LMFDB - Embedded newform 1344.4.b.h.895.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(895,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1344.895");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$79.2985670477$
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} - x^{11} - 2x^{10} - 6x^{9} + 56x^{7} - 448x^{6} + 448x^{5} - 3072x^{3} - 8192x^{2} - 32768x + 262144 $ x^12 - x^11 - 2x^10 - 6x^9 + 56x^7 - 448x^6 + 448x^5 - 3072x^3 - 8192x^2 - 32768x + 262144
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{30} $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			895.4
Root			$-1.72458 + 2.24184i$ of defining polynomial
Character	$\chi$	$=$	1344.895
Dual form			1344.4.b.h.895.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -6.58775i q^{5} +(15.1925 - 10.5918i) q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} -6.58775i q^{5} +(15.1925 - 10.5918i) q^{7} +9.00000 q^{9} +54.2380i q^{11} -40.9722i q^{13} -19.7632i q^{15} -69.4988i q^{17} -160.189 q^{19} +(45.5776 - 31.7754i) q^{21} -87.8244i q^{23} +81.6016 q^{25} +27.0000 q^{27} -236.369 q^{29} -131.598 q^{31} +162.714i q^{33} +(-69.7762 - 100.085i) q^{35} +23.6257 q^{37} -122.916i q^{39} +112.106i q^{41} -194.017i q^{43} -59.2897i q^{45} -269.914 q^{47} +(118.627 - 321.833i) q^{49} -208.496i q^{51} +120.407 q^{53} +357.307 q^{55} -480.567 q^{57} +338.109 q^{59} -267.146i q^{61} +(136.733 - 95.3263i) q^{63} -269.914 q^{65} +275.691i q^{67} -263.473i q^{69} -270.482i q^{71} -1237.57i q^{73} +244.805 q^{75} +(574.479 + 824.014i) q^{77} +691.966i q^{79} +81.0000 q^{81} -430.482 q^{83} -457.841 q^{85} -709.108 q^{87} +1220.75i q^{89} +(-433.969 - 622.472i) q^{91} -394.795 q^{93} +1055.28i q^{95} -381.884i q^{97} +488.142i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 36 q^{3} + 10 q^{7} + 108 q^{9}+O(q^{10}) $ 12 * q + 36 * q^3 + 10 * q^7 + 108 * q^9	$ 12 q + 36 q^{3} + 10 q^{7} + 108 q^{9} + 84 q^{19} + 30 q^{21} - 216 q^{25} + 324 q^{27} - 200 q^{29} - 384 q^{31} - 84 q^{35} + 244 q^{37} + 280 q^{47} - 424 q^{49} + 16 q^{53} + 212 q^{55} + 252 q^{57} - 1168 q^{59} + 90 q^{63} + 280 q^{65} - 648 q^{75} - 968 q^{77} + 972 q^{81} + 968 q^{83} + 852 q^{85} - 600 q^{87} - 1648 q^{91} - 1152 q^{93}+O(q^{100}) $ 12 * q + 36 * q^3 + 10 * q^7 + 108 * q^9 + 84 * q^19 + 30 * q^21 - 216 * q^25 + 324 * q^27 - 200 * q^29 - 384 * q^31 - 84 * q^35 + 244 * q^37 + 280 * q^47 - 424 * q^49 + 16 * q^53 + 212 * q^55 + 252 * q^57 - 1168 * q^59 + 90 * q^63 + 280 * q^65 - 648 * q^75 - 968 * q^77 + 972 * q^81 + 968 * q^83 + 852 * q^85 - 600 * q^87 - 1648 * q^91 - 1152 * q^93

Display 100 coefficients 10 coefficients

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
$3$	3.00000			0.577350
$3$	3.00000			0.577350
$4$		0			0
$5$		−	6.58775i		−	0.589226i	−0.955617	−	0.294613i	$-0.904809\pi$
$5$		−	6.58775i		−	0.589226i	0.955617	−	0.294613i	$-0.0951908\pi$
$6$		0			0
$7$	15.1925	−	10.5918i	0.820321	−	0.571904i
$7$	15.1925	−	10.5918i	0.820321	−	0.571904i
$8$		0			0
$9$	9.00000			0.333333
$10$		0			0
$11$			54.2380i			1.48667i	0.668919	+	0.743335i	$0.266757\pi$
$11$			54.2380i			1.48667i	−0.668919	+	0.743335i	$0.733243\pi$
$12$		0			0
$13$		−	40.9722i		−	0.874125i	−0.899431	−	0.437063i	$-0.856019\pi$
$13$		−	40.9722i		−	0.874125i	0.899431	−	0.437063i	$-0.143981\pi$
$14$		0			0
$15$		−	19.7632i		−	0.340190i
$16$		0			0
$17$		−	69.4988i		−	0.991526i	−0.868458	−	0.495763i	$-0.834889\pi$
$17$		−	69.4988i		−	0.991526i	0.868458	−	0.495763i	$-0.165111\pi$
$18$		0			0
$19$	−160.189			−1.93420			−0.967102	−	0.254389i	$-0.918126\pi$
$19$	−160.189			−1.93420			−0.967102	+	0.254389i	$0.918126\pi$
$20$		0			0
$21$	45.5776	−	31.7754i	0.473612	−	0.330189i
$22$		0			0
$23$		−	87.8244i		−	0.796202i	−0.917342	−	0.398101i	$-0.869669\pi$
$23$		−	87.8244i		−	0.796202i	0.917342	−	0.398101i	$-0.130331\pi$
$24$		0			0
$25$	81.6016			0.652812
$26$		0			0
$27$	27.0000			0.192450
$28$		0			0
$29$	−236.369			−1.51354			−0.756771	−	0.653680i	$-0.773224\pi$
$29$	−236.369			−1.51354			−0.756771	+	0.653680i	$0.773224\pi$
$30$		0			0
$31$	−131.598			−0.762443			−0.381222	−	0.924484i	$-0.624497\pi$
$31$	−131.598			−0.762443			−0.381222	+	0.924484i	$0.624497\pi$
$32$		0			0
$33$			162.714i			0.858330i
$34$		0			0
$35$	−69.7762	−	100.085i	−0.336981	−	0.483354i
$36$		0			0
$37$	23.6257			0.104974			0.0524871	−	0.998622i	$-0.483285\pi$
$37$	23.6257			0.104974			0.0524871	+	0.998622i	$0.483285\pi$
$38$		0			0
$39$		−	122.916i		−	0.504677i
$40$		0			0
$41$			112.106i			0.427024i	0.976940	+	0.213512i	$0.0684903\pi$
$41$			112.106i			0.427024i	−0.976940	+	0.213512i	$0.931510\pi$
$42$		0			0
$43$		−	194.017i		−	0.688076i	−0.938956	−	0.344038i	$-0.888205\pi$
$43$		−	194.017i		−	0.688076i	0.938956	−	0.344038i	$-0.111795\pi$
$44$		0			0
$45$		−	59.2897i		−	0.196409i
$46$		0			0
$47$	−269.914			−0.837682			−0.418841	−	0.908060i	$-0.637564\pi$
$47$	−269.914			−0.837682			−0.418841	+	0.908060i	$0.637564\pi$
$48$		0			0
$49$	118.627	−	321.833i	0.345852	−	0.938289i
$50$		0			0
$51$		−	208.496i		−	0.572458i
$52$		0			0
$53$	120.407			0.312059			0.156030	−	0.987752i	$-0.450130\pi$
$53$	120.407			0.312059			0.156030	+	0.987752i	$0.450130\pi$
$54$		0			0
$55$	357.307			0.875985
$56$		0			0
$57$	−480.567			−1.11671
$58$		0			0
$59$	338.109			0.746069			0.373035	−	0.927817i	$-0.378317\pi$
$59$	338.109			0.746069			0.373035	+	0.927817i	$0.378317\pi$
$60$		0			0
$61$		−	267.146i		−	0.560729i	−0.959894	−	0.280365i	$-0.909545\pi$
$61$		−	267.146i		−	0.560729i	0.959894	−	0.280365i	$-0.0904554\pi$
$62$		0			0
