Properties

Label 105.3.r.a
Level 105
Weight 3
Character orbit 105.r
Analytic conductor 2.861
Analytic rank 0
Dimension 32
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 105.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.86104277578\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32q + 32q^{4} - 6q^{5} - 48q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 32q + 32q^{4} - 6q^{5} - 48q^{9} + 78q^{10} - 28q^{11} + 60q^{14} - 24q^{15} - 40q^{16} - 60q^{19} + 12q^{21} - 34q^{25} - 96q^{26} - 88q^{29} + 84q^{31} - 170q^{35} - 192q^{36} + 36q^{39} + 330q^{40} + 320q^{44} + 18q^{45} - 60q^{46} + 356q^{49} + 12q^{51} - 468q^{56} - 804q^{59} - 198q^{60} + 336q^{61} - 400q^{64} - 46q^{65} - 108q^{66} - 438q^{70} + 344q^{71} + 900q^{74} + 144q^{75} - 20q^{79} + 1140q^{80} - 144q^{81} + 780q^{84} + 304q^{85} + 144q^{86} + 24q^{89} - 224q^{91} - 924q^{94} - 342q^{95} + 900q^{96} + 168q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −3.26825 1.88692i 0.866025 + 1.50000i 5.12097 + 8.86977i −3.80462 + 3.24421i 6.53650i −1.16497 6.90238i 23.5561i −1.50000 + 2.59808i 18.5560 3.42385i
19.2 −2.95153 1.70407i −0.866025 1.50000i 3.80769 + 6.59511i −3.57204 3.49865i 5.90306i −5.14807 + 4.74314i 12.3217i −1.50000 + 2.59808i 4.58104 + 16.4134i
19.3 −2.53945 1.46615i −0.866025 1.50000i 2.29920 + 3.98234i 1.01132 + 4.89665i 5.07890i 6.93149 + 0.976973i 1.75471i −1.50000 + 2.59808i 4.61104 13.9176i
19.4 −2.08943 1.20633i 0.866025 + 1.50000i 0.910488 + 1.57701i 4.48539 + 2.20936i 4.17887i −6.98737 + 0.420230i 5.25727i −1.50000 + 2.59808i −6.70671 10.0272i
19.5 −1.71634 0.990928i −0.866025 1.50000i −0.0361245 0.0625695i 4.45717 2.26576i 3.43267i −2.11222 6.67372i 8.07061i −1.50000 + 2.59808i −9.89520 0.527928i
19.6 −1.32532 0.765175i 0.866025 + 1.50000i −0.829016 1.43590i −3.72620 + 3.33398i 2.65064i 6.34933 + 2.94719i 8.65876i −1.50000 + 2.59808i 7.48949 1.56739i
19.7 −0.952353 0.549841i 0.866025 + 1.50000i −1.39535 2.41681i −2.62055 4.25825i 1.90471i −6.08554 + 3.45923i 7.46761i −1.50000 + 2.59808i 0.154330 + 5.49625i
19.8 −0.428040 0.247129i −0.866025 1.50000i −1.87785 3.25254i −2.75389 + 4.17326i 0.856079i −5.82770 + 3.87787i 3.83332i −1.50000 + 2.59808i 2.21011 1.10575i
19.9 0.428040 + 0.247129i 0.866025 + 1.50000i −1.87785 3.25254i 2.23720 4.47157i 0.856079i 5.82770 3.87787i 3.83332i −1.50000 + 2.59808i 2.06266 1.36113i
19.10 0.952353 + 0.549841i −0.866025 1.50000i −1.39535 2.41681i −4.99803 0.140341i 1.90471i 6.08554 3.45923i 7.46761i −1.50000 + 2.59808i −4.68273 2.88178i
19.11 1.32532 + 0.765175i −0.866025 1.50000i −0.829016 1.43590i 1.02421 4.89398i 2.65064i −6.34933 2.94719i 8.65876i −1.50000 + 2.59808i 5.10215 5.70239i
