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Properties

Label 25.16.b.a
Level $25$
Weight $16$
Character orbit 25.b
Analytic conductor $35.673$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [25,16,Mod(24,25)]
 
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([1]))
 
N = Newforms(chi, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.24");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(35.6733762750\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 108 \beta q^{2} + 1674 \beta q^{3} - 13888 q^{4} - 723168 q^{6} + 1411228 \beta q^{7} + 2039040 \beta q^{8} + 3139803 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 108 \beta q^{2} + 1674 \beta q^{3} - 13888 q^{4} - 723168 q^{6} + 1411228 \beta q^{7} + 2039040 \beta q^{8} + 3139803 q^{9} + 20586852 q^{11} - 23248512 \beta q^{12} + 95036669 \beta q^{13} - 609650496 q^{14} - 1335947264 q^{16} + 823263993 \beta q^{17} + 339098724 \beta q^{18} - 1563257180 q^{19} - 9449582688 q^{21} + 2223380016 \beta q^{22} - 4725558036 \beta q^{23} - 13653411840 q^{24} - 41055841008 q^{26} + 29276100540 \beta q^{27} - 19599134464 \beta q^{28} + 36902568330 q^{29} + 71588483552 q^{31} - 77467041792 \beta q^{32} + 34462390248 \beta q^{33} - 355650044976 q^{34} - 43605584064 q^{36} - 516826040777 \beta q^{37} - 168831775440 \beta q^{38} - 636365535624 q^{39} + 1641974018202 q^{41} - 1020554930304 \beta q^{42} + 246201554654 \beta q^{43} - 285910200576 q^{44} + 2041441071552 q^{46} - 1705342476312 \beta q^{47} - 2236375719936 \beta q^{48} - 3218696361993 q^{49} - 5512575697128 q^{51} - 1319869259072 \beta q^{52} - 3398575827951 \beta q^{53} - 12647275433280 q^{54} - 11510201364480 q^{56} - 2616892519320 \beta q^{57} + 3985477379640 \beta q^{58} - 9858856815540 q^{59} + 4931842626902 q^{61} + 7731556223616 \beta q^{62} + 4430977908084 \beta q^{63} - 10310557892608 q^{64} - 14887752587136 q^{66} - 14418913312682 \beta q^{67} - 11433490334784 \beta q^{68} + 31642336609056 q^{69} + 125050114914552 q^{71} + 6402183909120 \beta q^{72} + 41085727756739 \beta q^{73} + 223268849615664 q^{74} + 21710515715840 q^{76} + 29052741974256 \beta q^{77} - 68727477847392 \beta q^{78} + 25413078694480 q^{79} - 150980027970519 q^{81} + 177333193965816 \beta q^{82} + 140868365445234 \beta q^{83} + 131235804370944 q^{84} - 106359071610528 q^{86} + 61774899384420 \beta q^{87} + 41977414702080 \beta q^{88} - 715618564776810 q^{89} - 536473633278128 q^{91} + 65628550003968 \beta q^{92} + 119839121466048 \beta q^{93} + 736707949766784 q^{94} + 518719311839232 q^{96} + 306393068040913 \beta q^{97} - 347619207095244 \beta q^{98} + 64638659670156 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 27776 q^{4} - 1446336 q^{6} + 6279606 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 27776 q^{4} - 1446336 q^{6} + 6279606 q^{9} + 41173704 q^{11} - 1219300992 q^{14} - 2671894528 q^{16} - 3126514360 q^{19} - 18899165376 q^{21} - 27306823680 q^{24} - 82111682016 q^{26} + 73805136660 q^{29} + 143176967104 q^{31} - 711300089952 q^{34} - 87211168128 q^{36} - 1272731071248 q^{39} + 3283948036404 q^{41} - 571820401152 q^{44} + 4082882143104 q^{46} - 6437392723986 q^{49} - 11025151394256 q^{51} - 25294550866560 q^{54} - 23020402728960 q^{56} - 19717713631080 q^{59} + 9863685253804 q^{61} - 20621115785216 q^{64} - 29775505174272 q^{66} + 63284673218112 q^{69} + 250100229829104 q^{71} + 446537699231328 q^{74} + 43421031431680 q^{76} + 50826157388960 q^{79} - 301960055941038 q^{81} + 262471608741888 q^{84} - 212718143221056 q^{86} - 14\!\cdots\!20 q^{89}+ \cdots + 129277319340312 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1
− 1.00000i
1.00000i
− 216.000i − 3348.00i −13888.0 0 −723168. − 2.82246e6i − 4.07808e6i 3.13980e6 0
24.2 216.000i 3348.00i −13888.0 0 −723168. 2.82246e6i 4.07808e6i 3.13980e6 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 25.16.b.a 2
5.b even 2 1 inner 25.16.b.a 2
5.c odd 4 1 1.16.a.a ✓ 1
5.c odd 4 1 25.16.a.a 1
15.e even 4 1 9.16.a.a 1
20.e even 4 1 16.16.a.d 1
35.f even 4 1 49.16.a.a 1
35.k even 12 2 49.16.c.b 2
35.l odd 12 2 49.16.c.c 2
40.i odd 4 1 64.16.a.i 1
40.k even 4 1 64.16.a.c 1
55.e even 4 1 121.16.a.a 1
60.l odd 4 1 144.16.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.16.a.a ✓ 1 5.c odd 4 1
9.16.a.a 1 15.e even 4 1
16.16.a.d 1 20.e even 4 1
25.16.a.a 1 5.c odd 4 1
25.16.b.a 2 1.a even 1 1 trivial
25.16.b.a 2 5.b even 2 1 inner
49.16.a.a 1 35.f even 4 1
49.16.c.b 2 35.k even 12 2
49.16.c.c 2 35.l odd 12 2
64.16.a.c 1 40.k even 4 1
64.16.a.i 1 40.i odd 4 1
121.16.a.a 1 55.e even 4 1
144.16.a.f 1 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 46656 \) acting on \(S_{16}^{\mathrm{new}}(25, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 46656 \) Copy content Toggle raw display
$3$ \( T^{2} + 11209104 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 7966257871936 \) Copy content Toggle raw display
$11$ \( (T - 20586852)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 36\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{2} + 27\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T + 1563257180)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 89\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( (T - 36902568330)^{2} \) Copy content Toggle raw display
$31$ \( (T - 71588483552)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 10\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T - 1641974018202)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 24\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{2} + 11\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{2} + 46\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( (T + 9858856815540)^{2} \) Copy content Toggle raw display
$61$ \( (T - 4931842626902)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 83\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T - 125050114914552)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 67\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( (T - 25413078694480)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 79\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T + 715618564776810)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 37\!\cdots\!76 \) Copy content Toggle raw display
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