Properties

Label 4025.2.a.o
Level 4025
Weight 2
Character orbit 4025.a
Self dual yes
Analytic conductor 32.140
Analytic rank 0
Dimension 4
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.2777.1
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{3} ) q^{2} + ( 1 + \beta_{1} ) q^{3} + ( 1 - \beta_{2} - \beta_{3} ) q^{4} + ( 1 - \beta_{2} - \beta_{3} ) q^{6} - q^{7} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{8} + ( 3 \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{3} ) q^{2} + ( 1 + \beta_{1} ) q^{3} + ( 1 - \beta_{2} - \beta_{3} ) q^{4} + ( 1 - \beta_{2} - \beta_{3} ) q^{6} - q^{7} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{8} + ( 3 \beta_{1} + \beta_{2} ) q^{9} + ( -2 + 2 \beta_{1} - \beta_{3} ) q^{11} + ( 2 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{12} + ( 1 - \beta_{2} + 2 \beta_{3} ) q^{13} + ( -1 + \beta_{3} ) q^{14} + ( 2 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{16} + ( 2 - 2 \beta_{2} - \beta_{3} ) q^{17} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{18} + ( -1 - 3 \beta_{2} + 3 \beta_{3} ) q^{19} + ( -1 - \beta_{1} ) q^{21} + ( -3 \beta_{2} + 2 \beta_{3} ) q^{22} + q^{23} + ( 6 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{24} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{26} + ( 2 + 4 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{27} + ( -1 + \beta_{2} + \beta_{3} ) q^{28} + ( \beta_{1} - \beta_{3} ) q^{29} + ( 1 - 4 \beta_{1} + \beta_{2} - \beta_{3} ) q^{31} + ( 3 - \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{32} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{33} + ( 6 + 2 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{34} + ( -1 - 4 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{36} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{37} + ( -4 + 3 \beta_{1} - 2 \beta_{3} ) q^{38} + ( 2 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{39} + ( -1 + \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{41} + ( -1 + \beta_{2} + \beta_{3} ) q^{42} + ( 1 + 2 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{43} + ( 3 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{44} + ( 1 - \beta_{3} ) q^{46} + ( 5 + \beta_{1} + \beta_{2} ) q^{47} + ( 7 + 3 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{48} + q^{49} + ( 4 - \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{51} + ( -1 - \beta_{1} - \beta_{3} ) q^{52} + ( 4 + 3 \beta_{1} + 5 \beta_{2} - 4 \beta_{3} ) q^{53} + ( -4 - 4 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{54} + ( -2 - \beta_{1} + 2 \beta_{2} ) q^{56} + ( 2 - \beta_{1} ) q^{57} + ( 2 - 2 \beta_{2} ) q^{58} + ( -9 + \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{59} + ( \beta_{1} - \beta_{2} ) q^{61} + ( 2 - \beta_{1} + 4 \beta_{2} ) q^{62} + ( -3 \beta_{1} - \beta_{2} ) q^{63} + ( 6 - 3 \beta_{1} - \beta_{2} ) q^{64} + ( 3 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{66} + ( 9 - 3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{67} + ( 13 + 