LMFDB - Embedded newform 1152.4.l.a.863.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,4,Mod(287,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 2]))

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

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

chi := DirichletCharacter("1152.287");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1152.l (of order $4$, degree $2$, 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:	$67.9702003266$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			863.11
Character	$\chi$	$=$	1152.863
Dual form			1152.4.l.a.287.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.22357 - 3.22357i) q^{5} +13.1030 q^{7} +O(q^{10})$	$q+(-3.22357 - 3.22357i) q^{5} +13.1030 q^{7} +(-3.39235 + 3.39235i) q^{11} +(54.1435 + 54.1435i) q^{13} -70.7730i q^{17} +(32.5595 - 32.5595i) q^{19} -16.4197i q^{23} -104.217i q^{25} +(-28.0742 + 28.0742i) q^{29} +174.864i q^{31} +(-42.2385 - 42.2385i) q^{35} +(-116.834 + 116.834i) q^{37} -19.6290 q^{41} +(94.2456 + 94.2456i) q^{43} +372.603 q^{47} -171.310 q^{49} +(162.999 + 162.999i) q^{53} +21.8710 q^{55} +(-610.346 + 610.346i) q^{59} +(531.488 + 531.488i) q^{61} -349.071i q^{65} +(562.092 - 562.092i) q^{67} -1166.76i q^{71} +308.564i q^{73} +(-44.4501 + 44.4501i) q^{77} -1170.83i q^{79} +(-469.366 - 469.366i) q^{83} +(-228.142 + 228.142i) q^{85} +1534.65 q^{89} +(709.445 + 709.445i) q^{91} -209.916 q^{95} -139.670 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q - 48 q^{19} + 864 q^{43} + 2352 q^{49} + 576 q^{55} - 1824 q^{61} + 816 q^{67} + 480 q^{85} - 3600 q^{91}+O(q^{100}) $ 48 * q - 48 * q^19 + 864 * q^43 + 2352 * q^49 + 576 * q^55 - 1824 * q^61 + 816 * q^67 + 480 * q^85 - 3600 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{3}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−3.22357	−	3.22357i	−0.288325	−	0.288325i	0.548093	−	0.836418i	$-0.315354\pi$
$5$	−3.22357	−	3.22357i	−0.288325	−	0.288325i	−0.836418	+	0.548093i	$0.815354\pi$
$6$		0			0
$7$	13.1030			0.707497			0.353749	−	0.935341i	$-0.384907\pi$
$7$	13.1030			0.707497			0.353749	+	0.935341i	$0.384907\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.39235	+	3.39235i	−0.0929847	+	0.0929847i	−0.752069	−	0.659084i	$-0.770944\pi$
$11$	−3.39235	+	3.39235i	−0.0929847	+	0.0929847i	0.659084	+	0.752069i	$0.270944\pi$
$12$		0			0
$13$	54.1435	+	54.1435i	1.15513	+	1.15513i	0.985509	+	0.169623i	$0.0542550\pi$
$13$	54.1435	+	54.1435i	1.15513	+	1.15513i	0.169623	+	0.985509i	$0.445745\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	70.7730i		−	1.00970i	−0.863206	−	0.504852i	$-0.831547\pi$
$17$		−	70.7730i		−	1.00970i	0.863206	−	0.504852i	$-0.168453\pi$
$18$		0			0
$19$	32.5595	−	32.5595i	0.393140	−	0.393140i	−0.482665	−	0.875805i	$-0.660331\pi$
$19$	32.5595	−	32.5595i	0.393140	−	0.393140i	0.875805	+	0.482665i	$0.160331\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	16.4197i		−	0.148858i	−0.997226	−	0.0744291i	$-0.976287\pi$
$23$		−	16.4197i		−	0.148858i	0.997226	−	0.0744291i	$-0.0237135\pi$
$24$		0			0
$25$		−	104.217i		−	0.833738i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−28.0742	+	28.0742i	−0.179767	+	0.179767i	−0.791254	−	0.611487i	$-0.790572\pi$
$29$	−28.0742	+	28.0742i	−0.179767	+	0.179767i	0.611487	+	0.791254i	$0.290572\pi$
$30$		0			0
$31$			174.864i			1.01311i	0.862207	+	0.506556i	$0.169082\pi$
$31$			174.864i			1.01311i	−0.862207	+	0.506556i	$0.830918\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−42.2385	−	42.2385i	−0.203989	−	0.203989i
$36$		0			0
$37$	−116.834	+	116.834i	−0.519117	+	0.519117i	−0.917304	−	0.398187i	$-0.869640\pi$
$37$	−116.834	+	116.834i	−0.519117	+	0.519117i	0.398187	+	0.917304i	$0.369640\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−19.6290			−0.0747692			−0.0373846	−	0.999301i	$-0.511903\pi$
$41$	−19.6290			−0.0747692			−0.0373846	+	0.999301i	$0.511903\pi$
$42$		0			0
$43$	94.2456	+	94.2456i	0.334240	+	0.334240i	0.854194	−	0.519954i	$-0.174051\pi$
$43$	94.2456	+	94.2456i	0.334240	+	0.334240i	−0.519954	+	0.854194i	$0.674051\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	372.603			1.15638			0.578189	−	0.815903i	$-0.303760\pi$
$47$	372.603			1.15638			0.578189	+	0.815903i	$0.303760\pi$
$48$		0			0
$49$	−171.310			−0.499447
$50$		0			0
$51$		0			0
$52$		0			0
$53$	162.999	+	162.999i	0.422445	+	0.422445i	0.886045	−	0.463600i	$-0.153442\pi$
$53$	162.999	+	162.999i	0.422445	+	0.422445i	−0.463600	+	0.886045i	$0.653442\pi$
$54$		0			0
$55$	21.8710			0.0536196
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−610.346	+	610.346i	−1.34678	+	1.34678i	−0.457655	+	0.889130i	$0.651310\pi$
$59$	−610.346	+	610.346i	−1.34678	+	1.34678i	−0.889130	+	0.457655i	$0.848690\pi$
$60$		0			0
$61$	531.488	+	531.488i	1.11557	+	1.11557i	0.992383	+	0.123191i	$0.0393128\pi$
$61$	531.488	+	531.488i	1.11557	+	1.11557i	0.123191	+	0.992383i	$0.460687\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	349.071i		−	0.666106i
$66$		0			0
$67$	562.092	−	562.092i	1.02493	−	1.02493i	0.0252516	−	0.999681i	$-0.491961\pi$
$67$	562.092	−	562.092i	1.02493	−	1.02493i	0.999681	−	0.0252516i	$-0.00803869\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	1166.76i		−	1.95027i	−0.221612	−	0.975135i	$-0.571132\pi$
$71$		−	1166.76i		−	1.95027i	0.221612	−	0.975135i	$-0.428868\pi$
$72$		0			0
$73$			308.564i			0.494721i	0.968923	+	0.247361i	$0.0795633\pi$
$73$			308.564i			0.494721i	−0.968923	+	0.247361i	$0.920437\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−44.4501	+	44.4501i	−0.0657865	+	0.0657865i
$78$		0			0
$79$		−	1170.83i		−	1.66746i	−0.552176	−	0.833728i	$-0.686202\pi$
