LMFDB - Embedded newform 600.4.f.d.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,4,Mod(49,600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("600.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	600.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$35.4011460034$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{37}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	600.49
Dual form			600.4.f.d.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -9.00000 q^{9} +4.00000 q^{11} +54.0000i q^{13} -114.000i q^{17} -44.0000 q^{19} +96.0000i q^{23} +27.0000i q^{27} -134.000 q^{29} -272.000 q^{31} -12.0000i q^{33} +98.0000i q^{37} +162.000 q^{39} -6.00000 q^{41} +12.0000i q^{43} +200.000i q^{47} +343.000 q^{49} -342.000 q^{51} +654.000i q^{53} +132.000i q^{57} -36.0000 q^{59} -442.000 q^{61} +188.000i q^{67} +288.000 q^{69} -632.000 q^{71} -390.000i q^{73} -688.000 q^{79} +81.0000 q^{81} +1188.00i q^{83} +402.000i q^{87} +694.000 q^{89} +816.000i q^{93} +1726.00i q^{97} -36.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{9} + 8 q^{11} - 88 q^{19} - 268 q^{29} - 544 q^{31} + 324 q^{39} - 12 q^{41} + 686 q^{49} - 684 q^{51} - 72 q^{59} - 884 q^{61} + 576 q^{69} - 1264 q^{71} - 1376 q^{79} + 162 q^{81} + 1388 q^{89}+ \cdots - 72 q^{99}+O(q^{100}) $ 2 * q - 18 * q^9 + 8 * q^11 - 88 * q^19 - 268 * q^29 - 544 * q^31 + 324 * q^39 - 12 * q^41 + 686 * q^49 - 684 * q^51 - 72 * q^59 - 884 * q^61 + 576 * q^69 - 1264 * q^71 - 1376 * q^79 + 162 * q^81 + 1388 * q^89 - 72 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$.

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	4.00000	0.109640	0.0548202	−	0.998496i	$-0.482541\pi$
$11$	4.00000	0.109640	0.0548202	+	0.998496i	$0.482541\pi$
$12$	0	0
$13$	54.0000i	1.15207i	0.817425	+	0.576035i	$0.195401\pi$
$13$	54.0000i	1.15207i	−0.817425	+	0.576035i	$0.804599\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 114.000i	− 1.62642i	−0.581974	−	0.813208i	$-0.697719\pi$
$17$	− 114.000i	− 1.62642i	0.581974	−	0.813208i	$-0.302281\pi$
$18$	0	0
$19$	−44.0000	−0.531279	−0.265639	−	0.964072i	$-0.585583\pi$
$19$	−44.0000	−0.531279	−0.265639	+	0.964072i	$0.585583\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	96.0000i	0.870321i	0.900353	+	0.435161i	$0.143308\pi$
$23$	96.0000i	0.870321i	−0.900353	+	0.435161i	$0.856692\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	−134.000	−0.858041	−0.429020	−	0.903295i	$-0.641141\pi$
$29$	−134.000	−0.858041	−0.429020	+	0.903295i	$0.641141\pi$
$30$	0	0
$31$	−272.000	−1.57589	−0.787946	−	0.615745i	$-0.788855\pi$
$31$	−272.000	−1.57589	−0.787946	+	0.615745i	$0.788855\pi$
$32$	0	0
$33$	− 12.0000i	− 0.0633010i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	98.0000i	0.435435i	0.976012	+	0.217718i	$0.0698612\pi$
$37$	98.0000i	0.435435i	−0.976012	+	0.217718i	$0.930139\pi$
$38$	0	0
$39$	162.000	0.665148
$40$	0	0
$41$	−6.00000	−0.0228547	−0.0114273	−	0.999935i	$-0.503638\pi$
$41$	−6.00000	−0.0228547	−0.0114273	+	0.999935i	$0.503638\pi$
$42$	0	0
$43$	12.0000i	0.0425577i	0.999774	+	0.0212789i	$0.00677379\pi$
$43$	12.0000i	0.0425577i	−0.999774	+	0.0212789i	$0.993226\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	200.000i	0.620702i	0.950622	+	0.310351i	$0.100447\pi$
$47$	200.000i	0.620702i	−0.950622	+	0.310351i	$0.899553\pi$
$48$	0	0
$49$	343.000	1.00000
$50$	0	0
$51$	−342.000	−0.939011
$52$	0	0
$53$	654.000i	1.69498i	0.530813	+	0.847489i	$0.321887\pi$
$53$	654.000i	1.69498i	−0.530813	+	0.847489i	$0.678113\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	132.000i	0.306734i
$58$	0	0
$59$	−36.0000	−0.0794373	−0.0397187	−	0.999211i	$-0.512646\pi$
$59$	−36.0000	−0.0794373	−0.0397187	+	0.999211i	$0.512646\pi$
$60$	0	0
$61$	−442.000	−0.927743	−0.463871	−	0.885903i	$-0.653540\pi$
$61$	−442.000	−0.927743	−0.463871	+	0.885903i	$0.653540\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	188.000i	0.342804i	0.985201	+	0.171402i	$0.0548297\pi$
$67$	188.000i	0.342804i	−0.985201	+	0.171402i	$0.945170\pi$
$68$	0	0
$69$	288.000	0.502480
$70$	0	0
$71$	−632.000	−1.05640	−0.528201	−	0.849119i	$-0.677133\pi$
$71$	−632.000	−1.05640	−0.528201	+	0.849119i	$0.677133\pi$
$72$	0	0
$73$	− 390.000i	− 0.625288i	−0.949870	−	0.312644i	$-0.898785\pi$
$73$	− 390.000i	− 0.625288i	0.949870	−	0.312644i	$-0.101215\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−688.000	−0.979823	−0.489912	−	0.871772i	$-0.662971\pi$
$79$	−688.000	−0.979823	−0.489912	+	0.871772i	$0.662971\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1188.00i	1.57108i	0.618809	+	0.785542i	$0.287616\pi$
