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Properties

Label 225.6.b.e.199.1
Level $225$
Weight $6$
Character 225.199
Analytic conductor $36.086$
Analytic rank $0$
Dimension $2$
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] = [225,6,Mod(199,225)] 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("225.199"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 225.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,56,0,0,0,0,0,0,296] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(36.0863594579\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 5)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 225.199
Dual form 225.6.b.e.199.2

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} +28.0000 q^{4} +192.000i q^{7} -120.000i q^{8} +148.000 q^{11} -286.000i q^{13} +384.000 q^{14} +656.000 q^{16} +1678.00i q^{17} -1060.00 q^{19} -296.000i q^{22} +2976.00i q^{23} -572.000 q^{26} +5376.00i q^{28} -3410.00 q^{29} -2448.00 q^{31} -5152.00i q^{32} +3356.00 q^{34} +182.000i q^{37} +2120.00i q^{38} +9398.00 q^{41} +1244.00i q^{43} +4144.00 q^{44} +5952.00 q^{46} +12088.0i q^{47} -20057.0 q^{49} -8008.00i q^{52} +23846.0i q^{53} +23040.0 q^{56} +6820.00i q^{58} -20020.0 q^{59} +32302.0 q^{61} +4896.00i q^{62} +10688.0 q^{64} +60972.0i q^{67} +46984.0i q^{68} +32648.0 q^{71} +38774.0i q^{73} +364.000 q^{74} -29680.0 q^{76} +28416.0i q^{77} +33360.0 q^{79} -18796.0i q^{82} +16716.0i q^{83} +2488.00 q^{86} -17760.0i q^{88} +101370. q^{89} +54912.0 q^{91} +83328.0i q^{92} +24176.0 q^{94} -119038. i q^{97} +40114.0i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 56 q^{4} + 296 q^{11} + 768 q^{14} + 1312 q^{16} - 2120 q^{19} - 1144 q^{26} - 6820 q^{29} - 4896 q^{31} + 6712 q^{34} + 18796 q^{41} + 8288 q^{44} + 11904 q^{46} - 40114 q^{49} + 46080 q^{56} - 40040 q^{59}+ \cdots + 48352 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.00000i − 0.353553i −0.984251 − 0.176777i \(-0.943433\pi\)
0.984251 − 0.176777i \(-0.0565670\pi\)
\(3\) 0 0
\(4\) 28.0000 0.875000
\(5\) 0 0
\(6\) 0 0
\(7\) 192.000i 1.48100i 0.672054 + 0.740502i \(0.265412\pi\)
−0.672054 + 0.740502i \(0.734588\pi\)
\(8\) − 120.000i − 0.662913i
\(9\) 0 0
\(10\) 0 0
\(11\) 148.000 0.368791 0.184395 − 0.982852i \(-0.440967\pi\)
0.184395 + 0.982852i \(0.440967\pi\)
\(12\) 0 0
\(13\) − 286.000i − 0.469362i −0.972072 − 0.234681i \(-0.924595\pi\)
0.972072 − 0.234681i \(-0.0754045\pi\)
\(14\) 384.000 0.523614
\(15\) 0 0
\(16\) 656.000 0.640625
\(17\) 1678.00i 1.40822i 0.710092 + 0.704109i \(0.248653\pi\)
−0.710092 + 0.704109i \(0.751347\pi\)
\(18\) 0 0
\(19\) −1060.00 −0.673631 −0.336815 − 0.941571i \(-0.609350\pi\)
−0.336815 + 0.941571i \(0.609350\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 296.000i − 0.130387i
\(23\) 2976.00i 1.17304i 0.809934 + 0.586521i \(0.199503\pi\)
−0.809934 + 0.586521i \(0.800497\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −572.000 −0.165944
\(27\) 0 0
\(28\) 5376.00i 1.29588i
\(29\) −3410.00 −0.752938 −0.376469 − 0.926429i \(-0.622862\pi\)
−0.376469 + 0.926429i \(0.622862\pi\)
\(30\) 0 0
\(31\) −2448.00 −0.457517 −0.228758 − 0.973483i \(-0.573467\pi\)
−0.228758 + 0.973483i \(0.573467\pi\)
\(32\) − 5152.00i − 0.889408i
\(33\) 0 0
\(34\) 3356.00 0.497880
\(35\) 0 0
\(36\) 0 0
\(37\) 182.000i 0.0218558i 0.999940 + 0.0109279i \(0.00347853\pi\)
