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Properties

Label 225.6.b.c.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 = [2,0,0,32,0,0,0,0,0,0,868] 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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
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.c.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)
 
\(f(q)\) \(=\) \(q-4.00000i q^{2} +16.0000 q^{4} -225.000i q^{7} -192.000i q^{8} +434.000 q^{11} -613.000i q^{13} -900.000 q^{14} -256.000 q^{16} +878.000i q^{17} +731.000 q^{19} -1736.00i q^{22} +2850.00i q^{23} -2452.00 q^{26} -3600.00i q^{28} -7582.00 q^{29} +2175.00 q^{31} -5120.00i q^{32} +3512.00 q^{34} -9310.00i q^{37} -2924.00i q^{38} +12040.0 q^{41} +1121.00i q^{43} +6944.00 q^{44} +11400.0 q^{46} -29878.0i q^{47} -33818.0 q^{49} -9808.00i q^{52} +5740.00i q^{53} -43200.0 q^{56} +30328.0i q^{58} -5174.00 q^{59} -38717.0 q^{61} -8700.00i q^{62} -28672.0 q^{64} +31707.0i q^{67} +14048.0i q^{68} -64472.0 q^{71} +19790.0i q^{73} -37240.0 q^{74} +11696.0 q^{76} -97650.0i q^{77} +105000. q^{79} -48160.0i q^{82} -3318.00i q^{83} +4484.00 q^{86} -83328.0i q^{88} -65376.0 q^{89} -137925. q^{91} +45600.0i q^{92} -119512. q^{94} -89143.0i q^{97} +135272. i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 32 q^{4} + 868 q^{11} - 1800 q^{14} - 512 q^{16} + 1462 q^{19} - 4904 q^{26} - 15164 q^{29} + 4350 q^{31} + 7024 q^{34} + 24080 q^{41} + 13888 q^{44} + 22800 q^{46} - 67636 q^{49} - 86400 q^{56} - 10348 q^{59}+ \cdots - 239024 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 4.00000i − 0.707107i −0.935414 − 0.353553i \(-0.884973\pi\)
0.935414 − 0.353553i \(-0.115027\pi\)
\(3\) 0 0
\(4\) 16.0000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) − 225.000i − 1.73555i −0.496956 − 0.867776i \(-0.665549\pi\)
0.496956 − 0.867776i \(-0.334451\pi\)
\(8\) − 192.000i − 1.06066i
\(9\) 0 0
\(10\) 0 0
\(11\) 434.000 1.08145 0.540727 − 0.841198i \(-0.318149\pi\)
0.540727 + 0.841198i \(0.318149\pi\)
\(12\) 0 0
\(13\) − 613.000i − 1.00601i −0.864284 − 0.503005i \(-0.832228\pi\)
0.864284 − 0.503005i \(-0.167772\pi\)
\(14\) −900.000 −1.22722
\(15\) 0 0
\(16\) −256.000 −0.250000
\(17\) 878.000i 0.736838i 0.929660 + 0.368419i \(0.120101\pi\)
−0.929660 + 0.368419i \(0.879899\pi\)
\(18\) 0 0
\(19\) 731.000 0.464551 0.232275 − 0.972650i \(-0.425383\pi\)
0.232275 + 0.972650i \(0.425383\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 1736.00i − 0.764703i
\(23\) 2850.00i 1.12338i 0.827349 + 0.561688i \(0.189848\pi\)
−0.827349 + 0.561688i \(0.810152\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2452.00 −0.711356
\(27\) 0 0
\(28\) − 3600.00i − 0.867776i
\(29\) −7582.00 −1.67413 −0.837064 − 0.547105i \(-0.815730\pi\)
−0.837064 + 0.547105i \(0.815730\pi\)
\(30\) 0 0
\(31\) 2175.00 0.406495 0.203247 − 0.979127i \(-0.434850\pi\)
0.203247 + 0.979127i \(0.434850\pi\)
\(32\) − 5120.00i − 0.883883i
\(33\) 0 0
\(34\) 3512.00 0.521023
