LMFDB - Embedded newform 960.4.f.q.769.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [960,4,Mod(769,960)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("960.769"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(960, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 4, names="a")

Level:	$N$	$=$	$960 = 2^{6} \cdot 3 \cdot 5$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	960.f (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-6,0,0,0,-36,0,84] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$56.6418336055$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{41})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 21x^{2} + 100$ x^4 + 21*x^2 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2\cdot 3^{2}$
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.3
Root			$-2.70156i$ of defining polynomial
Character	$\chi$	$=$	960.769
Dual form			960.4.f.q.769.1

$q$ -expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} +(-11.1047 - 1.29844i) q^{5} -16.2094i q^{7} -9.00000 q^{9} +40.2094 q^{11} +19.7906i q^{13} +(3.89531 - 33.3141i) q^{15} +83.0156i q^{17} -48.8375 q^{19} +48.6281 q^{21} -1.61250i q^{23} +(121.628 + 28.8375i) q^{25} -27.0000i q^{27} -24.5344 q^{29} -12.4187 q^{31} +120.628i q^{33} +(-21.0469 + 180.000i) q^{35} -325.884i q^{37} -59.3719 q^{39} -242.419 q^{41} +367.350i q^{43} +(99.9422 + 11.6859i) q^{45} -204.544i q^{47} +80.2562 q^{49} -249.047 q^{51} -61.5281i q^{53} +(-446.512 - 52.2094i) q^{55} -146.512i q^{57} -112.209 q^{59} -477.350 q^{61} +145.884i q^{63} +(25.6969 - 219.769i) q^{65} -558.094i q^{67} +4.83749 q^{69} +558.281 q^{71} -1011.77i q^{73} +(-86.5125 + 364.884i) q^{75} -651.769i q^{77} -1150.47 q^{79} +81.0000 q^{81} -1157.92i q^{83} +(107.791 - 921.862i) q^{85} -73.6032i q^{87} -96.9751 q^{89} +320.794 q^{91} -37.2562i q^{93} +(542.325 + 63.4124i) q^{95} -1152.37i q^{97} -361.884 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 6 q^{5} - 36 q^{9} + 84 q^{11} + 54 q^{15} + 112 q^{19} - 36 q^{21} + 256 q^{25} - 636 q^{29} + 104 q^{31} + 300 q^{35} - 468 q^{39} - 816 q^{41} + 54 q^{45} - 140 q^{49} - 612 q^{51} - 864 q^{55}+ \cdots - 756 q^{99}+O(q^{100})$ 4 * q - 6 * q^5 - 36 * q^9 + 84 * q^11 + 54 * q^15 + 112 * q^19 - 36 * q^21 + 256 * q^25 - 636 * q^29 + 104 * q^31 + 300 * q^35 - 468 * q^39 - 816 * q^41 + 54 * q^45 - 140 * q^49 - 612 * q^51 - 864 * q^55 - 372 * q^59 - 680 * q^61 + 948 * q^65 - 288 * q^69 - 72 * q^71 + 576 * q^75 - 760 * q^79 + 324 * q^81 + 508 * q^85 - 2232 * q^89 - 1944 * q^91 + 2784 * q^95 - 756 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$ .

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$		0			0
$2$		0			0
$3$			3.00000i			0.577350i
$3$			3.00000i			0.577350i
$4$		0			0
$5$	−11.1047	−	1.29844i	−0.993233	−	0.116136i
$5$	−11.1047	−	1.29844i	−0.993233	−	0.116136i
$6$		0			0
$7$		−	16.2094i		−	0.875224i	−0.899164	−	0.437612i	$-0.855824\pi$
$7$		−	16.2094i		−	0.875224i	0.899164	−	0.437612i	$-0.144176\pi$
$8$		0			0
$9$	−9.00000			−0.333333
$10$		0			0
$11$	40.2094			1.10214			0.551072	−	0.834458i	$-0.314219\pi$
$11$	40.2094			1.10214			0.551072	+	0.834458i	$0.314219\pi$
$12$		0			0
$13$			19.7906i			0.422226i	0.977462	+	0.211113i	$0.0677087\pi$
$13$			19.7906i			0.422226i	−0.977462	+	0.211113i	$0.932291\pi$
$14$		0			0
$15$	3.89531	−	33.3141i	0.0670510	−	0.573444i
$16$		0			0
$17$			83.0156i			1.18437i	0.805803	+	0.592184i	$0.201734\pi$
$17$			83.0156i			1.18437i	−0.805803	+	0.592184i	$0.798266\pi$
$18$		0			0
$19$	−48.8375			−0.589689			−0.294844	−	0.955545i	$-0.595268\pi$
$19$	−48.8375			−0.589689			−0.294844	+	0.955545i	$0.595268\pi$
$20$		0			0
$21$	48.6281			0.505311
$22$		0			0
$23$		−	1.61250i		−	0.0146186i	−0.999973	−	0.00730932i	$-0.997673\pi$
$23$		−	1.61250i		−	0.0146186i	0.999973	−	0.00730932i	$-0.00232665\pi$
$24$		0			0
$25$	121.628	+	28.8375i	0.973025	+	0.230700i
$26$		0			0
$27$		−	27.0000i		−	0.192450i
$28$		0			0
$29$	−24.5344			−0.157101			−0.0785504	−	0.996910i	$-0.525029\pi$
$29$	−24.5344			−0.157101			−0.0785504	+	0.996910i	$0.525029\pi$
$30$		0			0
$31$	−12.4187			−0.0719507			−0.0359754	−	0.999353i	$-0.511454\pi$
$31$	−12.4187			−0.0719507			−0.0359754	+	0.999353i	$0.511454\pi$
$32$		0			0
$33$			120.628i			0.636323i
$34$		0			0
$35$	−21.0469	+	180.000i	−0.101645	+	0.869302i
$36$		0			0
$37$		−	325.884i		−	1.44797i	−0.689813	−	0.723987i	$-0.742307\pi$
$37$		−	325.884i		−	1.44797i	0.689813	−	0.723987i	$-0.257693\pi$
$38$		0			0
$39$	−59.3719			−0.243772
$40$		0			0
$41$	−242.419			−0.923401			−0.461701	−	0.887036i	$-0.652761\pi$
$41$	−242.419			−0.923401			−0.461701	+	0.887036i	$0.652761\pi$
$42$		0			0
$43$			367.350i			1.30280i	0.758735	+	0.651399i	$0.225818\pi$
$43$			367.350i			1.30280i	−0.758735	+	0.651399i	$0.774182\pi$
$44$		0			0
$45$	99.9422	+	11.6859i	0.331078	+	0.0387119i
$46$		0			0
$47$		−	204.544i		−	0.634804i	−0.948291	−	0.317402i	$-0.897190\pi$
$47$		−	204.544i		−	0.634804i	0.948291	−	0.317402i	$-0.102810\pi$
$48$		0			0
$49$	80.2562			0.233983
$50$		0			0
$51$	−249.047			−0.683795
$52$		0			0
$53$		−	61.5281i		−	0.159463i	−0.996816	−	0.0797314i	$-0.974594\pi$
$53$		−	61.5281i		−	0.159463i	0.996816	−	0.0797314i	$-0.0254063\pi$
$54$		0			0
$55$	−446.512	−	52.2094i	−1.09469	−	0.127998i
$56$		0			0
$57$		−	146.512i		−	0.340457i
$58$		0			0
$59$	−112.209			−0.247600			−0.123800	−	0.992307i	$-0.539508\pi$
$59$	−112.209			−0.247600			−0.123800	+	0.992307i	$0.539508\pi$
$60$		0			0
