LMFDB - Embedded newform 2320.2.d.f.929.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2320,2,Mod(929,2320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2320, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2320.929");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2320 = 2^{4} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2320.d (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:	$18.5252932689$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 145)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			929.3
Root			$-0.517638i$ of defining polynomial
Character	$\chi$	$=$	2320.929
Dual form			2320.2.d.f.929.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{3} +(-1.73205 - 1.41421i) q^{5} +2.44949i q^{7} +1.00000 q^{9} +1.26795 q^{11} +1.79315i q^{13} +(2.00000 - 2.44949i) q^{15} +1.41421i q^{17} -3.26795 q^{19} -3.46410 q^{21} -6.31319i q^{23} +(1.00000 + 4.89898i) q^{25} +5.65685i q^{27} -1.00000 q^{29} +8.73205 q^{31} +1.79315i q^{33} +(3.46410 - 4.24264i) q^{35} +9.14162i q^{37} -2.53590 q^{39} -6.92820 q^{41} +9.14162i q^{43} +(-1.73205 - 1.41421i) q^{45} -1.41421i q^{47} +1.00000 q^{49} -2.00000 q^{51} -5.93426i q^{53} +(-2.19615 - 1.79315i) q^{55} -4.62158i q^{57} -10.3923 q^{59} +2.92820 q^{61} +2.44949i q^{63} +(2.53590 - 3.10583i) q^{65} -4.24264i q^{67} +8.92820 q^{69} -3.46410 q^{71} +7.34847i q^{73} +(-6.92820 + 1.41421i) q^{75} +3.10583i q^{77} -4.19615 q^{79} -5.00000 q^{81} +10.1769i q^{83} +(2.00000 - 2.44949i) q^{85} -1.41421i q^{87} -10.3923 q^{89} -4.39230 q^{91} +12.3490i q^{93} +(5.66025 + 4.62158i) q^{95} -10.9348i q^{97} +1.26795 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{9} + 12 q^{11} + 8 q^{15} - 20 q^{19} + 4 q^{25} - 4 q^{29} + 28 q^{31} - 24 q^{39} + 4 q^{49} - 8 q^{51} + 12 q^{55} - 16 q^{61} + 24 q^{65} + 8 q^{69} + 4 q^{79} - 20 q^{81} + 8 q^{85} + 24 q^{91}+ \cdots + 12 q^{99}+O(q^{100}) $ 4 * q + 4 * q^9 + 12 * q^11 + 8 * q^15 - 20 * q^19 + 4 * q^25 - 4 * q^29 + 28 * q^31 - 24 * q^39 + 4 * q^49 - 8 * q^51 + 12 * q^55 - 16 * q^61 + 24 * q^65 + 8 * q^69 + 4 * q^79 - 20 * q^81 + 8 * q^85 + 24 * q^91 - 12 * q^95 + 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2320\mathbb{Z}\right)^\times$.

$n$	$321$	$581$	$1857$	$2031$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$			1.41421i			0.816497i	0.912871	+	0.408248i	$0.133860\pi$
$3$			1.41421i			0.816497i	−0.912871	+	0.408248i	$0.866140\pi$
$4$		0			0
$5$	−1.73205	−	1.41421i	−0.774597	−	0.632456i
$5$	−1.73205	−	1.41421i	−0.774597	−	0.632456i
$6$		0			0
$7$			2.44949i			0.925820i	0.886405	+	0.462910i	$0.153195\pi$
$7$			2.44949i			0.925820i	−0.886405	+	0.462910i	$0.846805\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$	1.26795			0.382301			0.191151	−	0.981561i	$-0.438778\pi$
$11$	1.26795			0.382301			0.191151	+	0.981561i	$0.438778\pi$
$12$		0			0
$13$			1.79315i			0.497331i	0.968589	+	0.248665i	$0.0799919\pi$
$13$			1.79315i			0.497331i	−0.968589	+	0.248665i	$0.920008\pi$
$14$		0			0
$15$	2.00000	−	2.44949i	0.516398	−	0.632456i
$16$		0			0
$17$			1.41421i			0.342997i	0.985184	+	0.171499i	$0.0548609\pi$
$17$			1.41421i			0.342997i	−0.985184	+	0.171499i	$0.945139\pi$
$18$		0			0
$19$	−3.26795			−0.749719			−0.374859	−	0.927082i	$-0.622309\pi$
$19$	−3.26795			−0.749719			−0.374859	+	0.927082i	$0.622309\pi$
$20$		0			0
$21$	−3.46410			−0.755929
$22$		0			0
$23$		−	6.31319i		−	1.31639i	−0.752847	−	0.658196i	$-0.771320\pi$
$23$		−	6.31319i		−	1.31639i	0.752847	−	0.658196i	$-0.228680\pi$
$24$		0			0
$25$	1.00000	+	4.89898i	0.200000	+	0.979796i
$26$		0			0
$27$			5.65685i			1.08866i
$28$		0			0
$29$	−1.00000			−0.185695
$29$	−1.00000			−0.185695
$30$		0			0
$31$	8.73205			1.56832			0.784161	−	0.620557i	$-0.213093\pi$
$31$	8.73205			1.56832			0.784161	+	0.620557i	$0.213093\pi$
$32$		0			0
$33$			1.79315i			0.312148i
$34$		0			0
$35$	3.46410	−	4.24264i	0.585540	−	0.717137i
$36$		0			0
$37$			9.14162i			1.50287i	0.659805	+	0.751437i	$0.270639\pi$
$37$			9.14162i			1.50287i	−0.659805	+	0.751437i	$0.729361\pi$
$38$		0			0
$39$	−2.53590			−0.406069
$40$		0			0
$41$	−6.92820			−1.08200			−0.541002	−	0.841021i	$-0.681955\pi$
$41$	−6.92820			−1.08200			−0.541002	+	0.841021i	$0.681955\pi$
$42$		0			0
$43$			9.14162i			1.39408i	0.717030	+	0.697042i	$0.245501\pi$
$43$			9.14162i			1.39408i	−0.717030	+	0.697042i	$0.754499\pi$
$44$		0			0
$45$	−1.73205	−	1.41421i	−0.258199	−	0.210819i
$46$		0			0
$47$		−	1.41421i		−	0.206284i	−0.994667	−	0.103142i	$-0.967110\pi$
$47$		−	1.41421i		−	0.206284i	0.994667	−	0.103142i	$-0.0328896\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$	−2.00000			−0.280056
$52$		0			0
$53$		−	5.93426i		−	0.815133i	−0.913176	−	0.407566i	$-0.866378\pi$
$53$		−	5.93426i		−	0.815133i	0.913176	−	0.407566i	$-0.133622\pi$
$54$		0			0
$55$	−2.19615	−	1.79315i	−0.296129	−	0.241788i
$56$		0			0
$57$		−	4.62158i		−	0.612143i
$58$		0			0
$59$	−10.3923			−1.35296			−0.676481	−	0.736460i	$-0.736496\pi$
$59$	−10.3923			−1.35296			−0.676481	+	0.736460i	$0.736496\pi$
$60$		0			0
$61$	2.92820			0.374918			0.187459	−	0.982272i	$-0.439975\pi$
$61$	2.92820			0.374918			0.187459	+	0.982272i	$0.439975\pi$
$62$		0			0
$63$			2.44949i			0.308607i
$64$		0			0
$65$	2.53590	−	3.10583i	0.314539	−	0.385231i
$66$		0			0
$67$		−	4.24264i		−	0.518321i	−0.965834	−	0.259161i	$-0.916554\pi$
$67$		−	4.24264i		−	0.518321i	0.965834	−	0.259161i	$-0.0834459\pi$
$68$		0			0
$69$	8.92820			1.07483
$70$		0			0
$71$	−3.46410			−0.411113			−0.205557	−	0.978645i	$-0.565900\pi$
