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Properties

Label 343.4.a.d
Level $343$
Weight $4$
Character orbit 343.a
Self dual yes
Analytic conductor $20.238$
Analytic rank $1$
Dimension $18$
CM no
Inner twists $1$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [343,4,Mod(1,343)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("343.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(343, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 343 = 7^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 343.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,2,-22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(20.2376551320\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 2 x^{17} - 101 x^{16} + 200 x^{15} + 4071 x^{14} - 7805 x^{13} - 84126 x^{12} + 151605 x^{11} + \cdots + 1016000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3}\cdot 7^{10} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{5} - 1) q^{3} + (\beta_{2} + 3) q^{4} + (\beta_{7} - 3) q^{5} + ( - \beta_{11} - \beta_{10} - \beta_{8} + \cdots - 2) q^{6} + ( - \beta_{15} + \beta_{14} - \beta_{12} + \cdots - 2) q^{8}+ \cdots + ( - 33 \beta_{17} + 12 \beta_{16} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 2 q^{2} - 22 q^{3} + 62 q^{4} - 62 q^{5} - 41 q^{6} - 18 q^{8} + 138 q^{9} - 78 q^{10} - 26 q^{11} - 245 q^{12} - 168 q^{13} + 118 q^{15} + 358 q^{16} - 568 q^{17} + 106 q^{18} - 188 q^{19} - 756 q^{20}+ \cdots + 512 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 2 x^{17} - 101 x^{16} + 200 x^{15} + 4071 x^{14} - 7805 x^{13} - 84126 x^{12} + 151605 x^{11} + \cdots + 1016000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 23\!\cdots\!51 \nu^{17} + \cdots - 10\!\cdots\!00 ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 33\!\cdots\!99 \nu^{17} + \cdots - 15\!\cdots\!00 ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12\!\cdots\!23 \nu^{17} + \cdots - 16\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 14\!\cdots\!03 \nu^{17} + \cdots - 47\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 14\!\cdots\!87 \nu^{17} + \cdots - 17\!\cdots\!80 ) / 12\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 71\!\cdots\!31 \nu^{17} + \cdots - 23\!\cdots\!00 ) / 32\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 14\!\cdots\!41 \nu^{17} + \cdots + 22\!\cdots\!00 ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 16\!\cdots\!37 \nu^{17} + \cdots + 10\!\cdots\!00 ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 17\!\cdots\!81 \nu^{17} + \cdots - 39\!\cdots\!00 ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 18\!\cdots\!43 \nu^{17} + \cdots + 75\!\cdots\!00 ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 21\!\cdots\!79 \nu^{17} + \cdots + 38\!\cdots\!00 ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 24\!\cdots\!52 \nu^{17} + \cdots + 31\!\cdots\!00 ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 29\!\cdots\!06 \nu^{17} + \cdots + 66\!\cdots\!00 ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 63\!\cdots\!23 \nu^{17} + \cdots - 48\!\cdots\!80 ) / 12\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 34\!\cdots\!17 \nu^{17} + \cdots + 62\!\cdots\!00 ) / 63\!\cdots\!00 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} + \beta_{14} - \beta_{12} + \beta_{11} - \beta_{5} + 20\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{16} + 2 \beta_{15} - 2 \beta_{14} + 7 \beta_{13} - \beta_{12} - \beta_{11} + \beta_{9} + \cdots + 218 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{17} - 31 \beta_{15} + 37 \beta_{14} + \beta_{13} - 36 \beta_{12} + 34 \beta_{11} - 4 \beta_{10} + \cdots - 98 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 13 \beta_{17} + 51 \beta_{16} + 71 \beta_{15} - 74 \beta_{14} + 288 \beta_{13} - 35 \beta_{12} + \cdots + 5041 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 110 \beta_{17} - 15 \beta_{16} - 807 \beta_{15} + 1114 \beta_{14} - 56 \beta_{13} - 1090 \beta_{12} + \cdots - 3650 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 693 \beta_{17} + 1831 \beta_{16} + 2021 \beta_{15} - 2326 \beta_{14} + 9238 \beta_{13} - 822 \beta_{12} + \cdots + 123684 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4336 \beta_{17} - 988 \beta_{16} - 20418 \beta_{15} + 31633 \beta_{14} - 5263 \beta_{13} - 31872 \beta_{12} + \cdots - 122533 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 26057 \beta_{17} + 58472 \beta_{16} + 54875 \beta_{15} - 71374 \beta_{14} + 274364 \beta_{13} + \cdots + 3131381 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 146731 \beta_{17} - 44914 \beta_{16} - 518141 \beta_{15} + 883545 \beta_{14} - 261087 \beta_{13} + \cdots - 3937902 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 850452 \beta_{17} + 1768107 \beta_{16} + 1484352 \beta_{15} - 2182207 \beta_{14} + 7908172 \beta_{13} + \cdots + 80979868 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 4553269 \beta_{17} - 1755615 \beta_{16} - 13297343 \beta_{15} + 24628290 \beta_{14} - 10478402 \beta_{13} + \cdots - 123975789 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 25752354 \beta_{17} + 51896802 \beta_{16} + 40510195 \beta_{15} - 66593358 \beta_{14} + \cdots + 2129654820 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 133842163 \beta_{17} - 63336605 \beta_{16} - 345892682 \beta_{15} + 688670155 \beta_{14} + \cdots - 3859505719 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 745104727 \beta_{17} + 1497043309 \beta_{16} + 1118955202 \beta_{15} - 2027204508 \beta_{14} + \cdots + 56820565294 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 3794694459 \beta_{17} - 2175670832 \beta_{16} - 9123330594 \beta_{15} + 19352137840 \beta_{14} + \cdots - 119281361965 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−5.44142
−4.90727
−4.63107
−3.07925
−2.75863
−1.94473
−1.61688
−1.12427
0.0692297
0.636124
1.03544
1.84535
2.34352
3.32527
3.57405
4.60928
5.02026
5.04500
−5.44142 −5.91546 21.6090 −19.1243 32.1885 0 −74.0523 7.99264 104.063
1.2 −4.90727 6.31818 16.0813 −1.18684 −31.0050 0 −39.6574 12.9194 5.82415
1.3 −4.63107 −3.52490 13.4468 11.2651 16.3241 0 −25.2245 −14.5751 −52.1695
1.4 −3.07925 4.29765 1.48178 −3.69765 −13.2335 0 20.0712 −8.53022 11.3860
1.5 −2.75863 −2.77741 −0.389940 12.4015 7.66185 0 23.1448 −19.2860 −34.2111
1.6 −1.94473 −4.36255 −4.21802 −15.0942 8.48399 0 23.7608 −7.96812 29.3541
1.7 −1.61688 7.70213 −5.38571 −6.75947 −12.4534 0 21.6431 32.3228 10.9292
1.8 −1.12427 −9.41375 −6.73601 −19.7412 10.5836 0 16.5673 61.6187 22.1944
1.9 0.0692297 −9.16909 −7.99521 1.11591 −0.634773 0 −1.10734 57.0722 0.0772543
1.10 0.636124 1.52022 −7.59535 18.5469 0.967051 0 −9.92057 −24.6889 11.7981
1.11 1.03544 1.36048 −6.92787 4.67490 1.40869 0 −15.4569 −25.1491 4.84057
1.12 1.84535 2.92703 −4.59470 6.48305 5.40139 0 −23.2416 −18.4325 11.9635
1.13 2.34352 8.38956 −2.50791 −14.9744 19.6611 0 −24.6255 43.3847 −35.0928
1.14 3.32527 3.00606 3.05744 −11.3122 9.99597 0 −16.4354 −17.9636 −37.6163
1.15 3.57405 −9.71974 4.77383 6.68093 −34.7388 0 −11.5305 67.4733 23.8780
1.16 4.60928 −4.88914 13.2455 −4.77784 −22.5354 0 24.1779 −3.09632 −22.0224
1.17 5.02026 −0.802131 17.2030 −20.0815 −4.02691 0 46.2015 −26.3566 −100.814
1.18 5.04500 −6.94713 17.4520 −6.41867 −35.0483 0 47.6854 21.2627 −32.3822
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \( -1 \)