$61$	−477.350			−1.00194			−0.500970	−	0.865464i	$-0.667023\pi$
$61$	−477.350			−1.00194			−0.500970	+	0.865464i	$0.667023\pi$
$62$		0			0
$63$			145.884i			0.291741i
$64$		0			0
$65$	25.6969	−	219.769i	0.0490355	−	0.419369i
$66$		0			0
$67$		−	558.094i		−	1.01764i	−0.860872	−	0.508821i	$-0.830082\pi$
$67$		−	558.094i		−	1.01764i	0.860872	−	0.508821i	$-0.169918\pi$
$68$		0			0
$69$	4.83749			0.00844008
$70$		0			0
$71$	558.281			0.933180			0.466590	−	0.884474i	$-0.345482\pi$
$71$	558.281			0.933180			0.466590	+	0.884474i	$0.345482\pi$
$72$		0			0
$73$		−	1011.77i		−	1.62217i	−0.584927	−	0.811086i	$-0.698877\pi$
$73$		−	1011.77i		−	1.62217i	0.584927	−	0.811086i	$-0.301123\pi$
$74$		0			0
$75$	−86.5125	+	364.884i	−0.133195	+	0.561776i
$76$		0			0
$77$		−	651.769i		−	0.964623i
$78$		0			0
$79$	−1150.47			−1.63845			−0.819227	−	0.573470i	$-0.805597\pi$
$79$	−1150.47			−1.63845			−0.819227	+	0.573470i	$0.805597\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$		−	1157.92i		−	1.53131i	−0.643251	−	0.765655i	$-0.722415\pi$
$83$		−	1157.92i		−	1.53131i	0.643251	−	0.765655i	$-0.277585\pi$
$84$		0			0
$85$	107.791	−	921.862i	0.137547	−	1.17635i
$86$		0			0
$87$		−	73.6032i		−	0.0907022i
$88$		0			0
$89$	−96.9751			−0.115498			−0.0577491	−	0.998331i	$-0.518392\pi$
$89$	−96.9751			−0.115498			−0.0577491	+	0.998331i	$0.518392\pi$
$90$		0			0
$91$	320.794			0.369542
$92$		0			0
$93$		−	37.2562i		−	0.0415408i
$94$		0			0
$95$	542.325	+	63.4124i	0.585699	+	0.0684840i
$96$		0			0
$97$		−	1152.37i		−	1.20625i	−0.797648	−	0.603123i	$-0.793923\pi$
$97$		−	1152.37i		−	1.20625i	0.797648	−	0.603123i	$-0.206077\pi$
$98$		0			0
$99$	−361.884			−0.367381
$100$		0			0
$101$	1156.49			1.13936			0.569679	−	0.821867i	$-0.307068\pi$
$101$	1156.49			1.13936			0.569679	+	0.821867i	$0.307068\pi$
$102$		0			0
$103$			1333.70i			1.27585i	0.770096	+	0.637927i	$0.220208\pi$
$103$			1333.70i			1.27585i	−0.770096	+	0.637927i	$0.779792\pi$
$104$		0			0
$105$	−540.000	−	63.1406i	−0.501891	−	0.0586847i
$106$		0			0
$107$			798.263i			0.721224i	0.932716	+	0.360612i	$0.117432\pi$
$107$			798.263i			0.721224i	−0.932716	+	0.360612i	$0.882568\pi$
$108$		0			0
$109$	−985.119			−0.865663			−0.432831	−	0.901475i	$-0.642485\pi$
$109$	−985.119			−0.865663			−0.432831	+	0.901475i	$0.642485\pi$
$110$		0			0
$111$	977.653			0.835988
$112$		0			0
$113$		−	1888.25i		−	1.57196i	−0.618253	−	0.785979i	$-0.712159\pi$
$113$		−	1888.25i		−	1.57196i	0.618253	−	0.785979i	$-0.287841\pi$
$114$		0			0
$115$	−2.09373	+	17.9063i	−0.00169775	+	0.0145197i
$116$		0			0
$117$		−	178.116i		−	0.140742i
$118$		0			0
$119$	1345.63			1.03659
$120$		0			0
$121$	285.794			0.214721
$122$		0			0
$123$		−	727.256i		−	0.533126i
$124$		0			0
$125$	−1313.20	−	478.158i	−0.939648	−	0.342142i
$126$		0			0
$127$		−	620.859i		−	0.433798i	−0.976194	−	0.216899i	$-0.930406\pi$
$127$		−	620.859i		−	0.433798i	0.976194	−	0.216899i	$-0.0695943\pi$
$128$		0			0
$129$	−1102.05			−0.752171
$130$		0			0
$131$	2588.35			1.72630			0.863151	−	0.504947i	$-0.168488\pi$
$131$	2588.35			1.72630			0.863151	+	0.504947i	$0.168488\pi$
$132$		0			0
$133$			791.625i			0.516110i
$134$		0			0
$135$	−35.0578	+	299.827i	−0.0223503	+	0.191148i
$136$		0			0
$137$		−	1656.29i		−	1.03289i	−0.856319	−	0.516447i	$-0.827254\pi$
$137$		−	1656.29i		−	1.03289i	0.856319	−	0.516447i	$-0.172746\pi$
$138$		0			0
$139$	−153.256			−0.0935182			−0.0467591	−	0.998906i	$-0.514889\pi$
$139$	−153.256			−0.0935182			−0.0467591	+	0.998906i	$0.514889\pi$
$140$		0			0
$141$	613.631			0.366504
$142$		0			0
$143$			795.769i			0.465353i
$144$		0			0
$145$	272.447	+	31.8564i	0.156038	+	0.0182450i
$146$		0			0
$147$			240.769i			0.135090i
$148$		0			0
$149$	1483.38			0.815591			0.407795	−	0.913073i	$-0.366298\pi$
$149$	1483.38			0.815591			0.407795	+	0.913073i	$0.366298\pi$
$150$		0			0
$151$	−394.281			−0.212491			−0.106246	−	0.994340i	$-0.533883\pi$
$151$	−394.281			−0.212491			−0.106246	+	0.994340i	$0.533883\pi$
$152$		0			0
$153$		−	747.141i		−	0.394789i
$154$		0			0
$155$	137.906	+	16.1250i	0.0714639	+	0.00835606i
$156$		0			0
$157$		−	1727.05i		−	0.877922i	−0.898506	−	0.438961i	$-0.855347\pi$
$157$		−	1727.05i		−	0.877922i	0.898506	−	0.438961i	$-0.144653\pi$
$158$		0			0
$159$	184.584			0.0920659
$160$		0			0
$161$	−26.1376			−0.0127946
$162$		0			0
$163$		−	2034.28i		−	0.977529i	−0.872416	−	0.488764i	$-0.837448\pi$
$163$		−	2034.28i		−	0.977529i	0.872416	−	0.488764i	$-0.162552\pi$
$164$		0			0
$165$	156.628	−	1339.54i	0.0738999	−	0.632017i
$166$		0			0
$167$			192.900i			0.0893835i	0.999001	+	0.0446918i	$0.0142306\pi$
$167$			192.900i			0.0893835i	−0.999001	+	0.0446918i	$0.985769\pi$
$168$		0			0
$169$	1805.33			0.821726
$170$		0			0
$171$	439.537			0.196563
$172$		0			0
$173$		−	1239.91i		−	0.544905i	−0.962169	−	0.272452i	$-0.912165\pi$
$173$		−	1239.91i		−	0.544905i	0.962169	−	0.272452i	$-0.0878347\pi$
$174$		0			0
$175$	467.438	−	1971.52i	0.201914	−	0.851615i
$176$		0			0
$177$		−	336.628i		−	0.142952i
$178$		0			0
$179$	−2636.86			−1.10105			−0.550525	−	0.834818i	$-0.685573\pi$
$179$	−2636.86			−1.10105			−0.550525	+	0.834818i	$0.685573\pi$
$180$		0			0
$181$	−3317.58			−1.36240			−0.681199	−	0.732099i	$-0.738541\pi$
$181$	−3317.58			−1.36240			−0.681199	+	0.732099i	$0.738541\pi$
$182$		0			0
$183$		−	1432.05i		−	0.578471i
$184$		0			0
$185$	−423.141	+	3618.84i	−0.168162	+	1.43818i
$186$		0			0
$187$			3338.01i			1.30534i
$188$		0			0
