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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [49,7,Mod(3,49)] 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("49.3"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 49.h (of order \(42\), degree \(12\), 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: \(11.2726500974\)
Analytic rank: \(0\)
Dimension: \(324\)
Relative dimension: \(27\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label 3.13
Character \(\chi\) \(=\) 49.3
Dual form 49.7.h.a.33.13

$q$-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+(-0.399133 + 1.01698i) q^{2} +(-2.92834 - 0.219449i) q^{3} +(46.0404 + 42.7192i) q^{4} +(-88.1806 + 129.337i) q^{5} +(1.39197 - 2.89046i) q^{6} +(305.864 - 155.229i) q^{7} +(-124.816 + 60.1083i) q^{8} +(-712.331 - 107.367i) q^{9} +(-96.3370 - 141.300i) q^{10} +(-432.166 + 65.1386i) q^{11} +(-125.447 - 135.200i) q^{12} +(-815.921 + 650.675i) q^{13} +(35.7831 + 373.014i) q^{14} +(286.606 - 359.393i) q^{15} +(289.076 + 3857.45i) q^{16} +(-442.178 + 1433.50i) q^{17} +(393.504 - 681.569i) q^{18} +(-4142.83 + 2391.87i) q^{19} +(-9585.06 + 2187.73i) q^{20} +(-929.740 + 387.441i) q^{21} +(106.248 - 465.501i) q^{22} +(-21640.3 + 6675.14i) q^{23} +(378.695 - 148.627i) q^{24} +(-3243.84 - 8265.18i) q^{25} +(-336.059 - 1089.48i) q^{26} +(4149.46 + 947.087i) q^{27} +(20713.4 + 5919.51i) q^{28} +(-6015.65 - 26356.3i) q^{29} +(251.099 + 434.917i) q^{30} +(41054.0 + 23702.5i) q^{31} +(-12510.7 - 3859.03i) q^{32} +(1279.82 - 95.9096i) q^{33} +(-1281.35 - 1021.84i) q^{34} +(-6894.47 + 53247.8i) q^{35} +(-28209.4 - 35373.4i) q^{36} +(2748.21 - 2549.97i) q^{37} +(-778.926 - 5167.83i) q^{38} +(2532.09 - 1726.35i) q^{39} +(3232.13 - 21443.8i) q^{40} +(1606.45 + 3335.83i) q^{41} +(-22.9276 - 1100.16i) q^{42} +(55682.5 + 26815.3i) q^{43} +(-22679.8 - 15462.8i) q^{44} +(76700.3 - 82663.2i) q^{45} +(1848.90 - 24671.9i) q^{46} +(124277. + 48775.2i) q^{47} -11359.4i q^{48} +(69457.1 - 94957.9i) q^{49} +9700.21 q^{50} +(1609.43 - 4100.76i) q^{51} +(-65361.7 - 4898.18i) q^{52} +(40970.8 + 38015.4i) q^{53} +(-2619.35 + 3841.89i) q^{54} +(29683.8 - 61639.1i) q^{55} +(-28846.3 + 37760.0i) q^{56} +(12656.5 - 6095.06i) q^{57} +(29204.8 + 4401.91i) q^{58} +(-78295.2 - 114838. i) q^{59} +(28548.4 - 4302.98i) q^{60} +(47403.4 + 51088.7i) q^{61} +(-40490.9 + 32290.4i) q^{62} +(-234543. + 77734.5i) q^{63} +(-145439. + 182374. i) q^{64} +(-12208.1 - 162906. i) q^{65} +(-413.283 + 1339.83i) q^{66} +(-52298.5 + 90583.7i) q^{67} +(-81596.3 + 47109.6i) q^{68} +(64835.0 - 14798.2i) q^{69} +(-51399.9 - 28264.5i) q^{70} +(-116564. + 510699. i) q^{71} +(95364.0 - 