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Properties

Label 805.2.p.b
Level $805$
Weight $2$
Character orbit 805.p
Analytic conductor $6.428$
Analytic rank $0$
Dimension $64$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [805,2,Mod(206,805)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(805, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("805.206");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 805 = 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 805.p (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: \(6.42795736271\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 2 q^{2} - 34 q^{4} + 32 q^{5} - 3 q^{7} - 12 q^{8} + 34 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 2 q^{2} - 34 q^{4} + 32 q^{5} - 3 q^{7} - 12 q^{8} + 34 q^{9} - 2 q^{10} - 12 q^{11} + 12 q^{14} - 46 q^{16} - q^{17} - 18 q^{18} + 4 q^{19} - 68 q^{20} - 26 q^{21} + 16 q^{23} + 18 q^{24} - 32 q^{25} + 12 q^{26} - 2 q^{28} + 30 q^{29} + 6 q^{30} - 3 q^{31} + 4 q^{32} - 10 q^{33} + 16 q^{34} - 3 q^{35} - 56 q^{36} - 39 q^{37} - 22 q^{38} + 12 q^{39} - 6 q^{40} + 50 q^{42} + 48 q^{44} - 34 q^{45} - 26 q^{46} + 36 q^{47} - 19 q^{49} - 4 q^{50} + 30 q^{51} - 33 q^{53} + 54 q^{54} - 62 q^{56} - 4 q^{58} + 27 q^{59} + 10 q^{61} + 45 q^{63} + 124 q^{64} + 2 q^{66} + 28 q^{68} - 62 q^{69} + 6 q^{70} - 22 q^{71} - 26 q^{72} + 18 q^{73} + 24 q^{74} + 72 q^{76} - 40 q^{77} + 80 q^{78} + 46 q^{80} - 24 q^{81} - 102 q^{82} + 22 q^{83} - 76 q^{84} - 2 q^{85} - 138 q^{86} - 42 q^{87} - 60 q^{88} - 18 q^{89} - 36 q^{90} - 42 q^{91} + 42 q^{92} + 10 q^{93} - 84 q^{94} - 4 q^{95} + 60 q^{96} - 36 q^{97} + 62 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
206.1 −1.37415 − 2.38010i −0.708535 − 0.409073i −2.77657 + 4.80917i 0.500000 + 0.866025i 2.24851i −1.75590 + 1.97910i 9.76512 −1.16532 − 2.01839i 1.37415 − 2.38010i
206.2 −1.32261 − 2.29083i 2.39292 + 1.38155i −2.49861 + 4.32773i 0.500000 + 0.866025i − 7.30903i 0.971243 + 2.46103i 7.92835 2.31736 + 4.01379i 1.32261 − 2.29083i
206.3 −1.30476 − 2.25991i −1.67396 − 0.966460i −2.40479 + 4.16521i 0.500000 + 0.866025i 5.04398i −0.0791150 − 2.64457i 7.33162 0.368089 + 0.637550i 1.30476 − 2.25991i
206.4 −1.06469 − 1.84410i −0.246709 − 0.142438i −1.26714 + 2.19475i 0.500000 + 0.866025i 0.606609i 2.37090 + 1.17423i 1.13767 −1.45942 − 2.52779i 1.06469 − 1.84410i
206.5 −1.00378 − 1.73861i 2.46971 + 1.42589i −1.01517 + 1.75832i 0.500000 + 0.866025i − 5.72513i −2.64463 − 0.0770837i 0.0608924 2.56631 + 4.44498i 1.00378 − 1.73861i
206.6 −0.992940 − 1.71982i −1.81894 − 1.05016i −0.971861 + 1.68331i 0.500000 + 0.866025i 4.17100i −0.654076 − 2.56363i −0.111761 0.705687 + 1.22229i 0.992940 − 1.71982i
206.7 −0.958524 − 1.66021i 1.30460 + 0.753212i −0.837536 + 1.45066i 0.500000 + 0.866025i − 2.88789i 0.697059 − 2.55228i −0.622902 −0.365342 − 0.632791i 0.958524 − 1.66021i
