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Properties

Label 696.2.t.g
Level $696$
Weight $2$
Character orbit 696.t
Analytic conductor $5.558$
Analytic rank $0$
Dimension $208$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [696,2,Mod(365,696)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(696, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("696.365");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 696 = 2^{3} \cdot 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 696.t (of order \(4\), 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: \(5.55758798068\)
Analytic rank: \(0\)
Dimension: \(208\)
Relative dimension: \(104\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 208 q - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 208 q - 16 q^{7} + 4 q^{10} - 2 q^{12} + 20 q^{15} + 24 q^{16} - 26 q^{18} + 44 q^{24} - 144 q^{25} - 8 q^{31} + 72 q^{36} + 44 q^{39} + 96 q^{40} - 52 q^{46} - 22 q^{48} - 48 q^{49} - 164 q^{52} - 4 q^{54} - 40 q^{55} + 48 q^{58} - 118 q^{60} + 66 q^{66} - 20 q^{70} - 70 q^{72} - 128 q^{73} + 40 q^{76} + 80 q^{78} + 8 q^{79} + 104 q^{81} + 40 q^{82} - 88 q^{84} - 64 q^{87} - 112 q^{88} - 46 q^{90} - 116 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
365.1 −1.41406 + 0.0210575i 0.120342 + 1.72787i 1.99911 − 0.0595530i − 1.71056i −0.206555 − 2.44077i −2.33051 −2.82561 + 0.126308i −2.97104 + 0.415869i 0.0360201 + 2.41883i
365.2 −1.41364 + 0.0403506i −0.683450 − 1.59151i 1.99674 − 0.114082i − 0.184926i 1.03037 + 2.22224i 4.38455 −2.81807 + 0.241841i −2.06579 + 2.17543i 0.00746188 + 0.261418i
365.3 −1.41348 − 0.0454891i −1.27747 + 1.16964i 1.99586 + 0.128596i − 1.69698i 1.85889 − 1.59516i 1.41883 −2.81526 − 0.272558i 0.263879 − 2.98837i −0.0771943 + 2.39866i
365.4 −1.41293 + 0.0601854i 0.648663 − 1.60600i 1.99276 − 0.170076i 0.515169i −0.819860 + 2.30821i 0.570962 −2.80539 + 0.360240i −2.15847 − 2.08351i −0.0310057 − 0.727900i
365.5 −1.40857 + 0.126265i −1.35825 + 1.07478i 1.96811 − 0.355705i 2.70554i 1.77747 − 1.68540i −1.72384 −2.72731 + 0.749537i 0.689675 − 2.91965i −0.341615 − 3.81093i
365.6 −1.40182 − 0.186780i 1.69253 + 0.367909i 1.93023 + 0.523666i 2.95738i −2.30391 − 0.831874i 2.92704 −2.60803 − 1.09462i 2.72929 + 1.24539i 0.552380 − 4.14573i
365.7 −1.40133 + 0.190463i 1.72373 + 0.169528i 1.92745 − 0.533802i − 3.69536i −2.44781 + 0.0907425i −4.39400 −2.59932 + 1.11514i 2.94252 + 0.584442i 0.703829 + 5.17842i
365.8 −1.36866 + 0.356045i 1.60865 + 0.642058i 1.74646 − 0.974611i 1.46659i −2.43030 − 0.306006i 0.778811 −2.04331 + 1.95573i 2.17552 + 2.06569i −0.522172 − 2.00726i
365.9 −1.36726 + 0.361401i −1.36679 − 1.06390i 1.73878 − 0.988255i 4.12894i 2.25325 + 0.960661i −0.676720 −2.02020 + 1.97959i 0.736244 + 2.90825i −1.49220 − 5.64532i
365.10 −1.36400 + 0.373515i 1.30958 − 1.13358i 1.72097 − 1.01895i 2.17248i −1.36285 + 2.03535i −3.83730 −1.96681 + 2.03265i 0.429979 − 2.96903i −0.811454 − 2.96325i
365.11 −1.34473 + 0.437852i −1.68576 − 0.397767i 1.61657 − 1.17758i − 2.70932i 2.44104 − 0.203226i 0.541761 −1.65824 + 2.29134i 2.68356 + 1.34108i 1.18628 + 3.64330i
