LMFDB - Embedded newform 120.4.f.a.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [120,4,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, 4, names="a")

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

chi := DirichletCharacter("120.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 120 = 2^{3} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
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:	$7.08022920069$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	120.49
Dual form			120.4.f.a.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +(-10.0000 - 5.00000i) q^{5} +10.0000i q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} +(-10.0000 - 5.00000i) q^{5} +10.0000i q^{7} -9.00000 q^{9} -46.0000 q^{11} +34.0000i q^{13} +(-15.0000 + 30.0000i) q^{15} +66.0000i q^{17} -104.000 q^{19} +30.0000 q^{21} -164.000i q^{23} +(75.0000 + 100.000i) q^{25} +27.0000i q^{27} -224.000 q^{29} -72.0000 q^{31} +138.000i q^{33} +(50.0000 - 100.000i) q^{35} -22.0000i q^{37} +102.000 q^{39} +194.000 q^{41} -108.000i q^{43} +(90.0000 + 45.0000i) q^{45} -480.000i q^{47} +243.000 q^{49} +198.000 q^{51} -286.000i q^{53} +(460.000 + 230.000i) q^{55} +312.000i q^{57} -426.000 q^{59} +698.000 q^{61} -90.0000i q^{63} +(170.000 - 340.000i) q^{65} +328.000i q^{67} -492.000 q^{69} +188.000 q^{71} +740.000i q^{73} +(300.000 - 225.000i) q^{75} -460.000i q^{77} -1168.00 q^{79} +81.0000 q^{81} -412.000i q^{83} +(330.000 - 660.000i) q^{85} +672.000i q^{87} -1206.00 q^{89} -340.000 q^{91} +216.000i q^{93} +(1040.00 + 520.000i) q^{95} -1384.00i q^{97} +414.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 20 q^{5} - 18 q^{9}+O(q^{10}) $ 2 * q - 20 * q^5 - 18 * q^9	$ 2 q - 20 q^{5} - 18 q^{9} - 92 q^{11} - 30 q^{15} - 208 q^{19} + 60 q^{21} + 150 q^{25} - 448 q^{29} - 144 q^{31} + 100 q^{35} + 204 q^{39} + 388 q^{41} + 180 q^{45} + 486 q^{49} + 396 q^{51} + 920 q^{55} - 852 q^{59} + 1396 q^{61} + 340 q^{65} - 984 q^{69} + 376 q^{71} + 600 q^{75} - 2336 q^{79} + 162 q^{81} + 660 q^{85} - 2412 q^{89} - 680 q^{91} + 2080 q^{95} + 828 q^{99}+O(q^{100}) $ 2 * q - 20 * q^5 - 18 * q^9 - 92 * q^11 - 30 * q^15 - 208 * q^19 + 60 * q^21 + 150 * q^25 - 448 * q^29 - 144 * q^31 + 100 * q^35 + 204 * q^39 + 388 * q^41 + 180 * q^45 + 486 * q^49 + 396 * q^51 + 920 * q^55 - 852 * q^59 + 1396 * q^61 + 340 * q^65 - 984 * q^69 + 376 * q^71 + 600 * q^75 - 2336 * q^79 + 162 * q^81 + 660 * q^85 - 2412 * q^89 - 680 * q^91 + 2080 * q^95 + 828 * 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
$3$		−	3.00000i		−	0.577350i
$3$		−	3.00000i		−	0.577350i
$4$		0			0
$5$	−10.0000	−	5.00000i	−0.894427	−	0.447214i
$5$	−10.0000	−	5.00000i	−0.894427	−	0.447214i
$6$		0			0
$7$			10.0000i			0.539949i	0.962867	+	0.269975i	$0.0870153\pi$
$7$			10.0000i			0.539949i	−0.962867	+	0.269975i	$0.912985\pi$
$8$		0			0
$9$	−9.00000			−0.333333
$10$		0			0
$11$	−46.0000			−1.26087			−0.630433	−	0.776244i	$-0.717123\pi$
$11$	−46.0000			−1.26087			−0.630433	+	0.776244i	$0.717123\pi$
$12$		0			0
$13$			34.0000i			0.725377i	0.931910	+	0.362689i	$0.118141\pi$
$13$			34.0000i			0.725377i	−0.931910	+	0.362689i	$0.881859\pi$
$14$		0			0
$15$	−15.0000	+	30.0000i	−0.258199	+	0.516398i
$16$		0			0
$17$			66.0000i			0.941609i	0.882238	+	0.470804i	$0.156036\pi$
$17$			66.0000i			0.941609i	−0.882238	+	0.470804i	$0.843964\pi$
$18$		0			0
$19$	−104.000			−1.25575			−0.627875	−	0.778314i	$-0.716075\pi$
$19$	−104.000			−1.25575			−0.627875	+	0.778314i	$0.716075\pi$
$20$		0			0
$21$	30.0000			0.311740
$22$		0			0
$23$		−	164.000i		−	1.48680i	−0.668848	−	0.743399i	$-0.733212\pi$
$23$		−	164.000i		−	1.48680i	0.668848	−	0.743399i	$-0.266788\pi$
$24$		0			0
$25$	75.0000	+	100.000i	0.600000	+	0.800000i
$26$		0			0
$27$			27.0000i			0.192450i
$28$		0			0
$29$	−224.000			−1.43434			−0.717168	−	0.696900i	$-0.754562\pi$
$29$	−224.000			−1.43434			−0.717168	+	0.696900i	$0.754562\pi$
$30$		0			0
$31$	−72.0000			−0.417148			−0.208574	−	0.978007i	$-0.566882\pi$
$31$	−72.0000			−0.417148			−0.208574	+	0.978007i	$0.566882\pi$
$32$		0			0
$33$			138.000i			0.727961i
$34$		0			0
$35$	50.0000	−	100.000i	0.241473	−	0.482945i
$36$		0			0
$37$		−	22.0000i		−	0.0977507i	−0.998805	−	0.0488754i	$-0.984436\pi$
$37$		−	22.0000i		−	0.0977507i	0.998805	−	0.0488754i	$-0.0155637\pi$
$38$		0			0
$39$	102.000			0.418797
$40$		0			0
$41$	194.000			0.738969			0.369484	−	0.929237i	$-0.379534\pi$
$41$	194.000			0.738969			0.369484	+	0.929237i	$0.379534\pi$
$42$		0			0
$43$		−	108.000i		−	0.383020i	−0.981491	−	0.191510i	$-0.938662\pi$
$43$		−	108.000i		−	0.383020i	0.981491	−	0.191510i	$-0.0613384\pi$
$44$		0			0
$45$	90.0000	+	45.0000i	0.298142	+	0.149071i
$46$		0			0
$47$		−	480.000i		−	1.48969i	−0.667240	−	0.744843i	$-0.732525\pi$
$47$		−	480.000i		−	1.48969i	0.667240	−	0.744843i	$-0.267475\pi$
$48$		0			0
$49$	243.000			0.708455
$50$		0			0
$51$	198.000			0.543638
$52$		0			0
$53$		−	286.000i		−	0.741229i	−0.928787	−	0.370614i	$-0.879147\pi$
$53$		−	286.000i		−	0.741229i	0.928787	−	0.370614i	$-0.120853\pi$
$54$		0			0
$55$	460.000	+	230.000i	1.12775	+	0.563876i
$56$		0			0
$57$			312.000i			0.725007i
$58$		0			0
$59$	−426.000			−0.940008			−0.470004	−	0.882664i	$-0.655748\pi$
$59$	−426.000			−0.940008			−0.470004	+	0.882664i	$0.655748\pi$
$60$		0			0
$61$	698.000			1.46508			0.732539	−	0.680725i	$-0.238335\pi$
$61$	698.000			1.46508			0.732539	+	0.680725i	$0.238335\pi$
$62$		0			0
$63$		−	90.0000i		−	0.179983i
$64$		0			0
$65$	170.000	−	340.000i	0.324399	−	0.648797i
$66$		0			0
$67$			328.000i			0.598083i	0.954240	+	0.299042i	$0.0966669\pi$
$67$			328.000i			0.598083i	−0.954240	+	0.299042i	$0.903333\pi$
