Properties

Label 2475.4.a.o
Level $2475$
Weight $4$
Character orbit 2475.a
Self dual yes
Analytic conductor $146.030$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + \beta q^{4} + ( -2 + 2 \beta ) q^{7} + ( 8 - 7 \beta ) q^{8} +O(q^{10})\) \( q + \beta q^{2} + \beta q^{4} + ( -2 + 2 \beta ) q^{7} + ( 8 - 7 \beta ) q^{8} -11 q^{11} + ( 42 - 8 \beta ) q^{13} + 16 q^{14} + ( -56 - 7 \beta ) q^{16} + ( -28 + 30 \beta ) q^{17} + ( -18 - 18 \beta ) q^{19} -11 \beta q^{22} + 112 q^{23} + ( -64 + 34 \beta ) q^{26} + 16 q^{28} + ( -88 - 46 \beta ) q^{29} + ( -72 + 104 \beta ) q^{31} + ( -120 - 7 \beta ) q^{32} + ( 240 + 2 \beta ) q^{34} + ( 46 - 44 \beta ) q^{37} + ( -144 - 36 \beta ) q^{38} + ( 248 - 2 \beta ) q^{41} + ( -10 + 86 \beta ) q^{43} -11 \beta q^{44} + 112 \beta q^{46} + ( -8 - 48 \beta ) q^{47} + ( -307 - 4 \beta ) q^{49} + ( -64 + 34 \beta ) q^{52} + ( 22 - 128 \beta ) q^{53} + ( -128 + 16 \beta ) q^{56} + ( -368 - 134 \beta ) q^{58} -196 q^{59} + ( -526 - 52 \beta ) q^{61} + ( 832 + 32 \beta ) q^{62} + ( 392 - 71 \beta ) q^{64} + ( -388 - 152 \beta ) q^{67} + ( 240 + 2 \beta ) q^{68} + ( -136 - 184 \beta ) q^{71} + ( 170 + 252 \beta ) q^{73} + ( -352 + 2 \beta ) q^{74} + ( -144 - 36 \beta ) q^{76} + ( 22 - 22 \beta ) q^{77} + ( -78 - 74 \beta ) q^{79} + ( -16 + 246 \beta ) q^{82} + ( 336 - 324 \beta ) q^{83} + ( 688 + 76 \beta ) q^{86} + ( -88 + 77 \beta ) q^{88} + ( -482 - 8 \beta ) q^{89} + ( -212 + 84 \beta ) q^{91} + 112 \beta q^{92} + ( -384 - 56 \beta ) q^{94} + ( 802 - 420 \beta ) q^{97} + ( -32 - 311 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + q^{4} - 2q^{7} + 9q^{8} + O(q^{10}) \) \( 2q + q^{2} + q^{4} - 2q^{7} + 9q^{8} - 22q^{11} + 76q^{13} + 32q^{14} - 119q^{16} - 26q^{17} - 54q^{19} - 11q^{22} + 224q^{23} - 94q^{26} + 32q^{28} - 222q^{29} - 40q^{31} - 247q^{32} + 482q^{34} + 48q^{37} - 324q^{38} + 494q^{41} + 66q^{43} - 11q^{44} + 112q^{46} - 64q^{47} - 618q^{49} - 94q^{52} - 84q^{53} - 240q^{56} - 870q^{58} - 392q^{59} - 1104q^{61} + 1696q^{62} + 713q^{64} - 928q^{67} + 482q^{68} - 456q^{71} + 592q^{73} - 702q^{74} - 324q^{76} + 22q^{77} - 230q^{79} + 214q^{82} + 348q^{83} + 1452q^{86} - 99q^{88} - 972q^{89} - 340q^{91} + 112q^{92} - 824q^{94} + 1184q^{97} - 375q^{98} + O(q^{100}) \)

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
−2.37228
3.37228
−2.37228 0 −2.37228 0 0 −6.74456 24.6060 0 0
1.2 3.37228 0 3.37228 0 0 4.74456 −15.6060 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.o 2
3.b odd 2 1 825.4.a.k 2
5.b even 2 1 99.4.a.e 2
15.d odd 2 1 33.4.a.d 2
15.e even 4 2 825.4.c.i 4
20.d odd 2 1 1584.4.a.x 2
55.d odd 2 1 1089.4.a.t 2
60.h even 2 1 528.4.a.o 2
105.g even 2 1 1617.4.a.j 2
120.i odd 2 1 2112.4.a.ba 2
120.m even 2 1 2112.4.a.bh 2
165.d even 2 1 363.4.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.d 2 15.d odd 2 1
99.4.a.e 2 5.b even 2 1
363.4.a.j 2 165.d even 2 1
528.4.a.o 2 60.h even 2 1
825.4.a.k 2 3.b odd 2 1
825.4.c.i 4 15.e even 4 2
1089.4.a.t 2 55.d odd 2 1
1584.4.a.x 2 20.d odd 2 1
1617.4.a.j 2 105.g even 2 1
2112.4.a.ba 2 120.i odd 2 1
2112.4.a.bh 2 120.m even 2 1
2475.4.a.o 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2475))\):

\( T_{2}^{2} - T_{2} - 8 \)
\( T_{7}^{2} + 2 T_{7} - 32 \)
\( T_{29}^{2} + 222 T_{29} - 5136 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -8 - T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( -32 + 2 T + T^{2} \)
$11$ \( ( 11 + T )^{2} \)
$13$ \( 916 - 76 T + T^{2} \)
$17$ \( -7256 + 26 T + T^{2} \)
$19$ \( -1944 + 54 T + T^{2} \)
$23$ \( ( -112 + T )^{2} \)
$29$ \( -5136 + 222 T + T^{2} \)
$31$ \( -88832 + 40 T + T^{2} \)
$37$ \( -15396 - 48 T + T^{2} \)
$41$ \( 60976 - 494 T + T^{2} \)
$43$ \( -59928 - 66 T + T^{2} \)
$47$ \( -17984 + 64 T + T^{2} \)
$53$ \( -133404 + 84 T + T^{2} \)
$59$ \( ( 196 + T )^{2} \)
$61$ \( 282396 + 1104 T + T^{2} \)
$67$ \( 24688 + 928 T + T^{2} \)
$71$ \( -227328 + 456 T + T^{2} \)
$73$ \( -436292 - 592 T + T^{2} \)
$79$ \( -31952 + 230 T + T^{2} \)
$83$ \( -835776 - 348 T + T^{2} \)
$89$ \( 235668 + 972 T + T^{2} \)
$97$ \( -1104836 - 1184 T + T^{2} \)
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