LMFDB - Embedded newform 240.5.e.b.31.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [240,5,Mod(31,240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("240.31");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	240.e (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:	$24.8087911401$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 2x^{2} + x + 1 $ x^4 - x^3 + 2*x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			31.2
Root			$-0.309017 + 0.535233i$ of defining polynomial
Character	$\chi$	$=$	240.31
Dual form			240.5.e.b.31.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.19615i q^{3} +11.1803 q^{5} +7.55292i q^{7} -27.0000 q^{9} +O(q^{10})$	$q-5.19615i q^{3} +11.1803 q^{5} +7.55292i q^{7} -27.0000 q^{9} +42.1939i q^{11} +155.082 q^{13} -58.0948i q^{15} +309.246 q^{17} +139.813i q^{19} +39.2461 q^{21} -67.3516i q^{23} +125.000 q^{25} +140.296i q^{27} -70.4257 q^{29} -636.145i q^{31} +219.246 q^{33} +84.4442i q^{35} +1006.56 q^{37} -805.830i q^{39} +2155.97 q^{41} -2020.88i q^{43} -301.869 q^{45} -1122.82i q^{47} +2343.95 q^{49} -1606.89i q^{51} +974.754 q^{53} +471.743i q^{55} +726.492 q^{57} -5387.70i q^{59} +2299.12 q^{61} -203.929i q^{63} +1733.87 q^{65} +5380.60i q^{67} -349.969 q^{69} -375.372i q^{71} +3796.70 q^{73} -649.519i q^{75} -318.687 q^{77} +2042.46i q^{79} +729.000 q^{81} -4249.60i q^{83} +3457.48 q^{85} +365.943i q^{87} -2328.29 q^{89} +1171.32i q^{91} -3305.51 q^{93} +1563.16i q^{95} +4131.31 q^{97} -1139.24i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 108 q^{9}+O(q^{10}) $ 4 * q - 108 * q^9	$ 4 q - 108 q^{9} + 352 q^{13} + 432 q^{17} - 648 q^{21} + 500 q^{25} - 2160 q^{29} + 72 q^{33} - 1072 q^{37} + 2184 q^{41} - 284 q^{49} + 4704 q^{53} + 1296 q^{57} - 1000 q^{61} + 3000 q^{65} + 5040 q^{69} - 13256 q^{73} - 5568 q^{77} + 2916 q^{81} + 9000 q^{85} - 34536 q^{89} - 14832 q^{93} + 12232 q^{97}+O(q^{100}) $ 4 * q - 108 * q^9 + 352 * q^13 + 432 * q^17 - 648 * q^21 + 500 * q^25 - 2160 * q^29 + 72 * q^33 - 1072 * q^37 + 2184 * q^41 - 284 * q^49 + 4704 * q^53 + 1296 * q^57 - 1000 * q^61 + 3000 * q^65 + 5040 * q^69 - 13256 * q^73 - 5568 * q^77 + 2916 * q^81 + 9000 * q^85 - 34536 * q^89 - 14832 * q^93 + 12232 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	− 5.19615i	− 0.577350i
$3$	− 5.19615i	− 0.577350i
$4$	0	0
$5$	11.1803	0.447214
$5$	11.1803	0.447214
$6$	0	0
$7$	7.55292i	0.154141i	0.997026	+	0.0770706i	$0.0245567\pi$
$7$	7.55292i	0.154141i	−0.997026	+	0.0770706i	$0.975443\pi$
$8$	0	0
$9$	−27.0000	−0.333333
$10$	0	0
$11$	42.1939i	0.348710i	0.984683	+	0.174355i	$0.0557841\pi$
$11$	42.1939i	0.348710i	−0.984683	+	0.174355i	$0.944216\pi$
$12$	0	0
$13$	155.082	0.917645	0.458823	−	0.888528i	$-0.348271\pi$
$13$	155.082	0.917645	0.458823	+	0.888528i	$0.348271\pi$
$14$	0	0
$15$	− 58.0948i	− 0.258199i
$16$	0	0
$17$	309.246	1.07006	0.535028	−	0.844834i	$-0.320301\pi$
$17$	309.246	1.07006	0.535028	+	0.844834i	$0.320301\pi$
$18$	0	0
$19$	139.813i	0.387295i	0.981071	+	0.193648i	$0.0620318\pi$
$19$	139.813i	0.387295i	−0.981071	+	0.193648i	$0.937968\pi$
$20$	0	0
$21$	39.2461	0.0889935
$22$	0	0
$23$	− 67.3516i	− 0.127319i	−0.997972	−	0.0636593i	$-0.979723\pi$
$23$	− 67.3516i	− 0.127319i	0.997972	−	0.0636593i	$-0.0202771\pi$
$24$	0	0
$25$	125.000	0.200000
$26$	0	0
$27$	140.296i	0.192450i
$28$	0	0
$29$	−70.4257	−0.0837405	−0.0418702	−	0.999123i	$-0.513332\pi$
$29$	−70.4257	−0.0837405	−0.0418702	+	0.999123i	$0.513332\pi$
$30$	0	0
$31$	− 636.145i	− 0.661962i	−0.943637	−	0.330981i	$-0.892620\pi$
$31$	− 636.145i	− 0.661962i	0.943637	−	0.330981i	$-0.107380\pi$
$32$	0	0
$33$	219.246	0.201328
$34$	0	0
$35$	84.4442i	0.0689340i
$36$	0	0
$37$	1006.56	0.735251	0.367626	−	0.929974i	$-0.380171\pi$
$37$	1006.56	0.735251	0.367626	+	0.929974i	$0.380171\pi$
$38$	0	0
$39$	− 805.830i	− 0.529803i
$40$	0	0
$41$	2155.97	1.28255	0.641276	−	0.767311i	$-0.278405\pi$
$41$	2155.97	1.28255	0.641276	+	0.767311i	$0.278405\pi$
$42$	0	0
$43$	− 2020.88i	− 1.09296i	−0.837473	−	0.546479i	$-0.815968\pi$
$43$	− 2020.88i	− 1.09296i	0.837473	−	0.546479i	$-0.184032\pi$
$44$	0	0
$45$	−301.869	−0.149071
$46$	0	0
$47$	− 1122.82i	− 0.508295i	−0.967165	−	0.254148i	$-0.918205\pi$
$47$	− 1122.82i	− 0.508295i	0.967165	−	0.254148i	$-0.0817949\pi$
$48$	0	0
$49$	2343.95	0.976240
$50$	0	0
$51$	− 1606.89i	− 0.617797i
$52$	0	0
$53$	974.754	0.347011	0.173505	−	0.984833i	$-0.444491\pi$
$53$	974.754	0.347011	0.173505	+	0.984833i	$0.444491\pi$
$54$	0	0
$55$	471.743i	0.155948i
$56$	0	0
$57$	726.492	0.223605
$58$	0	0
$59$	− 5387.70i	− 1.54775i	−0.633341	−	0.773873i	$-0.718317\pi$
$59$	− 5387.70i	− 1.54775i	0.633341	−	0.773873i	$-0.281683\pi$
$60$	0	0
$61$	2299.12	0.617876	0.308938	−	0.951082i	$-0.400026\pi$
$61$	2299.12	0.617876	0.308938	+	0.951082i	$0.400026\pi$
$62$	0	0
$63$	− 203.929i	− 0.0513804i
$64$	0	0
$65$	1733.87	0.410383
$66$	0	0
$67$	5380.60i	1.19862i	0.800517	+	0.599310i	$0.204558\pi$
