LMFDB - Embedded newform 1632.2.l.a.1393.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1632,2,Mod(1393,1632)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1632, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1632.1393");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1632 = 2^{5} \cdot 3 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1632.l (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$13.0315856099$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - x^{16} - 2x^{14} + 2x^{12} - 4x^{11} + 4x^{10} + 8x^{8} - 16x^{7} + 16x^{6} - 64x^{4} - 128x^{2} + 512 $ x^18 - x^16 - 2x^14 + 2x^12 - 4x^11 + 4x^10 + 8x^8 - 16x^7 + 16x^6 - 64x^4 - 128*x^2 + 512
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{17} $
Twist minimal:	no (minimal twist has level 408)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1393.4
Root			$-0.0535843 + 1.41320i$ of defining polynomial
Character	$\chi$	$=$	1632.1393
Dual form			1632.2.l.a.1393.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.17378 q^{5} +2.51539i q^{7} +1.00000 q^{9} +5.40062 q^{11} -3.61678i q^{13} +3.17378 q^{15} +(-4.09928 - 0.442631i) q^{17} -6.87329i q^{19} -2.51539i q^{21} +7.54740i q^{23} +5.07289 q^{25} -1.00000 q^{27} +3.32236 q^{29} -1.30617i q^{31} -5.40062 q^{33} -7.98329i q^{35} -6.65467 q^{37} +3.61678i q^{39} +5.04206i q^{41} +5.85844i q^{43} -3.17378 q^{45} +8.02838 q^{47} +0.672834 q^{49} +(4.09928 + 0.442631i) q^{51} +7.67778i q^{53} -17.1404 q^{55} +6.87329i q^{57} +4.23300i q^{59} -3.88181 q^{61} +2.51539i q^{63} +11.4789i q^{65} +10.8844i q^{67} -7.54740i q^{69} -4.23091i q^{71} +0.674683i q^{73} -5.07289 q^{75} +13.5846i q^{77} +1.43576i q^{79} +1.00000 q^{81} +16.1494i q^{83} +(13.0102 + 1.40482i) q^{85} -3.32236 q^{87} +9.77068 q^{89} +9.09761 q^{91} +1.30617i q^{93} +21.8143i q^{95} +0.954796i q^{97} +5.40062 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 18 q^{3} - 4 q^{5} + 18 q^{9} + 4 q^{15} - 2 q^{17} + 22 q^{25} - 18 q^{27} + 12 q^{29} - 16 q^{37} - 4 q^{45} - 18 q^{49} + 2 q^{51} + 16 q^{55} - 16 q^{61} - 22 q^{75} + 18 q^{81} + 16 q^{85} - 12 q^{87}+ \cdots + 20 q^{89}+O(q^{100}) $ 18 * q - 18 * q^3 - 4 * q^5 + 18 * q^9 + 4 * q^15 - 2 * q^17 + 22 * q^25 - 18 * q^27 + 12 * q^29 - 16 * q^37 - 4 * q^45 - 18 * q^49 + 2 * q^51 + 16 * q^55 - 16 * q^61 - 22 * q^75 + 18 * q^81 + 16 * q^85 - 12 * q^87 + 20 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1632\mathbb{Z}\right)^\times$.

$n$	$511$	$545$	$613$	$1057$
$\chi(n)$	$1$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$	−3.17378			−1.41936			−0.709679	−	0.704525i	$-0.751160\pi$
$5$	−3.17378			−1.41936			−0.709679	+	0.704525i	$0.751160\pi$
$6$		0			0
$7$			2.51539i			0.950727i	0.879790	+	0.475363i	$0.157683\pi$
$7$			2.51539i			0.950727i	−0.879790	+	0.475363i	$0.842317\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$	5.40062			1.62835			0.814174	−	0.580621i	$-0.197190\pi$
$11$	5.40062			1.62835			0.814174	+	0.580621i	$0.197190\pi$
$12$		0			0
$13$		−	3.61678i		−	1.00312i	−0.865124	−	0.501558i	$-0.832761\pi$
$13$		−	3.61678i		−	1.00312i	0.865124	−	0.501558i	$-0.167239\pi$
$14$		0			0
$15$	3.17378			0.819467
$16$		0			0
$17$	−4.09928	−	0.442631i	−0.994221	−	0.107354i
$17$	−4.09928	−	0.442631i	−0.994221	−	0.107354i
$18$		0			0
$19$		−	6.87329i		−	1.57684i	−0.615137	−	0.788420i	$-0.710899\pi$
$19$		−	6.87329i		−	1.57684i	0.615137	−	0.788420i	$-0.289101\pi$
$20$		0			0
$21$		−	2.51539i		−	0.548902i
$22$		0			0
$23$			7.54740i			1.57374i	0.617118	+	0.786871i	$0.288300\pi$
$23$			7.54740i			1.57374i	−0.617118	+	0.786871i	$0.711700\pi$
$24$		0			0
$25$	5.07289			1.01458
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	3.32236			0.616947			0.308474	−	0.951233i	$-0.400182\pi$
$29$	3.32236			0.616947			0.308474	+	0.951233i	$0.400182\pi$
$30$		0			0
$31$		−	1.30617i		−	0.234596i	−0.993097	−	0.117298i	$-0.962577\pi$
$31$		−	1.30617i		−	0.234596i	0.993097	−	0.117298i	$-0.0374232\pi$
$32$		0			0
$33$	−5.40062			−0.940127
$34$		0			0
$35$		−	7.98329i		−	1.34942i
$36$		0			0
$37$	−6.65467			−1.09402			−0.547010	−	0.837126i	$-0.684234\pi$
$37$	−6.65467			−1.09402			−0.547010	+	0.837126i	$0.684234\pi$
$38$		0			0
$39$			3.61678i			0.579149i
$40$		0			0
$41$			5.04206i			0.787437i	0.919231	+	0.393718i	$0.128811\pi$
$41$			5.04206i			0.787437i	−0.919231	+	0.393718i	$0.871189\pi$
$42$		0			0
$43$			5.85844i			0.893403i	0.894683	+	0.446702i	$0.147401\pi$
$43$			5.85844i			0.893403i	−0.894683	+	0.446702i	$0.852599\pi$
$44$		0			0
$45$	−3.17378			−0.473120
$46$		0			0
$47$	8.02838			1.17106			0.585530	−	0.810651i	$-0.300886\pi$
$47$	8.02838			1.17106			0.585530	+	0.810651i	$0.300886\pi$
$48$		0			0
$49$	0.672834			0.0961191
$50$		0			0
$51$	4.09928	+	0.442631i	0.574014	+	0.0619808i
$52$		0			0
$53$			7.67778i			1.05462i	0.849672	+	0.527312i	$0.176800\pi$
$53$			7.67778i			1.05462i	−0.849672	+	0.527312i	$0.823200\pi$
$54$		0			0
$55$	−17.1404			−2.31121
$56$		0			0
$57$			6.87329i			0.910389i
$58$		0			0
$59$			4.23300i			0.551090i	0.961288	+	0.275545i	$0.0888582\pi$
$59$			4.23300i			0.551090i	−0.961288	+	0.275545i	$0.911142\pi$
$60$		0			0
$61$	−3.88181			−0.497015			−0.248507	−	0.968630i	$-0.579940\pi$
$61$	−3.88181			−0.497015			−0.248507	+	0.968630i	$0.579940\pi$
$62$		0			0
$63$			2.51539i			0.316909i
$64$		0			0
$65$			11.4789i			1.42378i
$66$		0			0
$67$			10.8844i			1.32974i	0.746960	+	0.664869i	$0.231512\pi$
