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Properties

Label 806.2.bi.a
Level $806$
Weight $2$
Character orbit 806.bi
Analytic conductor $6.436$
Analytic rank $0$
Dimension $152$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [806,2,Mod(133,806)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(806, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([10, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("806.133");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 806 = 2 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 806.bi (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: \(6.43594240292\)
Analytic rank: \(0\)
Dimension: \(152\)
Relative dimension: \(19\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 152 q - 19 q^{2} - 2 q^{3} + 19 q^{4} + 4 q^{6} + 3 q^{7} + 38 q^{8} - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 152 q - 19 q^{2} - 2 q^{3} + 19 q^{4} + 4 q^{6} + 3 q^{7} + 38 q^{8} - 28 q^{9} - 10 q^{11} + q^{12} + 12 q^{13} - 3 q^{14} + 32 q^{15} + 19 q^{16} + 6 q^{17} - 24 q^{18} + 5 q^{19} + 34 q^{21} - 5 q^{22} - 15 q^{23} + 2 q^{24} - 80 q^{25} - 3 q^{26} + 13 q^{27} - 11 q^{28} + 24 q^{29} - 7 q^{30} + 76 q^{32} - 21 q^{33} + 24 q^{34} - 8 q^{35} - 76 q^{36} - 34 q^{37} + 20 q^{38} + 4 q^{39} - 13 q^{41} - 34 q^{42} - 18 q^{43} - 10 q^{44} - 21 q^{45} + 25 q^{46} - 10 q^{47} + q^{48} + 24 q^{49} + 50 q^{50} + 75 q^{51} - 15 q^{52} - 81 q^{53} + 29 q^{54} + 37 q^{55} + 2 q^{56} + 47 q^{57} + 11 q^{58} + 10 q^{59} - 19 q^{60} - 24 q^{61} + 27 q^{62} - 43 q^{63} - 38 q^{64} - 19 q^{65} + 38 q^{66} + 6 q^{67} + 18 q^{68} + 60 q^{69} - 16 q^{70} - 51 q^{71} + 48 q^{72} + 6 q^{73} + 33 q^{74} + 2 q^{75} + 10 q^{76} - q^{77} - 26 q^{78} + 76 q^{79} - 118 q^{81} - 21 q^{82} - 91 q^{83} - 8 q^{84} + 48 q^{85} - 7 q^{86} + 19 q^{87} - 10 q^{88} - 14 q^{89} - 89 q^{90} + 167 q^{91} + 20 q^{92} - 122 q^{93} - 10 q^{94} - 38 q^{95} - q^{96} + 5 q^{97} - 112 q^{98} + 25 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
133.1 0.978148 − 0.207912i −0.946080 − 2.91173i 0.913545 − 0.406737i 0.361195 + 0.625608i −1.53079 − 2.65140i 0.0218575 − 0.207960i 0.809017 − 0.587785i −5.15607 + 3.74611i 0.483373 + 0.536840i
133.2 0.978148 − 0.207912i −0.887998 − 2.73298i 0.913545 − 0.406737i −1.48416 − 2.57064i −1.43681 − 2.48863i 0.425350 − 4.04693i 0.809017 − 0.587785i −4.25357 + 3.09040i −1.98619 − 2.20589i
133.3 0.978148 − 0.207912i −0.860563 − 2.64854i 0.913545 − 0.406737i 1.64162 + 2.84337i −1.39242 − 2.41174i −0.346090 + 3.29283i 0.809017 − 0.587785i −3.84714 + 2.79511i 2.19692 + 2.43992i
133.4 0.978148 − 0.207912i −0.578157 − 1.77938i 0.913545 − 0.406737i 0.574779 + 0.995547i −0.935478 − 1.62029i −0.207382 + 1.97311i 0.809017 − 0.587785i −0.404892 + 0.294171i 0.769205 + 0.854288i
133.5 0.978148 − 0.207912i −0.515169 − 1.58553i 0.913545 − 0.406737i −0.631271 − 1.09339i −0.833561 − 1.44377i 0.139957 − 1.33160i 0.809017 − 0.587785i 0.178553 − 0.129726i −0.844806 − 0.938252i
133.6 0.978148 − 0.207912i −0.488129 − 1.50231i 0.913545 − 0.406737i −1.92455 − 3.33342i −0.789810 − 1.36799i 0.00139299 − 0.0132534i 0.809017 − 0.587785i 0.408393 − 0.296715i −2.57555 − 2.86044i
133.7 0.978148 − 0.207912i −0.292938 − 0.901570i 0.913545 − 0.406737i 1.81404 + 3.14201i −0.473983 − 0.820963i 0.333358 − 3.17169i 0.809017 − 0.587785i 1.70003 − 1.23515i 2.42766 + 2.69619i
