# Properties

 Label 525.2.g.f Level 525 Weight 2 Character orbit 525.g Analytic conductor 4.192 Analytic rank 0 Dimension 16 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ = $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 525.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}\cdot 3^{2}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + \beta_{4} q^{3} + ( 1 - \beta_{5} ) q^{4} -\beta_{8} q^{6} + ( -\beta_{4} - \beta_{6} + \beta_{9} ) q^{7} + ( \beta_{2} + \beta_{11} ) q^{8} + ( -1 - \beta_{5} - \beta_{13} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} + \beta_{4} q^{3} + ( 1 - \beta_{5} ) q^{4} -\beta_{8} q^{6} + ( -\beta_{4} - \beta_{6} + \beta_{9} ) q^{7} + ( \beta_{2} + \beta_{11} ) q^{8} + ( -1 - \beta_{5} - \beta_{13} ) q^{9} + ( \beta_{10} + \beta_{13} ) q^{11} + ( 2 \beta_{4} + \beta_{6} + \beta_{15} ) q^{12} + ( -\beta_{4} - \beta_{6} ) q^{13} + ( \beta_{3} + \beta_{8} - \beta_{13} ) q^{14} -\beta_{5} q^{16} + ( -\beta_{12} - \beta_{15} ) q^{17} + ( -\beta_{1} + \beta_{2} + 4 \beta_{9} + \beta_{11} ) q^{18} + ( \beta_{3} + 2 \beta_{7} - \beta_{8} ) q^{19} + ( -1 + \beta_{5} + \beta_{7} + \beta_{10} ) q^{21} + ( 3 \beta_{1} - 5 \beta_{9} ) q^{22} + ( -\beta_{2} + \beta_{11} ) q^{23} + ( -\beta_{3} - \beta_{7} - \beta_{8} + 2 \beta_{14} ) q^{24} + ( \beta_{3} + \beta_{8} ) q^{26} + ( \beta_{6} + \beta_{12} + \beta_{15} ) q^{27} + ( -\beta_{1} - 3 \beta_{4} - \beta_{6} + 2 \beta_{9} + \beta_{12} - \beta_{15} ) q^{28} + ( \beta_{10} - \beta_{13} ) q^{29} + ( -\beta_{3} - 2 \beta_{7} + \beta_{8} + \beta_{14} ) q^{31} -\beta_{11} q^{32} + ( \beta_{4} + 2 \beta_{6} - \beta_{12} - \beta_{15} ) q^{33} + ( -\beta_{3} - 2 \beta_{7} + \beta_{8} - 3 \beta_{14} ) q^{34} + ( 4 - \beta_{5} - \beta_{10} - 3 \beta_{13} ) q^{36} + ( -2 \beta_{1} - 3 \beta_{9} ) q^{37} + ( 3 \beta_{4} - \beta_{6} + 3 \beta_{12} + \beta_{15} ) q^{38} + ( -1 + \beta_{5} + \beta_{10} ) q^{39} + ( \beta_{3} - 4 \beta_{7} - \beta_{8} + \beta_{14} ) q^{41} + ( 2 \beta_{1} - 3 \beta_{2} - \beta_{9} - \beta_{11} + \beta_{12} ) q^{42} + ( -2 \beta_{1} - \beta_{9} ) q^{43} + ( \beta_{10} + 6 \beta_{13} ) q^{44} + ( -4 - \beta_{5} ) q^{46} + ( -\beta_{12} - \beta_{15} ) q^{47} + ( \beta_{4} + \beta_{6} + \beta_{15} ) q^{48} + ( 5 - 2 \beta_{14} ) q^{49} + ( -1 - 2 \beta_{5} + \beta_{10} + 3 \beta_{13} ) q^{51} + ( -3 \beta_{4} - \beta_{6} + \beta_{12} - \beta_{15} ) q^{52} + ( -4 \beta_{2} - 2 \beta_{11} ) q^{53} + ( 