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Properties

Label 169.2.e.b
Level $169$
Weight $2$
Character orbit 169.e
Analytic conductor $1.349$
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] = [169,2,Mod(23,169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 169.e (of order \(6\), degree \(2\), not 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: \(1.34947179416\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.17213603549184.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{10} + \beta_{8} - \beta_1) q^{2} + (\beta_{9} + \beta_{4} - \beta_{3} + 1) q^{3} + ( - \beta_{9} + \beta_{4}) q^{4} + ( - \beta_{11} - 2 \beta_{10} - \beta_{8} + 2 \beta_{6} - \beta_{2}) q^{5} - \beta_1 q^{6} + (\beta_{6} - \beta_{2} + \beta_1) q^{7} + ( - 2 \beta_{11} - \beta_{8} - 2 \beta_{2}) q^{8} + (\beta_{9} - 2 \beta_{7} + 2 \beta_{4} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{10} + \beta_{8} - \beta_1) q^{2} + (\beta_{9} + \beta_{4} - \beta_{3} + 1) q^{3} + ( - \beta_{9} + \beta_{4}) q^{4} + ( - \beta_{11} - 2 \beta_{10} - \beta_{8} + 2 \beta_{6} - \beta_{2}) q^{5} - \beta_1 q^{6} + (\beta_{6} - \beta_{2} + \beta_1) q^{7} + ( - 2 \beta_{11} - \beta_{8} - 2 \beta_{2}) q^{8} + (\beta_{9} - 2 \beta_{7} + 2 \beta_{4} + 2) q^{9} + (2 \beta_{7} - \beta_{5} - \beta_{4}) q^{10} + (\beta_{11} - 2 \beta_{10} + \beta_{8} - \beta_1) q^{11} + ( - \beta_{5} + \beta_{3}) q^{12} + (2 \beta_{5} + 2 \beta_{3} - 3) q^{14} + (\beta_{11} + \beta_{10}) q^{15} + (\beta_{9} - \beta_{7} - \beta_{5} - \beta_{3} + 1) q^{16} + (\beta_{9} + \beta_{7} - 2 \beta_{4} - 1) q^{17} + ( - \beta_{11} + 3 \beta_{10} + \beta_{8} - 3 \beta_{6} - \beta_{2}) q^{18} + ( - \beta_{6} - 2 \beta_{2} + \beta_1) q^{19} + ( - \beta_{6} + 3 \beta_{2}) q^{20} + (2 \beta_{11} + \beta_{10} + 3 \beta_{8} - \beta_{6} + 2 \beta_{2}) q^{21} + (2 \beta_{7} - 3 \beta_{4} - 2) q^{22} + (2 \beta_{9} - 3 \beta_{7} + 2 \beta_{4} - 2 \beta_{3} + 2) q^{23} + ( - \beta_{11} + 2 \beta_{10} - 2 \beta_{8} + 2 \beta_1) q^{24} + ( - \beta_{5} - 3 \beta_{3} + 3) q^{25} + ( - 2 \beta_{5} - 3 \beta_{3} + 2) q^{27} + ( - 5 \beta_{10} - \beta_{8} + \beta_1) q^{28} + ( - 5 \beta_{9} + 3 \beta_{7} + 2 \beta_{5} - 3 \beta_{4} + 5 \beta_{3} - 5) q^{29} + \beta_{9} q^{30} + (5 \beta_{11} + \beta_{10} + 3 \beta_{8} - \beta_{6} + 5 \beta_{2}) q^{31} + (5 \beta_{2} + 2 \beta_1) q^{32} + ( - \beta_{6} - 3 \beta_{2} - 4 \beta_1) q^{33} + (3 \beta_{11} - 4 \beta_{10} - 2 \beta_{8} + 4 \beta_{6} + 3 \beta_{2}) q^{34} + ( - 4 \beta_{9} + 3 \beta_{7} - \beta_{4} - 3) q^{35} + ( - 4 \beta_{9} - \beta_{7} + 4 \beta_{5} + 4 \beta_{3} - 4) q^{36} + ( - 2 \beta_{11} + 3 \beta_{10} - \beta_{8} + \beta_1) q^{37} + (6 \beta_{5} + 3 \beta_{3} - 7) q^{38} + ( - 3 \beta_{5} - 3 \beta_{3} + 1) q^{40} + ( - 4 \beta_{11} - \beta_{10} - 6 \beta_{8} + 6 \beta_1) q^{41} + ( - \beta_{9} - 2 \beta_{7} - \beta_{4} + \beta_{3} - 1) q^{42} + (2 \beta_{9} - 6 \beta_{7} + 3 \beta_{4} + 6) q^{43} + (\beta_{11} - \beta_{10} - 4 \beta_{8} + \beta_{6} + \beta_{2}) q^{44} + (5 \beta_{6} - 2 \beta_{2} - 2 \beta_1) q^{45} + ( - 3 \beta_{6} + \beta_1) q^{46} + (\beta_{11} + 6 \beta_{10} - \beta_{8} - 6 \beta_{6} + \beta_{2}) q^{47} + ( - \beta_{9} + \beta_{7} - \beta_{4} - 1) q^{48} + ( - \beta_{9} - \beta_{7} + \beta_{5} + \beta_{3} - 1) q^{49} + ( - \beta_{11} + \beta_{10} + 3 \beta_{8} - 3 \beta_1) q^{50} + (2 \beta_{5} - \beta_{3}) q^{51} + ( - 3 \beta_{5} + 4 \beta_{3}) q^{53} + ( - 2 \beta_{11} + \beta_{10} + 2 \beta_{8} - 2 \beta_1) q^{54} + (\beta_{9} - 3 \beta_{7} + \beta_{5} + 2 \beta_{4} - \beta_{3} + 1) q^{55} + ( - 3 \beta_{9} + 4 \beta_{7} - 5 \beta_{4} - 4) q^{56} + (2 \beta_{10} + \beta_{8} - 2 \beta_{6}) q^{57} + (\beta_{6} - 2 \beta_{2} + 2 \beta_1) q^{58} + (5 \beta_{6} + 4 \beta_{2}) q^{59} + ( - \beta_{11} - 3 \beta_{10} - \beta_{8} + 3 \beta_{6} - \beta_{2}) q^{60} + (6 \beta_{9} - \beta_{7} + \beta_{4} + 1) q^{61} + (2 \beta_{9} + \beta_{7} - 6 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} + 2) q^{62} + (5 \beta_{11} + \beta_{10} + 2 \beta_{8} - 2 \beta_1) q^{63} + ( - 6 \beta_{5} - \beta_{3} + 6) q^{64} + (3 \beta_{5} - \beta_{3} + 1) q^{66} + (6 \beta_{11} + 4 \beta_{10} + 5 \beta_{8} - 5 \beta_1) q^{67} + (3 \beta_{9} + 7 \beta_{7} - 6 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + 3) q^{68} + ( - \beta_{9} + 2 \beta_{7} + \beta_{4} - 2) q^{69} + ( - 3 \beta_{11} + \beta_{8} - 3 \beta_{2}) q^{70} + ( - 10 \beta_{6} + 3 \beta_{2}) q^{71} + (\beta_{6} - 2 \beta_{2} + 3 \beta_1) q^{72} + ( - 6 \beta_{11} - 4 \beta_{10} + 3 \beta_{8} + 4 \beta_{6} - 6 \beta_{2}) q^{73} + ( - \beta_{9} - 4 \beta_{7} + 5 \beta_{4} + 4) q^{74} + (2 \beta_{9} + 3 \beta_{7} + 2 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + 2) q^{75} + (2 \beta_{11} - 8 \beta_{10} - 5 \beta_{8} + 5 \beta_1) q^{76} + ( - 5 \beta_{5} - 1) q^{77} + (9 \beta_{5} + \beta_{3} - 5) q^{79} + (3 \beta_{11} + 3 \beta_{10} + \beta_{8} - \beta_1) q^{80} + (3 \beta_{9} - 3 \beta_{7} + 4 \beta_{5} + 7 \beta_{4} - 3 \beta_{3} + 3) q^{81} + (2 \beta_{9} + 3 \beta_{7} + 3 \beta_{4} - 3) q^{82} + ( - 9 \beta_{11} - 6 \beta_{10} - 2 \beta_{8} + 6 \beta_{6} - 9 \beta_{2}) q^{83} + ( - 4 \beta_{2} - 3 \beta_1) q^{84} + ( - \beta_{6} - 3 \beta_{2} + \beta_1) q^{85} + ( - \beta_{11} + 7 \beta_{10} + 4 \beta_{8} - 7 \beta_{6} - \beta_{2}) q^{86} + ( - 5 \beta_{7} - 3 \beta_{4} + 5) q^{87} + (5 \beta_{9} + 2 \beta_{7} - \beta_{5} + 4 \beta_{4} - 5 \beta_{3} + 5) q^{88} + ( - 7 \beta_{11} - 6 \beta_{10}) q^{89} + ( - 3 \beta_{5} + 5) q^{90} + (4 \beta_{5} + 5 \beta_{3} - 3) q^{92} + (2 \beta_{11} - 5 \beta_{10} + 5 \beta_{8} - 5 \beta_1) q^{93} + (2 \beta_{9} - 4 \beta_{7} + 3 \beta_{5} + 5 \beta_{4} - 2 \beta_{3} + 2) q^{94} + ( - 4 \beta_{9} + \beta_{4}) q^{95} + (2 \beta_{11} - 5 \beta_{10} + 4 \beta_{8} + 5 \beta_{6} + 2 \beta_{2}) q^{96} + (5 \beta_{6} - \beta_{2} - 7 \beta_1) q^{97} + ( - 2 \beta_{6} - \beta_{2} + 2 \beta_1) q^{98} + ( - 2 \beta_{11} - 3 \beta_{10} - 6 \beta_{8} + 