Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [49,7,Mod(19,49)] 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("49.19"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 49.d (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.2726500974\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 31.1
Root \(-0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 49.31
Dual form 49.7.d.c.19.1

$q$-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+(-4.12132 - 7.13834i) q^{2} +(23.0772 + 13.3236i) q^{3} +(-1.97056 + 3.41311i) q^{4} +(-68.5660 + 39.5866i) q^{5} -219.643i q^{6} -495.044 q^{8} +(-9.46299 - 16.3904i) q^{9} +(565.165 + 326.298i) q^{10} +(854.459 - 1479.97i) q^{11} +(-90.9500 + 52.5100i) q^{12} -3129.09i q^{13} -2109.75 q^{15} +(2166.35 + 3752.23i) q^{16} +(-3529.96 - 2038.03i) q^{17} +(-78.0000 + 135.100i) q^{18} +(-5085.89 + 2936.34i) q^{19} -312.032i q^{20} -14086.0 q^{22} +(-6660.75 - 11536.8i) q^{23} +(-11424.2 - 6595.77i) q^{24} +(-4678.30 + 8103.05i) q^{25} +(-22336.5 + 12896.0i) q^{26} -19930.1i q^{27} +6510.23 q^{29} +(8694.94 + 15060.1i) q^{30} +(10386.6 + 5996.69i) q^{31} +(2015.04 - 3490.16i) q^{32} +(39437.0 - 22769.0i) q^{33} +33597.4i q^{34} +74.5896 q^{36} +(2320.45 + 4019.14i) q^{37} +(41921.2 + 24203.2i) q^{38} +(41690.8 - 72210.6i) q^{39} +(33943.2 - 19597.1i) q^{40} -19308.8i q^{41} +91636.4 q^{43} +(3367.53 + 5832.73i) q^{44} +(1297.68 + 749.215i) q^{45} +(-54902.2 + 95093.3i) q^{46} +(55800.2 - 32216.2i) q^{47} +115454. i q^{48} +77123.1 q^{50} +(-54307.7 - 94063.7i) q^{51} +(10680.0 + 6166.07i) q^{52} +(-74799.9 + 129557. i) q^{53} +(-142268. + 82138.5i) q^{54} +135301. i q^{55} -156491. q^{57} +(-26830.7 - 46472.2i) q^{58} +(-52855.0 - 30515.9i) q^{59} +(4157.39 - 7200.80i) q^{60} +(85403.4 - 49307.7i) q^{61} -98857.1i q^{62} +244074. q^{64} +(123870. + 214549. i) q^{65} +(-325065. - 187676. i) q^{66} +(155906. - 270038. i) q^{67} +(13912.0 - 8032.11i) q^{68} -354981. i q^{69} -401209. q^{71} +(4684.59 + 8113.95i) q^{72} +(582322. + 336204. i) q^{73} +(19126.6 - 33128.3i) q^{74} +(-215924. + 124664. i) q^{75} -23145.0i q^{76} -687285. q^{78} +(160076. + 277260. i) q^{79} +(-297076. - 171517. i) q^{80} +(258643. - 447983. i) q^{81} +(-137833. + 79577.6i) q^{82} -832356. i q^{83} +322714. q^{85} +(-377663. - 654131. i) q^{86} +(150238. + 86739.7i) q^{87} +(-422995. + 732648. i) q^{88} +(-328654. + 189748. i) q^{89} -12351.0i q^{90} +52501.7 q^{92} +(159795. + 276773. i) q^{93} +(-459941. - 265547. i) q^{94} +(232480. - 402666. i) q^{95} +(93003.0 - 53695.3i) q^{96} +1.05514e6i q^{97} -32342.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{2} - 18 q^{3} + 60 q^{4} + 150 q^{5} - 928 q^{8} + 624 q^{9} + 1200 q^{10} + 1882 q^{11} - 4284 q^{12} - 16500 q^{15} + 248 q^{16} - 13458 q^{17} - 312 q^{18} - 18078 q^{19} - 28088 q^{22} + 2470 q^{23}+ \cdots + 157872 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.12132 − 7.13834i −0.515165 − 0.892292i −0.999845 − 0.0176005i \(-0.994397\pi\)
0.484680 − 0.874692i \(-0.338936\pi\)
