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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [425,2,Mod(69,425)] 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("425.69"); 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([9, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.r (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: \(160\)
Relative dimension: \(40\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label 69.18
Character \(\chi\) \(=\) 425.69
Dual form 425.2.r.a.154.18

$q$-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+(-0.292236 + 0.0949534i) q^{2} +(-0.973705 - 1.34019i) q^{3} +(-1.54165 + 1.12007i) q^{4} +(1.39711 - 1.74587i) q^{5} +(0.411808 + 0.299196i) q^{6} +1.62592i q^{7} +(0.705395 - 0.970893i) q^{8} +(0.0790435 - 0.243271i) q^{9} +(-0.242510 + 0.642869i) q^{10} +(-0.389672 - 1.19929i) q^{11} +(3.00222 + 0.975481i) q^{12} +(-2.15907 - 0.701526i) q^{13} +(-0.154386 - 0.475153i) q^{14} +(-3.70018 - 0.172428i) q^{15} +(1.06376 - 3.27392i) q^{16} +(-0.587785 + 0.809017i) q^{17} +0.0785981i q^{18} +(-3.60186 - 2.61690i) q^{19} +(-0.198347 + 4.25639i) q^{20} +(2.17904 - 1.58316i) q^{21} +(0.227753 + 0.313474i) q^{22} +(-3.44716 + 1.12005i) q^{23} -1.98803 q^{24} +(-1.09616 - 4.87836i) q^{25} +0.697573 q^{26} +(-5.12946 + 1.66666i) q^{27} +(-1.82115 - 2.50659i) q^{28} +(-0.213184 + 0.154887i) q^{29} +(1.09770 - 0.300955i) q^{30} +(-6.37905 - 4.63465i) q^{31} +3.45795i q^{32} +(-1.22785 + 1.68998i) q^{33} +(0.0949534 - 0.292236i) q^{34} +(2.83865 + 2.27159i) q^{35} +(0.150624 + 0.463572i) q^{36} +(-6.04607 - 1.96449i) q^{37} +(1.30108 + 0.422746i) q^{38} +(1.16212 + 3.57665i) q^{39} +(-0.709542 - 2.58798i) q^{40} +(2.73636 - 8.42164i) q^{41} +(-0.486468 + 0.669566i) q^{42} -0.812241i q^{43} +(1.94402 + 1.41242i) q^{44} +(-0.314288 - 0.477877i) q^{45} +(0.901034 - 0.654639i) q^{46} +(5.32740 + 7.33254i) q^{47} +(-5.42347 + 1.76219i) q^{48} +4.35639 q^{49} +(0.783555 + 1.32155i) q^{50} +1.65657 q^{51} +(4.11429 - 1.33681i) q^{52} +(-5.64476 - 7.76935i) q^{53} +(1.34076 - 0.974119i) q^{54} +(-2.63822 - 0.995218i) q^{55} +(1.57859 + 1.14691i) q^{56} +7.37526i q^{57} +(0.0475931 - 0.0655062i) q^{58} +(1.36411 - 4.19831i) q^{59} +(5.89750 - 3.87865i) q^{60} +(2.24358 + 6.90501i) q^{61} +(2.30427 + 0.748701i) q^{62} +(0.395538 + 0.128518i) q^{63} +(1.79918 + 5.53731i) q^{64} +(-4.24124 + 2.78936i) q^{65} +(0.198352 - 0.610463i) q^{66} +(-4.75656 + 6.54685i) q^{67} -1.90558i q^{68} +(4.85760 + 3.52925i) q^{69} +(-1.04525 - 0.394302i) q^{70} +(1.46562 - 1.06483i) q^{71} +(-0.180433 - 0.248345i) q^{72} +(15.3611 - 4.99113i) q^{73} +1.95342 q^{74} +(-5.47060 + 6.21915i) q^{75} +8.48392 q^{76} +(1.94994 - 0.633574i) q^{77} +(-0.679230 - 0.934880i) q^{78} +(13.2522 - 9.62832i) q^{79} +(-4.22966 - 6.43123i) q^{80} +(6.60740 + 4.80056i) q^{81} +2.72094i q^{82} +(-1.68587 + 2.32040i) q^{83} +(-1.58605 + 4.88137i) q^{84} +(0.591241 + 2.15649i) q^{85} +(0.0771250 + 0.237366i) q^{86} +(0.415156 + 0.134893i) q^{87} +(-1.43925 - 0.467641i) q^{88} +(4.18423 + 12.8777i) q^{89} +(0.137222 + 0.109810i) q^{90} +(1.14062 - 3.51048i) q^{91} +(4.05977 - 5.58780i) q^{92} +13.0619i q^{93} +(-2.25311 - 1.63698i) q^{94} +(-9.60098 + 2.63229i) q^{95} +(4.63431 - 3.36702i) q^{96} +(-4.19887 - 5.77924i) q^{97} +(-1.27310 + 0.413654i) q^{98} -0.322552 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 160 q + 40 q^{4} - 8 q^{5} + 4 q^{6} - 30 q^{8} + 36 q^{9} - 6 q^{10} + 8 q^{11} - 40 q^{12} - 20 q^{14} - 40 q^{15} - 64 q^{16} + 6 q^{19} + 2 q^{20} - 50 q^{22} + 20 q^{23} + 20 q^{24} + 32 q^{25} + 20 q^{26}+ \cdots + 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(52\) \(326\)
\(\chi(n)\) \(e\left(\frac{9}{10}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.292236 + 0.0949534i −0.206642 + 0.0671422i −0.410509 − 0.911856i \(-0.634649\pi\)
0.203867 + 0.978999i \(0.434649\pi\)
\(3\) −0.973705 − 1.34019i −0.562169 − 0.773759i 0.429431 − 0.903099i \(-0.358714\pi\)
−0.991600 + 0.129341i \(0.958714\pi\)
\(4\) −1.54165 + 1.12007i −0.770824 + 0.560036i
\(5\) 1.39711 − 1.74587i 0.624807 − 0.780779i
\(6\) 0.411808 + 0.299196i 0.168120 + 0.122146i
\(7\) 1.62592i 0.614539i 0.951622 + 0.307270i \(0.0994154\pi\)
−0.951622 + 0.307270i \(0.900585\pi\)
\(8\) 0.705395 − 0.970893i 0.249395 − 0.343263i
\(9\) 0.0790435 − 0.243271i 0.0263478 − 0.0810903i
\(10\) −0.242510 + 0.642869i −0.0766885 + 0.203293i
\(11\) −0.389672 − 1.19929i −0.117490 − 0.361598i 0.874968 − 0.484181i \(-0.160882\pi\)
−0.992458 + 0.122583i \(0.960882\pi\)
\(12\) 3.00222 + 0.975481i 0.866666 + 0.281597i
\(13\) −2.15907 − 0.701526i −0.598820 − 0.194568i −0.00610566 − 0.999981i \(-0.501944\pi\)
−0.592714 + 0.805413i \(0.701944\pi\)
\(14\) −0.154386 − 0.475153i −0.0412615 − 0.126990i
\(15\) −3.70018 − 0.172428i −0.955382 − 0.0445206i
\(16\) 1.06376 − 3.27392i 0.265940 − 0.818480i
\(17\) −0.587785 + 0.809017i −0.142559 + 0.196215i
\(18\) 0.0785981i 0.0185257i
\(19\) −3.60186 − 2.61690i −0.826323 − 0.600359i 0.0921939 − 0.995741i \(-0.470612\pi\)
−0.918517 + 0.395383i \(0.870612\pi\)
\(20\) −0.198347 + 4.25639i −0.0443518 + 0.951758i
