Properties

Label 26.6.a.c
Level $26$
Weight $6$
Character orbit 26.a
Self dual yes
Analytic conductor $4.170$
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] = [26,6,Mod(1,26)]
 
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([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("26.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 26 = 2 \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 26.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: \(4.16997931514\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{849}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 212 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{849})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 q^{2} + ( - \beta + 5) q^{3} + 16 q^{4} + (3 \beta + 35) q^{5} + ( - 4 \beta + 20) q^{6} + (9 \beta + 73) q^{7} + 64 q^{8} + ( - 9 \beta - 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + ( - \beta + 5) q^{3} + 16 q^{4} + (3 \beta + 35) q^{5} + ( - 4 \beta + 20) q^{6} + (9 \beta + 73) q^{7} + 64 q^{8} + ( - 9 \beta - 6) q^{9} + (12 \beta + 140) q^{10} + ( - 24 \beta - 98) q^{11} + ( - 16 \beta + 80) q^{12} - 169 q^{13} + (36 \beta + 292) q^{14} + ( - 23 \beta - 461) q^{15} + 256 q^{16} + (129 \beta - 159) q^{17} + ( - 36 \beta - 24) q^{18} + ( - 60 \beta - 1218) q^{19} + (48 \beta + 560) q^{20} + ( - 37 \beta - 1543) q^{21} + ( - 96 \beta - 392) q^{22} + ( - 108 \beta - 1468) q^{23} + ( - 64 \beta + 320) q^{24} + (219 \beta + 8) q^{25} - 676 q^{26} + (213 \beta + 663) q^{27} + (144 \beta + 1168) q^{28} + ( - 264 \beta + 1082) q^{29} + ( - 92 \beta - 1844) q^{30} + ( - 378 \beta + 1588) q^{31} + 1024 q^{32} + (2 \beta + 4598) q^{33} + (516 \beta - 636) q^{34} + (561 \beta + 8279) q^{35} + ( - 144 \beta - 96) q^{36} + ( - 177 \beta + 8991) q^{37} + ( - 240 \beta - 4872) q^{38} + (169 \beta - 845) q^{39} + (192 \beta + 2240) q^{40} + ( - 462 \beta + 6048) q^{41} + ( - 148 \beta - 6172) q^{42} + ( - 219 \beta - 1925) q^{43} + ( - 384 \beta - 1568) q^{44} + ( - 360 \beta - 5934) q^{45} + ( - 432 \beta - 5872) q^{46} + (405 \beta - 12947) q^{47} + ( - 256 \beta + 1280) q^{48} + (1395 \beta + 5694) q^{49} + (876 \beta + 32) q^{50} + (675 \beta - 28143) q^{51} - 2704 q^{52} + ( - 798 \beta - 1908) q^{53} + (852 \beta + 2652) q^{54} + ( - 1206 \beta - 18694) q^{55} + (576 \beta + 4672) q^{56} + (978 \beta + 6630) q^{57} + ( - 1056 \beta + 4328) q^{58} + ( - 456 \beta - 11482) q^{59} + ( - 368 \beta - 7376) q^{60} + ( - 402 \beta + 48616) q^{61} + ( - 1512 \beta + 6352) q^{62} + ( - 792 \beta - 17610) q^{63} + 4096 q^{64} + ( - 507 \beta - 5915) q^{65} + (8 \beta + 18392) q^{66} + ( - 276 \beta + 36358) q^{67} + (2064 \beta - 2544) q^{68} + (1036 \beta + 15556) q^{69} + (2244 \beta + 33116) q^{70} + (3219 \beta - 19949) q^{71} + ( - 576 \beta - 384) q^{72} + ( - 3084 \beta + 25118) q^{73} + ( - 708 \beta + 35964) q^{74} + (868 \beta - 46388) q^{75} + ( - 960 \beta - 19488) q^{76} + ( - 2850 \beta - 52946) q^{77} + (676 \beta - 3380) q^{78} + (1056 \beta + 25484) q^{79} + (768 \beta + 8960) q^{80} + (2376 \beta - 40383) q^{81} + ( - 1848 \beta + 24192) q^{82} + ( - 1134 \beta - 18312) q^{83} + ( - 592 \beta - 24688) q^{84} + (4425 \beta + 76479) q^{85} + ( - 876 \beta - 7700) q^{86} + ( - 2138 \beta + 61378) q^{87} + ( - 1536 \beta - 6272) q^{88} + (2820 \beta - 53946) q^{89} + ( - 1440 \beta - 23736) q^{90} + ( - 1521 \beta - 12337) q^{91} + ( - 1728 \beta - 23488) q^{92} + ( - 3100 \beta + 88076) q^{93} + (1620 \beta - 51788) q^{94} + ( - 5934 \beta - 80790) q^{95} + ( - 1024 \beta + 5120) q^{96} + (5724 \beta + 49334) q^{97} + (5580 \beta + 22776) q^{98} + (1242 \beta + 46380) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{2} + 9 q^{3} + 32 q^{4} + 73 q^{5} + 36 q^{6} + 155 q^{7} + 128 q^{8} - 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{2} + 9 q^{3} + 32 q^{4} + 73 q^{5} + 36 q^{6} + 155 q^{7} + 128 q^{8} - 21 q^{9} + 292 q^{10} - 220 q^{11} + 144 q^{12} - 338 q^{13} + 620 q^{14} - 945 q^{15} + 512 q^{16} - 189 q^{17} - 84 q^{18} - 2496 q^{19} + 1168 q^{20} - 3123 q^{21} - 880 q^{22} - 3044 q^{23} + 576 q^{24} + 235 q^{25} - 1352 q^{26} + 1539 q^{27} + 2480 q^{28} + 1900 q^{29} - 3780 q^{30} + 2798 q^{31} + 2048 q^{32} + 9198 q^{33} - 756 q^{34} + 17119 q^{35} - 336 q^{36} + 17805 q^{37} - 9984 q^{38} - 1521 q^{39} + 4672 q^{40} + 11634 q^{41} - 12492 q^{42} - 4069 q^{43} - 3520 q^{44} - 12228 q^{45} - 12176 q^{46} - 25489 q^{47} + 2304 q^{48} + 12783 q^{49} + 940 q^{50} - 55611 q^{51} - 5408 q^{52} - 4614 q^{53} + 6156 q^{54} - 38594 q^{55} + 9920 q^{56} + 14238 q^{57} + 7600 q^{58} - 23420 q^{59} - 15120 q^{60} + 96830 q^{61} + 11192 q^{62} - 36012 q^{63} + 8192 q^{64} - 12337 q^{65} + 36792 q^{66} + 72440 q^{67} - 3024 q^{68} + 32148 q^{69} + 68476 q^{70} - 36679 q^{71} - 1344 q^{72} + 47152 q^{73} + 71220 q^{74} - 91908 q^{75} - 39936 q^{76} - 108742 q^{77} - 6084 q^{78} + 52024 q^{79} + 18688 q^{80} - 78390 q^{81} + 46536 q^{82} - 37758 q^{83} - 49968 q^{84} + 157383 q^{85} - 16276 q^{86} + 120618 q^{87} - 14080 q^{88} - 105072 q^{89} - 48912 q^{90} - 26195 q^{91} - 48704 q^{92} + 173052 q^{93} - 101956 q^{94} - 167514 q^{95} + 9216 q^{96} + 104392 q^{97} + 51132 q^{98} + 94002 q^{99}+O(q^{100}) \) 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
15.0688
−14.0688
4.00000 −10.0688 16.0000 80.2064 −40.2752 208.619 64.0000 −141.619 320.826
1.2 4.00000 19.0688 16.0000 −7.20641 76.2752 −53.6192 64.0000 120.619 −28.8256
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(13\) \(1\)

