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Properties

Label 806.2.e.d
Level $806$
Weight $2$
Character orbit 806.e
Analytic conductor $6.436$
Analytic rank $0$
Dimension $18$
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(521,806)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(806, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("806.521");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 806 = 2 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 806.e (of order \(3\), 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: \(6.43594240292\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - x^{17} + 17 x^{16} - 4 x^{15} + 184 x^{14} - 22 x^{13} + 1092 x^{12} + 160 x^{11} + 4522 x^{10} + \cdots + 361 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} + (\beta_{12} - \beta_{8}) q^{5} - \beta_1 q^{6} + (\beta_{12} - \beta_{10} + \cdots + \beta_{2}) q^{7}+ \cdots + (\beta_{9} + \beta_{5} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_1 q^{3} + q^{4} + (\beta_{12} - \beta_{8}) q^{5} - \beta_1 q^{6} + (\beta_{12} - \beta_{10} + \cdots + \beta_{2}) q^{7}+ \cdots + ( - \beta_{17} + 2 \beta_{16} + \cdots - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 18 q^{2} + q^{3} + 18 q^{4} - q^{6} - 2 q^{7} - 18 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 18 q^{2} + q^{3} + 18 q^{4} - q^{6} - 2 q^{7} - 18 q^{8} - 6 q^{9} + 8 q^{11} + q^{12} - 9 q^{13} + 2 q^{14} + 6 q^{15} + 18 q^{16} + 6 q^{18} - 11 q^{19} - 5 q^{21} - 8 q^{22} + 32 q^{23} - q^{24} - 13 q^{25} + 9 q^{26} - 26 q^{27} - 2 q^{28} - 18 q^{29} - 6 q^{30} + 10 q^{31} - 18 q^{32} + 4 q^{33} - 24 q^{35} - 6 q^{36} + 11 q^{38} - 2 q^{39} + 2 q^{41} + 5 q^{42} + q^{43} + 8 q^{44} + 18 q^{45} - 32 q^{46} + 62 q^{47} + q^{48} - 25 q^{49} + 13 q^{50} - 2 q^{51} - 9 q^{52} - 22 q^{53} + 26 q^{54} - 38 q^{55} + 2 q^{56} - 3 q^{57} + 18 q^{58} - 11 q^{59} + 6 q^{60} + 44 q^{61} - 10 q^{62} - 72 q^{63} + 18 q^{64} - 4 q^{66} + 6 q^{67} - 2 q^{69} + 24 q^{70} - 26 q^{71} + 6 q^{72} + 5 q^{73} + 16 q^{75} - 11 q^{76} - 44 q^{77} + 2 q^{78} + 21 q^{79} + 7 q^{81} - 2 q^{82} - 17 q^{83} - 5 q^{84} + 38 q^{85} - q^{86} - 26 q^{87} - 8 q^{88} - 56 q^{89} - 18 q^{90} + 4 q^{91} + 32 q^{92} - 32 q^{93} - 62 q^{94} + 22 q^{95} - q^{96} - 26 q^{97} + 25 q^{98} + 29 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - x^{17} + 17 x^{16} - 4 x^{15} + 184 x^{14} - 22 x^{13} + 1092 x^{12} + 160 x^{11} + 4522 x^{10} + \cdots + 361 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10\!\cdots\!94 \nu^{17} + \cdots - 10\!\cdots\!76 ) / 27\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 24\!\cdots\!92 \nu^{17} + \cdots + 84\!\cdots\!08 ) / 27\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 54\!\cdots\!64 \nu^{17} + \cdots + 82\!\cdots\!99 ) / 81\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 44\!\cdots\!32 \nu^{17} + \cdots + 27\!\cdots\!25 ) / 51\!\cdots\!29 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 87\!\cdots\!02 \nu^{17} + \cdots + 51\!\cdots\!29 ) / 81\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 11\!\cdots\!84 \nu^{17} + \cdots + 65\!\cdots\!05 ) / 81\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 15\!\cdots\!22 \nu^{17} + \cdots + 66\!\cdots\!43 ) / 81\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 17\!\cdots\!28 \nu^{17} + \cdots + 19\!\cdots\!16 ) / 51\!\cdots\!29 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 91\!\cdots\!06 \nu^{17} + \cdots - 21\!\cdots\!35 ) / 15\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 94\!\cdots\!08 \nu^{17} + \cdots - 17\!\cdots\!63 ) / 15\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 11\!\cdots\!16 \nu^{17} + \cdots + 19\!\cdots\!59 ) / 15\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 70\!\cdots\!85 \nu^{17} + \cdots - 27\!\cdots\!80 ) / 81\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 13\!\cdots\!08 \nu^{17} + \cdots - 32\!\cdots\!33 ) / 15\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 83\!\cdots\!91 \nu^{17} + \cdots + 33\!\cdots\!21 ) / 81\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 51\!\cdots\!00 \nu^{17} + \cdots - 10\!