LMFDB - Embedded newform 1960.2.g.e.1569.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1960,2,Mod(1569,1960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1960.1569");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1960 = 2^{3} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1960.g (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:	$15.6506787962$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 25x^{10} + 230x^{8} + 950x^{6} + 1657x^{4} + 785x^{2} + 64 $ x^12 + 25x^10 + 230x^8 + 950x^6 + 1657x^4 + 785*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1569.2
Root			$-2.54431i$ of defining polynomial
Character	$\chi$	$=$	1960.1569
Dual form			1960.2.g.e.1569.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.54431i q^{3} +(1.65511 + 1.50353i) q^{5} -3.47349 q^{9} +O(q^{10})$	$q-2.54431i q^{3} +(1.65511 + 1.50353i) q^{5} -3.47349 q^{9} -3.91228 q^{11} +3.47349i q^{13} +(3.82544 - 4.21111i) q^{15} +5.08262i q^{17} -5.25188 q^{19} +6.01780i q^{23} +(0.478784 + 4.97702i) q^{25} +1.20471i q^{27} +10.2660 q^{29} -2.04350 q^{31} +9.95405i q^{33} +0.405001i q^{37} +8.83763 q^{39} +2.57534 q^{41} +6.15769i q^{43} +(-5.74901 - 5.22251i) q^{45} +0.480556i q^{47} +12.9317 q^{51} -0.466428i q^{53} +(-6.47526 - 5.88224i) q^{55} +13.3624i q^{57} -10.3113 q^{59} -8.08088 q^{61} +(-5.22251 + 5.74901i) q^{65} +4.15169i q^{67} +15.3111 q^{69} -4.28658 q^{71} +4.81483i q^{73} +(12.6631 - 1.21817i) q^{75} +12.6639 q^{79} -7.35533 q^{81} +8.42280i q^{83} +(-7.64188 + 8.41230i) q^{85} -26.1199i q^{87} +2.38595 q^{89} +5.19930i q^{93} +(-8.69245 - 7.89637i) q^{95} -1.32080i q^{97} +13.5893 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 14 q^{9}+O(q^{10}) $ 12 * q - 14 * q^9	$ 12 q - 14 q^{9} + 2 q^{11} + 6 q^{15} - 10 q^{19} + 2 q^{25} + 6 q^{29} + 4 q^{31} - 20 q^{39} - 12 q^{41} - 8 q^{45} + 6 q^{55} - 48 q^{59} - 18 q^{61} + 26 q^{65} + 30 q^{69} + 8 q^{71} - 14 q^{75} + 44 q^{79} - 12 q^{81} - 22 q^{85} + 30 q^{89} + 26 q^{95} - 22 q^{99}+O(q^{100}) $ 12 * q - 14 * q^9 + 2 * q^11 + 6 * q^15 - 10 * q^19 + 2 * q^25 + 6 * q^29 + 4 * q^31 - 20 * q^39 - 12 * q^41 - 8 * q^45 + 6 * q^55 - 48 * q^59 - 18 * q^61 + 26 * q^65 + 30 * q^69 + 8 * q^71 - 14 * q^75 + 44 * q^79 - 12 * q^81 - 22 * q^85 + 30 * q^89 + 26 * q^95 - 22 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1960\mathbb{Z}\right)^\times$.

$n$	$981$	$1081$	$1177$	$1471$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$		−	2.54431i		−	1.46896i	−0.678633	−	0.734478i	$-0.737427\pi$
$3$		−	2.54431i		−	1.46896i	0.678633	−	0.734478i	$-0.262573\pi$
$4$		0			0
$5$	1.65511	+	1.50353i	0.740188	+	0.672400i
$5$	1.65511	+	1.50353i	0.740188	+	0.672400i
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−3.47349			−1.15783
$10$		0			0
$11$	−3.91228			−1.17960			−0.589799	−	0.807550i	$-0.700793\pi$
$11$	−3.91228			−1.17960			−0.589799	+	0.807550i	$0.700793\pi$
$12$		0			0
$13$			3.47349i			0.963373i	0.876344	+	0.481687i	$0.159976\pi$
$13$			3.47349i			0.963373i	−0.876344	+	0.481687i	$0.840024\pi$
$14$		0			0
$15$	3.82544	−	4.21111i	0.987726	−	1.08730i
$16$		0			0
$17$			5.08262i			1.23272i	0.787466	+	0.616358i	$0.211393\pi$
$17$			5.08262i			1.23272i	−0.787466	+	0.616358i	$0.788607\pi$
$18$		0			0
$19$	−5.25188			−1.20486			−0.602432	−	0.798170i	$-0.705802\pi$
$19$	−5.25188			−1.20486			−0.602432	+	0.798170i	$0.705802\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			6.01780i			1.25480i	0.778698	+	0.627399i	$0.215880\pi$
$23$			6.01780i			1.25480i	−0.778698	+	0.627399i	$0.784120\pi$
$24$		0			0
$25$	0.478784	+	4.97702i	0.0957567	+	0.995405i
$26$		0			0
$27$			1.20471i			0.231846i
$28$		0			0
$29$	10.2660			1.90635			0.953175	−	0.302419i	$-0.0977942\pi$
$29$	10.2660			1.90635			0.953175	+	0.302419i	$0.0977942\pi$
$30$		0			0
$31$	−2.04350			−0.367024			−0.183512	−	0.983017i	$-0.558747\pi$
$31$	−2.04350			−0.367024			−0.183512	+	0.983017i	$0.558747\pi$
$32$		0			0
$33$			9.95405i			1.73278i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			0.405001i			0.0665818i	0.999446	+	0.0332909i	$0.0105988\pi$
$37$			0.405001i			0.0665818i	−0.999446	+	0.0332909i	$0.989401\pi$
$38$		0			0
$39$	8.83763			1.41515
$40$		0			0
$41$	2.57534			0.402200			0.201100	−	0.979571i	$-0.435548\pi$
$41$	2.57534			0.402200			0.201100	+	0.979571i	$0.435548\pi$
$42$		0			0
$43$			6.15769i			0.939038i	0.882922	+	0.469519i	$0.155573\pi$
$43$			6.15769i			0.939038i	−0.882922	+	0.469519i	$0.844427\pi$
$44$		0			0
$45$	−5.74901	−	5.22251i	−0.857012	−	0.778525i
$46$		0			0
$47$			0.480556i			0.0700963i	0.999386	+	0.0350481i	$0.0111585\pi$
$47$			0.480556i			0.0700963i	−0.999386	+	0.0350481i	$0.988842\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	12.9317			1.81081
$52$		0			0
$53$		−	0.466428i		−	0.0640688i	−0.999487	−	0.0320344i	$-0.989801\pi$
$53$		−	0.466428i		−	0.0640688i	0.999487	−	0.0320344i	$-0.0101986\pi$
$54$		0			0
$55$	−6.47526	−	5.88224i	−0.873124	−	0.793162i
$56$		0			0
$57$			13.3624i			1.76989i
$58$		0			0
$59$	−10.3113			−1.34242			−0.671208	−	0.741269i	$-0.734224\pi$
$59$	−10.3113			−1.34242			−0.671208	+	0.741269i	$0.734224\pi$
$60$		0			0
$61$	−8.08088			−1.03465			−0.517325	−	0.855789i	$-0.673072\pi$
$61$	−8.08088			−1.03465			−0.517325	+	0.855789i	$0.673072\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−5.22251	+	5.74901i	−0.647772	+	0.713077i
$66$		0			0
$67$			4.15169i			0.507210i	0.967308	+	0.253605i	$0.0816163\pi$
$67$			4.15169i			0.507210i	−0.967308	+	0.253605i	$0.918384\pi$
$68$		0			0
