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Properties

Label 12.13.d.a
Level $12$
Weight $13$
Character orbit 12.d
Analytic conductor $10.968$
Analytic rank $0$
Dimension $12$
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] = [12,13,Mod(7,12)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(12, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("12.7");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 12.d (of order \(2\), degree \(1\), 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: \(10.9679258073\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 1570 x^{10} - 4077 x^{9} + 1884069 x^{8} - 3551868 x^{7} + 881574992 x^{6} + \cdots + 104882177440000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{57}\cdot 3^{25} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

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

\(f(q)\) \(=\) \( q + (\beta_{2} - 8) q^{2} + (\beta_{2} + \beta_1) q^{3} + ( - \beta_{3} - 8 \beta_{2} + \cdots - 386) q^{4}+ \cdots - 177147 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 8) q^{2} + (\beta_{2} + \beta_1) q^{3} + ( - \beta_{3} - 8 \beta_{2} + \cdots - 386) q^{4}+ \cdots + ( - 354294 \beta_{11} + \cdots - 889277940) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 90 q^{2} - 4692 q^{4} - 10296 q^{5} - 48114 q^{6} - 648000 q^{8} - 2125764 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 90 q^{2} - 4692 q^{4} - 10296 q^{5} - 48114 q^{6} - 648000 q^{8} - 2125764 q^{9} + 923028 q^{10} - 3018060 q^{12} + 2094840 q^{13} + 15389208 q^{14} - 61526928 q^{16} - 12097800 q^{17} + 15943230 q^{18} - 377644248 q^{20} - 75057840 q^{21} + 482545560 q^{22} - 332039088 q^{24} + 1058748132 q^{25} + 243358236 q^{26} - 185369520 q^{28} + 1997608680 q^{29} - 577761660 q^{30} + 1733536800 q^{32} + 322101360 q^{33} - 7816269348 q^{34} + 831173724 q^{36} - 960170280 q^{37} - 8280525240 q^{38} + 3985807104 q^{40} + 5806392696 q^{41} - 1390844520 q^{42} + 4989496464 q^{44} + 1823905512 q^{45} + 4149450240 q^{46} - 7791843600 q^{48} - 60479071668 q^{49} + 68552901522 q^{50} - 31090133640 q^{52} + 42482511720 q^{53} + 8523250758 q^{54} - 38053468224 q^{56} - 58319941680 q^{57} + 159666562500 q^{58} - 19968517224 q^{60} + 137368568088 q^{61} - 27876030840 q^{62} + 188355529344 q^{64} - 328250713392 q^{65} - 136719325224 q^{66} + 77938316280 q^{68} + 214339017024 q^{69} - 454939318704 q^{70} + 114791256000 q^{72} - 804477880680 q^{73} - 502785766548 q^{74} + 143972453808 q^{76} + 1383727360320 q^{77} - 72052158420 q^{78} - 417712547808 q^{80} + 376572715308 q^{81} + 460673773020 q^{82} + 28008331632 q^{84} + 1437981718224 q^{85} + 1255416205464 q^{86} + 47622991680 q^{88} - 1422946205928 q^{89} - 163511641116 q^{90} - 3462722444160 q^{92} + 1056734080560 q^{93} + 847910842896 q^{94} + 341032101984 q^{96} - 4056673857000 q^{97} + 1702751294790 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 1570 x^{10} - 4077 x^{9} + 1884069 x^{8} - 3551868 x^{7} + 881574992 x^{6} + \cdots + 104882177440000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 15\!\cdots\!17 \nu^{11} + \cdots - 22\!\cdots\!00 ) / 85\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 23\!\cdots\!61 \nu^{11} + \cdots - 91\!\cdots\!00 ) / 85\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 49\!\cdots\!67 \nu^{11} + \cdots + 44\!\cdots\!00 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 28\!\cdots\!41 \nu^{11} + \cdots + 54\!\cdots\!00 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 24\!\cdots\!44 \nu^{11} + \cdots - 19\!\cdots\!00 ) / 85\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 38\!\cdots\!71 \nu^{11} + \cdots + 10\!\cdots\!00 ) / 85\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 45\!\cdots\!91 \nu^{11} + \cdots - 29\!\cdots\!00 ) / 85\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 12\!\cdots\!66 \nu^{11} + \cdots + 51\!\cdots\!00 ) / 85\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 29\!\cdots\!31 \nu^{11} + \cdots - 18\!\cdots\!00 ) / 85\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 49\!\cdots\!09 \nu^{11} + \cdots + 38\!\cdots\!00 ) / 85\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 32\!\cdots\!73 \nu^{11} + \cdots - 94\!\cdots\!00 ) / 42\!\cdots\!00 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( 8 \beta_{11} + 20 \beta_{10} + 22 \beta_{9} + 22 \beta_{8} - 90 \beta_{7} + 144 \beta_{6} + \cdots + 129832 ) / 373248 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 80 \beta_{11} - 286 \beta_{10} - 401 \beta_{9} + 355 \beta_{8} + 567 \beta_{7} + 4572 \beta_{6} + \cdots - 97635602 ) / 373248 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 4028 \beta_{11} - 14930 \beta_{10} - 7351 \beta_{9} - 23551 \beta_{8} - 167631 \beta_{7} + \cdots - 84683362 ) / 373248 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 3660 \beta_{11} + 147396 \beta_{10} + 9726 \beta_{9} + 570354 \beta_{8} + 431998 \beta_{7} + \cdots - 27007880872 ) / 124416 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 4995128 \beta_{11} - 7412198 \beta_{10} - 5144365 \beta_{9} + 1707263 \beta_{8} + \cdots - 184700568850 ) / 373248 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 71870492 \beta_{11} - 89038526 \beta_{10} + 315768383 \beta_{9} - 2296611649 \beta_{8} + \cdots + 146255720944826 ) / 373248 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 9032756612 \beta_{11} + 24633390032 \beta_{10} + 6792482104 \beta_{9} + 39768341236 \beta_{8} + \cdots - 378740060495780 ) / 373248 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 6806105008 \beta_{11} - 130871998130 \beta_{10} - 154570173319 \beta_{9} + 185237328869 \beta_{8} + \cdots - 23\!