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Properties

Label 136.2.h.a
Level $136$
Weight $2$
Character orbit 136.h
Analytic conductor $1.086$
Analytic rank $0$
Dimension $16$
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] = [136,2,Mod(101,136)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(136, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("136.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 136.h (of order \(2\), degree \(1\), 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: \(1.08596546749\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 8x^{14} + 20x^{12} + 36x^{10} + 240x^{8} - 156x^{6} + 268x^{4} + 136x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + \beta_{12} q^{3} - \beta_{3} q^{4} - \beta_{6} q^{5} - \beta_{10} q^{6} + \beta_{9} q^{7} + \beta_{8} q^{8} + (\beta_{8} - \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + \beta_{12} q^{3} - \beta_{3} q^{4} - \beta_{6} q^{5} - \beta_{10} q^{6} + \beta_{9} q^{7} + \beta_{8} q^{8} + (\beta_{8} - \beta_{2} + 1) q^{9} + \beta_1 q^{10} - \beta_{7} q^{11} + (\beta_{14} - \beta_{12} - \beta_{9}) q^{12} + ( - \beta_{15} - \beta_{8}) q^{13} + ( - \beta_{13} - \beta_{6}) q^{14} + ( - \beta_{8} + \beta_{5} - \beta_{4} + \cdots - 1) q^{15}+ \cdots + ( - \beta_{12} + 2 \beta_{7}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{2} - 6 q^{4} - 2 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{2} - 6 q^{4} - 2 q^{8} + 8 q^{9} - 8 q^{15} - 14 q^{16} - 2 q^{18} - 16 q^{30} - 2 q^{32} - 8 q^{33} + 18 q^{34} - 22 q^{36} + 36 q^{38} - 24 q^{47} - 8 q^{49} + 34 q^{50} - 8 q^{55} - 16 q^{60} - 30 q^{64} - 32 q^{66} + 38 q^{68} + 40 q^{70} + 70 q^{72} + 4 q^{76} - 24 q^{81} + 72 q^{84} + 4 q^{86} - 40 q^{87} - 24 q^{89} - 16 q^{94} + 34 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 8x^{14} + 20x^{12} + 36x^{10} + 240x^{8} - 156x^{6} + 268x^{4} + 136x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1122577 \nu^{14} - 14068890 \nu^{12} - 66487248 \nu^{10} - 169343412 \nu^{8} + \cdots - 581192704 ) / 388027784 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1923537 \nu^{14} - 13519726 \nu^{12} - 25288536 \nu^{10} - 46964780 \nu^{8} + \cdots + 151276224 ) / 388027784 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 511505 \nu^{14} - 4434484 \nu^{12} - 12763476 \nu^{10} - 23013159 \nu^{8} - 130749772 \nu^{6} + \cdots - 96545102 ) / 97006946 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2530173 \nu^{14} + 18755430 \nu^{12} + 39114176 \nu^{10} + 69125296 \nu^{8} + 597246024 \nu^{6} + \cdots + 450774640 ) / 388027784 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3391627 \nu^{15} - 26976512 \nu^{13} - 68170444 \nu^{11} - 131353936 \nu^{9} + \cdots + 49963176 \nu ) / 388027784 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3710839 \nu^{15} - 33502210 \nu^{13} - 107383932 \nu^{11} - 227489312 \nu^{9} + \cdots - 14322168 \nu ) / 388027784 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 4466259 \nu^{14} + 35818322 \nu^{12} + 84156112 \nu^{10} + 113648052 \nu^{8} + 941650648 \nu^{6} + \cdots + 997692448 ) / 388027784 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2689779 \nu^{15} - 22018279 \nu^{13} - 58720920 \nu^{11} - 117192984 \nu^{9} + \cdots - 870945096 \nu ) / 194013892 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1397209 \nu^{15} + 12471803 \nu^{13} + 37660840 \nu^{11} + 70747866 \nu^{9} + \cdots + 588735180 \nu ) / 97006946 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 4414637 \nu^{14} - 35845480 \nu^{12} - 93697396 \nu^{10} - 177380254 \nu^{8} + \cdots - 337140920 ) / 194013892 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 7976257 \nu^{15} - 58299578 \nu^{13} - 116440672 \nu^{11} - 185835896 \nu^{9} + \cdots + 205818936 \nu ) / 388027784 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 9891357 \nu^{15} - 76989604 \nu^{13} - 183627060 \nu^{11} - 339496648 \nu^{9} + \cdots - 1023743432 \nu ) / 388027784 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 10838191 \nu^{15} + 87123872 \nu^{13} + 214521736 \nu^{11} + 348654544 \nu^{9} + \cdots + 1007435080 \nu ) / 388027784 