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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 33.12
Character \(\chi\) \(=\) 49.33
Dual form 49.7.h.a.3.12

$q$-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+(-1.34783 - 3.43420i) q^{2} +(-1.28182 + 0.0960589i) q^{3} +(36.9382 - 34.2736i) q^{4} +(103.584 + 151.930i) q^{5} +(2.05755 + 4.27255i) q^{6} +(-113.036 + 323.839i) q^{7} +(-380.217 - 183.103i) q^{8} +(-719.224 + 108.406i) q^{9} +(382.145 - 560.504i) q^{10} +(48.9320 + 7.37532i) q^{11} +(-44.0557 + 47.4808i) q^{12} +(2757.90 + 2199.35i) q^{13} +(1264.48 - 48.2909i) q^{14} +(-147.370 - 184.796i) q^{15} +(124.653 - 1663.38i) q^{16} +(2579.08 + 8361.19i) q^{17} +(1341.67 + 2323.85i) q^{18} +(-3791.76 - 2189.17i) q^{19} +(9033.41 + 2061.82i) q^{20} +(113.784 - 425.961i) q^{21} +(-40.6235 - 177.983i) q^{22} +(7691.75 + 2372.59i) q^{23} +(504.957 + 198.181i) q^{24} +(-6644.61 + 16930.2i) q^{25} +(3835.85 - 12435.5i) q^{26} +(1825.07 - 416.560i) q^{27} +(6923.81 + 15836.2i) q^{28} +(3896.52 - 17071.8i) q^{29} +(-435.999 + 755.172i) q^{30} +(41317.6 - 23854.7i) q^{31} +(-31689.1 + 9774.79i) q^{32} +(-63.4304 - 4.75345i) q^{33} +(25237.9 - 20126.5i) q^{34} +(-60909.7 + 16371.1i) q^{35} +(-22851.4 + 28654.7i) q^{36} +(2631.41 + 2441.59i) q^{37} +(-2407.43 + 15972.3i) q^{38} +(-3746.39 - 2554.24i) q^{39} +(-11565.6 - 76733.0i) q^{40} +(-19639.6 + 40782.0i) q^{41} +(-1616.20 + 183.365i) q^{42} +(-42487.0 + 20460.7i) q^{43} +(2060.24 - 1404.65i) q^{44} +(-90970.3 - 98042.6i) q^{45} +(-2219.18 - 29612.9i) q^{46} +(-50203.3 + 19703.3i) q^{47} +2144.13i q^{48} +(-92094.8 - 73210.9i) q^{49} +67097.5 q^{50} +(-4109.08 - 10469.8i) q^{51} +(177252. - 13283.2i) q^{52} +(119199. - 110600. i) q^{53} +(-3890.43 - 5706.21i) q^{54} +(3948.05 + 8198.22i) q^{55} +(102274. - 102432. i) q^{56} +(5070.63 + 2441.89i) q^{57} +(-63879.8 + 9628.32i) q^{58} +(-44229.8 + 64873.2i) q^{59} +(-11777.2 - 1775.13i) q^{60} +(-200091. + 215647. i) q^{61} +(-137611. - 109741. i) q^{62} +(46192.1 - 245167. i) q^{63} +(9719.20 + 12187.5i) q^{64} +(-48472.9 + 646826. i) q^{65} +(69.1687 + 224.240i) q^{66} +(-90123.1 - 156098. i) q^{67} +(381835. + 220453. i) q^{68} +(-10087.3 - 2302.36i) q^{69} +(138317. + 187111. i) q^{70} +(15339.5 + 67206.9i) q^{71} +(293311. + 90474.3i) q^{72} +(-28472.9 - 11174.8i) q^{73} +(4838.23 - 12327.6i) q^{74} +(6890.88 - 22339.7i) q^{75} +(-215092. + 49093.3i) q^{76} +(-7919.49 + 15012.4i) q^{77} +(-3722.32 + 16308.5i) q^{78} +(324415. - 561904. i) q^{79} +(265630. - 