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Properties

Label 345.2.c.a
Level $345$
Weight $2$
Character orbit 345.c
Analytic conductor $2.755$
Analytic rank $0$
Dimension $16$
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] = [345,2,Mod(206,345)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(345, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("345.206");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 345 = 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 345.c (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: \(2.75483886973\)
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: \( x^{16} + 23x^{14} + 212x^{12} + 999x^{10} + 2537x^{8} + 3336x^{6} + 1976x^{4} + 416x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 23x^{14} + 212x^{12} + 999x^{10} + 2537x^{8} + 3336x^{6} + 1976x^{4} + 416x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{14} - 18\nu^{12} - 113\nu^{10} - 263\nu^{8} - 16\nu^{6} + 551\nu^{4} + 210\nu^{2} - 26 ) / 18 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{15} - 2 \nu^{14} - 21 \nu^{13} - 42 \nu^{12} - 170 \nu^{11} - 340 \nu^{10} - 659 \nu^{9} + \cdots - 4 ) / 36 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{15} + \nu^{14} - 21 \nu^{13} + 21 \nu^{12} - 170 \nu^{11} + 170 \nu^{10} - 659 \nu^{9} + \cdots - 52 ) / 36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{14} - 27 \nu^{12} - 284 \nu^{10} - 1469 \nu^{8} + 18 \nu^{7} - 3823 \nu^{6} + 180 \nu^{5} + \cdots + 100 ) / 36 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2\nu^{14} + 45\nu^{12} + 397\nu^{10} + 1732\nu^{8} + 3857\nu^{6} + 4019\nu^{4} + 1470\nu^{2} + 52 ) / 18 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - \nu^{15} - 23 \nu^{13} + 2 \nu^{12} - 208 \nu^{11} + 38 \nu^{10} - 929 \nu^{9} + 270 \nu^{8} + \cdots + 20 ) / 12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - \nu^{15} + 4 \nu^{14} - 21 \nu^{13} + 84 \nu^{12} - 170 \nu^{11} + 689 \nu^{10} - 659 \nu^{9} + \cdots + 8 ) / 18 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 7 \nu^{15} - \nu^{14} - 153 \nu^{13} - 27 \nu^{12} - 1322 \nu^{11} - 284 \nu^{10} - 5729 \nu^{9} + \cdots - 8 ) / 36 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 6 \nu^{15} + \nu^{14} + 132 \nu^{13} + 21 \nu^{12} + 1152 \nu^{11} + 170 \nu^{10} + 5088 \nu^{9} + \cdots - 52 ) / 36 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -5\nu^{15} - 117\nu^{13} - 1096\nu^{11} - 5221\nu^{9} - 13211\nu^{7} - 16676\nu^{5} - 8490\nu^{3} - 1156\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 5\nu^{15} + 117\nu^{13} + 1096\nu^{11} + 5221\nu^{9} + 13211\nu^{7} + 16712\nu^{5} + 8778\nu^{3} + 1624\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( \nu^{15} + 22 \nu^{13} - \nu^{12} + 192 \nu^{11} - 19 \nu^{10} + 845 \nu^{9} - 135 \nu^{8} + 1965 \nu^{7} + \cdots - 10 ) / 6 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} + \beta_{11} + \beta_{8} - \beta_{6} + \beta_{4} - 5\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{14} + \beta_{13} - 8\beta_{3} + 27\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -10\beta_{15} - 10\beta_{11} - 9\beta_{8} + \beta_{7} + 9\beta_{6} + \beta_{5} - 10\beta_{4} + 25\beta_{2} - 72 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -10\beta_{14} - 10\beta_{13} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + 53\beta_{3} - 151\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 74 \beta_{15} + \beta_{14} + 75 \beta_{11} + \beta_{9} + 63 \beta_{8} - 11 \beta_{7} - 66 \beta_{6} + \cdots + 390 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - \beta_{15} + 74 \beta_{14} + 74 \beta_{13} + 2 \beta_{12} + \beta_{11} - 14 \beta_{7} + \cdots + 861 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 492 \beta_{15} - 14 \beta_{14} + \beta_{13} - 507 \beta_{11} + 2 \beta_{10} - 15 \beta_{9} + \cdots - 2184 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 19 \beta_{15} - 491 \beta_{14} - 492 \beta_{13} - 34 \beta_{12} - 16 \beta_{11} + 3 \beta_{9} + \cdots - 4966 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 3113 \beta_{15} + 132 \beta_{14} - 19 \beta_{13} + 3264 \beta_{11} - 38 \beta_{10} + 151 \beta_{9} + \cdots + 12500 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 229 \beta_{15} + 3098 \beta_{14} + 3113 \beta_{13} + 376 \beta_{12} + 170 \beta_{11} + \cdots + 28844 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 19189 \beta_{15} - 1057 \beta_{14} + 229 \beta_{13} - 20475 \beta_{11} + 458 \beta_{10} + \cdots - 72568 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 2238 \beta_{15} - 19050 \beta_{14} - 19189 \beta_{13} - 3434 \beta_{12} - 1515 \beta_{11} + \cdots - 168278 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/345\mathbb{Z}\right)^\times\).

