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Properties

Label 85.2.e.a
Level $85$
Weight $2$
Character orbit 85.e
Analytic conductor $0.679$
Analytic rank $0$
Dimension $12$
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] = [85,2,Mod(21,85)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(85, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("85.21");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 85 = 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 85.e (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: \(0.678728417181\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 18x^{10} + 83x^{8} + 152x^{6} + 111x^{4} + 22x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{10} q^{2} + (\beta_{6} - \beta_{3}) q^{3} + ( - \beta_{8} - \beta_{5} + \beta_{4} + \cdots - 1) q^{4}+ \cdots + ( - \beta_{11} + \beta_{10} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{10} q^{2} + (\beta_{6} - \beta_{3}) q^{3} + ( - \beta_{8} - \beta_{5} + \beta_{4} + \cdots - 1) q^{4}+ \cdots + (3 \beta_{11} + \beta_{8} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{3} - 12 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{3} - 12 q^{4} - 4 q^{10} - 4 q^{11} - 8 q^{12} - 4 q^{14} + 4 q^{16} + 12 q^{17} + 28 q^{18} - 8 q^{20} - 16 q^{21} + 20 q^{22} + 12 q^{23} + 4 q^{24} - 4 q^{27} + 4 q^{28} - 12 q^{29} - 8 q^{30} - 16 q^{33} - 12 q^{34} + 16 q^{35} + 12 q^{37} + 24 q^{38} - 20 q^{39} - 8 q^{40} - 24 q^{41} + 8 q^{44} + 8 q^{45} - 24 q^{46} - 48 q^{47} - 20 q^{48} + 4 q^{50} + 32 q^{51} - 56 q^{52} + 28 q^{54} + 40 q^{56} + 36 q^{58} + 40 q^{61} + 40 q^{62} + 12 q^{63} + 28 q^{64} + 4 q^{65} - 8 q^{67} - 40 q^{68} + 28 q^{71} + 20 q^{72} - 48 q^{73} + 28 q^{74} + 4 q^{75} - 92 q^{78} - 8 q^{79} + 16 q^{80} + 28 q^{81} + 40 q^{82} - 4 q^{85} - 96 q^{86} - 72 q^{88} + 24 q^{89} - 16 q^{90} - 36 q^{91} - 16 q^{92} - 8 q^{95} - 32 q^{96} + 4 q^{97} + 44 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 18x^{10} + 83x^{8} + 152x^{6} + 111x^{4} + 22x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} - \nu^{10} + 15 \nu^{9} - 19 \nu^{8} + 34 \nu^{7} - 98 \nu^{6} - 16 \nu^{5} - 188 \nu^{4} + \cdots - 13 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} + \nu^{10} + 15 \nu^{9} + 19 \nu^{8} + 34 \nu^{7} + 98 \nu^{6} - 16 \nu^{5} + 188 \nu^{4} + \cdots + 13 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{10} - 18\nu^{8} - 84\nu^{6} - 166\nu^{4} - 133\nu^{2} - 18 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{11} + 3 \nu^{10} - 19 \nu^{9} + 49 \nu^{8} - 98 \nu^{7} + 168 \nu^{6} - 188 \nu^{5} + 186 \nu^{4} + \cdots + 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} + 3 \nu^{10} + 19 \nu^{9} + 49 \nu^{8} + 98 \nu^{7} + 168 \nu^{6} + 188 \nu^{5} + 186 \nu^{4} + \cdots + 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{11} + 37\nu^{9} + 182\nu^{7} + 354\nu^{5} + 258\nu^{3} + 31\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{11} + 5 \nu^{10} + 19 \nu^{9} + 81 \nu^{8} + 98 \nu^{7} + 268 \nu^{6} + 188 \nu^{5} + 258 \nu^{4} + \cdots - 3 ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 9 \nu^{11} - \nu^{10} + 159 \nu^{9} - 15 \nu^{8} + 696 \nu^{7} - 34 \nu^{6} + 1170 \nu^{5} + 16 \nu^{4} + \cdots + 19 ) / 8 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -5\nu^{11} - 87\nu^{9} - 366\nu^{7} - 592\nu^{5} - 369\nu^{3} - 61\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 