LMFDB - Embedded newform 2178.4.a.bx.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2178,4,Mod(1,2178)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2178.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2178, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2178 = 2 \cdot 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2178.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-8,0,16,6,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$128.506159993$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.5157648.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 123x^{2} - 132x + 2148 $ x^4 - 123x^2 - 132x + 2148
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 726)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-5.62469$ of defining polynomial
Character	$\chi$	$=$	2178.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} +6.27815 q^{5} -32.0906 q^{7} -8.00000 q^{8} -12.5563 q^{10} +24.1742 q^{13} +64.1812 q^{14} +16.0000 q^{16} -106.291 q^{17} +120.360 q^{19} +25.1126 q^{20} +71.8007 q^{23} -85.5849 q^{25} -48.3484 q^{26} -128.362 q^{28} -66.9125 q^{29} +132.668 q^{31} -32.0000 q^{32} +212.582 q^{34} -201.470 q^{35} +297.528 q^{37} -240.719 q^{38} -50.2252 q^{40} -249.537 q^{41} +218.221 q^{43} -143.601 q^{46} -420.061 q^{47} +686.808 q^{49} +171.170 q^{50} +96.6967 q^{52} +722.272 q^{53} +256.725 q^{56} +133.825 q^{58} +298.900 q^{59} +335.510 q^{61} -265.337 q^{62} +64.0000 q^{64} +151.769 q^{65} -95.1320 q^{67} -425.164 q^{68} +402.939 q^{70} -271.181 q^{71} -983.985 q^{73} -595.056 q^{74} +481.439 q^{76} +1130.58 q^{79} +100.450 q^{80} +499.074 q^{82} -1045.22 q^{83} -667.311 q^{85} -436.441 q^{86} -359.298 q^{89} -775.764 q^{91} +287.203 q^{92} +840.122 q^{94} +755.636 q^{95} -1329.67 q^{97} -1373.62 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} + 16 q^{4} + 6 q^{5} + 12 q^{7} - 32 q^{8} - 12 q^{10} + 12 q^{13} - 24 q^{14} + 64 q^{16} - 108 q^{17} + 192 q^{19} + 24 q^{20} - 156 q^{23} + 286 q^{25} - 24 q^{26} + 48 q^{28} - 408 q^{29}+ \cdots - 1348 q^{98}+O(q^{100}) $ 4 * q - 8 * q^2 + 16 * q^4 + 6 * q^5 + 12 * q^7 - 32 * q^8 - 12 * q^10 + 12 * q^13 - 24 * q^14 + 64 * q^16 - 108 * q^17 + 192 * q^19 + 24 * q^20 - 156 * q^23 + 286 * q^25 - 24 * q^26 + 48 * q^28 - 408 * q^29 - 26 * q^31 - 128 * q^32 + 216 * q^34 - 588 * q^35 - 224 * q^37 - 384 * q^38 - 48 * q^40 - 348 * q^41 + 540 * q^43 + 312 * q^46 - 132 * q^47 + 674 * q^49 - 572 * q^50 + 48 * q^52 + 1470 * q^53 - 96 * q^56 + 816 * q^58 + 684 * q^59 + 1320 * q^61 + 52 * q^62 + 256 * q^64 - 2556 * q^65 - 530 * q^67 - 432 * q^68 + 1176 * q^70 - 936 * q^71 - 2352 * q^73 + 448 * q^74 + 768 * q^76 - 192 * q^79 + 96 * q^80 + 696 * q^82 - 3192 * q^83 - 2340 * q^85 - 1080 * q^86 - 726 * q^89 + 360 * q^91 - 624 * q^92 + 264 * q^94 + 2064 * q^95 - 100 * q^97 - 1348 * q^98

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	6.27815	0.561534	0.280767	−	0.959776i	$-0.409411\pi$
$5$	6.27815	0.561534	0.280767	+	0.959776i	$0.409411\pi$
$6$	0	0
$7$	−32.0906	−1.73273	−0.866365	−	0.499411i	$-0.833550\pi$
$7$	−32.0906	−1.73273	−0.866365	+	0.499411i	$0.833550\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−12.5563	−0.397065
$11$	0	0
$11$	0	0
$12$	0	0
$13$	24.1742	0.515747	0.257874	−	0.966179i	$-0.416978\pi$
$13$	24.1742	0.515747	0.257874	+	0.966179i	$0.416978\pi$
$14$	64.1812	1.22523
$15$	0	0
$16$	16.0000	0.250000
$17$	−106.291	−1.51643	−0.758217	−	0.652002i	$-0.773929\pi$
$17$	−106.291	−1.51643	−0.758217	+	0.652002i	$0.773929\pi$
$18$	0	0
$19$	120.360	1.45328	0.726642	−	0.687016i	$-0.241080\pi$
$19$	120.360	1.45328	0.726642	+	0.687016i	$0.241080\pi$
$20$	25.1126	0.280767
$21$	0	0
$22$	0	0
$23$	71.8007	0.650934	0.325467	−	0.945553i	$-0.394479\pi$
$23$	71.8007	0.650934	0.325467	+	0.945553i	$0.394479\pi$
$24$	0	0
$25$	−85.5849	−0.684679
$26$	−48.3484	−0.364688
$27$	0	0
$28$	−128.362	−0.866365
$29$	−66.9125	−0.428460	−0.214230	−	0.976783i	$-0.568724\pi$
$29$	−66.9125	−0.428460	−0.214230	+	0.976783i	$0.568724\pi$
$30$	0	0
$31$	132.668	0.768644	0.384322	−	0.923199i	$-0.374435\pi$
$31$	132.668	0.768644	0.384322	+	0.923199i	$0.374435\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	212.582	1.07228
$35$	−201.470	−0.972988
$36$	0	0
$37$	297.528	1.32198	0.660990	−	0.750394i	$-0.270136\pi$
$37$	297.528	1.32198	0.660990	+	0.750394i	$0.270136\pi$
$38$	−240.719	−1.02763
$39$	0	0
$40$	−50.2252	−0.198532
$41$	−249.537	−0.950515	−0.475258	−	0.879847i	$-0.657645\pi$
$41$	−249.537	−0.950515	−0.475258	+	0.879847i	$0.657645\pi$
$42$	0	0
$43$	218.221	0.773915	0.386957	−	0.922098i	$-0.373526\pi$
$43$	218.221	0.773915	0.386957	+	0.922098i	$0.373526\pi$
$44$	0	0
$45$	0	0
$46$	−143.601	−0.460280
$47$	−420.061	−1.30366	−0.651832	−	0.758363i	$-0.725999\pi$
$47$	−420.061	−1.30366	−0.651832	+	0.758363i	$0.725999\pi$
$48$	0	0
$49$	686.808	2.00235
$50$	171.170	0.484141
$51$	0	0
$52$	96.6967	0.257874
$53$	722.272	1.87192	0.935960	−	0.352107i	$-0.114535\pi$
