LMFDB - Embedded newform 800.4.c.i.449.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,4,Mod(449,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("800.449");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	800.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$47.2015280046$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.4
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	800.449
Dual form			800.4.c.i.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.89898i q^{3} +20.4949i q^{7} -52.1918 q^{9} +O(q^{10})$	$q+8.89898i q^{3} +20.4949i q^{7} -52.1918 q^{9} -61.3939 q^{11} -45.1918i q^{13} -115.576i q^{17} +64.8082 q^{19} -182.384 q^{21} +6.11123i q^{23} -224.182i q^{27} +224.384 q^{29} -58.6061 q^{31} -546.343i q^{33} -99.6163i q^{37} +402.161 q^{39} -145.959 q^{41} -6.73776i q^{43} +203.687i q^{47} -77.0408 q^{49} +1028.50 q^{51} -275.576i q^{53} +576.727i q^{57} +262.020 q^{59} -790.767 q^{61} -1069.67i q^{63} +141.748i q^{67} -54.3837 q^{69} -1043.98 q^{71} +1057.88i q^{73} -1258.26i q^{77} -826.624 q^{79} +585.808 q^{81} -1026.25i q^{83} +1996.79i q^{87} +154.849 q^{89} +926.202 q^{91} -521.535i q^{93} +414.041i q^{97} +3204.26 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 52 q^{9}+O(q^{10}) $ 4 * q - 52 * q^9	$ 4 q - 52 q^{9} - 128 q^{11} + 416 q^{19} - 416 q^{21} + 584 q^{29} - 352 q^{31} + 864 q^{39} + 200 q^{41} - 1092 q^{49} + 2272 q^{51} + 1440 q^{59} - 2536 q^{61} + 96 q^{69} + 96 q^{71} + 64 q^{79} + 2500 q^{81} + 1560 q^{89} + 3744 q^{91} + 6272 q^{99}+O(q^{100}) $ 4 * q - 52 * q^9 - 128 * q^11 + 416 * q^19 - 416 * q^21 + 584 * q^29 - 352 * q^31 + 864 * q^39 + 200 * q^41 - 1092 * q^49 + 2272 * q^51 + 1440 * q^59 - 2536 * q^61 + 96 * q^69 + 96 * q^71 + 64 * q^79 + 2500 * q^81 + 1560 * q^89 + 3744 * q^91 + 6272 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	8.89898i	1.71261i	0.516471	+	0.856305i	$0.327245\pi$
$3$	8.89898i	1.71261i	−0.516471	+	0.856305i	$0.672755\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	20.4949i	1.10662i	0.832975	+	0.553310i	$0.186636\pi$
$7$	20.4949i	1.10662i	−0.832975	+	0.553310i	$0.813364\pi$
$8$	0	0
$9$	−52.1918	−1.93303
$10$	0	0
$11$	−61.3939	−1.68281	−0.841407	−	0.540402i	$-0.818272\pi$
$11$	−61.3939	−1.68281	−0.841407	+	0.540402i	$0.818272\pi$
$12$	0	0
$13$	− 45.1918i	− 0.964151i	−0.876130	−	0.482075i	$-0.839883\pi$
$13$	− 45.1918i	− 0.964151i	0.876130	−	0.482075i	$-0.160117\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 115.576i	− 1.64889i	−0.565940	−	0.824446i	$-0.691487\pi$
$17$	− 115.576i	− 1.64889i	0.565940	−	0.824446i	$-0.308513\pi$
$18$	0	0
$19$	64.8082	0.782527	0.391263	−	0.920279i	$-0.372038\pi$
$19$	64.8082	0.782527	0.391263	+	0.920279i	$0.372038\pi$
$20$	0	0
$21$	−182.384	−1.89521
$22$	0	0
$23$	6.11123i	0.0554034i	0.999616	+	0.0277017i	$0.00881886\pi$
$23$	6.11123i	0.0554034i	−0.999616	+	0.0277017i	$0.991181\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 224.182i	− 1.59792i
$28$	0	0
$29$	224.384	1.43679	0.718397	−	0.695634i	$-0.244876\pi$
$29$	224.384	1.43679	0.718397	+	0.695634i	$0.244876\pi$
$30$	0	0
$31$	−58.6061	−0.339547	−0.169774	−	0.985483i	$-0.554304\pi$
$31$	−58.6061	−0.339547	−0.169774	+	0.985483i	$0.554304\pi$
$32$	0	0
$33$	− 546.343i	− 2.88200i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 99.6163i	− 0.442617i	−0.975204	−	0.221308i	$-0.928967\pi$
$37$	− 99.6163i	− 0.442617i	0.975204	−	0.221308i	$-0.0710327\pi$
$38$	0	0
$39$	402.161	1.65121
$40$	0	0
$41$	−145.959	−0.555975	−0.277988	−	0.960585i	$-0.589667\pi$
$41$	−145.959	−0.555975	−0.277988	+	0.960585i	$0.589667\pi$
$42$	0	0
$43$	− 6.73776i	− 0.0238953i	−0.999929	−	0.0119477i	$-0.996197\pi$
$43$	− 6.73776i	− 0.0238953i	0.999929	−	0.0119477i	$-0.00380315\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	203.687i	0.632144i	0.948735	+	0.316072i	$0.102364\pi$
$47$	203.687i	0.632144i	−0.948735	+	0.316072i	$0.897636\pi$
$48$	0	0
$49$	−77.0408	−0.224609
$50$	0	0
$51$	1028.50	2.82391
$52$	0	0
$53$	− 275.576i	− 0.714211i	−0.934064	−	0.357106i	$-0.883764\pi$
$53$	− 275.576i	− 0.714211i	0.934064	−	0.357106i	$-0.116236\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	576.727i	1.34016i
$58$	0	0
$59$	262.020	0.578172	0.289086	−	0.957303i	$-0.406649\pi$
$59$	262.020	0.578172	0.289086	+	0.957303i	$0.406649\pi$
$60$	0	0
$61$	−790.767	−1.65979	−0.829897	−	0.557917i	$-0.811601\pi$
$61$	−790.767	−1.65979	−0.829897	+	0.557917i	$0.811601\pi$
$62$	0	0
$63$	− 1069.67i	− 2.13913i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	141.748i	0.258467i	0.991614	+	0.129233i	$0.0412516\pi$
$67$	141.748i	0.258467i	−0.991614	+	0.129233i	$0.958748\pi$
$68$	0	0
$69$	−54.3837	−0.0948844
$70$	0	0
$71$	−1043.98	−1.74503	−0.872516	−	0.488585i	$-0.837513\pi$
$71$	−1043.98	−1.74503	−0.872516	+	0.488585i	$0.837513\pi$
$72$	0	0
