LMFDB - Embedded newform 480.4.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [480,4,Mod(1,480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(480, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("480.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 480 = 2^{5} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	480.a (trivial)

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:	yes
Analytic conductor:	$28.3209168028$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.70156$ of defining polynomial
Character	$\chi$	$=$	480.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -5.00000 q^{5} +6.80625 q^{7} +9.00000 q^{9} -39.2250 q^{11} +78.4187 q^{13} +15.0000 q^{15} -95.2562 q^{17} +133.256 q^{19} -20.4187 q^{21} -66.8062 q^{23} +25.0000 q^{25} -27.0000 q^{27} +99.6750 q^{29} -322.031 q^{31} +117.675 q^{33} -34.0312 q^{35} +108.744 q^{37} -235.256 q^{39} +278.187 q^{41} -381.675 q^{43} -45.0000 q^{45} -211.519 q^{47} -296.675 q^{49} +285.769 q^{51} -411.862 q^{53} +196.125 q^{55} -399.769 q^{57} -447.225 q^{59} +158.837 q^{61} +61.2562 q^{63} -392.094 q^{65} -455.475 q^{67} +200.419 q^{69} -630.450 q^{71} +58.8375 q^{73} -75.0000 q^{75} -266.975 q^{77} -1257.32 q^{79} +81.0000 q^{81} +229.800 q^{83} +476.281 q^{85} -299.025 q^{87} -1178.19 q^{89} +533.737 q^{91} +966.094 q^{93} -666.281 q^{95} +1692.05 q^{97} -353.025 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} - 10 q^{5} - 12 q^{7} + 18 q^{9} + 24 q^{11} + 80 q^{13} + 30 q^{15} + 40 q^{17} + 36 q^{19} + 36 q^{21} - 108 q^{23} + 50 q^{25} - 54 q^{27} - 108 q^{29} - 516 q^{31} - 72 q^{33} + 60 q^{35}+ \cdots + 216 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 - 10 * q^5 - 12 * q^7 + 18 * q^9 + 24 * q^11 + 80 * q^13 + 30 * q^15 + 40 * q^17 + 36 * q^19 + 36 * q^21 - 108 * q^23 + 50 * q^25 - 54 * q^27 - 108 * q^29 - 516 * q^31 - 72 * q^33 + 60 * q^35 + 448 * q^37 - 240 * q^39 - 212 * q^41 - 456 * q^43 - 90 * q^45 - 756 * q^47 - 286 * q^49 - 120 * q^51 + 252 * q^53 - 120 * q^55 - 108 * q^57 - 792 * q^59 + 164 * q^61 - 108 * q^63 - 400 * q^65 + 216 * q^67 + 324 * q^69 - 1056 * q^71 - 36 * q^73 - 150 * q^75 - 1456 * q^77 - 2028 * q^79 + 162 * q^81 - 360 * q^83 - 200 * q^85 + 324 * q^87 - 1588 * q^89 + 504 * q^91 + 1548 * q^93 - 180 * q^95 + 1540 * q^97 + 216 * q^99

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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	6.80625	0.367503	0.183751	−	0.982973i	$-0.441176\pi$
$7$	6.80625	0.367503	0.183751	+	0.982973i	$0.441176\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−39.2250	−1.07516	−0.537581	−	0.843212i	$-0.680662\pi$
$11$	−39.2250	−1.07516	−0.537581	+	0.843212i	$0.680662\pi$
$12$	0	0
$13$	78.4187	1.67303	0.836517	−	0.547941i	$-0.184588\pi$
$13$	78.4187	1.67303	0.836517	+	0.547941i	$0.184588\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	−95.2562	−1.35900	−0.679501	−	0.733675i	$-0.737804\pi$
$17$	−95.2562	−1.35900	−0.679501	+	0.733675i	$0.737804\pi$
$18$	0	0
$19$	133.256	1.60900	0.804502	−	0.593950i	$-0.202432\pi$
$19$	133.256	1.60900	0.804502	+	0.593950i	$0.202432\pi$
$20$	0	0
$21$	−20.4187	−0.212178
$22$	0	0
$23$	−66.8062	−0.605655	−0.302828	−	0.953045i	$-0.597931\pi$
$23$	−66.8062	−0.605655	−0.302828	+	0.953045i	$0.597931\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	99.6750	0.638248	0.319124	−	0.947713i	$-0.396611\pi$
$29$	99.6750	0.638248	0.319124	+	0.947713i	$0.396611\pi$
$30$	0	0
$31$	−322.031	−1.86576	−0.932879	−	0.360189i	$-0.882712\pi$
$31$	−322.031	−1.86576	−0.932879	+	0.360189i	$0.882712\pi$
$32$	0	0
$33$	117.675	0.620745
$34$	0	0
$35$	−34.0312	−0.164352
$36$	0	0
$37$	108.744	0.483172	0.241586	−	0.970379i	$-0.422332\pi$
$37$	108.744	0.483172	0.241586	+	0.970379i	$0.422332\pi$
$38$	0	0
$39$	−235.256	−0.965927
$40$	0	0
$41$	278.187	1.05965	0.529824	−	0.848108i	$-0.322258\pi$
$41$	278.187	1.05965	0.529824	+	0.848108i	$0.322258\pi$
$42$	0	0
$43$	−381.675	−1.35360	−0.676801	−	0.736166i	$-0.736634\pi$
$43$	−381.675	−1.35360	−0.676801	+	0.736166i	$0.736634\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	−211.519	−0.656451	−0.328225	−	0.944599i	$-0.606451\pi$
$47$	−211.519	−0.656451	−0.328225	+	0.944599i	$0.606451\pi$
$48$	0	0
$49$	−296.675	−0.864942
$50$	0	0
$51$	285.769	0.784620
$52$	0	0
$53$	−411.862	−1.06743	−0.533714	−	0.845665i	$-0.679204\pi$
$53$	−411.862	−1.06743	−0.533714	+	0.845665i	$0.679204\pi$
$54$	0	0
$55$	196.125	0.480827
$56$	0	0
$57$	−399.769	−0.928959
$58$	0	0
$59$	−447.225	−0.986843	−0.493421	−	0.869790i	$-0.664254\pi$
$59$	−447.225	−0.986843	−0.493421	+	0.869790i	$0.664254\pi$
$60$	0	0
$61$	158.837	0.333394	0.166697	−	0.986008i	$-0.446690\pi$
$61$	158.837	0.333394	0.166697	+	0.986008i	$0.446690\pi$
