↑

Properties

Label 790.2.k.a.103.3
Level $790$
Weight $2$
Character 790.103
Analytic conductor $6.308$
Analytic rank $0$
Dimension $160$
Inner twists $4$

Related objects

Downloads

Learn more

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(103,790)] 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("790.103"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(790, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([9, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 790 = 2 \cdot 5 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 790.k (of order \(12\), 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: \(6.30818175968\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(40\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 103.3
Character \(\chi\) \(=\) 790.103
Dual form 790.2.k.a.767.3

$q$-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+(-0.258819 - 0.965926i) q^{2} +(-0.602390 - 2.24815i) q^{3} +(-0.866025 + 0.500000i) q^{4} +(-0.404988 - 2.19909i) q^{5} +(-2.01563 + 1.16373i) q^{6} +(-0.954993 + 3.56408i) q^{7} +(0.707107 + 0.707107i) q^{8} +(-2.09322 + 1.20852i) q^{9} +(-2.01934 + 0.960354i) q^{10} +(2.43554 + 4.21848i) q^{11} +(1.64576 + 1.64576i) q^{12} +(-1.45654 + 0.390279i) q^{13} +3.68981 q^{14} +(-4.69991 + 2.23518i) q^{15} +(0.500000 - 0.866025i) q^{16} +(1.48372 + 1.48372i) q^{17} +(1.70911 + 1.70911i) q^{18} +(-6.49688 + 3.75098i) q^{19} +(1.45027 + 1.70197i) q^{20} +8.58787 q^{21} +(3.44437 - 3.44437i) q^{22} +(-1.54189 + 5.75442i) q^{23} +(1.16373 - 2.01563i) q^{24} +(-4.67197 + 1.78121i) q^{25} +(0.753962 + 1.30590i) q^{26} +(-0.959404 - 0.959404i) q^{27} +(-0.954993 - 3.56408i) q^{28} +(0.424914 + 0.735972i) q^{29} +(3.37545 + 3.96126i) q^{30} +(3.16669 + 5.48488i) q^{31} +(-0.965926 - 0.258819i) q^{32} +(8.01662 - 8.01662i) q^{33} +(1.04915 - 1.81717i) q^{34} +(8.22449 + 0.656703i) q^{35} +(1.20852 - 2.09322i) q^{36} +(-2.66320 + 0.713604i) q^{37} +(5.30468 + 5.30468i) q^{38} +(1.75481 + 3.03942i) q^{39} +(1.26862 - 1.84136i) q^{40} -2.14439i q^{41} +(-2.22270 - 8.29524i) q^{42} +(-1.58066 - 5.89909i) q^{43} +(-4.21848 - 2.43554i) q^{44} +(3.50538 + 4.11374i) q^{45} +5.95742 q^{46} +(1.84157 - 6.87283i) q^{47} +(-2.24815 - 0.602390i) q^{48} +(-5.72850 - 3.30735i) q^{49} +(2.92971 + 4.05177i) q^{50} +(2.44184 - 4.22939i) q^{51} +(1.06626 - 1.06626i) q^{52} +(-12.9457 - 3.46878i) q^{53} +(-0.678401 + 1.17503i) q^{54} +(8.29044 - 7.06440i) q^{55} +(-3.19547 + 1.84491i) q^{56} +(12.3464 + 12.3464i) q^{57} +(0.600919 - 0.600919i) q^{58} +(-0.912055 + 1.57973i) q^{59} +(2.95266 - 4.28568i) q^{60} +4.66635i q^{61} +(4.47838 - 4.47838i) q^{62} +(-2.30826 - 8.61455i) q^{63} +1.00000i q^{64} +(1.44814 + 3.04501i) q^{65} +(-9.81832 - 5.66861i) q^{66} +(5.52343 - 5.52343i) q^{67} +(-2.02679 - 0.543078i) q^{68} +13.8656 q^{69} +(-1.49433 - 8.11422i) q^{70} +11.7597i q^{71} +(-2.33469 - 0.625577i) q^{72} +(2.57425 - 9.60724i) q^{73} +(1.37858 + 2.38776i) q^{74} +(6.81876 + 9.43030i) q^{75} +(3.75098 - 6.49688i) q^{76} +(-17.3609 + 4.65185i) q^{77} +(2.48168 - 2.48168i) q^{78} +(-8.56127 - 2.38845i) q^{79} +(-2.10696 - 0.748814i) q^{80} +(-5.20451 + 9.01448i) q^{81} +(-2.07132 + 0.555008i) q^{82} +(1.04188 - 3.88835i) q^{83} +(-7.43731 + 4.29393i) q^{84} +(2.66193 - 3.86371i) q^{85} +(-5.28898 + 3.05359i) q^{86} +(1.39861 - 1.39861i) q^{87} +(-1.26073 + 4.70510i) q^{88} +14.6803i q^{89} +(3.06631 - 4.45065i) q^{90} -5.56395i q^{91} +(-1.54189 - 5.75442i) q^{92} +(10.4232 - 10.4232i) q^{93} -7.11527 q^{94} +(10.8799 + 12.7681i) q^{95} +2.32745i q^{96} +(-3.03269 + 3.03269i) q^{97} +(-1.71201 + 6.38931i) q^{98} +(-10.1963 - 5.88681i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 160 q - 12 q^{7} - 8 q^{10} - 16 q^{11} + 80 q^{16} - 32 q^{18} + 48 q^{21} - 8 q^{22} - 12 q^{28} - 24 q^{31} - 36 q^{35} + 72 q^{36} - 36 q^{37} + 8 q^{38} + 20 q^{42} - 48 q^{43} - 4 q^{45} + 16 q^{46}+ \cdots + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(161\) \(317\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.258819 − 0.965926i −0.183013 − 0.683013i
\(3\) −0.602390 − 2.24815i −0.347790 − 1.29797i −0.889319 − 0.457287i \(-0.848821\pi\)
0.541529 − 0.840682i \(-0.317846\pi\)
\(4\) −0.866025 + 0.500000i −0.433013 + 0.250000i
\(5\) −0.404988 − 2.19909i −0.181116 − 0.983462i
\(6\) −2.01563 + 1.16373i −0.822879 + 0.475090i
\(7\) −0.954993 + 3.56408i −0.360954 + 1.34710i 0.511871 + 0.859062i \(0.328953\pi\)
−0.872824 + 0.488034i \(0.837714\pi\)
\(8\) 0.707107 + 0.707107i 0.250000 + 0.250000i
\(9\) −2.09322 + 1.20852i −0.697741 + 0.402841i
\(10\) −2.01934 + 0.960354i −0.638570 + 0.303691i
\(11\) 2.43554 + 4.21848i 0.734343 + 1.27192i 0.955011 + 0.296570i \(0.0958429\pi\)
−0.220668 + 0.975349i \(0.570824\pi\)
\(12\) 1.64576 + 1.64576i 0.475090 + 0.475090i
\(13\) −1.45654 + 0.390279i −0.403972 + 0.108244i −0.455083 − 0.890449i \(-0.650390\pi\)
0.0511109 + 0.998693i \(0.483724\pi\)
\(14\) 3.68981 0.986143
\(15\) −4.69991 + 2.23518i −1.21351 + 0.577121i
\(16\) 0.500000 − 0.866025i 0.125000 − 0.216506i
\(17\) 1.48372 + 1.48372i 0.359854 + 0.359854i 0.863759 − 0.503905i \(-0.168104\pi\)
−0.503905 + 0.863759i \(0.668104\pi\)
\(18\) 1.70911 + 1.70911i 0.402841 + 0.402841i
\(19\) −6.49688 + 3.75098i −1.49049 + 0.860533i −0.999941 − 0.0108825i \(-0.996536\pi\)
