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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [425,2,Mod(1,425)] 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("425.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.39364208590\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.1893456.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 10x^{3} + 10x^{2} + 23x - 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.96189\) of defining polynomial
Character \(\chi\) \(=\) 425.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.150980 q^{2} +1.96189 q^{3} -1.97720 q^{4} -0.296207 q^{6} +1.54475 q^{7} +0.600480 q^{8} +0.849020 q^{9} +4.56006 q^{11} -3.87906 q^{12} +1.09756 q^{13} -0.233227 q^{14} +3.86375 q^{16} -1.00000 q^{17} -0.128185 q^{18} +4.67524 q^{19} +3.03063 q^{21} -0.688480 q^{22} -0.529434 q^{23} +1.17808 q^{24} -0.165710 q^{26} -4.21999 q^{27} -3.05428 q^{28} +8.06670 q^{29} -4.78005 q^{31} -1.78431 q^{32} +8.94634 q^{33} +0.150980 q^{34} -1.67869 q^{36} +5.27917 q^{37} -0.705870 q^{38} +2.15329 q^{39} -0.751460 q^{41} -0.457565 q^{42} -9.49340 q^{43} -9.01617 q^{44} +0.0799341 q^{46} -10.7419 q^{47} +7.58026 q^{48} -4.61376 q^{49} -1.96189 q^{51} -2.17010 q^{52} -0.0227951 q^{53} +0.637136 q^{54} +0.927590 q^{56} +9.17232 q^{57} -1.21791 q^{58} -3.56962 q^{59} +3.92378 q^{61} +0.721694 q^{62} +1.31152 q^{63} -7.45810 q^{64} -1.35072 q^{66} +9.75929 q^{67} +1.97720 q^{68} -1.03869 q^{69} +1.21216 q^{71} +0.509819 q^{72} -10.1135 q^{73} -0.797051 q^{74} -9.24392 q^{76} +7.04414 q^{77} -0.325105 q^{78} +14.4151 q^{79} -10.8262 q^{81} +0.113456 q^{82} +5.08949 q^{83} -5.99217 q^{84} +1.43332 q^{86} +15.8260 q^{87} +2.73822 q^{88} -17.6123 q^{89} +1.69545 q^{91} +1.04680 q^{92} -9.37794 q^{93} +1.62182 q^{94} -3.50062 q^{96} +6.78753 q^{97} +0.696587 q^{98} +3.87158 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{2} - q^{3} + 11 q^{4} + 3 q^{6} - q^{7} + 9 q^{8} + 6 q^{9} + 4 q^{11} - 17 q^{12} + 3 q^{13} - 7 q^{14} + 27 q^{16} - 5 q^{17} + 22 q^{18} + 6 q^{19} - 5 q^{21} - 18 q^{22} - 4 q^{23} - 19 q^{24}+ \cdots - 14 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.150980 −0.106759 −0.0533796 − 0.998574i \(-0.516999\pi\)
−0.0533796 + 0.998574i \(0.516999\pi\)
\(3\) 1.96189 1.13270 0.566349 − 0.824165i \(-0.308355\pi\)
0.566349 + 0.824165i \(0.308355\pi\)
\(4\) −1.97720 −0.988602
\(5\) 0 0
\(6\) −0.296207 −0.120926
\(7\) 1.54475 0.583859 0.291930 − 0.956440i \(-0.405703\pi\)
0.291930 + 0.956440i \(0.405703\pi\)
\(8\) 0.600480 0.212302
\(9\) 0.849020 0.283007
\(10\) 0 0
\(11\) 4.56006 1.37491 0.687455 − 0.726227i \(-0.258728\pi\)
0.687455 + 0.726227i \(0.258728\pi\)
\(12\) −3.87906 −1.11979
\(13\) 1.09756 0.304408 0.152204 − 0.988349i \(-0.451363\pi\)
0.152204 + 0.988349i \(0.451363\pi\)
\(14\) −0.233227 −0.0623324
\(15\) 0 0
\(16\) 3.86375 0.965937
\(17\) −1.00000 −0.242536
\(18\) −0.128185 −0.0302136
\(19\) 4.67524 1.07257 0.536287 − 0.844036i \(-0.319826\pi\)
0.536287 + 0.844036i \(0.319826\pi\)
\(20\) 0 0
\(21\) 3.03063 0.661337
\(22\) −0.688480 −0.146784
\(23\) −0.529434 −0.110395 −0.0551973 − 0.998475i \(-0.517579\pi\)
−0.0551973 + 0.998475i \(0.517579\pi\)
\(24\) 1.17808 0.240474
\(25\) 0 0
\(26\) −0.165710 −0.0324984
\(27\) −4.21999 −0.812138
\(28\) −3.05428 −0.577205
\(29\) 8.06670 1.49795 0.748974 − 0.662599i \(-0.230547\pi\)
0.748974 + 0.662599i \(0.230547\pi\)
\(30\) 0 0
