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Properties

Label 37.4.c.a
Level $37$
Weight $4$
Character orbit 37.c
Analytic conductor $2.183$
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] = [37,4,Mod(10,37)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(37, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("37.10");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 37.c (of order \(3\), degree \(2\), 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: \(2.18307067021\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 2 x^{17} + 59 x^{16} - 54 x^{15} + 2186 x^{14} - 1424 x^{13} + 46875 x^{12} - 4582 x^{11} + \cdots + 5308416 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{11} + \beta_{6}) q^{3} + ( - \beta_{15} - 4 \beta_{7} + \cdots - 4) q^{4}+ \cdots + (\beta_{16} + \beta_{14} + \cdots - 3 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{11} + \beta_{6}) q^{3} + ( - \beta_{15} - 4 \beta_{7} + \cdots - 4) q^{4}+ \cdots + (17 \beta_{16} - 56 \beta_{15} + \cdots + 76 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 2 q^{2} + q^{3} - 42 q^{4} + 3 q^{5} + 12 q^{6} - 25 q^{7} - 120 q^{8} - 98 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 2 q^{2} + q^{3} - 42 q^{4} + 3 q^{5} + 12 q^{6} - 25 q^{7} - 120 q^{8} - 98 q^{9} + 48 q^{10} + 76 q^{11} + 156 q^{12} + 85 q^{13} + 16 q^{14} + 15 q^{15} - 118 q^{16} - 11 q^{17} + 312 q^{18} + 91 q^{19} + 272 q^{20} - 13 q^{21} - 50 q^{22} + 448 q^{23} - 576 q^{24} - 420 q^{25} + 276 q^{26} - 1094 q^{27} - 402 q^{28} - 368 q^{29} - 710 q^{30} + 540 q^{31} + 332 q^{32} - 296 q^{33} - 46 q^{34} + 749 q^{35} + 2156 q^{36} - 992 q^{37} + 1936 q^{38} - 401 q^{39} - 362 q^{40} - 999 q^{41} + 254 q^{42} - 692 q^{43} + 850 q^{44} + 904 q^{45} - 1164 q^{46} + 1156 q^{47} - 2272 q^{48} - 430 q^{49} - 96 q^{50} + 2226 q^{51} + 644 q^{52} - 63 q^{53} - 2526 q^{54} - 26 q^{55} - 586 q^{56} - 189 q^{57} + 2278 q^{58} - 409 q^{59} - 416 q^{60} + 455 q^{61} - 2226 q^{62} + 4480 q^{63} - 2852 q^{64} - 445 q^{65} - 6824 q^{66} - 1889 q^{67} + 5620 q^{68} + 440 q^{69} - 520 q^{70} + 327 q^{71} + 5532 q^{72} + 644 q^{73} + 838 q^{74} + 5312 q^{75} + 2582 q^{76} - 2088 q^{77} - 1722 q^{78} + 1571 q^{79} - 408 q^{80} + 1099 q^{81} + 5068 q^{82} - 3673 q^{83} - 8744 q^{84} - 3618 q^{85} - 1706 q^{86} - 1158 q^{87} - 2236 q^{88} + 169 q^{89} - 2498 q^{90} + 2311 q^{91} - 1452 q^{92} - 4232 q^{93} + 1746 q^{94} + 65 q^{95} - 6712 q^{96} + 972 q^{97} - 1380 q^{98} - 3358 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 2 x^{17} + 59 x^{16} - 54 x^{15} + 2186 x^{14} - 1424 x^{13} + 46875 x^{12} - 4582 x^{11} + \cdots + 5308416 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 45\!\cdots\!49 \nu^{17} + \cdots - 52\!\cdots\!68 ) / 42\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 12\!\cdots\!67 \nu^{17} + \cdots - 92\!\cdots\!04 ) / 42\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 47\!\cdots\!71 \nu^{17} + \cdots + 25\!\cdots\!28 ) / 95\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 24\!\cdots\!67 \nu^{17} + \cdots + 37\!\cdots\!64 ) / 28\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11\!\cdots\!55 \nu^{17} + \cdots - 28\!\cdots\!16 ) / 34\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 15\!\cdots\!21 \nu^{17} + \cdots - 10\!\cdots\!12 ) / 38\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 21\!\cdots\!51 \nu^{17} + \cdots - 16\!\cdots\!24 ) / 34\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 22\!\cdots\!15 \nu^{17} + \cdots + 10\!\cdots\!12 ) / 17\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 24\!\cdots\!61 \nu^{17} + \cdots - 13\!\cdots\!24 ) / 85\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 69\!\cdots\!49 \nu^{17} + \cdots - 59\!\cdots\!60 ) / 19\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 17\!\cdots\!13 \nu^{17} + \cdots + 22\!\cdots\!60 ) / 43\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 87\!\cdots\!27 \nu^{17} + \cdots - 83\!\cdots\!32 ) / 19\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 19\!\cdots\!35 \nu^{17} + \cdots - 25\!