$189$	−437.653			−0.168437
$190$		0			0
$191$	−624.506			−0.236585			−0.118292	−	0.992979i	$-0.537742\pi$
$191$	−624.506			−0.236585			−0.118292	+	0.992979i	$0.537742\pi$
$192$		0			0
$193$			436.144i			0.162665i	0.996687	+	0.0813324i	$0.0259175\pi$
$193$			436.144i			0.162665i	−0.996687	+	0.0813324i	$0.974082\pi$
$194$		0			0
$195$	659.306	+	77.0907i	0.242123	+	0.0283107i
$196$		0			0
$197$		−	3355.81i		−	1.21366i	−0.794831	−	0.606831i	$-0.792440\pi$
$197$		−	3355.81i		−	1.21366i	0.794831	−	0.606831i	$-0.207560\pi$
$198$		0			0
$199$	−3799.77			−1.35356			−0.676780	−	0.736185i	$-0.736625\pi$
$199$	−3799.77			−1.35356			−0.676780	+	0.736185i	$0.736625\pi$
$200$		0			0
$201$	1674.28			0.587536
$202$		0			0
$203$			397.687i			0.137498i
$204$		0			0
$205$	2691.98	+	314.766i	0.917153	+	0.107240i
$206$		0			0
$207$			14.5125i			0.00487288i
$208$		0			0
$209$	−1963.72			−0.649922
$210$		0			0
$211$	−2365.27			−0.771715			−0.385857	−	0.922558i	$-0.626094\pi$
$211$	−2365.27			−0.771715			−0.385857	+	0.922558i	$0.626094\pi$
$212$		0			0
$213$			1674.84i			0.538772i
$214$		0			0
$215$	476.981	−	4079.31i	0.151302	−	1.29398i
$216$		0			0
$217$			201.300i			0.0629730i
$218$		0			0
$219$	3035.31			0.936562
$220$		0			0
$221$	−1642.93			−0.500070
$222$		0			0
$223$			3328.58i			0.999545i	0.866157	+	0.499772i	$0.166583\pi$
$223$			3328.58i			0.999545i	−0.866157	+	0.499772i	$0.833417\pi$
$224$		0			0
$225$	−1094.65	−	259.537i	−0.324342	−	0.0769000i
$226$		0			0
$227$			527.100i			0.154118i	0.997027	+	0.0770592i	$0.0245530\pi$
$227$			527.100i			0.154118i	−0.997027	+	0.0770592i	$0.975447\pi$
$228$		0			0
$229$	2566.06			0.740479			0.370240	−	0.928936i	$-0.379276\pi$
$229$	2566.06			0.740479			0.370240	+	0.928936i	$0.379276\pi$
$230$		0			0
$231$	1955.31			0.556925
$232$		0			0
$233$		−	5534.99i		−	1.55626i	−0.628101	−	0.778132i	$-0.716168\pi$
$233$		−	5534.99i		−	1.55626i	0.628101	−	0.778132i	$-0.283832\pi$
$234$		0			0
$235$	−265.587	+	2271.39i	−0.0737234	+	0.630508i
$236$		0			0
$237$		−	3451.41i		−	0.945962i
$238$		0			0
$239$	−1010.01			−0.273355			−0.136678	−	0.990616i	$-0.543642\pi$
$239$	−1010.01			−0.273355			−0.136678	+	0.990616i	$0.543642\pi$
$240$		0			0
$241$	−4074.29			−1.08900			−0.544498	−	0.838762i	$-0.683280\pi$
$241$	−4074.29			−1.08900			−0.544498	+	0.838762i	$0.683280\pi$
$242$		0			0
$243$			243.000i			0.0641500i
$244$		0			0
$245$	−891.220	−	104.208i	−0.232400	−	0.0271738i
$246$		0			0
$247$		−	966.525i		−	0.248982i
$248$		0			0
$249$	3473.77			0.884103
$250$		0			0
$251$	−1773.98			−0.446107			−0.223054	−	0.974806i	$-0.571602\pi$
$251$	−1773.98			−0.446107			−0.223054	+	0.974806i	$0.571602\pi$
$252$		0			0
$253$		−	64.8375i		−	0.0161119i
$254$		0			0
$255$	2765.59	+	323.372i	0.679168	+	0.0794131i
$256$		0			0
$257$			662.784i			0.160869i	0.996760	+	0.0804345i	$0.0256308\pi$
$257$			662.784i			0.160869i	−0.996760	+	0.0804345i	$0.974369\pi$
$258$		0			0
$259$	−5282.38			−1.26730
$260$		0			0
$261$	220.810			0.0523669
$262$		0			0
$263$		−	712.312i		−	0.167008i	−0.996507	−	0.0835039i	$-0.973389\pi$
$263$		−	712.312i		−	0.167008i	0.996507	−	0.0835039i	$-0.0266111\pi$
$264$		0			0
$265$	−79.8904	+	683.250i	−0.0185194	+	0.158384i
$266$		0			0
$267$		−	290.925i		−	0.0666829i
$268$		0			0
$269$	−3136.41			−0.710894			−0.355447	−	0.934696i	$-0.615671\pi$
$269$	−3136.41			−0.710894			−0.355447	+	0.934696i	$0.615671\pi$
$270$		0			0
$271$	−2275.69			−0.510105			−0.255053	−	0.966927i	$-0.582093\pi$
$271$	−2275.69			−0.510105			−0.255053	+	0.966927i	$0.582093\pi$
$272$		0			0
$273$			962.381i			0.213355i
$274$		0			0
$275$	4890.59	+	1159.54i	1.07241	+	0.254264i
$276$		0			0
$277$		−	5171.00i		−	1.12164i	−0.827937	−	0.560821i	$-0.810485\pi$
$277$		−	5171.00i		−	1.12164i	0.827937	−	0.560821i	$-0.189515\pi$
$278$		0			0
$279$	111.769			0.0239836
$280$		0			0
$281$	2240.14			0.475571			0.237785	−	0.971318i	$-0.423578\pi$
$281$	2240.14			0.475571			0.237785	+	0.971318i	$0.423578\pi$
$282$		0			0
$283$			225.244i			0.0473123i	0.999720	+	0.0236561i	$0.00753068\pi$
$283$			225.244i			0.0473123i	−0.999720	+	0.0236561i	$0.992469\pi$
$284$		0			0
$285$	−190.237	+	1626.98i	−0.0395393	+	0.338153i
$286$		0			0
$287$			3929.46i			0.808183i
$288$		0			0
$289$	−1978.59			−0.402726
$290$		0			0
$291$	3457.12			0.696427
$292$		0			0
$293$			1139.86i			0.227274i	0.993522	+	0.113637i	$0.0362501\pi$
$293$			1139.86i			0.227274i	−0.993522	+	0.113637i	$0.963750\pi$
$294$		0			0
$295$	1246.05	+	145.697i	0.245925	+	0.0287553i
$296$		0			0
$297$		−	1085.65i		−	0.212108i
$298$		0			0
$299$	31.9123			0.00617237
$300$		0			0
$301$	5954.51			1.14024
$302$		0			0
$303$			3469.47i			0.657808i
$304$		0			0
$305$	5300.82	+	619.809i	0.995161	+	0.116361i
$306$		0			0
$307$		−	5244.86i		−	0.975049i	−0.873109	−	0.487525i	$-0.837900\pi$
$307$		−	5244.86i		−	0.975049i	0.873109	−	0.487525i	$-0.162100\pi$
$308$		0			0
$309$	−4001.09			−0.736615
$310$		0			0
$311$	−5188.26			−0.945977			−0.472989	−	0.881068i	$-0.656825\pi$
$311$	−5188.26			−0.945977			−0.472989	+	0.881068i	$0.656825\pi$
$312$		0			0
$313$		−	486.656i		−	0.0878832i	−0.999034	−	0.0439416i	$-0.986008\pi$
$313$		−	486.656i		−	0.0878832i	0.999034	−	0.0439416i	$-0.0139915\pi$
$314$		0			0
$315$	189.422	−	1620.00i	0.0338816	−	0.289767i
$316$		0			0
$317$			4218.87i			0.747493i	0.927531	+	0.373747i	$0.121927\pi$
$317$			4218.87i			0.747493i	−0.927531	+	0.373747i	$0.878073\pi$
$318$		0			0
$319$	−986.512			−0.173148
