LMFDB - Embedded newform 576.4.k.a.145.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,4,Mod(145,576)]

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([0, 3, 0]))

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

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

chi := DirichletCharacter("576.145");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	576.k (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:	$33.9851001633$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - x^{8} + 6x^{7} + 14x^{6} - 80x^{5} + 56x^{4} + 96x^{3} - 64x^{2} - 512x + 1024 $ x^10 - 2x^9 - x^8 + 6x^7 + 14x^6 - 80x^5 + 56x^4 + 96x^3 - 64x^2 - 512x + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			145.5
Root			$-1.56339 - 1.24732i$ of defining polynomial
Character	$\chi$	$=$	576.145
Dual form			576.4.k.a.433.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(12.6449 - 12.6449i) q^{5} +13.8754i q^{7} +O(q^{10})$	$q+(12.6449 - 12.6449i) q^{5} +13.8754i q^{7} +(1.54694 - 1.54694i) q^{11} +(32.7875 + 32.7875i) q^{13} -18.6531 q^{17} +(86.4042 + 86.4042i) q^{19} +134.006i q^{23} -194.786i q^{25} +(59.7949 + 59.7949i) q^{29} +31.5391 q^{31} +(175.453 + 175.453i) q^{35} +(89.1866 - 89.1866i) q^{37} -210.504i q^{41} +(-119.402 + 119.402i) q^{43} -182.902 q^{47} +150.474 q^{49} +(26.1644 - 26.1644i) q^{53} -39.1219i q^{55} +(441.584 - 441.584i) q^{59} +(-174.485 - 174.485i) q^{61} +829.188 q^{65} +(-91.7562 - 91.7562i) q^{67} -348.360i q^{71} +299.436i q^{73} +(21.4644 + 21.4644i) q^{77} +943.487 q^{79} +(313.272 + 313.272i) q^{83} +(-235.866 + 235.866i) q^{85} +1412.35i q^{89} +(-454.939 + 454.939i) q^{91} +2185.14 q^{95} +1515.29 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{5}+O(q^{10}) $ 10 * q + 2 * q^5	$ 10 q + 2 q^{5} + 18 q^{11} - 2 q^{13} + 4 q^{17} + 26 q^{19} + 202 q^{29} - 368 q^{31} + 476 q^{35} - 10 q^{37} + 838 q^{43} - 944 q^{47} + 94 q^{49} + 378 q^{53} + 1706 q^{59} + 910 q^{61} + 492 q^{65} - 1942 q^{67} + 268 q^{77} + 4416 q^{79} - 2562 q^{83} - 12 q^{85} - 3332 q^{91} + 6900 q^{95} - 4 q^{97}+O(q^{100}) $ 10 * q + 2 * q^5 + 18 * q^11 - 2 * q^13 + 4 * q^17 + 26 * q^19 + 202 * q^29 - 368 * q^31 + 476 * q^35 - 10 * q^37 + 838 * q^43 - 944 * q^47 + 94 * q^49 + 378 * q^53 + 1706 * q^59 + 910 * q^61 + 492 * q^65 - 1942 * q^67 + 268 * q^77 + 4416 * q^79 - 2562 * q^83 - 12 * q^85 - 3332 * q^91 + 6900 * q^95 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$65$	$127$	$325$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$

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
$5$	12.6449	−	12.6449i	1.13099	−	1.13099i	0.140981	−	0.990012i	$-0.454974\pi$
$5$	12.6449	−	12.6449i	1.13099	−	1.13099i	0.990012	−	0.140981i	$-0.0450257\pi$
$6$		0			0
$7$			13.8754i			0.749200i	0.927187	+	0.374600i	$0.122220\pi$
$7$			13.8754i			0.749200i	−0.927187	+	0.374600i	$0.877780\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.54694	−	1.54694i	0.0424019	−	0.0424019i	−0.685588	−	0.727990i	$-0.740455\pi$
$11$	1.54694	−	1.54694i	0.0424019	−	0.0424019i	0.727990	+	0.685588i	$0.240455\pi$
$12$		0			0
$13$	32.7875	+	32.7875i	0.699509	+	0.699509i	0.964304	−	0.264796i	$-0.0853046\pi$
$13$	32.7875	+	32.7875i	0.699509	+	0.699509i	−0.264796	+	0.964304i	$0.585305\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−18.6531			−0.266119			−0.133060	−	0.991108i	$-0.542480\pi$
$17$	−18.6531			−0.266119			−0.133060	+	0.991108i	$0.542480\pi$
$18$		0			0
$19$	86.4042	+	86.4042i	1.04329	+	1.04329i	0.999020	+	0.0442688i	$0.0140958\pi$
$19$	86.4042	+	86.4042i	1.04329	+	1.04329i	0.0442688	+	0.999020i	$0.485904\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			134.006i			1.21488i	0.794367	+	0.607438i	$0.207803\pi$
$23$			134.006i			1.21488i	−0.794367	+	0.607438i	$0.792197\pi$
$24$		0			0
$25$		−	194.786i		−	1.55829i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	59.7949	+	59.7949i	0.382884	+	0.382884i	0.872140	−	0.489256i	$-0.162732\pi$
$29$	59.7949	+	59.7949i	0.382884	+	0.382884i	−0.489256	+	0.872140i	$0.662732\pi$
$30$		0			0
$31$	31.5391			0.182729			0.0913645	−	0.995818i	$-0.470877\pi$
$31$	31.5391			0.182729			0.0913645	+	0.995818i	$0.470877\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	175.453	+	175.453i	0.847340	+	0.847340i
$36$		0			0
$37$	89.1866	−	89.1866i	0.396275	−	0.396275i	−0.480642	−	0.876917i	$-0.659596\pi$
$37$	89.1866	−	89.1866i	0.396275	−	0.396275i	0.876917	+	0.480642i	$0.159596\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	210.504i		−	0.801834i	−0.916114	−	0.400917i	$-0.868692\pi$
$41$		−	210.504i		−	0.801834i	0.916114	−	0.400917i	$-0.131308\pi$
$42$		0			0
$43$	−119.402	+	119.402i	−0.423456	+	0.423456i	−0.886392	−	0.462936i	$-0.846796\pi$
$43$	−119.402	+	119.402i	−0.423456	+	0.423456i	0.462936	+	0.886392i	$0.346796\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−182.902			−0.567638			−0.283819	−	0.958878i	$-0.591602\pi$
$47$	−182.902			−0.567638			−0.283819	+	0.958878i	$0.591602\pi$
$48$		0			0
$49$	150.474			0.438699
$50$		0			0
$51$		0			0
$52$		0			0
$53$	26.1644	−	26.1644i	0.0678104	−	0.0678104i	−0.672388	−	0.740199i	$-0.734731\pi$
$53$	26.1644	−	26.1644i	0.0678104	−	0.0678104i	0.740199	+	0.672388i	$0.234731\pi$
$54$		0			0
$55$		−	39.1219i		−	0.0959125i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	441.584	−	441.584i	0.974395	−	0.974395i	−0.0252856	−	0.999680i	$-0.508050\pi$
$59$	441.584	−	441.584i	0.974395	−	0.974395i	0.999680	+	0.0252856i	$0.00804951\pi$
$60$		0			0
$61$	−174.485	−	174.485i	−0.366238	−	0.366238i	0.499865	−	0.866103i	$-0.333383\pi$
$61$	−174.485	−	174.485i	−0.366238	−	0.366238i	−0.866103	+	0.499865i	$0.833383\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	829.188			1.58228
$66$		0			0
$67$	−91.7562	−	91.7562i	−0.167311	−	0.167311i	0.618486	−	0.785796i	$-0.287747\pi$