$63$	136.733	−	95.3263i	0.273440	−	0.190635i
$64$		0			0
$65$	−269.914			−0.515058
$66$		0			0
$67$			275.691i			0.502702i	0.967896	+	0.251351i	$0.0808749\pi$
$67$			275.691i			0.502702i	−0.967896	+	0.251351i	$0.919125\pi$
$68$		0			0
$69$		−	263.473i		−	0.459687i
$70$		0			0
$71$		−	270.482i		−	0.452117i	−0.974114	−	0.226058i	$-0.927416\pi$
$71$		−	270.482i		−	0.452117i	0.974114	−	0.226058i	$-0.0725840\pi$
$72$		0			0
$73$		−	1237.57i		−	1.98420i	−0.125469	−	0.992098i	$-0.540044\pi$
$73$		−	1237.57i		−	1.98420i	0.125469	−	0.992098i	$-0.459956\pi$
$74$		0			0
$75$	244.805			0.376901
$76$		0			0
$77$	574.479	+	824.014i	0.850233	+	1.21955i
$78$		0			0
$79$			691.966i			0.985472i	0.870179	+	0.492736i	$0.164003\pi$
$79$			691.966i			0.985472i	−0.870179	+	0.492736i	$0.835997\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$	−430.482			−0.569296			−0.284648	−	0.958632i	$-0.591877\pi$
$83$	−430.482			−0.569296			−0.284648	+	0.958632i	$0.591877\pi$
$84$		0			0
$85$	−457.841			−0.584233
$86$		0			0
$87$	−709.108			−0.873844
$88$		0			0
$89$			1220.75i			1.45392i	0.686680	+	0.726960i	$0.259067\pi$
$89$			1220.75i			1.45392i	−0.686680	+	0.726960i	$0.740933\pi$
$90$		0			0
$91$	−433.969	−	622.472i	−0.499916	−	0.717063i
$92$		0			0
$93$	−394.795			−0.440197
$94$		0			0
$95$			1055.28i			1.13968i
$96$		0			0
$97$		−	381.884i		−	0.399736i	−0.979823	−	0.199868i	$-0.935949\pi$
$97$		−	381.884i		−	0.399736i	0.979823	−	0.199868i	$-0.0640513\pi$
$98$		0			0
$99$			488.142i			0.495557i
$100$		0			0
$101$		−	696.862i		−	0.686538i	−0.939237	−	0.343269i	$-0.888466\pi$
$101$		−	696.862i		−	0.686538i	0.939237	−	0.343269i	$-0.111534\pi$
$102$		0			0
$103$	−621.333			−0.594386			−0.297193	−	0.954817i	$-0.596050\pi$
$103$	−621.333			−0.594386			−0.297193	+	0.954817i	$0.596050\pi$
$104$		0			0
$105$	−209.329	−	300.254i	−0.194556	−	0.279065i
$106$		0			0
$107$			295.518i			0.266999i	0.991049	+	0.133499i	$0.0426214\pi$
$107$			295.518i			0.266999i	−0.991049	+	0.133499i	$0.957379\pi$
$108$		0			0
$109$	−669.548			−0.588358			−0.294179	−	0.955750i	$-0.595046\pi$
$109$	−669.548			−0.588358			−0.294179	+	0.955750i	$0.595046\pi$
$110$		0			0
$111$	70.8772			0.0606069
$112$		0			0
$113$	−1093.94			−0.910703			−0.455352	−	0.890312i	$-0.650486\pi$
$113$	−1093.94			−0.910703			−0.455352	+	0.890312i	$0.650486\pi$
$114$		0			0
$115$	−578.565			−0.469143
$116$		0			0
$117$		−	368.749i		−	0.291375i
$118$		0			0
$119$	−736.118	−	1055.86i	−0.567058	−	0.813369i
$120$		0			0
$121$	−1610.76			−1.21019
$122$		0			0
$123$			336.318i			0.246543i
$124$		0			0
$125$		−	1361.04i		−	0.973880i
$126$		0			0
$127$			837.266i			0.585003i	0.956265	+	0.292501i	$0.0944876\pi$
$127$			837.266i			0.585003i	−0.956265	+	0.292501i	$0.905512\pi$
$128$		0			0
$129$		−	582.050i		−	0.397261i
$130$		0			0
$131$	1821.00			1.21452			0.607258	−	0.794504i	$-0.292269\pi$
$131$	1821.00			1.21452			0.607258	+	0.794504i	$0.292269\pi$
$132$		0			0
$133$	−2433.68	+	1696.69i	−1.58667	+	1.10618i
$134$		0			0
$135$		−	177.869i		−	0.113397i
$136$		0			0
$137$	321.471			0.200475			0.100238	−	0.994964i	$-0.468040\pi$
$137$	321.471			0.200475			0.100238	+	0.994964i	$0.468040\pi$
$138$		0			0
$139$	−1339.35			−0.817284			−0.408642	−	0.912695i	$-0.633998\pi$
$139$	−1339.35			−0.817284			−0.408642	+	0.912695i	$0.633998\pi$
$140$		0			0
$141$	−809.743			−0.483636
$142$		0			0
$143$	2222.25			1.29954
$144$		0			0
$145$			1557.14i			0.891818i
$146$		0			0
$147$	355.881	−	965.500i	0.199678	−	0.541722i
$148$		0			0
$149$	−2748.96			−1.51144			−0.755718	−	0.654898i	$-0.772712\pi$
$149$	−2748.96			−1.51144			−0.755718	+	0.654898i	$0.772712\pi$
$150$		0			0
$151$		−	1950.99i		−	1.05145i	−0.850655	−	0.525725i	$-0.823794\pi$
$151$		−	1950.99i		−	1.05145i	0.850655	−	0.525725i	$-0.176206\pi$
$152$		0			0
$153$		−	625.489i		−	0.330509i
$154$		0			0
$155$			866.936i			0.449252i
$156$		0			0
$157$		−	3019.82i		−	1.53508i	−0.641000	−	0.767540i	$-0.721480\pi$
$157$		−	3019.82i		−	1.53508i	0.641000	−	0.767540i	$-0.278520\pi$
$158$		0			0
$159$	361.220			0.180168
$160$		0			0
$161$	−930.219	−	1334.28i	−0.455351	−	0.653141i
$162$		0			0
$163$			2882.36i			1.38506i	0.721391	+	0.692528i	$0.243503\pi$
$163$			2882.36i			1.38506i	−0.721391	+	0.692528i	$0.756497\pi$
$164$		0			0
$165$	1071.92			0.505750
$166$		0			0
$167$	2512.97			1.16443			0.582213	−	0.813036i	$-0.302187\pi$
$167$	2512.97			1.16443			0.582213	+	0.813036i	$0.302187\pi$
$168$		0			0
$169$	518.282			0.235905
$170$		0			0
$171$	−1441.70			−0.644735
$172$		0			0
$173$		−	1307.12i		−	0.574442i	−0.957864	−	0.287221i	$-0.907269\pi$
$173$		−	1307.12i		−	0.574442i	0.957864	−	0.287221i	$-0.0927314\pi$
$174$		0			0
$175$	1239.74	−	864.308i	0.535515	−	0.373346i
$176$		0			0
$177$	1014.33			0.430743
$178$		0			0
$179$		−	3555.19i		−	1.48451i	−0.670117	−	0.742256i	$-0.733756\pi$
$179$		−	3555.19i		−	1.48451i	0.670117	−	0.742256i	$-0.266244\pi$
$180$		0			0
$181$			1746.96i			0.717406i	0.933452	+	0.358703i	$0.116781\pi$
$181$			1746.96i			0.717406i	−0.933452	+	0.358703i	$0.883219\pi$
$182$		0			0
$183$		−	801.437i		−	0.323737i
$184$		0			0
$185$		−	155.640i		−	0.0618536i
$186$		0			0
$187$	3769.48			1.47407
$188$		0			0
$189$	410.199	−	285.979i	0.157871	−	0.110063i
$190$		0			0
$191$		−	1935.70i		−	0.733310i	−0.930357	−	0.366655i	$-0.880503\pi$
$191$		−	1935.70i		−	0.733310i	0.930357	−	0.366655i	$-0.119497\pi$
$192$		0			0
$193$	−3842.27			−1.43302			−0.716509	−	0.697578i	$-0.754261\pi$
$193$	−3842.27			−1.43302			−0.716509	+	0.697578i	$0.754261\pi$
$194$		0			0
