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Properties

Label 790.2.p.a.89.8
Level $790$
Weight $2$
Character 790.89
Analytic conductor $6.308$
Analytic rank $0$
Dimension $480$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [790,2,Mod(89,790)] 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("790.89"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(790, base_ring=CyclotomicField(26)) chi = DirichletCharacter(H, H._module([13, 22])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 790 = 2 \cdot 5 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 790.p (of order \(26\), degree \(12\), 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: \(6.30818175968\)
Analytic rank: \(0\)
Dimension: \(480\)
Relative dimension: \(40\) over \(\Q(\zeta_{26})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{26}]$

Embedding invariants

Embedding label 89.8
Character \(\chi\) \(=\) 790.89
Dual form 790.2.p.a.719.8

$q$-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+(-0.935016 + 0.354605i) q^{2} +(-0.554697 - 1.05689i) q^{3} +(0.748511 - 0.663123i) q^{4} +(1.22445 - 1.87102i) q^{5} +(0.893428 + 0.791508i) q^{6} +(1.12968 + 2.15243i) q^{7} +(-0.464723 + 0.885456i) q^{8} +(0.894874 - 1.29645i) q^{9} +(-0.481405 + 2.18363i) q^{10} +(-0.327636 + 2.69833i) q^{11} +(-1.11604 - 0.423259i) q^{12} +(3.61771 + 4.08355i) q^{13} +(-1.81954 - 1.61197i) q^{14} +(-2.65666 - 0.256252i) q^{15} +(0.120537 - 0.992709i) q^{16} +(1.65469 + 1.86776i) q^{17} +(-0.376994 + 1.52953i) q^{18} +(1.46180 + 0.360301i) q^{19} +(-0.324205 - 2.21244i) q^{20} +(1.64824 - 2.38789i) q^{21} +(-0.650495 - 2.63916i) q^{22} +9.36558i q^{23} +1.19361 q^{24} +(-2.00146 - 4.58194i) q^{25} +(-4.83066 - 2.53533i) q^{26} +(-5.42130 - 0.658264i) q^{27} +(2.27291 + 0.862000i) q^{28} +(4.45921 + 6.46028i) q^{29} +(2.57489 - 0.702464i) q^{30} +(-2.36955 - 6.24798i) q^{31} +(0.239316 + 0.970942i) q^{32} +(3.03357 - 1.15048i) q^{33} +(-2.20948 - 1.15962i) q^{34} +(5.41049 + 0.521877i) q^{35} +(-0.189882 - 1.56382i) q^{36} +(1.55446 - 6.30669i) q^{37} +(-1.49457 + 0.181474i) q^{38} +(2.30912 - 6.08864i) q^{39} +(1.08768 + 1.95370i) q^{40} +(-0.147554 - 1.21521i) q^{41} +(-0.694376 + 2.81720i) q^{42} +(-1.58225 + 0.192120i) q^{43} +(1.54408 + 2.23699i) q^{44} +(-1.32996 - 3.26176i) q^{45} +(-3.32108 - 8.75697i) q^{46} +(-0.245517 - 0.996103i) q^{47} +(-1.11604 + 0.423259i) q^{48} +(0.619675 - 0.897754i) q^{49} +(3.49617 + 3.57446i) q^{50} +(1.05616 - 2.78486i) q^{51} +(5.41578 + 0.657595i) q^{52} +(-0.296774 + 0.565455i) q^{53} +(5.30242 - 1.30693i) q^{54} +(4.64746 + 3.91698i) q^{55} -2.43087 q^{56} +(-0.430058 - 1.74481i) q^{57} +(-6.46028 - 4.45921i) q^{58} +(0.965118 + 0.855020i) q^{59} +(-2.15846 + 1.56988i) q^{60} +(6.70639 + 1.65298i) q^{61} +(4.43113 + 