LMFDB - Embedded newform 1680.4.a.bg.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,4,Mod(1,1680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1680.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1680.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:	$99.1232088096$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{65}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 16 $ x^2 - x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.53113$ of defining polynomial
Character	$\chi$	$=$	1680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +2.93774 q^{11} -19.0623 q^{13} -15.0000 q^{15} +122.498 q^{17} -107.436 q^{19} -21.0000 q^{21} -210.623 q^{23} +25.0000 q^{25} -27.0000 q^{27} +95.4942 q^{29} +94.3074 q^{31} -8.81323 q^{33} +35.0000 q^{35} +97.1206 q^{37} +57.1868 q^{39} -491.113 q^{41} +43.0039 q^{43} +45.0000 q^{45} -473.494 q^{47} +49.0000 q^{49} -367.494 q^{51} -183.677 q^{53} +14.6887 q^{55} +322.307 q^{57} +760.615 q^{59} -198.747 q^{61} +63.0000 q^{63} -95.3113 q^{65} +309.992 q^{67} +631.868 q^{69} -665.693 q^{71} +621.288 q^{73} -75.0000 q^{75} +20.5642 q^{77} +24.7626 q^{79} +81.0000 q^{81} +406.724 q^{83} +612.490 q^{85} -286.483 q^{87} +261.751 q^{89} -133.436 q^{91} -282.922 q^{93} -537.179 q^{95} -1004.77 q^{97} +26.4397 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 + 10 * q^5 + 14 * q^7 + 18 * q^9	$ 2 q - 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9} + 22 q^{11} - 22 q^{13} - 30 q^{15} + 116 q^{17} - 102 q^{19} - 42 q^{21} - 260 q^{23} + 50 q^{25} - 54 q^{27} - 196 q^{29} - 150 q^{31} - 66 q^{33} + 70 q^{35} - 96 q^{37} + 66 q^{39} - 176 q^{41} + 344 q^{43} + 90 q^{45} - 560 q^{47} + 98 q^{49} - 348 q^{51} + 326 q^{53} + 110 q^{55} + 306 q^{57} + 844 q^{59} - 204 q^{61} + 126 q^{63} - 110 q^{65} + 104 q^{67} + 780 q^{69} - 1670 q^{71} - 386 q^{73} - 150 q^{75} + 154 q^{77} + 888 q^{79} + 162 q^{81} - 928 q^{83} + 580 q^{85} + 588 q^{87} + 588 q^{89} - 154 q^{91} + 450 q^{93} - 510 q^{95} + 522 q^{97} + 198 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 10 * q^5 + 14 * q^7 + 18 * q^9 + 22 * q^11 - 22 * q^13 - 30 * q^15 + 116 * q^17 - 102 * q^19 - 42 * q^21 - 260 * q^23 + 50 * q^25 - 54 * q^27 - 196 * q^29 - 150 * q^31 - 66 * q^33 + 70 * q^35 - 96 * q^37 + 66 * q^39 - 176 * q^41 + 344 * q^43 + 90 * q^45 - 560 * q^47 + 98 * q^49 - 348 * q^51 + 326 * q^53 + 110 * q^55 + 306 * q^57 + 844 * q^59 - 204 * q^61 + 126 * q^63 - 110 * q^65 + 104 * q^67 + 780 * q^69 - 1670 * q^71 - 386 * q^73 - 150 * q^75 + 154 * q^77 + 888 * q^79 + 162 * q^81 - 928 * q^83 + 580 * q^85 + 588 * q^87 + 588 * q^89 - 154 * q^91 + 450 * q^93 - 510 * q^95 + 522 * q^97 + 198 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−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
$11$	2.93774	0.0805239	0.0402619	−	0.999189i	$-0.487181\pi$
$11$	2.93774	0.0805239	0.0402619	+	0.999189i	$0.487181\pi$
$12$	0	0
$13$	−19.0623	−0.406686	−0.203343	−	0.979108i	$-0.565181\pi$
$13$	−19.0623	−0.406686	−0.203343	+	0.979108i	$0.565181\pi$
$14$	0	0
$15$	−15.0000	−0.258199
$16$	0	0
$17$	122.498	1.74766	0.873828	−	0.486236i	$-0.161630\pi$
$17$	122.498	1.74766	0.873828	+	0.486236i	$0.161630\pi$
$18$	0	0
$19$	−107.436	−1.29723	−0.648617	−	0.761115i	$-0.724652\pi$
$19$	−107.436	−1.29723	−0.648617	+	0.761115i	$0.724652\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	−210.623	−1.90947	−0.954736	−	0.297455i	$-0.903862\pi$
$23$	−210.623	−1.90947	−0.954736	+	0.297455i	$0.903862\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	95.4942	0.611477	0.305738	−	0.952116i	$-0.401097\pi$
$29$	95.4942	0.611477	0.305738	+	0.952116i	$0.401097\pi$
$30$	0	0
$31$	94.3074	0.546391	0.273195	−	0.961959i	$-0.411919\pi$
$31$	94.3074	0.546391	0.273195	+	0.961959i	$0.411919\pi$
$32$	0	0
$33$	−8.81323	−0.0464905
$34$	0	0
$35$	35.0000	0.169031
$36$	0	0
$37$	97.1206	0.431528	0.215764	−	0.976446i	$-0.430776\pi$
$37$	97.1206	0.431528	0.215764	+	0.976446i	$0.430776\pi$
$38$	0	0
$39$	57.1868	0.234800
$40$	0	0
$41$	−491.113	−1.87071	−0.935353	−	0.353716i	$-0.884918\pi$
$41$	−491.113	−1.87071	−0.935353	+	0.353716i	$0.884918\pi$
$42$	0	0
$43$	43.0039	0.152512	0.0762562	−	0.997088i	$-0.475703\pi$
$43$	43.0039	0.152512	0.0762562	+	0.997088i	$0.475703\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	−473.494	−1.46949	−0.734747	−	0.678341i	$-0.762699\pi$
$47$	−473.494	−1.46949	−0.734747	+	0.678341i	$0.762699\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−367.494	−1.00901
$52$	0	0
$53$	−183.677	−0.476038	−0.238019	−	0.971261i	$-0.576498\pi$
$53$	−183.677	−0.476038	−0.238019	+	0.971261i	$0.576498\pi$
$54$	0	0
$55$	14.6887	0.0360114
$56$	0	0
$57$	322.307	0.748959
$58$	0	0
$59$	760.615	1.67837	0.839183	−	0.543849i	$-0.183034\pi$
$59$	760.615	1.67837	0.839183	+	0.543849i	$0.183034\pi$
