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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [49,7,Mod(48,49)] 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("49.48"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 49.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] 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: \(11.2726500974\)
Analytic rank: \(0\)
Dimension: \(4\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 7 \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 48.1
Root \(-0.707107 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 49.48
Dual form 49.7.b.b.48.2

$q$-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)
 
\(f(q)\) \(=\) \(q-0.242641 q^{2} -37.0395i q^{3} -63.9411 q^{4} -165.776i q^{5} +8.98729i q^{6} +31.0437 q^{8} -642.926 q^{9} +40.2239i q^{10} -173.082 q^{11} +2368.35i q^{12} +1963.14i q^{13} -6140.25 q^{15} +4084.70 q^{16} -3693.93i q^{17} +156.000 q^{18} +4564.66i q^{19} +10599.9i q^{20} +41.9967 q^{22} -15791.5 q^{23} -1149.84i q^{24} -11856.6 q^{25} -476.337i q^{26} -3188.14i q^{27} -23782.2 q^{29} +1489.88 q^{30} -2061.80i q^{31} -2977.91 q^{32} +6410.88i q^{33} +896.298i q^{34} +41109.4 q^{36} +48510.9 q^{37} -1107.57i q^{38} +72713.6 q^{39} -5146.30i q^{40} -26437.9i q^{41} +68471.6 q^{43} +11067.1 q^{44} +106582. i q^{45} +3831.66 q^{46} -140362. i q^{47} -151295. i q^{48} +2876.89 q^{50} -136821. q^{51} -125525. i q^{52} -254130. q^{53} +773.573i q^{54} +28692.8i q^{55} +169073. q^{57} +5770.54 q^{58} +97200.4i q^{59} +392615. q^{60} +17392.5i q^{61} +500.276i q^{62} -260698. q^{64} +325440. q^{65} -1555.54i q^{66} -120682. q^{67} +236194. i q^{68} +584910. i q^{69} -339555. q^{71} -19958.8 q^{72} -111351. i q^{73} -11770.7 q^{74} +439163. i q^{75} -291869. i q^{76} -17643.3 q^{78} -615838. q^{79} -677144. i q^{80} -586780. q^{81} +6414.91i q^{82} +383668. i q^{83} -612364. q^{85} -16614.0 q^{86} +880882. i q^{87} -5373.11 q^{88} -771685. i q^{89} -25861.0i q^{90} +1.00973e6 q^{92} -76367.9 q^{93} +34057.5i q^{94} +756709. q^{95} +110300. i q^{96} -292263. i q^{97} +111279. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 16 q^{2} - 120 q^{4} - 928 q^{8} - 1248 q^{9} - 3764 q^{11} - 16500 q^{15} - 496 q^{16} + 624 q^{18} - 28088 q^{22} - 4940 q^{23} - 5000 q^{25} - 34544 q^{29} - 31800 q^{30} - 14016 q^{32} + 82368 q^{36}+ \cdots + 157872 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.242641 −0.0303301 −0.0151650 − 0.999885i \(-0.504827\pi\)
−0.0151650 + 0.999885i \(0.504827\pi\)
\(3\) − 37.0395i − 1.37183i −0.727680 − 0.685917i \(-0.759401\pi\)
0.727680 − 0.685917i \(-0.240599\pi\)
\(4\) −63.9411 −0.999080
\(5\) − 165.776i − 1.32621i −0.748528 − 0.663103i \(-0.769239\pi\)
0.748528 − 0.663103i \(-0.230761\pi\)
\(6\) 8.98729i 0.0416078i
\(7\) 0 0
\(8\) 31.0437 0.0606323
\(9\) −642.926 −0.881929
\(10\) 40.2239i 0.0402239i
\(11\) −173.082 −0.130039 −0.0650195 − 0.997884i \(-0.520711\pi\)
−0.0650195 + 0.997884i \(0.520711\pi\)
\(12\) 2368.35i 1.37057i
\(13\) 1963.14i 0.893553i 0.894646 + 0.446777i \(0.147428\pi\)
−0.894646 + 0.446777i \(0.852572\pi\)
\(14\) 0 0
\(15\) −6140.25 −1.81933
\(16\) 4084.70 0.997241
\(17\) − 3693.93i − 0.751868i −0.926646 − 0.375934i \(-0.877322\pi\)
0.926646 − 0.375934i \(-0.122678\pi\)
\(18\) 156.000 0.0267490
\(19\) 4564.66i 0.665499i 0.943015 + 0.332749i \(0.107976\pi\)
−0.943015 + 0.332749i \(0.892024\pi\)
\(20\) 10599.9i 1.32499i
\(21\) 0 0
\(22\) 41.9967 0.00394410
\(23\) −15791.5 −1.29790 −0.648948 − 0.760833i \(-0.724791\pi\)
−0.648948 + 0.760833i \(0.724791\pi\)
\(24\) − 1149.84i − 0.0831774i
\(25\) −11856.6 −0.758823
\(26\) − 476.337i − 0.0271015i
\(27\) − 3188.14i − 0.161974i
\(28\) 0 0
