Properties

Label 600.2.y.d
Level $600$
Weight $2$
Character orbit 600.y
Analytic conductor $4.791$
Analytic rank $0$
Dimension $12$
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] = [600,2,Mod(121,600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(600, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("600.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 600.y (of order \(5\), degree \(4\), 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: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 6 x^{10} + x^{9} - 14 x^{8} + 10 x^{7} + 35 x^{6} - 110 x^{5} + 230 x^{4} - 325 x^{3} + 300 x^{2} - 250 x + 125 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + ( - \beta_{11} - \beta_{10} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{5} + (\beta_{8} - \beta_{7} + \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1) q^{7} - \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{3} + ( - \beta_{11} - \beta_{10} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{5} + (\beta_{8} - \beta_{7} + \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1) q^{7} - \beta_{3} q^{9} + ( - \beta_{11} + 2 \beta_{9} + 2 \beta_{7} - 2 \beta_{5} + \beta_{2}) q^{11} + ( - \beta_{10} + \beta_{5} + \beta_{4} + \beta_{2}) q^{13} + ( - \beta_{9} - \beta_{6} + 1) q^{15} + ( - 2 \beta_{10} - \beta_{6} + \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} - 2) q^{17} + (\beta_{9} + \beta_{7} - 2 \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_1 + 2) q^{19} + ( - \beta_{11} + \beta_{9} - \beta_{8} + \beta_{3} - 1) q^{21} + (2 \beta_{9} + \beta_{8} + 2 \beta_{7} + \beta_{4} - \beta_1 - 2) q^{23} + ( - \beta_{11} - \beta_{10} + \beta_{9} + 2 \beta_{8} - \beta_{6} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + \cdots - 1) q^{25}+ \cdots + (\beta_{11} + \beta_{10} - \beta_{2} - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{3} - 4 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 3 q^{3} - 4 q^{7} - 3 q^{9} + 10 q^{11} + 10 q^{13} + 5 q^{15} - 9 q^{17} + 13 q^{19} - q^{21} - 13 q^{23} + 3 q^{27} + q^{29} + 3 q^{31} + 5 q^{33} - 20 q^{35} - 2 q^{37} - 10 q^{39} + 12 q^{41} + 8 q^{43} + 23 q^{47} - 12 q^{49} - 26 q^{51} - 14 q^{53} + 15 q^{55} + 22 q^{57} + 30 q^{59} + 12 q^{61} + q^{63} + 10 q^{65} - 2 q^{67} - 12 q^{69} + 25 q^{71} + 12 q^{73} - 20 q^{75} - 33 q^{77} + 34 q^{79} - 3 q^{81} + 16 q^{83} - 15 q^{85} - 6 q^{87} + 13 q^{89} - 7 q^{91} - 18 q^{93} + 5 q^{95} + 28 q^{97} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 6 x^{10} + x^{9} - 14 x^{8} + 10 x^{7} + 35 x^{6} - 110 x^{5} + 230 x^{4} - 325 x^{3} + 300 x^{2} - 250 x + 125 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 8500 \nu^{11} + 27691 \nu^{10} - 20739 \nu^{9} - 55289 \nu^{8} + 111206 \nu^{7} + 70641 \nu^{6} - 396950 \nu^{5} + 602875 \nu^{4} - 982560 \nu^{3} + \cdots + 675625 ) / 374525 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8797 \nu^{11} - 32897 \nu^{10} + 41413 \nu^{9} + 21743 \nu^{8} - 123267 \nu^{7} + 62971 \nu^{6} + 289305 \nu^{5} - 899230 \nu^{4} + 1845250 \nu^{3} - 2461995 \nu^{2} + \cdots - 1238725 ) / 374525 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 40926 \nu^{11} - 112175 \nu^{10} + 101300 \nu^{9} + 175115 \nu^{8} - 345870 \nu^{7} - 63756 \nu^{6} + 1348465 \nu^{5} - 2678090 \nu^{4} + 5881015 \nu^{3} + \cdots - 4105300 ) / 374525 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 42604 \nu^{11} + 116457 \nu^{10} - 103013 \nu^{9} - 180213 \nu^{8} + 353992 \nu^{7} + 75931 \nu^{6} - 1393785 \nu^{5} + 2785270 \nu^{4} - 6054855 \nu^{3} + \cdots + 4795375 ) / 374525 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 67540 \nu^{11} - 184923 \nu^{10} + 163897 \nu^{9} + 284782 \nu^{8} - 564428 \nu^{7} - 116598 \nu^{6} + 2231415 \nu^{5} - 4422365 \nu^{4} + 9666380 \nu^{3} + \cdots - 7323600 ) / 374525 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 102504 \nu^{11} - 303172 \nu^{10} + 308798 \nu^{9} + 405938 \nu^{8} - 1025517 \nu^{7} + 57319 \nu^{6} + 3586600 \nu^{5} - 7705180 \nu^{4} + 15939430 \nu^{3} + \cdots - 11045025 ) / 374525 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 132868 \nu^{11} - 375458 \nu^{10} + 364552 \nu^{9} + 534927 \nu^{8} - 1217313 \nu^{7} - 53541 \nu^{6} + 4501195 \nu^{5} - 9364890 \nu^{4} + 19877275 \nu^{3} + \cdots - 14403300 ) / 374525 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 136311 \nu^{11} - 386732 \nu^{10} + 370398 \nu^{9} + 564408 \nu^{8} - 1256242 \nu^{7} - 81583 \nu^{6} + 4691080 \nu^{5} - 9591220 \nu^{4} + 20149035 \nu^{3} + \cdots - 14601675 ) / 374525 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 154033 \nu^{11} + 447428 \nu^{10} - 442987 \nu^{9} - 633032 \nu^{8} + 1498533 \nu^{7} + 26626 \nu^{6} - 5410370 \nu^{5} + 11237145 \nu^{4} + \cdots + 16535300 ) / 374525 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 155098 \nu^{11} - 444623 \nu^{10} + 433467 \nu^{9} + 643337 \nu^{8} - 1489778 \nu^{7} - 32386 \nu^{6} + 5432570 \nu^{5} - 11273445 \nu^{4} + 23226025 \nu^{3} + \cdots - 16066775 ) / 374525 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 158566 \nu^{11} + 460173 \nu^{10} - 458557 \nu^{9} - 645092 \nu^{8} + 1543428 \nu^{7} - 10416 \nu^{6} - 5566125 \nu^{5} + 11700245 \nu^{4} + \cdots + 16705350 ) / 374525 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2 \beta_{11} + \beta_{10} - 4 \beta_{9} - \beta_{8} - \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta _1 + 4 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{11} + \beta_{10} - 4 \beta_{9} - 6 \beta_{8} - \beta_{7} + 3 \beta_{6} + \beta_{5} - 6 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + 2 \beta _1 + 4 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{11} + \beta_{10} - 4 \beta_{9} - 11 \beta_{8} - \beta_{7} - 2 \beta_{6} + 16 \beta_{5} + 19 \beta_{4} + 8 \beta_{3} + \beta_{2} + 2 \beta _1 - 16 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 17 \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + 4 \beta_{7} + 23 \beta_{6} + 11 \beta_{5} + 29 \beta_{4} + 13 \beta_{3} - 14 \beta_{2} - 3 \beta _1 - 21 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 8 \beta_{11} - 9 \beta_{10} + \beta_{9} + 14 \beta_{8} - 6 \beta_{7} - 2 \beta_{6} + 46 \beta_{5} + 69 \beta_{4} - 17 \beta_{3} - 9 \beta_{2} + 2 \beta _1 - 21 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 57 \beta_{11} + 31 \beta_{10} + 6 \beta_{9} + 44 \beta_{8} - 36 \beta_{7} + 48 \beta_{6} + \beta_{5} + 24 \beta_{4} + 8 \beta_{3} - 14 \beta_{2} + 27 \beta _1 - 26 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 103 \beta_{11} - 69 \beta_{10} - 44 \beta_{9} + 19 \beta_{8} - 96 \beta_{7} - 52 \beta_{6} + 11 \beta_{5} - 101 \beta_{4} - 32 \beta_{3} - 9 \beta_{2} + 67 \beta _1 + 99 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 63 \beta_{11} + 21 \beta_{10} - 89 \beta_{9} - \beta_{8} - 141 \beta_{7} - 142 \beta_{6} - 119 \beta_{5} - 46 \beta_{4} + 303 \beta_{3} + 96 \beta_{2} + 82 \beta _1 - 256 ) / 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 333 \beta_{11} - 299 \beta_{10} + 76 \beta_{9} - 26 \beta_{8} + 154 \beta_{7} - 52 \beta_{6} - 539 \beta_{5} - 371 \beta_{4} + 363 \beta_{3} - 164 \beta_{2} - 103 \beta _1 - 121 ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 508 \beta_{11} - 84 \beta_{10} + 576 \beta_{9} + 264 \beta_{8} + 669 \beta_{7} - 677 \beta_{6} - 454 \beta_{5} + 944 \beta_{4} + 733 \beta_{3} + 366 \beta_{2} - 398 \beta _1 - 1321 ) / 5 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 782 \beta_{11} + 131 \beta_{10} + 1906 \beta_{9} + 1319 \beta_{8} + 1764 \beta_{7} + 1048 \beta_{6} - 2374 \beta_{5} - 976 \beta_{4} - 192 \beta_{3} - 564 \beta_{2} - 1073 \beta _1 + 799 ) / 5 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
