LMFDB - Embedded newform 1440.3.j.c.1279.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1440,3,Mod(1279,1440)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1440.1279"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1440.j (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,-4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	no
Analytic conductor:	$39.2371580679$
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} - 6 x^{11} + 49 x^{10} - 190 x^{9} + 792 x^{8} - 2094 x^{7} + 5517 x^{6} - 9954 x^{5} + \cdots + 5584 $ x^12 - 6x^11 + 49x^10 - 190x^9 + 792x^8 - 2094x^7 + 5517x^6 - 9954x^5 + 17189x^4 - 19884x^3 + 20996x^2 - 12416*x + 5584
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{25} $
Twist minimal:	no (minimal twist has level 480)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1279.7
Root			$0.500000 - 1.80359i$ of defining polynomial
Character	$\chi$	$=$	1440.1279
Dual form			1440.3.j.c.1279.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.621940 - 4.96117i) q^{5} +9.43574 q^{7} +3.73048i q^{11} -12.5929i q^{13} -30.0484i q^{17} -10.7454i q^{19} -38.0158 q^{23} +(-24.2264 + 6.17109i) q^{25} +27.0083 q^{29} +16.9678i q^{31} +(-5.86846 - 46.8123i) q^{35} +45.2400i q^{37} -68.6923 q^{41} +81.0133 q^{43} +17.1188 q^{47} +40.0331 q^{49} -50.3769i q^{53} +(18.5075 - 2.32013i) q^{55} -63.3848i q^{59} -12.1092 q^{61} +(-62.4756 + 7.83204i) q^{65} -10.4760 q^{67} -1.92327i q^{71} +31.3735i q^{73} +35.1998i q^{77} -69.1535i q^{79} +67.3300 q^{83} +(-149.075 + 18.6883i) q^{85} -63.1761 q^{89} -118.823i q^{91} +(-53.3096 + 6.68297i) q^{95} +51.3786i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{5} - 80 q^{23} + 28 q^{25} + 40 q^{29} - 144 q^{35} - 136 q^{41} + 224 q^{43} + 208 q^{47} + 212 q^{49} + 192 q^{55} + 40 q^{61} - 96 q^{65} - 352 q^{67} - 64 q^{85} + 8 q^{89} - 176 q^{95}+O(q^{100}) $ 12 * q - 4 * q^5 - 80 * q^23 + 28 * q^25 + 40 * q^29 - 144 * q^35 - 136 * q^41 + 224 * q^43 + 208 * q^47 + 212 * q^49 + 192 * q^55 + 40 * q^61 - 96 * q^65 - 352 * q^67 - 64 * q^85 + 8 * q^89 - 176 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$991$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.621940	−	4.96117i	−0.124388	−	0.992234i
$5$	−0.621940	−	4.96117i	−0.124388	−	0.992234i
$6$		0			0
$7$	9.43574			1.34796			0.673981	−	0.738748i	$-0.264583\pi$
$7$	9.43574			1.34796			0.673981	+	0.738748i	$0.264583\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			3.73048i			0.339134i	0.985519	+	0.169567i	$0.0542370\pi$
$11$			3.73048i			0.339134i	−0.985519	+	0.169567i	$0.945763\pi$
$12$		0			0
$13$		−	12.5929i		−	0.968686i	−0.874878	−	0.484343i	$-0.839059\pi$
$13$		−	12.5929i		−	0.968686i	0.874878	−	0.484343i	$-0.160941\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	30.0484i		−	1.76755i	−0.467907	−	0.883777i	$-0.654992\pi$
$17$		−	30.0484i		−	1.76755i	0.467907	−	0.883777i	$-0.345008\pi$
$18$		0			0
$19$		−	10.7454i		−	0.565546i	−0.959187	−	0.282773i	$-0.908746\pi$
$19$		−	10.7454i		−	0.565546i	0.959187	−	0.282773i	$-0.0912543\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−38.0158			−1.65286			−0.826430	−	0.563040i	$-0.809632\pi$
$23$	−38.0158			−1.65286			−0.826430	+	0.563040i	$0.809632\pi$
$24$		0			0
$25$	−24.2264	+	6.17109i	−0.969055	+	0.246844i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	27.0083			0.931321			0.465661	−	0.884963i	$-0.345817\pi$
$29$	27.0083			0.931321			0.465661	+	0.884963i	$0.345817\pi$
$30$		0			0
$31$			16.9678i			0.547348i	0.961823	+	0.273674i	$0.0882389\pi$
$31$			16.9678i			0.547348i	−0.961823	+	0.273674i	$0.911761\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−5.86846	−	46.8123i	−0.167670	−	1.33749i
$36$		0			0
$37$			45.2400i			1.22270i	0.791359	+	0.611352i	$0.209374\pi$
$37$			45.2400i			1.22270i	−0.791359	+	0.611352i	$0.790626\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−68.6923			−1.67542			−0.837711	−	0.546113i	$-0.816107\pi$
$41$	−68.6923			−1.67542			−0.837711	+	0.546113i	$0.816107\pi$
$42$		0			0
$43$	81.0133			1.88403			0.942016	−	0.335569i	$-0.108929\pi$
$43$	81.0133			1.88403			0.942016	+	0.335569i	$0.108929\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	17.1188			0.364230			0.182115	−	0.983277i	$-0.441706\pi$
$47$	17.1188			0.364230			0.182115	+	0.983277i	$0.441706\pi$
$48$		0			0
$49$	40.0331			0.817003
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	50.3769i		−	0.950507i	−0.879849	−	0.475254i	$-0.842356\pi$
$53$		−	50.3769i		−	0.950507i	0.879849	−	0.475254i	$-0.157644\pi$
$54$		0			0
$55$	18.5075	−	2.32013i	0.336501	−	0.0421842i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	63.3848i		−	1.07432i	−0.843481	−	0.537160i	$-0.819497\pi$
$59$		−	63.3848i		−	1.07432i	0.843481	−	0.537160i	$-0.180503\pi$
$60$		0			0
$61$	−12.1092			−0.198512			−0.0992560	−	0.995062i	$-0.531646\pi$
$61$	−12.1092			−0.198512			−0.0992560	+	0.995062i	$0.531646\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−62.4756	+	7.83204i	−0.961163	+	0.120493i
$66$		0			0
$67$	−10.4760			−0.156358			−0.0781790	−	0.996939i	$-0.524911\pi$
$67$	−10.4760			−0.156358			−0.0781790	+	0.996939i	$0.524911\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	1.92327i		−	0.0270883i	−0.999908	−	0.0135442i	$-0.995689\pi$
