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Properties

Label 294.2.d.b.293.1
Level $294$
Weight $2$
Character 294.293
Analytic conductor $2.348$
Analytic rank $0$
Dimension $8$
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] = [294,2,Mod(293,294)] 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("294.293"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(294, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 294.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.34760181943\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 293.1
Root \(-0.382683 + 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 294.293
Dual form 294.2.d.b.293.5

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +(-0.923880 + 1.46508i) q^{3} -1.00000 q^{4} -1.53073 q^{5} +(1.46508 + 0.923880i) q^{6} +1.00000i q^{8} +(-1.29289 - 2.70711i) q^{9} +1.53073i q^{10} -6.24264i q^{11} +(0.923880 - 1.46508i) q^{12} -5.22625i q^{13} +(1.41421 - 2.24264i) q^{15} +1.00000 q^{16} -2.29610 q^{17} +(-2.70711 + 1.29289i) q^{18} +1.84776i q^{19} +1.53073 q^{20} -6.24264 q^{22} +2.82843i q^{23} +(-1.46508 - 0.923880i) q^{24} -2.65685 q^{25} -5.22625 q^{26} +(5.16059 + 0.606854i) q^{27} +0.828427i q^{29} +(-2.24264 - 1.41421i) q^{30} -6.75699i q^{31} -1.00000i q^{32} +(9.14594 + 5.76745i) q^{33} +2.29610i q^{34} +(1.29289 + 2.70711i) q^{36} -2.00000 q^{37} +1.84776 q^{38} +(7.65685 + 4.82843i) q^{39} -1.53073i q^{40} -9.23880 q^{41} +5.07107 q^{43} +6.24264i q^{44} +(1.97908 + 4.14386i) q^{45} +2.82843 q^{46} -3.69552 q^{47} +(-0.923880 + 1.46508i) q^{48} +2.65685i q^{50} +(2.12132 - 3.36396i) q^{51} +5.22625i q^{52} +6.48528i q^{53} +(0.606854 - 5.16059i) q^{54} +9.55582i q^{55} +(-2.70711 - 1.70711i) q^{57} +0.828427 q^{58} +4.46088 q^{59} +(-1.41421 + 2.24264i) q^{60} -5.86030i q^{61} -6.75699 q^{62} -1.00000 q^{64} +8.00000i q^{65} +(5.76745 - 9.14594i) q^{66} +6.48528 q^{67} +2.29610 q^{68} +(-4.14386 - 2.61313i) q^{69} -5.17157i q^{71} +(2.70711 - 1.29289i) q^{72} -0.765367i q^{73} +2.00000i q^{74} +(2.45461 - 3.89249i) q^{75} -1.84776i q^{76} +(4.82843 - 7.65685i) q^{78} +9.17157 q^{79} -1.53073 q^{80} +(-5.65685 + 7.00000i) q^{81} +9.23880i q^{82} +0.131316 q^{83} +3.51472 q^{85} -5.07107i q^{86} +(-1.21371 - 0.765367i) q^{87} +6.24264 q^{88} -6.17733 q^{89} +(4.14386 - 1.97908i) q^{90} -2.82843i q^{92} +(9.89949 + 6.24264i) q^{93} +3.69552i q^{94} -2.82843i q^{95} +(1.46508 + 0.923880i) q^{96} -2.48181i q^{97} +(-16.8995 + 8.07107i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} - 16 q^{9} + 8 q^{16} - 16 q^{18} - 16 q^{22} + 24 q^{25} + 16 q^{30} + 16 q^{36} - 16 q^{37} + 16 q^{39} - 16 q^{43} - 16 q^{57} - 16 q^{58} - 8 q^{64} - 16 q^{67} + 16 q^{72} + 16 q^{78}+ \cdots - 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/294\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(199\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) −0.923880 + 1.46508i −0.533402 + 0.845862i
