LMFDB - Embedded newform 1344.3.f.h.769.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1344,3,Mod(769,1344)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1344, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1344.769"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1344.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,4,0,-24,0,-16,0,0,0,-24,0,0,0,0,0,-12,0,-96,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$36.6213475300$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.551252791296.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 17x^{6} + 26x^{5} + 270x^{4} - 302x^{3} - 1007x^{2} - 3502x + 10609 $ x^8 - 2x^7 - 17x^6 + 26x^5 + 270x^4 - 302x^3 - 1007x^2 - 3502*x + 10609
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.2
Root			$-1.39023 - 2.19650i$ of defining polynomial
Character	$\chi$	$=$	1344.769
Dual form			1344.3.f.h.769.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} -3.76315i q^{5} +(-2.78047 - 6.42409i) q^{7} -3.00000 q^{9} +19.3926 q^{11} -10.3961i q^{13} -6.51796 q^{15} -13.8088i q^{17} -3.87797i q^{19} +(-11.1269 + 4.81591i) q^{21} -23.4580 q^{23} +10.8387 q^{25} +5.19615i q^{27} +26.7787 q^{29} -51.0424i q^{31} -33.5890i q^{33} +(-24.1748 + 10.4633i) q^{35} +55.2961 q^{37} -18.0065 q^{39} +4.59089i q^{41} +7.68092 q^{43} +11.2894i q^{45} +78.2491i q^{47} +(-33.5380 + 35.7240i) q^{49} -23.9176 q^{51} +6.58689 q^{53} -72.9772i q^{55} -6.71685 q^{57} -93.9754i q^{59} +33.9447i q^{61} +(8.34141 + 19.2723i) q^{63} -39.1219 q^{65} -111.481 q^{67} +40.6304i q^{69} -32.6658 q^{71} -76.3129i q^{73} -18.7732i q^{75} +(-53.9205 - 124.580i) q^{77} -60.5380 q^{79} +9.00000 q^{81} +37.8324i q^{83} -51.9647 q^{85} -46.3821i q^{87} +2.36600i q^{89} +(-66.7852 + 28.9059i) q^{91} -88.4080 q^{93} -14.5934 q^{95} +131.563i q^{97} -58.1778 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{7} - 24 q^{9} - 16 q^{11} - 24 q^{15} - 12 q^{21} - 96 q^{23} + 64 q^{29} - 120 q^{35} + 40 q^{37} + 48 q^{39} + 136 q^{43} + 112 q^{49} - 24 q^{51} - 112 q^{53} - 72 q^{57} - 12 q^{63} - 208 q^{65}+ \cdots + 48 q^{99}+O(q^{100}) $ 8 * q + 4 * q^7 - 24 * q^9 - 16 * q^11 - 24 * q^15 - 12 * q^21 - 96 * q^23 + 64 * q^29 - 120 * q^35 + 40 * q^37 + 48 * q^39 + 136 * q^43 + 112 * q^49 - 24 * q^51 - 112 * q^53 - 72 * q^57 - 12 * q^63 - 208 * q^65 - 8 * q^67 - 16 * q^71 - 288 * q^77 - 104 * q^79 + 72 * q^81 + 424 * q^85 - 192 * q^91 - 144 * q^93 + 240 * q^95 + 48 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$		−	1.73205i		−	0.577350i
$3$		−	1.73205i		−	0.577350i
$4$		0			0
$5$		−	3.76315i		−	0.752630i	−0.926492	−	0.376315i	$-0.877191\pi$
$5$		−	3.76315i		−	0.752630i	0.926492	−	0.376315i	$-0.122809\pi$
$6$		0			0
$7$	−2.78047	−	6.42409i	−0.397210	−	0.917728i
$7$	−2.78047	−	6.42409i	−0.397210	−	0.917728i
$8$		0			0
$9$	−3.00000			−0.333333
$10$		0			0
$11$	19.3926			1.76296			0.881482	−	0.472217i	$-0.156546\pi$
$11$	19.3926			1.76296			0.881482	+	0.472217i	$0.156546\pi$
$12$		0			0
$13$		−	10.3961i		−	0.799696i	−0.916581	−	0.399848i	$-0.869063\pi$
$13$		−	10.3961i		−	0.799696i	0.916581	−	0.399848i	$-0.130937\pi$
$14$		0			0
$15$	−6.51796			−0.434531
$16$		0			0
$17$		−	13.8088i		−	0.812285i	−0.913810	−	0.406142i	$-0.866874\pi$
$17$		−	13.8088i		−	0.812285i	0.913810	−	0.406142i	$-0.133126\pi$
$18$		0			0
$19$		−	3.87797i		−	0.204104i	−0.994779	−	0.102052i	$-0.967459\pi$
$19$		−	3.87797i		−	0.204104i	0.994779	−	0.102052i	$-0.0325408\pi$
$20$		0			0
$21$	−11.1269	+	4.81591i	−0.529850	+	0.229329i
$22$		0			0
$23$	−23.4580			−1.01991			−0.509956	−	0.860201i	$-0.670338\pi$
$23$	−23.4580			−1.01991			−0.509956	+	0.860201i	$0.670338\pi$
$24$		0			0
$25$	10.8387			0.433549
$26$		0			0
$27$			5.19615i			0.192450i
$28$		0			0
$29$	26.7787			0.923404			0.461702	−	0.887035i	$-0.347239\pi$
$29$	26.7787			0.923404			0.461702	+	0.887035i	$0.347239\pi$
$30$		0			0
$31$		−	51.0424i		−	1.64653i	−0.567658	−	0.823265i	$-0.692150\pi$
$31$		−	51.0424i		−	1.64653i	0.567658	−	0.823265i	$-0.307850\pi$
$32$		0			0
$33$		−	33.5890i		−	1.01785i
$34$		0			0
$35$	−24.1748	+	10.4633i	−0.690709	+	0.298952i
$36$		0			0
$37$	55.2961			1.49449			0.747245	−	0.664549i	$-0.231376\pi$
$37$	55.2961			1.49449			0.747245	+	0.664549i	$0.231376\pi$
$38$		0			0
$39$	−18.0065			−0.461705
$40$		0			0
$41$			4.59089i			0.111973i	0.998432	+	0.0559865i	$0.0178304\pi$
$41$			4.59089i			0.111973i	−0.998432	+	0.0559865i	$0.982170\pi$
$42$		0			0
$43$	7.68092			0.178626			0.0893130	−	0.996004i	$-0.471533\pi$
$43$	7.68092			0.178626			0.0893130	+	0.996004i	$0.471533\pi$
$44$		0			0
$45$			11.2894i			0.250877i
$46$		0			0
$47$			78.2491i			1.66488i	0.554119	+	0.832438i	$0.313055\pi$
$47$			78.2491i			1.66488i	−0.554119	+	0.832438i	$0.686945\pi$
$48$		0			0
$49$	−33.5380	+	35.7240i	−0.684449	+	0.729061i
$50$		0			0
$51$	−23.9176			−0.468973
$52$		0			0
$53$	6.58689			0.124281			0.0621405	−	0.998067i	$-0.480207\pi$
$53$	6.58689			0.124281			0.0621405	+	0.998067i	$0.480207\pi$
$54$		0			0
$55$		−	72.9772i		−	1.32686i
$56$		0			0
$57$	−6.71685			−0.117839
$58$		0			0
$59$		−	93.9754i		−	1.59280i	−0.604768	−	0.796402i	$-0.706734\pi$
$59$		−	93.9754i		−	1.59280i	0.604768	−	0.796402i	$-0.293266\pi$
$60$		0			0
$61$			33.9447i			0.556470i	0.960513	+	0.278235i	$0.0897495\pi$
$61$			33.9447i			0.556470i	−0.960513	+	0.278235i	$0.910251\pi$
$62$		0			0
$63$	8.34141	+	19.2723i	0.132403	+	0.305909i
$64$		0			0
$65$	−39.1219			−0.601875
$66$		0			0
$67$	−111.481			−1.66390			−0.831948	−	0.554854i	$-0.812774\pi$
