# Properties

 Label 1050.2.bc.e Level 1050 Weight 2 Character orbit 1050.bc Analytic conductor 8.384 Analytic rank 0 Dimension 16 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1050.bc (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.38429221223$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} -\beta_{11} q^{3} + ( \beta_{8} - \beta_{9} ) q^{4} -\beta_{9} q^{6} + ( -\beta_{1} + \beta_{7} + \beta_{11} + \beta_{13} ) q^{7} + ( \beta_{6} - \beta_{13} ) q^{8} -\beta_{8} q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} -\beta_{11} q^{3} + ( \beta_{8} - \beta_{9} ) q^{4} -\beta_{9} q^{6} + ( -\beta_{1} + \beta_{7} + \beta_{11} + \beta_{13} ) q^{7} + ( \beta_{6} - \beta_{13} ) q^{8} -\beta_{8} q^{9} + ( \beta_{3} - \beta_{4} - \beta_{5} + \beta_{9} ) q^{11} + \beta_{6} q^{12} + ( \beta_{2} + \beta_{7} + 2 \beta_{11} + \beta_{14} ) q^{13} -\beta_{10} q^{14} -\beta_{5} q^{16} + ( -\beta_{1} - \beta_{6} + \beta_{7} + \beta_{11} - \beta_{12} - \beta_{14} ) q^{17} + \beta_{13} q^{18} + ( -1 + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{8} + 3 \beta_{9} + \beta_{10} ) q^{19} + ( \beta_{3} + \beta_{9} ) q^{21} + ( -\beta_{1} - \beta_{6} + \beta_{7} + \beta_{11} - \beta_{12} ) q^{22} + ( -\beta_{1} - \beta_{6} - \beta_{12} ) q^{23} + ( -1 - \beta_{5} ) q^{24} + ( \beta_{3} + 2 \beta_{8} - \beta_{15} ) q^{26} + ( -\beta_{6} + \beta_{13} ) q^{27} + ( -\beta_{2} + \beta_{14} ) q^{28} + ( -1 - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{15} ) q^{29} + ( 1 - \beta_{3} - \beta_{4} - \beta_{8} - \beta_{10} + \beta_{15} ) q^{31} -\beta_{11} q^{32} + ( -\beta_{6} + \beta_{7} + \beta_{11} - \beta_{12} - \beta_{14} ) q^{33} + ( 1 + \beta_{4} - \beta_{10} ) q^{34} - q^{36} + ( -\beta_{1} + 6 \beta_{2} + 2 \beta_{7} + 2 \beta_{11} - \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{37} + ( \beta_{1} - 3 \beta_{6} - \beta_{7} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{38} + ( -1 + \beta_{3} + \beta_{4} + 2 \beta_{8} - \beta_{9} + \beta_{10} - \beta_{15} ) q^{39} + ( 1 - \beta_{4} - 6 \beta_{9} - \beta_{10} ) q^{41} + ( -\beta_{6} + \beta_{11} - \beta_{12} ) q^{42} + ( -\beta_{1} + 7 \beta_{6} + \beta_{7} + \beta_{11} + \beta_{12} - 6 \beta_{13} ) q^{43} + ( 1 + \beta_{4} + \beta_{5} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{15} ) q^{44} + ( 1 - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{15} ) q^{46} + ( -\beta_{1} + \beta_{7} + \beta_{11} + \beta_{14} ) q^{47} + ( -\beta_{2} - \beta_{11} ) q^{48} + ( -7 + \beta_{4} + \beta_{8} + \beta_{10} ) q^{49} + ( \beta_{3} - 