$67$	−91.7562	−	91.7562i	−0.167311	−	0.167311i	−0.785796	+	0.618486i	$0.787747\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	348.360i		−	0.582291i	−0.956679	−	0.291146i	$-0.905964\pi$
$71$		−	348.360i		−	0.582291i	0.956679	−	0.291146i	$-0.0940364\pi$
$72$		0			0
$73$			299.436i			0.480087i	0.970762	+	0.240043i	$0.0771617\pi$
$73$			299.436i			0.480087i	−0.970762	+	0.240043i	$0.922838\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	21.4644	+	21.4644i	0.0317675	+	0.0317675i
$78$		0			0
$79$	943.487			1.34368			0.671839	−	0.740697i	$-0.265505\pi$
$79$	943.487			1.34368			0.671839	+	0.740697i	$0.265505\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	313.272	+	313.272i	0.414290	+	0.414290i	0.883230	−	0.468940i	$-0.155364\pi$
$83$	313.272	+	313.272i	0.414290	+	0.414290i	−0.468940	+	0.883230i	$0.655364\pi$
$84$		0			0
$85$	−235.866	+	235.866i	−0.300979	+	0.300979i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			1412.35i			1.68212i	0.540942	+	0.841060i	$0.318068\pi$
$89$			1412.35i			1.68212i	−0.540942	+	0.841060i	$0.681932\pi$
$90$		0			0
$91$	−454.939	+	454.939i	−0.524072	+	0.524072i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2185.14			2.35990
$96$		0			0
$97$	1515.29			1.58613			0.793063	−	0.609140i	$-0.208485\pi$
$97$	1515.29			1.58613			0.793063	+	0.609140i	$0.208485\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	573.202	−	573.202i	0.564711	−	0.564711i	−0.365931	−	0.930642i	$-0.619249\pi$
$101$	573.202	−	573.202i	0.564711	−	0.564711i	0.930642	+	0.365931i	$0.119249\pi$
$102$		0			0
$103$		−	1021.00i		−	0.976717i	−0.872643	−	0.488359i	$-0.837596\pi$
$103$		−	1021.00i		−	0.976717i	0.872643	−	0.488359i	$-0.162404\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1240.79	+	1240.79i	−1.12105	+	1.12105i	−0.129462	+	0.991584i	$0.541325\pi$
$107$	−1240.79	+	1240.79i	−1.12105	+	1.12105i	−0.991584	+	0.129462i	$0.958675\pi$
$108$		0			0
$109$	−108.629	−	108.629i	−0.0954565	−	0.0954565i	0.657766	−	0.753222i	$-0.271502\pi$
$109$	−108.629	−	108.629i	−0.0954565	−	0.0954565i	−0.753222	+	0.657766i	$0.771502\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1722.22			1.43374			0.716870	−	0.697207i	$-0.245574\pi$
$113$	1722.22			1.43374			0.716870	+	0.697207i	$0.245574\pi$
$114$		0			0
$115$	1694.49	+	1694.49i	1.37402	+	1.37402i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	258.818i		−	0.199377i
$120$		0			0
$121$			1326.21i			0.996404i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−882.442	−	882.442i	−0.631424	−	0.631424i
$126$		0			0
$127$	−699.127			−0.488484			−0.244242	−	0.969714i	$-0.578539\pi$
$127$	−699.127			−0.488484			−0.244242	+	0.969714i	$0.578539\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	197.970	+	197.970i	0.132036	+	0.132036i	0.770036	−	0.638000i	$-0.220238\pi$
$131$	197.970	+	197.970i	0.132036	+	0.132036i	−0.638000	+	0.770036i	$0.720238\pi$
$132$		0			0
$133$	−1198.89	+	1198.89i	−0.781632	+	0.781632i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	271.386i		−	0.169242i	−0.996413	−	0.0846209i	$-0.973032\pi$
$137$		−	271.386i		−	0.169242i	0.996413	−	0.0846209i	$-0.0269679\pi$
$138$		0			0
$139$	−459.937	+	459.937i	−0.280657	+	0.280657i	−0.833371	−	0.552714i	$-0.813592\pi$
$139$	−459.937	+	459.937i	−0.280657	+	0.280657i	0.552714	+	0.833371i	$0.313592\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	101.441			0.0593210
$144$		0			0
$145$	1512.20			0.866079
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−605.772	+	605.772i	−0.333066	+	0.333066i	−0.853750	−	0.520684i	$-0.825677\pi$
$149$	−605.772	+	605.772i	−0.333066	+	0.333066i	0.520684	+	0.853750i	$0.325677\pi$
$150$		0			0
$151$		−	3534.47i		−	1.90484i	−0.304785	−	0.952421i	$-0.598585\pi$
$151$		−	3534.47i		−	1.90484i	0.304785	−	0.952421i	$-0.401415\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	398.809	−	398.809i	0.206665	−	0.206665i
$156$		0			0
$157$	−1233.54	−	1233.54i	−0.627051	−	0.627051i	0.320274	−	0.947325i	$-0.396225\pi$
$157$	−1233.54	−	1233.54i	−0.627051	−	0.627051i	−0.947325	+	0.320274i	$0.896225\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1859.38			−0.910186
$162$		0			0
$163$	−2569.36	−	2569.36i	−1.23465	−	1.23465i	−0.962158	−	0.272494i	$-0.912152\pi$
$163$	−2569.36	−	2569.36i	−1.23465	−	1.23465i	−0.272494	−	0.962158i	$-0.587848\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	3048.39i		−	1.41252i	−0.707950	−	0.706262i	$-0.750380\pi$
$167$		−	3048.39i		−	1.41252i	0.707950	−	0.706262i	$-0.249620\pi$
$168$		0			0
$169$		−	46.9618i		−	0.0213754i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1522.05	+	1522.05i	0.668898	+	0.668898i	0.957461	−	0.288563i	$-0.0931773\pi$
$173$	1522.05	+	1522.05i	0.668898	+	0.668898i	−0.288563	+	0.957461i	$0.593177\pi$
$174$		0			0
$175$	2702.74			1.16747
$176$		0			0
$177$		0			0
$178$		0			0
$179$	302.258	+	302.258i	0.126211	+	0.126211i	0.767391	−	0.641180i	$-0.221555\pi$
$179$	302.258	+	302.258i	0.126211	+	0.126211i	−0.641180	+	0.767391i	$0.721555\pi$
$180$		0			0
$181$	1696.94	−	1696.94i	0.696865	−	0.696865i	−0.266868	−	0.963733i	$-0.585989\pi$
$181$	1696.94	−	1696.94i	0.696865	−	0.696865i	0.963733	+	0.266868i	$0.0859889\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	2255.51i		−	0.896370i
$186$		0			0
$187$	−28.8552	+	28.8552i	−0.0112840	+	0.0112840i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4035.31			−1.52872			−0.764358	−	0.644792i	$-0.776944\pi$
$191$	−4035.31			−1.52872			−0.764358	+	0.644792i	$0.776944\pi$
$192$		0			0
$193$	−886.172			−0.330508			−0.165254	−	0.986251i	$-0.552844\pi$
$193$	−886.172			−0.330508			−0.165254	+	0.986251i	$0.552844\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−3270.12	+	3270.12i	−1.18267	+	1.18267i	−0.203624	+	0.979049i	$0.565272\pi$
