LMFDB - Embedded newform 1856.4.a.bg.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1856,4,Mod(1,1856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1856, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1856.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1856 = 2^{6} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1856.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:	$109.507544971$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 138x^{7} + 394x^{6} + 5872x^{5} - 10822x^{4} - 85158x^{3} + 30654x^{2} + 439999x + 396802 $ x^9 - 4x^8 - 138x^7 + 394x^6 + 5872x^5 - 10822x^4 - 85158x^3 + 30654x^2 + 439999x + 396802
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 928)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$9.74094$ of defining polynomial
Character	$\chi$	$=$	1856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.74094 q^{3} -17.5483 q^{5} -10.7391 q^{7} +67.8859 q^{9} +O(q^{10})$	$q+9.74094 q^{3} -17.5483 q^{5} -10.7391 q^{7} +67.8859 q^{9} +5.60196 q^{11} -17.1047 q^{13} -170.937 q^{15} -100.388 q^{17} +76.5128 q^{19} -104.609 q^{21} +76.5764 q^{23} +182.943 q^{25} +398.267 q^{27} +29.0000 q^{29} +181.082 q^{31} +54.5684 q^{33} +188.454 q^{35} +175.750 q^{37} -166.616 q^{39} +153.953 q^{41} -441.338 q^{43} -1191.28 q^{45} -431.277 q^{47} -227.671 q^{49} -977.874 q^{51} +178.303 q^{53} -98.3049 q^{55} +745.307 q^{57} -765.343 q^{59} -611.248 q^{61} -729.036 q^{63} +300.158 q^{65} -729.647 q^{67} +745.926 q^{69} -552.229 q^{71} -753.042 q^{73} +1782.04 q^{75} -60.1603 q^{77} -917.466 q^{79} +2046.58 q^{81} +225.355 q^{83} +1761.64 q^{85} +282.487 q^{87} +86.6665 q^{89} +183.690 q^{91} +1763.90 q^{93} -1342.67 q^{95} -1674.10 q^{97} +380.294 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 4 q^{3} - 10 q^{5} - 12 q^{7} + 49 q^{9}+O(q^{10}) $ 9 * q + 4 * q^3 - 10 * q^5 - 12 * q^7 + 49 * q^9	$ 9 q + 4 q^{3} - 10 q^{5} - 12 q^{7} + 49 q^{9} + 64 q^{11} - 70 q^{13} - 170 q^{15} - 66 q^{17} + 42 q^{19} - 76 q^{21} - 40 q^{23} + 111 q^{25} + 322 q^{27} + 261 q^{29} + 64 q^{31} - 52 q^{33} + 496 q^{35} + 54 q^{37} - 590 q^{39} - 378 q^{41} - 32 q^{43} - 1046 q^{45} - 1164 q^{47} - 351 q^{49} + 376 q^{51} - 278 q^{53} - 614 q^{55} + 28 q^{57} + 640 q^{59} - 1054 q^{61} - 1660 q^{63} - 708 q^{65} + 1184 q^{67} - 188 q^{69} - 1988 q^{71} - 750 q^{73} + 3126 q^{75} - 1260 q^{77} - 2916 q^{79} + 293 q^{81} + 2832 q^{83} - 56 q^{85} + 116 q^{87} - 370 q^{89} + 3016 q^{91} + 1696 q^{93} - 4412 q^{95} - 2234 q^{97} + 4118 q^{99}+O(q^{100}) $ 9 * q + 4 * q^3 - 10 * q^5 - 12 * q^7 + 49 * q^9 + 64 * q^11 - 70 * q^13 - 170 * q^15 - 66 * q^17 + 42 * q^19 - 76 * q^21 - 40 * q^23 + 111 * q^25 + 322 * q^27 + 261 * q^29 + 64 * q^31 - 52 * q^33 + 496 * q^35 + 54 * q^37 - 590 * q^39 - 378 * q^41 - 32 * q^43 - 1046 * q^45 - 1164 * q^47 - 351 * q^49 + 376 * q^51 - 278 * q^53 - 614 * q^55 + 28 * q^57 + 640 * q^59 - 1054 * q^61 - 1660 * q^63 - 708 * q^65 + 1184 * q^67 - 188 * q^69 - 1988 * q^71 - 750 * q^73 + 3126 * q^75 - 1260 * q^77 - 2916 * q^79 + 293 * q^81 + 2832 * q^83 - 56 * q^85 + 116 * q^87 - 370 * q^89 + 3016 * q^91 + 1696 * q^93 - 4412 * q^95 - 2234 * q^97 + 4118 * q^99

Display 100 coefficients 10 coefficients

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
$3$	9.74094	1.87464	0.937322	−	0.348464i	$-0.113297\pi$
$3$	9.74094	1.87464	0.937322	+	0.348464i	$0.113297\pi$
$4$	0	0
$5$	−17.5483	−1.56957	−0.784784	−	0.619770i	$-0.787226\pi$
$5$	−17.5483	−1.56957	−0.784784	+	0.619770i	$0.787226\pi$
$6$	0	0
$7$	−10.7391	−0.579859	−0.289930	−	0.957048i	$-0.593632\pi$
$7$	−10.7391	−0.579859	−0.289930	+	0.957048i	$0.593632\pi$
$8$	0	0
$9$	67.8859	2.51429
$10$	0	0
$11$	5.60196	0.153550	0.0767752	−	0.997048i	$-0.475538\pi$
$11$	5.60196	0.153550	0.0767752	+	0.997048i	$0.475538\pi$
$12$	0	0
$13$	−17.1047	−0.364922	−0.182461	−	0.983213i	$-0.558406\pi$
$13$	−17.1047	−0.364922	−0.182461	+	0.983213i	$0.558406\pi$
$14$	0	0
$15$	−170.937	−2.94238
$16$	0	0
$17$	−100.388	−1.43222	−0.716108	−	0.697989i	$-0.754078\pi$
$17$	−100.388	−1.43222	−0.716108	+	0.697989i	$0.754078\pi$
$18$	0	0
$19$	76.5128	0.923855	0.461927	−	0.886918i	$-0.347158\pi$
$19$	76.5128	0.923855	0.461927	+	0.886918i	$0.347158\pi$
$20$	0	0
$21$	−104.609	−1.08703
$22$	0	0
$23$	76.5764	0.694230	0.347115	−	0.937823i	$-0.387161\pi$
$23$	76.5764	0.694230	0.347115	+	0.937823i	$0.387161\pi$
$24$	0	0
$25$	182.943	1.46354
$26$	0	0
$27$	398.267	2.83876
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	181.082	1.04914	0.524568	−	0.851369i	$-0.324227\pi$
$31$	181.082	1.04914	0.524568	+	0.851369i	$0.324227\pi$
$32$	0	0
$33$	54.5684	0.287852
$34$	0	0
$35$	188.454	0.910128
$36$	0	0
$37$	175.750	0.780893	0.390447	−	0.920626i	$-0.372321\pi$
$37$	175.750	0.780893	0.390447	+	0.920626i	$0.372321\pi$
$38$	0	0
$39$	−166.616	−0.684100
$40$	0	0