2 \beta_{5} - \beta_{8} + 2 \beta_{9} + \beta_{15} ) q^{51} + ( -\beta_{6} + \beta_{7} + \beta_{11} - \beta_{12} - \beta_{13} ) q^{52} + ( -\beta_{1} + 6 \beta_{13} - \beta_{14} ) q^{53} + \beta_{5} q^{54} + ( \beta_{5} - \beta_{15} ) q^{56} + ( -\beta_{6} - \beta_{7} - \beta_{11} - \beta_{13} - \beta_{14} ) q^{57} + ( 2 \beta_{1} + \beta_{6} - \beta_{7} - \beta_{11} + \beta_{12} - \beta_{14} ) q^{58} + ( -6 - 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - 2 \beta_{9} ) q^{59} + ( -2 - \beta_{3} - \beta_{5} + 5 \beta_{8} + \beta_{15} ) q^{61} + ( -\beta_{1} + \beta_{6} + \beta_{12} + \beta_{14} ) q^{62} + ( \beta_{2} - \beta_{6} + \beta_{11} - \beta_{12} - \beta_{14} ) q^{63} -\beta_{9} q^{64} + ( \beta_{3} + \beta_{4} - \beta_{5} + \beta_{9} ) q^{66} + ( -2 \beta_{1} + \beta_{6} + 3 \beta_{7} - 3 \beta_{11} + 3 \beta_{12} + \beta_{13} + 2 \beta_{14} ) q^{67} + ( \beta_{1} - \beta_{7} - \beta_{11} + \beta_{14} ) q^{68} + ( 1 + \beta_{4} - \beta_{10} ) q^{69} + ( 6 + \beta_{3} + \beta_{4} + \beta_{8} - \beta_{15} ) q^{71} -\beta_{2} q^{72} + ( -\beta_{1} - 11 \beta_{6} + \beta_{7} - 4 \beta_{11} + \beta_{12} + 6 \beta_{13} + \beta_{14} ) q^{73} + ( 2 + \beta_{3} + \beta_{4} - 3 \beta_{5} + 5 \beta_{8} - 4 \beta_{9} - \beta_{10} + \beta_{15} ) q^{74} + ( \beta_{4} + 2 \beta_{5} + \beta_{10} ) q^{76} + ( \beta_{1} + \beta_{6} - \beta_{7} + 5 \beta_{11} + \beta_{12} + 6 \beta_{13} ) q^{77} + ( \beta_{1} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{78} + ( 1 + \beta_{4} + \beta_{5} - 6 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{15} ) q^{79} + ( 1 + \beta_{5} ) q^{81} + ( -\beta_{1} + 6 \beta_{6} + \beta_{7} + \beta_{11} + \beta_{14} ) q^{82} + ( -\beta_{1} + 6 \beta_{2} - \beta_{6} - \beta_{7} + 5 \beta_{11} - \beta_{12} - 2 \beta_{14} ) q^{83} + ( \beta_{4} + \beta_{5} + \beta_{8} - \beta_{15} ) q^{84} + ( 1 - \beta_{4} - 7 \beta_{5} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{15} ) q^{86} + ( \beta_{1} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{11} + 2 \beta_{12} + \beta_{14} ) q^{87} + ( \beta_{1} + \beta_{14} ) q^{88} + ( -1 - \beta_{3} + \beta_{4} + 9 \beta_{5} - 4 \beta_{8} + 9 \beta_{9} + \beta_{10} - 2 \beta_{15} ) q^{89} + ( -2 \beta_{3} + 6 \beta_{5} - 7 \beta_{8} + 5 \beta_{9} - \beta_{10} + \beta_{15} ) q^{91} + ( \beta_{1} + \beta_{6} + \beta_{12} + \beta_{14} ) q^{92} + ( -\beta_{1} + \beta_{7} + \beta_{11} + \beta_{14} ) q^{93} + ( 1 + \beta_{5} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{15} ) q^{94} -\beta_{8} q^{96} + ( -2 \beta_{1} - 3 \beta_{6} + 4 \beta_{7} + 4 \beta_{11} + 2 \beta_{12} + 5 \beta_{13} - 2 \beta_{14} ) q^{97} + ( \beta_{1} - 6 \beta_{2} - \beta_{7} - \beta_{11} - \beta_{13} - \beta_{14} ) q^{98} + ( -1 + \beta_{3} - 2 \beta_{5} - \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q - 4q^{11} - 8q^{14} + 8q^{16} + 4q^{19} - 4q^{21} - 8q^{24} + 16q^{34} - 16q^{36} + 12q^{44} + 8q^{46} - 96q^{49} + 8q^{51} - 8q^{54} - 4q^{56} - 40q^{59} - 24q^{61} + 12q^{66} + 16q^{69} + 104q^{71} + 48q^{74} + 12q^{79} + 8q^{81} + 4q^{84} + 52q^{86} - 60q^{89} - 52q^{91} + 4q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 28 x^{14} + 519 x^{12} - 5404 x^{10} + 40705 x^{8} - 194544 x^{6} + 672624 x^{4} - 1306368 x^{2} + 1679616$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-2107 \nu^{15} + 12398488 \nu^{13} - 340572477 \nu^{11} + 5822728240 \nu^{9} - 54184962811 \nu^{7} + 320563651380 \nu^{5} - 1048992541632 \nu^{3} + 1559363223744 \nu$$$$)/ 1091452828032$$ $$\beta_{3}$$ $$=$$ $$($$$$-2107 \nu^{14} + 12398488 \nu^{12} - 340572477 \nu^{10} + 5822728240 \nu^{8} - 54184962811 \nu^{6} + 320563651380 \nu^{4} - 1048992541632 \nu^{2} + 1559363223744$$$$)/ 181908804672$$ $$\beta_{4}$$ $$=$$ $$($$$$11299 \nu^{14} - 1267888 \nu^{12} + 26810709 \nu^{10} - 398280184 \nu^{8} + 2897647075 \nu^{6} - 15792050700 \nu^{4} + 39264450624 \nu^{2} - 87800666688$$$$)/ 12993486048$$ $$\beta_{5}$$ $$=$$ $$($$$$-236299 \nu^{14} + 6070207 \nu^{12} - 109750641 \nu^{10} + 1048998769 \nu^{8} - 7670501839 \nu^{6} + 32313724443 \nu^{4} - 122817017232 \nu^{2} + 100335652560$$$$)/ 136431603504$$ $$\beta_{6}$$ $$=$$ $$($$$$-90523 \nu^{15} - 908495 \nu^{13} + 32000682 \nu^{11} - 823985897 \nu^{9} + 6704863382 \nu^{7} - 45329453883 \nu^{5} + 123446110131 \nu^{3} - 348009515532 \nu$$$$)/ 204647405256$$ $$\beta_{7}$$ $$=$$ $$($$$$17575 \nu^{15} - 458203 \nu^{13} + 5317761 \nu^{11} - 14543173 \nu^{9} - 479450177 \nu^{7} + 5273830425 \nu^{5} - 35554785300 \nu^{3} + 94833934272 \nu$$$$)/ 38980458144$$ $$\beta_{8}$$ $$=$$ $$($$$$-961463 \nu^{14} + 19357688 \nu^{12} - 307019433 \nu^{10} + 1771996688 \nu^{8} - 7211681519 \nu^{6} - 31002566364 \nu^{4} + 163015417944 \nu^{2} - 667398916800$$$$)/ 272863207008$$ $$\beta_{9}$$ $$=$$ $$($$$$15973 \nu^{14} - 483532 \nu^{12} + 9009267 \nu^{10} - 98147980 \nu^{8} + 718144501 \nu^{6} - 3406609584 \nu^{4} + 9299398752 \nu^{2} - 14125104000$$$$)/ 3389605056$$ $$\beta_{10}$$ $$=$$ $$($$$$443203 \nu^{14} - 17467576 \nu^{12} + 339648981 \nu^{10} - 4086660592 \nu^{8} + 29388081667 \nu^{6} - 134626595796 \nu^{4} + 322801951680 \nu^{2} - 