$60$	0	0
$61$	−198.747	−0.417163	−0.208582	−	0.978005i	$-0.566885\pi$
$61$	−198.747	−0.417163	−0.208582	+	0.978005i	$0.566885\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	−95.3113	−0.181876
$66$	0	0
$67$	309.992	0.565247	0.282624	−	0.959231i	$-0.408795\pi$
$67$	309.992	0.565247	0.282624	+	0.959231i	$0.408795\pi$
$68$	0	0
$69$	631.868	1.10243
$70$	0	0
$71$	−665.693	−1.11272	−0.556360	−	0.830941i	$-0.687803\pi$
$71$	−665.693	−1.11272	−0.556360	+	0.830941i	$0.687803\pi$
$72$	0	0
$73$	621.288	0.996113	0.498057	−	0.867145i	$-0.334047\pi$
$73$	621.288	0.996113	0.498057	+	0.867145i	$0.334047\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	20.5642	0.0304352
$78$	0	0
$79$	24.7626	0.0352659	0.0176330	−	0.999845i	$-0.494387\pi$
$79$	24.7626	0.0352659	0.0176330	+	0.999845i	$0.494387\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	406.724	0.537876	0.268938	−	0.963157i	$-0.413327\pi$
$83$	406.724	0.537876	0.268938	+	0.963157i	$0.413327\pi$
$84$	0	0
$85$	612.490	0.781575
$86$	0	0
$87$	−286.483	−0.353036
$88$	0	0
$89$	261.751	0.311748	0.155874	−	0.987777i	$-0.450181\pi$
$89$	261.751	0.311748	0.155874	+	0.987777i	$0.450181\pi$
$90$	0	0
$91$	−133.436	−0.153713
$92$	0	0
$93$	−282.922	−0.315459
$94$	0	0
$95$	−537.179	−0.580141
$96$	0	0
$97$	−1004.77	−1.05175	−0.525873	−	0.850563i	$-0.676261\pi$
$97$	−1004.77	−1.05175	−0.525873	+	0.850563i	$0.676261\pi$
$98$	0	0
$99$	26.4397	0.0268413
$100$	0	0
$101$	−128.872	−0.126962	−0.0634812	−	0.997983i	$-0.520220\pi$
$101$	−128.872	−0.126962	−0.0634812	+	0.997983i	$0.520220\pi$
$102$	0	0
$103$	−806.008	−0.771051	−0.385526	−	0.922697i	$-0.625980\pi$
$103$	−806.008	−0.771051	−0.385526	+	0.922697i	$0.625980\pi$
$104$	0	0
$105$	−105.000	−0.0975900
$106$	0	0
$107$	769.712	0.695429	0.347714	−	0.937600i	$-0.386958\pi$
$107$	769.712	0.695429	0.347714	+	0.937600i	$0.386958\pi$
$108$	0	0
$109$	−780.856	−0.686169	−0.343085	−	0.939304i	$-0.611472\pi$
$109$	−780.856	−0.686169	−0.343085	+	0.939304i	$0.611472\pi$
$110$	0	0
$111$	−291.362	−0.249143
$112$	0	0
$113$	−1115.65	−0.928771	−0.464386	−	0.885633i	$-0.653725\pi$
$113$	−1115.65	−0.928771	−0.464386	+	0.885633i	$0.653725\pi$
$114$	0	0
$115$	−1053.11	−0.853942
$116$	0	0
$117$	−171.560	−0.135562
$118$	0	0
$119$	857.486	0.660552
$120$	0	0
$121$	−1322.37	−0.993516
$122$	0	0
$123$	1473.34	1.08005
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	1875.98	1.31076	0.655381	−	0.755299i	$-0.272508\pi$
$127$	1875.98	1.31076	0.655381	+	0.755299i	$0.272508\pi$
$128$	0	0
$129$	−129.012	−0.0880530
$130$	0	0
$131$	−364.203	−0.242905	−0.121452	−	0.992597i	$-0.538755\pi$
$131$	−364.203	−0.242905	−0.121452	+	0.992597i	$0.538755\pi$
$132$	0	0
$133$	−752.051	−0.490309
$134$	0	0
$135$	−135.000	−0.0860663
$136$	0	0
$137$	1603.13	0.999743	0.499872	−	0.866099i	$-0.333380\pi$
$137$	1603.13	0.999743	0.499872	+	0.866099i	$0.333380\pi$
$138$	0	0
$139$	−2431.12	−1.48349	−0.741746	−	0.670681i	$-0.766002\pi$
$139$	−2431.12	−1.48349	−0.741746	+	0.670681i	$0.766002\pi$
$140$	0	0
$141$	1420.48	0.848413
$142$	0	0
$143$	−56.0000	−0.0327479
$144$	0	0
$145$	477.471	0.273461
$146$	0	0
$147$	−147.000	−0.0824786
$148$	0	0
$149$	2341.57	1.28744	0.643722	−	0.765260i	$-0.277389\pi$
$149$	2341.57	1.28744	0.643722	+	0.765260i	$0.277389\pi$
$150$	0	0
$151$	2104.07	1.13395	0.566976	−	0.823734i	$-0.308113\pi$
$151$	2104.07	1.13395	0.566976	+	0.823734i	$0.308113\pi$
$152$	0	0
$153$	1102.48	0.582552
$154$	0	0
$155$	471.537	0.244353
$156$	0	0
$157$	−593.467	−0.301680	−0.150840	−	0.988558i	$-0.548198\pi$
$157$	−593.467	−0.301680	−0.150840	+	0.988558i	$0.548198\pi$
$158$	0	0
$159$	551.031	0.274840
$160$	0	0
$161$	−1474.36	−0.721712
$162$	0	0
$163$	−2178.71	−1.04693	−0.523465	−	0.852047i	$-0.675361\pi$
$163$	−2178.71	−1.04693	−0.523465	+	0.852047i	$0.675361\pi$
$164$	0	0
$165$	−44.0661	−0.0207912
$166$	0	0
$167$	−799.502	−0.370463	−0.185231	−	0.982695i	$-0.559303\pi$
$167$	−799.502	−0.370463	−0.185231	+	0.982695i	$0.559303\pi$
$168$	0	0
$169$	−1833.63	−0.834606
$170$	0	0
$171$	−966.922	−0.432412
$172$	0	0
$173$	−1444.36	−0.634754	−0.317377	−	0.948299i	$-0.602802\pi$
$173$	−1444.36	−0.634754	−0.317377	+	0.948299i	$0.602802\pi$
$174$	0	0
$175$	175.000	0.0755929
$176$	0	0
$177$	−2281.84	−0.969005
$178$	0	0
$179$	−3343.49	−1.39611	−0.698056	−	0.716043i	$-0.745952\pi$
$179$	−3343.49	−1.39611	−0.698056	+	0.716043i	$0.745952\pi$
$180$	0	0
$181$	2251.81	0.924729	0.462365	−	0.886690i	$-0.347001\pi$
$181$	2251.81	0.924729	0.462365	+	0.886690i	$0.347001\pi$
$182$	0	0
$183$	596.241	0.240849
$184$	0	0
$185$	485.603	0.192985
$186$	0	0
$187$	359.868	0.140728
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	1001.93	0.379565	0.189782	−	0.981826i	$-0.439222\pi$
$191$	1001.93	0.379565	0.189782	+	0.981826i	$0.439222\pi$
$192$	0	0
$193$	−4054.97	−1.51235	−0.756173	−	0.654372i	$-0.772933\pi$
