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Properties

Label 930.2.j.a.497.2
Level $930$
Weight $2$
Character 930.497
Analytic conductor $7.426$
Analytic rank $0$
Dimension $4$
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] = [930,2,Mod(497,930)] 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("930.497"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(930, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,4,0,0,-4,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.42608738798\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 497.2
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 930.497
Dual form 930.2.j.a.683.2

$q$-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+(0.707107 - 0.707107i) q^{2} +(0.292893 - 1.70711i) q^{3} -1.00000i q^{4} +(-2.12132 - 0.707107i) q^{5} +(-1.00000 - 1.41421i) q^{6} +(-0.707107 - 0.707107i) q^{8} +(-2.82843 - 1.00000i) q^{9} +(-2.00000 + 1.00000i) q^{10} -4.24264i q^{11} +(-1.70711 - 0.292893i) q^{12} +(-3.00000 + 3.00000i) q^{13} +(-1.82843 + 3.41421i) q^{15} -1.00000 q^{16} +(1.41421 - 1.41421i) q^{17} +(-2.70711 + 1.29289i) q^{18} +4.00000i q^{19} +(-0.707107 + 2.12132i) q^{20} +(-3.00000 - 3.00000i) q^{22} +(-1.41421 + 1.00000i) q^{24} +(4.00000 + 3.00000i) q^{25} +4.24264i q^{26} +(-2.53553 + 4.53553i) q^{27} -2.82843 q^{29} +(1.12132 + 3.70711i) q^{30} -1.00000 q^{31} +(-0.707107 + 0.707107i) q^{32} +(-7.24264 - 1.24264i) q^{33} -2.00000i q^{34} +(-1.00000 + 2.82843i) q^{36} +(-5.00000 - 5.00000i) q^{37} +(2.82843 + 2.82843i) q^{38} +(4.24264 + 6.00000i) q^{39} +(1.00000 + 2.00000i) q^{40} +(8.00000 - 8.00000i) q^{43} -4.24264 q^{44} +(5.29289 + 4.12132i) q^{45} +(-4.24264 + 4.24264i) q^{47} +(-0.292893 + 1.70711i) q^{48} -7.00000i q^{49} +(4.94975 - 0.707107i) q^{50} +(-2.00000 - 2.82843i) q^{51} +(3.00000 + 3.00000i) q^{52} +(-4.24264 - 4.24264i) q^{53} +(1.41421 + 5.00000i) q^{54} +(-3.00000 + 9.00000i) q^{55} +(6.82843 + 1.17157i) q^{57} +(-2.00000 + 2.00000i) q^{58} -5.65685 q^{59} +(3.41421 + 1.82843i) q^{60} -8.00000 q^{61} +(-0.707107 + 0.707107i) q^{62} +1.00000i q^{64} +(8.48528 - 4.24264i) q^{65} +(-6.00000 + 4.24264i) q^{66} +(-1.00000 - 1.00000i) q^{67} +(-1.41421 - 1.41421i) q^{68} +1.41421i q^{71} +(1.29289 + 2.70711i) q^{72} +(2.00000 - 2.00000i) q^{73} -7.07107 q^{74} +(6.29289 - 5.94975i) q^{75} +4.00000 q^{76} +(7.24264 + 1.24264i) q^{78} -10.0000i q^{79} +(2.12132 + 0.707107i) q^{80} +(7.00000 + 5.65685i) q^{81} +(12.7279 + 12.7279i) q^{83} +(-4.00000 + 2.00000i) q^{85} -11.3137i q^{86} +(-0.828427 + 4.82843i) q^{87} +(-3.00000 + 3.00000i) q^{88} -7.07107 q^{89} +(6.65685 - 0.828427i) q^{90} +(-0.292893 + 1.70711i) q^{93} +6.00000i q^{94} +(2.82843 - 8.48528i) q^{95} +(1.00000 + 1.41421i) q^{96} +(3.00000 + 3.00000i) q^{97} +(-4.94975 - 4.94975i) q^{98} +(-4.24264 + 12.0000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{6} - 8 q^{10} - 4 q^{12} - 12 q^{13} + 4 q^{15} - 4 q^{16} - 8 q^{18} - 12 q^{22} + 16 q^{25} + 4 q^{27} - 4 q^{30} - 4 q^{31} - 12 q^{33} - 4 q^{36} - 20 q^{37} + 4 q^{40} + 32 q^{43}+ \cdots + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/930\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.707107 − 0.707107i 0.500000 − 0.500000i
