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Properties

Label 615.2.j.b.91.1
Level $615$
Weight $2$
Character 615.91
Analytic conductor $4.911$
Analytic rank $0$
Dimension $4$
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] = [615,2,Mod(91,615)] 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("615.91"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(615, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 615 = 3 \cdot 5 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 615.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.91079972431\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 91.1
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 615.91
Dual form 615.2.j.b.196.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.41421i q^{2} +(-0.707107 - 0.707107i) q^{3} +1.00000i q^{5} +(-1.00000 + 1.00000i) q^{6} +(2.00000 + 2.00000i) q^{7} -2.82843i q^{8} +1.00000i q^{9} +1.41421 q^{10} +(-2.12132 - 2.12132i) q^{11} +(1.82843 + 1.82843i) q^{13} +(2.82843 - 2.82843i) q^{14} +(0.707107 - 0.707107i) q^{15} -4.00000 q^{16} +(2.29289 - 2.29289i) q^{17} +1.41421 q^{18} +(5.00000 - 5.00000i) q^{19} -2.82843i q^{21} +(-3.00000 + 3.00000i) q^{22} +5.65685 q^{23} +(-2.00000 + 2.00000i) q^{24} -1.00000 q^{25} +(2.58579 - 2.58579i) q^{26} +(0.707107 - 0.707107i) q^{27} +(-3.53553 - 3.53553i) q^{29} +(-1.00000 - 1.00000i) q^{30} +6.65685 q^{31} +3.00000i q^{33} +(-3.24264 - 3.24264i) q^{34} +(-2.00000 + 2.00000i) q^{35} -8.41421 q^{37} +(-7.07107 - 7.07107i) q^{38} -2.58579i q^{39} +2.82843 q^{40} +(6.12132 - 1.87868i) q^{41} -4.00000 q^{42} -1.24264i q^{43} -1.00000 q^{45} -8.00000i q^{46} +(-7.70711 + 7.70711i) q^{47} +(2.82843 + 2.82843i) q^{48} +1.00000i q^{49} +1.41421i q^{50} -3.24264 q^{51} +(3.41421 + 3.41421i) q^{53} +(-1.00000 - 1.00000i) q^{54} +(2.12132 - 2.12132i) q^{55} +(5.65685 - 5.65685i) q^{56} -7.07107 q^{57} +(-5.00000 + 5.00000i) q^{58} -9.07107 q^{59} -3.34315i q^{61} -9.41421i q^{62} +(-2.00000 + 2.00000i) q^{63} -8.00000 q^{64} +(-1.82843 + 1.82843i) q^{65} +4.24264 q^{66} +(-5.07107 + 5.07107i) q^{67} +(-4.00000 - 4.00000i) q^{69} +(2.82843 + 2.82843i) q^{70} +(1.29289 + 1.29289i) q^{71} +2.82843 q^{72} +11.7279i q^{73} +11.8995i q^{74} +(0.707107 + 0.707107i) q^{75} -8.48528i q^{77} -3.65685 q^{78} +(11.4142 + 11.4142i) q^{79} -4.00000i q^{80} -1.00000 q^{81} +(-2.65685 - 8.65685i) q^{82} +2.24264 q^{83} +(2.29289 + 2.29289i) q^{85} -1.75736 q^{86} +5.00000i q^{87} +(-6.00000 + 6.00000i) q^{88} +(-7.65685 - 7.65685i) q^{89} +1.41421i q^{90} +7.31371i q^{91} +(-4.70711 - 4.70711i) q^{93} +(10.8995 + 10.8995i) q^{94} +(5.00000 + 5.00000i) q^{95} +(0.585786 - 0.585786i) q^{97} +1.41421 q^{98} +(2.12132 - 2.12132i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{6} + 8 q^{7} - 4 q^{13} - 16 q^{16} + 12 q^{17} + 20 q^{19} - 12 q^{22} - 8 q^{24} - 4 q^{25} + 16 q^{26} - 4 q^{30} + 4 q^{31} + 4 q^{34} - 8 q^{35} - 28 q^{37} + 16 q^{41} - 16 q^{42} - 4 q^{45}+ \cdots + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/615\mathbb{Z}\right)^\times\).

