# Properties

 Label 525.2.q.f Level 525 Weight 2 Character orbit 525.q Analytic conductor 4.192 Analytic rank 0 Dimension 16 CM no Inner twists 4

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 525.q (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( -\beta_{1} + \beta_{7} - \beta_{8} + \beta_{12} - \beta_{13} + \beta_{15} ) q^{3} + ( \beta_{5} + \beta_{9} ) q^{4} + ( \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{9} + \beta_{11} ) q^{6} + ( -\beta_{1} + 2 \beta_{7} + \beta_{10} + \beta_{15} ) q^{7} + ( \beta_{1} - \beta_{7} - \beta_{10} - \beta_{12} + \beta_{13} ) q^{8} + ( 1 - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} - \beta_{11} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( -\beta_{1} + \beta_{7} - \beta_{8} + \beta_{12} - \beta_{13} + \beta_{15} ) q^{3} + ( \beta_{5} + \beta_{9} ) q^{4} + ( \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{9} + \beta_{11} ) q^{6} + ( -\beta_{1} + 2 \beta_{7} + \beta_{10} + \beta_{15} ) q^{7} + ( \beta_{1} - \beta_{7} - \beta_{10} - \beta_{12} + \beta_{13} ) q^{8} + ( 1 - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} - \beta_{11} ) q^{9} + ( \beta_{3} - \beta_{5} + \beta_{9} ) q^{11} + ( -\beta_{1} - \beta_{7} + 2 \beta_{8} + \beta_{10} - 3 \beta_{12} + \beta_{14} ) q^{12} + ( \beta_{14} + \beta_{15} ) q^{13} + ( 1 + \beta_{2} + \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - \beta_{9} + 2 \beta_{11} ) q^{14} + ( 1 + \beta_{2} - 2 \beta_{3} + \beta_{6} + \beta_{9} + \beta_{11} ) q^{16} + ( -3 \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{17} + ( -\beta_{1} + 2 \beta_{8} - \beta_{10} - \beta_{12} - \beta_{13} - 2 \beta_{14} ) q^{18} + ( -1 - \beta_{4} + \beta_{5} + \beta_{11} ) q^{19} + ( 1 + \beta_{2} - \beta_{3} + \beta_{5} - 2 \beta_{6} - \beta_{9} ) q^{21} + ( \beta_{1} + \beta_{7} - 3 \beta_{8} + 2 \beta_{10} - 2 \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{22} + ( \beta_{1} + 2 \beta_{10} - 4 \beta_{12} + 2 \beta_{13} - 2 \beta_{15} ) q^{23} + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{24} + ( \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{11} ) q^{26} + ( \beta_{1} - \beta_{7} - 4 \beta_{8} - 2 \beta_{10} + 4 \beta_{12} - 3 \beta_{14} + \beta_{15} ) q^{27} + ( -3 \beta_{7} + 3 \beta_{8} + \beta_{10} - 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{15} ) q^{28} + ( 3 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + 4 \beta_{6} - \beta_{9} - \beta_{11} ) q^{29} + ( -3 \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{6} + 3 \beta_{9} - \beta_{11} ) q^{31} + ( -3 \beta_{7} + 6 \beta_{8} - 3 \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{32} + ( \beta_{1} + 2 \beta_{7} + 2 \beta_{8} + 4 \beta_{10} - 2 \beta_{12} - \beta_{13} + \beta_{14} ) q^{33} + ( -1 - 3 \beta_{2} + 3 \beta_{3} - \beta_{5} - 2 \beta_{6} - \beta_{9} - \beta_{11} ) q^{34} + ( -1 + \beta_{2} + 4 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{9} ) q^{36} + ( \beta_{1} - 2 \beta_{7} + 3 \beta_{8} + 2 \beta_{10} - 3 \beta_{12} - 2 \beta_{13} + 2 \beta_{15} ) q^{37} + ( 3 \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{38} + ( 2 - 2 \beta_{2} + \beta_{3} + \beta_{6} - 2 \beta_{11} ) q^{39} + ( -3 - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{9} - \beta_{11} ) q^{41} + ( -4 \beta_{1} + \beta_{7} + 4 \beta_{8} - \beta_{10} - \beta_{12} - 2 \beta_{13} - \beta_{14} ) q^{42} + ( -3 \beta_{8} - \beta_{10} + \beta_{12} - \beta_{13} ) q^{43} + ( 2 - 6 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{11} ) q^{44} + ( -8 \beta_{2} + 4 \beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{9} ) q^{46} + ( -2 \beta_{8} - 2 \beta_{10} + 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{15} ) q^{47} + ( -\beta_{1} + 4 \beta_{8} + \beta_{10} + \beta_{14} + \beta_{15} ) q^{48} + ( -3 + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{9} + 3 \beta_{11} ) q^{49} + ( -1 - 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{5} + \beta_{6} - 3 \beta_{9} - 3 \beta_{11} ) q^{51} + ( 3 \beta_{7} + 3 \beta_{8} + \beta_{10} - 2 \beta_{12} - \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{52} + ( -4 \beta_{7} - 4 \beta_{8} - 4 \beta_{10} + 4 \beta_{12} - 4 \beta_{15} ) q^{53} + ( -2 + 7 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{11} ) q^{54} + ( -\beta_{3} - 3 \beta_{4} + \beta_{5} + 3 \beta_{9} + \beta_{11} ) q^{56} + ( -\beta_{1} - \beta_{7} + 3 \beta_{8} - \beta_{10} - \beta_{12} + \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{57} + ( -2 \beta_{1} + 4 \beta_{7} + 2 \beta_{10} - 5 \beta_{13} - 3 \beta_{14} + 2 \beta_{15} ) q^{58} + ( -2 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - 4 \beta_{6} + \beta_{9} - 2 \beta_{11} ) q^{59} + ( 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - 4 \beta_{9} + \beta_{11} ) q^{61} + ( 4 \beta_{1} + 4 \beta_{7} - 4 \beta_{8} - \beta_{10} + \beta_{12} - \beta_{13} - 4 \beta_{14} + 4 \beta_{15} ) q^{62} + ( -\beta_{1} + \beta_{7} - 5 \beta_{8} - 4 \beta_{10} + 2 \beta_{12} - \beta_{13} - 2 \beta_{14} + 4 \beta_{15} ) q^{63} + ( -1 - \beta_{2} - \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{6} + 4 \beta_{9} - 4 \beta_{11} ) q^{64} + ( -4 - 6 \beta_{2} + 7 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 6 \beta_{6} - 4 \beta_{9} + \beta_{11} ) q^{66} + ( 2 \beta_{1} - \beta_{7} + \beta_{8} + \beta_{10} - \beta_{15} ) q^{67} + ( -2 \beta_{1} + 4 \beta_{7} + 2 \beta_{8} + 2 \beta_{10} - 2 \beta_{12} + 2 \beta_{13} + 4 \beta_{14} + 2 \beta_{15} ) q^{68} + ( -6 - \beta_{2} + 6 \beta_{3} + \beta_{4} + 3 \beta_{5} - 6 \beta_{6} - 3 \beta_{9} + 3 \beta_{11} ) q^{69} + ( -2 + \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{6} - \beta_{9} - \beta_{11} ) q^{71} + ( -\beta_{1} - 3 \beta_{7} + 2 \beta_{8} - \beta_{10} - \beta_{12} + \beta_{13} + 2 \beta_{14} - 4 \beta_{15} ) q^{72} + ( 3 \beta_{7} - 2 \beta_{8} + \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{73} + ( -12 + 7 \beta_{3} - 2 \beta_{4} + \beta_{5} - 8 \beta_{6} - \beta_{9} - 2 \beta_{11} ) q^{74} + ( 1 + 3 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{9} + \beta_{11} ) q^{76} + ( 2 \beta_{1} - 4 \beta_{8} + 3 \beta_{10} - \beta_{12} - 3 \beta_{13} - 2 \beta_{14} ) q^{77} + ( 3 \beta_{1} - 3 \beta_{7} - 3 \beta_{8} - 3 \beta_{10} + 3 \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{78} + ( 11 + 2 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 8 \beta_{6} + 2 \beta_{9} - \beta_{11} ) q^{79} + ( -5 + \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{9} + \beta_{11} ) q^{81} + ( 6 \beta_{1} - 2 \beta_{8} - 2 \beta_{10} + 2 \beta_{12} - \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{82} + ( \beta_{1} + \beta_{7} - 6 \beta_{8} - 2 \beta_{14} + 2 \beta_{15} ) q^{83} + ( -3 + 3 \beta_{2} + 3 \beta_{3} + \beta_{4} + 6 \beta_{5} + \beta_{6} + \beta_{9} + 4 \beta_{11} ) q^{84} + ( 1 - 2 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{9} ) q^{86} + ( 3 \beta_{1} - 4 \beta_{7} - 3 \beta_{8} - 4 \beta_{10} + 10 \beta_{12} - 3 \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{87} + ( 4 \beta_{1} - 2 \beta_{7} - 4 \beta_{8} - 4 \beta_{10} + 4 \beta_{13} - 4 \beta_{14} ) q^{88} + ( 3 + 4 \beta_{2} - 8 \beta_{3} + \beta_{4} + \beta_{5} + 4 \beta_{6} + 5 \beta_{9} + 6 \beta_{11} ) q^{89} + ( -\beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} - 4 \beta_{6} + 3 \beta_{9} ) q^{91} + ( 3 \beta_{1} - 3 \beta_{7} - 8 \beta_{8} - 5 \beta_{10} + 11 \beta_{12} + 5 \beta_{13} ) q^{92} + ( 4 \beta_{1} - 2 \beta_{7} + 3 \beta_{8} + 4 \beta_{10} + \beta_{12} + 3 \beta_{13} + 3 \beta_{14} ) q^{93} + ( 8 - 6 \beta_{3} - 2 \beta_{5} + 4 \beta_{6} + 2 \beta_{9} ) q^{94} + ( 2 + 4 \beta_{2} + \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + 4 \beta_{6} + 3 \beta_{9} + \beta_{11} ) q^{96} + ( -3 \beta_{1} + 3 \beta_{7} - \beta_{8} - 2 \beta_{10} + 2 \beta_{13} + 3 \beta_{14} + 3 \beta_{15} ) q^{97} + ( -8 \beta_{1} + 3 \beta_{7} + 8 \beta_{8} + \beta_{10} - 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{98} + ( -\beta_{2} + 3 \beta_{3} - \beta_{4} - 2 \beta_{5} - 11 \beta_{6} + \beta_{9} - 3 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 6q^{4} + 10q^{6} + 10q^{9} + O(q^{10})$$ $$16q - 6q^{4} + 10q^{6} + 10q^{9} + 24q^{14} + 2q^{16} - 18q^{19} + 38q^{21} - 32q^{24} - 12q^{26} - 42q^{31} + 18q^{36} + 6q^{39} - 60q^{41} - 14q^{46} + 8q^{49} - 12q^{51} - 34q^{54} - 42q^{56} + 24q^{59} + 30q^{61} - 76q^{64} + 44q^{66} + 26q^{69} - 108q^{74} + 58q^{79} - 82q^{81} + 6q^{84} + 18q^{86} + 6q^{89} - 6q^{91} + 48q^{94} - 6q^{96} + 68q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 11 x^{14} + 85 x^{12} + 332 x^{10} + 940 x^{8} + 1064 x^{6} + 880 x^{4} + 128 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-13 \nu^{14} - 127 \nu^{12} - 1059 \nu^{10} - 4226 \nu^{8} - 14630 \nu^{6} - 20976 \nu^{4} - 28864 \nu^{2} + 3416$$$$)/17208$$ $$\beta_{3}$$ $$=$$ $$($$$$8072 \nu^{14} + 83435 \nu^{12} + 619005 \nu^{10} + 2199943 \nu^{8} + 5759872 \nu^{6} + 4542924 \nu^{4} + 4197620 \nu^{2} + 