LMFDB - Embedded newform 1856.4.a.bi.1.2

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:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4 x^{9} - 135 x^{8} + 788 x^{7} + 3323 x^{6} - 26136 x^{5} + 2315 x^{4} + 188664 x^{3} + \cdots + 128592 $ x^10 - 4x^9 - 135x^8 + 788x^7 + 3323x^6 - 26136x^5 + 2315x^4 + 188664x^3 - 265632x^2 - 18800*x + 128592
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 928)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.25115$ 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)

$f(q)$	$=$	$q-7.53335 q^{3} +7.78444 q^{5} -5.94345 q^{7} +29.7513 q^{9} +O(q^{10})$	$q-7.53335 q^{3} +7.78444 q^{5} -5.94345 q^{7} +29.7513 q^{9} -21.2550 q^{11} +74.0611 q^{13} -58.6429 q^{15} +109.045 q^{17} +96.9901 q^{19} +44.7741 q^{21} +16.9726 q^{23} -64.4025 q^{25} -20.7268 q^{27} +29.0000 q^{29} -216.077 q^{31} +160.121 q^{33} -46.2664 q^{35} +138.521 q^{37} -557.928 q^{39} +344.314 q^{41} +61.6012 q^{43} +231.598 q^{45} +77.0472 q^{47} -307.675 q^{49} -821.474 q^{51} -19.8393 q^{53} -165.458 q^{55} -730.660 q^{57} +732.490 q^{59} -310.023 q^{61} -176.826 q^{63} +576.524 q^{65} +765.685 q^{67} -127.860 q^{69} -74.3601 q^{71} -421.694 q^{73} +485.166 q^{75} +126.328 q^{77} -1116.13 q^{79} -647.144 q^{81} -1085.21 q^{83} +848.855 q^{85} -218.467 q^{87} -547.936 q^{89} -440.178 q^{91} +1627.78 q^{93} +755.014 q^{95} -29.9308 q^{97} -632.364 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 4 q^{5} + 50 q^{9}+O(q^{10}) $ 10 * q + 4 * q^5 + 50 * q^9	$ 10 q + 4 q^{5} + 50 q^{9} + 220 q^{13} + 76 q^{17} + 104 q^{21} + 446 q^{25} + 290 q^{29} - 1120 q^{33} + 1708 q^{37} - 980 q^{41} - 348 q^{45} + 1146 q^{49} + 44 q^{53} - 40 q^{57} + 1492 q^{61} - 2016 q^{65} + 2328 q^{69} - 3100 q^{73} + 3016 q^{77} - 2174 q^{81} + 9440 q^{85} + 636 q^{89} + 2536 q^{93} + 620 q^{97}+O(q^{100}) $ 10 * q + 4 * q^5 + 50 * q^9 + 220 * q^13 + 76 * q^17 + 104 * q^21 + 446 * q^25 + 290 * q^29 - 1120 * q^33 + 1708 * q^37 - 980 * q^41 - 348 * q^45 + 1146 * q^49 + 44 * q^53 - 40 * q^57 + 1492 * q^61 - 2016 * q^65 + 2328 * q^69 - 3100 * q^73 + 3016 * q^77 - 2174 * q^81 + 9440 * q^85 + 636 * q^89 + 2536 * q^93 + 620 * q^97

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
$3$	−7.53335	−1.44979	−0.724897	−	0.688857i	$-0.758113\pi$
$3$	−7.53335	−1.44979	−0.724897	+	0.688857i	$0.758113\pi$
$4$	0	0
$5$	7.78444	0.696262	0.348131	−	0.937446i	$-0.386817\pi$
$5$	7.78444	0.696262	0.348131	+	0.937446i	$0.386817\pi$
$6$	0	0
$7$	−5.94345	−0.320916	−0.160458	−	0.987043i	$-0.551297\pi$
$7$	−5.94345	−0.320916	−0.160458	+	0.987043i	$0.551297\pi$
$8$	0	0
$9$	29.7513	1.10190
$10$	0	0
$11$	−21.2550	−0.582601	−0.291301	−	0.956632i	$-0.594088\pi$
$11$	−21.2550	−0.582601	−0.291301	+	0.956632i	$0.594088\pi$
$12$	0	0
$13$	74.0611	1.58007	0.790033	−	0.613064i	$-0.210063\pi$
$13$	74.0611	1.58007	0.790033	+	0.613064i	$0.210063\pi$
$14$	0	0
$15$	−58.6429	−1.00944
$16$	0	0
$17$	109.045	1.55572	0.777862	−	0.628435i	$-0.216304\pi$
$17$	109.045	1.55572	0.777862	+	0.628435i	$0.216304\pi$
$18$	0	0
$19$	96.9901	1.17111	0.585554	−	0.810633i	$-0.300877\pi$
$19$	96.9901	1.17111	0.585554	+	0.810633i	$0.300877\pi$
$20$	0	0
$21$	44.7741	0.465262
$22$	0	0
$23$	16.9726	0.153871	0.0769354	−	0.997036i	$-0.475486\pi$
$23$	16.9726	0.153871	0.0769354	+	0.997036i	$0.475486\pi$
$24$	0	0
$25$	−64.4025	−0.515220
$26$	0	0
$27$	−20.7268	−0.147736
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	−216.077	−1.25189	−0.625944	−	0.779868i	$-0.715286\pi$
$31$	−216.077	−1.25189	−0.625944	+	0.779868i	$0.715286\pi$
$32$	0	0
$33$	160.121	0.844652
$34$	0	0
$35$	−46.2664	−0.223442
$36$	0	0
$37$	138.521	0.615480	0.307740	−	0.951470i	$-0.400427\pi$
$37$	138.521	0.615480	0.307740	+	0.951470i	$0.400427\pi$
$38$	0	0
$39$	−557.928	−2.29077
$40$	0	0
$41$	344.314	1.31153	0.655767	−	0.754964i	$-0.272346\pi$
$41$	344.314	1.31153	0.655767	+	0.754964i	$0.272346\pi$
$42$	0	0
$43$	61.6012	0.218467	0.109234	−	0.994016i	$-0.465160\pi$
$43$	61.6012	0.218467	0.109234	+	0.994016i	$0.465160\pi$
$44$	0	0
$45$	231.598	0.767212
$46$	0	0
$47$	77.0472	0.239117	0.119558	−	0.992827i	$-0.461852\pi$
$47$	77.0472	0.239117	0.119558	+	0.992827i	$0.461852\pi$
$48$	0	0
$49$	−307.675	−0.897013
$50$	0	0
$51$	−821.474	−2.25548
$52$	0	0
$53$	−19.8393	−0.0514176	−0.0257088	−	0.999669i	$-0.508184\pi$
$53$	−19.8393	−0.0514176	−0.0257088	+	0.999669i	$0.508184\pi$
$54$	0	0
$55$	−165.458	−0.405643
$56$	0	0
$57$	−730.660	−1.69787
$58$	0	0
$59$	732.490	1.61631	0.808153	−	0.588973i	$-0.200468\pi$
$59$	732.490	1.61631	0.808153	+	0.588973i	$0.200468\pi$
$60$	0	0
$61$	−310.023	−0.650728	−0.325364	−	0.945589i	$-0.605487\pi$
