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Properties

Label 1369.2.b.g
Level $1369$
Weight $2$
Character orbit 1369.b
Analytic conductor $10.932$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1369,2,Mod(1368,1369)]
 
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([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1369.1368");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1369 = 37^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1369.b (of order \(2\), degree \(1\), not minimal)

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: no
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: \( 2^{9} \)
Twist minimal: no (minimal twist has level 37)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{12} + \beta_1) q^{2} + ( - \beta_{15} + \beta_{7}) q^{3} + ( - \beta_{10} + \beta_{6} - 1) q^{4} + \beta_{16} q^{5} + (\beta_{17} - \beta_{16} + \cdots - 2 \beta_1) q^{6}+ \cdots + (\beta_{15} - \beta_{7} - \beta_{5} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{12} + \beta_1) q^{2} + ( - \beta_{15} + \beta_{7}) q^{3} + ( - \beta_{10} + \beta_{6} - 1) q^{4} + \beta_{16} q^{5} + (\beta_{17} - \beta_{16} + \cdots - 2 \beta_1) q^{6}+ \cdots + (2 \beta_{15} - \beta_{10} + \cdots + \beta_{5}) q^{99}+O(q^{100}) \) 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
\(\beta_{4}\)\(=\) \( ( - 183 \nu^{17} - 5230 \nu^{15} - 53379 \nu^{13} - 254586 \nu^{11} - 604198 \nu^{9} - 676800 \nu^{7} + \cdots + 33232 \nu ) / 512 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 75 \nu^{16} - 2182 \nu^{14} - 22983 \nu^{12} - 115746 \nu^{10} - 303438 \nu^{8} - 418400 \nu^{6} + \cdots - 14768 ) / 256 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 81 \nu^{16} + 2338 \nu^{14} + 24293 \nu^{12} + 119606 \nu^{10} + 301482 \nu^{8} + 385472 \nu^{6} + \cdots + 1872 ) / 256 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 153 \nu^{16} - 4530 \nu^{14} - 49165 \nu^{12} - 259878 \nu^{10} - 735738 \nu^{8} - 1142912 \nu^{6} + \cdots - 64208 ) / 256 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 279 \nu^{17} + 8094 \nu^{15} + 84867 \nu^{13} + 424554 \nu^{11} + 1102022 \nu^{9} + 1494432 \nu^{7} + \cdots + 37104 \nu ) / 256 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 257 \nu^{16} - 7570 \nu^{14} - 81461 \nu^{12} - 424998 \nu^{10} - 1180138 \nu^{8} - 1783584 \nu^{6} + \cdots - 92432 ) / 256 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 299 \nu^{16} + 8742 \nu^{14} + 92903 \nu^{12} + 475202 \nu^{10} + 1280014 \nu^{8} + 1849184 \nu^{6} + \cdots + 79152 ) / 256 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 169 \nu^{17} + 4958 \nu^{15} + 52997 \nu^{13} + 273634 \nu^{11} + 748082 \nu^{9} + 1105720 \nu^{7} + \cdots + 52064 \nu ) / 128 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 691 \nu^{17} - 20262 \nu^{15} - 216367 \nu^{13} - 1114850 \nu^{11} - 3034622 \nu^{9} + \cdots - 185072 \nu ) / 512 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 533 \nu^{16} + 15610 \nu^{14} + 166361 \nu^{12} + 854686 \nu^{10} + 2317234 \nu^{8} + 3377568 \nu^{6} + \cdots + 146000 ) / 256 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 305 \nu^{17} + 8962 \nu^{15} + 96037 \nu^{13} + 497622 \nu^{11} + 1366570 \nu^{9} + 2028800 \nu^{7} + \cdots + 91728 \nu ) / 128 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 189 \nu^{16} + 5534 \nu^{14} + 58953 \nu^{12} + 302642 \nu^{10} + 819354 \nu^{8} + 1190968 \nu^{6} + \cdots + 49536 ) / 64 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 721 \nu^{17} + 21138 \nu^{15} + 225669 \nu^{13} + 1162566 \nu^{11} + 3165194 \nu^{9} + \cdots + 204048 \nu ) / 128 \) 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
\(\nu^{2}\)\(=\) \( \beta_{13} - \beta_{10} + \beta_{7} - \beta_{6} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{16} + 2\beta_{14} - \beta_{11} + 3\beta_{8} + 3\beta_{4} + \beta_{2} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} - 16\beta_{13} + 11\beta_{10} - 2\beta_{9} - 17\beta_{7} + 11\beta_{6} - 4\beta_{5} - 3\beta_{3} + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5 \beta_{17} + 13 \beta_{16} - 27 \beta_{14} + 11 \beta_{12} + 6 \beta_{11} - 48 \beta_{8} + \cdots + 102 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 26 \beta_{15} + 228 \beta_{13} - 129 \beta_{10} + 39 \beta_{9} + 229 \beta_{7} - 121 \beta_{6} + \cdots - 227 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 93 \beta_{17} - 171 \beta_{16} + 343 \beta_{14} - 227 \beta_{12} - 24 \beta_{11} + 664 \beta_{8} + \cdots - 1257 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 436 \beta_{15} - 3091 \beta_{13} + 1600 \beta_{10} - 593 \beta_{9} - 2956 \beta_{7} + 1420 \beta_{6} + \cdots + 2670 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1365 \beta_{17} + 2254 \beta_{16} - 4385 \beta_{14} + 3495 \beta_{12} - 53 \beta_{11} + \cdots + 15966 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 6319 \beta_{15} + 40949 \beta_{13} - 20351 \beta_{10} + 8271 \beta_{9} + 38021 \beta_{7} + \cdots - 33190 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 18648 \beta_{17} - 29539 \beta_{16} + 56470 \beta_{14} - 48792 \beta_{12} + 3061 \beta_{11} + \cdots - 205274 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 86255 \beta_{15} - 536705 \beta_{13} + 261744 \beta_{10} - 111080 \beta_{9} - 490530 \beta_{7} + \cdots + 422583 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 247697 \beta_{17} + 385382 \beta_{16} - 730237 \beta_{14} + 654495 \beta_{12} - 54805 \beta_{11} + \cdots + 2653181 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 1146323 \beta_{15} + 6999282 \beta_{13} - 3383418 \beta_{10} + 1465641 \beta_{9} + 6346312 \beta_{7} + \cdots - 5437795 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 3250014 \beta_{17} - 5015426 \beta_{16} + 9462658 \beta_{14} - 8626314 \beta_{12} + \cdots - 34374833 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 15047460 \beta_{15} - 91064599 \beta_{13} + 43837491 \beta_{10} - 19175960 \beta_{9} - 82246475 \beta_{7} + \cdots + 70309501 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 42405000 \beta_{17} + 65190479 \beta_{16} - 122743770 \beta_{14} + 112787088 \beta_{12} + \cdots + 445857747 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1369\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1368.1
− 0.660907i
− 1.23399i
− 1.77531i
− 3.60322i
− 0.752039i
− 1.92581i
0.834738i
− 2.47983i
0.885952i
− 0.885952i
2.47983i
− 0.834738i
1.92581i
0.752039i
3.60322i
1.77531i
1.23399i
0.660907i
− 2.63052i 2.04069 −4.91965 0.997045i − 5.36808i −2.07000 7.68021i 1.16441 2.62275
1368.2 − 2.51956i −1.26457 −4.34819 1.99266i 3.18617i −3.19228 5.91640i −1.40086 5.02062
1368.3 − 2.45935i −2.27777 −4.04838 − 2.88499i 5.60182i 1.44829 5.03769i 2.18823 −7.09518
1368.4 − 2.31765i −3.08507 −3.37150 − 0.741123i 7.15011i 1.09325 3.17865i 6.51765 −1.71766
1368.5 − 1.43608i 2.75684 −0.0623251 2.56411i − 3.95904i 3.41890 − 2.78266i 4.60014 3.68226
1368.6 − 1.24177i 0.747614 0.458006 2.69675i − 0.928365i −1.33510 − 3.05228i −2.44107 3.34874
1368.7 − 1.13488i −3.00707 0.712053 2.95427i 3.41266i 1.63893 − 3.07785i 6.04248 3.35274
1368.8 − 0.510210i 1.15117 1.73969 1.53522i − 0.587340i 0.551684 − 1.90802i −1.67480 0.783286
1368.9 − 0.399624i −0.0618332 1.84030 − 2.49621i 0.0247100i 4.44633 − 1.53467i −2.99618 −0.997546
1368.10 0.399624i −0.0618332 1.84030 2.49621i − 0.0247100i 4.44633 1.53467i −2.99618 −0.997546
1368.11 0.510210i 1.15117 1.73969 − 1.53522i 0.587340i 0.551684 1.90802i −1.67480 0.783286
1368.12 1.13488i −3.00707 0.712053 − 2.95427i − 3.41266i 1.63893 3.07785i 6.04248 3.35274
1368.13 1.24177i 0.747614 0.458006 − 2.69675i 0.928365i −1.33510 3.05228i −2.44107 3.34874
1368.14 1.43608i 2.75684 −0.0623251 − 2.56411i 3.95904i 3.41890 2.78266i 4.60014 3.68226
1368.15 2.31765i −3.08507 −3.37150 0.741123i − 7.15011i 1.09325 − 3.17865i 6.51765 −1.71766
1368.16 2.45935i −2.27777 −4.04838 2.88499i − 5.60182i 1.44829 − 5.03769i 2.18823 −7.09518
1368.17 2.51956i −1.26457 −4.34819 − 1.99266i − 3.18617i −3.19228 − 5.91640i −1.40086 5.02062
1368.18 2.63052i 2.04069 −4.91965 − 0.997045i 5.36808i −2.07000 − 7.68021i 1.16441 2.62275
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1368.18
Significant digits:
Format:

