↑

Properties

Label 7.11.d.a
Level $7$
Weight $11$
Character orbit 7.d
Analytic conductor $4.448$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7,11,Mod(3,7)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7.3");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 7.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: \(4.44750076872\)
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: \( x^{12} + 4910 x^{10} - 54096 x^{9} + 18202260 x^{8} - 174741840 x^{7} + 27563858336 x^{6} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{4}\cdot 7^{6} \)
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.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{2} - \beta_1 - 2) q^{2} + (\beta_{5} - \beta_{3} - 14 \beta_{2} + \cdots - 27) q^{3}+ \cdots + (\beta_{11} + 2 \beta_{10} + \cdots + 30536) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{2} - \beta_1 - 2) q^{2} + (\beta_{5} - \beta_{3} - 14 \beta_{2} + \cdots - 27) q^{3}+ \cdots + (395922 \beta_{11} + 2076288 \beta_{10} + \cdots + 1670545776) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} - 246 q^{3} - 3700 q^{4} + 3330 q^{5} - 12572 q^{7} - 93504 q^{8} + 182964 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} - 246 q^{3} - 3700 q^{4} + 3330 q^{5} - 12572 q^{7} - 93504 q^{8} + 182964 q^{9} - 445740 q^{10} + 82566 q^{11} + 1581468 q^{12} - 934752 q^{14} - 729900 q^{15} - 230152 q^{16} + 389178 q^{17} + 378144 q^{18} - 5517150 q^{19} + 11878986 q^{21} - 19846048 q^{22} - 8131674 q^{23} + 31899384 q^{24} + 6794220 q^{25} + 37885176 q^{26} - 64544228 q^{28} - 86390616 q^{29} + 69082560 q^{30} - 44673798 q^{31} + 82297536 q^{32} + 293090418 q^{33} - 192479070 q^{35} - 895531584 q^{36} + 83071214 q^{37} + 528863040 q^{38} + 45216696 q^{39} + 840782640 q^{40} - 267874488 q^{42} - 975196744 q^{43} - 123891084 q^{44} + 960213420 q^{45} + 348447776 q^{46} + 782650314 q^{47} - 531271524 q^{49} - 3793600080 q^{50} + 90752418 q^{51} + 1019252472 q^{52} + 417566766 q^{53} + 4229214768 q^{54} - 2911004880 q^{56} - 3616618428 q^{57} + 2420166368 q^{58} + 456895242 q^{59} + 1019560500 q^{60} + 1469956122 q^{61} + 962168676 q^{63} - 1697861600 q^{64} - 480821880 q^{65} - 875442996 q^{66} + 1667792862 q^{67} - 3587950548 q^{68} + 4620799680 q^{70} + 9772297272 q^{71} - 5075622576 q^{72} - 3308125038 q^{73} - 8441358900 q^{74} - 13137007020 q^{75} + 1048649574 q^{77} + 25651166688 q^{78} - 2397038922 q^{79} + 1013632920 q^{80} + 394526610 q^{81} - 22926766968 q^{82} + 16162834044 q^{84} + 20859080820 q^{85} - 840264960 q^{86} - 31859197092 q^{87} - 3607027256 q^{88} - 2675986686 q^{89} + 17848798800 q^{91} + 58847183592 q^{92} + 8231670714 q^{93} - 26955870432 q^{94} - 13809374190 q^{95} - 30016813680 q^{96} - 50762451012 q^{98} + 20038367016 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 4910 x^{10} - 54096 x^{9} + 18202260 x^{8} - 174741840 x^{7} + 27563858336 x^{6} + \cdots + 11\!\cdots\!56 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11\!\cdots\!99 \nu^{11} + \cdots - 23\!\cdots\!24 ) / 18\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\!\cdots\!67 \nu^{11} + \cdots + 62\!\cdots\!68 ) / 82\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13\!\cdots\!11 \nu^{11} + \cdots - 11\!\cdots\!32 ) / 27\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 14\!\cdots\!73 \nu^{11} + \cdots - 23\!\cdots\!64 ) / 27\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10\!\cdots\!69 \nu^{11} + \cdots - 67\!\cdots\!28 ) / 40\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 97\!\cdots\!11 \nu^{11} + \cdots - 12\!\cdots\!64 ) / 91\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 38\!\cdots\!69 \nu^{11} + \cdots - 13\!\cdots\!84 ) / 27\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 77\!\cdots\!11 \nu^{11} + \cdots - 18\!\cdots\!68 ) / 27\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 83\!\cdots\!89 \nu^{11} + \cdots + 38\!\cdots\!96 ) / 27\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 63\!\cdots\!85 \nu^{11} + \cdots + 80\!\cdots\!56 ) / 13\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{6} - \beta_{5} + 2\beta_{4} - 7\beta_{3} + 1638\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{11} - 10 \beta_{10} - \beta_{9} + 5 \beta_{8} - 4 \beta_{7} + 22 \beta_{6} - 18 \beta_{5} + \cdots + 13527 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 28 \beta_{11} + 28 \beta_{10} - 28 \beta_{9} + 28 \beta_{8} - 3166 \beta_{7} + 28 \beta_{6} + \cdots - 4094764 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 5266 \beta_{11} + 17594 \beta_{10} + 10532 \beta_{9} - 35188 \beta_{8} + 85994 \beta_{7} + \cdots - 120604 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 329056 \beta_{11} - 545440 \beta_{10} - 164528 \beta_{9} + 272720 \beta_{8} - 108192 \beta_{7} + \cdots + 11911045820 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 20707564 \beta_{11} + 57614684 \beta_{10} - 20707564 \beta_{9} + 57614684 \beta_{8} + \cdots - 333681636572 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 742967568 \beta_{11} + 1439893392 \beta_{10} + 1485935136 \beta_{9} - 2879786784 \beta_{8} + \cdots - 69777900616 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 149865263696 \beta_{11} - 381193186960 \beta_{10} - 74932631848 \beta_{9} + 190596593480 \beta_{8} + \cdots + 13\!\cdots\!28 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 3045872833024 \beta_{11} + 6310088788864 \beta_{10} - 3045872833024 \beta_{9} + 6310088788864 \beta_{8} + \cdots - 12\!\cdots\!72 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 264521302345648 \beta_{11} + 642124690551152 \beta_{10} + 529042604691296 \beta_{9} + \cdots - 99\!\cdots\!44 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(1 + \beta_{2}\)

