LMFDB - Embedded newform 2394.2.f.b.2015.7

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			2015.7
Character	$\chi$	$=$	2394.2015
Dual form			2394.2.f.b.2015.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -1.26222 q^{5} +(1.09287 - 2.40949i) q^{7} +1.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} -1.26222 q^{5} +(1.09287 - 2.40949i) q^{7} +1.00000i q^{8} +1.26222i q^{10} +2.94604i q^{11} +0.580865i q^{13} +(-2.40949 - 1.09287i) q^{14} +1.00000 q^{16} +0.124939 q^{17} -1.00000i q^{19} +1.26222 q^{20} +2.94604 q^{22} +5.45717i q^{23} -3.40679 q^{25} +0.580865 q^{26} +(-1.09287 + 2.40949i) q^{28} +3.23794i q^{29} +2.94306i q^{31} -1.00000i q^{32} -0.124939i q^{34} +(-1.37945 + 3.04131i) q^{35} -0.135864 q^{37} -1.00000 q^{38} -1.26222i q^{40} +5.43121 q^{41} -2.47700 q^{43} -2.94604i q^{44} +5.45717 q^{46} +6.95074 q^{47} +(-4.61127 - 5.26651i) q^{49} +3.40679i q^{50} -0.580865i q^{52} +5.46970i q^{53} -3.71856i q^{55} +(2.40949 + 1.09287i) q^{56} +3.23794 q^{58} -6.29094 q^{59} -0.467276i q^{61} +2.94306 q^{62} -1.00000 q^{64} -0.733182i q^{65} +5.93507 q^{67} -0.124939 q^{68} +(3.04131 + 1.37945i) q^{70} +16.1645i q^{71} +11.4154i q^{73} +0.135864i q^{74} +1.00000i q^{76} +(7.09845 + 3.21964i) q^{77} +0.468124 q^{79} -1.26222 q^{80} -5.43121i q^{82} +9.79950 q^{83} -0.157701 q^{85} +2.47700i q^{86} -2.94604 q^{88} +9.24616 q^{89} +(1.39959 + 0.634810i) q^{91} -5.45717i q^{92} -6.95074i q^{94} +1.26222i q^{95} -7.54319i q^{97} +(-5.26651 + 4.61127i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−1.26222			−0.564484			−0.282242	−	0.959343i	$-0.591078\pi$
$5$	−1.26222			−0.564484			−0.282242	+	0.959343i	$0.591078\pi$
$6$		0			0
$7$	1.09287	−	2.40949i	0.413066	−	0.910701i
$7$	1.09287	−	2.40949i	0.413066	−	0.910701i
$8$			1.00000i			0.353553i
$9$		0			0
$10$			1.26222i			0.399150i
$11$			2.94604i			0.888264i	0.895961	+	0.444132i	$0.146488\pi$
$11$			2.94604i			0.888264i	−0.895961	+	0.444132i	$0.853512\pi$
$12$		0			0
$13$			0.580865i			0.161103i	0.996750	+	0.0805515i	$0.0256682\pi$
$13$			0.580865i			0.161103i	−0.996750	+	0.0805515i	$0.974332\pi$
$14$	−2.40949	−	1.09287i	−0.643963	−	0.292082i
$15$		0			0
$16$	1.00000			0.250000
$17$	0.124939			0.0303021			0.0151510	−	0.999885i	$-0.495177\pi$
$17$	0.124939			0.0303021			0.0151510	+	0.999885i	$0.495177\pi$
$18$		0			0
$19$		−	1.00000i		−	0.229416i
$19$		−	1.00000i		−	0.229416i
$20$	1.26222			0.282242
$21$		0			0
$22$	2.94604			0.628098
$23$			5.45717i			1.13790i	0.822372	+	0.568950i	$0.192650\pi$
$23$			5.45717i			1.13790i	−0.822372	+	0.568950i	$0.807350\pi$
$24$		0			0
$25$	−3.40679			−0.681358
$26$	0.580865			0.113917
$27$		0			0
$28$	−1.09287	+	2.40949i	−0.206533	+	0.455351i
$29$			3.23794i			0.601270i	0.953739	+	0.300635i	$0.0971986\pi$
$29$			3.23794i			0.601270i	−0.953739	+	0.300635i	$0.902801\pi$
$30$		0			0
$31$			2.94306i			0.528589i	0.964442	+	0.264294i	$0.0851391\pi$
$31$			2.94306i			0.528589i	−0.964442	+	0.264294i	$0.914861\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$		−	0.124939i		−	0.0214268i
$35$	−1.37945	+	3.04131i	−0.233169	+	0.514076i
$36$		0			0
$37$	−0.135864			−0.0223358			−0.0111679	−	0.999938i	$-0.503555\pi$
$37$	−0.135864			−0.0223358			−0.0111679	+	0.999938i	$0.503555\pi$
$38$	−1.00000			−0.162221
$39$		0			0
$40$		−	1.26222i		−	0.199575i
$41$	5.43121			0.848213			0.424106	−	0.905612i	$-0.360588\pi$
$41$	5.43121			0.848213			0.424106	+	0.905612i	$0.360588\pi$
$42$		0			0
$43$	−2.47700			−0.377739			−0.188869	−	0.982002i	$-0.560482\pi$
$43$	−2.47700			−0.377739			−0.188869	+	0.982002i	$0.560482\pi$
$44$		−	2.94604i		−	0.444132i
$45$		0			0
$46$	5.45717			0.804616
$47$	6.95074			1.01387			0.506935	−	0.861984i	$-0.330778\pi$
$47$	6.95074			1.01387			0.506935	+	0.861984i	$0.330778\pi$
$48$		0			0
$49$	−4.61127	−	5.26651i	−0.658753	−	0.752359i
$50$			3.40679i			0.481793i
$51$		0			0
$52$		−	0.580865i		−	0.0805515i
$53$			5.46970i			0.751321i	0.926757	+	0.375661i	$0.122584\pi$
$53$			5.46970i			0.751321i	−0.926757	+	0.375661i	$0.877416\pi$
$54$		0			0
$55$		−	3.71856i		−	0.501411i
$56$	2.40949	+	1.09287i	0.321981	+	0.146041i
$57$		0			0
$58$	3.23794			0.425162
$59$	−6.29094			−0.819011			−0.409505	−	0.912308i	$-0.634299\pi$
$59$	−6.29094			−0.819011			−0.409505	+	0.912308i	$0.634299\pi$
$60$		0			0
$61$		−	0.467276i		−	0.0598286i	−0.999552	−	0.0299143i	$-0.990477\pi$
$61$		−	0.467276i		−	0.0598286i	0.999552	−	0.0299143i	$-0.00952343\pi$
$62$	2.94306			0.373769
$63$		0			0
$64$	−1.00000			−0.125000
$65$		−	0.733182i		−	0.0909401i
$66$		0			0
$67$	5.93507			0.725084			0.362542	−	0.931967i	$-0.381909\pi$
$67$	5.93507			0.725084			0.362542	+	0.931967i	$0.381909\pi$
$68$	−0.124939			−0.0151510
$69$		0			0
$70$	3.04131	+	1.37945i	0.363507	+	0.164875i
$71$			16.1645i			1.91837i	0.282775	+	0.959186i	$0.408745\pi$
$71$			16.1645i			1.91837i	−0.282775	+	0.959186i	$0.591255\pi$
$72$		0			0
$73$			11.4154i			1.33608i	0.744127	+	0.668038i	$0.232866\pi$
$73$			11.4154i			1.33608i	−0.744127	+	0.668038i	$0.767134\pi$
$74$			0.135864i			0.0157938i
$75$		0			0
$76$			1.00000i			0.114708i
$77$	7.09845	+	3.21964i	0.808943	+	0.366912i
$78$		0			0
$79$	0.468124			0.0526680			0.0263340	−	0.999653i	$-0.491617\pi$
$79$	0.468124			0.0526680			0.0263340	+	0.999653i	$0.491617\pi$
$80$	−1.26222			−0.141121
$81$		0			0
$82$		−	5.43121i		−	0.599777i
$83$	9.79950			1.07564			0.537818	−	0.843061i	$-0.319249\pi$
$83$	9.79950			1.07564			0.537818	+	0.843061i	$0.319249\pi$
$84$		0			0
$85$	−0.157701			−0.0171050
$86$			2.47700i			0.267102i
$87$		0			0
$88$	−2.94604			−0.314049
$89$	9.24616			0.980091			0.490045	−	0.871697i	$-0.336980\pi$