\cdots\!10 ) / 15\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 64\!\cdots\!44 \nu^{17} + \cdots + 12\!\cdots\!97 ) / 15\!\cdots\!87 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + 4\beta_{5} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} + \beta_{7} - 6\beta_{3} - \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} - \beta_{12} - 2\beta_{10} - 8\beta_{9} - 26\beta_{5} - 8\beta_{2} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{17} + 2 \beta_{16} - 2 \beta_{15} + 2 \beta_{14} - 2 \beta_{13} - 12 \beta_{12} - 10 \beta_{11} + \cdots + 27 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -3\beta_{15} - 5\beta_{13} + 12\beta_{8} - 4\beta_{7} - 31\beta_{6} + 13\beta_{4} + 34\beta_{3} + 63\beta_{2} + 199 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 31 \beta_{17} - 36 \beta_{16} - 30 \beta_{14} + 109 \beta_{12} + 90 \beta_{11} + 60 \beta_{10} + \cdots + 358 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 60 \beta_{17} - 96 \beta_{16} + 60 \beta_{15} - 139 \beta_{14} + 96 \beta_{13} + 128 \beta_{12} + \cdots - 1662 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 362 \beta_{15} + 458 \beta_{13} - 941 \beta_{8} + 803 \beta_{7} + 818 \beta_{6} - 351 \beta_{4} + \cdots - 3083 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 818 \beta_{17} + 1276 \beta_{16} + 1408 \beta_{14} - 1349 \beta_{12} - 1026 \beta_{11} + \cdots - 4988 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3840 \beta_{17} + 5116 \beta_{16} - 3840 \beta_{15} + 3781 \beta_{14} - 5116 \beta_{13} - 8178 \beta_{12} + \cdots + 31516 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 9585 \beta_{15} - 14701 \beta_{13} + 14113 \beta_{8} - 12099 \beta_{7} - 39083 \beta_{6} + \cdots + 133250 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 39083 \beta_{17} - 53784 \beta_{16} - 39214 \beta_{14} + 72659 \beta_{12} + 66467 \beta_{11} + \cdots + 253770 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 104282 \beta_{17} - 158066 \beta_{16} + 104282 \beta_{15} - 137727 \beta_{14} + 158066 \beta_{13} + \cdots - 1241756 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 390059 \beta_{15} + 548125 \beta_{13} - 660841 \beta_{8} + 618828 \beta_{7} + 1089796 \beta_{6} + \cdots - 3207230 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 1089796 \beta_{17} + 1637921 \beta_{16} + 1352171 \beta_{14} - 1496926 \beta_{12} - 1409409 \beta_{11} + \cdots - 5840830 \beta_1 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 3855774 \beta_{17} + 5493695 \beta_{16} - 3855774 \beta_{15} + 4000529 \beta_{14} - 5493695 \beta_{13} + \cdots + 32074927 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/806\mathbb{Z}\right)^\times\).

\(n\) \(249\) \(313\)
\(\chi(n)\) \(1\) \(-1 + \beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
521.1
−1.32653 − 2.29762i
−0.897902 − 1.55521i
−0.884842 − 1.53259i
−0.443633 − 0.768396i
0.210052 + 0.363820i
0.242485 + 0.419996i
0.818842 + 1.41828i
1.21404 + 2.10278i
1.56749 + 2.71497i
−1.32653 + 2.29762i
−0.897902 + 1.55521i
−0.884842 + 1.53259i
−0.443633 + 0.768396i
0.210052 − 0.363820i
0.242485 − 0.419996i
0.818842 − 1.41828i
1.21404 − 2.10278i
1.56749 − 2.71497i
−1.00000 −1.32653 − 2.29762i 1.00000 1.45464 − 2.51951i 1.32653 + 2.29762i −1.12718 − 1.95233i −1.00000 −2.01936 + 3.49764i −1.45464 + 2.51951i
521.2 −1.00000 −0.897902 − 1.55521i 1.00000 −1.54316 + 2.67283i 0.897902 + 1.55521i 1.93348 + 3.34888i −1.00000 −0.112457 + 0.194781i 1.54316 − 2.67283i
521.3 −1.00000 −0.884842 − 1.53259i 1.00000 0.333587 − 0.577790i 0.884842 + 1.53259i 1.45763 + 2.52469i −1.00000 −0.0658892 + 0.114123i −0.333587 + 0.577790i
521.4 −1.00000 −0.443633 − 0.768396i 1.00000 −0.638056 + 1.10515i 0.443633 + 0.768396i −2.38820 − 4.13648i −1.00000 1.10638 − 1.91630i 0.638056 − 1.10515i
521.5 −1.00000 0.210052 + 0.363820i 1.00000 −1.44309 + 2.49951i −0.210052 − 0.363820i −0.0987916 − 0.171112i −1.00000 1.41176 − 2.44523i 1.44309 − 2.49951i
521.6 −1.00000 0.242485 + 0.419996i 1.00000 −0.560575 + 0.970944i −0.242485 − 0.419996i −1.84881 − 3.20224i −1.00000 1.38240 − 2.39439i 0.560575 − 0.970944i
