Properties

Label 2475.4.a.s
Level $2475$
Weight $4$
Character orbit 2475.a
Self dual yes
Analytic conductor $146.030$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2475.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(146.029727264\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.23612.1
Defining polynomial: \(x^{3} - x^{2} - 20 x + 26\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -1 - \beta_{1} ) q^{2} + ( 7 + \beta_{1} + \beta_{2} ) q^{4} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{7} + ( -15 - 3 \beta_{1} - 4 \beta_{2} ) q^{8} +O(q^{10})\) \( q + ( -1 - \beta_{1} ) q^{2} + ( 7 + \beta_{1} + \beta_{2} ) q^{4} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{7} + ( -15 - 3 \beta_{1} - 4 \beta_{2} ) q^{8} -11 q^{11} + ( 4 - 12 \beta_{1} - 2 \beta_{2} ) q^{13} + ( 22 - 10 \beta_{1} - 4 \beta_{2} ) q^{14} + ( 9 + 23 \beta_{1} + 7 \beta_{2} ) q^{16} + ( -76 + 10 \beta_{1} + 6 \beta_{2} ) q^{17} + ( 48 + 2 \beta_{1} + 4 \beta_{2} ) q^{19} + ( 11 + 11 \beta_{1} ) q^{22} + ( -60 - 20 \beta_{1} - 8 \beta_{2} ) q^{23} + ( 168 + 4 \beta_{1} + 18 \beta_{2} ) q^{26} + ( 110 + 10 \beta_{1} + 6 \beta_{2} ) q^{28} + ( -34 + 34 \beta_{1} - 20 \beta_{2} ) q^{29} + ( -24 + 4 \beta_{1} - 16 \beta_{2} ) q^{31} + ( -225 - 13 \beta_{1} - 12 \beta_{2} ) q^{32} + ( -76 + 52 \beta_{1} - 28 \beta_{2} ) q^{34} + ( 122 + 24 \beta_{1} + 4 \beta_{2} ) q^{37} + ( -84 - 64 \beta_{1} - 14 \beta_{2} ) q^{38} + ( 66 - 2 \beta_{1} + 20 \beta_{2} ) q^{41} + ( 150 + 74 \beta_{1} - 14 \beta_{2} ) q^{43} + ( -77 - 11 \beta_{1} - 11 \beta_{2} ) q^{44} + ( 356 + 92 \beta_{1} + 44 \beta_{2} ) q^{46} + ( -36 + 48 \beta_{1} + 28 \beta_{2} ) q^{47} + ( -51 - 4 \beta_{1} - 20 \beta_{2} ) q^{49} + ( -292 - 144 \beta_{1} - 42 \beta_{2} ) q^{52} + ( -42 - 32 \beta_{1} + 52 \beta_{2} ) q^{53} + ( -438 - 54 \beta_{1} + 4 \beta_{2} ) q^{56} + ( -402 + 114 \beta_{1} + 26 \beta_{2} ) q^{58} + ( 308 + 120 \beta_{1} + 4 \beta_{2} ) q^{59} + ( 218 - 12 \beta_{1} - 44 \beta_{2} ) q^{61} + ( 88 \beta_{1} + 44 \beta_{2} ) q^{62} + ( 359 + 89 \beta_{1} - 7 \beta_{2} ) q^{64} + ( 128 - 148 \beta_{1} + 28 \beta_{2} ) q^{67} + ( 12 + 108 \beta_{1} - 16 \beta_{2} ) q^{68} + ( 216 - 104 \beta_{1} + 16 \beta_{2} ) q^{71} + ( -356 + 168 \beta_{1} - 14 \beta_{2} ) q^{73} + ( -466 - 138 \beta_{1} - 36 \beta_{2} ) q^{74} + ( 624 + 124 \beta_{1} + 74 \beta_{2} ) q^{76} + ( -22 + 22 \beta_{1} - 22 \beta_{2} ) q^{77} + ( -524 - 14 \beta_{1} - 52 \beta_{2} ) q^{79} + ( -78 - 146 \beta_{1} - 58 \beta_{2} ) q^{82} + ( -582 + 164 \beta_{1} + 70 \beta_{2} ) q^{83} + ( -1158 - 94 \beta_{1} - 32 \beta_{2} ) q^{86} + ( 165 + 33 \beta_{1} + 44 \beta_{2} ) q^{88} + ( 730 - 68 \beta_{1} ) q^{89} + ( 56 - 176 \beta_{1} + 4 \beta_{2} ) q^{91} + ( -1252 - 372 \beta_{1} - 160 \beta_{2} ) q^{92} + ( -692 - 76 \beta_{1} - 132 \beta_{2} ) q^{94} + ( -286 + 240 \beta_{1} - 64 \beta_{2} ) q^{97} + ( 147 + 131 \beta_{1} + 64 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 4q^{2} + 22q^{4} + 4q^{7} - 48q^{8} + O(q^{10}) \) \( 3q - 4q^{2} + 22q^{4} + 4q^{7} - 48q^{8} - 33q^{11} + 56q^{14} + 50q^{16} - 218q^{17} + 146q^{19} + 44q^{22} - 200q^{23} + 508q^{26} + 340q^{28} - 68q^{29} - 68q^{31} - 688q^{32} - 176q^{34} + 390q^{37} - 316q^{38} + 196q^{41} + 524q^{43} - 242q^{44} + 1160q^{46} - 60q^{47} - 157q^{49} - 1020q^{52} - 158q^{53} - 1368q^{56} - 1092q^{58} + 1044q^{59} + 642q^{61} + 88q^{62} + 1166q^{64} + 236q^{67} + 144q^{68} + 544q^{71} - 900q^{73} - 1536q^{74} + 1996q^{76} - 44q^{77} - 1586q^{79} - 380q^{82} - 1582q^{83} - 3568q^{86} + 528q^{88} + 2122q^{89} - 8q^{91} - 4128q^{92} - 2152q^{94} - 618q^{97} + 572q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 20 x + 26\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + \nu - 14 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - \beta_{1} + 14\)