$89$	9.24616			0.980091			0.490045	+	0.871697i	$0.336980\pi$
$90$		0			0
$91$	1.39959	+	0.634810i	0.146717	+	0.0665462i
$92$		−	5.45717i		−	0.568950i
$93$		0			0
$94$		−	6.95074i		−	0.716915i
$95$			1.26222i			0.129501i
$96$		0			0
$97$		−	7.54319i		−	0.765895i	−0.923770	−	0.382948i	$-0.874909\pi$
$97$		−	7.54319i		−	0.765895i	0.923770	−	0.382948i	$-0.125091\pi$
$98$	−5.26651	+	4.61127i	−0.531998	+	0.465809i
$99$		0			0
$100$	3.40679			0.340679
$101$	4.60873			0.458586			0.229293	−	0.973357i	$-0.426359\pi$
$101$	4.60873			0.458586			0.229293	+	0.973357i	$0.426359\pi$
$102$		0			0
$103$		−	11.8587i		−	1.16848i	−0.811582	−	0.584238i	$-0.801393\pi$
$103$		−	11.8587i		−	1.16848i	0.811582	−	0.584238i	$-0.198607\pi$
$104$	−0.580865			−0.0569585
$105$		0			0
$106$	5.46970			0.531264
$107$			2.48007i			0.239757i	0.992789	+	0.119879i	$0.0382505\pi$
$107$			2.48007i			0.239757i	−0.992789	+	0.119879i	$0.961749\pi$
$108$		0			0
$109$	6.64142			0.636132			0.318066	−	0.948069i	$-0.396967\pi$
$109$	6.64142			0.636132			0.318066	+	0.948069i	$0.396967\pi$
$110$	−3.71856			−0.354551
$111$		0			0
$112$	1.09287	−	2.40949i	0.103266	−	0.227675i
$113$			5.59170i			0.526023i	0.964793	+	0.263012i	$0.0847158\pi$
$113$			5.59170i			0.526023i	−0.964793	+	0.263012i	$0.915284\pi$
$114$		0			0
$115$		−	6.88818i		−	0.642326i
$116$		−	3.23794i		−	0.300635i
$117$		0			0
$118$			6.29094i			0.579128i
$119$	0.136542	−	0.301038i	0.0125168	−	0.0275961i
$120$		0			0
$121$	2.32085			0.210986
$122$	−0.467276			−0.0423052
$123$		0			0
$124$		−	2.94306i		−	0.264294i
$125$	10.6113			0.949099
$126$		0			0
$127$	9.37751			0.832119			0.416060	−	0.909337i	$-0.363411\pi$
$127$	9.37751			0.832119			0.416060	+	0.909337i	$0.363411\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	−0.733182			−0.0643043
$131$	−14.9045			−1.30221			−0.651107	−	0.758986i	$-0.725695\pi$
$131$	−14.9045			−1.30221			−0.651107	+	0.758986i	$0.725695\pi$
$132$		0			0
$133$	−2.40949	−	1.09287i	−0.208929	−	0.0947638i
$134$		−	5.93507i		−	0.512712i
$135$		0			0
$136$			0.124939i			0.0107134i
$137$		−	1.93955i		−	0.165707i	−0.996562	−	0.0828533i	$-0.973597\pi$
$137$		−	1.93955i		−	0.165707i	0.996562	−	0.0828533i	$-0.0264033\pi$
$138$		0			0
$139$			12.8881i			1.09316i	0.837408	+	0.546579i	$0.184070\pi$
$139$			12.8881i			1.09316i	−0.837408	+	0.546579i	$0.815930\pi$
$140$	1.37945	−	3.04131i	0.116584	−	0.257038i
$141$		0			0
$142$	16.1645			1.35649
$143$	−1.71125			−0.143102
$144$		0			0
$145$		−	4.08700i		−	0.339407i
$146$	11.4154			0.944749
$147$		0			0
$148$	0.135864			0.0111679
$149$		−	4.62005i		−	0.378489i	−0.981930	−	0.189244i	$-0.939396\pi$
$149$		−	4.62005i		−	0.378489i	0.981930	−	0.189244i	$-0.0606039\pi$
$150$		0			0
$151$	8.86316			0.721274			0.360637	−	0.932706i	$-0.382559\pi$
$151$	8.86316			0.721274			0.360637	+	0.932706i	$0.382559\pi$
$152$	1.00000			0.0811107
$153$		0			0
$154$	3.21964	−	7.09845i	0.259446	−	0.572009i
$155$		−	3.71480i		−	0.298380i
$156$		0			0
$157$			13.9759i			1.11540i	0.830043	+	0.557699i	$0.188316\pi$
$157$			13.9759i			1.11540i	−0.830043	+	0.557699i	$0.811684\pi$
$158$		−	0.468124i		−	0.0372419i
$159$		0			0
$160$			1.26222i			0.0997876i
$161$	13.1490	+	5.96398i	1.03629	+	0.470027i
$162$		0			0
$163$	18.0963			1.41741			0.708707	−	0.705503i	$-0.249279\pi$
$163$	18.0963			1.41741			0.708707	+	0.705503i	$0.249279\pi$
$164$	−5.43121			−0.424106
$165$		0			0
$166$		−	9.79950i		−	0.760589i
$167$	−0.135153			−0.0104585			−0.00522924	−	0.999986i	$-0.501665\pi$
$167$	−0.135153			−0.0104585			−0.00522924	+	0.999986i	$0.501665\pi$
$168$		0			0
$169$	12.6626			0.974046
$170$			0.157701i			0.0120951i
$171$		0			0
$172$	2.47700			0.188869
$173$	−11.2899			−0.858353			−0.429176	−	0.903221i	$-0.641196\pi$
$173$	−11.2899			−0.858353			−0.429176	+	0.903221i	$0.641196\pi$
$174$		0			0
$175$	−3.72318	+	8.20862i	−0.281446	+	0.620514i
$176$			2.94604i			0.222066i
$177$		0			0
$178$		−	9.24616i		−	0.693029i
$179$			16.1068i			1.20388i	0.798543	+	0.601938i	$0.205605\pi$
$179$			16.1068i			1.20388i	−0.798543	+	0.601938i	$0.794395\pi$
$180$		0			0
$181$			4.83840i			0.359635i	0.983700	+	0.179818i	$0.0575507\pi$
$181$			4.83840i			0.359635i	−0.983700	+	0.179818i	$0.942449\pi$
$182$	0.634810	−	1.39959i	0.0470552	−	0.103744i
$183$		0			0
$184$	−5.45717			−0.402308
$185$	0.171490			0.0126082
$186$		0			0
$187$			0.368074i			0.0269163i
$188$	−6.95074			−0.506935
$189$		0			0
$190$	1.26222			0.0915714
$191$			0.947510i			0.0685594i	0.999412	+	0.0342797i	$0.0109137\pi$
$191$			0.947510i			0.0685594i	−0.999412	+	0.0342797i	$0.989086\pi$
$192$		0			0
$193$	9.93469			0.715115			0.357557	−	0.933891i	$-0.383610\pi$
$193$	9.93469			0.715115			0.357557	+	0.933891i	$0.383610\pi$
$194$	−7.54319			−0.541570
$195$		0			0
$196$	4.61127	+	5.26651i	0.329377	+	0.376180i
$197$			22.2190i			1.58304i	0.611145	+	0.791519i	$0.290709\pi$
$197$			22.2190i			1.58304i	−0.611145	+	0.791519i	$0.709291\pi$
$198$		0			0
$199$			4.47254i			0.317050i	0.987355	+	0.158525i	$0.0506738\pi$
$199$			4.47254i			0.317050i	−0.987355	+	0.158525i	$0.949326\pi$
$200$		−	3.40679i		−	0.240896i
$201$		0			0
$202$		−	4.60873i		−	0.324269i
$203$	7.80178	+	3.53864i	0.547577	+	0.248364i
$204$		0			0
$205$	−6.85541			−0.478802
$206$	−11.8587			−0.826238
$207$		0			0
$208$			0.580865i			0.0402758i
$209$	2.94604			0.203782
$210$		0			0
$211$	−10.8585			−0.747528			−0.373764	−	0.927524i	$-0.621933\pi$
$211$	−10.8585			−0.747528			−0.373764	+	0.927524i	$0.621933\pi$
$212$		−	5.46970i		−	0.375661i
$213$		0			0
$214$	2.48007			0.169534
$215$	3.12653			0.213227
$216$		0			0
$217$	7.09127	+	3.21638i	0.481387	+	0.218342i
$218$		−	6.64142i		−	0.449814i
$219$		0			0
$220$			3.71856i			0.250705i