2 \beta_{7} - \beta_{8} + 3 \beta_{14} ) q^{54} + ( 4 \beta_{7} + 2 \beta_{8} - \beta_{10} - \beta_{13} - \beta_{14} ) q^{56} + ( -2 \beta_{1} - 3 \beta_{2} + 7 \beta_{9} + \beta_{11} ) q^{57} + ( \beta_{1} + 3 \beta_{9} ) q^{58} + ( -4 \beta_{7} - 2 \beta_{8} + \beta_{14} ) q^{59} + ( \beta_{3} + 2 \beta_{7} - \beta_{8} - \beta_{14} ) q^{61} + ( -3 \beta_{4} + \beta_{6} - 2 \beta_{12} ) q^{62} + ( -\beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} - 2 \beta_{15} ) q^{63} + ( 1 + 4 \beta_{5} ) q^{64} + ( -3 \beta_{3} - 2 \beta_{7} - 3 \beta_{14} ) q^{66} + ( -4 \beta_{1} + 7 \beta_{9} ) q^{67} + ( -3 \beta_{4} + \beta_{6} - 4 \beta_{12} - 2 \beta_{15} ) q^{68} + ( -\beta_{3} - \beta_{7} + \beta_{8} + 2 \beta_{14} ) q^{69} + ( \beta_{10} + \beta_{13} ) q^{71} + ( -3 \beta_{1} + 4 \beta_{2} + 5 \beta_{9} - \beta_{11} ) q^{72} + ( \beta_{4} - 3 \beta_{6} - 2 \beta_{12} + 2 \beta_{15} ) q^{73} + ( -2 \beta_{10} + \beta_{13} ) q^{74} + ( 2 \beta_{3} + 4 \beta_{7} - 2 \beta_{8} + 5 \beta_{14} ) q^{76} + ( \beta_{2} + 3 \beta_{4} - \beta_{6} + \beta_{11} + 2 \beta_{12} ) q^{77} + ( 2 \beta_{1} - 3 \beta_{2} - \beta_{9} - \beta_{11} ) q^{78} + ( 3 - 2 \beta_{5} ) q^{79} + ( 3 + 2 \beta_{5} - 2 \beta_{10} - 2 \beta_{13} ) q^{81} + ( 3 \beta_{4} - \beta_{6} - 2 \beta_{12} + 2 \beta_{15} ) q^{82} + ( -3 \beta_{4} + \beta_{6} + \beta_{12} + 3 \beta_{15} ) q^{83} + ( -6 + \beta_{3} + 3 \beta_{5} + \beta_{7} + 3 \beta_{13} + \beta_{14} ) q^{84} + ( -2 \beta_{10} - \beta_{13} ) q^{86} + ( \beta_{4} + 2 \beta_{6} + \beta_{12} - \beta_{15} ) q^{87} + ( 2 \beta_{1} - 15 \beta_{9} ) q^{88} + ( 4 \beta_{7} + 2 \beta_{8} - \beta_{14} ) q^{89} + ( 6 - \beta_{14} ) q^{91} -\beta_{11} q^{92} + ( \beta_{1} + 3 \beta_{2} - 5 \beta_{9} - 2 \beta_{11} ) q^{93} + ( -\beta_{3} - 2 \beta_{7} + \beta_{8} - 3 \beta_{14} ) q^{94} + ( \beta_{3} + \beta_{7} - 2 \beta_{14} ) q^{96} + ( -\beta_{4} - 3 \beta_{6} - \beta_{12} + \beta_{15} ) q^{97} + ( 5 \beta_{2} - 2 \beta_{12} - 2 \beta_{15} ) q^{98} + ( 2 - 3 \beta_{5} - \beta_{10} + 4 \beta_{13} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 24q^{4} - 8q^{9} + O(q^{10})$$ $$16q + 24q^{4} - 8q^{9} + 8q^{16} - 24q^{21} + 72q^{36} - 24q^{39} - 56q^{46} + 80q^{49} - 16q^{64} + 64q^{79} + 32q^{81} - 120q^{84} + 96q^{91} + 56q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 32 x^{14} + 386 x^{12} + 2208 x^{10} + 6263 x^{8} + 8496 x^{6} + 4790 x^{4} + 704 