Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [520,2,Mod(101,520)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(520, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 3, 0, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("520.101");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 520 = 2^{3} \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 520.ca (of order \(6\), degree \(2\), minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Self dual: | no |
Analytic conductor: | \(4.15222090511\) |
Analytic rank: | \(0\) |
Dimension: | \(56\) |
Relative dimension: | \(28\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.
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.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
101.1 | −1.39780 | + | 0.214856i | −1.78301 | + | 1.02942i | 1.90767 | − | 0.600650i | −1.00000 | 2.27111 | − | 1.82202i | 1.47046 | + | 0.848970i | −2.53749 | + | 1.24946i | 0.619423 | − | 1.07287i | 1.39780 | − | 0.214856i | ||
101.2 | −1.38390 | − | 0.291221i | 1.44998 | − | 0.837149i | 1.83038 | + | 0.806043i | −1.00000 | −2.25043 | + | 0.736268i | 1.28278 | + | 0.740615i | −2.29833 | − | 1.64853i | −0.0983634 | + | 0.170370i | 1.38390 | + | 0.291221i | ||
101.3 | −1.37890 | − | 0.314043i | 1.00502 | − | 0.580249i | 1.80275 | + | 0.866072i | −1.00000 | −1.56805 | + | 0.484488i | −2.74313 | − | 1.58374i | −2.21384 | − | 1.76037i | −0.826622 | + | 1.43175i | 1.37890 | + | 0.314043i | ||
101.4 | −1.24568 | − | 0.669541i | −2.59349 | + | 1.49735i | 1.10343 | + | 1.66806i | −1.00000 | 4.23319 | − | 0.128773i | −0.668489 | − | 0.385952i | −0.257683 | − | 2.81666i | 2.98412 | − | 5.16865i | 1.24568 | + | 0.669541i | ||
101.5 | −1.13099 | + | 0.849042i | −0.176551 | + | 0.101932i | 0.558256 | − | 1.92051i | −1.00000 | 0.113132 | − | 0.265182i | 0.820502 | + | 0.473717i | 0.999213 | + | 2.64605i | −1.47922 | + | 2.56208i | 1.13099 | − | 0.849042i | ||
101.6 | −1.09478 | + | 0.895236i | −0.815452 | + | 0.470802i | 0.397104 | − | 1.96018i | −1.00000 | 0.471265 | − | 1.24545i | −2.82007 | − | 1.62817i | 1.32008 | + | 2.50148i | −1.05669 | + | 1.83024i | 1.09478 | − | 0.895236i | ||
101.7 | −1.00973 | − | 0.990170i | 2.90546 | − | 1.67747i | 0.0391258 | + | 1.99962i | −1.00000 | −4.59472 | − | 1.18310i | 0.132211 | + | 0.0763322i | 1.94045 | − | 2.05782i | 4.12781 | − | 7.14957i | 1.00973 | + | 0.990170i | ||
101.8 | −0.936579 | − | 1.05963i | 0.700368 | − | 0.404358i | −0.245639 | + | 1.98486i | −1.00000 | −1.08442 | − | 0.363419i | 3.38125 | + | 1.95217i | 2.33328 | − | 1.59869i | −1.17299 | + | 2.03168i | 0.936579 | + | 1.05963i | ||
101.9 | −0.819242 | + | 1.15275i | 2.50839 | − | 1.44822i | −0.657684 | − | 1.88877i | −1.00000 | −0.385538 | + | 4.07801i | −2.08112 | − | 1.20154i | 2.71609 | + | 0.789211i | 2.69469 | − | 4.66734i | 0.819242 | − | 1.15275i | ||
101.10 | −0.683296 | + | 1.23819i | −2.80828 | + | 1.62136i | −1.06621 | − | 1.69210i | −1.00000 | −0.0886625 | − | 4.58505i | 3.76770 | + | 2.17528i | 2.82367 | − | 0.163970i | 3.75763 | − | 6.50841i | 0.683296 | − | 1.23819i | ||
