# Properties

 Label 1050.2.bc.f 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: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 28 x^{14} + 519 x^{12} - 5404 x^{10} + 40705 x^{8} - 194544 x^{6} + 672624 x^{4} - 1306368 x^{2} + 1679616$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{11} q^{2} + \beta_{2} q^{3} -\beta_{8} q^{4} + \beta_{9} q^{6} -\beta_{12} q^{7} + ( \beta_{6} - \beta_{13} ) q^{8} + ( \beta_{8} - \beta_{9} ) q^{9} +O(q^{10})$$ $$q + \beta_{11} q^{2} + \beta_{2} q^{3} -\beta_{8} q^{4} + \beta_{9} q^{6} -\beta_{12} q^{7} + ( \beta_{6} - \beta_{13} ) q^{8} + ( \beta_{8} - \beta_{9} ) q^{9} + ( -1 + \beta_{4} + \beta_{5} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{15} ) q^{11} + \beta_{13} q^{12} + ( -\beta_{2} - \beta_{7} - 2 \beta_{11} - \beta_{14} ) q^{13} + ( 1 - \beta_{4} ) q^{14} + ( 1 + \beta_{5} ) q^{16} + ( -\beta_{1} - \beta_{6} - \beta_{12} ) q^{17} -\beta_{6} q^{18} + ( -1 - \beta_{5} + 2 \beta_{9} + \beta_{10} + \beta_{15} ) q^{19} + ( -1 + \beta_{4} + \beta_{8} - \beta_{9} - \beta_{15} ) q^{21} + ( -\beta_{1} - \beta_{6} + \beta_{7} + \beta_{11} - \beta_{12} ) q^{22} + ( \beta_{1} + \beta_{6} - \beta_{7} - \beta_{11} + \beta_{12} + \beta_{14} ) q^{23} -\beta_{5} q^{24} + ( -1 + \beta_{3} + \beta_{4} + 2 \beta_{8} - \beta_{9} + \beta_{10} - \beta_{15} ) q^{26} + ( \beta_{6} - \beta_{13} ) q^{27} -\beta_{7} q^{28} + ( -1 - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{15} ) q^{29} + ( -\beta_{3} - \beta_{8} + \beta_{15} ) q^{31} -\beta_{2} q^{32} + ( \beta_{1} - \beta_{14} ) q^{33} + ( -1 - \beta_{4} + \beta_{10} ) q^{34} - q^{36} + ( -\beta_{1} - 3 \beta_{6} - \beta_{7} + 5 \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{37} + ( \beta_{1} - 2 \beta_{6} - \beta_{12} + 2 \beta_{13} ) q^{38} + ( -\beta_{3} - 2 \beta_{8} + \beta_{15} ) q^{39} + ( -1 + \beta_{4} + 6 \beta_{9} + \beta_{10} ) q^{41} + ( -\beta_{1} + \beta_{2} + \beta_{7} + \beta_{11} ) q^{42} + ( -\beta_{1} + 7 \beta_{6} + \beta_{7} + \beta_{11} + \beta_{12} - 6 \beta_{13} ) q^{43} + ( -\beta_{3} - \beta_{4} + \beta_{5} - \beta_{9} ) q^{44} + ( \beta_{3} - 2 \beta_{5} - \beta_{8} + 2 \beta_{9} + \beta_{15} ) q^{46} + ( -\beta_{6} + \beta_{7} + \beta_{11} - \beta_{12} ) q^{47} + ( \beta_{2} + \beta_{11} ) q^{48} + ( 5 + \beta_{4} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{49} + ( 1 - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{15} ) q^{51} + ( -\beta_{1} - \beta_{6} + \beta_{7} + \beta_{11} + \beta_{14} ) q^{52} + ( -5 \beta_{6} + \beta_{7} + \beta_{11} + \beta_{12} + \beta_{14} ) q^{53} + ( 1 + \beta_{5} ) q^{54} + ( -1 + \beta_{3} - \beta_{5} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{56} + ( -\beta_{6} - \beta_{7} - \beta_{11} - \beta_{13} - \beta_{14} ) q^{57} + ( -\beta_{1} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{11} - 2 \beta_{12} - \beta_{14} ) q^{58} + ( -2 + 2 \beta_{4} - 4 \beta_{5} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{15} ) q^{59} + ( -\beta_{3} - \beta_{4} + \beta_{5} + 5 \beta_{8} - 6 \beta_{9} - \beta_{10} + \beta_{15} ) q^{61} + ( \beta_{1} - \beta_{6} - \beta_{12} - \beta_{14} ) q^{62} + ( \beta_{1} - \beta_{2} - \beta_{11} ) q^{63} -\beta_{9} q^{64} + ( 1 + \beta_{4} + \beta_{5} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{15} ) q^{66} + ( 3 \beta_{1} - 6 \beta_{2} - \beta_{6} - \beta_{7} - \beta_{11} - 2 \beta_{12} - 2 \beta_{13} + \beta_{14} ) q^{67} + ( -\beta_{6} - \beta_{7} - \beta_{11} - \beta_{12} ) q^{68} + ( -1 - \beta_{4} + \beta_{10} ) q^{69} + ( 6 + \beta_{3} + \beta_{4} + \beta_{8} - \beta_{15} ) q^{71} -\beta_{11} q^{72} + ( -\beta_{1} + 5 \beta_{2} - 5 \beta_{6} + \beta_{12} + 12 \beta_{13} ) q^{73} + ( -4 + \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 7 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + \beta_{15} ) q^{74} + ( -\beta_{4} - 2 \beta_{5} - \beta_{10} ) q^{76} + ( \beta_{1} + 6 \beta_{2} + 7 \beta_{6} - \beta_{7} - \beta_{11} + \beta_{12} ) q^{77} + ( \beta_{1} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{78} + ( -\beta_{3} - \beta_{4} + \beta_{5} + 7 \beta_{8} - 8 \beta_{9} ) q^{79} -\beta_{5} q^{81} + ( -\beta_{6} + \beta_{7} + \beta_{11} - \beta_{12} + 6 \beta_{13} ) q^{82} + ( \beta_{1} - 6 \beta_{2} + \beta_{6} + \beta_{7} - 5 \beta_{11} + \beta_{12} + 2 \beta_{14} ) q^{83} + ( -\beta_{3} + \beta_{5} ) q^{84} + ( 6 - \beta_{3} + \beta_{4} + 7 \beta_{5} - \beta_{9} ) q^{86} + ( 2 \beta_{1} + \beta_{6} - \beta_{7} - \beta_{11} + \beta_{12} - \beta_{14} ) q^{87} + ( -\beta_{6} - \beta_{7} - \beta_{11} - \beta_{12} - \beta_{14} ) q^{88} + ( 7 - 2 \beta_{3} + 2 \beta_{4} + 9 \beta_{5} + 7 \beta_{8} + 3 \beta_{9} - \beta_{10} - \beta_{15} ) q^{89} + ( -5 - \beta_{3} - \beta_{4} - 6 \beta_{5} - 8 \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{15} ) q^{91} + ( \beta_{1} + \beta_{6} + \beta_{12} + \beta_{14} ) q^{92} + ( \beta_{6} - \beta_{7} - \beta_{11} + \beta_{12} ) q^{93} + ( 1 - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{9} - \beta_{10} ) q^{94} + ( -\beta_{8} + \beta_{9} ) 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_{7} + 7 \beta_{11} - \beta_{12} ) 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 $$\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$$.

