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Properties

Label 1008.2.v.c.323.3
Level $1008$
Weight $2$
Character 1008.323
Analytic conductor $8.049$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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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(323,1008)] 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("1008.323"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3, 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.v (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.653473922154496.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4x^{10} + 13x^{8} - 28x^{6} + 52x^{4} - 64x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 323.3
Root \(-0.892524 + 1.09700i\) of defining polynomial
Character \(\chi\) \(=\) 1008.323
Dual form 1008.2.v.c.827.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.892524 + 1.09700i) q^{2} +(-0.406803 - 1.95819i) q^{4} +(1.41421 + 1.41421i) q^{5} -1.00000 q^{7} +(2.51121 + 1.30147i) q^{8} +(-2.81361 + 0.289169i) q^{10} +(3.97904 - 3.97904i) q^{11} +(-1.10278 - 1.10278i) q^{13} +(0.892524 - 1.09700i) q^{14} +(-3.66902 + 1.59320i) q^{16} -6.39852i q^{17} +(2.19399 - 3.34461i) q^{20} +(0.813607 + 7.91638i) q^{22} -2.97377i q^{23} -1.00000i q^{25} +(2.19399 - 0.225488i) q^{26} +(0.406803 + 1.95819i) q^{28} +(-5.53860 + 5.53860i) q^{29} -2.20555i q^{31} +(1.52696 - 5.44687i) q^{32} +(7.01916 + 5.71083i) q^{34} +(-1.41421 - 1.41421i) q^{35} +(-1.00000 + 1.00000i) q^{37} +(1.71083 + 5.39194i) q^{40} +2.08676 q^{41} +(-4.68111 - 4.68111i) q^{43} +(-9.41041 - 6.17303i) q^{44} +(3.26222 + 2.65416i) q^{46} +8.77597 q^{47} +1.00000 q^{49} +(1.09700 + 0.892524i) q^{50} +(-1.71083 + 2.60806i) q^{52} +(3.83369 + 3.83369i) q^{53} +11.2544 q^{55} +(-2.51121 - 1.30147i) q^{56} +(-1.13249 - 11.0192i) q^{58} +(9.51764 - 9.51764i) q^{59} +(-5.10278 - 5.10278i) q^{61} +(2.41948 + 1.96851i) q^{62} +(4.61235 + 6.53653i) q^{64} -3.11912i q^{65} +(6.10278 - 6.10278i) q^{67} +(-12.5295 + 2.60294i) q^{68} +(2.81361 - 0.289169i) q^{70} -6.62009i q^{71} +7.04888i q^{73} +(-0.204473 - 1.98952i) q^{74} +(-3.97904 + 3.97904i) q^{77} +4.41110i q^{79} +(-7.44190 - 2.93566i) q^{80} +(-1.86248 + 2.28917i) q^{82} +(10.0448 + 10.0448i) q^{83} +(9.04888 - 9.04888i) q^{85} +(9.31317 - 0.957161i) q^{86} +(15.1708 - 4.81361i) q^{88} +13.6912 q^{89} +(1.10278 + 1.10278i) q^{91} +(-5.82321 + 1.20974i) q^{92} +(-7.83276 + 9.62721i) q^{94} -15.5678 q^{97} +(-0.892524 + 1.09700i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 8 q^{4} - 12 q^{7} - 8 q^{10} + 16 q^{13} - 20 q^{16} - 16 q^{22} - 8 q^{28} - 12 q^{37} + 24 q^{40} - 20 q^{43} + 28 q^{46} + 12 q^{49} - 24 q^{52} + 32 q^{55} - 24 q^{58} - 32 q^{61} - 16 q^{64}+ \cdots - 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.892524 + 1.09700i −0.631109 + 0.775694i