−0.490546 + 0.871415i \(0.663203\pi\)
\(20\) 1.45027 + 1.70197i 0.324291 + 0.380572i
\(21\) 8.58787 1.87403
\(22\) 3.44437 − 3.44437i 0.734343 − 0.734343i
\(23\) −1.54189 + 5.75442i −0.321507 + 1.19988i 0.596270 + 0.802784i \(0.296649\pi\)
−0.917777 + 0.397096i \(0.870018\pi\)
\(24\) 1.16373 − 2.01563i 0.237545 − 0.411440i
\(25\) −4.67197 + 1.78121i −0.934394 + 0.356241i
\(26\) 0.753962 + 1.30590i 0.147864 + 0.256108i
\(27\) −0.959404 − 0.959404i −0.184637 − 0.184637i
\(28\) −0.954993 − 3.56408i −0.180477 − 0.673548i
\(29\) 0.424914 + 0.735972i 0.0789045 + 0.136667i 0.902778 − 0.430108i \(-0.141524\pi\)
−0.823873 + 0.566774i \(0.808191\pi\)
\(30\) 3.37545 + 3.96126i 0.616269 + 0.723224i
\(31\) 3.16669 + 5.48488i 0.568755 + 0.985113i 0.996689 + 0.0813029i \(0.0259081\pi\)
−0.427934 + 0.903810i \(0.640759\pi\)
\(32\) −0.965926 − 0.258819i −0.170753 − 0.0457532i
\(33\) 8.01662 − 8.01662i 1.39552 − 1.39552i
\(34\) 1.04915 − 1.81717i 0.179927 − 0.311643i
\(35\) 8.22449 + 0.656703i 1.39019 + 0.111003i
\(36\) 1.20852 − 2.09322i 0.201420 − 0.348870i
\(37\) −2.66320 + 0.713604i −0.437828 + 0.117316i −0.470999 − 0.882134i \(-0.656106\pi\)
0.0331703 + 0.999450i \(0.489440\pi\)
\(38\) 5.30468 + 5.30468i 0.860533 + 0.860533i
\(39\) 1.75481 + 3.03942i 0.280995 + 0.486697i
\(40\) 1.26862 − 1.84136i 0.200586 − 0.291144i
\(41\) − 2.14439i − 0.334897i −0.985881 − 0.167448i \(-0.946447\pi\)
0.985881 − 0.167448i \(-0.0535527\pi\)
\(42\) −2.22270 − 8.29524i −0.342971 − 1.27998i
\(43\) −1.58066 − 5.89909i −0.241048 − 0.899602i −0.975329 − 0.220757i \(-0.929147\pi\)
0.734281 − 0.678845i \(-0.237519\pi\)
\(44\) −4.21848 − 2.43554i −0.635960 − 0.367171i
\(45\) 3.50538 + 4.11374i 0.522551 + 0.613240i
\(46\) 5.95742 0.878373
\(47\) 1.84157 − 6.87283i 0.268620 − 1.00250i −0.691377 − 0.722495i \(-0.742995\pi\)
0.959997 − 0.280010i \(-0.0903379\pi\)
\(48\) −2.24815 − 0.602390i −0.324492 − 0.0869474i
\(49\) −5.72850 − 3.30735i −0.818357 − 0.472479i
\(50\) 2.92971 + 4.05177i 0.414323 + 0.573006i
\(51\) 2.44184 − 4.22939i 0.341926 − 0.592233i
\(52\) 1.06626 − 1.06626i 0.147864 − 0.147864i
\(53\) −12.9457 − 3.46878i −1.77822 − 0.476473i −0.787965 − 0.615720i \(-0.788865\pi\)
−0.990258 + 0.139246i \(0.955532\pi\)
\(54\) −0.678401 + 1.17503i −0.0923187 + 0.159901i
\(55\) 8.29044 − 7.06440i 1.11788 − 0.952563i
\(56\) −3.19547 + 1.84491i −0.427013 + 0.246536i
\(57\) 12.3464 + 12.3464i 1.63532 + 1.63532i
\(58\) 0.600919 − 0.600919i 0.0789045 − 0.0789045i
\(59\) −0.912055 + 1.57973i −0.118739 + 0.205663i −0.919268 − 0.393631i \(-0.871219\pi\)
0.800529 + 0.599294i \(0.204552\pi\)
\(60\) 2.95266 − 4.28568i 0.381186 − 0.553279i
