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Properties

Label 89.6.a.a
Level $89$
Weight $6$
Character orbit 89.a
Self dual yes
Analytic conductor $14.274$
Analytic rank $1$
Dimension $15$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [89,6,Mod(1,89)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(89, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("89.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 89 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 89.a (trivial)

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: yes
Analytic conductor: \(14.2741599634\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 6 x^{14} - 284 x^{13} + 1504 x^{12} + 31190 x^{11} - 139900 x^{10} - 1690580 x^{9} + \cdots - 7150464000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + ( - \beta_{6} - 1) q^{3} + (\beta_{2} - \beta_1 + 9) q^{4} + (\beta_{9} + \beta_{6} - 2 \beta_1 - 1) q^{5} + ( - \beta_{11} - \beta_{9} + 2 \beta_{6} + \cdots - 10) q^{6}+ \cdots + (2 \beta_{14} + \beta_{12} + \beta_{11} + \cdots + 55) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + ( - \beta_{6} - 1) q^{3} + (\beta_{2} - \beta_1 + 9) q^{4} + (\beta_{9} + \beta_{6} - 2 \beta_1 - 1) q^{5} + ( - \beta_{11} - \beta_{9} + 2 \beta_{6} + \cdots - 10) q^{6}+ \cdots + ( - 274 \beta_{14} + 140 \beta_{13} + \cdots - 46290) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 9 q^{2} - 9 q^{3} + 127 q^{4} - 39 q^{5} - 177 q^{6} - 540 q^{7} - 417 q^{8} + 758 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - 9 q^{2} - 9 q^{3} + 127 q^{4} - 39 q^{5} - 177 q^{6} - 540 q^{7} - 417 q^{8} + 758 q^{9} - 1165 q^{10} - 1450 q^{11} - 681 q^{12} - 778 q^{13} - 452 q^{14} - 2859 q^{15} + 87 q^{16} - 2135 q^{17} - 4544 q^{18} - 6085 q^{19} - 2113 q^{20} - 6532 q^{21} - 5358 q^{22} - 9461 q^{23} - 24297 q^{24} - 5000 q^{25} - 19682 q^{26} - 19755 q^{27} - 30828 q^{28} - 6128 q^{29} - 28211 q^{30} - 26917 q^{31} - 20073 q^{32} - 19874 q^{33} + 5263 q^{34} - 22700 q^{35} + 268 q^{36} - 8222 q^{37} + 25367 q^{38} - 30978 q^{39} - 7897 q^{40} - 4168 q^{41} + 51208 q^{42} - 20479 q^{43} - 23302 q^{44} - 2882 q^{45} - 13149 q^{46} + 5444 q^{47} + 72335 q^{48} - 6241 q^{49} + 52166 q^{50} - 37203 q^{51} + 55150 q^{52} + 35139 q^{53} + 103581 q^{54} - 26990 q^{55} + 151932 q^{56} + 43959 q^{57} + 7340 q^{58} - 30468 q^{59} + 252789 q^{60} + 39346 q^{61} + 200615 q^{62} + 2880 q^{63} + 19375 q^{64} + 108370 q^{65} + 279378 q^{66} - 41164 q^{67} + 12395 q^{68} + 31423 q^{69} + 185064 q^{70} - 48730 q^{71} + 101004 q^{72} - 110811 q^{73} + 116618 q^{74} + 146476 q^{75} + 3551 q^{76} + 147336 q^{77} + 240254 q^{78} - 292854 q^{79} + 310559 q^{80} + 45527 q^{81} + 258164 q^{82} + 107492 q^{83} + 19208 q^{84} - 253477 q^{85} + 263941 q^{86} + 142368 q^{87} + 175338 q^{88} - 118815 q^{89} + 69810 q^{90} - 302856 q^{91} + 333107 q^{92} - 421937 q^{93} - 300900 q^{94} - 366143 q^{95} + 26351 q^{96} - 378853 q^{97} - 83261 q^{98} - 690828 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 6 x^{14} - 284 x^{13} + 1504 x^{12} + 31190 x^{11} - 139900 x^{10} - 1690580 x^{9} + \cdots - 7150464000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 30\!\cdots\!83 \nu^{14} + \cdots + 12\!\cdots\!00 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 46\!\cdots\!51 \nu^{14} + \cdots - 15\!\cdots\!80 ) / 86\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 21\!\cdots\!83 \nu^{14} + \cdots + 91\!\cdots\!00 ) / 86\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 21\!\cdots\!03 \nu^{14} + \cdots + 94\!\cdots\!00 ) / 43\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13\!\cdots\!01 \nu^{14} + \cdots - 57\!\cdots\!00 ) / 86\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 14\!\cdots\!21 \nu^{14} + \cdots - 46\!\cdots\!00 ) / 86\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 30\!\cdots\!87 \nu^{14} + \cdots + 10\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 84\!\cdots\!89 \nu^{14} + \cdots + 22\!\cdots\!00 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 37\!\cdots\!65 \nu^{14} + \cdots - 11\!\cdots\!80 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 31\!\cdots\!07 \nu^{14} + \cdots + 92\!\cdots\!00 ) / 86\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 87\!\cdots\!19 \nu^{14} + \cdots - 30\!\cdots\!00 ) / 17\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 55\!\cdots\!21 \nu^{14} + \cdots - 16\!