$197$	−3270.12	+	3270.12i	−1.18267	+	1.18267i	−0.979049	+	0.203624i	$0.934728\pi$
$198$		0			0
$199$			222.513i			0.0792639i	0.999214	+	0.0396319i	$0.0126185\pi$
$199$			222.513i			0.0792639i	−0.999214	+	0.0396319i	$0.987381\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−829.677	+	829.677i	−0.286857	+	0.286857i
$204$		0			0
$205$	−2661.80	−	2661.80i	−0.906868	−	0.906868i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	267.325			0.0884748
$210$		0			0
$211$	3527.46	+	3527.46i	1.15090	+	1.15090i	0.986372	+	0.164529i	$0.0526105\pi$
$211$	3527.46	+	3527.46i	1.15090	+	1.15090i	0.164529	+	0.986372i	$0.447390\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			3019.65i			0.957852i
$216$		0			0
$217$			437.618i			0.136901i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−611.587	−	611.587i	−0.186153	−	0.186153i
$222$		0			0
$223$	−5841.90			−1.75427			−0.877136	−	0.480242i	$-0.840549\pi$
$223$	−5841.90			−1.75427			−0.877136	+	0.480242i	$0.840549\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1129.54	−	1129.54i	−0.330265	−	0.330265i	0.522422	−	0.852687i	$-0.325029\pi$
$227$	−1129.54	−	1129.54i	−0.330265	−	0.330265i	−0.852687	+	0.522422i	$0.825029\pi$
$228$		0			0
$229$	−2905.40	+	2905.40i	−0.838403	+	0.838403i	−0.988649	−	0.150246i	$-0.951994\pi$
$229$	−2905.40	+	2905.40i	−0.838403	+	0.838403i	0.150246	+	0.988649i	$0.451994\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	734.054i		−	0.206393i	−0.994661	−	0.103196i	$-0.967093\pi$
$233$		−	734.054i		−	0.206393i	0.994661	−	0.103196i	$-0.0329070\pi$
$234$		0			0
$235$	−2312.78	+	2312.78i	−0.641995	+	0.641995i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−511.807			−0.138519			−0.0692595	−	0.997599i	$-0.522064\pi$
$239$	−511.807			−0.138519			−0.0692595	+	0.997599i	$0.522064\pi$
$240$		0			0
$241$	5920.31			1.58241			0.791204	−	0.611552i	$-0.209455\pi$
$241$	5920.31			1.58241			0.791204	+	0.611552i	$0.209455\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1902.73	−	1902.73i	0.496166	−	0.496166i
$246$		0			0
$247$			5665.95i			1.45958i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−309.332	+	309.332i	−0.0777883	+	0.0777883i	−0.744930	−	0.667142i	$-0.767517\pi$
$251$	−309.332	+	309.332i	−0.0777883	+	0.0777883i	0.667142	+	0.744930i	$0.267517\pi$
$252$		0			0
$253$	207.300	+	207.300i	0.0515131	+	0.0515131i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−323.723			−0.0785730			−0.0392865	−	0.999228i	$-0.512509\pi$
$257$	−323.723			−0.0785730			−0.0392865	+	0.999228i	$0.512509\pi$
$258$		0			0
$259$	1237.50	+	1237.50i	0.296890	+	0.296890i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	2689.15i		−	0.630495i	−0.949009	−	0.315248i	$-0.897912\pi$
$263$		−	2689.15i		−	0.630495i	0.949009	−	0.315248i	$-0.102088\pi$
$264$		0			0
$265$		−	661.691i		−	0.153386i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4703.78	−	4703.78i	−1.06615	−	1.06615i	−0.997651	−	0.0684995i	$-0.978179\pi$
$269$	−4703.78	−	4703.78i	−1.06615	−	1.06615i	−0.0684995	−	0.997651i	$-0.521821\pi$
$270$		0			0
$271$	2018.97			0.452561			0.226280	−	0.974062i	$-0.427343\pi$
$271$	2018.97			0.452561			0.226280	+	0.974062i	$0.427343\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−301.324	−	301.324i	−0.0660745	−	0.0660745i
$276$		0			0
$277$	−3080.60	+	3080.60i	−0.668215	+	0.668215i	−0.957303	−	0.289088i	$-0.906648\pi$
$277$	−3080.60	+	3080.60i	−0.668215	+	0.668215i	0.289088	+	0.957303i	$0.406648\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			3893.51i			0.826575i	0.910601	+	0.413287i	$0.135619\pi$
$281$			3893.51i			0.826575i	−0.910601	+	0.413287i	$0.864381\pi$
$282$		0			0
$283$	−2026.38	+	2026.38i	−0.425639	+	0.425639i	−0.887140	−	0.461501i	$-0.847311\pi$
$283$	−2026.38	+	2026.38i	−0.425639	+	0.425639i	0.461501	+	0.887140i	$0.347311\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2920.82			0.600734
$288$		0			0
$289$	−4565.06			−0.929180
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1001.68	−	1001.68i	0.199724	−	0.199724i	−0.600158	−	0.799882i	$-0.704896\pi$
$293$	1001.68	−	1001.68i	0.199724	−	0.199724i	0.799882	+	0.600158i	$0.204896\pi$
$294$		0			0
$295$		−	11167.6i		−	2.20407i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4393.72	+	4393.72i	−0.849817	+	0.849817i
$300$		0			0
$301$	−1656.75	−	1656.75i	−0.317253	−	0.317253i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−4412.69			−0.828425
$306$		0			0
$307$	−2966.54	−	2966.54i	−0.551497	−	0.551497i	0.375376	−	0.926873i	$-0.377514\pi$
$307$	−2966.54	−	2966.54i	−0.551497	−	0.551497i	−0.926873	+	0.375376i	$0.877514\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			2911.18i			0.530797i	0.964139	+	0.265399i	$0.0855036\pi$
$311$			2911.18i			0.530797i	−0.964139	+	0.265399i	$0.914496\pi$
$312$		0			0
$313$		−	8287.74i		−	1.49665i	−0.663333	−	0.748324i	$-0.730859\pi$
$313$		−	8287.74i		−	1.49665i	0.663333	−	0.748324i	$-0.269141\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−5742.18	−	5742.18i	−1.01739	−	1.01739i	−0.999846	−	0.0175452i	$-0.994415\pi$
$317$	−5742.18	−	5742.18i	−1.01739	−	1.01739i	−0.0175452	−	0.999846i	$-0.505585\pi$
$318$		0			0
$319$	184.999			0.0324700
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1611.70	−	1611.70i	−0.277639	−	0.277639i
$324$		0			0
$325$	6386.56	−	6386.56i	1.09004	−	1.09004i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	2537.84i		−	0.425275i
$330$		0			0
$331$	−4499.27	+	4499.27i	−0.747136	+	0.747136i	−0.973940	−	0.226804i	$-0.927172\pi$
$331$	−4499.27	+	4499.27i	−0.747136	+	0.747136i	0.226804	+	0.973940i	$0.427172\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2320.50			−0.378454
$336$		0			0
$337$	−5860.06			−0.947234			−0.473617	−	0.880731i	$-0.657052\pi$
$337$	−5860.06			−0.947234			−0.473617	+	0.880731i	$0.657052\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	48.7893	−	48.7893i	0.00774806	−	0.00774806i