$62$	0	0
$63$	61.2562	0.122501
$64$	0	0
$65$	−392.094	−0.748204
$66$	0	0
$67$	−455.475	−0.830524	−0.415262	−	0.909702i	$-0.636310\pi$
$67$	−455.475	−0.830524	−0.415262	+	0.909702i	$0.636310\pi$
$68$	0	0
$69$	200.419	0.349675
$70$	0	0
$71$	−630.450	−1.05381	−0.526906	−	0.849924i	$-0.676648\pi$
$71$	−630.450	−1.05381	−0.526906	+	0.849924i	$0.676648\pi$
$72$	0	0
$73$	58.8375	0.0943343	0.0471672	−	0.998887i	$-0.484981\pi$
$73$	58.8375	0.0943343	0.0471672	+	0.998887i	$0.484981\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	−266.975	−0.395125
$78$	0	0
$79$	−1257.32	−1.79063	−0.895313	−	0.445438i	$-0.853048\pi$
$79$	−1257.32	−1.79063	−0.895313	+	0.445438i	$0.853048\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	229.800	0.303901	0.151951	−	0.988388i	$-0.451444\pi$
$83$	229.800	0.303901	0.151951	+	0.988388i	$0.451444\pi$
$84$	0	0
$85$	476.281	0.607764
$86$	0	0
$87$	−299.025	−0.368492
$88$	0	0
$89$	−1178.19	−1.40323	−0.701616	−	0.712555i	$-0.747538\pi$
$89$	−1178.19	−1.40323	−0.701616	+	0.712555i	$0.747538\pi$
$90$	0	0
$91$	533.737	0.614845
$92$	0	0
$93$	966.094	1.07720
$94$	0	0
$95$	−666.281	−0.719568
$96$	0	0
$97$	1692.05	1.77115	0.885575	−	0.464496i	$-0.153764\pi$
$97$	1692.05	1.77115	0.885575	+	0.464496i	$0.153764\pi$
$98$	0	0
$99$	−353.025	−0.358387
$100$	0	0
$101$	−791.675	−0.779947	−0.389973	−	0.920826i	$-0.627516\pi$
$101$	−791.675	−0.779947	−0.389973	+	0.920826i	$0.627516\pi$
$102$	0	0
$103$	−362.494	−0.346773	−0.173386	−	0.984854i	$-0.555471\pi$
$103$	−362.494	−0.346773	−0.173386	+	0.984854i	$0.555471\pi$
$104$	0	0
$105$	102.094	0.0948888
$106$	0	0
$107$	2114.36	1.91031	0.955155	−	0.296107i	$-0.0956885\pi$
$107$	2114.36	1.91031	0.955155	+	0.296107i	$0.0956885\pi$
$108$	0	0
$109$	−2149.72	−1.88905	−0.944524	−	0.328442i	$-0.893477\pi$
$109$	−2149.72	−1.88905	−0.944524	+	0.328442i	$0.893477\pi$
$110$	0	0
$111$	−326.231	−0.278959
$112$	0	0
$113$	577.119	0.480449	0.240225	−	0.970717i	$-0.422779\pi$
$113$	577.119	0.480449	0.240225	+	0.970717i	$0.422779\pi$
$114$	0	0
$115$	334.031	0.270857
$116$	0	0
$117$	705.769	0.557678
$118$	0	0
$119$	−648.338	−0.499437
$120$	0	0
$121$	207.600	0.155973
$122$	0	0
$123$	−834.562	−0.611788
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−55.5188	−0.0387913	−0.0193956	−	0.999812i	$-0.506174\pi$
$127$	−55.5188	−0.0387913	−0.0193956	+	0.999812i	$0.506174\pi$
$128$	0	0
$129$	1145.02	0.781503
$130$	0	0
$131$	−1577.06	−1.05182	−0.525911	−	0.850540i	$-0.676275\pi$
$131$	−1577.06	−1.05182	−0.525911	+	0.850540i	$0.676275\pi$
$132$	0	0
$133$	906.975	0.591314
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	−1133.12	−0.706634	−0.353317	−	0.935504i	$-0.614946\pi$
$137$	−1133.12	−0.706634	−0.353317	+	0.935504i	$0.614946\pi$
$138$	0	0
$139$	425.194	0.259457	0.129728	−	0.991550i	$-0.458589\pi$
$139$	425.194	0.259457	0.129728	+	0.991550i	$0.458589\pi$
$140$	0	0
$141$	634.556	0.379002
$142$	0	0
$143$	−3075.97	−1.79878
$144$	0	0
$145$	−498.375	−0.285433
$146$	0	0
$147$	890.025	0.499374
$148$	0	0
$149$	−136.975	−0.0753116	−0.0376558	−	0.999291i	$-0.511989\pi$
$149$	−136.975	−0.0753116	−0.0376558	+	0.999291i	$0.511989\pi$
$150$	0	0
$151$	−493.631	−0.266034	−0.133017	−	0.991114i	$-0.542466\pi$
$151$	−493.631	−0.266034	−0.133017	+	0.991114i	$0.542466\pi$
$152$	0	0
$153$	−857.306	−0.453001
$154$	0	0
$155$	1610.16	0.834393
$156$	0	0
$157$	−597.394	−0.303677	−0.151838	−	0.988405i	$-0.548519\pi$
$157$	−597.394	−0.303677	−0.151838	+	0.988405i	$0.548519\pi$
$158$	0	0
$159$	1235.59	0.616280
$160$	0	0
$161$	−454.700	−0.222580
$162$	0	0
$163$	2272.35	1.09193	0.545964	−	0.837809i	$-0.316164\pi$
$163$	2272.35	1.09193	0.545964	+	0.837809i	$0.316164\pi$
$164$	0	0
$165$	−588.375	−0.277606
$166$	0	0
$167$	2092.82	0.969744	0.484872	−	0.874585i	$-0.338866\pi$
$167$	2092.82	0.969744	0.484872	+	0.874585i	$0.338866\pi$
$168$	0	0
$169$	3952.50	1.79904
$170$	0	0
$171$	1199.31	0.536335
$172$	0	0
$173$	58.7498	0.0258189	0.0129094	−	0.999917i	$-0.495891\pi$
$173$	58.7498	0.0258189	0.0129094	+	0.999917i	$0.495891\pi$
$174$	0	0
$175$	170.156	0.0735006
$176$	0	0
$177$	1341.67	0.569754
$178$	0	0
$179$	2348.96	0.980836	0.490418	−	0.871487i	$-0.336844\pi$
$179$	2348.96	0.980836	0.490418	+	0.871487i	$0.336844\pi$
$180$	0	0
$181$	2103.12	0.863669	0.431834	−	0.901953i	$-0.357867\pi$
$181$	2103.12	0.863669	0.431834	+	0.901953i	$0.357867\pi$
$182$	0	0
$183$	−476.512	−0.192485
$184$	0	0
$185$	−543.719	−0.216081
$186$	0	0
$187$	3736.43	1.46115
$188$	0	0
$189$	−183.769	−0.0707260
$190$	0	0
$191$	3203.17	1.21347	0.606737	−	0.794903i	$-0.292478\pi$
$191$	3203.17	1.21347	0.606737	+	0.794903i	$0.292478\pi$
$192$	0	0
$193$	471.113	0.175707	0.0878534	−	0.996133i	$-0.471999\pi$