$71$		−	1.92327i		−	0.0270883i	0.999908	−	0.0135442i	$-0.00431137\pi$
$72$		0			0
$73$			31.3735i			0.429775i	0.976639	+	0.214887i	$0.0689384\pi$
$73$			31.3735i			0.429775i	−0.976639	+	0.214887i	$0.931062\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			35.1998i			0.457140i
$78$		0			0
$79$		−	69.1535i		−	0.875361i	−0.899131	−	0.437681i	$-0.855800\pi$
$79$		−	69.1535i		−	0.875361i	0.899131	−	0.437681i	$-0.144200\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	67.3300			0.811205			0.405603	−	0.914050i	$-0.367062\pi$
$83$	67.3300			0.811205			0.405603	+	0.914050i	$0.367062\pi$
$84$		0			0
$85$	−149.075	+	18.6883i	−1.75383	+	0.219862i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−63.1761			−0.709844			−0.354922	−	0.934896i	$-0.615493\pi$
$89$	−63.1761			−0.709844			−0.354922	+	0.934896i	$0.615493\pi$
$90$		0			0
$91$		−	118.823i		−	1.30575i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−53.3096	+	6.68297i	−0.561154	+	0.0703471i
$96$		0			0
$97$			51.3786i			0.529676i	0.964293	+	0.264838i	$0.0853185\pi$
$97$			51.3786i			0.529676i	−0.964293	+	0.264838i	$0.914681\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−68.0951			−0.674209			−0.337105	−	0.941467i	$-0.609448\pi$
$101$	−68.0951			−0.674209			−0.337105	+	0.941467i	$0.609448\pi$
$102$		0			0
$103$	−155.320			−1.50796			−0.753982	−	0.656895i	$-0.771869\pi$
$103$	−155.320			−1.50796			−0.753982	+	0.656895i	$0.771869\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−21.7681			−0.203440			−0.101720	−	0.994813i	$-0.532435\pi$
$107$	−21.7681			−0.203440			−0.101720	+	0.994813i	$0.532435\pi$
$108$		0			0
$109$	−126.784			−1.16315			−0.581577	−	0.813492i	$-0.697564\pi$
$109$	−126.784			−1.16315			−0.581577	+	0.813492i	$0.697564\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			80.0613i			0.708507i	0.935149	+	0.354253i	$0.115265\pi$
$113$			80.0613i			0.708507i	−0.935149	+	0.354253i	$0.884735\pi$
$114$		0			0
$115$	23.6435	+	188.603i	0.205596	+	1.64002i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	283.529i		−	2.38260i
$120$		0			0
$121$	107.084			0.884988
$122$		0			0
$123$		0			0
$124$		0			0
$125$	45.6832	+	116.353i	0.365465	+	0.930825i
$126$		0			0
$127$	−156.543			−1.23262			−0.616312	−	0.787502i	$-0.711374\pi$
$127$	−156.543			−1.23262			−0.616312	+	0.787502i	$0.711374\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	35.5220i		−	0.271160i	−0.990766	−	0.135580i	$-0.956710\pi$
$131$		−	35.5220i		−	0.271160i	0.990766	−	0.135580i	$-0.0432898\pi$
$132$		0			0
$133$		−	101.391i		−	0.762335i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	105.233i		−	0.768122i	−0.923308	−	0.384061i	$-0.874525\pi$
$137$		−	105.233i		−	0.768122i	0.923308	−	0.384061i	$-0.125475\pi$
$138$		0			0
$139$		−	181.835i		−	1.30817i	−0.756423	−	0.654083i	$-0.773055\pi$
$139$		−	181.835i		−	1.30817i	0.756423	−	0.654083i	$-0.226945\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	46.9776			0.328515
$144$		0			0
$145$	−16.7975	−	133.993i	−0.115845	−	0.924088i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−132.241			−0.887527			−0.443763	−	0.896144i	$-0.646357\pi$
$149$	−132.241			−0.887527			−0.443763	+	0.896144i	$0.646357\pi$
$150$		0			0
$151$		−	250.226i		−	1.65713i	−0.559895	−	0.828564i	$-0.689158\pi$
$151$		−	250.226i		−	1.65713i	0.559895	−	0.828564i	$-0.310842\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	84.1800	−	10.5529i	0.543097	−	0.0680834i
$156$		0			0
$157$		−	283.974i		−	1.80875i	−0.426735	−	0.904377i	$-0.640336\pi$
$157$		−	283.974i		−	1.80875i	0.426735	−	0.904377i	$-0.359664\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−358.707			−2.22799
$162$		0			0
$163$	142.259			0.872753			0.436376	−	0.899764i	$-0.356262\pi$
$163$	142.259			0.872753			0.436376	+	0.899764i	$0.356262\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	59.4298			0.355867			0.177934	−	0.984042i	$-0.443059\pi$
$167$	59.4298			0.355867			0.177934	+	0.984042i	$0.443059\pi$
$168$		0			0
$169$	10.4183			0.0616470
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	235.203i		−	1.35956i	−0.733418	−	0.679778i	$-0.762076\pi$
$173$		−	235.203i		−	1.35956i	0.733418	−	0.679778i	$-0.237924\pi$
$174$		0			0
$175$	−228.594	+	58.2288i	−1.30625	+	0.332736i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	52.4818i		−	0.293194i	−0.989196	−	0.146597i	$-0.953168\pi$
$179$		−	52.4818i		−	0.293194i	0.989196	−	0.146597i	$-0.0468321\pi$
$180$		0			0
$181$	−14.9662			−0.0826864			−0.0413432	−	0.999145i	$-0.513164\pi$
$181$	−14.9662			−0.0826864			−0.0413432	+	0.999145i	$0.513164\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	224.443	−	28.1366i	1.21321	−	0.152089i
$186$		0			0
$187$	112.095			0.599439
$188$		0			0
$189$		0			0
$190$		0			0
$191$			208.113i			1.08960i	0.838567	+	0.544798i	$0.183394\pi$
$191$			208.113i			1.08960i	−0.838567	+	0.544798i	$0.816606\pi$
$192$		0			0
$193$		−	303.975i		−	1.57500i	−0.616316	−	0.787499i	$-0.711375\pi$
$193$		−	303.975i		−	1.57500i	0.616316	−	0.787499i	$-0.288625\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			318.751i			1.61803i	0.587790	+	0.809014i	$0.299998\pi$
$197$			318.751i			1.61803i	−0.587790	+	0.809014i	$0.700002\pi$
$198$		0			0