101.11 | −0.567872 | − | 1.29519i | −0.654624 | + | 0.377948i | −1.35504 | + | 1.47101i | −1.00000 | 0.861257 | + | 0.633238i | 0.518819 | + | 0.299540i | 2.67472 | + | 0.919699i | −1.21431 | + | 2.10325i | 0.567872 | + | 1.29519i | ||
101.12 | −0.316120 | + | 1.37843i | 0.855456 | − | 0.493898i | −1.80014 | − | 0.871498i | −1.00000 | 0.410377 | + | 1.33532i | −1.13840 | − | 0.657256i | 1.77036 | − | 2.20586i | −1.01213 | + | 1.75306i | 0.316120 | − | 1.37843i | ||
101.13 | −0.137476 | + | 1.40752i | −1.00895 | + | 0.582519i | −1.96220 | − | 0.386999i | −1.00000 | −0.681198 | − | 1.50020i | 0.786953 | + | 0.454347i | 0.814463 | − | 2.70863i | −0.821344 | + | 1.42261i | 0.137476 | − | 1.40752i | ||
101.14 | 0.0259673 | − | 1.41398i | 1.48693 | − | 0.858477i | −1.99865 | − | 0.0734343i | −1.00000 | −1.17525 | − | 2.12477i | −3.27445 | − | 1.89050i | −0.155734 | + | 2.82414i | −0.0260346 | + | 0.0450933i | −0.0259673 | + | 1.41398i | ||
101.15 | 0.0710713 | − | 1.41243i | −2.41987 | + | 1.39711i | −1.98990 | − | 0.200766i | −1.00000 | 1.80134 | + | 3.51719i | −3.43922 | − | 1.98564i | −0.424992 | + | 2.79632i | 2.40385 | − | 4.16359i | −0.0710713 | + | 1.41243i | ||
101.16 | 0.164552 | + | 1.40461i | 1.79930 | − | 1.03883i | −1.94585 | + | 0.462263i | −1.00000 | 1.75522 | + | 2.35637i | 3.65440 | + | 2.10987i | −0.969491 | − | 2.65708i | 0.658316 | − | 1.14024i | −0.164552 | − | 1.40461i | ||
101.17 | 0.237097 | − | 1.39420i | −0.158158 | + | 0.0913126i | −1.88757 | − | 0.661121i | −1.00000 | 0.0898089 | + | 0.242153i | 2.61251 | + | 1.50833i | −1.36927 | + | 2.47489i | −1.48332 | + | 2.56919i | −0.237097 | + | 1.39420i | ||
101.18 | 0.491578 | + | 1.32603i | −2.26293 | + | 1.30650i | −1.51670 | + | 1.30369i | −1.00000 | −2.84486 | − | 2.35846i | −3.24721 | − | 1.87478i | −2.47431 | − | 1.37032i | 1.91389 | − | 3.31496i | −0.491578 | − | 1.32603i | ||
101.19 | 0.738148 | − | 1.20629i | 2.77854 | − | 1.60419i | −0.910276 | − | 1.78084i | −1.00000 | 0.115852 | − | 4.53585i | 1.83925 | + | 1.06189i | −2.82013 | − | 0.216468i | 3.64684 | − | 6.31652i | −0.738148 | + | 1.20629i | ||
101.20 | 0.795042 | + | 1.16958i | 0.127180 | − | 0.0734274i | −0.735818 | + | 1.85972i | −1.00000 | 0.186992 | + | 0.0903689i | −2.93738 | − | 1.69590i | −2.76009 | + | 0.617962i | −1.48922 | + | 2.57940i | −0.795042 | − | 1.16958i | ||
See all 56 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
104.s | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 520.2.ca.a | ✓ | 56 |
8.b | even | 2 | 1 | 520.2.ca.b | yes | 56 | |
13.e | even | 6 | 1 | 520.2.ca.b | yes | 56 | |
104.s | even | 6 | 1 | inner | 520.2.ca.a | ✓ | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
520.2.ca.a | ✓ | 56 | 1.a | even | 1 | 1 | trivial |
520.2.ca.a | ✓ | 56 | 104.s | even | 6 | 1 | inner |
520.2.ca.b | yes | 56 | 8.b | even | 2 | 1 | |
520.2.ca.b | yes | 56 | 13.e | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{56} - 56 T_{3}^{54} + 1764 T_{3}^{52} + 36 T_{3}^{51} - 38040 T_{3}^{50} - 1500 T_{3}^{49} + \cdots + 61465600 \) acting on \(S_{2}^{\mathrm{new}}(520, [\chi])\).