\(21\) 2.17904 − 1.58316i 0.475505 − 0.345475i
\(22\) 0.227753 + 0.313474i 0.0485570 + 0.0668330i
\(23\) −3.44716 + 1.12005i −0.718783 + 0.233547i −0.645495 − 0.763764i \(-0.723349\pi\)
−0.0732875 + 0.997311i \(0.523349\pi\)
\(24\) −1.98803 −0.405804
\(25\) −1.09616 − 4.87836i −0.219232 − 0.975673i
\(26\) 0.697573 0.136805
\(27\) −5.12946 + 1.66666i −0.987164 + 0.320749i
\(28\) −1.82115 − 2.50659i −0.344164 − 0.473702i
\(29\) −0.213184 + 0.154887i −0.0395873 + 0.0287618i −0.607403 − 0.794394i \(-0.707789\pi\)
0.567816 + 0.823156i \(0.307789\pi\)
\(30\) 1.09770 − 0.300955i 0.200412 − 0.0549466i
\(31\) −6.37905 − 4.63465i −1.14571 − 0.832407i −0.157806 − 0.987470i \(-0.550442\pi\)
−0.987905 + 0.155063i \(0.950442\pi\)
\(32\) 3.45795i 0.611284i
\(33\) −1.22785 + 1.68998i −0.213740 + 0.294189i
\(34\) 0.0949534 − 0.292236i 0.0162844 − 0.0501181i
\(35\) 2.83865 + 2.27159i 0.479819 + 0.383969i
\(36\) 0.150624 + 0.463572i 0.0251040 + 0.0772621i
\(37\) −6.04607 − 1.96449i −0.993968 − 0.322960i −0.233515 − 0.972353i \(-0.575023\pi\)
−0.760453 + 0.649393i \(0.775023\pi\)
\(38\) 1.30108 + 0.422746i 0.211063 + 0.0685784i
\(39\) 1.16212 + 3.57665i 0.186089 + 0.572722i
\(40\) −0.709542 − 2.58798i −0.112188 − 0.409195i
\(41\) 2.73636 − 8.42164i 0.427347 − 1.31524i −0.473382 − 0.880857i \(-0.656967\pi\)
0.900729 − 0.434381i \(-0.143033\pi\)
\(42\) −0.486468 + 0.669566i −0.0750636 + 0.103316i
\(43\) − 0.812241i − 0.123866i −0.998080 − 0.0619328i \(-0.980274\pi\)
0.998080 − 0.0619328i \(-0.0197264\pi\)
\(44\) 1.94402 + 1.41242i 0.293073 + 0.212930i
\(45\) −0.314288 − 0.477877i −0.0468513 − 0.0712376i
\(46\) 0.901034 − 0.654639i 0.132850 − 0.0965213i
\(47\) 5.32740 + 7.33254i 0.777081 + 1.06956i 0.995598 + 0.0937274i \(0.0298782\pi\)
−0.218517 + 0.975833i \(0.570122\pi\)
\(48\) −5.42347 + 1.76219i −0.782810 + 0.254350i
\(49\) 4.35639 0.622341
\(50\) 0.783555 + 1.32155i 0.110811 + 0.186896i
\(51\) 1.65657 0.231966
\(52\) 4.11429 − 1.33681i 0.570550 − 0.185383i
\(53\) −5.64476 − 7.76935i −0.775368 − 1.06720i −0.995778 − 0.0917948i \(-0.970740\pi\)
0.220410 − 0.975407i \(-0.429260\pi\)
\(54\) 1.34076 − 0.974119i 0.182454 − 0.132561i
\(55\) −2.63822 − 0.995218i −0.355737 − 0.134195i
\(56\) 1.57859 + 1.14691i 0.210948 + 0.153263i
\(57\) 7.37526i 0.976877i
\(58\) 0.0475931 − 0.0655062i 0.00624927 − 0.00860139i
\(59\) 1.36411 − 4.19831i 0.177593 − 0.546574i −0.822150 − 0.569271i \(-0.807225\pi\)
0.999742 + 0.0226975i \(0.00722545\pi\)
\(60\) 5.89750 − 3.87865i 0.761364 − 0.500731i
\(61\) 2.24358 + 6.90501i 0.287260 + 0.884097i 0.985712 + 0.168439i \(0.0538727\pi\)