$199$			180.444i			0.906756i	0.891318	+	0.453378i	$0.149781\pi$
$199$			180.444i			0.906756i	−0.891318	+	0.453378i	$0.850219\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	254.843			1.25539
$204$		0			0
$205$	42.7225	+	340.794i	0.208402	+	1.66241i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	40.0854			0.191796
$210$		0			0
$211$			47.1478i			0.223449i	0.993739	+	0.111725i	$0.0356375\pi$
$211$			47.1478i			0.223449i	−0.993739	+	0.111725i	$0.964363\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−50.3854	−	401.921i	−0.234351	−	1.86940i
$216$		0			0
$217$			160.103i			0.737804i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−378.398			−1.71221
$222$		0			0
$223$	281.448			1.26210			0.631050	−	0.775742i	$-0.282624\pi$
$223$	281.448			1.26210			0.631050	+	0.775742i	$0.282624\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	241.837			1.06536			0.532682	−	0.846316i	$-0.321184\pi$
$227$	241.837			1.06536			0.532682	+	0.846316i	$0.321184\pi$
$228$		0			0
$229$	103.354			0.451328			0.225664	−	0.974205i	$-0.427545\pi$
$229$	103.354			0.451328			0.225664	+	0.974205i	$0.427545\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			356.103i			1.52834i	0.645015	+	0.764170i	$0.276851\pi$
$233$			356.103i			1.52834i	−0.645015	+	0.764170i	$0.723149\pi$
$234$		0			0
$235$	−10.6469	−	84.9292i	−0.0453058	−	0.361401i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	61.3511i		−	0.256699i	−0.991729	−	0.128350i	$-0.959032\pi$
$239$		−	61.3511i		−	0.256699i	0.991729	−	0.128350i	$-0.0409680\pi$
$240$		0			0
$241$	−101.184			−0.419851			−0.209925	−	0.977717i	$-0.567322\pi$
$241$	−101.184			−0.419851			−0.209925	+	0.977717i	$0.567322\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−24.8982	−	198.611i	−0.101625	−	0.810658i
$246$		0			0
$247$	−135.316			−0.547837
$248$		0			0
$249$		0			0
$250$		0			0
$251$			115.712i			0.461005i	0.973072	+	0.230502i	$0.0740370\pi$
$251$			115.712i			0.461005i	−0.973072	+	0.230502i	$0.925963\pi$
$252$		0			0
$253$		−	141.817i		−	0.560542i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	159.654i		−	0.621222i	−0.950537	−	0.310611i	$-0.899466\pi$
$257$		−	159.654i		−	0.621222i	0.950537	−	0.310611i	$-0.100534\pi$
$258$		0			0
$259$			426.873i			1.64816i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	6.73896			0.0256234			0.0128117	−	0.999918i	$-0.495922\pi$
$263$	6.73896			0.0256234			0.0128117	+	0.999918i	$0.495922\pi$
$264$		0			0
$265$	−249.928	+	31.3314i	−0.943125	+	0.118232i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	333.834			1.24102			0.620509	−	0.784199i	$-0.286926\pi$
$269$	333.834			1.24102			0.620509	+	0.784199i	$0.286926\pi$
$270$		0			0
$271$		−	485.690i		−	1.79221i	−0.443839	−	0.896107i	$-0.646384\pi$
$271$		−	485.690i		−	1.79221i	0.443839	−	0.896107i	$-0.353616\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−23.0211	−	90.3760i	−0.0837132	−	0.328640i
$276$		0			0
$277$			132.443i			0.478134i	0.971003	+	0.239067i	$0.0768415\pi$
$277$			132.443i			0.478134i	−0.971003	+	0.239067i	$0.923158\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	265.883			0.946202			0.473101	−	0.881008i	$-0.343135\pi$
$281$	265.883			0.946202			0.473101	+	0.881008i	$0.343135\pi$
$282$		0			0
$283$	352.065			1.24405			0.622024	−	0.782998i	$-0.286311\pi$
$283$	352.065			1.24405			0.622024	+	0.782998i	$0.286311\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−648.163			−2.25841
$288$		0			0
$289$	−613.908			−2.12425
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	246.313i		−	0.840657i	−0.907372	−	0.420329i	$-0.861915\pi$
$293$		−	246.313i		−	0.840657i	0.907372	−	0.420329i	$-0.138085\pi$
$294$		0			0
$295$	−314.463	+	39.4215i	−1.06598	+	0.133632i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			478.730i			1.60110i
$300$		0			0
$301$	764.421			2.53960
$302$		0			0
$303$		0			0
$304$		0			0
$305$	7.53121	+	60.0760i	0.0246925	+	0.196970i
$306$		0			0
$307$	479.633			1.56232			0.781162	−	0.624329i	$-0.214627\pi$
$307$	479.633			1.56232			0.781162	+	0.624329i	$0.214627\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			238.106i			0.765613i	0.923828	+	0.382807i	$0.125042\pi$
$311$			238.106i			0.765613i	−0.923828	+	0.382807i	$0.874958\pi$
$312$		0			0
$313$			158.297i			0.505741i	0.967500	+	0.252870i	$0.0813746\pi$
$313$			158.297i			0.505741i	−0.967500	+	0.252870i	$0.918625\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			172.575i			0.544399i	0.962241	+	0.272200i	$0.0877511\pi$
$317$			172.575i			0.544399i	−0.962241	+	0.272200i	$0.912249\pi$
$318$		0			0
$319$			100.754i			0.315843i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−322.882			−0.999634
$324$		0			0
$325$	77.7121	+	305.081i	0.239114	+	0.938711i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	161.528			0.490968
$330$		0			0
$331$		−	394.897i		−	1.19304i	−0.802598	−	0.596521i	$-0.796549\pi$
$331$		−	394.897i		−	1.19304i	0.802598	−	0.596521i	$-0.203451\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	6.51543	+	51.9731i	0.0194490	+	0.155144i
$336$		0			0
$337$			552.712i			1.64010i	0.572295	+	0.820048i	$0.306053\pi$
$337$			552.712i			1.64010i	−0.572295	+	0.820048i	$0.693947\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−63.2979			−0.185624
