LMFDB - Embedded newform 120.6.f.a.49.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [120,6,Mod(49,120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(120, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("120.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 120 = 2^{3} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	120.f (of order $2$, degree $1$, 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:	$19.2460583776$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.25787221056.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 61x^{4} + 852x^{2} + 576 $ x^6 + 61x^4 + 852x^2 + 576
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.6
Root			$0.843753i$ of defining polynomial
Character	$\chi$	$=$	120.49
Dual form			120.6.f.a.49.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000i q^{3} +(42.6733 + 36.1108i) q^{5} +168.485i q^{7} -81.0000 q^{9} +O(q^{10})$	$q+9.00000i q^{3} +(42.6733 + 36.1108i) q^{5} +168.485i q^{7} -81.0000 q^{9} +10.5832 q^{11} +98.9988i q^{13} +(-324.997 + 384.060i) q^{15} -974.359i q^{17} +482.724 q^{19} -1516.37 q^{21} +2624.53i q^{23} +(517.018 + 3081.93i) q^{25} -729.000i q^{27} -6083.00 q^{29} -7163.79 q^{31} +95.2486i q^{33} +(-6084.14 + 7189.82i) q^{35} -634.887i q^{37} -890.989 q^{39} +1440.32 q^{41} +1258.09i q^{43} +(-3456.54 - 2924.98i) q^{45} +25807.4i q^{47} -11580.3 q^{49} +8769.23 q^{51} -4893.61i q^{53} +(451.619 + 382.167i) q^{55} +4344.52i q^{57} +39053.9 q^{59} -47099.7 q^{61} -13647.3i q^{63} +(-3574.93 + 4224.60i) q^{65} +23682.2i q^{67} -23620.8 q^{69} +55289.0 q^{71} -71561.5i q^{73} +(-27737.4 + 4653.16i) q^{75} +1783.11i q^{77} +68077.8 q^{79} +6561.00 q^{81} +6650.01i q^{83} +(35184.9 - 41579.1i) q^{85} -54747.0i q^{87} +76344.8 q^{89} -16679.8 q^{91} -64474.1i q^{93} +(20599.4 + 17431.6i) q^{95} -147219. i q^{97} -857.237 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 50 q^{5} - 486 q^{9}+O(q^{10}) $ 6 * q + 50 * q^5 - 486 * q^9	$ 6 q + 50 q^{5} - 486 q^{9} - 664 q^{11} + 540 q^{15} + 288 q^{19} - 1872 q^{21} + 3750 q^{25} - 1892 q^{29} - 10248 q^{31} - 18880 q^{35} + 1584 q^{39} - 1324 q^{41} - 4050 q^{45} + 28410 q^{49} + 20088 q^{51} - 47160 q^{55} - 10296 q^{59} - 52116 q^{61} - 13480 q^{65} + 51624 q^{69} - 28288 q^{71} - 41400 q^{75} + 220200 q^{79} + 39366 q^{81} + 95880 q^{85} + 351420 q^{89} - 17088 q^{91} - 349120 q^{95} + 53784 q^{99}+O(q^{100}) $ 6 * q + 50 * q^5 - 486 * q^9 - 664 * q^11 + 540 * q^15 + 288 * q^19 - 1872 * q^21 + 3750 * q^25 - 1892 * q^29 - 10248 * q^31 - 18880 * q^35 + 1584 * q^39 - 1324 * q^41 - 4050 * q^45 + 28410 * q^49 + 20088 * q^51 - 47160 * q^55 - 10296 * q^59 - 52116 * q^61 - 13480 * q^65 + 51624 * q^69 - 28288 * q^71 - 41400 * q^75 + 220200 * q^79 + 39366 * q^81 + 95880 * q^85 + 351420 * q^89 - 17088 * q^91 - 349120 * q^95 + 53784 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/120\mathbb{Z}\right)^\times$.

$n$	$31$	$41$	$61$	$97$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$			9.00000i			0.577350i
$3$			9.00000i			0.577350i
$4$		0			0
$5$	42.6733	+	36.1108i	0.763363	+	0.645970i
$5$	42.6733	+	36.1108i	0.763363	+	0.645970i
$6$		0			0
$7$			168.485i			1.29962i	0.760096	+	0.649811i	$0.225152\pi$
$7$			168.485i			1.29962i	−0.760096	+	0.649811i	$0.774848\pi$
$8$		0			0
$9$	−81.0000			−0.333333
$10$		0			0
$11$	10.5832			0.0263715			0.0131857	−	0.999913i	$-0.495803\pi$
$11$	10.5832			0.0263715			0.0131857	+	0.999913i	$0.495803\pi$
$12$		0			0
$13$			98.9988i			0.162469i	0.996695	+	0.0812347i	$0.0258863\pi$
$13$			98.9988i			0.162469i	−0.996695	+	0.0812347i	$0.974114\pi$
$14$		0			0
$15$	−324.997	+	384.060i	−0.372951	+	0.440728i
$16$		0			0
$17$		−	974.359i		−	0.817705i	−0.912601	−	0.408852i	$-0.865929\pi$
$17$		−	974.359i		−	0.817705i	0.912601	−	0.408852i	$-0.134071\pi$
$18$		0			0
$19$	482.724			0.306772			0.153386	−	0.988166i	$-0.450982\pi$
$19$	482.724			0.306772			0.153386	+	0.988166i	$0.450982\pi$
$20$		0			0
$21$	−1516.37			−0.750337
$22$		0			0
$23$			2624.53i			1.03450i	0.855833	+	0.517252i	$0.173045\pi$
$23$			2624.53i			1.03450i	−0.855833	+	0.517252i	$0.826955\pi$
$24$		0			0
$25$	517.018	+	3081.93i	0.165446	+	0.986219i
$26$		0			0
$27$		−	729.000i		−	0.192450i
$28$		0			0
$29$	−6083.00			−1.34314			−0.671572	−	0.740939i	$-0.734381\pi$
$29$	−6083.00			−1.34314			−0.671572	+	0.740939i	$0.734381\pi$
$30$		0			0
$31$	−7163.79			−1.33887			−0.669435	−	0.742870i	$-0.733464\pi$
$31$	−7163.79			−1.33887			−0.669435	+	0.742870i	$0.733464\pi$
$32$		0			0
$33$			95.2486i			0.0152256i
$34$		0			0
$35$	−6084.14	+	7189.82i	−0.839516	+	0.992083i
$36$		0			0
$37$		−	634.887i		−	0.0762417i	−0.999273	−	0.0381208i	$-0.987863\pi$
$37$		−	634.887i		−	0.0762417i	0.999273	−	0.0381208i	$-0.0121372\pi$
$38$		0			0
$39$	−890.989			−0.0938018
$40$		0			0
$41$	1440.32			0.133814			0.0669069	−	0.997759i	$-0.478687\pi$
$41$	1440.32			0.133814			0.0669069	+	0.997759i	$0.478687\pi$
$42$		0			0
$43$			1258.09i			0.103763i	0.998653	+	0.0518813i	$0.0165218\pi$
$43$			1258.09i			0.103763i	−0.998653	+	0.0518813i	$0.983478\pi$
$44$		0			0
$45$	−3456.54	−	2924.98i	−0.254454	−	0.215323i
$46$		0			0
$47$			25807.4i			1.70412i	0.523447	+	0.852058i	$0.324646\pi$
$47$			25807.4i			1.70412i	−0.523447	+	0.852058i	$0.675354\pi$
$48$		0			0
$49$	−11580.3			−0.689016
$50$		0			0
$51$	8769.23			0.472102
$52$		0			0
$53$		−	4893.61i		−	0.239298i	−0.992816	−	0.119649i	$-0.961823\pi$
$53$		−	4893.61i		−	0.239298i	0.992816	−	0.119649i	$-0.0381770\pi$
$54$		0			0
$55$	451.619	+	382.167i	0.0201310	+	0.0170352i
$56$		0			0
$57$			4344.52i			0.177115i
$58$		0			0
$59$	39053.9			1.46061			0.730305	−	0.683121i	$-0.239378\pi$
$59$	39053.9			1.46061			0.730305	+	0.683121i	$0.239378\pi$
$60$		0			0
$61$	−47099.7			−1.62067			−0.810333	−	0.585970i	$-0.800714\pi$
