LMFDB - Embedded newform 120.4.f.b.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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.b.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +(-2.00000 + 11.0000i) q^{5} +10.0000i q^{7} -9.00000 q^{9} -14.0000 q^{11} +82.0000i q^{13} +(33.0000 + 6.00000i) q^{15} +18.0000i q^{17} +136.000 q^{19} +30.0000 q^{21} +140.000i q^{23} +(-117.000 - 44.0000i) q^{25} +27.0000i q^{27} -112.000 q^{29} +72.0000 q^{31} +42.0000i q^{33} +(-110.000 - 20.0000i) q^{35} +26.0000i q^{37} +246.000 q^{39} -446.000 q^{41} -396.000i q^{43} +(18.0000 - 99.0000i) q^{45} -144.000i q^{47} +243.000 q^{49} +54.0000 q^{51} -158.000i q^{53} +(28.0000 - 154.000i) q^{55} -408.000i q^{57} +342.000 q^{59} +314.000 q^{61} -90.0000i q^{63} +(-902.000 - 164.000i) q^{65} -152.000i q^{67} +420.000 q^{69} -932.000 q^{71} +548.000i q^{73} +(-132.000 + 351.000i) q^{75} -140.000i q^{77} +512.000 q^{79} +81.0000 q^{81} -284.000i q^{83} +(-198.000 - 36.0000i) q^{85} +336.000i q^{87} +810.000 q^{89} -820.000 q^{91} -216.000i q^{93} +(-272.000 + 1496.00i) q^{95} +1304.00i q^{97} +126.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} - 18 q^{9} - 28 q^{11} + 66 q^{15} + 272 q^{19} + 60 q^{21} - 234 q^{25} - 224 q^{29} + 144 q^{31} - 220 q^{35} + 492 q^{39} - 892 q^{41} + 36 q^{45} + 486 q^{49} + 108 q^{51} + 56 q^{55}+ \cdots + 252 q^{99}+O(q^{100}) $ 2 * q - 4 * q^5 - 18 * q^9 - 28 * q^11 + 66 * q^15 + 272 * q^19 + 60 * q^21 - 234 * q^25 - 224 * q^29 + 144 * q^31 - 220 * q^35 + 492 * q^39 - 892 * q^41 + 36 * q^45 + 486 * q^49 + 108 * q^51 + 56 * q^55 + 684 * q^59 + 628 * q^61 - 1804 * q^65 + 840 * q^69 - 1864 * q^71 - 264 * q^75 + 1024 * q^79 + 162 * q^81 - 396 * q^85 + 1620 * q^89 - 1640 * q^91 - 544 * q^95 + 252 * q^99

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
$5$	−2.00000	+	11.0000i	−0.178885	+	0.983870i
$5$	−2.00000	+	11.0000i	−0.178885	+	0.983870i
$6$		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
$11$	−14.0000			−0.383742			−0.191871	−	0.981420i	$-0.561455\pi$
$11$	−14.0000			−0.383742			−0.191871	+	0.981420i	$0.561455\pi$
$12$		0			0
$13$			82.0000i			1.74944i	0.484629	+	0.874720i	$0.338954\pi$
$13$			82.0000i			1.74944i	−0.484629	+	0.874720i	$0.661046\pi$
$14$		0			0
$15$	33.0000	+	6.00000i	0.568038	+	0.103280i
$16$		0			0
$17$			18.0000i			0.256802i	0.991722	+	0.128401i	$0.0409845\pi$
$17$			18.0000i			0.256802i	−0.991722	+	0.128401i	$0.959015\pi$
$18$		0			0
$19$	136.000			1.64213			0.821067	−	0.570832i	$-0.193379\pi$
$19$	136.000			1.64213			0.821067	+	0.570832i	$0.193379\pi$
$20$		0			0
$21$	30.0000			0.311740
$22$		0			0
$23$			140.000i			1.26922i	0.772833	+	0.634609i	$0.218839\pi$
$23$			140.000i			1.26922i	−0.772833	+	0.634609i	$0.781161\pi$
$24$		0			0
$25$	−117.000	−	44.0000i	−0.936000	−	0.352000i
$26$		0			0
$27$			27.0000i			0.192450i
$28$		0			0
$29$	−112.000			−0.717168			−0.358584	−	0.933497i	$-0.616740\pi$
$29$	−112.000			−0.717168			−0.358584	+	0.933497i	$0.616740\pi$
$30$		0			0
$31$	72.0000			0.417148			0.208574	−	0.978007i	$-0.433118\pi$
$31$	72.0000			0.417148			0.208574	+	0.978007i	$0.433118\pi$
$32$		0			0
$33$			42.0000i			0.221553i
$34$		0			0
$35$	−110.000	−	20.0000i	−0.531240	−	0.0965891i
$36$		0			0
$37$			26.0000i			0.115524i	0.998330	+	0.0577618i	$0.0183964\pi$
$37$			26.0000i			0.115524i	−0.998330	+	0.0577618i	$0.981604\pi$
$38$		0			0
$39$	246.000			1.01004
$40$		0			0
$41$	−446.000			−1.69887			−0.849433	−	0.527697i	$-0.823056\pi$
$41$	−446.000			−1.69887			−0.849433	+	0.527697i	$0.823056\pi$
$42$		0			0
$43$		−	396.000i		−	1.40441i	−0.711977	−	0.702203i	$-0.752200\pi$
$43$		−	396.000i		−	1.40441i	0.711977	−	0.702203i	$-0.247800\pi$
$44$		0			0
$45$	18.0000	−	99.0000i	0.0596285	−	0.327957i
$46$		0			0
$47$		−	144.000i		−	0.446906i	−0.974715	−	0.223453i	$-0.928267\pi$
$47$		−	144.000i		−	0.446906i	0.974715	−	0.223453i	$-0.0717328\pi$
$48$		0			0
$49$	243.000			0.708455
$50$		0			0
$51$	54.0000			0.148265
$52$		0			0
$53$		−	158.000i		−	0.409490i	−0.978815	−	0.204745i	$-0.934363\pi$
$53$		−	158.000i		−	0.409490i	0.978815	−	0.204745i	$-0.0656365\pi$
$54$		0			0
$55$	28.0000	−	154.000i	0.0686458	−	0.377552i
$56$		0			0
$57$		−	408.000i		−	0.948086i
$58$		0			0
$59$	342.000			0.754654			0.377327	−	0.926080i	$-0.376843\pi$
$59$	342.000			0.754654			0.377327	+	0.926080i	$0.376843\pi$
$60$		0			0
$61$	314.000			0.659075			0.329538	−	0.944142i	$-0.393107\pi$
$61$	314.000			0.659075			0.329538	+	0.944142i	$0.393107\pi$
$62$		0			0
$63$		−	90.0000i		−	0.179983i
$64$		0			0
$65$	−902.000	−	164.000i	−1.72122	−	0.312949i
$66$		0			0
$67$		−	152.000i		−	0.277161i	−0.990351	−	0.138580i	$-0.955746\pi$
$67$		−	152.000i		−	0.277161i	0.990351	−	0.138580i	$-0.0442539\pi$
$68$		0			0
$69$	420.000			0.732783
$70$		0			0
$71$	−932.000			−1.55786			−0.778930	−	0.627111i	$-0.784237\pi$
$71$	−932.000			−1.55786			−0.778930	+	0.627111i	$0.784237\pi$
$72$		0			0
$73$			548.000i			0.878610i	0.898338	+	0.439305i	$0.144775\pi$
$73$			548.000i			0.878610i	−0.898338	+	0.439305i	$0.855225\pi$
$74$		0			0
$75$	−132.000	+	351.000i	−0.203227	+	0.540400i
$76$		0			0
$77$		−	140.000i		−	0.207201i
$78$		0			0
$79$	512.000			0.729171			0.364585	−	0.931170i	$-0.381211\pi$
$79$	512.000			0.729171			0.364585	+	0.931170i	$0.381211\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$		−	284.000i		−	0.375579i	−0.982209	−	0.187789i	$-0.939868\pi$