153362. i) q^{80} +(504380. - 155581. i) q^{81} +(166524. + 12479.3i) q^{82} +(177781. - 141775. i) q^{83} +(-10396.3 - 19634.0i) q^{84} +(-1.00316e6 + 1.25793e6i) q^{85} +(127531. + 118332. i) q^{86} +(-3354.73 + 22257.2i) q^{87} +(-17254.4 - 11763.8i) q^{88} +(-42335.6 - 280878. i) q^{89} +(-214086. + 444555. i) q^{90} +(-1.02398e6 + 644511. i) q^{91} +(365437. - 175985. i) q^{92} +(-50670.1 + 34546.3i) q^{93} +(135330. + 145852. i) q^{94} +(-60165.0 - 802846. i) q^{95} +(39680.6 - 15573.5i) q^{96} -590161. i q^{97} +(-127293. + 414948. i) q^{98} -35992.6 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 \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{41}{42}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.34783 − 3.43420i −0.168478 − 0.429275i 0.821552 − 0.570134i \(-0.193109\pi\)
−0.990030 + 0.140859i \(0.955014\pi\)
\(3\) −1.28182 + 0.0960589i −0.0474747 + 0.00355774i −0.0984463 − 0.995142i \(-0.531387\pi\)
0.0509716 + 0.998700i \(0.483768\pi\)
\(4\) 36.9382 − 34.2736i 0.577159 − 0.535526i
\(5\) 103.584 + 151.930i 0.828673 + 1.21544i 0.974110 + 0.226073i \(0.0725888\pi\)
−0.145437 + 0.989368i \(0.546459\pi\)
\(6\) 2.05755 + 4.27255i 0.00952570 + 0.0197803i
\(7\) −113.036 + 323.839i −0.329550 + 0.944138i
\(8\) −380.217 − 183.103i −0.742611 − 0.357623i
\(9\) −719.224 + 108.406i −0.986590 + 0.148704i
\(10\) 382.145 − 560.504i 0.382145 − 0.560504i
\(11\) 48.9320 + 7.37532i 0.0367634 + 0.00554119i 0.167398 − 0.985889i \(-0.446463\pi\)
−0.130635 + 0.991431i \(0.541702\pi\)
\(12\) −44.0557 + 47.4808i −0.0254952 + 0.0274773i
\(13\) 2757.90 + 2199.35i 1.25530 + 1.00107i 0.999409 + 0.0343648i \(0.0109408\pi\)
0.255893 + 0.966705i \(0.417631\pi\)
\(14\) 1264.48 − 48.2909i 0.460817 − 0.0175987i
\(15\) −147.370 − 184.796i −0.0436652 − 0.0547545i
\(16\) 124.653 − 1663.38i 0.0304330 − 0.406100i
\(17\) 2579.08 + 8361.19i 0.524951 + 1.70185i 0.697270 + 0.716808i \(0.254398\pi\)
−0.172319 + 0.985041i \(0.555126\pi\)
\(18\) 1341.67 + 2323.85i 0.230054 + 0.398465i
\(19\) −3791.76 − 2189.17i −0.552815 − 0.319168i 0.197442 − 0.980315i \(-0.436737\pi\)
−0.750257 + 0.661147i \(0.770070\pi\)
\(20\) 9033.41 + 2061.82i 1.12918 + 0.257727i
\(21\) 113.784 − 425.961i 0.0122863 − 0.0459951i
\(22\) −40.6235 − 177.983i −0.00381513 − 0.0167152i
\(23\) 7691.75 + 2372.59i 0.632181 + 0.195002i 0.594253 − 0.804278i \(-0.297448\pi\)
0.0379285 + 0.999280i \(0.487924\pi\)
\(24\) 504.957 + 198.181i 0.0365276 + 0.0143360i
\(25\) −6644.61 + 16930.2i −0.425255 + 1.08353i