$53$	722.272	1.87192	0.935960	+	0.352107i	$0.114535\pi$
$54$	0	0
$55$	0	0
$56$	256.725	0.612613
$57$	0	0
$58$	133.825	0.302967
$59$	298.900	0.659550	0.329775	−	0.944059i	$-0.393027\pi$
$59$	298.900	0.659550	0.329775	+	0.944059i	$0.393027\pi$
$60$	0	0
$61$	335.510	0.704224	0.352112	−	0.935958i	$-0.385464\pi$
$61$	335.510	0.704224	0.352112	+	0.935958i	$0.385464\pi$
$62$	−265.337	−0.543513
$63$	0	0
$64$	64.0000	0.125000
$65$	151.769	0.289610
$66$	0	0
$67$	−95.1320	−0.173466	−0.0867330	−	0.996232i	$-0.527643\pi$
$67$	−95.1320	−0.173466	−0.0867330	+	0.996232i	$0.527643\pi$
$68$	−425.164	−0.758217
$69$	0	0
$70$	402.939	0.688006
$71$	−271.181	−0.453285	−0.226642	−	0.973978i	$-0.572775\pi$
$71$	−271.181	−0.453285	−0.226642	+	0.973978i	$0.572775\pi$
$72$	0	0
$73$	−983.985	−1.57763	−0.788813	−	0.614633i	$-0.789304\pi$
$73$	−983.985	−1.57763	−0.788813	+	0.614633i	$0.789304\pi$
$74$	−595.056	−0.934782
$75$	0	0
$76$	481.439	0.726642
$77$	0	0
$78$	0	0
$79$	1130.58	1.61013	0.805064	−	0.593188i	$-0.202131\pi$
$79$	1130.58	1.61013	0.805064	+	0.593188i	$0.202131\pi$
$80$	100.450	0.140384
$81$	0	0
$82$	499.074	0.672116
$83$	−1045.22	−1.38227	−0.691133	−	0.722727i	$-0.742888\pi$
$83$	−1045.22	−1.38227	−0.691133	+	0.722727i	$0.742888\pi$
$84$	0	0
$85$	−667.311	−0.851530
$86$	−436.441	−0.547240
$87$	0	0
$88$	0	0
$89$	−359.298	−0.427927	−0.213964	−	0.976842i	$-0.568637\pi$
$89$	−359.298	−0.427927	−0.213964	+	0.976842i	$0.568637\pi$
$90$	0	0
$91$	−775.764	−0.893651
$92$	287.203	0.325467
$93$	0	0
$94$	840.122	0.921830
$95$	755.636	0.816070
$96$	0	0
$97$	−1329.67	−1.39183	−0.695914	−	0.718126i	$-0.745000\pi$
$97$	−1329.67	−1.39183	−0.695914	+	0.718126i	$0.745000\pi$
$98$	−1373.62	−1.41588
$99$	0	0
$100$	−342.340	−0.342340
$101$	1395.96	1.37528	0.687640	−	0.726052i	$-0.258647\pi$
$101$	1395.96	1.37528	0.687640	+	0.726052i	$0.258647\pi$
$102$	0	0
$103$	−162.379	−0.155337	−0.0776683	−	0.996979i	$-0.524748\pi$
$103$	−162.379	−0.155337	−0.0776683	+	0.996979i	$0.524748\pi$
$104$	−193.393	−0.182344
$105$	0	0
$106$	−1444.54	−1.32365
$107$	−1.34003	−0.00121071	−0.000605353	−	1.00000i	$-0.500193\pi$
$107$	−1.34003	−0.00121071	−0.000605353		1.00000i	$0.500193\pi$
$108$	0	0
$109$	−999.993	−0.878733	−0.439367	−	0.898308i	$-0.644797\pi$
$109$	−999.993	−0.878733	−0.439367	+	0.898308i	$0.644797\pi$
$110$	0	0
$111$	0	0
$112$	−513.450	−0.433183
$113$	913.603	0.760571	0.380286	−	0.924869i	$-0.375826\pi$
$113$	913.603	0.760571	0.380286	+	0.924869i	$0.375826\pi$
$114$	0	0
$115$	450.775	0.365522
$116$	−267.650	−0.214230
$117$	0	0
$118$	−597.800	−0.466373
$119$	3410.95	2.62757
$120$	0	0
$121$	0	0
$122$	−671.020	−0.497961
$123$	0	0
$124$	530.674	0.384322
$125$	−1322.08	−0.946005
$126$	0	0
$127$	106.551	0.0744479	0.0372240	−	0.999307i	$-0.488149\pi$
$127$	106.551	0.0744479	0.0372240	+	0.999307i	$0.488149\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	−303.538	−0.204785
$131$	−2807.45	−1.87243	−0.936214	−	0.351432i	$-0.885695\pi$
$131$	−2807.45	−1.87243	−0.936214	+	0.351432i	$0.885695\pi$
$132$	0	0
$133$	−3862.42	−2.51815
$134$	190.264	0.122659
$135$	0	0
$136$	850.329	0.536140
$137$	−1248.74	−0.778736	−0.389368	−	0.921082i	$-0.627307\pi$
$137$	−1248.74	−0.778736	−0.389368	+	0.921082i	$0.627307\pi$
$138$	0	0
$139$	2211.65	1.34957	0.674783	−	0.738016i	$-0.264237\pi$
$139$	2211.65	1.34957	0.674783	+	0.738016i	$0.264237\pi$
$140$	−805.878	−0.486494
$141$	0	0
$142$	542.361	0.320521
$143$	0	0
$144$	0	0
$145$	−420.087	−0.240595
$146$	1967.97	1.11555
$147$	0	0
$148$	1190.11	0.660990
$149$	−520.288	−0.286065	−0.143032	−	0.989718i	$-0.545685\pi$
$149$	−520.288	−0.286065	−0.143032	+	0.989718i	$0.545685\pi$
$150$	0	0
$151$	−3313.99	−1.78602	−0.893008	−	0.450041i	$-0.851409\pi$
$151$	−3313.99	−1.78602	−0.893008	+	0.450041i	$0.851409\pi$
$152$	−962.878	−0.513814
$153$	0	0
$154$	0	0
$155$	832.912	0.431620
$156$	0	0
$157$	1067.49	0.542645	0.271322	−	0.962489i	$-0.412539\pi$
$157$	1067.49	0.542645	0.271322	+	0.962489i	$0.412539\pi$
$158$	−2261.16	−1.13853
$159$	0	0
$160$	−200.901	−0.0992662
$161$	−2304.13	−1.12789
$162$	0	0
$163$	−1174.05	−0.564162	−0.282081	−	0.959391i	$-0.591025\pi$
$163$	−1174.05	−0.564162	−0.282081	+	0.959391i	$0.591025\pi$
$164$	−998.148	−0.475258
$165$	0	0
$166$	2090.45	0.977410
$167$	322.625	0.149494	0.0747469	−	0.997203i	$-0.476185\pi$
$167$	322.625	0.149494	0.0747469	+	0.997203i	$0.476185\pi$
$168$	0	0
$169$	−1612.61	−0.734005
$170$	1334.62	0.602122
$171$	0	0
$172$	872.882	0.386957
$173$	−2307.74	−1.01419	−0.507094	−	0.861891i	$-0.669280\pi$
$173$	−2307.74	−1.01419	−0.507094	+	0.861891i	$0.669280\pi$
$174$	0	0
$175$	2746.47	1.18636
$176$	0	0
$177$	0	0
$178$	718.596	0.302590
$179$	−524.523	−0.219021	−0.109510	−	0.993986i	$-0.534928\pi$
$179$	−524.523	−0.219021	−0.109510	+	0.993986i	$0.534928\pi$
$180$	0	0
$181$	1033.47	0.424405	0.212203	−	0.977226i	$-0.431936\pi$
$181$	1033.47	0.424405	0.212203	+	0.977226i	$0.431936\pi$
$182$	1551.53	0.631906
$183$	0	0
$184$	−574.406	−0.230140
$185$	1867.92	0.742338
$186$	0	0