$342$		0			0
$343$			6847.14i			1.07787i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2029.11	+	2029.11i	−0.313915	+	0.313915i	−0.846424	−	0.532509i	$-0.821249\pi$
$347$	−2029.11	+	2029.11i	−0.313915	+	0.313915i	0.532509	+	0.846424i	$0.321249\pi$
$348$		0			0
$349$	−1943.26	−	1943.26i	−0.298053	−	0.298053i	0.542198	−	0.840251i	$-0.317592\pi$
$349$	−1943.26	−	1943.26i	−0.298053	−	0.298053i	−0.840251	+	0.542198i	$0.817592\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	7548.63			1.13817			0.569084	−	0.822280i	$-0.307298\pi$
$353$	7548.63			1.13817			0.569084	+	0.822280i	$0.307298\pi$
$354$		0			0
$355$	−4404.97	−	4404.97i	−0.658568	−	0.658568i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	5554.15i		−	0.816537i	−0.912862	−	0.408269i	$-0.866133\pi$
$359$		−	5554.15i		−	0.816537i	0.912862	−	0.408269i	$-0.133867\pi$
$360$		0			0
$361$			8072.37i			1.17690i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3786.34	+	3786.34i	0.542975	+	0.542975i
$366$		0			0
$367$	3610.98			0.513601			0.256800	−	0.966464i	$-0.417332\pi$
$367$	3610.98			0.513601			0.256800	+	0.966464i	$0.417332\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	363.040	+	363.040i	0.0508035	+	0.0508035i
$372$		0			0
$373$	1215.49	−	1215.49i	0.168728	−	0.168728i	−0.617692	−	0.786420i	$-0.711932\pi$
$373$	1215.49	−	1215.49i	0.168728	−	0.168728i	0.786420	+	0.617692i	$0.211932\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			3921.05i			0.535661i
$378$		0			0
$379$	7347.81	−	7347.81i	0.995861	−	0.995861i	−0.00413018	−	0.999991i	$-0.501315\pi$
$379$	7347.81	−	7347.81i	0.995861	−	0.995861i	0.999991	+	0.00413018i	$0.00131468\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−7668.98			−1.02315			−0.511575	−	0.859238i	$-0.670938\pi$
$383$	−7668.98			−1.02315			−0.511575	+	0.859238i	$0.670938\pi$
$384$		0			0
$385$	542.831			0.0718577
$386$		0			0
$387$		0			0
$388$		0			0
$389$	200.924	−	200.924i	0.0261884	−	0.0261884i	−0.693891	−	0.720080i	$-0.744105\pi$
$389$	200.924	−	200.924i	0.0261884	−	0.0261884i	0.720080	+	0.693891i	$0.244105\pi$
$390$		0			0
$391$		−	2499.62i		−	0.323302i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	11930.3	−	11930.3i	1.51969	−	1.51969i
$396$		0			0
$397$	6512.21	+	6512.21i	0.823271	+	0.823271i	0.986576	−	0.163305i	$-0.0522153\pi$
$397$	6512.21	+	6512.21i	0.823271	+	0.823271i	−0.163305	+	0.986576i	$0.552215\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−5565.10			−0.693036			−0.346518	−	0.938043i	$-0.612636\pi$
$401$	−5565.10			−0.693036			−0.346518	+	0.938043i	$0.612636\pi$
$402$		0			0
$403$	1034.09	+	1034.09i	0.127820	+	0.127820i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	275.933i		−	0.0336057i
$408$		0			0
$409$			12077.6i			1.46014i	0.683370	+	0.730072i	$0.260514\pi$
$409$			12077.6i			1.46014i	−0.683370	+	0.730072i	$0.739486\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	6127.14	+	6127.14i	0.730017	+	0.730017i
$414$		0			0
$415$	7922.58			0.937119
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1453.03	−	1453.03i	−0.169415	−	0.169415i	0.617307	−	0.786722i	$-0.288224\pi$
$419$	−1453.03	−	1453.03i	−0.169415	−	0.169415i	−0.786722	+	0.617307i	$0.788224\pi$
$420$		0			0
$421$	−4822.25	+	4822.25i	−0.558247	+	0.558247i	−0.928808	−	0.370561i	$-0.879165\pi$
$421$	−4822.25	+	4822.25i	−0.558247	+	0.558247i	0.370561	+	0.928808i	$0.379165\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			3633.36i			0.414692i
$426$		0			0
$427$	2421.04	−	2421.04i	0.274385	−	0.274385i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	12519.2			1.39914			0.699571	−	0.714563i	$-0.253374\pi$
$431$	12519.2			1.39914			0.699571	+	0.714563i	$0.253374\pi$
$432$		0			0
$433$	−2921.40			−0.324235			−0.162117	−	0.986771i	$-0.551832\pi$
$433$	−2921.40			−0.324235			−0.162117	+	0.986771i	$0.551832\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−11578.7	+	11578.7i	−1.26747	+	1.26747i
$438$		0			0
$439$			1140.50i			0.123993i	0.998076	+	0.0619967i	$0.0197468\pi$
$439$			1140.50i			0.123993i	−0.998076	+	0.0619967i	$0.980253\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1843.05	+	1843.05i	−0.197665	+	0.197665i	−0.798999	−	0.601333i	$-0.794637\pi$
$443$	−1843.05	+	1843.05i	−0.197665	+	0.197665i	0.601333	+	0.798999i	$0.294637\pi$
$444$		0			0
$445$	17859.0	+	17859.0i	1.90247	+	1.90247i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1752.13			−0.184161			−0.0920805	−	0.995752i	$-0.529352\pi$
$449$	−1752.13			−0.184161			−0.0920805	+	0.995752i	$0.529352\pi$
$450$		0			0
$451$	−325.637	−	325.637i	−0.0339993	−	0.0339993i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			11505.3i			1.18544i
$456$		0			0
$457$		−	12875.6i		−	1.31794i	−0.752171	−	0.658968i	$-0.770993\pi$
$457$		−	12875.6i		−	1.31794i	0.752171	−	0.658968i	$-0.229007\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	13679.7	+	13679.7i	1.38205	+	1.38205i	0.840968	+	0.541085i	$0.181986\pi$
$461$	13679.7	+	13679.7i	1.38205	+	1.38205i	0.541085	+	0.840968i	$0.318014\pi$
$462$		0			0
$463$	−15002.4			−1.50588			−0.752938	−	0.658091i	$-0.771364\pi$
$463$	−15002.4			−1.50588			−0.752938	+	0.658091i	$0.771364\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	9669.42	+	9669.42i	0.958131	+	0.958131i	0.999158	−	0.0410271i	$-0.0130630\pi$
$467$	9669.42	+	9669.42i	0.958131	+	0.958131i	−0.0410271	+	0.999158i	$0.513063\pi$
$468$		0			0
$469$	1273.15	−	1273.15i	0.125349	−	0.125349i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			369.416i			0.0359107i
$474$		0			0
$475$	16830.4	−	16830.4i	1.62575	−	1.62575i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−3072.68			−0.293099			−0.146550	−	0.989203i	$-0.546817\pi$
$479$	−3072.68			−0.293099			−0.146550	+	0.989203i	$0.546817\pi$