\(26\) 3835.85 − 12435.5i 0.218244 − 0.707529i
\(27\) 1825.07 − 416.560i 0.0927232 − 0.0211635i
\(28\) 6923.81 + 15836.2i 0.315407 + 0.721401i
\(29\) 3896.52 − 17071.8i 0.159765 − 0.699978i −0.830058 − 0.557677i \(-0.811693\pi\)
0.989823 − 0.142301i \(-0.0454501\pi\)
\(30\) −435.999 + 755.172i −0.0161481 + 0.0279693i
\(31\) 41317.6 − 23854.7i 1.38692 − 0.800736i 0.393949 − 0.919132i \(-0.371109\pi\)
0.992966 + 0.118396i \(0.0377752\pi\)
\(32\) −31689.1 + 9774.79i −0.967074 + 0.298303i
\(33\) −63.4304 − 4.75345i −0.00176504 − 0.000132272i
\(34\) 25237.9 − 20126.5i 0.642119 − 0.512073i
\(35\) −60909.7 + 16371.1i −1.42063 + 0.381833i
\(36\) −22851.4 + 28654.7i −0.489784 + 0.614170i
\(37\) 2631.41 + 2441.59i 0.0519497 + 0.0482022i 0.705720 − 0.708490i \(-0.250623\pi\)
−0.653771 + 0.756693i \(0.726814\pi\)
\(38\) −2407.43 + 15972.3i −0.0438737 + 0.291083i
\(39\) −3746.39 − 2554.24i −0.0631566 − 0.0430595i
\(40\) −11565.6 − 76733.0i −0.180713 − 1.19895i
\(41\) −19639.6 + 40782.0i −0.284958 + 0.591721i −0.993485 − 0.113963i \(-0.963646\pi\)
0.708527 + 0.705683i \(0.249360\pi\)
\(42\) −1616.20 + 183.365i −0.0218145 + 0.00247496i
\(43\) −42487.0 + 20460.7i −0.534381 + 0.257344i −0.681556 − 0.731766i \(-0.738696\pi\)
0.147175 + 0.989110i \(0.452982\pi\)
\(44\) 2060.24 − 1404.65i 0.0241858 − 0.0164896i
\(45\) −90970.3 − 98042.6i −0.998302 − 1.07591i
\(46\) −2219.18 − 29612.9i −0.0227991 − 0.304233i
\(47\) −50203.3 + 19703.3i −0.483547 + 0.189778i −0.594577 − 0.804039i \(-0.702680\pi\)
0.111030 + 0.993817i \(0.464585\pi\)
\(48\) 2144.13i 0.0193877i
\(49\) −92094.8 − 73210.9i −0.782793 − 0.622282i
\(50\) 67097.5 0.536780
\(51\) −4109.08 − 10469.8i −0.0309766 − 0.0789271i
\(52\) 177252. − 13283.2i 1.26061 − 0.0944695i
\(53\) 119199. − 110600.i 0.800654 − 0.742898i −0.169463 − 0.985537i \(-0.554203\pi\)
0.970117 + 0.242638i \(0.0780128\pi\)
\(54\) −3890.43 − 5706.21i −0.0247068 − 0.0362382i
\(55\) 3948.05 + 8198.22i 0.0237298 + 0.0492755i
\(56\) 102274. − 102432.i 0.582373 − 0.583273i
\(57\) 5070.63 + 2441.89i 0.0273802 + 0.0131856i
\(58\) −63879.8 + 9628.32i −0.327400 + 0.0493477i
\(59\) −44229.8 + 64873.2i −0.215357 + 0.315871i −0.918682 − 0.394998i \(-0.870745\pi\)
0.703325 + 0.710868i \(0.251698\pi\)
\(60\) −11777.2 − 1775.13i −0.0545242 − 0.00821820i
\(61\) −200091. + 215647.i −0.881531 + 0.950065i −0.999075 − 0.0430030i \(-0.986307\pi\)
0.117544 + 0.993068i \(0.462498\pi\)
\(62\) −137611. − 109741.i −0.577401 − 0.460462i
\(63\) 46192.1 − 245167.i 0.184734 − 0.980482i