365.12 −1.31554 − 0.518995i 1.09403 + 1.34279i 1.46129 + 1.36552i − 0.317921i −0.742344 − 2.33429i −0.841194 −1.21369 − 2.55479i −0.606177 + 2.93812i −0.164999 + 0.418237i
365.13 −1.29557 − 0.567011i 1.59504 − 0.675154i 1.35700 + 1.46920i 0.668659i −2.44931 − 0.0296986i 0.121228 −0.925029 − 2.67289i 2.08833 − 2.15380i 0.379137 − 0.866294i
365.14 −1.28359 − 0.593631i −1.72318 − 0.175031i 1.29521 + 1.52396i 2.19946i 2.10796 + 1.24760i 3.15028 −0.757845 − 2.72501i 2.93873 + 0.603222i 1.30567 − 2.82320i
365.15 −1.27976 − 0.601852i −1.68965 + 0.380895i 1.27555 + 1.54045i − 3.70911i 2.39158 + 0.529467i −1.96163 −0.705268 − 2.73909i 2.70984 − 1.28716i −2.23233 + 4.74675i
365.16 −1.27768 + 0.606244i −1.33100 − 1.10835i 1.26494 − 1.54917i − 1.88643i 2.37253 + 0.609198i −3.99657 −0.677007 + 2.74621i 0.543143 + 2.95042i 1.14364 + 2.41025i
365.17 −1.24707 + 0.666946i 1.23118 + 1.21828i 1.11037 − 1.66346i − 2.82661i −2.34789 − 0.698151i 3.88321 −0.275267 + 2.81500i 0.0315900 + 2.99983i 1.88520 + 3.52498i
365.18 −1.23619 − 0.686901i −0.558752 + 1.63945i 1.05633 + 1.69828i 2.84970i 1.81686 − 1.64286i −1.96652 −0.139276 − 2.82500i −2.37559 − 1.83209i 1.95746 − 3.52277i
365.19 −1.21383 + 0.725691i 0.466862 + 1.66794i 0.946745 − 1.76172i 2.30375i −1.77710 − 1.68580i −1.01899 0.129284 + 2.82547i −2.56408 + 1.55740i −1.67181 − 2.79635i
365.20 −1.19450 + 0.757085i 0.745490 − 1.56341i 0.853646 − 1.80867i − 2.64820i 0.293147 + 2.43189i −0.0707996 0.349639 + 2.80673i −1.88849 − 2.33101i 2.00491 + 3.16326i
See next 80 embeddings (of 208 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 365.104
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner
29.c odd 4 1 inner
87.f even 4 1 inner
232.l odd 4 1 inner
696.t even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 696.2.t.g ✓ 208
3.b odd 2 1 inner 696.2.t.g ✓ 208
8.b even 2 1 inner 696.2.t.g ✓ 208
24.h odd 2 1 inner 696.2.t.g ✓ 208
29.c odd 4 1 inner 696.2.t.g ✓ 208
87.f even 4 1 inner 696.2.t.g ✓ 208
232.l odd 4 1 inner 696.2.t.g ✓ 208
696.t even 4 1 inner 696.2.t.g ✓ 208
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
696.2.t.g ✓ 208 1.a even 1 1 trivial
696.2.t.g ✓ 208 3.b odd 2 1 inner
696.2.t.g ✓ 208 8.b even 2 1 inner
696.2.t.g ✓ 208 24.h odd 2 1 inner
696.2.t.g ✓ 208 29.c odd 4 1 inner
696.2.t.g ✓ 208 87.f even 4 1 inner
696.2.t.g ✓ 208 232.l odd 4 1 inner
696.2.t.g ✓ 208 696.t even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(696, [\chi])\):

\( T_{5}^{52} + 148 T_{5}^{50} + 10191 T_{5}^{48} + 434018 T_{5}^{46} + 12819193 T_{5}^{44} + \cdots + 8997306368 \) Copy content Toggle raw display
\( T_{11}^{104} + 7474 T_{11}^{100} + 24816577 T_{11}^{96} + 48433801828 T_{11}^{92} + \cdots + 70\!\cdots\!24 \) Copy content Toggle raw display
\( T_{17}^{104} + 14912 T_{17}^{100} + 99596210 T_{17}^{96} + 395706958148 T_{17}^{92} + \cdots + 18\!\cdots\!84 \) Copy content Toggle raw display
\( T_{53}^{52} - 1280 T_{53}^{50} + 748209 T_{53}^{48} - 264812646 T_{53}^{46} + 63446678434 T_{53}^{44} + \cdots + 33\!\cdots\!52 \) Copy content Toggle raw display