$79$		−	1170.83i		−	1.66746i	0.552176	−	0.833728i	$-0.313798\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−469.366	−	469.366i	−0.620719	−	0.620719i	0.324996	−	0.945715i	$-0.394637\pi$
$83$	−469.366	−	469.366i	−0.620719	−	0.620719i	−0.945715	+	0.324996i	$0.894637\pi$
$84$		0			0
$85$	−228.142	+	228.142i	−0.291123	+	0.291123i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1534.65			1.82778			0.913892	−	0.405957i	$-0.133062\pi$
$89$	1534.65			1.82778			0.913892	+	0.405957i	$0.133062\pi$
$90$		0			0
$91$	709.445	+	709.445i	0.817253	+	0.817253i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−209.916			−0.226704
$96$		0			0
$97$	−139.670			−0.146200			−0.0730998	−	0.997325i	$-0.523289\pi$
$97$	−139.670			−0.146200			−0.0730998	+	0.997325i	$0.523289\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	525.800	+	525.800i	0.518011	+	0.518011i	0.916969	−	0.398958i	$-0.130628\pi$
$101$	525.800	+	525.800i	0.518011	+	0.518011i	−0.398958	+	0.916969i	$0.630628\pi$
$102$		0			0
$103$	221.106			0.211517			0.105758	−	0.994392i	$-0.466273\pi$
$103$	221.106			0.211517			0.105758	+	0.994392i	$0.466273\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	824.395	−	824.395i	0.744834	−	0.744834i	−0.228670	−	0.973504i	$-0.573438\pi$
$107$	824.395	−	824.395i	0.744834	−	0.744834i	0.973504	+	0.228670i	$0.0734376\pi$
$108$		0			0
$109$	675.079	+	675.079i	0.593219	+	0.593219i	0.938500	−	0.345281i	$-0.112216\pi$
$109$	675.079	+	675.079i	0.593219	+	0.593219i	−0.345281	+	0.938500i	$0.612216\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	1052.43i		−	0.876141i	−0.898941	−	0.438070i	$-0.855662\pi$
$113$		−	1052.43i		−	0.876141i	0.898941	−	0.438070i	$-0.144338\pi$
$114$		0			0
$115$	−52.9300	+	52.9300i	−0.0429195	+	0.0429195i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	927.341i		−	0.714363i
$120$		0			0
$121$			1307.98i			0.982708i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−738.897	+	738.897i	−0.528712	+	0.528712i
$126$		0			0
$127$		−	103.084i		−	0.0720254i	−0.999351	−	0.0360127i	$-0.988534\pi$
$127$		−	103.084i		−	0.0720254i	0.999351	−	0.0360127i	$-0.0114657\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	535.858	+	535.858i	0.357390	+	0.357390i	0.862850	−	0.505460i	$-0.168677\pi$
$131$	535.858	+	535.858i	0.357390	+	0.357390i	−0.505460	+	0.862850i	$0.668677\pi$
$132$		0			0
$133$	426.628	−	426.628i	0.278146	−	0.278146i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2093.02			1.30525			0.652625	−	0.757681i	$-0.273668\pi$
$137$	2093.02			1.30525			0.652625	+	0.757681i	$0.273668\pi$
$138$		0			0
$139$	1367.29	+	1367.29i	0.834331	+	0.834331i	0.988106	−	0.153775i	$-0.0491429\pi$
$139$	1367.29	+	1367.29i	0.834331	+	0.834331i	−0.153775	+	0.988106i	$0.549143\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−367.348			−0.214819
$144$		0			0
$145$	180.998			0.103663
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−713.205	−	713.205i	−0.392135	−	0.392135i	0.483313	−	0.875448i	$-0.339433\pi$
$149$	−713.205	−	713.205i	−0.392135	−	0.392135i	−0.875448	+	0.483313i	$0.839433\pi$
$150$		0			0
$151$	3184.13			1.71603			0.858017	−	0.513622i	$-0.171696\pi$
$151$	3184.13			1.71603			0.858017	+	0.513622i	$0.171696\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	563.686	−	563.686i	0.292105	−	0.292105i
$156$		0			0
$157$	1517.94	+	1517.94i	0.771622	+	0.771622i	0.978390	−	0.206768i	$-0.0662945\pi$
$157$	1517.94	+	1517.94i	0.771622	+	0.771622i	−0.206768	+	0.978390i	$0.566294\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	215.148i		−	0.105317i
$162$		0			0
$163$	1433.43	−	1433.43i	0.688802	−	0.688802i	−0.273165	−	0.961967i	$-0.588070\pi$
$163$	1433.43	−	1433.43i	0.688802	−	0.688802i	0.961967	+	0.273165i	$0.0880705\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	1574.93i		−	0.729771i	−0.931052	−	0.364886i	$-0.881108\pi$
$167$		−	1574.93i		−	0.729771i	0.931052	−	0.364886i	$-0.118892\pi$
$168$		0			0
$169$			3666.05i			1.66866i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	2323.05	−	2323.05i	1.02092	−	1.02092i	0.0211385	−	0.999777i	$-0.493271\pi$
$173$	2323.05	−	2323.05i	1.02092	−	1.02092i	0.999777	−	0.0211385i	$-0.00672911\pi$
$174$		0			0
$175$		−	1365.56i		−	0.589867i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−2499.16	−	2499.16i	−1.04355	−	1.04355i	−0.999007	−	0.0445458i	$-0.985816\pi$
$179$	−2499.16	−	2499.16i	−1.04355	−	1.04355i	−0.0445458	−	0.999007i	$-0.514184\pi$
$180$		0			0
$181$	−1207.94	+	1207.94i	−0.496053	+	0.496053i	−0.910207	−	0.414154i	$-0.864078\pi$
$181$	−1207.94	+	1207.94i	−0.496053	+	0.496053i	0.414154	+	0.910207i	$0.364078\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	753.243			0.299349
$186$		0			0
$187$	240.087	+	240.087i	0.0938871	+	0.0938871i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1303.81			−0.493930			−0.246965	−	0.969024i	$-0.579433\pi$
$191$	−1303.81			−0.493930			−0.246965	+	0.969024i	$0.579433\pi$
$192$		0			0
$193$	3208.56			1.19667			0.598335	−	0.801246i	$-0.295829\pi$
$193$	3208.56			1.19667			0.598335	+	0.801246i	$0.295829\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1476.04	+	1476.04i	0.533824	+	0.533824i	0.921708	−	0.387884i	$-0.126794\pi$
$197$	1476.04	+	1476.04i	0.533824	+	0.533824i	−0.387884	+	0.921708i	$0.626794\pi$
$198$		0			0
$199$	1698.58			0.605072			0.302536	−	0.953138i	$-0.402167\pi$
$199$	1698.58			0.605072			0.302536	+	0.953138i	$0.402167\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−367.857	+	367.857i	−0.127185	+	0.127185i
$204$		0			0
$205$	63.2755	+	63.2755i	0.0215578	+	0.0215578i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			220.907i			0.0731121i