\(29\) −23782.2 −0.975121 −0.487561 − 0.873089i \(-0.662113\pi\)
−0.487561 + 0.873089i \(0.662113\pi\)
\(30\) 1489.88 0.0551806
\(31\) − 2061.80i − 0.0692087i −0.999401 − 0.0346044i \(-0.988983\pi\)
0.999401 − 0.0346044i \(-0.0110171\pi\)
\(32\) −2977.91 −0.0908787
\(33\) 6410.88i 0.178392i
\(34\) 896.298i 0.0228042i
\(35\) 0 0
\(36\) 41109.4 0.881117
\(37\) 48510.9 0.957710 0.478855 − 0.877894i \(-0.341052\pi\)
0.478855 + 0.877894i \(0.341052\pi\)
\(38\) − 1107.57i − 0.0201846i
\(39\) 72713.6 1.22581
\(40\) − 5146.30i − 0.0804109i
\(41\) − 26437.9i − 0.383597i −0.981434 − 0.191799i \(-0.938568\pi\)
0.981434 − 0.191799i \(-0.0614320\pi\)
\(42\) 0 0
\(43\) 68471.6 0.861202 0.430601 − 0.902542i \(-0.358302\pi\)
0.430601 + 0.902542i \(0.358302\pi\)
\(44\) 11067.1 0.129919
\(45\) 106582.i 1.16962i
\(46\) 3831.66 0.0393653
\(47\) − 140362.i − 1.35193i −0.736932 − 0.675967i \(-0.763726\pi\)
0.736932 − 0.675967i \(-0.236274\pi\)
\(48\) − 151295.i − 1.36805i
\(49\) 0 0
\(50\) 2876.89 0.0230152
\(51\) −136821. −1.03144
\(52\) − 125525.i − 0.892731i
\(53\) −254130. −1.70698 −0.853489 − 0.521110i \(-0.825518\pi\)
−0.853489 + 0.521110i \(0.825518\pi\)
\(54\) 773.573i 0.00491270i
\(55\) 28692.8i 0.172459i
\(56\) 0 0
\(57\) 169073. 0.912954
\(58\) 5770.54 0.0295755
\(59\) 97200.4i 0.473273i 0.971598 + 0.236637i \(0.0760451\pi\)
−0.971598 + 0.236637i \(0.923955\pi\)
\(60\) 392615. 1.81766
\(61\) 17392.5i 0.0766255i 0.999266 + 0.0383127i \(0.0121983\pi\)
−0.999266 + 0.0383127i \(0.987802\pi\)
\(62\) 500.276i 0.00209911i
\(63\) 0 0
\(64\) −260698. −0.994485
\(65\) 325440. 1.18504
\(66\) − 1555.54i − 0.00541065i
\(67\) −120682. −0.401251 −0.200626 − 0.979668i \(-0.564297\pi\)
−0.200626 + 0.979668i \(0.564297\pi\)
\(68\) 236194.i 0.751177i
\(69\) 584910.i 1.78050i
\(70\) 0 0
\(71\) −339555. −0.948713 −0.474357 − 0.880333i \(-0.657319\pi\)
−0.474357 + 0.880333i \(0.657319\pi\)
\(72\) −19958.8 −0.0534733
\(73\) − 111351.i − 0.286237i −0.989706 − 0.143118i \(-0.954287\pi\)
0.989706 − 0.143118i \(-0.0457130\pi\)
\(74\) −11770.7 −0.0290474
\(75\) 439163.i 1.04098i
\(76\) − 291869.i − 0.664886i
\(77\) 0 0
\(78\) −17643.3 −0.0371788
\(79\) −615838. −1.24907 −0.624533 − 0.780998i \(-0.714711\pi\)
−0.624533 + 0.780998i \(0.714711\pi\)
\(80\) − 677144.i − 1.32255i
\(81\) −586780. −1.10413
\(82\) 6414.91i 0.0116345i
\(83\) 383668.i 0.670999i 0.942040 + 0.335499i \(0.108905\pi\)
−0.942040 + 0.335499i \(0.891095\pi\)
\(84\) 0 0
\(85\) −612364. −0.997133
\(86\) −16614.0 −0.0261203
\(87\) 880882.i 1.33770i
\(88\) −5373.11 −0.00788457
\(89\) − 771685.i − 1.09464i −0.836924 − 0.547318i \(-0.815649\pi\)
0.836924 − 0.547318i \(-0.184351\pi\)
\(90\) − 25861.0i − 0.0354746i
\(91\) 0 0
\(92\) 1.00973e6 1.29670
\(93\) −76367.9 −0.0949429
\(94\) 34057.5i 0.0410042i
\(95\) 756709. 0.882588
\(96\) 110300.i 0.124670i
\(97\) − 292263.i − 0.320227i −0.987099 − 0.160113i \(-0.948814\pi\)
0.987099 − 0.160113i \(-0.0511860\pi\)
\(98\) 0 0
\(99\) 111279. 0.114685
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.b.b.48.1 4
3.2 odd 2 441.7.d.b.244.4 4
7.2 even 3 7.7.d.b.3.2 ✓ 4
7.3 odd 6 7.7.d.b.5.2 yes 4
7.4 even 3 49.7.d.c.19.2 4
7.5 odd 6 49.7.d.c.31.2 4
7.6 odd 2 inner 49.7.b.b.48.2 4
21.2 odd 6 63.7.m.b.10.1 4
21.17 even 6 63.7.m.b.19.1 4
21.20 even 2 441.7.d.b.244.3 4
28.3 even 6 112.7.s.b.33.1 4
28.23 odd 6 112.7.s.b.17.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.7.d.b.3.2 ✓ 4 7.2 even 3
7.7.d.b.5.2 yes 4 7.3 odd 6
49.7.b.b.48.1 4 1.1 even 1 trivial
49.7.b.b.48.2 4 7.6 odd 2 inner
49.7.d.c.19.2 4 7.4 even 3
49.7.d.c.31.2 4 7.5 odd 6
63.7.m.b.10.1 4 21.2 odd 6
63.7.m.b.19.1 4 21.17 even 6
112.7.s.b.17.1 4 28.23 odd 6
112.7.s.b.33.1 4 28.3 even 6
441.7.d.b.244.3 4 21.20 even 2
441.7.d.b.244.4 4 3.2 odd 2