Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1008,2,Mod(337,1008)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1008.337"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.r (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,3,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 337.2
Root \(0.939693 + 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 1008.337
Dual form 1008.2.r.i.673.2

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.592396 + 1.62760i) q^{3} +(0.326352 - 0.565258i) q^{5} +(0.500000 + 0.866025i) q^{7} +(-2.29813 - 1.92836i) q^{9} +(1.70574 + 2.95442i) q^{11} +(0.152704 - 0.264490i) q^{13} +(0.726682 + 0.866025i) q^{15} -0.226682 q^{17} +2.16250 q^{19} +(-1.70574 + 0.300767i) q^{21} +(-3.35117 + 5.80439i) q^{23} +(2.28699 + 3.96118i) q^{25} +(4.50000 - 2.59808i) q^{27} +(0.254900 + 0.441500i) q^{29} +(-2.85844 + 4.95096i) q^{31} +(-5.81908 + 1.02606i) q^{33} +0.652704 q^{35} -2.28312 q^{37} +(0.340022 + 0.405223i) q^{39} +(-0.479055 + 0.829748i) q^{41} +(-3.85844 - 6.68302i) q^{43} +(-1.84002 + 0.669713i) q^{45} +(4.14543 + 7.18009i) q^{47} +(-0.500000 + 0.866025i) q^{49} +(0.134285 - 0.368946i) q^{51} -3.18479 q^{53} +2.22668 q^{55} +(-1.28106 + 3.51968i) q^{57} +(-6.07785 + 10.5271i) q^{59} +(1.75490 + 3.03958i) q^{61} +(0.520945 - 2.95442i) q^{63} +(-0.0996702 - 0.172634i) q^{65} +(2.87211 - 4.97464i) q^{67} +(-7.46198 - 8.89284i) q^{69} -14.0273 q^{71} +12.4192 q^{73} +(-7.80200 + 1.37570i) q^{75} +(-1.70574 + 2.95442i) q^{77} +(-5.98293 - 10.3627i) q^{79} +(1.56283 + 8.86327i) q^{81} +(2.13563 + 3.69902i) q^{83} +(-0.0739780 + 0.128134i) q^{85} +(-0.869585 + 0.153331i) q^{87} -1.19934 q^{89} +0.305407 q^{91} +(-6.36484 - 7.58532i) q^{93} +(0.705737 - 1.22237i) q^{95} +(7.48932 + 12.9719i) q^{97} +(1.77719 - 10.0789i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{5} + 3 q^{7} + 3 q^{13} - 9 q^{15} + 12 q^{17} + 18 q^{19} + 6 q^{23} + 6 q^{25} + 27 q^{27} + 3 q^{29} - 9 q^{31} - 18 q^{33} + 6 q^{35} - 30 q^{37} - 18 q^{39} - 6 q^{41} - 15 q^{43} + 9 q^{45}+ \cdots - 15 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.592396 + 1.62760i −0.342020 + 0.939693i
\(4\) 0 0
\(5\) 0.326352 − 0.565258i 0.145949 − 0.252791i −0.783778 − 0.621042i \(-0.786710\pi\)
0.929727 + 0.368251i \(0.120043\pi\)
\(6\) 0 0
\(7\) 0.500000 + 0.866025i 0.188982 + 0.327327i
\(8\) 0 0
\(9\) −2.29813 − 1.92836i −0.766044 − 0.642788i
\(10\) 0 0
