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Properties

Label 49.7.h.a.3.4
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 3.4
Character \(\chi\) \(=\) 49.3
Dual form 49.7.h.a.33.4

$q$-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+(-4.73204 + 12.0571i) q^{2} +(-20.1535 - 1.51030i) q^{3} +(-76.0649 - 70.5779i) q^{4} +(43.5678 - 63.9023i) q^{5} +(113.577 - 235.845i) q^{6} +(342.839 + 10.5003i) q^{7} +(464.042 - 223.471i) q^{8} +(-316.975 - 47.7763i) q^{9} +(564.308 + 827.688i) q^{10} +(1954.99 - 294.667i) q^{11} +(1426.38 + 1537.27i) q^{12} +(-744.349 + 593.598i) q^{13} +(-1748.93 + 4083.94i) q^{14} +(-974.556 + 1222.05i) q^{15} +(2.25471 + 30.0870i) q^{16} +(304.791 - 988.107i) q^{17} +(2075.98 - 3595.70i) q^{18} +(-2142.80 + 1237.15i) q^{19} +(-7824.07 + 1785.79i) q^{20} +(-6893.56 - 729.406i) q^{21} +(-5698.28 + 24965.8i) q^{22} +(2384.01 - 735.369i) q^{23} +(-9689.59 + 3802.88i) q^{24} +(3523.11 + 8976.73i) q^{25} +(-3634.75 - 11783.6i) q^{26} +(20679.7 + 4720.01i) q^{27} +(-25336.9 - 24995.6i) q^{28} +(7850.27 + 34394.3i) q^{29} +(-10122.7 - 17533.1i) q^{30} +(21319.6 + 12308.9i) q^{31} +(31125.2 + 9600.85i) q^{32} +(-39844.9 + 2985.96i) q^{33} +(10471.4 + 8350.65i) q^{34} +(15607.8 - 21450.7i) q^{35} +(20738.7 + 26005.5i) q^{36} +(45738.0 - 42438.6i) q^{37} +(-4776.51 - 31690.1i) q^{38} +(15897.7 - 10838.9i) q^{39} +(5937.01 - 39389.5i) q^{40} +(-21442.0 - 44524.8i) q^{41} +(41415.1 - 79664.4i) q^{42} +(107907. + 51965.2i) q^{43} +(-169503. - 115565. i) q^{44} +(-16862.9 + 18173.9i) q^{45} +(-2414.85 + 32223.9i) q^{46} +(-4029.84 - 1581.60i) q^{47} -609.764i q^{48} +(117428. + 7199.82i) q^{49} -124904. q^{50} +(-7634.94 + 19453.5i) q^{51} +(98513.7 + 7382.58i) q^{52} +(-25370.1 - 23540.0i) q^{53} +(-154767. + 227001. i) q^{54} +(66344.7 - 137766. i) q^{55} +(161438. - 71742.1i) q^{56} +(45053.4 - 21696.6i) q^{57} +(-451841. - 68104.1i) q^{58} +(-159216. - 233527. i) q^{59} +(160380. - 24173.3i) q^{60} +(182038. + 196190. i) q^{61} +(-249295. + 198806. i) q^{62} +(-108170. - 19707.9i) q^{63} +(-264248. + 331356. i) q^{64} +(5502.62 + 73427.4i) q^{65} +(152546. - 494542. i) q^{66} +(31587.9 - 54711.9i) q^{67} +(-92922.4 + 53648.8i) q^{68} +(-49156.8 + 11219.7i) q^{69} +(184776. + 289689. i) q^{70} +(-66040.1 + 289341. i) q^{71} +(-157766. + 48664.5i) q^{72} +(609797. - 239328. i) q^{73} +(295251. + 752286. i) q^{74} +(-57445.5 - 186234. i) q^{75} +(250307. + 57130.9i) q^{76} +(673341. - 