$195$	−809.743			−0.297369
$196$		0			0
$197$	−490.616			−0.177436			−0.0887181	−	0.996057i	$-0.528277\pi$
$197$	−490.616			−0.177436			−0.0887181	+	0.996057i	$0.528277\pi$
$198$		0			0
$199$	−3630.14			−1.29314			−0.646568	−	0.762857i	$-0.723796\pi$
$199$	−3630.14			−1.29314			−0.646568	+	0.762857i	$0.723796\pi$
$200$		0			0
$201$			827.074i			0.290235i
$202$		0			0
$203$	−3591.05	+	2503.58i	−1.24159	+	0.865600i
$204$		0			0
$205$	738.525			0.251614
$206$		0			0
$207$		−	790.419i		−	0.265401i
$208$		0			0
$209$		−	8688.33i		−	2.87552i
$210$		0			0
$211$		−	2486.62i		−	0.811308i	−0.914027	−	0.405654i	$-0.867044\pi$
$211$		−	2486.62i		−	0.811308i	0.914027	−	0.405654i	$-0.132956\pi$
$212$		0			0
$213$		−	811.446i		−	0.261030i
$214$		0			0
$215$	−1278.13			−0.405432
$216$		0			0
$217$	−1999.31	+	1393.86i	−0.625448	+	0.436044i
$218$		0			0
$219$		−	3712.70i		−	1.14558i
$220$		0			0
$221$	−2847.52			−0.866718
$222$		0			0
$223$	−992.723			−0.298106			−0.149053	−	0.988829i	$-0.547623\pi$
$223$	−992.723			−0.298106			−0.149053	+	0.988829i	$0.547623\pi$
$224$		0			0
$225$	734.414			0.217604
$226$		0			0
$227$	5666.03			1.65668			0.828342	−	0.560223i	$-0.189284\pi$
$227$	5666.03			1.65668			0.828342	+	0.560223i	$0.189284\pi$
$228$		0			0
$229$		−	6488.31i		−	1.87231i	−0.351583	−	0.936157i	$-0.614357\pi$
$229$		−	6488.31i		−	1.87231i	0.351583	−	0.936157i	$-0.385643\pi$
$230$		0			0
$231$	1723.44	+	2472.04i	0.490882	+	0.704106i
$232$		0			0
$233$	−3657.30			−1.02832			−0.514158	−	0.857696i	$-0.671895\pi$
$233$	−3657.30			−1.02832			−0.514158	+	0.857696i	$0.671895\pi$
$234$		0			0
$235$			1778.13i			0.493584i
$236$		0			0
$237$			2075.90i			0.568962i
$238$		0			0
$239$			5806.11i			1.57141i	0.618604	+	0.785703i	$0.287698\pi$
$239$			5806.11i			1.57141i	−0.618604	+	0.785703i	$0.712302\pi$
$240$		0			0
$241$			584.133i			0.156130i	0.996948	+	0.0780649i	$0.0248741\pi$
$241$			584.133i			0.156130i	−0.996948	+	0.0780649i	$0.975126\pi$
$242$		0			0
$243$	243.000			0.0641500
$244$		0			0
$245$	−2120.16	−	781.486i	−0.552865	−	0.203785i
$246$		0			0
$247$			6563.29i			1.69074i
$248$		0			0
$249$	−1291.45			−0.328683
$250$		0			0
$251$	3459.63			0.870000			0.435000	−	0.900430i	$-0.356748\pi$
$251$	3459.63			0.870000			0.435000	+	0.900430i	$0.356748\pi$
$252$		0			0
$253$	4763.42			1.18369
$254$		0			0
$255$	−1373.52			−0.337307
$256$		0			0
$257$		−	1501.55i		−	0.364453i	−0.983257	−	0.182226i	$-0.941670\pi$
$257$		−	1501.55i		−	0.364453i	0.983257	−	0.182226i	$-0.0583304\pi$
$258$		0			0
$259$	358.935	−	250.239i	0.0861125	−	0.0600352i
$260$		0			0
$261$	−2127.33			−0.504514
$262$		0			0
$263$			5235.13i			1.22742i	0.789531	+	0.613711i	$0.210324\pi$
$263$			5235.13i			1.22742i	−0.789531	+	0.613711i	$0.789676\pi$
$264$		0			0
$265$		−	793.210i		−	0.183874i
$266$		0			0
$267$			3662.24i			0.839421i
$268$		0			0
$269$			34.2010i			0.00775193i	0.999992	+	0.00387597i	$0.00123376\pi$
$269$			34.2010i			0.00775193i	−0.999992	+	0.00387597i	$0.998766\pi$
$270$		0			0
$271$	653.935			0.146582			0.0732910	−	0.997311i	$-0.476650\pi$
$271$	653.935			0.146582			0.0732910	+	0.997311i	$0.476650\pi$
$272$		0			0
$273$	−1301.91	−	1867.41i	−0.288627	−	0.413997i
$274$		0			0
$275$			4425.91i			0.970517i
$276$		0			0
$277$	−2250.58			−0.488174			−0.244087	−	0.969753i	$-0.578488\pi$
$277$	−2250.58			−0.488174			−0.244087	+	0.969753i	$0.578488\pi$
$278$		0			0
$279$	−1184.38			−0.254148
$280$		0			0
$281$	4213.47			0.894500			0.447250	−	0.894409i	$-0.352403\pi$
$281$	4213.47			0.894500			0.447250	+	0.894409i	$0.352403\pi$
$282$		0			0
$283$	−2069.17			−0.434627			−0.217313	−	0.976102i	$-0.569729\pi$
$283$	−2069.17			−0.434627			−0.217313	+	0.976102i	$0.569729\pi$
$284$		0			0
$285$			3165.85i			0.657997i
$286$		0			0
$287$	1187.40	+	1703.17i	0.244217	+	0.350297i
$288$		0			0
$289$	82.9156			0.0168768
$290$		0			0
$291$		−	1145.65i		−	0.230788i
$292$		0			0
$293$			2027.49i			0.404257i	0.979359	+	0.202128i	$0.0647859\pi$
$293$			2027.49i			0.404257i	−0.979359	+	0.202128i	$0.935214\pi$
$294$		0			0
$295$		−	2227.38i		−	0.439604i
$296$		0			0
$297$			1464.43i			0.286110i
$298$		0			0
$299$	−3598.35			−0.695981
$300$		0			0
$301$	−2054.99	−	2947.61i	−0.393513	−	0.564443i
$302$		0			0
$303$		−	2090.59i		−	0.396373i
$304$		0			0
$305$	−1759.89			−0.330396
$306$		0			0
$307$	−167.377			−0.0311162			−0.0155581	−	0.999879i	$-0.504953\pi$
$307$	−167.377			−0.0311162			−0.0155581	+	0.999879i	$0.504953\pi$
$308$		0			0
$309$	−1864.00			−0.343169
$310$		0			0
$311$	4720.36			0.860666			0.430333	−	0.902670i	$-0.358396\pi$
$311$	4720.36			0.860666			0.430333	+	0.902670i	$0.358396\pi$
$312$		0			0
$313$		−	6008.08i		−	1.08497i	−0.840064	−	0.542487i	$-0.817483\pi$
$313$		−	6008.08i		−	1.08497i	0.840064	−	0.542487i	$-0.182517\pi$
$314$		0			0
$315$	−627.986	−	900.762i	−0.112327	−	0.161118i
$316$		0			0
$317$	1790.11			0.317170			0.158585	−	0.987345i	$-0.449307\pi$
$317$	1790.11			0.317170			0.158585	+	0.987345i	$0.449307\pi$
$318$		0			0
$319$		−	12820.2i		−	2.25014i
$320$		0			0
$321$			886.555i			0.154152i
$322$		0			0
$323$			11132.9i			1.91781i
$324$		0			0
$325$		−	3343.39i		−	0.570640i
$326$		0			0
$327$	−2008.64			−0.339689
$328$		0			0
$329$	−4100.69	+	2858.88i	−0.687168	+	0.479074i
$330$		0			0
$331$		−	5059.04i		−	0.840090i	−0.907503	−	0.420045i	$-0.862014\pi$
$331$		−	5059.04i		−	0.840090i	0.907503	−	0.420045i	$-0.137986\pi$
$332$		0			0
$333$	212.632			0.0349914
$334$		0			0
$335$	1816.19			0.296205
$336$		0			0
$337$	−12120.3			−1.95915			−0.979574	−	0.201084i	$-0.935554\pi$
