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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [425,2,Mod(84,425)] 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("425.84"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([7, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.q (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.39364208590\)
Analytic rank: \(0\)
Dimension: \(176\)
Relative dimension: \(44\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label 84.2
Character \(\chi\) \(=\) 425.84
Dual form 425.2.q.a.339.2

$q$-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+(-1.61286 + 2.21990i) q^{2} +(0.861202 + 2.65051i) q^{3} +(-1.70864 - 5.25866i) q^{4} +(-0.297302 + 2.21622i) q^{5} +(-7.27287 - 2.36310i) q^{6} -1.90625 q^{7} +(9.21020 + 2.99258i) q^{8} +(-3.85647 + 2.80189i) q^{9} +(-4.44028 - 4.23442i) q^{10} +(-2.64178 + 3.63610i) q^{11} +(12.4666 - 9.05754i) q^{12} +(-0.219466 - 0.302069i) q^{13} +(3.07450 - 4.23169i) q^{14} +(-6.13013 + 1.12061i) q^{15} +(-12.5514 + 9.11912i) q^{16} +(4.03362 + 0.854352i) q^{17} -13.0800i q^{18} +(2.03005 - 6.24784i) q^{19} +(12.1623 - 2.22331i) q^{20} +(-1.64167 - 5.05253i) q^{21} +(-3.81099 - 11.7290i) q^{22} +(-1.16500 - 0.846425i) q^{23} +26.9889i q^{24} +(-4.82322 - 1.31777i) q^{25} +1.02453 q^{26} +(-3.98367 - 2.89430i) q^{27} +(3.25710 + 10.0243i) q^{28} +(-3.44009 + 1.11775i) q^{29} +(7.39938 - 15.4157i) q^{30} +(5.87547 + 1.90906i) q^{31} -23.2024i q^{32} +(-11.9126 - 3.87065i) q^{33} +(-8.40222 + 7.57630i) q^{34} +(0.566732 - 4.22466i) q^{35} +(21.3235 + 15.4924i) q^{36} +(2.09235 - 1.52018i) q^{37} +(10.5954 + 14.5834i) q^{38} +(0.611631 - 0.841838i) q^{39} +(-9.37041 + 19.5221i) q^{40} +(1.09109 + 1.50176i) q^{41} +(13.8639 + 4.50465i) q^{42} +12.2854i q^{43} +(23.6349 + 7.67943i) q^{44} +(-5.06306 - 9.37978i) q^{45} +(3.75796 - 1.22104i) q^{46} +(0.0802572 - 0.0260771i) q^{47} +(-34.9796 - 25.4142i) q^{48} -3.36622 q^{49} +(10.7045 - 8.58172i) q^{50} +(1.20930 + 11.4269i) q^{51} +(-1.21349 + 1.67022i) q^{52} +(0.319741 - 0.103890i) q^{53} +(12.8502 - 4.17527i) q^{54} +(-7.27298 - 6.93578i) q^{55} +(-17.5569 - 5.70460i) q^{56} +18.3082 q^{57} +(3.06706 - 9.43944i) q^{58} +(-1.06587 + 0.774396i) q^{59} +(16.3671 + 30.3216i) q^{60} +(5.79643 - 7.97811i) q^{61} +(-13.7142 + 9.96395i) q^{62} +(7.35140 - 5.34110i) q^{63} +(26.4043 + 19.1838i) q^{64} +(0.734697 - 0.396578i) q^{65} +(27.8058 - 20.2021i) q^{66} +(3.59511 + 1.16812i) q^{67} +(-2.39926 - 22.6712i) q^{68} +(1.24015 - 3.81679i) q^{69} +(8.46428 + 8.07185i) q^{70} +(-10.4209 + 3.38595i) q^{71} +(-43.9038 + 14.2652i) q^{72} +(1.80355 + 1.31036i) q^{73} +7.09664i q^{74} +(-0.661009 - 13.9189i) q^{75} -36.3239 q^{76} +(5.03589 - 6.93131i) q^{77} +(0.882327 + 2.71552i) q^{78} +(-9.56517 + 3.10791i) q^{79} +(-16.4784 - 30.5277i) q^{80} +(-0.178492 + 0.549342i) q^{81} -5.09354 q^{82} +(-1.19600 - 0.388603i) q^{83} +(-23.7645 + 17.2659i) q^{84} +(-3.09263 + 8.68537i) q^{85} +(-27.2723 - 19.8145i) q^{86} +(-5.92523 - 8.15537i) q^{87} +(-35.2127 + 25.5835i) q^{88} +(-5.55007 - 4.03236i) q^{89} +(28.9882 + 3.88872i) q^{90} +(0.418356 + 0.575818i) q^{91} +(-2.46048 + 7.57259i) q^{92} +17.2171i q^{93} +(-0.0715544 + 0.220222i) q^{94} +(13.2430 + 6.35652i) q^{95} +(61.4981 - 19.9819i) q^{96} +(2.90335 + 8.93560i) q^{97} +(5.42922 - 7.47268i) q^{98} -21.4245i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 176 q - 10 q^{2} + 38 q^{4} - 10 q^{8} - 50 q^{9} - 10 q^{13} + 6 q^{15} - 58 q^{16} - 8 q^{19} - 44 q^{21} - 20 q^{25} - 64 q^{26} + 78 q^{30} - 20 q^{33} - 23 q^{34} - 46 q^{35} + 52 q^{36} - 10 q^{38}+ \cdots - 250 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(326\)
\(\chi(n)\) \(e\left(\frac{7}{10}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.61286 + 2.21990i −1.14046 + 1.56971i −0.373988 + 0.927434i \(0.622010\pi\)
−0.766473 + 0.642276i \(0.777990\pi\)
\(3\) 0.861202 + 2.65051i 0.497215 + 1.53027i 0.813476 + 0.581599i \(0.197573\pi\)
−0.316260 + 0.948672i \(0.602427\pi\)
\(4\) −1.70864 − 5.25866i −0.854321 − 2.62933i
\(5\) −0.297302 + 2.21622i −0.132957 + 0.991122i
\(6\) −7.27287 − 2.36310i −2.96914 − 0.964731i
\(7\) −1.90625 −0.720494 −0.360247 − 0.932857i \(-0.617308\pi\)
−0.360247 + 0.932857i \(0.617308\pi\)
\(8\) 9.21020 + 2.99258i 3.25630 + 1.05804i
\(9\) −3.85647 + 2.80189i −1.28549 + 0.933964i
\(10\) −4.44028 − 4.23442i −1.40414 − 1.33904i
\(11\) −2.64178 + 3.63610i −0.796527 + 1.09633i 0.196737 + 0.980456i \(0.436965\pi\)
−0.993264 + 0.115869i \(0.963035\pi\)
\(12\) 12.4666 − 9.05754i 3.59881 − 2.61469i
\(13\) −0.219466 − 0.302069i −0.0608688 − 0.0837787i 0.777497 − 0.628886i \(-0.216489\pi\)
−0.838366 + 0.545107i \(0.816489\pi\)
\(14\) 3.07450 − 4.23169i 0.821696 − 1.13097i
\(15\) −6.13013 + 1.12061i −1.58279 + 0.289340i
\(16\) −12.5514 + 9.11912i −3.13785 + 2.27978i
\(17\) 4.03362 + 0.854352i 0.978296 + 0.207211i
\(18\) − 13.0800i − 3.08300i
\(19\) 2.03005 − 6.24784i 0.465725 − 1.43335i −0.392344 − 0.919818i \(-0.628336\pi\)
0.858069 − 0.513535i \(-0.171664\pi\)
\(20\) 12.1623 − 2.22331i 2.71957 − 0.497147i
\(21\) −1.64167 − 5.05253i −0.358241 − 1.10255i
\(22\) −3.81099 − 11.7290i −0.812505 − 2.50063i
