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Properties

Label 845.2.k.c
Level $845$
Weight $2$
Character orbit 845.k
Analytic conductor $6.747$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [845,2,Mod(268,845)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(845, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("845.268");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.k (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: \(6.74735897080\)
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: \( x^{12} + 20x^{10} + 124x^{8} + 300x^{6} + 292x^{4} + 88x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_{10} q^{2} + \beta_{6} q^{3} - \beta_{11} q^{4} + ( - \beta_{9} + \beta_1) q^{5} + (\beta_{10} - \beta_{9} - \beta_{4}) q^{6} + ( - \beta_{9} + \beta_{8} + \cdots - \beta_1) q^{7}+ \cdots + (\beta_{6} + \beta_{3} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{10} q^{2} + \beta_{6} q^{3} - \beta_{11} q^{4} + ( - \beta_{9} + \beta_1) q^{5} + (\beta_{10} - \beta_{9} - \beta_{4}) q^{6} + ( - \beta_{9} + \beta_{8} + \cdots - \beta_1) q^{7}+ \cdots + (2 \beta_{10} - 7 \beta_{9} + \cdots + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{3} + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{3} + 4 q^{4} - 12 q^{10} + 8 q^{12} + 4 q^{16} + 8 q^{17} + 32 q^{22} + 8 q^{23} - 20 q^{25} - 20 q^{27} - 4 q^{30} + 4 q^{35} + 28 q^{38} - 32 q^{40} - 28 q^{42} - 8 q^{43} - 36 q^{48} - 20 q^{49} + 44 q^{53} - 12 q^{55} - 64 q^{61} + 64 q^{62} - 36 q^{64} + 72 q^{66} + 48 q^{68} + 16 q^{69} - 52 q^{75} + 60 q^{77} + 12 q^{81} - 12 q^{82} + 92 q^{87} + 12 q^{88} - 76 q^{90} - 108 q^{92} - 64 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 20x^{10} + 124x^{8} + 300x^{6} + 292x^{4} + 88x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3\nu^{11} + 39\nu^{9} - 5\nu^{7} - 924\nu^{5} - 1548\nu^{3} - 262\nu ) / 52 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{11} - 65\nu^{9} - 223\nu^{7} + 166\nu^{5} + 816\nu^{3} + 150\nu ) / 52 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} - 28 \nu^{10} - 520 \nu^{8} - 240 \nu^{7} - 2744 \nu^{6} - 1530 \nu^{5} - 4740 \nu^{4} + \cdots - 328 ) / 104 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 18\nu^{11} + 338\nu^{9} + 1829\nu^{7} + 3348\nu^{5} + 2074\nu^{3} + 378\nu ) / 52 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 18\nu^{11} + 338\nu^{9} + 1829\nu^{7} + 3348\nu^{5} + 2074\nu^{3} + 326\nu ) / 52 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} + 28 \nu^{10} + 520 \nu^{8} - 240 \nu^{7} + 2744 \nu^{6} - 1530 \nu^{5} + 4740 \nu^{4} + \cdots + 328 ) / 104 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 21\nu^{10} + 377\nu^{8} + 1824\nu^{6} + 2424\nu^{4} + 526\nu^{2} + 64 ) / 52 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 9 \nu^{11} + 36 \nu^{10} + 156 \nu^{9} + 650 \nu^{8} + 674 \nu^{7} + 3190 \nu^{6} + 426 \nu^{5} + \cdots - 76 ) / 104 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 9 \nu^{11} + 36 \nu^{10} - 156 \nu^{9} + 650 \nu^{8} - 674 \nu^{7} + 3190 \nu^{6} - 426 \nu^{5} + \cdots - 76 ) / 104 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 22\nu^{10} + 403\nu^{8} + 2052\nu^{6} + 3182\nu^{4} + 1258\nu^{2} + 124 ) / 52 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -22\nu^{10} - 