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Properties

Label 33.7.b.a
Level $33$
Weight $7$
Character orbit 33.b
Analytic conductor $7.592$
Analytic rank $0$
Dimension $20$
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] = [33,7,Mod(23,33)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(33, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("33.23");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 33.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: \(7.59178475946\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 1026 x^{18} + 441321 x^{16} + 103808124 x^{14} + 14594358456 x^{12} + 1256133373152 x^{10} + \cdots + 32\!\cdots\!64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{17}\cdot 11^{10} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} + 1) q^{3} + (\beta_{2} - 39) q^{4} + \beta_{6} q^{5} + (\beta_{8} - 2 \beta_1 + 14) q^{6} + ( - \beta_{4} + \beta_{3} + 8) q^{7} + (\beta_{5} - 3 \beta_{3} - 37 \beta_1 + 1) q^{8} + (\beta_{11} + \beta_{6} - \beta_{3} + \cdots - 53) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{3} + 1) q^{3} + (\beta_{2} - 39) q^{4} + \beta_{6} q^{5} + (\beta_{8} - 2 \beta_1 + 14) q^{6} + ( - \beta_{4} + \beta_{3} + 8) q^{7} + (\beta_{5} - 3 \beta_{3} - 37 \beta_1 + 1) q^{8} + (\beta_{11} + \beta_{6} - \beta_{3} + \cdots - 53) q^{9}+ \cdots + ( - 54 \beta_{19} + 27 \beta_{18} + \cdots + 11006) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 16 q^{3} - 772 q^{4} + 286 q^{6} + 160 q^{7} - 1072 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 16 q^{3} - 772 q^{4} + 286 q^{6} + 160 q^{7} - 1072 q^{9} + 996 q^{10} + 6092 q^{12} + 808 q^{13} - 3032 q^{15} + 28004 q^{16} + 18686 q^{18} + 5920 q^{19} - 20888 q^{21} - 48096 q^{24} - 100612 q^{25} - 17624 q^{27} - 33296 q^{28} + 109582 q^{30} - 90896 q^{31} - 21296 q^{33} + 68928 q^{34} - 28988 q^{36} + 239656 q^{37} - 15416 q^{39} + 34632 q^{40} + 150364 q^{42} - 125840 q^{43} - 242428 q^{45} + 244380 q^{46} + 305492 q^{48} - 186204 q^{49} - 21992 q^{51} - 120368 q^{52} - 777728 q^{54} - 191664 q^{55} - 255840 q^{57} + 601176 q^{58} + 970736 q^{60} + 1108360 q^{61} + 574088 q^{63} - 2533132 q^{64} + 465850 q^{66} + 617728 q^{67} + 323804 q^{69} - 238680 q^{70} - 2031648 q^{72} - 1379960 q^{73} + 2481512 q^{75} + 4678408 q^{76} - 1556840 q^{78} + 347152 q^{79} - 1086136 q^{81} - 1760328 q^{82} - 345760 q^{84} - 4097232 q^{85} - 2983056 q^{87} - 622908 q^{88} - 4093630 q^{90} + 979616 q^{91} + 2363236 q^{93} - 217752 q^{94} + 8811824 q^{96} - 3139256 q^{97} + 212960 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 1026 x^{18} + 441321 x^{16} + 103808124 x^{14} + 14594358456 x^{12} + 1256133373152 x^{10} + \cdots + 32\!\cdots\!64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 103 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 18\!\cdots\!53 \nu^{19} + \cdots - 91\!\cdots\!40 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 18\!\cdots\!53 \nu^{19} + \cdots - 15\!\cdots\!60 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 18\!\cdots\!53 \nu^{19} + \cdots - 12\!\cdots\!80 ) / 34\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 77\!\cdots\!15 \nu^{19} + \cdots - 22\!\cdots\!08 \nu ) / 86\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 14\!\cdots\!85 \nu^{19} + \cdots + 56\!\cdots\!64 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 14\!\cdots\!79 \nu^{19} + \cdots + 25\!\cdots\!12 ) / 59\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 28\!\cdots\!17 \nu^{19} + \cdots - 75\!\cdots\!36 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 10\!\cdots\!41 \nu^{19} + \cdots + 60\!\cdots\!64 ) / 34\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 34\!