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Properties

Label 320.6.a.v
Level 320
Weight 6
Character orbit 320.a
Self dual yes
Analytic conductor 51.323
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 320.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.3228223402\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{70}) \)
Defining polynomial: \(x^{2} - 70\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 160)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{70}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 4 + \beta ) q^{3} -25 q^{5} + ( -52 + \beta ) q^{7} + ( 53 + 8 \beta ) q^{9} +O(q^{10})\) \( q + ( 4 + \beta ) q^{3} -25 q^{5} + ( -52 + \beta ) q^{7} + ( 53 + 8 \beta ) q^{9} + ( 160 - 10 \beta ) q^{11} + ( 50 - 40 \beta ) q^{13} + ( -100 - 25 \beta ) q^{15} + ( 290 + 40 \beta ) q^{17} + ( 360 - 40 \beta ) q^{19} + ( 72 - 48 \beta ) q^{21} + ( -844 - 97 \beta ) q^{23} + 625 q^{25} + ( 1480 - 158 \beta ) q^{27} + ( -54 + 80 \beta ) q^{29} + ( -4920 - 130 \beta ) q^{31} + ( -2160 + 120 \beta ) q^{33} + ( 1300 - 25 \beta ) q^{35} + ( -3270 + 560 \beta ) q^{37} + ( -11000 - 110 \beta ) q^{39} + ( -5310 - 808 \beta ) q^{41} + ( 12836 - 579 \beta ) q^{43} + ( -1325 - 200 \beta ) q^{45} + ( -14148 - 383 \beta ) q^{47} + ( -13823 - 104 \beta ) q^{49} + ( 12360 + 450 \beta ) q^{51} + ( -15670 - 1080 \beta ) q^{53} + ( -4000 + 250 \beta ) q^{55} + ( -9760 + 200 \beta ) q^{57} + ( 15400 - 20 \beta ) q^{59} + ( -12270 + 1184 \beta ) q^{61} + ( -516 - 363 \beta ) q^{63} + ( -1250 + 1000 \beta ) q^{65} + ( 17292 + 495 \beta ) q^{67} + ( -30536 - 1232 \beta ) q^{69} + ( 6200 + 2990 \beta ) q^{71} + ( -3590 + 3720 \beta ) q^{73} + ( 2500 + 625 \beta ) q^{75} + ( -11120 + 680 \beta ) q^{77} + ( -35920 + 220 \beta ) q^{79} + ( -51199 - 1096 \beta ) q^{81} + ( -15964 + 2817 \beta ) q^{83} + ( -7250 - 1000 \beta ) q^{85} + ( 22184 + 266 \beta ) q^{87} + ( -20374 - 3280 \beta ) q^{89} + ( -13800 + 2130 \beta ) q^{91} + ( -56080 - 5440 \beta ) q^{93} + ( -9000 + 1000 \beta ) q^{95} + ( -95070 - 4840 \beta ) q^{97} + ( -13920 + 750 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8q^{3} - 50q^{5} - 104q^{7} + 106q^{9} + O(q^{10}) \) \( 2q + 8q^{3} - 50q^{5} - 104q^{7} + 106q^{9} + 320q^{11} + 100q^{13} - 200q^{15} + 580q^{17} + 720q^{19} + 144q^{21} - 1688q^{23} + 1250q^{25} + 2960q^{27} - 108q^{29} - 9840q^{31} - 4320q^{33} + 2600q^{35} - 6540q^{37} - 22000q^{39} - 10620q^{41} + 25672q^{43} - 2650q^{45} - 28296q^{47} - 27646q^{49} + 24720q^{51} - 31340q^{53} - 8000q^{55} - 19520q^{57} + 30800q^{59} - 24540q^{61} - 1032q^{63} - 2500q^{65} + 34584q^{67} - 61072q^{69} + 12400q^{71} - 7180q^{73} + 5000q^{75} - 22240q^{77} - 71840q^{79} - 102398q^{81} - 31928q^{83} - 14500q^{85} + 44368q^{87} - 40748q^{89} - 27600q^{91} - 112160q^{93} - 18000q^{95} - 190140q^{97} - 27840q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−8.36660
