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Properties

Label 55.4.b.b
Level $55$
Weight $4$
Character orbit 55.b
Analytic conductor $3.245$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [55,4,Mod(34,55)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(55, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("55.34");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 55 = 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 55.b (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: \(3.24510505032\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 72x^{8} + 1771x^{6} + 17056x^{4} + 52892x^{2} + 3136 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3}\cdot 5 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} - 6) q^{4} + ( - \beta_{4} + 1) q^{5} + (\beta_{7} + \beta_{6} + \beta_{4} + \cdots + 2) q^{6}+ \cdots + (\beta_{9} - \beta_{7} + \beta_{5} + \cdots - 14) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} - 6) q^{4} + ( - \beta_{4} + 1) q^{5} + (\beta_{7} + \beta_{6} + \beta_{4} + \cdots + 2) q^{6}+ \cdots + (11 \beta_{9} - 11 \beta_{7} + \cdots - 154) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 64 q^{4} + 14 q^{5} + 26 q^{6} - 152 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 64 q^{4} + 14 q^{5} + 26 q^{6} - 152 q^{9} + 36 q^{10} + 110 q^{11} - 34 q^{14} - 176 q^{15} + 468 q^{16} - 90 q^{19} - 310 q^{20} + 302 q^{21} - 206 q^{24} - 232 q^{25} + 392 q^{26} + 58 q^{29} - 1390 q^{30} + 1242 q^{31} + 66 q^{34} - 318 q^{35} + 1786 q^{36} + 384 q^{39} - 2066 q^{40} + 416 q^{41} - 704 q^{44} - 332 q^{45} + 1816 q^{46} - 1980 q^{49} - 1030 q^{50} + 510 q^{51} - 1882 q^{54} + 154 q^{55} + 2626 q^{56} - 476 q^{59} + 274 q^{60} + 1650 q^{61} - 2576 q^{64} + 2032 q^{65} + 286 q^{66} - 4492 q^{69} - 2172 q^{70} - 498 q^{71} + 5374 q^{74} + 2134 q^{75} + 1410 q^{76} + 416 q^{79} + 1850 q^{80} - 710 q^{81} + 9018 q^{84} + 370 q^{85} - 6872 q^{86} - 1918 q^{89} - 3056 q^{90} + 1384 q^{91} + 2860 q^{94} - 1700 q^{95} - 10114 q^{96} - 1672 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 72x^{8} + 1771x^{6} + 17056x^{4} + 52892x^{2} + 3136 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 17\nu^{9} + 1210\nu^{7} + 27923\nu^{5} + 223018\nu^{3} + 428120\nu ) / 40376 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 269 \nu^{9} + 42 \nu^{8} + 20334 \nu^{7} + 6552 \nu^{6} + 526155 \nu^{5} + 220990 \nu^{4} + \cdots + 3228512 ) / 403760 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 184 \nu^{9} + 497 \nu^{8} + 14284 \nu^{7} + 27062 \nu^{6} + 386540 \nu^{5} + 428015 \nu^{4} + \cdots - 1297128 ) / 201880 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 319 \nu^{9} + 42 \nu^{8} + 20924 \nu^{7} + 6552 \nu^{6} + 447965 \nu^{5} + 220990 \nu^{4} + \cdots + 3430392 ) / 201880 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 737 \nu^{9} + 14 \nu^{8} - 50082 \nu^{7} + 2184 \nu^{6} - 1142855 \nu^{5} + 6370 \nu^{4} + \cdots - 4711056 ) / 403760 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 737 \nu^{9} - 1036 \nu^{8} - 50082 \nu^{7} - 60676 \nu^{6} - 1142855 \nu^{5} - 1077020 \nu^{4} + \cdots - 634256 ) / 403760 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 687 \nu^{9} + 539 \nu^{8} - 49492 \nu^{7} + 33614 \nu^{6} - 1221045 \nu^{5} + 649005 \nu^{4} + \cdots + 1931384 ) / 201880 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{6} + \beta_{5} - \beta_{4} - 23\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{9} + 2\beta_{8} - 6\beta_{7} - 2\beta_{6} + 4\beta_{4} - 29\beta_{2} + 322 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{9} - 39\beta_{8} - 37\beta_{6} - 37\beta_{5} + 39\beta_{4} - 28\beta_{3} + 593\beta _1 + 37 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 78 \beta_{9} - 76 \beta_{8} + 260 \beta_{7} + 106 \beta_{6} - 2 \beta_{5} - 124 \beta_{4} + \cdots - 8264 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 108\beta_{9} + 1303\beta_{8} + 1127\beta_{6} + 1195\beta_{5} - 1167\beta_{4} + 1552\beta_{3} - 16123\beta _1 - 1127 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2606 \beta_{9} + 2294 \beta_{8} - 8990 \beta_{7} - 4090 \beta_{6} + 312 \beta_{5} + 3104 \beta_{4} + \cdots + 222706 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 4402 \beta_{9} - 41803 \beta_{8} - 32561 \beta_{6} - 37401 \beta_{5} + 32123 \beta_{4} - 62100 \beta_{3} + \cdots + 32561 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/55\mathbb{Z}\right)^\times\).

