LMFDB - Embedded newform 840.4.a.i.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [840,4,Mod(1,840)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("840.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	840.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-6,0,10,0,-14,0,18,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$49.5616044048$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	840.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +5.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} +47.1918 q^{11} -54.9898 q^{13} -15.0000 q^{15} -13.0102 q^{17} -121.171 q^{19} +21.0000 q^{21} +145.798 q^{23} +25.0000 q^{25} -27.0000 q^{27} +159.353 q^{29} -286.969 q^{31} -141.576 q^{33} -35.0000 q^{35} +59.7571 q^{37} +164.969 q^{39} -293.192 q^{41} +355.373 q^{43} +45.0000 q^{45} +498.606 q^{47} +49.0000 q^{49} +39.0306 q^{51} -424.706 q^{53} +235.959 q^{55} +363.514 q^{57} -161.212 q^{59} -720.929 q^{61} -63.0000 q^{63} -274.949 q^{65} +316.908 q^{67} -437.394 q^{69} -390.969 q^{71} -1045.70 q^{73} -75.0000 q^{75} -330.343 q^{77} -968.727 q^{79} +81.0000 q^{81} +454.624 q^{83} -65.0510 q^{85} -478.059 q^{87} -352.343 q^{89} +384.929 q^{91} +860.908 q^{93} -605.857 q^{95} +969.614 q^{97} +424.727 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 10 q^{5} - 14 q^{7} + 18 q^{9} + 16 q^{11} - 12 q^{13} - 30 q^{15} - 124 q^{17} + 32 q^{19} + 42 q^{21} + 272 q^{23} + 50 q^{25} - 54 q^{27} - 132 q^{29} - 280 q^{31} - 48 q^{33} - 70 q^{35}+ \cdots + 144 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 10 * q^5 - 14 * q^7 + 18 * q^9 + 16 * q^11 - 12 * q^13 - 30 * q^15 - 124 * q^17 + 32 * q^19 + 42 * q^21 + 272 * q^23 + 50 * q^25 - 54 * q^27 - 132 * q^29 - 280 * q^31 - 48 * q^33 - 70 * q^35 - 292 * q^37 + 36 * q^39 - 508 * q^41 + 456 * q^43 + 90 * q^45 + 1056 * q^47 + 98 * q^49 + 372 * q^51 + 52 * q^53 + 80 * q^55 - 96 * q^57 - 440 * q^59 - 756 * q^61 - 126 * q^63 - 60 * q^65 - 248 * q^67 - 816 * q^69 - 488 * q^71 - 1092 * q^73 - 150 * q^75 - 112 * q^77 - 1232 * q^79 + 162 * q^81 - 776 * q^83 - 620 * q^85 + 396 * q^87 - 156 * q^89 + 84 * q^91 + 840 * q^93 + 160 * q^95 + 156 * q^97 + 144 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	47.1918	1.29353	0.646767	−	0.762688i	$-0.276121\pi$
$11$	47.1918	1.29353	0.646767	+	0.762688i	$0.276121\pi$
$12$	0	0
$13$	−54.9898	−1.17319	−0.586593	−	0.809882i	$-0.699531\pi$
$13$	−54.9898	−1.17319	−0.586593	+	0.809882i	$0.699531\pi$
$14$	0	0
$15$	−15.0000	−0.258199
$16$	0	0
$17$	−13.0102	−0.185614	−0.0928070	−	0.995684i	$-0.529584\pi$
$17$	−13.0102	−0.185614	−0.0928070	+	0.995684i	$0.529584\pi$
$18$	0	0
$19$	−121.171	−1.46309	−0.731543	−	0.681795i	$-0.761199\pi$
$19$	−121.171	−1.46309	−0.731543	+	0.681795i	$0.761199\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	0	0
$23$	145.798	1.32178	0.660891	−	0.750482i	$-0.270179\pi$
$23$	145.798	1.32178	0.660891	+	0.750482i	$0.270179\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	159.353	1.02038	0.510192	−	0.860061i	$-0.329574\pi$
$29$	159.353	1.02038	0.510192	+	0.860061i	$0.329574\pi$
$30$	0	0
$31$	−286.969	−1.66262	−0.831310	−	0.555809i	$-0.812409\pi$
$31$	−286.969	−1.66262	−0.831310	+	0.555809i	$0.812409\pi$
$32$	0	0
$33$	−141.576	−0.746822
$34$	0	0
$35$	−35.0000	−0.169031
$36$	0	0
$37$	59.7571	0.265514	0.132757	−	0.991149i	$-0.457617\pi$
$37$	59.7571	0.265514	0.132757	+	0.991149i	$0.457617\pi$
$38$	0	0
$39$	164.969	0.677340
$40$	0	0
$41$	−293.192	−1.11680	−0.558401	−	0.829571i	$-0.688585\pi$
$41$	−293.192	−1.11680	−0.558401	+	0.829571i	$0.688585\pi$
$42$	0	0
$43$	355.373	1.26032	0.630162	−	0.776464i	$-0.282988\pi$
$43$	355.373	1.26032	0.630162	+	0.776464i	$0.282988\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	498.606	1.54743	0.773715	−	0.633534i	$-0.218396\pi$
$47$	498.606	1.54743	0.773715	+	0.633534i	$0.218396\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	39.0306	0.107164
$52$	0	0
$53$	−424.706	−1.10071	−0.550357	−	0.834929i	$-0.685508\pi$
$53$	−424.706	−1.10071	−0.550357	+	0.834929i	$0.685508\pi$
$54$	0	0
$55$	235.959	0.578486
$56$	0	0
$57$	363.514	0.844713
$58$	0	0
$59$	−161.212	−0.355730	−0.177865	−	0.984055i	$-0.556919\pi$
$59$	−161.212	−0.355730	−0.177865	+	0.984055i	$0.556919\pi$
$60$	0	0
$61$	−720.929	−1.51320	−0.756602	−	0.653876i	$-0.773142\pi$
$61$	−720.929	−1.51320	−0.756602	+	0.653876i	$0.773142\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	−274.949	−0.524665
$66$	0	0
$67$	316.908	0.577858	0.288929	−	0.957351i	$-0.406701\pi$
$67$	316.908	0.577858	0.288929	+	0.957351i	$0.406701\pi$
$68$	0	0
$69$	−437.394	−0.763131
$70$	0	0