$73$	1057.88i	1.69610i	0.529917	+	0.848049i	$0.322223\pi$
$73$	1057.88i	1.69610i	−0.529917	+	0.848049i	$0.677777\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1258.26i	− 1.86224i
$78$	0	0
$79$	−826.624	−1.17725	−0.588624	−	0.808407i	$-0.700330\pi$
$79$	−826.624	−1.17725	−0.588624	+	0.808407i	$0.700330\pi$
$80$	0	0
$81$	585.808	0.803578
$82$	0	0
$83$	− 1026.25i	− 1.35718i	−0.734518	−	0.678589i	$-0.762592\pi$
$83$	− 1026.25i	− 1.35718i	0.734518	−	0.678589i	$-0.237408\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1996.79i	2.46067i
$88$	0	0
$89$	154.849	0.184427	0.0922133	−	0.995739i	$-0.470606\pi$
$89$	154.849	0.184427	0.0922133	+	0.995739i	$0.470606\pi$
$90$	0	0
$91$	926.202	1.06695
$92$	0	0
$93$	− 521.535i	− 0.581512i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	414.041i	0.433397i	0.976239	+	0.216698i	$0.0695288\pi$
$97$	414.041i	0.433397i	−0.976239	+	0.216698i	$0.930471\pi$
$98$	0	0
$99$	3204.26	3.25293
$100$	0	0
$101$	1776.99	1.75066	0.875331	−	0.483524i	$-0.160643\pi$
$101$	1776.99	1.75066	0.875331	+	0.483524i	$0.160643\pi$
$102$	0	0
$103$	− 767.991i	− 0.734683i	−0.930086	−	0.367342i	$-0.880268\pi$
$103$	− 767.991i	− 0.734683i	0.930086	−	0.367342i	$-0.119732\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1649.00i	− 1.48986i	−0.667144	−	0.744929i	$-0.732483\pi$
$107$	− 1649.00i	− 1.48986i	0.667144	−	0.744929i	$-0.267517\pi$
$108$	0	0
$109$	−884.220	−0.777000	−0.388500	−	0.921449i	$-0.627007\pi$
$109$	−884.220	−0.777000	−0.388500	+	0.921449i	$0.627007\pi$
$110$	0	0
$111$	886.484	0.758030
$112$	0	0
$113$	1138.85i	0.948088i	0.880501	+	0.474044i	$0.157206\pi$
$113$	1138.85i	0.948088i	−0.880501	+	0.474044i	$0.842794\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2358.64i	1.86373i
$118$	0	0
$119$	2368.71	1.82470
$120$	0	0
$121$	2438.21	1.83186
$122$	0	0
$123$	− 1298.89i	− 0.952169i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 1701.60i	− 1.18892i	−0.804125	−	0.594460i	$-0.797366\pi$
$127$	− 1701.60i	− 1.18892i	0.804125	−	0.594460i	$-0.202634\pi$
$128$	0	0
$129$	59.9592	0.0409233
$130$	0	0
$131$	1540.99	1.02776	0.513880	−	0.857862i	$-0.328208\pi$
$131$	1540.99	1.02776	0.513880	+	0.857862i	$0.328208\pi$
$132$	0	0
$133$	1328.24i	0.865960i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1391.53i	− 0.867787i	−0.900964	−	0.433894i	$-0.857139\pi$
$137$	− 1391.53i	− 0.867787i	0.900964	−	0.433894i	$-0.142861\pi$
$138$	0	0
$139$	156.649	0.0955885	0.0477942	−	0.998857i	$-0.484781\pi$
$139$	156.649	0.0955885	0.0477942	+	0.998857i	$0.484781\pi$
$140$	0	0
$141$	−1812.60	−1.08262
$142$	0	0
$143$	2774.50i	1.62249i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 685.585i	− 0.384667i
$148$	0	0
$149$	−214.245	−0.117796	−0.0588981	−	0.998264i	$-0.518759\pi$
$149$	−214.245	−0.117796	−0.0588981	+	0.998264i	$0.518759\pi$
$150$	0	0
$151$	35.0510	0.0188901	0.00944507	−	0.999955i	$-0.496993\pi$
$151$	35.0510	0.0188901	0.00944507	+	0.999955i	$0.496993\pi$
$152$	0	0
$153$	6032.10i	3.18736i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	349.592i	0.177710i	0.996045	+	0.0888550i	$0.0283208\pi$
$157$	349.592i	0.177710i	−0.996045	+	0.0888550i	$0.971679\pi$
$158$	0	0
$159$	2452.34	1.22317
$160$	0	0
$161$	−125.249	−0.0613106
$162$	0	0
$163$	− 2029.04i	− 0.975010i	−0.873120	−	0.487505i	$-0.837907\pi$
$163$	− 2029.04i	− 0.975010i	0.873120	−	0.487505i	$-0.162093\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 214.619i	− 0.0994476i	−0.998763	−	0.0497238i	$-0.984166\pi$
$167$	− 214.619i	− 0.0994476i	0.998763	−	0.0497238i	$-0.0158341\pi$
$168$	0	0
$169$	154.698	0.0704133
$170$	0	0
$171$	−3382.46	−1.51265
$172$	0	0
$173$	− 3424.99i	− 1.50518i	−0.658487	−	0.752592i	$-0.728803\pi$
$173$	− 3424.99i	− 1.50518i	0.658487	−	0.752592i	$-0.271197\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2331.71i	0.990183i
$178$	0	0
$179$	2030.10	0.847692	0.423846	−	0.905734i	$-0.360680\pi$
$179$	2030.10	0.847692	0.423846	+	0.905734i	$0.360680\pi$
$180$	0	0
$181$	204.139	0.0838316	0.0419158	−	0.999121i	$-0.486654\pi$
$181$	204.139	0.0838316	0.0419158	+	0.999121i	$0.486654\pi$
$182$	0	0
$183$	− 7037.02i	− 2.84258i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	7095.63i	2.77478i
$188$	0	0
$189$	4594.58	1.76829
$190$	0	0
$191$	−617.353	−0.233875	−0.116937	−	0.993139i	$-0.537308\pi$
$191$	−617.353	−0.233875	−0.116937	+	0.993139i	$0.537308\pi$
$192$	0	0
$193$	− 3071.33i	− 1.14549i	−0.819734	−	0.572744i	$-0.805879\pi$
$193$	− 3071.33i	− 1.14549i	0.819734	−	0.572744i	$-0.194121\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 812.098i	− 0.293703i	−0.989159	−	0.146852i	$-0.953086\pi$
$197$	− 812.098i	− 0.293703i	0.989159	−	0.146852i	$-0.0469140\pi$
$198$	0	0
$199$	1065.93	0.379709	0.189855	−	0.981812i	$-0.439198\pi$
$199$	1065.93	0.379709	0.189855	+	0.981812i	$0.439198\pi$
$200$	0	0
$201$	−1261.41	−0.442653