\(23\) −1.16500 − 0.846425i −0.242920 − 0.176492i 0.459663 − 0.888093i \(-0.347970\pi\)
−0.702583 + 0.711602i \(0.747970\pi\)
\(24\) 26.9889i 5.50909i
\(25\) −4.82322 − 1.31777i −0.964645 − 0.263554i
\(26\) 1.02453 0.200927
\(27\) −3.98367 − 2.89430i −0.766657 − 0.557009i
\(28\) 3.25710 + 10.0243i 0.615533 + 1.89442i
\(29\) −3.44009 + 1.11775i −0.638809 + 0.207561i −0.610473 − 0.792037i \(-0.709021\pi\)
−0.0283354 + 0.999598i \(0.509021\pi\)
\(30\) 7.39938 − 15.4157i 1.35093 − 2.81451i
\(31\) 5.87547 + 1.90906i 1.05527 + 0.342876i 0.784733 − 0.619834i \(-0.212800\pi\)
0.270533 + 0.962711i \(0.412800\pi\)
\(32\) − 23.2024i − 4.10164i
\(33\) −11.9126 − 3.87065i −2.07372 − 0.673793i
\(34\) −8.40222 + 7.57630i −1.44097 + 1.29933i
\(35\) 0.566732 − 4.22466i 0.0957951 − 0.714098i
\(36\) 21.3235 + 15.4924i 3.55392 + 2.58207i
\(37\) 2.09235 − 1.52018i 0.343980 − 0.249916i −0.402359 − 0.915482i \(-0.631810\pi\)
0.746339 + 0.665566i \(0.231810\pi\)
\(38\) 10.5954 + 14.5834i 1.71881 + 2.36574i
\(39\) 0.611631 − 0.841838i 0.0979393 − 0.134802i
\(40\) −9.37041 + 19.5221i −1.48159 + 3.08671i
\(41\) 1.09109 + 1.50176i 0.170400 + 0.234536i 0.885673 − 0.464310i \(-0.153698\pi\)
−0.715273 + 0.698845i \(0.753698\pi\)
\(42\) 13.8639 + 4.50465i 2.13925 + 0.695083i
\(43\) 12.2854i 1.87350i 0.350000 + 0.936750i \(0.386182\pi\)
−0.350000 + 0.936750i \(0.613818\pi\)
\(44\) 23.6349 + 7.67943i 3.56309 + 1.15772i
\(45\) −5.06306 − 9.37978i −0.754756 − 1.39826i
\(46\) 3.75796 − 1.22104i 0.554082 − 0.180032i
\(47\) 0.0802572 − 0.0260771i 0.0117067 − 0.00380374i −0.303158 − 0.952940i \(-0.598041\pi\)
0.314864 + 0.949137i \(0.398041\pi\)
\(48\) −34.9796 − 25.4142i −5.04887 − 3.66822i
\(49\) −3.36622 −0.480888
\(50\) 10.7045 − 8.58172i 1.51384 − 1.21364i
\(51\) 1.20930 + 11.4269i 0.169335 + 1.60009i
\(52\) −1.21349 + 1.67022i −0.168280 + 0.231618i
\(53\) 0.319741 − 0.103890i 0.0439198 − 0.0142704i −0.286975 − 0.957938i \(-0.592650\pi\)
0.330894 + 0.943668i \(0.392650\pi\)
\(54\) 12.8502 − 4.17527i 1.74869 − 0.568182i
\(55\) −7.27298 − 6.93578i −0.980688 − 0.935220i
\(56\) −17.5569 − 5.70460i −2.34614 − 0.762309i
\(57\) 18.3082 2.42498
\(58\) 3.06706 − 9.43944i 0.402725 − 1.23946i
\(59\) −1.06587 + 0.774396i −0.138764 + 0.100818i −0.655002 − 0.755627i \(-0.727332\pi\)
0.516238 + 0.856445i \(0.327332\pi\)
\(60\) 16.3671 + 30.3216i 2.11298 + 3.91450i
\(61\) 5.79643 − 7.97811i 0.742157 − 1.02149i −0.256334 − 0.966588i \(-0.582515\pi\)
0.998492 − 0.0549038i \(-0.0174852\pi\)
\(62\) −13.7142 + 9.96395i −1.74171 + 1.26542i
\(63\) 7.35140 − 5.34110i 0.926189 − 0.672916i