$71$	−390.969	−0.653514	−0.326757	−	0.945108i	$-0.605956\pi$
$71$	−390.969	−0.653514	−0.326757	+	0.945108i	$0.605956\pi$
$72$	0	0
$73$	−1045.70	−1.67657	−0.838284	−	0.545234i	$-0.816441\pi$
$73$	−1045.70	−1.67657	−0.838284	+	0.545234i	$0.816441\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	−330.343	−0.488910
$78$	0	0
$79$	−968.727	−1.37962	−0.689812	−	0.723989i	$-0.742307\pi$
$79$	−968.727	−1.37962	−0.689812	+	0.723989i	$0.742307\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	454.624	0.601223	0.300612	−	0.953747i	$-0.402809\pi$
$83$	454.624	0.601223	0.300612	+	0.953747i	$0.402809\pi$
$84$	0	0
$85$	−65.0510	−0.0830091
$86$	0	0
$87$	−478.059	−0.589119
$88$	0	0
$89$	−352.343	−0.419643	−0.209822	−	0.977740i	$-0.567288\pi$
$89$	−352.343	−0.419643	−0.209822	+	0.977740i	$0.567288\pi$
$90$	0	0
$91$	384.929	0.443423
$92$	0	0
$93$	860.908	0.959914
$94$	0	0
$95$	−605.857	−0.654312
$96$	0	0
$97$	969.614	1.01494	0.507471	−	0.861669i	$-0.330580\pi$
$97$	969.614	1.01494	0.507471	+	0.861669i	$0.330580\pi$
$98$	0	0
$99$	424.727	0.431178
$100$	0	0
$101$	−396.061	−0.390194	−0.195097	−	0.980784i	$-0.562502\pi$
$101$	−396.061	−0.390194	−0.195097	+	0.980784i	$0.562502\pi$
$102$	0	0
$103$	870.220	0.832479	0.416240	−	0.909255i	$-0.363348\pi$
$103$	870.220	0.832479	0.416240	+	0.909255i	$0.363348\pi$
$104$	0	0
$105$	105.000	0.0975900
$106$	0	0
$107$	−179.010	−0.161734	−0.0808671	−	0.996725i	$-0.525769\pi$
$107$	−179.010	−0.161734	−0.0808671	+	0.996725i	$0.525769\pi$
$108$	0	0
$109$	−1972.26	−1.73310	−0.866552	−	0.499087i	$-0.833669\pi$
$109$	−1972.26	−1.73310	−0.866552	+	0.499087i	$0.833669\pi$
$110$	0	0
$111$	−179.271	−0.153295
$112$	0	0
$113$	−1574.85	−1.31106	−0.655528	−	0.755171i	$-0.727554\pi$
$113$	−1574.85	−1.31106	−0.655528	+	0.755171i	$0.727554\pi$
$114$	0	0
$115$	728.990	0.591119
$116$	0	0
$117$	−494.908	−0.391062
$118$	0	0
$119$	91.0714	0.0701555
$120$	0	0
$121$	896.069	0.673230
$122$	0	0
$123$	879.576	0.644786
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−2684.36	−1.87558	−0.937789	−	0.347206i	$-0.887130\pi$
$127$	−2684.36	−1.87558	−0.937789	+	0.347206i	$0.887130\pi$
$128$	0	0
$129$	−1066.12	−0.727649
$130$	0	0
$131$	−1956.08	−1.30461	−0.652304	−	0.757957i	$-0.726197\pi$
$131$	−1956.08	−1.30461	−0.652304	+	0.757957i	$0.726197\pi$
$132$	0	0
$133$	848.200	0.552994
$134$	0	0
$135$	−135.000	−0.0860663
$136$	0	0
$137$	296.624	0.184981	0.0924903	−	0.995714i	$-0.470517\pi$
$137$	296.624	0.184981	0.0924903	+	0.995714i	$0.470517\pi$
$138$	0	0
$139$	−1347.35	−0.822164	−0.411082	−	0.911598i	$-0.634849\pi$
$139$	−1347.35	−0.822164	−0.411082	+	0.911598i	$0.634849\pi$
$140$	0	0
$141$	−1495.82	−0.893409
$142$	0	0
$143$	−2595.07	−1.51756
$144$	0	0
$145$	796.765	0.456329
$146$	0	0
$147$	−147.000	−0.0824786
$148$	0	0
$149$	−448.447	−0.246565	−0.123283	−	0.992372i	$-0.539342\pi$
$149$	−448.447	−0.246565	−0.123283	+	0.992372i	$0.539342\pi$
$150$	0	0
$151$	2846.54	1.53409	0.767046	−	0.641592i	$-0.221726\pi$
$151$	2846.54	1.53409	0.767046	+	0.641592i	$0.221726\pi$
$152$	0	0
$153$	−117.092	−0.0618713
$154$	0	0
$155$	−1434.85	−0.743546
$156$	0	0
$157$	−1691.07	−0.859630	−0.429815	−	0.902917i	$-0.641421\pi$
$157$	−1691.07	−0.859630	−0.429815	+	0.902917i	$0.641421\pi$
$158$	0	0
$159$	1274.12	0.635498
$160$	0	0
$161$	−1020.59	−0.499586
$162$	0	0
$163$	−2468.50	−1.18618	−0.593092	−	0.805135i	$-0.702093\pi$
$163$	−2468.50	−1.18618	−0.593092	+	0.805135i	$0.702093\pi$
$164$	0	0
$165$	−707.878	−0.333989
$166$	0	0
$167$	48.8265	0.0226246	0.0113123	−	0.999936i	$-0.496399\pi$
$167$	48.8265	0.0226246	0.0113123	+	0.999936i	$0.496399\pi$
$168$	0	0
$169$	826.878	0.376367
$170$	0	0
$171$	−1090.54	−0.487695
$172$	0	0
$173$	−1075.62	−0.472703	−0.236351	−	0.971668i	$-0.575952\pi$
$173$	−1075.62	−0.472703	−0.236351	+	0.971668i	$0.575952\pi$
$174$	0	0
$175$	−175.000	−0.0755929
$176$	0	0
$177$	483.637	0.205381
$178$	0	0
$179$	2098.34	0.876187	0.438094	−	0.898929i	$-0.355654\pi$
$179$	2098.34	0.876187	0.438094	+	0.898929i	$0.355654\pi$
$180$	0	0
$181$	−4233.81	−1.73866	−0.869328	−	0.494236i	$-0.835448\pi$
$181$	−4233.81	−1.73866	−0.869328	+	0.494236i	$0.835448\pi$
$182$	0	0
$183$	2162.79	0.873649
$184$	0	0
$185$	298.786	0.118741
$186$	0	0
$187$	−613.975	−0.240098
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	1199.53	0.454425	0.227212	−	0.973845i	$-0.427039\pi$
$191$	1199.53	0.454425	0.227212	+	0.973845i	$0.427039\pi$
$192$	0	0
$193$	−2403.61	−0.896455	−0.448227	−	0.893920i	$-0.647945\pi$
$193$	−2403.61	−0.896455	−0.448227	+	0.893920i	$0.647945\pi$
$194$	0	0
$195$	824.847	0.302915
$196$	0	0
$197$	−3547.57	−1.28301	−0.641507	−	0.767117i	$-0.721690\pi$