$71$	−3.46410			−0.411113			−0.205557	+	0.978645i	$0.565900\pi$
$72$		0			0
$73$			7.34847i			0.860073i	0.902811	+	0.430037i	$0.141499\pi$
$73$			7.34847i			0.860073i	−0.902811	+	0.430037i	$0.858501\pi$
$74$		0			0
$75$	−6.92820	+	1.41421i	−0.800000	+	0.163299i
$76$		0			0
$77$			3.10583i			0.353942i
$78$		0			0
$79$	−4.19615			−0.472104			−0.236052	−	0.971740i	$-0.575854\pi$
$79$	−4.19615			−0.472104			−0.236052	+	0.971740i	$0.575854\pi$
$80$		0			0
$81$	−5.00000			−0.555556
$82$		0			0
$83$			10.1769i			1.11706i	0.829484	+	0.558530i	$0.188634\pi$
$83$			10.1769i			1.11706i	−0.829484	+	0.558530i	$0.811366\pi$
$84$		0			0
$85$	2.00000	−	2.44949i	0.216930	−	0.265684i
$86$		0			0
$87$		−	1.41421i		−	0.151620i
$88$		0			0
$89$	−10.3923			−1.10158			−0.550791	−	0.834643i	$-0.685674\pi$
$89$	−10.3923			−1.10158			−0.550791	+	0.834643i	$0.685674\pi$
$90$		0			0
$91$	−4.39230			−0.460439
$92$		0			0
$93$			12.3490i			1.28053i
$94$		0			0
$95$	5.66025	+	4.62158i	0.580730	+	0.474164i
$96$		0			0
$97$		−	10.9348i		−	1.11026i	−0.831764	−	0.555129i	$-0.812669\pi$
$97$		−	10.9348i		−	1.11026i	0.831764	−	0.555129i	$-0.187331\pi$
$98$		0			0
$99$	1.26795			0.127434
$100$		0			0
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$			9.14162i			0.900751i	0.892839	+	0.450375i	$0.148710\pi$
$103$			9.14162i			0.900751i	−0.892839	+	0.450375i	$0.851290\pi$
$104$		0			0
$105$	6.00000	+	4.89898i	0.585540	+	0.478091i
$106$		0			0
$107$			18.6622i			1.80414i	0.431589	+	0.902070i	$0.357953\pi$
$107$			18.6622i			1.80414i	−0.431589	+	0.902070i	$0.642047\pi$
$108$		0			0
$109$	−17.8564			−1.71033			−0.855167	−	0.518353i	$-0.826545\pi$
$109$	−17.8564			−1.71033			−0.855167	+	0.518353i	$0.826545\pi$
$110$		0			0
$111$	−12.9282			−1.22709
$112$		0			0
$113$			13.0053i			1.22344i	0.791075	+	0.611719i	$0.209522\pi$
$113$			13.0053i			1.22344i	−0.791075	+	0.611719i	$0.790478\pi$
$114$		0			0
$115$	−8.92820	+	10.9348i	−0.832559	+	1.01967i
$116$		0			0
$117$			1.79315i			0.165777i
$118$		0			0
$119$	−3.46410			−0.317554
$120$		0			0
$121$	−9.39230			−0.853846
$122$		0			0
$123$		−	9.79796i		−	0.883452i
$124$		0			0
$125$	5.19615	−	9.89949i	0.464758	−	0.885438i
$126$		0			0
$127$		−	4.24264i		−	0.376473i	−0.982124	−	0.188237i	$-0.939723\pi$
$127$		−	4.24264i		−	0.376473i	0.982124	−	0.188237i	$-0.0602772\pi$
$128$		0			0
$129$	−12.9282			−1.13826
$130$		0			0
$131$	4.73205			0.413441			0.206721	−	0.978400i	$-0.433721\pi$
$131$	4.73205			0.413441			0.206721	+	0.978400i	$0.433721\pi$
$132$		0			0
$133$		−	8.00481i		−	0.694105i
$134$		0			0
$135$	8.00000	−	9.79796i	0.688530	−	0.843274i
$136$		0			0
$137$			14.7985i			1.26432i	0.774838	+	0.632159i	$0.217831\pi$
$137$			14.7985i			1.26432i	−0.774838	+	0.632159i	$0.782169\pi$
$138$		0			0
$139$	−8.00000			−0.678551			−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000			−0.678551			−0.339276	+	0.940687i	$0.610182\pi$
$140$		0			0
$141$	2.00000			0.168430
$142$		0			0
$143$			2.27362i			0.190130i
$144$		0			0
$145$	1.73205	+	1.41421i	0.143839	+	0.117444i
$146$		0			0
$147$			1.41421i			0.116642i
$148$		0			0
$149$	0.928203			0.0760414			0.0380207	−	0.999277i	$-0.487895\pi$
$149$	0.928203			0.0760414			0.0380207	+	0.999277i	$0.487895\pi$
$150$		0			0
$151$	18.7846			1.52867			0.764335	−	0.644819i	$-0.223067\pi$
$151$	18.7846			1.52867			0.764335	+	0.644819i	$0.223067\pi$
$152$		0			0
$153$			1.41421i			0.114332i
$154$		0			0
$155$	−15.1244	−	12.3490i	−1.21482	−	0.991894i
$156$		0			0
$157$			10.9348i			0.872690i	0.899780	+	0.436345i	$0.143727\pi$
$157$			10.9348i			0.872690i	−0.899780	+	0.436345i	$0.856273\pi$
$158$		0			0
$159$	8.39230			0.665553
$160$		0			0
$161$	15.4641			1.21874
$162$		0			0
$163$			14.0406i			1.09974i	0.835249	+	0.549872i	$0.185324\pi$
$163$			14.0406i			1.09974i	−0.835249	+	0.549872i	$0.814676\pi$
$164$		0			0
$165$	2.53590	−	3.10583i	0.197419	−	0.241788i
$166$		0			0
$167$		−	11.2122i		−	0.867624i	−0.901003	−	0.433812i	$-0.857168\pi$
$167$		−	11.2122i		−	0.867624i	0.901003	−	0.433812i	$-0.142832\pi$
$168$		0			0
$169$	9.78461			0.752662
$170$		0			0
$171$	−3.26795			−0.249906
$172$		0			0
$173$		−	4.62158i		−	0.351372i	−0.984446	−	0.175686i	$-0.943786\pi$
$173$		−	4.62158i		−	0.351372i	0.984446	−	0.175686i	$-0.0562144\pi$
$174$		0			0
$175$	−12.0000	+	2.44949i	−0.907115	+	0.185164i
$176$		0			0
$177$		−	14.6969i		−	1.10469i
$178$		0			0
$179$	2.53590			0.189542			0.0947710	−	0.995499i	$-0.469788\pi$
$179$	2.53590			0.189542			0.0947710	+	0.995499i	$0.469788\pi$
$180$		0			0
$181$	−16.0000			−1.18927			−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000			−1.18927			−0.594635	+	0.803996i	$0.702704\pi$
$182$		0			0
$183$			4.14110i			0.306119i
$184$		0			0
$185$	12.9282	−	15.8338i	0.950500	−	1.16412i
$186$		0			0
$187$			1.79315i			0.131128i
$188$		0			0
$189$	−13.8564			−1.00791
$190$		0			0
$191$	−15.1244			−1.09436			−0.547180	−	0.837015i	$-0.684299\pi$
$191$	−15.1244			−1.09436			−0.547180	+	0.837015i	$0.684299\pi$
$192$		0			0
$193$			0.656339i			0.0472443i	0.999721	+	0.0236222i	$0.00751986\pi$
$193$			0.656339i			0.0472443i	−0.999721	+	0.0236222i	$0.992480\pi$
$194$		0			0
$195$	4.39230	+	3.58630i	0.314539	+	0.256820i
$196$		0			0
$197$		−	22.4243i		−	1.59767i	−0.601551	−	0.798834i	$-0.705450\pi$
$197$		−	22.4243i		−	1.59767i	0.601551	−	0.798834i	$-0.294550\pi$
$198$		0			0
$199$	−14.0000			−0.992434			−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000			−0.992434			−0.496217	+	0.868199i	$0.665278\pi$
