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Properties

Label 790.2.m.c.21.1
Level $790$
Weight $2$
Character 790.21
Analytic conductor $6.308$
Analytic rank $0$
Dimension $48$
Inner twists $2$

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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(21,790)] 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("790.21"); 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([0, 18])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 790 = 2 \cdot 5 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 790.m (of order \(13\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,4,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.30818175968\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(4\) over \(\Q(\zeta_{13})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{13}]$

Embedding invariants

Embedding label 21.1
Character \(\chi\) \(=\) 790.21
Dual form 790.2.m.c.301.1

$q$-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+(0.748511 - 0.663123i) q^{2} +(-0.965499 + 1.39877i) q^{3} +(0.120537 - 0.992709i) q^{4} +(-0.970942 + 0.239316i) q^{5} +(0.204868 + 1.68724i) q^{6} +(-1.42706 + 2.06746i) q^{7} +(-0.568065 - 0.822984i) q^{8} +(0.0394546 + 0.104033i) q^{9} +(-0.568065 + 0.822984i) q^{10} +(-3.35988 - 0.828136i) q^{11} +(1.27219 + 1.12706i) q^{12} +(0.468909 - 3.86181i) q^{13} +(0.302806 + 2.49383i) q^{14} +(0.602697 - 1.58918i) q^{15} +(-0.970942 - 0.239316i) q^{16} +(0.639072 - 5.26323i) q^{17} +(0.0985189 + 0.0517067i) q^{18} +(-4.40630 - 2.31260i) q^{19} +(0.120537 + 0.992709i) q^{20} +(-1.51406 - 3.99226i) q^{21} +(-3.06406 + 1.60814i) q^{22} +1.61529 q^{23} +1.69963 q^{24} +(0.885456 - 0.464723i) q^{25} +(-2.20987 - 3.20155i) q^{26} +(-5.13433 - 1.26550i) q^{27} +(1.88037 + 1.66586i) q^{28} +(-0.376482 + 0.992702i) q^{29} +(-0.602697 - 1.58918i) q^{30} +(2.94386 - 2.60803i) q^{31} +(-0.885456 + 0.464723i) q^{32} +(4.40233 - 3.90012i) q^{33} +(-3.01182 - 4.36337i) q^{34} +(0.890821 - 2.34890i) q^{35} +(0.108030 - 0.0266271i) q^{36} +(-7.94810 - 4.17148i) q^{37} +(-4.83170 + 1.19091i) q^{38} +(4.94904 + 4.38447i) q^{39} +(0.748511 + 0.663123i) q^{40} +(1.00279 - 0.247165i) q^{41} +(-3.78065 - 1.98424i) q^{42} +(-7.82388 + 1.92841i) q^{43} +(-1.22709 + 3.23556i) q^{44} +(-0.0632048 - 0.0915680i) q^{45} +(1.20907 - 1.07114i) q^{46} +(-5.00522 + 2.62694i) q^{47} +(1.27219 - 1.12706i) q^{48} +(0.244360 + 0.644323i) q^{49} +(0.354605 - 0.935016i) q^{50} +(6.74501 + 5.97556i) q^{51} +(-3.77713 - 0.930980i) q^{52} +(-0.442838 - 0.641562i) q^{53} +(-4.68228 + 2.45745i) q^{54} +3.46043 q^{55} +2.51215 q^{56} +(7.48907 - 3.93057i) q^{57} +(0.376482 + 0.992702i) q^{58} +(1.29194 + 10.6401i) q^{59} +(-1.50495 - 0.789857i) q^{60} +(4.49612 + 2.35975i) q^{61} +(0.474065 - 3.90428i) q^{62} +(-0.271388 - 0.0668912i) q^{63} +(-0.354605 + 0.935016i) q^{64} +(0.468909 + 3.86181i) q^{65} +(0.708931 - 5.83857i) q^{66} +(-8.35759 - 7.40418i) q^{67} +(-5.14783 - 1.26883i) q^{68} +(-1.55956 + 2.25942i) q^{69} +(-0.890821 - 2.34890i) q^{70} +(-1.93552 - 2.80409i) q^{71} +(0.0632048 - 0.0915680i) q^{72} +(0.107846 + 0.888192i) q^{73} +(-8.71544 + 2.14816i) q^{74} +(-0.204868 + 1.68724i) q^{75} +(-2.82686 + 4.09542i) q^{76} +(6.50690 - 5.76461i) q^{77} +6.61185 q^{78} +(3.47636 - 8.18015i) q^{79} +1.00000 q^{80} +(6.47748 - 5.73855i) q^{81} +(0.586696 - 0.849976i) q^{82} +(-0.637875 + 5.25337i) q^{83} +(-4.14565 + 1.02181i) q^{84} +(0.639072 + 5.26323i) q^{85} +(-4.57748 + 6.63163i) q^{86} +(-1.02507 - 1.48506i) q^{87} +(1.22709 + 3.23556i) q^{88} +(-0.248768 + 0.360403i) q^{89} +(-0.108030 - 0.0266271i) q^{90} +(7.31497 + 6.48050i) q^{91} +(0.194702 - 1.60352i) q^{92} +(0.805734 + 6.63582i) q^{93} +(-2.00448 + 5.28537i) q^{94} +(4.83170 + 1.19091i) q^{95} +(0.204868 - 1.68724i) q^{96} +(-11.1076 - 5.82971i) q^{97} +(0.610171 + 0.320243i) q^{98} +(-0.0464090 - 0.382213i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 4 q^{2} - q^{3} - 4 q^{4} - 4 q^{5} + q^{6} - 9 q^{7} + 4 q^{8} - 21 q^{9} + 4 q^{10} + 2 q^{11} - q^{12} + 17 q^{13} - 4 q^{14} - 14 q^{15} - 4 q^{16} - 6 q^{17} + 21 q^{18} - 34 q^{19} - 4 q^{20}+ \cdots - 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(161\) \(317\)
\(\chi(n)\) \(e\left(\frac{9}{13}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.748511 − 0.663123i 0.529277 − 0.468899i
\(3\) −0.965499 + 1.39877i −0.557431 + 0.807578i −0.995833 − 0.0912009i \(-0.970929\pi\)
0.438401 + 0.898779i \(0.355545\pi\)
\(4\) 0.120537 − 0.992709i 0.0602683 − 0.496354i
\(5\) −0.970942 + 0.239316i −0.434218 + 0.107025i
\(6\) 0.204868 + 1.68724i 0.0836368 + 0.688811i
\(7\) −1.42706 + 2.06746i −0.539379 + 0.781426i −0.994014 − 0.109252i \(-0.965154\pi\)
0.454635 + 0.890678i \(0.349770\pi\)
\(8\) −0.568065 − 0.822984i −0.200841 − 0.290969i
\(9\) 0.0394546 + 0.104033i 0.0131515 + 0.0346777i
\(10\) −0.568065 + 0.822984i −0.179638 + 0.260250i
\(11\) −3.35988 − 0.828136i −1.01304 − 0.249692i −0.302351 − 0.953197i \(-0.597771\pi\)
−0.710691 + 0.703504i \(0.751618\pi\)
\(12\) 1.27219 + 1.12706i 0.367250 + 0.325355i
\(13\) 0.468909 − 3.86181i 0.130052 − 1.07107i −0.770840 − 0.637029i \(-0.780163\pi\)
0.900892 − 0.434044i \(-0.142914\pi\)
\(14\) 0.302806 + 2.49383i 0.0809284 + 0.666505i
\(15\) 0.602697 − 1.58918i 0.155616 − 0.410325i
\(16\) −0.970942 − 0.239316i −0.242735 − 0.0598289i
\(17\) 0.639072 − 5.26323i 0.154998 − 1.27652i −0.682826 − 0.730581i \(-0.739249\pi\)
0.837824 − 0.545941i \(-0.183828\pi\)
