Properties

Label 603.4.a.c
Level $603$
Weight $4$
Character orbit 603.a
Self dual yes
Analytic conductor $35.578$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [603,4,Mod(1,603)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(603, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("603.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 603 = 3^{2} \cdot 67 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 603.a (trivial)

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: yes
Analytic conductor: \(35.5781517335\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 38x^{5} + 18x^{4} + 373x^{3} - 151x^{2} - 956x + 498 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 201)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + \beta_1 + 3) q^{4} + ( - \beta_{5} + \beta_{2} - 1) q^{5} + ( - \beta_{6} - 2 \beta_{5} + \beta_{4} - 4) q^{7} + ( - 2 \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 + 6) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + \beta_1 + 3) q^{4} + ( - \beta_{5} + \beta_{2} - 1) q^{5} + ( - \beta_{6} - 2 \beta_{5} + \beta_{4} - 4) q^{7} + ( - 2 \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 + 6) q^{8} + ( - 2 \beta_{6} + \beta_{5} + \beta_{4} - 3 \beta_{3} - \beta_{2} + 2 \beta_1 - 7) q^{10} + ( - 2 \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{2} - 3 \beta_1 + 19) q^{11} + ( - \beta_{6} + 3 \beta_{5} - 4 \beta_{4} - \beta_{3} + \beta_{2} + 8 \beta_1 + 1) q^{13} + (\beta_{6} - \beta_{5} - \beta_{4} - 5 \beta_{3} + 2 \beta_{2} - 5 \beta_1 + 3) q^{14} + ( - 5 \beta_{5} + 7 \beta_{4} + 5 \beta_{2} + 20 \beta_1 + 10) q^{16} + ( - 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 5 \beta_{2} - 13 \beta_1 - 12) q^{17} + (2 \beta_{6} - 4 \beta_{5} - 2 \beta_{4} + 8 \beta_{3} + 5 \beta_{2} + 3 \beta_1 - 20) q^{19} + (4 \beta_{6} - 6 \beta_{5} + 5 \beta_{4} + 4 \beta_{3} + 7 \beta_{2} + 3 \beta_1 + 52) q^{20} + ( - 2 \beta_{6} + 3 \beta_{5} + \beta_{4} - 2 \beta_{3} + 35 \beta_1 - 33) q^{22} + (5 \beta_{6} - 6 \beta_{5} - 8 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - \beta_1 + 60) q^{23} + (8 \beta_{6} - 2 \beta_{4} + 7 \beta_{3} + 3 \beta_{2} - \beta_1 - 21) q^{25} + ( - \beta_{6} + \beta_{5} - 2 \beta_{4} + 5 \beta_{3} + 7 \beta_{2} + 20 \beta_1 + 87) q^{26} + (3 \beta_{6} - \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + 7 \beta_1 - 37) q^{28} + ( - \beta_{6} - \beta_{5} + 10 \beta_{4} + 13 \beta_{3} - 10 \beta_{2} + 15 \beta_1 + 33) q^{29} + ( - 5 \beta_{6} - 7 \beta_{5} - 8 \beta_{4} - 10 \beta_{3} + 6 \beta_{2} + \cdots - 71) q^{31}+ \cdots + (62 \beta_{6} - 16 \beta_{5} - 65 \beta_{4} + 108 \beta_{3} - 5 \beta_{2} + \cdots - 64) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + q^{2} + 21 q^{4} - 11 q^{5} - 33 q^{7} + 45 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + q^{2} + 21 q^{4} - 11 q^{5} - 33 q^{7} + 45 q^{8} - 51 q^{10} + 130 q^{11} + 16 q^{13} - 5 q^{14} + 77 q^{16} - 90 q^{17} - 132 q^{19} + 359 q^{20} - 192 q^{22} + 399 q^{23} - 132 q^{25} + 638 q^{26} - 245 q^{28} + 302 q^{29} - 555 q^{31} + 1031 q^{32} - 832 q^{34} + 775 q^{35} + 297 q^{37} - 98 q^{38} + 305 q^{40} + 717 q^{41} - 245 q^{43} + 1766 q^{44} - 497 q^{46} + 1072 q^{47} + 314 q^{49} - 454 q^{50} + 1344 q^{52} - 265 q^{53} + 1096 q^{55} + 477 q^{56} + 1610 q^{58} + 255 q^{59} + 418 q^{61} + 191 q^{62} + 1889 q^{64} - 262 q^{65} + 469 q^{67} - 3720 q^{68} + 1309 q^{70} + 1194 q^{71} + 995 q^{73} - 259 q^{74} + 1506 q^{76} - 230 q^{77} - 2640 q^{79} + 1949 q^{80} + 1535 q^{82} + 2579 q^{83} - 562 q^{85} - 1991 q^{86} + 3624 q^{88} + 1604 q^{89} - 2116 q^{91} + 351 q^{92} + 6178 q^{94} + 3028 q^{95} - 808 q^{97} - 258 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 38x^{5} + 18x^{4} + 373x^{3} - 151x^{2} - 956x + 498 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - 47\nu^{5} + 27\nu^{4} + 1339\nu^{3} - 408\nu^{2} - 6314\nu + 1966 ) / 466 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -25\nu^{6} + 10\nu^{5} + 956\nu^{4} + 