$		0			0
$244$		−	6.24942i		−	0.400078i
$245$	29.4461	+	3.24819i	1.88125	+	0.207519i
$246$		0			0
$247$	3.49874			0.222620
$248$	1.21499			0.0771517
$249$		0			0
$250$			33.4791i			2.11740i
$251$	0.322392			0.0203492			0.0101746	−	0.999948i	$-0.496761\pi$
$251$	0.322392			0.0203492			0.0101746	+	0.999948i	$0.496761\pi$
$252$		0			0
$253$	0.996677			0.0626605
$254$		−	11.9614i		−	0.750525i
$255$		0			0
$256$	1.00000			0.0625000
$257$	17.4765			1.09015			0.545076	−	0.838387i	$-0.316501\pi$
$257$	17.4765			1.09015			0.545076	+	0.838387i	$0.316501\pi$
$258$		0			0
$259$	−16.6302	−	0.914463i	−1.03335	−	0.0568220i
$260$			14.8070i			0.918294i
$261$		0			0
$262$			15.0295i			0.928529i
$263$			18.0430i			1.11258i	0.830989	+	0.556289i	$0.187775\pi$
$263$			18.0430i			1.11258i	−0.830989	+	0.556289i	$0.812225\pi$
$264$		0			0
$265$		−	44.4994i		−	2.73358i
$266$	−2.64176	−	0.145265i	−0.161977	−	0.00890678i
$267$		0			0
$268$	8.07646			0.493348
$269$	−10.3160			−0.628979			−0.314490	−	0.949261i	$-0.601833\pi$
$269$	−10.3160			−0.628979			−0.314490	+	0.949261i	$0.601833\pi$
$270$		0			0
$271$			6.52466i			0.396345i	0.980167	+	0.198172i	$0.0635006\pi$
$271$			6.52466i			0.396345i	−0.980167	+	0.198172i	$0.936499\pi$
$272$	0.180186			0.0109254
$273$		0			0
$274$	−4.35267			−0.262954
$275$			11.3066i			0.681816i
$276$		0			0
$277$	−8.95738			−0.538197			−0.269098	−	0.963113i	$-0.586726\pi$
$277$	−8.95738			−0.538197			−0.269098	+	0.963113i	$0.586726\pi$
$278$	−11.1373			−0.667970
$279$		0			0
$280$	0.614778	−	11.1802i	0.0367400	−	0.668146i
$281$			19.4917i			1.16278i	0.813625	+	0.581389i	$0.197491\pi$
$281$			19.4917i			1.16278i	−0.813625	+	0.581389i	$0.802509\pi$
$282$		0			0
$283$		−	7.67409i		−	0.456177i	−0.973640	−	0.228089i	$-0.926752\pi$
$283$		−	7.67409i		−	0.456177i	0.973640	−	0.228089i	$-0.0732476\pi$
$284$			3.26508i			0.193747i
$285$		0			0
$286$			3.06404i			0.181180i
$287$	21.6789	+	1.19208i	1.27966	+	0.0703661i
$288$		0			0
$289$	−16.9675			−0.998090
$290$	−41.6721			−2.44707
$291$		0			0
$292$			16.2323i			0.949923i
$293$	−25.5924			−1.49513			−0.747563	−	0.664191i	$-0.768776\pi$
$293$	−25.5924			−1.49513			−0.747563	+	0.664191i	$0.768776\pi$
$294$		0			0
$295$	55.3434			3.22222
$296$			6.29513i			0.365897i
$297$		0			0
$298$	15.8320			0.917121
$299$	−3.98184			−0.230276
$300$		0			0
$301$	0.672876	+	0.0370001i	0.0387839	+	0.00213265i
$302$		−	13.0465i		−	0.750740i
$303$		0			0
$304$			1.00000i			0.0573539i
$305$			26.4482i			1.51442i
$306$		0			0
$307$			5.43925i			0.310434i	0.987880	+	0.155217i	$0.0496077\pi$
$307$			5.43925i			0.310434i	−0.987880	+	0.155217i	$0.950392\pi$
$308$	0.127217	−	2.31353i	0.00724884	−	0.131826i
$309$		0			0
$310$	−5.14196			−0.292043
$311$	12.6262			0.715966			0.357983	−	0.933728i	$-0.383465\pi$
$311$	12.6262			0.715966			0.357983	+	0.933728i	$0.383465\pi$
$312$		0			0
$313$		−	0.732999i		−	0.0414315i	−0.999785	−	0.0207158i	$-0.993405\pi$
$313$		−	0.732999i		−	0.0414315i	0.999785	−	0.0207158i	$-0.00659451\pi$
$314$	−9.88029			−0.557577
$315$		0			0
$316$	12.6056			0.709122
$317$		−	9.57289i		−	0.537667i	−0.963187	−	0.268834i	$-0.913362\pi$
$317$		−	9.57289i		−	0.537667i	0.963187	−	0.268834i	$-0.0866382\pi$
$318$		0			0
$319$	−8.62324			−0.482809
$320$	−4.23211			−0.236582
$321$		0			0
$322$	3.00653	+	0.165323i	0.167547	+	0.00921310i
$323$			0.180186i			0.0100258i
$324$		0			0
$325$		−	45.1713i		−	2.50565i
$326$		−	16.6370i		−	0.921440i
$327$		0			0
$328$		−	8.20621i		−	0.453112i
$329$	−23.4646	−	1.29027i	−1.29364	−	0.0711349i
$330$		0			0
$331$	−23.9687			−1.31744			−0.658720	−	0.752388i	$-0.728902\pi$
$331$	−23.9687			−1.31744			−0.658720	+	0.752388i	$0.728902\pi$
$332$	−10.6139			−0.582513
$333$		0			0
$334$		−	22.2000i		−	1.21473i
$335$	−34.1804			−1.86748
$336$		0			0
$337$	−6.53938			−0.356223			−0.178111	−	0.984010i	$-0.556999\pi$
$337$	−6.53938			−0.356223			−0.178111	+	0.984010i	$0.556999\pi$
$338$			0.758811i			0.0412739i
$339$		0			0
$340$	−0.762568			−0.0413561
$341$	−1.06403			−0.0576205
$342$		0			0
$343$	18.2693	+	3.03831i	0.986451	+	0.164053i
$344$		−	0.254708i		−	0.0137329i
$345$		0			0
$346$		−	0.122870i		−	0.00660553i
