# Properties

 Label 700.2.g.l Level $700$ Weight $2$ Character orbit 700.g Analytic conductor $5.590$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$700 = 2^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 700.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.58952814149$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 17 x^{12} - 104 x^{10} + 713 x^{8} + 238 x^{6} + 1004 x^{4} - 152 x^{2} + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: no (minimal twist has level 140) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + \beta_{10} q^{3} + ( 1 - \beta_{4} ) q^{4} + \beta_{14} q^{6} + ( -\beta_{7} + \beta_{8} ) q^{7} + ( \beta_{2} - \beta_{6} ) q^{8} + ( -\beta_{3} - \beta_{4} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} + \beta_{10} q^{3} + ( 1 - \beta_{4} ) q^{4} + \beta_{14} q^{6} + ( -\beta_{7} + \beta_{8} ) q^{7} + ( \beta_{2} - \beta_{6} ) q^{8} + ( -\beta_{3} - \beta_{4} ) q^{9} + ( \beta_{3} - \beta_{9} ) q^{11} + ( -\beta_{8} + \beta_{10} + \beta_{12} ) q^{12} + ( \beta_{12} - \beta_{15} ) q^{13} + ( \beta_{9} - \beta_{11} ) q^{14} + ( -2 \beta_{3} - 2 \beta_{4} + \beta_{9} ) q^{16} + ( \beta_{8} + \beta_{10} - \beta_{12} - \beta_{15} ) q^{17} + ( \beta_{5} - \beta_{6} ) q^{18} + ( -\beta_{1} - \beta_{11} - \beta_{13} + 2 \beta_{14} ) q^{19} + ( -1 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{11} + \beta_{13} ) q^{21} + ( -\beta_{6} + 2 \beta_{7} ) q^{22} + ( \beta_{2} - \beta_{5} - 2 \beta_{7} ) q^{23} + ( \beta_{11} + \beta_{13} + \beta_{14} ) q^{24} + ( 2 \beta_{1} + \beta_{13} - \beta_{14} ) q^{26} + ( -\beta_{8} - 2 \beta_{10} ) q^{27} + ( -\beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{10} + \beta_{12} - \beta_{15} ) q^{28} + ( 1 + \beta_{3} + \beta_{4} ) q^{29} + ( -2 \beta_{1} - 2 \beta_{11} - \beta_{13} ) q^{31} + ( \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{32} + ( \beta_{8} + \beta_{10} - \beta_{12} - \beta_{15} ) q^{33} + ( 2 \beta_{1} - \beta_{11} - \beta_{13} ) q^{34} + ( 2 - 2 \beta_{3} + \beta_{9} ) q^{36} + ( -3 \beta_{2} - \beta_{5} ) q^{37} + ( \beta_{8} + 2 \beta_{10} + 2 \beta_{12} - \beta_{15} ) q^{38} + ( \beta_{3} - \beta_{9} ) q^{39} -3 \beta_{13} q^{41} + ( -\beta_{2} - \beta_{5} + \beta_{6} - \beta_{8} - 2 \beta_{10} + 2 \beta_{12} + \beta_{15} ) q^{42} + ( -\beta_{2} + \beta_{5} ) q^{43} + ( -1 - 2 \beta_{3} - \beta_{4} - \beta_{9} ) q^{44} + ( -2 - 2 \beta_{4} + 2 \beta_{9} ) q^{46} + ( -\beta_{8} + 4 \beta_{10} ) q^{47} + ( -3 \beta_{8} + 2 \beta_{12} + \beta_{15} ) q^{48} + ( -1 + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{13} ) q^{49} + ( \beta_{3} + 2 \beta_{4} - 3 \beta_{9} ) q^{51} + ( -3 \beta_{8} - \beta_{10} - \beta_{12} ) q^{52} + ( 2 \beta_{2} + 2 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{53} + ( \beta_{11} - 2 \beta_{14} ) q^{54} + ( -2 + 2 \beta_{1} + 2 \beta_{3} + \beta_{9} + \beta_{13} - 2 \beta_{14} ) q^{56} + ( 3 \beta_{2} + \beta_{5} ) q^{57} + ( \beta_{2} - \beta_{5} + \beta_{6} ) q^{58} + ( \beta_{1} + \beta_{11} + 2 \beta_{13} - 6 \beta_{14} ) q^{59} + ( -\beta_{1} + \beta_{11} - \beta_{13} ) q^{61} + ( 4 \beta_{8} - 2 \beta_{10} + 2 \beta_{12} - 2 \beta_{15} ) q^{62} + ( \beta_{2} - \beta_{5} - \beta_{7} - 2 \beta_{8} - 2 \beta_{10} ) q^{63} + ( 2 - 2 \beta_{3} + 3 \beta_{9} ) q^{64} + ( 2 \beta_{1} - \beta_{11} - \beta_{13} ) q^{66} + 2 \beta_{7} q^{67} + ( 3 \beta_{10} - 3 \beta_{12} - \beta_{15} ) q^{68} + ( -2 \beta_{1} + 2 \beta_{11} + 3 \beta_{13} ) q^{69} + ( -2 \beta_{3} + 4 \beta_{4} - 2 \beta_{9} ) q^{71} + ( 2 \beta_{2} + \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{72} + ( -2 \beta_{8} - 2 \beta_{10} + 2 \beta_{12} + 2 \beta_{15} ) q^{73} + ( -6 + 2 \beta_{4} ) q^{74} + ( 2 \beta_{1} - \beta_{11} + 2 \beta_{13} + \beta_{14} ) q^{76} + ( 3 \beta_{2} + \beta_{5} - \beta_{8} - \beta_{10} + 3 \beta_{12} - \beta_{15} ) q^{77} + ( -\beta_{6} + 2 \beta_{7} ) q^{78} + ( \beta_{3} + 2 \beta_{4} - 3 \beta_{9} ) q^{79} + ( -5 + 4 \beta_{3} + 4 \beta_{4} ) q^{81} + ( 6 \beta_{8} + 6 \beta_{10} - 6 \beta_{12} ) q^{82} + ( 5 \beta_{8} + \beta_{10} ) q^{83} + ( -3 - 2 \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{9} + \beta_{11} + 2 \beta_{13} - \beta_{14} ) q^{84} + ( 2 + 2 \beta_{4} ) q^{86} + \beta_{8} q^{87} + ( -\beta_{2} + 3 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{88} -\beta_{13} q^{89} + ( -\beta_{1} + 3 \beta_{3} - 2 \beta_{4} - \beta_{9} - \beta_{11} - \beta_{13} + 2 \beta_{14} ) q^{91} + ( -2 \beta_{2} - 2 \beta_{5} - 4 \beta_{7} ) q^{92} + ( -5 \beta_{2} - 3 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} ) q^{93} + ( \beta_{11} + 4 \beta_{14} ) q^{94} + ( -2 \beta_{1} + 3 \beta_{11} + 2 \beta_{13} + \beta_{14} ) q^{96} + ( -3 \beta_{8} - 3 \beta_{10} + 3 \beta_{12} + 3 \beta_{15} ) q^{97} + ( -\beta_{2} - 2 \beta_{5} + 2 \beta_{6} + 4 \beta_{8} + 4 \beta_{10} - 4 \beta_{12} ) q^{98} + ( -2 \beta_{3} + 2 \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 12q^{4} - 8q^{9} + O(q^{10})$$ $$16q + 12q^{4} - 8q^{9} + 4q^{14} - 12q^{16} - 8q^{21} + 24q^{29} + 28q^{36} - 32q^{44} - 32q^{46} - 20q^{56} + 36q^{64} - 88q^{74} - 48q^{81} - 40q^{84} + 40q^{86} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 17 x^{12} - 104 x^{10} + 713 x^{8} + 238 x^{6} + 1004 x^{4} - 152 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-1739863 \nu^{15} - 15207626 \nu^{13} + 18409195 \nu^{11} + 443618586 \nu^{9} + 534122349 \nu^{7} - 10160239700 \nu^{5} - 13245387916 \nu^{3} - 15043137568 \nu$$$$)/ 3284689024$$ $$\beta_{2}$$ $$=$$ $$($$$$59557735 \nu^{14} - 533150 \nu^{12} - 1053470715 \nu^{10} - 6261947986 \nu^{8} + 43163441379 \nu^{6} + 18860238060 \nu^{4} + 35114940892 \nu^{2} - 37634726944$$$$)/ 26277512192$$ $$\beta_{3}$$ $$=$$ $$($$$$-107882115 \nu^{14} - 77610858 \nu^{12} + 1877392743 \nu^{10} + 12516347930 \nu^{8} - 70055295983 \nu^{6} - 85825593884 \nu^{4} - 89210141260 \nu^{2} - 17971784544$$$$)/ 26277512192$$ $$\beta_{4}$$ $$=$$ $$($$$$157920899 \nu^{14} - 22764694 \nu^{12} - 2755245415 \nu^{10} - 15956221978 \nu^{8} + 116401708015 \nu^{6} + 28168725020 \nu^{4} + 100553570892 \nu^{2} - 26166657184$$$$)/ 26277512192$$ $$\beta_{5}$$ $$=$$ $$($$$$-158998017 \nu^{14} - 50557646 \nu^{12} + 2795857485 \nu^{10} + 17374927454 \nu^{8} - 109290094885 \nu^{6} - 81527049396 \nu^{4} - 100430658244 \nu^{2} - 32954999072$$$$)/ 26277512192$$ $$\beta_{6}$$ $$=$$ $$($$$$-111135863 \nu^{14} + 35207038 \nu^{12} + 1894572235 \nu^{10} + 10887840882 \nu^{8} - 82988272691 \nu^{6} - 677905516 \nu^{4} - 86864792220 \nu^{2} + 16400486944$$$$)/ 13138756096$$ $$\beta_{7}$$ $$=$$ $$($$$$-58429897 \nu^{14} - 4454734 \nu^{12} + 986022581 \nu^{10} + 6149655454 \nu^{8} - 41069887437 \nu^{6} - 16730498212 \nu^{4} - 64300182404 \nu^{2} + 4129925088$$$$)/ 6569378048$$ $$\beta_{8}$$ $$=$$ $$($$$$54968795 \nu^{15} - 11713990 \nu^{13} - 949732575 \nu^{11} - 5499231466 \nu^{9} + 40692711847 \nu^{7} + 6048096764 \nu^{5} + 38674762028 \nu^{3} - 15359983776 \nu$$$$)/ 6569378048$$ $$\beta_{9}$$ $$=$$ $$($$$$-305660455 \nu^{14} - 81621346 \nu^{12} + 5174279611 \nu^{10} + 33239289426 \nu^{8} - 209185441443 \nu^{6} - 127577110892 \nu^{4} - 347003644892 \nu^{2} + 150348832$$$$)/ 26277512192$$ $$\beta_{10}$$ $$=$$ $$($$$$-146835231 \nu^{15} + 25824526 \nu^{13} + 2503166931 \nu^{11} + 14845603426 \nu^{9} - 107472197627 \nu^{7} - 17672134092 \nu^{5} - 137231331644 \nu^{3} + 55358945312 \nu$$$$)/ 13138756096$$ $$\beta_{11}$$ $$=$$ $$($$$$171891871 \nu^{15} - 47519246 \nu^{13} - 2943150035 \nu^{11} - 17039166818 \nu^{9} + 127754563067 \nu^{7} + 8119855564 \nu^{5} + 144633757372 \nu^{3} - 39715733280 \nu$$$$)/ 13138756096$$ $$\beta_{12}$$ $$=$$ $$($$$$-354912961 \nu^{15} - 119115470 \nu^{13} + 6034586637 \nu^{11} + 39017889374 \nu^{9} - 240529045221 \nu^{7} - 170796167476 \nu^{5} - 394053088964 \nu^{3} - 16283111712 \nu$$$$)/ 26277512192$$ $$\beta_{13}$$ $$=$$ $$($$$$74983 \nu^{15} + 31874 \nu^{13} - 1281915 \nu^{11} - 8349682 \nu^{9} + 50321507 \nu^{7} + 41578380 \nu^{5} + 78299292 \nu^{3} + 9807584 \nu$$$$)/4257536$$ $$\beta_{14}$$ $$=$$ $$($$$$525461139 \nu^{15} - 22220726 \nu^{13} - 8971096375 \nu^{11} - 54288962554 \nu^{9} + 377737969983 \nu^{7} + 113217186140 \nu^{5} + 498241271180 \nu^{3} - 137015149728 \nu$$$$)/ 26277512192$$ $$\beta_{15}$$ $$=$$ $$($$$$-529467953 \nu^{15} - 68532718 \nu^{13} + 8933979069 \nu^{11} + 56185200254 \nu^{9} - 369146557717 \nu^{7} - 168419959540 \nu^{5} - 598285870468 \nu^{3} - 33389699360 \nu$$$$)/ 26277512192$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} - \beta_{10} - \beta_{8}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{9} + 2 \beta_{6} + 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$-\beta_{15} + \beta_{14} + 2 \beta_{10} - 2 \beta_{8} + 2 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$($$$$11 \beta_{9} - 16 \beta_{7} + 4 \beta_{6} - \beta_{4} - 4 \beta_{3} + 4 \beta_{2} + 7$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-11 \beta_{15} - 28 \beta_{14} + 9 \beta_{13} + 5 \beta_{12} - 10 \beta_{11} - 33 \beta_{10} + 2 \beta_{8} + 14 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-13 \beta_{9} + 24 \beta_{7} - 6 \beta_{6} + 30 \beta_{5} + 14 \beta_{4} - 21 \beta_{3} + 10 \beta_{2} + 44$$ $$\nu^{7}$$ $$=$$ $$($$$$3 \beta_{15} + 60 \beta_{14} + 113 \beta_{13} + 199 \beta_{12} + 40 \beta_{11} - 15 \beta_{10} - 136 \beta_{8} - 10 \beta_{1}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$109 \beta_{9} - 208 \beta_{7} + 176 \beta_{6} - 120 \beta_{5} + 303 \beta_{4} + 86 \beta_{3} - 568 \beta_{2} - 693$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$-267 \beta_{15} + 130 \beta_{14} - 247 \beta_{13} - 239 \beta_{12} - 343 \beta_{11} + 413 \beta_{10} + 287 \beta_{8} + 294 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$($$$$1241 \beta_{9} - 948 \beta_{7} - 1260 \beta_{6} + 1182 \beta_{5} - 1955 \beta_{4} - 2440 \beta_{3} + 1866 \beta_{2} + 2701$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$225 \beta_{15} - 2808 \beta_{14} + 2487 \beta_{13} + 3957 \beta_{12} - 622 \beta_{11} - 6345 \beta_{10} + 778 \beta_{8} - 1974 \beta_{1}$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$-4228 \beta_{9} + 6376 \beta_{7} - 700 \beta_{6} + 1976 \beta_{5} + 3935 \beta_{4} - 397 \beta_{3} - 5168 \beta_{2} - 4387$$ $$\nu^{13}$$ $$=$$ $$($$$$-475 \beta_{15} + 30988 \beta_{14} - 3947 \beta_{13} + 7849 \beta_{12} - 1324 \beta_{11} + 36311 \beta_{10} - 3780 \beta_{8} - 1986 \beta_{1}$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$49285 \beta_{9} - 68244 \beta_{7} + 2456 \beta_{6} - 35158 \beta_{5} - 24397 \beta_{4} - 3598 \beta_{3} - 40506 \beta_{2} - 89305$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$-24620 \beta_{15} - 29462 \beta_{14} - 45828 \beta_{13} - 62684 \beta_{12} - 53945 \beta_{11} - 3806 \beta_{10} + 86647 \beta_{8} + 13672 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/700\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$351$$ $$477$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 −0.409646 + 0.286988i 0.409646 − 0.286988i −0.409646 − 0.286988i 0.409646 + 0.286988i −0.645096 − 0.854135i 0.645096 + 0.854135i −0.645096 + 0.854135i 0.645096 − 0.854135i 2.15532 + 0.457057i −2.15532 − 0.457057i 2.15532 − 0.457057i −2.15532 + 0.457057i 0.877859 + 2.23141i −0.877859 − 2.23141i 0.877859 − 2.23141i −0.877859 + 2.23141i
−1.37491 0.331077i −2.13578 1.78078 + 0.910404i 0 2.93651 + 0.707107i 1.19935 2.35829i −2.14700 1.84130i 1.56155 0
251.2 −1.37491 0.331077i 2.13578 1.78078 + 0.910404i 0 −2.93651 0.707107i −1.19935 2.35829i −2.14700 1.84130i 1.56155 0
251.3 −1.37491 + 0.331077i −2.13578 1.78078 0.910404i 0 2.93651 0.707107i 1.19935 + 2.35829i −2.14700 + 1.84130i 1.56155 0
251.4 −1.37491 + 0.331077i 2.13578 1.78078 0.910404i 0 −2.93651 + 0.707107i −1.19935 + 2.35829i −2.14700 + 1.84130i 1.56155 0
251.5 −0.927153 1.06789i −0.662153 −0.280776 + 1.98019i 0 0.613917 + 0.707107i −2.35829 + 1.19935i 2.37495 1.53610i −2.56155 0
251.6 −0.927153 1.06789i 0.662153 −0.280776 + 1.98019i 0 −0.613917 0.707107i 2.35829 + 1.19935i 2.37495 1.53610i −2.56155 0
251.7 −0.927153 + 1.06789i −0.662153 −0.280776 1.98019i 0 0.613917 0.707107i −2.35829 1.19935i 2.37495 + 1.53610i −2.56155 0
251.8 −0.927153 + 1.06789i 0.662153 −0.280776 1.98019i 0 −0.613917 + 0.707107i 2.35829 1.19935i 2.37495 + 1.53610i −2.56155 0
251.9 0.927153 1.06789i −0.662153 −0.280776 1.98019i 0 −0.613917 + 0.707107i −2.35829 + 1.19935i −2.37495 1.53610i −2.56155 0
251.10 0.927153 1.06789i 0.662153 −0.280776 1.98019i 0 0.613917 0.707107i 2.35829 + 1.19935i −2.37495 1.53610i −2.56155 0
251.11 0.927153 + 1.06789i −0.662153 −0.280776 + 1.98019i 0 −0.613917 0.707107i −2.35829 1.19935i −2.37495 + 1.53610i −2.56155 0
251.12 0.927153 + 1.06789i 0.662153 −0.280776 + 1.98019i 0 0.613917 + 0.707107i 2.35829 1.19935i −2.37495 + 1.53610i −2.56155 0
251.13 1.37491 0.331077i −2.13578 1.78078 0.910404i 0 −2.93651 + 0.707107i 1.19935 2.35829i 2.14700 1.84130i 1.56155 0
251.14 1.37491 0.331077i 2.13578 1.78078 0.910404i 0 2.93651 0.707107i −1.19935 2.35829i 2.14700 1.84130i 1.56155 0
251.15 1.37491 + 0.331077i −2.13578 1.78078 + 0.910404i 0 −2.93651 0.707107i 1.19935 + 2.35829i 2.14700 + 1.84130i 1.56155 0
251.16 1.37491 + 0.331077i 2.13578 1.78078 + 0.910404i 0 2.93651 + 0.707107i −1.19935 + 2.35829i 2.14700 + 1.84130i 1.56155 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 251.16 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
