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Properties

Label 12.9.d.a
Level $12$
Weight $9$
Character orbit 12.d
Analytic conductor $4.889$
Analytic rank $0$
Dimension $8$
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] = [12,9,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, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("12.7");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
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: \(4.88854332073\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} - 40x^{6} - 395x^{5} + 403x^{4} + 8998x^{3} + 74584x^{2} + 217224x + 269328 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{10} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{2} + ( - \beta_{3} - \beta_1) q^{3} + ( - \beta_{3} + \beta_{2} - 6) q^{4} + (2 \beta_{5} - \beta_{4} - \beta_{3} + \cdots - 37) q^{5}+ \cdots - 2187 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{2} + ( - \beta_{3} - \beta_1) q^{3} + ( - \beta_{3} + \beta_{2} - 6) q^{4} + (2 \beta_{5} - \beta_{4} - \beta_{3} + \cdots - 37) q^{5}+ \cdots + (26244 \beta_{7} + 17496 \beta_{6} + \cdots - 170586) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{2} - 52 q^{4} - 336 q^{5} + 1134 q^{6} - 12960 q^{8} - 17496 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{2} - 52 q^{4} - 336 q^{5} + 1134 q^{6} - 12960 q^{8} - 17496 q^{9} + 36628 q^{10} - 11340 q^{12} - 2864 q^{13} + 52728 q^{14} + 99440 q^{16} - 193200 q^{17} - 13122 q^{18} + 335592 q^{20} + 121824 q^{21} - 556968 q^{22} + 221616 q^{24} - 579048 q^{25} + 21564 q^{26} - 594672 q^{28} + 2063472 q^{29} + 46980 q^{30} - 3602784 q^{32} - 920160 q^{33} + 1568476 q^{34} + 113724 q^{36} + 7470352 q^{37} + 3659400 q^{38} + 1749184 q^{40} - 8865456 q^{41} - 5288328 q^{42} + 2395920 q^{44} + 734832 q^{45} - 13649856 q^{46} + 10916208 q^{48} - 18923896 q^{49} + 14581842 q^{50} + 18592888 q^{52} + 8706672 q^{53} - 2480058 q^{54} - 45565632 q^{56} - 2325024 q^{57} - 8816444 q^{58} + 28348056 q^{60} + 13457296 q^{61} + 80783976 q^{62} + 1268864 q^{64} + 7293408 q^{65} - 51205608 q^{66} - 117288264 q^{68} - 8636544 q^{69} - 60373104 q^{70} + 28343520 q^{72} + 94738960 q^{73} + 119548428 q^{74} + 144621360 q^{76} - 56971392 q^{77} - 140630580 q^{78} - 163857888 q^{80} + 38263752 q^{81} - 188383460 q^{82} + 199712304 q^{84} - 201200416 q^{85} + 240327384 q^{86} + 156323520 q^{88} + 188992272 q^{89} - 80105436 q^{90} - 387657984 q^{92} - 54802656 q^{93} - 38749872 q^{94} + 246092256 q^{96} - 123291632 q^{97} + 691081830 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} - 40x^{6} - 395x^{5} + 403x^{4} + 8998x^{3} + 74584x^{2} + 217224x + 269328 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 13071742 \nu^{7} - 40510221 \nu^{6} + 365720344 \nu^{5} + 4477637544 \nu^{4} + \cdots - 864197730939 ) / 48999153875 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 442676109 \nu^{7} + 14876179533 \nu^{6} - 20899272462 \nu^{5} - 244299281437 \nu^{4} + \cdots + 257458199902472 ) / 783986462000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 16258001 \nu^{7} - 126478257 \nu^{6} - 1086101402 \nu^{5} - 4794860127 \nu^{4} + \cdots + 2529327015672 ) / 22399613200 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 909414791 \nu^{7} + 1589452833 \nu^{6} + 7394147338 \nu^{5} - 139320190137 \nu^{4} + \cdots - 157888370611128 ) / 391993231000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1924762227 \nu^{7} + 1444346301 \nu^{6} + 32579772786 \nu^{5} - 555685145789 \nu^{4} + \cdots - 343266156881416 ) / 783986462000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2088295459 \nu^{7} + 1743279843 \nu^{6} + 40266101998 \nu^{5} + 817766792173 \nu^{4} + \cdots - 153540313829368 ) / 156797292400 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 844419621 \nu^{7} + 4157547477 \nu^{6} + 19453004322 \nu^{5} + 248245173547 \nu^{4} + \cdots - 74886881534632 ) / 55999033000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{7} - 5\beta_{6} - 12\beta_{5} + 8\beta_{4} - 7\beta_{2} + 54\beta _1 + 335 ) / 864 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 12\beta_{7} - 11\beta_{6} - 12\beta_{5} - 22\beta_{4} - 16\beta_{3} - 