$193$	471.113	0.175707	0.0878534	+	0.996133i	$0.471999\pi$
$194$	0	0
$195$	1176.28	0.431976
$196$	0	0
$197$	−1535.21	−0.555225	−0.277613	−	0.960693i	$-0.589543\pi$
$197$	−1535.21	−0.555225	−0.277613	+	0.960693i	$0.589543\pi$
$198$	0	0
$199$	−4361.98	−1.55383	−0.776916	−	0.629604i	$-0.783217\pi$
$199$	−4361.98	−1.55383	−0.776916	+	0.629604i	$0.783217\pi$
$200$	0	0
$201$	1366.42	0.479503
$202$	0	0
$203$	678.413	0.234558
$204$	0	0
$205$	−1390.94	−0.473889
$206$	0	0
$207$	−601.256	−0.201885
$208$	0	0
$209$	−5226.98	−1.72994
$210$	0	0
$211$	385.444	0.125758	0.0628792	−	0.998021i	$-0.479972\pi$
$211$	385.444	0.125758	0.0628792	+	0.998021i	$0.479972\pi$
$212$	0	0
$213$	1891.35	0.608419
$214$	0	0
$215$	1908.37	0.605349
$216$	0	0
$217$	−2191.82	−0.685672
$218$	0	0
$219$	−176.512	−0.0544640
$220$	0	0
$221$	−7469.87	−2.27366
$222$	0	0
$223$	922.744	0.277092	0.138546	−	0.990356i	$-0.455757\pi$
$223$	922.744	0.277092	0.138546	+	0.990356i	$0.455757\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	1390.35	0.406523	0.203262	−	0.979124i	$-0.434846\pi$
$227$	1390.35	0.406523	0.203262	+	0.979124i	$0.434846\pi$
$228$	0	0
$229$	750.837	0.216667	0.108333	−	0.994115i	$-0.465449\pi$
$229$	750.837	0.216667	0.108333	+	0.994115i	$0.465449\pi$
$230$	0	0
$231$	800.925	0.228126
$232$	0	0
$233$	−4457.39	−1.25328	−0.626639	−	0.779310i	$-0.715570\pi$
$233$	−4457.39	−1.25328	−0.626639	+	0.779310i	$0.715570\pi$
$234$	0	0
$235$	1057.59	0.293574
$236$	0	0
$237$	3771.96	1.03382
$238$	0	0
$239$	−6388.69	−1.72908	−0.864539	−	0.502565i	$-0.832390\pi$
$239$	−6388.69	−1.72908	−0.864539	+	0.502565i	$0.832390\pi$
$240$	0	0
$241$	5719.02	1.52861	0.764305	−	0.644855i	$-0.223082\pi$
$241$	5719.02	1.52861	0.764305	+	0.644855i	$0.223082\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	1483.37	0.386814
$246$	0	0
$247$	10449.8	2.69192
$248$	0	0
$249$	−689.400	−0.175458
$250$	0	0
$251$	3449.44	0.867436	0.433718	−	0.901049i	$-0.357201\pi$
$251$	3449.44	0.867436	0.433718	+	0.901049i	$0.357201\pi$
$252$	0	0
$253$	2620.47	0.651177
$254$	0	0
$255$	−1428.84	−0.350893
$256$	0	0
$257$	−1012.83	−0.245831	−0.122916	−	0.992417i	$-0.539224\pi$
$257$	−1012.83	−0.245831	−0.122916	+	0.992417i	$0.539224\pi$
$258$	0	0
$259$	740.137	0.177567
$260$	0	0
$261$	897.075	0.212749
$262$	0	0
$263$	6716.31	1.57470	0.787348	−	0.616509i	$-0.211453\pi$
$263$	6716.31	1.57470	0.787348	+	0.616509i	$0.211453\pi$
$264$	0	0
$265$	2059.31	0.477368
$266$	0	0
$267$	3534.56	0.810156
$268$	0	0
$269$	1299.67	0.294582	0.147291	−	0.989093i	$-0.452945\pi$
$269$	1299.67	0.294582	0.147291	+	0.989093i	$0.452945\pi$
$270$	0	0
$271$	−1690.11	−0.378844	−0.189422	−	0.981896i	$-0.560661\pi$
$271$	−1690.11	−0.378844	−0.189422	+	0.981896i	$0.560661\pi$
$272$	0	0
$273$	−1601.21	−0.354981
$274$	0	0
$275$	−980.625	−0.215032
$276$	0	0
$277$	−6230.89	−1.35155	−0.675773	−	0.737110i	$-0.736190\pi$
$277$	−6230.89	−1.35155	−0.675773	+	0.737110i	$0.736190\pi$
$278$	0	0
$279$	−2898.28	−0.621920
$280$	0	0
$281$	−4759.76	−1.01048	−0.505238	−	0.862980i	$-0.668595\pi$
$281$	−4759.76	−1.01048	−0.505238	+	0.862980i	$0.668595\pi$
$282$	0	0
$283$	−2665.95	−0.559980	−0.279990	−	0.960003i	$-0.590331\pi$
$283$	−2665.95	−0.559980	−0.279990	+	0.960003i	$0.590331\pi$
$284$	0	0
$285$	1998.84	0.415443
$286$	0	0
$287$	1893.41	0.389424
$288$	0	0
$289$	4160.75	0.846886
$290$	0	0
$291$	−5076.15	−1.02257
$292$	0	0
$293$	517.063	0.103096	0.0515480	−	0.998671i	$-0.483584\pi$
$293$	517.063	0.103096	0.0515480	+	0.998671i	$0.483584\pi$
$294$	0	0
$295$	2236.12	0.441330
$296$	0	0
$297$	1059.07	0.206915
$298$	0	0
$299$	−5238.86	−1.01328
$300$	0	0
$301$	−2597.77	−0.497453
$302$	0	0
$303$	2375.02	0.450302
$304$	0	0
$305$	−794.187	−0.149098
$306$	0	0
$307$	123.038	0.0228734	0.0114367	−	0.999935i	$-0.496360\pi$
$307$	123.038	0.0228734	0.0114367	+	0.999935i	$0.496360\pi$
$308$	0	0
$309$	1087.48	0.200209
$310$	0	0
$311$	−699.938	−0.127620	−0.0638100	−	0.997962i	$-0.520325\pi$
$311$	−699.938	−0.127620	−0.0638100	+	0.997962i	$0.520325\pi$
$312$	0	0
$313$	1268.89	0.229143	0.114571	−	0.993415i	$-0.463450\pi$
$313$	1268.89	0.229143	0.114571	+	0.993415i	$0.463450\pi$
$314$	0	0
$315$	−306.281	−0.0547841
$316$	0	0
$317$	2507.21	0.444224	0.222112	−	0.975021i	$-0.428705\pi$
$317$	2507.21	0.444224	0.222112	+	0.975021i	$0.428705\pi$
$318$	0	0
$319$	−3909.75	−0.686219
$320$	0	0
$321$	−6343.09	−1.10292
$322$	0	0
$323$	−12693.5	−2.18664
$324$	0	0
$325$	1960.47	0.334607
$326$	0	0
$327$	6449.17	1.09064
$328$	0	0
$329$	−1439.65	−0.241248
$330$	0	0
$331$	−4639.26	−0.770382	−0.385191	−	0.922837i	$-0.625864\pi$
$331$	−4639.26	−0.770382	−0.385191	+	0.922837i	$0.625864\pi$
$332$	0	0
$333$	978.694	0.161057
$334$	0	0
$335$	2277.37	0.371422
