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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1040,2,Mod(577,1040)] 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("1040.577"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1040, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.bg (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,0,0,-2,0,0,0,0,0,-2,0,4,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.30444181021\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 41 x^{18} + 640 x^{16} + 4888 x^{14} + 19956 x^{12} + 45364 x^{10} + 57952 x^{8} + 41120 x^{6} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 520)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 593.3
Root \(2.54561i\) of defining polynomial
Character \(\chi\) \(=\) 1040.593
Dual form 1040.2.bg.p.577.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.70333 - 1.70333i) q^{3} +(-0.846581 - 2.06961i) q^{5} +3.32755i q^{7} +2.80266i q^{9} +(-1.53449 + 1.53449i) q^{11} +(3.52862 - 0.740837i) q^{13} +(-2.08323 + 4.96724i) q^{15} +(2.17503 + 2.17503i) q^{17} +(-0.135956 + 0.135956i) q^{19} +(5.66792 - 5.66792i) q^{21} +(-0.566521 + 0.566521i) q^{23} +(-3.56660 + 3.50419i) q^{25} +(-0.336128 + 0.336128i) q^{27} +5.39293i q^{29} +(4.97938 + 4.97938i) q^{31} +5.22750 q^{33} +(6.88674 - 2.81704i) q^{35} -3.05564i q^{37} +(-7.27229 - 4.74851i) q^{39} +(4.38255 + 4.38255i) q^{41} +(7.49966 - 7.49966i) q^{43} +(5.80043 - 2.37268i) q^{45} -11.3946i q^{47} -4.07259 q^{49} -7.40960i q^{51} +(-7.33548 - 7.33548i) q^{53} +(4.47488 + 1.87674i) q^{55} +0.463157 q^{57} +(0.300627 + 0.300627i) q^{59} +13.6424 q^{61} -9.32601 q^{63} +(-4.52051 - 6.67570i) q^{65} +10.2005 q^{67} +1.92994 q^{69} +(-4.56710 - 4.56710i) q^{71} -1.21555 q^{73} +(12.0439 + 0.106304i) q^{75} +(-5.10611 - 5.10611i) q^{77} +16.2706i q^{79} +9.55307 q^{81} +2.36951i q^{83} +(2.66014 - 6.34282i) q^{85} +(9.18594 - 9.18594i) q^{87} +(-4.78668 - 4.78668i) q^{89} +(2.46517 + 11.7417i) q^{91} -16.9630i q^{93} +(0.396476 + 0.166279i) q^{95} +1.71744 q^{97} +(-4.30067 - 4.30067i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{5} - 2 q^{11} + 4 q^{13} + 16 q^{15} - 6 q^{17} + 2 q^{19} + 6 q^{21} - 14 q^{23} + 14 q^{25} - 6 q^{27} - 10 q^{31} - 12 q^{33} + 16 q^{35} - 32 q^{39} + 28 q^{41} + 14 q^{45} - 16 q^{49} + 20 q^{53}+ \cdots + 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{1}{4}\right)\) \(1\)

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
\(3\) −1.70333 − 1.70333i −0.983418 − 0.983418i 0.0164469 − 0.999865i \(-0.494765\pi\)
−0.999865 + 0.0164469i \(0.994765\pi\)
\(4\) 0 0
\(5\) −0.846581 − 2.06961i −0.378603 − 0.925559i
\(6\) 0 0
\(7\) 3.32755i 1.25770i 0.777528 + 0.628848i \(0.216473\pi\)
−0.777528 + 0.628848i \(0.783527\pi\)
\(8\) 0 0
\(9\) 2.80266i 0.934221i
\(10\) 0 0
\(11\) −1.53449 + 1.53449i −0.462667 + 0.462667i −0.899529 − 0.436861i \(-0.856090\pi\)
0.436861 + 0.899529i \(0.356090\pi\)
\(12\) 0 0
\(13\) 3.52862 − 0.740837i 0.978663 − 0.205471i
\(14\) 0 0
\(15\) −2.08323 + 4.96724i −0.537887 + 1.28254i
\(16\) 0 0
\(17\) 2.17503 + 2.17503i 0.527523 + 0.527523i 0.919833 − 0.392310i \(-0.128324\pi\)
−0.392310 + 0.919833i \(0.628324\pi\)
\(18\) 0 0
\(19\) −0.135956 + 0.135956i −0.0311906 + 0.0311906i −0.722530 − 0.691340i \(-0.757021\pi\)
0.691340 + 0.722530i \(0.257021\pi\)
\(20\) 0 0
\(21\) 5.66792 − 5.66792i 1.23684 − 1.23684i
\(22\) 0 0
\(23\) −0.566521 + 0.566521i −0.118128 + 0.118128i −0.763700 − 0.645572i \(-0.776619\pi\)
0.645572 + 0.763700i \(0.276619\pi\)
\(24\) 0 0
\(25\) −3.56660 + 3.50419i −0.713320 + 0.700838i
\(26\) 0 0
\(27\) −0.336128 + 0.336128i −0.0646879 + 0.0646879i