$480$		0			0
$481$	5848.41			0.554396
$482$		0			0
$483$		0			0
$484$		0			0
$485$	19160.7	−	19160.7i	1.79390	−	1.79390i
$486$		0			0
$487$			8689.64i			0.808553i	0.914637	+	0.404276i	$0.132477\pi$
$487$			8689.64i			0.808553i	−0.914637	+	0.404276i	$0.867523\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	11194.3	−	11194.3i	1.02891	−	1.02891i	0.0293379	−	0.999570i	$-0.490660\pi$
$491$	11194.3	−	11194.3i	1.02891	−	1.02891i	0.999570	−	0.0293379i	$-0.00933987\pi$
$492$		0			0
$493$	−1115.36	−	1115.36i	−0.101893	−	0.101893i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	4833.62			0.436253
$498$		0			0
$499$	1632.72	+	1632.72i	0.146474	+	0.146474i	0.776541	−	0.630067i	$-0.216973\pi$
$499$	1632.72	+	1632.72i	0.146474	+	0.146474i	−0.630067	+	0.776541i	$0.716973\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			6901.81i			0.611802i	0.952063	+	0.305901i	$0.0989577\pi$
$503$			6901.81i			0.611802i	−0.952063	+	0.305901i	$0.901042\pi$
$504$		0			0
$505$		−	14496.2i		−	1.27737i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−92.9712	−	92.9712i	−0.00809603	−	0.00809603i	0.703047	−	0.711143i	$-0.251822\pi$
$509$	−92.9712	−	92.9712i	−0.00809603	−	0.00809603i	−0.711143	+	0.703047i	$0.751822\pi$
$510$		0			0
$511$	−4154.79			−0.359681
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−12910.4	−	12910.4i	−1.10466	−	1.10466i
$516$		0			0
$517$	−282.939	+	282.939i	−0.0240689	+	0.0240689i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	11931.3i		−	1.00330i	−0.865071	−	0.501649i	$-0.832727\pi$
$521$		−	11931.3i		−	1.00330i	0.865071	−	0.501649i	$-0.167273\pi$
$522$		0			0
$523$	9702.46	−	9702.46i	0.811203	−	0.811203i	−0.173611	−	0.984814i	$-0.555544\pi$
$523$	9702.46	−	9702.46i	0.811203	−	0.811203i	0.984814	+	0.173611i	$0.0555437\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−588.301			−0.0486277
$528$		0			0
$529$	−5790.59			−0.475926
$530$		0			0
$531$		0			0
$532$		0			0
$533$	6901.89	−	6901.89i	0.560889	−	0.560889i
$534$		0			0
$535$			31379.4i			2.53579i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	232.774	−	232.774i	0.0186017	−	0.0186017i
$540$		0			0
$541$	−8556.67	−	8556.67i	−0.680000	−	0.680000i	0.280000	−	0.960000i	$-0.409666\pi$
$541$	−8556.67	−	8556.67i	−0.680000	−	0.680000i	−0.960000	+	0.280000i	$0.909666\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2747.20			−0.215921
$546$		0			0
$547$	45.1953	+	45.1953i	0.00353274	+	0.00353274i	0.708871	−	0.705338i	$-0.249205\pi$
$547$	45.1953	+	45.1953i	0.00353274	+	0.00353274i	−0.705338	+	0.708871i	$0.749205\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			10333.1i			0.798917i
$552$		0			0
$553$			13091.2i			1.00668i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−4279.60	−	4279.60i	−0.325552	−	0.325552i	0.525340	−	0.850892i	$-0.323938\pi$
$557$	−4279.60	−	4279.60i	−0.325552	−	0.325552i	−0.850892	+	0.525340i	$0.823938\pi$
$558$		0			0
$559$	−7829.77			−0.592422
$560$		0			0
$561$		0			0
$562$		0			0
$563$	14593.9	+	14593.9i	1.09247	+	1.09247i	0.995265	+	0.0972023i	$0.0309894\pi$
$563$	14593.9	+	14593.9i	1.09247	+	1.09247i	0.0972023	+	0.995265i	$0.469011\pi$
$564$		0			0
$565$	21777.3	−	21777.3i	1.62155	−	1.62155i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	21728.1i		−	1.60086i	−0.599425	−	0.800431i	$-0.704604\pi$
$569$		−	21728.1i		−	1.60086i	0.599425	−	0.800431i	$-0.295396\pi$
$570$		0			0
$571$	−16078.0	+	16078.0i	−1.17836	+	1.17836i	−0.198202	+	0.980161i	$0.563510\pi$
$571$	−16078.0	+	16078.0i	−1.17836	+	1.17836i	−0.980161	+	0.198202i	$0.936490\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	26102.5			1.89313
$576$		0			0
$577$	−26648.2			−1.92267			−0.961335	−	0.275383i	$-0.911195\pi$
$577$	−26648.2			−1.92267			−0.961335	+	0.275383i	$0.911195\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−4346.77	+	4346.77i	−0.310386	+	0.310386i
$582$		0			0
$583$		−	80.9495i		−	0.00575058i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1342.62	−	1342.62i	0.0944050	−	0.0944050i	−0.658327	−	0.752732i	$-0.728736\pi$
$587$	1342.62	−	1342.62i	0.0944050	−	0.0944050i	0.752732	+	0.658327i	$0.228736\pi$
$588$		0			0
$589$	2725.11	+	2725.11i	0.190639	+	0.190639i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−4474.79			−0.309878			−0.154939	−	0.987924i	$-0.549518\pi$
$593$	−4474.79			−0.309878			−0.154939	+	0.987924i	$0.549518\pi$
$594$		0			0
$595$	−3272.73	−	3272.73i	−0.225494	−	0.225494i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	12603.8i		−	0.859725i	−0.902894	−	0.429863i	$-0.858562\pi$
$599$		−	12603.8i		−	0.859725i	0.902894	−	0.429863i	$-0.141438\pi$
$600$		0			0
$601$		−	7220.64i		−	0.490077i	−0.969513	−	0.245038i	$-0.921199\pi$
$601$		−	7220.64i		−	0.490077i	0.969513	−	0.245038i	$-0.0788006\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	16769.8	+	16769.8i	1.12693	+	1.12693i
$606$		0			0
$607$	13695.6			0.915796			0.457898	−	0.889005i	$-0.348603\pi$
$607$	13695.6			0.915796			0.457898	+	0.889005i	$0.348603\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−5996.90	−	5996.90i	−0.397068	−	0.397068i
$612$		0			0
$613$	−2358.34	+	2358.34i	−0.155387	+	0.155387i	−0.780519	−	0.625132i	$-0.785045\pi$
$613$	−2358.34	+	2358.34i	−0.155387	+	0.155387i	0.625132	+	0.780519i	$0.285045\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			4186.39i			0.273157i	0.990629	+	0.136579i	$0.0436106\pi$
$617$			4186.39i			0.273157i	−0.990629	+	0.136579i	$0.956389\pi$
$618$		0			0
$619$	4800.50	−	4800.50i	0.311710	−	0.311710i	−0.533862	−	0.845572i	$-0.679260\pi$
$619$	4800.50	−	4800.50i	0.311710	−	0.311710i	0.845572	+	0.533862i	$0.179260\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−19596.9			−1.26024
$624$		0			0
$625$	2031.54			0.130018
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1663.60	+	1663.60i	−0.105457	+	0.105457i
