# Properties

 Label 630.2.bv.c Level $630$ Weight $2$ Character orbit 630.bv Analytic conductor $5.031$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$630 = 2 \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 630.bv (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.03057532734$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 10 x^{14} + 61 x^{12} + 266 x^{10} + 852 x^{8} + 1438 x^{6} + 1933 x^{4} + 3038 x^{2} + 2401$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{13} q^{2} + ( \beta_{11} - \beta_{14} ) q^{4} + ( \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} ) q^{5} + ( 1 - \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} - \beta_{10} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{7} + \beta_{7} q^{8} +O(q^{10})$$ $$q + \beta_{13} q^{2} + ( \beta_{11} - \beta_{14} ) q^{4} + ( \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} ) q^{5} + ( 1 - \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} - \beta_{10} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{7} + \beta_{7} q^{8} + ( -1 + \beta_{2} - \beta_{4} + \beta_{13} ) q^{10} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{7} - \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{13} + 2 \beta_{15} ) q^{11} + ( \beta_{3} - \beta_{9} ) q^{13} + ( \beta_{1} - \beta_{7} - \beta_{9} - \beta_{10} + \beta_{13} - \beta_{15} ) q^{14} + ( 1 + \beta_{6} ) q^{16} + ( 3 + \beta_{1} - \beta_{2} + \beta_{6} - 2 \beta_{11} - \beta_{14} ) q^{17} + ( -\beta_{7} + 2 \beta_{8} - 3 \beta_{11} + \beta_{12} - \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{19} + ( -\beta_{1} - \beta_{4} - \beta_{10} + \beta_{11} - \beta_{12} - 2 \beta_{14} ) q^{20} + ( \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + 2 \beta_{8} - \beta_{10} - \beta_{12} ) q^{22} + ( 1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{11} + \beta_{12} + 2 \beta_{14} + \beta_{15} ) q^{23} + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{5} - 3 \beta_{6} + \beta_{11} + 2 \beta_{12} - 2 \beta_{15} ) q^{25} + ( -1 + \beta_{5} ) q^{26} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{7} + \beta_{9} + \beta_{15} ) q^{28} + ( -\beta_{1} - 4 \beta_{7} + 4 \beta_{8} + \beta_{11} + \beta_{12} ) q^{29} + ( 2 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - \beta_{8} + \beta_{13} + \beta_{15} ) q^{31} + ( -\beta_{8} + \beta_{13} ) q^{32} + ( -\beta_{3} + \beta_{4} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{13} - 2 \beta_{15} ) q^{34} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{11} + 3 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{35} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} + \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{13} + \beta_{14} ) q^{37} + ( -2 + \beta_{3} - \beta_{6} - 3 \beta_{7} - \beta_{11} - \beta_{14} - 2 \beta_{15} ) q^{38} + ( -\beta_{9} - \beta_{11} - \beta_{12} - \beta_{15} ) q^{40} + ( -1 - 2 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 4 \beta_{13} + 4 \beta_{15} ) q^{41} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{7} + \beta_{9} + \beta_{12} ) q^{43} + ( 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{13} + 2 \beta_{15} ) q^{44} + ( -\beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{13} + 2 \beta_{15} ) q^{46} + ( -\beta_{5} + \beta_{6} + 5 \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} - 2 \beta_{13} - \beta_{14} ) q^{47} + ( -\beta_{1} + 2 \beta_{7} + \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 3 \beta_{11} - 2 \beta_{12} - 3 \beta_{13} - 3 \beta_{14} + 3 \beta_{15} ) q^{49} + ( 2 + \beta_{3} - \beta_{4} + 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{50} + ( -\beta_{4} - \beta_{8} ) q^{52} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{5} - 5 \beta_{6} - \beta_{7} - 2 \beta_{9} + 4 \beta_{11} - 2 \beta_{12} - 5 \beta_{14} ) q^{53} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{12} + 6 \beta_{13} + 2 \beta_{14} + 6 \beta_{15} ) q^{55} + ( 1 - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} + \beta_{13} + \beta_{15} ) q^{56} + ( -4 + 2 \beta_{3} - 4 \beta_{6} - \beta_{9} - 4 \beta_{14} - \beta_{15} ) q^{58} + ( -2 \beta_{1} + 4 \beta_{7} - 2 \beta_{8} - 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{59} + ( -\beta_{5} + \beta_{6} - 2 \beta_{7} - 4 \beta_{8} + 2 \beta_{13} + 2 \beta_{15} ) q^{61} + ( 1 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} - 2 \beta_{10} + \beta_{11} - 2 \beta_{12} + 4 \beta_{13} - 2 \beta_{14} ) q^{62} + \beta_{11} q^{64} + ( 1 + 2 \beta_{1} + \beta_{6} - \beta_{7} - \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{65} + ( \beta_{4} - 4 \beta_{6} - 4 \beta_{8} + 2 \beta_{10} - 4 \beta_{11} + 3 \beta_{13} + 4 \beta_{14} ) q^{67} + ( 2 - \beta_{5} - \beta_{6} + 3 \beta_{11} + \beta_{12} - \beta_{14} ) q^{68} + ( \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} - \beta_{9} - \beta_{10} + 3 \beta_{11} + 2 \beta_{13} - \beta_{14} - 4 \beta_{15} ) q^{70} + ( -2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - 4 \beta_{7} - 4 \beta_{8} - \beta_{9} + \beta_{10} ) q^{71} + ( -1 - \beta_{1} - \beta_{2} + 6 \beta_{3} - \beta_{6} - 2 \beta_{7} - \beta_{14} - 4 \beta_{15} ) q^{73} + ( \beta_{1} - \beta_{3} + \beta_{4} + \beta_{7} + 2 \beta_{9} + 2 \beta_{10} - \beta_{11} + 2 \beta_{12} - \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{74} + ( -2 - \beta_{2} + \beta_{5} - 4 \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{13} - 2 \beta_{15} ) q^{76} + ( -3 - \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - 4 \beta_{6} - 3 \beta_{8} - 4 \beta_{10} - \beta_{11} - \beta_{12} - 2 \beta_{13} + 4 \beta_{14} ) q^{77} + ( -4 \beta_{3} + 4 \beta_{4} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 4 \beta_{13} - 2 \beta_{14} + 4 \beta_{15} ) q^{79} + ( 1 + \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} ) q^{80} + ( -2 + 2 \beta_{6} + 2 \beta_{10} + 4 \beta_{11} - \beta_{13} - 2 \beta_{14} ) q^{82} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{5} + 4 \beta_{6} - 5 \beta_{7} - 3 \beta_{9} - 2 \beta_{11} - 2 \beta_{12} + 4 \beta_{14} - 10 \beta_{15} ) q^{83} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - 6 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{12} ) q^{85} + ( -2 - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{13} - 2 \beta_{15} ) q^{86} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + 2 \beta_{9} - \beta_{11} + 2 \beta_{12} + 2 \beta_{14} ) q^{88} + ( 3 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 3 \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{89} + ( -\beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} - \beta_{13} - \beta_{15} ) q^{91} + ( \beta_{1} - \beta_{2} - \beta_{5} - \beta_{8} - \beta_{12} ) q^{92} + ( \beta_{1} - \beta_{3} + \beta_{4} + \beta_{7} - 3 \beta_{11} - \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{94} + ( -1 + 3 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 4 \beta_{10} - 4 \beta_{11} + 6 \beta_{12} - 4 \beta_{13} + 7 \beta_{14} + 2 \beta_{15} ) q^{95} + ( -1 - 2 \beta_{6} + 4 \beta_{8} - \beta_{11} - 8 \beta_{13} + 2 \beta_{14} ) q^{97} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 4 \beta_{6} + 2 \beta_{7} - \beta_{9} - 5 \beta_{11} + 4 \beta_{14} + \beta_{15} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 12q^{5} + 8q^{7} + O(q^{10})$$ $$16q + 12q^{5} + 8q^{7} - 12q^{10} + 12q^{11} + 8q^{16} + 36q^{17} - 8q^{22} + 4q^{23} + 12q^{25} - 12q^{26} + 4q^{28} + 24q^{31} - 8q^{35} + 4q^{37} - 24q^{38} - 8q^{43} - 8q^{46} - 12q^{47} + 32q^{50} + 28q^{53} + 4q^{56} - 32q^{58} - 12q^{61} + 8q^{65} + 32q^{67} + 36q^{68} - 12q^{70} - 16q^{71} - 12q^{73} - 16q^{77} + 12q^{80} - 48q^{82} + 24q^{85} - 12q^{86} - 4q^{88} - 16q^{91} - 8q^{92} - 20q^{95} - 40q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 10 x^{14} + 61 x^{12} + 266 x^{10} + 852 x^{8} + 1438 x^{6} + 1933 x^{4} + 3038 x^{2} + 2401$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$171 \nu^{14} + 2802 \nu^{12} + 20266 \nu^{10} + 96110 \nu^{8} + 343988 \nu^{6} + 866714 \nu^{4} + 786719 \nu^{2} + 1302910$$$$)/1020740$$ $$\beta_{3}$$ $$=$$ $$($$$$-6279 \nu^{15} - 6023 \nu^{14} - 94829 \nu^{13} - 319853 \nu^{12} - 291088 \nu^{11} - 2335656 \nu^{10} - 1140202 \nu^{9} - 13637974 \nu^{8} - 2242730 \nu^{7} - 52276470 \nu^{6} - 2623012 \nu^{5} - 131660244 \nu^{4} - 4504619 \nu^{3} - 120098303 \nu^{2} - 163267041 \nu - 204554077$$$$)/ 224562800$$ $$\beta_{4}$$ $$=$$ $$($$$$6279 \nu^{15} - 6023 \nu^{14} + 94829 \nu^{13} - 319853 \nu^{12} + 291088 \nu^{11} - 2335656 \nu^{10} + 1140202 \nu^{9} - 13637974 \nu^{8} + 2242730 \nu^{7} - 52276470 \nu^{6} + 2623012 \nu^{5} - 131660244 \nu^{4} + 4504619 \nu^{3} - 120098303 \nu^{2} + 163267041 \nu - 204554077$$$$)/ 224562800$$ $$\beta_{5}$$ $$=$$ $$($$$$657161 \nu^{14} + 6218761 \nu^{12} + 35853172 \nu^{10} + 144339958 \nu^{8} + 433352890 \nu^{6} + 516320528 \nu^{4} + 189804581 \nu^{2} + 1019822349$$$$)/ 785969800$$ $$\beta_{6}$$ $$=$$ $$($$$$-623003 \nu^{14} - 5366258 \nu^{12} - 30938216 \nu^{10} - 125675214 \nu^{8} - 373946420 \nu^{6} - 445540384 \nu^{4} - 843429583 \nu^{2} - 1273004222$$$$)/ 392984900$$ $$\beta_{7}$$ $$=$$ $$($$$$4174573 \nu^{15} + 307671 \nu^{14} + 41450603 \nu^{13} + 4646621 \nu^{12} + 238976156 \nu^{11} + 14263312 \nu^{10} + 995989274 \nu^{9} + 55869898 \nu^{8} + 2888475470 \nu^{7} + 109893770 \nu^{6} + 3441488944 \nu^{5} + 128527588 \nu^{4} + 1618097653 \nu^{3} + 220726331 \nu^{2} + 6797535927 \nu + 8000085009$$$$)/ 11003577200$$ $$\beta_{8}$$ $$=$$ $$($$$$-4174573 \nu^{15} + 307671 \nu^{14} - 41450603 \nu^{13} + 4646621 \nu^{12} - 238976156 \nu^{11} + 14263312 \nu^{10} - 995989274 \nu^{9} + 55869898 \nu^{8} - 2888475470 \nu^{7} + 109893770 \nu^{6} - 3441488944 \nu^{5} + 128527588 \nu^{4} - 1618097653 \nu^{3} + 220726331 \nu^{2} - 6797535927 \nu + 8000085009$$$$)/ 11003577200$$ $$\beta_{9}$$ $$=$$ $$($$$$-5413027 \nu^{15} - 6673919 \nu^{14} - 53207257 \nu^{13} - 67697189 \nu^{12} - 316254784 \nu^{11} - 390296228 \nu^{10} - 1397075246 \nu^{9} - 1650388502 \nu^{8} - 4444289310 \nu^{7} - 4717462610 \nu^{6} - 7454251516 \nu^{5} - 5620645072 \nu^{4} - 10077798427 \nu^{3} - 2889806479 \nu^{2} - 15782597033 \nu - 11101746201$$$$)/ 11003577200$$ $$\beta_{10}$$ $$=$$ $$($$$$-5413027 \nu^{15} + 6673919 \nu^{14} - 53207257 \nu^{13} + 67697189 \nu^{12} - 316254784 \nu^{11} + 390296228 \nu^{10} - 1397075246 \nu^{9} + 1650388502 \nu^{8} - 4444289310 \nu^{7} + 4717462610 \nu^{6} - 7454251516 \nu^{5} + 5620645072 \nu^{4} - 10077798427 \nu^{3} + 2889806479 \nu^{2} - 15782597033 \nu + 11101746201$$$$)/ 11003577200$$ $$\beta_{11}$$ $$=$$ $$($$$$26590 \nu^{15} + 257521 \nu^{13} + 1484692 \nu^{11} + 6079906 \nu^{9} + 17945290 \nu^{7} + 21381008 \nu^{5} + 8929484 \nu^{3} + 42231189 \nu$$$$)/50016260$$ $$\beta_{12}$$ $$=$$ $$($$$$-154017 \nu^{15} - 1553547 \nu^{13} - 9597064 \nu^{11} - 41588666 \nu^{9} - 133651610 \nu^{7} - 226254436 \nu^{5} - 303303017 \nu^{3} - 477500443 \nu$$$$)/ 239208200$$ $$\beta_{13}$$ $$=$$ $$($$$$7738601 \nu^{15} - 12843957 \nu^{14} + 58163261 \nu^{13} - 106723897 \nu^{12} + 313306372 \nu^{11} - 615297844 \nu^{10} + 1167320938 \nu^{9} - 2508526286 \nu^{8} + 3102633990 \nu^{7} - 7437029530 \nu^{6} + 1535163128 \nu^{5} - 8860887056 \nu^{4} + 6928535561 \nu^{3} - 16785607657 \nu^{2} + 9719272549 \nu - 