$193$	−4054.97	−1.51235	−0.756173	+	0.654372i	$0.772933\pi$
$194$	0	0
$195$	285.934	0.105006
$196$	0	0
$197$	−5140.23	−1.85902	−0.929508	−	0.368802i	$-0.879768\pi$
$197$	−5140.23	−1.85902	−0.929508	+	0.368802i	$0.879768\pi$
$198$	0	0
$199$	−585.631	−0.208614	−0.104307	−	0.994545i	$-0.533263\pi$
$199$	−585.631	−0.208614	−0.104307	+	0.994545i	$0.533263\pi$
$200$	0	0
$201$	−929.977	−0.326346
$202$	0	0
$203$	668.459	0.231116
$204$	0	0
$205$	−2455.56	−0.836605
$206$	0	0
$207$	−1895.60	−0.636490
$208$	0	0
$209$	−315.619	−0.104458
$210$	0	0
$211$	1055.16	0.344266	0.172133	−	0.985074i	$-0.444934\pi$
$211$	1055.16	0.344266	0.172133	+	0.985074i	$0.444934\pi$
$212$	0	0
$213$	1997.08	0.642430
$214$	0	0
$215$	215.019	0.0682056
$216$	0	0
$217$	660.152	0.206516
$218$	0	0
$219$	−1863.86	−0.575106
$220$	0	0
$221$	−2335.09	−0.710747
$222$	0	0
$223$	−4675.85	−1.40412	−0.702059	−	0.712119i	$-0.747736\pi$
$223$	−4675.85	−1.40412	−0.702059	+	0.712119i	$0.747736\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−5443.11	−1.59151	−0.795754	−	0.605621i	$-0.792925\pi$
$227$	−5443.11	−1.59151	−0.795754	+	0.605621i	$0.792925\pi$
$228$	0	0
$229$	−536.303	−0.154759	−0.0773797	−	0.997002i	$-0.524655\pi$
$229$	−536.303	−0.154759	−0.0773797	+	0.997002i	$0.524655\pi$
$230$	0	0
$231$	−61.6926	−0.0175717
$232$	0	0
$233$	−183.490	−0.0515916	−0.0257958	−	0.999667i	$-0.508212\pi$
$233$	−183.490	−0.0515916	−0.0257958	+	0.999667i	$0.508212\pi$
$234$	0	0
$235$	−2367.47	−0.657178
$236$	0	0
$237$	−74.2878	−0.0203608
$238$	0	0
$239$	−643.218	−0.174085	−0.0870425	−	0.996205i	$-0.527742\pi$
$239$	−643.218	−0.174085	−0.0870425	+	0.996205i	$0.527742\pi$
$240$	0	0
$241$	−5755.61	−1.53839	−0.769194	−	0.639015i	$-0.779342\pi$
$241$	−5755.61	−1.53839	−0.769194	+	0.639015i	$0.779342\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	245.000	0.0638877
$246$	0	0
$247$	2047.97	0.527567
$248$	0	0
$249$	−1220.17	−0.310543
$250$	0	0
$251$	5132.27	1.29062	0.645311	−	0.763920i	$-0.276728\pi$
$251$	5132.27	1.29062	0.645311	+	0.763920i	$0.276728\pi$
$252$	0	0
$253$	−618.755	−0.153758
$254$	0	0
$255$	−1837.47	−0.451243
$256$	0	0
$257$	5041.74	1.22372	0.611859	−	0.790967i	$-0.290422\pi$
$257$	5041.74	1.22372	0.611859	+	0.790967i	$0.290422\pi$
$258$	0	0
$259$	679.844	0.163102
$260$	0	0
$261$	859.448	0.203826
$262$	0	0
$263$	−7577.00	−1.77649	−0.888246	−	0.459367i	$-0.848076\pi$
$263$	−7577.00	−1.77649	−0.888246	+	0.459367i	$0.848076\pi$
$264$	0	0
$265$	−918.385	−0.212890
$266$	0	0
$267$	−785.253	−0.179988
$268$	0	0
$269$	1023.10	0.231893	0.115947	−	0.993255i	$-0.463010\pi$
$269$	1023.10	0.231893	0.115947	+	0.993255i	$0.463010\pi$
$270$	0	0
$271$	2251.98	0.504790	0.252395	−	0.967624i	$-0.418782\pi$
$271$	2251.98	0.504790	0.252395	+	0.967624i	$0.418782\pi$
$272$	0	0
$273$	400.307	0.0887462
$274$	0	0
$275$	73.4436	0.0161048
$276$	0	0
$277$	−8630.72	−1.87209	−0.936047	−	0.351875i	$-0.885544\pi$
$277$	−8630.72	−1.87209	−0.936047	+	0.351875i	$0.885544\pi$
$278$	0	0
$279$	848.767	0.182130
$280$	0	0
$281$	−7521.62	−1.59680	−0.798402	−	0.602124i	$-0.794321\pi$
$281$	−7521.62	−1.59680	−0.798402	+	0.602124i	$0.794321\pi$
$282$	0	0
$283$	−14.8169	−0.00311226	−0.00155613	−	0.999999i	$-0.500495\pi$
$283$	−14.8169	−0.00311226	−0.00155613	+	0.999999i	$0.500495\pi$
$284$	0	0
$285$	1611.54	0.334945
$286$	0	0
$287$	−3437.79	−0.707060
$288$	0	0
$289$	10092.8	2.05430
$290$	0	0
$291$	3014.32	0.607226
$292$	0	0
$293$	6913.39	1.37844	0.689222	−	0.724550i	$-0.257952\pi$
$293$	6913.39	1.37844	0.689222	+	0.724550i	$0.257952\pi$
$294$	0	0
$295$	3803.07	0.750588
$296$	0	0
$297$	−79.3190	−0.0154968
$298$	0	0
$299$	4014.94	0.776555
$300$	0	0
$301$	301.027	0.0576442
$302$	0	0
$303$	386.615	0.0733018
$304$	0	0
$305$	−993.735	−0.186561
$306$	0	0
$307$	7644.12	1.42108	0.710542	−	0.703655i	$-0.248450\pi$
$307$	7644.12	1.42108	0.710542	+	0.703655i	$0.248450\pi$
$308$	0	0
$309$	2418.02	0.445167
$310$	0	0
$311$	−7593.99	−1.38462	−0.692308	−	0.721602i	$-0.743406\pi$
$311$	−7593.99	−1.38462	−0.692308	+	0.721602i	$0.743406\pi$
$312$	0	0
$313$	−9127.84	−1.64836	−0.824179	−	0.566329i	$-0.808363\pi$
$313$	−9127.84	−1.64836	−0.824179	+	0.566329i	$0.808363\pi$
$314$	0	0
$315$	315.000	0.0563436
$316$	0	0
$317$	−4929.81	−0.873456	−0.436728	−	0.899593i	$-0.643863\pi$
$317$	−4929.81	−0.873456	−0.436728	+	0.899593i	$0.643863\pi$
$318$	0	0
$319$	280.537	0.0492385
$320$	0	0
$321$	−2309.14	−0.401506
$322$	0	0
$323$	−13160.7	−2.26712
$324$	0	0
$325$	−476.556	−0.0813372
$326$	0	0
$327$	2342.57	0.396160
$328$	0	0
$329$	−3314.46	−0.555417
$330$	0	0
$331$	−1221.67	−0.202867	−0.101433	−	0.994842i	$-0.532343\pi$
$331$	−1221.67	−0.202867	−0.101433	+	0.994842i	$0.532343\pi$
$332$	0	0
$333$	874.086	0.143843
$334$	0	0
$335$	1549.96	0.252786
$336$	0	0