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 26.6.a.c 2
3.b odd 2 1 234.6.a.h 2
4.b odd 2 1 208.6.a.g 2
5.b even 2 1 650.6.a.b 2
5.c odd 4 2 650.6.b.h 4
8.b even 2 1 832.6.a.k 2
8.d odd 2 1 832.6.a.m 2
13.b even 2 1 338.6.a.f 2
13.d odd 4 2 338.6.b.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.6.a.c 2 1.a even 1 1 trivial
208.6.a.g 2 4.b odd 2 1
234.6.a.h 2 3.b odd 2 1
338.6.a.f 2 13.b even 2 1
338.6.b.b 4 13.d odd 4 2
650.6.a.b 2 5.b even 2 1
650.6.b.h 4 5.c odd 4 2
832.6.a.k 2 8.b even 2 1
832.6.a.m 2 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 9T_{3} - 192 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(26))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 9T - 192 \) Copy content Toggle raw display
$5$ \( T^{2} - 73T - 578 \) Copy content Toggle raw display
$7$ \( T^{2} - 155T - 11186 \) Copy content Toggle raw display
$11$ \( T^{2} + 220T - 110156 \) Copy content Toggle raw display
$13$ \( (T + 169)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 189 T - 3523122 \) Copy content Toggle raw display
$19$ \( T^{2} + 2496 T + 793404 \) Copy content Toggle raw display
$23$ \( T^{2} + 3044 T - 159200 \) Copy content Toggle raw display
$29$ \( T^{2} - 1900 T - 13890476 \) Copy content Toggle raw display
$31$ \( T^{2} - 2798 T - 28369928 \) Copy content Toggle raw display
$37$ \( T^{2} - 17805 T + 72604926 \) Copy content Toggle raw display
$41$ \( T^{2} - 11634 T - 11466000 \) Copy content Toggle raw display
$43$ \( T^{2} + 4069 T - 6040532 \) Copy content Toggle raw display
$47$ \( T^{2} + 25489 T + 127607974 \) Copy content Toggle raw display
$53$ \( T^{2} + 4614 T - 129839400 \) Copy content Toggle raw display
$59$ \( T^{2} + 23420 T + 92989684 \) Copy content Toggle raw display
$61$ \( T^{2} - 96830 T + 2309711776 \) Copy content Toggle raw display
$67$ \( T^{2} - 72440 T + 1295720044 \) Copy content Toggle raw display
$71$ \( T^{2} + 36679 T - 1862988962 \) Copy content Toggle raw display
$73$ \( T^{2} - 47152 T - 1462893860 \) Copy content Toggle raw display
$79$ \( T^{2} - 52024 T + 439936528 \) Copy content Toggle raw display
$83$ \( T^{2} + 37758 T + 83472480 \) Copy content Toggle raw display
$89$ \( T^{2} + 105072 T + 1072134396 \) Copy content Toggle raw display
$97$ \( T^{2} - 104392 T - 4229773940 \) Copy content Toggle raw display
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