\(18\) 0.0985189 + 0.0517067i 0.0232211 + 0.0121874i
\(19\) −4.40630 − 2.31260i −1.01087 − 0.530548i −0.123865 − 0.992299i \(-0.539529\pi\)
−0.887009 + 0.461751i \(0.847221\pi\)
\(20\) 0.120537 + 0.992709i 0.0269528 + 0.221976i
\(21\) −1.51406 − 3.99226i −0.330396 − 0.871182i
\(22\) −3.06406 + 1.60814i −0.653260 + 0.342858i
\(23\) 1.61529 0.336812 0.168406 − 0.985718i \(-0.446138\pi\)
0.168406 + 0.985718i \(0.446138\pi\)
\(24\) 1.69963 0.346935
\(25\) 0.885456 − 0.464723i 0.177091 − 0.0929446i
\(26\) −2.20987 − 3.20155i −0.433391 − 0.627876i
\(27\) −5.13433 − 1.26550i −0.988103 − 0.243545i
\(28\) 1.88037 + 1.66586i 0.355357 + 0.314819i
\(29\) −0.376482 + 0.992702i −0.0699110 + 0.184340i −0.965418 − 0.260708i \(-0.916044\pi\)
0.895507 + 0.445048i \(0.146813\pi\)
\(30\) −0.602697 − 1.58918i −0.110037 − 0.290143i
\(31\) 2.94386 − 2.60803i 0.528732 − 0.468416i −0.356019 − 0.934479i \(-0.615866\pi\)
0.884752 + 0.466063i \(0.154328\pi\)
\(32\) −0.885456 + 0.464723i −0.156528 + 0.0821522i
\(33\) 4.40233 − 3.90012i 0.766347 − 0.678925i
\(34\) −3.01182 − 4.36337i −0.516522 − 0.748312i
\(35\) 0.890821 − 2.34890i 0.150576 − 0.397037i
\(36\) 0.108030 − 0.0266271i 0.0180051 − 0.00443785i
\(37\) −7.94810 − 4.17148i −1.30666 − 0.685788i −0.339656 − 0.940550i \(-0.610311\pi\)
−0.967004 + 0.254762i \(0.918003\pi\)
\(38\) −4.83170 + 1.19091i −0.783806 + 0.193191i
\(39\) 4.94904 + 4.38447i 0.792481 + 0.702077i
\(40\) 0.748511 + 0.663123i 0.118350 + 0.104849i
\(41\) 1.00279 − 0.247165i 0.156609 − 0.0386006i −0.160232 − 0.987079i \(-0.551224\pi\)
0.316841 + 0.948479i \(0.397378\pi\)
\(42\) −3.78065 − 1.98424i −0.583367 − 0.306175i
\(43\) −7.82388 + 1.92841i −1.19313 + 0.294080i −0.785380 − 0.619014i \(-0.787533\pi\)
−0.407750 + 0.913094i \(0.633686\pi\)
\(44\) −1.22709 + 3.23556i −0.184990 + 0.487779i
\(45\) −0.0632048 − 0.0915680i −0.00942202 − 0.0136502i
\(46\) 1.20907 − 1.07114i 0.178267 − 0.157931i
\(47\) −5.00522 + 2.62694i −0.730086 + 0.383179i −0.788436 − 0.615117i \(-0.789109\pi\)
0.0583493 + 0.998296i \(0.481416\pi\)
\(48\) 1.27219 − 1.12706i 0.183625 − 0.162677i
\(49\) 0.244360 + 0.644323i 0.0349085 + 0.0920462i
\(50\) 0.354605 − 0.935016i 0.0501487 − 0.132231i
\(51\) 6.74501 + 5.97556i 0.944491 + 0.836746i
\(52\) −3.77713 − 0.930980i −0.523794 − 0.129104i
\(53\) −0.442838 − 0.641562i −0.0608285 − 0.0881253i 0.791389 − 0.611313i \(-0.209358\pi\)
−0.852218 + 0.523187i \(0.824743\pi\)
\(54\) −4.68228 + 2.45745i −0.637178 + 0.334417i
\(55\) 3.46043 0.466605
\(56\) 2.51215 0.335700
\(57\) 7.48907 − 3.93057i 0.991952 − 0.520616i
\(58\) 0.376482 + 0.992702i 0.0494345 + 0.130348i
\(59\) 1.29194 + 10.6401i 0.168196 + 1.38522i 0.795268 + 0.606258i \(0.207330\pi\)
−0.627072 + 0.778962i \(0.715747\pi\)