\(64\) 26.4043 + 19.1838i 3.30053 + 2.39798i
\(65\) 0.734697 − 0.396578i 0.0911279 − 0.0491894i
\(66\) 27.8058 − 20.2021i 3.42266 − 2.48671i
\(67\) 3.59511 + 1.16812i 0.439213 + 0.142709i 0.520272 − 0.854000i \(-0.325830\pi\)
−0.0810596 + 0.996709i \(0.525830\pi\)
\(68\) −2.39926 − 22.6712i −0.290954 − 2.74929i
\(69\) 1.24015 − 3.81679i 0.149297 − 0.459488i
\(70\) 8.46428 + 8.07185i 1.01168 + 0.964771i
\(71\) −10.4209 + 3.38595i −1.23673 + 0.401838i −0.853149 − 0.521668i \(-0.825310\pi\)
−0.383583 + 0.923506i \(0.625310\pi\)
\(72\) −43.9038 + 14.2652i −5.17411 + 1.68117i
\(73\) 1.80355 + 1.31036i 0.211090 + 0.153366i 0.688307 − 0.725420i \(-0.258354\pi\)
−0.477217 + 0.878786i \(0.658354\pi\)
\(74\) 7.09664i 0.824968i
\(75\) −0.661009 − 13.9189i −0.0763268 − 1.60721i
\(76\) −36.3239 −4.16664
\(77\) 5.03589 − 6.93131i 0.573893 − 0.789896i
\(78\) 0.882327 + 2.71552i 0.0999039 + 0.307473i
\(79\) −9.56517 + 3.10791i −1.07617 + 0.349667i −0.792886 − 0.609370i \(-0.791422\pi\)
−0.283279 + 0.959037i \(0.591422\pi\)
\(80\) −16.4784 − 30.5277i −1.84234 − 3.41310i
\(81\) −0.178492 + 0.549342i −0.0198325 + 0.0610380i
\(82\) −5.09354 −0.562488
\(83\) −1.19600 − 0.388603i −0.131278 − 0.0426547i 0.242641 − 0.970116i \(-0.421986\pi\)
−0.373919 + 0.927461i \(0.621986\pi\)
\(84\) −23.7645 + 17.2659i −2.59292 + 1.88387i
\(85\) −3.09263 + 8.68537i −0.335443 + 0.942061i
\(86\) −27.2723 − 19.8145i −2.94085 − 2.13665i
\(87\) −5.92523 − 8.15537i −0.635251 − 0.874348i
\(88\) −35.2127 + 25.5835i −3.75368 + 2.72721i
\(89\) −5.55007 − 4.03236i −0.588307 − 0.427430i 0.253403 − 0.967361i \(-0.418450\pi\)
−0.841709 + 0.539931i \(0.818450\pi\)
\(90\) 28.9882 + 3.88872i 3.05563 + 0.409907i
\(91\) 0.418356 + 0.575818i 0.0438556 + 0.0603621i
\(92\) −2.46048 + 7.57259i −0.256523 + 0.789497i
\(93\) 17.2171i 1.78533i
\(94\) −0.0715544 + 0.220222i −0.00738028 + 0.0227142i
\(95\) 13.2430 + 6.35652i 1.35871 + 0.652165i
\(96\) 61.4981 − 19.9819i 6.27662 − 2.03940i
\(97\) 2.90335 + 8.93560i 0.294791 + 0.907273i 0.983292 + 0.182037i \(0.0582691\pi\)
−0.688501 + 0.725235i \(0.741731\pi\)
\(98\) 5.42922 − 7.47268i 0.548434 − 0.754854i
\(99\) − 21.4245i − 2.15324i
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.q.a.84.2 yes 176
17.16 even 2 inner 425.2.q.a.84.1 ✓ 176
25.14 even 10 inner 425.2.q.a.339.1 yes 176
425.339 even 10 inner 425.2.q.a.339.2 yes 176
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
425.2.q.a.84.1 ✓ 176 17.16 even 2 inner
425.2.q.a.84.2 yes 176 1.1 even 1 trivial
425.2.q.a.339.1 yes 176 25.14 even 10 inner
425.2.q.a.339.2 yes 176 425.339 even 10 inner