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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [25,22,Mod(24,25)] 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("25.24"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(25, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(69.8693360718\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 24.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 25.24
Dual form 25.22.b.a.24.1

$q$-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)
 
\(f(q)\) \(=\) \(q+288.000i q^{2} -128844. i q^{3} +2.01421e6 q^{4} +3.71071e7 q^{6} +7.68079e8i q^{7} +1.18407e9i q^{8} -6.14042e9 q^{9} -9.47249e10 q^{11} -2.59519e11i q^{12} -8.06218e10i q^{13} -2.21207e11 q^{14} +3.88309e12 q^{16} -3.05228e12i q^{17} -1.76844e12i q^{18} +7.92079e12 q^{19} +9.89623e13 q^{21} -2.72808e13i q^{22} -7.38454e13i q^{23} +1.52561e14 q^{24} +2.32191e13 q^{26} -5.56597e14i q^{27} +1.54707e15i q^{28} +4.25303e15 q^{29} +1.90054e15 q^{31} +3.60151e15i q^{32} +1.22047e16i q^{33} +8.79057e14 q^{34} -1.23681e16 q^{36} -2.21914e16i q^{37} +2.28119e15i q^{38} -1.03876e16 q^{39} -2.06228e16 q^{41} +2.85012e16i q^{42} -1.93606e17i q^{43} -1.90796e17 q^{44} +2.12675e16 q^{46} -1.46961e17i q^{47} -5.00313e17i q^{48} -3.13992e16 q^{49} -3.93268e17 q^{51} -1.62389e17i q^{52} +2.03827e18i q^{53} +1.60300e17 q^{54} -9.09460e17 q^{56} -1.02055e18i q^{57} +1.22487e18i q^{58} +5.97588e18 q^{59} +6.19062e18 q^{61} +5.47356e17i q^{62} -4.71633e18i q^{63} +7.10619e18 q^{64} -3.51496e18 q^{66} -1.69613e19i q^{67} -6.14793e18i q^{68} -9.51454e18 q^{69} -5.63276e18 q^{71} -7.27070e18i q^{72} -4.32848e19i q^{73} +6.39113e18 q^{74} +1.59541e19 q^{76} -7.27562e19i q^{77} -2.99164e18i q^{78} +5.12649e19 q^{79} -1.35945e20 q^{81} -5.93937e18i q^{82} +4.89119e19i q^{83} +1.99331e20 q^{84} +5.57585e19 q^{86} -5.47978e20i q^{87} -1.12161e20i q^{88} +5.04303e20 q^{89} +6.19239e19 q^{91} -1.48740e20i q^{92} -2.44873e20i q^{93} +4.23246e19 q^{94} +4.64033e20 q^{96} -8.08275e20i q^{97} -9.04297e18i q^{98} +5.81651e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4028416 q^{4} + 74214144 q^{6} - 12280846266 q^{9} - 189449858376 q^{11} - 442413393408 q^{14} + 7766175383552 q^{16} + 15841576703480 q^{19} + 197924691875904 q^{21} + 305121063075840 q^{24} + 46438150921344 q^{26}+ \cdots + 11\!\cdots\!08 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 288.000i 0.198874i 0.995044 + 0.0994369i \(0.0317041\pi\)
