Properties

Label 26.4.c.b
Level $26$
Weight $4$
Character orbit 26.c
Analytic conductor $1.534$
Analytic rank $0$
Dimension $4$
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] = [26,4,Mod(3,26)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(26, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("26.3");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 26 = 2 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 26.c (of order \(3\), degree \(2\), 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: \(1.53404966015\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{217})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 55x^{2} + 54x + 2916 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_{2} q^{2} + (\beta_{2} + \beta_1) q^{3} + (4 \beta_{2} - 4) q^{4} + ( - \beta_{3} - 3) q^{5} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 4) q^{6} + (\beta_{3} - 23 \beta_{2} + \beta_1 + 22) q^{7} + 8 q^{8} + (3 \beta_{3} + 28 \beta_{2} + 3 \beta_1 - 31) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{2} q^{2} + (\beta_{2} + \beta_1) q^{3} + (4 \beta_{2} - 4) q^{4} + ( - \beta_{3} - 3) q^{5} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 4) q^{6} + (\beta_{3} - 23 \beta_{2} + \beta_1 + 22) q^{7} + 8 q^{8} + (3 \beta_{3} + 28 \beta_{2} + 3 \beta_1 - 31) q^{9} + (8 \beta_{2} - 2 \beta_1) q^{10} + (\beta_{2} - 7 \beta_1) q^{11} + (4 \beta_{3} - 8) q^{12} + (5 \beta_{3} - 6 \beta_{2} + 7 \beta_1 - 10) q^{13} + ( - 2 \beta_{3} - 44) q^{14} + (50 \beta_{2} - 2 \beta_1) q^{15} - 16 \beta_{2} q^{16} + ( - 8 \beta_{3} - 61 \beta_{2} - 8 \beta_1 + 69) q^{17} + ( - 6 \beta_{3} + 62) q^{18} + (\beta_{3} + \beta_{2} + \beta_1 - 2) q^{19} + (4 \beta_{3} - 16 \beta_{2} + 4 \beta_1 + 12) q^{20} + ( - 21 \beta_{3} - 10) q^{21} + (14 \beta_{3} - 2 \beta_{2} + 14 \beta_1 - 12) q^{22} + ( - 15 \beta_{2} - 3 \beta_1) q^{23} + (8 \beta_{2} + 8 \beta_1) q^{24} + (7 \beta_{3} - 62) q^{25} + ( - 14 \beta_{3} + 22 \beta_{2} - 4 \beta_1 + 2) q^{26} + (7 \beta_{3} - 170) q^{27} + (92 \beta_{2} - 4 \beta_1) q^{28} + (93 \beta_{2} + 12 \beta_1) q^{29} + (4 \beta_{3} - 100 \beta_{2} + 4 \beta_1 + 96) q^{30} + (24 \beta_{3} + 128) q^{31} + (32 \beta_{2} - 32) q^{32} + ( - 13 \beta_{3} - 377 \beta_{2} - 13 \beta_1 + 390) q^{33} + (16 \beta_{3} - 138) q^{34} + ( - 26 \beta_{3} + 146 \beta_{2} - 26 \beta_1 - 120) q^{35} + ( - 112 \beta_{2} - 12 \beta_1) q^{36} + (353 \beta_{2} - 4 \beta_1) q^{37} + ( - 2 \beta_{3} + 4) q^{38} + (8 \beta_{3} + 97 \beta_{2} - 7 \beta_1 - 380) q^{39} + ( - 8 \beta_{3} - 24) q^{40} + ( - 131 \beta_{2} + 20 \beta_1) q^{41} + (62 \beta_{2} - 42 \beta_1) q^{42} + (25 \beta_{3} - 59 \beta_{2} + 25 \beta_1 + 