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Properties

Label 595.2.i.i
Level $595$
Weight $2$
Character orbit 595.i
Analytic conductor $4.751$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [595,2,Mod(86,595)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(595, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("595.86");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 595 = 5 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 595.i (of order \(3\), 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: \(4.75109892027\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} + 9 x^{12} - 2 x^{11} + 49 x^{10} - 5 x^{9} + 150 x^{8} + 33 x^{7} + 309 x^{6} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{13} - \beta_{8}) q^{3} + \beta_{12} q^{4} + ( - \beta_{8} - 1) q^{5} + (\beta_{6} + \beta_{4} + \beta_{3} + \cdots - 1) q^{6}+ \cdots + (2 \beta_{13} - \beta_{11} - 2 \beta_{8} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{13} - \beta_{8}) q^{3} + \beta_{12} q^{4} + ( - \beta_{8} - 1) q^{5} + (\beta_{6} + \beta_{4} + \beta_{3} + \cdots - 1) q^{6}+ \cdots + (\beta_{7} - 2 \beta_{6} + \cdots - 7 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + q^{2} + 6 q^{3} - 3 q^{4} - 7 q^{5} - 6 q^{6} - q^{7} - 12 q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + q^{2} + 6 q^{3} - 3 q^{4} - 7 q^{5} - 6 q^{6} - q^{7} - 12 q^{8} - 9 q^{9} + q^{10} + 3 q^{11} + 8 q^{12} + 12 q^{13} - 12 q^{15} + 13 q^{16} - 7 q^{17} - q^{18} + 14 q^{19} + 6 q^{20} - 5 q^{21} + 24 q^{22} + 3 q^{23} - 17 q^{24} - 7 q^{25} + 7 q^{26} - 78 q^{27} + 3 q^{28} - 2 q^{29} + 3 q^{30} + 10 q^{31} - 7 q^{32} - 14 q^{33} - 2 q^{34} + 5 q^{35} + 48 q^{36} - 8 q^{37} - 7 q^{38} + 17 q^{39} + 6 q^{40} - 24 q^{41} - 41 q^{42} - 8 q^{43} + 13 q^{44} - 9 q^{45} - 6 q^{46} + 22 q^{47} - 14 q^{48} + 17 q^{49} - 2 q^{50} + 6 q^{51} - 5 q^{52} - 16 q^{53} + 11 q^{54} - 6 q^{55} + 12 q^{56} + 74 q^{57} + 22 q^{58} + 47 q^{59} + 8 q^{60} - 26 q^{61} - 58 q^{62} + 14 q^{63} - 8 q^{64} - 6 q^{65} + 43 q^{66} + 20 q^{67} - 3 q^{68} - 42 q^{69} - 3 q^{70} - 8 q^{71} + 23 q^{72} + 19 q^{73} - 2 q^{74} + 6 q^{75} - 38 q^{76} - 31 q^{77} + 134 q^{78} + 46 q^{79} + 13 q^{80} - 47 q^{81} - 27 q^{82} - 68 q^{83} - 46 q^{84} + 14 q^{85} - 8 q^{86} - 20 q^{87} - 15 q^{88} + 26 q^{89} + 2 q^{90} - 54 q^{91} + 104 q^{92} + 13 q^{93} - 27 q^{94} + 14 q^{95} - 18 q^{96} - 38 q^{97} - 35 q^{98} - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} + 9 x^{12} - 2 x^{11} + 49 x^{10} - 5 x^{9} + 150 x^{8} + 33 x^{7} + 309 x^{6} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 181683923 \nu^{13} - 288466271 \nu^{12} - 511879049 \nu^{11} - 3970516527 \nu^{10} + \cdots + 21517347968 ) / 163316421962 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 470150194 \nu^{13} - 1123276258 \nu^{12} + 4333884373 \nu^{11} - 5820939330 \nu^{10} + \cdots - 327359579616 ) / 163316421962 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 851224369 \nu^{13} - 439204020 \nu^{12} + 5213737414 \nu^{11} + 3543570169 \nu^{10} + \cdots + 218002763208 ) / 163316421962 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 966639493 \nu^{13} + 4072541940 \nu^{12} - 11988916954 \nu^{11} + 28433039363 \nu^{10} + \cdots + 283892586500 ) / 163316421962 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1015201963 \nu^{13} + 257877445 \nu^{12} - 5869521520 \nu^{11} - 6090610251 \nu^{10} + \cdots + 228595201558 ) / 163316421962 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1614597122 \nu^{13} - 8718441049 \nu^{12} + 24699749612 \nu^{11} - 72494168558 \nu^{10} + \cdots - 513119260496 ) / 163316421962 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 5379336992 \nu^{13} + 5197653069 \nu^{12} - 48702499199 \nu^{11} + 10246794935 \nu^{10} + \cdots - 172389899894 ) / 163316421962 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 5644138732 \nu^{13} + 8543473356 \nu^{12} - 51645067802 \nu^{11} + 