80495.5i) q^{77} +(55456.3 + 242970. i) q^{78} +(58702.2 + 101675. i) q^{79} +(2020.86 + 1166.74i) q^{80} +(-186337. - 57477.5i) q^{81} +(638302. - 47834.1i) q^{82} +(-564425. - 450114. i) q^{83} +(472878. + 542015. i) q^{84} +(-49863.2 - 62526.5i) q^{85} +(-1.13717e6 + 1.05514e6i) q^{86} +(-106265. - 705022. i) q^{87} +(841348. - 573622. i) q^{88} +(-32620.6 + 216423. i) q^{89} +(-139328. - 289317. i) q^{90} +(-261425. + 195693. i) q^{91} +(-233240. - 112323. i) q^{92} +(-411076. - 280267. i) q^{93} +(38138.8 - 41103.8i) q^{94} +(-14300.7 + 190829. i) q^{95} +(-612782. - 240499. i) q^{96} -1.48393e6i q^{97} +(-642485. + 1.38177e6i) q^{98} -633760. 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.73204 + 12.0571i −0.591505 + 1.50713i 0.250750 + 0.968052i \(0.419323\pi\)
−0.842255 + 0.539079i \(0.818772\pi\)
\(3\) −20.1535 − 1.51030i −0.746426 − 0.0559369i −0.303914 − 0.952699i \(-0.598294\pi\)
−0.442512 + 0.896763i \(0.645913\pi\)
\(4\) −76.0649 − 70.5779i −1.18851 − 1.10278i
\(5\) 43.5678 − 63.9023i 0.348543 − 0.511218i −0.611435 − 0.791295i \(-0.709407\pi\)
0.959978 + 0.280076i \(0.0903598\pi\)
\(6\) 113.577 − 235.845i 0.525819 − 1.09188i
\(7\) 342.839 + 10.5003i 0.999531 + 0.0306131i
\(8\) 464.042 − 223.471i 0.906333 − 0.436467i
\(9\) −316.975 − 47.7763i −0.434808 − 0.0655367i
\(10\) 564.308 + 827.688i 0.564308 + 0.827688i
\(11\) 1954.99 − 294.667i 1.46881 − 0.221388i 0.634604 − 0.772837i \(-0.281163\pi\)
0.834208 + 0.551450i \(0.185925\pi\)
\(12\) 1426.38 + 1537.27i 0.825452 + 0.889626i
\(13\) −744.349 + 593.598i −0.338802 + 0.270186i −0.778080 − 0.628166i \(-0.783806\pi\)
0.439277 + 0.898352i \(0.355235\pi\)
\(14\) −1748.93 + 4083.94i −0.637366 + 1.48832i
\(15\) −974.556 + 1222.05i −0.288757 + 0.362090i
\(16\) 2.25471 + 30.0870i 0.000550466 + 0.00734546i
\(17\) 304.791 − 988.107i 0.0620376 − 0.201121i −0.919248 − 0.393679i \(-0.871202\pi\)
0.981286 + 0.192558i \(0.0616784\pi\)
\(18\) 2075.98 − 3595.70i 0.355963 − 0.616547i
\(19\) −2142.80 + 1237.15i −0.312407 + 0.180368i −0.648003 − 0.761638i \(-0.724396\pi\)
0.335596 + 0.942006i \(0.391062\pi\)
\(20\) −7824.07 + 1785.79i −0.978009 + 0.223224i
\(21\) −6893.56 − 729.406i −0.744364 − 0.0787611i
\(22\) −5698.28 + 24965.8i −0.535150 + 2.34464i
\(23\) 2384.01 − 735.369i 0.195941 − 0.0604396i −0.195232 − 0.980757i \(-0.562546\pi\)
0.391172 + 0.920318i \(0.372070\pi\)
\(24\) −9689.59 + 3802.88i −0.700925 + 0.275093i
\(25\) 3523.11 + 8976.73i 0.225479 + 0.574511i
