Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [414,2,Mod(55,414)] 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("414.55"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(414, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 10])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 414.i (of order \(11\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,-1,0,-1,0] 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: \(3.30580664368\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\Q(\zeta_{22})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 138)
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 307.1
Root \(-0.415415 + 0.909632i\) of defining polynomial
Character \(\chi\) \(=\) 414.307
Dual form 414.2.i.b.325.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.142315 + 0.989821i) q^{2} +(-0.959493 - 0.281733i) q^{4} +(-1.59283 - 1.83823i) q^{5} +(1.56130 + 1.00339i) q^{7} +(0.415415 - 0.909632i) q^{8} +(2.04620 - 1.31501i) q^{10} +(0.790323 + 5.49682i) q^{11} +(0.966031 - 0.620830i) q^{13} +(-1.21537 + 1.40261i) q^{14} +(0.841254 + 0.540641i) q^{16} +(7.20366 - 2.11519i) q^{17} +(4.29408 + 1.26086i) q^{19} +(1.01042 + 2.21252i) q^{20} -5.55334 q^{22} +(2.45093 + 4.12225i) q^{23} +(-0.130388 + 0.906870i) q^{25} +(0.477031 + 1.04455i) q^{26} +(-1.21537 - 1.40261i) q^{28} +(2.81913 - 0.827771i) q^{29} +(-0.863019 + 1.88975i) q^{31} +(-0.654861 + 0.755750i) q^{32} +(1.06847 + 7.43136i) q^{34} +(-0.642438 - 4.46825i) q^{35} +(-5.06718 + 5.84784i) q^{37} +(-1.85913 + 4.07093i) q^{38} +(-2.33380 + 0.685265i) q^{40} +(0.450833 + 0.520289i) q^{41} +(-3.55862 - 7.79229i) q^{43} +(0.790323 - 5.49682i) q^{44} +(-4.42909 + 1.83933i) q^{46} -7.09992 q^{47} +(-1.47703 - 3.23425i) q^{49} +(-0.879083 - 0.258122i) q^{50} +(-1.10181 + 0.323520i) q^{52} +(-4.53311 - 2.91326i) q^{53} +(8.84555 - 10.2083i) q^{55} +(1.56130 - 1.00339i) q^{56} +(0.418141 + 2.90824i) q^{58} +(3.39399 - 2.18119i) q^{59} +(-0.157543 + 0.344971i) q^{61} +(-1.74769 - 1.12317i) q^{62} +(-0.654861 - 0.755750i) q^{64} +(-2.67995 - 0.786905i) q^{65} +(-0.420243 + 2.92285i) q^{67} -7.50778 q^{68} +4.51420 q^{70} +(-2.14473 + 14.9169i) q^{71} +(11.0242 + 3.23701i) q^{73} +(-5.06718 - 5.84784i) q^{74} +(-3.76492 - 2.41956i) q^{76} +(-4.28150 + 9.37518i) q^{77} +(8.86394 - 5.69651i) q^{79} +(-0.346156 - 2.40757i) q^{80} +(-0.579153 + 0.372199i) q^{82} +(6.84228 - 7.89641i) q^{83} +(-15.3624 - 9.87283i) q^{85} +(8.21942 - 2.41344i) q^{86} +(5.32839 + 1.56456i) q^{88} +(-4.47642 - 9.80199i) q^{89} +2.13120 q^{91} +(-1.19028 - 4.64578i) q^{92} +(1.01042 - 7.02765i) q^{94} +(-4.52201 - 9.90183i) q^{95} +(6.51094 + 7.51402i) q^{97} +(3.41153 - 1.00172i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{2} - q^{4} - 2 q^{7} - q^{8} + 11 q^{10} - 11 q^{11} - 13 q^{13} - 13 q^{14} - q^{16} - 2 q^{19} + 11 q^{20} - 22 q^{22} + 10 q^{23} + 5 q^{25} + 9 q^{26} - 13 q^{28} + 27 q^{29} - 18 q^{31}+ \cdots + 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/414\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(235\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.142315 + 0.989821i −0.100632 + 0.699909i
