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Properties

Label 790.2.j.a.339.12
Level $790$
Weight $2$
Character 790.339
Analytic conductor $6.308$
Analytic rank $0$
Dimension $80$
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(339,790)] 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("790.339"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(790, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 790 = 2 \cdot 5 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 790.j (of order \(6\), degree \(2\), 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: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 339.12
Character \(\chi\) \(=\) 790.339
Dual form 790.2.j.a.529.12

$q$-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+(-0.866025 + 0.500000i) q^{2} +(0.512034 - 0.295623i) q^{3} +(0.500000 - 0.866025i) q^{4} +(0.479545 + 2.18404i) q^{5} +(-0.295623 + 0.512034i) q^{6} +(0.189884 + 0.109629i) q^{7} +1.00000i q^{8} +(-1.32521 + 2.29534i) q^{9} +(-1.50732 - 1.65166i) q^{10} +(2.87007 - 4.97111i) q^{11} -0.591246i q^{12} +(5.60385 - 3.23539i) q^{13} -0.219259 q^{14} +(0.891195 + 0.976538i) q^{15} +(-0.500000 - 0.866025i) q^{16} -5.66513i q^{17} -2.65043i q^{18} +(-1.65921 + 2.87384i) q^{19} +(2.13121 + 0.676723i) q^{20} +0.129636 q^{21} +5.74015i q^{22} +(5.83895 + 3.37112i) q^{23} +(0.295623 + 0.512034i) q^{24} +(-4.54007 + 2.09469i) q^{25} +(-3.23539 + 5.60385i) q^{26} +3.34079i q^{27} +(0.189884 - 0.109629i) q^{28} +(-4.38884 + 7.60169i) q^{29} +(-1.26007 - 0.400109i) q^{30} +(-1.81778 + 3.14848i) q^{31} +(0.866025 + 0.500000i) q^{32} -3.39384i q^{33} +(2.83257 + 4.90615i) q^{34} +(-0.148377 + 0.467286i) q^{35} +(1.32521 + 2.29534i) q^{36} +(7.45149 - 4.30212i) q^{37} -3.31842i q^{38} +(1.91291 - 3.31325i) q^{39} +(-2.18404 + 0.479545i) q^{40} +7.23689 q^{41} +(-0.112268 + 0.0648179i) q^{42} +(-1.12683 + 0.650574i) q^{43} +(-2.87007 - 4.97111i) q^{44} +(-5.64861 - 1.79360i) q^{45} -6.74224 q^{46} +(7.60849 + 4.39276i) q^{47} +(-0.512034 - 0.295623i) q^{48} +(-3.47596 - 6.02054i) q^{49} +(2.88447 - 4.08409i) q^{50} +(-1.67474 - 2.90074i) q^{51} -6.47077i q^{52} +(0.477821 + 0.275870i) q^{53} +(-1.67040 - 2.89321i) q^{54} +(12.2335 + 3.88449i) q^{55} +(-0.109629 + 0.189884i) q^{56} +1.96200i q^{57} -8.77767i q^{58} +(4.75003 + 8.22730i) q^{59} +(1.29130 - 0.283529i) q^{60} -2.67292 q^{61} -3.63555i q^{62} +(-0.503273 + 0.290565i) q^{63} -1.00000 q^{64} +(9.75352 + 10.6875i) q^{65} +(1.69692 + 2.93915i) q^{66} +4.68197i q^{67} +(-4.90615 - 2.83257i) q^{68} +3.98632 q^{69} +(-0.105144 - 0.478870i) q^{70} +12.5161 q^{71} +(-2.29534 - 1.32521i) q^{72} +(-11.1004 - 6.40882i) q^{73} +(-4.30212 + 7.45149i) q^{74} +(-1.70543 + 2.41470i) q^{75} +(1.65921 + 2.87384i) q^{76} +(1.08996 - 0.629289i) q^{77} +3.82582i q^{78} +(-6.50770 - 6.05391i) q^{79} +(1.65166 - 1.50732i) q^{80} +(-2.98803 - 5.17542i) q^{81} +(-6.26733 + 3.61844i) q^{82} +(-1.32504 - 0.765015i) q^{83} +(0.0648179 - 0.112268i) q^{84} +(12.3729 - 2.71669i) q^{85} +(0.650574 - 1.12683i) q^{86} +5.18976i q^{87} +(4.97111 + 2.87007i) q^{88} -1.78850 q^{89} +(5.78865 - 1.27100i) q^{90} +1.41877 q^{91} +(5.83895 - 3.37112i) q^{92} +2.14951i q^{93} -8.78553 q^{94} +(-7.07224 - 2.24565i) q^{95} +0.591246 q^{96} +5.06050i q^{97} +(6.02054 + 3.47596i) q^{98} +(7.60693 + 13.1756i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q + 40 q^{4} + 4 q^{6} + 36 q^{9} - 4 q^{10} - 8 q^{11} + 16 q^{14} - 8 q^{15} - 40 q^{16} - 8 q^{19} - 72 q^{21} - 4 q^{24} + 6 q^{25} - 8 q^{29} - 10 q^{30} - 16 q^{31} - 2 q^{35} - 36 q^{36} + 24 q^{39}+ \cdots + 52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 + 0.500000i −0.612372 + 0.353553i
