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Properties

Label 225.3.j.d
Level $225$
Weight $3$
Character orbit 225.j
Analytic conductor $6.131$
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] = [225,3,Mod(101,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.101");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 225.j (of order \(6\), 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: \(6.13080594811\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 48x^{14} + 912x^{12} + 8767x^{10} + 45114x^{8} + 120426x^{6} + 147724x^{4} + 65727x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1) q^{2} - \beta_{3} q^{3} + (\beta_{14} - \beta_{9} + \cdots - \beta_{3}) q^{4}+ \cdots + ( - \beta_{15} + \beta_{14} + \beta_{10} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1) q^{2} - \beta_{3} q^{3} + (\beta_{14} - \beta_{9} + \cdots - \beta_{3}) q^{4}+ \cdots + (3 \beta_{15} - 2 \beta_{14} + \cdots + 27) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{3} + 16 q^{4} - q^{6} + q^{7} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{3} + 16 q^{4} - q^{6} + q^{7} - 10 q^{9} + 36 q^{11} + 79 q^{12} + 10 q^{13} + 81 q^{14} - 32 q^{16} - 79 q^{18} + 26 q^{19} - 15 q^{21} - 12 q^{22} - 27 q^{23} - 102 q^{24} - 88 q^{27} + 16 q^{28} - 135 q^{29} - 16 q^{31} + 108 q^{32} + 76 q^{33} - 18 q^{34} - 161 q^{36} + 22 q^{37} + 126 q^{38} + 43 q^{39} - 234 q^{41} - 165 q^{42} + 28 q^{43} - 108 q^{46} - 108 q^{47} + 424 q^{48} - 93 q^{49} + 182 q^{51} - 89 q^{52} + 362 q^{54} + 522 q^{56} + 212 q^{57} - 90 q^{58} + 297 q^{59} + 41 q^{61} - 501 q^{63} - 122 q^{64} + 109 q^{66} + 112 q^{67} - 819 q^{68} - 3 q^{69} - 393 q^{72} - 134 q^{73} - 486 q^{74} + 170 q^{76} - 54 q^{77} + 115 q^{78} + 11 q^{79} - 274 q^{81} - 102 q^{82} + 666 q^{83} - 465 q^{84} - 378 q^{86} + 233 q^{87} + 72 q^{88} + 58 q^{91} + 684 q^{92} + 555 q^{93} + 201 q^{94} + 298 q^{96} + 199 q^{97} + 304 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 48x^{14} + 912x^{12} + 8767x^{10} + 45114x^{8} + 120426x^{6} + 147724x^{4} + 65727x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5 \nu^{14} - 131 \nu^{12} - 401 \nu^{10} + 11822 \nu^{8} + 80996 \nu^{6} + 15710 \nu^{4} + \cdots - 12393 ) / 390960 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 125 \nu^{15} + 517 \nu^{14} + 5085 \nu^{13} + 22885 \nu^{12} + 71565 \nu^{11} + 388549 \nu^{10} + \cdots - 1824741 ) / 781920 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 153 \nu^{15} + 7339 \nu^{13} + 139405 \nu^{11} + 1340950 \nu^{9} + 6914264 \nu^{7} + 18506174 \nu^{5} + \cdots - 195480 ) / 390960 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 244 \nu^{15} + 83 \nu^{14} + 12040 \nu^{13} + 3695 \nu^{12} + 233728 \nu^{11} + 61391 \nu^{10} + \cdots + 184221 ) / 390960 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 517 \nu^{15} - 925 \nu^{14} + 22885 \nu^{13} - 42697 \nu^{12} + 388549 \nu^{11} - 771397 \nu^{10} + \cdots - 34911 ) / 781920 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 755 \nu^{15} - 1091 \nu^{14} + 36795 \nu^{13} - 50087 \nu^{12} + 712875 \nu^{11} - 894179 \nu^{10} + \cdots - 403353 ) / 781920 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1855 \nu^{14} - 87697 \nu^{12} - 1621387 \nu^{10} - 14823206 \nu^{8} - 69215228 \nu^{6} + \cdots + 171909 ) / 390960 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 440 \nu^{15} + 287 \nu^{14} - 20940 \nu^{13} + 13601 \nu^{12} - 392220 \nu^{11} + 252815 \nu^{10} + \cdots + 3459807 ) / 390960 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 287 \nu^{15} + 911 \nu^{14} + 13601 \nu^{13} + 41027 \nu^{12} + 252815 \nu^{11} + 712499 \nu^{10} + \cdots + 367713 ) / 390960 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 335 \nu^{15} + 19 \nu^{14} + 15655 \nu^{13} + 1439 \nu^{12} + 285335 \nu^{11} + 39027 \nu^{10} + \cdots + 1350945 ) / 260640 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1091 \nu^{15} - 565 \nu^{14} + 50087 \nu^{13} - 24577 \nu^{12} + 894179 \nu^{11} - 408037 \nu^{10} + \cdots + 36369 ) / 781920 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 545 \nu^{15} + 555 \nu^{14} + 26225 \nu^{13} + 25763 \nu^{12} + 499105 \nu^{11} + 466603 \nu^{10} + \cdots + 877149 ) / 260640 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 913 \nu^{15} + 43903 \nu^{13} + 836029 \nu^{11} + 8057522 \nu^{9} + 41566580 \nu^{7} + \cdots + 1172880 ) / 390960 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1705 \nu^{15} - 925 \nu^{14} - 79785 \nu^{13} - 43783 \nu^{12} - 1458765 \nu^{11} - 810493 \nu^{10} + \cdots + 92151 ) / 390960 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{7} + \beta_{3} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} - \beta_{12} - \beta_{10} + \beta_{6} - 2\beta_{4} + \beta_{3} - 10\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} - 12 \beta_{9} + \beta_{8} - 17 \beta_{7} + \cdots + 58 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 6 \beta_{14} - 14 \beta_{13} + 18 \beta_{12} - \beta_{11} + 18 \beta_{10} + 3 \beta_{9} + 9 \beta_{7} + \cdots + 19 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 43 \beta_{13} - 26 \beta_{12} + 29 \beta_{11} + 26 \beta_{10} + 149 \beta_{9} - 19 \beta_{8} + \cdots - 658 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2 \beta_{15} + 164 \beta_{14} + 183 \beta_{13} - 272 \beta_{12} + 24 \beta_{11} - 272 \beta_{10} + \cdots - 376 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 708 \beta_{13} + 468 \beta_{12} - 543 \beta_{11} - 468 \beta_{10} - 1925 \beta_{9} + 288 \beta_{8} + \cdots + 8130 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 90 \beta_{15} - 3240 \beta_{14} - 2423 \beta_{13} + 3932 \beta_{12} - 471 \beta_{11} + 3932 \beta_{10} + \cdots + 6902 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 10702 \beta_{13} - 7469 \beta_{12} + 8906 \beta_{11} + 7469 \beta_{10} + 25629 \beta_{9} + \cdots - 105881 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2466 \beta_{15} + 56442 \beta_{14} + 32725 \beta_{13} - 56196 \beta_{12} + 8600 \beta_{11} + \cdots - 117773 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 156455 \beta_{13} + 113419 \beta_{12} - 138313 \beta_{11} - 113419 \beta_{10} - 348943 \beta_{9} + \cdots + 1426733 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 53764 \beta_{15} - 922858 \beta_{14} - 449901 \beta_{13} + 802591 \beta_{12} - 149478 \beta_{11} + \cdots + 1911674 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 2257107 \beta_{13} - 1684062 \beta_{12} + 2092305 \beta_{11} + 1684062 \beta_{10} + 4831984 \beta_{9} + \cdots - 19669191 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1030446 \beta_{15} + 14551200 \beta_{14} + 6275002 \beta_{13} - 11494387 \beta_{12} + 2498610 \beta_{11} + \cdots - 30038617 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-\beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
− 3.82667i
− 2.45272i
− 2.21755i
− 0.0351539i
1.07659i
1.15749i
2.93619i
3.36183i
3.82667i
2.45272i
2.21755i
0.0351539i
− 1.07659i
− 1.15749i
− 2.93619i
− 3.36183i
−3.31400 − 1.91334i 0.962657 − 2.84135i 5.32172 + 9.21749i 0 −8.62671 + 7.57435i −2.98785 + 5.17511i − 25.4223i −7.14658 − 5.47050i 0
101.2 −2.12412 − 1.22636i −2.87327 + 0.862751i 1.00793 + 1.74578i 0 7.16121 + 1.69108i 1.26608 − 2.19292i 4.86656i 7.51132 − 4.95783i 0
101.3 −1.92045 − 1.10877i 2.24090 + 1.99458i 0.458755 + 0.794587i 0 −2.09201 − 6.31515i −1.17362 + 2.03278i 6.83556i 1.04331 + 8.93932i 0
101.4 −0.0304442 − 0.0175770i 2.35128 − 1.86319i −1.99938 − 3.46303i 0 −0.104332 + 0.0153951i 0.368467 − 0.638204i 0.281188i 2.05703 − 8.76177i 0
101.5 0.932351 + 0.538293i −1.15284 + 2.76965i −1.42048 − 2.46035i 0 −2.56574 + 1.96172i 4.76431 − 8.25202i − 7.36488i −6.34191 − 6.38593i 0
