LMFDB - Embedded newform 576.5.m.a.271.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,5,Mod(271,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 0]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("576.271");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	576.m (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:	$59.5410987363$
Analytic rank:	$0$
Dimension:	$14$
Relative dimension:	$7$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 4 x^{13} + 15 x^{12} - 34 x^{11} + 62 x^{10} - 312 x^{9} + 1432 x^{8} - 4960 x^{7} + \cdots + 2097152 $ x^14 - 4x^13 + 15x^12 - 34x^11 + 62x^10 - 312x^9 + 1432x^8 - 4960x^7 + 11456x^6 - 19968x^5 + 31744x^4 - 139264x^3 + 491520x^2 - 1048576*x + 2097152
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{42} $
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			271.3
Root			$-2.15805 + 1.82834i$ of defining polynomial
Character	$\chi$	$=$	576.271
Dual form			576.5.m.a.559.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.72348 - 4.72348i) q^{5} -45.3712 q^{7} +O(q^{10})$	$q+(-4.72348 - 4.72348i) q^{5} -45.3712 q^{7} +(110.228 - 110.228i) q^{11} +(-157.128 + 157.128i) q^{13} +378.592 q^{17} +(-203.127 - 203.127i) q^{19} -740.444 q^{23} -580.377i q^{25} +(-82.6244 + 82.6244i) q^{29} -286.217i q^{31} +(214.310 + 214.310i) q^{35} +(1470.39 + 1470.39i) q^{37} +1301.21i q^{41} +(366.979 - 366.979i) q^{43} +751.307i q^{47} -342.455 q^{49} +(1929.33 + 1929.33i) q^{53} -1041.32 q^{55} +(-1357.57 + 1357.57i) q^{59} +(-1835.78 + 1835.78i) q^{61} +1484.39 q^{65} +(-2205.02 - 2205.02i) q^{67} +8970.40 q^{71} +9350.46i q^{73} +(-5001.17 + 5001.17i) q^{77} +2860.08i q^{79} +(-1036.63 - 1036.63i) q^{83} +(-1788.27 - 1788.27i) q^{85} +5171.06i q^{89} +(7129.10 - 7129.10i) q^{91} +1918.94i q^{95} +8539.92 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 2 q^{5} + 4 q^{7}+O(q^{10}) $ 14 * q + 2 * q^5 + 4 * q^7	$ 14 q + 2 q^{5} + 4 q^{7} + 94 q^{11} - 2 q^{13} + 4 q^{17} + 706 q^{19} + 1148 q^{23} - 862 q^{29} + 1340 q^{35} - 1826 q^{37} - 1694 q^{43} + 682 q^{49} + 482 q^{53} + 11780 q^{55} - 2786 q^{59} - 3778 q^{61} + 2020 q^{65} - 7998 q^{67} + 19964 q^{71} + 9508 q^{77} - 17282 q^{83} + 9948 q^{85} + 28036 q^{91} - 4 q^{97}+O(q^{100}) $ 14 * q + 2 * q^5 + 4 * q^7 + 94 * q^11 - 2 * q^13 + 4 * q^17 + 706 * q^19 + 1148 * q^23 - 862 * q^29 + 1340 * q^35 - 1826 * q^37 - 1694 * q^43 + 682 * q^49 + 482 * q^53 + 11780 * q^55 - 2786 * q^59 - 3778 * q^61 + 2020 * q^65 - 7998 * q^67 + 19964 * q^71 + 9508 * q^77 - 17282 * q^83 + 9948 * q^85 + 28036 * q^91 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{4}\right)$

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
$5$	−4.72348	−	4.72348i	−0.188939	−	0.188939i	0.606298	−	0.795237i	$-0.292654\pi$
$5$	−4.72348	−	4.72348i	−0.188939	−	0.188939i	−0.795237	+	0.606298i	$0.792654\pi$
$6$		0			0
$7$	−45.3712			−0.925943			−0.462971	−	0.886373i	$-0.653217\pi$
$7$	−45.3712			−0.925943			−0.462971	+	0.886373i	$0.653217\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	110.228	−	110.228i	0.910974	−	0.910974i	−0.0853752	−	0.996349i	$-0.527209\pi$
$11$	110.228	−	110.228i	0.910974	−	0.910974i	0.996349	+	0.0853752i	$0.0272089\pi$
$12$		0			0
$13$	−157.128	+	157.128i	−0.929754	+	0.929754i	−0.997690	−	0.0679357i	$-0.978359\pi$
$13$	−157.128	+	157.128i	−0.929754	+	0.929754i	0.0679357	+	0.997690i	$0.478359\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	378.592			1.31001			0.655003	−	0.755626i	$-0.272667\pi$
$17$	378.592			1.31001			0.655003	+	0.755626i	$0.272667\pi$
$18$		0			0
$19$	−203.127	−	203.127i	−0.562680	−	0.562680i	0.367388	−	0.930068i	$-0.380252\pi$
$19$	−203.127	−	203.127i	−0.562680	−	0.562680i	−0.930068	+	0.367388i	$0.880252\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−740.444			−1.39971			−0.699853	−	0.714287i	$-0.746751\pi$
$23$	−740.444			−1.39971			−0.699853	+	0.714287i	$0.746751\pi$
$24$		0			0
$25$		−	580.377i		−	0.928604i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−82.6244	+	82.6244i	−0.0982455	+	0.0982455i	−0.754521	−	0.656276i	$-0.772131\pi$
$29$	−82.6244	+	82.6244i	−0.0982455	+	0.0982455i	0.656276	+	0.754521i	$0.272131\pi$
$30$		0			0
$31$		−	286.217i		−	0.297833i	−0.988850	−	0.148916i	$-0.952421\pi$
$31$		−	286.217i		−	0.297833i	0.988850	−	0.148916i	$-0.0475785\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	214.310	+	214.310i	0.174947	+	0.174947i
$36$		0			0
$37$	1470.39	+	1470.39i	1.07406	+	1.07406i	0.997028	+	0.0770352i	$0.0245454\pi$
$37$	1470.39	+	1470.39i	1.07406	+	1.07406i	0.0770352	+	0.997028i	$0.475455\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			1301.21i			0.774070i	0.922065	+	0.387035i	$0.126501\pi$
$41$			1301.21i			0.774070i	−0.922065	+	0.387035i	$0.873499\pi$
$42$		0			0
$43$	366.979	−	366.979i	0.198474	−	0.198474i	−0.600871	−	0.799346i	$-0.705180\pi$
$43$	366.979	−	366.979i	0.198474	−	0.198474i	0.799346	+	0.600871i	$0.205180\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			751.307i			0.340112i	0.985434	+	0.170056i	$0.0543948\pi$
$47$			751.307i			0.340112i	−0.985434	+	0.170056i	$0.945605\pi$
$48$		0			0
$49$	−342.455			−0.142630
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1929.33	+	1929.33i	0.686838	+	0.686838i	0.961532	−	0.274693i	$-0.0885764\pi$
$53$	1929.33	+	1929.33i	0.686838	+	0.686838i	−0.274693	+	0.961532i	$0.588576\pi$
$54$		0			0
$55$	−1041.32			−0.344238
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1357.57	+	1357.57i	−0.389994	+	0.389994i	−0.874685	−	0.484691i	$-0.838932\pi$
$59$	−1357.57	+	1357.57i	−0.389994	+	0.389994i	0.484691	+	0.874685i	$0.338932\pi$
$60$		0			0
$61$	−1835.78	+	1835.78i	−0.493356	+	0.493356i	−0.909362	−	0.416006i	$-0.863430\pi$