$83$	1188.00i	1.57108i	−0.618809	+	0.785542i	$0.712384\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	402.000i	0.495390i
$88$	0	0
$89$	694.000	0.826560	0.413280	−	0.910604i	$-0.364383\pi$
$89$	694.000	0.826560	0.413280	+	0.910604i	$0.364383\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	816.000i	0.909841i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1726.00i	1.80669i	0.428917	+	0.903344i	$0.358895\pi$
$97$	1726.00i	1.80669i	−0.428917	+	0.903344i	$0.641105\pi$
$98$	0	0
$99$	−36.0000	−0.0365468
$100$	0	0
$101$	1182.00	1.16449	0.582245	−	0.813014i	$-0.302175\pi$
$101$	1182.00	1.16449	0.582245	+	0.813014i	$0.302175\pi$
$102$	0	0
$103$	1968.00i	1.88265i	0.337503	+	0.941324i	$0.390418\pi$
$103$	1968.00i	1.88265i	−0.337503	+	0.941324i	$0.609582\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 796.000i	− 0.719180i	−0.933110	−	0.359590i	$-0.882917\pi$
$107$	− 796.000i	− 0.719180i	0.933110	−	0.359590i	$-0.117083\pi$
$108$	0	0
$109$	−342.000	−0.300529	−0.150264	−	0.988646i	$-0.548013\pi$
$109$	−342.000	−0.300529	−0.150264	+	0.988646i	$0.548013\pi$
$110$	0	0
$111$	294.000	0.251399
$112$	0	0
$113$	114.000i	0.0949046i	0.998874	+	0.0474523i	$0.0151102\pi$
$113$	114.000i	0.0949046i	−0.998874	+	0.0474523i	$0.984890\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 486.000i	− 0.384023i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1315.00	−0.987979
$122$	0	0
$123$	18.0000i	0.0131952i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 2344.00i	− 1.63777i	−0.573960	−	0.818883i	$-0.694594\pi$
$127$	− 2344.00i	− 1.63777i	0.573960	−	0.818883i	$-0.305406\pi$
$128$	0	0
$129$	36.0000	0.0245707
$130$	0	0
$131$	−2164.00	−1.44328	−0.721640	−	0.692269i	$-0.756611\pi$
$131$	−2164.00	−1.44328	−0.721640	+	0.692269i	$0.756611\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2822.00i	1.75985i	0.475111	+	0.879926i	$0.342408\pi$
$137$	2822.00i	1.75985i	−0.475111	+	0.879926i	$0.657592\pi$
$138$	0	0
$139$	−1972.00	−1.20333	−0.601665	−	0.798749i	$-0.705496\pi$
$139$	−1972.00	−1.20333	−0.601665	+	0.798749i	$0.705496\pi$
$140$	0	0
$141$	600.000	0.358363
$142$	0	0
$143$	216.000i	0.126313i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1029.00i	− 0.577350i
$148$	0	0
$149$	1394.00	0.766449	0.383225	−	0.923655i	$-0.374814\pi$
$149$	1394.00	0.766449	0.383225	+	0.923655i	$0.374814\pi$
$150$	0	0
$151$	−2216.00	−1.19427	−0.597137	−	0.802139i	$-0.703695\pi$
$151$	−2216.00	−1.19427	−0.597137	+	0.802139i	$0.703695\pi$
$152$	0	0
$153$	1026.00i	0.542138i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	954.000i	0.484952i	0.970157	+	0.242476i	$0.0779596\pi$
$157$	954.000i	0.484952i	−0.970157	+	0.242476i	$0.922040\pi$
$158$	0	0
$159$	1962.00	0.978596
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 3404.00i	− 1.63572i	−0.575419	−	0.817858i	$-0.695161\pi$
$163$	− 3404.00i	− 1.63572i	0.575419	−	0.817858i	$-0.304839\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	832.000i	0.385522i	0.981246	+	0.192761i	$0.0617441\pi$
$167$	832.000i	0.385522i	−0.981246	+	0.192761i	$0.938256\pi$
$168$	0	0
$169$	−719.000	−0.327264
$170$	0	0
$171$	396.000	0.177093
$172$	0	0
$173$	− 362.000i	− 0.159089i	−0.996831	−	0.0795444i	$-0.974653\pi$
$173$	− 362.000i	− 0.159089i	0.996831	−	0.0795444i	$-0.0253465\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	108.000i	0.0458631i
$178$	0	0
$179$	3252.00	1.35791	0.678955	−	0.734180i	$-0.262433\pi$
$179$	3252.00	1.35791	0.678955	+	0.734180i	$0.262433\pi$
$180$	0	0
$181$	3086.00	1.26730	0.633648	−	0.773621i	$-0.281557\pi$
$181$	3086.00	1.26730	0.633648	+	0.773621i	$0.281557\pi$
$182$	0	0
$183$	1326.00i	0.535632i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 456.000i	− 0.178321i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4080.00	−1.54565	−0.772823	−	0.634621i	$-0.781156\pi$
$191$	−4080.00	−1.54565	−0.772823	+	0.634621i	$0.781156\pi$
$192$	0	0
$193$	− 2654.00i	− 0.989840i	−0.868939	−	0.494920i	$-0.835197\pi$
$193$	− 2654.00i	− 0.989840i	0.868939	−	0.494920i	$-0.164803\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 1534.00i	− 0.554787i	−0.960756	−	0.277393i	$-0.910529\pi$
$197$	− 1534.00i	− 0.554787i	0.960756	−	0.277393i	$-0.0894705\pi$
$198$	0	0
$199$	−4344.00	−1.54743	−0.773714	−	0.633536i	$-0.781603\pi$
$199$	−4344.00	−1.54743	−0.773714	+	0.633536i	$0.781603\pi$
$200$	0	0
$201$	564.000	0.197918
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 864.000i	− 0.290107i
$208$	0	0
$209$	−176.000	−0.0582496
$210$	0	0
$211$	−1380.00	−0.450252	−0.225126	−	0.974330i	$-0.572279\pi$
