LMFDB - Embedded newform 3024.2.t.l.1873.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3024,2,Mod(289,3024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3024, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4, 2]))

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

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

chi := DirichletCharacter("3024.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$3024 = 2^{4} \cdot 3^{3} \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3024.t (of order $3$ , 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:	$24.1467615712$
Analytic rank:	$0$
Dimension:	$22$
Relative dimension:	$11$ over $\Q(\zeta_{3})$
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1873.9
Character	$\chi$	$=$	3024.1873
Dual form			3024.2.t.l.289.9

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.52290 q^{5} +(1.07705 - 2.41660i) q^{7} -5.71296 q^{11} +(-2.45245 + 4.24777i) q^{13} +(-2.49483 + 4.32118i) q^{17} +(0.00383929 + 0.00664984i) q^{19} +0.667754 q^{23} +1.36505 q^{25} +(-3.85082 - 6.66981i) q^{29} +(-3.88302 - 6.72560i) q^{31} +(2.71729 - 6.09686i) q^{35} +(-3.19562 - 5.53498i) q^{37} +(-5.21159 + 9.02673i) q^{41} +(-4.42935 - 7.67185i) q^{43} +(-1.08052 + 1.87152i) q^{47} +(-4.67994 - 5.20559i) q^{49} +(3.69858 - 6.40613i) q^{53} -14.4133 q^{55} +(0.261797 + 0.453446i) q^{59} +(4.49541 - 7.78628i) q^{61} +(-6.18730 + 10.7167i) q^{65} +(-2.54791 - 4.41311i) q^{67} -5.68471 q^{71} +(-1.52062 + 2.63379i) q^{73} +(-6.15314 + 13.8060i) q^{77} +(3.08115 - 5.33671i) q^{79} +(-0.258726 - 0.448126i) q^{83} +(-6.29422 + 10.9019i) q^{85} +(-1.19093 - 2.06274i) q^{89} +(7.62377 + 10.5017i) q^{91} +(0.00968615 + 0.0167769i) q^{95} +(4.32994 + 7.49968i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$22 q + 6 q^{5} - 7 q^{7} + 6 q^{11} - 3 q^{13} - 7 q^{17} + q^{19} - 4 q^{23} + 20 q^{25} - 9 q^{29} + 4 q^{31} + 14 q^{35} + 2 q^{37} - 16 q^{41} + 5 q^{47} - 15 q^{49} - 11 q^{53} - 22 q^{55} - 19 q^{59}+ \cdots - 29 q^{97}+O(q^{100})$ 22 * q + 6 * q^5 - 7 * q^7 + 6 * q^11 - 3 * q^13 - 7 * q^17 + q^19 - 4 * q^23 + 20 * q^25 - 9 * q^29 + 4 * q^31 + 14 * q^35 + 2 * q^37 - 16 * q^41 + 5 * q^47 - 15 * q^49 - 11 * q^53 - 22 * q^55 - 19 * q^59 - 13 * q^61 - 13 * q^65 - 26 * q^67 - 48 * q^71 - 35 * q^73 + 4 * q^77 - 10 * q^79 - 28 * q^83 - 20 * q^85 - 6 * q^89 + 37 * q^91 + 12 * q^95 - 29 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$ .

$n$	$757$	$785$	$1135$	$2593$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	2.52290			1.12828			0.564139	−	0.825680i	$-0.309208\pi$
$5$	2.52290			1.12828			0.564139	+	0.825680i	$0.309208\pi$
$6$		0			0
$7$	1.07705	−	2.41660i	0.407086	−	0.913390i
$7$	1.07705	−	2.41660i	0.407086	−	0.913390i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−5.71296			−1.72252			−0.861262	−	0.508161i	$-0.830325\pi$
$11$	−5.71296			−1.72252			−0.861262	+	0.508161i	$0.830325\pi$
$12$		0			0
$13$	−2.45245	+	4.24777i	−0.680188	+	1.17812i	0.294735	+	0.955579i	$0.404769\pi$
$13$	−2.45245	+	4.24777i	−0.680188	+	1.17812i	−0.974923	+	0.222541i	$0.928565\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.49483	+	4.32118i	−0.605086	+	1.04804i	0.386952	+	0.922100i	$0.373528\pi$
$17$	−2.49483	+	4.32118i	−0.605086	+	1.04804i	−0.992038	+	0.125939i	$0.959806\pi$
$18$		0			0
$19$	0.00383929	+	0.00664984i	0.000880793	+	0.00152558i	0.866465	−	0.499237i	$-0.166386\pi$
$19$	0.00383929	+	0.00664984i	0.000880793	+	0.00152558i	−0.865585	+	0.500763i	$0.833053\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.667754			0.139236			0.0696181	−	0.997574i	$-0.477822\pi$
$23$	0.667754			0.139236			0.0696181	+	0.997574i	$0.477822\pi$
$24$		0			0
$25$	1.36505			0.273010
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.85082	−	6.66981i	−0.715079	−	1.23855i	−0.962929	−	0.269754i	$-0.913058\pi$
$29$	−3.85082	−	6.66981i	−0.715079	−	1.23855i	0.247851	−	0.968798i	$-0.420276\pi$
$30$		0			0
$31$	−3.88302	−	6.72560i	−0.697412	−	1.20795i	−0.969361	−	0.245641i	$-0.921002\pi$
$31$	−3.88302	−	6.72560i	−0.697412	−	1.20795i	0.271949	−	0.962312i	$-0.412332\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.71729	−	6.09686i	0.459306	−	1.03056i
$36$		0			0
$37$	−3.19562	−	5.53498i	−0.525357	−	0.909946i	−0.999564	−	0.0295319i	$-0.990598\pi$
$37$	−3.19562	−	5.53498i	−0.525357	−	0.909946i	0.474207	−	0.880414i	$-0.342735\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−5.21159	+	9.02673i	−0.813913	+	1.40974i	0.0961931	+	0.995363i	$0.469333\pi$
$41$	−5.21159	+	9.02673i	−0.813913	+	1.40974i	−0.910106	+	0.414376i	$0.864000\pi$
$42$		0			0
$43$	−4.42935	−	7.67185i	−0.675469	−	1.16995i	−0.976332	−	0.216279i	$-0.930608\pi$
$43$	−4.42935	−	7.67185i	−0.675469	−	1.16995i	0.300863	−	0.953668i	$-0.402725\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.08052	+	1.87152i	−0.157610	+	0.272989i	−0.934006	−	0.357256i	$-0.883712\pi$
$47$	−1.08052	+	1.87152i	−0.157610	+	0.272989i	0.776396	+	0.630245i	$0.217046\pi$
$48$		0			0
$49$	−4.67994	−	5.20559i	−0.668562	−	0.743656i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.69858	−	6.40613i	0.508039	−	0.879950i	−0.491917	−	0.870642i	$-0.663704\pi$
$53$	3.69858	−	6.40613i	0.508039	−	0.879950i	0.999957	−	0.00930815i	$-0.00296292\pi$
$54$		0			0
$55$	−14.4133			−1.94348
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.261797	+	0.453446i	0.0340831	+	0.0590336i	0.882564	−	0.470193i	$-0.155816\pi$
$59$	0.261797	+	0.453446i	0.0340831	+	0.0590336i	−0.848481	+	0.529226i	$0.822482\pi$
$60$		0			0
$61$	4.49541	−	7.78628i	0.575578	−	0.996931i	−0.420400	−	0.907339i	$-0.638110\pi$
$61$	4.49541	−	7.78628i	0.575578	−	0.996931i	0.995979	−	0.0895919i	$-0.0285563\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−6.18730	+	10.7167i	−0.767441	+	1.32925i
$66$		0			0