1.03979 1.59275i
−1.74662 + 0.753236i
1.70682 + 0.839517i
0.0830190 + 1.17264i
1.17529 0.0257946i
−0.258306 1.14684i
0.0830190 1.17264i
1.17529 + 0.0257946i
−0.258306 + 1.14684i
1.03979 + 1.59275i
−1.74662 0.753236i
1.70682 0.839517i
0 0.809017 + 0.587785i 0 −1.87239 1.22235i 0 −2.19700 0 0.309017 + 0.951057i 0
121.2 0 0.809017 + 0.587785i 0 0.885482 + 2.05327i 0 2.29387 0 0.309017 + 0.951057i 0
121.3 0 0.809017 + 0.587785i 0 0.986912 2.00649i 0 −1.09688 0 0.309017 + 0.951057i 0
241.1 0 −0.309017 0.951057i 0 −2.23049 + 0.157911i 0 3.52598 0 −0.809017 + 0.587785i 0
241.2 0 −0.309017 0.951057i 0 0.0490643 + 2.23553i 0 −1.25468 0 −0.809017 + 0.587785i 0
241.3 0 −0.309017 0.951057i 0 2.18142 0.491328i 0 −3.27131 0 −0.809017 + 0.587785i 0
361.1 0 −0.309017 + 0.951057i 0 −2.23049 0.157911i 0 3.52598 0 −0.809017 0.587785i 0
361.2 0 −0.309017 + 0.951057i 0 0.0490643 2.23553i 0 −1.25468 0 −0.809017 0.587785i 0
361.3 0 −0.309017 + 0.951057i 0 2.18142 + 0.491328i 0 −3.27131 0 −0.809017 0.587785i 0
481.1 0 0.809017 0.587785i 0 −1.87239 + 1.22235i 0 −2.19700 0 0.309017 0.951057i 0
481.2 0 0.809017 0.587785i 0 0.885482 2.05327i 0 2.29387 0 0.309017 0.951057i 0
481.3 0 0.809017 0.587785i 0 0.986912 + 2.00649i 0 −1.09688 0 0.309017 0.951057i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.2.y.d 12
25.d even 5 1 inner 600.2.y.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.y.d 12 1.a even 1 1 trivial
600.2.y.d 12 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} + 2T_{7}^{5} - 16T_{7}^{4} - 37T_{7}^{3} + 41T_{7}^{2} + 140T_{7} + 80 \) acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{12} + 15 T^{9} - 20 T^{8} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( (T^{6} + 2 T^{5} - 16 T^{4} - 37 T^{3} + \cdots + 80)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} - 10 T^{11} + 87 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( T^{12} - 10 T^{11} + 47 T^{10} + \cdots + 361 \) Copy content Toggle raw display
$17$ \( T^{12} + 9 T^{11} + 107 T^{10} + \cdots + 4116841 \) Copy content Toggle raw display
$19$ \( T^{12} - 13 T^{11} + 69 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( T^{12} + 13 T^{11} + 158 T^{10} + \cdots + 160000 \) Copy content Toggle raw display
$29$ \( T^{12} - T^{11} + 44 T^{10} - 50 T^{9} + \cdots + 36481 \) Copy content Toggle raw display
$31$ \( T^{12} - 3 T^{11} + 34 T^{10} + \cdots + 160000 \) Copy content Toggle raw display
$37$ \( T^{12} + 2 T^{11} - 6 T^{10} + \cdots + 990025 \) Copy content Toggle raw display
$41$ \( T^{12} - 12 T^{11} + 49 T^{10} + \cdots + 44342281 \) Copy content Toggle raw display
$43$ \( (T^{6} - 4 T^{5} - 109 T^{4} + 682 T^{3} + \cdots - 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} - 23 T^{11} + \cdots + 11337990400 \) Copy content Toggle raw display
$53$ \( T^{12} + 14 T^{11} + \cdots + 2663076025 \) Copy content Toggle raw display
$59$ \( T^{12} - 30 T^{11} + \cdots + 1097994496 \) Copy content Toggle raw display
$61$ \( T^{12} - 12 T^{11} + 189 T^{10} + \cdots + 58997761 \) Copy content Toggle raw display
$67$ \( T^{12} + 2 T^{11} + \cdots + 15332382976 \) Copy content Toggle raw display
$71$ \( T^{12} - 25 T^{11} + 343 T^{10} + \cdots + 90935296 \) Copy content Toggle raw display
$73$ \( T^{12} - 12 T^{11} + \cdots + 3713805481 \) Copy content Toggle raw display
$79$ \( T^{12} - 34 T^{11} + \cdots + 130627307776 \) Copy content Toggle raw display
$83$ \( T^{12} - 16 T^{11} + 461 T^{10} + \cdots + 65536 \) Copy content Toggle raw display
$89$ \( T^{12} - 13 T^{11} + 263 T^{10} + \cdots + 1575025 \) Copy content Toggle raw display
$97$ \( T^{12} - 28 T^{11} + \cdots + 264355081 \) Copy content Toggle raw display
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