$83$		−	284.000i		−	0.375579i	0.982209	−	0.187789i	$-0.0601323\pi$
$84$		0			0
$85$	−198.000	−	36.0000i	−0.252660	−	0.0459382i
$86$		0			0
$87$			336.000i			0.414057i
$88$		0			0
$89$	810.000			0.964717			0.482359	−	0.875974i	$-0.339780\pi$
$89$	810.000			0.964717			0.482359	+	0.875974i	$0.339780\pi$
$90$		0			0
$91$	−820.000			−0.944608
$92$		0			0
$93$		−	216.000i		−	0.240840i
$94$		0			0
$95$	−272.000	+	1496.00i	−0.293754	+	1.61565i
$96$		0			0
$97$			1304.00i			1.36496i	0.730904	+	0.682480i	$0.239099\pi$
$97$			1304.00i			1.36496i	−0.730904	+	0.682480i	$0.760901\pi$
$98$		0			0
$99$	126.000			0.127914
$100$		0			0
$101$	936.000			0.922133			0.461067	−	0.887365i	$-0.347467\pi$
$101$	936.000			0.922133			0.461067	+	0.887365i	$0.347467\pi$
$102$		0			0
$103$		−	1450.00i		−	1.38711i	−0.720402	−	0.693557i	$-0.756043\pi$
$103$		−	1450.00i		−	1.38711i	0.720402	−	0.693557i	$-0.243957\pi$
$104$		0			0
$105$	−60.0000	+	330.000i	−0.0557657	+	0.306711i
$106$		0			0
$107$			1292.00i			1.16731i	0.812001	+	0.583656i	$0.198378\pi$
$107$			1292.00i			1.16731i	−0.812001	+	0.583656i	$0.801622\pi$
$108$		0			0
$109$	2142.00			1.88226			0.941130	−	0.338044i	$-0.109765\pi$
$109$	2142.00			1.88226			0.941130	+	0.338044i	$0.109765\pi$
$110$		0			0
$111$	78.0000			0.0666976
$112$		0			0
$113$			1418.00i			1.18048i	0.807228	+	0.590240i	$0.200967\pi$
$113$			1418.00i			1.18048i	−0.807228	+	0.590240i	$0.799033\pi$
$114$		0			0
$115$	−1540.00	−	280.000i	−1.24875	−	0.227045i
$116$		0			0
$117$		−	738.000i		−	0.583146i
$118$		0			0
$119$	−180.000			−0.138660
$120$		0			0
$121$	−1135.00			−0.852742
$122$		0			0
$123$			1338.00i			0.980841i
$124$		0			0
$125$	718.000	−	1199.00i	0.513759	−	0.857935i
$126$		0			0
$127$			1674.00i			1.16963i	0.811165	+	0.584817i	$0.198834\pi$
$127$			1674.00i			1.16963i	−0.811165	+	0.584817i	$0.801166\pi$
$128$		0			0
$129$	−1188.00			−0.810834
$130$		0			0
$131$	1134.00			0.756321			0.378160	−	0.925740i	$-0.376557\pi$
$131$	1134.00			0.756321			0.378160	+	0.925740i	$0.376557\pi$
$132$		0			0
$133$			1360.00i			0.886669i
$134$		0			0
$135$	−297.000	−	54.0000i	−0.189346	−	0.0344265i
$136$		0			0
$137$		−	2866.00i		−	1.78729i	−0.448773	−	0.893646i	$-0.648139\pi$
$137$		−	2866.00i		−	1.78729i	0.448773	−	0.893646i	$-0.351861\pi$
$138$		0			0
$139$	764.000			0.466199			0.233099	−	0.972453i	$-0.425113\pi$
$139$	764.000			0.466199			0.233099	+	0.972453i	$0.425113\pi$
$140$		0			0
$141$	−432.000			−0.258021
$142$		0			0
$143$		−	1148.00i		−	0.671333i
$144$		0			0
$145$	224.000	−	1232.00i	0.128291	−	0.705600i
$146$		0			0
$147$		−	729.000i		−	0.409027i
$148$		0			0
$149$	3060.00			1.68245			0.841225	−	0.540686i	$-0.181835\pi$
$149$	3060.00			1.68245			0.841225	+	0.540686i	$0.181835\pi$
$150$		0			0
$151$	−1304.00			−0.702768			−0.351384	−	0.936231i	$-0.614289\pi$
$151$	−1304.00			−0.702768			−0.351384	+	0.936231i	$0.614289\pi$
$152$		0			0
$153$		−	162.000i		−	0.0856008i
$154$		0			0
$155$	−144.000	+	792.000i	−0.0746217	+	0.410419i
$156$		0			0
$157$			1582.00i			0.804187i	0.915599	+	0.402093i	$0.131717\pi$
$157$			1582.00i			0.804187i	−0.915599	+	0.402093i	$0.868283\pi$
$158$		0			0
$159$	−474.000			−0.236419
$160$		0			0
$161$	−1400.00			−0.685313
$162$		0			0
$163$		−	3232.00i		−	1.55307i	−0.630077	−	0.776533i	$-0.716976\pi$
$163$		−	3232.00i		−	1.55307i	0.630077	−	0.776533i	$-0.283024\pi$
$164$		0			0
$165$	−462.000	−	84.0000i	−0.217980	−	0.0396327i
$166$		0			0
$167$		−	2988.00i		−	1.38454i	−0.721638	−	0.692271i	$-0.756610\pi$
$167$		−	2988.00i		−	1.38454i	0.721638	−	0.692271i	$-0.243390\pi$
$168$		0			0
$169$	−4527.00			−2.06054
$170$		0			0
$171$	−1224.00			−0.547378
$172$		0			0
$173$			918.000i			0.403435i	0.979444	+	0.201717i	$0.0646523\pi$
$173$			918.000i			0.403435i	−0.979444	+	0.201717i	$0.935348\pi$
$174$		0			0
$175$	440.000	−	1170.00i	0.190062	−	0.505392i
$176$		0			0
$177$		−	1026.00i		−	0.435700i
$178$		0			0
$179$	1238.00			0.516941			0.258471	−	0.966019i	$-0.416781\pi$
$179$	1238.00			0.516941			0.258471	+	0.966019i	$0.416781\pi$
$180$		0			0
$181$	1450.00			0.595457			0.297728	−	0.954651i	$-0.403771\pi$
$181$	1450.00			0.595457			0.297728	+	0.954651i	$0.403771\pi$
$182$		0			0
$183$		−	942.000i		−	0.380517i
$184$		0			0
$185$	−286.000	−	52.0000i	−0.113660	−	0.0206655i
$186$		0			0
$187$		−	252.000i		−	0.0985458i
$188$		0			0
$189$	−270.000			−0.103913
$190$		0			0
$191$	4860.00			1.84114			0.920569	−	0.390581i	$-0.127726\pi$
$191$	4860.00			1.84114			0.920569	+	0.390581i	$0.127726\pi$
$192$		0			0
$193$			1412.00i			0.526622i	0.964711	+	0.263311i	$0.0848145\pi$
$193$			1412.00i			0.526622i	−0.964711	+	0.263311i	$0.915186\pi$
$194$		0			0
$195$	−492.000	+	2706.00i	−0.180681	+	0.993747i
$196$		0			0
$197$		−	1170.00i		−	0.423142i	−0.977363	−	0.211571i	$-0.932142\pi$
$197$		−	1170.00i		−	0.423142i	0.977363	−	0.211571i	$-0.0678580\pi$
$198$		0			0
$199$	−2080.00			−0.740941			−0.370471	−	0.928844i	$-0.620804\pi$
$199$	−2080.00			−0.740941			−0.370471	+	0.928844i	$0.620804\pi$
$200$		0			0
$201$	−456.000			−0.160019
$202$		0			0
$203$		−	1120.00i		−	0.387234i
$204$		0			0
$205$	892.000	−	4906.00i	0.303902	−	1.67146i
$206$		0			0
$207$		−	1260.00i		−	0.423073i
$208$		0			0
$209$	−1904.00			−0.630155
$210$		0			0
$211$	2692.00			0.878317			0.439159	−	0.898410i	$-0.355277\pi$