\(3\) 0.292893 − 1.70711i 0.169102 − 0.985599i
\(4\) − 1.00000i − 0.500000i
\(5\) −2.12132 − 0.707107i −0.948683 − 0.316228i
\(6\) −1.00000 − 1.41421i −0.408248 − 0.577350i
\(7\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) −0.707107 − 0.707107i −0.250000 − 0.250000i
\(9\) −2.82843 − 1.00000i −0.942809 − 0.333333i
\(10\) −2.00000 + 1.00000i −0.632456 + 0.316228i
\(11\) − 4.24264i − 1.27920i −0.768706 − 0.639602i \(-0.779099\pi\)
0.768706 − 0.639602i \(-0.220901\pi\)
\(12\) −1.70711 − 0.292893i −0.492799 − 0.0845510i
\(13\) −3.00000 + 3.00000i −0.832050 + 0.832050i −0.987797 − 0.155747i \(-0.950222\pi\)
0.155747 + 0.987797i \(0.450222\pi\)
\(14\) 0 0
\(15\) −1.82843 + 3.41421i −0.472098 + 0.881546i
\(16\) −1.00000 −0.250000
\(17\) 1.41421 − 1.41421i 0.342997 − 0.342997i −0.514496 − 0.857493i \(-0.672021\pi\)
0.857493 + 0.514496i \(0.172021\pi\)
\(18\) −2.70711 + 1.29289i −0.638071 + 0.304738i
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) −0.707107 + 2.12132i −0.158114 + 0.474342i
\(21\) 0 0
\(22\) −3.00000 − 3.00000i −0.639602 − 0.639602i
\(23\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(24\) −1.41421 + 1.00000i −0.288675 + 0.204124i
\(25\) 4.00000 + 3.00000i 0.800000 + 0.600000i
\(26\) 4.24264i 0.832050i
\(27\) −2.53553 + 4.53553i −0.487964 + 0.872864i
\(28\) 0 0
\(29\) −2.82843 −0.525226 −0.262613 − 0.964901i \(-0.584584\pi\)
−0.262613 + 0.964901i \(0.584584\pi\)
\(30\) 1.12132 + 3.70711i 0.204724 + 0.676822i
\(31\) −1.00000 −0.179605
\(32\) −0.707107 + 0.707107i −0.125000 + 0.125000i
\(33\) −7.24264 − 1.24264i −1.26078 − 0.216316i
\(34\) − 2.00000i − 0.342997i
\(35\) 0 0
\(36\) −1.00000 + 2.82843i −0.166667 + 0.471405i
\(37\) −5.00000 − 5.00000i −0.821995 − 0.821995i 0.164399 − 0.986394i \(-0.447432\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 2.82843 + 2.82843i 0.458831 + 0.458831i
\(39\) 4.24264 + 6.00000i 0.679366 + 0.960769i
\(40\) 1.00000 + 2.00000i 0.158114 + 0.316228i
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 8.00000 − 8.00000i 1.21999 − 1.21999i 0.252353 − 0.967635i \(-0.418795\pi\)
0.967635 − 0.252353i \(-0.0812046\pi\)
\(44\) −4.24264 −0.639602
\(45\) 5.29289 + 4.12132i 0.789018 + 0.614370i
\(46\) 0 0
\(47\) −4.24264 + 4.24264i −0.618853 + 0.618853i −0.945237 − 0.326384i \(-0.894170\pi\)
0.326384 + 0.945237i \(0.394170\pi\)
\(48\) −0.292893 + 1.70711i −0.0422755 + 0.246400i
\(49\) − 7.00000i − 1.00000i
\(50\) 4.94975 − 0.707107i 0.700000 − 0.100000i
\(51\) −2.00000 − 2.82843i −0.280056 − 0.396059i