$336$	0	0
$337$	5643.79	0.912275	0.456137	−	0.889909i	$-0.349233\pi$
$337$	5643.79	0.912275	0.456137	+	0.889909i	$0.349233\pi$
$338$	0	0
$339$	−1731.36	−0.277387
$340$	0	0
$341$	12631.7	2.00599
$342$	0	0
$343$	−4353.79	−0.685371
$344$	0	0
$345$	−1002.09	−0.156379
$346$	0	0
$347$	11753.5	1.81833	0.909167	−	0.416433i	$-0.136720\pi$
$347$	11753.5	1.81833	0.909167	+	0.416433i	$0.136720\pi$
$348$	0	0
$349$	476.812	0.0731322	0.0365661	−	0.999331i	$-0.488358\pi$
$349$	476.812	0.0731322	0.0365661	+	0.999331i	$0.488358\pi$
$350$	0	0
$351$	−2117.31	−0.321976
$352$	0	0
$353$	−11743.4	−1.77064	−0.885321	−	0.464981i	$-0.846061\pi$
$353$	−11743.4	−1.77064	−0.885321	+	0.464981i	$0.846061\pi$
$354$	0	0
$355$	3152.25	0.471279
$356$	0	0
$357$	1945.01	0.288350
$358$	0	0
$359$	−4552.87	−0.669336	−0.334668	−	0.942336i	$-0.608624\pi$
$359$	−4552.87	−0.669336	−0.334668	+	0.942336i	$0.608624\pi$
$360$	0	0
$361$	10898.2	1.58889
$362$	0	0
$363$	−622.800	−0.0900511
$364$	0	0
$365$	−294.187	−0.0421876
$366$	0	0
$367$	−792.056	−0.112657	−0.0563283	−	0.998412i	$-0.517939\pi$
$367$	−792.056	−0.112657	−0.0563283	+	0.998412i	$0.517939\pi$
$368$	0	0
$369$	2503.69	0.353216
$370$	0	0
$371$	−2803.24	−0.392283
$372$	0	0
$373$	−10232.0	−1.42036	−0.710179	−	0.704021i	$-0.751386\pi$
$373$	−10232.0	−1.42036	−0.710179	+	0.704021i	$0.751386\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	7816.39	1.06781
$378$	0	0
$379$	−2064.39	−0.279791	−0.139895	−	0.990166i	$-0.544677\pi$
$379$	−2064.39	−0.279791	−0.139895	+	0.990166i	$0.544677\pi$
$380$	0	0
$381$	166.556	0.0223962
$382$	0	0
$383$	2202.51	0.293845	0.146923	−	0.989148i	$-0.453063\pi$
$383$	2202.51	0.293845	0.146923	+	0.989148i	$0.453063\pi$
$384$	0	0
$385$	1334.88	0.176705
$386$	0	0
$387$	−3435.07	−0.451201
$388$	0	0
$389$	1898.55	0.247456	0.123728	−	0.992316i	$-0.460515\pi$
$389$	1898.55	0.247456	0.123728	+	0.992316i	$0.460515\pi$
$390$	0	0
$391$	6363.71	0.823086
$392$	0	0
$393$	4731.19	0.607269
$394$	0	0
$395$	6286.59	0.800792
$396$	0	0
$397$	−5070.61	−0.641024	−0.320512	−	0.947244i	$-0.603855\pi$
$397$	−5070.61	−0.641024	−0.320512	+	0.947244i	$0.603855\pi$
$398$	0	0
$399$	−2720.93	−0.341395
$400$	0	0
$401$	−4768.98	−0.593893	−0.296947	−	0.954894i	$-0.595968\pi$
$401$	−4768.98	−0.593893	−0.296947	+	0.954894i	$0.595968\pi$
$402$	0	0
$403$	−25253.3	−3.12148
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	−4265.47	−0.519488
$408$	0	0
$409$	−7003.60	−0.846713	−0.423357	−	0.905963i	$-0.639148\pi$
$409$	−7003.60	−0.846713	−0.423357	+	0.905963i	$0.639148\pi$
$410$	0	0
$411$	3399.36	0.407975
$412$	0	0
$413$	−3043.92	−0.362668
$414$	0	0
$415$	−1149.00	−0.135909
$416$	0	0
$417$	−1275.58	−0.149797
$418$	0	0
$419$	−6643.09	−0.774549	−0.387275	−	0.921964i	$-0.626583\pi$
$419$	−6643.09	−0.774549	−0.387275	+	0.921964i	$0.626583\pi$
$420$	0	0
$421$	−13952.5	−1.61521	−0.807603	−	0.589727i	$-0.799235\pi$
$421$	−13952.5	−1.61521	−0.807603	+	0.589727i	$0.799235\pi$
$422$	0	0
$423$	−1903.67	−0.218817
$424$	0	0
$425$	−2381.41	−0.271800
$426$	0	0
$427$	1081.09	0.122523
$428$	0	0
$429$	9227.92	1.03853
$430$	0	0
$431$	−1549.27	−0.173146	−0.0865730	−	0.996246i	$-0.527592\pi$
$431$	−1549.27	−0.173146	−0.0865730	+	0.996246i	$0.527592\pi$
$432$	0	0
$433$	7022.39	0.779387	0.389693	−	0.920945i	$-0.372581\pi$
$433$	7022.39	0.779387	0.389693	+	0.920945i	$0.372581\pi$
$434$	0	0
$435$	1495.12	0.164795
$436$	0	0
$437$	−8902.35	−0.974501
$438$	0	0
$439$	−3107.31	−0.337822	−0.168911	−	0.985631i	$-0.554025\pi$
$439$	−3107.31	−0.337822	−0.168911	+	0.985631i	$0.554025\pi$
$440$	0	0
$441$	−2670.07	−0.288314
$442$	0	0
$443$	−15463.8	−1.65848	−0.829241	−	0.558892i	$-0.811227\pi$
$443$	−15463.8	−1.65848	−0.829241	+	0.558892i	$0.811227\pi$
$444$	0	0
$445$	5890.94	0.627544
$446$	0	0
$447$	410.925	0.0434812
$448$	0	0
$449$	14043.0	1.47601	0.738005	−	0.674795i	$-0.235768\pi$
$449$	14043.0	1.47601	0.738005	+	0.674795i	$0.235768\pi$
$450$	0	0
$451$	−10911.9	−1.13929
$452$	0	0
$453$	1480.89	0.153595
$454$	0	0
$455$	−2668.69	−0.274967
$456$	0	0
$457$	−13860.5	−1.41875	−0.709375	−	0.704831i	$-0.751023\pi$
$457$	−13860.5	−1.41875	−0.709375	+	0.704831i	$0.751023\pi$
$458$	0	0
$459$	2571.92	0.261540
$460$	0	0
$461$	−7593.75	−0.767193	−0.383597	−	0.923501i	$-0.625315\pi$
$461$	−7593.75	−0.767193	−0.383597	+	0.923501i	$0.625315\pi$
$462$	0	0
$463$	6630.96	0.665587	0.332793	−	0.943000i	$-0.392009\pi$
$463$	6630.96	0.665587	0.332793	+	0.943000i	$0.392009\pi$
$464$	0	0
$465$	−4830.47	−0.481737
$466$	0	0
$467$	−6699.79	−0.663874	−0.331937	−	0.943302i	$-0.607702\pi$
$467$	−6699.79	−0.663874	−0.331937	+	0.943302i	$0.607702\pi$
$468$	0	0
$469$	−3100.08	−0.305220
$470$	0	0
$471$	1792.18	0.175328
$472$	0	0
$473$	14971.2	1.45534
