Properties

Label 4025.2.a.k
Level 4025
Weight 2
Character orbit 4025.a
Self dual yes
Analytic conductor 32.140
Analytic rank 1
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1397868136\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 805)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a root \(\beta\) of the polynomial \(x^{3} - x^{2} - 2 x + 1\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta ) q^{2} + ( 3 + \beta - 2 \beta^{2} ) q^{3} + ( -1 - 2 \beta + \beta^{2} ) q^{4} + ( 1 + 2 \beta - \beta^{2} ) q^{6} + q^{7} + ( -2 - \beta + 2 \beta^{2} ) q^{8} + ( 6 + 2 \beta - 3 \beta^{2} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta ) q^{2} + ( 3 + \beta - 2 \beta^{2} ) q^{3} + ( -1 - 2 \beta + \beta^{2} ) q^{4} + ( 1 + 2 \beta - \beta^{2} ) q^{6} + q^{7} + ( -2 - \beta + 2 \beta^{2} ) q^{8} + ( 6 + 2 \beta - 3 \beta^{2} ) q^{9} + ( -2 - \beta ) q^{11} + ( -6 + \beta + 2 \beta^{2} ) q^{12} + ( -7 + \beta + 3 \beta^{2} ) q^{13} + ( 1 - \beta ) q^{14} + ( 2 + \beta - \beta^{2} ) q^{16} + ( -4 - 3 \beta + 2 \beta^{2} ) q^{17} + ( 3 + 2 \beta - 2 \beta^{2} ) q^{18} + ( -3 - 2 \beta + \beta^{2} ) q^{19} + ( 3 + \beta - 2 \beta^{2} ) q^{21} + ( -2 + \beta + \beta^{2} ) q^{22} + q^{23} + ( -6 - \beta + \beta^{2} ) q^{24} + ( -4 + 2 \beta - \beta^{2} ) q^{26} + ( 10 + \beta - 2 \beta^{2} ) q^{27} + ( -1 - 2 \beta + \beta^{2} ) q^{28} + ( -8 + 2 \beta + 2 \beta^{2} ) q^{29} + ( -3 + \beta^{2} ) q^{31} + ( 5 + 3 \beta - 5 \beta^{2} ) q^{32} + ( -8 - \beta + 5 \beta^{2} ) q^{33} + ( -2 - 3 \beta + 3 \beta^{2} ) q^{34} + ( -11 - \beta + 4 \beta^{2} ) q^{36} -3 \beta q^{37} + ( -2 - \beta + 2 \beta^{2} ) q^{38} + ( -16 - 8 \beta + 7 \beta^{2} ) q^{39} + ( 5 + 6 \beta - 4 \beta^{2} ) q^{41} + ( 1 + 2 \beta - \beta^{2} ) q^{42} + ( -1 + \beta ) q^{43} + ( 3 + 3 \beta - \beta^{2} ) q^{44} + ( 1 - \beta ) q^{46} + ( -11 - 4 \beta + 5 \beta^{2} ) q^{47} + ( 7 + \beta - 3 \beta^{2} ) q^{48} + q^{49} + ( -16 - \beta + 7 \beta^{2} ) q^{51} + ( 9 + 6 \beta - 8 \beta^{2} ) q^{52} + ( 8 + 10 \beta - 5 \beta^{2} ) q^{53} + ( 8 - 5 \beta - \beta^{2} ) q^{54} + ( -2 - \beta + 2 \beta^{2} ) q^{56} + ( -12 - \beta + 6 \beta^{2} ) q^{57} + ( -6 + 6 \beta - 2 \beta^{2} ) q^{58} + ( -11 - 3 \beta + 3 \beta^{2} ) q^{59} + ( 4 + 4 \beta - 5 \beta^{2} ) q^{61} + ( -2 + \beta ) q^{62} + ( 6 + 2 \beta - 3 \beta^{2} ) q^{63} + ( -4 + 6 \beta - \beta^{2} ) q^{64} + ( -3 - 3 \beta + \beta^{2} ) q^{66} + ( -13 + \beta + 5 \beta^{2} ) q^{67} + ( 9 - \beta - \beta^{2} ) q^{68} + ( 3 + \beta - 2 \beta^{2} ) q^{69} + ( -9 + 6 \beta + \beta^{2} ) q^{71} + ( -13 - 2 \beta + 5 \beta^{2} ) q^{72} + ( 10 + 4 \beta - 5 \beta^{2} ) q^{73} + ( -3 \beta + 3 \beta^{2} ) q^{74} + ( 6 + \beta - \beta^{2} ) q^{76} + ( -2 - \beta ) q^{77} + ( -9 - 6 \beta + 8 \beta^{2} ) q^{78} + ( -12 + 3 \beta + 3 \beta^{2} ) q^{79} + ( 12 + 3 \beta - 8 \beta^{2} ) q^{81} + ( 1 + 9 \beta - 6 \beta^{2} ) q^{82} + ( -1 - 3 \beta + 7 \beta^{2} ) q^{83} + ( -6 + \beta + 2 \beta^{2} ) q^{84} + ( -1 + 2 \beta - \beta^{2} ) q^{86} + ( -18 - 10 \beta + 10 \beta^{2} ) q^{87} + ( 6 - 5 \beta^{2} ) q^{88} + ( -4 - \beta + 4 \beta^{2} ) q^{89} + ( -7 + \beta + 3 \beta^{2} ) q^{91} + ( -1 - 2 \beta + \beta^{2} ) q^{92} + ( -8 - 3 \beta + 4 \beta^{2} ) q^{93} + ( -6 - 3 \beta + 4 \beta^{2} ) q^{94} + ( 16 + 2 \beta - 3 \beta^{2} ) q^{96} + ( 13 - 8 \beta^{2} ) q^{97} + ( 1 - \beta ) q^{98} + ( -15 - 4 \beta + 7 \beta^{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 2q^{2} + 3q^{7} + 3q^{8} + 5q^{9} + O(q^{10}) \) \( 3q + 2q^{2} + 3q^{7} + 3q^{8} + 5q^{9} - 7q^{11} - 7q^{12} - 5q^{13} + 2q^{14} + 2q^{16} - 5q^{17} + q^{18} - 6q^{19} + 3q^{23} - 14q^{24} - 15q^{26} + 21q^{27} - 12q^{29} - 4q^{31} - 7q^{32} + 6q^{34} - 14q^{36} - 3q^{37} + 3q^{38} - 21q^{39} + q^{41} - 2q^{43} + 7q^{44} + 2q^{46} - 12q^{47} + 7q^{48} + 3q^{49} - 14q^{51} - 7q^{52} + 9q^{53} + 14q^{54} + 3q^{56} - 7q^{57} - 22q^{58} - 21q^{59} - 9q^{61} - 5q^{62} + 5q^{63} - 11q^{64} - 7q^{66} - 13q^{67} + 21q^{68} - 16q^{71} - 16q^{72} + 9q^{73} + 12q^{74} + 14q^{76} - 7q^{77} + 7q^{78} - 18q^{79} - q^{81} - 18q^{82} + 29q^{83} - 7q^{84} - 6q^{86} - 14q^{87} - 7q^{88} + 7q^{89} - 5q^{91} - 7q^{93} - q^{94} + 35q^{96} - q^{97} + 2q^{98} - 14q^{99} + O(q^{100}) \)

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
1.80194
0.445042
−1.24698
−0.801938 −1.69202 −1.35690 0 1.35690 1.00000 2.69202 −0.137063 0
1.2 0.554958 3.04892 −1.69202 0 1.69202 1.00000 −2.04892 6.29590 0
1.3 2.24698 −1.35690 3.04892 0 −3.04892 1.00000 2.35690 −1.15883 0
\(n\): 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 4025.2.a.k 3
5.b even 2 1 805.2.a.f 3
15.d odd 2 1 7245.2.a.ba 3
