# Properties

 Label 8034.2.a.bb Level $8034$ Weight $2$ Character orbit 8034.a Self dual yes Analytic conductor $64.152$ Analytic rank $1$ Dimension $14$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8034 = 2 \cdot 3 \cdot 13 \cdot 103$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8034.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.1518129839$$ Analytic rank: $$1$$ Dimension: $$14$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ Defining polynomial: $$x^{14} - 6 x^{13} - 29 x^{12} + 207 x^{11} + 269 x^{10} - 2601 x^{9} - 847 x^{8} + 14851 x^{7} + 678 x^{6} - 39390 x^{5} - 3280 x^{4} + 42456 x^{3} + 10816 x^{2} - 7296 x - 2048$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{13}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} -\beta_{1} q^{5} - q^{6} + \beta_{6} q^{7} - q^{8} + q^{9} +O(q^{10})$$ $$q - q^{2} + q^{3} + q^{4} -\beta_{1} q^{5} - q^{6} + \beta_{6} q^{7} - q^{8} + q^{9} + \beta_{1} q^{10} + ( -1 + \beta_{9} ) q^{11} + q^{12} - q^{13} -\beta_{6} q^{14} -\beta_{1} q^{15} + q^{16} + ( -1 - \beta_{6} - \beta_{10} ) q^{17} - q^{18} + ( \beta_{1} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{19} -\beta_{1} q^{20} + \beta_{6} q^{21} + ( 1 - \beta_{9} ) q^{22} + ( -1 - \beta_{5} - \beta_{6} ) q^{23} - q^{24} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{11} + \beta_{13} ) q^{25} + q^{26} + q^{27} + \beta_{6} q^{28} + ( -\beta_{7} + \beta_{11} - \beta_{12} ) q^{29} + \beta_{1} q^{30} + ( \beta_{2} + \beta_{5} + \beta_{6} + \beta_{11} ) q^{31} - q^{32} + ( -1 + \beta_{9} ) q^{33} + ( 1 + \beta_{6} + \beta_{10} ) q^{34} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{9} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{35} + q^{36} + ( -1 + \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} - \beta_{11} + \beta_{12} ) q^{37} + ( -\beta_{1} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{38} - q^{39} + \beta_{1} q^{40} + ( -3 + \beta_{1} - \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} ) q^{41} -\beta_{6} q^{42} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} - \beta_{13} ) q^{43} + ( -1 + \beta_{9} ) q^{44} -\beta_{1} q^{45} + ( 1 + \beta_{5} + \beta_{6} ) q^{46} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} - \beta_{13} ) q^{47} + q^{48} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{10} - \beta_{11} ) q^{49} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{11} - \beta_{13} ) q^{50} + ( -1 - \beta_{6} - \beta_{10} ) q^{51} - q^{52} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} - \beta_{13} ) q^{53} - q^{54} + ( -1 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{12} + \beta_{13} ) q^{55} -\beta_{6} q^{56} + ( \beta_{1} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{57} + ( \beta_{7} - \beta_{11} + \beta_{12} ) q^{58} + ( 1 - \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{13} ) q^{59} -\beta_{1} q^{60} + ( -1 + \beta_{1} + \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} ) q^{61} + ( -\beta_{2} - \beta_{5} - \beta_{6} - \beta_{11} ) q^{62} + \beta_{6} q^{63} + q^{64} + \beta_{1} q^{65} + ( 