x^{2} + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-1982 \nu^{15} - 58390 \nu^{13} - 615152 \nu^{11} - 2769307 \nu^{9} - 4960972 \nu^{7} - 2968343 \nu^{5} - 3899254 \nu^{3} - 5898695 \nu$$$$)/730740$$ $$\beta_{2}$$ $$=$$ $$($$$$2111 \nu^{14} + 65570 \nu^{12} + 756456 \nu^{10} + 4045936 \nu^{8} + 10451886 \nu^{6} + 12974084 \nu^{4} + 7508717 \nu^{2} + 1240590$$$$)/365370$$ $$\beta_{3}$$ $$=$$ $$($$$$8094 \nu^{15} - 3709 \nu^{14} + 263853 \nu^{13} - 106269 \nu^{12} + 3275277 \nu^{11} - 1050680 \nu^{10} + 19628064 \nu^{9} - 3908927 \nu^{8} + 60246093 \nu^{7} - 1676950 \nu^{6} + 93937026 \nu^{5} + 16015631 \nu^{4} + 68352861 \nu^{3} + 17337039 \nu^{2} + 15258957 \nu - 2302010$$$$)/1461480$$ $$\beta_{4}$$ $$=$$ $$($$$$-8537 \nu^{15} + 7764 \nu^{14} - 271988 \nu^{13} + 254549 \nu^{12} - 3258831 \nu^{11} + 3176075 \nu^{10} - 18434731 \nu^{9} + 19023522 \nu^{8} - 51200493 \nu^{7} + 57068335 \nu^{6} - 65827487 \nu^{5} + 81045624 \nu^{4} - 29287814 \nu^{3} + 44382761 \nu^{2} + 3579531 \nu + 4385015$$$$)/1461480$$ $$\beta_{5}$$ $$=$$ $$($$$$-448 \nu^{14} - 14550 \nu^{12} - 179552 \nu^{10} - 1064513 \nu^{8} - 3188560 \nu^{6} - 4632502 \nu^{4} - 2662080 \nu^{2} - 229922$$$$)/36537$$ $$\beta_{6}$$ $$=$$ $$($$$$8537 \nu^{15} - 24142 \nu^{14} + 271988 \nu^{13} - 758087 \nu^{12} + 3258831 \nu^{11} - 8868475 \nu^{10} + 18434731 \nu^{9} - 48131636 \nu^{8} + 51200493 \nu^{7} - 124193435 \nu^{6} + 65827487 \nu^{5} - 140933662 \nu^{4} + 29287814 \nu^{3} - 54181403 \nu^{2} - 3579531 \nu - 1164895$$$$)/1461480$$ $$\beta_{7}$$ $$=$$ $$($$$$32251 \nu^{15} - 11986 \nu^{14} + 1029006 \nu^{13} - 385689 \nu^{12} + 12354197 \nu^{11} - 4688987 \nu^{10} + 70112261 \nu^{9} - 27115394 \nu^{8} + 196162879 \nu^{7} - 77972107 \nu^{6} + 259803589 \nu^{5} - 106993792 \nu^{4} + 140138538 \nu^{3} - 58669455 \nu^{2} + 16546109 \nu - 3943235$$$$)/1461480$$ $$\beta_{8}$$ $$=$$ $$($$$$-2698 \nu^{15} - 9227 \nu^{14} - 87951 \nu^{13} - 292549 \nu^{12} - 1091759 \nu^{11} - 3476218 \nu^{10} - 6542688 \nu^{9} - 19379905 \nu^{8} - 20082031 \nu^{7} - 52540388 \nu^{6} - 31312342 \nu^{5} - 65990651 \nu^{4} - 22784287 \nu^{3} - 33333957 \nu^{2} - 5086319 \nu - 3396160$$$$)/487160$$ $$\beta_{9}$$ $$=$$ $$($$$$20394 \nu^{15} + 650497 \nu^{13} + 7806514 \nu^{11} + 44273496 \nu^{9} + 123681686 \nu^{7} + 162815538 \nu^{5} + 84713176 \nu^{3} + 7214029 \nu$$$$)/730740$$ $$\beta_{10}$$ $$=$$ $$($$$$21827 \nu^{15} + 686544 \nu^{13} + 8051137 \nu^{11} + 43834216 \nu^{9} + 