\(31\) −4.78005 −0.858522 −0.429261 − 0.903180i \(-0.641226\pi\)
−0.429261 + 0.903180i \(0.641226\pi\)
\(32\) −1.78431 −0.315425
\(33\) 8.94634 1.55736
\(34\) 0.150980 0.0258929
\(35\) 0 0
\(36\) −1.67869 −0.279781
\(37\) 5.27917 0.867889 0.433945 − 0.900939i \(-0.357121\pi\)
0.433945 + 0.900939i \(0.357121\pi\)
\(38\) −0.705870 −0.114507
\(39\) 2.15329 0.344803
\(40\) 0 0
\(41\) −0.751460 −0.117358 −0.0586792 − 0.998277i \(-0.518689\pi\)
−0.0586792 + 0.998277i \(0.518689\pi\)
\(42\) −0.457565 −0.0706038
\(43\) −9.49340 −1.44773 −0.723865 − 0.689941i \(-0.757636\pi\)
−0.723865 + 0.689941i \(0.757636\pi\)
\(44\) −9.01617 −1.35924
\(45\) 0 0
\(46\) 0.0799341 0.0117856
\(47\) −10.7419 −1.56687 −0.783437 − 0.621472i \(-0.786535\pi\)
−0.783437 + 0.621472i \(0.786535\pi\)
\(48\) 7.58026 1.09412
\(49\) −4.61376 −0.659108
\(50\) 0 0
\(51\) −1.96189 −0.274720
\(52\) −2.17010 −0.300939
\(53\) −0.0227951 −0.00313115 −0.00156557 − 0.999999i \(-0.500498\pi\)
−0.00156557 + 0.999999i \(0.500498\pi\)
\(54\) 0.637136 0.0867032
\(55\) 0 0
\(56\) 0.927590 0.123954
\(57\) 9.17232 1.21490
\(58\) −1.21791 −0.159920
\(59\) −3.56962 −0.464725 −0.232362 − 0.972629i \(-0.574646\pi\)
−0.232362 + 0.972629i \(0.574646\pi\)
\(60\) 0 0
\(61\) 3.92378 0.502389 0.251195 − 0.967937i \(-0.419177\pi\)
0.251195 + 0.967937i \(0.419177\pi\)
\(62\) 0.721694 0.0916552
\(63\) 1.31152 0.165236
\(64\) −7.45810 −0.932263
\(65\) 0 0
\(66\) −1.35072 −0.166262
\(67\) 9.75929 1.19229 0.596144 − 0.802878i \(-0.296699\pi\)
0.596144 + 0.802878i \(0.296699\pi\)
\(68\) 1.97720 0.239771
\(69\) −1.03869 −0.125044
\(70\) 0 0
\(71\) 1.21216 0.143857 0.0719285 − 0.997410i \(-0.477085\pi\)
0.0719285 + 0.997410i \(0.477085\pi\)
\(72\) 0.509819 0.0600828
\(73\) −10.1135 −1.18369 −0.591845 − 0.806052i \(-0.701600\pi\)
−0.591845 + 0.806052i \(0.701600\pi\)
\(74\) −0.797051 −0.0926552
\(75\) 0 0
\(76\) −9.24392 −1.06035
\(77\) 7.04414 0.802754
\(78\) −0.325105 −0.0368109
\(79\) 14.4151 1.62183 0.810913 − 0.585166i \(-0.198971\pi\)
0.810913 + 0.585166i \(0.198971\pi\)
\(80\) 0 0
\(81\) −10.8262 −1.20291
\(82\) 0.113456 0.0125291
\(83\) 5.08949 0.558644 0.279322 − 0.960197i \(-0.409890\pi\)
0.279322 + 0.960197i \(0.409890\pi\)
\(84\) −5.99217 −0.653799
\(85\) 0 0
\(86\) 1.43332 0.154559
\(87\) 15.8260 1.69672
\(88\) 2.73822 0.291896
\(89\) −17.6123 −1.86690 −0.933450 − 0.358707i \(-0.883218\pi\)
−0.933450 + 0.358707i \(0.883218\pi\)
\(90\) 0 0
\(91\) 1.69545 0.177732
\(92\) 1.04680 0.109136
\(93\) −9.37794 −0.972447
\(94\) 1.62182 0.167278
\(95\) 0 0
\(96\) −3.50062 −0.357281
\(97\) 6.78753 0.689170 0.344585 − 0.938755i \(-0.388020\pi\)
0.344585 + 0.938755i \(0.388020\pi\)
\(98\) 0.696587 0.0703659
\(99\) 3.87158 0.389108
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 425.2.a.j.1.3 yes 5
3.2 odd 2 3825.2.a.bl.1.3 5
4.3 odd 2 6800.2.a.cd.1.2 5
5.2 odd 4 425.2.b.f.324.5 10
5.3 odd 4 425.2.b.f.324.6 10
5.4 even 2 425.2.a.i.1.3 ✓ 5
15.14 odd 2 3825.2.a.bq.1.3 5
17.16 even 2 7225.2.a.y.1.3 5
20.19 odd 2 6800.2.a.bz.1.4 5
85.84 even 2 7225.2.a.x.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
425.2.a.i.1.3 ✓ 5 5.4 even 2
425.2.a.j.1.3 yes 5 1.1 even 1 trivial
425.2.b.f.324.5 10 5.2 odd 4
425.2.b.f.324.6 10 5.3 odd 4
3825.2.a.bl.1.3 5 3.2 odd 2
3825.2.a.bq.1.3 5 15.14 odd 2
6800.2.a.bz.1.4 5 20.19 odd 2
6800.2.a.cd.1.2 5 4.3 odd 2
7225.2.a.x.1.3 5 85.84 even 2
7225.2.a.y.1.3 5 17.16 even 2