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Properties

Label 76.7.h.a
Level $76$
Weight $7$
Character orbit 76.h
Analytic conductor $17.484$
Analytic rank $0$
Dimension $20$
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] = [76,7,Mod(65,76)]
 
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, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.65");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 76.h (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: \(17.4841103551\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 9460 x^{18} + 36670708 x^{16} + 75655761912 x^{14} + 90488614064544 x^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{8}\cdot 19^{4} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{3} + (\beta_{6} + 6 \beta_{3} - 6) q^{5} + (\beta_{10} - \beta_{9} + 23) q^{7} + (\beta_{8} - 2 \beta_{5} + \cdots - 6 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{3} + (\beta_{6} + 6 \beta_{3} - 6) q^{5} + (\beta_{10} - \beta_{9} + 23) q^{7} + (\beta_{8} - 2 \beta_{5} + \cdots - 6 \beta_1) q^{9}+ \cdots + (62 \beta_{19} + 31 \beta_{18} + \cdots + 121) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 30 q^{3} - 56 q^{5} + 464 q^{7} + 2200 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 30 q^{3} - 56 q^{5} + 464 q^{7} + 2200 q^{9} - 3644 q^{11} - 7140 q^{13} + 9168 q^{15} + 1132 q^{17} + 2110 q^{19} - 8748 q^{21} + 832 q^{23} - 27698 q^{25} - 10920 q^{29} - 30306 q^{33} + 4172 q^{35} + 81144 q^{39} + 109206 q^{41} + 110740 q^{43} - 785440 q^{45} + 107080 q^{47} + 136092 q^{49} + 199872 q^{51} + 254796 q^{53} + 354840 q^{55} + 212268 q^{57} - 610638 q^{59} + 47864 q^{61} - 254476 q^{63} - 839562 q^{67} + 366660 q^{71} + 854482 q^{73} + 763088 q^{77} + 1718592 q^{79} - 1054142 q^{81} + 439612 q^{83} - 400236 q^{85} - 1604736 q^{87} + 478032 q^{89} + 599856 q^{91} + 829380 q^{93} - 1055660 q^{95} - 191286 q^{97} - 2336728 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 9460 x^{18} + 36670708 x^{16} + 75655761912 x^{14} + 90488614064544 x^{12} + \cdots + 10\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 59\!\cdots\!99 \nu^{18} + \cdots - 12\!\cdots\!00 ) / 65\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 38\!\cdots\!87 \nu^{19} + \cdots + 10\!\cdots\!60 ) / 21\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 15\!\cdots\!65 \nu^{19} + \cdots + 30\!\cdots\!60 ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 43\!\cdots\!61 \nu^{19} + \cdots - 25\!\cdots\!40 ) / 20\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 43\!\cdots\!61 \nu^{19} + \cdots - 25\!\cdots\!40 ) / 20\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 16\!\cdots\!73 \nu^{19} + \cdots + 32\!\cdots\!20 ) / 43\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 16\!\cdots\!73 \nu^{19} + \cdots - 32\!\cdots\!20 ) / 43\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 46\!\cdots\!49 \nu^{19} + \cdots - 99\!\cdots\!20 ) / 43\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 46\!\cdots\!49 \nu^{19} + \cdots + 99\!\cdots\!20 ) / 43\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 37\!\cdots\!71 \nu^{19} + \cdots + 33\!\cdots\!00 ) / 21\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 14\!\cdots\!65 \nu^{19} + \cdots - 13\!\cdots\!60 ) / 43\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 14\!\cdots\!65 \nu^{19} + \cdots + 13\!\cdots\!60 ) / 43\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 32\!\cdots\!58 \nu^{19} + \cdots - 10\!\cdots\!00 ) / 36\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 14\!\cdots\!03 \nu^{19} + \cdots - 23\!\cdots\!00 ) / 86\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 14\!\cdots\!03 \nu^{19} + \cdots + 23\!\cdots\!00 ) / 86\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 79\!\cdots\!25 \nu^{19} + \cdots - 21\!\cdots\!20 ) / 43\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 20\!\cdots\!01 \nu^{19} + \cdots + 69\!\cdots\!40 ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 54\!\cdots\!79 \nu^{19} + \cdots - 58\!\cdots\!20 ) / 21\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{8} + \beta_{7} + 2\beta_{6} + 2\beta_{5} + 2\beta_{2} + \beta _1 - 947 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{19} - 3 \beta_{17} - \beta_{16} + \beta_{15} + 3 \beta_{13} + \beta_{12} + 5 \beta_{11} + \cdots - 1456 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 16 \beta_{19} - 32 \beta_{18} + 20 \beta_{17} + 8 \beta_{16} + 8 \beta_{15} - 40 \beta_{14} + \cdots + 1618536 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4774 \beta_{19} + 7164 \beta_{17} + 2416 \beta_{16} - 2358 \beta_{15} - 8460 \beta_{13} - 3686 \beta_{12} + \cdots + 6017272 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 49732 \beta_{19} + 99464 \beta_{18} - 76612 \beta_{17} - 18396 \beta_{16} - 31336 \beta_{15} + \cdots - 3293632188 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 10308332 \beta_{19} - 15814644 \beta_{17} - 5343468 \beta_{16} + 4964864 \beta_{15} + \cdots - 23088666492 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 137328540 \beta_{19} - 274657080 \beta_{18} + 237782024 \beta_{17} + 26751916 \beta_{16} + \cdots + 7244138989596 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 22852332736 \beta_{19} + 36056290812 \beta_{17} + 12160586612 \beta_{16} - 10691746124 \beta_{15} + \cdots + 77627402517440 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 377013904280 \beta_{19} + 754027808560 \beta_{18} - 687239052304 \beta_{17} - 18988277608 \beta_{16} + \cdots - 16\!