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Properties

Label 925.2.bb.a
Level $925$
Weight $2$
Character orbit 925.bb
Analytic conductor $7.386$
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] = [925,2,Mod(151,925)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(925, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("925.151");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.bb (of order \(18\), degree \(6\), 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: \(7.38616218697\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{18})\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 37)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{11} - \beta_{4}) q^{2} + ( - \beta_{16} + \beta_{12} + 1) q^{3} + ( - \beta_{15} + 2 \beta_{11} + \cdots - \beta_1) q^{4}+ \cdots + (\beta_{17} + \beta_{14} - \beta_{13} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{11} - \beta_{4}) q^{2} + ( - \beta_{16} + \beta_{12} + 1) q^{3} + ( - \beta_{15} + 2 \beta_{11} + \cdots - \beta_1) q^{4}+ \cdots + ( - 2 \beta_{17} + \beta_{16} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 9 q^{2} + 9 q^{3} - 3 q^{4} + 3 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 9 q^{2} + 9 q^{3} - 3 q^{4} + 3 q^{7} - 3 q^{9} + 9 q^{11} - 3 q^{12} - 9 q^{13} + 9 q^{14} - 3 q^{16} + 15 q^{17} - 6 q^{18} + 6 q^{19} - 12 q^{21} + 15 q^{22} + 9 q^{23} + 45 q^{24} - 15 q^{26} - 21 q^{27} + 27 q^{28} - 18 q^{29} - 6 q^{33} - 33 q^{34} - 72 q^{36} - 6 q^{37} - 54 q^{38} - 6 q^{39} - 24 q^{41} - 21 q^{42} + 18 q^{46} + 36 q^{47} - 3 q^{48} + 21 q^{49} + 81 q^{51} - 51 q^{52} + 39 q^{53} + 45 q^{54} + 81 q^{56} - 15 q^{57} - 33 q^{58} - 6 q^{59} + 42 q^{61} + 24 q^{62} + 27 q^{63} + 6 q^{64} - 81 q^{66} - 36 q^{69} - 9 q^{71} + 63 q^{72} + 54 q^{73} + 15 q^{74} - 69 q^{76} - 33 q^{77} + 45 q^{78} - 6 q^{79} - 45 q^{81} - 27 q^{82} + 24 q^{83} - 24 q^{84} - 42 q^{86} - 21 q^{87} - 54 q^{88} + 18 q^{89} - 3 q^{91} - 66 q^{92} - 66 q^{93} + 18 q^{94} + 15 q^{96} + 9 q^{97} + 45 q^{98} + 60 q^{99}+O(q^{100}) \) 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
\(\beta_{2}\)\(=\) \( ( - 69 \nu^{16} - 2010 \nu^{14} - 21225 \nu^{12} - 107422 \nu^{10} - 284274 \nu^{8} - 398496 \nu^{6} + \cdots - 13392 ) / 128 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 21 \nu^{17} - 57 \nu^{16} + 600 \nu^{15} - 1678 \nu^{14} + 6125 \nu^{13} - 18037 \nu^{12} + \cdots - 19808 ) / 128 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 153 \nu^{17} + 234 \nu^{16} - 4530 \nu^{15} + 6868 \nu^{14} - 49165 \nu^{13} + 73458 \nu^{12} + \cdots + 65824 ) / 512 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 279 \nu^{17} + 8094 \nu^{15} + 84867 \nu^{13} + 424554 \nu^{11} + 1102022 \nu^{9} + 1494432 \nu^{7} + \cdots - 256 ) / 512 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 691 \nu^{17} - 306 \nu^{16} - 20262 \nu^{15} - 9060 \nu^{14} - 216367 \nu^{13} - 98330 \nu^{12} + \cdots - 128416 ) / 1024 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 691 \nu^{17} + 306 \nu^{16} - 20262 \nu^{15} + 9060 \nu^{14} - 216367 \nu^{13} + 98330 \nu^{12} + \cdots + 128416 ) / 1024 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 127 \nu^{17} - 229 \nu^{16} - 3758 \nu^{15} - 6714 \nu^{14} - 40747 \nu^{13} - 71689 \nu^{12} + \cdots - 65488 ) / 