↑

Properties

Label 1369.2.a.m
Level $1369$
Weight $2$
Character orbit 1369.a
Self dual yes
Analytic conductor $10.932$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1369,2,Mod(1,1369)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1369, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1369.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1369 = 37^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1369.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: \(10.9315200367\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 30x^{16} + 333x^{14} - 1826x^{12} + 5490x^{10} - 9432x^{8} + 9385x^{6} - 5316x^{4} + 1584x^{2} - 192 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 37)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{13} + \beta_1) q^{2} + (\beta_{14} - \beta_{7}) q^{3} + ( - \beta_{9} - \beta_{4} + 1) q^{4} + \beta_{16} q^{5} + ( - \beta_{17} - \beta_{16} + \cdots + 2 \beta_1) q^{6}+ \cdots + (\beta_{14} - \beta_{7} - \beta_{5} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{13} + \beta_1) q^{2} + (\beta_{14} - \beta_{7}) q^{3} + ( - \beta_{9} - \beta_{4} + 1) q^{4} + \beta_{16} q^{5} + ( - \beta_{17} - \beta_{16} + \cdots + 2 \beta_1) q^{6}+ \cdots + ( - 2 \beta_{14} - \beta_{9} + \cdots + \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 6 q^{3} + 24 q^{4} + 12 q^{7} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 6 q^{3} + 24 q^{4} + 12 q^{7} + 24 q^{9} + 18 q^{10} + 18 q^{11} + 30 q^{12} + 36 q^{16} + 6 q^{21} + 30 q^{26} + 42 q^{27} - 48 q^{28} - 48 q^{30} + 6 q^{33} - 36 q^{34} + 72 q^{36} + 54 q^{38} - 42 q^{40} + 60 q^{41} + 90 q^{44} + 30 q^{46} + 72 q^{47} + 6 q^{48} - 18 q^{49} + 42 q^{53} - 30 q^{58} + 42 q^{62} + 54 q^{63} + 12 q^{64} + 36 q^{65} - 60 q^{67} - 48 q^{70} + 72 q^{71} + 54 q^{73} + 18 q^{75} - 6 q^{77} - 90 q^{78} + 18 q^{81} + 12 q^{83} + 48 q^{84} + 12 q^{85} + 48 q^{86} + 96 q^{95} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 30x^{16} + 333x^{14} - 1826x^{12} + 5490x^{10} - 9432x^{8} + 9385x^{6} - 5316x^{4} + 1584x^{2} - 192 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5 \nu^{17} + 186 \nu^{15} - 2697 \nu^{13} + 19742 \nu^{11} - 78994 \nu^{9} + 175008 \nu^{7} + \cdots - 27088 \nu ) / 512 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 59 \nu^{16} + 1622 \nu^{14} - 15383 \nu^{12} + 63538 \nu^{10} - 106766 \nu^{8} + 8576 \nu^{6} + \cdots + 31504 ) / 256 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 81 \nu^{16} - 2338 \nu^{14} + 24293 \nu^{12} - 119606 \nu^{10} + 301482 \nu^{8} - 385472 \nu^{6} + \cdots + 1872 ) / 256 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 75 \nu^{16} + 2182 \nu^{14} - 22983 \nu^{12} + 115746 \nu^{10} - 303438 \nu^{8} + 418400 \nu^{6} + \cdots - 14768 ) / 256 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 183 \nu^{17} - 5230 \nu^{15} + 53379 \nu^{13} - 254586 \nu^{11} + 604198 \nu^{9} - 676800 \nu^{7} + \cdots - 33232 \nu ) / 512 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 153 \nu^{16} + 4530 \nu^{14} - 49165 \nu^{12} + 259878 \nu^{10} - 735738 \nu^{8} + 1142912 \nu^{6} + \cdots - 64208 ) / 256 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 257 \nu^{16} + 7570 \nu^{14} - 81461 \nu^{12} + 424998 \nu^{10} - 1180138 \nu^{8} + 1783584 \nu^{6} + \cdots - 92432 ) / 256 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 299 \nu^{16} + 8742 \nu^{14} - 92903 \nu^{12} + 475202 \nu^{10} - 1280014 \nu^{8} + 1849184 \nu^{6} + \cdots - 79152 ) / 256 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 279 \nu^{17} + 8094 \nu^{15} - 84867 \nu^{13} + 424554 \nu^{11} - 1102022 \nu^{9} + \cdots - 37104 \nu ) / 256 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 533 \nu^{16} + 15610 \nu^{14} - 166361 \nu^{12} + 854686 \nu^{10} - 2317234 \nu^{8} + 3377568 \nu^{6} + \cdots - 146000 ) / 256 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 169 \nu^{17} - 4958 \nu^{15} + 52997 \nu^{13} - 273634 \nu^{11} + 748082 \nu^{9} - 1105720 \nu^{7} + \cdots + 52064 \nu ) / 128 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 