$67$	5380.60i	1.19862i	−0.800517	+	0.599310i	$0.795442\pi$
$68$	0	0
$69$	−349.969	−0.0735074
$70$	0	0
$71$	− 375.372i	− 0.0744639i	−0.999307	−	0.0372319i	$-0.988146\pi$
$71$	− 375.372i	− 0.0744639i	0.999307	−	0.0372319i	$-0.0118540\pi$
$72$	0	0
$73$	3796.70	0.712459	0.356230	−	0.934398i	$-0.384062\pi$
$73$	3796.70	0.712459	0.356230	+	0.934398i	$0.384062\pi$
$74$	0	0
$75$	− 649.519i	− 0.115470i
$76$	0	0
$77$	−318.687	−0.0537506
$78$	0	0
$79$	2042.46i	0.327265i	0.986521	+	0.163633i	$0.0523212\pi$
$79$	2042.46i	0.327265i	−0.986521	+	0.163633i	$0.947679\pi$
$80$	0	0
$81$	729.000	0.111111
$82$	0	0
$83$	− 4249.60i	− 0.616867i	−0.951246	−	0.308434i	$-0.900195\pi$
$83$	− 4249.60i	− 0.616867i	0.951246	−	0.308434i	$-0.0998048\pi$
$84$	0	0
$85$	3457.48	0.478543
$86$	0	0
$87$	365.943i	0.0483476i
$88$	0	0
$89$	−2328.29	−0.293939	−0.146969	−	0.989141i	$-0.546952\pi$
$89$	−2328.29	−0.293939	−0.146969	+	0.989141i	$0.546952\pi$
$90$	0	0
$91$	1171.32i	0.141447i
$92$	0	0
$93$	−3305.51	−0.382184
$94$	0	0
$95$	1563.16i	0.173204i
$96$	0	0
$97$	4131.31	0.439081	0.219540	−	0.975603i	$-0.429544\pi$
$97$	4131.31	0.439081	0.219540	+	0.975603i	$0.429544\pi$
$98$	0	0
$99$	− 1139.24i	− 0.116237i
$100$	0	0
$101$	541.761	0.0531086	0.0265543	−	0.999647i	$-0.491547\pi$
$101$	541.761	0.0531086	0.0265543	+	0.999647i	$0.491547\pi$
$102$	0	0
$103$	13331.5i	1.25662i	0.777961	+	0.628312i	$0.216254\pi$
$103$	13331.5i	1.25662i	−0.777961	+	0.628312i	$0.783746\pi$
$104$	0	0
$105$	438.785	0.0397991
$106$	0	0
$107$	4262.77i	0.372327i	0.982519	+	0.186163i	$0.0596054\pi$
$107$	4262.77i	0.372327i	−0.982519	+	0.186163i	$0.940395\pi$
$108$	0	0
$109$	−10679.6	−0.898885	−0.449442	−	0.893309i	$-0.648377\pi$
$109$	−10679.6	−0.898885	−0.449442	+	0.893309i	$0.648377\pi$
$110$	0	0
$111$	− 5230.23i	− 0.424497i
$112$	0	0
$113$	−15950.0	−1.24912	−0.624559	−	0.780978i	$-0.714721\pi$
$113$	−15950.0	−1.24912	−0.624559	+	0.780978i	$0.714721\pi$
$114$	0	0
$115$	− 753.013i	− 0.0569386i
$116$	0	0
$117$	−4187.22	−0.305882
$118$	0	0
$119$	2335.71i	0.164940i
$120$	0	0
$121$	12860.7	0.878401
$122$	0	0
$123$	− 11202.7i	− 0.740481i
$124$	0	0
$125$	1397.54	0.0894427
$126$	0	0
$127$	27501.0i	1.70506i	0.522676	+	0.852531i	$0.324934\pi$
$127$	27501.0i	1.70506i	−0.522676	+	0.852531i	$0.675066\pi$
$128$	0	0
$129$	−10500.8	−0.631020
$130$	0	0
$131$	− 28249.8i	− 1.64616i	−0.567925	−	0.823080i	$-0.692254\pi$
$131$	− 28249.8i	− 1.64616i	0.567925	−	0.823080i	$-0.307746\pi$
$132$	0	0
$133$	−1056.00	−0.0596981
$134$	0	0
$135$	1568.56i	0.0860663i
$136$	0	0
$137$	−29445.5	−1.56884	−0.784418	−	0.620233i	$-0.787038\pi$
$137$	−29445.5	−1.56884	−0.784418	+	0.620233i	$0.787038\pi$
$138$	0	0
$139$	9904.62i	0.512635i	0.966593	+	0.256317i	$0.0825092\pi$
$139$	9904.62i	0.512635i	−0.966593	+	0.256317i	$0.917491\pi$
$140$	0	0
$141$	−5834.37	−0.293465
$142$	0	0
$143$	6543.52i	0.319992i
$144$	0	0
$145$	−787.384	−0.0374499
$146$	0	0
$147$	− 12179.5i	− 0.563633i
$148$	0	0
$149$	−21108.4	−0.950786	−0.475393	−	0.879774i	$-0.657694\pi$
$149$	−21108.4	−0.950786	−0.475393	+	0.879774i	$0.657694\pi$
$150$	0	0
$151$	28405.9i	1.24582i	0.782294	+	0.622909i	$0.214049\pi$
$151$	28405.9i	1.24582i	−0.782294	+	0.622909i	$0.785951\pi$
$152$	0	0
$153$	−8349.65	−0.356685
$154$	0	0
$155$	− 7112.32i	− 0.296038i
$156$	0	0
$157$	−34023.0	−1.38030	−0.690150	−	0.723666i	$-0.742455\pi$
$157$	−34023.0	−1.38030	−0.690150	+	0.723666i	$0.742455\pi$
$158$	0	0
$159$	− 5064.97i	− 0.200347i
$160$	0	0
$161$	508.701	0.0196250
$162$	0	0
$163$	13506.1i	0.508342i	0.967159	+	0.254171i	$0.0818026\pi$
$163$	13506.1i	0.508342i	−0.967159	+	0.254171i	$0.918197\pi$
$164$	0	0
$165$	2451.25	0.0900366
$166$	0	0
$167$	27727.9i	0.994225i	0.867686	+	0.497112i	$0.165606\pi$
$167$	27727.9i	0.994225i	−0.867686	+	0.497112i	$0.834394\pi$
$168$	0	0
$169$	−4510.56	−0.157927
$170$	0	0
$171$	− 3774.96i	− 0.129098i
$172$	0	0
$173$	11724.8	0.391753	0.195876	−	0.980629i	$-0.437245\pi$
$173$	11724.8	0.391753	0.195876	+	0.980629i	$0.437245\pi$
$174$	0	0
$175$	944.115i	0.0308282i
$176$	0	0
$177$	−27995.3	−0.893592
$178$	0	0
$179$	7742.02i	0.241629i	0.992675	+	0.120814i	$0.0385506\pi$
$179$	7742.02i	0.241629i	−0.992675	+	0.120814i	$0.961449\pi$
$180$	0	0
$181$	53287.5	1.62655	0.813276	−	0.581878i	$-0.197682\pi$
$181$	53287.5	1.62655	0.813276	+	0.581878i	$0.197682\pi$
$182$	0	0
$183$	− 11946.6i	− 0.356731i
$184$	0	0
$185$	11253.7	0.328814
$186$	0	0
$187$	13048.3i	0.373139i
$188$	0	0
$189$	−1059.65	−0.0296645
$190$	0	0
$191$	− 16109.4i	− 0.441584i	−0.975321	−	0.220792i	$-0.929136\pi$
$191$	− 16109.4i	− 0.441584i	0.975321	−	0.220792i	$-0.0708641\pi$
$192$	0	0
$193$	−7109.66	−0.190869	−0.0954343	−	0.995436i	$-0.530424\pi$
$193$	−7109.66	−0.190869	−0.0954343	+	0.995436i	$0.530424\pi$
$194$	0	0
$195$	− 9009.45i	− 0.236935i
$196$	0	0
$197$	−24346.0	−0.627329	−0.313664	−	0.949534i	$-0.601557\pi$
