Properties

Label 13.6.a.a
Level $13$
Weight $6$
Character orbit 13.a
Self dual yes
Analytic conductor $2.085$
Analytic rank $1$
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] = [13,6,Mod(1,13)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(13, 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("13.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 13.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: \(2.08498965757\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 2) q^{2} + (6 \beta - 17) q^{3} + (5 \beta - 24) q^{4} + ( - 40 \beta - 1) q^{5} + ( - \beta + 10) q^{6} + (70 \beta - 53) q^{7} + (41 \beta + 92) q^{8} + ( - 168 \beta + 190) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 2) q^{2} + (6 \beta - 17) q^{3} + (5 \beta - 24) q^{4} + ( - 40 \beta - 1) q^{5} + ( - \beta + 10) q^{6} + (70 \beta - 53) q^{7} + (41 \beta + 92) q^{8} + ( - 168 \beta + 190) q^{9} + (121 \beta + 162) q^{10} + (84 \beta - 230) q^{11} + ( - 199 \beta + 528) q^{12} - 169 q^{13} + ( - 157 \beta - 174) q^{14} + (434 \beta - 943) q^{15} + ( - 375 \beta + 420) q^{16} + (128 \beta - 1379) q^{17} + (314 \beta + 292) q^{18} + ( - 28 \beta - 142) q^{19} + (755 \beta - 776) q^{20} + ( - 1088 \beta + 2581) q^{21} + ( - 22 \beta + 124) q^{22} + ( - 1416 \beta - 604) q^{23} + (101 \beta - 580) q^{24} + (1680 \beta + 3276) q^{25} + (169 \beta + 338) q^{26} + (1530 \beta - 3131) q^{27} + ( - 1595 \beta + 2672) q^{28} + ( - 48 \beta - 382) q^{29} + ( - 359 \beta + 150) q^{30} + (448 \beta + 3636) q^{31} + ( - 607 \beta - 2284) q^{32} + ( - 2304 \beta + 5926) q^{33} + (995 \beta + 2246) q^{34} + ( - 750 \beta - 11147) q^{35} + (4142 \beta - 7920) q^{36} + (1840 \beta - 9349) q^{37} + (226 \beta + 396) q^{38} + ( - 1014 \beta + 2873) q^{39} + ( - 5361 \beta - 6652) q^{40} + (1952 \beta + 2944) q^{41} + (683 \beta - 810) q^{42} + ( - 4718 \beta + 3569) q^{43} + ( - 2746 \beta + 7200) q^{44} + ( - 712 \beta + 26690) q^{45} + (4852 \beta + 6872) q^{46} + (9670 \beta + 151) q^{47} + (6645 \beta - 16140) q^{48} + ( - 2520 \beta + 5602) q^{49} + ( - 8316 \beta - 13272) q^{50} + ( - 9682 \beta + 26515) q^{51} + ( - 845 \beta + 4056) q^{52} + (6816 \beta - 25268) q^{53} + ( - 1459 \beta + 142) q^{54} + (5756 \beta - 13210) q^{55} + (7137 \beta + 6604) q^{56} + ( - 544 \beta + 1742) q^{57} + (526 \beta + 956) q^{58} + ( - 8668 \beta - 15134) q^{59} + ( - 12961 \beta + 31312) q^{60} + ( - 9296 \beta + 5640) q^{61} + ( - 4980 \beta - 9064) q^{62} + (10444 \beta - 57110) q^{63} + (16105 \beta - 6444) q^{64} + (6760 \beta + 169) q^{65} + (986 \beta - 2636) q^{66} + (196 \beta - 35062) q^{67} + ( - 9327 \beta + 35656) q^{68} + (11952 \beta - 23716) q^{69} + (13397 \beta + 25294) q^{70} + (3666 \beta + 31865) q^{71} + ( - 14554 \beta - 10072) q^{72} + ( - 18560 \beta + 46486) q^{73} + (3829 \beta + 11338) q^{74} + (1176 \beta - 15372) q^{75} + ( - 178 \beta + 2848) q^{76} + ( - 14672 \beta + 35710) q^{77} + (169 \beta - 1690) q^{78} + ( - 9736 \beta - 22780) q^{79} + ( - 1425 \beta + 59580) q^{80} + (5208 \beta + 43777) q^{81} + ( - 8800 \beta - 13696) q^{82} + ( - 17920 \beta - 28896) q^{83} + (33577 \beta - 83704) q^{84} + (49912 \beta - 19101) q^{85} + (10585 \beta + 11734) q^{86} + ( - 1764 \beta + 5342) q^{87} + (1742 \beta - 7384) q^{88} + ( - 23504 \beta + 70006) q^{89} + ( - 24554 \beta - 50532) q^{90} + ( - 11830 \beta + 8957) q^{91} + (23884 \beta - 13824) q^{92} + (16888 \beta - 51060) q^{93} + ( - 29161 \beta - 38982) q^{94} + (6828 \beta + 4622) q^{95} + ( - 7027 \beta + 24260) q^{96} + ( - 58720 \beta + 11982) q^{97} + (1958 \beta - 1124) q^{98} + (40488 \beta - 100148) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 5 q^{2} - 28 q^{3} - 43 q^{4} - 42 q^{5} + 19 q^{6} - 36 q^{7} + 225 q^{8} + 212 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 5 q^{2} - 28 q^{3} - 43 q^{4} - 42 q^{5} + 19 q^{6} - 36 q^{7} + 225 q^{8} + 212 q^{9} + 445 q^{10} - 376 q^{11} + 857 q^{12} - 338 q^{13} - 505 q^{14} - 