101.6 1.00241 + 0.578744i −2.59312 − 1.50856i −1.33011 − 2.30382i 0 −1.72631 − 3.01295i −5.34713 + 9.26150i − 7.70913i 4.44852 + 7.82373i 0
101.7 2.54282 + 1.46810i 2.37072 + 1.83839i 2.31062 + 4.00210i 0 3.32938 + 8.15514i −3.26469 + 5.65462i 1.82406i 2.24064 + 8.71662i 0
101.8 2.91143 + 1.68092i −0.306336 − 2.98432i 3.65096 + 6.32364i 0 4.12451 − 9.20356i 6.87444 − 11.9069i 11.1005i −8.81232 + 1.82841i 0
176.1 −3.31400 + 1.91334i 0.962657 + 2.84135i 5.32172 − 9.21749i 0 −8.62671 − 7.57435i −2.98785 − 5.17511i 25.4223i −7.14658 + 5.47050i 0
176.2 −2.12412 + 1.22636i −2.87327 − 0.862751i 1.00793 − 1.74578i 0 7.16121 − 1.69108i 1.26608 + 2.19292i − 4.86656i 7.51132 + 4.95783i 0
176.3 −1.92045 + 1.10877i 2.24090 − 1.99458i 0.458755 − 0.794587i 0 −2.09201 + 6.31515i −1.17362 − 2.03278i − 6.83556i 1.04331 − 8.93932i 0
176.4 −0.0304442 + 0.0175770i 2.35128 + 1.86319i −1.99938 + 3.46303i 0 −0.104332 − 0.0153951i 0.368467 + 0.638204i − 0.281188i 2.05703 + 8.76177i 0
176.5 0.932351 − 0.538293i −1.15284 − 2.76965i −1.42048 + 2.46035i 0 −2.56574 − 1.96172i 4.76431 + 8.25202i 7.36488i −6.34191 + 6.38593i 0
176.6 1.00241 − 0.578744i −2.59312 + 1.50856i −1.33011 + 2.30382i 0 −1.72631 + 3.01295i −5.34713 − 9.26150i 7.70913i 4.44852 − 7.82373i 0
176.7 2.54282 − 1.46810i 2.37072 − 1.83839i 2.31062 − 4.00210i 0 3.32938 − 8.15514i −3.26469 − 5.65462i − 1.82406i 2.24064 − 8.71662i 0
176.8 2.91143 − 1.68092i −0.306336 + 2.98432i 3.65096 − 6.32364i 0 4.12451 + 9.20356i 6.87444 + 11.9069i − 11.1005i −8.81232 − 1.82841i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.3.j.d yes 16
3.b odd 2 1 675.3.j.d 16
5.b even 2 1 225.3.j.c ✓ 16
5.c odd 4 2 225.3.i.c 32
9.c even 3 1 675.3.j.d 16
9.d odd 6 1 inner 225.3.j.d yes 16
15.d odd 2 1 675.3.j.c 16
15.e even 4 2 675.3.i.b 32
45.h odd 6 1 225.3.j.c ✓ 16
45.j even 6 1 675.3.j.c 16
45.k odd 12 2 675.3.i.b 32
45.l even 12 2 225.3.i.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.3.i.c 32 5.c odd 4 2
225.3.i.c 32 45.l even 12 2
225.3.j.c ✓ 16 5.b even 2 1
225.3.j.c ✓ 16 45.h odd 6 1
225.3.j.d yes 16 1.a even 1 1 trivial
225.3.j.d yes 16 9.d odd 6 1 inner
675.3.i.b 32 15.e even 4 2
675.3.i.b 32 45.k odd 12 2
675.3.j.c 16 15.d odd 2 1
675.3.j.c 16 45.j even 6 1
675.3.j.d 16 3.b odd 2 1
675.3.j.d 16 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 24 T_{2}^{14} + 408 T_{2}^{12} - 108 T_{2}^{11} - 3329 T_{2}^{10} + 1089 T_{2}^{9} + \cdots + 81 \) acting on \(S_{3}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 24 T^{14} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{16} - 2 T^{15} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 57329598096 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 38424724717824 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 59\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{8} - 13 T^{7} + \cdots - 851603306)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 50\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 74\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 63\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( (T^{8} - 11 T^{7} + \cdots - 5085928700)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 59\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 76\!\cdots\!49 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 13\!\cdots\!41 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 59\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 9213055191250)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 42\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 37\!\cdots\!01 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 92\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 12\!\cdots\!25 \) Copy content Toggle raw display
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