$202$	0	0
$203$	4598.72i	1.58998i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 318.956i	− 0.107097i
$208$	0	0
$209$	−3978.82	−1.31685
$210$	0	0
$211$	−1222.24	−0.398779	−0.199390	−	0.979920i	$-0.563896\pi$
$211$	−1222.24	−0.398779	−0.199390	+	0.979920i	$0.563896\pi$
$212$	0	0
$213$	− 9290.33i	− 2.98856i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 1201.13i	− 0.375750i
$218$	0	0
$219$	−9414.03	−2.90475
$220$	0	0
$221$	−5223.07	−1.58978
$222$	0	0
$223$	− 414.374i	− 0.124433i	−0.998063	−	0.0622165i	$-0.980183\pi$
$223$	− 414.374i	− 0.124433i	0.998063	−	0.0622165i	$-0.0198169\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 2293.88i	− 0.670707i	−0.942092	−	0.335353i	$-0.891144\pi$
$227$	− 2293.88i	− 0.670707i	0.942092	−	0.335353i	$-0.108856\pi$
$228$	0	0
$229$	−3214.72	−0.927662	−0.463831	−	0.885924i	$-0.653525\pi$
$229$	−3214.72	−0.927662	−0.463831	+	0.885924i	$0.653525\pi$
$230$	0	0
$231$	11197.2	3.18928
$232$	0	0
$233$	− 3598.60i	− 1.01181i	−0.862589	−	0.505905i	$-0.831159\pi$
$233$	− 3598.60i	− 1.01181i	0.862589	−	0.505905i	$-0.168841\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 7356.11i	− 2.01616i
$238$	0	0
$239$	−2462.58	−0.666491	−0.333245	−	0.942840i	$-0.608144\pi$
$239$	−2462.58	−0.666491	−0.333245	+	0.942840i	$0.608144\pi$
$240$	0	0
$241$	4153.27	1.11011	0.555054	−	0.831815i	$-0.312698\pi$
$241$	4153.27	1.11011	0.555054	+	0.831815i	$0.312698\pi$
$242$	0	0
$243$	− 839.809i	− 0.221703i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 2928.80i	− 0.754474i
$248$	0	0
$249$	9132.60	2.32432
$250$	0	0
$251$	−1335.26	−0.335779	−0.167890	−	0.985806i	$-0.553695\pi$
$251$	−1335.26	−0.335779	−0.167890	+	0.985806i	$0.553695\pi$
$252$	0	0
$253$	− 375.192i	− 0.0932336i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 530.359i	− 0.128727i	−0.997927	−	0.0643636i	$-0.979498\pi$
$257$	− 530.359i	− 0.128727i	0.997927	−	0.0643636i	$-0.0205017\pi$
$258$	0	0
$259$	2041.63	0.489809
$260$	0	0
$261$	−11711.0	−2.77737
$262$	0	0
$263$	2234.23i	0.523836i	0.965090	+	0.261918i	$0.0843549\pi$
$263$	2234.23i	0.523836i	−0.965090	+	0.261918i	$0.915645\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1378.00i	0.315851i
$268$	0	0
$269$	−5096.09	−1.15507	−0.577535	−	0.816366i	$-0.695985\pi$
$269$	−5096.09	−1.15507	−0.577535	+	0.816366i	$0.695985\pi$
$270$	0	0
$271$	1353.39	0.303367	0.151684	−	0.988429i	$-0.451530\pi$
$271$	1353.39	0.303367	0.151684	+	0.988429i	$0.451530\pi$
$272$	0	0
$273$	8242.25i	1.82727i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 8879.57i	− 1.92607i	−0.269376	−	0.963035i	$-0.586817\pi$
$277$	− 8879.57i	− 1.92607i	0.269376	−	0.963035i	$-0.413183\pi$
$278$	0	0
$279$	3058.76	0.656356
$280$	0	0
$281$	−1145.55	−0.243195	−0.121598	−	0.992579i	$-0.538802\pi$
$281$	−1145.55	−0.243195	−0.121598	+	0.992579i	$0.538802\pi$
$282$	0	0
$283$	− 385.399i	− 0.0809526i	−0.999180	−	0.0404763i	$-0.987112\pi$
$283$	− 385.399i	− 0.0809526i	0.999180	−	0.0404763i	$-0.0128875\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 2991.42i	− 0.615254i
$288$	0	0
$289$	−8444.70	−1.71885
$290$	0	0
$291$	−3684.54	−0.742239
$292$	0	0
$293$	− 3812.30i	− 0.760126i	−0.924961	−	0.380063i	$-0.875902\pi$
$293$	− 3812.30i	− 0.760126i	0.924961	−	0.380063i	$-0.124098\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	13763.4i	2.68900i
$298$	0	0
$299$	276.178	0.0534172
$300$	0	0
$301$	138.090	0.0264430
$302$	0	0
$303$	15813.4i	2.99820i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	6897.45i	1.28228i	0.767426	+	0.641138i	$0.221537\pi$
$307$	6897.45i	1.28228i	−0.767426	+	0.641138i	$0.778463\pi$
$308$	0	0
$309$	6834.33	1.25823
$310$	0	0
$311$	−8125.39	−1.48151	−0.740753	−	0.671778i	$-0.765531\pi$
$311$	−8125.39	−1.48151	−0.740753	+	0.671778i	$0.765531\pi$
$312$	0	0
$313$	− 2052.66i	− 0.370681i	−0.982674	−	0.185341i	$-0.940661\pi$
$313$	− 2052.66i	− 0.370681i	0.982674	−	0.185341i	$-0.0593389\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4175.12i	0.739741i	0.929083	+	0.369871i	$0.120598\pi$
$317$	4175.12i	0.739741i	−0.929083	+	0.369871i	$0.879402\pi$
$318$	0	0
$319$	−13775.8	−2.41786
$320$	0	0
$321$	14674.4	2.55154
$322$	0	0
$323$	− 7490.24i	− 1.29030i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 7868.66i	− 1.33070i
$328$	0	0
$329$	−4174.54	−0.699543
$330$	0	0
$331$	−11738.2	−1.94921	−0.974607	−	0.223923i	$-0.928114\pi$
$331$	−11738.2	−1.94921	−0.974607	+	0.223923i	$0.928114\pi$
$332$	0	0
$333$	5199.16i	0.855592i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 1698.85i	− 0.274606i	−0.990529	−	0.137303i	$-0.956157\pi$
$337$	− 1698.85i	− 0.274606i	0.990529	−	0.137303i	$-0.0438434\pi$
$338$	0	0
$339$	−10134.6	−1.62370
$340$	0	0
$341$	3598.06	0.571395
$342$	0	0
$343$	5450.81i	0.858064i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 9682.53i	− 1.49794i	−0.662604	−	0.748970i	$-0.730549\pi$