\(60\) −1.50495 − 0.789857i −0.194288 − 0.101970i
\(61\) 4.49612 + 2.35975i 0.575670 + 0.302135i 0.727319 − 0.686300i \(-0.240766\pi\)
−0.151649 + 0.988434i \(0.548458\pi\)
\(62\) 0.474065 − 3.90428i 0.0602063 − 0.495844i
\(63\) −0.271388 − 0.0668912i −0.0341917 − 0.00842750i
\(64\) −0.354605 + 0.935016i −0.0443256 + 0.116877i
\(65\) 0.468909 + 3.86181i 0.0581610 + 0.478999i
\(66\) 0.708931 − 5.83857i 0.0872634 − 0.718678i
\(67\) −8.35759 − 7.40418i −1.02104 − 0.904564i −0.0255750 − 0.999673i \(-0.508142\pi\)
−0.995467 + 0.0951087i \(0.969680\pi\)
\(68\) −5.14783 − 1.26883i −0.624266 − 0.153868i
\(69\) −1.55956 + 2.25942i −0.187750 + 0.272002i
\(70\) −0.890821 − 2.34890i −0.106473 − 0.280747i
\(71\) −1.93552 − 2.80409i −0.229704 − 0.332784i 0.691136 − 0.722725i \(-0.257111\pi\)
−0.920840 + 0.389941i \(0.872495\pi\)
\(72\) 0.0632048 − 0.0915680i 0.00744876 − 0.0107914i
\(73\) 0.107846 + 0.888192i 0.0126224 + 0.103955i 0.997658 − 0.0683939i \(-0.0217874\pi\)
−0.985036 + 0.172349i \(0.944864\pi\)
\(74\) −8.71544 + 2.14816i −1.01315 + 0.249719i
\(75\) −0.204868 + 1.68724i −0.0236561 + 0.194825i
\(76\) −2.82686 + 4.09542i −0.324263 + 0.469777i
\(77\) 6.50690 − 5.76461i 0.741530 − 0.656938i
\(78\) 6.61185 0.748645
\(79\) 3.47636 − 8.18015i 0.391121 − 0.920339i
\(80\) 1.00000 0.111803
\(81\) 6.47748 − 5.73855i 0.719721 − 0.637617i
\(82\) 0.586696 − 0.849976i 0.0647897 − 0.0938641i
\(83\) −0.637875 + 5.25337i −0.0700159 + 0.576633i 0.914910 + 0.403657i \(0.132261\pi\)
−0.984926 + 0.172975i \(0.944662\pi\)
\(84\) −4.14565 + 1.02181i −0.452328 + 0.111489i
\(85\) 0.639072 + 5.26323i 0.0693171 + 0.570878i
\(86\) −4.57748 + 6.63163i −0.493603 + 0.715107i
\(87\) −1.02507 − 1.48506i −0.109899 − 0.159215i
\(88\) 1.22709 + 3.23556i 0.130808 + 0.344912i
\(89\) −0.248768 + 0.360403i −0.0263694 + 0.0382026i −0.835947 − 0.548810i \(-0.815081\pi\)
0.809577 + 0.587013i \(0.199696\pi\)
\(90\) −0.108030 − 0.0266271i −0.0113874 − 0.00280674i
\(91\) 7.31497 + 6.48050i 0.766817 + 0.679341i
\(92\) 0.194702 − 1.60352i 0.0202991 − 0.167178i
\(93\) 0.805734 + 6.63582i 0.0835507 + 0.688102i
\(94\) −2.00448 + 5.28537i −0.206746 + 0.545144i
\(95\) 4.83170 + 1.19091i 0.495722 + 0.122185i
\(96\) 0.204868 − 1.68724i 0.0209092 − 0.172203i
\(97\) −11.1076 − 5.82971i −1.12780 − 0.591918i −0.205606 − 0.978635i \(-0.565917\pi\)
−0.922198 + 0.386717i \(0.873609\pi\)
\(98\) 0.610171 + 0.320243i 0.0616366 + 0.0323494i
\(99\) −0.0464090 − 0.382213i −0.00466428 − 0.0384138i
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.m.c.21.1 ✓ 48
79.64 even 13 inner 790.2.m.c.301.1 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
790.2.m.c.21.1 ✓ 48 1.1 even 1 trivial
790.2.m.c.301.1 yes 48 79.64 even 13 inner