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Properties

Label 21.8.c.a
Level $21$
Weight $8$
Character orbit 21.c
Analytic conductor $6.560$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [21,8,Mod(20,21)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("21.20"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(21, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 21.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.56008553517\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 52 x^{14} - 43704 x^{12} + 7634268 x^{10} + 4505094126 x^{8} + 450795891132 x^{6} + \cdots + 12\!\cdots\!01 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{19}\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_1 q^{3} + (\beta_{3} - 48) q^{4} + \beta_{6} q^{5} + \beta_{7} q^{6} + (\beta_{9} + 2 \beta_1 - 81) q^{7} + (\beta_{4} - 34 \beta_{2}) q^{8} + (\beta_{5} - 4 \beta_{3} + 3 \beta_{2} - 59) q^{9}+ \cdots + (678 \beta_{12} - 675 \beta_{11} + \cdots - 1355276) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 772 q^{4} - 1288 q^{7} - 936 q^{9} - 11664 q^{15} - 4220 q^{16} - 8172 q^{18} - 56952 q^{21} + 115440 q^{22} + 97648 q^{25} + 192724 q^{28} + 180396 q^{30} - 1179792 q^{36} - 916016 q^{37} + 149760 q^{39}+ \cdots - 21577824 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 52 x^{14} - 43704 x^{12} + 7634268 x^{10} + 4505094126 x^{8} + 450795891132 x^{6} + \cdots + 12\!\cdots\!01 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 3\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1529401 \nu^{14} + 348110567 \nu^{12} + 128982501459 \nu^{10} - 4848734614437 \nu^{8} + \cdots - 10\!\cdots\!31 ) / 95\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8167 \nu^{14} + 1920592 \nu^{12} + 108277137 \nu^{10} + 17117768952 \nu^{8} + \cdots - 53\!\cdots\!08 ) / 24\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 33607429 \nu^{14} + 19192704683 \nu^{12} - 8837524634409 \nu^{10} + \cdots - 10\!\cdots\!39 ) / 47\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 395326849 \nu^{14} + 86597063137 \nu^{12} + 3897865327509 \nu^{10} + 843380776967373 \nu^{8} + \cdots + 15\!\cdots\!99 ) / 28\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 134683 \nu^{15} - 10289093 \nu^{13} + 5052446775 \nu^{11} - 142335417249 \nu^{9} + \cdots + 41\!\cdots\!05 \nu ) / 44\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1529401 \nu^{15} + 348110567 \nu^{13} + 128982501459 \nu^{11} + \cdots - 10\!\cdots\!31 \nu ) / 31\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 71957345 \nu^{15} - 255382470 \nu^{14} - 19973269475 \nu^{13} - 551912348052 \nu^{12} + \cdots + 38\!\cdots\!52 ) / 14\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 71957345 \nu^{15} + 255382470 \nu^{14} - 19973269475 \nu^{13} + 551912348052 \nu^{12} + \cdots - 38\!\cdots\!32 ) / 14\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{15} + 52 \nu^{13} - 43704 \nu^{11} + 7634268 \nu^{9} + 4505094126 \nu^{7} + \cdots + 10\!\cdots\!48 \nu ) / 18\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 796358599 \nu^{14} - 59843566553 \nu^{12} - 28127813590701 \nu^{10} + \cdots - 86\!\cdots\!11 ) / 95\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1427213171 \nu^{14} + 59449626563 \nu^{12} - 116146887620049 \nu^{10} + \cdots - 12\!\cdots\!99 ) / 14\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 17355901 \nu^{15} - 10351432835 \nu^{13} + 1067334031089 \nu^{11} + \cdots + 24\!\cdots\!99 \nu ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 6269150447 \nu^{15} + 595558437937 \nu^{13} + 185901872951205 \nu^{11} + \cdots + 65\!\cdots\!75 \nu ) / 23\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 37321843 \nu^{15} - 23994034867 \nu^{13} - 3919982982255 \nu^{11} + \cdots - 73\!\cdots\!85 \nu ) / 10\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 4\beta_{3} + 3\beta_{2} - 59 ) / 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{14} + 5\beta_{13} - \beta_{10} - 5\beta_{9} - 5\beta_{8} + 5\beta_{7} - 11\beta_{6} - 15\beta _1 + 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 21 \beta_{12} + 36 \beta_{11} - 180 \beta_{9} + 180 \beta_{8} - 28 \beta_{5} + 51 \beta_{4} + \cdots + 101450 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 144 \beta_{15} - 403 \beta_{14} + 289 \beta_{13} + 1600 \beta_{10} - 361 \beta_{9} - 361 \beta_{8} + \cdots + 361 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 9429 \beta_{12} + 5256 \beta_{11} + 50400 \beta_{9} - 50400 \beta_{8} - 3197 \beta_{5} - 34773 \beta_{4} + \cdots - 33699326 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 32112 \beta_{15} + 148720 \beta_{14} - 3184 \beta_{13} + 733253 \beta_{10} - 11144 \beta_{9} + \cdots + 11144 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2825400 \beta_{12} - 4712760 \beta_{11} - 10487520 \beta_{9} + 10487520 \beta_{8} - 3895888 \beta_{5} + \cdots - 13633968079 ) / 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 4999248 \beta_{15} - 12553840 \beta_{14} + 2189632 \beta_{13} + 179586136 \beta_{10} + \cdots - 68804456 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 