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Properties

Label 37.6.b.a
Level $37$
Weight $6$
Character orbit 37.b
Analytic conductor $5.934$
Analytic rank $0$
Dimension $16$
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] = [37,6,Mod(36,37)]
 
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([1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("37.36");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 37.b (of order \(2\), degree \(1\), 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: \(5.93420133308\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 390 x^{14} + 60701 x^{12} + 4799932 x^{10} + 203487156 x^{8} + 4519465040 x^{6} + \cdots + 178006118400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{4} - 1) q^{3} + (\beta_{2} - 17) q^{4} - \beta_{7} q^{5} + ( - \beta_{6} - 3 \beta_1) q^{6} + (\beta_{5} - \beta_{4} - \beta_{2} + 12) q^{7} + (\beta_{3} - 15 \beta_1) q^{8} + (\beta_{9} + \beta_{5} + \beta_{4} + \cdots + 88) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{4} - 1) q^{3} + (\beta_{2} - 17) q^{4} - \beta_{7} q^{5} + ( - \beta_{6} - 3 \beta_1) q^{6} + (\beta_{5} - \beta_{4} - \beta_{2} + 12) q^{7} + (\beta_{3} - 15 \beta_1) q^{8} + (\beta_{9} + \beta_{5} + \beta_{4} + \cdots + 88) q^{9}+ \cdots + ( - 216 \beta_{11} + 55 \beta_{10} + \cdots - 37160) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 18 q^{3} - 268 q^{4} + 190 q^{7} + 1394 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 18 q^{3} - 268 q^{4} + 190 q^{7} + 1394 q^{9} - 74 q^{10} - 1110 q^{11} + 1402 q^{12} + 2900 q^{16} - 7010 q^{21} - 12052 q^{25} + 4902 q^{26} + 4266 q^{27} - 16824 q^{28} + 19280 q^{30} - 2478 q^{33} + 20556 q^{34} - 51402 q^{36} - 11400 q^{37} + 12108 q^{38} + 16966 q^{40} + 3918 q^{41} + 125394 q^{44} + 17470 q^{46} + 3822 q^{47} - 78034 q^{48} - 32618 q^{49} - 24126 q^{53} - 164718 q^{58} - 81426 q^{62} + 219268 q^{63} + 158076 q^{64} + 98976 q^{65} + 23560 q^{67} - 222404 q^{70} - 50046 q^{71} - 196274 q^{73} + 141216 q^{74} + 214054 q^{75} - 239574 q^{77} - 90822 q^{78} + 317312 q^{81} - 215814 q^{83} + 438572 q^{84} - 346472 q^{85} + 197640 q^{86} - 857612 q^{90} - 132504 q^{95} - 574860 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 390 x^{14} + 60701 x^{12} + 4799932 x^{10} + 203487156 x^{8} + 4519465040 x^{6} + \cdots + 178006118400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 49 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 79\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 674098535 \nu^{14} - 245321490958 \nu^{12} - 34551894466035 \nu^{10} + \cdots + 28\!\cdots\!36 ) / 28\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4384882567 \nu^{14} - 1545295917902 \nu^{12} - 210316644411027 \nu^{10} + \cdots - 30\!\cdots\!20 ) / 28\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 674098535 \nu^{15} + 245321490958 \nu^{13} + 34551894466035 \nu^{11} + \cdots - 33\!\cdots\!92 \nu ) / 28\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7252561847 \nu^{15} + 2718118676590 \nu^{13} + 401348460870147 \nu^{11} + \cdots + 38\!\cdots\!48 \nu ) / 28\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10784407991 \nu^{14} + 4028491445614 \nu^{12} + 594960727700739 \nu^{10} + \cdots + 61\!\cdots\!80 ) / 28\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 15874352675 \nu^{14} + 6039075706054 \nu^{12} + 893613182617311 \nu^{10} + \cdots + 21\!\cdots\!48 ) / 28\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 9773844473 \nu^{14} + 3608835414034 \nu^{12} + 522067677879597 \nu^{10} + \cdots + 12\!\cdots\!88 ) / 14\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 8183270029 \nu^{14} - 3039557947162 \nu^{12} - 441895753596305 \nu^{10} + \cdots - 97\!\cdots\!56 ) / 93\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 9610708493 \nu^{15} - 3250606019610 \nu^{13} - 414880002053393 \nu^{11} + \cdots + 23\!\cdots\!88 \nu ) / 93\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 647921873 \nu^{15} + 259554609880 \nu^{13} + 41451572192793 \nu^{11} + \cdots + 12\!\cdots\!12 \nu ) / 43\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 5239827795 \nu^{15} + 1952240948390 \nu^{13} + 281406349626831 \nu^{11} + \cdots + 17\!\cdots\!56 \nu ) / 18\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 108115966759 \nu^{15} - 40418014870670 \nu^{13} + \cdots - 17\!