403\nu^{8} - 2052\nu^{6} - 3182\nu^{4} - 1232\nu^{2} - 46 ) / 26 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{5} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + 2\beta_{10} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{8} + 7\beta_{5} - 7\beta_{4} - 3\beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -12\beta_{11} - 20\beta_{10} - 4\beta_{9} - 4\beta_{8} + 4\beta_{7} - \beta_{6} + \beta_{3} + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -17\beta_{9} + 17\beta_{8} - 5\beta_{6} - 66\beta_{5} + 64\beta_{4} - 5\beta_{3} + 40\beta_{2} + 16\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 138\beta_{11} + 200\beta_{10} + 62\beta_{9} + 62\beta_{8} - 56\beta_{7} + 22\beta_{6} - 22\beta_{3} - 212 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 222\beta_{9} - 222\beta_{8} + 84\beta_{6} + 680\beta_{5} - 644\beta_{4} + 84\beta_{3} - 462\beta_{2} - 194\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -1550\beta_{11} - 2080\beta_{10} - 768\beta_{9} - 768\beta_{8} + 656\beta_{7} - 306\beta_{6} + 306\beta_{3} + 2218 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2624 \beta_{9} + 2624 \beta_{8} - 1074 \beta_{6} - 7258 \beta_{5} + 6786 \beta_{4} + \cdots + 2206 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 17200 \beta_{11} + 22228 \beta_{10} + 8864 \beta_{9} + 8864 \beta_{8} - 7372 \beta_{7} + 3698 \beta_{6} + \cdots - 23872 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 29762 \beta_{9} - 29762 \beta_{8} + 12562 \beta_{6} + 78684 \beta_{5} - 73104 \beta_{4} + \cdots - 24572 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\) \(171\) \(677\)
\(\chi(n)\) \(-\beta_{5}\) \(\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
268.1
3.31384i
2.23432i
1.35014i
0.649863i
− 0.234321i
− 1.31384i
− 3.31384i
− 2.23432i
− 1.35014i
− 0.649863i
0.234321i
1.31384i
−2.31384 0.229681 + 0.229681i 3.35386 1.78239 + 1.35021i −0.531446 − 0.531446i 1.68313i −3.13261 − 2.89449i −4.12417 − 3.12417i
268.2 −1.23432 1.91004 + 1.91004i −0.476452 −1.12329 − 1.93345i −2.35761 − 2.35761i − 2.67073i 3.05674 4.29654i 1.38650 + 2.38650i
268.3 −0.350137 −1.13973 − 1.13973i −1.87740 0.749198 − 2.10682i 0.399061 + 0.399061i 4.00428i 1.35762 − 0.402049i −0.262322 + 0.737678i
268.4 0.350137 −1.13973 − 1.13973i −1.87740 −0.749198 + 2.10682i −0.399061 − 0.399061i − 4.00428i −1.35762 − 0.402049i −0.262322 + 0.737678i
268.5 1.23432 1.91004 + 1.91004i −0.476452 1.12329 + 1.93345i 2.35761 + 2.35761i 2.67073i −3.05674 4.29654i 1.38650 + 2.38650i
268.6 2.31384 0.229681 + 0.229681i 3.35386 −1.78239 − 1.35021i 0.531446 + 0.531446i − 1.68313i 3.13261 − 2.89449i −4.12417 − 3.12417i
577.1 −2.31384 0.229681 − 0.229681i 3.35386 1.78239 − 1.35021i −0.531446 + 0.531446i − 1.68313i −3.13261 2.89449i −4.12417 + 3.12417i
577.2 −1.23432 1.91004 − 1.91004i −0.476452 −1.12329 + 1.93345i −2.35761 + 2.35761i 2.67073i 3.05674 − 4.29654i 1.38650 − 2.38650i
577.3 −0.350137 −1.13973 + 1.13973i −1.87740 0.749198 + 2.10682i 0.399061 − 0.399061i − 4.00428i 1.35762 0.402049i −0.262322 − 0.737678i
577.4 0.350137 −1.13973 + 1.13973i −1.87740 −0.749198 − 2.10682i −0.399061 + 0.399061i 4.00428i −1.35762 0.402049i −0.262322 − 0.737678i
577.5 1.23432 1.91004 − 1.91004i −0.476452 1.12329 − 1.93345i 2.35761 − 2.35761i − 2.67073i −3.05674 − 4.29654i 1.38650 − 2.38650i
