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Properties

Label 385.2.bg.a
Level $385$
Weight $2$
Character orbit 385.bg
Analytic conductor $3.074$
Analytic rank $0$
Dimension $128$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [385,2,Mod(16,385)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(385, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 10, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("385.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 385 = 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 385.bg (of order \(15\), degree \(8\), 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: \(3.07424047782\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(16\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128 q + 3 q^{3} + 20 q^{4} - 16 q^{5} - 6 q^{7} + 20 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q + 3 q^{3} + 20 q^{4} - 16 q^{5} - 6 q^{7} + 20 q^{8} + 13 q^{9} - 10 q^{10} + 4 q^{11} + 6 q^{12} + 14 q^{13} + 13 q^{14} + 6 q^{15} + 22 q^{16} + 23 q^{17} - 33 q^{18} - 2 q^{19} + 30 q^{20} - 16 q^{21} - 2 q^{22} + 8 q^{23} + 30 q^{24} + 16 q^{25} + 16 q^{26} - 42 q^{27} + 27 q^{28} - 16 q^{29} - 8 q^{31} + 80 q^{32} - 4 q^{33} - 96 q^{34} + 2 q^{35} - 68 q^{36} - 2 q^{37} - 15 q^{38} - 44 q^{39} + 10 q^{40} - 8 q^{41} + 120 q^{42} - 252 q^{43} + 5 q^{44} + 62 q^{45} - 15 q^{46} - 18 q^{47} - 222 q^{48} + 24 q^{49} - 56 q^{51} + 26 q^{52} - 8 q^{53} + 66 q^{54} - 2 q^{55} - 70 q^{56} + 36 q^{57} + 17 q^{58} - 34 q^{59} + 29 q^{60} - 36 q^{61} + 44 q^{62} - 39 q^{63} + 44 q^{64} + 12 q^{65} - 60 q^{66} + 10 q^{67} + 78 q^{68} + 108 q^{69} - 13 q^{70} - 39 q^{72} - 3 q^{73} - 13 q^{74} - 2 q^{75} + 36 q^{76} + 13 q^{77} + 40 q^{78} - 69 q^{79} - 7 q^{80} + 71 q^{81} - 6 q^{82} + 48 q^{83} + 2 q^{84} - 4 q^{85} - 29 q^{86} - 122 q^{87} - 17 q^{88} + 24 q^{90} - 47 q^{91} + 28 q^{92} + 33 q^{93} - 21 q^{94} + 2 q^{95} + 15 q^{96} - 58 q^{97} - 90 q^{98} + 100 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −2.48362 − 0.527910i −0.154959 + 1.47433i 4.06260 + 1.80878i −0.669131 − 0.743145i 1.16317 − 3.57988i −2.63782 + 0.204771i −5.02672 − 3.65212i 0.784795 + 0.166813i 1.26955 + 2.19893i
16.2 −2.19313 − 0.466165i −0.134536 + 1.28002i 2.76543 + 1.23125i −0.669131 − 0.743145i 0.891757 − 2.74454i 2.11950 + 1.58358i −1.86315 − 1.35366i 1.31408 + 0.279317i 1.12106 + 1.94174i
16.3 −1.85877 − 0.395095i 0.0857569 − 0.815923i 1.47185 + 0.655310i −0.669131 − 0.743145i −0.481769 + 1.48273i 1.60824 − 2.10085i 0.597821 + 0.434342i 2.27607 + 0.483793i 0.950150 + 1.64571i
16.4 −1.73586 − 0.368968i 0.324493 − 3.08734i 1.04998 + 0.467481i −0.669131 − 0.743145i −1.70240 + 5.23946i 0.427265 + 2.61102i 1.22130 + 0.887324i −6.49193 − 1.37990i 0.887320 + 1.53688i
16.5 −1.29673 − 0.275629i 0.0805266 − 0.766160i −0.221545 − 0.0986380i −0.669131 − 0.743145i −0.315598 + 0.971309i −1.82828 − 1.91243i 2.40513 + 1.74743i 2.35393 + 0.500343i 0.662852 + 1.14809i
16.6 −0.948062 − 0.201517i −0.310072 + 2.95014i −0.968879 − 0.431373i −0.669131 − 0.743145i 0.888470 − 2.73443i −2.08004 + 1.63506i 2.39989 + 1.74363i −5.67274 − 1.20578i 0.484621 + 0.839388i
16.7 −0.0856524 − 0.0182060i −0.191136 + 1.81854i −1.82009 − 0.810355i −0.669131 − 0.743145i 0.0494794 − 0.152282i 1.21114 − 2.35226i 0.282826 + 0.205485i −0.336098 − 0.0714399i 0.0437829 + 0.0758343i
