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Properties

Label 207.2.e.a
Level $207$
Weight $2$
Character orbit 207.e
Analytic conductor $1.653$
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] = [207,2,Mod(70,207)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(207, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("207.70");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 207.e (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: \(1.65290332184\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + 11 x^{14} - 4 x^{13} + 77 x^{12} - 23 x^{11} + 282 x^{10} - 20 x^{9} + 714 x^{8} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} + 11 x^{14} - 4 x^{13} + 77 x^{12} - 23 x^{11} + 282 x^{10} - 20 x^{9} + 714 x^{8} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 31947983255046 \nu^{15} + 42919659097387 \nu^{14} + 258062907341319 \nu^{13} + \cdots + 728003956668958 ) / 16\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 32757646990073 \nu^{15} + 13127424455656 \nu^{14} - 306857950311650 \nu^{13} + \cdots - 58735636998065 ) / 16\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5200501304412 \nu^{15} + 4452646939339 \nu^{14} - 56235121512033 \nu^{13} + \cdots - 17021059454526 ) / 183783253466319 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 73079656886606 \nu^{15} - 113026380971527 \nu^{14} + 865876960331447 \nu^{13} + \cdots - 873177360246325 ) / 16\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 78395034498225 \nu^{15} + 43997326853966 \nu^{14} - 828322771548213 \nu^{13} + \cdots - 72323184817453 ) / 16\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 17021059454526 \nu^{15} + 22221560758938 \nu^{14} - 191684300939125 \nu^{13} + \cdots + 146231304071929 ) / 183783253466319 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 188019210094466 \nu^{15} - 191728399443810 \nu^{14} + \cdots - 10\!\cdots\!30 ) / 16\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 71800346416648 \nu^{15} - 36992778742313 \nu^{14} + 757895490967495 \nu^{13} + \cdots + 303543958771561 ) / 551349760398957 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 294055553321939 \nu^{15} - 225920511853787 \nu^{14} + \cdots + 606759053012914 ) / 16\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 25470814281398 \nu^{15} + 19482094978601 \nu^{14} - 273755182787744 \nu^{13} + \cdots - 212296429968886 ) / 127234560092067 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 414034473240770 \nu^{15} - 371664445216354 \nu^{14} + \cdots + 23\!\cdots\!44 ) / 16\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 50315323998505 \nu^{15} + 65694289440315 \nu^{14} - 566826900911792 \nu^{13} + \cdots + 433493410911375 ) / 183783253466319 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 569304536944556 \nu^{15} - 574879394129945 \nu^{14} + \cdots + 581632567333651 ) / 16\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 576035226230213 \nu^{15} - 583612929658436 \nu^{14} + \cdots + 54\!\cdots\!56 ) / 16\!\cdots\!71 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} - \beta_{14} - \beta_{13} + 3\beta_{7} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{14} + \beta_{12} - 4\beta_{4} - \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{13} + 2\beta_{8} - 14\beta_{7} + \beta_{6} - 2\beta_{5} - \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{15} + 2 \beta_{14} + 2 \beta_{13} - 8 \beta_{12} - 9 \beta_{11} - 2 \beta_{10} + 9 \beta_{8} + \cdots + 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 27 \beta_{15} + 38 \beta_{14} - 20 \beta_{12} - 17 \beta_{11} - \beta_{10} - 7 \beta_{9} - 9 \beta_{8} + \cdots + 75 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 43 \beta_{14} - 24 \beta_{13} + 43 \beta_{11} - 66 \beta_{8} + 93 \beta_{7} + 56 \beta_{5} + \cdots + 101 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 157 \beta_{15} - 220 \beta_{14} - 157 \beta_{13} + 153 \beta_{12} + 146 \beta_{11} + 16 \beta_{10} + \cdots - 439 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 210 \beta_{15} - 513 \beta_{14} + 386 \beta_{12} + 196 \beta_{11} + 149 \beta_{10} + 4 \beta_{9} + \cdots - 714 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 257 \beta_{14} + 962 \beta_{13} - 257 \beta_{11} + 1052 \beta_{8} - 2718 \beta_{7} + 221 \beta_{6} + \cdots - 775 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1633 \beta_{15} + 2055 \beta_{14} + 1633 \beta_{13} - 2657 \beta_{12} - 3093 \beta_{11} - 1010 \beta_{10} + \cdots + 5224 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 6103 \beta_{15} + 10829 \beta_{14} - 7397 \beta_{12} - 5259 \beta_{11} - 1418 \beta_{10} + \cdots + 17423 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 9944 \beta_{14} - 12005 \beta_{13} + 9944 \beta_{11} - 20883 \beta_{8} + 37199 \beta_{7} + \cdots + 22108 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 39614 \beta_{15} - 56194 \beta_{14} - 39614 \beta_{13} + 50229 \beta_{12} + 51155 \beta_{11} + \cdots - 114102 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 85634 \beta_{15} - 176403 \beta_{14} + 125344 \beta_{12} + 77641 \beta_{11} + 43338 \beta_{10} + \cdots - 260613 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/207\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(47\)
\(\chi(n)\) \(1\) \(-1 + \beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
70.1
1.30401 − 2.25862i
0.886875 − 1.53611i
0.735672 − 1.27422i
0.124128 − 0.214996i
−0.0750467 + 0.129985i
−0.521903 + 0.903963i
−0.877333 + 1.51959i
−1.07641 + 1.86439i
1.30401 + 2.25862i
0.886875 + 1.53611i
0.735672 + 1.27422i
0.124128 + 0.214996i
−0.0750467 − 0.129985i
−0.521903 − 0.903963i
−0.877333 − 1.51959i
−1.07641 − 1.86439i
−1.30401 + 2.25862i 1.44644 − 0.952792i −2.40091 − 4.15849i 1.60524 + 2.78036i 0.265817 + 4.50941i 1.48649 − 2.57468i 7.30721 1.18437 − 2.75631i −8.37303
70.2 −0.886875 + 1.53611i −1.71504 − 0.242159i −0.573096 − 0.992632i −0.609417 − 1.05554i 1.89301 − 2.41973i 1.67541 − 2.90190i −1.51444 2.88272 + 0.830624i 2.16191
70.3 −0.735672 + 1.27422i −0.166413 + 1.72404i −0.0824255 − 0.142765i −0.273800 − 0.474236i −2.07438 − 1.48037i −1.37902 + 2.38853i −2.70013 −2.94461 − 0.573803i 0.805708
70.4 −0.124128 + 0.214996i −1.13764 + 1.30605i 0.969185 + 1.67868i 0.711034 + 1.23155i −0.139583 − 0.406705i 0.504599 − 0.873991i −0.977722 −0.411544 − 2.97164i −0.353036
70.5 0.0750467 − 0.129985i 1.72681 + 0.134662i 0.988736 + 1.71254i −0.948287 − 1.64248i 0.147095 − 0.214353i −0.390608 + 0.676553i 0.596992 2.96373 + 0.465072i −0.284663
70.6 0.521903 − 0.903963i −1.59379 − 0.678113i 0.455234 + 0.788488i 1.22861 + 2.12802i −1.44479 + 1.08682i 1.08399 − 1.87752i 3.03797 2.08032 + 2.16154i 2.56487
70.7 0.877333 − 1.51959i −0.212476 − 1.71897i −0.539426 − 0.934314i −1.07428 − 1.86071i −2.79853 − 1.18523i −0.638951 + 1.10670i 1.61631 −2.90971 + 0.730480i −3.77000
70.8 1.07641 − 1.86439i 1.15211 + 1.29331i −1.31730 − 2.28163i −0.639106 − 1.10696i 3.65137 − 0.755860i 1.15808 − 2.00586i −1.36617 −0.345283 + 2.98006i −2.75175
139.1 −1.30401 − 2.25862i 1.44644 + 0.952792i −2.40091 + 4.15849i 1.60524 − 2.78036i 0.265817 − 4.50941i 1.48649 + 2.57468i 7.30721 1.18437 + 2.75631i −8.37303
139.2 −0.886875 − 1.53611i −1.71504 + 0.242159i −0.573096 + 0.992632i −0.609417 + 1.05554i 1.89301 + 2.41973i 1.67541 + 2.90190i −1.51444 2.88272 − 0.830624i 2.16191