$320$		0			0
$321$	−2394.79			−0.416399
$322$		0			0
$323$		−	4054.27i		−	0.698408i
$324$		0			0
$325$	−570.712	+	2407.10i	−0.0974074	+	0.410836i
$326$		0			0
$327$		−	2955.36i		−	0.499791i
$328$		0			0
$329$	−3315.53			−0.555595
$330$		0			0
$331$	−7439.94			−1.23546			−0.617728	−	0.786392i	$-0.711947\pi$
$331$	−7439.94			−1.23546			−0.617728	+	0.786392i	$0.711947\pi$
$332$		0			0
$333$			2932.96i			0.482658i
$334$		0			0
$335$	−724.650	+	6197.46i	−0.118185	+	1.01076i
$336$		0			0
$337$		−	6555.39i		−	1.05963i	−0.848113	−	0.529815i	$-0.822261\pi$
$337$		−	6555.39i		−	1.05963i	0.848113	−	0.529815i	$-0.177739\pi$
$338$		0			0
$339$	5664.74			0.907571
$340$		0			0
$341$	−499.350			−0.0793000
$342$		0			0
$343$		−	6860.72i		−	1.08001i
$344$		0			0
$345$	−53.7188	−	6.28118i	−0.00838297	−	0.000980195i
$346$		0			0
$347$		−	1950.56i		−	0.301763i	−0.988552	−	0.150881i	$-0.951789\pi$
$347$		−	1950.56i		−	0.301763i	0.988552	−	0.150881i	$-0.0482112\pi$
$348$		0			0
$349$	−1426.74			−0.218830			−0.109415	−	0.993996i	$-0.534898\pi$
$349$	−1426.74			−0.218830			−0.109415	+	0.993996i	$0.534898\pi$
$350$		0			0
$351$	534.347			0.0812573
$352$		0			0
$353$			7078.96i			1.06735i	0.845689	+	0.533676i	$0.179190\pi$
$353$			7078.96i			1.06735i	−0.845689	+	0.533676i	$0.820810\pi$
$354$		0			0
$355$	−6199.54	−	724.893i	−0.926866	−	0.108376i
$356$		0			0
$357$			4036.89i			0.598474i
$358$		0			0
$359$	5409.79			0.795314			0.397657	−	0.917534i	$-0.369823\pi$
$359$	5409.79			0.795314			0.397657	+	0.917534i	$0.369823\pi$
$360$		0			0
$361$	−4473.90			−0.652267
$362$		0			0
$363$			857.381i			0.123969i
$364$		0			0
$365$	−1313.72	+	11235.4i	−0.188392	+	1.61120i
$366$		0			0
$367$			4940.09i			0.702645i	0.936255	+	0.351322i	$0.114268\pi$
$367$			4940.09i			0.702645i	−0.936255	+	0.351322i	$0.885732\pi$
$368$		0			0
$369$	2181.77			0.307800
$370$		0			0
$371$	−997.332			−0.139566
$372$		0			0
$373$			12891.9i			1.78959i	0.446473	+	0.894797i	$0.352680\pi$
$373$			12891.9i			1.78959i	−0.446473	+	0.894797i	$0.647320\pi$
$374$		0			0
$375$	1434.47	−	3939.60i	0.197536	−	0.542506i
$376$		0			0
$377$		−	485.551i		−	0.0663320i
$378$		0			0
$379$	−9475.15			−1.28418			−0.642092	−	0.766627i	$-0.721933\pi$
$379$	−9475.15			−1.28418			−0.642092	+	0.766627i	$0.721933\pi$
$380$		0			0
$381$	1862.58			0.250453
$382$		0			0
$383$			5800.97i			0.773931i	0.922094	+	0.386966i	$0.126477\pi$
$383$			5800.97i			0.773931i	−0.922094	+	0.386966i	$0.873523\pi$
$384$		0			0
$385$	−846.281	+	7237.69i	−0.112027	+	0.958095i
$386$		0			0
$387$		−	3306.15i		−	0.434266i
$388$		0			0
$389$	13779.7			1.79603			0.898016	−	0.439962i	$-0.145008\pi$
$389$	13779.7			1.79603			0.898016	+	0.439962i	$0.145008\pi$
$390$		0			0
$391$	133.862			0.0173138
$392$		0			0
$393$			7765.06i			0.996680i
$394$		0			0
$395$	12775.6	+	1493.81i	1.62737	+	0.190283i
$396$		0			0
$397$			2816.46i			0.356056i	0.984025	+	0.178028i	$0.0569718\pi$
$397$			2816.46i			0.356056i	−0.984025	+	0.178028i	$0.943028\pi$
$398$		0			0
$399$	−2374.88			−0.297976
$400$		0			0
$401$	11986.4			1.49270			0.746352	−	0.665551i	$-0.231804\pi$
$401$	11986.4			1.49270			0.746352	+	0.665551i	$0.231804\pi$
$402$		0			0
$403$		−	245.775i		−	0.0303794i
$404$		0			0
$405$	−899.480	−	105.173i	−0.110359	−	0.0129040i
$406$		0			0
$407$		−	13103.6i		−	1.59588i
$408$		0			0
$409$	3339.07			0.403683			0.201841	−	0.979418i	$-0.435307\pi$
$409$	3339.07			0.403683			0.201841	+	0.979418i	$0.435307\pi$
$410$		0			0
$411$	4968.87			0.596342
$412$		0			0
$413$			1818.84i			0.216706i
$414$		0			0
$415$	−1503.49	+	12858.4i	−0.177840	+	1.52095i
$416$		0			0
$417$		−	459.769i		−	0.0539927i
$418$		0			0
$419$	1688.52			0.196873			0.0984363	−	0.995143i	$-0.468616\pi$
$419$	1688.52			0.196873			0.0984363	+	0.995143i	$0.468616\pi$
$420$		0			0
$421$	2664.27			0.308429			0.154214	−	0.988037i	$-0.450715\pi$
$421$	2664.27			0.308429			0.154214	+	0.988037i	$0.450715\pi$
$422$		0			0
$423$			1840.89i			0.211601i
$424$		0			0
$425$	−2393.96	+	10097.0i	−0.273233	+	1.15242i
$426$		0			0
$427$			7737.54i			0.876923i
$428$		0			0
$429$	−2387.31			−0.268672
$430$		0			0
$431$	12266.0			1.37084			0.685420	−	0.728148i	$-0.259619\pi$
$431$	12266.0			1.37084			0.685420	+	0.728148i	$0.259619\pi$
$432$		0			0
$433$			15647.3i			1.73664i	0.496008	+	0.868318i	$0.334799\pi$
$433$			15647.3i			1.73664i	−0.496008	+	0.868318i	$0.665201\pi$
$434$		0			0
$435$	−95.5691	+	817.340i	−0.0105338	+	0.0900884i
$436$		0			0
$437$			78.7503i			0.00862045i
$438$		0			0
$439$	−16131.0			−1.75373			−0.876867	−	0.480733i	$-0.840371\pi$
$439$	−16131.0			−1.75373			−0.876867	+	0.480733i	$0.840371\pi$
$440$		0			0
$441$	−722.306			−0.0779944
$442$		0			0
$443$			10053.7i			1.07825i	0.842225	+	0.539127i	$0.181246\pi$
$443$			10053.7i			1.07825i	−0.842225	+	0.539127i	$0.818754\pi$
$444$		0			0
$445$	1076.88	+	125.916i	0.114717	+	0.0134135i
$446$		0			0
$447$			4450.13i			0.470882i
$448$		0			0
$449$	−7477.71			−0.785957			−0.392979	−	0.919548i	$-0.628555\pi$
$449$	−7477.71			−0.785957			−0.392979	+	0.919548i	$0.628555\pi$
$450$		0			0
$451$	−9747.51			−1.01772
$452$		0			0
$453$		−	1182.84i		−	0.122682i
$454$		0			0
$455$	−3562.31	−	416.531i	−0.367041	−	0.0429170i
$456$		0			0
$457$			1363.46i			0.139562i	0.997562	+	0.0697812i	$0.0222301\pi$
$457$			1363.46i			0.139562i	−0.997562	+	0.0697812i	$0.977770\pi$
$458$		0			0
$459$	2241.42			0.227932
$460$		0			0
$461$	−5276.77			−0.533109			−0.266555	−	0.963820i	$-0.585885\pi$