577.6 2.31384 0.229681 − 0.229681i 3.35386 −1.78239 + 1.35021i 0.531446 − 0.531446i 1.68313i 3.13261 2.89449i −4.12417 + 3.12417i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 268.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
65.f even 4 1 inner
65.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.k.c yes 12
5.c odd 4 1 845.2.f.c ✓ 12
13.b even 2 1 inner 845.2.k.c yes 12
13.c even 3 2 845.2.o.h 24
13.d odd 4 2 845.2.f.c ✓ 12
13.e even 6 2 845.2.o.h 24
13.f odd 12 4 845.2.t.h 24
65.f even 4 1 inner 845.2.k.c yes 12
65.h odd 4 1 845.2.f.c ✓ 12
65.k even 4 1 inner 845.2.k.c yes 12
65.o even 12 2 845.2.o.h 24
65.q odd 12 2 845.2.t.h 24
65.r odd 12 2 845.2.t.h 24
65.t even 12 2 845.2.o.h 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
845.2.f.c ✓ 12 5.c odd 4 1
845.2.f.c ✓ 12 13.d odd 4 2
845.2.f.c ✓ 12 65.h odd 4 1
845.2.k.c yes 12 1.a even 1 1 trivial
845.2.k.c yes 12 13.b even 2 1 inner
845.2.k.c yes 12 65.f even 4 1 inner
845.2.k.c yes 12 65.k even 4 1 inner
845.2.o.h 24 13.c even 3 2
845.2.o.h 24 13.e even 6 2
845.2.o.h 24 65.o even 12 2
845.2.o.h 24 65.t even 12 2
845.2.t.h 24 13.f odd 12 4
845.2.t.h 24 65.q odd 12 2
845.2.t.h 24 65.r odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 7T_{2}^{4} + 9T_{2}^{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 7 T^{4} + 9 T^{2} - 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{6} - 2 T^{5} + 2 T^{4} + \cdots + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + 10 T^{10} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( (T^{6} + 26 T^{4} + \cdots + 324)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 660 T^{8} + \cdots + 9253764 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( (T^{6} - 4 T^{5} + \cdots + 2592)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 760 T^{8} + \cdots + 26244 \) Copy content Toggle raw display
$23$ \( (T^{6} - 4 T^{5} + \cdots + 46818)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 124 T^{4} + \cdots + 54756)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 749554884 \) Copy content Toggle raw display
$37$ \( (T^{6} + 158 T^{4} + \cdots + 8100)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 748 T^{8} + \cdots + 1827904 \) Copy content Toggle raw display
$43$ \( (T^{6} + 4 T^{5} + \cdots + 1922)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 98 T^{4} + \cdots + 4356)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 22 T^{5} + \cdots + 648)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 13584 T^{8} + \cdots + 26244 \) Copy content Toggle raw display
$61$ \( (T^{3} + 16 T^{2} + \cdots - 530)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} - 90 T^{4} + \cdots - 2916)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 749554884 \) Copy content Toggle raw display
$73$ \( (T^{6} - 98 T^{4} + \cdots - 324)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 92 T^{4} + \cdots + 10816)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 238 T^{4} + \cdots + 131044)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 32400000000 \) Copy content Toggle raw display
$97$ \( (T^{6} - 360 T^{4} + \cdots - 1411344)^{2} \) Copy content Toggle raw display
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