LMFDB - Embedded newform 240.5.l.a.161.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [240,5,Mod(161,240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(240, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("240.161");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	240.l (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:	$24.8087911401$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-5}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 5 $ x^2 + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.1
Root			$-2.23607i$ of defining polynomial
Character	$\chi$	$=$	240.161
Dual form			240.5.l.a.161.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-6.00000 - 6.70820i) q^{3} -11.1803i q^{5} -74.0000 q^{7} +(-9.00000 + 80.4984i) q^{9} +O(q^{10})$	\(q+(-6.00000 - 6.70820i) q^{3} -11.1803i q^{5} -74.0000 q^{7} +(-9.00000 + 80.4984i) q^{9} -120.748i q^{11} -16.0000 q^{13} +(-75.0000 + 67.0820i) q^{15} +174.413i q^{17} -374.000 q^{19} +(444.000 + 496.407i) q^{21} +657.404i q^{23} -125.000 q^{25} +(594.000 - 422.617i) q^{27} -1381.89i q^{29} +1426.00 q^{31} +(-810.000 + 724.486i) q^{33} +827.345i q^{35} +272.000 q^{37} +(96.0000 + 107.331i) q^{39} -778.152i q^{41} +1936.00 q^{43} +(900.000 + 100.623i) q^{45} +3206.52i q^{47} +3075.00 q^{49} +(1170.00 - 1046.48i) q^{51} +5487.31i q^{53} -1350.00 q^{55} +(2244.00 + 2508.87i) q^{57} +4816.49i q^{59} +2234.00 q^{61} +(666.000 - 5956.89i) q^{63} +178.885i q^{65} +5416.00 q^{67} +(4410.00 - 3944.42i) q^{69} -1556.30i q^{71} -538.000 q^{73} +(750.000 + 838.525i) q^{75} +8935.33i q^{77} -9962.00 q^{79} +(-6399.00 - 1448.97i) q^{81} -4655.49i q^{83} +1950.00 q^{85} +(-9270.00 + 8291.34i) q^{87} -5259.23i q^{89} +1184.00 q^{91} +(-8556.00 - 9565.90i) q^{93} +4181.45i q^{95} -10726.0 q^{97} +(9720.00 + 1086.73i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 12 q^{3} - 148 q^{7} - 18 q^{9}+O(q^{10}) $ 2 * q - 12 * q^3 - 148 * q^7 - 18 * q^9	$ 2 q - 12 q^{3} - 148 q^{7} - 18 q^{9} - 32 q^{13} - 150 q^{15} - 748 q^{19} + 888 q^{21} - 250 q^{25} + 1188 q^{27} + 2852 q^{31} - 1620 q^{33} + 544 q^{37} + 192 q^{39} + 3872 q^{43} + 1800 q^{45} + 6150 q^{49} + 2340 q^{51} - 2700 q^{55} + 4488 q^{57} + 4468 q^{61} + 1332 q^{63} + 10832 q^{67} + 8820 q^{69} - 1076 q^{73} + 1500 q^{75} - 19924 q^{79} - 12798 q^{81} + 3900 q^{85} - 18540 q^{87} + 2368 q^{91} - 17112 q^{93} - 21452 q^{97} + 19440 q^{99}+O(q^{100}) $ 2 * q - 12 * q^3 - 148 * q^7 - 18 * q^9 - 32 * q^13 - 150 * q^15 - 748 * q^19 + 888 * q^21 - 250 * q^25 + 1188 * q^27 + 2852 * q^31 - 1620 * q^33 + 544 * q^37 + 192 * q^39 + 3872 * q^43 + 1800 * q^45 + 6150 * q^49 + 2340 * q^51 - 2700 * q^55 + 4488 * q^57 + 4468 * q^61 + 1332 * q^63 + 10832 * q^67 + 8820 * q^69 - 1076 * q^73 + 1500 * q^75 - 19924 * q^79 - 12798 * q^81 + 3900 * q^85 - 18540 * q^87 + 2368 * q^91 - 17112 * q^93 - 21452 * q^97 + 19440 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/240\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$181$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	−6.00000	−	6.70820i	−0.666667	−	0.745356i
$3$	−6.00000	−	6.70820i	−0.666667	−	0.745356i
$4$		0			0
$5$		−	11.1803i		−	0.447214i
$5$		−	11.1803i		−	0.447214i
$6$		0			0
$7$	−74.0000			−1.51020			−0.755102	−	0.655607i	$-0.772413\pi$
$7$	−74.0000			−1.51020			−0.755102	+	0.655607i	$0.772413\pi$
$8$		0			0
$9$	−9.00000	+	80.4984i	−0.111111	+	0.993808i
$10$		0			0
$11$		−	120.748i		−	0.997915i	−0.866627	−	0.498957i	$-0.833716\pi$
$11$		−	120.748i		−	0.997915i	0.866627	−	0.498957i	$-0.166284\pi$
$12$		0			0
$13$	−16.0000			−0.0946746			−0.0473373	−	0.998879i	$-0.515074\pi$
$13$	−16.0000			−0.0946746			−0.0473373	+	0.998879i	$0.515074\pi$
$14$		0			0
$15$	−75.0000	+	67.0820i	−0.333333	+	0.298142i
$16$		0			0
$17$			174.413i			0.603506i	0.953386	+	0.301753i	$0.0975718\pi$
$17$			174.413i			0.603506i	−0.953386	+	0.301753i	$0.902428\pi$
$18$		0			0
$19$	−374.000			−1.03601			−0.518006	−	0.855377i	$-0.673325\pi$
$19$	−374.000			−1.03601			−0.518006	+	0.855377i	$0.673325\pi$
$20$		0			0
$21$	444.000	+	496.407i	1.00680	+	1.12564i
$22$		0			0
$23$			657.404i			1.24273i	0.783521	+	0.621365i	$0.213421\pi$
$23$			657.404i			1.24273i	−0.783521	+	0.621365i	$0.786579\pi$
$24$		0			0
$25$	−125.000			−0.200000
$26$		0			0
$27$	594.000	−	422.617i	0.814815	−	0.579721i
$28$		0			0
$29$		−	1381.89i		−	1.64315i	−0.570100	−	0.821576i	$-0.693095\pi$
$29$		−	1381.89i		−	1.64315i	0.570100	−	0.821576i	$-0.306905\pi$
$30$		0			0
$31$	1426.00			1.48387			0.741935	−	0.670471i	$-0.233908\pi$
$31$	1426.00			1.48387			0.741935	+	0.670471i	$0.233908\pi$
$32$		0			0
$33$	−810.000	+	724.486i	−0.743802	+	0.665276i
$34$		0			0
$35$			827.345i			0.675384i
$36$		0			0
$37$	272.000			0.198685			0.0993426	−	0.995053i	$-0.468326\pi$
$37$	272.000			0.198685			0.0993426	+	0.995053i	$0.468326\pi$
$38$		0			0
$39$	96.0000	+	107.331i	0.0631164	+	0.0705662i
$40$		0			0
$41$		−	778.152i		−	0.462910i	−0.972846	−	0.231455i	$-0.925651\pi$
$41$		−	778.152i		−	0.462910i	0.972846	−	0.231455i	$-0.0743486\pi$
$42$		0			0
$43$	1936.00			1.04705			0.523526	−	0.852010i	$-0.324616\pi$
$43$	1936.00			1.04705			0.523526	+	0.852010i	$0.324616\pi$
$44$		0			0
$45$	900.000	+	100.623i	0.444444	+	0.0496904i
$46$		0			0
$47$			3206.52i			1.45157i	0.687921	+	0.725786i	$0.258524\pi$
$47$			3206.52i			1.45157i	−0.687921	+	0.725786i	$0.741476\pi$
$48$		0			0
$49$	3075.00			1.28072
$50$		0			0
$51$	1170.00	−	1046.48i	0.449827	−	0.402337i
$52$		0			0
$53$			5487.31i			1.95347i	0.214439	+	0.976737i	$0.431208\pi$
$53$			5487.31i			1.95347i	−0.214439	+	0.976737i	$0.568792\pi$
$54$		0			0
$55$	−1350.00			−0.446281
$56$		0			0
$57$	2244.00	+	2508.87i	0.690674	+	0.772197i
$58$		0			0
$59$			4816.49i			1.38365i	0.722065	+	0.691826i	$0.243193\pi$
$59$			4816.49i			1.38365i	−0.722065	+	0.691826i	$0.756807\pi$
