Properties

Label 315.2.cb.a
Level 315
Weight 2
Character orbit 315.cb
Analytic conductor 2.515
Analytic rank 0
Dimension 176
CM no
Inner twists 8

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Newspace parameters

Level: \( N \) = \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 315.cb (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(176\)
Relative dimension: \(44\) over \(\Q(\zeta_{12})\)
Coefficient ring index: multiple of None
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.

\(\operatorname{Tr}(f)(q) = \) \( 176q - 4q^{2} - 2q^{7} - 32q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 176q - 4q^{2} - 2q^{7} - 32q^{8} - 12q^{11} - 20q^{15} + 56q^{16} - 8q^{18} + 12q^{22} - 12q^{23} - 4q^{25} - 32q^{28} - 56q^{30} + 48q^{32} - 8q^{35} - 80q^{36} - 16q^{37} + 38q^{42} - 4q^{43} - 80q^{46} - 76q^{50} - 28q^{51} + 64q^{53} - 52q^{56} - 112q^{57} - 44q^{58} - 40q^{60} + 24q^{63} + 20q^{65} - 4q^{67} + 18q^{70} - 64q^{71} - 20q^{72} + 26q^{77} + 76q^{78} - 64q^{81} - 4q^{85} + 80q^{86} - 60q^{88} - 16q^{91} - 68q^{92} + 88q^{93} + 40q^{95} - 120q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −2.62442 0.703211i −1.23789 + 1.21146i 4.66101 + 2.69104i −1.95271 + 1.08946i 4.10064 2.30889i −1.75017 1.98416i −6.49767 6.49767i 0.0647207 2.99930i 5.89084 1.48604i
13.2 −2.62442 0.703211i 1.23789 1.21146i 4.66101 + 2.69104i 1.95271 1.08946i −4.10064 + 2.30889i −2.50777 + 0.843251i −6.49767 6.49767i 0.0647207 2.99930i −5.89084 + 1.48604i
13.3 −2.51063 0.672721i −1.39463 1.02713i 4.11865 + 2.37791i 0.00805112 2.23605i 2.81044 + 3.51694i 2.62706 0.313957i −5.06493 5.06493i 0.890010 + 2.86494i −1.52445 + 5.60848i
13.4 −2.51063 0.672721i 1.39463 + 1.02713i 4.11865 + 2.37791i −0.00805112 + 2.23605i −2.81044 3.51694i 2.11812 + 1.58542i −5.06493 5.06493i 0.890010 + 2.86494i 1.52445 5.60848i
13.5 −2.12971 0.570655i −0.712199 1.57885i 2.47798 + 1.43066i 0.544352 + 2.16880i 0.615799 + 3.76892i −0.222357 2.63639i −1.34285 1.34285i −1.98555 + 2.24891i 0.0783216 4.92955i
13.6 −2.12971 0.570655i 0.712199 + 1.57885i 2.47798 + 1.43066i −0.544352 2.16880i −0.615799 3.76892i −1.51076 + 2.17200i −1.34285 1.34285i −1.98555 + 2.24891i −0.0783216 + 4.92955i
13.7 −2.08198 0.557866i −0.911321 + 1.47292i 2.29139 + 1.32293i 2.22883 + 0.179731i 2.71905 2.55820i 2.62753 + 0.309985i −0.984377 0.984377i −1.33899 2.68461i −4.54013 1.61759i
13.8 −2.08198 0.557866i 0.911321 1.47292i 2.29139 + 1.32293i −2.22883 0.179731i −2.71905 + 2.55820i 2.43050 + 1.04531i −0.984377 0.984377i −1.33899 2.68461i 4.54013 + 1.61759i
13.9 −1.73908 0.465984i −1.63791 0.563241i 1.07519 + 0.620762i −2.23593 + 0.0247189i 2.58599 + 1.74276i −1.24960 + 2.33206i 0.965609 + 0.965609i 2.36552 + 1.84508i 3.89997 + 0.998920i
13.10 −1.73908 0.465984i 1.63791 + 0.563241i 1.07519 + 0.620762i 2.23593 0.0247189i −2.58599 1.74276i 0.0838463 2.64442i 0.965609 + 0.965609i 2.36552 + 1.84508i −3.89997 0.998920i
13.11 −1.66459 0.446025i −1.52423 + 0.822634i 0.839860 + 0.484894i 1.24718 1.85595i 2.90413 0.689501i −2.31427 1.28225i 1.25538 + 1.25538i 1.64655 2.50776i −2.90384 + 2.53311i