$342$		0			0
$343$	−84.6090			−0.246674
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−342.973			−0.988395			−0.494197	−	0.869350i	$-0.664538\pi$
$347$	−342.973			−0.988395			−0.494197	+	0.869350i	$0.664538\pi$
$348$		0			0
$349$	−21.4938			−0.0615869			−0.0307935	−	0.999526i	$-0.509803\pi$
$349$	−21.4938			−0.0615869			−0.0307935	+	0.999526i	$0.509803\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	623.505i		−	1.76630i	−0.469086	−	0.883152i	$-0.655417\pi$
$353$		−	623.505i		−	1.76630i	0.469086	−	0.883152i	$-0.344583\pi$
$354$		0			0
$355$	−9.54167	+	1.19616i	−0.0268780	+	0.00336946i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			283.468i			0.789605i	0.918766	+	0.394802i	$0.129187\pi$
$359$			283.468i			0.789605i	−0.918766	+	0.394802i	$0.870813\pi$
$360$		0			0
$361$	245.537			0.680158
$362$		0			0
$363$		0			0
$364$		0			0
$365$	155.649	−	19.5124i	0.426437	−	0.0534588i
$366$		0			0
$367$	−542.611			−1.47850			−0.739252	−	0.673429i	$-0.764821\pi$
$367$	−542.611			−1.47850			−0.739252	+	0.673429i	$0.764821\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	475.343i		−	1.28125i
$372$		0			0
$373$			248.299i			0.665682i	0.942983	+	0.332841i	$0.108007\pi$
$373$			248.299i			0.665682i	−0.942983	+	0.332841i	$0.891993\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	340.114i		−	0.902158i
$378$		0			0
$379$			304.847i			0.804345i	0.915564	+	0.402173i	$0.131745\pi$
$379$			304.847i			0.804345i	−0.915564	+	0.402173i	$0.868255\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	3.78168			0.00987383			0.00493692	−	0.999988i	$-0.498429\pi$
$383$	3.78168			0.00987383			0.00493692	+	0.999988i	$0.498429\pi$
$384$		0			0
$385$	174.632	−	21.8922i	0.453590	−	0.0568627i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−394.970			−1.01535			−0.507674	−	0.861549i	$-0.669495\pi$
$389$	−394.970			−1.01535			−0.507674	+	0.861549i	$0.669495\pi$
$390$		0			0
$391$			1142.31i			2.92152i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−343.082	+	43.0093i	−0.868563	+	0.108884i
$396$		0			0
$397$		−	287.721i		−	0.724739i	−0.932035	−	0.362369i	$-0.881968\pi$
$397$		−	287.721i		−	0.724739i	0.932035	−	0.362369i	$-0.118032\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	640.735			1.59784			0.798922	−	0.601435i	$-0.205404\pi$
$401$	640.735			1.59784			0.798922	+	0.601435i	$0.205404\pi$
$402$		0			0
$403$	213.674			0.530208
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−168.767			−0.414661
$408$		0			0
$409$	8.13388			0.0198872			0.00994361	−	0.999951i	$-0.496835\pi$
$409$	8.13388			0.0198872			0.00994361	+	0.999951i	$0.496835\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	598.083i		−	1.44814i
$414$		0			0
$415$	−41.8752	−	334.036i	−0.100904	−	0.804905i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	335.898i		−	0.801665i	−0.916151	−	0.400832i	$-0.868721\pi$
$419$		−	335.898i		−	0.801665i	0.916151	−	0.400832i	$-0.131279\pi$
$420$		0			0
$421$	−101.764			−0.241720			−0.120860	−	0.992670i	$-0.538565\pi$
$421$	−101.764			−0.241720			−0.120860	+	0.992670i	$0.538565\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	185.432	+	727.965i	0.436310	+	1.71286i
$426$		0			0
$427$	−114.260			−0.267587
$428$		0			0
$429$		0			0
$430$		0			0
$431$			709.379i			1.64589i	0.568120	+	0.822946i	$0.307671\pi$
$431$			709.379i			1.64589i	−0.568120	+	0.822946i	$0.692329\pi$
$432$		0			0
$433$		−	775.129i		−	1.79014i	−0.445929	−	0.895068i	$-0.647127\pi$
$433$		−	775.129i		−	1.79014i	0.445929	−	0.895068i	$-0.352873\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			408.494i			0.934768i
$438$		0			0
$439$			49.7622i			0.113354i	0.998393	+	0.0566768i	$0.0180505\pi$
$439$			49.7622i			0.113354i	−0.998393	+	0.0566768i	$0.981950\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	625.947			1.41297			0.706487	−	0.707726i	$-0.250279\pi$
$443$	625.947			1.41297			0.706487	+	0.707726i	$0.250279\pi$
$444$		0			0
$445$	39.2917	+	313.427i	0.0882960	+	0.704331i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	444.155			0.989209			0.494605	−	0.869118i	$-0.335313\pi$
$449$	444.155			0.989209			0.494605	+	0.869118i	$0.335313\pi$
$450$		0			0
$451$		−	256.255i		−	0.568194i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−589.503	+	73.9010i	−1.29561	+	0.162420i
$456$		0			0
$457$			28.9532i			0.0633550i	0.999498	+	0.0316775i	$0.0100849\pi$
$457$			28.9532i			0.0633550i	−0.999498	+	0.0316775i	$0.989915\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	651.815			1.41392			0.706958	−	0.707256i	$-0.250067\pi$
$461$	651.815			1.41392			0.706958	+	0.707256i	$0.250067\pi$
$462$		0			0
$463$	329.121			0.710845			0.355422	−	0.934706i	$-0.384337\pi$
$463$	329.121			0.710845			0.355422	+	0.934706i	$0.384337\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	760.999			1.62955			0.814775	−	0.579778i	$-0.196861\pi$
$467$	760.999			1.62955			0.814775	+	0.579778i	$0.196861\pi$
$468$		0			0
$469$	−98.8486			−0.210765
$470$		0			0
$471$		0			0
$472$		0			0
$473$			302.219i			0.638940i
$474$		0			0
$475$	66.3107	+	260.322i	0.139602	+	0.548045i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			536.008i			1.11902i	0.828825	+	0.559508i	$0.189010\pi$