29415.9i) q^{72} +(39467.1 - 15489.7i) q^{73} +(1496.35 + 3812.65i) q^{74} +(7685.30 + 24915.1i) q^{75} +(-292916. - 66856.2i) q^{76} +(-122073. + 87008.1i) q^{77} +(745.012 + 3264.11i) q^{78} +(-218397. - 378275. i) q^{79} +(-524402. - 302764. i) q^{80} +(489880. + 151108. i) q^{81} +(-4033.65 + 302.280i) q^{82} +(748002. + 596512. i) q^{83} +(-59356.8 - 21879.9i) q^{84} +(-146414. - 183597. i) q^{85} +(-49495.2 + 45924.9i) q^{86} +(11832.0 + 78500.4i) q^{87} +(50025.9 - 34107.1i) q^{88} +(-4500.81 + 29860.9i) q^{89} +(53452.8 + 110996. i) q^{90} +(-148558. + 325673. i) q^{91} +(-1.28148e6 - 617130. i) q^{92} +(-115019. - 78418.4i) q^{93} +(-99206.3 + 106919. i) q^{94} +(55960.3 - 746739. i) q^{95} +(35788.7 + 14046.0i) q^{96} +1.42501e6i q^{97} +(68847.2 + 108537. i) q^{98} +314839. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 324 q - 13 q^{2} - 11 q^{3} + 819 q^{4} - 179 q^{5} + 770 q^{6} + 392 q^{7} + 828 q^{8} - 1160 q^{9} - 2594 q^{10} - 5305 q^{11} + 7497 q^{12} - 14 q^{13} - 11403 q^{14} - 6196 q^{15} + 27903 q^{16} - 5107 q^{17}+ \cdots - 4449616 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{1}{42}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.399133 + 1.01698i −0.0498917 + 0.127122i −0.953647 − 0.300927i \(-0.902704\pi\)
0.903755 + 0.428049i \(0.140799\pi\)
\(3\) −2.92834 − 0.219449i −0.108457 − 0.00812774i 0.0203910 − 0.999792i \(-0.493509\pi\)
−0.128848 + 0.991664i \(0.541128\pi\)
\(4\) 46.0404 + 42.7192i 0.719381 + 0.667488i
\(5\) −88.1806 + 129.337i −0.705445 + 1.03470i 0.291498 + 0.956571i \(0.405846\pi\)
−0.996943 + 0.0781265i \(0.975106\pi\)
\(6\) 1.39197 − 2.89046i 0.00644432 − 0.0133818i
\(7\) 305.864 − 155.229i 0.891733 − 0.452562i
\(8\) −124.816 + 60.1083i −0.243782 + 0.117399i
\(9\) −712.331 − 107.367i −0.977134 − 0.147279i
\(10\) −96.3370 − 141.300i −0.0963370 − 0.141300i
\(11\) −432.166 + 65.1386i −0.324693 + 0.0489396i −0.309366 − 0.950943i \(-0.600117\pi\)
−0.0153266 + 0.999883i \(0.504879\pi\)
\(12\) −125.447 − 135.200i −0.0725968 − 0.0782408i
\(13\) −815.921 + 650.675i −0.371380 + 0.296165i −0.791339 − 0.611378i \(-0.790616\pi\)
0.419959 + 0.907543i \(0.362044\pi\)
\(14\) 35.7831 + 373.014i 0.0130405 + 0.135938i
\(15\) 286.606 − 359.393i 0.0849203 − 0.106487i
\(16\) 289.076 + 3857.45i 0.0705751 + 0.941760i
\(17\) −442.178 + 1433.50i −0.0900016 + 0.291778i −0.989371 − 0.145411i \(-0.953550\pi\)
0.899370 + 0.437189i \(0.144026\pi\)
\(18\) 393.504 − 681.569i 0.0674733 − 0.116867i
\(19\) −4142.83 + 2391.87i −0.604000 + 0.348719i −0.770613 − 0.637303i \(-0.780050\pi\)
0.166614 + 0.986022i \(0.446717\pi\)
\(20\) −9585.06 + 2187.73i −1.19813 + 0.273466i
