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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 339.15
Character \(\chi\) \(=\) 425.339
Dual form 425.2.q.a.84.15

$q$-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+(-0.677444 - 0.932422i) q^{2} +(-1.03346 + 3.18065i) q^{3} +(0.207554 - 0.638786i) q^{4} +(2.01329 + 0.972965i) q^{5} +(3.66582 - 1.19110i) q^{6} -4.10770 q^{7} +(-2.92848 + 0.951521i) q^{8} +(-6.62147 - 4.81078i) q^{9} +(-0.456678 - 2.53636i) q^{10} +(-1.41394 - 1.94612i) q^{11} +(1.81726 + 1.32032i) q^{12} +(0.302563 - 0.416442i) q^{13} +(2.78274 + 3.83011i) q^{14} +(-5.17531 + 5.39806i) q^{15} +(1.78433 + 1.29639i) q^{16} +(2.65157 - 3.15740i) q^{17} +9.43303i q^{18} +(-0.992682 - 3.05516i) q^{19} +(1.03938 - 1.08412i) q^{20} +(4.24513 - 13.0652i) q^{21} +(-0.856742 + 2.63678i) q^{22} +(-0.444953 + 0.323277i) q^{23} -10.2978i q^{24} +(3.10668 + 3.91772i) q^{25} -0.593269 q^{26} +(14.0275 - 10.1916i) q^{27} +(-0.852571 + 2.62394i) q^{28} +(-1.70891 - 0.555258i) q^{29} +(8.53925 + 1.16869i) q^{30} +(-4.70334 + 1.52821i) q^{31} +3.61639i q^{32} +(7.65119 - 2.48602i) q^{33} +(-4.74032 - 0.333424i) q^{34} +(-8.26999 - 3.99665i) q^{35} +(-4.44737 + 3.23121i) q^{36} +(-7.70860 - 5.60063i) q^{37} +(-2.17621 + 2.99530i) q^{38} +(1.01187 + 1.39272i) q^{39} +(-6.82168 - 0.933619i) q^{40} +(0.481169 - 0.662273i) q^{41} +(-15.0581 + 4.89267i) q^{42} +3.79889i q^{43} +(-1.53663 + 0.499280i) q^{44} +(-8.65022 - 16.1279i) q^{45} +(0.602861 + 0.195881i) q^{46} +(-7.90580 - 2.56875i) q^{47} +(-5.96740 + 4.33557i) q^{48} +9.87320 q^{49} +(1.54837 - 5.55077i) q^{50} +(7.30230 + 11.6968i) q^{51} +(-0.203219 - 0.279707i) q^{52} +(-5.69406 - 1.85011i) q^{53} +(-19.0057 - 6.17534i) q^{54} +(-0.953166 - 5.29383i) q^{55} +(12.0293 - 3.90856i) q^{56} +10.7433 q^{57} +(0.639955 + 1.96958i) q^{58} +(-1.99351 - 1.44837i) q^{59} +(2.37405 + 4.42631i) q^{60} +(4.29318 + 5.90906i) q^{61} +(4.61118 + 3.35022i) q^{62} +(27.1990 + 19.7612i) q^{63} +(6.94066 - 5.04269i) q^{64} +(1.01433 - 0.544036i) q^{65} +(-7.50128 - 5.45000i) q^{66} +(-9.58242 + 3.11352i) q^{67} +(-1.46656 - 2.34912i) q^{68} +(-0.568392 - 1.74933i) q^{69} +(1.87590 + 10.4186i) q^{70} +(-13.8098 - 4.48708i) q^{71} +(23.9684 + 7.78780i) q^{72} +(-0.226203 + 0.164346i) q^{73} +10.9818i q^{74} +(-15.6715 + 5.83247i) q^{75} -2.15763 q^{76} +(5.80805 + 7.99410i) q^{77} +(0.613118 - 1.88698i) q^{78} +(16.0405 + 5.21188i) q^{79} +(2.33103 + 