$211$	2692.00			0.878317			0.439159	+	0.898410i	$0.355277\pi$
$212$		0			0
$213$			2796.00i			0.899431i
$214$		0			0
$215$	4356.00	+	792.000i	1.38175	+	0.251228i
$216$		0			0
$217$			720.000i			0.225239i
$218$		0			0
$219$	1644.00			0.507266
$220$		0			0
$221$	−1476.00			−0.449260
$222$		0			0
$223$			846.000i			0.254046i	0.991900	+	0.127023i	$0.0405423\pi$
$223$			846.000i			0.254046i	−0.991900	+	0.127023i	$0.959458\pi$
$224$		0			0
$225$	1053.00	+	396.000i	0.312000	+	0.117333i
$226$		0			0
$227$			2884.00i			0.843250i	0.906770	+	0.421625i	$0.138540\pi$
$227$			2884.00i			0.843250i	−0.906770	+	0.421625i	$0.861460\pi$
$228$		0			0
$229$	−3150.00			−0.908986			−0.454493	−	0.890750i	$-0.650180\pi$
$229$	−3150.00			−0.908986			−0.454493	+	0.890750i	$0.650180\pi$
$230$		0			0
$231$	−420.000			−0.119628
$232$		0			0
$233$			4014.00i			1.12861i	0.825567	+	0.564304i	$0.190856\pi$
$233$			4014.00i			1.12861i	−0.825567	+	0.564304i	$0.809144\pi$
$234$		0			0
$235$	1584.00	+	288.000i	0.439697	+	0.0799449i
$236$		0			0
$237$		−	1536.00i		−	0.420987i
$238$		0			0
$239$	−4900.00			−1.32617			−0.663085	−	0.748544i	$-0.730753\pi$
$239$	−4900.00			−1.32617			−0.663085	+	0.748544i	$0.730753\pi$
$240$		0			0
$241$	−2314.00			−0.618497			−0.309249	−	0.950981i	$-0.600078\pi$
$241$	−2314.00			−0.618497			−0.309249	+	0.950981i	$0.600078\pi$
$242$		0			0
$243$		−	243.000i		−	0.0641500i
$244$		0			0
$245$	−486.000	+	2673.00i	−0.126732	+	0.697027i
$246$		0			0
$247$			11152.0i			2.87281i
$248$		0			0
$249$	−852.000			−0.216841
$250$		0			0
$251$	−2002.00			−0.503447			−0.251723	−	0.967799i	$-0.580997\pi$
$251$	−2002.00			−0.503447			−0.251723	+	0.967799i	$0.580997\pi$
$252$		0			0
$253$		−	1960.00i		−	0.487052i
$254$		0			0
$255$	−108.000	+	594.000i	−0.0265224	+	0.145873i
$256$		0			0
$257$		−	450.000i		−	0.109223i	−0.998508	−	0.0546113i	$-0.982608\pi$
$257$		−	450.000i		−	0.109223i	0.998508	−	0.0546113i	$-0.0173920\pi$
$258$		0			0
$259$	−260.000			−0.0623769
$260$		0			0
$261$	1008.00			0.239056
$262$		0			0
$263$		−	180.000i		−	0.0422026i	−0.999777	−	0.0211013i	$-0.993283\pi$
$263$		−	180.000i		−	0.0422026i	0.999777	−	0.0211013i	$-0.00671725\pi$
$264$		0			0
$265$	1738.00	+	316.000i	0.402885	+	0.0732518i
$266$		0			0
$267$		−	2430.00i		−	0.556980i
$268$		0			0
$269$	−2448.00			−0.554859			−0.277430	−	0.960746i	$-0.589483\pi$
$269$	−2448.00			−0.554859			−0.277430	+	0.960746i	$0.589483\pi$
$270$		0			0
$271$	6776.00			1.51887			0.759433	−	0.650586i	$-0.225476\pi$
$271$	6776.00			1.51887			0.759433	+	0.650586i	$0.225476\pi$
$272$		0			0
$273$			2460.00i			0.545370i
$274$		0			0
$275$	1638.00	+	616.000i	0.359182	+	0.135077i
$276$		0			0
$277$			6426.00i			1.39387i	0.717136	+	0.696933i	$0.245453\pi$
$277$			6426.00i			1.39387i	−0.717136	+	0.696933i	$0.754547\pi$
$278$		0			0
$279$	−648.000			−0.139049
$280$		0			0
$281$	2718.00			0.577019			0.288509	−	0.957477i	$-0.406840\pi$
$281$	2718.00			0.577019			0.288509	+	0.957477i	$0.406840\pi$
$282$		0			0
$283$		−	6048.00i		−	1.27038i	−0.772358	−	0.635188i	$-0.780923\pi$
$283$		−	6048.00i		−	1.27038i	0.772358	−	0.635188i	$-0.219077\pi$
$284$		0			0
$285$	4488.00	+	816.000i	0.932794	+	0.169599i
$286$		0			0
$287$		−	4460.00i		−	0.917301i
$288$		0			0
$289$	4589.00			0.934053
$290$		0			0
$291$	3912.00			0.788060
$292$		0			0
$293$		−	1246.00i		−	0.248437i	−0.992255	−	0.124219i	$-0.960358\pi$
$293$		−	1246.00i		−	0.248437i	0.992255	−	0.124219i	$-0.0396424\pi$
$294$		0			0
$295$	−684.000	+	3762.00i	−0.134997	+	0.742482i
$296$		0			0
$297$		−	378.000i		−	0.0738511i
$298$		0			0
$299$	−11480.0			−2.22042
$300$		0			0
$301$	3960.00			0.758308
$302$		0			0
$303$		−	2808.00i		−	0.532394i
$304$		0			0
$305$	−628.000	+	3454.00i	−0.117899	+	0.648444i
$306$		0			0
$307$			1244.00i			0.231267i	0.993292	+	0.115633i	$0.0368897\pi$
$307$			1244.00i			0.231267i	−0.993292	+	0.115633i	$0.963110\pi$
$308$		0			0
$309$	−4350.00			−0.800851
$310$		0			0
$311$	−6372.00			−1.16181			−0.580905	−	0.813971i	$-0.697301\pi$
$311$	−6372.00			−1.16181			−0.580905	+	0.813971i	$0.697301\pi$
$312$		0			0
$313$		−	5500.00i		−	0.993222i	−0.867973	−	0.496611i	$-0.834578\pi$
$313$		−	5500.00i		−	0.993222i	0.867973	−	0.496611i	$-0.165422\pi$
$314$		0			0
$315$	990.000	+	180.000i	0.177080	+	0.0321964i
$316$		0			0
$317$			378.000i			0.0669735i	0.999439	+	0.0334867i	$0.0106612\pi$
$317$			378.000i			0.0669735i	−0.999439	+	0.0334867i	$0.989339\pi$
$318$		0			0
$319$	1568.00			0.275207
$320$		0			0
$321$	3876.00			0.673948
$322$		0			0
$323$			2448.00i			0.421704i
$324$		0			0
$325$	3608.00	−	9594.00i	0.615803	−	1.63747i
$326$		0			0
$327$		−	6426.00i		−	1.08672i
$328$		0			0
$329$	1440.00			0.241306
$330$		0			0
$331$	−11888.0			−1.97409			−0.987045	−	0.160446i	$-0.948707\pi$
$331$	−11888.0			−1.97409			−0.987045	+	0.160446i	$0.948707\pi$
$332$		0			0
$333$		−	234.000i		−	0.0385079i
$334$		0			0
$335$	1672.00	+	304.000i	0.272690	+	0.0495800i
$336$		0			0
$337$		−	9116.00i		−	1.47353i	−0.676148	−	0.736766i	$-0.736352\pi$
$337$		−	9116.00i		−	1.47353i	0.676148	−	0.736766i	$-0.263648\pi$
$338$		0			0
$339$	4254.00			0.681550
$340$		0			0
$341$	−1008.00			−0.160077
$342$		0			0
$343$			5860.00i			0.922479i
$344$		0			0
$345$	−840.000	+	4620.00i	−0.131084	+	0.720964i
$346$		0			0
$347$		−	4676.00i		−	0.723403i	−0.932294	−	0.361701i	$-0.882196\pi$