$337$	−12120.3			−1.95915			−0.979574	+	0.201084i	$0.935554\pi$
$338$		0			0
$339$	−3281.83			−0.525795
$340$		0			0
$341$		−	7137.63i		−	1.13350i
$342$		0			0
$343$	−1606.55	−	6145.94i	−0.252902	−	0.967492i
$344$		0			0
$345$	−1735.69			−0.270860
$346$		0			0
$347$		−	6366.23i		−	0.984892i	−0.870343	−	0.492446i	$-0.836103\pi$
$347$		−	6366.23i		−	0.984892i	0.870343	−	0.492446i	$-0.163897\pi$
$348$		0			0
$349$			4912.76i			0.753508i	0.926313	+	0.376754i	$0.122960\pi$
$349$			4912.76i			0.753508i	−0.926313	+	0.376754i	$0.877040\pi$
$350$		0			0
$351$		−	1106.25i		−	0.168226i
$352$		0			0
$353$			5847.50i			0.881674i	0.897587	+	0.440837i	$0.145318\pi$
$353$			5847.50i			0.881674i	−0.897587	+	0.440837i	$0.854682\pi$
$354$		0			0
$355$	−1781.87			−0.266399
$356$		0			0
$357$	−2208.35	−	3167.59i	−0.327391	−	0.469599i
$358$		0			0
$359$		−	7074.15i		−	1.04000i	−0.854167	−	0.519999i	$-0.825932\pi$
$359$		−	7074.15i		−	1.04000i	0.854167	−	0.519999i	$-0.174068\pi$
$360$		0			0
$361$	18801.5			2.74114
$362$		0			0
$363$	−4832.29			−0.698704
$364$		0			0
$365$	−8152.78			−1.16914
$366$		0			0
$367$	3730.67			0.530625			0.265312	−	0.964163i	$-0.414525\pi$
$367$	3730.67			0.530625			0.265312	+	0.964163i	$0.414525\pi$
$368$		0			0
$369$			1008.95i			0.142341i
$370$		0			0
$371$	1829.29	−	1275.33i	0.255989	−	0.178468i
$372$		0			0
$373$	−9966.13			−1.38345			−0.691725	−	0.722161i	$-0.743149\pi$
$373$	−9966.13			−1.38345			−0.691725	+	0.722161i	$0.743149\pi$
$374$		0			0
$375$		−	4083.12i		−	0.562270i
$376$		0			0
$377$			9684.57i			1.32303i
$378$		0			0
$379$		−	11391.4i		−	1.54390i	−0.635686	−	0.771948i	$-0.719283\pi$
$379$		−	11391.4i		−	1.54390i	0.635686	−	0.771948i	$-0.280717\pi$
$380$		0			0
$381$			2511.80i			0.337752i
$382$		0			0
$383$	3889.70			0.518941			0.259471	−	0.965751i	$-0.416452\pi$
$383$	3889.70			0.518941			0.259471	+	0.965751i	$0.416452\pi$
$384$		0			0
$385$	5428.40	−	3784.52i	0.718589	−	0.500980i
$386$		0			0
$387$		−	1746.15i		−	0.229359i
$388$		0			0
$389$	9689.51			1.26292			0.631462	−	0.775407i	$-0.282455\pi$
$389$	9689.51			1.26292			0.631462	+	0.775407i	$0.282455\pi$
$390$		0			0
$391$	−6103.69			−0.789455
$392$		0			0
$393$	5463.01			0.701201
$394$		0			0
$395$	4558.50			0.580666
$396$		0			0
$397$			10767.0i			1.36116i	0.732672	+	0.680582i	$0.238273\pi$
$397$			10767.0i			1.36116i	−0.732672	+	0.680582i	$0.761727\pi$
$398$		0			0
$399$	−7301.04	+	5090.07i	−0.916063	+	0.638653i
$400$		0			0
$401$	−4053.14			−0.504749			−0.252374	−	0.967630i	$-0.581211\pi$
$401$	−4053.14			−0.504749			−0.252374	+	0.967630i	$0.581211\pi$
$402$		0			0
$403$			5391.86i			0.666471i
$404$		0			0
$405$		−	533.608i		−	0.0654696i
$406$		0			0
$407$			1281.41i			0.156062i
$408$		0			0
$409$		−	1517.85i		−	0.183503i	−0.995782	−	0.0917517i	$-0.970753\pi$
$409$		−	1517.85i		−	0.183503i	0.995782	−	0.0917517i	$-0.0292466\pi$
$410$		0			0
$411$	964.413			0.115744
$412$		0			0
$413$	5136.74	−	3581.19i	0.612016	−	0.426680i
$414$		0			0
$415$			2835.91i			0.335444i
$416$		0			0
$417$	−4018.06			−0.471859
$418$		0			0
$419$	−1429.41			−0.166662			−0.0833310	−	0.996522i	$-0.526556\pi$
$419$	−1429.41			−0.166662			−0.0833310	+	0.996522i	$0.526556\pi$
$420$		0			0
$421$	924.052			0.106973			0.0534864	−	0.998569i	$-0.482967\pi$
$421$	924.052			0.106973			0.0534864	+	0.998569i	$0.482967\pi$
$422$		0			0
$423$	−2429.23			−0.279227
$424$		0			0
$425$		−	5671.21i		−	0.647280i
$426$		0			0
$427$	−2829.56	−	4058.62i	−0.320683	−	0.459978i
$428$		0			0
$429$	6666.75			0.750288
$430$		0			0
$431$			4099.49i			0.458157i	0.973408	+	0.229078i	$0.0735712\pi$
$431$			4099.49i			0.458157i	−0.973408	+	0.229078i	$0.926429\pi$
$432$		0			0
$433$			6350.16i			0.704779i	0.935853	+	0.352390i	$0.114631\pi$
$433$			6350.16i			0.704779i	−0.935853	+	0.352390i	$0.885369\pi$
$434$		0			0
$435$			4671.43i			0.514892i
$436$		0			0
$437$			14068.5i			1.54002i
$438$		0			0
$439$	5385.88			0.585545			0.292772	−	0.956182i	$-0.405422\pi$
$439$	5385.88			0.585545			0.292772	+	0.956182i	$0.405422\pi$
$440$		0			0
$441$	1067.64	−	2896.50i	0.115284	−	0.312763i
$442$		0			0
$443$		−	1259.12i		−	0.135040i	−0.997718	−	0.0675200i	$-0.978491\pi$
$443$		−	1259.12i		−	0.135040i	0.997718	−	0.0675200i	$-0.0215087\pi$
$444$		0			0
$445$	8041.97			0.856688
$446$		0			0
$447$	−8246.89			−0.872627
$448$		0			0
$449$	434.975			0.0457187			0.0228594	−	0.999739i	$-0.492723\pi$
$449$	434.975			0.0457187			0.0228594	+	0.999739i	$0.492723\pi$
$450$		0			0
$451$	−6080.40			−0.634845
$452$		0			0
$453$		−	5852.96i		−	0.607055i
$454$		0			0
$455$	−4100.69	+	2858.88i	−0.422512	+	0.294564i
$456$		0			0
$457$	−3172.32			−0.324715			−0.162357	−	0.986732i	$-0.551910\pi$
$457$	−3172.32			−0.324715			−0.162357	+	0.986732i	$0.551910\pi$
$458$		0			0
$459$		−	1876.47i		−	0.190819i
$460$		0			0
$461$		−	12644.2i		−	1.27744i	−0.769441	−	0.638718i	$-0.779465\pi$
$461$		−	12644.2i		−	1.27744i	0.769441	−	0.638718i	$-0.220535\pi$
$462$		0			0
$463$			13829.8i			1.38818i	0.719890	+	0.694089i	$0.244192\pi$
$463$			13829.8i			1.38818i	−0.719890	+	0.694089i	$0.755808\pi$
$464$		0			0
$465$			2600.81i			0.259376i
$466$		0			0
$467$	17043.7			1.68884			0.844419	−	0.535684i	$-0.179946\pi$
$467$	17043.7			1.68884			0.844419	+	0.535684i	$0.179946\pi$
$468$		0			0
$469$	2920.07	+	4188.45i	0.287497	+	0.412377i
$470$		0			0
$471$		−	9059.45i		−	0.886279i
$472$		0			0
$473$	10523.1			1.02294
$474$		0			0
$475$	−13071.7			−1.26267
$476$		0			0
$477$	1083.66			0.104020
$478$		0			0
