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Properties

Label 1682.4.a.p
Level $1682$
Weight $4$
Character orbit 1682.a
Self dual yes
Analytic conductor $99.241$
Analytic rank $1$
Dimension $9$
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] = [1682,4,Mod(1,1682)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1682, 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("1682.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1682 = 2 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1682.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: \(99.2412126297\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} - 122x^{7} + 45x^{6} + 4645x^{5} + 345x^{4} - 56415x^{3} + 3296x^{2} + 186266x - 139229 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 58)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + (\beta_1 - 1) q^{3} + 4 q^{4} + (\beta_{8} - \beta_{5} + 2 \beta_{3} - 1) q^{5} + (2 \beta_1 - 2) q^{6} + ( - \beta_{8} + \beta_{6} + 3 \beta_{5} + \cdots + 2) q^{7}+ \cdots + (\beta_{7} - 8 \beta_{5} + \beta_{4} + \cdots + 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + (\beta_1 - 1) q^{3} + 4 q^{4} + (\beta_{8} - \beta_{5} + 2 \beta_{3} - 1) q^{5} + (2 \beta_1 - 2) q^{6} + ( - \beta_{8} + \beta_{6} + 3 \beta_{5} + \cdots + 2) q^{7}+ \cdots + (\beta_{8} - 16 \beta_{7} + 35 \beta_{6} + \cdots - 291) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 18 q^{2} - 8 q^{3} + 36 q^{4} - 15 q^{5} - 16 q^{6} + 9 q^{7} + 72 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 18 q^{2} - 8 q^{3} + 36 q^{4} - 15 q^{5} - 16 q^{6} + 9 q^{7} + 72 q^{8} + 9 q^{9} - 30 q^{10} - 65 q^{11} - 32 q^{12} - 34 q^{13} + 18 q^{14} + 39 q^{15} + 144 q^{16} - 129 q^{17} + 18 q^{18} - 130 q^{19} - 60 q^{20} - 98 q^{21} - 130 q^{22} + 28 q^{23} - 64 q^{24} - 66 q^{25} - 68 q^{26} - 77 q^{27} + 36 q^{28} + 78 q^{30} - 175 q^{31} + 288 q^{32} + 98 q^{33} - 258 q^{34} - 643 q^{35} + 36 q^{36} - 531 q^{37} - 260 q^{38} + 208 q^{39} - 120 q^{40} + 58 q^{41} - 196 q^{42} - 627 q^{43} - 260 q^{44} - 45 q^{45} + 56 q^{46} - 648 q^{47} - 128 q^{48} - 450 q^{49} - 132 q^{50} - 1067 q^{51} - 136 q^{52} - 478 q^{53} - 154 q^{54} - 368 q^{55} + 72 q^{56} - 509 q^{57} + 1129 q^{59} + 156 q^{60} - 802 q^{61} - 350 q^{62} - 1491 q^{63} + 576 q^{64} - 757 q^{65} + 196 q^{66} + 2261 q^{67} - 516 q^{68} - 3655 q^{69} - 1286 q^{70} - 2417 q^{71} + 72 q^{72} - 1092 q^{73} - 1062 q^{74} - 1275 q^{75} - 520 q^{76} + 2371 q^{77} + 416 q^{78} - 964 q^{79} - 240 q^{80} - 1807 q^{81} + 116 q^{82} + 432 q^{83} - 392 q^{84} + 991 q^{85} - 1254 q^{86} - 520 q^{88} - 3145 q^{89} - 90 q^{90} + 1898 q^{91} + 112 q^{92} + 2087 q^{93} - 1296 q^{94} - 1516 q^{95} - 256 q^{96} - 938 q^{97} - 900 q^{98} - 2600 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - x^{8} - 122x^{7} + 45x^{6} + 4645x^{5} + 345x^{4} - 56415x^{3} + 3296x^{2} + 186266x - 139229 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 327686269 \nu^{8} - 1651090589 \nu^{7} + 61441826109 \nu^{6} + 178642109370 \nu^{5} + \cdots - 86592481480337 ) / 21050376122793 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 621942853 \nu^{8} - 294256584 \nu^{7} - 74225937477 \nu^{6} - 33454397724 \nu^{5} + \cdots + 62248605005172 ) / 21050376122793 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2272415348 \nu^{8} + 38309436176 \nu^{7} - 324318426387 \nu^{6} - 4377767706048 \nu^{5} + \cdots + 5181403671074 ) / 63151128368379 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2750788360 \nu^{8} - 260486947 \nu^{7} + 333316385688 \nu^{6} + 224622832830 \nu^{5} + \cdots - 313428071771041 ) / 63151128368379 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3994334114 \nu^{8} - 7233065999 \nu^{7} + 532733787357 \nu^{6} + 859626816339 \nu^{5} + \cdots - 642766957015451 ) / 63151128368379 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 37439811196 \nu^{8} - 45003256774 \nu^{7} + 4695567343131 \nu^{6} + \cdots - 61\!\cdots\!69 ) / 63151128368379 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 16530929528 \nu^{8} + 8992764157 \nu^{7} + 1975821379926 \nu^{6} + 273346731159 \nu^{5} + \cdots - 13\!\cdots\!71 ) / 21050376122793 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - 8\beta_{5} + \beta_{4} + 6\beta_{3} - 2\beta_{2} + \beta _1 + 31 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + 2\beta_{7} - 11\beta_{6} + \beta_{5} + \beta_{4} + 45\beta_{3} + 3\beta_{2} + 48\beta _1 + 30 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 13 \beta_{8} + 62 \beta_{7} - 11 \beta_{6} - 678 \beta_{5} + 53 \beta_{4} + 444 \beta_{3} - 110 \beta_{2} + \cdots + 1588 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 69 \beta_{8} + 181 \beta_{7} - 928 \beta_{6} - 484 \beta_{5} + 83 \beta_{4} + 2799 \beta_{3} + \cdots + 2867 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1138 \beta_{8} + 3747 \beta_{7} - 1482 \beta_{6} - 45567 \beta_{5} + 3048 \beta_{4} + 28817 \beta_{3} + \cdots + 89073 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4790 \beta_{8} + 13443 \beta_{7} - 64912 \beta_{6} - 68810 \beta_{5} + 7447 \beta_{4} + 157653 \beta_{3} + \cdots + 218272 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 79561 \beta_{8} + 228812 \beta_{7} - 148175 \beta_{6} - 2922787 \beta_{5} + 182082 \beta_{4} + \cdots + 5153318 \) 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
−7.28712
−6.16278
−3.85349
−2.81607
1.02627
1.36778
3.54802
7.11460
8.06278
2.00000 −8.28712 4.00000 3.20915 −16.5742 0.161510 8.00000 41.6763 6.41830
1.2 2.00000 −7.16278 4.00000 −12.7144 −14.3256 −22.4980 8.00000 24.3054 −25.4288
1.3 2.00000 −4.85349 4.00000 −12.2077 −9.70698 33.1249 8.00000 −3.44362 −24.4154
1.4 2.00000 −3.81607 4.00000 17.3356 −7.63214 0.532638 8.00000 −12.4376 34.6712
1.5 2.00000 0.0262741 4.00000 −9.87153 0.0525483 5.66160 8.00000 −26.9993 −19.7431
1.6 2.00000 0.367777 4.00000 11.1293 0.735553 −9.81213 8.00000 −26.8647 22.2587
1.7 2.00000 2.54802 4.00000 −12.5617 5.09604 24.4550 8.00000 −20.5076 −25.1234
1.8 2.00000 6.11460 4.00000 −5.05180 12.2292 −6.26623 8.00000 10.3884 −10.1036
1.9 2.00000 7.06278 4.00000 5.73307 14.1256 −16.3593 8.00000 22.8829 11.4661
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