34) q^{43} + ( - 28 \beta_{3} + 24) q^{44} + (19 \beta_{3} + 50 \beta_{2} + 19 \beta_1 - 69) q^{45} + (6 \beta_{3} + 30 \beta_{2} + 6 \beta_1 - 36) q^{46} + (56 \beta_{3} + 84) q^{47} + ( - 16 \beta_{3} - 16 \beta_{2} - 16 \beta_1 + 32) q^{48} + ( - 240 \beta_{2} + 45 \beta_1) q^{49} + (110 \beta_{2} + 14 \beta_1) q^{50} + ( - 77 \beta_{3} + 570) q^{51} + (8 \beta_{3} - 20 \beta_{2} - 20 \beta_1 + 36) q^{52} + ( - \beta_{3} - 267) q^{53} + (326 \beta_{2} + 14 \beta_1) q^{54} + ( - 382 \beta_{2} + 22 \beta_1) q^{55} + (8 \beta_{3} - 184 \beta_{2} + 8 \beta_1 + 176) q^{56} + (3 \beta_{3} - 58) q^{57} + ( - 24 \beta_{3} - 186 \beta_{2} - 24 \beta_1 + 210) q^{58} + ( - 7 \beta_{3} - 191 \beta_{2} - 7 \beta_1 + 198) q^{59} + ( - 8 \beta_{3} - 192) q^{60} + ( - 32 \beta_{3} + 343 \beta_{2} - 32 \beta_1 - 311) q^{61} + ( - 304 \beta_{2} + 48 \beta_1) q^{62} + (482 \beta_{2} + 38 \beta_1) q^{63} + 64 q^{64} + ( - 10 \beta_{3} + 402 \beta_{2} - 27 \beta_1 - 240) q^{65} + (26 \beta_{3} - 780) q^{66} + (85 \beta_{2} - 63 \beta_1) q^{67} + (244 \beta_{2} + 32 \beta_1) q^{68} + ( - 21 \beta_{3} - 177 \beta_{2} - 21 \beta_1 + 198) q^{69} + (52 \beta_{3} + 240) q^{70} + ( - 19 \beta_{3} - 275 \beta_{2} - 19 \beta_1 + 294) q^{71} + (24 \beta_{3} + 224 \beta_{2} + 24 \beta_1 - 248) q^{72} + (15 \beta_{3} - 319) q^{73} + (8 \beta_{3} - 706 \beta_{2} + 8 \beta_1 + 698) q^{74} + ( - 433 \beta_{2} - 69 \beta_1) q^{75} + ( - 4 \beta_{2} - 4 \beta_1) q^{76} + (155 \beta_{3} + 246) q^{77} + (14 \beta_{3} + 550 \beta_{2} + 30 \beta_1 + 180) q^{78} + ( - 48 \beta_{3} + 404) q^{79} + (64 \beta_{2} - 16 \beta_1) q^{80} + (215 \beta_{2} - 96 \beta_1) q^{81} + ( - 40 \beta_{3} + 262 \beta_{2} - 40 \beta_1 - 222) q^{82} + ( - 36 \beta_{3} - 864) q^{83} + (84 \beta_{3} - 124 \beta_{2} + 84 \beta_1 + 40) q^{84} + ( - 37 \beta_{3} - 188 \beta_{2} - 37 \beta_1 + 225) q^{85} + ( - 50 \beta_{3} - 68) q^{86} + (117 \beta_{3} + 741 \beta_{2} + 117 \beta_1 - 858) q^{87} + (8 \beta_{2} - 56 \beta_1) q^{88} + (451 \beta_{2} - 31 \beta_1) q^{89} + ( - 38 \beta_{3} + 138) q^{90} + ( - 55 \beta_{3} - 155 \beta_{2} + 105 \beta_1 - 306) q^{91} + ( - 12 \beta_{3} + 72) q^{92} + ( - 1144 \beta_{2} + 104 \beta_1) q^{93} + ( - 280 \beta_{2} + 112 \beta_1) q^{94} + ( - 2 \beta_{3} + 50 \beta_{2} - 2 \beta_1 - 48) q^{95} + (32 \beta_{3} - 64) q^{96} + ( - 41 \beta_{3} - 419 \beta_{2} - 41 \beta_1 + 460) q^{97} + ( - 90 \beta_{3} + 480 \beta_{2} - 90 \beta_1 - 390) q^{98} + ( - 214 \beta_{3} + 1320) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 3 q^{3} - 8 q^{4} - 14 q^{5} + 6 q^{6} + 45 q^{7} + 32 q^{8} - 59 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 3 q^{3} - 8 q^{4} - 14 