$210$		0			0
$211$	3141.93	−	3141.93i	1.02511	−	1.02511i	0.0254381	−	0.999676i	$-0.491902\pi$
$211$	3141.93	−	3141.93i	1.02511	−	1.02511i	0.999676	−	0.0254381i	$-0.00809808\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	607.614i		−	0.192739i
$216$		0			0
$217$			2291.25i			0.716774i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3831.90	−	3831.90i	1.16634	−	1.16634i
$222$		0			0
$223$			869.633i			0.261143i	0.991439	+	0.130572i	$0.0416812\pi$
$223$			869.633i			0.261143i	−0.991439	+	0.130572i	$0.958319\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1203.65	−	1203.65i	−0.351935	−	0.351935i	0.508894	−	0.860829i	$-0.330055\pi$
$227$	−1203.65	−	1203.65i	−0.351935	−	0.351935i	−0.860829	+	0.508894i	$0.830055\pi$
$228$		0			0
$229$	−3001.80	+	3001.80i	−0.866220	+	0.866220i	−0.992052	−	0.125832i	$-0.959840\pi$
$229$	−3001.80	+	3001.80i	−0.866220	+	0.866220i	0.125832	+	0.992052i	$0.459840\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5006.57			−1.40769			−0.703844	−	0.710355i	$-0.748534\pi$
$233$	−5006.57			−1.40769			−0.703844	+	0.710355i	$0.748534\pi$
$234$		0			0
$235$	−1201.11	−	1201.11i	−0.333412	−	0.333412i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−659.973			−0.178620			−0.0893098	−	0.996004i	$-0.528466\pi$
$239$	−659.973			−0.178620			−0.0893098	+	0.996004i	$0.528466\pi$
$240$		0			0
$241$	−1088.72			−0.290998			−0.145499	−	0.989358i	$-0.546479\pi$
$241$	−1088.72			−0.290998			−0.145499	+	0.989358i	$0.546479\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	552.231	+	552.231i	0.144003	+	0.144003i
$246$		0			0
$247$	3525.77			0.908258
$248$		0			0
$249$		0			0
$250$		0			0
$251$	3317.58	−	3317.58i	0.834277	−	0.834277i	−0.153821	−	0.988099i	$-0.549158\pi$
$251$	3317.58	−	3317.58i	0.834277	−	0.834277i	0.988099	+	0.153821i	$0.0491581\pi$
$252$		0			0
$253$	55.7013	+	55.7013i	0.0138415	+	0.0138415i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			7233.11i			1.75560i	0.479028	+	0.877800i	$0.340989\pi$
$257$			7233.11i			1.75560i	−0.479028	+	0.877800i	$0.659011\pi$
$258$		0			0
$259$	−1530.88	+	1530.88i	−0.367274	+	0.367274i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			7081.78i			1.66039i	0.557476	+	0.830193i	$0.311770\pi$
$263$			7081.78i			1.66039i	−0.557476	+	0.830193i	$0.688230\pi$
$264$		0			0
$265$		−	1050.88i		−	0.243603i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1277.73	−	1277.73i	0.289607	−	0.289607i	−0.547318	−	0.836925i	$-0.684351\pi$
$269$	1277.73	−	1277.73i	0.289607	−	0.289607i	0.836925	+	0.547318i	$0.184351\pi$
$270$		0			0
$271$			5785.88i			1.29693i	0.761246	+	0.648463i	$0.224588\pi$
$271$			5785.88i			1.29693i	−0.761246	+	0.648463i	$0.775412\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	353.541	+	353.541i	0.0775249	+	0.0775249i
$276$		0			0
$277$	−2243.55	+	2243.55i	−0.486648	+	0.486648i	−0.907247	−	0.420598i	$-0.861820\pi$
$277$	−2243.55	+	2243.55i	−0.486648	+	0.486648i	0.420598	+	0.907247i	$0.361820\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	645.762			0.137092			0.0685461	−	0.997648i	$-0.478164\pi$
$281$	645.762			0.137092			0.0685461	+	0.997648i	$0.478164\pi$
$282$		0			0
$283$	−4732.85	−	4732.85i	−0.994130	−	0.994130i	0.00585324	−	0.999983i	$-0.498137\pi$
$283$	−4732.85	−	4732.85i	−0.994130	−	0.994130i	−0.999983	+	0.00585324i	$0.998137\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−257.200			−0.0528990
$288$		0			0
$289$	−95.8192			−0.0195032
$290$		0			0
$291$		0			0
$292$		0			0
$293$	648.055	+	648.055i	0.129214	+	0.129214i	0.768756	−	0.639542i	$-0.220876\pi$
$293$	648.055	+	648.055i	0.129214	+	0.129214i	−0.639542	+	0.768756i	$0.720876\pi$
$294$		0			0
$295$	3934.99			0.776623
$296$		0			0
$297$		0			0
$298$		0			0
$299$	889.020	−	889.020i	0.171951	−	0.171951i
$300$		0			0
$301$	1234.90	+	1234.90i	0.236474	+	0.236474i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	3426.57i		−	0.643295i
$306$		0			0
$307$	3406.50	−	3406.50i	0.633288	−	0.633288i	−0.315604	−	0.948891i	$-0.602207\pi$
$307$	3406.50	−	3406.50i	0.633288	−	0.633288i	0.948891	+	0.315604i	$0.102207\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	7811.27i		−	1.42423i	−0.702061	−	0.712117i	$-0.747737\pi$
$311$		−	7811.27i		−	1.42423i	0.702061	−	0.712117i	$-0.252263\pi$
$312$		0			0
$313$		−	3161.73i		−	0.570963i	−0.958384	−	0.285481i	$-0.907847\pi$
$313$		−	3161.73i		−	0.570963i	0.958384	−	0.285481i	$-0.0921535\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−318.480	+	318.480i	−0.0564278	+	0.0564278i	−0.734758	−	0.678330i	$-0.762704\pi$
$317$	−318.480	+	318.480i	−0.0564278	+	0.0564278i	0.678330	+	0.734758i	$0.262704\pi$
$318$		0			0
$319$		−	190.475i		−	0.0334312i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2304.33	−	2304.33i	−0.396955	−	0.396955i
$324$		0			0
$325$	5642.69	−	5642.69i	0.963077	−	0.963077i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	4882.23			0.818134
$330$		0			0
$331$	2065.35	+	2065.35i	0.342966	+	0.342966i	0.857481	−	0.514515i	$-0.172028\pi$
$331$	2065.35	+	2065.35i	0.342966	+	0.342966i	−0.514515	+	0.857481i	$0.672028\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3623.89			−0.591027
$336$		0			0
$337$	6794.49			1.09828			0.549138	−	0.835731i	$-0.314956\pi$
$337$	6794.49			1.09828			0.549138	+	0.835731i	$0.314956\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−593.199	−	593.199i	−0.0942040	−	0.0942040i
$342$		0			0
$343$	−6739.03			−1.06086
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3999.59	−	3999.59i	0.618759	−	0.618759i	−0.326454	−	0.945213i	$-0.605854\pi$
$347$	3999.59	−	3999.59i	0.618759	−	0.618759i	0.945213	+	0.326454i	$0.105854\pi$
$348$		0			0