1131680$$$$)/2357496$$ $$\beta_{4}$$ $$=$$ $$($$$$-1699 \nu^{14} - 18896 \nu^{12} - 149820 \nu^{10} - 612405 \nu^{8} - 1879598 \nu^{6} - 2699154 \nu^{4} - 3252680 \nu^{2} - 749364$$$$)/392916$$ $$\beta_{5}$$ $$=$$ $$($$$$-9357 \nu^{14} - 91521 \nu^{12} - 672151 \nu^{10} - 2181786 \nu^{8} - 5398232 \nu^{6} - 1173004 \nu^{4} - 1016664 \nu^{2} + 2648608$$$$)/1571664$$ $$\beta_{6}$$ $$=$$ $$($$$$15229 \nu^{14} + 164807 \nu^{12} + 1267309 \nu^{10} + 4851724 \nu^{8} + 13616468 \nu^{6} + 14343892 \nu^{4} + 12681936 \nu^{2} + 272784$$$$)/1571664$$ $$\beta_{7}$$ $$=$$ $$($$$$15229 \nu^{15} + 164807 \nu^{13} + 1267309 \nu^{11} + 4851724 \nu^{9} + 13616468 \nu^{7} + 14343892 \nu^{5} + 12681936 \nu^{3} + 1844448 \nu$$$$)/1571664$$ $$\beta_{8}$$ $$=$$ $$($$$$427 \nu^{15} + 4723 \nu^{13} + 36549 \nu^{11} + 143882 \nu^{9} + 409832 \nu^{7} + 483588 \nu^{5} + 417712 \nu^{3} + 112384 \nu$$$$)/34416$$ $$\beta_{9}$$ $$=$$ $$($$$$-21101 \nu^{14} - 238093 \nu^{12} - 1862467 \nu^{10} - 7521662 \nu^{8} - 21834704 \nu^{6} - 27514780 \nu^{4} - 22775544 \nu^{2} - 3194176$$$$)/1571664$$ $$\beta_{10}$$ $$=$$ $$($$$$-30473 \nu^{15} - 308177 \nu^{13} - 2306547 \nu^{11} - 7971934 \nu^{9} - 20873260 \nu^{7} - 11665404 \nu^{5} - 10957532 \nu^{3} + 8495368 \nu$$$$)/2357496$$ $$\beta_{11}$$ $$=$$ $$($$$$-13411 \nu^{14} - 150080 \nu^{12} - 1163096 \nu^{10} - 4612487 \nu^{8} - 13036508 \nu^{6} - 15171236 \nu^{4} - 11170948 \nu^{2} - 1684584$$$$)/785832$$ $$\beta_{12}$$ $$=$$ $$($$$$-82961 \nu^{15} - 892865 \nu^{13} - 6850017 \nu^{11} - 26014876 \nu^{9} - 72425530 \nu^{7} - 72742140 \nu^{5} - 58172408 \nu^{3} + 5684968 \nu$$$$)/4714992$$ $$\beta_{13}$$ $$=$$ $$($$$$14088 \nu^{15} + 160481 \nu^{13} + 1257754 \nu^{11} + 5136186 \nu^{9} + 15012495 \nu^{7} + 19847572 \nu^{5} + 17571096 \nu^{3} + 4461244 \nu$$$$)/785832$$ $$\beta_{14}$$ $$=$$ $$($$$$29781 \nu^{15} + 319213 \nu^{13} + 2439960 \nu^{11} + 9179749 \nu^{9} + 25256949 \nu^{7} + 24011448 \nu^{5} + 18177892 \nu^{3} - 1445260 \nu$$$$)/785832$$ $$\beta_{15}$$ $$=$$ $$($$$$40479 \nu^{15} + 445749 \nu^{13} + 3442334 \nu^{11} + 13442445 \nu^{9} + 37944739 \nu^{7} + 42566624 \nu^{5} + 33653328 \nu^{3} + 2711404 \nu$$$$)/785832$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{9} + 2 \beta_{6} + \beta_{5}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{13} + \beta_{12} + \beta_{10} + 5 \beta_{7} - 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-5 \beta_{11} + \beta_{9} - 13 \beta_{6} - 6 \beta_{5} - 6 \beta_{4} - 2 \beta_{3} + \beta_{2} - 7$$ $$\nu^{5}$$ $$=$$ $$-7 \beta_{15} + 7 \beta_{14} + 7 \beta_{13} - 5 \beta_{12} + 10 \beta_{8} - 25 \beta_{7}$$ $$\nu^{6}$$ $$=$$ $$32 \beta_{11} - 32 \beta_{9} + 25 \beta_{6} + 7 \beta_{5} + 25 \beta_{4} + 9 \beta_{3} + 9 \beta_{2} + 29$$ $$\nu^{7}$$ $$=$$ $$43 \beta_{15} - 41 \beta_{14} - 2 \beta_{13} + 22 \beta_{12} - 43 \beta_{10} - 64 \beta_{8} + 125 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-41 \beta_{11} + 127 \beta_{9} + 168 \beta_{6} + 127 \beta_{5} + 41 \beta_{4} + 63 \beta_{3} - 126 \beta_{2}$$ $$\nu^{9}$$ $$=$$ $$-22 \beta_{15} - 22 \beta_{14} - 231 \beta_{13} - 65 \beta_{12} + 231 \beta_{10} + 148 \beta_{8} + 631 \beta_{7} - 631 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-653 \beta_{11} + 231 \beta_{9} - 1453 \beta_{6} - 884 \beta_{5} - 884 \beta_{4} - 802 \beta_{3} + 401 \beta_{2} - 569$$ $$\nu^{11}$$ $$=$$ $$-1285 \beta_{15} + 1455 \beta_{14} + 1455 \beta_{13} - 483 \beta_{12} + 170 \beta_{10} + 796 \beta_{8} - 3221 \beta_{7}$$ $$\nu^{12}$$ $$=$$ $$4676 \beta_{11} - 4676 \beta_{9} + 3391 \beta_{6} + 1285 \beta_{5} + 3391 \beta_{4} + 2427 \beta_{3} + 2427 \beta_{2} + 2587$$ $$\nu^{13}$$ $$=$$ $$8245 \beta_{15} - 7103 \beta_{14} - 1142 \beta_{13} + 7138 \beta_{12} - 8245 \beta_{10} - 9352 \beta_{8} + 16615 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-7103 \beta_{11} + 17757 \beta_{9} + 19024 \beta_{6} + 17757 \beta_{5} + 7103 \beta_{4} + 14241 \beta_{3} - 28482 \beta_{2}$$ $$\nu^{15}$$ $$=$$ $$-7138 \beta_{15} - 7138 \beta_{14} - 39101 \beta_{13} - 32139 \beta_{12} + 39101 \beta_{10} + 35620 \beta_{8} + 86501 \beta_{7} - 86501 \beta_{1}$$

## Character values

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

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1 + \beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
299.1
 1.16543 − 2.01859i 1.03144 − 1.78651i 0.539169 − 0.933868i 0.192865 − 0.334053i −0.192865 + 0.334053i −0.539169 + 0.933868i −1.03144 + 1.78651i −1.16543 + 2.01859i 1.16543 + 2.01859i 1.03144 + 1.78651i 0.539169 + 0.933868i 0.192865 + 0.334053i −0.192865 − 0.334053i −0.539169 − 0.933868i −1.03144 − 1.78651i −1.16543 − 2.01859i
−1.16543 + 2.01859i −1.23297 1.21646i −1.71646 2.97300i 0 3.89248 1.07116i −2.39840 1.11699i 3.33995 0.0404447 + 2.99973i 0
299.2 −1.03144 + 1.78651i −1.61429 + 0.627739i −1.12774 1.95330i 0 0.543588 3.53142i −2.64573 0.00953166i 0.527019 2.21189 2.02671i 0
299.3 −0.539169 + 0.933868i 1.46840 + 0.918594i 0.418594 + 0.725026i 0 −1.64956 + 0.876010i 0.929227 + 2.47720i −3.05945 1.31237 + 2.69772i 0
299.4 −0.192865 + 0.334053i −0.983691 + 1.42561i 0.925606 + 1.60320i 0 −0.286507 0.603555i −1.17656 2.36975i −1.48553 −1.06470 2.80471i 0
299.5 0.192865 0.334053i 0.983691 1.42561i 0.925606 + 1.60320i 0 −0.286507 0.603555i 1.17656 + 2.36975i 1.48553 −1.06470 2.80471i 0
299.6 0.539169 0.933868i −1.46840 0.918594i 0.418594 + 0.725026i 0 −1.64956 + 0.876010i −0.929227 2.47720i 3.05945 1.31237 + 2.69772i 0
299.7 1.03144 1.78651i 1.61429 0.627739i −1.12774 1.95330i 0 0.543588 3.53142i 2.64573 + 0.00953166i −0.527019 2.21189 2.02671i 0
299.8 1.16543 2.01859i 1.23297 + 1.21646i −1.71646 2.97300i 0 3.89248 1.07116i 2.39840 + 1.11699i −3.33995 0.0404447 + 2.99973i 0
374.1 −1.16543 2.01859i −1.23297 + 1.21646i −1.71646 + 2.97300i 0 3.89248 + 1.07116i −2.39840 + 1.11699i 3.33995 0.0404447 2.99973i 0
374.2 −1.03144 1.78651i −1.61429 0.627739i −1.12774 + 1.95330i 0 0.543588 + 3.53142i −2.64573 + 0.00953166i 0.527019 2.21189 + 2.02671i 0