242518237920$$$$)/ 90954402336$$ $$\beta_{11}$$ $$=$$ $$($$$$-198541 \nu^{15} + 5115945 \nu^{13} - 85575203 \nu^{11} + 733266583 \nu^{9} - 3994950813 \nu^{7} + 9236878637 \nu^{5} + 1083154212 \nu^{3} - 63434342592 \nu$$$$)/ 90954402336$$ $$\beta_{12}$$ $$=$$ $$($$$$7603 \nu^{15} - 127492 \nu^{13} + 1857273 \nu^{11} - 4730788 \nu^{9} - 12141473 \nu^{7} + 642448512 \nu^{5} - 2301711228 \nu^{3} + 7949027664 \nu$$$$)/ 1917071712$$ $$\beta_{13}$$ $$=$$ $$($$$$-14621479 \nu^{15} + 351463840 \nu^{13} - 6005325921 \nu^{11} + 51247418824 \nu^{9} - 328567062775 \nu^{7} + 1107271381356 \nu^{5} - 2905816074768 \nu^{3} + 1746136865088 \nu$$$$)/ 3274358484096$$ $$\beta_{14}$$ $$=$$ $$($$$$15973 \nu^{15} - 483532 \nu^{13} + 9009267 \nu^{11} - 98147980 \nu^{9} + 718144501 \nu^{7} - 3406609584 \nu^{5} + 9299398752 \nu^{3} - 14125104000 \nu$$$$)/ 3389605056$$ $$\beta_{15}$$ $$=$$ $$($$$$-14621479 \nu^{14} + 351463840 \nu^{12} - 6005325921 \nu^{10} + 51247418824 \nu^{8} - 328567062775 \nu^{6} + 1107271381356 \nu^{4} - 2905816074768 \nu^{2} + 1746136865088$$$$)/ 545726414016$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{15} + 2 \beta_{9} - \beta_{8} - 8 \beta_{5} + \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{14} + 6 \beta_{13} + \beta_{12} - 8 \beta_{11} - 8 \beta_{7} + \beta_{6} + 6 \beta_{2} + 8 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$15 \beta_{15} + 15 \beta_{10} + 37 \beta_{9} - 9 \beta_{8} - 79 \beta_{5} - 16 \beta_{4} + 16 \beta_{3} - 63$$ $$\nu^{5}$$ $$=$$ $$22 \beta_{14} + 90 \beta_{13} - 22 \beta_{12} - 169 \beta_{11} - 79 \beta_{7} - 118 \beta_{6} + 6 \beta_{2}$$ $$\nu^{6}$$ $$=$$ $$-28 \beta_{15} + 219 \beta_{10} - 132 \beta_{9} + 292 \beta_{8} - 191 \beta_{4} + 28 \beta_{3} - 693$$ $$\nu^{7}$$ $$=$$ $$-351 \beta_{14} - 168 \beta_{13} - 702 \beta_{12} - 1314 \beta_{11} - 1848 \beta_{6} - 1146 \beta_{2} - 884 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-2900 \beta_{15} + 519 \beta_{10} - 9493 \beta_{9} + 5006 \beta_{8} + 10585 \beta_{5} + 519 \beta_{4} - 2381 \beta_{3} - 519$$ $$\nu^{9}$$ $$=$$ $$-10012 \beta_{14} - 17400 \beta_{13} - 5006 \beta_{12} + 7471 \beta_{11} + 10585 \beta_{7} - 1892 \beta_{6} - 17400 \beta_{2} - 10585 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-29877 \beta_{15} - 29877 \beta_{10} - 97910 \beta_{9} - 159 \beta_{8} + 131384 \beta_{5} + 37997 \beta_{4} - 37997 \beta_{3} + 93387$$ $$\nu^{11}$$ $$=$$ $$-68033 \beta_{14} - 179262 \beta_{13} + 68033 \beta_{12} + 310646 \beta_{11} + 131384 \beta_{7} + 296015 \beta_{6} - 48720 \beta_{2}$$ $$\nu^{12}$$ $$=$$ $$116753 \beta_{15} - 495432 \beta_{10} + 408198 \beta_{9} - 933149 \beta_{8} + 378679 \beta_{4} - 116753 \beta_{3} + 1283736$$ $$\nu^{13}$$ $$=$$ $$903630 \beta_{14} + 700518 \beta_{13} + 1807260 \beta_{12} + 2972592 \beta_{11} + 4079334 \beta_{6} + 2272074 \beta_{2} + 1662415 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$6442267 \beta_{15} - 1604148 \beta_{10} + 22123946 \beta_{9} - 11864047 \beta_{8} - 21254876 \beta_{5} - 1604148 \beta_{4} + 4838119 \beta_{3} + 1604148$$ $$\nu^{15}$$ $$=$$ $$23728094 \beta_{14} + 38653602 \beta_{13} + 11864047 \beta_{12} - 11629988 \beta_{11} - 21254876 \beta_{7} + 2239159 \beta_{6} + 38653602 \beta_{2} + 21254876 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$701$$ $$\chi(n)$$ $$\beta_{9}$$ $$-\beta_{5}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
157.1
 3.11681 + 1.79949i −1.44378 − 0.833568i 1.44378 + 0.833568i −3.11681 − 1.79949i 1.90899 + 1.10215i −2.35727 − 1.36097i 2.35727 + 1.36097i −1.90899 − 1.10215i −2.35727 + 1.36097i 1.90899 − 1.10215i −1.90899 + 1.10215i 2.35727 − 1.36097i −1.44378 + 0.833568i 3.11681 − 1.79949i −3.11681 + 1.79949i 1.44378 − 0.833568i
−0.258819 + 0.965926i −0.965926 + 0.258819i −0.866025 0.500000i 0 1.00000i 0.258819 2.63306i 0.707107 0.707107i 0.866025 0.500000i 0
157.2 −0.258819 + 0.965926i −0.965926 + 0.258819i −0.866025 0.500000i 0 1.00000i 0.258819 + 2.63306i 0.707107 0.707107i 0.866025 0.500000i 0
157.3 0.258819 0.965926i 0.965926 0.258819i −0.866025 0.500000i 0 1.00000i −0.258819 2.63306i −0.707107 + 0.707107i 0.866025 0.500000i 0
157.4 0.258819 0.965926i 0.965926 0.258819i −0.866025 0.500000i 0 1.00000i −0.258819 + 2.63306i −0.707107 + 0.707107i 0.866025 0.500000i 0
493.1 −0.965926 0.258819i −0.258819 0.965926i 0.866025 + 0.500000i 0 1.00000i 0.965926 2.46313i −0.707107 0.707107i −0.866025 + 0.500000i 0
493.2 −0.965926 0.258819i −0.258819 0.965926i 0.866025 + 0.500000i 0 1.00000i 0.965926 + 2.46313i −0.707107 0.707107i −0.866025 + 0.500000i 0
493.3 0.965926 + 0.258819i 0.258819 + 0.965926i 0.866025 + 0.500000i 0 1.00000i −0.965926 2.46313i 0.707107 + 0.707107i −0.866025 + 0.500000i 0
493.4 0.965926 + 0.258819i 0.258819 + 0.965926i 0.866025 + 0.500000i 0 1.00000i −0.965926 + 2.46313i 0.707107 + 0.707107i −0.866025 + 0.500000i 0
607.1 −0.965926 + 0.258819i −0.258819 + 0.965926i 0.866025 0.500000i 0 1.00000i 0.965926 2.46313i −0.707107 + 0.707107i −0.866025 0.500000i 0
607.2 −0.965926 + 0.258819i −0.258819 + 0.965926i 0.866025 0.500000i 0 1.00000i 0.965926 + 2.46313i −0.707107 + 0.707107i −0.866025 0.500000i 0