\(3\) 0 0
\(4\) −0.406803 − 1.95819i −0.203402 − 0.979095i
\(5\) 1.41421 + 1.41421i 0.632456 + 0.632456i 0.948683 − 0.316228i \(-0.102416\pi\)
−0.316228 + 0.948683i \(0.602416\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 2.51121 + 1.30147i 0.887847 + 0.460139i
\(9\) 0 0
\(10\) −2.81361 + 0.289169i −0.889740 + 0.0914431i
\(11\) 3.97904 − 3.97904i 1.19973 − 1.19973i 0.225477 − 0.974248i \(-0.427606\pi\)
0.974248 − 0.225477i \(-0.0723942\pi\)
\(12\) 0 0
\(13\) −1.10278 − 1.10278i −0.305855 − 0.305855i 0.537444 − 0.843299i \(-0.319390\pi\)
−0.843299 + 0.537444i \(0.819390\pi\)
\(14\) 0.892524 − 1.09700i 0.238537 − 0.293185i
\(15\) 0 0
\(16\) −3.66902 + 1.59320i −0.917256 + 0.398299i
\(17\) − 6.39852i − 1.55187i −0.630813 − 0.775935i \(-0.717279\pi\)
0.630813 − 0.775935i \(-0.282721\pi\)
\(18\) 0 0
\(19\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(20\) 2.19399 − 3.34461i 0.490592 − 0.747877i
\(21\) 0 0
\(22\) 0.813607 + 7.91638i 0.173461 + 1.68778i
\(23\) − 2.97377i − 0.620075i −0.950724 − 0.310037i \(-0.899658\pi\)
0.950724 − 0.310037i \(-0.100342\pi\)
\(24\) 0 0
\(25\) − 1.00000i − 0.200000i
\(26\) 2.19399 − 0.225488i 0.430277 − 0.0442218i
\(27\) 0 0
\(28\) 0.406803 + 1.95819i 0.0768786 + 0.370063i
\(29\) −5.53860 + 5.53860i −1.02849 + 1.02849i −0.0289102 + 0.999582i \(0.509204\pi\)
−0.999582 + 0.0289102i \(0.990796\pi\)
\(30\) 0 0
\(31\) − 2.20555i − 0.396128i −0.980189 − 0.198064i \(-0.936535\pi\)
0.980189 − 0.198064i \(-0.0634655\pi\)
\(32\) 1.52696 − 5.44687i 0.269930 − 0.962880i
\(33\) 0 0
\(34\) 7.01916 + 5.71083i 1.20378 + 0.979400i
\(35\) −1.41421 − 1.41421i −0.239046 − 0.239046i
\(36\) 0 0
\(37\) −1.00000 + 1.00000i −0.164399 + 0.164399i −0.784512 − 0.620113i \(-0.787087\pi\)
0.620113 + 0.784512i \(0.287087\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 1.71083 + 5.39194i 0.270506 + 0.852541i
\(41\) 2.08676 0.325897 0.162949 − 0.986635i \(-0.447900\pi\)
0.162949 + 0.986635i \(0.447900\pi\)
\(42\) 0 0
\(43\) −4.68111 − 4.68111i −0.713863 − 0.713863i 0.253478 − 0.967341i \(-0.418425\pi\)
−0.967341 + 0.253478i \(0.918425\pi\)
\(44\) −9.41041 − 6.17303i −1.41867 − 0.930620i
\(45\) 0 0
\(46\) 3.26222 + 2.65416i 0.480988 + 0.391335i
\(47\) 8.77597 1.28011 0.640054 − 0.768330i \(-0.278912\pi\)
0.640054 + 0.768330i \(0.278912\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.09700 + 0.892524i 0.155139 + 0.126222i
\(51\) 0 0
\(52\) −1.71083 + 2.60806i −0.237250 + 0.361672i
\(53\) 3.83369 + 3.83369i 0.526598 + 0.526598i 0.919556 − 0.392958i \(-0.128548\pi\)
−0.392958 + 0.919556i \(0.628548\pi\)
\(54\) 0 0
\(55\) 11.2544 1.51755