\(21\) −929.740 + 387.441i −0.100393 + 0.0418358i
\(22\) 106.248 − 465.501i 0.00997818 − 0.0437173i
\(23\) −21640.3 + 6675.14i −1.77860 + 0.548627i −0.997158 − 0.0753447i \(-0.975994\pi\)
−0.781447 + 0.623972i \(0.785518\pi\)
\(24\) 378.695 − 148.627i 0.0273940 − 0.0107514i
\(25\) −3243.84 − 8265.18i −0.207606 − 0.528971i
\(26\) −336.059 − 1089.48i −0.0191204 − 0.0619867i
\(27\) 4149.46 + 947.087i 0.210814 + 0.0481170i
\(28\) 20713.4 + 5919.51i 0.943575 + 0.269657i
\(29\) −6015.65 − 26356.3i −0.246654 − 1.08066i −0.934823 − 0.355113i \(-0.884442\pi\)
0.688169 − 0.725551i \(-0.258415\pi\)
\(30\) 251.099 + 434.917i 0.00929998 + 0.0161080i
\(31\) 41054.0 + 23702.5i 1.37807 + 0.795627i 0.991927 − 0.126813i \(-0.0404748\pi\)
0.386140 + 0.922440i \(0.373808\pi\)
\(32\) −12510.7 − 3859.03i −0.381796 − 0.117768i
\(33\) 1279.82 − 95.9096i 0.0356130 − 0.00266883i
\(34\) −1281.35 − 1021.84i −0.0326010 − 0.0259985i
\(35\) −6894.47 + 53247.8i −0.160804 + 1.24193i
\(36\) −28209.4 − 35373.4i −0.604625 − 0.758175i
\(37\) 2748.21 − 2549.97i 0.0542557 − 0.0503419i −0.652579 − 0.757721i \(-0.726313\pi\)
0.706835 + 0.707379i \(0.250122\pi\)
\(38\) −778.926 − 5167.83i −0.0141953 − 0.0941798i
\(39\) 2532.09 − 1726.35i 0.0426859 − 0.0291028i
\(40\) 3232.13 − 21443.8i 0.0505020 − 0.335059i
\(41\) 1606.45 + 3335.83i 0.0233086 + 0.0484008i 0.912298 − 0.409527i \(-0.134306\pi\)
−0.888989 + 0.457928i \(0.848592\pi\)
\(42\) −22.9276 − 1100.16i −0.000309464 − 0.0148494i
\(43\) 55682.5 + 26815.3i 0.700347 + 0.337270i 0.749945 − 0.661500i \(-0.230080\pi\)
−0.0495980 + 0.998769i \(0.515794\pi\)
\(44\) −22679.8 − 15462.8i −0.266244 − 0.181522i
\(45\) 76700.3 − 82663.2i 0.841704 − 0.907141i
\(46\) 1848.90 − 24671.9i 0.0189951 − 0.253472i
\(47\) 124277. + 48775.2i 1.19701 + 0.469792i 0.878323 − 0.478068i \(-0.158663\pi\)
0.318687 + 0.947860i \(0.396758\pi\)
\(48\) − 11359.4i − 0.102714i
\(49\) 69457.1 − 94957.9i 0.590376 − 0.807129i
\(50\) 9700.21 0.0776017
\(51\) 1609.43 − 4100.76i 0.0121328 − 0.0309139i
\(52\) −65361.7 − 4898.18i −0.464850 − 0.0348357i
\(53\) 40970.8 + 38015.4i 0.275199 + 0.255347i 0.805682 − 0.592348i \(-0.201799\pi\)
−0.530483 + 0.847696i \(0.677989\pi\)
\(54\) −2619.35 + 3841.89i −0.0166346 + 0.0243985i
\(55\) 29683.8 − 61639.1i 0.178415 − 0.370483i
\(56\) −28846.3 + 37760.0i −0.164258 + 0.215015i
\(57\) 12656.5 − 6095.06i 0.0683423 − 0.0329119i
\(58\) 29204.8 + 4401.91i 0.149682 + 0.0225609i
\(59\) −78295.2 − 114838.i −0.381223 − 0.559151i 0.586890 − 0.809667i \(-0.300352\pi\)
−0.968113 + 0.250516i \(0.919400\pi\)
\(60\) 28548.4 − 4302.98i 0.132169 − 0.0199212i
\(61\) 47403.4 + 51088.7i 0.208843 + 0.225079i 0.828783 − 0.559570i \(-0.189034\pi\)
