Properties

Label 25.8.a.e
Level $25$
Weight $8$
Character orbit 25.a
Self dual yes
Analytic conductor $7.810$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{649}) \)
Defining polynomial: \(x^{2} - x - 162\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{649})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 8 - \beta ) q^{2} + ( 21 - 2 \beta ) q^{3} + ( 98 - 15 \beta ) q^{4} + ( 492 - 35 \beta ) q^{6} + ( -342 + 84 \beta ) q^{7} + ( 2190 - 75 \beta ) q^{8} + ( -1098 - 80 \beta ) q^{9} +O(q^{10})\) \( q + ( 8 - \beta ) q^{2} + ( 21 - 2 \beta ) q^{3} + ( 98 - 15 \beta ) q^{4} + ( 492 - 35 \beta ) q^{6} + ( -342 + 84 \beta ) q^{7} + ( 2190 - 75 \beta ) q^{8} + ( -1098 - 80 \beta ) q^{9} + ( 2047 + 250 \beta ) q^{11} + ( 6918 - 481 \beta ) q^{12} + ( 8636 + 408 \beta ) q^{13} + ( -16344 + 930 \beta ) q^{14} + ( 17126 - 795 \beta ) q^{16} + ( 3083 + 704 \beta ) q^{17} + ( 4176 + 538 \beta ) q^{18} + ( 9655 - 1110 \beta ) q^{19} + ( -34398 + 2280 \beta ) q^{21} + ( -24124 - 297 \beta ) q^{22} + ( 13926 - 6732 \beta ) q^{23} + ( 70290 - 5805 \beta ) q^{24} + ( 2992 - 5780 \beta ) q^{26} + ( -43065 + 5050 \beta ) q^{27} + ( -237636 + 12102 \beta ) q^{28} + ( 21320 + 13160 \beta ) q^{29} + ( -153138 + 4500 \beta ) q^{31} + ( -14522 - 13091 \beta ) q^{32} + ( -38013 + 656 \beta ) q^{33} + ( -89384 + 1845 \beta ) q^{34} + ( 86796 + 9830 \beta ) q^{36} + ( 310558 - 11256 \beta ) q^{37} + ( 257060 - 17425 \beta ) q^{38} + ( 49164 - 9520 \beta ) q^{39} + ( -55243 + 2000 \beta ) q^{41} + ( -644544 + 50358 \beta ) q^{42} + ( 473156 + 20088 \beta ) q^{43} + ( -406894 - 9955 \beta ) q^{44} + ( 1201992 - 61050 \beta ) q^{46} + ( 887108 + 13664 \beta ) q^{47} + ( 617226 - 49357 \beta ) q^{48} + ( 436493 - 50400 \beta ) q^{49} + ( -163353 + 7210 \beta ) q^{51} + ( -145112 - 95676 \beta ) q^{52} + ( 43346 + 44048 \beta ) q^{53} + ( -1162620 + 78415 \beta ) q^{54} + ( -1769580 + 203310 \beta ) q^{56} + ( 562395 - 40400 \beta ) q^{57} + ( -1961360 + 70800 \beta ) q^{58} + ( 990040 + 87520 \beta ) q^{59} + ( 403522 - 225000 \beta ) q^{61} + ( -1954104 + 184638 \beta ) q^{62} + ( -713124 - 71592 \beta ) q^{63} + ( -187562 + 24645 \beta ) q^{64} + ( -410376 + 42605 \beta ) q^{66} + ( 164583 - 73446 \beta ) q^{67} + ( -1408586 + 12187 \beta ) q^{68} + ( 2473614 - 155760 \beta ) q^{69} + ( -2389108 + 50000 \beta ) q^{71} + ( -1432620 - 86850 \beta ) q^{72} + ( -627499 - 84432 \beta ) q^{73} + ( 4307936 - 389350 \beta ) q^{74} + ( 3643490 - 236955 \beta ) q^{76} + ( 2701926 + 107448 \beta ) q^{77} + ( 1935552 - 115804 \beta ) q^{78} + ( -3637230 + 88260 \beta ) q^{79} + ( -139239 + 357040 \beta ) q^{81} + ( -765944 + 69243 \beta ) q^{82} + ( 5990091 + 69378 \beta ) q^{83} + ( -8911404 + 705210 \beta ) q^{84} + ( 530992 - 332540 \beta ) q^{86} + ( -3816120 + 207400 \beta ) q^{87} + ( 1445430 + 375225 \beta ) q^{88} + ( -3216465 + 442080 \beta ) q^{89} + ( 2598552 + 620160 \beta ) q^{91} + ( 17723508 - 767646 \beta ) q^{92} + ( -4673898 + 391776 \beta ) q^{93} + ( 4883296 - 791460 \beta ) q^{94} + ( 3936522 - 219685 \beta ) q^{96} + ( -8498642 - 122736 \beta ) q^{97} + ( 11656744 - 789293 \beta ) q^{98} + ( -5487606 - 458260 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 15q^{2} + 40q^{3} + 181q^{4} + 949q^{6} - 600q^{7} + 4305q^{8} - 2276q^{9} + O(q^{10}) \) \( 2q + 15q^{2} + 40q^{3} + 181q^{4} + 949q^{6} - 600q^{7} + 4305q^{8} - 2276q^{9} + 4344q^{11} + 13355q^{12} + 17680q^{13} - 31758q^{14} + 33457q^{16} + 6870q^{17} + 8890q^{18} + 18200q^{19} - 66516q^{21} - 48545q^{22} + 21120q^{23} + 134775q^{24} + 204q^{26} - 81080q^{27} - 463170q^{28} + 55800q^{29} - 301776q^{31} - 42135q^{32} - 75370q^{33} - 176923q^{34} + 183422q^{36} + 609860q^{37} + 496695q^{38} + 88808q^{39} - 108486q^{41} - 1238730q^{42} + 966400q^{43} - 823743q^{44} + 2342934q^{46} + 1787880q^{47} + 1185095q^{48} + 822586q^{49} - 319496q^{51} - 385900q^{52} + 130740q^{53} - 2246825q^{54} - 3335850q^{56} + 1084390q^{57} - 3851920q^{58} + 2067600q^{59} + 582044q^{61} - 3723570q^{62} - 1497840q^{63} - 350479q^{64} - 778147q^{66} + 255720q^{67} - 2804985q^{68} + 4791468q^{69} - 4728216q^{71} - 2952090q^{72} - 1339430q^{73} + 8226522q^{74} + 7050025q^{76} + 5511300q^{77} + 3755300q^{78} - 7186200q^{79} + 78562q^{81} - 1462645q^{82} + 12049560q^{83} - 17117598q^{84} + 729444q^{86} - 7424840q^{87} + 3266085q^{88} - 5990850q^{89} + 5817264q^{91} + 34679370q^{92} - 8956020q^{93} + 8975132q^{94} + 7653359q^{96} - 17120020q^{97} + 22524195q^{98} - 11433472q^{99} + O(q^{100}) \)

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
13.2377
−12.2377
−5.23774 −5.47548 −100.566 0 28.6791 769.970 1197.17 −2157.02 0
1.2 20.2377 45.4755 281.566 0 920.321 −1369.97 3107.83 −118.981 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 25.8.a.e yes 2
3.b odd 2 1 225.8.a.k 2
4.b odd 2 1 400.8.a.v 2
5.b even 2 1 25.8.a.c 2
5.c odd 4 2 25.8.b.b 4
15.d odd 2 1 225.8.a.v 2
15.e even 4 2 225.8.b.l 4
20.d odd 2 1 400.8.a.bd 2
20.e even 4 2 400.8.c.s 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.8.a.c 2 5.b even 2 1
25.8.a.e yes 2 1.a even 1 1 trivial
25.8.b.b 4 5.c odd 4 2
225.8.a.k 2 3.b odd 2 1
225.8.a.v 2 15.d odd 2 1
225.8.b.l 4 15.e even 4 2
400.8.a.v 2 4.b odd 2 1
