↑

Properties

Label 25.6.a.d.1.2
Level $25$
Weight $6$
Character 25.1
Self dual yes
Analytic conductor $4.010$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [25,6,Mod(1,25)] 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("25.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(25, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 25.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(4.00959549532\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{241}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 60 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-7.26209\) of defining polynomial
Character \(\chi\) \(=\) 25.1

$q$-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)
 
\(f(q)\) \(=\) \(q+10.2621 q^{2} -5.52417 q^{3} +73.3104 q^{4} -56.6896 q^{6} +68.9517 q^{7} +423.931 q^{8} -212.483 q^{9} -486.104 q^{11} -404.980 q^{12} -428.387 q^{13} +707.588 q^{14} +2004.49 q^{16} +1800.64 q^{17} -2180.52 q^{18} -1046.65 q^{19} -380.901 q^{21} -4988.45 q^{22} +686.855 q^{23} -2341.87 q^{24} -4396.14 q^{26} +2516.17 q^{27} +5054.88 q^{28} -1339.03 q^{29} +7990.30 q^{31} +7004.41 q^{32} +2685.33 q^{33} +18478.4 q^{34} -15577.3 q^{36} -1970.64 q^{37} -10740.9 q^{38} +2366.48 q^{39} +10772.2 q^{41} -3908.84 q^{42} +15017.7 q^{43} -35636.5 q^{44} +7048.57 q^{46} +895.337 q^{47} -11073.1 q^{48} -12052.7 q^{49} -9947.07 q^{51} -31405.2 q^{52} -19327.1 q^{53} +25821.2 q^{54} +29230.8 q^{56} +5781.90 q^{57} -13741.3 q^{58} +21193.7 q^{59} -27722.2 q^{61} +81997.1 q^{62} -14651.1 q^{63} +7736.31 q^{64} +27557.0 q^{66} -7719.33 q^{67} +132006. q^{68} -3794.31 q^{69} -51410.1 q^{71} -90078.4 q^{72} -43776.4 q^{73} -20222.9 q^{74} -76730.7 q^{76} -33517.7 q^{77} +24285.1 q^{78} -6225.68 q^{79} +37733.7 q^{81} +110545. q^{82} +52949.9 q^{83} -27924.0 q^{84} +154113. q^{86} +7397.05 q^{87} -206075. q^{88} +44631.2 q^{89} -29538.0 q^{91} +50353.6 q^{92} -44139.8 q^{93} +9188.03 q^{94} -38693.6 q^{96} -148018. q^{97} -123686. q^{98} +103289. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{2} + 20 q^{3} + 69 q^{4} - 191 q^{6} + 200 q^{7} + 615 q^{8} + 196 q^{9} - 196 q^{11} - 515 q^{12} - 360 q^{13} + 18 q^{14} + 1137 q^{16} + 1490 q^{17} - 4330 q^{18} - 3180 q^{19} + 2964 q^{21}+ \cdots + 221792 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 10.2621 1.81410 0.907049 − 0.421025i \(-0.138330\pi\)
0.907049 + 0.421025i \(0.138330\pi\)
\(3\) −5.52417 −0.354376 −0.177188 − 0.984177i \(-0.556700\pi\)
−0.177188 + 0.984177i \(0.556700\pi\)
\(4\) 73.3104 2.29095
\(5\) 0 0
\(6\) −56.6896 −0.642873
\(7\) 68.9517 0.531863 0.265931 − 0.963992i \(-0.414321\pi\)
0.265931 + 0.963992i \(0.414321\pi\)
\(8\) 423.931 2.34191
\(9\) −212.483 −0.874418
\(10\) 0 0
\(11\) −486.104 −1.21129 −0.605645 − 0.795735i \(-0.707085\pi\)
−0.605645 + 0.795735i \(0.707085\pi\)
\(12\) −404.980 −0.811858
\(13\) −428.387 −0.703036 −0.351518 − 0.936181i \(-0.614334\pi\)
−0.351518 + 0.936181i \(0.614334\pi\)
\(14\) 707.588 0.964851
\(15\) 0 0
\(16\) 2004.49 1.95751
\(17\) 1800.64 1.51114 0.755571 − 0.655066i \(-0.227359\pi\)
0.755571 + 0.655066i \(0.227359\pi\)
\(18\) −2180.52 −1.58628
\(19\) −1046.65 −0.665149 −0.332575 − 0.943077i \(-0.607917\pi\)
−0.332575 + 0.943077i \(0.607917\pi\)
\(20\) 0 0
\(21\) −380.901 −0.188479
\(22\) −4988.45 −2.19740
\(23\) 686.855 0.270736 0.135368 − 0.990795i \(-0.456778\pi\)
0.135368 + 0.990795i \(0.456778\pi\)
\(24\) −2341.87 −0.829917
\(25\) 0 0
\(26\) −4396.14 −1.27538
\(27\) 2516.17 0.664249
\(28\) 5054.88 1.21847