1 - \beta_{9} ) q^{66} + ( \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{11} - \beta_{12} - 2 \beta_{13} ) q^{67} + ( -1 - \beta_{6} - \beta_{10} ) q^{68} + ( -1 - \beta_{5} - \beta_{6} ) q^{69} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{9} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{70} + ( -2 + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{8} - 3 \beta_{9} - \beta_{11} + 3 \beta_{12} + \beta_{13} ) q^{71} - q^{72} + ( -2 + 2 \beta_{1} - \beta_{4} - \beta_{6} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} ) q^{73} + ( 1 - \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} + \beta_{11} - \beta_{12} ) q^{74} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{11} + \beta_{13} ) q^{75} + ( \beta_{1} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{76} + ( -\beta_{7} - \beta_{8} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{77} + q^{78} + ( -1 + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{7} - 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{79} -\beta_{1} q^{80} + q^{81} + ( 3 - \beta_{1} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} ) q^{82} + ( -3 + \beta_{1} + \beta_{3} + \beta_{8} - \beta_{9} - \beta_{11} + \beta_{13} ) q^{83} + \beta_{6} q^{84} + ( -1 - \beta_{2} + 3 \beta_{3} - \beta_{4} - 3 \beta_{5} + 2 \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{13} ) q^{85} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} + \beta_{13} ) q^{86} + ( -\beta_{7} + \beta_{11} - \beta_{12} ) q^{87} + ( 1 - \beta_{9} ) q^{88} + ( -3 + \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{11} + \beta_{12} ) q^{89} + \beta_{1} q^{90} -\beta_{6} q^{91} + ( -1 - \beta_{5} - \beta_{6} ) q^{92} + ( \beta_{2} + \beta_{5} + \beta_{6} + \beta_{11} ) q^{93} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{8} - \beta_{9} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{94} + ( -2 + \beta_{2} - 2 \beta_{3} - \beta_{4} + 3 \beta_{5} + \beta_{6} + \beta_{7} - 3 \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} - 2 \beta_{13} ) q^{95} - q^{96} + ( 3 - \beta_{1} + \beta_{4} - \beta_{7} + \beta_{9} + 2 \beta_{10} - 2 \beta_{12} ) q^{97} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} - \beta_{10} + \beta_{11} ) q^{98} + ( -1 + \beta_{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14q - 14q^{2} + 14q^{3} + 14q^{4} - 6q^{5} - 14q^{6} - 4q^{7} - 14q^{8} + 14q^{9} + O(q^{10})$$ $$14q - 14q^{2} + 14q^{3} + 14q^{4} - 6q^{5} - 14q^{6} - 4q^{7} - 14q^{8} + 14q^{9} + 6q^{10} - 8q^{11} + 14q^{12} - 14q^{13} + 4q^{14} - 6q^{15} + 14q^{16} - 4q^{17} - 14q^{18} - q^{19} - 6q^{20} - 4q^{21} + 8q^{22} - 9q^{23} - 14q^{24} + 24q^{25} + 14q^{26} + 14q^{27} - 4q^{28} - 10q^{29} + 6q^{30} - 5q^{31} - 14q^{32} - 8q^{33} + 4q^{34} - 16q^{35} + 14q^{36} - 4q^{37} + q^{38} - 14q^{39} + 6q^{40} - 24q^{41} + 4q^{42} - 8q^{44} - 6q^{45} + 9q^{46} - 32q^{47} + 14q^{48} + 24q^{49} - 24q^{50} - 4q^{51} - 14q^{52} - 5q^{53} - 14q^{54} - 