$461$	−5276.77			−0.533109			−0.266555	+	0.963820i	$0.585885\pi$
$462$		0			0
$463$		−	5740.02i		−	0.576159i	−0.957607	−	0.288079i	$-0.906983\pi$
$463$		−	5740.02i		−	0.576159i	0.957607	−	0.288079i	$-0.0930167\pi$
$464$		0			0
$465$	−48.3749	+	413.719i	−0.00482437	+	0.0412597i
$466$		0			0
$467$		−	6233.36i		−	0.617657i	−0.951118	−	0.308828i	$-0.900063\pi$
$467$		−	6233.36i		−	0.617657i	0.951118	−	0.308828i	$-0.0999368\pi$
$468$		0			0
$469$	−9046.35			−0.890664
$470$		0			0
$471$	5181.16			0.506869
$472$		0			0
$473$			14770.9i			1.43587i
$474$		0			0
$475$	−5940.01	−	1408.35i	−0.573782	−	0.136041i
$476$		0			0
$477$			553.753i			0.0531543i
$478$		0			0
$479$	19688.2			1.87803			0.939013	−	0.343881i	$-0.111742\pi$
$479$	19688.2			1.87803			0.939013	+	0.343881i	$0.111742\pi$
$480$		0			0
$481$	6449.46			0.611372
$482$		0			0
$483$		−	78.4127i		−	0.00738696i
$484$		0			0
$485$	−1496.29	+	12796.8i	−0.140088	+	1.19808i
$486$		0			0
$487$		−	3955.08i		−	0.368012i	−0.982925	−	0.184006i	$-0.941093\pi$
$487$		−	3955.08i		−	0.368012i	0.982925	−	0.184006i	$-0.0589065\pi$
$488$		0			0
$489$	6102.84			0.564377
$490$		0			0
$491$	13893.5			1.27699			0.638497	−	0.769624i	$-0.279557\pi$
$491$	13893.5			1.27699			0.638497	+	0.769624i	$0.279557\pi$
$492$		0			0
$493$		−	2036.74i		−	0.186065i
$494$		0			0
$495$	4018.61	+	469.884i	0.364895	+	0.0426661i
$496$		0			0
$497$		−	9049.39i		−	0.816741i
$498$		0			0
$499$	13523.7			1.21324			0.606618	−	0.794993i	$-0.292526\pi$
$499$	13523.7			1.21324			0.606618	+	0.794993i	$0.292526\pi$
$500$		0			0
$501$	−578.700			−0.0516056
$502$		0			0
$503$		−	13135.4i		−	1.16437i	−0.813057	−	0.582184i	$-0.802198\pi$
$503$		−	13135.4i		−	1.16437i	0.813057	−	0.582184i	$-0.197802\pi$
$504$		0			0
$505$	−12842.5	−	1501.63i	−1.13165	−	0.132320i
$506$		0			0
$507$			5415.99i			0.474423i
$508$		0			0
$509$	−2222.71			−0.193556			−0.0967778	−	0.995306i	$-0.530854\pi$
$509$	−2222.71			−0.193556			−0.0967778	+	0.995306i	$0.530854\pi$
$510$		0			0
$511$	−16400.1			−1.41976
$512$		0			0
$513$			1318.61i			0.113486i
$514$		0			0
$515$	1731.72	−	14810.3i	0.148172	−	1.26722i
$516$		0			0
$517$		−	8224.57i		−	0.699645i
$518$		0			0
$519$	3719.73			0.314601
$520$		0			0
$521$	−4916.42			−0.413421			−0.206710	−	0.978402i	$-0.566276\pi$
$521$	−4916.42			−0.413421			−0.206710	+	0.978402i	$0.566276\pi$
$522$		0			0
$523$			17743.4i			1.48349i	0.670681	+	0.741746i	$0.266002\pi$
$523$			17743.4i			1.48349i	−0.670681	+	0.741746i	$0.733998\pi$
$524$		0			0
$525$	5914.55	+	1402.31i	0.491680	+	0.116575i
$526$		0			0
$527$		−	1030.95i		−	0.0852161i
$528$		0			0
$529$	12164.4			0.999786
$530$		0			0
$531$	1009.88			0.0825334
$532$		0			0
$533$		−	4797.62i		−	0.389884i
$534$		0			0
$535$	1036.49	−	8864.46i	0.0837599	−	0.716344i
$536$		0			0
$537$		−	7910.58i		−	0.635692i
$538$		0			0
$539$	3227.05			0.257883
$540$		0			0
$541$	12671.3			1.00699			0.503495	−	0.863998i	$-0.332047\pi$
$541$	12671.3			1.00699			0.503495	+	0.863998i	$0.332047\pi$
$542$		0			0
$543$		−	9952.74i		−	0.786580i
$544$		0			0
$545$	10939.4	+	1279.12i	0.859805	+	0.100534i
$546$		0			0
$547$		−	5250.90i		−	0.410443i	−0.978716	−	0.205221i	$-0.934209\pi$
$547$		−	5250.90i		−	0.410443i	0.978716	−	0.205221i	$-0.0657915\pi$
$548$		0			0
$549$	4296.15			0.333980
$550$		0			0
$551$	1198.20			0.0926406
$552$		0			0
$553$			18648.4i			1.43401i
$554$		0			0
$555$	−10856.5	−	1269.42i	−0.830332	−	0.0970882i
$556$		0			0
$557$		−	25830.2i		−	1.96492i	−0.186465	−	0.982462i	$-0.559703\pi$
$557$		−	25830.2i		−	1.96492i	0.186465	−	0.982462i	$-0.440297\pi$
$558$		0			0
$559$	−7270.09			−0.550075
$560$		0			0
$561$	−10014.0			−0.753640
$562$		0			0
$563$			2021.14i			0.151298i	0.997135	+	0.0756490i	$0.0241029\pi$
$563$			2021.14i			0.151298i	−0.997135	+	0.0756490i	$0.975897\pi$
$564$		0			0
$565$	−2451.77	+	20968.4i	−0.182561	+	1.56132i
$566$		0			0
$567$		−	1312.96i		−	0.0972471i
$568$		0			0
$569$	−8706.51			−0.641469			−0.320734	−	0.947169i	$-0.603930\pi$
$569$	−8706.51			−0.641469			−0.320734	+	0.947169i	$0.603930\pi$
$570$		0			0
$571$	12194.5			0.893740			0.446870	−	0.894599i	$-0.352539\pi$
$571$	12194.5			0.893740			0.446870	+	0.894599i	$0.352539\pi$
$572$		0			0
$573$		−	1873.52i		−	0.136592i
$574$		0			0
$575$	46.5004	−	196.125i	0.00337252	−	0.0142243i
$576$		0			0
$577$		−	15264.0i		−	1.10130i	−0.834737	−	0.550649i	$-0.814380\pi$
$577$		−	15264.0i		−	1.10130i	0.834737	−	0.550649i	$-0.185620\pi$
$578$		0			0
$579$	−1308.43			−0.0939146
$580$		0			0
$581$	−18769.2			−1.34024
$582$		0			0
$583$		−	2474.01i		−	0.175751i
$584$		0			0
$585$	−231.272	+	1977.92i	−0.0163452	+	0.139790i
$586$		0			0
$587$			8456.89i			0.594639i	0.954778	+	0.297319i	$0.0960926\pi$
$587$			8456.89i			0.594639i	−0.954778	+	0.297319i	$0.903907\pi$
$588$		0			0
$589$	606.500			0.0424285
$590$		0			0
$591$	10067.4			0.700708
$592$		0			0
$593$			1225.23i			0.0848467i	0.999100	+	0.0424234i	$0.0135078\pi$
$593$			1225.23i			0.0848467i	−0.999100	+	0.0424234i	$0.986492\pi$
$594$		0			0
$595$	−14942.8	−	1747.22i	−1.02957	−	0.120385i
$596$		0			0
$597$		−	11399.3i		−	0.781478i
$598$		0			0
$599$	16060.0			1.09548			0.547741	−	0.836648i	$-0.315488\pi$
$599$	16060.0			1.09548			0.547741	+	0.836648i	$0.315488\pi$
$600$		0			0
$601$	9699.93			0.658350			0.329175	−	0.944269i	$-0.393229\pi$
$601$	9699.93			0.658350			0.329175	+	0.944269i	$0.393229\pi$
$602$		0			0
$603$			5022.84i			0.339214i