$221$			0.0725725i			0.00488176i
$222$		0			0
$223$		−	1.40924i		−	0.0943698i	−0.998886	−	0.0471849i	$-0.984975\pi$
$223$		−	1.40924i		−	0.0943698i	0.998886	−	0.0471849i	$-0.0150250\pi$
$224$	−2.40949	−	1.09287i	−0.160991	−	0.0730204i
$225$		0			0
$226$	5.59170			0.371955
$227$	−13.8910			−0.921977			−0.460989	−	0.887406i	$-0.652505\pi$
$227$	−13.8910			−0.921977			−0.460989	+	0.887406i	$0.652505\pi$
$228$		0			0
$229$		−	16.9038i		−	1.11703i	−0.829493	−	0.558517i	$-0.811370\pi$
$229$		−	16.9038i		−	1.11703i	0.829493	−	0.558517i	$-0.188630\pi$
$230$	−6.88818			−0.454193
$231$		0			0
$232$	−3.23794			−0.212581
$233$			15.0715i			0.987367i	0.869642	+	0.493684i	$0.164350\pi$
$233$			15.0715i			0.987367i	−0.869642	+	0.493684i	$0.835650\pi$
$234$		0			0
$235$	−8.77340			−0.572313
$236$	6.29094			0.409505
$237$		0			0
$238$	−0.301038	−	0.136542i	−0.0195134	−	0.00885068i
$239$		−	24.3817i		−	1.57712i	−0.614955	−	0.788562i	$-0.710826\pi$
$239$		−	24.3817i		−	1.57712i	0.614955	−	0.788562i	$-0.289174\pi$
$240$		0			0
$241$		−	3.30720i		−	0.213036i	−0.994311	−	0.106518i	$-0.966030\pi$
$241$		−	3.30720i		−	0.213036i	0.994311	−	0.106518i	$-0.0339701\pi$
$242$		−	2.32085i		−	0.149190i
$243$		0			0
$244$			0.467276i			0.0299143i
$245$	5.82046	+	6.64752i	0.371855	+	0.424694i
$246$		0			0
$247$	0.580865			0.0369596
$248$	−2.94306			−0.186884
$249$		0			0
$250$		−	10.6113i		−	0.671115i
$251$	−7.03123			−0.443807			−0.221904	−	0.975069i	$-0.571227\pi$
$251$	−7.03123			−0.443807			−0.221904	+	0.975069i	$0.571227\pi$
$252$		0			0
$253$	−16.0771			−1.01076
$254$		−	9.37751i		−	0.588397i
$255$		0			0
$256$	1.00000			0.0625000
$257$	−14.4173			−0.899327			−0.449664	−	0.893198i	$-0.648456\pi$
$257$	−14.4173			−0.899327			−0.449664	+	0.893198i	$0.648456\pi$
$258$		0			0
$259$	−0.148481	+	0.327362i	−0.00922617	+	0.0203413i
$260$			0.733182i			0.0454700i
$261$		0			0
$262$			14.9045i			0.920805i
$263$		−	3.36860i		−	0.207717i	−0.994592	−	0.103858i	$-0.966881\pi$
$263$		−	3.36860i		−	0.207717i	0.994592	−	0.103858i	$-0.0331188\pi$
$264$		0			0
$265$		−	6.90399i		−	0.424109i
$266$	−1.09287	+	2.40949i	−0.0670081	+	0.147735i
$267$		0			0
$268$	−5.93507			−0.362542
$269$	9.65894			0.588916			0.294458	−	0.955664i	$-0.404861\pi$
$269$	9.65894			0.588916			0.294458	+	0.955664i	$0.404861\pi$
$270$		0			0
$271$		−	19.6946i		−	1.19636i	−0.801362	−	0.598180i	$-0.795891\pi$
$271$		−	19.6946i		−	1.19636i	0.801362	−	0.598180i	$-0.204109\pi$
$272$	0.124939			0.00757552
$273$		0			0
$274$	−1.93955			−0.117172
$275$		−	10.0365i		−	0.605226i
$276$		0			0
$277$	−6.21766			−0.373583			−0.186791	−	0.982400i	$-0.559809\pi$
$277$	−6.21766			−0.373583			−0.186791	+	0.982400i	$0.559809\pi$
$278$	12.8881			0.772979
$279$		0			0
$280$	−3.04131	−	1.37945i	−0.181753	−	0.0824377i
$281$			13.0046i			0.775791i	0.921703	+	0.387896i	$0.126798\pi$
$281$			13.0046i			0.775791i	−0.921703	+	0.387896i	$0.873202\pi$
$282$		0			0
$283$			28.9851i			1.72299i	0.507770	+	0.861493i	$0.330470\pi$
$283$			28.9851i			1.72299i	−0.507770	+	0.861493i	$0.669530\pi$
$284$		−	16.1645i		−	0.959186i
$285$		0			0
$286$			1.71125i			0.101188i
$287$	5.93561	−	13.0864i	0.350368	−	0.772469i
$288$		0			0
$289$	−16.9844			−0.999082
$290$	−4.08700			−0.239997
$291$		0			0
$292$		−	11.4154i		−	0.668038i
$293$	−11.3255			−0.661644			−0.330822	−	0.943693i	$-0.607326\pi$
$293$	−11.3255			−0.661644			−0.330822	+	0.943693i	$0.607326\pi$
$294$		0			0
$295$	7.94058			0.462318
$296$		−	0.135864i		−	0.00789691i
$297$		0			0
$298$	−4.62005			−0.267632
$299$	−3.16988			−0.183319
$300$		0			0
$301$	−2.70704	+	5.96830i	−0.156031	+	0.344007i
$302$		−	8.86316i		−	0.510017i
$303$		0			0
$304$		−	1.00000i		−	0.0573539i
$305$			0.589807i			0.0337723i
$306$		0			0
$307$		−	20.2990i		−	1.15852i	−0.815141	−	0.579262i	$-0.803341\pi$
$307$		−	20.2990i		−	1.15852i	0.815141	−	0.579262i	$-0.196659\pi$
$308$	−7.09845	−	3.21964i	−0.404472	−	0.183456i
$309$		0			0
$310$	−3.71480			−0.210986
$311$	−25.2405			−1.43126			−0.715630	−	0.698480i	$-0.753860\pi$
$311$	−25.2405			−1.43126			−0.715630	+	0.698480i	$0.753860\pi$
$312$		0			0
$313$			7.13134i			0.403087i	0.979480	+	0.201544i	$0.0645958\pi$
$313$			7.13134i			0.403087i	−0.979480	+	0.201544i	$0.935404\pi$
$314$	13.9759			0.788705
$315$		0			0
$316$	−0.468124			−0.0263340
$317$		−	8.59004i		−	0.482465i	−0.970467	−	0.241232i	$-0.922448\pi$
$317$		−	8.59004i		−	0.482465i	0.970467	−	0.241232i	$-0.0775516\pi$
$318$		0			0
$319$	−9.53910			−0.534087
$320$	1.26222			0.0705605
$321$		0			0
$322$	5.96398	−	13.1490i	0.332360	−	0.732765i
$323$		−	0.124939i		−	0.00695177i
$324$		0			0
$325$		−	1.97889i		−	0.109769i
$326$		−	18.0963i		−	1.00226i
$327$		0			0
$328$			5.43121i			0.299889i
$329$	7.59626	−	16.7477i	0.418795	−	0.923333i
$330$		0			0
$331$	19.4273			1.06782			0.533909	−	0.845542i	$-0.320722\pi$
$331$	19.4273			1.06782			0.533909	+	0.845542i	$0.320722\pi$
$332$	−9.79950			−0.537818
$333$		0			0
$334$			0.135153i			0.00739526i
$335$	−7.49139			−0.409298
$336$		0			0
$337$	0.433336			0.0236053			0.0118026	−	0.999930i	$-0.496243\pi$
$337$	0.433336			0.0236053			0.0118026	+	0.999930i	$0.496243\pi$
$338$		−	12.6626i		−	0.688754i
$339$		0			0
$340$	0.157701			0.00855251
$341$	−8.67037			−0.469527
$342$		0			0
$343$	−17.7291	+	5.35520i	−0.957283	+	0.289153i
$344$		−	2.47700i		−	0.133551i
$345$		0			0
$346$			11.2899i			0.606947i
$347$		−	10.9231i		−	0.586380i	−0.956054	−	0.293190i	$-0.905283\pi$
$347$		−	10.9231i		−	0.586380i	0.956054	−	0.293190i	$-0.0947169\pi$
$348$		0			0
$349$		−	15.8940i		−	0.850789i	−0.905008	−	0.425394i	$-0.860135\pi$
$349$		−	15.8940i		−	0.850789i	0.905008	−	0.425394i	$-0.139865\pi$