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

## 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
 −1.90899 + 1.10215i 2.35727 − 1.36097i −2.35727 + 1.36097i 1.90899 − 1.10215i 1.44378 − 0.833568i −3.11681 + 1.79949i 3.11681 − 1.79949i −1.44378 + 0.833568i 1.44378 + 0.833568i −3.11681 − 1.79949i 3.11681 + 1.79949i −1.44378 − 0.833568i −1.90899 − 1.10215i 2.35727 + 1.36097i −2.35727 − 1.36097i 1.90899 + 1.10215i
−0.258819 + 0.965926i 0.965926 0.258819i −0.866025 0.500000i 0 1.00000i −2.46313 0.965926i 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 2.46313 0.965926i 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 −2.46313 + 0.965926i −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 2.46313 + 0.965926i −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 −2.63306 + 0.258819i −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 2.63306 + 0.258819i −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 −2.63306 0.258819i 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 2.63306 0.258819i 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 −2.63306 0.258819i −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 2.63306 0.258819i −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 −2.63306 + 0.258819i 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 2.63306 + 0.258819i 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 −2.46313 + 0.965926i 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 2.46313 + 0.965926i 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 −2.46313 0.965926i −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 2.46313 0.965926i −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.f yes 16
5.b even 2 1 inner 1050.2.bc.f yes 16
5.c odd 4 2 1050.2.bc.e 16
7.d odd 6 1 1050.2.bc.e 16
35.i odd 6 1 1050.2.bc.e 16
35.k even 12 2 inner 1050.2.bc.f yes 16