$68$		0			0
$69$	−492.000			−0.858403
$70$		0			0
$71$	188.000			0.314246			0.157123	−	0.987579i	$-0.449778\pi$
$71$	188.000			0.314246			0.157123	+	0.987579i	$0.449778\pi$
$72$		0			0
$73$			740.000i			1.18644i	0.805039	+	0.593222i	$0.202144\pi$
$73$			740.000i			1.18644i	−0.805039	+	0.593222i	$0.797856\pi$
$74$		0			0
$75$	300.000	−	225.000i	0.461880	−	0.346410i
$76$		0			0
$77$		−	460.000i		−	0.680803i
$78$		0			0
$79$	−1168.00			−1.66342			−0.831711	−	0.555209i	$-0.812638\pi$
$79$	−1168.00			−1.66342			−0.831711	+	0.555209i	$0.812638\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$		−	412.000i		−	0.544854i	−0.962176	−	0.272427i	$-0.912174\pi$
$83$		−	412.000i		−	0.544854i	0.962176	−	0.272427i	$-0.0878263\pi$
$84$		0			0
$85$	330.000	−	660.000i	0.421100	−	0.842201i
$86$		0			0
$87$			672.000i			0.828115i
$88$		0			0
$89$	−1206.00			−1.43636			−0.718178	−	0.695859i	$-0.755024\pi$
$89$	−1206.00			−1.43636			−0.718178	+	0.695859i	$0.755024\pi$
$90$		0			0
$91$	−340.000			−0.391667
$92$		0			0
$93$			216.000i			0.240840i
$94$		0			0
$95$	1040.00	+	520.000i	1.12318	+	0.561588i
$96$		0			0
$97$		−	1384.00i		−	1.44870i	−0.689432	−	0.724350i	$-0.742140\pi$
$97$		−	1384.00i		−	1.44870i	0.689432	−	0.724350i	$-0.257860\pi$
$98$		0			0
$99$	414.000			0.420289
$100$		0			0
$101$	−1128.00			−1.11129			−0.555645	−	0.831420i	$-0.687528\pi$
$101$	−1128.00			−1.11129			−0.555645	+	0.831420i	$0.687528\pi$
$102$		0			0
$103$			758.000i			0.725126i	0.931959	+	0.362563i	$0.118098\pi$
$103$			758.000i			0.725126i	−0.931959	+	0.362563i	$0.881902\pi$
$104$		0			0
$105$	−300.000	−	150.000i	−0.278829	−	0.139414i
$106$		0			0
$107$			1324.00i			1.19622i	0.801413	+	0.598112i	$0.204082\pi$
$107$			1324.00i			1.19622i	−0.801413	+	0.598112i	$0.795918\pi$
$108$		0			0
$109$	−1602.00			−1.40774			−0.703871	−	0.710328i	$-0.748546\pi$
$109$	−1602.00			−1.40774			−0.703871	+	0.710328i	$0.748546\pi$
$110$		0			0
$111$	−66.0000			−0.0564364
$112$		0			0
$113$			2074.00i			1.72660i	0.504693	+	0.863299i	$0.331606\pi$
$113$			2074.00i			1.72660i	−0.504693	+	0.863299i	$0.668394\pi$
$114$		0			0
$115$	−820.000	+	1640.00i	−0.664916	+	1.32983i
$116$		0			0
$117$		−	306.000i		−	0.241792i
$118$		0			0
$119$	−660.000			−0.508421
$120$		0			0
$121$	785.000			0.589782
$122$		0			0
$123$		−	582.000i		−	0.426644i
$124$		0			0
$125$	−250.000	−	1375.00i	−0.178885	−	0.983870i
$126$		0			0
$127$		−	534.000i		−	0.373109i	−0.982445	−	0.186554i	$-0.940268\pi$
$127$		−	534.000i		−	0.373109i	0.982445	−	0.186554i	$-0.0597321\pi$
$128$		0			0
$129$	−324.000			−0.221137
$130$		0			0
$131$	1806.00			1.20451			0.602256	−	0.798303i	$-0.294269\pi$
$131$	1806.00			1.20451			0.602256	+	0.798303i	$0.294269\pi$
$132$		0			0
$133$		−	1040.00i		−	0.678041i
$134$		0			0
$135$	135.000	−	270.000i	0.0860663	−	0.172133i
$136$		0			0
$137$			1822.00i			1.13623i	0.822948	+	0.568117i	$0.192328\pi$
$137$			1822.00i			1.13623i	−0.822948	+	0.568117i	$0.807672\pi$
$138$		0			0
$139$	−532.000			−0.324631			−0.162315	−	0.986739i	$-0.551896\pi$
$139$	−532.000			−0.324631			−0.162315	+	0.986739i	$0.551896\pi$
$140$		0			0
$141$	−1440.00			−0.860070
$142$		0			0
$143$		−	1564.00i		−	0.914603i
$144$		0			0
$145$	2240.00	+	1120.00i	1.28291	+	0.641455i
$146$		0			0
$147$		−	729.000i		−	0.409027i
$148$		0			0
$149$	1284.00			0.705969			0.352984	−	0.935629i	$-0.385167\pi$
$149$	1284.00			0.705969			0.352984	+	0.935629i	$0.385167\pi$
$150$		0			0
$151$	184.000			0.0991636			0.0495818	−	0.998770i	$-0.484211\pi$
$151$	184.000			0.0991636			0.0495818	+	0.998770i	$0.484211\pi$
$152$		0			0
$153$		−	594.000i		−	0.313870i
$154$		0			0
$155$	720.000	+	360.000i	0.373108	+	0.186554i
$156$		0			0
$157$		−	3746.00i		−	1.90423i	−0.305748	−	0.952113i	$-0.598906\pi$
$157$		−	3746.00i		−	1.90423i	0.305748	−	0.952113i	$-0.401094\pi$
$158$		0			0
$159$	−858.000			−0.427949
$160$		0			0
$161$	1640.00			0.802796
$162$		0			0
$163$		−	1504.00i		−	0.722714i	−0.932428	−	0.361357i	$-0.882314\pi$
$163$		−	1504.00i		−	0.722714i	0.932428	−	0.361357i	$-0.117686\pi$
$164$		0			0
$165$	690.000	−	1380.00i	0.325554	−	0.651108i
$166$		0			0
$167$			3012.00i			1.39566i	0.716262	+	0.697831i	$0.245851\pi$
$167$			3012.00i			1.39566i	−0.716262	+	0.697831i	$0.754149\pi$
$168$		0			0
$169$	1041.00			0.473828
$170$		0			0
$171$	936.000			0.418583
$172$		0			0
$173$			438.000i			0.192489i	0.995358	+	0.0962443i	$0.0306830\pi$
$173$			438.000i			0.192489i	−0.995358	+	0.0962443i	$0.969317\pi$
$174$		0			0
$175$	−1000.00	+	750.000i	−0.431959	+	0.323970i
$176$		0			0
$177$			1278.00i			0.542714i
$178$		0			0
$179$	1462.00			0.610475			0.305237	−	0.952276i	$-0.401264\pi$
$179$	1462.00			0.610475			0.305237	+	0.952276i	$0.401264\pi$
$180$		0			0
$181$	586.000			0.240647			0.120323	−	0.992735i	$-0.461607\pi$
$181$	586.000			0.240647			0.120323	+	0.992735i	$0.461607\pi$
$182$		0			0
$183$		−	2094.00i		−	0.845863i
$184$		0			0
$185$	−110.000	+	220.000i	−0.0437155	+	0.0874309i
$186$		0			0
$187$		−	3036.00i		−	1.18724i
$188$		0			0
$189$	−270.000			−0.103913
$190$		0			0
$191$	60.0000			0.0227301			0.0113650	−	0.999935i	$-0.496382\pi$
$191$	60.0000			0.0227301			0.0113650	+	0.999935i	$0.496382\pi$
$192$		0			0
$193$			4676.00i			1.74397i	0.489534	+	0.871984i	$0.337167\pi$
$193$			4676.00i			1.74397i	−0.489534	+	0.871984i	$0.662833\pi$
$194$		0			0
$195$	−1020.00	−	510.000i	−0.374583	−	0.187292i
$196$		0			0
$197$			2286.00i			0.826755i	0.910560	+	0.413378i	$0.135651\pi$
$197$			2286.00i			0.826755i	−0.910560	+	0.413378i	$0.864349\pi$
$198$		0			0
$199$	3536.00			1.25960			0.629800	−	0.776757i	$-0.283137\pi$
$199$	3536.00			1.25960			0.629800	+	0.776757i	$0.283137\pi$
