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Properties

Label 109.2.c.a
Level $109$
Weight $2$
Character orbit 109.c
Analytic conductor $0.870$
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] = [109,2,Mod(45,109)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(109, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("109.45");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 109 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 109.c (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: \(0.870369382032\)
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: \( x^{14} - 2 x^{13} + 11 x^{12} - 10 x^{11} + 61 x^{10} - 52 x^{9} + 198 x^{8} - 81 x^{7} + 339 x^{6} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} + 11 x^{12} - 10 x^{11} + 61 x^{10} - 52 x^{9} + 198 x^{8} - 81 x^{7} + 339 x^{6} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 4731899081 \nu^{13} + 14230905240 \nu^{12} - 55838681512 \nu^{11} + 88553579148 \nu^{10} + \cdots + 8505268300 ) / 84589624237 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4840492823 \nu^{13} + 1509735572 \nu^{12} + 31089164151 \nu^{11} + 74231215789 \nu^{10} + \cdots + 4698960209 ) / 84589624237 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5082962037 \nu^{13} - 9740517412 \nu^{12} + 54491860056 \nu^{11} - 44888873505 \nu^{10} + \cdots - 158725373499 ) / 84589624237 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5190244615 \nu^{13} + 15644119590 \nu^{12} - 67439283162 \nu^{11} + 109196305125 \nu^{10} + \cdots - 5223183487 ) / 84589624237 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5263630360 \nu^{13} - 10346592397 \nu^{12} + 57293858975 \nu^{11} - 49834304681 \nu^{10} + \cdots + 5190244615 ) / 84589624237 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 7243323757 \nu^{13} + 3343686285 \nu^{12} - 59837846539 \nu^{11} - 43265993607 \nu^{10} + \cdots - 7657078318 ) / 84589624237 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 8115423852 \nu^{13} - 23970694743 \nu^{12} + 99172149433 \nu^{11} - 157051355696 \nu^{10} + \cdots - 73489998596 ) / 84589624237 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 9721885954 \nu^{13} + 29072511621 \nu^{12} - 124007319675 \nu^{11} + 198887936264 \nu^{10} + \cdots + 132234061377 ) / 84589624237 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 9955082568 \nu^{13} - 29867516829 \nu^{12} + 128937819459 \nu^{11} - 208224697180 \nu^{10} + \cdots - 153649919463 ) / 84589624237 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 10199820907 \nu^{13} - 30682164195 \nu^{12} + 132076567405 \nu^{11} - 213447179074 \nu^{10} + \cdots - 153469251140 ) / 84589624237 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 21235189763 \nu^{13} - 41992444573 \nu^{12} + 231977434819 \nu^{11} - 204282649900 \nu^{10} + \cdots + 184676596574 ) / 84589624237 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 58131250987 \nu^{13} - 111550192312 \nu^{12} + 625409293841 \nu^{11} - 521942474438 \nu^{10} + \cdots + 45057678826 ) / 84589624237 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{6} + 2\beta_{5} - \beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{12} + 5\beta_{6} - \beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{11} + \beta_{10} - \beta_{9} - 8\beta_{5} - 7\beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{13} + 7 \beta_{12} - 9 \beta_{11} + 7 \beta_{10} - \beta_{7} - 26 \beta_{6} - 3 \beta_{5} + \cdots - 26 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{13} + 10\beta_{12} + 9\beta_{9} + 2\beta_{8} - 2\beta_{7} - 45\beta_{6} + 36\beta_{4} - 9\beta_{3} + 37 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 67\beta_{11} - 45\beta_{10} + 13\beta_{9} + 11\beta_{8} + 30\beta_{5} + 9\beta_{2} + 143\beta _1 + 30 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 13 \beta_{13} - 80 \beta_{12} + 221 \beta_{11} - 80 \beta_{10} + 24 \beta_{7} + 283 \beta_{6} + \cdots + 283 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 65 \beta_{13} - 286 \beta_{12} - 117 \beta_{9} - 89 \beta_{8} + 89 \beta_{7} + 821 \beta_{6} + \cdots - 226 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -1378\beta_{11} + 582\beta_{10} - 440\beta_{9} - 206\beta_{8} - 983\beta_{5} - 117\beta_{2} - 1771\beta _1 - 983 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 440 \beta_{13} + 1818 \beta_{12} - 3116 \beta_{11} + 1818 \beta_{10} - 646 \beta_{7} + \cdots - 4864 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 905 \beta_{13} + 4021 \beta_{12} + 2904 \beta_{9} + 1551 \beta_{8} - 1551 \beta_{7} - 11094 \beta_{6} + \cdots + 5487 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 20475 \beta_{11} - 11577 \beta_{10} + 6477 \beta_{9} + 4455 \beta_{8} + 10202 \beta_{5} + 2904 \beta_{2} + \cdots + 10202 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/109\mathbb{Z}\right)^\times\).