$197$	−24346.0	−0.627329	−0.313664	+	0.949534i	$0.601557\pi$
$198$	0	0
$199$	65664.6i	1.65816i	0.559133	+	0.829078i	$0.311134\pi$
$199$	65664.6i	1.65816i	−0.559133	+	0.829078i	$0.688866\pi$
$200$	0	0
$201$	27958.4	0.692023
$202$	0	0
$203$	− 531.920i	− 0.0129079i
$204$	0	0
$205$	24104.5	0.573574
$206$	0	0
$207$	1818.49i	0.0424395i
$208$	0	0
$209$	−5899.28	−0.135054
$210$	0	0
$211$	− 37519.7i	− 0.842742i	−0.906888	−	0.421371i	$-0.861549\pi$
$211$	− 37519.7i	− 0.842742i	0.906888	−	0.421371i	$-0.138451\pi$
$212$	0	0
$213$	−1950.49	−0.0429917
$214$	0	0
$215$	− 22594.1i	− 0.488786i
$216$	0	0
$217$	4804.75	0.102036
$218$	0	0
$219$	− 19728.2i	− 0.411339i
$220$	0	0
$221$	47958.5	0.981932
$222$	0	0
$223$	31119.5i	0.625782i	0.949789	+	0.312891i	$0.101298\pi$
$223$	31119.5i	0.625782i	−0.949789	+	0.312891i	$0.898702\pi$
$224$	0	0
$225$	−3375.00	−0.0666667
$226$	0	0
$227$	− 69883.2i	− 1.35619i	−0.734973	−	0.678096i	$-0.762805\pi$
$227$	− 69883.2i	− 1.35619i	0.734973	−	0.678096i	$-0.237195\pi$
$228$	0	0
$229$	41898.3	0.798961	0.399480	−	0.916742i	$-0.369191\pi$
$229$	41898.3	0.798961	0.399480	+	0.916742i	$0.369191\pi$
$230$	0	0
$231$	1655.95i	0.0310329i
$232$	0	0
$233$	−10900.7	−0.200790	−0.100395	−	0.994948i	$-0.532011\pi$
$233$	−10900.7	−0.200790	−0.100395	+	0.994948i	$0.532011\pi$
$234$	0	0
$235$	− 12553.6i	− 0.227317i
$236$	0	0
$237$	10612.9	0.188947
$238$	0	0
$239$	628.198i	0.0109977i	0.999985	+	0.00549884i	$0.00175034\pi$
$239$	628.198i	0.0109977i	−0.999985	+	0.00549884i	$0.998250\pi$
$240$	0	0
$241$	107576.	1.85217	0.926083	−	0.377319i	$-0.123154\pi$
$241$	107576.	1.85217	0.926083	+	0.377319i	$0.123154\pi$
$242$	0	0
$243$	− 3788.00i	− 0.0641500i
$244$	0	0
$245$	26206.2	0.436588
$246$	0	0
$247$	21682.6i	0.355399i
$248$	0	0
$249$	−22081.6	−0.356149
$250$	0	0
$251$	− 111242.i	− 1.76573i	−0.469631	−	0.882863i	$-0.655613\pi$
$251$	− 111242.i	− 1.76573i	0.469631	−	0.882863i	$-0.344387\pi$
$252$	0	0
$253$	2841.83	0.0443973
$254$	0	0
$255$	− 17965.6i	− 0.276287i
$256$	0	0
$257$	−66628.9	−1.00878	−0.504390	−	0.863476i	$-0.668283\pi$
$257$	−66628.9	−1.00878	−0.504390	+	0.863476i	$0.668283\pi$
$258$	0	0
$259$	7602.46i	0.113332i
$260$	0	0
$261$	1901.49	0.0279135
$262$	0	0
$263$	45734.4i	0.661198i	0.943771	+	0.330599i	$0.107251\pi$
$263$	45734.4i	0.661198i	−0.943771	+	0.330599i	$0.892749\pi$
$264$	0	0
$265$	10898.1	0.155188
$266$	0	0
$267$	12098.1i	0.169706i
$268$	0	0
$269$	−24305.3	−0.335890	−0.167945	−	0.985796i	$-0.553713\pi$
$269$	−24305.3	−0.335890	−0.167945	+	0.985796i	$0.553713\pi$
$270$	0	0
$271$	− 43275.2i	− 0.589251i	−0.955613	−	0.294626i	$-0.904805\pi$
$271$	− 43275.2i	− 0.589251i	0.955613	−	0.294626i	$-0.0951950\pi$
$272$	0	0
$273$	6086.37	0.0816644
$274$	0	0
$275$	5274.24i	0.0697420i
$276$	0	0
$277$	−81134.7	−1.05742	−0.528710	−	0.848803i	$-0.677324\pi$
$277$	−81134.7	−1.05742	−0.528710	+	0.848803i	$0.677324\pi$
$278$	0	0
$279$	17175.9i	0.220654i
$280$	0	0
$281$	143161.	1.81306	0.906529	−	0.422143i	$-0.138722\pi$
$281$	143161.	1.81306	0.906529	+	0.422143i	$0.138722\pi$
$282$	0	0
$283$	− 108367.i	− 1.35308i	−0.736405	−	0.676540i	$-0.763478\pi$
$283$	− 108367.i	− 1.35308i	0.736405	−	0.676540i	$-0.236522\pi$
$284$	0	0
$285$	8122.43	0.0999991
$286$	0	0
$287$	16283.9i	0.197694i
$288$	0	0
$289$	12112.2	0.145019
$290$	0	0
$291$	− 21466.9i	− 0.253504i
$292$	0	0
$293$	126204.	1.47007	0.735036	−	0.678029i	$-0.237165\pi$
$293$	126204.	1.47007	0.735036	+	0.678029i	$0.237165\pi$
$294$	0	0
$295$	− 60236.4i	− 0.692173i
$296$	0	0
$297$	−5919.65	−0.0671093
$298$	0	0
$299$	− 10445.0i	− 0.116833i
$300$	0	0
$301$	15263.5	0.168470
$302$	0	0
$303$	− 2815.07i	− 0.0306623i
$304$	0	0
$305$	25704.9	0.276323
$306$	0	0
$307$	23505.3i	0.249396i	0.992195	+	0.124698i	$0.0397962\pi$
$307$	23505.3i	0.249396i	−0.992195	+	0.124698i	$0.960204\pi$
$308$	0	0
$309$	69272.6	0.725512
$310$	0	0
$311$	− 13436.9i	− 0.138924i	−0.997585	−	0.0694621i	$-0.977872\pi$
$311$	− 13436.9i	− 0.138924i	0.997585	−	0.0694621i	$-0.0221283\pi$
$312$	0	0
$313$	36101.4	0.368498	0.184249	−	0.982880i	$-0.441015\pi$
$313$	36101.4	0.368498	0.184249	+	0.982880i	$0.441015\pi$
$314$	0	0
$315$	− 2279.99i	− 0.0229780i
$316$	0	0
$317$	−108921.	−1.08391	−0.541954	−	0.840408i	$-0.682315\pi$
$317$	−108921.	−1.08391	−0.541954	+	0.840408i	$0.682315\pi$
$318$	0	0
$319$	− 2971.54i	− 0.0292012i
$320$	0	0
$321$	22150.0	0.214963
$322$	0	0
$323$	43236.8i	0.414427i
$324$	0	0
$325$	19385.3	0.183529
$326$	0	0
$327$	55493.1i	0.518971i
$328$	0	0
$329$	8480.60	0.0783493
$330$	0	0
$331$	− 52575.0i	− 0.479869i	−0.970789	−	0.239935i	$-0.922874\pi$
$331$	− 52575.0i	− 0.479869i	0.970789	−	0.239935i	$-0.0771260\pi$
$332$	0	0
$333$	−27177.1	−0.245084
$334$	0	0
$335$	60157.0i	0.536039i
$336$	0	0
$337$	−41672.3	−0.366933	−0.183467	−	0.983026i	$-0.558732\pi$
$337$	−41672.3	−0.366933	−0.183467	+	0.983026i	$0.558732\pi$
$338$	0	0
$339$	82878.5i	0.721178i
$340$	0	0
$341$	26841.5	0.230833