$350$	8.20862	+	3.72318i	0.438769	+	0.199012i
$351$		0			0
$352$	2.94604			0.157024
$353$	7.06470			0.376016			0.188008	−	0.982167i	$-0.439797\pi$
$353$	7.06470			0.376016			0.188008	+	0.982167i	$0.439797\pi$
$354$		0			0
$355$		−	20.4032i		−	1.08289i
$356$	−9.24616			−0.490045
$357$		0			0
$358$	16.1068			0.851269
$359$		−	16.2159i		−	0.855841i	−0.903816	−	0.427921i	$-0.859246\pi$
$359$		−	16.2159i		−	0.855841i	0.903816	−	0.427921i	$-0.140754\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$	4.83840			0.254300
$363$		0			0
$364$	−1.39959	−	0.634810i	−0.0733584	−	0.0332731i
$365$		−	14.4088i		−	0.754193i
$366$		0			0
$367$			30.8759i			1.61171i	0.592113	+	0.805855i	$0.298294\pi$
$367$			30.8759i			1.61171i	−0.592113	+	0.805855i	$0.701706\pi$
$368$			5.45717i			0.284475i
$369$		0			0
$370$		−	0.171490i		−	0.00891535i
$371$	13.1792	+	5.97767i	0.684229	+	0.310345i
$372$		0			0
$373$	−27.9801			−1.44876			−0.724378	−	0.689403i	$-0.757873\pi$
$373$	−27.9801			−1.44876			−0.724378	+	0.689403i	$0.757873\pi$
$374$	0.368074			0.0190327
$375$		0			0
$376$			6.95074i			0.358457i
$377$	−1.88081			−0.0968664
$378$		0			0
$379$	−21.6929			−1.11429			−0.557145	−	0.830415i	$-0.688103\pi$
$379$	−21.6929			−1.11429			−0.557145	+	0.830415i	$0.688103\pi$
$380$		−	1.26222i		−	0.0647507i
$381$		0			0
$382$	0.947510			0.0484788
$383$	23.2099			1.18597			0.592985	−	0.805213i	$-0.297949\pi$
$383$	23.2099			1.18597			0.592985	+	0.805213i	$0.297949\pi$
$384$		0			0
$385$	−8.95983	−	4.06390i	−0.456635	−	0.207116i
$386$		−	9.93469i		−	0.505663i
$387$		0			0
$388$			7.54319i			0.382948i
$389$			16.4672i			0.834920i	0.908695	+	0.417460i	$0.137080\pi$
$389$			16.4672i			0.834920i	−0.908695	+	0.417460i	$0.862920\pi$
$390$		0			0
$391$			0.681812i			0.0344807i
$392$	5.26651	−	4.61127i	0.265999	−	0.232904i
$393$		0			0
$394$	22.2190			1.11938
$395$	−0.590877			−0.0297303
$396$		0			0
$397$		−	1.81132i		−	0.0909077i	−0.998966	−	0.0454539i	$-0.985527\pi$
$397$		−	1.81132i		−	0.0909077i	0.998966	−	0.0454539i	$-0.0144734\pi$
$398$	4.47254			0.224188
$399$		0			0
$400$	−3.40679			−0.170340
$401$		−	4.95375i		−	0.247378i	−0.992321	−	0.123689i	$-0.960527\pi$
$401$		−	4.95375i		−	0.247378i	0.992321	−	0.123689i	$-0.0394726\pi$
$402$		0			0
$403$	−1.70952			−0.0851573
$404$	−4.60873			−0.229293
$405$		0			0
$406$	3.53864	−	7.80178i	0.175620	−	0.387196i
$407$		−	0.400259i		−	0.0198401i
$408$		0			0
$409$			30.2420i			1.49537i	0.664054	+	0.747685i	$0.268835\pi$
$409$			30.2420i			1.49537i	−0.664054	+	0.747685i	$0.731165\pi$
$410$			6.85541i			0.338564i
$411$		0			0
$412$			11.8587i			0.584238i
$413$	−6.87518	+	15.1580i	−0.338305	+	0.745874i
$414$		0			0
$415$	−12.3692			−0.607179
$416$	0.580865			0.0284793
$417$		0			0
$418$		−	2.94604i		−	0.144096i
$419$	4.16455			0.203452			0.101726	−	0.994812i	$-0.467564\pi$
$419$	4.16455			0.203452			0.101726	+	0.994812i	$0.467564\pi$
$420$		0			0
$421$	−29.1382			−1.42011			−0.710054	−	0.704147i	$-0.751330\pi$
$421$	−29.1382			−1.42011			−0.710054	+	0.704147i	$0.751330\pi$
$422$			10.8585i			0.528582i
$423$		0			0
$424$	−5.46970			−0.265632
$425$	−0.425640			−0.0206466
$426$		0			0
$427$	−1.12590	−	0.510672i	−0.0544859	−	0.0247131i
$428$		−	2.48007i		−	0.119879i
$429$		0			0
$430$		−	3.12653i		−	0.150775i
$431$		−	18.8415i		−	0.907562i	−0.891113	−	0.453781i	$-0.850075\pi$
$431$		−	18.8415i		−	0.907562i	0.891113	−	0.453781i	$-0.149925\pi$
$432$		0			0
$433$			31.2388i			1.50124i	0.660733	+	0.750621i	$0.270246\pi$
$433$			31.2388i			1.50124i	−0.660733	+	0.750621i	$0.729754\pi$
$434$	3.21638	−	7.09127i	0.154391	−	0.340392i
$435$		0			0
$436$	−6.64142			−0.318066
$437$	5.45717			0.261052
$438$		0			0
$439$		−	23.8375i		−	1.13770i	−0.822440	−	0.568852i	$-0.807388\pi$
$439$		−	23.8375i		−	1.13770i	0.822440	−	0.568852i	$-0.192612\pi$
$440$	3.71856			0.177275
$441$		0			0
$442$	0.0725725			0.00345192
$443$		−	0.210570i		−	0.0100045i	−0.999987	−	0.00500224i	$-0.998408\pi$
$443$		−	0.210570i		−	0.0100045i	0.999987	−	0.00500224i	$-0.00159227\pi$
$444$		0			0
$445$	−11.6707			−0.553245
$446$	−1.40924			−0.0667295
$447$		0			0
$448$	−1.09287	+	2.40949i	−0.0516332	+	0.113838i
$449$		−	3.49441i		−	0.164911i	−0.996595	−	0.0824557i	$-0.973724\pi$
$449$		−	3.49441i		−	0.164911i	0.996595	−	0.0824557i	$-0.0262763\pi$
$450$		0			0
$451$			16.0006i			0.753437i
$452$		−	5.59170i		−	0.263012i
$453$		0			0
$454$			13.8910i			0.651936i
$455$	−1.76659	−	0.801272i	−0.0828192	−	0.0375642i
$456$		0			0
$457$	1.81078			0.0847047			0.0423523	−	0.999103i	$-0.486515\pi$
$457$	1.81078			0.0847047			0.0423523	+	0.999103i	$0.486515\pi$
$458$	−16.9038			−0.789863
$459$		0			0
$460$			6.88818i			0.321163i
$461$	−10.2919			−0.479343			−0.239672	−	0.970854i	$-0.577040\pi$
$461$	−10.2919			−0.479343			−0.239672	+	0.970854i	$0.577040\pi$
$462$		0			0
$463$	−0.465421			−0.0216299			−0.0108150	−	0.999942i	$-0.503443\pi$
$463$	−0.465421			−0.0216299			−0.0108150	+	0.999942i	$0.503443\pi$
$464$			3.23794i			0.150318i
$465$		0			0
$466$	15.0715			0.698174
$467$	10.0215			0.463740			0.231870	−	0.972747i	$-0.425516\pi$
$467$	10.0215			0.463740			0.231870	+	0.972747i	$0.425516\pi$
$468$		0			0
$469$	6.48626	−	14.3005i	0.299507	−	0.660335i
$470$			8.77340i			0.404687i
$471$		0			0
$472$		−	6.29094i		−	0.289564i
$473$		−	7.29734i		−	0.335532i
$474$		0			0
$475$			3.40679i			0.156314i
$476$	−0.136542	+	0.301038i	−0.00625838	+	0.0137981i
$477$		0			0
$478$	−24.3817			−1.11520
$479$	−42.7360			−1.95266			−0.976330	−	0.216287i	$-0.930605\pi$
$479$	−42.7360			−1.95266			−0.976330	+	0.216287i	$0.930605\pi$
$480$		0			0
$481$		−	0.0789184i		−	0.00359837i
$482$	−3.30720			−0.150639
$483$		0			0