\(61\) 4.66635i 0.597465i 0.954337 + 0.298732i \(0.0965638\pi\)
−0.954337 + 0.298732i \(0.903436\pi\)
\(62\) 4.47838 − 4.47838i 0.568755 − 0.568755i
\(63\) −2.30826 − 8.61455i −0.290814 − 1.08533i
\(64\) 1.00000i 0.125000i
\(65\) 1.44814 + 3.04501i 0.179620 + 0.377687i
\(66\) −9.81832 − 5.66861i −1.20855 − 0.697758i
\(67\) 5.52343 − 5.52343i 0.674794 − 0.674794i −0.284023 − 0.958817i \(-0.591669\pi\)
0.958817 + 0.284023i \(0.0916693\pi\)
\(68\) −2.02679 − 0.543078i −0.245785 − 0.0658578i
\(69\) 13.8656 1.66922
\(70\) −1.49433 − 8.11422i −0.178606 − 0.969834i
\(71\) 11.7597i 1.39562i 0.716281 + 0.697812i \(0.245843\pi\)
−0.716281 + 0.697812i \(0.754157\pi\)
\(72\) −2.33469 − 0.625577i −0.275145 − 0.0737250i
\(73\) 2.57425 − 9.60724i 0.301293 − 1.12444i −0.634796 − 0.772680i \(-0.718916\pi\)
0.936089 − 0.351762i \(-0.114417\pi\)
\(74\) 1.37858 + 2.38776i 0.160256 + 0.277572i
\(75\) 6.81876 + 9.43030i 0.787363 + 1.08892i
\(76\) 3.75098 − 6.49688i 0.430266 − 0.745243i
\(77\) −17.3609 + 4.65185i −1.97846 + 0.530127i
\(78\) 2.48168 − 2.48168i 0.280995 − 0.280995i
\(79\) −8.56127 − 2.38845i −0.963218 − 0.268721i
\(80\) −2.10696 − 0.748814i −0.235565 − 0.0837199i
\(81\) −5.20451 + 9.01448i −0.578279 + 1.00161i
\(82\) −2.07132 + 0.555008i −0.228739 + 0.0612903i
\(83\) 1.04188 − 3.88835i 0.114361 − 0.426802i −0.884877 − 0.465825i \(-0.845758\pi\)
0.999238 + 0.0390225i \(0.0124244\pi\)
\(84\) −7.43731 + 4.29393i −0.811477 + 0.468507i
\(85\) 2.66193 − 3.86371i 0.288727 − 0.419078i
\(86\) −5.28898 + 3.05359i −0.570325 + 0.329277i
\(87\) 1.39861 − 1.39861i 0.149947 − 0.149947i
\(88\) −1.26073 + 4.70510i −0.134394 + 0.501566i
\(89\) 14.6803i 1.55611i 0.628195 + 0.778056i \(0.283794\pi\)
−0.628195 + 0.778056i \(0.716206\pi\)
\(90\) 3.06631 − 4.45065i 0.323218 − 0.469139i
\(91\) − 5.56395i − 0.583261i
\(92\) −1.54189 − 5.75442i −0.160753 − 0.599940i
\(93\) 10.4232 − 10.4232i 1.08084 − 1.08084i
\(94\) −7.11527 −0.733884
\(95\) 10.8799 + 12.7681i 1.11625 + 1.30998i
\(96\) 2.32745i 0.237545i
\(97\) −3.03269 + 3.03269i −0.307923 + 0.307923i −0.844103 − 0.536180i \(-0.819867\pi\)
0.536180 + 0.844103i \(0.319867\pi\)
\(98\) −1.71201 + 6.38931i −0.172939 + 0.645418i
\(99\) −10.1963 − 5.88681i −1.02476 − 0.591647i
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.k.a.103.3 ✓ 160
5.2 odd 4 inner 790.2.k.a.577.23 yes 160
79.56 odd 6 inner 790.2.k.a.293.23 yes 160
395.372 even 12 inner 790.2.k.a.767.3 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
790.2.k.a.103.3 ✓ 160 1.1 even 1 trivial
790.2.k.a.293.23 yes 160 79.56 odd 6 inner
790.2.k.a.577.23 yes 160 5.2 odd 4 inner
790.2.k.a.767.3 yes 160 395.372 even 12 inner