−0.999940 + 0.0109279i \(0.996521\pi\)
\(38\) 2120.00i 0.238164i
\(39\) 0 0
\(40\) 0 0
\(41\) 9398.00 0.873124 0.436562 − 0.899674i \(-0.356196\pi\)
0.436562 + 0.899674i \(0.356196\pi\)
\(42\) 0 0
\(43\) 1244.00i 0.102600i 0.998683 + 0.0513002i \(0.0163365\pi\)
−0.998683 + 0.0513002i \(0.983663\pi\)
\(44\) 4144.00 0.322692
\(45\) 0 0
\(46\) 5952.00 0.414733
\(47\) 12088.0i 0.798196i 0.916908 + 0.399098i \(0.130677\pi\)
−0.916908 + 0.399098i \(0.869323\pi\)
\(48\) 0 0
\(49\) −20057.0 −1.19337
\(50\) 0 0
\(51\) 0 0
\(52\) − 8008.00i − 0.410691i
\(53\) 23846.0i 1.16607i 0.812446 + 0.583037i \(0.198136\pi\)
−0.812446 + 0.583037i \(0.801864\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 23040.0 0.981776
\(57\) 0 0
\(58\) 6820.00i 0.266204i
\(59\) −20020.0 −0.748745 −0.374373 − 0.927278i \(-0.622142\pi\)
−0.374373 + 0.927278i \(0.622142\pi\)
\(60\) 0 0
\(61\) 32302.0 1.11149 0.555744 − 0.831353i \(-0.312433\pi\)
0.555744 + 0.831353i \(0.312433\pi\)
\(62\) 4896.00i 0.161757i
\(63\) 0 0
\(64\) 10688.0 0.326172
\(65\) 0 0
\(66\) 0 0
\(67\) 60972.0i 1.65937i 0.558231 + 0.829685i \(0.311480\pi\)
−0.558231 + 0.829685i \(0.688520\pi\)
\(68\) 46984.0i 1.23219i
\(69\) 0 0
\(70\) 0 0
\(71\) 32648.0 0.768618 0.384309 − 0.923204i \(-0.374440\pi\)
0.384309 + 0.923204i \(0.374440\pi\)
\(72\) 0 0
\(73\) 38774.0i 0.851596i 0.904818 + 0.425798i \(0.140007\pi\)
−0.904818 + 0.425798i \(0.859993\pi\)
\(74\) 364.000 0.00772720
\(75\) 0 0
\(76\) −29680.0 −0.589427
\(77\) 28416.0i 0.546180i
\(78\) 0 0
\(79\) 33360.0 0.601393 0.300696 − 0.953720i \(-0.402781\pi\)
0.300696 + 0.953720i \(0.402781\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 18796.0i − 0.308696i
\(83\) 16716.0i 0.266340i 0.991093 + 0.133170i \(0.0425157\pi\)
−0.991093 + 0.133170i \(0.957484\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2488.00 0.0362747
\(87\) 0 0
\(88\) − 17760.0i − 0.244476i
\(89\) 101370. 1.35655 0.678273 − 0.734810i \(-0.262729\pi\)
0.678273 + 0.734810i \(0.262729\pi\)
\(90\) 0 0
\(91\) 54912.0 0.695126
\(92\) 83328.0i 1.02641i
\(93\) 0 0
\(94\) 24176.0 0.282205
\(95\) 0 0
\(96\) 0 0
\(97\) − 119038.i − 1.28457i −0.766468 − 0.642283i \(-0.777987\pi\)
0.766468 − 0.642283i \(-0.222013\pi\)
\(98\) 40114.0i 0.421921i
\(99\) 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 225.6.b.e.199.1 2
3.2 odd 2 25.6.b.a.24.2 2
5.2 odd 4 225.6.a.f.1.1 1
5.3 odd 4 45.6.a.b.1.1 1
5.4 even 2 inner 225.6.b.e.199.2 2
12.11 even 2 400.6.c.j.49.1 2
15.2 even 4 25.6.a.a.1.1 1
15.8 even 4 5.6.a.a.1.1 ✓ 1
15.14 odd 2 25.6.b.a.24.1 2
20.3 even 4 720.6.a.a.1.1 1
60.23 odd 4 80.6.a.e.1.1 1
60.47 odd 4 400.6.a.g.1.1 1
60.59 even 2 400.6.c.j.49.2 2
105.83 odd 4 245.6.a.b.1.1 1
120.53 even 4 320.6.a.j.1.1 1
120.83 odd 4 320.6.a.g.1.1 1
165.98 odd 4 605.6.a.a.1.1 1
195.38 even 4 845.6.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.6.a.a.1.1 ✓ 1 15.8 even 4
25.6.a.a.1.1 1 15.2 even 4
25.6.b.a.24.1 2 15.14 odd 2
25.6.b.a.24.2 2 3.2 odd 2
45.6.a.b.1.1 1 5.3 odd 4
80.6.a.e.1.1 1 60.23 odd 4
225.6.a.f.1.1 1 5.2 odd 4
225.6.b.e.199.1 2 1.1 even 1 trivial
225.6.b.e.199.2 2 5.4 even 2 inner
245.6.a.b.1.1 1 105.83 odd 4
320.6.a.g.1.1 1 120.83 odd 4
320.6.a.j.1.1 1 120.53 even 4
400.6.a.g.1.1 1 60.47 odd 4
400.6.c.j.49.1 2 12.11 even 2
400.6.c.j.49.2 2 60.59 even 2
605.6.a.a.1.1 1 165.98 odd 4
720.6.a.a.1.1 1 20.3 even 4
845.6.a.b.1.1 1 195.38 even 4