$347$		−	4676.00i		−	0.723403i	0.932294	−	0.361701i	$-0.117804\pi$
$348$		0			0
$349$	−11906.0			−1.82611			−0.913057	−	0.407833i	$-0.866285\pi$
$349$	−11906.0			−1.82611			−0.913057	+	0.407833i	$0.866285\pi$
$350$		0			0
$351$	−2214.00			−0.336680
$352$		0			0
$353$			2142.00i			0.322966i	0.986875	+	0.161483i	$0.0516278\pi$
$353$			2142.00i			0.322966i	−0.986875	+	0.161483i	$0.948372\pi$
$354$		0			0
$355$	1864.00	−	10252.0i	0.278678	−	1.53273i
$356$		0			0
$357$			540.000i			0.0800555i
$358$		0			0
$359$	−9824.00			−1.44426			−0.722132	−	0.691755i	$-0.756838\pi$
$359$	−9824.00			−1.44426			−0.722132	+	0.691755i	$0.756838\pi$
$360$		0			0
$361$	11637.0			1.69660
$362$		0			0
$363$			3405.00i			0.492331i
$364$		0			0
$365$	−6028.00	−	1096.00i	−0.864438	−	0.157171i
$366$		0			0
$367$			5354.00i			0.761516i	0.924675	+	0.380758i	$0.124337\pi$
$367$			5354.00i			0.761516i	−0.924675	+	0.380758i	$0.875663\pi$
$368$		0			0
$369$	4014.00			0.566289
$370$		0			0
$371$	1580.00			0.221104
$372$		0			0
$373$		−	7694.00i		−	1.06804i	−0.845471	−	0.534022i	$-0.820680\pi$
$373$		−	7694.00i		−	1.06804i	0.845471	−	0.534022i	$-0.179320\pi$
$374$		0			0
$375$	−3597.00	−	2154.00i	−0.495329	−	0.296619i
$376$		0			0
$377$		−	9184.00i		−	1.25464i
$378$		0			0
$379$	6004.00			0.813733			0.406866	−	0.913488i	$-0.366621\pi$
$379$	6004.00			0.813733			0.406866	+	0.913488i	$0.366621\pi$
$380$		0			0
$381$	5022.00			0.675288
$382$		0			0
$383$		−	9432.00i		−	1.25836i	−0.777259	−	0.629181i	$-0.783390\pi$
$383$		−	9432.00i		−	1.25836i	0.777259	−	0.629181i	$-0.216610\pi$
$384$		0			0
$385$	1540.00	+	280.000i	0.203859	+	0.0370653i
$386$		0			0
$387$			3564.00i			0.468135i
$388$		0			0
$389$	6156.00			0.802369			0.401185	−	0.915997i	$-0.368599\pi$
$389$	6156.00			0.802369			0.401185	+	0.915997i	$0.368599\pi$
$390$		0			0
$391$	−2520.00			−0.325938
$392$		0			0
$393$		−	3402.00i		−	0.436662i
$394$		0			0
$395$	−1024.00	+	5632.00i	−0.130438	+	0.717409i
$396$		0			0
$397$			7866.00i			0.994416i	0.867631	+	0.497208i	$0.165641\pi$
$397$			7866.00i			0.994416i	−0.867631	+	0.497208i	$0.834359\pi$
$398$		0			0
$399$	4080.00			0.511918
$400$		0			0
$401$	−2074.00			−0.258281			−0.129140	−	0.991626i	$-0.541222\pi$
$401$	−2074.00			−0.258281			−0.129140	+	0.991626i	$0.541222\pi$
$402$		0			0
$403$			5904.00i			0.729775i
$404$		0			0
$405$	−162.000	+	891.000i	−0.0198762	+	0.109319i
$406$		0			0
$407$		−	364.000i		−	0.0443312i
$408$		0			0
$409$	2746.00			0.331983			0.165991	−	0.986127i	$-0.446918\pi$
$409$	2746.00			0.331983			0.165991	+	0.986127i	$0.446918\pi$
$410$		0			0
$411$	−8598.00			−1.03189
$412$		0			0
$413$			3420.00i			0.407475i
$414$		0			0
$415$	3124.00	+	568.000i	0.369521	+	0.0671856i
$416$		0			0
$417$		−	2292.00i		−	0.269160i
$418$		0			0
$419$	6426.00			0.749238			0.374619	−	0.927179i	$-0.377774\pi$
$419$	6426.00			0.749238			0.374619	+	0.927179i	$0.377774\pi$
$420$		0			0
$421$	−10610.0			−1.22827			−0.614133	−	0.789203i	$-0.710494\pi$
$421$	−10610.0			−1.22827			−0.614133	+	0.789203i	$0.710494\pi$
$422$		0			0
$423$			1296.00i			0.148969i
$424$		0			0
$425$	792.000	−	2106.00i	0.0903945	−	0.240367i
$426$		0			0
$427$			3140.00i			0.355867i
$428$		0			0
$429$	−3444.00			−0.387594
$430$		0			0
$431$	8384.00			0.936991			0.468495	−	0.883466i	$-0.344796\pi$
$431$	8384.00			0.936991			0.468495	+	0.883466i	$0.344796\pi$
$432$		0			0
$433$		−	1980.00i		−	0.219752i	−0.993945	−	0.109876i	$-0.964955\pi$
$433$		−	1980.00i		−	0.219752i	0.993945	−	0.109876i	$-0.0350454\pi$
$434$		0			0
$435$	−3696.00	−	672.000i	−0.407378	−	0.0740688i
$436$		0			0
$437$			19040.0i			2.08423i
$438$		0			0
$439$	−864.000			−0.0939327			−0.0469664	−	0.998896i	$-0.514955\pi$
$439$	−864.000			−0.0939327			−0.0469664	+	0.998896i	$0.514955\pi$
$440$		0			0
$441$	−2187.00			−0.236152
$442$		0			0
$443$			13212.0i			1.41698i	0.705722	+	0.708489i	$0.250623\pi$
$443$			13212.0i			1.41698i	−0.705722	+	0.708489i	$0.749377\pi$
$444$		0			0
$445$	−1620.00	+	8910.00i	−0.172574	+	0.949156i
$446$		0			0
$447$		−	9180.00i		−	0.971363i
$448$		0			0
$449$	16290.0			1.71219			0.856094	−	0.516820i	$-0.172884\pi$
$449$	16290.0			1.71219			0.856094	+	0.516820i	$0.172884\pi$
$450$		0			0
$451$	6244.00			0.651926
$452$		0			0
$453$			3912.00i			0.405743i
$454$		0			0
$455$	1640.00	−	9020.00i	0.168977	−	0.929372i
$456$		0			0
$457$			6336.00i			0.648546i	0.945964	+	0.324273i	$0.105120\pi$
$457$			6336.00i			0.648546i	−0.945964	+	0.324273i	$0.894880\pi$
$458$		0			0
$459$	−486.000			−0.0494217
$460$		0			0
$461$	−13716.0			−1.38572			−0.692861	−	0.721071i	$-0.743650\pi$
$461$	−13716.0			−1.38572			−0.692861	+	0.721071i	$0.743650\pi$
$462$		0			0
$463$			14626.0i			1.46809i	0.679098	+	0.734047i	$0.262371\pi$
$463$			14626.0i			1.46809i	−0.679098	+	0.734047i	$0.737629\pi$
$464$		0			0
$465$	2376.00	+	432.000i	0.236956	+	0.0430828i
$466$		0			0
$467$		−	5796.00i		−	0.574319i	−0.957883	−	0.287159i	$-0.907289\pi$
$467$		−	5796.00i		−	0.574319i	0.957883	−	0.287159i	$-0.0927109\pi$
$468$		0			0
$469$	1520.00			0.149653
$470$		0			0
$471$	4746.00			0.464298
$472$		0			0
$473$			5544.00i			0.538929i
$474$		0			0
$475$	−15912.0	−	5984.00i	−1.53704	−	0.578031i
$476$		0			0
$477$			1422.00i			0.136497i
$478$		0			0
$479$	8348.00			0.796305			0.398152	−	0.917319i	$-0.369652\pi$
