Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8,10,Mod(5,8)] 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("8.5"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 8.b (of order \(2\), degree \(1\), 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: \(4.12028668931\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 59x^{6} - 313x^{5} - 315x^{4} - 92091x^{3} + 1261649x^{2} - 16074123x + 251007534 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{28}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 5.6
Root \(-2.43481 - 11.2224i\) of defining polynomial
Character \(\chi\) \(=\) 8.5
Dual form 8.10.b.a.5.5

$q$-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+(2.86961 + 22.4447i) q^{2} -247.414i q^{3} +(-495.531 + 128.815i) q^{4} -1417.55i q^{5} +(5553.14 - 709.983i) q^{6} +5087.57 q^{7} +(-4313.20 - 10752.4i) q^{8} -41530.8 q^{9} +(31816.6 - 4067.83i) q^{10} +14811.2i q^{11} +(31870.7 + 122601. i) q^{12} -64089.8i q^{13} +(14599.3 + 114189. i) q^{14} -350723. q^{15} +(228957. - 127664. i) q^{16} +251342. q^{17} +(-119177. - 932147. i) q^{18} +511568. i q^{19} +(182602. + 702441. i) q^{20} -1.25874e6i q^{21} +(-332432. + 42502.3i) q^{22} +1.96905e6 q^{23} +(-2.66030e6 + 1.06715e6i) q^{24} -56329.6 q^{25} +(1.43848e6 - 183913. i) q^{26} +5.40545e6i q^{27} +(-2.52105e6 + 655356. i) q^{28} -2.16116e6i q^{29} +(-1.00644e6 - 7.87187e6i) q^{30} -3.03997e6 q^{31} +(3.52240e6 + 4.77253e6i) q^{32} +3.66449e6 q^{33} +(721253. + 5.64129e6i) q^{34} -7.21189e6i q^{35} +(2.05798e7 - 5.34980e6i) q^{36} -8.74315e6i q^{37} +(-1.14820e7 + 1.46800e6i) q^{38} -1.58567e7 q^{39} +(-1.52421e7 + 6.11419e6i) q^{40} -1.48418e7 q^{41} +(2.82520e7 - 3.61208e6i) q^{42} -153307. i q^{43} +(-1.90790e6 - 7.33938e6i) q^{44} +5.88721e7i q^{45} +(5.65042e6 + 4.41949e7i) q^{46} +5.17595e7 q^{47} +(-3.15858e7 - 5.66473e7i) q^{48} -1.44703e7 q^{49} +(-161644. - 1.26430e6i) q^{50} -6.21855e7i q^{51} +(8.25574e6 + 3.17584e7i) q^{52} -4.26277e7i q^{53} +(-1.21324e8 + 1.55116e7i) q^{54} +2.09956e7 q^{55} +(-2.19437e7 - 5.47035e7i) q^{56} +1.26569e8 q^{57} +(4.85067e7 - 6.20170e6i) q^{58} -1.15231e7i q^{59} +(1.73794e8 - 4.51784e7i) q^{60} +1.95800e8i q^{61} +(-8.72353e6 - 6.82312e7i) q^{62} -2.11291e8 q^{63} +(-9.70103e7 + 9.27545e7i) q^{64} -9.08506e7 q^{65} +(1.05157e7 + 8.22485e7i) q^{66} +1.69718e8i q^{67} +(-1.24547e8 + 3.23766e7i) q^{68} -4.87172e8i q^{69} +(1.61869e8 - 2.06953e7i) q^{70} +2.23552e8 q^{71} +(1.79131e8 + 4.46555e8i) q^{72} -4.10130e8 