8q^{55} + 4q^{56} - q^{57} + 10q^{58} - 13q^{59} - 6q^{60} + 2q^{61} + 5q^{62} - 4q^{63} + 14q^{64} + 6q^{65} + 8q^{66} - 16q^{67} - 4q^{68} - 9q^{69} + 16q^{70} - 29q^{71} - 14q^{72} + 4q^{74} + 24q^{75} - q^{76} - 9q^{77} + 14q^{78} - 21q^{79} - 6q^{80} + 14q^{81} + 24q^{82} - 40q^{83} - 4q^{84} - 7q^{85} - 10q^{87} + 8q^{88} - 48q^{89} + 6q^{90} + 4q^{91} - 9q^{92} - 5q^{93} + 32q^{94} - 26q^{95} - 14q^{96} + 18q^{97} - 24q^{98} - 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} - 6 x^{13} - 29 x^{12} + 207 x^{11} + 269 x^{10} - 2601 x^{9} - 847 x^{8} + 14851 x^{7} + 678 x^{6} - 39390 x^{5} - 3280 x^{4} + 42456 x^{3} + 10816 x^{2} - 7296 x - 2048$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$17211124339 \nu^{13} + 48838930211 \nu^{12} - 1083388884247 \nu^{11} - 1278700159484 \nu^{10} + 21896289725388 \nu^{9} + 8353374678764 \nu^{8} - 175798632679800 \nu^{7} + 21533806125636 \nu^{6} + 467406199485803 \nu^{5} - 351416224121502 \nu^{4} + 243900222298282 \nu^{3} + 769318123450260 \nu^{2} - 1166460932605168 \nu + 118321024199296$$$$)/ 165587393151104$$ $$\beta_{3}$$ $$=$$ $$($$$$-88374380939 \nu^{13} + 341475795883 \nu^{12} + 3738713297463 \nu^{11} - 13680291423778 \nu^{10} - 60950374089966 \nu^{9} + 203902612897310 \nu^{8} + 480284475172790 \nu^{7} - 1384162777559138 \nu^{6} - 1866871026854413 \nu^{5} + 4166130863706398 \nu^{4} + 3191603596475186 \nu^{3} - 4474034344329852 \nu^{2} - 1493240738430736 \nu + 823088103896320$$$$)/ 165587393151104$$ $$\beta_{4}$$ $$=$$ $$($$$$-406907271083 \nu^{13} + 2304692170158 \nu^{12} + 11004973389543 \nu^{11} - 75016193619169 \nu^{10} - 82526311620203 \nu^{9} + 861495599563087 \nu^{8} + 21324319463753 \nu^{7} - 4294335680262885 \nu^{6} + 1434806545518450 \nu^{5} + 9639491681032898 \nu^{4} - 3273111940991512 \nu^{3} - 10523459905050216 \nu^{2} + 2930798928626144 \nu + 3910063812049920$$$$)/ 662349572604416$$ $$\beta_{5}$$ $$=$$ $$($$$$113401884803 \nu^{13} - 883549115785 \nu^{12} - 2323893522087 \nu^{11} + 29836393456848 \nu^{10} - 852019922224 \nu^{9} - 364296872907848 \nu^{8} + 263847579785132 \nu^{7} + 2001792019065248 \nu^{6} - 1727833814125777 \nu^{5} - 5028642484757934 \nu^{4} + 3738614769502962 \nu^{3} + 5009595167008196 \nu^{2} - 2681914257270768 \nu - 1049016746216704$$$$)/ 165587393151104$$ $$\beta_{6}$$ $$=$$ $$($$$$736531555801 \nu^{13} - 3946026327910 \nu^{12} - 23999796454509 \nu^{11} + 138185242548231 \nu^{10} + 283131019369069 \nu^{9} - 1765360355157129 \nu^{8} - 1550184318259343 \nu^{7} + 10222081364164867 \nu^{6} + 4328705466068094 \nu^{5} - 27089707111959846 \nu^{4} - 6353469675911824 \nu^{3} + 27368166174349864 \nu^{2} + 3859293469715616 \nu - 3240111054661120$$$$)/ 662349572604416$$ $$\beta_{7}$$ $$=$$ $$($$$$1012895827421 \nu^{13} - 5401357076630 \nu^{12} - 30269645318433 \nu^{11} + 182412406997691 \nu^{10} + 294059953736009 \nu^{9} - 2216963035494389 \nu^{8} - 971532240127523 \nu^{7} + 11978958293225447 \nu^{6} + 149793013139294 \nu^{5} - 28653365696975870 \nu^{4} + 2303549139588352 \nu^{3} + 24319897537607016 \nu^{2} - 796238968943584 \nu - 804113198053376$$$$)/ 662349572604416$$ $$\beta_{8}$$ $$=$$ $$($$$$-161946406606 \nu^{13} + 993100465781 \nu^{12} + 4568699998010 \nu^{11} - 33836085473159 \nu^{10} - 39248816007673 \nu^{9} + 415409487608205 \nu^{8} + 80934994695967 \nu^{7} - 2274108863937751 \nu^{6} + 260702734356941 \nu^{5} + 5633078350696404 \nu^{4} - 672606967463486 \nu^{3} - 5548224775659980 \nu^{2} - 339551833683136 \nu + 821129125068864$$$$)/ 82793696575552$$ $$\beta_{9}$$ $$=$$ $$($$$$-339042867519 \nu^{13} + 2193728871065 \nu^{12} + 8513285647275 \nu^{11} - 73385354065404 \nu^{10} - 47754006020300 \nu^{9} + 882287181466036 \nu^{8} - 215439634992760 \nu^{7} - 4744161357889636 \nu^{6} + 2237814174755737 \nu^{5} + 11755366222327694 \nu^{4} - 3969078366908066 \nu^{3} - 12224135566363796 \nu^{2} + 525183514160720 \nu + 2100534074540032$$$$)/ 165587393151104$$ $$\beta_{10}$$ $$=$$ $$($$$$-2206595272859 \nu^{13} + 14412109193718 \nu^{12} + 54849889967511 \nu^{11} - 474790047743417 \nu^{10} - 307896247879923 \nu^{9} + 5544743002953591 \nu^{8} - 1212936140282751 \nu^{7} - 28137851717749085 \nu^{6} + 12427081549409178 \nu^{5} + 62275142320830146 \nu^{4} - 19724650983886152 \nu^{3} - 53935123111369352 \nu^{2} + 943866195323488 \nu + 6787275004993536$$$$)/ 662349572604416$$ $$\beta_{11}$$ $$=$$ $$($$$$-2312493819663 \nu^{13} + 15946662074178 \nu^{12} + 56053557441147 \nu^{11} - 536833570831273 \nu^{10} - 265156769412035 \nu^{9} + 6499156388583719 \nu^{8} - 2119780149955167 \nu^{7} - 35077355093355085 \nu^{6} + 18558566709942790 \nu^{5} + 85869291262886602 \nu^{4} - 35303846107753504 \nu^{3} - 83761116115370808 \nu^{2} + 9764388104189088 \nu + 10136073926334976$$$$)/ 662349572604416$$ $$\beta_{12}$$ $$=$$ $$($$$$-155979666211 \nu^{13} + 1092105641867 \nu^{12} + 3542117432817 \nu^{11} - 36139437565114 \nu^{10} - 10207803685344 \nu^{9} + 425935692445960 \nu^{8} - 227141704110620 \nu^{7} - 2204586185386584 \nu^{6} + 1611592294003569 \nu^{5} + 5085371693596568 \nu^{4} - 2867839136841206 \nu^{3} - 4698153606819416 \nu^{2} + 774956085111656 \nu + 564375014073344$$$$)/ 41396848287776$$ $$\beta_{13}$$ $$=$$ $$($$$$2734835840775 \nu^{13} - 17117209536866 \nu^{12} - 75341966167987 \nu^{11} + 586439935888449 \nu^{10} + 596543424673451 \nu^{9} - 7281353341457903 \nu^{8} - 504552281455193 \nu^{7} + 40700141428094181 \nu^{6} - 9221457804581926 \nu^{5} - 103939479614198202 \nu^{4} + 23513032611045888 \nu^{3} + 105396875559095672 \nu^{2} - 9119618453491232 \nu - 17591589253353984$$$$)/ 662349572604416$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{13} + \beta_{11} + \beta_{3} - \beta_{2} + \beta_{1} + 7$$ $$\nu^{3}$$ $$=$$ $$\beta_{13} - 2 \beta_{12} + 3 \beta_{11} + \beta_{8} - \beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{4} + 10 \beta_{1} + 7$$ $$\nu^{4}$$ $$=$$ $$13 \beta_{13} - 4 \beta_{12} + 19 \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + 5 \beta_{6} + 4 \beta_{5} + 4 \beta_{4} + 14 \beta_{3} - 14 \beta_{2} + 17 \beta_{1} + 