$347$		−	7.41154i		−	0.397872i	−0.980012	−	0.198936i	$-0.936251\pi$
$347$		−	7.41154i		−	0.397872i	0.980012	−	0.198936i	$-0.0637486\pi$
$348$		0			0
$349$		−	13.2771i		−	0.710705i	−0.934732	−	0.355353i	$-0.884361\pi$
$349$		−	13.2771i		−	0.710705i	0.934732	−	0.355353i	$-0.115639\pi$
$350$	−1.87548	+	34.1071i	−0.100249	+	1.82310i
$351$		0			0
$352$	−0.875754			−0.0466779
$353$	7.04350			0.374888			0.187444	−	0.982275i	$-0.439980\pi$
$353$	7.04350			0.374888			0.187444	+	0.982275i	$0.439980\pi$
$354$		0			0
$355$		−	13.8182i		−	0.733393i
$356$	−1.87911			−0.0995925
$357$		0			0
$358$	−6.05614			−0.320077
$359$		−	32.2091i		−	1.69993i	−0.526837	−	0.849966i	$-0.676622\pi$
$359$		−	32.2091i		−	1.69993i	0.526837	−	0.849966i	$-0.323378\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$	−8.02414			−0.421740
$363$		0			0
$364$	−0.508245	+	9.24284i	−0.0266393	+	0.484456i
$365$		−	68.6968i		−	3.59575i
$366$		0			0
$367$			23.7850i			1.24157i	0.783981	+	0.620785i	$0.213186\pi$
$367$			23.7850i			1.24157i	−0.783981	+	0.620785i	$0.786814\pi$
$368$		−	1.13808i		−	0.0593264i
$369$		0			0
$370$		−	26.6417i		−	1.38503i
$371$	1.52742	−	27.7774i	0.0792999	−	1.44213i
$372$		0			0
$373$	16.3365			0.845870			0.422935	−	0.906160i	$-0.361000\pi$
$373$	16.3365			0.845870			0.422935	+	0.906160i	$0.361000\pi$
$374$	−0.157799			−0.00815960
$375$		0			0
$376$			8.88217i			0.458063i
$377$	34.4509			1.77431
$378$		0			0
$379$	4.94720			0.254121			0.127060	−	0.991895i	$-0.459446\pi$
$379$	4.94720			0.254121			0.127060	+	0.991895i	$0.459446\pi$
$380$		−	4.23211i		−	0.217103i
$381$		0			0
$382$	−17.3111			−0.885713
$383$	−24.4256			−1.24809			−0.624045	−	0.781388i	$-0.714512\pi$
$383$	−24.4256			−1.24809			−0.624045	+	0.781388i	$0.714512\pi$
$384$		0			0
$385$	−0.538395	+	9.79112i	−0.0274391	+	0.499002i
$386$		−	10.0773i		−	0.512920i
$387$		0			0
$388$		−	6.10878i		−	0.310126i
$389$			8.72327i			0.442287i	0.975241	+	0.221144i	$0.0709790\pi$
$389$			8.72327i			0.442287i	−0.975241	+	0.221144i	$0.929021\pi$
$390$		0			0
$391$		−	0.205066i		−	0.0103706i
$392$	0.767512	−	6.95780i	0.0387652	−	0.351422i
$393$		0			0
$394$	−21.0687			−1.06142
$395$	−53.3484			−2.68425
$396$		0			0
$397$		−	36.1479i		−	1.81421i	−0.420904	−	0.907105i	$-0.638287\pi$
$397$		−	36.1479i		−	1.81421i	0.420904	−	0.907105i	$-0.361713\pi$
$398$	9.30112			0.466223
$399$		0			0
$400$	12.9107			0.645537
$401$		−	35.3134i		−	1.76347i	−0.471749	−	0.881733i	$-0.656377\pi$
$401$		−	35.3134i		−	1.76347i	0.471749	−	0.881733i	$-0.343623\pi$
$402$		0			0
$403$	4.25092			0.211754
$404$	0.696515			0.0346529
$405$		0			0
$406$	−26.0125	−	1.43038i	−1.29098	−	0.0709883i
$407$		−	5.51299i		−	0.273269i
$408$		0			0
$409$		−	29.8515i		−	1.47606i	−0.674767	−	0.738031i	$-0.735756\pi$
$409$		−	29.8515i		−	1.47606i	0.674767	−	0.738031i	$-0.264244\pi$
$410$			34.7296i			1.71517i
$411$		0			0
$412$		−	9.23113i		−	0.454785i
$413$	34.5464	+	1.89964i	1.69992	+	0.0934750i
$414$		0			0
$415$	44.9191			2.20499
$416$	3.49874			0.171540
$417$		0			0
$418$		−	0.875754i		−	0.0428345i
$419$	−35.3705			−1.72796			−0.863981	−	0.503525i	$-0.832036\pi$
$419$	−35.3705			−1.72796			−0.863981	+	0.503525i	$0.832036\pi$
$420$		0			0
$421$	0.362549			0.0176695			0.00883477	−	0.999961i	$-0.497188\pi$
$421$	0.362549			0.0176695			0.00883477	+	0.999961i	$0.497188\pi$
$422$		−	6.46105i		−	0.314519i
$423$		0			0
$424$	−10.5147			−0.510640
$425$	2.32634			0.112844
$426$		0			0
$427$	−0.907824	+	16.5095i	−0.0439327	+	0.798949i
$428$			14.2578i			0.689178i
$429$		0			0
$430$			1.07795i			0.0519834i
$431$		−	21.2205i		−	1.02216i	−0.859534	−	0.511078i	$-0.829246\pi$
$431$		−	21.2205i		−	1.02216i	0.859534	−	0.511078i	$-0.170754\pi$
$432$		0			0
$433$		−	23.9881i		−	1.15279i	−0.817170	−	0.576396i	$-0.804458\pi$
$433$		−	23.9881i		−	1.15279i	0.817170	−	0.576396i	$-0.195542\pi$
$434$	−3.20970	−	0.176495i	−0.154071	−	0.00847205i
$435$		0			0
$436$	0.289836			0.0138806
$437$	1.13808			0.0544417
$438$		0			0
$439$			16.6706i			0.795646i	0.917462	+	0.397823i	$0.130234\pi$
$439$			16.6706i			0.795646i	−0.917462	+	0.397823i	$0.869766\pi$
$440$	3.70629			0.176690
$441$		0			0
$442$	0.630426			0.0299863
$443$			32.9295i			1.56453i	0.622948	+	0.782264i	$0.285935\pi$