$347$	− 9682.53i	− 1.49794i	0.662604	−	0.748970i	$-0.269451\pi$
$348$	0	0
$349$	10755.1	1.64959	0.824796	−	0.565430i	$-0.191290\pi$
$349$	10755.1	1.64959	0.824796	+	0.565430i	$0.191290\pi$
$350$	0	0
$351$	−10131.2	−1.54063
$352$	0	0
$353$	7044.94i	1.06222i	0.847302	+	0.531111i	$0.178225\pi$
$353$	7044.94i	1.06222i	−0.847302	+	0.531111i	$0.821775\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	21079.1i	3.12500i
$358$	0	0
$359$	805.576	0.118431	0.0592154	−	0.998245i	$-0.481140\pi$
$359$	805.576	0.118431	0.0592154	+	0.998245i	$0.481140\pi$
$360$	0	0
$361$	−2658.90	−0.387652
$362$	0	0
$363$	21697.6i	3.13726i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	11157.2i	1.58692i	0.608623	+	0.793460i	$0.291722\pi$
$367$	11157.2i	1.58692i	−0.608623	+	0.793460i	$0.708278\pi$
$368$	0	0
$369$	7617.88	1.07472
$370$	0	0
$371$	5647.89	0.790361
$372$	0	0
$373$	4634.24i	0.643303i	0.946858	+	0.321652i	$0.104238\pi$
$373$	4634.24i	0.643303i	−0.946858	+	0.321652i	$0.895762\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 10140.3i	− 1.38529i
$378$	0	0
$379$	−754.678	−0.102283	−0.0511414	−	0.998691i	$-0.516286\pi$
$379$	−754.678	−0.102283	−0.0511414	+	0.998691i	$0.516286\pi$
$380$	0	0
$381$	15142.5	2.03616
$382$	0	0
$383$	− 984.709i	− 0.131374i	−0.997840	−	0.0656871i	$-0.979076\pi$
$383$	− 984.709i	− 0.131374i	0.997840	−	0.0656871i	$-0.0209239\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	351.656i	0.0461904i
$388$	0	0
$389$	3259.21	0.424803	0.212402	−	0.977182i	$-0.431872\pi$
$389$	3259.21	0.424803	0.212402	+	0.977182i	$0.431872\pi$
$390$	0	0
$391$	706.308	0.0913543
$392$	0	0
$393$	13713.2i	1.76015i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 13713.2i	− 1.73361i	−0.498644	−	0.866807i	$-0.666169\pi$
$397$	− 13713.2i	− 1.73361i	0.498644	−	0.866807i	$-0.333831\pi$
$398$	0	0
$399$	−11820.0	−1.48305
$400$	0	0
$401$	−671.306	−0.0835996	−0.0417998	−	0.999126i	$-0.513309\pi$
$401$	−671.306	−0.0835996	−0.0417998	+	0.999126i	$0.513309\pi$
$402$	0	0
$403$	2648.52i	0.327375i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6115.83i	0.744842i
$408$	0	0
$409$	−12957.4	−1.56651	−0.783253	−	0.621703i	$-0.786441\pi$
$409$	−12957.4	−1.56651	−0.783253	+	0.621703i	$0.786441\pi$
$410$	0	0
$411$	12383.2	1.48618
$412$	0	0
$413$	5370.08i	0.639817i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1394.02i	0.163706i
$418$	0	0
$419$	14023.2	1.63503	0.817516	−	0.575906i	$-0.195351\pi$
$419$	14023.2	1.63503	0.817516	+	0.575906i	$0.195351\pi$
$420$	0	0
$421$	−4825.76	−0.558653	−0.279326	−	0.960196i	$-0.590111\pi$
$421$	−4825.76	−0.558653	−0.279326	+	0.960196i	$0.590111\pi$
$422$	0	0
$423$	− 10630.8i	− 1.22195i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 16206.7i	− 1.83676i
$428$	0	0
$429$	−24690.2	−2.77868
$430$	0	0
$431$	−3901.30	−0.436007	−0.218003	−	0.975948i	$-0.569954\pi$
$431$	−3901.30	−0.436007	−0.218003	+	0.975948i	$0.569954\pi$
$432$	0	0
$433$	− 1027.04i	− 0.113987i	−0.998375	−	0.0569933i	$-0.981849\pi$
$433$	− 1027.04i	− 0.113987i	0.998375	−	0.0569933i	$-0.0181514\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	396.057i	0.0433547i
$438$	0	0
$439$	−255.290	−0.0277547	−0.0138774	−	0.999904i	$-0.504417\pi$
$439$	−255.290	−0.0277547	−0.0138774	+	0.999904i	$0.504417\pi$
$440$	0	0
$441$	4020.90	0.434176
$442$	0	0
$443$	6551.19i	0.702611i	0.936261	+	0.351305i	$0.114262\pi$
$443$	6551.19i	0.702611i	−0.936261	+	0.351305i	$0.885738\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 1906.56i	− 0.201739i
$448$	0	0
$449$	2364.38	0.248512	0.124256	−	0.992250i	$-0.460346\pi$
$449$	2364.38	0.248512	0.124256	+	0.992250i	$0.460346\pi$
$450$	0	0
$451$	8961.00	0.935603
$452$	0	0
$453$	311.918i	0.0323514i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4440.07i	0.454481i	0.973839	+	0.227241i	$0.0729704\pi$
$457$	4440.07i	0.454481i	−0.973839	+	0.227241i	$0.927030\pi$
$458$	0	0
$459$	−25909.9	−2.63479
$460$	0	0
$461$	−891.518	−0.0900697	−0.0450349	−	0.998985i	$-0.514340\pi$
$461$	−891.518	−0.0900697	−0.0450349	+	0.998985i	$0.514340\pi$
$462$	0	0
$463$	− 3294.85i	− 0.330723i	−0.986233	−	0.165361i	$-0.947121\pi$
$463$	− 3294.85i	− 0.330723i	0.986233	−	0.165361i	$-0.0528790\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1488.22i	0.147466i	0.997278	+	0.0737329i	$0.0234912\pi$
$467$	1488.22i	0.147466i	−0.997278	+	0.0737329i	$0.976509\pi$
$468$	0	0
$469$	−2905.11	−0.286025
$470$	0	0
$471$	−3111.01	−0.304348
$472$	0	0
$473$	413.657i	0.0402114i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	14382.8i	1.38059i
$478$	0	0
$479$	−17534.7	−1.67261	−0.836304	−	0.548266i	$-0.815288\pi$
$479$	−17534.7	−1.67261	−0.836304	+	0.548266i	$0.815288\pi$
$480$	0	0
$481$	−4501.84	−0.426749
$482$	0	0
$483$	− 1114.59i	− 0.105001i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	10113.8i	0.941067i	0.882382	+	0.470533i	$0.155938\pi$