$67$	−2.54791	−	4.41311i	−0.311277	−	0.539147i	0.667362	−	0.744733i	$-0.267423\pi$
$67$	−2.54791	−	4.41311i	−0.311277	−	0.539147i	−0.978639	+	0.205586i	$0.934090\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−5.68471			−0.674651			−0.337325	−	0.941388i	$-0.609522\pi$
$71$	−5.68471			−0.674651			−0.337325	+	0.941388i	$0.609522\pi$
$72$		0			0
$73$	−1.52062	+	2.63379i	−0.177975	+	0.308262i	−0.941187	−	0.337887i	$-0.890288\pi$
$73$	−1.52062	+	2.63379i	−0.177975	+	0.308262i	0.763212	+	0.646148i	$0.223621\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6.15314	+	13.8060i	−0.701215	+	1.57334i
$78$		0			0
$79$	3.08115	−	5.33671i	0.346657	−	0.600427i	−0.638997	−	0.769209i	$-0.720650\pi$
$79$	3.08115	−	5.33671i	0.346657	−	0.600427i	0.985653	+	0.168783i	$0.0539836\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−0.258726	−	0.448126i	−0.0283988	−	0.0491882i	0.851477	−	0.524392i	$-0.175708\pi$
$83$	−0.258726	−	0.448126i	−0.0283988	−	0.0491882i	−0.879876	+	0.475204i	$0.842374\pi$
$84$		0			0
$85$	−6.29422	+	10.9019i	−0.682704	+	1.18248i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.19093	−	2.06274i	−0.126238	−	0.218650i	0.795978	−	0.605325i	$-0.206957\pi$
$89$	−1.19093	−	2.06274i	−0.126238	−	0.218650i	−0.922216	+	0.386675i	$0.873624\pi$
$90$		0			0
$91$	7.62377	+	10.5017i	0.799188	+	1.10087i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.00968615	+	0.0167769i	0.000993778	+	0.00172127i
$96$		0			0
$97$	4.32994	+	7.49968i	0.439639	+	0.761477i	0.997662	−	0.0683485i	$-0.0217730\pi$
$97$	4.32994	+	7.49968i	0.439639	+	0.761477i	−0.558022	+	0.829826i	$0.688440\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	9.33566			0.928933			0.464466	−	0.885591i	$-0.346246\pi$
$101$	9.33566			0.928933			0.464466	+	0.885591i	$0.346246\pi$
$102$		0			0
$103$	−16.2185			−1.59806			−0.799029	−	0.601293i	$-0.794653\pi$
$103$	−16.2185			−1.59806			−0.799029	+	0.601293i	$0.794653\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4.50171	+	7.79718i	0.435196	+	0.753782i	0.997312	−	0.0732767i	$-0.0233456\pi$
$107$	4.50171	+	7.79718i	0.435196	+	0.753782i	−0.562115	+	0.827059i	$0.690012\pi$
$108$		0			0
$109$	3.71563	−	6.43566i	0.355893	−	0.616424i	−0.631378	−	0.775476i	$-0.717510\pi$
$109$	3.71563	−	6.43566i	0.355893	−	0.616424i	0.987270	+	0.159051i	$0.0508435\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−7.14642	+	12.3780i	−0.672278	+	1.16442i	0.304978	+	0.952359i	$0.401351\pi$
$113$	−7.14642	+	12.3780i	−0.672278	+	1.16442i	−0.977256	+	0.212061i	$0.931982\pi$
$114$		0			0
$115$	1.68468			0.157097
$116$		0			0
$117$		0			0
$118$		0			0
$119$	7.75551	+	10.6831i	0.710947	+	0.979321i
$120$		0			0
$121$	21.6380			1.96709
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.17064			−0.820247
$126$		0			0
$127$	1.96011			0.173932			0.0869660	−	0.996211i	$-0.472283\pi$
$127$	1.96011			0.173932			0.0869660	+	0.996211i	$0.472283\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.98825			−0.348455			−0.174227	−	0.984705i	$-0.555743\pi$
$131$	−3.98825			−0.348455			−0.174227	+	0.984705i	$0.555743\pi$
$132$		0			0
$133$	0.0202051	−	0.00211583i	0.00175201	−	0.000183466i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	7.40843			0.632945			0.316473	−	0.948602i	$-0.397501\pi$
$137$	7.40843			0.632945			0.316473	+	0.948602i	$0.397501\pi$
$138$		0			0
$139$	−6.92660	+	11.9972i	−0.587507	+	1.01759i	0.407051	+	0.913405i	$0.366557\pi$
$139$	−6.92660	+	11.9972i	−0.587507	+	1.01759i	−0.994558	+	0.104186i	$0.966776\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	14.0108	−	24.2674i	1.17164	−	2.02934i
$144$		0			0
$145$	−9.71524	−	16.8273i	−0.806807	−	1.39743i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	14.1040			1.15545			0.577724	−	0.816232i	$-0.303941\pi$
$149$	14.1040			1.15545			0.577724	+	0.816232i	$0.303941\pi$
$150$		0			0
$151$	10.6005			0.862660			0.431330	−	0.902194i	$-0.358044\pi$
$151$	10.6005			0.862660			0.431330	+	0.902194i	$0.358044\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−9.79650	−	16.9680i	−0.786874	−	1.36291i
$156$		0			0
$157$	0.129779	+	0.224784i	0.0103575	+	0.0179397i	0.871158	−	0.491003i	$-0.163370\pi$
$157$	0.129779	+	0.224784i	0.0103575	+	0.0179397i	−0.860800	+	0.508943i	$0.830036\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	0.719203	−	1.61370i	0.0566811	−	0.127177i
$162$		0			0
$163$	−6.31882	−	10.9445i	−0.494928	−	0.857241i	0.505055	−	0.863087i	$-0.331472\pi$
$163$	−6.31882	−	10.9445i	−0.494928	−	0.857241i	−0.999983	+	0.00584647i	$0.998139\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−5.74959	+	9.95859i	−0.444917	+	0.770619i	−0.998046	−	0.0624765i	$-0.980100\pi$
$167$	−5.74959	+	9.95859i	−0.444917	+	0.770619i	0.553129	+	0.833095i	$0.313433\pi$
$168$		0			0
$169$	−5.52905	−	9.57659i	−0.425311	−	0.736661i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−7.90471	+	13.6914i	−0.600984	+	1.04094i	0.391688	+	0.920098i	$0.371891\pi$
$173$	−7.90471	+	13.6914i	−0.600984	+	1.04094i	−0.992672	+	0.120837i	$0.961442\pi$
$174$		0			0
$175$	1.47022	−	3.29878i	0.111138	−	0.249364i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−8.49849	+	14.7198i	−0.635207	+	1.10021i	0.351265	+	0.936276i	$0.385752\pi$
$179$	−8.49849	+	14.7198i	−0.635207	+	1.10021i	−0.986471	+	0.163934i	$0.947582\pi$
$180$		0			0
$181$	6.35841			0.472617			0.236308	−	0.971678i	$-0.424062\pi$
$181$	6.35841			0.472617			0.236308	+	0.971678i	$0.424062\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−8.06225	−	13.9642i	−0.592749	−	1.02667i
$186$		0			0
$187$	14.2529	−	24.6867i	1.04227	−	1.80527i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2.07047	−	3.58616i	0.149814	−	0.259485i	−0.781345	−	0.624100i	$-0.785466\pi$
$191$	2.07047	−	3.58616i	0.149814	−	0.259485i	0.931159	+	0.364614i	$0.118799\pi$
$192$		0			0
