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Properties

Label 49.7.h.a.3.15
Level $49$
Weight $7$
Character 49.3
Analytic conductor $11.273$
Analytic rank $0$
Dimension $324$
Inner twists $2$

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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(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 3.15
Character \(\chi\) \(=\) 49.3
Dual form 49.7.h.a.33.15

$q$-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+(0.472580 - 1.20411i) q^{2} +(-31.3476 - 2.34917i) q^{3} +(45.6888 + 42.3930i) q^{4} +(70.8216 - 103.876i) q^{5} +(-17.6429 + 36.6359i) q^{6} +(-336.701 + 65.4302i) q^{7} +(147.225 - 70.8999i) q^{8} +(256.293 + 38.6300i) q^{9} +(-91.6099 - 134.367i) q^{10} +(1694.46 - 255.398i) q^{11} +(-1332.64 - 1436.25i) q^{12} +(-3178.30 + 2534.61i) q^{13} +(-80.3329 + 436.348i) q^{14} +(-2464.11 + 3089.89i) q^{15} +(282.296 + 3766.98i) q^{16} +(-1759.07 + 5702.77i) q^{17} +(167.634 - 290.350i) q^{18} +(-9125.01 + 5268.33i) q^{19} +(7639.37 - 1743.64i) q^{20} +(10708.5 - 1260.11i) q^{21} +(493.238 - 2161.02i) q^{22} +(-4838.71 + 1492.54i) q^{23} +(-4781.70 + 1876.68i) q^{24} +(-66.1162 - 168.461i) q^{25} +(1549.96 + 5024.84i) q^{26} +(14398.5 + 3286.36i) q^{27} +(-18157.3 - 11284.4i) q^{28} +(-4510.15 - 19760.2i) q^{29} +(2556.10 + 4427.29i) q^{30} +(-4091.98 - 2362.51i) q^{31} +(14662.7 + 4522.86i) q^{32} +(-53717.1 + 4025.54i) q^{33} +(6035.49 + 4813.14i) q^{34} +(-17049.1 + 39609.2i) q^{35} +(10072.1 + 12630.0i) q^{36} +(3069.05 - 2847.66i) q^{37} +(2031.37 + 13477.3i) q^{38} +(105586. - 71987.5i) q^{39} +(3061.91 - 20314.4i) q^{40} +(14822.9 + 30780.1i) q^{41} +(3543.30 - 13489.7i) q^{42} +(-63649.5 - 30652.0i) q^{43} +(88244.7 + 60164.2i) q^{44} +(22163.8 - 23886.9i) q^{45} +(-489.483 + 6531.70i) q^{46} +(-156171. - 61292.7i) q^{47} -118749. i q^{48} +(109087. - 44060.9i) q^{49} -234.092 q^{50} +(68539.4 - 174636. i) q^{51} +(-252662. - 18934.4i) q^{52} +(159637. + 148122. i) q^{53} +(10761.6 - 15784.3i) q^{54} +(93474.4 - 194101. i) q^{55} +(-44931.9 + 33505.1i) q^{56} +(298423. - 143713. i) q^{57} +(-25925.0 - 3907.56i) q^{58} +(-65782.3 - 96484.9i) q^{59} +(-243572. + 36712.5i) q^{60} +(266462. + 287178. i) q^{61} +(-4778.51 + 3810.74i) q^{62} +(-88821.8 + 3762.55i) q^{63} +(-138361. + 173499. i) q^{64} +(38193.4 + 509655. i) q^{65} +(-20538.4 + 66583.8i) q^{66} +(-164135. + 284290. i) q^{67} +(-322127. + 185980. i) q^{68} +(155188. - 35420.6i) q^{69} +(39636.9 + 39247.5i) q^{70} +(43478.0 - 190490. i) q^{71} +(40471.6 - 12483.8i) q^{72} +(337497. - 132458. i) q^{73} +(-1978.54 - 5041.23i) q^{74} +(1676.84 + 5436.17i) q^{75} +(-640250. - 146133. i) q^{76} +(-553815. + 196862. i) q^{77} +(-36783.2 - 161158. i) q^{78} +(-18936.6 - 32799.1i) q^{79} +(411292. + 237460. i) q^{80} +(-624190. - 192537. i) q^{81} +(44067.7 - 3302.42i) q^{82} +(-562383. - 448486. i) q^{83} +(542677. + 396391. i) q^{84} +(467802. + 586605. i) q^{85} +(-66988.0 + 62155.8i) q^{86} +(94961.8 + 630030. i) q^{87} +(231359. - 157738. i) q^{88} +(139449. - 925185. i) q^{89} +(-18288.4 - 37976.2i) q^{90} +(904299. - 1.06136e6i) q^{91} +(-284348. - 136935. i) q^{92} +(122724. + 83671.5i) q^{93} +(-147607. + 159082. i) q^{94} +(-98994.0 + 1.32098e6i) q^{95} +(-449016. - 176226. i) q^{96} +245920. i q^{97} +(-1502.15 - 152175. i) q^{98} +444143. 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.472580 − 1.20411i 0.0590725 − 0.150514i −0.898290 − 0.439402i \(-0.855190\pi\)
0.957363 + 0.288888i \(0.0932855\pi\)
\(3\) −31.3476 − 2.34917i −1.16102 − 0.0870065i −0.519748 − 0.854320i \(-0.673974\pi\)
−0.641273 + 0.767313i \(0.721593\pi\)
\(4\) 45.6888 + 42.3930i 0.713887 + 0.662390i
\(5\) 70.8216 − 103.876i 0.566573 − 0.831010i −0.430875 − 0.902412i \(-0.641795\pi\)
0.997448 + 0.0714019i \(0.0227473\pi\)
\(6\) −17.6429 + 36.6359i −0.0816801 + 0.169610i
\(7\) −336.701 + 65.4302i −0.981637 + 0.190759i
\(8\) 147.225 − 70.8999i 0.287549 − 0.138476i
\(9\) 256.293 + 38.6300i 0.351568 + 0.0529903i
\(10\) −91.6099 − 134.367i −0.0916099 − 0.134367i
\(11\) 1694.46 − 255.398i 1.27307 − 0.191884i 0.522480 − 0.852652i \(-0.325007\pi\)
0.750591 + 0.660767i \(0.229769\pi\)
\(12\) −1332.64 − 1436.25i −0.771205 − 0.831161i
\(13\) −3178.30 + 2534.61i −1.44666 + 1.15367i −0.486682 + 0.873579i \(0.661793\pi\)
−0.959974 + 0.280090i \(0.909636\pi\)
\(14\) −80.3329 + 436.348i −0.0292758 + 0.159019i
\(15\) −2464.11 + 3089.89i −0.730106 + 0.915524i
\(16\) 282.296 + 3766.98i 0.0689199 + 0.919672i
\(17\) −1759.07 + 5702.77i −0.358045 + 1.16075i 0.580111 + 0.814537i \(0.303009\pi\)
−0.938155 + 0.346214i \(0.887467\pi\)
\(18\) 167.634 − 290.350i 0.0287438 − 0.0497857i
\(19\) −9125.01 + 5268.33i −1.33037 + 0.768090i −0.985356 − 0.170508i \(-0.945459\pi\)
−0.345014 + 0.938598i \(0.612126\pi\)
\(20\) 7639.37 − 1743.64i 0.954922 − 0.217955i
\(21\) 10708.5 − 1260.11i 1.15630 − 0.136066i
\(22\) 493.238 − 2161.02i 0.0463221 − 0.202950i
\(23\) −4838.71 + 1492.54i −0.397691 + 0.122671i −0.487148 − 0.873319i \(-0.661963\pi\)
0.0894574 + 0.995991i \(0.471487\pi\)