$67$	−111.481			−1.66390			−0.831948	+	0.554854i	$0.812774\pi$
$68$		0			0
$69$			40.6304i			0.588846i
$70$		0			0
$71$	−32.6658			−0.460082			−0.230041	−	0.973181i	$-0.573886\pi$
$71$	−32.6658			−0.460082			−0.230041	+	0.973181i	$0.573886\pi$
$72$		0			0
$73$		−	76.3129i		−	1.04538i	−0.852522	−	0.522691i	$-0.824928\pi$
$73$		−	76.3129i		−	1.04538i	0.852522	−	0.522691i	$-0.175072\pi$
$74$		0			0
$75$		−	18.7732i		−	0.250310i
$76$		0			0
$77$	−53.9205	−	124.580i	−0.700267	−	1.61792i
$78$		0			0
$79$	−60.5380			−0.766304			−0.383152	−	0.923685i	$-0.625161\pi$
$79$	−60.5380			−0.766304			−0.383152	+	0.923685i	$0.625161\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$			37.8324i			0.455813i	0.973683	+	0.227906i	$0.0731880\pi$
$83$			37.8324i			0.455813i	−0.973683	+	0.227906i	$0.926812\pi$
$84$		0			0
$85$	−51.9647			−0.611350
$86$		0			0
$87$		−	46.3821i		−	0.533128i
$88$		0			0
$89$			2.36600i			0.0265842i	0.999912	+	0.0132921i	$0.00423114\pi$
$89$			2.36600i			0.0265842i	−0.999912	+	0.0132921i	$0.995769\pi$
$90$		0			0
$91$	−66.7852	+	28.9059i	−0.733903	+	0.317647i
$92$		0			0
$93$	−88.4080			−0.950624
$94$		0			0
$95$	−14.5934			−0.153615
$96$		0			0
$97$			131.563i			1.35632i	0.734913	+	0.678161i	$0.237223\pi$
$97$			131.563i			1.35632i	−0.734913	+	0.678161i	$0.762777\pi$
$98$		0			0
$99$	−58.1778			−0.587655
$100$		0			0
$101$			15.6865i			0.155312i	0.996980	+	0.0776559i	$0.0247435\pi$
$101$			15.6865i			0.155312i	−0.996980	+	0.0776559i	$0.975256\pi$
$102$		0			0
$103$			158.794i			1.54169i	0.637021	+	0.770846i	$0.280166\pi$
$103$			158.794i			1.54169i	−0.637021	+	0.770846i	$0.719834\pi$
$104$		0			0
$105$	18.1230	+	41.8720i	0.172600	+	0.398781i
$106$		0			0
$107$	−54.4863			−0.509218			−0.254609	−	0.967044i	$-0.581947\pi$
$107$	−54.4863			−0.509218			−0.254609	+	0.967044i	$0.581947\pi$
$108$		0			0
$109$	−46.3852			−0.425552			−0.212776	−	0.977101i	$-0.568251\pi$
$109$	−46.3852			−0.425552			−0.212776	+	0.977101i	$0.568251\pi$
$110$		0			0
$111$		−	95.7757i		−	0.862844i
$112$		0			0
$113$	43.3245			0.383402			0.191701	−	0.981453i	$-0.438600\pi$
$113$	43.3245			0.383402			0.191701	+	0.981453i	$0.438600\pi$
$114$		0			0
$115$			88.2758i			0.767616i
$116$		0			0
$117$			31.1882i			0.266565i
$118$		0			0
$119$	−88.7093	+	38.3951i	−0.745457	+	0.322648i
$120$		0			0
$121$	255.073			2.10804
$122$		0			0
$123$	7.95165			0.0646476
$124$		0			0
$125$		−	134.866i		−	1.07893i
$126$		0			0
$127$	−150.320			−1.18362			−0.591812	−	0.806076i	$-0.701587\pi$
$127$	−150.320			−1.18362			−0.591812	+	0.806076i	$0.701587\pi$
$128$		0			0
$129$		−	13.3037i		−	0.103130i
$130$		0			0
$131$		−	201.373i		−	1.53720i	−0.639732	−	0.768598i	$-0.720955\pi$
$131$		−	201.373i		−	1.53720i	0.639732	−	0.768598i	$-0.279045\pi$
$132$		0			0
$133$	−24.9125	+	10.7826i	−0.187312	+	0.0810720i
$134$		0			0
$135$	19.5539			0.144844
$136$		0			0
$137$	−5.63627			−0.0411406			−0.0205703	−	0.999788i	$-0.506548\pi$
$137$	−5.63627			−0.0411406			−0.0205703	+	0.999788i	$0.506548\pi$
$138$		0			0
$139$		−	81.3364i		−	0.585154i	−0.956242	−	0.292577i	$-0.905487\pi$
$139$		−	81.3364i		−	0.585154i	0.956242	−	0.292577i	$-0.0945127\pi$
$140$		0			0
$141$	135.531			0.961216
$142$		0			0
$143$		−	201.607i		−	1.40984i
$144$		0			0
$145$		−	100.772i		−	0.694981i
$146$		0			0
$147$	61.8758	+	58.0895i	0.420924	+	0.395167i
$148$		0			0
$149$	56.3232			0.378008			0.189004	−	0.981976i	$-0.439474\pi$
$149$	56.3232			0.378008			0.189004	+	0.981976i	$0.439474\pi$
$150$		0			0
$151$	65.8620			0.436172			0.218086	−	0.975930i	$-0.430019\pi$
$151$	65.8620			0.436172			0.218086	+	0.975930i	$0.430019\pi$
$152$		0			0
$153$			41.4265i			0.270762i
$154$		0			0
$155$	−192.080			−1.23923
$156$		0			0
$157$			268.600i			1.71083i	0.517945	+	0.855414i	$0.326697\pi$
$157$			268.600i			1.71083i	−0.517945	+	0.855414i	$0.673303\pi$
$158$		0			0
$159$		−	11.4088i		−	0.0717537i
$160$		0			0
$161$	65.2242	+	150.696i	0.405119	+	0.936001i
$162$		0			0
$163$	−185.843			−1.14014			−0.570069	−	0.821597i	$-0.693084\pi$
$163$	−185.843			−1.14014			−0.570069	+	0.821597i	$0.693084\pi$
$164$		0			0
$165$	−126.400			−0.766062
$166$		0			0
$167$		−	73.9811i		−	0.443000i	−0.975160	−	0.221500i	$-0.928905\pi$
$167$		−	73.9811i		−	0.443000i	0.975160	−	0.221500i	$-0.0710954\pi$
$168$		0			0
$169$	60.9221			0.360486
$170$		0			0
$171$			11.6339i			0.0680346i
$172$		0			0
$173$		−	156.300i		−	0.903468i	−0.892153	−	0.451734i	$-0.850806\pi$
$173$		−	156.300i		−	0.903468i	0.892153	−	0.451734i	$-0.149194\pi$
$174$		0			0
$175$	−30.1367	−	69.6290i	−0.172210	−	0.397880i
$176$		0			0
$177$	−162.770			−0.919606
$178$		0			0
$179$	7.47589			0.0417648			0.0208824	−	0.999782i	$-0.493352\pi$
$179$	7.47589			0.0417648			0.0208824	+	0.999782i	$0.493352\pi$
$180$		0			0
$181$			199.003i			1.09946i	0.835342	+	0.549731i	$0.185270\pi$
$181$			199.003i			1.09946i	−0.835342	+	0.549731i	$0.814730\pi$
$182$		0			0
$183$	58.7939			0.321278
$184$		0			0
$185$		−	208.088i		−	1.12480i
$186$		0			0
$187$		−	267.790i		−	1.43203i
$188$		0			0
$189$	33.3806	−	14.4477i	0.176617	−	0.0764431i
$190$		0			0
$191$	−230.401			−1.20629			−0.603144	−	0.797632i	$-0.706086\pi$
$191$	−230.401			−1.20629			−0.603144	+	0.797632i	$0.706086\pi$
$192$		0			0
$193$	−385.249			−1.99611			−0.998055	−	0.0623370i	$-0.980145\pi$
$193$	−385.249			−1.99611			−0.998055	+	0.0623370i	$0.980145\pi$
$194$		0			0
$195$			67.7611i			0.347493i
$196$		0			0
$197$	325.670			1.65315			0.826575	−	0.562827i	$-0.190286\pi$
$197$	325.670			1.65315			0.826575	+	0.562827i	$0.190286\pi$