113317871 \nu^{7} + 127014752 \nu^{5} + 43497930 \nu^{3} - 4103552 \nu$$$$)/730740$$ $$\beta_{11}$$ $$=$$ $$($$$$14023 \nu^{14} + 441570 \nu^{12} + 5194823 \nu^{10} + 28523978 \nu^{8} + 75452983 \nu^{6} + 90432082 \nu^{4} + 40009446 \nu^{2} + 2564660$$$$)/365370$$ $$\beta_{12}$$ $$=$$ $$($$$$77004 \nu^{15} - 23717 \nu^{14} + 2472421 \nu^{13} - 760867 \nu^{12} + 29962597 \nu^{11} - 9198350 \nu^{10} + 172446432 \nu^{9} - 52601101 \nu^{8} + 492261521 \nu^{7} - 147699220 \nu^{6} + 667354674 \nu^{5} - 192035267 \nu^{4} + 362884003 \nu^{3} - 93664843 \nu^{2} + 41055973 \nu - 7159970$$$$)/1461480$$ $$\beta_{13}$$ $$=$$ $$($$$$-38677 \nu^{15} - 1235424 \nu^{13} - 14856572 \nu^{11} - 84501056 \nu^{9} - 236911486 \nu^{7} - 312656992 \nu^{5} - 162283005 \nu^{3} - 15379688 \nu$$$$)/730740$$ $$\beta_{14}$$ $$=$$ $$($$$$88 \nu^{15} + 2799 \nu^{13} + 33437 \nu^{11} + 188138 \nu^{9} + 518065 \nu^{7} + 664072 \nu^{5} + 330615 \nu^{3} + 27821 \nu$$$$)/1140$$ $$\beta_{15}$$ $$=$$ $$($$$$94078 \nu^{15} + 23717 \nu^{14} + 3016397 \nu^{13} + 760867 \nu^{12} + 36480259 \nu^{11} + 9198350 \nu^{10} + 209315894 \nu^{9} + 52601101 \nu^{8} + 594662507 \nu^{7} + 147699220 \nu^{6} + 799009648 \nu^{5} + 192035267 \nu^{4} + 421459631 \nu^{3} + 93664843 \nu^{2} + 33896911 \nu + 7159970$$$$)/1461480$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{15} - \beta_{14} + \beta_{12} + 6 \beta_{9} + \beta_{8} - 2 \beta_{7} - \beta_{6} + 3 \beta_{4} - \beta_{3}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{15} - \beta_{14} - \beta_{12} + \beta_{8} + 4 \beta_{7} + \beta_{6} + 3 \beta_{4} - \beta_{3} + 3 \beta_{2} - 12$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$4 \beta_{15} + 5 \beta_{14} - 18 \beta_{13} - 10 \beta_{12} - 60 \beta_{9} - 8 \beta_{8} + 16 \beta_{7} + 7 \beta_{6} - 21 \beta_{4} + 8 \beta_{3}$$$$)/6$$ $$\nu^{4}$$ $$=$$ $$($$$$-12 \beta_{15} + 10 \beta_{14} + 12 \beta_{12} - 16 \beta_{8} - 40 \beta_{7} - 6 \beta_{6} - 3 \beta_{5} - 30 \beta_{4} + 4 \beta_{3} - 36 \beta_{2} + 93$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-11 \beta_{15} - 36 \beta_{14} + 240 \beta_{13} + 113 \beta_{12} + 576 \beta_{9} + 81 \beta_{8} - 162 \beta_{7} - 62 \beta_{6} + 186 \beta_{4} - 81 \beta_{3} - 30 \beta_{1}$$$$)/6$$ $$\nu^{6}$$ $$=$$ $$($$$$131 \beta_{15} - 94 \beta_{14} - 131 \beta_{12} + 3 \beta_{11} + 187 \beta_{8} + 376 \beta_{7} + 56 \beta_{6} + 72 \beta_{5} + 318 \beta_{4} - \beta_{3} + 411 \beta_{2} - 