$211$	−1380.00	−0.450252	−0.225126	+	0.974330i	$0.572279\pi$
$212$	0	0
$213$	1896.00i	0.609914i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1170.00	−0.361010
$220$	0	0
$221$	6156.00	1.87374
$222$	0	0
$223$	− 5224.00i	− 1.56872i	−0.620305	−	0.784361i	$-0.712991\pi$
$223$	− 5224.00i	− 1.56872i	0.620305	−	0.784361i	$-0.287009\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 3364.00i	− 0.983597i	−0.870709	−	0.491799i	$-0.836340\pi$
$227$	− 3364.00i	− 0.983597i	0.870709	−	0.491799i	$-0.163660\pi$
$228$	0	0
$229$	−3998.00	−1.15369	−0.576846	−	0.816853i	$-0.695717\pi$
$229$	−3998.00	−1.15369	−0.576846	+	0.816853i	$0.695717\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 3590.00i	− 1.00939i	−0.863297	−	0.504697i	$-0.831604\pi$
$233$	− 3590.00i	− 1.00939i	0.863297	−	0.504697i	$-0.168396\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2064.00i	0.565701i
$238$	0	0
$239$	1104.00	0.298794	0.149397	−	0.988777i	$-0.452267\pi$
$239$	1104.00	0.298794	0.149397	+	0.988777i	$0.452267\pi$
$240$	0	0
$241$	1618.00	0.432467	0.216233	−	0.976342i	$-0.430623\pi$
$241$	1618.00	0.432467	0.216233	+	0.976342i	$0.430623\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 2376.00i	− 0.612070i
$248$	0	0
$249$	3564.00	0.907066
$250$	0	0
$251$	5780.00	1.45351	0.726754	−	0.686898i	$-0.241028\pi$
$251$	5780.00	1.45351	0.726754	+	0.686898i	$0.241028\pi$
$252$	0	0
$253$	384.000i	0.0954224i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 2594.00i	− 0.629608i	−0.949157	−	0.314804i	$-0.898061\pi$
$257$	− 2594.00i	− 0.629608i	0.949157	−	0.314804i	$-0.101939\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1206.00	0.286014
$262$	0	0
$263$	3696.00i	0.866559i	0.901260	+	0.433280i	$0.142644\pi$
$263$	3696.00i	0.866559i	−0.901260	+	0.433280i	$0.857356\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 2082.00i	− 0.477215i
$268$	0	0
$269$	2250.00	0.509981	0.254991	−	0.966944i	$-0.417928\pi$
$269$	2250.00	0.509981	0.254991	+	0.966944i	$0.417928\pi$
$270$	0	0
$271$	2208.00	0.494932	0.247466	−	0.968897i	$-0.420402\pi$
$271$	2208.00	0.494932	0.247466	+	0.968897i	$0.420402\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1682.00i	0.364843i	0.983220	+	0.182422i	$0.0583936\pi$
$277$	1682.00i	0.364843i	−0.983220	+	0.182422i	$0.941606\pi$
$278$	0	0
$279$	2448.00	0.525297
$280$	0	0
$281$	7306.00	1.55103	0.775515	−	0.631329i	$-0.217490\pi$
$281$	7306.00	1.55103	0.775515	+	0.631329i	$0.217490\pi$
$282$	0	0
$283$	− 8164.00i	− 1.71484i	−0.514618	−	0.857419i	$-0.672066\pi$
$283$	− 8164.00i	− 1.71484i	0.514618	−	0.857419i	$-0.327934\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−8083.00	−1.64523
$290$	0	0
$291$	5178.00	1.04309
$292$	0	0
$293$	− 514.000i	− 0.102485i	−0.998686	−	0.0512427i	$-0.983682\pi$
$293$	− 514.000i	− 0.102485i	0.998686	−	0.0512427i	$-0.0163182\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	108.000i	0.0211003i
$298$	0	0
$299$	−5184.00	−1.00267
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 3546.00i	− 0.672318i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2476.00i	0.460302i	0.973155	+	0.230151i	$0.0739220\pi$
$307$	2476.00i	0.460302i	−0.973155	+	0.230151i	$0.926078\pi$
$308$	0	0
$309$	5904.00	1.08695
$310$	0	0
$311$	2296.00	0.418631	0.209315	−	0.977848i	$-0.432876\pi$
$311$	2296.00	0.418631	0.209315	+	0.977848i	$0.432876\pi$
$312$	0	0
$313$	− 9878.00i	− 1.78383i	−0.452207	−	0.891913i	$-0.649363\pi$
$313$	− 9878.00i	− 1.78383i	0.452207	−	0.891913i	$-0.350637\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2138.00i	0.378808i	0.981899	+	0.189404i	$0.0606555\pi$
$317$	2138.00i	0.378808i	−0.981899	+	0.189404i	$0.939344\pi$
$318$	0	0
$319$	−536.000	−0.0940760
$320$	0	0
$321$	−2388.00	−0.415219
$322$	0	0
$323$	5016.00i	0.864080i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1026.00i	0.173510i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−6460.00	−1.07273	−0.536365	−	0.843986i	$-0.680203\pi$
$331$	−6460.00	−1.07273	−0.536365	+	0.843986i	$0.680203\pi$
$332$	0	0
$333$	− 882.000i	− 0.145145i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 626.000i	− 0.101188i	−0.998719	−	0.0505941i	$-0.983889\pi$
$337$	− 626.000i	− 0.101188i	0.998719	−	0.0505941i	$-0.0161115\pi$
$338$	0	0
$339$	342.000	0.0547932
$340$	0	0
$341$	−1088.00	−0.172782
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 876.000i	− 0.135522i	−0.997702	−	0.0677610i	$-0.978414\pi$
$347$	− 876.000i	− 0.135522i	0.997702	−	0.0677610i	$-0.0215855\pi$
$348$	0	0
$349$	9850.00	1.51077	0.755385	−	0.655282i	$-0.227450\pi$
$349$	9850.00	1.51077	0.755385	+	0.655282i	$0.227450\pi$