$187$	0	0
$188$	−1680.24	−0.651832
$189$	0	0
$190$	−1511.27	−0.577048
$191$	2693.88	1.02054	0.510268	−	0.860016i	$-0.329546\pi$
$191$	2693.88	1.02054	0.510268	+	0.860016i	$0.329546\pi$
$192$	0	0
$193$	3518.61	1.31231	0.656153	−	0.754628i	$-0.272183\pi$
$193$	3518.61	1.31231	0.656153	+	0.754628i	$0.272183\pi$
$194$	2659.33	0.984170
$195$	0	0
$196$	2747.23	1.00118
$197$	4868.79	1.76085	0.880424	−	0.474187i	$-0.157258\pi$
$197$	4868.79	1.76085	0.880424	+	0.474187i	$0.157258\pi$
$198$	0	0
$199$	−4200.44	−1.49629	−0.748144	−	0.663536i	$-0.769055\pi$
$199$	−4200.44	−1.49629	−0.748144	+	0.663536i	$0.769055\pi$
$200$	684.679	0.242071
$201$	0	0
$202$	−2791.92	−0.972470
$203$	2147.26	0.742406
$204$	0	0
$205$	−1566.63	−0.533747
$206$	324.758	0.109840
$207$	0	0
$208$	386.787	0.128937
$209$	0	0
$210$	0	0
$211$	2367.53	0.772454	0.386227	−	0.922404i	$-0.373778\pi$
$211$	2367.53	0.772454	0.386227	+	0.922404i	$0.373778\pi$
$212$	2889.09	0.935960
$213$	0	0
$214$	2.68006	0.000856099 0
$215$	1370.02	0.434580
$216$	0	0
$217$	−4257.41	−1.33185
$218$	1999.99	0.621358
$219$	0	0
$220$	0	0
$221$	−2569.50	−0.782096
$222$	0	0
$223$	1363.27	0.409379	0.204690	−	0.978827i	$-0.434382\pi$
$223$	1363.27	0.409379	0.204690	+	0.978827i	$0.434382\pi$
$224$	1026.90	0.306306
$225$	0	0
$226$	−1827.21	−0.537805
$227$	−6105.77	−1.78526	−0.892630	−	0.450790i	$-0.851142\pi$
$227$	−6105.77	−1.78526	−0.892630	+	0.450790i	$0.851142\pi$
$228$	0	0
$229$	4444.16	1.28244	0.641219	−	0.767358i	$-0.278429\pi$
$229$	4444.16	1.28244	0.641219	+	0.767358i	$0.278429\pi$
$230$	−901.551	−0.258463
$231$	0	0
$232$	535.300	0.151484
$233$	−3243.92	−0.912087	−0.456043	−	0.889958i	$-0.650734\pi$
$233$	−3243.92	−0.912087	−0.456043	+	0.889958i	$0.650734\pi$
$234$	0	0
$235$	−2637.21	−0.732052
$236$	1195.60	0.329775
$237$	0	0
$238$	−6821.89	−1.85797
$239$	−2152.12	−0.582464	−0.291232	−	0.956652i	$-0.594065\pi$
$239$	−2152.12	−0.582464	−0.291232	+	0.956652i	$0.594065\pi$
$240$	0	0
$241$	1454.40	0.388740	0.194370	−	0.980928i	$-0.437734\pi$
$241$	1454.40	0.388740	0.194370	+	0.980928i	$0.437734\pi$
$242$	0	0
$243$	0	0
$244$	1342.04	0.352112
$245$	4311.88	1.12439
$246$	0	0
$247$	2909.60	0.749527
$248$	−1061.35	−0.271757
$249$	0	0
$250$	2644.17	0.668927
$251$	−4398.73	−1.10616	−0.553079	−	0.833129i	$-0.686547\pi$
$251$	−4398.73	−1.10616	−0.553079	+	0.833129i	$0.686547\pi$
$252$	0	0
$253$	0	0
$254$	−213.102	−0.0526426
$255$	0	0
$256$	256.000	0.0625000
$257$	2018.47	0.489918	0.244959	−	0.969533i	$-0.421225\pi$
$257$	2018.47	0.489918	0.244959	+	0.969533i	$0.421225\pi$
$258$	0	0
$259$	−9547.85	−2.29064
$260$	607.076	0.144805
$261$	0	0
$262$	5614.90	1.32401
$263$	4304.49	1.00922	0.504612	−	0.863346i	$-0.331635\pi$
$263$	4304.49	1.00922	0.504612	+	0.863346i	$0.331635\pi$
$264$	0	0
$265$	4534.53	1.05115
$266$	7724.83	1.78060
$267$	0	0
$268$	−380.528	−0.0867330
$269$	−185.624	−0.0420733	−0.0210366	−	0.999779i	$-0.506697\pi$
$269$	−185.624	−0.0420733	−0.0210366	+	0.999779i	$0.506697\pi$
$270$	0	0
$271$	−6717.78	−1.50582	−0.752908	−	0.658126i	$-0.771349\pi$
$271$	−6717.78	−1.50582	−0.752908	+	0.658126i	$0.771349\pi$
$272$	−1700.66	−0.379108
$273$	0	0
$274$	2497.48	0.550650
$275$	0	0
$276$	0	0
$277$	−873.692	−0.189513	−0.0947564	−	0.995500i	$-0.530207\pi$
$277$	−873.692	−0.189513	−0.0947564	+	0.995500i	$0.530207\pi$
$278$	−4423.30	−0.954288
$279$	0	0
$280$	1611.76	0.344003
$281$	7130.47	1.51377	0.756883	−	0.653551i	$-0.226721\pi$
$281$	7130.47	1.51377	0.756883	+	0.653551i	$0.226721\pi$
$282$	0	0
$283$	−4293.78	−0.901903	−0.450952	−	0.892548i	$-0.648915\pi$
$283$	−4293.78	−0.901903	−0.450952	+	0.892548i	$0.648915\pi$
$284$	−1084.72	−0.226642
$285$	0	0
$286$	0	0
$287$	8007.79	1.64699
$288$	0	0
$289$	6384.79	1.29957
$290$	840.173	0.170126
$291$	0	0
$292$	−3935.94	−0.788813
$293$	−4621.40	−0.921451	−0.460726	−	0.887543i	$-0.652411\pi$
$293$	−4621.40	−0.921451	−0.460726	+	0.887543i	$0.652411\pi$
$294$	0	0
$295$	1876.54	0.370360
$296$	−2380.22	−0.467391
$297$	0	0
$298$	1040.58	0.202278
$299$	1735.72	0.335717
$300$	0	0
$301$	−7002.83	−1.34099
$302$	6627.97	1.26290
$303$	0	0
$304$	1925.76	0.363321
$305$	2106.38	0.395446
$306$	0	0
$307$	−7095.92	−1.31917	−0.659586	−	0.751630i	$-0.729268\pi$
$307$	−7095.92	−1.31917	−0.659586	+	0.751630i	$0.729268\pi$
$308$	0	0
$309$	0	0
$310$	−1665.82	−0.305201
$311$	−9557.20	−1.74257	−0.871285	−	0.490777i	$-0.836713\pi$
$311$	−9557.20	−1.74257	−0.871285	+	0.490777i	$0.836713\pi$
$312$	0	0
$313$	2639.27	0.476615	0.238307	−	0.971190i	$-0.423407\pi$
$313$	2639.27	0.476615	0.238307	+	0.971190i	$0.423407\pi$
$314$	−2134.99	−0.383708
$315$	0	0
$316$	4522.32	0.805064
$317$	7919.47	1.40316	0.701580	−	0.712590i	$-0.252478\pi$
$317$	7919.47	1.40316	0.701580	+	0.712590i	$0.252478\pi$
$318$	0	0
$319$	0	0
$320$	401.801	0.0701918
$321$	0	0
$322$	4608.26	0.797541
$323$	−12793.2	−2.20381
$324$	0	0
$325$	−2068.94	−0.353121
$326$	2348.09	0.398923
$327$	0	0
$328$	1996.30	0.336058
$329$	13480.0	2.25890
$330$	0	0