$337$	−8744.83	−1.41354	−0.706768	−	0.707446i	$-0.749847\pi$
$337$	−8744.83	−1.41354	−0.706768	+	0.707446i	$0.749847\pi$
$338$	0	0
$339$	3346.94	0.536226
$340$	0	0
$341$	277.051	0.0439975
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	3159.34	0.493023
$346$	0	0
$347$	−4589.56	−0.710031	−0.355015	−	0.934860i	$-0.615524\pi$
$347$	−4589.56	−0.710031	−0.355015	+	0.934860i	$0.615524\pi$
$348$	0	0
$349$	−3989.89	−0.611960	−0.305980	−	0.952038i	$-0.598984\pi$
$349$	−3989.89	−0.611960	−0.305980	+	0.952038i	$0.598984\pi$
$350$	0	0
$351$	514.681	0.0782668
$352$	0	0
$353$	2416.35	0.364333	0.182166	−	0.983268i	$-0.441689\pi$
$353$	2416.35	0.364333	0.182166	+	0.983268i	$0.441689\pi$
$354$	0	0
$355$	−3328.46	−0.497624
$356$	0	0
$357$	−2572.46	−0.381370
$358$	0	0
$359$	2756.24	0.405206	0.202603	−	0.979261i	$-0.435060\pi$
$359$	2756.24	0.405206	0.202603	+	0.979261i	$0.435060\pi$
$360$	0	0
$361$	4683.45	0.682818
$362$	0	0
$363$	3967.11	0.573607
$364$	0	0
$365$	3106.44	0.445475
$366$	0	0
$367$	−11112.8	−1.58061	−0.790307	−	0.612711i	$-0.790079\pi$
$367$	−11112.8	−1.58061	−0.790307	+	0.612711i	$0.790079\pi$
$368$	0	0
$369$	−4420.02	−0.623569
$370$	0	0
$371$	−1285.74	−0.179925
$372$	0	0
$373$	6091.09	0.845535	0.422768	−	0.906238i	$-0.361059\pi$
$373$	6091.09	0.845535	0.422768	+	0.906238i	$0.361059\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	0	0
$377$	−1820.33	−0.248679
$378$	0	0
$379$	−3984.29	−0.539998	−0.269999	−	0.962861i	$-0.587023\pi$
$379$	−3984.29	−0.539998	−0.269999	+	0.962861i	$0.587023\pi$
$380$	0	0
$381$	−5627.95	−0.756768
$382$	0	0
$383$	−318.475	−0.0424890	−0.0212445	−	0.999774i	$-0.506763\pi$
$383$	−318.475	−0.0424890	−0.0212445	+	0.999774i	$0.506763\pi$
$384$	0	0
$385$	102.821	0.0136110
$386$	0	0
$387$	387.035	0.0508374
$388$	0	0
$389$	3885.46	0.506429	0.253214	−	0.967410i	$-0.418512\pi$
$389$	3885.46	0.506429	0.253214	+	0.967410i	$0.418512\pi$
$390$	0	0
$391$	−25800.9	−3.33710
$392$	0	0
$393$	1092.61	0.140241
$394$	0	0
$395$	123.813	0.0157714
$396$	0	0
$397$	4806.04	0.607578	0.303789	−	0.952739i	$-0.401748\pi$
$397$	4806.04	0.607578	0.303789	+	0.952739i	$0.401748\pi$
$398$	0	0
$399$	2256.15	0.283080
$400$	0	0
$401$	3618.59	0.450633	0.225316	−	0.974286i	$-0.427658\pi$
$401$	3618.59	0.450633	0.225316	+	0.974286i	$0.427658\pi$
$402$	0	0
$403$	−1797.71	−0.222209
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	285.315	0.0347483
$408$	0	0
$409$	−2109.05	−0.254978	−0.127489	−	0.991840i	$-0.540692\pi$
$409$	−2109.05	−0.254978	−0.127489	+	0.991840i	$0.540692\pi$
$410$	0	0
$411$	−4809.40	−0.577202
$412$	0	0
$413$	5324.30	0.634363
$414$	0	0
$415$	2033.62	0.240546
$416$	0	0
$417$	7293.37	0.856494
$418$	0	0
$419$	6905.91	0.805193	0.402597	−	0.915377i	$-0.368108\pi$
$419$	6905.91	0.805193	0.402597	+	0.915377i	$0.368108\pi$
$420$	0	0
$421$	−9647.54	−1.11685	−0.558423	−	0.829556i	$-0.688593\pi$
$421$	−9647.54	−1.11685	−0.558423	+	0.829556i	$0.688593\pi$
$422$	0	0
$423$	−4261.45	−0.489831
$424$	0	0
$425$	3062.45	0.349531
$426$	0	0
$427$	−1391.23	−0.157673
$428$	0	0
$429$	168.000	0.0189070
$430$	0	0
$431$	13002.7	1.45318	0.726589	−	0.687073i	$-0.241105\pi$
$431$	13002.7	1.45318	0.726589	+	0.687073i	$0.241105\pi$
$432$	0	0
$433$	7356.07	0.816420	0.408210	−	0.912888i	$-0.366153\pi$
$433$	7356.07	0.816420	0.408210	+	0.912888i	$0.366153\pi$
$434$	0	0
$435$	−1432.41	−0.157883
$436$	0	0
$437$	22628.4	2.47703
$438$	0	0
$439$	−6909.21	−0.751159	−0.375579	−	0.926790i	$-0.622556\pi$
$439$	−6909.21	−0.751159	−0.375579	+	0.926790i	$0.622556\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	14812.6	1.58864	0.794318	−	0.607502i	$-0.207828\pi$
$443$	14812.6	1.58864	0.794318	+	0.607502i	$0.207828\pi$
$444$	0	0
$445$	1308.75	0.139418
$446$	0	0
$447$	−7024.72	−0.743306
$448$	0	0
$449$	−10654.5	−1.11986	−0.559932	−	0.828538i	$-0.689173\pi$
$449$	−10654.5	−1.11986	−0.559932	+	0.828538i	$0.689173\pi$
$450$	0	0
$451$	−1442.76	−0.150636
$452$	0	0
$453$	−6312.21	−0.654688
$454$	0	0
$455$	−667.179	−0.0687425
$456$	0	0
$457$	−5855.16	−0.599328	−0.299664	−	0.954045i	$-0.596875\pi$
$457$	−5855.16	−0.599328	−0.299664	+	0.954045i	$0.596875\pi$
$458$	0	0
$459$	−3307.45	−0.336336
$460$	0	0
$461$	3204.74	0.323774	0.161887	−	0.986809i	$-0.448242\pi$
$461$	3204.74	0.323774	0.161887	+	0.986809i	$0.448242\pi$
$462$	0	0
$463$	−371.658	−0.0373054	−0.0186527	−	0.999826i	$-0.505938\pi$
$463$	−371.658	−0.0373054	−0.0186527	+	0.999826i	$0.505938\pi$
$464$	0	0
$465$	−1414.61	−0.141077
$466$	0	0
$467$	19752.3	1.95723	0.978614	−	0.205703i	$-0.0659482\pi$
$467$	19752.3	1.95723	0.978614	+	0.205703i	$0.0659482\pi$
$468$	0	0
$469$	2169.95	0.213643
$470$	0	0
$471$	1780.40	0.174175
$472$	0	0
$473$	126.334	0.0122809
$474$	0	0
$475$	−2685.90	−0.259447
$476$	0	0
$477$	−1653.09	−0.158679
$478$	0	0
$479$	20762.0	1.98046	0.990232	−	0.139433i	$-0.0445279\pi$
