Properties

Label 21.22.c.b
Level $21$
Weight $22$
Character orbit 21.c
Analytic conductor $58.690$
Analytic rank $0$
Dimension $52$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 21.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(58.6902423003\)
Analytic rank: \(0\)
Dimension: \(52\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 52q - 58720260q^{4} + 1422287188q^{7} + 10150644696q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 52q - 58720260q^{4} + 1422287188q^{7} + 10150644696q^{9} - 687153165468q^{15} + 36241782390404q^{16} + 36759093002772q^{18} + 79412641656180q^{21} + 228880341648112q^{22} + 4502168959229812q^{25} - 328465100861772q^{28} - 2422487119598196q^{30} + 94669723666911312q^{36} - 111253244635261912q^{37} - 198389110134084780q^{39} - 125754675449474820q^{42} + 808208758088766656q^{43} - 1124214920076830456q^{46} + 760563680048831596q^{49} + 4341973540932052200q^{51} - 13080454499124088812q^{57} - 10814528487339424568q^{58} + 13643758017961015644q^{60} - 38182310822257441896q^{63} - 30480555033993680900q^{64} - 25547251957658589040q^{67} + 73727169165159410040q^{70} + 39846964674222889428q^{72} + 45848199464273536260q^{78} + 199210197493083963080q^{79} + 235499606962845306588q^{81} - 373919224753720758924q^{84} - 570842672309470848672q^{85} - 1060053674241923431168q^{88} - 923077800435233893128q^{91} - 2832166256981601869904q^{93} + 1525182290356701596928q^{99} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1 1223.76i −94027.5 40239.2i 599559. 3.45471e7 −4.92432e7 + 1.15067e8i −6.99978e8 + 2.61872e8i 3.30013e9i 7.22197e9 + 7.56717e9i 4.22775e10i
20.2 1223.76i −94027.5 + 40239.2i 599559. 3.45471e7 −4.92432e7 1.15067e8i −6.99978e8 2.61872e8i 3.30013e9i 7.22197e9 7.56717e9i 4.22775e10i
20.3 247.072i 16209.6 + 100983.i 2.03611e6 −3.48005e7 2.49501e7 4.00494e6i 7.14474e8 2.19256e8i 1.02121e9i −9.93485e9 + 3.27379e9i 8.59823e9i
20.4 247.072i 16209.6 100983.i 2.03611e6 −3.48005e7 2.49501e7 + 4.00494e6i 7.14474e8 + 2.19256e8i 1.02121e9i −9.93485e9 3.27379e9i 8.59823e9i
20.5 2329.72i 52907.9 87527.8i −3.33046e6 −3.45099e7 −2.03916e8 1.23261e8i 4.88555e7 7.45761e8i 2.87327e9i −4.86187e9 9.26181e9i 8.03985e10i
20.6 2329.72i 52907.9 + 87527.8i −3.33046e6 −3.45099e7 −2.03916e8 + 1.23261e8i 4.88555e7 + 7.45761e8i 2.87327e9i −4.86187e9 + 9.26181e9i 8.03985e10i
20.7 367.125i −91325.3 46043.9i 1.96237e6 −3.23352e7 −1.69039e7 + 3.35278e7i −4.79207e8 5.73504e8i 1.49035e9i 6.22027e9 + 8.40995e9i 1.18711e10i
20.8 367.125i −91325.3 + 46043.9i 1.96237e6 −3.23352e7 −1.69039e7 3.35278e7i −4.79207e8 + 5.73504e8i 1.49035e9i 6.22027e9 8.40995e9i 1.18711e10i
