Properties

Label 25.8.a.d
Level 25
Weight 8
Character orbit 25.a
Self dual yes
Analytic conductor 7.810
Analytic rank 1
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 25.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.80962563710\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{29}) \)
Defining polynomial: \(x^{2} - x - 7\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{29}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} + 3 \beta q^{3} -12 q^{4} -348 q^{6} + 39 \beta q^{7} + 140 \beta q^{8} -1143 q^{9} +O(q^{10})\) \( q -\beta q^{2} + 3 \beta q^{3} -12 q^{4} -348 q^{6} + 39 \beta q^{7} + 140 \beta q^{8} -1143 q^{9} -6828 q^{11} -36 \beta q^{12} -942 \beta q^{13} -4524 q^{14} -14704 q^{16} -1456 \beta q^{17} + 1143 \beta q^{18} -6860 q^{19} + 13572 q^{21} + 6828 \beta q^{22} + 2713 \beta q^{23} + 48720 q^{24} + 109272 q^{26} -9990 \beta q^{27} -468 \beta q^{28} -25590 q^{29} + 82112 q^{31} -3216 \beta q^{32} -20484 \beta q^{33} + 168896 q^{34} + 13716 q^{36} + 20754 \beta q^{37} + 6860 \beta q^{38} -327816 q^{39} -533118 q^{41} -13572 \beta q^{42} + 65823 \beta q^{43} + 81936 q^{44} -314708 q^{46} -541 \beta q^{47} -44112 \beta q^{48} -647107 q^{49} -506688 q^{51} + 11304 \beta q^{52} -54722 \beta q^{53} + 1158840 q^{54} + 633360 q^{56} -20580 \beta q^{57} + 25590 \beta q^{58} -1438980 q^{59} + 1381022 q^{61} -82112 \beta q^{62} -44577 \beta q^{63} + 2255168 q^{64} + 2376144 q^{66} + 252069 \beta q^{67} + 17472 \beta q^{68} + 944124 q^{69} -481608 q^{71} -160020 \beta q^{72} + 137988 \beta q^{73} -2407464 q^{74} + 82320 q^{76} -266292 \beta q^{77} + 327816 \beta q^{78} + 1059760 q^{79} -976779 q^{81} + 533118 \beta q^{82} -241757 \beta q^{83} -162864 q^{84} -7635468 q^{86} -76770 \beta q^{87} -955920 \beta q^{88} -5644170 q^{89} -4261608 q^{91} -32556 \beta q^{92} + 246336 \beta q^{93} + 62756 q^{94} -1119168 q^{96} -1115016 \beta q^{97} + 647107 \beta q^{98} + 7804404 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 24q^{4} - 696q^{6} - 2286q^{9} + O(q^{10}) \) \( 2q - 24q^{4} - 696q^{6} - 2286q^{9} - 13656q^{11} - 9048q^{14} - 29408q^{16} - 13720q^{19} + 27144q^{21} + 97440q^{24} + 218544q^{26} - 51180q^{29} + 164224q^{31} + 337792q^{34} + 27432q^{36} - 655632q^{39} - 1066236q^{41} + 163872q^{44} - 629416q^{46} - 1294214q^{49} - 1013376q^{51} + 2317680q^{54} + 1266720q^{56} - 2877960q^{59} + 2762044q^{61} + 4510336q^{64} + 4752288q^{66} + 1888248q^{69} - 963216q^{71} - 4814928q^{74} + 164640q^{76} + 2119520q^{79} - 1953558q^{81} - 325728q^{84} - 15270936q^{86} - 11288340q^{89} - 8523216q^{91} + 125512q^{94} - 2238336q^{96} + 15608808q^{99} + O(q^{100}) \)

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
3.19258
−2.19258
−10.7703 32.3110 −12.0000 0 −348.000 420.043 1507.85 −1143.00 0
1.2 10.7703 −32.3110 −12.0000 0 −348.000 −420.043 −1507.85 −1143.00 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.8.a.d 2
3.b odd 2 1 225.8.a.n 2
4.b odd 2 1 400.8.a.y 2
5.b even 2 1 inner 25.8.a.d 2
5.c odd 4 2 5.8.b.a 2
15.d odd 2 1 225.8.a.n 2
15.e even 4 2 45.8.b.a 2
20.d odd 2 1 400.8.a.y 2
20.e even 4 2 80.8.c.a 2
40.i odd 4 2 320.8.c.d 2
40.k even 4 2 320.8.c.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.8.b.a 2 5.c odd 4 2
25.8.a.d 2 1.a even 1 1 trivial
25.8.a.d 2 5.b even 2 1 inner
45.8.b.a 2 15.e even 4 2
80.8.c.a 2 20.e even 4 2
225.8.a.n 2 3.b odd 2 1
225.8.a.n 2 15.d odd 2 1
320.8.c.c 2 40.k even 4 2
320.8.c.d 2 40.i odd 4 2
400.8.a.y 2 4.b odd 2 1
400.8.a.y 2 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 116 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(25))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 140 T^{2} + 16384 T^{4} \)
$3$ \( 1 + 3330 T^{2} + 4782969 T^{4} \)
$5$ 1
$7$ \( 1 + 1470650 T^{2} + 678223072849 T^{4} \)
$11$ \( ( 1 + 6828 T + 19487171 T^{2} )^{2} \)
$13$ \( 1 + 22562810 T^{2} + 3937376385699289 T^{4} \)
$17$ \( 1 + 574764770 T^{2} + 168377826559400929 T^{4} \)
$19$ \( ( 1 + 6860 T + 893871739 T^{2} )^{2} \)
$23$ \( 1 + 5955848090 T^{2} + 11592836324538749809 T^{4} \)
$29$ \( ( 1 + 25590 T + 17249876309 T^{2} )^{2} \)
$31$ \( ( 1 - 82112 T + 27512614111 T^{2} )^{2} \)
$37$ \( 1 + 139899246410 T^{2} + \)\(90\!\cdots\!89\)\( T^{4} \)
$41$ \( ( 1 + 533118 T + 194754273881 T^{2} )^{2} \)
$43$ \( 1 + 41047812050 T^{2} + \)\(73\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 + 1013212289930 T^{2} + \)\(25\!\cdots\!69\)\( T^{4} \)
$53$ \( 1 + 2002060594730 T^{2} + \)\(13\!\cdots\!69\)\( T^{4} \)
$59$ \( ( 1 + 1438980 T + 2488651484819 T^{2} )^{2} \)
$61$ \( ( 1 - 1381022 T + 3142742836021 T^{2} )^{2} \)
$67$ \( 1 + 4750924642370 T^{2} + \)\(36\!\cdots\!29\)\( T^{4} \)
$71$ \( ( 1 + 481608 T + 9095120158391 T^{2} )^{2} \)
$73$ \( 1 + 19886077213490 T^{2} + \)\(12\!\cdots\!09\)\( T^{4} \)
$79$ \( ( 1 - 1059760 T + 19203908986159 T^{2} )^{2} \)
$83$ \( 1 + 47492314121570 T^{2} + \)\(73\!\cdots\!29\)\( T^{4} \)
$89$ \( ( 1 + 5644170 T + 44231334895529 T^{2} )^{2} \)
$97$ \( 1 + 17378330046530 T^{2} + \)\(65\!\cdots\!69\)\( T^{4} \)
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