$60$		0			0
$61$	2234.00			0.600376			0.300188	−	0.953880i	$-0.402951\pi$
$61$	2234.00			0.600376			0.300188	+	0.953880i	$0.402951\pi$
$62$		0			0
$63$	666.000	−	5956.89i	0.167800	−	1.50085i
$64$		0			0
$65$			178.885i			0.0423397i
$66$		0			0
$67$	5416.00			1.20650			0.603252	−	0.797550i	$-0.293871\pi$
$67$	5416.00			1.20650			0.603252	+	0.797550i	$0.293871\pi$
$68$		0			0
$69$	4410.00	−	3944.42i	0.926276	−	0.828486i
$70$		0			0
$71$		−	1556.30i		−	0.308729i	−0.988014	−	0.154365i	$-0.950667\pi$
$71$		−	1556.30i		−	0.308729i	0.988014	−	0.154365i	$-0.0493330\pi$
$72$		0			0
$73$	−538.000			−0.100957			−0.0504785	−	0.998725i	$-0.516075\pi$
$73$	−538.000			−0.100957			−0.0504785	+	0.998725i	$0.516075\pi$
$74$		0			0
$75$	750.000	+	838.525i	0.133333	+	0.149071i
$76$		0			0
$77$			8935.33i			1.50705i
$78$		0			0
$79$	−9962.00			−1.59622			−0.798109	−	0.602513i	$-0.794166\pi$
$79$	−9962.00			−1.59622			−0.798109	+	0.602513i	$0.794166\pi$
$80$		0			0
$81$	−6399.00	−	1448.97i	−0.975309	−	0.220846i
$82$		0			0
$83$		−	4655.49i		−	0.675787i	−0.941184	−	0.337893i	$-0.890286\pi$
$83$		−	4655.49i		−	0.675787i	0.941184	−	0.337893i	$-0.109714\pi$
$84$		0			0
$85$	1950.00			0.269896
$86$		0			0
$87$	−9270.00	+	8291.34i	−1.22473	+	1.09543i
$88$		0			0
$89$		−	5259.23i		−	0.663961i	−0.943286	−	0.331980i	$-0.892283\pi$
$89$		−	5259.23i		−	0.663961i	0.943286	−	0.331980i	$-0.107717\pi$
$90$		0			0
$91$	1184.00			0.142978
$92$		0			0
$93$	−8556.00	−	9565.90i	−0.989247	−	1.10601i
$94$		0			0
$95$			4181.45i			0.463318i
$96$		0			0
$97$	−10726.0			−1.13997			−0.569986	−	0.821654i	$-0.693051\pi$
$97$	−10726.0			−1.13997			−0.569986	+	0.821654i	$0.693051\pi$
$98$		0			0
$99$	9720.00	+	1086.73i	0.991736	+	0.110879i
$100$		0			0
$101$			9968.39i			0.977197i	0.872509	+	0.488599i	$0.162492\pi$
$101$			9968.39i			0.977197i	−0.872509	+	0.488599i	$0.837508\pi$
$102$		0			0
$103$	9238.00			0.870770			0.435385	−	0.900244i	$-0.356612\pi$
$103$	9238.00			0.870770			0.435385	+	0.900244i	$0.356612\pi$
$104$		0			0
$105$	5550.00	−	4964.07i	0.503401	−	0.450256i
$106$		0			0
$107$			4655.49i			0.406629i	0.979114	+	0.203314i	$0.0651714\pi$
$107$			4655.49i			0.406629i	−0.979114	+	0.203314i	$0.934829\pi$
$108$		0			0
$109$	−5506.00			−0.463429			−0.231715	−	0.972784i	$-0.574434\pi$
$109$	−5506.00			−0.463429			−0.231715	+	0.972784i	$0.574434\pi$
$110$		0			0
$111$	−1632.00	−	1824.63i	−0.132457	−	0.148091i
$112$		0			0
$113$		−	7794.93i		−	0.610458i	−0.952279	−	0.305229i	$-0.901267\pi$
$113$		−	7794.93i		−	0.610458i	0.952279	−	0.305229i	$-0.0987329\pi$
$114$		0			0
$115$	7350.00			0.555766
$116$		0			0
$117$	144.000	−	1287.98i	0.0105194	−	0.0940883i
$118$		0			0
$119$		−	12906.6i		−	0.911418i
$120$		0			0
$121$	61.0000			0.00416638
$122$		0			0
$123$	−5220.00	+	4668.91i	−0.345033	+	0.308607i
$124$		0			0
$125$			1397.54i			0.0894427i
$126$		0			0
$127$	−3782.00			−0.234484			−0.117242	−	0.993103i	$-0.537405\pi$
$127$	−3782.00			−0.234484			−0.117242	+	0.993103i	$0.537405\pi$
$128$		0			0
$129$	−11616.0	−	12987.1i	−0.698035	−	0.780427i
$130$		0			0
$131$		−	10988.0i		−	0.640291i	−0.947368	−	0.320146i	$-0.896268\pi$
$131$		−	10988.0i		−	0.640291i	0.947368	−	0.320146i	$-0.103732\pi$
$132$		0			0
$133$	27676.0			1.56459
$134$		0			0
$135$	−4725.00	−	6641.12i	−0.259259	−	0.364396i
$136$		0			0
$137$			18796.4i			1.00146i	0.865604	+	0.500730i	$0.166935\pi$
$137$			18796.4i			1.00146i	−0.865604	+	0.500730i	$0.833065\pi$
$138$		0			0
$139$	−38102.0			−1.97205			−0.986026	−	0.166594i	$-0.946723\pi$
$139$	−38102.0			−1.97205			−0.986026	+	0.166594i	$0.946723\pi$
$140$		0			0
$141$	21510.0	−	19239.1i	1.08194	−	0.967714i
$142$		0			0
$143$			1931.96i			0.0944771i
$144$		0			0
$145$	−15450.0			−0.734839
$146$		0			0
$147$	−18450.0	−	20627.7i	−0.853811	−	0.954590i
$148$		0			0
$149$			29395.3i			1.32406i	0.749479	+	0.662028i	$0.230304\pi$
$149$			29395.3i			1.32406i	−0.749479	+	0.662028i	$0.769696\pi$
$150$		0			0
$151$	11086.0			0.486207			0.243103	−	0.970000i	$-0.421835\pi$
$151$	11086.0			0.486207			0.243103	+	0.970000i	$0.421835\pi$
$152$		0			0
$153$	−14040.0	−	1569.72i	−0.599769	−	0.0670562i
$154$		0			0
$155$		−	15943.2i		−	0.663607i
$156$		0			0
$157$	−24556.0			−0.996227			−0.498114	−	0.867112i	$-0.665974\pi$
$157$	−24556.0			−0.996227			−0.498114	+	0.867112i	$0.665974\pi$
$158$		0			0
$159$	36810.0	−	32923.9i	1.45603	−	1.30232i
$160$		0			0
$161$		−	48647.9i		−	1.87678i
$162$		0			0
$163$	−41084.0			−1.54631			−0.773157	−	0.634215i	$-0.781323\pi$
$163$	−41084.0			−1.54631			−0.773157	+	0.634215i	$0.781323\pi$
$164$		0			0
$165$	8100.00	+	9056.08i	0.297521	+	0.332638i
$166$		0			0
$167$			46890.3i			1.68132i	0.541563	+	0.840660i	$0.317833\pi$
$167$			46890.3i			1.68132i	−0.541563	+	0.840660i	$0.682167\pi$
$168$		0			0
$169$	−28305.0			−0.991037
$170$		0			0
$171$	3366.00	−	30106.4i	0.115112	−	1.02960i
$172$		0			0
$173$			10558.7i			0.352792i	0.984319	+	0.176396i	$0.0564439\pi$
$173$			10558.7i			0.352792i	−0.984319	+	0.176396i	$0.943556\pi$
$174$		0			0
$175$	9250.00			0.302041
$176$		0			0
$177$	32310.0	−	28898.9i	1.03131	−	0.922434i
$178$		0			0
$179$			21748.0i			0.678755i	0.940650	+	0.339378i	$0.110216\pi$
$179$			21748.0i			0.678755i	−0.940650	+	0.339378i	$0.889784\pi$
$180$		0			0
$181$	39194.0			1.19636			0.598181	−	0.801361i	$-0.295890\pi$
$181$	39194.0			1.19636			0.598181	+	0.801361i	$0.295890\pi$
$182$		0			0
$183$	−13404.0	−	14986.1i	−0.400251	−	0.447494i
$184$		0			0
$185$		−	3041.05i		−	0.0888547i
$186$		0			0
$187$	21060.0			0.602248
$188$		0			0
$189$	−43956.0	+	31273.6i	−1.23054	+	0.875498i
$190$		0			0
$191$			28630.6i			0.784809i	0.919793	+	0.392404i	$0.128357\pi$
$191$			28630.6i			0.784809i	−0.919793	+	0.392404i	$0.871643\pi$
$192$		0			0
$193$	−11926.0			−0.320170			−0.160085	−	0.987103i	$-0.551177\pi$
$193$	−11926.0			−0.320170			−0.160085	+	0.987103i	$0.551177\pi$