$67$			10.8844i			1.32974i	−0.746960	+	0.664869i	$0.768488\pi$
$68$		0			0
$69$		−	7.54740i		−	0.908600i
$70$		0			0
$71$		−	4.23091i		−	0.502116i	−0.967972	−	0.251058i	$-0.919221\pi$
$71$		−	4.23091i		−	0.502116i	0.967972	−	0.251058i	$-0.0807785\pi$
$72$		0			0
$73$			0.674683i			0.0789656i	0.999220	+	0.0394828i	$0.0125710\pi$
$73$			0.674683i			0.0789656i	−0.999220	+	0.0394828i	$0.987429\pi$
$74$		0			0
$75$	−5.07289			−0.585767
$76$		0			0
$77$			13.5846i			1.54811i
$78$		0			0
$79$			1.43576i			0.161536i	0.996733	+	0.0807680i	$0.0257373\pi$
$79$			1.43576i			0.161536i	−0.996733	+	0.0807680i	$0.974263\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			16.1494i			1.77262i	0.463088	+	0.886312i	$0.346741\pi$
$83$			16.1494i			1.77262i	−0.463088	+	0.886312i	$0.653259\pi$
$84$		0			0
$85$	13.0102	+	1.40482i	1.41116	+	0.152374i
$86$		0			0
$87$	−3.32236			−0.356195
$88$		0			0
$89$	9.77068			1.03569			0.517845	−	0.855475i	$-0.326734\pi$
$89$	9.77068			1.03569			0.517845	+	0.855475i	$0.326734\pi$
$90$		0			0
$91$	9.09761			0.953688
$92$		0			0
$93$			1.30617i			0.135444i
$94$		0			0
$95$			21.8143i			2.23810i
$96$		0			0
$97$			0.954796i			0.0969449i	0.998825	+	0.0484724i	$0.0154353\pi$
$97$			0.954796i			0.0969449i	−0.998825	+	0.0484724i	$0.984565\pi$
$98$		0			0
$99$	5.40062			0.542783
$100$		0			0
$101$			11.7795i			1.17210i	0.810275	+	0.586050i	$0.199318\pi$
$101$			11.7795i			1.17210i	−0.810275	+	0.586050i	$0.800682\pi$
$102$		0			0
$103$	3.63788			0.358451			0.179225	−	0.983808i	$-0.442641\pi$
$103$	3.63788			0.358451			0.179225	+	0.983808i	$0.442641\pi$
$104$		0			0
$105$			7.98329i			0.779089i
$106$		0			0
$107$	2.78476			0.269213			0.134606	−	0.990899i	$-0.457023\pi$
$107$	2.78476			0.269213			0.134606	+	0.990899i	$0.457023\pi$
$108$		0			0
$109$	−14.7517			−1.41296			−0.706479	−	0.707734i	$-0.749718\pi$
$109$	−14.7517			−1.41296			−0.706479	+	0.707734i	$0.749718\pi$
$110$		0			0
$111$	6.65467			0.631633
$112$		0			0
$113$			7.57157i			0.712274i	0.934434	+	0.356137i	$0.115906\pi$
$113$			7.57157i			0.712274i	−0.934434	+	0.356137i	$0.884094\pi$
$114$		0			0
$115$		−	23.9538i		−	2.23370i
$116$		0			0
$117$		−	3.61678i		−	0.334372i
$118$		0			0
$119$	1.11339	−	10.3113i	0.102064	−	0.945232i
$120$		0			0
$121$	18.1667			1.65152
$122$		0			0
$123$		−	5.04206i		−	0.454627i
$124$		0			0
$125$	−0.231351			−0.0206927
$126$		0			0
$127$	8.80507			0.781324			0.390662	−	0.920534i	$-0.372246\pi$
$127$	8.80507			0.781324			0.390662	+	0.920534i	$0.372246\pi$
$128$		0			0
$129$		−	5.85844i		−	0.515807i
$130$		0			0
$131$	−21.0088			−1.83555			−0.917773	−	0.397106i	$-0.870014\pi$
$131$	−21.0088			−1.83555			−0.917773	+	0.397106i	$0.870014\pi$
$132$		0			0
$133$	17.2890			1.49914
$134$		0			0
$135$	3.17378			0.273156
$136$		0			0
$137$	−15.9533			−1.36298			−0.681490	−	0.731828i	$-0.738668\pi$
$137$	−15.9533			−1.36298			−0.681490	+	0.731828i	$0.738668\pi$
$138$		0			0
$139$	−6.61325			−0.560929			−0.280464	−	0.959864i	$-0.590488\pi$
$139$	−6.61325			−0.560929			−0.280464	+	0.959864i	$0.590488\pi$
$140$		0			0
$141$	−8.02838			−0.676112
$142$		0			0
$143$		−	19.5329i		−	1.63342i
$144$		0			0
$145$	−10.5445			−0.875670
$146$		0			0
$147$	−0.672834			−0.0554944
$148$		0			0
$149$		−	15.3325i		−	1.25609i	−0.778178	−	0.628044i	$-0.783856\pi$
$149$		−	15.3325i		−	1.25609i	0.778178	−	0.628044i	$-0.216144\pi$
$150$		0			0
$151$	1.10540			0.0899565			0.0449782	−	0.998988i	$-0.485678\pi$
$151$	1.10540			0.0899565			0.0449782	+	0.998988i	$0.485678\pi$
$152$		0			0
$153$	−4.09928	−	0.442631i	−0.331407	−	0.0357846i
$154$		0			0
$155$			4.14551i			0.332975i
$156$		0			0
$157$		−	17.6004i		−	1.40467i	−0.711849	−	0.702333i	$-0.752142\pi$
$157$		−	17.6004i		−	1.40467i	0.711849	−	0.702333i	$-0.247858\pi$
$158$		0			0
$159$		−	7.67778i		−	0.608888i
$160$		0			0
$161$	−18.9846			−1.49620
$162$		0			0
$163$	0.345904			0.0270933			0.0135467	−	0.999908i	$-0.495688\pi$
$163$	0.345904			0.0270933			0.0135467	+	0.999908i	$0.495688\pi$
$164$		0			0
$165$	17.1404			1.33438
$166$		0			0
$167$			20.7946i			1.60913i	0.593861	+	0.804567i	$0.297603\pi$
$167$			20.7946i			1.60913i	−0.593861	+	0.804567i	$0.702397\pi$
$168$		0			0
$169$	−0.0811240			−0.00624031
$170$		0			0
$171$		−	6.87329i		−	0.525614i
$172$		0			0
$173$	8.28576			0.629954			0.314977	−	0.949099i	$-0.398003\pi$
$173$	8.28576			0.629954			0.314977	+	0.949099i	$0.398003\pi$
$174$		0			0
$175$			12.7603i			0.964587i
$176$		0			0
$177$		−	4.23300i		−	0.318172i
$178$		0			0
$179$			17.2101i			1.28634i	0.765722	+	0.643172i	$0.222382\pi$
$179$			17.2101i			1.28634i	−0.765722	+	0.643172i	$0.777618\pi$
$180$		0			0
$181$	1.76280			0.131028			0.0655141	−	0.997852i	$-0.479131\pi$
$181$	1.76280			0.131028			0.0655141	+	0.997852i	$0.479131\pi$
$182$		0			0
$183$	3.88181			0.286952
$184$		0			0
$185$	21.1205			1.55281
$186$		0			0
$187$	−22.1386	−	2.39048i	−1.61894	−	0.174809i
$188$		0			0
$189$		−	2.51539i		−	0.182967i
$190$		0			0
$191$	−4.73825			−0.342848			−0.171424	−	0.985197i	$-0.554837\pi$
$191$	−4.73825			−0.342848			−0.171424	+	0.985197i	$0.554837\pi$
$192$		0			0
$193$			20.9493i			1.50796i	0.656897	+	0.753980i	$0.271869\pi$
$193$			20.9493i			1.50796i	−0.656897	+	0.753980i	$0.728131\pi$
$194$		0			0
$195$		−	11.4789i		−	0.822020i
$196$		0			0
$197$	8.35170			0.595034			0.297517	−	0.954716i	$-0.403841\pi$
$197$	8.35170			0.595034			0.297517	+	0.954716i	$0.403841\pi$