$487$	10113.8i	0.941067i	−0.882382	+	0.470533i	$0.844062\pi$
$488$	0	0
$489$	18056.4	1.66981
$490$	0	0
$491$	17855.8	1.64119	0.820594	−	0.571512i	$-0.193643\pi$
$491$	17855.8	1.64119	0.820594	+	0.571512i	$0.193643\pi$
$492$	0	0
$493$	− 25933.3i	− 2.36912i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 21396.2i	− 1.93109i
$498$	0	0
$499$	−2985.84	−0.267865	−0.133932	−	0.990990i	$-0.542760\pi$
$499$	−2985.84	−0.267865	−0.133932	+	0.990990i	$0.542760\pi$
$500$	0	0
$501$	1909.89	0.170315
$502$	0	0
$503$	− 2159.75i	− 0.191449i	−0.995408	−	0.0957243i	$-0.969483\pi$
$503$	− 2159.75i	− 0.191449i	0.995408	−	0.0957243i	$-0.0305167\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1376.65i	0.120590i
$508$	0	0
$509$	−16880.6	−1.46998	−0.734989	−	0.678079i	$-0.762813\pi$
$509$	−16880.6	−1.46998	−0.734989	+	0.678079i	$0.762813\pi$
$510$	0	0
$511$	−21681.1	−1.87694
$512$	0	0
$513$	− 14528.8i	− 1.25041i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 12505.1i	− 1.06378i
$518$	0	0
$519$	30478.9	2.57779
$520$	0	0
$521$	3915.76	0.329276	0.164638	−	0.986354i	$-0.447354\pi$
$521$	3915.76	0.329276	0.164638	+	0.986354i	$0.447354\pi$
$522$	0	0
$523$	− 14478.8i	− 1.21054i	−0.796019	−	0.605271i	$-0.793065\pi$
$523$	− 14478.8i	− 1.21054i	0.796019	−	0.605271i	$-0.206935\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6773.43i	0.559877i
$528$	0	0
$529$	12129.7	0.996930
$530$	0	0
$531$	−13675.3	−1.11762
$532$	0	0
$533$	6596.16i	0.536044i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	18065.8i	1.45177i
$538$	0	0
$539$	4729.83	0.377975
$540$	0	0
$541$	−11827.6	−0.939940	−0.469970	−	0.882682i	$-0.655735\pi$
$541$	−11827.6	−0.939940	−0.469970	+	0.882682i	$0.655735\pi$
$542$	0	0
$543$	1816.63i	0.143571i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	6446.22i	0.503877i	0.967743	+	0.251938i	$0.0810680\pi$
$547$	6446.22i	0.503877i	−0.967743	+	0.251938i	$0.918932\pi$
$548$	0	0
$549$	41271.6	3.20843
$550$	0	0
$551$	14541.9	1.12433
$552$	0	0
$553$	− 16941.6i	− 1.30277i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 14419.5i	− 1.09690i	−0.836183	−	0.548451i	$-0.815218\pi$
$557$	− 14419.5i	− 1.09690i	0.836183	−	0.548451i	$-0.184782\pi$
$558$	0	0
$559$	−304.492	−0.0230387
$560$	0	0
$561$	−63143.9	−4.75211
$562$	0	0
$563$	817.038i	0.0611617i	0.999532	+	0.0305808i	$0.00973570\pi$
$563$	817.038i	0.0611617i	−0.999532	+	0.0305808i	$0.990264\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	12006.1i	0.889256i
$568$	0	0
$569$	16795.5	1.23744	0.618720	−	0.785612i	$-0.287652\pi$
$569$	16795.5	1.23744	0.618720	+	0.785612i	$0.287652\pi$
$570$	0	0
$571$	−15342.6	−1.12447	−0.562233	−	0.826979i	$-0.690057\pi$
$571$	−15342.6	−1.12447	−0.562233	+	0.826979i	$0.690057\pi$
$572$	0	0
$573$	− 5493.81i	− 0.400536i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1287.65i	0.0929039i	0.998921	+	0.0464519i	$0.0147914\pi$
$577$	1287.65i	0.0929039i	−0.998921	+	0.0464519i	$0.985209\pi$
$578$	0	0
$579$	27331.7	1.96177
$580$	0	0
$581$	21032.9	1.50188
$582$	0	0
$583$	16918.6i	1.20188i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2074.66i	0.145878i	0.997336	+	0.0729391i	$0.0232379\pi$
$587$	2074.66i	0.145878i	−0.997336	+	0.0729391i	$0.976762\pi$
$588$	0	0
$589$	−3798.16	−0.265705
$590$	0	0
$591$	7226.84	0.502999
$592$	0	0
$593$	− 16263.0i	− 1.12621i	−0.826386	−	0.563104i	$-0.809607\pi$
$593$	− 16263.0i	− 1.12621i	0.826386	−	0.563104i	$-0.190393\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9485.73i	0.650293i
$598$	0	0
$599$	−22423.7	−1.52956	−0.764780	−	0.644292i	$-0.777152\pi$
$599$	−22423.7	−1.52956	−0.764780	+	0.644292i	$0.777152\pi$
$600$	0	0
$601$	−2915.58	−0.197885	−0.0989424	−	0.995093i	$-0.531546\pi$
$601$	−2915.58	−0.197885	−0.0989424	+	0.995093i	$0.531546\pi$
$602$	0	0
$603$	− 7398.09i	− 0.499624i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 11897.3i	− 0.795546i	−0.917484	−	0.397773i	$-0.869783\pi$
$607$	− 11897.3i	− 0.795546i	0.917484	−	0.397773i	$-0.130217\pi$
$608$	0	0
$609$	−40923.9	−2.72302
$610$	0	0
$611$	9204.98	0.609482
$612$	0	0
$613$	22791.7i	1.50171i	0.660466	+	0.750856i	$0.270359\pi$
$613$	22791.7i	1.50171i	−0.660466	+	0.750856i	$0.729641\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11894.0i	0.776071i	0.921644	+	0.388036i	$0.126846\pi$
$617$	11894.0i	0.776071i	−0.921644	+	0.388036i	$0.873154\pi$
$618$	0	0
$619$	15567.8	1.01086	0.505431	−	0.862867i	$-0.331333\pi$
$619$	15567.8	1.01086	0.505431	+	0.862867i	$0.331333\pi$
$620$	0	0
$621$	1370.02	0.0885301
$622$	0	0
$623$	3173.61i	0.204090i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 35407.5i	− 2.25524i
$628$	0	0
$629$	−11513.2	−0.729828
$630$	0	0
$631$	−23681.8	−1.49407	−0.747034	−	0.664786i	$-0.768523\pi$
$631$	−23681.8	−1.49407	−0.747034	+	0.664786i	$0.768523\pi$
$632$	0	0