$194$		0			0
$195$	1200.00	−	1073.31i	0.0315582	−	0.0282265i
$196$		0			0
$197$		−	8975.58i		−	0.231276i	−0.993291	−	0.115638i	$-0.963109\pi$
$197$		−	8975.58i		−	0.231276i	0.993291	−	0.115638i	$-0.0368912\pi$
$198$		0			0
$199$	31906.0			0.805687			0.402843	−	0.915269i	$-0.368022\pi$
$199$	31906.0			0.805687			0.402843	+	0.915269i	$0.368022\pi$
$200$		0			0
$201$	−32496.0	−	36331.6i	−0.804337	−	0.899276i
$202$		0			0
$203$			102260.i			2.48149i
$204$		0			0
$205$	−8700.00			−0.207020
$206$		0			0
$207$	−52920.0	−	5916.64i	−1.23503	−	0.138081i
$208$		0			0
$209$			45159.6i			1.03385i
$210$		0			0
$211$	−74.0000			−0.00166214			−0.000831068	−	1.00000i	$-0.500265\pi$
$211$	−74.0000			−0.00166214			−0.000831068		1.00000i	$0.500265\pi$
$212$		0			0
$213$	−10440.0	+	9337.82i	−0.230113	+	0.205819i
$214$		0			0
$215$		−	21645.1i		−	0.468256i
$216$		0			0
$217$	−105524.			−2.24095
$218$		0			0
$219$	3228.00	+	3609.01i	0.0673047	+	0.0752489i
$220$		0			0
$221$		−	2790.61i		−	0.0571367i
$222$		0			0
$223$	−25334.0			−0.509441			−0.254721	−	0.967015i	$-0.581983\pi$
$223$	−25334.0			−0.509441			−0.254721	+	0.967015i	$0.581983\pi$
$224$		0			0
$225$	1125.00	−	10062.3i	0.0222222	−	0.198762i
$226$		0			0
$227$		−	33742.3i		−	0.654821i	−0.944882	−	0.327410i	$-0.893824\pi$
$227$		−	33742.3i		−	0.654821i	0.944882	−	0.327410i	$-0.106176\pi$
$228$		0			0
$229$	92174.0			1.75767			0.878835	−	0.477125i	$-0.158321\pi$
$229$	92174.0			1.75767			0.878835	+	0.477125i	$0.158321\pi$
$230$		0			0
$231$	59940.0	−	53612.0i	1.12329	−	1.00470i
$232$		0			0
$233$		−	21318.7i		−	0.392689i	−0.980535	−	0.196344i	$-0.937093\pi$
$233$		−	21318.7i		−	0.392689i	0.980535	−	0.196344i	$-0.0629070\pi$
$234$		0			0
$235$	35850.0			0.649163
$236$		0			0
$237$	59772.0	+	66827.1i	1.06415	+	1.18975i
$238$		0			0
$239$			70302.0i			1.23076i	0.788232	+	0.615378i	$0.210996\pi$
$239$			70302.0i			1.23076i	−0.788232	+	0.615378i	$0.789004\pi$
$240$		0			0
$241$	77414.0			1.33286			0.666431	−	0.745566i	$-0.267821\pi$
$241$	77414.0			1.33286			0.666431	+	0.745566i	$0.267821\pi$
$242$		0			0
$243$	28674.0	+	51619.6i	0.485597	+	0.874183i
$244$		0			0
$245$		−	34379.5i		−	0.572754i
$246$		0			0
$247$	5984.00			0.0980839
$248$		0			0
$249$	−31230.0	+	27933.0i	−0.503702	+	0.450524i
$250$		0			0
$251$		−	84161.1i		−	1.33587i	−0.744220	−	0.667935i	$-0.767178\pi$
$251$		−	84161.1i		−	1.33587i	0.744220	−	0.667935i	$-0.232822\pi$
$252$		0			0
$253$	79380.0			1.24014
$254$		0			0
$255$	−11700.0	−	13081.0i	−0.179931	−	0.201169i
$256$		0			0
$257$			24806.9i			0.375584i	0.982209	+	0.187792i	$0.0601331\pi$
$257$			24806.9i			0.375584i	−0.982209	+	0.187792i	$0.939867\pi$
$258$		0			0
$259$	−20128.0			−0.300055
$260$		0			0
$261$	111240.	+	12437.0i	1.63298	+	0.182572i
$262$		0			0
$263$		−	94223.4i		−	1.36222i	−0.732181	−	0.681110i	$-0.761497\pi$
$263$		−	94223.4i		−	1.36222i	0.732181	−	0.681110i	$-0.238503\pi$
$264$		0			0
$265$	61350.0			0.873621
$266$		0			0
$267$	−35280.0	+	31555.4i	−0.494887	+	0.442640i
$268$		0			0
$269$			81102.2i			1.12080i	0.828222	+	0.560400i	$0.189353\pi$
$269$			81102.2i			1.12080i	−0.828222	+	0.560400i	$0.810647\pi$
$270$		0			0
$271$	38158.0			0.519574			0.259787	−	0.965666i	$-0.416348\pi$
$271$	38158.0			0.519574			0.259787	+	0.965666i	$0.416348\pi$
$272$		0			0
$273$	−7104.00	−	7942.51i	−0.0953186	−	0.106569i
$274$		0			0
$275$			15093.5i			0.199583i
$276$		0			0
$277$	7064.00			0.0920643			0.0460321	−	0.998940i	$-0.485342\pi$
$277$	7064.00			0.0920643			0.0460321	+	0.998940i	$0.485342\pi$
$278$		0			0
$279$	−12834.0	+	114791.i	−0.164875	+	1.47468i
$280$		0			0
$281$			73602.4i			0.932136i	0.884749	+	0.466068i	$0.154330\pi$
$281$			73602.4i			0.932136i	−0.884749	+	0.466068i	$0.845670\pi$
$282$		0			0
$283$	117616.			1.46857			0.734283	−	0.678843i	$-0.237518\pi$
$283$	117616.			1.46857			0.734283	+	0.678843i	$0.237518\pi$
$284$		0			0
$285$	28050.0	−	25088.7i	0.345337	−	0.308879i
$286$		0			0
$287$			57583.2i			0.699089i
$288$		0			0
$289$	53101.0			0.635780
$290$		0			0
$291$	64356.0	+	71952.2i	0.759982	+	0.849685i
$292$		0			0
$293$			91674.3i			1.06786i	0.845530	+	0.533928i	$0.179285\pi$
$293$			91674.3i			1.06786i	−0.845530	+	0.533928i	$0.820715\pi$
$294$		0			0
$295$	53850.0			0.618788
$296$		0			0
$297$	−51030.0	−	71724.1i	−0.578512	−	0.813116i
$298$		0			0
$299$		−	10518.5i		−	0.117655i
$300$		0			0
$301$	−143264.			−1.58126
$302$		0			0
$303$	66870.0	−	59810.3i	0.728360	−	0.651465i
$304$		0			0
$305$		−	24976.9i		−	0.268496i
$306$		0			0
$307$	−44984.0			−0.477289			−0.238644	−	0.971107i	$-0.576703\pi$
$307$	−44984.0			−0.477289			−0.238644	+	0.971107i	$0.576703\pi$
$308$		0			0
$309$	−55428.0	−	61970.4i	−0.580513	−	0.649034i
$310$		0			0
$311$			58549.2i			0.605341i	0.953095	+	0.302671i	$0.0978782\pi$
$311$			58549.2i			0.605341i	−0.953095	+	0.302671i	$0.902122\pi$
$312$		0			0
$313$	−79546.0			−0.811951			−0.405975	−	0.913884i	$-0.633068\pi$
$313$	−79546.0			−0.811951			−0.405975	+	0.913884i	$0.633068\pi$
$314$		0			0
$315$	−66600.0	−	7446.11i	−0.671202	−	0.0750426i
$316$		0			0
$317$			103534.i			1.03031i	0.857098	+	0.515153i	$0.172265\pi$
$317$			103534.i			1.03031i	−0.857098	+	0.515153i	$0.827735\pi$
$318$		0			0
$319$	−166860.			−1.63972
$320$		0			0
$321$	31230.0	−	27933.0i	0.303083	−	0.271086i
$322$		0			0
$323$		−	65230.6i		−	0.625239i
$324$		0			0
$325$	2000.00			0.0189349
$326$		0			0
$327$	33036.0	+	36935.4i	0.308953	+	0.345420i
$328$		0			0
$329$		−	237283.i		−	2.19217i
$330$		0			0
$331$	60238.0			0.549812			0.274906	−	0.961471i	$-0.411353\pi$
$331$	60238.0			0.549812			0.274906	+	0.961471i	$0.411353\pi$
$332$		0			0
$333$	−2448.00	+	21895.6i	−0.0220761	+	0.197455i
$334$		0			0
$335$		−	60552.7i		−	0.539565i
$336$		0			0
$337$	−113686.			−1.00103			−0.500515	−	0.865728i	$-0.666856\pi$
