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Properties

Label 14.9.d.a
Level $14$
Weight $9$
Character orbit 14.d
Analytic conductor $5.703$
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] = [14,9,Mod(3,14)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(14, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("14.3");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 14.d (of order \(6\), 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: \(5.70330054086\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 1771 x^{10} + 26038 x^{9} + 2442597 x^{8} + 26522276 x^{7} + 1175865280 x^{6} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{4}\cdot 7^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_{2}) q^{2} + (\beta_{4} + \beta_{2} - 9 \beta_1 + 18) q^{3} - 128 \beta_1 q^{4} + (\beta_{9} + \beta_{7} - 2 \beta_{5} + \cdots + 93) q^{5}+ \cdots + (\beta_{11} - 2 \beta_{10} + \cdots + 3933) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_{2}) q^{2} + (\beta_{4} + \beta_{2} - 9 \beta_1 + 18) q^{3} - 128 \beta_1 q^{4} + (\beta_{9} + \beta_{7} - 2 \beta_{5} + \cdots + 93) q^{5}+ \cdots + (30465 \beta_{11} + 58524 \beta_{10} + \cdots - 28914951) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 162 q^{3} - 768 q^{4} + 1674 q^{5} - 1308 q^{7} + 23604 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 162 q^{3} - 768 q^{4} + 1674 q^{5} - 1308 q^{7} + 23604 q^{9} + 17664 q^{10} + 10302 q^{11} - 20736 q^{12} + 56832 q^{14} - 255468 q^{15} - 98304 q^{16} + 173178 q^{17} + 16896 q^{18} + 405978 q^{19} - 656910 q^{21} - 941568 q^{22} + 158934 q^{23} + 98304 q^{24} + 838668 q^{25} + 1958400 q^{26} - 255744 q^{28} - 4355256 q^{29} + 916992 q^{30} + 4520250 q^{31} + 4954482 q^{33} - 5270790 q^{35} - 6042624 q^{36} + 134214 q^{37} + 1278720 q^{38} + 1335384 q^{39} - 2260992 q^{40} + 6660096 q^{42} - 12961896 q^{43} + 1318656 q^{44} - 8415396 q^{45} + 2345472 q^{46} + 18385002 q^{47} - 3659172 q^{49} + 2970624 q^{50} - 2673894 q^{51} - 3369984 q^{52} - 16540506 q^{53} - 19646208 q^{54} + 4325376 q^{56} + 100263780 q^{57} + 9176064 q^{58} + 31163922 q^{59} + 16349952 q^{60} - 85390158 q^{61} + 4361988 q^{63} + 25165824 q^{64} - 46506264 q^{65} - 111873024 q^{66} - 37750362 q^{67} - 22166784 q^{68} + 92031744 q^{70} + 45506424 q^{71} + 2162688 q^{72} + 9414786 q^{73} + 58837248 q^{74} + 9837540 q^{75} - 100614066 q^{77} - 25463808 q^{78} + 59730294 q^{79} - 27426816 q^{80} - 89677422 q^{81} - 93259776 q^{82} + 122616576 q^{84} - 64652220 q^{85} - 15144960 q^{86} + 334229724 q^{87} + 60260352 q^{88} + 323014482 q^{89} - 266861424 q^{91} - 40687104 q^{92} - 11119662 q^{93} - 443440128 q^{94} - 175918350 q^{95} - 12582912 q^{96} + 472166400 q^{98} - 346906296 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 1771 x^{10} + 26038 x^{9} + 2442597 x^{8} + 26522276 x^{7} + 1175865280 x^{6} + \cdots + 36\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 24\!\cdots\!14 \nu^{11} + \cdots + 10\!\cdots\!00 ) / 12\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 20\!\cdots\!84 \nu^{11} + \cdots + 15\!\cdots\!00 ) / 14\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 41\!\cdots\!32 \nu^{11} + \cdots + 11\!\cdots\!00 ) / 22\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 41\!\cdots\!63 \nu^{11} + \cdots + 17\!\cdots\!00 ) / 44\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11\!\cdots\!87 \nu^{11} + \cdots - 24\!\cdots\!00 ) / 14\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 54\!\cdots\!58 \nu^{11} + \cdots + 77\!