$474$	0	0
$475$	3331.41	0.321801
$476$	0	0
$477$	−3706.76	−0.355809
$478$	0	0
$479$	−19984.2	−1.90627	−0.953133	−	0.302552i	$-0.902161\pi$
$479$	−19984.2	−1.90627	−0.953133	+	0.302552i	$0.902161\pi$
$480$	0	0
$481$	8527.55	0.808363
$482$	0	0
$483$	1364.10	0.128507
$484$	0	0
$485$	−8460.25	−0.792083
$486$	0	0
$487$	20569.1	1.91392	0.956958	−	0.290227i	$-0.0937309\pi$
$487$	20569.1	1.91392	0.956958	+	0.290227i	$0.0937309\pi$
$488$	0	0
$489$	−6817.05	−0.630425
$490$	0	0
$491$	−19131.8	−1.75847	−0.879233	−	0.476391i	$-0.841945\pi$
$491$	−19131.8	−1.75847	−0.879233	+	0.476391i	$0.841945\pi$
$492$	0	0
$493$	−9494.66	−0.867380
$494$	0	0
$495$	1765.12	0.160276
$496$	0	0
$497$	−4291.00	−0.387279
$498$	0	0
$499$	−16396.9	−1.47100	−0.735498	−	0.677526i	$-0.763052\pi$
$499$	−16396.9	−1.47100	−0.735498	+	0.677526i	$0.763052\pi$
$500$	0	0
$501$	−6278.46	−0.559882
$502$	0	0
$503$	7013.16	0.621672	0.310836	−	0.950464i	$-0.399391\pi$
$503$	7013.16	0.621672	0.310836	+	0.950464i	$0.399391\pi$
$504$	0	0
$505$	3958.37	0.348803
$506$	0	0
$507$	−11857.5	−1.03868
$508$	0	0
$509$	12431.8	1.08257	0.541287	−	0.840838i	$-0.317937\pi$
$509$	12431.8	1.08257	0.541287	+	0.840838i	$0.317937\pi$
$510$	0	0
$511$	400.463	0.0346681
$512$	0	0
$513$	−3597.92	−0.309653
$514$	0	0
$515$	1812.47	0.155081
$516$	0	0
$517$	8296.82	0.705791
$518$	0	0
$519$	−176.249	−0.0149065
$520$	0	0
$521$	−6707.07	−0.563997	−0.281998	−	0.959415i	$-0.590997\pi$
$521$	−6707.07	−0.563997	−0.281998	+	0.959415i	$0.590997\pi$
$522$	0	0
$523$	16645.5	1.39169	0.695846	−	0.718191i	$-0.255030\pi$
$523$	16645.5	1.39169	0.695846	+	0.718191i	$0.255030\pi$
$524$	0	0
$525$	−510.469	−0.0424356
$526$	0	0
$527$	30675.5	2.53557
$528$	0	0
$529$	−7703.93	−0.633182
$530$	0	0
$531$	−4025.02	−0.328948
$532$	0	0
$533$	21815.1	1.77283
$534$	0	0
$535$	−10571.8	−0.854316
$536$	0	0
$537$	−7046.89	−0.566286
$538$	0	0
$539$	11637.1	0.929952
$540$	0	0
$541$	15644.5	1.24327	0.621636	−	0.783306i	$-0.286468\pi$
$541$	15644.5	1.24327	0.621636	+	0.783306i	$0.286468\pi$
$542$	0	0
$543$	−6309.37	−0.498639
$544$	0	0
$545$	10748.6	0.844808
$546$	0	0
$547$	11548.0	0.902667	0.451334	−	0.892355i	$-0.350948\pi$
$547$	11548.0	0.902667	0.451334	+	0.892355i	$0.350948\pi$
$548$	0	0
$549$	1429.54	0.111131
$550$	0	0
$551$	13282.3	1.02694
$552$	0	0
$553$	−8557.62	−0.658060
$554$	0	0
$555$	1631.16	0.124754
$556$	0	0
$557$	25455.4	1.93641	0.968206	−	0.250156i	$-0.0804820\pi$
$557$	25455.4	1.93641	0.968206	+	0.250156i	$0.0804820\pi$
$558$	0	0
$559$	−29930.5	−2.26462
$560$	0	0
$561$	−11209.3	−0.843594
$562$	0	0
$563$	9573.34	0.716639	0.358320	−	0.933599i	$-0.383350\pi$
$563$	9573.34	0.716639	0.358320	+	0.933599i	$0.383350\pi$
$564$	0	0
$565$	−2885.59	−0.214863
$566$	0	0
$567$	551.306	0.0408337
$568$	0	0
$569$	13986.4	1.03048	0.515238	−	0.857047i	$-0.327704\pi$
$569$	13986.4	1.03048	0.515238	+	0.857047i	$0.327704\pi$
$570$	0	0
$571$	7881.02	0.577601	0.288801	−	0.957389i	$-0.406744\pi$
$571$	7881.02	0.577601	0.288801	+	0.957389i	$0.406744\pi$
$572$	0	0
$573$	−9609.52	−0.700600
$574$	0	0
$575$	−1670.16	−0.121131
$576$	0	0
$577$	9304.52	0.671321	0.335661	−	0.941983i	$-0.391040\pi$
$577$	9304.52	0.671321	0.335661	+	0.941983i	$0.391040\pi$
$578$	0	0
$579$	−1413.34	−0.101444
$580$	0	0
$581$	1564.08	0.111685
$582$	0	0
$583$	16155.3	1.14766
$584$	0	0
$585$	−3528.84	−0.249401
$586$	0	0
$587$	−3335.66	−0.234544	−0.117272	−	0.993100i	$-0.537415\pi$
$587$	−3335.66	−0.234544	−0.117272	+	0.993100i	$0.537415\pi$
$588$	0	0
$589$	−42912.7	−3.00201
$590$	0	0
$591$	4605.64	0.320559
$592$	0	0
$593$	−14241.4	−0.986210	−0.493105	−	0.869970i	$-0.664138\pi$
$593$	−14241.4	−0.986210	−0.493105	+	0.869970i	$0.664138\pi$
$594$	0	0
$595$	3241.69	0.223355
$596$	0	0
$597$	13085.9	0.897105
$598$	0	0
$599$	13333.6	0.909508	0.454754	−	0.890617i	$-0.349727\pi$
$599$	13333.6	0.909508	0.454754	+	0.890617i	$0.349727\pi$
$600$	0	0
$601$	13080.5	0.887795	0.443897	−	0.896078i	$-0.353595\pi$
$601$	13080.5	0.887795	0.443897	+	0.896078i	$0.353595\pi$
$602$	0	0
$603$	−4099.27	−0.276841
$604$	0	0
$605$	−1038.00	−0.0697533
$606$	0	0
$607$	−20502.2	−1.37094	−0.685469	−	0.728102i	$-0.740403\pi$
$607$	−20502.2	−1.37094	−0.685469	+	0.728102i	$0.740403\pi$
$608$	0	0
$609$	−2035.24	−0.135422
$610$	0	0
$611$	−16587.0	−1.09826
$612$	0	0
$613$	−11003.5	−0.725004	−0.362502	−	0.931983i	$-0.618077\pi$
$613$	−11003.5	−0.725004	−0.362502	+	0.931983i	$0.618077\pi$
$614$	0	0
$615$	4172.81	0.273600
$616$	0	0
$617$	13387.8	0.873538	0.436769	−	0.899574i	$-0.356123\pi$
$617$	13387.8	0.873538	0.436769	+	0.899574i	$0.356123\pi$
$618$	0	0
$619$	25240.6	1.63894	0.819471	−	0.573121i	$-0.194267\pi$
$619$	25240.6	1.63894	0.819471	+	0.573121i	$0.194267\pi$
$620$	0	0