−0.995044 + 0.0994369i \(0.968296\pi\)
\(3\) − 128844.i − 1.25977i −0.776689 − 0.629885i \(-0.783102\pi\)
0.776689 − 0.629885i \(-0.216898\pi\)
\(4\) 2.01421e6 0.960449
\(5\) 0 0
\(6\) 3.71071e7 0.250535
\(7\) 7.68079e8i 1.02772i 0.857873 + 0.513862i \(0.171786\pi\)
−0.857873 + 0.513862i \(0.828214\pi\)
\(8\) 1.18407e9i 0.389882i
\(9\) −6.14042e9 −0.587019
\(10\) 0 0
\(11\) −9.47249e10 −1.10114 −0.550568 − 0.834790i \(-0.685589\pi\)
−0.550568 + 0.834790i \(0.685589\pi\)
\(12\) − 2.59519e11i − 1.20994i
\(13\) − 8.06218e10i − 0.162199i −0.996706 − 0.0810993i \(-0.974157\pi\)
0.996706 − 0.0810993i \(-0.0258431\pi\)
\(14\) −2.21207e11 −0.204387
\(15\) 0 0
\(16\) 3.88309e12 0.882912
\(17\) − 3.05228e12i − 0.367207i −0.983000 − 0.183604i \(-0.941224\pi\)
0.983000 − 0.183604i \(-0.0587763\pi\)
\(18\) − 1.76844e12i − 0.116743i
\(19\) 7.92079e12 0.296385 0.148192 − 0.988959i \(-0.452655\pi\)
0.148192 + 0.988959i \(0.452655\pi\)
\(20\) 0 0
\(21\) 9.89623e13 1.29469
\(22\) − 2.72808e13i − 0.218987i
\(23\) − 7.38454e13i − 0.371690i −0.982579 − 0.185845i \(-0.940498\pi\)
0.982579 − 0.185845i \(-0.0595023\pi\)
\(24\) 1.52561e14 0.491161
\(25\) 0 0
\(26\) 2.32191e13 0.0322571
\(27\) − 5.56597e14i − 0.520261i
\(28\) 1.54707e15i 0.987076i
\(29\) 4.25303e15 1.87724 0.938620 − 0.344954i \(-0.112105\pi\)
0.938620 + 0.344954i \(0.112105\pi\)
\(30\) 0 0
\(31\) 1.90054e15 0.416466 0.208233 − 0.978079i \(-0.433229\pi\)
0.208233 + 0.978079i \(0.433229\pi\)
\(32\) 3.60151e15i 0.565470i
\(33\) 1.22047e16i 1.38718i
\(34\) 8.79057e14 0.0730279
\(35\) 0 0
\(36\) −1.23681e16 −0.563802
\(37\) − 2.21914e16i − 0.758695i −0.925254 − 0.379347i \(-0.876149\pi\)
0.925254 − 0.379347i \(-0.123851\pi\)
\(38\) 2.28119e15i 0.0589431i
\(39\) −1.03876e16 −0.204333
\(40\) 0 0
\(41\) −2.06228e16 −0.239948 −0.119974 − 0.992777i \(-0.538281\pi\)
−0.119974 + 0.992777i \(0.538281\pi\)
\(42\) 2.85012e16i 0.257481i
\(43\) − 1.93606e17i − 1.36615i −0.730346 − 0.683077i \(-0.760641\pi\)
0.730346 − 0.683077i \(-0.239359\pi\)
\(44\) −1.90796e17 −1.05759
\(45\) 0 0
\(46\) 2.12675e16 0.0739195
\(47\) − 1.46961e17i − 0.407543i −0.979019 − 0.203771i \(-0.934680\pi\)
0.979019 − 0.203771i \(-0.0653199\pi\)
\(48\) − 5.00313e17i − 1.11227i
\(49\) −3.13992e16 −0.0562160
\(50\) 0 0
\(51\) −3.93268e17 −0.462596
\(52\) − 1.62389e17i − 0.155784i
\(53\) 2.03827e18i 1.60090i 0.599399 + 0.800450i \(0.295406\pi\)
−0.599399 + 0.800450i \(0.704594\pi\)
\(54\) 1.60300e17 0.103466
\(55\) 0 0