\cdots\!00 ) / 44\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 15\!\cdots\!52 \nu^{11} + \cdots + 57\!\cdots\!00 ) / 89\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 21\!\cdots\!74 \nu^{11} + \cdots + 62\!\cdots\!00 ) / 44\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 21\!\cdots\!04 \nu^{11} + \cdots - 53\!\cdots\!50 ) / 44\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 17\!\cdots\!06 \nu^{11} + \cdots - 72\!\cdots\!50 ) / 29\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 33\!\cdots\!77 \nu^{11} + \cdots + 23\!\cdots\!50 ) / 44\!\cdots\!50 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 4 \beta_{11} + \beta_{10} + \beta_{9} + 2 \beta_{8} + 2 \beta_{7} - 5 \beta_{6} - 35 \beta_{5} + \cdots + 3 ) / 336 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 6 \beta_{11} - 49 \beta_{10} + 108 \beta_{9} + 49 \beta_{8} - 90 \beta_{7} - 24 \beta_{6} + \cdots - 198061 ) / 336 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 4450 \beta_{11} - 3438 \beta_{10} - 1302 \beta_{9} - 1719 \beta_{8} - 5274 \beta_{7} + \cdots - 2483606 ) / 336 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 37984 \beta_{11} - 38339 \beta_{10} - 166368 \beta_{9} - 76678 \beta_{8} - 97428 \beta_{7} + \cdots - 57331 ) / 168 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 580054 \beta_{11} + 1425607 \beta_{10} - 3624328 \beta_{9} - 1425607 \beta_{8} + 1884166 \beta_{7} + \cdots + 2917289727 ) / 168 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 13549640 \beta_{11} + 17760170 \beta_{10} + 15048932 \beta_{9} + 8880085 \beta_{8} + 38227648 \beta_{7} + \cdots + 21328938000 ) / 24 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3581059988 \beta_{11} + 2441634696 \beta_{10} + 10040555165 \beta_{9} + 4883269392 \beta_{8} + \cdots + 4232164690 ) / 168 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 8497690459 \beta_{11} - 25897143496 \beta_{10} + 67862605303 \beta_{9} + 25897143496 \beta_{8} + \cdots - 59901453828933 ) / 42 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 7353839084605 \beta_{11} - 8363749520578 \beta_{10} - 6546479969827 \beta_{9} + \cdots - 94\!\cdots\!11 ) / 168 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 117706616186580 \beta_{11} - 87304514607604 \beta_{10} - 369567931992129 \beta_{9} + \cdots - 146157822700894 ) / 84 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 24\!\cdots\!83 \beta_{11} + \cdots + 16\!\cdots\!42 ) / 168 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/14\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(1 - \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
3.92504 − 6.79836i
−12.5688 + 21.7698i
9.14374 − 15.8374i
20.6104 − 35.6982i
−9.54988 + 16.5409i
−10.5605 + 18.2913i
3.92504 + 6.79836i
−12.5688 − 21.7698i
9.14374 + 15.8374i
20.6104 + 35.6982i
−9.54988 − 16.5409i
−10.5605 − 18.2913i
−5.65685 − 9.79796i −90.8670 − 52.4621i −64.0000 + 110.851i 32.9005 − 18.9951i 1187.08i 1399.84 + 1950.71i 1448.15 2224.04 + 3852.15i −372.227 − 214.905i
3.2 −5.65685 − 9.79796i 24.5958 + 14.2004i −64.0000 + 110.851i 753.641 − 435.115i − 321.318i −1836.02 − 1547.20i 1448.15 −2877.20 − 4983.45i −8526.48 − 4922.77i
3.3 −5.65685 − 9.79796i 123.742 + 71.4423i −64.0000 + 110.851i −758.365 + 437.842i − 1616.56i 855.887 + 2243.27i 1448.15 6927.51 + 11998.8i 8579.92 + 4953.62i
3.4 5.65685 + 9.79796i −81.1886 − 46.8743i −64.0000 + 110.851i 928.109 − 535.844i − 1060.64i 2212.86 − 931.703i −1448.15 1113.90 + 1929.33i 10500.4 + 6062.38i