$349$	−1482.98	−	1482.98i	−0.227456	−	0.227456i	0.584173	−	0.811629i	$-0.301419\pi$
$349$	−1482.98	−	1482.98i	−0.227456	−	0.227456i	−0.811629	+	0.584173i	$0.801419\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	6753.68i		−	1.01831i	−0.860676	−	0.509153i	$-0.829959\pi$
$353$		−	6753.68i		−	1.01831i	0.860676	−	0.509153i	$-0.170041\pi$
$354$		0			0
$355$	−3761.14	+	3761.14i	−0.562311	+	0.562311i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			8234.72i			1.21062i	0.795991	+	0.605309i	$0.206950\pi$
$359$			8234.72i			1.21062i	−0.795991	+	0.605309i	$0.793050\pi$
$360$		0			0
$361$			4738.76i			0.690882i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	994.677	−	994.677i	0.142640	−	0.142640i
$366$		0			0
$367$		−	5584.58i		−	0.794312i	−0.917751	−	0.397156i	$-0.869997\pi$
$367$		−	5584.58i		−	0.794312i	0.917751	−	0.397156i	$-0.130003\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2135.78	+	2135.78i	0.298879	+	0.298879i
$372$		0			0
$373$	3202.91	−	3202.91i	0.444613	−	0.444613i	−0.448946	−	0.893559i	$-0.648200\pi$
$373$	3202.91	−	3202.91i	0.444613	−	0.444613i	0.893559	+	0.448946i	$0.148200\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−3040.08			−0.415310
$378$		0			0
$379$	−21.6142	−	21.6142i	−0.00292941	−	0.00292941i	0.705641	−	0.708570i	$-0.250659\pi$
$379$	−21.6142	−	21.6142i	−0.00292941	−	0.00292941i	−0.708570	+	0.705641i	$0.750659\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4947.98			0.660131			0.330065	−	0.943958i	$-0.392929\pi$
$383$	4947.98			0.660131			0.330065	+	0.943958i	$0.392929\pi$
$384$		0			0
$385$	286.576			0.0379357
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4708.60	−	4708.60i	−0.613716	−	0.613716i	0.330196	−	0.943912i	$-0.392885\pi$
$389$	−4708.60	−	4708.60i	−0.613716	−	0.613716i	−0.943912	+	0.330196i	$0.892885\pi$
$390$		0			0
$391$	−1162.07			−0.150303
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3774.26	+	3774.26i	−0.480769	+	0.480769i
$396$		0			0
$397$	−7418.04	−	7418.04i	−0.937786	−	0.937786i	0.0603890	−	0.998175i	$-0.480766\pi$
$397$	−7418.04	−	7418.04i	−0.937786	−	0.937786i	−0.998175	+	0.0603890i	$0.980766\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			13363.8i			1.66423i	0.554600	+	0.832117i	$0.312871\pi$
$401$			13363.8i			1.66423i	−0.554600	+	0.832117i	$0.687129\pi$
$402$		0			0
$403$	−9467.75	+	9467.75i	−1.17028	+	1.17028i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	792.682i		−	0.0965400i
$408$		0			0
$409$		−	1784.64i		−	0.215758i	−0.994164	−	0.107879i	$-0.965594\pi$
$409$		−	1784.64i		−	0.215758i	0.994164	−	0.107879i	$-0.0344059\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−7997.39	+	7997.39i	−0.952847	+	0.952847i
$414$		0			0
$415$			3026.07i			0.357937i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	10663.3	+	10663.3i	1.24328	+	1.24328i	0.958632	+	0.284650i	$0.0918774\pi$
$419$	10663.3	+	10663.3i	1.24328	+	1.24328i	0.284650	+	0.958632i	$0.408123\pi$
$420$		0			0
$421$	−1980.40	+	1980.40i	−0.229261	+	0.229261i	−0.812384	−	0.583123i	$-0.801831\pi$
$421$	−1980.40	+	1980.40i	−0.229261	+	0.229261i	0.583123	+	0.812384i	$0.301831\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−7375.77			−0.841829
$426$		0			0
$427$	6964.10	+	6964.10i	0.789266	+	0.789266i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	3750.79			0.419186			0.209593	−	0.977789i	$-0.432786\pi$
$431$	3750.79			0.419186			0.209593	+	0.977789i	$0.432786\pi$
$432$		0			0
$433$	−14507.9			−1.61017			−0.805086	−	0.593159i	$-0.797881\pi$
$433$	−14507.9			−1.61017			−0.805086	+	0.593159i	$0.797881\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−534.617	−	534.617i	−0.0585222	−	0.0585222i
$438$		0			0
$439$	−13109.7			−1.42526			−0.712632	−	0.701538i	$-0.752497\pi$
$439$	−13109.7			−1.42526			−0.712632	+	0.701538i	$0.752497\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−9118.69	+	9118.69i	−0.977973	+	0.977973i	−0.999763	−	0.0217894i	$-0.993064\pi$
$443$	−9118.69	+	9118.69i	−0.977973	+	0.977973i	0.0217894	+	0.999763i	$0.493064\pi$
$444$		0			0
$445$	−4947.06	−	4947.06i	−0.526995	−	0.526995i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	3154.04i		−	0.331511i	−0.986167	−	0.165756i	$-0.946994\pi$
$449$		−	3154.04i		−	0.331511i	0.986167	−	0.165756i	$-0.0530063\pi$
$450$		0			0
$451$	66.5885	−	66.5885i	0.00695240	−	0.00695240i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	4573.89i		−	0.471268i
$456$		0			0
$457$		−	14705.8i		−	1.50527i	−0.658440	−	0.752633i	$-0.728783\pi$
$457$		−	14705.8i		−	1.50527i	0.658440	−	0.752633i	$-0.271217\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−9633.21	+	9633.21i	−0.973240	+	0.973240i	−0.999651	−	0.0264114i	$-0.991592\pi$
$461$	−9633.21	+	9633.21i	−0.973240	+	0.973240i	0.0264114	+	0.999651i	$0.491592\pi$
$462$		0			0
$463$		−	13731.1i		−	1.37827i	−0.724632	−	0.689136i	$-0.757990\pi$
$463$		−	13731.1i		−	1.37827i	0.724632	−	0.689136i	$-0.242010\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1307.20	+	1307.20i	0.129529	+	0.129529i	0.768899	−	0.639370i	$-0.220805\pi$
$467$	1307.20	+	1307.20i	0.129529	+	0.129529i	−0.639370	+	0.768899i	$0.720805\pi$
$468$		0			0
$469$	7365.11	−	7365.11i	0.725137	−	0.725137i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−639.428			−0.0621584
$474$		0			0
$475$	−3393.26	−	3393.26i	−0.327776	−	0.327776i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5950.85			−0.567644			−0.283822	−	0.958877i	$-0.591602\pi$
$479$	−5950.85			−0.567644			−0.283822	+	0.958877i	$0.591602\pi$
$480$		0			0
$481$	−12651.6			−1.19930
$482$		0			0
$483$		0			0
$484$		0			0
$485$	450.237	+	450.237i	0.0421530	+	0.0421530i