$337$	−113686.			−1.00103			−0.500515	+	0.865728i	$0.666856\pi$
$338$		0			0
$339$	−52290.0	+	46769.6i	−0.455008	+	0.406972i
$340$		0			0
$341$		−	172186.i		−	1.48078i
$342$		0			0
$343$	−49876.0			−0.423939
$344$		0			0
$345$	−44100.0	−	49305.3i	−0.370510	−	0.414243i
$346$		0			0
$347$			18689.1i			0.155213i	0.996984	+	0.0776066i	$0.0247278\pi$
$347$			18689.1i			0.155213i	−0.996984	+	0.0776066i	$0.975272\pi$
$348$		0			0
$349$	15182.0			0.124646			0.0623230	−	0.998056i	$-0.480149\pi$
$349$	15182.0			0.124646			0.0623230	+	0.998056i	$0.480149\pi$
$350$		0			0
$351$	−9504.00	+	6761.87i	−0.0771422	+	0.0548849i
$352$		0			0
$353$			158622.i			1.27296i	0.771293	+	0.636480i	$0.219610\pi$
$353$			158622.i			1.27296i	−0.771293	+	0.636480i	$0.780390\pi$
$354$		0			0
$355$	−17400.0			−0.138068
$356$		0			0
$357$	−86580.0	+	77439.5i	−0.679331	+	0.607612i
$358$		0			0
$359$			138109.i			1.07160i	0.844346	+	0.535799i	$0.179989\pi$
$359$			138109.i			1.07160i	−0.844346	+	0.535799i	$0.820011\pi$
$360$		0			0
$361$	9555.00			0.0733190
$362$		0			0
$363$	−366.000	−	409.200i	−0.00277759	−	0.00310544i
$364$		0			0
$365$			6015.02i			0.0451494i
$366$		0			0
$367$	−58694.0			−0.435774			−0.217887	−	0.975974i	$-0.569916\pi$
$367$	−58694.0			−0.435774			−0.217887	+	0.975974i	$0.569916\pi$
$368$		0			0
$369$	62640.0	+	7003.36i	0.460044	+	0.0514344i
$370$		0			0
$371$		−	406061.i		−	2.95015i
$372$		0			0
$373$	−91036.0			−0.654328			−0.327164	−	0.944968i	$-0.606093\pi$
$373$	−91036.0			−0.654328			−0.327164	+	0.944968i	$0.606093\pi$
$374$		0			0
$375$	9375.00	−	8385.25i	0.0666667	−	0.0596285i
$376$		0			0
$377$			22110.2i			0.155565i
$378$		0			0
$379$	76546.0			0.532898			0.266449	−	0.963849i	$-0.414150\pi$
$379$	76546.0			0.532898			0.266449	+	0.963849i	$0.414150\pi$
$380$		0			0
$381$	22692.0	+	25370.4i	0.156323	+	0.174774i
$382$		0			0
$383$		−	19145.2i		−	0.130516i	−0.997868	−	0.0652578i	$-0.979213\pi$
$383$		−	19145.2i		−	0.130516i	0.997868	−	0.0652578i	$-0.0207870\pi$
$384$		0			0
$385$	99900.0			0.673975
$386$		0			0
$387$	−17424.0	+	155845.i	−0.116339	+	1.04057i
$388$		0			0
$389$			104259.i			0.688992i	0.938788	+	0.344496i	$0.111950\pi$
$389$			104259.i			0.688992i	−0.938788	+	0.344496i	$0.888050\pi$
$390$		0			0
$391$	−114660.			−0.749995
$392$		0			0
$393$	−73710.0	+	65928.2i	−0.477245	+	0.426861i
$394$		0			0
$395$			111379.i			0.713851i
$396$		0			0
$397$	89804.0			0.569790			0.284895	−	0.958559i	$-0.408041\pi$
$397$	89804.0			0.569790			0.284895	+	0.958559i	$0.408041\pi$
$398$		0			0
$399$	−166056.	−	185656.i	−1.04306	−	1.16618i
$400$		0			0
$401$		−	231567.i		−	1.44009i	−0.693930	−	0.720043i	$-0.744122\pi$
$401$		−	231567.i		−	1.44009i	0.693930	−	0.720043i	$-0.255878\pi$
$402$		0			0
$403$	−22816.0			−0.140485
$404$		0			0
$405$	−16200.0	+	71543.0i	−0.0987654	+	0.436171i
$406$		0			0
$407$		−	32843.4i		−	0.198271i
$408$		0			0
$409$	−69766.0			−0.417059			−0.208529	−	0.978016i	$-0.566868\pi$
$409$	−69766.0			−0.417059			−0.208529	+	0.978016i	$0.566868\pi$
$410$		0			0
$411$	126090.	−	112778.i	0.746444	−	0.667639i
$412$		0			0
$413$		−	356420.i		−	2.08960i
$414$		0			0
$415$	−52050.0			−0.302221
$416$		0			0
$417$	228612.	+	255596.i	1.31470	+	1.46988i
$418$		0			0
$419$		−	286937.i		−	1.63440i	−0.576355	−	0.817199i	$-0.695525\pi$
$419$		−	286937.i		−	1.63440i	0.576355	−	0.817199i	$-0.304475\pi$
$420$		0			0
$421$	90602.0			0.511180			0.255590	−	0.966785i	$-0.417730\pi$
$421$	90602.0			0.511180			0.255590	+	0.966785i	$0.417730\pi$
$422$		0			0
$423$	−258120.	−	28858.7i	−1.44258	−	0.161286i
$424$		0			0
$425$		−	21801.7i		−	0.120701i
$426$		0			0
$427$	−165316.			−0.906691
$428$		0			0
$429$	12960.0	−	11591.8i	0.0704191	−	0.0629848i
$430$		0			0
$431$		−	18756.1i		−	0.100969i	−0.998725	−	0.0504846i	$-0.983923\pi$
$431$		−	18756.1i		−	0.100969i	0.998725	−	0.0504846i	$-0.0160766\pi$
$432$		0			0
$433$	15554.0			0.0829595			0.0414798	−	0.999139i	$-0.486793\pi$
$433$	15554.0			0.0829595			0.0414798	+	0.999139i	$0.486793\pi$
$434$		0			0
$435$	92700.0	+	103642.i	0.489893	+	0.547717i
$436$		0			0
$437$		−	245869.i		−	1.28748i
$438$		0			0
$439$	134866.			0.699799			0.349900	−	0.936787i	$-0.386216\pi$
$439$	134866.			0.699799			0.349900	+	0.936787i	$0.386216\pi$
$440$		0			0
$441$	−27675.0	+	247533.i	−0.142302	+	1.27279i
$442$		0			0
$443$			94599.1i			0.482036i	0.970521	+	0.241018i	$0.0774813\pi$
$443$			94599.1i			0.482036i	−0.970521	+	0.241018i	$0.922519\pi$
$444$		0			0
$445$	−58800.0			−0.296932
$446$		0			0
$447$	197190.	−	176372.i	0.986892	−	0.882703i
$448$		0			0
$449$		−	227784.i		−	1.12987i	−0.825134	−	0.564937i	$-0.808900\pi$
$449$		−	227784.i		−	1.12987i	0.825134	−	0.564937i	$-0.191100\pi$
$450$		0			0
$451$	−93960.0			−0.461945
$452$		0			0
$453$	−66516.0	−	74367.1i	−0.324138	−	0.362397i
$454$		0			0
$455$		−	13237.5i		−	0.0639417i
$456$		0			0
$457$	−123778.			−0.592667			−0.296334	−	0.955084i	$-0.595764\pi$
$457$	−123778.			−0.592667			−0.296334	+	0.955084i	$0.595764\pi$
$458$		0			0
$459$	73710.0	+	103602.i	0.349865	+	0.491746i
$460$		0			0
$461$		−	63902.4i		−	0.300687i	−0.988634	−	0.150344i	$-0.951962\pi$
$461$		−	63902.4i		−	0.300687i	0.988634	−	0.150344i	$-0.0480380\pi$
$462$		0			0
$463$	143698.			0.670330			0.335165	−	0.942159i	$-0.391208\pi$
$463$	143698.			0.670330			0.335165	+	0.942159i	$0.391208\pi$
$464$		0			0
$465$	−106950.	+	95659.0i	−0.494624	+	0.442405i
$466$		0			0
$467$			12168.7i			0.0557969i	0.999611	+	0.0278984i	$0.00888150\pi$
$467$			12168.7i			0.0557969i	−0.999611	+	0.0278984i	$0.991119\pi$
$468$		0			0
$469$	−400784.			−1.82207
$470$		0			0
$471$	147336.	+	164727.i	0.664151	+	0.742544i
$472$		0			0
$473$		−	233767.i		−	1.04487i
$474$		0			0
$475$	46750.0			0.207202
$476$		0			0
$477$	−441720.	−	49385.8i	−1.94138	−	0.217053i
$478$		0			0
$479$		−	146561.i		−	0.638774i	−0.947624	−	0.319387i	$-0.896523\pi$