5.00171i) q^{62} +(3.80144 + 0.461578i) q^{63} +(-0.568065 - 0.822984i) q^{64} +(12.0701 - 1.76872i) q^{65} +(-2.42847 + 2.15144i) q^{66} +(3.27685 + 1.24274i) q^{67} +(2.47711 + 0.300775i) q^{68} +(9.89835 - 5.19506i) q^{69} +(-5.24395 + 1.43062i) q^{70} +(4.44353 + 2.33215i) q^{71} +(0.732079 + 1.39486i) q^{72} +(-0.950125 + 1.07247i) q^{73} +(0.782939 + 6.44808i) q^{74} +(-3.73239 + 4.65690i) q^{75} +(1.33310 - 0.699663i) q^{76} +(-6.17809 + 2.34304i) q^{77} +6.51180i q^{78} +(6.86371 - 5.64708i) q^{79} +(-1.70979 - 1.44105i) q^{80} +(0.635635 + 1.67603i) q^{81} +(0.568886 + 1.08392i) q^{82} +(-8.21549 - 9.27337i) q^{83} +(-0.349738 - 2.88035i) q^{84} +(5.52070 - 0.808990i) q^{85} +(1.41131 - 0.740711i) q^{86} +(4.35427 - 8.29637i) q^{87} +(-2.23699 - 1.54408i) q^{88} +(-7.71418 + 4.04872i) q^{89} +(2.40017 + 2.57819i) q^{90} +(-4.70269 + 12.4000i) q^{91} +(6.21053 + 7.01024i) q^{92} +(-5.28903 + 5.97008i) q^{93} +(0.582786 + 0.844311i) q^{94} +(2.46403 - 2.29389i) q^{95} +(0.893428 - 0.791508i) q^{96} +(1.19489 - 4.84785i) q^{97} +(-0.261058 + 1.05915i) q^{98} +(3.20505 + 2.83943i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 480 q + 40 q^{4} + 4 q^{6} + 48 q^{9} - 4 q^{10} - 8 q^{11} - 8 q^{14} - 8 q^{15} - 40 q^{16} + 16 q^{19} + 52 q^{21} + 48 q^{24} - 14 q^{25} - 20 q^{29} + 20 q^{30} - 68 q^{31} + 70 q^{35} - 48 q^{36}+ \cdots + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/790\mathbb{Z}\right)^\times\).

\(n\) \(161\) \(317\)
\(\chi(n)\) \(e\left(\frac{11}{13}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.935016 + 0.354605i −0.661156 + 0.250744i
\(3\) −0.554697 − 1.05689i −0.320254 − 0.610194i 0.671106 − 0.741362i \(-0.265820\pi\)
−0.991360 + 0.131168i \(0.958127\pi\)
\(4\) 0.748511 − 0.663123i 0.374255 − 0.331561i
\(5\) 1.22445 − 1.87102i 0.547590 − 0.836747i
\(6\) 0.893428 + 0.791508i 0.364740 + 0.323132i
\(7\) 1.12968 + 2.15243i 0.426980 + 0.813543i 0.999982 − 0.00601139i \(-0.00191350\pi\)
−0.573002 + 0.819554i \(0.694221\pi\)
\(8\) −0.464723 + 0.885456i −0.164304 + 0.313056i
\(9\) 0.894874 − 1.29645i 0.298291 − 0.432149i
\(10\) −0.481405 + 2.18363i −0.152234 + 0.690525i
\(11\) −0.327636 + 2.69833i −0.0987861 + 0.813577i 0.856142 + 0.516740i \(0.172855\pi\)
−0.954928 + 0.296837i \(0.904068\pi\)
\(12\) −1.11604 − 0.423259i −0.322174 − 0.122184i
\(13\) 3.61771 + 4.08355i 1.00337 + 1.13257i 0.991163 + 0.132647i \(0.0423475\pi\)
0.0122079 + 0.999925i \(0.496114\pi\)
\(14\) −1.81954 − 1.61197i −0.486291 − 0.430816i
\(15\) −2.65666 − 0.256252i −0.685946 − 0.0661639i
\(16\) 0.120537 − 0.992709i 0.0301342 − 0.248177i
\(17\) 1.65469 + 1.86776i 0.401321 + 0.452998i 0.914080 − 0.405535i \(-0.132915\pi\)
−0.512758 + 0.858533i \(0.671376\pi\)
\(18\) −0.376994 + 1.52953i −0.0888584 + 0.360513i