$198$		0			0
$199$			12.3310i			0.874121i	0.899432	+	0.437060i	$0.143980\pi$
$199$			12.3310i			0.874121i	−0.899432	+	0.437060i	$0.856020\pi$
$200$		0			0
$201$		−	10.8844i		−	0.767724i
$202$		0			0
$203$			8.35703i			0.586548i
$204$		0			0
$205$		−	16.0024i		−	1.11766i
$206$		0			0
$207$			7.54740i			0.524581i
$208$		0			0
$209$		−	37.1200i		−	2.56765i
$210$		0			0
$211$	16.0368			1.10402			0.552010	−	0.833837i	$-0.313861\pi$
$211$	16.0368			1.10402			0.552010	+	0.833837i	$0.313861\pi$
$212$		0			0
$213$			4.23091i			0.289897i
$214$		0			0
$215$		−	18.5934i		−	1.26806i
$216$		0			0
$217$	3.28553			0.223036
$218$		0			0
$219$		−	0.674683i		−	0.0455908i
$220$		0			0
$221$	−1.60090	+	14.8262i	−0.107688	+	0.997318i
$222$		0			0
$223$	−10.0104			−0.670349			−0.335174	−	0.942156i	$-0.608795\pi$
$223$	−10.0104			−0.670349			−0.335174	+	0.942156i	$0.608795\pi$
$224$		0			0
$225$	5.07289			0.338193
$226$		0			0
$227$	−14.0466			−0.932305			−0.466152	−	0.884704i	$-0.654360\pi$
$227$	−14.0466			−0.932305			−0.466152	+	0.884704i	$0.654360\pi$
$228$		0			0
$229$		−	3.35868i		−	0.221948i	−0.993823	−	0.110974i	$-0.964603\pi$
$229$		−	3.35868i		−	0.221948i	0.993823	−	0.110974i	$-0.0353970\pi$
$230$		0			0
$231$		−	13.5846i		−	0.893804i
$232$		0			0
$233$		−	15.1152i		−	0.990230i	−0.868827	−	0.495115i	$-0.835126\pi$
$233$		−	15.1152i		−	0.990230i	0.868827	−	0.495115i	$-0.164874\pi$
$234$		0			0
$235$	−25.4803			−1.66215
$236$		0			0
$237$		−	1.43576i		−	0.0932628i
$238$		0			0
$239$	−11.3025			−0.731101			−0.365550	−	0.930792i	$-0.619119\pi$
$239$	−11.3025			−0.731101			−0.365550	+	0.930792i	$0.619119\pi$
$240$		0			0
$241$			23.8864i			1.53866i	0.638851	+	0.769330i	$0.279410\pi$
$241$			23.8864i			1.53866i	−0.638851	+	0.769330i	$0.720590\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$	−2.13543			−0.136427
$246$		0			0
$247$	−24.8592			−1.58175
$248$		0			0
$249$		−	16.1494i		−	1.02343i
$250$		0			0
$251$			7.16691i			0.452371i	0.974084	+	0.226186i	$0.0726256\pi$
$251$			7.16691i			0.452371i	−0.974084	+	0.226186i	$0.927374\pi$
$252$		0			0
$253$			40.7606i			2.56260i
$254$		0			0
$255$	−13.0102	−	1.40482i	−0.814731	−	0.0879729i
$256$		0			0
$257$	−5.46601			−0.340960			−0.170480	−	0.985361i	$-0.554532\pi$
$257$	−5.46601			−0.340960			−0.170480	+	0.985361i	$0.554532\pi$
$258$		0			0
$259$		−	16.7391i		−	1.04011i
$260$		0			0
$261$	3.32236			0.205649
$262$		0			0
$263$	15.2442			0.939997			0.469999	−	0.882667i	$-0.344254\pi$
$263$	15.2442			0.939997			0.469999	+	0.882667i	$0.344254\pi$
$264$		0			0
$265$		−	24.3676i		−	1.49689i
$266$		0			0
$267$	−9.77068			−0.597956
$268$		0			0
$269$	−6.61819			−0.403518			−0.201759	−	0.979435i	$-0.564666\pi$
$269$	−6.61819			−0.403518			−0.201759	+	0.979435i	$0.564666\pi$
$270$		0			0
$271$	11.5920			0.704166			0.352083	−	0.935969i	$-0.385473\pi$
$271$	11.5920			0.704166			0.352083	+	0.935969i	$0.385473\pi$
$272$		0			0
$273$	−9.09761			−0.550612
$274$		0			0
$275$	27.3968			1.65209
$276$		0			0
$277$	16.6679			1.00148			0.500738	−	0.865599i	$-0.333062\pi$
$277$	16.6679			1.00148			0.500738	+	0.865599i	$0.333062\pi$
$278$		0			0
$279$		−	1.30617i		−	0.0781985i
$280$		0			0
$281$	−6.51175			−0.388458			−0.194229	−	0.980956i	$-0.562221\pi$
$281$	−6.51175			−0.388458			−0.194229	+	0.980956i	$0.562221\pi$
$282$		0			0
$283$	21.2924			1.26570			0.632850	−	0.774275i	$-0.281885\pi$
$283$	21.2924			1.26570			0.632850	+	0.774275i	$0.281885\pi$
$284$		0			0
$285$		−	21.8143i		−	1.29217i
$286$		0			0
$287$	−12.6827			−0.748637
$288$		0			0
$289$	16.6082	+	3.62894i	0.976950	+	0.213467i
$290$		0			0
$291$		−	0.954796i		−	0.0559712i
$292$		0			0
$293$			6.20078i			0.362253i	0.983460	+	0.181127i	$0.0579744\pi$
$293$			6.20078i			0.362253i	−0.983460	+	0.181127i	$0.942026\pi$
$294$		0			0
$295$		−	13.4346i		−	0.782194i
$296$		0			0
$297$	−5.40062			−0.313376
$298$		0			0
$299$	27.2973			1.57864
$300$		0			0
$301$	−14.7362			−0.849382
$302$		0			0
$303$		−	11.7795i		−	0.676712i
$304$		0			0
$305$	12.3200			0.705442
$306$		0			0
$307$		−	3.96275i		−	0.226166i	−0.993586	−	0.113083i	$-0.963927\pi$
$307$		−	3.96275i		−	0.226166i	0.993586	−	0.113083i	$-0.0360726\pi$
$308$		0			0
$309$	−3.63788			−0.206952
$310$		0			0
$311$		−	6.26994i		−	0.355535i	−0.984072	−	0.177768i	$-0.943112\pi$
$311$		−	6.26994i		−	0.355535i	0.984072	−	0.177768i	$-0.0568876\pi$
$312$		0			0
$313$		−	10.7509i		−	0.607675i	−0.952724	−	0.303838i	$-0.901732\pi$
$313$		−	10.7509i		−	0.607675i	0.952724	−	0.303838i	$-0.0982681\pi$
$314$		0			0
$315$		−	7.98329i		−	0.449807i
$316$		0			0
$317$	−17.0890			−0.959815			−0.479908	−	0.877319i	$-0.659330\pi$
$317$	−17.0890			−0.959815			−0.479908	+	0.877319i	$0.659330\pi$
$318$		0			0
$319$	17.9428			1.00461
$320$		0			0
$321$	−2.78476			−0.155430
$322$		0			0
$323$	−3.04233	+	28.1755i	−0.169280	+	1.56773i
$324$		0			0
$325$		−	18.3476i		−	1.01774i
$326$		0			0
$327$	14.7517			0.815772
$328$		0			0
$329$			20.1945i			1.11336i
$330$		0			0
$331$		−	7.05539i		−	0.387799i	−0.981021	−	0.193900i	$-0.937886\pi$
$331$		−	7.05539i		−	0.387799i	0.981021	−	0.193900i	$-0.0621136\pi$
$332$		0			0
$333$	−6.65467			−0.364673
$334$		0			0
$335$		−	34.5446i		−	1.88737i
$336$		0			0
$337$		−	19.8274i		−	1.08007i	−0.841643	−	0.540035i	$-0.818411\pi$
$337$		−	19.8274i		−	1.08007i	0.841643	−	0.540035i	$-0.181589\pi$
$338$		0			0
$339$		−	7.57157i		−	0.411231i