\(24\) −4781.70 + 1876.68i −0.345899 + 0.135755i
\(25\) −66.1162 − 168.461i −0.00423144 − 0.0107815i
\(26\) 1549.96 + 5024.84i 0.0881861 + 0.285892i
\(27\) 14398.5 + 3286.36i 0.731518 + 0.166964i
\(28\) −18157.3 − 11284.4i −0.827134 − 0.514047i
\(29\) −4510.15 − 19760.2i −0.184925 − 0.810211i −0.979240 − 0.202704i \(-0.935027\pi\)
0.794315 − 0.607507i \(-0.207830\pi\)
\(30\) 2556.10 + 4427.29i 0.0946702 + 0.163974i
\(31\) −4091.98 − 2362.51i −0.137356 − 0.0793026i 0.429747 − 0.902949i \(-0.358603\pi\)
−0.567103 + 0.823647i \(0.691936\pi\)
\(32\) 14662.7 + 4522.86i 0.447471 + 0.138027i
\(33\) −53717.1 + 4025.54i −1.49476 + 0.112016i
\(34\) 6035.49 + 4813.14i 0.153559 + 0.122459i
\(35\) −17049.1 + 39609.2i −0.397647 + 0.923829i
\(36\) 10072.1 + 12630.0i 0.215879 + 0.270704i
\(37\) 3069.05 − 2847.66i 0.0605897 − 0.0562190i −0.649297 − 0.760535i \(-0.724937\pi\)
0.709886 + 0.704316i \(0.248746\pi\)
\(38\) 2031.37 + 13477.3i 0.0370201 + 0.245613i
\(39\) 105586. − 71987.5i 1.77997 − 1.21357i
\(40\) 3061.91 − 20314.4i 0.0478423 − 0.317413i
\(41\) 14822.9 + 30780.1i 0.215071 + 0.446599i 0.980394 − 0.197047i \(-0.0631351\pi\)
−0.765323 + 0.643646i \(0.777421\pi\)
\(42\) 3543.30 − 13489.7i 0.0478255 − 0.182077i
\(43\) −63649.5 − 30652.0i −0.800553 − 0.385526i −0.0115636 − 0.999933i \(-0.503681\pi\)
−0.788989 + 0.614407i \(0.789395\pi\)
\(44\) 88244.7 + 60164.2i 1.03593 + 0.706286i
\(45\) 22163.8 − 23886.9i 0.243224 − 0.262133i
\(46\) −489.483 + 6531.70i −0.00502880 + 0.0671047i
\(47\) −156171. − 61292.7i −1.50421 − 0.590358i −0.536828 − 0.843692i \(-0.680378\pi\)
−0.967379 + 0.253334i \(0.918473\pi\)
\(48\) − 118749.i − 1.07376i
\(49\) 109087. − 44060.9i 0.927222 − 0.374512i
\(50\) −234.092 −0.00187273
\(51\) 68539.4 − 174636.i 0.516690 − 1.31650i
\(52\) −252662. − 18934.4i −1.79693 − 0.134661i
\(53\) 159637. + 148122.i 1.07228 + 0.994928i 0.999997 − 0.00245658i \(-0.000781955\pi\)
0.0722802 + 0.997384i \(0.476972\pi\)
\(54\) 10761.6 − 15784.3i 0.0683431 − 0.100241i
\(55\) 93474.4 − 194101.i 0.561829 − 1.16665i
\(56\) −44931.9 + 33505.1i −0.255853 + 0.190786i
\(57\) 298423. − 143713.i 1.61142 − 0.776017i
\(58\) −25925.0 − 3907.56i −0.132872 − 0.0200273i
\(59\) −65782.3 − 96484.9i −0.320297 − 0.469790i 0.631997 − 0.774971i \(-0.282235\pi\)
−0.952294 + 0.305181i \(0.901283\pi\)
\(60\) −243572. + 36712.5i −1.12765 + 0.169965i
\(61\) 266462. + 287178.i 1.17394 + 1.26521i 0.957346 + 0.288945i \(0.0933044\pi\)
0.216594 + 0.976262i \(0.430505\pi\)
\(62\) −4778.51 + 3810.74i −0.0200502 + 0.0159895i
