Properties

Label 13.7.d.a
Level $13$
Weight $7$
Character orbit 13.d
Analytic conductor $2.991$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [13,7,Mod(5,13)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(13, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.5");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 13.d (of order \(4\), 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: \(2.99070308706\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 18 x^{10} + 488 x^{9} + 36205 x^{8} - 155430 x^{7} + 399962 x^{6} + 9502784 x^{5} + \cdots + 56070144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 5^{2}\cdot 13^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_{7} - \beta_{3} + \cdots + \beta_1) q^{3}+ \cdots + ( - \beta_{11} + \beta_{10} + \cdots + 159) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + ( - \beta_{7} - \beta_{3} + \cdots + \beta_1) q^{3}+ \cdots + (2014 \beta_{11} + 1103 \beta_{10} + \cdots + 247870) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{2} - 4 q^{3} + 108 q^{5} - 640 q^{6} + 398 q^{7} - 912 q^{8} + 1940 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{2} - 4 q^{3} + 108 q^{5} - 640 q^{6} + 398 q^{7} - 912 q^{8} + 1940 q^{9} + 1686 q^{11} - 3926 q^{13} + 5484 q^{14} - 15268 q^{15} + 2132 q^{16} + 15254 q^{18} + 1766 q^{19} - 19044 q^{20} + 3428 q^{21} + 28832 q^{22} + 31608 q^{24} - 58266 q^{26} - 20464 q^{27} + 4092 q^{28} - 90108 q^{29} + 61014 q^{31} - 64932 q^{32} - 44452 q^{33} + 259896 q^{34} + 158772 q^{35} - 40212 q^{37} + 137852 q^{39} - 104196 q^{40} - 190416 q^{41} - 959204 q^{42} + 489372 q^{44} - 151444 q^{45} - 44412 q^{46} + 562446 q^{47} + 930308 q^{48} + 82422 q^{50} - 578500 q^{52} + 509136 q^{53} - 871432 q^{54} - 1264036 q^{55} + 939908 q^{57} - 1019980 q^{58} - 994458 q^{59} + 2407804 q^{60} + 1013696 q^{61} + 865778 q^{63} - 1130064 q^{65} - 418352 q^{66} - 1442386 q^{67} - 2313132 q^{68} + 2958968 q^{70} - 655866 q^{71} - 1706508 q^{72} + 2588228 q^{73} + 3373752 q^{74} + 246984 q^{76} + 77480 q^{78} - 75316 q^{79} - 2685408 q^{80} - 4016140 q^{81} + 894966 q^{83} - 3220504 q^{84} + 105396 q^{85} + 3704832 q^{86} + 2109064 q^{87} - 977376 q^{89} + 1088750 q^{91} + 3682872 q^{92} - 216268 q^{93} - 6238300 q^{94} + 896384 q^{96} + 983388 q^{97} + 1039302 q^{98} + 2894714 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 18 x^{10} + 488 x^{9} + 36205 x^{8} - 155430 x^{7} + 399962 x^{6} + 9502784 x^{5} + \cdots + 56070144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 35\!\cdots\!91 \nu^{11} + \cdots - 84\!\cdots\!84 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 32\!\cdots\!99 \nu^{11} + \cdots - 39\!\cdots\!76 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 22\!\cdots\!13 \nu^{11} + \cdots + 29\!\cdots\!12 ) / 55\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 92\!\cdots\!59 \nu^{11} + \cdots + 26\!\cdots\!16 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12\!\cdots\!41 \nu^{11} + \cdots - 13\!\cdots\!84 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 88\!\cdots\!42 \nu^{11} + \cdots - 51\!\cdots\!92 ) / 92\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 88\!\cdots\!22 \nu^{11} + \cdots - 77\!\cdots\!28 ) / 92\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 65\!\cdots\!49 \nu^{11} + \cdots + 78\!\cdots\!72 ) / 66\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 12\!\cdots\!07 \nu^{11} + \cdots - 69\!\cdots\!68 ) / 83\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 18\!\cdots\!17 \nu^{11} + \cdots - 10\!