# Properties

 Label 4025.2.a.ba Level $4025$ Weight $2$ Character orbit 4025.a Self dual yes Analytic conductor $32.140$ Analytic rank $1$ Dimension $14$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4025 = 5^{2} \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4025.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.1397868136$$ Analytic rank: $$1$$ Dimension: $$14$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ Defining polynomial: $$x^{14} - x^{13} - 22 x^{12} + 18 x^{11} + 187 x^{10} - 118 x^{9} - 772 x^{8} + 346 x^{7} + 1581 x^{6} - 443 x^{5} - 1429 x^{4} + 193 x^{3} + 386 x^{2} - 3 x - 5$$ x^14 - x^13 - 22*x^12 + 18*x^11 + 187*x^10 - 118*x^9 - 772*x^8 + 346*x^7 + 1581*x^6 - 443*x^5 - 1429*x^4 + 193*x^3 + 386*x^2 - 3*x - 5 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3^{3}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{7} + \beta_1) q^{6} - q^{7} + ( - \beta_{3} - \beta_1 - 1) q^{8} + ( - \beta_{10} + \beta_{4} + 1) q^{9}+O(q^{10})$$ q - b1 * q^2 - b4 * q^3 + (b2 + 1) * q^4 + (b7 + b1) * q^6 - q^7 + (-b3 - b1 - 1) * q^8 + (-b10 + b4 + 1) * q^9 $$q - \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{7} + \beta_1) q^{6} - q^{7} + ( - \beta_{3} - \beta_1 - 1) q^{8} + ( - \beta_{10} + \beta_{4} + 1) q^{9} - \beta_{13} q^{11} + ( - \beta_{13} + \beta_{12} - 2 \beta_{9} + \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \cdots + 2 \beta_1) q^{12}+ \cdots + ( - \beta_{13} + \beta_{12} + 3 \beta_{8} + 2 \beta_{7} + \beta_{6} + 2 \beta_{4} - 2 \beta_{3} + \cdots - 2) q^{99}+O(q^{100})$$ q - b1 * q^2 - b4 * q^3 + (b2 + 1) * q^4 + (b7 + b1) * q^6 - q^7 + (-b3 - b1 - 1) * q^8 + (-b10 + b4 + 1) * q^9 - b13 * q^11 + (-b13 + b12 - 2*b9 + b7 + b6 - b5 - b4 - b3 - b2 + 2*b1) * q^12 + (-b9 + b6 - b2 + b1 - 1) * q^13 + b1 * q^14 + (b13 + b10 + 2*b9 - b7 + b4 + b3 + b2) * q^16 + (b13 - b12 + 2*b9 - b7 + b3 - b1 - 1) * q^17 + (-b7 - b6 + b4 + b3 - 2*b1 - 1) * q^18 + (b12 + b8 - b6 - b1) * q^19 + b4 * q^21 + (-b12 - b7 - b6 + 2*b5 - b4 - b2 - b1) * q^22 + q^23 + (2*b13 - b12 + 2*b9 - b8 - b7 - b6 + b5 + b4 + 2*b3) * q^24 + (-b12 + b11 + b3 - b2 + 2*b1 + 1) * q^26 + (b13 + 2*b10 + b9 - b7 - b6 + b5 - b4 - 3) * q^27 + (-b2 - 1) * q^28 + (-b12 - b8 - b7 + b4 + b2) * q^29 + (b12 - b11 - 2*b9 + 2*b7 - b5 + b4 - b3 - b2 + 2*b1) * q^31 + (b10 - 2*b9 + b6 - b5 - 2*b3 - b2 + b1 - 2) * q^32 + (b11 + b10 - b9 + b6 - b5 + b3 + b1 - 