88$$ $$\nu^{5}$$ $$=$$ $$26 \beta_{13} - 37 \beta_{12} + 65 \beta_{11} - 3 \beta_{10} - 5 \beta_{9} + 18 \beta_{8} - 20 \beta_{7} + 33 \beta_{6} + 16 \beta_{5} + 31 \beta_{4} + 5 \beta_{3} - 2 \beta_{2} + 123 \beta_{1} + 148$$ $$\nu^{6}$$ $$=$$ $$167 \beta_{13} - 103 \beta_{12} + 335 \beta_{11} + 18 \beta_{10} - 41 \beta_{9} - 37 \beta_{8} - 17 \beta_{7} + 111 \beta_{6} + 113 \beta_{5} + 108 \beta_{4} + 177 \beta_{3} - 177 \beta_{2} + 276 \beta_{1} + 1263$$ $$\nu^{7}$$ $$=$$ $$479 \beta_{13} - 611 \beta_{12} + 1200 \beta_{11} - 93 \beta_{10} - 162 \beta_{9} + 242 \beta_{8} - 350 \beta_{7} + 483 \beta_{6} + 315 \beta_{5} + 679 \beta_{4} + 107 \beta_{3} - 18 \beta_{2} + 1682 \beta_{1} + 2702$$ $$\nu^{8}$$ $$=$$ $$2222 \beta_{13} - 2060 \beta_{12} + 5775 \beta_{11} + 203 \beta_{10} - 1006 \beta_{9} - 885 \beta_{8} - 542 \beta_{7} + 1884 \beta_{6} + 2415 \beta_{5} + 2350 \beta_{4} + 2137 \beta_{3} - 2159 \beta_{2} + 4443 \beta_{1} + 19087$$ $$\nu^{9}$$ $$=$$ $$7842 \beta_{13} - 10023 \beta_{12} + 21287 \beta_{11} - 2124 \beta_{10} - 3977 \beta_{9} + 2676 \beta_{8} - 5991 \beta_{7} + 6985 \beta_{6} + 6668 \beta_{5} + 13401 \beta_{4} + 1459 \beta_{3} + 285 \beta_{2} + 24370 \beta_{1} + 47413$$ $$\nu^{10}$$ $$=$$ $$30616 \beta_{13} - 37773 \beta_{12} + 98893 \beta_{11} + 986 \beta_{10} - 21437 \beta_{9} - 18052 \beta_{8} - 12588 \beta_{7} + 29338 \beta_{6} + 46972 \beta_{5} + 47547 \beta_{4} + 24544 \beta_{3} - 25202 \beta_{2} + 71435 \beta_{1} + 296729$$ $$\nu^{11}$$ $$=$$ $$122139 \beta_{13} - 166504 \beta_{12} + 373041 \beta_{11} - 43026 \beta_{10} - 86704 \beta_{9} + 20835 \beta_{8} - 102258 \beta_{7} + 101217 \beta_{6} + 138595 \beta_{5} + 252965 \beta_{4} + 12729 \beta_{3} + 16621 \beta_{2} + 365621 \beta_{1} + 819867$$ $$\nu^{12}$$ $$=$$ $$434104 \beta_{13} - 667453 \beta_{12} + 1693145 \beta_{11} - 27689 \beta_{10} - 430192 \beta_{9} - 343448 \beta_{8} - 259336 \beta_{7} + 440956 \beta_{6} + 876463 \beta_{5} + 925564 \beta_{4} + 258691 \beta_{3} - 270723 \beta_{2} + 1151467 \beta_{1} + 4704078$$ $$\nu^{13}$$ $$=$$ $$1860057 \beta_{13} - 2802582 \beta_{12} + 6513435 \beta_{11} - 823424 \beta_{10} - 1769865 \beta_{9} - 35291 \beta_{8} - 1747648 \beta_{7} + 1471030 \beta_{6} + 2780674 \beta_{5} + 4665904 \beta_{4} - 25839 \beta_{3} + 487738 \beta_{2} + 5617405 \beta_{1} + 14098951$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 4.19123 3.81560 3.75223 2.57230 2.22430 1.73527 0.453790 −0.290663 −0.480706 −1.34813 −1.50270 −2.36221 −3.07028 −3.69005
−1.00000 1.00000 1.00000 −4.19123 −1.00000 −2.48485 −1.00000 1.00000 4.19123
1.2 −1.00000 1.00000 1.00000 −3.81560 −1.00000 4.94203 −1.00000 1.00000 3.81560
1.3 −1.00000 1.00000 1.00000 −3.75223 −1.00000 1.91373 −1.00000 1.00000 3.75223
1.4 −1.00000 1.00000 1.00000 −2.57230 −1.00000 −4.81484 −1.00000 1.00000 2.57230
1.5 −1.00000 1.00000 1.00000 −2.22430 −1.00000 1.37131 −1.00000 1.00000 2.22430
1.6 −1.00000 1.00000 1.00000 −1.73527 −1.00000 −3.07502 −1.00000 1.00000 1.73527