$479$	20762.0	1.98046	0.990232	+	0.139433i	$0.0445279\pi$
$480$	0	0
$481$	−1851.34	−0.175496
$482$	0	0
$483$	4423.07	0.416681
$484$	0	0
$485$	−5023.87	−0.470355
$486$	0	0
$487$	−17647.6	−1.64207	−0.821035	−	0.570878i	$-0.806603\pi$
$487$	−17647.6	−1.64207	−0.821035	+	0.570878i	$0.806603\pi$
$488$	0	0
$489$	6536.12	0.604445
$490$	0	0
$491$	5637.46	0.518157	0.259078	−	0.965856i	$-0.416581\pi$
$491$	5637.46	0.518157	0.259078	+	0.965856i	$0.416581\pi$
$492$	0	0
$493$	11697.9	1.06865
$494$	0	0
$495$	132.198	0.0120038
$496$	0	0
$497$	−4659.85	−0.420569
$498$	0	0
$499$	17474.1	1.56764	0.783818	−	0.620991i	$-0.213270\pi$
$499$	17474.1	1.56764	0.783818	+	0.620991i	$0.213270\pi$
$500$	0	0
$501$	2398.51	0.213887
$502$	0	0
$503$	−7444.81	−0.659936	−0.329968	−	0.943992i	$-0.607038\pi$
$503$	−7444.81	−0.659936	−0.329968	+	0.943992i	$0.607038\pi$
$504$	0	0
$505$	−644.358	−0.0567793
$506$	0	0
$507$	5500.89	0.481860
$508$	0	0
$509$	−3384.48	−0.294724	−0.147362	−	0.989083i	$-0.547078\pi$
$509$	−3384.48	−0.294724	−0.147362	+	0.989083i	$0.547078\pi$
$510$	0	0
$511$	4349.02	0.376495
$512$	0	0
$513$	2900.77	0.249653
$514$	0	0
$515$	−4030.04	−0.344825
$516$	0	0
$517$	−1391.00	−0.118329
$518$	0	0
$519$	4333.07	0.366476
$520$	0	0
$521$	2973.12	0.250009	0.125005	−	0.992156i	$-0.460105\pi$
$521$	2973.12	0.250009	0.125005	+	0.992156i	$0.460105\pi$
$522$	0	0
$523$	−2689.02	−0.224823	−0.112412	−	0.993662i	$-0.535858\pi$
$523$	−2689.02	−0.224823	−0.112412	+	0.993662i	$0.535858\pi$
$524$	0	0
$525$	−525.000	−0.0436436
$526$	0	0
$527$	11552.5	0.954903
$528$	0	0
$529$	32194.9	2.64608
$530$	0	0
$531$	6845.53	0.559455
$532$	0	0
$533$	9361.72	0.760790
$534$	0	0
$535$	3848.56	0.311005
$536$	0	0
$537$	10030.5	0.806046
$538$	0	0
$539$	143.949	0.0115034
$540$	0	0
$541$	−14429.5	−1.14671	−0.573356	−	0.819306i	$-0.694359\pi$
$541$	−14429.5	−1.14671	−0.573356	+	0.819306i	$0.694359\pi$
$542$	0	0
$543$	−6755.44	−0.533893
$544$	0	0
$545$	−3904.28	−0.306864
$546$	0	0
$547$	−13811.2	−1.07957	−0.539784	−	0.841804i	$-0.681494\pi$
$547$	−13811.2	−1.07957	−0.539784	+	0.841804i	$0.681494\pi$
$548$	0	0
$549$	−1788.72	−0.139054
$550$	0	0
$551$	−10259.5	−0.793229
$552$	0	0
$553$	173.338	0.0133293
$554$	0	0
$555$	−1456.81	−0.111420
$556$	0	0
$557$	−6033.26	−0.458954	−0.229477	−	0.973314i	$-0.573702\pi$
$557$	−6033.26	−0.458954	−0.229477	+	0.973314i	$0.573702\pi$
$558$	0	0
$559$	−819.751	−0.0620246
$560$	0	0
$561$	−1079.60	−0.0812493
$562$	0	0
$563$	6958.47	0.520896	0.260448	−	0.965488i	$-0.416130\pi$
$563$	6958.47	0.520896	0.260448	+	0.965488i	$0.416130\pi$
$564$	0	0
$565$	−5578.23	−0.415359
$566$	0	0
$567$	567.000	0.0419961
$568$	0	0
$569$	−13396.4	−0.987009	−0.493505	−	0.869743i	$-0.664284\pi$
$569$	−13396.4	−0.987009	−0.493505	+	0.869743i	$0.664284\pi$
$570$	0	0
$571$	8055.84	0.590414	0.295207	−	0.955433i	$-0.404611\pi$
$571$	8055.84	0.590414	0.295207	+	0.955433i	$0.404611\pi$
$572$	0	0
$573$	−3005.78	−0.219142
$574$	0	0
$575$	−5265.56	−0.381894
$576$	0	0
$577$	−21456.9	−1.54812	−0.774059	−	0.633114i	$-0.781777\pi$
$577$	−21456.9	−1.54812	−0.774059	+	0.633114i	$0.781777\pi$
$578$	0	0
$579$	12164.9	0.873153
$580$	0	0
$581$	2847.07	0.203298
$582$	0	0
$583$	−539.596	−0.0383324
$584$	0	0
$585$	−857.802	−0.0606252
$586$	0	0
$587$	−20156.3	−1.41728	−0.708638	−	0.705572i	$-0.750690\pi$
$587$	−20156.3	−1.41728	−0.708638	+	0.705572i	$0.750690\pi$
$588$	0	0
$589$	−10132.0	−0.708797
$590$	0	0
$591$	15420.7	1.07330
$592$	0	0
$593$	599.307	0.0415018	0.0207509	−	0.999785i	$-0.493394\pi$
$593$	599.307	0.0415018	0.0207509	+	0.999785i	$0.493394\pi$
$594$	0	0
$595$	4287.43	0.295408
$596$	0	0
$597$	1756.89	0.120444
$598$	0	0
$599$	5493.05	0.374691	0.187346	−	0.982294i	$-0.440012\pi$
$599$	5493.05	0.374691	0.187346	+	0.982294i	$0.440012\pi$
$600$	0	0
$601$	24292.8	1.64879	0.824396	−	0.566014i	$-0.191515\pi$
$601$	24292.8	1.64879	0.824396	+	0.566014i	$0.191515\pi$
$602$	0	0
$603$	2789.93	0.188416
$604$	0	0
$605$	−6611.85	−0.444314
$606$	0	0
$607$	−3029.50	−0.202576	−0.101288	−	0.994857i	$-0.532296\pi$
$607$	−3029.50	−0.202576	−0.101288	+	0.994857i	$0.532296\pi$
$608$	0	0
$609$	−2005.38	−0.133435
$610$	0	0
$611$	9025.87	0.597623
$612$	0	0
$613$	−19339.6	−1.27426	−0.637129	−	0.770757i	$-0.719878\pi$
$613$	−19339.6	−1.27426	−0.637129	+	0.770757i	$0.719878\pi$
$614$	0	0
$615$	7366.69	0.483014
$616$	0	0
$617$	−5743.91	−0.374783	−0.187391	−	0.982285i	$-0.560003\pi$
$617$	−5743.91	−0.374783	−0.187391	+	0.982285i	$0.560003\pi$
$618$	0	0
$619$	8243.35	0.535264	0.267632	−	0.963521i	$-0.413759\pi$
$619$	8243.35	0.535264	0.267632	+	0.963521i	$0.413759\pi$
$620$	0	0
$621$	5686.81	0.367478
$622$	0	0
$623$	1832.26	0.117830
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	946.856	0.0603091
$628$	0	0
$629$	11897.1	0.754162
$630$	0	0