$479$			536.008i			1.11902i	−0.828825	+	0.559508i	$0.810990\pi$
$480$		0			0
$481$	569.704			1.18442
$482$		0			0
$483$		0			0
$484$		0			0
$485$	254.898	−	31.9544i	0.525563	−	0.0658853i
$486$		0			0
$487$	22.2364			0.0456600			0.0228300	−	0.999739i	$-0.492732\pi$
$487$	22.2364			0.0456600			0.0228300	+	0.999739i	$0.492732\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			762.724i			1.55341i	0.629865	+	0.776705i	$0.283110\pi$
$491$			762.724i			1.55341i	−0.629865	+	0.776705i	$0.716890\pi$
$492$		0			0
$493$		−	811.558i		−	1.64616i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	18.1475i		−	0.0365141i
$498$		0			0
$499$			162.461i			0.325572i	0.986661	+	0.162786i	$0.0520481\pi$
$499$			162.461i			0.325572i	−0.986661	+	0.162786i	$0.947952\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	517.965			1.02975			0.514876	−	0.857265i	$-0.327838\pi$
$503$	517.965			1.02975			0.514876	+	0.857265i	$0.327838\pi$
$504$		0			0
$505$	42.3511	+	337.831i	0.0838635	+	0.668973i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−143.469			−0.281865			−0.140932	−	0.990019i	$-0.545010\pi$
$509$	−143.469			−0.281865			−0.140932	+	0.990019i	$0.545010\pi$
$510$		0			0
$511$			296.033i			0.579320i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	96.5998	+	770.570i	0.187572	+	1.49625i
$516$		0			0
$517$			63.8613i			0.123523i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−542.459			−1.04119			−0.520594	−	0.853804i	$-0.674290\pi$
$521$	−542.459			−1.04119			−0.520594	+	0.853804i	$0.674290\pi$
$522$		0			0
$523$	572.228			1.09413			0.547063	−	0.837091i	$-0.315746\pi$
$523$	572.228			1.09413			0.547063	+	0.837091i	$0.315746\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	509.855			0.967467
$528$		0			0
$529$	916.199			1.73194
$530$		0			0
$531$		0			0
$532$		0			0
$533$			865.037i			1.62296i
$534$		0			0
$535$	13.5384	+	107.995i	0.0253055	+	0.201860i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			149.343i			0.277074i
$540$		0			0
$541$	−875.914			−1.61906			−0.809532	−	0.587075i	$-0.800279\pi$
$541$	−875.914			−1.61906			−0.809532	+	0.587075i	$0.800279\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	78.8518	+	628.995i	0.144682	+	1.15412i
$546$		0			0
$547$	−776.471			−1.41951			−0.709754	−	0.704449i	$-0.751194\pi$
$547$	−776.471			−1.41951			−0.709754	+	0.704449i	$0.751194\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	290.215i		−	0.526705i
$552$		0			0
$553$		−	652.514i		−	1.17995i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			417.952i			0.750363i	0.926951	+	0.375182i	$0.122420\pi$
$557$			417.952i			0.750363i	−0.926951	+	0.375182i	$0.877580\pi$
$558$		0			0
$559$		−	1020.19i		−	1.82504i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	131.170			0.232984			0.116492	−	0.993192i	$-0.462835\pi$
$563$	131.170			0.232984			0.116492	+	0.993192i	$0.462835\pi$
$564$		0			0
$565$	397.197	−	49.7933i	0.703004	−	0.0881297i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−241.675			−0.424736			−0.212368	−	0.977190i	$-0.568118\pi$
$569$	−241.675			−0.424736			−0.212368	+	0.977190i	$0.568118\pi$
$570$		0			0
$571$			311.504i			0.545542i	0.962079	+	0.272771i	$0.0879401\pi$
$571$			311.504i			0.545542i	−0.962079	+	0.272771i	$0.912060\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	920.985	−	234.599i	1.60171	−	0.407998i
$576$		0			0
$577$			293.075i			0.507929i	0.967214	+	0.253964i	$0.0817346\pi$
$577$			293.075i			0.507929i	−0.967214	+	0.253964i	$0.918265\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	635.308			1.09347
$582$		0			0
$583$	187.930			0.322350
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−528.326			−0.900044			−0.450022	−	0.893018i	$-0.648584\pi$
$587$	−528.326			−0.900044			−0.450022	+	0.893018i	$0.648584\pi$
$588$		0			0
$589$	182.325			0.309550
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	216.255i		−	0.364680i	−0.983236	−	0.182340i	$-0.941633\pi$
$593$		−	216.255i		−	0.364680i	0.983236	−	0.182340i	$-0.0583671\pi$
$594$		0			0
$595$	−1406.64	+	176.338i	−2.36409	+	0.296366i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	300.236i		−	0.501229i	−0.968087	−	0.250615i	$-0.919367\pi$
$599$		−	300.236i		−	0.501229i	0.968087	−	0.250615i	$-0.0806327\pi$
$600$		0			0
$601$	−387.274			−0.644383			−0.322191	−	0.946675i	$-0.604419\pi$
$601$	−387.274			−0.644383			−0.322191	+	0.946675i	$0.604419\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−66.5995	−	531.259i	−0.110082	−	0.878115i
$606$		0			0
$607$	−160.885			−0.265049			−0.132524	−	0.991180i	$-0.542308\pi$
$607$	−160.885			−0.265049			−0.132524	+	0.991180i	$0.542308\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	215.576i		−	0.352824i
$612$		0			0
$613$		−	150.755i		−	0.245930i	−0.992411	−	0.122965i	$-0.960760\pi$
$613$		−	150.755i		−	0.245930i	0.992411	−	0.122965i	$-0.0392404\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	252.185i		−	0.408728i	−0.978895	−	0.204364i	$-0.934487\pi$
$617$		−	252.185i		−	0.408728i	0.978895	−	0.204364i	$-0.0655127\pi$
$618$		0			0
$619$		−	22.1097i		−	0.0357184i	−0.999841	−	0.0178592i	$-0.994315\pi$
$619$		−	22.1097i		−	0.0357184i	0.999841	−	0.0178592i	$-0.00568506\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−596.113			−0.956843