$486$		0			0
$487$	−13651.9			−1.27028			−0.635141	−	0.772396i	$-0.719058\pi$
$487$	−13651.9			−1.27028			−0.635141	+	0.772396i	$0.719058\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−9699.69	+	9699.69i	−0.891530	+	0.891530i	−0.994667	−	0.103137i	$-0.967112\pi$
$491$	−9699.69	+	9699.69i	−0.891530	+	0.891530i	0.103137	+	0.994667i	$0.467112\pi$
$492$		0			0
$493$	1986.90	+	1986.90i	0.181512	+	0.181512i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	15288.1i		−	1.37981i
$498$		0			0
$499$	−9376.88	+	9376.88i	−0.841216	+	0.841216i	−0.989017	−	0.147801i	$-0.952780\pi$
$499$	−9376.88	+	9376.88i	−0.841216	+	0.841216i	0.147801	+	0.989017i	$0.452780\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			9495.92i			0.841753i	0.907118	+	0.420877i	$0.138277\pi$
$503$			9495.92i			0.841753i	−0.907118	+	0.420877i	$0.861723\pi$
$504$		0			0
$505$		−	3389.91i		−	0.298711i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−5671.64	+	5671.64i	−0.493892	+	0.493892i	−0.909530	−	0.415638i	$-0.863558\pi$
$509$	−5671.64	+	5671.64i	−0.493892	+	0.493892i	0.415638	+	0.909530i	$0.363558\pi$
$510$		0			0
$511$			4043.12i			0.350014i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−712.750	−	712.750i	−0.0609855	−	0.0609855i
$516$		0			0
$517$	−1264.00	+	1264.00i	−0.107525	+	0.107525i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	11983.5			1.00769			0.503846	−	0.863794i	$-0.331918\pi$
$521$	11983.5			1.00769			0.503846	+	0.863794i	$0.331918\pi$
$522$		0			0
$523$	−1886.67	−	1886.67i	−0.157741	−	0.157741i	0.623824	−	0.781565i	$-0.285578\pi$
$523$	−1886.67	−	1886.67i	−0.157741	−	0.157741i	−0.781565	+	0.623824i	$0.785578\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	12375.6			1.02294
$528$		0			0
$529$	11897.4			0.977841
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1062.78	−	1062.78i	−0.0863683	−	0.0863683i
$534$		0			0
$535$	−5314.99			−0.429508
$536$		0			0
$537$		0			0
$538$		0			0
$539$	581.145	−	581.145i	0.0464410	−	0.0464410i
$540$		0			0
$541$	2569.34	+	2569.34i	0.204186	+	0.204186i	0.801791	−	0.597605i	$-0.203881\pi$
$541$	2569.34	+	2569.34i	0.204186	+	0.204186i	−0.597605	+	0.801791i	$0.703881\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	4352.33i		−	0.342079i
$546$		0			0
$547$	10127.9	−	10127.9i	0.791662	−	0.791662i	−0.190102	−	0.981764i	$-0.560882\pi$
$547$	10127.9	−	10127.9i	0.791662	−	0.791662i	0.981764	+	0.190102i	$0.0608820\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			1828.17i			0.141348i
$552$		0			0
$553$		−	15341.5i		−	1.17972i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−11050.4	+	11050.4i	−0.840612	+	0.840612i	−0.988938	−	0.148327i	$-0.952611\pi$
$557$	−11050.4	+	11050.4i	−0.840612	+	0.840612i	0.148327	+	0.988938i	$0.452611\pi$
$558$		0			0
$559$			10205.6i			0.772183i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3100.96	−	3100.96i	−0.232131	−	0.232131i	0.581450	−	0.813582i	$-0.302485\pi$
$563$	−3100.96	−	3100.96i	−0.232131	−	0.232131i	−0.813582	+	0.581450i	$0.802485\pi$
$564$		0			0
$565$	−3392.57	+	3392.57i	−0.252613	+	0.252613i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−12626.6			−0.930288			−0.465144	−	0.885235i	$-0.653997\pi$
$569$	−12626.6			−0.930288			−0.465144	+	0.885235i	$0.653997\pi$
$570$		0			0
$571$	−5777.87	−	5777.87i	−0.423461	−	0.423461i	0.462932	−	0.886394i	$-0.346797\pi$
$571$	−5777.87	−	5777.87i	−0.423461	−	0.423461i	−0.886394	+	0.462932i	$0.846797\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1711.21			−0.124109
$576$		0			0
$577$	−9644.17			−0.695827			−0.347913	−	0.937527i	$-0.613110\pi$
$577$	−9644.17			−0.695827			−0.347913	+	0.937527i	$0.613110\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−6150.13	−	6150.13i	−0.439157	−	0.439157i
$582$		0			0
$583$	−1105.90			−0.0785619
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−6832.33	+	6832.33i	−0.480410	+	0.480410i	−0.905262	−	0.424853i	$-0.860326\pi$
$587$	−6832.33	+	6832.33i	−0.480410	+	0.480410i	0.424853	+	0.905262i	$0.360326\pi$
$588$		0			0
$589$	5693.48	+	5693.48i	0.398295	+	0.398295i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			3807.51i			0.263669i	0.991272	+	0.131835i	$0.0420868\pi$
$593$			3807.51i			0.263669i	−0.991272	+	0.131835i	$0.957913\pi$
$594$		0			0
$595$	−2989.35	+	2989.35i	−0.205969	+	0.205969i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			1731.22i			0.118090i	0.998255	+	0.0590449i	$0.0188055\pi$
$599$			1731.22i			0.118090i	−0.998255	+	0.0590449i	$0.981194\pi$
$600$		0			0
$601$		−	21704.9i		−	1.47315i	−0.676356	−	0.736575i	$-0.736442\pi$
$601$		−	21704.9i		−	1.47315i	0.676356	−	0.736575i	$-0.263558\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	4216.38	−	4216.38i	0.283339	−	0.283339i
$606$		0			0
$607$			19231.2i			1.28595i	0.765888	+	0.642974i	$0.222300\pi$
$607$			19231.2i			1.28595i	−0.765888	+	0.642974i	$0.777700\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	20174.1	+	20174.1i	1.33577	+	1.33577i
$612$		0			0
$613$	9854.85	−	9854.85i	0.649321	−	0.649321i	−0.303508	−	0.952829i	$-0.598158\pi$
$613$	9854.85	−	9854.85i	0.649321	−	0.649321i	0.952829	+	0.303508i	$0.0981579\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−17948.8			−1.17114			−0.585569	−	0.810622i	$-0.699129\pi$
$617$	−17948.8			−1.17114			−0.585569	+	0.810622i	$0.699129\pi$
$618$		0			0
$619$	−6084.79	−	6084.79i	−0.395102	−	0.395102i	0.481399	−	0.876501i	$-0.340129\pi$
$619$	−6084.79	−	6084.79i	−0.395102	−	0.395102i	−0.876501	+	0.481399i	$0.840129\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	20108.6			1.29315
$624$		0			0
$625$	−8263.38			−0.528856
$626$		0			0
$627$		0			0
$628$		0			0