$41$	153.953	0.586426	0.293213	−	0.956047i	$-0.405276\pi$
$41$	153.953	0.586426	0.293213	+	0.956047i	$0.405276\pi$
$42$	0	0
$43$	−441.338	−1.56520	−0.782598	−	0.622528i	$-0.786106\pi$
$43$	−441.338	−1.56520	−0.782598	+	0.622528i	$0.786106\pi$
$44$	0	0
$45$	−1191.28	−3.94635
$46$	0	0
$47$	−431.277	−1.33847	−0.669236	−	0.743050i	$-0.733378\pi$
$47$	−431.277	−1.33847	−0.669236	+	0.743050i	$0.733378\pi$
$48$	0	0
$49$	−227.671	−0.663763
$50$	0	0
$51$	−977.874	−2.68490
$52$	0	0
$53$	178.303	0.462108	0.231054	−	0.972941i	$-0.425783\pi$
$53$	178.303	0.462108	0.231054	+	0.972941i	$0.425783\pi$
$54$	0	0
$55$	−98.3049	−0.241008
$56$	0	0
$57$	745.307	1.73190
$58$	0	0
$59$	−765.343	−1.68880	−0.844399	−	0.535714i	$-0.820042\pi$
$59$	−765.343	−1.68880	−0.844399	+	0.535714i	$0.820042\pi$
$60$	0	0
$61$	−611.248	−1.28299	−0.641495	−	0.767128i	$-0.721685\pi$
$61$	−611.248	−1.28299	−0.641495	+	0.767128i	$0.721685\pi$
$62$	0	0
$63$	−729.036	−1.45794
$64$	0	0
$65$	300.158	0.572770
$66$	0	0
$67$	−729.647	−1.33046	−0.665228	−	0.746641i	$-0.731666\pi$
$67$	−729.647	−1.33046	−0.665228	+	0.746641i	$0.731666\pi$
$68$	0	0
$69$	745.926	1.30143
$70$	0	0
$71$	−552.229	−0.923063	−0.461532	−	0.887124i	$-0.652700\pi$
$71$	−552.229	−0.923063	−0.461532	+	0.887124i	$0.652700\pi$
$72$	0	0
$73$	−753.042	−1.20736	−0.603678	−	0.797229i	$-0.706299\pi$
$73$	−753.042	−1.20736	−0.603678	+	0.797229i	$0.706299\pi$
$74$	0	0
$75$	1782.04	2.74362
$76$	0	0
$77$	−60.1603	−0.0890376
$78$	0	0
$79$	−917.466	−1.30662	−0.653310	−	0.757091i	$-0.726620\pi$
$79$	−917.466	−1.30662	−0.653310	+	0.757091i	$0.726620\pi$
$80$	0	0
$81$	2046.58	2.80737
$82$	0	0
$83$	225.355	0.298023	0.149012	−	0.988835i	$-0.452391\pi$
$83$	225.355	0.298023	0.149012	+	0.988835i	$0.452391\pi$
$84$	0	0
$85$	1761.64	2.24796
$86$	0	0
$87$	282.487	0.348113
$88$	0	0
$89$	86.6665	0.103221	0.0516103	−	0.998667i	$-0.483565\pi$
$89$	86.6665	0.103221	0.0516103	+	0.998667i	$0.483565\pi$
$90$	0	0
$91$	183.690	0.211604
$92$	0	0
$93$	1763.90	1.96676
$94$	0	0
$95$	−1342.67	−1.45005
$96$	0	0
$97$	−1674.10	−1.75236	−0.876180	−	0.481984i	$-0.839916\pi$
$97$	−1674.10	−1.75236	−0.876180	+	0.481984i	$0.839916\pi$
$98$	0	0
$99$	380.294	0.386071
$100$	0	0
$101$	852.396	0.839768	0.419884	−	0.907578i	$-0.362071\pi$
$101$	852.396	0.839768	0.419884	+	0.907578i	$0.362071\pi$
$102$	0	0
$103$	−547.790	−0.524032	−0.262016	−	0.965063i	$-0.584387\pi$
$103$	−547.790	−0.524032	−0.262016	+	0.965063i	$0.584387\pi$
$104$	0	0
$105$	1835.72	1.70617
$106$	0	0
$107$	−313.407	−0.283161	−0.141580	−	0.989927i	$-0.545218\pi$
$107$	−313.407	−0.283161	−0.141580	+	0.989927i	$0.545218\pi$
$108$	0	0
$109$	−990.739	−0.870602	−0.435301	−	0.900285i	$-0.643358\pi$
$109$	−990.739	−0.870602	−0.435301	+	0.900285i	$0.643358\pi$
$110$	0	0
$111$	1711.97	1.46390
$112$	0	0
$113$	−619.659	−0.515864	−0.257932	−	0.966163i	$-0.583041\pi$
$113$	−619.659	−0.515864	−0.257932	+	0.966163i	$0.583041\pi$
$114$	0	0
$115$	−1343.79	−1.08964
$116$	0	0
$117$	−1161.17	−0.917521
$118$	0	0
$119$	1078.08	0.830484
$120$	0	0
$121$	−1299.62	−0.976422
$122$	0	0
$123$	1499.65	1.09934
$124$	0	0
$125$	−1016.80	−0.727562
$126$	0	0
$127$	−1120.64	−0.782997	−0.391499	−	0.920179i	$-0.628043\pi$
$127$	−1120.64	−0.782997	−0.391499	+	0.920179i	$0.628043\pi$
$128$	0	0
$129$	−4299.05	−2.93419
$130$	0	0
$131$	−260.572	−0.173789	−0.0868943	−	0.996218i	$-0.527694\pi$
$131$	−260.572	−0.173789	−0.0868943	+	0.996218i	$0.527694\pi$
$132$	0	0
$133$	−821.682	−0.535706
$134$	0	0
$135$	−6988.91	−4.45563
$136$	0	0
$137$	2696.57	1.68163	0.840815	−	0.541322i	$-0.182076\pi$
$137$	2696.57	1.68163	0.840815	+	0.541322i	$0.182076\pi$
$138$	0	0
$139$	1058.83	0.646104	0.323052	−	0.946381i	$-0.395291\pi$
$139$	1058.83	0.646104	0.323052	+	0.946381i	$0.395291\pi$
$140$	0	0
$141$	−4201.04	−2.50916
$142$	0	0
$143$	−95.8198	−0.0560340
$144$	0	0
$145$	−508.901	−0.291461
$146$	0	0
$147$	−2217.73	−1.24432
$148$	0	0
$149$	2996.14	1.64734	0.823670	−	0.567069i	$-0.191923\pi$
$149$	2996.14	1.64734	0.823670	+	0.567069i	$0.191923\pi$
$150$	0	0
$151$	720.557	0.388332	0.194166	−	0.980969i	$-0.437800\pi$
$151$	720.557	0.388332	0.194166	+	0.980969i	$0.437800\pi$
$152$	0	0
$153$	−6814.93	−3.60101
$154$	0	0
$155$	−3177.67	−1.64669
$156$	0	0
$157$	168.540	0.0856748	0.0428374	−	0.999082i	$-0.486360\pi$
$157$	168.540	0.0856748	0.0428374	+	0.999082i	$0.486360\pi$
$158$	0	0
$159$	1736.83	0.866289
$160$	0	0
$161$	−822.365	−0.402556
$162$	0	0
$163$	3883.20	1.86599	0.932994	−	0.359892i	$-0.117187\pi$
$163$	3883.20	1.86599	0.932994	+	0.359892i	$0.117187\pi$
$164$	0	0
$165$	−957.582	−0.451804
$166$	0	0
$167$	−3170.09	−1.46891	−0.734457	−	0.678655i	$-0.762563\pi$
$167$	−3170.09	−1.46891	−0.734457	+	0.678655i	$0.762563\pi$
$168$	0	0
$169$	−1904.43	−0.866832
$170$	0	0
$171$	5194.14	2.32284
$172$	0	0
$173$	3717.86	1.63389	0.816946	−	0.576713i	$-0.195665\pi$