133.8 0.978148 − 0.207912i −0.262401 − 0.807587i 0.913545 − 0.406737i −0.0241035 − 0.0417485i −0.424574 − 0.735383i −0.0505658 + 0.481102i 0.809017 − 0.587785i 1.84371 − 1.33953i −0.0322568 − 0.0358248i
133.9 0.978148 − 0.207912i −0.203302 − 0.625700i 0.913545 − 0.406737i 0.608097 + 1.05325i −0.328950 − 0.569758i 0.392108 − 3.73066i 0.809017 − 0.587785i 2.07688 − 1.50894i 0.813793 + 0.903808i
133.10 0.978148 − 0.207912i 0.207922 + 0.639918i 0.913545 − 0.406737i −0.316912 − 0.548908i 0.336425 + 0.582705i −0.217596 + 2.07028i 0.809017 − 0.587785i 2.06079 − 1.49725i −0.424111 − 0.471023i
133.11 0.978148 − 0.207912i 0.262763 + 0.808702i 0.913545 − 0.406737i −1.69745 − 2.94008i 0.425160 + 0.736398i 0.319985 − 3.04445i 0.809017 − 0.587785i 1.84210 − 1.33836i −2.27164 − 2.52291i
133.12 0.978148 − 0.207912i 0.263428 + 0.810747i 0.913545 − 0.406737i −1.64465 − 2.84862i 0.426235 + 0.738261i −0.466938 + 4.44261i 0.809017 − 0.587785i 1.83913 − 1.33621i −2.20097 − 2.44443i
133.13 0.978148 − 0.207912i 0.326065 + 1.00352i 0.913545 − 0.406737i 1.59865 + 2.76894i 0.527584 + 0.913802i −0.205203 + 1.95238i 0.809017 − 0.587785i 1.52631 − 1.10893i 2.13941 + 2.37605i
133.14 0.978148 − 0.207912i 0.389712 + 1.19941i 0.913545 − 0.406737i −0.817925 − 1.41669i 0.630567 + 1.09217i 0.172738 − 1.64349i 0.809017 − 0.587785i 1.14034 − 0.828509i −1.09460 − 1.21567i
133.15 0.978148 − 0.207912i 0.547884 + 1.68621i 0.913545 − 0.406737i 1.52308 + 2.63805i 0.886495 + 1.53545i 0.437263 − 4.16028i 0.809017 − 0.587785i −0.116089 + 0.0843438i 2.03828 + 2.26373i
133.16 0.978148 − 0.207912i 0.674310 + 2.07531i 0.913545 − 0.406737i 0.735480 + 1.27389i 1.09106 + 1.88977i −0.285129 + 2.71282i 0.809017 − 0.587785i −1.42518 + 1.03545i 0.984265 + 1.09314i
133.17 0.978148 − 0.207912i 0.827540 + 2.54691i 0.913545 − 0.406737i −0.0389726 − 0.0675026i 1.33899 + 2.31919i −0.272490 + 2.59257i 0.809017 − 0.587785i −3.37485 + 2.45198i −0.0521556 − 0.0579246i
133.18 0.978148 − 0.207912i 0.868421 + 2.67272i 0.913545 − 0.406737i −1.41461 − 2.45018i 1.40513 + 2.43376i 0.421488 − 4.01019i 0.809017 − 0.587785i −3.96225 + 2.87874i −1.89312 − 2.10252i
133.19 0.978148 − 0.207912i 0.975710 + 3.00293i 0.913545 − 0.406737i 1.13768 + 1.97052i 1.57873 + 2.73444i 0.190344 − 1.81100i 0.809017 − 0.587785i −5.63850 + 4.09661i 1.52251 + 1.69092i
237.1 −0.669131 + 0.743145i −0.973641 + 2.99656i −0.104528 − 0.994522i −2.11246 − 3.65889i −1.57538 − 2.72865i 4.09844 + 1.82474i 0.809017 + 0.587785i −5.60434 − 4.07179i 4.13259 + 0.878410i
See next 80 embeddings (of 152 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 133.19
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
403.bj even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 806.2.bi.a ✓ 152
13.c even 3 1 806.2.bj.a yes 152
31.g even 15 1 806.2.bj.a yes 152
403.bj even 15 1 inner 806.2.bi.a ✓ 152
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
806.2.bi.a ✓ 152 1.a even 1 1 trivial
806.2.bi.a ✓ 152 403.bj even 15 1 inner
806.2.bj.a yes 152 13.c even 3 1
806.2.bj.a yes 152 31.g even 15 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{152} + 2 T_{3}^{151} + 73 T_{3}^{150} + 143 T_{3}^{149} + 2971 T_{3}^{148} + \cdots + 59\!\cdots\!01 \) acting on \(S_{2}^{\mathrm{new}}(806, [\chi])\). Copy content Toggle raw display