374.3 −0.539169 0.933868i 1.46840 0.918594i 0.418594 0.725026i 0 −1.64956 0.876010i 0.929227 2.47720i −3.05945 1.31237 2.69772i 0
374.4 −0.192865 0.334053i −0.983691 1.42561i 0.925606 1.60320i 0 −0.286507 + 0.603555i −1.17656 + 2.36975i −1.48553 −1.06470 + 2.80471i 0
374.5 0.192865 + 0.334053i 0.983691 + 1.42561i 0.925606 1.60320i 0 −0.286507 + 0.603555i 1.17656 2.36975i 1.48553 −1.06470 + 2.80471i 0
374.6 0.539169 + 0.933868i −1.46840 + 0.918594i 0.418594 0.725026i 0 −1.64956 0.876010i −0.929227 + 2.47720i 3.05945 1.31237 2.69772i 0
374.7 1.03144 + 1.78651i 1.61429 + 0.627739i −1.12774 + 1.95330i 0 0.543588 + 3.53142i 2.64573 0.00953166i −0.527019 2.21189 + 2.02671i 0
374.8 1.16543 + 2.01859i 1.23297 1.21646i −1.71646 + 2.97300i 0 3.89248 + 1.07116i 2.39840 1.11699i −3.33995 0.0404447 2.99973i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 374.8 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
21.g even 6 1 inner
105.p even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.q.f 16
3.b odd 2 1 525.2.q.e 16
5.b even 2 1 inner 525.2.q.f 16
5.c odd 4 1 105.2.s.c 8
5.c odd 4 1 525.2.t.g 8
7.d odd 6 1 525.2.q.e 16
15.d odd 2 1 525.2.q.e 16
15.e even 4 1 105.2.s.d yes 8
15.e even 4 1 525.2.t.f 8
21.g even 6 1 inner 525.2.q.f 16
35.f even 4 1 735.2.s.k 8
35.i odd 6 1 525.2.q.e 16
35.k even 12 1 105.2.s.d yes 8
35.k even 12 1 525.2.t.f 8
35.k even 12 1 735.2.b.c 8
35.l odd 12 1 735.2.b.d 8
35.l odd 12 1 735.2.s.l 8
105.k odd 4 1 735.2.s.l 8
105.p even 6 1 inner 525.2.q.f 16
105.w odd 12 1 105.2.s.c 8
105.w odd 12 1 525.2.t.g 8
105.w odd 12 1 735.2.b.d 8
105.x even 12 1 735.2.b.c 8
105.x even 12 1 735.2.s.k 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.s.c 8 5.c odd 4 1
105.2.s.c 8 105.w odd 12 1
105.2.s.d yes 8 15.e even 4 1
105.2.s.d yes 8 35.k even 12 1
525.2.q.e 16 3.b odd 2 1
525.2.q.e 16 7.d odd 6 1
525.2.q.e 16 15.d odd 2 1
525.2.q.e 16 35.i odd 6 1
525.2.q.f 16 1.a even 1 1 trivial
525.2.q.f 16 5.b even 2 1 inner
525.2.q.f 16 21.g even 6 1 inner
525.2.q.f 16 105.p even 6 1 inner
525.2.t.f 8 15.e even 4 1
525.2.t.f 8 35.k even 12 1
525.2.t.g 8 5.c odd 4 1
525.2.t.g 8 105.w odd 12 1
735.2.b.c 8 35.k even 12 1
735.2.b.c 8 105.x even 12 1
735.2.b.d 8 35.l odd 12 1
735.2.b.d 8 105.w odd 12 1
735.2.s.k 8 35.f even 4 1
735.2.s.k 8 105.x even 12 1
735.2.s.l 8 35.l odd 12 1
735.2.s.l 8 105.k odd 4 1

## Hecke kernels

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

 $$T_{2}^{16} + \cdots$$ $$T_{11}^{8} - 28 T_{11}^{6} + 636 T_{11}^{4} - 168 T_{11}^{3} - 4132 T_{11}^{2} + 888 T_{11} + 21904$$ $$T_{13}^{8} - 21 T_{13}^{6} + 123 T_{13}^{4} - 135 T_{13}^{2} + 36$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 5 T^{2} + 9 T^{4} + 8 T^{6} - 64 T^{8} + 120 T^{10} - 32 T^{12} - 368 T^{14} + 1104 T^{16} - 1472 T^{18} - 512 T^{20} + 7680 T^{22} - 16384 T^{24} + 8192 T^{26} + 36864 T^{28} - 81920 T^{30} + 65536 T^{32}$$
$3$ $$1 - 5 T^{2} + 33 T^{4} - 110 T^{6} + 430 T^{8} - 990 T^{10} + 2673 T^{12} - 3645 T^{14} + 6561 T^{16}$$