\(26\) −3634.75 − 11783.6i −0.206802 − 0.670436i
\(27\) 20679.7 + 4720.01i 1.05064 + 0.239801i
\(28\) −25336.9 − 24995.6i −1.15420 − 1.13865i
\(29\) 7850.27 + 34394.3i 0.321878 + 1.41024i 0.834208 + 0.551449i \(0.185925\pi\)
−0.512331 + 0.858788i \(0.671218\pi\)
\(30\) −10122.7 − 17533.1i −0.374916 − 0.649374i
\(31\) 21319.6 + 12308.9i 0.715641 + 0.413175i 0.813146 − 0.582060i \(-0.197753\pi\)
−0.0975055 + 0.995235i \(0.531086\pi\)
\(32\) 31125.2 + 9600.85i 0.949866 + 0.292995i
\(33\) −39844.9 + 2985.96i −1.10874 + 0.0830889i
\(34\) 10471.4 + 8350.65i 0.266420 + 0.212463i
\(35\) 15607.8 − 21450.7i 0.364029 − 0.500309i
\(36\) 20738.7 + 26005.5i 0.444502 + 0.557388i
\(37\) 45738.0 − 42438.6i 0.902966 − 0.837830i −0.0843760 − 0.996434i \(-0.526890\pi\)
0.987342 + 0.158604i \(0.0506992\pi\)
\(38\) −4776.51 − 31690.1i −0.0870482 − 0.577527i
\(39\) 15897.7 − 10838.9i 0.268004 − 0.182722i
\(40\) 5937.01 − 39389.5i 0.0927659 − 0.615461i
\(41\) −21442.0 − 44524.8i −0.311110 − 0.646026i 0.685520 − 0.728054i \(-0.259575\pi\)
−0.996630 + 0.0820274i \(0.973860\pi\)
\(42\) 41415.1 − 79664.4i 0.558999 − 1.07527i
\(43\) 107907. + 51965.2i 1.35720 + 0.653593i 0.964011 − 0.265864i \(-0.0856572\pi\)
0.393190 + 0.919457i \(0.371371\pi\)
\(44\) −169503. − 115565.i −1.98985 − 1.35665i
\(45\) −16862.9 + 18173.9i −0.185053 + 0.199439i
\(46\) −2414.85 + 32223.9i −0.0248094 + 0.331058i
\(47\) −4029.84 − 1581.60i −0.0388145 − 0.0152336i 0.345854 − 0.938288i \(-0.387589\pi\)
−0.384669 + 0.923055i \(0.625684\pi\)
\(48\) − 609.764i − 0.00551364i
\(49\) 117428. + 7199.82i 0.998126 + 0.0611974i
\(50\) −124904. −0.999236
\(51\) −7634.94 + 19453.5i −0.0575566 + 0.146652i
\(52\) 98513.7 + 7382.58i 0.700627 + 0.0525047i
\(53\) −25370.1 − 23540.0i −0.170410 − 0.158117i 0.590358 − 0.807142i \(-0.298987\pi\)
−0.760767 + 0.649025i \(0.775177\pi\)
\(54\) −154767. + 227001.i −0.982870 + 1.44161i
\(55\) 66344.7 − 137766.i 0.398766 − 0.828047i
\(56\) 161438. − 71742.1i 0.919270 − 0.408517i
\(57\) 45053.4 − 21696.6i 0.243278 − 0.117156i
\(58\) −451841. − 68104.1i −2.31581 − 0.349052i
\(59\) −159216. − 233527.i −0.775232 − 1.13706i −0.987154 − 0.159770i \(-0.948925\pi\)
0.211923 − 0.977286i \(-0.432028\pi\)
\(60\) 160380. − 24173.3i 0.742498 − 0.111914i
\(61\) 182038. + 196190.i 0.801995 + 0.864345i 0.993106 − 0.117224i \(-0.0373995\pi\)
−0.191111 + 0.981568i \(0.561209\pi\)
\(62\) −249295. + 198806.i −1.04601 + 0.834169i
\(63\) −108170. − 19707.9i −0.432598 − 0.0788168i