$193$	3.84793	+	6.66481i	0.276980	+	0.479743i	0.970633	−	0.240566i	$-0.0773331\pi$
$193$	3.84793	+	6.66481i	0.276980	+	0.479743i	−0.693653	+	0.720310i	$0.744000\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−3.29508			−0.234765			−0.117383	−	0.993087i	$-0.537450\pi$
$197$	−3.29508			−0.234765			−0.117383	+	0.993087i	$0.537450\pi$
$198$		0			0
$199$	−8.08840	+	14.0095i	−0.573371	+	0.993108i	0.422845	+	0.906202i	$0.361031\pi$
$199$	−8.08840	+	14.0095i	−0.573371	+	0.993108i	−0.996216	+	0.0869063i	$0.972302\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−20.2658	+	2.12219i	−1.42238	+	0.148948i
$204$		0			0
$205$	−13.1483	+	22.7736i	−0.918319	+	1.59058i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.0219337	−	0.0379903i	−0.00151719	−	0.00262784i
$210$		0			0
$211$	13.9633	−	24.1851i	0.961273	−	1.66497i	0.241961	−	0.970286i	$-0.422209\pi$
$211$	13.9633	−	24.1851i	0.961273	−	1.66497i	0.719312	−	0.694687i	$-0.244457\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−11.1748	−	19.3554i	−0.762116	−	1.32002i
$216$		0			0
$217$	−20.4353	+	2.13994i	−1.38724	+	0.145268i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−12.2369	−	21.1950i	−0.823144	−	1.42573i
$222$		0			0
$223$	10.1652	+	17.6066i	0.680711	+	1.17903i	0.974764	+	0.223237i	$0.0716623\pi$
$223$	10.1652	+	17.6066i	0.680711	+	1.17903i	−0.294054	+	0.955789i	$0.595004\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−5.68939			−0.377618			−0.188809	−	0.982014i	$-0.560463\pi$
$227$	−5.68939			−0.377618			−0.188809	+	0.982014i	$0.560463\pi$
$228$		0			0
$229$	14.8542			0.981590			0.490795	−	0.871275i	$-0.336706\pi$
$229$	14.8542			0.981590			0.490795	+	0.871275i	$0.336706\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6.70652	−	11.6160i	−0.439358	−	0.760991i	0.558282	−	0.829652i	$-0.311461\pi$
$233$	−6.70652	−	11.6160i	−0.439358	−	0.760991i	−0.997640	+	0.0686603i	$0.978128\pi$
$234$		0			0
$235$	−2.72605	+	4.72166i	−0.177828	+	0.308007i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−9.33123	+	16.1622i	−0.603587	+	1.04544i	0.388686	+	0.921370i	$0.372929\pi$
$239$	−9.33123	+	16.1622i	−0.603587	+	1.04544i	−0.992273	+	0.124073i	$0.960404\pi$
$240$		0			0
$241$	−21.4160			−1.37952			−0.689762	−	0.724036i	$-0.742285\pi$
$241$	−21.4160			−1.37952			−0.689762	+	0.724036i	$0.742285\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−11.8070	−	13.1332i	−0.754324	−	0.839050i
$246$		0			0
$247$	−0.0376627			−0.00239642
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−0.462898			−0.0292179			−0.0146089	−	0.999893i	$-0.504650\pi$
$251$	−0.462898			−0.0292179			−0.0146089	+	0.999893i	$0.504650\pi$
$252$		0			0
$253$	−3.81485			−0.239838
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−0.802512			−0.0500593			−0.0250297	−	0.999687i	$-0.507968\pi$
$257$	−0.802512			−0.0500593			−0.0250297	+	0.999687i	$0.507968\pi$
$258$		0			0
$259$	−16.8177	+	1.76111i	−1.04500	+	0.109430i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	0.201387			0.0124180			0.00620902	−	0.999981i	$-0.498024\pi$
$263$	0.201387			0.0124180			0.00620902	+	0.999981i	$0.498024\pi$
$264$		0			0
$265$	9.33117	−	16.1621i	0.573209	−	0.992828i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	11.1773	−	19.3596i	0.681490	−	1.18038i	−0.293036	−	0.956101i	$-0.594666\pi$
$269$	11.1773	−	19.3596i	0.681490	−	1.18038i	0.974526	−	0.224274i	$-0.0720011\pi$
$270$		0			0
$271$	1.78925	+	3.09907i	0.108689	+	0.188255i	0.915240	−	0.402910i	$-0.132001\pi$
$271$	1.78925	+	3.09907i	0.108689	+	0.188255i	−0.806550	+	0.591166i	$0.798668\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−7.79847			−0.470266
$276$		0			0
$277$	−10.1067			−0.607254			−0.303627	−	0.952791i	$-0.598198\pi$
$277$	−10.1067			−0.607254			−0.303627	+	0.952791i	$0.598198\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−12.7114	−	22.0167i	−0.758296	−	1.31341i	−0.943719	−	0.330749i	$-0.892699\pi$
$281$	−12.7114	−	22.0167i	−0.758296	−	1.31341i	0.185422	−	0.982659i	$-0.440635\pi$
$282$		0			0
$283$	1.93833	+	3.35728i	0.115222	+	0.199570i	0.917868	−	0.396885i	$-0.129909\pi$
$283$	1.93833	+	3.35728i	0.115222	+	0.199570i	−0.802647	+	0.596455i	$0.796575\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	16.2009	+	22.3166i	0.956308	+	1.31730i
$288$		0			0
$289$	−3.94838	−	6.83879i	−0.232257	−	0.402282i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.428834	+	0.742762i	−0.0250527	+	0.0433926i	−0.878280	−	0.478147i	$-0.841309\pi$
$293$	−0.428834	+	0.742762i	−0.0250527	+	0.0433926i	0.853227	+	0.521539i	$0.174642\pi$
$294$		0			0
$295$	0.660489	+	1.14400i	0.0384551	+	0.0666063i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.63763	+	2.83647i	−0.0947068	+	0.164037i
$300$		0			0
$301$	−23.3104	+	2.44102i	−1.34359	+	0.140698i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	11.3415	−	19.6440i	0.649412	−	1.12481i
$306$		0			0
$307$	0.717950			0.0409756			0.0204878	−	0.999790i	$-0.493478\pi$
$307$	0.717950			0.0409756			0.0204878	+	0.999790i	$0.493478\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	4.72606	+	8.18578i	0.267990	+	0.464173i	0.968343	−	0.249624i	$-0.0803071\pi$
$311$	4.72606	+	8.18578i	0.267990	+	0.464173i	−0.700352	+	0.713797i	$0.746974\pi$
$312$		0			0
$313$	11.6317	−	20.1467i	0.657464	−	1.13876i	−0.323806	−	0.946124i	$-0.604962\pi$
$313$	11.6317	−	20.1467i	0.657464	−	1.13876i	0.981270	−	0.192638i	$-0.0617043\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6.61771	−	11.4622i	0.371687	−	0.643781i	−0.618138	−	0.786070i	$-0.712113\pi$
$317$	6.61771	−	11.4622i	0.371687	−	0.643781i	0.989825	+	0.142288i	$0.0454460\pi$
$318$		0			0
$319$	21.9996	+	38.1044i	1.23174	+	2.13344i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−0.0383135			−0.00213182
$324$		0			0
$325$	−3.34772	+	5.79841i	−0.185698	+	0.321638i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3.35894	+	4.62690i	0.185184	+	0.255089i