$200$		0			0
$201$	984.000			0.345304
$202$		0			0
$203$		−	2240.00i		−	0.774469i
$204$		0			0
$205$	−1940.00	−	970.000i	−0.660954	−	0.330477i
$206$		0			0
$207$			1476.00i			0.495599i
$208$		0			0
$209$	4784.00			1.58333
$210$		0			0
$211$	−3500.00			−1.14194			−0.570971	−	0.820970i	$-0.693433\pi$
$211$	−3500.00			−1.14194			−0.570971	+	0.820970i	$0.693433\pi$
$212$		0			0
$213$		−	564.000i		−	0.181430i
$214$		0			0
$215$	−540.000	+	1080.00i	−0.171292	+	0.342583i
$216$		0			0
$217$		−	720.000i		−	0.225239i
$218$		0			0
$219$	2220.00			0.684994
$220$		0			0
$221$	−2244.00			−0.683022
$222$		0			0
$223$		−	5874.00i		−	1.76391i	−0.471333	−	0.881955i	$-0.656227\pi$
$223$		−	5874.00i		−	1.76391i	0.471333	−	0.881955i	$-0.343773\pi$
$224$		0			0
$225$	−675.000	−	900.000i	−0.200000	−	0.266667i
$226$		0			0
$227$		−	124.000i		−	0.0362563i	−0.999836	−	0.0181281i	$-0.994229\pi$
$227$		−	124.000i		−	0.0362563i	0.999836	−	0.0181281i	$-0.00577068\pi$
$228$		0			0
$229$	1362.00			0.393028			0.196514	−	0.980501i	$-0.437038\pi$
$229$	1362.00			0.393028			0.196514	+	0.980501i	$0.437038\pi$
$230$		0			0
$231$	−1380.00			−0.393062
$232$		0			0
$233$			3870.00i			1.08812i	0.839046	+	0.544060i	$0.183114\pi$
$233$			3870.00i			1.08812i	−0.839046	+	0.544060i	$0.816886\pi$
$234$		0			0
$235$	−2400.00	+	4800.00i	−0.666207	+	1.33241i
$236$		0			0
$237$			3504.00i			0.960377i
$238$		0			0
$239$	−6116.00			−1.65528			−0.827638	−	0.561262i	$-0.810316\pi$
$239$	−6116.00			−1.65528			−0.827638	+	0.561262i	$0.810316\pi$
$240$		0			0
$241$	−5962.00			−1.59355			−0.796776	−	0.604274i	$-0.793463\pi$
$241$	−5962.00			−1.59355			−0.796776	+	0.604274i	$0.793463\pi$
$242$		0			0
$243$		−	243.000i		−	0.0641500i
$244$		0			0
$245$	−2430.00	−	1215.00i	−0.633661	−	0.316831i
$246$		0			0
$247$		−	3536.00i		−	0.910892i
$248$		0			0
$249$	−1236.00			−0.314572
$250$		0			0
$251$	−1490.00			−0.374693			−0.187347	−	0.982294i	$-0.559989\pi$
$251$	−1490.00			−0.374693			−0.187347	+	0.982294i	$0.559989\pi$
$252$		0			0
$253$			7544.00i			1.87465i
$254$		0			0
$255$	−1980.00	−	990.000i	−0.486245	−	0.243122i
$256$		0			0
$257$		−	5394.00i		−	1.30922i	−0.755969	−	0.654608i	$-0.772834\pi$
$257$		−	5394.00i		−	1.30922i	0.755969	−	0.654608i	$-0.227166\pi$
$258$		0			0
$259$	220.000			0.0527804
$260$		0			0
$261$	2016.00			0.478112
$262$		0			0
$263$			636.000i			0.149116i	0.997217	+	0.0745579i	$0.0237546\pi$
$263$			636.000i			0.149116i	−0.997217	+	0.0745579i	$0.976245\pi$
$264$		0			0
$265$	−1430.00	+	2860.00i	−0.331488	+	0.662975i
$266$		0			0
$267$			3618.00i			0.829281i
$268$		0			0
$269$	3360.00			0.761572			0.380786	−	0.924663i	$-0.375653\pi$
$269$	3360.00			0.761572			0.380786	+	0.924663i	$0.375653\pi$
$270$		0			0
$271$	5768.00			1.29292			0.646459	−	0.762948i	$-0.276249\pi$
$271$	5768.00			1.29292			0.646459	+	0.762948i	$0.276249\pi$
$272$		0			0
$273$			1020.00i			0.226129i
$274$		0			0
$275$	−3450.00	−	4600.00i	−0.756519	−	1.00869i
$276$		0			0
$277$		−	1398.00i		−	0.303241i	−0.988439	−	0.151620i	$-0.951551\pi$
$277$		−	1398.00i		−	0.303241i	0.988439	−	0.151620i	$-0.0484491\pi$
$278$		0			0
$279$	648.000			0.139049
$280$		0			0
$281$	−4194.00			−0.890367			−0.445183	−	0.895439i	$-0.646862\pi$
$281$	−4194.00			−0.890367			−0.445183	+	0.895439i	$0.646862\pi$
$282$		0			0
$283$			8256.00i			1.73416i	0.498166	+	0.867082i	$0.334007\pi$
$283$			8256.00i			1.73416i	−0.498166	+	0.867082i	$0.665993\pi$
$284$		0			0
$285$	1560.00	−	3120.00i	0.324233	−	0.648466i
$286$		0			0
$287$			1940.00i			0.399006i
$288$		0			0
$289$	557.000			0.113373
$290$		0			0
$291$	−4152.00			−0.836407
$292$		0			0
$293$		−	5534.00i		−	1.10341i	−0.834039	−	0.551706i	$-0.813977\pi$
$293$		−	5534.00i		−	1.10341i	0.834039	−	0.551706i	$-0.186023\pi$
$294$		0			0
$295$	4260.00	+	2130.00i	0.840769	+	0.420384i
$296$		0			0
$297$		−	1242.00i		−	0.242654i
$298$		0			0
$299$	5576.00			1.07849
$300$		0			0
$301$	1080.00			0.206811
$302$		0			0
$303$			3384.00i			0.641603i
$304$		0			0
$305$	−6980.00	−	3490.00i	−1.31041	−	0.655203i
$306$		0			0
$307$		−	484.000i		−	0.0899783i	−0.998987	−	0.0449892i	$-0.985675\pi$
$307$		−	484.000i		−	0.0899783i	0.998987	−	0.0449892i	$-0.0143253\pi$
$308$		0			0
$309$	2274.00			0.418652
$310$		0			0
$311$	−2724.00			−0.496668			−0.248334	−	0.968674i	$-0.579883\pi$
$311$	−2724.00			−0.496668			−0.248334	+	0.968674i	$0.579883\pi$
$312$		0			0
$313$		−	5308.00i		−	0.958549i	−0.877665	−	0.479275i	$-0.840900\pi$
$313$		−	5308.00i		−	0.958549i	0.877665	−	0.479275i	$-0.159100\pi$
$314$		0			0
$315$	−450.000	+	900.000i	−0.0804909	+	0.160982i
$316$		0			0
$317$			4218.00i			0.747339i	0.927562	+	0.373670i	$0.121901\pi$
$317$			4218.00i			0.747339i	−0.927562	+	0.373670i	$0.878099\pi$
$318$		0			0
$319$	10304.0			1.80851
$320$		0			0
$321$	3972.00			0.690640
$322$		0			0
$323$		−	6864.00i		−	1.18242i
$324$		0			0
$325$	−3400.00	+	2550.00i	−0.580302	+	0.435226i
$326$		0			0
$327$			4806.00i			0.812760i
$328$		0			0
$329$	4800.00			0.804354
$330$		0			0
$331$	−4640.00			−0.770506			−0.385253	−	0.922811i	$-0.625886\pi$
$331$	−4640.00			−0.770506			−0.385253	+	0.922811i	$0.625886\pi$
$332$		0			0
$333$			198.000i			0.0325836i
$334$		0			0
$335$	1640.00	−	3280.00i	0.267471	−	0.534942i
$336$		0			0
$337$		−	8156.00i		−	1.31835i	−0.751987	−	0.659177i	$-0.770905\pi$
$337$		−	8156.00i		−	1.31835i	0.751987	−	0.659177i	$-0.229095\pi$
$338$		0			0
$339$	6222.00			0.996851
$340$		0			0
$341$	3312.00			0.525967
$342$		0			0
$343$			5860.00i			0.922479i
$344$		0			0
$345$	4920.00	+	2460.00i	0.767779	+	0.383890i
$346$		0			0
$347$			4124.00i			0.638006i	0.947754	+	0.319003i	$0.103348\pi$
