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Properties

Label 1334.4.a.d
Level $1334$
Weight $4$
Character orbit 1334.a
Self dual yes
Analytic conductor $78.709$
Analytic rank $1$
Dimension $19$
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] = [1334,4,Mod(1,1334)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1334, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1334.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1334 = 2 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1334.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: \(78.7085479477\)
Analytic rank: \(1\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{19} - 5 x^{18} - 327 x^{17} + 1564 x^{16} + 43869 x^{15} - 203270 x^{14} - 3103297 x^{13} + \cdots - 37580243456 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{18}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} - \beta_1 q^{3} + 4 q^{4} - \beta_{6} q^{5} + 2 \beta_1 q^{6} + ( - \beta_{3} - 1) q^{7} - 8 q^{8} + (\beta_{2} + 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} - \beta_1 q^{3} + 4 q^{4} - \beta_{6} q^{5} + 2 \beta_1 q^{6} + ( - \beta_{3} - 1) q^{7} - 8 q^{8} + (\beta_{2} + 9) q^{9} + 2 \beta_{6} q^{10} + (\beta_{12} + \beta_{6} + 2) q^{11} - 4 \beta_1 q^{12} + ( - \beta_{17} - \beta_{12} + \beta_{11} + \cdots - 8) q^{13}+ \cdots + (14 \beta_{18} + 2 \beta_{17} + \cdots - 130) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q - 38 q^{2} - 5 q^{3} + 76 q^{4} + 10 q^{6} - 13 q^{7} - 152 q^{8} + 166 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 19 q - 38 q^{2} - 5 q^{3} + 76 q^{4} + 10 q^{6} - 13 q^{7} - 152 q^{8} + 166 q^{9} + 30 q^{11} - 20 q^{12} - 151 q^{13} + 26 q^{14} + 2 q^{15} + 304 q^{16} - 56 q^{17} - 332 q^{18} - 266 q^{19} - 239 q^{21} - 60 q^{22} + 437 q^{23} + 40 q^{24} + 395 q^{25} + 302 q^{26} - 68 q^{27} - 52 q^{28} - 551 q^{29} - 4 q^{30} - 622 q^{31} - 608 q^{32} - 291 q^{33} + 112 q^{34} + 21 q^{35} + 664 q^{36} - 183 q^{37} + 532 q^{38} - 212 q^{39} - 395 q^{41} + 478 q^{42} - 291 q^{43} + 120 q^{44} - 1018 q^{45} - 874 q^{46} - 241 q^{47} - 80 q^{48} + 1394 q^{49} - 790 q^{50} - 184 q^{51} - 604 q^{52} - 631 q^{53} + 136 q^{54} - 1765 q^{55} + 104 q^{56} + 144 q^{57} + 1102 q^{58} - 457 q^{59} + 8 q^{60} - 630 q^{61} + 1244 q^{62} + 125 q^{63} + 1216 q^{64} + 137 q^{65} + 582 q^{66} + 1421 q^{67} - 224 q^{68} - 115 q^{69} - 42 q^{70} + 983 q^{71} - 1328 q^{72} - 2532 q^{73} + 366 q^{74} - 1081 q^{75} - 1064 q^{76} - 595 q^{77} + 424 q^{78} - 4361 q^{79} - 721 q^{81} + 790 q^{82} + 1422 q^{83} - 956 q^{84} + 722 q^{85} + 582 q^{86} + 145 q^{87} - 240 q^{88} - 580 q^{89} + 2036 q^{90} - 447 q^{91} + 1748 q^{92} + 2106 q^{93} + 482 q^{94} - 310 q^{95} + 160 q^{96} - 2016 q^{97} - 2788 q^{98} - 1972 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{19} - 5 x^{18} - 327 x^{17} + 1564 x^{16} + 43869 x^{15} - 203270 x^{14} - 3103297 x^{13} + \cdots - 37580243456 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 36 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 30\!\cdots\!44 \nu^{18} + \cdots + 41\!\cdots\!28 ) / 30\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 16\!\cdots\!91 \nu^{18} + \cdots - 16\!\cdots\!32 ) / 12\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 20\!\cdots\!11 \nu^{18} + \cdots - 44\!\cdots\!68 ) / 67\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 17\!\cdots\!91 \nu^{18} + \cdots - 16\!\cdots\!68 ) / 58\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 73\!\cdots\!11 \nu^{18} + \cdots + 72\!\cdots\!36 ) / 24\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 17\!\cdots\!63 \nu^{18} + \cdots - 14\!\cdots\!64 ) / 48\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 29\!\cdots\!13 \nu^{18} + \cdots - 63\!\cdots\!96 ) / 58\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 14\!