\cdots\!34 ) / 124416 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 3890527022236 \beta_{11} - 17094744642106 \beta_{10} + 675981456253 \beta_{9} + \cdots + 12\!\cdots\!74 ) / 373248 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 103575226611500 \beta_{11} + 539346476025884 \beta_{10} + 255009375397138 \beta_{9} + \cdots - 65\!\cdots\!72 ) / 373248 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 51\!\cdots\!48 \beta_{11} + \cdots - 89\!\cdots\!74 ) / 373248 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
14.6572 + 25.3870i
14.6572 − 25.3870i
2.19768 − 3.80649i
2.19768 + 3.80649i
13.5291 + 23.4332i
13.5291 − 23.4332i
−16.4016 + 28.4085i
−16.4016 − 28.4085i
−2.17114 − 3.76053i
−2.17114 + 3.76053i
−10.3112 − 17.8596i
−10.3112 + 17.8596i
−62.3881 − 14.2733i − 420.888i 3688.55 + 1780.96i −21622.0 −6007.45 + 26258.4i − 155240.i −204701. − 163759.i −177147. 1.34896e6 + 308617.i
7.2 −62.3881 + 14.2733i 420.888i 3688.55 − 1780.96i −21622.0 −6007.45 − 26258.4i 155240.i −204701. + 163759.i −177147. 1.34896e6 − 308617.i
7.3 −46.3006 − 44.1844i 420.888i 191.485 + 4091.52i 3884.37 18596.7 − 19487.4i − 57100.7i 171915. − 197900.i −177147. −179848. − 171628.i
7.4 −46.3006 + 44.1844i − 420.888i 191.485 − 4091.52i 3884.37 18596.7 + 19487.4i 57100.7i 171915. + 197900.i −177147. −179848. + 171628.i
7.5 −35.2919 − 53.3899i − 420.888i −1604.96 + 3768.46i 8409.28 −22471.2 + 14854.0i 192052.i 257840. − 47307.4i −177147. −296780. − 448971.i
7.6 −35.2919 + 53.3899i 420.888i −1604.96 − 3768.46i 8409.28 −22471.2 − 14854.0i − 192052.i 257840. + 47307.4i −177147. −296780. + 448971.i
7.7 18.1780 − 61.3642i 420.888i −3435.12 − 2230.95i −204.488 25827.5 + 7650.89i 121613.i −199344. + 170239.i −177147. −3717.17 + 12548.2i
7.8 18.1780 + 61.3642i − 420.888i −3435.12 + 2230.95i −204.488 25827.5 − 7650.89i − 121613.i −199344. − 170239.i −177147. −3717.17 − 12548.2i
7.9 29.4812 − 56.8054i − 420.888i −2357.72 − 3349.39i 28943.6 −23908.8 − 12408.3i − 157943.i −259772. + 35187.3i −177147. 853291. − 1.64415e6i
7.10 29.4812 + 56.8054i 420.888i −2357.72 + 3349.39i 28943.6 −23908.8 + 12408.3i 157943.i −259772. − 35187.3i −177147. 853291. + 1.64415e6i
7.11 51.3214 − 38.2376i − 420.888i 1171.77 − 3924.81i −24558.7 −16093.8 − 21600.6i 96476.2i −89938.5 − 246233.i −177147. −1.26039e6 + 939066.i
7.12 51.3214 + 38.2376i 420.888i 1171.77 + 3924.81i −24558.7 −16093.8 + 21600.6i − 96476.2i −89938.5 + 246233.i −177147. −1.26039e6 − 939066.i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 12.13.d.a ✓ 12
3.b odd 2 1 36.13.d.d 12
4.b odd 2 1 inner 12.13.d.a ✓ 12
8.b even 2 1 192.13.g.e 12
8.d odd 2 1 192.13.g.e 12
12.b even 2 1 36.13.d.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.13.d.a ✓ 12 1.a even 1 1 trivial
12.13.d.a ✓ 12 4.b odd 2 1 inner
36.13.d.d 12 3.b odd 2 1
36.13.d.d 12 12.b even 2 1
192.13.g.e 12 8.b even 2 1
192.13.g.e 12 8.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{13}^{\mathrm{new}}(12, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 47\!\cdots\!96 \) Copy content Toggle raw display
$3$ \( (T^{2} + 177147)^{6} \) Copy content Toggle raw display
$5$ \( (T^{6} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 99\!\cdots\!44 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 13\!\cdots\!84)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 66\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 22\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 21\!\cdots\!04)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 29\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots - 31\!\cdots\!44)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 16\!\cdots\!44)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 95\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 32\!\cdots\!44)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 62\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 15\!\cdots\!16)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 44\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 39\!\cdots\!76)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 53\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 54\!\cdots\!56)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 44\!\cdots\!16)^{2} \) Copy content Toggle raw display
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