$197$	−3547.57	−1.28301	−0.641507	+	0.767117i	$0.721690\pi$
$198$	0	0
$199$	4727.32	1.68398	0.841988	−	0.539497i	$-0.181385\pi$
$199$	4727.32	1.68398	0.841988	+	0.539497i	$0.181385\pi$
$200$	0	0
$201$	−950.724	−0.333627
$202$	0	0
$203$	−1115.47	−0.385669
$204$	0	0
$205$	−1465.96	−0.499449
$206$	0	0
$207$	1312.18	0.440594
$208$	0	0
$209$	−5718.30	−1.89255
$210$	0	0
$211$	−1961.86	−0.640094	−0.320047	−	0.947402i	$-0.603699\pi$
$211$	−1961.86	−0.640094	−0.320047	+	0.947402i	$0.603699\pi$
$212$	0	0
$213$	1172.91	0.377307
$214$	0	0
$215$	1776.87	0.563634
$216$	0	0
$217$	2008.79	0.628411
$218$	0	0
$219$	3137.09	0.967967
$220$	0	0
$221$	715.429	0.217760
$222$	0	0
$223$	−1872.81	−0.562388	−0.281194	−	0.959651i	$-0.590730\pi$
$223$	−1872.81	−0.562388	−0.281194	+	0.959651i	$0.590730\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	6346.49	1.85565	0.927823	−	0.373021i	$-0.121678\pi$
$227$	6346.49	1.85565	0.927823	+	0.373021i	$0.121678\pi$
$228$	0	0
$229$	5470.66	1.57865	0.789327	−	0.613973i	$-0.210430\pi$
$229$	5470.66	1.57865	0.789327	+	0.613973i	$0.210430\pi$
$230$	0	0
$231$	991.029	0.282272
$232$	0	0
$233$	−2202.27	−0.619209	−0.309605	−	0.950865i	$-0.600197\pi$
$233$	−2202.27	−0.619209	−0.309605	+	0.950865i	$0.600197\pi$
$234$	0	0
$235$	2493.03	0.692031
$236$	0	0
$237$	2906.18	0.796526
$238$	0	0
$239$	−7195.97	−1.94757	−0.973784	−	0.227474i	$-0.926953\pi$
$239$	−7195.97	−1.94757	−0.973784	+	0.227474i	$0.926953\pi$
$240$	0	0
$241$	1073.99	0.287062	0.143531	−	0.989646i	$-0.454154\pi$
$241$	1073.99	0.287062	0.143531	+	0.989646i	$0.454154\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	245.000	0.0638877
$246$	0	0
$247$	6663.19	1.71647
$248$	0	0
$249$	−1363.87	−0.347116
$250$	0	0
$251$	3019.36	0.759283	0.379641	−	0.925134i	$-0.376047\pi$
$251$	3019.36	0.759283	0.379641	+	0.925134i	$0.376047\pi$
$252$	0	0
$253$	6880.47	1.70977
$254$	0	0
$255$	195.153	0.0479253
$256$	0	0
$257$	−789.137	−0.191537	−0.0957685	−	0.995404i	$-0.530531\pi$
$257$	−789.137	−0.191537	−0.0957685	+	0.995404i	$0.530531\pi$
$258$	0	0
$259$	−418.300	−0.100355
$260$	0	0
$261$	1434.18	0.340128
$262$	0	0
$263$	−1695.81	−0.397597	−0.198799	−	0.980040i	$-0.563704\pi$
$263$	−1695.81	−0.397597	−0.198799	+	0.980040i	$0.563704\pi$
$264$	0	0
$265$	−2123.53	−0.492255
$266$	0	0
$267$	1057.03	0.242281
$268$	0	0
$269$	4534.24	1.02772	0.513861	−	0.857873i	$-0.328215\pi$
$269$	4534.24	1.02772	0.513861	+	0.857873i	$0.328215\pi$
$270$	0	0
$271$	1613.68	0.361712	0.180856	−	0.983510i	$-0.442113\pi$
$271$	1613.68	0.361712	0.180856	+	0.983510i	$0.442113\pi$
$272$	0	0
$273$	−1154.79	−0.256010
$274$	0	0
$275$	1179.80	0.258707
$276$	0	0
$277$	7146.50	1.55015	0.775075	−	0.631869i	$-0.217712\pi$
$277$	7146.50	1.55015	0.775075	+	0.631869i	$0.217712\pi$
$278$	0	0
$279$	−2582.72	−0.554207
$280$	0	0
$281$	−156.874	−0.0333035	−0.0166518	−	0.999861i	$-0.505301\pi$
$281$	−156.874	−0.0333035	−0.0166518	+	0.999861i	$0.505301\pi$
$282$	0	0
$283$	−431.151	−0.0905628	−0.0452814	−	0.998974i	$-0.514418\pi$
$283$	−431.151	−0.0905628	−0.0452814	+	0.998974i	$0.514418\pi$
$284$	0	0
$285$	1817.57	0.377767
$286$	0	0
$287$	2052.34	0.422111
$288$	0	0
$289$	−4743.73	−0.965547
$290$	0	0
$291$	−2908.84	−0.585977
$292$	0	0
$293$	2891.45	0.576520	0.288260	−	0.957552i	$-0.406923\pi$
$293$	2891.45	0.576520	0.288260	+	0.957552i	$0.406923\pi$
$294$	0	0
$295$	−806.061	−0.159087
$296$	0	0
$297$	−1274.18	−0.248941
$298$	0	0
$299$	−8017.40	−1.55070
$300$	0	0
$301$	−2487.61	−0.476358
$302$	0	0
$303$	1188.18	0.225278
$304$	0	0
$305$	−3604.64	−0.676725
$306$	0	0
$307$	6735.27	1.25212	0.626062	−	0.779773i	$-0.284666\pi$
$307$	6735.27	1.25212	0.626062	+	0.779773i	$0.284666\pi$
$308$	0	0
$309$	−2610.66	−0.480632
$310$	0	0
$311$	−7984.07	−1.45574	−0.727869	−	0.685716i	$-0.759489\pi$
$311$	−7984.07	−1.45574	−0.727869	+	0.685716i	$0.759489\pi$
$312$	0	0
$313$	195.994	0.0353937	0.0176968	−	0.999843i	$-0.494367\pi$
$313$	195.994	0.0353937	0.0176968	+	0.999843i	$0.494367\pi$
$314$	0	0
$315$	−315.000	−0.0563436
$316$	0	0
$317$	7615.69	1.34934	0.674669	−	0.738121i	$-0.264286\pi$
$317$	7615.69	1.34934	0.674669	+	0.738121i	$0.264286\pi$
$318$	0	0
$319$	7520.16	1.31990
$320$	0	0
$321$	537.031	0.0933773
$322$	0	0
$323$	1576.47	0.271569
$324$	0	0
$325$	−1374.74	−0.234637
$326$	0	0
$327$	5916.78	1.00061
$328$	0	0
$329$	−3490.24	−0.584873
$330$	0	0
$331$	−4143.51	−0.688061	−0.344030	−	0.938959i	$-0.611792\pi$
$331$	−4143.51	−0.688061	−0.344030	+	0.938959i	$0.611792\pi$
$332$	0	0
$333$	537.814	0.0885046
$334$	0	0
$335$	1584.54	0.258426
$336$	0	0
$337$	497.963	0.0804920	0.0402460	−	0.999190i	$-0.487186\pi$
$337$	497.963	0.0804920	0.0402460	+	0.999190i	$0.487186\pi$