$479$	8348.00			0.796305			0.398152	+	0.917319i	$0.369652\pi$
$480$		0			0
$481$	−2132.00			−0.202102
$482$		0			0
$483$			4200.00i			0.395666i
$484$		0			0
$485$	−14344.0	−	2608.00i	−1.34294	−	0.244172i
$486$		0			0
$487$		−	1898.00i		−	0.176605i	−0.996094	−	0.0883025i	$-0.971856\pi$
$487$		−	1898.00i		−	0.176605i	0.996094	−	0.0883025i	$-0.0281442\pi$
$488$		0			0
$489$	−9696.00			−0.896663
$490$		0			0
$491$	16654.0			1.53072			0.765361	−	0.643601i	$-0.222560\pi$
$491$	16654.0			1.53072			0.765361	+	0.643601i	$0.222560\pi$
$492$		0			0
$493$		−	2016.00i		−	0.184171i
$494$		0			0
$495$	−252.000	+	1386.00i	−0.0228819	+	0.125851i
$496$		0			0
$497$		−	9320.00i		−	0.841165i
$498$		0			0
$499$	10600.0			0.950944			0.475472	−	0.879731i	$-0.342277\pi$
$499$	10600.0			0.950944			0.475472	+	0.879731i	$0.342277\pi$
$500$		0			0
$501$	−8964.00			−0.799365
$502$		0			0
$503$			10048.0i			0.890692i	0.895359	+	0.445346i	$0.146919\pi$
$503$			10048.0i			0.890692i	−0.895359	+	0.445346i	$0.853081\pi$
$504$		0			0
$505$	−1872.00	+	10296.0i	−0.164956	+	0.907259i
$506$		0			0
$507$			13581.0i			1.18965i
$508$		0			0
$509$	15088.0			1.31388			0.656939	−	0.753944i	$-0.271851\pi$
$509$	15088.0			1.31388			0.656939	+	0.753944i	$0.271851\pi$
$510$		0			0
$511$	−5480.00			−0.474405
$512$		0			0
$513$			3672.00i			0.316029i
$514$		0			0
$515$	15950.0	+	2900.00i	1.36474	+	0.248135i
$516$		0			0
$517$			2016.00i			0.171496i
$518$		0			0
$519$	2754.00			0.232923
$520$		0			0
$521$	8582.00			0.721659			0.360829	−	0.932632i	$-0.382494\pi$
$521$	8582.00			0.721659			0.360829	+	0.932632i	$0.382494\pi$
$522$		0			0
$523$		−	16928.0i		−	1.41532i	−0.706556	−	0.707658i	$-0.749752\pi$
$523$		−	16928.0i		−	1.41532i	0.706556	−	0.707658i	$-0.250248\pi$
$524$		0			0
$525$	−3510.00	−	1320.00i	−0.291788	−	0.109732i
$526$		0			0
$527$			1296.00i			0.107125i
$528$		0			0
$529$	−7433.00			−0.610915
$530$		0			0
$531$	−3078.00			−0.251551
$532$		0			0
$533$		−	36572.0i		−	2.97206i
$534$		0			0
$535$	−14212.0	−	2584.00i	−1.14848	−	0.208815i
$536$		0			0
$537$		−	3714.00i		−	0.298456i
$538$		0			0
$539$	−3402.00			−0.271864
$540$		0			0
$541$	−11150.0			−0.886092			−0.443046	−	0.896499i	$-0.646102\pi$
$541$	−11150.0			−0.886092			−0.443046	+	0.896499i	$0.646102\pi$
$542$		0			0
$543$		−	4350.00i		−	0.343787i
$544$		0			0
$545$	−4284.00	+	23562.0i	−0.336709	+	1.85190i
$546$		0			0
$547$		−	19628.0i		−	1.53425i	−0.641500	−	0.767123i	$-0.721688\pi$
$547$		−	19628.0i		−	1.53425i	0.641500	−	0.767123i	$-0.278312\pi$
$548$		0			0
$549$	−2826.00			−0.219692
$550$		0			0
$551$	−15232.0			−1.17769
$552$		0			0
$553$			5120.00i			0.393715i
$554$		0			0
$555$	−156.000	+	858.000i	−0.0119312	+	0.0656218i
$556$		0			0
$557$			8694.00i			0.661358i	0.943743	+	0.330679i	$0.107278\pi$
$557$			8694.00i			0.661358i	−0.943743	+	0.330679i	$0.892722\pi$
$558$		0			0
$559$	32472.0			2.45692
$560$		0			0
$561$	−756.000			−0.0568954
$562$		0			0
$563$			828.000i			0.0619823i	0.999520	+	0.0309912i	$0.00986637\pi$
$563$			828.000i			0.0619823i	−0.999520	+	0.0309912i	$0.990134\pi$
$564$		0			0
$565$	−15598.0	−	2836.00i	−1.16144	−	0.211171i
$566$		0			0
$567$			810.000i			0.0599944i
$568$		0			0
$569$	17514.0			1.29038			0.645189	−	0.764023i	$-0.276779\pi$
$569$	17514.0			1.29038			0.645189	+	0.764023i	$0.276779\pi$
$570$		0			0
$571$	−5552.00			−0.406907			−0.203454	−	0.979085i	$-0.565217\pi$
$571$	−5552.00			−0.406907			−0.203454	+	0.979085i	$0.565217\pi$
$572$		0			0
$573$		−	14580.0i		−	1.06298i
$574$		0			0
$575$	6160.00	−	16380.0i	0.446765	−	1.18799i
$576$		0			0
$577$		−	12896.0i		−	0.930446i	−0.885193	−	0.465223i	$-0.845974\pi$
$577$		−	12896.0i		−	0.930446i	0.885193	−	0.465223i	$-0.154026\pi$
$578$		0			0
$579$	4236.00			0.304045
$580$		0			0
$581$	2840.00			0.202794
$582$		0			0
$583$			2212.00i			0.157138i
$584$		0			0
$585$	8118.00	+	1476.00i	0.573740	+	0.104316i
$586$		0			0
$587$		−	1588.00i		−	0.111659i	−0.998440	−	0.0558295i	$-0.982220\pi$
$587$		−	1588.00i		−	0.111659i	0.998440	−	0.0558295i	$-0.0177803\pi$
$588$		0			0
$589$	9792.00			0.685012
$590$		0			0
$591$	−3510.00			−0.244301
$592$		0			0
$593$		−	22262.0i		−	1.54164i	−0.637055	−	0.770819i	$-0.719848\pi$
$593$		−	22262.0i		−	1.54164i	0.637055	−	0.770819i	$-0.280152\pi$
$594$		0			0
$595$	360.000	−	1980.00i	0.0248043	−	0.136424i
$596$		0			0
$597$			6240.00i			0.427783i
$598$		0			0
$599$	−4640.00			−0.316503			−0.158251	−	0.987399i	$-0.550586\pi$
$599$	−4640.00			−0.316503			−0.158251	+	0.987399i	$0.550586\pi$
$600$		0			0
$601$	−2574.00			−0.174702			−0.0873508	−	0.996178i	$-0.527840\pi$
$601$	−2574.00			−0.174702			−0.0873508	+	0.996178i	$0.527840\pi$
$602$		0			0
$603$			1368.00i			0.0923868i
$604$		0			0
$605$	2270.00	−	12485.0i	0.152543	−	0.838987i
$606$		0			0
$607$		−	11170.0i		−	0.746913i	−0.927648	−	0.373457i	$-0.878172\pi$
$607$		−	11170.0i		−	0.746913i	0.927648	−	0.373457i	$-0.121828\pi$
$608$		0			0
$609$	−3360.00			−0.223570
$610$		0			0
$611$	11808.0			0.781834
$612$		0			0
$613$			3646.00i			0.240229i	0.992760	+	0.120115i	$0.0383262\pi$
$613$			3646.00i			0.240229i	−0.992760	+	0.120115i	$0.961674\pi$
$614$		0			0
$615$	−14718.0	−	2676.00i	−0.965020	−	0.175458i
$616$		0			0
$617$		−	7646.00i		−	0.498892i	−0.968389	−	0.249446i	$-0.919751\pi$
$617$		−	7646.00i		−	0.498892i	0.968389	−	0.249446i	$-0.0802485\pi$