17501784573$$$$)/ 11003577200$$ $$\beta_{14}$$ $$=$$ $$($$$$-5179651 \nu^{15} - 39396031 \nu^{13} - 212402552 \nu^{11} - 802913678 \nu^{9} - 2161262850 \nu^{7} - 1033713348 \nu^{5} - 4832044771 \nu^{3} - 6839548919 \nu$$$$)/ 5501788600$$ $$\beta_{15}$$ $$=$$ $$($$$$-850941 \nu^{15} - 939402 \nu^{14} - 7115276 \nu^{13} - 7955037 \nu^{12} - 39448752 \nu^{11} - 44968654 \nu^{10} - 154522158 \nu^{9} - 183171156 \nu^{8} - 427936390 \nu^{7} - 539065950 \nu^{6} - 355475148 \nu^{5} - 642101046 \nu^{4} - 610473801 \nu^{3} - 1214738142 \nu^{2} - 1179772034 \nu - 1821562113$$$$)/ 785969800$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{15} + \beta_{13} + \beta_{7} - 2 \beta_{6} - \beta_{5} - 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{15} - 2 \beta_{14} - \beta_{13} - 2 \beta_{12} - 3 \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + 2 \beta_{7} + \beta_{4} - \beta_{3} - 2 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-5 \beta_{15} - 5 \beta_{13} + 5 \beta_{8} + 8 \beta_{6} + 2 \beta_{4} + 2 \beta_{3} + 5 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$-14 \beta_{15} + 22 \beta_{14} + 14 \beta_{13} + 3 \beta_{12} - 3 \beta_{10} - 3 \beta_{9} - 14 \beta_{7}$$ $$\nu^{6}$$ $$=$$ $$14 \beta_{15} + 14 \beta_{13} - 14 \beta_{10} + 14 \beta_{9} - 12 \beta_{8} + 2 \beta_{7} - 22 \beta_{6} + 22 \beta_{5} - 14 \beta_{4} - 14 \beta_{3} - 22 \beta_{2} - 7$$ $$\nu^{7}$$ $$=$$ $$64 \beta_{15} - 94 \beta_{14} - 64 \beta_{13} + 94 \beta_{11} + 64 \beta_{8} + 2 \beta_{4} - 2 \beta_{3} - 7 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-75 \beta_{15} - 75 \beta_{13} + 62 \beta_{10} - 62 \beta_{9} - 75 \beta_{7} + 112 \beta_{6} - 87 \beta_{5} + 112$$ $$\nu^{9}$$ $$=$$ $$-75 \beta_{15} + 112 \beta_{14} + 75 \beta_{13} + 112 \beta_{12} - 385 \beta_{11} - 75 \beta_{10} - 75 \beta_{9} - 273 \beta_{8} + 198 \beta_{7} - 75 \beta_{4} + 75 \beta_{3} + 112 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$337 \beta_{15} + 337 \beta_{13} - 337 \beta_{8} - 486 \beta_{6} + 198 \beta_{4} + 198 \beta_{3} + 273 \beta_{2}$$ $$\nu^{11}$$ $$=$$ $$-332 \beta_{15} + 456 \beta_{14} + 332 \beta_{13} - 759 \beta_{12} + 535 \beta_{10} + 535 \beta_{9} - 332 \beta_{7}$$ $$\nu^{12}$$ $$=$$ $$332 \beta_{15} + 332 \beta_{13} - 332 \beta_{10} + 332 \beta_{9} + 2364 \beta_{8} + 2696 \beta_{7} - 456 \beta_{6} + 456 \beta_{5} - 332 \beta_{4} - 332 \beta_{3} - 456 \beta_{2} - 3803$$ $$\nu^{13}$$ $$=$$ $$1452 \beta_{15} - 2032 \beta_{14} - 1452 \beta_{13} + 2032 \beta_{11} + 1452 \beta_{8} + 2696 \beta_{4} - 2696 \beta_{3} - 3803 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-10647 \beta_{15} - 10647 \beta_{13} - 1244 \beta_{10} + 1244 \beta_{9} - 10647 \beta_{7} + 15030 \beta_{6} + 1771 \beta_{5} + 15030$$ $$\nu^{15}$$ $$=$$ $$-10647 \beta_{15} + 15030 \beta_{14} + 10647 \beta_{13} + 15030 \beta_{12} + 7801 \beta_{11} - 10647 \beta_{10} - 10647 \beta_{9} + 5503 \beta_{8} - 16150 \beta_{7} - 10647 \beta_{4} + 10647 \beta_{3} + 15030 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/630\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$281$$ $$451$$ $$\chi(n)$$ $$-\beta_{11}$$ $$1$$ $$1 + \beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