$61$	−47099.7			−1.62067			−0.810333	+	0.585970i	$0.800714\pi$
$62$		0			0
$63$		−	13647.3i		−	0.433207i
$64$		0			0
$65$	−3574.93	+	4224.60i	−0.104950	+	0.124023i
$66$		0			0
$67$			23682.2i			0.644518i	0.946651	+	0.322259i	$0.104442\pi$
$67$			23682.2i			0.644518i	−0.946651	+	0.322259i	$0.895558\pi$
$68$		0			0
$69$	−23620.8			−0.597272
$70$		0			0
$71$	55289.0			1.30165			0.650823	−	0.759229i	$-0.274424\pi$
$71$	55289.0			1.30165			0.650823	+	0.759229i	$0.274424\pi$
$72$		0			0
$73$		−	71561.5i		−	1.57171i	−0.618411	−	0.785855i	$-0.712223\pi$
$73$		−	71561.5i		−	1.57171i	0.618411	−	0.785855i	$-0.287777\pi$
$74$		0			0
$75$	−27737.4	+	4653.16i	−0.569394	+	0.0955201i
$76$		0			0
$77$			1783.11i			0.0342729i
$78$		0			0
$79$	68077.8			1.22726			0.613632	−	0.789592i	$-0.289708\pi$
$79$	68077.8			1.22726			0.613632	+	0.789592i	$0.289708\pi$
$80$		0			0
$81$	6561.00			0.111111
$82$		0			0
$83$			6650.01i			0.105956i	0.998596	+	0.0529782i	$0.0168714\pi$
$83$			6650.01i			0.105956i	−0.998596	+	0.0529782i	$0.983129\pi$
$84$		0			0
$85$	35184.9	−	41579.1i	0.528213	−	0.624205i
$86$		0			0
$87$		−	54747.0i		−	0.775465i
$88$		0			0
$89$	76344.8			1.02165			0.510827	−	0.859683i	$-0.329339\pi$
$89$	76344.8			1.02165			0.510827	+	0.859683i	$0.329339\pi$
$90$		0			0
$91$	−16679.8			−0.211149
$92$		0			0
$93$		−	64474.1i		−	0.772997i
$94$		0			0
$95$	20599.4	+	17431.6i	0.234178	+	0.198165i
$96$		0			0
$97$		−	147219.i		−	1.58867i	−0.607480	−	0.794335i	$-0.707819\pi$
$97$		−	147219.i		−	1.58867i	0.607480	−	0.794335i	$-0.292181\pi$
$98$		0			0
$99$	−857.237			−0.00879049
$100$		0			0
$101$	17758.0			0.173217			0.0866083	−	0.996242i	$-0.472397\pi$
$101$	17758.0			0.173217			0.0866083	+	0.996242i	$0.472397\pi$
$102$		0			0
$103$		−	2414.15i		−	0.0224219i	−0.999937	−	0.0112109i	$-0.996431\pi$
$103$		−	2414.15i		−	0.0224219i	0.999937	−	0.0112109i	$-0.00356862\pi$
$104$		0			0
$105$	−64708.4	−	54757.3i	−0.572779	−	0.484695i
$106$		0			0
$107$			92188.9i			0.778429i	0.921147	+	0.389215i	$0.127254\pi$
$107$			92188.9i			0.778429i	−0.921147	+	0.389215i	$0.872746\pi$
$108$		0			0
$109$	221512.			1.78579			0.892896	−	0.450263i	$-0.148670\pi$
$109$	221512.			1.78579			0.892896	+	0.450263i	$0.148670\pi$
$110$		0			0
$111$	5713.99			0.0440181
$112$		0			0
$113$			177162.i			1.30519i	0.757705	+	0.652597i	$0.226320\pi$
$113$			177162.i			1.30519i	−0.757705	+	0.652597i	$0.773680\pi$
$114$		0			0
$115$	−94774.1	+	111997.i	−0.668259	+	0.789703i
$116$		0			0
$117$		−	8018.90i		−	0.0541565i
$118$		0			0
$119$	164165.			1.06271
$120$		0			0
$121$	−160939.			−0.999305
$122$		0			0
$123$			12962.9i			0.0772574i
$124$		0			0
$125$	−89228.3	+	150186.i	−0.510773	+	0.859716i
$126$		0			0
$127$		−	161397.i		−	0.887947i	−0.896040	−	0.443973i	$-0.853569\pi$
$127$		−	161397.i		−	0.887947i	0.896040	−	0.443973i	$-0.146431\pi$
$128$		0			0
$129$	−11322.8			−0.0599074
$130$		0			0
$131$	236198.			1.20254			0.601269	−	0.799047i	$-0.294662\pi$
$131$	236198.			1.20254			0.601269	+	0.799047i	$0.294662\pi$
$132$		0			0
$133$			81331.9i			0.398687i
$134$		0			0
$135$	26324.8	−	31108.8i	0.124317	−	0.146909i
$136$		0			0
$137$			8498.24i			0.0386837i	0.999813	+	0.0193418i	$0.00615708\pi$
$137$			8498.24i			0.0386837i	−0.999813	+	0.0193418i	$0.993843\pi$
$138$		0			0
$139$	−305893.			−1.34287			−0.671433	−	0.741065i	$-0.734321\pi$
$139$	−305893.			−1.34287			−0.671433	+	0.741065i	$0.734321\pi$
$140$		0			0
$141$	−232266.			−0.983872
$142$		0			0
$143$			1047.72i			0.00428456i
$144$		0			0
$145$	−259582.	−	219662.i	−1.02531	−	0.867631i
$146$		0			0
$147$		−	104223.i		−	0.397804i
$148$		0			0
$149$	395616.			1.45985			0.729924	−	0.683528i	$-0.239555\pi$
$149$	395616.			1.45985			0.729924	+	0.683528i	$0.239555\pi$
$150$		0			0
$151$	−398843.			−1.42351			−0.711754	−	0.702429i	$-0.752099\pi$
$151$	−398843.			−1.42351			−0.711754	+	0.702429i	$0.752099\pi$
$152$		0			0
$153$			78923.0i			0.272568i
$154$		0			0
$155$	−305702.	−	258690.i	−1.02204	−	0.864870i
$156$		0			0
$157$			334415.i			1.08277i	0.840775	+	0.541385i	$0.182100\pi$
$157$			334415.i			1.08277i	−0.840775	+	0.541385i	$0.817900\pi$
$158$		0			0
$159$	44042.5			0.138159
$160$		0			0
$161$	−442195.			−1.34446
$162$		0			0
$163$			73669.6i			0.217180i	0.994087	+	0.108590i	$0.0346335\pi$
$163$			73669.6i			0.217180i	−0.994087	+	0.108590i	$0.965366\pi$
$164$		0			0
$165$	−3439.50	+	4064.57i	−0.00983526	+	0.0116226i
$166$		0			0
$167$			525651.i			1.45850i	0.684248	+	0.729250i	$0.260131\pi$
$167$			525651.i			1.45850i	−0.684248	+	0.729250i	$0.739869\pi$
$168$		0			0
$169$	361492.			0.973604
$170$		0			0
$171$	−39100.7			−0.102257
$172$		0			0
$173$			732557.i			1.86091i	0.366401	+	0.930457i	$0.380590\pi$
$173$			732557.i			1.86091i	−0.366401	+	0.930457i	$0.619410\pi$
$174$		0			0
$175$	−519261.	+	87109.9i	−1.28171	+	0.215017i
$176$		0			0
$177$			351485.i			0.843284i
$178$		0			0
$179$	353522.			0.824677			0.412338	−	0.911031i	$-0.364712\pi$
$179$	353522.			0.824677			0.412338	+	0.911031i	$0.364712\pi$
$180$		0			0
$181$	665873.			1.51076			0.755380	−	0.655288i	$-0.227453\pi$
$181$	665873.			1.51076			0.755380	+	0.655288i	$0.227453\pi$
$182$		0			0
$183$		−	423897.i		−	0.935692i
$184$		0			0
$185$	22926.3	−	27092.7i	0.0492498	−	0.0582001i
$186$		0			0
$187$		−	10311.8i		−	0.0215641i
$188$		0			0
$189$	122826.			0.250112
$190$		0			0
$191$	−901873.			−1.78880			−0.894400	−	0.447267i	$-0.852397\pi$
$191$	−901873.			−1.78880			−0.894400	+	0.447267i	$0.852397\pi$
$192$		0			0
$193$			133364.i			0.257719i	0.991663	+	0.128859i	$0.0411316\pi$
$193$			133364.i			0.257719i	−0.991663	+	0.128859i	$0.958868\pi$
$194$		0			0
$195$	−38021.4	−	32174.4i	−0.0716048	−	0.0605931i
$196$		0			0
$197$		−	163286.i		−	0.299766i	−0.988704	−	0.149883i	$-0.952110\pi$