$629$	8268.68	+	8268.68i	0.524155	+	0.524155i
$630$		0			0
$631$	−10974.2			−0.692352			−0.346176	−	0.938170i	$-0.612520\pi$
$631$	−10974.2			−0.692352			−0.346176	+	0.938170i	$0.612520\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−332.298	+	332.298i	−0.0207667	+	0.0207667i
$636$		0			0
$637$	−9275.36	−	9275.36i	−0.576928	−	0.576928i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			2520.18i			0.155290i	0.996981	+	0.0776451i	$0.0247401\pi$
$641$			2520.18i			0.155290i	−0.996981	+	0.0776451i	$0.975260\pi$
$642$		0			0
$643$	−9626.14	+	9626.14i	−0.590386	+	0.590386i	−0.937736	−	0.347350i	$-0.887082\pi$
$643$	−9626.14	+	9626.14i	−0.590386	+	0.590386i	0.347350	+	0.937736i	$0.387082\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			1795.90i			0.109126i	0.998510	+	0.0545628i	$0.0173765\pi$
$647$			1795.90i			0.109126i	−0.998510	+	0.0545628i	$0.982623\pi$
$648$		0			0
$649$		−	4141.02i		−	0.250461i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−2175.09	+	2175.09i	−0.130349	+	0.130349i	−0.769271	−	0.638922i	$-0.779380\pi$
$653$	−2175.09	+	2175.09i	−0.130349	+	0.130349i	0.638922	+	0.769271i	$0.279380\pi$
$654$		0			0
$655$		−	3454.75i		−	0.206089i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−16223.9	−	16223.9i	−0.959017	−	0.959017i	0.0401752	−	0.999193i	$-0.487208\pi$
$659$	−16223.9	−	16223.9i	−0.959017	−	0.959017i	−0.999193	+	0.0401752i	$0.987208\pi$
$660$		0			0
$661$	2953.68	−	2953.68i	0.173805	−	0.173805i	−0.614844	−	0.788649i	$-0.710781\pi$
$661$	2953.68	−	2953.68i	0.173805	−	0.173805i	0.788649	+	0.614844i	$0.210781\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2750.53			−0.160393
$666$		0			0
$667$	460.970	+	460.970i	0.0267598	+	0.0267598i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−3605.99			−0.207463
$672$		0			0
$673$	13136.0			0.752385			0.376193	−	0.926542i	$-0.377233\pi$
$673$	13136.0			0.752385			0.376193	+	0.926542i	$0.377233\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−2954.97	−	2954.97i	−0.167753	−	0.167753i	0.618238	−	0.785991i	$-0.287847\pi$
$677$	−2954.97	−	2954.97i	−0.167753	−	0.167753i	−0.785991	+	0.618238i	$0.787847\pi$
$678$		0			0
$679$	−1830.10			−0.103436
$680$		0			0
$681$		0			0
$682$		0			0
$683$	6641.03	−	6641.03i	0.372053	−	0.372053i	−0.496172	−	0.868224i	$-0.665261\pi$
$683$	6641.03	−	6641.03i	0.372053	−	0.372053i	0.868224	+	0.496172i	$0.165261\pi$
$684$		0			0
$685$	−6747.01	−	6747.01i	−0.376336	−	0.376336i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			17650.7i			0.975960i
$690$		0			0
$691$	−17821.2	+	17821.2i	−0.981115	+	0.981115i	−0.999825	−	0.0187097i	$-0.994044\pi$
$691$	−17821.2	+	17821.2i	−0.981115	+	0.981115i	0.0187097	+	0.999825i	$0.494044\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	8815.11i		−	0.481117i
$696$		0			0
$697$			1389.20i			0.0754948i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	19022.9	−	19022.9i	1.02494	−	1.02494i	0.0252614	−	0.999681i	$-0.491958\pi$
$701$	19022.9	−	19022.9i	1.02494	−	1.02494i	0.999681	−	0.0252614i	$-0.00804182\pi$
$702$		0			0
$703$			7608.10i			0.408172i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6889.58	+	6889.58i	0.366491	+	0.366491i
$708$		0			0
$709$	−15338.6	+	15338.6i	−0.812485	+	0.812485i	−0.985006	−	0.172521i	$-0.944809\pi$
$709$	−15338.6	+	15338.6i	−0.812485	+	0.812485i	0.172521	+	0.985006i	$0.444809\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2871.21			0.150810
$714$		0			0
$715$	1184.17	+	1184.17i	0.0619377	+	0.0619377i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	24937.9			1.29350			0.646751	−	0.762701i	$-0.276127\pi$
$719$	24937.9			1.29350			0.646751	+	0.762701i	$0.276127\pi$
$720$		0			0
$721$	2897.16			0.149647
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2925.82	+	2925.82i	0.149879	+	0.149879i
$726$		0			0
$727$	−4479.75			−0.228535			−0.114267	−	0.993450i	$-0.536452\pi$
$727$	−4479.75			−0.228535			−0.114267	+	0.993450i	$0.536452\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	6670.05	−	6670.05i	0.337484	−	0.337484i
$732$		0			0
$733$	−7481.44	−	7481.44i	−0.376989	−	0.376989i	0.493025	−	0.870015i	$-0.335891\pi$
$733$	−7481.44	−	7481.44i	−0.376989	−	0.376989i	−0.870015	+	0.493025i	$0.835891\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			3813.63i			0.190606i
$738$		0			0
$739$	24827.3	−	24827.3i	1.23584	−	1.23584i	0.274160	−	0.961684i	$-0.411600\pi$
$739$	24827.3	−	24827.3i	1.23584	−	1.23584i	0.961684	−	0.274160i	$-0.0883998\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			16682.5i			0.823716i	0.911248	+	0.411858i	$0.135120\pi$
$743$			16682.5i			0.823716i	−0.911248	+	0.411858i	$0.864880\pi$
$744$		0			0
$745$			4598.13i			0.226124i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	10802.1	−	10802.1i	0.526968	−	0.526968i
$750$		0			0
$751$		−	21972.3i		−	1.06762i	−0.845605	−	0.533809i	$-0.820760\pi$
$751$		−	21972.3i		−	1.06762i	0.845605	−	0.533809i	$-0.179240\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−10264.3	−	10264.3i	−0.494775	−	0.494775i
$756$		0			0
$757$	13715.8	−	13715.8i	0.658530	−	0.658530i	−0.296502	−	0.955032i	$-0.595820\pi$
$757$	13715.8	−	13715.8i	0.658530	−	0.658530i	0.955032	+	0.296502i	$0.0958201\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−3346.07			−0.159389			−0.0796944	−	0.996819i	$-0.525394\pi$
$761$	−3346.07			−0.159389			−0.0796944	+	0.996819i	$0.525394\pi$
$762$		0			0
$763$	8845.59	+	8845.59i	0.419701	+	0.419701i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−66092.6			−3.11143
$768$		0			0
$769$	18053.3			0.846578			0.423289	−	0.905995i	$-0.360876\pi$
