LMFDB - Embedded newform 960.3.j.e.319.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [960,3,Mod(319,960)] 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("960.319"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(960, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 1])) N = Newforms(chi, 3, names="a")

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

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,-4,0,0,0,24,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$26.1581053786$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.389136420864.4
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 5x^{6} + 24x^{4} + 80x^{2} + 256 $ x^8 + 5x^6 + 24x^4 + 80*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			319.8
Root			$0.656712 + 1.88911i$ of defining polynomial
Character	$\chi$	$=$	960.319
Dual form			960.3.j.e.319.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} +(3.27492 + 3.77822i) q^{5} -9.55505 q^{7} +3.00000 q^{9} +9.92480i q^{11} -7.55643i q^{13} +(5.67232 + 6.54406i) q^{15} -17.1903i q^{17} +26.1762i q^{19} -16.5498 q^{21} +1.67451 q^{23} +(-3.54983 + 24.7467i) q^{25} +5.19615 q^{27} +0.350497 q^{29} +46.0258i q^{31} +17.1903i q^{33} +(-31.2920 - 36.1010i) q^{35} +22.6693i q^{37} -13.0881i q^{39} -77.2990 q^{41} -41.7994 q^{43} +(9.82475 + 11.3346i) q^{45} -14.0866 q^{47} +42.2990 q^{49} -29.7744i q^{51} +22.6693i q^{53} +(-37.4980 + 32.5029i) q^{55} +45.3386i q^{57} +94.7802i q^{59} -38.0000 q^{61} -28.6652 q^{63} +(28.5498 - 24.7467i) q^{65} -29.8477 q^{67} +2.90033 q^{69} -7.19630i q^{71} -34.3805i q^{73} +(-6.14849 + 42.8625i) q^{75} -94.8320i q^{77} -46.0258i q^{79} +9.00000 q^{81} +24.1336 q^{83} +(64.9485 - 56.2967i) q^{85} +0.607078 q^{87} +100.199 q^{89} +72.2021i q^{91} +79.7191i q^{93} +(-98.8995 + 85.7250i) q^{95} -131.861i q^{97} +29.7744i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{5} + 24 q^{9} - 72 q^{21} + 32 q^{25} + 184 q^{29} - 256 q^{41} - 12 q^{45} - 24 q^{49} - 304 q^{61} + 168 q^{65} + 144 q^{69} + 72 q^{81} - 24 q^{85} + 560 q^{89}+O(q^{100}) $ 8 * q - 4 * q^5 + 24 * q^9 - 72 * q^21 + 32 * q^25 + 184 * q^29 - 256 * q^41 - 12 * q^45 - 24 * q^49 - 304 * q^61 + 168 * q^65 + 144 * q^69 + 72 * q^81 - 24 * q^85 + 560 * q^89