$200$		0			0
$201$	6.00000			0.423207
$202$		0			0
$203$		−	2.44949i		−	0.171920i
$204$		0			0
$205$	12.0000	+	9.79796i	0.838116	+	0.684319i
$206$		0			0
$207$		−	6.31319i		−	0.438797i
$208$		0			0
$209$	−4.14359			−0.286618
$210$		0			0
$211$	14.7321			1.01420			0.507098	−	0.861888i	$-0.330718\pi$
$211$	14.7321			1.01420			0.507098	+	0.861888i	$0.330718\pi$
$212$		0			0
$213$		−	4.89898i		−	0.335673i
$214$		0			0
$215$	12.9282	−	15.8338i	0.881696	−	1.07985i
$216$		0			0
$217$			21.3891i			1.45198i
$218$		0			0
$219$	−10.3923			−0.702247
$220$		0			0
$221$	−2.53590			−0.170583
$222$		0			0
$223$		−	15.3533i		−	1.02813i	−0.857751	−	0.514066i	$-0.828139\pi$
$223$		−	15.3533i		−	1.02813i	0.857751	−	0.514066i	$-0.171861\pi$
$224$		0			0
$225$	1.00000	+	4.89898i	0.0666667	+	0.326599i
$226$		0			0
$227$			20.4553i			1.35767i	0.734292	+	0.678834i	$0.237514\pi$
$227$			20.4553i			1.35767i	−0.734292	+	0.678834i	$0.762486\pi$
$228$		0			0
$229$	−18.7846			−1.24132			−0.620661	−	0.784079i	$-0.713136\pi$
$229$	−18.7846			−1.24132			−0.620661	+	0.784079i	$0.713136\pi$
$230$		0			0
$231$	−4.39230			−0.288992
$232$		0			0
$233$			3.38323i			0.221643i	0.993840	+	0.110821i	$0.0353481\pi$
$233$			3.38323i			0.221643i	−0.993840	+	0.110821i	$0.964652\pi$
$234$		0			0
$235$	−2.00000	+	2.44949i	−0.130466	+	0.159787i
$236$		0			0
$237$		−	5.93426i		−	0.385471i
$238$		0			0
$239$	10.3923			0.672222			0.336111	−	0.941822i	$-0.390888\pi$
$239$	10.3923			0.672222			0.336111	+	0.941822i	$0.390888\pi$
$240$		0			0
$241$	11.4641			0.738468			0.369234	−	0.929337i	$-0.379620\pi$
$241$	11.4641			0.738468			0.369234	+	0.929337i	$0.379620\pi$
$242$		0			0
$243$			9.89949i			0.635053i
$244$		0			0
$245$	−1.73205	−	1.41421i	−0.110657	−	0.0903508i
$246$		0			0
$247$		−	5.85993i		−	0.372858i
$248$		0			0
$249$	−14.3923			−0.912075
$250$		0			0
$251$	−15.1244			−0.954641			−0.477320	−	0.878729i	$-0.658392\pi$
$251$	−15.1244			−0.954641			−0.477320	+	0.878729i	$0.658392\pi$
$252$		0			0
$253$		−	8.00481i		−	0.503258i
$254$		0			0
$255$	3.46410	+	2.82843i	0.216930	+	0.177123i
$256$		0			0
$257$		−	9.52056i		−	0.593876i	−0.954897	−	0.296938i	$-0.904035\pi$
$257$		−	9.52056i		−	0.593876i	0.954897	−	0.296938i	$-0.0959655\pi$
$258$		0			0
$259$	−22.3923			−1.39139
$260$		0			0
$261$	−1.00000			−0.0618984
$262$		0			0
$263$			4.79744i			0.295823i	0.989001	+	0.147912i	$0.0472551\pi$
$263$			4.79744i			0.295823i	−0.989001	+	0.147912i	$0.952745\pi$
$264$		0			0
$265$	−8.39230	+	10.2784i	−0.515535	+	0.631399i
$266$		0			0
$267$		−	14.6969i		−	0.899438i
$268$		0			0
$269$	12.0000			0.731653			0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000			0.731653			0.365826	+	0.930683i	$0.380786\pi$
$270$		0			0
$271$	26.7321			1.62386			0.811928	−	0.583757i	$-0.198418\pi$
$271$	26.7321			1.62386			0.811928	+	0.583757i	$0.198418\pi$
$272$		0			0
$273$		−	6.21166i		−	0.375947i
$274$		0			0
$275$	1.26795	+	6.21166i	0.0764602	+	0.374577i
$276$		0			0
$277$			18.7637i			1.12740i	0.825979	+	0.563701i	$0.190623\pi$
$277$			18.7637i			1.12740i	−0.825979	+	0.563701i	$0.809377\pi$
$278$		0			0
$279$	8.73205			0.522774
$280$		0			0
$281$	32.7846			1.95577			0.977883	−	0.209153i	$-0.0670707\pi$
$281$	32.7846			1.95577			0.977883	+	0.209153i	$0.0670707\pi$
$282$		0			0
$283$		−	27.4249i		−	1.63024i	−0.579293	−	0.815119i	$-0.696671\pi$
$283$		−	27.4249i		−	1.63024i	0.579293	−	0.815119i	$-0.303329\pi$
$284$		0			0
$285$	−6.53590	+	8.00481i	−0.387153	+	0.474164i
$286$		0			0
$287$		−	16.9706i		−	1.00174i
$288$		0			0
$289$	15.0000			0.882353
$290$		0			0
$291$	15.4641			0.906522
$292$		0			0
$293$		−	30.7338i		−	1.79549i	−0.440520	−	0.897743i	$-0.645206\pi$
$293$		−	30.7338i		−	1.79549i	0.440520	−	0.897743i	$-0.354794\pi$
$294$		0			0
$295$	18.0000	+	14.6969i	1.04800	+	0.855689i
$296$		0			0
$297$			7.17260i			0.416197i
$298$		0			0
$299$	11.3205			0.654682
$300$		0			0
$301$	−22.3923			−1.29067
$302$		0			0
$303$		−	8.48528i		−	0.487467i
$304$		0			0
$305$	−5.07180	−	4.14110i	−0.290410	−	0.237119i
$306$		0			0
$307$		−	10.4543i		−	0.596658i	−0.954463	−	0.298329i	$-0.903571\pi$
$307$		−	10.4543i		−	0.596658i	0.954463	−	0.298329i	$-0.0964293\pi$
$308$		0			0
$309$	−12.9282			−0.735460
$310$		0			0
$311$	6.58846			0.373597			0.186799	−	0.982398i	$-0.440189\pi$
$311$	6.58846			0.373597			0.186799	+	0.982398i	$0.440189\pi$
$312$		0			0
$313$			23.6627i			1.33749i	0.743490	+	0.668747i	$0.233169\pi$
$313$			23.6627i			1.33749i	−0.743490	+	0.668747i	$0.766831\pi$
$314$		0			0
$315$	3.46410	−	4.24264i	0.195180	−	0.239046i
$316$		0			0
$317$		−	5.75839i		−	0.323423i	−0.986838	−	0.161712i	$-0.948299\pi$
$317$		−	5.75839i		−	0.323423i	0.986838	−	0.161712i	$-0.0517015\pi$
$318$		0			0
$319$	−1.26795			−0.0709915
$320$		0			0
$321$	−26.3923			−1.47307
$322$		0			0
$323$		−	4.62158i		−	0.257151i
$324$		0			0
$325$	−8.78461	+	1.79315i	−0.487282	+	0.0994661i
$326$		0			0
$327$		−	25.2528i		−	1.39648i
$328$		0			0
$329$	3.46410			0.190982
$330$		0			0
$331$	18.1962			1.00015			0.500075	−	0.865982i	$-0.333306\pi$
$331$	18.1962			1.00015			0.500075	+	0.865982i	$0.333306\pi$
$332$		0			0
$333$			9.14162i			0.500958i
$334$		0			0
$335$	−6.00000	+	7.34847i	−0.327815	+	0.401490i
$336$		0			0
$337$		−	27.9053i		−	1.52010i	−0.649864	−	0.760050i	$-0.725174\pi$
$337$		−	27.9053i		−	1.52010i	0.649864	−	0.760050i	$-0.274826\pi$
$338$		0			0
$339$	−18.3923			−0.998933
$340$		0			0
$341$	11.0718			0.599571
$342$		0			0