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

## 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

$p$ $F_p(T)$
$2$ $$( 1 - T^{4} + T^{8} )^{2}$$
$3$ $$( 1 - T^{4} + T^{8} )^{2}$$
$5$ $$T^{16}$$
$7$ $$( 2401 - 1176 T^{2} + 239 T^{4} - 24 T^{6} + T^{8} )^{2}$$
$11$ $$( 20736 - 3456 T + 4320 T^{2} + 48 T^{3} + 580 T^{4} - 4 T^{5} + 30 T^{6} + 2 T^{7} + T^{8} )^{2}$$
$13$ $$4096 + 11618176 T^{4} + 283569 T^{8} + 1714 T^{12} + T^{16}$$
$17$ $$1679616 + 3359232 T^{2} + 2642544 T^{4} + 806112 T^{6} + 12481 T^{8} - 29856 T^{10} + 3383 T^{12} - 96 T^{14} + T^{16}$$
$19$ $$( 4356 + 6732 T + 8490 T^{2} + 3222 T^{3} + 1111 T^{4} + 146 T^{5} + 33 T^{6} + 2 T^{7} + T^{8} )^{2}$$
$23$ $$1679616 - 3359232 T^{2} + 2642544 T^{4} - 806112 T^{6} + 12481 T^{8} + 29856 T^{10} + 3383 T^{12} + 96 T^{14} + T^{16}$$
$29$ $$( 1679616 + 217728 T^{2} + 9540 T^{4} + 168 T^{6} + T^{8} )^{2}$$
$31$ $$( 1296 + 1296 T - 468 T^{2} - 900 T^{3} + 589 T^{4} - 25 T^{6} + T^{8} )^{2}$$
$37$ $$403540761128976 + 9079520681520 T^{2} - 177243394212 T^{4} - 5520031740 T^{6} + 123645285 T^{8} + 439668 T^{10} - 11781 T^{12} - 36 T^{14} + T^{16}$$
$41$ $$( 46656 + 88992 T^{2} + 7657 T^{4} + 170 T^{6} + T^{8} )^{2}$$
$43$ $$393460125696 + 20712319488 T^{4} + 108864016 T^{8} + 22232 T^{12} + T^{16}$$
$47$ $$1679616 + 4478976 T^{2} + 4384368 T^{4} + 1074816 T^{6} + 12481 T^{8} - 22392 T^{10} + 2039 T^{12} - 72 T^{14} + T^{16}$$
$53$ $$2176782336 - 4111699968 T^{2} + 2281851648 T^{4} + 579882240 T^{6} + 41487184 T^{8} - 394800 T^{10} - 5380 T^{12} + 60 T^{14} + T^{16}$$
$59$ $$( 20736 + 89856 T + 383616 T^{2} + 30720 T^{3} + 13936 T^{4} - 2048 T^{5} + 360 T^{6} - 20 T^{7} + T^{8} )^{2}$$
$61$ $$( 181476 + 370620 T + 280842 T^{2} + 58290 T^{3} + 1435 T^{4} - 804 T^{5} - 19 T^{6} + 12 T^{7} + T^{8} )^{2}$$
$67$ $$251928510529536 + 43280213704704 T^{2} + 2001885841152 T^{4} - 81871689600 T^{6} + 754742737 T^{8} + 4323600 T^{10} - 23113 T^{12} - 144 T^{14} + T^{16}$$
$71$ $$( 792 - 780 T + 229 T^{2} - 26 T^{3} + T^{4} )^{4}$$
$73$ $$49763656400896 - 66999713071104 T^{2} + 30048781290496 T^{4} + 26602956864 T^{6} - 1822760223 T^{8} - 1613376 T^{10} + 107791 T^{12} + 576 T^{14} + T^{16}$$
$79$ $$( 3200521 - 654774 T - 195074 T^{2} + 49044 T^{3} + 15435 T^{4} - 804 T^{5} - 122 T^{6} + 6 T^{7} + T^{8} )^{2}$$
$83$ $$2176782336 + 11008203264 T^{4} + 176298768 T^{8} + 27288 T^{12} + T^{16}$$
$89$ $$( 382124304 - 59816880 T + 11416140 T^{2} - 851580 T^{3} + 122373 T^{4} - 9270 T^{5} + 795 T^{6} - 30 T^{7} + T^{8} )^{2}$$
$97$ $$103682919315361 + 86181352416484 T^{4} + 10954318566 T^{8} + 210340 T^{12} + T^{16}$$