$198$		0			0
$199$			54.1376i			0.272048i	0.990706	+	0.136024i	$0.0434325\pi$
$199$			54.1376i			0.272048i	−0.990706	+	0.136024i	$0.956568\pi$
$200$		0			0
$201$			193.091i			0.960651i
$202$		0			0
$203$	−74.4574	−	172.029i	−0.366785	−	0.847434i
$204$		0			0
$205$	17.2762			0.0842741
$206$		0			0
$207$	70.3739			0.339971
$208$		0			0
$209$		−	75.2040i		−	0.359828i
$210$		0			0
$211$	285.042			1.35091			0.675455	−	0.737402i	$-0.263947\pi$
$211$	285.042			1.35091			0.675455	+	0.737402i	$0.263947\pi$
$212$		0			0
$213$			56.5788i			0.265628i
$214$		0			0
$215$		−	28.9044i		−	0.134439i
$216$		0			0
$217$	−327.901	+	141.922i	−1.51107	+	0.654018i
$218$		0			0
$219$	−132.178			−0.603551
$220$		0			0
$221$	−143.557			−0.649581
$222$		0			0
$223$		−	4.66059i		−	0.0208995i	−0.999945	−	0.0104497i	$-0.996674\pi$
$223$		−	4.66059i		−	0.0208995i	0.999945	−	0.0104497i	$-0.00332632\pi$
$224$		0			0
$225$	−32.5162			−0.144516
$226$		0			0
$227$			234.981i			1.03516i	0.855636	+	0.517579i	$0.173166\pi$
$227$			234.981i			1.03516i	−0.855636	+	0.517579i	$0.826834\pi$
$228$		0			0
$229$		−	278.190i		−	1.21480i	−0.794395	−	0.607402i	$-0.792212\pi$
$229$		−	278.190i		−	1.21480i	0.794395	−	0.607402i	$-0.207788\pi$
$230$		0			0
$231$	−215.779	+	93.3931i	−0.934107	+	0.404299i
$232$		0			0
$233$	−209.817			−0.900504			−0.450252	−	0.892902i	$-0.648666\pi$
$233$	−209.817			−0.900504			−0.450252	+	0.892902i	$0.648666\pi$
$234$		0			0
$235$	294.463			1.25303
$236$		0			0
$237$			104.855i			0.442426i
$238$		0			0
$239$	325.282			1.36101			0.680507	−	0.732741i	$-0.261759\pi$
$239$	325.282			1.36101			0.680507	+	0.732741i	$0.261759\pi$
$240$		0			0
$241$		−	125.858i		−	0.522231i	−0.965308	−	0.261115i	$-0.915910\pi$
$241$		−	125.858i		−	0.522231i	0.965308	−	0.261115i	$-0.0840903\pi$
$242$		0			0
$243$		−	15.5885i		−	0.0641500i
$244$		0			0
$245$	134.435	+	126.208i	0.548713	+	0.515136i
$246$		0			0
$247$	−40.3156			−0.163221
$248$		0			0
$249$	65.5277			0.263164
$250$		0			0
$251$			267.651i			1.06634i	0.846008	+	0.533170i	$0.178999\pi$
$251$			267.651i			1.06634i	−0.846008	+	0.533170i	$0.821001\pi$
$252$		0			0
$253$	−454.911			−1.79807
$254$		0			0
$255$			90.0055i			0.352963i
$256$		0			0
$257$		−	508.452i		−	1.97841i	−0.146531	−	0.989206i	$-0.546811\pi$
$257$		−	508.452i		−	1.97841i	0.146531	−	0.989206i	$-0.453189\pi$
$258$		0			0
$259$	−153.749	−	355.228i	−0.593626	−	1.37154i
$260$		0			0
$261$	−80.3362			−0.307801
$262$		0			0
$263$	−265.869			−1.01091			−0.505454	−	0.862854i	$-0.668675\pi$
$263$	−265.869			−1.01091			−0.505454	+	0.862854i	$0.668675\pi$
$264$		0			0
$265$		−	24.7875i		−	0.0935376i
$266$		0			0
$267$	4.09803			0.0153484
$268$		0			0
$269$		−	201.938i		−	0.750698i	−0.926883	−	0.375349i	$-0.877523\pi$
$269$		−	201.938i		−	0.750698i	0.926883	−	0.375349i	$-0.122477\pi$
$270$		0			0
$271$		−	302.239i		−	1.11527i	−0.830085	−	0.557637i	$-0.811708\pi$
$271$		−	302.239i		−	1.11527i	0.830085	−	0.557637i	$-0.188292\pi$
$272$		0			0
$273$	50.0665	+	115.675i	0.183394	+	0.423719i
$274$		0			0
$275$	210.191			0.764331
$276$		0			0
$277$	359.698			1.29855			0.649274	−	0.760554i	$-0.275073\pi$
$277$	359.698			1.29855			0.649274	+	0.760554i	$0.275073\pi$
$278$		0			0
$279$			153.127i			0.548843i
$280$		0			0
$281$	533.204			1.89752			0.948762	−	0.315992i	$-0.102337\pi$
$281$	533.204			1.89752			0.948762	+	0.315992i	$0.102337\pi$
$282$		0			0
$283$			514.640i			1.81852i	0.416233	+	0.909258i	$0.363350\pi$
$283$			514.640i			1.81852i	−0.416233	+	0.909258i	$0.636650\pi$
$284$		0			0
$285$			25.2765i			0.0886894i
$286$		0			0
$287$	29.4923	−	12.7648i	0.102761	−	0.0444767i
$288$		0			0
$289$	98.3158			0.340193
$290$		0			0
$291$	227.874			0.783073
$292$		0			0
$293$			393.368i			1.34255i	0.741208	+	0.671276i	$0.234253\pi$
$293$			393.368i			1.34255i	−0.741208	+	0.671276i	$0.765747\pi$
$294$		0			0
$295$	−353.643			−1.19879
$296$		0			0
$297$			100.767i			0.339283i
$298$		0			0
$299$			243.870i			0.815620i
$300$		0			0
$301$	−21.3566	−	49.3430i	−0.0709520	−	0.163930i
$302$		0			0
$303$	27.1698			0.0896693
$304$		0			0
$305$	127.739			0.418816
$306$		0			0
$307$			149.373i			0.486557i	0.969957	+	0.243278i	$0.0782229\pi$
$307$			149.373i			0.486557i	−0.969957	+	0.243278i	$0.921777\pi$
$308$		0			0
$309$	275.040			0.890097
$310$		0			0
$311$		−	198.689i		−	0.638873i	−0.947608	−	0.319436i	$-0.896506\pi$
$311$		−	198.689i		−	0.638873i	0.947608	−	0.319436i	$-0.103494\pi$
$312$		0			0
$313$			547.057i			1.74779i	0.486118	+	0.873893i	$0.338412\pi$
$313$			547.057i			1.74779i	−0.486118	+	0.873893i	$0.661588\pi$
$314$		0			0
$315$	72.5245	−	31.3899i	0.230236	−	0.0996506i
$316$		0			0
$317$	182.012			0.574171			0.287086	−	0.957905i	$-0.407314\pi$
$317$	182.012			0.574171			0.287086	+	0.957905i	$0.407314\pi$
$318$		0			0
$319$	519.309			1.62793
$320$		0			0
$321$			94.3730i			0.293997i
$322$		0			0
$323$	−53.5503			−0.165790
$324$		0			0
$325$		−	112.680i		−	0.346707i
$326$		0			0
$327$			80.3416i			0.245693i
$328$		0			0
$329$	502.680	−	217.569i	1.52790	−	0.661305i
$330$		0			0
$331$	94.5420			0.285625			0.142813	−	0.989750i	$-0.454385\pi$
$331$	94.5420			0.285625			0.142813	+	0.989750i	$0.454385\pi$
$332$		0			0
$333$	−165.888			−0.498163
$334$		0			0
$335$			419.519i			1.25230i
$336$		0			0
$337$	−41.6805			−0.123681			−0.0618406	−	0.998086i	$-0.519697\pi$
$337$	−41.6805			−0.123681			−0.0618406	+	0.998086i	$0.519697\pi$
$338$		0			0
$339$		−	75.0402i		−	0.221357i
$340$		0			0
$341$		−	989.845i		−	2.90277i