$61$	−1835.78	+	1835.78i	−0.493356	+	0.493356i	0.416006	+	0.909362i	$0.363430\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1484.39			0.351334
$66$		0			0
$67$	−2205.02	−	2205.02i	−0.491204	−	0.491204i	0.417481	−	0.908686i	$-0.362913\pi$
$67$	−2205.02	−	2205.02i	−0.491204	−	0.491204i	−0.908686	+	0.417481i	$0.862913\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	8970.40			1.77949			0.889744	−	0.456460i	$-0.150883\pi$
$71$	8970.40			1.77949			0.889744	+	0.456460i	$0.150883\pi$
$72$		0			0
$73$			9350.46i			1.75464i	0.479909	+	0.877318i	$0.340670\pi$
$73$			9350.46i			1.75464i	−0.479909	+	0.877318i	$0.659330\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−5001.17	+	5001.17i	−0.843509	+	0.843509i
$78$		0			0
$79$			2860.08i			0.458273i	0.973394	+	0.229137i	$0.0735903\pi$
$79$			2860.08i			0.458273i	−0.973394	+	0.229137i	$0.926410\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1036.63	−	1036.63i	−0.150477	−	0.150477i	0.627854	−	0.778331i	$-0.283933\pi$
$83$	−1036.63	−	1036.63i	−0.150477	−	0.150477i	−0.778331	+	0.627854i	$0.783933\pi$
$84$		0			0
$85$	−1788.27	−	1788.27i	−0.247512	−	0.247512i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			5171.06i			0.652830i	0.945227	+	0.326415i	$0.105841\pi$
$89$			5171.06i			0.652830i	−0.945227	+	0.326415i	$0.894159\pi$
$90$		0			0
$91$	7129.10	−	7129.10i	0.860899	−	0.860899i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			1918.94i			0.212625i
$96$		0			0
$97$	8539.92			0.907633			0.453817	−	0.891095i	$-0.350062\pi$
$97$	8539.92			0.907633			0.453817	+	0.891095i	$0.350062\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−537.642	−	537.642i	−0.0527049	−	0.0527049i	0.680263	−	0.732968i	$-0.261866\pi$
$101$	−537.642	−	537.642i	−0.0527049	−	0.0527049i	−0.732968	+	0.680263i	$0.761866\pi$
$102$		0			0
$103$	5543.86			0.522562			0.261281	−	0.965263i	$-0.415855\pi$
$103$	5543.86			0.522562			0.261281	+	0.965263i	$0.415855\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2989.60	+	2989.60i	−0.261123	+	0.261123i	−0.825510	−	0.564387i	$-0.809113\pi$
$107$	−2989.60	+	2989.60i	−0.261123	+	0.261123i	0.564387	+	0.825510i	$0.309113\pi$
$108$		0			0
$109$	−3353.83	+	3353.83i	−0.282285	+	0.282285i	−0.834020	−	0.551734i	$-0.813966\pi$
$109$	−3353.83	+	3353.83i	−0.282285	+	0.282285i	0.551734	+	0.834020i	$0.313966\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3193.81			0.250122			0.125061	−	0.992149i	$-0.460087\pi$
$113$	3193.81			0.250122			0.125061	+	0.992149i	$0.460087\pi$
$114$		0			0
$115$	3497.48	+	3497.48i	0.264459	+	0.264459i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−17177.2			−1.21299
$120$		0			0
$121$		−	9659.34i		−	0.659746i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−5693.58	+	5693.58i	−0.364389	+	0.364389i
$126$		0			0
$127$			22900.5i			1.41983i	0.704287	+	0.709915i	$0.251267\pi$
$127$			22900.5i			1.41983i	−0.704287	+	0.709915i	$0.748733\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	19255.9	+	19255.9i	1.12207	+	1.12207i	0.991429	+	0.130644i	$0.0417045\pi$
$131$	19255.9	+	19255.9i	1.12207	+	1.12207i	0.130644	+	0.991429i	$0.458296\pi$
$132$		0			0
$133$	9216.13	+	9216.13i	0.521009	+	0.521009i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			19938.2i			1.06229i	0.847280	+	0.531147i	$0.178239\pi$
$137$			19938.2i			1.06229i	−0.847280	+	0.531147i	$0.821761\pi$
$138$		0			0
$139$	−25175.4	+	25175.4i	−1.30301	+	1.30301i	−0.376649	+	0.926356i	$0.622924\pi$
$139$	−25175.4	+	25175.4i	−1.30301	+	1.30301i	−0.926356	+	0.376649i	$0.877076\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			34639.8i			1.69396i
$144$		0			0
$145$	780.550			0.0371249
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3216.99	+	3216.99i	0.144903	+	0.144903i	0.775837	−	0.630934i	$-0.217328\pi$
$149$	3216.99	+	3216.99i	0.144903	+	0.144903i	−0.630934	+	0.775837i	$0.717328\pi$
$150$		0			0
$151$	14770.5			0.647802			0.323901	−	0.946091i	$-0.395006\pi$
$151$	14770.5			0.647802			0.323901	+	0.946091i	$0.395006\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1351.94	+	1351.94i	−0.0562723	+	0.0562723i
$156$		0			0
$157$	−18014.6	+	18014.6i	−0.730845	+	0.730845i	−0.970787	−	0.239942i	$-0.922872\pi$
$157$	−18014.6	+	18014.6i	−0.730845	+	0.730845i	0.239942	+	0.970787i	$0.422872\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	33594.8			1.29605
$162$		0			0
$163$	−15083.2	−	15083.2i	−0.567700	−	0.567700i	0.363784	−	0.931483i	$-0.381485\pi$
$163$	−15083.2	−	15083.2i	−0.567700	−	0.567700i	−0.931483	+	0.363784i	$0.881485\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	23733.6			0.851001			0.425501	−	0.904958i	$-0.360098\pi$
$167$	23733.6			0.851001			0.425501	+	0.904958i	$0.360098\pi$
$168$		0			0
$169$		−	20817.7i		−	0.728885i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	12072.3	−	12072.3i	0.403363	−	0.403363i	−0.476053	−	0.879416i	$-0.657933\pi$
$173$	12072.3	−	12072.3i	0.403363	−	0.403363i	0.879416	+	0.476053i	$0.157933\pi$
$174$		0			0
$175$			26332.4i			0.859834i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−28111.5	−	28111.5i	−0.877361	−	0.877361i	0.115900	−	0.993261i	$-0.463025\pi$
$179$	−28111.5	−	28111.5i	−0.877361	−	0.877361i	−0.993261	+	0.115900i	$0.963025\pi$
$180$		0			0
$181$	−15906.7	−	15906.7i	−0.485539	−	0.485539i	0.421356	−	0.906895i	$-0.361554\pi$
$181$	−15906.7	−	15906.7i	−0.485539	−	0.485539i	−0.906895	+	0.421356i	$0.861554\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	13890.8i		−	0.405866i
$186$		0			0
$187$	41731.4	−	41731.4i	1.19338	−	1.19338i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			28038.6i			0.768581i	0.923212	+	0.384290i	$0.125554\pi$
$191$			28038.6i			0.768581i	−0.923212	+	0.384290i	$0.874446\pi$
$192$		0			0
$193$	56713.1			1.52254			0.761270	−	0.648436i	$-0.224576\pi$