$340$		0			0
$341$		−	7.05414i		−	0.382003i
$342$		0			0
$343$			19.3001i			1.04211i
$344$		0			0
$345$			23.9538i			1.28963i
$346$		0			0
$347$	24.3591			1.30767			0.653833	−	0.756639i	$-0.273160\pi$
$347$	24.3591			1.30767			0.653833	+	0.756639i	$0.273160\pi$
$348$		0			0
$349$			1.20575i			0.0645421i	0.999479	+	0.0322710i	$0.0102740\pi$
$349$			1.20575i			0.0645421i	−0.999479	+	0.0322710i	$0.989726\pi$
$350$		0			0
$351$			3.61678i			0.193050i
$352$		0			0
$353$	11.4575			0.609820			0.304910	−	0.952381i	$-0.401374\pi$
$353$	11.4575			0.609820			0.304910	+	0.952381i	$0.401374\pi$
$354$		0			0
$355$			13.4280i			0.712683i
$356$		0			0
$357$	−1.11339	+	10.3113i	−0.0589268	+	0.545730i
$358$		0			0
$359$	31.0234			1.63735			0.818677	−	0.574254i	$-0.194708\pi$
$359$	31.0234			1.63735			0.818677	+	0.574254i	$0.194708\pi$
$360$		0			0
$361$	−28.2421			−1.48643
$362$		0			0
$363$	−18.1667			−0.953504
$364$		0			0
$365$		−	2.14130i		−	0.112081i
$366$		0			0
$367$		−	9.00308i		−	0.469957i	−0.972001	−	0.234978i	$-0.924498\pi$
$367$		−	9.00308i		−	0.469957i	0.972001	−	0.234978i	$-0.0755020\pi$
$368$		0			0
$369$			5.04206i			0.262479i
$370$		0			0
$371$	−19.3126			−1.00266
$372$		0			0
$373$		−	23.7413i		−	1.22928i	−0.788809	−	0.614638i	$-0.789302\pi$
$373$		−	23.7413i		−	1.22928i	0.788809	−	0.614638i	$-0.210698\pi$
$374$		0			0
$375$	0.231351			0.0119469
$376$		0			0
$377$		−	12.0163i		−	0.618869i
$378$		0			0
$379$	10.7431			0.551835			0.275917	−	0.961181i	$-0.411018\pi$
$379$	10.7431			0.551835			0.275917	+	0.961181i	$0.411018\pi$
$380$		0			0
$381$	−8.80507			−0.451098
$382$		0			0
$383$	13.2678			0.677952			0.338976	−	0.940795i	$-0.389919\pi$
$383$	13.2678			0.677952			0.338976	+	0.940795i	$0.389919\pi$
$384$		0			0
$385$		−	43.1147i		−	2.19733i
$386$		0			0
$387$			5.85844i			0.297801i
$388$		0			0
$389$			32.5854i			1.65214i	0.563565	+	0.826072i	$0.309429\pi$
$389$			32.5854i			1.65214i	−0.563565	+	0.826072i	$0.690571\pi$
$390$		0			0
$391$	3.34071	−	30.9389i	0.168947	−	1.56465i
$392$		0			0
$393$	21.0088			1.05975
$394$		0			0
$395$		−	4.55680i		−	0.229277i
$396$		0			0
$397$	−13.2519			−0.665092			−0.332546	−	0.943087i	$-0.607908\pi$
$397$	−13.2519			−0.665092			−0.332546	+	0.943087i	$0.607908\pi$
$398$		0			0
$399$	−17.2890			−0.865531
$400$		0			0
$401$			12.2827i			0.613366i	0.951812	+	0.306683i	$0.0992192\pi$
$401$			12.2827i			0.613366i	−0.951812	+	0.306683i	$0.900781\pi$
$402$		0			0
$403$	−4.72415			−0.235326
$404$		0			0
$405$	−3.17378			−0.157707
$406$		0			0
$407$	−35.9393			−1.78145
$408$		0			0
$409$	−16.8564			−0.833493			−0.416747	−	0.909023i	$-0.636830\pi$
$409$	−16.8564			−0.833493			−0.416747	+	0.909023i	$0.636830\pi$
$410$		0			0
$411$	15.9533			0.786916
$412$		0			0
$413$	−10.6476			−0.523936
$414$		0			0
$415$		−	51.2546i		−	2.51599i
$416$		0			0
$417$	6.61325			0.323852
$418$		0			0
$419$	−7.38718			−0.360887			−0.180444	−	0.983585i	$-0.557753\pi$
$419$	−7.38718			−0.360887			−0.180444	+	0.983585i	$0.557753\pi$
$420$		0			0
$421$			20.6976i			1.00874i	0.863489	+	0.504369i	$0.168275\pi$
$421$			20.6976i			1.00874i	−0.863489	+	0.504369i	$0.831725\pi$
$422$		0			0
$423$	8.02838			0.390353
$424$		0			0
$425$	−20.7952	−	2.24542i	−1.00872	−	0.108919i
$426$		0			0
$427$		−	9.76425i		−	0.472525i
$428$		0			0
$429$			19.5329i			0.943056i
$430$		0			0
$431$			13.1852i			0.635107i	0.948240	+	0.317554i	$0.102861\pi$
$431$			13.1852i			0.635107i	−0.948240	+	0.317554i	$0.897139\pi$
$432$		0			0
$433$	14.7426			0.708485			0.354243	−	0.935154i	$-0.384739\pi$
$433$	14.7426			0.708485			0.354243	+	0.935154i	$0.384739\pi$
$434$		0			0
$435$	10.5445			0.505568
$436$		0			0
$437$	51.8755			2.48154
$438$		0			0
$439$		−	6.15820i		−	0.293915i	−0.989143	−	0.146958i	$-0.953052\pi$
$439$		−	6.15820i		−	0.293915i	0.989143	−	0.146958i	$-0.0469481\pi$
$440$		0			0
$441$	0.672834			0.0320397
$442$		0			0
$443$			2.64879i			0.125848i	0.998018	+	0.0629239i	$0.0200425\pi$
$443$			2.64879i			0.125848i	−0.998018	+	0.0629239i	$0.979957\pi$
$444$		0			0
$445$	−31.0100			−1.47002
$446$		0			0
$447$			15.3325i			0.725203i
$448$		0			0
$449$		−	33.2045i		−	1.56702i	−0.621381	−	0.783509i	$-0.713428\pi$
$449$		−	33.2045i		−	1.56702i	0.621381	−	0.783509i	$-0.286572\pi$
$450$		0			0
$451$			27.2302i			1.28222i
$452$		0			0
$453$	−1.10540			−0.0519364
$454$		0			0
$455$	−28.8738			−1.35363
$456$		0			0
$457$	−21.2316			−0.993171			−0.496585	−	0.867988i	$-0.665413\pi$
$457$	−21.2316			−0.993171			−0.496585	+	0.867988i	$0.665413\pi$
$458$		0			0
$459$	4.09928	+	0.442631i	0.191338	+	0.0206603i
$460$		0			0
$461$		−	35.0837i		−	1.63401i	−0.576631	−	0.817004i	$-0.695633\pi$
$461$		−	35.0837i		−	1.63401i	0.576631	−	0.817004i	$-0.304367\pi$
$462$		0			0
$463$	22.4625			1.04392			0.521960	−	0.852970i	$-0.325201\pi$
$463$	22.4625			1.04392			0.521960	+	0.852970i	$0.325201\pi$
$464$		0			0
$465$		−	4.14551i		−	0.192243i
$466$		0			0
$467$		−	14.3658i		−	0.664772i	−0.943143	−	0.332386i	$-0.892146\pi$
$467$		−	14.3658i		−	0.664772i	0.943143	−	0.332386i	$-0.107854\pi$
$468$		0			0
$469$	−27.3784			−1.26422
$470$		0			0
$471$			17.6004i			0.810984i
$472$		0			0
$473$			31.6392i			1.45477i
$474$		0			0
$475$		−	34.8675i		−	1.59983i
$476$		0			0
$477$			7.67778i			0.351541i
$478$		0			0
$479$		−	7.01618i		−	0.320577i	−0.987070	−	0.160289i	$-0.948758\pi$
$479$		−	7.01618i		−	0.320577i	0.987070	−	0.160289i	$-0.0512425\pi$