13.12 −1.66459 0.446025i 1.52423 0.822634i 0.839860 + 0.484894i −1.24718 + 1.85595i −2.90413 + 0.689501i −2.64534 0.0466691i 1.25538 + 1.25538i 1.64655 2.50776i 2.90384 2.53311i
13.13 −1.44975 0.388459i −0.181894 1.72247i 0.218820 + 0.126336i 2.12297 + 0.702131i −0.405409 + 2.56781i 0.203685 + 2.63790i 1.85442 + 1.85442i −2.93383 + 0.626616i −2.80503 1.84260i
13.14 −1.44975 0.388459i 0.181894 + 1.72247i 0.218820 + 0.126336i −2.12297 0.702131i 0.405409 2.56781i 1.49535 2.18265i 1.85442 + 1.85442i −2.93383 + 0.626616i 2.80503 + 1.84260i
13.15 −1.01873 0.272968i −0.709229 + 1.58019i −0.768749 0.443838i −0.354370 + 2.20781i 1.15386 1.41619i −0.541135 + 2.58982i 2.15352 + 2.15352i −1.99399 2.24143i 0.963669 2.15243i
13.16 −1.01873 0.272968i 0.709229 1.58019i −0.768749 0.443838i 0.354370 2.20781i −1.15386 + 1.41619i 0.826274 2.51342i 2.15352 + 2.15352i −1.99399 2.24143i −0.963669 + 2.15243i
13.17 −0.511726 0.137117i −1.55469 0.763502i −1.48899 0.859668i 2.01511 0.969187i 0.690887 + 0.603878i 2.55829 + 0.674650i 1.39330 + 1.39330i 1.83413 + 2.37402i −1.16408 + 0.219653i
13.18 −0.511726 0.137117i 1.55469 + 0.763502i −1.48899 0.859668i −2.01511 + 0.969187i −0.690887 0.603878i 2.55287 + 0.694881i 1.39330 + 1.39330i 1.83413 + 2.37402i 1.16408 0.219653i
13.19 −0.461159 0.123567i −1.73179 0.0298722i −1.53465 0.886032i 1.48035 + 1.67588i 0.794941 + 0.227769i −2.29050 1.32425i 1.27342 + 1.27342i 2.99822 + 0.103465i −0.475591 0.955770i
13.20 −0.461159 0.123567i 1.73179 + 0.0298722i −1.53465 0.886032i −1.48035 1.67588i −0.794941 0.227769i −2.64575 + 0.00158185i 1.27342 + 1.27342i 2.99822 + 0.103465i 0.475591 + 0.955770i
See next 80 embeddings (of 176 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 223.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
9.c even 3 1 inner
35.f even 4 1 inner
45.k odd 12 1 inner
63.l odd 6 1 inner
315.cb even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.cb.a 176
3.b odd 2 1 945.2.ce.a 176
5.c odd 4 1 inner 315.2.cb.a 176
7.b odd 2 1 inner 315.2.cb.a 176
9.c even 3 1 inner 315.2.cb.a 176
9.d odd 6 1 945.2.ce.a 176
15.e even 4 1 945.2.ce.a 176
21.c even 2 1 945.2.ce.a 176
35.f even 4 1 inner 315.2.cb.a 176
45.k odd 12 1 inner 315.2.cb.a 176
45.l even 12 1 945.2.ce.a 176
63.l odd 6 1 inner 315.2.cb.a 176
63.o even 6 1 945.2.ce.a 176
105.k odd 4 1 945.2.ce.a 176
315.cb even 12 1 inner 315.2.cb.a 176
315.cf odd 12 1 945.2.ce.a 176
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.cb.a 176 1.a even 1 1 trivial
315.2.cb.a 176 5.c odd 4 1 inner
315.2.cb.a 176 7.b odd 2 1 inner
315.2.cb.a 176 9.c even 3 1 inner
315.2.cb.a 176 35.f even 4 1 inner
315.2.cb.a 176 45.k odd 12 1 inner
315.2.cb.a 176 63.l odd 6 1 inner
315.2.cb.a 176 315.cb even 12 1 inner
945.2.ce.a 176 3.b odd 2 1
945.2.ce.a 176 9.d odd 6 1
945.2.ce.a 176 15.e even 4 1
945.2.ce.a 176 21.c even 2 1
945.2.ce.a 176 45.l even 12 1
945.2.ce.a 176 63.o even 6 1
945.2.ce.a 176 105.k odd 4 1
945.2.ce.a 176 315.cf odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(315, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database