206.8 −0.827267 − 1.43287i 1.10377 + 0.637262i −0.368741 + 0.638678i 0.500000 + 0.866025i − 2.10874i −2.51706 + 0.815116i −2.08888 −0.687794 − 1.19129i 0.827267 − 1.43287i
206.9 −0.789539 − 1.36752i −0.576533 − 0.332861i −0.246743 + 0.427371i 0.500000 + 0.866025i 1.05123i 0.114652 + 2.64327i −2.37890 −1.27841 − 2.21427i 0.789539 − 1.36752i
206.10 −0.495730 − 0.858629i −2.78618 − 1.60860i 0.508504 − 0.880756i 0.500000 + 0.866025i 3.18973i 1.47466 + 2.19667i −2.99124 3.67520 + 6.36563i 0.495730 − 0.858629i
206.11 −0.477169 − 0.826482i −0.426860 − 0.246448i 0.544619 − 0.943307i 0.500000 + 0.866025i 0.470389i −2.20137 − 1.46764i −2.94818 −1.37853 − 2.38768i 0.477169 − 0.826482i
206.12 −0.471261 − 0.816248i −2.12288 − 1.22564i 0.555826 − 0.962719i 0.500000 + 0.866025i 2.31039i 2.30347 − 1.30154i −2.93280 1.50441 + 2.60571i 0.471261 − 0.816248i
206.13 −0.322496 − 0.558580i 0.702468 + 0.405570i 0.791992 − 1.37177i 0.500000 + 0.866025i − 0.523179i 2.11060 − 1.59542i −2.31164 −1.17103 − 2.02828i 0.322496 − 0.558580i
206.14 −0.321106 − 0.556172i 2.14301 + 1.23727i 0.793782 − 1.37487i 0.500000 + 0.866025i − 1.58918i 1.70587 + 2.02238i −2.30398 1.56167 + 2.70489i 0.321106 − 0.556172i
206.15 −0.0204929 − 0.0354947i 1.19870 + 0.692069i 0.999160 − 1.73060i 0.500000 + 0.866025i − 0.0567299i −1.05882 − 2.42465i −0.163874 −0.542081 − 0.938912i 0.0204929 − 0.0354947i
206.16 0.0296987 + 0.0514396i −1.50278 − 0.867630i 0.998236 − 1.72900i 0.500000 + 0.866025i − 0.103070i −0.0361863 + 2.64550i 0.237380 0.00556458 + 0.00963813i −0.0296987 + 0.0514396i
206.17 0.0605633 + 0.104899i 2.74857 + 1.58689i 0.992664 − 1.71934i 0.500000 + 0.866025i 0.384428i −2.51580 + 0.818987i 0.482729 3.53641 + 6.12525i −0.0605633 + 0.104899i
206.18 0.157097 + 0.272100i −0.498094 − 0.287574i 0.950641 − 1.64656i 0.500000 + 0.866025i − 0.180708i −0.939845 + 2.47319i 1.22576 −1.33460 − 2.31160i −0.157097 + 0.272100i
206.19 0.353959 + 0.613075i −1.84959 − 1.06786i 0.749426 − 1.29804i 0.500000 + 0.866025i − 1.51192i 1.78038 − 1.95710i 2.47690 0.780653 + 1.35213i −0.353959 + 0.613075i
206.20 0.361068 + 0.625389i 2.25889 + 1.30417i 0.739259 − 1.28043i 0.500000 + 0.866025i 1.88358i 1.90529 − 1.83572i 2.51197 1.90173 + 3.29389i −0.361068 + 0.625389i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 206.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
161.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 805.2.p.b yes 64
7.d odd 6 1 805.2.p.a ✓ 64
23.b odd 2 1 805.2.p.a ✓ 64
161.g even 6 1 inner 805.2.p.b yes 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.p.a ✓ 64 7.d odd 6 1
805.2.p.a ✓ 64 23.b odd 2 1
805.2.p.b yes 64 1.a even 1 1 trivial
805.2.p.b yes 64 161.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{64} + 12 T_{11}^{63} - 115 T_{11}^{62} - 1956 T_{11}^{61} + 7038 T_{11}^{60} + \cdots + 87\!\cdots\!96 \) acting on \(S_{2}^{\mathrm{new}}(805, [\chi])\). Copy content Toggle raw display