3.5 5.65685 + 9.79796i −21.7328 − 12.5474i −64.0000 + 110.851i −492.158 + 284.147i − 283.915i −1740.97 + 1653.43i −1448.15 −2965.62 − 5136.61i −5568.13 − 3214.76i
3.6 5.65685 + 9.79796i 126.451 + 73.0064i −64.0000 + 110.851i 372.872 − 215.278i 1651.95i −1545.60 − 1837.37i −1448.15 7379.38 + 12781.5i 4218.56 + 2435.59i
5.1 −5.65685 + 9.79796i −90.8670 + 52.4621i −64.0000 − 110.851i 32.9005 + 18.9951i − 1187.08i 1399.84 − 1950.71i 1448.15 2224.04 − 3852.15i −372.227 + 214.905i
5.2 −5.65685 + 9.79796i 24.5958 − 14.2004i −64.0000 − 110.851i 753.641 + 435.115i 321.318i −1836.02 + 1547.20i 1448.15 −2877.20 + 4983.45i −8526.48 + 4922.77i
5.3 −5.65685 + 9.79796i 123.742 − 71.4423i −64.0000 − 110.851i −758.365 − 437.842i 1616.56i 855.887 − 2243.27i 1448.15 6927.51 − 11998.8i 8579.92 − 4953.62i
5.4 5.65685 − 9.79796i −81.1886 + 46.8743i −64.0000 − 110.851i 928.109 + 535.844i 1060.64i 2212.86 + 931.703i −1448.15 1113.90 − 1929.33i 10500.4 − 6062.38i
5.5 5.65685 − 9.79796i −21.7328 + 12.5474i −64.0000 − 110.851i −492.158 − 284.147i 283.915i −1740.97 − 1653.43i −1448.15 −2965.62 + 5136.61i −5568.13 + 3214.76i
5.6 5.65685 − 9.79796i 126.451 − 73.0064i −64.0000 − 110.851i 372.872 + 215.278i − 1651.95i −1545.60 + 1837.37i −1448.15 7379.38 − 12781.5i 4218.56 − 2435.59i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 14.9.d.a ✓ 12
3.b odd 2 1 126.9.n.b 12
4.b odd 2 1 112.9.s.c 12
7.b odd 2 1 98.9.d.b 12
7.c even 3 1 98.9.b.c 12
7.c even 3 1 98.9.d.b 12
7.d odd 6 1 inner 14.9.d.a ✓ 12
7.d odd 6 1 98.9.b.c 12
21.g even 6 1 126.9.n.b 12
28.f even 6 1 112.9.s.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.9.d.a ✓ 12 1.a even 1 1 trivial
14.9.d.a ✓ 12 7.d odd 6 1 inner
98.9.b.c 12 7.c even 3 1
98.9.b.c 12 7.d odd 6 1
98.9.d.b 12 7.b odd 2 1
98.9.d.b 12 7.c even 3 1
112.9.s.c 12 4.b odd 2 1
112.9.s.c 12 28.f even 6 1
126.9.n.b 12 3.b odd 2 1
126.9.n.b 12 21.g even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(14, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 128 T^{2} + 16384)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 21\!\cdots\!81 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 57\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 36\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 48\!\cdots\!81 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 35\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 30\!\cdots\!01 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 94\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 25\!\cdots\!41 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 79\!\cdots\!12)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 95\!\cdots\!29 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 20\!\cdots\!21 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 50\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 32\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 85\!\cdots\!49 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 10\!\cdots\!41 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 28\!\cdots\!01 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 13\!\cdots\!69 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 23\!\cdots\!89 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 21\!\cdots\!64)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 26\!\cdots\!49 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 92\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 13\!\cdots\!21 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 52\!\cdots\!16 \) Copy content Toggle raw display
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