35.c odd 2 1 5635.2.a.r 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.a.f 3 5.b even 2 1
4025.2.a.k 3 1.a even 1 1 trivial
5635.2.a.r 3 35.c odd 2 1
7245.2.a.ba 3 15.d odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(7\) \(-1\)
\(23\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4025))\):

\( T_{2}^{3} - 2 T_{2}^{2} - T_{2} + 1 \)
\( T_{3}^{3} - 7 T_{3} - 7 \)
\( T_{11}^{3} + 7 T_{11}^{2} + 14 T_{11} + 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 5 T^{2} - 7 T^{3} + 10 T^{4} - 8 T^{5} + 8 T^{6} \)
$3$ \( 1 + 2 T^{2} - 7 T^{3} + 6 T^{4} + 27 T^{6} \)
$5$ 1
$7$ \( ( 1 - T )^{3} \)
$11$ \( 1 + 7 T + 47 T^{2} + 161 T^{3} + 517 T^{4} + 847 T^{5} + 1331 T^{6} \)
$13$ \( 1 + 5 T + 17 T^{2} + 33 T^{3} + 221 T^{4} + 845 T^{5} + 2197 T^{6} \)
$17$ \( 1 + 5 T + 43 T^{2} + 129 T^{3} + 731 T^{4} + 1445 T^{5} + 4913 T^{6} \)
$19$ \( 1 + 6 T + 62 T^{2} + 215 T^{3} + 1178 T^{4} + 2166 T^{5} + 6859 T^{6} \)
$23$ \( ( 1 - T )^{3} \)
$29$ \( 1 + 12 T + 107 T^{2} + 592 T^{3} + 3103 T^{4} + 10092 T^{5} + 24389 T^{6} \)
$31$ \( 1 + 4 T + 96 T^{2} + 247 T^{3} + 2976 T^{4} + 3844 T^{5} + 29791 T^{6} \)
$37$ \( 1 + 3 T + 93 T^{2} + 195 T^{3} + 3441 T^{4} + 4107 T^{5} + 50653 T^{6} \)
$41$ \( 1 - T + 58 T^{2} + 87 T^{3} + 2378 T^{4} - 1681 T^{5} + 68921 T^{6} \)
$43$ \( 1 + 2 T + 128 T^{2} + 171 T^{3} + 5504 T^{4} + 3698 T^{5} + 79507 T^{6} \)
$47$ \( 1 + 12 T + 140 T^{2} + 1087 T^{3} + 6580 T^{4} + 26508 T^{5} + 103823 T^{6} \)
$53$ \( 1 - 9 T + 11 T^{2} + 419 T^{3} + 583 T^{4} - 25281 T^{5} + 148877 T^{6} \)
$59$ \( 1 + 21 T + 303 T^{2} + 2681 T^{3} + 17877 T^{4} + 73101 T^{5} + 205379 T^{6} \)
$61$ \( 1 + 9 T + 161 T^{2} + 887 T^{3} + 9821 T^{4} + 33489 T^{5} + 226981 T^{6} \)
$67$ \( 1 + 13 T + 185 T^{2} + 1365 T^{3} + 12395 T^{4} + 58357 T^{5} + 300763 T^{6} \)
$71$ \( 1 + 16 T + 198 T^{2} + 1809 T^{3} + 14058 T^{4} + 80656 T^{5} + 357911 T^{6} \)
$73$ \( 1 - 9 T + 197 T^{2} - 1285 T^{3} + 14381 T^{4} - 47961 T^{5} + 389017 T^{6} \)
$79$ \( 1 + 18 T + 282 T^{2} + 2493 T^{3} + 22278 T^{4} + 112338 T^{5} + 493039 T^{6} \)
$83$ \( 1 - 29 T + 443 T^{2} - 4603 T^{3} + 36769 T^{4} - 199781 T^{5} + 571787 T^{6} \)
$89$ \( 1 - 7 T + 253 T^{2} - 1155 T^{3} + 22517 T^{4} - 55447 T^{5} + 704969 T^{6} \)
$97$ \( 1 + T + 142 T^{2} + 277 T^{3} + 13774 T^{4} + 9409 T^{5} + 912673 T^{6} \)
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