$443$			32.9295i			1.56453i	−0.622948	+	0.782264i	$0.714065\pi$
$444$		0			0
$445$	7.95259			0.376989
$446$	−1.18887			−0.0562947
$447$		0			0
$448$	−2.64176	−	0.145265i	−0.124811	−	0.00686314i
$449$			28.5975i			1.34960i	0.738002	+	0.674799i	$0.235770\pi$
$449$			28.5975i			1.34960i	−0.738002	+	0.674799i	$0.764230\pi$
$450$		0			0
$451$			7.18663i			0.338405i
$452$			3.45428i			0.162476i
$453$		0			0
$454$			8.96755i			0.420868i
$455$	2.15095	−	39.1167i	0.100838	−	1.83382i
$456$		0			0
$457$	−30.0918			−1.40764			−0.703818	−	0.710380i	$-0.748523\pi$
$457$	−30.0918			−1.40764			−0.703818	+	0.710380i	$0.748523\pi$
$458$	12.1250			0.566566
$459$		0			0
$460$			4.81647i			0.224569i
$461$	16.1424			0.751824			0.375912	−	0.926655i	$-0.377329\pi$
$461$	16.1424			0.751824			0.375912	+	0.926655i	$0.377329\pi$
$462$		0			0
$463$	−39.4522			−1.83350			−0.916749	−	0.399464i	$-0.869196\pi$
$463$	−39.4522			−1.83350			−0.916749	+	0.399464i	$0.869196\pi$
$464$			9.84665i			0.457119i
$465$		0			0
$466$	20.3722			0.943726
$467$	10.9473			0.506580			0.253290	−	0.967390i	$-0.418487\pi$
$467$	10.9473			0.506580			0.253290	+	0.967390i	$0.418487\pi$
$468$		0			0
$469$	−21.3361	−	1.17323i	−0.985208	−	0.0541746i
$470$		−	37.5903i		−	1.73391i
$471$		0			0
$472$		−	13.0770i		−	0.601919i
$473$			0.223061i			0.0102564i
$474$		0			0
$475$			12.9107i			0.592385i
$476$	−0.476009	−	0.0261748i	−0.0218179	−	0.00119972i
$477$		0			0
$478$	−15.3450			−0.701866
$479$	−29.4474			−1.34548			−0.672742	−	0.739877i	$-0.734884\pi$
$479$	−29.4474			−1.34548			−0.672742	+	0.739877i	$0.734884\pi$
$480$		0			0
$481$			22.0250i			1.00425i
$482$	6.49921			0.296031
$483$		0			0
$484$	−10.2331			−0.465139
$485$			25.8530i			1.17392i
$486$		0			0
$487$	−7.55471			−0.342337			−0.171168	−	0.985242i	$-0.554754\pi$
$487$	−7.55471			−0.342337			−0.171168	+	0.985242i	$0.554754\pi$
$488$	6.24942			0.282898
$489$		0			0
$490$	−3.24819	+	29.4461i	−0.146738	+	1.33024i
$491$		−	10.6059i		−	0.478636i	−0.970941	−	0.239318i	$-0.923076\pi$
$491$		−	10.6059i		−	0.478636i	0.970941	−	0.239318i	$-0.0769238\pi$
$492$		0			0
$493$			1.77423i			0.0799074i
$494$			3.49874i			0.157416i
$495$		0			0
$496$			1.21499i			0.0545545i
$497$	0.474303	−	8.62557i	0.0212754	−	0.386910i
$498$		0			0
$499$	15.7791			0.706372			0.353186	−	0.935553i	$-0.385098\pi$
$499$	15.7791			0.706372			0.353186	+	0.935553i	$0.385098\pi$
$500$	−33.4791			−1.49723
$501$		0			0
$502$			0.322392i			0.0143891i
$503$	21.9497			0.978690			0.489345	−	0.872090i	$-0.337236\pi$
$503$	21.9497			0.978690			0.489345	+	0.872090i	$0.337236\pi$
$504$		0			0
$505$	−2.94772			−0.131172
$506$			0.996677i			0.0443077i
$507$		0			0
$508$	11.9614			0.530701
$509$	2.56396			0.113645			0.0568227	−	0.998384i	$-0.481903\pi$
$509$	2.56396			0.113645			0.0568227	+	0.998384i	$0.481903\pi$
$510$		0			0
$511$	2.35799	−	42.8818i	0.104311	−	1.89698i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$			17.4765i			0.770854i
$515$			39.0671i			1.72150i
$516$		0			0
$517$		−	7.77860i		−	0.342102i
$518$	0.914463	−	16.6302i	0.0401792	−	0.730690i
$519$		0			0
$520$	−14.8070			−0.649332
$521$	34.0584			1.49212			0.746062	−	0.665876i	$-0.231942\pi$
$521$	34.0584			1.49212			0.746062	+	0.665876i	$0.231942\pi$
$522$		0			0
$523$			42.0859i			1.84029i	0.391581	+	0.920143i	$0.371928\pi$
$523$			42.0859i			1.84029i	−0.391581	+	0.920143i	$0.628072\pi$
$524$	−15.0295			−0.656569
$525$		0			0
$526$	−18.0430			−0.786712
$527$			0.218924i			0.00953649i
$528$		0			0
$529$	21.7048			0.943686
$530$	44.4994			1.93293
$531$		0			0
$532$	0.145265	−	2.64176i	0.00629805	−	0.114535i
$533$		−	28.7114i		−	1.24363i
$534$		0			0
$535$		−	60.3407i		−	2.60875i
$536$			8.07646i			0.348850i
$537$		0			0
$538$		−	10.3160i		−	0.444756i
$539$	−0.672152	+	6.09332i	−0.0289516	+	0.262458i
$540$		0			0
$541$	18.6033			0.799819			0.399909	−	0.916555i	$-0.369042\pi$
$541$	18.6033			0.799819			0.399909	+	0.916555i	$0.369042\pi$
$542$	−6.52466			−0.280258
$543$		0			0
$544$			0.180186i			0.00772543i
$545$	−1.22662			−0.0525425
$546$		0			0
$547$	−11.5450			−0.493629			−0.246814	−	0.969063i	$-0.579384\pi$
$547$	−11.5450			−0.493629			−0.246814	+	0.969063i	$0.579384\pi$
