Properties

Label 13.6.a.b
Level $13$
Weight $6$
Character orbit 13.a
Self dual yes
Analytic conductor $2.085$
Analytic rank $0$
Dimension $3$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.168897.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 100x + 256 \) 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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 2) q^{2} + ( - \beta_{2} - \beta_1 + 3) q^{3} + (4 \beta_{2} + \beta_1 + 40) q^{4} + ( - \beta_{2} - 7 \beta_1 + 21) q^{5} + ( - 10 \beta_{2} - \beta_1 - 66) q^{6} + ( - 3 \beta_{2} - 3 \beta_1 - 19) q^{7} + (28 \beta_{2} + 27 \beta_1 + 100) q^{8} + (\beta_{2} + 7 \beta_1 - 66) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 2) q^{2} + ( - \beta_{2} - \beta_1 + 3) q^{3} + (4 \beta_{2} + \beta_1 + 40) q^{4} + ( - \beta_{2} - 7 \beta_1 + 21) q^{5} + ( - 10 \beta_{2} - \beta_1 - 66) q^{6} + ( - 3 \beta_{2} - 3 \beta_1 - 19) q^{7} + (28 \beta_{2} + 27 \beta_1 + 100) q^{8} + (\beta_{2} + 7 \beta_1 - 66) q^{9} + ( - 34 \beta_{2} + 23 \beta_1 - 438) q^{10} + (30 \beta_{2} - 26 \beta_1 + 194) q^{11} + ( - 32 \beta_{2} - 83 \beta_1 - 336) q^{12} + 169 q^{13} + ( - 30 \beta_{2} - 31 \beta_1 - 254) q^{14} + (31 \beta_{2} - 17 \beta_1 + 663) q^{15} + (148 \beta_{2} + 181 \beta_1 + 868) q^{16} + ( - 93 \beta_{2} + 77 \beta_1 + 277) q^{17} + (34 \beta_{2} - 68 \beta_1 + 348) q^{18} + ( - 134 \beta_{2} + 178 \beta_1 - 10) q^{19} + ( - 80 \beta_{2} - 407 \beta_1 - 120) q^{20} + (31 \beta_{2} + 49 \beta_1 + 447) q^{21} + (76 \beta_{2} + 370 \beta_1 - 1260) q^{22} + (152 \beta_{2} - 72 \beta_1 + 1232) q^{23} + ( - 204 \beta_{2} - 381 \beta_1 - 4332) q^{24} + (205 \beta_{2} - 365 \beta_1 + 796) q^{25} + (169 \beta_1 + 338) q^{26} + (257 \beta_{2} + 305 \beta_1 - 1527) q^{27} + ( - 208 \beta_{2} - 277 \beta_1 - 2128) q^{28} + ( - 584 \beta_{2} + 56 \beta_1 - 2938) q^{29} + (118 \beta_{2} + 835 \beta_1 + 294) q^{30} + ( - 268 \beta_{2} - 148 \beta_1 - 820) q^{31} + (716 \beta_{2} + 563 \beta_1 + 11436) q^{32} + (134 \beta_{2} - 550 \beta_1 - 426) q^{33} + ( - 250 \beta_{2} - 265 \beta_1 + 5418) q^{34} + (121 \beta_{2} + 145 \beta_1 + 1401) q^{35} + ( - 100 \beta_{2} + 362 \beta_1 - 1680) q^{36} + (419 \beta_{2} - 451 \beta_1 - 6727) q^{37} + ( - 92 \beta_{2} - 858 \beta_1 + 11548) q^{38} + ( - 169 \beta_{2} - 169 \beta_1 + 507) q^{39} + ( - 1020 \beta_{2} - 849 \beta_1 - 14220) q^{40} + ( - 250 \beta_{2} + 858 \beta_1 - 3952) q^{41} + (382 \beta_{2} + 553 \beta_1 + 4350) q^{42} + (697 \beta_{2} - 431 \beta_1 + 821) q^{43} + (976 \beta_{2} - 418 \beta_1 + 16736) q^{44} + ( - 160 \beta_{2} + 680 \beta_1 - 4866) q^{45} + (624 \beta_{2} + 2064 \beta_1 - 1824) q^{46} + ( - 807 \beta_{2} + 33 \beta_1 + 11409) q^{47} + ( - 1724 \beta_{2} - 2315 \beta_1 - 24636) q^{48} + (177 \beta_{2} + 231 \beta_1 - 14934) q^{49} + ( - 230 \beta_{2} + 2186 \beta_1 - 22408) q^{50} + ( - 1265 \beta_{2} + 823 \beta_1 + 4215) q^{51} + (676 \beta_{2} + 169 \beta_1 + 6760) q^{52} + (2154 \beta_{2} + 822 \beta_1 - 4464) q^{53} + (2762 \beta_{2} - 547 \beta_1 + 18714) q^{54} + (578 \beta_{2} - 3550 \beta_1 + 12954) q^{55} + ( - 1396 \beta_{2} - 1899 \beta_1 - 15796) q^{56} + ( - 1950 \beta_{2} + 1662 \beta_1 + 18) q^{57} + ( - 3280 \beta_{2} - 5914 \beta_1 - 4404) q^{58} + ( - 950 \beta_{2} + 1458 \beta_1 + 20830) q^{59} + (3056 \beta_{2} + 593 \beta_1 + 