$330$		0			0
$331$	−15.2165	+	26.3558i	−0.836375	+	1.44864i	0.0565316	+	0.998401i	$0.481996\pi$
$331$	−15.2165	+	26.3558i	−0.836375	+	1.44864i	−0.892906	+	0.450243i	$0.851338\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−6.42813	−	11.1339i	−0.351206	−	0.608307i
$336$		0			0
$337$	−0.767420	+	1.32921i	−0.0418041	+	0.0724067i	−0.886170	−	0.463360i	$-0.846644\pi$
$337$	−0.767420	+	1.32921i	−0.0418041	+	0.0724067i	0.844366	+	0.535766i	$0.179977\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	22.1836	+	38.4231i	1.20131	+	2.08073i
$342$		0			0
$343$	−17.6204	+	5.70287i	−0.951410	+	0.307926i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	14.3036	+	24.7745i	0.767856	+	1.32997i	0.938723	+	0.344672i	$0.112010\pi$
$347$	14.3036	+	24.7745i	0.767856	+	1.32997i	−0.170867	+	0.985294i	$0.554657\pi$
$348$		0			0
$349$	−9.05123	−	15.6772i	−0.484501	−	0.839181i	0.515340	−	0.856986i	$-0.327666\pi$
$349$	−9.05123	−	15.6772i	−0.484501	−	0.839181i	−0.999841	+	0.0178047i	$0.994332\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−14.5908			−0.776591			−0.388295	−	0.921535i	$-0.626936\pi$
$353$	−14.5908			−0.776591			−0.388295	+	0.921535i	$0.626936\pi$
$354$		0			0
$355$	−14.3420			−0.761193
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.05831	−	1.83304i	−0.0558554	−	0.0967443i	0.836746	−	0.547592i	$-0.184455\pi$
$359$	−1.05831	−	1.83304i	−0.0558554	−	0.0967443i	−0.892601	+	0.450847i	$0.851122\pi$
$360$		0			0
$361$	9.49997	−	16.4544i	0.499998	−	0.866023i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3.83638	+	6.64480i	−0.200805	+	0.347805i
$366$		0			0
$367$	−6.66209			−0.347758			−0.173879	−	0.984767i	$-0.555630\pi$
$367$	−6.66209			−0.347758			−0.173879	+	0.984767i	$0.555630\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−11.4975	−	15.8377i	−0.596922	−	0.822253i
$372$		0			0
$373$	−12.4983			−0.647138			−0.323569	−	0.946205i	$-0.604883\pi$
$373$	−12.4983			−0.647138			−0.323569	+	0.946205i	$0.604883\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	37.7758			1.94555
$378$		0			0
$379$	19.5504			1.00423			0.502117	−	0.864800i	$-0.332555\pi$
$379$	19.5504			1.00423			0.502117	+	0.864800i	$0.332555\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2.67480			−0.136676			−0.0683379	−	0.997662i	$-0.521770\pi$
$383$	−2.67480			−0.136676			−0.0683379	+	0.997662i	$0.521770\pi$
$384$		0			0
$385$	−15.5238	+	34.8311i	−0.791165	+	1.77516i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.96310			0.200937			0.100469	−	0.994940i	$-0.467966\pi$
$389$	3.96310			0.200937			0.100469	+	0.994940i	$0.467966\pi$
$390$		0			0
$391$	−1.66593	+	2.88548i	−0.0842499	+	0.145925i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	7.77345	−	13.4640i	0.391125	−	0.677448i
$396$		0			0
$397$	10.2978	+	17.8362i	0.516829	+	0.895175i	0.999809	+	0.0195431i	$0.00622114\pi$
$397$	10.2978	+	17.8362i	0.516829	+	0.895175i	−0.482980	+	0.875632i	$0.660446\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3.83957			0.191739			0.0958696	−	0.995394i	$-0.469437\pi$
$401$	3.83957			0.191739			0.0958696	+	0.995394i	$0.469437\pi$
$402$		0			0
$403$	38.0917			1.89748
$404$		0			0
$405$		0			0
$406$		0			0
$407$	18.2565	+	31.6212i	0.904940	+	1.56740i
$408$		0			0
$409$	14.7113	+	25.4808i	0.727428	+	1.25994i	0.957967	+	0.286880i	$0.0926180\pi$
$409$	14.7113	+	25.4808i	0.727428	+	1.25994i	−0.230538	+	0.973063i	$0.574049\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1.37777	−	0.144276i	0.0677954	−	0.00709938i
$414$		0			0
$415$	−0.652741	−	1.13058i	−0.0320418	−	0.0554980i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4.40821	−	7.63525i	0.215355	−	0.373006i	−0.738027	−	0.674771i	$-0.764242\pi$
$419$	4.40821	−	7.63525i	0.215355	−	0.373006i	0.953382	+	0.301765i	$0.0975757\pi$
$420$		0			0
$421$	−17.6437	−	30.5597i	−0.859899	−	1.48939i	−0.872024	−	0.489462i	$-0.837193\pi$
$421$	−17.6437	−	30.5597i	−0.859899	−	1.48939i	0.0121255	−	0.999926i	$-0.496140\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.40557	+	5.89861i	−0.165194	+	0.286125i
$426$		0			0
$427$	−13.9746	−	19.2498i	−0.676277	−	0.931564i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−12.8099	+	22.1873i	−0.617030	+	1.06873i	0.372995	+	0.927833i	$0.378331\pi$
$431$	−12.8099	+	22.1873i	−0.617030	+	1.06873i	−0.990025	+	0.140893i	$0.955003\pi$
$432$		0			0
$433$	16.8556			0.810030			0.405015	−	0.914310i	$-0.367266\pi$
$433$	16.8556			0.810030			0.405015	+	0.914310i	$0.367266\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.00256370	+	0.00444045i	0.000122638	+	0.000212416i
$438$		0			0
$439$	15.4596	−	26.7768i	0.737846	−	1.27799i	−0.215618	−	0.976478i	$-0.569176\pi$
$439$	15.4596	−	26.7768i	0.737846	−	1.27799i	0.953463	−	0.301509i	$-0.0974902\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4.65544	−	8.06345i	0.221186	−	0.383106i	−0.733982	−	0.679169i	$-0.762340\pi$
$443$	4.65544	−	8.06345i	0.221186	−	0.383106i	0.955169	+	0.296063i	$0.0956737\pi$
$444$		0			0
$445$	−3.00459	−	5.20410i	−0.142431	−	0.246698i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−23.8055			−1.12345			−0.561724	−	0.827324i	$-0.689862\pi$
$449$	−23.8055			−1.12345			−0.561724	+	0.827324i	$0.689862\pi$
$450$		0			0
$451$	29.7736	−	51.5694i	1.40198	−	2.42831i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	19.2340	+	26.4947i	0.901706	+	1.24209i
$456$		0			0
$457$	6.90552	−	11.9607i	0.323027	−	0.559498i	−0.658084	−	0.752944i	$-0.728633\pi$
$457$	6.90552	−	11.9607i	0.323027	−	0.559498i	0.981111	+	0.193446i	$0.0619663\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	0.00256407	+	0.00444110i	0.000119421	+	0.000206843i	0.866085	−	0.499897i	$-0.166629\pi$
$461$	0.00256407	+	0.00444110i	0.000119421	+	0.000206843i	−0.865966	+	0.500103i	$0.833295\pi$