$338$	0	0
$339$	4724.55	0.756939
$340$	0	0
$341$	−13542.6	−2.15066
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−2186.97	−0.341283
$346$	0	0
$347$	3285.68	0.508313	0.254156	−	0.967163i	$-0.418202\pi$
$347$	3285.68	0.508313	0.254156	+	0.967163i	$0.418202\pi$
$348$	0	0
$349$	−406.377	−0.0623292	−0.0311646	−	0.999514i	$-0.509922\pi$
$349$	−406.377	−0.0623292	−0.0311646	+	0.999514i	$0.509922\pi$
$350$	0	0
$351$	1484.72	0.225780
$352$	0	0
$353$	−4286.74	−0.646346	−0.323173	−	0.946340i	$-0.604749\pi$
$353$	−4286.74	−0.646346	−0.323173	+	0.946340i	$0.604749\pi$
$354$	0	0
$355$	−1954.85	−0.292261
$356$	0	0
$357$	−273.214	−0.0405043
$358$	0	0
$359$	4524.65	0.665187	0.332594	−	0.943070i	$-0.392076\pi$
$359$	4524.65	0.665187	0.332594	+	0.943070i	$0.392076\pi$
$360$	0	0
$361$	7823.51	1.14062
$362$	0	0
$363$	−2688.21	−0.388690
$364$	0	0
$365$	−5228.48	−0.749784
$366$	0	0
$367$	−643.678	−0.0915523	−0.0457762	−	0.998952i	$-0.514576\pi$
$367$	−643.678	−0.0915523	−0.0457762	+	0.998952i	$0.514576\pi$
$368$	0	0
$369$	−2638.73	−0.372267
$370$	0	0
$371$	2972.94	0.416031
$372$	0	0
$373$	5980.51	0.830185	0.415093	−	0.909779i	$-0.363749\pi$
$373$	5980.51	0.830185	0.415093	+	0.909779i	$0.363749\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	0	0
$377$	−8762.79	−1.19710
$378$	0	0
$379$	9531.01	1.29176	0.645878	−	0.763441i	$-0.276492\pi$
$379$	9531.01	1.29176	0.645878	+	0.763441i	$0.276492\pi$
$380$	0	0
$381$	8053.08	1.08287
$382$	0	0
$383$	5683.65	0.758279	0.379140	−	0.925339i	$-0.376220\pi$
$383$	5683.65	0.758279	0.379140	+	0.925339i	$0.376220\pi$
$384$	0	0
$385$	−1651.71	−0.218647
$386$	0	0
$387$	3198.36	0.420108
$388$	0	0
$389$	12787.0	1.66665	0.833323	−	0.552787i	$-0.186436\pi$
$389$	12787.0	1.66665	0.833323	+	0.552787i	$0.186436\pi$
$390$	0	0
$391$	−1896.86	−0.245341
$392$	0	0
$393$	5868.24	0.753216
$394$	0	0
$395$	−4843.63	−0.616986
$396$	0	0
$397$	−84.7735	−0.0107170	−0.00535852	−	0.999986i	$-0.501706\pi$
$397$	−84.7735	−0.0107170	−0.00535852	+	0.999986i	$0.501706\pi$
$398$	0	0
$399$	−2544.60	−0.319272
$400$	0	0
$401$	−2166.77	−0.269834	−0.134917	−	0.990857i	$-0.543077\pi$
$401$	−2166.77	−0.269834	−0.134917	+	0.990857i	$0.543077\pi$
$402$	0	0
$403$	15780.4	1.95056
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	2820.05	0.343451
$408$	0	0
$409$	14072.1	1.70127	0.850635	−	0.525757i	$-0.176218\pi$
$409$	14072.1	1.70127	0.850635	+	0.525757i	$0.176218\pi$
$410$	0	0
$411$	−889.873	−0.106799
$412$	0	0
$413$	1128.49	0.134453
$414$	0	0
$415$	2273.12	0.268875
$416$	0	0
$417$	4042.05	0.474677
$418$	0	0
$419$	14548.4	1.69627	0.848133	−	0.529783i	$-0.177726\pi$
$419$	14548.4	1.69627	0.848133	+	0.529783i	$0.177726\pi$
$420$	0	0
$421$	−11270.5	−1.30473	−0.652364	−	0.757905i	$-0.726223\pi$
$421$	−11270.5	−1.30473	−0.652364	+	0.757905i	$0.726223\pi$
$422$	0	0
$423$	4487.46	0.515810
$424$	0	0
$425$	−325.255	−0.0371228
$426$	0	0
$427$	5046.50	0.571937
$428$	0	0
$429$	7785.21	0.876162
$430$	0	0
$431$	−13711.3	−1.53236	−0.766182	−	0.642623i	$-0.777846\pi$
$431$	−13711.3	−1.53236	−0.766182	+	0.642623i	$0.777846\pi$
$432$	0	0
$433$	−2459.18	−0.272935	−0.136467	−	0.990645i	$-0.543575\pi$
$433$	−2459.18	−0.272935	−0.136467	+	0.990645i	$0.543575\pi$
$434$	0	0
$435$	−2390.30	−0.263462
$436$	0	0
$437$	−17666.5	−1.93388
$438$	0	0
$439$	−12562.0	−1.36572	−0.682860	−	0.730549i	$-0.739264\pi$
$439$	−12562.0	−1.36572	−0.682860	+	0.730549i	$0.739264\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	−8985.09	−0.963644	−0.481822	−	0.876269i	$-0.660025\pi$
$443$	−8985.09	−0.963644	−0.481822	+	0.876269i	$0.660025\pi$
$444$	0	0
$445$	−1761.71	−0.187670
$446$	0	0
$447$	1345.34	0.142354
$448$	0	0
$449$	−7814.36	−0.821341	−0.410671	−	0.911784i	$-0.634705\pi$
$449$	−7814.36	−0.821341	−0.410671	+	0.911784i	$0.634705\pi$
$450$	0	0
$451$	−13836.3	−1.44462
$452$	0	0
$453$	−8539.62	−0.885709
$454$	0	0
$455$	1924.64	0.198305
$456$	0	0
$457$	5783.38	0.591980	0.295990	−	0.955191i	$-0.404350\pi$
$457$	5783.38	0.591980	0.295990	+	0.955191i	$0.404350\pi$
$458$	0	0
$459$	351.276	0.0357214
$460$	0	0
$461$	−3169.08	−0.320171	−0.160086	−	0.987103i	$-0.551177\pi$
$461$	−3169.08	−0.320171	−0.160086	+	0.987103i	$0.551177\pi$
$462$	0	0
$463$	−19069.3	−1.91409	−0.957045	−	0.289941i	$-0.906364\pi$
$463$	−19069.3	−1.91409	−0.957045	+	0.289941i	$0.906364\pi$
$464$	0	0
$465$	4304.54	0.429287
$466$	0	0
$467$	19764.9	1.95848	0.979239	−	0.202707i	$-0.0649739\pi$
$467$	19764.9	1.95848	0.979239	+	0.202707i	$0.0649739\pi$
$468$	0	0
$469$	−2218.36	−0.218410
$470$	0	0
$471$	5073.20	0.496307
$472$	0	0
$473$	16770.7	1.63027
$474$	0	0
$475$	−3029.29	−0.292617
$476$	0	0
