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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1008,5,Mod(449,1008)] 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("1008.449"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 1008.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,0,0,0,0,0,0,-848] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(104.196922789\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 864x^{10} + 264408x^{8} + 36891992x^{6} + 2458364040x^{4} + 71189190576x^{2} + 596584767321 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{4}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.8
Root \(-3.68150i\) of defining polynomial
Character \(\chi\) \(=\) 1008.449
Dual form 1008.5.d.e.449.5

$q$-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)
 
\(f(q)\) \(=\) \(q+2.07209i q^{5} -18.5203 q^{7} +98.9042i q^{11} -70.7044 q^{13} +83.3192i q^{17} +593.349 q^{19} -969.044i q^{23} +620.706 q^{25} -884.253i q^{29} -1615.59 q^{31} -38.3757i q^{35} -2207.99 q^{37} +1057.21i q^{41} +2657.76 q^{43} +2201.29i q^{47} +343.000 q^{49} +4062.63i q^{53} -204.939 q^{55} +445.984i q^{59} +1689.80 q^{61} -146.506i q^{65} +5343.36 q^{67} -3870.26i q^{71} -8113.31 q^{73} -1831.73i q^{77} +3901.83 q^{79} -9195.05i q^{83} -172.645 q^{85} +12837.6i q^{89} +1309.46 q^{91} +1229.47i q^{95} +196.114 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 848 q^{13} + 96 q^{19} - 3036 q^{25} - 2464 q^{31} + 1688 q^{37} - 4400 q^{43} + 4116 q^{49} - 7840 q^{55} - 5048 q^{61} + 14224 q^{67} + 4928 q^{73} + 15040 q^{79} - 13848 q^{85} + 5488 q^{91} + 16352 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 0
\(4\) 0 0
\(5\) 2.07209i 0.0828838i 0.999141 + 0.0414419i \(0.0131951\pi\)
−0.999141 + 0.0414419i \(0.986805\pi\)
\(6\) 0 0
\(7\) −18.5203 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 98.9042i 0.817390i 0.912671 + 0.408695i \(0.134016\pi\)
−0.912671 + 0.408695i \(0.865984\pi\)
\(12\) 0 0
\(13\) −70.7044 −0.418369 −0.209185 − 0.977876i \(-0.567081\pi\)
−0.209185 + 0.977876i \(0.567081\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 83.3192i 0.288302i 0.989556 + 0.144151i \(0.0460451\pi\)
−0.989556 + 0.144151i \(0.953955\pi\)
\(18\) 0 0
\(19\) 593.349 1.64362 0.821812 − 0.569758i \(-0.192963\pi\)
0.821812 + 0.569758i \(0.192963\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 969.044i − 1.83184i −0.401359 − 0.915921i \(-0.631462\pi\)
0.401359 − 0.915921i \(-0.368538\pi\)
\(24\) 0 0
\(25\) 620.706 0.993130
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 884.253i − 1.05143i −0.850661 − 0.525715i \(-0.823798\pi\)
0.850661 − 0.525715i \(-0.176202\pi\)
\(30\) 0 0
\(31\) −1615.59 −1.68116 −0.840578 − 0.541691i \(-0.817784\pi\)
−0.840578 + 0.541691i \(0.817784\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 38.3757i − 0.0313271i
\(36\) 0 0
\(37\) −2207.99 −1.61285 −0.806425 − 0.591336i \(-0.798601\pi\)
−0.806425 + 0.591336i \(0.798601\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1057.21i 0.628919i 0.949271 + 0.314459i \(0.101823\pi\)
−0.949271 + 0.314459i \(0.898177\pi\)
\(42\) 0 0
\(43\) 2657.76 1.43740 0.718702 − 0.695318i \(-0.244736\pi\)
0.718702 + 0.695318i \(0.244736\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2201.29i 0.996511i 0.867030 + 0.498256i \(0.166026\pi\)
−0.867030 + 0.498256i \(0.833974\pi\)
\(48\) 0 0
\(49\) 343.000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4062.63i 1.44629i 0.690696 + 0.723145i \(0.257304\pi\)
−0.690696 + 0.723145i \(0.742696\pi\)
\(54\) 0 0
\(55\) −204.939 −0.0677484
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 445.984i 0.128120i 0.997946 + 0.0640598i \(0.0204048\pi\)
−0.997946 + 0.0640598i \(0.979595\pi\)
\(60\) 0 0
\(61\) 1689.80 0.454126 0.227063 − 0.973880i \(-0.427088\pi\)
0.227063 + 0.973880i \(0.427088\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 146.506i − 0.0346760i
\(66\) 0 0
\(67\) 5343.36 1.19032 0.595161 − 0.803606i \(-0.297088\pi\)
0.595161 + 0.803606i \(0.297088\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 3870.26i − 0.767757i −0.923384 − 0.383879i \(-0.874588\pi\)
0.923384 − 0.383879i \(-0.125412\pi\)
\(72\) 0 0
\(73\) −8113.31 −1.52248 −0.761241 − 0.648469i \(-0.775410\pi\)
−0.761241 + 0.648469i \(0.775410\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1831.73i − 0.308945i
\(78\) 0 0
\(79\) 3901.83 0.625192 0.312596 − 0.949886i \(-0.398801\pi\)
0.312596 + 0.949886i \(0.398801\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 9195.05i − 1.33474i −0.744724 − 0.667372i \(-0.767419\pi\)
0.744724 − 0.667372i \(-0.232581\pi\)
\(84\) 0 0
\(85\) −172.645 −0.0238955
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12837.6i 1.62071i 0.585942 + 0.810353i \(0.300725\pi\)
−0.585942 + 0.810353i \(0.699275\pi\)
\(90\) 0 0
\(91\) 1309.46 0.158129
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1229.47i 0.136230i
\(96\) 0 0
\(97\) 196.114 0.0208432 0.0104216 − 0.999946i \(-0.496683\pi\)
0.0104216 + 0.999946i \(0.496683\pi\)
\(98\) 0 0
\(99\) 0 0
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 1008.5.d.e.449.8 12
3.2 odd 2 inner 1008.5.d.e.449.5 12
4.3 odd 2 504.5.d.a.449.8 yes 12
12.11 even 2 504.5.d.a.449.5 ✓ 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.5.d.a.449.5 ✓ 12 12.11 even 2
504.5.d.a.449.8 yes 12 4.3 odd 2
1008.5.d.e.449.5 12 3.2 odd 2 inner
1008.5.d.e.449.8 12 1.1 even 1 trivial