$479$	4131.98			0.394144			0.197072	−	0.980389i	$-0.436857\pi$
$479$	4131.98			0.394144			0.197072	+	0.980389i	$0.436857\pi$
$480$		0			0
$481$		−	967.997i		−	0.0917607i
$482$		0			0
$483$	−2790.66	−	4002.83i	−0.262897	−	0.377091i
$484$		0			0
$485$	−2515.75			−0.235535
$486$		0			0
$487$			11356.0i			1.05665i	0.849042	+	0.528325i	$0.177180\pi$
$487$			11356.0i			1.05665i	−0.849042	+	0.528325i	$0.822820\pi$
$488$		0			0
$489$			8647.09i			0.799663i
$490$		0			0
$491$			7964.89i			0.732079i	0.930599	+	0.366039i	$0.119286\pi$
$491$			7964.89i			0.732079i	−0.930599	+	0.366039i	$0.880714\pi$
$492$		0			0
$493$			16427.4i			1.50072i
$494$		0			0
$495$	3215.76			0.291995
$496$		0			0
$497$	−2864.89	−	4109.31i	−0.258567	−	0.370881i
$498$		0			0
$499$		−	2086.90i		−	0.187219i	−0.995609	−	0.0936096i	$-0.970159\pi$
$499$		−	2086.90i		−	0.187219i	0.995609	−	0.0936096i	$-0.0298406\pi$
$500$		0			0
$501$	7538.90			0.672282
$502$		0			0
$503$	−10945.8			−0.970277			−0.485138	−	0.874437i	$-0.661231\pi$
$503$	−10945.8			−0.970277			−0.485138	+	0.874437i	$0.661231\pi$
$504$		0			0
$505$	−4590.75			−0.404526
$506$		0			0
$507$	1554.85			0.136200
$508$		0			0
$509$			21490.9i			1.87145i	0.352732	+	0.935724i	$0.385253\pi$
$509$			21490.9i			1.87145i	−0.352732	+	0.935724i	$0.614747\pi$
$510$		0			0
$511$	−13108.1	−	18801.8i	−1.13477	−	1.62768i
$512$		0			0
$513$	−4325.10			−0.372238
$514$		0			0
$515$			4093.18i			0.350228i
$516$		0			0
$517$		−	14639.6i		−	1.24536i
$518$		0			0
$519$		−	3921.36i		−	0.331654i
$520$		0			0
$521$			6155.93i			0.517651i	0.965924	+	0.258826i	$0.0833355\pi$
$521$			6155.93i			0.517651i	−0.965924	+	0.258826i	$0.916665\pi$
$522$		0			0
$523$	−5018.12			−0.419555			−0.209777	−	0.977749i	$-0.567274\pi$
$523$	−5018.12			−0.419555			−0.209777	+	0.977749i	$0.567274\pi$
$524$		0			0
$525$	3719.21	−	2592.92i	0.309180	−	0.215551i
$526$		0			0
$527$			9145.92i			0.755982i
$528$		0			0
$529$	4453.88			0.366062
$530$		0			0
$531$	3042.98			0.248690
$532$		0			0
$533$	4593.22			0.373273
$534$		0			0
$535$	1946.80			0.157323
$536$		0			0
$537$		−	10665.6i		−	0.857083i
$538$		0			0
$539$	17455.6	+	6434.10i	1.39493	+	0.514168i
$540$		0			0
$541$	1260.99			0.100211			0.0501055	−	0.998744i	$-0.484044\pi$
$541$	1260.99			0.100211			0.0501055	+	0.998744i	$0.484044\pi$
$542$		0			0
$543$			5240.88i			0.414194i
$544$		0			0
$545$			4410.81i			0.346676i
$546$		0			0
$547$		−	16180.1i		−	1.26474i	−0.774666	−	0.632370i	$-0.782082\pi$
$547$		−	16180.1i		−	1.26474i	0.774666	−	0.632370i	$-0.217918\pi$
$548$		0			0
$549$		−	2404.31i		−	0.186910i
$550$		0			0
$551$	37863.8			2.92750
$552$		0			0
$553$	7329.17	+	10512.7i	0.563595	+	0.808403i
$554$		0			0
$555$		−	466.921i		−	0.0357112i
$556$		0			0
$557$	21384.1			1.62670			0.813351	−	0.581774i	$-0.197641\pi$
$557$	21384.1			1.62670			0.813351	+	0.581774i	$0.197641\pi$
$558$		0			0
$559$	−7949.28			−0.601464
$560$		0			0
$561$	11308.4			0.851056
$562$		0			0
$563$	−2029.82			−0.151948			−0.0759739	−	0.997110i	$-0.524207\pi$
$563$	−2029.82			−0.151948			−0.0759739	+	0.997110i	$0.524207\pi$
$564$		0			0
$565$			7206.62i			0.536610i
$566$		0			0
$567$	1230.60	−	857.937i	0.0911467	−	0.0635449i
$568$		0			0
$569$	21597.4			1.59123			0.795616	−	0.605801i	$-0.207147\pi$
$569$	21597.4			1.59123			0.795616	+	0.605801i	$0.207147\pi$
$570$		0			0
$571$			13250.6i			0.971137i	0.874199	+	0.485568i	$0.161387\pi$
$571$			13250.6i			0.971137i	−0.874199	+	0.485568i	$0.838613\pi$
$572$		0			0
$573$		−	5807.10i		−	0.423377i
$574$		0			0
$575$		−	7166.61i		−	0.519771i
$576$		0			0
$577$		−	17602.4i		−	1.27001i	−0.772507	−	0.635006i	$-0.780998\pi$
$577$		−	17602.4i		−	1.27001i	0.772507	−	0.635006i	$-0.219002\pi$
$578$		0			0
$579$	−11526.8			−0.827353
$580$		0			0
$581$	−6540.12	+	4559.59i	−0.467005	+	0.325583i
$582$		0			0
$583$			6530.63i			0.463930i
$584$		0			0
$585$	−2429.23			−0.171686
$586$		0			0
$587$	10190.1			0.716510			0.358255	−	0.933624i	$-0.383372\pi$
$587$	10190.1			0.716510			0.358255	+	0.933624i	$0.383372\pi$
$588$		0			0
$589$	21080.6			1.47472
$590$		0			0
$591$	−1471.85			−0.102443
$592$		0			0
$593$			1211.86i			0.0839213i	0.999119	+	0.0419606i	$0.0133604\pi$
$593$			1211.86i			0.0839213i	−0.999119	+	0.0419606i	$0.986640\pi$
$594$		0			0
$595$	−6955.77	+	4849.36i	−0.479258	+	0.334125i
$596$		0			0
$597$	−10890.4			−0.746592
$598$		0			0
$599$			9213.72i			0.628485i	0.949343	+	0.314243i	$0.101751\pi$
$599$			9213.72i			0.628485i	−0.949343	+	0.314243i	$0.898249\pi$
$600$		0			0
$601$			17051.3i			1.15730i	0.815575	+	0.578651i	$0.196421\pi$
$601$			17051.3i			1.15730i	−0.815575	+	0.578651i	$0.803579\pi$
$602$		0			0
$603$			2481.22i			0.167567i
$604$		0			0
$605$			10611.3i			0.713076i
$606$		0			0
$607$	9836.13			0.657721			0.328860	−	0.944379i	$-0.393335\pi$
$607$	9836.13			0.657721			0.328860	+	0.944379i	$0.393335\pi$
$608$		0			0
$609$	−10773.2	+	7510.74i	−0.716832	+	0.499755i
$610$		0			0
$611$			11059.0i			0.732239i
$612$		0			0
$613$	−10697.4			−0.704837			−0.352418	−	0.935843i	$-0.614641\pi$
$613$	−10697.4			−0.704837			−0.352418	+	0.935843i	$0.614641\pi$
$614$		0			0
$615$	2215.58			0.145269
$616$		0			0
$617$	18439.5			1.20316			0.601578	−	0.798814i	$-0.294539\pi$
$617$	18439.5			1.20316			0.601578	+	0.798814i	$0.294539\pi$
$618$		0			0
$619$	26672.5			1.73192			0.865960	−	0.500113i	$-0.166708\pi$
$619$	26672.5			1.73192			0.865960	+	0.500113i	$0.166708\pi$
$620$		0			0
$621$		−	2371.26i		−	0.153229i
$622$		0			0
$623$	12929.9	+	18546.3i	0.831503	+	1.19268i
$624$		0			0
$625$	1234.01			0.0789766