$5$ 
$7$ $$1 - 4 T^{2} - 26 T^{4} - 244 T^{6} + 3907 T^{8} - 11956 T^{10} - 62426 T^{12} - 470596 T^{14} + 5764801 T^{16}$$
$11$ $$( 1 + 16 T^{2} - 2 T^{4} + 30 T^{5} + 268 T^{6} + 1548 T^{7} + 21079 T^{8} + 17028 T^{9} + 32428 T^{10} + 39930 T^{11} - 29282 T^{12} + 28344976 T^{14} + 214358881 T^{16} )^{2}$$
$13$ $$( 1 + 83 T^{2} + 3217 T^{4} + 76058 T^{6} + 1197778 T^{8} + 12853802 T^{10} + 91880737 T^{12} + 400625147 T^{14} + 815730721 T^{16} )^{2}$$
$17$ $$1 + 76 T^{2} + 3024 T^{4} + 74528 T^{6} + 1179962 T^{8} + 10465632 T^{10} - 4424000 T^{12} - 1753904276 T^{14} - 37267323453 T^{16} - 506878335764 T^{18} - 369496904000 T^{20} + 252614914528608 T^{22} + 8231128701597242 T^{24} + 150247993412663072 T^{26} + 1761849645382797264 T^{28} + 12796714818514470604 T^{30} + 48661191875666868481 T^{32}$$
$19$ $$( 1 + 9 T + 100 T^{2} + 657 T^{3} + 4723 T^{4} + 26244 T^{5} + 148996 T^{6} + 704196 T^{7} + 3331528 T^{8} + 13379724 T^{9} + 53787556 T^{10} + 180007596 T^{11} + 615506083 T^{12} + 1626797043 T^{13} + 4704588100 T^{14} + 8044845651 T^{15} + 16983563041 T^{16} )^{2}$$
$23$ $$1 - 53 T^{2} + 399 T^{4} + 18308 T^{6} - 55297 T^{8} - 3303501 T^{10} - 170695622 T^{12} - 2962951637 T^{14} + 281105182152 T^{16} - 1567401415973 T^{18} - 47767633556102 T^{20} - 489036707347389 T^{22} - 4330362553083457 T^{24} + 758436567299485892 T^{26} + 8743935148376108079 T^{28} -$$$$61\!\cdots\!77$$$$T^{30} +$$$$61\!\cdots\!61$$$$T^{32}$$
$29$ $$( 1 - 53 T^{2} + 3250 T^{4} - 128951 T^{6} + 4063174 T^{8} - 108447791 T^{10} + 2298663250 T^{12} - 31525636013 T^{14} + 500246412961 T^{16} )^{2}$$
$31$ $$( 1 + 21 T + 262 T^{2} + 2415 T^{3} + 17293 T^{4} + 101304 T^{5} + 505090 T^{6} + 2328618 T^{7} + 11769748 T^{8} + 72187158 T^{9} + 485391490 T^{10} + 3017947464 T^{11} + 15970448653 T^{12} + 69139399665 T^{13} + 232525964422 T^{14} + 577764896331 T^{15} + 852891037441 T^{16} )^{2}$$
$37$ $$1 + 97 T^{2} + 5532 T^{4} + 186305 T^{6} + 3327797 T^{8} + 12794040 T^{10} + 1243533586 T^{12} + 183679377646 T^{14} + 10522943320200 T^{16} + 251457067997374 T^{18} + 2330582149071346 T^{20} + 32826006305802360 T^{22} + 11688818589319942037 T^{24} +$$$$89\!\cdots\!45$$$$T^{26} +$$$$36\!\cdots\!92$$$$T^{28} +$$$$87\!\cdots\!33$$$$T^{30} +$$$$12\!\cdots\!41$$$$T^{32}$$
$41$ $$( 1 + 15 T + 218 T^{2} + 1791 T^{3} + 14136 T^{4} + 73431 T^{5} + 366458 T^{6} + 1033815 T^{7} + 2825761 T^{8} )^{4}$$
$43$ $$( 1 - 304 T^{2} + 41758 T^{4} - 3395548 T^{6} + 179098699 T^{8} - 6278368252 T^{10} + 142762292158 T^{12} - 1921694366896 T^{14} + 11688200277601 T^{16} )^{2}$$
$47$ $$1 + 268 T^{2} + 36732 T^{4} + 3628040 T^{6} + 294880202 T^{8} + 20635855860 T^{10} + 1272375350416 T^{12} + 70342526859604 T^{14} + 3494142383383395 T^{16} + 155386641832865236 T^{18} + 6208785822293297296 T^{20} +$$$$22\!\cdots\!40$$$$T^{22} +$$$$70\!\cdots\!22$$$$T^{24} +$$$$19\!\cdots\!60$$$$T^{26} +$$$$42\!\cdots\!12$$$$T^{28} +$$$$68\!\cdots\!92$$$$T^{30} +$$$$56\!\cdots\!21$$$$T^{32}$$
