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Properties

Label 702.2.bc.a.557.3
Level $702$
Weight $2$
Character 702.557
Analytic conductor $5.605$
Analytic rank $0$
Dimension $56$
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] = [702,2,Mod(305,702)] 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("702.305"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(702, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([2, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 702 = 2 \cdot 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 702.bc (of order \(12\), degree \(4\), not 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: \(5.60549822189\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 234)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 557.3
Character \(\chi\) \(=\) 702.557
Dual form 702.2.bc.a.305.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.258819 - 0.965926i) q^{2} +(-0.866025 + 0.500000i) q^{4} +(-0.266463 - 0.994452i) q^{5} +(-0.248284 + 0.248284i) q^{7} +(0.707107 + 0.707107i) q^{8} +(-0.891601 + 0.514766i) q^{10} +(-3.50922 + 0.940292i) q^{11} +(-3.13452 + 1.78180i) q^{13} +(0.304085 + 0.175564i) q^{14} +(0.500000 - 0.866025i) q^{16} +(-1.67236 + 2.89660i) q^{17} +(-0.969723 + 0.259837i) q^{19} +(0.727989 + 0.727989i) q^{20} +(1.81650 + 3.14628i) q^{22} -1.47052 q^{23} +(3.41219 - 1.97003i) q^{25} +(2.53236 + 2.56655i) q^{26} +(0.0908784 - 0.339163i) q^{28} +(-5.04092 - 2.91038i) q^{29} +(-5.54424 + 1.48558i) q^{31} +(-0.965926 - 0.258819i) q^{32} +(3.23074 + 0.865675i) q^{34} +(0.313065 + 0.180748i) q^{35} +(4.96682 + 1.33085i) q^{37} +(0.501966 + 0.869430i) q^{38} +(0.514766 - 0.891601i) q^{40} +(-6.60980 + 6.60980i) q^{41} +2.73886i q^{43} +(2.56893 - 2.56893i) q^{44} +(0.380599 + 1.42041i) q^{46} +(-2.19251 + 8.18256i) q^{47} +6.87671i q^{49} +(-2.78605 - 2.78605i) q^{50} +(1.82367 - 3.11034i) q^{52} -2.67501i q^{53} +(1.87015 + 3.23920i) q^{55} -0.351127 q^{56} +(-1.50652 + 5.62241i) q^{58} +(-1.25408 + 4.68030i) q^{59} +5.72649 q^{61} +(2.86991 + 4.97083i) q^{62} +1.00000i q^{64} +(2.60715 + 2.64234i) q^{65} +(-10.6589 - 10.6589i) q^{67} -3.34471i q^{68} +(0.0935622 - 0.349179i) q^{70} +(-1.53985 - 5.74681i) q^{71} +(2.47165 - 2.47165i) q^{73} -5.14203i q^{74} +(0.709887 - 0.709887i) q^{76} +(0.637824 - 1.10474i) q^{77} +(-1.76544 - 3.05783i) q^{79} +(-0.994452 - 0.266463i) q^{80} +(8.09532 + 4.67384i) q^{82} +(8.34680 + 2.23652i) q^{83} +(3.32615 + 0.891240i) q^{85} +(2.64553 - 0.708868i) q^{86} +(-3.14628 - 1.81650i) q^{88} +(-2.68520 + 10.0213i) q^{89} +(0.335858 - 1.22064i) q^{91} +(1.27351 - 0.735260i) q^{92} +8.47121 q^{94} +(0.516790 + 0.895107i) q^{95} +(-10.4441 - 10.4441i) q^{97} +(6.64239 - 1.77982i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q - 8 q^{7} + 24 q^{11} + 28 q^{16} - 8 q^{19} - 4 q^{28} - 4 q^{31} + 24 q^{35} - 4 q^{37} - 36 q^{38} + 48 q^{41} + 36 q^{47} + 24 q^{50} + 8 q^{52} + 36 q^{65} + 28 q^{67} + 24 q^{71} + 28 q^{73}+ \cdots - 48 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/702\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(677\)