\(3\) 0.512034 − 0.295623i 0.295623 − 0.170678i −0.344852 − 0.938657i \(-0.612071\pi\)
0.640475 + 0.767979i \(0.278738\pi\)
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) 0.479545 + 2.18404i 0.214459 + 0.976733i
\(6\) −0.295623 + 0.512034i −0.120687 + 0.209037i
\(7\) 0.189884 + 0.109629i 0.0717693 + 0.0414360i 0.535455 − 0.844564i \(-0.320140\pi\)
−0.463686 + 0.886000i \(0.653473\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.32521 + 2.29534i −0.441738 + 0.765113i
\(10\) −1.50732 − 1.65166i −0.476656 − 0.522302i
\(11\) 2.87007 − 4.97111i 0.865360 − 1.49885i −0.00132919 − 0.999999i \(-0.500423\pi\)
0.866689 − 0.498848i \(-0.166244\pi\)
\(12\) − 0.591246i − 0.170678i
\(13\) 5.60385 − 3.23539i 1.55423 − 0.897335i 0.556439 − 0.830888i \(-0.312167\pi\)
0.997790 − 0.0664464i \(-0.0211661\pi\)
\(14\) −0.219259 −0.0585994
\(15\) 0.891195 + 0.976538i 0.230106 + 0.252141i
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) − 5.66513i − 1.37400i −0.726659 − 0.686998i \(-0.758928\pi\)
0.726659 − 0.686998i \(-0.241072\pi\)
\(18\) − 2.65043i − 0.624712i
\(19\) −1.65921 + 2.87384i −0.380649 + 0.659303i −0.991155 − 0.132708i \(-0.957633\pi\)
0.610506 + 0.792012i \(0.290966\pi\)
\(20\) 2.13121 + 0.676723i 0.476553 + 0.151320i
\(21\) 0.129636 0.0282888
\(22\) 5.74015i 1.22380i
\(23\) 5.83895 + 3.37112i 1.21751 + 0.702927i 0.964384 − 0.264507i \(-0.0852093\pi\)
0.253122 + 0.967434i \(0.418543\pi\)
\(24\) 0.295623 + 0.512034i 0.0603437 + 0.104518i
\(25\) −4.54007 + 2.09469i −0.908015 + 0.418938i
\(26\) −3.23539 + 5.60385i −0.634511 + 1.09901i
\(27\) 3.34079i 0.642935i
\(28\) 0.189884 − 0.109629i 0.0358846 − 0.0207180i
\(29\) −4.38884 + 7.60169i −0.814986 + 1.41160i 0.0943513 + 0.995539i \(0.469922\pi\)
−0.909338 + 0.416059i \(0.863411\pi\)
\(30\) −1.26007 − 0.400109i −0.230056 − 0.0730496i
\(31\) −1.81778 + 3.14848i −0.326482 + 0.565484i −0.981811 − 0.189860i \(-0.939197\pi\)
0.655329 + 0.755344i \(0.272530\pi\)
\(32\) 0.866025 + 0.500000i 0.153093 + 0.0883883i
\(33\) − 3.39384i − 0.590791i
\(34\) 2.83257 + 4.90615i 0.485781 + 0.841398i
\(35\) −0.148377 + 0.467286i −0.0250804 + 0.0789858i
\(36\) 1.32521 + 2.29534i 0.220869 + 0.382556i
\(37\) 7.45149 − 4.30212i 1.22502 − 0.707264i 0.259034 − 0.965868i \(-0.416596\pi\)
0.965983 + 0.258604i \(0.0832626\pi\)
\(38\) − 3.31842i − 0.538319i
\(39\) 1.91291 − 3.31325i 0.306310 − 0.530545i
\(40\) −2.18404 + 0.479545i −0.345327 + 0.0758227i
\(41\) 7.23689 1.13021 0.565106 − 0.825018i \(-0.308835\pi\)
0.565106 + 0.825018i \(0.308835\pi\)
\(42\) −0.112268 + 0.0648179i −0.0173233 + 0.0100016i
\(43\) −1.12683 + 0.650574i −0.171840 + 0.0992116i −0.583453 − 0.812147i \(-0.698299\pi\)
0.411613 + 0.911359i \(0.364965\pi\)
\(44\) −2.87007 − 4.97111i −0.432680 − 0.749424i
\(45\) −5.64861 − 1.79360i −0.842046 − 0.267375i
\(46\) −6.74224 −0.994089
\(47\) 7.60849 + 4.39276i 1.10981 + 0.640750i 0.938781 − 0.344515i \(-0.111957\pi\)
