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Properties

Label 147.6.c.c
Level $147$
Weight $6$
Character orbit 147.c
Analytic conductor $23.576$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [147,6,Mod(146,147)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(147, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.146");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 147.c (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: \(23.5764215125\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 171 x^{14} + 21495 x^{12} - 1128902 x^{10} + 42970860 x^{8} - 655075344 x^{6} + \cdots + 90230547456 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{8}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 21)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + \beta_{5} q^{3} + ( - \beta_{2} - 11) q^{4} + \beta_{8} q^{5} + ( - \beta_{10} + \beta_{7} + \cdots + 4 \beta_1) q^{6}+ \cdots + ( - 2 \beta_{14} + \beta_{13} + \cdots + 76) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + \beta_{5} q^{3} + ( - \beta_{2} - 11) q^{4} + \beta_{8} q^{5} + ( - \beta_{10} + \beta_{7} + \cdots + 4 \beta_1) q^{6}+ \cdots + ( - 42 \beta_{15} + 193 \beta_{14} + \cdots + 13949) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 172 q^{4} + 1212 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 172 q^{4} + 1212 q^{9} + 1188 q^{15} + 5716 q^{16} + 876 q^{18} - 21900 q^{22} + 13156 q^{25} - 900 q^{30} - 15132 q^{36} + 20932 q^{37} + 34836 q^{39} + 111052 q^{43} - 163392 q^{46} - 63192 q^{51} - 31368 q^{57} + 83412 q^{58} - 120132 q^{60} - 158884 q^{64} + 204404 q^{67} - 661728 q^{72} - 277512 q^{78} + 502616 q^{79} - 358524 q^{81} + 205152 q^{85} + 719028 q^{88} - 35352 q^{93} + 215472 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 171 x^{14} + 21495 x^{12} - 1128902 x^{10} + 42970860 x^{8} - 655075344 x^{6} + \cdots + 90230547456 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 12\!\cdots\!65 \nu^{14} + \cdots + 21\!\cdots\!64 ) / 64\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 32290359027 \nu^{14} + 5080229838581 \nu^{12} - 625594991538609 \nu^{10} + \cdots - 55\!\cdots\!48 ) / 14\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13230933495217 \nu^{15} + \cdots + 93\!\cdots\!12 \nu ) / 25\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13230933495217 \nu^{14} + \cdots - 22\!\cdots\!28 ) / 18\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 42\!\cdots\!10 \nu^{15} + \cdots - 48\!\cdots\!96 ) / 32\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 250787622883333 \nu^{14} + \cdots - 74\!\cdots\!16 ) / 14\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 37\!\cdots\!85 \nu^{15} + \cdots - 40\!\cdots\!92 ) / 25\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 25\!\cdots\!69 \nu^{15} + \cdots - 86\!\cdots\!68 \nu ) / 12\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 16\!\cdots\!51 \nu^{14} + \cdots + 26\!\cdots\!92 ) / 80\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 21\!\cdots\!43 \nu^{15} + \cdots - 97\!\cdots\!92 ) / 64\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 464108966034257 \nu^{14} + \cdots - 59\!\cdots\!44 ) / 14\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 10\!\cdots\!45 \nu^{15} + \cdots + 40\!\cdots\!92 ) / 25\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 16\!\cdots\!29 \nu^{15} + \cdots + 51\!\cdots\!80 ) / 21\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 32\!\cdots\!39 \nu^{15} + \cdots + 33\!\cdots\!76 ) / 86\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 61\!\cdots\!77 \nu^{15} + \cdots + 33\!\cdots\!76 ) / 86\!\cdots\!88 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{12} + \beta_{10} - \beta_{7} - \beta_{5} - 7\beta_{3} ) / 14 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 7\beta_{12} - 7\beta_{7} - 14\beta_{5} + 40\beta_{4} + 7\beta_{2} + 7\beta _1 + 301 ) / 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - 3\beta_{14} - \beta_{11} - 80\beta_{3} - \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 651 \beta_{12} - 14 \beta_{11} - 182 \beta_{9} - 651 \beta_{7} + 112 \beta_{6} - 1302 \beta_{5} + \cdots - 24255 ) / 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 973 \beta_{15} - 2499 \beta_{14} + 476 \beta_{13} + 3000 \beta_{12} - 1001 \beta_{11} - 11260 \beta_{10} + \cdots - 525 ) / 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -566\beta_{11} + 2524\beta_{6} - 11445\beta_{2} - 333005 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 111489 \beta_{15} + 271719 \beta_{14} - 78596 \beta_{13} + 279656 \beta_{12} + 119413 \beta_{11} + \cdots + 40817 ) / 14 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 6320615 \beta_{12} - 569058 \beta_{11} + 2600178 \beta_{9} + 6320615 \beta_{7} + 2153676 \beta_{6} + \cdots - 238072499 ) / 14 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1744655 \beta_{15} + 4165881 \beta_{14} - 1393260 \beta_{13} + 1907243 \beta_{11} - 696630 \beta_{6} + \cdots + 513983 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 658696647 \beta_{12} + 67988914 \beta_{11} + 280792162 \beta_{9} + 658696647 \beta_{7} + \cdots + 24936699795 ) / 14 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1317846089 \beta_{15} + 3119998287 \beta_{14} - 1105495636 \beta_{13} - 2936793576 \beta_{12} + \cdots + 348328281 ) / 14 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1085940862\beta_{11} - 3779909252\beta_{6} + 13721538933\beta_{2} + 377261441413 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 141367916025 \beta_{15} - 333469461087 \beta_{14} + 121040631124 \beta_{13} - 310243081576 \beta_{12} + \cdots - 35530456969 ) / 14 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 7397466827527 \beta_{12} + 827711044818 \beta_{11} - 3221809636290 \beta_{9} - 7397466827527 \beta_{7} + \cdots + 281013171864019 ) / 14 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 2161024276975 \beta_{15} - 5089578052089 \beta_{14} + 1866472518732 \beta_{13} - 2397513146923 \beta_{11} + \cdots - 531040628191 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\) \(50\) \(52\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