$342$		0			0
$343$	322.746	+	116.122i	0.940949	+	0.338547i
$344$		0			0
$345$	152.898			0.443183
$346$		0			0
$347$	266.888			0.769129			0.384564	−	0.923098i	$-0.374352\pi$
$347$	266.888			0.769129			0.384564	+	0.923098i	$0.374352\pi$
$348$		0			0
$349$		−	121.506i		−	0.348154i	−0.984732	−	0.174077i	$-0.944306\pi$
$349$		−	121.506i		−	0.348154i	0.984732	−	0.174077i	$-0.0556941\pi$
$350$		0			0
$351$	54.0195			0.153902
$352$		0			0
$353$		−	655.237i		−	1.85619i	−0.372337	−	0.928097i	$-0.621444\pi$
$353$		−	655.237i		−	1.85619i	0.372337	−	0.928097i	$-0.378556\pi$
$354$		0			0
$355$			122.926i			0.346271i
$356$		0			0
$357$	66.5022	+	153.649i	0.186281	+	0.430390i
$358$		0			0
$359$	220.525			0.614277			0.307138	−	0.951665i	$-0.400629\pi$
$359$	220.525			0.614277			0.307138	+	0.951665i	$0.400629\pi$
$360$		0			0
$361$	345.961			0.958342
$362$		0			0
$363$		−	441.800i		−	1.21708i
$364$		0			0
$365$	−287.177			−0.786785
$366$		0			0
$367$			324.086i			0.883068i	0.897244	+	0.441534i	$0.145566\pi$
$367$			324.086i			0.883068i	−0.897244	+	0.441534i	$0.854434\pi$
$368$		0			0
$369$		−	13.7727i		−	0.0373243i
$370$		0			0
$371$	−18.3147	−	42.3148i	−0.0493656	−	0.114056i
$372$		0			0
$373$	−108.432			−0.290702			−0.145351	−	0.989380i	$-0.546431\pi$
$373$	−108.432			−0.290702			−0.145351	+	0.989380i	$0.546431\pi$
$374$		0			0
$375$	−233.595			−0.622921
$376$		0			0
$377$		−	278.393i		−	0.738443i
$378$		0			0
$379$	10.3008			0.0271788			0.0135894	−	0.999908i	$-0.495674\pi$
$379$	10.3008			0.0271788			0.0135894	+	0.999908i	$0.495674\pi$
$380$		0			0
$381$			260.362i			0.683365i
$382$		0			0
$383$			60.5108i			0.157992i	0.996875	+	0.0789958i	$0.0251714\pi$
$383$			60.5108i			0.157992i	−0.996875	+	0.0789958i	$0.974829\pi$
$384$		0			0
$385$	−468.813	+	202.911i	−1.21770	+	0.527041i
$386$		0			0
$387$	−23.0428			−0.0595420
$388$		0			0
$389$	237.516			0.610581			0.305290	−	0.952259i	$-0.401246\pi$
$389$	237.516			0.610581			0.305290	+	0.952259i	$0.401246\pi$
$390$		0			0
$391$			323.927i			0.828459i
$392$		0			0
$393$	−348.788			−0.887500
$394$		0			0
$395$			227.813i			0.576743i
$396$		0			0
$397$		−	246.791i		−	0.621640i	−0.950469	−	0.310820i	$-0.899396\pi$
$397$		−	246.791i		−	0.621640i	0.950469	−	0.310820i	$-0.100604\pi$
$398$		0			0
$399$	18.6760	+	43.1497i	0.0468070	+	0.108144i
$400$		0			0
$401$	−292.374			−0.729112			−0.364556	−	0.931181i	$-0.618779\pi$
$401$	−292.374			−0.729112			−0.364556	+	0.931181i	$0.618779\pi$
$402$		0			0
$403$	−530.639			−1.31672
$404$		0			0
$405$		−	33.8683i		−	0.0836255i
$406$		0			0
$407$	1072.34			2.63473
$408$		0			0
$409$			537.456i			1.31407i	0.753859	+	0.657037i	$0.228190\pi$
$409$			537.456i			1.31407i	−0.753859	+	0.657037i	$0.771810\pi$
$410$		0			0
$411$			9.76230i			0.0237526i
$412$		0			0
$413$	−603.707	+	261.296i	−1.46176	+	0.632677i
$414$		0			0
$415$	142.369			0.343058
$416$		0			0
$417$	−140.879			−0.337839
$418$		0			0
$419$		−	369.290i		−	0.881361i	−0.897664	−	0.440680i	$-0.854737\pi$
$419$		−	369.290i		−	0.881361i	0.897664	−	0.440680i	$-0.145263\pi$
$420$		0			0
$421$	1.84724			0.00438774			0.00219387	−	0.999998i	$-0.499302\pi$
$421$	1.84724			0.00438774			0.00219387	+	0.999998i	$0.499302\pi$
$422$		0			0
$423$		−	234.747i		−	0.554958i
$424$		0			0
$425$		−	149.670i		−	0.352165i
$426$		0			0
$427$	218.064	−	94.3822i	0.510688	−	0.221036i
$428$		0			0
$429$	−349.193			−0.813969
$430$		0			0
$431$	473.957			1.09967			0.549834	−	0.835274i	$-0.314691\pi$
$431$	473.957			1.09967			0.549834	+	0.835274i	$0.314691\pi$
$432$		0			0
$433$			115.253i			0.266174i	0.991104	+	0.133087i	$0.0424889\pi$
$433$			115.253i			0.266174i	−0.991104	+	0.133087i	$0.957511\pi$
$434$		0			0
$435$	−174.543			−0.401248
$436$		0			0
$437$			90.9694i			0.208168i
$438$		0			0
$439$		−	274.329i		−	0.624894i	−0.949935	−	0.312447i	$-0.898851\pi$
$439$		−	274.329i		−	0.624894i	0.949935	−	0.312447i	$-0.101149\pi$
$440$		0			0
$441$	100.614	−	107.172i	0.228150	−	0.243020i
$442$		0			0
$443$	171.563			0.387275			0.193637	−	0.981073i	$-0.437971\pi$
$443$	171.563			0.387275			0.193637	+	0.981073i	$0.437971\pi$
$444$		0			0
$445$	8.90360			0.0200081
$446$		0			0
$447$		−	97.5547i		−	0.218243i
$448$		0			0
$449$	−380.428			−0.847278			−0.423639	−	0.905831i	$-0.639248\pi$
$449$	−380.428			−0.847278			−0.423639	+	0.905831i	$0.639248\pi$
$450$		0			0
$451$			89.0293i			0.197404i
$452$		0			0
$453$		−	114.076i		−	0.251824i
$454$		0			0
$455$	108.777	+	251.323i	0.239071	+	0.552357i
$456$		0			0
$457$	150.525			0.329375			0.164688	−	0.986346i	$-0.447338\pi$
$457$	150.525			0.329375			0.164688	+	0.986346i	$0.447338\pi$
$458$		0			0
$459$	71.7529			0.156324
$460$		0			0
$461$		−	17.3349i		−	0.0376029i	−0.999823	−	0.0188014i	$-0.994015\pi$
$461$		−	17.3349i		−	0.0376029i	0.999823	−	0.0188014i	$-0.00598504\pi$
$462$		0			0
$463$	−79.1458			−0.170941			−0.0854706	−	0.996341i	$-0.527239\pi$
$463$	−79.1458			−0.170941			−0.0854706	+	0.996341i	$0.527239\pi$
$464$		0			0
$465$			332.692i			0.715468i
$466$		0			0
$467$			470.490i			1.00747i	0.863857	+	0.503736i	$0.168042\pi$
$467$			470.490i			1.00747i	−0.863857	+	0.503736i	$0.831958\pi$
$468$		0			0
$469$	309.969	+	716.165i	0.660916	+	1.52700i
$470$		0			0
$471$	465.229			0.987747
$472$		0			0
$473$	148.953			0.314911
$474$		0			0
$475$		−	42.0323i		−	0.0884890i
$476$		0			0
$477$	−19.7607			−0.0414270
$478$		0			0
$479$			245.422i			0.512364i	0.966629	+	0.256182i	$0.0824646\pi$
$479$			245.422i			0.512364i	−0.966629	+	0.256182i	$0.917535\pi$