$193$	56713.1			1.52254			0.761270	+	0.648436i	$0.224576\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	10179.3	+	10179.3i	0.262292	+	0.262292i	0.825985	−	0.563692i	$-0.190620\pi$
$197$	10179.3	+	10179.3i	0.262292	+	0.262292i	−0.563692	+	0.825985i	$0.690620\pi$
$198$		0			0
$199$	−31787.3			−0.802689			−0.401344	−	0.915927i	$-0.631457\pi$
$199$	−31787.3			−0.802689			−0.401344	+	0.915927i	$0.631457\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3748.77	−	3748.77i	0.0909697	−	0.0909697i
$204$		0			0
$205$	6146.25	−	6146.25i	0.146252	−	0.146252i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−44780.6			−1.02517
$210$		0			0
$211$	10699.8	+	10699.8i	0.240332	+	0.240332i	0.816988	−	0.576655i	$-0.195642\pi$
$211$	10699.8	+	10699.8i	0.240332	+	0.240332i	−0.576655	+	0.816988i	$0.695642\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−3466.84			−0.0749992
$216$		0			0
$217$			12986.0i			0.275776i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−59487.5	+	59487.5i	−1.21798	+	1.21798i
$222$		0			0
$223$			91700.4i			1.84400i	0.387187	+	0.922001i	$0.373447\pi$
$223$			91700.4i			1.84400i	−0.387187	+	0.922001i	$0.626553\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	17499.5	+	17499.5i	0.339605	+	0.339605i	0.856218	−	0.516614i	$-0.172808\pi$
$227$	17499.5	+	17499.5i	0.339605	+	0.339605i	−0.516614	+	0.856218i	$0.672808\pi$
$228$		0			0
$229$	−63859.8	−	63859.8i	−1.21775	−	1.21775i	−0.968420	−	0.249326i	$-0.919791\pi$
$229$	−63859.8	−	63859.8i	−1.21775	−	1.21775i	−0.249326	−	0.968420i	$-0.580209\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	63891.7i		−	1.17688i	−0.808540	−	0.588441i	$-0.799742\pi$
$233$		−	63891.7i		−	1.17688i	0.808540	−	0.588441i	$-0.200258\pi$
$234$		0			0
$235$	3548.79	−	3548.79i	0.0642605	−	0.0642605i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			12096.4i			0.211768i	0.994378	+	0.105884i	$0.0337672\pi$
$239$			12096.4i			0.211768i	−0.994378	+	0.105884i	$0.966233\pi$
$240$		0			0
$241$	27978.6			0.481717			0.240859	−	0.970560i	$-0.422571\pi$
$241$	27978.6			0.481717			0.240859	+	0.970560i	$0.422571\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1617.58	+	1617.58i	0.0269484	+	0.0269484i
$246$		0			0
$247$	63834.2			1.04631
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1996.06	−	1996.06i	0.0316830	−	0.0316830i	−0.691088	−	0.722771i	$-0.742868\pi$
$251$	1996.06	−	1996.06i	0.0316830	−	0.0316830i	0.722771	+	0.691088i	$0.242868\pi$
$252$		0			0
$253$	−81617.5	+	81617.5i	−1.27509	+	1.27509i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	73510.4			1.11297			0.556484	−	0.830859i	$-0.312150\pi$
$257$	73510.4			1.11297			0.556484	+	0.830859i	$0.312150\pi$
$258$		0			0
$259$	−66713.5	−	66713.5i	−0.994521	−	0.994521i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	68109.3			0.984680			0.492340	−	0.870403i	$-0.336142\pi$
$263$	68109.3			0.984680			0.492340	+	0.870403i	$0.336142\pi$
$264$		0			0
$265$		−	18226.3i		−	0.259542i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	15056.1	−	15056.1i	0.208069	−	0.208069i	−0.595377	−	0.803446i	$-0.702997\pi$
$269$	15056.1	−	15056.1i	0.208069	−	0.208069i	0.803446	+	0.595377i	$0.202997\pi$
$270$		0			0
$271$		−	104320.i		−	1.42046i	−0.703968	−	0.710232i	$-0.748590\pi$
$271$		−	104320.i		−	1.42046i	0.703968	−	0.710232i	$-0.251410\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−63973.7	−	63973.7i	−0.845934	−	0.845934i
$276$		0			0
$277$	2136.36	+	2136.36i	0.0278430	+	0.0278430i	0.720891	−	0.693048i	$-0.243733\pi$
$277$	2136.36	+	2136.36i	0.0278430	+	0.0278430i	−0.693048	+	0.720891i	$0.743733\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			36783.7i			0.465846i	0.972495	+	0.232923i	$0.0748290\pi$
$281$			36783.7i			0.465846i	−0.972495	+	0.232923i	$0.925171\pi$
$282$		0			0
$283$	−14172.0	+	14172.0i	−0.176953	+	0.176953i	−0.790026	−	0.613073i	$-0.789933\pi$
$283$	−14172.0	+	14172.0i	−0.176953	+	0.176953i	0.613073	+	0.790026i	$0.289933\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	59037.5i		−	0.716744i
$288$		0			0
$289$	59810.8			0.716117
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−80721.8	−	80721.8i	−0.940276	−	0.940276i	0.0580384	−	0.998314i	$-0.481515\pi$
$293$	−80721.8	−	80721.8i	−0.940276	−	0.940276i	−0.998314	+	0.0580384i	$0.981515\pi$
$294$		0			0
$295$	12824.9			0.147370
$296$		0			0
$297$		0			0
$298$		0			0
$299$	116345.	−	116345.i	1.30138	−	1.30138i
$300$		0			0
$301$	−16650.3	+	16650.3i	−0.183776	+	0.183776i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	17342.5			0.186429
$306$		0			0
$307$	−113714.	−	113714.i	−1.20653	−	1.20653i	−0.972143	−	0.234389i	$-0.924691\pi$
$307$	−113714.	−	113714.i	−1.20653	−	1.20653i	−0.234389	−	0.972143i	$-0.575309\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−93801.8			−0.969819			−0.484909	−	0.874564i	$-0.661147\pi$
$311$	−93801.8			−0.969819			−0.484909	+	0.874564i	$0.661147\pi$
$312$		0			0
$313$		−	9229.52i		−	0.0942086i	−0.998890	−	0.0471043i	$-0.985001\pi$
$313$		−	9229.52i		−	0.0942086i	0.998890	−	0.0471043i	$-0.0149993\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	8447.44	−	8447.44i	0.0840634	−	0.0840634i	−0.663825	−	0.747888i	$-0.731068\pi$
$317$	8447.44	−	8447.44i	0.0840634	−	0.0840634i	0.747888	+	0.663825i	$0.231068\pi$
$318$		0			0
$319$			18215.0i			0.178998i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−76902.4	−	76902.4i	−0.737114	−	0.737114i
$324$		0			0
$325$	91193.8	+	91193.8i	0.863373	+	0.863373i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	34087.7i		−	0.314924i
$330$		0			0
$331$	46092.1	−	46092.1i	0.420698	−	0.420698i	−0.464746	−	0.885444i	$-0.653854\pi$
$331$	46092.1	−	46092.1i	0.420698	−	0.420698i	0.885444	+	0.464746i	$0.153854\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			20830.7i			0.185616i
