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Properties

Label 605.2.b.h
Level $605$
Weight $2$
Character orbit 605.b
Analytic conductor $4.831$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [605,2,Mod(364,605)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(605, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("605.364");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.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: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + x^{10} + 34 x^{9} - 123 x^{8} - 20 x^{7} + 516 x^{6} - 668 x^{5} - 67 x^{4} + \cdots + 1089 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{9} q^{2} - \beta_{8} q^{3} + ( - \beta_{6} - 2) q^{4} + \beta_{4} q^{5} + (\beta_{7} - \beta_{3} - 2 \beta_1) q^{6} + (\beta_{9} + \beta_{7}) q^{7} + ( - \beta_{10} - 2 \beta_{9}) q^{8} + ( - \beta_{6} - \beta_{4} + \beta_{2} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{2} - \beta_{8} q^{3} + ( - \beta_{6} - 2) q^{4} + \beta_{4} q^{5} + (\beta_{7} - \beta_{3} - 2 \beta_1) q^{6} + (\beta_{9} + \beta_{7}) q^{7} + ( - \beta_{10} - 2 \beta_{9}) q^{8} + ( - \beta_{6} - \beta_{4} + \beta_{2} - 3) q^{9} + ( - \beta_{10} - \beta_{9} + \cdots + \beta_1) q^{10}+ \cdots + (3 \beta_{10} + 3 \beta_{9} - \beta_{7}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 20 q^{4} - 6 q^{5} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 20 q^{4} - 6 q^{5} - 20 q^{9} - 32 q^{14} + 20 q^{15} + 36 q^{16} + 26 q^{20} - 10 q^{25} + 20 q^{26} + 8 q^{31} + 12 q^{34} + 92 q^{36} - 18 q^{45} - 4 q^{49} + 48 q^{56} - 32 q^{59} - 92 q^{60} + 28 q^{64} - 16 q^{69} - 12 q^{70} + 112 q^{71} + 36 q^{75} - 106 q^{80} - 20 q^{81} - 56 q^{86} - 108 q^{89} + 72 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + x^{10} + 34 x^{9} - 123 x^{8} - 20 x^{7} + 516 x^{6} - 668 x^{5} - 67 x^{4} + \cdots + 1089 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4845953018 \nu^{11} + 963958706563 \nu^{10} - 4797665688052 \nu^{9} + \cdots + 26\!\cdots\!08 ) / 308731269222837 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3079103815 \nu^{11} + 42281880539 \nu^{10} - 121107594570 \nu^{9} + \cdots + 156378153801710 ) / 34303474358093 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 66674 \nu^{11} - 682376 \nu^{10} + 2180609 \nu^{9} + 29294 \nu^{8} - 21496194 \nu^{7} + \cdots - 625607220 ) / 167498433 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 17157797861 \nu^{11} - 108022004876 \nu^{10} + 240501508768 \nu^{9} + 116703163235 \nu^{8} + \cdots - 48489496588954 ) / 34303474358093 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 344078139086 \nu^{11} - 1840864299167 \nu^{10} + 2869558436669 \nu^{9} + \cdots + 993528750572394 ) / 308731269222837 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 38582738806 \nu^{11} + 312787367384 \nu^{10} - 856308492056 \nu^{9} + \cdots + 227364595507030 ) / 34303474358093 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 403353306232 \nu^{11} - 1768122412744 \nu^{10} + 1528746626863 \nu^{9} + \cdots + 13\!\cdots\!55 ) / 308731269222837 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 231664336675 \nu^{11} - 1055801231693 \nu^{10} + 791158199715 \nu^{9} + \cdots + 364041010452140 ) / 34303474358093 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2127198780859 \nu^{11} + 9706814485894 \nu^{10} - 7553443773796 \nu^{9} + \cdots - 36\!