# Properties

 Label 1064.2.q.n Level $1064$ Weight $2$ Character orbit 1064.q Analytic conductor $8.496$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1064 = 2^{3} \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1064.q (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.49608277506$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 15 x^{14} - 2 x^{13} + 159 x^{12} - 19 x^{11} + 839 x^{10} - 62 x^{9} + 3204 x^{8} + 8 x^{7} + 4560 x^{6} + 1376 x^{5} + 4688 x^{4} + 736 x^{3} + 1280 x^{2} - 128 x + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + \beta_{10} q^{5} + \beta_{11} q^{7} + ( -\beta_{2} + \beta_{8} + \beta_{12} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + \beta_{10} q^{5} + \beta_{11} q^{7} + ( -\beta_{2} + \beta_{8} + \beta_{12} ) q^{9} + ( -2 + 2 \beta_{2} + \beta_{7} + \beta_{8} + \beta_{13} ) q^{11} + ( -\beta_{2} - \beta_{8} + \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{13} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{8} + \beta_{10} + 2 \beta_{11} ) q^{15} + ( -1 + \beta_{3} + \beta_{5} - \beta_{7} - \beta_{9} - \beta_{15} ) q^{17} -\beta_{2} q^{19} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} - 2 \beta_{11} - \beta_{12} + \beta_{13} ) q^{21} + ( -4 \beta_{2} - \beta_{4} + \beta_{5} + \beta_{11} + \beta_{12} + \beta_{14} ) q^{23} + ( -2 + 3 \beta_{2} + \beta_{3} + \beta_{8} - \beta_{11} + \beta_{13} + \beta_{15} ) q^{25} + ( 1 - \beta_{5} - \beta_{8} + \beta_{9} - \beta_{10} ) q^{27} + ( 2 + 2 \beta_{2} - 2 \beta_{5} - \beta_{7} - \beta_{9} + \beta_{10} - 3 \beta_{11} - \beta_{12} + \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{29} + ( \beta_{2} - \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{11} + \beta_{13} + \beta_{15} ) q^{31} + ( -2 \beta_{2} + 2 \beta_{4} - \beta_{6} + \beta_{8} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{33} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{9} + \beta_{13} - \beta_{14} ) q^{35} + ( -\beta_{2} + 2 \beta_{4} - \beta_{5} + 2 \beta_{10} - \beta_{11} - \beta_{12} - 2 \beta_{14} ) q^{37} + ( 2 - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{15} ) q^{39} -2 q^{41} + ( 2 - \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{8} + 2 \beta_{11} + \beta_{14} - \beta_{15} ) q^{43} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{5} - \beta_{8} + \beta_{11} - \beta_{13} + \beta_{15} ) q^{45} + ( 5 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{10} - 3 \beta_{11} + 2 \beta_{13} ) q^{47} + ( \beta_{1} - 3 \beta_{2} + \beta_{4} + \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{49} + ( \beta_{2} - \beta_{4} - \beta_{5} + \beta_{8} - 3 \beta_{11} + 2 \beta_{13} ) q^{51} + ( -2 - 3 \beta_{1} + 3 \beta_{7} + 2 \beta_{9} + 2 \beta_{11} - 2 \beta_{15} ) q^{53} + ( 2 + \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} + 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - \beta_{14} + \beta_{15} ) q^{55} + ( -\beta_{1} - \beta_{4} ) q^{57} + ( -2 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{13} - \beta_{15} ) q^{59} + ( 2 \beta_{2} + \beta_{4} + \beta_{6} + \beta_{8} - 3 \beta_{10} + \beta_{12} + 