8.36660
0 −12.7332 0 −25.0000 0 −68.7332 0 −80.8656 0
1.2 0 20.7332 0 −25.0000 0 −35.2668 0 186.866 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.6.a.v 2
4.b odd 2 1 320.6.a.r 2
8.b even 2 1 160.6.a.a ✓ 2
8.d odd 2 1 160.6.a.e yes 2
40.e odd 2 1 800.6.a.g 2
40.f even 2 1 800.6.a.l 2
40.i odd 4 2 800.6.c.f 4
40.k even 4 2 800.6.c.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.a.a ✓ 2 8.b even 2 1
160.6.a.e yes 2 8.d odd 2 1
320.6.a.r 2 4.b odd 2 1
320.6.a.v 2 1.a even 1 1 trivial
800.6.a.g 2 40.e odd 2 1
800.6.a.l 2 40.f even 2 1
800.6.c.f 4 40.i odd 4 2
800.6.c.g 4 40.k even 4 2

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 8 T_{3} - 264 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(320))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 8 T + 222 T^{2} - 1944 T^{3} + 59049 T^{4} \)
$5$ \( ( 1 + 25 T )^{2} \)
$7$ \( 1 + 104 T + 36038 T^{2} + 1747928 T^{3} + 282475249 T^{4} \)
$11$ \( 1 - 320 T + 319702 T^{2} - 51536320 T^{3} + 25937424601 T^{4} \)
$13$ \( 1 - 100 T + 297086 T^{2} - 37129300 T^{3} + 137858491849 T^{4} \)
$17$ \( 1 - 580 T + 2475814 T^{2} - 823517060 T^{3} + 2015993900449 T^{4} \)
$19$ \( 1 - 720 T + 4633798 T^{2} - 1782791280 T^{3} + 6131066257801 T^{4} \)
$23$ \( 1 + 1688 T + 10950502 T^{2} + 10864546984 T^{3} + 41426511213649 T^{4} \)
$29$ \( 1 + 108 T + 39233214 T^{2} + 2215204092 T^{3} + 420707233300201 T^{4} \)
$31$ \( 1 + 9840 T + 76732702 T^{2} + 281710845840 T^{3} + 819628286980801 T^{4} \)
$37$ \( 1 + 6540 T + 61572814 T^{2} + 453509478780 T^{3} + 4808584372417849 T^{4} \)
$41$ \( 1 + 10620 T + 77106582 T^{2} + 1230392854620 T^{3} + 13422659310152401 T^{4} \)
$43$ \( 1 - 25672 T + 364912302 T^{2} - 3774000748696 T^{3} + 21611482313284249 T^{4} \)
$47$ \( 1 + 28296 T + 617782998 T^{2} + 6489546318072 T^{3} + 52599132235830049 T^{4} \)
$53$ \( 1 + 31340 T + 755347886 T^{2} + 13106246750620 T^{3} + 174887470365513049 T^{4} \)
$59$ \( 1 - 30800 T + 1666896598 T^{2} - 22019668409200 T^{3} + 511116753300641401 T^{4} \)
$61$ \( 1 + 24540 T + 1447225822 T^{2} + 20726393226540 T^{3} + 713342911662882601 T^{4} \)
$67$ \( 1 - 34584 T + 2930656478 T^{2} - 46692726700488 T^{3} + 1822837804551761449 T^{4} \)
$71$ \( 1 - 12400 T + 1143670702 T^{2} - 22372443952400 T^{3} + 3255243551009881201 T^{4} \)
$73$ \( 1 + 7180 T + 284279286 T^{2} + 14884654037740 T^{3} + 4297625829703557649 T^{4} \)
$79$ \( 1 + 71840 T + 7430807198 T^{2} + 221055731704160 T^{3} + 9468276082626847201 T^{4} \)
$83$ \( 1 + 31928 T + 5910993662 T^{2} + 125765689649704 T^{3} + 15516041187205853449 T^{4} \)
$89$ \( 1 + 40748 T + 8570866774 T^{2} + 227539254427852 T^{3} + 31181719929966183601 T^{4} \)
$97$ \( 1 + 190140 T + 19653817414 T^{2} + 1632796876465980 T^{3} + 73742412689492826049 T^{4} \)
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