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.b.g 18
37.b even 2 1 inner 1369.2.b.g 18
37.d odd 4 2 1369.2.a.m 18
37.f even 9 1 37.2.h.a ✓ 18
37.h even 18 1 37.2.h.a ✓ 18
111.n odd 18 1 333.2.bl.d 18
111.p odd 18 1 333.2.bl.d 18
148.o odd 18 1 592.2.bq.d 18
148.p odd 18 1 592.2.bq.d 18
185.v even 18 1 925.2.bb.a 18
185.x even 18 1 925.2.bb.a 18
185.y odd 36 2 925.2.ba.a 36
185.bd odd 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.f even 9 1
37.2.h.a ✓ 18 37.h even 18 1
333.2.bl.d 18 111.n odd 18 1
333.2.bl.d 18 111.p odd 18 1
592.2.bq.d 18 148.o odd 18 1
592.2.bq.d 18 148.p odd 18 1
925.2.ba.a 36 185.y odd 36 2
925.2.ba.a 36 185.bd odd 36 2
925.2.bb.a 18 185.v even 18 1
925.2.bb.a 18 185.x even 18 1
1369.2.a.m 18 37.d odd 4 2
1369.2.b.g 18 1.a even 1 1 trivial
1369.2.b.g 18 37.b even 2 1 inner

Hecke kernels

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

\( T_{2}^{18} + 30 T_{2}^{16} + 369 T_{2}^{14} + 2396 T_{2}^{12} + 8832 T_{2}^{10} + 18612 T_{2}^{8} + \cdots + 243 \) 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
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