$477$	−3822.36	−0.366905
$478$	0	0
$479$	5303.50	0.505894	0.252947	−	0.967480i	$-0.418600\pi$
$479$	5303.50	0.505894	0.252947	+	0.967480i	$0.418600\pi$
$480$	0	0
$481$	−3286.03	−0.311497
$482$	0	0
$483$	3061.76	0.288436
$484$	0	0
$485$	4848.07	0.453896
$486$	0	0
$487$	−11565.7	−1.07616	−0.538081	−	0.842893i	$-0.680851\pi$
$487$	−11565.7	−1.07616	−0.538081	+	0.842893i	$0.680851\pi$
$488$	0	0
$489$	7405.50	0.684843
$490$	0	0
$491$	11837.5	1.08803	0.544013	−	0.839077i	$-0.316904\pi$
$491$	11837.5	1.08803	0.544013	+	0.839077i	$0.316904\pi$
$492$	0	0
$493$	−2073.22	−0.189397
$494$	0	0
$495$	2123.63	0.192829
$496$	0	0
$497$	2736.79	0.247005
$498$	0	0
$499$	14408.7	1.29263	0.646314	−	0.763071i	$-0.276309\pi$
$499$	14408.7	1.29263	0.646314	+	0.763071i	$0.276309\pi$
$500$	0	0
$501$	−146.480	−0.0130623
$502$	0	0
$503$	−3333.20	−0.295467	−0.147734	−	0.989027i	$-0.547198\pi$
$503$	−3333.20	−0.295467	−0.147734	+	0.989027i	$0.547198\pi$
$504$	0	0
$505$	−1980.31	−0.174500
$506$	0	0
$507$	−2480.63	−0.217295
$508$	0	0
$509$	−15270.2	−1.32974	−0.664871	−	0.746958i	$-0.731514\pi$
$509$	−15270.2	−1.32974	−0.664871	+	0.746958i	$0.731514\pi$
$510$	0	0
$511$	7319.87	0.633683
$512$	0	0
$513$	3271.63	0.281571
$514$	0	0
$515$	4351.10	0.372296
$516$	0	0
$517$	23530.1	2.00165
$518$	0	0
$519$	3226.85	0.272915
$520$	0	0
$521$	−10683.9	−0.898410	−0.449205	−	0.893429i	$-0.648293\pi$
$521$	−10683.9	−0.898410	−0.449205	+	0.893429i	$0.648293\pi$
$522$	0	0
$523$	−2376.98	−0.198735	−0.0993673	−	0.995051i	$-0.531682\pi$
$523$	−2376.98	−0.198735	−0.0993673	+	0.995051i	$0.531682\pi$
$524$	0	0
$525$	525.000	0.0436436
$526$	0	0
$527$	3733.53	0.308606
$528$	0	0
$529$	9090.04	0.747107
$530$	0	0
$531$	−1450.91	−0.118577
$532$	0	0
$533$	16122.6	1.31022
$534$	0	0
$535$	−895.051	−0.0723298
$536$	0	0
$537$	−6295.03	−0.505867
$538$	0	0
$539$	2312.40	0.184791
$540$	0	0
$541$	14797.6	1.17597	0.587986	−	0.808871i	$-0.299921\pi$
$541$	14797.6	1.17597	0.587986	+	0.808871i	$0.299921\pi$
$542$	0	0
$543$	12701.4	1.00381
$544$	0	0
$545$	−9861.31	−0.775068
$546$	0	0
$547$	11757.1	0.919006	0.459503	−	0.888176i	$-0.348028\pi$
$547$	11757.1	0.919006	0.459503	+	0.888176i	$0.348028\pi$
$548$	0	0
$549$	−6488.36	−0.504401
$550$	0	0
$551$	−19309.0	−1.49291
$552$	0	0
$553$	6781.09	0.521449
$554$	0	0
$555$	−896.357	−0.0685554
$556$	0	0
$557$	−18075.4	−1.37501	−0.687505	−	0.726179i	$-0.741294\pi$
$557$	−18075.4	−1.37501	−0.687505	+	0.726179i	$0.741294\pi$
$558$	0	0
$559$	−19541.9	−1.47860
$560$	0	0
$561$	1841.93	0.138621
$562$	0	0
$563$	8061.61	0.603475	0.301737	−	0.953391i	$-0.402433\pi$
$563$	8061.61	0.603475	0.301737	+	0.953391i	$0.402433\pi$
$564$	0	0
$565$	−7874.24	−0.586322
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	−330.559	−0.0243546	−0.0121773	−	0.999926i	$-0.503876\pi$
$569$	−330.559	−0.0243546	−0.0121773	+	0.999926i	$0.503876\pi$
$570$	0	0
$571$	9981.68	0.731559	0.365780	−	0.930701i	$-0.380802\pi$
$571$	9981.68	0.731559	0.365780	+	0.930701i	$0.380802\pi$
$572$	0	0
$573$	−3598.60	−0.262362
$574$	0	0
$575$	3644.95	0.264356
$576$	0	0
$577$	24960.4	1.80090	0.900448	−	0.434964i	$-0.143239\pi$
$577$	24960.4	1.80090	0.900448	+	0.434964i	$0.143239\pi$
$578$	0	0
$579$	7210.84	0.517568
$580$	0	0
$581$	−3182.37	−0.227241
$582$	0	0
$583$	−20042.7	−1.42381
$584$	0	0
$585$	−2474.54	−0.174888
$586$	0	0
$587$	−24359.7	−1.71283	−0.856415	−	0.516288i	$-0.827313\pi$
$587$	−24359.7	−1.71283	−0.856415	+	0.516288i	$0.827313\pi$
$588$	0	0
$589$	34772.5	2.43256
$590$	0	0
$591$	10642.7	0.740748
$592$	0	0
$593$	−1046.62	−0.0724780	−0.0362390	−	0.999343i	$-0.511538\pi$
$593$	−1046.62	−0.0724780	−0.0362390	+	0.999343i	$0.511538\pi$
$594$	0	0
$595$	455.357	0.0313745
$596$	0	0
$597$	−14182.0	−0.972244
$598$	0	0
$599$	−13879.2	−0.946727	−0.473364	−	0.880867i	$-0.656960\pi$
$599$	−13879.2	−0.946727	−0.473364	+	0.880867i	$0.656960\pi$
$600$	0	0
$601$	−433.702	−0.0294361	−0.0147180	−	0.999892i	$-0.504685\pi$
$601$	−433.702	−0.0294361	−0.0147180	+	0.999892i	$0.504685\pi$
$602$	0	0
$603$	2852.17	0.192619
$604$	0	0
$605$	4480.35	0.301078
$606$	0	0
$607$	−22916.0	−1.53234	−0.766169	−	0.642639i	$-0.777840\pi$
$607$	−22916.0	−1.53234	−0.766169	+	0.642639i	$0.777840\pi$
$608$	0	0
$609$	3346.41	0.222666
$610$	0	0
$611$	−27418.2	−1.81542
$612$	0	0
$613$	−742.986	−0.0489542	−0.0244771	−	0.999700i	$-0.507792\pi$
$613$	−742.986	−0.0489542	−0.0244771	+	0.999700i	$0.507792\pi$
$614$	0	0
$615$	4397.88	0.288357
$616$	0	0
$617$	−18919.9	−1.23450	−0.617252	−	0.786766i	$-0.711754\pi$
$617$	−18919.9	−1.23450	−0.617252	+	0.786766i	$0.711754\pi$
$618$	0	0
$619$	−10500.0	−0.681791	−0.340896	−	0.940101i	$-0.610730\pi$
