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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [25,8,Mod(1,25)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(25, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 25.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(7.80962563710\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{649}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 162 \) Copy content Toggle raw display
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

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

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 \( T_{2}^{2} - 15T_{2} - 106 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(25))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 15T - 106 \) Copy content Toggle raw display
$3$ \( T^{2} - 40T - 249 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 600 T - 1054836 \) Copy content Toggle raw display
$11$ \( T^{2} - 4344 T - 5423041 \) Copy content Toggle raw display
$13$ \( T^{2} - 17680 T + 51136816 \) Copy content Toggle raw display
$17$ \( T^{2} - 6870 T - 68614471 \) Copy content Toggle raw display
$19$ \( T^{2} - 18200 T - 117098225 \) Copy content Toggle raw display
$23$ \( T^{2} - 21120 T - 7241627844 \) Copy content Toggle raw display
$29$ \( T^{2} - 55800 T - 27320953600 \) Copy content Toggle raw display
$31$ \( T^{2} + 301776 T + 19481626044 \) Copy content Toggle raw display
$37$ \( T^{2} - 609860 T + 72425629684 \) Copy content Toggle raw display
$41$ \( T^{2} + 108486 T + 2293303049 \) Copy content Toggle raw display
$43$ \( T^{2} - 966400 T + 168009863536 \) Copy content Toggle raw display
$47$ \( T^{2} - 1787880 T + 768835854224 \) Copy content Toggle raw display
$53$ \( T^{2} - 130740 T - 310528480924 \) Copy content Toggle raw display
$59$ \( T^{2} - 2067600 T - 174052062400 \) Copy content Toggle raw display
$61$ \( T^{2} - 582044 T - 8129212445516 \) Copy content Toggle raw display
$67$ \( T^{2} - 255720 T - 858879415521 \) Copy content Toggle raw display
$71$ \( T^{2} + 4728216 T + 5183381635664 \) Copy content Toggle raw display
$73$ \( T^{2} + 1339430 T - 708123554519 \) Copy content Toggle raw display
$79$ \( T^{2} + 7186200 T + 11646468081900 \) Copy content Toggle raw display
$83$ \( T^{2} - 12049560 T + 35517015006471 \) Copy content Toggle raw display
$89$ \( T^{2} + 5990850 T - 22736713427775 \) Copy content Toggle raw display
$97$ \( T^{2} + 17120020 T + 70829616805924 \) Copy content Toggle raw display
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