$462$		0			0
$463$	−12.9682	+	22.4616i	−0.602685	+	1.04388i	0.389728	+	0.920930i	$0.372569\pi$
$463$	−12.9682	+	22.4616i	−0.602685	+	1.04388i	−0.992413	+	0.122951i	$0.960764\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−12.0484	−	20.8684i	−0.557532	−	0.965673i	−0.997702	−	0.0677588i	$-0.978415\pi$
$467$	−12.0484	−	20.8684i	−0.557532	−	0.965673i	0.440170	−	0.897914i	$-0.354918\pi$
$468$		0			0
$469$	−13.4090	+	1.40416i	−0.619168	+	0.0648379i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	25.3047	+	43.8290i	1.16351	+	2.01526i
$474$		0			0
$475$	0.00524081	+	0.00907735i	0.000240465	+	0.000416497i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	14.7823			0.675420			0.337710	−	0.941250i	$-0.390348\pi$
$479$	14.7823			0.675420			0.337710	+	0.941250i	$0.390348\pi$
$480$		0			0
$481$	31.3485			1.42937
$482$		0			0
$483$		0			0
$484$		0			0
$485$	10.9240	+	18.9210i	0.496035	+	0.859158i
$486$		0			0
$487$	−9.38360	+	16.2529i	−0.425211	+	0.736488i	−0.996440	−	0.0843033i	$-0.973134\pi$
$487$	−9.38360	+	16.2529i	−0.425211	+	0.736488i	0.571229	+	0.820791i	$0.306467\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	18.2871	−	31.6741i	0.825284	−	1.42943i	−0.0764182	−	0.997076i	$-0.524348\pi$
$491$	18.2871	−	31.6741i	0.825284	−	1.42943i	0.901702	−	0.432358i	$-0.142318\pi$
$492$		0			0
$493$	38.4286			1.73074
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.12270	+	13.7377i	−0.274641	+	0.616219i
$498$		0			0
$499$	−4.63182			−0.207349			−0.103674	−	0.994611i	$-0.533060\pi$
$499$	−4.63182			−0.207349			−0.103674	+	0.994611i	$0.533060\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	16.4143			0.731879			0.365940	−	0.930639i	$-0.380748\pi$
$503$	16.4143			0.731879			0.365940	+	0.930639i	$0.380748\pi$
$504$		0			0
$505$	23.5530			1.04809
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−10.4834			−0.464668			−0.232334	−	0.972636i	$-0.574636\pi$
$509$	−10.4834			−0.464668			−0.232334	+	0.972636i	$0.574636\pi$
$510$		0			0
$511$	4.72705	+	6.51145i	0.209112	+	0.288050i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−40.9178			−1.80305
$516$		0			0
$517$	6.17298	−	10.6919i	0.271487	−	0.470230i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	11.0087	−	19.0675i	0.482298	−	0.835364i	−0.517496	−	0.855686i	$-0.673136\pi$
$521$	11.0087	−	19.0675i	0.482298	−	0.835364i	0.999793	+	0.0203215i	$0.00646899\pi$
$522$		0			0
$523$	1.18541	+	2.05320i	0.0518346	+	0.0897801i	0.890778	−	0.454438i	$-0.150160\pi$
$523$	1.18541	+	2.05320i	0.0518346	+	0.0897801i	−0.838944	+	0.544218i	$0.816826\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	38.7500			1.68798
$528$		0			0
$529$	−22.5541			−0.980613
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−25.5623	−	44.2753i	−1.10723	−	1.91777i
$534$		0			0
$535$	11.3574	+	19.6715i	0.491022	+	0.850475i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	26.7363	+	29.7394i	1.15161	+	1.28097i
$540$		0			0
$541$	6.65209	+	11.5218i	0.285996	+	0.495359i	0.972850	−	0.231436i	$-0.0743423\pi$
$541$	6.65209	+	11.5218i	0.285996	+	0.495359i	−0.686854	+	0.726795i	$0.741009\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	9.37418	−	16.2366i	0.401546	−	0.695498i
$546$		0			0
$547$	2.43685	+	4.22074i	0.104192	+	0.180466i	0.913408	−	0.407046i	$-0.133441\pi$
$547$	2.43685	+	4.22074i	0.104192	+	0.180466i	−0.809216	+	0.587512i	$0.800108\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	0.0295688	−	0.0512146i	0.00125967	−	0.00218182i
$552$		0			0
$553$	−9.57816	−	13.1938i	−0.407305	−	0.561058i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−7.09601	+	12.2907i	−0.300668	+	0.520772i	−0.976287	−	0.216479i	$-0.930543\pi$
$557$	−7.09601	+	12.2907i	−0.300668	+	0.520772i	0.675620	+	0.737250i	$0.263876\pi$
$558$		0			0
$559$	43.4511			1.83778
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3.51985	−	6.09657i	−0.148344	−	0.256940i	0.782271	−	0.622938i	$-0.214061\pi$
$563$	−3.51985	−	6.09657i	−0.148344	−	0.256940i	−0.930616	+	0.365998i	$0.880728\pi$
$564$		0			0
$565$	−18.0297	+	31.2284i	−0.758516	+	1.31379i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−9.15081	+	15.8497i	−0.383622	+	0.664453i	−0.991577	−	0.129519i	$-0.958657\pi$
$569$	−9.15081	+	15.8497i	−0.383622	+	0.664453i	0.607955	+	0.793972i	$0.291990\pi$
$570$		0			0
$571$	−15.2192	−	26.3604i	−0.636902	−	1.10315i	−0.986109	−	0.166102i	$-0.946882\pi$
$571$	−15.2192	−	26.3604i	−0.636902	−	1.10315i	0.349206	−	0.937046i	$-0.386451\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.911516			0.0380128
$576$		0			0
$577$	5.65385	−	9.79275i	0.235373	−	0.407678i	−0.724008	−	0.689791i	$-0.757702\pi$
$577$	5.65385	−	9.79275i	0.235373	−	0.407678i	0.959381	+	0.282114i	$0.0910356\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1.36160	+	0.142584i	−0.0564888	+	0.00591538i
$582$		0			0
$583$	−21.1299	+	36.5980i	−0.875110	+	1.51573i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−9.89755	−	17.1431i	−0.408516	−	0.707570i	0.586208	−	0.810161i	$-0.300620\pi$
$587$	−9.89755	−	17.1431i	−0.408516	−	0.707570i	−0.994724	+	0.102591i	$0.967287\pi$
$588$		0			0
$589$	0.0298161	−	0.0516430i	0.00122855	−	0.00212791i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−2.69067	−	4.66038i	−0.110493	−	0.191379i	0.805476	−	0.592628i	$-0.201910\pi$
$593$	−2.69067	−	4.66038i	−0.110493	−	0.191379i	−0.915969	+	0.401249i	$0.868576\pi$
$594$		0			0
$595$	19.5664	+	26.9525i	0.802145	+	1.10495i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1.12979	−	1.95686i	−0.0461622	−	0.0799552i	0.842021	−	0.539445i	$-0.181366\pi$
$599$	−1.12979	−	1.95686i	−0.0461622	−	0.0799552i	−0.888183	+	0.459489i	$0.848032\pi$
$600$		0			0
$601$	18.1873	+	31.5013i	0.741875	+	1.28496i	0.951641	+	0.307213i	$0.0993964\pi$
$601$	18.1873	+	31.5013i	0.741875	+	1.28496i	−0.209766	+	0.977752i	$0.567270\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	54.5905			2.21942
