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Properties

Label 76.6.e.a
Level $76$
Weight $6$
Character orbit 76.e
Analytic conductor $12.189$
Analytic rank $0$
Dimension $18$
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] = [76,6,Mod(45,76)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(76, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.45");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 76.e (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: \(12.1891703058\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 2 x^{17} + 1540 x^{16} - 768 x^{15} + 1608492 x^{14} - 1027368 x^{13} + 897054160 x^{12} + \cdots + 80\!\cdots\!36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{3}\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_{2} + \beta_1) q^{3} + (\beta_{8} + \beta_{6} + \beta_{2}) q^{5} + ( - \beta_{9} + \beta_{4} - 2 \beta_{3} + 18) q^{7} + ( - \beta_{10} - \beta_{7} + \beta_{6} + \cdots - 99) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_{2} + \beta_1) q^{3} + (\beta_{8} + \beta_{6} + \beta_{2}) q^{5} + ( - \beta_{9} + \beta_{4} - 2 \beta_{3} + 18) q^{7} + ( - \beta_{10} - \beta_{7} + \beta_{6} + \cdots - 99) q^{9}+ \cdots + ( - 34 \beta_{17} - 34 \beta_{16} + \cdots - 21070) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 11 q^{3} + 11 q^{5} + 336 q^{7} - 902 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 11 q^{3} + 11 q^{5} + 336 q^{7} - 902 q^{9} - 320 q^{11} + 227 q^{13} - 101 q^{15} + 179 q^{17} - 868 q^{19} - 5700 q^{21} - 3425 q^{23} - 7054 q^{25} + 14722 q^{27} - 7349 q^{29} - 9960 q^{31} - 2998 q^{33} + 15888 q^{35} + 26444 q^{37} - 30246 q^{39} - 7311 q^{41} - 8283 q^{43} - 62164 q^{45} + 37603 q^{47} + 124738 q^{49} + 47227 q^{51} - 20337 q^{53} + 716 q^{55} - 57555 q^{57} - 74455 q^{59} - 7569 q^{61} - 52544 q^{63} + 188998 q^{65} - 26177 q^{67} + 116282 q^{69} - 53463 q^{71} - 14103 q^{73} + 120912 q^{75} - 31960 q^{77} + 31825 q^{79} - 21137 q^{81} + 82600 q^{83} - 50787 q^{85} - 339766 q^{87} - 155197 q^{89} - 2800 q^{91} - 46460 q^{93} + 49315 q^{95} + 111241 q^{97} - 193544 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 2 x^{17} + 1540 x^{16} - 768 x^{15} + 1608492 x^{14} - 1027368 x^{13} + 897054160 x^{12} + \cdots + 80\!\cdots\!36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 47\!\cdots\!39 \nu^{17} + \cdots + 71\!\cdots\!56 ) / 80\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 21\!\cdots\!33 \nu^{17} + \cdots - 10\!\cdots\!88 ) / 21\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10\!\cdots\!29 \nu^{17} + \cdots + 28\!\cdots\!72 ) / 71\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 23\!\cdots\!79 \nu^{17} + \cdots + 19\!\cdots\!88 ) / 79\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 53\!\cdots\!01 \nu^{17} + \cdots + 57\!\cdots\!00 ) / 15\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18\!\cdots\!65 \nu^{17} + \cdots - 15\!\cdots\!36 ) / 51\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 20\!\cdots\!93 \nu^{17} + \cdots - 28\!\cdots\!92 ) / 51\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 45\!\cdots\!89 \nu^{17} + \cdots + 13\!\cdots\!64 ) / 71\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 24\!\cdots\!04 \nu^{17} + \cdots + 57\!\cdots\!84 ) / 19\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 28\!\cdots\!65 \nu^{17} + \cdots - 96\!\cdots\!76 ) / 14\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 14\!\cdots\!81 \nu^{17} + \cdots - 11\!\cdots\!68 ) / 51\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 96\!\cdots\!23 \nu^{17} + \cdots - 36\!\cdots\!56 ) / 17\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 26\!\cdots\!83 \nu^{17} + \cdots + 65\!\cdots\!36 ) / 46\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 53\!\cdots\!31 \nu^{17} + \cdots - 40\!\cdots\!32 ) / 53\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 10\!\cdots\!37 \nu^{17} + \cdots + 59\!\cdots\!76 ) / 71\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 30\!\cdots\!91 \nu^{17} + \cdots + 57\!\cdots\!52 ) / 71\!\cdots\!72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - \beta_{8} - \beta_{6} + \beta_{3} - 341\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} - 4 \beta_{14} + 4 \beta_{13} - \beta_{12} - \beta_{11} + 15 \beta_{9} - 15 \beta_{8} + \cdots - 251 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 14 \beta_{17} + 32 \beta_{16} - 50 \beta_{15} - 18 \beta_{14} + 28 \beta_{11} - 792 \beta_{10} + \cdots - 188900 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 1468 \beta_{17} + 1540 \beta_{16} + 3260 \beta_{15} + 734 \beta_{14} - 3260 \beta_{13} + 1082 \beta_{12} + \cdots - 734 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 15280 \beta_{17} + 5816 \beta_{14} + 61060 \beta_{13} + 51260 \beta_{12} - 15280 \beta_{11} + \cdots + 121678584 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 452052 \beta_{17} - 1202040 \beta_{16} - 2091648 \beta_{15} + 1654092 \beta_{14} + 904104 \beta_{11} + \cdots + 793104240 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 26067120 \beta_{17} - 32425248 \beta_{16} + 