Properties

Label 76.5.h.a
Level $76$
Weight $5$
Character orbit 76.h
Analytic conductor $7.856$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [76,5,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, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.65");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
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: \(7.85611719437\)
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: \( x^{12} - 6 x^{11} + 631 x^{10} - 3100 x^{9} + 142264 x^{8} - 550522 x^{7} + 14083117 x^{6} - 40335478 x^{5} + 638031136 x^{4} - 1209472584 x^{3} + \cdots + 90728724573 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_1) q^{3} + ( - \beta_{8} - \beta_{5} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{5} + (\beta_{4} + \beta_{3} - 4) q^{7} + ( - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + 22 \beta_{5} - \beta_{3} + 2 \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_1) q^{3} + ( - \beta_{8} - \beta_{5} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{5} + (\beta_{4} + \beta_{3} - 4) q^{7} + ( - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + 22 \beta_{5} - \beta_{3} + 2 \beta_1 - 1) q^{9} + (\beta_{11} - \beta_{10} + \beta_{3} - \beta_{2} + 1) q^{11} + (\beta_{11} - \beta_{10} + \beta_{7} + 5 \beta_{5} - \beta_{3} - 11) q^{13} + ( - 2 \beta_{10} - \beta_{9} + 3 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + 41 \beta_{5} + \cdots - 79) q^{15}+ \cdots + (29 \beta_{10} + 38 \beta_{9} - 46 \beta_{8} + 29 \beta_{7} - 29 \beta_{6} + \cdots - 170) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{3} + 9 q^{5} - 52 q^{7} + 136 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{3} + 9 q^{5} - 52 q^{7} + 136 q^{9} + 6 q^{11} - 93 q^{13} - 741 q^{15} - 483 q^{17} - 533 q^{19} + 972 q^{21} + 531 q^{23} - 217 q^{25} + 2025 q^{29} - 75 q^{33} - 1128 q^{35} - 2250 q^{39} - 1692 q^{41} - 63 q^{43} + 7976 q^{45} - 3471 q^{47} + 420 q^{49} + 6741 q^{51} - 3771 q^{53} - 2014 q^{55} + 7617 q^{57} - 9594 q^{59} + 1229 q^{61} + 1514 q^{63} + 7590 q^{67} + 963 q^{71} - 2838 q^{73} - 15408 q^{77} + 11073 q^{79} + 2086 q^{81} - 14202 q^{83} + 9455 q^{85} - 39510 q^{87} + 6525 q^{89} - 7686 q^{91} - 5316 q^{93} + 1521 q^{95} - 34110 q^{97} + 13220 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 631 x^{10} - 3100 x^{9} + 142264 x^{8} - 550522 x^{7} + 14083117 x^{6} - 40335478 x^{5} + 638031136 x^{4} - 1209472584 x^{3} + \cdots + 90728724573 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 101041319944 \nu^{11} - 35886079446026 \nu^{10} + 120597901725160 \nu^{9} + \cdots - 13\!\cdots\!78 ) / 16\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 162445852 \nu^{11} - 5113227722 \nu^{10} - 69019867025 \nu^{9} - 3210361481925 \nu^{8} - 6648576376768 \nu^{7} + \cdots - 28\!\cdots\!57 ) / 25\!\cdots\!61 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 303123959832 \nu^{11} - 2063916106978 \nu^{10} + 186817067049800 \nu^{9} + \cdots + 18\!\cdots\!09 ) / 48\!\cdots\!26 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 324891704 \nu^{11} - 1786904372 \nu^{10} + 198106533130 \nu^{9} - 878077616295 \nu^{8} + 42131954279636 \nu^{7} + \cdots - 58\!\cdots\!35 ) / 25\!\cdots\!61 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1328468937428 \nu^{11} - 47554498066674 \nu^{10} - 659027444905734 \nu^{9} + \cdots - 60\!\cdots\!40 ) / 48\!\cdots\!26 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1328468937428 \nu^{11} - 62167656378382 \nu^{10} + \cdots - 97\!