Properties

Label 21.30.c.a
Level $21$
Weight $30$
Character orbit 21.c
Analytic conductor $111.884$
Analytic rank $0$
Dimension $76$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 30 \)
Character orbit: \([\chi]\) \(=\) 21.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(111.883889004\)
Analytic rank: \(0\)
Dimension: \(76\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 76 q - 20401094660 q^{4} + 1962751543872 q^{7} - 89581451225964 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 76 q - 20401094660 q^{4} + 1962751543872 q^{7} - 89581451225964 q^{9} - 116668222909200048 q^{15} + 4964525139713333892 q^{16} + 1542346202347854996 q^{18} + 33840646982361357348 q^{21} - 9870996616978150672 q^{22} + \)\(32\!\cdots\!12\)\( q^{25} - \)\(31\!\cdots\!64\)\( q^{28} + \)\(51\!\cdots\!44\)\( q^{30} - \)\(85\!\cdots\!52\)\( q^{36} - \)\(17\!\cdots\!76\)\( q^{37} + \)\(38\!\cdots\!96\)\( q^{39} + \)\(80\!\cdots\!56\)\( q^{42} + \)\(23\!\cdots\!20\)\( q^{43} - \)\(10\!\cdots\!00\)\( q^{46} - \)\(51\!\cdots\!40\)\( q^{49} - \)\(96\!\cdots\!00\)\( q^{51} + \)\(20\!\cdots\!48\)\( q^{57} - \)\(40\!\cdots\!72\)\( q^{58} - \)\(16\!\cdots\!96\)\( q^{60} - \)\(36\!\cdots\!84\)\( q^{63} - \)\(16\!\cdots\!28\)\( q^{64} + \)\(48\!\cdots\!64\)\( q^{67} - \)\(65\!\cdots\!20\)\( q^{70} + \)\(19\!\cdots\!24\)\( q^{72} - \)\(80\!\cdots\!36\)\( q^{78} - \)\(11\!\cdots\!52\)\( q^{79} + \)\(18\!\cdots\!32\)\( q^{81} - \)\(72\!\cdots\!52\)\( q^{84} + \)\(14\!\cdots\!48\)\( q^{85} - \)\(16\!\cdots\!84\)\( q^{88} + \)\(44\!\cdots\!40\)\( q^{91} + \)\(22\!\cdots\!32\)\( q^{93} - \)\(23\!\cdots\!24\)\( q^{99} + O(q^{100}) \)

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 27459.9i −7.86336e6 2.60729e6i −2.17175e8 −2.60713e10 −7.15959e10 + 2.15927e11i −1.78867e12 1.43352e11i 8.77881e12i 5.50345e13 + 4.10041e13i 7.15915e14i
20.2 27459.9i −7.86336e6 + 2.60729e6i −2.17175e8 −2.60713e10 −7.15959e10 2.15927e11i −1.78867e12 + 1.43352e11i 8.77881e12i 5.50345e13 4.10041e13i 7.15915e14i
20.3 40537.3i −896293. + 8.23572e6i −1.10640e9 2.53522e10 3.33854e11 + 3.63333e10i 3.38830e11 1.76213e12i 2.30873e13i −6.70237e13 1.47632e13i 1.02771e15i
20.4 40537.3i −896293. 8.23572e6i −1.10640e9 2.53522e10 3.33854e11 3.63333e10i 3.38830e11 + 1.76213e12i 2.30873e13i −6.70237e13 + 1.47632e13i 1.02771e15i
20.5 38457.7i 8.27750e6 336727.i −9.42122e8 −2.26488e10 −1.29498e10 3.18333e11i 6.92104e11 1.65557e12i 1.55850e13i 6.84036e13 5.57452e12i 8.71020e14i
20.6 38457.7i 8.27750e6 + 336727.i −9.42122e8 −2.26488e10 −1.29498e10 + 3.18333e11i 6.92104e11 + 1.65557e12i 1.55850e13i 6.84036e13 + 5.57452e12i 8.71020e14i