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 343.4.a.d ✓ 18
7.b odd 2 1 343.4.a.e yes 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
343.4.a.d ✓ 18 1.a even 1 1 trivial
343.4.a.e yes 18 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(343))\):

\( T_{2}^{18} - 2 T_{2}^{17} - 101 T_{2}^{16} + 200 T_{2}^{15} + 4071 T_{2}^{14} - 7805 T_{2}^{13} + \cdots + 1016000 \) Copy content Toggle raw display
\( T_{3}^{18} + 22 T_{3}^{17} - 70 T_{3}^{16} - 4458 T_{3}^{15} - 12962 T_{3}^{14} + 335346 T_{3}^{13} + \cdots + 184391395349 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} - 2 T^{17} + \cdots + 1016000 \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots + 184391395349 \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots - 10\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{18} \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 31\!\cdots\!43 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 41\!\cdots\!67 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 64\!\cdots\!83 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 16\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 20\!\cdots\!75 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 54\!\cdots\!53 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 15\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 18\!\cdots\!25 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 49\!\cdots\!75 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 28\!\cdots\!75 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 31\!\cdots\!37 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots - 19\!\cdots\!75 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots - 26\!\cdots\!75 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots - 47\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 20\!\cdots\!75 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots - 10\!\cdots\!37 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots - 73\!\cdots\!69 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 14\!\cdots\!83 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 99\!\cdots\!73 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 23\!\cdots\!75 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots - 48\!\cdots\!25 \) Copy content Toggle raw display
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