$69$	15.3111			1.84324
$70$		0			0
$71$	−4.28658			−0.508724			−0.254362	−	0.967109i	$-0.581865\pi$
$71$	−4.28658			−0.508724			−0.254362	+	0.967109i	$0.581865\pi$
$72$		0			0
$73$			4.81483i			0.563533i	0.959483	+	0.281767i	$0.0909204\pi$
$73$			4.81483i			0.563533i	−0.959483	+	0.281767i	$0.909080\pi$
$74$		0			0
$75$	12.6631	−	1.21817i	1.46221	−	0.140662i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	12.6639			1.42481			0.712403	−	0.701771i	$-0.247607\pi$
$79$	12.6639			1.42481			0.712403	+	0.701771i	$0.247607\pi$
$80$		0			0
$81$	−7.35533			−0.817259
$82$		0			0
$83$			8.42280i			0.924522i	0.886744	+	0.462261i	$0.152962\pi$
$83$			8.42280i			0.924522i	−0.886744	+	0.462261i	$0.847038\pi$
$84$		0			0
$85$	−7.64188	+	8.41230i	−0.828878	+	0.912442i
$86$		0			0
$87$		−	26.1199i		−	2.80034i
$88$		0			0
$89$	2.38595			0.252910			0.126455	−	0.991972i	$-0.459640\pi$
$89$	2.38595			0.252910			0.126455	+	0.991972i	$0.459640\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$			5.19930i			0.539142i
$94$		0			0
$95$	−8.69245	−	7.89637i	−0.891826	−	0.810151i
$96$		0			0
$97$		−	1.32080i		−	0.134107i	−0.997749	−	0.0670537i	$-0.978640\pi$
$97$		−	1.32080i		−	0.134107i	0.997749	−	0.0670537i	$-0.0213599\pi$
$98$		0			0
$99$	13.5893			1.36577
$100$		0			0
$101$	−7.75166			−0.771319			−0.385660	−	0.922641i	$-0.626026\pi$
$101$	−7.75166			−0.771319			−0.385660	+	0.922641i	$0.626026\pi$
$102$		0			0
$103$			2.20364i			0.217131i	0.994089	+	0.108565i	$0.0346257\pi$
$103$			2.20364i			0.217131i	−0.994089	+	0.108565i	$0.965374\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	10.8235i		−	1.04635i	−0.852225	−	0.523176i	$-0.824747\pi$
$107$		−	10.8235i		−	1.04635i	0.852225	−	0.523176i	$-0.175253\pi$
$108$		0			0
$109$	11.0433			1.05776			0.528880	−	0.848697i	$-0.322612\pi$
$109$	11.0433			1.05776			0.528880	+	0.848697i	$0.322612\pi$
$110$		0			0
$111$	1.03045			0.0978057
$112$		0			0
$113$		−	7.29167i		−	0.685942i	−0.939346	−	0.342971i	$-0.888567\pi$
$113$		−	7.29167i		−	0.685942i	0.939346	−	0.342971i	$-0.111433\pi$
$114$		0			0
$115$	−9.04795	+	9.96012i	−0.843726	+	0.928786i
$116$		0			0
$117$		−	12.0651i		−	1.11542i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	4.30597			0.391452
$122$		0			0
$123$		−	6.55244i		−	0.590814i
$124$		0			0
$125$	−6.69067	+	8.95739i	−0.598432	+	0.801174i
$126$		0			0
$127$			7.66305i			0.679986i	0.940428	+	0.339993i	$0.110425\pi$
$127$			7.66305i			0.679986i	−0.940428	+	0.339993i	$0.889575\pi$
$128$		0			0
$129$	15.6670			1.37941
$130$		0			0
$131$	17.2119			1.50381			0.751906	−	0.659270i	$-0.229135\pi$
$131$	17.2119			1.50381			0.751906	+	0.659270i	$0.229135\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−1.81132	+	1.99393i	−0.155893	+	0.171610i
$136$		0			0
$137$		−	5.26072i		−	0.449454i	−0.974422	−	0.224727i	$-0.927851\pi$
$137$		−	5.26072i		−	0.449454i	0.974422	−	0.224727i	$-0.0721490\pi$
$138$		0			0
$139$	−15.2148			−1.29050			−0.645250	−	0.763971i	$-0.723247\pi$
$139$	−15.2148			−1.29050			−0.645250	+	0.763971i	$0.723247\pi$
$140$		0			0
$141$	1.22268			0.102968
$142$		0			0
$143$		−	13.5893i		−	1.13639i
$144$		0			0
$145$	16.9914	+	15.4353i	1.41106	+	1.28183i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.01721			0.0833335			0.0416667	−	0.999132i	$-0.486733\pi$
$149$	1.01721			0.0833335			0.0416667	+	0.999132i	$0.486733\pi$
$150$		0			0
$151$	13.2583			1.07894			0.539472	−	0.842004i	$-0.318624\pi$
$151$	13.2583			1.07894			0.539472	+	0.842004i	$0.318624\pi$
$152$		0			0
$153$		−	17.6544i		−	1.42728i
$154$		0			0
$155$	−3.38222	−	3.07247i	−0.271667	−	0.246787i
$156$		0			0
$157$		−	17.7722i		−	1.41838i	−0.705019	−	0.709189i	$-0.749061\pi$
$157$		−	17.7722i		−	1.41838i	0.705019	−	0.709189i	$-0.250939\pi$
$158$		0			0
$159$	−1.18674			−0.0941143
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	2.91952i		−	0.228675i	−0.993442	−	0.114337i	$-0.963526\pi$
$163$		−	2.91952i		−	0.228675i	0.993442	−	0.114337i	$-0.0364745\pi$
$164$		0			0
$165$	−14.9662	+	16.4751i	−1.16512	+	1.28258i
$166$		0			0
$167$			9.21710i			0.713240i	0.934249	+	0.356620i	$0.116071\pi$
$167$			9.21710i			0.713240i	−0.934249	+	0.356620i	$0.883929\pi$
$168$		0			0
$169$	0.934854			0.0719118
$170$		0			0
$171$	18.2424			1.39503
$172$		0			0
$173$		−	21.6803i		−	1.64832i	−0.566354	−	0.824162i	$-0.691647\pi$
$173$		−	21.6803i		−	1.64832i	0.566354	−	0.824162i	$-0.308353\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			26.2351i			1.97195i
$178$		0			0
$179$	24.9965			1.86833			0.934164	−	0.356843i	$-0.116147\pi$
$179$	24.9965			1.86833			0.934164	+	0.356843i	$0.116147\pi$
$180$		0			0
$181$	4.17900			0.310623			0.155311	−	0.987866i	$-0.450362\pi$
$181$	4.17900			0.310623			0.155311	+	0.987866i	$0.450362\pi$
$182$		0			0
$183$			20.5602i			1.51986i
$184$		0			0
$185$	−0.608932	+	0.670322i	−0.0447696	+	0.0492830i
$186$		0			0
$187$		−	19.8846i		−	1.45411i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−8.97169			−0.649169			−0.324585	−	0.945857i	$-0.605224\pi$
$191$	−8.97169			−0.649169			−0.324585	+	0.945857i	$0.605224\pi$
$192$		0			0
$193$			18.2218i			1.31164i	0.754919	+	0.655818i	$0.227676\pi$
$193$			18.2218i			1.31164i	−0.754919	+	0.655818i	$0.772324\pi$
$194$		0			0
$195$	14.6272	+	13.2877i	1.04748	+	0.951549i
$196$		0			0
$197$			23.5632i			1.67881i	0.543509	+	0.839403i	$0.317095\pi$
$197$			23.5632i			1.67881i	−0.543509	+	0.839403i	$0.682905\pi$
$198$		0			0
$199$	20.7163			1.46854			0.734271	−	0.678856i	$-0.237524\pi$