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 12834559 \nu^{14} - 99013618 \nu^{12} - 221895280 \nu^{10} - 344990604 \nu^{8} + \cdots - 1418449856 ) / 388027784 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + \beta_{8} - \beta_{5} - 3\beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{14} - \beta_{13} - 2\beta_{12} + 3\beta_{10} + 2\beta_{9} + 3\beta_{6} - 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{15} - 3\beta_{11} + 4\beta_{5} - \beta_{4} + 5\beta_{3} + 3\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5\beta_{14} - 2\beta_{13} + 9\beta_{12} - 7\beta_{10} - 2\beta_{9} + 3\beta_{7} - 11\beta_{6} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -13\beta_{15} + 16\beta_{11} - 12\beta_{8} - 13\beta_{5} - 13\beta_{4} - 12\beta_{3} - 10\beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -12\beta_{14} + 13\beta_{13} - 38\beta_{12} + 13\beta_{10} - 5\beta_{9} - 27\beta_{7} + 72\beta_{6} - 13\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 68\beta_{15} - 53\beta_{11} + 95\beta_{8} + 109\beta_{5} + 114\beta_{4} + 27\beta_{3} + 9\beta_{2} - 98 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( \beta_{14} - 42\beta_{13} + 137\beta_{12} + 26\beta_{10} + 18\beta_{9} + 189\beta_{7} - 389\beta_{6} - 103\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -337\beta_{15} + 60\beta_{11} - 671\beta_{8} - 627\beta_{5} - 627\beta_{4} + 171\beta_{3} + 148\beta_{2} + 857 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 276 \beta_{14} + 232 \beta_{13} - 398 \beta_{12} - 610 \beta_{10} - 202 \beta_{9} - 1120 \beta_{7} + \cdots + 1322 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1346 \beta_{15} + 812 \beta_{11} + 4020 \beta_{8} + 2292 \beta_{5} + 3408 \beta_{4} - 2900 \beta_{3} + \cdots - 4928 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 3152 \beta_{14} - 1094 \beta_{13} - 460 \beta_{12} + 5826 \beta_{10} + 2058 \beta_{9} + \cdots - 9842 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 3052 \beta_{15} - 10250 \beta_{11} - 19406 \beta_{8} - 5826 \beta_{5} - 15868 \beta_{4} + \cdots + 24820 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 23600 \beta_{14} + 2314 \beta_{13} + 17496 \beta_{12} - 39954 \beta_{10} - 12460 \beta_{9} + \cdots + 60402 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/136\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(103\) \(105\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
0.970055 + 0.510012i
−0.970055 − 0.510012i
0.970055 − 0.510012i
−0.970055 + 0.510012i
−0.289081 − 0.578468i
0.289081 + 0.578468i
−0.289081 + 0.578468i
0.289081 − 0.578468i
−0.476789 + 2.28924i
0.476789 − 2.28924i
−0.476789 − 2.28924i
0.476789 + 2.28924i
1.32216 − 1.07919i
−1.32216 + 1.07919i
1.32216 + 1.07919i
−1.32216 − 1.07919i
−1.25175 − 0.658116i −0.909242 1.13377 + 1.64760i −1.54992 1.13815 + 0.598386i − 3.57366i −0.334887 − 2.80853i −2.17328 1.94011 + 1.02002i
101.2 −1.25175 − 0.658116i 0.909242 1.13377 + 1.64760i 1.54992 −1.13815 − 0.598386i 3.57366i −0.334887 − 2.80853i −2.17328 −1.94011 − 1.02002i
101.3 −1.25175 + 0.658116i −0.909242 1.13377 − 1.64760i −1.54992 1.13815 − 0.598386i 3.57366i −0.334887 + 2.80853i −2.17328 1.94011 − 1.02002i
101.4 −1.25175 + 0.658116i 0.909242 1.13377 − 1.64760i 1.54992 −1.13815 + 0.598386i − 3.57366i −0.334887 + 2.80853i −2.17328 −1.94011 + 1.02002i
101.5 −0.632188 − 1.26504i −2.88107 −1.20068 + 1.59949i 0.914541 1.82138 + 3.64469i 2.61974i 2.78248 + 0.507728i 5.30059 −0.578162 − 1.15694i
101.6 −0.632188 − 1.26504i 2.88107 −1.20068 + 1.59949i −0.914541 −1.82138 − 3.64469i − 2.61974i 2.78248 + 0.507728i 5.30059 0.578162 + 1.15694i
101.7 −0.632188 + 1.26504i −2.88107 −1.20068 − 1.59949i 0.914541 1.82138 − 3.64469i − 2.61974i 2.78248 − 0.507728i 5.30059 −0.578162 + 1.15694i
101.8 −0.632188 + 1.26504i 2.88107 −1.20068 − 1.59949i −0.914541 −1.82138 + 3.64469i 2.61974i 2.78248 − 0.507728i 5.30059 0.578162 − 1.15694i
101.9 0.288356 − 1.38450i −1.14379 −1.83370 − 0.798461i −3.30694 −0.329818 + 1.58358i − 1.93801i −1.63423 + 2.30853i −1.69175 −0.953577 + 4.57847i
101.10 0.288356 − 1.38450i 1.14379 −1.83370 − 0.798461i 3.30694 0.329818 − 1.58358i 1.93801i −1.63423 + 2.30853i −1.69175 0.953577 − 4.57847i
