Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [225,10,Mod(1,225)] 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("225.1"); S:= CuspForms(chi, 10); 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, 0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 225.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,2,0,1792,0,0,13036] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(115.883063137\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 1546x^{2} + 152x + 559560 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3\cdot 5^{2}\cdot 23 \)
Twist minimal: no (minimal twist has level 75)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(33.1381\) of defining polynomial
Character \(\chi\) \(=\) 225.1

$q$-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)
 
\(f(q)\) \(=\) \(q+44.2736 q^{2} +1448.15 q^{4} +3879.20 q^{7} +41447.0 q^{8} -84809.6 q^{11} +58662.8 q^{13} +171746. q^{14} +1.09355e6 q^{16} -176199. q^{17} +800041. q^{19} -3.75483e6 q^{22} +1.65723e6 q^{23} +2.59721e6 q^{26} +5.61768e6 q^{28} +4.70073e6 q^{29} +5.05198e6 q^{31} +2.71947e7 q^{32} -7.80099e6 q^{34} +4.68753e6 q^{37} +3.54207e7 q^{38} +4.71307e6 q^{41} -1.65548e7 q^{43} -1.22817e8 q^{44} +7.33718e7 q^{46} -1.79860e7 q^{47} -2.53054e7 q^{49} +8.49527e7 q^{52} -1.50727e7 q^{53} +1.60781e8 q^{56} +2.08118e8 q^{58} -1.15379e7 q^{59} -3.48684e7 q^{61} +2.23669e8 q^{62} +6.44110e8 q^{64} +1.51756e7 q^{67} -2.55164e8 q^{68} +3.98346e7 q^{71} -7.02902e7 q^{73} +2.07534e8 q^{74} +1.15858e9 q^{76} -3.28993e8 q^{77} +1.16781e8 q^{79} +2.08665e8 q^{82} -7.69619e8 q^{83} -7.32941e8 q^{86} -3.51510e9 q^{88} +9.57076e8 q^{89} +2.27564e8 q^{91} +2.39993e9 q^{92} -7.96305e8 q^{94} +4.93418e8 q^{97} -1.12036e9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 1792 q^{4} + 13036 q^{7} + 24636 q^{8} - 104696 q^{11} + 140812 q^{13} + 181062 q^{14} + 1319800 q^{16} - 489352 q^{17} + 393308 q^{19} - 793956 q^{22} - 2326488 q^{23} + 6477502 q^{26}+ \cdots - 591217148 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 44.2736 1.95664 0.978318 − 0.207107i \(-0.0664047\pi\)
0.978318 + 0.207107i \(0.0664047\pi\)
\(3\) 0 0
\(4\) 1448.15 2.82843
\(5\) 0 0
\(6\) 0 0
\(7\) 3879.20 0.610662 0.305331 − 0.952246i \(-0.401233\pi\)
0.305331 + 0.952246i \(0.401233\pi\)
\(8\) 41447.0 3.57757
\(9\) 0 0
\(10\) 0 0
\(11\) −84809.6 −1.74654 −0.873269 − 0.487239i \(-0.838004\pi\)
−0.873269 + 0.487239i \(0.838004\pi\)
\(12\) 0 0
\(13\) 58662.8 0.569662 0.284831 − 0.958578i \(-0.408062\pi\)
0.284831 + 0.958578i \(0.408062\pi\)
\(14\) 171746. 1.19484
\(15\) 0 0
\(16\) 1.09355e6 4.17157
\(17\) −176199. −0.511663 −0.255832 − 0.966721i \(-0.582349\pi\)
−0.255832 + 0.966721i \(0.582349\pi\)
\(18\) 0 0
\(19\) 800041. 1.40838 0.704192 − 0.710009i \(-0.251309\pi\)
0.704192 + 0.710009i \(0.251309\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −3.75483e6 −3.41734
\(23\) 1.65723e6 1.23483 0.617417 − 0.786636i \(-0.288179\pi\)
0.617417 + 0.786636i \(0.288179\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.59721e6 1.11462
\(27\) 0 0
\(28\) 5.61768e6 1.72721
\(29\) 4.70073e6 1.23417 0.617084 − 0.786897i \(-0.288314\pi\)