$61$	−310.023	−0.650728	−0.325364	+	0.945589i	$0.605487\pi$
$62$	0	0
$63$	−176.826	−0.353618
$64$	0	0
$65$	576.524	1.10014
$66$	0	0
$67$	765.685	1.39617	0.698084	−	0.716016i	$-0.254036\pi$
$67$	765.685	1.39617	0.698084	+	0.716016i	$0.254036\pi$
$68$	0	0
$69$	−127.860	−0.223081
$70$	0	0
$71$	−74.3601	−0.124295	−0.0621473	−	0.998067i	$-0.519795\pi$
$71$	−74.3601	−0.124295	−0.0621473	+	0.998067i	$0.519795\pi$
$72$	0	0
$73$	−421.694	−0.676104	−0.338052	−	0.941127i	$-0.609768\pi$
$73$	−421.694	−0.676104	−0.338052	+	0.941127i	$0.609768\pi$
$74$	0	0
$75$	485.166	0.746962
$76$	0	0
$77$	126.328	0.186966
$78$	0	0
$79$	−1116.13	−1.58954	−0.794771	−	0.606909i	$-0.792409\pi$
$79$	−1116.13	−1.58954	−0.794771	+	0.606909i	$0.792409\pi$
$80$	0	0
$81$	−647.144	−0.887715
$82$	0	0
$83$	−1085.21	−1.43514	−0.717571	−	0.696485i	$-0.754746\pi$
$83$	−1085.21	−1.43514	−0.717571	+	0.696485i	$0.754746\pi$
$84$	0	0
$85$	848.855	1.08319
$86$	0	0
$87$	−218.467	−0.269220
$88$	0	0
$89$	−547.936	−0.652597	−0.326298	−	0.945267i	$-0.605801\pi$
$89$	−547.936	−0.652597	−0.326298	+	0.945267i	$0.605801\pi$
$90$	0	0
$91$	−440.178	−0.507069
$92$	0	0
$93$	1627.78	1.81498
$94$	0	0
$95$	755.014	0.815398
$96$	0	0
$97$	−29.9308	−0.0313300	−0.0156650	−	0.999877i	$-0.504987\pi$
$97$	−29.9308	−0.0313300	−0.0156650	+	0.999877i	$0.504987\pi$
$98$	0	0
$99$	−632.364	−0.641969
$100$	0	0
$101$	1404.10	1.38330	0.691650	−	0.722233i	$-0.256884\pi$
$101$	1404.10	1.38330	0.691650	+	0.722233i	$0.256884\pi$
$102$	0	0
$103$	976.294	0.933953	0.466976	−	0.884270i	$-0.345343\pi$
$103$	976.294	0.933953	0.466976	+	0.884270i	$0.345343\pi$
$104$	0	0
$105$	348.541	0.323944
$106$	0	0
$107$	−841.796	−0.760556	−0.380278	−	0.924872i	$-0.624172\pi$
$107$	−841.796	−0.760556	−0.380278	+	0.924872i	$0.624172\pi$
$108$	0	0
$109$	1575.81	1.38473	0.692365	−	0.721547i	$-0.256569\pi$
$109$	1575.81	1.38473	0.692365	+	0.721547i	$0.256569\pi$
$110$	0	0
$111$	−1043.53	−0.892319
$112$	0	0
$113$	−629.239	−0.523839	−0.261920	−	0.965090i	$-0.584356\pi$
$113$	−629.239	−0.523839	−0.261920	+	0.965090i	$0.584356\pi$
$114$	0	0
$115$	132.122	0.107134
$116$	0	0
$117$	2203.42	1.74108
$118$	0	0
$119$	−648.104	−0.499257
$120$	0	0
$121$	−879.226	−0.660576
$122$	0	0
$123$	−2593.84	−1.90145
$124$	0	0
$125$	−1474.39	−1.05499
$126$	0	0
$127$	1964.51	1.37262	0.686309	−	0.727310i	$-0.259230\pi$
$127$	1964.51	1.37262	0.686309	+	0.727310i	$0.259230\pi$
$128$	0	0
$129$	−464.063	−0.316733
$130$	0	0
$131$	−1680.11	−1.12055	−0.560274	−	0.828308i	$-0.689304\pi$
$131$	−1680.11	−1.12055	−0.560274	+	0.828308i	$0.689304\pi$
$132$	0	0
$133$	−576.456	−0.375828
$134$	0	0
$135$	−161.346	−0.102863
$136$	0	0
$137$	329.621	0.205558	0.102779	−	0.994704i	$-0.467227\pi$
$137$	329.621	0.205558	0.102779	+	0.994704i	$0.467227\pi$
$138$	0	0
$139$	−3255.25	−1.98638	−0.993189	−	0.116516i	$-0.962827\pi$
$139$	−3255.25	−1.98638	−0.993189	+	0.116516i	$0.962827\pi$
$140$	0	0
$141$	−580.424	−0.346670
$142$	0	0
$143$	−1574.17	−0.920548
$144$	0	0
$145$	225.749	0.129293
$146$	0	0
$147$	2317.83	1.30048
$148$	0	0
$149$	−299.289	−0.164555	−0.0822776	−	0.996609i	$-0.526219\pi$
$149$	−299.289	−0.164555	−0.0822776	+	0.996609i	$0.526219\pi$
$150$	0	0
$151$	495.386	0.266980	0.133490	−	0.991050i	$-0.457382\pi$
$151$	495.386	0.266980	0.133490	+	0.991050i	$0.457382\pi$
$152$	0	0
$153$	3244.24	1.71425
$154$	0	0
$155$	−1682.04	−0.871642
$156$	0	0
$157$	−1916.87	−0.974414	−0.487207	−	0.873286i	$-0.661984\pi$
$157$	−1916.87	−0.974414	−0.487207	+	0.873286i	$0.661984\pi$
$158$	0	0
$159$	149.456	0.0745449
$160$	0	0
$161$	−100.876	−0.0493796
$162$	0	0
$163$	2540.97	1.22101	0.610503	−	0.792014i	$-0.290967\pi$
$163$	2540.97	1.22101	0.610503	+	0.792014i	$0.290967\pi$
$164$	0	0
$165$	1246.45	0.588098
$166$	0	0
$167$	1311.27	0.607597	0.303799	−	0.952736i	$-0.401745\pi$
$167$	1311.27	0.607597	0.303799	+	0.952736i	$0.401745\pi$
$168$	0	0
$169$	3288.05	1.49661
$170$	0	0
$171$	2885.59	1.29045
$172$	0	0
$173$	1164.21	0.511637	0.255818	−	0.966725i	$-0.417655\pi$
$173$	1164.21	0.511637	0.255818	+	0.966725i	$0.417655\pi$
$174$	0	0
$175$	382.773	0.165342
$176$	0	0
$177$	−5518.10	−2.34331
$178$	0	0
$179$	2407.25	1.00518	0.502588	−	0.864526i	$-0.332381\pi$
$179$	2407.25	1.00518	0.502588	+	0.864526i	$0.332381\pi$
$180$	0	0
$181$	3278.67	1.34642	0.673208	−	0.739453i	$-0.264916\pi$
$181$	3278.67	1.34642	0.673208	+	0.739453i	$0.264916\pi$
$182$	0	0
$183$	2335.51	0.943421
$184$	0	0
$185$	1078.31	0.428535
$186$	0	0
$187$	−2317.75	−0.906367
$188$	0	0
$189$	123.189	0.0474109
$190$	0	0
$191$	681.430	0.258149	0.129075	−	0.991635i	$-0.458799\pi$
$191$	681.430	0.258149	0.129075	+	0.991635i	$0.458799\pi$
$192$	0	0