$604$		0			0
$605$	−3173.65	−	371.085i	−0.213268	−	0.0249368i
$606$		0			0
$607$			23661.2i			1.58217i	0.611703	+	0.791087i	$0.290485\pi$
$607$			23661.2i			1.58217i	−0.611703	+	0.791087i	$0.709515\pi$
$608$		0			0
$609$	−1193.06			−0.0793847
$610$		0			0
$611$	4048.05			0.268030
$612$		0			0
$613$		−	8085.63i		−	0.532749i	−0.963870	−	0.266375i	$-0.914174\pi$
$613$		−	8085.63i		−	0.532749i	0.963870	−	0.266375i	$-0.0858258\pi$
$614$		0			0
$615$	−944.297	+	8075.95i	−0.0619150	+	0.529518i
$616$		0			0
$617$		−	11035.1i		−	0.720029i	−0.932947	−	0.360014i	$-0.882772\pi$
$617$		−	11035.1i		−	0.720029i	0.932947	−	0.360014i	$-0.117228\pi$
$618$		0			0
$619$	−16826.3			−1.09258			−0.546290	−	0.837596i	$-0.683960\pi$
$619$	−16826.3			−1.09258			−0.546290	+	0.837596i	$0.683960\pi$
$620$		0			0
$621$	−43.5374			−0.00281336
$622$		0			0
$623$			1571.90i			0.101087i
$624$		0			0
$625$	13961.8	+	7014.90i	0.893555	+	0.448954i
$626$		0			0
$627$		−	5891.17i		−	0.375233i
$628$		0			0
$629$	27053.5			1.71493
$630$		0			0
$631$	−3705.91			−0.233803			−0.116902	−	0.993144i	$-0.537296\pi$
$631$	−3705.91			−0.233803			−0.116902	+	0.993144i	$0.537296\pi$
$632$		0			0
$633$		−	7095.81i		−	0.445550i
$634$		0			0
$635$	−806.147	+	6894.45i	−0.0503795	+	0.430863i
$636$		0			0
$637$			1588.32i			0.0987937i
$638$		0			0
$639$	−5024.53			−0.311060
$640$		0			0
$641$	−24597.4			−1.51566			−0.757829	−	0.652453i	$-0.773740\pi$
$641$	−24597.4			−1.51566			−0.757829	+	0.652453i	$0.773740\pi$
$642$		0			0
$643$		−	21479.5i		−	1.31737i	−0.752419	−	0.658685i	$-0.771113\pi$
$643$		−	21479.5i		−	1.31737i	0.752419	−	0.658685i	$-0.228887\pi$
$644$		0			0
$645$	12237.9	+	1430.94i	0.747081	+	0.0873540i
$646$		0			0
$647$			27119.7i			1.64789i	0.566668	+	0.823946i	$0.308232\pi$
$647$			27119.7i			1.64789i	−0.566668	+	0.823946i	$0.691768\pi$
$648$		0			0
$649$	−4511.87			−0.272891
$650$		0			0
$651$	−603.900			−0.0363575
$652$		0			0
$653$			18476.4i			1.10725i	0.832765	+	0.553627i	$0.186757\pi$
$653$			18476.4i			1.10725i	−0.832765	+	0.553627i	$0.813243\pi$
$654$		0			0
$655$	−28742.8	−	3360.82i	−1.71462	−	0.200485i
$656$		0			0
$657$			9105.92i			0.540724i
$658$		0			0
$659$	−19273.5			−1.13928			−0.569641	−	0.821894i	$-0.692918\pi$
$659$	−19273.5			−1.13928			−0.569641	+	0.821894i	$0.692918\pi$
$660$		0			0
$661$	−25605.3			−1.50670			−0.753352	−	0.657618i	$-0.771564\pi$
$661$	−25605.3			−1.50670			−0.753352	+	0.657618i	$0.771564\pi$
$662$		0			0
$663$		−	4928.79i		−	0.288716i
$664$		0			0
$665$	1027.88	−	8790.75i	0.0599388	−	0.512617i
$666$		0			0
$667$			39.5616i			0.00229660i
$668$		0			0
$669$	−9985.75			−0.577087
$670$		0			0
$671$	−19193.9			−1.10428
$672$		0			0
$673$		−	7855.52i		−	0.449938i	−0.974366	−	0.224969i	$-0.927772\pi$
$673$		−	7855.52i		−	0.449938i	0.974366	−	0.224969i	$-0.0722280\pi$
$674$		0			0
$675$	778.612	−	3283.96i	0.0443982	−	0.187259i
$676$		0			0
$677$			6763.09i			0.383939i	0.981401	+	0.191970i	$0.0614875\pi$
$677$			6763.09i			0.383939i	−0.981401	+	0.191970i	$0.938512\pi$
$678$		0			0
$679$	−18679.3			−1.05574
$680$		0			0
$681$	−1581.30			−0.0889803
$682$		0			0
$683$			15608.6i			0.874447i	0.899353	+	0.437224i	$0.144038\pi$
$683$			15608.6i			0.874447i	−0.899353	+	0.437224i	$0.855962\pi$
$684$		0			0
$685$	−2150.59	+	18392.6i	−0.119956	+	1.02590i
$686$		0			0
$687$			7698.17i			0.427516i
$688$		0			0
$689$	1217.68			0.0673293
$690$		0			0
$691$	−6203.15			−0.341504			−0.170752	−	0.985314i	$-0.554620\pi$
$691$	−6203.15			−0.341504			−0.170752	+	0.985314i	$0.554620\pi$
$692$		0			0
$693$			5865.92i			0.321541i
$694$		0			0
$695$	1701.86	+	198.994i	0.0928854	+	0.0108608i
$696$		0			0
$697$		−	20124.5i		−	1.09365i
$698$		0			0
$699$	16605.0			0.898509
$700$		0			0
$701$	16507.9			0.889435			0.444718	−	0.895671i	$-0.353304\pi$
$701$	16507.9			0.889435			0.444718	+	0.895671i	$0.353304\pi$
$702$		0			0
$703$			15915.4i			0.853854i
$704$		0			0
$705$	−6814.18	−	796.762i	−0.364024	−	0.0425642i
$706$		0			0
$707$		−	18746.0i		−	0.997193i
$708$		0			0
$709$	−25539.6			−1.35283			−0.676416	−	0.736520i	$-0.736468\pi$
$709$	−25539.6			−1.35283			−0.676416	+	0.736520i	$0.736468\pi$
$710$		0			0
$711$	10354.2			0.546151
$712$		0			0
$713$			20.0252i			0.00105182i
$714$		0			0
$715$	1033.26	−	8836.76i	0.0540442	−	0.462204i
$716$		0			0
$717$		−	3030.02i		−	0.157822i
$718$		0			0
$719$	7353.45			0.381415			0.190708	−	0.981647i	$-0.438922\pi$
$719$	7353.45			0.381415			0.190708	+	0.981647i	$0.438922\pi$
$720$		0			0
$721$	21618.4			1.11666
$722$		0			0
$723$		−	12222.9i		−	0.628732i
$724$		0			0
$725$	−2984.07	−	707.510i	−0.152863	−	0.0362431i
$726$		0			0
$727$		−	21696.5i		−	1.10685i	−0.832900	−	0.553424i	$-0.813321\pi$
$727$		−	21696.5i		−	1.10685i	0.832900	−	0.553424i	$-0.186679\pi$
$728$		0			0
$729$	−729.000			−0.0370370
$730$		0			0
$731$	−30495.8			−1.54299
$732$		0			0
$733$			90.2714i			0.00454877i	0.999997	+	0.00227439i	$0.000723960\pi$
$733$			90.2714i			0.00454877i	−0.999997	+	0.00227439i	$0.999276\pi$
$734$		0			0
$735$	312.623	−	2673.66i	0.0156888	−	0.134176i
$736$		0			0
$737$		−	22440.6i		−	1.12159i
$738$		0			0
$739$	14273.1			0.710479			0.355239	−	0.934775i	$-0.384399\pi$
$739$	14273.1			0.710479			0.355239	+	0.934775i	$0.384399\pi$
$740$		0			0
$741$	2899.57			0.143750
$742$		0			0
$743$		−	15866.6i		−	0.783429i	−0.920087	−	0.391715i	$-0.871882\pi$
$743$		−	15866.6i		−	0.783429i	0.920087	−	0.391715i	$-0.128118\pi$
$744$		0			0