$173$	3717.86	1.63389	0.816946	+	0.576713i	$0.195665\pi$
$174$	0	0
$175$	−1964.65	−0.848649
$176$	0	0
$177$	−7455.16	−3.16590
$178$	0	0
$179$	−686.938	−0.286839	−0.143419	−	0.989662i	$-0.545810\pi$
$179$	−686.938	−0.286839	−0.143419	+	0.989662i	$0.545810\pi$
$180$	0	0
$181$	−3013.30	−1.23744	−0.618721	−	0.785611i	$-0.712349\pi$
$181$	−3013.30	−1.23744	−0.618721	+	0.785611i	$0.712349\pi$
$182$	0	0
$183$	−5954.13	−2.40515
$184$	0	0
$185$	−3084.11	−1.22566
$186$	0	0
$187$	−562.370	−0.219917
$188$	0	0
$189$	−4277.05	−1.64608
$190$	0	0
$191$	2784.80	1.05498	0.527489	−	0.849562i	$-0.323134\pi$
$191$	2784.80	1.05498	0.527489	+	0.849562i	$0.323134\pi$
$192$	0	0
$193$	3735.88	1.39334	0.696669	−	0.717393i	$-0.254665\pi$
$193$	3735.88	1.39334	0.696669	+	0.717393i	$0.254665\pi$
$194$	0	0
$195$	2923.82	1.07374
$196$	0	0
$197$	−3655.07	−1.32189	−0.660947	−	0.750432i	$-0.729845\pi$
$197$	−3655.07	−1.32189	−0.660947	+	0.750432i	$0.729845\pi$
$198$	0	0
$199$	−4094.62	−1.45859	−0.729296	−	0.684198i	$-0.760152\pi$
$199$	−4094.62	−1.45859	−0.729296	+	0.684198i	$0.760152\pi$
$200$	0	0
$201$	−7107.44	−2.49413
$202$	0	0
$203$	−311.435	−0.107677
$204$	0	0
$205$	−2701.62	−0.920435
$206$	0	0
$207$	5198.46	1.74550
$208$	0	0
$209$	428.622	0.141858
$210$	0	0
$211$	2520.40	0.822328	0.411164	−	0.911561i	$-0.365122\pi$
$211$	2520.40	0.822328	0.411164	+	0.911561i	$0.365122\pi$
$212$	0	0
$213$	−5379.23	−1.73042
$214$	0	0
$215$	7744.73	2.45668
$216$	0	0
$217$	−1944.66	−0.608351
$218$	0	0
$219$	−7335.34	−2.26336
$220$	0	0
$221$	1717.11	0.522648
$222$	0	0
$223$	−4042.81	−1.21402	−0.607010	−	0.794694i	$-0.707631\pi$
$223$	−4042.81	−1.21402	−0.607010	+	0.794694i	$0.707631\pi$
$224$	0	0
$225$	12419.2	3.67977
$226$	0	0
$227$	4866.97	1.42305	0.711525	−	0.702661i	$-0.248005\pi$
$227$	4866.97	1.42305	0.711525	+	0.702661i	$0.248005\pi$
$228$	0	0
$229$	4283.66	1.23612	0.618062	−	0.786130i	$-0.287918\pi$
$229$	4283.66	1.23612	0.618062	+	0.786130i	$0.287918\pi$
$230$	0	0
$231$	−586.017	−0.166914
$232$	0	0
$233$	4160.74	1.16987	0.584933	−	0.811081i	$-0.301121\pi$
$233$	4160.74	1.16987	0.584933	+	0.811081i	$0.301121\pi$
$234$	0	0
$235$	7568.17	2.10082
$236$	0	0
$237$	−8936.98	−2.44945
$238$	0	0
$239$	626.044	0.169437	0.0847185	−	0.996405i	$-0.473001\pi$
$239$	626.044	0.169437	0.0847185	+	0.996405i	$0.473001\pi$
$240$	0	0
$241$	2243.06	0.599537	0.299768	−	0.954012i	$-0.403091\pi$
$241$	2243.06	0.599537	0.299768	+	0.954012i	$0.403091\pi$
$242$	0	0
$243$	9182.36	2.42407
$244$	0	0
$245$	3995.24	1.04182
$246$	0	0
$247$	−1308.73	−0.337135
$248$	0	0
$249$	2195.17	0.558688
$250$	0	0
$251$	−59.2693	−0.0149046	−0.00745228	−	0.999972i	$-0.502372\pi$
$251$	−59.2693	−0.0149046	−0.00745228	+	0.999972i	$0.502372\pi$
$252$	0	0
$253$	428.978	0.106599
$254$	0	0
$255$	17160.0	4.21413
$256$	0	0
$257$	−2380.35	−0.577752	−0.288876	−	0.957367i	$-0.593282\pi$
$257$	−2380.35	−0.577752	−0.288876	+	0.957367i	$0.593282\pi$
$258$	0	0
$259$	−1887.40	−0.452808
$260$	0	0
$261$	1968.69	0.466892
$262$	0	0
$263$	−4998.97	−1.17205	−0.586027	−	0.810292i	$-0.699309\pi$
$263$	−4998.97	−1.17205	−0.586027	+	0.810292i	$0.699309\pi$
$264$	0	0
$265$	−3128.91	−0.725310
$266$	0	0
$267$	844.213	0.193502
$268$	0	0
$269$	−3620.61	−0.820642	−0.410321	−	0.911941i	$-0.634583\pi$
$269$	−3620.61	−0.820642	−0.410321	+	0.911941i	$0.634583\pi$
$270$	0	0
$271$	2530.93	0.567318	0.283659	−	0.958925i	$-0.408452\pi$
$271$	2530.93	0.567318	0.283659	+	0.958925i	$0.408452\pi$
$272$	0	0
$273$	1789.31	0.396681
$274$	0	0
$275$	1024.84	0.224728
$276$	0	0
$277$	5251.71	1.13915	0.569575	−	0.821939i	$-0.307108\pi$
$277$	5251.71	1.13915	0.569575	+	0.821939i	$0.307108\pi$
$278$	0	0
$279$	12292.9	2.63783
$280$	0	0
$281$	−4074.81	−0.865063	−0.432532	−	0.901619i	$-0.642380\pi$
$281$	−4074.81	−0.865063	−0.432532	+	0.901619i	$0.642380\pi$
$282$	0	0
$283$	3546.84	0.745009	0.372504	−	0.928030i	$-0.378499\pi$
$283$	3546.84	0.745009	0.372504	+	0.928030i	$0.378499\pi$
$284$	0	0
$285$	−13078.9	−2.71833
$286$	0	0
$287$	−1653.33	−0.340044
$288$	0	0
$289$	5164.76	1.05124
$290$	0	0
$291$	−16307.3	−3.28505
$292$	0	0
$293$	−6412.44	−1.27856	−0.639281	−	0.768973i	$-0.720768\pi$
$293$	−6412.44	−1.27856	−0.639281	+	0.768973i	$0.720768\pi$
$294$	0	0
$295$	13430.5	2.65068
$296$	0	0
$297$	2231.08	0.435893
$298$	0	0
$299$	−1309.82	−0.253340
$300$	0	0
$301$	4739.59	0.907593
$302$	0	0
$303$	8303.14	1.57427
$304$	0	0
$305$	10726.4	2.01374
$306$	0	0
$307$	−4426.08	−0.822833	−0.411417	−	0.911447i	$-0.634966\pi$
$307$	−4426.08	−0.822833	−0.411417	+	0.911447i	$0.634966\pi$
$308$	0	0
$309$	−5335.99	−0.982374
$310$	0	0
$311$	−6175.65	−1.12601	−0.563005	−	0.826453i	$-0.690355\pi$
$311$	−6175.65	−1.12601	−0.563005	+	0.826453i	$0.690355\pi$
$312$	0	0
$313$	3684.41	0.665352	0.332676	−	0.943041i	$-0.392048\pi$
$313$	3684.41	0.665352	0.332676	+	0.943041i	$0.392048\pi$
$314$	0	0
$315$	12793.3	2.28833
$316$	0	0