$769$	18053.3			0.846578			0.423289	+	0.905995i	$0.360876\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	8674.66	+	8674.66i	0.403630	+	0.403630i	0.879510	−	0.475880i	$-0.157870\pi$
$773$	8674.66	+	8674.66i	0.403630	+	0.403630i	−0.475880	+	0.879510i	$0.657870\pi$
$774$		0			0
$775$	18223.8			0.844670
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−639.111	+	639.111i	−0.0293948	+	0.0293948i
$780$		0			0
$781$	3958.07	+	3958.07i	0.181345	+	0.181345i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	9786.36i		−	0.444956i
$786$		0			0
$787$	6512.95	−	6512.95i	0.294996	−	0.294996i	−0.544054	−	0.839050i	$-0.683111\pi$
$787$	6512.95	−	6512.95i	0.294996	−	0.294996i	0.839050	+	0.544054i	$0.183111\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	13790.0i		−	0.619867i
$792$		0			0
$793$			57553.3i			2.57727i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−10831.9	+	10831.9i	−0.481413	+	0.481413i	−0.905583	−	0.424169i	$-0.860566\pi$
$797$	−10831.9	+	10831.9i	−0.481413	+	0.481413i	0.424169	+	0.905583i	$0.360566\pi$
$798$		0			0
$799$		−	26370.2i		−	1.16760i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1046.76	−	1046.76i	−0.0460015	−	0.0460015i
$804$		0			0
$805$	−693.543	+	693.543i	−0.0303654	+	0.0303654i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	18515.3			0.804651			0.402326	−	0.915497i	$-0.368202\pi$
$809$	18515.3			0.804651			0.402326	+	0.915497i	$0.368202\pi$
$810$		0			0
$811$	−15062.3	−	15062.3i	−0.652167	−	0.652167i	0.301347	−	0.953514i	$-0.402564\pi$
$811$	−15062.3	−	15062.3i	−0.652167	−	0.652167i	−0.953514	+	0.301347i	$0.902564\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−9241.51			−0.397197
$816$		0			0
$817$	6137.18			0.262806
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−24601.7	−	24601.7i	−1.04580	−	1.04580i	−0.998899	−	0.0469030i	$-0.985065\pi$
$821$	−24601.7	−	24601.7i	−1.04580	−	1.04580i	−0.0469030	−	0.998899i	$-0.514935\pi$
$822$		0			0
$823$	24417.7			1.03420			0.517101	−	0.855924i	$-0.327011\pi$
$823$	24417.7			1.03420			0.517101	+	0.855924i	$0.327011\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	26807.5	−	26807.5i	1.12719	−	1.12719i	0.136560	−	0.990632i	$-0.456396\pi$
$827$	26807.5	−	26807.5i	1.12719	−	1.12719i	0.990632	−	0.136560i	$-0.0436045\pi$
$828$		0			0
$829$	3214.77	+	3214.77i	0.134685	+	0.134685i	0.771235	−	0.636550i	$-0.219639\pi$
$829$	3214.77	+	3214.77i	0.134685	+	0.134685i	−0.636550	+	0.771235i	$0.719639\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			12124.2i			0.504294i
$834$		0			0
$835$	−5076.90	+	5076.90i	−0.210411	+	0.210411i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			40899.9i			1.68298i	0.540271	+	0.841491i	$0.318322\pi$
$839$			40899.9i			1.68298i	−0.540271	+	0.841491i	$0.681678\pi$
$840$		0			0
$841$			22812.7i			0.935367i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	11817.8	−	11817.8i	0.481116	−	0.481116i
$846$		0			0
$847$			17138.6i			0.695263i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1918.37	+	1918.37i	0.0772749	+	0.0772749i
$852$		0			0
$853$	−21139.7	+	21139.7i	−0.848545	+	0.848545i	−0.989952	−	0.141407i	$-0.954837\pi$
$853$	−21139.7	+	21139.7i	−0.848545	+	0.848545i	0.141407	+	0.989952i	$0.454837\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	2133.86			0.0850540			0.0425270	−	0.999095i	$-0.486459\pi$
$857$	2133.86			0.0850540			0.0425270	+	0.999095i	$0.486459\pi$
$858$		0			0
$859$	−14755.9	−	14755.9i	−0.586107	−	0.586107i	0.350468	−	0.936575i	$-0.386023\pi$
$859$	−14755.9	−	14755.9i	−0.586107	−	0.586107i	−0.936575	+	0.350468i	$0.886023\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−13261.8			−0.523101			−0.261550	−	0.965190i	$-0.584234\pi$
$863$	−13261.8			−0.523101			−0.261550	+	0.965190i	$0.584234\pi$
$864$		0			0
$865$	−14977.0			−0.588710
$866$		0			0
$867$		0			0
$868$		0			0
$869$	3971.88	+	3971.88i	0.155048	+	0.155048i
$870$		0			0
$871$	60867.3			2.36787
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−9681.80	+	9681.80i	−0.374062	+	0.374062i
$876$		0			0
$877$	−17125.7	−	17125.7i	−0.659399	−	0.659399i	0.295839	−	0.955238i	$-0.404401\pi$
$877$	−17125.7	−	17125.7i	−0.659399	−	0.659399i	−0.955238	+	0.295839i	$0.904401\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			34029.0i			1.30132i	0.759368	+	0.650662i	$0.225508\pi$
$881$			34029.0i			1.30132i	−0.759368	+	0.650662i	$0.774492\pi$
$882$		0			0
$883$	7536.29	−	7536.29i	0.287221	−	0.287221i	−0.548759	−	0.835981i	$-0.684900\pi$
$883$	7536.29	−	7536.29i	0.287221	−	0.287221i	0.835981	+	0.548759i	$0.184900\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	11775.4i		−	0.445749i	−0.974847	−	0.222875i	$-0.928456\pi$
$887$		−	11775.4i		−	0.445749i	0.974847	−	0.222875i	$-0.0715441\pi$
$888$		0			0
$889$		−	1350.71i		−	0.0509578i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	12131.8	−	12131.8i	0.454619	−	0.454619i
$894$		0			0
$895$			16112.4i			0.601764i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−4909.17	−	4909.17i	−0.182124	−	0.182124i
$900$		0			0
$901$	11535.9	−	11535.9i	0.426545	−	0.426545i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	7787.77			0.286049
$906$		0			0
$907$	24197.1	+	24197.1i	0.885833	+	0.885833i	0.994120	−	0.108287i	$-0.0345365\pi$
$907$	24197.1	+	24197.1i	0.885833	+	0.885833i	−0.108287	+	0.994120i	$0.534537\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−15777.4			−0.573798			−0.286899	−	0.957961i	$-0.592624\pi$
$911$	−15777.4			−0.573798			−0.286899	+	0.957961i	$0.592624\pi$
$912$		0			0
$913$	3184.51			0.115435
$914$		0			0
$915$		0			0
$916$		0			0
$917$	7021.37	+	7021.37i	0.252853	+	0.252853i
$918$		0			0
$919$	−24261.8			−0.870863			−0.435432	−	0.900222i	$-0.643404\pi$