$484$	−2.32085			−0.105493
$485$			9.52120i			0.432335i
$486$		0			0
$487$	7.59226			0.344038			0.172019	−	0.985094i	$-0.444971\pi$
$487$	7.59226			0.344038			0.172019	+	0.985094i	$0.444971\pi$
$488$	0.467276			0.0211526
$489$		0			0
$490$	6.64752	−	5.82046i	0.300304	−	0.262942i
$491$		−	22.2349i		−	1.00345i	−0.865028	−	0.501724i	$-0.832699\pi$
$491$		−	22.2349i		−	1.00345i	0.865028	−	0.501724i	$-0.167301\pi$
$492$		0			0
$493$			0.404544i			0.0182197i
$494$		−	0.580865i		−	0.0261344i
$495$		0			0
$496$			2.94306i			0.132147i
$497$	38.9482	+	17.6657i	1.74706	+	0.792414i
$498$		0			0
$499$	−19.3476			−0.866117			−0.433058	−	0.901366i	$-0.642566\pi$
$499$	−19.3476			−0.866117			−0.433058	+	0.901366i	$0.642566\pi$
$500$	−10.6113			−0.474550
$501$		0			0
$502$			7.03123i			0.313819i
$503$	−9.02726			−0.402506			−0.201253	−	0.979539i	$-0.564501\pi$
$503$	−9.02726			−0.402506			−0.201253	+	0.979539i	$0.564501\pi$
$504$		0			0
$505$	−5.81725			−0.258864
$506$			16.0771i			0.714712i
$507$		0			0
$508$	−9.37751			−0.416060
$509$	−3.32324			−0.147300			−0.0736501	−	0.997284i	$-0.523465\pi$
$509$	−3.32324			−0.147300			−0.0736501	+	0.997284i	$0.523465\pi$
$510$		0			0
$511$	27.5054	+	12.4756i	1.21677	+	0.551887i
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$			14.4173i			0.635920i
$515$			14.9684i			0.659586i
$516$		0			0
$517$			20.4772i			0.900585i
$518$	0.327362	+	0.148481i	0.0143834	+	0.00652389i
$519$		0			0
$520$	0.733182			0.0321522
$521$	−17.4482			−0.764419			−0.382209	−	0.924076i	$-0.624837\pi$
$521$	−17.4482			−0.764419			−0.382209	+	0.924076i	$0.624837\pi$
$522$		0			0
$523$			31.3099i			1.36909i	0.728972	+	0.684543i	$0.239998\pi$
$523$			31.3099i			1.36909i	−0.728972	+	0.684543i	$0.760002\pi$
$524$	14.9045			0.651107
$525$		0			0
$526$	−3.36860			−0.146878
$527$			0.367702i			0.0160173i
$528$		0			0
$529$	−6.78075			−0.294815
$530$	−6.90399			−0.299890
$531$		0			0
$532$	2.40949	+	1.09287i	0.104465	+	0.0473819i
$533$			3.15480i			0.136650i
$534$		0			0
$535$		−	3.13040i		−	0.135339i
$536$			5.93507i			0.256356i
$537$		0			0
$538$		−	9.65894i		−	0.416427i
$539$	15.5154	−	13.5850i	0.668294	−	0.585147i
$540$		0			0
$541$	−9.85184			−0.423564			−0.211782	−	0.977317i	$-0.567927\pi$
$541$	−9.85184			−0.423564			−0.211782	+	0.977317i	$0.567927\pi$
$542$	−19.6946			−0.845954
$543$		0			0
$544$		−	0.124939i		−	0.00535670i
$545$	−8.38296			−0.359086
$546$		0			0
$547$	17.0186			0.727663			0.363832	−	0.931465i	$-0.381468\pi$
$547$	17.0186			0.727663			0.363832	+	0.931465i	$0.381468\pi$
$548$			1.93955i			0.0828533i
$549$		0			0
$550$	−10.0365			−0.427959
$551$	3.23794			0.137941
$552$		0			0
$553$	0.511598	−	1.12794i	0.0217554	−	0.0479648i
$554$			6.21766i			0.264163i
$555$		0			0
$556$		−	12.8881i		−	0.546579i
$557$			40.5732i			1.71914i	0.511015	+	0.859572i	$0.329270\pi$
$557$			40.5732i			1.71914i	−0.511015	+	0.859572i	$0.670730\pi$
$558$		0			0
$559$		−	1.43880i		−	0.0608549i
$560$	−1.37945	+	3.04131i	−0.0582922	+	0.128519i
$561$		0			0
$562$	13.0046			0.548567
$563$	8.53357			0.359647			0.179824	−	0.983699i	$-0.442447\pi$
$563$	8.53357			0.359647			0.179824	+	0.983699i	$0.442447\pi$
$564$		0			0
$565$		−	7.05798i		−	0.296932i
$566$	28.9851			1.21834
$567$		0			0
$568$	−16.1645			−0.678247
$569$			19.3163i			0.809779i	0.914366	+	0.404890i	$0.132690\pi$
$569$			19.3163i			0.809779i	−0.914366	+	0.404890i	$0.867310\pi$
$570$		0			0
$571$	−9.05058			−0.378755			−0.189378	−	0.981904i	$-0.560647\pi$
$571$	−9.05058			−0.378755			−0.189378	+	0.981904i	$0.560647\pi$
$572$	1.71125			0.0715510
$573$		0			0
$574$	−13.0864	−	5.93561i	−0.546218	−	0.247747i
$575$		−	18.5914i		−	0.775317i
$576$		0			0
$577$		−	5.53186i		−	0.230294i	−0.993348	−	0.115147i	$-0.963266\pi$
$577$		−	5.53186i		−	0.230294i	0.993348	−	0.115147i	$-0.0367340\pi$
$578$			16.9844i			0.706458i
$579$		0			0
$580$			4.08700i			0.169704i
$581$	10.7096	−	23.6118i	0.444308	−	0.979582i
$582$		0			0
$583$	−16.1140			−0.667372
$584$	−11.4154			−0.472374
$585$		0			0
$586$			11.3255i			0.467853i
$587$	−15.1982			−0.627296			−0.313648	−	0.949539i	$-0.601551\pi$
$587$	−15.1982			−0.627296			−0.313648	+	0.949539i	$0.601551\pi$
$588$		0			0
$589$	2.94306			0.121267
$590$		−	7.94058i		−	0.326908i
$591$		0			0
$592$	−0.135864			−0.00558396
$593$	18.0338			0.740557			0.370279	−	0.928921i	$-0.379262\pi$
$593$	18.0338			0.740557			0.370279	+	0.928921i	$0.379262\pi$
$594$		0			0
$595$	−0.172346	+	0.379978i	−0.00706550	+	0.0155776i
$596$			4.62005i			0.189244i
$597$		0			0
$598$			3.16988i			0.129626i
$599$		−	33.6893i		−	1.37651i	−0.725471	−	0.688253i	$-0.758378\pi$
$599$		−	33.6893i		−	1.37651i	0.725471	−	0.688253i	$-0.241622\pi$
$600$		0			0
$601$			14.8721i			0.606644i	0.952888	+	0.303322i	$0.0980957\pi$
$601$			14.8721i			0.606644i	−0.952888	+	0.303322i	$0.901904\pi$
$602$	5.96830	+	2.70704i	0.243250	+	0.110331i
$603$		0			0
$604$	−8.86316			−0.360637
$605$	−2.92943			−0.119098
$606$		0			0
$607$		−	42.4857i		−	1.72444i	−0.506533	−	0.862221i	$-0.669073\pi$
$607$		−	42.4857i		−	1.72444i	0.506533	−	0.862221i	$-0.330927\pi$
$608$	−1.00000			−0.0405554
$609$		0			0
$610$	0.589807			0.0238806
$611$			4.03745i			0.163338i
$612$		0			0
$613$	−19.0605			−0.769845			−0.384923	−	0.922949i	$-0.625772\pi$
$613$	−19.0605			−0.769845			−0.384923	+	0.922949i	$0.625772\pi$
$614$	−20.2990			−0.819200
$615$		0			0
$616$	−3.21964	+	7.09845i	−0.129723	+	0.286005i
$617$			6.67373i			0.268674i	0.990936	+	0.134337i	$0.0428905\pi$
$617$			6.67373i			0.268674i	−0.990936	+	0.134337i	$0.957110\pi$
$618$		0			0
$619$			4.62968i			0.186083i	0.995662	+	0.0930414i	$0.0296589\pi$
$619$			4.62968i			0.186083i	−0.995662	+	0.0930414i	$0.970341\pi$