\(11\) 1.70574 + 2.95442i 0.514299 + 0.890792i 0.999862 + 0.0165906i \(0.00528120\pi\)
−0.485563 + 0.874202i \(0.661385\pi\)
\(12\) 0 0
\(13\) 0.152704 − 0.264490i 0.0423524 − 0.0733565i −0.844072 − 0.536230i \(-0.819848\pi\)
0.886425 + 0.462873i \(0.153181\pi\)
\(14\) 0 0
\(15\) 0.726682 + 0.866025i 0.187628 + 0.223607i
\(16\) 0 0
\(17\) −0.226682 −0.0549784 −0.0274892 − 0.999622i \(-0.508751\pi\)
−0.0274892 + 0.999622i \(0.508751\pi\)
\(18\) 0 0
\(19\) 2.16250 0.496112 0.248056 − 0.968746i \(-0.420208\pi\)
0.248056 + 0.968746i \(0.420208\pi\)
\(20\) 0 0
\(21\) −1.70574 + 0.300767i −0.372222 + 0.0656328i
\(22\) 0 0
\(23\) −3.35117 + 5.80439i −0.698767 + 1.21030i 0.270128 + 0.962824i \(0.412934\pi\)
−0.968894 + 0.247475i \(0.920399\pi\)
\(24\) 0 0
\(25\) 2.28699 + 3.96118i 0.457398 + 0.792236i
\(26\) 0 0
\(27\) 4.50000 − 2.59808i 0.866025 − 0.500000i
\(28\) 0 0
\(29\) 0.254900 + 0.441500i 0.0473338 + 0.0819845i 0.888722 − 0.458447i \(-0.151594\pi\)
−0.841388 + 0.540432i \(0.818261\pi\)
\(30\) 0 0
\(31\) −2.85844 + 4.95096i −0.513391 + 0.889219i 0.486488 + 0.873687i \(0.338278\pi\)
−0.999879 + 0.0155324i \(0.995056\pi\)
\(32\) 0 0
\(33\) −5.81908 + 1.02606i −1.01297 + 0.178614i
\(34\) 0 0
\(35\) 0.652704 0.110327
\(36\) 0 0
\(37\) −2.28312 −0.375342 −0.187671 − 0.982232i \(-0.560094\pi\)
−0.187671 + 0.982232i \(0.560094\pi\)
\(38\) 0 0
\(39\) 0.340022 + 0.405223i 0.0544472 + 0.0648876i
\(40\) 0 0
\(41\) −0.479055 + 0.829748i −0.0748159 + 0.129585i −0.901006 − 0.433806i \(-0.857170\pi\)
0.826190 + 0.563391i \(0.190504\pi\)
\(42\) 0 0
\(43\) −3.85844 − 6.68302i −0.588407 − 1.01915i −0.994441 − 0.105293i \(-0.966422\pi\)
0.406035 − 0.913858i \(-0.366911\pi\)
\(44\) 0 0
\(45\) −1.84002 + 0.669713i −0.274294 + 0.0998350i
\(46\) 0 0
\(47\) 4.14543 + 7.18009i 0.604673 + 1.04732i 0.992103 + 0.125425i \(0.0400296\pi\)
−0.387430 + 0.921899i \(0.626637\pi\)
\(48\) 0 0
\(49\) −0.500000 + 0.866025i −0.0714286 + 0.123718i
\(50\) 0 0
\(51\) 0.134285 − 0.368946i 0.0188037 − 0.0516628i
\(52\) 0 0
\(53\) −3.18479 −0.437465 −0.218732 − 0.975785i \(-0.570192\pi\)
−0.218732 + 0.975785i \(0.570192\pi\)
\(54\) 0 0
\(55\) 2.22668 0.300246
\(56\) 0 0
\(57\) −1.28106 + 3.51968i −0.169680 + 0.466193i
\(58\) 0 0
\(59\) −6.07785 + 10.5271i −0.791268 + 1.37052i 0.133913 + 0.990993i \(0.457246\pi\)
−0.925182 + 0.379524i \(0.876088\pi\)
\(60\) 0 0
\(61\) 1.75490 + 3.03958i 0.224692 + 0.389178i 0.956227 − 0.292626i \(-0.0945291\pi\)