\(19\) 1.46180 + 0.360301i 0.335360 + 0.0826588i 0.403400 − 0.915024i \(-0.367828\pi\)
−0.0680400 + 0.997683i \(0.521675\pi\)
\(20\) −0.324205 − 2.21244i −0.0724946 − 0.494717i
\(21\) 1.64824 − 2.38789i 0.359676 − 0.521081i
\(22\) −0.650495 − 2.63916i −0.138686 − 0.562671i
\(23\) 9.36558i 1.95286i 0.215839 + 0.976429i \(0.430751\pi\)
−0.215839 + 0.976429i \(0.569249\pi\)
\(24\) 1.19361 0.243644
\(25\) −2.00146 − 4.58194i −0.400291 − 0.916388i
\(26\) −4.83066 − 2.53533i −0.947370 − 0.497218i
\(27\) −5.42130 − 0.658264i −1.04333 − 0.126683i
\(28\) 2.27291 + 0.862000i 0.429539 + 0.162903i
\(29\) 4.45921 + 6.46028i 0.828054 + 1.19964i 0.977513 + 0.210876i \(0.0676315\pi\)
−0.149459 + 0.988768i \(0.547753\pi\)
\(30\) 2.57489 − 0.702464i 0.470108 − 0.128252i
\(31\) −2.36955 − 6.24798i −0.425583 − 1.12217i −0.960748 − 0.277423i \(-0.910520\pi\)
0.535164 − 0.844748i \(-0.320250\pi\)
\(32\) 0.239316 + 0.970942i 0.0423054 + 0.171640i
\(33\) 3.03357 − 1.15048i 0.528076 − 0.200273i
\(34\) −2.20948 − 1.15962i −0.378922 − 0.198874i
\(35\) 5.41049 + 0.521877i 0.914539 + 0.0882133i
\(36\) −0.189882 − 1.56382i −0.0316469 − 0.260636i
\(37\) 1.55446 − 6.30669i 0.255552 − 1.03681i −0.693188 − 0.720757i \(-0.743795\pi\)
0.948740 − 0.316058i \(-0.102359\pi\)
\(38\) −1.49457 + 0.181474i −0.242451 + 0.0294389i
\(39\) 2.30912 − 6.08864i 0.369754 − 0.974962i
\(40\) 1.08768 + 1.95370i 0.171977 + 0.308907i
\(41\) −0.147554 − 1.21521i −0.0230440 − 0.189785i 0.976641 − 0.214877i \(-0.0689351\pi\)
−0.999685 + 0.0250925i \(0.992012\pi\)
\(42\) −0.694376 + 2.81720i −0.107145 + 0.434703i
\(43\) −1.58225 + 0.192120i −0.241292 + 0.0292981i −0.240289 − 0.970701i \(-0.577242\pi\)
−0.00100256 + 0.999999i \(0.500319\pi\)
\(44\) 1.54408 + 2.23699i 0.232779 + 0.337239i
\(45\) −1.32996 − 3.26176i −0.198259 − 0.486235i
\(46\) −3.32108 − 8.75697i −0.489666 − 1.29114i
\(47\) −0.245517 − 0.996103i −0.0358124 − 0.145297i 0.950314 − 0.311293i \(-0.100762\pi\)
−0.986126 + 0.165997i \(0.946916\pi\)
\(48\) −1.11604 + 0.423259i −0.161087 + 0.0610922i
\(49\) 0.619675 − 0.897754i 0.0885250 − 0.128251i
\(50\) 3.49617 + 3.57446i 0.494433 + 0.505505i
\(51\) 1.05616 − 2.78486i 0.147892 − 0.389958i
\(52\) 5.41578 + 0.657595i 0.751034 + 0.0911921i
\(53\) −0.296774 + 0.565455i −0.0407650 + 0.0776713i −0.904986 − 0.425440i \(-0.860119\pi\)
0.864221 + 0.503112i \(0.167812\pi\)
\(54\) 5.30242 − 1.30693i 0.721568 − 0.177851i
\(55\) 4.64746 + 3.91698i 0.626664 + 0.528165i
\(56\) −2.43087 −0.324839
\(57\) −0.430058 − 1.74481i −0.0569626 − 0.231106i
\(58\) −6.46028 − 4.45921i −0.848276 − 0.585523i
\(59\) 0.965118 + 0.855020i 0.125648 + 0.111314i 0.723607 − 0.690213i \(-0.242483\pi\)
−0.597959 + 0.801527i \(0.704021\pi\)
\(60\) −2.15846 + 1.56988i −0.278656 + 0.202671i
\(61\) 6.70639 + 1.65298i 0.858666 + 0.211642i 0.643981 − 0.765041i \(-0.277281\pi\)