$631$	4376.56	0.276114	0.138057	−	0.990424i	$-0.455914\pi$
$631$	4376.56	0.276114	0.138057	+	0.990424i	$0.455914\pi$
$632$	0	0
$633$	−3165.48	−0.198762
$634$	0	0
$635$	9379.92	0.586190
$636$	0	0
$637$	−934.051	−0.0580980
$638$	0	0
$639$	−5991.23	−0.370907
$640$	0	0
$641$	11836.6	0.729357	0.364678	−	0.931133i	$-0.381179\pi$
$641$	11836.6	0.729357	0.364678	+	0.931133i	$0.381179\pi$
$642$	0	0
$643$	−1448.21	−0.0888209	−0.0444104	−	0.999013i	$-0.514141\pi$
$643$	−1448.21	−0.0888209	−0.0444104	+	0.999013i	$0.514141\pi$
$644$	0	0
$645$	−645.058	−0.0393785
$646$	0	0
$647$	8732.95	0.530646	0.265323	−	0.964160i	$-0.414521\pi$
$647$	8732.95	0.530646	0.265323	+	0.964160i	$0.414521\pi$
$648$	0	0
$649$	2234.49	0.135149
$650$	0	0
$651$	−1980.46	−0.119232
$652$	0	0
$653$	−21978.4	−1.31712	−0.658562	−	0.752527i	$-0.728835\pi$
$653$	−21978.4	−1.31712	−0.658562	+	0.752527i	$0.728835\pi$
$654$	0	0
$655$	−1821.01	−0.108630
$656$	0	0
$657$	5591.59	0.332038
$658$	0	0
$659$	27761.7	1.64103	0.820516	−	0.571623i	$-0.193686\pi$
$659$	27761.7	1.64103	0.820516	+	0.571623i	$0.193686\pi$
$660$	0	0
$661$	−8573.72	−0.504507	−0.252254	−	0.967661i	$-0.581172\pi$
$661$	−8573.72	−0.504507	−0.252254	+	0.967661i	$0.581172\pi$
$662$	0	0
$663$	7005.27	0.410350
$664$	0	0
$665$	−3760.25	−0.219273
$666$	0	0
$667$	−20113.2	−1.16760
$668$	0	0
$669$	14027.6	0.810668
$670$	0	0
$671$	−583.868	−0.0335916
$672$	0	0
$673$	27159.2	1.55559	0.777795	−	0.628518i	$-0.216338\pi$
$673$	27159.2	1.55559	0.777795	+	0.628518i	$0.216338\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	−1392.30	−0.0790404	−0.0395202	−	0.999219i	$-0.512583\pi$
$677$	−1392.30	−0.0790404	−0.0395202	+	0.999219i	$0.512583\pi$
$678$	0	0
$679$	−7033.42	−0.397523
$680$	0	0
$681$	16329.3	0.918857
$682$	0	0
$683$	8675.09	0.486007	0.243004	−	0.970025i	$-0.421867\pi$
$683$	8675.09	0.486007	0.243004	+	0.970025i	$0.421867\pi$
$684$	0	0
$685$	8015.66	0.447099
$686$	0	0
$687$	1608.91	0.0893504
$688$	0	0
$689$	3501.30	0.193598
$690$	0	0
$691$	21426.0	1.17957	0.589785	−	0.807561i	$-0.299213\pi$
$691$	21426.0	1.17957	0.589785	+	0.807561i	$0.299213\pi$
$692$	0	0
$693$	185.078	0.0101451
$694$	0	0
$695$	−12155.6	−0.663438
$696$	0	0
$697$	−60160.4	−3.26935
$698$	0	0
$699$	550.470	0.0297864
$700$	0	0
$701$	24840.5	1.33839	0.669197	−	0.743085i	$-0.266638\pi$
$701$	24840.5	1.33839	0.669197	+	0.743085i	$0.266638\pi$
$702$	0	0
$703$	−10434.2	−0.559793
$704$	0	0
$705$	7102.41	0.379422
$706$	0	0
$707$	−902.101	−0.0479873
$708$	0	0
$709$	12525.0	0.663450	0.331725	−	0.943376i	$-0.392369\pi$
$709$	12525.0	0.663450	0.331725	+	0.943376i	$0.392369\pi$
$710$	0	0
$711$	222.863	0.0117553
$712$	0	0
$713$	−19863.3	−1.04332
$714$	0	0
$715$	−280.000	−0.0146453
$716$	0	0
$717$	1929.65	0.100508
$718$	0	0
$719$	−28085.0	−1.45674	−0.728369	−	0.685185i	$-0.759721\pi$
$719$	−28085.0	−1.45674	−0.728369	+	0.685185i	$0.759721\pi$
$720$	0	0
$721$	−5642.05	−0.291430
$722$	0	0
$723$	17266.8	0.888189
$724$	0	0
$725$	2387.35	0.122295
$726$	0	0
$727$	14326.2	0.730851	0.365426	−	0.930841i	$-0.380923\pi$
$727$	14326.2	0.730851	0.365426	+	0.930841i	$0.380923\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	5267.89	0.266539
$732$	0	0
$733$	6727.85	0.339016	0.169508	−	0.985529i	$-0.445782\pi$
$733$	6727.85	0.339016	0.169508	+	0.985529i	$0.445782\pi$
$734$	0	0
$735$	−735.000	−0.0368856
$736$	0	0
$737$	910.677	0.0455159
$738$	0	0
$739$	3418.51	0.170165	0.0850826	−	0.996374i	$-0.472885\pi$
$739$	3418.51	0.170165	0.0850826	+	0.996374i	$0.472885\pi$
$740$	0	0
$741$	−6143.91	−0.304591
$742$	0	0
$743$	−8095.50	−0.399724	−0.199862	−	0.979824i	$-0.564049\pi$
$743$	−8095.50	−0.399724	−0.199862	+	0.979824i	$0.564049\pi$
$744$	0	0
$745$	11707.9	0.575762
$746$	0	0
$747$	3660.51	0.179292
$748$	0	0
$749$	5387.99	0.262847
$750$	0	0
$751$	−13446.8	−0.653371	−0.326686	−	0.945133i	$-0.605932\pi$
$751$	−13446.8	−0.653371	−0.326686	+	0.945133i	$0.605932\pi$
$752$	0	0
$753$	−15396.8	−0.745141
$754$	0	0
$755$	10520.4	0.507119
$756$	0	0
$757$	−2593.24	−0.124508	−0.0622541	−	0.998060i	$-0.519829\pi$
$757$	−2593.24	−0.124508	−0.0622541	+	0.998060i	$0.519829\pi$
$758$	0	0
$759$	1856.26	0.0887722
$760$	0	0
$761$	27079.4	1.28992	0.644959	−	0.764217i	$-0.276875\pi$
$761$	27079.4	1.28992	0.644959	+	0.764217i	$0.276875\pi$
$762$	0	0
$763$	−5465.99	−0.259348
$764$	0	0
$765$	5512.41	0.260525
$766$	0	0
$767$	−14499.0	−0.682568
$768$	0	0
$769$	2138.72	0.100292	0.0501458	−	0.998742i	$-0.484031\pi$
$769$	2138.72	0.100292	0.0501458	+	0.998742i	$0.484031\pi$
$770$	0	0
$771$	−15125.2	−0.706513
$772$	0	0
$773$	25864.0	1.20345	0.601724	−	0.798704i	$-0.294481\pi$
$773$	25864.0	1.20345	0.601724	+	0.798704i	$0.294481\pi$
$774$	0	0
$775$	2357.69	0.109278
$776$	0	0
$777$	−2039.53	−0.0941671
$778$	0	0
$779$	52763.1	2.42675
$780$	0	0
$781$	−1955.63	−0.0896006
$782$	0	0
$783$	−2578.34	−0.117679