310\nu^{3} - 9139\nu^{2} - 5250\nu + 17954 ) / 932 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -35\nu^{6} + 14\nu^{5} + 1152\nu^{4} + 434\nu^{3} - 7389\nu^{2} - 4554\nu + 4818 ) / 932 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -87\nu^{6} + 128\nu^{5} + 3010\nu^{4} - 2556\nu^{3} - 23919\nu^{2} + 13418\nu + 32134 ) / 1864 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{6} + \beta_{5} + 2\beta_{4} - \beta_{3} + \beta_{2} + 18\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{5} + 7\beta_{4} + 29\beta_{2} + 44\beta _1 + 210 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -58\beta_{6} + 22\beta_{5} + 67\beta_{4} - 39\beta_{3} + 53\beta_{2} + 427\beta _1 + 343 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -48\beta_{6} - 170\beta_{5} + 282\beta_{4} - 28\beta_{3} + 777\beta_{2} + 1501\beta _1 + 4939 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−4.44080
−3.26131
−2.11972
0.535294
1.85609
3.01477
5.41567
−4.44080 0 11.7207 15.6745 0 −4.28763 −16.5228 0 −69.6071
1.2 −3.26131 0 2.63612 −7.83356 0 −27.8825 17.4933 0 25.5476
1.3 −2.11972 0 −3.50678 −1.76139 0 22.2326 24.3912 0 3.73365
1.4 0.535294 0 −7.71346 −12.7037 0 −8.67573 −8.41132 0 −6.80020
1.5 1.85609 0 −4.55494 4.35854 0 17.6123 −23.3031 0 8.08983
1.6 3.01477 0 1.08886 −14.7190 0 −32.2394 −20.8355 0 −44.3744
1.7 5.41567 0 21.3295 5.98460 0 0.240383 72.1883 0 32.4106
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(67\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 603.4.a.c 7
3.b odd 2 1 201.4.a.c 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.4.a.c 7 3.b odd 2 1
603.4.a.c 7 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} - T_{2}^{6} - 38T_{2}^{5} + 18T_{2}^{4} + 373T_{2}^{3} - 151T_{2}^{2} - 956T_{2} + 498 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(603))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} - T^{6} - 38 T^{5} + 18 T^{4} + \cdots + 498 \) Copy content Toggle raw display
$3$ \( T^{7} \) Copy content Toggle raw display
$5$ \( T^{7} + 11 T^{6} - 311 T^{5} + \cdots - 1054848 \) Copy content Toggle raw display
$7$ \( T^{7} + 33 T^{6} - 813 T^{5} + \cdots - 3147392 \) Copy content Toggle raw display
$11$ \( T^{7} - 130 T^{6} + \cdots - 3855752448 \) Copy content Toggle raw display
$13$ \( T^{7} - 16 T^{6} + \cdots + 36060533248 \) Copy content Toggle raw display
$17$ \( T^{7} + 90 T^{6} + \cdots - 47177769168 \) Copy content Toggle raw display
$19$ \( T^{7} + 132 T^{6} + \cdots + 2352293777728 \) Copy content Toggle raw display
$23$ \( T^{7} - 399 T^{6} + \cdots - 176992942356 \) Copy content Toggle raw display
$29$ \( T^{7} + \cdots + 154337966321664 \) Copy content Toggle raw display
$31$ \( T^{7} + 555 T^{6} + \cdots + 6536744991936 \) Copy content Toggle raw display
$37$ \( T^{7} - 297 T^{6} + \cdots + 18\!\cdots\!52 \) Copy content Toggle raw display
$41$ \( T^{7} - 717 T^{6} + \cdots - 17\!\cdots\!28 \) Copy content Toggle raw display
$43$ \( T^{7} + 245 T^{6} + \cdots + 15\!\cdots\!28 \) Copy content Toggle raw display
$47$ \( T^{7} - 1072 T^{6} + \cdots + 50\!\cdots\!08 \) Copy content Toggle raw display
$53$ \( T^{7} + 265 T^{6} + \cdots - 42\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{7} - 255 T^{6} + \cdots - 18\!\cdots\!88 \) Copy content Toggle raw display
$61$ \( T^{7} - 418 T^{6} + \cdots + 34\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( (T - 67)^{7} \) Copy content Toggle raw display
$71$ \( T^{7} - 1194 T^{6} + \cdots + 10\!\cdots\!92 \) Copy content Toggle raw display
$73$ \( T^{7} - 995 T^{6} + \cdots + 22\!\cdots\!52 \) Copy content Toggle raw display
$79$ \( T^{7} + 2640 T^{6} + \cdots + 18\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( T^{7} - 2579 T^{6} + \cdots - 50\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{7} - 1604 T^{6} + \cdots + 19\!\cdots\!92 \) Copy content Toggle raw display
$97$ \( T^{7} + 808 T^{6} + \cdots - 88\!\cdots\!16 \) Copy content Toggle raw display
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