$197$		−	163286.i		−	0.299766i	0.988704	−	0.149883i	$-0.0478898\pi$
$198$		0			0
$199$	567475.			1.01581			0.507906	−	0.861412i	$-0.330420\pi$
$199$	567475.			1.01581			0.507906	+	0.861412i	$0.330420\pi$
$200$		0			0
$201$	−213140.			−0.372113
$202$		0			0
$203$		−	1.02490e6i		−	1.74558i
$204$		0			0
$205$	61463.4	+	52011.3i	0.102148	+	0.0864397i
$206$		0			0
$207$		−	212587.i		−	0.344835i
$208$		0			0
$209$	5108.75			0.00809002
$210$		0			0
$211$	355737.			0.550077			0.275038	−	0.961433i	$-0.411309\pi$
$211$	355737.			0.550077			0.275038	+	0.961433i	$0.411309\pi$
$212$		0			0
$213$			497601.i			0.751506i
$214$		0			0
$215$	−45430.7	+	53686.9i	−0.0670276	+	0.0792086i
$216$		0			0
$217$		−	1.20699e6i		−	1.74002i
$218$		0			0
$219$	644053.			0.907427
$220$		0			0
$221$	96460.4			0.132852
$222$		0			0
$223$			146827.i			0.197717i	0.995102	+	0.0988585i	$0.0315191\pi$
$223$			146827.i			0.197717i	−0.995102	+	0.0988585i	$0.968481\pi$
$224$		0			0
$225$	−41878.4	−	249637.i	−0.0551486	−	0.328740i
$226$		0			0
$227$		−	781767.i		−	1.00696i	−0.864007	−	0.503480i	$-0.832053\pi$
$227$		−	781767.i		−	1.00696i	0.864007	−	0.503480i	$-0.167947\pi$
$228$		0			0
$229$	1.02791e6			1.29529			0.647643	−	0.761944i	$-0.275755\pi$
$229$	1.02791e6			1.29529			0.647643	+	0.761944i	$0.275755\pi$
$230$		0			0
$231$	−16048.0			−0.0197875
$232$		0			0
$233$		−	769122.i		−	0.928123i	−0.885803	−	0.464061i	$-0.846392\pi$
$233$		−	769122.i		−	0.928123i	0.885803	−	0.464061i	$-0.153608\pi$
$234$		0			0
$235$	−931925.	+	1.10129e6i	−1.10081	+	1.30086i
$236$		0			0
$237$			612700.i			0.708561i
$238$		0			0
$239$	1.03405e6			1.17097			0.585485	−	0.810683i	$-0.300904\pi$
$239$	1.03405e6			1.17097			0.585485	+	0.810683i	$0.300904\pi$
$240$		0			0
$241$	554994.			0.615525			0.307762	−	0.951463i	$-0.400420\pi$
$241$	554994.			0.615525			0.307762	+	0.951463i	$0.400420\pi$
$242$		0			0
$243$			59049.0i			0.0641500i
$244$		0			0
$245$	−494169.	−	418174.i	−0.525969	−	0.445084i
$246$		0			0
$247$			47789.1i			0.0498410i
$248$		0			0
$249$	−59850.1			−0.0611740
$250$		0			0
$251$	265109.			0.265607			0.132804	−	0.991142i	$-0.457602\pi$
$251$	265109.			0.265607			0.132804	+	0.991142i	$0.457602\pi$
$252$		0			0
$253$			27775.9i			0.0272814i
$254$		0			0
$255$	374212.	+	316664.i	0.360385	+	0.304964i
$256$		0			0
$257$		−	783952.i		−	0.740383i	−0.928955	−	0.370192i	$-0.879292\pi$
$257$		−	783952.i		−	0.740383i	0.928955	−	0.370192i	$-0.120708\pi$
$258$		0			0
$259$	106969.			0.0990853
$260$		0			0
$261$	492723.			0.447715
$262$		0			0
$263$		−	1.66434e6i		−	1.48372i	−0.670555	−	0.741859i	$-0.733944\pi$
$263$		−	1.66434e6i		−	1.48372i	0.670555	−	0.741859i	$-0.266056\pi$
$264$		0			0
$265$	176712.	−	208826.i	0.154580	−	0.182672i
$266$		0			0
$267$			687103.i			0.589853i
$268$		0			0
$269$	−301598.			−0.254125			−0.127063	−	0.991895i	$-0.540555\pi$
$269$	−301598.			−0.254125			−0.127063	+	0.991895i	$0.540555\pi$
$270$		0			0
$271$	−1.19814e6			−0.991024			−0.495512	−	0.868601i	$-0.665020\pi$
$271$	−1.19814e6			−0.991024			−0.495512	+	0.868601i	$0.665020\pi$
$272$		0			0
$273$		−	150119.i		−	0.121907i
$274$		0			0
$275$	5471.69	+	32616.6i	0.00436305	+	0.0260080i
$276$		0			0
$277$		−	565558.i		−	0.442872i	−0.975175	−	0.221436i	$-0.928926\pi$
$277$		−	565558.i		−	0.442872i	0.975175	−	0.221436i	$-0.0710743\pi$
$278$		0			0
$279$	580267.			0.446290
$280$		0			0
$281$	−903797.			−0.682818			−0.341409	−	0.939915i	$-0.610904\pi$
$281$	−903797.			−0.682818			−0.341409	+	0.939915i	$0.610904\pi$
$282$		0			0
$283$		−	714771.i		−	0.530519i	−0.964177	−	0.265259i	$-0.914542\pi$
$283$		−	714771.i		−	0.530519i	0.964177	−	0.265259i	$-0.0854576\pi$
$284$		0			0
$285$	−156884.	+	185395.i	−0.114411	+	0.135203i
$286$		0			0
$287$			242674.i			0.173907i
$288$		0			0
$289$	470482.			0.331359
$290$		0			0
$291$	1.32497e6			0.917219
$292$		0			0
$293$			1.18693e6i			0.807709i	0.914823	+	0.403855i	$0.132330\pi$
$293$			1.18693e6i			0.807709i	−0.914823	+	0.403855i	$0.867670\pi$
$294$		0			0
$295$	1.66656e6	+	1.41027e6i	1.11498	+	0.943510i
$296$		0			0
$297$		−	7715.13i		−	0.00507519i
$298$		0			0
$299$	−259826.			−0.168075
$300$		0			0
$301$	−211970.			−0.134852
$302$		0			0
$303$			159822.i			0.100007i
$304$		0			0
$305$	−2.00990e6	−	1.70081e6i	−1.23716	−	1.04690i
$306$		0			0
$307$			2.34189e6i			1.41814i	0.705137	+	0.709071i	$0.250885\pi$
$307$			2.34189e6i			1.41814i	−0.705137	+	0.709071i	$0.749115\pi$
$308$		0			0
$309$	21727.4			0.0129453
$310$		0			0
$311$	−211330.			−0.123897			−0.0619483	−	0.998079i	$-0.519731\pi$
$311$	−211330.			−0.123897			−0.0619483	+	0.998079i	$0.519731\pi$
$312$		0			0
$313$			1.85595e6i			1.07079i	0.844602	+	0.535395i	$0.179837\pi$
$313$			1.85595e6i			1.07079i	−0.844602	+	0.535395i	$0.820163\pi$
$314$		0			0
$315$	492815.	−	582375.i	0.279839	−	0.330694i
$316$		0			0
$317$			2.25437e6i			1.26002i	0.776588	+	0.630008i	$0.216948\pi$
$317$			2.25437e6i			1.26002i	−0.776588	+	0.630008i	$0.783052\pi$
$318$		0			0
$319$	−64377.5			−0.0354207
$320$		0			0
$321$	−829700.			−0.449426
$322$		0			0
$323$		−	470346.i		−	0.250849i
$324$		0			0
$325$	−305108.	+	51184.2i	−0.160230	+	0.0268799i
$326$		0			0
$327$			1.99361e6i			1.03103i
$328$		0			0
$329$	−4.34816e6			−2.21471
$330$		0			0
$331$	−1.34981e6			−0.677180			−0.338590	−	0.940934i	$-0.609950\pi$
$331$	−1.34981e6			−0.677180			−0.338590	+	0.940934i	$0.609950\pi$
$332$		0			0
$333$			51425.9i			0.0254139i
$334$		0			0
$335$	−855184.	+	1.01060e6i	−0.416340	+	0.492001i
$336$		0			0
$337$		−	3.70157e6i		−	1.77546i	−0.460363	−	0.887731i	$-0.652281\pi$
$337$		−	3.70157e6i		−	1.77546i	0.460363	−	0.887731i	$-0.347719\pi$
$338$		0			0
$339$	−1.59446e6			−0.753554
$340$		0			0
$341$	−75815.6			−0.0353080