$784$	0	0
$785$	−2967.33	−0.134916
$786$	0	0
$787$	32371.3	1.46621	0.733107	−	0.680113i	$-0.238069\pi$
$787$	32371.3	1.46621	0.733107	+	0.680113i	$0.238069\pi$
$788$	0	0
$789$	22731.0	1.02566
$790$	0	0
$791$	−7809.52	−0.351043
$792$	0	0
$793$	3788.57	0.169654
$794$	0	0
$795$	2755.16	0.122912
$796$	0	0
$797$	2024.33	0.0899691	0.0449845	−	0.998988i	$-0.485676\pi$
$797$	2024.33	0.0899691	0.0449845	+	0.998988i	$0.485676\pi$
$798$	0	0
$799$	−58002.1	−2.56817
$800$	0	0
$801$	2355.76	0.103916
$802$	0	0
$803$	1825.18	0.0802109
$804$	0	0
$805$	−7371.79	−0.322760
$806$	0	0
$807$	−3069.29	−0.133884
$808$	0	0
$809$	12391.7	0.538526	0.269263	−	0.963067i	$-0.413220\pi$
$809$	12391.7	0.538526	0.269263	+	0.963067i	$0.413220\pi$
$810$	0	0
$811$	−14654.5	−0.634511	−0.317256	−	0.948340i	$-0.602761\pi$
$811$	−14654.5	−0.634511	−0.317256	+	0.948340i	$0.602761\pi$
$812$	0	0
$813$	−6755.94	−0.291441
$814$	0	0
$815$	−10893.5	−0.468201
$816$	0	0
$817$	−4620.16	−0.197844
$818$	0	0
$819$	−1200.92	−0.0512376
$820$	0	0
$821$	23887.9	1.01546	0.507731	−	0.861516i	$-0.330485\pi$
$821$	23887.9	1.01546	0.507731	+	0.861516i	$0.330485\pi$
$822$	0	0
$823$	4008.41	0.169774	0.0848871	−	0.996391i	$-0.472947\pi$
$823$	4008.41	0.169774	0.0848871	+	0.996391i	$0.472947\pi$
$824$	0	0
$825$	−220.331	−0.00929810
$826$	0	0
$827$	45110.4	1.89679	0.948394	−	0.317096i	$-0.102708\pi$
$827$	45110.4	1.89679	0.948394	+	0.317096i	$0.102708\pi$
$828$	0	0
$829$	16165.4	0.677260	0.338630	−	0.940920i	$-0.390036\pi$
$829$	16165.4	0.677260	0.338630	+	0.940920i	$0.390036\pi$
$830$	0	0
$831$	25892.2	1.08085
$832$	0	0
$833$	6002.41	0.249665
$834$	0	0
$835$	−3997.51	−0.165676
$836$	0	0
$837$	−2546.30	−0.105153
$838$	0	0
$839$	25244.4	1.03878	0.519388	−	0.854538i	$-0.326160\pi$
$839$	25244.4	1.03878	0.519388	+	0.854538i	$0.326160\pi$
$840$	0	0
$841$	−15269.9	−0.626096
$842$	0	0
$843$	22564.9	0.921916
$844$	0	0
$845$	−9168.15	−0.373247
$846$	0	0
$847$	−9256.59	−0.375514
$848$	0	0
$849$	44.4506	0.00179687
$850$	0	0
$851$	−20455.8	−0.823990
$852$	0	0
$853$	−30168.1	−1.21094	−0.605472	−	0.795867i	$-0.707016\pi$
$853$	−30168.1	−1.21094	−0.605472	+	0.795867i	$0.707016\pi$
$854$	0	0
$855$	−4834.61	−0.193380
$856$	0	0
$857$	−13393.6	−0.533857	−0.266929	−	0.963716i	$-0.586009\pi$
$857$	−13393.6	−0.533857	−0.266929	+	0.963716i	$0.586009\pi$
$858$	0	0
$859$	−19060.4	−0.757081	−0.378541	−	0.925585i	$-0.623574\pi$
$859$	−19060.4	−0.757081	−0.378541	+	0.925585i	$0.623574\pi$
$860$	0	0
$861$	10313.4	0.408222
$862$	0	0
$863$	9466.86	0.373413	0.186707	−	0.982416i	$-0.440219\pi$
$863$	9466.86	0.373413	0.186707	+	0.982416i	$0.440219\pi$
$864$	0	0
$865$	−7221.79	−0.283871
$866$	0	0
$867$	−30278.3	−1.18605
$868$	0	0
$869$	72.7461	0.00283975
$870$	0	0
$871$	−5909.15	−0.229878
$872$	0	0
$873$	−9042.97	−0.350582
$874$	0	0
$875$	875.000	0.0338062
$876$	0	0
$877$	37740.6	1.45315	0.726573	−	0.687090i	$-0.241112\pi$
$877$	37740.6	1.45315	0.726573	+	0.687090i	$0.241112\pi$
$878$	0	0
$879$	−20740.2	−0.795845
$880$	0	0
$881$	25991.5	0.993957	0.496979	−	0.867763i	$-0.334443\pi$
$881$	25991.5	0.993957	0.496979	+	0.867763i	$0.334443\pi$
$882$	0	0
$883$	−39420.3	−1.50238	−0.751189	−	0.660087i	$-0.770519\pi$
$883$	−39420.3	−1.50238	−0.751189	+	0.660087i	$0.770519\pi$
$884$	0	0
$885$	−11409.2	−0.433352
$886$	0	0
$887$	−46005.2	−1.74149	−0.870745	−	0.491735i	$-0.836363\pi$
$887$	−46005.2	−1.74149	−0.870745	+	0.491735i	$0.836363\pi$
$888$	0	0
$889$	13131.9	0.495421
$890$	0	0
$891$	237.957	0.00894710
$892$	0	0
$893$	50870.2	1.90628
$894$	0	0
$895$	−16717.5	−0.624361
$896$	0	0
$897$	−12044.8	−0.448345
$898$	0	0
$899$	9005.81	0.334105
$900$	0	0
$901$	−22500.1	−0.831950
$902$	0	0
$903$	−903.081	−0.0332809
$904$	0	0
$905$	11259.1	0.413551
$906$	0	0
$907$	2838.97	0.103932	0.0519661	−	0.998649i	$-0.483451\pi$
$907$	2838.97	0.103932	0.0519661	+	0.998649i	$0.483451\pi$
$908$	0	0
$909$	−1159.84	−0.0423208
$910$	0	0
$911$	−39890.9	−1.45076	−0.725382	−	0.688347i	$-0.758337\pi$
$911$	−39890.9	−1.45076	−0.725382	+	0.688347i	$0.758337\pi$
$912$	0	0
$913$	1194.85	0.0433119
$914$	0	0
$915$	2981.21	0.107711
$916$	0	0
$917$	−2549.42	−0.0918094
$918$	0	0
$919$	646.475	0.0232048	0.0116024	−	0.999933i	$-0.496307\pi$
$919$	646.475	0.0232048	0.0116024	+	0.999933i	$0.496307\pi$
$920$	0	0
$921$	−22932.4	−0.820463
$922$	0	0
$923$	12689.6	0.452528
$924$	0	0
$925$	2428.02	0.0863056
$926$	0	0
$927$	−7254.07	−0.257017
$928$	0	0
$929$	51188.2	1.80778	0.903892	−	0.427760i	$-0.140697\pi$
$929$	51188.2	1.80778	0.903892	+	0.427760i	$0.140697\pi$
$930$	0	0
$931$	−5264.35	−0.185319
$932$	0	0
$933$	22782.0	0.799409
$934$	0	0
$935$	1799.34	0.0629355
$936$	0	0
$937$	29786.1	1.03849	0.519247	−	0.854624i	$-0.326212\pi$
$937$	29786.1	1.03849	0.519247	+	0.854624i	$0.326212\pi$
$938$	0	0
$939$	27383.5	0.951680
$940$	0	0