$548$		−	4.35267i		−	0.185937i
$549$		0			0
$550$	−11.3066			−0.482117
$551$	−9.84665			−0.419481
$552$		0			0
$553$	−33.3011	−	1.83116i	−1.41611	−	0.0778689i
$554$		−	8.95738i		−	0.380562i
$555$		0			0
$556$		−	11.1373i		−	0.472326i
$557$		−	14.2940i		−	0.605657i	−0.953045	−	0.302828i	$-0.902069\pi$
$557$		−	14.2940i		−	0.605657i	0.953045	−	0.302828i	$-0.0979309\pi$
$558$		0			0
$559$		−	0.891156i		−	0.0376919i
$560$	11.1802	+	0.614778i	0.472450	+	0.0259791i
$561$		0			0
$562$	−19.4917			−0.822209
$563$	−37.0385			−1.56098			−0.780492	−	0.625165i	$-0.785032\pi$
$563$	−37.0385			−1.56098			−0.780492	+	0.625165i	$0.785032\pi$
$564$		0			0
$565$		−	14.6189i		−	0.615022i
$566$	7.67409			0.322566
$567$		0			0
$568$	−3.26508			−0.137000
$569$			29.8324i			1.25064i	0.780368	+	0.625320i	$0.215032\pi$
$569$			29.8324i			1.25064i	−0.780368	+	0.625320i	$0.784968\pi$
$570$		0			0
$571$	−24.7185			−1.03444			−0.517219	−	0.855853i	$-0.673033\pi$
$571$	−24.7185			−1.03444			−0.517219	+	0.855853i	$0.673033\pi$
$572$	−3.06404			−0.128114
$573$		0			0
$574$	−1.19208	+	21.6789i	−0.0497564	+	0.904858i
$575$		−	14.6934i		−	0.612758i
$576$		0			0
$577$		−	40.1002i		−	1.66939i	−0.550710	−	0.834697i	$-0.685643\pi$
$577$		−	40.1002i		−	1.66939i	0.550710	−	0.834697i	$-0.314357\pi$
$578$		−	16.9675i		−	0.705756i
$579$		0			0
$580$		−	41.6721i		−	1.73034i
$581$	28.0394	+	1.54183i	1.16327	+	0.0639659i
$582$		0			0
$583$	9.20832			0.381369
$584$	−16.2323			−0.671697
$585$		0			0
$586$		−	25.5924i		−	1.05721i
$587$	11.6361			0.480272			0.240136	−	0.970739i	$-0.422808\pi$
$587$	11.6361			0.480272			0.240136	+	0.970739i	$0.422808\pi$
$588$		0			0
$589$	−1.21499			−0.0500627
$590$			55.3434i			2.27845i
$591$		0			0
$592$	−6.29513			−0.258728
$593$	−36.8701			−1.51407			−0.757037	−	0.653372i	$-0.773354\pi$
$593$	−36.8701			−1.51407			−0.757037	+	0.653372i	$0.773354\pi$
$594$		0			0
$595$	2.01452	+	0.110775i	0.0825874	+	0.00454132i
$596$			15.8320i			0.648502i
$597$		0			0
$598$		−	3.98184i		−	0.162830i
$599$		−	27.3962i		−	1.11938i	−0.828703	−	0.559688i	$-0.810921\pi$
$599$		−	27.3962i		−	1.11938i	0.828703	−	0.559688i	$-0.189079\pi$
$600$		0			0
$601$			22.8361i			0.931503i	0.884916	+	0.465751i	$0.154216\pi$
$601$			22.8361i			0.931503i	−0.884916	+	0.465751i	$0.845784\pi$
$602$	−0.0370001	+	0.672876i	−0.00150801	+	0.0274244i
$603$		0			0
$604$	13.0465			0.530853
$605$	43.3074			1.76070
$606$		0			0
$607$			41.7091i			1.69292i	0.532454	+	0.846459i	$0.321270\pi$
$607$			41.7091i			1.69292i	−0.532454	+	0.846459i	$0.678730\pi$
$608$	−1.00000			−0.0405554
$609$		0			0
$610$	−26.4482			−1.07086
$611$			31.0764i			1.25722i
$612$		0			0
$613$	−30.2043			−1.21994			−0.609971	−	0.792424i	$-0.708819\pi$
$613$	−30.2043			−1.21994			−0.609971	+	0.792424i	$0.708819\pi$
$614$	−5.43925			−0.219510
$615$		0			0
$616$	2.31353	+	0.127217i	0.0932149	+	0.00512571i
$617$			3.97164i			0.159892i	0.996799	+	0.0799461i	$0.0254748\pi$
$617$			3.97164i			0.159892i	−0.996799	+	0.0799461i	$0.974525\pi$
$618$		0			0
$619$		−	37.4070i		−	1.50351i	−0.659441	−	0.751756i	$-0.729207\pi$
$619$		−	37.4070i		−	1.50351i	0.659441	−	0.751756i	$-0.270793\pi$
$620$		−	5.14196i		−	0.206506i
$621$		0			0
$622$			12.6262i			0.506264i
$623$	4.96415	+	0.272969i	0.198885	+	0.0109363i
$624$		0			0
$625$	77.1335			3.08534
$626$	0.732999			0.0292965
$627$		0			0
$628$		−	9.88029i		−	0.394267i
$629$	−1.13430			−0.0452274
$630$		0			0
$631$	28.7678			1.14523			0.572614	−	0.819825i	$-0.305929\pi$
$631$	28.7678			1.14523			0.572614	+	0.819825i	$0.305929\pi$
$632$			12.6056i			0.501425i
$633$		0			0
$634$	9.57289			0.380188
$635$	−50.6219			−2.00887
$636$		0			0
$637$	2.68533	−	24.3435i	0.106396	−	0.964526i
$638$		−	8.62324i		−	0.341397i
$639$		0			0
$640$		−	4.23211i		−	0.167289i
$641$		−	38.6594i		−	1.52695i	−0.645836	−	0.763476i	$-0.723491\pi$
$641$		−	38.6594i		−	1.52695i	0.645836	−	0.763476i	$-0.276509\pi$
$642$		0			0
$643$		−	46.5874i		−	1.83723i	−0.395156	−	0.918614i	$-0.629309\pi$
$643$		−	46.5874i		−	1.83723i	0.395156	−	0.918614i	$-0.370691\pi$
$644$	−0.165323	+	3.00653i	−0.00651465	+	0.118474i
$645$		0			0
$646$	−0.180186			−0.00708934