$479$		−	146561.i		−	0.638774i	0.947624	−	0.319387i	$-0.103477\pi$
$480$		0			0
$481$	−4352.00			−0.0188104
$482$		0			0
$483$	−326340.	+	291887.i	−1.39887	+	1.25118i
$484$		0			0
$485$			119920.i			0.509811i
$486$		0			0
$487$	−70874.0			−0.298833			−0.149417	−	0.988774i	$-0.547740\pi$
$487$	−70874.0			−0.298833			−0.149417	+	0.988774i	$0.547740\pi$
$488$		0			0
$489$	246504.	+	275600.i	1.03088	+	1.15255i
$490$		0			0
$491$			289191.i			1.19956i	0.800166	+	0.599779i	$0.204745\pi$
$491$			289191.i			1.19956i	−0.800166	+	0.599779i	$0.795255\pi$
$492$		0			0
$493$	241020.			0.991652
$494$		0			0
$495$	12150.0	−	108673.i	0.0495868	−	0.443518i
$496$		0			0
$497$			115166.i			0.466244i
$498$		0			0
$499$	278986.			1.12042			0.560211	−	0.828350i	$-0.310720\pi$
$499$	278986.			1.12042			0.560211	+	0.828350i	$0.310720\pi$
$500$		0			0
$501$	314550.	−	281342.i	1.25318	−	1.12088i
$502$		0			0
$503$			41496.9i			0.164014i	0.996632	+	0.0820069i	$0.0261329\pi$
$503$			41496.9i			0.164014i	−0.996632	+	0.0820069i	$0.973867\pi$
$504$		0			0
$505$	111450.			0.437016
$506$		0			0
$507$	169830.	+	189876.i	0.660691	+	0.738675i
$508$		0			0
$509$		−	56925.8i		−	0.219722i	−0.993947	−	0.109861i	$-0.964959\pi$
$509$		−	56925.8i		−	0.219722i	0.993947	−	0.109861i	$-0.0350406\pi$
$510$		0			0
$511$	39812.0			0.152466
$512$		0			0
$513$	−222156.	+	158059.i	−0.844157	+	0.600598i
$514$		0			0
$515$		−	103284.i		−	0.389420i
$516$		0			0
$517$	387180.			1.44854
$518$		0			0
$519$	70830.0	−	63352.3i	0.262956	−	0.235195i
$520$		0			0
$521$			243239.i			0.896104i	0.894007	+	0.448052i	$0.147882\pi$
$521$			243239.i			0.896104i	−0.894007	+	0.448052i	$0.852118\pi$
$522$		0			0
$523$	197896.			0.723492			0.361746	−	0.932277i	$-0.382181\pi$
$523$	197896.			0.723492			0.361746	+	0.932277i	$0.382181\pi$
$524$		0			0
$525$	−55500.0	−	62050.9i	−0.201361	−	0.225128i
$526$		0			0
$527$			248713.i			0.895525i
$528$		0			0
$529$	−152339.			−0.544377
$530$		0			0
$531$	−387720.	−	43348.4i	−1.37508	−	0.153739i
$532$		0			0
$533$			12450.4i			0.0438258i
$534$		0			0
$535$	52050.0			0.181850
$536$		0			0
$537$	145890.	−	130488.i	0.505914	−	0.452504i
$538$		0			0
$539$		−	371299.i		−	1.27805i
$540$		0			0
$541$	−572926.			−1.95751			−0.978755	−	0.205033i	$-0.934270\pi$
$541$	−572926.			−1.95751			−0.978755	+	0.205033i	$0.934270\pi$
$542$		0			0
$543$	−235164.	−	262921.i	−0.797574	−	0.891715i
$544$		0			0
$545$			61559.0i			0.207252i
$546$		0			0
$547$	−456992.			−1.52733			−0.763667	−	0.645610i	$-0.776603\pi$
$547$	−456992.			−1.52733			−0.763667	+	0.645610i	$0.776603\pi$
$548$		0			0
$549$	−20106.0	+	179834.i	−0.0667085	+	0.596659i
$550$		0			0
$551$			516827.i			1.70232i
$552$		0			0
$553$	737188.			2.41062
$554$		0			0
$555$	−20400.0	+	18246.3i	−0.0662284	+	0.0592365i
$556$		0			0
$557$			99160.7i			0.319616i	0.987148	+	0.159808i	$0.0510876\pi$
$557$			99160.7i			0.319616i	−0.987148	+	0.159808i	$0.948912\pi$
$558$		0			0
$559$	−30976.0			−0.0991292
$560$		0			0
$561$	−126360.	−	141275.i	−0.401498	−	0.448889i
$562$		0			0
$563$		−	396817.i		−	1.25191i	−0.779859	−	0.625956i	$-0.784709\pi$
$563$		−	396817.i		−	1.25191i	0.779859	−	0.625956i	$-0.215291\pi$
$564$		0			0
$565$	−87150.0			−0.273005
$566$		0			0
$567$	473526.	+	107224.i	1.47292	+	0.333523i
$568$		0			0
$569$			523777.i			1.61779i	0.587955	+	0.808894i	$0.299933\pi$
$569$			523777.i			1.61779i	−0.587955	+	0.808894i	$0.700067\pi$
$570$		0			0
$571$	−363194.			−1.11395			−0.556976	−	0.830529i	$-0.688038\pi$
$571$	−363194.			−1.11395			−0.556976	+	0.830529i	$0.688038\pi$
$572$		0			0
$573$	192060.	−	171784.i	0.584962	−	0.523206i
$574$		0			0
$575$		−	82175.5i		−	0.248546i
$576$		0			0
$577$	508874.			1.52848			0.764238	−	0.644934i	$-0.223115\pi$
$577$	508874.			1.52848			0.764238	+	0.644934i	$0.223115\pi$
$578$		0			0
$579$	71556.0	+	80002.0i	0.213446	+	0.238640i
$580$		0			0
$581$			344507.i			1.02058i
$582$		0			0
$583$	662580.			1.94940
$584$		0			0
$585$	−14400.0	−	1609.97i	−0.0420776	−	0.00470442i
$586$		0			0
$587$		−	462879.i		−	1.34336i	−0.740842	−	0.671679i	$-0.765573\pi$
$587$		−	462879.i		−	1.34336i	0.740842	−	0.671679i	$-0.234427\pi$
$588$		0			0
$589$	−533324.			−1.53731
$590$		0			0
$591$	−60210.0	+	53853.5i	−0.172383	+	0.154184i
$592$		0			0
$593$			364859.i			1.03757i	0.854906	+	0.518783i	$0.173615\pi$
$593$			364859.i			1.03757i	−0.854906	+	0.518783i	$0.826385\pi$
$594$		0			0
$595$	−144300.			−0.407598
$596$		0			0
$597$	−191436.	−	214032.i	−0.537124	−	0.600523i
$598$		0			0
$599$			199717.i			0.556622i	0.960491	+	0.278311i	$0.0897746\pi$
$599$			199717.i			0.556622i	−0.960491	+	0.278311i	$0.910225\pi$
$600$		0			0
$601$	82034.0			0.227115			0.113557	−	0.993531i	$-0.463775\pi$
$601$	82034.0			0.227115			0.113557	+	0.993531i	$0.463775\pi$
$602$		0			0
$603$	−48744.0	+	435980.i	−0.134056	+	1.19903i
$604$		0			0
$605$		−	682.001i		−	0.00186326i
$606$		0			0
$607$	79258.0			0.215113			0.107556	−	0.994199i	$-0.465697\pi$
$607$	79258.0			0.215113			0.107556	+	0.994199i	$0.465697\pi$
$608$		0			0
$609$	685980.	−	613559.i	1.84960	−	1.65433i
$610$		0			0
$611$		−	51304.3i		−	0.137427i
$612$		0			0
$613$	218972.			0.582730			0.291365	−	0.956612i	$-0.405891\pi$
$613$	218972.			0.582730			0.291365	+	0.956612i	$0.405891\pi$
$614$		0			0
$615$	52200.0	+	58361.4i	0.138013	+	0.154303i
$616$		0			0
$617$			96289.6i			0.252935i	0.991971	+	0.126467i	$0.0403639\pi$
$617$			96289.6i			0.252935i	−0.991971	+	0.126467i	$0.959636\pi$
$618$		0			0
$619$	−267014.			−0.696872			−0.348436	−	0.937333i	$-0.613287\pi$
$619$	−267014.			−0.696872			−0.348436	+	0.937333i	$0.613287\pi$
$620$		0			0
$621$	277830.	+	390498.i	0.720437	+	1.01259i
$622$		0			0
$623$			389183.i			1.00272i
$624$		0			0
$625$	15625.0			0.0400000
$626$		0			0
$627$	302940.	−	270958.i	0.770587	−	0.689234i
$628$		0			0
$629$			47440.4i			0.119908i