\(64\) 9719.20 + 12187.5i 0.0370758 + 0.0464916i
\(65\) −48472.9 + 646826.i −0.176506 + 2.35531i
\(66\) 69.1687 + 224.240i 0.000240590 + 0.000779975i
\(67\) −90123.1 − 156098.i −0.299648 − 0.519006i 0.676407 − 0.736528i \(-0.263536\pi\)
−0.976055 + 0.217522i \(0.930203\pi\)
\(68\) 381835. + 220453.i 1.21436 + 0.701114i
\(69\) −10087.3 − 2302.36i −0.0307064 − 0.00700853i
\(70\) 138317. + 187111.i 0.403257 + 0.545512i
\(71\) 15339.5 + 67206.9i 0.0428585 + 0.187775i 0.991826 − 0.127599i \(-0.0407271\pi\)
−0.948967 + 0.315374i \(0.897870\pi\)
\(72\) 293311. + 90474.3i 0.785833 + 0.242397i
\(73\) −28472.9 − 11174.8i −0.0731918 − 0.0287257i 0.328463 − 0.944517i \(-0.393469\pi\)
−0.401654 + 0.915791i \(0.631565\pi\)
\(74\) 4838.23 − 12327.6i 0.0119397 − 0.0304217i
\(75\) 6890.88 − 22339.7i 0.0163339 − 0.0529533i
\(76\) −215092. + 49093.3i −0.489985 + 0.111836i
\(77\) −7919.49 + 15012.4i −0.0173470 + 0.0328836i
\(78\) −3722.32 + 16308.5i −0.00784385 + 0.0343662i
\(79\) 324415. − 561904.i 0.657991 − 1.13967i −0.323143 − 0.946350i \(-0.604740\pi\)
0.981135 − 0.193325i \(-0.0619270\pi\)
\(80\) 265630. − 153362.i 0.518809 − 0.299535i
\(81\) 504380. − 155581.i 0.949080 − 0.292752i
\(82\) 166524. + 12479.3i 0.302020 + 0.0226333i
\(83\) 177781. − 141775.i 0.310921 − 0.247951i −0.455579 − 0.890195i \(-0.650568\pi\)
0.766500 + 0.642244i \(0.221996\pi\)
\(84\) −10396.3 − 19634.0i −0.0175404 − 0.0331261i
\(85\) −1.00316e6 + 1.25793e6i −1.63348 + 2.04832i
\(86\) 127531. + 118332.i 0.200503 + 0.186040i
\(87\) −3354.73 + 22257.2i −0.00509448 + 0.0337997i
\(88\) −17254.4 − 11763.8i −0.0253192 − 0.0172624i
\(89\) −42335.6 − 280878.i −0.0600532 − 0.398427i −0.998582 − 0.0532270i \(-0.983049\pi\)
0.938529 − 0.345200i \(-0.112189\pi\)
\(90\) −214086. + 444555.i −0.293671 + 0.609814i
\(91\) −1.02398e6 + 644511.i −1.35883 + 0.855275i
\(92\) 365437. − 175985.i 0.469298 − 0.226002i
\(93\) −50670.1 + 34546.3i −0.0629946 + 0.0429490i
\(94\) 135330. + 145852.i 0.162934 + 0.175601i
\(95\) −60165.0 − 802846.i −0.0701735 − 0.936400i
\(96\) 39680.6 − 15573.5i 0.0448502 − 0.0176024i
\(97\) − 590161.i − 0.646629i −0.946292 − 0.323315i \(-0.895203\pi\)
0.946292 − 0.323315i \(-0.104797\pi\)
\(98\) −127293. + 414948.i −0.135247 + 0.440875i
\(99\) −35992.6 −0.0370944
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.33.12 yes 324
49.3 odd 42 inner 49.7.h.a.3.12 ✓ 324
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.7.h.a.3.12 ✓ 324 49.3 odd 42 inner
49.7.h.a.33.12 yes 324 1.1 even 1 trivial