$331$	235.958	0.0391826	0.0195913	−	0.999808i	$-0.493763\pi$
$331$	235.958	0.0391826	0.0195913	+	0.999808i	$0.493763\pi$
$332$	−4180.89	−0.691133
$333$	0	0
$334$	−645.250	−0.105708
$335$	−597.253	−0.0974071
$336$	0	0
$337$	−3734.29	−0.603619	−0.301810	−	0.953368i	$-0.597591\pi$
$337$	−3734.29	−0.603619	−0.301810	+	0.953368i	$0.597591\pi$
$338$	3225.22	0.519020
$339$	0	0
$340$	−2669.24	−0.425765
$341$	0	0
$342$	0	0
$343$	−11033.0	−1.73681
$344$	−1745.76	−0.273620
$345$	0	0
$346$	4615.49	0.717139
$347$	−6149.01	−0.951286	−0.475643	−	0.879638i	$-0.657785\pi$
$347$	−6149.01	−0.951286	−0.475643	+	0.879638i	$0.657785\pi$
$348$	0	0
$349$	−7026.80	−1.07775	−0.538877	−	0.842384i	$-0.681151\pi$
$349$	−7026.80	−1.07775	−0.538877	+	0.842384i	$0.681151\pi$
$350$	−5492.94	−0.838886
$351$	0	0
$352$	0	0
$353$	−11895.9	−1.79363	−0.896816	−	0.442403i	$-0.854126\pi$
$353$	−11895.9	−1.79363	−0.896816	+	0.442403i	$0.854126\pi$
$354$	0	0
$355$	−1702.51	−0.254535
$356$	−1437.19	−0.213964
$357$	0	0
$358$	1049.05	0.154871
$359$	−804.806	−0.118318	−0.0591588	−	0.998249i	$-0.518842\pi$
$359$	−804.806	−0.118318	−0.0591588	+	0.998249i	$0.518842\pi$
$360$	0	0
$361$	7627.46	1.11204
$362$	−2066.94	−0.300100
$363$	0	0
$364$	−3103.06	−0.446825
$365$	−6177.60	−0.885891
$366$	0	0
$367$	−10844.9	−1.54251	−0.771255	−	0.636527i	$-0.780371\pi$
$367$	−10844.9	−1.54251	−0.771255	+	0.636527i	$0.780371\pi$
$368$	1148.81	0.162733
$369$	0	0
$370$	−3735.85	−0.524912
$371$	−23178.2	−3.24353
$372$	0	0
$373$	−2421.16	−0.336094	−0.168047	−	0.985779i	$-0.553746\pi$
$373$	−2421.16	−0.336094	−0.168047	+	0.985779i	$0.553746\pi$
$374$	0	0
$375$	0	0
$376$	3360.49	0.460915
$377$	−1617.56	−0.220977
$378$	0	0
$379$	−4921.73	−0.667051	−0.333526	−	0.942741i	$-0.608238\pi$
$379$	−4921.73	−0.667051	−0.333526	+	0.942741i	$0.608238\pi$
$380$	3022.54	0.408035
$381$	0	0
$382$	−5387.76	−0.721627
$383$	−8091.03	−1.07946	−0.539729	−	0.841839i	$-0.681473\pi$
$383$	−8091.03	−1.07946	−0.539729	+	0.841839i	$0.681473\pi$
$384$	0	0
$385$	0	0
$386$	−7037.22	−0.927940
$387$	0	0
$388$	−5318.67	−0.695914
$389$	13081.1	1.70499	0.852494	−	0.522737i	$-0.175089\pi$
$389$	13081.1	1.70499	0.852494	+	0.522737i	$0.175089\pi$
$390$	0	0
$391$	−7631.77	−0.987098
$392$	−5494.46	−0.707939
$393$	0	0
$394$	−9737.58	−1.24511
$395$	7097.94	0.904142
$396$	0	0
$397$	−7707.71	−0.974405	−0.487203	−	0.873289i	$-0.661983\pi$
$397$	−7707.71	−0.974405	−0.487203	+	0.873289i	$0.661983\pi$
$398$	8400.88	1.05804
$399$	0	0
$400$	−1369.36	−0.171170
$401$	7838.19	0.976110	0.488055	−	0.872813i	$-0.337706\pi$
$401$	7838.19	0.976110	0.488055	+	0.872813i	$0.337706\pi$
$402$	0	0
$403$	3207.15	0.396426
$404$	5583.84	0.687640
$405$	0	0
$406$	−4294.53	−0.524960
$407$	0	0
$408$	0	0
$409$	2367.43	0.286215	0.143107	−	0.989707i	$-0.454291\pi$
$409$	2367.43	0.286215	0.143107	+	0.989707i	$0.454291\pi$
$410$	3133.26	0.377416
$411$	0	0
$412$	−649.516	−0.0776683
$413$	−9591.89	−1.14282
$414$	0	0
$415$	−6562.06	−0.776190
$416$	−773.574	−0.0911721
$417$	0	0
$418$	0	0
$419$	−1221.38	−0.142407	−0.0712033	−	0.997462i	$-0.522684\pi$
$419$	−1221.38	−0.142407	−0.0712033	+	0.997462i	$0.522684\pi$
$420$	0	0
$421$	−3696.33	−0.427905	−0.213952	−	0.976844i	$-0.568634\pi$
$421$	−3696.33	−0.427905	−0.213952	+	0.976844i	$0.568634\pi$
$422$	−4735.07	−0.546207
$423$	0	0
$424$	−5778.18	−0.661824
$425$	9096.91	1.03827
$426$	0	0
$427$	−10766.7	−1.22023
$428$	−5.36012	−0.000605353 0
$429$	0	0
$430$	−2740.04	−0.307294
$431$	−7884.17	−0.881130	−0.440565	−	0.897721i	$-0.645222\pi$
$431$	−7884.17	−0.881130	−0.440565	+	0.897721i	$0.645222\pi$
$432$	0	0
$433$	−16140.1	−1.79133	−0.895665	−	0.444729i	$-0.853300\pi$
$433$	−16140.1	−1.79133	−0.895665	+	0.444729i	$0.853300\pi$
$434$	8514.82	0.941762
$435$	0	0
$436$	−3999.97	−0.439367
$437$	8641.91	0.945992
$438$	0	0
$439$	1026.67	0.111618	0.0558092	−	0.998441i	$-0.482226\pi$
$439$	1026.67	0.111618	0.0558092	+	0.998441i	$0.482226\pi$
$440$	0	0
$441$	0	0
$442$	5139.00	0.553026
$443$	−9365.88	−1.00448	−0.502242	−	0.864727i	$-0.667491\pi$
$443$	−9365.88	−1.00448	−0.502242	+	0.864727i	$0.667491\pi$
$444$	0	0
$445$	−2255.73	−0.240296
$446$	−2726.55	−0.289475
$447$	0	0
$448$	−2053.80	−0.216591
$449$	−14727.5	−1.54795	−0.773977	−	0.633213i	$-0.781736\pi$
$449$	−14727.5	−1.54795	−0.773977	+	0.633213i	$0.781736\pi$
$450$	0	0
$451$	0	0
$452$	3654.41	0.380286
$453$	0	0
$454$	12211.5	1.26237
$455$	−4870.36	−0.501816
$456$	0	0
$457$	9061.30	0.927505	0.463752	−	0.885965i	$-0.346503\pi$
$457$	9061.30	0.927505	0.463752	+	0.885965i	$0.346503\pi$
$458$	−8888.31	−0.906820
$459$	0	0
$460$	1803.10	0.182761
$461$	−12699.2	−1.28300	−0.641498	−	0.767125i	$-0.721687\pi$
$461$	−12699.2	−1.28300	−0.641498	+	0.767125i	$0.721687\pi$
$462$	0	0
$463$	−16296.0	−1.63572	−0.817861	−	0.575416i	$-0.804840\pi$
$463$	−16296.0	−1.63572	−0.817861	+	0.575416i	$0.804840\pi$
$464$	−1070.60	−0.107115
$465$	0	0
$466$	6487.84	0.644943
$467$	4636.89	0.459464	0.229732	−	0.973254i	$-0.426215\pi$
