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Properties

Label 21.9.f.b
Level $21$
Weight $9$
Character orbit 21.f
Analytic conductor $8.555$
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] = [21,9,Mod(10,21)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(21, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.10");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 21.f (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: \(8.55495081128\)
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: \( x^{12} - 3 x^{11} + 1331 x^{10} + 1482 x^{9} + 1359381 x^{8} + 1222227 x^{7} + 474169529 x^{6} + \cdots + 556282980265104 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{5}\cdot 7^{5} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_{2} + 1) q^{2} + (27 \beta_{2} + 54) q^{3} + (\beta_{5} + 186 \beta_{2} - \beta_1) q^{4} + (\beta_{8} + \beta_{6} + \beta_{4} + \cdots + 76) q^{5}+ \cdots + (2187 \beta_{2} + 2187) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_{2} + 1) q^{2} + (27 \beta_{2} + 54) q^{3} + (\beta_{5} + 186 \beta_{2} - \beta_1) q^{4} + (\beta_{8} + \beta_{6} + \beta_{4} + \cdots + 76) q^{5}+ \cdots + (4374 \beta_{11} + 6561 \beta_{10} + \cdots + 6679098) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{2} + 486 q^{3} - 1117 q^{4} + 1389 q^{5} - 1226 q^{7} + 3558 q^{8} + 13122 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 3 q^{2} + 486 q^{3} - 1117 q^{4} + 1389 q^{5} - 1226 q^{7} + 3558 q^{8} + 13122 q^{9} - 19731 q^{10} + 18081 q^{11} - 90477 q^{12} - 175188 q^{14} + 75006 q^{15} - 298849 q^{16} + 63306 q^{17} - 6561 q^{18} + 476265 q^{19} - 154953 q^{21} + 419258 q^{22} - 66384 q^{23} + 144099 q^{24} + 792375 q^{25} + 1890762 q^{26} - 4229651 q^{28} + 671106 q^{29} - 532737 q^{30} + 717240 q^{31} + 895149 q^{32} + 1464561 q^{33} + 3454614 q^{35} - 4885758 q^{36} + 2289443 q^{37} - 6275988 q^{38} - 930933 q^{39} + 8773863 q^{40} - 4123629 q^{42} - 10119958 q^{43} + 14123901 q^{44} + 3037743 q^{45} - 19926544 q^{46} - 4198782 q^{47} + 8565642 q^{49} - 1162248 q^{50} + 1709262 q^{51} - 26384844 q^{52} - 209511 q^{53} - 531441 q^{54} + 70081743 q^{56} + 25718310 q^{57} + 1938935 q^{58} - 97052259 q^{59} - 51464727 q^{60} - 24195864 q^{61} - 9869931 q^{63} + 272849458 q^{64} + 30440562 q^{65} + 16979949 q^{66} - 57546057 q^{67} - 32476050 q^{68} - 108189027 q^{70} + 224566620 q^{71} + 3890673 q^{72} - 193344135 q^{73} - 71015400 q^{74} + 64182375 q^{75} + 92041617 q^{77} + 102101148 q^{78} - 29314488 q^{79} - 735358533 q^{80} - 28697814 q^{81} + 321118926 q^{82} - 120939804 q^{84} + 248647620 q^{85} + 127268268 q^{86} + 27179793 q^{87} - 487070897 q^{88} + 119863098 q^{89} + 100090965 q^{91} + 364039200 q^{92} + 19365480 q^{93} - 191086092 q^{94} - 95584488 q^{95} + 72507069 q^{96} + 24200217 q^{98} + 79086294 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 1331 x^{10} + 1482 x^{9} + 1359381 x^{8} + 1222227 x^{7} + 474169529 x^{6} + \cdots + 556282980265104 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 26\!\cdots\!73 \nu^{11} + \cdots - 90\!\cdots\!64 ) / 74\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13\!\cdots\!96 \nu^{11} + \cdots + 18\!\cdots\!88 ) / 94\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19\!\cdots\!82 \nu^{11} + \cdots - 11\!\cdots\!13 ) / 31\!\cdots\!33 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\!\cdots\!69 \nu^{11} + \cdots + 37\!\cdots\!52 ) / 82\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 92\!\cdots\!16 \nu^{11} + \cdots + 24\!\cdots\!76 ) / 35\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 44\!\cdots\!50 \nu^{11} + \cdots + 43\!\cdots\!60 ) / 11\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 19\!\cdots\!27 \nu^{11} + \cdots - 29\!\cdots\!92 ) / 11\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 95\!\cdots\!82 \nu^{11} + \cdots - 49\!\cdots\!36 ) / 35\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 18\!\cdots\!86 \nu^{11} + \cdots + 17\!\cdots\!68 ) / 59\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 37\!\cdots\!79 \nu^{11} + \cdots + 12\!\cdots\!52 ) / 35\!\cdots\!68 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} - \beta_{4} - 3\beta_{3} - 441\beta_{2} - 441 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - \beta_{10} - \beta_{9} - 3\beta_{8} - 2\beta_{6} - \beta_{5} - 2\beta_{4} - 739\beta_{3} - 740\beta _1 - 998 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 33 \beta_{10} + 8 \beta_{9} - 50 \beta_{8} + \beta_{7} - 91 \beta_{6} + 1004 \beta_{5} + \cdots - 3745 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 1142 \beta_{11} + 144 \beta_{10} + 1862 \beta_{9} + 1430 \beta_{8} + 1142 \beta_{7} - 710 \beta_{6} + \cdots + 1293844 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 3506 \beta_{11} + 58546 \beta_{10} + 47538 \beta_{9} + 139158 \beta_{8} + 115364 \beta_{6} + \cdots + 283224229 