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Properties

Label 990.2.m.d.307.1
Level $990$
Weight $2$
Character 990.307
Analytic conductor $7.905$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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] = [990,2,Mod(307,990)] 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("990.307"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(990, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 990.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.90518980011\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 110)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 307.1
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 990.307
Dual form 990.2.m.d.703.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.707107 + 0.707107i) q^{2} -1.00000i q^{4} +(0.707107 - 2.12132i) q^{5} +(-2.12132 + 2.12132i) q^{7} +(0.707107 + 0.707107i) q^{8} +(1.00000 + 2.00000i) q^{10} +(1.41421 - 3.00000i) q^{11} +(-3.00000 - 3.00000i) q^{13} -3.00000i q^{14} -1.00000 q^{16} +(-5.12132 + 5.12132i) q^{17} +3.00000 q^{19} +(-2.12132 - 0.707107i) q^{20} +(1.12132 + 3.12132i) q^{22} +(0.171573 - 0.171573i) q^{23} +(-4.00000 - 3.00000i) q^{25} +4.24264 q^{26} +(2.12132 + 2.12132i) q^{28} -1.24264 q^{29} -7.24264 q^{31} +(0.707107 - 0.707107i) q^{32} -7.24264i q^{34} +(3.00000 + 6.00000i) q^{35} +(0.121320 + 0.121320i) q^{37} +(-2.12132 + 2.12132i) q^{38} +(2.00000 - 1.00000i) q^{40} -1.75736i q^{41} +(-1.24264 - 1.24264i) q^{43} +(-3.00000 - 1.41421i) q^{44} +0.242641i q^{46} +(-4.41421 - 4.41421i) q^{47} -2.00000i q^{49} +(4.94975 - 0.707107i) q^{50} +(-3.00000 + 3.00000i) q^{52} +(-9.53553 + 9.53553i) q^{53} +(-5.36396 - 5.12132i) q^{55} -3.00000 q^{56} +(0.878680 - 0.878680i) q^{58} -1.41421i q^{59} +7.24264i q^{61} +(5.12132 - 5.12132i) q^{62} +1.00000i q^{64} +(-8.48528 + 4.24264i) q^{65} +(-4.00000 - 4.00000i) q^{67} +(5.12132 + 5.12132i) q^{68} +(-6.36396 - 2.12132i) q^{70} +1.24264 q^{71} +(-6.00000 - 6.00000i) q^{73} -0.171573 q^{74} -3.00000i q^{76} +(3.36396 + 9.36396i) q^{77} -10.2426 q^{79} +(-0.707107 + 2.12132i) q^{80} +(1.24264 + 1.24264i) q^{82} +(-7.24264 - 7.24264i) q^{83} +(7.24264 + 14.4853i) q^{85} +1.75736 q^{86} +(3.12132 - 1.12132i) q^{88} -5.48528i q^{89} +12.7279 q^{91} +(-0.171573 - 0.171573i) q^{92} +6.24264 q^{94} +(2.12132 - 6.36396i) q^{95} +(-2.24264 - 2.24264i) q^{97} +(1.41421 + 1.41421i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{10} - 12 q^{13} - 4 q^{16} - 12 q^{17} + 12 q^{19} - 4 q^{22} + 12 q^{23} - 16 q^{25} + 12 q^{29} - 12 q^{31} + 12 q^{35} - 8 q^{37} + 8 q^{40} + 12 q^{43} - 12 q^{44} - 12 q^{47} - 12 q^{52}+ \cdots + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/990\mathbb{Z}\right)^\times\).