20.7 21897.4i −3.41396e6 + 7.54819e6i 5.73727e7 −2.19660e10 1.65286e11 + 7.47571e10i 7.76630e11 1.61764e12i 1.30124e13i −4.53201e13 5.15385e13i 4.81000e14i
20.8 21897.4i −3.41396e6 7.54819e6i 5.73727e7 −2.19660e10 1.65286e11 7.47571e10i 7.76630e11 + 1.61764e12i 1.30124e13i −4.53201e13 + 5.15385e13i 4.81000e14i
20.9 1978.96i 6.63697e6 + 4.95792e6i 5.32955e8 −2.10087e10 9.81152e9 1.31343e10i 1.76916e12 2.99987e11i 2.11714e12i 1.94684e13 + 6.58112e13i 4.15753e13i
20.10 1978.96i 6.63697e6 4.95792e6i 5.32955e8 −2.10087e10 9.81152e9 + 1.31343e10i 1.76916e12 + 2.99987e11i 2.11714e12i 1.94684e13 6.58112e13i 4.15753e13i
20.11 23469.1i −4.70121e6 6.82122e6i −1.39279e7 1.98757e10 −1.60088e11 + 1.10333e11i −7.54484e11 1.62808e12i 1.22730e13i −2.44277e13 + 6.41359e13i 4.66464e14i
20.12 23469.1i −4.70121e6 + 6.82122e6i −1.39279e7 1.98757e10 −1.60088e11 1.10333e11i −7.54484e11 + 1.62808e12i 1.22730e13i −2.44277e13 6.41359e13i 4.66464e14i
20.13 42628.6i −4.19005e6 + 7.14660e6i −1.28033e9 −1.62384e10 3.04650e11 + 1.78616e11i 1.66265e12 + 6.74901e11i 3.16925e13i −3.35174e13 5.98892e13i 6.92219e14i
20.14 42628.6i −4.19005e6 7.14660e6i −1.28033e9 −1.62384e10 3.04650e11 1.78616e11i 1.66265e12 6.74901e11i 3.16925e13i −3.35174e13 + 5.98892e13i 6.92219e14i
20.15 18452.1i −2.07416e6 + 8.02049e6i 1.96390e8 1.42232e10 1.47995e11 + 3.82726e10i −7.69210e11 + 1.62118e12i 1.35302e13i −6.00261e13 3.32715e13i 2.62448e14i
20.16 18452.1i −2.07416e6 8.02049e6i 1.96390e8 1.42232e10 1.47995e11 3.82726e10i −7.69210e11 1.62118e12i 1.35302e13i −6.00261e13 + 3.32715e13i 2.62448e14i
20.17 3217.48i 7.29657e6 3.92306e6i 5.26519e8 1.29709e10 −1.26224e10 2.34766e10i 5.86304e11 1.69592e12i 3.42144e12i 3.78496e13 5.72498e13i 4.17336e13i
20.18 3217.48i 7.29657e6 + 3.92306e6i 5.26519e8 1.29709e10 −1.26224e10 + 2.34766e10i 5.86304e11 + 1.69592e12i 3.42144e12i 3.78496e13 + 5.72498e13i 4.17336e13i
20.19 1646.15i 2.25912e6 7.97037e6i 5.34161e8 1.37077e10 −1.31204e10 3.71885e9i −1.61859e12 + 7.74652e11i 1.76308e12i −5.84231e13 3.60121e13i 2.25649e13i
20.20 1646.15i 2.25912e6 + 7.97037e6i 5.34161e8 1.37077e10 −1.31204e10 + 3.71885e9i −1.61859e12 7.74652e11i 1.76308e12i −5.84231e13 + 3.60121e13i 2.25649e13i
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 20.76
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 21.30.c.a 76
3.b odd 2 1 inner 21.30.c.a 76
7.b odd 2 1 inner 21.30.c.a 76
21.c even 2 1 inner 21.30.c.a 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.30.c.a 76 1.a even 1 1 trivial
21.30.c.a 76 3.b odd 2 1 inner
21.30.c.a 76 7.b odd 2 1 inner
21.30.c.a 76 21.c even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{30}^{\mathrm{new}}(21, [\chi])\).