$630$		0			0
$631$		−	16106.3i		−	1.01614i	−0.861316	−	0.508069i	$-0.830360\pi$
$631$		−	16106.3i		−	1.01614i	0.861316	−	0.508069i	$-0.169640\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−8840.39	+	8840.39i	−0.552473	+	0.552473i
$636$		0			0
$637$	4933.66	+	4933.66i	0.306874	+	0.306874i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	6682.21			0.411749			0.205875	−	0.978578i	$-0.433996\pi$
$641$	6682.21			0.411749			0.205875	+	0.978578i	$0.433996\pi$
$642$		0			0
$643$	4983.47	+	4983.47i	0.305644	+	0.305644i	0.843217	−	0.537573i	$-0.180659\pi$
$643$	4983.47	+	4983.47i	0.305644	+	0.305644i	−0.537573	+	0.843217i	$0.680659\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	5078.45i		−	0.308585i	−0.988025	−	0.154292i	$-0.950690\pi$
$647$		−	5078.45i		−	0.308585i	0.988025	−	0.154292i	$-0.0493098\pi$
$648$		0			0
$649$		−	1366.21i		−	0.0826324i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−6189.91	−	6189.91i	−0.370949	−	0.370949i	0.496874	−	0.867823i	$-0.334481\pi$
$653$	−6189.91	−	6189.91i	−0.370949	−	0.370949i	−0.867823	+	0.496874i	$0.834481\pi$
$654$		0			0
$655$	5006.62			0.298664
$656$		0			0
$657$		0			0
$658$		0			0
$659$	5751.19	+	5751.19i	0.339962	+	0.339962i	0.856353	−	0.516391i	$-0.172725\pi$
$659$	5751.19	+	5751.19i	0.339962	+	0.339962i	−0.516391	+	0.856353i	$0.672725\pi$
$660$		0			0
$661$	−6305.38	+	6305.38i	−0.371030	+	0.371030i	−0.867852	−	0.496822i	$-0.834500\pi$
$661$	−6305.38	+	6305.38i	−0.371030	+	0.371030i	0.496822	+	0.867852i	$0.334500\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			30319.7i			1.76804i
$666$		0			0
$667$	−8012.87	+	8012.87i	−0.465157	+	0.465157i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−539.837			−0.0310584
$672$		0			0
$673$	14664.4			0.839925			0.419963	−	0.907541i	$-0.362043\pi$
$673$	14664.4			0.839925			0.419963	+	0.907541i	$0.362043\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−5795.16	+	5795.16i	−0.328990	+	0.328990i	−0.852202	−	0.523212i	$-0.824733\pi$
$677$	−5795.16	+	5795.16i	−0.328990	+	0.328990i	0.523212	+	0.852202i	$0.324733\pi$
$678$		0			0
$679$			21025.2i			1.18833i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−13134.1	+	13134.1i	−0.735816	+	0.735816i	−0.971765	−	0.235950i	$-0.924180\pi$
$683$	−13134.1	+	13134.1i	−0.735816	+	0.735816i	0.235950	+	0.971765i	$0.424180\pi$
$684$		0			0
$685$	−3431.65	−	3431.65i	−0.191411	−	0.191411i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1715.73			0.0948679
$690$		0			0
$691$	−14324.0	−	14324.0i	−0.788583	−	0.788583i	0.192679	−	0.981262i	$-0.438282\pi$
$691$	−14324.0	−	14324.0i	−0.788583	−	0.788583i	−0.981262	+	0.192679i	$0.938282\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			11631.7i			0.634843i
$696$		0			0
$697$			3926.54i			0.213383i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−15987.2	−	15987.2i	−0.861382	−	0.861382i	0.130117	−	0.991499i	$-0.458465\pi$
$701$	−15987.2	−	15987.2i	−0.861382	−	0.861382i	−0.991499	+	0.130117i	$0.958465\pi$
$702$		0			0
$703$	15412.2			0.826859
$704$		0			0
$705$		0			0
$706$		0			0
$707$	7953.40	+	7953.40i	0.423081	+	0.423081i
$708$		0			0
$709$	19580.4	−	19580.4i	1.03718	−	1.03718i	0.0378960	−	0.999282i	$-0.487934\pi$
$709$	19580.4	−	19580.4i	1.03718	−	1.03718i	0.999282	−	0.0378960i	$-0.0120656\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			4226.43i			0.221993i
$714$		0			0
$715$	1282.71	−	1282.71i	0.0670916	−	0.0670916i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−2111.24			−0.109507			−0.0547537	−	0.998500i	$-0.517437\pi$
$719$	−2111.24			−0.109507			−0.0547537	+	0.998500i	$0.517437\pi$
$720$		0			0
$721$	14166.7			0.731757
$722$		0			0
$723$		0			0
$724$		0			0
$725$	11647.2	−	11647.2i	0.596645	−	0.596645i
$726$		0			0
$727$		−	14763.6i		−	0.753164i	−0.926383	−	0.376582i	$-0.877099\pi$
$727$		−	14763.6i		−	0.753164i	0.926383	−	0.376582i	$-0.122901\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	2227.21	−	2227.21i	0.112690	−	0.112690i
$732$		0			0
$733$	3419.77	+	3419.77i	0.172322	+	0.172322i	0.787999	−	0.615677i	$-0.211117\pi$
$733$	3419.77	+	3419.77i	0.172322	+	0.172322i	−0.615677	+	0.787999i	$0.711117\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−283.883			−0.0141886
$738$		0			0
$739$	11324.8	+	11324.8i	0.563723	+	0.563723i	0.930363	−	0.366640i	$-0.119492\pi$
$739$	11324.8	+	11324.8i	0.563723	+	0.563723i	−0.366640	+	0.930363i	$0.619492\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	20313.3i		−	1.00299i	−0.865160	−	0.501495i	$-0.832783\pi$
$743$		−	20313.3i		−	1.00299i	0.865160	−	0.501495i	$-0.167217\pi$
$744$		0			0
$745$			15319.9i			0.753391i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−17216.5	−	17216.5i	−0.839888	−	0.839888i
$750$		0			0
$751$	−28755.3			−1.39720			−0.698598	−	0.715514i	$-0.746192\pi$
$751$	−28755.3			−1.39720			−0.698598	+	0.715514i	$0.746192\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−44693.0	−	44693.0i	−2.15436	−	2.15436i
$756$		0			0
$757$	−23006.0	+	23006.0i	−1.10458	+	1.10458i	−0.110730	+	0.993850i	$0.535319\pi$
$757$	−23006.0	+	23006.0i	−1.10458	+	1.10458i	−0.993850	+	0.110730i	$0.964681\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	9298.53i		−	0.442932i	−0.975168	−	0.221466i	$-0.928916\pi$
$761$		−	9298.53i		−	0.442932i	0.975168	−	0.221466i	$-0.0710843\pi$
$762$		0			0
$763$	1507.27	−	1507.27i	0.0715160	−	0.0715160i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	28956.8			1.36319
$768$		0			0
$769$	−20402.0			−0.956717			−0.478358	−	0.878165i	$-0.658768\pi$
$769$	−20402.0			−0.956717			−0.478358	+	0.878165i	$0.658768\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−7337.03	+	7337.03i	−0.341390	+	0.341390i	−0.856890	−	0.515500i	$-0.827606\pi$
$773$	−7337.03	+	7337.03i	−0.341390	+	0.341390i	0.515500	+	0.856890i	$0.327606\pi$