400.8.a.bd 2 20.d odd 2 1
400.8.c.s 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 15 T_{2} - 106 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(25))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 15 T + 150 T^{2} - 1920 T^{3} + 16384 T^{4} \)
$3$ \( 1 - 40 T + 4125 T^{2} - 87480 T^{3} + 4782969 T^{4} \)
$5$ 1
$7$ \( 1 + 600 T + 592250 T^{2} + 494125800 T^{3} + 678223072849 T^{4} \)
$11$ \( 1 - 4344 T + 33551301 T^{2} - 84652270824 T^{3} + 379749833583241 T^{4} \)
$13$ \( 1 - 17680 T + 176633850 T^{2} - 1109393780560 T^{3} + 3937376385699289 T^{4} \)
$17$ \( 1 - 6870 T + 752062875 T^{2} - 2819026683510 T^{3} + 168377826559400929 T^{4} \)
$19$ \( 1 - 18200 T + 1670645253 T^{2} - 16268465649800 T^{3} + 799006685782884121 T^{4} \)
$23$ \( 1 - 21120 T - 431976950 T^{2} - 71909913440640 T^{3} + 11592836324538749809 T^{4} \)
$29$ \( 1 - 55800 T + 7178799018 T^{2} - 962543098042200 T^{3} + \)\(29\!\cdots\!81\)\( T^{4} \)
$31$ \( 1 + 301776 T + 74506854266 T^{2} + 8302646635961136 T^{3} + \)\(75\!\cdots\!21\)\( T^{4} \)
$37$ \( 1 - 609860 T + 262289383950 T^{2} - 57895154588331380 T^{3} + \)\(90\!\cdots\!89\)\( T^{4} \)
$41$ \( 1 + 108486 T + 391801850811 T^{2} + 21128112156254166 T^{3} + \)\(37\!\cdots\!61\)\( T^{4} \)
$43$ \( 1 - 966400 T + 711647085750 T^{2} - 262685505773804800 T^{3} + \)\(73\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 - 1787880 T + 1782082095150 T^{2} - 905781344613388440 T^{3} + \)\(25\!\cdots\!69\)\( T^{4} \)
$53$ \( 1 - 130740 T + 2038893798750 T^{2} - 153581734422289380 T^{3} + \)\(13\!\cdots\!69\)\( T^{4} \)
$59$ \( 1 - 2067600 T + 4803250907238 T^{2} - 5145535810011764400 T^{3} + \)\(61\!\cdots\!61\)\( T^{4} \)
$61$ \( 1 - 582044 T - 1843726773474 T^{2} - 1829214611249006924 T^{3} + \)\(98\!\cdots\!41\)\( T^{4} \)
$67$ \( 1 - 255720 T + 11262543795125 T^{2} - 1549845171713197560 T^{3} + \)\(36\!\cdots\!29\)\( T^{4} \)
$71$ \( 1 + 4728216 T + 23373621952446 T^{2} + 43003692654826860456 T^{3} + \)\(82\!\cdots\!81\)\( T^{4} \)
$73$ \( 1 + 1339430 T + 21386673483675 T^{2} + 14797216998434094710 T^{3} + \)\(12\!\cdots\!09\)\( T^{4} \)
$79$ \( 1 + 7186200 T + 50054286054218 T^{2} + \)\(13\!\cdots\!00\)\( T^{3} + \)\(36\!\cdots\!81\)\( T^{4} \)
$83$ \( 1 - 12049560 T + 89789116985725 T^{2} - \)\(32\!\cdots\!20\)\( T^{3} + \)\(73\!\cdots\!29\)\( T^{4} \)
$89$ \( 1 + 5990850 T + 65725956363283 T^{2} + \)\(26\!\cdots\!50\)\( T^{3} + \)\(19\!\cdots\!41\)\( T^{4} \)
$97$ \( 1 + 17120020 T + 232426185762150 T^{2} + \)\(13\!\cdots\!60\)\( T^{3} + \)\(65\!\cdots\!69\)\( T^{4} \)
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