$317$	2552.35	0.452222	0.226111	−	0.974102i	$-0.427399\pi$
$317$	2552.35	0.452222	0.226111	+	0.974102i	$0.427399\pi$
$318$	0	0
$319$	162.457	0.0285136
$320$	0	0
$321$	−3052.88	−0.530826
$322$	0	0
$323$	−7680.97	−1.32316
$324$	0	0
$325$	−3129.18	−0.534079
$326$	0	0
$327$	−9650.73	−1.63207
$328$	0	0
$329$	4631.54	0.776125
$330$	0	0
$331$	−2316.72	−0.384709	−0.192354	−	0.981326i	$-0.561612\pi$
$331$	−2316.72	−0.384709	−0.192354	+	0.981326i	$0.561612\pi$
$332$	0	0
$333$	11930.9	1.96339
$334$	0	0
$335$	12804.1	2.08824
$336$	0	0
$337$	−9573.91	−1.54755	−0.773774	−	0.633461i	$-0.781634\pi$
$337$	−9573.91	−1.54755	−0.773774	+	0.633461i	$0.781634\pi$
$338$	0	0
$339$	−6036.06	−0.967062
$340$	0	0
$341$	1014.41	0.161095
$342$	0	0
$343$	6128.52	0.964749
$344$	0	0
$345$	−13089.7	−2.04269
$346$	0	0
$347$	6553.28	1.01383	0.506914	−	0.861997i	$-0.330786\pi$
$347$	6553.28	1.01383	0.506914	+	0.861997i	$0.330786\pi$
$348$	0	0
$349$	1434.33	0.219994	0.109997	−	0.993932i	$-0.464916\pi$
$349$	1434.33	0.219994	0.109997	+	0.993932i	$0.464916\pi$
$350$	0	0
$351$	−6812.24	−1.03593
$352$	0	0
$353$	9404.29	1.41796	0.708980	−	0.705229i	$-0.249156\pi$
$353$	9404.29	1.41796	0.708980	+	0.705229i	$0.249156\pi$
$354$	0	0
$355$	9690.68	1.44881
$356$	0	0
$357$	10501.5	1.55686
$358$	0	0
$359$	−694.109	−0.102044	−0.0510218	−	0.998698i	$-0.516248\pi$
$359$	−694.109	−0.102044	−0.0510218	+	0.998698i	$0.516248\pi$
$360$	0	0
$361$	−1004.79	−0.146492
$362$	0	0
$363$	−12659.5	−1.83044
$364$	0	0
$365$	13214.6	1.89503
$366$	0	0
$367$	−4686.76	−0.666613	−0.333306	−	0.942819i	$-0.608164\pi$
$367$	−4686.76	−0.666613	−0.333306	+	0.942819i	$0.608164\pi$
$368$	0	0
$369$	10451.3	1.47445
$370$	0	0
$371$	−1914.82	−0.267958
$372$	0	0
$373$	35.4816	0.00492539	0.00246269	−	0.999997i	$-0.499216\pi$
$373$	35.4816	0.00492539	0.00246269	+	0.999997i	$0.499216\pi$
$374$	0	0
$375$	−9904.57	−1.36392
$376$	0	0
$377$	−496.036	−0.0677644
$378$	0	0
$379$	5942.37	0.805380	0.402690	−	0.915336i	$-0.368075\pi$
$379$	5942.37	0.805380	0.402690	+	0.915336i	$0.368075\pi$
$380$	0	0
$381$	−10916.1	−1.46784
$382$	0	0
$383$	−3028.49	−0.404044	−0.202022	−	0.979381i	$-0.564751\pi$
$383$	−3028.49	−0.404044	−0.202022	+	0.979381i	$0.564751\pi$
$384$	0	0
$385$	1055.71	0.139751
$386$	0	0
$387$	−29960.6	−3.93536
$388$	0	0
$389$	−3661.34	−0.477217	−0.238608	−	0.971116i	$-0.576691\pi$
$389$	−3661.34	−0.477217	−0.238608	+	0.971116i	$0.576691\pi$
$390$	0	0
$391$	−7687.36	−0.994288
$392$	0	0
$393$	−2538.22	−0.325792
$394$	0	0
$395$	16100.0	2.05083
$396$	0	0
$397$	11965.3	1.51264	0.756321	−	0.654201i	$-0.226995\pi$
$397$	11965.3	1.51264	0.756321	+	0.654201i	$0.226995\pi$
$398$	0	0
$399$	−8003.96	−1.00426
$400$	0	0
$401$	−11152.4	−1.38884	−0.694421	−	0.719569i	$-0.744339\pi$
$401$	−11152.4	−1.38884	−0.694421	+	0.719569i	$0.744339\pi$
$402$	0	0
$403$	−3097.35	−0.382853
$404$	0	0
$405$	−35913.9	−4.40636
$406$	0	0
$407$	984.542	0.119907
$408$	0	0
$409$	−14431.6	−1.74474	−0.872369	−	0.488848i	$-0.837417\pi$
$409$	−14431.6	−1.74474	−0.872369	+	0.488848i	$0.837417\pi$
$410$	0	0
$411$	26267.1	3.15246
$412$	0	0
$413$	8219.12	0.979266
$414$	0	0
$415$	−3954.60	−0.467768
$416$	0	0
$417$	10314.0	1.21122
$418$	0	0
$419$	−8648.14	−1.00833	−0.504164	−	0.863608i	$-0.668199\pi$
$419$	−8648.14	−1.00833	−0.504164	+	0.863608i	$0.668199\pi$
$420$	0	0
$421$	−14692.1	−1.70083	−0.850417	−	0.526109i	$-0.823651\pi$
$421$	−14692.1	−1.70083	−0.850417	+	0.526109i	$0.823651\pi$
$422$	0	0
$423$	−29277.6	−3.36531
$424$	0	0
$425$	−18365.3	−2.09611
$426$	0	0
$427$	6564.28	0.743953
$428$	0	0
$429$	−933.375	−0.105044
$430$	0	0
$431$	4245.71	0.474498	0.237249	−	0.971449i	$-0.423754\pi$
$431$	4245.71	0.474498	0.237249	+	0.971449i	$0.423754\pi$
$432$	0	0
$433$	−11639.1	−1.29178	−0.645891	−	0.763430i	$-0.723514\pi$
$433$	−11639.1	−1.29178	−0.645891	+	0.763430i	$0.723514\pi$
$434$	0	0
$435$	−4957.17	−0.546387
$436$	0	0
$437$	5859.08	0.641368
$438$	0	0
$439$	14134.4	1.53667	0.768336	−	0.640046i	$-0.221085\pi$
$439$	14134.4	1.53667	0.768336	+	0.640046i	$0.221085\pi$
$440$	0	0
$441$	−15455.6	−1.66889
$442$	0	0
$443$	4837.32	0.518799	0.259400	−	0.965770i	$-0.416475\pi$
$443$	4837.32	0.518799	0.259400	+	0.965770i	$0.416475\pi$
$444$	0	0
$445$	−1520.85	−0.162012
$446$	0	0
$447$	29185.3	3.08818
$448$	0	0
$449$	−12734.9	−1.33852	−0.669259	−	0.743029i	$-0.733389\pi$
$449$	−12734.9	−1.33852	−0.669259	+	0.743029i	$0.733389\pi$
$450$	0	0
$451$	862.440	0.0900459
$452$	0	0
$453$	7018.90	0.727984
$454$	0	0
$455$	−3223.44	−0.332126
$456$	0	0
$457$	−11682.9	−1.19585	−0.597923	−	0.801553i	$-0.704007\pi$
$457$	−11682.9	−1.19585	−0.597923	+	0.801553i	$0.704007\pi$
$458$	0	0
$459$	−39981.3	−4.06572
$460$	0	0
$461$	−3885.85	−0.392585	−0.196293	−	0.980545i	$-0.562890\pi$
$461$	−3885.85	−0.392585	−0.196293	+	0.980545i	$0.562890\pi$
$462$	0	0
$463$	16345.5	1.64069	0.820344	−	0.571870i	$-0.193782\pi$