\(3\) 23.0772 + 13.3236i 0.854710 + 0.493467i 0.862237 − 0.506505i \(-0.169063\pi\)
−0.00752738 + 0.999972i \(0.502396\pi\)
\(4\) −1.97056 + 3.41311i −0.0307900 + 0.0533299i
\(5\) −68.5660 + 39.5866i −0.548528 + 0.316693i −0.748528 − 0.663103i \(-0.769239\pi\)
0.200000 + 0.979796i \(0.435906\pi\)
\(6\) − 219.643i − 1.01687i
\(7\) 0 0
\(8\) −495.044 −0.966882
\(9\) −9.46299 − 16.3904i −0.0129808 − 0.0224834i
\(10\) 565.165 + 326.298i 0.565165 + 0.326298i
\(11\) 854.459 − 1479.97i 0.641968 − 1.11192i −0.343025 − 0.939326i \(-0.611452\pi\)
0.984993 − 0.172594i \(-0.0552150\pi\)
\(12\) −90.9500 + 52.5100i −0.0526331 + 0.0303877i
\(13\) − 3129.09i − 1.42426i −0.702049 − 0.712129i \(-0.747731\pi\)
0.702049 − 0.712129i \(-0.252269\pi\)
\(14\) 0 0
\(15\) −2109.75 −0.625110
\(16\) 2166.35 + 3752.23i 0.528894 + 0.916071i
\(17\) −3529.96 − 2038.03i −0.718494 − 0.414823i 0.0957039 − 0.995410i \(-0.469490\pi\)
−0.814198 + 0.580587i \(0.802823\pi\)
\(18\) −78.0000 + 135.100i −0.0133745 + 0.0231653i
\(19\) −5085.89 + 2936.34i −0.741492 + 0.428101i −0.822611 − 0.568604i \(-0.807484\pi\)
0.0811196 + 0.996704i \(0.474150\pi\)
\(20\) − 312.032i − 0.0390039i
\(21\) 0 0
\(22\) −14086.0 −1.32288
\(23\) −6660.75 − 11536.8i −0.547444 − 0.948201i −0.998449 − 0.0556791i \(-0.982268\pi\)
0.451005 − 0.892522i \(-0.351066\pi\)
\(24\) −11424.2 − 6595.77i −0.826404 − 0.477124i
\(25\) −4678.30 + 8103.05i −0.299411 + 0.518596i
\(26\) −22336.5 + 12896.0i −1.27085 + 0.733728i
\(27\) − 19930.1i − 1.01256i
\(28\) 0 0
\(29\) 6510.23 0.266933 0.133466 − 0.991053i \(-0.457389\pi\)
0.133466 + 0.991053i \(0.457389\pi\)
\(30\) 8694.94 + 15060.1i 0.322035 + 0.557781i
\(31\) 10386.6 + 5996.69i 0.348648 + 0.201292i 0.664090 − 0.747653i \(-0.268819\pi\)
−0.315442 + 0.948945i \(0.602153\pi\)
\(32\) 2015.04 − 3490.16i 0.0614943 − 0.106511i
\(33\) 39437.0 − 22769.0i 1.09739 − 0.633580i
\(34\) 33597.4i 0.854809i
\(35\) 0 0
\(36\) 74.5896 0.00159871
\(37\) 2320.45 + 4019.14i 0.0458107 + 0.0793465i 0.888022 − 0.459802i \(-0.152080\pi\)
−0.842211 + 0.539148i \(0.818746\pi\)
\(38\) 41921.2 + 24203.2i 0.763981 + 0.441085i
\(39\) 41690.8 − 72210.6i 0.702824 − 1.21733i
\(40\) 33943.2 − 19597.1i 0.530362 − 0.306205i
\(41\) − 19308.8i − 0.280158i −0.990140 − 0.140079i \(-0.955264\pi\)
0.990140 − 0.140079i \(-0.0447357\pi\)
\(42\) 0 0
\(43\) 91636.4 1.15256 0.576279 − 0.817253i \(-0.304504\pi\)
0.576279 + 0.817253i \(0.304504\pi\)
\(44\) 3367.53 + 5832.73i 0.0395324 + 0.0684722i
\(45\) 1297.68 + 749.215i 0.0142406 + 0.00822184i
\(46\) −54902.2 + 95093.3i −0.564048 + 0.976960i
\(47\) 55800.2 − 32216.2i 0.537455 − 0.310300i −0.206592 − 0.978427i \(-0.566237\pi\)
0.744047 + 0.668128i \(0.232904\pi\)
\(48\) 115454.i 1.04397i
\(49\) 0 0
\(50\) 77123.1 0.616985
\(51\) −54307.7 − 94063.7i −0.409403 − 0.709106i
\(52\) 10680.0 + 6166.07i 0.0759555 + 0.0438529i
\(53\) −74799.9 + 129557.i −0.502428 + 0.870230i 0.497568 + 0.867425i \(0.334226\pi\)
−0.999996 + 0.00280549i \(0.999107\pi\)
\(54\) −142268. + 82138.5i −0.903496 + 0.521634i
\(55\) 135301.i 0.813226i
\(56\) 0 0