$626$		0			0
$627$		−	26065.0i		−	1.66018i
$628$		0			0
$629$		−	1641.96i		−	0.104085i
$630$		0			0
$631$		−	10491.0i		−	0.661870i	−0.943653	−	0.330935i	$-0.892636\pi$
$631$		−	10491.0i		−	0.661870i	0.943653	−	0.330935i	$-0.107364\pi$
$632$		0			0
$633$		−	7459.86i		−	0.468409i
$634$		0			0
$635$	5515.70			0.344699
$636$		0			0
$637$	−13186.2	−	4860.41i	−0.820183	−	0.302318i
$638$		0			0
$639$		−	2434.34i		−	0.150706i
$640$		0			0
$641$	2839.23			0.174950			0.0874749	−	0.996167i	$-0.472120\pi$
$641$	2839.23			0.174950			0.0874749	+	0.996167i	$0.472120\pi$
$642$		0			0
$643$	6153.86			0.377426			0.188713	−	0.982032i	$-0.439568\pi$
$643$	6153.86			0.377426			0.188713	+	0.982032i	$0.439568\pi$
$644$		0			0
$645$	−3834.40			−0.234076
$646$		0			0
$647$	−3054.45			−0.185599			−0.0927996	−	0.995685i	$-0.529582\pi$
$647$	−3054.45			−0.185599			−0.0927996	+	0.995685i	$0.529582\pi$
$648$		0			0
$649$			18338.4i			1.10916i
$650$		0			0
$651$	−5997.94	+	4181.59i	−0.361103	+	0.251750i
$652$		0			0
$653$	17522.7			1.05010			0.525052	−	0.851070i	$-0.324046\pi$
$653$	17522.7			1.05010			0.525052	+	0.851070i	$0.324046\pi$
$654$		0			0
$655$		−	11996.3i		−	0.715625i
$656$		0			0
$657$		−	11138.1i		−	0.661398i
$658$		0			0
$659$			131.768i			0.00778899i	0.999992	+	0.00389450i	$0.00123966\pi$
$659$			131.768i			0.00778899i	−0.999992	+	0.00389450i	$0.998760\pi$
$660$		0			0
$661$		−	21150.0i		−	1.24454i	−0.782803	−	0.622270i	$-0.786211\pi$
$661$		−	21150.0i		−	1.24454i	0.782803	−	0.622270i	$-0.213789\pi$
$662$		0			0
$663$	−8542.55			−0.500400
$664$		0			0
$665$	11177.4	+	16032.5i	0.651790	+	0.934906i
$666$		0			0
$667$			20759.0i			1.20508i
$668$		0			0
$669$	−2978.17			−0.172112
$670$		0			0
$671$	14489.5			0.833620
$672$		0			0
$673$	11238.3			0.643693			0.321846	−	0.946792i	$-0.395697\pi$
$673$	11238.3			0.643693			0.321846	+	0.946792i	$0.395697\pi$
$674$		0			0
$675$	2203.24			0.125634
$676$		0			0
$677$		−	4728.28i		−	0.268423i	−0.990953	−	0.134212i	$-0.957150\pi$
$677$		−	4728.28i		−	0.268423i	0.990953	−	0.134212i	$-0.0428502\pi$
$678$		0			0
$679$	−4044.84	−	5801.78i	−0.228611	−	0.327912i
$680$		0			0
$681$	16998.1			0.956487
$682$		0			0
$683$			26608.6i			1.49070i	0.666673	+	0.745350i	$0.267718\pi$
$683$			26608.6i			1.49070i	−0.666673	+	0.745350i	$0.732282\pi$
$684$		0			0
$685$		−	2117.77i		−	0.118125i
$686$		0			0
$687$		−	19464.9i		−	1.08098i
$688$		0			0
$689$		−	4933.33i		−	0.272779i
$690$		0			0
$691$	−9292.47			−0.511580			−0.255790	−	0.966732i	$-0.582336\pi$
$691$	−9292.47			−0.511580			−0.255790	+	0.966732i	$0.582336\pi$
$692$		0			0
$693$	5170.31	+	7416.12i	0.283411	+	0.406516i
$694$		0			0
$695$			8823.33i			0.481565i
$696$		0			0
$697$	7791.22			0.423406
$698$		0			0
$699$	−10971.9			−0.593698
$700$		0			0
$701$	13950.7			0.751657			0.375829	−	0.926689i	$-0.377358\pi$
$701$	13950.7			0.751657			0.375829	+	0.926689i	$0.377358\pi$
$702$		0			0
$703$	−3784.58			−0.203042
$704$		0			0
$705$			5334.38i			0.284971i
$706$		0			0
$707$	−7381.03	−	10587.1i	−0.392634	−	0.563181i
$708$		0			0
$709$	29363.4			1.55538			0.777691	−	0.628646i	$-0.216391\pi$
$709$	29363.4			1.55538			0.777691	+	0.628646i	$0.216391\pi$
$710$		0			0
$711$			6227.69i			0.328491i
$712$		0			0
$713$			11557.5i			0.607059i
$714$		0			0
$715$		−	14639.6i		−	0.765721i
$716$		0			0
$717$			17418.3i			0.907251i
$718$		0			0
$719$	8612.18			0.446704			0.223352	−	0.974738i	$-0.428300\pi$
$719$	8612.18			0.446704			0.223352	+	0.974738i	$0.428300\pi$
$720$		0			0
$721$	−9439.63	+	6581.04i	−0.487587	+	0.339932i
$722$		0			0
$723$			1752.40i			0.0901416i
$724$		0			0
$725$	−19288.1			−0.988059
$726$		0			0
$727$	−19226.6			−0.980846			−0.490423	−	0.871484i	$-0.663158\pi$
$727$	−19226.6			−0.980846			−0.490423	+	0.871484i	$0.663158\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	−13483.9			−0.682245
$732$		0			0
$733$			33053.9i			1.66559i	0.553584	+	0.832793i	$0.313260\pi$
$733$			33053.9i			1.66559i	−0.553584	+	0.832793i	$0.686740\pi$
$734$		0			0
$735$	−6360.47	−	2344.46i	−0.319197	−	0.117655i
$736$		0			0
$737$	−14953.0			−0.747353
$738$		0			0
$739$		−	4071.71i		−	0.202679i	−0.994852	−	0.101340i	$-0.967687\pi$
$739$		−	4071.71i		−	0.202679i	0.994852	−	0.101340i	$-0.0323129\pi$
$740$		0			0
$741$			19689.9i			0.976147i
$742$		0			0
$743$		−	807.693i		−	0.0398807i	−0.999801	−	0.0199404i	$-0.993652\pi$
$743$		−	807.693i		−	0.0398807i	0.999801	−	0.0199404i	$-0.00634763\pi$
$744$		0			0
$745$			18109.5i			0.890577i
$746$		0			0
$747$	−3874.34			−0.189765
$748$		0			0
$749$	3130.07	+	4489.68i	0.152698	+	0.219024i
$750$		0			0
$751$		−	12604.7i		−	0.612454i	−0.951959	−	0.306227i	$-0.900933\pi$
$751$		−	12604.7i		−	0.612454i	0.951959	−	0.306227i	$-0.0990666\pi$
$752$		0			0
$753$	10378.9			0.502295
$754$		0			0
$755$	−12852.6			−0.619542
$756$		0			0
$757$	18825.2			0.903848			0.451924	−	0.892056i	$-0.350738\pi$
$757$	18825.2			0.903848			0.451924	+	0.892056i	$0.350738\pi$
$758$		0			0
$759$	14290.3			0.683404
$760$		0			0
$761$		−	7341.96i		−	0.349732i	−0.984592	−	0.174866i	$-0.944051\pi$
$761$		−	7341.96i		−	0.349732i	0.984592	−	0.174866i	$-0.0559492\pi$
$762$		0			0
$763$	−10172.1	+	7091.72i	−0.482642	+	0.336484i
$764$		0			0
$765$	−4120.57			−0.194744
$766$		0			0
$767$		−	13853.1i		−	0.652158i
$768$		0			0
$769$		−	37425.8i		−	1.75502i	−0.479560	−	0.877509i	$-0.659204\pi$
$769$		−	37425.8i		−	1.75502i	0.479560	−	0.877509i	$-0.340796\pi$
$770$		0			0
$771$		−	4504.66i		−	0.210417i
$772$		0			0
$773$		−	36961.6i		−	1.71982i	−0.510449	−	0.859908i	$-0.670521\pi$