$343$			19.5959i			1.05808i
$344$		0			0
$345$	−15.4641	−	12.6264i	−0.832559	−	0.679782i
$346$		0			0
$347$		−	21.4906i		−	1.15368i	−0.816859	−	0.576838i	$-0.804286\pi$
$347$		−	21.4906i		−	1.15368i	0.816859	−	0.576838i	$-0.195714\pi$
$348$		0			0
$349$	22.7846			1.21963			0.609816	−	0.792543i	$-0.291243\pi$
$349$	22.7846			1.21963			0.609816	+	0.792543i	$0.291243\pi$
$350$		0			0
$351$	−10.1436			−0.541425
$352$		0			0
$353$			31.9449i			1.70026i	0.526576	+	0.850128i	$0.323475\pi$
$353$			31.9449i			1.70026i	−0.526576	+	0.850128i	$0.676525\pi$
$354$		0			0
$355$	6.00000	+	4.89898i	0.318447	+	0.260011i
$356$		0			0
$357$		−	4.89898i		−	0.259281i
$358$		0			0
$359$	−7.26795			−0.383588			−0.191794	−	0.981435i	$-0.561430\pi$
$359$	−7.26795			−0.383588			−0.191794	+	0.981435i	$0.561430\pi$
$360$		0			0
$361$	−8.32051			−0.437921
$362$		0			0
$363$		−	13.2827i		−	0.697162i
$364$		0			0
$365$	10.3923	−	12.7279i	0.543958	−	0.666210i
$366$		0			0
$367$			33.6365i			1.75581i	0.478833	+	0.877906i	$0.341060\pi$
$367$			33.6365i			1.75581i	−0.478833	+	0.877906i	$0.658940\pi$
$368$		0			0
$369$	−6.92820			−0.360668
$370$		0			0
$371$	14.5359			0.754666
$372$		0			0
$373$			19.5959i			1.01464i	0.861758	+	0.507319i	$0.169363\pi$
$373$			19.5959i			1.01464i	−0.861758	+	0.507319i	$0.830637\pi$
$374$		0			0
$375$	14.0000	+	7.34847i	0.722957	+	0.379473i
$376$		0			0
$377$		−	1.79315i		−	0.0923520i
$378$		0			0
$379$	−16.1962			−0.831940			−0.415970	−	0.909378i	$-0.636558\pi$
$379$	−16.1962			−0.831940			−0.415970	+	0.909378i	$0.636558\pi$
$380$		0			0
$381$	6.00000			0.307389
$382$		0			0
$383$		−	0.933740i		−	0.0477119i	−0.999715	−	0.0238559i	$-0.992406\pi$
$383$		−	0.933740i		−	0.0477119i	0.999715	−	0.0238559i	$-0.00759430\pi$
$384$		0			0
$385$	4.39230	−	5.37945i	0.223853	−	0.274162i
$386$		0			0
$387$			9.14162i			0.464695i
$388$		0			0
$389$	20.7846			1.05382			0.526911	−	0.849921i	$-0.323350\pi$
$389$	20.7846			1.05382			0.526911	+	0.849921i	$0.323350\pi$
$390$		0			0
$391$	8.92820			0.451519
$392$		0			0
$393$			6.69213i			0.337573i
$394$		0			0
$395$	7.26795	+	5.93426i	0.365690	+	0.298585i
$396$		0			0
$397$			4.89898i			0.245873i	0.992415	+	0.122936i	$0.0392311\pi$
$397$			4.89898i			0.245873i	−0.992415	+	0.122936i	$0.960769\pi$
$398$		0			0
$399$	11.3205			0.566734
$400$		0			0
$401$	37.8564			1.89046			0.945229	−	0.326407i	$-0.105838\pi$
$401$	37.8564			1.89046			0.945229	+	0.326407i	$0.105838\pi$
$402$		0			0
$403$			15.6579i			0.779975i
$404$		0			0
$405$	8.66025	+	7.07107i	0.430331	+	0.351364i
$406$		0			0
$407$			11.5911i			0.574550i
$408$		0			0
$409$	−9.07180			−0.448571			−0.224286	−	0.974523i	$-0.572005\pi$
$409$	−9.07180			−0.448571			−0.224286	+	0.974523i	$0.572005\pi$
$410$		0			0
$411$	−20.9282			−1.03231
$412$		0			0
$413$		−	25.4558i		−	1.25260i
$414$		0			0
$415$	14.3923	−	17.6269i	0.706490	−	0.865271i
$416$		0			0
$417$		−	11.3137i		−	0.554035i
$418$		0			0
$419$	−6.00000			−0.293119			−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000			−0.293119			−0.146560	+	0.989202i	$0.546820\pi$
$420$		0			0
$421$	5.46410			0.266304			0.133152	−	0.991096i	$-0.457490\pi$
$421$	5.46410			0.266304			0.133152	+	0.991096i	$0.457490\pi$
$422$		0			0
$423$		−	1.41421i		−	0.0687614i
$424$		0			0
$425$	−6.92820	+	1.41421i	−0.336067	+	0.0685994i
$426$		0			0
$427$			7.17260i			0.347107i
$428$		0			0
$429$	−3.21539			−0.155241
$430$		0			0
$431$	37.1769			1.79075			0.895374	−	0.445314i	$-0.146908\pi$
$431$	37.1769			1.79075			0.895374	+	0.445314i	$0.146908\pi$
$432$		0			0
$433$			2.92996i			0.140805i	0.997519	+	0.0704025i	$0.0224284\pi$
$433$			2.92996i			0.140805i	−0.997519	+	0.0704025i	$0.977572\pi$
$434$		0			0
$435$	−2.00000	+	2.44949i	−0.0958927	+	0.117444i
$436$		0			0
$437$			20.6312i			0.986924i
$438$		0			0
$439$	−7.07180			−0.337518			−0.168759	−	0.985657i	$-0.553976\pi$
$439$	−7.07180			−0.337518			−0.168759	+	0.985657i	$0.553976\pi$
$440$		0			0
$441$	1.00000			0.0476190
$442$		0			0
$443$			18.1817i			0.863839i	0.901912	+	0.431919i	$0.142164\pi$
$443$			18.1817i			0.863839i	−0.901912	+	0.431919i	$0.857836\pi$
$444$		0			0
$445$	18.0000	+	14.6969i	0.853282	+	0.696702i
$446$		0			0
$447$			1.31268i			0.0620875i
$448$		0			0
$449$	25.8564			1.22024			0.610120	−	0.792309i	$-0.291121\pi$
$449$	25.8564			1.22024			0.610120	+	0.792309i	$0.291121\pi$
$450$		0			0
$451$	−8.78461			−0.413651
$452$		0			0
$453$			26.5654i			1.24815i
$454$		0			0
$455$	7.60770	+	6.21166i	0.356654	+	0.291207i
$456$		0			0
$457$		−	23.6627i		−	1.10689i	−0.832884	−	0.553447i	$-0.813312\pi$
$457$		−	23.6627i		−	1.10689i	0.832884	−	0.553447i	$-0.186688\pi$
$458$		0			0
$459$	−8.00000			−0.373408
$460$		0			0
$461$	24.9282			1.16102			0.580511	−	0.814252i	$-0.302853\pi$
$461$	24.9282			1.16102			0.580511	+	0.814252i	$0.302853\pi$
$462$		0			0
$463$			6.86800i			0.319183i	0.987183	+	0.159591i	$0.0510177\pi$
$463$			6.86800i			0.319183i	−0.987183	+	0.159591i	$0.948982\pi$
$464$		0			0
$465$	17.4641	−	21.3891i	0.809878	−	0.991894i
$466$		0			0
$467$		−	21.0101i		−	0.972233i	−0.873894	−	0.486116i	$-0.838413\pi$
$467$		−	21.0101i		−	0.972233i	0.873894	−	0.486116i	$-0.161587\pi$
$468$		0			0
$469$	10.3923			0.479872
$470$		0			0
$471$	−15.4641			−0.712548
$472$		0			0
$473$			11.5911i			0.532960i
$474$		0			0
$475$	−3.26795	−	16.0096i	−0.149944	−	0.734572i
$476$		0			0
$477$		−	5.93426i		−	0.271711i
$478$		0			0
$479$	5.66025			0.258624			0.129312	−	0.991604i	$-0.458723\pi$