639881400 \beta_{12} + 620839368 \beta_{11} + 1698829920 \beta_{9} - 1698829920 \beta_{8} + \cdots - 3803666996099 ) / 9 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1502193456 \beta_{15} + 687508081 \beta_{14} - 7005608011 \beta_{13} - 31334745337 \beta_{10} + \cdots - 12305901043 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 42466390797 \beta_{12} - 315034972884 \beta_{11} + 507085297740 \beta_{9} - 507085297740 \beta_{8} + \cdots + 457323117920978 ) / 9 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 1034258243040 \beta_{15} + 492273870557 \beta_{14} - 3354086391743 \beta_{13} + 2771425350616 \beta_{10} + \cdots - 2096074761311 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 35058358697187 \beta_{12} + 39586028381712 \beta_{11} - 17971717870080 \beta_{9} + \cdots - 72\!\cdots\!98 ) / 9 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 183287026029600 \beta_{15} - 235652437939040 \beta_{14} + 497519496815072 \beta_{13} + \cdots + 645440486770832 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1
−14.8588 + 4.71353i
14.8588 − 4.71353i
−7.14497 − 13.8546i
7.14497 + 13.8546i
−1.39256 − 15.5261i
1.39256 + 15.5261i
−14.1148 + 6.61608i
14.1148 − 6.61608i
−14.1148 − 6.61608i
14.1148 + 6.61608i
−1.39256 + 15.5261i
1.39256 − 15.5261i
−7.14497 + 13.8546i
7.14497 − 13.8546i
−14.8588 − 4.71353i
14.8588 + 4.71353i
− 19.4285i −44.5763 + 14.1406i −249.468 167.310 274.731 + 866.052i −693.192 + 585.685i 2359.95i 1787.09 − 1260.67i − 3250.59i
20.2 − 19.4285i 44.5763 − 14.1406i −249.468 −167.310 −274.731 − 866.052i −693.192 − 585.685i 2359.95i 1787.09 − 1260.67i 3250.59i
20.3 − 15.4383i −21.4349 − 41.5637i −110.341 −281.593 −641.674 + 330.919i 907.482 + 4.45738i − 272.618i −1268.09 + 1781.83i 4347.33i
20.4 − 15.4383i 21.4349 + 41.5637i −110.341 281.593 641.674 − 330.919i 907.482 − 4.45738i − 272.618i −1268.09 + 1781.83i − 4347.33i
20.5 − 8.08481i −4.17769 − 46.5784i 62.6358 479.042 −376.578 + 33.7758i −752.366 − 507.434i − 1541.25i −2152.09 + 389.180i − 3872.96i
20.6 − 8.08481i 4.17769 + 46.5784i 62.6358 −479.042 376.578 − 33.7758i −752.366 + 507.434i − 1541.25i −2152.09 + 389.180i 3872.96i
20.7 − 4.88121i −42.3444 + 19.8482i 104.174 −11.9828 96.8835 + 206.692i 216.076 − 881.393i − 1133.29i 1399.09 − 1680.92i 58.4908i
20.8 − 4.88121i 42.3444 − 19.8482i 104.174 11.9828 −96.8835 − 206.692i 216.076 + 881.393i − 1133.29i 1399.09 − 1680.92i − 58.4908i
20.9 4.88121i −42.3444 − 19.8482i 104.174 −11.9828 96.8835 − 206.692i 216.076 + 881.393i 1133.29i 1399.09 + 1680.92i − 58.4908i
20.10 4.88121i 42.3444 + 19.8482i 104.174 11.9828 −96.8835 + 206.692i 216.076 − 881.393i 1133.29i 1399.09 + 1680.92i 58.4908i
20.11 8.08481i −4.17769 + 46.5784i 62.6358 479.042 −376.578 − 33.7758i −752.366 + 507.434i 1541.25i −2152.09 − 389.180i 3872.96i
20.12 8.08481i 4.17769 − 46.5784i 62.6358 −479.042 376.578 + 33.7758i −752.366 − 507.434i 1541.25i −2152.09 − 389.180i − 3872.96i
20.13 15.4383i −21.4349 + 41.5637i −110.341 −281.593 −641.674 − 330.919i 907.482 − 4.45738i 272.618i −1268.09 − 1781.83i − 4347.33i
20.14 15.4383i 21.4349 − 41.5637i −110.341 281.593 641.674 + 330.919i 907.482 + 4.45738i 272.618i −1268.09 − 1781.83i 4347.33i
20.15 19.4285i −44.5763 − 14.1406i −249.468 167.310 274.731 − 866.052i −693.192 − 585.685i − 2359.95i 1787.09 + 1260.67i 3250.59i
20.16 19.4285i 44.5763 + 14.1406i −249.468 −167.310 −274.731 + 866.052i −693.192 + 585.685i − 2359.95i 1787.09 + 1260.67i − 3250.59i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 20.16
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 21.8.c.a ✓ 16
3.b odd 2 1 inner 21.8.c.a ✓ 16
7.b odd 2 1 inner 21.8.c.a ✓ 16
21.c even 2 1 inner 21.8.c.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.8.c.a ✓ 16 1.a even 1 1 trivial
21.8.c.a ✓ 16 3.b odd 2 1 inner
21.8.c.a ✓ 16 7.b odd 2 1 inner
21.8.c.a ✓ 16 21.c even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 705 T^{6} + \cdots + 140112000)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 52\!\cdots\!21 \) Copy content Toggle raw display
$5$ \( (T^{8} + \cdots + 73\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 45\!\cdots\!01)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 26\!\cdots\!32)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 84\!\cdots\!40)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 13\!\cdots\!20)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 18\!\cdots\!00)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 62\!\cdots\!20)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 66\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 80\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 79\!\cdots\!40)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 83\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 65\!\cdots\!92)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 18\!\cdots\!60)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 16\!\cdots\!60)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 15\!\cdots\!52)^{2} \) Copy content Toggle raw display
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