840$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$-91 \beta_{15} + 387 \beta_{14} - 2730 \beta_{13} - 1253 \beta_{12} - 42 \beta_{10} - 5760 \beta_{9} - 861 \beta_{8} + 1722 \beta_{7} + 581 \beta_{6} - 1743 \beta_{4} + 861 \beta_{3} + 588 \beta_{1}$$$$)/6$$ $$\nu^{8}$$ $$=$$ $$-468 \beta_{15} + 300 \beta_{14} + 468 \beta_{12} - 40 \beta_{11} - 692 \beta_{8} - 1200 \beta_{7} - 228 \beta_{6} - 383 \beta_{5} - 1164 \beta_{4} - 92 \beta_{3} - 1512 \beta_{2} + 2694$$ $$\nu^{9}$$ $$=$$ $$($$$$2534 \beta_{15} - 4945 \beta_{14} + 29952 \beta_{13} + 13780 \beta_{12} + 1152 \beta_{10} + 59406 \beta_{9} + 9172 \beta_{8} - 18344 \beta_{7} - 5623 \beta_{6} + 16869 \beta_{4} - 9172 \beta_{3} - 8586 \beta_{1}$$$$)/6$$ $$\nu^{10}$$ $$=$$ $$($$$$14956 \beta_{15} - 8827 \beta_{14} - 14956 \beta_{12} + 2541 \beta_{11} + 22828 \beta_{8} + 35308 \beta_{7} + 8941 \beta_{6} + 15660 \beta_{5} + 38853 \beta_{4} + 5174 \beta_{3} + 49404 \beta_{2} - 80472$$$$)/3$$ $$\nu^{11}$$ $$=$$ $$($$$$-39881 \beta_{15} + 64502 \beta_{14} - 325512 \beta_{13} - 151333 \beta_{12} - 21054 \beta_{10} - 624300 \beta_{9} - 97571 \beta_{8} + 195142 \beta_{7} + 55726 \beta_{6} - 167178 \beta_{4} + 97571 \beta_{3} + 112068 \beta_{1}$$$$)/6$$ $$\nu^{12}$$ $$=$$ $$($$$$-159138 \beta_{15} + 88336 \beta_{14} + 159138 \beta_{12} - 41796 \beta_{11} - 250798 \beta_{8} - 353344 \beta_{7} - 115560 \beta_{6} - 197874 \beta_{5} - 433836 \beta_{4} - 74126 \beta_{3} - 536112 \beta_{2} + 818847$$$$)/3$$ $$\nu^{13}$$ $$=$$ $$($$$$538339 \beta_{15} - 822339 \beta_{14} + 3531840 \beta_{13} + 1663073 \beta_{12} + 321360 \beta_{10} + 6638586 \beta_{9} + 1038753 \beta_{8} - 2077506 \beta_{7} - 562367 \beta_{6} + 1687101 \beta_{4} - 1038753 \beta_{3} - 1383252 \beta_{1}$$$$)/6$$ $$\nu^{14}$$ $$=$$ $$($$$$1696163 \beta_{15} - 898177 \beta_{14} - 1696163 \beta_{12} + 602826 \beta_{11} + 2757643 \beta_{8} + 3592708 \beta_{7} + 1452509 \beta_{6} + 2398704 \beta_{5} + 4844835 \beta_{4} + 961289 \beta_{3} + 5818659 \beta_{2} - 8459568$$$$)/3$$ $$\nu^{15}$$ $$=$$ $$($$$$-6782926 \beta_{15} + 10194465 \beta_{14} - 38360154 \beta_{13} - 18295316 \beta_{12} - 4437276 \beta_{10} - 71158980 \beta_{9} - 11087598 \beta_{8} + 22175196 \beta_{7} + 5756195 \beta_{6} - 17268585 \beta_{4} + 11087598 \beta_{3} + 16535880 \beta_{1}$$$$)/6$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
524.1
 − 3.32519i 3.32519i − 1.32519i 1.32519i − 3.02470i 3.02470i − 1.02470i 1.02470i − 0.378617i 0.378617i − 2.37862i 2.37862i 1.77038i − 1.77038i − 0.229617i 0.229617i
