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Properties

Label 79.2.c.a
Level $79$
Weight $2$
Character orbit 79.c
Analytic conductor $0.631$
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] = [79,2,Mod(23,79)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(79, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("79.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 79.c (of order \(3\), 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: \(0.630818175968\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 10 x^{10} - 12 x^{9} + 58 x^{8} - 69 x^{7} + 144 x^{6} - 34 x^{5} + 58 x^{4} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{9} + \beta_{3} + \beta_1) q^{3} + (\beta_{11} - \beta_{8} + \beta_{7} + \cdots - 1) q^{4}+ \cdots + (\beta_{10} + \beta_{8} + \beta_{7} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{9} + \beta_{3} + \beta_1) q^{3} + (\beta_{11} - \beta_{8} + \beta_{7} + \cdots - 1) q^{4}+ \cdots + ( - 4 \beta_{10} - 4 \beta_{9} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} - 4 q^{4} - q^{5} + 9 q^{6} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{2} - 4 q^{4} - q^{5} + 9 q^{6} - 8 q^{9} - 20 q^{10} + 3 q^{11} - 10 q^{12} - 3 q^{13} + 4 q^{14} + 16 q^{15} - 8 q^{16} - 22 q^{17} + 32 q^{18} - 6 q^{19} + 24 q^{20} - 6 q^{21} - 16 q^{22} - 10 q^{23} + 8 q^{24} - 7 q^{25} + 11 q^{26} - 12 q^{27} - 7 q^{28} + 15 q^{29} - 13 q^{30} - 13 q^{31} + 3 q^{32} + 14 q^{33} + 5 q^{34} + 20 q^{35} - q^{36} + 14 q^{37} - 38 q^{38} + 17 q^{39} + 29 q^{40} - 8 q^{41} - 17 q^{42} - 8 q^{43} + 15 q^{44} - 11 q^{45} + 12 q^{46} + 6 q^{47} + 16 q^{48} + 4 q^{49} - 42 q^{50} - 20 q^{51} + 48 q^{52} + 14 q^{53} + 4 q^{54} + 21 q^{55} + 5 q^{56} - 20 q^{57} + 40 q^{58} + 7 q^{59} - 45 q^{60} - 20 q^{61} + 38 q^{62} + 36 q^{63} - 44 q^{64} - 10 q^{65} + 12 q^{66} + 20 q^{67} - 3 q^{68} - 86 q^{69} + 2 q^{70} - 36 q^{71} - 21 q^{72} + 46 q^{73} + 20 q^{74} + 63 q^{75} - 34 q^{76} - 4 q^{77} - 26 q^{78} - 34 q^{79} - 30 q^{80} + 18 q^{81} - 21 q^{82} + 6 q^{83} - 23 q^{84} - 8 q^{85} - 36 q^{86} - 30 q^{87} - 50 q^{88} - 10 q^{89} + 30 q^{90} + 52 q^{91} - 14 q^{92} + 18 q^{93} + 40 q^{94} - 9 q^{95} - 80 q^{96} + 44 q^{97} + 15 q^{98} + 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 10 x^{10} - 12 x^{9} + 58 x^{8} - 69 x^{7} + 144 x^{6} - 34 x^{5} + 58 x^{4} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 380064 \nu^{11} - 1387786 \nu^{10} + 3331282 \nu^{9} - 7014772 \nu^{8} + 12040546 \nu^{7} + \cdots - 19816263 ) / 107922775 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1058608 \nu^{11} - 766892 \nu^{10} + 5956659 \nu^{9} + 4686616 \nu^{8} + 27058212 \nu^{7} + \cdots + 23463989 ) / 21584555 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5855788 \nu^{11} - 6380013 \nu^{10} - 23127994 \nu^{9} - 108970326 \nu^{8} - 127465057 \nu^{7} + \cdots - 147913429 ) / 107922775 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1372702 \nu^{11} + 1092317 \nu^{10} + 5782656 \nu^{9} + 22391744 \nu^{8} + 30723308 \nu^{7} + \cdots + 90449111 ) / 21584555 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1752766 \nu^{11} - 295469 \nu^{10} + 9113938 \nu^{9} + 15376972 \nu^{8} + 42763854 \nu^{7} + \cdots + 49048293 ) / 21584555 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 19816263 \nu^{11} + 39252462 \nu^{10} - 196774844 \nu^{9} + 234463874 \nu^{8} + \cdots + 6483446 ) / 107922775 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 4560387 \nu^{11} - 13043143 \nu^{10} + 53800076 \nu^{9} - 93237626 \nu^{8} + 312548393 \nu^{7} + \cdots - 2811374 ) / 21584555 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 39434944 \nu^{11} - 81332956 \nu^{10} + 391073897 \nu^{9} - 483220437 \nu^{8} + 2236763366 \nu^{7} + \cdots + 102700327 ) / 107922775 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 79898392 \nu^{11} - 179619883 \nu^{10} + 839335846 \nu^{9} - 1157289691 \nu^{8} + \cdots - 33259614 ) / 107922775 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 81623066 \nu^{11} - 183442459 \nu^{10} + 856870908 \nu^{9} - 1179582918 \nu^{8} + \cdots - 33887272 ) / 107922775 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - \beta_{8} + 3\beta_{7} + \beta_{5} + \beta_{4} - \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{5} - 5\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{11} - \beta_{10} + \beta_{9} + 6\beta_{8} - 16\beta_{7} - \beta_{3} - \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{11} - 8\beta_{10} + 2\beta_{9} - 8\beta_{6} + 7\beta_{5} + 29\beta_{2} - 36\beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -10\beta_{6} - 26\beta_{5} - 34\beta_{4} + 10\beta_{3} + 37\beta_{2} + 81 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -43\beta_{11} + 54\beta_{10} - 20\beta_{9} - \beta_{8} + 5\beta_{7} + 20\beta_{3} + 215\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 140 \beta_{11} + 75 \beta_{10} - 74 \beta_{9} - 195 \beta_{8} + 527 \beta_{7} + 75 \beta_{6} + \cdots - 452 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 344\beta_{6} - 252\beta_{5} + 18\beta_{4} - 149\beta_{3} - 1029\beta_{2} + 272 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -770\beta_{11} - 511\beta_{10} + 493\beta_{9} + 1132\beta_{8} - 3080\beta_{7} - 493\beta_{3} - 693\beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1443 \beta_{11} - 2136 \beta_{10} + 1004 \beta_{9} + 200 \beta_{8} - 713 \beta_{7} - 2136 \beta_{6} + \cdots - 1423 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/79\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1 + \beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
1.23998 + 2.14771i
0.887972 + 1.53801i
0.430452 + 0.745564i
−0.103002 − 0.178405i
−0.270006 − 0.467665i
−1.18539 − 2.05316i
1.23998 − 2.14771i
0.887972 − 1.53801i
0.430452 − 0.745564i
−0.103002 + 0.178405i
−0.270006 + 0.467665i
−1.18539 + 2.05316i
−1.23998 − 2.14771i 1.27561 + 2.20942i −2.07509 + 3.59417i 2.11345 − 3.66061i 3.16345 − 5.47925i 0.0143665 − 0.0248836i 5.33237 −1.75434 + 3.03861i −10.4826
23.2 −0.887972 − 1.53801i −0.960957 − 1.66443i −0.576988 + 0.999373i 0.183420 − 0.317692i −1.70661 + 2.95593i −1.04110 + 1.80323i −1.50249 −0.346876 + 0.600806i −0.651487
23.3 −0.430452 − 0.745564i 1.31487 + 2.27742i 0.629423 − 1.09019i −1.77976 + 3.08263i 1.13198 − 1.96064i 1.02731 − 1.77936i −2.80555 −1.95777 + 3.39095i 3.06440
23.4 0.103002 + 0.178405i 0.259947 + 0.450241i 0.978781 − 1.69530i 0.345354 − 0.598171i −0.0535501 + 0.0927514i −1.76185 + 3.05162i 0.815274 1.36486 − 2.36400i 0.142289
23.5 0.270006 + 0.467665i −1.37362 − 2.37918i 0.854193 − 1.47951i −1.19830 + 2.07551i 0.741774 − 1.28479i 2.04369 − 3.53977i 2.00258 −2.27368 + 3.93813i −1.29419
23.6 1.18539 + 2.05316i −0.515844 − 0.893468i −1.81032 + 3.13556i −0.164177 + 0.284363i 1.22296 − 2.11822i −0.282417 + 0.489160i −3.84217 0.967810 − 1.67630i −0.778458
55.1 −1.23998 + 2.14771i 1.27561 − 2.20942i −2.07509 − 3.59417i 2.11345 + 3.66061i 3.16345 + 5.47925i 0.0143665 + 0.0248836i 5.33237 −1.75434 − 3.03861i −10.4826