$336$		0			0
$337$	39317.8			0.346201			0.173101	−	0.984904i	$-0.444621\pi$
$337$	39317.8			0.346201			0.173101	+	0.984904i	$0.444621\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−31549.1	−	31549.1i	−0.271318	−	0.271318i
$342$		0			0
$343$	124474.			1.05801
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−67694.0	+	67694.0i	−0.562201	+	0.562201i	−0.929932	−	0.367731i	$-0.880135\pi$
$347$	−67694.0	+	67694.0i	−0.562201	+	0.562201i	0.367731	+	0.929932i	$0.380135\pi$
$348$		0			0
$349$	−7662.83	+	7662.83i	−0.0629127	+	0.0629127i	−0.737863	−	0.674950i	$-0.764165\pi$
$349$	−7662.83	+	7662.83i	−0.0629127	+	0.0629127i	0.674950	+	0.737863i	$0.264165\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−70618.0			−0.566716			−0.283358	−	0.959014i	$-0.591449\pi$
$353$	−70618.0			−0.566716			−0.283358	+	0.959014i	$0.591449\pi$
$354$		0			0
$355$	−42371.5	−	42371.5i	−0.336215	−	0.336215i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−168144.			−1.30465			−0.652323	−	0.757941i	$-0.726206\pi$
$359$	−168144.			−1.30465			−0.652323	+	0.757941i	$0.726206\pi$
$360$		0			0
$361$		−	47799.6i		−	0.366783i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	44166.7	−	44166.7i	0.331520	−	0.331520i
$366$		0			0
$367$			82596.4i			0.613238i	0.951832	+	0.306619i	$0.0991977\pi$
$367$			82596.4i			0.613238i	−0.951832	+	0.306619i	$0.900802\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−87536.0	−	87536.0i	−0.635973	−	0.635973i
$372$		0			0
$373$	−76631.0	−	76631.0i	−0.550791	−	0.550791i	0.375878	−	0.926669i	$-0.377341\pi$
$373$	−76631.0	−	76631.0i	−0.550791	−	0.550791i	−0.926669	+	0.375878i	$0.877341\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	25965.3i		−	0.182688i
$378$		0			0
$379$	95935.8	−	95935.8i	0.667886	−	0.667886i	−0.289340	−	0.957226i	$-0.593436\pi$
$379$	95935.8	−	95935.8i	0.667886	−	0.667886i	0.957226	+	0.289340i	$0.0934360\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	168708.i		−	1.15011i	−0.818116	−	0.575054i	$-0.804981\pi$
$383$		−	168708.i		−	1.15011i	0.818116	−	0.575054i	$-0.195019\pi$
$384$		0			0
$385$	47245.9			0.318744
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−66490.0	−	66490.0i	−0.439397	−	0.439397i	0.452412	−	0.891809i	$-0.350564\pi$
$389$	−66490.0	−	66490.0i	−0.439397	−	0.439397i	−0.891809	+	0.452412i	$0.850564\pi$
$390$		0			0
$391$	−280326.			−1.83362
$392$		0			0
$393$		0			0
$394$		0			0
$395$	13509.6	−	13509.6i	0.0865859	−	0.0865859i
$396$		0			0
$397$	18578.8	−	18578.8i	0.117879	−	0.117879i	−0.645706	−	0.763586i	$-0.723437\pi$
$397$	18578.8	−	18578.8i	0.117879	−	0.117879i	0.763586	+	0.645706i	$0.223437\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	129108.			0.802908			0.401454	−	0.915879i	$-0.368505\pi$
$401$	129108.			0.802908			0.401454	+	0.915879i	$0.368505\pi$
$402$		0			0
$403$	44972.9	+	44972.9i	0.276911	+	0.276911i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	324156.			1.95689
$408$		0			0
$409$		−	17952.4i		−	0.107319i	−0.998559	−	0.0536594i	$-0.982911\pi$
$409$		−	17952.4i		−	0.107319i	0.998559	−	0.0536594i	$-0.0170885\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	61594.5	−	61594.5i	0.361112	−	0.361112i
$414$		0			0
$415$			9793.04i			0.0568619i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	56887.3	+	56887.3i	0.324031	+	0.324031i	0.850311	−	0.526280i	$-0.176414\pi$
$419$	56887.3	+	56887.3i	0.324031	+	0.324031i	−0.526280	+	0.850311i	$0.676414\pi$
$420$		0			0
$421$	−144162.	−	144162.i	−0.813365	−	0.813365i	0.171772	−	0.985137i	$-0.445051\pi$
$421$	−144162.	−	144162.i	−0.813365	−	0.813365i	−0.985137	+	0.171772i	$0.945051\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	219726.i		−	1.21648i
$426$		0			0
$427$	83291.4	−	83291.4i	0.456819	−	0.456819i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			18641.5i			0.100352i	0.998740	+	0.0501760i	$0.0159782\pi$
$431$			18641.5i			0.100352i	−0.998740	+	0.0501760i	$0.984022\pi$
$432$		0			0
$433$	2193.97			0.0117019			0.00585094	−	0.999983i	$-0.498138\pi$
$433$	2193.97			0.0117019			0.00585094	+	0.999983i	$0.498138\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	150404.	+	150404.i	0.787586	+	0.787586i
$438$		0			0
$439$	−118115.			−0.612880			−0.306440	−	0.951890i	$-0.599138\pi$
$439$	−118115.			−0.612880			−0.306440	+	0.951890i	$0.599138\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	37885.8	−	37885.8i	0.193050	−	0.193050i	−0.603963	−	0.797013i	$-0.706412\pi$
$443$	37885.8	−	37885.8i	0.193050	−	0.193050i	0.797013	+	0.603963i	$0.206412\pi$
$444$		0			0
$445$	24425.4	−	24425.4i	0.123345	−	0.123345i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	85186.6			0.422550			0.211275	−	0.977427i	$-0.432238\pi$
$449$	85186.6			0.422550			0.211275	+	0.977427i	$0.432238\pi$
$450$		0			0
$451$	143430.	+	143430.i	0.705157	+	0.705157i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−67348.4			−0.325315
$456$		0			0
$457$		−	96935.7i		−	0.464142i	−0.972699	−	0.232071i	$-0.925450\pi$
$457$		−	96935.7i		−	0.464142i	0.972699	−	0.232071i	$-0.0745502\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−295014.	+	295014.i	−1.38816	+	1.38816i	−0.558988	+	0.829176i	$0.688810\pi$
$461$	−295014.	+	295014.i	−1.38816	+	1.38816i	−0.829176	+	0.558988i	$0.811190\pi$
$462$		0			0
$463$			224348.i			1.04655i	0.852164	+	0.523275i	$0.175290\pi$
$463$			224348.i			1.04655i	−0.852164	+	0.523275i	$0.824710\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	189556.	+	189556.i	0.869168	+	0.869168i	0.992380	−	0.123212i	$-0.0393196\pi$
$467$	189556.	+	189556.i	0.869168	+	0.869168i	−0.123212	+	0.992380i	$0.539320\pi$
$468$		0			0
$469$	100044.	+	100044.i	0.454827	+	0.454827i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	80902.6i		−	0.361610i