\cdots\!40 ) / 42\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 62\!\cdots\!83 \nu^{17} + \cdots - 15\!\cdots\!40 ) / 12\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 73\!\cdots\!65 \nu^{17} + \cdots + 19\!\cdots\!88 ) / 76\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 44\!\cdots\!69 \nu^{17} + \cdots + 32\!\cdots\!48 ) / 45\!\cdots\!16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{15} - 12\beta_{7} + \beta_{3} + \beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} - 2\beta_{6} + 20\beta_{3} - 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{16} + 25 \beta_{15} + 2 \beta_{14} + \beta_{13} + 2 \beta_{12} - 4 \beta_{11} + 2 \beta_{8} + \cdots - 32 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6 \beta_{16} + 8 \beta_{15} + 6 \beta_{14} + 30 \beta_{13} + 6 \beta_{12} - 82 \beta_{11} + 30 \beta_{9} + \cdots + 226 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 50\beta_{9} - 88\beta_{8} + 236\beta_{6} + 88\beta_{5} + 72\beta_{4} - 935\beta_{3} + 593\beta_{2} + 5184 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 16 \beta_{17} - 276 \beta_{16} - 480 \beta_{15} - 324 \beta_{14} - 787 \beta_{13} - 308 \beta_{12} + \cdots + 10546 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 48 \beta_{17} - 2158 \beta_{16} - 14265 \beta_{15} - 3134 \beta_{14} - 1803 \beta_{13} - 2958 \beta_{12} + \cdots - 119828 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 976 \beta_{10} - 20208 \beta_{9} + 11474 \beta_{8} - 79458 \beta_{6} - 12706 \beta_{5} - 9426 \beta_{4} + \cdots - 227386 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 3280 \beta_{17} + 61412 \beta_{16} + 350993 \beta_{15} + 101668 \beta_{14} + 57388 \beta_{13} + \cdots - 786477 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 40256 \beta_{17} + 289888 \beta_{16} + 683936 \beta_{15} + 437440 \beta_{14} + 520373 \beta_{13} + \cdots + 6911902 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 147552 \beta_{10} + 1725141 \beta_{9} - 2677354 \beta_{8} + 9745812 \beta_{6} + 3128266 \beta_{5} + \cdots + 70954252 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1411040 \beta_{17} - 8509886 \beta_{16} - 22110216 \beta_{15} - 14046238 \beta_{14} + \cdots + 168093883 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 5536352 \beta_{17} - 47772416 \beta_{16} - 226652305 \beta_{15} - 93178928 \beta_{14} + \cdots - 1798409456 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 45406512 \beta_{10} - 356630519 \beta_{9} + 353048972 \beta_{8} - 1865786554 \beta_{6} + \cdots - 6041385718 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 188183120 \beta_{17} + 1328545958 \beta_{16} + 5919296441 \beta_{15} + 2718307158 \beta_{14} + \cdots - 18491424226 \beta_1 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 1389761200 \beta_{17} + 6921794026 \beta_{16} + 20473140840 \beta_{15} + 12945298842 \beta_{14} + \cdots + 175082673130 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1 - \beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
−2.29711 + 3.97872i
−2.22320 + 3.85069i
−1.18500 + 2.05247i
−0.877263 + 1.51946i
0.108844 − 0.188523i
0.695130 − 1.20400i
1.87727 − 3.25153i
2.25557 − 3.90676i
2.64576 − 4.58259i
−2.29711 − 3.97872i
−2.22320 − 3.85069i
−1.18500 − 2.05247i
−0.877263 − 1.51946i
0.108844 + 0.188523i
0.695130 + 1.20400i
1.87727 + 3.25153i
2.25557 + 3.90676i
2.64576 + 4.58259i
−2.29711 + 3.97872i −2.23093 − 3.86409i −6.55347 − 11.3509i −5.11264 − 8.85535i 20.4988 0.224385 + 0.388646i 23.4625 3.54587 − 6.14163i 46.9773
10.2 −2.22320 + 3.85069i 2.65861 + 4.60485i −5.88523 − 10.1935i 8.36602 + 14.4904i −23.6425 −14.1618 − 24.5289i 16.7650 −0.636450 + 1.10236i −74.3973
10.3 −1.18500 + 2.05247i 3.66231 + 6.34331i 1.19157 + 2.06386i −7.12563 − 12.3420i −17.3593 8.28951 + 14.3579i −24.6080 −13.3251 + 23.0797i 33.7754
10.4 −0.877263 + 1.51946i −2.09463 − 3.62801i 2.46082 + 4.26226i 6.70214 + 11.6084i 7.35019 11.2358 + 19.4610i −22.6714 4.72501 − 8.18396i −23.5182
10.5 0.108844 − 0.188523i −4.07974 − 7.06632i 3.97631 + 6.88716i −6.20388 − 10.7454i −1.77622 −14.4299 − 24.9934i 3.47269 −19.7886 + 34.2749i −2.70102
