Properties

Label 304.3.r.a
Level $304$
Weight $3$
Character orbit 304.r
Analytic conductor $8.283$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 304.r (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.28340003655\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{3} + \beta_{2} q^{5} + ( -2 + 4 \beta_{1} - 2 \beta_{3} ) q^{7} + ( -4 - 2 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{3} + \beta_{2} q^{5} + ( -2 + 4 \beta_{1} - 2 \beta_{3} ) q^{7} + ( -4 - 2 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{9} + ( 10 - 4 \beta_{1} + 2 \beta_{3} ) q^{11} + ( -5 + 6 \beta_{1} - 5 \beta_{2} ) q^{13} + ( -1 + \beta_{1} - \beta_{2} ) q^{15} + ( 2 \beta_{1} + 7 \beta_{2} + 2 \beta_{3} ) q^{17} + ( -5 + 6 \beta_{1} + 2 \beta_{2} + 9 \beta_{3} ) q^{19} + ( 12 - 8 \beta_{1} - 6 \beta_{2} + 8 \beta_{3} ) q^{21} + ( 5 - 9 \beta_{1} - 5 \beta_{2} + 18 \beta_{3} ) q^{23} + ( 24 - 24 \beta_{2} ) q^{25} + ( 9 - 18 \beta_{2} + 7 \beta_{3} ) q^{27} + ( 11 + 18 \beta_{1} + 11 \beta_{2} ) q^{29} + ( -18 + 36 \beta_{2} ) q^{31} + ( -28 + 16 \beta_{1} + 14 \beta_{2} - 16 \beta_{3} ) q^{33} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{35} + ( 2 - 4 \beta_{2} + 18 \beta_{3} ) q^{37} + ( 27 - 22 \beta_{1} + 11 \beta_{3} ) q^{39} + ( 6 - 42 \beta_{1} - 3 \beta_{2} + 42 \beta_{3} ) q^{41} + ( 23 \beta_{1} - 19 \beta_{2} + 23 \beta_{3} ) q^{43} + ( -4 - 4 \beta_{1} + 2 \beta_{3} ) q^{45} + ( 35 + 19 \beta_{1} - 35 \beta_{2} - 38 \beta_{3} ) q^{47} + ( -21 - 16 \beta_{1} + 8 \beta_{3} ) q^{49} + ( -3 + \beta_{1} - 3 \beta_{2} ) q^{51} + ( -7 + 48 \beta_{1} - 7 \beta_{2} ) q^{53} + ( -2 \beta_{1} + 10 \beta_{2} - 2 \beta_{3} ) q^{55} + ( 20 - 24 \beta_{1} + 11 \beta_{2} + 2 \beta_{3} ) q^{57} + ( -14 + 33 \beta_{1} + 7 \beta_{2} - 33 \beta_{3} ) q^{59} + ( 37 + 16 \beta_{1} - 37 \beta_{2} - 32 \beta_{3} ) q^{61} + ( -16 - 4 \beta_{1} + 16 \beta_{2} + 8 \beta_{3} ) q^{63} + ( 5 - 10 \beta_{2} + 6 \beta_{3} ) q^{65} + ( -17 - 63 \beta_{1} - 17 \beta_{2} ) q^{67} + ( -23 + 46 \beta_{2} - 32 \beta_{3} ) q^{69} + ( -102 + 9 \beta_{1} + 51 \beta_{2} - 9 \beta_{3} ) q^{71} + ( -32 \beta_{1} + 49 \beta_{2} - 32 \beta_{3} ) q^{73} + ( -24 + 48 \beta_{2} - 24 \beta_{3} ) q^{75} + ( -44 + 48 \beta_{1} - 24 \beta_{3} ) q^{77} + ( 42 + 21 \beta_{1} - 21 \beta_{2} - 21 \beta_{3} ) q^{79} + ( -34 \beta_{1} + 5 \beta_{2} - 34 \beta_{3} ) q^{81} + ( 16 + 12 \beta_{1} - 6 \beta_{3} ) q^{83} + ( -7 - 2 \beta_{1} + 7 \beta_{2} + 4 \beta_{3} ) q^{85} + ( 3 - 14 \beta_{1} + 7 \beta_{3} ) q^{87} + ( -1 - 6 \beta_{1} - \beta_{2} ) q^{89} + ( 34 - 42 \beta_{1} + 34 \beta_{2} ) q^{91} + ( 18 \beta_{1} - 54 \beta_{2} + 18 \beta_{3} ) q^{93} + ( -2 - 9 \beta_{1} - 3 \beta_{2} + 15 \beta_{3} ) q^{95} + ( -162 - 6 \beta_{1} + 81 \beta_{2} + 6 \beta_{3} ) q^{97} + ( -16 - 12 \beta_{1} + 16 \beta_{2} + 24 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 6q^{3} + 2q^{5} - 8q^{7} - 8q^{9} + O(q^{10}) \) \( 4q - 