36624) q^{60} + (2330 \beta_{2} + 1910 \beta_1 - 4888) q^{61} + ( - 2200 \beta_{2} - 2012 \beta_1 - 12776) q^{62} + (14 \beta_{2} - 10 \beta_1 - 546) q^{63} + (1812 \beta_{2} + 8661 \beta_1 + 36244) q^{64} + ( - 169 \beta_{2} - 1183 \beta_1 + 3549) q^{65} + ( - 1396 \beta_{2} + 794 \beta_1 - 37716) q^{66} + ( - 1666 \beta_{2} + 1478 \beta_1 + 18218) q^{67} + (416 \beta_{2} + 1969 \beta_1 - 17048) q^{68} + ( - 48 \beta_{2} - 2976 \beta_1 - 5712) q^{69} + (1306 \beta_{2} + 1861 \beta_1 + 13146) q^{70} + (3127 \beta_{2} - 633 \beta_1 + 26071) q^{71} + ( - 240 \beta_{2} - 366 \beta_1 + 9720) q^{72} + ( - 2568 \beta_{2} - 2712 \beta_1 - 13438) q^{73} + (710 \beta_{2} - 4181 \beta_1 - 42446) q^{74} + (2944 \beta_{2} - 3416 \beta_1 + 8988) q^{75} + (304 \beta_{2} + 6250 \beta_1 - 35296) q^{76} + ( - 438 \beta_{2} - 922 \beta_1 - 6710) q^{77} + ( - 1690 \beta_{2} - 169 \beta_1 - 11154) q^{78} + ( - 3576 \beta_{2} - 2856 \beta_1 - 19672) q^{79} + ( - 6956 \beta_{2} - 5447 \beta_1 - 86412) q^{80} + ( - 128 \beta_{2} - 2696 \beta_1 - 35175) q^{81} + (1932 \beta_{2} - 6060 \beta_1 + 49440) q^{82} + (4840 \beta_{2} + 3808 \beta_1 + 31428) q^{83} + (3512 \beta_{2} + 4139 \beta_1 + 33528) q^{84} + ( - 2555 \beta_{2} + 4723 \beta_1 - 19983) q^{85} + (2458 \beta_{2} + 4737 \beta_1 - 24878) q^{86} + (154 \beta_{2} + 9418 \beta_1 + 43218) q^{87} + (1752 \beta_{2} + 10194 \beta_1 + 49272) q^{88} + ( - 432 \beta_{2} - 8864 \beta_1 + 14186) q^{89} + (1760 \beta_{2} - 6346 \beta_1 + 35868) q^{90} + ( - 507 \beta_{2} - 507 \beta_1 - 3211) q^{91} + (7136 \beta_{2} + 1536 \beta_1 + 99776) q^{92} + (932 \beta_{2} + 3620 \beta_1 + 33924) q^{93} + ( - 4710 \beta_{2} + 7341 \beta_1 + 21834) q^{94} + ( - 5418 \beta_{2} + 12150 \beta_1 - 69570) q^{95} + ( - 13076 \beta_{2} - 18749 \beta_1 - 74964) q^{96} + (500 \beta_{2} - 1396 \beta_1 + 25910) q^{97} + (1986 \beta_{2} - 14280 \beta_1 - 13452) q^{98} + ( - 1928 \beta_{2} + 4720 \beta_1 - 21684) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 7 q^{2} + 8 q^{3} + 121 q^{4} + 56 q^{5} - 199 q^{6} - 60 q^{7} + 327 q^{8} - 191 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 7 q^{2} + 8 q^{3} + 121 q^{4} + 56 q^{5} - 199 q^{6} - 60 q^{7} + 327 q^{8} - 191 q^{9} - 1291 q^{10} + 556 q^{11} - 1091 q^{12} + 507 q^{13} - 793 q^{14} + 1972 q^{15} + 2785 q^{16} + 908 q^{17} + 976 q^{18} + 148 q^{19} - 767 q^{20} + 1390 q^{21} - 3410 q^{22} + 3624 q^{23} - 13377 q^{24} + 2023 q^{25} + 1183 q^{26} - 4276 q^{27} - 6661 q^{28} - 8758 q^{29} + 1717 q^{30} - 2608 q^{31} + 34871 q^{32} - 1828 q^{33} + 15989 q^{34} + 4348 q^{35} - 4678 q^{36} - 20632 q^{37} + 33786 q^{38} + 1352 q^{39} - 43509 q^{40} - 10998 q^{41} + 13603 q^{42} + 2032 q^{43} + 49790 q^{44} - 13918 q^{45} - 3408 q^{46} + 34260 q^{47} - 76223 q^{48} - 44571 q^{49} - 65038 q^{50} + 13468 q^{51} + 20449 q^{52} - 12570 q^{53} + 55595 q^{54} + 35312 q^{55} - 49287 q^{56} + 1716 q^{57} - 19126 q^{58} + 63948 q^{59} + 110465 q^{60} - 12754 q^{61} - 40340 q^{62} - 1648 q^{63} + 117393 q^{64} + 9464 q^{65} - 112354 q^{66} + 56132 q^{67} - 49175 q^{68} - 20112 q^{69} + 41299 q^{70} + 77580 q^{71} + 28794 q^{72} - 43026 q^{73} - 131519 q^{74} + 23548 q^{75} - 99638 q^{76} - 21052 q^{77} - 33631 q^{78} - 61872 q^{79} - 264683 q^{80} - 108221 q^{81} + 142260 q^{82} + 98092 q^{83} + 104723 q^{84} - 55226 q^{85} - 69897 q^{86} + 139072 q^{87} + 158010 q^{88} + 33694 q^{89} + 101258 q^{90} - 10140 q^{91} + 300864 q^{92} + 105392 q^{93} + 72843 q^{94} - 196560 q^{95} - 243641 q^{96} + 76334 q^{97} - 54636 q^{98} - 60332 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 100x + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} + 3\nu - 68 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 4\beta_{2} - 3\beta _1 + 68 \) 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
