# Properties

 Label 784.2.x.n Level $784$ Weight $2$ Character orbit 784.x Analytic conductor $6.260$ Analytic rank $0$ Dimension $40$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 784.x (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.26027151847$$ Analytic rank: $$0$$ Dimension: $$40$$ Relative dimension: $$10$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$40 q + 8 q^{4}+O(q^{10})$$ 40 * q + 8 * q^4 $$\operatorname{Tr}(f)(q) =$$ $$40 q + 8 q^{4} - 4 q^{11} - 32 q^{15} - 16 q^{18} - 8 q^{29} - 8 q^{30} + 40 q^{32} + 80 q^{36} + 20 q^{37} + 120 q^{43} - 56 q^{44} + 64 q^{46} - 112 q^{50} + 16 q^{51} - 28 q^{53} + 72 q^{58} + 24 q^{60} - 64 q^{64} - 16 q^{65} - 12 q^{67} - 16 q^{72} + 16 q^{74} - 176 q^{78} + 72 q^{79} - 12 q^{81} + 64 q^{85} + 40 q^{86} - 80 q^{88} - 48 q^{92} - 48 q^{93} - 64 q^{95} - 216 q^{99}+O(q^{100})$$ 40 * q + 8 * q^4 - 4 * q^11 - 32 * q^15 - 16 * q^18 - 8 * q^29 - 8 * q^30 + 40 * q^32 + 80 * q^36 + 20 * q^37 + 120 * q^43 - 56 * q^44 + 64 * q^46 - 112 * q^50 + 16 * q^51 - 28 * q^53 + 72 * q^58 + 24 * q^60 - 64 * q^64 - 16 * q^65 - 12 * q^67 - 16 * q^72 + 16 * q^74 - 176 * q^78 + 72 * q^79 - 12 * q^81 + 64 * q^85 + 40 * q^86 - 80 * q^88 - 48 * q^92 - 48 * q^93 - 64 * q^95 - 216 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
165.1 −1.21510 0.723563i −0.608516 2.27101i 0.952912 + 1.75840i −0.400455 + 1.49452i −0.903818 + 3.19980i 0 0.114433 2.82611i −2.18914 + 1.26390i 1.56797 1.52623i
165.2 −1.21510 0.723563i 0.608516 + 2.27101i 0.952912 + 1.75840i 0.400455 1.49452i 0.903818 3.19980i 0 0.114433 2.82611i −2.18914 + 1.26390i −1.56797 + 1.52623i
165.3 −0.953167 + 1.04474i −0.554848 2.07072i −0.182947 1.99162i −0.265812 + 0.992025i 2.69222 + 1.39407i 0 2.25509 + 1.70721i −1.38196 + 0.797875i −0.783041 1.22327i
165.4 −0.953167 + 1.04474i 0.554848 + 2.07072i −0.182947 1.99162i 0.265812 0.992025i −2.69222 1.39407i 0 2.25509 + 1.70721i −1.38196 + 0.797875i 0.783041 + 1.22327i
165.5 0.164410 + 1.40462i −0.338847 1.26459i −1.94594 + 0.461868i 0.958008 3.57533i 1.72057 0.683864i 0 −0.968683 2.65738i 1.11370 0.642992i 5.17951 + 0.757821i
165.6 0.164410 + 1.40462i 0.338847 + 1.26459i −1.94594 + 0.461868i −0.958008 + 3.57533i −1.72057 + 0.683864i 0 −0.968683 2.65738i 1.11370 0.642992i −5.17951 0.757821i
165.7 0.674626 1.24293i −0.839505 3.13308i −1.08976 1.67703i 0.734404 2.74083i −4.46055 1.07021i 0 −2.81961 + 0.223131i −6.51332 + 3.76047i −2.91122 2.76185i
165.8 0.674626 1.24293i 0.839505 + 3.13308i −1.08976 1.67703i −0.734404 + 2.74083i 4.46055 + 1.07021i 0 −2.81961 + 0.223131i −6.51332 + 3.76047i 2.91122 + 2.76185i
165.9 1.32923 0.482865i −0.331601 1.23755i 1.53368 1.28367i −0.850527 + 3.17421i −1.03834 1.48487i 0 1.41877 2.44685i 1.17650 0.679252i 0.402172 + 4.62993i
165.10 1.32923 0.482865i 0.331601 + 1.23755i 1.53368 1.28367i 0.850527 3.17421i 1.03834 + 1.48487i 0 1.41877 2.44685i 1.17650 0.679252i −0.402172 4.62993i
373.1 −1.41372 + 0.0372230i −3.13308 0.839505i 1.99723 0.105246i 2.74083 0.734404i 4.46055 + 1.07021i 0 −2.81961 + 0.223131i 6.51332 + 3.76047i −3.84744 + 1.14027i
373.2 −1.41372 + 0.0372230i 3.13308 + 0.839505i 1.99723 0.105246i −2.74083 + 0.734404i −4.46055 1.07021i 0 −2.81961 + 0.223131i 6.51332 + 3.76047i 3.84744 1.14027i