2 \beta_{14} ) q^{61} + ( -4 + \beta_{1} - 2 \beta_{2} + 2 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{12} + 2 \beta_{13} - \beta_{15} ) q^{63} + ( 2 \beta_{2} - \beta_{4} + \beta_{5} - \beta_{8} + \beta_{10} + 2 \beta_{11} - \beta_{13} + 3 \beta_{14} ) q^{65} + ( -\beta_{1} - \beta_{2} - \beta_{7} - \beta_{8} + \beta_{11} - \beta_{13} - \beta_{15} ) q^{67} + ( -1 - 4 \beta_{1} + \beta_{2} - 4 \beta_{4} - \beta_{5} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{69} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} - 2 \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{71} + ( 1 + 3 \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{5} - \beta_{8} + \beta_{11} - \beta_{13} + 2 \beta_{15} ) q^{73} + ( 4 \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} + 3 \beta_{10} - 3 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} ) q^{75} + ( 4 - 2 \beta_{2} - \beta_{4} - 2 \beta_{5} - 3 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - 3 \beta_{12} - \beta_{13} - \beta_{14} ) q^{77} + ( -2 \beta_{2} - 4 \beta_{4} - \beta_{8} + 2 \beta_{10} - \beta_{12} - \beta_{14} ) q^{79} + ( 2 + \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{7} + \beta_{9} ) q^{81} + ( -6 + 2 \beta_{1} + 2 \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{83} + ( 3 - 2 \beta_{1} - 4 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{6} + 2 \beta_{7} - \beta_{8} + 3 \beta_{9} - 2 \beta_{10} + 4 \beta_{11} + 2 \beta_{12} - 4 \beta_{13} + 3 \beta_{14} - 3 \beta_{15} ) q^{85} + ( -4 + 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{7} + 2 \beta_{9} + 2 \beta_{11} - 2 \beta_{15} ) q^{87} + ( 3 \beta_{4} - \beta_{5} + \beta_{8} + \beta_{10} - 2 \beta_{11} + \beta_{13} - 3 \beta_{14} ) q^{89} + ( \beta_{1} - \beta_{3} - \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{11} + 2 \beta_{13} + 3 \beta_{15} ) q^{91} + ( 2 \beta_{2} - 2 \beta_{5} + \beta_{6} + 2 \beta_{8} - 2 \beta_{10} - 3 \beta_{11} + \beta_{13} - \beta_{14} ) q^{93} -\beta_{9} q^{95} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - 3 \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} - 3 \beta_{11} - 3 \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{97} + ( 5 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{11} - \beta_{12} + 3 \beta_{13} - \beta_{14} + \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + q^{5} + 5q^{7} - 6q^{9} + O(q^{10})$$ $$16q + q^{5} + 5q^{7} - 6q^{9} - 9q^{11} + 16q^{15} - 4q^{17} - 8q^{19} - 2q^{21} - 25q^{23} - 15q^{25} + 6q^{27} + 12q^{29} - 8q^{33} + 5q^{35} - 13q^{37} + 11q^{39} - 32q^{41} + 34q^{43} - 17q^{45} + 24q^{47} - 13q^{49} - 5q^{51} - 2q^{53} + 10q^{55} - 2q^{59} + 13q^{61} - 52q^{63} + 26q^{65} - 2q^{67} - 22q^{69} + 20q^{71} - 5q^{73} + 20q^{75} + 28q^{77} - 16q^{79} + 12q^{81} - 86q^{83} + 48q^{85} - 20q^{87} - 8q^{89} - 34q^{91} - 2q^{93} + q^{95} - 24q^{97} + 74q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 15 x^{14} - 2 x^{13} + 159 x^{12} - 19 x^{11} + 839 x^{10} - 62 x^{9} + 3204 x^{8} + 8 x^{7} + 4560 x^{6} + 1376 x^{5} + 4688 x^{4} + 736 x^{3} + 1280 x^{2} - 128 x + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-887991240725 \nu^{15} - 1621149621626 \nu^{14} - 13171357172271 \nu^{13} - 22061790698884 \nu^{12} - 135681102976851 \nu^{11} - 234620536245287 \nu^{10} - 691104095414161 \nu^{9} - 1240172855303516 \nu^{8} - 2623214172465900 \nu^{7} - 4897476918238464 \nu^{6} - 3589595465252720 \nu^{5} - 7490464984730624 \nu^{4} - 5626733575860240 \nu^{3} - 7592361290310144 \nu^{2} - 1066686719972864 \nu - 233244093841280$$$$)/ 1593234061512192$$ $$\beta_{3}$$ $$=$$ $$($$$$14068115277 \nu^{15} + 65538410882 \nu^{14} + 235055440711 \nu^{13} + 953328046732 \nu^{12} + 2435537980379 \nu^{11} + 10037162516759 \nu^{10} + 14036480431633 \nu^{9} + 53431360026644 \nu^{8} + 58410208152940 \nu^{7} + 207318990934624 \nu^{6} + 132337661273488 \nu^{5} + 326444577847296 \nu^{4} + 243008252638736 \nu^{3} + 338317725714176 \nu^{2} + 182455911050240 \nu + 81247815120256$$$$)/ 23090348717568$$ $$\beta_{4}$$ $$=$$ $$($$$$-810574810813 \nu^{15} + 74255719302 \nu^{14} - 11918886590167 \nu^{13} + 2754752149212 \nu^{12} - 125746184909531 \nu^{11} + 26960277777057 \nu^{10} - 647614156114233 \nu^{9} + 110954881408500 \nu^{8} - 2445186494156332 \nu^{7} + 229822296226640 \nu^{6} - 3134294518746512 \nu^{5} - 731915319670720 \nu^{4} - 3469399868568272 \nu^{3} + 34971034077568 \nu^{2} - 970070517083136 \nu + 113662878812800$$$$)/ 796617030756096$$ $$\beta_{5}$$ $$=$$ $$($$$$-204905039787 \nu^{15} + 164538036370 \nu^{14} - 3309757518145 \nu^{13} + 2678729722988 \nu^{12} - 36330971679053 \nu^{11} + 27603120276079 \nu^{10} - 210471150872935 \nu^{9} + 125141579564692 \nu^{8} - 839841355419604 \nu^{7} + 393069278434784 \nu^{6} - 1562079233957104 \nu^{5} - 38130599038080 \nu^{4} - 1305166508577008 \nu^{3} - 121800031476992 \nu^{2} - 951893337440768 \nu - 35169887843968$$$$)/ 144839460137472$$ $$\beta_{6}$$ $$=$$ $$($$$$-2689825004519 \nu^{15} - 5441931926454 \nu^{14} - 41601999321077 \nu^{13} - 74457739040052 \nu^{12} - 432681058698385 \nu^{11} - 788348746957845 \nu^{10} - 2324795793996291 \nu^{9} - 4164946945743468 \nu^{8} - 9028198334938436 \nu^{7} - 16426930474636256 \nu^{6} - 14880934278277936 \nu^{5} - 25301719258064000 \nu^{4} - 19365128459647408 \nu^{3} - 23404554005083648 \nu^{2} - 3711273657583104 \nu - 770795160447616$$$$)/ 1593234061512192$$ $$\beta_{7}$$ $$=$$ $$($$$$425431690537 \nu^{15} + 750640917806 \nu^{14} + 6302277954239 \nu^{13} + 10247092847008 \nu^{12} + 64950712395523 \nu^{11} + 109195740583175 \nu^{10} + 330377236922557 \nu^{9} + 582112627729628 \nu^{8} + 1252530362554664 \nu^{7} + 2308256804479040 \nu^{6} + 1698938827624368 \nu^{5} + 3662588781234544 \nu^{4} + 2655478264221136 \nu^{3} + 3580160077276672 \nu^{2} + 530866034229248 \nu + 861362289784704$$$$)/ 199154257689024$$ $$\beta_{8}$$ $$=$$ $$($$$$351188739807 \nu^{15} + 33860631830 \nu^{14} + 5457395599933 \nu^{13} - 225714105596 \nu^{12} + 58645372805081 \nu^{11} - 2408221942867 \nu^{10} + 324075334315243 \nu^{9} - 5736755156452 \nu^{8} + 1283648427035908 \nu^{7} + 40506159472960 \nu^{6} + 2200066100140336 \nu^{5} + 393946120082304 \nu^{4} + 2573217916964144 \nu^{3} + 176245078812416 \nu^{2} + 977494501291520 \nu - 127746035785088$$$$)/ 144839460137472$$ $$\beta_{9}$$ $$=$$ $$($$$$-88198852655 \nu^{15} - 65002779306 \nu^{14} - 1257457268597 \nu^{13} - 836729119824 \nu^{12} - 12973171701001 \nu^{11} - 9357603859449 \nu^{10} - 63000513650151 \nu^{9} - 55741803266736 \nu^{8} - 229375750636916 \nu^{7} - 238837003395824 \nu^{6} - 214367384973280 \nu^{5} - 500568422777600 \nu^{4} - 272058697425520 \nu^{3} - 347332382905984 \nu^{2} + 80173165426176 \nu - 83894204762752$$$$)/ 34635523076352$$ $$\beta_{10}$$ $$=$$ $$($$$$2030314903703 \nu^{15} - 3031790914950 \nu^{14} + 30779792913437 \nu^{13} - 49264868546880 \nu^{12} + 333364954141009 \nu^{11} - 515558496262335 \nu^{10} + 1811850113799231 \nu^{9} - 2604517297831248 \nu^{8} + 6954537806399252 \nu^{7} - 9256849763956384 \nu^{6} + 10366389494085856 \nu^{5} - 9078567171603712 \nu^{4} + 6788149393737328 \nu^{3} - 8497918771647872 \nu^{2} + 2743897321585152 \nu - 918889371704192$$$$)/ 796617030756096$$ $$\beta_{11}$$ $$=$$ $$($$$$-5244700706093 \nu^{15} + 2346883868598 \nu^{14} - 77164507985591 \nu^{13} + 45953130310332 \nu^{12} - 813699122229115 \nu^{11} + 472486928172657 \nu^{10} - 4179379074795177 \nu^{9} + 2274150901209828 \nu^{8} - 15423304307469836 \nu^{7} + 7307910961291648 \nu^{6} - 17942216498989552 \nu^{5} + 2649634451897728 \nu^{4} - 10209227561493520 \nu^{3} + 5820835873481216 \nu^{2} + 5705608627616256 \nu + 1314601402099328$$$$)/ 1593234061512192$$ $$\beta_{12}$$ $$=$$ $$($$$$-7415041100777 \nu^{15} - 6857065436634 \nu^{14} - 112716780288347 \nu^{13} - 85764307633980 \nu^{12} - 1187823512763295 \nu^{11} - 911991703609611 \nu^{10} - 6329245059124317 \nu^{9} - 4897587114493092 \nu^{8} - 24612989387258588 \nu^{7} - 20035475427156416 \nu^{6} - 38559108962554576 \nu^{5} - 34295267259827840 \nu^{4} - 50812331390046544 \nu^{3} - 30714906966664960 \nu^{2} - 15019186394098176 \nu + 472230018270848$$$$)/ 1593234061512192$$ $$\beta_{13}$$ $$=$$ $$($$$$-4771753121831 \nu^{15} - 582473115970 \nu^{14} - 69446380585885 \nu^{13} + 1023801855592 \nu^{12} - 726252196117697 \nu^{11} - 1844402747305 \nu^{10} - 3661473258291431 \nu^{9} - 183234561054040 \nu^{8} - 13555293653234356 \nu^{7} - 1730392506840640 \nu^{6} - 15387289160498112 \nu^{5} - 8116566088565696 \nu^{4} - 15510066748381616 \nu^{3} - 1666632002964608 \nu^{2} + 526633776519680 \nu + 518001185001600$$$$)/ 796617030756096$$ $$\beta_{14}$$ $$=$$ $$($$$$2700796668517 \nu^{15} - 4121934292162 \nu^{14} + 42850402073615 \nu^{13} - 66543603564584 \nu^{12} + 470421547551451 \nu^{11} - 699548588005093 \nu^{10} + 2682057305391013 \nu^{9} - 3533109813540112 \nu^{8} + 10511025807388268 \nu^{7} - 12492104630391544 \nu^{6} + 17971408524512784 \nu^{5} - 11685298902396128 \nu^{4} + 10976605479526576 \nu^{3} - 7853459524596416 \nu^{2} + 4197912374516480 \nu - 1287743730750336$$$$)/ 398308515378048$$ $$\beta_{15}$$ $$=$$ $$($$$$-4572753596683 \nu^{15} - 60070110278 \nu^{14} - 65505508351409 \nu^{13} + 5707585000580 \nu^{12} - 680749049937037 \nu^{11} + 35042677420231 \nu^{10} - 3343191330072847 \nu^{9} - 198112420450724 \nu^{8} - 12051988426735124 \nu^{7} - 2209924722238976 \nu^{6} - 11128677529052688 \nu^{5} - 12839819484107008 \nu^{4} - 8663083660596592 \nu^{3} - 4284953758961152 \nu^{2} + 1981015106403328 \nu - 1605614924870784$$$$)/ 531078020504064$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{12} + \beta_{8} - 4 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{10} + \beta_{9} - \beta_{8} - \beta_{5} - 6 \beta_{4} - 6 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{9} + 8 \beta_{7} - \beta_{3} + 25 \beta_{2} + \beta_{1} - 25$$ $$\nu^{5}$$ $$=$$ $$\beta_{14} - 10 \beta_{13} + 3 \beta_{12} + 12 \beta_{11} + 7 \beta_{10} + \beta_{8} - \beta_{6} + 2 \beta_{5} + 40 \beta_{4} - 15 \beta_{2}$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{15} - 2 \beta_{14} + 11 \beta_{13} - 62 \beta_{12} - 19 \beta_{11} - \beta_{10} - 12 \beta_{9} - 61 \beta_{8} - 62 \beta_{7} + 13 \beta_{6} - 5 \beta_{5} - 13 \beta_{4} + 13 \beta_{3} + 15 \beta_{2} - 13 \beta_{1} + 174$$ $$\nu^{7}$$ $$=$$ $$-15 \beta_{15} + 84 \beta_{13} - 58 \beta_{11} - 27 \beta_{9} + 84 \beta_{8} + 45 \beta_{7} + 73 \beta_{5} - 16 \beta_{3} + 130 \beta_{2} + 277 \beta_{1} - 145$$ $$\nu^{8}$$ $$=$$ $$31 \beta_{14} - 150 \beta_{13} + 481 \beta_{12} + 212 \beta_{11} + 18 \beta_{10} + 419 \beta_{8} - 130 \beta_{6} + 62 \beta_{5} + 137 \beta_{4} - 1388 \beta_{2}$$ $$\nu^{9}$$ $$=$$ $$161 \beta_{15} - 161 \beta_{14} + 82 \beta_{13} - 499 \beta_{12} - 606 \beta_{11} - 260 \beta_{10} + 77 \beta_{9} - 916 \beta_{8} - 499 \beta_{7} + 183 \beta_{6} - 840 \beta_{5} - 1964 \beta_{4} + 183 \beta_{3} + 344 \beta_{2} - 1964 \beta_{1} + 1369$$ $$\nu^{10}$$ $$=$$ $$-344 \beta_{15} + 603 \beta_{13} + 81 \beta_{11} + 969 \beta_{9} + 603 \beta_{8} + 3748 \beta_{7} + 263 \beta_{5} - 1180 \beta_{3} + 8959 \beta_{2} + 1342 \beta_{1} - 9303$$ $$\nu^{11}$$ $$=$$ $$1524 \beta_{14} - 6371 \beta_{13} + 4912 \beta_{12} + 8783 \beta_{11} + 1592 \beta_{10} + 2500 \beta_{8} - 1835 \beta_{6} + 2412 \beta_{5} + 14199 \beta_{4} - 14210 \beta_{2}$$ $$\nu^{12}$$ $$=$$ $$3359 \beta_{15} - 3359 \beta_{14} + 6948 \beta_{13} - 29353 \beta_{12} - 20176 \beta_{11} - 2085 \beta_{10} - 8118 \beta_{9} - 28909 \beta_{8} - 29353 \beta_{7} + 10203 \beta_{6} - 9529 \beta_{5} - 12596 \beta_{4} + 10203 \beta_{3} + 13562 \beta_{2} - 12596 \beta_{1} + 70018$$ $$\nu^{13}$$ $$=$$ $$-13562 \beta_{15} + 43145 \beta_{13} - 21909 \beta_{11} + 7196 \beta_{9} + 43145 \beta_{8} + 45446 \beta_{7} + 35471 \beta_{5} - 17203 \beta_{3} + 95384 \beta_{2} + 104463 \beta_{1} - 108946$$ $$\nu^{14}$$ $$=$$ $$30765 \beta_{14} - 115323 \beta_{13} + 231047 \beta_{12} + 175193 \beta_{11} + 18909 \beta_{10} + 171177 \beta_{8} - 85812 \beta_{6} + 59870 \beta_{5} + 114699 \beta_{4} - 620647 \beta_{2}$$ $$\nu^{15}$$ $$=$$ $$116577 \beta_{15} - 116577 \beta_{14} + 89381 \beta_{13} - 405215 \beta_{12} - 453833 \beta_{11} - 65202 \beta_{10} - 89828 \beta_{9} - 565790 \beta_{8} - 405215 \beta_{7} + 155030 \beta_{6} - 459378 \beta_{5} - 780775 \beta_{4} + 155030 \beta_{3} + 271607 \beta_{2} - 780775 \beta_{1} + 941906$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1064\mathbb{Z}\right)^\times$$.