$480$		0			0
$481$			24.0685i			1.09743i
$482$		0			0
$483$	18.9846			0.863830
$484$		0			0
$485$		−	3.03032i		−	0.137600i
$486$		0			0
$487$		−	24.5769i		−	1.11368i	−0.830619	−	0.556842i	$-0.812013\pi$
$487$		−	24.5769i		−	1.11368i	0.830619	−	0.556842i	$-0.187987\pi$
$488$		0			0
$489$	−0.345904			−0.0156423
$490$		0			0
$491$			7.60956i			0.343415i	0.985148	+	0.171707i	$0.0549283\pi$
$491$			7.60956i			0.343415i	−0.985148	+	0.171707i	$0.945072\pi$
$492$		0			0
$493$	−13.6193	−	1.47058i	−0.613382	−	0.0662317i
$494$		0			0
$495$	−17.1404			−0.770403
$496$		0			0
$497$	10.6424			0.477375
$498$		0			0
$499$	16.2432			0.727144			0.363572	−	0.931566i	$-0.381557\pi$
$499$	16.2432			0.727144			0.363572	+	0.931566i	$0.381557\pi$
$500$		0			0
$501$		−	20.7946i		−	0.929034i
$502$		0			0
$503$			26.3137i			1.17327i	0.809851	+	0.586636i	$0.199548\pi$
$503$			26.3137i			1.17327i	−0.809851	+	0.586636i	$0.800452\pi$
$504$		0			0
$505$		−	37.3854i		−	1.66363i
$506$		0			0
$507$	0.0811240			0.00360284
$508$		0			0
$509$			7.93657i			0.351782i	0.984410	+	0.175891i	$0.0562806\pi$
$509$			7.93657i			0.351782i	−0.984410	+	0.175891i	$0.943719\pi$
$510$		0			0
$511$	−1.69709			−0.0750747
$512$		0			0
$513$			6.87329i			0.303463i
$514$		0			0
$515$	−11.5458			−0.508770
$516$		0			0
$517$	43.3582			1.90689
$518$		0			0
$519$	−8.28576			−0.363704
$520$		0			0
$521$			9.50069i			0.416233i	0.978104	+	0.208116i	$0.0667333\pi$
$521$			9.50069i			0.416233i	−0.978104	+	0.208116i	$0.933267\pi$
$522$		0			0
$523$		−	3.17435i		−	0.138805i	−0.997589	−	0.0694024i	$-0.977891\pi$
$523$		−	3.17435i		−	0.138805i	0.997589	−	0.0694024i	$-0.0221092\pi$
$524$		0			0
$525$		−	12.7603i		−	0.556905i
$526$		0			0
$527$	−0.578153	+	5.35437i	−0.0251847	+	0.233240i
$528$		0			0
$529$	−33.9632			−1.47666
$530$		0			0
$531$			4.23300i			0.183697i
$532$		0			0
$533$	18.2360			0.789890
$534$		0			0
$535$	−8.83821			−0.382109
$536$		0			0
$537$		−	17.2101i		−	0.742671i
$538$		0			0
$539$	3.63372			0.156515
$540$		0			0
$541$	19.5892			0.842205			0.421103	−	0.907013i	$-0.361643\pi$
$541$	19.5892			0.842205			0.421103	+	0.907013i	$0.361643\pi$
$542$		0			0
$543$	−1.76280			−0.0756491
$544$		0			0
$545$	46.8187			2.00549
$546$		0			0
$547$	17.5753			0.751467			0.375734	−	0.926728i	$-0.377391\pi$
$547$	17.5753			0.751467			0.375734	+	0.926728i	$0.377391\pi$
$548$		0			0
$549$	−3.88181			−0.165672
$550$		0			0
$551$		−	22.8356i		−	0.972828i
$552$		0			0
$553$	−3.61150			−0.153576
$554$		0			0
$555$	−21.1205			−0.896514
$556$		0			0
$557$			30.3013i			1.28391i	0.766744	+	0.641953i	$0.221875\pi$
$557$			30.3013i			1.28391i	−0.766744	+	0.641953i	$0.778125\pi$
$558$		0			0
$559$	21.1887			0.896186
$560$		0			0
$561$	22.1386	+	2.39048i	0.934694	+	0.100926i
$562$		0			0
$563$			2.46670i			0.103959i	0.998648	+	0.0519794i	$0.0165530\pi$
$563$			2.46670i			0.103959i	−0.998648	+	0.0519794i	$0.983447\pi$
$564$		0			0
$565$		−	24.0305i		−	1.01097i
$566$		0			0
$567$			2.51539i			0.105636i
$568$		0			0
$569$	21.6017			0.905590			0.452795	−	0.891615i	$-0.350427\pi$
$569$	21.6017			0.905590			0.452795	+	0.891615i	$0.350427\pi$
$570$		0			0
$571$	10.9931			0.460046			0.230023	−	0.973185i	$-0.426120\pi$
$571$	10.9931			0.460046			0.230023	+	0.973185i	$0.426120\pi$
$572$		0			0
$573$	4.73825			0.197943
$574$		0			0
$575$			38.2872i			1.59668i
$576$		0			0
$577$	10.7397			0.447100			0.223550	−	0.974692i	$-0.428235\pi$
$577$	10.7397			0.447100			0.223550	+	0.974692i	$0.428235\pi$
$578$		0			0
$579$		−	20.9493i		−	0.870621i
$580$		0			0
$581$	−40.6219			−1.68528
$582$		0			0
$583$			41.4648i			1.71730i
$584$		0			0
$585$			11.4789i			0.474593i
$586$		0			0
$587$		−	35.1504i		−	1.45081i	−0.688321	−	0.725406i	$-0.741652\pi$
$587$		−	35.1504i		−	1.45081i	0.688321	−	0.725406i	$-0.258348\pi$
$588$		0			0
$589$	−8.97771			−0.369920
$590$		0			0
$591$	−8.35170			−0.343543
$592$		0			0
$593$	−30.3326			−1.24561			−0.622805	−	0.782377i	$-0.714007\pi$
$593$	−30.3326			−1.24561			−0.622805	+	0.782377i	$0.714007\pi$
$594$		0			0
$595$	−3.53365	+	32.7257i	−0.144866	+	1.34162i
$596$		0			0
$597$		−	12.3310i		−	0.504674i
$598$		0			0
$599$	−31.3182			−1.27963			−0.639814	−	0.768530i	$-0.720988\pi$
$599$	−31.3182			−1.27963			−0.639814	+	0.768530i	$0.720988\pi$
$600$		0			0
$601$			30.7936i			1.25610i	0.778174	+	0.628048i	$0.216146\pi$
$601$			30.7936i			1.25610i	−0.778174	+	0.628048i	$0.783854\pi$
$602$		0			0
$603$			10.8844i			0.443246i
$604$		0			0
$605$	−57.6571			−2.34410
$606$		0			0
$607$		−	2.10908i		−	0.0856050i	−0.999084	−	0.0428025i	$-0.986371\pi$
$607$		−	2.10908i		−	0.0856050i	0.999084	−	0.0428025i	$-0.0136286\pi$
$608$		0			0
$609$		−	8.35703i		−	0.338644i
$610$		0			0
$611$		−	29.0369i		−	1.17471i
$612$		0			0
$613$			18.9528i			0.765497i	0.923853	+	0.382749i	$0.125022\pi$
$613$			18.9528i			0.765497i	−0.923853	+	0.382749i	$0.874978\pi$
$614$		0			0
$615$			16.0024i			0.645279i
$616$		0			0
$617$		−	7.76650i		−	0.312668i	−0.987704	−	0.156334i	$-0.950032\pi$
$617$		−	7.76650i		−	0.312668i	0.987704	−	0.156334i	$-0.0499676\pi$
$618$		0			0
$619$	−17.1983			−0.691258			−0.345629	−	0.938371i	$-0.612334\pi$
$619$	−17.1983			−0.691258			−0.345629	+	0.938371i	$0.612334\pi$
$620$		0			0
$621$		−	7.54740i		−	0.302867i
$622$		0			0
$623$			24.5770i			0.984658i
$624$		0			0
$625$	−24.6302			−0.985209
$626$		0			0
$627$			37.1200i			1.48243i
$628$		0			0