$621$	1803.77	0.116558
$622$	0	0
$623$	−8019.04	−0.515692
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	15680.9	0.998781
$628$	0	0
$629$	−10358.5	−0.656632
$630$	0	0
$631$	23458.3	1.47997	0.739985	−	0.672623i	$-0.234832\pi$
$631$	23458.3	1.47997	0.739985	+	0.672623i	$0.234832\pi$
$632$	0	0
$633$	−1156.33	−0.0726067
$634$	0	0
$635$	277.594	0.0173480
$636$	0	0
$637$	−23264.9	−1.44708
$638$	0	0
$639$	−5674.05	−0.351271
$640$	0	0
$641$	1962.14	0.120904	0.0604522	−	0.998171i	$-0.480746\pi$
$641$	1962.14	0.120904	0.0604522	+	0.998171i	$0.480746\pi$
$642$	0	0
$643$	−30832.4	−1.89100	−0.945499	−	0.325624i	$-0.894426\pi$
$643$	−30832.4	−1.89100	−0.945499	+	0.325624i	$0.894426\pi$
$644$	0	0
$645$	−5725.12	−0.349499
$646$	0	0
$647$	19561.1	1.18861	0.594303	−	0.804241i	$-0.297428\pi$
$647$	19561.1	1.18861	0.594303	+	0.804241i	$0.297428\pi$
$648$	0	0
$649$	17542.4	1.06102
$650$	0	0
$651$	6575.47	0.395873
$652$	0	0
$653$	32695.6	1.95939	0.979693	−	0.200505i	$-0.0642583\pi$
$653$	32695.6	1.95939	0.979693	+	0.200505i	$0.0642583\pi$
$654$	0	0
$655$	7885.31	0.470389
$656$	0	0
$657$	529.537	0.0314448
$658$	0	0
$659$	29419.3	1.73902	0.869509	−	0.493918i	$-0.164436\pi$
$659$	29419.3	1.73902	0.869509	+	0.493918i	$0.164436\pi$
$660$	0	0
$661$	25674.1	1.51075	0.755375	−	0.655293i	$-0.227455\pi$
$661$	25674.1	1.51075	0.755375	+	0.655293i	$0.227455\pi$
$662$	0	0
$663$	22409.6	1.31270
$664$	0	0
$665$	−4534.88	−0.264443
$666$	0	0
$667$	−6658.91	−0.386558
$668$	0	0
$669$	−2768.23	−0.159979
$670$	0	0
$671$	−6230.40	−0.358453
$672$	0	0
$673$	19112.3	1.09469	0.547343	−	0.836908i	$-0.315639\pi$
$673$	19112.3	1.09469	0.547343	+	0.836908i	$0.315639\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	3091.32	0.175494	0.0877469	−	0.996143i	$-0.472033\pi$
$677$	3091.32	0.175494	0.0877469	+	0.996143i	$0.472033\pi$
$678$	0	0
$679$	11516.5	0.650903
$680$	0	0
$681$	−4171.05	−0.234706
$682$	0	0
$683$	13351.7	0.748006	0.374003	−	0.927427i	$-0.377985\pi$
$683$	13351.7	0.748006	0.374003	+	0.927427i	$0.377985\pi$
$684$	0	0
$685$	5665.59	0.316016
$686$	0	0
$687$	−2252.51	−0.125093
$688$	0	0
$689$	−32297.7	−1.78584
$690$	0	0
$691$	10622.7	0.584815	0.292408	−	0.956294i	$-0.405544\pi$
$691$	10622.7	0.584815	0.292408	+	0.956294i	$0.405544\pi$
$692$	0	0
$693$	−2402.78	−0.131708
$694$	0	0
$695$	−2125.97	−0.116033
$696$	0	0
$697$	−26499.1	−1.44006
$698$	0	0
$699$	13372.2	0.723580
$700$	0	0
$701$	18023.4	0.971092	0.485546	−	0.874211i	$-0.338621\pi$
$701$	18023.4	0.971092	0.485546	+	0.874211i	$0.338621\pi$
$702$	0	0
$703$	14490.8	0.777426
$704$	0	0
$705$	−3172.78	−0.169495
$706$	0	0
$707$	−5388.34	−0.286633
$708$	0	0
$709$	14437.1	0.764732	0.382366	−	0.924011i	$-0.375109\pi$
$709$	14437.1	0.764732	0.382366	+	0.924011i	$0.375109\pi$
$710$	0	0
$711$	−11315.9	−0.596875
$712$	0	0
$713$	21513.7	1.13001
$714$	0	0
$715$	15379.9	0.804440
$716$	0	0
$717$	19166.1	0.998284
$718$	0	0
$719$	−31328.7	−1.62498	−0.812491	−	0.582974i	$-0.801889\pi$
$719$	−31328.7	−1.62498	−0.812491	+	0.582974i	$0.801889\pi$
$720$	0	0
$721$	−2467.22	−0.127440
$722$	0	0
$723$	−17157.1	−0.882543
$724$	0	0
$725$	2491.87	0.127650
$726$	0	0
$727$	−19209.6	−0.979978	−0.489989	−	0.871729i	$-0.662999\pi$
$727$	−19209.6	−0.979978	−0.489989	+	0.871729i	$0.662999\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	36356.9	1.83955
$732$	0	0
$733$	−14683.0	−0.739874	−0.369937	−	0.929057i	$-0.620621\pi$
$733$	−14683.0	−0.739874	−0.369937	+	0.929057i	$0.620621\pi$
$734$	0	0
$735$	−4450.12	−0.223327
$736$	0	0
$737$	17866.0	0.892948
$738$	0	0
$739$	25001.5	1.24451	0.622256	−	0.782814i	$-0.286216\pi$
$739$	25001.5	1.24451	0.622256	+	0.782814i	$0.286216\pi$
$740$	0	0
$741$	−31349.4	−1.55418
$742$	0	0
$743$	5512.07	0.272165	0.136082	−	0.990698i	$-0.456549\pi$
$743$	5512.07	0.272165	0.136082	+	0.990698i	$0.456549\pi$
$744$	0	0
$745$	684.875	0.0336804
$746$	0	0
$747$	2068.20	0.101300
$748$	0	0
$749$	14390.9	0.702044
$750$	0	0
$751$	−9405.13	−0.456988	−0.228494	−	0.973545i	$-0.573380\pi$
$751$	−9405.13	−0.456988	−0.228494	+	0.973545i	$0.573380\pi$
$752$	0	0
$753$	−10348.3	−0.500815
$754$	0	0
$755$	2468.16	0.118974
$756$	0	0
$757$	−37411.5	−1.79623	−0.898113	−	0.439765i	$-0.855062\pi$
$757$	−37411.5	−1.79623	−0.898113	+	0.439765i	$0.855062\pi$
$758$	0	0
$759$	−7861.42	−0.375957
$760$	0	0
$761$	−23465.7	−1.11778	−0.558891	−	0.829241i	$-0.688773\pi$
$761$	−23465.7	−1.11778	−0.558891	+	0.829241i	$0.688773\pi$
$762$	0	0
$763$	−14631.6	−0.694231
$764$	0	0
$765$	4286.53	0.202588
$766$	0	0
$767$	−35070.8	−1.65102
$768$	0	0
$769$	−25677.3	−1.20409	−0.602046	−	0.798461i	$-0.705648\pi$
$769$	−25677.3	−1.20409	−0.602046	+	0.798461i	$0.705648\pi$
$770$	0	0
$771$	3038.49	0.141931
$772$	0	0