$193$	2756.45	1.02805	0.514025	−	0.857775i	$-0.328154\pi$
$193$	2756.45	1.02805	0.514025	+	0.857775i	$0.328154\pi$
$194$	0	0
$195$	−4343.16	−1.59497
$196$	0	0
$197$	2690.53	0.973059	0.486529	−	0.873664i	$-0.338263\pi$
$197$	2690.53	0.973059	0.486529	+	0.873664i	$0.338263\pi$
$198$	0	0
$199$	−1698.55	−0.605061	−0.302530	−	0.953140i	$-0.597831\pi$
$199$	−1698.55	−0.605061	−0.302530	+	0.953140i	$0.597831\pi$
$200$	0	0
$201$	−5768.17	−2.02416
$202$	0	0
$203$	−172.360	−0.0595926
$204$	0	0
$205$	2680.29	0.913170
$206$	0	0
$207$	504.957	0.169550
$208$	0	0
$209$	−2061.52	−0.682289
$210$	0	0
$211$	5693.97	1.85777	0.928885	−	0.370369i	$-0.120769\pi$
$211$	5693.97	1.85777	0.928885	+	0.370369i	$0.120769\pi$
$212$	0	0
$213$	560.180	0.180202
$214$	0	0
$215$	479.531	0.152110
$216$	0	0
$217$	1284.24	0.401751
$218$	0	0
$219$	3176.77	0.980211
$220$	0	0
$221$	8076.00	2.45815
$222$	0	0
$223$	−2497.17	−0.749877	−0.374939	−	0.927050i	$-0.622336\pi$
$223$	−2497.17	−0.749877	−0.374939	+	0.927050i	$0.622336\pi$
$224$	0	0
$225$	−1916.06	−0.567721
$226$	0	0
$227$	380.211	0.111169	0.0555847	−	0.998454i	$-0.482298\pi$
$227$	380.211	0.111169	0.0555847	+	0.998454i	$0.482298\pi$
$228$	0	0
$229$	−6036.43	−1.74192	−0.870958	−	0.491358i	$-0.836501\pi$
$229$	−6036.43	−1.74192	−0.870958	+	0.491358i	$0.836501\pi$
$230$	0	0
$231$	−951.672	−0.271062
$232$	0	0
$233$	323.182	0.0908684	0.0454342	−	0.998967i	$-0.485533\pi$
$233$	323.182	0.0908684	0.0454342	+	0.998967i	$0.485533\pi$
$234$	0	0
$235$	599.770	0.166488
$236$	0	0
$237$	8408.16	2.30451
$238$	0	0
$239$	912.481	0.246960	0.123480	−	0.992347i	$-0.460594\pi$
$239$	912.481	0.246960	0.123480	+	0.992347i	$0.460594\pi$
$240$	0	0
$241$	2041.34	0.545618	0.272809	−	0.962068i	$-0.412047\pi$
$241$	2041.34	0.545618	0.272809	+	0.962068i	$0.412047\pi$
$242$	0	0
$243$	5434.78	1.43474
$244$	0	0
$245$	−2395.08	−0.624556
$246$	0	0
$247$	7183.20	1.85043
$248$	0	0
$249$	8175.23	2.08066
$250$	0	0
$251$	−2501.26	−0.628997	−0.314499	−	0.949258i	$-0.601836\pi$
$251$	−2501.26	−0.628997	−0.314499	+	0.949258i	$0.601836\pi$
$252$	0	0
$253$	−360.752	−0.0896453
$254$	0	0
$255$	−6394.72	−1.57040
$256$	0	0
$257$	1170.26	0.284043	0.142021	−	0.989864i	$-0.454640\pi$
$257$	1170.26	0.284043	0.142021	+	0.989864i	$0.454640\pi$
$258$	0	0
$259$	−823.295	−0.197518
$260$	0	0
$261$	862.789	0.204618
$262$	0	0
$263$	2106.70	0.493935	0.246967	−	0.969024i	$-0.420566\pi$
$263$	2106.70	0.493935	0.246967	+	0.969024i	$0.420566\pi$
$264$	0	0
$265$	−154.438	−0.0358001
$266$	0	0
$267$	4127.79	0.946130
$268$	0	0
$269$	1802.45	0.408539	0.204270	−	0.978915i	$-0.434518\pi$
$269$	1802.45	0.408539	0.204270	+	0.978915i	$0.434518\pi$
$270$	0	0
$271$	3411.58	0.764718	0.382359	−	0.924014i	$-0.375112\pi$
$271$	3411.58	0.764718	0.382359	+	0.924014i	$0.375112\pi$
$272$	0	0
$273$	3316.02	0.735145
$274$	0	0
$275$	1368.87	0.300168
$276$	0	0
$277$	−1921.08	−0.416701	−0.208351	−	0.978054i	$-0.566810\pi$
$277$	−1921.08	−0.416701	−0.208351	+	0.978054i	$0.566810\pi$
$278$	0	0
$279$	−6428.57	−1.37946
$280$	0	0
$281$	−1445.68	−0.306912	−0.153456	−	0.988155i	$-0.549040\pi$
$281$	−1445.68	−0.306912	−0.153456	+	0.988155i	$0.549040\pi$
$282$	0	0
$283$	−2207.60	−0.463703	−0.231852	−	0.972751i	$-0.574478\pi$
$283$	−2207.60	−0.463703	−0.231852	+	0.972751i	$0.574478\pi$
$284$	0	0
$285$	−5687.78	−1.18216
$286$	0	0
$287$	−2046.41	−0.420892
$288$	0	0
$289$	6977.83	1.42028
$290$	0	0
$291$	225.479	0.0454221
$292$	0	0
$293$	1676.45	0.334263	0.167132	−	0.985935i	$-0.446550\pi$
$293$	1676.45	0.334263	0.167132	+	0.985935i	$0.446550\pi$
$294$	0	0
$295$	5702.02	1.12537
$296$	0	0
$297$	440.547	0.0860712
$298$	0	0
$299$	1257.01	0.243126
$300$	0	0
$301$	−366.124	−0.0701097
$302$	0	0
$303$	−10577.6	−2.00550
$304$	0	0
$305$	−2413.36	−0.453077
$306$	0	0
$307$	7593.57	1.41169	0.705843	−	0.708368i	$-0.250568\pi$
$307$	7593.57	1.41169	0.705843	+	0.708368i	$0.250568\pi$
$308$	0	0
$309$	−7354.77	−1.35404
$310$	0	0
$311$	9386.31	1.71141	0.855706	−	0.517463i	$-0.173123\pi$
$311$	9386.31	1.71141	0.855706	+	0.517463i	$0.173123\pi$
$312$	0	0
$313$	5805.88	1.04846	0.524229	−	0.851577i	$-0.324354\pi$
$313$	5805.88	1.04846	0.524229	+	0.851577i	$0.324354\pi$
$314$	0	0
$315$	−1376.49	−0.246211
$316$	0	0
$317$	10025.7	1.77634	0.888172	−	0.459510i	$-0.151975\pi$
$317$	10025.7	1.77634	0.888172	+	0.459510i	$0.151975\pi$
$318$	0	0
$319$	−616.394	−0.108186
$320$	0	0
$321$	6341.54	1.10265
$322$	0	0
$323$	10576.3	1.82192
$324$	0	0
$325$	−4769.72	−0.814081
$326$	0	0
$327$	−11871.2	−2.00757
$328$	0	0
$329$	−457.926	−0.0767365
$330$	0	0
$331$	−6125.19	−1.01713	−0.508566	−	0.861023i	$-0.669824\pi$
$331$	−6125.19	−1.01713	−0.508566	+	0.861023i	$0.669824\pi$
