Properties

Label 26.6.b.a
Level $26$
Weight $6$
Character orbit 26.b
Analytic conductor $4.170$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [26,6,Mod(25,26)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(26, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("26.25");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 26 = 2 \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 26.b (of order \(2\), degree \(1\), 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: \(4.16997931514\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 i q^{2} - 13 q^{3} - 16 q^{4} - 51 i q^{5} - 52 i q^{6} - 105 i q^{7} - 64 i q^{8} - 74 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 4 i q^{2} - 13 q^{3} - 16 q^{4} - 51 i q^{5} - 52 i q^{6} - 105 i q^{7} - 64 i q^{8} - 74 q^{9} + 204 q^{10} - 120 i q^{11} + 208 q^{12} + ( - 117 i - 598) q^{13} + 420 q^{14} + 663 i q^{15} + 256 q^{16} - 1101 q^{17} - 296 i q^{18} + 1170 i q^{19} + 816 i q^{20} + 1365 i q^{21} + 480 q^{22} + 1050 q^{23} + 832 i q^{24} + 524 q^{25} + ( - 2392 i + 468) q^{26} + 4121 q^{27} + 1680 i q^{28} - 4104 q^{29} - 2652 q^{30} - 9624 i q^{31} + 1024 i q^{32} + 1560 i q^{33} - 4404 i q^{34} - 5355 q^{35} + 1184 q^{36} - 8709 i q^{37} - 4680 q^{38} + (1521 i + 7774) q^{39} - 3264 q^{40} + 9480 i q^{41} - 5460 q^{42} + 9995 q^{43} + 1920 i q^{44} + 3774 i q^{45} + 4200 i q^{46} + 2943 i q^{47} - 3328 q^{48} + 5782 q^{49} + 2096 i q^{50} + 14313 q^{51} + (1872 i + 9568) q^{52} - 750 q^{53} + 16484 i q^{54} - 6120 q^{55} - 6720 q^{56} - 15210 i q^{57} - 16416 i q^{58} + 40938 i q^{59} - 10608 i q^{60} - 57920 q^{61} + 38496 q^{62} + 7770 i q^{63} - 4096 q^{64} + (30498 i - 5967) q^{65} - 6240 q^{66} - 22812 i q^{67} + 17616 q^{68} - 13650 q^{69} - 21420 i q^{70} - 63741 i q^{71} + 4736 i q^{72} - 58866 i q^{73} + 34836 q^{74} - 6812 q^{75} - 18720 i q^{76} - 12600 q^{77} + (31096 i - 6084) q^{78} + 63202 q^{79} - 13056 i q^{80} - 35591 q^{81} - 37920 q^{82} - 55458 i q^{83} - 21840 i q^{84} + 56151 i q^{85} + 39980 i q^{86} + 53352 q^{87} - 7680 q^{88} + 104778 i q^{89} - 15096 q^{90} + (62790 i - 12285) q^{91} - 16800 q^{92} + 125112 i q^{93} - 11772 q^{94} + 59670 q^{95} - 13312 i q^{96} - 160452 i q^{97} + 23128 i q^{98} + 8880 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 26 q^{3} - 32 q^{4} - 148 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 26 q^{3} - 32 q^{4} - 148 q^{9} + 408 q^{10} + 416 q^{12} - 1196 q^{13} + 840 q^{14} + 512 q^{16} - 2202 q^{17} + 960 q^{22} + 2100 q^{23} + 1048 q^{25} + 936 q^{26} + 8242 q^{27} - 8208 q^{29} - 5304 q^{30} - 10710 q^{35} + 2368 q^{36} - 9360 q^{38} + 15548 q^{39} - 6528 q^{40} - 10920 q^{42} + 19990 q^{43} - 6656 q^{48} + 11564 q^{49} + 28626 q^{51} + 19136 q^{52} - 1500 q^{53} - 12240 q^{55} - 13440 q^{56} - 115840 q^{61} + 76992 q^{62} - 8192 q^{64} - 11934 q^{65} - 12480 q^{66} + 35232 q^{68} - 27300 q^{69} + 69672 q^{74} - 13624 q^{75} - 25200 q^{77} - 12168 q^{78} + 126404 q^{79} - 71182 q^{81} - 75840 q^{82} + 106704 q^{87} - 15360 q^{88} - 30192 q^{90} - 24570 q^{91} - 33600 q^{92} - 23544 q^{94} + 119340 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/26\mathbb{Z}\right)^\times\).

\(n\) \(15\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
1.00000i
1.00000i
4.00000i −13.0000 −16.0000 51.0000i 52.0000i 105.000i 64.0000i −74.0000 204.000
25.2 4.00000i −13.0000 −16.0000 51.0000i 52.0000i 105.000i 64.0000i −74.0000 204.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 26.6.b.a 2
3.b odd 2 1 234.6.b.b 2
4.b odd 2 1 208.6.f.b 2
13.b even 2 1 inner 26.6.b.a 2
13.d odd 4 1 338.6.a.a 1
13.d odd 4 1 338.6.a.c 1
39.d odd 2 1 234.6.b.b 2
52.b odd 2 1 208.6.f.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.6.b.a 2 1.a even 1 1 trivial
26.6.b.a 2 13.b even 2 1 inner
208.6.f.b 2 4.b odd 2 1
208.6.f.b 2 52.b odd 2 1
234.6.b.b 2 3.b odd 2 1
234.6.b.b 2 39.d odd 2 1
338.6.a.a 1 13.d odd 4 1
338.6.a.c 1 13.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 13 \) acting on \(S_{6}^{\mathrm{new}}(26, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 16 \) Copy content Toggle raw display
$3$ \( (T + 13)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2601 \) Copy content Toggle raw display
$7$ \( T^{2} + 11025 \) Copy content Toggle raw display
$11$ \( T^{2} + 14400 \) Copy content Toggle raw display
$13$ \( T^{2} + 1196 T + 371293 \) Copy content Toggle raw display
$17$ \( (T + 1101)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 1368900 \) Copy content Toggle raw display
$23$ \( (T - 1050)^{2} \) Copy content Toggle raw display
$29$ \( (T + 4104)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 92621376 \) Copy content Toggle raw display
$37$ \( T^{2} + 75846681 \) Copy content Toggle raw display
$41$ \( T^{2} + 89870400 \) Copy content Toggle raw display
$43$ \( (T - 9995)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 8661249 \) Copy content Toggle raw display
$53$ \( (T + 750)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 1675919844 \) Copy content Toggle raw display
$61$ \( (T + 57920)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 520387344 \) Copy content Toggle raw display
$71$ \( T^{2} + 4062915081 \) Copy content Toggle raw display
$73$ \( T^{2} + 3465205956 \) Copy content Toggle raw display
$79$ \( (T - 63202)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 3075589764 \) Copy content Toggle raw display
$89$ \( T^{2} + 10978429284 \) Copy content Toggle raw display
$97$ \( T^{2} + 25744844304 \) Copy content Toggle raw display
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