73.1
 −0.587308 + 2.01725i 1.45333 − 1.51725i 1.01089 − 0.750919i −0.144868 + 1.25092i −0.587308 − 2.01725i 1.45333 + 1.51725i 1.01089 + 0.750919i −0.144868 − 1.25092i −1.01089 − 0.750919i 0.144868 + 1.25092i −1.45333 − 1.51725i 0.587308 + 2.01725i −1.01089 + 0.750919i 0.144868 − 1.25092i −1.45333 + 1.51725i 0.587308 − 2.01725i
−0.965926 0.258819i 0 0.866025 + 0.500000i 1.38266 1.75735i 0 2.58583 0.559876i −0.707107 0.707107i 0 −1.79038 + 1.33961i
73.2 −0.965926 0.258819i 0 0.866025 + 0.500000i 1.79038 + 1.33961i 0 −2.55176 0.698943i −0.707107 0.707107i 0 −1.38266 1.75735i
73.3 0.965926 + 0.258819i 0 0.866025 + 0.500000i −2.20382 0.378409i 0 0.126334 2.64273i 0.707107 + 0.707107i 0 −2.03078 0.935904i
73.4 0.965926 + 0.258819i 0 0.866025 + 0.500000i 2.03078 0.935904i 0 1.83959 + 1.90155i 0.707107 + 0.707107i 0 2.20382 0.378409i
397.1 −0.965926 + 0.258819i 0 0.866025 0.500000i 1.38266 + 1.75735i 0 2.58583 + 0.559876i −0.707107 + 0.707107i 0 −1.79038 1.33961i
397.2 −0.965926 + 0.258819i 0 0.866025 0.500000i 1.79038 1.33961i 0 −2.55176 + 0.698943i −0.707107 + 0.707107i 0 −1.38266 + 1.75735i
397.3 0.965926 0.258819i 0 0.866025 0.500000i −2.20382 + 0.378409i 0 0.126334 + 2.64273i 0.707107 0.707107i 0 −2.03078 + 0.935904i
397.4 0.965926 0.258819i 0 0.866025 0.500000i 2.03078 + 0.935904i 0 1.83959 1.90155i 0.707107 0.707107i 0 2.20382 + 0.378409i
523.1 −0.258819 0.965926i 0 −0.866025 + 0.500000i −0.774197 2.09777i 0 2.64273 0.126334i 0.707107 + 0.707107i 0 −1.82591 + 1.29076i
523.2 −0.258819 0.965926i 0 −0.866025 + 0.500000i 1.82591 + 1.29076i 0 −1.90155 1.83959i 0.707107 + 0.707107i 0 0.774197 2.09777i
523.3 0.258819 + 0.965926i 0 −0.866025 + 0.500000i −0.264946 + 2.22032i 0 0.698943 + 2.55176i −0.707107 0.707107i 0 −2.21323 + 0.318742i
523.4 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 2.21323 + 0.318742i 0 0.559876 2.58583i −0.707107 0.707107i 0 0.264946 + 2.22032i
577.1 −0.258819 + 0.965926i 0 −0.866025 0.500000i −0.774197 + 2.09777i 0 2.64273 + 0.126334i 0.707107 0.707107i 0 −1.82591 1.29076i
577.2 −0.258819 + 0.965926i 0 −0.866025 0.500000i 1.82591 1.29076i 0 −1.90155 + 1.83959i 0.707107 0.707107i 0 0.774197 + 2.09777i
577.3 0.258819 0.965926i 0 −0.866025 0.500000i −0.264946 2.22032i 0 0.698943 2.55176i −0.707107 + 0.707107i 0 −2.21323 0.318742i
577.4 0.258819 0.965926i 0 −0.866025 0.500000i 2.21323 0.318742i 0 0.559876 + 2.58583i −0.707107 + 0.707107i 0 0.264946 2.22032i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 577.4 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.c odd 4 1 inner