$630$		0			0
$631$	−54134.0			−0.135960			−0.0679800	−	0.997687i	$-0.521655\pi$
$631$	−54134.0			−0.135960			−0.0679800	+	0.997687i	$0.521655\pi$
$632$		0			0
$633$	444.000	+	496.407i	0.00110809	+	0.00123888i
$634$		0			0
$635$			42284.0i			0.104865i
$636$		0			0
$637$	−49200.0			−0.121251
$638$		0			0
$639$	125280.	+	14006.7i	0.306817	+	0.0343032i
$640$		0			0
$641$			558498.i			1.35927i	0.733550	+	0.679635i	$0.237862\pi$
$641$			558498.i			1.35927i	−0.733550	+	0.679635i	$0.762138\pi$
$642$		0			0
$643$	305428.			0.738732			0.369366	−	0.929284i	$-0.379575\pi$
$643$	305428.			0.738732			0.369366	+	0.929284i	$0.379575\pi$
$644$		0			0
$645$	−145200.	+	129871.i	−0.349017	+	0.312171i
$646$		0			0
$647$		−	51908.1i		−	0.124001i	−0.998076	−	0.0620007i	$-0.980252\pi$
$647$		−	51908.1i		−	0.124001i	0.998076	−	0.0620007i	$-0.0197481\pi$
$648$		0			0
$649$	581580.			1.38077
$650$		0			0
$651$	633144.	+	707877.i	1.49397	+	1.67030i
$652$		0			0
$653$		−	474793.i		−	1.11347i	−0.830690	−	0.556735i	$-0.812054\pi$
$653$		−	474793.i		−	1.11347i	0.830690	−	0.556735i	$-0.187946\pi$
$654$		0			0
$655$	−122850.			−0.286347
$656$		0			0
$657$	4842.00	−	43308.2i	0.0112174	−	0.100332i
$658$		0			0
$659$			711888.i			1.63923i	0.572912	+	0.819617i	$0.305814\pi$
$659$			711888.i			1.63923i	−0.572912	+	0.819617i	$0.694186\pi$
$660$		0			0
$661$	−247198.			−0.565773			−0.282886	−	0.959153i	$-0.591292\pi$
$661$	−247198.			−0.565773			−0.282886	+	0.959153i	$0.591292\pi$
$662$		0			0
$663$	−18720.0	+	16743.7i	−0.0425872	+	0.0380911i
$664$		0			0
$665$		−	309427.i		−	0.699705i
$666$		0			0
$667$	908460.			2.04199
$668$		0			0
$669$	152004.	+	169946.i	0.339627	+	0.379715i
$670$		0			0
$671$		−	269750.i		−	0.599124i
$672$		0			0
$673$	−805438.			−1.77829			−0.889144	−	0.457628i	$-0.848699\pi$
$673$	−805438.			−1.77829			−0.889144	+	0.457628i	$0.848699\pi$
$674$		0			0
$675$	−74250.0	+	52827.1i	−0.162963	+	0.115944i
$676$		0			0
$677$		−	821124.i		−	1.79156i	−0.444497	−	0.895781i	$-0.646617\pi$
$677$		−	821124.i		−	1.79156i	0.444497	−	0.895781i	$-0.353383\pi$
$678$		0			0
$679$	793724.			1.72159
$680$		0			0
$681$	−226350.	+	202454.i	−0.488075	+	0.436547i
$682$		0			0
$683$		−	72703.5i		−	0.155853i	−0.996959	−	0.0779263i	$-0.975170\pi$
$683$		−	72703.5i		−	0.155853i	0.996959	−	0.0779263i	$-0.0248299\pi$
$684$		0			0
$685$	210150.			0.447866
$686$		0			0
$687$	−553044.	−	618322.i	−1.17178	−	1.31009i
$688$		0			0
$689$		−	87797.0i		−	0.184944i
$690$		0			0
$691$	−250802.			−0.525261			−0.262630	−	0.964897i	$-0.584590\pi$
$691$	−250802.			−0.525261			−0.262630	+	0.964897i	$0.584590\pi$
$692$		0			0
$693$	−719280.	−	80417.9i	−1.49772	−	0.167451i
$694$		0			0
$695$			425993.i			0.881928i
$696$		0			0
$697$	135720.			0.279369
$698$		0			0
$699$	−143010.	+	127912.i	−0.292693	+	0.261792i
$700$		0			0
$701$		−	236759.i		−	0.481805i	−0.970549	−	0.240902i	$-0.922557\pi$
$701$		−	236759.i		−	0.481805i	0.970549	−	0.240902i	$-0.0774434\pi$
$702$		0			0
$703$	−101728.			−0.205840
$704$		0			0
$705$	−215100.	−	240489.i	−0.432775	−	0.483857i
$706$		0			0
$707$		−	737661.i		−	1.47577i
$708$		0			0
$709$	25094.0			0.0499203			0.0249602	−	0.999688i	$-0.492054\pi$
$709$	25094.0			0.0499203			0.0249602	+	0.999688i	$0.492054\pi$
$710$		0			0
$711$	89658.0	−	801926.i	0.177358	−	1.58633i
$712$		0			0
$713$			937458.i			1.84405i
$714$		0			0
$715$	21600.0			0.0422515
$716$		0			0
$717$	471600.	−	421812.i	0.917351	−	0.820504i
$718$		0			0
$719$			140819.i			0.272397i	0.990682	+	0.136198i	$0.0434885\pi$
$719$			140819.i			0.272397i	−0.990682	+	0.136198i	$0.956512\pi$
$720$		0			0
$721$	−683612.			−1.31504
$722$		0			0
$723$	−464484.	−	519309.i	−0.888575	−	0.993457i
$724$		0			0
$725$			172736.i			0.328630i
$726$		0			0
$727$	−806354.			−1.52566			−0.762829	−	0.646601i	$-0.776190\pi$
$727$	−806354.			−1.52566			−0.762829	+	0.646601i	$0.776190\pi$
$728$		0			0
$729$	174231.	−	502069.i	0.327846	−	0.944731i
$730$		0			0
$731$			337664.i			0.631903i
$732$		0			0
$733$	280112.			0.521343			0.260672	−	0.965428i	$-0.416056\pi$
$733$	280112.			0.521343			0.260672	+	0.965428i	$0.416056\pi$
$734$		0			0
$735$	−230625.	+	206277.i	−0.426905	+	0.381836i
$736$		0			0
$737$		−	653969.i		−	1.20399i
$738$		0			0
$739$	−765902.			−1.40244			−0.701220	−	0.712945i	$-0.747361\pi$
$739$	−765902.			−1.40244			−0.701220	+	0.712945i	$0.747361\pi$
$740$		0			0
$741$	−35904.0	−	40141.9i	−0.0653893	−	0.0731074i
$742$		0			0
$743$			107398.i			0.194545i	0.995258	+	0.0972725i	$0.0310118\pi$
$743$			107398.i			0.194545i	−0.995258	+	0.0972725i	$0.968988\pi$
$744$		0			0
$745$	328650.			0.592135
$746$		0			0
$747$	374760.	+	41899.4i	0.671602	+	0.0750874i
$748$		0			0
$749$		−	344507.i		−	0.614093i
$750$		0			0
$751$	−1.06734e6			−1.89245			−0.946223	−	0.323514i	$-0.895136\pi$
$751$	−1.06734e6			−1.89245			−0.946223	+	0.323514i	$0.895136\pi$
$752$		0			0
$753$	−564570.	+	504967.i	−0.995698	+	0.890580i
$754$		0			0
$755$		−	123945.i		−	0.217438i
$756$		0			0
$757$	−526576.			−0.918902			−0.459451	−	0.888203i	$-0.651954\pi$
$757$	−526576.			−0.918902			−0.459451	+	0.888203i	$0.651954\pi$
$758$		0			0
$759$	−476280.	−	532497.i	−0.826759	−	0.924344i
$760$		0			0
$761$			415721.i			0.717848i	0.933367	+	0.358924i	$0.116856\pi$
$761$			415721.i			0.717848i	−0.933367	+	0.358924i	$0.883144\pi$
$762$		0			0
$763$	407444.			0.699872
$764$		0			0
$765$	−17550.0	+	156972.i	−0.0299885	+	0.268225i
$766$		0			0
$767$		−	77063.8i		−	0.130997i
$768$		0			0
$769$	728174.			1.23135			0.615676	−	0.787999i	$-0.288883\pi$
$769$	728174.			1.23135			0.615676	+	0.787999i	$0.288883\pi$
$770$		0			0
$771$	166410.	−	148842.i	0.279944	−	0.250389i
$772$		0			0
$773$			247439.i			0.414103i	0.978330	+	0.207052i	$0.0663868\pi$
$773$			247439.i			0.414103i	−0.978330	+	0.207052i	$0.933613\pi$
$774$		0			0
$775$	−178250.			−0.296774