q^{5} + 6 q^{6} + 45 q^{7} + 32 q^{8} - 59 q^{9} + 14 q^{10} - 5 q^{11} - 24 q^{12} - 35 q^{13} - 180 q^{14} + 98 q^{15} - 32 q^{16} + 130 q^{17} + 236 q^{18} - 3 q^{19} + 28 q^{20} - 82 q^{21} - 10 q^{22} - 33 q^{23} + 24 q^{24} - 234 q^{25} + 20 q^{26} - 666 q^{27} + 180 q^{28} + 198 q^{29} + 196 q^{30} + 560 q^{31} - 64 q^{32} + 767 q^{33} - 520 q^{34} - 266 q^{35} - 236 q^{36} + 702 q^{37} + 12 q^{38} - 1317 q^{39} - 112 q^{40} - 242 q^{41} + 82 q^{42} + 93 q^{43} + 40 q^{44} - 119 q^{45} - 66 q^{46} + 448 q^{47} + 48 q^{48} - 435 q^{49} + 234 q^{50} + 2126 q^{51} + 100 q^{52} - 1070 q^{53} + 666 q^{54} - 742 q^{55} + 360 q^{56} - 226 q^{57} + 396 q^{58} + 389 q^{59} - 784 q^{60} - 654 q^{61} - 560 q^{62} + 1002 q^{63} + 256 q^{64} - 203 q^{65} - 3068 q^{66} + 107 q^{67} + 520 q^{68} + 375 q^{69} + 1064 q^{70} + 569 q^{71} - 472 q^{72} - 1246 q^{73} + 1404 q^{74} - 935 q^{75} - 12 q^{76} + 1294 q^{77} + 1878 q^{78} + 1520 q^{79} + 112 q^{80} + 334 q^{81} - 484 q^{82} - 3528 q^{83} + 164 q^{84} + 413 q^{85} - 372 q^{86} - 1599 q^{87} - 40 q^{88} + 871 q^{89} + 476 q^{90} - 1539 q^{91} + 264 q^{92} - 2184 q^{93} - 448 q^{94} - 98 q^{95} - 192 q^{96} + 879 q^{97} - 870 q^{98} + 4852 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 55x^{2} + 54x + 2916 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 55\nu^{2} - 55\nu + 2916 ) / 2970 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 109 ) / 55 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 54\beta_{2} + \beta _1 - 55 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 55\beta_{3} - 109 \) Copy content Toggle raw display

Character values

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

\(n\) \(15\)
\(\chi(n)\) \(-1 + \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
−3.43273 5.94566i
3.93273 + 6.81169i
−3.43273 + 5.94566i
3.93273 6.81169i
−1.00000 1.73205i −2.93273 5.07964i −2.00000 + 3.46410i −10.8655 −5.86546 + 10.1593i 14.9327 25.8642i 8.00000 −3.70181 + 6.41172i 10.8655 + 18.8195i
3.2 −1.00000 1.73205i 4.43273 + 7.67771i −2.00000 + 3.46410i 3.86546 8.86546 15.3554i 7.56727 13.1069i 8.00000 −25.7982 + 44.6838i −3.86546 6.69517i
9.1 −1.00000 + 1.73205i −2.93273 + 5.07964i −2.00000 3.46410i −10.8655 −5.86546 10.1593i 14.9327 + 25.8642i 8.00000 −3.70181 6.41172i 10.8655 18.8195i
9.2 −1.00000 + 1.73205i 4.43273 7.67771i −2.00000 3.46410i 3.86546 8.86546 + 15.3554i 7.56727 + 13.1069i 8.00000 −25.7982 44.6838i −3.86546 + 6.69517i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 26.4.c.b 4
3.b odd 2 1 234.4.h.h 4
4.b odd 2 1 208.4.i.d 4
13.b even 2 1 338.4.c.j 4
13.c even 3 1 inner 26.4.c.b 4
13.c even 3 1 338.4.a.h 2
13.d odd 4 2 338.4.e.f 8