$619$	−10500.0	−0.681791	−0.340896	+	0.940101i	$0.610730\pi$
$620$	0	0
$621$	−3936.54	−0.254377
$622$	0	0
$623$	2466.40	0.158610
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	17154.9	1.09267
$628$	0	0
$629$	−777.453	−0.0492831
$630$	0	0
$631$	−8462.27	−0.533879	−0.266940	−	0.963713i	$-0.586012\pi$
$631$	−8462.27	−0.533879	−0.266940	+	0.963713i	$0.586012\pi$
$632$	0	0
$633$	5885.57	0.369558
$634$	0	0
$635$	−13421.8	−0.838784
$636$	0	0
$637$	−2694.50	−0.167598
$638$	0	0
$639$	−3518.72	−0.217838
$640$	0	0
$641$	−1454.86	−0.0896468	−0.0448234	−	0.998995i	$-0.514273\pi$
$641$	−1454.86	−0.0896468	−0.0448234	+	0.998995i	$0.514273\pi$
$642$	0	0
$643$	−6402.64	−0.392683	−0.196342	−	0.980536i	$-0.562906\pi$
$643$	−6402.64	−0.392683	−0.196342	+	0.980536i	$0.562906\pi$
$644$	0	0
$645$	−5330.60	−0.325414
$646$	0	0
$647$	22610.4	1.37389	0.686946	−	0.726708i	$-0.258951\pi$
$647$	22610.4	1.37389	0.686946	+	0.726708i	$0.258951\pi$
$648$	0	0
$649$	−7607.90	−0.460148
$650$	0	0
$651$	−6026.36	−0.362813
$652$	0	0
$653$	19767.7	1.18464	0.592319	−	0.805704i	$-0.298213\pi$
$653$	19767.7	1.18464	0.592319	+	0.805704i	$0.298213\pi$
$654$	0	0
$655$	−9780.41	−0.583438
$656$	0	0
$657$	−9411.26	−0.558856
$658$	0	0
$659$	−10127.1	−0.598630	−0.299315	−	0.954154i	$-0.596758\pi$
$659$	−10127.1	−0.598630	−0.299315	+	0.954154i	$0.596758\pi$
$660$	0	0
$661$	10454.7	0.615192	0.307596	−	0.951517i	$-0.400476\pi$
$661$	10454.7	0.615192	0.307596	+	0.951517i	$0.400476\pi$
$662$	0	0
$663$	−2146.29	−0.125724
$664$	0	0
$665$	4241.00	0.247307
$666$	0	0
$667$	23233.4	1.34872
$668$	0	0
$669$	5618.42	0.324695
$670$	0	0
$671$	−34021.9	−1.95738
$672$	0	0
$673$	−6038.29	−0.345853	−0.172927	−	0.984935i	$-0.555322\pi$
$673$	−6038.29	−0.345853	−0.172927	+	0.984935i	$0.555322\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	8055.89	0.457331	0.228666	−	0.973505i	$-0.426564\pi$
$677$	8055.89	0.457331	0.228666	+	0.973505i	$0.426564\pi$
$678$	0	0
$679$	−6787.30	−0.383612
$680$	0	0
$681$	−19039.5	−1.07136
$682$	0	0
$683$	23195.9	1.29951	0.649757	−	0.760142i	$-0.274871\pi$
$683$	23195.9	1.29951	0.649757	+	0.760142i	$0.274871\pi$
$684$	0	0
$685$	1483.12	0.0827258
$686$	0	0
$687$	−16412.0	−0.911436
$688$	0	0
$689$	23354.5	1.29134
$690$	0	0
$691$	−9896.53	−0.544836	−0.272418	−	0.962179i	$-0.587823\pi$
$691$	−9896.53	−0.544836	−0.272418	+	0.962179i	$0.587823\pi$
$692$	0	0
$693$	−2973.09	−0.162970
$694$	0	0
$695$	−6736.75	−0.367683
$696$	0	0
$697$	3814.49	0.207294
$698$	0	0
$699$	6606.82	0.357501
$700$	0	0
$701$	−20459.5	−1.10234	−0.551172	−	0.834391i	$-0.685819\pi$
$701$	−20459.5	−1.10234	−0.551172	+	0.834391i	$0.685819\pi$
$702$	0	0
$703$	−7240.86	−0.388470
$704$	0	0
$705$	−7479.09	−0.399545
$706$	0	0
$707$	2772.43	0.147479
$708$	0	0
$709$	−26221.8	−1.38897	−0.694485	−	0.719507i	$-0.744368\pi$
$709$	−26221.8	−1.38897	−0.694485	+	0.719507i	$0.744368\pi$
$710$	0	0
$711$	−8718.54	−0.459874
$712$	0	0
$713$	−41839.6	−2.19762
$714$	0	0
$715$	−12975.3	−0.678672
$716$	0	0
$717$	21587.9	1.12443
$718$	0	0
$719$	6875.76	0.356638	0.178319	−	0.983973i	$-0.442934\pi$
$719$	6875.76	0.356638	0.178319	+	0.983973i	$0.442934\pi$
$720$	0	0
$721$	−6091.54	−0.314648
$722$	0	0
$723$	−3221.98	−0.165735
$724$	0	0
$725$	3983.83	0.204077
$726$	0	0
$727$	30650.8	1.56365	0.781825	−	0.623497i	$-0.214289\pi$
$727$	30650.8	1.56365	0.781825	+	0.623497i	$0.214289\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−4623.48	−0.233934
$732$	0	0
$733$	−17526.1	−0.883138	−0.441569	−	0.897227i	$-0.645578\pi$
$733$	−17526.1	−0.883138	−0.441569	+	0.897227i	$0.645578\pi$
$734$	0	0
$735$	−735.000	−0.0368856
$736$	0	0
$737$	14955.5	0.747479
$738$	0	0
$739$	2894.14	0.144063	0.0720316	−	0.997402i	$-0.477052\pi$
$739$	2894.14	0.144063	0.0720316	+	0.997402i	$0.477052\pi$
$740$	0	0
$741$	−19989.6	−0.991006
$742$	0	0
$743$	−37962.8	−1.87445	−0.937227	−	0.348720i	$-0.886616\pi$
$743$	−37962.8	−1.87445	−0.937227	+	0.348720i	$0.886616\pi$
$744$	0	0
$745$	−2242.23	−0.110267
$746$	0	0
$747$	4091.62	0.200408
$748$	0	0
$749$	1253.07	0.0611298
$750$	0	0
$751$	−7549.90	−0.366844	−0.183422	−	0.983034i	$-0.558717\pi$
$751$	−7549.90	−0.366844	−0.183422	+	0.983034i	$0.558717\pi$
$752$	0	0
$753$	−9058.07	−0.438372
$754$	0	0
$755$	14232.7	0.686067
$756$	0	0
$757$	9984.20	0.479369	0.239684	−	0.970851i	$-0.422956\pi$
$757$	9984.20	0.479369	0.239684	+	0.970851i	$0.422956\pi$
$758$	0	0
$759$	−20641.4	−0.987136
$760$	0	0
$761$	5804.62	0.276501	0.138251	−	0.990397i	$-0.455852\pi$
$761$	5804.62	0.276501	0.138251	+	0.990397i	$0.455852\pi$
$762$	0	0
$763$	13805.8	0.655052
$764$	0	0
$765$	−585.459	−0.0276697
$766$	0	0
$767$	8865.03	0.417337