16.8 −0.0138669 − 0.00294751i −0.00623667 + 0.0593379i −1.82691 − 0.813392i −0.669131 − 0.743145i 0.000261382 0 0.000804452i −1.70824 + 2.02037i 0.0458745 + 0.0333298i 2.93096 + 0.622995i 0.00708836 + 0.0122774i
16.9 0.899593 + 0.191214i 0.00620345 − 0.0590219i −1.05439 − 0.469443i −0.669131 − 0.743145i 0.0168664 − 0.0519095i −1.51002 − 2.17252i −2.34684 − 1.70508i 2.93100 + 0.623003i −0.459845 − 0.796475i
16.10 1.00398 + 0.213403i 0.279619 − 2.66040i −0.864648 − 0.384966i −0.669131 − 0.743145i 0.848472 − 2.61133i 2.64455 + 0.0798230i −2.44671 − 1.77764i −4.06510 − 0.864063i −0.513207 − 0.888900i
16.11 1.27591 + 0.271203i 0.276533 − 2.63103i −0.272693 − 0.121411i −0.669131 − 0.743145i 1.06638 − 3.28197i −2.31419 + 1.28239i −2.42559 − 1.76230i −3.91143 − 0.831400i −0.652208 − 1.12966i
16.12 1.52639 + 0.324444i −0.319852 + 3.04319i 0.397510 + 0.176983i −0.669131 − 0.743145i −1.47557 + 4.54132i 1.59067 + 2.11418i −1.97559 − 1.43535i −6.22427 − 1.32301i −0.780245 − 1.35142i
16.13 1.91336 + 0.406697i −0.00809976 + 0.0770640i 1.66844 + 0.742838i −0.669131 − 0.743145i −0.0468394 + 0.144157i 2.64565 − 0.0229757i −0.274827 − 0.199673i 2.92857 + 0.622487i −0.978051 − 1.69403i
16.14 2.41002 + 0.512266i −0.00445547 + 0.0423909i 3.71869 + 1.65567i −0.669131 − 0.743145i −0.0324532 + 0.0998806i 0.537679 + 2.59054i 4.12736 + 2.99871i 2.93267 + 0.623357i −1.23193 − 2.13377i
16.15 2.47196 + 0.525431i 0.228008 − 2.16935i 4.00741 + 1.78421i −0.669131 − 0.743145i 1.70347 − 5.24275i −1.75901 − 1.97633i 4.87959 + 3.54523i −1.71967 − 0.365527i −1.26359 − 2.18860i
16.16 2.65346 + 0.564011i −0.320924 + 3.05338i 4.89566 + 2.17969i −0.669131 − 0.743145i −2.57370 + 7.92104i −0.256108 − 2.63333i 7.37178 + 5.35591i −6.28572 − 1.33607i −1.35637 − 2.34930i
81.1 −0.267970 + 2.54956i −0.289753 − 0.321804i −4.47216 − 0.950588i −0.913545 − 0.406737i 0.898104 − 0.652510i −1.48138 − 2.19215i 2.03759 − 6.27107i 0.293985 − 2.79708i 1.28180 − 2.22015i
81.2 −0.248098 + 2.36050i −0.446775 − 0.496193i −3.55411 − 0.755449i −0.913545 − 0.406737i 1.28211 − 0.931506i −0.591520 + 2.57878i 1.19810 − 3.68738i 0.266985 − 2.54019i 1.18675 − 2.05551i
81.3 −0.211129 + 2.00876i 1.98285 + 2.20217i −2.03426 − 0.432395i −0.913545 − 0.406737i −4.84228 + 3.51812i −2.64106 + 0.157528i 0.0497481 − 0.153109i −0.604305 + 5.74958i 1.00991 − 1.74922i
81.4 −0.200627 + 1.90883i −1.92283 − 2.13552i −1.64710 − 0.350102i −0.913545 − 0.406737i 4.46213 − 3.24193i 2.30574 − 1.29752i −0.187483 + 0.577013i −0.549585 + 5.22896i 0.959674 − 1.66220i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
11.c even 5 1 inner
77.m even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 385.2.bg.a ✓ 128
7.c even 3 1 inner 385.2.bg.a ✓ 128
11.c even 5 1 inner 385.2.bg.a ✓ 128
77.m even 15 1 inner 385.2.bg.a ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.bg.a ✓ 128 1.a even 1 1 trivial
385.2.bg.a ✓ 128 7.c even 3 1 inner
385.2.bg.a ✓ 128 11.c even 5 1 inner
385.2.bg.a ✓ 128 77.m even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{128} - 26 T_{2}^{126} - 20 T_{2}^{125} + 278 T_{2}^{124} + 416 T_{2}^{123} - 710 T_{2}^{122} + \cdots + 923521 \) acting on \(S_{2}^{\mathrm{new}}(385, [\chi])\). Copy content Toggle raw display