$773$		−	36961.6i		−	1.71982i	0.510449	−	0.859908i	$-0.329479\pi$
$774$		0			0
$775$	−10738.6			−0.497733
$776$		0			0
$777$	1076.81	−	750.718i	0.0497171	−	0.0346613i
$778$		0			0
$779$		−	17958.1i		−	0.825952i
$780$		0			0
$781$	14670.4			0.672149
$782$		0			0
$783$	−6381.98			−0.291281
$784$		0			0
$785$	−19893.8			−0.904510
$786$		0			0
$787$	−22096.0			−1.00081			−0.500406	−	0.865791i	$-0.666816\pi$
$787$	−22096.0			−1.00081			−0.500406	+	0.865791i	$0.666816\pi$
$788$		0			0
$789$			15705.4i			0.708652i
$790$		0			0
$791$	−16619.8	+	11586.8i	−0.747068	+	0.520835i
$792$		0			0
$793$	−10945.5			−0.490148
$794$		0			0
$795$		−	2379.63i		−	0.106159i
$796$		0			0
$797$			40538.6i			1.80170i	0.434134	+	0.900848i	$0.357054\pi$
$797$			40538.6i			1.80170i	−0.434134	+	0.900848i	$0.642946\pi$
$798$		0			0
$799$			18758.7i			0.830583i
$800$		0			0
$801$			10986.7i			0.484640i
$802$		0			0
$803$	67123.2			2.94985
$804$		0			0
$805$	−8789.88	+	6128.05i	−0.384848	+	0.268305i
$806$		0			0
$807$			102.603i			0.00447558i
$808$		0			0
$809$	−15624.5			−0.679022			−0.339511	−	0.940602i	$-0.610262\pi$
$809$	−15624.5			−0.679022			−0.339511	+	0.940602i	$0.610262\pi$
$810$		0			0
$811$	−4997.93			−0.216401			−0.108200	−	0.994129i	$-0.534509\pi$
$811$	−4997.93			−0.216401			−0.108200	+	0.994129i	$0.534509\pi$
$812$		0			0
$813$	1961.80			0.0846292
$814$		0			0
$815$	18988.3			0.816111
$816$		0			0
$817$			31079.3i			1.33088i
$818$		0			0
$819$	−3905.72	−	5602.24i	−0.166639	−	0.239021i
$820$		0			0
$821$	−45367.1			−1.92853			−0.964266	−	0.264937i	$-0.914649\pi$
$821$	−45367.1			−1.92853			−0.964266	+	0.264937i	$0.914649\pi$
$822$		0			0
$823$			3913.77i			0.165766i	0.996559	+	0.0828829i	$0.0264128\pi$
$823$			3913.77i			0.165766i	−0.996559	+	0.0828829i	$0.973587\pi$
$824$		0			0
$825$			13277.7i			0.560328i
$826$		0			0
$827$			7368.11i			0.309812i	0.987929	+	0.154906i	$0.0495074\pi$
$827$			7368.11i			0.309812i	−0.987929	+	0.154906i	$0.950493\pi$
$828$		0			0
$829$		−	25074.8i		−	1.05052i	−0.850941	−	0.525261i	$-0.823968\pi$
$829$		−	25074.8i		−	1.05052i	0.850941	−	0.525261i	$-0.176032\pi$
$830$		0			0
$831$	−6751.74			−0.281848
$832$		0			0
$833$	−22367.0	−	8244.44i	−0.930338	−	0.342921i
$834$		0			0
$835$		−	16554.8i		−	0.686110i
$836$		0			0
$837$	−3553.15			−0.146732
$838$		0			0
$839$	−11380.1			−0.468279			−0.234139	−	0.972203i	$-0.575227\pi$
$839$	−11380.1			−0.468279			−0.234139	+	0.972203i	$0.575227\pi$
$840$		0			0
$841$	31481.5			1.29081
$842$		0			0
$843$	12640.4			0.516440
$844$		0			0
$845$		−	3414.31i		−	0.139001i
$846$		0			0
$847$	−24471.6	+	17060.9i	−0.992744	+	0.692113i
$848$		0			0
$849$	−6207.51			−0.250932
$850$		0			0
$851$		−	2074.92i		−	0.0835807i
$852$		0			0
$853$		−	2040.57i		−	0.0819082i	−0.999161	−	0.0409541i	$-0.986960\pi$
$853$		−	2040.57i		−	0.0819082i	0.999161	−	0.0409541i	$-0.0130397\pi$
$854$		0			0
$855$			9497.56i			0.379895i
$856$		0			0
$857$		−	14755.7i		−	0.588152i	−0.955782	−	0.294076i	$-0.904988\pi$
$857$		−	14755.7i		−	0.588152i	0.955782	−	0.294076i	$-0.0950120\pi$
$858$		0			0
$859$	44194.2			1.75540			0.877698	−	0.479213i	$-0.159078\pi$
$859$	44194.2			1.75540			0.877698	+	0.479213i	$0.159078\pi$
$860$		0			0
$861$	3562.21	+	5109.52i	0.140999	+	0.202244i
$862$		0			0
$863$			47866.0i			1.88804i	0.329888	+	0.944020i	$0.392989\pi$
$863$			47866.0i			1.88804i	−0.329888	+	0.944020i	$0.607011\pi$
$864$		0			0
$865$	−8610.97			−0.338476
$866$		0			0
$867$	248.747			0.00974381
$868$		0			0
$869$	−37530.9			−1.46507
$870$		0			0
$871$	11295.7			0.439425
$872$		0			0
$873$		−	3436.95i		−	0.133245i
$874$		0			0
$875$	−14415.9	−	20677.7i	−0.556966	−	0.798894i
$876$		0			0
$877$	25430.0			0.979146			0.489573	−	0.871962i	$-0.337153\pi$
$877$	25430.0			0.979146			0.489573	+	0.871962i	$0.337153\pi$
$878$		0			0
$879$			6082.47i			0.233398i
$880$		0			0
$881$		−	13602.8i		−	0.520192i	−0.965583	−	0.260096i	$-0.916246\pi$
$881$		−	13602.8i		−	0.520192i	0.965583	−	0.260096i	$-0.0837541\pi$
$882$		0			0
$883$		−	16042.8i		−	0.611419i	−0.952125	−	0.305710i	$-0.901106\pi$
$883$		−	16042.8i		−	0.611419i	0.952125	−	0.305710i	$-0.0988937\pi$
$884$		0			0
$885$		−	6682.14i		−	0.253805i
$886$		0			0
$887$	45834.8			1.73504			0.867520	−	0.497402i	$-0.165713\pi$
$887$	45834.8			1.73504			0.867520	+	0.497402i	$0.165713\pi$
$888$		0			0
$889$	8868.17	+	12720.2i	0.334565	+	0.479890i
$890$		0			0
$891$			4393.28i			0.165186i
$892$		0			0
$893$	43237.3			1.62025
$894$		0			0
$895$	−23420.7			−0.874713
$896$		0			0
$897$	−10795.1			−0.401825
$898$		0			0
$899$	31105.8			1.15399
$900$		0			0
$901$		−	8368.13i		−	0.309415i
$902$		0			0
$903$	−6164.96	−	8842.82i	−0.227195	−	0.325881i
$904$		0			0
$905$	11508.5			0.422714
$906$		0			0
$907$		−	11789.5i		−	0.431603i	−0.976437	−	0.215801i	$-0.930764\pi$
$907$		−	11789.5i		−	0.431603i	0.976437	−	0.215801i	$-0.0692363\pi$
$908$		0			0
$909$		−	6271.76i		−	0.228846i
$910$		0			0
$911$		−	34980.3i		−	1.27217i	−0.771618	−	0.636087i	$-0.780552\pi$
$911$		−	34980.3i		−	1.27217i	0.771618	−	0.636087i	$-0.219448\pi$
$912$		0			0
$913$		−	23348.5i		−	0.846356i
$914$		0			0
$915$	−5279.67			−0.190754
$916$		0			0
$917$	27665.7	−	19287.7i	0.996293	−	0.694587i
$918$		0			0
$919$		−	34836.9i		−	1.25045i	−0.780444	−	0.625225i	$-0.785007\pi$
$919$		−	34836.9i		−	1.25045i	0.780444	−	0.625225i	$-0.214993\pi$
$920$		0			0
$921$	−502.130			−0.0179650
$922$		0			0
$923$	−11082.2			−0.395207
$924$		0			0
$925$	1927.90			0.0685285
$926$		0			0
$927$	−5591.99			−0.198129
$928$		0			0