$624$		0			0
$625$	548.835	−	299.007i	0.878136	−	0.478410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1359.39			2.16120
$630$		0			0
$631$		−	33.7222i		−	0.0534424i	−0.999643	−	0.0267212i	$-0.991493\pi$
$631$		−	33.7222i		−	0.0534424i	0.999643	−	0.0267212i	$-0.00850664\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	97.3605	+	776.638i	0.153324	+	1.22305i
$636$		0			0
$637$		−	504.134i		−	0.791419i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	492.458			0.768265			0.384133	−	0.923278i	$-0.374501\pi$
$641$	492.458			0.768265			0.384133	+	0.923278i	$0.374501\pi$
$642$		0			0
$643$	981.203			1.52598			0.762988	−	0.646413i	$-0.223732\pi$
$643$	981.203			1.52598			0.762988	+	0.646413i	$0.223732\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	635.264			0.981861			0.490930	−	0.871199i	$-0.336657\pi$
$647$	635.264			0.981861			0.490930	+	0.871199i	$0.336657\pi$
$648$		0			0
$649$	236.456			0.364339
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	12.2032i		−	0.0186879i	−0.999956	−	0.00934396i	$-0.997026\pi$
$653$		−	12.2032i		−	0.0186879i	0.999956	−	0.00934396i	$-0.00297432\pi$
$654$		0			0
$655$	−176.231	+	22.0925i	−0.269054	+	0.0337290i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			420.257i			0.637720i	0.947802	+	0.318860i	$0.103300\pi$
$659$			420.257i			0.637720i	−0.947802	+	0.318860i	$0.896700\pi$
$660$		0			0
$661$	−344.639			−0.521390			−0.260695	−	0.965421i	$-0.583952\pi$
$661$	−344.639			−0.521390			−0.260695	+	0.965421i	$0.583952\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−503.016	+	63.0588i	−0.756414	+	0.0948252i
$666$		0			0
$667$	−1026.74			−1.53934
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	45.1733i		−	0.0673223i
$672$		0			0
$673$		−	1307.73i		−	1.94314i	−0.236747	−	0.971571i	$-0.576081\pi$
$673$		−	1307.73i		−	1.94314i	0.236747	−	0.971571i	$-0.423919\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			247.644i			0.365796i	0.983132	+	0.182898i	$0.0585479\pi$
$677$			247.644i			0.365796i	−0.983132	+	0.182898i	$0.941452\pi$
$678$		0			0
$679$			484.795i			0.713984i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	242.635			0.355249			0.177624	−	0.984098i	$-0.443159\pi$
$683$	242.635			0.355249			0.177624	+	0.984098i	$0.443159\pi$
$684$		0			0
$685$	−522.077	+	65.4484i	−0.762157	+	0.0955451i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−634.392			−0.920743
$690$		0			0
$691$			753.499i			1.09045i	0.838291	+	0.545224i	$0.183555\pi$
$691$			753.499i			1.09045i	−0.838291	+	0.545224i	$0.816445\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−902.114	+	113.090i	−1.29801	+	0.162720i
$696$		0			0
$697$			2064.10i			2.96140i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−367.412			−0.524126			−0.262063	−	0.965051i	$-0.584403\pi$
$701$	−367.412			−0.524126			−0.262063	+	0.965051i	$0.584403\pi$
$702$		0			0
$703$	486.121			0.691495
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−642.528			−0.908809
$708$		0			0
$709$	394.799			0.556839			0.278419	−	0.960460i	$-0.410190\pi$
$709$	394.799			0.556839			0.278419	+	0.960460i	$0.410190\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	645.043i		−	0.904689i
$714$		0			0
$715$	−29.2172	−	233.064i	−0.0408633	−	0.325964i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	475.466i		−	0.661288i	−0.943756	−	0.330644i	$-0.892734\pi$
$719$		−	475.466i		−	0.661288i	0.943756	−	0.330644i	$-0.107266\pi$
$720$		0			0
$721$	−1465.56			−2.03268
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−654.314	+	166.671i	−0.902502	+	0.229891i
$726$		0			0
$727$	452.918			0.622996			0.311498	−	0.950247i	$-0.399169\pi$
$727$	452.918			0.622996			0.311498	+	0.950247i	$0.399169\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	2434.32i		−	3.33013i
$732$		0			0
$733$		−	92.6649i		−	0.126419i	−0.998000	−	0.0632094i	$-0.979866\pi$
$733$		−	92.6649i		−	0.126419i	0.998000	−	0.0632094i	$-0.0201336\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	39.0804i		−	0.0530264i
$738$		0			0
$739$		−	1373.49i		−	1.85857i	−0.369360	−	0.929286i	$-0.620423\pi$
$739$		−	1373.49i		−	1.85857i	0.369360	−	0.929286i	$-0.379577\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	423.832			0.570434			0.285217	−	0.958463i	$-0.407934\pi$
$743$	423.832			0.570434			0.285217	+	0.958463i	$0.407934\pi$
$744$		0			0
$745$	82.2462	+	656.072i	0.110398	+	0.880634i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−205.398			−0.274230
$750$		0			0
$751$			781.600i			1.04075i	0.853939	+	0.520373i	$0.174207\pi$
$751$			781.600i			1.04075i	−0.853939	+	0.520373i	$0.825793\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1241.41	+	155.626i	−1.64426	+	0.206127i
$756$		0			0
$757$			122.603i			0.161959i	0.996716	+	0.0809793i	$0.0258048\pi$
$757$			122.603i			0.161959i	−0.996716	+	0.0809793i	$0.974195\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−481.169			−0.632285			−0.316142	−	0.948712i	$-0.602388\pi$
$761$	−481.169			−0.632285			−0.316142	+	0.948712i	$0.602388\pi$
$762$		0			0
$763$	−1196.30			−1.56789
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−798.200			−1.04068
$768$		0			0
$769$	1369.26			1.78057			0.890287	−	0.455400i	$-0.150504\pi$
