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Properties

Label 147.7.b.d
Level $147$
Weight $7$
Character orbit 147.b
Analytic conductor $33.818$
Analytic rank $0$
Dimension $14$
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] = [147,7,Mod(50,147)]
 
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, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.50");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 147.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: \(33.8179502921\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 639 x^{12} + 153726 x^{10} + 17511500 x^{8} + 970722648 x^{6} + 23787564192 x^{4} + \cdots + 236051988480 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{11}\cdot 7^{2} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{5} q^{3} + (\beta_{2} - 27) q^{4} + ( - \beta_{3} + \beta_1) q^{5} + (\beta_{7} - \beta_{2} + 12) q^{6} + (\beta_{10} - 5 \beta_{5} + \cdots - 29 \beta_1) q^{8}+ \cdots + ( - \beta_{8} + 2 \beta_{2} + \cdots - 73) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{5} q^{3} + (\beta_{2} - 27) q^{4} + ( - \beta_{3} + \beta_1) q^{5} + (\beta_{7} - \beta_{2} + 12) q^{6} + (\beta_{10} - 5 \beta_{5} + \cdots - 29 \beta_1) q^{8}+ \cdots + ( - 153 \beta_{13} - 1103 \beta_{12} + \cdots + 17745) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - q^{3} - 382 q^{4} + 178 q^{6} - 1031 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - q^{3} - 382 q^{4} + 178 q^{6} - 1031 q^{9} - 930 q^{10} - 404 q^{12} - 1816 q^{13} + 2725 q^{15} + 13706 q^{16} - 1612 q^{18} + 6830 q^{19} - 13674 q^{22} - 54498 q^{24} + 7640 q^{25} - 6634 q^{27} - 32380 q^{30} + 69410 q^{31} - 85459 q^{33} + 15672 q^{34} + 199514 q^{36} + 71830 q^{37} - 104570 q^{39} + 230250 q^{40} - 249452 q^{43} - 85375 q^{45} + 342660 q^{46} + 162364 q^{48} - 42177 q^{51} + 218000 q^{52} - 402122 q^{54} - 115830 q^{55} + 393391 q^{57} - 459522 q^{58} - 1021150 q^{60} + 153158 q^{61} - 699706 q^{64} + 80896 q^{66} + 977098 q^{67} + 760371 q^{69} + 1001352 q^{72} - 488350 q^{73} + 1434610 q^{75} - 3098140 q^{76} + 122588 q^{78} - 380762 q^{79} - 893531 q^{81} - 3665004 q^{82} - 456270 q^{85} + 1452416 q^{87} + 1522482 q^{88} + 5856910 q^{90} + 1690071 q^{93} - 3034044 q^{94} + 7549346 q^{96} - 3606088 q^{97} + 217933 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 639 x^{12} + 153726 x^{10} + 17511500 x^{8} + 970722648 x^{6} + 23787564192 x^{4} + \cdots + 236051988480 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 91 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3061061 \nu^{13} - 1990313427 \nu^{11} - 485527460718 \nu^{9} + \cdots - 38\!\cdots\!20 \nu ) / 41\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5710907 \nu^{13} - 383301402 \nu^{12} + 3608149791 \nu^{11} - 229780253142 \nu^{10} + \cdots - 22\!\cdots\!08 ) / 20\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5710907 \nu^{13} + 10645998 \nu^{12} + 3608149791 \nu^{11} + 5699223090 \nu^{10} + \cdots - 44\!\cdots\!84 ) / 20\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4998127 \nu^{13} - 67721808 \nu^{12} + 501378429 \nu^{11} - 43679488776 \nu^{10} + \cdots + 19\!\cdots\!40 ) / 13\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 161303 \nu^{13} + 623027 \nu^{12} - 86351865 \nu^{11} + 394052415 \nu^{10} + \cdots + 45\!\cdots\!32 ) / 313325903213568 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1313791 \nu^{13} - 9674544 \nu^{12} + 1013940813 \nu^{11} - 6239926968 \nu^{10} + \cdots + 27\!