1.7 −1.00000 1.00000 1.00000 −0.453790 −1.00000 3.87720 −1.00000 1.00000 0.453790
1.8 −1.00000 1.00000 1.00000 0.290663 −1.00000 −3.15490 −1.00000 1.00000 −0.290663
1.9 −1.00000 1.00000 1.00000 0.480706 −1.00000 0.765483 −1.00000 1.00000 −0.480706
1.10 −1.00000 1.00000 1.00000 1.34813 −1.00000 2.71387 −1.00000 1.00000 −1.34813
1.11 −1.00000 1.00000 1.00000 1.50270 −1.00000 0.210711 −1.00000 1.00000 −1.50270
1.12 −1.00000 1.00000 1.00000 2.36221 −1.00000 −4.26027 −1.00000 1.00000 −2.36221
1.13 −1.00000 1.00000 1.00000 3.07028 −1.00000 −1.20257 −1.00000 1.00000 −3.07028
1.14 −1.00000 1.00000 1.00000 3.69005 −1.00000 −0.801879 −1.00000 1.00000 −3.69005
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$13$$ $$1$$
$$103$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8034.2.a.bb 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.bb 14 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8034))$$:

 $$T_{5}^{14} + \cdots$$ $$T_{7}^{14} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{14}$$
$3$ $$( -1 + T )^{14}$$
$5$ $$-2048 + 7296 T + 10816 T^{2} - 42456 T^{3} - 3280 T^{4} + 39390 T^{5} + 678 T^{6} - 14851 T^{7} - 847 T^{8} + 2601 T^{9} + 269 T^{10} - 207 T^{11} - 29 T^{12} + 6 T^{13} + T^{14}$$
$7$ $$-10496 + 47936 T + 37432 T^{2} - 129678 T^{3} - 49359 T^{4} + 101391 T^{5} + 27271 T^{6} - 30365 T^{7} - 7497 T^{8} + 3920 T^{9} + 963 T^{10} - 214 T^{11} - 53 T^{12} + 4 T^{13} + T^{14}$$
$11$ $$-234648 - 602086 T - 326372 T^{2} + 368853 T^{3} + 428958 T^{4} + 10126 T^{5} - 139847 T^{6} - 41170 T^{7} + 14463 T^{8} + 7608 T^{9} - 92 T^{10} - 451 T^{11} - 39 T^{12} + 8 T^{13} + T^{14}$$
$13$ $$( 1 + T )^{14}$$
$17$ $$-309504 + 999040 T + 11295184 T^{2} - 6574000 T^{3} - 6921032 T^{4} + 2526320 T^{5} + 1504661 T^{6} - 313111 T^{7} - 138347 T^{8} + 17117 T^{9} + 6066 T^{10} - 429 T^{11} - 126 T^{12} + 4 T^{13} + T^{14}$$
$19$ $$1835136 + 6635520 T + 494640 T^{2} - 15053904 T^{3} - 5989872 T^{4} + 5363056 T^{5} + 2045895 T^{6} - 575279 T^{7} - 224110 T^{8} + 20191 T^{9} + 9215 T^{10} - 264 T^{11} - 160 T^{12} + T^{13} + T^{14}$$
$23$ $$-3987232 - 11441704 T - 5814820 T^{2} + 7039194 T^{3} + 6157233 T^{4} - 742315 T^{5} - 1732985 T^{6} - 253390 T^{7} + 155028 T^{8} + 49205 T^{9} + 469 T^{10} - 1297 T^{11} - 104 T^{12} + 9 T^{13} + T^{14}$$
$29$ $$-151832576 - 537265152 T - 487211056 T^{2} - 75467136 T^{3} + 79563240 T^{4} + 27253336 T^{5} - 3787783 T^{6} - 2336094 T^{7} - 3160 T^{8} + 88288 T^{9} + 4977 T^{10} - 1541 T^{11} - 130 T^{12} + 10 T^{13} + T^{14}$$
$31$ $$-4772992 - 35737184 T + 88449472 T^{2} - 28157220 T^{3} - 34636918 T^{4} + 12770236 T^{5} + 4670089 T^{6} - 1411635 T^{7} - 328090 T^{8} + 59669 T^{9} + 11689 T^{10} - 1000 T^{11} - 188 T^{12} + 5 T^{13} + T^{14}$$
$37$ $$1682701888 - 689045036 T - 1706500494 T^{2} - 42060906 T^{3} + 507556633 T^{4} + 161750034 T^{5} - 5998037 T^{6} - 8925114 T^{7} - 682595 T^{8} + 177023 T^{9} + 22411 T^{10} - 1455 T^{11} - 253 T^{12} + 4 T^{13} + T^{14}$$