$618$		0			0
$619$	−26668.0			−1.73163			−0.865814	−	0.500366i	$-0.833199\pi$
$619$	−26668.0			−1.73163			−0.865814	+	0.500366i	$0.833199\pi$
$620$		0			0
$621$	−3780.00			−0.244261
$622$		0			0
$623$			8100.00i			0.520898i
$624$		0			0
$625$	11753.0	+	10296.0i	0.752192	+	0.658944i
$626$		0			0
$627$			5712.00i			0.363820i
$628$		0			0
$629$	−468.000			−0.0296667
$630$		0			0
$631$	7712.00			0.486545			0.243272	−	0.969958i	$-0.421779\pi$
$631$	7712.00			0.486545			0.243272	+	0.969958i	$0.421779\pi$
$632$		0			0
$633$		−	8076.00i		−	0.507097i
$634$		0			0
$635$	−18414.0	−	3348.00i	−1.15077	−	0.209230i
$636$		0			0
$637$			19926.0i			1.23940i
$638$		0			0
$639$	8388.00			0.519287
$640$		0			0
$641$	−22302.0			−1.37422			−0.687111	−	0.726553i	$-0.741121\pi$
$641$	−22302.0			−1.37422			−0.687111	+	0.726553i	$0.741121\pi$
$642$		0			0
$643$		−	2232.00i		−	0.136892i	−0.997655	−	0.0684459i	$-0.978196\pi$
$643$		−	2232.00i		−	0.136892i	0.997655	−	0.0684459i	$-0.0218041\pi$
$644$		0			0
$645$	2376.00	−	13068.0i	0.145046	−	0.797755i
$646$		0			0
$647$			9464.00i			0.575067i	0.957771	+	0.287533i	$0.0928352\pi$
$647$			9464.00i			0.575067i	−0.957771	+	0.287533i	$0.907165\pi$
$648$		0			0
$649$	−4788.00			−0.289592
$650$		0			0
$651$	2160.00			0.130042
$652$		0			0
$653$			4878.00i			0.292329i	0.989260	+	0.146165i	$0.0466929\pi$
$653$			4878.00i			0.292329i	−0.989260	+	0.146165i	$0.953307\pi$
$654$		0			0
$655$	−2268.00	+	12474.0i	−0.135295	+	0.744121i
$656$		0			0
$657$		−	4932.00i		−	0.292870i
$658$		0			0
$659$	−10206.0			−0.603292			−0.301646	−	0.953420i	$-0.597536\pi$
$659$	−10206.0			−0.603292			−0.301646	+	0.953420i	$0.597536\pi$
$660$		0			0
$661$	20906.0			1.23018			0.615090	−	0.788457i	$-0.289120\pi$
$661$	20906.0			1.23018			0.615090	+	0.788457i	$0.289120\pi$
$662$		0			0
$663$			4428.00i			0.259380i
$664$		0			0
$665$	−14960.0	−	2720.00i	−0.872367	−	0.158612i
$666$		0			0
$667$		−	15680.0i		−	0.910243i
$668$		0			0
$669$	2538.00			0.146674
$670$		0			0
$671$	−4396.00			−0.252915
$672$		0			0
$673$			6812.00i			0.390168i	0.980787	+	0.195084i	$0.0624980\pi$
$673$			6812.00i			0.390168i	−0.980787	+	0.195084i	$0.937502\pi$
$674$		0			0
$675$	1188.00	−	3159.00i	0.0677424	−	0.180133i
$676$		0			0
$677$			10026.0i			0.569174i	0.958650	+	0.284587i	$0.0918564\pi$
$677$			10026.0i			0.569174i	−0.958650	+	0.284587i	$0.908144\pi$
$678$		0			0
$679$	−13040.0			−0.737009
$680$		0			0
$681$	8652.00			0.486851
$682$		0			0
$683$			21236.0i			1.18971i	0.803832	+	0.594856i	$0.202791\pi$
$683$			21236.0i			1.18971i	−0.803832	+	0.594856i	$0.797209\pi$
$684$		0			0
$685$	31526.0	+	5732.00i	1.75846	+	0.319720i
$686$		0			0
$687$			9450.00i			0.524803i
$688$		0			0
$689$	12956.0			0.716378
$690$		0			0
$691$	−11520.0			−0.634213			−0.317107	−	0.948390i	$-0.602711\pi$
$691$	−11520.0			−0.634213			−0.317107	+	0.948390i	$0.602711\pi$
$692$		0			0
$693$			1260.00i			0.0690670i
$694$		0			0
$695$	−1528.00	+	8404.00i	−0.0833962	+	0.458679i
$696$		0			0
$697$		−	8028.00i		−	0.436273i
$698$		0			0
$699$	12042.0			0.651603
$700$		0			0
$701$	−400.000			−0.0215518			−0.0107759	−	0.999942i	$-0.503430\pi$
$701$	−400.000			−0.0215518			−0.0107759	+	0.999942i	$0.503430\pi$
$702$		0			0
$703$			3536.00i			0.189705i
$704$		0			0
$705$	864.000	−	4752.00i	0.0461562	−	0.253859i
$706$		0			0
$707$			9360.00i			0.497905i
$708$		0			0
$709$	5930.00			0.314113			0.157056	−	0.987590i	$-0.449800\pi$
$709$	5930.00			0.314113			0.157056	+	0.987590i	$0.449800\pi$
$710$		0			0
$711$	−4608.00			−0.243057
$712$		0			0
$713$			10080.0i			0.529452i
$714$		0			0
$715$	12628.0	+	2296.00i	0.660504	+	0.120092i
$716$		0			0
$717$			14700.0i			0.765665i
$718$		0			0
$719$	−7160.00			−0.371381			−0.185691	−	0.982608i	$-0.559452\pi$
$719$	−7160.00			−0.371381			−0.185691	+	0.982608i	$0.559452\pi$
$720$		0			0
$721$	14500.0			0.748971
$722$		0			0
$723$			6942.00i			0.357090i
$724$		0			0
$725$	13104.0	+	4928.00i	0.671269	+	0.252443i
$726$		0			0
$727$		−	8874.00i		−	0.452708i	−0.974045	−	0.226354i	$-0.927319\pi$
$727$		−	8874.00i		−	0.452708i	0.974045	−	0.226354i	$-0.0726806\pi$
$728$		0			0
$729$	−729.000			−0.0370370
$730$		0			0
$731$	7128.00			0.360655
$732$		0			0
$733$		−	5562.00i		−	0.280269i	−0.990132	−	0.140134i	$-0.955247\pi$
$733$		−	5562.00i		−	0.280269i	0.990132	−	0.140134i	$-0.0447535\pi$
$734$		0			0
$735$	8019.00	+	1458.00i	0.402429	+	0.0731689i
$736$		0			0
$737$			2128.00i			0.106358i
$738$		0			0
$739$	12096.0			0.602109			0.301055	−	0.953607i	$-0.402661\pi$
$739$	12096.0			0.602109			0.301055	+	0.953607i	$0.402661\pi$
$740$		0			0
$741$	33456.0			1.65862
$742$		0			0
$743$		−	21312.0i		−	1.05230i	−0.850391	−	0.526152i	$-0.823634\pi$
$743$		−	21312.0i		−	1.05230i	0.850391	−	0.526152i	$-0.176366\pi$
$744$		0			0
$745$	−6120.00	+	33660.0i	−0.300966	+	1.65531i
$746$		0			0
$747$			2556.00i			0.125193i
$748$		0			0
$749$	−12920.0			−0.630289
$750$		0			0
$751$	−6832.00			−0.331962			−0.165981	−	0.986129i	$-0.553079\pi$
$751$	−6832.00			−0.331962			−0.165981	+	0.986129i	$0.553079\pi$
$752$		0			0
$753$			6006.00i			0.290665i
$754$		0			0
$755$	2608.00	−	14344.0i	0.125715	−	0.691433i
$756$		0			0
$757$		−	7174.00i		−	0.344443i	−0.985058	−	0.172222i	$-0.944905\pi$
$757$		−	7174.00i		−	0.344443i	0.985058	−	0.172222i	$-0.0550945\pi$
$758$		0			0
$759$	−5880.00			−0.281200