$479$	5.66025			0.258624			0.129312	+	0.991604i	$0.458723\pi$
$480$		0			0
$481$	−16.3923			−0.747425
$482$		0			0
$483$			21.8695i			0.995099i
$484$		0			0
$485$	−15.4641	+	18.9396i	−0.702189	+	0.860002i
$486$		0			0
$487$			10.9348i			0.495502i	0.968824	+	0.247751i	$0.0796915\pi$
$487$			10.9348i			0.495502i	−0.968824	+	0.247751i	$0.920309\pi$
$488$		0			0
$489$	−19.8564			−0.897938
$490$		0			0
$491$	−7.26795			−0.327998			−0.163999	−	0.986461i	$-0.552439\pi$
$491$	−7.26795			−0.327998			−0.163999	+	0.986461i	$0.552439\pi$
$492$		0			0
$493$		−	1.41421i		−	0.0636930i
$494$		0			0
$495$	−2.19615	−	1.79315i	−0.0987097	−	0.0805961i
$496$		0			0
$497$		−	8.48528i		−	0.380617i
$498$		0			0
$499$	−1.07180			−0.0479802			−0.0239901	−	0.999712i	$-0.507637\pi$
$499$	−1.07180			−0.0479802			−0.0239901	+	0.999712i	$0.507637\pi$
$500$		0			0
$501$	15.8564			0.708412
$502$		0			0
$503$		−	12.1731i		−	0.542773i	−0.962471	−	0.271386i	$-0.912518\pi$
$503$		−	12.1731i		−	0.542773i	0.962471	−	0.271386i	$-0.0874821\pi$
$504$		0			0
$505$	10.3923	+	8.48528i	0.462451	+	0.377590i
$506$		0			0
$507$			13.8375i			0.614546i
$508$		0			0
$509$	−33.4641			−1.48327			−0.741635	−	0.670804i	$-0.765949\pi$
$509$	−33.4641			−1.48327			−0.741635	+	0.670804i	$0.765949\pi$
$510$		0			0
$511$	−18.0000			−0.796273
$512$		0			0
$513$		−	18.4863i		−	0.816191i
$514$		0			0
$515$	12.9282	−	15.8338i	0.569685	−	0.697718i
$516$		0			0
$517$		−	1.79315i		−	0.0788627i
$518$		0			0
$519$	6.53590			0.286894
$520$		0			0
$521$	4.39230			0.192430			0.0962152	−	0.995361i	$-0.469326\pi$
$521$	4.39230			0.192430			0.0962152	+	0.995361i	$0.469326\pi$
$522$		0			0
$523$			1.61729i			0.0707190i	0.999375	+	0.0353595i	$0.0112576\pi$
$523$			1.61729i			0.0707190i	−0.999375	+	0.0353595i	$0.988742\pi$
$524$		0			0
$525$	−3.46410	−	16.9706i	−0.151186	−	0.740656i
$526$		0			0
$527$			12.3490i			0.537930i
$528$		0			0
$529$	−16.8564			−0.732887
$530$		0			0
$531$	−10.3923			−0.450988
$532$		0			0
$533$		−	12.4233i		−	0.538113i
$534$		0			0
$535$	26.3923	−	32.3238i	1.14104	−	1.39748i
$536$		0			0
$537$			3.58630i			0.154760i
$538$		0			0
$539$	1.26795			0.0546144
$540$		0			0
$541$	42.3923			1.82259			0.911294	−	0.411757i	$-0.135085\pi$
$541$	42.3923			1.82259			0.911294	+	0.411757i	$0.135085\pi$
$542$		0			0
$543$		−	22.6274i		−	0.971035i
$544$		0			0
$545$	30.9282	+	25.2528i	1.32482	+	1.08171i
$546$		0			0
$547$			2.44949i			0.104733i	0.998628	+	0.0523663i	$0.0166763\pi$
$547$			2.44949i			0.104733i	−0.998628	+	0.0523663i	$0.983324\pi$
$548$		0			0
$549$	2.92820			0.124973
$550$		0			0
$551$	3.26795			0.139219
$552$		0			0
$553$		−	10.2784i		−	0.437083i
$554$		0			0
$555$	22.3923	+	18.2832i	0.950500	+	0.776080i
$556$		0			0
$557$		−	0.554803i		−	0.0235078i	−0.999931	−	0.0117539i	$-0.996259\pi$
$557$		−	0.554803i		−	0.0235078i	0.999931	−	0.0117539i	$-0.00374146\pi$
$558$		0			0
$559$	−16.3923			−0.693321
$560$		0			0
$561$	−2.53590			−0.107066
$562$		0			0
$563$			13.2827i			0.559800i	0.960029	+	0.279900i	$0.0903013\pi$
$563$			13.2827i			0.559800i	−0.960029	+	0.279900i	$0.909699\pi$
$564$		0			0
$565$	18.3923	−	22.5259i	0.773770	−	0.947671i
$566$		0			0
$567$		−	12.2474i		−	0.514344i
$568$		0			0
$569$	−21.4641			−0.899822			−0.449911	−	0.893073i	$-0.648544\pi$
$569$	−21.4641			−0.899822			−0.449911	+	0.893073i	$0.648544\pi$
$570$		0			0
$571$	39.3205			1.64551			0.822756	−	0.568395i	$-0.192435\pi$
$571$	39.3205			1.64551			0.822756	+	0.568395i	$0.192435\pi$
$572$		0			0
$573$		−	21.3891i		−	0.893541i
$574$		0			0
$575$	30.9282	−	6.31319i	1.28980	−	0.263278i
$576$		0			0
$577$			9.14162i			0.380571i	0.981729	+	0.190285i	$0.0609413\pi$
$577$			9.14162i			0.380571i	−0.981729	+	0.190285i	$0.939059\pi$
$578$		0			0
$579$	−0.928203			−0.0385748
$580$		0			0
$581$	−24.9282			−1.03420
$582$		0			0
$583$		−	7.52433i		−	0.311626i
$584$		0			0
$585$	2.53590	−	3.10583i	0.104846	−	0.128410i
$586$		0			0
$587$			15.5563i			0.642079i	0.947066	+	0.321040i	$0.104032\pi$
$587$			15.5563i			0.642079i	−0.947066	+	0.321040i	$0.895968\pi$
$588$		0			0
$589$	−28.5359			−1.17580
$590$		0			0
$591$	31.7128			1.30449
$592$		0			0
$593$		−	25.5302i		−	1.04840i	−0.851596	−	0.524199i	$-0.824365\pi$
$593$		−	25.5302i		−	1.04840i	0.851596	−	0.524199i	$-0.175635\pi$
$594$		0			0
$595$	6.00000	+	4.89898i	0.245976	+	0.200839i
$596$		0			0
$597$		−	19.7990i		−	0.810319i
$598$		0			0
$599$	1.26795			0.0518070			0.0259035	−	0.999664i	$-0.491754\pi$
$599$	1.26795			0.0518070			0.0259035	+	0.999664i	$0.491754\pi$
$600$		0			0
$601$	16.7846			0.684659			0.342329	−	0.939580i	$-0.388784\pi$
$601$	16.7846			0.684659			0.342329	+	0.939580i	$0.388784\pi$
$602$		0			0
$603$		−	4.24264i		−	0.172774i
$604$		0			0
$605$	16.2679	+	13.2827i	0.661386	+	0.540020i
$606$		0			0
$607$		−	16.3142i		−	0.662174i	−0.943600	−	0.331087i	$-0.892585\pi$
$607$		−	16.3142i		−	0.662174i	0.943600	−	0.331087i	$-0.107415\pi$
$608$		0			0
$609$	3.46410			0.140372
$610$		0			0
$611$	2.53590			0.102591
$612$		0			0
$613$		−	18.2832i		−	0.738453i	−0.929339	−	0.369227i	$-0.879623\pi$
$613$		−	18.2832i		−	0.738453i	0.929339	−	0.369227i	$-0.120377\pi$
$614$		0			0
$615$	−13.8564	+	16.9706i	−0.558744	+	0.684319i
$616$		0			0
$617$		−	32.0464i		−	1.29014i	−0.764123	−	0.645071i	$-0.776828\pi$
$617$		−	32.0464i		−	1.29014i	0.764123	−	0.645071i	$-0.223172\pi$
$618$		0			0
$619$	−23.8038			−0.956757			−0.478379	−	0.878154i	$-0.658775\pi$
$619$	−23.8038			−0.956757			−0.478379	+	0.878154i	$0.658775\pi$