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

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 1682.4.a.p 9
29.b even 2 1 1682.4.a.o 9
29.e even 14 2 58.4.d.a ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.4.d.a ✓ 18 29.e even 14 2
1682.4.a.o 9 29.b even 2 1
1682.4.a.p 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{9} + 8 T_{3}^{8} - 94 T_{3}^{7} - 753 T_{3}^{6} + 2423 T_{3}^{5} + 20031 T_{3}^{4} - 11927 T_{3}^{3} + \cdots - 1169 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1682))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{9} \) Copy content Toggle raw display
$3$ \( T^{9} + 8 T^{8} + \cdots - 1169 \) Copy content Toggle raw display
$5$ \( T^{9} + 15 T^{8} + \cdots + 345139759 \) Copy content Toggle raw display
$7$ \( T^{9} - 9 T^{8} + \cdots - 8928331 \) Copy content Toggle raw display
$11$ \( T^{9} + \cdots - 1808248030469 \) Copy content Toggle raw display
$13$ \( T^{9} + \cdots + 1335780916651 \) Copy content Toggle raw display
$17$ \( T^{9} + \cdots + 142668951628288 \) Copy content Toggle raw display
$19$ \( T^{9} + \cdots + 43\!\cdots\!23 \) Copy content Toggle raw display
$23$ \( T^{9} + \cdots + 11\!\cdots\!63 \) Copy content Toggle raw display
$29$ \( T^{9} \) Copy content Toggle raw display
$31$ \( T^{9} + \cdots + 71\!\cdots\!77 \) Copy content Toggle raw display
$37$ \( T^{9} + \cdots + 13\!\cdots\!73 \) Copy content Toggle raw display
$41$ \( T^{9} + \cdots - 14\!\cdots\!88 \) Copy content Toggle raw display
$43$ \( T^{9} + \cdots + 15\!\cdots\!03 \) Copy content Toggle raw display
$47$ \( T^{9} + \cdots - 97\!\cdots\!63 \) Copy content Toggle raw display
$53$ \( T^{9} + \cdots - 69\!\cdots\!83 \) Copy content Toggle raw display
$59$ \( T^{9} + \cdots - 63\!\cdots\!32 \) Copy content Toggle raw display
$61$ \( T^{9} + \cdots + 11\!\cdots\!07 \) Copy content Toggle raw display
$67$ \( T^{9} + \cdots + 21\!\cdots\!09 \) Copy content Toggle raw display
$71$ \( T^{9} + \cdots - 17\!\cdots\!69 \) Copy content Toggle raw display
$73$ \( T^{9} + \cdots - 12\!\cdots\!33 \) Copy content Toggle raw display
$79$ \( T^{9} + \cdots + 39\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{9} + \cdots + 52\!\cdots\!49 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots + 10\!\cdots\!21 \) Copy content Toggle raw display
$97$ \( T^{9} + \cdots - 37\!\cdots\!52 \) Copy content Toggle raw display
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