9 \nu^{11} - \nu^{10} - 159 \nu^{9} - 15 \nu^{8} - 696 \nu^{7} - 34 \nu^{6} - 1170 \nu^{5} + \cdots + 19 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{9} - \beta_{8} + 2\beta_{6} + \beta_{5} + \beta_{4} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{11} - 3\beta_{10} - 2\beta_{9} + 3\beta_{6} - 3\beta_{5} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -9\beta_{11} - 9\beta_{9} + 16\beta_{8} - 28\beta_{6} - 12\beta_{5} - 12\beta_{4} - \beta_{3} + \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 28 \beta_{11} + 40 \beta_{10} + 28 \beta_{9} - 4 \beta_{7} - 41 \beta_{6} + 41 \beta_{5} + \cdots + 61 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 102 \beta_{11} + 102 \beta_{9} - 203 \beta_{8} + 348 \beta_{6} + 145 \beta_{5} + 143 \beta_{4} + \cdots - 21 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 348 \beta_{11} - 493 \beta_{10} - 348 \beta_{9} + 60 \beta_{7} + 504 \beta_{6} - 504 \beta_{5} + \cdots - 718 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1222 \beta_{11} - 1222 \beta_{9} + 2482 \beta_{8} - 4240 \beta_{6} - 1758 \beta_{5} - 1726 \beta_{4} + \cdots + 225 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 4240 \beta_{11} + 5998 \beta_{10} + 4240 \beta_{9} - 756 \beta_{7} - 6122 \beta_{6} + \cdots + 8667 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 14789 \beta_{11} + 14789 \beta_{9} - 30147 \beta_{8} + 51470 \beta_{6} + 21323 \beta_{5} + 20911 \beta_{4} + \cdots - 2669 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 51470 \beta_{11} - 72793 \beta_{10} - 51470 \beta_{9} + 9236 \beta_{7} + 74263 \beta_{6} + \cdots - 105040 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/85\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(71\)
\(\chi(n)\) \(1\) \(-\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1
− 1.19804i
− 0.455023i
1.35757i
− 3.48265i
1.52346i
0.254679i
− 0.254679i
− 1.52346i
3.48265i
− 1.35757i
0.455023i
1.19804i
− 2.24891i 0.140032 − 0.140032i −3.05761 0.707107 − 0.707107i −0.314920 − 0.314920i −1.33807 − 1.33807i 2.37848i 2.96078i −1.59022 − 1.59022i
21.2 − 0.783476i 0.385357 − 0.385357i 1.38617 −0.707107 + 0.707107i −0.301918 − 0.301918i −0.840380 − 0.840380i − 2.65298i 2.70300i 0.554001 + 0.554001i
21.3 − 0.677603i −1.66705 + 1.66705i 1.54085 0.707107 − 0.707107i 1.12960 + 1.12960i 3.02462 + 3.02462i − 2.39929i − 2.55814i −0.479138 − 0.479138i
21.4 1.12708i −1.75550 + 1.75550i 0.729699 −0.707107 + 0.707107i −1.97858 − 1.97858i −1.72715 − 1.72715i 3.07658i − 3.16356i −0.796963 − 0.796963i
21.5 2.07061i 1.78436 − 1.78436i −2.28744 −0.707107 + 0.707107i 3.69471 + 3.69471i −0.260895 − 0.260895i − 0.595174i − 3.36786i −1.46414 − 1.46414i
21.6 2.51230i −0.887192 + 0.887192i −4.31167 0.707107 − 0.707107i −2.22889 − 2.22889i 1.14187 + 1.14187i − 5.80761i 1.42578i 1.77647 + 1.77647i
81.1 − 2.51230i −0.887192 − 0.887192i −4.31167 0.707107 + 0.707107i −2.22889 + 2.22889i 1.14187 − 1.14187i 5.80761i − 1.42578i 1.77647 − 1.77647i
81.2 − 2.07061i 1.78436 + 1.78436i −2.28744 −0.707107 − 0.707107i 3.69471 − 3.69471i −0.260895 + 0.260895i 0.595174i 3.36786i −1.46414 + 1.46414i
81.3 − 1.12708i −1.75550 − 1.75550i 0.729699 −0.707107 − 0.707107i −1.97858 + 1.97858i −1.72715 + 1.72715i − 3.07658i 3.16356i −0.796963 + 0.796963i
81.4 0.677603i −1.66705 − 1.66705i 1.54085 0.707107 + 0.707107i 1.12960 − 1.12960i 3.02462 − 3.02462i 2.39929i 2.55814i −0.479138 + 0.479138i