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
4.26150
1.32906
−4.59056
−5.26150 0 19.6833 0 0 10.3207 −61.4719 0 0
1.2 −2.32906 0 −2.57547 0 0 −22.4672 24.6309 0 0
1.3 3.59056 0 4.89212 0 0 16.1465 −11.1590 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(11\) \(1\)

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 2475.4.a.s 3
3.b odd 2 1 825.4.a.s 3
5.b even 2 1 495.4.a.l 3
15.d odd 2 1 165.4.a.d 3
15.e even 4 2 825.4.c.l 6
165.d even 2 1 1815.4.a.s 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.d 3 15.d odd 2 1
495.4.a.l 3 5.b even 2 1
825.4.a.s 3 3.b odd 2 1
825.4.c.l 6 15.e even 4 2
1815.4.a.s 3 165.d even 2 1
2475.4.a.s 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2475))\):

\( T_{2}^{3} + 4 T_{2}^{2} - 15 T_{2} - 44 \)
\( T_{7}^{3} - 4 T_{7}^{2} - 428 T_{7} + 3744 \)
\( T_{29}^{3} + 68 T_{29}^{2} - 54364 T_{29} - 3163056 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -44 - 15 T + 4 T^{2} + T^{3} \)
$3$ \( T^{3} \)
$5$ \( T^{3} \)
$7$ \( 3744 - 428 T - 4 T^{2} + T^{3} \)
$11$ \( ( 11 + T )^{3} \)
$13$ \( 34144 - 3560 T + T^{3} \)
$17$ \( -235104 + 9680 T + 218 T^{2} + T^{3} \)
$19$ \( -30960 + 5376 T - 146 T^{2} + T^{3} \)
$23$ \( 1664 - 2672 T + 200 T^{2} + T^{3} \)
$29$ \( -3163056 - 54364 T + 68 T^{2} + T^{3} \)
$31$ \( -1812096 - 23232 T + 68 T^{2} + T^{3} \)
$37$ \( -618952 + 36460 T - 390 T^{2} + T^{3} \)
$41$ \( 4364208 - 26076 T - 196 T^{2} + T^{3} \)
$43$ \( 31273920 - 28668 T - 524 T^{2} + T^{3} \)
$47$ \( -20966976 - 135920 T + 60 T^{2} + T^{3} \)
$53$ \( 39574952 - 260852 T + 158 T^{2} + T^{3} \)
$59$ \( 84227264 + 64144 T - 1044 T^{2} + T^{3} \)
$61$ \( 22757384 - 60548 T - 642 T^{2} + T^{3} \)
$67$ \( -87537664 - 462208 T - 236 T^{2} + T^{3} \)
$71$ \( -6553600 - 129728 T - 544 T^{2} + T^{3} \)
$73$ \( 5609344 - 299576 T + 900 T^{2} + T^{3} \)
$79$ \( -14694992 + 562208 T + 1586 T^{2} + T^{3} \)
$83$ \( -924645384 - 307644 T + 1582 T^{2} + T^{3} \)
$89$ \( -293444632 + 1406940 T - 2122 T^{2} + T^{3} \)
$97$ \( 223543736 - 1291700 T + 618 T^{2} + T^{3} \)
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