$776$		0			0
$777$	120768.	+	135023.i	0.200037	+	0.223648i
$778$		0			0
$779$			291029.i			0.479580i
$780$		0			0
$781$	−187920.			−0.308085
$782$		0			0
$783$	−584010.	−	820843.i	−0.952570	−	1.33886i
$784$		0			0
$785$			274544.i			0.445526i
$786$		0			0
$787$	165016.			0.266426			0.133213	−	0.991087i	$-0.457471\pi$
$787$	165016.			0.266426			0.133213	+	0.991087i	$0.457471\pi$
$788$		0			0
$789$	−632070.	+	565341.i	−1.01534	+	0.908147i
$790$		0			0
$791$			576825.i			0.921916i
$792$		0			0
$793$	−35744.0			−0.0568404
$794$		0			0
$795$	−368100.	−	411548.i	−0.582414	−	0.651158i
$796$		0			0
$797$			659510.i			1.03826i	0.854696	+	0.519129i	$0.173743\pi$
$797$			659510.i			1.03826i	−0.854696	+	0.519129i	$0.826257\pi$
$798$		0			0
$799$	−559260.			−0.876032
$800$		0			0
$801$	423360.	+	47333.1i	0.659849	+	0.0737734i
$802$		0			0
$803$			64962.2i			0.100746i
$804$		0			0
$805$	−543900.			−0.839319
$806$		0			0
$807$	544050.	−	486613.i	0.835395	−	0.747200i
$808$		0			0
$809$		−	1.17168e6i		−	1.79025i	−0.445820	−	0.895123i	$-0.647088\pi$
$809$		−	1.17168e6i		−	1.79025i	0.445820	−	0.895123i	$-0.352912\pi$
$810$		0			0
$811$	70486.0			0.107167			0.0535835	−	0.998563i	$-0.482936\pi$
$811$	70486.0			0.107167			0.0535835	+	0.998563i	$0.482936\pi$
$812$		0			0
$813$	−228948.	−	255972.i	−0.346382	−	0.387267i
$814$		0			0
$815$			459333.i			0.691532i
$816$		0			0
$817$	−724064.			−1.08476
$818$		0			0
$819$	−10656.0	+	95310.2i	−0.0158864	+	0.142093i
$820$		0			0
$821$		−	458962.i		−	0.680911i	−0.940261	−	0.340455i	$-0.889419\pi$
$821$		−	458962.i		−	0.680911i	0.940261	−	0.340455i	$-0.110581\pi$
$822$		0			0
$823$	53926.0			0.0796157			0.0398078	−	0.999207i	$-0.487325\pi$
$823$	53926.0			0.0796157			0.0398078	+	0.999207i	$0.487325\pi$
$824$		0			0
$825$	101250.	−	90560.8i	0.148760	−	0.133055i
$826$		0			0
$827$		−	29610.0i		−	0.0432940i	−0.999766	−	0.0216470i	$-0.993109\pi$
$827$		−	29610.0i		−	0.0432940i	0.999766	−	0.0216470i	$-0.00689099\pi$
$828$		0			0
$829$	416342.			0.605817			0.302908	−	0.953020i	$-0.402042\pi$
$829$	416342.			0.605817			0.302908	+	0.953020i	$0.402042\pi$
$830$		0			0
$831$	−42384.0	−	47386.8i	−0.0613762	−	0.0686207i
$832$		0			0
$833$			536321.i			0.772920i
$834$		0			0
$835$	524250.			0.751909
$836$		0			0
$837$	847044.	−	602652.i	1.20908	−	0.860232i
$838$		0			0
$839$			549456.i			0.780564i	0.920695	+	0.390282i	$0.127623\pi$
$839$			549456.i			0.780564i	−0.920695	+	0.390282i	$0.872377\pi$
$840$		0			0
$841$	−1.20234e6			−1.69995
$842$		0			0
$843$	493740.	−	441614.i	0.694773	−	0.621424i
$844$		0			0
$845$			316460.i			0.443205i
$846$		0			0
$847$	−4514.00			−0.00629209
$848$		0			0
$849$	−705696.	−	788992.i	−0.979044	−	1.09460i
$850$		0			0
$851$			178814.i			0.246912i
$852$		0			0
$853$	1.02736e6			1.41197			0.705986	−	0.708225i	$-0.250504\pi$
$853$	1.02736e6			1.41197			0.705986	+	0.708225i	$0.250504\pi$
$854$		0			0
$855$	−336600.	−	37633.0i	−0.460449	−	0.0514798i
$856$		0			0
$857$		−	717764.i		−	0.977283i	−0.872485	−	0.488641i	$-0.837493\pi$
$857$		−	717764.i		−	0.977283i	0.872485	−	0.488641i	$-0.162507\pi$
$858$		0			0
$859$	507298.			0.687507			0.343753	−	0.939060i	$-0.388302\pi$
$859$	507298.			0.687507			0.343753	+	0.939060i	$0.388302\pi$
$860$		0			0
$861$	386280.	−	345499.i	0.521070	−	0.466059i
$862$		0			0
$863$		−	162593.i		−	0.218314i	−0.994025	−	0.109157i	$-0.965185\pi$
$863$		−	162593.i		−	0.218314i	0.994025	−	0.109157i	$-0.0348151\pi$
$864$		0			0
$865$	118050.			0.157773
$866$		0			0
$867$	−318606.	−	356212.i	−0.423853	−	0.473883i
$868$		0			0
$869$			1.20289e6i			1.59289i
$870$		0			0
$871$	−86656.0			−0.114225
$872$		0			0
$873$	96534.0	−	863426.i	0.126664	−	1.13291i
$874$		0			0
$875$		−	103418.i		−	0.135077i
$876$		0			0
$877$	541844.			0.704490			0.352245	−	0.935908i	$-0.385418\pi$
$877$	541844.			0.704490			0.352245	+	0.935908i	$0.385418\pi$
$878$		0			0
$879$	614970.	−	550046.i	0.795932	−	0.711904i
$880$		0			0
$881$			448376.i			0.577685i	0.957377	+	0.288842i	$0.0932703\pi$
$881$			448376.i			0.577685i	−0.957377	+	0.288842i	$0.906730\pi$
$882$		0			0
$883$	−322124.			−0.413144			−0.206572	−	0.978431i	$-0.566231\pi$
$883$	−322124.			−0.413144			−0.206572	+	0.978431i	$0.566231\pi$
$884$		0			0
$885$	−323100.	−	361237.i	−0.412525	−	0.461217i
$886$		0			0
$887$			1.00708e6i			1.28001i	0.768369	+	0.640007i	$0.221069\pi$
$887$			1.00708e6i			1.28001i	−0.768369	+	0.640007i	$0.778931\pi$
$888$		0			0
$889$	279868.			0.354119
$890$		0			0
$891$	−174960.	+	772664.i	−0.220386	+	0.973275i
$892$		0			0
$893$		−	1.19924e6i		−	1.50384i
$894$		0			0
$895$	243150.			0.303549
$896$		0			0
$897$	−70560.0	+	63110.8i	−0.0876948	+	0.0784366i
$898$		0			0
$899$		−	1.97058e6i		−	2.43822i
$900$		0			0
$901$	−957060.			−1.17893
$902$		0			0
$903$	859584.	+	961044.i	1.05418	+	1.17860i
$904$		0			0
$905$		−	438202.i		−	0.535029i
$906$		0			0
$907$	455788.			0.554049			0.277025	−	0.960863i	$-0.410652\pi$
$907$	455788.			0.554049			0.277025	+	0.960863i	$0.410652\pi$
$908$		0			0
$909$	−802440.	−	89715.5i	−0.971147	−	0.108577i
$910$		0			0
$911$			316708.i			0.381612i	0.981628	+	0.190806i	$0.0611101\pi$
$911$			316708.i			0.381612i	−0.981628	+	0.190806i	$0.938890\pi$
$912$		0			0
$913$	−562140.			−0.674377
$914$		0			0
$915$	−167550.	+	149861.i	−0.200125	+	0.178998i
$916$		0			0
$917$			813115.i			0.966970i
$918$		0			0
$919$	702286.			0.831540			0.415770	−	0.909470i	$-0.363512\pi$
$919$	702286.			0.831540			0.415770	+	0.909470i	$0.363512\pi$
$920$		0			0
$921$	269904.	+	301762.i	0.318193	+	0.355750i
$922$		0			0
$923$			24900.9i			0.0292288i
$924$		0			0
$925$	−34000.0			−0.0397370
$926$		0			0
$927$	−83142.0	+	743645.i	−0.0967522	+	0.865378i
$928$		0			0
$929$		−	513285.i		−	0.594740i	−0.954762	−	0.297370i	$-0.903891\pi$
$929$		−	513285.i		−	0.594740i	0.954762	−	0.297370i	$-0.0961095\pi$