$347$			4124.00i			0.638006i	−0.947754	+	0.319003i	$0.896652\pi$
$348$		0			0
$349$	−3650.00			−0.559828			−0.279914	−	0.960025i	$-0.590306\pi$
$349$	−3650.00			−0.559828			−0.279914	+	0.960025i	$0.590306\pi$
$350$		0			0
$351$	−918.000			−0.139599
$352$		0			0
$353$		−	6834.00i		−	1.03042i	−0.857065	−	0.515208i	$-0.827715\pi$
$353$		−	6834.00i		−	1.03042i	0.857065	−	0.515208i	$-0.172285\pi$
$354$		0			0
$355$	−1880.00	−	940.000i	−0.281071	−	0.140535i
$356$		0			0
$357$			1980.00i			0.293537i
$358$		0			0
$359$	−3904.00			−0.573942			−0.286971	−	0.957939i	$-0.592648\pi$
$359$	−3904.00			−0.573942			−0.286971	+	0.957939i	$0.592648\pi$
$360$		0			0
$361$	3957.00			0.576906
$362$		0			0
$363$		−	2355.00i		−	0.340511i
$364$		0			0
$365$	3700.00	−	7400.00i	0.530594	−	1.06119i
$366$		0			0
$367$		−	13174.0i		−	1.87378i	−0.349624	−	0.936890i	$-0.613691\pi$
$367$		−	13174.0i		−	1.87378i	0.349624	−	0.936890i	$-0.386309\pi$
$368$		0			0
$369$	−1746.00			−0.246323
$370$		0			0
$371$	2860.00			0.400226
$372$		0			0
$373$			1090.00i			0.151308i	0.997134	+	0.0756542i	$0.0241045\pi$
$373$			1090.00i			0.151308i	−0.997134	+	0.0756542i	$0.975895\pi$
$374$		0			0
$375$	−4125.00	+	750.000i	−0.568038	+	0.103280i
$376$		0			0
$377$		−	7616.00i		−	1.04043i
$378$		0			0
$379$	9220.00			1.24960			0.624802	−	0.780784i	$-0.285180\pi$
$379$	9220.00			1.24960			0.624802	+	0.780784i	$0.285180\pi$
$380$		0			0
$381$	−1602.00			−0.215415
$382$		0			0
$383$			3960.00i			0.528320i	0.964479	+	0.264160i	$0.0850947\pi$
$383$			3960.00i			0.528320i	−0.964479	+	0.264160i	$0.914905\pi$
$384$		0			0
$385$	−2300.00	+	4600.00i	−0.304465	+	0.608929i
$386$		0			0
$387$			972.000i			0.127673i
$388$		0			0
$389$	1788.00			0.233047			0.116523	−	0.993188i	$-0.462825\pi$
$389$	1788.00			0.233047			0.116523	+	0.993188i	$0.462825\pi$
$390$		0			0
$391$	10824.0			1.39998
$392$		0			0
$393$		−	5418.00i		−	0.695425i
$394$		0			0
$395$	11680.0	+	5840.00i	1.48781	+	0.743905i
$396$		0			0
$397$			9642.00i			1.21894i	0.792810	+	0.609469i	$0.208617\pi$
$397$			9642.00i			1.21894i	−0.792810	+	0.609469i	$0.791383\pi$
$398$		0			0
$399$	−3120.00			−0.391467
$400$		0			0
$401$	−410.000			−0.0510584			−0.0255292	−	0.999674i	$-0.508127\pi$
$401$	−410.000			−0.0510584			−0.0255292	+	0.999674i	$0.508127\pi$
$402$		0			0
$403$		−	2448.00i		−	0.302589i
$404$		0			0
$405$	−810.000	−	405.000i	−0.0993808	−	0.0496904i
$406$		0			0
$407$			1012.00i			0.123251i
$408$		0			0
$409$	−13766.0			−1.66427			−0.832133	−	0.554576i	$-0.812880\pi$
$409$	−13766.0			−1.66427			−0.832133	+	0.554576i	$0.812880\pi$
$410$		0			0
$411$	5466.00			0.656005
$412$		0			0
$413$		−	4260.00i		−	0.507557i
$414$		0			0
$415$	−2060.00	+	4120.00i	−0.243666	+	0.487332i
$416$		0			0
$417$			1596.00i			0.187426i
$418$		0			0
$419$	−16998.0			−1.98188			−0.990939	−	0.134315i	$-0.957117\pi$
$419$	−16998.0			−1.98188			−0.990939	+	0.134315i	$0.957117\pi$
$420$		0			0
$421$	−2450.00			−0.283624			−0.141812	−	0.989894i	$-0.545293\pi$
$421$	−2450.00			−0.283624			−0.141812	+	0.989894i	$0.545293\pi$
$422$		0			0
$423$			4320.00i			0.496562i
$424$		0			0
$425$	−6600.00	+	4950.00i	−0.753287	+	0.564965i
$426$		0			0
$427$			6980.00i			0.791068i
$428$		0			0
$429$	−4692.00			−0.528046
$430$		0			0
$431$	−9248.00			−1.03355			−0.516776	−	0.856121i	$-0.672868\pi$
$431$	−9248.00			−1.03355			−0.516776	+	0.856121i	$0.672868\pi$
$432$		0			0
$433$			5028.00i			0.558038i	0.960286	+	0.279019i	$0.0900092\pi$
$433$			5028.00i			0.558038i	−0.960286	+	0.279019i	$0.909991\pi$
$434$		0			0
$435$	3360.00	−	6720.00i	0.370344	−	0.740688i
$436$		0			0
$437$			17056.0i			1.86705i
$438$		0			0
$439$	−3120.00			−0.339202			−0.169601	−	0.985513i	$-0.554248\pi$
$439$	−3120.00			−0.339202			−0.169601	+	0.985513i	$0.554248\pi$
$440$		0			0
$441$	−2187.00			−0.236152
$442$		0			0
$443$			8220.00i			0.881589i	0.897608	+	0.440795i	$0.145303\pi$
$443$			8220.00i			0.881589i	−0.897608	+	0.440795i	$0.854697\pi$
$444$		0			0
$445$	12060.0	+	6030.00i	1.28472	+	0.642358i
$446$		0			0
$447$		−	3852.00i		−	0.407591i
$448$		0			0
$449$	5826.00			0.612352			0.306176	−	0.951975i	$-0.400950\pi$
$449$	5826.00			0.612352			0.306176	+	0.951975i	$0.400950\pi$
$450$		0			0
$451$	−8924.00			−0.931740
$452$		0			0
$453$		−	552.000i		−	0.0572521i
$454$		0			0
$455$	3400.00	+	1700.00i	0.350317	+	0.175159i
$456$		0			0
$457$		−	16896.0i		−	1.72946i	−0.502240	−	0.864728i	$-0.667491\pi$
$457$		−	16896.0i		−	1.72946i	0.502240	−	0.864728i	$-0.332509\pi$
$458$		0			0
$459$	−1782.00			−0.181213
$460$		0			0
$461$	−996.000			−0.100625			−0.0503127	−	0.998734i	$-0.516022\pi$
$461$	−996.000			−0.100625			−0.0503127	+	0.998734i	$0.516022\pi$
$462$		0			0
$463$		−	10046.0i		−	1.00837i	−0.863594	−	0.504187i	$-0.831792\pi$
$463$		−	10046.0i		−	1.00837i	0.863594	−	0.504187i	$-0.168208\pi$
$464$		0			0
$465$	1080.00	−	2160.00i	0.107707	−	0.215414i
$466$		0			0
$467$		−	8388.00i		−	0.831157i	−0.909557	−	0.415579i	$-0.863579\pi$
$467$		−	8388.00i		−	0.831157i	0.909557	−	0.415579i	$-0.136421\pi$
$468$		0			0
$469$	−3280.00			−0.322935
$470$		0			0
$471$	−11238.0			−1.09940
$472$		0			0
$473$			4968.00i			0.482936i
$474$		0			0
$475$	−7800.00	−	10400.0i	−0.753450	−	1.00460i
$476$		0			0
$477$			2574.00i			0.247076i
$478$		0			0
$479$	−11396.0			−1.08705			−0.543525	−	0.839393i	$-0.682911\pi$
$479$	−11396.0			−1.08705			−0.543525	+	0.839393i	$0.682911\pi$
$480$		0			0
$481$	748.000			0.0709062
$482$		0			0
$483$		−	4920.00i		−	0.463494i
$484$		0			0
$485$	−6920.00	+	13840.0i	−0.647878	+	1.29576i
$486$		0			0
$487$			6454.00i			0.600531i	0.953856	+	0.300266i	$0.0970753\pi$