$342$		0			0
$343$			880623.i			0.404161i
$344$		0			0
$345$	−1.00798e6	−	852966.i	−0.455935	−	0.385820i
$346$		0			0
$347$			1.69975e6i			0.757812i	0.925435	+	0.378906i	$0.123700\pi$
$347$			1.69975e6i			0.757812i	−0.925435	+	0.378906i	$0.876300\pi$
$348$		0			0
$349$	160277.			0.0704382			0.0352191	−	0.999380i	$-0.488787\pi$
$349$	160277.			0.0704382			0.0352191	+	0.999380i	$0.488787\pi$
$350$		0			0
$351$	72170.1			0.0312673
$352$		0			0
$353$		−	1.59927e6i		−	0.683103i	−0.939863	−	0.341552i	$-0.889048\pi$
$353$		−	1.59927e6i		−	0.683103i	0.939863	−	0.341552i	$-0.110952\pi$
$354$		0			0
$355$	2.35936e6	+	1.99653e6i	0.993629	+	0.840825i
$356$		0			0
$357$			1.47749e6i			0.613554i
$358$		0			0
$359$	−1.20287e6			−0.492586			−0.246293	−	0.969195i	$-0.579212\pi$
$359$	−1.20287e6			−0.492586			−0.246293	+	0.969195i	$0.579212\pi$
$360$		0			0
$361$	−2.24308e6			−0.905891
$362$		0			0
$363$		−	1.44845e6i		−	0.576949i
$364$		0			0
$365$	2.58414e6	−	3.05376e6i	1.01528	−	1.19978i
$366$		0			0
$367$		−	1.03609e6i		−	0.401544i	−0.979638	−	0.200772i	$-0.935655\pi$
$367$		−	1.03609e6i		−	0.401544i	0.979638	−	0.200772i	$-0.0643450\pi$
$368$		0			0
$369$	−116666.			−0.0446046
$370$		0			0
$371$	824502.			0.310997
$372$		0			0
$373$			4.12251e6i			1.53423i	0.641512	+	0.767113i	$0.278307\pi$
$373$			4.12251e6i			1.53423i	−0.641512	+	0.767113i	$0.721693\pi$
$374$		0			0
$375$	−1.35168e6	−	803055.i	−0.496357	−	0.294895i
$376$		0			0
$377$		−	602210.i		−	0.218220i
$378$		0			0
$379$	−2.87944e6			−1.02970			−0.514849	−	0.857281i	$-0.672152\pi$
$379$	−2.87944e6			−1.02970			−0.514849	+	0.857281i	$0.672152\pi$
$380$		0			0
$381$	1.45258e6			0.512656
$382$		0			0
$383$		−	486001.i		−	0.169294i	−0.996411	−	0.0846468i	$-0.973024\pi$
$383$		−	486001.i		−	0.169294i	0.996411	−	0.0846468i	$-0.0269762\pi$
$384$		0			0
$385$	−64389.5	+	76091.1i	−0.0221393	+	0.0261627i
$386$		0			0
$387$		−	101905.i		−	0.0345876i
$388$		0			0
$389$	2.43116e6			0.814592			0.407296	−	0.913296i	$-0.366472\pi$
$389$	2.43116e6			0.814592			0.407296	+	0.913296i	$0.366472\pi$
$390$		0			0
$391$	2.55724e6			0.845920
$392$		0			0
$393$			2.12579e6i			0.694286i
$394$		0			0
$395$	2.90510e6	+	2.45834e6i	0.936847	+	0.792775i
$396$		0			0
$397$		−	4.09895e6i		−	1.30526i	−0.757678	−	0.652629i	$-0.773666\pi$
$397$		−	4.09895e6i		−	1.30526i	0.757678	−	0.652629i	$-0.226334\pi$
$398$		0			0
$399$	−731987.			−0.230182
$400$		0			0
$401$	3.20180e6			0.994335			0.497167	−	0.867655i	$-0.334373\pi$
$401$	3.20180e6			0.994335			0.497167	+	0.867655i	$0.334373\pi$
$402$		0			0
$403$		−	709207.i		−	0.217526i
$404$		0			0
$405$	279979.	+	236923.i	0.0848181	+	0.0717744i
$406$		0			0
$407$		−	6719.12i		−	0.00201060i
$408$		0			0
$409$	−4.16809e6			−1.23205			−0.616026	−	0.787725i	$-0.711259\pi$
$409$	−4.16809e6			−1.23205			−0.616026	+	0.787725i	$0.711259\pi$
$410$		0			0
$411$	−76484.2			−0.0223340
$412$		0			0
$413$			6.58001e6i			1.89824i
$414$		0			0
$415$	−240137.	+	283778.i	−0.0684446	+	0.0808832i
$416$		0			0
$417$		−	2.75304e6i		−	0.775304i
$418$		0			0
$419$	−2.92197e6			−0.813093			−0.406547	−	0.913630i	$-0.633267\pi$
$419$	−2.92197e6			−0.813093			−0.406547	+	0.913630i	$0.633267\pi$
$420$		0			0
$421$	3.01627e6			0.829402			0.414701	−	0.909958i	$-0.363886\pi$
$421$	3.01627e6			0.829402			0.414701	+	0.909958i	$0.363886\pi$
$422$		0			0
$423$		−	2.09040e6i		−	0.568039i
$424$		0			0
$425$	3.00291e6	−	503761.i	0.806436	−	0.135286i
$426$		0			0
$427$		−	7.93560e6i		−	2.10625i
$428$		0			0
$429$	−9429.50			−0.00247369
$430$		0			0
$431$	1.78958e6			0.464042			0.232021	−	0.972711i	$-0.425466\pi$
$431$	1.78958e6			0.464042			0.232021	+	0.972711i	$0.425466\pi$
$432$		0			0
$433$			460836.i			0.118121i	0.998254	+	0.0590604i	$0.0188105\pi$
$433$			460836.i			0.118121i	−0.998254	+	0.0590604i	$0.981190\pi$
$434$		0			0
$435$	1.97696e6	−	2.33623e6i	0.500927	−	0.591961i
$436$		0			0
$437$			1.26693e6i			0.317357i
$438$		0			0
$439$	−68566.3			−0.0169805			−0.00849023	−	0.999964i	$-0.502703\pi$
$439$	−68566.3			−0.0169805			−0.00849023	+	0.999964i	$0.502703\pi$
$440$		0			0
$441$	938004.			0.229672
$442$		0			0
$443$			246132.i			0.0595880i	0.999556	+	0.0297940i	$0.00948513\pi$
$443$			246132.i			0.0595880i	−0.999556	+	0.0297940i	$0.990515\pi$
$444$		0			0
$445$	3.25788e6	+	2.75687e6i	0.779893	+	0.659958i
$446$		0			0
$447$			3.56054e6i			0.842844i
$448$		0			0
$449$	1.40455e6			0.328792			0.164396	−	0.986394i	$-0.447433\pi$
$449$	1.40455e6			0.328792			0.164396	+	0.986394i	$0.447433\pi$
$450$		0			0
$451$	15243.2			0.00352886
$452$		0			0
$453$		−	3.58959e6i		−	0.821862i
$454$		0			0
$455$	−711784.	−	602323.i	−0.161183	−	0.136396i
$456$		0			0
$457$		−	831955.i		−	0.186341i	−0.995650	−	0.0931707i	$-0.970300\pi$
$457$		−	831955.i		−	0.186341i	0.995650	−	0.0931707i	$-0.0297002\pi$
$458$		0			0
$459$	−710307.			−0.157367
$460$		0			0
$461$	4.65324e6			1.01977			0.509886	−	0.860242i	$-0.329688\pi$
$461$	4.65324e6			1.01977			0.509886	+	0.860242i	$0.329688\pi$
$462$		0			0
$463$		−	8.38346e6i		−	1.81748i	−0.417358	−	0.908742i	$-0.637044\pi$
$463$		−	8.38346e6i		−	1.81748i	0.417358	−	0.908742i	$-0.362956\pi$
$464$		0			0
$465$	2.32821e6	−	2.75132e6i	0.499333	−	0.590077i
$466$		0			0
$467$			3.73089e6i			0.791626i	0.918331	+	0.395813i	$0.129537\pi$
$467$			3.73089e6i			0.791626i	−0.918331	+	0.395813i	$0.870463\pi$
$468$		0			0
$469$	−3.99011e6			−0.837630
$470$		0			0
$471$	−3.00973e6			−0.625138
$472$		0			0
$473$			13314.6i			0.00273637i
$474$		0			0
$475$	249577.	+	1.48772e6i	0.0507540	+	0.302544i
$476$		0			0
$477$			396383.i			0.0797661i
$478$		0			0
$479$	6.08718e6			1.21221			0.606105	−	0.795385i	$-0.292731\pi$
$479$	6.08718e6			1.21221			0.606105	+	0.795385i	$0.292731\pi$
$480$		0			0
$481$	62853.1			0.0123869
