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Properties

Label 89.4.a.d
Level $89$
Weight $4$
Character orbit 89.a
Self dual yes
Analytic conductor $5.251$
Analytic rank $0$
Dimension $14$
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] = [89,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("89.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 89 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(5.25116999051\)
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: \( x^{14} - 3 x^{13} - 91 x^{12} + 253 x^{11} + 3175 x^{10} - 7585 x^{9} - 54593 x^{8} + 96727 x^{7} + \cdots - 589824 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} + 6) q^{4} + (\beta_{12} + \beta_1) q^{5} + (\beta_{13} - \beta_{12} + \cdots + \beta_1) q^{6}+ \cdots + ( - \beta_{13} + \beta_{12} + \cdots + 12) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} + 6) q^{4} + (\beta_{12} + \beta_1) q^{5} + (\beta_{13} - \beta_{12} + \cdots + \beta_1) q^{6}+ \cdots + ( - 54 \beta_{13} + 14 \beta_{12} + \cdots + 114) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 3 q^{2} + 6 q^{3} + 79 q^{4} + 6 q^{5} + 15 q^{6} + 76 q^{7} + 39 q^{8} + 164 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 3 q^{2} + 6 q^{3} + 79 q^{4} + 6 q^{5} + 15 q^{6} + 76 q^{7} + 39 q^{8} + 164 q^{9} + 99 q^{10} + 98 q^{11} + 111 q^{12} + 48 q^{13} + 76 q^{14} + 144 q^{15} + 495 q^{16} + 82 q^{17} + 220 q^{18} + 362 q^{19} + 143 q^{20} + 260 q^{21} + 362 q^{22} + 162 q^{23} - 569 q^{24} + 372 q^{25} - 874 q^{26} - 486 q^{27} + 228 q^{28} - 48 q^{29} - 1787 q^{30} + 556 q^{31} - 1665 q^{32} - 424 q^{33} - 769 q^{34} - 48 q^{35} - 676 q^{36} + 156 q^{37} - 1873 q^{38} + 738 q^{39} - 921 q^{40} - 92 q^{41} - 2864 q^{42} + 856 q^{43} - 198 q^{44} - 316 q^{45} + 211 q^{46} - 342 q^{47} - 361 q^{48} + 1826 q^{49} - 702 q^{50} + 1492 q^{51} - 930 q^{52} - 314 q^{53} - 1427 q^{54} + 1286 q^{55} - 1340 q^{56} - 434 q^{57} + 788 q^{58} + 732 q^{59} - 2019 q^{60} + 628 q^{61} - 401 q^{62} + 1542 q^{63} + 3015 q^{64} - 284 q^{65} - 1454 q^{66} + 922 q^{67} + 1211 q^{68} + 638 q^{69} - 832 q^{70} + 2788 q^{71} + 1988 q^{72} + 1982 q^{73} + 314 q^{74} - 80 q^{75} + 2807 q^{76} + 1328 q^{77} + 2606 q^{78} + 4636 q^{79} + 1903 q^{80} + 2474 q^{81} + 828 q^{82} + 304 q^{83} + 2232 q^{84} + 3050 q^{85} + 453 q^{86} + 3566 q^{87} + 4314 q^{88} - 1246 q^{89} + 1714 q^{90} + 2048 q^{91} + 91 q^{92} - 3438 q^{93} + 3612 q^{94} + 1776 q^{95} - 3049 q^{96} - 1366 q^{97} - 4553 q^{98} + 1108 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 3 x^{13} - 91 x^{12} + 253 x^{11} + 3175 x^{10} - 7585 x^{9} - 54593 x^{8} + 96727 x^{7} + \cdots - 589824 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1975 \nu^{13} + 5069373 \nu^{12} - 1964107 \nu^{11} - 409252067 \nu^{10} + \cdots + 282060709888 ) / 26686164992 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 78219 \nu^{13} - 1996345 \nu^{12} - 18881145 \nu^{11} + 202934327 \nu^{10} + \cdots + 56432812032 ) / 26686164992 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 253021 \nu^{13} - 6493047 \nu^{12} + 4424817 \nu^{11} + 459676713 \nu^{10} + \cdots - 180029341696 ) / 26686164992 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 449019 \nu^{13} - 3034065 \nu^{12} - 12996889 \nu^{11} + 175460559 \nu^{10} + \cdots - 92043747328 ) / 26686164992 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 775577 \nu^{13} + 5338155 \nu^{12} + 71325811 \nu^{11} - 477745333 \nu^{10} + \cdots + 484006821888 ) / 26686164992 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 286941 \nu^{13} - 1091893 \nu^{12} - 18977895 \nu^{11} + 75763463 \nu^{10} + \cdots + 152344094720 ) / 6671541248 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 959513 \nu^{13} - 13763903 \nu^{12} - 70090319 \nu^{11} + 1119757105 \nu^{10} + \cdots - 968971276288 ) / 13343082496 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2051191 \nu^{13} - 14582685 \nu^{12} - 166560317 \nu^{11} + 1187615955 \nu^{10} + \cdots - 302511013888 ) / 26686164992 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2973135 \nu^{13} + 13581245 \nu^{12} + 254866053 \nu^{11} - 1177262691 \nu^{10} + \cdots + 1415096283136 ) / 26686164992 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 386876 \nu^{13} + 904193 \nu^{12} + 27746019 \nu^{11} - 61746087 \nu^{10} + \cdots - 41085627392 ) / 3335770624 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 2362627 \nu^{13} - 2575725 \nu^{12} - 204458881 \nu^{11} + 236086187 \nu^{10} + \cdots - 437243652096 ) / 13343082496 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{12} + \beta_{11} - \beta_{8} - 2\beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} + 21\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{13} - \beta_{12} - \beta_{11} + 2 \beta_{10} - 2 \beta_{9} + \beta_{8} - 2 \beta_{7} + \cdots + 313 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6 \beta_{13} - 30 \beta_{12} + 44 \beta_{11} - 6 \beta_{10} + 