146.1
8.95007 + 5.16733i
−8.95007 + 5.16733i
−5.89199 + 3.40174i
5.89199 + 3.40174i
3.16536 + 1.82752i
−3.16536 + 1.82752i
1.84692 + 1.06632i
−1.84692 + 1.06632i
1.84692 − 1.06632i
−1.84692 − 1.06632i
3.16536 − 1.82752i
−3.16536 − 1.82752i
−5.89199 − 3.40174i
5.89199 − 3.40174i
8.95007 − 5.16733i
−8.95007 − 5.16733i
− 10.3347i −12.5210 − 9.28579i −74.8050 −31.7408 −95.9654 + 129.400i 0 442.375i 70.5484 + 232.534i 328.031i
146.2 − 10.3347i 12.5210 + 9.28579i −74.8050 31.7408 95.9654 − 129.400i 0 442.375i 70.5484 + 232.534i − 328.031i
146.3 − 6.80349i −12.4320 + 9.40448i −14.2874 −14.9141 63.9832 + 84.5813i 0 − 120.507i 66.1117 − 233.834i 101.468i
146.4 − 6.80349i 12.4320 − 9.40448i −14.2874 14.9141 −63.9832 − 84.5813i 0 − 120.507i 66.1117 − 233.834i − 101.468i
146.5 − 3.65505i −15.5635 + 0.882561i 18.6406 74.3244 3.22580 + 56.8851i 0 − 185.094i 241.442 − 27.4714i − 271.659i
146.6 − 3.65505i 15.5635 − 0.882561i 18.6406 −74.3244 −3.22580 − 56.8851i 0 − 185.094i 241.442 − 27.4714i 271.659i
146.7 − 2.13264i −9.16236 + 12.6115i 27.4518 −95.0525 26.8959 + 19.5400i 0 − 126.790i −75.1022 − 231.103i 202.713i
146.8 − 2.13264i 9.16236 − 12.6115i 27.4518 95.0525 −26.8959 − 19.5400i 0 − 126.790i −75.1022 − 231.103i − 202.713i
146.9 2.13264i −9.16236 − 12.6115i 27.4518 −95.0525 26.8959 − 19.5400i 0 126.790i −75.1022 + 231.103i − 202.713i
146.10 2.13264i 9.16236 + 12.6115i 27.4518 95.0525 −26.8959 + 19.5400i 0 126.790i −75.1022 + 231.103i 202.713i
146.11 3.65505i −15.5635 − 0.882561i 18.6406 74.3244 3.22580 − 56.8851i 0 185.094i 241.442 + 27.4714i 271.659i
146.12 3.65505i 15.5635 + 0.882561i 18.6406 −74.3244 −3.22580 + 56.8851i 0 185.094i 241.442 + 27.4714i − 271.659i
146.13 6.80349i −12.4320 − 9.40448i −14.2874 −14.9141 63.9832 − 84.5813i 0 120.507i 66.1117 + 233.834i − 101.468i
146.14 6.80349i 12.4320 + 9.40448i −14.2874 14.9141 −63.9832 + 84.5813i 0 120.507i 66.1117 + 233.834i 101.468i
146.15 10.3347i −12.5210 + 9.28579i −74.8050 −31.7408 −95.9654 − 129.400i 0 − 442.375i 70.5484 − 232.534i − 328.031i
146.16 10.3347i 12.5210 − 9.28579i −74.8050 31.7408 95.9654 + 129.400i 0 − 442.375i 70.5484 − 232.534i 328.031i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 146.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.6.c.c 16
3.b odd 2 1 inner 147.6.c.c 16
7.b odd 2 1 inner 147.6.c.c 16
7.c even 3 1 21.6.g.c ✓ 16
7.d odd 6 1 21.6.g.c ✓ 16
21.c even 2 1 inner 147.6.c.c 16
21.g even 6 1 21.6.g.c ✓ 16
21.h odd 6 1 21.6.g.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.g.c ✓ 16 7.c even 3 1
21.6.g.c ✓ 16 7.d odd 6 1
21.6.g.c ✓ 16 21.g even 6 1
21.6.g.c ✓ 16 21.h odd 6 1
147.6.c.c 16 1.a even 1 1 trivial
147.6.c.c 16 3.b odd 2 1 inner
147.6.c.c 16 7.b odd 2 1 inner
147.6.c.c 16 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 171T_{2}^{6} + 7746T_{2}^{4} + 97832T_{2}^{2} + 300384 \) acting on \(S_{6}^{\mathrm{new}}(147, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 171 T^{6} + \cdots + 300384)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 12\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( (T^{8} + \cdots + 11184619211136)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 50\!\cdots\!44)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 39\!\cdots\!16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 79\!\cdots\!44)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 29\!\cdots\!24)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 25\!\cdots\!96)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 48\!\cdots\!04)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 62\!\cdots\!61)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 244571248924934)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 13\!\cdots\!44)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 52\!\cdots\!64)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 20\!\cdots\!96)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 33\!\cdots\!84)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 16\!\cdots\!96)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 12\!\cdots\!76)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots - 49\!\cdots\!86)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 25\!\cdots\!96)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 10\!\cdots\!36)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 28\!\cdots\!53)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 56\!\cdots\!04)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 56\!\cdots\!96)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 60\!\cdots\!24)^{2} \) Copy content Toggle raw display
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