$633$	− 10876.7i	− 0.682953i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3481.62i	0.216557i
$638$	0	0
$639$	54487.1	3.37320
$640$	0	0
$641$	21351.6	1.31566	0.657831	−	0.753166i	$-0.271474\pi$
$641$	21351.6	1.31566	0.657831	+	0.753166i	$0.271474\pi$
$642$	0	0
$643$	− 7196.04i	− 0.441344i	−0.975348	−	0.220672i	$-0.929175\pi$
$643$	− 7196.04i	− 0.441344i	0.975348	−	0.220672i	$-0.0708250\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 20897.9i	− 1.26983i	−0.772582	−	0.634915i	$-0.781035\pi$
$647$	− 20897.9i	− 1.26983i	0.772582	−	0.634915i	$-0.218965\pi$
$648$	0	0
$649$	−16086.4	−0.972956
$650$	0	0
$651$	10688.8	0.643513
$652$	0	0
$653$	− 8655.43i	− 0.518703i	−0.965783	−	0.259352i	$-0.916491\pi$
$653$	− 8655.43i	− 0.518703i	0.965783	−	0.259352i	$-0.0835088\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 55212.6i	− 3.27861i
$658$	0	0
$659$	13004.3	0.768705	0.384352	−	0.923186i	$-0.374425\pi$
$659$	13004.3	0.768705	0.384352	+	0.923186i	$0.374425\pi$
$660$	0	0
$661$	−19135.2	−1.12598	−0.562991	−	0.826463i	$-0.690349\pi$
$661$	−19135.2	−1.12598	−0.562991	+	0.826463i	$0.690349\pi$
$662$	0	0
$663$	− 46480.0i	− 2.72267i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1371.26i	0.0796033i
$668$	0	0
$669$	3687.51	0.213105
$670$	0	0
$671$	48548.3	2.79312
$672$	0	0
$673$	− 24101.3i	− 1.38044i	−0.723599	−	0.690221i	$-0.757513\pi$
$673$	− 24101.3i	− 1.38044i	0.723599	−	0.690221i	$-0.242487\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	17892.9i	1.01577i	0.861424	+	0.507887i	$0.169573\pi$
$677$	17892.9i	1.01577i	−0.861424	+	0.507887i	$0.830427\pi$
$678$	0	0
$679$	−8485.72	−0.479606
$680$	0	0
$681$	20413.2	1.14866
$682$	0	0
$683$	− 27545.4i	− 1.54319i	−0.636117	−	0.771593i	$-0.719460\pi$
$683$	− 27545.4i	− 1.54319i	0.636117	−	0.771593i	$-0.280540\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 28607.7i	− 1.58872i
$688$	0	0
$689$	−12453.8	−0.688608
$690$	0	0
$691$	−11360.1	−0.625408	−0.312704	−	0.949851i	$-0.601235\pi$
$691$	−11360.1	−0.625408	−0.312704	+	0.949851i	$0.601235\pi$
$692$	0	0
$693$	65671.0i	3.59976i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	16869.3i	0.916744i
$698$	0	0
$699$	32023.8	1.73284
$700$	0	0
$701$	−148.008	−0.00797459	−0.00398729	−	0.999992i	$-0.501269\pi$
$701$	−148.008	−0.00797459	−0.00398729	+	0.999992i	$0.501269\pi$
$702$	0	0
$703$	− 6455.95i	− 0.346360i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	36419.2i	1.93732i
$708$	0	0
$709$	−2401.61	−0.127213	−0.0636067	−	0.997975i	$-0.520260\pi$
$709$	−2401.61	−0.127213	−0.0636067	+	0.997975i	$0.520260\pi$
$710$	0	0
$711$	43143.0	2.27566
$712$	0	0
$713$	− 358.155i	− 0.0188121i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 21914.5i	− 1.14144i
$718$	0	0
$719$	13003.4	0.674472	0.337236	−	0.941420i	$-0.390508\pi$
$719$	13003.4	0.674472	0.337236	+	0.941420i	$0.390508\pi$
$720$	0	0
$721$	15739.9	0.813016
$722$	0	0
$723$	36959.9i	1.90118i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	36506.5i	1.86238i	0.364533	+	0.931190i	$0.381229\pi$
$727$	36506.5i	1.86238i	−0.364533	+	0.931190i	$0.618771\pi$
$728$	0	0
$729$	23290.3	1.18327
$730$	0	0
$731$	−778.720	−0.0394008
$732$	0	0
$733$	18661.5i	0.940350i	0.882573	+	0.470175i	$0.155809\pi$
$733$	18661.5i	0.940350i	−0.882573	+	0.470175i	$0.844191\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 8702.46i	− 0.434951i
$738$	0	0
$739$	−27675.0	−1.37759	−0.688797	−	0.724954i	$-0.741861\pi$
$739$	−27675.0	−1.37759	−0.688797	+	0.724954i	$0.741861\pi$
$740$	0	0
$741$	26063.3	1.29212
$742$	0	0
$743$	17622.0i	0.870103i	0.900406	+	0.435052i	$0.143270\pi$
$743$	17622.0i	0.870103i	−0.900406	+	0.435052i	$0.856730\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	53562.0i	2.62347i
$748$	0	0
$749$	33796.1	1.64871
$750$	0	0
$751$	−34581.0	−1.68027	−0.840133	−	0.542381i	$-0.817523\pi$
$751$	−34581.0	−1.68027	−0.840133	+	0.542381i	$0.817523\pi$
$752$	0	0
$753$	− 11882.4i	− 0.575058i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	33994.4i	1.63216i	0.577936	+	0.816082i	$0.303858\pi$
$757$	33994.4i	1.63216i	−0.577936	+	0.816082i	$0.696142\pi$
$758$	0	0
$759$	3338.82	0.159673
$760$	0	0
$761$	−34917.8	−1.66330	−0.831649	−	0.555302i	$-0.812603\pi$
$761$	−34917.8	−1.66330	−0.831649	+	0.555302i	$0.812603\pi$
$762$	0	0
$763$	− 18122.0i	− 0.859844i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 11841.2i	− 0.557445i
$768$	0	0
$769$	20714.9	0.971388	0.485694	−	0.874129i	$-0.338567\pi$
$769$	20714.9	0.971388	0.485694	+	0.874129i	$0.338567\pi$
$770$	0	0
$771$	4719.66	0.220459
$772$	0	0
$773$	12461.4i	0.579826i	0.957053	+	0.289913i	$0.0936264\pi$
$773$	12461.4i	0.579826i	−0.957053	+	0.289913i	$0.906374\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	18168.4i	0.838851i
$778$	0	0
$779$	−9459.35	−0.435066
$780$	0	0
$781$	64093.8	2.93657
$782$	0	0
