Properties

Label 405.4.a.n
Level $405$
Weight $4$
Character orbit 405.a
Self dual yes
Analytic conductor $23.896$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 405.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.8957735523\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \( x^{7} - 2x^{6} - 44x^{5} + 74x^{4} + 479x^{3} - 460x^{2} - 1200x + 288 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 45)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 5) q^{4} + 5 q^{5} + ( - \beta_{5} + 3) q^{7} + (\beta_{3} + 7 \beta_1 + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 5) q^{4} + 5 q^{5} + ( - \beta_{5} + 3) q^{7} + (\beta_{3} + 7 \beta_1 + 1) q^{8} + 5 \beta_1 q^{10} + (\beta_{4} + 3 \beta_1 + 2) q^{11} + (\beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} - 4 \beta_1 + 15) q^{13} + (\beta_{6} - \beta_{5} - 2 \beta_{4} - 2 \beta_{2} + 7 \beta_1 - 4) q^{14} + ( - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - \beta_{3} + 7 \beta_{2} - 6 \beta_1 + 46) q^{16} + ( - 2 \beta_{6} + 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 22) q^{17} + ( - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 40) q^{19} + (5 \beta_{2} + 25) q^{20} + (\beta_{6} + 3 \beta_{5} + 2 \beta_{4} + \beta_{3} + 5 \beta_{2} - \beta_1 + 44) q^{22} + (\beta_{5} - 2 \beta_{3} + 2 \beta_{2} + 6 \beta_1 + 11) q^{23} + 25 q^{25} + ( - 4 \beta_{5} - 2 \beta_{4} - \beta_{3} - 10 \beta_{2} + 32 \beta_1 - 59) q^{26} + ( - \beta_{6} + \beta_{5} - 4 \beta_{4} - 5 \beta_{3} - \beta_{2} - 14 \beta_1 + 52) q^{28} + (3 \beta_{6} - \beta_{5} - 5 \beta_{2} + 15 \beta_1 - 46) q^{29} + (\beta_{6} - \beta_{5} - \beta_{4} - 4 \beta_{3} - 9 \beta_{2} - 8 \beta_1 + 37) q^{31} + (2 \beta_{6} + 6 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + 53 \beta_1 - 58) q^{32} + (3 \beta_{6} + \beta_{5} - 2 \beta_{4} + 7 \beta_{3} - 7 \beta_{2} + 45 \beta_1 + 10) q^{34} + ( - 5 \beta_{5} + 15) q^{35} + ( - \beta_{6} - \beta_{5} + 5 \beta_{4} - 11 \beta_{2} - 14 \beta_1 + 61) q^{37} + ( - 5 \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} + 17 \beta_{2} + 49 \beta_1 + 26) q^{38} + (5 \beta_{3} + 35 \beta_1 + 5) q^{40} + ( - \beta_{6} - 9 \beta_{5} + \beta_{4} + 2 \beta_{3} - 5 \beta_{2} + 4 \beta_1 - 8) q^{41} + (5 \beta_{4} - 2 \beta_{3} - 12 \beta_{2} - 29 \beta_1 + 82) q^{43} + ( - 3 \beta_{6} + 11 \beta_{5} + 6 \beta_{4} + 5 \beta_{3} + 15 \beta_{2} + \cdots - 6) q^{44}+ \cdots + (5 \beta_{6} - 9 \beta_{5} - 34 \beta_{4} - 11 \beta_{3} + 25 \beta_{2} + \cdots + 188) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 2 q^{2} + 36 q^{4} + 35 q^{5} + 22 q^{7} + 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 2 q^{2} + 36 q^{4} + 35 q^{5} + 22 q^{7} + 18 q^{8} + 10 q^{10} + 23 q^{11} + 96 q^{13} - 21 q^{14} + 324 q^{16} + 161 q^{17} + 279 q^{19} + 180 q^{20} + 311 q^{22} + 96 q^{23} + 175 q^{25} - 358 q^{26} + 337 q^{28} - 296 q^{29} + 244 q^{31} - 314 q^{32} + 125 q^{34} + 110 q^{35} + 404 q^{37} + 305 q^{38} + 90 q^{40} - 47 q^{41} + 525 q^{43} + 55 q^{44} + 717 q^{46} + 164 q^{47} + 1225 q^{49} + 50 q^{50} + 1682 q^{52} + 506 q^{53} + 115 q^{55} - 981 q^{56} + 1183 q^{58} - 85 q^{59} + 828 q^{61} - 786 q^{62} + 2236 q^{64} + 480 q^{65} + 1093 q^{67} + 2473 q^{68} - 105 q^{70} + 328 q^{71} + 2085 q^{73} - 1316 q^{74} + 2789 q^{76} + 24 q^{77} + 2110 q^{79} + 1620 q^{80} - 62 q^{82} + 1290 q^{83} + 805 q^{85} - 2569 q^{86} + 2271 q^{88} - 3048 q^{89} + 3338 q^{91} + 2763 q^{92} - 517 q^{94} + 1395 q^{95} + 1787 q^{97} + 1279 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 2x^{6} - 44x^{5} + 74x^{4} + 479x^{3} - 460x^{2} - 1200x + 288 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 23\nu - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - \nu^{5} - 39\nu^{4} + 35\nu^{3} + 310\nu^{2} - 114\nu - 240 ) / 12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 4\nu^{5} + 42\nu^{4} - 140\nu^{3} - 397\nu^{2} + 756\nu + 672 ) / 24 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 2\nu^{5} - 48\nu^{4} - 82\nu^{3} + 595\nu^{2} + 732\nu - 1104 ) / 24 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 23\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{6} + 2\beta_{5} + 2\beta_{4} - \beta_{3} + 31\beta_{2} - 6\beta _1 + 294 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{6} + 6\beta_{5} + 2\beta_{4} + 36\beta_{3} - 2\beta_{2} + 597\beta _1 - 26 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -76\beta_{6} + 84\beta_{5} + 92\beta_{4} - 38\beta_{3} + 897\beta_{2} - 328\beta _1 + 7615 \) Copy content Toggle raw display