13.e even 6 1 338.4.a.g 2
13.e even 6 1 338.4.c.j 4
13.f odd 12 2 338.4.b.e 4
13.f odd 12 2 338.4.e.f 8
39.i odd 6 1 234.4.h.h 4
52.j odd 6 1 208.4.i.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.c.b 4 1.a even 1 1 trivial
26.4.c.b 4 13.c even 3 1 inner
208.4.i.d 4 4.b odd 2 1
208.4.i.d 4 52.j odd 6 1
234.4.h.h 4 3.b odd 2 1
234.4.h.h 4 39.i odd 6 1
338.4.a.g 2 13.e even 6 1
338.4.a.h 2 13.c even 3 1
338.4.b.e 4 13.f odd 12 2
338.4.c.j 4 13.b even 2 1
338.4.c.j 4 13.e even 6 1
338.4.e.f 8 13.d odd 4 2
338.4.e.f 8 13.f odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 3T_{3}^{3} + 61T_{3}^{2} + 156T_{3} + 2704 \) acting on \(S_{4}^{\mathrm{new}}(26, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - 3 T^{3} + 61 T^{2} + \cdots + 2704 \) Copy content Toggle raw display
$5$ \( (T^{2} + 7 T - 42)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 45 T^{3} + 1573 T^{2} + \cdots + 204304 \) Copy content Toggle raw display
$11$ \( T^{4} + 5 T^{3} + 2677 T^{2} + \cdots + 7033104 \) Copy content Toggle raw display
$13$ \( T^{4} + 35 T^{3} + 4212 T^{2} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{4} - 130 T^{3} + 16147 T^{2} + \cdots + 567009 \) Copy content Toggle raw display
$19$ \( T^{4} + 3 T^{3} + 61 T^{2} + \cdots + 2704 \) Copy content Toggle raw display
$23$ \( T^{4} + 33 T^{3} + 1305 T^{2} + \cdots + 46656 \) Copy content Toggle raw display
$29$ \( T^{4} - 198 T^{3} + 37215 T^{2} + \cdots + 3956121 \) Copy content Toggle raw display
$31$ \( (T^{2} - 280 T - 11648)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 702 T^{3} + \cdots + 14965362889 \) Copy content Toggle raw display
$41$ \( T^{4} + 242 T^{3} + \cdots + 49829481 \) Copy content Toggle raw display
$43$ \( T^{4} - 93 T^{3} + \cdots + 1007681536 \) Copy content Toggle raw display
$47$ \( (T^{2} - 224 T - 157584)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 535 T + 71502)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 389 T^{3} + \cdots + 1237069584 \) Copy content Toggle raw display
$61$ \( T^{4} + 654 T^{3} + \cdots + 2639596129 \) Copy content Toggle raw display
$67$ \( T^{4} - 107 T^{3} + \cdots + 45137551936 \) Copy content Toggle raw display
$71$ \( T^{4} - 569 T^{3} + \cdots + 3764558736 \) Copy content Toggle raw display
$73$ \( (T^{2} + 623 T + 84826)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 760 T + 19408)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 1764 T + 707616)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 871 T^{3} + \cdots + 18913400676 \) Copy content Toggle raw display
$97$ \( T^{4} - 879 T^{3} + \cdots + 10397065156 \) Copy content Toggle raw display
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