691 \nu^{17} - 20262 \nu^{15} + 216367 \nu^{13} - 1114850 \nu^{11} + 3034622 \nu^{9} + \cdots + 185072 \nu ) / 512 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 189 \nu^{16} - 5534 \nu^{14} + 58953 \nu^{12} - 302642 \nu^{10} + 819354 \nu^{8} - 1190968 \nu^{6} + \cdots + 49536 ) / 64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 305 \nu^{17} - 8962 \nu^{15} + 96037 \nu^{13} - 497622 \nu^{11} + 1366570 \nu^{9} - 2028800 \nu^{7} + \cdots + 91728 \nu ) / 128 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 721 \nu^{17} + 21138 \nu^{15} - 225669 \nu^{13} + 1162566 \nu^{11} - 3165194 \nu^{9} + \cdots - 204048 \nu ) / 128 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 837 \nu^{17} - 24538 \nu^{15} + 261961 \nu^{13} - 1349598 \nu^{11} + 3675442 \nu^{9} + \cdots + 242064 \nu ) / 128 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - \beta_{9} - \beta_{7} + \beta_{4} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{16} - 2\beta_{15} + \beta_{12} + 3\beta_{10} + 3\beta_{6} - \beta_{2} + 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} + 16\beta_{11} - 11\beta_{9} - 2\beta_{8} - 17\beta_{7} - 4\beta_{5} + 11\beta_{4} - 3\beta_{3} + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5 \beta_{17} - 13 \beta_{16} - 27 \beta_{15} - 11 \beta_{13} + 6 \beta_{12} + 48 \beta_{10} + \cdots + 102 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 26 \beta_{14} + 228 \beta_{11} - 129 \beta_{9} - 39 \beta_{8} - 229 \beta_{7} - 70 \beta_{5} + \cdots + 227 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 93 \beta_{17} - 171 \beta_{16} - 343 \beta_{15} - 227 \beta_{13} + 24 \beta_{12} + 664 \beta_{10} + \cdots + 1257 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 436 \beta_{14} + 3091 \beta_{11} - 1600 \beta_{9} - 593 \beta_{8} - 2956 \beta_{7} - 990 \beta_{5} + \cdots + 2670 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1365 \beta_{17} - 2254 \beta_{16} - 4385 \beta_{15} - 3495 \beta_{13} - 53 \beta_{12} + \cdots + 15966 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 6319 \beta_{14} + 40949 \beta_{11} - 20351 \beta_{9} - 8271 \beta_{8} - 38021 \beta_{7} - 13246 \beta_{5} + \cdots + 33190 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 18648 \beta_{17} - 29539 \beta_{16} - 56470 \beta_{15} - 48792 \beta_{13} - 3061 \beta_{12} + \cdots + 205274 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 86255 \beta_{14} + 536705 \beta_{11} - 261744 \beta_{9} - 111080 \beta_{8} - 490530 \beta_{7} + \cdots + 422583 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 247697 \beta_{17} - 385382 \beta_{16} - 730237 \beta_{15} - 654495 \beta_{13} - 54805 \beta_{12} + \cdots + 2653181 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 1146323 \beta_{14} + 6999282 \beta_{11} - 3383418 \beta_{9} - 1465641 \beta_{8} - 6346312 \beta_{7} + \cdots + 5437795 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 3250014 \beta_{17} - 5015426 \beta_{16} - 9462658 \beta_{15} - 8626314 \beta_{13} + \cdots + 34374833 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 15047460 \beta_{14} + 91064599 \beta_{11} - 43837491 \beta_{9} - 19175960 \beta_{8} - 82246475 \beta_{7} + \cdots + 70309501 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 42405000 \beta_{17} - 65190479 \beta_{16} - 122743770 \beta_{15} - 112787088 \beta_{13} + \cdots + 445857747 \beta_1 \) 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
−0.660907
−1.23399
−1.77531
−3.60322
−0.752039
−1.92581
0.834738
−2.47983
0.885952
−0.885952
2.47983
−0.834738
1.92581
0.752039
3.60322
1.77531
1.23399
0.660907
−2.63052 −2.04069 4.91965 −0.997045 5.36808 −2.07000 −7.68021 1.16441 2.62275
1.2 −2.51956 1.26457 4.34819 −1.99266 −3.18617 −3.19228 −5.91640 −1.40086 5.02062
1.3 −2.45935 2.27777 4.04838 2.88499 −5.60182 1.44829 −5.03769 2.18823 −7.09518
1.4 −2.31765 3.08507 3.37150 0.741123 −7.15011 1.09325 −3.17865 6.51765 −1.71766
1.5 −1.43608 −2.75684 0.0623251 −2.56411 3.95904 3.41890 2.78266 4.60014 3.68226
1.6 −1.24177 −0.747614 −0.458006 −2.69675 0.928365 −1.33510 3.05228 −2.44107 3.34874