\(n\) \(397\) \(541\) \(551\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.707107 + 0.707107i −0.500000 + 0.500000i
\(3\) 0 0
\(4\) − 1.00000i − 0.500000i
\(5\) 0.707107 − 2.12132i 0.316228 − 0.948683i
\(6\) 0 0
\(7\) −2.12132 + 2.12132i −0.801784 + 0.801784i −0.983374 − 0.181591i \(-0.941875\pi\)
0.181591 + 0.983374i \(0.441875\pi\)
\(8\) 0.707107 + 0.707107i 0.250000 + 0.250000i
\(9\) 0 0
\(10\) 1.00000 + 2.00000i 0.316228 + 0.632456i
\(11\) 1.41421 − 3.00000i 0.426401 − 0.904534i
\(12\) 0 0
\(13\) −3.00000 − 3.00000i −0.832050 − 0.832050i 0.155747 − 0.987797i \(-0.450222\pi\)
−0.987797 + 0.155747i \(0.950222\pi\)
\(14\) − 3.00000i − 0.801784i
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −5.12132 + 5.12132i −1.24210 + 1.24210i −0.282975 + 0.959127i \(0.591322\pi\)
−0.959127 + 0.282975i \(0.908678\pi\)
\(18\) 0 0
\(19\) 3.00000 0.688247 0.344124 − 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) −2.12132 − 0.707107i −0.474342 − 0.158114i
\(21\) 0 0
\(22\) 1.12132 + 3.12132i 0.239066 + 0.665468i
\(23\) 0.171573 − 0.171573i 0.0357754 − 0.0357754i −0.688993 − 0.724768i \(-0.741947\pi\)
0.724768 + 0.688993i \(0.241947\pi\)
\(24\) 0 0
\(25\) −4.00000 − 3.00000i −0.800000 − 0.600000i
\(26\) 4.24264 0.832050
\(27\) 0 0
\(28\) 2.12132 + 2.12132i 0.400892 + 0.400892i
\(29\) −1.24264 −0.230753 −0.115376 − 0.993322i \(-0.536807\pi\)
−0.115376 + 0.993322i \(0.536807\pi\)
\(30\) 0 0
\(31\) −7.24264 −1.30082 −0.650408 − 0.759585i \(-0.725402\pi\)
−0.650408 + 0.759585i \(0.725402\pi\)
\(32\) 0.707107 − 0.707107i 0.125000 − 0.125000i
\(33\) 0 0
\(34\) − 7.24264i − 1.24210i
\(35\) 3.00000 + 6.00000i 0.507093 + 1.01419i
\(36\) 0 0
\(37\) 0.121320 + 0.121320i 0.0199449 + 0.0199449i 0.717009 − 0.697064i \(-0.245511\pi\)
−0.697064 + 0.717009i \(0.745511\pi\)
\(38\) −2.12132 + 2.12132i −0.344124 + 0.344124i
\(39\) 0 0
\(40\) 2.00000 − 1.00000i 0.316228 − 0.158114i
\(41\) − 1.75736i − 0.274453i −0.990540 − 0.137227i \(-0.956181\pi\)
0.990540 − 0.137227i \(-0.0438189\pi\)
\(42\) 0 0
\(43\) −1.24264 − 1.24264i −0.189501 − 0.189501i 0.605979 − 0.795480i \(-0.292781\pi\)
−0.795480 + 0.605979i \(0.792781\pi\)
\(44\) −3.00000 − 1.41421i −0.452267 − 0.213201i
\(45\) 0 0
\(46\) 0.242641i 0.0357754i
\(47\) −4.41421 − 4.41421i −0.643879 − 0.643879i 0.307628 − 0.951507i \(-0.400465\pi\)
−0.951507 + 0.307628i \(0.900465\pi\)
\(48\) 0 0
\(49\) − 2.00000i − 0.285714i
\(50\) 4.94975 − 0.707107i 0.700000 − 0.100000i
\(51\) 0 0
\(52\) −3.00000 + 3.00000i −0.416025 + 0.416025i
\(53\) −9.53553 + 9.53553i −1.30981 + 1.30981i −0.388254 + 0.921552i \(0.626922\pi\)
−0.921552 + 0.388254i \(0.873078\pi\)
\(54\) 0 0
\(55\) −5.36396 − 5.12132i −0.723276 − 0.690559i
\(56\) −3.00000 −0.400892
\(57\) 0 0
\(58\) 0.878680 − 0.878680i 0.115376 − 0.115376i
\(59\) − 1.41421i − 0.184115i −0.995754 − 0.0920575i \(-0.970656\pi\)
0.995754 − 0.0920575i \(-0.0293443\pi\)
\(60\) 0 0