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1980795 \beta_{10} - 213456 \beta_{9} + 2768022 \beta_{8} - 1126971 \beta_{7} + 4195617 \beta_{6} + \cdots + 593763343 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 5693475 \beta_{11} - 11877864 \beta_{10} - 65082795 \beta_{9} - 73091667 \beta_{8} - 5693475 \beta_{7} + \cdots - 259845486819 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1077958192 \beta_{11} - 2295970672 \beta_{10} - 2052368176 \beta_{9} - 4562761872 \beta_{8} + \cdots - 1812479642816 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 55296351348 \beta_{10} + 11967736640 \beta_{9} - 81075419624 \beta_{8} + 7425404788 \beta_{7} + \cdots - 5442078503659 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1023600157301 \beta_{11} + 257272951392 \beta_{10} + 2309964914261 \beta_{9} + 1789287974453 \beta_{8} + \cdots + 20\!\cdots\!90 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\) \(8\) \(10\)
\(\chi(n)\) \(1\) \(1 + \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
15.7115 − 27.2131i
9.75345 − 16.8935i
3.88954 − 6.73687i
−5.65790 + 9.79977i
−7.37103 + 12.7670i
−14.8255 + 25.6786i
15.7115 + 27.2131i
9.75345 + 16.8935i
3.88954 + 6.73687i
−5.65790 − 9.79977i
−7.37103 − 12.7670i
−14.8255 − 25.6786i
−15.2115 − 26.3471i 40.5000 + 23.3827i −334.779 + 579.854i 890.914 − 514.369i − 1422.74i 2247.80 + 843.914i 12581.7 1093.50 + 1894.00i −27104.3 − 15648.6i
10.2 −9.25345 − 16.0274i 40.5000 + 23.3827i −43.2526 + 74.9157i −255.810 + 147.692i − 865.482i −2399.31 + 89.9896i −3136.82 1093.50 + 1894.00i 4734.26 + 2733.32i
10.3 −3.38954 − 5.87085i 40.5000 + 23.3827i 105.022 − 181.904i −231.653 + 133.745i − 317.026i 2354.84 − 468.558i −3159.35 1093.50 + 1894.00i 1570.39 + 906.668i
10.4 6.15790 + 10.6658i 40.5000 + 23.3827i 52.1605 − 90.3446i 599.787 − 346.287i 575.953i −419.022 − 2364.15i 4437.64 1093.50 + 1894.00i 7386.86 + 4264.81i
10.5 7.87103 + 13.6330i 40.5000 + 23.3827i 4.09363 − 7.09038i −872.651 + 503.826i 736.184i −1325.41 + 2002.02i 4158.85 1093.50 + 1894.00i −13737.3 − 7931.26i
10.6 15.3255 + 26.5446i 40.5000 + 23.3827i −341.745 + 591.919i 563.914 − 325.576i 1433.41i −1071.89 + 2148.45i −13103.0 1093.50 + 1894.00i 17284.6 + 9979.26i
19.1 −15.2115 + 26.3471i 40.5000 − 23.3827i −334.779 − 579.854i 890.914 + 514.369i 1422.74i 2247.80 − 843.914i 12581.7 1093.50 − 1894.00i −27104.3 + 15648.6i
19.2 −9.25345 + 16.0274i 40.5000 − 23.3827i −43.2526 − 74.9157i −255.810 − 147.692i 865.482i −2399.31 − 89.9896i −3136.82 1093.50 − 1894.00i 4734.26 − 2733.32i
19.3 −3.38954 + 5.87085i 40.5000 − 23.3827i 105.022 + 181.904i −231.653 − 133.745i 317.026i 2354.84 + 468.558i −3159.35 1093.50 − 1894.00i 1570.39 − 906.668i
19.4 6.15790 − 10.6658i 40.5000 − 23.3827i 52.1605 + 90.3446i 599.787 + 346.287i − 575.953i −419.022 + 2364.15i 4437.64 1093.50 − 1894.00i 7386.86 − 4264.81i
19.5 7.87103 − 13.6330i 40.5000 − 23.3827i 4.09363 + 7.09038i −872.651 − 503.826i − 736.184i −1325.41 − 2002.02i 4158.85 1093.50 − 1894.00i −13737.3 + 7931.26i
19.6 15.3255 − 26.5446i 40.5000 − 23.3827i −341.745 − 591.919i 563.914 + 325.576i − 1433.41i −1071.89 − 2148.45i −13103.0 1093.50 − 1894.00i 17284.6 − 9979.26i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.9.f.b ✓ 12
3.b odd 2 1 63.9.m.d 12
7.c even 3 1 147.9.d.b 12
7.d odd 6 1 inner 21.9.f.b ✓ 12
7.d odd 6 1 147.9.d.b 12
21.g even 6 1 63.9.m.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.9.f.b ✓ 12 1.a even 1 1 trivial
21.9.f.b ✓ 12 7.d odd 6 1 inner
63.9.m.d 12 3.b odd 2 1
63.9.m.d 12 21.g even 6 1
147.9.d.b 12 7.c even 3 1
147.9.d.b 12 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 3 T_{2}^{11} + 1331 T_{2}^{10} - 4146 T_{2}^{9} + 1372044 T_{2}^{8} + \cdots + 514460785065984 \) acting on \(S_{9}^{\mathrm{new}}(21, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 514460785065984 \) Copy content Toggle raw display
$3$ \( (T^{2} - 81 T + 2187)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 36\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 90\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 95\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 19\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 60\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 92\!\cdots\!80)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 61\!\cdots\!69 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 52\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 39\!\cdots\!96)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 35\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 15\!\cdots\!12)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 39\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 78\!\cdots\!81 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 55\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 21\!\cdots\!84 \) Copy content Toggle raw display
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