$487$			6454.00i			0.600531i	−0.953856	+	0.300266i	$0.902925\pi$
$488$		0			0
$489$	−4512.00			−0.417259
$490$		0			0
$491$	18638.0			1.71308			0.856539	−	0.516083i	$-0.172610\pi$
$491$	18638.0			1.71308			0.856539	+	0.516083i	$0.172610\pi$
$492$		0			0
$493$		−	14784.0i		−	1.35058i
$494$		0			0
$495$	−4140.00	−	2070.00i	−0.375917	−	0.187959i
$496$		0			0
$497$			1880.00i			0.169677i
$498$		0			0
$499$	−17768.0			−1.59400			−0.796999	−	0.603981i	$-0.793580\pi$
$499$	−17768.0			−1.59400			−0.796999	+	0.603981i	$0.793580\pi$
$500$		0			0
$501$	9036.00			0.805786
$502$		0			0
$503$			7952.00i			0.704895i	0.935832	+	0.352447i	$0.114650\pi$
$503$			7952.00i			0.704895i	−0.935832	+	0.352447i	$0.885350\pi$
$504$		0			0
$505$	11280.0	+	5640.00i	0.993967	+	0.496984i
$506$		0			0
$507$		−	3123.00i		−	0.273565i
$508$		0			0
$509$	12896.0			1.12300			0.561498	−	0.827478i	$-0.310225\pi$
$509$	12896.0			1.12300			0.561498	+	0.827478i	$0.310225\pi$
$510$		0			0
$511$	−7400.00			−0.640620
$512$		0			0
$513$		−	2808.00i		−	0.241669i
$514$		0			0
$515$	3790.00	−	7580.00i	0.324286	−	0.648572i
$516$		0			0
$517$			22080.0i			1.87829i
$518$		0			0
$519$	1314.00			0.111133
$520$		0			0
$521$	−2714.00			−0.228220			−0.114110	−	0.993468i	$-0.536402\pi$
$521$	−2714.00			−0.228220			−0.114110	+	0.993468i	$0.536402\pi$
$522$		0			0
$523$			13792.0i			1.15312i	0.817055	+	0.576560i	$0.195605\pi$
$523$			13792.0i			1.15312i	−0.817055	+	0.576560i	$0.804395\pi$
$524$		0			0
$525$	2250.00	+	3000.00i	0.187044	+	0.249392i
$526$		0			0
$527$		−	4752.00i		−	0.392790i
$528$		0			0
$529$	−14729.0			−1.21057
$530$		0			0
$531$	3834.00			0.313336
$532$		0			0
$533$			6596.00i			0.536031i
$534$		0			0
$535$	6620.00	−	13240.0i	0.534967	−	1.06993i
$536$		0			0
$537$		−	4386.00i		−	0.352458i
$538$		0			0
$539$	−11178.0			−0.893266
$540$		0			0
$541$	6802.00			0.540556			0.270278	−	0.962782i	$-0.412884\pi$
$541$	6802.00			0.540556			0.270278	+	0.962782i	$0.412884\pi$
$542$		0			0
$543$		−	1758.00i		−	0.138937i
$544$		0			0
$545$	16020.0	+	8010.00i	1.25912	+	0.629561i
$546$		0			0
$547$		−	18188.0i		−	1.42169i	−0.703350	−	0.710843i	$-0.748313\pi$
$547$		−	18188.0i		−	1.42169i	0.703350	−	0.710843i	$-0.251687\pi$
$548$		0			0
$549$	−6282.00			−0.488359
$550$		0			0
$551$	23296.0			1.80117
$552$		0			0
$553$		−	11680.0i		−	0.898163i
$554$		0			0
$555$	660.000	+	330.000i	0.0504783	+	0.0252391i
$556$		0			0
$557$			21462.0i			1.63263i	0.577608	+	0.816314i	$0.303986\pi$
$557$			21462.0i			1.63263i	−0.577608	+	0.816314i	$0.696014\pi$
$558$		0			0
$559$	3672.00			0.277834
$560$		0			0
$561$	−9108.00			−0.685455
$562$		0			0
$563$			17244.0i			1.29085i	0.763824	+	0.645424i	$0.223319\pi$
$563$			17244.0i			1.29085i	−0.763824	+	0.645424i	$0.776681\pi$
$564$		0			0
$565$	10370.0	−	20740.0i	0.772158	−	1.54432i
$566$		0			0
$567$			810.000i			0.0599944i
$568$		0			0
$569$	−8790.00			−0.647620			−0.323810	−	0.946122i	$-0.604964\pi$
$569$	−8790.00			−0.647620			−0.323810	+	0.946122i	$0.604964\pi$
$570$		0			0
$571$	−5984.00			−0.438568			−0.219284	−	0.975661i	$-0.570372\pi$
$571$	−5984.00			−0.438568			−0.219284	+	0.975661i	$0.570372\pi$
$572$		0			0
$573$		−	180.000i		−	0.0131232i
$574$		0			0
$575$	16400.0	−	12300.0i	1.18944	−	0.892079i
$576$		0			0
$577$		−	9344.00i		−	0.674170i	−0.941474	−	0.337085i	$-0.890559\pi$
$577$		−	9344.00i		−	0.674170i	0.941474	−	0.337085i	$-0.109441\pi$
$578$		0			0
$579$	14028.0			1.00688
$580$		0			0
$581$	4120.00			0.294193
$582$		0			0
$583$			13156.0i			0.934590i
$584$		0			0
$585$	−1530.00	+	3060.00i	−0.108133	+	0.216266i
$586$		0			0
$587$		−	6932.00i		−	0.487418i	−0.969848	−	0.243709i	$-0.921636\pi$
$587$		−	6932.00i		−	0.487418i	0.969848	−	0.243709i	$-0.0783642\pi$
$588$		0			0
$589$	7488.00			0.523833
$590$		0			0
$591$	6858.00			0.477327
$592$		0			0
$593$		−	9382.00i		−	0.649701i	−0.945765	−	0.324850i	$-0.894686\pi$
$593$		−	9382.00i		−	0.649701i	0.945765	−	0.324850i	$-0.105314\pi$
$594$		0			0
$595$	6600.00	+	3300.00i	0.454746	+	0.227373i
$596$		0			0
$597$		−	10608.0i		−	0.727230i
$598$		0			0
$599$	−4096.00			−0.279396			−0.139698	−	0.990194i	$-0.544613\pi$
$599$	−4096.00			−0.279396			−0.139698	+	0.990194i	$0.544613\pi$
$600$		0			0
$601$	22962.0			1.55847			0.779234	−	0.626733i	$-0.215608\pi$
$601$	22962.0			1.55847			0.779234	+	0.626733i	$0.215608\pi$
$602$		0			0
$603$		−	2952.00i		−	0.199361i
$604$		0			0
$605$	−7850.00	−	3925.00i	−0.527517	−	0.263759i
$606$		0			0
$607$		−	3490.00i		−	0.233369i	−0.993169	−	0.116684i	$-0.962773\pi$
$607$		−	3490.00i		−	0.233369i	0.993169	−	0.116684i	$-0.0372266\pi$
$608$		0			0
$609$	−6720.00			−0.447140
$610$		0			0
$611$	16320.0			1.08058
$612$		0			0
$613$		−	6386.00i		−	0.420764i	−0.977619	−	0.210382i	$-0.932529\pi$
$613$		−	6386.00i		−	0.420764i	0.977619	−	0.210382i	$-0.0674707\pi$
$614$		0			0
$615$	−2910.00	+	5820.00i	−0.190801	+	0.381602i
$616$		0			0
$617$		−	19534.0i		−	1.27457i	−0.770629	−	0.637285i	$-0.780058\pi$
$617$		−	19534.0i		−	1.27457i	0.770629	−	0.637285i	$-0.219942\pi$
$618$		0			0
$619$	−8764.00			−0.569071			−0.284535	−	0.958666i	$-0.591839\pi$
$619$	−8764.00			−0.569071			−0.284535	+	0.958666i	$0.591839\pi$
$620$		0			0
$621$	4428.00			0.286134
$622$		0			0
$623$		−	12060.0i		−	0.775560i
$624$		0			0
$625$	−4375.00	+	15000.0i	−0.280000	+	0.960000i
$626$		0			0
$627$		−	14352.0i		−	0.914137i
$628$		0			0
$629$	1452.00			0.0920430
$630$		0			0
$631$	7856.00			0.495630			0.247815	−	0.968807i	$-0.420288\pi$
$631$	7856.00			0.495630			0.247815	+	0.968807i	$0.420288\pi$
$632$		0			0
$633$			10500.0i			0.659301i
$634$		0			0
$635$	−2670.00	+	5340.00i	−0.166859	+	0.333719i