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
−5.38503
−3.04174
−1.57021
0.225250
2.19444
4.26178
5.31551
−5.38503 0 20.9986 5.00000 0 12.5702 −69.9976 0 −26.9252
1.2 −3.04174 0 1.25219 5.00000 0 −13.7122 20.5251 0 −15.2087
1.3 −1.57021 0 −5.53444 5.00000 0 34.2398 21.2519 0 −7.85104
1.4 0.225250 0 −7.94926 5.00000 0 −31.1940 −3.59257 0 1.12625
1.5 2.19444 0 −3.18442 5.00000 0 2.76605 −24.5436 0 10.9722
1.6 4.26178 0 10.1628 5.00000 0 30.7639 9.21718 0 21.3089
1.7 5.31551 0 20.2546 5.00000 0 −13.4337 65.1396 0 26.5775
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)

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 405.4.a.n 7
3.b odd 2 1 405.4.a.m 7
5.b even 2 1 2025.4.a.ba 7
9.c even 3 2 135.4.e.c 14
9.d odd 6 2 45.4.e.c 14
15.d odd 2 1 2025.4.a.bb 7
45.h odd 6 2 225.4.e.d 14
45.l even 12 4 225.4.k.d 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.e.c 14 9.d odd 6 2
135.4.e.c 14 9.c even 3 2
225.4.e.d 14 45.h odd 6 2
225.4.k.d 28 45.l even 12 4
405.4.a.m 7 3.b odd 2 1
405.4.a.n 7 1.a even 1 1 trivial
2025.4.a.ba 7 5.b even 2 1
2025.4.a.bb 7 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} - 2T_{2}^{6} - 44T_{2}^{5} + 74T_{2}^{4} + 479T_{2}^{3} - 460T_{2}^{2} - 1200T_{2} + 288 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(405))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} - 2 T^{6} - 44 T^{5} + 74 T^{4} + \cdots + 288 \) Copy content Toggle raw display
$3$ \( T^{7} \) Copy content Toggle raw display
$5$ \( (T - 5)^{7} \) Copy content Toggle raw display
$7$ \( T^{7} - 22 T^{6} + \cdots + 210450744 \) Copy content Toggle raw display
$11$ \( T^{7} - 23 T^{6} + \cdots - 20019444768 \) Copy content Toggle raw display
$13$ \( T^{7} - 96 T^{6} + \cdots + 159234392576 \) Copy content Toggle raw display
$17$ \( T^{7} - 161 T^{6} + \cdots + 60588009792 \) Copy content Toggle raw display
$19$ \( T^{7} - 279 T^{6} + \cdots - 1377989598400 \) Copy content Toggle raw display
$23$ \( T^{7} - 96 T^{6} + \cdots - 4061042235738 \) Copy content Toggle raw display
$29$ \( T^{7} + 296 T^{6} + \cdots - 12128150971026 \) Copy content Toggle raw display
$31$ \( T^{7} - 244 T^{6} + \cdots + 29481260805504 \) Copy content Toggle raw display
$37$ \( T^{7} - 404 T^{6} + \cdots - 83646911884544 \) Copy content Toggle raw display
$41$ \( T^{7} + 47 T^{6} + \cdots + 15\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( T^{7} - 525 T^{6} + \cdots - 13\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{7} - 164 T^{6} + \cdots + 65\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{7} - 506 T^{6} + \cdots + 11\!\cdots\!92 \) Copy content Toggle raw display
$59$ \( T^{7} + 85 T^{6} + \cdots - 59\!\cdots\!08 \) Copy content Toggle raw display
$61$ \( T^{7} - 828 T^{6} + \cdots + 12\!\cdots\!46 \) Copy content Toggle raw display
$67$ \( T^{7} - 1093 T^{6} + \cdots + 29\!\cdots\!03 \) Copy content Toggle raw display
$71$ \( T^{7} - 328 T^{6} + \cdots + 32\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( T^{7} - 2085 T^{6} + \cdots + 82\!\cdots\!12 \) Copy content Toggle raw display
$79$ \( T^{7} - 2110 T^{6} + \cdots + 16\!\cdots\!08 \) Copy content Toggle raw display
$83$ \( T^{7} - 1290 T^{6} + \cdots - 66\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{7} + 3048 T^{6} + \cdots + 32\!\cdots\!50 \) Copy content Toggle raw display
$97$ \( T^{7} - 1787 T^{6} + \cdots + 47\!\cdots\!76 \) Copy content Toggle raw display
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