4.34610i) q^{80} +(10.3316 + 31.7973i) q^{81} -0.943483 q^{82} +(0.180325 - 0.0585910i) q^{83} +(-7.46476 - 5.42346i) q^{84} +(8.41042 - 3.77687i) q^{85} +(3.54216 - 2.57353i) q^{86} +(3.53216 - 4.86161i) q^{87} +(5.99248 + 4.35379i) q^{88} +(0.532227 - 0.386686i) q^{89} +(-9.17801 + 18.9914i) q^{90} +(-1.24284 + 1.71062i) q^{91} +(0.114153 + 0.351327i) q^{92} -16.5390i q^{93} +(2.96058 + 9.11172i) q^{94} +(0.974007 - 7.11677i) q^{95} +(-11.5025 - 3.73738i) q^{96} +(-0.579938 + 1.78487i) q^{97} +(-6.68854 - 9.20599i) q^{98} +19.6884i 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 \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(326\)
\(\chi(n)\) \(e\left(\frac{3}{10}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.677444 − 0.932422i −0.479025 − 0.659322i 0.499292 − 0.866434i \(-0.333593\pi\)
−0.978317 + 0.207112i \(0.933593\pi\)
\(3\) −1.03346 + 3.18065i −0.596666 + 1.83635i −0.0504168 + 0.998728i \(0.516055\pi\)
−0.546250 + 0.837622i \(0.683945\pi\)
\(4\) 0.207554 − 0.638786i 0.103777 − 0.319393i
\(5\) 2.01329 + 0.972965i 0.900371 + 0.435123i
\(6\) 3.66582 − 1.19110i 1.49656 − 0.486263i
\(7\) −4.10770 −1.55256 −0.776282 − 0.630385i \(-0.782897\pi\)
−0.776282 + 0.630385i \(0.782897\pi\)
\(8\) −2.92848 + 0.951521i −1.03537 + 0.336413i
\(9\) −6.62147 − 4.81078i −2.20716 − 1.60359i
\(10\) −0.456678 − 2.53636i −0.144414 − 0.802069i
\(11\) −1.41394 − 1.94612i −0.426320 − 0.586779i 0.540784 − 0.841161i \(-0.318128\pi\)
−0.967104 + 0.254383i \(0.918128\pi\)
\(12\) 1.81726 + 1.32032i 0.524597 + 0.381142i
\(13\) 0.302563 − 0.416442i 0.0839158 − 0.115500i −0.764995 − 0.644036i \(-0.777259\pi\)
0.848911 + 0.528536i \(0.177259\pi\)
\(14\) 2.78274 + 3.83011i 0.743718 + 1.02364i
\(15\) −5.17531 + 5.39806i −1.33626 + 1.39377i
\(16\) 1.78433 + 1.29639i 0.446083 + 0.324098i
\(17\) 2.65157 − 3.15740i 0.643101 − 0.765781i
\(18\) 9.43303i 2.22339i
\(19\) −0.992682 − 3.05516i −0.227737 − 0.700902i −0.998002 − 0.0631785i \(-0.979876\pi\)
0.770265 − 0.637723i \(-0.220124\pi\)
\(20\) 1.03938 − 1.08412i 0.232413 − 0.242417i
\(21\) 4.24513 − 13.0652i 0.926363 − 2.85105i
\(22\) −0.856742 + 2.63678i −0.182658 + 0.562163i
\(23\) −0.444953 + 0.323277i −0.0927790 + 0.0674079i −0.633208 − 0.773982i \(-0.718262\pi\)
0.540429 + 0.841390i \(0.318262\pi\)
\(24\) − 10.2978i − 2.10204i
\(25\) 3.10668 + 3.91772i 0.621336 + 0.783544i
\(26\) −0.593269 −0.116350
\(27\) 14.0275 − 10.1916i 2.69960 − 1.96137i
\(28\) −0.852571 + 2.62394i −0.161121 + 0.495879i