\(\chi(n)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{5}{6}\right)\)

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 0
\(4\) −0.866025 + 0.500000i −0.433013 + 0.250000i
\(5\) −0.266463 − 0.994452i −0.119166 − 0.444732i 0.880399 − 0.474234i \(-0.157275\pi\)
−0.999565 + 0.0295011i \(0.990608\pi\)
\(6\) 0 0
\(7\) −0.248284 + 0.248284i −0.0938426 + 0.0938426i −0.752470 − 0.658627i \(-0.771138\pi\)
0.658627 + 0.752470i \(0.271138\pi\)
\(8\) 0.707107 + 0.707107i 0.250000 + 0.250000i
\(9\) 0 0
\(10\) −0.891601 + 0.514766i −0.281949 + 0.162783i
\(11\) −3.50922 + 0.940292i −1.05807 + 0.283509i −0.745582 − 0.666413i \(-0.767829\pi\)
−0.312487 + 0.949922i \(0.601162\pi\)
\(12\) 0 0
\(13\) −3.13452 + 1.78180i −0.869358 + 0.494182i
\(14\) 0.304085 + 0.175564i 0.0812701 + 0.0469213i
\(15\) 0 0
\(16\) 0.500000 − 0.866025i 0.125000 − 0.216506i
\(17\) −1.67236 + 2.89660i −0.405606 + 0.702530i −0.994392 − 0.105759i \(-0.966273\pi\)
0.588786 + 0.808289i \(0.299606\pi\)
\(18\) 0 0
\(19\) −0.969723 + 0.259837i −0.222470 + 0.0596106i −0.368332 − 0.929694i \(-0.620071\pi\)
0.145862 + 0.989305i \(0.453404\pi\)
\(20\) 0.727989 + 0.727989i 0.162783 + 0.162783i
\(21\) 0 0
\(22\) 1.81650 + 3.14628i 0.387280 + 0.670789i
\(23\) −1.47052 −0.306625 −0.153312 − 0.988178i \(-0.548994\pi\)
−0.153312 + 0.988178i \(0.548994\pi\)
\(24\) 0 0
\(25\) 3.41219 − 1.97003i 0.682439 − 0.394006i
\(26\) 2.53236 + 2.56655i 0.496636 + 0.503341i
\(27\) 0 0
\(28\) 0.0908784 − 0.339163i 0.0171744 − 0.0640957i
\(29\) −5.04092 − 2.91038i −0.936075 − 0.540443i −0.0473474 − 0.998878i \(-0.515077\pi\)
−0.888728 + 0.458435i \(0.848410\pi\)
\(30\) 0 0
\(31\) −5.54424 + 1.48558i −0.995775 + 0.266817i −0.719675 − 0.694312i \(-0.755709\pi\)
−0.276101 + 0.961129i \(0.589042\pi\)
\(32\) −0.965926 − 0.258819i −0.170753 − 0.0457532i
\(33\) 0 0
\(34\) 3.23074 + 0.865675i 0.554068 + 0.148462i
\(35\) 0.313065 + 0.180748i 0.0529177 + 0.0305520i
\(36\) 0 0
\(37\) 4.96682 + 1.33085i 0.816540 + 0.218791i 0.642833 − 0.766006i \(-0.277759\pi\)
0.173707 + 0.984797i \(0.444426\pi\)
\(38\) 0.501966 + 0.869430i 0.0814296 + 0.141040i
\(39\) 0 0
\(40\) 0.514766 − 0.891601i 0.0813917 − 0.140975i
\(41\) −6.60980 + 6.60980i −1.03228 + 1.03228i −0.0328164 + 0.999461i \(0.510448\pi\)
−0.999461 + 0.0328164i \(0.989552\pi\)
\(42\) 0 0
\(43\) 2.73886i 0.417672i 0.977951 + 0.208836i \(0.0669674\pi\)
−0.977951 + 0.208836i \(0.933033\pi\)
\(44\) 2.56893 − 2.56893i 0.387280 − 0.387280i
\(45\) 0 0
\(46\) 0.380599 + 1.42041i 0.0561162 + 0.209429i
\(47\) −2.19251 + 8.18256i −0.319811 + 1.19355i 0.599616 + 0.800288i \(0.295320\pi\)
−0.919427 + 0.393262i \(0.871347\pi\)
\(48\) 0 0
\(49\) 6.87671i 0.982387i
\(50\) −2.78605 − 2.78605i −0.394006 − 0.394006i
\(51\) 0 0
\(52\) 1.82367 − 3.11034i 0.252898 − 0.431327i