$783$	− 50302.7i	− 2.29588i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	32821.0i	1.48658i	0.668967	+	0.743292i	$0.266737\pi$
$787$	32821.0i	1.48658i	−0.668967	+	0.743292i	$0.733263\pi$
$788$	0	0
$789$	−19882.4	−0.897126
$790$	0	0
$791$	−23340.6	−1.04917
$792$	0	0
$793$	35736.2i	1.60029i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7777.06i	0.345643i	0.984953	+	0.172822i	$0.0552884\pi$
$797$	7777.06i	0.345643i	−0.984953	+	0.172822i	$0.944712\pi$
$798$	0	0
$799$	23541.2	1.04234
$800$	0	0
$801$	−8081.85	−0.356502
$802$	0	0
$803$	− 64947.2i	− 2.85422i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 45350.0i	− 1.97819i
$808$	0	0
$809$	−20407.4	−0.886880	−0.443440	−	0.896304i	$-0.646242\pi$
$809$	−20407.4	−0.886880	−0.443440	+	0.896304i	$0.646242\pi$
$810$	0	0
$811$	−25876.6	−1.12041	−0.560205	−	0.828354i	$-0.689277\pi$
$811$	−25876.6	−1.12041	−0.560205	+	0.828354i	$0.689277\pi$
$812$	0	0
$813$	12043.8i	0.519550i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 436.662i	− 0.0186987i
$818$	0	0
$819$	−48340.2	−2.06245
$820$	0	0
$821$	1100.58	0.0467850	0.0233925	−	0.999726i	$-0.492553\pi$
$821$	1100.58	0.0467850	0.0233925	+	0.999726i	$0.492553\pi$
$822$	0	0
$823$	36245.7i	1.53517i	0.640948	+	0.767585i	$0.278542\pi$
$823$	36245.7i	1.53517i	−0.640948	+	0.767585i	$0.721458\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1996.72i	0.0839572i	0.999119	+	0.0419786i	$0.0133661\pi$
$827$	1996.72i	0.0839572i	−0.999119	+	0.0419786i	$0.986634\pi$
$828$	0	0
$829$	−43393.8	−1.81801	−0.909004	−	0.416788i	$-0.863156\pi$
$829$	−43393.8	−1.81801	−0.909004	+	0.416788i	$0.863156\pi$
$830$	0	0
$831$	79019.1	3.29861
$832$	0	0
$833$	8904.03i	0.370356i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	13138.4i	0.542569i
$838$	0	0
$839$	3424.53	0.140915	0.0704576	−	0.997515i	$-0.477554\pi$
$839$	3424.53	0.140915	0.0704576	+	0.997515i	$0.477554\pi$
$840$	0	0
$841$	25959.0	1.06437
$842$	0	0
$843$	− 10194.2i	− 0.416498i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	49970.8i	2.02718i
$848$	0	0
$849$	3429.66	0.138640
$850$	0	0
$851$	608.778	0.0245225
$852$	0	0
$853$	5173.63i	0.207669i	0.994595	+	0.103835i	$0.0331113\pi$
$853$	5173.63i	0.207669i	−0.994595	+	0.103835i	$0.966889\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	28323.8i	1.12896i	0.825445	+	0.564482i	$0.190924\pi$
$857$	28323.8i	1.12896i	−0.825445	+	0.564482i	$0.809076\pi$
$858$	0	0
$859$	42970.7	1.70680	0.853401	−	0.521255i	$-0.174536\pi$
$859$	42970.7	1.70680	0.853401	+	0.521255i	$0.174536\pi$
$860$	0	0
$861$	26620.6	1.05369
$862$	0	0
$863$	− 12664.1i	− 0.499527i	−0.968307	−	0.249764i	$-0.919647\pi$
$863$	− 12664.1i	− 0.499527i	0.968307	−	0.249764i	$-0.0803529\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 75149.2i	− 2.94371i
$868$	0	0
$869$	50749.7	1.98109
$870$	0	0
$871$	6405.85	0.249201
$872$	0	0
$873$	− 21609.6i	− 0.837769i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 7403.14i	− 0.285047i	−0.989791	−	0.142524i	$-0.954478\pi$
$877$	− 7403.14i	− 0.285047i	0.989791	−	0.142524i	$-0.0455217\pi$
$878$	0	0
$879$	33925.6	1.30180
$880$	0	0
$881$	−5691.74	−0.217661	−0.108831	−	0.994060i	$-0.534711\pi$
$881$	−5691.74	−0.217661	−0.108831	+	0.994060i	$0.534711\pi$
$882$	0	0
$883$	13915.1i	0.530329i	0.964203	+	0.265164i	$0.0854263\pi$
$883$	13915.1i	0.530329i	−0.964203	+	0.265164i	$0.914574\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 35631.2i	− 1.34879i	−0.738370	−	0.674396i	$-0.764404\pi$
$887$	− 35631.2i	− 1.34879i	0.738370	−	0.674396i	$-0.235596\pi$
$888$	0	0
$889$	34874.2	1.31568
$890$	0	0
$891$	−35965.0	−1.35227
$892$	0	0
$893$	13200.6i	0.494670i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2457.70i	0.0914829i
$898$	0	0
$899$	−13150.3	−0.487859
$900$	0	0
$901$	−31849.8	−1.17766
$902$	0	0
$903$	1228.86i	0.0452866i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	22734.4i	0.832285i	0.909299	+	0.416143i	$0.136618\pi$
$907$	22734.4i	0.832285i	−0.909299	+	0.416143i	$0.863382\pi$
$908$	0	0
$909$	−92744.3	−3.38408
$910$	0	0
$911$	32714.8	1.18978	0.594890	−	0.803807i	$-0.297195\pi$
$911$	32714.8	1.18978	0.594890	+	0.803807i	$0.297195\pi$
$912$	0	0
$913$	63005.6i	2.28388i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	31582.3i	1.13734i
$918$	0	0
$919$	−41077.0	−1.47443	−0.737217	−	0.675656i	$-0.763861\pi$
$919$	−41077.0	−1.47443	−0.737217	+	0.675656i	$0.763861\pi$
$920$	0	0
$921$	−61380.3	−2.19604
$922$	0	0
$923$	47179.3i	1.68247i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	40082.9i	1.42017i
$928$	0	0
$929$	−27678.8	−0.977515	−0.488758	−	0.872420i	$-0.662550\pi$
$929$	−27678.8	−0.977515	−0.488758	+	0.872420i	$0.662550\pi$
$930$	0	0
$931$	−4992.87	−0.175762
$932$	0	0
$933$	− 72307.6i	− 2.53724i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 17120.7i	− 0.596915i	−0.954423	−	0.298457i	$-0.903528\pi$
