Properties

Label 2475.4.a.n
Level $2475$
Weight $4$
Character orbit 2475.a
Self dual yes
Analytic conductor $146.030$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} + ( -4 + \beta ) q^{4} + ( 4 - 4 \beta ) q^{7} + ( -4 + 11 \beta ) q^{8} +O(q^{10})\) \( q -\beta q^{2} + ( -4 + \beta ) q^{4} + ( 4 - 4 \beta ) q^{7} + ( -4 + 11 \beta ) q^{8} + 11 q^{11} + ( 44 + 2 \beta ) q^{13} + 16 q^{14} + ( -12 - 15 \beta ) q^{16} + ( -30 + 44 \beta ) q^{17} + ( -74 - 22 \beta ) q^{19} -11 \beta q^{22} + ( -32 - 60 \beta ) q^{23} + ( -8 - 46 \beta ) q^{26} + ( -32 + 16 \beta ) q^{28} + ( 96 - 34 \beta ) q^{29} + ( 36 - 12 \beta ) q^{31} + ( 92 - 61 \beta ) q^{32} + ( -176 - 14 \beta ) q^{34} + ( 130 + 112 \beta ) q^{37} + ( 88 + 96 \beta ) q^{38} + ( -96 + 154 \beta ) q^{41} + ( 196 + 124 \beta ) q^{43} + ( -44 + 11 \beta ) q^{44} + ( 240 + 92 \beta ) q^{46} + ( 4 + 216 \beta ) q^{47} + ( -263 - 16 \beta ) q^{49} + ( -168 + 38 \beta ) q^{52} + ( 334 - 196 \beta ) q^{53} + ( -192 + 16 \beta ) q^{56} + ( 136 - 62 \beta ) q^{58} + ( -4 - 240 \beta ) q^{59} + ( -146 + 364 \beta ) q^{61} + ( 48 - 24 \beta ) q^{62} + ( 340 + 89 \beta ) q^{64} + ( 380 - 16 \beta ) q^{67} + ( 296 - 162 \beta ) q^{68} + ( -1008 - 44 \beta ) q^{71} + ( 272 - 58 \beta ) q^{73} + ( -448 - 242 \beta ) q^{74} + ( 208 - 8 \beta ) q^{76} + ( 44 - 44 \beta ) q^{77} + ( 474 - 306 \beta ) q^{79} + ( -616 - 58 \beta ) q^{82} + ( 70 - 426 \beta ) q^{83} + ( -496 - 320 \beta ) q^{86} + ( -44 + 121 \beta ) q^{88} + ( -186 + 128 \beta ) q^{89} + ( 144 - 176 \beta ) q^{91} + ( -112 + 148 \beta ) q^{92} + ( -864 - 220 \beta ) q^{94} + ( 298 - 428 \beta ) q^{97} + ( 64 + 279 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - 7q^{4} + 4q^{7} + 3q^{8} + O(q^{10}) \) \( 2q - q^{2} - 7q^{4} + 4q^{7} + 3q^{8} + 22q^{11} + 90q^{13} + 32q^{14} - 39q^{16} - 16q^{17} - 170q^{19} - 11q^{22} - 124q^{23} - 62q^{26} - 48q^{28} + 158q^{29} + 60q^{31} + 123q^{32} - 366q^{34} + 372q^{37} + 272q^{38} - 38q^{41} + 516q^{43} - 77q^{44} + 572q^{46} + 224q^{47} - 542q^{49} - 298q^{52} + 472q^{53} - 368q^{56} + 210q^{58} - 248q^{59} + 72q^{61} + 72q^{62} + 769q^{64} + 744q^{67} + 430q^{68} - 2060q^{71} + 486q^{73} - 1138q^{74} + 408q^{76} + 44q^{77} + 642q^{79} - 1290q^{82} - 286q^{83} - 1312q^{86} + 33q^{88} - 244q^{89} + 112q^{91} - 76q^{92} - 1948q^{94} + 168q^{97} + 407q^{98} + O(q^{100}) \)

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
2.56155
−1.56155
−2.56155 0 −1.43845 0 0 −6.24621 24.1771 0 0
1.2 1.56155 0 −5.56155 0 0 10.2462 −21.1771 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.n 2
3.b odd 2 1 825.4.a.m 2
5.b even 2 1 495.4.a.d 2
15.d odd 2 1 165.4.a.c 2
15.e even 4 2 825.4.c.j 4
165.d even 2 1 1815.4.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.c 2 15.d odd 2 1
495.4.a.d 2 5.b even 2 1
825.4.a.m 2 3.b odd 2 1
825.4.c.j 4 15.e even 4 2
1815.4.a.n 2 165.d even 2 1
2475.4.a.n 2 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2475))\):

\( T_{2}^{2} + T_{2} - 4 \)
\( T_{7}^{2} - 4 T_{7} - 64 \)
\( T_{29}^{2} - 158 T_{29} + 1328 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 + T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( -64 - 4 T + T^{2} \)
$11$ \( ( -11 + T )^{2} \)
$13$ \( 2008 - 90 T + T^{2} \)
$17$ \( -8164 + 16 T + T^{2} \)
$19$ \( 5168 + 170 T + T^{2} \)
$23$ \( -11456 + 124 T + T^{2} \)
$29$ \( 1328 - 158 T + T^{2} \)
$31$ \( 288 - 60 T + T^{2} \)
$37$ \( -18716 - 372 T + T^{2} \)
$41$ \( -100432 + 38 T + T^{2} \)
$43$ \( 1216 - 516 T + T^{2} \)
$47$ \( -185744 - 224 T + T^{2} \)
$53$ \( -107572 - 472 T + T^{2} \)
$59$ \( -229424 + 248 T + T^{2} \)
$61$ \( -561812 - 72 T + T^{2} \)
$67$ \( 137296 - 744 T + T^{2} \)
$71$ \( 1052672 + 2060 T + T^{2} \)
$73$ \( 44752 - 486 T + T^{2} \)
$79$ \( -294912 - 642 T + T^{2} \)
$83$ \( -750824 + 286 T + T^{2} \)
$89$ \( -54748 + 244 T + T^{2} \)
$97$ \( -771476 - 168 T + T^{2} \)
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