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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1332,2,Mod(445,1332)] 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("1332.445"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1332, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1332 = 2^{2} \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1332.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [34,0,1,0,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.6360735492\)
Analytic rank: \(0\)
Dimension: \(34\)
Relative dimension: \(17\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 445.9
Character \(\chi\) \(=\) 1332.445
Dual form 1332.2.i.c.889.9

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.0706845 + 1.73061i) q^{3} +(-1.19957 + 2.07772i) q^{5} +(-1.81720 - 3.14748i) q^{7} +(-2.99001 + 0.244654i) q^{9} +(-2.18505 - 3.78463i) q^{11} +(-0.0910477 + 0.157699i) q^{13} +(-3.68051 - 1.92913i) q^{15} +6.44469 q^{17} +1.21049 q^{19} +(5.31860 - 3.36733i) q^{21} +(1.94282 - 3.36506i) q^{23} +(-0.377943 - 0.654616i) q^{25} +(-0.634748 - 5.15724i) q^{27} +(3.00240 + 5.20030i) q^{29} +(2.37541 - 4.11434i) q^{31} +(6.39525 - 4.04899i) q^{33} +8.71943 q^{35} -1.00000 q^{37} +(-0.279351 - 0.146421i) q^{39} +(2.18122 - 3.77798i) q^{41} +(-2.91557 - 5.04992i) q^{43} +(3.07840 - 6.50587i) q^{45} +(3.66254 + 6.34371i) q^{47} +(-3.10441 + 5.37699i) q^{49} +(0.455539 + 11.1532i) q^{51} +0.267092 q^{53} +10.4845 q^{55} +(0.0855631 + 2.09489i) q^{57} +(-0.173807 + 0.301042i) q^{59} +(-7.30607 - 12.6545i) q^{61} +(6.20348 + 8.96639i) q^{63} +(-0.218436 - 0.378343i) q^{65} +(-5.62379 + 9.74069i) q^{67} +(5.96094 + 3.12440i) q^{69} +5.71014 q^{71} -0.0249932 q^{73} +(1.10617 - 0.700342i) q^{75} +(-7.94135 + 13.7548i) q^{77} +(-3.71473 - 6.43409i) q^{79} +(8.88029 - 1.46304i) q^{81} +(-4.97807 - 8.62227i) q^{83} +(-7.73086 + 13.3902i) q^{85} +(-8.78746 + 5.56355i) q^{87} +13.7030 q^{89} +0.661806 q^{91} +(7.28821 + 3.82009i) q^{93} +(-1.45207 + 2.51506i) q^{95} +(3.31019 + 5.73341i) q^{97} +(7.45925 + 10.7815i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 34 q + q^{3} - 7 q^{5} - q^{7} + 13 q^{9} - 4 q^{11} + 3 q^{13} - 5 q^{15} + 12 q^{17} - 2 q^{19} + 7 q^{21} - 13 q^{23} - 22 q^{25} - 17 q^{27} - 13 q^{29} - 7 q^{31} - q^{33} - 8 q^{35} - 34 q^{37}+ \cdots - 55 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1332\mathbb{Z}\right)^\times\).

\(n\) \(667\) \(1037\) \(1297\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\)