5.b even 2 1 inner
7.b odd 2 1 inner
20.d odd 2 1 inner
28.d even 2 1 inner
35.c odd 2 1 inner
140.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.2.g.l 16
4.b odd 2 1 inner 700.2.g.l 16
5.b even 2 1 inner 700.2.g.l 16
5.c odd 4 2 140.2.c.b 16
7.b odd 2 1 inner 700.2.g.l 16
20.d odd 2 1 inner 700.2.g.l 16
20.e even 4 2 140.2.c.b 16
28.d even 2 1 inner 700.2.g.l 16
35.c odd 2 1 inner 700.2.g.l 16
35.f even 4 2 140.2.c.b 16
35.k even 12 4 980.2.s.f 32
35.l odd 12 4 980.2.s.f 32
40.i odd 4 2 2240.2.e.f 16
40.k even 4 2 2240.2.e.f 16
140.c even 2 1 inner 700.2.g.l 16
140.j odd 4 2 140.2.c.b 16
140.w even 12 4 980.2.s.f 32
140.x odd 12 4 980.2.s.f 32
280.s even 4 2 2240.2.e.f 16
280.y odd 4 2 2240.2.e.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.c.b 16 5.c odd 4 2
140.2.c.b 16 20.e even 4 2
140.2.c.b 16 35.f even 4 2
140.2.c.b 16 140.j odd 4 2
700.2.g.l 16 1.a even 1 1 trivial
700.2.g.l 16 4.b odd 2 1 inner
700.2.g.l 16 5.b even 2 1 inner
700.2.g.l 16 7.b odd 2 1 inner
700.2.g.l 16 20.d odd 2 1 inner
700.2.g.l 16 28.d even 2 1 inner
700.2.g.l 16 35.c odd 2 1 inner
700.2.g.l 16 140.c even 2 1 inner
980.2.s.f 32 35.k even 12 4
980.2.s.f 32 35.l odd 12 4
980.2.s.f 32 140.w even 12 4
980.2.s.f 32 140.x odd 12 4
2240.2.e.f 16 40.i odd 4 2
2240.2.e.f 16 40.k even 4 2
2240.2.e.f 16 280.s even 4 2
2240.2.e.f 16 280.y odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(700, [\chi])$$:

 $$T_{3}^{4} - 5 T_{3}^{2} + 2$$ $$T_{11}^{4} + 15 T_{11}^{2} + 52$$ $$T_{19}^{4} - 38 T_{19}^{2} + 208$$ $$T_{37}^{4} - 44 T_{37}^{2} + 416$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 16 - 12 T^{2} + 6 T^{4} - 3 T^{6} + T^{8} )^{2}$$
$3$ $$( 2 - 5 T^{2} + T^{4} )^{4}$$
$5$ $$T^{16}$$
$7$ $$( 2401 + 30 T^{4} + T^{8} )^{2}$$
$11$ $$( 52 + 15 T^{2} + T^{4} )^{4}$$
$13$ $$( 26 + 23 T^{2} + T^{4} )^{4}$$
$17$ $$( 104 + 29 T^{2} + T^{4} )^{4}$$
$19$ $$( 208 - 38 T^{2} + T^{4} )^{4}$$
$23$ $$( 128 + 40 T^{2} + T^{4} )^{4}$$
$29$ $$( -2 - 3 T + T^{2} )^{8}$$
$31$ $$( 3328 - 120 T^{2} + T^{4} )^{4}$$
$37$ $$( 416 - 44 T^{2} + T^{4} )^{4}$$
$41$ $$( 72 + T^{2} )^{8}$$
$43$ $$( 32 + 20 T^{2} + T^{4} )^{4}$$
$47$ $$( 8 - 95 T^{2} + T^{4} )^{4}$$
$53$ $$( 6656 - 164 T^{2} + T^{4} )^{4}$$
$59$ $$( 13312 - 270 T^{2} + T^{4} )^{4}$$
$61$ $$( 16 + 42 T^{2} + T^{4} )^{4}$$
$67$ $$( 128 + 28 T^{2} + T^{4} )^{4}$$
$71$ $$( 13312 + 236 T^{2} + T^{4} )^{4}$$
$73$ $$( 1664 + 116 T^{2} + T^{4} )^{4}$$
$79$ $$( 208 + 115 T^{2} + T^{4} )^{4}$$
$83$ $$( 2312 - 170 T^{2} + T^{4} )^{4}$$
$89$ $$( 8 + T^{2} )^{8}$$
$97$ $$( 8424 + 261 T^{2} + T^{4} )^{4}$$