 $$n$$ $$533$$ $$799$$ $$913$$ $$1009$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
305.1
 −1.42832 + 2.47393i −1.19508 + 2.06994i −0.456148 + 0.790072i −0.342094 + 0.592525i 0.218288 − 0.378086i 0.670757 − 1.16179i 1.23497 − 2.13902i 1.29764 − 2.24758i −1.42832 − 2.47393i −1.19508 − 2.06994i −0.456148 − 0.790072i −0.342094 − 0.592525i 0.218288 + 0.378086i 0.670757 + 1.16179i 1.23497 + 2.13902i 1.29764 + 2.24758i
0 −1.42832 + 2.47393i 0 −0.413955 0.716991i 0 0.222612 2.63637i 0 −2.58022 4.46907i 0
305.2 0 −1.19508 + 2.06994i 0 −1.62589 2.81613i 0 2.05789 + 1.66286i 0 −1.35644 2.34943i 0
305.3 0 −0.456148 + 0.790072i 0 0.832641 + 1.44218i 0 −0.462738 + 2.60497i 0 1.08386 + 1.87730i 0
305.4 0 −0.342094 + 0.592525i 0 1.82548 + 3.16183i 0 −1.95459 + 1.78314i 0 1.26594 + 2.19268i 0
305.5 0 0.218288 0.378086i 0 0.149820 + 0.259496i 0 1.68999 2.03567i 0 1.40470 + 2.43301i 0
305.6 0 0.670757 1.16179i 0 −1.45434 2.51900i 0 −2.59748 + 0.503078i 0 0.600170 + 1.03952i 0
305.7 0 1.23497 2.13902i 0 2.01936 + 3.49763i 0 1.24923 2.33226i 0 −1.55028 2.68516i 0
305.8 0 1.29764 2.24758i 0 −0.833107 1.44298i 0 2.29509 + 1.31627i 0 −1.86773 3.23500i 0
457.1 0 −1.42832 2.47393i 0 −0.413955 + 0.716991i 0 0.222612 + 2.63637i 0 −2.58022 + 4.46907i 0
457.2 0 −1.19508 2.06994i 0 −1.62589 + 2.81613i 0 2.05789 1.66286i 0 −1.35644 + 2.34943i 0
457.3 0 −0.456148 0.790072i 0 0.832641 1.44218i 0 −0.462738 2.60497i 0 1.08386 1.87730i 0
457.4 0 −0.342094 0.592525i 0 1.82548 3.16183i 0 −1.95459 1.78314i 0 1.26594 2.19268i 0
457.5 0 0.218288 + 0.378086i 0 0.149820 0.259496i 0 1.68999 + 2.03567i 0 1.40470 2.43301i 0
457.6 0 0.670757 + 1.16179i 0 −1.45434 + 2.51900i 0 −2.59748 0.503078i 0 0.600170 1.03952i 0
457.7 0 1.23497 + 2.13902i 0 2.01936 3.49763i 0 1.24923 + 2.33226i 0 −1.55028 + 2.68516i 0
457.8 0 1.29764 + 2.24758i 0 −0.833107 + 1.44298i 0 2.29509 1.31627i 0 −1.86773 + 3.23500i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 457.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1064.2.q.n 16
7.c even 3 1 inner 1064.2.q.n 16
7.c even 3 1 7448.2.a.bq 8
7.d odd 6 1 7448.2.a.br 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1064.2.q.n 16 1.a even 1 1 trivial
1064.2.q.n 16 7.c even 3 1 inner
7448.2.a.bq 8 7.c even 3 1
7448.2.a.br 8 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1064, [\chi])$$:

 $$T_{3}^{16} + \cdots$$ $$T_{5}^{16} - \cdots$$ $$T_{11}^{16} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$256 - 128 T + 1280 T^{2} + 736 T^{3} + 4688 T^{4} + 1376 T^{5} + 4560 T^{6} + 8 T^{7} + 3204 T^{8} - 62 T^{9} + 839 T^{10} - 19 T^{11} + 159 T^{12} - 2 T^{13} + 15 T^{14} + T^{16}$$
$5$ $$9216 - 18432 T + 81408 T^{2} + 79104 T^{3} + 203968 T^{4} + 61984 T^{5} + 102256 T^{6} + 24984 T^{7} + 36576 T^{8} + 5078 T^{9} + 5199 T^{10} + 253 T^{11} + 516 T^{12} + 9 T^{13} + 28 T^{14} - T^{15} + T^{16}$$
$7$ $$5764801 - 4117715 T + 2235331 T^{2} - 420175 T^{3} + 43218 T^{4} + 67914 T^{5} - 13720 T^{6} + 5460 T^{7} + 1213 T^{8} + 780 T^{9} - 280 T^{10} + 198 T^{11} + 18 T^{12} - 25 T^{13} + 19 T^{14} - 5 T^{15} + T^{16}$$
$11$ $$64 + 96 T + 2352 T^{2} - 6816 T^{3} + 68948 T^{4} - 72372 T^{5} + 203949 T^{6} + 255093 T^{7} + 280827 T^{8} + 134394 T^{9} + 53829 T^{10} + 11535 T^{11} + 2717 T^{12} + 414 T^{13} + 87 T^{14} + 9 T^{15} + T^{16}$$
$13$ $$( -16 - 488 T - 516 T^{2} + 200 T^{3} + 310 T^{4} + T^{5} - 41 T^{6} + T^{8} )^{2}$$
$17$ $$606735424 + 1208445920 T + 2065336288 T^{2} + 1009989912 T^{3} + 567942608 T^{4} + 102851294 T^{5} + 60998967 T^{6} + 8515393 T^{7} + 4188508 T^{8} + 380573 T^{9} + 180015 T^{10} + 14023 T^{11} + 5327 T^{12} + 270 T^{13} + 94 T^{14} + 4 T^{15} + T^{16}$$
$19$ $$( 1 + T + T^{2} )^{8}$$
$23$ $$77986561 + 75416740 T + 130960101 T^{2} + 36414878 T^{3} + 85676027 T^{4} + 37522851 T^{5} + 25970073 T^{6} + 11911293 T^{7} + 6398682 T^{8} + 2457595 T^{9} + 768631 T^{10} + 174879 T^{11} + 31137 T^{12} + 4041 T^{13} + 396 T^{14} + 25 T^{15} + T^{16}$$
$29$ $$( 6672 + 3024 T - 19204 T^{2} - 420 T^{3} + 3630 T^{4} + 329 T^{5} - 105 T^{6} - 6 T^{7} + T^{8} )^{2}$$
$31$ $$87310336 - 27508736 T + 100761600 T^{2} + 36192256 T^{3} + 77845504 T^{4} + 16314880 T^{5} + 17780224 T^{6} + 1834496 T^{7} + 2828736 T^{8} + 145760 T^{9} + 188144 T^{10} - 11984 T^{11} + 8056 T^{12} - 232 T^{13} + 100 T^{14} + T^{16}$$
$37$ $$287296000000 + 45246976000 T + 76066381056 T^{2} + 30139982080 T^{3} + 18607317904 T^{4} + 5418952176 T^{5} + 1700873504 T^{6} + 333436136 T^{7} + 76346496 T^{8} + 11907200 T^{9} + 2181449 T^{10} + 238629 T^{11} + 31114 T^{12} + 2077 T^{13} + 258 T^{14} + 13 T^{15} + T^{16}$$
$41$ $$( 2 + T )^{16}$$
$43$ $$( -51712 - 8256 T + 33288 T^{2} - 2180 T^{3} - 4964 T^{4} + 1081 T^{5} + 11 T^{6} - 17 T^{7} + T^{8} )^{2}$$
$47$ $$173633329 - 1241642356 T + 7571546752 T^{2} - 10665360256 T^{3} + 14929688586 T^{4} - 417792676 T^{5} + 5682087856 T^{6} - 2139405144 T^{7} + 653744491 T^{8} - 105661260 T^{9} + 14332496 T^{10} - 1274112 T^{11} + 113786 T^{12} - 7448 T^{13} + 560 T^{14} - 24 T^{15} + T^{16}$$