\(3\) 0 0
\(4\) −0.959493 − 0.281733i −0.479746 − 0.140866i
\(5\) −1.59283 − 1.83823i −0.712337 − 0.822080i 0.278027 − 0.960573i \(-0.410320\pi\)
−0.990363 + 0.138493i \(0.955774\pi\)
\(6\) 0 0
\(7\) 1.56130 + 1.00339i 0.590116 + 0.379245i 0.801354 − 0.598190i \(-0.204113\pi\)
−0.211238 + 0.977435i \(0.567750\pi\)
\(8\) 0.415415 − 0.909632i 0.146871 − 0.321603i
\(9\) 0 0
\(10\) 2.04620 − 1.31501i 0.647065 − 0.415844i
\(11\) 0.790323 + 5.49682i 0.238291 + 1.65735i 0.660479 + 0.750844i \(0.270353\pi\)
−0.422188 + 0.906508i \(0.638738\pi\)
\(12\) 0 0
\(13\) 0.966031 − 0.620830i 0.267929 − 0.172187i −0.399777 − 0.916612i \(-0.630913\pi\)
0.667706 + 0.744425i \(0.267276\pi\)
\(14\) −1.21537 + 1.40261i −0.324821 + 0.374864i
\(15\) 0 0
\(16\) 0.841254 + 0.540641i 0.210313 + 0.135160i
\(17\) 7.20366 − 2.11519i 1.74714 − 0.513008i 0.757043 − 0.653365i \(-0.226643\pi\)
0.990101 + 0.140357i \(0.0448250\pi\)
\(18\) 0 0
\(19\) 4.29408 + 1.26086i 0.985130 + 0.289260i 0.734340 − 0.678781i \(-0.237492\pi\)
0.250789 + 0.968042i \(0.419310\pi\)
\(20\) 1.01042 + 2.21252i 0.225938 + 0.494734i
\(21\) 0 0
\(22\) −5.55334 −1.18398
\(23\) 2.45093 + 4.12225i 0.511054 + 0.859548i
\(24\) 0 0
\(25\) −0.130388 + 0.906870i −0.0260776 + 0.181374i
\(26\) 0.477031 + 1.04455i 0.0935534 + 0.204853i
\(27\) 0 0
\(28\) −1.21537 − 1.40261i −0.229683 − 0.265069i
\(29\) 2.81913 − 0.827771i 0.523499 − 0.153713i −0.00929715 − 0.999957i \(-0.502959\pi\)
0.532796 + 0.846244i \(0.321141\pi\)
\(30\) 0 0
\(31\) −0.863019 + 1.88975i −0.155003 + 0.339409i −0.971163 − 0.238417i \(-0.923371\pi\)
0.816160 + 0.577826i \(0.196099\pi\)
\(32\) −0.654861 + 0.755750i −0.115764 + 0.133599i
\(33\) 0 0
\(34\) 1.06847 + 7.43136i 0.183241 + 1.27447i
\(35\) −0.642438 − 4.46825i −0.108592 − 0.755273i
\(36\) 0 0
\(37\) −5.06718 + 5.84784i −0.833039 + 0.961379i −0.999696 − 0.0246361i \(-0.992157\pi\)
0.166657 + 0.986015i \(0.446703\pi\)
\(38\) −1.85913 + 4.07093i −0.301591 + 0.660393i
\(39\) 0 0
\(40\) −2.33380 + 0.685265i −0.369006 + 0.108350i
\(41\) 0.450833 + 0.520289i 0.0704083 + 0.0812555i 0.789861 − 0.613286i \(-0.210153\pi\)
−0.719453 + 0.694542i \(0.755607\pi\)
\(42\) 0 0
\(43\) −3.55862 − 7.79229i −0.542684 − 1.18831i −0.960115 − 0.279604i \(-0.909797\pi\)
0.417431 − 0.908709i \(-0.362931\pi\)
\(44\) 0.790323 − 5.49682i 0.119146 − 0.828676i
\(45\) 0 0
\(46\) −4.42909 + 1.83933i −0.653034 + 0.271194i
\(47\) −7.09992 −1.03563 −0.517815 − 0.855493i \(-0.673254\pi\)
−0.517815 + 0.855493i \(0.673254\pi\)
\(48\) 0 0
\(49\) −1.47703 − 3.23425i −0.211004 − 0.462035i
\(50\) −0.879083 − 0.258122i −0.124321 − 0.0365040i
\(51\) 0 0
\(52\) −1.10181 + 0.323520i −0.152793 + 0.0448641i
\(53\) −4.53311 − 2.91326i −0.622671 − 0.400166i 0.190919 − 0.981606i \(-0.438853\pi\)