$350$	0	0
$351$	−1458.00	−0.221716
$352$	0	0
$353$	− 8894.00i	− 1.34102i	−0.741901	−	0.670510i	$-0.766075\pi$
$353$	− 8894.00i	− 1.34102i	0.741901	−	0.670510i	$-0.233925\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1464.00	−0.215228	−0.107614	−	0.994193i	$-0.534321\pi$
$359$	−1464.00	−0.215228	−0.107614	+	0.994193i	$0.534321\pi$
$360$	0	0
$361$	−4923.00	−0.717743
$362$	0	0
$363$	3945.00i	0.570410i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	7016.00i	0.997908i	0.866628	+	0.498954i	$0.166282\pi$
$367$	7016.00i	0.997908i	−0.866628	+	0.498954i	$0.833718\pi$
$368$	0	0
$369$	54.0000	0.00761823
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 1010.00i	− 0.140203i	−0.997540	−	0.0701016i	$-0.977668\pi$
$373$	− 1010.00i	− 0.140203i	0.997540	−	0.0701016i	$-0.0223324\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 7236.00i	− 0.988522i
$378$	0	0
$379$	−4900.00	−0.664106	−0.332053	−	0.943261i	$-0.607741\pi$
$379$	−4900.00	−0.664106	−0.332053	+	0.943261i	$0.607741\pi$
$380$	0	0
$381$	−7032.00	−0.945565
$382$	0	0
$383$	7800.00i	1.04063i	0.853974	+	0.520315i	$0.174186\pi$
$383$	7800.00i	1.04063i	−0.853974	+	0.520315i	$0.825814\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 108.000i	− 0.0141859i
$388$	0	0
$389$	12258.0	1.59770	0.798850	−	0.601530i	$-0.205442\pi$
$389$	12258.0	1.59770	0.798850	+	0.601530i	$0.205442\pi$
$390$	0	0
$391$	10944.0	1.41550
$392$	0	0
$393$	6492.00i	0.833278i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 5558.00i	− 0.702640i	−0.936255	−	0.351320i	$-0.885733\pi$
$397$	− 5558.00i	− 0.702640i	0.936255	−	0.351320i	$-0.114267\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1970.00	0.245329	0.122665	−	0.992448i	$-0.460856\pi$
$401$	1970.00	0.245329	0.122665	+	0.992448i	$0.460856\pi$
$402$	0	0
$403$	− 14688.0i	− 1.81554i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	392.000i	0.0477413i
$408$	0	0
$409$	−15626.0	−1.88913	−0.944567	−	0.328318i	$-0.893518\pi$
$409$	−15626.0	−1.88913	−0.944567	+	0.328318i	$0.893518\pi$
$410$	0	0
$411$	8466.00	1.01605
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5916.00i	0.694743i
$418$	0	0
$419$	5412.00	0.631011	0.315505	−	0.948924i	$-0.397826\pi$
$419$	5412.00	0.631011	0.315505	+	0.948924i	$0.397826\pi$
$420$	0	0
$421$	−10690.0	−1.23753	−0.618763	−	0.785577i	$-0.712366\pi$
$421$	−10690.0	−1.23753	−0.618763	+	0.785577i	$0.712366\pi$
$422$	0	0
$423$	− 1800.00i	− 0.206901i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	648.000	0.0729271
$430$	0	0
$431$	−14048.0	−1.57000	−0.784998	−	0.619498i	$-0.787336\pi$
$431$	−14048.0	−1.57000	−0.784998	+	0.619498i	$0.787336\pi$
$432$	0	0
$433$	17778.0i	1.97311i	0.163433	+	0.986554i	$0.447743\pi$
$433$	17778.0i	1.97311i	−0.163433	+	0.986554i	$0.552257\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 4224.00i	− 0.462383i
$438$	0	0
$439$	−7240.00	−0.787122	−0.393561	−	0.919299i	$-0.628757\pi$
$439$	−7240.00	−0.787122	−0.393561	+	0.919299i	$0.628757\pi$
$440$	0	0
$441$	−3087.00	−0.333333
$442$	0	0
$443$	11740.0i	1.25911i	0.776957	+	0.629553i	$0.216762\pi$
$443$	11740.0i	1.25911i	−0.776957	+	0.629553i	$0.783238\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 4182.00i	− 0.442510i
$448$	0	0
$449$	−15234.0	−1.60120	−0.800598	−	0.599202i	$-0.795485\pi$
$449$	−15234.0	−1.60120	−0.800598	+	0.599202i	$0.795485\pi$
$450$	0	0
$451$	−24.0000	−0.00250580
$452$	0	0
$453$	6648.00i	0.689515i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 3866.00i	− 0.395720i	−0.980230	−	0.197860i	$-0.936601\pi$
$457$	− 3866.00i	− 0.395720i	0.980230	−	0.197860i	$-0.0633991\pi$
$458$	0	0
$459$	3078.00	0.313004
$460$	0	0
$461$	−1706.00	−0.172356	−0.0861782	−	0.996280i	$-0.527465\pi$
$461$	−1706.00	−0.172356	−0.0861782	+	0.996280i	$0.527465\pi$
$462$	0	0
$463$	3944.00i	0.395882i	0.980214	+	0.197941i	$0.0634254\pi$
$463$	3944.00i	0.395882i	−0.980214	+	0.197941i	$0.936575\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	9452.00i	0.936588i	0.883573	+	0.468294i	$0.155131\pi$
$467$	9452.00i	0.936588i	−0.883573	+	0.468294i	$0.844869\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	2862.00	0.279987
$472$	0	0
$473$	48.0000i	0.00466605i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 5886.00i	− 0.564993i
$478$	0	0
$479$	12544.0	1.19656	0.598278	−	0.801289i	$-0.295852\pi$
$479$	12544.0	1.19656	0.598278	+	0.801289i	$0.295852\pi$
$480$	0	0
$481$	−5292.00	−0.501652
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 7936.00i	− 0.738428i	−0.929344	−	0.369214i	$-0.879627\pi$
