# Properties

 Label 100.4.l.b Level $100$ Weight $4$ Character orbit 100.l Analytic conductor $5.900$ Analytic rank $0$ Dimension $336$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$100 = 2^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 100.l (of order $$20$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.90019100057$$ Analytic rank: $$0$$ Dimension: $$336$$ Relative dimension: $$42$$ over $$\Q(\zeta_{20})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$336 q - 4 q^{2} - 10 q^{4} - 20 q^{5} - 6 q^{6} + 2 q^{8} - 20 q^{9}+O(q^{10})$$ 336 * q - 4 * q^2 - 10 * q^4 - 20 * q^5 - 6 * q^6 + 2 * q^8 - 20 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$336 q - 4 q^{2} - 10 q^{4} - 20 q^{5} - 6 q^{6} + 2 q^{8} - 20 q^{9} + 100 q^{10} + 70 q^{12} - 136 q^{13} - 10 q^{14} - 134 q^{16} + 312 q^{17} - 748 q^{18} - 1030 q^{20} - 12 q^{21} - 370 q^{22} - 360 q^{25} - 312 q^{26} + 870 q^{28} - 20 q^{29} + 1230 q^{30} + 1646 q^{32} - 100 q^{33} + 90 q^{34} + 170 q^{36} + 1452 q^{37} + 880 q^{38} + 620 q^{40} + 932 q^{41} - 470 q^{42} - 1340 q^{44} - 1200 q^{45} - 6 q^{46} - 3400 q^{48} - 2850 q^{50} - 2948 q^{52} + 3484 q^{53} - 3780 q^{54} - 6 q^{56} + 940 q^{57} + 24 q^{58} + 2810 q^{60} - 948 q^{61} + 2900 q^{62} + 4820 q^{64} - 2160 q^{65} - 870 q^{66} + 834 q^{68} - 20 q^{69} + 3030 q^{70} + 2756 q^{72} - 1456 q^{73} + 240 q^{76} - 3140 q^{77} - 3460 q^{78} - 1850 q^{80} + 2904 q^{81} - 6938 q^{82} - 11290 q^{84} + 900 q^{85} - 6 q^{86} - 1570 q^{88} - 6940 q^{89} + 2090 q^{90} + 6130 q^{92} - 1300 q^{93} + 11030 q^{94} - 1746 q^{96} - 13848 q^{97} + 11952 q^{98}+O(q^{100})$$ 336 * q - 4 * q^2 - 10 * q^4 - 20 * q^5 - 6 * q^6 + 2 * q^8 - 20 * q^9 + 100 * q^10 + 70 * q^12 - 136 * q^13 - 10 * q^14 - 134 * q^16 + 312 * q^17 - 748 * q^18 - 1030 * q^20 - 12 * q^21 - 370 * q^22 - 360 * q^25 - 312 * q^26 + 870 * q^28 - 20 * q^29 + 1230 * q^30 + 1646 * q^32 - 100 * q^33 + 90 * q^34 + 170 * q^36 + 1452 * q^37 + 880 * q^38 + 620 * q^40 + 932 * q^41 - 470 * q^42 - 1340 * q^44 - 1200 * q^45 - 6 * q^46 - 3400 * q^48 - 2850 * q^50 - 2948 * q^52 + 3484 * q^53 - 3780 * q^54 - 6 * q^56 + 940 * q^57 + 24 * q^58 + 2810 * q^60 - 948 * q^61 + 2900 * q^62 + 4820 * q^64 - 2160 * q^65 - 870 * q^66 + 834 * q^68 - 20 * q^69 + 3030 * q^70 + 2756 * q^72 - 1456 * q^73 + 240 * q^76 - 3140 * q^77 - 3460 * q^78 - 1850 * q^80 + 2904 * q^81 - 6938 * q^82 - 11290 * q^84 + 900 * q^85 - 6 * q^86 - 1570 * q^88 - 6940 * q^89 + 2090 * q^90 + 6130 * q^92 - 1300 * q^93 + 11030 * q^94 - 1746 * q^96 - 13848 * q^97 + 11952 * q^98