$774$		0			0
$775$		−	6143.40i		−	0.284745i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	18188.4	−	18188.4i	0.836544	−	0.836544i
$780$		0			0
$781$	−538.893	−	538.893i	−0.0246903	−	0.0246903i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−31195.9			−1.41838
$786$		0			0
$787$	11928.6	+	11928.6i	0.540292	+	0.540292i	0.923615	−	0.383323i	$-0.125220\pi$
$787$	11928.6	+	11928.6i	0.540292	+	0.540292i	−0.383323	+	0.923615i	$0.625220\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			23896.4i			1.07416i
$792$		0			0
$793$		−	11441.8i		−	0.512373i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	6576.18	+	6576.18i	0.292271	+	0.292271i	0.837977	−	0.545706i	$-0.183738\pi$
$797$	6576.18	+	6576.18i	0.292271	+	0.292271i	−0.545706	+	0.837977i	$0.683738\pi$
$798$		0			0
$799$	3411.68			0.151060
$800$		0			0
$801$		0			0
$802$		0			0
$803$	463.211	+	463.211i	0.0203566	+	0.0203566i
$804$		0			0
$805$	−23511.7	+	23511.7i	−1.02941	+	1.02941i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			29320.9i			1.27425i	0.770760	+	0.637126i	$0.219877\pi$
$809$			29320.9i			1.27425i	−0.770760	+	0.637126i	$0.780123\pi$
$810$		0			0
$811$	14487.9	−	14487.9i	0.627297	−	0.627297i	−0.320090	−	0.947387i	$-0.603713\pi$
$811$	14487.9	−	14487.9i	0.627297	−	0.627297i	0.947387	+	0.320090i	$0.103713\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−64978.7			−2.79276
$816$		0			0
$817$	−20633.6			−0.883574
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−20259.2	+	20259.2i	−0.861208	+	0.861208i	−0.991479	−	0.130270i	$-0.958415\pi$
$821$	−20259.2	+	20259.2i	−0.861208	+	0.861208i	0.130270	+	0.991479i	$0.458415\pi$
$822$		0			0
$823$		−	24605.9i		−	1.04217i	−0.853504	−	0.521086i	$-0.825527\pi$
$823$		−	24605.9i		−	1.04217i	0.853504	−	0.521086i	$-0.174473\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−24095.8	+	24095.8i	−1.01317	+	1.01317i	−0.0132601	+	0.999912i	$0.504221\pi$
$827$	−24095.8	+	24095.8i	−1.01317	+	1.01317i	−0.999912	+	0.0132601i	$0.995779\pi$
$828$		0			0
$829$	−914.616	−	914.616i	−0.0383184	−	0.0383184i	0.687688	−	0.726006i	$-0.258626\pi$
$829$	−914.616	−	914.616i	−0.0383184	−	0.0383184i	−0.726006	+	0.687688i	$0.758626\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2806.80			−0.116746
$834$		0			0
$835$	−38546.6	−	38546.6i	−1.59756	−	1.59756i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	28847.5i		−	1.18704i	−0.804819	−	0.593521i	$-0.797737\pi$
$839$		−	28847.5i		−	1.18704i	0.804819	−	0.593521i	$-0.202263\pi$
$840$		0			0
$841$		−	17238.1i		−	0.706800i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−593.827	−	593.827i	−0.0241755	−	0.0241755i
$846$		0			0
$847$	−18401.7			−0.746506
$848$		0			0
$849$		0			0
$850$		0			0
$851$	11951.5	+	11951.5i	0.481426	+	0.481426i
$852$		0			0
$853$	41.5562	−	41.5562i	0.00166806	−	0.00166806i	−0.706272	−	0.707940i	$-0.749625\pi$
$853$	41.5562	−	41.5562i	0.00166806	−	0.00166806i	0.707940	+	0.706272i	$0.249625\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	20953.6i		−	0.835194i	−0.908632	−	0.417597i	$-0.862872\pi$
$857$		−	20953.6i		−	0.835194i	0.908632	−	0.417597i	$-0.137128\pi$
$858$		0			0
$859$	−29316.3	+	29316.3i	−1.16444	+	1.16444i	−0.180953	+	0.983492i	$0.557918\pi$
$859$	−29316.3	+	29316.3i	−1.16444	+	1.16444i	−0.983492	+	0.180953i	$0.942082\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−3389.59			−0.133700			−0.0668499	−	0.997763i	$-0.521295\pi$
$863$	−3389.59			−0.133700			−0.0668499	+	0.997763i	$0.521295\pi$
$864$		0			0
$865$	38492.3			1.51304
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1459.52	−	1459.52i	0.0569745	−	0.0569745i
$870$		0			0
$871$		−	6016.91i		−	0.234070i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	12244.2	−	12244.2i	0.473063	−	0.473063i
$876$		0			0
$877$	−13912.5	−	13912.5i	−0.535681	−	0.535681i	0.386576	−	0.922257i	$-0.373658\pi$
$877$	−13912.5	−	13912.5i	−0.535681	−	0.535681i	−0.922257	+	0.386576i	$0.873658\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1497.48			0.0572660			0.0286330	−	0.999590i	$-0.490885\pi$
$881$	1497.48			0.0572660			0.0286330	+	0.999590i	$0.490885\pi$
$882$		0			0
$883$	4143.81	+	4143.81i	0.157928	+	0.157928i	0.781648	−	0.623720i	$-0.214379\pi$
$883$	4143.81	+	4143.81i	0.157928	+	0.157928i	−0.623720	+	0.781648i	$0.714379\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			18058.0i			0.683573i	0.939778	+	0.341787i	$0.111032\pi$
$887$			18058.0i			0.683573i	−0.939778	+	0.341787i	$0.888968\pi$
$888$		0			0
$889$		−	9700.66i		−	0.365973i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−15803.5	−	15803.5i	−0.592211	−	0.592211i
$894$		0			0
$895$	7644.03			0.285488
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1885.88	+	1885.88i	0.0699640	+	0.0699640i
$900$		0			0
$901$	−488.045	+	488.045i	−0.0180457	+	0.0180457i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	42915.2i		−	1.57630i
$906$		0			0
$907$	21423.5	−	21423.5i	0.784295	−	0.784295i	−0.196257	−	0.980552i	$-0.562879\pi$
$907$	21423.5	−	21423.5i	0.784295	−	0.784295i	0.980552	+	0.196257i	$0.0628788\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−31977.7			−1.16297			−0.581487	−	0.813556i	$-0.697529\pi$
$911$	−31977.7			−1.16297			−0.581487	+	0.813556i	$0.697529\pi$
$912$		0			0
$913$	969.228			0.0351334
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2746.91	+	2746.91i	−0.0989215	+	0.0989215i
$918$		0			0
$919$		−	40696.7i		−	1.46078i	−0.683029	−	0.730391i	$-0.739338\pi$
$919$		−	40696.7i		−	1.46078i	0.683029	−	0.730391i	$-0.260662\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	11421.8	−	11421.8i	0.407318	−	0.407318i
$924$		0			0
$925$	−17372.4	−	17372.4i	−0.617513	−	0.617513i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	11467.5			0.404989			0.202495	−	0.979283i	$-0.435095\pi$