$773$	−24591.8	−1.14425	−0.572125	−	0.820167i	$-0.693881\pi$
$773$	−24591.8	−1.14425	−0.572125	+	0.820167i	$0.693881\pi$
$774$	0	0
$775$	−8050.78	−0.373152
$776$	0	0
$777$	−2220.41	−0.102518
$778$	0	0
$779$	37070.2	1.70498
$780$	0	0
$781$	24729.4	1.13302
$782$	0	0
$783$	−2691.22	−0.122831
$784$	0	0
$785$	2986.97	0.135808
$786$	0	0
$787$	35055.7	1.58780	0.793902	−	0.608046i	$-0.208046\pi$
$787$	35055.7	1.58780	0.793902	+	0.608046i	$0.208046\pi$
$788$	0	0
$789$	−20148.9	−0.909151
$790$	0	0
$791$	3928.01	0.176566
$792$	0	0
$793$	12455.8	0.557780
$794$	0	0
$795$	−6177.94	−0.275609
$796$	0	0
$797$	15901.1	0.706709	0.353354	−	0.935490i	$-0.385041\pi$
$797$	15901.1	0.706709	0.353354	+	0.935490i	$0.385041\pi$
$798$	0	0
$799$	20148.5	0.892118
$800$	0	0
$801$	−10603.7	−0.467744
$802$	0	0
$803$	−2307.90	−0.101425
$804$	0	0
$805$	2273.50	0.0995408
$806$	0	0
$807$	−3899.02	−0.170077
$808$	0	0
$809$	14097.0	0.612640	0.306320	−	0.951929i	$-0.400902\pi$
$809$	14097.0	0.612640	0.306320	+	0.951929i	$0.400902\pi$
$810$	0	0
$811$	−6846.50	−0.296441	−0.148220	−	0.988954i	$-0.547354\pi$
$811$	−6846.50	−0.296441	−0.148220	+	0.988954i	$0.547354\pi$
$812$	0	0
$813$	5070.32	0.218725
$814$	0	0
$815$	−11361.7	−0.488325
$816$	0	0
$817$	−50860.6	−2.17795
$818$	0	0
$819$	4803.64	0.204948
$820$	0	0
$821$	−3531.25	−0.150111	−0.0750557	−	0.997179i	$-0.523913\pi$
$821$	−3531.25	−0.150111	−0.0750557	+	0.997179i	$0.523913\pi$
$822$	0	0
$823$	20412.7	0.864572	0.432286	−	0.901736i	$-0.357707\pi$
$823$	20412.7	0.864572	0.432286	+	0.901736i	$0.357707\pi$
$824$	0	0
$825$	2941.87	0.124149
$826$	0	0
$827$	6429.97	0.270365	0.135183	−	0.990821i	$-0.456838\pi$
$827$	6429.97	0.270365	0.135183	+	0.990821i	$0.456838\pi$
$828$	0	0
$829$	−15491.8	−0.649039	−0.324520	−	0.945879i	$-0.605203\pi$
$829$	−15491.8	−0.649039	−0.324520	+	0.945879i	$0.605203\pi$
$830$	0	0
$831$	18692.7	0.780315
$832$	0	0
$833$	28260.1	1.17546
$834$	0	0
$835$	−10464.1	−0.433683
$836$	0	0
$837$	8694.84	0.359065
$838$	0	0
$839$	−8101.80	−0.333379	−0.166690	−	0.986009i	$-0.553308\pi$
$839$	−8101.80	−0.333379	−0.166690	+	0.986009i	$0.553308\pi$
$840$	0	0
$841$	−14453.9	−0.592640
$842$	0	0
$843$	14279.3	0.583398
$844$	0	0
$845$	−19762.5	−0.804557
$846$	0	0
$847$	1412.98	0.0573205
$848$	0	0
$849$	7997.85	0.323304
$850$	0	0
$851$	−7264.76	−0.292636
$852$	0	0
$853$	18343.8	0.736317	0.368159	−	0.929763i	$-0.379988\pi$
$853$	18343.8	0.736317	0.368159	+	0.929763i	$0.379988\pi$
$854$	0	0
$855$	−5996.53	−0.239856
$856$	0	0
$857$	34069.4	1.35798	0.678991	−	0.734147i	$-0.262418\pi$
$857$	34069.4	1.35798	0.678991	+	0.734147i	$0.262418\pi$
$858$	0	0
$859$	22555.2	0.895894	0.447947	−	0.894060i	$-0.352155\pi$
$859$	22555.2	0.895894	0.447947	+	0.894060i	$0.352155\pi$
$860$	0	0
$861$	−5680.24	−0.224834
$862$	0	0
$863$	26094.0	1.02926	0.514630	−	0.857413i	$-0.327929\pi$
$863$	26094.0	1.02926	0.514630	+	0.857413i	$0.327929\pi$
$864$	0	0
$865$	−293.749	−0.0115466
$866$	0	0
$867$	−12482.3	−0.488950
$868$	0	0
$869$	49318.3	1.92521
$870$	0	0
$871$	−35717.8	−1.38950
$872$	0	0
$873$	15228.4	0.590384
$874$	0	0
$875$	−850.781	−0.0328705
$876$	0	0
$877$	−32275.2	−1.24271	−0.621355	−	0.783530i	$-0.713417\pi$
$877$	−32275.2	−1.24271	−0.621355	+	0.783530i	$0.713417\pi$
$878$	0	0
$879$	−1551.19	−0.0595225
$880$	0	0
$881$	32969.3	1.26080	0.630399	−	0.776271i	$-0.282891\pi$
$881$	32969.3	1.26080	0.630399	+	0.776271i	$0.282891\pi$
$882$	0	0
$883$	19708.7	0.751134	0.375567	−	0.926795i	$-0.377448\pi$
$883$	19708.7	0.751134	0.375567	+	0.926795i	$0.377448\pi$
$884$	0	0
$885$	−6708.37	−0.254802
$886$	0	0
$887$	−29868.2	−1.13064	−0.565319	−	0.824872i	$-0.691247\pi$
$887$	−29868.2	−1.13064	−0.565319	+	0.824872i	$0.691247\pi$
$888$	0	0
$889$	−377.875	−0.0142559
$890$	0	0
$891$	−3177.22	−0.119462
$892$	0	0
$893$	−28186.2	−1.05623
$894$	0	0
$895$	−11744.8	−0.438643
$896$	0	0
$897$	15716.6	0.585018
$898$	0	0
$899$	−32098.5	−1.19082
$900$	0	0
$901$	39232.5	1.45064
$902$	0	0
$903$	7793.32	0.287204
$904$	0	0
$905$	−10515.6	−0.386244
$906$	0	0
$907$	−25106.4	−0.919124	−0.459562	−	0.888146i	$-0.651994\pi$
$907$	−25106.4	−0.919124	−0.459562	+	0.888146i	$0.651994\pi$
$908$	0	0
$909$	−7125.07	−0.259982
$910$	0	0
$911$	17411.2	0.633215	0.316608	−	0.948557i	$-0.397456\pi$
$911$	17411.2	0.633215	0.316608	+	0.948557i	$0.397456\pi$
$912$	0	0
$913$	−9013.90	−0.326743
$914$	0	0
$915$	2382.56	0.0860821
$916$	0	0
$917$	−10733.9	−0.386547
$918$	0	0
$919$	−38193.1	−1.37092	−0.685460	−	0.728111i	$-0.740399\pi$
$919$	−38193.1	−1.37092	−0.685460	+	0.728111i	$0.740399\pi$
$920$	0	0
$921$	−369.113	−0.0132059
$922$	0	0
$923$	−49439.1	−1.76306
$924$	0	0
$925$	2718.59	0.0966344
$926$	0	0
$927$	−3262.44	−0.115591
$928$	0	0