$474$		0			0
$475$	−117891.	+	117891.i	−0.522506	+	0.522506i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			132796.i			0.578781i	0.957211	+	0.289390i	$0.0934526\pi$
$479$			132796.i			0.578781i	−0.957211	+	0.289390i	$0.906547\pi$
$480$		0			0
$481$	−462081.			−1.99723
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−40338.2	−	40338.2i	−0.171488	−	0.171488i
$486$		0			0
$487$	283936.			1.19719			0.598595	−	0.801052i	$-0.295726\pi$
$487$	283936.			1.19719			0.598595	+	0.801052i	$0.295726\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−111725.	+	111725.i	−0.463435	+	0.463435i	−0.899780	−	0.436345i	$-0.856273\pi$
$491$	−111725.	+	111725.i	−0.463435	+	0.463435i	0.436345	+	0.899780i	$0.356273\pi$
$492$		0			0
$493$	−31280.9	+	31280.9i	−0.128702	+	0.128702i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−406998.			−1.64770
$498$		0			0
$499$	−61595.1	−	61595.1i	−0.247369	−	0.247369i	0.572521	−	0.819890i	$-0.305965\pi$
$499$	−61595.1	−	61595.1i	−0.247369	−	0.247369i	−0.819890	+	0.572521i	$0.805965\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−70874.1			−0.280125			−0.140062	−	0.990143i	$-0.544730\pi$
$503$	−70874.1			−0.280125			−0.140062	+	0.990143i	$0.544730\pi$
$504$		0			0
$505$			5079.09i			0.0199160i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	128181.	−	128181.i	0.494754	−	0.494754i	−0.415046	−	0.909800i	$-0.636235\pi$
$509$	128181.	−	128181.i	0.494754	−	0.494754i	0.909800	+	0.415046i	$0.136235\pi$
$510$		0			0
$511$		−	424241.i		−	1.62469i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−26186.3	−	26186.3i	−0.0987324	−	0.0987324i
$516$		0			0
$517$	82814.9	+	82814.9i	0.309833	+	0.309833i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			494250.i			1.82084i	0.413688	+	0.910419i	$0.364241\pi$
$521$			494250.i			1.82084i	−0.413688	+	0.910419i	$0.635759\pi$
$522$		0			0
$523$	−179670.	+	179670.i	−0.656859	+	0.656859i	−0.954636	−	0.297776i	$-0.903755\pi$
$523$	−179670.	+	179670.i	−0.656859	+	0.656859i	0.297776	+	0.954636i	$0.403755\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	108360.i		−	0.390163i
$528$		0			0
$529$	268417.			0.959175
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−204457.	−	204457.i	−0.719694	−	0.719694i
$534$		0			0
$535$	28242.7			0.0986729
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−37748.1	+	37748.1i	−0.129932	+	0.129932i
$540$		0			0
$541$	300961.	−	300961.i	1.02829	−	1.02829i	0.0287034	−	0.999588i	$-0.490862\pi$
$541$	300961.	−	300961.i	1.02829	−	1.02829i	0.999588	−	0.0287034i	$-0.00913784\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	31683.6			0.106670
$546$		0			0
$547$	159057.	+	159057.i	0.531591	+	0.531591i	0.921046	−	0.389455i	$-0.127337\pi$
$547$	159057.	+	159057.i	0.531591	+	0.531591i	−0.389455	+	0.921046i	$0.627337\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	33566.6			0.110561
$552$		0			0
$553$		−	129765.i		−	0.424335i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	394524.	−	394524.i	1.27164	−	1.27164i	0.326409	−	0.945229i	$-0.394161\pi$
$557$	394524.	−	394524.i	1.27164	−	1.27164i	0.945229	−	0.326409i	$-0.105839\pi$
$558$		0			0
$559$			115326.i			0.369065i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	169317.	+	169317.i	0.534177	+	0.534177i	0.921813	−	0.387636i	$-0.126708\pi$
$563$	169317.	+	169317.i	0.534177	+	0.534177i	−0.387636	+	0.921813i	$0.626708\pi$
$564$		0			0
$565$	−15085.9	−	15085.9i	−0.0472579	−	0.0472579i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			165645.i			0.511626i	0.966726	+	0.255813i	$0.0823432\pi$
$569$			165645.i			0.511626i	−0.966726	+	0.255813i	$0.917657\pi$
$570$		0			0
$571$	−335641.	+	335641.i	−1.02945	+	1.02945i	−0.0298925	+	0.999553i	$0.509517\pi$
$571$	−335641.	+	335641.i	−1.02945	+	1.02945i	−0.999553	+	0.0298925i	$0.990483\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			429737.i			1.29977i
$576$		0			0
$577$	−99615.9			−0.299211			−0.149605	−	0.988746i	$-0.547800\pi$
$577$	−99615.9			−0.299211			−0.149605	+	0.988746i	$0.547800\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	47033.3	+	47033.3i	0.139333	+	0.139333i
$582$		0			0
$583$	425331.			1.25138
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−45023.9	+	45023.9i	−0.130667	+	0.130667i	−0.769416	−	0.638748i	$-0.779452\pi$
$587$	−45023.9	+	45023.9i	−0.130667	+	0.130667i	0.638748	+	0.769416i	$0.279452\pi$
$588$		0			0
$589$	−58138.5	+	58138.5i	−0.167584	+	0.167584i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	385101.			1.09513			0.547564	−	0.836764i	$-0.315555\pi$
$593$	385101.			1.09513			0.547564	+	0.836764i	$0.315555\pi$
$594$		0			0
$595$	81136.1	+	81136.1i	0.229182	+	0.229182i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	209331.			0.583417			0.291709	−	0.956507i	$-0.405776\pi$
$599$	209331.			0.583417			0.291709	+	0.956507i	$0.405776\pi$
$600$		0			0
$601$			515555.i			1.42734i	0.700484	+	0.713668i	$0.252967\pi$
$601$			515555.i			1.42734i	−0.700484	+	0.713668i	$0.747033\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−45625.7	+	45625.7i	−0.124652	+	0.124652i
$606$		0			0
$607$		−	30292.7i		−	0.0822167i	−0.999155	−	0.0411084i	$-0.986911\pi$
$607$		−	30292.7i		−	0.0822167i	0.999155	−	0.0411084i	$-0.0130889\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−118052.	−	118052.i	−0.316220	−	0.316220i
$612$		0			0
$613$	−99803.4	−	99803.4i	−0.265598	−	0.265598i	0.561726	−	0.827324i	$-0.310138\pi$
$613$	−99803.4	−	99803.4i	−0.265598	−	0.265598i	−0.827324	+	0.561726i	$0.810138\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			102196.i			0.268451i	0.990951	+	0.134226i	$0.0428547\pi$
$617$			102196.i			0.268451i	−0.990951	+	0.134226i	$0.957145\pi$
$618$		0			0
$619$	105904.	−	105904.i	0.276395	−	0.276395i	−0.555273	−	0.831668i	$-0.687386\pi$