$487$	− 7936.00i	− 0.738428i	0.929344	−	0.369214i	$-0.120373\pi$
$488$	0	0
$489$	−10212.0	−0.944382
$490$	0	0
$491$	−8412.00	−0.773174	−0.386587	−	0.922253i	$-0.626346\pi$
$491$	−8412.00	−0.773174	−0.386587	+	0.922253i	$0.626346\pi$
$492$	0	0
$493$	15276.0i	1.39553i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	15092.0	1.35393	0.676965	−	0.736016i	$-0.263295\pi$
$499$	15092.0	1.35393	0.676965	+	0.736016i	$0.263295\pi$
$500$	0	0
$501$	2496.00	0.222581
$502$	0	0
$503$	6112.00i	0.541790i	0.962609	+	0.270895i	$0.0873197\pi$
$503$	6112.00i	0.541790i	−0.962609	+	0.270895i	$0.912680\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2157.00i	0.188946i
$508$	0	0
$509$	−2534.00	−0.220663	−0.110332	−	0.993895i	$-0.535191\pi$
$509$	−2534.00	−0.220663	−0.110332	+	0.993895i	$0.535191\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	− 1188.00i	− 0.102245i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	800.000i	0.0680541i
$518$	0	0
$519$	−1086.00	−0.0918499
$520$	0	0
$521$	−9894.00	−0.831985	−0.415992	−	0.909368i	$-0.636566\pi$
$521$	−9894.00	−0.831985	−0.415992	+	0.909368i	$0.636566\pi$
$522$	0	0
$523$	16172.0i	1.35211i	0.736852	+	0.676054i	$0.236311\pi$
$523$	16172.0i	1.35211i	−0.736852	+	0.676054i	$0.763689\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	31008.0i	2.56305i
$528$	0	0
$529$	2951.00	0.242541
$530$	0	0
$531$	324.000	0.0264791
$532$	0	0
$533$	− 324.000i	− 0.0263302i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 9756.00i	− 0.783990i
$538$	0	0
$539$	1372.00	0.109640
$540$	0	0
$541$	−6138.00	−0.487788	−0.243894	−	0.969802i	$-0.578425\pi$
$541$	−6138.00	−0.487788	−0.243894	+	0.969802i	$0.578425\pi$
$542$	0	0
$543$	− 9258.00i	− 0.731674i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	21852.0i	1.70809i	0.520201	+	0.854044i	$0.325857\pi$
$547$	21852.0i	1.70809i	−0.520201	+	0.854044i	$0.674143\pi$
$548$	0	0
$549$	3978.00	0.309248
$550$	0	0
$551$	5896.00	0.455859
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1962.00i	0.149251i	0.997212	+	0.0746253i	$0.0237761\pi$
$557$	1962.00i	0.149251i	−0.997212	+	0.0746253i	$0.976224\pi$
$558$	0	0
$559$	−648.000	−0.0490295
$560$	0	0
$561$	−1368.00	−0.102954
$562$	0	0
$563$	− 10876.0i	− 0.814154i	−0.913394	−	0.407077i	$-0.866548\pi$
$563$	− 10876.0i	− 0.814154i	0.913394	−	0.407077i	$-0.133452\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5610.00	−0.413328	−0.206664	−	0.978412i	$-0.566261\pi$
$569$	−5610.00	−0.413328	−0.206664	+	0.978412i	$0.566261\pi$
$570$	0	0
$571$	5076.00	0.372021	0.186010	−	0.982548i	$-0.440444\pi$
$571$	5076.00	0.372021	0.186010	+	0.982548i	$0.440444\pi$
$572$	0	0
$573$	12240.0i	0.892379i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	6526.00i	0.470851i	0.971892	+	0.235425i	$0.0756483\pi$
$577$	6526.00i	0.470851i	−0.971892	+	0.235425i	$0.924352\pi$
$578$	0	0
$579$	−7962.00	−0.571484
$580$	0	0
$581$	0	0
$582$	0	0
$583$	2616.00i	0.185838i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 2332.00i	− 0.163973i	−0.996633	−	0.0819863i	$-0.973874\pi$
$587$	− 2332.00i	− 0.163973i	0.996633	−	0.0819863i	$-0.0261264\pi$
$588$	0	0
$589$	11968.0	0.837237
$590$	0	0
$591$	−4602.00	−0.320306
$592$	0	0
$593$	− 9582.00i	− 0.663551i	−0.943358	−	0.331775i	$-0.892352\pi$
$593$	− 9582.00i	− 0.663551i	0.943358	−	0.331775i	$-0.107648\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	13032.0i	0.893407i
$598$	0	0
$599$	17624.0	1.20217	0.601083	−	0.799187i	$-0.294736\pi$
$599$	17624.0	1.20217	0.601083	+	0.799187i	$0.294736\pi$
$600$	0	0
$601$	−21238.0	−1.44146	−0.720729	−	0.693217i	$-0.756193\pi$
$601$	−21238.0	−1.44146	−0.720729	+	0.693217i	$0.756193\pi$
$602$	0	0
$603$	− 1692.00i	− 0.114268i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 13000.0i	− 0.869281i	−0.900604	−	0.434641i	$-0.856875\pi$
$607$	− 13000.0i	− 0.869281i	0.900604	−	0.434641i	$-0.143125\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−10800.0	−0.715092
$612$	0	0
$613$	9214.00i	0.607096i	0.952816	+	0.303548i	$0.0981713\pi$
$613$	9214.00i	0.607096i	−0.952816	+	0.303548i	$0.901829\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 4474.00i	− 0.291923i	−0.989290	−	0.145961i	$-0.953372\pi$
$617$	− 4474.00i	− 0.291923i	0.989290	−	0.145961i	$-0.0466276\pi$
$618$	0	0
$619$	12556.0	0.815296	0.407648	−	0.913139i	$-0.366349\pi$
$619$	12556.0	0.815296	0.407648	+	0.913139i	$0.366349\pi$
$620$	0	0
$621$	−2592.00	−0.167493
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	528.000i	0.0336304i
$628$	0	0
$629$	11172.0	0.708198
$630$	0	0
$631$	26936.0	1.69937	0.849687	−	0.527287i	$-0.176791\pi$
