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Properties

Label 69.3.b.a
Level $69$
Weight $3$
Character orbit 69.b
Analytic conductor $1.880$
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] = [69,3,Mod(47,69)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(69, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("69.47");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 69.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: \(1.88011382409\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 40x^{12} + 598x^{10} + 4207x^{8} + 14465x^{6} + 23786x^{4} + 17144x^{2} + 3887 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{8} q^{3} + (\beta_{2} - 2) q^{4} + \beta_{3} q^{5} + (\beta_{4} + 1) q^{6} + \beta_{5} q^{7} + (\beta_{13} + \beta_{12} + \cdots - 2 \beta_1) q^{8}+ \cdots + (\beta_{13} - \beta_{9} + \beta_{7} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{8} q^{3} + (\beta_{2} - 2) q^{4} + \beta_{3} q^{5} + (\beta_{4} + 1) q^{6} + \beta_{5} q^{7} + (\beta_{13} + \beta_{12} + \cdots - 2 \beta_1) q^{8}+ \cdots + ( - \beta_{13} - 4 \beta_{12} + \cdots + 16) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 4 q^{3} - 24 q^{4} + 11 q^{6} - 4 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 4 q^{3} - 24 q^{4} + 11 q^{6} - 4 q^{7} - 8 q^{9} - 8 q^{10} + 19 q^{12} - 14 q^{15} + 72 q^{16} - 31 q^{18} + 8 q^{19} - 2 q^{21} - 84 q^{22} - 44 q^{24} + 38 q^{25} + 62 q^{27} + 76 q^{28} + 62 q^{30} - 144 q^{31} + 90 q^{33} - 68 q^{34} + 3 q^{36} + 48 q^{37} - 78 q^{39} + 120 q^{40} - 76 q^{42} - 48 q^{43} - 18 q^{45} - 317 q^{48} - 30 q^{49} + 18 q^{51} - 6 q^{52} + 312 q^{54} + 232 q^{55} + 76 q^{57} + 66 q^{58} - 36 q^{60} - 140 q^{61} - 206 q^{63} - 346 q^{64} + 398 q^{66} + 204 q^{67} + 80 q^{70} + 384 q^{72} - 224 q^{73} - 80 q^{75} + 100 q^{76} - 341 q^{78} - 344 q^{79} - 232 q^{81} - 62 q^{82} - 330 q^{84} + 480 q^{85} + 86 q^{87} + 436 q^{88} - 514 q^{90} - 172 q^{91} + 62 q^{93} + 514 q^{94} + 609 q^{96} - 24 q^{97} + 234 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 40x^{12} + 598x^{10} + 4207x^{8} + 14465x^{6} + 23786x^{4} + 17144x^{2} + 3887 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -32\nu^{13} - 1839\nu^{11} - 39104\nu^{9} - 384237\nu^{7} - 1758889\nu^{5} - 3225120\nu^{3} - 1623643\nu ) / 88920 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9 \nu^{13} + 40 \nu^{12} + 348 \nu^{11} + 1515 \nu^{10} + 5013 \nu^{9} + 20950 \nu^{8} + 34029 \nu^{7} + \cdots + 81365 ) / 6840 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 409\nu^{12} + 15783\nu^{10} + 221923\nu^{8} + 1396044\nu^{6} + 3838193\nu^{4} + 3932805\nu^{2} + 990011 ) / 13680 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 178 \nu^{13} + 1820 \nu^{12} + 6756 \nu^{11} + 70785 \nu^{10} + 91546 \nu^{9} + 1008800 \nu^{8} + \cdots + 5975125 ) / 88920 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 178 \nu^{13} + 2379 \nu^{12} + 6756 \nu^{11} + 90753 \nu^{10} + 91546 \nu^{9} + 1258413 \nu^{8} + \cdots + 6384261 ) / 88920 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 295 \nu^{13} + 117 \nu^{12} - 11280 \nu^{11} + 4524 \nu^{10} - 156715 \nu^{9} + 65169 \nu^{8} + \cdots + 799578 ) / 88920 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -131\nu^{13} - 5097\nu^{11} - 73385\nu^{9} - 489744\nu^{7} - 1548127\nu^{5} - 2150079\nu^{3} - 913741\nu ) / 35568 