\(57\) −156491. −0.845014
\(58\) −26830.7 − 46472.2i −0.137515 − 0.238182i
\(59\) −52855.0 − 30515.9i −0.257354 − 0.148583i 0.365773 − 0.930704i \(-0.380805\pi\)
−0.623127 + 0.782121i \(0.714138\pi\)
\(60\) 4157.39 − 7200.80i 0.0192472 − 0.0333371i
\(61\) 85403.4 − 49307.7i 0.376258 − 0.217233i −0.299931 − 0.953961i \(-0.596964\pi\)
0.676189 + 0.736728i \(0.263630\pi\)
\(62\) − 98857.1i − 0.414794i
\(63\) 0 0
\(64\) 244074. 0.931069
\(65\) 123870. + 214549.i 0.451052 + 0.781245i
\(66\) −325065. − 187676.i −1.13068 − 0.652796i
\(67\) 155906. − 270038.i 0.518369 − 0.897842i −0.481403 − 0.876499i \(-0.659873\pi\)
0.999772 − 0.0213423i \(-0.00679398\pi\)
\(68\) 13912.0 − 8032.11i 0.0442449 − 0.0255448i
\(69\) − 354981.i − 1.08058i
\(70\) 0 0
\(71\) −401209. −1.12097 −0.560487 − 0.828163i \(-0.689386\pi\)
−0.560487 + 0.828163i \(0.689386\pi\)
\(72\) 4684.59 + 8113.95i 0.0125509 + 0.0217388i
\(73\) 582322. + 336204.i 1.49691 + 0.864239i 0.999994 − 0.00356186i \(-0.00113378\pi\)
0.496912 + 0.867801i \(0.334467\pi\)
\(74\) 19126.6 − 33128.3i 0.0472002 − 0.0817531i
\(75\) −215924. + 124664.i −0.511819 + 0.295499i
\(76\) − 23145.0i − 0.0527249i
\(77\) 0 0
\(78\) −687285. −1.44828
\(79\) 160076. + 277260.i 0.324672 + 0.562348i 0.981446 − 0.191739i \(-0.0614128\pi\)
−0.656774 + 0.754087i \(0.728079\pi\)
\(80\) −297076. − 171517.i −0.580226 − 0.334994i
\(81\) 258643. − 447983.i 0.486682 − 0.842958i
\(82\) −137833. + 79577.6i −0.249983 + 0.144328i
\(83\) − 832356.i − 1.45571i −0.685731 − 0.727855i \(-0.740517\pi\)
0.685731 − 0.727855i \(-0.259483\pi\)
\(84\) 0 0
\(85\) 322714. 0.525486
\(86\) −377663. − 654131.i −0.593757 − 1.02842i
\(87\) 150238. + 86739.7i 0.228150 + 0.131723i
\(88\) −422995. + 732648.i −0.620707 + 1.07510i
\(89\) −328654. + 189748.i −0.466196 + 0.269158i −0.714646 − 0.699486i \(-0.753412\pi\)
0.248450 + 0.968645i \(0.420079\pi\)
\(90\) − 12351.0i − 0.0169424i
\(91\) 0 0
\(92\) 52501.7 0.0674233
\(93\) 159795. + 276773.i 0.198662 + 0.344092i
\(94\) −459941. − 265547.i −0.553756 − 0.319711i
\(95\) 232480. − 402666.i 0.271153 − 0.469650i
\(96\) 93003.0 − 53695.3i 0.105119 − 0.0606908i
\(97\) 1.05514e6i 1.15610i 0.816001 + 0.578050i \(0.196186\pi\)
−0.816001 + 0.578050i \(0.803814\pi\)
\(98\) 0 0
\(99\) −32342.9 −0.0333330
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 49.7.d.c.31.1 4
7.2 even 3 7.7.d.b.5.1 yes 4
7.3 odd 6 49.7.b.b.48.4 4
7.4 even 3 49.7.b.b.48.3 4
7.5 odd 6 inner 49.7.d.c.19.1 4
7.6 odd 2 7.7.d.b.3.1 ✓ 4
21.2 odd 6 63.7.m.b.19.2 4
21.11 odd 6 441.7.d.b.244.2 4
21.17 even 6 441.7.d.b.244.1 4
21.20 even 2 63.7.m.b.10.2 4
28.23 odd 6 112.7.s.b.33.2 4
28.27 even 2 112.7.s.b.17.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.7.d.b.3.1 ✓ 4 7.6 odd 2
7.7.d.b.5.1 yes 4 7.2 even 3
49.7.b.b.48.3 4 7.4 even 3
49.7.b.b.48.4 4 7.3 odd 6
49.7.d.c.19.1 4 7.5 odd 6 inner
49.7.d.c.31.1 4 1.1 even 1 trivial
63.7.m.b.10.2 4 21.20 even 2
63.7.m.b.19.2 4 21.2 odd 6
112.7.s.b.17.2 4 28.27 even 2
112.7.s.b.33.2 4 28.23 odd 6
441.7.d.b.244.1 4 21.17 even 6
441.7.d.b.244.2 4 21.11 odd 6