256 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 667 \nu^{17} + 554 \nu^{16} + 19846 \nu^{15} + 16052 \nu^{14} + 217111 \nu^{13} + 167986 \nu^{12} + \cdots + 91552 ) / 1024 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 681 \nu^{17} + 754 \nu^{16} - 20018 \nu^{15} + 22180 \nu^{14} - 214685 \nu^{13} + 238170 \nu^{12} + \cdots + 259232 ) / 1024 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 169 \nu^{17} + 343 \nu^{16} + 4958 \nu^{15} + 10070 \nu^{14} + 52997 \nu^{13} + 107763 \nu^{12} + \cdots + 105104 ) / 256 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 169 \nu^{17} - 343 \nu^{16} + 4958 \nu^{15} - 10070 \nu^{14} + 52997 \nu^{13} - 107763 \nu^{12} + \cdots - 105104 ) / 256 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1037 \nu^{17} - 2286 \nu^{16} + 30218 \nu^{15} - 67068 \nu^{14} + 319281 \nu^{13} - 716870 \nu^{12} + \cdots - 657888 ) / 1024 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 691 \nu^{17} - 3330 \nu^{16} - 20262 \nu^{15} - 97604 \nu^{14} - 216367 \nu^{13} - 1041578 \nu^{12} + \cdots - 920992 ) / 1024 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 2523 \nu^{17} + 1818 \nu^{16} + 73766 \nu^{15} + 53332 \nu^{14} + 783895 \nu^{13} + 569954 \nu^{12} + \cdots + 524704 ) / 1024 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 837 \nu^{17} - 106 \nu^{16} + 24538 \nu^{15} - 3076 \nu^{14} + 261961 \nu^{13} - 32274 \nu^{12} + \cdots - 16224 ) / 256 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 890 \nu^{17} - 601 \nu^{16} - 26096 \nu^{15} - 17578 \nu^{14} - 278666 \nu^{13} - 186909 \nu^{12} + \cdots - 148720 ) / 256 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{17} - \beta_{15} + \beta_{10} - \beta_{9} + \beta_{6} + \beta_{5} - 2\beta_{4} - \beta_{3} - \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{16} - 3 \beta_{15} - \beta_{14} - \beta_{13} + 3 \beta_{12} + \beta_{11} + \beta_{10} + \beta_{9} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8 \beta_{17} - 3 \beta_{16} + 11 \beta_{15} - \beta_{14} + 3 \beta_{13} + 3 \beta_{12} + \beta_{11} + \cdots + 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 5 \beta_{17} - 9 \beta_{16} + 40 \beta_{15} + 13 \beta_{14} + 19 \beta_{13} - 57 \beta_{12} + \cdots - 53 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 75 \beta_{17} + 54 \beta_{16} - 129 \beta_{15} + 26 \beta_{14} - 54 \beta_{13} - 68 \beta_{12} + \cdots - 278 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 93 \beta_{17} + 79 \beta_{16} - 514 \beta_{15} - 171 \beta_{14} - 265 \beta_{13} + 819 \beta_{12} + \cdots + 765 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 828 \beta_{17} - 772 \beta_{16} + 1600 \beta_{15} - 436 \beta_{14} + 772 \beta_{13} + 1078 \beta_{12} + \cdots + 3100 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1365 \beta_{17} - 766 \beta_{16} + 6639 \beta_{15} + 2254 \beta_{14} + 3496 \beta_{13} - 10992 \beta_{12} + \cdots - 10402 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 9974 \beta_{17} + 10377 \beta_{16} - 20351 \beta_{15} + 6319 \beta_{14} - 10377 \beta_{13} + \cdots - 37415 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 18648 \beta_{17} + 8283 \beta_{16} - 86009 \beta_{15} - 29539 \beta_{14} - 45579 \beta_{13} + \cdots + 138011 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 