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
\(3\) 0.0706845 + 1.73061i 0.0408097 + 0.999167i
\(4\) 0 0
\(5\) −1.19957 + 2.07772i −0.536465 + 0.929184i 0.462626 + 0.886553i \(0.346907\pi\)
−0.999091 + 0.0426306i \(0.986426\pi\)
\(6\) 0 0
\(7\) −1.81720 − 3.14748i −0.686836 − 1.18963i −0.972856 − 0.231411i \(-0.925666\pi\)
0.286020 − 0.958224i \(-0.407668\pi\)
\(8\) 0 0
\(9\) −2.99001 + 0.244654i −0.996669 + 0.0815514i
\(10\) 0 0
\(11\) −2.18505 − 3.78463i −0.658819 − 1.14111i −0.980922 − 0.194403i \(-0.937723\pi\)
0.322103 − 0.946705i \(-0.395610\pi\)
\(12\) 0 0
\(13\) −0.0910477 + 0.157699i −0.0252521 + 0.0437379i −0.878375 − 0.477972i \(-0.841372\pi\)
0.853123 + 0.521710i \(0.174706\pi\)
\(14\) 0 0
\(15\) −3.68051 − 1.92913i −0.950303 − 0.498098i
\(16\) 0 0
\(17\) 6.44469 1.56307 0.781533 − 0.623864i \(-0.214438\pi\)
0.781533 + 0.623864i \(0.214438\pi\)
\(18\) 0 0
\(19\) 1.21049 0.277706 0.138853 − 0.990313i \(-0.455658\pi\)
0.138853 + 0.990313i \(0.455658\pi\)
\(20\) 0 0
\(21\) 5.31860 − 3.36733i 1.16061 − 0.734812i
\(22\) 0 0
\(23\) 1.94282 − 3.36506i 0.405106 − 0.701665i −0.589228 − 0.807967i \(-0.700568\pi\)
0.994334 + 0.106303i \(0.0339012\pi\)
\(24\) 0 0
\(25\) −0.377943 − 0.654616i −0.0755886 − 0.130923i
\(26\) 0 0
\(27\) −0.634748 − 5.15724i −0.122157 − 0.992511i
\(28\) 0 0
\(29\) 3.00240 + 5.20030i 0.557531 + 0.965672i 0.997702 + 0.0677581i \(0.0215846\pi\)
−0.440171 + 0.897914i \(0.645082\pi\)
\(30\) 0 0
\(31\) 2.37541 − 4.11434i 0.426637 − 0.738957i −0.569935 − 0.821690i \(-0.693032\pi\)
0.996572 + 0.0827330i \(0.0263649\pi\)
\(32\) 0 0
\(33\) 6.39525 − 4.04899i 1.11327 − 0.704838i
\(34\) 0 0
\(35\) 8.71943 1.47385
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) −0.279351 − 0.146421i −0.0447320 − 0.0234461i
\(40\) 0 0
\(41\) 2.18122 − 3.77798i 0.340649 − 0.590021i −0.643905 − 0.765106i \(-0.722687\pi\)
0.984553 + 0.175085i \(0.0560200\pi\)
\(42\) 0 0
\(43\) −2.91557 − 5.04992i −0.444621 − 0.770106i 0.553405 − 0.832912i \(-0.313328\pi\)
−0.998026 + 0.0628065i \(0.979995\pi\)
\(44\) 0 0
\(45\) 3.07840 − 6.50587i 0.458901 − 0.969838i
\(46\) 0 0
\(47\) 3.66254 + 6.34371i 0.534237 + 0.925326i 0.999200 + 0.0399955i \(0.0127344\pi\)
−0.464963 + 0.885330i \(0.653932\pi\)
\(48\) 0 0
\(49\) −3.10441 + 5.37699i −0.443487 + 0.768142i
\(50\) 0 0
\(51\) 0.455539 + 11.1532i 0.0637883 + 1.56176i
\(52\) 0 0
\(53\) 0.267092 0.0366879 0.0183440 − 0.999832i \(-0.494161\pi\)
0.0183440 + 0.999832i \(0.494161\pi\)
\(54\) 0 0
\(55\) 10.4845 1.41373
\(56\) 0 0
\(57\) 0.0855631 + 2.09489i 0.0113331 + 0.277475i
\(58\) 0 0
\(59\) −0.173807 + 0.301042i −0.0226277 + 0.0391923i −0.877118 − 0.480276i \(-0.840537\pi\)
0.854490 + 0.519468i \(0.173870\pi\)
\(60\) 0 0
\(61\) −7.30607 − 12.6545i −0.935447 − 1.62024i −0.773835 − 0.633387i \(-0.781664\pi\)
−0.161612 − 0.986854i \(-0.551669\pi\)
\(62\) 0 0
\(63\) 6.20348 + 8.96639i 0.781564 + 1.12966i
\(64\) 0 0
\(65\) −0.218436 − 0.378343i −0.0270937 − 0.0469277i
\(66\) 0 0
\(67\) −5.62379 + 9.74069i −0.687055 + 1.19001i 0.285731 + 0.958310i \(0.407764\pi\)
−0.972786 + 0.231705i \(0.925570\pi\)
\(68\) 0 0
\(69\) 5.96094 + 3.12440i 0.717612 + 0.376134i
\(70\) 0 0
\(71\) 5.71014 0.677668 0.338834 − 0.940846i \(-0.389967\pi\)
0.338834 + 0.940846i \(0.389967\pi\)
\(72\) 0 0
\(73\) −0.0249932 −0.00292523 −0.00146262 − 0.999999i \(-0.500466\pi\)
−0.00146262 + 0.999999i \(0.500466\pi\)
\(74\) 0 0
\(75\) 1.10617 − 0.700342i 0.127729 − 0.0808686i
\(76\) 0 0
\(77\) −7.94135 + 13.7548i −0.905001 + 1.56751i
\(78\) 0 0
\(79\) −3.71473 − 6.43409i −0.417939 − 0.723892i 0.577793 − 0.816183i \(-0.303914\pi\)
−0.995732 + 0.0922915i \(0.970581\pi\)
\(80\) 0 0
\(81\) 8.88029 − 1.46304i 0.986699 − 0.162560i
\(82\) 0 0
\(83\) −4.97807 − 8.62227i −0.546414 − 0.946417i −0.998516 − 0.0544507i \(-0.982659\pi\)
0.452103 − 0.891966i \(-0.350674\pi\)
\(84\) 0 0
\(85\) −7.73086 + 13.3902i −0.838530 + 1.45238i
\(86\) 0 0
\(87\) −8.78746 + 5.56355i −0.942115 + 0.596475i
\(88\) 0 0
\(89\) 13.7030 1.45251 0.726256 − 0.687424i \(-0.241259\pi\)
0.726256 + 0.687424i \(0.241259\pi\)
\(90\) 0 0
\(91\) 0.661806 0.0693762
\(92\) 0 0
\(93\) 7.28821 + 3.82009i 0.755752 + 0.396125i
\(94\) 0 0
\(95\) −1.45207 + 2.51506i −0.148980 + 0.258040i
\(96\) 0 0
\(97\) 3.31019 + 5.73341i 0.336099 + 0.582140i 0.983695 − 0.179843i \(-0.0575591\pi\)
−0.647597 + 0.761983i \(0.724226\pi\)
\(98\) 0 0
\(99\) 7.45925 + 10.7815i 0.749683 + 1.08358i
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 1332.2.i.c.445.9 ✓ 34
3.2 odd 2 3996.2.i.d.1333.13 34
9.2 odd 6 3996.2.i.d.2665.13 34
9.7 even 3 inner 1332.2.i.c.889.9 yes 34
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1332.2.i.c.445.9 ✓ 34 1.1 even 1 trivial
1332.2.i.c.889.9 yes 34 9.7 even 3 inner
3996.2.i.d.1333.13 34 3.2 odd 2
3996.2.i.d.2665.13 34 9.2 odd 6