0.617084 + 0.786897i \(0.288314\pi\)
\(30\) 0 0
\(31\) 5.05198e6 0.982503 0.491251 − 0.871018i \(-0.336540\pi\)
0.491251 + 0.871018i \(0.336540\pi\)
\(32\) 2.71947e7 4.58469
\(33\) 0 0
\(34\) −7.80099e6 −1.00114
\(35\) 0 0
\(36\) 0 0
\(37\) 4.68753e6 0.411184 0.205592 − 0.978638i \(-0.434088\pi\)
0.205592 + 0.978638i \(0.434088\pi\)
\(38\) 3.54207e7 2.75570
\(39\) 0 0
\(40\) 0 0
\(41\) 4.71307e6 0.260481 0.130241 − 0.991482i \(-0.458425\pi\)
0.130241 + 0.991482i \(0.458425\pi\)
\(42\) 0 0
\(43\) −1.65548e7 −0.738441 −0.369221 − 0.929342i \(-0.620375\pi\)
−0.369221 + 0.929342i \(0.620375\pi\)
\(44\) −1.22817e8 −4.93995
\(45\) 0 0
\(46\) 7.33718e7 2.41612
\(47\) −1.79860e7 −0.537643 −0.268821 − 0.963190i \(-0.586634\pi\)
−0.268821 + 0.963190i \(0.586634\pi\)
\(48\) 0 0
\(49\) −2.53054e7 −0.627092
\(50\) 0 0
\(51\) 0 0
\(52\) 8.49527e7 1.61125
\(53\) −1.50727e7 −0.262391 −0.131195 − 0.991357i \(-0.541882\pi\)
−0.131195 + 0.991357i \(0.541882\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.60781e8 2.18468
\(57\) 0 0
\(58\) 2.08118e8 2.41482
\(59\) −1.15379e7 −0.123963 −0.0619814 − 0.998077i \(-0.519742\pi\)
−0.0619814 + 0.998077i \(0.519742\pi\)
\(60\) 0 0
\(61\) −3.48684e7 −0.322439 −0.161219 − 0.986919i \(-0.551543\pi\)
−0.161219 + 0.986919i \(0.551543\pi\)
\(62\) 2.23669e8 1.92240
\(63\) 0 0
\(64\) 6.44110e8 4.79899
\(65\) 0 0
\(66\) 0 0
\(67\) 1.51756e7 0.0920044 0.0460022 − 0.998941i \(-0.485352\pi\)
0.0460022 + 0.998941i \(0.485352\pi\)
\(68\) −2.55164e8 −1.44720
\(69\) 0 0
\(70\) 0 0
\(71\) 3.98346e7 0.186036 0.0930182 − 0.995664i \(-0.470349\pi\)
0.0930182 + 0.995664i \(0.470349\pi\)
\(72\) 0 0
\(73\) −7.02902e7 −0.289696 −0.144848 − 0.989454i \(-0.546269\pi\)
−0.144848 + 0.989454i \(0.546269\pi\)
\(74\) 2.07534e8 0.804538
\(75\) 0 0
\(76\) 1.15858e9 3.98351
\(77\) −3.28993e8 −1.06654
\(78\) 0 0
\(79\) 1.16781e8 0.337327 0.168663 − 0.985674i \(-0.446055\pi\)
0.168663 + 0.985674i \(0.446055\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 2.08665e8 0.509667
\(83\) −7.69619e8 −1.78002 −0.890009 − 0.455943i \(-0.849302\pi\)
−0.890009 + 0.455943i \(0.849302\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −7.32941e8 −1.44486
\(87\) 0 0
\(88\) −3.51510e9 −6.24836
\(89\) 9.57076e8 1.61693 0.808466 − 0.588543i \(-0.200298\pi\)
0.808466 + 0.588543i \(0.200298\pi\)
\(90\) 0 0
\(91\) 2.27564e8 0.347871
\(92\) 2.39993e9 3.49264
\(93\) 0 0
\(94\) −7.96305e8 −1.05197
\(95\) 0 0
\(96\) 0 0
\(97\) 4.93418e8 0.565903 0.282952 − 0.959134i \(-0.408686\pi\)
0.282952 + 0.959134i \(0.408686\pi\)
\(98\) −1.12036e9 −1.22699
\(99\) 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 225.10.a.t.1.4 4
3.2 odd 2 75.10.a.j.1.1 ✓ 4
5.2 odd 4 225.10.b.n.199.8 8
5.3 odd 4 225.10.b.n.199.1 8
5.4 even 2 225.10.a.r.1.1 4
15.2 even 4 75.10.b.h.49.1 8
15.8 even 4 75.10.b.h.49.8 8
15.14 odd 2 75.10.a.k.1.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.10.a.j.1.1 ✓ 4 3.2 odd 2
75.10.a.k.1.4 yes 4 15.14 odd 2
75.10.b.h.49.1 8 15.2 even 4
75.10.b.h.49.8 8 15.8 even 4
225.10.a.r.1.1 4 5.4 even 2
225.10.a.t.1.4 4 1.1 even 1 trivial
225.10.b.n.199.1 8 5.3 odd 4
225.10.b.n.199.8 8 5.2 odd 4