$342$	0	0
$343$	35838.2i	0.304620i
$344$	0	0
$345$	−3912.77	−0.0328735
$346$	0	0
$347$	74320.1i	0.617230i	0.951187	+	0.308615i	$0.0998655\pi$
$347$	74320.1i	0.617230i	−0.951187	+	0.308615i	$0.900135\pi$
$348$	0	0
$349$	−218841.	−1.79671	−0.898354	−	0.439271i	$-0.855237\pi$
$349$	−218841.	−1.79671	−0.898354	+	0.439271i	$0.855237\pi$
$350$	0	0
$351$	21757.4i	0.176601i
$352$	0	0
$353$	−77609.4	−0.622823	−0.311412	−	0.950275i	$-0.600802\pi$
$353$	−77609.4	−0.622823	−0.311412	+	0.950275i	$0.600802\pi$
$354$	0	0
$355$	− 4196.79i	− 0.0333013i
$356$	0	0
$357$	12136.7	0.0952280
$358$	0	0
$359$	65657.9i	0.509446i	0.967014	+	0.254723i	$0.0819843\pi$
$359$	65657.9i	0.509446i	−0.967014	+	0.254723i	$0.918016\pi$
$360$	0	0
$361$	110773.	0.850003
$362$	0	0
$363$	− 66826.0i	− 0.507145i
$364$	0	0
$365$	42448.4	0.318622
$366$	0	0
$367$	40397.8i	0.299934i	0.988691	+	0.149967i	$0.0479167\pi$
$367$	40397.8i	0.299934i	−0.988691	+	0.149967i	$0.952083\pi$
$368$	0	0
$369$	−58211.2	−0.427517
$370$	0	0
$371$	7362.24i	0.0534887i
$372$	0	0
$373$	−164447.	−1.18197	−0.590986	−	0.806682i	$-0.701261\pi$
$373$	−164447.	−1.18197	−0.590986	+	0.806682i	$0.701261\pi$
$374$	0	0
$375$	− 7261.84i	− 0.0516398i
$376$	0	0
$377$	−10921.8	−0.0768440
$378$	0	0
$379$	− 112948.i	− 0.786322i	−0.919470	−	0.393161i	$-0.871382\pi$
$379$	− 112948.i	− 0.786322i	0.919470	−	0.393161i	$-0.128618\pi$
$380$	0	0
$381$	142899.	0.984418
$382$	0	0
$383$	202001.i	1.37707i	0.725202	+	0.688536i	$0.241746\pi$
$383$	202001.i	1.37707i	−0.725202	+	0.688536i	$0.758254\pi$
$384$	0	0
$385$	−3563.03	−0.0240380
$386$	0	0
$387$	54563.7i	0.364319i
$388$	0	0
$389$	−113658.	−0.751102	−0.375551	−	0.926802i	$-0.622547\pi$
$389$	−113658.	−0.751102	−0.375551	+	0.926802i	$0.622547\pi$
$390$	0	0
$391$	− 20828.2i	− 0.136238i
$392$	0	0
$393$	−146790.	−0.950411
$394$	0	0
$395$	22835.4i	0.146358i
$396$	0	0
$397$	−156656.	−0.993956	−0.496978	−	0.867763i	$-0.665557\pi$
$397$	−156656.	−0.993956	−0.496978	+	0.867763i	$0.665557\pi$
$398$	0	0
$399$	5487.14i	0.0344667i
$400$	0	0
$401$	−95988.5	−0.596940	−0.298470	−	0.954419i	$-0.596476\pi$
$401$	−95988.5	−0.596940	−0.298470	+	0.954419i	$0.596476\pi$
$402$	0	0
$403$	− 98654.7i	− 0.607446i
$404$	0	0
$405$	8150.47	0.0496904
$406$	0	0
$407$	42470.7i	0.256390i
$408$	0	0
$409$	−83555.3	−0.499491	−0.249745	−	0.968312i	$-0.580347\pi$
$409$	−83555.3	−0.499491	−0.249745	+	0.968312i	$0.580347\pi$
$410$	0	0
$411$	153003.i	0.905767i
$412$	0	0
$413$	40692.9	0.238571
$414$	0	0
$415$	− 47512.0i	− 0.275872i
$416$	0	0
$417$	51465.9	0.295970
$418$	0	0
$419$	210190.i	1.19725i	0.801029	+	0.598625i	$0.204286\pi$
$419$	210190.i	1.19725i	−0.801029	+	0.598625i	$0.795714\pi$
$420$	0	0
$421$	−238232.	−1.34411	−0.672055	−	0.740501i	$-0.734588\pi$
$421$	−238232.	−1.34411	−0.672055	+	0.740501i	$0.734588\pi$
$422$	0	0
$423$	30316.3i	0.169432i
$424$	0	0
$425$	38655.8	0.214011
$426$	0	0
$427$	17365.0i	0.0952402i
$428$	0	0
$429$	34001.1	0.184748
$430$	0	0
$431$	− 253122.i	− 1.36262i	−0.731994	−	0.681311i	$-0.761410\pi$
$431$	− 253122.i	− 1.36262i	0.731994	−	0.681311i	$-0.238590\pi$
$432$	0	0
$433$	−202531.	−1.08023	−0.540115	−	0.841591i	$-0.681619\pi$
$433$	−202531.	−1.08023	−0.540115	+	0.841591i	$0.681619\pi$
$434$	0	0
$435$	4091.36i	0.0216217i
$436$	0	0
$437$	9416.66	0.0493099
$438$	0	0
$439$	− 285147.i	− 1.47959i	−0.672835	−	0.739793i	$-0.734924\pi$
$439$	− 285147.i	− 1.47959i	0.672835	−	0.739793i	$-0.265076\pi$
$440$	0	0
$441$	−63286.7	−0.325413
$442$	0	0
$443$	102353.i	0.521547i	0.965400	+	0.260774i	$0.0839776\pi$
$443$	102353.i	0.521547i	−0.965400	+	0.260774i	$0.916022\pi$
$444$	0	0
$445$	−26031.1	−0.131453
$446$	0	0
$447$	109682.i	0.548937i
$448$	0	0
$449$	−97139.8	−0.481842	−0.240921	−	0.970545i	$-0.577449\pi$
$449$	−97139.8	−0.481842	−0.240921	+	0.970545i	$0.577449\pi$
$450$	0	0
$451$	90968.8i	0.447239i
$452$	0	0
$453$	147601.	0.719273
$454$	0	0
$455$	13095.8i	0.0632570i
$456$	0	0
$457$	31356.4	0.150139	0.0750695	−	0.997178i	$-0.476082\pi$
$457$	31356.4	0.150139	0.0750695	+	0.997178i	$0.476082\pi$
$458$	0	0
$459$	43386.0i	0.205932i
$460$	0	0
$461$	−139066.	−0.654365	−0.327183	−	0.944961i	$-0.606099\pi$
$461$	−139066.	−0.654365	−0.327183	+	0.944961i	$0.606099\pi$
$462$	0	0
$463$	− 384625.i	− 1.79422i	−0.441808	−	0.897109i	$-0.645663\pi$
$463$	− 384625.i	− 1.79422i	0.441808	−	0.897109i	$-0.354337\pi$
$464$	0	0
$465$	−36956.7	−0.170918
$466$	0	0
$467$	223769.i	1.02605i	0.858375	+	0.513023i	$0.171474\pi$
$467$	223769.i	1.02605i	−0.858375	+	0.513023i	$0.828526\pi$
$468$	0	0
$469$	−40639.3	−0.184757
$470$	0	0
$471$	176789.i	0.796916i
$472$	0	0
$473$	85268.9	0.381126
$474$	0	0
$475$	17476.7i	0.0774590i
$476$	0	0
$477$	−26318.4	−0.115670
$478$	0	0
$479$	− 400422.i	− 1.74521i	−0.488430	−	0.872603i	$-0.662430\pi$
$479$	− 400422.i	− 1.74521i	0.488430	−	0.872603i	$-0.337570\pi$
$480$	0	0
$481$	156099.	0.674700
$482$	0	0
$483$	− 2643.29i	− 0.0113305i