0.214685 + 0.976683i \(0.431128\pi\)
\(62\) 4.43113 + 5.00171i 0.562754 + 0.635218i
\(63\) 3.80144 + 0.461578i 0.478936 + 0.0581534i
\(64\) −0.568065 − 0.822984i −0.0710081 − 0.102873i
\(65\) 12.0701 − 1.76872i 1.49711 − 0.219383i
\(66\) −2.42847 + 2.15144i −0.298924 + 0.264823i
\(67\) 3.27685 + 1.24274i 0.400330 + 0.151825i 0.546549 − 0.837427i \(-0.315941\pi\)
−0.146219 + 0.989252i \(0.546710\pi\)
\(68\) 2.47711 + 0.300775i 0.300393 + 0.0364743i
\(69\) 9.89835 − 5.19506i 1.19162 − 0.625411i
\(70\) −5.24395 + 1.43062i −0.626772 + 0.170992i
\(71\) 4.44353 + 2.33215i 0.527350 + 0.276775i 0.707315 − 0.706898i \(-0.249906\pi\)
−0.179965 + 0.983673i \(0.557598\pi\)
\(72\) 0.732079 + 1.39486i 0.0862764 + 0.164386i
\(73\) −0.950125 + 1.07247i −0.111204 + 0.125523i −0.801477 − 0.598026i \(-0.795952\pi\)
0.690273 + 0.723549i \(0.257490\pi\)
\(74\) 0.782939 + 6.44808i 0.0910148 + 0.749574i
\(75\) −3.73239 + 4.65690i −0.430979 + 0.537732i
\(76\) 1.33310 − 0.699663i 0.152917 − 0.0802568i
\(77\) −6.17809 + 2.34304i −0.704059 + 0.267014i
\(78\) 6.51180i 0.737316i
\(79\) 6.86371 − 5.64708i 0.772228 − 0.635346i
\(80\) −1.70979 − 1.44105i −0.191160 − 0.161114i
\(81\) 0.635635 + 1.67603i 0.0706262 + 0.186226i
\(82\) 0.568886 + 1.08392i 0.0628229 + 0.119699i
\(83\) −8.21549 − 9.27337i −0.901767 − 1.01788i −0.999742 − 0.0226956i \(-0.992775\pi\)
0.0979755 − 0.995189i \(-0.468763\pi\)
\(84\) −0.349738 − 2.88035i −0.0381596 − 0.314272i
\(85\) 5.52070 − 0.808990i 0.598804 − 0.0877473i
\(86\) 1.41131 − 0.740711i 0.152185 − 0.0798729i
\(87\) 4.35427 − 8.29637i 0.466827 − 0.889465i
\(88\) −2.23699 − 1.54408i −0.238464 − 0.164600i
\(89\) −7.71418 + 4.04872i −0.817702 + 0.429163i −0.821067 − 0.570831i \(-0.806621\pi\)
0.00336550 + 0.999994i \(0.498929\pi\)
\(90\) 2.40017 + 2.57819i 0.253000 + 0.271765i
\(91\) −4.70269 + 12.4000i −0.492976 + 1.29987i
\(92\) 6.21053 + 7.01024i 0.647492 + 0.730868i
\(93\) −5.28903 + 5.97008i −0.548447 + 0.619069i
\(94\) 0.582786 + 0.844311i 0.0601097 + 0.0870840i
\(95\) 2.46403 − 2.29389i 0.252804 − 0.235348i
\(96\) 0.893428 − 0.791508i 0.0911851 − 0.0807829i
\(97\) 1.19489 − 4.84785i 0.121322 − 0.492225i −0.878571 − 0.477612i \(-0.841502\pi\)
0.999893 − 0.0146124i \(-0.00465143\pi\)
\(98\) −0.261058 + 1.05915i −0.0263709 + 0.106991i
\(99\) 3.20505 + 2.83943i 0.322120 + 0.285373i
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 790.2.p.a.89.8 ✓ 480
5.4 even 2 inner 790.2.p.a.89.33 yes 480
79.8 even 13 inner 790.2.p.a.719.33 yes 480
395.324 even 26 inner 790.2.p.a.719.8 yes 480
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
790.2.p.a.89.8 ✓ 480 1.1 even 1 trivial
790.2.p.a.89.33 yes 480 5.4 even 2 inner
790.2.p.a.719.8 yes 480 395.324 even 26 inner
790.2.p.a.719.33 yes 480 79.8 even 13 inner