\cdots\!07 ) / 48\!\cdots\!26 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 4824263493526 \nu^{11} - 42886275115864 \nu^{10} + \cdots - 64\!\cdots\!65 ) / 16\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 22328146096228 \nu^{11} - 128607403047369 \nu^{10} + \cdots + 27\!\cdots\!15 ) / 48\!\cdots\!26 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 43277988338263 \nu^{11} + 40046604044395 \nu^{10} + \cdots + 10\!\cdots\!51 ) / 48\!\cdots\!26 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 43277988338263 \nu^{11} - 516104475765288 \nu^{10} + \cdots - 12\!\cdots\!18 ) / 48\!\cdots\!26 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} - \beta_{4} - \beta_{2} + \beta _1 - 103 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 3 \beta_{11} - 3 \beta_{10} + 17 \beta_{9} - 11 \beta_{8} + \beta_{7} + 2 \beta_{6} + 67 \beta_{5} - 10 \beta_{4} - 14 \beta_{3} + 4 \beta_{2} - 181 \beta _1 - 111 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 26 \beta_{11} + 14 \beta_{10} + 34 \beta_{9} - 22 \beta_{8} - 250 \beta_{7} - 248 \beta_{6} + 279 \beta_{5} + 234 \beta_{4} + 184 \beta_{3} + 340 \beta_{2} - 508 \beta _1 + 17950 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 964 \beta_{11} + 1064 \beta_{10} - 4760 \beta_{9} + 2966 \beta_{8} - 212 \beta_{7} - 1038 \beta_{6} - 41715 \beta_{5} + 3010 \beta_{4} + 4393 \beta_{3} - 658 \beta_{2} + 37256 \beta _1 + 51008 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9523 \beta_{11} - 3409 \beta_{10} - 14365 \beta_{9} + 8953 \beta_{8} + 56977 \beta_{7} + 54494 \beta_{6} - 183628 \beta_{5} - 49772 \beta_{4} - 80211 \beta_{3} - 83682 \beta_{2} + 170824 \beta _1 - 3639920 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 258177 \beta_{11} - 303789 \beta_{10} + 1135779 \beta_{9} - 644067 \beta_{8} + 65256 \beta_{7} + 329271 \beta_{6} + 14039235 \beta_{5} - 769437 \beta_{4} - 1213323 \beta_{3} + \cdots - 16938933 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2726304 \beta_{11} + 449880 \beta_{10} + 4610232 \beta_{9} - 2618100 \beta_{8} - 12798344 \beta_{7} - 11730692 \beta_{6} + 74171253 \beta_{5} + 10627604 \beta_{4} + \cdots + 765988175 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 65592264 \beta_{11} + 80159184 \beta_{10} - 260711524 \beta_{9} + 128003272 \beta_{8} - 21438716 \beta_{7} - 91314364 \beta_{6} - 3935592755 \beta_{5} + \cdots + 4926306888 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 730967377 \beta_{11} + 14905901 \beta_{10} - 1338235085 \beta_{9} + 659714891 \beta_{8} + 2863528856 \beta_{7} + 2506125835 \beta_{6} - 24856554570 \beta_{5} + \cdots - 162574704089 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 16201138664 \beta_{11} - 20273509714 \beta_{10} + 58997836558 \beta_{9} - 24135753784 \beta_{8} + 6631724131 \beta_{7} + 23939300133 \beta_{6} + \cdots - 1340587725328 \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(39\)
\(\chi(n)\) \(\beta_{5}\) \(1\)

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
0.500000 + 14.3199i
0.500000 + 9.58497i
0.500000 + 4.44379i
0.500000 4.96177i
0.500000 6.42649i
0.500000 15.2283i
0.500000 14.3199i
0.500000 9.58497i
0.500000 4.44379i
0.500000 + 4.96177i
0.500000 + 6.42649i
0.500000 + 15.2283i
0 −13.1514 7.59296i 0 19.9265 34.5138i 0 −53.3663 0 74.8062 + 129.568i 0
65.2 0 −9.05083 5.22550i 0 −12.7505 + 22.0845i 0 27.7589 0 14.1116 + 24.4421i 0