$620$			3.71480i			0.149190i
$621$		0			0
$622$			25.2405i			1.01205i
$623$	10.1048	−	22.2785i	0.404842	−	0.892570i
$624$		0			0
$625$	3.64017			0.145607
$626$	7.13134			0.285026
$627$		0			0
$628$		−	13.9759i		−	0.557699i
$629$	−0.0169746			−0.000676822
$630$		0			0
$631$	32.3219			1.28671			0.643356	−	0.765567i	$-0.277541\pi$
$631$	32.3219			1.28671			0.643356	+	0.765567i	$0.277541\pi$
$632$			0.468124i			0.0186210i
$633$		0			0
$634$	−8.59004			−0.341154
$635$	−11.8365			−0.469718
$636$		0			0
$637$	3.05914	−	2.67853i	0.121207	−	0.106127i
$638$			9.53910i			0.377656i
$639$		0			0
$640$		−	1.26222i		−	0.0498938i
$641$			17.7180i			0.699818i	0.936784	+	0.349909i	$0.113788\pi$
$641$			17.7180i			0.699818i	−0.936784	+	0.349909i	$0.886212\pi$
$642$		0			0
$643$			35.3274i			1.39318i	0.717471	+	0.696589i	$0.245300\pi$
$643$			35.3274i			1.39318i	−0.717471	+	0.696589i	$0.754700\pi$
$644$	−13.1490	−	5.96398i	−0.518143	−	0.235014i
$645$		0			0
$646$	−0.124939			−0.00491565
$647$	20.4309			0.803222			0.401611	−	0.915810i	$-0.368450\pi$
$647$	20.4309			0.803222			0.401611	+	0.915810i	$0.368450\pi$
$648$		0			0
$649$		−	18.5334i		−	0.727498i
$650$	−1.97889			−0.0776183
$651$		0			0
$652$	−18.0963			−0.708707
$653$		−	16.8668i		−	0.660051i	−0.943972	−	0.330025i	$-0.892943\pi$
$653$		−	16.8668i		−	0.660051i	0.943972	−	0.330025i	$-0.107057\pi$
$654$		0			0
$655$	18.8129			0.735079
$656$	5.43121			0.212053
$657$		0			0
$658$	−16.7477	−	7.59626i	−0.652895	−	0.296133i
$659$			5.01860i			0.195497i	0.995211	+	0.0977484i	$0.0311640\pi$
$659$			5.01860i			0.195497i	−0.995211	+	0.0977484i	$0.968836\pi$
$660$		0			0
$661$			24.2491i			0.943180i	0.881818	+	0.471590i	$0.156320\pi$
$661$			24.2491i			0.943180i	−0.881818	+	0.471590i	$0.843680\pi$
$662$		−	19.4273i		−	0.755062i
$663$		0			0
$664$			9.79950i			0.380295i
$665$	3.04131	+	1.37945i	0.117937	+	0.0534926i
$666$		0			0
$667$	−17.6700			−0.684185
$668$	0.135153			0.00522924
$669$		0			0
$670$			7.49139i			0.289418i
$671$	1.37661			0.0531436
$672$		0			0
$673$	−31.4830			−1.21358			−0.606791	−	0.794862i	$-0.707543\pi$
$673$	−31.4830			−1.21358			−0.606791	+	0.794862i	$0.707543\pi$
$674$		−	0.433336i		−	0.0166915i
$675$		0			0
$676$	−12.6626			−0.487023
$677$	−18.2121			−0.699948			−0.349974	−	0.936760i	$-0.613809\pi$
$677$	−18.2121			−0.699948			−0.349974	+	0.936760i	$0.613809\pi$
$678$		0			0
$679$	−18.1752	−	8.24372i	−0.697501	−	0.316365i
$680$		−	0.157701i		−	0.00604754i
$681$		0			0
$682$			8.67037i			0.332006i
$683$		−	5.61612i		−	0.214895i	−0.994211	−	0.107447i	$-0.965732\pi$
$683$		−	5.61612i		−	0.214895i	0.994211	−	0.107447i	$-0.0342678\pi$
$684$		0			0
$685$			2.44814i			0.0935387i
$686$	5.35520	+	17.7291i	0.204462	+	0.676901i
$687$		0			0
$688$	−2.47700			−0.0944347
$689$	−3.17716			−0.121040
$690$		0			0
$691$		−	21.0472i		−	0.800674i	−0.916368	−	0.400337i	$-0.868893\pi$
$691$		−	21.0472i		−	0.800674i	0.916368	−	0.400337i	$-0.131107\pi$
$692$	11.2899			0.429176
$693$		0			0
$694$	−10.9231			−0.414634
$695$		−	16.2677i		−	0.617070i
$696$		0			0
$697$	0.678568			0.0257026
$698$	−15.8940			−0.601598
$699$		0			0
$700$	3.72318	−	8.20862i	0.140723	−	0.310257i
$701$			27.9599i			1.05603i	0.849235	+	0.528016i	$0.177064\pi$
$701$			27.9599i			1.05603i	−0.849235	+	0.528016i	$0.822936\pi$
$702$		0			0
$703$			0.135864i			0.00512419i
$704$		−	2.94604i		−	0.111033i
$705$		0			0
$706$		−	7.06470i		−	0.265884i
$707$	5.03675	−	11.1047i	0.189426	−	0.417635i
$708$		0			0
$709$	1.62154			0.0608982			0.0304491	−	0.999536i	$-0.490306\pi$
$709$	1.62154			0.0608982			0.0304491	+	0.999536i	$0.490306\pi$
$710$	−20.4032			−0.765719
$711$		0			0
$712$			9.24616i			0.346514i
$713$	−16.0608			−0.601481
$714$		0			0
$715$	2.15998			0.0807788
$716$		−	16.1068i		−	0.601938i
$717$		0			0
$718$	−16.2159			−0.605171
$719$	8.80165			0.328246			0.164123	−	0.986440i	$-0.447521\pi$
$719$	8.80165			0.328246			0.164123	+	0.986440i	$0.447521\pi$
$720$		0			0
$721$	−28.5735	−	12.9601i	−1.06413	−	0.482658i
$722$			1.00000i			0.0372161i
$723$		0			0
$724$		−	4.83840i		−	0.179818i
$725$		−	11.0310i		−	0.409680i
$726$		0			0
$727$		−	14.6658i		−	0.543926i	−0.962308	−	0.271963i	$-0.912327\pi$
$727$		−	14.6658i		−	0.543926i	0.962308	−	0.271963i	$-0.0876728\pi$
$728$	−0.634810	+	1.39959i	−0.0235276	+	0.0518722i
$729$		0			0
$730$	−14.4088			−0.533295
$731$	−0.309473			−0.0114463
$732$		0			0
$733$		−	22.9641i		−	0.848200i	−0.905615	−	0.424100i	$-0.860591\pi$
$733$		−	22.9641i		−	0.848200i	0.905615	−	0.424100i	$-0.139409\pi$
$734$	30.8759			1.13965
$735$		0			0
$736$	5.45717			0.201154
$737$			17.4849i			0.644066i
$738$		0			0
$739$	27.8443			1.02427			0.512135	−	0.858905i	$-0.328855\pi$
$739$	27.8443			1.02427			0.512135	+	0.858905i	$0.328855\pi$
$740$	−0.171490			−0.00630411
$741$		0			0
$742$	5.97767	−	13.1792i	0.219447	−	0.483823i
$743$		−	31.0494i		−	1.13909i	−0.821959	−	0.569547i	$-0.807119\pi$
$743$		−	31.0494i		−	1.13909i	0.821959	−	0.569547i	$-0.192881\pi$
$744$		0			0
$745$			5.83153i			0.213651i
$746$			27.9801i			1.02443i
$747$		0			0
$748$		−	0.368074i		−	0.0134581i
$749$	5.97569	+	2.71039i	0.218347	+	0.0990355i
$750$		0			0
$751$	9.69902			0.353922			0.176961	−	0.984218i	$-0.443373\pi$
$751$	9.69902			0.353922			0.176961	+	0.984218i	$0.443373\pi$
$752$	6.95074			0.253468
$753$		0			0
$754$			1.88081i			0.0684949i
$755$	−11.1873			−0.407147
$756$		0			0
$757$	14.3310			0.520870			0.260435	−	0.965491i	$-0.416134\pi$
$757$	14.3310			0.520870			0.260435	+	0.965491i	$0.416134\pi$
$758$			21.6929i			0.787923i
$759$		0			0
$760$	−1.26222			−0.0457857
$761$	35.6888			1.29372			0.646860	−	0.762609i	$-0.276082\pi$