$480$		0			0
$481$		−	574.861i		−	1.19514i
$482$		0			0
$483$	261.014	−	112.972i	0.540401	−	0.233896i
$484$		0			0
$485$	495.092			1.02081
$486$		0			0
$487$	748.094			1.53613			0.768064	−	0.640373i	$-0.221220\pi$
$487$	748.094			1.53613			0.768064	+	0.640373i	$0.221220\pi$
$488$		0			0
$489$			321.889i			0.658259i
$490$		0			0
$491$	528.089			1.07554			0.537769	−	0.843092i	$-0.319267\pi$
$491$	528.089			1.07554			0.537769	+	0.843092i	$0.319267\pi$
$492$		0			0
$493$		−	369.783i		−	0.750068i
$494$		0			0
$495$			218.932i			0.442286i
$496$		0			0
$497$	90.8262	+	209.848i	0.182749	+	0.422230i
$498$		0			0
$499$	174.083			0.348863			0.174431	−	0.984669i	$-0.444191\pi$
$499$	174.083			0.348863			0.174431	+	0.984669i	$0.444191\pi$
$500$		0			0
$501$	−128.139			−0.255766
$502$		0			0
$503$		−	645.875i		−	1.28404i	−0.766686	−	0.642022i	$-0.778096\pi$
$503$		−	645.875i		−	1.28404i	0.766686	−	0.642022i	$-0.221904\pi$
$504$		0			0
$505$	59.0306			0.116892
$506$		0			0
$507$		−	105.520i		−	0.208127i
$508$		0			0
$509$		−	293.593i		−	0.576803i	−0.957510	−	0.288401i	$-0.906876\pi$
$509$		−	293.593i		−	0.576803i	0.957510	−	0.288401i	$-0.0931237\pi$
$510$		0			0
$511$	−490.241	+	212.185i	−0.959376	+	0.415236i
$512$		0			0
$513$	20.1505			0.0392798
$514$		0			0
$515$	597.567			1.16032
$516$		0			0
$517$			1517.46i			2.93512i
$518$		0			0
$519$	−270.719			−0.521617
$520$		0			0
$521$			324.494i			0.622829i	0.950274	+	0.311415i	$0.100803\pi$
$521$			324.494i			0.622829i	−0.950274	+	0.311415i	$0.899197\pi$
$522$		0			0
$523$			162.076i			0.309897i	0.987923	+	0.154949i	$0.0495212\pi$
$523$			162.076i			0.309897i	−0.987923	+	0.154949i	$0.950479\pi$
$524$		0			0
$525$	−120.601	+	52.1983i	−0.229716	+	0.0994254i
$526$		0			0
$527$	−704.837			−1.33745
$528$		0			0
$529$	21.2764			0.0402201
$530$		0			0
$531$			281.926i			0.530934i
$532$		0			0
$533$	47.7271			0.0895443
$534$		0			0
$535$			205.040i			0.383252i
$536$		0			0
$537$		−	12.9486i		−	0.0241129i
$538$		0			0
$539$	−650.389	+	692.781i	−1.20666	+	1.28531i
$540$		0			0
$541$	−440.049			−0.813400			−0.406700	−	0.913562i	$-0.633321\pi$
$541$	−440.049			−0.813400			−0.406700	+	0.913562i	$0.633321\pi$
$542$		0			0
$543$	344.683			0.634775
$544$		0			0
$545$			174.554i			0.320283i
$546$		0			0
$547$	1067.12			1.95085			0.975427	−	0.220321i	$-0.0707106\pi$
$547$	1067.12			1.95085			0.975427	+	0.220321i	$0.0707106\pi$
$548$		0			0
$549$		−	101.834i		−	0.185490i
$550$		0			0
$551$		−	103.847i		−	0.188470i
$552$		0			0
$553$	168.324	+	388.902i	0.304383	+	0.703258i
$554$		0			0
$555$	−360.418			−0.649402
$556$		0			0
$557$	118.784			0.213257			0.106628	−	0.994299i	$-0.465994\pi$
$557$	118.784			0.213257			0.106628	+	0.994299i	$0.465994\pi$
$558$		0			0
$559$		−	79.8512i		−	0.142847i
$560$		0			0
$561$	−463.825			−0.826783
$562$		0			0
$563$		−	939.344i		−	1.66846i	−0.551415	−	0.834231i	$-0.685912\pi$
$563$		−	939.344i		−	1.66846i	0.551415	−	0.834231i	$-0.314088\pi$
$564$		0			0
$565$		−	163.036i		−	0.288560i
$566$		0			0
$567$	−25.0242	−	57.8169i	−0.0441344	−	0.101970i
$568$		0			0
$569$	658.156			1.15669			0.578345	−	0.815793i	$-0.303699\pi$
$569$	658.156			1.15669			0.578345	+	0.815793i	$0.303699\pi$
$570$		0			0
$571$	609.733			1.06783			0.533916	−	0.845537i	$-0.320720\pi$
$571$	609.733			1.06783			0.533916	+	0.845537i	$0.320720\pi$
$572$		0			0
$573$			399.066i			0.696451i
$574$		0			0
$575$	−254.254			−0.442182
$576$		0			0
$577$		−	228.215i		−	0.395521i	−0.980250	−	0.197760i	$-0.936633\pi$
$577$		−	228.215i		−	0.395521i	0.980250	−	0.197760i	$-0.0633668\pi$
$578$		0			0
$579$			667.271i			1.15245i
$580$		0			0
$581$	243.039	−	105.192i	0.418312	−	0.181053i
$582$		0			0
$583$	127.737			0.219103
$584$		0			0
$585$	117.366			0.200625
$586$		0			0
$587$		−	469.966i		−	0.800624i	−0.916379	−	0.400312i	$-0.868902\pi$
$587$		−	469.966i		−	0.800624i	0.916379	−	0.400312i	$-0.131098\pi$
$588$		0			0
$589$	−197.941			−0.336063
$590$		0			0
$591$		−	564.078i		−	0.954446i
$592$		0			0
$593$		−	324.759i		−	0.547654i	−0.961779	−	0.273827i	$-0.911710\pi$
$593$		−	324.759i		−	0.547654i	0.961779	−	0.273827i	$-0.0882896\pi$
$594$		0			0
$595$	144.486	+	333.826i	0.242834	+	0.561053i
$596$		0			0
$597$	93.7691			0.157067
$598$		0			0
$599$	−439.936			−0.734451			−0.367225	−	0.930132i	$-0.619692\pi$
$599$	−439.936			−0.734451			−0.367225	+	0.930132i	$0.619692\pi$
$600$		0			0
$601$			762.072i			1.26801i	0.773330	+	0.634003i	$0.218589\pi$
$601$			762.072i			1.26801i	−0.773330	+	0.634003i	$0.781411\pi$
$602$		0			0
$603$	334.443			0.554632
$604$		0			0
$605$		−	959.878i		−	1.58658i
$606$		0			0
$607$			96.1044i			0.158327i	0.996862	+	0.0791634i	$0.0252249\pi$
$607$			96.1044i			0.158327i	−0.996862	+	0.0791634i	$0.974775\pi$
$608$		0			0
$609$	−297.963	+	128.964i	−0.489266	+	0.211764i
$610$		0			0
$611$	813.482			1.33139
$612$		0			0
$613$	99.6647			0.162585			0.0812926	−	0.996690i	$-0.474095\pi$
$613$	99.6647			0.162585			0.0812926	+	0.996690i	$0.474095\pi$
$614$		0			0
$615$		−	29.9232i		−	0.0486557i
$616$		0			0
$617$	−492.064			−0.797510			−0.398755	−	0.917058i	$-0.630558\pi$
$617$	−492.064			−0.797510			−0.398755	+	0.917058i	$0.630558\pi$
$618$		0			0
$619$		−	1057.98i		−	1.70917i	−0.519309	−	0.854587i	$-0.673810\pi$
$619$		−	1057.98i		−	1.70917i	0.519309	−	0.854587i	$-0.326190\pi$
$620$		0			0
$621$		−	121.891i		−	0.196282i
$622$		0			0
$623$	15.1994	−	6.57858i	0.0243971	−	0.0105595i
$624$		0			0
$625$	−236.554			−0.378487
$626$		0			0
$627$	−130.257			−0.207747
$628$		0			0