\(56\) −9.09460e17 −0.400691
\(57\) − 1.02055e18i − 0.373376i
\(58\) 1.22487e18i 0.373334i
\(59\) 5.97588e18 1.52214 0.761072 − 0.648667i \(-0.224673\pi\)
0.761072 + 0.648667i \(0.224673\pi\)
\(60\) 0 0
\(61\) 6.19062e18 1.11114 0.555572 − 0.831468i \(-0.312499\pi\)
0.555572 + 0.831468i \(0.312499\pi\)
\(62\) 5.47356e17i 0.0828242i
\(63\) − 4.71633e18i − 0.603293i
\(64\) 7.10619e18 0.770455
\(65\) 0 0
\(66\) −3.51496e18 −0.275873
\(67\) − 1.69613e19i − 1.13677i −0.822761 − 0.568387i \(-0.807568\pi\)
0.822761 − 0.568387i \(-0.192432\pi\)
\(68\) − 6.14793e18i − 0.352684i
\(69\) −9.51454e18 −0.468244
\(70\) 0 0
\(71\) −5.63276e18 −0.205357 −0.102678 − 0.994715i \(-0.532741\pi\)
−0.102678 + 0.994715i \(0.532741\pi\)
\(72\) − 7.27070e18i − 0.228868i
\(73\) − 4.32848e19i − 1.17881i −0.807837 − 0.589407i \(-0.799362\pi\)
0.807837 − 0.589407i \(-0.200638\pi\)
\(74\) 6.39113e18 0.150885
\(75\) 0 0
\(76\) 1.59541e19 0.284662
\(77\) − 7.27562e19i − 1.13166i
\(78\) − 2.99164e18i − 0.0406365i
\(79\) 5.12649e19 0.609166 0.304583 − 0.952486i \(-0.401483\pi\)
0.304583 + 0.952486i \(0.401483\pi\)
\(80\) 0 0
\(81\) −1.35945e20 −1.24243
\(82\) − 5.93937e18i − 0.0477194i
\(83\) 4.89119e19i 0.346014i 0.984921 + 0.173007i \(0.0553484\pi\)
−0.984921 + 0.173007i \(0.944652\pi\)
\(84\) 1.99331e20 1.24349
\(85\) 0 0
\(86\) 5.57585e19 0.271692
\(87\) − 5.47978e20i − 2.36489i
\(88\) − 1.12161e20i − 0.429313i
\(89\) 5.04303e20 1.71434 0.857170 − 0.515034i \(-0.172221\pi\)
0.857170 + 0.515034i \(0.172221\pi\)
\(90\) 0 0
\(91\) 6.19239e19 0.166695
\(92\) − 1.48740e20i − 0.356990i
\(93\) − 2.44873e20i − 0.524651i
\(94\) 4.23246e19 0.0810496
\(95\) 0 0
\(96\) 4.64033e20 0.712362
\(97\) − 8.08275e20i − 1.11290i −0.830881 − 0.556450i \(-0.812163\pi\)
0.830881 − 0.556450i \(-0.187837\pi\)
\(98\) − 9.04297e18i − 0.0111799i
\(99\) 5.81651e20 0.646388
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 25.22.b.a.24.2 2
5.2 odd 4 1.22.a.a.1.1 ✓ 1
5.3 odd 4 25.22.a.a.1.1 1
5.4 even 2 inner 25.22.b.a.24.1 2
15.2 even 4 9.22.a.c.1.1 1
20.7 even 4 16.22.a.c.1.1 1
35.27 even 4 49.22.a.a.1.1 1
40.27 even 4 64.22.a.a.1.1 1
40.37 odd 4 64.22.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.22.a.a.1.1 ✓ 1 5.2 odd 4
9.22.a.c.1.1 1 15.2 even 4
16.22.a.c.1.1 1 20.7 even 4
25.22.a.a.1.1 1 5.3 odd 4
25.22.b.a.24.1 2 5.4 even 2 inner
25.22.b.a.24.2 2 1.1 even 1 trivial
49.22.a.a.1.1 1 35.27 even 4
64.22.a.a.1.1 1 40.27 even 4
64.22.a.g.1.1 1 40.37 odd 4