$761$	35.6888			1.29372			0.646860	+	0.762609i	$0.276082\pi$
$762$		0			0
$763$	7.25820	−	16.0024i	0.262765	−	0.579326i
$764$		−	0.947510i		−	0.0342797i
$765$		0			0
$766$		−	23.2099i		−	0.838608i
$767$		−	3.65419i		−	0.131945i
$768$		0			0
$769$			44.8001i			1.61553i	0.589502	+	0.807767i	$0.299324\pi$
$769$			44.8001i			1.61553i	−0.589502	+	0.807767i	$0.700676\pi$
$770$	−4.06390	+	8.95983i	−0.146453	+	0.322890i
$771$		0			0
$772$	−9.93469			−0.357557
$773$	−6.42581			−0.231120			−0.115560	−	0.993300i	$-0.536866\pi$
$773$	−6.42581			−0.231120			−0.115560	+	0.993300i	$0.536866\pi$
$774$		0			0
$775$		−	10.0264i		−	0.360158i
$776$	7.54319			0.270785
$777$		0			0
$778$	16.4672			0.590377
$779$		−	5.43121i		−	0.194593i
$780$		0			0
$781$	−47.6212			−1.70402
$782$	0.681812			0.0243815
$783$		0			0
$784$	−4.61127	−	5.26651i	−0.164688	−	0.188090i
$785$		−	17.6407i		−	0.629624i
$786$		0			0
$787$			9.06830i			0.323250i	0.986852	+	0.161625i	$0.0516735\pi$
$787$			9.06830i			0.323250i	−0.986852	+	0.161625i	$0.948327\pi$
$788$		−	22.2190i		−	0.791519i
$789$		0			0
$790$			0.590877i			0.0210225i
$791$	13.4731	+	6.11100i	0.479050	+	0.217282i
$792$		0			0
$793$	0.271424			0.00963857
$794$	−1.81132			−0.0642815
$795$		0			0
$796$		−	4.47254i		−	0.158525i
$797$	−36.6622			−1.29864			−0.649322	−	0.760514i	$-0.724947\pi$
$797$	−36.6622			−1.29864			−0.649322	+	0.760514i	$0.724947\pi$
$798$		0			0
$799$	0.868417			0.0307224
$800$			3.40679i			0.120448i
$801$		0			0
$802$	−4.95375			−0.174923
$803$	−33.6303			−1.18679
$804$		0			0
$805$	−16.5970	−	7.52788i	−0.584967	−	0.265323i
$806$			1.70952i			0.0602153i
$807$		0			0
$808$			4.60873i			0.162135i
$809$		−	34.6653i		−	1.21877i	−0.792876	−	0.609383i	$-0.791417\pi$
$809$		−	34.6653i		−	1.21877i	0.792876	−	0.609383i	$-0.208583\pi$
$810$		0			0
$811$		−	35.5440i		−	1.24812i	−0.781377	−	0.624060i	$-0.785482\pi$
$811$		−	35.5440i		−	1.24812i	0.781377	−	0.624060i	$-0.214518\pi$
$812$	−7.80178	−	3.53864i	−0.273789	−	0.124182i
$813$		0			0
$814$	−0.400259			−0.0140291
$815$	−22.8416			−0.800107
$816$		0			0
$817$			2.47700i			0.0866592i
$818$	30.2420			1.05739
$819$		0			0
$820$	6.85541			0.239401
$821$			43.7225i			1.52593i	0.646441	+	0.762964i	$0.276257\pi$
$821$			43.7225i			1.52593i	−0.646441	+	0.762964i	$0.723743\pi$
$822$		0			0
$823$	−10.6982			−0.372915			−0.186457	−	0.982463i	$-0.559701\pi$
$823$	−10.6982			−0.372915			−0.186457	+	0.982463i	$0.559701\pi$
$824$	11.8587			0.413119
$825$		0			0
$826$	15.1580	+	6.87518i	0.527413	+	0.239218i
$827$			20.1660i			0.701239i	0.936518	+	0.350620i	$0.114029\pi$
$827$			20.1660i			0.701239i	−0.936518	+	0.350620i	$0.885971\pi$
$828$		0			0
$829$			20.3025i			0.705134i	0.935787	+	0.352567i	$0.114691\pi$
$829$			20.3025i			0.705134i	−0.935787	+	0.352567i	$0.885309\pi$
$830$			12.3692i			0.429340i
$831$		0			0
$832$		−	0.580865i		−	0.0201379i
$833$	−0.576126	−	0.657991i	−0.0199616	−	0.0227980i
$834$		0			0
$835$	0.170594			0.00590364
$836$	−2.94604			−0.101891
$837$		0			0
$838$		−	4.16455i		−	0.143862i
$839$	0.533787			0.0184284			0.00921419	−	0.999958i	$-0.497067\pi$
$839$	0.533787			0.0184284			0.00921419	+	0.999958i	$0.497067\pi$
$840$		0			0
$841$	18.5158			0.638474
$842$			29.1382i			1.00417i
$843$		0			0
$844$	10.8585			0.373764
$845$	−15.9830			−0.549833
$846$		0			0
$847$	2.53639	−	5.59206i	0.0871513	−	0.192146i
$848$			5.46970i			0.187830i
$849$		0			0
$850$			0.425640i			0.0145993i
$851$		−	0.741431i		−	0.0254159i
$852$		0			0
$853$			54.1595i			1.85439i	0.374584	+	0.927193i	$0.377785\pi$
$853$			54.1595i			1.85439i	−0.374584	+	0.927193i	$0.622215\pi$
$854$	−0.510672	+	1.12590i	−0.0174748	+	0.0385274i
$855$		0			0
$856$	−2.48007			−0.0847670
$857$	27.5095			0.939707			0.469854	−	0.882744i	$-0.344307\pi$
$857$	27.5095			0.939707			0.469854	+	0.882744i	$0.344307\pi$
$858$		0			0
$859$		−	16.2347i		−	0.553921i	−0.960881	−	0.276960i	$-0.910673\pi$
$859$		−	16.2347i		−	0.553921i	0.960881	−	0.276960i	$-0.0893271\pi$
$860$	−3.12653			−0.106614
$861$		0			0
$862$	−18.8415			−0.641743
$863$		−	35.1989i		−	1.19818i	−0.800680	−	0.599092i	$-0.795528\pi$
$863$		−	35.1989i		−	1.19818i	0.800680	−	0.599092i	$-0.204472\pi$
$864$		0			0
$865$	14.2503			0.484526
$866$	31.2388			1.06154
$867$		0			0
$868$	−7.09127	−	3.21638i	−0.240693	−	0.109171i
$869$			1.37911i			0.0467831i
$870$		0			0
$871$			3.44748i			0.116813i
$872$			6.64142i			0.224907i
$873$		0			0
$874$		−	5.45717i		−	0.184592i
$875$	11.5967	−	25.5677i	0.392041	−	0.864346i
$876$		0			0
$877$	−19.7067			−0.665448			−0.332724	−	0.943024i	$-0.607968\pi$
$877$	−19.7067			−0.665448			−0.332724	+	0.943024i	$0.607968\pi$
$878$	−23.8375			−0.804478
$879$		0			0
$880$		−	3.71856i		−	0.125353i
$881$	3.71631			0.125206			0.0626029	−	0.998039i	$-0.480060\pi$
$881$	3.71631			0.125206			0.0626029	+	0.998039i	$0.480060\pi$
$882$		0			0
$883$	−3.15411			−0.106144			−0.0530722	−	0.998591i	$-0.516901\pi$
$883$	−3.15411			−0.106144			−0.0530722	+	0.998591i	$0.516901\pi$
$884$		−	0.0725725i		−	0.00244088i
$885$		0			0
$886$	−0.210570			−0.00707424
$887$	−11.5663			−0.388357			−0.194178	−	0.980966i	$-0.562204\pi$
$887$	−11.5663			−0.388357			−0.194178	+	0.980966i	$0.562204\pi$
$888$		0			0
$889$	10.2484	−	22.5950i	0.343720	−	0.757812i
$890$			11.6707i			0.391204i
$891$		0			0
$892$			1.40924i			0.0471849i
$893$		−	6.95074i		−	0.232598i
$894$		0			0
$895$		−	20.3303i		−	0.679568i
$896$	2.40949	+	1.09287i	0.0804954	+	0.0365102i
$897$		0			0
$898$	−3.49441			−0.116610
$899$	−9.52944			−0.317825
$900$		0			0
$901$			0.683377i			0.0227666i
$902$	16.0006			0.532761
$903$		0			0
$904$	−5.59170			−0.185977
$905$		−	6.10714i		−	0.203008i