$484$	0	0
$485$	46189.5	0.196363
$486$	0	0
$487$	− 63385.2i	− 0.267258i	−0.991031	−	0.133629i	$-0.957337\pi$
$487$	− 63385.2i	− 0.267258i	0.991031	−	0.133629i	$-0.0426630\pi$
$488$	0	0
$489$	70180.0	0.293491
$490$	0	0
$491$	− 162264.i	− 0.673068i	−0.941671	−	0.336534i	$-0.890745\pi$
$491$	− 162264.i	− 0.673068i	0.941671	−	0.336534i	$-0.109255\pi$
$492$	0	0
$493$	−21778.9	−0.0896070
$494$	0	0
$495$	− 12737.0i	− 0.0519826i
$496$	0	0
$497$	2835.16	0.0114780
$498$	0	0
$499$	390434.i	1.56800i	0.620761	+	0.784000i	$0.286824\pi$
$499$	390434.i	1.56800i	−0.620761	+	0.784000i	$0.713176\pi$
$500$	0	0
$501$	144079.	0.574016
$502$	0	0
$503$	381101.i	1.50627i	0.657864	+	0.753136i	$0.271460\pi$
$503$	381101.i	1.50627i	−0.657864	+	0.753136i	$0.728540\pi$
$504$	0	0
$505$	6057.07	0.0237509
$506$	0	0
$507$	23437.6i	0.0911794i
$508$	0	0
$509$	280832.	1.08396	0.541978	−	0.840393i	$-0.317676\pi$
$509$	280832.	1.08396	0.541978	+	0.840393i	$0.317676\pi$
$510$	0	0
$511$	28676.1i	0.109819i
$512$	0	0
$513$	−19615.3	−0.0745350
$514$	0	0
$515$	149051.i	0.561979i
$516$	0	0
$517$	47376.4	0.177248
$518$	0	0
$519$	− 60923.7i	− 0.226179i
$520$	0	0
$521$	−292319.	−1.07691	−0.538457	−	0.842653i	$-0.680993\pi$
$521$	−292319.	−1.07691	−0.538457	+	0.842653i	$0.680993\pi$
$522$	0	0
$523$	− 237839.i	− 0.869520i	−0.900546	−	0.434760i	$-0.856833\pi$
$523$	− 237839.i	− 0.869520i	0.900546	−	0.434760i	$-0.143167\pi$
$524$	0	0
$525$	4905.76	0.0177987
$526$	0	0
$527$	− 196725.i	− 0.708336i
$528$	0	0
$529$	275305.	0.983790
$530$	0	0
$531$	145468.i	0.515915i
$532$	0	0
$533$	334352.	1.17693
$534$	0	0
$535$	47659.2i	0.166510i
$536$	0	0
$537$	40228.7	0.139504
$538$	0	0
$539$	98900.6i	0.340425i
$540$	0	0
$541$	−79736.8	−0.272436	−0.136218	−	0.990679i	$-0.543495\pi$
$541$	−79736.8	−0.272436	−0.136218	+	0.990679i	$0.543495\pi$
$542$	0	0
$543$	− 276890.i	− 0.939090i
$544$	0	0
$545$	−119402.	−0.401993
$546$	0	0
$547$	− 48273.6i	− 0.161337i	−0.996741	−	0.0806687i	$-0.974294\pi$
$547$	− 48273.6i	− 0.161337i	0.996741	−	0.0806687i	$-0.0257056\pi$
$548$	0	0
$549$	−62076.2	−0.205959
$550$	0	0
$551$	− 9846.47i	− 0.0324323i
$552$	0	0
$553$	−15426.6	−0.0504451
$554$	0	0
$555$	− 58475.8i	− 0.189841i
$556$	0	0
$557$	439861.	1.41777	0.708885	−	0.705325i	$-0.249199\pi$
$557$	439861.	1.41777	0.708885	+	0.705325i	$0.249199\pi$
$558$	0	0
$559$	− 313402.i	− 1.00295i
$560$	0	0
$561$	67801.0	0.215432
$562$	0	0
$563$	338737.i	1.06868i	0.845271	+	0.534338i	$0.179439\pi$
$563$	338737.i	1.06868i	−0.845271	+	0.534338i	$0.820561\pi$
$564$	0	0
$565$	−178326.	−0.558622
$566$	0	0
$567$	5506.08i	0.0171268i
$568$	0	0
$569$	−22182.9	−0.0685162	−0.0342581	−	0.999413i	$-0.510907\pi$
$569$	−22182.9	−0.0685162	−0.0342581	+	0.999413i	$0.510907\pi$
$570$	0	0
$571$	− 33430.5i	− 0.102535i	−0.998685	−	0.0512673i	$-0.983674\pi$
$571$	− 33430.5i	− 0.102535i	0.998685	−	0.0512673i	$-0.0163261\pi$
$572$	0	0
$573$	−83706.9	−0.254948
$574$	0	0
$575$	− 8418.94i	− 0.0254637i
$576$	0	0
$577$	430002.	1.29157	0.645786	−	0.763518i	$-0.276530\pi$
$577$	430002.	1.29157	0.645786	+	0.763518i	$0.276530\pi$
$578$	0	0
$579$	36942.9i	0.110198i
$580$	0	0
$581$	32096.9	0.0950847
$582$	0	0
$583$	41128.7i	0.121006i
$584$	0	0
$585$	−46814.5	−0.136794
$586$	0	0
$587$	− 578882.i	− 1.68002i	−0.542572	−	0.840009i	$-0.682549\pi$
$587$	− 578882.i	− 1.68002i	0.542572	−	0.840009i	$-0.317451\pi$
$588$	0	0
$589$	88941.7	0.256374
$590$	0	0
$591$	126506.i	0.362189i
$592$	0	0
$593$	422609.	1.20179	0.600896	−	0.799327i	$-0.294811\pi$
$593$	422609.	1.20179	0.600896	+	0.799327i	$0.294811\pi$
$594$	0	0
$595$	26114.0i	0.0737633i
$596$	0	0
$597$	341204.	0.957337
$598$	0	0
$599$	− 203657.i	− 0.567603i	−0.958883	−	0.283802i	$-0.908404\pi$
$599$	− 203657.i	− 0.567603i	0.958883	−	0.283802i	$-0.0915957\pi$
$600$	0	0
$601$	191287.	0.529585	0.264792	−	0.964305i	$-0.414697\pi$
$601$	191287.	0.529585	0.264792	+	0.964305i	$0.414697\pi$
$602$	0	0
$603$	− 145276.i	− 0.399540i
$604$	0	0
$605$	143787.	0.392833
$606$	0	0
$607$	− 409449.i	− 1.11128i	−0.831424	−	0.555638i	$-0.812474\pi$
$607$	− 409449.i	− 1.11128i	0.831424	−	0.555638i	$-0.187526\pi$
$608$	0	0
$609$	−2763.94	−0.00745235
$610$	0	0
$611$	− 174130.i	− 0.466435i
$612$	0	0
$613$	−61242.7	−0.162980	−0.0814899	−	0.996674i	$-0.525968\pi$
$613$	−61242.7	−0.162980	−0.0814899	+	0.996674i	$0.525968\pi$
$614$	0	0
$615$	− 125250.i	− 0.331153i
$616$	0	0
$617$	−73134.4	−0.192111	−0.0960554	−	0.995376i	$-0.530623\pi$
$617$	−73134.4	−0.192111	−0.0960554	+	0.995376i	$0.530623\pi$
$618$	0	0
$619$	70181.6i	0.183165i	0.995798	+	0.0915824i	$0.0291925\pi$
$619$	70181.6i	0.183165i	−0.995798	+	0.0915824i	$0.970808\pi$
$620$	0	0
$621$	9449.16	0.0245025
$622$	0	0
$623$	− 17585.4i	− 0.0453081i
$624$	0	0
$625$	15625.0	0.0400000
$626$	0	0
$627$	30653.6i	0.0779733i
$628$	0	0
$629$	311274.	0.786760
$630$	0	0
$631$	378253.i	0.950001i	0.879986	+	0.475001i	$0.157552\pi$