$463$	16345.5	1.64069	0.820344	+	0.571870i	$0.193782\pi$
$464$	0	0
$465$	−30953.5	−3.08696
$466$	0	0
$467$	−4281.32	−0.424231	−0.212115	−	0.977245i	$-0.568035\pi$
$467$	−4281.32	−0.424231	−0.212115	+	0.977245i	$0.568035\pi$
$468$	0	0
$469$	7835.78	0.771477
$470$	0	0
$471$	1641.74	0.160610
$472$	0	0
$473$	−2472.36	−0.240336
$474$	0	0
$475$	13997.5	1.35210
$476$	0	0
$477$	12104.2	1.16188
$478$	0	0
$479$	7855.87	0.749361	0.374680	−	0.927154i	$-0.377752\pi$
$479$	7855.87	0.749361	0.374680	+	0.927154i	$0.377752\pi$
$480$	0	0
$481$	−3006.14	−0.284965
$482$	0	0
$483$	−8010.61	−0.754649
$484$	0	0
$485$	29377.6	2.75045
$486$	0	0
$487$	19096.3	1.77687	0.888436	−	0.459001i	$-0.151793\pi$
$487$	19096.3	1.77687	0.888436	+	0.459001i	$0.151793\pi$
$488$	0	0
$489$	37826.1	3.49806
$490$	0	0
$491$	3703.96	0.340443	0.170221	−	0.985406i	$-0.445552\pi$
$491$	3703.96	0.340443	0.170221	+	0.985406i	$0.445552\pi$
$492$	0	0
$493$	−2911.25	−0.265956
$494$	0	0
$495$	−6673.52	−0.605964
$496$	0	0
$497$	5930.46	0.535247
$498$	0	0
$499$	−18997.2	−1.70428	−0.852138	−	0.523317i	$-0.824694\pi$
$499$	−18997.2	−1.70428	−0.852138	+	0.523317i	$0.824694\pi$
$500$	0	0
$501$	−30879.6	−2.75369
$502$	0	0
$503$	−547.229	−0.0485084	−0.0242542	−	0.999706i	$-0.507721\pi$
$503$	−547.229	−0.0485084	−0.0242542	+	0.999706i	$0.507721\pi$
$504$	0	0
$505$	−14958.1	−1.31807
$506$	0	0
$507$	−18550.9	−1.62500
$508$	0	0
$509$	15507.8	1.35044	0.675218	−	0.737618i	$-0.264050\pi$
$509$	15507.8	1.35044	0.675218	+	0.737618i	$0.264050\pi$
$510$	0	0
$511$	8087.03	0.700096
$512$	0	0
$513$	30472.5	2.62260
$514$	0	0
$515$	9612.78	0.822504
$516$	0	0
$517$	−2416.00	−0.205523
$518$	0	0
$519$	36215.4	3.06297
$520$	0	0
$521$	−9184.57	−0.772329	−0.386164	−	0.922430i	$-0.626200\pi$
$521$	−9184.57	−0.772329	−0.386164	+	0.922430i	$0.626200\pi$
$522$	0	0
$523$	9005.24	0.752910	0.376455	−	0.926435i	$-0.377143\pi$
$523$	9005.24	0.752910	0.376455	+	0.926435i	$0.377143\pi$
$524$	0	0
$525$	−19137.5	−1.59091
$526$	0	0
$527$	−18178.4	−1.50259
$528$	0	0
$529$	−6303.05	−0.518045
$530$	0	0
$531$	−51956.0	−4.24613
$532$	0	0
$533$	−2633.32	−0.214000
$534$	0	0
$535$	5499.76	0.444440
$536$	0	0
$537$	−6691.42	−0.537721
$538$	0	0
$539$	−1275.40	−0.101921
$540$	0	0
$541$	−18272.7	−1.45214	−0.726069	−	0.687622i	$-0.758655\pi$
$541$	−18272.7	−1.45214	−0.726069	+	0.687622i	$0.758655\pi$
$542$	0	0
$543$	−29352.4	−2.31976
$544$	0	0
$545$	17385.8	1.36647
$546$	0	0
$547$	14680.1	1.14749	0.573743	−	0.819035i	$-0.305491\pi$
$547$	14680.1	1.14749	0.573743	+	0.819035i	$0.305491\pi$
$548$	0	0
$549$	−41495.1	−3.22581
$550$	0	0
$551$	2218.87	0.171556
$552$	0	0
$553$	9852.80	0.757656
$554$	0	0
$555$	−30042.1	−2.29769
$556$	0	0
$557$	2917.65	0.221947	0.110974	−	0.993823i	$-0.464603\pi$
$557$	2917.65	0.221947	0.110974	+	0.993823i	$0.464603\pi$
$558$	0	0
$559$	7548.95	0.571175
$560$	0	0
$561$	−5478.01	−0.412267
$562$	0	0
$563$	19604.1	1.46752	0.733760	−	0.679409i	$-0.237764\pi$
$563$	19604.1	1.46752	0.733760	+	0.679409i	$0.237764\pi$
$564$	0	0
$565$	10874.0	0.809684
$566$	0	0
$567$	−21978.5	−1.62788
$568$	0	0
$569$	181.689	0.0133863	0.00669315	−	0.999978i	$-0.497869\pi$
$569$	181.689	0.0133863	0.00669315	+	0.999978i	$0.497869\pi$
$570$	0	0
$571$	−6844.96	−0.501668	−0.250834	−	0.968030i	$-0.580705\pi$
$571$	−6844.96	−0.501668	−0.250834	+	0.968030i	$0.580705\pi$
$572$	0	0
$573$	27126.5	1.97771
$574$	0	0
$575$	14009.1	1.01604
$576$	0	0
$577$	11208.4	0.808686	0.404343	−	0.914607i	$-0.367500\pi$
$577$	11208.4	0.808686	0.404343	+	0.914607i	$0.367500\pi$
$578$	0	0
$579$	36390.9	2.61201
$580$	0	0
$581$	−2420.12	−0.172812
$582$	0	0
$583$	998.844	0.0709569
$584$	0	0
$585$	20376.5	1.44011
$586$	0	0
$587$	7609.36	0.535046	0.267523	−	0.963552i	$-0.413795\pi$
$587$	7609.36	0.535046	0.267523	+	0.963552i	$0.413795\pi$
$588$	0	0
$589$	13855.1	0.969249
$590$	0	0
$591$	−35603.9	−2.47808
$592$	0	0
$593$	10037.9	0.695124	0.347562	−	0.937657i	$-0.387010\pi$
$593$	10037.9	0.695124	0.347562	+	0.937657i	$0.387010\pi$
$594$	0	0
$595$	−18918.5	−1.30350
$596$	0	0
$597$	−39885.4	−2.73434
$598$	0	0
$599$	−8240.90	−0.562127	−0.281064	−	0.959689i	$-0.590687\pi$
$599$	−8240.90	−0.562127	−0.281064	+	0.959689i	$0.590687\pi$
$600$	0	0
$601$	13333.4	0.904957	0.452478	−	0.891775i	$-0.350540\pi$
$601$	13333.4	0.904957	0.452478	+	0.891775i	$0.350540\pi$
$602$	0	0
$603$	−49532.7	−3.34515
$604$	0	0
$605$	22806.1	1.53256
$606$	0	0
$607$	−21808.8	−1.45831	−0.729154	−	0.684350i	$-0.760086\pi$
$607$	−21808.8	−1.45831	−0.729154	+	0.684350i	$0.760086\pi$
$608$	0	0
$609$	−3033.67	−0.201856
$610$	0	0
$611$	7376.86	0.488438
$612$	0	0
$613$	12358.3	0.814271	0.407136	−	0.913368i	$-0.366528\pi$
$613$	12358.3	0.814271	0.407136	+	0.913368i	$0.366528\pi$
$614$	0	0
$615$	−26316.3	−1.72549
$616$	0	0
$617$	11601.1	0.756959	0.378479	−	0.925610i	$-0.376447\pi$