3 \beta_{6} - 2 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{3} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{3} + 6 q^{9} + 10 q^{10} - 20 q^{14} - 4 q^{16} - 4 q^{17} - 6 q^{22} - 10 q^{23} + 20 q^{25} + 4 q^{27} + 2 q^{29} - 2 q^{30} - 8 q^{35} - 14 q^{36} - 48 q^{38} - 12 q^{40} - 16 q^{42} + 26 q^{43} - 2 q^{48} - 8 q^{49} + 4 q^{51} + 4 q^{53} - 12 q^{55} - 8 q^{56} - 8 q^{61} + 2 q^{62} + 44 q^{64} + 20 q^{66} + 42 q^{68} - 12 q^{69} + 16 q^{74} + 30 q^{75} - 32 q^{77} - 20 q^{79} + 2 q^{81} - 28 q^{82} + 36 q^{87} + 30 q^{88} + 48 q^{90} - 10 q^{94} + 6 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -25\nu^{11} + 95\nu^{9} - 361\nu^{7} + 155\nu^{5} - 30\nu^{3} - 1563\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 25\nu^{10} - 95\nu^{8} + 361\nu^{6} - 155\nu^{4} + 30\nu^{2} + 1004 ) / 559 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{10} - 20\nu^{8} + 76\nu^{6} - 139\nu^{4} + 124\nu^{2} - 24 ) / 43 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 45\nu^{10} - 171\nu^{8} + 538\nu^{6} - 279\nu^{4} + 54\nu^{2} + 242 ) / 559 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 70\nu^{11} - 266\nu^{9} + 899\nu^{7} - 434\nu^{5} + 84\nu^{3} + 1246\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 114\nu^{10} - 545\nu^{8} + 2071\nu^{6} - 2831\nu^{4} + 3379\nu^{2} - 95 ) / 559 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 114\nu^{11} - 545\nu^{9} + 2071\nu^{7} - 2831\nu^{5} + 3379\nu^{3} - 95\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -128\nu^{10} + 710\nu^{8} - 2698\nu^{6} + 4483\nu^{4} - 4402\nu^{2} + 852 ) / 559 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -242\nu^{11} + 1255\nu^{9} - 4769\nu^{7} + 7314\nu^{5} - 7781\nu^{3} + 1506\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -317\nu^{11} + 1540\nu^{9} - 5852\nu^{7} + 8338\nu^{5} - 9548\nu^{3} + 1848\nu ) / 559 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{7} + \beta_{4} - \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + 3\beta_{8} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{9} + 2\beta_{7} + 4\beta_{4} - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{11} - \beta_{10} + 9\beta_{8} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -5\beta_{5} + 9\beta_{3} - 14 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -5\beta_{6} - 14\beta_{2} - 28\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -28\beta_{9} - 14\beta_{7} - 19\beta_{5} - 47\beta_{4} + 28\beta_{3} - 28 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -47\beta_{11} + 19\beta_{10} - 89\beta_{8} - 19\beta_{6} - 47\beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -89\beta_{9} - 42\beta_{7} - 155\beta_{4} + 42 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -155\beta_{11} + 66\beta_{10} - 286\beta_{8} + 286\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(1 - \beta_{7}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
1.07992 + 0.623490i
1.56052 + 0.900969i
−0.385418 − 0.222521i
0.385418 + 0.222521i