\cdots\!19 \nu^{18} + \cdots - 32\!\cdots\!08 ) / 24\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 92\!\cdots\!53 \nu^{18} + \cdots + 74\!\cdots\!24 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 27\!\cdots\!67 \nu^{18} + \cdots + 36\!\cdots\!24 ) / 30\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 14\!\cdots\!33 \nu^{18} + \cdots - 36\!\cdots\!96 ) / 16\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 22\!\cdots\!01 \nu^{18} + \cdots + 13\!\cdots\!60 ) / 24\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 12\!\cdots\!71 \nu^{18} + \cdots - 20\!\cdots\!32 ) / 12\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 28\!\cdots\!09 \nu^{18} + \cdots + 33\!\cdots\!48 ) / 24\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 48\!\cdots\!89 \nu^{18} + \cdots - 46\!\cdots\!88 ) / 24\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 84\!\cdots\!19 \nu^{18} + \cdots - 12\!\cdots\!48 ) / 40\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{17} - 2 \beta_{16} + 2 \beta_{15} + 2 \beta_{14} - 2 \beta_{13} + \beta_{12} + 2 \beta_{11} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{17} - 12 \beta_{16} - 3 \beta_{15} + 13 \beta_{14} - 10 \beta_{13} + 8 \beta_{12} + 12 \beta_{11} + \cdots + 2135 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 49 \beta_{18} - 53 \beta_{17} - 218 \beta_{16} + 191 \beta_{15} + 177 \beta_{14} - 208 \beta_{13} + \cdots + 1396 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 182 \beta_{18} + 649 \beta_{17} - 1622 \beta_{16} - 324 \beta_{15} + 1560 \beta_{14} - 1390 \beta_{13} + \cdots + 146439 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 7229 \beta_{18} - 316 \beta_{17} - 20566 \beta_{16} + 16132 \beta_{15} + 14712 \beta_{14} + \cdots + 223489 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 35284 \beta_{18} + 95827 \beta_{17} - 171249 \beta_{16} - 19357 \beta_{15} + 158303 \beta_{14} + \cdots + 10818094 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 816790 \beta_{18} + 331978 \beta_{17} - 1892769 \beta_{16} + 1365504 \beta_{15} + 1285079 \beta_{14} + \cdots + 26432508 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 4824552 \beta_{18} + 11296976 \beta_{17} - 16654611 \beta_{16} - 297527 \beta_{15} + 15211324 \beta_{14} + \cdots + 836935896 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 84581913 \beta_{18} + 57173808 \beta_{17} - 172976589 \beta_{16} + 118237705 \beta_{15} + \cdots + 2781246240 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 569767868 \beta_{18} + 1203503176 \beta_{17} - 1563157305 \beta_{16} + 118209178 \beta_{15} + \cdots + 66860594628 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 8432090277 \beta_{18} + 7202449243 \beta_{17} - 15740993811 \beta_{16} + 10465352211 \beta_{15} + \cdots + 276241617966 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 62207550536 \beta_{18} + 121445314473 \beta_{17} - 144173246525 \beta_{16} + 23625419387 \beta_{15} + \cdots + 5469487246969 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 823798148209 \beta_{18} + 804896129168 \beta_{17} - 1427735946795 \beta_{16} + 942159051383 \beta_{15} + \cdots + 26565733989815 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 6478123212561 \beta_{18} + 11871902340735 \beta_{17} - 13175438523004 \beta_{16} + \cdots + 455645468391581 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 79475241148774 \beta_{18} + 84549855890457 \beta_{17} - 129189566540883 \beta_{16} + \cdots + 25\!\cdots\!50 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 654199801798879 \beta_{18} + \cdots + 38\!\cdots\!42 \) 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
9.55575
9.03794
7.86787
6.31041
5.83619
4.23241
3.67612
3.42615
2.00501
−0.170763
−0.347281
−0.647345
−3.20044
−4.77505
−6.33858
−6.92609
−7.88757
−7.89870
−8.75601
−2.00000 −9.55575 4.00000 −15.6269 19.1115 28.1388 −8.00000 64.3123 31.2538