65.3 0 −4.59843 2.65491i 0 −1.89169 + 3.27650i 0 36.8385 0 −26.4029 45.7312i 0
65.4 0 3.54702 + 2.04787i 0 15.5891 27.0012i 0 −2.67956 0 −32.1124 55.6204i 0
65.5 0 4.81550 + 2.78023i 0 −13.2594 + 22.9660i 0 −84.4930 0 −25.0406 43.3717i 0
65.6 0 12.4381 + 7.18116i 0 −3.11411 + 5.39380i 0 49.9415 0 62.6382 + 108.493i 0
69.1 0 −13.1514 + 7.59296i 0 19.9265 + 34.5138i 0 −53.3663 0 74.8062 129.568i 0
69.2 0 −9.05083 + 5.22550i 0 −12.7505 22.0845i 0 27.7589 0 14.1116 24.4421i 0
69.3 0 −4.59843 + 2.65491i 0 −1.89169 3.27650i 0 36.8385 0 −26.4029 + 45.7312i 0
69.4 0 3.54702 2.04787i 0 15.5891 + 27.0012i 0 −2.67956 0 −32.1124 + 55.6204i 0
69.5 0 4.81550 2.78023i 0 −13.2594 22.9660i 0 −84.4930 0 −25.0406 + 43.3717i 0
69.6 0 12.4381 7.18116i 0 −3.11411 5.39380i 0 49.9415 0 62.6382 108.493i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.6
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 76.5.h.a 12
3.b odd 2 1 684.5.y.c 12
4.b odd 2 1 304.5.r.b 12
19.d odd 6 1 inner 76.5.h.a 12
57.f even 6 1 684.5.y.c 12
76.f even 6 1 304.5.r.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.5.h.a 12 1.a even 1 1 trivial
76.5.h.a 12 19.d odd 6 1 inner
304.5.r.b 12 4.b odd 2 1
304.5.r.b 12 76.f even 6 1
684.5.y.c 12 3.b odd 2 1
684.5.y.c 12 57.f even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(76, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 12 T^{11} + \cdots + 75979063449 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 392044989615876 \) Copy content Toggle raw display
$7$ \( (T^{6} + 26 T^{5} - 6970 T^{4} + \cdots - 617046128)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 3 T^{5} - 37505 T^{4} + \cdots - 781272216)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 93 T^{11} + \cdots + 50\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{12} + 483 T^{11} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{12} + 533 T^{11} + \cdots + 48\!\cdots\!21 \) Copy content Toggle raw display
$23$ \( T^{12} - 531 T^{11} + \cdots + 48\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{12} - 2025 T^{11} + \cdots + 43\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( T^{12} + 5102232 T^{10} + \cdots + 20\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( T^{12} + 13373304 T^{10} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{12} + 1692 T^{11} + \cdots + 48\!\cdots\!81 \) Copy content Toggle raw display
$43$ \( T^{12} + 63 T^{11} + \cdots + 68\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{12} + 3471 T^{11} + \cdots + 24\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{12} + 3771 T^{11} + \cdots + 80\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{12} + 9594 T^{11} + \cdots + 48\!\cdots\!09 \) Copy content Toggle raw display
$61$ \( T^{12} - 1229 T^{11} + \cdots + 74\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{12} - 7590 T^{11} + \cdots + 27\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( T^{12} - 963 T^{11} + \cdots + 22\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{12} + 2838 T^{11} + \cdots + 11\!\cdots\!89 \) Copy content Toggle raw display
$79$ \( T^{12} - 11073 T^{11} + \cdots + 19\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( (T^{6} + 7101 T^{5} + \cdots + 48\!\cdots\!04)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} - 6525 T^{11} + \cdots + 52\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{12} + 34110 T^{11} + \cdots + 49\!\cdots\!69 \) Copy content Toggle raw display
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