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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 84.9
Character \(\chi\) \(=\) 425.84
Dual form 425.2.q.a.339.9

$q$-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+(-1.14277 + 1.57289i) q^{2} +(-0.782761 - 2.40909i) q^{3} +(-0.550027 - 1.69281i) q^{4} +(1.07788 + 1.95913i) q^{5} +(4.68376 + 1.52185i) q^{6} +3.64131 q^{7} +(-0.406932 - 0.132220i) q^{8} +(-2.76396 + 2.00813i) q^{9} +(-4.31327 - 0.543447i) q^{10} +(-1.43874 + 1.98025i) q^{11} +(-3.64759 + 2.65013i) q^{12} +(-1.60261 - 2.20580i) q^{13} +(-4.16119 + 5.72739i) q^{14} +(3.87599 - 4.13024i) q^{15} +(3.55298 - 2.58139i) q^{16} +(3.51182 - 2.16036i) q^{17} -6.64225i q^{18} +(-0.374938 + 1.15394i) q^{19} +(2.72356 - 2.90222i) q^{20} +(-2.85028 - 8.77225i) q^{21} +(-1.47057 - 4.52595i) q^{22} +(7.60100 + 5.52245i) q^{23} +1.08383i q^{24} +(-2.67635 + 4.22341i) q^{25} +5.30091 q^{26} +(0.853404 + 0.620035i) q^{27} +(-2.00282 - 6.16404i) q^{28} +(-3.09952 + 1.00709i) q^{29} +(2.06705 + 10.8164i) q^{30} +(7.54089 + 2.45019i) q^{31} +7.68265i q^{32} +(5.89679 + 1.91598i) q^{33} +(-0.615200 + 7.99251i) q^{34} +(3.92490 + 7.13379i) q^{35} +(4.91963 + 3.57432i) q^{36} +(8.00851 - 5.81853i) q^{37} +(-1.38656 - 1.90843i) q^{38} +(-4.05952 + 5.58744i) q^{39} +(-0.179588 - 0.939748i) q^{40} +(2.97251 + 4.09130i) q^{41} +(17.0550 + 5.54152i) q^{42} -0.892274i q^{43} +(4.14352 + 1.34631i) q^{44} +(-6.91340 - 3.25041i) q^{45} +(-17.3725 + 5.64465i) q^{46} +(2.81477 - 0.914573i) q^{47} +(-8.99993 - 6.53883i) q^{48} +6.25914 q^{49} +(-3.58451 - 9.03601i) q^{50} +(-7.95341 - 6.76924i) q^{51} +(-2.85252 + 3.92616i) q^{52} +(11.5809 - 3.76285i) q^{53} +(-1.95050 + 0.633754i) q^{54} +(-5.43034 - 0.684192i) q^{55} +(-1.48176 - 0.481455i) q^{56} +3.07344 q^{57} +(1.95800 - 6.02609i) q^{58} +(-8.37777 + 6.08680i) q^{59} +(-9.12360 - 4.28957i) q^{60} +(1.74201 - 2.39767i) q^{61} +(-12.4714 + 9.06101i) q^{62} +(-10.0644 + 7.31223i) q^{63} +(-4.97803 - 3.61675i) q^{64} +(2.59402 - 5.51730i) q^{65} +(-9.75233 + 7.08548i) q^{66} +(-11.2277 - 3.64809i) q^{67} +(-5.58866 - 4.75658i) q^{68} +(7.35432 - 22.6343i) q^{69} +(-15.7059 - 1.97886i) q^{70} +(-3.57649 + 1.16207i) q^{71} +(1.39026 - 0.451722i) q^{72} +(2.32787 + 1.69130i) q^{73} +19.2458i q^{74} +(12.2695 + 3.14165i) q^{75} +2.15963 q^{76} +(-5.23888 + 7.21070i) q^{77} +(-4.14934 - 12.7704i) q^{78} +(7.12162 - 