3 \beta_{1} - 5 \beta_{2} - 7 \beta_{3} ) q^{68} + ( 1 + \beta_{1} ) q^{69} + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{71} + ( 5 + 4 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} ) q^{72} + ( -8 - \beta_{1} - 3 \beta_{2} + 4 \beta_{3} ) q^{73} + ( -2 \beta_{1} + 3 \beta_{2} ) q^{74} + ( 2 + \beta_{2} - 2 \beta_{3} ) q^{76} + ( 2 - 2 \beta_{1} + \beta_{3} ) q^{77} + ( -1 - \beta_{1} - \beta_{3} ) q^{78} + ( -8 + 3 \beta_{1} + \beta_{2} + 5 \beta_{3} ) q^{79} + ( 6 + 6 \beta_{1} + 6 \beta_{2} + 5 \beta_{3} ) q^{81} + ( 3 - 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{82} + ( 3 + \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{83} + ( -2 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{84} + ( -1 - 4 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{86} + ( 2 + \beta_{1} - \beta_{3} ) q^{87} + ( 6 + \beta_{1} + 5 \beta_{2} - 8 \beta_{3} ) q^{88} + ( -\beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{89} + ( -1 + \beta_{2} - 2 \beta_{3} ) q^{91} + ( 1 - \beta_{2} - \beta_{3} ) q^{92} + ( -8 - 7 \beta_{1} - 4 \beta_{2} ) q^{93} + ( 4 - \beta_{1} - 4 \beta_{3} ) q^{94} + ( 2 - 3 \beta_{1} - 5 \beta_{2} - 4 \beta_{3} ) q^{96} + ( -1 + 6 \beta_{1} + 4 \beta_{2} ) q^{97} + ( 1 - \beta_{3} ) q^{98} + ( 9 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} + 5q^{3} + 2q^{4} + 2q^{6} - 4q^{7} + 9q^{8} + 3q^{9} + O(q^{10}) \) \( 4q + 2q^{2} + 5q^{3} + 2q^{4} + 2q^{6} - 4q^{7} + 9q^{8} + 3q^{9} - 8q^{11} + 3q^{12} + 8q^{13} - 2q^{14} + 6q^{16} + 6q^{17} - 3q^{18} + 2q^{19} - 5q^{21} + 4q^{22} + 4q^{23} + 22q^{24} - 11q^{26} + 14q^{27} - 2q^{28} - q^{29} - 2q^{31} + 5q^{32} + 7q^{33} + 18q^{34} - 10q^{36} + 7q^{37} - 17q^{38} + 12q^{39} - 9q^{41} - 2q^{42} + 4q^{43} + 9q^{44} + 2q^{46} + 21q^{47} + 25q^{48} + 4q^{49} + 9q^{51} - 7q^{52} + 11q^{53} - 16q^{54} - 9q^{56} + 7q^{57} + 8q^{58} - 37q^{59} + q^{61} + 7q^{62} - 3q^{63} + 21q^{64} + 9q^{66} + 31q^{67} + 41q^{68} + 5q^{69} - 8q^{71} + 18q^{72} - 25q^{73} - 2q^{74} + 4q^{76} + 8q^{77} - 7q^{78} - 19q^{79} + 40q^{81} + 16q^{82} + 7q^{83} - 3q^{84} - 2q^{86} + 7q^{87} + 9q^{88} + 7q^{89} - 8q^{91} + 2q^{92} - 39q^{93} + 7q^{94} - 3q^{96} + 2q^{97} + 2q^{98} + 40q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 4 x^{2} + x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 3 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 4 \beta_{1} + 1\)

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
−0.679643
2.36234
−1.50848
0.825785
−1.26308 0.320357 −0.404635 0 −0.404635 −1.00000 3.03724 −2.89737 0
1.2 −0.515722 3.36234 −1.73403 0 −1.73403 −1.00000 1.92572 8.30533 0
1.3 1.18264 −0.508481 −0.601352 0 −0.601352 −1.00000 −3.07647 −2.74145 0
1.4 2.59615 1.82578 4.74002 0 4.74002 −1.00000 7.11351 0.333489 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.o 4
5.b even 2 1 805.2.a.g 4
15.d odd 2 1 7245.2.a.bf 4
35.c odd 2 1 5635.2.a.s 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.a.g 4 5.b even 2 1
4025.2.a.o 4 1.a even 1 1 trivial