$467$	4636.89	0.459464	0.229732	+	0.973254i	$0.426215\pi$
$468$	0	0
$469$	3052.84	0.300570
$470$	5274.41	0.517639
$471$	0	0
$472$	−2391.20	−0.233186
$473$	0	0
$474$	0	0
$475$	−10301.0	−0.995034
$476$	13643.8	1.31379
$477$	0	0
$478$	4304.24	0.411865
$479$	10495.0	1.00111	0.500553	−	0.865706i	$-0.333130\pi$
$479$	10495.0	1.00111	0.500553	+	0.865706i	$0.333130\pi$
$480$	0	0
$481$	7192.49	0.681808
$482$	−2908.81	−0.274881
$483$	0	0
$484$	0	0
$485$	−8347.84	−0.781559
$486$	0	0
$487$	−1027.10	−0.0955695	−0.0477847	−	0.998858i	$-0.515216\pi$
$487$	−1027.10	−0.0955695	−0.0477847	+	0.998858i	$0.515216\pi$
$488$	−2684.08	−0.248981
$489$	0	0
$490$	−8623.76	−0.795064
$491$	2020.52	0.185713	0.0928564	−	0.995680i	$-0.470400\pi$
$491$	2020.52	0.185713	0.0928564	+	0.995680i	$0.470400\pi$
$492$	0	0
$493$	7112.21	0.649732
$494$	−5819.20	−0.529996
$495$	0	0
$496$	2122.70	0.192161
$497$	8702.35	0.785420
$498$	0	0
$499$	21173.0	1.89946	0.949731	−	0.313066i	$-0.101356\pi$
$499$	21173.0	1.89946	0.949731	+	0.313066i	$0.101356\pi$
$500$	−5288.33	−0.473003
$501$	0	0
$502$	8797.46	0.782171
$503$	5391.50	0.477923	0.238961	−	0.971029i	$-0.423193\pi$
$503$	5391.50	0.477923	0.238961	+	0.971029i	$0.423193\pi$
$504$	0	0
$505$	8764.05	0.772267
$506$	0	0
$507$	0	0
$508$	426.205	0.0372240
$509$	−2012.03	−0.175209	−0.0876047	−	0.996155i	$-0.527921\pi$
$509$	−2012.03	−0.175209	−0.0876047	+	0.996155i	$0.527921\pi$
$510$	0	0
$511$	31576.7	2.73360
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−4036.95	−0.346425
$515$	−1019.44	−0.0872269
$516$	0	0
$517$	0	0
$518$	19095.7	1.61972
$519$	0	0
$520$	−1214.15	−0.102393
$521$	−13209.1	−1.11075	−0.555374	−	0.831600i	$-0.687425\pi$
$521$	−13209.1	−1.11075	−0.555374	+	0.831600i	$0.687425\pi$
$522$	0	0
$523$	−3584.21	−0.299668	−0.149834	−	0.988711i	$-0.547874\pi$
$523$	−3584.21	−0.299668	−0.149834	+	0.988711i	$0.547874\pi$
$524$	−11229.8	−0.936214
$525$	0	0
$526$	−8608.97	−0.713629
$527$	−14101.5	−1.16560
$528$	0	0
$529$	−7011.66	−0.576285
$530$	−9069.06	−0.743274
$531$	0	0
$532$	−15449.7	−1.25908
$533$	−6032.35	−0.490225
$534$	0	0
$535$	−8.41291	−0.000679854 0
$536$	761.056	0.0613295
$537$	0	0
$538$	371.249	0.0297503
$539$	0	0
$540$	0	0
$541$	4372.63	0.347494	0.173747	−	0.984790i	$-0.444413\pi$
$541$	4372.63	0.347494	0.173747	+	0.984790i	$0.444413\pi$
$542$	13435.6	1.06477
$543$	0	0
$544$	3401.31	0.268070
$545$	−6278.10	−0.493439
$546$	0	0
$547$	−15753.5	−1.23139	−0.615696	−	0.787984i	$-0.711125\pi$
$547$	−15753.5	−1.23139	−0.615696	+	0.787984i	$0.711125\pi$
$548$	−4994.95	−0.389368
$549$	0	0
$550$	0	0
$551$	−8053.57	−0.622675
$552$	0	0
$553$	−36281.0	−2.78992
$554$	1747.38	0.134006
$555$	0	0
$556$	8846.60	0.674783
$557$	13100.6	0.996568	0.498284	−	0.867014i	$-0.333964\pi$
$557$	13100.6	0.996568	0.498284	+	0.867014i	$0.333964\pi$
$558$	0	0
$559$	5275.30	0.399144
$560$	−3223.51	−0.243247
$561$	0	0
$562$	−14260.9	−1.07039
$563$	−11068.4	−0.828556	−0.414278	−	0.910150i	$-0.635966\pi$
$563$	−11068.4	−0.828556	−0.414278	+	0.910150i	$0.635966\pi$
$564$	0	0
$565$	5735.73	0.427087
$566$	8587.56	0.637742
$567$	0	0
$568$	2169.45	0.160260
$569$	624.533	0.0460137	0.0230068	−	0.999735i	$-0.492676\pi$
$569$	624.533	0.0460137	0.0230068	+	0.999735i	$0.492676\pi$
$570$	0	0
$571$	24257.0	1.77780	0.888899	−	0.458104i	$-0.151471\pi$
$571$	24257.0	1.77780	0.888899	+	0.458104i	$0.151471\pi$
$572$	0	0
$573$	0	0
$574$	−16015.6	−1.16460
$575$	−6145.05	−0.445681
$576$	0	0
$577$	−4484.35	−0.323546	−0.161773	−	0.986828i	$-0.551721\pi$
$577$	−4484.35	−0.323546	−0.161773	+	0.986828i	$0.551721\pi$
$578$	−12769.6	−0.918936
$579$	0	0
$580$	−1680.35	−0.120298
$581$	33541.9	2.39510
$582$	0	0
$583$	0	0
$584$	7871.88	0.557775
$585$	0	0
$586$	9242.80	0.651564
$587$	−18000.1	−1.26566	−0.632832	−	0.774289i	$-0.718108\pi$
$587$	−18000.1	−1.26566	−0.632832	+	0.774289i	$0.718108\pi$
$588$	0	0
$589$	15967.9	1.11706
$590$	−3753.08	−0.261884
$591$	0	0
$592$	4760.45	0.330495
$593$	21301.0	1.47509	0.737545	−	0.675297i	$-0.235985\pi$
$593$	21301.0	1.47509	0.737545	+	0.675297i	$0.235985\pi$
$594$	0	0
$595$	21414.4	1.47547
$596$	−2081.15	−0.143032
$597$	0	0
$598$	−3471.45	−0.237388
$599$	−12898.6	−0.879838	−0.439919	−	0.898037i	$-0.644993\pi$
$599$	−12898.6	−0.879838	−0.439919	+	0.898037i	$0.644993\pi$
$600$	0	0
$601$	−20347.0	−1.38099	−0.690493	−	0.723339i	$-0.742607\pi$
$601$	−20347.0	−1.38099	−0.690493	+	0.723339i	$0.742607\pi$
$602$	14005.7	0.948220
$603$	0	0
$604$	−13255.9	−0.893008
$605$	0	0
$606$	0	0
$607$	−1791.89	−0.119820	−0.0599098	−	0.998204i	$-0.519081\pi$
$607$	−1791.89	−0.119820	−0.0599098	+	0.998204i	$0.519081\pi$
$608$	−3851.51	−0.256907
$609$	0	0
$610$	−4212.76	−0.279622
$611$	−10154.6	−0.672361
$612$	0	0
$613$	5518.24	0.363588	0.181794	−	0.983337i	$-0.441810\pi$
$613$	5518.24	0.363588	0.181794	+	0.983337i	$0.441810\pi$
$614$	14191.8	0.932795
$615$	0	0
$616$	0	0
$617$	2771.91	0.180864	0.0904319	−	0.995903i	$-0.471175\pi$
$617$	2771.91	0.180864	0.0904319	+	0.995903i	$0.471175\pi$
$618$	0	0