$617$	11601.1	0.756959	0.378479	+	0.925610i	$0.376447\pi$
$618$	0	0
$619$	−5099.08	−0.331097	−0.165549	−	0.986202i	$-0.552939\pi$
$619$	−5099.08	−0.331097	−0.165549	+	0.986202i	$0.552939\pi$
$620$	0	0
$621$	30497.9	1.97075
$622$	0	0
$623$	−930.724	−0.0598534
$624$	0	0
$625$	−5024.77	−0.321585
$626$	0	0
$627$	4175.18	0.265934
$628$	0	0
$629$	−17643.2	−1.11841
$630$	0	0
$631$	−9754.48	−0.615404	−0.307702	−	0.951483i	$-0.599560\pi$
$631$	−9754.48	−0.615404	−0.307702	+	0.951483i	$0.599560\pi$
$632$	0	0
$633$	24551.0	1.54157
$634$	0	0
$635$	19665.3	1.22897
$636$	0	0
$637$	3894.24	0.242222
$638$	0	0
$639$	−37488.5	−2.32085
$640$	0	0
$641$	−16385.8	−1.00967	−0.504835	−	0.863216i	$-0.668447\pi$
$641$	−16385.8	−1.00967	−0.504835	+	0.863216i	$0.668447\pi$
$642$	0	0
$643$	−21455.9	−1.31592	−0.657961	−	0.753052i	$-0.728581\pi$
$643$	−21455.9	−1.31592	−0.657961	+	0.753052i	$0.728581\pi$
$644$	0	0
$645$	75440.9	4.60540
$646$	0	0
$647$	−8758.14	−0.532176	−0.266088	−	0.963949i	$-0.585731\pi$
$647$	−8758.14	−0.532176	−0.266088	+	0.963949i	$0.585731\pi$
$648$	0	0
$649$	−4287.42	−0.259316
$650$	0	0
$651$	−18942.8	−1.14044
$652$	0	0
$653$	12870.2	0.771287	0.385644	−	0.922648i	$-0.373979\pi$
$653$	12870.2	0.771287	0.385644	+	0.922648i	$0.373979\pi$
$654$	0	0
$655$	4572.60	0.272773
$656$	0	0
$657$	−51120.9	−3.03564
$658$	0	0
$659$	4743.37	0.280388	0.140194	−	0.990124i	$-0.455227\pi$
$659$	4743.37	0.280388	0.140194	+	0.990124i	$0.455227\pi$
$660$	0	0
$661$	−12612.2	−0.742142	−0.371071	−	0.928605i	$-0.621009\pi$
$661$	−12612.2	−0.742142	−0.371071	+	0.928605i	$0.621009\pi$
$662$	0	0
$663$	16726.2	0.979779
$664$	0	0
$665$	14419.1	0.840827
$666$	0	0
$667$	2220.72	0.128915
$668$	0	0
$669$	−39380.7	−2.27586
$670$	0	0
$671$	−3424.19	−0.197004
$672$	0	0
$673$	18411.8	1.05457	0.527283	−	0.849690i	$-0.323211\pi$
$673$	18411.8	1.05457	0.527283	+	0.849690i	$0.323211\pi$
$674$	0	0
$675$	72860.1	4.15465
$676$	0	0
$677$	1005.18	0.0570637	0.0285318	−	0.999593i	$-0.490917\pi$
$677$	1005.18	0.0570637	0.0285318	+	0.999593i	$0.490917\pi$
$678$	0	0
$679$	17978.4	1.01612
$680$	0	0
$681$	47408.9	2.66771
$682$	0	0
$683$	22900.5	1.28296	0.641482	−	0.767138i	$-0.278320\pi$
$683$	22900.5	1.28296	0.641482	+	0.767138i	$0.278320\pi$
$684$	0	0
$685$	−47320.2	−2.63943
$686$	0	0
$687$	41726.9	2.31729
$688$	0	0
$689$	−3049.81	−0.168634
$690$	0	0
$691$	−35804.5	−1.97116	−0.985578	−	0.169221i	$-0.945875\pi$
$691$	−35804.5	−1.97116	−0.985578	+	0.169221i	$0.945875\pi$
$692$	0	0
$693$	−4084.03	−0.223867
$694$	0	0
$695$	−18580.6	−1.01410
$696$	0	0
$697$	−15455.1	−0.839889
$698$	0	0
$699$	40529.5	2.19308
$700$	0	0
$701$	4711.14	0.253834	0.126917	−	0.991913i	$-0.459492\pi$
$701$	4711.14	0.253834	0.126917	+	0.991913i	$0.459492\pi$
$702$	0	0
$703$	13447.1	0.721432
$704$	0	0
$705$	73721.1	3.93830
$706$	0	0
$707$	−9154.00	−0.486947
$708$	0	0
$709$	−12122.9	−0.642149	−0.321075	−	0.947054i	$-0.604044\pi$
$709$	−12122.9	−0.642149	−0.321075	+	0.947054i	$0.604044\pi$
$710$	0	0
$711$	−62283.0	−3.28522
$712$	0	0
$713$	13866.6	0.728342
$714$	0	0
$715$	1681.48	0.0879491
$716$	0	0
$717$	6098.26	0.317634
$718$	0	0
$719$	5870.01	0.304471	0.152235	−	0.988344i	$-0.451353\pi$
$719$	5870.01	0.304471	0.152235	+	0.988344i	$0.451353\pi$
$720$	0	0
$721$	5882.79	0.303865
$722$	0	0
$723$	21849.5	1.12392
$724$	0	0
$725$	5305.34	0.271773
$726$	0	0
$727$	−11475.6	−0.585428	−0.292714	−	0.956200i	$-0.594558\pi$
$727$	−11475.6	−0.585428	−0.292714	+	0.956200i	$0.594558\pi$
$728$	0	0
$729$	34187.2	1.73689
$730$	0	0
$731$	44305.1	2.24170
$732$	0	0
$733$	−23960.0	−1.20734	−0.603672	−	0.797233i	$-0.706296\pi$
$733$	−23960.0	−1.20734	−0.603672	+	0.797233i	$0.706296\pi$
$734$	0	0
$735$	38917.3	1.95304
$736$	0	0
$737$	−4087.45	−0.204292
$738$	0	0
$739$	29560.7	1.47146	0.735728	−	0.677277i	$-0.236840\pi$
$739$	29560.7	1.47146	0.735728	+	0.677277i	$0.236840\pi$
$740$	0	0
$741$	−12748.2	−0.632009
$742$	0	0
$743$	−1422.99	−0.0702619	−0.0351309	−	0.999383i	$-0.511185\pi$
$743$	−1422.99	−0.0702619	−0.0351309	+	0.999383i	$0.511185\pi$
$744$	0	0
$745$	−52577.2	−2.58561
$746$	0	0
$747$	15298.4	0.749318
$748$	0	0
$749$	3365.72	0.164193
$750$	0	0
$751$	−9772.07	−0.474817	−0.237409	−	0.971410i	$-0.576298\pi$
$751$	−9772.07	−0.474817	−0.237409	+	0.971410i	$0.576298\pi$
$752$	0	0
$753$	−577.339	−0.0279408
$754$	0	0
$755$	−12644.6	−0.609513
$756$	0	0
$757$	−8632.13	−0.414452	−0.207226	−	0.978293i	$-0.566444\pi$
$757$	−8632.13	−0.414452	−0.207226	+	0.978293i	$0.566444\pi$
$758$	0	0
$759$	4178.65	0.199836
$760$	0	0
$761$	3147.37	0.149924	0.0749620	−	0.997186i	$-0.476116\pi$
$761$	3147.37	0.149924	0.0749620	+	0.997186i	$0.476116\pi$
$762$	0	0
$763$	10639.7	0.504827
$764$	0	0
$765$	119590.	5.65203
$766$	0	0
$767$	13091.0	0.616280
$768$	0	0
$769$	18736.6	0.878622	0.439311	−	0.898335i	$-0.355223\pi$
$769$	18736.6	0.878622	0.439311	+	0.898335i	$0.355223\pi$
$770$	0	0