125127 \beta_{17} - 136617 \beta_{16} + 261744 \beta_{15} - 86255 \beta_{14} + 136617 \beta_{13} + \cdots + 469379 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 247697 \beta_{17} - 97158 \beta_{16} + 1115619 \beta_{15} + 385382 \beta_{14} + 592552 \beta_{13} + \cdots - 1810496 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 1599045 \beta_{17} + 1784373 \beta_{16} - 3383418 \beta_{15} + 1146323 \beta_{14} - 1784373 \beta_{13} + \cdots - 5996963 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 3250014 \beta_{17} + 1197218 \beta_{16} - 14478084 \beta_{15} - 5015426 \beta_{14} - 7697246 \beta_{13} + \cdots + 23623192 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 20608451 \beta_{17} - 23229040 \beta_{16} + 43837491 \beta_{15} - 15047460 \beta_{14} + 23229040 \beta_{13} + \cdots + 77276212 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 42405000 \beta_{17} - 15148291 \beta_{16} + 187934249 \beta_{15} + 65190479 \beta_{14} + \cdots - 307435721 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/925\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(852\)
\(\chi(n)\) \(\beta_{11}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1
3.60322i
0.885952i
− 1.23399i
2.47983i
0.834738i
− 0.660907i
− 2.47983i
− 0.834738i
0.660907i
1.92581i
− 0.752039i
− 1.77531i
− 1.92581i
0.752039i
1.77531i
− 3.60322i
− 0.885952i
1.23399i
−1.48976 + 1.77542i 2.36330 − 1.98304i −0.585455 − 3.32028i 0 7.15011i 1.02732 + 0.373914i 2.75280 + 1.58933i 1.13178 − 6.41863i 0
151.2 0.256873 − 0.306129i 0.0473670 − 0.0397456i 0.319565 + 1.81234i 0 − 0.0247100i 4.17818 + 1.52073i 1.32907 + 0.767337i −0.520281 + 2.95066i 0
151.3 1.61954 − 1.93010i 0.968719 − 0.812851i −0.755055 − 4.28213i 0 − 3.18617i −2.99976 − 1.09182i −5.12375 − 2.95820i −0.243256 + 1.37957i 0
176.1 −0.502458 + 0.0885970i −0.199899 + 1.13369i −1.63477 + 0.595008i 0 − 0.587340i −0.422615 − 0.354616i 1.65240 − 0.954012i 1.57379 + 0.572814i 0
176.2 1.11764 − 0.197069i 0.522172 − 2.96139i −0.669111 + 0.243536i 0 − 3.41266i −1.25550 − 1.05349i −2.66549 + 1.53892i −5.67807 − 2.06665i 0
176.3 2.59056 − 0.456785i −0.354362 + 2.00969i 4.62296 − 1.68262i 0 5.36808i 1.58572 + 1.33057i 6.65125 − 3.84010i −1.09419 − 0.398252i 0
226.1 −0.502458 − 0.0885970i −0.199899 − 1.13369i −1.63477 − 0.595008i 0 0.587340i −0.422615 + 0.354616i 1.65240 + 0.954012i 1.57379 − 0.572814i 0
226.2 1.11764 + 0.197069i 0.522172 + 2.96139i −0.669111 − 0.243536i 0 3.41266i −1.25550 + 1.05349i −2.66549 − 1.53892i −5.67807 + 2.06665i 0
226.3 2.59056 + 0.456785i −0.354362 − 2.00969i 4.62296 + 1.68262i 0 − 5.36808i 1.58572 − 1.33057i 6.65125 + 3.84010i −1.09419 + 0.398252i 0
326.1 −0.424710 − 1.16688i 0.702528 + 0.255699i 0.350853 − 0.294401i 0 − 0.928365i 0.231838 + 1.31482i −2.64335 − 1.52614i −1.86997 − 1.56909i 0
326.2 0.491168 + 1.34947i 2.59058 + 0.942893i −0.0477438 + 0.0400618i 0 3.95904i −0.593686 − 3.36696i 2.40985 + 1.39133i 3.52391 + 2.95691i 0
326.3 0.841146 + 2.31103i −2.14040 − 0.779043i −3.10124 + 2.60225i 0 − 5.60182i −0.251492 − 1.42628i −4.36277 − 2.51884i 1.67628 + 1.40657i 0
576.1 −0.424710 + 1.16688i 0.702528 − 0.255699i 0.350853 + 0.294401i 0 0.928365i 0.231838 − 1.31482i −2.64335 + 1.52614i −1.86997 + 1.56909i 0