1452 q^{15} + 465 q^{16} - 2630 q^{17} + 898 q^{18} - 312 q^{19} - 797 q^{20} + 4074 q^{21} + 226 q^{22} - 2624 q^{23} - 1059 q^{24} + 8232 q^{25} + 845 q^{26} - 4732 q^{27} + 3749 q^{28} - 812 q^{29} - 59 q^{30} + 7720 q^{31} - 5175 q^{32} + 9548 q^{33} + 5487 q^{34} - 23044 q^{35} - 11698 q^{36} - 16858 q^{37} + 1018 q^{38} + 4732 q^{39} - 18665 q^{40} + 7840 q^{41} - 937 q^{42} + 2420 q^{43} + 11654 q^{44} + 52668 q^{45} + 18596 q^{46} + 9972 q^{47} - 25635 q^{48} + 8684 q^{49} - 34860 q^{50} + 43348 q^{51} + 7267 q^{52} - 43720 q^{53} - 1175 q^{54} - 20664 q^{55} + 20345 q^{56} + 2940 q^{57} + 2438 q^{58} - 38936 q^{59} + 49663 q^{60} + 1984 q^{61} - 23108 q^{62} - 103776 q^{63} + 3217 q^{64} + 7098 q^{65} - 4286 q^{66} - 69928 q^{67} + 61985 q^{68} - 35480 q^{69} + 63985 q^{70} + 67396 q^{71} - 34698 q^{72} + 74412 q^{73} + 26505 q^{74} - 29568 q^{75} + 5518 q^{76} + 56748 q^{77} - 3211 q^{78} - 55296 q^{79} + 117735 q^{80} + 92762 q^{81} - 36192 q^{82} - 75712 q^{83} - 133831 q^{84} + 11710 q^{85} + 34053 q^{86} + 8920 q^{87} - 13026 q^{88} + 116508 q^{89} - 125618 q^{90} + 6084 q^{91} - 3764 q^{92} - 85232 q^{93} - 107125 q^{94} + 16072 q^{95} + 41493 q^{96} - 34756 q^{97} - 290 q^{98} - 159808 q^{99}+O(q^{100}) \) 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
2.56155
−1.56155
−4.56155 −1.63068 −11.1922 −103.462 7.43845 126.309 197.024 −240.341 471.948
1.2 −0.438447 −26.3693 −31.8078 61.4621 11.5616 −162.309 27.9763 452.341 −26.9479
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \( +1 \)

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 13.6.a.a 2
3.b odd 2 1 117.6.a.c 2
4.b odd 2 1 208.6.a.h 2
5.b even 2 1 325.6.a.b 2
5.c odd 4 2 325.6.b.b 4
7.b odd 2 1 637.6.a.a 2
8.b even 2 1 832.6.a.p 2
8.d odd 2 1 832.6.a.i 2
13.b even 2 1 169.6.a.a 2
13.d odd 4 2 169.6.b.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.6.a.a 2 1.a even 1 1 trivial
117.6.a.c 2 3.b odd 2 1
169.6.a.a 2 13.b even 2 1
169.6.b.a 4 13.d odd 4 2
208.6.a.h 2 4.b odd 2 1
325.6.a.b 2 5.b even 2 1
325.6.b.b 4 5.c odd 4 2
637.6.a.a 2 7.b odd 2 1
832.6.a.i 2 8.d odd 2 1
832.6.a.p 2 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 5T_{2} + 2 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(13))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 5T + 2 \) Copy content Toggle raw display
$3$ \( T^{2} + 28T + 43 \) Copy content Toggle raw display
$5$ \( T^{2} + 42T - 6359 \) Copy content Toggle raw display
$7$ \( T^{2} + 36T - 20501 \) Copy content Toggle raw display
$11$ \( T^{2} + 376T + 5356 \) Copy content Toggle raw display
$13$ \( (T + 169)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2630 T + 1659593 \) Copy content Toggle raw display
$19$ \( T^{2} + 312T + 21004 \) Copy content Toggle raw display
$23$ \( T^{2} + 2624 T - 6800144 \) Copy content Toggle raw display
$29$ \( T^{2} + 812T + 155044 \) Copy content Toggle raw display
$31$ \( T^{2} - 7720 T + 14046608 \) Copy content Toggle raw display
$37$ \( T^{2} + 16858 T + 56659241 \) Copy content Toggle raw display
$41$ \( T^{2} - 7840 T - 827392 \) Copy content Toggle raw display
$43$ \( T^{2} - 2420 T - 93138877 \) Copy content Toggle raw display
$47$ \( T^{2} - 9972 T - 372552629 \) Copy content Toggle raw display
$53$ \( T^{2} + 43720 T + 280413712 \) Copy content Toggle raw display
$59$ \( T^{2} + 38936 T + 59682572 \) Copy content Toggle raw display
$61$ \( T^{2} - 1984 T - 366282304 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 1222318028 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 1078437091 \) Copy content Toggle raw display
$73$ \( T^{2} - 74412 T - 79726364 \) Copy content Toggle raw display
$79$ \( T^{2} + 55296 T + 361555696 \) Copy content Toggle raw display
$83$ \( T^{2} + 75712 T + 68289536 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 1045666948 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 14352168316 \) Copy content Toggle raw display
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