$771$	−23186.9	−1.08308
$772$	0	0
$773$	38518.9	1.79227	0.896137	−	0.443778i	$-0.146362\pi$
$773$	38518.9	1.79227	0.896137	+	0.443778i	$0.146362\pi$
$774$	0	0
$775$	33127.6	1.53546
$776$	0	0
$777$	−18385.0	−0.848855
$778$	0	0
$779$	11779.4	0.541772
$780$	0	0
$781$	−3093.56	−0.141737
$782$	0	0
$783$	11549.7	0.527144
$784$	0	0
$785$	−2957.59	−0.134472
$786$	0	0
$787$	6989.92	0.316600	0.158300	−	0.987391i	$-0.449399\pi$
$787$	6989.92	0.316600	0.158300	+	0.987391i	$0.449399\pi$
$788$	0	0
$789$	−48694.7	−2.19718
$790$	0	0
$791$	6654.61	0.299129
$792$	0	0
$793$	10455.2	0.468191
$794$	0	0
$795$	−30478.5	−1.35970
$796$	0	0
$797$	−34810.7	−1.54712	−0.773561	−	0.633721i	$-0.781527\pi$
$797$	−34810.7	−1.54712	−0.773561	+	0.633721i	$0.781527\pi$
$798$	0	0
$799$	43295.0	1.91698
$800$	0	0
$801$	5883.43	0.259527
$802$	0	0
$803$	−4218.51	−0.185390
$804$	0	0
$805$	14431.1	0.631838
$806$	0	0
$807$	−35268.2	−1.53841
$808$	0	0
$809$	28933.1	1.25740	0.628699	−	0.777649i	$-0.283588\pi$
$809$	28933.1	1.25740	0.628699	+	0.777649i	$0.283588\pi$
$810$	0	0
$811$	−817.545	−0.0353981	−0.0176991	−	0.999843i	$-0.505634\pi$
$811$	−817.545	−0.0353981	−0.0176991	+	0.999843i	$0.505634\pi$
$812$	0	0
$813$	24653.7	1.06352
$814$	0	0
$815$	−68143.6	−2.92879
$816$	0	0
$817$	−33768.0	−1.44601
$818$	0	0
$819$	12469.9	0.532033
$820$	0	0
$821$	−42229.2	−1.79514	−0.897570	−	0.440872i	$-0.854669\pi$
$821$	−42229.2	−1.79514	−0.897570	+	0.440872i	$0.854669\pi$
$822$	0	0
$823$	−2535.89	−0.107406	−0.0537032	−	0.998557i	$-0.517102\pi$
$823$	−2535.89	−0.107406	−0.0537032	+	0.998557i	$0.517102\pi$
$824$	0	0
$825$	9982.89	0.421284
$826$	0	0
$827$	1558.52	0.0655322	0.0327661	−	0.999463i	$-0.489568\pi$
$827$	1558.52	0.0655322	0.0327661	+	0.999463i	$0.489568\pi$
$828$	0	0
$829$	−41444.3	−1.73633	−0.868166	−	0.496274i	$-0.834701\pi$
$829$	−41444.3	−1.73633	−0.868166	+	0.496274i	$0.834701\pi$
$830$	0	0
$831$	51156.6	2.13550
$832$	0	0
$833$	22855.4	0.950653
$834$	0	0
$835$	55629.7	2.30556
$836$	0	0
$837$	72118.8	2.97825
$838$	0	0
$839$	33855.0	1.39309	0.696545	−	0.717513i	$-0.254720\pi$
$839$	33855.0	1.39309	0.696545	+	0.717513i	$0.254720\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−39692.5	−1.62169
$844$	0	0
$845$	33419.5	1.36055
$846$	0	0
$847$	13956.8	0.566187
$848$	0	0
$849$	34549.5	1.39663
$850$	0	0
$851$	13458.3	0.542120
$852$	0	0
$853$	22478.9	0.902300	0.451150	−	0.892448i	$-0.351014\pi$
$853$	22478.9	0.902300	0.451150	+	0.892448i	$0.351014\pi$
$854$	0	0
$855$	−91148.3	−3.64586
$856$	0	0
$857$	−17846.7	−0.711356	−0.355678	−	0.934609i	$-0.615750\pi$
$857$	−17846.7	−0.711356	−0.355678	+	0.934609i	$0.615750\pi$
$858$	0	0
$859$	−40534.8	−1.61005	−0.805023	−	0.593243i	$-0.797847\pi$
$859$	−40534.8	−1.61005	−0.805023	+	0.593243i	$0.797847\pi$
$860$	0	0
$861$	−16104.9	−0.637462
$862$	0	0
$863$	−20338.9	−0.802253	−0.401126	−	0.916023i	$-0.631381\pi$
$863$	−20338.9	−0.802253	−0.401126	+	0.916023i	$0.631381\pi$
$864$	0	0
$865$	−65242.1	−2.56451
$866$	0	0
$867$	50309.6	1.97071
$868$	0	0
$869$	−5139.61	−0.200632
$870$	0	0
$871$	12480.4	0.485513
$872$	0	0
$873$	−113648.	−4.40594
$874$	0	0
$875$	10919.5	0.421883
$876$	0	0
$877$	32024.7	1.23306	0.616532	−	0.787330i	$-0.288537\pi$
$877$	32024.7	1.23306	0.616532	+	0.787330i	$0.288537\pi$
$878$	0	0
$879$	−62463.2	−2.39685
$880$	0	0
$881$	8059.13	0.308194	0.154097	−	0.988056i	$-0.450753\pi$
$881$	8059.13	0.308194	0.154097	+	0.988056i	$0.450753\pi$
$882$	0	0
$883$	−39365.2	−1.50028	−0.750139	−	0.661280i	$-0.770013\pi$
$883$	−39365.2	−1.50028	−0.750139	+	0.661280i	$0.770013\pi$
$884$	0	0
$885$	130825.	4.96909
$886$	0	0
$887$	21288.8	0.805872	0.402936	−	0.915228i	$-0.367990\pi$
$887$	21288.8	0.805872	0.402936	+	0.915228i	$0.367990\pi$
$888$	0	0
$889$	12034.7	0.454028
$890$	0	0
$891$	11464.8	0.431073
$892$	0	0
$893$	−32998.2	−1.23655
$894$	0	0
$895$	12054.6	0.450213
$896$	0	0
$897$	−12758.8	−0.474922
$898$	0	0
$899$	5251.37	0.194820
$900$	0	0
$901$	−17899.5	−0.661839
$902$	0	0
$903$	46168.1	1.70141
$904$	0	0
$905$	52878.4	1.94225
$906$	0	0
$907$	−4804.03	−0.175871	−0.0879356	−	0.996126i	$-0.528027\pi$
$907$	−4804.03	−0.175871	−0.0879356	+	0.996126i	$0.528027\pi$
$908$	0	0
$909$	57865.7	2.11142
$910$	0	0
$911$	−6289.39	−0.228734	−0.114367	−	0.993439i	$-0.536484\pi$
$911$	−6289.39	−0.228734	−0.114367	+	0.993439i	$0.536484\pi$
$912$	0	0
$913$	1262.43	0.0457616
$914$	0	0
$915$	104485.	3.77504
$916$	0	0
$917$	2798.32	0.100773
$918$	0	0
$919$	411.825	0.0147822	0.00739111	−	0.999973i	$-0.497647\pi$
$919$	411.825	0.0147822	0.00739111	+	0.999973i	$0.497647\pi$
$920$	0	0
$921$	−43114.2	−1.54252
$922$	0	0
$923$	9445.71	0.336846
$924$	0	0
$925$	32152.1	1.14287
$926$	0	0
$927$	−37187.2	−1.31757
$928$	0	0
$929$	−10194.5	−0.360033	−0.180016	−	0.983664i	$-0.557615\pi$
$929$	−10194.5	−0.360033	−0.180016	+	0.983664i	$0.557615\pi$
$930$	0	0
$931$	−17419.7	−0.613221