607.3 0.965926 0.258819i 0.258819 0.965926i 0.866025 0.500000i 0 1.00000i −0.965926 2.46313i 0.707107 0.707107i −0.866025 0.500000i 0
607.4 0.965926 0.258819i 0.258819 0.965926i 0.866025 0.500000i 0 1.00000i −0.965926 + 2.46313i 0.707107 0.707107i −0.866025 0.500000i 0
943.1 −0.258819 0.965926i −0.965926 0.258819i −0.866025 + 0.500000i 0 1.00000i 0.258819 2.63306i 0.707107 + 0.707107i 0.866025 + 0.500000i 0
943.2 −0.258819 0.965926i −0.965926 0.258819i −0.866025 + 0.500000i 0 1.00000i 0.258819 + 2.63306i 0.707107 + 0.707107i 0.866025 + 0.500000i 0
943.3 0.258819 + 0.965926i 0.965926 + 0.258819i −0.866025 + 0.500000i 0 1.00000i −0.258819 2.63306i −0.707107 0.707107i 0.866025 + 0.500000i 0
943.4 0.258819 + 0.965926i 0.965926 + 0.258819i −0.866025 + 0.500000i 0 1.00000i −0.258819 + 2.63306i −0.707107 0.707107i 0.866025 + 0.500000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 943.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
35.k even 12 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.bc.e 16
5.b even 2 1 inner 1050.2.bc.e 16
5.c odd 4 2 1050.2.bc.f yes 16
7.d odd 6 1 1050.2.bc.f yes 16
35.i odd 6 1 1050.2.bc.f yes 16
35.k even 12 2 inner 1050.2.bc.e 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.2.bc.e 16 1.a even 1 1 trivial
1050.2.bc.e 16 5.b even 2 1 inner
1050.2.bc.e 16 35.k even 12 2 inner
1050.2.bc.f yes 16 5.c odd 4 2
1050.2.bc.f yes 16 7.d odd 6 1
1050.2.bc.f yes 16 35.i odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1050, [\chi])$$:

 $$T_{11}^{8} + \cdots$$ $$T_{13}^{16} + 1714 T_{13}^{12} + 283569 T_{13}^{8} + 11618176 T_{13}^{4} + 4096$$ $$T_{17}^{16} + \cdots$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{4} + T^{8} )^{2}$$
$3$ $$( 1 - T^{4} + T^{8} )^{2}$$
$5$ 1
$7$ $$( 1 + 24 T^{2} + 239 T^{4} + 1176 T^{6} + 2401 T^{8} )^{2}$$
$11$ $$( 1 + 2 T - 14 T^{2} - 48 T^{3} - 58 T^{4} + 26 T^{5} - 168 T^{6} + 2902 T^{7} + 26203 T^{8} + 31922 T^{9} - 20328 T^{10} + 34606 T^{11} - 849178 T^{12} - 7730448 T^{13} - 24801854 T^{14} + 38974342 T^{15} + 214358881 T^{16} )^{2}$$
$13$ $$1 + 50 T^{4} + 39793 T^{8} + 11442962 T^{12} + 523271620 T^{16} + 326822437682 T^{20} + 32460372580753 T^{24} + 1164904256124050 T^{28} + 665416609183179841 T^{32}$$
$17$ $$1 + 96 T^{2} + 4335 T^{4} + 121248 T^{6} + 2296193 T^{8} + 27677184 T^{10} + 58458882 T^{12} - 6525249024 T^{14} - 167337014754 T^{16} - 1885796967936 T^{18} + 4882544283522 T^{20} + 668059938525696 T^{22} + 16017685405722113 T^{24} + 244435228441640352 T^{26} + 2525667398391013935 T^{28} + 16164271349702489184 T^{30} + 48661191875666868481 T^{32}$$