$745$	−16472.4	−	1926.07i	−0.810072	−	0.0947193i
$746$		0			0
$747$			10421.3i			0.510437i
$748$		0			0
$749$	12939.3			0.631232
$750$		0			0
$751$	26776.9			1.30107			0.650534	−	0.759477i	$-0.274545\pi$
$751$	26776.9			1.30107			0.650534	+	0.759477i	$0.274545\pi$
$752$		0			0
$753$		−	5321.95i		−	0.257560i
$754$		0			0
$755$	4378.37	+	511.950i	0.211053	+	0.0246778i
$756$		0			0
$757$		−	30478.0i		−	1.46333i	−0.681663	−	0.731666i	$-0.738743\pi$
$757$		−	30478.0i		−	1.46333i	0.681663	−	0.731666i	$-0.261257\pi$
$758$		0			0
$759$	194.512			0.00930218
$760$		0			0
$761$	29104.7			1.38639			0.693195	−	0.720750i	$-0.256202\pi$
$761$	29104.7			1.38639			0.693195	+	0.720750i	$0.256202\pi$
$762$		0			0
$763$			15968.2i			0.757649i
$764$		0			0
$765$	−970.116	+	8296.76i	−0.0458492	+	0.392118i
$766$		0			0
$767$		−	2220.69i		−	0.104543i
$768$		0			0
$769$	−4170.65			−0.195575			−0.0977876	−	0.995207i	$-0.531177\pi$
$769$	−4170.65			−0.195575			−0.0977876	+	0.995207i	$0.531177\pi$
$770$		0			0
$771$	−1988.35			−0.0928778
$772$		0			0
$773$		−	17738.5i		−	0.825367i	−0.910874	−	0.412684i	$-0.864591\pi$
$773$		−	17738.5i		−	0.825367i	0.910874	−	0.412684i	$-0.135409\pi$
$774$		0			0
$775$	−1510.47	−	358.125i	−0.0700099	−	0.0165990i
$776$		0			0
$777$		−	15847.1i		−	0.731677i
$778$		0			0
$779$	11839.1			0.544519
$780$		0			0
$781$	22448.1			1.02850
$782$		0			0
$783$			662.429i			0.0302341i
$784$		0			0
$785$	−2242.47	+	19178.4i	−0.101958	+	0.871982i
$786$		0			0
$787$			3807.92i			0.172475i	0.996275	+	0.0862374i	$0.0274844\pi$
$787$			3807.92i			0.172475i	−0.996275	+	0.0862374i	$0.972516\pi$
$788$		0			0
$789$	2136.94			0.0964220
$790$		0			0
$791$	−30607.3			−1.37582
$792$		0			0
$793$		−	9447.06i		−	0.423045i
$794$		0			0
$795$	−2049.75	−	239.671i	−0.0914430	−	0.0106922i
$796$		0			0
$797$		−	23840.3i		−	1.05956i	−0.848136	−	0.529779i	$-0.822275\pi$
$797$		−	23840.3i		−	1.05956i	0.848136	−	0.529779i	$-0.177725\pi$
$798$		0			0
$799$	16980.3			0.751841
$800$		0			0
$801$	872.775			0.0384994
$802$		0			0
$803$		−	40682.6i		−	1.78787i
$804$		0			0
$805$	290.249	+	33.9380i	0.0127080	+	0.00148591i
$806$		0			0
$807$		−	9409.24i		−	0.410435i
$808$		0			0
$809$	1984.22			0.0862316			0.0431158	−	0.999070i	$-0.486272\pi$
$809$	1984.22			0.0862316			0.0431158	+	0.999070i	$0.486272\pi$
$810$		0			0
$811$	9713.78			0.420588			0.210294	−	0.977638i	$-0.432558\pi$
$811$	9713.78			0.420588			0.210294	+	0.977638i	$0.432558\pi$
$812$		0			0
$813$		−	6827.08i		−	0.294509i
$814$		0			0
$815$	−2641.39	+	22590.1i	−0.113526	+	0.970914i
$816$		0			0
$817$		−	17940.5i		−	0.768246i
$818$		0			0
$819$	−2887.14			−0.123181
$820$		0			0
$821$	19235.4			0.817686			0.408843	−	0.912605i	$-0.365932\pi$
$821$	19235.4			0.817686			0.408843	+	0.912605i	$0.365932\pi$
$822$		0			0
$823$			12717.6i			0.538650i	0.963049	+	0.269325i	$0.0868005\pi$
$823$			12717.6i			0.538650i	−0.963049	+	0.269325i	$0.913199\pi$
$824$		0			0
$825$	−3478.61	+	14671.8i	−0.146800	+	0.619158i
$826$		0			0
$827$			6744.75i			0.283601i	0.989895	+	0.141800i	$0.0452891\pi$
$827$			6744.75i			0.283601i	−0.989895	+	0.141800i	$0.954711\pi$
$828$		0			0
$829$	3404.22			0.142622			0.0713108	−	0.997454i	$-0.477282\pi$
$829$	3404.22			0.142622			0.0713108	+	0.997454i	$0.477282\pi$
$830$		0			0
$831$	15513.0			0.647581
$832$		0			0
$833$			6662.52i			0.277122i
$834$		0			0
$835$	250.469	−	2142.09i	0.0103806	−	0.0887787i
$836$		0			0
$837$			335.306i			0.0138469i
$838$		0			0
$839$	21361.9			0.879015			0.439508	−	0.898239i	$-0.355153\pi$
$839$	21361.9			0.879015			0.439508	+	0.898239i	$0.355153\pi$
$840$		0			0
$841$	−23787.1			−0.975319
$842$		0			0
$843$			6720.41i			0.274571i
$844$		0			0
$845$	−20047.6	−	2344.11i	−0.816165	−	0.0954318i
$846$		0			0
$847$		−	4632.54i		−	0.187929i
$848$		0			0
$849$	−675.732			−0.0273158
$850$		0			0
$851$	−525.488			−0.0211674
$852$		0			0
$853$		−	10728.9i		−	0.430657i	−0.976542	−	0.215328i	$-0.930918\pi$
$853$		−	10728.9i		−	0.430657i	0.976542	−	0.215328i	$-0.0690822\pi$
$854$		0			0
$855$	−4880.93	−	570.712i	−0.195233	−	0.0228280i
$856$		0			0
$857$		−	42895.2i		−	1.70977i	−0.518817	−	0.854885i	$-0.673627\pi$
$857$		−	42895.2i		−	1.70977i	0.518817	−	0.854885i	$-0.326373\pi$
$858$		0			0
$859$	35530.5			1.41127			0.705637	−	0.708574i	$-0.250661\pi$
$859$	35530.5			1.41127			0.705637	+	0.708574i	$0.250661\pi$
$860$		0			0
$861$	−11788.4			−0.466605
$862$		0			0
$863$		−	5704.35i		−	0.225004i	−0.993652	−	0.112502i	$-0.964114\pi$
$863$		−	5704.35i		−	0.225004i	0.993652	−	0.112502i	$-0.0358865\pi$
$864$		0			0
$865$	−1609.94	+	13768.8i	−0.0632830	+	0.541218i
$866$		0			0
$867$		−	5935.78i		−	0.232514i
$868$		0			0
$869$	−46259.6			−1.80581
$870$		0			0
$871$	11045.0			0.429674
$872$		0			0
$873$			10371.4i			0.402082i
$874$		0			0
$875$	−7750.64	+	21286.1i	−0.299451	+	0.822403i
$876$		0			0
$877$		−	50249.0i		−	1.93476i	−0.253324	−	0.967382i	$-0.581524\pi$
$877$		−	50249.0i		−	1.93476i	0.253324	−	0.967382i	$-0.418476\pi$
$878$		0			0
$879$	−3419.58			−0.131217
$880$		0			0
$881$	−26864.5			−1.02734			−0.513672	−	0.857987i	$-0.671715\pi$
$881$	−26864.5			−1.02734			−0.513672	+	0.857987i	$0.671715\pi$
$882$		0			0
$883$			18942.1i			0.721918i	0.932582	+	0.360959i	$0.117551\pi$
$883$			18942.1i			0.721918i	−0.932582	+	0.360959i	$0.882449\pi$
$884$		0			0
$885$	−437.091	+	3738.15i	−0.0166019	+	0.141985i
$886$		0			0
$887$		−	25344.8i		−	0.959409i	−0.877430	−	0.479705i	$-0.840744\pi$