\(56\) −2.51121 − 1.30147i −0.335575 − 0.173916i
\(57\) 0 0
\(58\) −1.13249 − 11.0192i −0.148704 − 1.44689i
\(59\) 9.51764 − 9.51764i 1.23909 − 1.23909i 0.278718 − 0.960373i \(-0.410090\pi\)
0.960373 − 0.278718i \(-0.0899096\pi\)
\(60\) 0 0
\(61\) −5.10278 − 5.10278i −0.653343 − 0.653343i 0.300453 − 0.953797i \(-0.402862\pi\)
−0.953797 + 0.300453i \(0.902862\pi\)
\(62\) 2.41948 + 1.96851i 0.307274 + 0.250000i
\(63\) 0 0
\(64\) 4.61235 + 6.53653i 0.576544 + 0.817066i
\(65\) − 3.11912i − 0.386879i
\(66\) 0 0
\(67\) 6.10278 − 6.10278i 0.745573 − 0.745573i −0.228072 − 0.973644i \(-0.573242\pi\)
0.973644 + 0.228072i \(0.0732421\pi\)
\(68\) −12.5295 + 2.60294i −1.51943 + 0.315653i
\(69\) 0 0
\(70\) 2.81361 − 0.289169i 0.336290 − 0.0345623i
\(71\) − 6.62009i − 0.785660i −0.919611 − 0.392830i \(-0.871496\pi\)
0.919611 − 0.392830i \(-0.128504\pi\)
\(72\) 0 0
\(73\) 7.04888i 0.825009i 0.910956 + 0.412504i \(0.135346\pi\)
−0.910956 + 0.412504i \(0.864654\pi\)
\(74\) −0.204473 − 1.98952i −0.0237695 − 0.231277i
\(75\) 0 0
\(76\) 0 0
\(77\) −3.97904 + 3.97904i −0.453454 + 0.453454i
\(78\) 0 0
\(79\) 4.41110i 0.496288i 0.968723 + 0.248144i \(0.0798205\pi\)
−0.968723 + 0.248144i \(0.920179\pi\)
\(80\) −7.44190 − 2.93566i −0.832030 − 0.328217i
\(81\) 0 0
\(82\) −1.86248 + 2.28917i −0.205677 + 0.252796i
\(83\) 10.0448 + 10.0448i 1.10256 + 1.10256i 0.994100 + 0.108464i \(0.0345933\pi\)
0.108464 + 0.994100i \(0.465407\pi\)
\(84\) 0 0
\(85\) 9.04888 − 9.04888i 0.981488 − 0.981488i
\(86\) 9.31317 − 0.957161i 1.00426 − 0.103213i
\(87\) 0 0
\(88\) 15.1708 − 4.81361i 1.61721 − 0.513132i
\(89\) 13.6912 1.45126 0.725630 − 0.688085i \(-0.241548\pi\)
0.725630 + 0.688085i \(0.241548\pi\)
\(90\) 0 0
\(91\) 1.10278 + 1.10278i 0.115602 + 0.115602i
\(92\) −5.82321 + 1.20974i −0.607112 + 0.126124i
\(93\) 0 0
\(94\) −7.83276 + 9.62721i −0.807888 + 0.992971i
\(95\) 0 0
\(96\) 0 0
\(97\) −15.5678 −1.58067 −0.790334 − 0.612676i \(-0.790093\pi\)
−0.790334 + 0.612676i \(0.790093\pi\)
\(98\) −0.892524 + 1.09700i −0.0901585 + 0.110813i
\(99\) 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 1008.2.v.c.323.3 ✓ 12
3.2 odd 2 inner 1008.2.v.c.323.4 yes 12
4.3 odd 2 4032.2.v.c.1583.4 12
12.11 even 2 4032.2.v.c.1583.3 12
16.5 even 4 4032.2.v.c.3599.3 12
16.11 odd 4 inner 1008.2.v.c.827.4 yes 12
48.5 odd 4 4032.2.v.c.3599.4 12
48.11 even 4 inner 1008.2.v.c.827.3 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1008.2.v.c.323.3 ✓ 12 1.1 even 1 trivial
1008.2.v.c.323.4 yes 12 3.2 odd 2 inner
1008.2.v.c.827.3 yes 12 48.11 even 4 inner
1008.2.v.c.827.4 yes 12 16.11 odd 4 inner
4032.2.v.c.1583.3 12 12.11 even 2
4032.2.v.c.1583.4 12 4.3 odd 2
4032.2.v.c.3599.3 12 16.5 even 4
4032.2.v.c.3599.4 12 48.5 odd 4