$631$	378253.i	0.950001i	−0.879986	+	0.475001i	$0.842448\pi$
$632$	0	0
$633$	−194958.	−0.486558
$634$	0	0
$635$	307470.i	0.762527i
$636$	0	0
$637$	363505.	0.895842
$638$	0	0
$639$	10135.1i	0.0248213i
$640$	0	0
$641$	−590384.	−1.43687	−0.718436	−	0.695593i	$-0.755142\pi$
$641$	−590384.	−1.43687	−0.718436	+	0.695593i	$0.755142\pi$
$642$	0	0
$643$	− 113645.i	− 0.274870i	−0.990511	−	0.137435i	$-0.956114\pi$
$643$	− 113645.i	− 0.274870i	0.990511	−	0.137435i	$-0.0438858\pi$
$644$	0	0
$645$	−117402.	−0.282201
$646$	0	0
$647$	− 696622.i	− 1.66413i	−0.554675	−	0.832067i	$-0.687157\pi$
$647$	− 696622.i	− 1.66413i	0.554675	−	0.832067i	$-0.312843\pi$
$648$	0	0
$649$	227328.	0.539715
$650$	0	0
$651$	− 24966.2i	− 0.0589103i
$652$	0	0
$653$	288300.	0.676111	0.338056	−	0.941126i	$-0.390231\pi$
$653$	288300.	0.676111	0.338056	+	0.941126i	$0.390231\pi$
$654$	0	0
$655$	− 315842.i	− 0.736185i
$656$	0	0
$657$	−102511.	−0.237486
$658$	0	0
$659$	135428.i	0.311844i	0.987769	+	0.155922i	$0.0498349\pi$
$659$	135428.i	0.311844i	−0.987769	+	0.155922i	$0.950165\pi$
$660$	0	0
$661$	−544657.	−1.24658	−0.623290	−	0.781991i	$-0.714204\pi$
$661$	−544657.	−1.24658	−0.623290	+	0.781991i	$0.714204\pi$
$662$	0	0
$663$	− 249200.i	− 0.566918i
$664$	0	0
$665$	−11806.4	−0.0266978
$666$	0	0
$667$	4743.28i	0.0106617i
$668$	0	0
$669$	161702.	0.361296
$670$	0	0
$671$	97008.8i	0.215460i
$672$	0	0
$673$	406407.	0.897287	0.448643	−	0.893711i	$-0.351907\pi$
$673$	406407.	0.897287	0.448643	+	0.893711i	$0.351907\pi$
$674$	0	0
$675$	17537.0i	0.0384900i
$676$	0	0
$677$	−296984.	−0.647971	−0.323985	−	0.946062i	$-0.605023\pi$
$677$	−296984.	−0.647971	−0.323985	+	0.946062i	$0.605023\pi$
$678$	0	0
$679$	31203.5i	0.0676805i
$680$	0	0
$681$	−363124.	−0.782998
$682$	0	0
$683$	171571.i	0.367792i	0.982946	+	0.183896i	$0.0588710\pi$
$683$	171571.i	0.367792i	−0.982946	+	0.183896i	$0.941129\pi$
$684$	0	0
$685$	−329210.	−0.701604
$686$	0	0
$687$	− 217710.i	− 0.461280i
$688$	0	0
$689$	151167.	0.318433
$690$	0	0
$691$	901619.i	1.88828i	0.329543	+	0.944141i	$0.393105\pi$
$691$	901619.i	1.88828i	−0.329543	+	0.944141i	$0.606895\pi$
$692$	0	0
$693$	8604.56	0.0179169
$694$	0	0
$695$	110737.i	0.229257i
$696$	0	0
$697$	666725.	1.37240
$698$	0	0
$699$	56641.6i	0.115926i
$700$	0	0
$701$	−743094.	−1.51219	−0.756097	−	0.654459i	$-0.772896\pi$
$701$	−743094.	−1.51219	−0.756097	+	0.654459i	$0.772896\pi$
$702$	0	0
$703$	140730.i	0.284759i
$704$	0	0
$705$	−65230.2	−0.131241
$706$	0	0
$707$	4091.87i	0.00818622i
$708$	0	0
$709$	38851.5	0.0772886	0.0386443	−	0.999253i	$-0.487696\pi$
$709$	38851.5	0.0772886	0.0386443	+	0.999253i	$0.487696\pi$
$710$	0	0
$711$	− 55146.5i	− 0.109088i
$712$	0	0
$713$	−42845.4	−0.0842801
$714$	0	0
$715$	73158.8i	0.143105i
$716$	0	0
$717$	3264.21	0.00634951
$718$	0	0
$719$	− 512224.i	− 0.990836i	−0.868655	−	0.495418i	$-0.835015\pi$
$719$	− 512224.i	− 0.990836i	0.868655	−	0.495418i	$-0.164985\pi$
$720$	0	0
$721$	−100692.	−0.193698
$722$	0	0
$723$	− 558980.i	− 1.06935i
$724$	0	0
$725$	−8803.22	−0.0167481
$726$	0	0
$727$	706059.i	1.33589i	0.744209	+	0.667947i	$0.232827\pi$
$727$	706059.i	1.33589i	−0.744209	+	0.667947i	$0.767173\pi$
$728$	0	0
$729$	−19683.0	−0.0370370
$730$	0	0
$731$	− 624949.i	− 1.16953i
$732$	0	0
$733$	−362969.	−0.675556	−0.337778	−	0.941226i	$-0.609675\pi$
$733$	−362969.	−0.675556	−0.337778	+	0.941226i	$0.609675\pi$
$734$	0	0
$735$	− 136171.i	− 0.252064i
$736$	0	0
$737$	−227029.	−0.417971
$738$	0	0
$739$	750600.i	1.37442i	0.726459	+	0.687210i	$0.241165\pi$
$739$	750600.i	1.37442i	−0.726459	+	0.687210i	$0.758835\pi$
$740$	0	0
$741$	112666.	0.205190
$742$	0	0
$743$	1.09620e6i	1.98570i	0.119380	+	0.992849i	$0.461909\pi$
$743$	1.09620e6i	1.98570i	−0.119380	+	0.992849i	$0.538091\pi$
$744$	0	0
$745$	−235999.	−0.425204
$746$	0	0
$747$	114739.i	0.205622i
$748$	0	0
$749$	−32196.4	−0.0573909
$750$	0	0
$751$	948570.i	1.68186i	0.541145	+	0.840929i	$0.317991\pi$
$751$	948570.i	1.68186i	−0.541145	+	0.840929i	$0.682009\pi$
$752$	0	0
$753$	−578033.	−1.01944
$754$	0	0
$755$	317588.i	0.557147i
$756$	0	0
$757$	−382901.	−0.668182	−0.334091	−	0.942541i	$-0.608429\pi$
$757$	−382901.	−0.668182	−0.334091	+	0.942541i	$0.608429\pi$
$758$	0	0
$759$	− 14766.6i	− 0.0256328i
$760$	0	0
$761$	−580800.	−1.00290	−0.501450	−	0.865187i	$-0.667200\pi$
$761$	−580800.	−1.00290	−0.501450	+	0.865187i	$0.667200\pi$
$762$	0	0
$763$	− 80662.5i	− 0.138555i
$764$	0	0
$765$	−93351.9	−0.159514
$766$	0	0
$767$	− 835536.i	− 1.42028i
$768$	0	0
$769$	911273.	1.54098	0.770488	−	0.637455i	$-0.220013\pi$
$769$	911273.	1.54098	0.770488	+	0.637455i	$0.220013\pi$
$770$	0	0
$771$	346214.i	0.582419i
$772$	0	0
$773$	−359266.	−0.601253	−0.300627	−	0.953742i	$-0.597196\pi$
$773$	−359266.	−0.601253	−0.300627	+	0.953742i	$0.597196\pi$
$774$	0	0
$775$	− 79518.2i	− 0.132392i
$776$	0	0
$777$	39503.5	0.0654325
$778$	0	0
$779$	301434.i	0.496726i
$780$	0	0
$781$	15838.4	0.0259663