2.31396i) q^{79} +(8.88695 + 4.17830i) q^{80} +(-2.34150 + 7.20641i) q^{81} -9.83208 q^{82} +(-1.51987 - 0.493836i) q^{83} +(-13.2820 + 9.64994i) q^{84} +(8.01773 + 4.55148i) q^{85} +(1.40345 + 1.01967i) q^{86} +(4.85237 + 6.67871i) q^{87} +(0.847296 - 0.615597i) q^{88} +(-6.62169 - 4.81094i) q^{89} +(13.0130 - 7.15955i) q^{90} +(-5.83559 - 8.03200i) q^{91} +(5.16770 - 15.9045i) q^{92} -20.0846i q^{93} +(-1.77811 + 5.47248i) q^{94} +(-2.66485 + 0.509259i) q^{95} +(18.5082 - 6.01368i) q^{96} +(0.0876674 + 0.269813i) q^{97} +(-7.15278 + 9.84496i) q^{98} -8.36249i 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.14277 + 1.57289i −0.808063 + 1.11220i 0.183556 + 0.983009i \(0.441239\pi\)
−0.991619 + 0.129194i \(0.958761\pi\)
\(3\) −0.782761 − 2.40909i −0.451927 − 1.39089i −0.874705 − 0.484656i \(-0.838945\pi\)
0.422777 − 0.906234i \(-0.361055\pi\)
\(4\) −0.550027 − 1.69281i −0.275013 − 0.846404i
\(5\) 1.07788 + 1.95913i 0.482043 + 0.876148i
\(6\) 4.68376 + 1.52185i 1.91214 + 0.621291i
\(7\) 3.64131 1.37629 0.688143 − 0.725575i \(-0.258426\pi\)
0.688143 + 0.725575i \(0.258426\pi\)
\(8\) −0.406932 − 0.132220i −0.143872 − 0.0467469i
\(9\) −2.76396 + 2.00813i −0.921319 + 0.669377i
\(10\) −4.31327 − 0.543447i −1.36398 − 0.171853i
\(11\) −1.43874 + 1.98025i −0.433795 + 0.597068i −0.968819 − 0.247769i \(-0.920303\pi\)
0.535024 + 0.844837i \(0.320303\pi\)
\(12\) −3.64759 + 2.65013i −1.05297 + 0.765026i
\(13\) −1.60261 − 2.20580i −0.444483 − 0.611779i 0.526718 − 0.850040i \(-0.323423\pi\)
−0.971201 + 0.238261i \(0.923423\pi\)
\(14\) −4.16119 + 5.72739i −1.11213 + 1.53071i
\(15\) 3.87599 − 4.13024i 1.00078 − 1.06642i
\(16\) 3.55298 − 2.58139i 0.888244 − 0.645347i
\(17\) 3.51182 − 2.16036i 0.851741 − 0.523964i
\(18\) − 6.64225i − 1.56559i
\(19\) −0.374938 + 1.15394i −0.0860167 + 0.264732i −0.984809 − 0.173643i \(-0.944446\pi\)
0.898792 + 0.438376i \(0.144446\pi\)
\(20\) 2.72356 − 2.90222i 0.609007 − 0.648955i
\(21\) −2.85028 − 8.77225i −0.621981 − 1.91426i
\(22\) −1.47057 − 4.52595i −0.313527 − 0.964937i
\(23\) 7.60100 + 5.52245i 1.58492 + 1.15151i 0.910769 + 0.412917i \(0.135490\pi\)
0.674150 + 0.738594i \(0.264510\pi\)
\(24\) 1.08383i 0.221236i
\(25\) −2.67635 + 4.22341i −0.535270 + 0.844681i
\(26\) 5.30091 1.03959
\(27\) 0.853404 + 0.620035i 0.164238 + 0.119326i
\(28\) −2.00282 − 6.16404i −0.378497 − 1.16489i
\(29\) −3.09952 + 1.00709i −0.575566 + 0.187013i −0.582313 − 0.812964i \(-0.697852\pi\)