$647$	−4.92889			−0.193775			−0.0968873	−	0.995295i	$-0.530889\pi$
$647$	−4.92889			−0.193775			−0.0968873	+	0.995295i	$0.530889\pi$
$648$		0			0
$649$			11.4523i			0.449541i
$650$	45.1713			1.77177
$651$		0			0
$652$	16.6370			0.651556
$653$			10.5757i			0.413859i	0.978356	+	0.206929i	$0.0663470\pi$
$653$			10.5757i			0.413859i	−0.978356	+	0.206929i	$0.933653\pi$
$654$		0			0
$655$	63.6067			2.48532
$656$	8.20621			0.320399
$657$		0			0
$658$	1.29027	−	23.4646i	0.0503000	−	0.914744i
$659$			3.59490i			0.140037i	0.997546	+	0.0700187i	$0.0223059\pi$
$659$			3.59490i			0.140037i	−0.997546	+	0.0700187i	$0.977694\pi$
$660$		0			0
$661$		−	4.55921i		−	0.177333i	−0.996061	−	0.0886664i	$-0.971740\pi$
$661$		−	4.55921i		−	0.177333i	0.996061	−	0.0886664i	$-0.0282605\pi$
$662$		−	23.9687i		−	0.931571i
$663$		0			0
$664$		−	10.6139i		−	0.411899i
$665$	−0.614778	+	11.1802i	−0.0238401	+	0.433550i
$666$		0			0
$667$	11.2063			0.433908
$668$	22.2000			0.858943
$669$		0			0
$670$		−	34.1804i		−	1.32051i
$671$	−5.47296			−0.211281
$672$		0			0
$673$	−15.5837			−0.600706			−0.300353	−	0.953828i	$-0.597105\pi$
$673$	−15.5837			−0.600706			−0.300353	+	0.953828i	$0.597105\pi$
$674$		−	6.53938i		−	0.251888i
$675$		0			0
$676$	−0.758811			−0.0291851
$677$	15.3765			0.590967			0.295484	−	0.955348i	$-0.404519\pi$
$677$	15.3765			0.590967			0.295484	+	0.955348i	$0.404519\pi$
$678$		0			0
$679$	−0.887393	+	16.1379i	−0.0340550	+	0.619317i
$680$		−	0.762568i		−	0.0292432i
$681$		0			0
$682$		−	1.06403i		−	0.0407438i
$683$		−	25.2842i		−	0.967474i	−0.875213	−	0.483737i	$-0.839279\pi$
$683$		−	25.2842i		−	0.967474i	0.875213	−	0.483737i	$-0.160721\pi$
$684$		0			0
$685$			18.4210i			0.703828i
$686$	−3.03831	+	18.2693i	−0.116003	+	0.697527i
$687$		0			0
$688$	0.254708			0.00971063
$689$	−36.7883			−1.40152
$690$		0			0
$691$			27.6193i			1.05069i	0.850890	+	0.525343i	$0.176063\pi$
$691$			27.6193i			1.05069i	−0.850890	+	0.525343i	$0.823937\pi$
$692$	0.122870			0.00467081
$693$		0			0
$694$	7.41154			0.281338
$695$			47.1342i			1.78790i
$696$		0			0
$697$	1.47865			0.0560078
$698$	13.2771			0.502544
$699$		0			0
$700$	−34.1071	−	1.87548i	−1.28913	−	0.0708865i
$701$			19.4636i			0.735129i	0.929998	+	0.367565i	$0.119808\pi$
$701$			19.4636i			0.735129i	−0.929998	+	0.367565i	$0.880192\pi$
$702$		0			0
$703$		−	6.29513i		−	0.237425i
$704$		−	0.875754i		−	0.0330062i
$705$		0			0
$706$			7.04350i			0.265086i
$707$	−1.84002	−	0.101179i	−0.0692012	−	0.00380524i
$708$		0			0
$709$	2.59303			0.0973834			0.0486917	−	0.998814i	$-0.484495\pi$
$709$	2.59303			0.0973834			0.0486917	+	0.998814i	$0.484495\pi$
$710$	13.8182			0.518587
$711$		0			0
$712$		−	1.87911i		−	0.0704226i
$713$	1.38275			0.0517844
$714$		0			0
$715$	12.9673			0.484951
$716$		−	6.05614i		−	0.226329i
$717$		0			0
$718$	32.2091			1.20203
$719$	34.0342			1.26926			0.634631	−	0.772816i	$-0.281152\pi$
$719$	34.0342			1.26926			0.634631	+	0.772816i	$0.281152\pi$
$720$		0			0
$721$	−1.34096	+	24.3864i	−0.0499401	+	0.908198i
$722$		−	1.00000i		−	0.0372161i
$723$		0			0
$724$		−	8.02414i		−	0.298215i
$725$			127.127i			4.72140i
$726$		0			0
$727$			30.2813i			1.12307i	0.827452	+	0.561536i	$0.189790\pi$
$727$			30.2813i			1.12307i	−0.827452	+	0.561536i	$0.810210\pi$
$728$	−9.24284	−	0.508245i	−0.342562	−	0.0188368i
$729$		0			0
$730$	68.6968			2.54258
$731$	0.0458948			0.00169748
$732$		0			0
$733$			11.7846i			0.435276i	0.976030	+	0.217638i	$0.0698352\pi$
$733$			11.7846i			0.435276i	−0.976030	+	0.217638i	$0.930165\pi$
$734$	−23.7850			−0.877922
$735$		0			0
$736$	1.13808			0.0419501
$737$		−	7.07299i		−	0.260537i
$738$		0			0
$739$	26.9854			0.992675			0.496338	−	0.868130i	$-0.334678\pi$
$739$	26.9854			0.992675			0.496338	+	0.868130i	$0.334678\pi$
$740$	26.6417			0.979367
$741$		0			0
$742$	27.7774	+	1.52742i	1.01974	+	0.0560735i
$743$		−	12.6827i		−	0.465283i	−0.972563	−	0.232641i	$-0.925263\pi$
$743$		−	12.6827i		−	0.465283i	0.972563	−	0.232641i	$-0.0747368\pi$
$744$		0			0
$745$		−	67.0025i		−	2.45478i
$746$			16.3365i			0.598120i
$747$		0			0
$748$		−	0.157799i		−	0.00576970i
$749$	2.07117	−	37.6658i	0.0756788	−	1.37628i
$750$		0			0
$751$	−24.2924			−0.886441			−0.443221	−	0.896413i	$-0.646164\pi$