6q^{3} + 2q^{5} - 8q^{7} - 8q^{9} + 40q^{11} - 30q^{13} - 6q^{15} + 14q^{17} - 16q^{19} + 36q^{21} + 10q^{23} + 48q^{25} + 66q^{29} - 84q^{33} - 4q^{35} + 108q^{39} + 18q^{41} - 38q^{43} - 16q^{45} + 70q^{47} - 84q^{49} - 18q^{51} - 42q^{53} + 20q^{55} + 102q^{57} - 42q^{59} + 74q^{61} - 32q^{63} - 102q^{67} - 306q^{71} + 98q^{73} - 176q^{77} + 126q^{79} + 10q^{81} + 64q^{83} - 14q^{85} + 12q^{87} - 6q^{89} + 204q^{91} - 108q^{93} - 14q^{95} - 486q^{97} - 32q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(1 - \beta_{2}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i
1.22474 + 0.707107i
0 −2.72474 1.57313i 0 0.500000 0.866025i 0 −6.89898 0 0.449490 + 0.778539i 0
65.2 0 −0.275255 0.158919i 0 0.500000 0.866025i 0 2.89898 0 −4.44949 7.70674i 0
145.1 0 −2.72474 + 1.57313i 0 0.500000 + 0.866025i 0 −6.89898 0 0.449490 0.778539i 0
145.2 0 −0.275255 + 0.158919i 0 0.500000 + 0.866025i 0 2.89898 0 −4.44949 + 7.70674i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.3.r.a 4
4.b odd 2 1 38.3.d.a 4
12.b even 2 1 342.3.m.a 4
19.d odd 6 1 inner 304.3.r.a 4
76.f even 6 1 38.3.d.a 4
76.f even 6 1 722.3.b.b 4
76.g odd 6 1 722.3.b.b 4
228.n odd 6 1 342.3.m.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.3.d.a 4 4.b odd 2 1
38.3.d.a 4 76.f even 6 1
304.3.r.a 4 1.a even 1 1 trivial
304.3.r.a 4 19.d odd 6 1 inner
342.3.m.a 4 12.b even 2 1
342.3.m.a 4 228.n odd 6 1
722.3.b.b 4 76.f even 6 1
722.3.b.b 4 76.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 6 T_{3}^{3} + 13 T_{3}^{2} + 6 T_{3} + 1 \) acting on \(S_{3}^{\mathrm{new}}(304, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 1 + 6 T + 13 T^{2} + 6 T^{3} + T^{4} \)
$5$ \( ( 1 - T + T^{2} )^{2} \)
$7$ \( ( -20 + 4 T + T^{2} )^{2} \)
$11$ \( ( 76 - 20 T + T^{2} )^{2} \)
$13$ \( 9 + 90 T + 303 T^{2} + 30 T^{3} + T^{4} \)
$17$ \( 625 - 350 T + 171 T^{2} - 14 T^{3} + T^{4} \)
$19$ \( 130321 + 5776 T + 570 T^{2} + 16 T^{3} + T^{4} \)
$23$ \( 212521 + 4610 T + 561 T^{2} - 10 T^{3} + T^{4} \)
$29$ \( 81225 + 18810 T + 1167 T^{2} - 66 T^{3} + T^{4} \)
$31$ \( ( 972 + T^{2} )^{2} \)
$37$ \( 404496 + 1320 T^{2} + T^{4} \)
$41$ \( 12257001 + 63018 T - 3393 T^{2} - 18 T^{3} + T^{4} \)
$43$ \( 7912969 - 106894 T + 4257 T^{2} + 38 T^{3} + T^{4} \)
$47$ \( 885481 + 65870 T + 5841 T^{2} - 70 T^{3} + T^{4} \)
$53$ \( 19900521 - 187362 T - 3873 T^{2} + 42 T^{3} + T^{4} \)
$59$ \( 4124961 - 85302 T - 1443 T^{2} + 42 T^{3} + T^{4} \)
$61$ \( 27889 + 12358 T + 5643 T^{2} - 74 T^{3} + T^{4} \)
$67$ \( 49999041 - 721242 T - 3603 T^{2} + 102 T^{3} + T^{4} \)
$71$ \( 58384881 + 2338146 T + 38853 T^{2} + 306 T^{3} + T^{4} \)
$73$ \( 14010049 + 366814 T + 13347 T^{2} - 98 T^{3} + T^{4} \)
$79$ \( 194481 - 55566 T + 5733 T^{2} - 126 T^{3} + T^{4} \)
$83$ \( ( 40 - 32 T + T^{2} )^{2} \)
$89$ \( 4761 - 414 T - 57 T^{2} + 6 T^{3} + T^{4} \)
$97$ \( 384591321 + 9530946 T + 98343 T^{2} + 486 T^{3} + T^{4} \)
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