0.00674718 + 0.999977i \(0.497852\pi\)
\(30\) 2.06705 + 10.8164i 0.377389 + 1.97480i
\(31\) 7.54089 + 2.45019i 1.35438 + 0.440066i 0.894165 − 0.447739i \(-0.147770\pi\)
0.460220 + 0.887805i \(0.347770\pi\)
\(32\) 7.68265i 1.35811i
\(33\) 5.89679 + 1.91598i 1.02650 + 0.333530i
\(34\) −0.615200 + 7.99251i −0.105506 + 1.37070i
\(35\) 3.92490 + 7.13379i 0.663429 + 1.20583i
\(36\) 4.91963 + 3.57432i 0.819938 + 0.595720i
\(37\) 8.00851 − 5.81853i 1.31659 − 0.956560i 0.316624 − 0.948551i \(-0.397451\pi\)
0.999968 − 0.00800840i \(-0.00254918\pi\)
\(38\) −1.38656 − 1.90843i −0.224929 − 0.309588i
\(39\) −4.05952 + 5.58744i −0.650043 + 0.894707i
\(40\) −0.179588 − 0.939748i −0.0283953 − 0.148587i
\(41\) 2.97251 + 4.09130i 0.464227 + 0.638954i 0.975379 − 0.220537i \(-0.0707808\pi\)
−0.511151 + 0.859491i \(0.670781\pi\)
\(42\) 17.0550 + 5.54152i 2.63165 + 0.855074i
\(43\) − 0.892274i − 0.136070i −0.997683 − 0.0680352i \(-0.978327\pi\)
0.997683 − 0.0680352i \(-0.0216730\pi\)
\(44\) 4.14352 + 1.34631i 0.624660 + 0.202964i
\(45\) −6.91340 − 3.25041i −1.03059 − 0.484543i
\(46\) −17.3725 + 5.64465i −2.56143 + 0.832259i
\(47\) 2.81477 − 0.914573i 0.410576 − 0.133404i −0.0964443 − 0.995338i \(-0.530747\pi\)
0.507020 + 0.861934i \(0.330747\pi\)
\(48\) −8.99993 − 6.53883i −1.29903 − 0.943799i
\(49\) 6.25914 0.894163
\(50\) −3.58451 − 9.03601i −0.506926 − 1.27788i
\(51\) −7.95341 − 6.76924i −1.11370 − 0.947884i
\(52\) −2.85252 + 3.92616i −0.395573 + 0.544460i
\(53\) 11.5809 − 3.76285i 1.59075 − 0.516868i 0.625956 − 0.779858i \(-0.284709\pi\)
0.964798 + 0.262990i \(0.0847088\pi\)
\(54\) −1.95050 + 0.633754i −0.265429 + 0.0862431i
\(55\) −5.43034 − 0.684192i −0.732227 − 0.0922564i
\(56\) −1.48176 − 0.481455i −0.198009 − 0.0643371i
\(57\) 3.07344 0.407087
\(58\) 1.95800 − 6.02609i 0.257097 − 0.791265i
\(59\) −8.37777 + 6.08680i −1.09069 + 0.792434i −0.979516 − 0.201366i \(-0.935462\pi\)
−0.111176 + 0.993801i \(0.535462\pi\)
\(60\) −9.12360 − 4.28957i −1.17785 − 0.553780i
\(61\) 1.74201 − 2.39767i 0.223042 − 0.306991i −0.682801 − 0.730604i \(-0.739238\pi\)
0.905843 + 0.423613i \(0.139238\pi\)
\(62\) −12.4714 + 9.06101i −1.58387 + 1.15075i
\(63\) −10.0644 + 7.31223i −1.26800 + 0.921255i
\(64\) −4.97803 − 3.61675i −0.622253 − 0.452094i
\(65\) 2.59402 − 5.51730i 0.321749 − 0.684337i
\(66\) −9.75233 + 7.08548i −1.20043 + 0.872163i