$482$		0			0
$483$		−	3.97976e6i		−	0.776227i
$484$		0			0
$485$	5.31619e6	−	6.28231e6i	1.02623	−	1.21273i
$486$		0			0
$487$		−	9.69393e6i		−	1.85216i	−0.377333	−	0.926078i	$-0.623159\pi$
$487$		−	9.69393e6i		−	1.85216i	0.377333	−	0.926078i	$-0.376841\pi$
$488$		0			0
$489$	−663026.			−0.125389
$490$		0			0
$491$	−1.37631e6			−0.257640			−0.128820	−	0.991668i	$-0.541119\pi$
$491$	−1.37631e6			−0.257640			−0.128820	+	0.991668i	$0.541119\pi$
$492$		0			0
$493$			5.92702e6i			1.09830i
$494$		0			0
$495$	−36581.1	−	30955.5i	−0.00671033	−	0.00567839i
$496$		0			0
$497$			9.31539e6i			1.69165i
$498$		0			0
$499$	−6.74202e6			−1.21210			−0.606050	−	0.795426i	$-0.707247\pi$
$499$	−6.74202e6			−1.21210			−0.606050	+	0.795426i	$0.707247\pi$
$500$		0			0
$501$	−4.73086e6			−0.842065
$502$		0			0
$503$		−	5.55162e6i		−	0.978363i	−0.872182	−	0.489182i	$-0.837296\pi$
$503$		−	5.55162e6i		−	0.978363i	0.872182	−	0.489182i	$-0.162704\pi$
$504$		0			0
$505$	757790.	+	641254.i	0.132227	+	0.111893i
$506$		0			0
$507$			3.25343e6i			0.562110i
$508$		0			0
$509$	−1.06644e7			−1.82450			−0.912249	−	0.409637i	$-0.865655\pi$
$509$	−1.06644e7			−1.82450			−0.912249	+	0.409637i	$0.865655\pi$
$510$		0			0
$511$	1.20571e7			2.04263
$512$		0			0
$513$		−	351906.i		−	0.0590382i
$514$		0			0
$515$	87177.0	−	103020.i	0.0144838	−	0.0171160i
$516$		0			0
$517$			273124.i			0.0449400i
$518$		0			0
$519$	−6.59302e6			−1.07440
$520$		0			0
$521$	2.61185e6			0.421554			0.210777	−	0.977534i	$-0.432401\pi$
$521$	2.61185e6			0.421554			0.210777	+	0.977534i	$0.432401\pi$
$522$		0			0
$523$		−	510350.i		−	0.0815856i	−0.999168	−	0.0407928i	$-0.987012\pi$
$523$		−	510350.i		−	0.0815856i	0.999168	−	0.0407928i	$-0.0129884\pi$
$524$		0			0
$525$	−783989.	−	4.67335e6i	−0.124140	−	0.739996i
$526$		0			0
$527$			6.98010e6i			1.09480i
$528$		0			0
$529$	−451834.			−0.0702005
$530$		0			0
$531$	−3.16337e6			−0.486870
$532$		0			0
$533$			142590.i			0.0217406i
$534$		0			0
$535$	−3.32902e6	+	3.93400e6i	−0.502842	+	0.594224i
$536$		0			0
$537$			3.18170e6i			0.476127i
$538$		0			0
$539$	−122556.			−0.0181704
$540$		0			0
$541$	−4.39256e6			−0.645245			−0.322623	−	0.946528i	$-0.604564\pi$
$541$	−4.39256e6			−0.645245			−0.322623	+	0.946528i	$0.604564\pi$
$542$		0			0
$543$			5.99286e6i			0.872237i
$544$		0			0
$545$	9.45264e6	+	7.99898e6i	1.36321	+	1.15357i
$546$		0			0
$547$			3.78496e6i			0.540870i	0.962738	+	0.270435i	$0.0871675\pi$
$547$			3.78496e6i			0.540870i	−0.962738	+	0.270435i	$0.912833\pi$
$548$		0			0
$549$	3.81507e6			0.540222
$550$		0			0
$551$	−2.93641e6			−0.412039
$552$		0			0
$553$			1.14701e7i			1.59498i
$554$		0			0
$555$	243835.	+	206337.i	0.0336018	+	0.0284344i
$556$		0			0
$557$			4.28905e6i			0.585765i	0.956149	+	0.292882i	$0.0946144\pi$
$557$			4.28905e6i			0.585765i	−0.956149	+	0.292882i	$0.905386\pi$
$558$		0			0
$559$	−124550.			−0.0168583
$560$		0			0
$561$	92806.3			0.0124500
$562$		0			0
$563$		−	2.03903e6i		−	0.271115i	−0.990770	−	0.135557i	$-0.956718\pi$
$563$		−	2.03903e6i		−	0.271115i	0.990770	−	0.135557i	$-0.0432825\pi$
$564$		0			0
$565$	−6.39747e6	+	7.56009e6i	−0.843116	+	0.996336i
$566$		0			0
$567$			1.10543e6i			0.144402i
$568$		0			0
$569$	2.72663e6			0.353057			0.176529	−	0.984296i	$-0.443513\pi$
$569$	2.72663e6			0.353057			0.176529	+	0.984296i	$0.443513\pi$
$570$		0			0
$571$	−3.04693e6			−0.391086			−0.195543	−	0.980695i	$-0.562647\pi$
$571$	−3.04693e6			−0.391086			−0.195543	+	0.980695i	$0.562647\pi$
$572$		0			0
$573$		−	8.11686e6i		−	1.03276i
$574$		0			0
$575$	−8.08864e6	+	1.35693e6i	−1.02025	+	0.171154i
$576$		0			0
$577$		−	1.31381e7i		−	1.64284i	−0.570326	−	0.821419i	$-0.693183\pi$
$577$		−	1.31381e7i		−	1.64284i	0.570326	−	0.821419i	$-0.306817\pi$
$578$		0			0
$579$	−1.20028e6			−0.148794
$580$		0			0
$581$	−1.12043e6			−0.137703
$582$		0			0
$583$		−	51789.9i		−	0.00631065i
$584$		0			0
$585$	289569.	−	342193.i	0.0349835	−	0.0413410i
$586$		0			0
$587$			2.11315e6i			0.253125i	0.991959	+	0.126562i	$0.0403944\pi$
$587$			2.11315e6i			0.253125i	−0.991959	+	0.126562i	$0.959606\pi$
$588$		0			0
$589$	−3.45813e6			−0.410727
$590$		0			0
$591$	1.46957e6			0.173070
$592$		0			0
$593$		−	2.19021e6i		−	0.255770i	−0.991789	−	0.127885i	$-0.959181\pi$
$593$		−	2.19021e6i		−	0.255770i	0.991789	−	0.127885i	$-0.0408189\pi$
$594$		0			0
$595$	7.00546e6	+	5.92814e6i	0.811231	+	0.686477i
$596$		0			0
$597$			5.10727e6i			0.586480i
$598$		0			0
$599$	−1.44998e7			−1.65118			−0.825590	−	0.564270i	$-0.809158\pi$
$599$	−1.44998e7			−1.65118			−0.825590	+	0.564270i	$0.809158\pi$
$600$		0			0
$601$	−1.72529e6			−0.194839			−0.0974197	−	0.995243i	$-0.531059\pi$
$601$	−1.72529e6			−0.194839			−0.0974197	+	0.995243i	$0.531059\pi$
$602$		0			0
$603$		−	1.91826e6i		−	0.214839i
$604$		0			0
$605$	−6.86780e6	−	5.81164e6i	−0.762832	−	0.645521i
$606$		0			0
$607$		−	5.79998e6i		−	0.638932i	−0.947598	−	0.319466i	$-0.896496\pi$
$607$		−	5.79998e6i		−	0.638932i	0.947598	−	0.319466i	$-0.103504\pi$
$608$		0			0
$609$	9.22407e6			1.00781
$610$		0			0
$611$	−2.55490e6			−0.276867
$612$		0			0
$613$		−	1.31838e7i		−	1.41707i	−0.705678	−	0.708533i	$-0.749357\pi$
$613$		−	1.31838e7i		−	1.41707i	0.705678	−	0.708533i	$-0.250643\pi$
$614$		0			0
$615$	−468102.	+	553170.i	−0.0499060	+	0.0589754i
$616$		0			0
$617$		−	9.13531e6i		−	0.966074i	−0.875600	−	0.483037i	$-0.839534\pi$
$617$		−	9.13531e6i		−	0.966074i	0.875600	−	0.483037i	$-0.160466\pi$
$618$		0			0
$619$	−3.58946e6			−0.376533			−0.188266	−	0.982118i	$-0.560287\pi$
$619$	−3.58946e6			−0.376533			−0.188266	+	0.982118i	$0.560287\pi$
$620$		0			0
$621$	1.91329e6			0.199091
$622$		0			0
$623$			1.28630e7i			1.32776i
$624$		0			0
$625$	−9.23101e6	+	3.18683e6i	−0.945255	+	0.326331i
$626$		0			0
$627$			45978.8i			0.00467077i