$937$	− 17120.7i	− 0.596915i	0.954423	−	0.298457i	$-0.0964720\pi$
$938$	0	0
$939$	18266.6	0.634832
$940$	0	0
$941$	−30687.2	−1.06310	−0.531549	−	0.847028i	$-0.678390\pi$
$941$	−30687.2	−1.06310	−0.531549	+	0.847028i	$0.678390\pi$
$942$	0	0
$943$	− 891.989i	− 0.0308029i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 36649.9i	− 1.25761i	−0.777561	−	0.628807i	$-0.783544\pi$
$947$	− 36649.9i	− 1.25761i	0.777561	−	0.628807i	$-0.216456\pi$
$948$	0	0
$949$	47807.4	1.63529
$950$	0	0
$951$	−37154.3	−1.26689
$952$	0	0
$953$	− 23768.8i	− 0.807919i	−0.914777	−	0.403959i	$-0.867634\pi$
$953$	− 23768.8i	− 0.807919i	0.914777	−	0.403959i	$-0.132366\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 122590.i	− 4.14084i
$958$	0	0
$959$	28519.4	0.960311
$960$	0	0
$961$	−26356.3	−0.884708
$962$	0	0
$963$	86064.3i	2.87994i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 36241.8i	− 1.20523i	−0.798032	−	0.602615i	$-0.794125\pi$
$967$	− 36241.8i	− 1.20523i	0.798032	−	0.602615i	$-0.205875\pi$
$968$	0	0
$969$	66655.5	2.20979
$970$	0	0
$971$	16837.4	0.556476	0.278238	−	0.960512i	$-0.410250\pi$
$971$	16837.4	0.556476	0.278238	+	0.960512i	$0.410250\pi$
$972$	0	0
$973$	3210.51i	0.105780i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 26322.4i	− 0.861954i	−0.902363	−	0.430977i	$-0.858169\pi$
$977$	− 26322.4i	− 0.861954i	0.902363	−	0.430977i	$-0.141831\pi$
$978$	0	0
$979$	−9506.78	−0.310355
$980$	0	0
$981$	46149.1	1.50196
$982$	0	0
$983$	42438.7i	1.37699i	0.725240	+	0.688497i	$0.241729\pi$
$983$	42438.7i	1.37699i	−0.725240	+	0.688497i	$0.758271\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 37149.1i	− 1.19804i
$988$	0	0
$989$	41.1760	0.00132388
$990$	0	0
$991$	−17890.3	−0.573464	−0.286732	−	0.958011i	$-0.592569\pi$
$991$	−17890.3	−0.573464	−0.286732	+	0.958011i	$0.592569\pi$
$992$	0	0
$993$	− 104458.i	− 3.33824i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	16189.9i	0.514283i	0.966374	+	0.257142i	$0.0827807\pi$
$997$	16189.9i	0.514283i	−0.966374	+	0.257142i	$0.917219\pi$
$998$	0	0
$999$	−22332.2	−0.707265

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.4.c.i.449.4		4
4.3	odd	2		800.4.c.k.449.1		4
5.2	odd	4		800.4.a.s.1.2		2
5.3	odd	4		160.4.a.c.1.1	✓	2
5.4	even	2	inner	800.4.c.i.449.1		4
15.8	even	4		1440.4.a.t.1.2		2
20.3	even	4		160.4.a.g.1.2	yes	2
20.7	even	4		800.4.a.m.1.1		2
20.19	odd	2		800.4.c.k.449.4		4
40.3	even	4		320.4.a.o.1.1		2
40.13	odd	4		320.4.a.s.1.2		2
40.27	even	4		1600.4.a.cn.1.2		2
40.37	odd	4		1600.4.a.cd.1.1		2
60.23	odd	4		1440.4.a.x.1.1		2
80.3	even	4		1280.4.d.q.641.4		4
80.13	odd	4		1280.4.d.x.641.1		4
80.43	even	4		1280.4.d.q.641.1		4
80.53	odd	4		1280.4.d.x.641.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.4.a.c.1.1	✓	2	5.3	odd	4
160.4.a.g.1.2	yes	2	20.3	even	4
320.4.a.o.1.1		2	40.3	even	4
320.4.a.s.1.2		2	40.13	odd	4
800.4.a.m.1.1		2	20.7	even	4
800.4.a.s.1.2		2	5.2	odd	4
800.4.c.i.449.1		4	5.4	even	2	inner
800.4.c.i.449.4		4	1.1	even	1	trivial
800.4.c.k.449.1		4	4.3	odd	2
800.4.c.k.449.4		4	20.19	odd	2
1280.4.d.q.641.1		4	80.43	even	4
1280.4.d.q.641.4		4	80.3	even	4
1280.4.d.x.641.1		4	80.13	odd	4
1280.4.d.x.641.4		4	80.53	odd	4
1440.4.a.t.1.2		2	15.8	even	4
1440.4.a.x.1.1		2	60.23	odd	4
1600.4.a.cd.1.1		2	40.37	odd	4
1600.4.a.cn.1.2		2	40.27	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	800.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+8.89898i q^{3} +20.4949i q^{7} -52.1918 q^{9} +O(q^{10})\)	\(q+8.89898i q^{3} +20.4949i q^{7} -52.1918 q^{9} -61.3939 q^{11} -45.1918i q^{13} -115.576i q^{17} +64.8082 q^{19} -182.384 q^{21} +6.11123i q^{23} -224.182i q^{27} +224.384 q^{29} -58.6061 q^{31} -546.343i q^{33} -99.6163i q^{37} +402.161 q^{39} -145.959 q^{41} -6.73776i q^{43} +203.687i q^{47} -77.0408 q^{49} +1028.50 q^{51} -275.576i q^{53} +576.727i q^{57} +262.020 q^{59} -790.767 q^{61} -1069.67i q^{63} +141.748i q^{67} -54.3837 q^{69} -1043.98 q^{71} +1057.88i q^{73} -1258.26i q^{77} -826.624 q^{79} +585.808 q^{81} -1026.25i q^{83} +1996.79i q^{87} +154.849 q^{89} +926.202 q^{91} -521.535i q^{93} +414.041i q^{97} +3204.26 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 52 q^{9}+O(q^{10}) \) 4 * q - 52 * q^9	\( 4 q - 52 q^{9} - 128 q^{11} + 416 q^{19} - 416 q^{21} + 584 q^{29} - 352 q^{31} + 864 q^{39} + 200 q^{41} - 1092 q^{49} + 2272 q^{51} + 1440 q^{59} - 2536 q^{61} + 96 q^{69} + 96 q^{71} + 64 q^{79} + 2500 q^{81} + 1560 q^{89} + 3744 q^{91} + 6272 q^{99}+O(q^{100}) \) 4 * q - 52 * q^9 - 128 * q^11 + 416 * q^19 - 416 * q^21 + 584 * q^29 - 352 * q^31 + 864 * q^39 + 200 * q^41 - 1092 * q^49 + 2272 * q^51 + 1440 * q^59 - 2536 * q^61 + 96 * q^69 + 96 * q^71 + 64 * q^79 + 2500 * q^81 + 1560 * q^89 + 3744 * q^91 + 6272 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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