$769$	1369.26			1.78057			0.890287	+	0.455400i	$0.150504\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	248.772i		−	0.321826i	−0.986969	−	0.160913i	$-0.948556\pi$
$773$		−	248.772i		−	0.321826i	0.986969	−	0.160913i	$-0.0514439\pi$
$774$		0			0
$775$	−104.710	−	411.068i	−0.135109	−	0.530410i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			738.125i			0.947529i
$780$		0			0
$781$	7.17472			0.00918659
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1408.84	+	176.615i	−1.79471	+	0.224987i
$786$		0			0
$787$	−73.4073			−0.0932749			−0.0466374	−	0.998912i	$-0.514851\pi$
$787$	−73.4073			−0.0932749			−0.0466374	+	0.998912i	$0.514851\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			755.437i			0.955040i
$792$		0			0
$793$			152.491i			0.192296i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	845.614i		−	1.06100i	−0.847686	−	0.530498i	$-0.822005\pi$
$797$		−	845.614i		−	1.06100i	0.847686	−	0.530498i	$-0.177995\pi$
$798$		0			0
$799$		−	514.393i		−	0.643796i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−117.038			−0.145751
$804$		0			0
$805$	223.094	+	1779.60i	0.277135	+	2.21069i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−46.2307			−0.0571455			−0.0285728	−	0.999592i	$-0.509096\pi$
$809$	−46.2307			−0.0571455			−0.0285728	+	0.999592i	$0.509096\pi$
$810$		0			0
$811$		−	614.528i		−	0.757742i	−0.925450	−	0.378871i	$-0.876312\pi$
$811$		−	614.528i		−	0.757742i	0.925450	−	0.378871i	$-0.123688\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−88.4763	−	705.769i	−0.108560	−	0.865975i
$816$		0			0
$817$		−	870.519i		−	1.06551i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1006.35			1.22576			0.612880	−	0.790176i	$-0.290011\pi$
$821$	1006.35			1.22576			0.612880	+	0.790176i	$0.290011\pi$
$822$		0			0
$823$	742.969			0.902757			0.451379	−	0.892332i	$-0.350932\pi$
$823$	742.969			0.902757			0.451379	+	0.892332i	$0.350932\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1080.66			−1.30672			−0.653361	−	0.757047i	$-0.726642\pi$
$827$	−1080.66			−1.30672			−0.653361	+	0.757047i	$0.726642\pi$
$828$		0			0
$829$	33.5681			0.0404922			0.0202461	−	0.999795i	$-0.493555\pi$
$829$	33.5681			0.0404922			0.0202461	+	0.999795i	$0.493555\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	1202.93i		−	1.44410i
$834$		0			0
$835$	−36.9618	−	294.841i	−0.0442656	−	0.353103i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	228.003i		−	0.271756i	−0.990726	−	0.135878i	$-0.956614\pi$
$839$		−	228.003i		−	0.271756i	0.990726	−	0.135878i	$-0.0433855\pi$
$840$		0			0
$841$	−111.551			−0.132641
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−6.47957	−	51.6871i	−0.00766813	−	0.0611682i
$846$		0			0
$847$	1010.41			1.19293
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	1719.83i		−	2.02096i
$852$		0			0
$853$			1499.35i			1.75774i	0.477065	+	0.878868i	$0.341701\pi$
$853$			1499.35i			1.75774i	−0.477065	+	0.878868i	$0.658299\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			1462.41i			1.70643i	0.521558	+	0.853216i	$0.325351\pi$
$857$			1462.41i			1.70643i	−0.521558	+	0.853216i	$0.674649\pi$
$858$		0			0
$859$			1082.73i			1.26045i	0.776412	+	0.630225i	$0.217037\pi$
$859$			1082.73i			1.26045i	−0.776412	+	0.630225i	$0.782963\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1359.98			1.57587			0.787935	−	0.615758i	$-0.211150\pi$
$863$	1359.98			1.57587			0.787935	+	0.615758i	$0.211150\pi$
$864$		0			0
$865$	−1166.88	+	146.282i	−1.34900	+	0.169112i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	257.976			0.296865
$870$		0			0
$871$			131.923i			0.151462i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	431.054	+	1097.88i	0.492634	+	1.25472i
$876$		0			0
$877$		−	814.748i		−	0.929018i	−0.885569	−	0.464509i	$-0.846231\pi$
$877$		−	814.748i		−	0.929018i	0.885569	−	0.464509i	$-0.153769\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	477.767			0.542301			0.271150	−	0.962537i	$-0.412596\pi$
$881$	477.767			0.542301			0.271150	+	0.962537i	$0.412596\pi$
$882$		0			0
$883$	1044.41			1.18280			0.591401	−	0.806378i	$-0.298575\pi$
$883$	1044.41			1.18280			0.591401	+	0.806378i	$0.298575\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1042.60			1.17543			0.587713	−	0.809069i	$-0.300028\pi$
$887$	1042.60			1.17543			0.587713	+	0.809069i	$0.300028\pi$
$888$		0			0
$889$	−1477.10			−1.66153
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	183.948i		−	0.205989i
$894$		0			0
$895$	−260.371	+	32.6405i	−0.290917	+	0.0364698i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			458.271i			0.509756i
$900$		0			0
$901$	−1513.75			−1.68007
$902$		0			0
$903$		0			0
$904$		0			0
$905$	9.30809	+	74.2500i	0.0102852	+	0.0820442i
$906$		0			0
$907$	615.703			0.678835			0.339417	−	0.940636i	$-0.389770\pi$
$907$	615.703			0.678835			0.339417	+	0.940636i	$0.389770\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	400.257i		−	0.439360i	−0.975572	−	0.219680i	$-0.929499\pi$
$911$		−	400.257i		−	0.439360i	0.975572	−	0.219680i	$-0.0705013\pi$
$912$		0			0
$913$			251.173i			0.275108i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	335.176i		−	0.365514i
$918$		0			0
$919$		−	1614.36i		−	1.75665i	−0.478066	−	0.878324i	$-0.658662\pi$