$628$		0			0
$629$	−618608.			−0.0623432
$630$		0			0
$631$	2.61289e6			0.261245			0.130623	−	0.991432i	$-0.458302\pi$
$631$	2.61289e6			0.261245			0.130623	+	0.991432i	$0.458302\pi$
$632$		0			0
$633$			3.20164e6i			0.317587i
$634$		0			0
$635$	5.82819e6	−	6.88735e6i	0.573587	−	0.677825i
$636$		0			0
$637$		−	1.14644e6i		−	0.111944i
$638$		0			0
$639$	−4.47841e6			−0.433882
$640$		0			0
$641$	−1.09445e7			−1.05209			−0.526043	−	0.850458i	$-0.676325\pi$
$641$	−1.09445e7			−1.05209			−0.526043	+	0.850458i	$0.676325\pi$
$642$		0			0
$643$			1.20567e7i			1.15001i	0.818149	+	0.575007i	$0.195001\pi$
$643$			1.20567e7i			1.15001i	−0.818149	+	0.575007i	$0.804999\pi$
$644$		0			0
$645$	−483182.	−	408876.i	−0.0457311	−	0.0386984i
$646$		0			0
$647$			1.10377e7i			1.03662i	0.855194	+	0.518308i	$0.173438\pi$
$647$			1.10377e7i			1.03662i	−0.855194	+	0.518308i	$0.826562\pi$
$648$		0			0
$649$	413314.			0.0385184
$650$		0			0
$651$	1.08629e7			1.00460
$652$		0			0
$653$		−	8.04814e6i		−	0.738606i	−0.929309	−	0.369303i	$-0.879596\pi$
$653$		−	8.04814e6i		−	0.738606i	0.929309	−	0.369303i	$-0.120404\pi$
$654$		0			0
$655$	1.00794e7	+	8.52932e6i	0.917973	+	0.776803i
$656$		0			0
$657$			5.79648e6i			0.523903i
$658$		0			0
$659$	2.18154e7			1.95681			0.978407	−	0.206688i	$-0.0662685\pi$
$659$	2.18154e7			1.95681			0.978407	+	0.206688i	$0.0662685\pi$
$660$		0			0
$661$	−1.41037e7			−1.25554			−0.627768	−	0.778400i	$-0.716031\pi$
$661$	−1.41037e7			−1.25554			−0.627768	+	0.778400i	$0.716031\pi$
$662$		0			0
$663$			868143.i			0.0767022i
$664$		0			0
$665$	−2.93696e6	+	3.47070e6i	−0.257540	+	0.304343i
$666$		0			0
$667$		−	1.59650e7i		−	1.38949i
$668$		0			0
$669$	−1.32144e6			−0.114152
$670$		0			0
$671$	−498464.			−0.0427393
$672$		0			0
$673$			1.92452e7i			1.63789i	0.573875	+	0.818943i	$0.305440\pi$
$673$			1.92452e7i			1.63789i	−0.573875	+	0.818943i	$0.694560\pi$
$674$		0			0
$675$	2.24673e6	−	376906.i	0.189798	−	0.0318400i
$676$		0			0
$677$		−	2.03479e7i		−	1.70627i	−0.521690	−	0.853135i	$-0.674698\pi$
$677$		−	2.03479e7i		−	1.70627i	0.521690	−	0.853135i	$-0.325302\pi$
$678$		0			0
$679$	2.48042e7			2.06467
$680$		0			0
$681$	7.03590e6			0.581369
$682$		0			0
$683$		−	1.75186e7i		−	1.43697i	−0.695542	−	0.718486i	$-0.744836\pi$
$683$		−	1.75186e7i		−	1.43697i	0.695542	−	0.718486i	$-0.255164\pi$
$684$		0			0
$685$	−306879.	+	362648.i	−0.0249885	+	0.0295297i
$686$		0			0
$687$			9.25117e6i			0.747834i
$688$		0			0
$689$	484462.			0.0388787
$690$		0			0
$691$	8.40517e6			0.669655			0.334828	−	0.942279i	$-0.391322\pi$
$691$	8.40517e6			0.669655			0.334828	+	0.942279i	$0.391322\pi$
$692$		0			0
$693$		−	144432.i		−	0.0114243i
$694$		0			0
$695$	−1.30535e7	−	1.10461e7i	−1.02509	−	0.867452i
$696$		0			0
$697$		−	1.40339e6i		−	0.109420i
$698$		0			0
$699$	6.92210e6			0.535852
$700$		0			0
$701$	1.53129e6			0.117696			0.0588480	−	0.998267i	$-0.481257\pi$
$701$	1.53129e6			0.117696			0.0588480	+	0.998267i	$0.481257\pi$
$702$		0			0
$703$		−	306475.i		−	0.0233888i
$704$		0			0
$705$	−9.91157e6	−	8.38733e6i	−0.751051	−	0.635552i
$706$		0			0
$707$			2.99195e6i			0.225116i
$708$		0			0
$709$	2.42089e7			1.80867			0.904336	−	0.426821i	$-0.140367\pi$
$709$	2.42089e7			1.80867			0.904336	+	0.426821i	$0.140367\pi$
$710$		0			0
$711$	−5.51430e6			−0.409088
$712$		0			0
$713$		−	1.88016e7i		−	1.38507i
$714$		0			0
$715$	−37834.1	+	44709.7i	−0.00276770	+	0.00327067i
$716$		0			0
$717$			9.30643e6i			0.676060i
$718$		0			0
$719$	1.80277e7			1.30052			0.650262	−	0.759710i	$-0.274659\pi$
$719$	1.80277e7			1.30052			0.650262	+	0.759710i	$0.274659\pi$
$720$		0			0
$721$	406749.			0.0291399
$722$		0			0
$723$			4.99494e6i			0.355373i
$724$		0			0
$725$	−3.14502e6	−	1.87474e7i	−0.222218	−	1.32463i
$726$		0			0
$727$		−	1.34886e7i		−	0.946523i	−0.880922	−	0.473261i	$-0.843077\pi$
$727$		−	1.34886e7i		−	0.946523i	0.880922	−	0.473261i	$-0.156923\pi$
$728$		0			0
$729$	−531441.			−0.0370370
$730$		0			0
$731$	1.22583e6			0.0848472
$732$		0			0
$733$		−	2.11271e7i		−	1.45238i	−0.687496	−	0.726188i	$-0.741290\pi$
$733$		−	2.11271e7i		−	1.45238i	0.687496	−	0.726188i	$-0.258710\pi$
$734$		0			0
$735$	3.76357e6	−	4.44752e6i	0.256969	−	0.303669i
$736$		0			0
$737$			250633.i			0.0169969i
$738$		0			0
$739$	−5.86024e6			−0.394734			−0.197367	−	0.980330i	$-0.563239\pi$
$739$	−5.86024e6			−0.394734			−0.197367	+	0.980330i	$0.563239\pi$
$740$		0			0
$741$	−430102.			−0.0287757
$742$		0			0
$743$			6.34482e6i			0.421645i	0.977524	+	0.210823i	$0.0676143\pi$
$743$			6.34482e6i			0.421645i	−0.977524	+	0.210823i	$0.932386\pi$
$744$		0			0
$745$	1.68822e7	+	1.42860e7i	1.11439	+	0.943018i
$746$		0			0
$747$		−	538651.i		−	0.0353188i
$748$		0			0
$749$	−1.55325e7			−1.01166
$750$		0			0
$751$	−8.54447e6			−0.552822			−0.276411	−	0.961039i	$-0.589145\pi$
$751$	−8.54447e6			−0.552822			−0.276411	+	0.961039i	$0.589145\pi$
$752$		0			0
$753$			2.38598e6i			0.153348i
$754$		0			0
$755$	−1.70199e7	−	1.44026e7i	−1.08665	−	0.919543i
$756$		0			0
$757$			9.65781e6i			0.612546i	0.951944	+	0.306273i	$0.0990820\pi$
$757$			9.65781e6i			0.612546i	−0.951944	+	0.306273i	$0.900918\pi$
$758$		0			0
$759$	−249983.			−0.0157509
$760$		0			0
$761$	2.45175e7			1.53467			0.767335	−	0.641247i	$-0.221583\pi$
$761$	2.45175e7			1.53467			0.767335	+	0.641247i	$0.221583\pi$
$762$		0			0
$763$			3.73215e7i			2.32085i
$764$		0			0
$765$	−2.84998e6	+	3.36791e6i	−0.176071	+	0.208068i
$766$		0			0
$767$			3.86629e6i			0.237305i
$768$		0			0
$769$	3.07433e7			1.87471			0.937355	−	0.348376i	$-0.113267\pi$
$769$	3.07433e7			1.87471			0.937355	+	0.348376i	$0.113267\pi$
$770$		0			0
$771$	7.05557e6			0.427461
$772$		0			0
$773$		−	3.09471e6i		−	0.186282i	−0.995653	−	0.0931411i	$-0.970309\pi$