19.12 1.71634 + 0.990928i 0.866025 + 1.50000i −0.0361245 0.0625695i 0.266380 + 4.99290i 3.43267i 2.11222 + 6.67372i 8.07061i −1.50000 + 2.59808i −4.49040 + 8.83346i
19.13 2.08943 + 1.20633i −0.866025 1.50000i 0.910488 + 1.57701i 4.15605 + 2.77979i 4.17887i 6.98737 0.420230i 5.25727i −1.50000 + 2.59808i 5.33045 + 10.8218i
19.14 2.53945 + 1.46615i 0.866025 + 1.50000i 2.29920 + 3.98234i 4.74629 1.57250i 5.07890i −6.93149 0.976973i 1.75471i −1.50000 + 2.59808i 14.3585 + 2.96550i
19.15 2.95153 + 1.70407i 0.866025 + 1.50000i 3.80769 + 6.59511i −4.81594 1.34415i 5.90306i 5.14807 4.74314i 12.3217i −1.50000 + 2.59808i −11.9239 12.1740i
19.16 3.26825 + 1.88692i −0.866025 1.50000i 5.12097 + 8.86977i 0.907258 4.91700i 6.53650i 1.16497 + 6.90238i 23.5561i −1.50000 + 2.59808i 12.2431 14.3580i
94.1 −3.26825 + 1.88692i 0.866025 1.50000i 5.12097 8.86977i −3.80462 3.24421i 6.53650i −1.16497 + 6.90238i 23.5561i −1.50000 2.59808i 18.5560 + 3.42385i
94.2 −2.95153 + 1.70407i −0.866025 + 1.50000i 3.80769 6.59511i −3.57204 + 3.49865i 5.90306i −5.14807 4.74314i 12.3217i −1.50000 2.59808i 4.58104 16.4134i
94.3 −2.53945 + 1.46615i −0.866025 + 1.50000i 2.29920 3.98234i 1.01132 4.89665i 5.07890i 6.93149 0.976973i 1.75471i −1.50000 2.59808i 4.61104 + 13.9176i
94.4 −2.08943 + 1.20633i 0.866025 1.50000i 0.910488 1.57701i 4.48539 2.20936i 4.17887i −6.98737 0.420230i 5.25727i −1.50000 2.59808i −6.70671 + 10.0272i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 94.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.d odd 6 1 inner
35.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.3.r.a 32
3.b odd 2 1 315.3.bi.e 32
5.b even 2 1 inner 105.3.r.a 32
5.c odd 4 1 525.3.o.p 16
5.c odd 4 1 525.3.o.q 16
7.c even 3 1 735.3.e.a 32
7.d odd 6 1 inner 105.3.r.a 32
7.d odd 6 1 735.3.e.a 32
15.d odd 2 1 315.3.bi.e 32
21.g even 6 1 315.3.bi.e 32
35.i odd 6 1 inner 105.3.r.a 32
35.i odd 6 1 735.3.e.a 32
35.j even 6 1 735.3.e.a 32
35.k even 12 1 525.3.o.p 16
35.k even 12 1 525.3.o.q 16
105.p even 6 1 315.3.bi.e 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.r.a 32 1.a even 1 1 trivial
105.3.r.a 32 5.b even 2 1 inner
105.3.r.a 32 7.d odd 6 1 inner
105.3.r.a 32 35.i odd 6 1 inner
315.3.bi.e 32 3.b odd 2 1
315.3.bi.e 32 15.d odd 2 1
315.3.bi.e 32 21.g even 6 1
315.3.bi.e 32 105.p even 6 1
525.3.o.p 16 5.c odd 4 1
525.3.o.p 16 35.k even 12 1
525.3.o.q 16 5.c odd 4 1
525.3.o.q 16 35.k even 12 1
735.3.e.a 32 7.c even 3 1
735.3.e.a 32 7.d odd 6 1
735.3.e.a 32 35.i odd 6 1
735.3.e.a 32 35.j even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(105, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database