1) * q^33 + (b10 - b9 + b6 - b5 - b3 + 3*b1 - 1) * q^34 + (-b11 + b9 - b7 - b6 + b5 + b4 + b3 - b1) * q^36 + (-b10 + b9 - b8 + b7 - b6 + b1 - 1) * q^37 + (b13 + b12 - 2*b11 - b10 + 2*b9 - 2*b6 + b5 + 3*b4 + b3 + b2 - 2*b1 - 2) * q^38 + (-b13 - b11 + b10 - b9 + b5 - 2*b3 + b2 + b1 - 1) * q^39 + (-b9 - b8 - b6 + b5 - b2) * q^41 + (-b7 - b1) * q^42 + (b13 - b11 + 2*b9 - b8 - b7 - b6 - b5 + 3*b4 + 2*b3 + b2 - 2*b1 - 3) * q^43 + (-b13 + b11 - b10 + 2*b8 - b7 + b5 - b4 + 2*b3 - b2 - b1 + 1) * q^44 - b1 * q^46 + (b13 + b10 + b9 + 2*b8 - b7 + b5 + b3 - b2 - 1) * q^47 + (-2*b13 + b11 - b10 - 2*b9 + b8 + b6 - 2*b5 - b3 - b2 - 1) * q^48 + q^49 + (b13 - b12 + b11 + 2*b9 + b8 - 2*b7 - b6 + b5 + b4 + b3 - b2 - 3*b1 + 2) * q^51 + (-b13 - b10 - b8 - 2*b4 - 2*b2 + 2*b1 - 2) * q^52 + (b11 + b10 - b9 + b6 + 2*b5 - b4 - 2*b3 - b2 + b1 - 1) * q^53 + (-b13 + b12 + b8 + b7 + 2*b6 - b5 - 2*b3 + 3*b1 - 3) * q^54 + (b3 + b1 + 1) * q^56 + (b13 - 2*b10 + 2*b9 - b8 + b7 - 2*b5 + b4 + 2*b3 + b2 + 2*b1 + 2) * q^57 + (-b12 + b11 + b9 - 2*b7 + b6 + b4 - 3*b1 - 1) * q^58 + (-b12 + b11 + b10 - b8 - b7 - b5 + b3 + 2*b1 - 1) * q^59 + (2*b11 - b9 + 2*b8 + b7 + 2*b6 - 2*b2 + 2*b1 + 1) * q^61 + (b13 + 2*b12 - b11 - b10 + 2*b9 + b7 - 2*b5 + 3*b3 + b2 + b1 + 1) * q^62 + (b10 - b4 - 1) * q^63 + (2*b13 - b12 - b10 + 3*b9 - b8 + b6 + 2*b4 + 2*b3 + 2*b2 + b1) * q^64 + (b13 - b12 - b10 - b9 - 2*b8 + b7 + 2*b6 - 2*b3 - 2*b2 + 4*b1 + 1) * q^66 + (-b13 + b11 + b10 - b9 - b3 + b1 - 4) * q^67 + (b13 + b12 + b10 - b8 + b7 + b6 + 2*b4 - b3 + 2*b1 - 5) * q^68 - b4 * q^69 + (-b13 + 3*b12 - b11 - 2*b9 + 2*b7 + b6 - b5 + b4 - 2*b3 + 4*b1 - 1) * q^71 + (-2*b13 - b10 - b9 + 2*b8 + b7 + b5 - 2*b4 - 2*b3 - 2*b2 + 1) * q^72 + (-2*b12 + 2*b11 + b10 + b8 - b7 + b6 + 2*b5 - 2*b4 - b3 - 2*b1 - 1) * q^73 + (-2*b13 + 2*b12 - 4*b9 + 2*b7 + b6 - 2*b5 - 3*b4 - b3 - b2 + 6*b1 - 1) * q^74 + (-3*b13 + b12 - b11 - b10 - 3*b9 + b8 - b6 - 2*b5 - b3 - 2*b1) * q^76 + b13 * q^77 + (-b12 + b11 + b10 + 4*b9 + 2*b8 - 3*b7 - b6 + 2*b5 - b4 + b3 + b2 - 8*b1 - 3) * q^78 + (-b11 - 2*b8 - 2*b7 + 2*b4 - 2) * q^79 + (-b13 - b12 - 2*b10 - b9 - b8 + b7 + 2*b6 - 2*b5 + 4*b4 - b3 - b2 + b1 + 1) * q^81 + (-3*b13 + b12 + b11 - 2*b10 - 2*b9 + b8 + b7 + b6 - b5 - b4 + b3 - b2 + 2*b1 + 4) * q^82 + (-b13 + b12 - 2*b11 - 2*b10 + b7 - 2*b6 - b5 + b4 + b2 - b1 - 1) * q^83 + (b13 - b12 + 2*b9 - b7 - b6 + b5 + b4 + b3 + b2 - 2*b1) * q^84 + (b12 - b11 - b10 - 2*b9 - 2*b5 + 2*b4 - 2*b3 + 2*b1 + 2) * q^86 + (-b13 + b12 + 2*b10 - 2*b9 + b8 + b7 + 2*b6 + b5 - 2*b4 - 3*b3 - 2*b2 + 4*b1 - 5) * q^87 + (-b13 - b11 - 2*b10 - b9 + b7 - 2*b6 + b5 + 3*b4 + b3 - 3*b2 - 3) * q^88 + (2*b12 - b9 - b8 + 2*b7 - 2*b5 + b4 + b2 + 2*b1 - 3) * q^89 + (b9 - b6 + b2 - b1 + 1) * q^91 + (b2 + 1) * q^92 + (b13 + b12 - 3*b11 + b9 - b8 - 2*b6 + b5 + 2*b4 + b3 + 2*b2 - 4*b1 - 4) * q^93 + (-b11 + b9 + b8 - b7 - b6 + b5 + b4 - 2*b2 - b1 - 6) * q^94 + (2*b13 - b12 - 2*b11 + 4*b9 - b8 - 2*b7 - b6 + 3*b5 + 4*b4 + 2*b3 + 2*b2 - 2) * q^96 + (-3*b12 + b9 - 2*b7 - b6 + 2*b5 - b4 + b2 - b1 - 1) * q^97 - b1 * q^98 + (-b13 + b12 + 3*b8 + 2*b7 + b6 + 2*b4 - 2*b3 + 2*b2 - b1 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14 q - q^{2} - 6 q^{3} + 17 q^{4} - 4 q^{6} - 14 q^{7} - 9 q^{8} + 18 q^{9}+O(q^{10})$$ 14 * q - q^2 - 6 * q^3 + 17 * q^4 - 4 * q^6 - 14 * q^7 - 9 * q^8 + 18 * q^9 $$14 q - q^{2} - 6 q^{3} + 17 q^{4} - 4 q^{6} - 14 q^{7} - 9 q^{8} + 18 q^{9} - 3 q^{11} - 11 q^{12} - 15 q^{13} + q^{14} + 23 q^{16} - 9 q^{17} - 17 q^{18} - 4 q^{19} + 6 q^{21} - 9 q^{22} + 14 q^{23} + 10 q^{24} - 5 q^{26} - 33 q^{27} - 17 q^{28} + 11 q^{29} - q^{31} - 24 q^{32} - 26 q^{33} - 6 q^{34} + 13 q^{36} - 18 q^{37} + 6 q^{38} + 6 q^{39} - 7 q^{41} + 4 q^{42} - 18 q^{43} - 16 q^{44} - q^{46} - 10 q^{47} - 40 q^{48} + 14 q^{49} + 28 q^{51} - 46 q^{52} - 5 q^{53} - 24 q^{54} + 9 q^{56} + 26 q^{57} - 2 q^{58} - 24 q^{59} - 6 q^{61} + 16 q^{62} - 18 q^{63} + 29 q^{64} + 27 q^{66} - 61 q^{67} - 35 q^{68} - 6 q^{69} + 11 q^{71} - 12 q^{72} - 28 q^{73} - 49 q^{74} - 27 q^{76} + 3 q^{77} - 38 q^{78} + 6 q^{79} + 26 q^{81} + 14 q^{82} - 16 q^{83} + 11 q^{84} + 46 q^{86} - 61 q^{87} - 58 q^{88} - 39 q^{89} + 15 q^{91} + 17 q^{92} - 21 q^{93} - 74 q^{94} + 41 q^{96} - 19 q^{97} - q^{98} - 9 q^{99}+O(q^{100})$$ 14 * q - q^2 - 6 * q^3 + 17 * q^4 - 4 * q^6 - 14 * q^7 - 9 * q^8 + 18 * q^9 - 3 * q^11 - 11 * q^12 - 15 * q^13 + q^14 + 23 * q^16 - 9 * q^17 - 17 * q^18 - 4 * q^19 + 6 * q^21 - 9 * q^22 + 14 * q^23 + 10 * q^24 - 5 * q^26 - 33 * q^27 - 17 * q^28 + 11 * q^29 - q^31 - 24 * q^32 - 26 * q^33 - 6 * q^34 + 13 * q^36 - 18 * q^37 + 6 * q^38 + 6 * q^39 - 7 * q^41 + 4 * q^42 - 18 * q^43 - 16 * q^44 - q^46 - 10 * q^47 - 40 * q^48 + 14 * q^49 + 28 * q^51 - 46 * q^52 - 5 * q^53 - 24 * q^54 + 9 * q^56 + 26 * q^57 - 2 * q^58 - 24 * q^59 - 6 * q^61 + 16 * q^62 - 18 * q^63 + 29 * q^64 + 27 * q^66 - 61 * q^67 - 35 * q^68 - 6 * q^69 + 11 * q^71 - 12 * q^72 - 28 * q^73 - 49 * q^74 - 27 * q^76 + 3 * q^77 - 38 * q^78 + 6 * q^79 + 26 * q^81 + 14 * q^82 - 16 * q^83 + 11 * q^84 + 46 * q^86 - 61 * q^87 - 58 * q^88 - 39 * q^89 + 15 * q^91 + 17 * q^92 - 21 * q^93 - 74 * q^94 + 41 * q^96 - 19 * q^97 - q^98 - 9 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} - x^{13} - 22 x^{12} + 18 x^{11} + 187 x^{10} - 118 x^{9} - 772 x^{8} + 346 x^{7} + 1581 x^{6} - 443 x^{5} - 1429 x^{4} + 193 x^{3} + 386 x^{2} - 3 x - 5$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 5\nu - 1$$ v^3 - 5*v - 1 $$\beta_{4}$$ $$=$$ $$( - 134 \nu^{13} + 37911 \nu^{12} + 9431 \nu^{11} - 833971 \nu^{10} - 178425 \nu^{9} + 6807185 \nu^{8} + 1395439 \nu^{7} - 25174921 \nu^{6} - 5051683 \nu^{5} + \cdots + 1719962 ) / 1242873$$ (-134*v^13 + 37911*v^12 + 9431*v^11 - 833971*v^10 - 178425*v^9 + 6807185*v^8 + 1395439*v^7 - 25174921*v^6 - 5051683*v^5 + 40706850*v^4 + 8180960*v^3 - 22002987*v^2 - 4823059*v + 1719962) / 1242873 $$\beta_{5}$$ $$=$$ $$( - 8617 \nu^{13} - 22078 \nu^{12} + 273594 \nu^{11} + 349093 \nu^{10} - 3046784 \nu^{9} - 1722211 \nu^{8} + 15312112 \nu^{7} + 2054981 \nu^{6} - 35751429 \nu^{5} + \cdots + 208671 ) / 414291$$ (-8617*v^13 - 22078*v^12 + 273594*v^11 + 349093*v^10 - 3046784*v^9 - 1722211*v^8 + 15312112*v^7 + 2054981*v^6 - 35751429*v^5 + 4401504*v^4 + 34327336*v^3 - 7553195*v^2 - 8567221*v + 208671) / 414291 $$\beta_{6}$$ $$=$$ $$( 3427 \nu^{13} - 14217 \nu^{12} - 50538 \nu^{11} + 246032 \nu^{10} + 208975 \nu^{9} - 1485425 \nu^{8} + 41160 \nu^{7} + 3684384 \nu^{6} - 1876665 \nu^{5} + \cdots - 136436 ) / 138097$$ (3427*v^13 - 14217*v^12 - 50538*v^11 + 246032*v^10 + 208975*v^9 - 1485425*v^8 + 41160*v^7 + 3684384*v^6 - 1876665*v^5 - 3337746*v^4 + 3436136*v^3 + 746869*v^2 - 1764856*v - 136436) / 138097 $$\beta_{7}$$ $$=$$ $$( 37777 \nu^{13} + 6483 \nu^{12} - 831559 \nu^{11} - 153367 \nu^{10} + 6791373 \nu^{9} + 1291991 \nu^{8} - 25128557 \nu^{7} - 4839829 \nu^{6} + 40647488 \nu^{5} + \cdots - 670 ) / 1242873$$ (37777*v^13 + 6483*v^12 - 831559*v^11 - 153367*v^10 + 6791373*v^9 + 1291991*v^8 - 25128557*v^7 - 4839829*v^6 + 40647488*v^5 + 7989474*v^4 - 21977125*v^3 - 4771335*v^2 + 476687*v - 670) / 1242873 $$\beta_{8}$$ $$=$$ $$( - 17266 \nu^{13} + 28793 \nu^{12} + 353631 \nu^{11} - 477956 \nu^{10} - 2836277 \nu^{9} + 2649746 \nu^{8} + 