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1 −2.81124 + 0.311368i −8.65906 + 1.37146i 7.80610 1.75066i 1.78395 + 11.0371i 23.9156 6.55165i 19.9642 + 19.9642i −21.3997 + 7.35208i 47.4198 15.4076i −8.45170 30.4724i
3.2 −2.76384 + 0.600991i 3.69203 0.584760i 7.27762 3.32209i −8.66395 + 7.06654i −9.85274 + 3.83506i 6.66746 + 6.66746i −18.1176 + 13.5555i −12.3894 + 4.02556i 19.6988 24.7377i
3.3 −2.75376 0.645594i 4.29532 0.680312i 7.16642 + 3.55563i 10.8663 2.63139i −12.2675 0.899617i 14.2732 + 14.2732i −17.4391 14.4179i −7.69155 + 2.49913i −31.6219 + 0.231015i
3.4 −2.74803 0.669566i 8.37146 1.32591i 7.10336 + 3.67998i −8.14258 7.66149i −23.8928 1.96161i −22.9461 22.9461i −17.0563 14.8689i 42.6448 13.8562i 17.2462 + 26.5060i
3.5 −2.74601 0.677799i −7.00032 + 1.10874i 7.08118 + 3.72249i 4.93198 10.0337i 19.9745 + 1.70019i −12.8348 12.8348i −16.9219 15.0216i 22.0967 7.17965i −20.3441 + 24.2098i
3.6 −2.62239 + 1.05975i 0.515751 0.0816869i 5.75386 5.55815i −0.945089 11.1403i −1.26593 + 0.760781i 5.78969 + 5.78969i −9.19863 + 20.6733i −25.4192 + 8.25920i 14.2843 + 28.2127i
3.7 −2.60734 + 1.09625i −2.84160 + 0.450065i 5.59648 5.71659i 7.98147 + 7.82918i 6.91564 4.28857i −22.0571 22.0571i −8.32514 + 21.0403i −17.8064 + 5.78565i −29.3932 11.6637i
3.8 −2.45543 1.40388i −3.77500 + 0.597901i 4.05822 + 6.89426i −11.1789 0.177504i 10.1086 + 3.83155i 9.86563 + 9.86563i −0.285929 22.6256i −11.7854 + 3.82931i 27.1998 + 16.1298i
3.9 −2.33331 1.59865i 0.894258 0.141637i 2.88864 + 7.46028i 2.87150 + 10.8053i −2.31301 1.09912i −10.5284 10.5284i 5.18627 22.0250i −24.8989 + 8.09014i 10.5738 29.8026i
3.10 −2.29464 + 1.65367i 9.44389 1.49577i 2.53077 7.58915i 10.1965 + 4.58592i −19.1969 + 19.0493i 0.499346 + 0.499346i 6.74271 + 21.5994i 61.2713 19.9082i −30.9810 + 6.33864i
3.11 −1.99108 + 2.00888i −6.98860 + 1.10689i −0.0711750 7.99968i −10.3657 4.18947i 11.6913 16.2431i −4.44930 4.44930i 16.2121 + 15.7851i 21.9368 7.12771i 29.0552 12.4819i
3.12 −1.53298 2.37697i 9.34224 1.47967i −3.29997 + 7.28767i −7.08642 + 8.64770i −17.8386 19.9379i 22.1183 + 22.1183i 22.3814 3.32790i 59.4095 19.3033i 31.4186 + 3.58748i
3.13 −1.47086 + 2.41590i −3.25037 + 0.514808i −3.67314 7.10690i 11.1335 1.02216i 3.53711 8.60978i 12.3581 + 12.3581i 22.5722 + 1.57932i −15.3787 + 4.99683i −13.9064 + 28.4009i
3.14 −1.41772 + 2.44746i 3.68208 0.583183i −3.98013 6.93964i −7.22245 + 8.53442i −3.79284 + 9.83853i −12.1513 12.1513i 22.6272 + 0.0972760i −12.4610 + 4.04881i −10.6482 29.7761i
3.15 −1.29383 2.51516i 6.30160 0.998076i −4.65203 + 6.50835i 10.5614 3.66829i −10.6635 14.5582i −13.0824 13.0824i 22.3884 + 3.27991i 13.0355 4.23550i −22.8910 21.8175i
3.16 −1.27431 2.52510i −6.30160 + 0.998076i −4.75225 + 6.43553i 10.5614 3.66829i 10.5505 + 14.6403i 13.0824 + 13.0824i 22.3062 + 3.79902i 13.0355 4.23550i −22.7213 21.9941i
3.17 −1.18699 + 2.56730i 7.22567 1.14443i −5.18210 6.09474i −7.02886 8.69454i −5.63871 + 19.9089i 11.6198 + 11.6198i 21.7982 6.06960i 25.2220 8.19513i 30.6647 7.72487i
3.18 −1.02195 2.63735i −9.34224 + 1.47967i −5.91124 + 5.39047i −7.08642 + 8.64770i 13.4497 + 23.1266i −22.1183 22.1183i 20.2576 + 10.0812i 59.4095 19.3033i 30.0490 + 9.85187i
3.19 −0.282842 + 2.81425i 1.74080 0.275715i −7.84000 1.59197i 5.30299 9.84268i 0.283561 + 4.97702i −20.0397 20.0397i 6.69769 21.6134i −22.7242 + 7.38353i 26.1998 + 17.7079i
3.20 −0.195449 + 2.82167i −6.08610 + 0.963944i −7.92360 1.10299i −4.53820 + 10.2179i −1.53040 17.3614i 1.35326 + 1.35326i 4.66092 22.1422i 10.4330 3.38987i −27.9444 14.8024i
See next 80 embeddings (of 336 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 87.42 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
25.f odd 20 1 inner
100.l even 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.4.l.b 336
4.b odd 2 1 inner 100.4.l.b 336
25.f odd 20 1 inner 100.4.l.b 336
100.l even 20 1 inner 100.4.l.b 336

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.4.l.b 336 1.a even 1 1 trivial
100.4.l.b 336 4.b odd 2 1 inner
100.4.l.b 336 25.f odd 20 1 inner
100.4.l.b 336 100.l even 20 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{336} + 10 T_{3}^{334} - 46603 T_{3}^{332} - 507400 T_{3}^{330} + 1221749268 T_{3}^{328} + 14149575790 T_{3}^{326} - 23873143283406 T_{3}^{324} - 277642087614780 T_{3}^{322} + \cdots + 72\!\cdots\!00$$ acting on $$S_{4}^{\mathrm{new}}(100, [\chi])$$.