$199$	20.7163			1.46854			0.734271	+	0.678856i	$0.237524\pi$
$200$		0			0
$201$	10.5632			0.745069
$202$		0			0
$203$		0			0
$204$		0			0
$205$	4.26247	+	3.87210i	0.297703	+	0.270439i
$206$		0			0
$207$		−	20.9028i		−	1.45284i
$208$		0			0
$209$	20.5469			1.42126
$210$		0			0
$211$	7.69865			0.529997			0.264998	−	0.964249i	$-0.414629\pi$
$211$	7.69865			0.529997			0.264998	+	0.964249i	$0.414629\pi$
$212$		0			0
$213$			10.9064i			0.747292i
$214$		0			0
$215$	−9.25828	+	10.1917i	−0.631409	+	0.695065i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	12.2504			0.827805
$220$		0			0
$221$	−17.6544			−1.18757
$222$		0			0
$223$			12.7136i			0.851366i	0.904872	+	0.425683i	$0.139966\pi$
$223$			12.7136i			0.851366i	−0.904872	+	0.425683i	$0.860034\pi$
$224$		0			0
$225$	−1.66305	−	17.2877i	−0.110870	−	1.15251i
$226$		0			0
$227$			10.7229i			0.711706i	0.934542	+	0.355853i	$0.115810\pi$
$227$			10.7229i			0.711706i	−0.934542	+	0.355853i	$0.884190\pi$
$228$		0			0
$229$	−24.7892			−1.63812			−0.819059	−	0.573709i	$-0.805504\pi$
$229$	−24.7892			−1.63812			−0.819059	+	0.573709i	$0.805504\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	22.7544i		−	1.49069i	−0.666679	−	0.745345i	$-0.732285\pi$
$233$		−	22.7544i		−	1.49069i	0.666679	−	0.745345i	$-0.267715\pi$
$234$		0			0
$235$	−0.722531	+	0.795373i	−0.0471327	+	0.0518844i
$236$		0			0
$237$		−	32.2210i		−	2.09298i
$238$		0			0
$239$	1.49071			0.0964258			0.0482129	−	0.998837i	$-0.484647\pi$
$239$	1.49071			0.0964258			0.0482129	+	0.998837i	$0.484647\pi$
$240$		0			0
$241$	−26.1677			−1.68561			−0.842806	−	0.538217i	$-0.819098\pi$
$241$	−26.1677			−1.68561			−0.842806	+	0.538217i	$0.819098\pi$
$242$		0			0
$243$			22.3283i			1.43236i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	18.2424i		−	1.16073i
$248$		0			0
$249$	21.4302			1.35808
$250$		0			0
$251$	−4.90277			−0.309460			−0.154730	−	0.987957i	$-0.549451\pi$
$251$	−4.90277			−0.309460			−0.154730	+	0.987957i	$0.549451\pi$
$252$		0			0
$253$		−	23.5433i		−	1.48016i
$254$		0			0
$255$	21.4035	+	19.4433i	1.34034	+	1.21759i
$256$		0			0
$257$		−	12.3399i		−	0.769739i	−0.922971	−	0.384870i	$-0.874246\pi$
$257$		−	12.3399i		−	0.769739i	0.922971	−	0.384870i	$-0.125754\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−35.6589			−2.20723
$262$		0			0
$263$			20.1247i			1.24094i	0.784229	+	0.620471i	$0.213059\pi$
$263$			20.1247i			1.24094i	−0.784229	+	0.620471i	$0.786941\pi$
$264$		0			0
$265$	0.701290	−	0.771990i	0.0430799	−	0.0474230i
$266$		0			0
$267$		−	6.07059i		−	0.371514i
$268$		0			0
$269$	−14.9394			−0.910874			−0.455437	−	0.890268i	$-0.650517\pi$
$269$	−14.9394			−0.910874			−0.455437	+	0.890268i	$0.650517\pi$
$270$		0			0
$271$	5.80932			0.352891			0.176446	−	0.984310i	$-0.443540\pi$
$271$	5.80932			0.352891			0.176446	+	0.984310i	$0.443540\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.87314	−	19.4715i	−0.112954	−	1.17418i
$276$		0			0
$277$		−	25.9390i		−	1.55853i	−0.626697	−	0.779263i	$-0.715594\pi$
$277$		−	25.9390i		−	1.55853i	0.626697	−	0.779263i	$-0.284406\pi$
$278$		0			0
$279$	7.09809			0.424952
$280$		0			0
$281$	−14.8840			−0.887904			−0.443952	−	0.896051i	$-0.646424\pi$
$281$	−14.8840			−0.887904			−0.443952	+	0.896051i	$0.646424\pi$
$282$		0			0
$283$			9.71540i			0.577520i	0.957401	+	0.288760i	$0.0932430\pi$
$283$			9.71540i			0.577520i	−0.957401	+	0.288760i	$0.906757\pi$
$284$		0			0
$285$	−20.0908	+	22.1162i	−1.19008	+	1.31005i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−8.83301			−0.519589
$290$		0			0
$291$	−3.36053			−0.196998
$292$		0			0
$293$		−	8.08596i		−	0.472387i	−0.971706	−	0.236194i	$-0.924100\pi$
$293$		−	8.08596i		−	0.472387i	0.971706	−	0.236194i	$-0.0758999\pi$
$294$		0			0
$295$	−17.0663	−	15.5034i	−0.993640	−	0.902640i
$296$		0			0
$297$		−	4.71316i		−	0.273485i
$298$		0			0
$299$	−20.9028			−1.20884
$300$		0			0
$301$		0			0
$302$		0			0
$303$			19.7226i			1.13303i
$304$		0			0
$305$	−13.3747	−	12.1499i	−0.765836	−	0.695699i
$306$		0			0
$307$		−	8.81809i		−	0.503275i	−0.967822	−	0.251637i	$-0.919031\pi$
$307$		−	8.81809i		−	0.503275i	0.967822	−	0.251637i	$-0.0809690\pi$
$308$		0			0
$309$	5.60673			0.318956
$310$		0			0
$311$	−9.66439			−0.548017			−0.274009	−	0.961727i	$-0.588350\pi$
$311$	−9.66439			−0.548017			−0.274009	+	0.961727i	$0.588350\pi$
$312$		0			0
$313$			27.1864i			1.53667i	0.640049	+	0.768334i	$0.278914\pi$
$313$			27.1864i			1.53667i	−0.640049	+	0.768334i	$0.721086\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	12.1583i		−	0.682876i	−0.939904	−	0.341438i	$-0.889086\pi$
$317$		−	12.1583i		−	0.682876i	0.939904	−	0.341438i	$-0.110914\pi$
$318$		0			0
$319$	−40.1635			−2.24873
$320$		0			0
$321$	−27.5384			−1.53704
$322$		0			0
$323$		−	26.6933i		−	1.48526i
$324$		0			0
$325$	−17.2877	+	1.66305i	−0.958946	+	0.0922495i
$326$		0			0
$327$		−	28.0976i		−	1.55380i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−3.42945			−0.188500			−0.0942499	−	0.995549i	$-0.530045\pi$
$331$	−3.42945			−0.188500			−0.0942499	+	0.995549i	$0.530045\pi$
$332$		0			0
$333$		−	1.40677i		−	0.0770904i
$334$		0			0
$335$	−6.24220	+	6.87151i	−0.341048	+	0.375431i
$336$		0			0
$337$			1.39770i			0.0761375i	0.999275	+	0.0380687i	$0.0121206\pi$
$337$			1.39770i			0.0761375i	−0.999275	+	0.0380687i	$0.987879\pi$
$338$		0			0
$339$	−18.5522			−1.00762
$340$		0			0
$341$	7.99477			0.432941
$342$		0			0
$343$		0			0
$344$		0			0
$345$	25.3416	+	23.0208i	1.36435	+	1.23940i
$346$		0			0
$347$		−	3.46302i		−	0.185904i	−0.995671	−	0.0929522i	$-0.970370\pi$