$919$		−	1614.36i		−	1.75665i	0.478066	−	0.878324i	$-0.341338\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−24.2196			−0.0262401
$924$		0			0
$925$	−279.180	−	1096.00i	−0.301817	−	1.18487i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−153.105			−0.164807			−0.0824033	−	0.996599i	$-0.526260\pi$
$929$	−153.105			−0.164807			−0.0824033	+	0.996599i	$0.526260\pi$
$930$		0			0
$931$		−	430.171i		−	0.462053i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−69.7163	−	556.122i	−0.0745629	−	0.594783i
$936$		0			0
$937$			296.081i			0.315989i	0.987440	+	0.157994i	$0.0505028\pi$
$937$			296.081i			0.315989i	−0.987440	+	0.157994i	$0.949497\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−129.275			−0.137380			−0.0686902	−	0.997638i	$-0.521882\pi$
$941$	−129.275			−0.137380			−0.0686902	+	0.997638i	$0.521882\pi$
$942$		0			0
$943$	2611.39			2.76924
$944$		0			0
$945$		0			0
$946$		0			0
$947$	141.017			0.148910			0.0744548	−	0.997224i	$-0.476278\pi$
$947$	141.017			0.148910			0.0744548	+	0.997224i	$0.476278\pi$
$948$		0			0
$949$	395.085			0.416317
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	806.942i		−	0.846738i	−0.905957	−	0.423369i	$-0.860847\pi$
$953$		−	806.942i		−	0.846738i	0.905957	−	0.423369i	$-0.139153\pi$
$954$		0			0
$955$	1032.48	−	129.434i	1.08113	−	0.135533i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	992.948i		−	1.03540i
$960$		0			0
$961$	673.095			0.700411
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1508.07	+	189.054i	−1.56277	+	0.195911i
$966$		0			0
$967$	889.210			0.919555			0.459778	−	0.888034i	$-0.347929\pi$
$967$	889.210			0.919555			0.459778	+	0.888034i	$0.347929\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1331.48i		−	1.37125i	−0.727955	−	0.685625i	$-0.759529\pi$
$971$		−	1331.48i		−	1.37125i	0.727955	−	0.685625i	$-0.240471\pi$
$972$		0			0
$973$		−	1715.75i		−	1.76336i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1024.81i		−	1.04894i	−0.851430	−	0.524468i	$-0.824264\pi$
$977$		−	1024.81i		−	1.04894i	0.851430	−	0.524468i	$-0.175736\pi$
$978$		0			0
$979$		−	235.677i		−	0.240733i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	566.908			0.576712			0.288356	−	0.957523i	$-0.406891\pi$
$983$	566.908			0.576712			0.288356	+	0.957523i	$0.406891\pi$
$984$		0			0
$985$	1581.38	−	198.244i	1.60546	−	0.201263i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−3079.78			−3.11404
$990$		0			0
$991$			1522.10i			1.53592i	0.640497	+	0.767961i	$0.278729\pi$
$991$			1522.10i			1.53592i	−0.640497	+	0.767961i	$0.721271\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	895.215	−	112.226i	0.899714	−	0.112789i
$996$		0			0
$997$			1574.81i			1.57954i	0.613400	+	0.789772i	$0.289801\pi$
$997$			1574.81i			1.57954i	−0.613400	+	0.789772i	$0.710199\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.3.j.c.1279.7		12
3.2	odd	2		480.3.j.b.319.4	yes	12
4.3	odd	2		1440.3.j.d.1279.7		12
5.4	even	2		1440.3.j.d.1279.8		12
12.11	even	2		480.3.j.a.319.10	yes	12
15.2	even	4		2400.3.e.h.1951.12		12
15.8	even	4		2400.3.e.i.1951.1		12
15.14	odd	2		480.3.j.a.319.9	✓	12
20.19	odd	2	inner	1440.3.j.c.1279.8		12
24.5	odd	2		960.3.j.g.319.9		12
24.11	even	2		960.3.j.f.319.3		12
60.23	odd	4		2400.3.e.i.1951.12		12
60.47	odd	4		2400.3.e.h.1951.1		12
60.59	even	2		480.3.j.b.319.3	yes	12
120.29	odd	2		960.3.j.f.319.4		12
120.59	even	2		960.3.j.g.319.10		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.3.j.a.319.9	✓	12	15.14	odd	2
480.3.j.a.319.10	yes	12	12.11	even	2
480.3.j.b.319.3	yes	12	60.59	even	2
480.3.j.b.319.4	yes	12	3.2	odd	2
960.3.j.f.319.3		12	24.11	even	2
960.3.j.f.319.4		12	120.29	odd	2
960.3.j.g.319.9		12	24.5	odd	2
960.3.j.g.319.10		12	120.59	even	2
1440.3.j.c.1279.7		12	1.1	even	1	trivial
1440.3.j.c.1279.8		12	20.19	odd	2	inner
1440.3.j.d.1279.7		12	4.3	odd	2
1440.3.j.d.1279.8		12	5.4	even	2
2400.3.e.h.1951.1		12	60.47	odd	4
2400.3.e.h.1951.12		12	15.2	even	4
2400.3.e.i.1951.1		12	15.8	even	4
2400.3.e.i.1951.12		12	60.23	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1440.j (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.621940 - 4.96117i) q^{5} +9.43574 q^{7} +3.73048i q^{11} -12.5929i q^{13} -30.0484i q^{17} -10.7454i q^{19} -38.0158 q^{23} +(-24.2264 + 6.17109i) q^{25} +27.0083 q^{29} +16.9678i q^{31} +(-5.86846 - 46.8123i) q^{35} +45.2400i q^{37} -68.6923 q^{41} +81.0133 q^{43} +17.1188 q^{47} +40.0331 q^{49} -50.3769i q^{53} +(18.5075 - 2.32013i) q^{55} -63.3848i q^{59} -12.1092 q^{61} +(-62.4756 + 7.83204i) q^{65} -10.4760 q^{67} -1.92327i q^{71} +31.3735i q^{73} +35.1998i q^{77} -69.1535i q^{79} +67.3300 q^{83} +(-149.075 + 18.6883i) q^{85} -63.1761 q^{89} -118.823i q^{91} +(-53.3096 + 6.68297i) q^{95} +51.3786i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{5} - 80 q^{23} + 28 q^{25} + 40 q^{29} - 144 q^{35} - 136 q^{41} + 224 q^{43} + 208 q^{47} + 212 q^{49} + 192 q^{55} + 40 q^{61} - 96 q^{65} - 352 q^{67} - 64 q^{85} + 8 q^{89} - 176 q^{95}+O(q^{100}) \) 12 * q - 4 * q^5 - 80 * q^23 + 28 * q^25 + 40 * q^29 - 144 * q^35 - 136 * q^41 + 224 * q^43 + 208 * q^47 + 212 * q^49 + 192 * q^55 + 40 * q^61 - 96 * q^65 - 352 * q^67 - 64 * q^85 + 8 * q^89 - 176 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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