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 707 \nu^{13} + 1469 \nu^{12} - 27084 \nu^{11} + 57213 \nu^{10} - 378599 \nu^{9} + 813293 \nu^{8} + \cdots + 4020991 ) / 88920 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2565 \nu^{13} - 169 \nu^{12} + 98895 \nu^{11} - 6123 \nu^{10} + 1389375 \nu^{9} - 76843 \nu^{8} + \cdots + 2452489 ) / 177840 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2752 \nu^{13} - 169 \nu^{12} - 106284 \nu^{11} - 6123 \nu^{10} - 1495624 \nu^{9} + \cdots + 2452489 ) / 177840 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 3407 \nu^{13} + 169 \nu^{12} + 131769 \nu^{11} + 6123 \nu^{10} + 1862549 \nu^{9} + 76843 \nu^{8} + \cdots - 2452489 ) / 177840 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + \beta_{12} + \beta_{9} - 10\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} + \beta_{11} + 2\beta_{10} - \beta_{9} - 3\beta_{8} + \beta_{7} - 2\beta_{5} + \beta_{4} - 15\beta_{2} + 60 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 14 \beta_{13} - 15 \beta_{12} - 3 \beta_{11} + \beta_{10} - 19 \beta_{9} - 3 \beta_{8} + 2 \beta_{7} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 20 \beta_{12} - 20 \beta_{11} - 49 \beta_{10} + 20 \beta_{9} + 76 \beta_{8} - 21 \beta_{7} + \cdots - 712 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 189 \beta_{13} + 210 \beta_{12} + 77 \beta_{11} - 18 \beta_{10} + 304 \beta_{9} + 93 \beta_{8} + \cdots + 31 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 326 \beta_{12} + 326 \beta_{11} + 899 \beta_{10} - 326 \beta_{9} - 1404 \beta_{8} + 355 \beta_{7} + \cdots + 9173 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2622 \beta_{13} - 2963 \beta_{12} - 1457 \beta_{11} + 258 \beta_{10} - 4693 \beta_{9} - 1923 \beta_{8} + \cdots - 641 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 5019 \beta_{12} - 5019 \beta_{11} - 14875 \beta_{10} + 5019 \beta_{9} + 23217 \beta_{8} + \cdots - 124369 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 37291 \beta_{13} + 42419 \beta_{12} + 24622 \beta_{11} - 3505 \beta_{10} + 71541 \beta_{9} + \cdots + 11370 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 75674 \beta_{12} + 75674 \beta_{11} + 234702 \beta_{10} - 75674 \beta_{9} - 365376 \beta_{8} + \cdots + 1744535 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 539654 \beta_{13} - 614842 \beta_{12} - 394216 \beta_{11} + 47320 \beta_{10} - 1083222 \beta_{9} + \cdots - 187382 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/69\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(47\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
− 3.86310i
− 3.07980i
− 2.92693i
− 1.90763i
− 1.29748i
− 1.13976i
− 0.634638i
0.634638i
1.13976i
1.29748i
1.90763i
2.92693i
3.07980i
3.86310i
− 3.86310i −2.26727 + 1.96456i −10.9236 − 1.45241i 7.58928 + 8.75871i −6.75781 26.7464i 1.28104 − 8.90836i −5.61080
47.2 − 3.07980i −0.479940 − 2.96136i −5.48516 0.529218i −9.12040 + 1.47812i 5.42850 4.57400i −8.53932 + 2.84255i 1.62989
47.3 − 2.92693i 2.91955 + 0.690107i −4.56694 − 2.77451i 2.01990 − 8.54532i −1.17249 1.65938i 8.04750 + 4.02960i −8.12081
47.4 − 1.90763i 0.570904 + 2.94518i 0.360941 9.03892i 5.61831 − 1.08907i 1.87935 − 8.31907i −8.34814 + 3.36283i 17.2429
47.5 − 1.29748i −2.20436 + 2.03489i 2.31655 − 5.31444i 2.64023 + 2.86012i 12.0394 − 8.19559i 0.718425 − 8.97128i −6.89538