$19$ $$( 1 - 2 T - 43 T^{2} - 70 T^{3} + 1187 T^{4} + 3200 T^{5} - 13284 T^{6} - 38196 T^{7} + 120370 T^{8} - 725724 T^{9} - 4795524 T^{10} + 21948800 T^{11} + 154691027 T^{12} - 173326930 T^{13} - 2022972883 T^{14} - 1787743478 T^{15} + 16983563041 T^{16} )^{2}$$
$23$ $$1 - 96 T^{2} + 4119 T^{4} - 100512 T^{6} + 1547777 T^{8} - 13526016 T^{10} - 398667390 T^{12} + 32311398144 T^{14} - 1026908911218 T^{16} + 17092729618176 T^{18} - 111563481084990 T^{20} - 2002335803188224 T^{22} + 121207941865270337 T^{24} - 4163861495106288288 T^{26} + 90266338035491702199 T^{28} -$$$$11\!\cdots\!64$$$$T^{30} +$$$$61\!\cdots\!61$$$$T^{32}$$
$29$ $$( 1 - 64 T^{2} + 3856 T^{4} - 135376 T^{6} + 4752862 T^{8} - 113851216 T^{10} + 2727275536 T^{12} - 38068692544 T^{14} + 500246412961 T^{16} )^{2}$$
$31$ $$( 1 + 99 T^{2} + 5549 T^{4} + 2448 T^{5} + 231102 T^{6} + 153072 T^{7} + 7750614 T^{8} + 4745232 T^{9} + 222089022 T^{10} + 72928368 T^{11} + 5124618029 T^{12} + 87862864419 T^{14} + 852891037441 T^{16} )^{2}$$
$37$ $$1 + 36 T^{2} + 4499 T^{4} + 146412 T^{6} + 10243837 T^{8} + 329458176 T^{10} + 19288003070 T^{12} + 581067044496 T^{14} + 30324317162902 T^{16} + 795480783915024 T^{18} + 36148823121674270 T^{20} + 845299542824169984 T^{22} + 35981266991815734877 T^{24} +$$$$70\!\cdots\!88$$$$T^{26} +$$$$29\!\cdots\!19$$$$T^{28} +$$$$32\!\cdots\!04$$$$T^{30} +$$$$12\!\cdots\!41$$$$T^{32}$$
$41$ $$( 1 - 158 T^{2} + 12905 T^{4} - 739782 T^{6} + 33449684 T^{8} - 1243573542 T^{10} + 36466445705 T^{12} - 750516470078 T^{14} + 7984925229121 T^{16} )^{2}$$
$43$ $$1 - 1504 T^{4} - 258116 T^{8} - 4620736288 T^{12} + 14886307959622 T^{16} - 15797377842150688 T^{20} - 3016911502853259716 T^{24} -$$$$60\!\cdots\!04$$$$T^{28} +$$$$13\!\cdots\!01$$$$T^{32}$$
$47$ $$1 + 72 T^{2} - 2097 T^{4} - 275400 T^{6} - 1208767 T^{8} + 434448 T^{10} - 23245686270 T^{12} + 566620331904 T^{14} + 118374567124638 T^{16} + 1251664313175936 T^{18} - 113431533623679870 T^{20} + 4683008541253392 T^{22} - 28782297544276858687 T^{24} -$$$$14\!\cdots\!00$$$$T^{26} -$$$$24\!\cdots\!77$$$$T^{28} +$$$$18\!\cdots\!68$$$$T^{30} +$$$$56\!\cdots\!21$$$$T^{32}$$
$53$ $$1 - 60 T^{2} + 2040 T^{4} - 50400 T^{6} + 8790530 T^{8} + 35607396 T^{10} - 20650486848 T^{12} + 1867209163140 T^{14} - 61066879873293 T^{16} + 5244990539260260 T^{18} - 162942274114893888 T^{20} + 789215183807310084 T^{22} +$$$$54\!\cdots\!30$$$$T^{24} -$$$$88\!\cdots\!00$$$$T^{26} +$$$$10\!\cdots\!40$$$$T^{28} -$$$$82\!\cdots\!40$$$$T^{30} +$$$$38\!\cdots\!21$$$$T^{32}$$