$629$	27.2793	+	2.94556i	1.08770	+	0.117447i
$630$		0			0
$631$	0.102276			0.00407152			0.00203576	−	0.999998i	$-0.499352\pi$
$631$	0.102276			0.00407152			0.00203576	+	0.999998i	$0.499352\pi$
$632$		0			0
$633$	−16.0368			−0.637407
$634$		0			0
$635$	−27.9454			−1.10898
$636$		0			0
$637$		−	2.43349i		−	0.0964185i
$638$		0			0
$639$		−	4.23091i		−	0.167372i
$640$		0			0
$641$		−	47.2480i		−	1.86618i	−0.359639	−	0.933091i	$-0.617100\pi$
$641$		−	47.2480i		−	1.86618i	0.359639	−	0.933091i	$-0.382900\pi$
$642$		0			0
$643$	−22.6820			−0.894491			−0.447245	−	0.894411i	$-0.647595\pi$
$643$	−22.6820			−0.894491			−0.447245	+	0.894411i	$0.647595\pi$
$644$		0			0
$645$			18.5934i			0.732115i
$646$		0			0
$647$	42.0519			1.65323			0.826615	−	0.562768i	$-0.190264\pi$
$647$	42.0519			1.65323			0.826615	+	0.562768i	$0.190264\pi$
$648$		0			0
$649$			22.8608i			0.897366i
$650$		0			0
$651$	−3.28553			−0.128770
$652$		0			0
$653$	−42.6516			−1.66909			−0.834544	−	0.550942i	$-0.814269\pi$
$653$	−42.6516			−1.66909			−0.834544	+	0.550942i	$0.814269\pi$
$654$		0			0
$655$	66.6773			2.60530
$656$		0			0
$657$			0.674683i			0.0263219i
$658$		0			0
$659$			1.41696i			0.0551970i	0.999619	+	0.0275985i	$0.00878600\pi$
$659$			1.41696i			0.0551970i	−0.999619	+	0.0275985i	$0.991214\pi$
$660$		0			0
$661$		−	11.2234i		−	0.436541i	−0.975888	−	0.218270i	$-0.929959\pi$
$661$		−	11.2234i		−	0.436541i	0.975888	−	0.218270i	$-0.0700415\pi$
$662$		0			0
$663$	1.60090	−	14.8262i	0.0621739	−	0.575802i
$664$		0			0
$665$	−54.8714			−2.12782
$666$		0			0
$667$			25.0752i			0.970916i
$668$		0			0
$669$	10.0104			0.387026
$670$		0			0
$671$	−20.9642			−0.809313
$672$		0			0
$673$		−	22.9936i		−	0.886337i	−0.896438	−	0.443169i	$-0.853854\pi$
$673$		−	22.9936i		−	0.886337i	0.896438	−	0.443169i	$-0.146146\pi$
$674$		0			0
$675$	−5.07289			−0.195256
$676$		0			0
$677$	39.4852			1.51754			0.758771	−	0.651358i	$-0.225800\pi$
$677$	39.4852			1.51754			0.758771	+	0.651358i	$0.225800\pi$
$678$		0			0
$679$	−2.40168			−0.0921681
$680$		0			0
$681$	14.0466			0.538266
$682$		0			0
$683$	20.6340			0.789537			0.394768	−	0.918781i	$-0.370825\pi$
$683$	20.6340			0.789537			0.394768	+	0.918781i	$0.370825\pi$
$684$		0			0
$685$	50.6322			1.93456
$686$		0			0
$687$			3.35868i			0.128142i
$688$		0			0
$689$	27.7689			1.05791
$690$		0			0
$691$	19.5064			0.742057			0.371028	−	0.928622i	$-0.379005\pi$
$691$	19.5064			0.742057			0.371028	+	0.928622i	$0.379005\pi$
$692$		0			0
$693$			13.5846i			0.516038i
$694$		0			0
$695$	20.9890			0.796159
$696$		0			0
$697$	2.23177	−	20.6688i	0.0845344	−	0.782886i
$698$		0			0
$699$			15.1152i			0.571710i
$700$		0			0
$701$			22.6742i			0.856392i	0.903686	+	0.428196i	$0.140851\pi$
$701$			22.6742i			0.856392i	−0.903686	+	0.428196i	$0.859149\pi$
$702$		0			0
$703$			45.7394i			1.72510i
$704$		0			0
$705$	25.4803			0.959645
$706$		0			0
$707$	−29.6299			−1.11435
$708$		0			0
$709$	2.38593			0.0896056			0.0448028	−	0.998996i	$-0.485734\pi$
$709$	2.38593			0.0896056			0.0448028	+	0.998996i	$0.485734\pi$
$710$		0			0
$711$			1.43576i			0.0538453i
$712$		0			0
$713$	9.85821			0.369193
$714$		0			0
$715$			61.9931i			2.31841i
$716$		0			0
$717$	11.3025			0.422101
$718$		0			0
$719$		−	29.3055i		−	1.09291i	−0.837489	−	0.546455i	$-0.815977\pi$
$719$		−	29.3055i		−	1.09291i	0.837489	−	0.546455i	$-0.184023\pi$
$720$		0			0
$721$			9.15067i			0.340789i
$722$		0			0
$723$		−	23.8864i		−	0.888346i
$724$		0			0
$725$	16.8540			0.625942
$726$		0			0
$727$	−21.2949			−0.789785			−0.394892	−	0.918727i	$-0.629218\pi$
$727$	−21.2949			−0.789785			−0.394892	+	0.918727i	$0.629218\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	2.59313	−	24.0154i	0.0959103	−	0.888240i
$732$		0			0
$733$			34.3060i			1.26712i	0.773693	+	0.633561i	$0.218407\pi$
$733$			34.3060i			1.26712i	−0.773693	+	0.633561i	$0.781593\pi$
$734$		0			0
$735$	2.13543			0.0787664
$736$		0			0
$737$			58.7823i			2.16527i
$738$		0			0
$739$			35.2799i			1.29779i	0.760877	+	0.648896i	$0.224769\pi$
$739$			35.2799i			1.29779i	−0.760877	+	0.648896i	$0.775231\pi$
$740$		0			0
$741$	24.8592			0.913225
$742$		0			0
$743$		−	19.0184i		−	0.697719i	−0.937175	−	0.348859i	$-0.886569\pi$
$743$		−	19.0184i		−	0.697719i	0.937175	−	0.348859i	$-0.113431\pi$
$744$		0			0
$745$			48.6620i			1.78284i
$746$		0			0
$747$			16.1494i			0.590875i
$748$		0			0
$749$			7.00474i			0.255948i
$750$		0			0
$751$			38.5164i			1.40548i	0.711446	+	0.702741i	$0.248041\pi$
$751$			38.5164i			1.40548i	−0.711446	+	0.702741i	$0.751959\pi$
$752$		0			0
$753$		−	7.16691i		−	0.261177i
$754$		0			0
$755$	−3.50831			−0.127681
$756$		0			0
$757$			13.2396i			0.481200i	0.970624	+	0.240600i	$0.0773442\pi$
$757$			13.2396i			0.481200i	−0.970624	+	0.240600i	$0.922656\pi$
$758$		0			0
$759$		−	40.7606i		−	1.47952i
$760$		0			0
$761$	23.2876			0.844173			0.422087	−	0.906556i	$-0.361298\pi$
$761$	23.2876			0.844173			0.422087	+	0.906556i	$0.361298\pi$
$762$		0			0
$763$		−	37.1063i		−	1.34334i
$764$		0			0
$765$	13.0102	+	1.40482i	0.470385	+	0.0507912i
$766$		0			0
$767$	15.3098			0.552806
$768$		0			0
$769$	18.6064			0.670964			0.335482	−	0.942047i	$-0.391101\pi$
$769$	18.6064			0.670964			0.335482	+	0.942047i	$0.391101\pi$
$770$		0			0
$771$	5.46601			0.196854
$772$		0			0
$773$		−	14.1210i		−	0.507898i	−0.967218	−	0.253949i	$-0.918270\pi$
$773$		−	14.1210i		−	0.507898i	0.967218	−	0.253949i	$-0.0817295\pi$
$774$		0			0