28097152861 \nu^{10} + \cdots - 152860680818 ) / 163316421962 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 6502613250 \nu^{13} + 6423461954 \nu^{12} - 57917022514 \nu^{11} + 13051123953 \nu^{10} + \cdots - 10227343016 ) / 163316421962 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 6999645473 \nu^{13} - 7003611994 \nu^{12} + 66024599079 \nu^{11} - 16475877953 \nu^{10} + \cdots + 12399797116 ) / 163316421962 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 5379336992 \nu^{13} - 5197653069 \nu^{12} + 48702499199 \nu^{11} - 10246794935 \nu^{10} + \cdots + 172389899894 ) / 81658210981 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 11019509203 \nu^{13} - 10713336603 \nu^{12} + 97870979994 \nu^{11} - 18360918139 \nu^{10} + \cdots + 297251998820 ) / 163316421962 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} + 2\beta_{8} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} - \beta_{3} + 3\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} - 5\beta_{12} - \beta_{10} - 7\beta_{8} - \beta_{5} - 5\beta_{3} - \beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} - 8\beta_{12} + \beta_{11} - 6\beta_{10} - 6\beta_{8} + 6\beta_{6} + \beta_{4} - 11\beta_{2} - 11\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{7} + 8\beta_{6} + 8\beta_{5} + \beta_{4} + 25\beta_{3} - 9\beta_{2} + 28 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 10 \beta_{13} + 50 \beta_{12} - 9 \beta_{11} + 32 \beta_{10} - \beta_{9} + 33 \beta_{8} + \beta_{7} + \cdots + 33 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 41 \beta_{13} + 138 \beta_{12} - 12 \beta_{11} + 59 \beta_{10} - 9 \beta_{9} + 122 \beta_{8} + \cdots + 52 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -12\beta_{7} - 158\beta_{6} - 83\beta_{5} - 59\beta_{4} - 278\beta_{3} + 222\beta_{2} - 180 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 229 \beta_{13} - 741 \beta_{12} + 95 \beta_{11} - 349 \beta_{10} + 59 \beta_{9} - 568 \beta_{8} + \cdots - 568 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 444 \beta_{13} - 1631 \beta_{12} + 347 \beta_{11} - 911 \beta_{10} + 95 \beta_{9} - 982 \beta_{8} + \cdots - 1019 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 347\beta_{7} + 1633\beta_{6} + 1605\beta_{5} + 634\beta_{4} + 3658\beta_{3} - 2144\beta_{2} + 2783 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 2614 \beta_{13} + 9040 \beta_{12} - 1952 \beta_{11} + 4916 \beta_{10} - 634 \beta_{9} + 5360 \beta_{8} + \cdots + 5360 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/595\mathbb{Z}\right)^\times\).

\(n\) \(71\) \(171\) \(477\)
\(\chi(n)\) \(1\) \(-1 - \beta_{8}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
86.1
−0.917797 + 1.58967i
−0.783519 + 1.35709i
−0.442527 + 0.766480i
−0.0720309 + 0.124761i
0.649484 − 1.12494i
0.899192 − 1.55745i
1.16720 − 2.02165i
−0.917797 − 1.58967i
−0.783519 − 1.35709i
−0.442527 − 0.766480i
−0.0720309 − 0.124761i
0.649484 + 1.12494i
0.899192 + 1.55745i
1.16720 + 2.02165i
−0.917797 + 1.58967i 1.69447 + 2.93491i −0.684703 − 1.18594i −0.500000 + 0.866025i −6.22073 −0.186095 − 2.63920i −1.15752 −4.24247 + 7.34817i −0.917797 − 1.58967i
86.2 −0.783519 + 1.35709i −0.0392325 − 0.0679526i −0.227804 − 0.394569i −0.500000 + 0.866025i 0.122957 −2.59379 − 0.521803i −2.42012 1.49692 − 2.59274i −0.783519 − 1.35709i
86.3 −0.442527 + 0.766480i −0.298260 − 0.516601i 0.608339 + 1.05367i −0.500000 + 0.866025i 0.527952 1.85483 + 1.88669i −2.84694 1.32208 − 2.28991i −0.442527 − 0.766480i
86.4 −0.0720309 + 0.124761i 1.31614 + 2.27962i 0.989623 + 1.71408i −0.500000 + 0.866025i −0.379211 2.34781 + 1.21975i −0.573257 −1.96445 + 3.40253i −0.0720309 − 0.124761i
86.5 0.649484 − 1.12494i −0.248031 − 0.429602i 0.156342 + 0.270792i −0.500000 + 0.866025i −0.644369 −1.64534 − 2.07192i 3.00410 1.37696 − 2.38497i 0.649484 + 1.12494i
86.6 0.899192 − 1.55745i −0.848163 − 1.46906i −0.617091 − 1.06883i −0.500000 + 0.866025i −3.05065 2.12948 − 1.57013i 1.37723 0.0612376 − 0.106067i 0.899192 + 1.55745i