$941$	−44817.4	−1.55261	−0.776304	−	0.630358i	$-0.782908\pi$
$941$	−44817.4	−1.55261	−0.776304	+	0.630358i	$0.782908\pi$
$942$	0	0
$943$	103439.	3.57206
$944$	0	0
$945$	−945.000	−0.0325300
$946$	0	0
$947$	−54697.1	−1.87689	−0.938446	−	0.345425i	$-0.887735\pi$
$947$	−54697.1	−1.87689	−0.938446	+	0.345425i	$0.887735\pi$
$948$	0	0
$949$	−11843.2	−0.405105
$950$	0	0
$951$	14789.4	0.504290
$952$	0	0
$953$	−7577.51	−0.257565	−0.128783	−	0.991673i	$-0.541107\pi$
$953$	−7577.51	−0.257565	−0.128783	+	0.991673i	$0.541107\pi$
$954$	0	0
$955$	5009.63	0.169746
$956$	0	0
$957$	−841.612	−0.0284278
$958$	0	0
$959$	11221.9	0.377868
$960$	0	0
$961$	−20897.1	−0.701457
$962$	0	0
$963$	6927.41	0.231810
$964$	0	0
$965$	−20274.8	−0.676342
$966$	0	0
$967$	−50779.0	−1.68867	−0.844334	−	0.535817i	$-0.820004\pi$
$967$	−50779.0	−1.68867	−0.844334	+	0.535817i	$0.820004\pi$
$968$	0	0
$969$	39482.0	1.30892
$970$	0	0
$971$	15313.2	0.506102	0.253051	−	0.967453i	$-0.418566\pi$
$971$	15313.2	0.506102	0.253051	+	0.967453i	$0.418566\pi$
$972$	0	0
$973$	−17017.9	−0.560707
$974$	0	0
$975$	1429.67	0.0469601
$976$	0	0
$977$	46620.4	1.52663	0.763316	−	0.646025i	$-0.223570\pi$
$977$	46620.4	1.52663	0.763316	+	0.646025i	$0.223570\pi$
$978$	0	0
$979$	768.957	0.0251031
$980$	0	0
$981$	−7027.70	−0.228723
$982$	0	0
$983$	2824.37	0.0916414	0.0458207	−	0.998950i	$-0.485410\pi$
$983$	2824.37	0.0916414	0.0458207	+	0.998950i	$0.485410\pi$
$984$	0	0
$985$	−25701.1	−0.831377
$986$	0	0
$987$	9943.38	0.320670
$988$	0	0
$989$	−9057.59	−0.291218
$990$	0	0
$991$	−16951.4	−0.543370	−0.271685	−	0.962386i	$-0.587581\pi$
$991$	−16951.4	−0.543370	−0.271685	+	0.962386i	$0.587581\pi$
$992$	0	0
$993$	3665.00	0.117125
$994$	0	0
$995$	−2928.15	−0.0932952
$996$	0	0
$997$	23847.8	0.757540	0.378770	−	0.925491i	$-0.376347\pi$
$997$	23847.8	0.757540	0.378770	+	0.925491i	$0.376347\pi$
$998$	0	0
$999$	−2622.26	−0.0830476

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.4.a.bg.1.1		2
4.3	odd	2		105.4.a.f.1.1	✓	2
12.11	even	2		315.4.a.i.1.2		2
20.3	even	4		525.4.d.h.274.3		4
20.7	even	4		525.4.d.h.274.2		4
20.19	odd	2		525.4.a.k.1.2		2
28.27	even	2		735.4.a.p.1.1		2
60.59	even	2		1575.4.a.w.1.1		2
84.83	odd	2		2205.4.a.z.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.f.1.1	✓	2	4.3	odd	2
315.4.a.i.1.2		2	12.11	even	2
525.4.a.k.1.2		2	20.19	odd	2
525.4.d.h.274.2		4	20.7	even	4
525.4.d.h.274.3		4	20.3	even	4
735.4.a.p.1.1		2	28.27	even	2
1575.4.a.w.1.1		2	60.59	even	2
1680.4.a.bg.1.1		2	1.1	even	1	trivial
2205.4.a.z.1.2		2	84.83	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1680.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +2.93774 q^{11} -19.0623 q^{13} -15.0000 q^{15} +122.498 q^{17} -107.436 q^{19} -21.0000 q^{21} -210.623 q^{23} +25.0000 q^{25} -27.0000 q^{27} +95.4942 q^{29} +94.3074 q^{31} -8.81323 q^{33} +35.0000 q^{35} +97.1206 q^{37} +57.1868 q^{39} -491.113 q^{41} +43.0039 q^{43} +45.0000 q^{45} -473.494 q^{47} +49.0000 q^{49} -367.494 q^{51} -183.677 q^{53} +14.6887 q^{55} +322.307 q^{57} +760.615 q^{59} -198.747 q^{61} +63.0000 q^{63} -95.3113 q^{65} +309.992 q^{67} +631.868 q^{69} -665.693 q^{71} +621.288 q^{73} -75.0000 q^{75} +20.5642 q^{77} +24.7626 q^{79} +81.0000 q^{81} +406.724 q^{83} +612.490 q^{85} -286.483 q^{87} +261.751 q^{89} -133.436 q^{91} -282.922 q^{93} -537.179 q^{95} -1004.77 q^{97} +26.4397 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 + 10 * q^5 + 14 * q^7 + 18 * q^9	\( 2 q - 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9} + 22 q^{11} - 22 q^{13} - 30 q^{15} + 116 q^{17} - 102 q^{19} - 42 q^{21} - 260 q^{23} + 50 q^{25} - 54 q^{27} - 196 q^{29} - 150 q^{31} - 66 q^{33} + 70 q^{35} - 96 q^{37} + 66 q^{39} - 176 q^{41} + 344 q^{43} + 90 q^{45} - 560 q^{47} + 98 q^{49} - 348 q^{51} + 326 q^{53} + 110 q^{55} + 306 q^{57} + 844 q^{59} - 204 q^{61} + 126 q^{63} - 110 q^{65} + 104 q^{67} + 780 q^{69} - 1670 q^{71} - 386 q^{73} - 150 q^{75} + 154 q^{77} + 888 q^{79} + 162 q^{81} - 928 q^{83} + 580 q^{85} + 588 q^{87} + 588 q^{89} - 154 q^{91} + 450 q^{93} - 510 q^{95} + 522 q^{97} + 198 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 10 * q^5 + 14 * q^7 + 18 * q^9 + 22 * q^11 - 22 * q^13 - 30 * q^15 + 116 * q^17 - 102 * q^19 - 42 * q^21 - 260 * q^23 + 50 * q^25 - 54 * q^27 - 196 * q^29 - 150 * q^31 - 66 * q^33 + 70 * q^35 - 96 * q^37 + 66 * q^39 - 176 * q^41 + 344 * q^43 + 90 * q^45 - 560 * q^47 + 98 * q^49 - 348 * q^51 + 326 * q^53 + 110 * q^55 + 306 * q^57 + 844 * q^59 - 204 * q^61 + 126 * q^63 - 110 * q^65 + 104 * q^67 + 780 * q^69 - 1670 * q^71 - 386 * q^73 - 150 * q^75 + 154 * q^77 + 888 * q^79 + 162 * q^81 - 928 * q^83 + 580 * q^85 + 588 * q^87 + 588 * q^89 - 154 * q^91 + 450 * q^93 - 510 * q^95 + 522 * q^97 + 198 * q^99

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