$606$		0			0
$607$	−16.2161			−0.658190			−0.329095	−	0.944297i	$-0.606744\pi$
$607$	−16.2161			−0.658190			−0.329095	+	0.944297i	$0.606744\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−5.29985	−	9.17961i	−0.214409	−	0.371367i
$612$		0			0
$613$	21.6357	−	37.4741i	0.873857	−	1.51357i	0.0158822	−	0.999874i	$-0.494944\pi$
$613$	21.6357	−	37.4741i	0.873857	−	1.51357i	0.857975	−	0.513691i	$-0.171722\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	5.92248	−	10.2580i	0.238430	−	0.412973i	−0.721834	−	0.692066i	$-0.756701\pi$
$617$	5.92248	−	10.2580i	0.238430	−	0.412973i	0.960264	+	0.279093i	$0.0900339\pi$
$618$		0			0
$619$	−40.3288			−1.62095			−0.810475	−	0.585773i	$-0.800791\pi$
$619$	−40.3288			−1.62095			−0.810475	+	0.585773i	$0.800791\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−6.26751	+	0.656320i	−0.251103	+	0.0262949i
$624$		0			0
$625$	−29.9619			−1.19848
$626$		0			0
$627$		0			0
$628$		0			0
$629$	31.8902			1.27154
$630$		0			0
$631$	−13.9489			−0.555298			−0.277649	−	0.960683i	$-0.589555\pi$
$631$	−13.9489			−0.555298			−0.277649	+	0.960683i	$0.589555\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4.94518			0.196243
$636$		0			0
$637$	33.5895	−	7.11283i	1.33086	−	0.281821i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−17.5395			−0.692768			−0.346384	−	0.938093i	$-0.612591\pi$
$641$	−17.5395			−0.692768			−0.346384	+	0.938093i	$0.612591\pi$
$642$		0			0
$643$	13.5329	−	23.4397i	0.533686	−	0.924371i	−0.465540	−	0.885027i	$-0.654140\pi$
$643$	13.5329	−	23.4397i	0.533686	−	0.924371i	0.999226	−	0.0393443i	$-0.0125269\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	11.3252	−	19.6159i	0.445240	−	0.771179i	−0.552828	−	0.833295i	$-0.686452\pi$
$647$	11.3252	−	19.6159i	0.445240	−	0.771179i	0.998069	+	0.0621160i	$0.0197849\pi$
$648$		0			0
$649$	−1.49564	−	2.59052i	−0.0587089	−	0.101687i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	0.784108			0.0306845			0.0153423	−	0.999882i	$-0.495116\pi$
$653$	0.784108			0.0306845			0.0153423	+	0.999882i	$0.495116\pi$
$654$		0			0
$655$	−10.0620			−0.393154
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−16.7219	−	28.9632i	−0.651392	−	1.12824i	−0.982785	−	0.184752i	$-0.940852\pi$
$659$	−16.7219	−	28.9632i	−0.651392	−	1.12824i	0.331393	−	0.943493i	$-0.392481\pi$
$660$		0			0
$661$	−1.53258	−	2.65450i	−0.0596104	−	0.103248i	0.834680	−	0.550735i	$-0.185652\pi$
$661$	−1.53258	−	2.65450i	−0.0596104	−	0.103248i	−0.894291	+	0.447487i	$0.852319\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.0509756	−	0.00533805i	0.00197675	−	0.000207001i
$666$		0			0
$667$	−2.57140	−	4.45379i	−0.0995649	−	0.172451i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−25.6821	+	44.4827i	−0.991447	+	1.71724i
$672$		0			0
$673$	14.4618	+	25.0487i	0.557463	+	0.965555i	0.997707	+	0.0676766i	$0.0215586\pi$
$673$	14.4618	+	25.0487i	0.557463	+	0.965555i	−0.440244	+	0.897878i	$0.645108\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−5.85818	+	10.1467i	−0.225148	+	0.389968i	−0.956364	−	0.292178i	$-0.905620\pi$
$677$	−5.85818	+	10.1467i	−0.225148	+	0.389968i	0.731216	+	0.682146i	$0.238953\pi$
$678$		0			0
$679$	22.7873	−	2.38624i	0.874497	−	0.0915753i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	20.7190	−	35.8864i	0.792791	−	1.37315i	−0.131441	−	0.991324i	$-0.541960\pi$
$683$	20.7190	−	35.8864i	0.792791	−	1.37315i	0.924232	−	0.381831i	$-0.124706\pi$
$684$		0			0
$685$	18.6908			0.714138
$686$		0			0
$687$		0			0
$688$		0			0
$689$	18.1412	+	31.4215i	0.691125	+	1.19706i
$690$		0			0
$691$	−3.45675	+	5.98727i	−0.131501	+	0.227766i	−0.924255	−	0.381775i	$-0.875313\pi$
$691$	−3.45675	+	5.98727i	−0.131501	+	0.227766i	0.792754	+	0.609541i	$0.208646\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−17.4752	+	30.2679i	−0.662871	+	1.14813i
$696$		0			0
$697$	−26.0041	−	45.0404i	−0.984974	−	1.70603i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−39.1954			−1.48039			−0.740195	−	0.672392i	$-0.765267\pi$
$701$	−39.1954			−1.48039			−0.740195	+	0.672392i	$0.765267\pi$
$702$		0			0
$703$	0.0245378	−	0.0425008i	0.000925462	−	0.00160295i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	10.0550	−	22.5606i	0.378155	−	0.848478i
$708$		0			0
$709$	−10.2436	+	17.7424i	−0.384705	+	0.666328i	−0.991728	−	0.128356i	$-0.959030\pi$
$709$	−10.2436	+	17.7424i	−0.384705	+	0.666328i	0.607023	+	0.794684i	$0.292363\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−2.59290	−	4.49104i	−0.0971050	−	0.168191i
$714$		0			0
$715$	35.3479	−	61.2243i	1.32193	−	2.28966i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	17.1300	+	29.6700i	0.638840	+	1.10650i	0.985688	+	0.168582i	$0.0539187\pi$
$719$	17.1300	+	29.6700i	0.638840	+	1.10650i	−0.346848	+	0.937921i	$0.612748\pi$
$720$		0			0
$721$	−17.4681	+	39.1937i	−0.650547	+	1.45965i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−5.25655	−	9.10461i	−0.195223	−	0.338137i
$726$		0			0
$727$	−7.18914	−	12.4520i	−0.266631	−	0.461818i	0.701359	−	0.712808i	$-0.252577\pi$
$727$	−7.18914	−	12.4520i	−0.266631	−	0.461818i	−0.967990	+	0.250991i	$0.919244\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	44.2019			1.63487
$732$		0			0
$733$	39.5773			1.46182			0.730911	−	0.682473i	$-0.239096\pi$
$733$	39.5773			1.46182			0.730911	+	0.682473i	$0.239096\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	14.5561	+	25.2119i	0.536182	+	0.928694i
$738$		0			0
$739$	−10.8407	+	18.7767i	−0.398783	+	0.690712i	−0.993576	−	0.113167i	$-0.963901\pi$
$739$	−10.8407	+	18.7767i	−0.398783	+	0.690712i	0.594793	+	0.803879i	$0.297234\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	16.5692	−	28.6987i	0.607864	−	1.05285i	−0.383727	−	0.923446i	$-0.625360\pi$
