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Properties

Label 3307.2.a.b
Level $3307$
Weight $2$
Character orbit 3307.a
Self dual yes
Analytic conductor $26.407$
Analytic rank $1$
Dimension $129$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3307,2,Mod(1,3307)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3307, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3307.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3307 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3307.a (trivial)

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: yes
Analytic conductor: \(26.4065279484\)
Analytic rank: \(1\)
Dimension: \(129\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 129 q - 29 q^{2} - 14 q^{3} + 119 q^{4} - 62 q^{5} - 11 q^{6} - 18 q^{7} - 84 q^{8} + 95 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 129 q - 29 q^{2} - 14 q^{3} + 119 q^{4} - 62 q^{5} - 11 q^{6} - 18 q^{7} - 84 q^{8} + 95 q^{9} - 40 q^{11} - 37 q^{12} - 58 q^{13} - 45 q^{14} - 51 q^{15} + 107 q^{16} - 53 q^{17} - 73 q^{18} - 21 q^{19} - 106 q^{20} - 64 q^{21} - 31 q^{22} - 86 q^{23} - 18 q^{24} + 91 q^{25} - 17 q^{26} - 44 q^{27} - 34 q^{28} - 98 q^{29} - 11 q^{30} - 14 q^{31} - 172 q^{32} - 73 q^{33} - 12 q^{34} - 54 q^{35} + 64 q^{36} - 113 q^{37} - 91 q^{38} - 28 q^{39} + 26 q^{40} - 72 q^{41} - 33 q^{42} - 35 q^{43} - 90 q^{44} - 160 q^{45} + 7 q^{46} - 92 q^{47} - 61 q^{48} + 93 q^{49} - 116 q^{50} - 38 q^{51} - 92 q^{52} - 298 q^{53} - 13 q^{54} - q^{55} - 120 q^{56} - 123 q^{57} - 43 q^{58} - 83 q^{59} - 127 q^{60} - 29 q^{61} - 37 q^{62} - 70 q^{63} + 108 q^{64} - 111 q^{65} - 10 q^{66} - 85 q^{67} - 113 q^{68} - 69 q^{69} + 9 q^{70} - 84 q^{71} - 190 q^{72} - 36 q^{73} - 21 q^{74} + 5 q^{75} - 195 q^{77} - 68 q^{78} - 13 q^{79} - 180 q^{80} + 29 q^{81} + 34 q^{82} - 103 q^{83} - 63 q^{84} - 66 q^{85} - 84 q^{86} - 50 q^{87} - 52 q^{88} - 133 q^{89} + 56 q^{90} - 28 q^{91} - 209 q^{92} - 256 q^{93} + 36 q^{94} - 60 q^{95} + 28 q^{96} - 74 q^{97} - 155 q^{98} - 76 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.78396 0.889532 5.75046 −0.545732 −2.47642 −0.794244 −10.4411 −2.20873 1.51930
1.2 −2.77406 2.90320 5.69541 −4.20865 −8.05366 4.27610 −10.2513 5.42858 11.6750
1.3 −2.77198 2.11471 5.68389 0.0519948 −5.86194 −0.963917 −10.2117 1.47200 −0.144129
1.4 −2.76517 −2.46317 5.64618 −1.86024 6.81109 3.61457 −10.0823 3.06721 5.14387
1.5 −2.76491 −0.495888 5.64471 −3.96201 1.37108 −5.06230 −10.0773 −2.75409 10.9546
1.6 −2.74995 −3.31135 5.56222 −1.75412 9.10604 −1.23435 −9.79593 7.96503 4.82374
1.7 −2.70220 −1.52671 5.30191 2.33445 4.12549 1.42500 −8.92242 −0.669142 −6.30817
1.8 −2.61223 −2.19468 4.82373 3.70085 5.73299 2.70663 −7.37623 1.81660 −9.66745
1.9 −2.57463 −1.09307 4.62873 −2.99373 2.81425 2.68118 −6.76801 −1.80520 7.70775
1.10 −2.57189 2.10907 4.61462 −0.250542 −5.42429 3.24250 −6.72451 1.44816 0.644366
1.11 −2.56234 −0.786924 4.56558 −3.18731 2.01637 3.88258 −6.57387 −2.38075 8.16696
1.12 −2.55943 0.0172770 4.55068 2.19921 −0.0442193 −2.28446 −6.52828 −2.99970 −5.62872
1.13 −2.55549 2.48625 4.53054 1.00890 −6.35360 −5.11275 −6.46678 3.18145 −2.57823
1.14 −2.53874 −2.56732 4.44520 3.42578 6.51775 −2.40942 −6.20773 3.59112 −8.69716
1.15 −2.50638 1.04190 4.28194 −2.44145 −2.61140 1.20445 −5.71940 −1.91444 6.11920
1.16 −2.44089 1.39457 3.95795 2.90842 −3.40399 4.10078 −4.77915 −1.05518 −7.09913
1.17 −2.39471 −2.55895 3.73464 −1.63356 6.12794 −2.52047 −4.15397 3.54820 3.91191
1.18 −2.37201 2.91880 3.62642 −3.69975 −6.92342 −3.44265 −3.85789 5.51941 8.77585
1.19 −2.22802 0.833421 2.96408 −0.696882 −1.85688 3.23280 −2.14798 −2.30541 1.55267
1.20 −2.14687 −1.69654 2.60904 0.419551 3.64225 −3.77943 −1.30752 −0.121745 −0.900720
See next 80 embeddings (of 129 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.129
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3307\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3307.2.a.b ✓ 129
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3307.2.a.b ✓ 129 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{129} + 29 T_{2}^{128} + 232 T_{2}^{127} - 1335 T_{2}^{126} - 30558 T_{2}^{125} - 77760 T_{2}^{124} + \cdots - 11502167 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3307))\). Copy content Toggle raw display