0.171031 + 0.985266i \(0.445290\pi\)
\(48\) −0.512034 − 0.295623i −0.0739057 − 0.0426695i
\(49\) −3.47596 − 6.02054i −0.496566 − 0.860078i
\(50\) 2.88447 − 4.08409i 0.407926 − 0.577578i
\(51\) −1.67474 − 2.90074i −0.234511 − 0.406185i
\(52\) − 6.47077i − 0.897335i
\(53\) 0.477821 + 0.275870i 0.0656338 + 0.0378937i 0.532458 − 0.846457i \(-0.321269\pi\)
−0.466824 + 0.884350i \(0.654602\pi\)
\(54\) −1.67040 − 2.89321i −0.227312 − 0.393716i
\(55\) 12.2335 + 3.88449i 1.64956 + 0.523784i
\(56\) −0.109629 + 0.189884i −0.0146498 + 0.0253743i
\(57\) 1.96200i 0.259873i
\(58\) − 8.77767i − 1.15256i
\(59\) 4.75003 + 8.22730i 0.618402 + 1.07110i 0.989777 + 0.142621i \(0.0455529\pi\)
−0.371376 + 0.928483i \(0.621114\pi\)
\(60\) 1.29130 − 0.283529i 0.166707 − 0.0366034i
\(61\) −2.67292 −0.342232 −0.171116 − 0.985251i \(-0.554737\pi\)
−0.171116 + 0.985251i \(0.554737\pi\)
\(62\) − 3.63555i − 0.461716i
\(63\) −0.503273 + 0.290565i −0.0634065 + 0.0366077i
\(64\) −1.00000 −0.125000
\(65\) 9.75352 + 10.6875i 1.20977 + 1.32563i
\(66\) 1.69692 + 2.93915i 0.208876 + 0.361784i
\(67\) 4.68197i 0.571994i 0.958231 + 0.285997i \(0.0923247\pi\)
−0.958231 + 0.285997i \(0.907675\pi\)
\(68\) −4.90615 − 2.83257i −0.594958 − 0.343499i
\(69\) 3.98632 0.479896
\(70\) −0.105144 − 0.478870i −0.0125672 − 0.0572359i
\(71\) 12.5161 1.48539 0.742696 − 0.669629i \(-0.233547\pi\)
0.742696 + 0.669629i \(0.233547\pi\)
\(72\) −2.29534 − 1.32521i −0.270508 − 0.156178i
\(73\) −11.1004 − 6.40882i −1.29920 − 0.750096i −0.318938 − 0.947776i \(-0.603326\pi\)
−0.980267 + 0.197680i \(0.936659\pi\)
\(74\) −4.30212 + 7.45149i −0.500111 + 0.866218i
\(75\) −1.70543 + 2.41470i −0.196926 + 0.278826i
\(76\) 1.65921 + 2.87384i 0.190325 + 0.329652i
\(77\) 1.08996 − 0.629289i 0.124213 − 0.0717141i
\(78\) 3.82582i 0.433188i
\(79\) −6.50770 − 6.05391i −0.732173 − 0.681118i
\(80\) 1.65166 − 1.50732i 0.184662 − 0.168523i
\(81\) −2.98803 − 5.17542i −0.332003 − 0.575047i
\(82\) −6.26733 + 3.61844i −0.692111 + 0.399590i
\(83\) −1.32504 − 0.765015i −0.145443 − 0.0839713i 0.425513 − 0.904952i \(-0.360094\pi\)
−0.570955 + 0.820981i \(0.693427\pi\)
\(84\) 0.0648179 − 0.112268i 0.00707221 − 0.0122494i
\(85\) 12.3729 − 2.71669i 1.34203 − 0.294666i
\(86\) 0.650574 − 1.12683i 0.0701532 − 0.121509i
\(87\) 5.18976i 0.556401i
\(88\) 4.97111 + 2.87007i 0.529923 + 0.305951i
\(89\) −1.78850 −0.189581 −0.0947904 − 0.995497i \(-0.530218\pi\)
−0.0947904 + 0.995497i \(0.530218\pi\)
\(90\) 5.78865 − 1.27100i 0.610177 − 0.133975i
\(91\) 1.41877 0.148728
\(92\) 5.83895 − 3.37112i 0.608753 − 0.351464i
\(93\) 2.14951i 0.222893i
\(94\) −8.78553 −0.906158
\(95\) −7.07224 − 2.24565i −0.725597 − 0.230399i
\(96\) 0.591246 0.0603437
\(97\) 5.06050i 0.513816i 0.966436 + 0.256908i \(0.0827037\pi\)
−0.966436 + 0.256908i \(0.917296\pi\)
\(98\) 6.02054 + 3.47596i 0.608167 + 0.351125i
\(99\) 7.60693 + 13.1756i 0.764525 + 1.32420i
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.j.a.339.12 ✓ 80
5.4 even 2 inner 790.2.j.a.339.29 yes 80
79.55 even 3 inner 790.2.j.a.529.29 yes 80
395.134 even 6 inner 790.2.j.a.529.12 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
790.2.j.a.339.12 ✓ 80 1.1 even 1 trivial
790.2.j.a.339.29 yes 80 5.4 even 2 inner
790.2.j.a.529.12 yes 80 395.134 even 6 inner
790.2.j.a.529.29 yes 80 79.55 even 3 inner