Properties

Label 1.24.a.a
Level $1$
Weight $24$
Character orbit 1.a
Self dual yes
Analytic conductor $3.352$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.35204037345\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{144169}) \)
Defining polynomial: \( x^{2} - x - 36042 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 12\sqrt{144169}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 540) q^{2} + (48 \beta + 169740) q^{3} + ( - 1080 \beta + 12663328) q^{4} + (15040 \beta + 36534510) q^{5} + ( - 143820 \beta - 904836528) q^{6} + (985824 \beta - 679592200) q^{7} + ( - 4857920 \beta + 24729511680) q^{8} + (16295040 \beta - 17499697083) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 540) q^{2} + (48 \beta + 169740) q^{3} + ( - 1080 \beta + 12663328) q^{4} + (15040 \beta + 36534510) q^{5} + ( - 143820 \beta - 904836528) q^{6} + (985824 \beta - 679592200) q^{7} + ( - 4857920 \beta + 24729511680) q^{8} + (16295040 \beta - 17499697083) q^{9} + ( - 28412910 \beta - 292506818040) q^{10} + ( - 38671600 \beta + 428400984132) q^{11} + (424520544 \beta + 1073257476480) q^{12} + ( - 1268350272 \beta + 2188054661030) q^{13} + (1211937160 \beta - 20833017264864) q^{14} + (4306546080 \beta + 21188669492520) q^{15} + ( - 18293091840 \beta + 7978293200896) q^{16} + (23522231424 \beta + 127014073798770) q^{17} + (26299018683 \beta - 347740341958260) q^{18} + ( - 137218594320 \beta + 2130300489980) q^{19} + (150999182320 \beta + 125434193734080) q^{20} + (134713340160 \beta + 867015818861472) q^{21} + ( - 449283648132 \beta + 10\!\cdots\!80) q^{22}+ \cdots + (76\!\cdots\!80 \beta - 20\!\cdots\!56) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 1080 q^{2} + 339480 q^{3} + 25326656 q^{4} + 73069020 q^{5} - 1809673056 q^{6} - 1359184400 q^{7} + 49459023360 q^{8} - 34999394166 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 1080 q^{2} + 339480 q^{3} + 25326656 q^{4} + 73069020 q^{5} - 1809673056 q^{6} - 1359184400 q^{7} + 49459023360 q^{8} - 34999394166 q^{9} - 585013636080 q^{10} + 856801968264 q^{11} + 2146514952960 q^{12} + 4376109322060 q^{13} - 41666034529728 q^{14} + 42377338985040 q^{15} + 15956586401792 q^{16} + 254028147597540 q^{17} - 695480683916520 q^{18} + 4260600979960 q^{19} + 250868387468160 q^{20} + 17\!\cdots\!44 q^{21}+ \cdots - 41\!\cdots\!12 q^{99}+O(q^{100}) \) 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
190.348
−189.348
−4016.35 388445. 7.74247e6 1.05062e8 −1.56013e9 3.81217e9 2.59512e9 5.67462e10 −4.21966e11
1.2 5096.35 −48964.9 1.75842e7 −3.19930e7 −2.49542e8 −5.17135e9 4.68639e10 −9.17456e10 −1.63048e11
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 1.24.a.a 2
3.b odd 2 1 9.24.a.b 2
4.b odd 2 1 16.24.a.b 2
5.b even 2 1 25.24.a.a 2
5.c odd 4 2 25.24.b.a 4
7.b odd 2 1 49.24.a.b 2
8.b even 2 1 64.24.a.d 2
8.d odd 2 1 64.24.a.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.24.a.a 2 1.a even 1 1 trivial
9.24.a.b 2 3.b odd 2 1
16.24.a.b 2 4.b odd 2 1
25.24.a.a 2 5.b even 2 1
25.24.b.a 4 5.c odd 4 2
49.24.a.b 2 7.b odd 2 1
64.24.a.d 2 8.b even 2 1
64.24.a.g 2 8.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{24}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 1080 T - 20468736 \) Copy content Toggle raw display
$3$ \( T^{2} - 339480 T - 19020146544 \) Copy content Toggle raw display
$5$ \( T^{2} - 73069020 T - 33\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{2} + 1359184400 T - 19\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( T^{2} - 856801968264 T + 15\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( T^{2} - 4376109322060 T - 28\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{2} - 254028147597540 T + 46\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{2} - 4260600979960 T - 39\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 85\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 18\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 42\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 51\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 43\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 21\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 48\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 28\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 13\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 40\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 17\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 40\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 11\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 93\!\cdots\!36 \) Copy content Toggle raw display
show more
show less