$775$		−	6.62608i		−	0.238016i
$776$		0			0
$777$			16.7391i			0.600510i
$778$		0			0
$779$	34.6555			1.24166
$780$		0			0
$781$		−	22.8495i		−	0.817620i
$782$		0			0
$783$	−3.32236			−0.118732
$784$		0			0
$785$			55.8599i			1.99372i
$786$		0			0
$787$	−36.6396			−1.30606			−0.653030	−	0.757332i	$-0.726503\pi$
$787$	−36.6396			−1.30606			−0.653030	+	0.757332i	$0.726503\pi$
$788$		0			0
$789$	−15.2442			−0.542708
$790$		0			0
$791$	−19.0454			−0.677177
$792$		0			0
$793$			14.0397i			0.498563i
$794$		0			0
$795$			24.3676i			0.864230i
$796$		0			0
$797$		−	38.0109i		−	1.34642i	−0.739453	−	0.673208i	$-0.764916\pi$
$797$		−	38.0109i		−	1.34642i	0.739453	−	0.673208i	$-0.235084\pi$
$798$		0			0
$799$	−32.9106	−	3.55361i	−1.16429	−	0.125718i
$800$		0			0
$801$	9.77068			0.345230
$802$		0			0
$803$			3.64370i			0.128584i
$804$		0			0
$805$	60.2531			2.12364
$806$		0			0
$807$	6.61819			0.232971
$808$		0			0
$809$		−	1.43730i		−	0.0505328i	−0.999681	−	0.0252664i	$-0.991957\pi$
$809$		−	1.43730i		−	0.0505328i	0.999681	−	0.0252664i	$-0.00804340\pi$
$810$		0			0
$811$	−32.8607			−1.15390			−0.576948	−	0.816781i	$-0.695756\pi$
$811$	−32.8607			−1.15390			−0.576948	+	0.816781i	$0.695756\pi$
$812$		0			0
$813$	−11.5920			−0.406550
$814$		0			0
$815$	−1.09783			−0.0384551
$816$		0			0
$817$	40.2667			1.40875
$818$		0			0
$819$	9.09761			0.317896
$820$		0			0
$821$	20.4356			0.713207			0.356604	−	0.934256i	$-0.383935\pi$
$821$	20.4356			0.713207			0.356604	+	0.934256i	$0.383935\pi$
$822$		0			0
$823$			35.9374i			1.25270i	0.779543	+	0.626349i	$0.215451\pi$
$823$			35.9374i			1.25270i	−0.779543	+	0.626349i	$0.784549\pi$
$824$		0			0
$825$	−27.3968			−0.953833
$826$		0			0
$827$	−54.9541			−1.91094			−0.955470	−	0.295089i	$-0.904651\pi$
$827$	−54.9541			−1.91094			−0.955470	+	0.295089i	$0.904651\pi$
$828$		0			0
$829$			15.4841i			0.537785i	0.963170	+	0.268893i	$0.0866577\pi$
$829$			15.4841i			0.537785i	−0.963170	+	0.268893i	$0.913342\pi$
$830$		0			0
$831$	−16.6679			−0.578203
$832$		0			0
$833$	−2.75813	−	0.297817i	−0.0955636	−	0.0103188i
$834$		0			0
$835$		−	65.9976i		−	2.28394i
$836$		0			0
$837$			1.30617i			0.0451479i
$838$		0			0
$839$			23.2368i			0.802224i	0.916029	+	0.401112i	$0.131376\pi$
$839$			23.2368i			0.802224i	−0.916029	+	0.401112i	$0.868624\pi$
$840$		0			0
$841$	−17.9619			−0.619376
$842$		0			0
$843$	6.51175			0.224276
$844$		0			0
$845$	0.257470			0.00885723
$846$		0			0
$847$			45.6962i			1.57014i
$848$		0			0
$849$	−21.2924			−0.730752
$850$		0			0
$851$		−	50.2254i		−	1.72171i
$852$		0			0
$853$	−34.7441			−1.18962			−0.594808	−	0.803868i	$-0.702772\pi$
$853$	−34.7441			−1.18962			−0.594808	+	0.803868i	$0.702772\pi$
$854$		0			0
$855$			21.8143i			0.746034i
$856$		0			0
$857$		−	40.0894i		−	1.36943i	−0.728811	−	0.684715i	$-0.759927\pi$
$857$		−	40.0894i		−	1.36943i	0.728811	−	0.684715i	$-0.240073\pi$
$858$		0			0
$859$			18.2273i			0.621906i	0.950425	+	0.310953i	$0.100648\pi$
$859$			18.2273i			0.621906i	−0.950425	+	0.310953i	$0.899352\pi$
$860$		0			0
$861$	12.6827			0.432226
$862$		0			0
$863$	−36.4069			−1.23930			−0.619652	−	0.784877i	$-0.712726\pi$
$863$	−36.4069			−1.23930			−0.619652	+	0.784877i	$0.712726\pi$
$864$		0			0
$865$	−26.2972			−0.894131
$866$		0			0
$867$	−16.6082	−	3.62894i	−0.564043	−	0.123245i
$868$		0			0
$869$			7.75401i			0.263037i
$870$		0			0
$871$	39.3664			1.33388
$872$		0			0
$873$			0.954796i			0.0323150i
$874$		0			0
$875$		−	0.581938i		−	0.0196731i
$876$		0			0
$877$	−34.0214			−1.14882			−0.574410	−	0.818568i	$-0.694768\pi$
$877$	−34.0214			−1.14882			−0.574410	+	0.818568i	$0.694768\pi$
$878$		0			0
$879$		−	6.20078i		−	0.209147i
$880$		0			0
$881$			56.8188i			1.91427i	0.289637	+	0.957137i	$0.406466\pi$
$881$			56.8188i			1.91427i	−0.289637	+	0.957137i	$0.593534\pi$
$882$		0			0
$883$		−	25.0090i		−	0.841619i	−0.907149	−	0.420810i	$-0.861746\pi$
$883$		−	25.0090i		−	0.841619i	0.907149	−	0.420810i	$-0.138254\pi$
$884$		0			0
$885$			13.4346i			0.451600i
$886$		0			0
$887$			20.9573i			0.703679i	0.936060	+	0.351839i	$0.114444\pi$
$887$			20.9573i			0.703679i	−0.936060	+	0.351839i	$0.885556\pi$
$888$		0			0
$889$			22.1482i			0.742825i
$890$		0			0
$891$	5.40062			0.180928
$892$		0			0
$893$		−	55.1814i		−	1.84657i
$894$		0			0
$895$		−	54.6211i		−	1.82578i
$896$		0			0
$897$	−27.2973			−0.911431
$898$		0			0
$899$		−	4.33958i		−	0.144733i
$900$		0			0
$901$	3.39843	−	31.4734i	0.113218	−	1.04853i
$902$		0			0
$903$	14.7362			0.490391
$904$		0			0
$905$	−5.59475			−0.185976
$906$		0			0
$907$	−27.8495			−0.924727			−0.462363	−	0.886691i	$-0.652998\pi$
$907$	−27.8495			−0.924727			−0.462363	+	0.886691i	$0.652998\pi$
$908$		0			0
$909$			11.7795i			0.390700i
$910$		0			0
$911$			21.9920i			0.728626i	0.931276	+	0.364313i	$0.118696\pi$
$911$			21.9920i			0.728626i	−0.931276	+	0.364313i	$0.881304\pi$
$912$		0			0
$913$			87.2166i			2.88645i
$914$		0			0
$915$	−12.3200			−0.407287
$916$		0			0
$917$		−	52.8452i		−	1.74510i
$918$		0			0
$919$	−11.9263			−0.393412			−0.196706	−	0.980462i	$-0.563025\pi$
$919$	−11.9263			−0.393412			−0.196706	+	0.980462i	$0.563025\pi$
$920$		0			0
$921$			3.96275i			0.130577i
$922$		0			0
$923$	−15.3023			−0.503680
$924$		0			0
$925$	−33.7584			−1.10997
$926$		0			0
$927$	3.63788			0.119484
$928$		0			0
$929$		−	33.2071i		−	1.08949i	−0.838602	−	0.544744i	$-0.816627\pi$
$929$		−	33.2071i		−	1.08949i	0.838602	−	0.544744i	$-0.183373\pi$