$631$	26936.0	1.69937	0.849687	+	0.527287i	$0.176791\pi$
$632$	0	0
$633$	4140.00i	0.259953i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	18522.0i	1.15207i
$638$	0	0
$639$	5688.00	0.352134
$640$	0	0
$641$	−19134.0	−1.17901	−0.589507	−	0.807764i	$-0.700678\pi$
$641$	−19134.0	−1.17901	−0.589507	+	0.807764i	$0.700678\pi$
$642$	0	0
$643$	12436.0i	0.762718i	0.924427	+	0.381359i	$0.124544\pi$
$643$	12436.0i	0.762718i	−0.924427	+	0.381359i	$0.875456\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2784.00i	0.169166i	0.996416	+	0.0845829i	$0.0269558\pi$
$647$	2784.00i	0.169166i	−0.996416	+	0.0845829i	$0.973044\pi$
$648$	0	0
$649$	−144.000	−0.00870954
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7318.00i	0.438554i	0.975663	+	0.219277i	$0.0703698\pi$
$653$	7318.00i	0.438554i	−0.975663	+	0.219277i	$0.929630\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3510.00i	0.208429i
$658$	0	0
$659$	−8108.00	−0.479276	−0.239638	−	0.970862i	$-0.577029\pi$
$659$	−8108.00	−0.479276	−0.239638	+	0.970862i	$0.577029\pi$
$660$	0	0
$661$	1230.00	0.0723774	0.0361887	−	0.999345i	$-0.488478\pi$
$661$	1230.00	0.0723774	0.0361887	+	0.999345i	$0.488478\pi$
$662$	0	0
$663$	− 18468.0i	− 1.08181i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 12864.0i	− 0.746771i
$668$	0	0
$669$	−15672.0	−0.905702
$670$	0	0
$671$	−1768.00	−0.101718
$672$	0	0
$673$	− 14078.0i	− 0.806340i	−0.915125	−	0.403170i	$-0.867908\pi$
$673$	− 14078.0i	− 0.806340i	0.915125	−	0.403170i	$-0.132092\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 25246.0i	− 1.43321i	−0.697480	−	0.716605i	$-0.745695\pi$
$677$	− 25246.0i	− 1.43321i	0.697480	−	0.716605i	$-0.254305\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−10092.0	−0.567880
$682$	0	0
$683$	24332.0i	1.36316i	0.731744	+	0.681580i	$0.238707\pi$
$683$	24332.0i	1.36316i	−0.731744	+	0.681580i	$0.761293\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	11994.0i	0.666084i
$688$	0	0
$689$	−35316.0	−1.95273
$690$	0	0
$691$	19036.0	1.04799	0.523997	−	0.851720i	$-0.324440\pi$
$691$	19036.0	1.04799	0.523997	+	0.851720i	$0.324440\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	684.000i	0.0371712i
$698$	0	0
$699$	−10770.0	−0.582774
$700$	0	0
$701$	28806.0	1.55205	0.776025	−	0.630702i	$-0.217233\pi$
$701$	28806.0	1.55205	0.776025	+	0.630702i	$0.217233\pi$
$702$	0	0
$703$	− 4312.00i	− 0.231337i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	25090.0	1.32902	0.664510	−	0.747280i	$-0.268640\pi$
$709$	25090.0	1.32902	0.664510	+	0.747280i	$0.268640\pi$
$710$	0	0
$711$	6192.00	0.326608
$712$	0	0
$713$	− 26112.0i	− 1.37153i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 3312.00i	− 0.172509i
$718$	0	0
$719$	−36432.0	−1.88969	−0.944843	−	0.327523i	$-0.893786\pi$
$719$	−36432.0	−1.88969	−0.944843	+	0.327523i	$0.893786\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 4854.00i	− 0.249685i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 21616.0i	− 1.10274i	−0.834260	−	0.551371i	$-0.814105\pi$
$727$	− 21616.0i	− 1.10274i	0.834260	−	0.551371i	$-0.185895\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	1368.00	0.0692166
$732$	0	0
$733$	28102.0i	1.41606i	0.706183	+	0.708029i	$0.250416\pi$
$733$	28102.0i	1.41606i	−0.706183	+	0.708029i	$0.749584\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	752.000i	0.0375852i
$738$	0	0
$739$	−764.000	−0.0380300	−0.0190150	−	0.999819i	$-0.506053\pi$
$739$	−764.000	−0.0380300	−0.0190150	+	0.999819i	$0.506053\pi$
$740$	0	0
$741$	−7128.00	−0.353379
$742$	0	0
$743$	6256.00i	0.308897i	0.988001	+	0.154448i	$0.0493600\pi$
$743$	6256.00i	0.308897i	−0.988001	+	0.154448i	$0.950640\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 10692.0i	− 0.523695i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	1184.00	0.0575297	0.0287648	−	0.999586i	$-0.490843\pi$
$751$	1184.00	0.0575297	0.0287648	+	0.999586i	$0.490843\pi$
$752$	0	0
$753$	− 17340.0i	− 0.839183i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 26446.0i	− 1.26974i	−0.772617	−	0.634872i	$-0.781053\pi$
$757$	− 26446.0i	− 1.26974i	0.772617	−	0.634872i	$-0.218947\pi$
$758$	0	0
$759$	1152.00	0.0550922
$760$	0	0
$761$	36778.0	1.75191	0.875954	−	0.482395i	$-0.160233\pi$
$761$	36778.0	1.75191	0.875954	+	0.482395i	$0.160233\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1944.00i	− 0.0915173i
$768$	0	0
$769$	10302.0	0.483094	0.241547	−	0.970389i	$-0.422345\pi$
$769$	10302.0	0.483094	0.241547	+	0.970389i	$0.422345\pi$
$770$	0	0
$771$	−7782.00	−0.363504
$772$	0	0
$773$	− 4674.00i	− 0.217480i	−0.994070	−	0.108740i	$-0.965318\pi$