−2.40651 −1.54779 0.777403i 3.79129 0 3.72476 + 1.87083i 2.44949 1.00000i −4.31075 1.79129 + 2.40651i 0
524.2 −2.40651 −1.54779 + 0.777403i 3.79129 0 3.72476 1.87083i 2.44949 + 1.00000i −4.31075 1.79129 2.40651i 0
524.3 −2.40651 1.54779 0.777403i 3.79129 0 −3.72476 + 1.87083i −2.44949 + 1.00000i −4.31075 1.79129 2.40651i 0
524.4 −2.40651 1.54779 + 0.777403i 3.79129 0 −3.72476 1.87083i −2.44949 1.00000i −4.31075 1.79129 + 2.40651i 0
524.5 −1.09941 −0.323042 1.70166i −0.791288 0 0.355157 + 1.87083i −2.44949 1.00000i 3.06878 −2.79129 + 1.09941i 0
524.6 −1.09941 −0.323042 + 1.70166i −0.791288 0 0.355157 1.87083i −2.44949 + 1.00000i 3.06878 −2.79129 1.09941i 0
524.7 −1.09941 0.323042 1.70166i −0.791288 0 −0.355157 + 1.87083i 2.44949 + 1.00000i 3.06878 −2.79129 1.09941i 0
524.8 −1.09941 0.323042 + 1.70166i −0.791288 0 −0.355157 1.87083i 2.44949 1.00000i 3.06878 −2.79129 + 1.09941i 0
524.9 1.09941 −0.323042 1.70166i −0.791288 0 −0.355157 1.87083i −2.44949 + 1.00000i −3.06878 −2.79129 + 1.09941i 0
524.10 1.09941 −0.323042 + 1.70166i −0.791288 0 −0.355157 + 1.87083i −2.44949 1.00000i −3.06878 −2.79129 1.09941i 0
524.11 1.09941 0.323042 1.70166i −0.791288 0 0.355157 1.87083i 2.44949 1.00000i −3.06878 −2.79129 1.09941i 0
524.12 1.09941 0.323042 + 1.70166i −0.791288 0 0.355157 + 1.87083i 2.44949 + 1.00000i −3.06878 −2.79129 + 1.09941i 0
524.13 2.40651 −1.54779 0.777403i 3.79129 0 −3.72476 1.87083i 2.44949 + 1.00000i 4.31075 1.79129 + 2.40651i 0
524.14 2.40651 −1.54779 + 0.777403i 3.79129 0 −3.72476 + 1.87083i 2.44949 1.00000i 4.31075 1.79129 2.40651i 0
524.15 2.40651 1.54779 0.777403i 3.79129 0 3.72476 1.87083i −2.44949 1.00000i 4.31075 1.79129 2.40651i 0
524.16 2.40651 1.54779 + 0.777403i 3.79129 0 3.72476 + 1.87083i −2.44949 + 1.00000i 4.31075 1.79129 + 2.40651i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 524.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
15.d odd 2 1 inner
21.c even 2 1 inner
35.c odd 2 1 inner
105.g even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.g.f 16
3.b odd 2 1 inner 525.2.g.f 16
5.b even 2 1 inner 525.2.g.f 16
5.c odd 4 1 525.2.b.h 8
5.c odd 4 1 525.2.b.i yes 8
7.b odd 2 1 inner 525.2.g.f 16
15.d odd 2 1 inner 525.2.g.f 16
15.e even 4 1 525.2.b.h 8
15.e even 4 1 525.2.b.i yes 8
21.c even 2 1 inner 525.2.g.f 16
35.c odd 2 1 inner 525.2.g.f 16
35.f even 4 1 525.2.b.h 8
35.f even 4 1 525.2.b.i yes 8