\(29\) −1.70891 − 0.555258i −0.317336 − 0.103109i 0.146018 − 0.989282i \(-0.453354\pi\)
−0.463354 + 0.886173i \(0.653354\pi\)
\(30\) 8.53925 + 1.16869i 1.55905 + 0.213372i
\(31\) −4.70334 + 1.52821i −0.844744 + 0.274474i −0.699243 − 0.714884i \(-0.746479\pi\)
−0.145501 + 0.989358i \(0.546479\pi\)
\(32\) 3.61639i 0.639293i
\(33\) 7.65119 − 2.48602i 1.33190 − 0.432761i
\(34\) −4.74032 − 0.333424i −0.812958 − 0.0571817i
\(35\) −8.26999 − 3.99665i −1.39788 − 0.675557i
\(36\) −4.44737 + 3.23121i −0.741229 + 0.538534i
\(37\) −7.70860 − 5.60063i −1.26729 − 0.920737i −0.268195 − 0.963365i \(-0.586427\pi\)
−0.999091 + 0.0426271i \(0.986427\pi\)
\(38\) −2.17621 + 2.99530i −0.353028 + 0.485902i
\(39\) 1.01187 + 1.39272i 0.162029 + 0.223014i
\(40\) −6.82168 − 0.933619i −1.07860 − 0.147618i
\(41\) 0.481169 − 0.662273i 0.0751460 − 0.103430i −0.769789 − 0.638298i \(-0.779639\pi\)
0.844935 + 0.534868i \(0.179639\pi\)
\(42\) −15.0581 + 4.89267i −2.32351 + 0.754955i
\(43\) 3.79889i 0.579325i 0.957129 + 0.289662i \(0.0935430\pi\)
−0.957129 + 0.289662i \(0.906457\pi\)
\(44\) −1.53663 + 0.499280i −0.231655 + 0.0752694i
\(45\) −8.65022 − 16.1279i −1.28950 − 2.40421i
\(46\) 0.602861 + 0.195881i 0.0888870 + 0.0288811i
\(47\) −7.90580 − 2.56875i −1.15318 − 0.374691i −0.330839 − 0.943687i \(-0.607332\pi\)
−0.822340 + 0.568996i \(0.807332\pi\)
\(48\) −5.96740 + 4.33557i −0.861320 + 0.625786i
\(49\) 9.87320 1.41046
\(50\) 1.54837 − 5.55077i 0.218972 − 0.784998i
\(51\) 7.30230 + 11.6968i 1.02253 + 1.63787i
\(52\) −0.203219 − 0.279707i −0.0281814 − 0.0387884i
\(53\) −5.69406 − 1.85011i −0.782139 − 0.254132i −0.109386 − 0.993999i \(-0.534888\pi\)
−0.672753 + 0.739867i \(0.734888\pi\)
\(54\) −19.0057 − 6.17534i −2.58635 − 0.840357i
\(55\) −0.953166 − 5.29383i −0.128525 − 0.713820i
\(56\) 12.0293 − 3.90856i 1.60748 − 0.522303i
\(57\) 10.7433 1.42298
\(58\) 0.639955 + 1.96958i 0.0840302 + 0.258618i
\(59\) −1.99351 − 1.44837i −0.259534 − 0.188562i 0.450408 − 0.892823i \(-0.351279\pi\)
−0.709941 + 0.704261i \(0.751279\pi\)
\(60\) 2.37405 + 4.42631i 0.306489 + 0.571434i
\(61\) 4.29318 + 5.90906i 0.549686 + 0.756577i 0.989969 − 0.141281i \(-0.0451222\pi\)
−0.440284 + 0.897859i \(0.645122\pi\)
\(62\) 4.61118 + 3.35022i 0.585620 + 0.425478i
\(63\) 27.1990 + 19.7612i 3.42675 + 2.48968i
\(64\) 6.94066 − 5.04269i 0.867583 − 0.630336i
\(65\) 1.01433 − 0.544036i 0.125812 − 0.0674794i
\(66\) −7.50128 − 5.45000i −0.923343 − 0.670848i