$332$	0	0
$333$	4121.19	0.678198
$334$	0	0
$335$	5960.43	0.972098
$336$	0	0
$337$	2311.21	0.373590	0.186795	−	0.982399i	$-0.440190\pi$
$337$	2311.21	0.373590	0.186795	+	0.982399i	$0.440190\pi$
$338$	0	0
$339$	4740.28	0.759459
$340$	0	0
$341$	4592.70	0.729352
$342$	0	0
$343$	3867.26	0.608782
$344$	0	0
$345$	−995.322	−0.155323
$346$	0	0
$347$	−2037.25	−0.315174	−0.157587	−	0.987505i	$-0.550371\pi$
$347$	−2037.25	−0.315174	−0.157587	+	0.987505i	$0.550371\pi$
$348$	0	0
$349$	7945.78	1.21870	0.609352	−	0.792900i	$-0.291430\pi$
$349$	7945.78	1.21870	0.609352	+	0.792900i	$0.291430\pi$
$350$	0	0
$351$	−1535.05	−0.233433
$352$	0	0
$353$	−6205.52	−0.935656	−0.467828	−	0.883820i	$-0.654963\pi$
$353$	−6205.52	−0.935656	−0.467828	+	0.883820i	$0.654963\pi$
$354$	0	0
$355$	−578.852	−0.0865416
$356$	0	0
$357$	4882.39	0.723820
$358$	0	0
$359$	5508.39	0.809810	0.404905	−	0.914359i	$-0.367305\pi$
$359$	5508.39	0.809810	0.404905	+	0.914359i	$0.367305\pi$
$360$	0	0
$361$	2548.09	0.371495
$362$	0	0
$363$	6623.52	0.957698
$364$	0	0
$365$	−3282.65	−0.470745
$366$	0	0
$367$	6852.56	0.974661	0.487330	−	0.873218i	$-0.337971\pi$
$367$	6852.56	0.974661	0.487330	+	0.873218i	$0.337971\pi$
$368$	0	0
$369$	10243.8	1.44518
$370$	0	0
$371$	117.914	0.0165007
$372$	0	0
$373$	5570.08	0.773211	0.386606	−	0.922245i	$-0.373647\pi$
$373$	5570.08	0.773211	0.386606	+	0.922245i	$0.373647\pi$
$374$	0	0
$375$	11107.1	1.52952
$376$	0	0
$377$	2147.77	0.293411
$378$	0	0
$379$	2259.15	0.306186	0.153093	−	0.988212i	$-0.451077\pi$
$379$	2259.15	0.306186	0.153093	+	0.988212i	$0.451077\pi$
$380$	0	0
$381$	−14799.4	−1.99001
$382$	0	0
$383$	−698.695	−0.0932158	−0.0466079	−	0.998913i	$-0.514841\pi$
$383$	−698.695	−0.0932158	−0.0466079	+	0.998913i	$0.514841\pi$
$384$	0	0
$385$	983.392	0.130177
$386$	0	0
$387$	1832.72	0.240730
$388$	0	0
$389$	−12080.5	−1.57456	−0.787282	−	0.616593i	$-0.788512\pi$
$389$	−12080.5	−1.57456	−0.787282	+	0.616593i	$0.788512\pi$
$390$	0	0
$391$	1850.78	0.239381
$392$	0	0
$393$	12656.8	1.62456
$394$	0	0
$395$	−8688.41	−1.10674
$396$	0	0
$397$	7898.20	0.998487	0.499244	−	0.866462i	$-0.333611\pi$
$397$	7898.20	0.998487	0.499244	+	0.866462i	$0.333611\pi$
$398$	0	0
$399$	4342.64	0.544872
$400$	0	0
$401$	9390.39	1.16941	0.584705	−	0.811246i	$-0.301210\pi$
$401$	9390.39	1.16941	0.584705	+	0.811246i	$0.301210\pi$
$402$	0	0
$403$	−16002.9	−1.97807
$404$	0	0
$405$	−5037.65	−0.618082
$406$	0	0
$407$	−2944.27	−0.358580
$408$	0	0
$409$	590.704	0.0714143	0.0357072	−	0.999362i	$-0.488632\pi$
$409$	590.704	0.0714143	0.0357072	+	0.999362i	$0.488632\pi$
$410$	0	0
$411$	−2483.15	−0.298017
$412$	0	0
$413$	−4353.52	−0.518698
$414$	0	0
$415$	−8447.72	−0.999235
$416$	0	0
$417$	24522.9	2.87984
$418$	0	0
$419$	971.182	0.113235	0.0566174	−	0.998396i	$-0.481968\pi$
$419$	971.182	0.113235	0.0566174	+	0.998396i	$0.481968\pi$
$420$	0	0
$421$	−12134.3	−1.40472	−0.702361	−	0.711821i	$-0.747871\pi$
$421$	−12134.3	−1.40472	−0.702361	+	0.711821i	$0.747871\pi$
$422$	0	0
$423$	2292.26	0.263483
$424$	0	0
$425$	−7022.77	−0.801540
$426$	0	0
$427$	1842.61	0.208829
$428$	0	0
$429$	11858.7	1.33461
$430$	0	0
$431$	5646.17	0.631013	0.315506	−	0.948923i	$-0.397826\pi$
$431$	5646.17	0.631013	0.315506	+	0.948923i	$0.397826\pi$
$432$	0	0
$433$	−12057.5	−1.33821	−0.669106	−	0.743167i	$-0.733323\pi$
$433$	−12057.5	−1.33821	−0.669106	+	0.743167i	$0.733323\pi$
$434$	0	0
$435$	−1700.64	−0.187447
$436$	0	0
$437$	1646.17	0.180199
$438$	0	0
$439$	−8456.68	−0.919397	−0.459699	−	0.888075i	$-0.652043\pi$
$439$	−8456.68	−0.919397	−0.459699	+	0.888075i	$0.652043\pi$
$440$	0	0
$441$	−9153.75	−0.988420
$442$	0	0
$443$	−11418.9	−1.22467	−0.612333	−	0.790600i	$-0.709769\pi$
$443$	−11418.9	−1.22467	−0.612333	+	0.790600i	$0.709769\pi$
$444$	0	0
$445$	−4265.38	−0.454378
$446$	0	0
$447$	2254.65	0.238571
$448$	0	0
$449$	14853.3	1.56118	0.780588	−	0.625045i	$-0.214920\pi$
$449$	14853.3	1.56118	0.780588	+	0.625045i	$0.214920\pi$
$450$	0	0
$451$	−7318.39	−0.764101
$452$	0	0
$453$	−3731.92	−0.387066
$454$	0	0
$455$	−3426.54	−0.353052
$456$	0	0
$457$	−6038.15	−0.618058	−0.309029	−	0.951053i	$-0.600004\pi$
$457$	−6038.15	−0.618058	−0.309029	+	0.951053i	$0.600004\pi$
$458$	0	0
$459$	−2260.15	−0.229836
$460$	0	0
$461$	3962.18	0.400298	0.200149	−	0.979765i	$-0.435857\pi$
$461$	3962.18	0.400298	0.200149	+	0.979765i	$0.435857\pi$
$462$	0	0
$463$	−16060.5	−1.61208	−0.806040	−	0.591862i	$-0.798393\pi$
$463$	−16060.5	−1.61208	−0.806040	+	0.591862i	$0.798393\pi$
$464$	0	0
$465$	12671.4	1.26370
$466$	0	0
$467$	−12619.7	−1.25047	−0.625237	−	0.780435i	$-0.714998\pi$
$467$	−12619.7	−1.25047	−0.625237	+	0.780435i	$0.714998\pi$