373.3 −1.08279 0.909711i −1.23755 0.331601i 0.344852 + 1.97004i −3.17421 + 0.850527i 1.03834 + 1.48487i 0 1.41877 2.44685i −1.17650 0.679252i 4.21072 + 1.96667i
373.4 −1.08279 0.909711i 1.23755 + 0.331601i 0.344852 + 1.97004i 3.17421 0.850527i −1.03834 1.48487i 0 1.41877 2.44685i −1.17650 0.679252i −4.21072 1.96667i
373.5 −0.0190769 + 1.41408i −2.27101 0.608516i −1.99927 0.0539526i −1.49452 + 0.400455i 0.903818 3.19980i 0 0.114433 2.82611i 2.18914 + 1.26390i −0.537766 2.12101i
373.6 −0.0190769 + 1.41408i 2.27101 + 0.608516i −1.99927 0.0539526i 1.49452 0.400455i −0.903818 + 3.19980i 0 0.114433 2.82611i 2.18914 + 1.26390i 0.537766 + 2.12101i
373.7 1.13424 0.844695i −1.26459 0.338847i 0.572980 1.91617i 3.57533 0.958008i −1.72057 + 0.683864i 0 −0.968683 2.65738i −1.11370 0.642992i 3.24604 4.10667i
373.8 1.13424 0.844695i 1.26459 + 0.338847i 0.572980 1.91617i −3.57533 + 0.958008i 1.72057 0.683864i 0 −0.968683 2.65738i −1.11370 0.642992i −3.24604 + 4.10667i
373.9 1.38135 + 0.303098i −2.07072 0.554848i 1.81626 + 0.837371i −0.992025 + 0.265812i −2.69222 1.39407i 0 2.25509 + 1.70721i 1.38196 + 0.797875i −1.45090 + 0.0664989i
373.10 1.38135 + 0.303098i 2.07072 + 0.554848i 1.81626 + 0.837371i 0.992025 0.265812i 2.69222 + 1.39407i 0 2.25509 + 1.70721i 1.38196 + 0.797875i 1.45090 0.0664989i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 765.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
16.e even 4 1 inner
112.l odd 4 1 inner
112.w even 12 1 inner
112.x odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.x.n 40
7.b odd 2 1 inner 784.2.x.n 40
7.c even 3 1 784.2.m.i 20
7.c even 3 1 inner 784.2.x.n 40
7.d odd 6 1 784.2.m.i 20
7.d odd 6 1 inner 784.2.x.n 40
16.e even 4 1 inner 784.2.x.n 40
112.l odd 4 1 inner 784.2.x.n 40
112.w even 12 1 784.2.m.i 20
112.w even 12 1 inner 784.2.x.n 40
112.x odd 12 1 784.2.m.i 20
112.x odd 12 1 inner 784.2.x.n 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
784.2.m.i 20 7.c even 3 1
784.2.m.i 20 7.d odd 6 1
784.2.m.i 20 112.w even 12 1
784.2.m.i 20 112.x odd 12 1
784.2.x.n 40 1.a even 1 1 trivial
784.2.x.n 40 7.b odd 2 1 inner
784.2.x.n 40 7.c even 3 1 inner
784.2.x.n 40 7.d odd 6 1 inner
784.2.x.n 40 16.e even 4 1 inner
784.2.x.n 40 112.l odd 4 1 inner
784.2.x.n 40 112.w even 12 1 inner
784.2.x.n 40 112.x odd 12 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(784, [\chi])$$:

 $$T_{3}^{40} - 168 T_{3}^{36} + 20936 T_{3}^{32} - 1007232 T_{3}^{28} + 34421424 T_{3}^{24} - 639742720 T_{3}^{20} + 8394086528 T_{3}^{16} - 40915214848 T_{3}^{12} + 143584211200 T_{3}^{8} + \cdots + 319794774016$$ T3^40 - 168*T3^36 + 20936*T3^32 - 1007232*T3^28 + 34421424*T3^24 - 639742720*T3^20 + 8394086528*T3^16 - 40915214848*T3^12 + 143584211200*T3^8 - 256033067008*T3^4 + 319794774016 $$T_{5}^{40} - 376 T_{5}^{36} + 97224 T_{5}^{32} - 13188608 T_{5}^{28} + 1297863856 T_{5}^{24} - 67841683456 T_{5}^{20} + 2467458621568 T_{5}^{16} - 16224454550016 T_{5}^{12} + \cdots + 81867462148096$$ T5^40 - 376*T5^36 + 97224*T5^32 - 13188608*T5^28 + 1297863856*T5^24 - 67841683456*T5^20 + 2467458621568*T5^16 - 16224454550016*T5^12 + 84101591847168*T5^8 - 90272317374464*T5^4 + 81867462148096