$619$	5500.30	0.357150	0.178575	−	0.983926i	$-0.442851\pi$
$619$	5500.30	0.357150	0.178575	+	0.983926i	$0.442851\pi$
$620$	3331.65	0.215810
$621$	0	0
$622$	19114.4	1.23218
$623$	11530.1	0.741482
$624$	0	0
$625$	2397.88	0.153464
$626$	−5278.54	−0.337018
$627$	0	0
$628$	4269.97	0.271322
$629$	−31624.6	−2.00470
$630$	0	0
$631$	−9506.23	−0.599741	−0.299871	−	0.953980i	$-0.596944\pi$
$631$	−9506.23	−0.599741	−0.299871	+	0.953980i	$0.596944\pi$
$632$	−9044.63	−0.569266
$633$	0	0
$634$	−15838.9	−0.992184
$635$	668.944	0.0418051
$636$	0	0
$637$	16603.0	1.03271
$638$	0	0
$639$	0	0
$640$	−803.603	−0.0496331
$641$	18945.3	1.16738	0.583692	−	0.811975i	$-0.301607\pi$
$641$	18945.3	1.16738	0.583692	+	0.811975i	$0.301607\pi$
$642$	0	0
$643$	−14788.0	−0.906969	−0.453485	−	0.891264i	$-0.649819\pi$
$643$	−14788.0	−0.906969	−0.453485	+	0.891264i	$0.649819\pi$
$644$	−9216.51	−0.563946
$645$	0	0
$646$	25586.3	1.55833
$647$	3523.14	0.214079	0.107039	−	0.994255i	$-0.465863\pi$
$647$	3523.14	0.214079	0.107039	+	0.994255i	$0.465863\pi$
$648$	0	0
$649$	0	0
$650$	4137.89	0.249694
$651$	0	0
$652$	−4696.18	−0.282081
$653$	−15408.4	−0.923395	−0.461698	−	0.887037i	$-0.652759\pi$
$653$	−15408.4	−0.923395	−0.461698	+	0.887037i	$0.652759\pi$
$654$	0	0
$655$	−17625.6	−1.05143
$656$	−3992.59	−0.237629
$657$	0	0
$658$	−26960.0	−1.59728
$659$	2280.11	0.134781	0.0673905	−	0.997727i	$-0.478533\pi$
$659$	2280.11	0.134781	0.0673905	+	0.997727i	$0.478533\pi$
$660$	0	0
$661$	15098.9	0.888473	0.444236	−	0.895910i	$-0.353475\pi$
$661$	15098.9	0.888473	0.444236	+	0.895910i	$0.353475\pi$
$662$	−471.917	−0.0277063
$663$	0	0
$664$	8361.79	0.488705
$665$	−24248.8	−1.41403
$666$	0	0
$667$	−4804.37	−0.278899
$668$	1290.50	0.0747469
$669$	0	0
$670$	1194.51	0.0688772
$671$	0	0
$672$	0	0
$673$	2188.38	0.125343	0.0626716	−	0.998034i	$-0.480038\pi$
$673$	2188.38	0.125343	0.0626716	+	0.998034i	$0.480038\pi$
$674$	7468.58	0.426823
$675$	0	0
$676$	−6450.44	−0.367002
$677$	−879.954	−0.0499548	−0.0249774	−	0.999688i	$-0.507951\pi$
$677$	−879.954	−0.0499548	−0.0249774	+	0.999688i	$0.507951\pi$
$678$	0	0
$679$	42669.8	2.41166
$680$	5338.49	0.301061
$681$	0	0
$682$	0	0
$683$	−16135.6	−0.903972	−0.451986	−	0.892025i	$-0.649284\pi$
$683$	−16135.6	−0.903972	−0.451986	+	0.892025i	$0.649284\pi$
$684$	0	0
$685$	−7839.76	−0.437287
$686$	22066.0	1.22811
$687$	0	0
$688$	3491.53	0.193479
$689$	17460.3	0.965437
$690$	0	0
$691$	−6909.92	−0.380414	−0.190207	−	0.981744i	$-0.560916\pi$
$691$	−6909.92	−0.380414	−0.190207	+	0.981744i	$0.560916\pi$
$692$	−9230.97	−0.507094
$693$	0	0
$694$	12298.0	0.672661
$695$	13885.1	0.757828
$696$	0	0
$697$	26523.6	1.44139
$698$	14053.6	0.762087
$699$	0	0
$700$	10985.9	0.593182
$701$	−15459.7	−0.832959	−0.416479	−	0.909145i	$-0.636736\pi$
$701$	−15459.7	−0.832959	−0.416479	+	0.909145i	$0.636736\pi$
$702$	0	0
$703$	35810.4	1.92121
$704$	0	0
$705$	0	0
$706$	23791.7	1.26829
$707$	−44797.2	−2.38299
$708$	0	0
$709$	−10097.4	−0.534861	−0.267431	−	0.963577i	$-0.586175\pi$
$709$	−10097.4	−0.534861	−0.267431	+	0.963577i	$0.586175\pi$
$710$	3405.02	0.179983
$711$	0	0
$712$	2874.38	0.151295
$713$	9525.69	0.500336
$714$	0	0
$715$	0	0
$716$	−2098.09	−0.109510
$717$	0	0
$718$	1609.61	0.0836632
$719$	3012.20	0.156239	0.0781197	−	0.996944i	$-0.475108\pi$
$719$	3012.20	0.156239	0.0781197	+	0.996944i	$0.475108\pi$
$720$	0	0
$721$	5210.84	0.269157
$722$	−15254.9	−0.786329
$723$	0	0
$724$	4133.89	0.212203
$725$	5726.70	0.293358
$726$	0	0
$727$	6256.59	0.319180	0.159590	−	0.987183i	$-0.448983\pi$
$727$	6256.59	0.319180	0.159590	+	0.987183i	$0.448983\pi$
$728$	6206.11	0.315953
$729$	0	0
$730$	12355.2	0.626420
$731$	−23194.9	−1.17359
$732$	0	0
$733$	−12249.4	−0.617245	−0.308622	−	0.951185i	$-0.599868\pi$
$733$	−12249.4	−0.617245	−0.308622	+	0.951185i	$0.599868\pi$
$734$	21689.9	1.09072
$735$	0	0
$736$	−2297.62	−0.115070
$737$	0	0
$738$	0	0
$739$	19117.1	0.951601	0.475800	−	0.879553i	$-0.342159\pi$
$739$	19117.1	0.951601	0.475800	+	0.879553i	$0.342159\pi$
$740$	7471.69	0.371169
$741$	0	0
$742$	46356.3	2.29352
$743$	11615.1	0.573510	0.286755	−	0.958004i	$-0.407423\pi$
$743$	11615.1	0.573510	0.286755	+	0.958004i	$0.407423\pi$
$744$	0	0
$745$	−3266.44	−0.160635
$746$	4842.33	0.237655
$747$	0	0
$748$	0	0
$749$	43.0024	0.00209783
$750$	0	0
$751$	31001.2	1.50633	0.753163	−	0.657833i	$-0.228527\pi$
$751$	31001.2	1.50633	0.753163	+	0.657833i	$0.228527\pi$
$752$	−6720.98	−0.325916
$753$	0	0
$754$	3235.11	0.156254
$755$	−20805.7	−1.00291
$756$	0	0
$757$	1830.43	0.0878837	0.0439418	−	0.999034i	$-0.486008\pi$
$757$	1830.43	0.0878837	0.0439418	+	0.999034i	$0.486008\pi$
$758$	9843.46	0.471676
$759$	0	0
$760$	−6045.09	−0.288524
$761$	27976.3	1.33264	0.666321	−	0.745665i	$-0.267868\pi$
$761$	27976.3	1.33264	0.666321	+	0.745665i	$0.267868\pi$
$762$	0	0
$763$	32090.4	1.52261
$764$	10775.5	0.510268
$765$	0	0
$766$	16182.1	0.763292
$767$	7225.66	0.340161
$768$	0	0
$769$	−11867.5	−0.556505	−0.278252	−	0.960508i	$-0.589755\pi$