$760$		0			0
$761$	−15394.0			−0.733288			−0.366644	−	0.930361i	$-0.619493\pi$
$761$	−15394.0			−0.733288			−0.366644	+	0.930361i	$0.619493\pi$
$762$		0			0
$763$			21420.0i			1.01633i
$764$		0			0
$765$	1782.00	+	324.000i	0.0842201	+	0.0153127i
$766$		0			0
$767$			28044.0i			1.32022i
$768$		0			0
$769$	−9730.00			−0.456271			−0.228136	−	0.973629i	$-0.573263\pi$
$769$	−9730.00			−0.456271			−0.228136	+	0.973629i	$0.573263\pi$
$770$		0			0
$771$	−1350.00			−0.0630597
$772$		0			0
$773$			24206.0i			1.12630i	0.826355	+	0.563150i	$0.190411\pi$
$773$			24206.0i			1.12630i	−0.826355	+	0.563150i	$0.809589\pi$
$774$		0			0
$775$	−8424.00	−	3168.00i	−0.390450	−	0.146836i
$776$		0			0
$777$			780.000i			0.0360133i
$778$		0			0
$779$	−60656.0			−2.78976
$780$		0			0
$781$	13048.0			0.597816
$782$		0			0
$783$		−	3024.00i		−	0.138019i
$784$		0			0
$785$	−17402.0	−	3164.00i	−0.791215	−	0.143857i
$786$		0			0
$787$		−	31312.0i		−	1.41824i	−0.705089	−	0.709118i	$-0.749093\pi$
$787$		−	31312.0i		−	1.41824i	0.705089	−	0.709118i	$-0.250907\pi$
$788$		0			0
$789$	−540.000			−0.0243657
$790$		0			0
$791$	−14180.0			−0.637399
$792$		0			0
$793$			25748.0i			1.15301i
$794$		0			0
$795$	948.000	−	5214.00i	0.0422919	−	0.232606i
$796$		0			0
$797$			10746.0i			0.477595i	0.971069	+	0.238797i	$0.0767531\pi$
$797$			10746.0i			0.477595i	−0.971069	+	0.238797i	$0.923247\pi$
$798$		0			0
$799$	2592.00			0.114766
$800$		0			0
$801$	−7290.00			−0.321572
$802$		0			0
$803$		−	7672.00i		−	0.337159i
$804$		0			0
$805$	2800.00	−	15400.0i	0.122593	−	0.674259i
$806$		0			0
$807$			7344.00i			0.320348i
$808$		0			0
$809$	−30546.0			−1.32749			−0.663745	−	0.747959i	$-0.731034\pi$
$809$	−30546.0			−1.32749			−0.663745	+	0.747959i	$0.731034\pi$
$810$		0			0
$811$	−2628.00			−0.113787			−0.0568937	−	0.998380i	$-0.518120\pi$
$811$	−2628.00			−0.113787			−0.0568937	+	0.998380i	$0.518120\pi$
$812$		0			0
$813$		−	20328.0i		−	0.876918i
$814$		0			0
$815$	35552.0	+	6464.00i	1.52802	+	0.277821i
$816$		0			0
$817$		−	53856.0i		−	2.30622i
$818$		0			0
$819$	7380.00			0.314869
$820$		0			0
$821$	26280.0			1.11715			0.558574	−	0.829455i	$-0.311349\pi$
$821$	26280.0			1.11715			0.558574	+	0.829455i	$0.311349\pi$
$822$		0			0
$823$			26146.0i			1.10740i	0.832715	+	0.553701i	$0.186785\pi$
$823$			26146.0i			1.10740i	−0.832715	+	0.553701i	$0.813215\pi$
$824$		0			0
$825$	1848.00	−	4914.00i	0.0779868	−	0.207374i
$826$		0			0
$827$			29268.0i			1.23065i	0.788273	+	0.615325i	$0.210975\pi$
$827$			29268.0i			1.23065i	−0.788273	+	0.615325i	$0.789025\pi$
$828$		0			0
$829$	−6202.00			−0.259836			−0.129918	−	0.991525i	$-0.541471\pi$
$829$	−6202.00			−0.259836			−0.129918	+	0.991525i	$0.541471\pi$
$830$		0			0
$831$	19278.0			0.804749
$832$		0			0
$833$			4374.00i			0.181933i
$834$		0			0
$835$	32868.0	+	5976.00i	1.36221	+	0.247674i
$836$		0			0
$837$			1944.00i			0.0802801i
$838$		0			0
$839$	−4680.00			−0.192576			−0.0962882	−	0.995353i	$-0.530697\pi$
$839$	−4680.00			−0.192576			−0.0962882	+	0.995353i	$0.530697\pi$
$840$		0			0
$841$	−11845.0			−0.485670
$842$		0			0
$843$		−	8154.00i		−	0.333142i
$844$		0			0
$845$	9054.00	−	49797.0i	0.368600	−	2.02730i
$846$		0			0
$847$		−	11350.0i		−	0.460438i
$848$		0			0
$849$	−18144.0			−0.733452
$850$		0			0
$851$	−3640.00			−0.146625
$852$		0			0
$853$		−	12122.0i		−	0.486576i	−0.969954	−	0.243288i	$-0.921774\pi$
$853$		−	12122.0i		−	0.486576i	0.969954	−	0.243288i	$-0.0782260\pi$
$854$		0			0
$855$	2448.00	−	13464.0i	0.0979179	−	0.538549i
$856$		0			0
$857$		−	26006.0i		−	1.03658i	−0.855205	−	0.518289i	$-0.826569\pi$
$857$		−	26006.0i		−	1.03658i	0.855205	−	0.518289i	$-0.173431\pi$
$858$		0			0
$859$	17684.0			0.702410			0.351205	−	0.936299i	$-0.385772\pi$
$859$	17684.0			0.702410			0.351205	+	0.936299i	$0.385772\pi$
$860$		0			0
$861$	−13380.0			−0.529604
$862$		0			0
$863$		−	15084.0i		−	0.594977i	−0.954725	−	0.297489i	$-0.903851\pi$
$863$		−	15084.0i		−	0.594977i	0.954725	−	0.297489i	$-0.0961490\pi$
$864$		0			0
$865$	−10098.0	−	1836.00i	−0.396928	−	0.0721686i
$866$		0			0
$867$		−	13767.0i		−	0.539275i
$868$		0			0
$869$	−7168.00			−0.279813
$870$		0			0
$871$	12464.0			0.484875
$872$		0			0
$873$		−	11736.0i		−	0.454987i
$874$		0			0
$875$	11990.0	+	7180.00i	0.463241	+	0.277404i
$876$		0			0
$877$			17614.0i			0.678201i	0.940750	+	0.339101i	$0.110123\pi$
$877$			17614.0i			0.678201i	−0.940750	+	0.339101i	$0.889877\pi$
$878$		0			0
$879$	−3738.00			−0.143435
$880$		0			0
$881$	−3298.00			−0.126121			−0.0630604	−	0.998010i	$-0.520086\pi$
$881$	−3298.00			−0.126121			−0.0630604	+	0.998010i	$0.520086\pi$
$882$		0			0
$883$		−	9496.00i		−	0.361909i	−0.983491	−	0.180955i	$-0.942081\pi$
$883$		−	9496.00i		−	0.361909i	0.983491	−	0.180955i	$-0.0579187\pi$
$884$		0			0
$885$	11286.0	+	2052.00i	0.428672	+	0.0779404i
$886$		0			0
$887$		−	50220.0i		−	1.90104i	−0.310663	−	0.950520i	$-0.600551\pi$
$887$		−	50220.0i		−	1.90104i	0.310663	−	0.950520i	$-0.399449\pi$
$888$		0			0
$889$	−16740.0			−0.631543
$890$		0			0
$891$	−1134.00			−0.0426380
$892$		0			0
$893$		−	19584.0i		−	0.733879i
$894$		0			0
$895$	−2476.00	+	13618.0i	−0.0924732	+	0.508603i
$896$		0			0
$897$			34440.0i			1.28196i
$898$		0			0
$899$	−8064.00			−0.299165
$900$		0			0
$901$	2844.00			0.105158
$902$		0			0
$903$		−	11880.0i		−	0.437809i
$904$		0			0
$905$	−2900.00	+	15950.0i	−0.106519	+	0.585852i