−0.813590 + 0.581439i \(0.802490\pi\)
\(54\) 0 0
\(55\) 8.84555 − 10.2083i 1.19273 − 1.37649i
\(56\) 1.56130 − 1.00339i 0.208638 − 0.134083i
\(57\) 0 0
\(58\) 0.418141 + 2.90824i 0.0549047 + 0.381870i
\(59\) 3.39399 − 2.18119i 0.441860 − 0.283966i −0.300734 − 0.953708i \(-0.597232\pi\)
0.742594 + 0.669742i \(0.233595\pi\)
\(60\) 0 0
\(61\) −0.157543 + 0.344971i −0.0201713 + 0.0441690i −0.919450 − 0.393208i \(-0.871365\pi\)
0.899278 + 0.437377i \(0.144092\pi\)
\(62\) −1.74769 − 1.12317i −0.221957 − 0.142643i
\(63\) 0 0
\(64\) −0.654861 − 0.755750i −0.0818576 − 0.0944687i
\(65\) −2.67995 − 0.786905i −0.332407 − 0.0976036i
\(66\) 0 0
\(67\) −0.420243 + 2.92285i −0.0513408 + 0.357083i 0.947915 + 0.318522i \(0.103187\pi\)
−0.999256 + 0.0385611i \(0.987723\pi\)
\(68\) −7.50778 −0.910452
\(69\) 0 0
\(70\) 4.51420 0.539550
\(71\) −2.14473 + 14.9169i −0.254532 + 1.77031i 0.315731 + 0.948849i \(0.397750\pi\)
−0.570264 + 0.821462i \(0.693159\pi\)
\(72\) 0 0
\(73\) 11.0242 + 3.23701i 1.29029 + 0.378863i 0.853685 − 0.520790i \(-0.174363\pi\)
0.436605 + 0.899653i \(0.356181\pi\)
\(74\) −5.06718 − 5.84784i −0.589048 − 0.679797i
\(75\) 0 0
\(76\) −3.76492 − 2.41956i −0.431866 − 0.277543i
\(77\) −4.28150 + 9.37518i −0.487923 + 1.06840i
\(78\) 0 0
\(79\) 8.86394 − 5.69651i 0.997272 − 0.640908i 0.0632029 − 0.998001i \(-0.479868\pi\)
0.934069 + 0.357093i \(0.116232\pi\)
\(80\) −0.346156 − 2.40757i −0.0387014 − 0.269174i
\(81\) 0 0
\(82\) −0.579153 + 0.372199i −0.0639568 + 0.0411025i
\(83\) 6.84228 − 7.89641i 0.751037 − 0.866743i −0.243631 − 0.969868i \(-0.578339\pi\)
0.994668 + 0.103125i \(0.0328841\pi\)
\(84\) 0 0
\(85\) −15.3624 − 9.87283i −1.66629 − 1.07086i
\(86\) 8.21942 − 2.41344i 0.886323 − 0.260248i
\(87\) 0 0
\(88\) 5.32839 + 1.56456i 0.568009 + 0.166782i
\(89\) −4.47642 − 9.80199i −0.474499 − 1.03901i −0.983939 − 0.178502i \(-0.942875\pi\)
0.509440 − 0.860506i \(-0.329853\pi\)
\(90\) 0 0
\(91\) 2.13120 0.223410
\(92\) −1.19028 − 4.64578i −0.124095 − 0.484356i
\(93\) 0 0
\(94\) 1.01042 − 7.02765i 0.104217 − 0.724847i
\(95\) −4.52201 − 9.90183i −0.463949 − 1.01591i
\(96\) 0 0
\(97\) 6.51094 + 7.51402i 0.661085 + 0.762933i 0.982954 − 0.183851i \(-0.0588564\pi\)
−0.321869 + 0.946784i \(0.604311\pi\)
\(98\) 3.41153 − 1.00172i 0.344616 − 0.101189i
\(99\) 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 414.2.i.b.307.1 10
3.2 odd 2 138.2.e.c.31.1 ✓ 10
23.3 even 11 inner 414.2.i.b.325.1 10
23.7 odd 22 9522.2.a.ca.1.1 5
23.16 even 11 9522.2.a.bv.1.5 5
69.26 odd 22 138.2.e.c.49.1 yes 10
69.53 even 22 3174.2.a.y.1.5 5
69.62 odd 22 3174.2.a.z.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.2.e.c.31.1 ✓ 10 3.2 odd 2
138.2.e.c.49.1 yes 10 69.26 odd 22
414.2.i.b.307.1 10 1.1 even 1 trivial
414.2.i.b.325.1 10 23.3 even 11 inner
3174.2.a.y.1.5 5 69.53 even 22
3174.2.a.z.1.1 5 69.62 odd 22
9522.2.a.bv.1.5 5 23.16 even 11
9522.2.a.ca.1.1 5 23.7 odd 22