$782$	0	0
$783$	− 9880.46i	− 0.0161159i
$784$	0	0
$785$	−380389.	−0.617289
$786$	0	0
$787$	356898.i	0.576228i	0.957596	+	0.288114i	$0.0930282\pi$
$787$	356898.i	0.576228i	−0.957596	+	0.288114i	$0.906972\pi$
$788$	0	0
$789$	237643.	0.381743
$790$	0	0
$791$	− 120469.i	− 0.192540i
$792$	0	0
$793$	356552.	0.566991
$794$	0	0
$795$	− 56628.1i	− 0.0895979i
$796$	0	0
$797$	706472.	1.11219	0.556094	−	0.831119i	$-0.312299\pi$
$797$	706472.	1.11219	0.556094	+	0.831119i	$0.312299\pi$
$798$	0	0
$799$	− 347229.i	− 0.543904i
$800$	0	0
$801$	62863.8	0.0979796
$802$	0	0
$803$	160198.i	0.248442i
$804$	0	0
$805$	5687.45	0.00877659
$806$	0	0
$807$	126294.i	0.193926i
$808$	0	0
$809$	−717993.	−1.09704	−0.548521	−	0.836137i	$-0.684809\pi$
$809$	−717993.	−1.09704	−0.548521	+	0.836137i	$0.684809\pi$
$810$	0	0
$811$	236072.i	0.358924i	0.983765	+	0.179462i	$0.0574357\pi$
$811$	236072.i	0.358924i	−0.983765	+	0.179462i	$0.942564\pi$
$812$	0	0
$813$	−224865.	−0.340204
$814$	0	0
$815$	151003.i	0.227337i
$816$	0	0
$817$	282546.	0.423297
$818$	0	0
$819$	− 31625.7i	− 0.0471490i
$820$	0	0
$821$	−837198.	−1.24206	−0.621029	−	0.783788i	$-0.713285\pi$
$821$	−837198.	−1.24206	−0.621029	+	0.783788i	$0.713285\pi$
$822$	0	0
$823$	607023.i	0.896201i	0.893983	+	0.448100i	$0.147899\pi$
$823$	607023.i	0.896201i	−0.893983	+	0.448100i	$0.852101\pi$
$824$	0	0
$825$	27405.8	0.0402656
$826$	0	0
$827$	− 1.03479e6i	− 1.51301i	−0.653991	−	0.756503i	$-0.726907\pi$
$827$	− 1.03479e6i	− 1.51301i	0.653991	−	0.756503i	$-0.273093\pi$
$828$	0	0
$829$	−572311.	−0.832766	−0.416383	−	0.909189i	$-0.636702\pi$
$829$	−572311.	−0.832766	−0.416383	+	0.909189i	$0.636702\pi$
$830$	0	0
$831$	421588.i	0.610501i
$832$	0	0
$833$	724858.	1.04463
$834$	0	0
$835$	310008.i	0.444631i
$836$	0	0
$837$	89248.7	0.127395
$838$	0	0
$839$	− 591684.i	− 0.840555i	−0.907396	−	0.420277i	$-0.861933\pi$
$839$	− 591684.i	− 0.840555i	0.907396	−	0.420277i	$-0.138067\pi$
$840$	0	0
$841$	−702321.	−0.992988
$842$	0	0
$843$	− 743886.i	− 1.04677i
$844$	0	0
$845$	−50429.6	−0.0706272
$846$	0	0
$847$	97135.6i	0.135398i
$848$	0	0
$849$	−563091.	−0.781202
$850$	0	0
$851$	− 67793.3i	− 0.0936112i
$852$	0	0
$853$	−773052.	−1.06245	−0.531227	−	0.847229i	$-0.678269\pi$
$853$	−773052.	−1.06245	−0.531227	+	0.847229i	$0.678269\pi$
$854$	0	0
$855$	− 42205.4i	− 0.0577345i
$856$	0	0
$857$	−247321.	−0.336744	−0.168372	−	0.985724i	$-0.553851\pi$
$857$	−247321.	−0.336744	−0.168372	+	0.985724i	$0.553851\pi$
$858$	0	0
$859$	826453.i	1.12004i	0.828481	+	0.560018i	$0.189206\pi$
$859$	826453.i	1.12004i	−0.828481	+	0.560018i	$0.810794\pi$
$860$	0	0
$861$	84613.4	0.114139
$862$	0	0
$863$	622351.i	0.835630i	0.908532	+	0.417815i	$0.137204\pi$
$863$	622351.i	0.835630i	−0.908532	+	0.417815i	$0.862796\pi$
$864$	0	0
$865$	131087.	0.175197
$866$	0	0
$867$	− 62936.6i	− 0.0837270i
$868$	0	0
$869$	−86179.5	−0.114121
$870$	0	0
$871$	834435.i	1.09991i
$872$	0	0
$873$	−111545.	−0.146360
$874$	0	0
$875$	10555.5i	0.0137868i
$876$	0	0
$877$	406609.	0.528662	0.264331	−	0.964432i	$-0.414849\pi$
$877$	406609.	0.528662	0.264331	+	0.964432i	$0.414849\pi$
$878$	0	0
$879$	− 655776.i	− 0.848746i
$880$	0	0
$881$	−734863.	−0.946791	−0.473396	−	0.880850i	$-0.656972\pi$
$881$	−734863.	−0.946791	−0.473396	+	0.880850i	$0.656972\pi$
$882$	0	0
$883$	− 1.30515e6i	− 1.67393i	−0.547253	−	0.836967i	$-0.684327\pi$
$883$	− 1.30515e6i	− 1.67393i	0.547253	−	0.836967i	$-0.315673\pi$
$884$	0	0
$885$	−312997.	−0.399626
$886$	0	0
$887$	1.33807e6i	1.70071i	0.526207	+	0.850357i	$0.323614\pi$
$887$	1.33807e6i	1.70071i	−0.526207	+	0.850357i	$0.676386\pi$
$888$	0	0
$889$	−207713.	−0.262820
$890$	0	0
$891$	30759.4i	0.0387456i
$892$	0	0
$893$	156986.	0.196860
$894$	0	0
$895$	86558.5i	0.108060i
$896$	0	0
$897$	−54273.9	−0.0674538
$898$	0	0
$899$	44801.0i	0.0554330i
$900$	0	0
$901$	301439.	0.371321
$902$	0	0
$903$	− 79311.7i	− 0.0972661i
$904$	0	0
$905$	595772.	0.727416
$906$	0	0
$907$	− 665923.i	− 0.809486i	−0.914431	−	0.404743i	$-0.867361\pi$
$907$	− 665923.i	− 0.809486i	0.914431	−	0.404743i	$-0.132639\pi$
$908$	0	0
$909$	−14627.5	−0.0177029
$910$	0	0
$911$	− 1.55452e6i	− 1.87309i	−0.350550	−	0.936544i	$-0.614005\pi$
$911$	− 1.55452e6i	− 1.87309i	0.350550	−	0.936544i	$-0.385995\pi$
$912$	0	0
$913$	179307.	0.215108
$914$	0	0
$915$	− 133567.i	− 0.159535i
$916$	0	0
$917$	213368.	0.253741
$918$	0	0
$919$	− 144382.i	− 0.170955i	−0.996340	−	0.0854774i	$-0.972758\pi$
$919$	− 144382.i	− 0.170955i	0.996340	−	0.0854774i	$-0.0272415\pi$
$920$	0	0
$921$	122137.	0.143989
$922$	0	0
$923$	− 58213.5i	− 0.0683314i
$924$	0	0
$925$	125820.	0.147050
$926$	0	0
$927$	− 359951.i	− 0.418875i
$928$	0	0
$929$	−60068.7	−0.0696012	−0.0348006	−	0.999394i	$-0.511080\pi$
$929$	−60068.7	−0.0696012	−0.0348006	+	0.999394i	$0.511080\pi$
$930$	0	0
$931$	327716.i	0.378093i
$932$	0	0
$933$	−69820.2	−0.0802080
$934$	0	0
$935$	145885.i	0.166873i
$936$	0	0