$768$	0	0
$769$	20782.5	0.974560	0.487280	−	0.873246i	$-0.337989\pi$
$769$	20782.5	0.974560	0.487280	+	0.873246i	$0.337989\pi$
$770$	0	0
$771$	2367.41	0.110584
$772$	0	0
$773$	40998.7	1.90766	0.953829	−	0.300349i	$-0.0971033\pi$
$773$	40998.7	1.90766	0.953829	+	0.300349i	$0.0971033\pi$
$774$	0	0
$775$	−7174.23	−0.332524
$776$	0	0
$777$	1254.90	0.0579399
$778$	0	0
$779$	35526.5	1.63398
$780$	0	0
$781$	−18450.6	−0.845343
$782$	0	0
$783$	−4302.53	−0.196373
$784$	0	0
$785$	−8455.34	−0.384438
$786$	0	0
$787$	−33578.0	−1.52087	−0.760436	−	0.649413i	$-0.775015\pi$
$787$	−33578.0	−1.52087	−0.760436	+	0.649413i	$0.775015\pi$
$788$	0	0
$789$	5087.43	0.229553
$790$	0	0
$791$	11023.9	0.495533
$792$	0	0
$793$	39643.7	1.77527
$794$	0	0
$795$	6370.59	0.284203
$796$	0	0
$797$	12041.7	0.535181	0.267591	−	0.963533i	$-0.413773\pi$
$797$	12041.7	0.535181	0.267591	+	0.963533i	$0.413773\pi$
$798$	0	0
$799$	−6486.97	−0.287225
$800$	0	0
$801$	−3171.09	−0.139881
$802$	0	0
$803$	−49348.3	−2.16870
$804$	0	0
$805$	−5102.93	−0.223422
$806$	0	0
$807$	−13602.7	−0.593356
$808$	0	0
$809$	14146.8	0.614801	0.307401	−	0.951580i	$-0.400541\pi$
$809$	14146.8	0.614801	0.307401	+	0.951580i	$0.400541\pi$
$810$	0	0
$811$	−18899.6	−0.818318	−0.409159	−	0.912463i	$-0.634178\pi$
$811$	−18899.6	−0.818318	−0.409159	+	0.912463i	$0.634178\pi$
$812$	0	0
$813$	−4841.04	−0.208835
$814$	0	0
$815$	−12342.5	−0.530477
$816$	0	0
$817$	−43061.1	−1.84396
$818$	0	0
$819$	3464.36	0.147808
$820$	0	0
$821$	11865.1	0.504378	0.252189	−	0.967678i	$-0.418850\pi$
$821$	11865.1	0.504378	0.252189	+	0.967678i	$0.418850\pi$
$822$	0	0
$823$	−21936.6	−0.929116	−0.464558	−	0.885543i	$-0.653787\pi$
$823$	−21936.6	−0.929116	−0.464558	+	0.885543i	$0.653787\pi$
$824$	0	0
$825$	−3539.39	−0.149364
$826$	0	0
$827$	−16012.5	−0.673290	−0.336645	−	0.941632i	$-0.609292\pi$
$827$	−16012.5	−0.673290	−0.336645	+	0.941632i	$0.609292\pi$
$828$	0	0
$829$	29366.5	1.23033	0.615164	−	0.788399i	$-0.289090\pi$
$829$	29366.5	1.23033	0.615164	+	0.788399i	$0.289090\pi$
$830$	0	0
$831$	−21439.5	−0.894980
$832$	0	0
$833$	−637.500	−0.0265163
$834$	0	0
$835$	244.133	0.0101180
$836$	0	0
$837$	7748.17	0.319971
$838$	0	0
$839$	39833.0	1.63908	0.819540	−	0.573022i	$-0.194229\pi$
$839$	39833.0	1.63908	0.819540	+	0.573022i	$0.194229\pi$
$840$	0	0
$841$	1004.40	0.0411824
$842$	0	0
$843$	470.621	0.0192278
$844$	0	0
$845$	4134.39	0.168316
$846$	0	0
$847$	−6272.49	−0.254457
$848$	0	0
$849$	1293.45	0.0522864
$850$	0	0
$851$	8712.47	0.350951
$852$	0	0
$853$	−13635.3	−0.547321	−0.273660	−	0.961826i	$-0.588234\pi$
$853$	−13635.3	−0.547321	−0.273660	+	0.961826i	$0.588234\pi$
$854$	0	0
$855$	−5452.71	−0.218104
$856$	0	0
$857$	41890.0	1.66970	0.834851	−	0.550476i	$-0.185554\pi$
$857$	41890.0	1.66970	0.834851	+	0.550476i	$0.185554\pi$
$858$	0	0
$859$	−42883.1	−1.70332	−0.851661	−	0.524093i	$-0.824405\pi$
$859$	−42883.1	−1.70332	−0.851661	+	0.524093i	$0.824405\pi$
$860$	0	0
$861$	−6157.03	−0.243706
$862$	0	0
$863$	33971.3	1.33997	0.669987	−	0.742373i	$-0.266300\pi$
$863$	33971.3	1.33997	0.669987	+	0.742373i	$0.266300\pi$
$864$	0	0
$865$	−5378.08	−0.211399
$866$	0	0
$867$	14231.2	0.557459
$868$	0	0
$869$	−45716.0	−1.78459
$870$	0	0
$871$	−17426.7	−0.677935
$872$	0	0
$873$	8726.53	0.338314
$874$	0	0
$875$	−875.000	−0.0338062
$876$	0	0
$877$	−19300.5	−0.743138	−0.371569	−	0.928405i	$-0.621180\pi$
$877$	−19300.5	−0.743138	−0.371569	+	0.928405i	$0.621180\pi$
$878$	0	0
$879$	−8674.35	−0.332854
$880$	0	0
$881$	13031.2	0.498332	0.249166	−	0.968461i	$-0.419843\pi$
$881$	13031.2	0.498332	0.249166	+	0.968461i	$0.419843\pi$
$882$	0	0
$883$	−13752.4	−0.524127	−0.262063	−	0.965051i	$-0.584403\pi$
$883$	−13752.4	−0.524127	−0.262063	+	0.965051i	$0.584403\pi$
$884$	0	0
$885$	2418.18	0.0918490
$886$	0	0
$887$	−30659.5	−1.16059	−0.580296	−	0.814406i	$-0.697063\pi$
$887$	−30659.5	−1.16059	−0.580296	+	0.814406i	$0.697063\pi$
$888$	0	0
$889$	18790.5	0.708902
$890$	0	0
$891$	3822.54	0.143726
$892$	0	0
$893$	−60416.8	−2.26402
$894$	0	0
$895$	10491.7	0.391843
$896$	0	0
$897$	24052.2	0.895295
$898$	0	0
$899$	−45729.4	−1.69651
$900$	0	0
$901$	5525.51	0.204308
$902$	0	0
$903$	7462.84	0.275025
$904$	0	0
$905$	−21169.1	−0.777550
$906$	0	0
$907$	11692.5	0.428051	0.214026	−	0.976828i	$-0.431342\pi$
$907$	11692.5	0.428051	0.214026	+	0.976828i	$0.431342\pi$
$908$	0	0
$909$	−3564.55	−0.130065
$910$	0	0
$911$	−38683.3	−1.40685	−0.703423	−	0.710772i	$-0.748346\pi$
$911$	−38683.3	−1.40685	−0.703423	+	0.710772i	$0.748346\pi$
$912$	0	0
$913$	21454.6	0.777703
$914$	0	0
$915$	10813.9	0.390708
$916$	0	0
$917$	13692.6	0.493095
$918$	0	0
$919$	−29118.2	−1.04518	−0.522590	−	0.852584i	$-0.675034\pi$