\(64\) −264248. + 331356.i −1.00803 + 1.26402i
\(65\) 5502.62 + 73427.4i 0.0200368 + 0.267373i
\(66\) 152546. − 494542.i 0.530602 − 1.72017i
\(67\) 31587.9 − 54711.9i 0.105026 − 0.181910i −0.808723 − 0.588190i \(-0.799841\pi\)
0.913749 + 0.406280i \(0.133174\pi\)
\(68\) −92922.4 + 53648.8i −0.295525 + 0.170621i
\(69\) −49156.8 + 11219.7i −0.149636 + 0.0341534i
\(70\) 184776. + 289689.i 0.538706 + 0.844575i
\(71\) −66040.1 + 289341.i −0.184515 + 0.808415i 0.794929 + 0.606702i \(0.207508\pi\)
−0.979445 + 0.201713i \(0.935349\pi\)
\(72\) −157766. + 48664.5i −0.422685 + 0.130381i
\(73\) 609797. − 239328.i 1.56753 − 0.615211i 0.586654 − 0.809837i \(-0.300445\pi\)
0.980877 + 0.194626i \(0.0623495\pi\)
\(74\) 295251. + 752286.i 0.728611 + 1.85647i
\(75\) −57445.5 − 186234.i −0.136167 − 0.441443i
\(76\) 250307. + 57130.9i 0.570206 + 0.130146i
\(77\) 673341. − 80495.5i 1.47490 − 0.176319i
\(78\) 55456.3 + 242970.i 0.116860 + 0.511999i
\(79\) 58702.2 + 101675.i 0.119062 + 0.206222i 0.919396 − 0.393333i \(-0.128678\pi\)
−0.800334 + 0.599554i \(0.795345\pi\)
\(80\) 2020.86 + 1166.74i 0.00394699 + 0.00227880i
\(81\) −186337. − 57477.5i −0.350627 − 0.108154i
\(82\) 638302. − 47834.1i 1.15767 − 0.0867553i
\(83\) −564425. − 450114.i −0.987124 − 0.787205i −0.0100166 − 0.999950i \(-0.503188\pi\)
−0.977108 + 0.212744i \(0.931760\pi\)
\(84\) 472878. + 542015.i 0.797831 + 0.914478i
\(85\) −49863.2 − 62526.5i −0.0811940 − 0.101814i
\(86\) −1.13717e6 + 1.05514e6i −1.78784 + 1.65888i
\(87\) −106265. − 705022.i −0.161374 − 1.07064i
\(88\) 841348. − 573622.i 1.23460 − 0.841739i
\(89\) −32620.6 + 216423.i −0.0462724 + 0.306997i 0.953708 + 0.300734i \(0.0972317\pi\)
−0.999980 + 0.00626298i \(0.998006\pi\)
\(90\) −139328. − 289317.i −0.191122 − 0.396868i
\(91\) −261425. + 195693.i −0.346915 + 0.259687i
\(92\) −233240. − 112323.i −0.299530 − 0.144246i
\(93\) −411076. − 280267.i −0.511061 − 0.348436i
\(94\) 38138.8 − 41103.8i 0.0459180 − 0.0494878i
\(95\) −14300.7 + 190829.i −0.0166796 + 0.222574i
\(96\) −612782. − 240499.i −0.692616 − 0.271832i
\(97\) − 1.48393e6i − 1.62592i −0.582321 − 0.812959i \(-0.697855\pi\)
0.582321 − 0.812959i \(-0.302145\pi\)
\(98\) −642485. + 1.38177e6i −0.682629 + 1.46811i
\(99\) −633760. −0.653160
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.3.4 ✓ 324
49.33 odd 42 inner 49.7.h.a.33.4 yes 324
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.7.h.a.3.4 ✓ 324 1.1 even 1 trivial
49.7.h.a.33.4 yes 324 49.33 odd 42 inner