\cdots\!20 ) / 19\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5710907 \nu^{13} + 45238710 \nu^{12} + 3608149791 \nu^{11} + 29546606034 \nu^{10} + \cdots - 21\!\cdots\!52 ) / 68\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 60170131 \nu^{13} + 106459980 \nu^{12} + 38071811337 \nu^{11} + 56992230900 \nu^{10} + \cdots - 44\!\cdots\!40 ) / 41\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 11421814 \nu^{13} + 26614995 \nu^{12} - 7216299582 \nu^{11} + 14248057725 \nu^{10} + \cdots - 11\!\cdots\!32 ) / 51\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 111072673 \nu^{13} - 79311144 \nu^{12} - 62016697311 \nu^{11} - 58436052840 \nu^{10} + \cdots - 11\!\cdots\!28 ) / 41\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 500342611 \nu^{13} - 10645998 \nu^{12} + 315022940583 \nu^{11} - 5699223090 \nu^{10} + \cdots + 44\!\cdots\!84 ) / 20\!\cdots\!88 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 91 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - 5\beta_{5} + \beta_{3} - 157\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{12} - 2 \beta_{11} - \beta_{9} + 4 \beta_{8} + 12 \beta_{7} + 4 \beta_{6} - 13 \beta_{5} + \cdots + 14341 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 13 \beta_{13} - 9 \beta_{12} - 52 \beta_{11} - 283 \beta_{10} + 6 \beta_{8} - 18 \beta_{7} + \cdots - 43 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 563 \beta_{12} + 370 \beta_{11} + 307 \beta_{9} - 1272 \beta_{8} - 4796 \beta_{7} - 1272 \beta_{6} + \cdots - 2663857 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 5839 \beta_{13} + 3859 \beta_{12} + 23868 \beta_{11} + 68715 \beta_{10} - 2610 \beta_{8} + 7718 \beta_{7} + \cdots + 20009 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 195379 \beta_{12} - 52154 \beta_{11} - 99315 \beta_{9} + 316352 \beta_{8} + 1414220 \beta_{7} + \cdots + 537425613 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1846335 \beta_{13} - 1259203 \beta_{12} - 7563356 \beta_{11} - 16125527 \beta_{10} + 884082 \beta_{8} + \cdots - 6304153 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 56401631 \beta_{12} + 4941970 \beta_{11} + 30798239 \beta_{9} - 73447664 \beta_{8} - 372501852 \beta_{7} + \cdots - 113675254073 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 509218427 \beta_{13} + 364789455 \beta_{12} + 2077666508 \beta_{11} + 3748145579 \beta_{10} + \cdots + 1712877053 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 14901642067 \beta_{12} + 252346870 \beta_{11} - 8821858835 \beta_{9} + 16698524880 \beta_{8} + \cdots + 24753349286165 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 131165587967 \beta_{13} - 98374273091 \beta_{12} - 532493786076 \beta_{11} - 868468429599 \beta_{10} + \cdots - 434119512985 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
50.1
− 15.2026i
− 12.8291i
− 10.6688i
− 8.57654i
− 6.57815i
− 3.33150i
− 1.24227i
1.24227i
3.33150i
6.57815i
8.57654i
10.6688i
12.8291i
15.2026i
− 15.2026i −0.357466 + 26.9976i −167.119 − 94.8761i 410.434 + 5.43441i 0 1567.67i −728.744 − 19.3015i −1442.36
50.2 − 12.8291i 12.9455 − 23.6942i −100.586 − 44.4166i −303.975 − 166.080i 0 469.371i −393.827 − 613.467i −569.826
50.3 − 10.6688i −24.2591 − 11.8530i −49.8239 176.484i −126.458 + 258.817i 0 − 151.243i 448.012 + 575.088i 1882.88
50.4 − 8.57654i 26.0037 + 7.26701i −9.55711 102.046i 62.3258 − 223.022i 0 − 466.932i 623.381 + 377.938i 875.203