$919$	−29118.2	−1.04518	−0.522590	+	0.852584i	$0.675034\pi$
$920$	0	0
$921$	−20205.8	−0.722914
$922$	0	0
$923$	21499.3	0.766694
$924$	0	0
$925$	1493.93	0.0531028
$926$	0	0
$927$	7831.98	0.277493
$928$	0	0
$929$	16538.1	0.584065	0.292032	−	0.956408i	$-0.405668\pi$
$929$	16538.1	0.584065	0.292032	+	0.956408i	$0.405668\pi$
$930$	0	0
$931$	−5937.40	−0.209012
$932$	0	0
$933$	23952.2	0.840471
$934$	0	0
$935$	−3069.88	−0.107375
$936$	0	0
$937$	−18629.9	−0.649534	−0.324767	−	0.945794i	$-0.605286\pi$
$937$	−18629.9	−0.649534	−0.324767	+	0.945794i	$0.605286\pi$
$938$	0	0
$939$	−587.981	−0.0204346
$940$	0	0
$941$	−24466.7	−0.847598	−0.423799	−	0.905756i	$-0.639304\pi$
$941$	−24466.7	−0.847598	−0.423799	+	0.905756i	$0.639304\pi$
$942$	0	0
$943$	−42746.8	−1.47617
$944$	0	0
$945$	945.000	0.0325300
$946$	0	0
$947$	14940.6	0.512676	0.256338	−	0.966587i	$-0.417484\pi$
$947$	14940.6	0.512676	0.256338	+	0.966587i	$0.417484\pi$
$948$	0	0
$949$	57502.6	1.96693
$950$	0	0
$951$	−22847.1	−0.779040
$952$	0	0
$953$	17893.4	0.608211	0.304106	−	0.952638i	$-0.401642\pi$
$953$	17893.4	0.608211	0.304106	+	0.952638i	$0.401642\pi$
$954$	0	0
$955$	5997.66	0.203225
$956$	0	0
$957$	−22560.5	−0.762045
$958$	0	0
$959$	−2076.37	−0.0699161
$960$	0	0
$961$	52560.4	1.76431
$962$	0	0
$963$	−1611.09	−0.0539114
$964$	0	0
$965$	−12018.1	−0.400907
$966$	0	0
$967$	27889.3	0.927466	0.463733	−	0.885975i	$-0.346510\pi$
$967$	27889.3	0.927466	0.463733	+	0.885975i	$0.346510\pi$
$968$	0	0
$969$	−4729.40	−0.156791
$970$	0	0
$971$	35825.1	1.18402	0.592009	−	0.805931i	$-0.298335\pi$
$971$	35825.1	1.18402	0.592009	+	0.805931i	$0.298335\pi$
$972$	0	0
$973$	9431.46	0.310749
$974$	0	0
$975$	4124.23	0.135468
$976$	0	0
$977$	−15539.0	−0.508841	−0.254420	−	0.967094i	$-0.581885\pi$
$977$	−15539.0	−0.508841	−0.254420	+	0.967094i	$0.581885\pi$
$978$	0	0
$979$	−16627.7	−0.542823
$980$	0	0
$981$	−17750.4	−0.577701
$982$	0	0
$983$	12668.3	0.411043	0.205521	−	0.978653i	$-0.434111\pi$
$983$	12668.3	0.411043	0.205521	+	0.978653i	$0.434111\pi$
$984$	0	0
$985$	−17737.8	−0.573781
$986$	0	0
$987$	10470.7	0.337677
$988$	0	0
$989$	51812.7	1.66587
$990$	0	0
$991$	−42239.4	−1.35396	−0.676982	−	0.736000i	$-0.736712\pi$
$991$	−42239.4	−1.35396	−0.676982	+	0.736000i	$0.736712\pi$
$992$	0	0
$993$	12430.5	0.397252
$994$	0	0
$995$	23636.6	0.753097
$996$	0	0
$997$	−28615.7	−0.908996	−0.454498	−	0.890748i	$-0.650181\pi$
$997$	−28615.7	−0.908996	−0.454498	+	0.890748i	$0.650181\pi$
$998$	0	0
$999$	−1613.44	−0.0510982

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	840.4.a.i.1.2	✓	2
4.3	odd	2		1680.4.a.bn.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.4.a.i.1.2	✓	2	1.1	even	1	trivial
1680.4.a.bn.1.1		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	840.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +5.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} +47.1918 q^{11} -54.9898 q^{13} -15.0000 q^{15} -13.0102 q^{17} -121.171 q^{19} +21.0000 q^{21} +145.798 q^{23} +25.0000 q^{25} -27.0000 q^{27} +159.353 q^{29} -286.969 q^{31} -141.576 q^{33} -35.0000 q^{35} +59.7571 q^{37} +164.969 q^{39} -293.192 q^{41} +355.373 q^{43} +45.0000 q^{45} +498.606 q^{47} +49.0000 q^{49} +39.0306 q^{51} -424.706 q^{53} +235.959 q^{55} +363.514 q^{57} -161.212 q^{59} -720.929 q^{61} -63.0000 q^{63} -274.949 q^{65} +316.908 q^{67} -437.394 q^{69} -390.969 q^{71} -1045.70 q^{73} -75.0000 q^{75} -330.343 q^{77} -968.727 q^{79} +81.0000 q^{81} +454.624 q^{83} -65.0510 q^{85} -478.059 q^{87} -352.343 q^{89} +384.929 q^{91} +860.908 q^{93} -605.857 q^{95} +969.614 q^{97} +424.727 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 10 q^{5} - 14 q^{7} + 18 q^{9} + 16 q^{11} - 12 q^{13} - 30 q^{15} - 124 q^{17} + 32 q^{19} + 42 q^{21} + 272 q^{23} + 50 q^{25} - 54 q^{27} - 132 q^{29} - 280 q^{31} - 48 q^{33} - 70 q^{35}+ \cdots + 144 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 10 * q^5 - 14 * q^7 + 18 * q^9 + 16 * q^11 - 12 * q^13 - 30 * q^15 - 124 * q^17 + 32 * q^19 + 42 * q^21 + 272 * q^23 + 50 * q^25 - 54 * q^27 - 132 * q^29 - 280 * q^31 - 48 * q^33 - 70 * q^35 - 292 * q^37 + 36 * q^39 - 508 * q^41 + 456 * q^43 + 90 * q^45 + 1056 * q^47 + 98 * q^49 + 372 * q^51 + 52 * q^53 + 80 * q^55 - 96 * q^57 - 440 * q^59 - 756 * q^61 - 126 * q^63 - 60 * q^65 - 248 * q^67 - 816 * q^69 - 488 * q^71 - 1092 * q^73 - 150 * q^75 - 112 * q^77 - 1232 * q^79 + 162 * q^81 - 776 * q^83 - 620 * q^85 + 396 * q^87 - 156 * q^89 + 84 * q^91 + 840 * q^93 + 160 * q^95 + 156 * q^97 + 144 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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