\cdots\!23 \nu^{19} + \cdots - 97\!\cdots\!08 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 76\!\cdots\!51 \nu^{19} + \cdots + 18\!\cdots\!80 ) / 20\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 18\!\cdots\!53 \nu^{19} + \cdots - 74\!\cdots\!20 ) / 43\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 14\!\cdots\!27 \nu^{19} + \cdots - 75\!\cdots\!12 ) / 34\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 96\!\cdots\!95 \nu^{19} + \cdots + 31\!\cdots\!60 ) / 16\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 29\!\cdots\!93 \nu^{19} + \cdots + 35\!\cdots\!88 ) / 34\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 60\!\cdots\!35 \nu^{19} + \cdots + 48\!\cdots\!12 ) / 34\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 10\!\cdots\!99 \nu^{19} + \cdots - 16\!\cdots\!28 ) / 20\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 34\!\cdots\!07 \nu^{19} + \cdots + 46\!\cdots\!64 ) / 52\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 103 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - 3\beta_{3} - 165\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 3 \beta_{17} - 2 \beta_{15} - 5 \beta_{14} + \beta_{13} + 2 \beta_{10} - \beta_{9} + 4 \beta_{8} + \cdots + 17107 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4 \beta_{19} + 6 \beta_{18} + 3 \beta_{17} - 20 \beta_{16} + 8 \beta_{15} + \beta_{14} + 17 \beta_{13} + \cdots - 415 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1129 \beta_{17} + 560 \beta_{15} + 1593 \beta_{14} - 323 \beta_{13} + 140 \beta_{11} - 788 \beta_{10} + \cdots - 3333125 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 938 \beta_{19} - 1952 \beta_{18} - 1409 \beta_{17} + 9250 \beta_{16} - 3750 \beta_{15} + \cdots + 138641 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 328131 \beta_{17} - 120258 \beta_{15} - 394481 \beta_{14} + 93045 \beta_{13} - 83594 \beta_{11} + \cdots + 709315871 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 135292 \beta_{19} + 442738 \beta_{18} + 497969 \beta_{17} - 3080732 \beta_{16} + 1210824 \beta_{15} + \cdots - 41316039 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 87691193 \beta_{17} + 23376692 \beta_{15} + 90193309 \beta_{14} - 26065059 \beta_{13} + \cdots - 158844273061 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 5986582 \beta_{19} - 79733948 \beta_{18} - 156760521 \beta_{17} + 906047582 \beta_{16} + \cdots + 11541186337 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 22630072363 \beta_{17} - 4261775030 \beta_{15} - 19962492901 \beta_{14} + 7177428821 \beta_{13} + \cdots + 36686162740695 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 4820394472 \beta_{19} + 10249514342 \beta_{18} + 46295830421 \beta_{17} - 250810612648 \beta_{16} + \cdots - 3105586855463 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 5745892287009 \beta_{17} + 724396369776 \beta_{15} + 4352303058137 \beta_{14} - 1949142583491 \beta_{13} + \cdots - 86\!\cdots\!93 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 2544515036342 \beta_{19} - 56507963560 \beta_{18} - 13132279560317 \beta_{17} + 67148844261794 \beta_{16} + \cdots + 817247420494337 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 14\!\cdots\!43 \beta_{17} - 109733308050138 \beta_{15} - 941841102783785 \beta_{14} + \cdots + 20\!\cdots\!79 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 913399405568316 \beta_{19} - 583315569631942 \beta_{18} + \cdots - 21\!\cdots\!51 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 36\!\cdots\!85 \beta_{17} + \cdots - 49\!\cdots\!89 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 28\!\cdots\!34 \beta_{19} + \cdots + 54\!\cdots\!65 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).