$636$		0			0
$637$			8262.00i			0.513897i
$638$		0			0
$639$	−1692.00			−0.104749
$640$		0			0
$641$	−22974.0			−1.41563			−0.707815	−	0.706398i	$-0.750319\pi$
$641$	−22974.0			−1.41563			−0.707815	+	0.706398i	$0.750319\pi$
$642$		0			0
$643$			6216.00i			0.381237i	0.981664	+	0.190618i	$0.0610493\pi$
$643$			6216.00i			0.381237i	−0.981664	+	0.190618i	$0.938951\pi$
$644$		0			0
$645$	3240.00	+	1620.00i	0.197791	+	0.0988953i
$646$		0			0
$647$			13384.0i			0.813260i	0.913593	+	0.406630i	$0.133296\pi$
$647$			13384.0i			0.813260i	−0.913593	+	0.406630i	$0.866704\pi$
$648$		0			0
$649$	19596.0			1.18522
$650$		0			0
$651$	−2160.00			−0.130042
$652$		0			0
$653$		−	12882.0i		−	0.771993i	−0.922500	−	0.385997i	$-0.873858\pi$
$653$		−	12882.0i		−	0.771993i	0.922500	−	0.385997i	$-0.126142\pi$
$654$		0			0
$655$	−18060.0	−	9030.00i	−1.07735	−	0.538674i
$656$		0			0
$657$		−	6660.00i		−	0.395482i
$658$		0			0
$659$	2082.00			0.123070			0.0615351	−	0.998105i	$-0.480400\pi$
$659$	2082.00			0.123070			0.0615351	+	0.998105i	$0.480400\pi$
$660$		0			0
$661$	−9430.00			−0.554893			−0.277447	−	0.960741i	$-0.589488\pi$
$661$	−9430.00			−0.554893			−0.277447	+	0.960741i	$0.589488\pi$
$662$		0			0
$663$			6732.00i			0.394343i
$664$		0			0
$665$	−5200.00	+	10400.0i	−0.303229	+	0.606458i
$666$		0			0
$667$			36736.0i			2.13257i
$668$		0			0
$669$	−17622.0			−1.01839
$670$		0			0
$671$	−32108.0			−1.84727
$672$		0			0
$673$		−	3268.00i		−	0.187180i	−0.995611	−	0.0935900i	$-0.970166\pi$
$673$		−	3268.00i		−	0.187180i	0.995611	−	0.0935900i	$-0.0298343\pi$
$674$		0			0
$675$	−2700.00	+	2025.00i	−0.153960	+	0.115470i
$676$		0			0
$677$		−	15606.0i		−	0.885949i	−0.896534	−	0.442974i	$-0.853923\pi$
$677$		−	15606.0i		−	0.885949i	0.896534	−	0.442974i	$-0.146077\pi$
$678$		0			0
$679$	13840.0			0.782225
$680$		0			0
$681$	−372.000			−0.0209326
$682$		0			0
$683$		−	428.000i		−	0.0239780i	−0.999928	−	0.0119890i	$-0.996184\pi$
$683$		−	428.000i		−	0.0239780i	0.999928	−	0.0119890i	$-0.00381631\pi$
$684$		0			0
$685$	9110.00	−	18220.0i	0.508139	−	1.01628i
$686$		0			0
$687$		−	4086.00i		−	0.226915i
$688$		0			0
$689$	9724.00			0.537670
$690$		0			0
$691$	−6384.00			−0.351460			−0.175730	−	0.984438i	$-0.556229\pi$
$691$	−6384.00			−0.351460			−0.175730	+	0.984438i	$0.556229\pi$
$692$		0			0
$693$			4140.00i			0.226934i
$694$		0			0
$695$	5320.00	+	2660.00i	0.290358	+	0.145179i
$696$		0			0
$697$			12804.0i			0.695819i
$698$		0			0
$699$	11610.0			0.628227
$700$		0			0
$701$	−12224.0			−0.658622			−0.329311	−	0.944221i	$-0.606816\pi$
$701$	−12224.0			−0.658622			−0.329311	+	0.944221i	$0.606816\pi$
$702$		0			0
$703$			2288.00i			0.122750i
$704$		0			0
$705$	14400.0	+	7200.00i	0.769270	+	0.384635i
$706$		0			0
$707$		−	11280.0i		−	0.600040i
$708$		0			0
$709$	−19510.0			−1.03345			−0.516723	−	0.856153i	$-0.672848\pi$
$709$	−19510.0			−1.03345			−0.516723	+	0.856153i	$0.672848\pi$
$710$		0			0
$711$	10512.0			0.554474
$712$		0			0
$713$			11808.0i			0.620215i
$714$		0			0
$715$	−7820.00	+	15640.0i	−0.409023	+	0.818046i
$716$		0			0
$717$			18348.0i			0.955674i
$718$		0			0
$719$	3368.00			0.174694			0.0873472	−	0.996178i	$-0.472161\pi$
$719$	3368.00			0.174694			0.0873472	+	0.996178i	$0.472161\pi$
$720$		0			0
$721$	−7580.00			−0.391531
$722$		0			0
$723$			17886.0i			0.920038i
$724$		0			0
$725$	−16800.0	−	22400.0i	−0.860602	−	1.14747i
$726$		0			0
$727$			22134.0i			1.12917i	0.825376	+	0.564584i	$0.190963\pi$
$727$			22134.0i			1.12917i	−0.825376	+	0.564584i	$0.809037\pi$
$728$		0			0
$729$	−729.000			−0.0370370
$730$		0			0
$731$	7128.00			0.360655
$732$		0			0
$733$		−	32298.0i		−	1.62750i	−0.581219	−	0.813748i	$-0.697424\pi$
$733$		−	32298.0i		−	1.62750i	0.581219	−	0.813748i	$-0.302576\pi$
$734$		0			0
$735$	−3645.00	+	7290.00i	−0.182922	+	0.365844i
$736$		0			0
$737$		−	15088.0i		−	0.754103i
$738$		0			0
$739$	−25104.0			−1.24962			−0.624808	−	0.780779i	$-0.714823\pi$
$739$	−25104.0			−1.24962			−0.624808	+	0.780779i	$0.714823\pi$
$740$		0			0
$741$	−10608.0			−0.525904
$742$		0			0
$743$			27696.0i			1.36752i	0.729707	+	0.683760i	$0.239657\pi$
$743$			27696.0i			1.36752i	−0.729707	+	0.683760i	$0.760343\pi$
$744$		0			0
$745$	−12840.0	−	6420.00i	−0.631438	−	0.315719i
$746$		0			0
$747$			3708.00i			0.181618i
$748$		0			0
$749$	−13240.0			−0.645900
$750$		0			0
$751$	−2176.00			−0.105730			−0.0528651	−	0.998602i	$-0.516835\pi$
$751$	−2176.00			−0.105730			−0.0528651	+	0.998602i	$0.516835\pi$
$752$		0			0
$753$			4470.00i			0.216329i
$754$		0			0
$755$	−1840.00	−	920.000i	−0.0886946	−	0.0443473i
$756$		0			0
$757$			25514.0i			1.22500i	0.790472	+	0.612498i	$0.209835\pi$
$757$			25514.0i			1.22500i	−0.790472	+	0.612498i	$0.790165\pi$
$758$		0			0
$759$	22632.0			1.08233
$760$		0			0
$761$	18238.0			0.868761			0.434380	−	0.900730i	$-0.356967\pi$
$761$	18238.0			0.868761			0.434380	+	0.900730i	$0.356967\pi$
$762$		0			0
$763$		−	16020.0i		−	0.760109i
$764$		0			0
$765$	−2970.00	+	5940.00i	−0.140367	+	0.280734i
$766$		0			0
$767$		−	14484.0i		−	0.681860i
$768$		0			0
$769$	14462.0			0.678170			0.339085	−	0.940756i	$-0.389882\pi$
$769$	14462.0			0.678170			0.339085	+	0.940756i	$0.389882\pi$
$770$		0			0
$771$	−16182.0			−0.755876
$772$		0			0
$773$		−	34034.0i		−	1.58359i	−0.610785	−	0.791797i	$-0.709146\pi$
$773$		−	34034.0i		−	1.58359i	0.610785	−	0.791797i	$-0.290854\pi$
$774$		0			0
$775$	−5400.00	−	7200.00i	−0.250289	−	0.333718i
$776$		0			0
$777$		−	660.000i		−	0.0304728i
$778$		0			0
$779$	−20176.0			−0.927959
$780$		0			0
$781$	−8648.00			−0.396222
$782$		0			0
$783$		−	6048.00i		−	0.276038i
$784$		0			0
