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Properties

Label 37.8.a.b
Level $37$
Weight $8$
Character orbit 37.a
Self dual yes
Analytic conductor $11.558$
Analytic rank $0$
Dimension $11$
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] = [37,8,Mod(1,37)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(37, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("37.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 37.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: \(11.5582459429\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 5 x^{10} - 1078 x^{9} + 4966 x^{8} + 379692 x^{7} - 1385588 x^{6} - 48765978 x^{5} + \cdots + 6680404080 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5} \)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{2} + (\beta_{2} + 11) q^{3} + (\beta_{3} + \beta_{2} + 2 \beta_1 + 71) q^{4} + ( - \beta_{5} + \beta_{2} - 6 \beta_1 + 37) q^{5} + (\beta_{5} + \beta_{4} + 5 \beta_{2} + \cdots + 38) q^{6}+ \cdots + ( - 3 \beta_{10} - \beta_{9} + \cdots + 864) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{2} + (\beta_{2} + 11) q^{3} + (\beta_{3} + \beta_{2} + 2 \beta_1 + 71) q^{4} + ( - \beta_{5} + \beta_{2} - 6 \beta_1 + 37) q^{5} + (\beta_{5} + \beta_{4} + 5 \beta_{2} + \cdots + 38) q^{6}+ \cdots + (17686 \beta_{10} + 10991 \beta_{9} + \cdots - 1304541) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 16 q^{2} + 121 q^{3} + 794 q^{4} + 376 q^{5} + 519 q^{6} + 2243 q^{7} + 3870 q^{8} + 9826 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 11 q + 16 q^{2} + 121 q^{3} + 794 q^{4} + 376 q^{5} + 519 q^{6} + 2243 q^{7} + 3870 q^{8} + 9826 q^{9} - 12629 q^{10} + 9415 q^{11} + 27955 q^{12} + 12512 q^{13} + 18260 q^{14} + 25714 q^{15} + 167866 q^{16} + 54312 q^{17} + 163911 q^{18} + 97192 q^{19} + 85625 q^{20} + 97795 q^{21} - 12345 q^{22} + 107342 q^{23} + 163119 q^{24} + 165051 q^{25} + 61531 q^{26} + 446611 q^{27} + 215454 q^{28} + 41748 q^{29} - 1080964 q^{30} - 272248 q^{31} + 593306 q^{32} - 216525 q^{33} - 923600 q^{34} + 436814 q^{35} - 456119 q^{36} - 557183 q^{37} - 175872 q^{38} - 1587326 q^{39} - 3206863 q^{40} + 525465 q^{41} - 3814396 q^{42} - 1376086 q^{43} - 1337377 q^{44} - 2315492 q^{45} - 2037327 q^{46} + 2269179 q^{47} + 1779791 q^{48} + 2282536 q^{49} - 3881347 q^{50} - 103604 q^{51} - 4200495 q^{52} - 346415 q^{53} + 6349248 q^{54} + 4169374 q^{55} - 4307934 q^{56} + 6170792 q^{57} - 1334849 q^{58} + 4598828 q^{59} - 4448200 q^{60} + 6208418 q^{61} + 4732115 q^{62} + 6882994 q^{63} + 12483426 q^{64} + 9330160 q^{65} - 5715150 q^{66} + 2199016 q^{67} + 8095824 q^{68} + 13516268 q^{69} - 6471708 q^{70} + 4653285 q^{71} + 12839097 q^{72} - 1080699 q^{73} - 810448 q^{74} + 16194855 q^{75} + 1331888 q^{76} + 22058153 q^{77} - 23968103 q^{78} - 1336084 q^{79} - 89443 q^{80} + 9585355 q^{81} + 9689125 q^{82} + 28551309 q^{83} - 37602282 q^{84} + 13256012 q^{85} - 47733694 q^{86} - 5826578 q^{87} - 58704117 q^{88} - 8994788 q^{89} - 46526086 q^{90} - 696642 q^{91} - 41894465 q^{92} - 9859184 q^{93} - 26180048 q^{94} + 124152 q^{95} - 19485621 q^{96} - 3968264 q^{97} - 7312590 q^{98} - 14172918 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - 5 x^{10} - 1078 x^{9} + 4966 x^{8} + 379692 x^{7} - 1385588 x^{6} - 48765978 x^{5} + \cdots + 6680404080 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11\!\cdots\!61 \nu^{10} + \cdots - 13\!\cdots\!52 ) / 28\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11\!\cdots\!61 \nu^{10} + \cdots - 43\!\cdots\!52 ) / 28\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 23\!\cdots\!05 \nu^{10} + \cdots + 15\!\cdots\!88 ) / 28\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12\!\cdots\!57 \nu^{10} + \cdots - 59\!\cdots\!28 ) / 14\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 20\!\cdots\!65 \nu^{10} + \cdots + 17\!\cdots\!64 ) / 15\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 34\!\cdots\!99 \nu^{10} + \cdots - 28\!\cdots\!28 ) / 14\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 16\!\cdots\!45 \nu^{10} + \cdots + 30\!\cdots\!60 ) / 31\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 25\!\cdots\!53 \nu^{10} + \cdots + 10\!\cdots\!88 ) / 35\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 68\!\cdots\!49 \nu^{10} + \cdots + 24\!\cdots\!08 ) / 71\!\cdots\!12 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 198 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{10} + 3\beta_{8} + 3\beta_{7} + 4\beta_{5} - \beta_{3} + 375\beta _1 - 42 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 11 \beta_{10} - 3 \beta_{9} + 17 \beta_{8} - 24 \beta_{7} + 16 \beta_{6} + 6 \beta_{5} + 8 \beta_{4} + \cdots + 74069 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 1078 \beta_{10} - 78 \beta_{9} + 1672 \beta_{8} + 1398 \beta_{7} - 492 \beta_{6} + 2204 \beta_{5} + \cdots - 91134 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 7358 \beta_{10} - 2782 \beta_{9} + 9956 \beta_{8} - 15934 \beta_{7} + 14532 \beta_{6} + \cdots + 31118420 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 492764 \beta_{10} - 33482 \beta_{9} + 798175 \beta_{8} + 625297 \beta_{7} - 428524 \beta_{6} + \cdots - 72197176 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 3700033 \beta_{10} - 1519241 \beta_{9} + 4550125 \beta_{8} - 8591574 \beta_{7} + 9873860 \beta_{6} + \cdots + 13538125867 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 213476112 \beta_{10} - 5774168 \beta_{9} + 365393820 \beta_{8} + 286873436 \beta_{7} + \cdots - 45428759148 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1671109388 \beta_{10} - 693474316 \beta_{9} + 1862539160 \beta_{8} - 4372551324 \beta_{7} + \cdots + 5995121970338 \) 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
−22.1395
−19.7203
−9.08402
−7.85832
−4.12221
0.259260
4.43681
7.16645
14.8861
20.2283
20.9474
−21.1395 34.2904 318.879 463.709 −724.881 1277.57 −4035.09 −1011.17 −9802.57
1.2 −18.7203 −14.9523 222.450 −4.96717 279.912 −1164.11 −1768.14 −1963.43 92.9870
1.3 −8.08402 72.5716 −62.6486 414.761 −586.670 49.3599 1541.21 3079.64 −3352.94
1.4 −6.85832 15.3600 −80.9635 −335.413 −105.343 735.210 1433.14 −1951.07 2300.37
1.5 −3.12221 −59.7866 −118.252 −342.676 186.666 −975.157 768.850 1387.44 1069.91
1.6 1.25926 −26.2543 −126.414 409.245 −33.0610 −873.009 −320.374 −1497.71 515.347
1.7 5.43681 85.9489 −98.4411 −143.945 467.288 1592.07 −1231.12 5200.21 −782.599
1.8 8.16645 −75.6574 −61.3091 −56.3977 −617.852 1198.36 −1545.98 3537.05 −460.569
1.9 15.8861 41.5699 124.370 303.592 660.386 157.297 −57.6738 −458.942 4822.91
1.10 21.2283 81.9649 322.641 −351.409 1739.98 −895.269 4131.89 4531.25 −7459.83
1.11 21.9474 −34.0550 353.689 19.5004 −747.419 1140.68 4953.28 −1027.26 427.983
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Atkin-Lehner signs