$53$ $$1 - 104 T^{2} + 3204 T^{4} + 8144 T^{6} - 2651158 T^{8} - 214376472 T^{10} + 18555084688 T^{12} + 949831131304 T^{14} - 128029390162317 T^{16} + 2668075647832936 T^{18} + 146408543184054928 T^{20} - 4751517542968956888 T^{22} -$$$$16\!\cdots\!38$$$$T^{24} +$$$$14\!\cdots\!56$$$$T^{26} +$$$$15\!\cdots\!64$$$$T^{28} -$$$$14\!\cdots\!76$$$$T^{30} +$$$$38\!\cdots\!21$$$$T^{32}$$
$59$ $$( 1 - 12 T - 80 T^{2} + 1164 T^{3} + 7690 T^{4} - 80082 T^{5} - 434420 T^{6} + 1772232 T^{7} + 28861927 T^{8} + 104561688 T^{9} - 1512216020 T^{10} - 16447161078 T^{11} + 93182506090 T^{12} + 832171884036 T^{13} - 3374442691280 T^{14} - 29863817817828 T^{15} + 146830437604321 T^{16} )^{2}$$
$61$ $$( 1 - 15 T + 223 T^{2} - 2220 T^{3} + 19711 T^{4} - 141723 T^{5} + 816310 T^{6} - 5175267 T^{7} + 31433836 T^{8} - 315691287 T^{9} + 3037489510 T^{10} - 32168428263 T^{11} + 272915371951 T^{12} - 1875003788220 T^{13} + 11489043482503 T^{14} - 47141142540315 T^{15} + 191707312997281 T^{16} )^{2}$$
$67$ $$1 + 484 T^{2} + 128562 T^{4} + 23999600 T^{6} + 3478000361 T^{8} + 410568162660 T^{10} + 40599889512562 T^{12} + 3416547531270040 T^{14} + 246775367829948324 T^{16} + 15336881867871209560 T^{18} +$$$$81\!\cdots\!02$$$$T^{20} +$$$$37\!\cdots\!40$$$$T^{22} +$$$$14\!\cdots\!01$$$$T^{24} +$$$$43\!\cdots\!00$$$$T^{26} +$$$$10\!\cdots\!82$$$$T^{28} +$$$$17\!\cdots\!36$$$$T^{30} +$$$$16\!\cdots\!81$$$$T^{32}$$
$71$ $$( 1 - 464 T^{2} + 99532 T^{4} - 12936548 T^{6} + 1114829374 T^{8} - 65213138468 T^{10} + 2529275433292 T^{12} - 59438531739344 T^{14} + 645753531245761 T^{16} )^{2}$$
$73$ $$1 - 335 T^{2} + 63708 T^{4} - 7606879 T^{6} + 587709533 T^{8} - 19770442248 T^{10} - 1697574900854 T^{12} + 348751235496070 T^{14} - 32202831466789560 T^{16} + 1858495333958557030 T^{18} - 48208141150002997814 T^{20} -$$$$29\!\cdots\!72$$$$T^{22} +$$$$47\!\cdots\!73$$$$T^{24} -$$$$32\!\cdots\!71$$$$T^{26} +$$$$14\!\cdots\!68$$$$T^{28} -$$$$40\!\cdots\!15$$$$T^{30} +$$$$65\!\cdots\!61$$$$T^{32}$$
$79$ $$( 1 - 29 T + 294 T^{2} - 1975 T^{3} + 27377 T^{4} - 260496 T^{5} + 598654 T^{6} - 2403434 T^{7} + 77714340 T^{8} - 189871286 T^{9} + 3736199614 T^{10} - 128434687344 T^{11} + 1066336367537 T^{12} - 6077186388025 T^{13} + 71467711923174 T^{14} - 556913360598611 T^{15} + 1517108809906561 T^{16} )^{2}$$
$83$ $$( 1 - 535 T^{2} + 130150 T^{4} - 19183249 T^{6} + 1908109846 T^{8} - 132153402361 T^{10} + 6176700478150 T^{12} - 174913099752415 T^{14} + 2252292232139041 T^{16} )^{2}$$
$89$ $$( 1 - 3 T - 53 T^{2} - 2820 T^{3} + 14227 T^{4} + 160275 T^{5} + 3467116 T^{6} - 26593569 T^{7} - 193500020 T^{8} - 2366827641 T^{9} + 27463025836 T^{10} + 112988906475 T^{11} + 892633862707 T^{12} - 15747047646180 T^{13} - 26340008420933 T^{14} - 132694004686587 T^{15} + 3936588805702081 T^{16} )^{2}$$
$97$ $$( 1 + 368 T^{2} + 81676 T^{4} + 12257504 T^{6} + 1385094598 T^{8} + 115330855136 T^{10} + 7230717554956 T^{12} + 306533697813872 T^{14} + 7837433594376961 T^{16} )^{2}$$
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