47.6 − 1.13976i −2.53605 − 1.60264i 2.70094 − 4.29888i −1.82663 + 2.89049i −9.25811 − 7.63748i 3.86309 + 8.12875i −4.89970
47.7 − 0.634638i 1.99717 − 2.23859i 3.59723 4.18173i −1.42070 − 1.26748i −4.15888 − 4.82149i −1.02261 − 8.94172i 2.65389
47.8 0.634638i 1.99717 + 2.23859i 3.59723 − 4.18173i −1.42070 + 1.26748i −4.15888 4.82149i −1.02261 + 8.94172i 2.65389
47.9 1.13976i −2.53605 + 1.60264i 2.70094 4.29888i −1.82663 − 2.89049i −9.25811 7.63748i 3.86309 − 8.12875i −4.89970
47.10 1.29748i −2.20436 − 2.03489i 2.31655 5.31444i 2.64023 − 2.86012i 12.0394 8.19559i 0.718425 + 8.97128i −6.89538
47.11 1.90763i 0.570904 − 2.94518i 0.360941 − 9.03892i 5.61831 + 1.08907i 1.87935 8.31907i −8.34814 − 3.36283i 17.2429
47.12 2.92693i 2.91955 − 0.690107i −4.56694 2.77451i 2.01990 + 8.54532i −1.17249 − 1.65938i 8.04750 − 4.02960i −8.12081
47.13 3.07980i −0.479940 + 2.96136i −5.48516 − 0.529218i −9.12040 − 1.47812i 5.42850 − 4.57400i −8.53932 − 2.84255i 1.62989
47.14 3.86310i −2.26727 − 1.96456i −10.9236 1.45241i 7.58928 − 8.75871i −6.75781 − 26.7464i 1.28104 + 8.90836i −5.61080
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 69.3.b.a ✓ 14
3.b odd 2 1 inner 69.3.b.a ✓ 14
4.b odd 2 1 1104.3.g.b 14
12.b even 2 1 1104.3.g.b 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.3.b.a ✓ 14 1.a even 1 1 trivial
69.3.b.a ✓ 14 3.b odd 2 1 inner
1104.3.g.b 14 4.b odd 2 1
1104.3.g.b 14 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(69, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 40 T^{12} + \cdots + 3887 \) Copy content Toggle raw display
$3$ \( T^{14} + 4 T^{13} + \cdots + 4782969 \) Copy content Toggle raw display
$5$ \( T^{14} + 156 T^{12} + \cdots + 3391488 \) Copy content Toggle raw display
$7$ \( (T^{7} + 2 T^{6} + \cdots - 37472)^{2} \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 44251280863232 \) Copy content Toggle raw display
$13$ \( (T^{7} - 263 T^{5} + \cdots - 839808)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 76\!\cdots\!52 \) Copy content Toggle raw display
$19$ \( (T^{7} - 4 T^{6} + \cdots - 369654688)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 23)^{7} \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 19\!\cdots\!92 \) Copy content Toggle raw display
$31$ \( (T^{7} + 72 T^{6} + \cdots - 1909922976)^{2} \) Copy content Toggle raw display
$37$ \( (T^{7} - 24 T^{6} + \cdots + 31114889632)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 24\!\cdots\!92 \) Copy content Toggle raw display
$43$ \( (T^{7} + 24 T^{6} + \cdots + 8823945600)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 13\!\cdots\!08 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 85\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 32\!\cdots\!12 \) Copy content Toggle raw display
$61$ \( (T^{7} + \cdots + 1148734281696)^{2} \) Copy content Toggle raw display
$67$ \( (T^{7} - 102 T^{6} + \cdots - 173963774816)^{2} \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 41\!\cdots\!68 \) Copy content Toggle raw display
$73$ \( (T^{7} + 112 T^{6} + \cdots + 921220683136)^{2} \) Copy content Toggle raw display
$79$ \( (T^{7} + 172 T^{6} + \cdots + 607693691232)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 16\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{7} + 12 T^{6} + \cdots + 144273643904)^{2} \) Copy content Toggle raw display
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