521.7 −1.00000 0.818842 + 1.41828i 1.00000 2.03480 − 3.52438i −0.818842 − 1.41828i −1.47299 − 2.55129i −1.00000 0.158996 − 0.275389i −2.03480 + 3.52438i
521.8 −1.00000 1.21404 + 2.10278i 1.00000 1.38127 − 2.39243i −1.21404 − 2.10278i 1.67693 + 2.90454i −1.00000 −1.44779 + 2.50764i −1.38127 + 2.39243i
521.9 −1.00000 1.56749 + 2.71497i 1.00000 −1.01941 + 1.76567i −1.56749 − 2.71497i 0.867926 + 1.50329i −1.00000 −3.41404 + 5.91329i 1.01941 − 1.76567i
625.1 −1.00000 −1.32653 + 2.29762i 1.00000 1.45464 + 2.51951i 1.32653 − 2.29762i −1.12718 + 1.95233i −1.00000 −2.01936 − 3.49764i −1.45464 − 2.51951i
625.2 −1.00000 −0.897902 + 1.55521i 1.00000 −1.54316 − 2.67283i 0.897902 − 1.55521i 1.93348 − 3.34888i −1.00000 −0.112457 − 0.194781i 1.54316 + 2.67283i
625.3 −1.00000 −0.884842 + 1.53259i 1.00000 0.333587 + 0.577790i 0.884842 − 1.53259i 1.45763 − 2.52469i −1.00000 −0.0658892 − 0.114123i −0.333587 − 0.577790i
625.4 −1.00000 −0.443633 + 0.768396i 1.00000 −0.638056 − 1.10515i 0.443633 − 0.768396i −2.38820 + 4.13648i −1.00000 1.10638 + 1.91630i 0.638056 + 1.10515i
625.5 −1.00000 0.210052 − 0.363820i 1.00000 −1.44309 − 2.49951i −0.210052 + 0.363820i −0.0987916 + 0.171112i −1.00000 1.41176 + 2.44523i 1.44309 + 2.49951i
625.6 −1.00000 0.242485 − 0.419996i 1.00000 −0.560575 − 0.970944i −0.242485 + 0.419996i −1.84881 + 3.20224i −1.00000 1.38240 + 2.39439i 0.560575 + 0.970944i
625.7 −1.00000 0.818842 − 1.41828i 1.00000 2.03480 + 3.52438i −0.818842 + 1.41828i −1.47299 + 2.55129i −1.00000 0.158996 + 0.275389i −2.03480 − 3.52438i
625.8 −1.00000 1.21404 − 2.10278i 1.00000 1.38127 + 2.39243i −1.21404 + 2.10278i 1.67693 − 2.90454i −1.00000 −1.44779 − 2.50764i −1.38127 − 2.39243i
625.9 −1.00000 1.56749 − 2.71497i 1.00000 −1.01941 − 1.76567i −1.56749 + 2.71497i 0.867926 − 1.50329i −1.00000 −3.41404 − 5.91329i 1.01941 + 1.76567i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 521.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 806.2.e.d ✓ 18
31.c even 3 1 inner 806.2.e.d ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
806.2.e.d ✓ 18 1.a even 1 1 trivial
806.2.e.d ✓ 18 31.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{18} - T_{3}^{17} + 17 T_{3}^{16} - 4 T_{3}^{15} + 184 T_{3}^{14} - 22 T_{3}^{13} + 1092 T_{3}^{12} + \cdots + 361 \) acting on \(S_{2}^{\mathrm{new}}(806, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{18} \) Copy content Toggle raw display
$3$ \( T^{18} - T^{17} + \cdots + 361 \) Copy content Toggle raw display
$5$ \( T^{18} + 29 T^{16} + \cdots + 321489 \) Copy content Toggle raw display
$7$ \( T^{18} + 2 T^{17} + \cdots + 2313441 \) Copy content Toggle raw display
$11$ \( T^{18} - 8 T^{17} + \cdots + 419904 \) Copy content Toggle raw display
$13$ \( (T^{2} + T + 1)^{9} \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 3602520441 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 330054846016 \) Copy content Toggle raw display
$23$ \( (T^{9} - 16 T^{8} + \cdots - 198936)^{2} \) Copy content Toggle raw display
$29$ \( (T^{9} + 9 T^{8} + \cdots - 33048)^{2} \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 26439622160671 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 69932215809 \) Copy content Toggle raw display
$41$ \( T^{18} - 2 T^{17} + \cdots + 419904 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 26\!\cdots\!01 \) Copy content Toggle raw display
$47$ \( (T^{9} - 31 T^{8} + \cdots - 67869)^{2} \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 29\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 166836303936 \) Copy content Toggle raw display
$61$ \( (T^{9} - 22 T^{8} + \cdots - 3916312)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 487410531045376 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 717792027472881 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 984891684744256 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 94111514176 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 42670338624 \) Copy content Toggle raw display
$89$ \( (T^{9} + 28 T^{8} + \cdots + 150738264)^{2} \) Copy content Toggle raw display
$97$ \( (T^{9} + 13 T^{8} + \cdots - 22664)^{2} \) Copy content Toggle raw display
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