$41$ $$104915276544 + 210419648240 T + 160608782240 T^{2} + 55236159984 T^{3} + 5799095716 T^{4} - 1505759727 T^{5} - 452716791 T^{6} - 15572283 T^{7} + 7404948 T^{8} + 788646 T^{9} - 31888 T^{10} - 7806 T^{11} - 144 T^{12} + 24 T^{13} + T^{14}$$
$43$ $$37069568 + 70439872 T + 16863824 T^{2} - 38574300 T^{3} - 21067316 T^{4} + 4789673 T^{5} + 4301139 T^{6} - 172893 T^{7} - 362017 T^{8} - 398 T^{9} + 14082 T^{10} + 72 T^{11} - 231 T^{12} + T^{14}$$
$47$ $$23612928 + 58066448 T + 36697796 T^{2} - 17094326 T^{3} - 26841546 T^{4} - 4954957 T^{5} + 4665832 T^{6} + 2080227 T^{7} - 33711 T^{8} - 169430 T^{9} - 33117 T^{10} - 1220 T^{11} + 279 T^{12} + 32 T^{13} + T^{14}$$
$53$ $$74233155072 + 31262083840 T - 22857178848 T^{2} - 11271279872 T^{3} + 1406327172 T^{4} + 1126459464 T^{5} + 25520409 T^{6} - 41977199 T^{7} - 3510817 T^{8} + 559384 T^{9} + 64390 T^{10} - 2931 T^{11} - 432 T^{12} + 5 T^{13} + T^{14}$$
$59$ $$10577117773824 + 3245690564608 T - 750693961216 T^{2} - 303345802112 T^{3} + 12764059744 T^{4} + 10785808360 T^{5} + 189623660 T^{6} - 187669387 T^{7} - 8525140 T^{8} + 1691978 T^{9} + 103042 T^{10} - 7533 T^{11} - 531 T^{12} + 13 T^{13} + T^{14}$$
$61$ $$67974144 + 939806848 T + 1947371232 T^{2} + 320249560 T^{3} - 440159950 T^{4} - 72651675 T^{5} + 38946361 T^{6} + 4678203 T^{7} - 1693865 T^{8} - 115944 T^{9} + 36162 T^{10} + 920 T^{11} - 323 T^{12} - 2 T^{13} + T^{14}$$
$67$ $$-6677658624 + 10250350976 T + 3573547808 T^{2} - 6381396436 T^{3} - 427069762 T^{4} + 1177421794 T^{5} - 14260867 T^{6} - 62606044 T^{7} - 1724665 T^{8} + 1101777 T^{9} + 53045 T^{10} - 7264 T^{11} - 419 T^{12} + 16 T^{13} + T^{14}$$
$71$ $$3192115560448 - 2336753307648 T - 1445743302144 T^{2} + 7255143360 T^{3} + 84312073272 T^{4} + 6541584446 T^{5} - 1782859060 T^{6} - 215713695 T^{7} + 14803498 T^{8} + 2701517 T^{9} - 20607 T^{10} - 14681 T^{11} - 289 T^{12} + 29 T^{13} + T^{14}$$
$73$ $$13746176 - 132041920 T - 324289392 T^{2} - 122136008 T^{3} + 157604410 T^{4} + 128697076 T^{5} + 16253503 T^{6} - 9253912 T^{7} - 2230314 T^{8} + 135777 T^{9} + 53039 T^{10} - 544 T^{11} - 428 T^{12} + T^{14}$$
$79$ $$562577408 - 13787470848 T - 26954790400 T^{2} - 11364085200 T^{3} + 2495914288 T^{4} + 1498225040 T^{5} - 55579368 T^{6} - 63394423 T^{7} - 642631 T^{8} + 1107441 T^{9} + 33309 T^{10} - 8138 T^{11} - 331 T^{12} + 21 T^{13} + T^{14}$$
$83$ $$-663287808 + 632765440 T + 2448060288 T^{2} + 790731584 T^{3} - 854959776 T^{4} - 429828848 T^{5} + 26106754 T^{6} + 37073803 T^{7} + 4095164 T^{8} - 561640 T^{9} - 136595 T^{10} - 6785 T^{11} + 327 T^{12} + 40 T^{13} + T^{14}$$
$89$ $$-416381859328 + 570119779712 T - 181902448744 T^{2} - 34923221168 T^{3} + 20456661694 T^{4} + 510298292 T^{5} - 810875313 T^{6} - 7850362 T^{7} + 15945248 T^{8} + 425299 T^{9} - 154083 T^{10} - 8194 T^{11} + 500 T^{12} + 48 T^{13} + T^{14}$$
$97$ $$6667509669504 + 10839648155392 T + 4881027050800 T^{2} + 298826499840 T^{3} - 207866858560 T^{4} - 23460577536 T^{5} + 3742450687 T^{6} + 429329488 T^{7} - 38299027 T^{8} - 3458315 T^{9} + 232331 T^{10} + 12864 T^{11} - 757 T^{12} - 18 T^{13} + T^{14}$$