\cdots\!72 ) / 308731269222837 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 255788416792 \nu^{11} + 1158943341170 \nu^{10} - 884228369606 \nu^{9} + \cdots - 386058541069797 ) / 34303474358093 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 446829251302 \nu^{11} + 2049366765782 \nu^{10} - 1565285873757 \nu^{9} + \cdots - 724073294811309 ) / 34303474358093 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} - \beta_{8} + \beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{9} - 2\beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} + 2\beta_{3} - 2\beta_{2} + 3\beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{11} - 2 \beta_{10} + 3 \beta_{9} - 3 \beta_{8} - 6 \beta_{7} + \beta_{6} + 5 \beta_{4} + \cdots - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 3 \beta_{11} - 4 \beta_{10} + 16 \beta_{9} + 5 \beta_{8} + 3 \beta_{7} - \beta_{6} - 4 \beta_{5} + \cdots + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 19 \beta_{11} - 45 \beta_{10} + 78 \beta_{9} - 10 \beta_{8} + 17 \beta_{7} + 50 \beta_{6} + \cdots + 195 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 50 \beta_{11} - 100 \beta_{10} + 90 \beta_{9} - 132 \beta_{8} + 141 \beta_{7} + 101 \beta_{6} + \cdots + 735 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 261 \beta_{11} - 325 \beta_{10} - 44 \beta_{9} - 914 \beta_{8} - 15 \beta_{7} + 248 \beta_{6} + \cdots + 1185 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 491 \beta_{11} - 396 \beta_{10} + 348 \beta_{9} - 1087 \beta_{8} + 92 \beta_{7} - 91 \beta_{6} + \cdots + 229 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 3350 \beta_{11} - 4108 \beta_{10} + 5953 \beta_{9} - 5265 \beta_{8} + 154 \beta_{7} - 539 \beta_{6} + \cdots - 3082 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 8420 \beta_{11} - 14920 \beta_{10} + 25746 \beta_{9} - 8578 \beta_{8} + 11169 \beta_{7} + 93 \beta_{6} + \cdots + 1945 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 24494 \beta_{11} - 52886 \beta_{10} + 65207 \beta_{9} - 44377 \beta_{8} + 37878 \beta_{7} + \cdots + 42832 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(486\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
364.1
−0.576643 + 0.233980i
1.15541 + 2.37379i
3.11392 − 0.445974i
1.38187 + 2.46878i
−2.40330 − 0.316825i
−0.671252 + 0.645175i
−0.671252 − 0.645175i
−2.40330 + 0.316825i
1.38187 − 2.46878i
3.11392 + 0.445974i
1.15541 − 2.37379i
−0.576643 − 0.233980i
− 2.60777i − 2.13980i −4.80044 −2.11084 + 0.737808i −5.58011 − 0.988879i 7.30289i −1.57876 1.92403 + 5.50457i
364.2 − 2.60777i 2.13980i −4.80044 −2.11084 − 0.737808i 5.58011 − 0.988879i 7.30289i −1.57876 −1.92403 + 5.50457i
364.3 − 2.02281i − 2.91475i −2.09174 1.20202 + 1.88551i −5.89598 − 3.21128i 0.185581i −5.49579 3.81402 − 2.43146i
364.4 − 2.02281i 2.91475i −2.09174 1.20202 − 1.88551i 5.89598 − 3.21128i 0.185581i −5.49579 −3.81402 − 2.43146i
364.5 − 0.328351i − 0.962000i 1.89219 −0.591185 − 2.15650i −0.315873 3.27259i − 1.27800i 2.07456 −0.708089 + 0.194116i
364.6 − 0.328351i 0.962000i 1.89219 −0.591185 + 2.15650i 0.315873 3.27259i − 1.27800i 2.07456 0.708089 + 0.194116i
364.7 0.328351i − 0.962000i 1.89219 −0.591185 − 2.15650i 0.315873 − 3.27259i 1.27800i 2.07456 0.708089 − 0.194116i
364.8 0.328351i 0.962000i 1.89219 −0.591185 + 2.15650i −0.315873 − 3.27259i 1.27800i 2.07456 −0.708089 − 0.194116i