20.9 1925.57i 78349.6 65739.6i −1.61068e6 2.77549e7 −1.26586e8 1.50868e8i 7.43581e8 + 7.50540e7i 9.36744e8i 1.81697e9 1.03013e10i 5.34441e10i
20.10 1925.57i 78349.6 + 65739.6i −1.61068e6 2.77549e7 −1.26586e8 + 1.50868e8i 7.43581e8 7.50540e7i 9.36744e8i 1.81697e9 + 1.03013e10i 5.34441e10i
20.11 1704.62i −65800.8 78298.2i −808569. −2.49846e7 −1.33469e8 + 1.12165e8i 7.21028e7 + 7.43873e8i 2.19654e9i −1.80087e9 + 1.03042e10i 4.25893e10i
20.12 1704.62i −65800.8 + 78298.2i −808569. −2.49846e7 −1.33469e8 1.12165e8i 7.21028e7 7.43873e8i 2.19654e9i −1.80087e9 1.03042e10i 4.25893e10i
20.13 2426.55i 100608. + 18395.7i −3.79100e6 −2.12980e7 4.46381e7 2.44130e8i 5.03992e8 + 5.51850e8i 4.11022e9i 9.78355e9 + 3.70150e9i 5.16808e10i
20.14 2426.55i 100608. 18395.7i −3.79100e6 −2.12980e7 4.46381e7 + 2.44130e8i 5.03992e8 5.51850e8i 4.11022e9i 9.78355e9 3.70150e9i 5.16808e10i
20.15 2502.20i −102208. + 3714.28i −4.16388e6 −1.82569e7 9.29389e6 + 2.55746e8i −7.47262e8 + 1.20672e7i 5.17136e9i 1.04328e10 7.59261e8i 4.56824e10i
20.16 2502.20i −102208. 3714.28i −4.16388e6 −1.82569e7 9.29389e6 2.55746e8i −7.47262e8 1.20672e7i 5.17136e9i 1.04328e10 + 7.59261e8i 4.56824e10i
20.17 2061.69i −759.014 102273.i −2.15341e6 1.95223e7 −2.10855e8 + 1.56485e6i −6.67328e8 3.36481e8i 1.15978e8i −1.04592e10 + 1.55253e8i 4.02489e10i
20.18 2061.69i −759.014 + 102273.i −2.15341e6 1.95223e7 −2.10855e8 1.56485e6i −6.67328e8 + 3.36481e8i 1.15978e8i −1.04592e10 1.55253e8i 4.02489e10i
20.19 1077.74i 64855.1 79083.3i 935630. −7.14030e6 −8.52312e7 6.98969e7i −3.76236e8 + 6.45749e8i 3.26855e9i −2.04799e9 1.02579e10i 7.69538e9i
20.20 1077.74i 64855.1 + 79083.3i 935630. −7.14030e6 −8.52312e7 + 6.98969e7i −3.76236e8 6.45749e8i 3.26855e9i −2.04799e9 + 1.02579e10i 7.69538e9i
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 20.52
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 21.22.c.b 52
3.b odd 2 1 inner 21.22.c.b 52
7.b odd 2 1 inner 21.22.c.b 52
21.c even 2 1 inner 21.22.c.b 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.22.c.b 52 1.a even 1 1 trivial
21.22.c.b 52 3.b odd 2 1 inner
21.22.c.b 52 7.b odd 2 1 inner
21.22.c.b 52 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(77\!\cdots\!68\)\( T_{2}^{22} + \)\(82\!\cdots\!96\)\( T_{2}^{20} + \)\(55\!\cdots\!20\)\( T_{2}^{18} + \)\(25\!\cdots\!76\)\( T_{2}^{16} + \)\(78\!\cdots\!20\)\( T_{2}^{14} + \)\(16\!\cdots\!48\)\( T_{2}^{12} + \)\(23\!\cdots\!88\)\( T_{2}^{10} + \)\(20\!\cdots\!20\)\( T_{2}^{8} + \)\(10\!\cdots\!44\)\( T_{2}^{6} + \)\(29\!\cdots\!16\)\( T_{2}^{4} + \)\(32\!\cdots\!12\)\( T_{2}^{2} + \)\(10\!\cdots\!00\)\( \)">\(T_{2}^{26} + \cdots\) acting on \(S_{22}^{\mathrm{new}}(21, [\chi])\).