105.g even 2 1 inner 525.2.g.f 16
105.k odd 4 1 525.2.b.h 8
105.k odd 4 1 525.2.b.i yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.b.h 8 5.c odd 4 1
525.2.b.h 8 15.e even 4 1
525.2.b.h 8 35.f even 4 1
525.2.b.h 8 105.k odd 4 1
525.2.b.i yes 8 5.c odd 4 1
525.2.b.i yes 8 15.e even 4 1
525.2.b.i yes 8 35.f even 4 1
525.2.b.i yes 8 105.k odd 4 1
525.2.g.f 16 1.a even 1 1 trivial
525.2.g.f 16 3.b odd 2 1 inner
525.2.g.f 16 5.b even 2 1 inner
525.2.g.f 16 7.b odd 2 1 inner
525.2.g.f 16 15.d odd 2 1 inner
525.2.g.f 16 21.c even 2 1 inner
525.2.g.f 16 35.c odd 2 1 inner
525.2.g.f 16 105.g even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(525, [\chi])$$:

 $$T_{2}^{4} - 7 T_{2}^{2} + 7$$ $$T_{41}^{4} - 126 T_{41}^{2} + 2268$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} + 3 T^{4} + 4 T^{6} + 16 T^{8} )^{4}$$
$3$ $$( 1 + 2 T^{2} - 2 T^{4} + 18 T^{6} + 81 T^{8} )^{2}$$
$5$ 1
$7$ $$( 1 - 10 T^{2} + 49 T^{4} )^{4}$$
$11$ $$( 1 - 16 T^{2} + 285 T^{4} - 1936 T^{6} + 14641 T^{8} )^{4}$$
$13$ $$( 1 + 20 T^{2} + 169 T^{4} )^{8}$$
$17$ $$( 1 - 26 T^{2} + 558 T^{4} - 7514 T^{6} + 83521 T^{8} )^{4}$$
$19$ $$( 1 - 10 T^{2} + 558 T^{4} - 3610 T^{6} + 130321 T^{8} )^{4}$$
$23$ $$( 1 + 64 T^{2} + 1893 T^{4} + 33856 T^{6} + 279841 T^{8} )^{4}$$
$29$ $$( 1 - 88 T^{2} + 3429 T^{4} - 74008 T^{6} + 707281 T^{8} )^{4}$$
$31$ $$( 1 - 58 T^{2} + 2574 T^{4} - 55738 T^{6} + 923521 T^{8} )^{4}$$
$37$ $$( 1 - 74 T^{2} + 2763 T^{4} - 101306 T^{6} + 1874161 T^{8} )^{4}$$
$41$ $$( 1 + 38 T^{2} + 2022 T^{4} + 63878 T^{6} + 2825761 T^{8} )^{4}$$
$43$ $$( 1 - 122 T^{2} + 7083 T^{4} - 225578 T^{6} + 3418801 T^{8} )^{4}$$
$47$ $$( 1 - 146 T^{2} + 9558 T^{4} - 322514 T^{6} + 4879681 T^{8} )^{4}$$
$53$ $$( 1 + 16 T^{2} - 1122 T^{4} + 44944 T^{6} + 7890481 T^{8} )^{4}$$
$59$ $$( 1 + 68 T^{2} + 7362 T^{4} + 236708 T^{6} + 12117361 T^{8} )^{4}$$
$61$ $$( 1 - 178 T^{2} + 15174 T^{4} - 662338 T^{6} + 13845841 T^{8} )^{4}$$
$67$ $$( 1 - 50 T^{2} + 1203 T^{4} - 224450 T^{6} + 20151121 T^{8} )^{4}$$
$71$ $$( 1 - 256 T^{2} + 26445 T^{4} - 1290496 T^{6} + 25411681 T^{8} )^{4}$$
$73$ $$( 1 - 8 T^{2} - 1422 T^{4} - 42632 T^{6} + 28398241 T^{8} )^{4}$$
$79$ $$( 1 - 8 T + 153 T^{2} - 632 T^{3} + 6241 T^{4} )^{8}$$
$83$ $$( 1 - 38 T^{2} + 4878 T^{4} - 261782 T^{6} + 47458321 T^{8} )^{4}$$
$89$ $$( 1 + 188 T^{2} + 23922 T^{4} + 1489148 T^{6} + 62742241 T^{8} )^{4}$$
$97$ $$( 1 + 250 T^{2} + 29718 T^{4} + 2352250 T^{6} + 88529281 T^{8} )^{4}$$