10.6 0.695130 − 1.20400i 1.89202 + 3.27708i 3.03359 + 5.25433i 1.59836 + 2.76845i 5.26081 −4.73151 − 8.19522i 19.5570 6.34050 − 10.9821i 4.44428
10.7 1.87727 − 3.25153i −1.14202 − 1.97804i −3.04829 − 5.27979i −6.80013 − 11.7782i −8.57553 13.2412 + 22.9344i 7.14647 10.8916 − 18.8648i −51.0627
10.8 2.25557 − 3.90676i −3.09491 − 5.36054i −6.17519 − 10.6957i 10.3474 + 17.9222i −27.9231 −7.44732 − 12.8991i −19.6252 −5.65694 + 9.79811i 93.3573
10.9 2.64576 − 4.58259i 4.92929 + 8.53779i −10.0001 − 17.3207i −0.271659 − 0.470527i 52.1669 −4.72036 − 8.17590i −63.4992 −35.0958 + 60.7878i −2.87498
26.1 −2.29711 − 3.97872i −2.23093 + 3.86409i −6.55347 + 11.3509i −5.11264 + 8.85535i 20.4988 0.224385 − 0.388646i 23.4625 3.54587 + 6.14163i 46.9773
26.2 −2.22320 − 3.85069i 2.65861 − 4.60485i −5.88523 + 10.1935i 8.36602 − 14.4904i −23.6425 −14.1618 + 24.5289i 16.7650 −0.636450 − 1.10236i −74.3973
26.3 −1.18500 − 2.05247i 3.66231 − 6.34331i 1.19157 − 2.06386i −7.12563 + 12.3420i −17.3593 8.28951 − 14.3579i −24.6080 −13.3251 − 23.0797i 33.7754
26.4 −0.877263 − 1.51946i −2.09463 + 3.62801i 2.46082 − 4.26226i 6.70214 − 11.6084i 7.35019 11.2358 − 19.4610i −22.6714 4.72501 + 8.18396i −23.5182
26.5 0.108844 + 0.188523i −4.07974 + 7.06632i 3.97631 − 6.88716i −6.20388 + 10.7454i −1.77622 −14.4299 + 24.9934i 3.47269 −19.7886 − 34.2749i −2.70102
26.6 0.695130 + 1.20400i 1.89202 − 3.27708i 3.03359 − 5.25433i 1.59836 − 2.76845i 5.26081 −4.73151 + 8.19522i 19.5570 6.34050 + 10.9821i 4.44428
26.7 1.87727 + 3.25153i −1.14202 + 1.97804i −3.04829 + 5.27979i −6.80013 + 11.7782i −8.57553 13.2412 − 22.9344i 7.14647 10.8916 + 18.8648i −51.0627
26.8 2.25557 + 3.90676i −3.09491 + 5.36054i −6.17519 + 10.6957i 10.3474 − 17.9222i −27.9231 −7.44732 + 12.8991i −19.6252 −5.65694 − 9.79811i 93.3573
26.9 2.64576 + 4.58259i 4.92929 − 8.53779i −10.0001 + 17.3207i −0.271659 + 0.470527i 52.1669 −4.72036 + 8.17590i −63.4992 −35.0958 − 60.7878i −2.87498
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.4.c.a ✓ 18
3.b odd 2 1 333.4.f.c 18
37.c even 3 1 inner 37.4.c.a ✓ 18
37.c even 3 1 1369.4.a.e 9
37.e even 6 1 1369.4.a.f 9
111.i odd 6 1 333.4.f.c 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.4.c.a ✓ 18 1.a even 1 1 trivial
37.4.c.a ✓ 18 37.c even 3 1 inner
333.4.f.c 18 3.b odd 2 1
333.4.f.c 18 111.i odd 6 1
1369.4.a.e 9 37.c even 3 1
1369.4.a.f 9 37.e even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(37, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} - 2 T^{17} + \cdots + 5308416 \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots + 9814836842496 \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 39\!\cdots\!21 \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( (T^{9} + \cdots + 1139821940736)^{2} \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 38\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 17\!\cdots\!21 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 57\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( (T^{9} + \cdots - 24\!\cdots\!28)^{2} \) Copy content Toggle raw display
$29$ \( (T^{9} + \cdots + 16\!\cdots\!42)^{2} \) Copy content Toggle raw display
$31$ \( (T^{9} + \cdots + 32\!\cdots\!48)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 21\!\cdots\!33 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 85\!\cdots\!69 \) Copy content Toggle raw display
$43$ \( (T^{9} + \cdots - 79\!\cdots\!08)^{2} \) Copy content Toggle raw display
$47$ \( (T^{9} + \cdots - 81\!\cdots\!88)^{2} \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 80\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 21\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 83\!\cdots\!69 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( (T^{9} + \cdots - 88\!\cdots\!88)^{2} \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 38\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 43\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 39\!\cdots\!09 \) Copy content Toggle raw display
$97$ \( (T^{9} + \cdots + 37\!\cdots\!58)^{2} \) Copy content Toggle raw display
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