Properties

Label 25.6.a.b
Level $25$
Weight $6$
Character orbit 25.a
Self dual yes
Analytic conductor $4.010$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [25,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
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: \(4.00959549532\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{241}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 60 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{241})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 2) q^{2} + (2 \beta - 11) q^{3} + (5 \beta + 32) q^{4} + (5 \beta - 98) q^{6} + (4 \beta - 102) q^{7} + ( - 15 \beta - 300) q^{8} + ( - 40 \beta + 118) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 2) q^{2} + (2 \beta - 11) q^{3} + (5 \beta + 32) q^{4} + (5 \beta - 98) q^{6} + (4 \beta - 102) q^{7} + ( - 15 \beta - 300) q^{8} + ( - 40 \beta + 118) q^{9} + ( - 50 \beta - 73) q^{11} + (19 \beta + 248) q^{12} + (32 \beta + 164) q^{13} + (90 \beta - 36) q^{14} + (185 \beta + 476) q^{16} + ( - 136 \beta - 677) q^{17} + (2 \beta + 2164) q^{18} + (70 \beta - 1625) q^{19} + ( - 240 \beta + 1602) q^{21} + (223 \beta + 3146) q^{22} + (12 \beta - 786) q^{23} + ( - 465 \beta + 1500) q^{24} + ( - 260 \beta - 2248) q^{26} + (110 \beta - 3425) q^{27} + ( - 362 \beta - 2064) q^{28} + (80 \beta - 2000) q^{29} + (1100 \beta - 1098) q^{31} + ( - 551 \beta - 2452) q^{32} + (304 \beta - 5197) q^{33} + (1085 \beta + 9514) q^{34} + ( - 890 \beta - 8224) q^{36} + (384 \beta - 1202) q^{37} + (1415 \beta - 950) q^{38} + (40 \beta + 2036) q^{39} + ( - 400 \beta + 14077) q^{41} + ( - 882 \beta + 11196) q^{42} + ( - 2128 \beta + 2564) q^{43} + ( - 2215 \beta - 17336) q^{44} + (750 \beta + 852) q^{46} + (1544 \beta - 13652) q^{47} + ( - 713 \beta + 16964) q^{48} + ( - 800 \beta - 5443) q^{49} + ( - 130 \beta - 8873) q^{51} + (2004 \beta + 14848) q^{52} + (752 \beta + 13114) q^{53} + (3095 \beta + 250) q^{54} + (270 \beta + 27000) q^{56} + ( - 3880 \beta + 26275) q^{57} + (1760 \beta - 800) q^{58} + (1960 \beta + 5000) q^{59} + ( - 2000 \beta - 11198) q^{61} + ( - 2202 \beta - 63804) q^{62} + (4392 \beta - 21636) q^{63} + ( - 1815 \beta + 22732) q^{64} + (4285 \beta - 7846) q^{66} + ( - 1586 \beta + 20823) q^{67} + ( - 8417 \beta - 62464) q^{68} + ( - 1680 \beta + 10086) q^{69} + ( - 1000 \beta - 43148) q^{71} + (10830 \beta + 600) q^{72} + (1112 \beta + 34589) q^{73} + (50 \beta - 20636) q^{74} + ( - 5535 \beta - 31000) q^{76} + (4608 \beta - 4554) q^{77} + ( - 2156 \beta - 6472) q^{78} + ( - 5020 \beta + 35250) q^{79} + (1880 \beta + 22201) q^{81} + ( - 12877 \beta - 4154) q^{82} + ( - 858 \beta - 45861) q^{83} + ( - 870 \beta - 20736) q^{84} + (3820 \beta + 122552) q^{86} + ( - 4720 \beta + 31600) q^{87} + (16845 \beta + 66900) q^{88} + (10440 \beta - 41625) q^{89} + ( - 2480 \beta - 9048) q^{91} + ( - 3486 \beta - 21552) q^{92} + ( - 12096 \beta + 144078) q^{93} + (9020 \beta - 65336) q^{94} + (55 \beta - 39148) q^{96} + (10944 \beta + 57598) q^{97} + (7843 \beta + 58886) q^{98} + ( - 980 \beta + 111386) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 5 q^{2} - 20 q^{3} + 69 q^{4} - 191 q^{6} - 200 q^{7} - 615 q^{8} + 196 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 5 q^{2} - 20 q^{3} + 69 q^{4} - 191 q^{6} - 200 q^{7} - 615 q^{8} + 196 q^{9} - 196 q^{11} + 515 q^{12} + 360 q^{13} + 18 q^{14} + 1137 q^{16} - 1490 q^{17} + 4330 q^{18} - 3180 q^{19} + 2964 q^{21} + 6515 q^{22} - 1560 q^{23} + 2535 q^{24} - 