\(n\) \(206\) \(211\) \(247\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.41421i − 1.00000i −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 − 0.500000i \(-0.166667\pi\)
\(3\) −0.707107 − 0.707107i −0.408248 − 0.408248i
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) −1.00000 + 1.00000i −0.408248 + 0.408248i
\(7\) 2.00000 + 2.00000i 0.755929 + 0.755929i 0.975579 − 0.219650i \(-0.0704915\pi\)
−0.219650 + 0.975579i \(0.570491\pi\)
\(8\) − 2.82843i − 1.00000i
\(9\) 1.00000i 0.333333i
\(10\) 1.41421 0.447214
\(11\) −2.12132 − 2.12132i −0.639602 − 0.639602i 0.310855 − 0.950457i \(-0.399385\pi\)
−0.950457 + 0.310855i \(0.899385\pi\)
\(12\) 0 0
\(13\) 1.82843 + 1.82843i 0.507114 + 0.507114i 0.913640 − 0.406525i \(-0.133260\pi\)
−0.406525 + 0.913640i \(0.633260\pi\)
\(14\) 2.82843 − 2.82843i 0.755929 − 0.755929i
\(15\) 0.707107 − 0.707107i 0.182574 − 0.182574i
\(16\) −4.00000 −1.00000
\(17\) 2.29289 − 2.29289i 0.556108 − 0.556108i −0.372089 − 0.928197i \(-0.621358\pi\)
0.928197 + 0.372089i \(0.121358\pi\)
\(18\) 1.41421 0.333333
\(19\) 5.00000 − 5.00000i 1.14708 − 1.14708i 0.159954 − 0.987124i \(-0.448865\pi\)
0.987124 − 0.159954i \(-0.0511347\pi\)
\(20\) 0 0
\(21\) − 2.82843i − 0.617213i
\(22\) −3.00000 + 3.00000i −0.639602 + 0.639602i
\(23\) 5.65685 1.17954 0.589768 − 0.807573i \(-0.299219\pi\)
0.589768 + 0.807573i \(0.299219\pi\)
\(24\) −2.00000 + 2.00000i −0.408248 + 0.408248i
\(25\) −1.00000 −0.200000
\(26\) 2.58579 − 2.58579i 0.507114 − 0.507114i
\(27\) 0.707107 − 0.707107i 0.136083 − 0.136083i
\(28\) 0 0
\(29\) −3.53553 − 3.53553i −0.656532 − 0.656532i 0.298026 − 0.954558i \(-0.403672\pi\)
−0.954558 + 0.298026i \(0.903672\pi\)
\(30\) −1.00000 − 1.00000i −0.182574 − 0.182574i
\(31\) 6.65685 1.19561 0.597803 − 0.801643i \(-0.296040\pi\)
0.597803 + 0.801643i \(0.296040\pi\)
\(32\) 0 0
\(33\) 3.00000i 0.522233i
\(34\) −3.24264 − 3.24264i −0.556108 − 0.556108i
\(35\) −2.00000 + 2.00000i −0.338062 + 0.338062i
\(36\) 0 0
\(37\) −8.41421 −1.38329 −0.691644 − 0.722238i \(-0.743113\pi\)
−0.691644 + 0.722238i \(0.743113\pi\)
\(38\) −7.07107 − 7.07107i −1.14708 − 1.14708i
\(39\) − 2.58579i − 0.414057i
\(40\) 2.82843 0.447214
\(41\) 6.12132 − 1.87868i 0.955990 − 0.293400i
\(42\) −4.00000 −0.617213
\(43\) − 1.24264i − 0.189501i −0.995501 − 0.0947505i \(-0.969795\pi\)
0.995501 − 0.0947505i \(-0.0302053\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) − 8.00000i − 1.17954i
\(47\) −7.70711 + 7.70711i −1.12420 + 1.12420i −0.133094 + 0.991103i \(0.542491\pi\)
−0.991103 + 0.133094i \(0.957509\pi\)
\(48\) 2.82843 + 2.82843i 0.408248 + 0.408248i
\(49\) 1.00000i 0.142857i
\(50\) 1.41421i 0.200000i
