Properties

Label 69.7.d.a
Level $69$
Weight $7$
Character orbit 69.d
Analytic conductor $15.874$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 69.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.8737317698\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24q - 20q^{2} + 816q^{4} - 324q^{6} - 940q^{8} + 5832q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q - 20q^{2} + 816q^{4} - 324q^{6} - 940q^{8} + 5832q^{9} + 384q^{13} + 29544q^{16} - 4860q^{18} + 29336q^{23} - 39204q^{24} - 61272q^{25} + 10088q^{26} + 64672q^{29} + 9696q^{31} - 319620q^{32} - 225744q^{35} + 198288q^{36} - 11664q^{39} + 135280q^{41} + 233232q^{46} - 74336q^{47} + 552096q^{48} - 722136q^{49} + 619324q^{50} + 1059720q^{52} - 78732q^{54} - 1019328q^{55} - 694344q^{58} + 1057648q^{59} - 488776q^{62} - 273888q^{64} - 23328q^{69} + 2785512q^{70} - 255392q^{71} - 228420q^{72} - 322560q^{73} - 365472q^{75} - 1002960q^{77} - 171072q^{78} + 1417176q^{81} - 5732712q^{82} - 2704704q^{85} + 611712q^{87} - 1611444q^{92} + 2484432q^{93} - 147720q^{94} - 1672656q^{95} - 1818612q^{96} + 9104212q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1 −15.4071 15.5885 173.378 161.383i −240.172 585.691i −1685.19 243.000 2486.44i
22.2 −15.4071 15.5885 173.378 161.383i −240.172 585.691i −1685.19 243.000 2486.44i
22.3 −13.6749 −15.5885 123.003 37.2861i 213.170 322.993i −806.853 243.000 509.883i
22.4 −13.6749 −15.5885 123.003 37.2861i 213.170 322.993i −806.853 243.000 509.883i
22.5 −10.6305 15.5885 49.0083 66.6032i −165.714 306.844i 159.370 243.000 708.028i
22.6 −10.6305 15.5885 49.0083 66.6032i −165.714 306.844i 159.370 243.000 708.028i
22.7 −8.53823 −15.5885 8.90142 233.104i 133.098 155.980i 470.445 243.000 1990.29i
22.8 −8.53823 −15.5885 8.90142 233.104i 133.098 155.980i 470.445 243.000 1990.29i
22.9 −4.73591 −15.5885 −41.5712 38.4146i 73.8255 655.803i 499.975 243.000 181.928i
22.10 −4.73591 −15.5885 −41.5712 38.4146i 73.8255 655.803i 499.975 243.000 181.928i
22.11 −3.69366 15.5885 −50.3569 218.943i −57.5785 261.444i 422.396 243.000 808.702i
22.12 −3.69366 15.5885 −50.3569 218.943i −57.5785 261.444i 422.396 243.000 808.702i
22.13 −0.368531 15.5885 −63.8642 113.928i −5.74483 569.901i 47.1219 243.000 41.9859i
22.14 −0.368531 15.5885 −63.8642 113.928i −5.74483 569.901i 47.1219 243.000 41.9859i
22.15 2.88321 −15.5885 −55.6871 133.948i −44.9448 59.3150i −345.083 243.000 386.201i
22.16 2.88321 −15.5885 −55.6871 133.948i −44.9448 59.3150i −345.083 243.000 386.201i
22.17 6.36312 15.5885 −23.5107 60.2242i 99.1913 233.613i −556.841 243.000 383.214i
22.18 6.36312 15.5885 −23.5107 60.2242i 99.1913 233.613i −556.841 243.000 383.214i
22.19 10.8996 −15.5885 54.8005 162.476i −169.907 219.782i −100.271 243.000 1770.91i
22.20 10.8996 −15.5885 54.8005 162.476i −169.907 219.782i −100.271 243.000 1770.91i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 69.7.d.a 24
3.b odd 2 1 207.7.d.e 24
23.b odd 2 1 inner 69.7.d.a 24
69.c even 2 1 207.7.d.e 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.7.d.a 24 1.a even 1 1 trivial
69.7.d.a 24 23.b odd 2 1 inner
207.7.d.e 24 3.b odd 2 1
207.7.d.e 24 69.c even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(69, [\chi])\).