$769$	−11867.5	−0.556505	−0.278252	+	0.960508i	$0.589755\pi$
$770$	0	0
$771$	0	0
$772$	14074.4	0.656153
$773$	5572.82	0.259302	0.129651	−	0.991560i	$-0.458614\pi$
$773$	5572.82	0.259302	0.129651	+	0.991560i	$0.458614\pi$
$774$	0	0
$775$	−11354.4	−0.526274
$776$	10637.3	0.492085
$777$	0	0
$778$	−26162.3	−1.20561
$779$	−30034.2	−1.38137
$780$	0	0
$781$	0	0
$782$	15263.5	0.697984
$783$	0	0
$784$	10988.9	0.500589
$785$	6701.88	0.304714
$786$	0	0
$787$	22315.3	1.01074	0.505371	−	0.862902i	$-0.331356\pi$
$787$	22315.3	1.01074	0.505371	+	0.862902i	$0.331356\pi$
$788$	19475.2	0.880424
$789$	0	0
$790$	−14195.9	−0.639325
$791$	−29318.1	−1.31787
$792$	0	0
$793$	8110.68	0.363201
$794$	15415.4	0.689009
$795$	0	0
$796$	−16801.8	−0.748144
$797$	−1682.22	−0.0747646	−0.0373823	−	0.999301i	$-0.511902\pi$
$797$	−1682.22	−0.0747646	−0.0373823	+	0.999301i	$0.511902\pi$
$798$	0	0
$799$	44648.8	1.97692
$800$	2738.72	0.121035
$801$	0	0
$802$	−15676.4	−0.690214
$803$	0	0
$804$	0	0
$805$	−14465.7	−0.633351
$806$	−6414.30	−0.280315
$807$	0	0
$808$	−11167.7	−0.486235
$809$	−5425.93	−0.235804	−0.117902	−	0.993025i	$-0.537617\pi$
$809$	−5425.93	−0.235804	−0.117902	+	0.993025i	$0.537617\pi$
$810$	0	0
$811$	3182.64	0.137802	0.0689010	−	0.997624i	$-0.478051\pi$
$811$	3182.64	0.137802	0.0689010	+	0.997624i	$0.478051\pi$
$812$	8589.06	0.371203
$813$	0	0
$814$	0	0
$815$	−7370.83	−0.316796
$816$	0	0
$817$	26265.0	1.12472
$818$	−4734.86	−0.202384
$819$	0	0
$820$	−6266.52	−0.266874
$821$	7927.50	0.336993	0.168497	−	0.985702i	$-0.446109\pi$
$821$	7927.50	0.336993	0.168497	+	0.985702i	$0.446109\pi$
$822$	0	0
$823$	−15276.4	−0.647026	−0.323513	−	0.946224i	$-0.604864\pi$
$823$	−15276.4	−0.647026	−0.323513	+	0.946224i	$0.604864\pi$
$824$	1299.03	0.0549198
$825$	0	0
$826$	19183.8	0.808098
$827$	−6276.71	−0.263921	−0.131960	−	0.991255i	$-0.542127\pi$
$827$	−6276.71	−0.263921	−0.131960	+	0.991255i	$0.542127\pi$
$828$	0	0
$829$	−5476.37	−0.229436	−0.114718	−	0.993398i	$-0.536596\pi$
$829$	−5476.37	−0.229436	−0.114718	+	0.993398i	$0.536596\pi$
$830$	13124.1	0.548850
$831$	0	0
$832$	1547.15	0.0644684
$833$	−73001.5	−3.03644
$834$	0	0
$835$	2025.49	0.0839459
$836$	0	0
$837$	0	0
$838$	2442.76	0.100697
$839$	6283.73	0.258568	0.129284	−	0.991608i	$-0.458732\pi$
$839$	6283.73	0.258568	0.129284	+	0.991608i	$0.458732\pi$
$840$	0	0
$841$	−19911.7	−0.816422
$842$	7392.66	0.302574
$843$	0	0
$844$	9470.13	0.386227
$845$	−10124.2	−0.412169
$846$	0	0
$847$	0	0
$848$	11556.4	0.467980
$849$	0	0
$850$	−18193.8	−0.734168
$851$	21362.7	0.860522
$852$	0	0
$853$	−13438.3	−0.539413	−0.269707	−	0.962943i	$-0.586927\pi$
$853$	−13438.3	−0.539413	−0.269707	+	0.962943i	$0.586927\pi$
$854$	21533.4	0.862833
$855$	0	0
$856$	10.7202	0.000428050 0
$857$	20901.6	0.833122	0.416561	−	0.909108i	$-0.363235\pi$
$857$	20901.6	0.833122	0.416561	+	0.909108i	$0.363235\pi$
$858$	0	0
$859$	7765.47	0.308445	0.154223	−	0.988036i	$-0.450713\pi$
$859$	7765.47	0.308445	0.154223	+	0.988036i	$0.450713\pi$
$860$	5480.08	0.217290
$861$	0	0
$862$	15768.3	0.623053
$863$	−49326.8	−1.94566	−0.972830	−	0.231521i	$-0.925630\pi$
$863$	−49326.8	−1.94566	−0.972830	+	0.231521i	$0.925630\pi$
$864$	0	0
$865$	−14488.3	−0.569501
$866$	32280.3	1.26666
$867$	0	0
$868$	−17029.6	−0.665926
$869$	0	0
$870$	0	0
$871$	−2299.74	−0.0894646
$872$	7999.94	0.310679
$873$	0	0
$874$	−17283.8	−0.668918
$875$	42426.4	1.63917
$876$	0	0
$877$	−4318.04	−0.166260	−0.0831300	−	0.996539i	$-0.526492\pi$
$877$	−4318.04	−0.166260	−0.0831300	+	0.996539i	$0.526492\pi$
$878$	−2053.35	−0.0789261
$879$	0	0
$880$	0	0
$881$	34689.4	1.32658	0.663290	−	0.748362i	$-0.269160\pi$
$881$	34689.4	1.32658	0.663290	+	0.748362i	$0.269160\pi$
$882$	0	0
$883$	−41300.1	−1.57402	−0.787010	−	0.616941i	$-0.788372\pi$
$883$	−41300.1	−1.57402	−0.787010	+	0.616941i	$0.788372\pi$
$884$	−10278.0	−0.391048
$885$	0	0
$886$	18731.8	0.710277
$887$	32373.5	1.22547	0.612737	−	0.790287i	$-0.290068\pi$
$887$	32373.5	1.22547	0.612737	+	0.790287i	$0.290068\pi$
$888$	0	0
$889$	−3419.29	−0.128998
$890$	4511.45	0.169915
$891$	0	0
$892$	5453.09	0.204690
$893$	−50558.4	−1.89460
$894$	0	0
$895$	−3293.03	−0.122988
$896$	4107.60	0.153153
$897$	0	0
$898$	29454.9	1.09457
$899$	−8877.18	−0.329333
$900$	0	0
$901$	−76771.1	−2.83864
$902$	0	0
$903$	0	0
$904$	−7308.83	−0.268903
$905$	6488.29	0.238318
$906$	0	0
$907$	−5922.77	−0.216827	−0.108414	−	0.994106i	$-0.534577\pi$
$907$	−5922.77	−0.216827	−0.108414	+	0.994106i	$0.534577\pi$
$908$	−24423.1	−0.892630
$909$	0	0
$910$	9740.72	0.354837
$911$	−1075.02	−0.0390965	−0.0195483	−	0.999809i	$-0.506223\pi$
$911$	−1075.02	−0.0390965	−0.0195483	+	0.999809i	$0.506223\pi$
$912$	0	0
$913$	0	0
$914$	−18122.6	−0.655845
$915$	0	0
$916$	17776.6	0.641219
$917$	90092.8	3.24441
$918$	0	0
$919$	11283.5	0.405013	0.202507	−	0.979281i	$-0.435091\pi$
$919$	11283.5	0.405013	0.202507	+	0.979281i	$0.435091\pi$
$920$	−3606.20	−0.129231
$921$	0	0
$922$	25398.4	0.907215
$923$	−6555.57	−0.233780
$924$	0	0
$925$	−25463.9	−0.905133