q^{73} +(1.96238e8 - 2.50895e7i) q^{74} +1.39367e7i q^{75} +(-6.58977e7 - 2.53497e8i) q^{76} +7.53528e7i q^{77} +(-4.55026e7 - 3.55899e8i) q^{78} +2.51240e7 q^{79} +(-1.80970e8 - 3.24559e8i) q^{80} +5.19935e8 q^{81} +(-4.25902e7 - 3.33120e8i) q^{82} +4.73174e8i q^{83} +(1.62144e8 + 6.23742e8i) q^{84} -3.56290e8i q^{85} +(3.44094e6 - 439932. i) q^{86} -5.34702e8 q^{87} +(1.59255e8 - 6.38836e7i) q^{88} +8.45560e8 q^{89} +(-1.32137e9 + 1.68940e8i) q^{90} -3.26061e8i q^{91} +(-9.75726e8 + 2.53644e8i) q^{92} +7.52131e8i q^{93} +(1.48530e8 + 1.16173e9i) q^{94} +7.25174e8 q^{95} +(1.18079e9 - 8.71491e8i) q^{96} -7.97179e8 q^{97} +(-4.15241e7 - 3.24781e8i) q^{98} -6.15119e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 18 q^{2} - 428 q^{4} + 4684 q^{6} + 4800 q^{7} - 3384 q^{8} - 39368 q^{9} + 26392 q^{10} + 54760 q^{12} - 72336 q^{14} - 163136 q^{15} + 185616 q^{16} - 102000 q^{17} - 23614 q^{18} + 1245264 q^{20}+ \cdots - 3062604162 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.86961 + 22.4447i 0.126820 + 0.991926i
\(3\) − 247.414i − 1.76351i −0.471704 − 0.881757i \(-0.656361\pi\)
0.471704 − 0.881757i \(-0.343639\pi\)
\(4\) −495.531 + 128.815i −0.967833 + 0.251592i
\(5\) − 1417.55i − 1.01432i −0.861853 − 0.507159i \(-0.830696\pi\)
0.861853 − 0.507159i \(-0.169304\pi\)
\(6\) 5553.14 − 709.983i 1.74928 − 0.223649i
\(7\) 5087.57 0.800883 0.400441 − 0.916322i \(-0.368857\pi\)
0.400441 + 0.916322i \(0.368857\pi\)
\(8\) −4313.20 − 10752.4i −0.372302 − 0.928112i
\(9\) −41530.8 −2.10998
\(10\) 31816.6 − 4067.83i 1.00613 − 0.128636i
\(11\) 14811.2i 0.305016i 0.988302 + 0.152508i \(0.0487349\pi\)
−0.988302 + 0.152508i \(0.951265\pi\)
\(12\) 31870.7 + 122601.i 0.443687 + 1.70679i
\(13\) − 64089.8i − 0.622363i −0.950351 − 0.311181i \(-0.899275\pi\)
0.950351 − 0.311181i \(-0.100725\pi\)
\(14\) 14599.3 + 114189.i 0.101568 + 0.794416i
\(15\) −350723. −1.78876
\(16\) 228957. − 127664.i 0.873403 − 0.486999i
\(17\) 251342. 0.729868 0.364934 − 0.931033i \(-0.381092\pi\)
0.364934 + 0.931033i \(0.381092\pi\)
\(18\) −119177. − 932147.i −0.267588 − 2.09295i
\(19\) 511568.i 0.900558i 0.892888 + 0.450279i \(0.148676\pi\)
−0.892888 + 0.450279i \(0.851324\pi\)
\(20\) 182602. + 702441.i 0.255195 + 0.981691i
\(21\) − 1.25874e6i − 1.41237i
\(22\) −332432. + 42502.3i −0.302553 + 0.0386821i
\(23\) 1.96905e6 1.46718 0.733588 − 0.679594i \(-0.237844\pi\)
0.733588 + 0.679594i \(0.237844\pi\)
\(24\) −2.66030e6 + 1.06715e6i −1.63674 + 0.656559i
\(25\) −56329.6 −0.0288408