$929$		−	8027.80i		−	0.283513i	−0.989902	−	0.141757i	$-0.954725\pi$
$929$		−	8027.80i		−	0.283513i	0.989902	−	0.141757i	$-0.0452750\pi$
$930$		0			0
$931$	−19002.8	+	51554.1i	−0.668948	+	1.81484i
$932$		0			0
$933$	14161.1			0.496906
$934$		0			0
$935$		−	24832.4i		−	0.868562i
$936$		0			0
$937$		−	31147.0i		−	1.08594i	−0.839751	−	0.542972i	$-0.817299\pi$
$937$		−	31147.0i		−	1.08594i	0.839751	−	0.542972i	$-0.182701\pi$
$938$		0			0
$939$		−	18024.3i		−	0.626410i
$940$		0			0
$941$		−	14386.4i		−	0.498389i	−0.968453	−	0.249195i	$-0.919834\pi$
$941$		−	14386.4i		−	0.498389i	0.968453	−	0.249195i	$-0.0801659\pi$
$942$		0			0
$943$	9845.63			0.339998
$944$		0			0
$945$	−1883.96	−	2702.29i	−0.0648520	−	0.0930216i
$946$		0			0
$947$		−	1481.87i		−	0.0508494i	−0.999677	−	0.0254247i	$-0.991906\pi$
$947$		−	1481.87i		−	0.0508494i	0.999677	−	0.0254247i	$-0.00809380\pi$
$948$		0			0
$949$	−50705.8			−1.73444
$950$		0			0
$951$	5370.34			0.183118
$952$		0			0
$953$	−4125.04			−0.140213			−0.0701065	−	0.997540i	$-0.522334\pi$
$953$	−4125.04			−0.140213			−0.0701065	+	0.997540i	$0.522334\pi$
$954$		0			0
$955$	−12751.9			−0.432086
$956$		0			0
$957$		−	38460.6i		−	1.29912i
$958$		0			0
$959$	4883.96	−	3404.96i	0.164454	−	0.114653i
$960$		0			0
$961$	−12472.9			−0.418680
$962$		0			0
$963$			2659.67i			0.0889995i
$964$		0			0
$965$			25311.9i			0.844372i
$966$		0			0
$967$			17872.4i			0.594351i	0.954823	+	0.297175i	$0.0960446\pi$
$967$			17872.4i			0.594351i	−0.954823	+	0.297175i	$0.903955\pi$
$968$		0			0
$969$			33398.8i			1.10725i
$970$		0			0
$971$	6473.23			0.213940			0.106970	−	0.994262i	$-0.465885\pi$
$971$	6473.23			0.213940			0.106970	+	0.994262i	$0.465885\pi$
$972$		0			0
$973$	−20348.2	+	14186.2i	−0.670435	+	0.467408i
$974$		0			0
$975$		−	10030.2i		−	0.329459i
$976$		0			0
$977$	32958.4			1.07926			0.539628	−	0.841904i	$-0.318565\pi$
$977$	32958.4			1.07926			0.539628	+	0.841904i	$0.318565\pi$
$978$		0			0
$979$	−66210.9			−2.16150
$980$		0			0
$981$	−6025.93			−0.196119
$982$		0			0
$983$	−11342.0			−0.368010			−0.184005	−	0.982925i	$-0.558906\pi$
$983$	−11342.0			−0.368010			−0.184005	+	0.982925i	$0.558906\pi$
$984$		0			0
$985$			3232.06i			0.104550i
$986$		0			0
$987$	−12302.1	+	8576.64i	−0.396736	+	0.276593i
$988$		0			0
$989$	−17039.4			−0.547847
$990$		0			0
$991$		−	15910.0i		−	0.509987i	−0.966943	−	0.254994i	$-0.917927\pi$
$991$		−	15910.0i		−	0.509987i	0.966943	−	0.254994i	$-0.0820733\pi$
$992$		0			0
$993$		−	15177.1i		−	0.485026i
$994$		0			0
$995$			23914.5i			0.761949i
$996$		0			0
$997$			3756.74i			0.119335i	0.998218	+	0.0596676i	$0.0190041\pi$
$997$			3756.74i			0.119335i	−0.998218	+	0.0596676i	$0.980996\pi$
$998$		0			0
$999$	637.895			0.0202023

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.b.h.895.4		12
4.3	odd	2		1344.4.b.g.895.4		12
7.6	odd	2		1344.4.b.g.895.9		12
8.3	odd	2		84.4.b.b.55.4	yes	12
8.5	even	2		84.4.b.a.55.3	✓	12
24.5	odd	2		252.4.b.f.55.10		12
24.11	even	2		252.4.b.e.55.9		12
28.27	even	2	inner	1344.4.b.h.895.9		12
56.13	odd	2		84.4.b.b.55.3	yes	12
56.27	even	2		84.4.b.a.55.4	yes	12
168.83	odd	2		252.4.b.f.55.9		12
168.125	even	2		252.4.b.e.55.10		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.4.b.a.55.3	✓	12	8.5	even	2
84.4.b.a.55.4	yes	12	56.27	even	2
84.4.b.b.55.3	yes	12	56.13	odd	2
84.4.b.b.55.4	yes	12	8.3	odd	2
252.4.b.e.55.9		12	24.11	even	2
252.4.b.e.55.10		12	168.125	even	2
252.4.b.f.55.9		12	168.83	odd	2
252.4.b.f.55.10		12	24.5	odd	2
1344.4.b.g.895.4		12	4.3	odd	2
1344.4.b.g.895.9		12	7.6	odd	2
1344.4.b.h.895.4		12	1.1	even	1	trivial
1344.4.b.h.895.9		12	28.27	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -6.58775i q^{5} +(15.1925 - 10.5918i) q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} -6.58775i q^{5} +(15.1925 - 10.5918i) q^{7} +9.00000 q^{9} +54.2380i q^{11} -40.9722i q^{13} -19.7632i q^{15} -69.4988i q^{17} -160.189 q^{19} +(45.5776 - 31.7754i) q^{21} -87.8244i q^{23} +81.6016 q^{25} +27.0000 q^{27} -236.369 q^{29} -131.598 q^{31} +162.714i q^{33} +(-69.7762 - 100.085i) q^{35} +23.6257 q^{37} -122.916i q^{39} +112.106i q^{41} -194.017i q^{43} -59.2897i q^{45} -269.914 q^{47} +(118.627 - 321.833i) q^{49} -208.496i q^{51} +120.407 q^{53} +357.307 q^{55} -480.567 q^{57} +338.109 q^{59} -267.146i q^{61} +(136.733 - 95.3263i) q^{63} -269.914 q^{65} +275.691i q^{67} -263.473i q^{69} -270.482i q^{71} -1237.57i q^{73} +244.805 q^{75} +(574.479 + 824.014i) q^{77} +691.966i q^{79} +81.0000 q^{81} -430.482 q^{83} -457.841 q^{85} -709.108 q^{87} +1220.75i q^{89} +(-433.969 - 622.472i) q^{91} -394.795 q^{93} +1055.28i q^{95} -381.884i q^{97} +488.142i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 36 q^{3} + 10 q^{7} + 108 q^{9}+O(q^{10}) \) 12 * q + 36 * q^3 + 10 * q^7 + 108 * q^9	\( 12 q + 36 q^{3} + 10 q^{7} + 108 q^{9} + 84 q^{19} + 30 q^{21} - 216 q^{25} + 324 q^{27} - 200 q^{29} - 384 q^{31} - 84 q^{35} + 244 q^{37} + 280 q^{47} - 424 q^{49} + 16 q^{53} + 212 q^{55} + 252 q^{57} - 1168 q^{59} + 90 q^{63} + 280 q^{65} - 648 q^{75} - 968 q^{77} + 972 q^{81} + 968 q^{83} + 852 q^{85} - 600 q^{87} - 1648 q^{91} - 1152 q^{93}+O(q^{100}) \) 12 * q + 36 * q^3 + 10 * q^7 + 108 * q^9 + 84 * q^19 + 30 * q^21 - 216 * q^25 + 324 * q^27 - 200 * q^29 - 384 * q^31 - 84 * q^35 + 244 * q^37 + 280 * q^47 - 424 * q^49 + 16 * q^53 + 212 * q^55 + 252 * q^57 - 1168 * q^59 + 90 * q^63 + 280 * q^65 - 648 * q^75 - 968 * q^77 + 972 * q^81 + 968 * q^83 + 852 * q^85 - 600 * q^87 - 1648 * q^91 - 1152 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more