$619$	105904.	−	105904.i	0.276395	−	0.276395i	0.831668	+	0.555273i	$0.187386\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	234617.i		−	0.604483i
$624$		0			0
$625$	−308949.			−0.790909
$626$		0			0
$627$		0			0
$628$		0			0
$629$	556679.	+	556679.i	1.40703	+	1.40703i
$630$		0			0
$631$	−692480.			−1.73920			−0.869598	−	0.493760i	$-0.835622\pi$
$631$	−692480.			−1.73920			−0.869598	+	0.493760i	$0.835622\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	108170.	−	108170.i	0.268262	−	0.268262i
$636$		0			0
$637$	53809.4	−	53809.4i	0.132611	−	0.132611i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−49519.0			−0.120519			−0.0602595	−	0.998183i	$-0.519193\pi$
$641$	−49519.0			−0.120519			−0.0602595	+	0.998183i	$0.519193\pi$
$642$		0			0
$643$	165494.	+	165494.i	0.400276	+	0.400276i	0.878330	−	0.478054i	$-0.158658\pi$
$643$	165494.	+	165494.i	0.400276	+	0.400276i	−0.478054	+	0.878330i	$0.658658\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−666565.			−1.59233			−0.796166	−	0.605078i	$-0.793142\pi$
$647$	−666565.			−1.59233			−0.796166	+	0.605078i	$0.793142\pi$
$648$		0			0
$649$			299284.i			0.710548i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−461118.	+	461118.i	−1.08140	+	1.08140i	−0.0850198	+	0.996379i	$0.527095\pi$
$653$	−461118.	+	461118.i	−1.08140	+	1.08140i	−0.996379	+	0.0850198i	$0.972905\pi$
$654$		0			0
$655$		−	181910.i		−	0.424008i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	326883.	+	326883.i	0.752698	+	0.752698i	0.974982	−	0.222284i	$-0.0713511\pi$
$659$	326883.	+	326883.i	0.752698	+	0.752698i	−0.222284	+	0.974982i	$0.571351\pi$
$660$		0			0
$661$	549615.	+	549615.i	1.25793	+	1.25793i	0.952080	+	0.305848i	$0.0989401\pi$
$661$	549615.	+	549615.i	1.25793	+	1.25793i	0.305848	+	0.952080i	$0.401060\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	87064.5i		−	0.196878i
$666$		0			0
$667$	61178.8	−	61178.8i	0.137515	−	0.137515i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			404708.i			0.898869i
$672$		0			0
$673$	191664.			0.423166			0.211583	−	0.977360i	$-0.432138\pi$
$673$	191664.			0.423166			0.211583	+	0.977360i	$0.432138\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	544201.	+	544201.i	1.18736	+	1.18736i	0.977795	+	0.209564i	$0.0672043\pi$
$677$	544201.	+	544201.i	1.18736	+	1.18736i	0.209564	+	0.977795i	$0.432796\pi$
$678$		0			0
$679$	−387466.			−0.840416
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−570769.	+	570769.i	−1.22354	+	1.22354i	−0.257179	+	0.966364i	$0.582793\pi$
$683$	−570769.	+	570769.i	−1.22354	+	1.22354i	−0.966364	+	0.257179i	$0.917207\pi$
$684$		0			0
$685$	94177.8	−	94177.8i	0.200709	−	0.200709i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−606305.			−1.27718
$690$		0			0
$691$	333477.	+	333477.i	0.698409	+	0.698409i	0.964067	−	0.265658i	$-0.0855893\pi$
$691$	333477.	+	333477.i	0.698409	+	0.698409i	−0.265658	+	0.964067i	$0.585589\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	237831.			0.492378
$696$		0			0
$697$			492628.i			1.01404i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−31503.2	+	31503.2i	−0.0641089	+	0.0641089i	−0.738434	−	0.674325i	$-0.764435\pi$
$701$	−31503.2	+	31503.2i	−0.0641089	+	0.0641089i	0.674325	+	0.738434i	$0.264435\pi$
$702$		0			0
$703$		−	597354.i		−	1.20871i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	24393.5	+	24393.5i	0.0488017	+	0.0488017i
$708$		0			0
$709$	609708.	+	609708.i	1.21291	+	1.21291i	0.970064	+	0.242847i	$0.0780815\pi$
$709$	609708.	+	609708.i	1.21291	+	1.21291i	0.242847	+	0.970064i	$0.421919\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			211928.i			0.416878i
$714$		0			0
$715$	163621.	−	163621.i	0.320056	−	0.320056i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	720908.i		−	1.39451i	−0.716823	−	0.697256i	$-0.754404\pi$
$719$		−	720908.i		−	1.39451i	0.716823	−	0.697256i	$-0.245596\pi$
$720$		0			0
$721$	−251531.			−0.483862
$722$		0			0
$723$		0			0
$724$		0			0
$725$	47953.4	+	47953.4i	0.0912311	+	0.0912311i
$726$		0			0
$727$	279904.			0.529591			0.264796	−	0.964305i	$-0.414696\pi$
$727$	279904.			0.529591			0.264796	+	0.964305i	$0.414696\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	138935.	−	138935.i	0.260003	−	0.260003i
$732$		0			0
$733$	−266306.	+	266306.i	−0.495648	+	0.495648i	−0.910080	−	0.414432i	$-0.863980\pi$
$733$	−266306.	+	266306.i	−0.495648	+	0.495648i	0.414432	+	0.910080i	$0.363980\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−486108.			−0.894948
$738$		0			0
$739$	594958.	+	594958.i	1.08943	+	1.08943i	0.995587	+	0.0938378i	$0.0299135\pi$
$739$	594958.	+	594958.i	1.08943	+	1.08943i	0.0938378	+	0.995587i	$0.470086\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	289367.			0.524170			0.262085	−	0.965045i	$-0.415590\pi$
$743$	289367.			0.524170			0.262085	+	0.965045i	$0.415590\pi$
$744$		0			0
$745$		−	30390.8i		−	0.0547557i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	135642.	−	135642.i	0.241785	−	0.241785i
$750$		0			0
$751$		−	935795.i		−	1.65921i	−0.558352	−	0.829604i	$-0.688566\pi$
$751$		−	935795.i		−	1.65921i	0.558352	−	0.829604i	$-0.311434\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−69768.4	−	69768.4i	−0.122395	−	0.122395i
$756$		0			0
$757$	−626956.	−	626956.i	−1.09407	−	1.09407i	−0.995089	−	0.0989818i	$-0.968441\pi$
$757$	−626956.	−	626956.i	−1.09407	−	1.09407i	−0.0989818	−	0.995089i	$-0.531559\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	253505.i		−	0.437741i	−0.975754	−	0.218871i	$-0.929763\pi$
$761$		−	253505.i		−	0.437741i	0.975754	−	0.218871i	$-0.0702372\pi$
$762$		0			0
$763$	152167.	−	152167.i	0.261380	−	0.261380i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	426625.i		−	0.725197i
$768$		0			0