$468$	0	0
$469$	−4550.81	−0.448053
$470$	0	0
$471$	14440.5	1.41270
$472$	0	0
$473$	−1309.33	−0.127279
$474$	0	0
$475$	−6246.40	−0.603378
$476$	0	0
$477$	−590.245	−0.0566571
$478$	0	0
$479$	−2228.38	−0.212562	−0.106281	−	0.994336i	$-0.533894\pi$
$479$	−2228.38	−0.212562	−0.106281	+	0.994336i	$0.533894\pi$
$480$	0	0
$481$	10259.0	0.972499
$482$	0	0
$483$	759.932	0.0715903
$484$	0	0
$485$	−232.995	−0.0218139
$486$	0	0
$487$	15583.9	1.45005	0.725026	−	0.688721i	$-0.241828\pi$
$487$	15583.9	1.45005	0.725026	+	0.688721i	$0.241828\pi$
$488$	0	0
$489$	−19142.0	−1.77021
$490$	0	0
$491$	15164.4	1.39380	0.696902	−	0.717166i	$-0.254561\pi$
$491$	15164.4	1.39380	0.696902	+	0.717166i	$0.254561\pi$
$492$	0	0
$493$	3162.31	0.288891
$494$	0	0
$495$	−4922.60	−0.446978
$496$	0	0
$497$	441.955	0.0398881
$498$	0	0
$499$	3382.78	0.303475	0.151737	−	0.988421i	$-0.451513\pi$
$499$	3382.78	0.303475	0.151737	+	0.988421i	$0.451513\pi$
$500$	0	0
$501$	−9878.22	−0.880891
$502$	0	0
$503$	8466.34	0.750488	0.375244	−	0.926926i	$-0.377559\pi$
$503$	8466.34	0.750488	0.375244	+	0.926926i	$0.377559\pi$
$504$	0	0
$505$	10930.1	0.963138
$506$	0	0
$507$	−24770.0	−2.16977
$508$	0	0
$509$	−16296.3	−1.41910	−0.709549	−	0.704656i	$-0.751101\pi$
$509$	−16296.3	−1.41910	−0.709549	+	0.704656i	$0.751101\pi$
$510$	0	0
$511$	2506.32	0.216973
$512$	0	0
$513$	−2010.29	−0.173015
$514$	0	0
$515$	7599.91	0.650276
$516$	0	0
$517$	−1637.64	−0.139310
$518$	0	0
$519$	−8770.39	−0.741768
$520$	0	0
$521$	−1187.61	−0.0998658	−0.0499329	−	0.998753i	$-0.515901\pi$
$521$	−1187.61	−0.0998658	−0.0499329	+	0.998753i	$0.515901\pi$
$522$	0	0
$523$	−6317.03	−0.528153	−0.264077	−	0.964502i	$-0.585067\pi$
$523$	−6317.03	−0.528153	−0.264077	+	0.964502i	$0.585067\pi$
$524$	0	0
$525$	−2883.56	−0.239712
$526$	0	0
$527$	−23562.1	−1.94759
$528$	0	0
$529$	−11878.9	−0.976324
$530$	0	0
$531$	21792.5	1.78101
$532$	0	0
$533$	25500.3	2.07231
$534$	0	0
$535$	−6552.91	−0.529546
$536$	0	0
$537$	−18134.7	−1.45730
$538$	0	0
$539$	6539.63	0.522601
$540$	0	0
$541$	14578.1	1.15853	0.579263	−	0.815140i	$-0.303340\pi$
$541$	14578.1	1.15853	0.579263	+	0.815140i	$0.303340\pi$
$542$	0	0
$543$	−24699.3	−1.95203
$544$	0	0
$545$	12266.8	0.964135
$546$	0	0
$547$	−7576.86	−0.592254	−0.296127	−	0.955149i	$-0.595695\pi$
$547$	−7576.86	−0.592254	−0.296127	+	0.955149i	$0.595695\pi$
$548$	0	0
$549$	−9223.61	−0.717038
$550$	0	0
$551$	2812.71	0.217469
$552$	0	0
$553$	6633.63	0.510110
$554$	0	0
$555$	−8123.29	−0.621288
$556$	0	0
$557$	−7868.81	−0.598586	−0.299293	−	0.954161i	$-0.596751\pi$
$557$	−7868.81	−0.598586	−0.299293	+	0.954161i	$0.596751\pi$
$558$	0	0
$559$	4562.26	0.345193
$560$	0	0
$561$	17460.4	1.31405
$562$	0	0
$563$	−17959.6	−1.34442	−0.672208	−	0.740363i	$-0.734654\pi$
$563$	−17959.6	−1.34442	−0.672208	+	0.740363i	$0.734654\pi$
$564$	0	0
$565$	−4898.28	−0.364729
$566$	0	0
$567$	3846.27	0.284882
$568$	0	0
$569$	−20873.8	−1.53792	−0.768958	−	0.639299i	$-0.779224\pi$
$569$	−20873.8	−1.53792	−0.768958	+	0.639299i	$0.779224\pi$
$570$	0	0
$571$	−16144.6	−1.18324	−0.591620	−	0.806217i	$-0.701512\pi$
$571$	−16144.6	−1.18324	−0.591620	+	0.806217i	$0.701512\pi$
$572$	0	0
$573$	−5133.45	−0.374263
$574$	0	0
$575$	−1093.08	−0.0792773
$576$	0	0
$577$	−12475.5	−0.900107	−0.450053	−	0.893002i	$-0.648595\pi$
$577$	−12475.5	−0.900107	−0.450053	+	0.893002i	$0.648595\pi$
$578$	0	0
$579$	−20765.3	−1.49046
$580$	0	0
$581$	6449.87	0.460560
$582$	0	0
$583$	421.683	0.0299560
$584$	0	0
$585$	17152.4	1.21224
$586$	0	0
$587$	−5322.95	−0.374279	−0.187140	−	0.982333i	$-0.559922\pi$
$587$	−5322.95	−0.374279	−0.187140	+	0.982333i	$0.559922\pi$
$588$	0	0
$589$	−20957.3	−1.46610
$590$	0	0
$591$	−20268.7	−1.41073
$592$	0	0
$593$	−23347.6	−1.61681	−0.808407	−	0.588624i	$-0.799670\pi$
$593$	−23347.6	−1.61681	−0.808407	+	0.588624i	$0.799670\pi$
$594$	0	0
$595$	−5045.13	−0.347613
$596$	0	0
$597$	12795.8	0.877213
$598$	0	0
$599$	27159.8	1.85262	0.926310	−	0.376763i	$-0.122963\pi$
$599$	27159.8	1.85262	0.926310	+	0.376763i	$0.122963\pi$
$600$	0	0
$601$	−11573.8	−0.785532	−0.392766	−	0.919638i	$-0.628482\pi$
$601$	−11573.8	−0.785532	−0.392766	+	0.919638i	$0.628482\pi$
$602$	0	0
$603$	22780.1	1.53844
$604$	0	0
$605$	−6844.29	−0.459934
$606$	0	0
$607$	12088.1	0.808302	0.404151	−	0.914692i	$-0.367567\pi$
$607$	12088.1	0.808302	0.404151	+	0.914692i	$0.367567\pi$
$608$	0	0
$609$	1298.45	0.0863970
$610$	0	0
$611$	5706.20	0.377821
$612$	0	0
$613$	−15903.2	−1.04784	−0.523918	−	0.851769i	$-0.675530\pi$
$613$	−15903.2	−1.04784	−0.523918	+	0.851769i	$0.675530\pi$
$614$	0	0
$615$	−20191.6	−1.32391
$616$	0	0
$617$	22075.0	1.44037	0.720184	−	0.693783i	$-0.244057\pi$