55\beta_{2} - 358\beta _1 + 4715 ) / 432 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 207\beta_{7} - 175\beta_{6} - 18\beta_{5} + 64\beta_{4} + 914\beta_{3} - 413\beta_{2} + 788\beta _1 + 169669 ) / 864 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1395 \beta_{7} - 1505 \beta_{6} - 3942 \beta_{5} + 1940 \beta_{4} + 374 \beta_{3} - 4999 \beta_{2} + \cdots + 843359 ) / 864 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6099 \beta_{7} - 6978 \beta_{6} - 7368 \beta_{5} - 2886 \beta_{4} - 5956 \beta_{3} - 22716 \beta_{2} + \cdots + 4054608 ) / 432 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 114921 \beta_{7} - 109501 \beta_{6} - 117126 \beta_{5} + 41656 \beta_{4} + 242822 \beta_{3} + \cdots + 70038967 ) / 864 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 815151 \beta_{7} - 862161 \beta_{6} - 1430598 \beta_{5} + 431916 \beta_{4} + 507302 \beta_{3} + \cdots + 525219687 ) / 864 \) 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
−1.97054 − 1.25304i
−1.97054 + 1.25304i
8.23534 + 0.0522875i
8.23534 − 0.0522875i
−1.11858 − 4.64627i
−1.11858 + 4.64627i
−3.64622 − 4.31154i
−3.64622 + 4.31154i
−15.8645 − 2.07809i 46.7654i 247.363 + 65.9356i −374.901 97.1827 − 741.908i − 4472.52i −3787.26 − 1560.08i −2187.00 5947.61 + 779.079i
7.2 −15.8645 + 2.07809i − 46.7654i 247.363 − 65.9356i −374.901 97.1827 + 741.908i 4472.52i −3787.26 + 1560.08i −2187.00 5947.61 − 779.079i
7.3 −2.26258 − 15.8392i 46.7654i −245.761 + 71.6749i −538.046 740.727 − 105.810i 3350.97i 1691.33 + 3730.50i −2187.00 1217.37 + 8522.23i
7.4 −2.26258 + 15.8392i − 46.7654i −245.761 − 71.6749i −538.046 740.727 + 105.810i − 3350.97i 1691.33 − 3730.50i −2187.00 1217.37 − 8522.23i
7.5 7.47945 − 14.1442i − 46.7654i −144.116 − 211.581i −159.249 −661.458 − 349.779i − 707.133i −4070.55 + 455.883i −2187.00 −1191.10 + 2252.45i
7.6 7.47945 + 14.1442i 46.7654i −144.116 + 211.581i −159.249 −661.458 + 349.779i 707.133i −4070.55 − 455.883i −2187.00 −1191.10 − 2252.45i
7.7 13.6476 − 8.35123i 46.7654i 116.514 − 227.948i 904.196 390.548 + 638.235i − 888.085i −313.513 − 4083.98i −2187.00 12340.1 − 7551.15i
7.8 13.6476 + 8.35123i − 46.7654i 116.514 + 227.948i 904.196 390.548 − 638.235i 888.085i −313.513 + 4083.98i −2187.00 12340.1 + 7551.15i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.8
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 12.9.d.a ✓ 8
3.b odd 2 1 36.9.d.c 8
4.b odd 2 1 inner 12.9.d.a ✓ 8
8.b even 2 1 192.9.g.e 8
8.d odd 2 1 192.9.g.e 8
12.b even 2 1 36.9.d.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.9.d.a ✓ 8 1.a even 1 1 trivial
12.9.d.a ✓ 8 4.b odd 2 1 inner
36.9.d.c 8 3.b odd 2 1
36.9.d.c 8 12.b even 2 1
192.9.g.e 8 8.b even 2 1
192.9.g.e 8 8.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(12, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + \cdots + 4294967296 \) Copy content Toggle raw display
$3$ \( (T^{2} + 2187)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} + 168 T^{3} + \cdots - 29045327600)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 88\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 71\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 46\!\cdots\!76)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 20\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 83\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 15\!\cdots\!24)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 17\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 48\!\cdots\!84)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 44\!\cdots\!92)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 21\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots - 39\!\cdots\!44)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 34\!\cdots\!04)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 42\!\cdots\!28)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 29\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 25\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 16\!\cdots\!88)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 30\!\cdots\!48)^{2} \) Copy content Toggle raw display
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