−0.698452 + 0.715657i \(0.746127\pi\)
\(62\) 2.30427 + 0.748701i 0.292642 + 0.0950852i
\(63\) 0.395538 + 0.128518i 0.0498332 + 0.0161918i
\(64\) 1.79918 + 5.53731i 0.224897 + 0.692163i
\(65\) −4.24124 + 2.78936i −0.526062 + 0.345978i
\(66\) 0.198352 − 0.610463i 0.0244154 − 0.0751428i
\(67\) −4.75656 + 6.54685i −0.581107 + 0.799825i −0.993816 − 0.111038i \(-0.964582\pi\)
0.412710 + 0.910863i \(0.364582\pi\)
\(68\) − 1.90558i − 0.231086i
\(69\) 4.85760 + 3.52925i 0.584786 + 0.424872i
\(70\) −1.04525 − 0.394302i −0.124932 − 0.0471281i
\(71\) 1.46562 − 1.06483i 0.173937 − 0.126372i −0.497411 − 0.867515i \(-0.665716\pi\)
0.671348 + 0.741143i \(0.265716\pi\)
\(72\) −0.180433 − 0.248345i −0.0212642 − 0.0292677i
\(73\) 15.3611 − 4.99113i 1.79788 − 0.584167i 0.798051 − 0.602590i \(-0.205864\pi\)
0.999830 + 0.0184224i \(0.00586437\pi\)
\(74\) 1.95342 0.227080
\(75\) −5.47060 + 6.21915i −0.631690 + 0.718125i
\(76\) 8.48392 0.973172
\(77\) 1.94994 − 0.633574i 0.222216 − 0.0722025i
\(78\) −0.679230 − 0.934880i −0.0769076 − 0.105854i
\(79\) 13.2522 − 9.62832i 1.49099 − 1.08327i 0.517192 − 0.855869i \(-0.326977\pi\)
0.973801 − 0.227401i \(-0.0730228\pi\)
\(80\) −4.22966 − 6.43123i −0.472891 − 0.719033i
\(81\) 6.60740 + 4.80056i 0.734156 + 0.533396i
\(82\) 2.72094i 0.300477i
\(83\) −1.68587 + 2.32040i −0.185048 + 0.254696i −0.891455 − 0.453109i \(-0.850315\pi\)
0.706407 + 0.707806i \(0.250315\pi\)
\(84\) −1.58605 + 4.88137i −0.173052 + 0.532601i
\(85\) 0.591241 + 2.15649i 0.0641291 + 0.233904i
\(86\) 0.0771250 + 0.237366i 0.00831660 + 0.0255959i
\(87\) 0.415156 + 0.134893i 0.0445094 + 0.0144620i
\(88\) −1.43925 − 0.467641i −0.153425 − 0.0498507i
\(89\) 4.18423 + 12.8777i 0.443527 + 1.36504i 0.884091 + 0.467315i \(0.154779\pi\)
−0.440564 + 0.897721i \(0.645221\pi\)
\(90\) 0.137222 + 0.109810i 0.0144645 + 0.0115750i
\(91\) 1.14062 − 3.51048i 0.119570 − 0.367998i
\(92\) 4.05977 − 5.58780i 0.423260 − 0.582568i
\(93\) 13.0619i 1.35446i
\(94\) −2.25311 − 1.63698i −0.232391 − 0.168842i
\(95\) −9.60098 + 2.63229i −0.985040 + 0.270067i
\(96\) 4.63431 − 3.36702i 0.472987 − 0.343645i
\(97\) −4.19887 − 5.77924i −0.426330 − 0.586793i 0.540776 − 0.841167i \(-0.318131\pi\)
−0.967106 + 0.254373i \(0.918131\pi\)
\(98\) −1.27310 + 0.413654i −0.128602 + 0.0417854i
\(99\) −0.322552 −0.0324177
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.r.a.69.18 ✓ 160
25.4 even 10 inner 425.2.r.a.154.18 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
425.2.r.a.69.18 ✓ 160 1.1 even 1 trivial
425.2.r.a.154.18 yes 160 25.4 even 10 inner