$347$		−	3.46302i		−	0.185904i	0.995671	−	0.0929522i	$-0.0296304\pi$
$348$		0			0
$349$	−19.2360			−1.02968			−0.514838	−	0.857287i	$-0.672148\pi$
$349$	−19.2360			−1.02968			−0.514838	+	0.857287i	$0.672148\pi$
$350$		0			0
$351$	−4.18454			−0.223354
$352$		0			0
$353$			17.2651i			0.918931i	0.888196	+	0.459465i	$0.151959\pi$
$353$			17.2651i			0.918931i	−0.888196	+	0.459465i	$0.848041\pi$
$354$		0			0
$355$	−7.09477	−	6.44501i	−0.376551	−	0.342066i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−0.867633			−0.0457919			−0.0228960	−	0.999738i	$-0.507289\pi$
$359$	−0.867633			−0.0457919			−0.0228960	+	0.999738i	$0.507289\pi$
$360$		0			0
$361$	8.58226			0.451698
$362$		0			0
$363$		−	10.9557i		−	0.575025i
$364$		0			0
$365$	−7.23925	+	7.96908i	−0.378920	+	0.417121i
$366$		0			0
$367$			16.4984i			0.861210i	0.902541	+	0.430605i	$0.141700\pi$
$367$			16.4984i			0.861210i	−0.902541	+	0.430605i	$0.858300\pi$
$368$		0			0
$369$	−8.94541			−0.465679
$370$		0			0
$371$		0			0
$372$		0			0
$373$			20.3871i			1.05560i	0.849368	+	0.527801i	$0.176983\pi$
$373$			20.3871i			1.05560i	−0.849368	+	0.527801i	$0.823017\pi$
$374$		0			0
$375$	22.7903	+	17.0231i	1.17689	+	0.879070i
$376$		0			0
$377$			35.6589i			1.83653i
$378$		0			0
$379$	−13.1652			−0.676251			−0.338126	−	0.941101i	$-0.609793\pi$
$379$	−13.1652			−0.676251			−0.338126	+	0.941101i	$0.609793\pi$
$380$		0			0
$381$	19.4971			0.998869
$382$		0			0
$383$			5.19162i			0.265280i	0.991164	+	0.132640i	$0.0423454\pi$
$383$			5.19162i			0.265280i	−0.991164	+	0.132640i	$0.957655\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	21.3887i		−	1.08725i
$388$		0			0
$389$	−31.1419			−1.57896			−0.789478	−	0.613779i	$-0.789649\pi$
$389$	−31.1419			−1.57896			−0.789478	+	0.613779i	$0.789649\pi$
$390$		0			0
$391$	−30.5862			−1.54681
$392$		0			0
$393$		−	43.7924i		−	2.20903i
$394$		0			0
$395$	20.9602	+	19.0406i	1.05462	+	0.958039i
$396$		0			0
$397$			15.1065i			0.758174i	0.925361	+	0.379087i	$0.123762\pi$
$397$			15.1065i			0.758174i	−0.925361	+	0.379087i	$0.876238\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	16.2471			0.811340			0.405670	−	0.914020i	$-0.367038\pi$
$401$	16.2471			0.811340			0.405670	+	0.914020i	$0.367038\pi$
$402$		0			0
$403$		−	7.09809i		−	0.353581i
$404$		0			0
$405$	−12.1739	−	11.0590i	−0.604925	−	0.549525i
$406$		0			0
$407$		−	1.58448i		−	0.0785397i
$408$		0			0
$409$	33.3206			1.64760			0.823798	−	0.566883i	$-0.191851\pi$
$409$	33.3206			1.64760			0.823798	+	0.566883i	$0.191851\pi$
$410$		0			0
$411$	−13.3849			−0.660228
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−12.6639	+	13.9407i	−0.621649	+	0.684320i
$416$		0			0
$417$			38.7110i			1.89569i
$418$		0			0
$419$	11.3290			0.553457			0.276729	−	0.960948i	$-0.410750\pi$
$419$	11.3290			0.553457			0.276729	+	0.960948i	$0.410750\pi$
$420$		0			0
$421$	12.5047			0.609440			0.304720	−	0.952442i	$-0.401437\pi$
$421$	12.5047			0.609440			0.304720	+	0.952442i	$0.401437\pi$
$422$		0			0
$423$		−	1.66921i		−	0.0811596i
$424$		0			0
$425$	−25.2963	+	2.43347i	−1.22705	+	0.118041i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−34.5753			−1.66931
$430$		0			0
$431$	−5.14713			−0.247928			−0.123964	−	0.992287i	$-0.539561\pi$
$431$	−5.14713			−0.247928			−0.123964	+	0.992287i	$0.539561\pi$
$432$		0			0
$433$			1.88420i			0.0905491i	0.998975	+	0.0452745i	$0.0144163\pi$
$433$			1.88420i			0.0905491i	−0.998975	+	0.0452745i	$0.985584\pi$
$434$		0			0
$435$	39.2721	−	43.2313i	1.88295	−	2.07278i
$436$		0			0
$437$		−	31.6048i		−	1.51186i
$438$		0			0
$439$	10.5904			0.505454			0.252727	−	0.967538i	$-0.418673\pi$
$439$	10.5904			0.505454			0.252727	+	0.967538i	$0.418673\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	24.3510i		−	1.15695i	−0.815700	−	0.578476i	$-0.803648\pi$
$443$		−	24.3510i		−	1.15695i	0.815700	−	0.578476i	$-0.196352\pi$
$444$		0			0
$445$	3.94901	+	3.58735i	0.187201	+	0.170057i
$446$		0			0
$447$		−	2.58811i		−	0.122413i
$448$		0			0
$449$	−20.2506			−0.955684			−0.477842	−	0.878446i	$-0.658581\pi$
$449$	−20.2506			−0.955684			−0.477842	+	0.878446i	$0.658581\pi$
$450$		0			0
$451$	−10.0754			−0.474434
$452$		0			0
$453$		−	33.7331i		−	1.58492i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	19.1864i		−	0.897503i	−0.893657	−	0.448752i	$-0.851869\pi$
$457$		−	19.1864i		−	0.897503i	0.893657	−	0.448752i	$-0.148131\pi$
$458$		0			0
$459$	−6.12307			−0.285801
$460$		0			0
$461$	13.4096			0.624548			0.312274	−	0.949992i	$-0.398909\pi$
$461$	13.4096			0.624548			0.312274	+	0.949992i	$0.398909\pi$
$462$		0			0
$463$			17.7864i			0.826604i	0.910594	+	0.413302i	$0.135625\pi$
$463$			17.7864i			0.826604i	−0.910594	+	0.413302i	$0.864375\pi$
$464$		0			0
$465$	−7.81731	+	8.60541i	−0.362519	+	0.399067i
$466$		0			0
$467$		−	15.2100i		−	0.703835i	−0.936031	−	0.351918i	$-0.885530\pi$
$467$		−	15.2100i		−	0.703835i	0.936031	−	0.351918i	$-0.114470\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−45.2180			−2.08353
$472$		0			0
$473$		−	24.0906i		−	1.10769i
$474$		0			0
$475$	−2.51451	−	26.1387i	−0.115374	−	1.19933i
$476$		0			0
$477$			1.62013i			0.0741809i
$478$		0			0
$479$	−12.5684			−0.574266			−0.287133	−	0.957891i	$-0.592702\pi$
$479$	−12.5684			−0.574266			−0.287133	+	0.957891i	$0.592702\pi$
$480$		0			0
$481$	−1.40677			−0.0641431
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.98587	−	2.18608i	0.0901738	−	0.0992647i
$486$		0			0
$487$			13.7901i			0.624888i	0.949936	+	0.312444i	$0.101148\pi$