139.3 −0.735672 − 1.27422i −0.166413 − 1.72404i −0.0824255 + 0.142765i −0.273800 + 0.474236i −2.07438 + 1.48037i −1.37902 − 2.38853i −2.70013 −2.94461 + 0.573803i 0.805708
139.4 −0.124128 − 0.214996i −1.13764 − 1.30605i 0.969185 − 1.67868i 0.711034 − 1.23155i −0.139583 + 0.406705i 0.504599 + 0.873991i −0.977722 −0.411544 + 2.97164i −0.353036
139.5 0.0750467 + 0.129985i 1.72681 − 0.134662i 0.988736 − 1.71254i −0.948287 + 1.64248i 0.147095 + 0.214353i −0.390608 − 0.676553i 0.596992 2.96373 − 0.465072i −0.284663
139.6 0.521903 + 0.903963i −1.59379 + 0.678113i 0.455234 − 0.788488i 1.22861 − 2.12802i −1.44479 − 1.08682i 1.08399 + 1.87752i 3.03797 2.08032 − 2.16154i 2.56487
139.7 0.877333 + 1.51959i −0.212476 + 1.71897i −0.539426 + 0.934314i −1.07428 + 1.86071i −2.79853 + 1.18523i −0.638951 − 1.10670i 1.61631 −2.90971 − 0.730480i −3.77000
139.8 1.07641 + 1.86439i 1.15211 − 1.29331i −1.31730 + 2.28163i −0.639106 + 1.10696i 3.65137 + 0.755860i 1.15808 + 2.00586i −1.36617 −0.345283 − 2.98006i −2.75175
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 70.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 207.2.e.a ✓ 16
3.b odd 2 1 621.2.e.a 16
9.c even 3 1 inner 207.2.e.a ✓ 16
9.c even 3 1 1863.2.a.f 8
9.d odd 6 1 621.2.e.a 16
9.d odd 6 1 1863.2.a.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
207.2.e.a ✓ 16 1.a even 1 1 trivial
207.2.e.a ✓ 16 9.c even 3 1 inner
621.2.e.a 16 3.b odd 2 1
621.2.e.a 16 9.d odd 6 1
1863.2.a.e 8 9.d odd 6 1
1863.2.a.f 8 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + T_{2}^{15} + 11 T_{2}^{14} + 4 T_{2}^{13} + 77 T_{2}^{12} + 23 T_{2}^{11} + 282 T_{2}^{10} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(207, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + T^{15} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{16} + T^{15} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{16} + 15 T^{14} + \cdots + 1521 \) Copy content Toggle raw display
$7$ \( T^{16} - 7 T^{15} + \cdots + 19321 \) Copy content Toggle raw display
$11$ \( T^{16} - 3 T^{15} + \cdots + 169 \) Copy content Toggle raw display
$13$ \( T^{16} - 15 T^{15} + \cdots + 729 \) Copy content Toggle raw display
$17$ \( (T^{8} + 6 T^{7} - 38 T^{6} + \cdots - 31)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 12 T^{7} + \cdots + 8147)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - T + 1)^{8} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 97626877209 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 491957751609 \) Copy content Toggle raw display
$37$ \( (T^{8} + 19 T^{7} + \cdots + 3736)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 416403055849 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 54554010624 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 5777672121 \) Copy content Toggle raw display
$53$ \( (T^{8} - 4 T^{7} + \cdots + 2141409)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 84145826241 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 201230296569 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 772784401 \) Copy content Toggle raw display
$71$ \( (T^{8} + 5 T^{7} + \cdots + 26539)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 48 T^{7} + \cdots - 2870281)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 18912765952161 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 19423439424 \) Copy content Toggle raw display
$89$ \( (T^{8} - 11 T^{7} + \cdots - 2989879)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 185553898381489 \) Copy content Toggle raw display
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