\cdots\!08 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{4} + 90\beta_{3} - \beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} + \beta_{5} + \beta_{4} + 203\beta_{3} - 145\beta_{2} - 203 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 8 \beta_{11} + \beta_{10} - 151 \beta_{8} - 237 \beta_{7} + 8 \beta_{6} + \beta_{5} + 229 \beta_{3} + \cdots - 12974 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 32 \beta_{11} + 271 \beta_{10} - 196 \beta_{9} - 227 \beta_{8} - 196 \beta_{7} - 227 \beta_{4} + \cdots - 18442 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2200 \beta_{11} + 531 \beta_{10} - 47025 \beta_{9} - 2200 \beta_{6} - 531 \beta_{5} - 23503 \beta_{4} + \cdots + 46621 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 79632 \beta_{9} + 47607 \beta_{8} + 79632 \beta_{7} - 12896 \beta_{6} - 56887 \beta_{5} + \cdots + 7589847 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 467992 \beta_{11} - 162311 \beta_{10} + 3755579 \beta_{8} + 8718005 \beta_{7} - 467992 \beta_{6} + \cdots + 311456790 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3532960 \beta_{11} - 10914595 \beta_{10} + 22071372 \beta_{9} + 9671615 \beta_{8} + 22071372 \beta_{7} + \cdots + 854289306 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 90849720 \beta_{11} - 40051887 \beta_{10} + 1567964169 \beta_{9} + 90849720 \beta_{6} + \cdots - 1779235357 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 5203970696 \beta_{9} - 1928696959 \beta_{8} - 5203970696 \beta_{7} + 822529632 \beta_{6} + \cdots - 259296282423 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
9.50218 + 9.50218i
7.85712 + 7.85712i
0.561862 + 0.561862i
0.388469 + 0.388469i
−6.57888 6.57888i
−8.73075 8.73075i
9.50218 9.50218i
7.85712 7.85712i
0.561862 0.561862i
0.388469 0.388469i
−6.57888 + 6.57888i
−8.73075 + 8.73075i
−8.50218 + 8.50218i −17.1101 80.5742i 73.6481 73.6481i 145.473 145.473i −214.232 214.232i 140.917 + 140.917i −436.244 1252.34i
5.2 −6.85712 + 6.85712i 38.7457 30.0401i −127.232 + 127.232i −265.684 + 265.684i 215.792 + 215.792i −232.867 232.867i 772.229 1744.89i
5.3 0.438138 0.438138i −27.8396 63.6161i −78.9164 + 78.9164i −12.1976 + 12.1976i −88.1833 88.1833i 55.9134 + 55.9134i 46.0426 69.1525i
5.4 0.611531 0.611531i 18.7379 63.2521i 134.607 134.607i 11.4588 11.4588i 241.226 + 241.226i 77.8186 + 77.8186i −377.891 164.633i
5.5 7.57888 7.57888i 26.7789 50.8787i −59.0322 + 59.0322i 202.954 202.954i −226.950 226.950i 99.4446 + 99.4446i −11.8881 894.795i
5.6 9.73075 9.73075i −41.3128 125.375i 110.925 110.925i −402.005 + 402.005i 271.347 + 271.347i −597.226 597.226i 977.751 2158.77i
8.1 −8.50218 8.50218i −17.1101 80.5742i 73.6481 + 73.6481i 145.473 + 145.473i −214.232 + 214.232i 140.917 140.917i −436.244 1252.34i
8.2 −6.85712 6.85712i 38.7457 30.0401i −127.232 127.232i −265.684 265.684i 215.792 215.792i −232.867 + 232.867i 772.229 1744.89i
8.3 0.438138 + 0.438138i −27.8396 63.6161i −78.9164 78.9164i −12.1976 12.1976i −88.1833 + 88.1833i 55.9134 55.9134i 46.0426 69.1525i
8.4 0.611531 + 0.611531i 18.7379 63.2521i 134.607 + 134.607i 11.4588 + 11.4588i 241.226 241.226i 77.8186 77.8186i −377.891 164.633i
8.5 7.57888 + 7.57888i 26.7789 50.8787i −59.0322 59.0322i 202.954 + 202.954i −226.950 + 226.950i 99.4446 99.4446i −11.8881 894.795i
8.6 9.73075 + 9.73075i −41.3128 125.375i 110.925 + 110.925i −402.005 402.005i 271.347 271.347i −597.226 + 597.226i 977.751 2158.77i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.7.d.a 12
3.b odd 2 1 117.7.j.b 12
4.b odd 2 1 208.7.t.c 12
13.d odd 4 1 inner 13.7.d.a 12
39.f even 4 1 117.7.j.b 12
52.f even 4 1 208.7.t.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.7.d.a 12 1.a even 1 1 trivial
13.7.d.a 12 13.d odd 4 1 inner
117.7.j.b 12 3.b odd 2 1
117.7.j.b 12 39.f even 4 1
208.7.t.c 12 4.b odd 2 1
208.7.t.c 12 52.f even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(13, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 6 T^{11} + \cdots + 84934656 \) Copy content Toggle raw display
$3$ \( (T^{6} + 2 T^{5} + \cdots - 382594752)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 89\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 12\!\cdots\!41 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 21\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 31\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 57\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 51\!\cdots\!36)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 49\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 48\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 14\!\cdots\!68)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 25\!\cdots\!64 \) Copy content Toggle raw display
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