\(63\) −88821.8 + 3762.55i −0.355220 + 0.0150474i
\(64\) −138361. + 173499.i −0.527806 + 0.661848i
\(65\) 38193.4 + 509655.i 0.139075 + 1.85582i
\(66\) −20538.4 + 66583.8i −0.0714389 + 0.231599i
\(67\) −164135. + 284290.i −0.545729 + 0.945230i 0.452832 + 0.891596i \(0.350414\pi\)
−0.998561 + 0.0536343i \(0.982919\pi\)
\(68\) −322127. + 185980.i −1.02447 + 0.591480i
\(69\) 155188. − 35420.6i 0.472401 − 0.107822i
\(70\) 39636.9 + 39247.5i 0.115559 + 0.114424i
\(71\) 43478.0 − 190490.i 0.121477 − 0.532226i −0.877168 − 0.480184i \(-0.840570\pi\)
0.998645 − 0.0520420i \(-0.0165730\pi\)
\(72\) 40471.6 − 12483.8i 0.108431 − 0.0334465i
\(73\) 337497. − 132458.i 0.867563 − 0.340493i 0.110532 − 0.993873i \(-0.464744\pi\)
0.757031 + 0.653379i \(0.226649\pi\)
\(74\) −1978.54 − 5041.23i −0.00488258 − 0.0124406i
\(75\) 1676.84 + 5436.17i 0.00397472 + 0.0128857i
\(76\) −640250. − 146133.i −1.45851 − 0.332895i
\(77\) −553815. + 196862.i −1.21309 + 0.431210i
\(78\) −36783.2 − 161158.i −0.0775114 − 0.339600i
\(79\) −18936.6 − 32799.1i −0.0384079 − 0.0665244i 0.846182 − 0.532893i \(-0.178895\pi\)
−0.884590 + 0.466369i \(0.845562\pi\)
\(80\) 411292. + 237460.i 0.803305 + 0.463788i
\(81\) −624190. − 192537.i −1.17452 − 0.362293i
\(82\) 44067.7 − 3302.42i 0.0799243 − 0.00598950i
\(83\) −562383. − 448486.i −0.983554 − 0.784358i −0.00707556 − 0.999975i \(-0.502252\pi\)
−0.976478 + 0.215617i \(0.930824\pi\)
\(84\) 542677. + 396391.i 0.915595 + 0.668785i
\(85\) 467802. + 586605.i 0.761738 + 0.955189i
\(86\) −66988.0 + 62155.8i −0.105318 + 0.0977206i
\(87\) 94961.8 + 630030.i 0.144209 + 0.956761i
\(88\) 231359. − 157738.i 0.339499 − 0.231466i
\(89\) 139449. − 925185.i 0.197809 − 1.31238i −0.640827 − 0.767685i \(-0.721408\pi\)
0.838636 − 0.544692i \(-0.183353\pi\)
\(90\) −18288.4 − 37976.2i −0.0250870 − 0.0520936i
\(91\) 904299. − 1.06136e6i 1.20002 − 1.40845i
\(92\) −284348. − 136935.i −0.365163 − 0.175853i
\(93\) 122724. + 83671.5i 0.152574 + 0.104023i
\(94\) −147607. + 159082.i −0.177714 + 0.191531i
\(95\) −98994.0 + 1.32098e6i −0.115462 + 1.54073i
\(96\) −449016. − 176226.i −0.507514 − 0.199185i
\(97\) 245920.i 0.269450i 0.990883 + 0.134725i \(0.0430151\pi\)
−0.990883 + 0.134725i \(0.956985\pi\)
\(98\) −1502.15 − 152175.i −0.00159601 − 0.161683i
\(99\) 444143. 0.457739
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.3.15 ✓ 324
49.33 odd 42 inner 49.7.h.a.33.15 yes 324
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.7.h.a.3.15 ✓ 324 1.1 even 1 trivial
49.7.h.a.33.15 yes 324 49.33 odd 42 inner