\(26\) 1.43848e6 − 183913.i 0.617337 − 0.0789281i
\(27\) 5.40545e6i 1.95747i
\(28\) −2.52105e6 + 655356.i −0.775121 + 0.201496i
\(29\) − 2.16116e6i − 0.567409i −0.958912 − 0.283705i \(-0.908436\pi\)
0.958912 − 0.283705i \(-0.0915636\pi\)
\(30\) −1.00644e6 − 7.87187e6i −0.226851 − 1.77432i
\(31\) −3.03997e6 −0.591210 −0.295605 − 0.955310i \(-0.595521\pi\)
−0.295605 + 0.955310i \(0.595521\pi\)
\(32\) 3.52240e6 + 4.77253e6i 0.593832 + 0.804589i
\(33\) 3.66449e6 0.537899
\(34\) 721253. + 5.64129e6i 0.0925620 + 0.723975i
\(35\) − 7.21189e6i − 0.812350i
\(36\) 2.05798e7 − 5.34980e6i 2.04211 − 0.530855i
\(37\) − 8.74315e6i − 0.766938i −0.923554 − 0.383469i \(-0.874729\pi\)
0.923554 − 0.383469i \(-0.125271\pi\)
\(38\) −1.14820e7 + 1.46800e6i −0.893287 + 0.114209i
\(39\) −1.58567e7 −1.09755
\(40\) −1.52421e7 + 6.11419e6i −0.941400 + 0.377632i
\(41\) −1.48418e7 −0.820274 −0.410137 − 0.912024i \(-0.634519\pi\)
−0.410137 + 0.912024i \(0.634519\pi\)
\(42\) 2.82520e7 − 3.61208e6i 1.40096 − 0.179117i
\(43\) − 153307.i − 0.00683840i −0.999994 − 0.00341920i \(-0.998912\pi\)
0.999994 − 0.00341920i \(-0.00108837\pi\)
\(44\) −1.90790e6 − 7.33938e6i −0.0767396 − 0.295204i
\(45\) 5.88721e7i 2.14019i
\(46\) 5.65042e6 + 4.41949e7i 0.186068 + 1.45533i
\(47\) 5.17595e7 1.54721 0.773605 − 0.633668i \(-0.218451\pi\)
0.773605 + 0.633668i \(0.218451\pi\)
\(48\) −3.15858e7 − 5.66473e7i −0.858829 − 1.54026i
\(49\) −1.44703e7 −0.358587
\(50\) −161644. − 1.26430e6i −0.00365759 − 0.0286079i
\(51\) − 6.21855e7i − 1.28713i
\(52\) 8.25574e6 + 3.17584e7i 0.156582 + 0.602343i
\(53\) − 4.26277e7i − 0.742080i −0.928617 − 0.371040i \(-0.879001\pi\)
0.928617 − 0.371040i \(-0.120999\pi\)
\(54\) −1.21324e8 + 1.55116e7i −1.94166 + 0.248247i
\(55\) 2.09956e7 0.309383
\(56\) −2.19437e7 − 5.47035e7i −0.298170 − 0.743309i
\(57\) 1.26569e8 1.58815
\(58\) 4.85067e7 − 6.20170e6i 0.562828 − 0.0719589i
\(59\) − 1.15231e7i − 0.123804i −0.998082 − 0.0619021i \(-0.980283\pi\)
0.998082 − 0.0619021i \(-0.0197167\pi\)
\(60\) 1.73794e8 − 4.51784e7i 1.73123 − 0.450039i
\(61\) 1.95800e8i 1.81062i 0.424748 + 0.905312i \(0.360363\pi\)
−0.424748 + 0.905312i \(0.639637\pi\)
\(62\) −8.72353e6 − 6.82312e7i −0.0749773 − 0.586436i
\(63\) −2.11291e8 −1.68985
\(64\) −9.70103e7 + 9.27545e7i −0.722783 + 0.691075i
\(65\) −9.08506e7 −0.631274
\(66\) 1.05157e7 + 8.22485e7i 0.0682165 + 0.533556i
\(67\) 1.69718e8i 1.02894i 0.857508 + 0.514470i \(0.172011\pi\)
−0.857508 + 0.514470i \(0.827989\pi\)
\(68\) −1.24547e8 + 3.23766e7i −0.706390 + 0.183629i