6 \beta_{9} - 28 \beta_{8} - 76 \beta_{7} + \cdots - 38 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 76 \beta_{13} - 26 \beta_{12} - 26 \beta_{11} + 84 \beta_{10} - 68 \beta_{9} + 58 \beta_{8} + \cdots + 7908 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 300 \beta_{13} - 907 \beta_{12} + 1615 \beta_{11} - 324 \beta_{10} + 356 \beta_{9} - 847 \beta_{8} + \cdots - 4727 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2422 \beta_{13} - 451 \beta_{12} - 507 \beta_{11} + 2958 \beta_{10} - 1838 \beta_{9} + 2491 \beta_{8} + \cdots + 211239 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 11506 \beta_{13} - 28424 \beta_{12} + 55242 \beta_{11} - 13562 \beta_{10} + 15354 \beta_{9} + \cdots - 226832 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 75392 \beta_{13} - 2748 \beta_{12} - 11428 \beta_{11} + 99312 \beta_{10} - 47744 \beta_{9} + \cdots + 5832194 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 401880 \beta_{13} - 902165 \beta_{12} + 1812957 \beta_{11} - 514920 \beta_{10} + 578568 \beta_{9} + \cdots - 8796849 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2360426 \beta_{13} + 246459 \beta_{12} - 412821 \beta_{11} + 3275706 \beta_{10} - 1280122 \beta_{9} + \cdots + 164663541 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 13441374 \beta_{13} - 28641202 \beta_{12} + 57942088 \beta_{11} - 18516430 \beta_{10} + 20253390 \beta_{9} + \cdots - 311767450 \) 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
−5.60559
−5.20974
−3.20738
−3.17706
−1.78138
−1.46954
−1.03956
0.226376
1.87657
3.46314
4.28593
4.61283
4.71825
5.30717
−5.60559 5.24773 23.4226 7.86224 −29.4166 32.8011 −86.4527 0.538618 −44.0725
1.2 −5.20974 −3.82753 19.1414 −6.91561 19.9404 −28.6225 −58.0438 −12.3500 36.0285
1.3 −3.20738 5.49046 2.28731 17.7123 −17.6100 −11.6867 18.3228 3.14517 −56.8102
1.4 −3.17706 −0.220560 2.09372 −21.5687 0.700731 24.9202 18.7646 −26.9514 68.5250
1.5 −1.78138 −9.66875 −4.82667 −18.0178 17.2238 −15.8173 22.8492 66.4847 32.0966
1.6 −1.46954 −4.30486 −5.84045 4.42831 6.32617 −10.9898 20.3391 −8.46817 −6.50758
1.7 −1.03956 9.18743 −6.91931 −3.63120 −9.55089 24.5824 15.5095 57.4089 3.77485
1.8 0.226376 −7.07943 −7.94875 2.27369 −1.60261 25.4263 −3.61041 23.1183 0.514707
1.9 1.87657 4.34789 −4.47850 15.2478 8.15910 13.6966 −23.4167 −8.09588 28.6135
1.10 3.46314 8.76574 3.99334 4.00079 30.3570 −18.0947 −13.8756 49.8382 13.8553
1.11 4.28593 0.535184 10.3692 7.51047 2.29376 16.6148 10.1541 −26.7136 32.1893
1.12 4.61283 6.11775 13.2782 −19.6946 28.2201 22.0558 24.3473 10.4269 −90.8479
1.13 4.71825 −9.42949 14.2619 12.6998 −44.4907 23.7446 29.5451 61.9152 59.9207
1.14 5.30717 0.838432 20.1661 4.09251 4.44970 −22.6308 64.5675 −26.2970 21.7197
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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 89.4.a.d ✓ 14
3.b odd 2 1 801.4.a.h 14
4.b odd 2 1 1424.4.a.i 14
5.b even 2 1 2225.4.a.e 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
89.4.a.d ✓ 14 1.a even 1 1 trivial
801.4.a.h 14 3.b odd 2 1
1424.4.a.i 14 4.b odd 2 1
2225.4.a.e 14 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} - 3 T_{2}^{13} - 91 T_{2}^{12} + 253 T_{2}^{11} + 3175 T_{2}^{10} - 7585 T_{2}^{9} + \cdots - 589824 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(89))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - 3 T^{13} + \cdots - 589824 \) Copy content Toggle raw display
$3$ \( T^{14} - 6 T^{13} + \cdots + 64962664 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots - 6417249096656 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 14\!\cdots\!72 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots - 14\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 35\!\cdots\!32 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 11\!\cdots\!92 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 15\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 30\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots - 61\!\cdots\!52 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots - 13\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots - 24\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots - 15\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 44\!\cdots\!68 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 83\!\cdots\!72 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 95\!\cdots\!92 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots - 30\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 66\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 20\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 91\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 62\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 14\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T + 89)^{14} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 73\!\cdots\!48 \) Copy content Toggle raw display
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