−0.731535 + 0.681804i \(0.761196\pi\)
\(62\) 0 0
\(63\) 0.520945 − 2.95442i 0.0656328 − 0.372222i
\(64\) 0 0
\(65\) −0.0996702 − 0.172634i −0.0123626 − 0.0214126i
\(66\) 0 0
\(67\) 2.87211 − 4.97464i 0.350884 − 0.607749i −0.635520 − 0.772084i \(-0.719214\pi\)
0.986405 + 0.164335i \(0.0525477\pi\)
\(68\) 0 0
\(69\) −7.46198 − 8.89284i −0.898317 − 1.07057i
\(70\) 0 0
\(71\) −14.0273 −1.66474 −0.832370 − 0.554221i \(-0.813016\pi\)
−0.832370 + 0.554221i \(0.813016\pi\)
\(72\) 0 0
\(73\) 12.4192 1.45356 0.726780 − 0.686871i \(-0.241016\pi\)
0.726780 + 0.686871i \(0.241016\pi\)
\(74\) 0 0
\(75\) −7.80200 + 1.37570i −0.900898 + 0.158853i
\(76\) 0 0
\(77\) −1.70574 + 2.95442i −0.194387 + 0.336688i
\(78\) 0 0
\(79\) −5.98293 − 10.3627i −0.673132 − 1.16590i −0.977011 − 0.213188i \(-0.931615\pi\)
0.303879 − 0.952710i \(-0.401718\pi\)
\(80\) 0 0
\(81\) 1.56283 + 8.86327i 0.173648 + 0.984808i
\(82\) 0 0
\(83\) 2.13563 + 3.69902i 0.234416 + 0.406020i 0.959103 − 0.283058i \(-0.0913490\pi\)
−0.724687 + 0.689078i \(0.758016\pi\)
\(84\) 0 0
\(85\) −0.0739780 + 0.128134i −0.00802404 + 0.0138980i
\(86\) 0 0
\(87\) −0.869585 + 0.153331i −0.0932293 + 0.0164388i
\(88\) 0 0
\(89\) −1.19934 −0.127130 −0.0635649 − 0.997978i \(-0.520247\pi\)
−0.0635649 + 0.997978i \(0.520247\pi\)
\(90\) 0 0
\(91\) 0.305407 0.0320154
\(92\) 0 0
\(93\) −6.36484 − 7.58532i −0.660003 − 0.786561i
\(94\) 0 0
\(95\) 0.705737 − 1.22237i 0.0724071 − 0.125413i
\(96\) 0 0
\(97\) 7.48932 + 12.9719i 0.760425 + 1.31710i 0.942631 + 0.333835i \(0.108343\pi\)
−0.182206 + 0.983260i \(0.558324\pi\)
\(98\) 0 0
\(99\) 1.77719 − 10.0789i 0.178614 − 1.01297i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1008.2.r.i.337.2 6
3.2 odd 2 3024.2.r.h.1009.2 6
4.3 odd 2 504.2.r.c.337.2 yes 6
9.2 odd 6 3024.2.r.h.2017.2 6
9.4 even 3 9072.2.a.br.1.2 3
9.5 odd 6 9072.2.a.cc.1.2 3
9.7 even 3 inner 1008.2.r.i.673.2 6
12.11 even 2 1512.2.r.c.1009.2 6
36.7 odd 6 504.2.r.c.169.2 ✓ 6
36.11 even 6 1512.2.r.c.505.2 6
36.23 even 6 4536.2.a.v.1.2 3
36.31 odd 6 4536.2.a.s.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.r.c.169.2 ✓ 6 36.7 odd 6
504.2.r.c.337.2 yes 6 4.3 odd 2
1008.2.r.i.337.2 6 1.1 even 1 trivial
1008.2.r.i.673.2 6 9.7 even 3 inner
1512.2.r.c.505.2 6 36.11 even 6
1512.2.r.c.1009.2 6 12.11 even 2
3024.2.r.h.1009.2 6 3.2 odd 2
3024.2.r.h.2017.2 6 9.2 odd 6
4536.2.a.s.1.2 3 36.31 odd 6
4536.2.a.v.1.2 3 36.23 even 6
9072.2.a.br.1.2 3 9.4 even 3
9072.2.a.cc.1.2 3 9.5 odd 6