$620$		0			0
$621$	35.7128			1.43311
$622$		0			0
$623$		−	25.4558i		−	1.01987i
$624$		0			0
$625$	−23.0000	+	9.79796i	−0.920000	+	0.391918i
$626$		0			0
$627$		−	5.85993i		−	0.234023i
$628$		0			0
$629$	−12.9282			−0.515481
$630$		0			0
$631$	−15.6077			−0.621333			−0.310666	−	0.950519i	$-0.600552\pi$
$631$	−15.6077			−0.621333			−0.310666	+	0.950519i	$0.600552\pi$
$632$		0			0
$633$			20.8343i			0.828088i
$634$		0			0
$635$	−6.00000	+	7.34847i	−0.238103	+	0.291615i
$636$		0			0
$637$			1.79315i			0.0710472i
$638$		0			0
$639$	−3.46410			−0.137038
$640$		0			0
$641$	34.6410			1.36824			0.684119	−	0.729370i	$-0.260187\pi$
$641$	34.6410			1.36824			0.684119	+	0.729370i	$0.260187\pi$
$642$		0			0
$643$		−	10.4543i		−	0.412277i	−0.978523	−	0.206139i	$-0.933910\pi$
$643$		−	10.4543i		−	0.412277i	0.978523	−	0.206139i	$-0.0660898\pi$
$644$		0			0
$645$	22.3923	+	18.2832i	0.881696	+	0.719902i
$646$		0			0
$647$		−	14.3180i		−	0.562899i	−0.959576	−	0.281449i	$-0.909185\pi$
$647$		−	14.3180i		−	0.562899i	0.959576	−	0.281449i	$-0.0908151\pi$
$648$		0			0
$649$	−13.1769			−0.517239
$650$		0			0
$651$	−30.2487			−1.18554
$652$		0			0
$653$			13.0053i			0.508938i	0.967081	+	0.254469i	$0.0819006\pi$
$653$			13.0053i			0.508938i	−0.967081	+	0.254469i	$0.918099\pi$
$654$		0			0
$655$	−8.19615	−	6.69213i	−0.320250	−	0.261483i
$656$		0			0
$657$			7.34847i			0.286691i
$658$		0			0
$659$	32.4449			1.26387			0.631936	−	0.775020i	$-0.282260\pi$
$659$	32.4449			1.26387			0.631936	+	0.775020i	$0.282260\pi$
$660$		0			0
$661$	−0.784610			−0.0305178			−0.0152589	−	0.999884i	$-0.504857\pi$
$661$	−0.784610			−0.0305178			−0.0152589	+	0.999884i	$0.504857\pi$
$662$		0			0
$663$		−	3.58630i		−	0.139280i
$664$		0			0
$665$	−11.3205	+	13.8647i	−0.438990	+	0.537651i
$666$		0			0
$667$			6.31319i			0.244448i
$668$		0			0
$669$	21.7128			0.839466
$670$		0			0
$671$	3.71281			0.143332
$672$		0			0
$673$		−	46.3644i		−	1.78722i	−0.448846	−	0.893609i	$-0.648165\pi$
$673$		−	46.3644i		−	1.78722i	0.448846	−	0.893609i	$-0.351835\pi$
$674$		0			0
$675$	−27.7128	+	5.65685i	−1.06667	+	0.217732i
$676$		0			0
$677$			32.2495i			1.23945i	0.784819	+	0.619725i	$0.212756\pi$
$677$			32.2495i			1.23945i	−0.784819	+	0.619725i	$0.787244\pi$
$678$		0			0
$679$	26.7846			1.02790
$680$		0			0
$681$	−28.9282			−1.10853
$682$		0			0
$683$		−	21.9711i		−	0.840700i	−0.907362	−	0.420350i	$-0.861907\pi$
$683$		−	21.9711i		−	0.840700i	0.907362	−	0.420350i	$-0.138093\pi$
$684$		0			0
$685$	20.9282	−	25.6317i	0.799626	−	0.979337i
$686$		0			0
$687$		−	26.5654i		−	1.01354i
$688$		0			0
$689$	10.6410			0.405390
$690$		0			0
$691$	−41.7128			−1.58683			−0.793415	−	0.608681i	$-0.791699\pi$
$691$	−41.7128			−1.58683			−0.793415	+	0.608681i	$0.791699\pi$
$692$		0			0
$693$			3.10583i			0.117981i
$694$		0			0
$695$	13.8564	+	11.3137i	0.525603	+	0.429153i
$696$		0			0
$697$		−	9.79796i		−	0.371124i
$698$		0			0
$699$	−4.78461			−0.180971
$700$		0			0
$701$	27.7128			1.04670			0.523349	−	0.852118i	$-0.324682\pi$
$701$	27.7128			1.04670			0.523349	+	0.852118i	$0.324682\pi$
$702$		0			0
$703$		−	29.8744i		−	1.12673i
$704$		0			0
$705$	−3.46410	−	2.82843i	−0.130466	−	0.106525i
$706$		0			0
$707$		−	14.6969i		−	0.552735i
$708$		0			0
$709$	−20.3923			−0.765849			−0.382925	−	0.923780i	$-0.625083\pi$
$709$	−20.3923			−0.765849			−0.382925	+	0.923780i	$0.625083\pi$
$710$		0			0
$711$	−4.19615			−0.157368
$712$		0			0
$713$		−	55.1271i		−	2.06453i
$714$		0			0
$715$	3.21539	−	3.93803i	0.120249	−	0.147274i
$716$		0			0
$717$			14.6969i			0.548867i
$718$		0			0
$719$	−19.8564			−0.740519			−0.370260	−	0.928928i	$-0.620731\pi$
$719$	−19.8564			−0.740519			−0.370260	+	0.928928i	$0.620731\pi$
$720$		0			0
$721$	−22.3923			−0.833933
$722$		0			0
$723$			16.2127i			0.602956i
$724$		0			0
$725$	−1.00000	−	4.89898i	−0.0371391	−	0.181944i
$726$		0			0
$727$			46.6690i			1.73086i	0.501031	+	0.865430i	$0.332954\pi$
$727$			46.6690i			1.73086i	−0.501031	+	0.865430i	$0.667046\pi$
$728$		0			0
$729$	−29.0000			−1.07407
$730$		0			0
$731$	−12.9282			−0.478167
$732$		0			0
$733$		−	12.7279i		−	0.470117i	−0.971981	−	0.235058i	$-0.924472\pi$
$733$		−	12.7279i		−	0.470117i	0.971981	−	0.235058i	$-0.0755281\pi$
$734$		0			0
$735$	2.00000	−	2.44949i	0.0737711	−	0.0903508i
$736$		0			0
$737$		−	5.37945i		−	0.198155i
$738$		0			0
$739$	−18.9808			−0.698219			−0.349109	−	0.937082i	$-0.613516\pi$
$739$	−18.9808			−0.698219			−0.349109	+	0.937082i	$0.613516\pi$
$740$		0			0
$741$	8.28719			0.304437
$742$		0			0
$743$		−	9.89949i		−	0.363177i	−0.983375	−	0.181589i	$-0.941876\pi$
$743$		−	9.89949i		−	0.363177i	0.983375	−	0.181589i	$-0.0581239\pi$
$744$		0			0
$745$	−1.60770	−	1.31268i	−0.0589014	−	0.0480928i
$746$		0			0
$747$			10.1769i			0.372353i
$748$		0			0
$749$	−45.7128			−1.67031
$750$		0			0
$751$	−49.9090			−1.82120			−0.910602	−	0.413284i	$-0.864382\pi$
$751$	−49.9090			−1.82120			−0.910602	+	0.413284i	$0.864382\pi$
$752$		0			0
$753$		−	21.3891i		−	0.779461i
$754$		0			0
$755$	−32.5359	−	26.5654i	−1.18410	−	0.966816i
$756$		0			0
$757$			10.9348i			0.397431i	0.980057	+	0.198716i	$0.0636770\pi$
$757$			10.9348i			0.397431i	−0.980057	+	0.198716i	$0.936323\pi$
$758$		0			0
$759$	11.3205			0.410908
$760$		0			0
$761$	−35.5692			−1.28938			−0.644692	−	0.764443i	$-0.723014\pi$
$761$	−35.5692			−1.28938			−0.644692	+	0.764443i	$0.723014\pi$
$762$		0			0
$763$		−	43.7391i		−	1.58346i
$764$		0			0
$765$	2.00000	−	2.44949i	0.0723102	−	0.0885615i
$766$		0			0