$773$	− 4674.00i	− 0.217480i	0.994070	−	0.108740i	$-0.0346816\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	264.000	0.0121422
$780$	0	0
$781$	−2528.00	−0.115825
$782$	0	0
$783$	− 3618.00i	− 0.165130i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	23084.0i	1.04556i	0.852468	+	0.522780i	$0.175105\pi$
$787$	23084.0i	1.04556i	−0.852468	+	0.522780i	$0.824895\pi$
$788$	0	0
$789$	11088.0	0.500308
$790$	0	0
$791$	0	0
$792$	0	0
$793$	− 23868.0i	− 1.06882i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 10694.0i	− 0.475283i	−0.971353	−	0.237642i	$-0.923626\pi$
$797$	− 10694.0i	− 0.475283i	0.971353	−	0.237642i	$-0.0763744\pi$
$798$	0	0
$799$	22800.0	1.00952
$800$	0	0
$801$	−6246.00	−0.275520
$802$	0	0
$803$	− 1560.00i	− 0.0685569i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 6750.00i	− 0.294438i
$808$	0	0
$809$	−9594.00	−0.416943	−0.208472	−	0.978028i	$-0.566849\pi$
$809$	−9594.00	−0.416943	−0.208472	+	0.978028i	$0.566849\pi$
$810$	0	0
$811$	10244.0	0.443546	0.221773	−	0.975098i	$-0.428816\pi$
$811$	10244.0	0.443546	0.221773	+	0.975098i	$0.428816\pi$
$812$	0	0
$813$	− 6624.00i	− 0.285749i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 528.000i	− 0.0226100i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1390.00	0.0590881	0.0295441	−	0.999563i	$-0.490594\pi$
$821$	1390.00	0.0590881	0.0295441	+	0.999563i	$0.490594\pi$
$822$	0	0
$823$	− 8448.00i	− 0.357811i	−0.983866	−	0.178906i	$-0.942744\pi$
$823$	− 8448.00i	− 0.357811i	0.983866	−	0.178906i	$-0.0572557\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 41484.0i	− 1.74430i	−0.489234	−	0.872152i	$-0.662724\pi$
$827$	− 41484.0i	− 1.74430i	0.489234	−	0.872152i	$-0.337276\pi$
$828$	0	0
$829$	31610.0	1.32432	0.662160	−	0.749363i	$-0.269640\pi$
$829$	31610.0	1.32432	0.662160	+	0.749363i	$0.269640\pi$
$830$	0	0
$831$	5046.00	0.210642
$832$	0	0
$833$	− 39102.0i	− 1.62642i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 7344.00i	− 0.303280i
$838$	0	0
$839$	−38264.0	−1.57452	−0.787259	−	0.616623i	$-0.788500\pi$
$839$	−38264.0	−1.57452	−0.787259	+	0.616623i	$0.788500\pi$
$840$	0	0
$841$	−6433.00	−0.263766
$842$	0	0
$843$	− 21918.0i	− 0.895488i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−24492.0	−0.990063
$850$	0	0
$851$	−9408.00	−0.378968
$852$	0	0
$853$	30350.0i	1.21825i	0.793076	+	0.609123i	$0.208479\pi$
$853$	30350.0i	1.21825i	−0.793076	+	0.609123i	$0.791521\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	12566.0i	0.500871i	0.968133	+	0.250435i	$0.0805738\pi$
$857$	12566.0i	0.500871i	−0.968133	+	0.250435i	$0.919426\pi$
$858$	0	0
$859$	−11812.0	−0.469174	−0.234587	−	0.972095i	$-0.575374\pi$
$859$	−11812.0	−0.469174	−0.234587	+	0.972095i	$0.575374\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 31496.0i	− 1.24234i	−0.783677	−	0.621168i	$-0.786658\pi$
$863$	− 31496.0i	− 1.24234i	0.783677	−	0.621168i	$-0.213342\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	24249.0i	0.949872i
$868$	0	0
$869$	−2752.00	−0.107428
$870$	0	0
$871$	−10152.0	−0.394934
$872$	0	0
$873$	− 15534.0i	− 0.602229i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 7414.00i	− 0.285465i	−0.989761	−	0.142733i	$-0.954411\pi$
$877$	− 7414.00i	− 0.285465i	0.989761	−	0.142733i	$-0.0455889\pi$
$878$	0	0
$879$	−1542.00	−0.0591699
$880$	0	0
$881$	−22190.0	−0.848581	−0.424291	−	0.905526i	$-0.639477\pi$
$881$	−22190.0	−0.848581	−0.424291	+	0.905526i	$0.639477\pi$
$882$	0	0
$883$	− 10172.0i	− 0.387673i	−0.981034	−	0.193836i	$-0.937907\pi$
$883$	− 10172.0i	− 0.387673i	0.981034	−	0.193836i	$-0.0620931\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	20784.0i	0.786763i	0.919375	+	0.393381i	$0.128695\pi$
$887$	20784.0i	0.786763i	−0.919375	+	0.393381i	$0.871305\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	324.000	0.0121823
$892$	0	0
$893$	− 8800.00i	− 0.329766i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	15552.0i	0.578892i
$898$	0	0
$899$	36448.0	1.35218
$900$	0	0
$901$	74556.0	2.75674
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	7652.00i	0.280133i	0.990142	+	0.140066i	$0.0447316\pi$
$907$	7652.00i	0.280133i	−0.990142	+	0.140066i	$0.955268\pi$
$908$	0	0
$909$	−10638.0	−0.388163
$910$	0	0
$911$	19296.0	0.701762	0.350881	−	0.936420i	$-0.385882\pi$
$911$	19296.0	0.701762	0.350881	+	0.936420i	$0.385882\pi$
$912$	0	0
$913$	4752.00i	0.172254i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	35896.0	1.28847	0.644233	−	0.764830i	$-0.277177\pi$
$919$	35896.0	1.28847	0.644233	+	0.764830i	$0.277177\pi$
$920$	0	0
$921$	7428.00	0.265756
$922$	0	0