\(4\) −1.00000 −0.500000
\(5\) −1.53073 −0.684565 −0.342282 − 0.939597i \(-0.611200\pi\)
−0.342282 + 0.939597i \(0.611200\pi\)
\(6\) 1.46508 + 0.923880i 0.598115 + 0.377172i
\(7\) 0 0
\(8\) 1.00000i 0.353553i
\(9\) −1.29289 − 2.70711i −0.430964 − 0.902369i
\(10\) 1.53073i 0.484061i
\(11\) − 6.24264i − 1.88223i −0.338091 − 0.941113i \(-0.609781\pi\)
0.338091 − 0.941113i \(-0.390219\pi\)
\(12\) 0.923880 − 1.46508i 0.266701 − 0.422931i
\(13\) − 5.22625i − 1.44950i −0.689011 − 0.724751i \(-0.741955\pi\)
0.689011 − 0.724751i \(-0.258045\pi\)
\(14\) 0 0
\(15\) 1.41421 − 2.24264i 0.365148 − 0.579047i
\(16\) 1.00000 0.250000
\(17\) −2.29610 −0.556886 −0.278443 − 0.960453i \(-0.589818\pi\)
−0.278443 + 0.960453i \(0.589818\pi\)
\(18\) −2.70711 + 1.29289i −0.638071 + 0.304738i
\(19\) 1.84776i 0.423905i 0.977280 + 0.211953i \(0.0679822\pi\)
−0.977280 + 0.211953i \(0.932018\pi\)
\(20\) 1.53073 0.342282
\(21\) 0 0
\(22\) −6.24264 −1.33094
\(23\) 2.82843i 0.589768i 0.955533 + 0.294884i \(0.0952810\pi\)
−0.955533 + 0.294884i \(0.904719\pi\)
\(24\) −1.46508 − 0.923880i −0.299057 − 0.188586i
\(25\) −2.65685 −0.531371
\(26\) −5.22625 −1.02495
\(27\) 5.16059 + 0.606854i 0.993157 + 0.116789i
\(28\) 0 0
\(29\) 0.828427i 0.153835i 0.997037 + 0.0769175i \(0.0245078\pi\)
−0.997037 + 0.0769175i \(0.975492\pi\)
\(30\) −2.24264 − 1.41421i −0.409448 − 0.258199i
\(31\) − 6.75699i − 1.21359i −0.794858 − 0.606795i \(-0.792455\pi\)
0.794858 − 0.606795i \(-0.207545\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 9.14594 + 5.76745i 1.59210 + 1.00398i
\(34\) 2.29610i 0.393778i
\(35\) 0 0
\(36\) 1.29289 + 2.70711i 0.215482 + 0.451184i
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 1.84776 0.299746
\(39\) 7.65685 + 4.82843i 1.22608 + 0.773167i
\(40\) − 1.53073i − 0.242030i
\(41\) −9.23880 −1.44286 −0.721429 − 0.692489i \(-0.756514\pi\)
−0.721429 + 0.692489i \(0.756514\pi\)
\(42\) 0 0
\(43\) 5.07107 0.773331 0.386665 − 0.922220i \(-0.373627\pi\)
0.386665 + 0.922220i \(0.373627\pi\)
\(44\) 6.24264i 0.941113i
\(45\) 1.97908 + 4.14386i 0.295023 + 0.617730i
\(46\) 2.82843 0.417029
\(47\) −3.69552 −0.539047 −0.269523 − 0.962994i \(-0.586866\pi\)
−0.269523 + 0.962994i \(0.586866\pi\)
\(48\) −0.923880 + 1.46508i −0.133351 + 0.211465i
\(49\) 0 0
\(50\) 2.65685i 0.375736i
\(51\) 2.12132 − 3.36396i 0.297044 − 0.471049i
\(52\) 5.22625i 0.724751i
\(53\) 6.48528i 0.890822i 0.895326 + 0.445411i \(0.146942\pi\)
−0.895326 + 0.445411i \(0.853058\pi\)
\(54\) 0.606854 − 5.16059i 0.0825824 − 0.702268i
\(55\) 9.55582i 1.28851i
\(56\) 0 0
\(57\) −2.70711 − 1.70711i −0.358565 − 0.226112i
\(58\) 0.828427 0.108778
\(59\) 4.46088 0.580758 0.290379 − 0.956912i \(-0.406219\pi\)
0.290379 + 0.956912i \(0.406219\pi\)
\(60\) −1.41421 + 2.24264i −0.182574 + 0.289524i
\(61\) − 5.86030i − 0.750335i −0.926957 − 0.375167i \(-0.877585\pi\)