$932$	0	0
$933$	−60156.7	−2.11087
$934$	0	0
$935$	9868.64	0.345175
$936$	0	0
$937$	29024.7	1.01195	0.505974	−	0.862549i	$-0.331133\pi$
$937$	29024.7	1.01195	0.505974	+	0.862549i	$0.331133\pi$
$938$	0	0
$939$	35889.6	1.24730
$940$	0	0
$941$	49630.5	1.71935	0.859675	−	0.510842i	$-0.170666\pi$
$941$	49630.5	1.71935	0.859675	+	0.510842i	$0.170666\pi$
$942$	0	0
$943$	11789.2	0.407114
$944$	0	0
$945$	75054.9	2.58364
$946$	0	0
$947$	2892.13	0.0992415	0.0496207	−	0.998768i	$-0.484199\pi$
$947$	2892.13	0.0992415	0.0496207	+	0.998768i	$0.484199\pi$
$948$	0	0
$949$	12880.6	0.440591
$950$	0	0
$951$	24862.3	0.847755
$952$	0	0
$953$	22833.4	0.776124	0.388062	−	0.921633i	$-0.373145\pi$
$953$	22833.4	0.776124	0.388062	+	0.921633i	$0.373145\pi$
$954$	0	0
$955$	−48868.5	−1.65586
$956$	0	0
$957$	1582.48	0.0534529
$958$	0	0
$959$	−28958.8	−0.975109
$960$	0	0
$961$	2999.55	0.100686
$962$	0	0
$963$	−21275.9	−0.711949
$964$	0	0
$965$	−65558.3	−2.18694
$966$	0	0
$967$	49187.4	1.63574	0.817869	−	0.575404i	$-0.195155\pi$
$967$	49187.4	1.63574	0.817869	+	0.575404i	$0.195155\pi$
$968$	0	0
$969$	−74819.9	−2.48046
$970$	0	0
$971$	−11678.0	−0.385957	−0.192979	−	0.981203i	$-0.561815\pi$
$971$	−11678.0	−0.385957	−0.192979	+	0.981203i	$0.561815\pi$
$972$	0	0
$973$	−11370.9	−0.374650
$974$	0	0
$975$	−30481.2	−1.00121
$976$	0	0
$977$	20277.2	0.663998	0.331999	−	0.943280i	$-0.392277\pi$
$977$	20277.2	0.663998	0.331999	+	0.943280i	$0.392277\pi$
$978$	0	0
$979$	485.502	0.0158496
$980$	0	0
$981$	−67257.2	−2.18895
$982$	0	0
$983$	33140.5	1.07530	0.537650	−	0.843168i	$-0.319312\pi$
$983$	33140.5	1.07530	0.537650	+	0.843168i	$0.319312\pi$
$984$	0	0
$985$	64140.3	2.07480
$986$	0	0
$987$	45115.6	1.45496
$988$	0	0
$989$	−33796.1	−1.08661
$990$	0	0
$991$	−10611.0	−0.340129	−0.170065	−	0.985433i	$-0.554398\pi$
$991$	−10611.0	−0.340129	−0.170065	+	0.985433i	$0.554398\pi$
$992$	0	0
$993$	−22567.1	−0.721192
$994$	0	0
$995$	71853.6	2.28936
$996$	0	0
$997$	6584.26	0.209153	0.104576	−	0.994517i	$-0.466651\pi$
$997$	6584.26	0.209153	0.104576	+	0.994517i	$0.466651\pi$
$998$	0	0
$999$	69995.3	2.21677

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.bg.1.9		9
4.3	odd	2		1856.4.a.bf.1.1		9
8.3	odd	2		928.4.a.e.1.9	yes	9
8.5	even	2		928.4.a.d.1.1	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.4.a.d.1.1	✓	9	8.5	even	2
928.4.a.e.1.9	yes	9	8.3	odd	2
1856.4.a.bf.1.1		9	4.3	odd	2
1856.4.a.bg.1.9		9	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)

\(f(q)\)	\(=\)	\(q+9.74094 q^{3} -17.5483 q^{5} -10.7391 q^{7} +67.8859 q^{9} +O(q^{10})\)	\(q+9.74094 q^{3} -17.5483 q^{5} -10.7391 q^{7} +67.8859 q^{9} +5.60196 q^{11} -17.1047 q^{13} -170.937 q^{15} -100.388 q^{17} +76.5128 q^{19} -104.609 q^{21} +76.5764 q^{23} +182.943 q^{25} +398.267 q^{27} +29.0000 q^{29} +181.082 q^{31} +54.5684 q^{33} +188.454 q^{35} +175.750 q^{37} -166.616 q^{39} +153.953 q^{41} -441.338 q^{43} -1191.28 q^{45} -431.277 q^{47} -227.671 q^{49} -977.874 q^{51} +178.303 q^{53} -98.3049 q^{55} +745.307 q^{57} -765.343 q^{59} -611.248 q^{61} -729.036 q^{63} +300.158 q^{65} -729.647 q^{67} +745.926 q^{69} -552.229 q^{71} -753.042 q^{73} +1782.04 q^{75} -60.1603 q^{77} -917.466 q^{79} +2046.58 q^{81} +225.355 q^{83} +1761.64 q^{85} +282.487 q^{87} +86.6665 q^{89} +183.690 q^{91} +1763.90 q^{93} -1342.67 q^{95} -1674.10 q^{97} +380.294 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 4 q^{3} - 10 q^{5} - 12 q^{7} + 49 q^{9}+O(q^{10}) \) 9 * q + 4 * q^3 - 10 * q^5 - 12 * q^7 + 49 * q^9	\( 9 q + 4 q^{3} - 10 q^{5} - 12 q^{7} + 49 q^{9} + 64 q^{11} - 70 q^{13} - 170 q^{15} - 66 q^{17} + 42 q^{19} - 76 q^{21} - 40 q^{23} + 111 q^{25} + 322 q^{27} + 261 q^{29} + 64 q^{31} - 52 q^{33} + 496 q^{35} + 54 q^{37} - 590 q^{39} - 378 q^{41} - 32 q^{43} - 1046 q^{45} - 1164 q^{47} - 351 q^{49} + 376 q^{51} - 278 q^{53} - 614 q^{55} + 28 q^{57} + 640 q^{59} - 1054 q^{61} - 1660 q^{63} - 708 q^{65} + 1184 q^{67} - 188 q^{69} - 1988 q^{71} - 750 q^{73} + 3126 q^{75} - 1260 q^{77} - 2916 q^{79} + 293 q^{81} + 2832 q^{83} - 56 q^{85} + 116 q^{87} - 370 q^{89} + 3016 q^{91} + 1696 q^{93} - 4412 q^{95} - 2234 q^{97} + 4118 q^{99}+O(q^{100}) \) 9 * q + 4 * q^3 - 10 * q^5 - 12 * q^7 + 49 * q^9 + 64 * q^11 - 70 * q^13 - 170 * q^15 - 66 * q^17 + 42 * q^19 - 76 * q^21 - 40 * q^23 + 111 * q^25 + 322 * q^27 + 261 * q^29 + 64 * q^31 - 52 * q^33 + 496 * q^35 + 54 * q^37 - 590 * q^39 - 378 * q^41 - 32 * q^43 - 1046 * q^45 - 1164 * q^47 - 351 * q^49 + 376 * q^51 - 278 * q^53 - 614 * q^55 + 28 * q^57 + 640 * q^59 - 1054 * q^61 - 1660 * q^63 - 708 * q^65 + 1184 * q^67 - 188 * q^69 - 1988 * q^71 - 750 * q^73 + 3126 * q^75 - 1260 * q^77 - 2916 * q^79 + 293 * q^81 + 2832 * q^83 - 56 * q^85 + 116 * q^87 - 370 * q^89 + 3016 * q^91 + 1696 * q^93 - 4412 * q^95 - 2234 * q^97 + 4118 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more