$751$	−24.2924			−0.886441			−0.443221	+	0.896413i	$0.646164\pi$
$752$	−8.88217			−0.323899
$753$		0			0
$754$			34.4509i			1.25463i
$755$	−55.2141			−2.00945
$756$		0			0
$757$	−5.35357			−0.194579			−0.0972895	−	0.995256i	$-0.531017\pi$
$757$	−5.35357			−0.194579			−0.0972895	+	0.995256i	$0.531017\pi$
$758$			4.94720i			0.179690i
$759$		0			0
$760$	4.23211			0.153515
$761$	−2.24818			−0.0814964			−0.0407482	−	0.999169i	$-0.512974\pi$
$761$	−2.24818			−0.0814964			−0.0407482	+	0.999169i	$0.512974\pi$
$762$		0			0
$763$	−0.765677	−	0.0421031i	−0.0277194	−	0.00152423i
$764$		−	17.3111i		−	0.626294i
$765$		0			0
$766$		−	24.4256i		−	0.882534i
$767$		−	45.7531i		−	1.65205i
$768$		0			0
$769$			25.4350i			0.917209i	0.888640	+	0.458605i	$0.151651\pi$
$769$			25.4350i			0.917209i	−0.888640	+	0.458605i	$0.848349\pi$
$770$	−9.79112	−	0.538395i	−0.352848	−	0.0194024i
$771$		0			0
$772$	10.0773			0.362689
$773$	−33.8884			−1.21888			−0.609441	−	0.792832i	$-0.708606\pi$
$773$	−33.8884			−1.21888			−0.609441	+	0.792832i	$0.708606\pi$
$774$		0			0
$775$			15.6864i			0.563471i
$776$	6.10878			0.219292
$777$		0			0
$778$	−8.72327			−0.312744
$779$			8.20621i			0.294018i
$780$		0			0
$781$	2.85941			0.102318
$782$	0.205066			0.00733315
$783$		0			0
$784$	6.95780	+	0.767512i	0.248493	+	0.0274111i
$785$			41.8145i			1.49242i
$786$		0			0
$787$		−	12.8267i		−	0.457223i	−0.973518	−	0.228611i	$-0.926582\pi$
$787$		−	12.8267i		−	0.457223i	0.973518	−	0.228611i	$-0.0734185\pi$
$788$		−	21.0687i		−	0.750540i
$789$		0			0
$790$		−	53.3484i		−	1.89805i
$791$	0.501787	−	9.12539i	0.0178415	−	0.324461i
$792$		0			0
$793$	21.8651			0.776453
$794$	36.1479			1.28284
$795$		0			0
$796$			9.30112i			0.329669i
$797$	−25.8239			−0.914728			−0.457364	−	0.889279i	$-0.651206\pi$
$797$	−25.8239			−0.914728			−0.457364	+	0.889279i	$0.651206\pi$
$798$		0			0
$799$	−1.60045			−0.0566197
$800$			12.9107i			0.456463i
$801$		0			0
$802$	35.3134			1.24696
$803$	14.2155			0.501654
$804$		0			0
$805$	0.699666	−	12.7240i	0.0246600	−	0.448461i
$806$			4.25092i			0.149732i
$807$		0			0
$808$			0.696515i			0.0245033i
$809$		−	6.00829i		−	0.211240i	−0.994407	−	0.105620i	$-0.966317\pi$
$809$		−	6.00829i		−	0.211240i	0.994407	−	0.105620i	$-0.0336828\pi$
$810$		0			0
$811$		−	35.7769i		−	1.25630i	−0.778093	−	0.628148i	$-0.783813\pi$
$811$		−	35.7769i		−	1.25630i	0.778093	−	0.628148i	$-0.216187\pi$
$812$	1.43038	−	26.0125i	0.0501963	−	0.912859i
$813$		0			0
$814$	5.51299			0.193230
$815$	−70.4097			−2.46634
$816$		0			0
$817$			0.254708i			0.00891109i
$818$	29.8515			1.04373
$819$		0			0
$820$	−34.7296			−1.21281
$821$			37.6807i			1.31507i	0.753426	+	0.657533i	$0.228400\pi$
$821$			37.6807i			1.31507i	−0.753426	+	0.657533i	$0.771600\pi$
$822$		0			0
$823$	1.31530			0.0458486			0.0229243	−	0.999737i	$-0.492702\pi$
$823$	1.31530			0.0458486			0.0229243	+	0.999737i	$0.492702\pi$
$824$	9.23113			0.321582
$825$		0			0
$826$	−1.89964	+	34.5464i	−0.0660968	+	1.20202i
$827$		−	26.4063i		−	0.918237i	−0.888375	−	0.459119i	$-0.848165\pi$
$827$		−	26.4063i		−	0.918237i	0.888375	−	0.459119i	$-0.151835\pi$
$828$		0			0
$829$			13.2112i			0.458842i	0.973327	+	0.229421i	$0.0736833\pi$
$829$			13.2112i			0.458842i	−0.973327	+	0.229421i	$0.926317\pi$
$830$			44.9191i			1.55917i
$831$		0			0
$832$			3.49874i			0.121297i
$833$	1.25370	+	0.138295i	0.0434381	+	0.00479165i
$834$		0			0
$835$	−93.9528			−3.25137
$836$	0.875754			0.0302886
$837$		0			0
$838$		−	35.3705i		−	1.22185i
$839$	−17.7815			−0.613885			−0.306943	−	0.951728i	$-0.599306\pi$
$839$	−17.7815			−0.613885			−0.306943	+	0.951728i	$0.599306\pi$
$840$		0			0
$841$	−67.9564			−2.34333
$842$			0.362549i			0.0124942i
$843$		0			0
$844$	6.46105			0.222399
$845$	3.21137			0.110475
$846$		0			0
$847$	27.0333	+	1.48651i	0.928874	+	0.0510770i
$848$		−	10.5147i		−	0.361077i
$849$		0			0
$850$			2.32634i			0.0797928i
$851$			7.16435i			0.245591i
$852$		0			0
$853$		−	46.0268i		−	1.57593i	−0.615721	−	0.787964i	$-0.711135\pi$
$853$		−	46.0268i		−	1.57593i	0.615721	−	0.787964i	$-0.288865\pi$
$854$	−16.5095	−	0.907824i	−0.564943	−	0.0310651i
$855$		0			0
$856$	−14.2578			−0.487323
$857$	26.3033			0.898504			0.449252	−	0.893405i	$-0.351690\pi$