0.926957 − 0.375167i \(-0.122415\pi\)
\(62\) −6.75699 −0.858138
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 8.00000i 0.992278i
\(66\) 5.76745 − 9.14594i 0.709924 − 1.12579i
\(67\) 6.48528 0.792303 0.396152 − 0.918185i \(-0.370345\pi\)
0.396152 + 0.918185i \(0.370345\pi\)
\(68\) 2.29610 0.278443
\(69\) −4.14386 − 2.61313i −0.498862 − 0.314583i
\(70\) 0 0
\(71\) − 5.17157i − 0.613753i −0.951749 − 0.306876i \(-0.900716\pi\)
0.951749 − 0.306876i \(-0.0992838\pi\)
\(72\) 2.70711 − 1.29289i 0.319036 − 0.152369i
\(73\) − 0.765367i − 0.0895794i −0.998996 − 0.0447897i \(-0.985738\pi\)
0.998996 − 0.0447897i \(-0.0142618\pi\)
\(74\) 2.00000i 0.232495i
\(75\) 2.45461 − 3.89249i 0.283434 − 0.449466i
\(76\) − 1.84776i − 0.211953i
\(77\) 0 0
\(78\) 4.82843 − 7.65685i 0.546712 − 0.866968i
\(79\) 9.17157 1.03188 0.515941 − 0.856624i \(-0.327442\pi\)
0.515941 + 0.856624i \(0.327442\pi\)
\(80\) −1.53073 −0.171141
\(81\) −5.65685 + 7.00000i −0.628539 + 0.777778i
\(82\) 9.23880i 1.02025i
\(83\) 0.131316 0.0144138 0.00720691 − 0.999974i \(-0.497706\pi\)
0.00720691 + 0.999974i \(0.497706\pi\)
\(84\) 0 0
\(85\) 3.51472 0.381225
\(86\) − 5.07107i − 0.546827i
\(87\) −1.21371 − 0.765367i −0.130123 − 0.0820559i
\(88\) 6.24264 0.665468
\(89\) −6.17733 −0.654795 −0.327398 − 0.944887i \(-0.606172\pi\)
−0.327398 + 0.944887i \(0.606172\pi\)
\(90\) 4.14386 − 1.97908i 0.436801 − 0.208613i
\(91\) 0 0
\(92\) − 2.82843i − 0.294884i
\(93\) 9.89949 + 6.24264i 1.02653 + 0.647332i
\(94\) 3.69552i 0.381164i
\(95\) − 2.82843i − 0.290191i
\(96\) 1.46508 + 0.923880i 0.149529 + 0.0942931i
\(97\) − 2.48181i − 0.251990i −0.992031 − 0.125995i \(-0.959788\pi\)
0.992031 − 0.125995i \(-0.0402123\pi\)
\(98\) 0 0
\(99\) −16.8995 + 8.07107i −1.69846 + 0.811173i
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 294.2.d.b.293.1 ✓ 8
3.2 odd 2 inner 294.2.d.b.293.8 yes 8
4.3 odd 2 2352.2.k.f.881.7 8
7.2 even 3 294.2.f.c.227.2 16
7.3 odd 6 294.2.f.c.215.5 16
7.4 even 3 294.2.f.c.215.8 16
7.5 odd 6 294.2.f.c.227.3 16
7.6 odd 2 inner 294.2.d.b.293.4 yes 8
12.11 even 2 2352.2.k.f.881.1 8
21.2 odd 6 294.2.f.c.227.5 16
21.5 even 6 294.2.f.c.227.8 16
21.11 odd 6 294.2.f.c.215.3 16
21.17 even 6 294.2.f.c.215.2 16
21.20 even 2 inner 294.2.d.b.293.5 yes 8
28.27 even 2 2352.2.k.f.881.2 8
84.83 odd 2 2352.2.k.f.881.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.2.d.b.293.1 ✓ 8 1.1 even 1 trivial
294.2.d.b.293.4 yes 8 7.6 odd 2 inner
294.2.d.b.293.5 yes 8 21.20 even 2 inner
294.2.d.b.293.8 yes 8 3.2 odd 2 inner
294.2.f.c.215.2 16 21.17 even 6
294.2.f.c.215.3 16 21.11 odd 6
294.2.f.c.215.5 16 7.3 odd 6
294.2.f.c.215.8 16 7.4 even 3
294.2.f.c.227.2 16 7.2 even 3
294.2.f.c.227.3 16 7.5 odd 6
294.2.f.c.227.5 16 21.2 odd 6
294.2.f.c.227.8 16 21.5 even 6
2352.2.k.f.881.1 8 12.11 even 2
2352.2.k.f.881.2 8 28.27 even 2
2352.2.k.f.881.7 8 4.3 odd 2
2352.2.k.f.881.8 8 84.83 odd 2