\cdots\!00 ) / 86\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 40 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{14} + 2\beta_{12} + 2\beta_{11} - \beta_{10} - 2\beta_{9} + 4\beta_{6} + 69\beta _1 + 28 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{14} + 2 \beta_{13} + 5 \beta_{12} + 6 \beta_{11} - \beta_{10} + 5 \beta_{9} - 3 \beta_{8} + \cdots + 2811 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 87 \beta_{14} - 24 \beta_{13} + 248 \beta_{12} + 246 \beta_{11} - 89 \beta_{10} - 240 \beta_{9} + \cdots + 3966 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 268 \beta_{14} + 186 \beta_{13} + 907 \beta_{12} + 1146 \beta_{11} - 9 \beta_{10} + 567 \beta_{9} + \cdots + 234269 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 6881 \beta_{14} - 3552 \beta_{13} + 25510 \beta_{12} + 26338 \beta_{11} - 6453 \beta_{10} + \cdots + 522656 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 28902 \beta_{14} + 13834 \beta_{13} + 121989 \beta_{12} + 157654 \beta_{11} + 10931 \beta_{10} + \cdots + 21018683 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 530139 \beta_{14} - 387992 \beta_{13} + 2523012 \beta_{12} + 2749342 \beta_{11} - 384481 \beta_{10} + \cdots + 64836962 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2913216 \beta_{14} + 935946 \beta_{13} + 14772671 \beta_{12} + 19327010 \beta_{11} + 2301247 \beta_{10} + \cdots + 1960794025 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 39739925 \beta_{14} - 38011936 \beta_{13} + 248845234 \beta_{12} + 286678090 \beta_{11} + \cdots + 7672124484 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 289198746 \beta_{14} + 57938522 \beta_{13} + 1699600489 \beta_{12} + 2247742462 \beta_{11} + \cdots + 187346425655 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 2845501007 \beta_{14} - 3548640632 \beta_{13} + 24745596592 \beta_{12} + 30030466150 \beta_{11} + \cdots + 878113484102 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 28937127284 \beta_{14} + 3090383738 \beta_{13} + 189827909443 \beta_{12} + 254155069418 \beta_{11} + \cdots + 18199338678821 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−9.42087
−8.81719
−6.93501
−4.65107
−4.13202
−2.20570
−2.11821
−0.669116
2.20628
3.80216
5.66348
5.94673
7.28939
9.71861
10.3225
−10.4209 20.4324 76.5946 61.3741 −212.923 −205.618 −464.715 174.481 −639.571
1.2 −9.81719 4.58474 64.3772 −14.0393 −45.0093 −33.0553 −317.854 −221.980 137.826
1.3 −7.93501 −26.0486 30.9644 −41.3475 206.696 −97.9153 8.21743 435.527 328.093
1.4 −5.65107 16.6685 −0.0653705 5.33349 −94.1949 −36.8345 181.204 34.8385 −30.1399
1.5 −5.13202 −18.1814 −5.66236 −30.2685 93.3071 169.544 193.284 87.5618 155.338
1.6 −3.20570 −5.45865 −21.7235 78.0732 17.4988 −102.652 172.221 −213.203 −250.279
1.7 −3.11821 12.6857 −22.2768 −33.3473 −39.5567 176.730 169.246 −82.0723 103.984
1.8 −1.66912 −28.6129 −29.2141 72.3310 47.7582 65.5654 102.173 575.697 −120.729
1.9 1.20628 15.3455 −30.5449 21.6112 18.5109 −140.172 −75.4464 −7.51687 26.0691
1.10 2.80216 26.2279 −24.1479 −103.047 73.4949 −7.69926 −157.336 444.904 −288.756
1.11 4.66348 −9.83909 −10.2520 33.0124 −45.8844 130.744 −197.041 −146.192 153.953
1.12 4.94673 5.71669 −7.52983 32.6924 28.2790 −242.429 −195.544 −210.319 161.721
1.13 6.28939 5.91733 7.55640 −60.4767 37.2164 −13.1679 −153.735 −207.985 −380.361
1.14 8.71861 −4.84108 44.0141 −75.5133 −42.2075 −87.8392 104.746 −219.564 −658.371
1.15 9.32255 −23.5971 54.9099 14.6123 −219.985 −115.201 213.578 313.823 136.224
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(89\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 89.6.a.a ✓ 15
3.b odd 2 1 801.6.a.a 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
89.6.a.a ✓ 15 1.a even 1 1 trivial
801.6.a.a 15 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{15} + 9 T_{2}^{14} - 263 T_{2}^{13} - 2279 T_{2}^{12} + 26267 T_{2}^{11} + 218227 T_{2}^{10} + \cdots - 15657808128 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(89))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} + \cdots - 15657808128 \) Copy content Toggle raw display
$3$ \( T^{15} + \cdots + 22\!\cdots\!72 \) Copy content Toggle raw display
$5$ \( T^{15} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$7$ \( T^{15} + \cdots + 22\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( T^{15} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{15} + \cdots + 80\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots - 38\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots - 97\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots - 48\!\cdots\!72 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 14\!\cdots\!32 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots + 31\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 39\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots - 75\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 63\!\cdots\!72 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 19\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 20\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots + 64\!\cdots\!60 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 52\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 18\!\cdots\!40 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 21\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T + 7921)^{15} \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots + 49\!\cdots\!32 \) Copy content Toggle raw display
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