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Properties

Label 84.4.b.a
Level $84$
Weight $4$
Character orbit 84.b
Analytic conductor $4.956$
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] = [84,4,Mod(55,84)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(84, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("84.55");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 84.b (of order \(2\), degree \(1\), 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: \(4.95616044048\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} - 2x^{10} - 6x^{9} + 56x^{7} - 448x^{6} + 448x^{5} - 3072x^{3} - 8192x^{2} - 32768x + 262144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{13} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 3 q^{3} + \beta_{2} q^{4} + ( - \beta_{8} - \beta_1) q^{5} - 3 \beta_1 q^{6} + (\beta_{5} - \beta_1 + 1) q^{7} + (\beta_{3} + 2) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 3 q^{3} + \beta_{2} q^{4} + ( - \beta_{8} - \beta_1) q^{5} - 3 \beta_1 q^{6} + (\beta_{5} - \beta_1 + 1) q^{7} + (\beta_{3} + 2) q^{8} + 9 q^{9} + (\beta_{11} - \beta_{8} + \beta_{5} + \cdots + 5) q^{10}+ \cdots + (9 \beta_{11} + 9 \beta_{10} + \cdots + 18 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + q^{2} - 36 q^{3} + 5 q^{4} - 3 q^{6} + 10 q^{7} + 25 q^{8} + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + q^{2} - 36 q^{3} + 5 q^{4} - 3 q^{6} + 10 q^{7} + 25 q^{8} + 108 q^{9} + 56 q^{10} - 15 q^{12} + 69 q^{14} + 41 q^{16} + 9 q^{18} - 84 q^{19} + 172 q^{20} - 30 q^{21} - 182 q^{22} - 75 q^{24} - 216 q^{25} + 300 q^{26} - 324 q^{27} + 309 q^{28} + 200 q^{29} - 168 q^{30} - 384 q^{31} - 159 q^{32} - 164 q^{34} + 84 q^{35} + 45 q^{36} - 244 q^{37} + 268 q^{38} - 316 q^{40} - 207 q^{42} + 190 q^{44} + 894 q^{46} + 280 q^{47} - 123 q^{48} - 424 q^{49} - 1771 q^{50} - 796 q^{52} - 16 q^{53} - 27 q^{54} + 212 q^{55} - 7 q^{56} + 252 q^{57} - 570 q^{58} + 1168 q^{59} - 516 q^{60} - 384 q^{62} + 90 q^{63} + 2705 q^{64} + 280 q^{65} + 546 q^{66} + 1552 q^{68} + 968 q^{70} + 225 q^{72} + 1622 q^{74} + 648 q^{75} - 788 q^{76} + 968 q^{77} - 900 q^{78} - 3060 q^{80} + 972 q^{81} - 2540 q^{82} - 968 q^{83} - 927 q^{84} - 852 q^{85} - 258 q^{86} - 600 q^{87} - 2186 q^{88} + 504 q^{90} + 1648 q^{91} + 4298 q^{92} + 1152 q^{93} + 4256 q^{94} + 477 q^{96} + 97 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} - 2x^{10} - 6x^{9} + 56x^{7} - 448x^{6} + 448x^{5} - 3072x^{3} - 8192x^{2} - 32768x + 262144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} - \nu^{10} - 2\nu^{9} - 6\nu^{8} + 56\nu^{6} - 448\nu^{5} + 448\nu^{4} - 3072\nu^{2} - 8192\nu - 32768 ) / 8192 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7 \nu^{11} + 11 \nu^{10} + 32 \nu^{9} + 50 \nu^{8} - 172 \nu^{7} + 904 \nu^{6} - 1232 \nu^{5} + \cdots - 786432 ) / 24576 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5 \nu^{11} - 2 \nu^{10} - 15 \nu^{9} - 84 \nu^{8} + 138 \nu^{7} + 104 \nu^{6} + 1272 \nu^{5} + \cdots + 274432 ) / 12288 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13 \nu^{11} - 47 \nu^{10} - 74 \nu^{9} - 26 \nu^{8} + 40 \nu^{7} - 1048 \nu^{6} + 5792 \nu^{5} + \cdots + 1671168 ) / 49152 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 6 \nu^{11} + 11 \nu^{10} + 19 \nu^{9} + 10 \nu^{8} + 58 \nu^{7} + 688 \nu^{6} - 1480 \nu^{5} + \cdots - 598016 ) / 12288 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 19 \nu^{11} + 25 \nu^{10} + 78 \nu^{9} - 42 \nu^{8} + 120 \nu^{7} + 488 \nu^{6} - 1824 \nu^{5} + \cdots - 1384448 ) / 24576 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 41 \nu^{11} - 75 \nu^{10} - 26 \nu^{9} - 194 \nu^{8} - 8 \nu^{7} - 2232 \nu^{6} + 5984 \nu^{5} + \cdots + 3129344 ) / 49152 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 29 \nu^{11} - 51 \nu^{10} - 86 \nu^{9} - 146 \nu^{8} - 416 \nu^{7} - 1944 \nu^{6} + \cdots + 2564096 ) / 24576 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} - 3\beta_{8} + \beta_{6} + 2\beta_{5} + \beta_{4} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + \beta_{8} + 6 \beta_{7} + \beta_{6} + 2 \beta_{5} - 9 \beta_{4} + \cdots - 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3 \beta_{11} + 2 \beta_{10} - 6 \beta_{9} + 23 \beta_{8} + 10 \beta_{7} + 3 \beta_{6} + 10 \beta_{5} + \cdots + 238 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 33 \beta_{11} + 14 \beta_{10} - 2 \beta_{9} + 7 \beta_{8} + 38 \beta_{7} + 11 \beta_{6} - 38 \beta_{5} + \cdots - 38 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 85 \beta_{11} + 6 \beta_{10} - 66 \beta_{9} - 69 \beta_{8} + 94 \beta_{7} - 41 \beta_{6} + 58 \beta_{5} + \cdots + 306 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 77 \beta_{11} + 302 \beta_{10} + 150 \beta_{9} - 5 \beta_{8} - 26 \beta_{7} - 57 \beta_{6} + \cdots + 4378 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 75 \beta_{11} - 514 \beta_{10} - 106 \beta_{9} - 477 \beta_{8} - 970 \beta_{7} + 319 \beta_{6} + \cdots + 12586 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 139 \beta_{11} - 882 \beta_{10} + 1030 \beta_{9} - 397 \beta_{8} + 1670 \beta_{7} - 209 \beta_{6} + \cdots + 34554 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/84\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(43\) \(73\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
−2.78362 − 0.501431i
−2.78362 + 0.501431i
−1.72458 − 2.24184i
−1.72458 + 2.24184i
−0.951271 − 2.66366i
−0.951271 + 2.66366i
0.965027 − 2.65871i
0.965027 + 2.65871i
2.16644 − 1.81839i
2.16644 + 1.81839i
2.82801 − 0.0488466i
2.82801 + 0.0488466i
−2.78362 − 0.501431i −3.00000 7.49713 + 2.79159i − 4.57514i 8.35087 + 1.50429i 2.93118 + 18.2868i −19.4694 − 11.5300i 9.00000 −2.29411 + 12.7355i
55.2 −2.78362 + 0.501431i −3.00000 7.49713 − 2.79159i 4.57514i 8.35087 − 1.50429i 2.93118 − 18.2868i −19.4694 + 11.5300i 9.00000 −2.29411 − 12.7355i
55.3 −1.72458 − 2.24184i −3.00000 −2.05167 + 7.73244i 6.58775i 5.17373 + 6.72551i 15.1925 − 10.5918i 20.8731 − 8.73568i 9.00000 14.7687 − 11.3611i
55.4 −1.72458 + 2.24184i −3.00000 −2.05167 − 7.73244i − 6.58775i 5.17373 − 6.72551i 15.1925 + 10.5918i 20.8731 + 8.73568i 9.00000 14.7687 + 11.3611i
55.5 −0.951271 − 2.66366i −3.00000 −6.19017 + 5.06772i − 10.8462i 2.85381 + 7.99098i −15.1344 + 10.6748i 19.3872 + 11.6677i 9.00000 −28.8905 + 10.3176i
55.6 −0.951271 + 2.66366i −3.00000 −6.19017 − 5.06772i 10.8462i 2.85381 − 7.99098i −15.1344 − 10.6748i 19.3872 − 11.6677i 9.00000 −28.8905 − 10.3176i
55.7 0.965027 − 2.65871i −3.00000 −6.13744 − 5.13145i 19.4608i −2.89508 + 7.97612i 1.95109 + 18.4172i −19.5658 + 11.3657i 9.00000 51.7405 + 18.7802i