−1.56052 − 0.900969i
−1.07992 − 0.623490i
1.07992 − 0.623490i
1.56052 − 0.900969i
−0.385418 + 0.222521i
0.385418 − 0.222521i
−1.56052 + 0.900969i
−1.07992 + 0.623490i
−1.94594 + 1.12349i 0.277479 + 0.480608i 1.52446 − 2.64044i − 1.44504i −1.07992 − 0.623490i 1.77441 + 1.02446i 2.35690i 1.34601 − 2.33136i 1.62349 + 2.81197i
23.2 −0.694498 + 0.400969i 1.12349 + 1.94594i −0.678448 + 1.17511i − 0.246980i −1.56052 − 0.900969i 2.04113 + 1.17845i − 2.69202i −1.02446 + 1.77441i 0.0990311 + 0.171527i
23.3 −0.480608 + 0.277479i −0.400969 − 0.694498i −0.846011 + 1.46533i − 2.80194i 0.385418 + 0.222521i −2.33136 − 1.34601i − 2.04892i 1.17845 − 2.04113i 0.777479 + 1.34663i
23.4 0.480608 − 0.277479i −0.400969 − 0.694498i −0.846011 + 1.46533i 2.80194i −0.385418 − 0.222521i 2.33136 + 1.34601i 2.04892i 1.17845 − 2.04113i 0.777479 + 1.34663i
23.5 0.694498 − 0.400969i 1.12349 + 1.94594i −0.678448 + 1.17511i 0.246980i 1.56052 + 0.900969i −2.04113 − 1.17845i 2.69202i −1.02446 + 1.77441i 0.0990311 + 0.171527i
23.6 1.94594 − 1.12349i 0.277479 + 0.480608i 1.52446 − 2.64044i 1.44504i 1.07992 + 0.623490i −1.77441 − 1.02446i − 2.35690i 1.34601 − 2.33136i 1.62349 + 2.81197i
147.1 −1.94594 − 1.12349i 0.277479 − 0.480608i 1.52446 + 2.64044i 1.44504i −1.07992 + 0.623490i 1.77441 − 1.02446i − 2.35690i 1.34601 + 2.33136i 1.62349 − 2.81197i
147.2 −0.694498 − 0.400969i 1.12349 − 1.94594i −0.678448 − 1.17511i 0.246980i −1.56052 + 0.900969i 2.04113 − 1.17845i 2.69202i −1.02446 − 1.77441i 0.0990311 − 0.171527i
147.3 −0.480608 − 0.277479i −0.400969 + 0.694498i −0.846011 − 1.46533i 2.80194i 0.385418 − 0.222521i −2.33136 + 1.34601i 2.04892i 1.17845 + 2.04113i 0.777479 − 1.34663i
147.4 0.480608 + 0.277479i −0.400969 + 0.694498i −0.846011 − 1.46533i − 2.80194i −0.385418 + 0.222521i 2.33136 − 1.34601i − 2.04892i 1.17845 + 2.04113i 0.777479 − 1.34663i
147.5 0.694498 + 0.400969i 1.12349 − 1.94594i −0.678448 − 1.17511i − 0.246980i 1.56052 − 0.900969i −2.04113 + 1.17845i − 2.69202i −1.02446 − 1.77441i 0.0990311 − 0.171527i
147.6 1.94594 + 1.12349i 0.277479 − 0.480608i 1.52446 + 2.64044i − 1.44504i 1.07992 − 0.623490i −1.77441 + 1.02446i 2.35690i 1.34601 + 2.33136i 1.62349 − 2.81197i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.2.e.b 12
13.b even 2 1 inner 169.2.e.b 12
13.c even 3 1 169.2.b.b 6
13.c even 3 1 inner 169.2.e.b 12
13.d odd 4 1 169.2.c.b 6
13.d odd 4 1 169.2.c.c 6
13.e even 6 1 169.2.b.b 6
13.e even 6 1 inner 169.2.e.b 12
13.f odd 12 1 169.2.a.b ✓ 3
13.f odd 12 1 169.2.a.c yes 3
13.f odd 12 1 169.2.c.b 6
13.f odd 12 1 169.2.c.c 6
39.h odd 6 1 1521.2.b.l 6
39.i odd 6 1 1521.2.b.l 6
39.k even 12 1 1521.2.a.o 3
39.k even 12 1 1521.2.a.r 3
52.i odd 6 1 2704.2.f.o 6
52.j odd 6 1 2704.2.f.o 6
52.l even 12 1 2704.2.a.z 3
52.l even 12 1 2704.2.a.ba 3
65.s odd 12 1 4225.2.a.bb 3
65.s odd 12 1 4225.2.a.bg 3
91.bc even 12 1 8281.2.a.bf 3
91.bc even 12 1 8281.2.a.bj 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.2.a.b ✓ 3 13.f odd 12 1
169.2.a.c yes 3 13.f odd 12 1