\cdots\!96 \nu ) / 28\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 49 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 79\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{10} + \beta_{8} - 13\beta_{4} - 102\beta_{2} + 3861 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{15} + \beta_{14} + 2\beta_{12} + 27\beta_{7} + 32\beta_{6} - 116\beta_{3} + 6971\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12\beta_{11} + 155\beta_{10} + 30\beta_{9} - 153\beta_{8} - 116\beta_{5} + 3071\beta_{4} + 9752\beta_{2} - 339353 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 676 \beta_{15} - 137 \beta_{14} + 24 \beta_{13} - 406 \beta_{12} - 4663 \beta_{7} + \cdots - 640347 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1612 \beta_{11} - 18471 \beta_{10} - 5614 \beta_{9} + 18837 \beta_{8} + 22500 \beta_{5} + \cdots + 31058425 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 85844 \beta_{15} + 14469 \beta_{14} - 3224 \beta_{13} + 55806 \beta_{12} + 580859 \beta_{7} + \cdots + 60132987 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 124924 \beta_{11} + 2015023 \beta_{10} + 729814 \beta_{9} - 2138981 \beta_{8} - 3077108 \beta_{5} + \cdots - 2907542169 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 9767636 \beta_{15} - 1410133 \beta_{14} + 249848 \beta_{13} - 6626222 \beta_{12} + \cdots - 5733574315 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 3656924 \beta_{11} - 210974255 \beta_{10} - 81252086 \beta_{9} + 232820453 \beta_{8} + \cdots + 276525368009 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1050093588 \beta_{15} + 133379093 \beta_{14} - 7313848 \beta_{13} + 735264334 \beta_{12} + \cdots + 552923669851 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 834774532 \beta_{11} + 21606729663 \beta_{10} + 8267141558 \beta_{9} - 24705433237 \beta_{8} + \cdots - 26612991785209 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 109158608916 \beta_{15} - 12504813573 \beta_{14} - 1669549064 \beta_{13} + \cdots - 53781326022347 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/37\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
36.1
− 10.0606i
− 9.87320i
− 8.69480i
− 8.11553i
− 4.68483i
− 4.32146i
− 2.85171i
− 1.04262i
1.04262i
2.85171i
4.32146i
4.68483i
8.11553i
8.69480i
9.87320i
10.0606i
− 10.0606i −25.7580 −69.2166 − 77.4051i 259.142i 168.370 374.424i 420.475 −778.746
36.2 − 9.87320i −2.75305 −65.4800 100.817i 27.1814i −37.8971 330.555i −235.421 995.382
36.3 − 8.69480i 28.4744 −43.5996 − 14.1700i − 247.579i 47.5239 100.856i 567.789 −123.205
36.4 − 8.11553i 2.07043 −33.8618 − 43.8828i − 16.8026i −152.059 15.1093i −238.713 −356.132
36.5 − 4.68483i 5.81584 10.0523 − 4.49750i − 27.2462i 210.637 − 197.008i −209.176 −21.0700
36.6 − 4.32146i −24.1711 13.3250 61.4962i 104.454i 41.8425 − 195.870i 341.241 265.753
36.7 − 2.85171i −12.3080 23.8677 − 38.2713i 35.0990i −96.3094 − 159.319i −91.5127 −109.139
36.8 − 1.04262i 19.6295 30.9129 86.4713i − 20.4662i −87.1072 − 65.5944i 142.318 90.1568
36.9 1.04262i 19.6295 30.9129 − 86.4713i 20.4662i −87.1072 65.5944i 142.318 90.1568
36.10 2.85171i −12.3080 23.8677 38.2713i − 35.0990i −96.3094 159.319i −91.5127 −109.139
36.11 4.32146i −24.1711 13.3250 − 61.4962i − 104.454i 41.8425 195.870i 341.241 265.753
36.12 4.68483i 5.81584 10.0523 4.49750i 27.2462i 210.637 197.008i −209.176 −21.0700
36.13 8.11553i 2.07043 −33.8618 43.8828i 16.8026i −152.059 − 15.1093i −238.713 −356.132
36.14 8.69480i 28.4744 −43.5996 14.1700i 247.579i 47.5239 − 100.856i 567.789 −123.205
36.15 9.87320i −2.75305 −65.4800 − 100.817i − 27.1814i −37.8971 − 330.555i −235.421 995.382
36.16 10.0606i −25.7580 −69.2166 77.4051i − 259.142i 168.370 − 374.424i 420.475 −778.746
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 36.16
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 37.6.b.a ✓ 16
3.b odd 2 1 333.6.c.d 16
4.b odd 2 1 592.6.g.c 16
37.b even 2 1 inner 37.6.b.a ✓ 16
111.d odd 2 1 333.6.c.d 16
148.b odd 2 1 592.6.g.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.6.b.a ✓ 16 1.a even 1 1 trivial
37.6.b.a ✓ 16 37.b even 2 1 inner
333.6.c.d 16 3.b odd 2 1
333.6.c.d 16 111.d odd 2 1
592.6.g.c 16 4.b odd 2 1
592.6.g.c 16 148.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(37, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 178006118400 \) Copy content Toggle raw display
$3$ \( (T^{8} + 9 T^{7} + \cdots + 141986196)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 19\!\cdots\!44 \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 34\!\cdots\!92)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 64\!\cdots\!64)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 85\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 13\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 53\!\cdots\!01 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 14\!\cdots\!82)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 44\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots - 24\!\cdots\!68)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots - 39\!\cdots\!16)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 84\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots - 47\!\cdots\!72)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots - 18\!\cdots\!72)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 51\!\cdots\!46)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 21\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots - 15\!\cdots\!40)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 81\!\cdots\!96 \) Copy content Toggle raw display
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