\(n\) \(6\)
\(\chi(n)\) \(-1 - \beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1
1.26204 + 2.18592i
1.04608 + 1.81187i
0.399171 + 0.691384i
0.180227 + 0.312162i
−0.104959 − 0.181794i
−0.787950 − 1.36477i
−0.994614 − 1.72272i
1.26204 − 2.18592i
1.04608 − 1.81187i
0.399171 − 0.691384i
0.180227 − 0.312162i
−0.104959 + 0.181794i
−0.787950 + 1.36477i
−0.994614 + 1.72272i
−2.52409 1.00491 − 1.74055i 4.37102 −0.306817 + 0.531423i −2.53648 + 4.39331i 0.836195 − 1.44833i −5.98465 −0.519683 − 0.900118i 0.774434 − 1.34136i
45.2 −2.09216 −0.555057 + 0.961388i 2.37715 1.29404 − 2.24135i 1.16127 − 2.01138i −2.02108 + 3.50062i −0.789063 0.883822 + 1.53083i −2.70735 + 4.68927i
45.3 −0.798342 −0.0346488 + 0.0600135i −1.36265 1.16095 − 2.01082i 0.0276616 − 0.0479113i 2.11661 − 3.66608i 2.68454 1.49760 + 2.59392i −0.926833 + 1.60532i
45.4 −0.360453 1.62469 − 2.81405i −1.87007 0.262453 − 0.454582i −0.585624 + 1.01433i −0.633883 + 1.09792i 1.39498 −3.77923 − 6.54583i −0.0946021 + 0.163856i
45.5 0.209918 −0.988709 + 1.71249i −1.95593 −0.893177 + 1.54703i −0.207548 + 0.359483i −0.258287 + 0.447367i −0.830420 −0.455093 − 0.788244i −0.187494 + 0.324749i
45.6 1.57590 0.635947 − 1.10149i 0.483462 −0.453226 + 0.785010i 1.00219 − 1.73584i 0.0831704 − 0.144055i −2.38991 0.691143 + 1.19710i −0.714239 + 1.23710i
45.7 1.98923 −1.18713 + 2.05617i 1.95703 1.43578 − 2.48684i −2.36147 + 4.09019i −1.12273 + 1.94462i −0.0854776 −1.31855 − 2.28380i 2.85609 − 4.94689i
63.1 −2.52409 1.00491 + 1.74055i 4.37102 −0.306817 − 0.531423i −2.53648 − 4.39331i 0.836195 + 1.44833i −5.98465 −0.519683 + 0.900118i 0.774434 + 1.34136i
63.2 −2.09216 −0.555057 − 0.961388i 2.37715 1.29404 + 2.24135i 1.16127 + 2.01138i −2.02108 − 3.50062i −0.789063 0.883822 − 1.53083i −2.70735 − 4.68927i
63.3 −0.798342 −0.0346488 − 0.0600135i −1.36265 1.16095 + 2.01082i 0.0276616 + 0.0479113i 2.11661 + 3.66608i 2.68454 1.49760 − 2.59392i −0.926833 − 1.60532i
63.4 −0.360453 1.62469 + 2.81405i −1.87007 0.262453 + 0.454582i −0.585624 − 1.01433i −0.633883 − 1.09792i 1.39498 −3.77923 + 6.54583i −0.0946021 − 0.163856i
63.5 0.209918 −0.988709 − 1.71249i −1.95593 −0.893177 − 1.54703i −0.207548 − 0.359483i −0.258287 − 0.447367i −0.830420 −0.455093 + 0.788244i −0.187494 − 0.324749i
63.6 1.57590 0.635947 + 1.10149i 0.483462 −0.453226 − 0.785010i 1.00219 + 1.73584i 0.0831704 + 0.144055i −2.38991 0.691143 − 1.19710i −0.714239 − 1.23710i
63.7 1.98923 −1.18713 − 2.05617i 1.95703 1.43578 + 2.48684i −2.36147 − 4.09019i −1.12273 − 1.94462i −0.0854776 −1.31855 + 2.28380i 2.85609 + 4.94689i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 45.7
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 109.2.c.a ✓ 14
3.b odd 2 1 981.2.h.d 14
109.c even 3 1 inner 109.2.c.a ✓ 14
327.i odd 6 1 981.2.h.d 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
109.2.c.a ✓ 14 1.a even 1 1 trivial
109.2.c.a ✓ 14 109.c even 3 1 inner
981.2.h.d 14 3.b odd 2 1
981.2.h.d 14 327.i odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(109, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{7} + 2 T^{6} - 7 T^{5} + \cdots - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{14} - T^{13} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( T^{14} - 5 T^{13} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( T^{14} + 2 T^{13} + \cdots + 49 \) Copy content Toggle raw display
$11$ \( T^{14} + 28 T^{12} + \cdots + 674041 \) Copy content Toggle raw display
$13$ \( T^{14} - 7 T^{13} + \cdots + 9 \) Copy content Toggle raw display
$17$ \( (T^{7} - 3 T^{6} + \cdots + 4222)^{2} \) Copy content Toggle raw display
$19$ \( (T^{7} - 7 T^{6} + \cdots - 1132)^{2} \) Copy content Toggle raw display
$23$ \( (T^{7} + 12 T^{6} + \cdots - 136)^{2} \) Copy content Toggle raw display
$29$ \( T^{14} + 5 T^{13} + \cdots + 113569 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 1430881929 \) Copy content Toggle raw display
$37$ \( T^{14} + 117 T^{12} + \cdots + 657721 \) Copy content Toggle raw display
$41$ \( (T^{7} - 5 T^{6} + \cdots - 46022)^{2} \) Copy content Toggle raw display
$43$ \( (T^{7} - 8 T^{6} + \cdots - 118368)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 37058945049 \) Copy content Toggle raw display
$53$ \( T^{14} + 5 T^{13} + \cdots + 45037521 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 26165327049 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 460705635009 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 289238049 \) Copy content Toggle raw display
$71$ \( (T^{7} + 20 T^{6} + \cdots + 8232)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 429194089 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 615188809 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 15256449289 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 56547413209 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 29815964929 \) Copy content Toggle raw display
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