$629$		−	763.576i		−	1.21395i
$630$		0			0
$631$	−689.786			−1.09316			−0.546582	−	0.837406i	$-0.684071\pi$
$631$	−689.786			−1.09316			−0.546582	+	0.837406i	$0.684071\pi$
$632$		0			0
$633$		−	493.707i		−	0.779948i
$634$		0			0
$635$			565.677i			0.890830i
$636$		0			0
$637$	371.388	+	348.663i	0.583027	+	0.547351i
$638$		0			0
$639$	97.9974			0.153361
$640$		0			0
$641$	−298.810			−0.466162			−0.233081	−	0.972457i	$-0.574881\pi$
$641$	−298.810			−0.466162			−0.233081	+	0.972457i	$0.574881\pi$
$642$		0			0
$643$			635.333i			0.988076i	0.869440	+	0.494038i	$0.164480\pi$
$643$			635.333i			0.988076i	−0.869440	+	0.494038i	$0.835520\pi$
$644$		0			0
$645$	−50.0639			−0.0776185
$646$		0			0
$647$		−	135.921i		−	0.210078i	−0.994468	−	0.105039i	$-0.966503\pi$
$647$		−	135.921i		−	0.210078i	0.994468	−	0.105039i	$-0.0334968\pi$
$648$		0			0
$649$		−	1822.43i		−	2.80806i
$650$		0			0
$651$	245.816	+	567.942i	0.377597	+	0.872414i
$652$		0			0
$653$	893.314			1.36802			0.684008	−	0.729475i	$-0.260235\pi$
$653$	893.314			1.36802			0.684008	+	0.729475i	$0.260235\pi$
$654$		0			0
$655$	−757.795			−1.15694
$656$		0			0
$657$			228.939i			0.348461i
$658$		0			0
$659$	−780.417			−1.18424			−0.592122	−	0.805848i	$-0.701710\pi$
$659$	−780.417			−1.18424			−0.592122	+	0.805848i	$0.701710\pi$
$660$		0			0
$661$			158.047i			0.239103i	0.992828	+	0.119552i	$0.0381457\pi$
$661$			158.047i			0.239103i	−0.992828	+	0.119552i	$0.961854\pi$
$662$		0			0
$663$			248.649i			0.375036i
$664$		0			0
$665$	40.5764	+	93.7493i	0.0610172	+	0.140976i
$666$		0			0
$667$	−628.175			−0.941791
$668$		0			0
$669$	−8.07237			−0.0120663
$670$		0			0
$671$			658.276i			0.981038i
$672$		0			0
$673$	−343.166			−0.509905			−0.254953	−	0.966954i	$-0.582060\pi$
$673$	−343.166			−0.509905			−0.254953	+	0.966954i	$0.582060\pi$
$674$		0			0
$675$			56.3196i			0.0834365i
$676$		0			0
$677$			392.542i			0.579825i	0.957053	+	0.289913i	$0.0936262\pi$
$677$			392.542i			0.579825i	−0.957053	+	0.289913i	$0.906374\pi$
$678$		0			0
$679$	845.175	−	365.807i	1.24473	−	0.538744i
$680$		0			0
$681$	406.998			0.597648
$682$		0			0
$683$	−785.575			−1.15018			−0.575091	−	0.818089i	$-0.695033\pi$
$683$	−785.575			−1.15018			−0.575091	+	0.818089i	$0.695033\pi$
$684$		0			0
$685$			21.2101i			0.0309637i
$686$		0			0
$687$	−481.839			−0.701367
$688$		0			0
$689$		−	68.4777i		−	0.0993871i
$690$		0			0
$691$		−	614.068i		−	0.888666i	−0.895862	−	0.444333i	$-0.853441\pi$
$691$		−	614.068i		−	0.888666i	0.895862	−	0.444333i	$-0.146559\pi$
$692$		0			0
$693$	161.762	+	373.740i	0.233422	+	0.539307i
$694$		0			0
$695$	−306.081			−0.440404
$696$		0			0
$697$	63.3949			0.0909539
$698$		0			0
$699$			363.414i			0.519906i
$700$		0			0
$701$	−873.078			−1.24548			−0.622738	−	0.782431i	$-0.713980\pi$
$701$	−873.078			−1.24548			−0.622738	+	0.782431i	$0.713980\pi$
$702$		0			0
$703$		−	214.437i		−	0.305031i
$704$		0			0
$705$		−	510.025i		−	0.723440i
$706$		0			0
$707$	100.771	−	43.6158i	0.142534	−	0.0616913i
$708$		0			0
$709$	−597.884			−0.843277			−0.421639	−	0.906764i	$-0.638545\pi$
$709$	−597.884			−0.843277			−0.421639	+	0.906764i	$0.638545\pi$
$710$		0			0
$711$	181.614			0.255435
$712$		0			0
$713$			1197.35i			1.67931i
$714$		0			0
$715$	−758.675			−1.06108
$716$		0			0
$717$		−	563.406i		−	0.785782i
$718$		0			0
$719$			862.986i			1.20026i	0.799903	+	0.600129i	$0.204884\pi$
$719$			862.986i			1.20026i	−0.799903	+	0.600129i	$0.795116\pi$
$720$		0			0
$721$	1020.11	−	441.523i	1.41485	−	0.612376i
$722$		0			0
$723$	−217.992			−0.301510
$724$		0			0
$725$	290.247			0.400341
$726$		0			0
$727$			259.735i			0.357270i	0.983915	+	0.178635i	$0.0571681\pi$
$727$			259.735i			0.357270i	−0.983915	+	0.178635i	$0.942832\pi$
$728$		0			0
$729$	−27.0000			−0.0370370
$730$		0			0
$731$		−	106.065i		−	0.145095i
$732$		0			0
$733$		−	216.829i		−	0.295810i	−0.989002	−	0.147905i	$-0.952747\pi$
$733$		−	216.829i		−	0.295810i	0.989002	−	0.147905i	$-0.0472531\pi$
$734$		0			0
$735$	218.599	−	232.848i	0.297414	−	0.316799i
$736$		0			0
$737$	−2161.91			−2.93339
$738$		0			0
$739$	−596.194			−0.806758			−0.403379	−	0.915033i	$-0.632164\pi$
$739$	−596.194			−0.806758			−0.403379	+	0.915033i	$0.632164\pi$
$740$		0			0
$741$			69.8287i			0.0942357i
$742$		0			0
$743$	307.869			0.414359			0.207179	−	0.978303i	$-0.433572\pi$
$743$	307.869			0.414359			0.207179	+	0.978303i	$0.433572\pi$
$744$		0			0
$745$		−	211.953i		−	0.284500i
$746$		0			0
$747$		−	113.497i		−	0.151938i
$748$		0			0
$749$	151.497	+	350.025i	0.202266	+	0.467323i
$750$		0			0
$751$	967.485			1.28826			0.644131	−	0.764915i	$-0.277219\pi$
$751$	967.485			1.28826			0.644131	+	0.764915i	$0.277219\pi$
$752$		0			0
$753$	463.586			0.615651
$754$		0			0
$755$		−	247.848i		−	0.328276i
$756$		0			0
$757$	1164.53			1.53834			0.769171	−	0.639042i	$-0.220669\pi$
$757$	1164.53			1.53834			0.769171	+	0.639042i	$0.220669\pi$
$758$		0			0
$759$			787.929i			1.03812i
$760$		0			0
$761$		−	1226.50i		−	1.61170i	−0.592119	−	0.805851i	$-0.701708\pi$
$761$		−	1226.50i		−	1.61170i	0.592119	−	0.805851i	$-0.298292\pi$
$762$		0			0
$763$	128.973	+	297.983i	0.169034	+	0.390541i
$764$		0			0
$765$	155.894			0.203783
$766$		0			0
$767$	−976.973			−1.27376
$768$		0			0
$769$			835.523i			1.08651i	0.839569	+	0.543253i	$0.182807\pi$
$769$			835.523i			1.08651i	−0.839569	+	0.543253i	$0.817193\pi$
$770$		0			0
$771$	−880.665			−1.14224
$772$		0			0
$773$		−	534.533i		−	0.691505i	−0.938326	−	0.345752i	$-0.887624\pi$
$773$		−	534.533i		−	0.691505i	0.938326	−	0.345752i	$-0.112376\pi$