\(n\) \(13\) \(23\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
− 15.8546i
− 14.5373i
− 12.6955i
− 11.1144i
− 9.36068i
− 9.18066i
− 8.58590i
− 5.13283i
− 2.50689i
− 0.582665i
0.582665i
2.50689i
5.13283i
8.58590i
9.18066i
9.36068i
11.1144i
12.6955i
14.5373i
15.8546i
− 15.8546i 10.1117 + 25.0350i −187.368 128.653i 396.920 − 160.317i 185.193 1955.95i −524.506 + 506.295i 2039.74
23.2 − 14.5373i −26.2741 − 6.21862i −147.333 − 220.756i −90.4019 + 381.955i 126.134 1211.44i 651.658 + 326.777i −3209.19
23.3 − 12.6955i 16.0155 − 21.7372i −97.1752 − 9.46491i −275.964 − 203.324i −297.499 421.175i −216.009 − 696.262i −120.162
23.4 − 11.1144i −19.8885 − 18.2606i −59.5295 223.873i −202.955 + 221.048i −152.100 − 49.6872i 62.1023 + 726.350i 2488.21
23.5 − 9.36068i 26.7168 + 3.90029i −23.6224 − 36.7722i 36.5094 − 250.088i 305.058 − 377.962i 698.575 + 208.407i −344.213
23.6 − 9.18066i −4.20063 + 26.6712i −20.2844 − 92.7007i 244.859 + 38.5645i −527.539 − 401.338i −693.709 − 224.072i −851.053
23.7 − 8.58590i −21.1258 + 16.8137i −9.71763 61.7994i 144.360 + 181.384i 427.595 − 466.063i 163.602 − 710.405i 530.603
23.8 − 5.13283i −8.59633 − 25.5950i 37.6541 − 122.121i −131.375 + 44.1235i 168.062 − 521.773i −581.206 + 440.046i −626.825
23.9 − 2.50689i 23.0874 + 13.9990i 57.7155 201.357i 35.0940 − 57.8777i −481.867 − 305.128i 337.056 + 646.401i 504.781
23.10 − 0.582665i 12.1539 − 24.1098i 63.6605 147.782i −14.0479 − 7.08168i 326.963 − 74.3834i −433.563 − 586.058i 86.1076
23.11 0.582665i 12.1539 + 24.1098i 63.6605 − 147.782i −14.0479 + 7.08168i 326.963 74.3834i −433.563 + 586.058i 86.1076
23.12 2.50689i 23.0874 − 13.9990i 57.7155 − 201.357i 35.0940 + 57.8777i −481.867 305.128i 337.056 − 646.401i 504.781
23.13 5.13283i −8.59633 + 25.5950i 37.6541 122.121i −131.375 − 44.1235i 168.062 521.773i −581.206 − 440.046i −626.825
23.14 8.58590i −21.1258 − 16.8137i −9.71763 − 61.7994i 144.360 − 181.384i 427.595 466.063i 163.602 + 710.405i 530.603
23.15 9.18066i −4.20063 − 26.6712i −20.2844 92.7007i 244.859 − 38.5645i −527.539 401.338i −693.709 + 224.072i −851.053
23.16 9.36068i 26.7168 − 3.90029i −23.6224 36.7722i 36.5094 + 250.088i 305.058 377.962i 698.575 − 208.407i −344.213
23.17 11.1144i −19.8885 + 18.2606i −59.5295 − 223.873i −202.955 − 221.048i −152.100 49.6872i 62.1023 − 726.350i 2488.21
23.18 12.6955i 16.0155 + 21.7372i −97.1752 9.46491i −275.964 + 203.324i −297.499 − 421.175i −216.009 + 696.262i −120.162
23.19 14.5373i −26.2741 + 6.21862i −147.333 220.756i −90.4019 − 381.955i 126.134 − 1211.44i 651.658 − 326.777i −3209.19
23.20 15.8546i 10.1117 − 25.0350i −187.368 − 128.653i 396.920 + 160.317i 185.193 − 1955.95i −524.506 − 506.295i 2039.74
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.7.b.a ✓ 20
3.b odd 2 1 inner 33.7.b.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.7.b.a ✓ 20 1.a even 1 1 trivial
33.7.b.a ✓ 20 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(33, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 32\!\cdots\!64 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 42\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 19\!\cdots\!32)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 161051)^{10} \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots - 24\!\cdots\!44)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 44\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 62\!\cdots\!84)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 43\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 58\!\cdots\!16)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots - 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 50\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots - 42\!\cdots\!68)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 62\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 29\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 88\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 12\!\cdots\!72)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 88\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 26\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 40\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 23\!\cdots\!80)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 70\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 32\!\cdots\!04)^{2} \) Copy content Toggle raw display
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