\(n\) \(12\) \(46\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1
− 5.44091i
− 4.89559i
− 3.54108i
− 2.41454i
− 0.245890i
0.245890i
2.41454i
3.54108i
4.89559i
5.44091i
− 5.44091i − 5.19344i −21.6035 9.86485 + 5.26163i −28.2570 − 18.8252i 74.0156i 0.0282092 28.6281 − 53.6738i
34.2 − 4.89559i 8.92429i −15.9668 −6.48536 + 9.10715i 43.6896 17.6628i 39.0021i −52.6430 44.5848 + 31.7496i
34.3 − 3.54108i 4.97406i −4.53923 2.19798 − 10.9622i 17.6135 − 25.4334i − 12.2549i 2.25876 −38.8178 − 7.78322i
34.4 − 2.41454i − 8.55947i 2.16999 8.10322 − 7.70310i −20.6672 35.7980i − 24.5559i −46.2646 −18.5995 − 19.5656i
34.5 − 0.245890i 2.52576i 7.93954 −6.68069 + 8.96484i 0.621060 10.5016i − 3.91938i 20.6205 2.20437 + 1.64272i
34.6 0.245890i − 2.52576i 7.93954 −6.68069 − 8.96484i 0.621060 − 10.5016i 3.91938i 20.6205 2.20437 − 1.64272i
34.7 2.41454i 8.55947i 2.16999 8.10322 + 7.70310i −20.6672 − 35.7980i 24.5559i −46.2646 −18.5995 + 19.5656i
34.8 3.54108i − 4.97406i −4.53923 2.19798 + 10.9622i 17.6135 25.4334i 12.2549i 2.25876 −38.8178 + 7.78322i
34.9 4.89559i − 8.92429i −15.9668 −6.48536 − 9.10715i 43.6896 − 17.6628i − 39.0021i −52.6430 44.5848 − 31.7496i
34.10 5.44091i 5.19344i −21.6035 9.86485 − 5.26163i −28.2570 18.8252i − 74.0156i 0.0282092 28.6281 + 53.6738i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 55.4.b.b ✓ 10
3.b odd 2 1 495.4.c.b 10
4.b odd 2 1 880.4.b.i 10
5.b even 2 1 inner 55.4.b.b ✓ 10
5.c odd 4 2 275.4.a.k 10
15.d odd 2 1 495.4.c.b 10
15.e even 4 2 2475.4.a.bw 10
20.d odd 2 1 880.4.b.i 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.4.b.b ✓ 10 1.a even 1 1 trivial
55.4.b.b ✓ 10 5.b even 2 1 inner
275.4.a.k 10 5.c odd 4 2
495.4.c.b 10 3.b odd 2 1
495.4.c.b 10 15.d odd 2 1
880.4.b.i 10 4.b odd 2 1
880.4.b.i 10 20.d odd 2 1
2475.4.a.bw 10 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 72T_{2}^{8} + 1771T_{2}^{6} + 17056T_{2}^{4} + 52892T_{2}^{2} + 3136 \) acting on \(S_{4}^{\mathrm{new}}(55, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 72 T^{8} + \cdots + 3136 \) Copy content Toggle raw display
$3$ \( T^{10} + 211 T^{8} + \cdots + 24840256 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 30517578125 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 10107210905856 \) Copy content Toggle raw display
$11$ \( (T - 11)^{10} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 243731546505216 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 292181315129344 \) Copy content Toggle raw display
$19$ \( (T^{5} + 45 T^{4} + \cdots - 1060640000)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 56\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{5} - 29 T^{4} + \cdots - 11998651200)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 621 T^{4} + \cdots + 198945702400)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T^{5} - 208 T^{4} + \cdots + 38905838592)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 17\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 28\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( (T^{5} + 238 T^{4} + \cdots + 80525414400)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 825 T^{4} + \cdots + 53510101312)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 38\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 5550175109664)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 32\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 17303484620800)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 64\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots - 41374767850500)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
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