$774$		0			0
$775$		−	553.234i		−	0.713851i
$776$		0			0
$777$	−615.272	+	266.301i	−0.791856	+	0.342730i
$778$		0			0
$779$	17.8033			0.0228541
$780$		0			0
$781$	−633.475			−0.811108
$782$		0			0
$783$			139.146i			0.177709i
$784$		0			0
$785$	1010.78			1.28762
$786$		0			0
$787$			199.536i			0.253540i	0.991932	+	0.126770i	$0.0404610\pi$
$787$			199.536i			0.253540i	−0.991932	+	0.126770i	$0.959539\pi$
$788$		0			0
$789$			460.498i			0.583648i
$790$		0			0
$791$	−120.462	−	278.320i	−0.152291	−	0.351859i
$792$		0			0
$793$	352.891			0.445007
$794$		0			0
$795$	−42.9331			−0.0540039
$796$		0			0
$797$			451.538i			0.566546i	0.959039	+	0.283273i	$0.0914203\pi$
$797$			451.538i			0.566546i	−0.959039	+	0.283273i	$0.908580\pi$
$798$		0			0
$799$	1080.53			1.35235
$800$		0			0
$801$		−	7.09799i		−	0.00886141i
$802$		0			0
$803$		−	1479.91i		−	1.84297i
$804$		0			0
$805$	567.092	−	245.448i	0.704462	−	0.304904i
$806$		0			0
$807$	−349.767			−0.433416
$808$		0			0
$809$	665.841			0.823042			0.411521	−	0.911400i	$-0.364998\pi$
$809$	665.841			0.823042			0.411521	+	0.911400i	$0.364998\pi$
$810$		0			0
$811$			84.5863i			0.104299i	0.998639	+	0.0521494i	$0.0166072\pi$
$811$			84.5863i			0.104299i	−0.998639	+	0.0521494i	$0.983393\pi$
$812$		0			0
$813$	−523.494			−0.643904
$814$		0			0
$815$			699.353i			0.858102i
$816$		0			0
$817$		−	29.7864i		−	0.0364583i
$818$		0			0
$819$	200.356	−	86.7177i	0.244634	−	0.105882i
$820$		0			0
$821$	91.2306			0.111121			0.0555607	−	0.998455i	$-0.482305\pi$
$821$	91.2306			0.111121			0.0555607	+	0.998455i	$0.482305\pi$
$822$		0			0
$823$	−1061.69			−1.29002			−0.645012	−	0.764173i	$-0.723148\pi$
$823$	−1061.69			−1.29002			−0.645012	+	0.764173i	$0.723148\pi$
$824$		0			0
$825$		−	364.062i		−	0.441287i
$826$		0			0
$827$	385.891			0.466615			0.233308	−	0.972403i	$-0.425045\pi$
$827$	385.891			0.466615			0.233308	+	0.972403i	$0.425045\pi$
$828$		0			0
$829$		−	135.122i		−	0.162994i	−0.996674	−	0.0814972i	$-0.974030\pi$
$829$		−	135.122i		−	0.162994i	0.996674	−	0.0814972i	$-0.0259702\pi$
$830$		0			0
$831$		−	623.015i		−	0.749717i
$832$		0			0
$833$	493.307	+	463.121i	0.592205	+	0.555967i
$834$		0			0
$835$	−278.402			−0.333415
$836$		0			0
$837$	265.224			0.316875
$838$		0			0
$839$		−	1376.59i		−	1.64075i	−0.571825	−	0.820376i	$-0.693764\pi$
$839$		−	1376.59i		−	1.64075i	0.571825	−	0.820376i	$-0.306236\pi$
$840$		0			0
$841$	−123.900			−0.147324
$842$		0			0
$843$		−	923.537i		−	1.09554i
$844$		0			0
$845$		−	229.259i		−	0.271312i
$846$		0			0
$847$	−709.223	−	1638.61i	−0.837336	−	1.93461i
$848$		0			0
$849$	891.383			1.04992
$850$		0			0
$851$	−1297.14			−1.52425
$852$		0			0
$853$		−	1172.84i		−	1.37496i	−0.726203	−	0.687480i	$-0.758717\pi$
$853$		−	1172.84i		−	1.37496i	0.726203	−	0.687480i	$-0.241283\pi$
$854$		0			0
$855$	43.7801			0.0512049
$856$		0			0
$857$			903.580i			1.05435i	0.849756	+	0.527176i	$0.176749\pi$
$857$			903.580i			1.05435i	−0.849756	+	0.527176i	$0.823251\pi$
$858$		0			0
$859$		−	16.2939i		−	0.0189684i	−0.999955	−	0.00948421i	$-0.996981\pi$
$859$		−	16.2939i		−	0.0189684i	0.999955	−	0.00948421i	$-0.00301896\pi$
$860$		0			0
$861$	−22.1093	−	51.0822i	−0.0256787	−	0.0593289i
$862$		0			0
$863$	27.5419			0.0319142			0.0159571	−	0.999873i	$-0.494920\pi$
$863$	27.5419			0.0319142			0.0159571	+	0.999873i	$0.494920\pi$
$864$		0			0
$865$	−588.180			−0.679977
$866$		0			0
$867$		−	170.288i		−	0.196411i
$868$		0			0
$869$	−1173.99			−1.35097
$870$		0			0
$871$			1158.96i			1.33061i
$872$		0			0
$873$		−	394.690i		−	0.452107i
$874$		0			0
$875$	−866.395	+	374.992i	−0.990165	+	0.428562i
$876$		0			0
$877$	−951.041			−1.08443			−0.542213	−	0.840241i	$-0.682413\pi$
$877$	−951.041			−1.08443			−0.542213	+	0.840241i	$0.682413\pi$
$878$		0			0
$879$	681.333			0.775123
$880$		0			0
$881$			495.395i			0.562309i	0.959662	+	0.281155i	$0.0907174\pi$
$881$			495.395i			0.562309i	−0.959662	+	0.281155i	$0.909283\pi$
$882$		0			0
$883$	−1443.19			−1.63442			−0.817209	−	0.576342i	$-0.804480\pi$
$883$	−1443.19			−1.63442			−0.817209	+	0.576342i	$0.804480\pi$
$884$		0			0
$885$			612.528i			0.692122i
$886$		0			0
$887$		−	534.307i		−	0.602375i	−0.953565	−	0.301188i	$-0.902617\pi$
$887$		−	534.307i		−	0.602375i	0.953565	−	0.301188i	$-0.0973830\pi$
$888$		0			0
$889$	417.961	+	965.671i	0.470147	+	1.08624i
$890$		0			0
$891$	174.533			0.195885
$892$		0			0
$893$	303.448			0.339807
$894$		0			0
$895$		−	28.1329i		−	0.0314334i
$896$		0			0
$897$	422.396			0.470898
$898$		0			0
$899$		−	1366.85i		−	1.52041i
$900$		0			0
$901$		−	90.9574i		−	0.100952i
$902$		0			0
$903$	−85.4645	+	36.9906i	−0.0946451	+	0.0409642i
$904$		0			0
$905$	748.876			0.827488
$906$		0			0
$907$	420.676			0.463810			0.231905	−	0.972738i	$-0.425504\pi$
$907$	420.676			0.463810			0.231905	+	0.972738i	$0.425504\pi$
$908$		0			0
$909$		−	47.0595i		−	0.0517706i
$910$		0			0
$911$	495.015			0.543376			0.271688	−	0.962385i	$-0.412418\pi$
$911$	495.015			0.543376			0.271688	+	0.962385i	$0.412418\pi$
$912$		0			0
$913$			733.670i			0.803581i
$914$		0			0
$915$		−	221.250i		−	0.241804i
$916$		0			0
$917$	−1293.64	+	559.910i	−1.41073	+	0.610589i
$918$		0			0
$919$	−881.176			−0.958842			−0.479421	−	0.877585i	$-0.659153\pi$
$919$	−881.176			−0.958842			−0.479421	+	0.877585i	$0.659153\pi$
$920$		0			0
$921$	258.721			0.280914
$922$		0			0
$923$			339.595i			0.367926i
$924$		0			0
$925$	599.339			0.647934
$926$		0			0
$927$		−	476.383i		−	0.513898i
$928$		0			0