$785$	−18730.0	+	37460.0i	−0.851595	+	1.70319i
$786$		0			0
$787$			22064.0i			0.999360i	0.866210	+	0.499680i	$0.166549\pi$
$787$			22064.0i			0.999360i	−0.866210	+	0.499680i	$0.833451\pi$
$788$		0			0
$789$	1908.00			0.0860920
$790$		0			0
$791$	−20740.0			−0.932275
$792$		0			0
$793$			23732.0i			1.06273i
$794$		0			0
$795$	8580.00	+	4290.00i	0.382769	+	0.191384i
$796$		0			0
$797$		−	23334.0i		−	1.03705i	−0.855061	−	0.518527i	$-0.826480\pi$
$797$		−	23334.0i		−	1.03705i	0.855061	−	0.518527i	$-0.173520\pi$
$798$		0			0
$799$	31680.0			1.40270
$800$		0			0
$801$	10854.0			0.478786
$802$		0			0
$803$		−	34040.0i		−	1.49595i
$804$		0			0
$805$	−16400.0	−	8200.00i	−0.718042	−	0.359021i
$806$		0			0
$807$		−	10080.0i		−	0.439694i
$808$		0			0
$809$	7566.00			0.328809			0.164404	−	0.986393i	$-0.447430\pi$
$809$	7566.00			0.328809			0.164404	+	0.986393i	$0.447430\pi$
$810$		0			0
$811$	5964.00			0.258230			0.129115	−	0.991630i	$-0.458786\pi$
$811$	5964.00			0.258230			0.129115	+	0.991630i	$0.458786\pi$
$812$		0			0
$813$		−	17304.0i		−	0.746467i
$814$		0			0
$815$	−7520.00	+	15040.0i	−0.323207	+	0.646415i
$816$		0			0
$817$			11232.0i			0.480977i
$818$		0			0
$819$	3060.00			0.130556
$820$		0			0
$821$	−11880.0			−0.505012			−0.252506	−	0.967595i	$-0.581255\pi$
$821$	−11880.0			−0.505012			−0.252506	+	0.967595i	$0.581255\pi$
$822$		0			0
$823$			1762.00i			0.0746287i	0.999304	+	0.0373144i	$0.0118803\pi$
$823$			1762.00i			0.0746287i	−0.999304	+	0.0373144i	$0.988120\pi$
$824$		0			0
$825$	−13800.0	+	10350.0i	−0.582369	+	0.436777i
$826$		0			0
$827$		−	14124.0i		−	0.593881i	−0.954896	−	0.296941i	$-0.904034\pi$
$827$		−	14124.0i		−	0.593881i	0.954896	−	0.296941i	$-0.0959663\pi$
$828$		0			0
$829$	21350.0			0.894471			0.447235	−	0.894416i	$-0.352409\pi$
$829$	21350.0			0.894471			0.447235	+	0.894416i	$0.352409\pi$
$830$		0			0
$831$	−4194.00			−0.175076
$832$		0			0
$833$			16038.0i			0.667087i
$834$		0			0
$835$	15060.0	−	30120.0i	0.624159	−	1.24832i
$836$		0			0
$837$		−	1944.00i		−	0.0802801i
$838$		0			0
$839$	26136.0			1.07546			0.537732	−	0.843116i	$-0.319281\pi$
$839$	26136.0			1.07546			0.537732	+	0.843116i	$0.319281\pi$
$840$		0			0
$841$	25787.0			1.05732
$842$		0			0
$843$			12582.0i			0.514053i
$844$		0			0
$845$	−10410.0	−	5205.00i	−0.423805	−	0.211902i
$846$		0			0
$847$			7850.00i			0.318452i
$848$		0			0
$849$	24768.0			1.00122
$850$		0			0
$851$	−3608.00			−0.145336
$852$		0			0
$853$			7030.00i			0.282184i	0.989997	+	0.141092i	$0.0450613\pi$
$853$			7030.00i			0.282184i	−0.989997	+	0.141092i	$0.954939\pi$
$854$		0			0
$855$	−9360.00	−	4680.00i	−0.374392	−	0.187196i
$856$		0			0
$857$		−	9574.00i		−	0.381612i	−0.981628	−	0.190806i	$-0.938890\pi$
$857$		−	9574.00i		−	0.381612i	0.981628	−	0.190806i	$-0.0611102\pi$
$858$		0			0
$859$	43748.0			1.73767			0.868837	−	0.495098i	$-0.164868\pi$
$859$	43748.0			1.73767			0.868837	+	0.495098i	$0.164868\pi$
$860$		0			0
$861$	5820.00			0.230366
$862$		0			0
$863$		−	41436.0i		−	1.63441i	−0.576345	−	0.817206i	$-0.695522\pi$
$863$		−	41436.0i		−	1.63441i	0.576345	−	0.817206i	$-0.304478\pi$
$864$		0			0
$865$	2190.00	−	4380.00i	0.0860835	−	0.172167i
$866$		0			0
$867$		−	1671.00i		−	0.0654558i
$868$		0			0
$869$	53728.0			2.09735
$870$		0			0
$871$	−11152.0			−0.433836
$872$		0			0
$873$			12456.0i			0.482900i
$874$		0			0
$875$	13750.0	−	2500.00i	0.531240	−	0.0965891i
$876$		0			0
$877$			4606.00i			0.177347i	0.996061	+	0.0886736i	$0.0282628\pi$
$877$			4606.00i			0.177347i	−0.996061	+	0.0886736i	$0.971737\pi$
$878$		0			0
$879$	−16602.0			−0.637055
$880$		0			0
$881$	−4610.00			−0.176294			−0.0881469	−	0.996107i	$-0.528094\pi$
$881$	−4610.00			−0.176294			−0.0881469	+	0.996107i	$0.528094\pi$
$882$		0			0
$883$		−	23512.0i		−	0.896084i	−0.894013	−	0.448042i	$-0.852122\pi$
$883$		−	23512.0i		−	0.896084i	0.894013	−	0.448042i	$-0.147878\pi$
$884$		0			0
$885$	6390.00	−	12780.0i	0.242709	−	0.485418i
$886$		0			0
$887$		−	30396.0i		−	1.15062i	−0.817936	−	0.575309i	$-0.804882\pi$
$887$		−	30396.0i		−	1.15062i	0.817936	−	0.575309i	$-0.195118\pi$
$888$		0			0
$889$	5340.00			0.201460
$890$		0			0
$891$	−3726.00			−0.140096
$892$		0			0
$893$			49920.0i			1.87067i
$894$		0			0
$895$	−14620.0	−	7310.00i	−0.546025	−	0.273013i
$896$		0			0
$897$		−	16728.0i		−	0.622666i
$898$		0			0
$899$	16128.0			0.598330
$900$		0			0
$901$	18876.0			0.697948
$902$		0			0
$903$		−	3240.00i		−	0.119402i
$904$		0			0
$905$	−5860.00	−	2930.00i	−0.215241	−	0.107620i
$906$		0			0
$907$			3452.00i			0.126375i	0.998002	+	0.0631873i	$0.0201266\pi$
$907$			3452.00i			0.126375i	−0.998002	+	0.0631873i	$0.979873\pi$
$908$		0			0
$909$	10152.0			0.370430
$910$		0			0
$911$	14256.0			0.518466			0.259233	−	0.965815i	$-0.416530\pi$
$911$	14256.0			0.518466			0.259233	+	0.965815i	$0.416530\pi$
$912$		0			0
$913$			18952.0i			0.686988i
$914$		0			0
$915$	−10470.0	+	20940.0i	−0.378281	+	0.756563i
$916$		0			0
$917$			18060.0i			0.650375i
$918$		0			0
$919$	−8064.00			−0.289452			−0.144726	−	0.989472i	$-0.546230\pi$
$919$	−8064.00			−0.289452			−0.144726	+	0.989472i	$0.546230\pi$
$920$		0			0
$921$	−1452.00			−0.0519490
$922$		0			0
$923$			6392.00i			0.227947i
$924$		0			0
$925$	2200.00	−	1650.00i	0.0782006	−	0.0586504i
$926$		0			0
$927$		−	6822.00i		−	0.241709i
$928$		0			0
$929$	40930.0			1.44550			0.722750	−	0.691109i	$-0.242878\pi$
$929$	40930.0			1.44550			0.722750	+	0.691109i	$0.242878\pi$
$930$		0			0
$931$	−25272.0			−0.889642
$932$		0			0
$933$			8172.00i			0.286752i
$934$		0			0
$935$	−15180.0	+	30360.0i	−0.530951	+	1.06190i
$936$		0			0
$937$			9016.00i			0.314344i	0.987571	+	0.157172i	$0.0502376\pi$