$906$		0			0
$907$	14.2545			0.473312			0.236656	−	0.971593i	$-0.423949\pi$
$907$	14.2545			0.473312			0.236656	+	0.971593i	$0.423949\pi$
$908$	13.8910			0.460989
$909$		0			0
$910$	−0.801272	+	1.76659i	−0.0265619	+	0.0585620i
$911$		−	24.0760i		−	0.797675i	−0.917022	−	0.398837i	$-0.869414\pi$
$911$		−	24.0760i		−	0.797675i	0.917022	−	0.398837i	$-0.130586\pi$
$912$		0			0
$913$			28.8697i			0.955449i
$914$		−	1.81078i		−	0.0598953i
$915$		0			0
$916$			16.9038i			0.558517i
$917$	−16.2887	+	35.9123i	−0.537900	+	1.18593i
$918$		0			0
$919$	50.2325			1.65702			0.828508	−	0.559977i	$-0.189190\pi$
$919$	50.2325			1.65702			0.828508	+	0.559977i	$0.189190\pi$
$920$	6.88818			0.227096
$921$		0			0
$922$			10.2919i			0.338947i
$923$	−9.38939			−0.309056
$924$		0			0
$925$	0.462859			0.0152187
$926$			0.465421i			0.0152947i
$927$		0			0
$928$	3.23794			0.106291
$929$	16.1259			0.529074			0.264537	−	0.964376i	$-0.414781\pi$
$929$	16.1259			0.529074			0.264537	+	0.964376i	$0.414781\pi$
$930$		0			0
$931$	−5.26651	+	4.61127i	−0.172603	+	0.151128i
$932$		−	15.0715i		−	0.493684i
$933$		0			0
$934$		−	10.0215i		−	0.327914i
$935$		−	0.464592i		−	0.0151938i
$936$		0			0
$937$		−	58.3192i		−	1.90521i	−0.304218	−	0.952603i	$-0.598395\pi$
$937$		−	58.3192i		−	1.90521i	0.304218	−	0.952603i	$-0.401605\pi$
$938$	−14.3005	−	6.48626i	−0.466927	−	0.211784i
$939$		0			0
$940$	8.77340			0.286157
$941$	−41.9844			−1.36865			−0.684326	−	0.729177i	$-0.739903\pi$
$941$	−41.9844			−1.36865			−0.684326	+	0.729177i	$0.739903\pi$
$942$		0			0
$943$			29.6391i			0.965181i
$944$	−6.29094			−0.204753
$945$		0			0
$946$	−7.29734			−0.237257
$947$			44.1729i			1.43543i	0.696339	+	0.717713i	$0.254811\pi$
$947$			44.1729i			1.43543i	−0.696339	+	0.717713i	$0.745189\pi$
$948$		0			0
$949$	−6.63083			−0.215246
$950$	3.40679			0.110531
$951$		0			0
$952$	0.301038	+	0.136542i	0.00975671	+	0.00442534i
$953$		−	22.4882i		−	0.728465i	−0.931308	−	0.364233i	$-0.881331\pi$
$953$		−	22.4882i		−	0.728465i	0.931308	−	0.364233i	$-0.118669\pi$
$954$		0			0
$955$		−	1.19597i		−	0.0387007i
$956$			24.3817i			0.788562i
$957$		0			0
$958$			42.7360i			1.38074i
$959$	−4.67331	−	2.11967i	−0.150909	−	0.0684477i
$960$		0			0
$961$	22.3384			0.720594
$962$	−0.0789184			−0.00254443
$963$		0			0
$964$			3.30720i			0.106518i
$965$	−12.5398			−0.403671
$966$		0			0
$967$	−0.677844			−0.0217980			−0.0108990	−	0.999941i	$-0.503469\pi$
$967$	−0.677844			−0.0217980			−0.0108990	+	0.999941i	$0.503469\pi$
$968$			2.32085i			0.0745950i
$969$		0			0
$970$	9.52120			0.305707
$971$	44.9323			1.44195			0.720973	−	0.692963i	$-0.243695\pi$
$971$	44.9323			1.44195			0.720973	+	0.692963i	$0.243695\pi$
$972$		0			0
$973$	31.0538	+	14.0851i	0.995540	+	0.451546i
$974$		−	7.59226i		−	0.243272i
$975$		0			0
$976$		−	0.467276i		−	0.0149571i
$977$			22.5723i			0.722151i	0.932537	+	0.361075i	$0.117590\pi$
$977$			22.5723i			0.722151i	−0.932537	+	0.361075i	$0.882410\pi$
$978$		0			0
$979$			27.2396i			0.870580i
$980$	−5.82046	−	6.64752i	−0.185928	−	0.212347i
$981$		0			0
$982$	−22.2349			−0.709545
$983$	−44.7270			−1.42657			−0.713285	−	0.700874i	$-0.752793\pi$
$983$	−44.7270			−1.42657			−0.713285	+	0.700874i	$0.752793\pi$
$984$		0			0
$985$		−	28.0453i		−	0.893599i
$986$	0.404544			0.0128833
$987$		0			0
$988$	−0.580865			−0.0184798
$989$		−	13.5174i		−	0.429829i
$990$		0			0
$991$	42.4945			1.34988			0.674941	−	0.737872i	$-0.264169\pi$
$991$	42.4945			1.34988			0.674941	+	0.737872i	$0.264169\pi$
$992$	2.94306			0.0934422
$993$		0			0
$994$	17.6657	−	38.9482i	0.560321	−	1.23536i
$995$		−	5.64534i		−	0.178969i
$996$		0			0
$997$			56.9184i			1.80262i	0.433171	+	0.901312i	$0.357394\pi$
$997$			56.9184i			1.80262i	−0.433171	+	0.901312i	$0.642606\pi$
$998$			19.3476i			0.612437i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2394.2.f.b.2015.7	yes	24
3.2	odd	2		2394.2.f.a.2015.8	yes	24
7.6	odd	2		2394.2.f.a.2015.7	✓	24
21.20	even	2	inner	2394.2.f.b.2015.8	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2394.2.f.a.2015.7	✓	24	7.6	odd	2
2394.2.f.a.2015.8	yes	24	3.2	odd	2
2394.2.f.b.2015.7	yes	24	1.1	even	1	trivial
2394.2.f.b.2015.8	yes	24	21.20	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -1.26222 q^{5} +(1.09287 - 2.40949i) q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -1.26222 q^{5} +(1.09287 - 2.40949i) q^{7} +1.00000i q^{8} +1.26222i q^{10} +2.94604i q^{11} +0.580865i q^{13} +(-2.40949 - 1.09287i) q^{14} +1.00000 q^{16} +0.124939 q^{17} -1.00000i q^{19} +1.26222 q^{20} +2.94604 q^{22} +5.45717i q^{23} -3.40679 q^{25} +0.580865 q^{26} +(-1.09287 + 2.40949i) q^{28} +3.23794i q^{29} +2.94306i q^{31} -1.00000i q^{32} -0.124939i q^{34} +(-1.37945 + 3.04131i) q^{35} -0.135864 q^{37} -1.00000 q^{38} -1.26222i q^{40} +5.43121 q^{41} -2.47700 q^{43} -2.94604i q^{44} +5.45717 q^{46} +6.95074 q^{47} +(-4.61127 - 5.26651i) q^{49} +3.40679i q^{50} -0.580865i q^{52} +5.46970i q^{53} -3.71856i q^{55} +(2.40949 + 1.09287i) q^{56} +3.23794 q^{58} -6.29094 q^{59} -0.467276i q^{61} +2.94306 q^{62} -1.00000 q^{64} -0.733182i q^{65} +5.93507 q^{67} -0.124939 q^{68} +(3.04131 + 1.37945i) q^{70} +16.1645i q^{71} +11.4154i q^{73} +0.135864i q^{74} +1.00000i q^{76} +(7.09845 + 3.21964i) q^{77} +0.468124 q^{79} -1.26222 q^{80} -5.43121i q^{82} +9.79950 q^{83} -0.157701 q^{85} +2.47700i q^{86} -2.94604 q^{88} +9.24616 q^{89} +(1.39959 + 0.634810i) q^{91} -5.45717i q^{92} -6.95074i q^{94} +1.26222i q^{95} -7.54319i q^{97} +(-5.26651 + 4.61127i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{4} + 4 q^{7}+O(q^{10}) \) 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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