$929$		−	1229.74i		−	1.32373i	−0.749625	−	0.661863i	$-0.769766\pi$
$929$		−	1229.74i		−	1.32373i	0.749625	−	0.661863i	$-0.230234\pi$
$930$		0			0
$931$	138.537	+	130.059i	0.148804	+	0.139699i
$932$		0			0
$933$	−344.140			−0.368853
$934$		0			0
$935$	−1007.73			−1.07779
$936$		0			0
$937$		−	43.8963i		−	0.0468477i	−0.999726	−	0.0234239i	$-0.992543\pi$
$937$		−	43.8963i		−	0.0468477i	0.999726	−	0.0234239i	$-0.00745673\pi$
$938$		0			0
$939$	947.531			1.00908
$940$		0			0
$941$		−	533.698i		−	0.567161i	−0.958948	−	0.283580i	$-0.908478\pi$
$941$		−	533.698i		−	0.567161i	0.958948	−	0.283580i	$-0.0915222\pi$
$942$		0			0
$943$		−	107.693i		−	0.114202i
$944$		0			0
$945$	−54.3690	−	125.616i	−0.0575333	−	0.132927i
$946$		0			0
$947$	1563.48			1.65098			0.825491	−	0.564415i	$-0.190898\pi$
$947$	1563.48			1.65098			0.825491	+	0.564415i	$0.190898\pi$
$948$		0			0
$949$	−793.352			−0.835988
$950$		0			0
$951$		−	315.255i		−	0.331498i
$952$		0			0
$953$	844.464			0.886111			0.443055	−	0.896494i	$-0.353894\pi$
$953$	844.464			0.886111			0.443055	+	0.896494i	$0.353894\pi$
$954$		0			0
$955$			867.033i			0.907888i
$956$		0			0
$957$		−	899.470i		−	0.939885i
$958$		0			0
$959$	15.6715	+	36.2079i	0.0163415	+	0.0377559i
$960$		0			0
$961$	−1644.33			−1.71106
$962$		0			0
$963$	163.459			0.169739
$964$		0			0
$965$			1449.75i			1.50233i
$966$		0			0
$967$	−1436.89			−1.48593			−0.742965	−	0.669330i	$-0.766581\pi$
$967$	−1436.89			−1.48593			−0.742965	+	0.669330i	$0.766581\pi$
$968$		0			0
$969$			92.7519i			0.0957192i
$970$		0			0
$971$		−	906.910i		−	0.933996i	−0.884258	−	0.466998i	$-0.845335\pi$
$971$		−	906.910i		−	0.933996i	0.884258	−	0.466998i	$-0.154665\pi$
$972$		0			0
$973$	−522.513	+	226.153i	−0.537012	+	0.232429i
$974$		0			0
$975$	−195.167			−0.200172
$976$		0			0
$977$	−797.054			−0.815817			−0.407909	−	0.913023i	$-0.633742\pi$
$977$	−797.054			−0.815817			−0.407909	+	0.913023i	$0.633742\pi$
$978$		0			0
$979$			45.8829i			0.0468671i
$980$		0			0
$981$	139.156			0.141851
$982$		0			0
$983$			876.423i			0.891580i	0.895138	+	0.445790i	$0.147077\pi$
$983$			876.423i			0.891580i	−0.895138	+	0.445790i	$0.852923\pi$
$984$		0			0
$985$		−	1225.55i		−	1.24421i
$986$		0			0
$987$	−376.841	−	870.667i	−0.381805	−	0.882135i
$988$		0			0
$989$	−180.179			−0.182183
$990$		0			0
$991$	407.126			0.410823			0.205412	−	0.978676i	$-0.434147\pi$
$991$	407.126			0.410823			0.205412	+	0.978676i	$0.434147\pi$
$992$		0			0
$993$		−	163.751i		−	0.164906i
$994$		0			0
$995$	203.728			0.204751
$996$		0			0
$997$		−	1801.91i		−	1.80733i	−0.428240	−	0.903665i	$-0.640866\pi$
$997$		−	1801.91i		−	1.80733i	0.428240	−	0.903665i	$-0.359134\pi$
$998$		0			0
$999$			287.327i			0.287615i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.3.f.h.769.2		8
4.3	odd	2		1344.3.f.g.769.6		8
7.6	odd	2	inner	1344.3.f.h.769.7		8
8.3	odd	2		168.3.f.a.97.3	✓	8
8.5	even	2		336.3.f.d.97.7		8
24.5	odd	2		1008.3.f.j.433.2		8
24.11	even	2		504.3.f.c.433.2		8
28.27	even	2		1344.3.f.g.769.3		8
56.3	even	6		1176.3.z.f.313.3		8
56.11	odd	6		1176.3.z.a.313.2		8
56.13	odd	2		336.3.f.d.97.2		8
56.19	even	6		1176.3.z.a.913.2		8
56.27	even	2		168.3.f.a.97.6	yes	8
56.51	odd	6		1176.3.z.f.913.3		8
168.83	odd	2		504.3.f.c.433.7		8
168.125	even	2		1008.3.f.j.433.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.3.f.a.97.3	✓	8	8.3	odd	2
168.3.f.a.97.6	yes	8	56.27	even	2
336.3.f.d.97.2		8	56.13	odd	2
336.3.f.d.97.7		8	8.5	even	2
504.3.f.c.433.2		8	24.11	even	2
504.3.f.c.433.7		8	168.83	odd	2
1008.3.f.j.433.2		8	24.5	odd	2
1008.3.f.j.433.7		8	168.125	even	2
1176.3.z.a.313.2		8	56.11	odd	6
1176.3.z.a.913.2		8	56.19	even	6
1176.3.z.f.313.3		8	56.3	even	6
1176.3.z.f.913.3		8	56.51	odd	6
1344.3.f.g.769.3		8	28.27	even	2
1344.3.f.g.769.6		8	4.3	odd	2
1344.3.f.h.769.2		8	1.1	even	1	trivial
1344.3.f.h.769.7		8	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1344.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} -3.76315i q^{5} +(-2.78047 - 6.42409i) q^{7} -3.00000 q^{9} +19.3926 q^{11} -10.3961i q^{13} -6.51796 q^{15} -13.8088i q^{17} -3.87797i q^{19} +(-11.1269 + 4.81591i) q^{21} -23.4580 q^{23} +10.8387 q^{25} +5.19615i q^{27} +26.7787 q^{29} -51.0424i q^{31} -33.5890i q^{33} +(-24.1748 + 10.4633i) q^{35} +55.2961 q^{37} -18.0065 q^{39} +4.59089i q^{41} +7.68092 q^{43} +11.2894i q^{45} +78.2491i q^{47} +(-33.5380 + 35.7240i) q^{49} -23.9176 q^{51} +6.58689 q^{53} -72.9772i q^{55} -6.71685 q^{57} -93.9754i q^{59} +33.9447i q^{61} +(8.34141 + 19.2723i) q^{63} -39.1219 q^{65} -111.481 q^{67} +40.6304i q^{69} -32.6658 q^{71} -76.3129i q^{73} -18.7732i q^{75} +(-53.9205 - 124.580i) q^{77} -60.5380 q^{79} +9.00000 q^{81} +37.8324i q^{83} -51.9647 q^{85} -46.3821i q^{87} +2.36600i q^{89} +(-66.7852 + 28.9059i) q^{91} -88.4080 q^{93} -14.5934 q^{95} +131.563i q^{97} -58.1778 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{7} - 24 q^{9} - 16 q^{11} - 24 q^{15} - 12 q^{21} - 96 q^{23} + 64 q^{29} - 120 q^{35} + 40 q^{37} + 48 q^{39} + 136 q^{43} + 112 q^{49} - 24 q^{51} - 112 q^{53} - 72 q^{57} - 12 q^{63} - 208 q^{65}+ \cdots + 48 q^{99}+O(q^{100}) \) 8 * q + 4 * q^7 - 24 * q^9 - 16 * q^11 - 24 * q^15 - 12 * q^21 - 96 * q^23 + 64 * q^29 - 120 * q^35 + 40 * q^37 + 48 * q^39 + 136 * q^43 + 112 * q^49 - 24 * q^51 - 112 * q^53 - 72 * q^57 - 12 * q^63 - 208 * q^65 - 8 * q^67 - 16 * q^71 - 288 * q^77 - 104 * q^79 + 72 * q^81 + 424 * q^85 - 192 * q^91 - 144 * q^93 + 240 * q^95 + 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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