\(53\) − 2.67501i − 0.367442i −0.982978 − 0.183721i \(-0.941186\pi\)
0.982978 − 0.183721i \(-0.0588142\pi\)
\(54\) 0 0
\(55\) 1.87015 + 3.23920i 0.252171 + 0.436773i
\(56\) −0.351127 −0.0469213
\(57\) 0 0
\(58\) −1.50652 + 5.62241i −0.197816 + 0.738259i
\(59\) −1.25408 + 4.68030i −0.163268 + 0.609323i 0.834987 + 0.550269i \(0.185475\pi\)
−0.998255 + 0.0590536i \(0.981192\pi\)
\(60\) 0 0
\(61\) 5.72649 0.733202 0.366601 − 0.930378i \(-0.380521\pi\)
0.366601 + 0.930378i \(0.380521\pi\)
\(62\) 2.86991 + 4.97083i 0.364479 + 0.631296i
\(63\) 0 0
\(64\) 1.00000i 0.125000i
\(65\) 2.60715 + 2.64234i 0.323377 + 0.327742i
\(66\) 0 0
\(67\) −10.6589 − 10.6589i −1.30219 − 1.30219i −0.926911 − 0.375282i \(-0.877546\pi\)
−0.375282 − 0.926911i \(-0.622454\pi\)
\(68\) − 3.34471i − 0.405606i
\(69\) 0 0
\(70\) 0.0935622 − 0.349179i 0.0111828 − 0.0417349i
\(71\) −1.53985 − 5.74681i −0.182747 − 0.682021i −0.995102 − 0.0988571i \(-0.968481\pi\)
0.812355 − 0.583164i \(-0.198185\pi\)
\(72\) 0 0
\(73\) 2.47165 − 2.47165i 0.289285 − 0.289285i −0.547513 − 0.836797i \(-0.684425\pi\)
0.836797 + 0.547513i \(0.184425\pi\)
\(74\) − 5.14203i − 0.597749i
\(75\) 0 0
\(76\) 0.709887 − 0.709887i 0.0814296 − 0.0814296i
\(77\) 0.637824 − 1.10474i 0.0726868 − 0.125897i
\(78\) 0 0
\(79\) −1.76544 − 3.05783i −0.198627 − 0.344033i 0.749456 − 0.662054i \(-0.230315\pi\)
−0.948084 + 0.318021i \(0.896982\pi\)
\(80\) −0.994452 − 0.266463i −0.111183 − 0.0297914i
\(81\) 0 0
\(82\) 8.09532 + 4.67384i 0.893979 + 0.516139i
\(83\) 8.34680 + 2.23652i 0.916181 + 0.245490i 0.685952 − 0.727647i \(-0.259386\pi\)
0.230228 + 0.973137i \(0.426053\pi\)
\(84\) 0 0
\(85\) 3.32615 + 0.891240i 0.360772 + 0.0966686i
\(86\) 2.64553 − 0.708868i 0.285275 − 0.0764392i
\(87\) 0 0
\(88\) −3.14628 − 1.81650i −0.335394 − 0.193640i
\(89\) −2.68520 + 10.0213i −0.284630 + 1.06225i 0.664479 + 0.747307i \(0.268653\pi\)
−0.949109 + 0.314947i \(0.898013\pi\)
\(90\) 0 0
\(91\) 0.335858 − 1.22064i 0.0352075 − 0.127958i
\(92\) 1.27351 − 0.735260i 0.132772 − 0.0766562i
\(93\) 0 0
\(94\) 8.47121 0.873739
\(95\) 0.516790 + 0.895107i 0.0530215 + 0.0918360i
\(96\) 0 0
\(97\) −10.4441 − 10.4441i −1.06044 − 1.06044i −0.998052 − 0.0623848i \(-0.980129\pi\)
−0.0623848 − 0.998052i \(-0.519871\pi\)
\(98\) 6.64239 − 1.77982i 0.670983 − 0.179789i
\(99\) 0 0
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 702.2.bc.a.557.3 56
3.2 odd 2 234.2.z.a.167.12 yes 56
9.2 odd 6 702.2.bb.a.89.10 56
9.7 even 3 234.2.y.a.11.5 ✓ 56
13.6 odd 12 702.2.bb.a.71.10 56
39.32 even 12 234.2.y.a.149.5 yes 56
117.97 odd 12 234.2.z.a.227.12 yes 56
117.110 even 12 inner 702.2.bc.a.305.3 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
234.2.y.a.11.5 ✓ 56 9.7 even 3
234.2.y.a.149.5 yes 56 39.32 even 12
234.2.z.a.167.12 yes 56 3.2 odd 2
234.2.z.a.227.12 yes 56 117.97 odd 12
702.2.bb.a.71.10 56 13.6 odd 12
702.2.bb.a.89.10 56 9.2 odd 6
702.2.bc.a.305.3 56 117.110 even 12 inner
702.2.bc.a.557.3 56 1.1 even 1 trivial