1.2 −2.00000 −9.03794 4.00000 8.80908 18.0759 18.0861 −8.00000 54.6843 −17.6182
1.3 −2.00000 −7.86787 4.00000 4.58634 15.7357 0.440519 −8.00000 34.9033 −9.17269
1.4 −2.00000 −6.31041 4.00000 −8.25016 12.6208 −34.1337 −8.00000 12.8212 16.5003
1.5 −2.00000 −5.83619 4.00000 11.6251 11.6724 −13.6316 −8.00000 7.06111 −23.2502
1.6 −2.00000 −4.23241 4.00000 −10.7502 8.46481 12.1405 −8.00000 −9.08675 21.5004
1.7 −2.00000 −3.67612 4.00000 18.7156 7.35224 −19.6479 −8.00000 −13.4861 −37.4311
1.8 −2.00000 −3.42615 4.00000 −21.2866 6.85230 −25.1113 −8.00000 −15.2615 42.5732
1.9 −2.00000 −2.00501 4.00000 −4.12964 4.01002 −10.8297 −8.00000 −22.9799 8.25927
1.10 −2.00000 0.170763 4.00000 18.6165 −0.341525 7.62274 −8.00000 −26.9708 −37.2330
1.11 −2.00000 0.347281 4.00000 11.2667 −0.694561 25.1624 −8.00000 −26.8794 −22.5334
1.12 −2.00000 0.647345 4.00000 −3.02681 −1.29469 32.0552 −8.00000 −26.5809 6.05362
1.13 −2.00000 3.20044 4.00000 7.27593 −6.40088 7.24632 −8.00000 −16.7572 −14.5519
1.14 −2.00000 4.77505 4.00000 −0.897863 −9.55010 −29.7883 −8.00000 −4.19889 1.79573
1.15 −2.00000 6.33858 4.00000 −6.43002 −12.6772 9.90564 −8.00000 13.1776 12.8600
1.16 −2.00000 6.92609 4.00000 −14.9620 −13.8522 13.3304 −8.00000 20.9708 29.9240
1.17 −2.00000 7.88757 4.00000 −16.1417 −15.7751 −15.3318 −8.00000 35.2137 32.2834
1.18 −2.00000 7.89870 4.00000 7.71017 −15.7974 10.8739 −8.00000 35.3895 −15.4203
1.19 −2.00000 8.75601 4.00000 12.8965 −17.5120 −29.5283 −8.00000 49.6678 −25.7929
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.19
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(23\) \(-1\)
\(29\) \(1\)

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 1334.4.a.d ✓ 19
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1334.4.a.d ✓ 19 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{19} + 5 T_{3}^{18} - 327 T_{3}^{17} - 1564 T_{3}^{16} + 43869 T_{3}^{15} + 203270 T_{3}^{14} + \cdots + 37580243456 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1334))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{19} \) Copy content Toggle raw display
$3$ \( T^{19} + \cdots + 37580243456 \) Copy content Toggle raw display
$5$ \( T^{19} + \cdots - 68\!\cdots\!80 \) Copy content Toggle raw display
$7$ \( T^{19} + \cdots - 58\!\cdots\!80 \) Copy content Toggle raw display
$11$ \( T^{19} + \cdots + 17\!\cdots\!80 \) Copy content Toggle raw display
$13$ \( T^{19} + \cdots + 15\!\cdots\!42 \) Copy content Toggle raw display
$17$ \( T^{19} + \cdots - 16\!\cdots\!72 \) Copy content Toggle raw display
$19$ \( T^{19} + \cdots + 27\!\cdots\!40 \) Copy content Toggle raw display
$23$ \( (T - 23)^{19} \) Copy content Toggle raw display
$29$ \( (T + 29)^{19} \) Copy content Toggle raw display
$31$ \( T^{19} + \cdots - 13\!\cdots\!20 \) Copy content Toggle raw display
$37$ \( T^{19} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{19} + \cdots + 87\!\cdots\!20 \) Copy content Toggle raw display
$43$ \( T^{19} + \cdots - 21\!\cdots\!98 \) Copy content Toggle raw display
$47$ \( T^{19} + \cdots + 62\!\cdots\!52 \) Copy content Toggle raw display
$53$ \( T^{19} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{19} + \cdots - 17\!\cdots\!40 \) Copy content Toggle raw display
$61$ \( T^{19} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{19} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{19} + \cdots + 43\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{19} + \cdots + 12\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{19} + \cdots + 61\!\cdots\!26 \) Copy content Toggle raw display
$83$ \( T^{19} + \cdots + 82\!\cdots\!80 \) Copy content Toggle raw display
$89$ \( T^{19} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{19} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
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