\cdots\!80 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 52844695637720 \beta_{19} - 85436071675344 \beta_{17} - 28820144485616 \beta_{16} + \cdots - 23\!\cdots\!84 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 10\!\cdots\!32 \beta_{19} + \cdots + 39\!\cdots\!84 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 12\!\cdots\!80 \beta_{19} + \cdots + 69\!\cdots\!48 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 27\!\cdots\!08 \beta_{19} + \cdots - 95\!\cdots\!96 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 31\!\cdots\!80 \beta_{19} + \cdots - 19\!\cdots\!88 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 74\!\cdots\!64 \beta_{19} + \cdots + 23\!\cdots\!32 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 77\!\cdots\!76 \beta_{19} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 19\!\cdots\!80 \beta_{19} + \cdots - 59\!\cdots\!40 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 19\!\cdots\!60 \beta_{19} + \cdots - 14\!\cdots\!68 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(21\) \(39\)
\(\chi(n)\) \(\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
− 50.9288i
− 35.1361i
− 25.0503i
− 21.8702i
3.67103i
4.04998i
17.5071i
21.3475i
42.7313i
43.6786i
50.9288i
35.1361i
25.0503i
21.8702i
− 3.67103i
− 4.04998i
− 17.5071i
− 21.3475i
− 42.7313i
− 43.6786i
0 −45.6057 − 26.3304i 0 −106.246 + 184.024i 0 −29.3098 0 1022.08 + 1770.30i 0
65.2 0 −31.9288 − 18.4341i 0 71.3406 − 123.566i 0 421.828 0 315.131 + 545.823i 0
65.3 0 −23.1942 − 13.3912i 0 16.0954 − 27.8780i 0 −416.808 0 −5.85193 − 10.1358i 0
65.4 0 −20.4402 − 11.8011i 0 15.3686 − 26.6192i 0 −77.2273 0 −85.9659 − 148.897i 0
65.5 0 1.67921 + 0.969492i 0 −72.6742 + 125.875i 0 621.925 0 −362.620 − 628.077i 0
65.6 0 2.00739 + 1.15896i 0 −67.5013 + 116.916i 0 −157.715 0 −361.814 − 626.680i 0
65.7 0 13.6616 + 7.88751i 0 93.3237 − 161.641i 0 −434.577 0 −240.074 − 415.821i 0
65.8 0 16.9875 + 9.80775i 0 72.3330 − 125.284i 0 479.542 0 −172.116 − 298.114i 0
65.9 0 35.5064 + 20.4996i 0 20.1615 − 34.9207i 0 53.8286 0 475.968 + 824.401i 0
65.10 0 36.3268 + 20.9733i 0 −70.2011 + 121.592i 0 −229.488 0 515.258 + 892.453i 0
69.1 0 −45.6057 + 26.3304i 0 −106.246 − 184.024i 0 −29.3098 0 1022.08 − 1770.30i 0
69.2 0 −31.9288 + 18.4341i 0 71.3406 + 123.566i 0 421.828 0 315.131 − 545.823i 0
69.3 0 −23.1942 + 13.3912i 0 16.0954 + 27.8780i 0 −416.808 0 −5.85193 + 10.1358i 0
69.4 0 −20.4402 + 11.8011i 0 15.3686 + 26.6192i 0 −77.2273 0 −85.9659 + 148.897i 0
69.5 0 1.67921 − 0.969492i 0 −72.6742 − 125.875i 0 621.925 0 −362.620 + 628.077i 0
69.6 0 2.00739 − 1.15896i 0 −67.5013 − 116.916i 0 −157.715 0 −361.814 + 626.680i 0
69.7 0 13.6616 − 7.88751i 0 93.3237 + 161.641i 0 −434.577 0 −240.074 + 415.821i 0
69.8 0 16.9875 − 9.80775i 0 72.3330 + 125.284i 0 479.542 0 −172.116 + 298.114i 0
69.9 0 35.5064 − 20.4996i 0 20.1615 + 34.9207i 0 53.8286 0 475.968 − 824.401i 0
69.10 0 36.3268 − 20.9733i 0 −70.2011 − 121.592i 0 −229.488 0 515.258 − 892.453i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.7.h.a ✓ 20
3.b odd 2 1 684.7.y.c 20
19.d odd 6 1 inner 76.7.h.a ✓ 20
57.f even 6 1 684.7.y.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.7.h.a ✓ 20 1.a even 1 1 trivial
76.7.h.a ✓ 20 19.d odd 6 1 inner
684.7.y.c 20 3.b odd 2 1
684.7.y.c 20 57.f even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(76, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 86\!\cdots\!25 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 80\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 10\!\cdots\!60)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots - 16\!\cdots\!64)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 53\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 27\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 77\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 26\!\cdots\!89 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 21\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 65\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 39\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 39\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 82\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 63\!\cdots\!25 \) Copy content Toggle raw display
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