86.7 1.16720 − 2.02165i 1.42307 + 2.46484i −1.72471 − 2.98728i −0.500000 + 0.866025i 6.64404 −2.40691 + 1.09855i −3.38351 −2.55028 + 4.41721i 1.16720 + 2.02165i
256.1 −0.917797 − 1.58967i 1.69447 − 2.93491i −0.684703 + 1.18594i −0.500000 − 0.866025i −6.22073 −0.186095 + 2.63920i −1.15752 −4.24247 − 7.34817i −0.917797 + 1.58967i
256.2 −0.783519 − 1.35709i −0.0392325 + 0.0679526i −0.227804 + 0.394569i −0.500000 − 0.866025i 0.122957 −2.59379 + 0.521803i −2.42012 1.49692 + 2.59274i −0.783519 + 1.35709i
256.3 −0.442527 − 0.766480i −0.298260 + 0.516601i 0.608339 − 1.05367i −0.500000 − 0.866025i 0.527952 1.85483 − 1.88669i −2.84694 1.32208 + 2.28991i −0.442527 + 0.766480i
256.4 −0.0720309 − 0.124761i 1.31614 − 2.27962i 0.989623 − 1.71408i −0.500000 − 0.866025i −0.379211 2.34781 − 1.21975i −0.573257 −1.96445 − 3.40253i −0.0720309 + 0.124761i
256.5 0.649484 + 1.12494i −0.248031 + 0.429602i 0.156342 − 0.270792i −0.500000 − 0.866025i −0.644369 −1.64534 + 2.07192i 3.00410 1.37696 + 2.38497i 0.649484 − 1.12494i
256.6 0.899192 + 1.55745i −0.848163 + 1.46906i −0.617091 + 1.06883i −0.500000 − 0.866025i −3.05065 2.12948 + 1.57013i 1.37723 0.0612376 + 0.106067i 0.899192 − 1.55745i
256.7 1.16720 + 2.02165i 1.42307 − 2.46484i −1.72471 + 2.98728i −0.500000 − 0.866025i 6.64404 −2.40691 − 1.09855i −3.38351 −2.55028 − 4.41721i 1.16720 − 2.02165i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 86.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 595.2.i.i ✓ 14
7.c even 3 1 inner 595.2.i.i ✓ 14
7.c even 3 1 4165.2.a.bk 7
7.d odd 6 1 4165.2.a.bl 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
595.2.i.i ✓ 14 1.a even 1 1 trivial
595.2.i.i ✓ 14 7.c even 3 1 inner
4165.2.a.bk 7 7.c even 3 1
4165.2.a.bl 7 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(595, [\chi])\):

\( T_{2}^{14} - T_{2}^{13} + 9 T_{2}^{12} - 2 T_{2}^{11} + 49 T_{2}^{10} - 5 T_{2}^{9} + 150 T_{2}^{8} + \cdots + 4 \) Copy content Toggle raw display
\( T_{11}^{14} - 3 T_{11}^{13} + 31 T_{11}^{12} + 449 T_{11}^{10} - 19 T_{11}^{9} + 3276 T_{11}^{8} + \cdots + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - T^{13} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{14} - 6 T^{13} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{7} \) Copy content Toggle raw display
$7$ \( T^{14} + T^{13} + \cdots + 823543 \) Copy content Toggle raw display
$11$ \( T^{14} - 3 T^{13} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( (T^{7} - 6 T^{6} - 33 T^{5} + \cdots + 16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + T + 1)^{7} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 119333776 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 142086400 \) Copy content Toggle raw display
$29$ \( (T^{7} + T^{6} - 36 T^{5} + \cdots - 20)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} - 10 T^{13} + \cdots + 84750436 \) Copy content Toggle raw display
$37$ \( T^{14} + 8 T^{13} + \cdots + 23088025 \) Copy content Toggle raw display
$41$ \( (T^{7} + 12 T^{6} + \cdots - 61884)^{2} \) Copy content Toggle raw display
$43$ \( (T^{7} + 4 T^{6} + \cdots - 6848)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 6682735504 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 1172377600 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 561335103729 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 930372004 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 61408804864 \) Copy content Toggle raw display
$71$ \( (T^{7} + 4 T^{6} + \cdots + 86400)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} - 19 T^{13} + \cdots + 16908544 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( (T^{7} + 34 T^{6} + \cdots + 122008)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 3080361001 \) Copy content Toggle raw display
$97$ \( (T^{7} + 19 T^{6} + \cdots - 13504)^{2} \) Copy content Toggle raw display
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