169.2.b.b 6 13.c even 3 1
169.2.b.b 6 13.e even 6 1
169.2.c.b 6 13.d odd 4 1
169.2.c.b 6 13.f odd 12 1
169.2.c.c 6 13.d odd 4 1
169.2.c.c 6 13.f odd 12 1
169.2.e.b 12 1.a even 1 1 trivial
169.2.e.b 12 13.b even 2 1 inner
169.2.e.b 12 13.c even 3 1 inner
169.2.e.b 12 13.e even 6 1 inner
1521.2.a.o 3 39.k even 12 1
1521.2.a.r 3 39.k even 12 1
1521.2.b.l 6 39.h odd 6 1
1521.2.b.l 6 39.i odd 6 1
2704.2.a.z 3 52.l even 12 1
2704.2.a.ba 3 52.l even 12 1
2704.2.f.o 6 52.i odd 6 1
2704.2.f.o 6 52.j odd 6 1
4225.2.a.bb 3 65.s odd 12 1
4225.2.a.bg 3 65.s odd 12 1
8281.2.a.bf 3 91.bc even 12 1
8281.2.a.bj 3 91.bc even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 6T_{2}^{10} + 31T_{2}^{8} - 28T_{2}^{6} + 19T_{2}^{4} - 5T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 6 T^{10} + 31 T^{8} - 28 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{6} - 2 T^{5} + 5 T^{4} + 3 T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} + 10 T^{4} + 17 T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} - 17 T^{10} + 195 T^{8} + \cdots + 28561 \) Copy content Toggle raw display
$11$ \( T^{12} - 26 T^{10} + 523 T^{8} + \cdots + 28561 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( (T^{6} + 2 T^{5} + 19 T^{4} - 56 T^{3} + \cdots + 169)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} - 38 T^{10} + 1315 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( (T^{6} + 5 T^{5} + 26 T^{4} + 21 T^{3} + \cdots + 169)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - T^{5} + 45 T^{4} + 210 T^{3} + \cdots + 6889)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 97 T^{4} + 2966 T^{2} + \cdots + 27889)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} - 62 T^{10} + 2859 T^{8} + \cdots + 707281 \) Copy content Toggle raw display
$41$ \( T^{12} - 147 T^{10} + 19894 T^{8} + \cdots + 5764801 \) Copy content Toggle raw display
$43$ \( (T^{6} - 13 T^{5} + 129 T^{4} - 546 T^{3} + \cdots + 169)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 122 T^{4} + 4189 T^{2} + \cdots + 27889)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - T^{2} - 86 T + 337)^{4} \) Copy content Toggle raw display
$59$ \( T^{12} - 195 T^{10} + 31174 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( (T^{6} + 4 T^{5} + 83 T^{4} + 210 T^{3} + \cdots + 57121)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} - 145 T^{10} + 15923 T^{8} + \cdots + 2825761 \) Copy content Toggle raw display
$71$ \( T^{12} - 285 T^{10} + \cdots + 89526025681 \) Copy content Toggle raw display
$73$ \( (T^{6} + 321 T^{4} + 30798 T^{2} + \cdots + 829921)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 5 T^{2} - 162 T + 127)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 329 T^{4} + 16758 T^{2} + \cdots + 41209)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} - 269 T^{10} + \cdots + 6234839521 \) Copy content Toggle raw display
$97$ \( T^{12} - 217 T^{10} + \cdots + 8208541201 \) Copy content Toggle raw display
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