$857$	26.3033			0.898504			0.449252	+	0.893405i	$0.351690\pi$
$858$		0			0
$859$		−	47.0322i		−	1.60472i	−0.596841	−	0.802360i	$-0.703578\pi$
$859$		−	47.0322i		−	1.60472i	0.596841	−	0.802360i	$-0.296422\pi$
$860$	−1.07795			−0.0367578
$861$		0			0
$862$	21.2205			0.722774
$863$		−	32.2633i		−	1.09826i	−0.835738	−	0.549128i	$-0.814960\pi$
$863$		−	32.2633i		−	1.09826i	0.835738	−	0.549128i	$-0.185040\pi$
$864$		0			0
$865$	−0.519999			−0.0176805
$866$	23.9881			0.815147
$867$		0			0
$868$	0.176495	−	3.20970i	0.00599064	−	0.108944i
$869$		−	11.0394i		−	0.374487i
$870$		0			0
$871$			28.2574i			0.957466i
$872$			0.289836i			0.00981508i
$873$		0			0
$874$			1.13808i			0.0384961i
$875$	88.4437	+	4.86335i	2.98994	+	0.164411i
$876$		0			0
$877$	−46.2269			−1.56097			−0.780486	−	0.625173i	$-0.785028\pi$
$877$	−46.2269			−1.56097			−0.780486	+	0.625173i	$0.785028\pi$
$878$	−16.6706			−0.562607
$879$		0			0
$880$			3.70629i			0.124939i
$881$	−36.7738			−1.23894			−0.619471	−	0.785019i	$-0.712653\pi$
$881$	−36.7738			−1.23894			−0.619471	+	0.785019i	$0.712653\pi$
$882$		0			0
$883$	8.71720			0.293357			0.146679	−	0.989184i	$-0.453142\pi$
$883$	8.71720			0.293357			0.146679	+	0.989184i	$0.453142\pi$
$884$			0.630426i			0.0212035i
$885$		0			0
$886$	−32.9295			−1.10629
$887$	−20.1170			−0.675462			−0.337731	−	0.941243i	$-0.609659\pi$
$887$	−20.1170			−0.675462			−0.337731	+	0.941243i	$0.609659\pi$
$888$		0			0
$889$	−31.5992	−	1.73758i	−1.05980	−	0.0582764i
$890$			7.95259i			0.266571i
$891$		0			0
$892$		−	1.18887i		−	0.0398064i
$893$		−	8.88217i		−	0.297231i
$894$		0			0
$895$			25.6302i			0.856725i
$896$	0.145265	−	2.64176i	0.00485297	−	0.0882550i
$897$		0			0
$898$	−28.5975			−0.954309
$899$	−11.9635			−0.399007
$900$		0			0
$901$		−	1.89461i		−	0.0631186i
$902$	−7.18663			−0.239289
$903$		0			0
$904$	−3.45428			−0.114888
$905$			33.9590i			1.12884i
$906$		0			0
$907$	26.9575			0.895109			0.447554	−	0.894257i	$-0.352295\pi$
$907$	26.9575			0.895109			0.447554	+	0.894257i	$0.352295\pi$
$908$	−8.96755			−0.297599
$909$		0			0
$910$	39.1167	+	2.15095i	1.29671	+	0.0713033i
$911$		−	22.3583i		−	0.740763i	−0.928880	−	0.370381i	$-0.879227\pi$
$911$		−	22.3583i		−	0.740763i	0.928880	−	0.370381i	$-0.120773\pi$
$912$		0			0
$913$			9.29516i			0.307625i
$914$		−	30.0918i		−	0.995350i
$915$		0			0
$916$			12.1250i			0.400623i
$917$	39.7045	+	2.18327i	1.31116	+	0.0720980i
$918$		0			0
$919$	23.2018			0.765355			0.382678	−	0.923882i	$-0.375002\pi$
$919$	23.2018			0.765355			0.382678	+	0.923882i	$0.375002\pi$
$920$	−4.81647			−0.158794
$921$		0			0
$922$			16.1424i			0.531620i
$923$	−11.4237			−0.376015
$924$		0			0
$925$	−81.2748			−2.67230
$926$		−	39.4522i		−	1.29648i
$927$		0			0
$928$	−9.84665			−0.323232
$929$	13.7521			0.451193			0.225597	−	0.974221i	$-0.427567\pi$
$929$	13.7521			0.451193			0.225597	+	0.974221i	$0.427567\pi$
$930$		0			0
$931$	−0.767512	+	6.95780i	−0.0251542	+	0.228033i
$932$			20.3722i			0.667315i
$933$		0			0
$934$			10.9473i			0.358206i
$935$			0.667823i			0.0218401i
$936$		0			0
$937$		−	13.6798i		−	0.446899i	−0.974716	−	0.223449i	$-0.928268\pi$
$937$		−	13.6798i		−	0.446899i	0.974716	−	0.223449i	$-0.0717317\pi$
$938$	1.17323	−	21.3361i	0.0383073	−	0.696647i
$939$		0			0
$940$	37.5903			1.22606
$941$	24.1573			0.787507			0.393753	−	0.919216i	$-0.371176\pi$
$941$	24.1573			0.787507			0.393753	+	0.919216i	$0.371176\pi$
$942$		0			0
$943$		−	9.33931i		−	0.304130i
$944$	13.0770			0.425621
$945$		0			0
$946$	−0.223061			−0.00725235
$947$			42.5128i			1.38148i	0.723104	+	0.690739i	$0.242715\pi$
$947$			42.5128i			1.38148i	−0.723104	+	0.690739i	$0.757285\pi$
$948$		0			0
$949$	−56.7926			−1.84356
$950$	−12.9107			−0.418880
$951$		0			0
$952$	0.0261748	−	0.476009i	0.000848331	−	0.0154276i
$953$			50.9995i			1.65204i	0.563644	+	0.826018i	$0.309399\pi$
$953$			50.9995i			1.65204i	−0.563644	+	0.826018i	$0.690601\pi$
$954$		0			0
$955$			73.2624i			2.37072i
$956$		−	15.3450i		−	0.496294i
$957$		0			0
$958$		−	29.4474i		−	0.951401i
$959$	−0.632291	+	11.4987i	−0.0204177	+	0.371312i
$960$		0			0
$961$	29.5238			0.952381
$962$	−22.0250			−0.710115
$963$		0			0
$964$			6.49921i			0.209325i