55.2 −0.887972 + 1.53801i −0.960957 + 1.66443i −0.576988 − 0.999373i 0.183420 + 0.317692i −1.70661 − 2.95593i −1.04110 − 1.80323i −1.50249 −0.346876 − 0.600806i −0.651487
55.3 −0.430452 + 0.745564i 1.31487 − 2.27742i 0.629423 + 1.09019i −1.77976 − 3.08263i 1.13198 + 1.96064i 1.02731 + 1.77936i −2.80555 −1.95777 − 3.39095i 3.06440
55.4 0.103002 − 0.178405i 0.259947 − 0.450241i 0.978781 + 1.69530i 0.345354 + 0.598171i −0.0535501 − 0.0927514i −1.76185 − 3.05162i 0.815274 1.36486 + 2.36400i 0.142289
55.5 0.270006 − 0.467665i −1.37362 + 2.37918i 0.854193 + 1.47951i −1.19830 − 2.07551i 0.741774 + 1.28479i 2.04369 + 3.53977i 2.00258 −2.27368 − 3.93813i −1.29419
55.6 1.18539 − 2.05316i −0.515844 + 0.893468i −1.81032 − 3.13556i −0.164177 − 0.284363i 1.22296 + 2.11822i −0.282417 − 0.489160i −3.84217 0.967810 + 1.67630i −0.778458
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
79.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 79.2.c.a ✓ 12
3.b odd 2 1 711.2.f.c 12
4.b odd 2 1 1264.2.i.h 12
79.c even 3 1 inner 79.2.c.a ✓ 12
79.c even 3 1 6241.2.a.j 6
79.d odd 6 1 6241.2.a.k 6
237.g odd 6 1 711.2.f.c 12
316.h odd 6 1 1264.2.i.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
79.2.c.a ✓ 12 1.a even 1 1 trivial
79.2.c.a ✓ 12 79.c even 3 1 inner
711.2.f.c 12 3.b odd 2 1
711.2.f.c 12 237.g odd 6 1
1264.2.i.h 12 4.b odd 2 1
1264.2.i.h 12 316.h odd 6 1
6241.2.a.j 6 79.c even 3 1
6241.2.a.k 6 79.d odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(79, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} + 13 T^{10} + \cdots + 361 \) Copy content Toggle raw display
$5$ \( T^{12} + T^{11} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{12} + 19 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{12} - 3 T^{11} + \cdots + 1923769 \) Copy content Toggle raw display
$13$ \( T^{12} + 3 T^{11} + \cdots + 9 \) Copy content Toggle raw display
$17$ \( (T^{6} + 11 T^{5} + 31 T^{4} + \cdots - 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 6 T^{11} + \cdots + 3644281 \) Copy content Toggle raw display
$23$ \( T^{12} + 10 T^{11} + \cdots + 5612161 \) Copy content Toggle raw display
$29$ \( T^{12} - 15 T^{11} + \cdots + 488601 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 115369081 \) Copy content Toggle raw display
$37$ \( T^{12} - 14 T^{11} + \cdots + 9916201 \) Copy content Toggle raw display
$41$ \( (T^{6} + 4 T^{5} + \cdots - 149825)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 8 T^{11} + \cdots + 7070281 \) Copy content Toggle raw display
$47$ \( T^{12} - 6 T^{11} + \cdots + 5536609 \) Copy content Toggle raw display
$53$ \( T^{12} - 14 T^{11} + \cdots + 5621641 \) Copy content Toggle raw display
$59$ \( T^{12} - 7 T^{11} + \cdots + 690561 \) Copy content Toggle raw display
$61$ \( (T^{6} + 10 T^{5} + \cdots + 131)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 10 T^{5} + \cdots + 1997)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 18 T^{5} + \cdots + 77307)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 518427361 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 243087455521 \) Copy content Toggle raw display
$83$ \( T^{12} - 6 T^{11} + \cdots + 36864 \) Copy content Toggle raw display
$89$ \( (T^{6} + 5 T^{5} + \cdots - 82669)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 22 T^{5} + \cdots + 619)^{2} \) Copy content Toggle raw display
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