\(67\) −11.2277 − 3.64809i −1.37168 − 0.445685i −0.471754 − 0.881730i \(-0.656379\pi\)
−0.899925 + 0.436045i \(0.856379\pi\)
\(68\) −5.58866 − 4.75658i −0.677725 − 0.576820i
\(69\) 7.35432 − 22.6343i 0.885357 − 2.72485i
\(70\) −15.7059 − 1.97886i −1.87722 − 0.236519i
\(71\) −3.57649 + 1.16207i −0.424451 + 0.137912i −0.513450 − 0.858119i \(-0.671633\pi\)
0.0889996 + 0.996032i \(0.471633\pi\)
\(72\) 1.39026 − 0.451722i 0.163843 − 0.0532359i
\(73\) 2.32787 + 1.69130i 0.272457 + 0.197951i 0.715620 − 0.698489i \(-0.246144\pi\)
−0.443164 + 0.896441i \(0.646144\pi\)
\(74\) 19.2458i 2.23728i
\(75\) 12.2695 + 3.14165i 1.41676 + 0.362766i
\(76\) 2.15963 0.247726
\(77\) −5.23888 + 7.21070i −0.597026 + 0.821736i
\(78\) −4.14934 − 12.7704i −0.469821 − 1.44596i
\(79\) 7.12162 − 2.31396i 0.801245 − 0.260340i 0.120360 − 0.992730i \(-0.461595\pi\)
0.680885 + 0.732390i \(0.261595\pi\)
\(80\) 8.88695 + 4.17830i 0.993591 + 0.467148i
\(81\) −2.34150 + 7.20641i −0.260167 + 0.800712i
\(82\) −9.83208 −1.08577
\(83\) −1.51987 − 0.493836i −0.166828 − 0.0542056i 0.224412 − 0.974494i \(-0.427954\pi\)
−0.391240 + 0.920289i \(0.627954\pi\)
\(84\) −13.2820 + 9.64994i −1.44919 + 1.05290i
\(85\) 8.01773 + 4.55148i 0.869645 + 0.493678i
\(86\) 1.40345 + 1.01967i 0.151338 + 0.109953i
\(87\) 4.85237 + 6.67871i 0.520228 + 0.716033i
\(88\) 0.847296 − 0.615597i 0.0903221 − 0.0656228i
\(89\) −6.62169 − 4.81094i −0.701898 − 0.509958i 0.178652 − 0.983912i \(-0.442826\pi\)
−0.880550 + 0.473954i \(0.842826\pi\)
\(90\) 13.0130 − 7.15955i 1.37169 − 0.754683i
\(91\) −5.83559 − 8.03200i −0.611736 − 0.841983i
\(92\) 5.16770 − 15.9045i 0.538770 − 1.65816i
\(93\) − 20.0846i − 2.08268i
\(94\) −1.77811 + 5.47248i −0.183399 + 0.564443i
\(95\) −2.66485 + 0.509259i −0.273408 + 0.0522489i
\(96\) 18.5082 − 6.01368i 1.88898 − 0.613768i
\(97\) 0.0876674 + 0.269813i 0.00890128 + 0.0273953i 0.955409 − 0.295287i \(-0.0954154\pi\)
−0.946507 + 0.322683i \(0.895415\pi\)
\(98\) −7.15278 + 9.84496i −0.722540 + 0.994491i
\(99\) − 8.36249i − 0.840462i
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.84.9 ✓ 176
17.16 even 2 inner 425.2.q.a.84.10 yes 176
25.14 even 10 inner 425.2.q.a.339.10 yes 176
425.339 even 10 inner 425.2.q.a.339.9 yes 176
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
425.2.q.a.84.9 ✓ 176 1.1 even 1 trivial
425.2.q.a.84.10 yes 176 17.16 even 2 inner
425.2.q.a.339.9 yes 176 425.339 even 10 inner
425.2.q.a.339.10 yes 176 25.14 even 10 inner