$617$	22075.0	1.44037	0.720184	+	0.693783i	$0.244057\pi$
$618$	0	0
$619$	28345.2	1.84053	0.920265	−	0.391295i	$-0.127973\pi$
$619$	28345.2	1.84053	0.920265	+	0.391295i	$0.127973\pi$
$620$	0	0
$621$	−351.787	−0.0227323
$622$	0	0
$623$	3256.63	0.209429
$624$	0	0
$625$	−3427.01	−0.219329
$626$	0	0
$627$	15530.2	0.989179
$628$	0	0
$629$	15105.1	0.957518
$630$	0	0
$631$	−13276.1	−0.837578	−0.418789	−	0.908084i	$-0.637545\pi$
$631$	−13276.1	−0.837578	−0.418789	+	0.908084i	$0.637545\pi$
$632$	0	0
$633$	−42894.7	−2.69338
$634$	0	0
$635$	15292.6	0.955701
$636$	0	0
$637$	−22786.8	−1.41734
$638$	0	0
$639$	−2212.31	−0.136960
$640$	0	0
$641$	7678.65	0.473149	0.236574	−	0.971613i	$-0.423975\pi$
$641$	7678.65	0.473149	0.236574	+	0.971613i	$0.423975\pi$
$642$	0	0
$643$	17537.3	1.07559	0.537793	−	0.843077i	$-0.319258\pi$
$643$	17537.3	1.07559	0.537793	+	0.843077i	$0.319258\pi$
$644$	0	0
$645$	−3612.48	−0.220529
$646$	0	0
$647$	10013.3	0.608447	0.304223	−	0.952601i	$-0.401603\pi$
$647$	10013.3	0.608447	0.304223	+	0.952601i	$0.401603\pi$
$648$	0	0
$649$	−15569.0	−0.941662
$650$	0	0
$651$	−9674.64	−0.582456
$652$	0	0
$653$	−15691.5	−0.940359	−0.470179	−	0.882571i	$-0.655811\pi$
$653$	−15691.5	−0.940359	−0.470179	+	0.882571i	$0.655811\pi$
$654$	0	0
$655$	−13078.7	−0.780194
$656$	0	0
$657$	−12546.0	−0.745000
$658$	0	0
$659$	22852.7	1.35086	0.675430	−	0.737424i	$-0.263958\pi$
$659$	22852.7	1.35086	0.675430	+	0.737424i	$0.263958\pi$
$660$	0	0
$661$	12895.6	0.758821	0.379411	−	0.925228i	$-0.376127\pi$
$661$	12895.6	0.758821	0.379411	+	0.925228i	$0.376127\pi$
$662$	0	0
$663$	−60839.3	−3.56381
$664$	0	0
$665$	−4487.39	−0.261674
$666$	0	0
$667$	492.205	0.0285731
$668$	0	0
$669$	18812.0	1.08717
$670$	0	0
$671$	6589.54	0.379115
$672$	0	0
$673$	−13093.4	−0.749945	−0.374973	−	0.927036i	$-0.622348\pi$
$673$	−13093.4	−0.749945	−0.374973	+	0.927036i	$0.622348\pi$
$674$	0	0
$675$	1334.86	0.0761165
$676$	0	0
$677$	13195.6	0.749111	0.374555	−	0.927205i	$-0.377795\pi$
$677$	13195.6	0.749111	0.374555	+	0.927205i	$0.377795\pi$
$678$	0	0
$679$	177.892	0.0100543
$680$	0	0
$681$	−2864.26	−0.161173
$682$	0	0
$683$	−12806.9	−0.717483	−0.358742	−	0.933437i	$-0.616794\pi$
$683$	−12806.9	−0.717483	−0.358742	+	0.933437i	$0.616794\pi$
$684$	0	0
$685$	2565.92	0.143122
$686$	0	0
$687$	45474.5	2.52542
$688$	0	0
$689$	−1469.32	−0.0812432
$690$	0	0
$691$	8300.30	0.456959	0.228479	−	0.973549i	$-0.426625\pi$
$691$	8300.30	0.456959	0.228479	+	0.973549i	$0.426625\pi$
$692$	0	0
$693$	3758.42	0.206018
$694$	0	0
$695$	−25340.3	−1.38304
$696$	0	0
$697$	37545.8	2.04038
$698$	0	0
$699$	−2434.64	−0.131740
$700$	0	0
$701$	9885.14	0.532606	0.266303	−	0.963889i	$-0.414198\pi$
$701$	9885.14	0.532606	0.266303	+	0.963889i	$0.414198\pi$
$702$	0	0
$703$	13435.2	0.720794
$704$	0	0
$705$	−4518.27	−0.241373
$706$	0	0
$707$	−8345.20	−0.443923
$708$	0	0
$709$	−8654.46	−0.458427	−0.229214	−	0.973376i	$-0.573615\pi$
$709$	−8654.46	−0.458427	−0.229214	+	0.973376i	$0.573615\pi$
$710$	0	0
$711$	−33206.2	−1.75152
$712$	0	0
$713$	−3667.38	−0.192629
$714$	0	0
$715$	−12254.0	−0.640943
$716$	0	0
$717$	−6874.04	−0.358041
$718$	0	0
$719$	−32870.4	−1.70495	−0.852475	−	0.522769i	$-0.824899\pi$
$719$	−32870.4	−1.70495	−0.852475	+	0.522769i	$0.824899\pi$
$720$	0	0
$721$	−5802.56	−0.299721
$722$	0	0
$723$	−15378.1	−0.791034
$724$	0	0
$725$	−1867.67	−0.0956739
$726$	0	0
$727$	14049.1	0.716715	0.358358	−	0.933584i	$-0.383337\pi$
$727$	14049.1	0.716715	0.358358	+	0.933584i	$0.383337\pi$
$728$	0	0
$729$	−23469.2	−1.19236
$730$	0	0
$731$	6717.31	0.339875
$732$	0	0
$733$	28327.1	1.42740	0.713700	−	0.700452i	$-0.247018\pi$
$733$	28327.1	1.42740	0.713700	+	0.700452i	$0.247018\pi$
$734$	0	0
$735$	18043.0	0.905477
$736$	0	0
$737$	−16274.6	−0.813409
$738$	0	0
$739$	−12694.9	−0.631919	−0.315960	−	0.948773i	$-0.602326\pi$
$739$	−12694.9	−0.631919	−0.315960	+	0.948773i	$0.602326\pi$
$740$	0	0
$741$	−54113.5	−2.68274
$742$	0	0
$743$	−7523.12	−0.371462	−0.185731	−	0.982601i	$-0.559465\pi$
$743$	−7523.12	−0.371462	−0.185731	+	0.982601i	$0.559465\pi$
$744$	0	0
$745$	−2329.80	−0.114573
$746$	0	0
$747$	−32286.3	−1.58139
$748$	0	0
$749$	5003.17	0.244075
$750$	0	0
$751$	−18543.4	−0.901011	−0.450506	−	0.892774i	$-0.648756\pi$
$751$	−18543.4	−0.901011	−0.450506	+	0.892774i	$0.648756\pi$
$752$	0	0
$753$	18842.9	0.911916
$754$	0	0
$755$	3856.31	0.185888
$756$	0	0
$757$	−36394.9	−1.74742	−0.873709	−	0.486448i	$-0.838292\pi$
$757$	−36394.9	−1.74742	−0.873709	+	0.486448i	$0.838292\pi$
$758$	0	0
$759$	2717.67	0.129967
$760$	0	0
$761$	8143.52	0.387914	0.193957	−	0.981010i	$-0.437868\pi$
$761$	8143.52	0.387914	0.193957	+	0.981010i	$0.437868\pi$