$887$		−	25344.8i		−	0.959409i	0.877430	−	0.479705i	$-0.159256\pi$
$888$		0			0
$889$	−10063.7			−0.379670
$890$		0			0
$891$	3256.96			0.122460
$892$		0			0
$893$			9989.40i			0.374337i
$894$		0			0
$895$	29281.5	+	3423.80i	1.09360	+	0.127871i
$896$		0			0
$897$			95.7370i			0.00356362i
$898$		0			0
$899$	304.686			0.0113035
$900$		0			0
$901$	5107.79			0.188863
$902$		0			0
$903$			17863.5i			0.658318i
$904$		0			0
$905$	36840.7	+	4307.67i	1.35318	+	0.158223i
$906$		0			0
$907$		−	4800.11i		−	0.175728i	−0.996132	−	0.0878639i	$-0.971996\pi$
$907$		−	4800.11i		−	0.175728i	0.996132	−	0.0878639i	$-0.0280041\pi$
$908$		0			0
$909$	−10408.4			−0.379786
$910$		0			0
$911$	−25731.7			−0.935819			−0.467909	−	0.883776i	$-0.654993\pi$
$911$	−25731.7			−0.935819			−0.467909	+	0.883776i	$0.654993\pi$
$912$		0			0
$913$		−	46559.4i		−	1.68772i
$914$		0			0
$915$	−1859.43	+	15902.5i	−0.0671812	+	0.574557i
$916$		0			0
$917$		−	41955.6i		−	1.51090i
$918$		0			0
$919$	12751.9			0.457722			0.228861	−	0.973459i	$-0.426500\pi$
$919$	12751.9			0.457722			0.228861	+	0.973459i	$0.426500\pi$
$920$		0			0
$921$	15734.6			0.562945
$922$		0			0
$923$			11048.7i			0.394012i
$924$		0			0
$925$	9397.69	−	39636.7i	0.334048	−	1.40892i
$926$		0			0
$927$		−	12003.3i		−	0.425285i
$928$		0			0
$929$	−15557.8			−0.549444			−0.274722	−	0.961524i	$-0.588586\pi$
$929$	−15557.8			−0.549444			−0.274722	+	0.961524i	$0.588586\pi$
$930$		0			0
$931$	−3919.51			−0.137977
$932$		0			0
$933$		−	15564.8i		−	0.546160i
$934$		0			0
$935$	4334.19	−	37067.5i	0.151597	−	1.29651i
$936$		0			0
$937$		−	23858.0i		−	0.831811i	−0.909408	−	0.415905i	$-0.863465\pi$
$937$		−	23858.0i		−	0.831811i	0.909408	−	0.415905i	$-0.136535\pi$
$938$		0			0
$939$	1459.97			0.0507394
$940$		0			0
$941$	−9748.00			−0.337700			−0.168850	−	0.985642i	$-0.554005\pi$
$941$	−9748.00			−0.337700			−0.168850	+	0.985642i	$0.554005\pi$
$942$		0			0
$943$			390.899i			0.0134989i
$944$		0			0
$945$	4860.00	+	568.265i	0.167297	+	0.0195616i
$946$		0			0
$947$			51537.0i			1.76845i	0.467057	+	0.884227i	$0.345314\pi$
$947$			51537.0i			1.76845i	−0.467057	+	0.884227i	$0.654686\pi$
$948$		0			0
$949$	20023.5			0.684923
$950$		0			0
$951$	−12656.6			−0.431566
$952$		0			0
$953$			5631.36i			0.191414i	0.995410	+	0.0957071i	$0.0305112\pi$
$953$			5631.36i			0.191414i	−0.995410	+	0.0957071i	$0.969489\pi$
$954$		0			0
$955$	6934.95	+	810.883i	0.234984	+	0.0274760i
$956$		0			0
$957$		−	2959.54i		−	0.0999668i
$958$		0			0
$959$	−26847.4			−0.904013
$960$		0			0
$961$	−29636.8			−0.994823
$962$		0			0
$963$		−	7184.36i		−	0.240408i
$964$		0			0
$965$	566.305	−	4843.24i	0.0188912	−	0.161564i
$966$		0			0
$967$		−	43360.9i		−	1.44198i	−0.692946	−	0.720989i	$-0.743688\pi$
$967$		−	43360.9i		−	1.44198i	0.692946	−	0.720989i	$-0.256312\pi$
$968$		0			0
$969$	12162.8			0.403226
$970$		0			0
$971$	−12920.0			−0.427007			−0.213503	−	0.976942i	$-0.568487\pi$
$971$	−12920.0			−0.427007			−0.213503	+	0.976942i	$0.568487\pi$
$972$		0			0
$973$			2484.19i			0.0818493i
$974$		0			0
$975$	−7221.29	−	1712.14i	−0.237196	−	0.0562382i
$976$		0			0
$977$		−	10650.4i		−	0.348759i	−0.984679	−	0.174379i	$-0.944208\pi$
$977$		−	10650.4i		−	0.348759i	0.984679	−	0.174379i	$-0.0557919\pi$
$978$		0			0
$979$	−3899.31			−0.127296
$980$		0			0
$981$	8866.07			0.288554
$982$		0			0
$983$			49450.3i			1.60450i	0.596991	+	0.802248i	$0.296363\pi$
$983$			49450.3i			1.60450i	−0.596991	+	0.802248i	$0.703637\pi$
$984$		0			0
$985$	−4357.31	+	37265.2i	−0.140950	+	1.20545i
$986$		0			0
$987$		−	9946.58i		−	0.320773i
$988$		0			0
$989$	592.351			0.0190452
$990$		0			0
$991$	9410.47			0.301648			0.150824	−	0.988561i	$-0.451807\pi$
$991$	9410.47			0.301648			0.150824	+	0.988561i	$0.451807\pi$
$992$		0			0
$993$		−	22319.8i		−	0.713291i
$994$		0			0
$995$	42195.2	+	4933.76i	1.34440	+	0.157197i
$996$		0			0
$997$		−	532.117i		−	0.0169030i	−0.999964	−	0.00845151i	$-0.997310\pi$
$997$		−	532.117i		−	0.0169030i	0.999964	−	0.00845151i	$-0.00269023\pi$
$998$		0			0
$999$	−8798.88			−0.278663

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.4.f.q.769.3		4
4.3	odd	2		960.4.f.p.769.1		4
5.4	even	2	inner	960.4.f.q.769.1		4
8.3	odd	2		240.4.f.f.49.4		4
8.5	even	2		15.4.b.a.4.4	yes	4
20.19	odd	2		960.4.f.p.769.3		4
24.5	odd	2		45.4.b.b.19.1		4
24.11	even	2		720.4.f.j.289.1		4
40.3	even	4		1200.4.a.bn.1.2		2
40.13	odd	4		75.4.a.f.1.2		2
40.19	odd	2		240.4.f.f.49.2		4
40.27	even	4		1200.4.a.bt.1.1		2
40.29	even	2		15.4.b.a.4.1	✓	4
40.37	odd	4		75.4.a.c.1.1		2
120.29	odd	2		45.4.b.b.19.4		4
120.53	even	4		225.4.a.i.1.1		2
120.59	even	2		720.4.f.j.289.2		4
120.77	even	4		225.4.a.o.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.4.b.a.4.1	✓	4	40.29	even	2
15.4.b.a.4.4	yes	4	8.5	even	2
45.4.b.b.19.1		4	24.5	odd	2
45.4.b.b.19.4		4	120.29	odd	2
75.4.a.c.1.1		2	40.37	odd	4
75.4.a.f.1.2		2	40.13	odd	4
225.4.a.i.1.1		2	120.53	even	4
225.4.a.o.1.2		2	120.77	even	4
240.4.f.f.49.2		4	40.19	odd	2
240.4.f.f.49.4		4	8.3	odd	2
720.4.f.j.289.1		4	24.11	even	2
720.4.f.j.289.2		4	120.59	even	2
960.4.f.p.769.1		4	4.3	odd	2
960.4.f.p.769.3		4	20.19	odd	2
960.4.f.q.769.1		4	5.4	even	2	inner
960.4.f.q.769.3		4	1.1	even	1	trivial
1200.4.a.bn.1.2		2	40.3	even	4
1200.4.a.bt.1.1		2	40.27	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	960.4.f.q.769.3

Level	$960$
Weight	$4$
Character	960.769
Analytic conductor	$56.642$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$