$487$			13.7901i			0.624888i	−0.949936	+	0.312444i	$0.898852\pi$
$488$		0			0
$489$	−7.42816			−0.335913
$490$		0			0
$491$	27.4423			1.23845			0.619226	−	0.785213i	$-0.287447\pi$
$491$	27.4423			1.23845			0.619226	+	0.785213i	$0.287447\pi$
$492$		0			0
$493$			52.1782i			2.34999i
$494$		0			0
$495$	22.4918	+	20.4319i	1.01093	+	0.918347i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	19.7379			0.883590			0.441795	−	0.897116i	$-0.354342\pi$
$499$	19.7379			0.883590			0.441795	+	0.897116i	$0.354342\pi$
$500$		0			0
$501$	23.4511			1.04772
$502$		0			0
$503$			5.00417i			0.223125i	0.993757	+	0.111562i	$0.0355855\pi$
$503$			5.00417i			0.223125i	−0.993757	+	0.111562i	$0.964415\pi$
$504$		0			0
$505$	−12.8299	−	11.6549i	−0.570921	−	0.518635i
$506$		0			0
$507$		−	2.37855i		−	0.105635i
$508$		0			0
$509$	22.7592			1.00878			0.504392	−	0.863475i	$-0.331717\pi$
$509$	22.7592			1.00878			0.504392	+	0.863475i	$0.331717\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		−	6.32699i		−	0.279343i
$514$		0			0
$515$	−3.31324	+	3.64726i	−0.145999	+	0.160718i
$516$		0			0
$517$		−	1.88007i		−	0.0826854i
$518$		0			0
$519$	−55.1614			−2.42131
$520$		0			0
$521$	17.5371			0.768315			0.384158	−	0.923267i	$-0.374492\pi$
$521$	17.5371			0.768315			0.384158	+	0.923267i	$0.374492\pi$
$522$		0			0
$523$		−	13.7901i		−	0.602998i	−0.953467	−	0.301499i	$-0.902513\pi$
$523$		−	13.7901i		−	0.602998i	0.953467	−	0.301499i	$-0.0974869\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	10.3863i		−	0.452436i
$528$		0			0
$529$	−13.2139			−0.574517
$530$		0			0
$531$	35.8162			1.55429
$532$		0			0
$533$			8.94541i			0.387469i
$534$		0			0
$535$	16.2735	−	17.9142i	0.703567	−	0.774497i
$536$		0			0
$537$		−	63.5988i		−	2.74449i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	46.0001			1.97770			0.988849	−	0.148922i	$-0.0475805\pi$
$541$	46.0001			1.97770			0.988849	+	0.148922i	$0.0475805\pi$
$542$		0			0
$543$		−	10.6327i		−	0.456291i
$544$		0			0
$545$	18.2779	+	16.6040i	0.782941	+	0.711237i
$546$		0			0
$547$			26.4411i			1.13054i	0.824906	+	0.565270i	$0.191228\pi$
$547$			26.4411i			1.13054i	−0.824906	+	0.565270i	$0.808772\pi$
$548$		0			0
$549$	28.0689			1.19795
$550$		0			0
$551$	−53.9159			−2.29689
$552$		0			0
$553$		0			0
$554$		0			0
$555$	1.70550	+	1.54931i	0.0723946	+	0.0657645i
$556$		0			0
$557$		−	31.7219i		−	1.34410i	−0.740506	−	0.672050i	$-0.765414\pi$
$557$		−	31.7219i		−	1.34410i	0.740506	−	0.672050i	$-0.234586\pi$
$558$		0			0
$559$	−21.3887			−0.904644
$560$		0			0
$561$	−50.5926			−2.13602
$562$		0			0
$563$		−	20.7249i		−	0.873448i	−0.899595	−	0.436724i	$-0.856139\pi$
$563$		−	20.7249i		−	0.873448i	0.899595	−	0.436724i	$-0.143861\pi$
$564$		0			0
$565$	10.9633	−	12.0685i	0.461228	−	0.507726i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−42.5454			−1.78360			−0.891799	−	0.452432i	$-0.850556\pi$
$569$	−42.5454			−1.78360			−0.891799	+	0.452432i	$0.850556\pi$
$570$		0			0
$571$	30.1062			1.25990			0.629952	−	0.776634i	$-0.283074\pi$
$571$	30.1062			1.25990			0.629952	+	0.776634i	$0.283074\pi$
$572$		0			0
$573$			22.8267i			0.953601i
$574$		0			0
$575$	−29.9507	+	2.88122i	−1.24903	+	0.120155i
$576$		0			0
$577$			10.2683i			0.427473i	0.976891	+	0.213736i	$0.0685634\pi$
$577$			10.2683i			0.427473i	−0.976891	+	0.213736i	$0.931437\pi$
$578$		0			0
$579$	46.3619			1.92674
$580$		0			0
$581$		0			0
$582$		0			0
$583$			1.82480i			0.0755755i
$584$		0			0
$585$	18.1403	−	19.9692i	0.750010	−	0.825623i
$586$		0			0
$587$		−	41.5546i		−	1.71514i	−0.514367	−	0.857570i	$-0.671973\pi$
$587$		−	41.5546i		−	1.71514i	0.514367	−	0.857570i	$-0.328027\pi$
$588$		0			0
$589$	10.7322			0.442214
$590$		0			0
$591$	59.9519			2.46609
$592$		0			0
$593$			23.9303i			0.982698i	0.870963	+	0.491349i	$0.163496\pi$
$593$			23.9303i			0.982698i	−0.870963	+	0.491349i	$0.836504\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	52.7087i		−	2.15722i
$598$		0			0
$599$	−9.08267			−0.371108			−0.185554	−	0.982634i	$-0.559408\pi$
$599$	−9.08267			−0.371108			−0.185554	+	0.982634i	$0.559408\pi$
$600$		0			0
$601$	29.8162			1.21623			0.608114	−	0.793850i	$-0.291926\pi$
$601$	29.8162			1.21623			0.608114	+	0.793850i	$0.291926\pi$
$602$		0			0
$603$		−	14.4209i		−	0.587263i
$604$		0			0
$605$	7.12686	+	6.47416i	0.289748	+	0.263212i
$606$		0			0
$607$		−	14.5480i		−	0.590486i	−0.955422	−	0.295243i	$-0.904599\pi$
$607$		−	14.5480i		−	0.590486i	0.955422	−	0.295243i	$-0.0954006\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1.66921			−0.0675289
$612$		0			0
$613$		−	10.3228i		−	0.416934i	−0.978029	−	0.208467i	$-0.933153\pi$
$613$		−	10.3228i		−	0.416934i	0.978029	−	0.208467i	$-0.0668475\pi$
$614$		0			0
$615$	9.85180	−	10.8450i	0.397263	−	0.437313i
$616$		0			0
$617$			13.0906i			0.527008i	0.964658	+	0.263504i	$0.0848782\pi$
$617$			13.0906i			0.527008i	−0.964658	+	0.263504i	$0.915122\pi$
$618$		0			0
$619$	−32.5588			−1.30865			−0.654325	−	0.756213i	$-0.727047\pi$
$619$	−32.5588			−1.30865			−0.654325	+	0.756213i	$0.727047\pi$
$620$		0			0
$621$	−7.24969			−0.290920
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−24.5415	+	4.76583i	−0.981661	+	0.190633i
$626$		0			0
$627$		−	52.2775i		−	2.08776i
$628$		0			0
$629$	−2.05847			−0.0820764
$630$		0			0
$631$	−13.2537			−0.527620			−0.263810	−	0.964575i	$-0.584979\pi$
$631$	−13.2537			−0.527620			−0.263810	+	0.964575i	$0.584979\pi$
$632$		0			0
$633$		−	19.5877i		−	0.778542i
$634$		0			0