$906$		0			0
$907$			20636.0i			0.755465i	0.925915	+	0.377733i	$0.123296\pi$
$907$			20636.0i			0.755465i	−0.925915	+	0.377733i	$0.876704\pi$
$908$		0			0
$909$	−8424.00			−0.307378
$910$		0			0
$911$	13680.0			0.497518			0.248759	−	0.968565i	$-0.419977\pi$
$911$	13680.0			0.497518			0.248759	+	0.968565i	$0.419977\pi$
$912$		0			0
$913$			3976.00i			0.144125i
$914$		0			0
$915$	10362.0	+	1884.00i	0.374379	+	0.0680690i
$916$		0			0
$917$			11340.0i			0.408375i
$918$		0			0
$919$	−21456.0			−0.770150			−0.385075	−	0.922885i	$-0.625824\pi$
$919$	−21456.0			−0.770150			−0.385075	+	0.922885i	$0.625824\pi$
$920$		0			0
$921$	3732.00			0.133522
$922$		0			0
$923$		−	76424.0i		−	2.72538i
$924$		0			0
$925$	1144.00	−	3042.00i	0.0406643	−	0.108130i
$926$		0			0
$927$			13050.0i			0.462371i
$928$		0			0
$929$	−16510.0			−0.583074			−0.291537	−	0.956560i	$-0.594167\pi$
$929$	−16510.0			−0.583074			−0.291537	+	0.956560i	$0.594167\pi$
$930$		0			0
$931$	33048.0			1.16338
$932$		0			0
$933$			19116.0i			0.670771i
$934$		0			0
$935$	2772.00	+	504.000i	0.0969562	+	0.0176284i
$936$		0			0
$937$		−	36296.0i		−	1.26546i	−0.774371	−	0.632731i	$-0.781934\pi$
$937$		−	36296.0i		−	1.26546i	0.774371	−	0.632731i	$-0.218066\pi$
$938$		0			0
$939$	−16500.0			−0.573437
$940$		0			0
$941$	13540.0			0.469066			0.234533	−	0.972108i	$-0.424644\pi$
$941$	13540.0			0.469066			0.234533	+	0.972108i	$0.424644\pi$
$942$		0			0
$943$		−	62440.0i		−	2.15623i
$944$		0			0
$945$	540.000	−	2970.00i	0.0185886	−	0.102237i
$946$		0			0
$947$			32940.0i			1.13031i	0.824984	+	0.565156i	$0.191184\pi$
$947$			32940.0i			1.13031i	−0.824984	+	0.565156i	$0.808816\pi$
$948$		0			0
$949$	−44936.0			−1.53708
$950$		0			0
$951$	1134.00			0.0386672
$952$		0			0
$953$			22482.0i			0.764180i	0.924125	+	0.382090i	$0.124796\pi$
$953$			22482.0i			0.764180i	−0.924125	+	0.382090i	$0.875204\pi$
$954$		0			0
$955$	−9720.00	+	53460.0i	−0.329353	+	1.81144i
$956$		0			0
$957$		−	4704.00i		−	0.158891i
$958$		0			0
$959$	28660.0			0.965047
$960$		0			0
$961$	−24607.0			−0.825988
$962$		0			0
$963$		−	11628.0i		−	0.389104i
$964$		0			0
$965$	−15532.0	−	2824.00i	−0.518127	−	0.0942049i
$966$		0			0
$967$		−	9566.00i		−	0.318120i	−0.987269	−	0.159060i	$-0.949154\pi$
$967$		−	9566.00i		−	0.318120i	0.987269	−	0.159060i	$-0.0508463\pi$
$968$		0			0
$969$	7344.00			0.243471
$970$		0			0
$971$	10062.0			0.332549			0.166274	−	0.986080i	$-0.446826\pi$
$971$	10062.0			0.332549			0.166274	+	0.986080i	$0.446826\pi$
$972$		0			0
$973$			7640.00i			0.251724i
$974$		0			0
$975$	−28782.0	−	10824.0i	−0.945397	−	0.355534i
$976$		0			0
$977$			48506.0i			1.58838i	0.607671	+	0.794189i	$0.292104\pi$
$977$			48506.0i			1.58838i	−0.607671	+	0.794189i	$0.707896\pi$
$978$		0			0
$979$	−11340.0			−0.370202
$980$		0			0
$981$	−19278.0			−0.627420
$982$		0			0
$983$			41144.0i			1.33498i	0.744617	+	0.667492i	$0.232632\pi$
$983$			41144.0i			1.33498i	−0.744617	+	0.667492i	$0.767368\pi$
$984$		0			0
$985$	12870.0	+	2340.00i	0.416317	+	0.0756940i
$986$		0			0
$987$		−	4320.00i		−	0.139318i
$988$		0			0
$989$	55440.0			1.78250
$990$		0			0
$991$	16120.0			0.516719			0.258360	−	0.966049i	$-0.416818\pi$
$991$	16120.0			0.516719			0.258360	+	0.966049i	$0.416818\pi$
$992$		0			0
$993$			35664.0i			1.13974i
$994$		0			0
$995$	4160.00	−	22880.0i	0.132544	−	0.728990i
$996$		0			0
$997$		−	36666.0i		−	1.16472i	−0.812932	−	0.582359i	$-0.802130\pi$
$997$		−	36666.0i		−	1.16472i	0.812932	−	0.582359i	$-0.197870\pi$
$998$		0			0
$999$	−702.000			−0.0222325

Twists

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

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

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +(-2.00000 + 11.0000i) q^{5} +10.0000i q^{7} -9.00000 q^{9} -14.0000 q^{11} +82.0000i q^{13} +(33.0000 + 6.00000i) q^{15} +18.0000i q^{17} +136.000 q^{19} +30.0000 q^{21} +140.000i q^{23} +(-117.000 - 44.0000i) q^{25} +27.0000i q^{27} -112.000 q^{29} +72.0000 q^{31} +42.0000i q^{33} +(-110.000 - 20.0000i) q^{35} +26.0000i q^{37} +246.000 q^{39} -446.000 q^{41} -396.000i q^{43} +(18.0000 - 99.0000i) q^{45} -144.000i q^{47} +243.000 q^{49} +54.0000 q^{51} -158.000i q^{53} +(28.0000 - 154.000i) q^{55} -408.000i q^{57} +342.000 q^{59} +314.000 q^{61} -90.0000i q^{63} +(-902.000 - 164.000i) q^{65} -152.000i q^{67} +420.000 q^{69} -932.000 q^{71} +548.000i q^{73} +(-132.000 + 351.000i) q^{75} -140.000i q^{77} +512.000 q^{79} +81.0000 q^{81} -284.000i q^{83} +(-198.000 - 36.0000i) q^{85} +336.000i q^{87} +810.000 q^{89} -820.000 q^{91} -216.000i q^{93} +(-272.000 + 1496.00i) q^{95} +1304.00i q^{97} +126.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} - 18 q^{9} - 28 q^{11} + 66 q^{15} + 272 q^{19} + 60 q^{21} - 234 q^{25} - 224 q^{29} + 144 q^{31} - 220 q^{35} + 492 q^{39} - 892 q^{41} + 36 q^{45} + 486 q^{49} + 108 q^{51} + 56 q^{55}+ \cdots + 252 q^{99}+O(q^{100}) \) 2 * q - 4 * q^5 - 18 * q^9 - 28 * q^11 + 66 * q^15 + 272 * q^19 + 60 * q^21 - 234 * q^25 - 224 * q^29 + 144 * q^31 - 220 * q^35 + 492 * q^39 - 892 * q^41 + 36 * q^45 + 486 * q^49 + 108 * q^51 + 56 * q^55 + 684 * q^59 + 628 * q^61 - 1804 * q^65 + 840 * q^69 - 1864 * q^71 - 264 * q^75 + 1024 * q^79 + 162 * q^81 - 396 * q^85 + 1620 * q^89 - 1640 * q^91 - 544 * q^95 + 252 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more