\(35\) 0 0
\(36\) 0 0
\(37\) − 9310.00i − 1.11801i −0.829165 − 0.559005i \(-0.811183\pi\)
0.829165 − 0.559005i \(-0.188817\pi\)
\(38\) − 2924.00i − 0.328487i
\(39\) 0 0
\(40\) 0 0
\(41\) 12040.0 1.11858 0.559290 − 0.828972i \(-0.311074\pi\)
0.559290 + 0.828972i \(0.311074\pi\)
\(42\) 0 0
\(43\) 1121.00i 0.0924559i 0.998931 + 0.0462279i \(0.0147201\pi\)
−0.998931 + 0.0462279i \(0.985280\pi\)
\(44\) 6944.00 0.540727
\(45\) 0 0
\(46\) 11400.0 0.794347
\(47\) − 29878.0i − 1.97291i −0.164038 − 0.986454i \(-0.552452\pi\)
0.164038 − 0.986454i \(-0.447548\pi\)
\(48\) 0 0
\(49\) −33818.0 −2.01214
\(50\) 0 0
\(51\) 0 0
\(52\) − 9808.00i − 0.503005i
\(53\) 5740.00i 0.280687i 0.990103 + 0.140343i \(0.0448207\pi\)
−0.990103 + 0.140343i \(0.955179\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −43200.0 −1.84083
\(57\) 0 0
\(58\) 30328.0i 1.18379i
\(59\) −5174.00 −0.193507 −0.0967534 − 0.995308i \(-0.530846\pi\)
−0.0967534 + 0.995308i \(0.530846\pi\)
\(60\) 0 0
\(61\) −38717.0 −1.33222 −0.666112 − 0.745852i \(-0.732043\pi\)
−0.666112 + 0.745852i \(0.732043\pi\)
\(62\) − 8700.00i − 0.287435i
\(63\) 0 0
\(64\) −28672.0 −0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) 31707.0i 0.862915i 0.902133 + 0.431458i \(0.142001\pi\)
−0.902133 + 0.431458i \(0.857999\pi\)
\(68\) 14048.0i 0.368419i
\(69\) 0 0
\(70\) 0 0
\(71\) −64472.0 −1.51784 −0.758919 − 0.651185i \(-0.774272\pi\)
−0.758919 + 0.651185i \(0.774272\pi\)
\(72\) 0 0
\(73\) 19790.0i 0.434649i 0.976099 + 0.217324i \(0.0697330\pi\)
−0.976099 + 0.217324i \(0.930267\pi\)
\(74\) −37240.0 −0.790552
\(75\) 0 0
\(76\) 11696.0 0.232275
\(77\) − 97650.0i − 1.87692i
\(78\) 0 0
\(79\) 105000. 1.89287 0.946437 − 0.322889i \(-0.104654\pi\)
0.946437 + 0.322889i \(0.104654\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 48160.0i − 0.790955i
\(83\) − 3318.00i − 0.0528666i −0.999651 − 0.0264333i \(-0.991585\pi\)
0.999651 − 0.0264333i \(-0.00841496\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4484.00 0.0653762
\(87\) 0 0
\(88\) − 83328.0i − 1.14706i
\(89\) −65376.0 −0.874870 −0.437435 − 0.899250i \(-0.644113\pi\)
−0.437435 + 0.899250i \(0.644113\pi\)
\(90\) 0 0
\(91\) −137925. −1.74598
\(92\) 45600.0i 0.561688i
\(93\) 0 0
\(94\) −119512. −1.39506
\(95\) 0 0
\(96\) 0 0
\(97\) − 89143.0i − 0.961962i −0.876731 − 0.480981i \(-0.840281\pi\)
0.876731 − 0.480981i \(-0.159719\pi\)
\(98\) 135272.i 1.42280i
\(99\) 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.c.199.1 2
3.2 odd 2 75.6.b.c.49.2 2
5.2 odd 4 225.6.a.g.1.1 1
5.3 odd 4 225.6.a.b.1.1 1
5.4 even 2 inner 225.6.b.c.199.2 2
15.2 even 4 75.6.a.b.1.1 ✓ 1
15.8 even 4 75.6.a.d.1.1 yes 1
15.14 odd 2 75.6.b.c.49.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.6.a.b.1.1 ✓ 1 15.2 even 4
75.6.a.d.1.1 yes 1 15.8 even 4
75.6.b.c.49.1 2 15.14 odd 2
75.6.b.c.49.2 2 3.2 odd 2
225.6.a.b.1.1 1 5.3 odd 4
225.6.a.g.1.1 1 5.2 odd 4
225.6.b.c.199.1 2 1.1 even 1 trivial
225.6.b.c.199.2 2 5.4 even 2 inner