55.8 0.965027 + 2.65871i −3.00000 −6.13744 + 5.13145i − 19.4608i −2.89508 − 7.97612i 1.95109 − 18.4172i −19.5658 − 11.3657i 9.00000 51.7405 − 18.7802i
55.9 2.16644 − 1.81839i −3.00000 1.38692 − 7.87886i − 4.47531i −6.49932 + 5.45517i −14.9825 − 10.8869i −11.3222 − 19.5910i 9.00000 −8.13786 − 9.69549i
55.10 2.16644 + 1.81839i −3.00000 1.38692 + 7.87886i 4.47531i −6.49932 − 5.45517i −14.9825 + 10.8869i −11.3222 + 19.5910i 9.00000 −8.13786 + 9.69549i
55.11 2.82801 − 0.0488466i −3.00000 7.99523 − 0.276277i 16.6517i −8.48402 + 0.146540i 15.0420 − 10.8045i 22.5971 − 1.17185i 9.00000 0.813380 + 47.0912i
55.12 2.82801 + 0.0488466i −3.00000 7.99523 + 0.276277i − 16.6517i −8.48402 − 0.146540i 15.0420 + 10.8045i 22.5971 + 1.17185i 9.00000 0.813380 − 47.0912i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.4.b.a ✓ 12
3.b odd 2 1 252.4.b.f 12
4.b odd 2 1 84.4.b.b yes 12
7.b odd 2 1 84.4.b.b yes 12
8.b even 2 1 1344.4.b.h 12
8.d odd 2 1 1344.4.b.g 12
12.b even 2 1 252.4.b.e 12
21.c even 2 1 252.4.b.e 12
28.d even 2 1 inner 84.4.b.a ✓ 12
56.e even 2 1 1344.4.b.h 12
56.h odd 2 1 1344.4.b.g 12
84.h odd 2 1 252.4.b.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.b.a ✓ 12 1.a even 1 1 trivial
84.4.b.a ✓ 12 28.d even 2 1 inner
84.4.b.b yes 12 4.b odd 2 1
84.4.b.b yes 12 7.b odd 2 1
252.4.b.e 12 12.b even 2 1
252.4.b.e 12 21.c even 2 1
252.4.b.f 12 3.b odd 2 1
252.4.b.f 12 84.h odd 2 1
1344.4.b.g 12 8.d odd 2 1
1344.4.b.g 12 56.h odd 2 1
1344.4.b.h 12 8.b even 2 1
1344.4.b.h 12 56.e even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{19}^{6} + 42T_{19}^{5} - 27084T_{19}^{4} - 1835880T_{19}^{3} + 99824480T_{19}^{2} + 8618904192T_{19} + 111463534080 \) acting on \(S_{4}^{\mathrm{new}}(84, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - T^{11} + \cdots + 262144 \) Copy content Toggle raw display
$3$ \( (T + 3)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 224760987648 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 16\!\cdots\!49 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 62\!\cdots\!08 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 59\!\cdots\!12 \) Copy content Toggle raw display
$19$ \( (T^{6} + 42 T^{5} + \cdots + 111463534080)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 17\!\cdots\!12 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 3698977199040)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 34547086393344)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 122 T^{5} + \cdots + 11485380096)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 47\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 118618190512128)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 24896048217408)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 19\!\cdots\!20)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 64\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 22\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 96\!\cdots\!88 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 17\!\cdots\!56)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 18\!\cdots\!72 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 29\!\cdots\!88 \) Copy content Toggle raw display
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