576.2 0.491168 − 1.34947i 2.59058 − 0.942893i −0.0477438 − 0.0400618i 0 − 3.95904i −0.593686 + 3.36696i 2.40985 − 1.39133i 3.52391 − 2.95691i 0
576.3 0.841146 − 2.31103i −2.14040 + 0.779043i −3.10124 − 2.60225i 0 5.60182i −0.251492 + 1.42628i −4.36277 + 2.51884i 1.67628 − 1.40657i 0
876.1 −1.48976 − 1.77542i 2.36330 + 1.98304i −0.585455 + 3.32028i 0 − 7.15011i 1.02732 − 0.373914i 2.75280 − 1.58933i 1.13178 + 6.41863i 0
876.2 0.256873 + 0.306129i 0.0473670 + 0.0397456i 0.319565 − 1.81234i 0 0.0247100i 4.17818 − 1.52073i 1.32907 − 0.767337i −0.520281 − 2.95066i 0
876.3 1.61954 + 1.93010i 0.968719 + 0.812851i −0.755055 + 4.28213i 0 3.18617i −2.99976 + 1.09182i −5.12375 + 2.95820i −0.243256 − 1.37957i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 151.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.h even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 925.2.bb.a 18
5.b even 2 1 37.2.h.a ✓ 18
5.c odd 4 2 925.2.ba.a 36
15.d odd 2 1 333.2.bl.d 18
20.d odd 2 1 592.2.bq.d 18
37.h even 18 1 inner 925.2.bb.a 18
185.v even 18 1 37.2.h.a ✓ 18
185.v even 18 1 1369.2.b.g 18
185.x even 18 1 1369.2.b.g 18
185.y odd 36 2 925.2.ba.a 36
185.ba odd 36 2 1369.2.a.m 18
555.bu odd 18 1 333.2.bl.d 18
740.bu odd 18 1 592.2.bq.d 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.h.a ✓ 18 5.b even 2 1
37.2.h.a ✓ 18 185.v even 18 1
333.2.bl.d 18 15.d odd 2 1
333.2.bl.d 18 555.bu odd 18 1
592.2.bq.d 18 20.d odd 2 1
592.2.bq.d 18 740.bu odd 18 1
925.2.ba.a 36 5.c odd 4 2
925.2.ba.a 36 185.y odd 36 2
925.2.bb.a 18 1.a even 1 1 trivial
925.2.bb.a 18 37.h even 18 1 inner
1369.2.a.m 18 185.ba odd 36 2
1369.2.b.g 18 185.v even 18 1
1369.2.b.g 18 185.x even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} - 9 T_{2}^{17} + 42 T_{2}^{16} - 135 T_{2}^{15} + 345 T_{2}^{14} - 837 T_{2}^{13} + \cdots + 243 \) acting on \(S_{2}^{\mathrm{new}}(925, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} - 9 T^{17} + \cdots + 243 \) Copy content Toggle raw display
$3$ \( T^{18} - 9 T^{17} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{18} \) Copy content Toggle raw display
$7$ \( T^{18} - 3 T^{17} + \cdots + 36864 \) Copy content Toggle raw display
$11$ \( T^{18} - 9 T^{17} + \cdots + 46656 \) Copy content Toggle raw display
$13$ \( T^{18} + 9 T^{17} + \cdots + 27 \) Copy content Toggle raw display
$17$ \( T^{18} - 15 T^{17} + \cdots + 23705163 \) Copy content Toggle raw display
$19$ \( T^{18} - 6 T^{17} + \cdots + 5614272 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 180910927872 \) Copy content Toggle raw display
$29$ \( T^{18} + 18 T^{17} + \cdots + 30509163 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 1142154432 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 129961739795077 \) Copy content Toggle raw display
$41$ \( T^{18} + 24 T^{17} + \cdots + 6561 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 74566243008 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 2176782336 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 1450162667529 \) 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
$67$ \( T^{18} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 20139015744 \) 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
$83$ \( T^{18} + \cdots + 4807480896 \) 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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