$59$ $$( 1 + 20 T + 124 T^{2} - 312 T^{3} - 7894 T^{4} - 53140 T^{5} - 123312 T^{6} + 3441412 T^{7} + 48399379 T^{8} + 203043308 T^{9} - 429249072 T^{10} - 10913840060 T^{11} - 95654447734 T^{12} - 223056381288 T^{13} + 5230386171484 T^{14} + 49773029696380 T^{15} + 146830437604321 T^{16} )^{2}$$
$61$ $$( 1 + 12 T + 225 T^{2} + 2124 T^{3} + 22907 T^{4} + 153816 T^{5} + 1374816 T^{6} + 7573500 T^{7} + 71943462 T^{8} + 461983500 T^{9} + 5115690336 T^{10} + 34913309496 T^{11} + 317166679787 T^{12} + 1793922543324 T^{13} + 11592084231225 T^{14} + 37712914032252 T^{15} + 191707312997281 T^{16} )^{2}$$
$67$ $$1 + 144 T^{2} + 15479 T^{4} + 1233648 T^{6} + 82019857 T^{8} + 5129471520 T^{10} + 354430386290 T^{12} + 27207762471936 T^{14} + 1909973041583038 T^{16} + 122135645736520704 T^{18} + 7142169600206531090 T^{20} +$$$$46\!\cdots\!80$$$$T^{22} +$$$$33\!\cdots\!37$$$$T^{24} +$$$$22\!\cdots\!52$$$$T^{26} +$$$$12\!\cdots\!19$$$$T^{28} +$$$$52\!\cdots\!76$$$$T^{30} +$$$$16\!\cdots\!81$$$$T^{32}$$
$71$ $$( 1 - 26 T + 513 T^{2} - 6318 T^{3} + 63556 T^{4} - 448578 T^{5} + 2586033 T^{6} - 9305686 T^{7} + 25411681 T^{8} )^{4}$$
$73$ $$1 - 576 T^{2} + 168527 T^{4} - 33370560 T^{6} + 5044720033 T^{8} - 621838271232 T^{10} + 64869439165922 T^{12} - 5836679136073728 T^{14} + 456532218503037886 T^{16} - 31103663116136896512 T^{18} +$$$$18\!\cdots\!02$$$$T^{20} -$$$$94\!\cdots\!48$$$$T^{22} +$$$$40\!\cdots\!73$$$$T^{24} -$$$$14\!\cdots\!40$$$$T^{26} +$$$$38\!\cdots\!67$$$$T^{28} -$$$$70\!\cdots\!84$$$$T^{30} +$$$$65\!\cdots\!61$$$$T^{32}$$
$79$ $$( 1 - 6 T + 194 T^{2} - 1092 T^{3} + 17173 T^{4} - 108768 T^{5} + 1220606 T^{6} - 9301122 T^{7} + 98127316 T^{8} - 734788638 T^{9} + 7617802046 T^{10} - 53626865952 T^{11} + 668889741013 T^{12} - 3360145587708 T^{13} + 47158966371074 T^{14} - 115223453916954 T^{15} + 1517108809906561 T^{16} )^{2}$$
$83$ $$1 + 2720 T^{4} - 86504132 T^{8} - 3906190240 T^{12} + 5664126472953286 T^{16} - 185381230296987040 T^{20} -$$$$19\!\cdots\!12$$$$T^{24} +$$$$29\!\cdots\!20$$$$T^{28} +$$$$50\!\cdots\!81$$$$T^{32}$$
$89$ $$( 1 + 30 T + 439 T^{2} + 3930 T^{3} + 17353 T^{4} - 77580 T^{5} - 2924786 T^{6} - 44484000 T^{7} - 479997158 T^{8} - 3959076000 T^{9} - 23167229906 T^{10} - 54691495020 T^{11} + 1088766108073 T^{12} + 21945353634570 T^{13} + 218174786731879 T^{14} + 1326940046865870 T^{15} + 3936588805702081 T^{16} )^{2}$$
$97$ $$1 + 4700 T^{4} - 3632918 T^{8} + 688067520560 T^{12} + 9198193610860915 T^{16} + 60914122874629517360 T^{20} -$$$$28\!\cdots\!98$$$$T^{24} +$$$$32\!\cdots\!00$$$$T^{28} +$$$$61\!\cdots\!21$$$$T^{32}$$