# Properties

 Label 33.7.g.a Level 33 Weight 7 Character orbit 33.g Analytic conductor 7.592 Analytic rank 0 Dimension 48 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 33.g (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.59178475946$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$12$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48q + 204q^{4} - 224q^{5} + 720q^{7} - 3200q^{8} - 2916q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q + 204q^{4} - 224q^{5} + 720q^{7} - 3200q^{8} - 2916q^{9} + 6664q^{11} - 1944q^{12} + 3600q^{13} - 14602q^{14} + 9396q^{15} - 13416q^{16} - 7280q^{17} - 12150q^{18} - 11958q^{20} - 19518q^{22} + 64704q^{23} + 85050q^{24} - 9492q^{25} + 89270q^{26} - 149370q^{28} - 128800q^{29} - 82620q^{30} + 79308q^{31} - 4212q^{33} + 355152q^{34} + 380480q^{35} + 49572q^{36} + 236112q^{37} - 318458q^{38} - 1045170q^{40} - 528360q^{41} + 44388q^{42} + 676754q^{44} - 54432q^{45} + 1215450q^{46} + 141720q^{47} + 66096q^{48} + 215268q^{49} - 679470q^{50} - 362880q^{51} - 1957890q^{52} - 313488q^{53} - 397656q^{55} + 960220q^{56} + 466560q^{57} + 198558q^{58} + 1236208q^{59} + 1247076q^{60} - 1409280q^{61} - 453280q^{62} - 174960q^{63} + 521784q^{64} - 1211112q^{66} + 930528q^{67} - 842240q^{68} - 82944q^{69} + 2765100q^{70} - 3520024q^{71} - 206550q^{72} + 201780q^{73} + 2051950q^{74} + 1318032q^{75} + 2434584q^{77} - 971352q^{78} - 321960q^{79} + 2395994q^{80} - 708588q^{81} + 2354178q^{82} - 4816400q^{83} - 3280500q^{84} - 1250040q^{85} - 3323560q^{86} - 1493928q^{88} + 4157744q^{89} + 1268460q^{90} - 1843512q^{91} - 4448550q^{92} + 4197096q^{93} + 14061810q^{94} + 14355680q^{95} + 1397250q^{96} + 2325888q^{97} - 810648q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 −8.29374 + 11.4154i 4.81710 14.8255i −41.7471 128.484i −55.3317 + 40.2008i 129.287 + 177.948i 511.454 166.181i 954.081 + 310.000i −196.591 142.832i 965.045i
7.2 −6.91015 + 9.51101i −4.81710 + 14.8255i −22.9320 70.5775i 174.946 127.106i −107.719 148.262i 472.320 153.466i 114.152 + 37.0902i −196.591 142.832i 2542.24i
7.3 −6.32244 + 8.70209i −4.81710 + 14.8255i −15.9760 49.1691i −92.2594 + 67.0304i −98.5570 135.652i −121.251 + 39.3968i −125.833 40.8857i −196.591 142.832i 1226.64i
7.4 −5.06354 + 6.96937i 4.81710 14.8255i −3.15555 9.71180i −8.79332 + 6.38872i 78.9328 + 108.642i −242.092 + 78.6605i −440.688 143.188i −196.591 142.832i 93.6335i
7.5 −1.16939 + 1.60953i −4.81710 + 14.8255i 18.5540 + 57.1033i −1.49745 + 1.08796i −18.2291 25.0902i −25.7895 + 8.37953i −234.702 76.2594i −196.591 142.832i 3.68244i
7.6 1.61569 2.22380i 4.81710 14.8255i 17.4422 + 53.6817i −125.017 + 90.8304i −25.1861 34.6656i −357.347 + 116.109i 314.870 + 102.307i −196.591 142.832i 424.767i
7.7 2.37250 3.26547i 4.81710 14.8255i 14.7426 + 45.3730i 26.5586 19.2960i −36.9836 50.9036i 457.293 148.583i 428.823 + 139.333i −196.591 142.832i 132.506i
7.8 4.99839 6.87969i −4.81710 + 14.8255i −2.56917 7.90708i −1.33123 + 0.967196i 77.9171 + 107.244i 525.675 170.802i 450.364 + 146.332i −196.591 142.832i 13.9929i
7.9 6.02470 8.29229i −4.81710 + 14.8255i −12.6880 39.0496i −154.787 + 112.460i 93.9158 + 129.264i −515.310 + 167.434i 223.631 + 72.6622i −196.591 142.832i 1961.08i
7.10 6.26570 8.62400i 4.81710 14.8255i −15.3373 47.2032i 161.919 117.641i −97.6727 134.435i −310.129 + 100.767i 145.660 + 47.3278i −196.591 142.832i 2133.50i
7.11 8.69356 11.9657i 4.81710 14.8255i −47.8219 147.181i −118.124 + 85.8219i −135.519 186.526i 185.111 60.1462i −1276.60 414.794i −196.591 142.832i 2159.53i
7.12 8.96906 12.3449i −4.81710 + 14.8255i −52.1743 160.576i 137.717 100.057i 139.814 + 192.437i −91.3553 + 29.6831i −1521.46 494.352i −196.591 142.832i 2597.52i
13.1 −14.5443 4.72574i −12.6113 + 9.16267i 137.428 + 99.8469i −61.8586 190.381i 226.723 73.6669i 168.802 232.336i −951.651 1309.84i 75.0911 231.107i 3061.29i
13.2 −12.0509 3.91558i 12.6113 9.16267i 78.1159 + 56.7545i −13.4634 41.4361i −187.855 + 61.0379i −335.737 + 462.103i −242.477 333.741i 75.0911 231.107i 552.060i
13.3 −9.54864 3.10254i −12.6113 + 9.16267i 29.7736 + 21.6318i 33.3013 + 102.491i 148.849 48.3638i −90.4877 + 124.546i 160.505 + 220.916i 75.0911 231.107i 1081.97i
13.4 −9.48025 3.08032i 12.6113 9.16267i 28.6097 + 20.7861i −12.6698 38.9936i −147.782 + 48.0174i 304.303 418.837i 167.785 + 230.936i 75.0911 231.107i 408.696i
13.5 −5.17091 1.68013i −12.6113 + 9.16267i −27.8616 20.2427i −25.6501 78.9429i 80.6065 26.1906i −29.3745 + 40.4305i 314.591 + 432.997i 75.0911 231.107i 451.302i
13.6 −2.69988 0.877245i 12.6113 9.16267i −45.2573 32.8813i 71.7465 + 220.813i −42.0870 + 13.6749i 166.082 228.592i 200.136 + 275.463i 75.0911 231.107i 659.108i
13.7 −0.674572 0.219182i 12.6113 9.16267i −51.3701 37.3225i −8.69665 26.7655i −10.5155 + 3.41670i −301.986 + 415.648i 53.1545 + 73.1609i 75.0911 231.107i 19.9614i
13.8 4.51299 + 1.46636i −12.6113 + 9.16267i −33.5602 24.3829i 6.35291 + 19.5523i −70.3505 + 22.8583i 355.989 489.977i −294.210 404.946i 75.0911 231.107i 97.5547i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 28.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.7.g.a 48
3.b odd 2 1 99.7.k.b 48
11.d odd 10 1 inner 33.7.g.a 48
33.f even 10 1 99.7.k.b 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.7.g.a 48 1.a even 1 1 trivial
33.7.g.a 48 11.d odd 10 1 inner
99.7.k.b 48 3.b odd 2 1
99.7.k.b 48 33.f even 10 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{7}^{\mathrm{new}}(33, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database