$930$		0			0
$931$	−1.15005e6			−1.32684
$932$		0			0
$933$	392760.	−	351295.i	0.451195	−	0.403561i
$934$		0			0
$935$		−	235458.i		−	0.269333i
$936$		0			0
$937$	−540346.			−0.615450			−0.307725	−	0.951475i	$-0.599568\pi$
$937$	−540346.			−0.615450			−0.307725	+	0.951475i	$0.599568\pi$
$938$		0			0
$939$	477276.	+	533611.i	0.541300	+	0.605192i
$940$		0			0
$941$			1.19327e6i			1.34759i	0.738917	+	0.673797i	$0.235338\pi$
$941$			1.19327e6i			1.34759i	−0.738917	+	0.673797i	$0.764662\pi$
$942$		0			0
$943$	511560.			0.575272
$944$		0			0
$945$	349650.	+	491443.i	0.391534	+	0.550313i
$946$		0			0
$947$			847447.i			0.944959i	0.881342	+	0.472479i	$0.156641\pi$
$947$			847447.i			0.944959i	−0.881342	+	0.472479i	$0.843359\pi$
$948$		0			0
$949$	8608.00			0.00955806
$950$		0			0
$951$	694530.	−	621207.i	0.767945	−	0.686871i
$952$		0			0
$953$		−	920164.i		−	1.01316i	−0.862192	−	0.506582i	$-0.830909\pi$
$953$		−	920164.i		−	1.01316i	0.862192	−	0.506582i	$-0.169091\pi$
$954$		0			0
$955$	320100.			0.350977
$956$		0			0
$957$	1.00116e6	+	1.11933e6i	1.09315	+	1.22218i
$958$		0			0
$959$		−	1.39093e6i		−	1.51241i
$960$		0			0
$961$	1.10996e6			1.20187
$962$		0			0
$963$	−374760.	−	41899.4i	−0.404111	−	0.0451810i
$964$		0			0
$965$			133337.i			0.143184i
$966$		0			0
$967$	1.13064e6			1.20912			0.604562	−	0.796558i	$-0.293348\pi$
$967$	1.13064e6			1.20912			0.604562	+	0.796558i	$0.293348\pi$
$968$		0			0
$969$	−437580.	+	391383.i	−0.466026	+	0.416826i
$970$		0			0
$971$		−	497816.i		−	0.527996i	−0.964523	−	0.263998i	$-0.914959\pi$
$971$		−	497816.i		−	0.527996i	0.964523	−	0.263998i	$-0.0850412\pi$
$972$		0			0
$973$	2.81955e6			2.97820
$974$		0			0
$975$	−12000.0	−	13416.4i	−0.0126233	−	0.0141132i
$976$		0			0
$977$			1.57403e6i			1.64901i	0.565856	+	0.824504i	$0.308546\pi$
$977$			1.57403e6i			1.64901i	−0.565856	+	0.824504i	$0.691454\pi$
$978$		0			0
$979$	−635040.			−0.662576
$980$		0			0
$981$	49554.0	−	443224.i	0.0514921	−	0.460559i
$982$		0			0
$983$			1.43589e6i			1.48599i	0.669300	+	0.742993i	$0.266594\pi$
$983$			1.43589e6i			1.48599i	−0.669300	+	0.742993i	$0.733406\pi$
$984$		0			0
$985$	−100350.			−0.103430
$986$		0			0
$987$	−1.59174e6	+	1.42370e6i	−1.63395	+	1.46145i
$988$		0			0
$989$			1.27273e6i			1.30120i
$990$		0			0
$991$	−754754.			−0.768525			−0.384263	−	0.923224i	$-0.625544\pi$
$991$	−754754.			−0.768525			−0.384263	+	0.923224i	$0.625544\pi$
$992$		0			0
$993$	−361428.	−	404089.i	−0.366542	−	0.409806i
$994$		0			0
$995$		−	356720.i		−	0.360314i
$996$		0			0
$997$	−855208.			−0.860362			−0.430181	−	0.902743i	$-0.641550\pi$
$997$	−855208.			−0.860362			−0.430181	+	0.902743i	$0.641550\pi$
$998$		0			0
$999$	161568.	−	114952.i	0.161892	−	0.115182i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.5.l.a.161.1		2
3.2	odd	2	inner	240.5.l.a.161.2		2
4.3	odd	2		60.5.g.a.41.2	yes	2
12.11	even	2		60.5.g.a.41.1	✓	2
20.3	even	4		300.5.b.c.149.2		4
20.7	even	4		300.5.b.c.149.3		4
20.19	odd	2		300.5.g.d.101.1		2
60.23	odd	4		300.5.b.c.149.4		4
60.47	odd	4		300.5.b.c.149.1		4
60.59	even	2		300.5.g.d.101.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.5.g.a.41.1	✓	2	12.11	even	2
60.5.g.a.41.2	yes	2	4.3	odd	2
240.5.l.a.161.1		2	1.1	even	1	trivial
240.5.l.a.161.2		2	3.2	odd	2	inner
300.5.b.c.149.1		4	60.47	odd	4
300.5.b.c.149.2		4	20.3	even	4
300.5.b.c.149.3		4	20.7	even	4
300.5.b.c.149.4		4	60.23	odd	4
300.5.g.d.101.1		2	20.19	odd	2
300.5.g.d.101.2		2	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:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	240.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-6.00000 - 6.70820i) q^{3} -11.1803i q^{5} -74.0000 q^{7} +(-9.00000 + 80.4984i) q^{9} +O(q^{10})\)	\(q+(-6.00000 - 6.70820i) q^{3} -11.1803i q^{5} -74.0000 q^{7} +(-9.00000 + 80.4984i) q^{9} -120.748i q^{11} -16.0000 q^{13} +(-75.0000 + 67.0820i) q^{15} +174.413i q^{17} -374.000 q^{19} +(444.000 + 496.407i) q^{21} +657.404i q^{23} -125.000 q^{25} +(594.000 - 422.617i) q^{27} -1381.89i q^{29} +1426.00 q^{31} +(-810.000 + 724.486i) q^{33} +827.345i q^{35} +272.000 q^{37} +(96.0000 + 107.331i) q^{39} -778.152i q^{41} +1936.00 q^{43} +(900.000 + 100.623i) q^{45} +3206.52i q^{47} +3075.00 q^{49} +(1170.00 - 1046.48i) q^{51} +5487.31i q^{53} -1350.00 q^{55} +(2244.00 + 2508.87i) q^{57} +4816.49i q^{59} +2234.00 q^{61} +(666.000 - 5956.89i) q^{63} +178.885i q^{65} +5416.00 q^{67} +(4410.00 - 3944.42i) q^{69} -1556.30i q^{71} -538.000 q^{73} +(750.000 + 838.525i) q^{75} +8935.33i q^{77} -9962.00 q^{79} +(-6399.00 - 1448.97i) q^{81} -4655.49i q^{83} +1950.00 q^{85} +(-9270.00 + 8291.34i) q^{87} -5259.23i q^{89} +1184.00 q^{91} +(-8556.00 - 9565.90i) q^{93} +4181.45i q^{95} -10726.0 q^{97} +(9720.00 + 1086.73i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 12 q^{3} - 148 q^{7} - 18 q^{9}+O(q^{10}) \) 2 * q - 12 * q^3 - 148 * q^7 - 18 * q^9	\( 2 q - 12 q^{3} - 148 q^{7} - 18 q^{9} - 32 q^{13} - 150 q^{15} - 748 q^{19} + 888 q^{21} - 250 q^{25} + 1188 q^{27} + 2852 q^{31} - 1620 q^{33} + 544 q^{37} + 192 q^{39} + 3872 q^{43} + 1800 q^{45} + 6150 q^{49} + 2340 q^{51} - 2700 q^{55} + 4488 q^{57} + 4468 q^{61} + 1332 q^{63} + 10832 q^{67} + 8820 q^{69} - 1076 q^{73} + 1500 q^{75} - 19924 q^{79} - 12798 q^{81} + 3900 q^{85} - 18540 q^{87} + 2368 q^{91} - 17112 q^{93} - 21452 q^{97} + 19440 q^{99}+O(q^{100}) \) 2 * q - 12 * q^3 - 148 * q^7 - 18 * q^9 - 32 * q^13 - 150 * q^15 - 748 * q^19 + 888 * q^21 - 250 * q^25 + 1188 * q^27 + 2852 * q^31 - 1620 * q^33 + 544 * q^37 + 192 * q^39 + 3872 * q^43 + 1800 * q^45 + 6150 * q^49 + 2340 * q^51 - 2700 * q^55 + 4488 * q^57 + 4468 * q^61 + 1332 * q^63 + 10832 * q^67 + 8820 * q^69 - 1076 * q^73 + 1500 * q^75 - 19924 * q^79 - 12798 * q^81 + 3900 * q^85 - 18540 * q^87 + 2368 * q^91 - 17112 * q^93 - 21452 * q^97 + 19440 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more