\(61\) 7.24264i 0.927325i 0.886012 + 0.463663i \(0.153465\pi\)
−0.886012 + 0.463663i \(0.846535\pi\)
\(62\) 5.12132 − 5.12132i 0.650408 − 0.650408i
\(63\) 0 0
\(64\) 1.00000i 0.125000i
\(65\) −8.48528 + 4.24264i −1.05247 + 0.526235i
\(66\) 0 0
\(67\) −4.00000 − 4.00000i −0.488678 − 0.488678i 0.419211 − 0.907889i \(-0.362307\pi\)
−0.907889 + 0.419211i \(0.862307\pi\)
\(68\) 5.12132 + 5.12132i 0.621051 + 0.621051i
\(69\) 0 0
\(70\) −6.36396 − 2.12132i −0.760639 − 0.253546i
\(71\) 1.24264 0.147474 0.0737372 − 0.997278i \(-0.476507\pi\)
0.0737372 + 0.997278i \(0.476507\pi\)
\(72\) 0 0
\(73\) −6.00000 − 6.00000i −0.702247 − 0.702247i 0.262646 − 0.964892i \(-0.415405\pi\)
−0.964892 + 0.262646i \(0.915405\pi\)
\(74\) −0.171573 −0.0199449
\(75\) 0 0
\(76\) − 3.00000i − 0.344124i
\(77\) 3.36396 + 9.36396i 0.383359 + 1.06712i
\(78\) 0 0
\(79\) −10.2426 −1.15239 −0.576194 − 0.817313i \(-0.695463\pi\)
−0.576194 + 0.817313i \(0.695463\pi\)
\(80\) −0.707107 + 2.12132i −0.0790569 + 0.237171i
\(81\) 0 0
\(82\) 1.24264 + 1.24264i 0.137227 + 0.137227i
\(83\) −7.24264 − 7.24264i −0.794983 − 0.794983i 0.187317 − 0.982300i \(-0.440021\pi\)
−0.982300 + 0.187317i \(0.940021\pi\)
\(84\) 0 0
\(85\) 7.24264 + 14.4853i 0.785575 + 1.57115i
\(86\) 1.75736 0.189501
\(87\) 0 0
\(88\) 3.12132 − 1.12132i 0.332734 − 0.119533i
\(89\) − 5.48528i − 0.581439i −0.956808 − 0.290719i \(-0.906105\pi\)
0.956808 − 0.290719i \(-0.0938946\pi\)
\(90\) 0 0
\(91\) 12.7279 1.33425
\(92\) −0.171573 − 0.171573i −0.0178877 − 0.0178877i
\(93\) 0 0
\(94\) 6.24264 0.643879
\(95\) 2.12132 − 6.36396i 0.217643 − 0.652929i
\(96\) 0 0
\(97\) −2.24264 − 2.24264i −0.227706 − 0.227706i 0.584028 − 0.811734i \(-0.301476\pi\)
−0.811734 + 0.584028i \(0.801476\pi\)
\(98\) 1.41421 + 1.41421i 0.142857 + 0.142857i
\(99\) 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 990.2.m.d.307.1 4
3.2 odd 2 110.2.f.c.87.2 yes 4
5.3 odd 4 990.2.m.c.703.2 4
11.10 odd 2 990.2.m.c.307.2 4
12.11 even 2 880.2.bd.b.417.1 4
15.2 even 4 550.2.f.a.43.2 4
15.8 even 4 110.2.f.b.43.1 ✓ 4
15.14 odd 2 550.2.f.b.307.1 4
33.32 even 2 110.2.f.b.87.1 yes 4
55.43 even 4 inner 990.2.m.d.703.1 4
60.23 odd 4 880.2.bd.c.593.1 4
132.131 odd 2 880.2.bd.c.417.1 4
165.32 odd 4 550.2.f.b.43.1 4
165.98 odd 4 110.2.f.c.43.2 yes 4
165.164 even 2 550.2.f.a.307.2 4
660.263 even 4 880.2.bd.b.593.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.2.f.b.43.1 ✓ 4 15.8 even 4
110.2.f.b.87.1 yes 4 33.32 even 2
110.2.f.c.43.2 yes 4 165.98 odd 4
110.2.f.c.87.2 yes 4 3.2 odd 2
550.2.f.a.43.2 4 15.2 even 4
550.2.f.a.307.2 4 165.164 even 2
550.2.f.b.43.1 4 165.32 odd 4
550.2.f.b.307.1 4 15.14 odd 2
880.2.bd.b.417.1 4 12.11 even 2
880.2.bd.b.593.1 4 660.263 even 4
880.2.bd.c.417.1 4 132.131 odd 2
880.2.bd.c.593.1 4 60.23 odd 4
990.2.m.c.307.2 4 11.10 odd 2
990.2.m.c.703.2 4 5.3 odd 4
990.2.m.d.307.1 4 1.1 even 1 trivial
990.2.m.d.703.1 4 55.43 even 4 inner