4756 q^{26} - 6740 q^{27} - 4490 q^{28} - 3920 q^{29} - 1096 q^{31} - 5455 q^{32} - 10090 q^{33} + 20113 q^{34} - 17338 q^{36} - 2020 q^{37} - 485 q^{38} + 4112 q^{39} + 27754 q^{41} + 21510 q^{42} + 3000 q^{43} - 36887 q^{44} + 2454 q^{46} - 25760 q^{47} + 33215 q^{48} - 11686 q^{49} - 17876 q^{51} + 31700 q^{52} + 26980 q^{53} + 3595 q^{54} + 54270 q^{56} + 48670 q^{57} + 160 q^{58} + 11960 q^{59} - 24396 q^{61} - 129810 q^{62} - 38880 q^{63} + 43649 q^{64} - 11407 q^{66} + 40060 q^{67} - 133345 q^{68} + 18492 q^{69} - 87296 q^{71} + 12030 q^{72} + 70290 q^{73} - 41222 q^{74} - 67535 q^{76} - 4500 q^{77} - 15100 q^{78} + 65480 q^{79} + 46282 q^{81} - 21185 q^{82} - 92580 q^{83} - 42342 q^{84} + 248924 q^{86} + 58480 q^{87} + 150645 q^{88} - 72810 q^{89} - 20576 q^{91} - 46590 q^{92} + 276060 q^{93} - 121652 q^{94} - 78241 q^{96} + 126140 q^{97} + 125615 q^{98} + 221792 q^{99}+O(q^{100}) \) 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
8.26209
−7.26209
−10.2621 5.52417 73.3104 0 −56.6896 −68.9517 −423.931 −212.483 0
1.2 5.26209 −25.5242 −4.31044 0 −134.310 −131.048 −191.069 408.483 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 25.6.a.b 2
3.b odd 2 1 225.6.a.s 2
4.b odd 2 1 400.6.a.w 2
5.b even 2 1 25.6.a.d yes 2
5.c odd 4 2 25.6.b.b 4
15.d odd 2 1 225.6.a.l 2
15.e even 4 2 225.6.b.i 4
20.d odd 2 1 400.6.a.o 2
20.e even 4 2 400.6.c.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.6.a.b 2 1.a even 1 1 trivial
25.6.a.d yes 2 5.b even 2 1
25.6.b.b 4 5.c odd 4 2
225.6.a.l 2 15.d odd 2 1
225.6.a.s 2 3.b odd 2 1
225.6.b.i 4 15.e even 4 2
400.6.a.o 2 20.d odd 2 1
400.6.a.w 2 4.b odd 2 1
400.6.c.n 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 5T_{2} - 54 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(25))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 5T - 54 \) Copy content Toggle raw display
$3$ \( T^{2} + 20T - 141 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 200T + 9036 \) Copy content Toggle raw display
$11$ \( T^{2} + 196T - 141021 \) Copy content Toggle raw display
$13$ \( T^{2} - 360T - 29296 \) Copy content Toggle raw display
$17$ \( T^{2} + 1490 T - 559359 \) Copy content Toggle raw display
$19$ \( T^{2} + 3180 T + 2232875 \) Copy content Toggle raw display
$23$ \( T^{2} + 1560 T + 599724 \) Copy content Toggle raw display
$29$ \( T^{2} + 3920 T + 3456000 \) Copy content Toggle raw display
$31$ \( T^{2} + 1096 T - 72602196 \) Copy content Toggle raw display
$37$ \( T^{2} + 2020 T - 7864124 \) Copy content Toggle raw display
$41$ \( T^{2} - 27754 T + 182931129 \) Copy content Toggle raw display
$43$ \( T^{2} - 3000 T - 270585136 \) Copy content Toggle raw display
$47$ \( T^{2} + 25760 T + 22262256 \) Copy content Toggle raw display
$53$ \( T^{2} - 26980 T + 147908484 \) Copy content Toggle raw display
$59$ \( T^{2} - 11960 T - 195696000 \) Copy content Toggle raw display
$61$ \( T^{2} + 24396 T - 92208796 \) Copy content Toggle raw display
$67$ \( T^{2} - 40060 T + 249648291 \) Copy content Toggle raw display
$71$ \( T^{2} + 87296 T + 1844897904 \) Copy content Toggle raw display
$73$ \( T^{2} - 70290 T + 1160669249 \) Copy content Toggle raw display
$79$ \( T^{2} - 65480 T - 446416500 \) Copy content Toggle raw display
$83$ \( T^{2} + 92580 T + 2098410219 \) Copy content Toggle raw display
$89$ \( T^{2} + 72810 T - 5241540375 \) Copy content Toggle raw display
$97$ \( T^{2} - 126140 T - 3238386044 \) Copy content Toggle raw display
show more
show less