$919$	−24261.8			−0.870863			−0.435432	+	0.900222i	$0.643404\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	63172.6	−	63172.6i	2.25282	−	2.25282i
$924$		0			0
$925$	12176.1	+	12176.1i	0.432808	+	0.432808i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	19578.0i		−	0.691423i	−0.938341	−	0.345712i	$-0.887638\pi$
$929$		−	19578.0i		−	0.691423i	0.938341	−	0.345712i	$-0.112362\pi$
$930$		0			0
$931$	−5577.79	+	5577.79i	−0.196353	+	0.196353i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	1547.87i		−	0.0541400i
$936$		0			0
$937$			23706.5i			0.826530i	0.910611	+	0.413265i	$0.135612\pi$
$937$			23706.5i			0.826530i	−0.910611	+	0.413265i	$0.864388\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	516.925	−	516.925i	0.0179078	−	0.0179078i	−0.698096	−	0.716004i	$-0.745969\pi$
$941$	516.925	−	516.925i	0.0179078	−	0.0179078i	0.716004	+	0.698096i	$0.245969\pi$
$942$		0			0
$943$			322.302i			0.0111300i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−9551.64	−	9551.64i	−0.327758	−	0.327758i	0.523976	−	0.851733i	$-0.324448\pi$
$947$	−9551.64	−	9551.64i	−0.327758	−	0.327758i	−0.851733	+	0.523976i	$0.824448\pi$
$948$		0			0
$949$	−16706.7	+	16706.7i	−0.571469	+	0.571469i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	12241.2			0.416089			0.208044	−	0.978119i	$-0.433290\pi$
$953$	12241.2			0.416089			0.208044	+	0.978119i	$0.433290\pi$
$954$		0			0
$955$	4202.94	+	4202.94i	0.142412	+	0.142412i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	27425.0			0.923461
$960$		0			0
$961$	−786.360			−0.0263959
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−10343.0	−	10343.0i	−0.345030	−	0.345030i
$966$		0			0
$967$	12983.6			0.431773			0.215886	−	0.976418i	$-0.430736\pi$
$967$	12983.6			0.431773			0.215886	+	0.976418i	$0.430736\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−35022.4	+	35022.4i	−1.15749	+	1.15749i	−0.172477	+	0.985014i	$0.555177\pi$
$971$	−35022.4	+	35022.4i	−1.15749	+	1.15749i	−0.985014	+	0.172477i	$0.944823\pi$
$972$		0			0
$973$	17915.7	+	17915.7i	0.590287	+	0.590287i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	43174.5i		−	1.41379i	−0.707317	−	0.706897i	$-0.750095\pi$
$977$		−	43174.5i		−	1.41379i	0.707317	−	0.706897i	$-0.249905\pi$
$978$		0			0
$979$	−5206.08	+	5206.08i	−0.169956	+	0.169956i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	43699.4i		−	1.41790i	−0.705260	−	0.708949i	$-0.749170\pi$
$983$		−	43699.4i		−	1.41790i	0.705260	−	0.708949i	$-0.250830\pi$
$984$		0			0
$985$		−	9516.21i		−	0.307829i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1547.48	−	1547.48i	0.0497544	−	0.0497544i
$990$		0			0
$991$			8324.37i			0.266834i	0.991060	+	0.133417i	$0.0425949\pi$
$991$			8324.37i			0.266834i	−0.991060	+	0.133417i	$0.957405\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−5475.50	−	5475.50i	−0.174457	−	0.174457i
$996$		0			0
$997$	563.783	−	563.783i	0.0179089	−	0.0179089i	−0.698096	−	0.716005i	$-0.745969\pi$
$997$	563.783	−	563.783i	0.0179089	−	0.0179089i	0.716005	+	0.698096i	$0.245969\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.4.l.a.863.11		48
3.2	odd	2	inner	1152.4.l.a.863.14		48
4.3	odd	2		1152.4.l.b.863.11		48
8.3	odd	2		144.4.l.a.35.8	✓	48
8.5	even	2		576.4.l.a.431.14		48
12.11	even	2		1152.4.l.b.863.14		48
16.3	odd	4		576.4.l.a.143.11		48
16.5	even	4		1152.4.l.b.287.14		48
16.11	odd	4	inner	1152.4.l.a.287.14		48
16.13	even	4		144.4.l.a.107.17	yes	48
24.5	odd	2		576.4.l.a.431.11		48
24.11	even	2		144.4.l.a.35.17	yes	48
48.5	odd	4		1152.4.l.b.287.11		48
48.11	even	4	inner	1152.4.l.a.287.11		48
48.29	odd	4		144.4.l.a.107.8	yes	48
48.35	even	4		576.4.l.a.143.14		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.4.l.a.35.8	✓	48	8.3	odd	2
144.4.l.a.35.17	yes	48	24.11	even	2
144.4.l.a.107.8	yes	48	48.29	odd	4
144.4.l.a.107.17	yes	48	16.13	even	4
576.4.l.a.143.11		48	16.3	odd	4
576.4.l.a.143.14		48	48.35	even	4
576.4.l.a.431.11		48	24.5	odd	2
576.4.l.a.431.14		48	8.5	even	2
1152.4.l.a.287.11		48	48.11	even	4	inner
1152.4.l.a.287.14		48	16.11	odd	4	inner
1152.4.l.a.863.11		48	1.1	even	1	trivial
1152.4.l.a.863.14		48	3.2	odd	2	inner
1152.4.l.b.287.11		48	48.5	odd	4
1152.4.l.b.287.14		48	16.5	even	4
1152.4.l.b.863.11		48	4.3	odd	2
1152.4.l.b.863.14		48	12.11	even	2

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:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1152.l (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-3.22357 - 3.22357i) q^{5} +13.1030 q^{7} +O(q^{10})\)	\(q+(-3.22357 - 3.22357i) q^{5} +13.1030 q^{7} +(-3.39235 + 3.39235i) q^{11} +(54.1435 + 54.1435i) q^{13} -70.7730i q^{17} +(32.5595 - 32.5595i) q^{19} -16.4197i q^{23} -104.217i q^{25} +(-28.0742 + 28.0742i) q^{29} +174.864i q^{31} +(-42.2385 - 42.2385i) q^{35} +(-116.834 + 116.834i) q^{37} -19.6290 q^{41} +(94.2456 + 94.2456i) q^{43} +372.603 q^{47} -171.310 q^{49} +(162.999 + 162.999i) q^{53} +21.8710 q^{55} +(-610.346 + 610.346i) q^{59} +(531.488 + 531.488i) q^{61} -349.071i q^{65} +(562.092 - 562.092i) q^{67} -1166.76i q^{71} +308.564i q^{73} +(-44.4501 + 44.4501i) q^{77} -1170.83i q^{79} +(-469.366 - 469.366i) q^{83} +(-228.142 + 228.142i) q^{85} +1534.65 q^{89} +(709.445 + 709.445i) q^{91} -209.916 q^{95} -139.670 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q - 48 q^{19} + 864 q^{43} + 2352 q^{49} + 576 q^{55} - 1824 q^{61} + 816 q^{67} + 480 q^{85} - 3600 q^{91}+O(q^{100}) \) 48 * q - 48 * q^19 + 864 * q^43 + 2352 * q^49 + 576 * q^55 - 1824 * q^61 + 816 * q^67 + 480 * q^85 - 3600 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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