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
25.2573 + 43.7468i
19.6758 + 34.0795i
10.3681 + 17.9580i
−5.54713 − 9.60792i
−19.6838 − 34.0934i
−30.0701 − 52.0830i
25.2573 − 43.7468i
19.6758 − 34.0795i
10.3681 − 17.9580i
−5.54713 + 9.60792i
−19.6838 + 34.0934i
−30.0701 + 52.0830i
−26.2573 − 45.4789i −275.972 − 159.333i −866.887 + 1501.49i 4280.03 − 2471.08i 16734.6i −15104.0 + 7371.90i 37273.4 21249.3 + 36804.9i −224764. − 129767.i
3.2 −20.6758 − 35.8115i 372.264 + 214.926i −342.976 + 594.052i 1538.44 − 888.218i − 17775.1i 9740.18 − 13696.9i −13978.8 62862.3 + 108881.i −63616.9 − 36729.2i
3.3 −11.3681 − 19.6901i −39.4996 − 22.8051i 253.534 − 439.134i −4891.74 + 2824.25i 1037.00i −1953.67 + 16693.1i −34810.6 −28484.4 − 49336.3i 111219. + 64212.5i
3.4 4.54713 + 7.87587i −104.508 − 60.3375i 470.647 − 815.185i 1827.87 − 1055.32i − 1097.45i 8971.10 − 14212.5i 17872.9 −22243.3 − 38526.5i 16623.2 + 9597.40i
3.5 18.6838 + 32.3614i 256.371 + 148.016i −186.172 + 322.459i −49.0009 + 28.2907i 11062.0i −15712.7 + 5965.47i 24350.9 14292.7 + 24755.8i −1831.05 − 1057.16i
3.6 29.0701 + 50.3510i −331.655 − 191.481i −1178.15 + 2040.61i −1040.60 + 600.792i − 22265.5i 7773.05 + 14901.5i −77459.9 43805.3 + 75873.0i −60500.9 − 34930.2i
5.1 −26.2573 + 45.4789i −275.972 + 159.333i −866.887 − 1501.49i 4280.03 + 2471.08i − 16734.6i −15104.0 − 7371.90i 37273.4 21249.3 − 36804.9i −224764. + 129767.i
5.2 −20.6758 + 35.8115i 372.264 − 214.926i −342.976 − 594.052i 1538.44 + 888.218i 17775.1i 9740.18 + 13696.9i −13978.8 62862.3 − 108881.i −63616.9 + 36729.2i
5.3 −11.3681 + 19.6901i −39.4996 + 22.8051i 253.534 + 439.134i −4891.74 − 2824.25i − 1037.00i −1953.67 − 16693.1i −34810.6 −28484.4 + 49336.3i 111219. − 64212.5i
5.4 4.54713 − 7.87587i −104.508 + 60.3375i 470.647 + 815.185i 1827.87 + 1055.32i 1097.45i 8971.10 + 14212.5i 17872.9 −22243.3 + 38526.5i 16623.2 − 9597.40i
5.5 18.6838 − 32.3614i 256.371 − 148.016i −186.172 − 322.459i −49.0009 − 28.2907i − 11062.0i −15712.7 − 5965.47i 24350.9 14292.7 − 24755.8i −1831.05 + 1057.16i
5.6 29.0701 − 50.3510i −331.655 + 191.481i −1178.15 − 2040.61i −1040.60 − 600.792i 22265.5i 7773.05 − 14901.5i −77459.9 43805.3 − 75873.0i −60500.9 + 34930.2i
\(n\): 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 7.11.d.a ✓ 12
3.b odd 2 1 63.11.m.c 12
4.b odd 2 1 112.11.s.b 12
7.b odd 2 1 49.11.d.c 12
7.c even 3 1 49.11.b.a 12
7.c even 3 1 49.11.d.c 12
7.d odd 6 1 inner 7.11.d.a ✓ 12
7.d odd 6 1 49.11.b.a 12
21.g even 6 1 63.11.m.c 12
28.f even 6 1 112.11.s.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.11.d.a ✓ 12 1.a even 1 1 trivial
7.11.d.a ✓ 12 7.d odd 6 1 inner
49.11.b.a 12 7.c even 3 1
49.11.b.a 12 7.d odd 6 1
49.11.d.c 12 7.b odd 2 1
49.11.d.c 12 7.c even 3 1
63.11.m.c 12 3.b odd 2 1
63.11.m.c 12 21.g even 6 1
112.11.s.b 12 4.b odd 2 1
112.11.s.b 12 28.f even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(7, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 95\!\cdots\!56 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 73\!\cdots\!09 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 50\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 50\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 73\!\cdots\!81 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 24\!\cdots\!29 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 11\!\cdots\!49 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 10\!\cdots\!16)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 35\!\cdots\!69 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 47\!\cdots\!21 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 13\!\cdots\!16)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 51\!\cdots\!29 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 27\!\cdots\!29 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 61\!\cdots\!49 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 45\!\cdots\!61 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 39\!\cdots\!04)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 28\!\cdots\!29 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 16\!\cdots\!01 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 18\!\cdots\!29 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
show more
show less