$773$		−	3.09471e6i		−	0.186282i	0.995653	−	0.0931411i	$-0.0296908\pi$
$774$		0			0
$775$	−3.70381e6	−	2.20783e7i	−0.221510	−	1.32042i
$776$		0			0
$777$			962723.i			0.0572069i
$778$		0			0
$779$	695280.			0.0410503
$780$		0			0
$781$	585134.			0.0343263
$782$		0			0
$783$			4.43451e6i			0.258488i
$784$		0			0
$785$	−1.20760e7	+	1.42706e7i	−0.699437	+	0.826547i
$786$		0			0
$787$			1.07040e7i			0.616041i	0.951380	+	0.308020i	$0.0996665\pi$
$787$			1.07040e7i			0.616041i	−0.951380	+	0.308020i	$0.900333\pi$
$788$		0			0
$789$	1.49790e7			0.856626
$790$		0			0
$791$	−2.98492e7			−1.69626
$792$		0			0
$793$		−	4.66281e6i		−	0.263309i
$794$		0			0
$795$	1.87944e6	+	1.59041e6i	0.105465	+	0.0892466i
$796$		0			0
$797$			1.17067e7i			0.652815i	0.945229	+	0.326407i	$0.105838\pi$
$797$			1.17067e7i			0.652815i	−0.945229	+	0.326407i	$0.894162\pi$
$798$		0			0
$799$	2.51456e7			1.39346
$800$		0			0
$801$	−6.18393e6			−0.340552
$802$		0			0
$803$		−	757348.i		−	0.0414483i
$804$		0			0
$805$	−1.88699e7	−	1.59680e7i	−1.02631	−	0.868484i
$806$		0			0
$807$		−	2.71438e6i		−	0.146719i
$808$		0			0
$809$	−2.16373e7			−1.16233			−0.581167	−	0.813784i	$-0.697404\pi$
$809$	−2.16373e7			−1.16233			−0.581167	+	0.813784i	$0.697404\pi$
$810$		0			0
$811$	−3.41109e7			−1.82113			−0.910566	−	0.413364i	$-0.864354\pi$
$811$	−3.41109e7			−1.82113			−0.910566	+	0.413364i	$0.864354\pi$
$812$		0			0
$813$		−	1.07833e7i		−	0.572168i
$814$		0			0
$815$	−2.66027e6	+	3.14372e6i	−0.140292	+	0.165787i
$816$		0			0
$817$			607311.i			0.0318314i
$818$		0			0
$819$	1.35107e6			0.0703829
$820$		0			0
$821$	−1.75077e7			−0.906506			−0.453253	−	0.891382i	$-0.649737\pi$
$821$	−1.75077e7			−0.906506			−0.453253	+	0.891382i	$0.649737\pi$
$822$		0			0
$823$		−	1.89754e6i		−	0.0976542i	−0.998807	−	0.0488271i	$-0.984452\pi$
$823$		−	1.89754e6i		−	0.0976542i	0.998807	−	0.0488271i	$-0.0155483\pi$
$824$		0			0
$825$	−293550.	+	49245.2i	−0.0150157	+	0.00251901i
$826$		0			0
$827$		−	2.57976e7i		−	1.31164i	−0.754917	−	0.655820i	$-0.772323\pi$
$827$		−	2.57976e7i		−	1.31164i	0.754917	−	0.655820i	$-0.227677\pi$
$828$		0			0
$829$	−1.64098e7			−0.829312			−0.414656	−	0.909978i	$-0.636098\pi$
$829$	−1.64098e7			−0.829312			−0.414656	+	0.909978i	$0.636098\pi$
$830$		0			0
$831$	5.09002e6			0.255692
$832$		0			0
$833$			1.12834e7i			0.563412i
$834$		0			0
$835$	−1.89817e7	+	2.24313e7i	−0.942147	+	1.11336i
$836$		0			0
$837$			5.22240e6i			0.257666i
$838$		0			0
$839$	3.03172e7			1.48691			0.743454	−	0.668787i	$-0.233186\pi$
$839$	3.03172e7			1.48691			0.743454	+	0.668787i	$0.233186\pi$
$840$		0			0
$841$	1.64917e7			0.804038
$842$		0			0
$843$		−	8.13417e6i		−	0.394225i
$844$		0			0
$845$	1.54261e7	+	1.30538e7i	0.743213	+	0.628919i
$846$		0			0
$847$		−	2.71159e7i		−	1.29872i
$848$		0			0
$849$	6.43294e6			0.306295
$850$		0			0
$851$	1.66628e6			0.0788724
$852$		0			0
$853$		−	7.03333e6i		−	0.330970i	−0.986212	−	0.165485i	$-0.947081\pi$
$853$		−	7.03333e6i		−	0.330970i	0.986212	−	0.165485i	$-0.0529189\pi$
$854$		0			0
$855$	−1.66855e6	−	1.41196e6i	−0.0780593	−	0.0660551i
$856$		0			0
$857$			2.02224e7i			0.940545i	0.882521	+	0.470273i	$0.155844\pi$
$857$			2.02224e7i			0.940545i	−0.882521	+	0.470273i	$0.844156\pi$
$858$		0			0
$859$	−3.42836e7			−1.58527			−0.792635	−	0.609696i	$-0.791291\pi$
$859$	−3.42836e7			−1.58527			−0.792635	+	0.609696i	$0.791291\pi$
$860$		0			0
$861$	−2.18406e6			−0.100405
$862$		0			0
$863$			1.45590e7i			0.665434i	0.943027	+	0.332717i	$0.107965\pi$
$863$			1.45590e7i			0.665434i	−0.943027	+	0.332717i	$0.892035\pi$
$864$		0			0
$865$	−2.64532e7	+	3.12606e7i	−1.20209	+	1.42055i
$866$		0			0
$867$			4.23434e6i			0.191310i
$868$		0			0
$869$	720479.			0.0323647
$870$		0			0
$871$	−2.34451e6			−0.104715
$872$		0			0
$873$			1.19247e7i			0.529557i
$874$		0			0
$875$	−2.53042e7	−	1.50337e7i	−1.11731	−	0.663811i
$876$		0			0
$877$		−	1.63585e7i		−	0.718197i	−0.933300	−	0.359099i	$-0.883084\pi$
$877$		−	1.63585e7i		−	0.718197i	0.933300	−	0.359099i	$-0.116916\pi$
$878$		0			0
$879$	−1.06823e7			−0.466331
$880$		0			0
$881$	4.41404e6			0.191600			0.0958002	−	0.995401i	$-0.469459\pi$
$881$	4.41404e6			0.191600			0.0958002	+	0.995401i	$0.469459\pi$
$882$		0			0
$883$			1.87834e7i			0.810724i	0.914156	+	0.405362i	$0.132855\pi$
$883$			1.87834e7i			0.810724i	−0.914156	+	0.405362i	$0.867145\pi$
$884$		0			0
$885$	−1.26924e7	+	1.49990e7i	−0.544736	+	0.643731i
$886$		0			0
$887$		−	1.18701e7i		−	0.506578i	−0.967391	−	0.253289i	$-0.918488\pi$
$887$		−	1.18701e7i		−	0.506578i	0.967391	−	0.253289i	$-0.0815124\pi$
$888$		0			0
$889$	2.71931e7			1.15399
$890$		0			0
$891$	69436.2			0.00293016
$892$		0			0
$893$			1.24578e7i			0.522774i
$894$		0			0
$895$	1.50859e7	+	1.27660e7i	0.629527	+	0.532716i
$896$		0			0
$897$		−	2.33843e6i		−	0.0970384i
$898$		0			0
$899$	4.35773e7			1.79830
$900$		0			0
$901$	−4.76813e6			−0.195675
$902$		0			0
$903$		−	1.90773e6i		−	0.0778570i
$904$		0			0
$905$	2.84150e7	+	2.40452e7i	1.15326	+	0.975905i
$906$		0			0
$907$			3.24034e7i			1.30789i	0.756541	+	0.653946i	$0.226888\pi$
$907$			3.24034e7i			1.30789i	−0.756541	+	0.653946i	$0.773112\pi$
$908$		0			0
$909$	−1.43839e6			−0.0577389
$910$		0			0
$911$	−2.71444e7			−1.08364			−0.541820	−	0.840495i	$-0.682265\pi$
$911$	−2.71444e7			−1.08364			−0.541820	+	0.840495i	$0.682265\pi$
$912$		0			0
$913$			70378.2i			0.00279423i
$914$		0			0
$915$	1.53073e7	−	1.80891e7i	0.604429	−	0.714272i
$916$		0			0
$917$			3.97960e7i			1.56284i
$918$		0			0
$919$	−4.03452e7			−1.57581			−0.787904	−	0.615799i	$-0.788833\pi$
$919$	−4.03452e7			−1.57581			−0.787904	+	0.615799i	$0.788833\pi$
$920$		0			0
$921$	−2.10770e7			−0.818765
$922$		0			0
$923$			5.47355e6i			0.211478i
$924$		0			0
$925$	1.95668e6	−	328248.i	0.0751910	−	0.0126139i
$926$		0			0