$635$	−11.5216	+	12.6832i	−0.457222	+	0.503317i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	14.8894			0.589016
$640$		0			0
$641$	0.703073			0.0277697			0.0138848	−	0.999904i	$-0.495580\pi$
$641$	0.703073			0.0277697			0.0138848	+	0.999904i	$0.495580\pi$
$642$		0			0
$643$			4.64163i			0.183048i	0.995803	+	0.0915240i	$0.0291738\pi$
$643$			4.64163i			0.183048i	−0.995803	+	0.0915240i	$0.970826\pi$
$644$		0			0
$645$	25.9307	+	23.5559i	1.02102	+	0.927512i
$646$		0			0
$647$		−	12.0354i		−	0.473161i	−0.971612	−	0.236580i	$-0.923973\pi$
$647$		−	12.0354i		−	0.473161i	0.971612	−	0.236580i	$-0.0760266\pi$
$648$		0			0
$649$	40.3407			1.58351
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	18.2464i		−	0.714036i	−0.934098	−	0.357018i	$-0.883794\pi$
$653$		−	18.2464i		−	0.714036i	0.934098	−	0.357018i	$-0.116206\pi$
$654$		0			0
$655$	28.4876	+	25.8787i	1.11310	+	1.01116i
$656$		0			0
$657$		−	16.7243i		−	0.652476i
$658$		0			0
$659$	−29.4231			−1.14616			−0.573081	−	0.819499i	$-0.694252\pi$
$659$	−29.4231			−1.14616			−0.573081	+	0.819499i	$0.694252\pi$
$660$		0			0
$661$	17.2957			0.672723			0.336361	−	0.941733i	$-0.390804\pi$
$661$	17.2957			0.672723			0.336361	+	0.941733i	$0.390804\pi$
$662$		0			0
$663$			44.9183i			1.74448i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			61.7788i			2.39208i
$668$		0			0
$669$	32.3473			1.25062
$670$		0			0
$671$	31.6147			1.22047
$672$		0			0
$673$			28.7525i			1.10833i	0.832407	+	0.554164i	$0.186962\pi$
$673$			28.7525i			1.10833i	−0.832407	+	0.554164i	$0.813038\pi$
$674$		0			0
$675$	−5.99586	+	0.576795i	−0.230781	+	0.0222008i
$676$		0			0
$677$			5.48594i			0.210842i	0.994428	+	0.105421i	$0.0336190\pi$
$677$			5.48594i			0.210842i	−0.994428	+	0.105421i	$0.966381\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	27.2824			1.04546
$682$		0			0
$683$			11.3768i			0.435319i	0.976025	+	0.217660i	$0.0698423\pi$
$683$			11.3768i			0.435319i	−0.976025	+	0.217660i	$0.930158\pi$
$684$		0			0
$685$	7.90967	−	8.70708i	0.302213	−	0.332681i
$686$		0			0
$687$			63.0714i			2.40632i
$688$		0			0
$689$	1.62013			0.0617222
$690$		0			0
$691$	45.3261			1.72429			0.862144	−	0.506664i	$-0.169121\pi$
$691$	45.3261			1.72429			0.862144	+	0.506664i	$0.169121\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−25.1821	−	22.8759i	−0.955213	−	0.867732i
$696$		0			0
$697$			13.0894i			0.495798i
$698$		0			0
$699$	−57.8941			−2.18976
$700$		0			0
$701$	4.87634			0.184177			0.0920884	−	0.995751i	$-0.470646\pi$
$701$	4.87634			0.184177			0.0920884	+	0.995751i	$0.470646\pi$
$702$		0			0
$703$		−	2.12702i		−	0.0802220i
$704$		0			0
$705$	2.02367	+	1.83834i	0.0762159	+	0.0692359i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	24.8887			0.934713			0.467357	−	0.884069i	$-0.345206\pi$
$709$	24.8887			0.934713			0.467357	+	0.884069i	$0.345206\pi$
$710$		0			0
$711$	−43.9881			−1.64968
$712$		0			0
$713$		−	12.2974i		−	0.460541i
$714$		0			0
$715$	20.4319	−	22.4918i	0.764111	−	0.841145i
$716$		0			0
$717$		−	3.79281i		−	0.141645i
$718$		0			0
$719$	−5.43573			−0.202719			−0.101359	−	0.994850i	$-0.532319\pi$
$719$	−5.43573			−0.202719			−0.101359	+	0.994850i	$0.532319\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$			66.5787i			2.47609i
$724$		0			0
$725$	4.91520	+	51.0942i	0.182546	+	1.89759i
$726$		0			0
$727$		−	17.1112i		−	0.634621i	−0.948322	−	0.317310i	$-0.897220\pi$
$727$		−	17.1112i		−	0.634621i	0.948322	−	0.317310i	$-0.102780\pi$
$728$		0			0
$729$	34.7441			1.28682
$730$		0			0
$731$	−31.2972			−1.15757
$732$		0			0
$733$			7.85744i			0.290221i	0.989415	+	0.145111i	$0.0463538\pi$
$733$			7.85744i			0.290221i	−0.989415	+	0.145111i	$0.953646\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	16.2426i		−	0.598304i
$738$		0			0
$739$	−51.0109			−1.87647			−0.938234	−	0.346002i	$-0.887539\pi$
$739$	−51.0109			−1.87647			−0.938234	+	0.346002i	$0.887539\pi$
$740$		0			0
$741$	−46.4142			−1.70507
$742$		0			0
$743$		−	6.82733i		−	0.250470i	−0.992127	−	0.125235i	$-0.960031\pi$
$743$		−	6.82733i		−	0.250470i	0.992127	−	0.125235i	$-0.0399686\pi$
$744$		0			0
$745$	1.68360	+	1.52942i	0.0616825	+	0.0560334i
$746$		0			0
$747$		−	29.2565i		−	1.07044i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	32.3570			1.18072			0.590361	−	0.807139i	$-0.298985\pi$
$751$	32.3570			1.18072			0.590361	+	0.807139i	$0.298985\pi$
$752$		0			0
$753$			12.4742i			0.454583i
$754$		0			0
$755$	21.9439	+	19.9342i	0.798621	+	0.725481i
$756$		0			0
$757$		−	29.1180i		−	1.05831i	−0.848524	−	0.529156i	$-0.822509\pi$
$757$		−	29.1180i		−	1.05831i	0.848524	−	0.529156i	$-0.177491\pi$
$758$		0			0
$759$	−59.9014			−2.17428
$760$		0			0
$761$	−28.1730			−1.02127			−0.510636	−	0.859797i	$-0.670590\pi$
$761$	−28.1730			−1.02127			−0.510636	+	0.859797i	$0.670590\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	26.5440	−	29.2200i	0.959701	−	1.05645i
$766$		0			0
$767$		−	35.8162i		−	1.29325i
$768$		0			0
$769$	16.5017			0.595065			0.297532	−	0.954712i	$-0.403836\pi$
$769$	16.5017			0.595065			0.297532	+	0.954712i	$0.403836\pi$
$770$		0			0
$771$	−31.3964			−1.13071
$772$		0			0
$773$		−	18.4891i		−	0.665005i	−0.943102	−	0.332503i	$-0.892107\pi$
$773$		−	18.4891i		−	0.665005i	0.943102	−	0.332503i	$-0.107893\pi$
$774$		0			0
$775$	−0.978396	−	10.1706i	−0.0351450	−	0.365338i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−13.5254			−0.484596
$780$		0			0
$781$	16.7703			0.600089
$782$		0			0
$783$			12.3675i			0.441980i
$784$		0			0
$785$	26.7211	−	29.4150i	0.953717	−	1.04987i
$786$		0			0