$743$	16.5692	−	28.6987i	0.607864	−	1.05285i	0.991592	−	0.129406i	$-0.0413069\pi$
$744$		0			0
$745$	35.5831			1.30366
$746$		0			0
$747$		0			0
$748$		0			0
$749$	23.6912	−	2.48089i	0.865659	−	0.0906499i
$750$		0			0
$751$	25.9324			0.946288			0.473144	−	0.880985i	$-0.343119\pi$
$751$	25.9324			0.946288			0.473144	+	0.880985i	$0.343119\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	26.7442			0.973320
$756$		0			0
$757$	−30.5846			−1.11162			−0.555808	−	0.831311i	$-0.687591\pi$
$757$	−30.5846			−1.11162			−0.555808	+	0.831311i	$0.687591\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	36.5295			1.32419			0.662097	−	0.749418i	$-0.269667\pi$
$761$	36.5295			1.32419			0.662097	+	0.749418i	$0.269667\pi$
$762$		0			0
$763$	−11.5505	−	15.9107i	−0.418157	−	0.576007i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.56818			−0.0927315
$768$		0			0
$769$	−21.3107	+	36.9113i	−0.768485	+	1.33105i	0.169900	+	0.985461i	$0.445656\pi$
$769$	−21.3107	+	36.9113i	−0.768485	+	1.33105i	−0.938384	+	0.345593i	$0.887678\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−16.1309	+	27.9395i	−0.580187	+	1.00491i	0.415270	+	0.909698i	$0.363687\pi$
$773$	−16.1309	+	27.9395i	−0.580187	+	1.00491i	−0.995457	+	0.0952148i	$0.969646\pi$
$774$		0			0
$775$	−5.30051	−	9.18076i	−0.190400	−	0.329783i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−0.0800351			−0.00286755
$780$		0			0
$781$	32.4765			1.16210
$782$		0			0
$783$		0			0
$784$		0			0
$785$	0.327420	+	0.567107i	0.0116861	+	0.0202409i
$786$		0			0
$787$	15.3838	+	26.6455i	0.548373	+	0.949810i	0.998386	+	0.0567879i	$0.0180859\pi$
$787$	15.3838	+	26.6455i	0.548373	+	0.949810i	−0.450013	+	0.893022i	$0.648581\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	22.2156	+	30.6017i	0.789895	+	1.08807i
$792$		0			0
$793$	22.0496	+	38.1910i	0.783003	+	1.35620i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	3.59378	−	6.22460i	0.127298	−	0.220487i	−0.795331	−	0.606176i	$-0.792703\pi$
$797$	3.59378	−	6.22460i	0.127298	−	0.220487i	0.922629	+	0.385689i	$0.126036\pi$
$798$		0			0
$799$	−5.39144	−	9.33824i	−0.190735	−	0.330363i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	8.68725	−	15.0468i	0.306566	−	0.530989i
$804$		0			0
$805$	1.81448	−	4.07120i	0.0639520	−	0.143491i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	23.3886	−	40.5103i	0.822301	−	1.42427i	−0.0816637	−	0.996660i	$-0.526023\pi$
$809$	23.3886	−	40.5103i	0.822301	−	1.42427i	0.903965	−	0.427607i	$-0.140643\pi$
$810$		0			0
$811$	−17.6946			−0.621341			−0.310671	−	0.950518i	$-0.600554\pi$
$811$	−17.6946			−0.621341			−0.310671	+	0.950518i	$0.600554\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−15.9418	−	27.6120i	−0.558416	−	0.967205i
$816$		0			0
$817$	0.0340111	−	0.0589089i	0.00118990	−	0.00206096i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−8.61075	+	14.9143i	−0.300517	+	0.520511i	−0.976253	−	0.216633i	$-0.930493\pi$
$821$	−8.61075	+	14.9143i	−0.300517	+	0.520511i	0.675736	+	0.737144i	$0.263826\pi$
$822$		0			0
$823$	5.77170	+	9.99688i	0.201189	+	0.348469i	0.948912	−	0.315542i	$-0.102186\pi$
$823$	5.77170	+	9.99688i	0.201189	+	0.348469i	−0.747723	+	0.664011i	$0.768853\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−20.1448			−0.700503			−0.350251	−	0.936656i	$-0.613904\pi$
$827$	−20.1448			−0.700503			−0.350251	+	0.936656i	$0.613904\pi$
$828$		0			0
$829$	−4.01358	+	6.95172i	−0.139397	+	0.241443i	−0.927269	−	0.374397i	$-0.877850\pi$
$829$	−4.01358	+	6.95172i	−0.139397	+	0.241443i	0.787871	+	0.615840i	$0.211183\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	34.1699	−	7.23574i	1.18392	−	0.250704i
$834$		0			0
$835$	−14.5057	+	25.1246i	−0.501990	+	0.869472i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−4.59341	−	7.95603i	−0.158582	−	0.274673i	0.775775	−	0.631009i	$-0.217359\pi$
$839$	−4.59341	−	7.95603i	−0.158582	−	0.274673i	−0.934358	+	0.356337i	$0.884026\pi$
$840$		0			0
$841$	−15.1576	+	26.2537i	−0.522675	+	0.905299i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−13.9493	−	24.1608i	−0.479869	−	0.831158i
$846$		0			0
$847$	23.3051	−	52.2904i	0.800774	−	1.79672i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2.13389	−	3.69600i	−0.0731488	−	0.126697i
$852$		0			0
$853$	−10.7925	−	18.6931i	−0.369527	−	0.640040i	0.619964	−	0.784630i	$-0.287147\pi$
$853$	−10.7925	−	18.6931i	−0.369527	−	0.640040i	−0.989492	+	0.144590i	$0.953814\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−24.0651			−0.822049			−0.411024	−	0.911624i	$-0.634829\pi$
$857$	−24.0651			−0.822049			−0.411024	+	0.911624i	$0.634829\pi$
$858$		0			0
$859$	16.3172			0.556737			0.278368	−	0.960474i	$-0.410206\pi$
$859$	16.3172			0.556737			0.278368	+	0.960474i	$0.410206\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−9.64675	−	16.7087i	−0.328379	−	0.568770i	0.653811	−	0.756658i	$-0.273169\pi$
$863$	−9.64675	−	16.7087i	−0.328379	−	0.568770i	−0.982190	+	0.187888i	$0.939836\pi$
$864$		0			0
$865$	−19.9428	+	34.5420i	−0.678077	+	1.17446i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−17.6025	+	30.4884i	−0.597124	+	1.03425i
$870$		0			0
$871$	24.9945			0.846907
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−9.87722	+	22.1618i	−0.333911	+	0.749205i
$876$		0			0
$877$	30.2928			1.02291			0.511457	−	0.859309i	$-0.329106\pi$
$877$	30.2928			1.02291			0.511457	+	0.859309i	$0.329106\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−19.4943			−0.656779			−0.328390	−	0.944542i	$-0.606506\pi$
$881$	−19.4943			−0.656779			−0.328390	+	0.944542i	$0.606506\pi$
$882$		0			0
$883$	−47.5302			−1.59952			−0.799759	−	0.600321i	$-0.795040\pi$
$883$	−47.5302			−1.59952			−0.799759	+	0.600321i	$0.795040\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−4.11892			−0.138300			−0.0691499	−	0.997606i	$-0.522029\pi$
$887$	−4.11892			−0.138300			−0.0691499	+	0.997606i	$0.522029\pi$