5635.2.a.s 4 35.c odd 2 1
7245.2.a.bf 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4025))\):

\( T_{2}^{4} - 2 T_{2}^{3} - 3 T_{2}^{2} + 3 T_{2} + 2 \)
\( T_{3}^{4} - 5 T_{3}^{3} + 5 T_{3}^{2} + 2 T_{3} - 1 \)
\( T_{11}^{4} + 8 T_{11}^{3} + 3 T_{11}^{2} - 51 T_{11} + 41 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 5 T^{2} - 9 T^{3} + 14 T^{4} - 18 T^{5} + 20 T^{6} - 16 T^{7} + 16 T^{8} \)
$3$ \( 1 - 5 T + 17 T^{2} - 43 T^{3} + 83 T^{4} - 129 T^{5} + 153 T^{6} - 135 T^{7} + 81 T^{8} \)
$5$ 1
$7$ \( ( 1 + T )^{4} \)
$11$ \( 1 + 8 T + 47 T^{2} + 213 T^{3} + 833 T^{4} + 2343 T^{5} + 5687 T^{6} + 10648 T^{7} + 14641 T^{8} \)
$13$ \( 1 - 8 T + 59 T^{2} - 291 T^{3} + 1197 T^{4} - 3783 T^{5} + 9971 T^{6} - 17576 T^{7} + 28561 T^{8} \)
$17$ \( 1 - 6 T + 51 T^{2} - 293 T^{3} + 1187 T^{4} - 4981 T^{5} + 14739 T^{6} - 29478 T^{7} + 83521 T^{8} \)
$19$ \( 1 - 2 T + 19 T^{2} - 83 T^{3} + 4 T^{4} - 1577 T^{5} + 6859 T^{6} - 13718 T^{7} + 130321 T^{8} \)
$23$ \( ( 1 - T )^{4} \)
$29$ \( 1 + T + 108 T^{2} + 83 T^{3} + 4590 T^{4} + 2407 T^{5} + 90828 T^{6} + 24389 T^{7} + 707281 T^{8} \)
$31$ \( 1 + 2 T + 43 T^{2} + 9 T^{3} + 878 T^{4} + 279 T^{5} + 41323 T^{6} + 59582 T^{7} + 923521 T^{8} \)
$37$ \( 1 - 7 T + 140 T^{2} - 718 T^{3} + 7578 T^{4} - 26566 T^{5} + 191660 T^{6} - 354571 T^{7} + 1874161 T^{8} \)
$41$ \( 1 + 9 T + 151 T^{2} + 1098 T^{3} + 9024 T^{4} + 45018 T^{5} + 253831 T^{6} + 620289 T^{7} + 2825761 T^{8} \)
$43$ \( 1 - 4 T + 97 T^{2} - 393 T^{3} + 5906 T^{4} - 16899 T^{5} + 179353 T^{6} - 318028 T^{7} + 3418801 T^{8} \)
$47$ \( 1 - 21 T + 345 T^{2} - 3461 T^{3} + 28589 T^{4} - 162667 T^{5} + 762105 T^{6} - 2180283 T^{7} + 4879681 T^{8} \)
$53$ \( 1 - 11 T + 102 T^{2} - 318 T^{3} + 2720 T^{4} - 16854 T^{5} + 286518 T^{6} - 1637647 T^{7} + 7890481 T^{8} \)
$59$ \( 1 + 37 T + 708 T^{2} + 8954 T^{3} + 80808 T^{4} + 528286 T^{5} + 2464548 T^{6} + 7599023 T^{7} + 12117361 T^{8} \)
$61$ \( 1 - T + 234 T^{2} - 170 T^{3} + 21104 T^{4} - 10370 T^{5} + 870714 T^{6} - 226981 T^{7} + 13845841 T^{8} \)
$67$ \( 1 - 31 T + 578 T^{2} - 7134 T^{3} + 67506 T^{4} - 477978 T^{5} + 2594642 T^{6} - 9323653 T^{7} + 20151121 T^{8} \)
$71$ \( 1 + 8 T + 281 T^{2} + 1677 T^{3} + 29804 T^{4} + 119067 T^{5} + 1416521 T^{6} + 2863288 T^{7} + 25411681 T^{8} \)
$73$ \( 1 + 25 T + 446 T^{2} + 5146 T^{3} + 50676 T^{4} + 375658 T^{5} + 2376734 T^{6} + 9725425 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 19 T + 275 T^{2} + 3079 T^{3} + 31569 T^{4} + 243241 T^{5} + 1716275 T^{6} + 9367741 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 7 T + 292 T^{2} - 1748 T^{3} + 34698 T^{4} - 145084 T^{5} + 2011588 T^{6} - 4002509 T^{7} + 47458321 T^{8} \)
$89$ \( 1 - 7 T + 254 T^{2} - 810 T^{3} + 26912 T^{4} - 72090 T^{5} + 2011934 T^{6} - 4934783 T^{7} + 62742241 T^{8} \)
$97$ \( 1 - 2 T + 176 T^{2} - 1868 T^{3} + 13257 T^{4} - 181196 T^{5} + 1655984 T^{6} - 1825346 T^{7} + 88529281 T^{8} \)
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