−0.619940 + 0.784649i \(0.712843\pi\)
\(62\) −40490.9 + 32290.4i −0.169896 + 0.135487i
\(63\) −234543. + 77734.5i −0.937996 + 0.310880i
\(64\) −145439. + 182374.i −0.554805 + 0.695703i
\(65\) −12208.1 − 162906.i −0.0444537 − 0.593194i
\(66\) −413.283 + 1339.83i −0.00143753 + 0.00466035i
\(67\) −52298.5 + 90583.7i −0.173886 + 0.301180i −0.939775 − 0.341793i \(-0.888966\pi\)
0.765889 + 0.642973i \(0.222299\pi\)
\(68\) −81596.3 + 47109.6i −0.259504 + 0.149825i
\(69\) 64835.0 − 14798.2i 0.197361 − 0.0450465i
\(70\) −51399.9 − 28264.5i −0.149854 − 0.0824038i
\(71\) −116564. + 510699.i −0.325678 + 1.42689i 0.501602 + 0.865098i \(0.332744\pi\)
−0.827280 + 0.561790i \(0.810113\pi\)
\(72\) 95364.0 − 29415.9i 0.255498 − 0.0788106i
\(73\) 39467.1 − 15489.7i 0.101454 − 0.0398176i −0.314071 − 0.949399i \(-0.601693\pi\)
0.415525 + 0.909582i \(0.363598\pi\)
\(74\) 1496.35 + 3812.65i 0.00369266 + 0.00940874i
\(75\) 7685.30 + 24915.1i 0.0182170 + 0.0590581i
\(76\) −292916. − 66856.2i −0.667272 − 0.152300i
\(77\) −122073. + 87008.1i −0.267391 + 0.190585i
\(78\) 745.012 + 3264.11i 0.00156993 + 0.00687830i
\(79\) −218397. − 378275.i −0.442961 − 0.767231i 0.554946 − 0.831886i \(-0.312739\pi\)
−0.997908 + 0.0646547i \(0.979405\pi\)
\(80\) −524402. − 302764.i −1.02422 − 0.591336i
\(81\) 489880. + 151108.i 0.921796 + 0.284336i
\(82\) −4033.65 + 302.280i −0.00731571 + 0.000548237i
\(83\) 748002. + 596512.i 1.30818 + 1.04324i 0.995634 + 0.0933392i \(0.0297541\pi\)
0.312548 + 0.949902i \(0.398817\pi\)
\(84\) −59356.8 − 21879.9i −0.100146 − 0.0369153i
\(85\) −146414. − 183597.i −0.238411 − 0.298958i
\(86\) −49495.2 + 45924.9i −0.0778159 + 0.0722026i
\(87\) 11832.0 + 78500.4i 0.0179681 + 0.119210i
\(88\) 50025.9 − 34107.1i 0.0734087 − 0.0500492i
\(89\) −4500.81 + 29860.9i −0.00638440 + 0.0423577i −0.991796 − 0.127829i \(-0.959199\pi\)
0.985412 + 0.170187i \(0.0544372\pi\)
\(90\) 53452.8 + 110996.i 0.0733235 + 0.152258i
\(91\) −148558. + 325673.i −0.197138 + 0.432173i
\(92\) −1.28148e6 − 617130.i −1.64570 − 0.792526i
\(93\) −115019. − 78418.4i −0.142994 − 0.0974920i
\(94\) −99206.3 + 106919.i −0.119442 + 0.128727i
\(95\) 55960.3 − 746739.i 0.0652694 − 0.870959i
\(96\) 35788.7 + 14046.0i 0.0404513 + 0.0158759i
\(97\) 1.42501e6i 1.56135i 0.624935 + 0.780677i \(0.285126\pi\)
−0.624935 + 0.780677i \(0.714874\pi\)
\(98\) 68847.2 + 108537.i 0.0731489 + 0.115319i
\(99\) 314839. 0.324476
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.h.a.3.13 ✓ 324
49.33 odd 42 inner 49.7.h.a.33.13 yes 324
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.7.h.a.3.13 ✓ 324 1.1 even 1 trivial
49.7.h.a.33.13 yes 324 49.33 odd 42 inner