\(69\) − 4.87172e8i − 2.58739i
\(70\) 1.61869e8 − 2.06953e7i 0.805790 − 0.103022i
\(71\) 2.23552e8 1.04404 0.522018 − 0.852934i \(-0.325179\pi\)
0.522018 + 0.852934i \(0.325179\pi\)
\(72\) 1.79131e8 + 4.46555e8i 0.785550 + 1.95830i
\(73\) −4.10130e8 −1.69032 −0.845159 − 0.534514i \(-0.820495\pi\)
−0.845159 + 0.534514i \(0.820495\pi\)
\(74\) 1.96238e8 − 2.50895e7i 0.760746 − 0.0972632i
\(75\) 1.39367e7i 0.0508611i
\(76\) −6.58977e7 − 2.53497e8i −0.226574 − 0.871590i
\(77\) 7.53528e7i 0.244282i
\(78\) −4.55026e7 − 3.55899e8i −0.139191 − 1.08868i
\(79\) 2.51240e7 0.0725716 0.0362858 − 0.999341i \(-0.488447\pi\)
0.0362858 + 0.999341i \(0.488447\pi\)
\(80\) −1.80970e8 − 3.24559e8i −0.493972 − 0.885908i
\(81\) 5.19935e8 1.34204
\(82\) −4.25902e7 − 3.33120e8i −0.104027 − 0.813651i
\(83\) 4.73174e8i 1.09438i 0.837008 + 0.547191i \(0.184303\pi\)
−0.837008 + 0.547191i \(0.815697\pi\)
\(84\) 1.62144e8 + 6.23742e8i 0.355341 + 1.36694i
\(85\) − 3.56290e8i − 0.740318i
\(86\) 3.44094e6 − 439932.i 0.00678318 − 0.000867247i
\(87\) −5.34702e8 −1.00063
\(88\) 1.59255e8 − 6.38836e7i 0.283089 − 0.113558i
\(89\) 8.45560e8 1.42853 0.714265 − 0.699875i \(-0.246761\pi\)
0.714265 + 0.699875i \(0.246761\pi\)
\(90\) −1.32137e9 + 1.68940e8i −2.12291 + 0.271420i
\(91\) − 3.26061e8i − 0.498439i
\(92\) −9.75726e8 + 2.53644e8i −1.41998 + 0.369130i
\(93\) 7.52131e8i 1.04261i
\(94\) 1.48530e8 + 1.16173e9i 0.196217 + 1.53472i
\(95\) 7.25174e8 0.913452
\(96\) 1.18079e9 − 8.71491e8i 1.41890 − 1.04723i
\(97\) −7.97179e8 −0.914288 −0.457144 − 0.889393i \(-0.651128\pi\)
−0.457144 + 0.889393i \(0.651128\pi\)
\(98\) −4.15241e7 − 3.24781e8i −0.0454760 − 0.355692i
\(99\) − 6.15119e8i − 0.643578i
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 8.10.b.a.5.6 yes 8
3.2 odd 2 72.10.d.b.37.3 8
4.3 odd 2 32.10.b.a.17.8 8
8.3 odd 2 32.10.b.a.17.1 8
8.5 even 2 inner 8.10.b.a.5.5 ✓ 8
12.11 even 2 288.10.d.b.145.7 8
16.3 odd 4 256.10.a.s.1.1 8
16.5 even 4 256.10.a.p.1.1 8
16.11 odd 4 256.10.a.s.1.8 8
16.13 even 4 256.10.a.p.1.8 8
24.5 odd 2 72.10.d.b.37.4 8
24.11 even 2 288.10.d.b.145.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.10.b.a.5.5 ✓ 8 8.5 even 2 inner
8.10.b.a.5.6 yes 8 1.1 even 1 trivial
32.10.b.a.17.1 8 8.3 odd 2
32.10.b.a.17.8 8 4.3 odd 2
72.10.d.b.37.3 8 3.2 odd 2
72.10.d.b.37.4 8 24.5 odd 2
256.10.a.p.1.1 8 16.5 even 4
256.10.a.p.1.8 8 16.13 even 4
256.10.a.s.1.1 8 16.3 odd 4
256.10.a.s.1.8 8 16.11 odd 4
288.10.d.b.145.2 8 24.11 even 2
288.10.d.b.145.7 8 12.11 even 2