$767$		−	18.6350i		−	0.672870i
$768$		0			0
$769$	19.0718			0.687747			0.343873	−	0.939016i	$-0.388261\pi$
$769$	19.0718			0.687747			0.343873	+	0.939016i	$0.388261\pi$
$770$		0			0
$771$	13.4641			0.484898
$772$		0			0
$773$			11.2122i			0.403274i	0.979460	+	0.201637i	$0.0646261\pi$
$773$			11.2122i			0.403274i	−0.979460	+	0.201637i	$0.935374\pi$
$774$		0			0
$775$	8.73205	+	42.7781i	0.313665	+	1.53664i
$776$		0			0
$777$		−	31.6675i		−	1.13607i
$778$		0			0
$779$	22.6410			0.811199
$780$		0			0
$781$	−4.39230			−0.157169
$782$		0			0
$783$		−	5.65685i		−	0.202159i
$784$		0			0
$785$	15.4641	−	18.9396i	0.551937	−	0.675983i
$786$		0			0
$787$		−	35.0779i		−	1.25039i	−0.780467	−	0.625197i	$-0.785019\pi$
$787$		−	35.0779i		−	1.25039i	0.780467	−	0.625197i	$-0.214981\pi$
$788$		0			0
$789$	−6.78461			−0.241539
$790$		0			0
$791$	−31.8564			−1.13268
$792$		0			0
$793$			5.25071i			0.186458i
$794$		0			0
$795$	−14.5359	−	11.8685i	−0.515535	−	0.420933i
$796$		0			0
$797$			29.0149i			1.02776i	0.857862	+	0.513881i	$0.171793\pi$
$797$			29.0149i			1.02776i	−0.857862	+	0.513881i	$0.828207\pi$
$798$		0			0
$799$	2.00000			0.0707549
$800$		0			0
$801$	−10.3923			−0.367194
$802$		0			0
$803$			9.31749i			0.328807i
$804$		0			0
$805$	−26.7846	−	21.8695i	−0.944033	−	0.770800i
$806$		0			0
$807$			16.9706i			0.597392i
$808$		0			0
$809$	−13.6077			−0.478421			−0.239211	−	0.970968i	$-0.576889\pi$
$809$	−13.6077			−0.478421			−0.239211	+	0.970968i	$0.576889\pi$
$810$		0			0
$811$	36.7846			1.29168			0.645841	−	0.763472i	$-0.276507\pi$
$811$	36.7846			1.29168			0.645841	+	0.763472i	$0.276507\pi$
$812$		0			0
$813$			37.8048i			1.32587i
$814$		0			0
$815$	19.8564	−	24.3190i	0.695540	−	0.851859i
$816$		0			0
$817$		−	29.8744i		−	1.04517i
$818$		0			0
$819$	−4.39230			−0.153480
$820$		0			0
$821$	9.21539			0.321619			0.160810	−	0.986985i	$-0.448589\pi$
$821$	9.21539			0.321619			0.160810	+	0.986985i	$0.448589\pi$
$822$		0			0
$823$		−	40.8091i		−	1.42252i	−0.702931	−	0.711258i	$-0.748126\pi$
$823$		−	40.8091i		−	1.42252i	0.702931	−	0.711258i	$-0.251874\pi$
$824$		0			0
$825$	−8.78461	+	1.79315i	−0.305841	+	0.0624295i
$826$		0			0
$827$		−	3.68784i		−	0.128239i	−0.997942	−	0.0641193i	$-0.979576\pi$
$827$		−	3.68784i		−	0.128239i	0.997942	−	0.0641193i	$-0.0204238\pi$
$828$		0			0
$829$	16.7846			0.582954			0.291477	−	0.956578i	$-0.405853\pi$
$829$	16.7846			0.582954			0.291477	+	0.956578i	$0.405853\pi$
$830$		0			0
$831$	−26.5359			−0.920520
$832$		0			0
$833$			1.41421i			0.0489996i
$834$		0			0
$835$	−15.8564	+	19.4201i	−0.548734	+	0.672059i
$836$		0			0
$837$			49.3959i			1.70737i
$838$		0			0
$839$	40.9808			1.41481			0.707407	−	0.706807i	$-0.249865\pi$
$839$	40.9808			1.41481			0.707407	+	0.706807i	$0.249865\pi$
$840$		0			0
$841$	1.00000			0.0344828
$842$		0			0
$843$			46.3644i			1.59688i
$844$		0			0
$845$	−16.9474	−	13.8375i	−0.583010	−	0.476025i
$846$		0			0
$847$		−	23.0064i		−	0.790508i
$848$		0			0
$849$	38.7846			1.33108
$850$		0			0
$851$	57.7128			1.97837
$852$		0			0
$853$		−	32.3238i		−	1.10675i	−0.832933	−	0.553374i	$-0.813340\pi$
$853$		−	32.3238i		−	1.10675i	0.832933	−	0.553374i	$-0.186660\pi$
$854$		0			0
$855$	5.66025	+	4.62158i	0.193577	+	0.158055i
$856$		0			0
$857$			33.2576i			1.13606i	0.823009	+	0.568029i	$0.192294\pi$
$857$			33.2576i			1.13606i	−0.823009	+	0.568029i	$0.807706\pi$
$858$		0			0
$859$	−18.7321			−0.639129			−0.319565	−	0.947564i	$-0.603537\pi$
$859$	−18.7321			−0.639129			−0.319565	+	0.947564i	$0.603537\pi$
$860$		0			0
$861$	24.0000			0.817918
$862$		0			0
$863$			1.69161i			0.0575832i	0.999585	+	0.0287916i	$0.00916592\pi$
$863$			1.69161i			0.0575832i	−0.999585	+	0.0287916i	$0.990834\pi$
$864$		0			0
$865$	−6.53590	+	8.00481i	−0.222227	+	0.272172i
$866$		0			0
$867$			21.2132i			0.720438i
$868$		0			0
$869$	−5.32051			−0.180486
$870$		0			0
$871$	7.60770			0.257777
$872$		0			0
$873$		−	10.9348i		−	0.370086i
$874$		0			0
$875$	24.2487	+	12.7279i	0.819756	+	0.430282i
$876$		0			0
$877$			24.9754i			0.843358i	0.906745	+	0.421679i	$0.138559\pi$
$877$			24.9754i			0.843358i	−0.906745	+	0.421679i	$0.861441\pi$
$878$		0			0
$879$	43.4641			1.46601
$880$		0			0
$881$	−50.5359			−1.70260			−0.851299	−	0.524681i	$-0.824185\pi$
$881$	−50.5359			−1.70260			−0.851299	+	0.524681i	$0.824185\pi$
$882$		0			0
$883$			1.48854i			0.0500935i	0.999686	+	0.0250467i	$0.00797346\pi$
$883$			1.48854i			0.0500935i	−0.999686	+	0.0250467i	$0.992027\pi$
$884$		0			0
$885$	−20.7846	+	25.4558i	−0.698667	+	0.855689i
$886$		0			0
$887$		−	21.0101i		−	0.705451i	−0.935727	−	0.352726i	$-0.885255\pi$
$887$		−	21.0101i		−	0.705451i	0.935727	−	0.352726i	$-0.114745\pi$
$888$		0			0
$889$	10.3923			0.348547
$890$		0			0
$891$	−6.33975			−0.212389
$892$		0			0
$893$			4.62158i			0.154655i
$894$		0			0
$895$	−4.39230	−	3.58630i	−0.146819	−	0.119877i
$896$		0			0
$897$			16.0096i			0.534546i
$898$		0			0
$899$	−8.73205			−0.291230
$900$		0			0
$901$	8.39230			0.279588
$902$		0			0
$903$		−	31.6675i		−	1.05383i
$904$		0			0
$905$	27.7128	+	22.6274i	0.921205	+	0.752161i
$906$		0			0
$907$			45.7081i			1.51771i	0.651258	+	0.758856i	$0.274242\pi$
$907$			45.7081i			1.51771i	−0.651258	+	0.758856i	$0.725758\pi$
$908$		0			0
$909$	−6.00000			−0.199007
$910$		0			0
$911$	9.80385			0.324816			0.162408	−	0.986724i	$-0.448074\pi$
$911$	9.80385			0.324816			0.162408	+	0.986724i	$0.448074\pi$
$912$		0			0
$913$			12.9038i			0.427053i
$914$		0			0
$915$	5.85641	−	7.17260i	0.193607	−	0.237119i
