Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,7,Mod(17,112)] 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("112.17"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 112.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-18,0,-150] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.7660573654\)
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 33.1
Root \(0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 112.33
Dual form 112.7.s.b.17.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-32.0772 + 18.5198i) q^{3} +(-143.566 - 82.8879i) q^{5} +(197.286 - 280.583i) q^{7} +(321.463 - 556.790i) q^{9} +(-86.5410 - 149.893i) q^{11} -1963.14i q^{13} +6140.25 q^{15} +(3199.04 - 1846.96i) q^{17} +(-3953.11 - 2282.33i) q^{19} +(-1132.05 + 12654.0i) q^{21} +(-7895.75 + 13675.8i) q^{23} +(5928.30 + 10268.1i) q^{25} -3188.14i q^{27} -23782.2 q^{29} +(-1785.57 + 1030.90i) q^{31} +(5551.98 + 3205.44i) q^{33} +(-51580.6 + 23929.6i) q^{35} +(-24255.5 + 42011.7i) q^{37} +(36356.8 + 62971.8i) q^{39} +26437.9i q^{41} -68471.6 q^{43} +(-92302.3 + 53290.8i) q^{45} +(121557. + 70180.9i) q^{47} +(-39805.2 - 110711. i) q^{49} +(-68410.7 + 118491. i) q^{51} +(127065. + 220083. i) q^{53} +28692.8i q^{55} +169073. q^{57} +(84178.0 - 48600.2i) q^{59} +(15062.4 + 8696.26i) q^{61} +(-92805.9 - 200044. i) q^{63} +(-162720. + 281840. i) q^{65} +(-60340.8 - 104513. i) q^{67} -584910. i q^{69} +339555. q^{71} +(96432.8 - 55675.5i) q^{73} +(-380326. - 219581. i) q^{75} +(-59131.0 - 5289.95i) q^{77} +(-307919. + 533332. i) q^{79} +(293390. + 508167. i) q^{81} +383668. i q^{83} -612364. q^{85} +(762866. - 440441. i) q^{87} +(-668299. - 385843. i) q^{89} +(-550824. - 387300. i) q^{91} +(38184.0 - 66136.6i) q^{93} +(378355. + 655329. i) q^{95} +292263. i q^{97} -111279. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 18 q^{3} - 150 q^{5} - 280 q^{7} + 624 q^{9} - 1882 q^{11} + 16500 q^{15} + 13458 q^{17} - 18078 q^{19} - 16170 q^{21} - 2470 q^{23} + 2500 q^{25} - 34544 q^{29} + 17202 q^{31} - 67770 q^{33} - 154350 q^{35}+ \cdots - 157872 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(1\)

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 0
\(3\) −32.0772 + 18.5198i −1.18804 + 0.685917i −0.957861 − 0.287231i \(-0.907265\pi\)
−0.230182 + 0.973148i \(0.573932\pi\)
\(4\) 0 0
\(5\) −143.566 − 82.8879i −1.14853 − 0.663103i −0.200000 − 0.979796i \(-0.564094\pi\)
−0.948528 + 0.316693i \(0.897428\pi\)
\(6\) 0 0
\(7\) 197.286 − 280.583i 0.575179 − 0.818028i
\(8\) 0 0
\(9\) 321.463 − 556.790i 0.440964 − 0.763773i
\(10\) 0 0
\(11\) −86.5410 − 149.893i −0.0650195 − 0.112617i 0.831683 − 0.555251i \(-0.187378\pi\)
−0.896703 + 0.442633i \(0.854044\pi\)
\(12\) 0 0
\(13\) − 1963.14i − 0.893553i −0.894646 − 0.446777i \(-0.852572\pi\)
0.894646 − 0.446777i \(-0.147428\pi\)
\(14\) 0 0
\(15\) 6140.25 1.81933
\(16\) 0 0
\(17\) 3199.04 − 1846.96i 0.651137 − 0.375934i −0.137755 − 0.990466i \(-0.543989\pi\)
0.788892 + 0.614532i \(0.210655\pi\)
\(18\) 0 0
\(19\) −3953.11 − 2282.33i −0.576339 − 0.332749i 0.183338 − 0.983050i \(-0.441310\pi\)
−0.759677 + 0.650301i \(0.774643\pi\)
\(20\) 0 0
\(21\) −1132.05 + 12654.0i −0.122238 + 1.36638i
\(22\) 0 0
\(23\) −7895.75 + 13675.8i −0.648948 + 1.12401i 0.334427 + 0.942422i \(0.391457\pi\)
−0.983375 + 0.181589i \(0.941876\pi\)
\(24\) 0 0
\(25\) 5928.30 + 10268.1i 0.379411 + 0.657160i
\(26\) 0 0
\(27\) − 3188.14i − 0.161974i
\(28\) 0 0
\(29\) −23782.2 −0.975121 −0.487561 − 0.873089i \(-0.662113\pi\)
−0.487561 + 0.873089i \(0.662113\pi\)
\(30\) 0 0
\(31\) −1785.57 + 1030.90i −0.0599365 + 0.0346044i −0.529669 − 0.848205i \(-0.677684\pi\)