\(28\) 0 0
\(29\) 5.39293i 1.00144i 0.865609 + 0.500721i \(0.166932\pi\)
−0.865609 + 0.500721i \(0.833068\pi\)
\(30\) 0 0
\(31\) 4.97938 + 4.97938i 0.894323 + 0.894323i 0.994927 − 0.100604i \(-0.0320775\pi\)
−0.100604 + 0.994927i \(0.532077\pi\)
\(32\) 0 0
\(33\) 5.22750 0.909991
\(34\) 0 0
\(35\) 6.88674 − 2.81704i 1.16407 − 0.476167i
\(36\) 0 0
\(37\) − 3.05564i − 0.502344i −0.967943 − 0.251172i \(-0.919184\pi\)
0.967943 − 0.251172i \(-0.0808159\pi\)
\(38\) 0 0
\(39\) −7.27229 − 4.74851i −1.16450 − 0.760371i
\(40\) 0 0
\(41\) 4.38255 + 4.38255i 0.684440 + 0.684440i 0.960997 − 0.276558i \(-0.0891937\pi\)
−0.276558 + 0.960997i \(0.589194\pi\)
\(42\) 0 0
\(43\) 7.49966 − 7.49966i 1.14369 − 1.14369i 0.155917 − 0.987770i \(-0.450167\pi\)
0.987770 − 0.155917i \(-0.0498331\pi\)
\(44\) 0 0
\(45\) 5.80043 − 2.37268i 0.864677 − 0.353699i
\(46\) 0 0
\(47\) − 11.3946i − 1.66207i −0.556217 − 0.831037i \(-0.687748\pi\)
0.556217 − 0.831037i \(-0.312252\pi\)
\(48\) 0 0
\(49\) −4.07259 −0.581799
\(50\) 0 0
\(51\) − 7.40960i − 1.03755i
\(52\) 0 0
\(53\) −7.33548 − 7.33548i −1.00761 − 1.00761i −0.999971 − 0.00763424i \(-0.997570\pi\)
−0.00763424 − 0.999971i \(-0.502430\pi\)
\(54\) 0 0
\(55\) 4.47488 + 1.87674i 0.603393 + 0.253059i
\(56\) 0 0
\(57\) 0.463157 0.0613467
\(58\) 0 0
\(59\) 0.300627 + 0.300627i 0.0391384 + 0.0391384i 0.726405 − 0.687267i \(-0.241190\pi\)
−0.687267 + 0.726405i \(0.741190\pi\)
\(60\) 0 0
\(61\) 13.6424 1.74673 0.873365 − 0.487067i \(-0.161933\pi\)
0.873365 + 0.487067i \(0.161933\pi\)
\(62\) 0 0
\(63\) −9.32601 −1.17497
\(64\) 0 0
\(65\) −4.52051 − 6.67570i −0.560700 − 0.828019i
\(66\) 0 0
\(67\) 10.2005 1.24619 0.623097 − 0.782145i \(-0.285874\pi\)
0.623097 + 0.782145i \(0.285874\pi\)
\(68\) 0 0
\(69\) 1.92994 0.232338
\(70\) 0 0
\(71\) −4.56710 − 4.56710i −0.542014 − 0.542014i 0.382105 − 0.924119i \(-0.375199\pi\)
−0.924119 + 0.382105i \(0.875199\pi\)
\(72\) 0 0
\(73\) −1.21555 −0.142270 −0.0711348 − 0.997467i \(-0.522662\pi\)
−0.0711348 + 0.997467i \(0.522662\pi\)
\(74\) 0 0
\(75\) 12.0439 + 0.106304i 1.39071 + 0.0122749i
\(76\) 0 0
\(77\) −5.10611 − 5.10611i −0.581895 − 0.581895i
\(78\) 0 0
\(79\) 16.2706i 1.83059i 0.402788 + 0.915293i \(0.368041\pi\)
−0.402788 + 0.915293i \(0.631959\pi\)
\(80\) 0 0
\(81\) 9.55307 1.06145
\(82\) 0 0
\(83\) 2.36951i 0.260088i 0.991508 + 0.130044i \(0.0415118\pi\)
−0.991508 + 0.130044i \(0.958488\pi\)
\(84\) 0 0
\(85\) 2.66014 − 6.34282i 0.288532 − 0.687975i
\(86\) 0 0
\(87\) 9.18594 − 9.18594i 0.984837 − 0.984837i
\(88\) 0 0
\(89\) −4.78668 − 4.78668i −0.507387 − 0.507387i 0.406337 − 0.913723i \(-0.366806\pi\)
−0.913723 + 0.406337i \(0.866806\pi\)
\(90\) 0 0
\(91\) 2.46517 + 11.7417i 0.258420 + 1.23086i
\(92\) 0 0
\(93\) − 16.9630i − 1.75899i
\(94\) 0 0
\(95\) 0.396476 + 0.166279i 0.0406775 + 0.0170599i
\(96\) 0 0
\(97\) 1.71744 0.174379 0.0871896 − 0.996192i \(-0.472211\pi\)
0.0871896 + 0.996192i \(0.472211\pi\)
\(98\) 0 0
\(99\) −4.30067 − 4.30067i −0.432234 − 0.432234i
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 1040.2.bg.p.593.3 20
4.3 odd 2 520.2.w.f.73.8 yes 20
5.2 odd 4 1040.2.cd.p.177.3 20
13.5 odd 4 1040.2.cd.p.993.3 20
20.7 even 4 520.2.bh.f.177.8 yes 20
52.31 even 4 520.2.bh.f.473.8 yes 20
65.57 even 4 inner 1040.2.bg.p.577.3 20
260.187 odd 4 520.2.w.f.57.8 ✓ 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
520.2.w.f.57.8 ✓ 20 260.187 odd 4
520.2.w.f.73.8 yes 20 4.3 odd 2
520.2.bh.f.177.8 yes 20 20.7 even 4
520.2.bh.f.473.8 yes 20 52.31 even 4
1040.2.bg.p.577.3 20 65.57 even 4 inner
1040.2.bg.p.593.3 20 1.1 even 1 trivial
1040.2.cd.p.177.3 20 5.2 odd 4
1040.2.cd.p.993.3 20 13.5 odd 4