11524948 \nu^{7} - 5168881 \nu^{6} + \cdots - 1259865 ) / 414291$$ (-17266*v^13 + 28793*v^12 + 353631*v^11 - 477956*v^10 - 2836277*v^9 + 2649746*v^8 + 11524948*v^7 - 5168881*v^6 - 25230657*v^5 + 449829*v^4 + 27355117*v^3 + 5168560*v^2 - 9402664*v - 1259865) / 414291 $$\beta_{9}$$ $$=$$ $$( 74561 \nu^{13} - 92562 \nu^{12} - 1571588 \nu^{11} + 1647493 \nu^{10} + 12628452 \nu^{9} - 10583948 \nu^{8} - 48372151 \nu^{7} + 29999800 \nu^{6} + \cdots + 3221839 ) / 1242873$$ (74561*v^13 - 92562*v^12 - 1571588*v^11 + 1647493*v^10 + 12628452*v^9 - 10583948*v^8 - 48372151*v^7 + 29999800*v^6 + 89700733*v^5 - 36175764*v^4 - 72909281*v^3 + 12237873*v^2 + 22678219*v + 3221839) / 1242873 $$\beta_{10}$$ $$=$$ $$( 92428 \nu^{13} - 123405 \nu^{12} - 1867552 \nu^{11} + 2127977 \nu^{10} + 14235777 \nu^{9} - 12965704 \nu^{8} - 51178406 \nu^{7} + 33005087 \nu^{6} + \cdots + 4568996 ) / 1242873$$ (92428*v^13 - 123405*v^12 - 1867552*v^11 + 2127977*v^10 + 14235777*v^9 - 12965704*v^8 - 51178406*v^7 + 33005087*v^6 + 87794291*v^5 - 29107302*v^4 - 61333048*v^3 - 3662787*v^2 + 6952400*v + 4568996) / 1242873 $$\beta_{11}$$ $$=$$ $$( - 114541 \nu^{13} + 106461 \nu^{12} + 2641672 \nu^{11} - 2117591 \nu^{10} - 23419830 \nu^{9} + 15754684 \nu^{8} + 99159767 \nu^{7} - 53798933 \nu^{6} + \cdots + 1997443 ) / 1242873$$ (-114541*v^13 + 106461*v^12 + 2641672*v^11 - 2117591*v^10 - 23419830*v^9 + 15754684*v^8 + 99159767*v^7 - 53798933*v^6 - 199755620*v^5 + 81640410*v^4 + 158892526*v^3 - 42632694*v^2 - 23191094*v + 1997443) / 1242873 $$\beta_{12}$$ $$=$$ $$( - 64550 \nu^{13} + 87137 \nu^{12} + 1331797 \nu^{11} - 1563334 \nu^{10} - 10454477 \nu^{9} + 10248993 \nu^{8} + 39148617 \nu^{7} - 30423831 \nu^{6} + \cdots + 1835824 ) / 414291$$ (-64550*v^13 + 87137*v^12 + 1331797*v^11 - 1563334*v^10 - 10454477*v^9 + 10248993*v^8 + 39148617*v^7 - 30423831*v^6 - 70995548*v^5 + 41000631*v^4 + 53938016*v^3 - 21439823*v^2 - 9671628*v + 1835824) / 414291 $$\beta_{13}$$ $$=$$ $$( - 203639 \nu^{13} + 277101 \nu^{12} + 4169738 \nu^{11} - 4742359 \nu^{10} - 32522883 \nu^{9} + 28618406 \nu^{8} + 121398712 \nu^{7} - 72669595 \nu^{6} + \cdots - 2790322 ) / 1242873$$ (-203639*v^13 + 277101*v^12 + 4169738*v^11 - 4742359*v^10 - 32522883*v^9 + 28618406*v^8 + 121398712*v^7 - 72669595*v^6 - 221496586*v^5 + 69984327*v^4 + 175750652*v^3 - 12281418*v^2 - 40794727*v - 2790322) / 1242873