$927$			195546.i			0.00747395i
$928$		0			0
$929$	8.39583e6			0.319172			0.159586	−	0.987184i	$-0.448984\pi$
$929$	8.39583e6			0.319172			0.159586	+	0.987184i	$0.448984\pi$
$930$		0			0
$931$	−5.59009e6			−0.211371
$932$		0			0
$933$		−	1.90197e6i		−	0.0715318i
$934$		0			0
$935$	372368.	−	440039.i	0.0139297	−	0.0164612i
$936$		0			0
$937$		−	2.03592e7i		−	0.757552i	−0.925488	−	0.378776i	$-0.876345\pi$
$937$		−	2.03592e7i		−	0.757552i	0.925488	−	0.378776i	$-0.123655\pi$
$938$		0			0
$939$	−1.67035e7			−0.618221
$940$		0			0
$941$	6.19895e6			0.228215			0.114107	−	0.993468i	$-0.463599\pi$
$941$	6.19895e6			0.228215			0.114107	+	0.993468i	$0.463599\pi$
$942$		0			0
$943$			3.78018e6i			0.138431i
$944$		0			0
$945$	5.24138e6	+	4.43534e6i	0.190926	+	0.161565i
$946$		0			0
$947$		−	9.65567e6i		−	0.349871i	−0.984580	−	0.174935i	$-0.944028\pi$
$947$		−	9.65567e6i		−	0.349871i	0.984580	−	0.174935i	$-0.0559716\pi$
$948$		0			0
$949$	7.08450e6			0.255355
$950$		0			0
$951$	−2.02893e7			−0.727471
$952$		0			0
$953$		−	1.69461e7i		−	0.604420i	−0.953241	−	0.302210i	$-0.902276\pi$
$953$		−	1.69461e7i		−	0.604420i	0.953241	−	0.302210i	$-0.0977244\pi$
$954$		0			0
$955$	−3.84859e7	−	3.25674e7i	−1.36550	−	1.15551i
$956$		0			0
$957$		−	579397.i		−	0.0204501i
$958$		0			0
$959$	−1.43183e6			−0.0502741
$960$		0			0
$961$	2.26907e7			0.792574
$962$		0			0
$963$		−	7.46730e6i		−	0.259476i
$964$		0			0
$965$	−4.81589e6	+	5.69109e6i	−0.166479	+	0.196733i
$966$		0			0
$967$			641135.i			0.0220487i	0.999939	+	0.0110244i	$0.00350924\pi$
$967$			641135.i			0.0220487i	−0.999939	+	0.0110244i	$0.996491\pi$
$968$		0			0
$969$	4.23312e6			0.144827
$970$		0			0
$971$	8.63536e6			0.293922			0.146961	−	0.989142i	$-0.453051\pi$
$971$	8.63536e6			0.293922			0.146961	+	0.989142i	$0.453051\pi$
$972$		0			0
$973$		−	5.15385e7i		−	1.74522i
$974$		0			0
$975$	−460657.	−	2.74597e6i	−0.0155191	−	0.0925091i
$976$		0			0
$977$			1.09793e7i			0.367992i	0.982927	+	0.183996i	$0.0589033\pi$
$977$			1.09793e7i			0.367992i	−0.982927	+	0.183996i	$0.941097\pi$
$978$		0			0
$979$	807970.			0.0269425
$980$		0			0
$981$	−1.79425e7			−0.595264
$982$		0			0
$983$		−	7.71273e6i		−	0.254580i	−0.991866	−	0.127290i	$-0.959372\pi$
$983$		−	7.71273e6i		−	0.254580i	0.991866	−	0.127290i	$-0.0406279\pi$
$984$		0			0
$985$	5.89639e6	−	6.96794e6i	0.193640	−	0.228831i
$986$		0			0
$987$		−	3.91335e7i		−	1.27866i
$988$		0			0
$989$	−3.30190e6			−0.107343
$990$		0			0
$991$	3.74467e7			1.21124			0.605618	−	0.795755i	$-0.292926\pi$
$991$	3.74467e7			1.21124			0.605618	+	0.795755i	$0.292926\pi$
$992$		0			0
$993$		−	1.21483e7i		−	0.390970i
$994$		0			0
$995$	2.42160e7	+	2.04920e7i	0.775434	+	0.656185i
$996$		0			0
$997$			5.30667e7i			1.69077i	0.534159	+	0.845384i	$0.320628\pi$
$997$			5.30667e7i			1.69077i	−0.534159	+	0.845384i	$0.679372\pi$
$998$		0			0
$999$	−462833.			−0.0146727

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	120.6.f.a.49.6	yes	6
3.2	odd	2		360.6.f.a.289.1		6
4.3	odd	2		240.6.f.e.49.3		6
5.2	odd	4		600.6.a.s.1.1		3
5.3	odd	4		600.6.a.r.1.3		3
5.4	even	2	inner	120.6.f.a.49.3	✓	6
12.11	even	2		720.6.f.l.289.1		6
15.14	odd	2		360.6.f.a.289.2		6
20.19	odd	2		240.6.f.e.49.6		6
60.59	even	2		720.6.f.l.289.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.6.f.a.49.3	✓	6	5.4	even	2	inner
120.6.f.a.49.6	yes	6	1.1	even	1	trivial
240.6.f.e.49.3		6	4.3	odd	2
240.6.f.e.49.6		6	20.19	odd	2
360.6.f.a.289.1		6	3.2	odd	2
360.6.f.a.289.2		6	15.14	odd	2
600.6.a.r.1.3		3	5.3	odd	4
600.6.a.s.1.1		3	5.2	odd	4
720.6.f.l.289.1		6	12.11	even	2
720.6.f.l.289.2		6	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	120.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+9.00000i q^{3} +(42.6733 + 36.1108i) q^{5} +168.485i q^{7} -81.0000 q^{9} +O(q^{10})\)	\(q+9.00000i q^{3} +(42.6733 + 36.1108i) q^{5} +168.485i q^{7} -81.0000 q^{9} +10.5832 q^{11} +98.9988i q^{13} +(-324.997 + 384.060i) q^{15} -974.359i q^{17} +482.724 q^{19} -1516.37 q^{21} +2624.53i q^{23} +(517.018 + 3081.93i) q^{25} -729.000i q^{27} -6083.00 q^{29} -7163.79 q^{31} +95.2486i q^{33} +(-6084.14 + 7189.82i) q^{35} -634.887i q^{37} -890.989 q^{39} +1440.32 q^{41} +1258.09i q^{43} +(-3456.54 - 2924.98i) q^{45} +25807.4i q^{47} -11580.3 q^{49} +8769.23 q^{51} -4893.61i q^{53} +(451.619 + 382.167i) q^{55} +4344.52i q^{57} +39053.9 q^{59} -47099.7 q^{61} -13647.3i q^{63} +(-3574.93 + 4224.60i) q^{65} +23682.2i q^{67} -23620.8 q^{69} +55289.0 q^{71} -71561.5i q^{73} +(-27737.4 + 4653.16i) q^{75} +1783.11i q^{77} +68077.8 q^{79} +6561.00 q^{81} +6650.01i q^{83} +(35184.9 - 41579.1i) q^{85} -54747.0i q^{87} +76344.8 q^{89} -16679.8 q^{91} -64474.1i q^{93} +(20599.4 + 17431.6i) q^{95} -147219. i q^{97} -857.237 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 50 q^{5} - 486 q^{9}+O(q^{10}) \) 6 * q + 50 * q^5 - 486 * q^9	\( 6 q + 50 q^{5} - 486 q^{9} - 664 q^{11} + 540 q^{15} + 288 q^{19} - 1872 q^{21} + 3750 q^{25} - 1892 q^{29} - 10248 q^{31} - 18880 q^{35} + 1584 q^{39} - 1324 q^{41} - 4050 q^{45} + 28410 q^{49} + 20088 q^{51} - 47160 q^{55} - 10296 q^{59} - 52116 q^{61} - 13480 q^{65} + 51624 q^{69} - 28288 q^{71} - 41400 q^{75} + 220200 q^{79} + 39366 q^{81} + 95880 q^{85} + 351420 q^{89} - 17088 q^{91} - 349120 q^{95} + 53784 q^{99}+O(q^{100}) \) 6 * q + 50 * q^5 - 486 * q^9 - 664 * q^11 + 540 * q^15 + 288 * q^19 - 1872 * q^21 + 3750 * q^25 - 1892 * q^29 - 10248 * q^31 - 18880 * q^35 + 1584 * q^39 - 1324 * q^41 - 4050 * q^45 + 28410 * q^49 + 20088 * q^51 - 47160 * q^55 - 10296 * q^59 - 52116 * q^61 - 13480 * q^65 + 51624 * q^69 - 28288 * q^71 - 41400 * q^75 + 220200 * q^79 + 39366 * q^81 + 95880 * q^85 + 351420 * q^89 - 17088 * q^91 - 349120 * q^95 + 53784 * q^99

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