$965$	−42.6482			−1.37289
$966$		0			0
$967$	39.0256			1.25498			0.627490	−	0.778625i	$-0.284082\pi$
$967$	39.0256			1.25498			0.627490	+	0.778625i	$0.284082\pi$
$968$		−	10.2331i		−	0.328903i
$969$		0			0
$970$	−25.8530			−0.830090
$971$	−16.6748			−0.535120			−0.267560	−	0.963541i	$-0.586217\pi$
$971$	−16.6748			−0.535120			−0.267560	+	0.963541i	$0.586217\pi$
$972$		0			0
$973$	−1.61786	+	29.4220i	−0.0518662	+	0.943227i
$974$		−	7.55471i		−	0.242069i
$975$		0			0
$976$			6.24942i			0.200039i
$977$		−	24.7723i		−	0.792537i	−0.918135	−	0.396269i	$-0.870305\pi$
$977$		−	24.7723i		−	0.792537i	0.918135	−	0.396269i	$-0.129695\pi$
$978$		0			0
$979$			1.64564i			0.0525948i
$980$	−29.4461	−	3.24819i	−0.940623	−	0.103760i
$981$		0			0
$982$	10.6059			0.338446
$983$	−29.3741			−0.936890			−0.468445	−	0.883493i	$-0.655186\pi$
$983$	−29.3741			−0.936890			−0.468445	+	0.883493i	$0.655186\pi$
$984$		0			0
$985$			89.1648i			2.84103i
$986$	−1.77423			−0.0565031
$987$		0			0
$988$	−3.49874			−0.111310
$989$		−	0.289877i		−	0.00921755i
$990$		0			0
$991$	22.9403			0.728723			0.364361	−	0.931258i	$-0.381287\pi$
$991$	22.9403			0.728723			0.364361	+	0.931258i	$0.381287\pi$
$992$	−1.21499			−0.0385759
$993$		0			0
$994$	8.62557	+	0.474303i	0.273586	+	0.0150440i
$995$		−	39.3633i		−	1.24790i
$996$		0			0
$997$			37.0368i			1.17297i	0.809961	+	0.586483i	$0.199488\pi$
$997$			37.0368i			1.17297i	−0.809961	+	0.586483i	$0.800512\pi$
$998$			15.7791i			0.499480i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2394.2.f.b.2015.4	yes	24
3.2	odd	2		2394.2.f.a.2015.3	✓	24
7.6	odd	2		2394.2.f.a.2015.4	yes	24
21.20	even	2	inner	2394.2.f.b.2015.3	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2394.2.f.a.2015.3	✓	24	3.2	odd	2
2394.2.f.a.2015.4	yes	24	7.6	odd	2
2394.2.f.b.2015.3	yes	24	21.20	even	2	inner
2394.2.f.b.2015.4	yes	24	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:	\( N \)	\(=\)	\( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2394.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +4.23211 q^{5} +(2.64176 + 0.145265i) q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +4.23211 q^{5} +(2.64176 + 0.145265i) q^{7} -1.00000i q^{8} +4.23211i q^{10} +0.875754i q^{11} -3.49874i q^{13} +(-0.145265 + 2.64176i) q^{14} +1.00000 q^{16} +0.180186 q^{17} +1.00000i q^{19} -4.23211 q^{20} -0.875754 q^{22} -1.13808i q^{23} +12.9107 q^{25} +3.49874 q^{26} +(-2.64176 - 0.145265i) q^{28} +9.84665i q^{29} +1.21499i q^{31} +1.00000i q^{32} +0.180186i q^{34} +(11.1802 + 0.614778i) q^{35} -6.29513 q^{37} -1.00000 q^{38} -4.23211i q^{40} +8.20621 q^{41} +0.254708 q^{43} -0.875754i q^{44} +1.13808 q^{46} -8.88217 q^{47} +(6.95780 + 0.767512i) q^{49} +12.9107i q^{50} +3.49874i q^{52} -10.5147i q^{53} +3.70629i q^{55} +(0.145265 - 2.64176i) q^{56} -9.84665 q^{58} +13.0770 q^{59} +6.24942i q^{61} -1.21499 q^{62} -1.00000 q^{64} -14.8070i q^{65} -8.07646 q^{67} -0.180186 q^{68} +(-0.614778 + 11.1802i) q^{70} -3.26508i q^{71} -16.2323i q^{73} -6.29513i q^{74} -1.00000i q^{76} +(-0.127217 + 2.31353i) q^{77} -12.6056 q^{79} +4.23211 q^{80} +8.20621i q^{82} +10.6139 q^{83} +0.762568 q^{85} +0.254708i q^{86} +0.875754 q^{88} +1.87911 q^{89} +(0.508245 - 9.24284i) q^{91} +1.13808i q^{92} -8.88217i q^{94} +4.23211i q^{95} +6.10878i q^{97} +(-0.767512 + 6.95780i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{4} + 4 q^{7}+O(q^{10}) \) 24 * q - 24 * q^4 + 4 * q^7	\( 24 q - 24 q^{4} + 4 q^{7} + 4 q^{14} + 24 q^{16} - 32 q^{17} - 8 q^{22} + 16 q^{25} - 4 q^{28} + 24 q^{35} - 24 q^{38} + 8 q^{41} + 16 q^{43} - 8 q^{46} - 16 q^{49} - 4 q^{56} + 16 q^{58} - 16 q^{59} + 16 q^{62} - 24 q^{64} - 24 q^{67} + 32 q^{68} - 16 q^{70} - 8 q^{77} - 40 q^{79} + 64 q^{83} + 40 q^{85} + 8 q^{88} + 64 q^{89} + 8 q^{91} + 16 q^{98}+O(q^{100}) \) 24 * q - 24 * q^4 + 4 * q^7 + 4 * q^14 + 24 * q^16 - 32 * q^17 - 8 * q^22 + 16 * q^25 - 4 * q^28 + 24 * q^35 - 24 * q^38 + 8 * q^41 + 16 * q^43 - 8 * q^46 - 16 * q^49 - 4 * q^56 + 16 * q^58 - 16 * q^59 + 16 * q^62 - 24 * q^64 - 24 * q^67 + 32 * q^68 - 16 * q^70 - 8 * q^77 - 40 * q^79 + 64 * q^83 + 40 * q^85 + 8 * q^88 + 64 * q^89 + 8 * q^91 + 16 * q^98

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