$53$ $$3340823595264 + 4807253805696 T + 5839660168704 T^{2} + 1887124703904 T^{3} + 636694665552 T^{4} + 82524224880 T^{5} + 22668604824 T^{6} + 2221348128 T^{7} + 551185828 T^{8} + 30434884 T^{9} + 6834509 T^{10} + 218893 T^{11} + 62851 T^{12} + 968 T^{13} + 299 T^{14} + 2 T^{15} + T^{16}$$
$59$ $$907401035776 + 279767760896 T + 659433750528 T^{2} - 141566542848 T^{3} + 341938046976 T^{4} - 4792838976 T^{5} + 15925395632 T^{6} - 770663144 T^{7} + 529005816 T^{8} - 16542422 T^{9} + 7195415 T^{10} - 77487 T^{11} + 70221 T^{12} - 312 T^{13} + 315 T^{14} + 2 T^{15} + T^{16}$$
$61$ $$15605006400 + 125309250720 T + 993377947536 T^{2} + 132296551536 T^{3} + 126420510964 T^{4} - 21590511816 T^{5} + 10586578105 T^{6} - 1214429725 T^{7} + 285006963 T^{8} - 25483470 T^{9} + 4907845 T^{10} - 355539 T^{11} + 50305 T^{12} - 2606 T^{13} + 327 T^{14} - 13 T^{15} + T^{16}$$
$67$ $$220463104 + 58916864 T + 153059328 T^{2} - 16384000 T^{3} + 62404096 T^{4} - 8507520 T^{5} + 14368400 T^{6} - 2766032 T^{7} + 2213844 T^{8} - 248132 T^{9} + 132053 T^{10} - 1917 T^{11} + 5287 T^{12} - 28 T^{13} + 87 T^{14} + 2 T^{15} + T^{16}$$
$71$ $$( -712576 + 651456 T + 99968 T^{2} - 147712 T^{3} + 12656 T^{4} + 2476 T^{5} - 240 T^{6} - 10 T^{7} + T^{8} )^{2}$$
$73$ $$80020528641 + 201916201410 T + 392949733221 T^{2} + 333665452260 T^{3} + 227204065171 T^{4} + 9627477119 T^{5} + 14766096571 T^{6} + 633170743 T^{7} + 678214162 T^{8} - 556465 T^{9} + 11540849 T^{10} + 445337 T^{11} + 109849 T^{12} + 1807 T^{13} + 382 T^{14} + 5 T^{15} + T^{16}$$
$79$ $$4275962265600 + 4808373166080 T + 5357447737344 T^{2} + 792686407680 T^{3} + 437752829952 T^{4} + 54180002304 T^{5} + 25006518528 T^{6} + 2285005056 T^{7} + 651124096 T^{8} + 38053504 T^{9} + 11120720 T^{10} + 573264 T^{11} + 91808 T^{12} + 3288 T^{13} + 476 T^{14} + 16 T^{15} + T^{16}$$
$83$ $$( 95768 + 192580 T - 42608 T^{2} - 86413 T^{3} - 9059 T^{4} + 2854 T^{5} + 626 T^{6} + 43 T^{7} + T^{8} )^{2}$$
$89$ $$262799769600 - 381601013760 T + 397297115136 T^{2} - 222612744192 T^{3} + 98733743104 T^{4} - 26523368960 T^{5} + 6626135296 T^{6} - 973843200 T^{7} + 237230592 T^{8} - 23344960 T^{9} + 5881680 T^{10} + 99136 T^{11} + 63648 T^{12} + 696 T^{13} + 328 T^{14} + 8 T^{15} + T^{16}$$
$97$ $$( 580864 - 820352 T + 135104 T^{2} + 117344 T^{3} + 4944 T^{4} - 2336 T^{5} - 168 T^{6} + 12 T^{7} + T^{8} )^{2}$$