$930$		0			0
$931$		−	4.62458i		−	0.151565i
$932$		0			0
$933$			6.26994i			0.205269i
$934$		0			0
$935$	70.2632	+	7.58687i	2.29785	+	0.248117i
$936$		0			0
$937$	30.7131			1.00335			0.501677	−	0.865055i	$-0.332717\pi$
$937$	30.7131			1.00335			0.501677	+	0.865055i	$0.332717\pi$
$938$		0			0
$939$			10.7509i			0.350842i
$940$		0			0
$941$	11.5870			0.377724			0.188862	−	0.982004i	$-0.439520\pi$
$941$	11.5870			0.377724			0.188862	+	0.982004i	$0.439520\pi$
$942$		0			0
$943$	−38.0544			−1.23922
$944$		0			0
$945$			7.98329i			0.259696i
$946$		0			0
$947$	27.4089			0.890669			0.445334	−	0.895364i	$-0.353085\pi$
$947$	27.4089			0.890669			0.445334	+	0.895364i	$0.353085\pi$
$948$		0			0
$949$	2.44018			0.0792116
$950$		0			0
$951$	17.0890			0.554150
$952$		0			0
$953$	−33.5870			−1.08799			−0.543994	−	0.839089i	$-0.683089\pi$
$953$	−33.5870			−1.08799			−0.543994	+	0.839089i	$0.683089\pi$
$954$		0			0
$955$	15.0382			0.486624
$956$		0			0
$957$	−17.9428			−0.580009
$958$		0			0
$959$		−	40.1286i		−	1.29582i
$960$		0			0
$961$	29.2939			0.944965
$962$		0			0
$963$	2.78476			0.0897375
$964$		0			0
$965$		−	66.4884i		−	2.14034i
$966$		0			0
$967$	−37.3601			−1.20142			−0.600710	−	0.799467i	$-0.705115\pi$
$967$	−37.3601			−1.20142			−0.600710	+	0.799467i	$0.705115\pi$
$968$		0			0
$969$	3.04233	−	28.1755i	0.0977338	−	0.905128i
$970$		0			0
$971$			7.46186i			0.239463i	0.992806	+	0.119731i	$0.0382033\pi$
$971$			7.46186i			0.239463i	−0.992806	+	0.119731i	$0.961797\pi$
$972$		0			0
$973$		−	16.6349i		−	0.533290i
$974$		0			0
$975$			18.3476i			0.587592i
$976$		0			0
$977$	−31.4037			−1.00469			−0.502346	−	0.864666i	$-0.667530\pi$
$977$	−31.4037			−1.00469			−0.502346	+	0.864666i	$0.667530\pi$
$978$		0			0
$979$	52.7677			1.68646
$980$		0			0
$981$	−14.7517			−0.470986
$982$		0			0
$983$			2.30014i			0.0733630i	0.999327	+	0.0366815i	$0.0116787\pi$
$983$			2.30014i			0.0733630i	−0.999327	+	0.0366815i	$0.988321\pi$
$984$		0			0
$985$	−26.5065			−0.844567
$986$		0			0
$987$		−	20.1945i		−	0.642797i
$988$		0			0
$989$	−44.2160			−1.40599
$990$		0			0
$991$		−	18.2755i		−	0.580539i	−0.956945	−	0.290269i	$-0.906255\pi$
$991$		−	18.2755i		−	0.580539i	0.956945	−	0.290269i	$-0.0937450\pi$
$992$		0			0
$993$			7.05539i			0.223896i
$994$		0			0
$995$		−	39.1359i		−	1.24069i
$996$		0			0
$997$	6.78517			0.214888			0.107444	−	0.994211i	$-0.465733\pi$
$997$	6.78517			0.214888			0.107444	+	0.994211i	$0.465733\pi$
$998$		0			0
$999$	6.65467			0.210544

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1632.2.l.a.1393.4		18
3.2	odd	2		4896.2.l.d.3025.16		18
4.3	odd	2		408.2.l.b.373.10	yes	18
8.3	odd	2		408.2.l.a.373.9	✓	18
8.5	even	2		1632.2.l.b.1393.16		18
12.11	even	2		1224.2.l.d.1189.9		18
17.16	even	2		1632.2.l.b.1393.15		18
24.5	odd	2		4896.2.l.c.3025.4		18
24.11	even	2		1224.2.l.c.1189.10		18
51.50	odd	2		4896.2.l.c.3025.3		18
68.67	odd	2		408.2.l.a.373.10	yes	18
136.67	odd	2		408.2.l.b.373.9	yes	18
136.101	even	2	inner	1632.2.l.a.1393.3		18
204.203	even	2		1224.2.l.c.1189.9		18
408.101	odd	2		4896.2.l.d.3025.15		18
408.203	even	2		1224.2.l.d.1189.10		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
408.2.l.a.373.9	✓	18	8.3	odd	2
408.2.l.a.373.10	yes	18	68.67	odd	2
408.2.l.b.373.9	yes	18	136.67	odd	2
408.2.l.b.373.10	yes	18	4.3	odd	2
1224.2.l.c.1189.9		18	204.203	even	2
1224.2.l.c.1189.10		18	24.11	even	2
1224.2.l.d.1189.9		18	12.11	even	2
1224.2.l.d.1189.10		18	408.203	even	2
1632.2.l.a.1393.3		18	136.101	even	2	inner
1632.2.l.a.1393.4		18	1.1	even	1	trivial
1632.2.l.b.1393.15		18	17.16	even	2
1632.2.l.b.1393.16		18	8.5	even	2
4896.2.l.c.3025.3		18	51.50	odd	2
4896.2.l.c.3025.4		18	24.5	odd	2
4896.2.l.d.3025.15		18	408.101	odd	2
4896.2.l.d.3025.16		18	3.2	odd	2

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:	\( N \)	\(=\)	\( 1632 = 2^{5} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1632.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.17378 q^{5} +2.51539i q^{7} +1.00000 q^{9} +5.40062 q^{11} -3.61678i q^{13} +3.17378 q^{15} +(-4.09928 - 0.442631i) q^{17} -6.87329i q^{19} -2.51539i q^{21} +7.54740i q^{23} +5.07289 q^{25} -1.00000 q^{27} +3.32236 q^{29} -1.30617i q^{31} -5.40062 q^{33} -7.98329i q^{35} -6.65467 q^{37} +3.61678i q^{39} +5.04206i q^{41} +5.85844i q^{43} -3.17378 q^{45} +8.02838 q^{47} +0.672834 q^{49} +(4.09928 + 0.442631i) q^{51} +7.67778i q^{53} -17.1404 q^{55} +6.87329i q^{57} +4.23300i q^{59} -3.88181 q^{61} +2.51539i q^{63} +11.4789i q^{65} +10.8844i q^{67} -7.54740i q^{69} -4.23091i q^{71} +0.674683i q^{73} -5.07289 q^{75} +13.5846i q^{77} +1.43576i q^{79} +1.00000 q^{81} +16.1494i q^{83} +(13.0102 + 1.40482i) q^{85} -3.32236 q^{87} +9.77068 q^{89} +9.09761 q^{91} +1.30617i q^{93} +21.8143i q^{95} +0.954796i q^{97} +5.40062 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 18 q^{3} - 4 q^{5} + 18 q^{9} + 4 q^{15} - 2 q^{17} + 22 q^{25} - 18 q^{27} + 12 q^{29} - 16 q^{37} - 4 q^{45} - 18 q^{49} + 2 q^{51} + 16 q^{55} - 16 q^{61} - 22 q^{75} + 18 q^{81} + 16 q^{85} - 12 q^{87}+ \cdots + 20 q^{89}+O(q^{100}) \) 18 * q - 18 * q^3 - 4 * q^5 + 18 * q^9 + 4 * q^15 - 2 * q^17 + 22 * q^25 - 18 * q^27 + 12 * q^29 - 16 * q^37 - 4 * q^45 - 18 * q^49 + 2 * q^51 + 16 * q^55 - 16 * q^61 - 22 * q^75 + 18 * q^81 + 16 * q^85 - 12 * q^87 + 20 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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