$762$	0	0
$763$	−9365.77	−0.444382
$764$	0	0
$765$	25254.6	1.19357
$766$	0	0
$767$	54249.0	2.55387
$768$	0	0
$769$	4954.01	0.232310	0.116155	−	0.993231i	$-0.462943\pi$
$769$	4954.01	0.232310	0.116155	+	0.993231i	$0.462943\pi$
$770$	0	0
$771$	−8816.00	−0.411803
$772$	0	0
$773$	25123.2	1.16897	0.584487	−	0.811403i	$-0.301296\pi$
$773$	25123.2	1.16897	0.584487	+	0.811403i	$0.301296\pi$
$774$	0	0
$775$	13915.9	0.644997
$776$	0	0
$777$	6202.16	0.286360
$778$	0	0
$779$	33395.1	1.53595
$780$	0	0
$781$	1580.52	0.0724142
$782$	0	0
$783$	−601.077	−0.0274339
$784$	0	0
$785$	−14921.8	−0.678447
$786$	0	0
$787$	43514.1	1.97092	0.985458	−	0.169921i	$-0.0543514\pi$
$787$	43514.1	1.97092	0.985458	+	0.169921i	$0.0543514\pi$
$788$	0	0
$789$	−15870.5	−0.716104
$790$	0	0
$791$	3739.85	0.168108
$792$	0	0
$793$	−22960.7	−1.02819
$794$	0	0
$795$	1163.43	0.0519028
$796$	0	0
$797$	10095.1	0.448667	0.224334	−	0.974512i	$-0.427979\pi$
$797$	10095.1	0.448667	0.224334	+	0.974512i	$0.427979\pi$
$798$	0	0
$799$	8401.62	0.372000
$800$	0	0
$801$	−16301.8	−0.719097
$802$	0	0
$803$	8963.10	0.393899
$804$	0	0
$805$	−785.261	−0.0343811
$806$	0	0
$807$	−13578.5	−0.592298
$808$	0	0
$809$	−9015.49	−0.391802	−0.195901	−	0.980624i	$-0.562763\pi$
$809$	−9015.49	−0.391802	−0.195901	+	0.980624i	$0.562763\pi$
$810$	0	0
$811$	37768.7	1.63531	0.817655	−	0.575708i	$-0.195273\pi$
$811$	37768.7	1.63531	0.817655	+	0.575708i	$0.195273\pi$
$812$	0	0
$813$	−25700.6	−1.10868
$814$	0	0
$815$	19780.0	0.850140
$816$	0	0
$817$	5974.71	0.255849
$818$	0	0
$819$	−13095.9	−0.558740
$820$	0	0
$821$	2803.90	0.119192	0.0595962	−	0.998223i	$-0.481019\pi$
$821$	2803.90	0.119192	0.0595962	+	0.998223i	$0.481019\pi$
$822$	0	0
$823$	42006.7	1.77917	0.889587	−	0.456765i	$-0.150992\pi$
$823$	42006.7	1.77917	0.889587	+	0.456765i	$0.150992\pi$
$824$	0	0
$825$	−10312.2	−0.435181
$826$	0	0
$827$	−12606.7	−0.530081	−0.265040	−	0.964237i	$-0.585385\pi$
$827$	−12606.7	−0.530081	−0.265040	+	0.964237i	$0.585385\pi$
$828$	0	0
$829$	43457.6	1.82068	0.910341	−	0.413860i	$-0.135820\pi$
$829$	43457.6	1.82068	0.910341	+	0.413860i	$0.135820\pi$
$830$	0	0
$831$	14472.1	0.604131
$832$	0	0
$833$	−33550.5	−1.39550
$834$	0	0
$835$	10207.5	0.423047
$836$	0	0
$837$	4478.58	0.184949
$838$	0	0
$839$	37900.8	1.55957	0.779786	−	0.626046i	$-0.215328\pi$
$839$	37900.8	1.55957	0.779786	+	0.626046i	$0.215328\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	10890.8	0.444959
$844$	0	0
$845$	25595.6	1.04203
$846$	0	0
$847$	5225.64	0.211989
$848$	0	0
$849$	16630.6	0.672274
$850$	0	0
$851$	2351.07	0.0947044
$852$	0	0
$853$	29288.9	1.17565	0.587827	−	0.808987i	$-0.299984\pi$
$853$	29288.9	1.17565	0.587827	+	0.808987i	$0.299984\pi$
$854$	0	0
$855$	22462.7	0.898488
$856$	0	0
$857$	−29392.4	−1.17156	−0.585778	−	0.810471i	$-0.699211\pi$
$857$	−29392.4	−1.17156	−0.585778	+	0.810471i	$0.699211\pi$
$858$	0	0
$859$	9084.28	0.360828	0.180414	−	0.983591i	$-0.442256\pi$
$859$	9084.28	0.360828	0.180414	+	0.983591i	$0.442256\pi$
$860$	0	0
$861$	15416.4	0.610207
$862$	0	0
$863$	−35765.3	−1.41074	−0.705368	−	0.708841i	$-0.749218\pi$
$863$	−35765.3	−1.41074	−0.705368	+	0.708841i	$0.749218\pi$
$864$	0	0
$865$	9062.72	0.356233
$866$	0	0
$867$	−52566.4	−2.05911
$868$	0	0
$869$	23723.2	0.926070
$870$	0	0
$871$	56707.5	2.20604
$872$	0	0
$873$	−890.482	−0.0345226
$874$	0	0
$875$	8762.98	0.338563
$876$	0	0
$877$	−12434.7	−0.478781	−0.239390	−	0.970923i	$-0.576948\pi$
$877$	−12434.7	−0.478781	−0.239390	+	0.970923i	$0.576948\pi$
$878$	0	0
$879$	−12629.3	−0.484613
$880$	0	0
$881$	9545.34	0.365029	0.182515	−	0.983203i	$-0.441576\pi$
$881$	9545.34	0.365029	0.182515	+	0.983203i	$0.441576\pi$
$882$	0	0
$883$	20819.9	0.793484	0.396742	−	0.917930i	$-0.370141\pi$
$883$	20819.9	0.793484	0.396742	+	0.917930i	$0.370141\pi$
$884$	0	0
$885$	−42955.3	−1.63156
$886$	0	0
$887$	−18636.1	−0.705454	−0.352727	−	0.935726i	$-0.614746\pi$
$887$	−18636.1	−0.705454	−0.352727	+	0.935726i	$0.614746\pi$
$888$	0	0
$889$	−11676.0	−0.440495
$890$	0	0
$891$	13755.0	0.517184
$892$	0	0
$893$	7472.82	0.280032
$894$	0	0
$895$	18739.1	0.699866
$896$	0	0
$897$	−9469.48	−0.352483
$898$	0	0
$899$	−6266.23	−0.232470
$900$	0	0
$901$	−2163.37	−0.0799916
$902$	0	0
$903$	2758.14	0.101645
$904$	0	0
$905$	25522.6	0.937458
$906$	0	0
$907$	32100.9	1.17518	0.587592	−	0.809157i	$-0.300076\pi$
$907$	32100.9	1.17518	0.587592	+	0.809157i	$0.300076\pi$
$908$	0	0
$909$	41773.9	1.52426
$910$	0	0
$911$	50169.6	1.82458	0.912290	−	0.409545i	$-0.134312\pi$
$911$	50169.6	1.82458	0.912290	+	0.409545i	$0.134312\pi$
$912$	0	0
$913$	23066.0	0.836116
$914$	0	0
$915$	18180.7	0.656868
$916$	0	0
$917$	9985.63	0.359602