$769$	−740462.			−1.25213			−0.626066	−	0.779770i	$-0.715336\pi$
$769$	−740462.			−1.25213			−0.626066	+	0.779770i	$0.715336\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−422057.	−	422057.i	−0.706337	−	0.706337i	0.259426	−	0.965763i	$-0.416467\pi$
$773$	−422057.	−	422057.i	−0.706337	−	0.706337i	−0.965763	+	0.259426i	$0.916467\pi$
$774$		0			0
$775$	−166114.			−0.276569
$776$		0			0
$777$		0			0
$778$		0			0
$779$	264312.	−	264312.i	0.435553	−	0.435553i
$780$		0			0
$781$	988787.	−	988787.i	1.62107	−	1.62107i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	170183.			0.276171
$786$		0			0
$787$	−658152.	−	658152.i	−1.06262	−	1.06262i	−0.997904	−	0.0647134i	$-0.979387\pi$
$787$	−658152.	−	658152.i	−1.06262	−	1.06262i	−0.0647134	−	0.997904i	$-0.520613\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−144907.			−0.231599
$792$		0			0
$793$		−	576906.i		−	0.917400i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	573702.	−	573702.i	0.903171	−	0.903171i	−0.0925382	−	0.995709i	$-0.529498\pi$
$797$	573702.	−	573702.i	0.903171	−	0.903171i	0.995709	+	0.0925382i	$0.0294980\pi$
$798$		0			0
$799$			284439.i			0.445549i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.03068e6	+	1.03068e6i	1.59843	+	1.59843i
$804$		0			0
$805$	−158685.	−	158685.i	−0.244874	−	0.244874i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			297398.i			0.454402i	0.973848	+	0.227201i	$0.0729575\pi$
$809$			297398.i			0.454402i	−0.973848	+	0.227201i	$0.927043\pi$
$810$		0			0
$811$	−481203.	+	481203.i	−0.731621	+	0.731621i	−0.970941	−	0.239320i	$-0.923076\pi$
$811$	−481203.	+	481203.i	−0.731621	+	0.731621i	0.239320	+	0.970941i	$0.423076\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			142491.i			0.214522i
$816$		0			0
$817$	−149087.			−0.223355
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−506859.	−	506859.i	−0.751971	−	0.751971i	0.222876	−	0.974847i	$-0.428456\pi$
$821$	−506859.	−	506859.i	−0.751971	−	0.751971i	−0.974847	+	0.222876i	$0.928456\pi$
$822$		0			0
$823$	−401221.			−0.592357			−0.296179	−	0.955133i	$-0.595712\pi$
$823$	−401221.			−0.592357			−0.296179	+	0.955133i	$0.595712\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	410169.	−	410169.i	0.599725	−	0.599725i	−0.340515	−	0.940239i	$-0.610601\pi$
$827$	410169.	−	410169.i	0.599725	−	0.599725i	0.940239	+	0.340515i	$0.110601\pi$
$828$		0			0
$829$	−328989.	+	328989.i	−0.478710	+	0.478710i	−0.904719	−	0.426009i	$-0.859919\pi$
$829$	−328989.	+	328989.i	−0.478710	+	0.478710i	0.426009	+	0.904719i	$0.359919\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−129651.			−0.186846
$834$		0			0
$835$	−112105.	−	112105.i	−0.160788	−	0.160788i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−114596.			−0.162797			−0.0813985	−	0.996682i	$-0.525939\pi$
$839$	−114596.			−0.162797			−0.0813985	+	0.996682i	$0.525939\pi$
$840$		0			0
$841$			693627.i			0.980696i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−98332.0	+	98332.0i	−0.137715	+	0.137715i
$846$		0			0
$847$			438256.i			0.610887i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1.08874e6	−	1.08874e6i	−1.50337	−	1.50337i
$852$		0			0
$853$	79602.6	+	79602.6i	0.109403	+	0.109403i	0.759689	−	0.650286i	$-0.225351\pi$
$853$	79602.6	+	79602.6i	0.109403	+	0.109403i	−0.650286	+	0.759689i	$0.725351\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			556304.i			0.757444i	0.925510	+	0.378722i	$0.123636\pi$
$857$			556304.i			0.757444i	−0.925510	+	0.378722i	$0.876364\pi$
$858$		0			0
$859$	−825505.	+	825505.i	−1.11875	+	1.11875i	−0.126825	+	0.991925i	$0.540479\pi$
$859$	−825505.	+	825505.i	−1.11875	+	1.11875i	−0.991925	+	0.126825i	$0.959521\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			800580.i			1.07494i	0.843284	+	0.537469i	$0.180619\pi$
$863$			800580.i			1.07494i	−0.843284	+	0.537469i	$0.819381\pi$
$864$		0			0
$865$	−114046.			−0.152422
$866$		0			0
$867$		0			0
$868$		0			0
$869$	315261.	+	315261.i	0.417475	+	0.417475i
$870$		0			0
$871$	692941.			0.913398
$872$		0			0
$873$		0			0
$874$		0			0
$875$	258325.	−	258325.i	0.337403	−	0.337403i
$876$		0			0
$877$	−684224.	+	684224.i	−0.889609	+	0.889609i	−0.994485	−	0.104876i	$-0.966555\pi$
$877$	−684224.	+	684224.i	−0.889609	+	0.889609i	0.104876	+	0.994485i	$0.466555\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	995272.			1.28230			0.641150	−	0.767415i	$-0.278458\pi$
$881$	995272.			1.28230			0.641150	+	0.767415i	$0.278458\pi$
$882$		0			0
$883$	−67491.2	−	67491.2i	−0.0865617	−	0.0865617i	0.662500	−	0.749062i	$-0.269495\pi$
$883$	−67491.2	−	67491.2i	−0.0865617	−	0.0865617i	−0.749062	+	0.662500i	$0.769495\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	180136.			0.228957			0.114479	−	0.993426i	$-0.463480\pi$
$887$	180136.			0.228957			0.114479	+	0.993426i	$0.463480\pi$
$888$		0			0
$889$		−	1.03902e6i		−	1.31468i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	152611.	−	152611.i	0.191374	−	0.191374i
$894$		0			0
$895$			265569.i			0.331536i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	23648.5	+	23648.5i	0.0292607	+	0.0292607i
$900$		0			0
$901$	730428.	+	730428.i	0.899763	+	0.899763i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			150271.i			0.183475i
$906$		0			0
$907$	893550.	−	893550.i	1.08619	−	1.08619i	0.0902686	−	0.995917i	$-0.471227\pi$
$907$	893550.	−	893550.i	1.08619	−	1.08619i	0.995917	−	0.0902686i	$-0.0287726\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	638798.i		−	0.769709i	−0.922977	−	0.384855i	$-0.874252\pi$
$911$		−	638798.i		−	0.769709i	0.922977	−	0.384855i	$-0.125748\pi$
$912$		0			0
$913$	−228532.			−0.274160
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−873663.	−	873663.i	−1.03898	−	1.03898i
$918$		0			0
$919$	1.28409e6			1.52043			0.760213	−	0.649674i	$-0.225095\pi$
