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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 84.15
Character \(\chi\) \(=\) 425.84
Dual form 425.2.q.a.339.15

$q$-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 \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.677444 + 0.932422i −0.479025 + 0.659322i −0.978317 − 0.207112i \(-0.933593\pi\)
0.499292 + 0.866434i \(0.333593\pi\)
\(3\) −1.03346 − 3.18065i −0.596666 − 1.83635i −0.546250 − 0.837622i \(-0.683945\pi\)
−0.0504168 − 0.998728i \(-0.516055\pi\)
\(4\) 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 −0.967104 − 0.254383i \(-0.918128\pi\)
0.540784 + 0.841161i \(0.318128\pi\)
\(12\) 1.81726 − 1.32032i 0.524597 − 0.381142i
\(13\) 0.302563 + 0.416442i 0.0839158 + 0.115500i 0.848911 − 0.528536i \(-0.177259\pi\)
−0.764995 + 0.644036i \(0.777259\pi\)
\(14\) 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 0.770265 + 0.637723i \(0.220124\pi\)
−0.998002 + 0.0631785i \(0.979876\pi\)
\(20\) 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 0.540429 − 0.841390i \(-0.318262\pi\)
−0.633208 + 0.773982i \(0.718262\pi\)
\(24\) 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 −0.463354 − 0.886173i \(-0.653354\pi\)
0.146018 + 0.989282i \(0.453354\pi\)
\(30\) 8.53925 − 1.16869i 1.55905 − 0.213372i
\(31\) −4.70334 − 1.52821i −0.844744 − 0.274474i −0.145501 − 0.989358i \(-0.546479\pi\)
−0.699243 + 0.714884i \(0.746479\pi\)
\(32\) − 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 −0.999091 − 0.0426271i \(-0.986427\pi\)
−0.268195 + 0.963365i \(0.586427\pi\)
\(38\) −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 0.844935 − 0.534868i \(-0.179639\pi\)
−0.769789 + 0.638298i \(0.779639\pi\)
\(42\) −15.0581 − 4.89267i −2.32351 − 0.754955i
\(43\) − 3.79889i − 0.579325i −0.957129 − 0.289662i \(-0.906457\pi\)
0.957129 − 0.289662i \(-0.0935430\pi\)
\(44\) −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 −0.822340 − 0.568996i \(-0.807332\pi\)
−0.330839 + 0.943687i \(0.607332\pi\)
\(48\) −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 −0.672753 − 0.739867i \(-0.734888\pi\)
−0.109386 + 0.993999i \(0.534888\pi\)
\(54\) −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 −0.709941 − 0.704261i \(-0.751279\pi\)
0.450408 + 0.892823i \(0.351279\pi\)
\(60\) 2.37405 − 4.42631i 0.306489 − 0.571434i
\(61\) 4.29318 − 5.90906i 0.549686 − 0.756577i −0.440284 − 0.897859i \(-0.645122\pi\)
0.989969 + 0.141281i \(0.0451222\pi\)
\(62\) 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 −0.341783 − 0.939779i \(-0.611031\pi\)
−0.828896 + 0.559402i \(0.811031\pi\)
\(68\) −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 −0.976297 − 0.216434i \(-0.930557\pi\)
−0.662624 + 0.748952i \(0.730557\pi\)
\(72\) 23.9684 − 7.78780i 2.82470 − 0.917801i
\(73\) −0.226203 − 0.164346i −0.0264750 − 0.0192352i 0.574469 − 0.818526i \(-0.305209\pi\)
−0.600944 + 0.799291i \(0.705209\pi\)
\(74\) − 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 0.804727 − 0.593645i \(-0.202312\pi\)
0.999974 + 0.00726275i \(0.00231183\pi\)
\(80\) 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 0.318897 − 0.947789i \(-0.396688\pi\)
−0.299104 + 0.954221i \(0.596688\pi\)
\(84\) −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 0.615636 − 0.788031i \(-0.288899\pi\)
−0.559220 + 0.829019i \(0.688899\pi\)
\(90\) −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 0.917288 − 0.398224i \(-0.130373\pi\)
−0.976172 + 0.216998i \(0.930373\pi\)
\(98\) −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.84.15 ✓ 176
17.16 even 2 inner 425.2.q.a.84.16 yes 176
25.14 even 10 inner 425.2.q.a.339.16 yes 176
425.339 even 10 inner 425.2.q.a.339.15 yes 176
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
425.2.q.a.84.15 ✓ 176 1.1 even 1 trivial
425.2.q.a.84.16 yes 176 17.16 even 2 inner
425.2.q.a.339.15 yes 176 425.339 even 10 inner
425.2.q.a.339.16 yes 176 25.14 even 10 inner