$937$	737197.	0.839662	0.419831	−	0.907602i	$-0.362089\pi$
$937$	737197.	0.839662	0.419831	+	0.907602i	$0.362089\pi$
$938$	0	0
$939$	− 187588.i	− 0.212753i
$940$	0	0
$941$	−111692.	−0.126137	−0.0630686	−	0.998009i	$-0.520089\pi$
$941$	−111692.	−0.126137	−0.0630686	+	0.998009i	$0.520089\pi$
$942$	0	0
$943$	− 145208.i	− 0.163293i
$944$	0	0
$945$	−11847.2	−0.0132664
$946$	0	0
$947$	− 281397.i	− 0.313775i	−0.987616	−	0.156888i	$-0.949854\pi$
$947$	− 281397.i	− 0.313775i	0.987616	−	0.156888i	$-0.0501461\pi$
$948$	0	0
$949$	588799.	0.653785
$950$	0	0
$951$	565969.i	0.625795i
$952$	0	0
$953$	903510.	0.994826	0.497413	−	0.867514i	$-0.334283\pi$
$953$	903510.	0.994826	0.497413	+	0.867514i	$0.334283\pi$
$954$	0	0
$955$	− 180109.i	− 0.197482i
$956$	0	0
$957$	−15440.6	−0.0168593
$958$	0	0
$959$	− 222399.i	− 0.241822i
$960$	0	0
$961$	518840.	0.561807
$962$	0	0
$963$	− 115095.i	− 0.124109i
$964$	0	0
$965$	−79488.4	−0.0853590
$966$	0	0
$967$	932369.i	0.997091i	0.866863	+	0.498546i	$0.166132\pi$
$967$	932369.i	0.997091i	−0.866863	+	0.498546i	$0.833868\pi$
$968$	0	0
$969$	224665.	0.239270
$970$	0	0
$971$	− 1.24612e6i	− 1.32166i	−0.750535	−	0.660831i	$-0.770204\pi$
$971$	− 1.24612e6i	− 1.32166i	0.750535	−	0.660831i	$-0.229796\pi$
$972$	0	0
$973$	−74808.8	−0.0790181
$974$	0	0
$975$	− 100729.i	− 0.105961i
$976$	0	0
$977$	−190362.	−0.199431	−0.0997153	−	0.995016i	$-0.531793\pi$
$977$	−190362.	−0.199431	−0.0997153	+	0.995016i	$0.531793\pi$
$978$	0	0
$979$	− 98239.6i	− 0.102499i
$980$	0	0
$981$	288351.	0.299628
$982$	0	0
$983$	1.31977e6i	1.36581i	0.730506	+	0.682907i	$0.239284\pi$
$983$	1.31977e6i	1.36581i	−0.730506	+	0.682907i	$0.760716\pi$
$984$	0	0
$985$	−272197.	−0.280550
$986$	0	0
$987$	− 44066.5i	− 0.0452350i
$988$	0	0
$989$	−136109.	−0.139154
$990$	0	0
$991$	− 113211.i	− 0.115277i	−0.998338	−	0.0576385i	$-0.981643\pi$
$991$	− 113211.i	− 0.115277i	0.998338	−	0.0576385i	$-0.0183571\pi$
$992$	0	0
$993$	−273188.	−0.277053
$994$	0	0
$995$	734153.i	0.741550i
$996$	0	0
$997$	−28812.1	−0.0289858	−0.0144929	−	0.999895i	$-0.504613\pi$
$997$	−28812.1	−0.0289858	−0.0144929	+	0.999895i	$0.504613\pi$
$998$	0	0
$999$	141216.i	0.141499i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.5.e.b.31.2	✓	4
3.2	odd	2		720.5.e.e.271.2		4
4.3	odd	2	inner	240.5.e.b.31.4	yes	4
5.2	odd	4		1200.5.j.c.799.2		8
5.3	odd	4		1200.5.j.c.799.8		8
5.4	even	2		1200.5.e.c.751.3		4
8.3	odd	2		960.5.e.a.511.1		4
8.5	even	2		960.5.e.a.511.3		4
12.11	even	2		720.5.e.e.271.1		4
20.3	even	4		1200.5.j.c.799.1		8
20.7	even	4		1200.5.j.c.799.7		8
20.19	odd	2		1200.5.e.c.751.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.5.e.b.31.2	✓	4	1.1	even	1	trivial
240.5.e.b.31.4	yes	4	4.3	odd	2	inner
720.5.e.e.271.1		4	12.11	even	2
720.5.e.e.271.2		4	3.2	odd	2
960.5.e.a.511.1		4	8.3	odd	2
960.5.e.a.511.3		4	8.5	even	2
1200.5.e.c.751.2		4	20.19	odd	2
1200.5.e.c.751.3		4	5.4	even	2
1200.5.j.c.799.1		8	20.3	even	4
1200.5.j.c.799.2		8	5.2	odd	4
1200.5.j.c.799.7		8	20.7	even	4
1200.5.j.c.799.8		8	5.3	odd	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:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	240.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.19615i q^{3} +11.1803 q^{5} +7.55292i q^{7} -27.0000 q^{9} +O(q^{10})\)	\(q-5.19615i q^{3} +11.1803 q^{5} +7.55292i q^{7} -27.0000 q^{9} +42.1939i q^{11} +155.082 q^{13} -58.0948i q^{15} +309.246 q^{17} +139.813i q^{19} +39.2461 q^{21} -67.3516i q^{23} +125.000 q^{25} +140.296i q^{27} -70.4257 q^{29} -636.145i q^{31} +219.246 q^{33} +84.4442i q^{35} +1006.56 q^{37} -805.830i q^{39} +2155.97 q^{41} -2020.88i q^{43} -301.869 q^{45} -1122.82i q^{47} +2343.95 q^{49} -1606.89i q^{51} +974.754 q^{53} +471.743i q^{55} +726.492 q^{57} -5387.70i q^{59} +2299.12 q^{61} -203.929i q^{63} +1733.87 q^{65} +5380.60i q^{67} -349.969 q^{69} -375.372i q^{71} +3796.70 q^{73} -649.519i q^{75} -318.687 q^{77} +2042.46i q^{79} +729.000 q^{81} -4249.60i q^{83} +3457.48 q^{85} +365.943i q^{87} -2328.29 q^{89} +1171.32i q^{91} -3305.51 q^{93} +1563.16i q^{95} +4131.31 q^{97} -1139.24i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 108 q^{9}+O(q^{10}) \) 4 * q - 108 * q^9	\( 4 q - 108 q^{9} + 352 q^{13} + 432 q^{17} - 648 q^{21} + 500 q^{25} - 2160 q^{29} + 72 q^{33} - 1072 q^{37} + 2184 q^{41} - 284 q^{49} + 4704 q^{53} + 1296 q^{57} - 1000 q^{61} + 3000 q^{65} + 5040 q^{69} - 13256 q^{73} - 5568 q^{77} + 2916 q^{81} + 9000 q^{85} - 34536 q^{89} - 14832 q^{93} + 12232 q^{97}+O(q^{100}) \) 4 * q - 108 * q^9 + 352 * q^13 + 432 * q^17 - 648 * q^21 + 500 * q^25 - 2160 * q^29 + 72 * q^33 - 1072 * q^37 + 2184 * q^41 - 284 * q^49 + 4704 * q^53 + 1296 * q^57 - 1000 * q^61 + 3000 * q^65 + 5040 * q^69 - 13256 * q^73 - 5568 * q^77 + 2916 * q^81 + 9000 * q^85 - 34536 * q^89 - 14832 * q^93 + 12232 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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