\(51\) −3.24264 −0.454061
\(52\) 0 0
\(53\) 3.41421 + 3.41421i 0.468978 + 0.468978i 0.901583 − 0.432605i \(-0.142406\pi\)
−0.432605 + 0.901583i \(0.642406\pi\)
\(54\) −1.00000 − 1.00000i −0.136083 − 0.136083i
\(55\) 2.12132 − 2.12132i 0.286039 − 0.286039i
\(56\) 5.65685 − 5.65685i 0.755929 − 0.755929i
\(57\) −7.07107 −0.936586
\(58\) −5.00000 + 5.00000i −0.656532 + 0.656532i
\(59\) −9.07107 −1.18095 −0.590476 − 0.807055i \(-0.701060\pi\)
−0.590476 + 0.807055i \(0.701060\pi\)
\(60\) 0 0
\(61\) − 3.34315i − 0.428046i −0.976829 − 0.214023i \(-0.931343\pi\)
0.976829 − 0.214023i \(-0.0686567\pi\)
\(62\) − 9.41421i − 1.19561i
\(63\) −2.00000 + 2.00000i −0.251976 + 0.251976i
\(64\) −8.00000 −1.00000
\(65\) −1.82843 + 1.82843i −0.226788 + 0.226788i
\(66\) 4.24264 0.522233
\(67\) −5.07107 + 5.07107i −0.619530 + 0.619530i −0.945411 − 0.325881i \(-0.894339\pi\)
0.325881 + 0.945411i \(0.394339\pi\)
\(68\) 0 0
\(69\) −4.00000 − 4.00000i −0.481543 − 0.481543i
\(70\) 2.82843 + 2.82843i 0.338062 + 0.338062i
\(71\) 1.29289 + 1.29289i 0.153438 + 0.153438i 0.779652 − 0.626213i \(-0.215396\pi\)
−0.626213 + 0.779652i \(0.715396\pi\)
\(72\) 2.82843 0.333333
\(73\) 11.7279i 1.37265i 0.727295 + 0.686325i \(0.240777\pi\)
−0.727295 + 0.686325i \(0.759223\pi\)
\(74\) 11.8995i 1.38329i
\(75\) 0.707107 + 0.707107i 0.0816497 + 0.0816497i
\(76\) 0 0
\(77\) − 8.48528i − 0.966988i
\(78\) −3.65685 −0.414057
\(79\) 11.4142 + 11.4142i 1.28420 + 1.28420i 0.938255 + 0.345944i \(0.112441\pi\)
0.345944 + 0.938255i \(0.387559\pi\)
\(80\) − 4.00000i − 0.447214i
\(81\) −1.00000 −0.111111
\(82\) −2.65685 − 8.65685i −0.293400 − 0.955990i
\(83\) 2.24264 0.246162 0.123081 − 0.992397i \(-0.460723\pi\)
0.123081 + 0.992397i \(0.460723\pi\)
\(84\) 0 0
\(85\) 2.29289 + 2.29289i 0.248699 + 0.248699i
\(86\) −1.75736 −0.189501
\(87\) 5.00000i 0.536056i
\(88\) −6.00000 + 6.00000i −0.639602 + 0.639602i
\(89\) −7.65685 − 7.65685i −0.811625 − 0.811625i 0.173253 − 0.984877i \(-0.444572\pi\)
−0.984877 + 0.173253i \(0.944572\pi\)
\(90\) 1.41421i 0.149071i
\(91\) 7.31371i 0.766685i
\(92\) 0 0
\(93\) −4.70711 − 4.70711i −0.488104 − 0.488104i
\(94\) 10.8995 + 10.8995i 1.12420 + 1.12420i
\(95\) 5.00000 + 5.00000i 0.512989 + 0.512989i
\(96\) 0 0
\(97\) 0.585786 − 0.585786i 0.0594776 − 0.0594776i −0.676742 − 0.736220i \(-0.736609\pi\)
0.736220 + 0.676742i \(0.236609\pi\)
\(98\) 1.41421 0.142857
\(99\) 2.12132 − 2.12132i 0.213201 − 0.213201i
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 615.2.j.b.91.1 ✓ 4
41.32 even 4 inner 615.2.j.b.196.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
615.2.j.b.91.1 ✓ 4 1.1 even 1 trivial
615.2.j.b.196.2 yes 4 41.32 even 4 inner