$926$	32592.0	1.15663
$927$	0	0
$928$	2141.20	0.0757418
$929$	8138.20	0.287412	0.143706	−	0.989620i	$-0.454098\pi$
$929$	8138.20	0.287412	0.143706	+	0.989620i	$0.454098\pi$
$930$	0	0
$931$	82664.0	2.90999
$932$	−12975.7	−0.456043
$933$	0	0
$934$	−9273.79	−0.324890
$935$	0	0
$936$	0	0
$937$	28286.7	0.986217	0.493109	−	0.869968i	$-0.335861\pi$
$937$	28286.7	0.986217	0.493109	+	0.869968i	$0.335861\pi$
$938$	−6105.69	−0.212535
$939$	0	0
$940$	−10548.8	−0.366026
$941$	−20864.3	−0.722803	−0.361402	−	0.932410i	$-0.617702\pi$
$941$	−20864.3	−0.722803	−0.361402	+	0.932410i	$0.617702\pi$
$942$	0	0
$943$	−17916.9	−0.618723
$944$	4782.40	0.164888
$945$	0	0
$946$	0	0
$947$	−18025.8	−0.618543	−0.309271	−	0.950974i	$-0.600085\pi$
$947$	−18025.8	−0.618543	−0.309271	+	0.950974i	$0.600085\pi$
$948$	0	0
$949$	−23787.0	−0.813656
$950$	20601.9	0.703595
$951$	0	0
$952$	−27287.6	−0.928987
$953$	−9938.82	−0.337828	−0.168914	−	0.985631i	$-0.554026\pi$
$953$	−9938.82	−0.337828	−0.168914	+	0.985631i	$0.554026\pi$
$954$	0	0
$955$	16912.6	0.573066
$956$	−8608.48	−0.291232
$957$	0	0
$958$	−20990.0	−0.707889
$959$	40072.8	1.34934
$960$	0	0
$961$	−12190.1	−0.409187
$962$	−14385.0	−0.482111
$963$	0	0
$964$	5817.62	0.194370
$965$	22090.3	0.736905
$966$	0	0
$967$	−35056.9	−1.16582	−0.582912	−	0.812535i	$-0.698087\pi$
$967$	−35056.9	−1.16582	−0.582912	+	0.812535i	$0.698087\pi$
$968$	0	0
$969$	0	0
$970$	16695.7	0.552646
$971$	34695.4	1.14668	0.573342	−	0.819316i	$-0.305647\pi$
$971$	34695.4	1.14668	0.573342	+	0.819316i	$0.305647\pi$
$972$	0	0
$973$	−70973.2	−2.33843
$974$	2054.20	0.0675778
$975$	0	0
$976$	5368.16	0.176056
$977$	−45795.6	−1.49962	−0.749812	−	0.661651i	$-0.769856\pi$
$977$	−45795.6	−1.49962	−0.749812	+	0.661651i	$0.769856\pi$
$978$	0	0
$979$	0	0
$980$	17247.5	0.562195
$981$	0	0
$982$	−4041.05	−0.131319
$983$	−26090.6	−0.846551	−0.423276	−	0.906001i	$-0.639120\pi$
$983$	−26090.6	−0.846551	−0.423276	+	0.906001i	$0.639120\pi$
$984$	0	0
$985$	30567.0	0.988777
$986$	−14224.4	−0.459430
$987$	0	0
$988$	11638.4	0.374764
$989$	15668.4	0.503767
$990$	0	0
$991$	−16965.7	−0.543827	−0.271913	−	0.962322i	$-0.587656\pi$
$991$	−16965.7	−0.543827	−0.271913	+	0.962322i	$0.587656\pi$
$992$	−4245.39	−0.135878
$993$	0	0
$994$	−17404.7	−0.555376
$995$	−26371.0	−0.840217
$996$	0	0
$997$	−27153.2	−0.862539	−0.431270	−	0.902223i	$-0.641934\pi$
$997$	−27153.2	−0.862539	−0.431270	+	0.902223i	$0.641934\pi$
$998$	−42345.9	−1.34312
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2178.4.a.bx.1.3		4
3.2	odd	2		726.4.a.y.1.2	yes	4
11.10	odd	2		2178.4.a.cb.1.3		4
33.32	even	2		726.4.a.v.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
726.4.a.v.1.2	✓	4	33.32	even	2
726.4.a.y.1.2	yes	4	3.2	odd	2
2178.4.a.bx.1.3		4	1.1	even	1	trivial
2178.4.a.cb.1.3		4	11.10	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2178.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} +6.27815 q^{5} -32.0906 q^{7} -8.00000 q^{8} -12.5563 q^{10} +24.1742 q^{13} +64.1812 q^{14} +16.0000 q^{16} -106.291 q^{17} +120.360 q^{19} +25.1126 q^{20} +71.8007 q^{23} -85.5849 q^{25} -48.3484 q^{26} -128.362 q^{28} -66.9125 q^{29} +132.668 q^{31} -32.0000 q^{32} +212.582 q^{34} -201.470 q^{35} +297.528 q^{37} -240.719 q^{38} -50.2252 q^{40} -249.537 q^{41} +218.221 q^{43} -143.601 q^{46} -420.061 q^{47} +686.808 q^{49} +171.170 q^{50} +96.6967 q^{52} +722.272 q^{53} +256.725 q^{56} +133.825 q^{58} +298.900 q^{59} +335.510 q^{61} -265.337 q^{62} +64.0000 q^{64} +151.769 q^{65} -95.1320 q^{67} -425.164 q^{68} +402.939 q^{70} -271.181 q^{71} -983.985 q^{73} -595.056 q^{74} +481.439 q^{76} +1130.58 q^{79} +100.450 q^{80} +499.074 q^{82} -1045.22 q^{83} -667.311 q^{85} -436.441 q^{86} -359.298 q^{89} -775.764 q^{91} +287.203 q^{92} +840.122 q^{94} +755.636 q^{95} -1329.67 q^{97} -1373.62 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} + 16 q^{4} + 6 q^{5} + 12 q^{7} - 32 q^{8} - 12 q^{10} + 12 q^{13} - 24 q^{14} + 64 q^{16} - 108 q^{17} + 192 q^{19} + 24 q^{20} - 156 q^{23} + 286 q^{25} - 24 q^{26} + 48 q^{28} - 408 q^{29}+ \cdots - 1348 q^{98}+O(q^{100}) \) 4 * q - 8 * q^2 + 16 * q^4 + 6 * q^5 + 12 * q^7 - 32 * q^8 - 12 * q^10 + 12 * q^13 - 24 * q^14 + 64 * q^16 - 108 * q^17 + 192 * q^19 + 24 * q^20 - 156 * q^23 + 286 * q^25 - 24 * q^26 + 48 * q^28 - 408 * q^29 - 26 * q^31 - 128 * q^32 + 216 * q^34 - 588 * q^35 - 224 * q^37 - 384 * q^38 - 48 * q^40 - 348 * q^41 + 540 * q^43 + 312 * q^46 - 132 * q^47 + 674 * q^49 - 572 * q^50 + 48 * q^52 + 1470 * q^53 - 96 * q^56 + 816 * q^58 + 684 * q^59 + 1320 * q^61 + 52 * q^62 + 256 * q^64 - 2556 * q^65 - 530 * q^67 - 432 * q^68 + 1176 * q^70 - 936 * q^71 - 2352 * q^73 + 448 * q^74 + 768 * q^76 - 192 * q^79 + 96 * q^80 + 696 * q^82 - 3192 * q^83 - 2340 * q^85 - 1080 * q^86 - 726 * q^89 + 360 * q^91 - 624 * q^92 + 264 * q^94 + 2064 * q^95 - 100 * q^97 - 1348 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

Properties

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