0.469732 + 0.882809i \(0.344350\pi\)
\(32\) 0 0
\(33\) 5551.98 + 3205.44i 0.154492 + 0.0891960i
\(34\) 0 0
\(35\) −51580.6 + 23929.6i −1.20305 + 0.558125i
\(36\) 0 0
\(37\) −24255.5 + 42011.7i −0.478855 + 0.829401i −0.999706 − 0.0242462i \(-0.992281\pi\)
0.520851 + 0.853648i \(0.325615\pi\)
\(38\) 0 0
\(39\) 36356.8 + 62971.8i 0.612903 + 1.06158i
\(40\) 0 0
\(41\) 26437.9i 0.383597i 0.981434 + 0.191799i \(0.0614320\pi\)
−0.981434 + 0.191799i \(0.938568\pi\)
\(42\) 0 0
\(43\) −68471.6 −0.861202 −0.430601 − 0.902542i \(-0.641698\pi\)
−0.430601 + 0.902542i \(0.641698\pi\)
\(44\) 0 0
\(45\) −92302.3 + 53290.8i −1.01292 + 0.584810i
\(46\) 0 0
\(47\) 121557. + 70180.9i 1.17081 + 0.675967i 0.953870 − 0.300219i \(-0.0970597\pi\)
0.216938 + 0.976185i \(0.430393\pi\)
\(48\) 0 0
\(49\) −39805.2 − 110711.i −0.338338 − 0.941024i
\(50\) 0 0
\(51\) −68410.7 + 118491.i −0.515719 + 0.893252i
\(52\) 0 0
\(53\) 127065. + 220083.i 0.853489 + 1.47829i 0.878039 + 0.478588i \(0.158851\pi\)
−0.0245502 + 0.999699i \(0.507815\pi\)
\(54\) 0 0
\(55\) 28692.8i 0.172459i
\(56\) 0 0
\(57\) 169073. 0.912954
\(58\) 0 0
\(59\) 84178.0 − 48600.2i 0.409867 − 0.236637i −0.280866 − 0.959747i \(-0.590622\pi\)
0.690732 + 0.723110i \(0.257288\pi\)
\(60\) 0 0
\(61\) 15062.4 + 8696.26i 0.0663596 + 0.0383127i 0.532813 − 0.846233i \(-0.321135\pi\)
−0.466453 + 0.884546i \(0.654468\pi\)
\(62\) 0 0
\(63\) −92805.9 − 200044.i −0.371154 − 0.800027i
\(64\) 0 0
\(65\) −162720. + 281840.i −0.592518 + 1.02627i
\(66\) 0 0
\(67\) −60340.8 − 104513.i −0.200626 − 0.347494i 0.748105 − 0.663581i \(-0.230964\pi\)
−0.948730 + 0.316087i \(0.897631\pi\)
\(68\) 0 0
\(69\) − 584910.i − 1.78050i
\(70\) 0 0
\(71\) 339555. 0.948713 0.474357 − 0.880333i \(-0.342681\pi\)
0.474357 + 0.880333i \(0.342681\pi\)
\(72\) 0 0
\(73\) 96432.8 − 55675.5i 0.247888 − 0.143118i −0.370909 − 0.928669i \(-0.620954\pi\)
0.618797 + 0.785551i \(0.287620\pi\)
\(74\) 0 0
\(75\) −380326. − 219581.i −0.901514 − 0.520489i
\(76\) 0 0
\(77\) −59131.0 − 5289.95i −0.129522 − 0.0115872i
\(78\) 0 0
\(79\) −307919. + 533332.i −0.624533 + 1.08172i 0.364098 + 0.931361i \(0.381377\pi\)
−0.988631 + 0.150362i \(0.951956\pi\)
\(80\) 0 0
\(81\) 293390. + 508167.i 0.552065 + 0.956205i
\(82\) 0 0
\(83\) 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
\(86\) 0 0
\(87\) 762866. − 440441.i 1.15849 − 0.668852i
\(88\) 0 0
\(89\) −668299. − 385843.i −0.947983 − 0.547318i −0.0555294 − 0.998457i \(-0.517685\pi\)
−0.892454 + 0.451139i \(0.851018\pi\)
\(90\) 0 0
\(91\) −550824. − 387300.i −0.730951 − 0.513953i
\(92\) 0 0
\(93\) 38184.0 − 66136.6i 0.0474714 − 0.0822229i
\(94\) 0 0
\(95\) 378355. + 655329.i 0.441294 + 0.764344i
\(96\) 0 0
\(97\) 292263.i 0.320227i 0.987099 + 0.160113i \(0.0511860\pi\)
−0.987099 + 0.160113i \(0.948814\pi\)
\(98\) 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 112.7.s.b.33.1 4
4.3 odd 2 7.7.d.b.5.2 yes 4
7.3 odd 6 inner 112.7.s.b.17.1 4
12.11 even 2 63.7.m.b.19.1 4
28.3 even 6 7.7.d.b.3.2 ✓ 4
28.11 odd 6 49.7.d.c.31.2 4
28.19 even 6 49.7.b.b.48.1 4
28.23 odd 6 49.7.b.b.48.2 4
28.27 even 2 49.7.d.c.19.2 4
84.23 even 6 441.7.d.b.244.3 4
84.47 odd 6 441.7.d.b.244.4 4
84.59 odd 6 63.7.m.b.10.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.7.d.b.3.2 ✓ 4 28.3 even 6
7.7.d.b.5.2 yes 4 4.3 odd 2
49.7.b.b.48.1 4 28.19 even 6
49.7.b.b.48.2 4 28.23 odd 6
49.7.d.c.19.2 4 28.27 even 2
49.7.d.c.31.2 4 28.11 odd 6
63.7.m.b.10.1 4 84.59 odd 6
63.7.m.b.19.1 4 12.11 even 2
112.7.s.b.17.1 4 7.3 odd 6 inner
112.7.s.b.33.1 4 1.1 even 1 trivial
441.7.d.b.244.3 4 84.23 even 6
441.7.d.b.244.4 4 84.47 odd 6