54202584 \beta_{15} + 13033560 \beta_{14} + \cdots - 13033560 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 260144008 \beta_{17} - 1498148416 \beta_{14} + 1233764272 \beta_{13} - 912669496 \beta_{12} + \cdots - 789149237528 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 10257646976 \beta_{17} + 27145364768 \beta_{16} - 43264584944 \beta_{15} - 16887717792 \beta_{14} + \cdots - 60579644611040 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 283435484512 \beta_{17} + 804464195872 \beta_{16} + 689690631680 \beta_{15} + 141717742256 \beta_{14} + \cdots - 141717742256 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 7813949482144 \beta_{17} + 6701339085632 \beta_{14} + 33095720229280 \beta_{13} + 36460271819072 \beta_{12} + \cdots + 45\!\cdots\!16 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 71334586816224 \beta_{17} - 661186383087936 \beta_{16} - 361921497433920 \beta_{15} + \cdots + 65\!\cdots\!04 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 11\!\cdots\!12 \beta_{17} + \cdots - 58\!\cdots\!56 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 30\!\cdots\!72 \beta_{17} + \cdots - 56\!\cdots\!48 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 44\!\cdots\!60 \beta_{17} + \cdots - 26\!\cdots\!96 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 14\!\cdots\!00 \beta_{17} + \cdots - 74\!\cdots\!00 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(39\)
\(\chi(n)\) \(-1 + \beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1
14.2764 − 24.7275i
11.1685 − 19.3444i
6.29505 − 10.9033i
3.49628 − 6.05573i
2.80322 − 4.85531i
−4.70426 + 8.14802i
−8.45017 + 14.6361i
−10.9539 + 18.9727i
−12.9311 + 22.3974i
14.2764 + 24.7275i
11.1685 + 19.3444i
6.29505 + 10.9033i
3.49628 + 6.05573i
2.80322 + 4.85531i
−4.70426 − 8.14802i
−8.45017 − 14.6361i
−10.9539 − 18.9727i
−12.9311 − 22.3974i
0 −14.7764 − 25.5935i 0 35.6401 + 61.7304i 0 252.315 0 −315.186 + 545.918i 0
45.2 0 −11.6685 − 20.2105i 0 −25.2470 − 43.7291i 0 −187.942 0 −150.808 + 261.208i 0
45.3 0 −6.79505 − 11.7694i 0 −47.4871 − 82.2500i 0 189.860 0 29.1546 − 50.4972i 0
45.4 0 −3.99628 − 6.92176i 0 31.6056 + 54.7425i 0 −80.2775 0 89.5595 − 155.122i 0
45.5 0 −3.30322 − 5.72134i 0 12.6095 + 21.8402i 0 −40.7176 0 99.6775 − 172.647i 0
45.6 0 4.20426 + 7.28199i 0 −18.6530 − 32.3080i 0 15.8772 0 86.1484 − 149.213i 0
45.7 0 7.95017 + 13.7701i 0 −15.4645 − 26.7852i 0 132.225 0 −4.91049 + 8.50521i 0
45.8 0 10.4539 + 18.1067i 0 50.5928 + 87.6293i 0 95.5451 0 −97.0688 + 168.128i 0
45.9 0 12.4311 + 21.5314i 0 −18.0963 − 31.3437i 0 −208.885 0 −187.566 + 324.875i 0
49.1 0 −14.7764 + 25.5935i 0 35.6401 − 61.7304i 0 252.315 0 −315.186 − 545.918i 0
49.2 0 −11.6685 + 20.2105i 0 −25.2470 + 43.7291i 0 −187.942 0 −150.808 − 261.208i 0
49.3 0 −6.79505 + 11.7694i 0 −47.4871 + 82.2500i 0 189.860 0 29.1546 + 50.4972i 0
49.4 0 −3.99628 + 6.92176i 0 31.6056 − 54.7425i 0 −80.2775 0 89.5595 + 155.122i 0
49.5 0 −3.30322 + 5.72134i 0 12.6095 − 21.8402i 0 −40.7176 0 99.6775 + 172.647i 0
49.6 0 4.20426 − 7.28199i 0 −18.6530 + 32.3080i 0 15.8772 0 86.1484 + 149.213i 0
49.7 0 7.95017 − 13.7701i 0 −15.4645 + 26.7852i 0 132.225 0 −4.91049 − 8.50521i 0
49.8 0 10.4539 − 18.1067i 0 50.5928 − 87.6293i 0 95.5451 0 −97.0688 − 168.128i 0
49.9 0 12.4311 − 21.5314i 0 −18.0963 + 31.3437i 0 −208.885 0 −187.566 − 324.875i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 45.9
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.6.e.a ✓ 18
3.b odd 2 1 684.6.k.f 18
4.b odd 2 1 304.6.i.d 18
19.c even 3 1 inner 76.6.e.a ✓ 18
57.h odd 6 1 684.6.k.f 18
76.g odd 6 1 304.6.i.d 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.6.e.a ✓ 18 1.a even 1 1 trivial
76.6.e.a ✓ 18 19.c even 3 1 inner
304.6.i.d 18 4.b odd 2 1
304.6.i.d 18 76.g odd 6 1
684.6.k.f 18 3.b odd 2 1
684.6.k.f 18 57.h odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(76, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots + 11\!\cdots\!89 \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 53\!\cdots\!96 \) Copy content Toggle raw display
$7$ \( (T^{9} + \cdots - 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( (T^{9} + \cdots + 15\!\cdots\!96)^{2} \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 60\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 34\!\cdots\!99 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 96\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( (T^{9} + \cdots - 91\!\cdots\!60)^{2} \) Copy content Toggle raw display
$37$ \( (T^{9} + \cdots - 70\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 33\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 50\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 21\!\cdots\!89 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 92\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 43\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( (T^{9} + \cdots + 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 86\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 93\!\cdots\!25 \) Copy content Toggle raw display
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