−10.6486
2.68079
8.96778
−8.64858 10.2870 42.7979 92.1784 −88.9676 2.86088 −93.3863 −137.178 −797.212
1.2 4.68079 13.5120 −10.0902 15.4272 63.2466 12.5359 −197.015 −60.4272 72.2115
1.3 10.9678 −15.7989 88.2923 −51.6056 −173.279 −75.3967 617.402 6.60562 −565.999
\(n\): 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.b 3
3.b odd 2 1 117.6.a.d 3
4.b odd 2 1 208.6.a.j 3
5.b even 2 1 325.6.a.c 3
5.c odd 4 2 325.6.b.c 6
7.b odd 2 1 637.6.a.b 3
8.b even 2 1 832.6.a.s 3
8.d odd 2 1 832.6.a.t 3
13.b even 2 1 169.6.a.b 3
13.d odd 4 2 169.6.b.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.6.a.b 3 1.a even 1 1 trivial
117.6.a.d 3 3.b odd 2 1
169.6.a.b 3 13.b even 2 1
169.6.b.b 6 13.d odd 4 2
208.6.a.j 3 4.b odd 2 1
325.6.a.c 3 5.b even 2 1
325.6.b.c 6 5.c odd 4 2
637.6.a.b 3 7.b odd 2 1
832.6.a.s 3 8.b even 2 1
832.6.a.t 3 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 7T_{2}^{2} - 84T_{2} + 444 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(13))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 7 T^{2} + \cdots + 444 \) Copy content Toggle raw display
$3$ \( T^{3} - 8 T^{2} + \cdots + 2196 \) Copy content Toggle raw display
$5$ \( T^{3} - 56 T^{2} + \cdots + 73386 \) Copy content Toggle raw display
$7$ \( T^{3} + 60 T^{2} + \cdots + 2704 \) Copy content Toggle raw display
$11$ \( T^{3} - 556 T^{2} + \cdots + 39698256 \) Copy content Toggle raw display
$13$ \( (T - 169)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - 908 T^{2} + \cdots + 77884638 \) Copy content Toggle raw display
$19$ \( T^{3} + \cdots + 1415854512 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots + 5045833728 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots - 221025174456 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots - 1607044480 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots + 46212896426 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots + 29456898048 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots + 281385762060 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots - 696870885384 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots - 4415410372608 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots - 1932677407728 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots - 18650455523968 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots + 2080268535536 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots + 37395101110464 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots + 5649650834008 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots - 2044988893184 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots + 17971240920768 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 28887869991912 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 12102379894216 \) Copy content Toggle raw display
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