$937$			9016.00i			0.314344i	−0.987571	+	0.157172i	$0.949762\pi$
$938$		0			0
$939$	−15924.0			−0.553419
$940$		0			0
$941$	−10444.0			−0.361812			−0.180906	−	0.983500i	$-0.557903\pi$
$941$	−10444.0			−0.361812			−0.180906	+	0.983500i	$0.557903\pi$
$942$		0			0
$943$		−	31816.0i		−	1.09870i
$944$		0			0
$945$	2700.00	+	1350.00i	0.0929429	+	0.0464714i
$946$		0			0
$947$			21516.0i			0.738306i	0.929369	+	0.369153i	$0.120352\pi$
$947$			21516.0i			0.738306i	−0.929369	+	0.369153i	$0.879648\pi$
$948$		0			0
$949$	−25160.0			−0.860620
$950$		0			0
$951$	12654.0			0.431476
$952$		0			0
$953$			28098.0i			0.955072i	0.878612	+	0.477536i	$0.158470\pi$
$953$			28098.0i			0.955072i	−0.878612	+	0.477536i	$0.841530\pi$
$954$		0			0
$955$	−600.000	−	300.000i	−0.0203304	−	0.0101652i
$956$		0			0
$957$		−	30912.0i		−	1.04414i
$958$		0			0
$959$	−18220.0			−0.613508
$960$		0			0
$961$	−24607.0			−0.825988
$962$		0			0
$963$		−	11916.0i		−	0.398741i
$964$		0			0
$965$	23380.0	−	46760.0i	0.779926	−	1.55985i
$966$		0			0
$967$		−	14558.0i		−	0.484130i	−0.970260	−	0.242065i	$-0.922175\pi$
$967$		−	14558.0i		−	0.484130i	0.970260	−	0.242065i	$-0.0778247\pi$
$968$		0			0
$969$	−20592.0			−0.682673
$970$		0			0
$971$	24846.0			0.821160			0.410580	−	0.911825i	$-0.365326\pi$
$971$	24846.0			0.821160			0.410580	+	0.911825i	$0.365326\pi$
$972$		0			0
$973$		−	5320.00i		−	0.175284i
$974$		0			0
$975$	7650.00	+	10200.0i	0.251278	+	0.335037i
$976$		0			0
$977$		−	12950.0i		−	0.424061i	−0.977263	−	0.212030i	$-0.931992\pi$
$977$		−	12950.0i		−	0.424061i	0.977263	−	0.212030i	$-0.0680076\pi$
$978$		0			0
$979$	55476.0			1.81105
$980$		0			0
$981$	14418.0			0.469247
$982$		0			0
$983$			26728.0i			0.867234i	0.901097	+	0.433617i	$0.142763\pi$
$983$			26728.0i			0.867234i	−0.901097	+	0.433617i	$0.857237\pi$
$984$		0			0
$985$	11430.0	−	22860.0i	0.369736	−	0.739472i
$986$		0			0
$987$		−	14400.0i		−	0.464394i
$988$		0			0
$989$	−17712.0			−0.569473
$990$		0			0
$991$	3880.00			0.124372			0.0621858	−	0.998065i	$-0.480193\pi$
$991$	3880.00			0.124372			0.0621858	+	0.998065i	$0.480193\pi$
$992$		0			0
$993$			13920.0i			0.444852i
$994$		0			0
$995$	−35360.0	−	17680.0i	−1.12662	−	0.563310i
$996$		0			0
$997$			18582.0i			0.590269i	0.955456	+	0.295134i	$0.0953644\pi$
$997$			18582.0i			0.590269i	−0.955456	+	0.295134i	$0.904636\pi$
$998$		0			0
$999$	594.000			0.0188121

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	120.4.f.a.49.1	✓	2
3.2	odd	2		360.4.f.c.289.2		2
4.3	odd	2		240.4.f.b.49.2		2
5.2	odd	4		600.4.a.b.1.1		1
5.3	odd	4		600.4.a.o.1.1		1
5.4	even	2	inner	120.4.f.a.49.2	yes	2
8.3	odd	2		960.4.f.g.769.1		2
8.5	even	2		960.4.f.l.769.2		2
12.11	even	2		720.4.f.g.289.2		2
15.2	even	4		1800.4.a.j.1.1		1
15.8	even	4		1800.4.a.y.1.1		1
15.14	odd	2		360.4.f.c.289.1		2
20.3	even	4		1200.4.a.f.1.1		1
20.7	even	4		1200.4.a.bh.1.1		1
20.19	odd	2		240.4.f.b.49.1		2
40.19	odd	2		960.4.f.g.769.2		2
40.29	even	2		960.4.f.l.769.1		2
60.59	even	2		720.4.f.g.289.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.4.f.a.49.1	✓	2	1.1	even	1	trivial
120.4.f.a.49.2	yes	2	5.4	even	2	inner
240.4.f.b.49.1		2	20.19	odd	2
240.4.f.b.49.2		2	4.3	odd	2
360.4.f.c.289.1		2	15.14	odd	2
360.4.f.c.289.2		2	3.2	odd	2
600.4.a.b.1.1		1	5.2	odd	4
600.4.a.o.1.1		1	5.3	odd	4
720.4.f.g.289.1		2	60.59	even	2
720.4.f.g.289.2		2	12.11	even	2
960.4.f.g.769.1		2	8.3	odd	2
960.4.f.g.769.2		2	40.19	odd	2
960.4.f.l.769.1		2	40.29	even	2
960.4.f.l.769.2		2	8.5	even	2
1200.4.a.f.1.1		1	20.3	even	4
1200.4.a.bh.1.1		1	20.7	even	4
1800.4.a.j.1.1		1	15.2	even	4
1800.4.a.y.1.1		1	15.8	even	4

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	120.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +(-10.0000 - 5.00000i) q^{5} +10.0000i q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} +(-10.0000 - 5.00000i) q^{5} +10.0000i q^{7} -9.00000 q^{9} -46.0000 q^{11} +34.0000i q^{13} +(-15.0000 + 30.0000i) q^{15} +66.0000i q^{17} -104.000 q^{19} +30.0000 q^{21} -164.000i q^{23} +(75.0000 + 100.000i) q^{25} +27.0000i q^{27} -224.000 q^{29} -72.0000 q^{31} +138.000i q^{33} +(50.0000 - 100.000i) q^{35} -22.0000i q^{37} +102.000 q^{39} +194.000 q^{41} -108.000i q^{43} +(90.0000 + 45.0000i) q^{45} -480.000i q^{47} +243.000 q^{49} +198.000 q^{51} -286.000i q^{53} +(460.000 + 230.000i) q^{55} +312.000i q^{57} -426.000 q^{59} +698.000 q^{61} -90.0000i q^{63} +(170.000 - 340.000i) q^{65} +328.000i q^{67} -492.000 q^{69} +188.000 q^{71} +740.000i q^{73} +(300.000 - 225.000i) q^{75} -460.000i q^{77} -1168.00 q^{79} +81.0000 q^{81} -412.000i q^{83} +(330.000 - 660.000i) q^{85} +672.000i q^{87} -1206.00 q^{89} -340.000 q^{91} +216.000i q^{93} +(1040.00 + 520.000i) q^{95} -1384.00i q^{97} +414.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 20 q^{5} - 18 q^{9}+O(q^{10}) \) 2 * q - 20 * q^5 - 18 * q^9	\( 2 q - 20 q^{5} - 18 q^{9} - 92 q^{11} - 30 q^{15} - 208 q^{19} + 60 q^{21} + 150 q^{25} - 448 q^{29} - 144 q^{31} + 100 q^{35} + 204 q^{39} + 388 q^{41} + 180 q^{45} + 486 q^{49} + 396 q^{51} + 920 q^{55} - 852 q^{59} + 1396 q^{61} + 340 q^{65} - 984 q^{69} + 376 q^{71} + 600 q^{75} - 2336 q^{79} + 162 q^{81} + 660 q^{85} - 2412 q^{89} - 680 q^{91} + 2080 q^{95} + 828 q^{99}+O(q^{100}) \) 2 * q - 20 * q^5 - 18 * q^9 - 92 * q^11 - 30 * q^15 - 208 * q^19 + 60 * q^21 + 150 * q^25 - 448 * q^29 - 144 * q^31 + 100 * q^35 + 204 * q^39 + 388 * q^41 + 180 * q^45 + 486 * q^49 + 396 * q^51 + 920 * q^55 - 852 * q^59 + 1396 * q^61 + 340 * q^65 - 984 * q^69 + 376 * q^71 + 600 * q^75 - 2336 * q^79 + 162 * q^81 + 660 * q^85 - 2412 * q^89 - 680 * q^91 + 2080 * q^95 + 828 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more