7.d odd 6 1 inner
35.k even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.bv.c 16
3.b odd 2 1 70.2.k.a 16
5.c odd 4 1 inner 630.2.bv.c 16
7.d odd 6 1 inner 630.2.bv.c 16
12.b even 2 1 560.2.ci.c 16
15.d odd 2 1 350.2.o.c 16
15.e even 4 1 70.2.k.a 16
15.e even 4 1 350.2.o.c 16
21.c even 2 1 490.2.l.c 16
21.g even 6 1 70.2.k.a 16
21.g even 6 1 490.2.g.c 16
21.h odd 6 1 490.2.g.c 16
21.h odd 6 1 490.2.l.c 16
35.k even 12 1 inner 630.2.bv.c 16
60.l odd 4 1 560.2.ci.c 16
84.j odd 6 1 560.2.ci.c 16
105.k odd 4 1 490.2.l.c 16
105.p even 6 1 350.2.o.c 16
105.w odd 12 1 70.2.k.a 16
105.w odd 12 1 350.2.o.c 16
105.w odd 12 1 490.2.g.c 16
105.x even 12 1 490.2.g.c 16
105.x even 12 1 490.2.l.c 16
420.br even 12 1 560.2.ci.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.k.a 16 3.b odd 2 1
70.2.k.a 16 15.e even 4 1
70.2.k.a 16 21.g even 6 1
70.2.k.a 16 105.w odd 12 1
350.2.o.c 16 15.d odd 2 1
350.2.o.c 16 15.e even 4 1
350.2.o.c 16 105.p even 6 1
350.2.o.c 16 105.w odd 12 1
490.2.g.c 16 21.g even 6 1
490.2.g.c 16 21.h odd 6 1
490.2.g.c 16 105.w odd 12 1
490.2.g.c 16 105.x even 12 1
490.2.l.c 16 21.c even 2 1
490.2.l.c 16 21.h odd 6 1
490.2.l.c 16 105.k odd 4 1
490.2.l.c 16 105.x even 12 1
560.2.ci.c 16 12.b even 2 1
560.2.ci.c 16 60.l odd 4 1
560.2.ci.c 16 84.j odd 6 1
560.2.ci.c 16 420.br even 12 1
630.2.bv.c 16 1.a even 1 1 trivial
630.2.bv.c 16 5.c odd 4 1 inner
630.2.bv.c 16 7.d odd 6 1 inner
630.2.bv.c 16 35.k even 12 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(630, [\chi])$$:

 $$T_{11}^{8} - \cdots$$ $$T_{13}^{16} + 90 T_{13}^{12} + 1361 T_{13}^{8} + 2280 T_{13}^{4} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{4} + T^{8} )^{2}$$
$3$ $$T^{16}$$
$5$ $$390625 - 937500 T + 1031250 T^{2} - 675000 T^{3} + 281250 T^{4} - 58500 T^{5} - 21600 T^{6} + 31260 T^{7} - 18241 T^{8} + 6252 T^{9} - 864 T^{10} - 468 T^{11} + 450 T^{12} - 216 T^{13} + 66 T^{14} - 12 T^{15} + T^{16}$$
$7$ $$5764801 - 6588344 T + 3764768 T^{2} - 1344560 T^{3} + 175273 T^{4} + 54880 T^{5} - 20384 T^{6} - 4872 T^{7} + 4944 T^{8} - 696 T^{9} - 416 T^{10} + 160 T^{11} + 73 T^{12} - 80 T^{13} + 32 T^{14} - 8 T^{15} + T^{16}$$
$11$ $$( 3844 - 5952 T + 8410 T^{2} - 1992 T^{3} + 807 T^{4} - 114 T^{5} + 49 T^{6} - 6 T^{7} + T^{8} )^{2}$$
$13$ $$16 + 2280 T^{4} + 1361 T^{8} + 90 T^{12} + T^{16}$$
$17$ $$9834496 - 50577408 T + 130056192 T^{2} - 222953472 T^{3} + 273629952 T^{4} - 243823104 T^{5} + 162570240 T^{6} - 83276928 T^{7} + 33047888 T^{8} - 10173024 T^{9} + 2449440 T^{10} - 463968 T^{11} + 68652 T^{12} - 7776 T^{13} + 648 T^{14} - 36 T^{15} + T^{16}$$
$19$ $$100000000 + 58000000 T^{2} + 23150000 T^{4} + 4844200 T^{6} + 730801 T^{8} + 53438 T^{10} + 2795 T^{12} + 62 T^{14} + T^{16}$$