$916$		0			0
$917$			11.5911i			0.382772i
$918$		0			0
$919$	43.9615			1.45016			0.725078	−	0.688666i	$-0.241803\pi$
$919$	43.9615			1.45016			0.725078	+	0.688666i	$0.241803\pi$
$920$		0			0
$921$	14.7846			0.487169
$922$		0			0
$923$		−	6.21166i		−	0.204459i
$924$		0			0
$925$	−44.7846	+	9.14162i	−1.47251	+	0.300575i
$926$		0			0
$927$			9.14162i			0.300250i
$928$		0			0
$929$	−55.8564			−1.83259			−0.916295	−	0.400505i	$-0.868835\pi$
$929$	−55.8564			−1.83259			−0.916295	+	0.400505i	$0.868835\pi$
$930$		0			0
$931$	−3.26795			−0.107103
$932$		0			0
$933$			9.31749i			0.305041i
$934$		0			0
$935$	2.53590	−	3.10583i	0.0829327	−	0.101571i
$936$		0			0
$937$			2.75410i			0.0899724i	0.998988	+	0.0449862i	$0.0143244\pi$
$937$			2.75410i			0.0899724i	−0.998988	+	0.0449862i	$0.985676\pi$
$938$		0			0
$939$	−33.4641			−1.09206
$940$		0			0
$941$	−7.85641			−0.256112			−0.128056	−	0.991767i	$-0.540874\pi$
$941$	−7.85641			−0.256112			−0.128056	+	0.991767i	$0.540874\pi$
$942$		0			0
$943$			43.7391i			1.42434i
$944$		0			0
$945$	24.0000	+	19.5959i	0.780720	+	0.637455i
$946$		0			0
$947$			27.9797i			0.909217i	0.890691	+	0.454608i	$0.150221\pi$
$947$			27.9797i			0.909217i	−0.890691	+	0.454608i	$0.849779\pi$
$948$		0			0
$949$	−13.1769			−0.427741
$950$		0			0
$951$	8.14359			0.264074
$952$		0			0
$953$			5.65685i			0.183243i	0.995794	+	0.0916217i	$0.0292051\pi$
$953$			5.65685i			0.183243i	−0.995794	+	0.0916217i	$0.970795\pi$
$954$		0			0
$955$	26.1962	+	21.3891i	0.847688	+	0.692134i
$956$		0			0
$957$		−	1.79315i		−	0.0579643i
$958$		0			0
$959$	−36.2487			−1.17053
$960$		0			0
$961$	45.2487			1.45964
$962$		0			0
$963$			18.6622i			0.601380i
$964$		0			0
$965$	0.928203	−	1.13681i	0.0298799	−	0.0365953i
$966$		0			0
$967$			1.96902i			0.0633193i	0.999499	+	0.0316596i	$0.0100793\pi$
$967$			1.96902i			0.0633193i	−0.999499	+	0.0316596i	$0.989921\pi$
$968$		0			0
$969$	6.53590			0.209963
$970$		0			0
$971$	49.5167			1.58907			0.794533	−	0.607221i	$-0.207716\pi$
$971$	49.5167			1.58907			0.794533	+	0.607221i	$0.207716\pi$
$972$		0			0
$973$		−	19.5959i		−	0.628216i
$974$		0			0
$975$	−2.53590	−	12.4233i	−0.0812137	−	0.397864i
$976$		0			0
$977$		−	22.9048i		−	0.732790i	−0.930459	−	0.366395i	$-0.880592\pi$
$977$		−	22.9048i		−	0.732790i	0.930459	−	0.366395i	$-0.119408\pi$
$978$		0			0
$979$	−13.1769			−0.421136
$980$		0			0
$981$	−17.8564			−0.570111
$982$		0			0
$983$			30.2533i			0.964930i	0.875915	+	0.482465i	$0.160258\pi$
$983$			30.2533i			0.964930i	−0.875915	+	0.482465i	$0.839742\pi$
$984$		0			0
$985$	−31.7128	+	38.8401i	−1.01045	+	1.23755i
$986$		0			0
$987$			4.89898i			0.155936i
$988$		0			0
$989$	57.7128			1.83516
$990$		0			0
$991$	−34.7846			−1.10497			−0.552485	−	0.833523i	$-0.686320\pi$
$991$	−34.7846			−1.10497			−0.552485	+	0.833523i	$0.686320\pi$
$992$		0			0
$993$			25.7332i			0.816620i
$994$		0			0
$995$	24.2487	+	19.7990i	0.768736	+	0.627670i
$996$		0			0
$997$			26.5927i			0.842198i	0.907015	+	0.421099i	$0.138355\pi$
$997$			26.5927i			0.842198i	−0.907015	+	0.421099i	$0.861645\pi$
$998$		0			0
$999$	−51.7128			−1.63612

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2320.2.d.f.929.3		4
4.3	odd	2		145.2.b.b.59.3	yes	4
5.4	even	2	inner	2320.2.d.f.929.1		4
12.11	even	2		1305.2.c.f.784.2		4
20.3	even	4		725.2.a.f.1.3		4
20.7	even	4		725.2.a.f.1.2		4
20.19	odd	2		145.2.b.b.59.2	✓	4
60.23	odd	4		6525.2.a.bj.1.2		4
60.47	odd	4		6525.2.a.bj.1.3		4
60.59	even	2		1305.2.c.f.784.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.b.b.59.2	✓	4	20.19	odd	2
145.2.b.b.59.3	yes	4	4.3	odd	2
725.2.a.f.1.2		4	20.7	even	4
725.2.a.f.1.3		4	20.3	even	4
1305.2.c.f.784.2		4	12.11	even	2
1305.2.c.f.784.3		4	60.59	even	2
2320.2.d.f.929.1		4	5.4	even	2	inner
2320.2.d.f.929.3		4	1.1	even	1	trivial
6525.2.a.bj.1.2		4	60.23	odd	4
6525.2.a.bj.1.3		4	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2320.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{3} +(-1.73205 - 1.41421i) q^{5} +2.44949i q^{7} +1.00000 q^{9} +1.26795 q^{11} +1.79315i q^{13} +(2.00000 - 2.44949i) q^{15} +1.41421i q^{17} -3.26795 q^{19} -3.46410 q^{21} -6.31319i q^{23} +(1.00000 + 4.89898i) q^{25} +5.65685i q^{27} -1.00000 q^{29} +8.73205 q^{31} +1.79315i q^{33} +(3.46410 - 4.24264i) q^{35} +9.14162i q^{37} -2.53590 q^{39} -6.92820 q^{41} +9.14162i q^{43} +(-1.73205 - 1.41421i) q^{45} -1.41421i q^{47} +1.00000 q^{49} -2.00000 q^{51} -5.93426i q^{53} +(-2.19615 - 1.79315i) q^{55} -4.62158i q^{57} -10.3923 q^{59} +2.92820 q^{61} +2.44949i q^{63} +(2.53590 - 3.10583i) q^{65} -4.24264i q^{67} +8.92820 q^{69} -3.46410 q^{71} +7.34847i q^{73} +(-6.92820 + 1.41421i) q^{75} +3.10583i q^{77} -4.19615 q^{79} -5.00000 q^{81} +10.1769i q^{83} +(2.00000 - 2.44949i) q^{85} -1.41421i q^{87} -10.3923 q^{89} -4.39230 q^{91} +12.3490i q^{93} +(5.66025 + 4.62158i) q^{95} -10.9348i q^{97} +1.26795 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{9} + 12 q^{11} + 8 q^{15} - 20 q^{19} + 4 q^{25} - 4 q^{29} + 28 q^{31} - 24 q^{39} + 4 q^{49} - 8 q^{51} + 12 q^{55} - 16 q^{61} + 24 q^{65} + 8 q^{69} + 4 q^{79} - 20 q^{81} + 8 q^{85} + 24 q^{91}+ \cdots + 12 q^{99}+O(q^{100}) \) 4 * q + 4 * q^9 + 12 * q^11 + 8 * q^15 - 20 * q^19 + 4 * q^25 - 4 * q^29 + 28 * q^31 - 24 * q^39 + 4 * q^49 - 8 * q^51 + 12 * q^55 - 16 * q^61 + 24 * q^65 + 8 * q^69 + 4 * q^79 - 20 * q^81 + 8 * q^85 + 24 * q^91 - 12 * q^95 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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