$787$		−	15.7443i		−	0.561224i	−0.959821	−	0.280612i	$-0.909463\pi$
$787$		−	15.7443i		−	0.561224i	0.959821	−	0.280612i	$-0.0905375\pi$
$788$		0			0
$789$	51.2034			1.82289
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	28.0689i		−	0.996755i
$794$		0			0
$795$	−1.96418	−	1.78430i	−0.0696623	−	0.0632824i
$796$		0			0
$797$			29.9307i			1.06020i	0.847935	+	0.530100i	$0.177846\pi$
$797$			29.9307i			1.06020i	−0.847935	+	0.530100i	$0.822154\pi$
$798$		0			0
$799$	−2.44248			−0.0864088
$800$		0			0
$801$	−8.28758			−0.292827
$802$		0			0
$803$		−	18.8370i		−	0.664743i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			38.0105i			1.33803i
$808$		0			0
$809$	−5.16658			−0.181647			−0.0908236	−	0.995867i	$-0.528950\pi$
$809$	−5.16658			−0.181647			−0.0908236	+	0.995867i	$0.528950\pi$
$810$		0			0
$811$	38.4698			1.35086			0.675430	−	0.737425i	$-0.263958\pi$
$811$	38.4698			1.35086			0.675430	+	0.737425i	$0.263958\pi$
$812$		0			0
$813$		−	14.7807i		−	0.518381i
$814$		0			0
$815$	4.38960	−	4.83213i	0.153761	−	0.169262i
$816$		0			0
$817$		−	32.3394i		−	1.13141i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−3.69551			−0.128974			−0.0644872	−	0.997919i	$-0.520541\pi$
$821$	−3.69551			−0.128974			−0.0644872	+	0.997919i	$0.520541\pi$
$822$		0			0
$823$			42.4030i			1.47807i	0.673665	+	0.739037i	$0.264719\pi$
$823$			42.4030i			1.47807i	−0.673665	+	0.739037i	$0.735281\pi$
$824$		0			0
$825$	−49.5415	+	4.76583i	−1.72481	+	0.165925i
$826$		0			0
$827$		−	40.0950i		−	1.39424i	−0.716954	−	0.697120i	$-0.754464\pi$
$827$		−	40.0950i		−	1.39424i	0.716954	−	0.697120i	$-0.245536\pi$
$828$		0			0
$829$	53.8264			1.86947			0.934734	−	0.355347i	$-0.115637\pi$
$829$	53.8264			1.86947			0.934734	+	0.355347i	$0.115637\pi$
$830$		0			0
$831$	−65.9968			−2.28941
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−13.8582	+	15.2553i	−0.479583	+	0.527932i
$836$		0			0
$837$		−	2.46183i		−	0.0850931i
$838$		0			0
$839$	−25.1002			−0.866556			−0.433278	−	0.901260i	$-0.642643\pi$
$839$	−25.1002			−0.866556			−0.433278	+	0.901260i	$0.642643\pi$
$840$		0			0
$841$	76.3910			2.63417
$842$		0			0
$843$			37.8694i			1.30429i
$844$		0			0
$845$	1.54729	+	1.40558i	0.0532283	+	0.0483535i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	24.7189			0.848352
$850$		0			0
$851$	−2.43721			−0.0835466
$852$		0			0
$853$			26.7320i			0.915286i	0.889136	+	0.457643i	$0.151306\pi$
$853$			26.7320i			0.915286i	−0.889136	+	0.457643i	$0.848694\pi$
$854$		0			0
$855$	30.1931	+	27.4280i	1.03258	+	0.938017i
$856$		0			0
$857$		−	46.5891i		−	1.59145i	−0.605657	−	0.795726i	$-0.707089\pi$
$857$		−	46.5891i		−	1.59145i	0.605657	−	0.795726i	$-0.292911\pi$
$858$		0			0
$859$	−18.6535			−0.636448			−0.318224	−	0.948016i	$-0.603086\pi$
$859$	−18.6535			−0.636448			−0.318224	+	0.948016i	$0.603086\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			17.8239i			0.606733i	0.952874	+	0.303367i	$0.0981107\pi$
$863$			17.8239i			0.606733i	−0.952874	+	0.303367i	$0.901889\pi$
$864$		0			0
$865$	32.5970	−	35.8833i	1.10833	−	1.22007i
$866$		0			0
$867$			22.4739i			0.763253i
$868$		0			0
$869$	−49.5450			−1.68070
$870$		0			0
$871$	−14.4209			−0.488633
$872$		0			0
$873$			4.58781i			0.155274i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			26.7793i			0.904273i	0.891949	+	0.452136i	$0.149338\pi$
$877$			26.7793i			0.904273i	−0.891949	+	0.452136i	$0.850662\pi$
$878$		0			0
$879$	−20.5732			−0.693916
$880$		0			0
$881$	−17.5843			−0.592429			−0.296215	−	0.955121i	$-0.595724\pi$
$881$	−17.5843			−0.592429			−0.296215	+	0.955121i	$0.595724\pi$
$882$		0			0
$883$			18.9638i			0.638182i	0.947724	+	0.319091i	$0.103378\pi$
$883$			18.9638i			0.638182i	−0.947724	+	0.319091i	$0.896622\pi$
$884$		0			0
$885$	−39.4453	+	43.4220i	−1.32594	+	1.45961i
$886$		0			0
$887$			11.1654i			0.374897i	0.982274	+	0.187448i	$0.0600217\pi$
$887$			11.1654i			0.374897i	−0.982274	+	0.187448i	$0.939978\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	28.7761			0.964037
$892$		0			0
$893$		−	2.52382i		−	0.0844565i
$894$		0			0
$895$	41.3720	+	37.5831i	1.38291	+	1.25626i
$896$		0			0
$897$			53.1830i			1.77573i
$898$		0			0
$899$	−20.9786			−0.699676
$900$		0			0
$901$	2.37068			0.0789787
$902$		0			0
$903$		0			0
$904$		0			0
$905$	6.91671	+	6.28326i	0.229919	+	0.208863i
$906$		0			0
$907$			38.1647i			1.26724i	0.773645	+	0.633619i	$0.218431\pi$
$907$			38.1647i			1.26724i	−0.773645	+	0.633619i	$0.781569\pi$
$908$		0			0
$909$	26.9253			0.893057
$910$		0			0
$911$	47.0251			1.55801			0.779006	−	0.627016i	$-0.215724\pi$
$911$	47.0251			1.55801			0.779006	+	0.627016i	$0.215724\pi$
$912$		0			0
$913$		−	32.9524i		−	1.09056i
$914$		0			0
$915$	−30.9130	+	34.0295i	−1.02195	+	1.12498i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−25.3879			−0.837469			−0.418734	−	0.908109i	$-0.637526\pi$
$919$	−25.3879			−0.837469			−0.418734	+	0.908109i	$0.637526\pi$
$920$		0			0
$921$	−22.4359			−0.739288
$922$		0			0
$923$		−	14.8894i		−	0.490091i
$924$		0			0
$925$	−2.01570	+	0.193908i	−0.0662758	+	0.00637565i
$926$		0			0
$927$		−	7.65432i		−	0.251401i
$928$		0			0
$929$	7.64709			0.250893			0.125446	−	0.992100i	$-0.459964\pi$
$929$	7.64709			0.250893			0.125446	+	0.992100i	$0.459964\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$			24.5892i			0.805013i
$934$		0			0
$935$	29.8972	−	32.9113i	0.977743	−	1.07631i
$936$		0			0
$937$		−	1.40923i		−	0.0460375i	−0.999735	−	0.0230188i	$-0.992672\pi$
$937$		−	1.40923i		−	0.0460375i	0.999735	−	0.0230188i	$-0.00732774\pi$