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 5\beta _1 + 1$$ b3 + 5*b1 + 1 $$\nu^{4}$$ $$=$$ $$\beta_{13} + \beta_{10} + 2\beta_{9} - \beta_{7} + \beta_{4} + \beta_{3} + 7\beta_{2} + 14$$ b13 + b10 + 2*b9 - b7 + b4 + b3 + 7*b2 + 14 $$\nu^{5}$$ $$=$$ $$-\beta_{10} + 2\beta_{9} - \beta_{6} + \beta_{5} + 10\beta_{3} + \beta_{2} + 27\beta _1 + 10$$ -b10 + 2*b9 - b6 + b5 + 10*b3 + b2 + 27*b1 + 10 $$\nu^{6}$$ $$=$$ $$12 \beta_{13} - \beta_{12} + 9 \beta_{10} + 23 \beta_{9} - \beta_{8} - 10 \beta_{7} + \beta_{6} + 12 \beta_{4} + 12 \beta_{3} + 48 \beta_{2} + \beta _1 + 76$$ 12*b13 - b12 + 9*b10 + 23*b9 - b8 - 10*b7 + b6 + 12*b4 + 12*b3 + 48*b2 + b1 + 76 $$\nu^{7}$$ $$=$$ $$3 \beta_{13} - \beta_{12} - 2 \beta_{11} - 12 \beta_{10} + 29 \beta_{9} - 2 \beta_{7} - 13 \beta_{6} + 15 \beta_{5} + 5 \beta_{4} + 83 \beta_{3} + 13 \beta_{2} + 155 \beta _1 + 78$$ 3*b13 - b12 - 2*b11 - 12*b10 + 29*b9 - 2*b7 - 13*b6 + 15*b5 + 5*b4 + 83*b3 + 13*b2 + 155*b1 + 78 $$\nu^{8}$$ $$=$$ $$111 \beta_{13} - 14 \beta_{12} - 2 \beta_{11} + 67 \beta_{10} + 205 \beta_{9} - 17 \beta_{8} - 78 \beta_{7} + 12 \beta_{6} + 4 \beta_{5} + 109 \beta_{4} + 110 \beta_{3} + 333 \beta_{2} + 17 \beta _1 + 453$$ 111*b13 - 14*b12 - 2*b11 + 67*b10 + 205*b9 - 17*b8 - 78*b7 + 12*b6 + 4*b5 + 109*b4 + 110*b3 + 333*b2 + 17*b1 + 453 $$\nu^{9}$$ $$=$$ $$57 \beta_{13} - 21 \beta_{12} - 33 \beta_{11} - 107 \beta_{10} + 307 \beta_{9} - 6 \beta_{8} - 37 \beta_{7} - 129 \beta_{6} + 161 \beta_{5} + 88 \beta_{4} + 654 \beta_{3} + 130 \beta_{2} + 932 \beta _1 + 567$$ 57*b13 - 21*b12 - 33*b11 - 107*b10 + 307*b9 - 6*b8 - 37*b7 - 129*b6 + 161*b5 + 88*b4 + 654*b3 + 130*b2 + 932*b1 + 567 $$\nu^{10}$$ $$=$$ $$945 \beta_{13} - 149 \beta_{12} - 38 \beta_{11} + 476 \beta_{10} + 1692 \beta_{9} - 194 \beta_{8} - 571 \beta_{7} + 93 \beta_{6} + 83 \beta_{5} + 906 \beta_{4} + 931 \beta_{3} + 2344 \beta_{2} + 186 \beta _1 + 2876$$ 945*b13 - 149*b12 - 38*b11 + 476*b10 + 1692*b9 - 194*b8 - 571*b7 + 93*b6 + 83*b5 + 906*b4 + 931*b3 + 2344*b2 + 186*b1 + 2876 $$\nu^{11}$$ $$=$$ $$720 \beta_{13} - 281 \beta_{12} - 370 \beta_{11} - 854 \beta_{10} + 2872 \beta_{9} - 121 \beta_{8} - 444 \beta_{7} - 1155 \beta_{6} + 1513 \beta_{5} + 1043 \beta_{4} + 5055 \beta_{3} + 1186 \beta_{2} + 5815 \beta _1 + 4049$$ 720*b13 - 281*b12 - 370*b11 - 854*b10 + 2872*b9 - 121*b8 - 444*b7 - 1155*b6 + 1513*b5 + 1043*b4 + 5055*b3 + 1186*b2 + 5815*b1 + 4049 $$\nu^{12}$$ $$=$$ $$7758 \beta_{13} - 1431 \beta_{12} - 479 \beta_{11} + 3338 \beta_{10} + 13556 \beta_{9} - 1883 \beta_{8} - 4128 \beta_{7} + 546 \beta_{6} + 1117 \beta_{5} + 7294 \beta_{4} + 7643 \beta_{3} + 16718 \beta_{2} + \cdots + 19068$$ 7758*b13 - 1431*b12 - 479*b11 + 3338*b10 + 13556*b9 - 1883*b8 - 4128*b7 + 546*b6 + 1117*b5 + 7294*b4 + 7643*b3 + 16718*b2 + 1716*b1 + 19068 $$\nu^{13}$$ $$=$$ $$7632 \beta_{13} - 3084 \beta_{12} - 3546 \beta_{11} - 6459 \beta_{10} + 25222 \beta_{9} - 1596 \beta_{8} - 4431 \beta_{7} - 9803 \beta_{6} + 13271 \beta_{5} + 10489 \beta_{4} + 38763 \beta_{3} + 10361 \beta_{2} + \cdots + 28967$$ 7632*b13 - 3084*b12 - 3546*b11 - 6459*b10 + 25222*b9 - 1596*b8 - 4431*b7 - 9803*b6 + 13271*b5 + 10489*b4 + 38763*b3 + 10361*b2 + 37406*b1 + 28967

## 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
 2.79074 2.52898 2.17410 1.82692 1.31493 0.704319 0.117364 −0.116065 −0.582976 −1.25067 −1.76659 −1.83943 −2.25329 −2.64833
−2.79074 −2.29281 5.78823 0 6.39865 −1.00000 −10.5720 2.25700 0
1.2 −2.52898 1.80521 4.39572 0 −4.56532 −1.00000 −6.05871 0.258773 0
1.3 −2.17410 −1.37249 2.72672 0 2.98394 −1.00000 −1.57997 −1.11626 0
1.4 −1.82692 2.52976 1.33763 0 −4.62167 −1.00000 1.21009 3.39970 0
1.5 −1.31493 −3.36299 −0.270967 0 4.42209 −1.00000 2.98616 8.30972 0
1.6 −0.704319 2.54639 −1.50393 0 −1.79347 −1.00000 2.46789 3.48411 0
1.7 −0.117364 −0.701283 −1.98623 0 0.0823052 −1.00000 0.467839 −2.50820 0
1.8 0.116065 −1.59146 −1.98653 0 −0.184713 −1.00000 −0.462695 −0.467244 0
1.9 0.582976 0.367598 −1.66014 0 0.214301 −1.00000 −2.13377 −2.86487 0
1.10 1.25067 −3.28603 −0.435826 0 −4.10974 −1.00000 −3.04641 7.79801 0
1.11 1.76659 0.991358 1.12083 0 1.75132 −1.00000 −1.55314 −2.01721 0
1.12 1.83943 1.64563 1.38349 0 3.02702 −1.00000 −1.13403 −0.291898 0
1.13 2.25329 −2.73097 3.07734 0 −6.15367 −1.00000 2.42755 4.45818 0
1.14 2.64833 −0.547903 5.01366 0 −1.45103 −1.00000 7.98118 −2.69980 0
 $$n$$: 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
$$5$$ $$-1$$
$$7$$ $$1$$
$$23$$ $$-1$$

## 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 4025.2.a.ba 14
5.b even 2 1 4025.2.a.bb yes 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4025.2.a.ba 14 1.a even 1 1 trivial
4025.2.a.bb yes 14 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4025))$$:

 $$T_{2}^{14} + T_{2}^{13} - 22 T_{2}^{12} - 18 T_{2}^{11} + 187 T_{2}^{10} + 118 T_{2}^{9} - 772 T_{2}^{8} - 346 T_{2}^{7} + 1581 T_{2}^{6} + 443 T_{2}^{5} - 1429 T_{2}^{4} - 193 T_{2}^{3} + 386 T_{2}^{2} + 3 T_{2} - 5$$ T2^14 + T2^13 - 22*T2^12 - 18*T2^11 + 187*T2^10 + 118*T2^9 - 772*T2^8 - 346*T2^7 + 1581*T2^6 + 443*T2^5 - 1429*T2^4 - 193*T2^3 + 386*T2^2 + 3*T2 - 5 $$T_{3}^{14} + 6 T_{3}^{13} - 12 T_{3}^{12} - 121 T_{3}^{11} - 8 T_{3}^{10} + 909 T_{3}^{9} + 632 T_{3}^{8} - 3128 T_{3}^{7} - 3000 T_{3}^{6} + 4860 T_{3}^{5} + 5298 T_{3}^{4} - 2694 T_{3}^{3} - 3258 T_{3}^{2} + 135 T_{3} + 405$$ T3^14 + 6*T3^13 - 12*T3^12 - 121*T3^11 - 8*T3^10 + 909*T3^9 + 632*T3^8 - 3128*T3^7 - 3000*T3^6 + 4860*T3^5 + 5298*T3^4 - 2694*T3^3 - 3258*T3^2 + 135*T3 + 405 $$T_{11}^{14} + 3 T_{11}^{13} - 98 T_{11}^{12} - 265 T_{11}^{11} + 3589 T_{11}^{10} + 8946 T_{11}^{9} - 61324 T_{11}^{8} - 150514 T_{11}^{7} + 483981 T_{11}^{6} + 1283456 T_{11}^{5} - 1262144 T_{11}^{4} - 4537424 T_{11}^{3} + \cdots - 30528$$ T11^14 + 3*T11^13 - 98*T11^12 - 265*T11^11 + 3589*T11^10 + 8946*T11^9 - 61324*T11^8 - 150514*T11^7 + 483981*T11^6 + 1283456*T11^5 - 1262144*T11^4 - 4537424*T11^3 - 1593248*T11^2 + 1189152*T11 - 30528

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{14} + T^{13} - 22 T^{12} - 18 T^{11} + \cdots - 5$$
$3$ $$T^{14} + 6 T^{13} - 12 T^{12} - 121 T^{11} + \cdots + 405$$
$5$ $$T^{14}$$
$7$ $$(T + 1)^{14}$$
$11$ $$T^{14} + 3 T^{13} - 98 T^{12} + \cdots - 30528$$
$13$ $$T^{14} + 15 T^{13} - 4 T^{12} + \cdots + 2762487$$
$17$ $$T^{14} + 9 T^{13} - 89 T^{12} + \cdots + 3043008$$
$19$ $$T^{14} + 4 T^{13} + \cdots + 1202640448$$
$23$ $$(T - 1)^{14}$$
$29$ $$T^{14} - 11 T^{13} - 109 T^{12} + \cdots + 3379103$$
$31$ $$T^{14} + T^{13} - 316 T^{12} + \cdots - 57840829$$
$37$ $$T^{14} + 18 T^{13} + \cdots - 6636147392$$
$41$ $$T^{14} + 7 T^{13} + \cdots - 1303165251$$
$43$ $$T^{14} + 18 T^{13} + \cdots - 4740109632$$
$47$ $$T^{14} + 10 T^{13} + \cdots - 1176896709$$
$53$ $$T^{14} + 5 T^{13} + \cdots + 2549303360$$
$59$ $$T^{14} + 24 T^{13} - 218 T^{12} + \cdots - 19782035$$
$61$ $$T^{14} + 6 T^{13} + \cdots + 14122038080$$
$67$ $$T^{14} + 61 T^{13} + 1483 T^{12} + \cdots - 17784256$$
$71$ $$T^{14} - 11 T^{13} + \cdots + 1585190508535$$
$73$ $$T^{14} + 28 T^{13} + \cdots - 20947706379$$
$79$ $$T^{14} - 6 T^{13} + \cdots + 65236518720$$
$83$ $$T^{14} + 16 T^{13} + \cdots + 6763300416$$
$89$ $$T^{14} + 39 T^{13} + \cdots - 138345595456$$
$97$ $$T^{14} + 19 T^{13} + \cdots + 1258422080$$