\(67\) −9.58242 + 3.11352i −1.17068 + 0.380377i −0.828896 − 0.559402i \(-0.811031\pi\)
−0.341783 + 0.939779i \(0.611031\pi\)
\(68\) −1.46656 − 2.34912i −0.177846 − 0.284873i
\(69\) −0.568392 − 1.74933i −0.0684264 − 0.210595i
\(70\) 1.87590 + 10.4186i 0.224213 + 1.24526i
\(71\) −13.8098 − 4.48708i −1.63892 − 0.532518i −0.662624 − 0.748952i \(-0.730557\pi\)
−0.976297 + 0.216434i \(0.930557\pi\)
\(72\) 23.9684 + 7.78780i 2.82470 + 0.917801i
\(73\) −0.226203 + 0.164346i −0.0264750 + 0.0192352i −0.600944 − 0.799291i \(-0.705209\pi\)
0.574469 + 0.818526i \(0.305209\pi\)
\(74\) 10.9818i 1.27661i
\(75\) −15.6715 + 5.83247i −1.80959 + 0.673476i
\(76\) −2.15763 −0.247497
\(77\) 5.80805 + 7.99410i 0.661889 + 0.911012i
\(78\) 0.613118 − 1.88698i 0.0694219 − 0.213659i
\(79\) 16.0405 + 5.21188i 1.80470 + 0.586383i 0.999974 − 0.00726275i \(-0.00231183\pi\)
0.804727 + 0.593645i \(0.202312\pi\)
\(80\) 2.33103 + 4.34610i 0.260617 + 0.485909i
\(81\) 10.3316 + 31.7973i 1.14795 + 3.53303i
\(82\) −0.943483 −0.104190
\(83\) 0.180325 − 0.0585910i 0.0197932 − 0.00643120i −0.299104 − 0.954221i \(-0.596688\pi\)
0.318897 + 0.947789i \(0.396688\pi\)
\(84\) −7.46476 − 5.42346i −0.814472 − 0.591748i
\(85\) 8.41042 − 3.77687i 0.912239 − 0.409659i
\(86\) 3.54216 − 2.57353i 0.381961 − 0.277511i
\(87\) 3.53216 − 4.86161i 0.378688 − 0.521219i
\(88\) 5.99248 + 4.35379i 0.638800 + 0.464116i
\(89\) 0.532227 − 0.386686i 0.0564160 − 0.0409886i −0.559220 − 0.829019i \(-0.688899\pi\)
0.615636 + 0.788031i \(0.288899\pi\)
\(90\) −9.17801 + 18.9914i −0.967447 + 2.00187i
\(91\) −1.24284 + 1.71062i −0.130285 + 0.179322i
\(92\) 0.114153 + 0.351327i 0.0119013 + 0.0366284i
\(93\) − 16.5390i − 1.71502i
\(94\) 2.96058 + 9.11172i 0.305360 + 0.939802i
\(95\) 0.974007 − 7.11677i 0.0999310 − 0.730165i
\(96\) −11.5025 − 3.73738i −1.17397 − 0.381445i
\(97\) −0.579938 + 1.78487i −0.0588838 + 0.181226i −0.976172 − 0.216998i \(-0.930373\pi\)
0.917288 + 0.398224i \(0.130373\pi\)
\(98\) −6.68854 − 9.20599i −0.675645 − 0.929945i
\(99\) 19.6884i 1.97875i
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.339.15 yes 176
17.16 even 2 inner 425.2.q.a.339.16 yes 176
25.9 even 10 inner 425.2.q.a.84.16 yes 176
425.84 even 10 inner 425.2.q.a.84.15 ✓ 176
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
425.2.q.a.84.15 ✓ 176 425.84 even 10 inner
425.2.q.a.84.16 yes 176 25.9 even 10 inner
425.2.q.a.339.15 yes 176 1.1 even 1 trivial
425.2.q.a.339.16 yes 176 17.16 even 2 inner