81.5 0.783476i 0.385357 + 0.385357i 1.38617 −0.707107 − 0.707107i −0.301918 + 0.301918i −0.840380 + 0.840380i 2.65298i − 2.70300i 0.554001 − 0.554001i
81.6 2.24891i 0.140032 + 0.140032i −3.05761 0.707107 + 0.707107i −0.314920 + 0.314920i −1.33807 + 1.33807i − 2.37848i − 2.96078i −1.59022 + 1.59022i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 85.2.e.a ✓ 12
3.b odd 2 1 765.2.k.b 12
4.b odd 2 1 1360.2.bt.d 12
5.b even 2 1 425.2.e.f 12
5.c odd 4 1 425.2.j.b 12
5.c odd 4 1 425.2.j.c 12
17.c even 4 1 inner 85.2.e.a ✓ 12
17.d even 8 1 1445.2.a.n 6
17.d even 8 1 1445.2.a.o 6
17.d even 8 2 1445.2.d.g 12
51.f odd 4 1 765.2.k.b 12
68.f odd 4 1 1360.2.bt.d 12
85.f odd 4 1 425.2.j.b 12
85.i odd 4 1 425.2.j.c 12
85.j even 4 1 425.2.e.f 12
85.m even 8 1 7225.2.a.z 6
85.m even 8 1 7225.2.a.bb 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.2.e.a ✓ 12 1.a even 1 1 trivial
85.2.e.a ✓ 12 17.c even 4 1 inner
425.2.e.f 12 5.b even 2 1
425.2.e.f 12 85.j even 4 1
425.2.j.b 12 5.c odd 4 1
425.2.j.b 12 85.f odd 4 1
425.2.j.c 12 5.c odd 4 1
425.2.j.c 12 85.i odd 4 1
765.2.k.b 12 3.b odd 2 1
765.2.k.b 12 51.f odd 4 1
1360.2.bt.d 12 4.b odd 2 1
1360.2.bt.d 12 68.f odd 4 1
1445.2.a.n 6 17.d even 8 1
1445.2.a.o 6 17.d even 8 1
1445.2.d.g 12 17.d even 8 2
7225.2.a.z 6 85.m even 8 1
7225.2.a.bb 6 85.m even 8 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(85, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 18 T^{10} + \cdots + 49 \) Copy content Toggle raw display
$3$ \( T^{12} + 4 T^{11} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T^{4} + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{12} + 28 T^{9} + \cdots + 196 \) Copy content Toggle raw display
$11$ \( T^{12} + 4 T^{11} + \cdots + 198916 \) Copy content Toggle raw display
$13$ \( (T^{6} - 58 T^{4} + \cdots - 316)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} - 12 T^{11} + \cdots + 24137569 \) Copy content Toggle raw display
$19$ \( T^{12} + 136 T^{10} + \cdots + 2166784 \) Copy content Toggle raw display
$23$ \( T^{12} - 12 T^{11} + \cdots + 196 \) Copy content Toggle raw display
$29$ \( T^{12} + 12 T^{11} + \cdots + 5345344 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 1645275844 \) Copy content Toggle raw display
$37$ \( T^{12} - 12 T^{11} + \cdots + 3655744 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 192876544 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 225120016 \) Copy content Toggle raw display
$47$ \( (T^{6} + 24 T^{5} + \cdots - 18076)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 17453580544 \) Copy content Toggle raw display
$59$ \( T^{12} + 240 T^{10} + \cdots + 802816 \) Copy content Toggle raw display
$61$ \( T^{12} - 40 T^{11} + \cdots + 984064 \) Copy content Toggle raw display
$67$ \( (T^{6} + 4 T^{5} - 46 T^{4} + \cdots - 92)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 29133710596 \) Copy content Toggle raw display
$73$ \( T^{12} + 48 T^{11} + \cdots + 541696 \) Copy content Toggle raw display
$79$ \( T^{12} + 8 T^{11} + \cdots + 15225604 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 125529616 \) Copy content Toggle raw display
$89$ \( (T^{6} - 12 T^{5} + \cdots - 31292)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 227086559296 \) Copy content Toggle raw display
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