$888$		0			0
$889$	2.11114	−	4.73682i	0.0708052	−	0.158868i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−0.0165937			−0.000555287
$894$		0			0
$895$	−21.4409	+	37.1367i	−0.716689	+	1.24134i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−29.9056	+	51.7981i	−0.997408	+	1.72756i
$900$		0			0
$901$	18.4547	+	31.9645i	0.614815	+	1.06489i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	16.0417			0.533243
$906$		0			0
$907$	−8.14168			−0.270340			−0.135170	−	0.990822i	$-0.543158\pi$
$907$	−8.14168			−0.270340			−0.135170	+	0.990822i	$0.543158\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−24.6454	−	42.6871i	−0.816540	−	1.41429i	−0.908217	−	0.418499i	$-0.862556\pi$
$911$	−24.6454	−	42.6871i	−0.816540	−	1.41429i	0.0916774	−	0.995789i	$-0.470777\pi$
$912$		0			0
$913$	1.47809	+	2.56013i	0.0489177	+	0.0847279i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−4.29553	+	9.63801i	−0.141851	+	0.318275i
$918$		0			0
$919$	4.71585	+	8.16809i	0.155561	+	0.269440i	0.933263	−	0.359193i	$-0.116948\pi$
$919$	4.71585	+	8.16809i	0.155561	+	0.269440i	−0.777702	+	0.628633i	$0.783615\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	13.9415	−	24.1473i	0.458889	−	0.794819i
$924$		0			0
$925$	−4.36218	−	7.55552i	−0.143428	−	0.248424i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	14.7369	−	25.5251i	0.483503	−	0.837452i	−0.516317	−	0.856397i	$-0.672697\pi$
$929$	14.7369	−	25.5251i	0.483503	−	0.837452i	0.999821	+	0.0189453i	$0.00603083\pi$
$930$		0			0
$931$	0.0166487	−	0.0511066i	0.000545641	−	0.00167495i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	35.9587	−	62.2823i	1.17597	−	2.03685i
$936$		0			0
$937$	−54.3451			−1.77538			−0.887688	−	0.460445i	$-0.847690\pi$
$937$	−54.3451			−1.77538			−0.887688	+	0.460445i	$0.847690\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.49087	−	2.58226i	−0.0486008	−	0.0841791i	0.840702	−	0.541499i	$-0.182143\pi$
$941$	−1.49087	−	2.58226i	−0.0486008	−	0.0841791i	−0.889302	+	0.457320i	$0.848810\pi$
$942$		0			0
$943$	−3.48005	+	6.02763i	−0.113326	+	0.196287i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−15.1646	+	26.2659i	−0.492783	+	0.853526i	−0.999965	−	0.00831303i	$-0.997354\pi$
$947$	−15.1646	+	26.2659i	−0.492783	+	0.853526i	0.507182	+	0.861839i	$0.330687\pi$
$948$		0			0
$949$	−7.45850	−	12.9185i	−0.242113	−	0.419352i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.380127			−0.0123135			−0.00615676	−	0.999981i	$-0.501960\pi$
$953$	−0.380127			−0.0123135			−0.00615676	+	0.999981i	$0.501960\pi$
$954$		0			0
$955$	5.22360	−	9.04754i	0.169032	−	0.292771i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	7.97924	−	17.9032i	0.257663	−	0.578126i
$960$		0			0
$961$	−14.6558	+	25.3845i	−0.472766	+	0.818855i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	9.70796	+	16.8147i	0.312510	+	0.541284i
$966$		0			0
$967$	22.6744	−	39.2732i	0.729160	−	1.26294i	−0.228078	−	0.973643i	$-0.573244\pi$
$967$	22.6744	−	39.2732i	0.729160	−	1.26294i	0.957239	−	0.289300i	$-0.0934224\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−27.3746	−	47.4141i	−0.878491	−	1.52159i	−0.852996	−	0.521917i	$-0.825217\pi$
$971$	−27.3746	−	47.4141i	−0.878491	−	1.52159i	−0.0254951	−	0.999675i	$-0.508116\pi$
$972$		0			0
$973$	21.5323	+	29.6604i	0.690292	+	0.950870i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.35656	+	2.34962i	0.0434001	+	0.0751711i	0.886909	−	0.461943i	$-0.152848\pi$
$977$	1.35656	+	2.34962i	0.0434001	+	0.0751711i	−0.843509	+	0.537114i	$0.819514\pi$
$978$		0			0
$979$	6.80371	+	11.7844i	0.217448	+	0.376630i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	42.7656			1.36401			0.682006	−	0.731347i	$-0.261108\pi$
$983$	42.7656			1.36401			0.682006	+	0.731347i	$0.261108\pi$
$984$		0			0
$985$	−8.31318			−0.264880
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2.95771	−	5.12291i	−0.0940498	−	0.162899i
$990$		0			0
$991$	−29.6731	+	51.3954i	−0.942598	+	1.63263i	−0.182107	+	0.983279i	$0.558292\pi$
$991$	−29.6731	+	51.3954i	−0.942598	+	1.63263i	−0.760491	+	0.649349i	$0.775042\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−20.4063	+	35.3447i	−0.646922	+	1.12050i
$996$		0			0
$997$	44.0641			1.39552			0.697762	−	0.716330i	$-0.254179\pi$
$997$	44.0641			1.39552			0.697762	+	0.716330i	$0.254179\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3024.2.t.l.1873.9		22
3.2	odd	2		1008.2.t.k.193.2		22
4.3	odd	2		1512.2.t.d.361.9		22
7.2	even	3		3024.2.q.k.2305.3		22
9.2	odd	6		1008.2.q.k.529.9		22
9.7	even	3		3024.2.q.k.2881.3		22
12.11	even	2		504.2.t.d.193.10	yes	22
21.2	odd	6		1008.2.q.k.625.9		22
28.23	odd	6		1512.2.q.c.793.3		22
36.7	odd	6		1512.2.q.c.1369.3		22
36.11	even	6		504.2.q.d.25.3	✓	22
63.2	odd	6		1008.2.t.k.961.2		22
63.16	even	3	inner	3024.2.t.l.289.9		22
84.23	even	6		504.2.q.d.121.3	yes	22
252.79	odd	6		1512.2.t.d.289.9		22
252.191	even	6		504.2.t.d.457.10	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.q.d.25.3	✓	22	36.11	even	6
504.2.q.d.121.3	yes	22	84.23	even	6
504.2.t.d.193.10	yes	22	12.11	even	2
504.2.t.d.457.10	yes	22	252.191	even	6
1008.2.q.k.529.9		22	9.2	odd	6
1008.2.q.k.625.9		22	21.2	odd	6
1008.2.t.k.193.2		22	3.2	odd	2
1008.2.t.k.961.2		22	63.2	odd	6
1512.2.q.c.793.3		22	28.23	odd	6
1512.2.q.c.1369.3		22	36.7	odd	6
1512.2.t.d.289.9		22	252.79	odd	6
1512.2.t.d.361.9		22	4.3	odd	2
3024.2.q.k.2305.3		22	7.2	even	3
3024.2.q.k.2881.3		22	9.7	even	3
3024.2.t.l.289.9		22	63.16	even	3	inner
3024.2.t.l.1873.9		22	1.1	even	1	trivial

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

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Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	3024.2.t.l.1873.9

Level	$3024$
Weight	$2$
Character	3024.1873
Analytic conductor	$24.147$
Analytic rank	$0$
Dimension	$22$
Inner twists	$2$