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

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 37.8.a.b ✓ 11
3.b odd 2 1 333.8.a.d 11
4.b odd 2 1 592.8.a.g 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.8.a.b ✓ 11 1.a even 1 1 trivial
333.8.a.d 11 3.b odd 2 1
592.8.a.g 11 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{11} - 16 T_{2}^{10} - 973 T_{2}^{9} + 14278 T_{2}^{8} + 302086 T_{2}^{7} - 3815344 T_{2}^{6} + \cdots + 28348180480 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(37))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{11} + \cdots + 28348180480 \) Copy content Toggle raw display
$3$ \( T^{11} + \cdots + 67\!\cdots\!84 \) Copy content Toggle raw display
$5$ \( T^{11} + \cdots - 75\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{11} + \cdots - 14\!\cdots\!64 \) Copy content Toggle raw display
$11$ \( T^{11} + \cdots + 11\!\cdots\!52 \) Copy content Toggle raw display
$13$ \( T^{11} + \cdots + 16\!\cdots\!72 \) Copy content Toggle raw display
$17$ \( T^{11} + \cdots + 27\!\cdots\!72 \) Copy content Toggle raw display
$19$ \( T^{11} + \cdots - 66\!\cdots\!60 \) Copy content Toggle raw display
$23$ \( T^{11} + \cdots + 10\!\cdots\!48 \) Copy content Toggle raw display
$29$ \( T^{11} + \cdots + 51\!\cdots\!60 \) Copy content Toggle raw display
$31$ \( T^{11} + \cdots - 68\!\cdots\!92 \) Copy content Toggle raw display
$37$ \( (T + 50653)^{11} \) Copy content Toggle raw display
$41$ \( T^{11} + \cdots - 36\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( T^{11} + \cdots + 32\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{11} + \cdots - 12\!\cdots\!12 \) Copy content Toggle raw display
$53$ \( T^{11} + \cdots + 14\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{11} + \cdots - 44\!\cdots\!40 \) Copy content Toggle raw display
$61$ \( T^{11} + \cdots - 16\!\cdots\!72 \) Copy content Toggle raw display
$67$ \( T^{11} + \cdots + 19\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{11} + \cdots + 32\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( T^{11} + \cdots + 83\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( T^{11} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{11} + \cdots - 64\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{11} + \cdots + 13\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( T^{11} + \cdots + 31\!\cdots\!28 \) Copy content Toggle raw display
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