\(n\) \(116\) \(166\) \(277\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
206.1
− 2.46267i
− 2.40520i
− 1.99340i
− 1.82617i
− 1.63417i
− 0.844960i
− 0.668713i
− 0.100453i
0.100453i
0.668713i
0.844960i
1.63417i
1.82617i
1.99340i
2.40520i
2.46267i
− 2.46267i −0.901990 + 1.47865i −4.06474 −1.00000 3.64143 + 2.22130i 2.88688i 5.08476i −1.37283 − 2.66746i 2.46267i
206.2 − 2.40520i −0.278220 − 1.70956i −3.78500 −1.00000 −4.11184 + 0.669175i 3.68408i 4.29329i −2.84519 + 0.951266i 2.40520i
206.3 − 1.99340i 1.49280 − 0.878375i −1.97365 −1.00000 −1.75095 − 2.97575i − 0.911383i − 0.0525236i 1.45691 − 2.62248i 1.99340i
206.4 − 1.82617i 0.863972 + 1.50118i −1.33488 −1.00000 2.74141 − 1.57776i − 4.90360i − 1.21462i −1.50710 + 2.59396i 1.82617i
206.5 − 1.63417i −1.69326 + 0.364501i −0.670506 −1.00000 0.595656 + 2.76708i − 0.579477i − 2.17262i 2.73428 − 1.23439i 1.63417i
206.6 − 0.844960i 1.57672 + 0.716899i 1.28604 −1.00000 0.605751 − 1.33227i 2.76231i − 2.77658i 1.97211 + 2.26070i 0.844960i
206.7 − 0.668713i 0.397101 − 1.68592i 1.55282 −1.00000 −1.12739 − 0.265547i − 1.46029i − 2.37582i −2.68462 − 1.33896i 0.668713i
206.8 − 0.100453i −1.45713 − 0.936367i 1.98991 −1.00000 −0.0940612 + 0.146373i 3.88831i − 0.400800i 1.24643 + 2.72881i 0.100453i
206.9 0.100453i −1.45713 + 0.936367i 1.98991 −1.00000 −0.0940612 − 0.146373i − 3.88831i 0.400800i 1.24643 − 2.72881i − 0.100453i
206.10 0.668713i 0.397101 + 1.68592i 1.55282 −1.00000 −1.12739 + 0.265547i 1.46029i 2.37582i −2.68462 + 1.33896i − 0.668713i
206.11 0.844960i 1.57672 − 0.716899i 1.28604 −1.00000 0.605751 + 1.33227i − 2.76231i 2.77658i 1.97211 − 2.26070i − 0.844960i
206.12 1.63417i −1.69326 − 0.364501i −0.670506 −1.00000 0.595656 − 2.76708i 0.579477i 2.17262i 2.73428 + 1.23439i − 1.63417i
206.13 1.82617i 0.863972 − 1.50118i −1.33488 −1.00000 2.74141 + 1.57776i 4.90360i 1.21462i −1.50710 − 2.59396i − 1.82617i
206.14 1.99340i 1.49280 + 0.878375i −1.97365 −1.00000 −1.75095 + 2.97575i 0.911383i 0.0525236i 1.45691 + 2.62248i − 1.99340i
206.15 2.40520i −0.278220 + 1.70956i −3.78500 −1.00000 −4.11184 − 0.669175i − 3.68408i − 4.29329i −2.84519 − 0.951266i − 2.40520i
206.16 2.46267i −0.901990 − 1.47865i −4.06474 −1.00000 3.64143 − 2.22130i − 2.88688i − 5.08476i −1.37283 + 2.66746i − 2.46267i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 206.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
69.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 345.2.c.a ✓ 16
3.b odd 2 1 345.2.c.b yes 16
23.b odd 2 1 345.2.c.b yes 16
69.c even 2 1 inner 345.2.c.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
345.2.c.a ✓ 16 1.a even 1 1 trivial
345.2.c.a ✓ 16 69.c even 2 1 inner
345.2.c.b yes 16 3.b odd 2 1
345.2.c.b yes 16 23.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{8} - 4T_{11}^{7} - 44T_{11}^{6} + 190T_{11}^{5} + 446T_{11}^{4} - 2348T_{11}^{3} - 216T_{11}^{2} + 6912T_{11} - 3456 \) acting on \(S_{2}^{\mathrm{new}}(345, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 23 T^{14} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{16} + T^{14} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( (T + 1)^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 72 T^{14} + \cdots + 186624 \) Copy content Toggle raw display
$11$ \( (T^{8} - 4 T^{7} + \cdots - 3456)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 31 T^{6} + \cdots + 72)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 4 T^{7} + \cdots + 3888)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 812934144 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 78310985281 \) Copy content Toggle raw display
$29$ \( T^{16} + 194 T^{14} + \cdots + 473344 \) Copy content Toggle raw display
$31$ \( (T^{8} + 4 T^{7} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 5612577951744 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 16236875776 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 256343740416 \) Copy content Toggle raw display
$47$ \( T^{16} + 306 T^{14} + \cdots + 24522304 \) Copy content Toggle raw display
$53$ \( (T^{8} + 6 T^{7} + \cdots + 15116544)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 2148780720384 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 1573840502784 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 69753498644736 \) Copy content Toggle raw display
$71$ \( T^{16} + 406 T^{14} + \cdots + 6130576 \) Copy content Toggle raw display
$73$ \( (T^{8} - 6 T^{7} + \cdots + 922072)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 50145432846336 \) Copy content Toggle raw display
$83$ \( (T^{8} - 12 T^{7} + \cdots - 3098304)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 22 T^{7} + \cdots + 186624)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 2797938671616 \) Copy content Toggle raw display
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