1.7 −1.13488 3.00707 −0.712053 −2.95427 −3.41266 1.63893 3.07785 6.04248 3.35274
1.8 −0.510210 −1.15117 −1.73969 −1.53522 0.587340 0.551684 1.90802 −1.67480 0.783286
1.9 −0.399624 0.0618332 −1.84030 2.49621 −0.0247100 4.44633 1.53467 −2.99618 −0.997546
1.10 0.399624 0.0618332 −1.84030 −2.49621 0.0247100 4.44633 −1.53467 −2.99618 −0.997546
1.11 0.510210 −1.15117 −1.73969 1.53522 −0.587340 0.551684 −1.90802 −1.67480 0.783286
1.12 1.13488 3.00707 −0.712053 2.95427 3.41266 1.63893 −3.07785 6.04248 3.35274
1.13 1.24177 −0.747614 −0.458006 2.69675 −0.928365 −1.33510 −3.05228 −2.44107 3.34874
1.14 1.43608 −2.75684 0.0623251 2.56411 −3.95904 3.41890 −2.78266 4.60014 3.68226
1.15 2.31765 3.08507 3.37150 −0.741123 7.15011 1.09325 3.17865 6.51765 −1.71766
1.16 2.45935 2.27777 4.04838 −2.88499 5.60182 1.44829 5.03769 2.18823 −7.09518
1.17 2.51956 1.26457 4.34819 1.99266 3.18617 −3.19228 5.91640 −1.40086 5.02062
1.18 2.63052 −2.04069 4.91965 0.997045 −5.36808 −2.07000 7.68021 1.16441 2.62275
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(37\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1369.2.a.m 18
37.b even 2 1 inner 1369.2.a.m 18
37.d odd 4 2 1369.2.b.g 18
37.i odd 36 2 37.2.h.a ✓ 18
111.q even 36 2 333.2.bl.d 18
148.q even 36 2 592.2.bq.d 18
185.z even 36 2 925.2.ba.a 36
185.ba odd 36 2 925.2.bb.a 18
185.bc even 36 2 925.2.ba.a 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.h.a ✓ 18 37.i odd 36 2
333.2.bl.d 18 111.q even 36 2
592.2.bq.d 18 148.q even 36 2
925.2.ba.a 36 185.z even 36 2
925.2.ba.a 36 185.bc even 36 2
925.2.bb.a 18 185.ba odd 36 2
1369.2.a.m 18 1.a even 1 1 trivial
1369.2.a.m 18 37.b even 2 1 inner
1369.2.b.g 18 37.d odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1369))\):

\( T_{2}^{18} - 30 T_{2}^{16} + 369 T_{2}^{14} - 2396 T_{2}^{12} + 8832 T_{2}^{10} - 18612 T_{2}^{8} + \cdots - 243 \) Copy content Toggle raw display
\( T_{3}^{9} - 3T_{3}^{8} - 15T_{3}^{7} + 41T_{3}^{6} + 75T_{3}^{5} - 159T_{3}^{4} - 157T_{3}^{3} + 162T_{3}^{2} + 120T_{3} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} - 30 T^{16} + \cdots - 243 \) Copy content Toggle raw display
$3$ \( (T^{9} - 3 T^{8} - 15 T^{7} + \cdots - 8)^{2} \) Copy content Toggle raw display
$5$ \( T^{18} - 45 T^{16} + \cdots - 110592 \) Copy content Toggle raw display
$7$ \( (T^{9} - 6 T^{8} + \cdots + 192)^{2} \) Copy content Toggle raw display
$11$ \( (T^{9} - 9 T^{8} + \cdots + 216)^{2} \) Copy content Toggle raw display
$13$ \( T^{18} - 63 T^{16} + \cdots - 27 \) Copy content Toggle raw display
$17$ \( T^{18} - 135 T^{16} + \cdots - 23705163 \) Copy content Toggle raw display
$19$ \( T^{18} - 120 T^{16} + \cdots - 5614272 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots - 180910927872 \) Copy content Toggle raw display
$29$ \( T^{18} - 222 T^{16} + \cdots - 30509163 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots - 1142154432 \) Copy content Toggle raw display
$37$ \( T^{18} \) Copy content Toggle raw display
$41$ \( (T^{9} - 30 T^{8} + \cdots - 81)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots - 74566243008 \) Copy content Toggle raw display
$47$ \( (T^{9} - 36 T^{8} + \cdots - 46656)^{2} \) Copy content Toggle raw display
$53$ \( (T^{9} - 21 T^{8} + \cdots + 1204227)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots - 1084930414272 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots - 24685456563 \) Copy content Toggle raw display
$67$ \( (T^{9} + 30 T^{8} + \cdots + 35581704)^{2} \) Copy content Toggle raw display
$71$ \( (T^{9} - 36 T^{8} + \cdots - 141912)^{2} \) Copy content Toggle raw display
$73$ \( (T^{9} - 27 T^{8} + \cdots - 8467983)^{2} \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots - 890793213988032 \) Copy content Toggle raw display
$83$ \( (T^{9} - 6 T^{8} + \cdots - 69336)^{2} \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots - 15282295962363 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots - 2765473961067 \) Copy content Toggle raw display
show more
show less