50.5 − 6.57815i −21.8612 + 15.8458i 20.7280 − 66.3489i 104.236 + 143.806i 0 − 557.353i 226.821 − 692.815i −436.453
50.6 − 3.33150i −9.87838 − 25.1280i 52.9011 − 192.095i −83.7139 + 32.9098i 0 − 389.456i −533.835 + 496.449i −639.963
50.7 − 1.24227i 16.9070 + 21.0512i 62.4568 − 108.250i 26.1514 − 21.0031i 0 − 157.094i −157.308 + 711.825i −134.476
50.8 1.24227i 16.9070 − 21.0512i 62.4568 108.250i 26.1514 + 21.0031i 0 157.094i −157.308 − 711.825i −134.476
50.9 3.33150i −9.87838 + 25.1280i 52.9011 192.095i −83.7139 − 32.9098i 0 389.456i −533.835 − 496.449i −639.963
50.10 6.57815i −21.8612 − 15.8458i 20.7280 66.3489i 104.236 − 143.806i 0 557.353i 226.821 + 692.815i −436.453
50.11 8.57654i 26.0037 − 7.26701i −9.55711 − 102.046i 62.3258 + 223.022i 0 466.932i 623.381 − 377.938i 875.203
50.12 10.6688i −24.2591 + 11.8530i −49.8239 − 176.484i −126.458 − 258.817i 0 151.243i 448.012 − 575.088i 1882.88
50.13 12.8291i 12.9455 + 23.6942i −100.586 44.4166i −303.975 + 166.080i 0 − 469.371i −393.827 + 613.467i −569.826
50.14 15.2026i −0.357466 − 26.9976i −167.119 94.8761i 410.434 − 5.43441i 0 − 1567.67i −728.744 + 19.3015i −1442.36
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 50.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.7.b.d 14
3.b odd 2 1 inner 147.7.b.d 14
7.b odd 2 1 147.7.b.e 14
7.d odd 6 2 21.7.h.a ✓ 28
21.c even 2 1 147.7.b.e 14
21.g even 6 2 21.7.h.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.7.h.a ✓ 28 7.d odd 6 2
21.7.h.a ✓ 28 21.g even 6 2
147.7.b.d 14 1.a even 1 1 trivial
147.7.b.d 14 3.b odd 2 1 inner
147.7.b.e 14 7.b odd 2 1
147.7.b.e 14 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(147, [\chi])\):

\( T_{2}^{14} + 639 T_{2}^{12} + 153726 T_{2}^{10} + 17511500 T_{2}^{8} + 970722648 T_{2}^{6} + \cdots + 236051988480 \) Copy content Toggle raw display
\( T_{13}^{7} + 908 T_{13}^{6} - 13041327 T_{13}^{5} - 6941153706 T_{13}^{4} + 31700420190228 T_{13}^{3} + \cdots + 50\!\cdots\!32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots + 236051988480 \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 10\!\cdots\!09 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{14} \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{7} + \cdots + 50\!\cdots\!32)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 26\!\cdots\!80 \) Copy content Toggle raw display
$19$ \( (T^{7} + \cdots + 85\!\cdots\!76)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 27\!\cdots\!20 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{7} + \cdots + 20\!\cdots\!43)^{2} \) Copy content Toggle raw display
$37$ \( (T^{7} + \cdots - 53\!\cdots\!76)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots - 11\!\cdots\!92)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 17\!\cdots\!20 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 13\!\cdots\!20 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 16\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( (T^{7} + \cdots + 33\!\cdots\!68)^{2} \) Copy content Toggle raw display
$67$ \( (T^{7} + \cdots - 55\!\cdots\!64)^{2} \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{7} + \cdots - 12\!\cdots\!64)^{2} \) Copy content Toggle raw display
$79$ \( (T^{7} + \cdots + 34\!\cdots\!97)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 91\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 20\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots + 30\!\cdots\!24)^{2} \) Copy content Toggle raw display
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