\(52\) 3.00000 + 3.00000i 0.416025 + 0.416025i
\(53\) −4.24264 − 4.24264i −0.582772 − 0.582772i 0.352892 − 0.935664i \(-0.385198\pi\)
−0.935664 + 0.352892i \(0.885198\pi\)
\(54\) 1.41421 + 5.00000i 0.192450 + 0.680414i
\(55\) −3.00000 + 9.00000i −0.404520 + 1.21356i
\(56\) 0 0
\(57\) 6.82843 + 1.17157i 0.904447 + 0.155179i
\(58\) −2.00000 + 2.00000i −0.262613 + 0.262613i
\(59\) −5.65685 −0.736460 −0.368230 − 0.929735i \(-0.620036\pi\)
−0.368230 + 0.929735i \(0.620036\pi\)
\(60\) 3.41421 + 1.82843i 0.440773 + 0.236049i
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) −0.707107 + 0.707107i −0.0898027 + 0.0898027i
\(63\) 0 0
\(64\) 1.00000i 0.125000i
\(65\) 8.48528 − 4.24264i 1.05247 − 0.526235i
\(66\) −6.00000 + 4.24264i −0.738549 + 0.522233i
\(67\) −1.00000 − 1.00000i −0.122169 − 0.122169i 0.643379 − 0.765548i \(-0.277532\pi\)
−0.765548 + 0.643379i \(0.777532\pi\)
\(68\) −1.41421 − 1.41421i −0.171499 − 0.171499i
\(69\) 0 0
\(70\) 0 0
\(71\) 1.41421i 0.167836i 0.996473 + 0.0839181i \(0.0267434\pi\)
−0.996473 + 0.0839181i \(0.973257\pi\)
\(72\) 1.29289 + 2.70711i 0.152369 + 0.319036i
\(73\) 2.00000 − 2.00000i 0.234082 − 0.234082i −0.580312 − 0.814394i \(-0.697069\pi\)
0.814394 + 0.580312i \(0.197069\pi\)
\(74\) −7.07107 −0.821995
\(75\) 6.29289 − 5.94975i 0.726641 − 0.687018i
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 7.24264 + 1.24264i 0.820068 + 0.140701i
\(79\) − 10.0000i − 1.12509i −0.826767 − 0.562544i \(-0.809823\pi\)
0.826767 − 0.562544i \(-0.190177\pi\)
\(80\) 2.12132 + 0.707107i 0.237171 + 0.0790569i
\(81\) 7.00000 + 5.65685i 0.777778 + 0.628539i
\(82\) 0 0
\(83\) 12.7279 + 12.7279i 1.39707 + 1.39707i 0.808300 + 0.588771i \(0.200388\pi\)
0.588771 + 0.808300i \(0.299612\pi\)
\(84\) 0 0
\(85\) −4.00000 + 2.00000i −0.433861 + 0.216930i
\(86\) − 11.3137i − 1.21999i
\(87\) −0.828427 + 4.82843i −0.0888167 + 0.517662i
\(88\) −3.00000 + 3.00000i −0.319801 + 0.319801i
\(89\) −7.07107 −0.749532 −0.374766 − 0.927119i \(-0.622277\pi\)
−0.374766 + 0.927119i \(0.622277\pi\)
\(90\) 6.65685 − 0.828427i 0.701694 − 0.0873239i
\(91\) 0 0
\(92\) 0 0
\(93\) −0.292893 + 1.70711i −0.0303716 + 0.177019i
\(94\) 6.00000i 0.618853i
\(95\) 2.82843 − 8.48528i 0.290191 − 0.870572i
\(96\) 1.00000 + 1.41421i 0.102062 + 0.144338i
\(97\) 3.00000 + 3.00000i 0.304604 + 0.304604i 0.842812 − 0.538208i \(-0.180899\pi\)
−0.538208 + 0.842812i \(0.680899\pi\)
\(98\) −4.94975 − 4.94975i −0.500000 − 0.500000i
\(99\) −4.24264 + 12.0000i −0.426401 + 1.20605i
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 930.2.j.a.497.2 yes 4
3.2 odd 2 inner 930.2.j.a.497.1 ✓ 4
5.3 odd 4 inner 930.2.j.a.683.1 yes 4
15.8 even 4 inner 930.2.j.a.683.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.j.a.497.1 ✓ 4 3.2 odd 2 inner
930.2.j.a.497.2 yes 4 1.1 even 1 trivial
930.2.j.a.683.1 yes 4 5.3 odd 4 inner
930.2.j.a.683.2 yes 4 15.8 even 4 inner