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: \( x^{2} - x - 4 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - \beta q^{2} + (\beta - 4) q^{4} + ( - 4 \beta + 4) q^{7} + (11 \beta - 4) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + (\beta - 4) q^{4} + ( - 4 \beta + 4) q^{7} + (11 \beta - 4) q^{8} + 11 q^{11} + (2 \beta + 44) q^{13} + 16 q^{14} + ( - 15 \beta - 12) q^{16} + (44 \beta - 30) q^{17} + ( - 22 \beta - 74) q^{19} - 11 \beta q^{22} + ( - 60 \beta - 32) q^{23} + ( - 46 \beta - 8) q^{26} + (16 \beta - 32) q^{28} + ( - 34 \beta + 96) q^{29} + ( - 12 \beta + 36) q^{31} + ( - 61 \beta + 92) q^{32} + ( - 14 \beta - 176) q^{34} + (112 \beta + 130) q^{37} + (96 \beta + 88) q^{38} + (154 \beta - 96) q^{41} + (124 \beta + 196) q^{43} + (11 \beta - 44) q^{44} + (92 \beta + 240) q^{46} + (216 \beta + 4) q^{47} + ( - 16 \beta - 263) q^{49} + (38 \beta - 168) q^{52} + ( - 196 \beta + 334) q^{53} + (16 \beta - 192) q^{56} + ( - 62 \beta + 136) q^{58} + ( - 240 \beta - 4) q^{59} + (364 \beta - 146) q^{61} + ( - 24 \beta + 48) q^{62} + (89 \beta + 340) q^{64} + ( - 16 \beta + 380) q^{67} + ( - 162 \beta + 296) q^{68} + ( - 44 \beta - 1008) q^{71} + ( - 58 \beta + 272) q^{73} + ( - 242 \beta - 448) q^{74} + ( - 8 \beta + 208) q^{76} + ( - 44 \beta + 44) q^{77} + ( - 306 \beta + 474) q^{79} + ( - 58 \beta - 616) q^{82} + ( - 426 \beta + 70) q^{83} + ( - 320 \beta - 496) q^{86} + (121 \beta - 44) q^{88} + (128 \beta - 186) q^{89} + ( - 176 \beta + 144) q^{91} + (148 \beta - 112) q^{92} + ( - 220 \beta - 864) q^{94} + ( - 428 \beta + 298) q^{97} + (279 \beta + 64) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 7 q^{4} + 4 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - 7 q^{4} + 4 q^{7} + 3 q^{8} + 22 q^{11} + 90 q^{13} + 32 q^{14} - 39 q^{16} - 16 q^{17} - 170 q^{19} - 11 q^{22} - 124 q^{23} - 62 q^{26} - 48 q^{28} + 158 q^{29} + 60 q^{31} + 123 q^{32} - 366 q^{34} + 372 q^{37} + 272 q^{38} - 38 q^{41} + 516 q^{43} - 77 q^{44} + 572 q^{46} + 224 q^{47} - 542 q^{49} - 298 q^{52} + 472 q^{53} - 368 q^{56} + 210 q^{58} - 248 q^{59} + 72 q^{61} + 72 q^{62} + 769 q^{64} + 744 q^{67} + 430 q^{68} - 2060 q^{71} + 486 q^{73} - 1138 q^{74} + 408 q^{76} + 44 q^{77} + 642 q^{79} - 1290 q^{82} - 286 q^{83} - 1312 q^{86} + 33 q^{88} - 244 q^{89} + 112 q^{91} - 76 q^{92} - 1948 q^{94} + 168 q^{97} + 407 q^{98}+O(q^{100}) \) 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
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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2475))\):

\( T_{2}^{2} + T_{2} - 4 \) Copy content Toggle raw display
\( T_{7}^{2} - 4T_{7} - 64 \) Copy content Toggle raw display
\( T_{29}^{2} - 158T_{29} + 1328 \) Copy content Toggle raw display

Hecke characteristic polynomials

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