101.11 0.288356 + 1.38450i −1.14379 −1.83370 + 0.798461i −3.30694 −0.329818 − 1.58358i 1.93801i −1.63423 − 2.30853i −1.69175 −0.953577 − 4.57847i
101.12 0.288356 + 1.38450i 1.14379 −1.83370 + 0.798461i 3.30694 0.329818 + 1.58358i − 1.93801i −1.63423 − 2.30853i −1.69175 0.953577 + 4.57847i
101.13 1.09558 − 0.894257i −1.88797 0.400610 − 1.95947i 2.41361 −2.06843 + 1.68833i − 2.57100i −1.31336 − 2.50501i 0.564439 2.64431 − 2.15839i
101.14 1.09558 − 0.894257i 1.88797 0.400610 − 1.95947i −2.41361 2.06843 − 1.68833i 2.57100i −1.31336 − 2.50501i 0.564439 −2.64431 + 2.15839i
101.15 1.09558 + 0.894257i −1.88797 0.400610 + 1.95947i 2.41361 −2.06843 − 1.68833i 2.57100i −1.31336 + 2.50501i 0.564439 2.64431 + 2.15839i
101.16 1.09558 + 0.894257i 1.88797 0.400610 + 1.95947i −2.41361 2.06843 + 1.68833i − 2.57100i −1.31336 + 2.50501i 0.564439 −2.64431 − 2.15839i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
17.b even 2 1 inner
136.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.2.h.a ✓ 16
3.b odd 2 1 1224.2.l.b 16
4.b odd 2 1 544.2.h.a 16
8.b even 2 1 inner 136.2.h.a ✓ 16
8.d odd 2 1 544.2.h.a 16
12.b even 2 1 4896.2.l.b 16
17.b even 2 1 inner 136.2.h.a ✓ 16
24.f even 2 1 4896.2.l.b 16
24.h odd 2 1 1224.2.l.b 16
51.c odd 2 1 1224.2.l.b 16
68.d odd 2 1 544.2.h.a 16
136.e odd 2 1 544.2.h.a 16
136.h even 2 1 inner 136.2.h.a ✓ 16
204.h even 2 1 4896.2.l.b 16
408.b odd 2 1 1224.2.l.b 16
408.h even 2 1 4896.2.l.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.h.a ✓ 16 1.a even 1 1 trivial
136.2.h.a ✓ 16 8.b even 2 1 inner
136.2.h.a ✓ 16 17.b even 2 1 inner
136.2.h.a ✓ 16 136.h even 2 1 inner
544.2.h.a 16 4.b odd 2 1
544.2.h.a 16 8.d odd 2 1
544.2.h.a 16 68.d odd 2 1
544.2.h.a 16 136.e odd 2 1
1224.2.l.b 16 3.b odd 2 1
1224.2.l.b 16 24.h odd 2 1
1224.2.l.b 16 51.c odd 2 1
1224.2.l.b 16 408.b odd 2 1
4896.2.l.b 16 12.b even 2 1
4896.2.l.b 16 24.f even 2 1
4896.2.l.b 16 204.h even 2 1
4896.2.l.b 16 408.h even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(136, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + T^{7} + 2 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} - 14 T^{6} + \cdots + 32)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} - 20 T^{6} + \cdots + 128)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 30 T^{6} + \cdots + 2176)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 42 T^{6} + \cdots + 32)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 56 T^{6} + \cdots + 4352)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 12 T^{6} + \cdots + 83521)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 80 T^{6} + \cdots + 4352)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 106 T^{6} + \cdots + 8704)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 132 T^{6} + \cdots + 445568)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 178 T^{6} + \cdots + 8704)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 164 T^{6} + \cdots + 682112)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 264 T^{6} + \cdots + 8912896)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 124 T^{6} + \cdots + 17408)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 6 T^{3} + \cdots - 256)^{4} \) Copy content Toggle raw display
$53$ \( (T^{8} + 208 T^{6} + \cdots + 69632)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 188 T^{6} + \cdots + 17408)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 164 T^{6} + \cdots + 15488)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 224 T^{6} + \cdots + 526592)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 182 T^{6} + \cdots + 34816)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 464 T^{6} + \cdots + 67403776)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 266 T^{6} + \cdots + 263296)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 268 T^{6} + \cdots + 5030912)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 6 T^{3} - 96 T^{2} + \cdots + 8)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + 424 T^{6} + \cdots + 117121024)^{2} \) Copy content Toggle raw display
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