364.9 2.02281i − 2.91475i −2.09174 1.20202 + 1.88551i 5.89598 3.21128i − 0.185581i −5.49579 −3.81402 + 2.43146i
364.10 2.02281i 2.91475i −2.09174 1.20202 − 1.88551i −5.89598 3.21128i − 0.185581i −5.49579 3.81402 + 2.43146i
364.11 2.60777i − 2.13980i −4.80044 −2.11084 + 0.737808i 5.58011 0.988879i − 7.30289i −1.57876 −1.92403 − 5.50457i
364.12 2.60777i 2.13980i −4.80044 −2.11084 − 0.737808i −5.58011 0.988879i − 7.30289i −1.57876 1.92403 − 5.50457i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 364.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.b odd 2 1 inner
55.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.b.h ✓ 12
5.b even 2 1 inner 605.2.b.h ✓ 12
5.c odd 4 2 3025.2.a.bo 12
11.b odd 2 1 inner 605.2.b.h ✓ 12
11.c even 5 4 605.2.j.k 48
11.d odd 10 4 605.2.j.k 48
55.d odd 2 1 inner 605.2.b.h ✓ 12
55.e even 4 2 3025.2.a.bo 12
55.h odd 10 4 605.2.j.k 48
55.j even 10 4 605.2.j.k 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.b.h ✓ 12 1.a even 1 1 trivial
605.2.b.h ✓ 12 5.b even 2 1 inner
605.2.b.h ✓ 12 11.b odd 2 1 inner
605.2.b.h ✓ 12 55.d odd 2 1 inner
605.2.j.k 48 11.c even 5 4
605.2.j.k 48 11.d odd 10 4
605.2.j.k 48 55.h odd 10 4
605.2.j.k 48 55.j even 10 4
3025.2.a.bo 12 5.c odd 4 2
3025.2.a.bo 12 55.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\):

\( T_{2}^{6} + 11T_{2}^{4} + 29T_{2}^{2} + 3 \) Copy content Toggle raw display
\( T_{19}^{6} - 72T_{19}^{4} + 1332T_{19}^{2} - 3888 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 11 T^{4} + 29 T^{2} + 3)^{2} \) Copy content Toggle raw display
$3$ \( (T^{6} + 14 T^{4} + \cdots + 36)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} + 3 T^{5} + \cdots + 125)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 22 T^{4} + \cdots + 108)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( (T^{6} + 37 T^{4} + \cdots + 108)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 17 T^{4} + \cdots + 48)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 72 T^{4} + \cdots - 3888)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 80 T^{4} + \cdots + 17424)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 75 T^{4} + \cdots - 3888)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 2 T^{2} - 38 T - 68)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + 51 T^{4} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 117 T^{4} + \cdots - 2187)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 130 T^{4} + \cdots + 432)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 14 T^{4} + \cdots + 36)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 47 T^{4} + \cdots + 36)^{2} \) Copy content Toggle raw display
$59$ \( (T^{3} + 8 T^{2} - 50 T + 24)^{4} \) Copy content Toggle raw display
$61$ \( (T^{6} - 90 T^{4} + \cdots - 1728)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 174 T^{4} + \cdots + 39204)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} - 28 T^{2} + \cdots + 48)^{4} \) Copy content Toggle raw display
$73$ \( (T^{6} + 292 T^{4} + \cdots + 442368)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 300 T^{4} + \cdots - 110592)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 296 T^{4} + \cdots + 591408)^{2} \) Copy content Toggle raw display
$89$ \( (T + 9)^{12} \) Copy content Toggle raw display
$97$ \( (T^{6} + 519 T^{4} + \cdots + 4743684)^{2} \) Copy content Toggle raw display
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