$919$	1.28409e6			1.52043			0.760213	+	0.649674i	$0.225095\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1.40950e6	+	1.40950e6i	−1.65449	+	1.65449i
$924$		0			0
$925$	853383.	−	853383.i	0.997380	−	0.997380i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−198494.			−0.229994			−0.114997	−	0.993366i	$-0.536686\pi$
$929$	−198494.			−0.229994			−0.114997	+	0.993366i	$0.536686\pi$
$930$		0			0
$931$	69562.0	+	69562.0i	0.0802551	+	0.0802551i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−394235.			−0.450953
$936$		0			0
$937$		−	389175.i		−	0.443267i	−0.975130	−	0.221634i	$-0.928861\pi$
$937$		−	389175.i		−	0.443267i	0.975130	−	0.221634i	$-0.0711389\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	481044.	−	481044.i	0.543257	−	0.543257i	−0.381225	−	0.924482i	$-0.624498\pi$
$941$	481044.	−	481044.i	0.543257	−	0.543257i	0.924482	+	0.381225i	$0.124498\pi$
$942$		0			0
$943$		−	963474.i		−	1.08347i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−399574.	−	399574.i	−0.445551	−	0.445551i	0.448321	−	0.893872i	$-0.352022\pi$
$947$	−399574.	−	399574.i	−0.445551	−	0.445551i	−0.893872	+	0.448321i	$0.852022\pi$
$948$		0			0
$949$	−1.46922e6	−	1.46922e6i	−1.63138	−	1.63138i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1.26137e6i		−	1.38885i	−0.719564	−	0.694427i	$-0.755658\pi$
$953$		−	1.26137e6i		−	1.38885i	0.719564	−	0.694427i	$-0.244342\pi$
$954$		0			0
$955$	132440.	−	132440.i	0.145215	−	0.145215i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	904620.i		−	0.983624i
$960$		0			0
$961$	841601.			0.911296
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−267883.	−	267883.i	−0.287668	−	0.287668i
$966$		0			0
$967$	−1.50478e6			−1.60924			−0.804619	−	0.593791i	$-0.797630\pi$
$967$	−1.50478e6			−1.60924			−0.804619	+	0.593791i	$0.797630\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	818605.	−	818605.i	0.868233	−	0.868233i	−0.124044	−	0.992277i	$-0.539586\pi$
$971$	818605.	−	818605.i	0.868233	−	0.868233i	0.992277	+	0.124044i	$0.0395864\pi$
$972$		0			0
$973$	1.14224e6	−	1.14224e6i	1.20651	−	1.20651i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−753210.			−0.789091			−0.394545	−	0.918876i	$-0.629098\pi$
$977$	−753210.			−0.789091			−0.394545	+	0.918876i	$0.629098\pi$
$978$		0			0
$979$	569995.	+	569995.i	0.594711	+	0.594711i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−460098.			−0.476149			−0.238075	−	0.971247i	$-0.576516\pi$
$983$	−460098.			−0.476149			−0.238075	+	0.971247i	$0.576516\pi$
$984$		0			0
$985$		−	96163.6i		−	0.0991147i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−271727.	+	271727.i	−0.277806	+	0.277806i
$990$		0			0
$991$			1.16135e6i			1.18254i	0.806472	+	0.591272i	$0.201374\pi$
$991$			1.16135e6i			1.18254i	−0.806472	+	0.591272i	$0.798626\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	150147.	+	150147.i	0.151659	+	0.151659i
$996$		0			0
$997$	271712.	+	271712.i	0.273349	+	0.273349i	0.830447	−	0.557098i	$-0.188085\pi$
$997$	271712.	+	271712.i	0.273349	+	0.273349i	−0.557098	+	0.830447i	$0.688085\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.5.m.a.271.3		14
3.2	odd	2		64.5.f.a.15.3		14
4.3	odd	2		144.5.m.a.91.4		14
12.11	even	2		16.5.f.a.11.4	yes	14
16.3	odd	4	inner	576.5.m.a.559.3		14
16.13	even	4		144.5.m.a.19.4		14
24.5	odd	2		128.5.f.a.31.5		14
24.11	even	2		128.5.f.b.31.3		14
48.5	odd	4		128.5.f.b.95.3		14
48.11	even	4		128.5.f.a.95.5		14
48.29	odd	4		16.5.f.a.3.4	✓	14
48.35	even	4		64.5.f.a.47.3		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.5.f.a.3.4	✓	14	48.29	odd	4
16.5.f.a.11.4	yes	14	12.11	even	2
64.5.f.a.15.3		14	3.2	odd	2
64.5.f.a.47.3		14	48.35	even	4
128.5.f.a.31.5		14	24.5	odd	2
128.5.f.a.95.5		14	48.11	even	4
128.5.f.b.31.3		14	24.11	even	2
128.5.f.b.95.3		14	48.5	odd	4
144.5.m.a.19.4		14	16.13	even	4
144.5.m.a.91.4		14	4.3	odd	2
576.5.m.a.271.3		14	1.1	even	1	trivial
576.5.m.a.559.3		14	16.3	odd	4	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:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	576.m (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.72348 - 4.72348i) q^{5} -45.3712 q^{7} +O(q^{10})\)	\(q+(-4.72348 - 4.72348i) q^{5} -45.3712 q^{7} +(110.228 - 110.228i) q^{11} +(-157.128 + 157.128i) q^{13} +378.592 q^{17} +(-203.127 - 203.127i) q^{19} -740.444 q^{23} -580.377i q^{25} +(-82.6244 + 82.6244i) q^{29} -286.217i q^{31} +(214.310 + 214.310i) q^{35} +(1470.39 + 1470.39i) q^{37} +1301.21i q^{41} +(366.979 - 366.979i) q^{43} +751.307i q^{47} -342.455 q^{49} +(1929.33 + 1929.33i) q^{53} -1041.32 q^{55} +(-1357.57 + 1357.57i) q^{59} +(-1835.78 + 1835.78i) q^{61} +1484.39 q^{65} +(-2205.02 - 2205.02i) q^{67} +8970.40 q^{71} +9350.46i q^{73} +(-5001.17 + 5001.17i) q^{77} +2860.08i q^{79} +(-1036.63 - 1036.63i) q^{83} +(-1788.27 - 1788.27i) q^{85} +5171.06i q^{89} +(7129.10 - 7129.10i) q^{91} +1918.94i q^{95} +8539.92 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 2 q^{5} + 4 q^{7}+O(q^{10}) \) 14 * q + 2 * q^5 + 4 * q^7	\( 14 q + 2 q^{5} + 4 q^{7} + 94 q^{11} - 2 q^{13} + 4 q^{17} + 706 q^{19} + 1148 q^{23} - 862 q^{29} + 1340 q^{35} - 1826 q^{37} - 1694 q^{43} + 682 q^{49} + 482 q^{53} + 11780 q^{55} - 2786 q^{59} - 3778 q^{61} + 2020 q^{65} - 7998 q^{67} + 19964 q^{71} + 9508 q^{77} - 17282 q^{83} + 9948 q^{85} + 28036 q^{91} - 4 q^{97}+O(q^{100}) \) 14 * q + 2 * q^5 + 4 * q^7 + 94 * q^11 - 2 * q^13 + 4 * q^17 + 706 * q^19 + 1148 * q^23 - 862 * q^29 + 1340 * q^35 - 1826 * q^37 - 1694 * q^43 + 682 * q^49 + 482 * q^53 + 11780 * q^55 - 2786 * q^59 - 3778 * q^61 + 2020 * q^65 - 7998 * q^67 + 19964 * q^71 + 9508 * q^77 - 17282 * q^83 + 9948 * q^85 + 28036 * q^91 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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