$23$ $$260144641 + 237612428 T + 108515912 T^{2} + 202471528 T^{3} + 80047982 T^{4} - 44050244 T^{5} + 5166336 T^{6} - 2575212 T^{7} + 443539 T^{8} + 48324 T^{9} - 5376 T^{10} + 3052 T^{11} - 658 T^{12} - 56 T^{13} + 8 T^{14} - 4 T^{15} + T^{16}$$
$29$ $$( 329476 + 90948 T^{2} + 7325 T^{4} + 162 T^{6} + T^{8} )^{2}$$
$31$ $$( 16 - 672 T + 9472 T^{2} - 2688 T^{3} - 412 T^{4} + 192 T^{5} + 32 T^{6} - 12 T^{7} + T^{8} )^{2}$$
$37$ $$65536 + 327680 T + 819200 T^{2} + 5005312 T^{3} + 12386048 T^{4} - 6952448 T^{5} + 1552512 T^{6} - 950256 T^{7} + 232225 T^{8} + 22860 T^{9} - 4536 T^{10} + 2296 T^{11} - 385 T^{12} - 56 T^{13} + 8 T^{14} - 4 T^{15} + T^{16}$$
$41$ $$( 18769 + 31468 T^{2} + 5174 T^{4} + 140 T^{6} + T^{8} )^{2}$$
$43$ $$( 784 + 896 T + 512 T^{2} - 976 T^{3} + 673 T^{4} - 140 T^{5} + 8 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$47$ $$9834496 - 71651328 T + 261015552 T^{2} - 633894912 T^{3} + 655934400 T^{4} + 400292352 T^{5} + 103800096 T^{6} + 18641688 T^{7} + 920753 T^{8} - 355332 T^{9} - 82584 T^{10} - 15288 T^{11} - 825 T^{12} + 288 T^{13} + 72 T^{14} + 12 T^{15} + T^{16}$$
$53$ $$41740124416 + 16252791808 T + 3164260352 T^{2} + 14817500160 T^{3} - 96585232 T^{4} - 3470092096 T^{5} + 1286198656 T^{6} - 363698592 T^{7} + 99156337 T^{8} - 19525980 T^{9} + 2989576 T^{10} - 431240 T^{11} + 52271 T^{12} - 4776 T^{13} + 392 T^{14} - 28 T^{15} + T^{16}$$
$59$ $$268435456 + 419430400 T^{2} + 581697536 T^{4} + 110116864 T^{6} + 16306432 T^{8} + 632192 T^{10} + 18608 T^{12} + 152 T^{14} + T^{16}$$
$61$ $$( 148996 - 78744 T - 5042 T^{2} + 9996 T^{3} + 1607 T^{4} - 294 T^{5} - 37 T^{6} + 6 T^{7} + T^{8} )^{2}$$
$67$ $$3429742096 + 3117947360 T + 1417248800 T^{2} + 5674148832 T^{3} + 1483484684 T^{4} - 3138391520 T^{5} + 1227548872 T^{6} - 476714964 T^{7} + 180649957 T^{8} - 39085404 T^{9} + 6081184 T^{10} - 939268 T^{11} + 112235 T^{12} - 8184 T^{13} + 512 T^{14} - 32 T^{15} + T^{16}$$
$71$ $$( -4424 - 1816 T - 190 T^{2} + 4 T^{3} + T^{4} )^{4}$$
$73$ $$1017603875209216 - 198908406202368 T + 19440056696832 T^{2} - 1266632589312 T^{3} - 616424309760 T^{4} + 46930397184 T^{5} + 3390958080 T^{6} + 1224624768 T^{7} + 171765008 T^{8} - 7027296 T^{9} - 295776 T^{10} - 125664 T^{11} - 16260 T^{12} + 288 T^{13} + 72 T^{14} + 12 T^{15} + T^{16}$$
$79$ $$61465600000000 - 5720064000000 T^{2} + 349926400000 T^{4} - 12457574400 T^{6} + 323248896 T^{8} - 5240832 T^{10} + 59680 T^{12} - 288 T^{14} + T^{16}$$
$83$ $$3812835757370896 + 5479636353768 T^{4} + 1373332049 T^{8} + 69978 T^{12} + T^{16}$$
$89$ $$9971220736 + 6274951040 T^{2} + 3225808304 T^{4} + 417079160 T^{6} + 40392625 T^{8} + 1250110 T^{10} + 28859 T^{12} + 190 T^{14} + T^{16}$$
$97$ $$( 3111696 + 8136 T^{4} + T^{8} )^{2}$$
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