$918$	0	0
$919$	−443.178	−0.0159076	−0.00795380	−	0.999968i	$-0.502532\pi$
$919$	−443.178	−0.0159076	−0.00795380	+	0.999968i	$0.502532\pi$
$920$	0	0
$921$	−57205.0	−2.04665
$922$	0	0
$923$	−5507.19	−0.196394
$924$	0	0
$925$	−8921.12	−0.317108
$926$	0	0
$927$	29046.1	1.02912
$928$	0	0
$929$	−23326.9	−0.823821	−0.411911	−	0.911224i	$-0.635138\pi$
$929$	−23326.9	−0.823821	−0.411911	+	0.911224i	$0.635138\pi$
$930$	0	0
$931$	−29841.5	−1.05050
$932$	0	0
$933$	−70710.4	−2.48119
$934$	0	0
$935$	−18042.4	−0.631069
$936$	0	0
$937$	−30862.8	−1.07603	−0.538017	−	0.842934i	$-0.680826\pi$
$937$	−30862.8	−1.07603	−0.538017	+	0.842934i	$0.680826\pi$
$938$	0	0
$939$	−43737.7	−1.52005
$940$	0	0
$941$	15893.4	0.550597	0.275298	−	0.961359i	$-0.411223\pi$
$941$	15893.4	0.550597	0.275298	+	0.961359i	$0.411223\pi$
$942$	0	0
$943$	5843.90	0.201807
$944$	0	0
$945$	958.954	0.0330104
$946$	0	0
$947$	−17620.9	−0.604650	−0.302325	−	0.953205i	$-0.597763\pi$
$947$	−17620.9	−0.604650	−0.302325	+	0.953205i	$0.597763\pi$
$948$	0	0
$949$	−31231.1	−1.06829
$950$	0	0
$951$	−75527.4	−2.57533
$952$	0	0
$953$	−55202.4	−1.87637	−0.938186	−	0.346132i	$-0.887495\pi$
$953$	−55202.4	−1.87637	−0.938186	+	0.346132i	$0.887495\pi$
$954$	0	0
$955$	5304.55	0.179740
$956$	0	0
$957$	4643.51	0.156848
$958$	0	0
$959$	−1959.09	−0.0659669
$960$	0	0
$961$	16898.2	0.567224
$962$	0	0
$963$	−25044.6	−0.838058
$964$	0	0
$965$	21457.4	0.715792
$966$	0	0
$967$	3325.66	0.110596	0.0552979	−	0.998470i	$-0.482389\pi$
$967$	3325.66	0.110596	0.0552979	+	0.998470i	$0.482389\pi$
$968$	0	0
$969$	−79674.9	−2.64141
$970$	0	0
$971$	−30495.2	−1.00786	−0.503932	−	0.863743i	$-0.668114\pi$
$971$	−30495.2	−1.00786	−0.503932	+	0.863743i	$0.668114\pi$
$972$	0	0
$973$	19347.4	0.637461
$974$	0	0
$975$	35932.0	1.18025
$976$	0	0
$977$	−16285.4	−0.533281	−0.266641	−	0.963796i	$-0.585914\pi$
$977$	−16285.4	−0.533281	−0.266641	+	0.963796i	$0.585914\pi$
$978$	0	0
$979$	11646.4	0.380204
$980$	0	0
$981$	46882.6	1.52584
$982$	0	0
$983$	51154.5	1.65979	0.829895	−	0.557919i	$-0.188400\pi$
$983$	51154.5	1.65979	0.829895	+	0.557919i	$0.188400\pi$
$984$	0	0
$985$	20944.3	0.677504
$986$	0	0
$987$	3449.72	0.111252
$988$	0	0
$989$	1045.53	0.0336158
$990$	0	0
$991$	3688.23	0.118224	0.0591122	−	0.998251i	$-0.481173\pi$
$991$	3688.23	0.118224	0.0591122	+	0.998251i	$0.481173\pi$
$992$	0	0
$993$	46143.2	1.47463
$994$	0	0
$995$	−13222.3	−0.421280
$996$	0	0
$997$	−14293.1	−0.454028	−0.227014	−	0.973892i	$-0.572896\pi$
$997$	−14293.1	−0.454028	−0.227014	+	0.973892i	$0.572896\pi$
$998$	0	0
$999$	−2871.10	−0.0909286

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.bi.1.2		10
4.3	odd	2	inner	1856.4.a.bi.1.9		10
8.3	odd	2		928.4.a.f.1.2	✓	10
8.5	even	2		928.4.a.f.1.9	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.4.a.f.1.2	✓	10	8.3	odd	2
928.4.a.f.1.9	yes	10	8.5	even	2
1856.4.a.bi.1.2		10	1.1	even	1	trivial
1856.4.a.bi.1.9		10	4.3	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-7.53335 q^{3} +7.78444 q^{5} -5.94345 q^{7} +29.7513 q^{9} +O(q^{10})\)	\(q-7.53335 q^{3} +7.78444 q^{5} -5.94345 q^{7} +29.7513 q^{9} -21.2550 q^{11} +74.0611 q^{13} -58.6429 q^{15} +109.045 q^{17} +96.9901 q^{19} +44.7741 q^{21} +16.9726 q^{23} -64.4025 q^{25} -20.7268 q^{27} +29.0000 q^{29} -216.077 q^{31} +160.121 q^{33} -46.2664 q^{35} +138.521 q^{37} -557.928 q^{39} +344.314 q^{41} +61.6012 q^{43} +231.598 q^{45} +77.0472 q^{47} -307.675 q^{49} -821.474 q^{51} -19.8393 q^{53} -165.458 q^{55} -730.660 q^{57} +732.490 q^{59} -310.023 q^{61} -176.826 q^{63} +576.524 q^{65} +765.685 q^{67} -127.860 q^{69} -74.3601 q^{71} -421.694 q^{73} +485.166 q^{75} +126.328 q^{77} -1116.13 q^{79} -647.144 q^{81} -1085.21 q^{83} +848.855 q^{85} -218.467 q^{87} -547.936 q^{89} -440.178 q^{91} +1627.78 q^{93} +755.014 q^{95} -29.9308 q^{97} -632.364 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 4 q^{5} + 50 q^{9}+O(q^{10}) \) 10 * q + 4 * q^5 + 50 * q^9	\( 10 q + 4 q^{5} + 50 q^{9} + 220 q^{13} + 76 q^{17} + 104 q^{21} + 446 q^{25} + 290 q^{29} - 1120 q^{33} + 1708 q^{37} - 980 q^{41} - 348 q^{45} + 1146 q^{49} + 44 q^{53} - 40 q^{57} + 1492 q^{61} - 2016 q^{65} + 2328 q^{69} - 3100 q^{73} + 3016 q^{77} - 2174 q^{81} + 9440 q^{85} + 636 q^{89} + 2536 q^{93} + 620 q^{97}+O(q^{100}) \) 10 * q + 4 * q^5 + 50 * q^9 + 220 * q^13 + 76 * q^17 + 104 * q^21 + 446 * q^25 + 290 * q^29 - 1120 * q^33 + 1708 * q^37 - 980 * q^41 - 348 * q^45 + 1146 * q^49 + 44 * q^53 - 40 * q^57 + 1492 * q^61 - 2016 * q^65 + 2328 * q^69 - 3100 * q^73 + 3016 * q^77 - 2174 * q^81 + 9440 * q^85 + 636 * q^89 + 2536 * q^93 + 620 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more