$938$		0			0
$939$	69.1706			2.25730
$940$		0			0
$941$	−52.4101			−1.70852			−0.854260	−	0.519845i	$-0.825990\pi$
$941$	−52.4101			−1.70852			−0.854260	+	0.519845i	$0.825990\pi$
$942$		0			0
$943$			15.4978i			0.504679i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	32.7229i		−	1.06335i	−0.846948	−	0.531675i	$-0.821563\pi$
$947$		−	32.7229i		−	1.06335i	0.846948	−	0.531675i	$-0.178437\pi$
$948$		0			0
$949$	−16.7243			−0.542893
$950$		0			0
$951$	−30.9344			−1.00312
$952$		0			0
$953$			41.7747i			1.35322i	0.736344	+	0.676608i	$0.236551\pi$
$953$			41.7747i			1.35322i	−0.736344	+	0.676608i	$0.763449\pi$
$954$		0			0
$955$	−14.8491	−	13.4892i	−0.480507	−	0.436501i
$956$		0			0
$957$			102.188i			3.30328i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−26.8241			−0.865293
$962$		0			0
$963$			37.5955i			1.21150i
$964$		0			0
$965$	−27.3971	+	30.1592i	−0.881944	+	0.970858i
$966$		0			0
$967$		−	6.47375i		−	0.208182i	−0.994568	−	0.104091i	$-0.966807\pi$
$967$		−	6.47375i		−	0.208182i	0.994568	−	0.104091i	$-0.0331933\pi$
$968$		0			0
$969$	−67.9159			−2.18177
$970$		0			0
$971$	33.1743			1.06462			0.532308	−	0.846551i	$-0.321325\pi$
$971$	33.1743			1.06462			0.532308	+	0.846551i	$0.321325\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	4.23131	+	43.9851i	0.135510	+	1.40865i
$976$		0			0
$977$			40.0711i			1.28199i	0.767546	+	0.640994i	$0.221478\pi$
$977$			40.0711i			1.28199i	−0.767546	+	0.640994i	$0.778522\pi$
$978$		0			0
$979$	−9.33452			−0.298333
$980$		0			0
$981$	−38.3589			−1.22471
$982$		0			0
$983$			0.852116i			0.0271783i	0.999908	+	0.0135891i	$0.00432569\pi$
$983$			0.852116i			0.0271783i	−0.999908	+	0.0135891i	$0.995674\pi$
$984$		0			0
$985$	−35.4280	+	38.9997i	−1.12883	+	1.24263i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−37.0557			−1.17830
$990$		0			0
$991$	−5.02760			−0.159707			−0.0798535	−	0.996807i	$-0.525445\pi$
$991$	−5.02760			−0.159707			−0.0798535	+	0.996807i	$0.525445\pi$
$992$		0			0
$993$			8.72558i			0.276898i
$994$		0			0
$995$	34.2878	+	31.1477i	1.08700	+	0.987448i
$996$		0			0
$997$		−	27.0291i		−	0.856021i	−0.903774	−	0.428011i	$-0.859215\pi$
$997$		−	27.0291i		−	0.856021i	0.903774	−	0.428011i	$-0.140785\pi$
$998$		0			0
$999$	−0.487908			−0.0154367

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1960.2.g.e.1569.2		12
5.2	odd	4		9800.2.a.cy.1.1		6
5.3	odd	4		9800.2.a.cw.1.6		6
5.4	even	2	inner	1960.2.g.e.1569.11		12
7.3	odd	6		280.2.bg.a.9.11	yes	24
7.5	odd	6		280.2.bg.a.249.2	yes	24
7.6	odd	2		1960.2.g.f.1569.11		12
28.3	even	6		560.2.bw.f.289.2		24
28.19	even	6		560.2.bw.f.529.11		24
35.3	even	12		1400.2.q.n.401.6		12
35.12	even	12		1400.2.q.o.1201.1		12
35.13	even	4		9800.2.a.cx.1.1		6
35.17	even	12		1400.2.q.o.401.1		12
35.19	odd	6		280.2.bg.a.249.11	yes	24
35.24	odd	6		280.2.bg.a.9.2	✓	24
35.27	even	4		9800.2.a.cv.1.6		6
35.33	even	12		1400.2.q.n.1201.6		12
35.34	odd	2		1960.2.g.f.1569.2		12
140.19	even	6		560.2.bw.f.529.2		24
140.59	even	6		560.2.bw.f.289.11		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.bg.a.9.2	✓	24	35.24	odd	6
280.2.bg.a.9.11	yes	24	7.3	odd	6
280.2.bg.a.249.2	yes	24	7.5	odd	6
280.2.bg.a.249.11	yes	24	35.19	odd	6
560.2.bw.f.289.2		24	28.3	even	6
560.2.bw.f.289.11		24	140.59	even	6
560.2.bw.f.529.2		24	140.19	even	6
560.2.bw.f.529.11		24	28.19	even	6
1400.2.q.n.401.6		12	35.3	even	12
1400.2.q.n.1201.6		12	35.33	even	12
1400.2.q.o.401.1		12	35.17	even	12
1400.2.q.o.1201.1		12	35.12	even	12
1960.2.g.e.1569.2		12	1.1	even	1	trivial
1960.2.g.e.1569.11		12	5.4	even	2	inner
1960.2.g.f.1569.2		12	35.34	odd	2
1960.2.g.f.1569.11		12	7.6	odd	2
9800.2.a.cv.1.6		6	35.27	even	4
9800.2.a.cw.1.6		6	5.3	odd	4
9800.2.a.cx.1.1		6	35.13	even	4
9800.2.a.cy.1.1		6	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1960.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.54431i q^{3} +(1.65511 + 1.50353i) q^{5} -3.47349 q^{9} +O(q^{10})\)	\(q-2.54431i q^{3} +(1.65511 + 1.50353i) q^{5} -3.47349 q^{9} -3.91228 q^{11} +3.47349i q^{13} +(3.82544 - 4.21111i) q^{15} +5.08262i q^{17} -5.25188 q^{19} +6.01780i q^{23} +(0.478784 + 4.97702i) q^{25} +1.20471i q^{27} +10.2660 q^{29} -2.04350 q^{31} +9.95405i q^{33} +0.405001i q^{37} +8.83763 q^{39} +2.57534 q^{41} +6.15769i q^{43} +(-5.74901 - 5.22251i) q^{45} +0.480556i q^{47} +12.9317 q^{51} -0.466428i q^{53} +(-6.47526 - 5.88224i) q^{55} +13.3624i q^{57} -10.3113 q^{59} -8.08088 q^{61} +(-5.22251 + 5.74901i) q^{65} +4.15169i q^{67} +15.3111 q^{69} -4.28658 q^{71} +4.81483i q^{73} +(12.6631 - 1.21817i) q^{75} +12.6639 q^{79} -7.35533 q^{81} +8.42280i q^{83} +(-7.64188 + 8.41230i) q^{85} -26.1199i q^{87} +2.38595 q^{89} +5.19930i q^{93} +(-8.69245 - 7.89637i) q^{95} -1.32080i q^{97} +13.5893 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 14 q^{9}+O(q^{10}) \) 12 * q - 14 * q^9	\( 12 q - 14 q^{9} + 2 q^{11} + 6 q^{15} - 10 q^{19} + 2 q^{25} + 6 q^{29} + 4 q^{31} - 20 q^{39} - 12 q^{41} - 8 q^{45} + 6 q^{55} - 48 q^{59} - 18 q^{61} + 26 q^{65} + 30 q^{69} + 8 q^{71} - 14 q^{75} + 44 q^{79} - 12 q^{81} - 22 q^{85} + 30 q^{89} + 26 q^{95} - 22 q^{99}+O(q^{100}) \) 12 * q - 14 * q^9 + 2 * q^11 + 6 * q^15 - 10 * q^19 + 2 * q^25 + 6 * q^29 + 4 * q^31 - 20 * q^39 - 12 * q^41 - 8 * q^45 + 6 * q^55 - 48 * q^59 - 18 * q^61 + 26 * q^65 + 30 * q^69 + 8 * q^71 - 14 * q^75 + 44 * q^79 - 12 * q^81 - 22 * q^85 + 30 * q^89 + 26 * q^95 - 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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