# 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 $4$

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## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{30}^{\mathrm{new}}(21, [\chi])$$.