$929$	−36084.2	−1.27437	−0.637183	−	0.770713i	$-0.719900\pi$
$929$	−36084.2	−1.27437	−0.637183	+	0.770713i	$0.719900\pi$
$930$	0	0
$931$	−39533.8	−1.39169
$932$	0	0
$933$	2099.81	0.0736814
$934$	0	0
$935$	−18682.1	−0.653445
$936$	0	0
$937$	−12854.3	−0.448166	−0.224083	−	0.974570i	$-0.571939\pi$
$937$	−12854.3	−0.448166	−0.224083	+	0.974570i	$0.571939\pi$
$938$	0	0
$939$	−3806.66	−0.132296
$940$	0	0
$941$	−40368.6	−1.39849	−0.699245	−	0.714882i	$-0.746480\pi$
$941$	−40368.6	−1.39849	−0.699245	+	0.714882i	$0.746480\pi$
$942$	0	0
$943$	−18584.7	−0.641781
$944$	0	0
$945$	918.844	0.0316296
$946$	0	0
$947$	−18858.1	−0.647103	−0.323551	−	0.946211i	$-0.604877\pi$
$947$	−18858.1	−0.647103	−0.323551	+	0.946211i	$0.604877\pi$
$948$	0	0
$949$	4613.96	0.157825
$950$	0	0
$951$	−7521.64	−0.256473
$952$	0	0
$953$	−37267.2	−1.26674	−0.633370	−	0.773849i	$-0.718329\pi$
$953$	−37267.2	−1.26674	−0.633370	+	0.773849i	$0.718329\pi$
$954$	0	0
$955$	−16015.9	−0.542682
$956$	0	0
$957$	11729.3	0.396189
$958$	0	0
$959$	−7712.29	−0.259690
$960$	0	0
$961$	73913.1	2.48106
$962$	0	0
$963$	19029.3	0.636770
$964$	0	0
$965$	−2355.56	−0.0785785
$966$	0	0
$967$	−33833.4	−1.12514	−0.562569	−	0.826750i	$-0.690187\pi$
$967$	−33833.4	−1.12514	−0.562569	+	0.826750i	$0.690187\pi$
$968$	0	0
$969$	38080.5	1.26246
$970$	0	0
$971$	42354.0	1.39980	0.699899	−	0.714242i	$-0.253228\pi$
$971$	42354.0	1.39980	0.699899	+	0.714242i	$0.253228\pi$
$972$	0	0
$973$	2893.97	0.0953510
$974$	0	0
$975$	−5881.41	−0.193185
$976$	0	0
$977$	26051.8	0.853092	0.426546	−	0.904466i	$-0.359730\pi$
$977$	26051.8	0.853092	0.426546	+	0.904466i	$0.359730\pi$
$978$	0	0
$979$	46214.4	1.50870
$980$	0	0
$981$	−19347.5	−0.629683
$982$	0	0
$983$	34810.4	1.12948	0.564740	−	0.825269i	$-0.308977\pi$
$983$	34810.4	1.12948	0.564740	+	0.825269i	$0.308977\pi$
$984$	0	0
$985$	7676.06	0.248304
$986$	0	0
$987$	4318.95	0.139284
$988$	0	0
$989$	25498.3	0.819816
$990$	0	0
$991$	15896.1	0.509542	0.254771	−	0.967001i	$-0.418000\pi$
$991$	15896.1	0.509542	0.254771	+	0.967001i	$0.418000\pi$
$992$	0	0
$993$	13917.8	0.444780
$994$	0	0
$995$	21809.9	0.694895
$996$	0	0
$997$	43981.7	1.39711	0.698553	−	0.715559i	$-0.253828\pi$
$997$	43981.7	1.39711	0.698553	+	0.715559i	$0.253828\pi$
$998$	0	0
$999$	−2936.08	−0.0929865

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.4.a.m.1.2	✓	2
3.2	odd	2		1440.4.a.z.1.2		2
4.3	odd	2		480.4.a.q.1.1	yes	2
5.4	even	2		2400.4.a.bc.1.1		2
8.3	odd	2		960.4.a.bm.1.1		2
8.5	even	2		960.4.a.bo.1.2		2
12.11	even	2		1440.4.a.bg.1.1		2
20.19	odd	2		2400.4.a.x.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.a.m.1.2	✓	2	1.1	even	1	trivial
480.4.a.q.1.1	yes	2	4.3	odd	2
960.4.a.bm.1.1		2	8.3	odd	2
960.4.a.bo.1.2		2	8.5	even	2
1440.4.a.z.1.2		2	3.2	odd	2
1440.4.a.bg.1.1		2	12.11	even	2
2400.4.a.x.1.2		2	20.19	odd	2
2400.4.a.bc.1.1		2	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	480.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -5.00000 q^{5} +6.80625 q^{7} +9.00000 q^{9} -39.2250 q^{11} +78.4187 q^{13} +15.0000 q^{15} -95.2562 q^{17} +133.256 q^{19} -20.4187 q^{21} -66.8062 q^{23} +25.0000 q^{25} -27.0000 q^{27} +99.6750 q^{29} -322.031 q^{31} +117.675 q^{33} -34.0312 q^{35} +108.744 q^{37} -235.256 q^{39} +278.187 q^{41} -381.675 q^{43} -45.0000 q^{45} -211.519 q^{47} -296.675 q^{49} +285.769 q^{51} -411.862 q^{53} +196.125 q^{55} -399.769 q^{57} -447.225 q^{59} +158.837 q^{61} +61.2562 q^{63} -392.094 q^{65} -455.475 q^{67} +200.419 q^{69} -630.450 q^{71} +58.8375 q^{73} -75.0000 q^{75} -266.975 q^{77} -1257.32 q^{79} +81.0000 q^{81} +229.800 q^{83} +476.281 q^{85} -299.025 q^{87} -1178.19 q^{89} +533.737 q^{91} +966.094 q^{93} -666.281 q^{95} +1692.05 q^{97} -353.025 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 10 q^{5} - 12 q^{7} + 18 q^{9} + 24 q^{11} + 80 q^{13} + 30 q^{15} + 40 q^{17} + 36 q^{19} + 36 q^{21} - 108 q^{23} + 50 q^{25} - 54 q^{27} - 108 q^{29} - 516 q^{31} - 72 q^{33} + 60 q^{35}+ \cdots + 216 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 - 10 * q^5 - 12 * q^7 + 18 * q^9 + 24 * q^11 + 80 * q^13 + 30 * q^15 + 40 * q^17 + 36 * q^19 + 36 * q^21 - 108 * q^23 + 50 * q^25 - 54 * q^27 - 108 * q^29 - 516 * q^31 - 72 * q^33 + 60 * q^35 + 448 * q^37 - 240 * q^39 - 212 * q^41 - 456 * q^43 - 90 * q^45 - 756 * q^47 - 286 * q^49 - 120 * q^51 + 252 * q^53 - 120 * q^55 - 108 * q^57 - 792 * q^59 + 164 * q^61 - 108 * q^63 - 400 * q^65 + 216 * q^67 + 324 * q^69 - 1056 * q^71 - 36 * q^73 - 150 * q^75 - 1456 * q^77 - 2028 * q^79 + 162 * q^81 - 360 * q^83 - 200 * q^85 + 324 * q^87 - 1588 * q^89 + 504 * q^91 + 1548 * q^93 - 180 * q^95 + 1540 * q^97 + 216 * q^99

Introduction

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