\(29\) −1339.03 −0.295663 −0.147831 − 0.989013i \(-0.547229\pi\)
−0.147831 + 0.989013i \(0.547229\pi\)
\(30\) 0 0
\(31\) 7990.30 1.49334 0.746670 − 0.665195i \(-0.231651\pi\)
0.746670 + 0.665195i \(0.231651\pi\)
\(32\) 7004.41 1.20920
\(33\) 2685.33 0.429252
\(34\) 18478.4 2.74136
\(35\) 0 0
\(36\) −15577.3 −2.00325
\(37\) −1970.64 −0.236648 −0.118324 − 0.992975i \(-0.537752\pi\)
−0.118324 + 0.992975i \(0.537752\pi\)
\(38\) −10740.9 −1.20665
\(39\) 2366.48 0.249139
\(40\) 0 0
\(41\) 10772.2 1.00079 0.500395 − 0.865797i \(-0.333188\pi\)
0.500395 + 0.865797i \(0.333188\pi\)
\(42\) −3908.84 −0.341920
\(43\) 15017.7 1.23861 0.619303 − 0.785152i \(-0.287415\pi\)
0.619303 + 0.785152i \(0.287415\pi\)
\(44\) −35636.5 −2.77500
\(45\) 0 0
\(46\) 7048.57 0.491141
\(47\) 895.337 0.0591210 0.0295605 − 0.999563i \(-0.490589\pi\)
0.0295605 + 0.999563i \(0.490589\pi\)
\(48\) −11073.1 −0.693693
\(49\) −12052.7 −0.717122
\(50\) 0 0
\(51\) −9947.07 −0.535513
\(52\) −31405.2 −1.61062
\(53\) −19327.1 −0.945098 −0.472549 − 0.881304i \(-0.656666\pi\)
−0.472549 + 0.881304i \(0.656666\pi\)
\(54\) 25821.2 1.20501
\(55\) 0 0
\(56\) 29230.8 1.24558
\(57\) 5781.90 0.235713
\(58\) −13741.3 −0.536361
\(59\) 21193.7 0.792641 0.396321 − 0.918112i \(-0.370287\pi\)
0.396321 + 0.918112i \(0.370287\pi\)
\(60\) 0 0
\(61\) −27722.2 −0.953900 −0.476950 − 0.878931i \(-0.658258\pi\)
−0.476950 + 0.878931i \(0.658258\pi\)
\(62\) 81997.1 2.70906
\(63\) −14651.1 −0.465070
\(64\) 7736.31 0.236093
\(65\) 0 0
\(66\) 27557.0 0.778705
\(67\) −7719.33 −0.210084 −0.105042 − 0.994468i \(-0.533498\pi\)
−0.105042 + 0.994468i \(0.533498\pi\)
\(68\) 132006. 3.46195
\(69\) −3794.31 −0.0959422
\(70\) 0 0
\(71\) −51410.1 −1.21033 −0.605163 − 0.796101i \(-0.706892\pi\)
−0.605163 + 0.796101i \(0.706892\pi\)
\(72\) −90078.4 −2.04781
\(73\) −43776.4 −0.961465 −0.480732 − 0.876867i \(-0.659629\pi\)
−0.480732 + 0.876867i \(0.659629\pi\)
\(74\) −20222.9 −0.429303
\(75\) 0 0
\(76\) −76730.7 −1.52382
\(77\) −33517.7 −0.644240
\(78\) 24285.1 0.451963
\(79\) −6225.68 −0.112233 −0.0561163 − 0.998424i \(-0.517872\pi\)
−0.0561163 + 0.998424i \(0.517872\pi\)
\(80\) 0 0
\(81\) 37733.7 0.639024
\(82\) 110545. 1.81553
\(83\) 52949.9 0.843664 0.421832 − 0.906674i \(-0.361387\pi\)
0.421832 + 0.906674i \(0.361387\pi\)
\(84\) −27924.0 −0.431797
\(85\) 0 0
\(86\) 154113. 2.24695
\(87\) 7397.05 0.104776
\(88\) −206075. −2.83673
\(89\) 44631.2 0.597260 0.298630 − 0.954369i \(-0.403470\pi\)
0.298630 + 0.954369i \(0.403470\pi\)
\(90\) 0 0
\(91\) −29538.0 −0.373919
\(92\) 50353.6 0.620242
\(93\) −44139.8 −0.529204
\(94\) 9188.03 0.107251
\(95\) 0 0
\(96\) −38693.6 −0.428510
\(97\) −148018. −1.59730 −0.798649 − 0.601797i \(-0.794452\pi\)
−0.798649 + 0.601797i \(0.794452\pi\)
\(98\) −123686. −1.30093
\(99\) 103289. 1.05917
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 25.6.a.d.1.2 yes 2
3.2 odd 2 225.6.a.l.1.1 2
4.3 odd 2 400.6.a.o.1.2 2
5.2 odd 4 25.6.b.b.24.4 4
5.3 odd 4 25.6.b.b.24.1 4
5.4 even 2 25.6.a.b.1.1 ✓ 2
15.2 even 4 225.6.b.i.199.1 4
15.8 even 4 225.6.b.i.199.4 4
15.14 odd 2 225.6.a.s.1.2 2
20.3 even 4 400.6.c.n.49.3 4
20.7 even 4 400.6.c.n.49.2 4
20.19 odd 2 400.6.a.w.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.6.a.b.1.1 ✓ 2 5.4 even 2
25.6.a.d.1.2 yes 2 1.1 even 1 trivial
25.6.b.b.24.1 4 5.3 odd 4
25.6.b.b.24.4 4 5.2 odd 4
225.6.a.l.1.1 2 3.2 odd 2
225.6.a.s.1.2 2 15.14 odd 2
225.6.b.i.199.1 4 15.2 even 4
225.6.b.i.199.4 4 15.8 even 4
400.6.a.o.1.2 2 4.3 odd 2
400.6.a.w.1.1 2 20.19 odd 2
400.6.c.n.49.2 4 20.7 even 4
400.6.c.n.49.3 4 20.3 even 4