$923$	− 34128.0i	− 1.21705i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 17712.0i	− 0.627550i
$928$	0	0
$929$	16350.0	0.577423	0.288712	−	0.957416i	$-0.406773\pi$
$929$	16350.0	0.577423	0.288712	+	0.957416i	$0.406773\pi$
$930$	0	0
$931$	−15092.0	−0.531279
$932$	0	0
$933$	− 6888.00i	− 0.241697i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	19686.0i	0.686354i	0.939271	+	0.343177i	$0.111503\pi$
$937$	19686.0i	0.686354i	−0.939271	+	0.343177i	$0.888497\pi$
$938$	0	0
$939$	−29634.0	−1.02989
$940$	0	0
$941$	56246.0	1.94853	0.974265	−	0.225405i	$-0.0723704\pi$
$941$	56246.0	1.94853	0.974265	+	0.225405i	$0.0723704\pi$
$942$	0	0
$943$	− 576.000i	− 0.0198909i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	11436.0i	0.392418i	0.980562	+	0.196209i	$0.0628631\pi$
$947$	11436.0i	0.392418i	−0.980562	+	0.196209i	$0.937137\pi$
$948$	0	0
$949$	21060.0	0.720376
$950$	0	0
$951$	6414.00	0.218705
$952$	0	0
$953$	− 22582.0i	− 0.767579i	−0.923421	−	0.383789i	$-0.874619\pi$
$953$	− 22582.0i	− 0.767579i	0.923421	−	0.383789i	$-0.125381\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1608.00i	0.0543148i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	44193.0	1.48343
$962$	0	0
$963$	7164.00i	0.239727i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2112.00i	0.0702351i	0.999383	+	0.0351175i	$0.0111806\pi$
$967$	2112.00i	0.0702351i	−0.999383	+	0.0351175i	$0.988819\pi$
$968$	0	0
$969$	15048.0	0.498877
$970$	0	0
$971$	−47964.0	−1.58521	−0.792605	−	0.609736i	$-0.791275\pi$
$971$	−47964.0	−1.58521	−0.792605	+	0.609736i	$0.791275\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	10510.0i	0.344160i	0.985083	+	0.172080i	$0.0550488\pi$
$977$	10510.0i	0.344160i	−0.985083	+	0.172080i	$0.944951\pi$
$978$	0	0
$979$	2776.00	0.0906245
$980$	0	0
$981$	3078.00	0.100176
$982$	0	0
$983$	11488.0i	0.372747i	0.982479	+	0.186373i	$0.0596734\pi$
$983$	11488.0i	0.372747i	−0.982479	+	0.186373i	$0.940327\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1152.00	−0.0370389
$990$	0	0
$991$	−23120.0	−0.741101	−0.370550	−	0.928812i	$-0.620831\pi$
$991$	−23120.0	−0.741101	−0.370550	+	0.928812i	$0.620831\pi$
$992$	0	0
$993$	19380.0i	0.619341i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 30078.0i	− 0.955446i	−0.878510	−	0.477723i	$-0.841462\pi$
$997$	− 30078.0i	− 0.955446i	0.878510	−	0.477723i	$-0.158538\pi$
$998$	0	0
$999$	−2646.00	−0.0837995

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.4.f.d.49.1		2
3.2	odd	2		1800.4.f.m.649.2		2
4.3	odd	2		1200.4.f.l.49.2		2
5.2	odd	4		120.4.a.d.1.1	✓	1
5.3	odd	4		600.4.a.m.1.1		1
5.4	even	2	inner	600.4.f.d.49.2		2
15.2	even	4		360.4.a.c.1.1		1
15.8	even	4		1800.4.a.s.1.1		1
15.14	odd	2		1800.4.f.m.649.1		2
20.3	even	4		1200.4.a.j.1.1		1
20.7	even	4		240.4.a.k.1.1		1
20.19	odd	2		1200.4.f.l.49.1		2
40.27	even	4		960.4.a.e.1.1		1
40.37	odd	4		960.4.a.x.1.1		1
60.47	odd	4		720.4.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.4.a.d.1.1	✓	1	5.2	odd	4
240.4.a.k.1.1		1	20.7	even	4
360.4.a.c.1.1		1	15.2	even	4
600.4.a.m.1.1		1	5.3	odd	4
600.4.f.d.49.1		2	1.1	even	1	trivial
600.4.f.d.49.2		2	5.4	even	2	inner
720.4.a.i.1.1		1	60.47	odd	4
960.4.a.e.1.1		1	40.27	even	4
960.4.a.x.1.1		1	40.37	odd	4
1200.4.a.j.1.1		1	20.3	even	4
1200.4.f.l.49.1		2	20.19	odd	2
1200.4.f.l.49.2		2	4.3	odd	2
1800.4.a.s.1.1		1	15.8	even	4
1800.4.f.m.649.1		2	15.14	odd	2
1800.4.f.m.649.2		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	600.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -9.00000 q^{9} +4.00000 q^{11} +54.0000i q^{13} -114.000i q^{17} -44.0000 q^{19} +96.0000i q^{23} +27.0000i q^{27} -134.000 q^{29} -272.000 q^{31} -12.0000i q^{33} +98.0000i q^{37} +162.000 q^{39} -6.00000 q^{41} +12.0000i q^{43} +200.000i q^{47} +343.000 q^{49} -342.000 q^{51} +654.000i q^{53} +132.000i q^{57} -36.0000 q^{59} -442.000 q^{61} +188.000i q^{67} +288.000 q^{69} -632.000 q^{71} -390.000i q^{73} -688.000 q^{79} +81.0000 q^{81} +1188.00i q^{83} +402.000i q^{87} +694.000 q^{89} +816.000i q^{93} +1726.00i q^{97} -36.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{9} + 8 q^{11} - 88 q^{19} - 268 q^{29} - 544 q^{31} + 324 q^{39} - 12 q^{41} + 686 q^{49} - 684 q^{51} - 72 q^{59} - 884 q^{61} + 576 q^{69} - 1264 q^{71} - 1376 q^{79} + 162 q^{81} + 1388 q^{89}+ \cdots - 72 q^{99}+O(q^{100}) \) 2 * q - 18 * q^9 + 8 * q^11 - 88 * q^19 - 268 * q^29 - 544 * q^31 + 324 * q^39 - 12 * q^41 + 686 * q^49 - 684 * q^51 - 72 * q^59 - 884 * q^61 + 576 * q^69 - 1264 * q^71 - 1376 * q^79 + 162 * q^81 + 1388 * q^89 - 72 * q^99

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