$929$	11467.5			0.404989			0.202495	+	0.979283i	$0.435095\pi$
$930$		0			0
$931$	13001.6	+	13001.6i	0.457690	+	0.457690i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			729.742i			0.0255242i
$936$		0			0
$937$			14100.2i			0.491603i	0.969320	+	0.245802i	$0.0790512\pi$
$937$			14100.2i			0.491603i	−0.969320	+	0.245802i	$0.920949\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	21058.7	+	21058.7i	0.729538	+	0.729538i	0.970528	−	0.240989i	$-0.0774719\pi$
$941$	21058.7	+	21058.7i	0.729538	+	0.729538i	−0.240989	+	0.970528i	$0.577472\pi$
$942$		0			0
$943$	28208.8			0.974129
$944$		0			0
$945$		0			0
$946$		0			0
$947$	24998.9	+	24998.9i	0.857820	+	0.857820i	0.991081	−	0.133261i	$-0.0425448\pi$
$947$	24998.9	+	24998.9i	0.857820	+	0.857820i	−0.133261	+	0.991081i	$0.542545\pi$
$948$		0			0
$949$	−9817.75	+	9817.75i	−0.335825	+	0.335825i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			6456.01i			0.219445i	0.993962	+	0.109722i	$0.0349961\pi$
$953$			6456.01i			0.219445i	−0.993962	+	0.109722i	$0.965004\pi$
$954$		0			0
$955$	−51026.0	+	51026.0i	−1.72897	+	1.72897i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	3765.59			0.126796
$960$		0			0
$961$	−28796.3			−0.966610
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−11205.5	+	11205.5i	−0.373802	+	0.373802i
$966$		0			0
$967$			15099.9i			0.502153i	0.967967	+	0.251076i	$0.0807845\pi$
$967$			15099.9i			0.502153i	−0.967967	+	0.251076i	$0.919215\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−6223.12	+	6223.12i	−0.205674	+	0.205674i	−0.802426	−	0.596752i	$-0.796458\pi$
$971$	−6223.12	+	6223.12i	−0.205674	+	0.205674i	0.596752	+	0.802426i	$0.296458\pi$
$972$		0			0
$973$	−6381.80	−	6381.80i	−0.210268	−	0.210268i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	34900.0			1.14284			0.571418	−	0.820659i	$-0.306393\pi$
$977$	34900.0			1.14284			0.571418	+	0.820659i	$0.306393\pi$
$978$		0			0
$979$	2184.82	+	2184.82i	0.0713251	+	0.0713251i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	21221.5i		−	0.688567i	−0.938866	−	0.344283i	$-0.888122\pi$
$983$		−	21221.5i		−	0.688567i	0.938866	−	0.344283i	$-0.111878\pi$
$984$		0			0
$985$			82700.7i			2.67519i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−16000.6	−	16000.6i	−0.514447	−	0.514447i
$990$		0			0
$991$	−23985.3			−0.768838			−0.384419	−	0.923159i	$-0.625598\pi$
$991$	−23985.3			−0.768838			−0.384419	+	0.923159i	$0.625598\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	2813.65	+	2813.65i	0.0896469	+	0.0896469i
$996$		0			0
$997$	10763.8	−	10763.8i	0.341919	−	0.341919i	−0.515169	−	0.857088i	$-0.672271\pi$
$997$	10763.8	−	10763.8i	0.341919	−	0.341919i	0.857088	+	0.515169i	$0.172271\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.4.k.a.145.5		10
3.2	odd	2		64.4.e.a.17.4		10
4.3	odd	2		144.4.k.a.109.1		10
12.11	even	2		16.4.e.a.13.5	yes	10
16.5	even	4	inner	576.4.k.a.433.5		10
16.11	odd	4		144.4.k.a.37.1		10
24.5	odd	2		128.4.e.a.33.2		10
24.11	even	2		128.4.e.b.33.4		10
48.5	odd	4		64.4.e.a.49.4		10
48.11	even	4		16.4.e.a.5.5	✓	10
48.29	odd	4		128.4.e.a.97.2		10
48.35	even	4		128.4.e.b.97.4		10
96.5	odd	8		1024.4.a.m.1.3		10
96.11	even	8		1024.4.a.n.1.3		10
96.29	odd	8		1024.4.b.k.513.8		10
96.35	even	8		1024.4.b.j.513.3		10
96.53	odd	8		1024.4.a.m.1.8		10
96.59	even	8		1024.4.a.n.1.8		10
96.77	odd	8		1024.4.b.k.513.3		10
96.83	even	8		1024.4.b.j.513.8		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.4.e.a.5.5	✓	10	48.11	even	4
16.4.e.a.13.5	yes	10	12.11	even	2
64.4.e.a.17.4		10	3.2	odd	2
64.4.e.a.49.4		10	48.5	odd	4
128.4.e.a.33.2		10	24.5	odd	2
128.4.e.a.97.2		10	48.29	odd	4
128.4.e.b.33.4		10	24.11	even	2
128.4.e.b.97.4		10	48.35	even	4
144.4.k.a.37.1		10	16.11	odd	4
144.4.k.a.109.1		10	4.3	odd	2
576.4.k.a.145.5		10	1.1	even	1	trivial
576.4.k.a.433.5		10	16.5	even	4	inner
1024.4.a.m.1.3		10	96.5	odd	8
1024.4.a.m.1.8		10	96.53	odd	8
1024.4.a.n.1.3		10	96.11	even	8
1024.4.a.n.1.8		10	96.59	even	8
1024.4.b.j.513.3		10	96.35	even	8
1024.4.b.j.513.8		10	96.83	even	8
1024.4.b.k.513.3		10	96.77	odd	8
1024.4.b.k.513.8		10	96.29	odd	8

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	576.k (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(12.6449 - 12.6449i) q^{5} +13.8754i q^{7} +O(q^{10})\)	\(q+(12.6449 - 12.6449i) q^{5} +13.8754i q^{7} +(1.54694 - 1.54694i) q^{11} +(32.7875 + 32.7875i) q^{13} -18.6531 q^{17} +(86.4042 + 86.4042i) q^{19} +134.006i q^{23} -194.786i q^{25} +(59.7949 + 59.7949i) q^{29} +31.5391 q^{31} +(175.453 + 175.453i) q^{35} +(89.1866 - 89.1866i) q^{37} -210.504i q^{41} +(-119.402 + 119.402i) q^{43} -182.902 q^{47} +150.474 q^{49} +(26.1644 - 26.1644i) q^{53} -39.1219i q^{55} +(441.584 - 441.584i) q^{59} +(-174.485 - 174.485i) q^{61} +829.188 q^{65} +(-91.7562 - 91.7562i) q^{67} -348.360i q^{71} +299.436i q^{73} +(21.4644 + 21.4644i) q^{77} +943.487 q^{79} +(313.272 + 313.272i) q^{83} +(-235.866 + 235.866i) q^{85} +1412.35i q^{89} +(-454.939 + 454.939i) q^{91} +2185.14 q^{95} +1515.29 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{5}+O(q^{10}) \) 10 * q + 2 * q^5	\( 10 q + 2 q^{5} + 18 q^{11} - 2 q^{13} + 4 q^{17} + 26 q^{19} + 202 q^{29} - 368 q^{31} + 476 q^{35} - 10 q^{37} + 838 q^{43} - 944 q^{47} + 94 q^{49} + 378 q^{53} + 1706 q^{59} + 910 q^{61} + 492 q^{65} - 1942 q^{67} + 268 q^{77} + 4416 q^{79} - 2562 q^{83} - 12 q^{85} - 3332 q^{91} + 6900 q^{95} - 4 q^{97}+O(q^{100}) \) 10 * q + 2 * q^5 + 18 * q^11 - 2 * q^13 + 4 * q^17 + 26 * q^19 + 202 * q^29 - 368 * q^31 + 476 * q^35 - 10 * q^37 + 838 * q^43 - 944 * q^47 + 94 * q^49 + 378 * q^53 + 1706 * q^59 + 910 * q^61 + 492 * q^65 - 1942 * q^67 + 268 * q^77 + 4416 * q^79 - 2562 * q^83 - 12 * q^85 - 3332 * q^91 + 6900 * q^95 - 4 * q^97

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