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
$3$	1.73205			0.577350
$3$	1.73205			0.577350
$4$		0			0
$5$	3.27492	+	3.77822i	0.654983	+	0.755643i
$5$	3.27492	+	3.77822i	0.654983	+	0.755643i
$6$		0			0
$7$	−9.55505			−1.36501			−0.682504	−	0.730882i	$-0.739109\pi$
$7$	−9.55505			−1.36501			−0.682504	+	0.730882i	$0.739109\pi$
$8$		0			0
$9$	3.00000			0.333333
$10$		0			0
$11$			9.92480i			0.902255i	0.892460	+	0.451127i	$0.148978\pi$
$11$			9.92480i			0.902255i	−0.892460	+	0.451127i	$0.851022\pi$
$12$		0			0
$13$		−	7.55643i		−	0.581264i	−0.956835	−	0.290632i	$-0.906134\pi$
$13$		−	7.55643i		−	0.581264i	0.956835	−	0.290632i	$-0.0938655\pi$
$14$		0			0
$15$	5.67232	+	6.54406i	0.378155	+	0.436271i
$16$		0			0
$17$		−	17.1903i		−	1.01119i	−0.862770	−	0.505596i	$-0.831273\pi$
$17$		−	17.1903i		−	1.01119i	0.862770	−	0.505596i	$-0.168727\pi$
$18$		0			0
$19$			26.1762i			1.37770i	0.724905	+	0.688849i	$0.241884\pi$
$19$			26.1762i			1.37770i	−0.724905	+	0.688849i	$0.758116\pi$
$20$		0			0
$21$	−16.5498			−0.788087
$22$		0			0
$23$	1.67451			0.0728047			0.0364023	−	0.999337i	$-0.488410\pi$
$23$	1.67451			0.0728047			0.0364023	+	0.999337i	$0.488410\pi$
$24$		0			0
$25$	−3.54983	+	24.7467i	−0.141993	+	0.989868i
$26$		0			0
$27$	5.19615			0.192450
$28$		0			0
$29$	0.350497			0.0120861			0.00604305	−	0.999982i	$-0.498076\pi$
$29$	0.350497			0.0120861			0.00604305	+	0.999982i	$0.498076\pi$
$30$		0			0
$31$			46.0258i			1.48470i	0.670010	+	0.742352i	$0.266290\pi$
$31$			46.0258i			1.48470i	−0.670010	+	0.742352i	$0.733710\pi$
$32$		0			0
$33$			17.1903i			0.520917i
$34$		0			0
$35$	−31.2920	−	36.1010i	−0.894057	−	1.03146i
$36$		0			0
$37$			22.6693i			0.612684i	0.951922	+	0.306342i	$0.0991051\pi$
$37$			22.6693i			0.612684i	−0.951922	+	0.306342i	$0.900895\pi$
$38$		0			0
$39$		−	13.0881i		−	0.335593i
$40$		0			0
$41$	−77.2990			−1.88534			−0.942671	−	0.333724i	$-0.891695\pi$
$41$	−77.2990			−1.88534			−0.942671	+	0.333724i	$0.891695\pi$
$42$		0			0
$43$	−41.7994			−0.972079			−0.486039	−	0.873937i	$-0.661559\pi$
$43$	−41.7994			−0.972079			−0.486039	+	0.873937i	$0.661559\pi$
$44$		0			0
$45$	9.82475	+	11.3346i	0.218328	+	0.251881i
$46$		0			0
$47$	−14.0866			−0.299715			−0.149857	−	0.988708i	$-0.547881\pi$
$47$	−14.0866			−0.299715			−0.149857	+	0.988708i	$0.547881\pi$
$48$		0			0
$49$	42.2990			0.863245
$50$		0			0
$51$		−	29.7744i		−	0.583812i
$52$		0			0
$53$			22.6693i			0.427723i	0.976864	+	0.213861i	$0.0686041\pi$
$53$			22.6693i			0.427723i	−0.976864	+	0.213861i	$0.931396\pi$
$54$		0			0
$55$	−37.4980	+	32.5029i	−0.681783	+	0.590962i
$56$		0			0
$57$			45.3386i			0.795414i
$58$		0			0
$59$			94.7802i			1.60644i	0.595680	+	0.803222i	$0.296883\pi$
$59$			94.7802i			1.60644i	−0.595680	+	0.803222i	$0.703117\pi$
$60$		0			0
$61$	−38.0000			−0.622951			−0.311475	−	0.950254i	$-0.600823\pi$
$61$	−38.0000			−0.622951			−0.311475	+	0.950254i	$0.600823\pi$
$62$		0			0
$63$	−28.6652			−0.455002
$64$		0			0
$65$	28.5498	−	24.7467i	0.439228	−	0.380718i
$66$		0			0
$67$	−29.8477			−0.445488			−0.222744	−	0.974877i	$-0.571501\pi$
$67$	−29.8477			−0.445488			−0.222744	+	0.974877i	$0.571501\pi$
$68$		0			0
$69$	2.90033			0.0420338
$70$		0			0
$71$		−	7.19630i		−	0.101356i	−0.998715	−	0.0506782i	$-0.983862\pi$
$71$		−	7.19630i		−	0.101356i	0.998715	−	0.0506782i	$-0.0161383\pi$
$72$		0			0
$73$		−	34.3805i		−	0.470966i	−0.971878	−	0.235483i	$-0.924333\pi$
$73$		−	34.3805i		−	0.470966i	0.971878	−	0.235483i	$-0.0756672\pi$
$74$		0			0
$75$	−6.14849	+	42.8625i	−0.0819799	+	0.571500i
$76$		0			0
$77$		−	94.8320i		−	1.23158i
$78$		0			0
$79$		−	46.0258i		−	0.582606i	−0.956631	−	0.291303i	$-0.905911\pi$
$79$		−	46.0258i		−	0.582606i	0.956631	−	0.291303i	$-0.0940887\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$	24.1336			0.290767			0.145383	−	0.989375i	$-0.453558\pi$
$83$	24.1336			0.290767			0.145383	+	0.989375i	$0.453558\pi$
$84$		0			0
$85$	64.9485	−	56.2967i	0.764100	−	0.662314i
$86$		0			0
$87$	0.607078			0.00697791
$88$		0			0
$89$	100.199			1.12584			0.562918	−	0.826513i	$-0.309679\pi$
$89$	100.199			1.12584			0.562918	+	0.826513i	$0.309679\pi$
$90$		0			0
$91$			72.2021i			0.793430i
$92$		0			0
$93$			79.7191i			0.857195i
$94$		0			0
$95$	−98.8995	+	85.7250i	−1.04105	+	0.902369i
$96$		0			0
$97$		−	131.861i		−	1.35939i	−0.733494	−	0.679696i	$-0.762112\pi$
$97$		−	131.861i		−	1.35939i	0.733494	−	0.679696i	$-0.237888\pi$
$98$		0			0
$99$			29.7744i			0.300752i
$100$		0			0
$101$	−29.4502			−0.291586			−0.145793	−	0.989315i	$-0.546573\pi$
$101$	−29.4502			−0.291586			−0.145793	+	0.989315i	$0.546573\pi$
$102$		0			0
$103$	−143.786			−1.39598			−0.697991	−	0.716107i	$-0.745922\pi$
$103$	−143.786			−1.39598			−0.697991	+	0.716107i	$0.745922\pi$
$104$		0			0
$105$	−54.1993	−	62.5289i	−0.516184	−	0.595513i
$106$		0			0
$107$	−35.1014			−0.328050			−0.164025	−	0.986456i	$-0.552448\pi$
$107$	−35.1014			−0.328050			−0.164025	+	0.986456i	$0.552448\pi$
$108$		0			0
$109$	151.498			1.38989			0.694947	−	0.719061i	$-0.255428\pi$
$109$	151.498			1.38989			0.694947	+	0.719061i	$0.255428\pi$
$110$		0			0
$111$			39.2644i			0.353733i
$112$		0			0
$113$			32.3031i			0.285868i	0.989732	+	0.142934i	$0.0456537\pi$
$113$			32.3031i			0.285868i	−0.989732	+	0.142934i	$0.954346\pi$
$114$		0			0
$115$	5.48387	+	6.32665i	0.0476858	+	0.0550143i
$116$		0			0
$117$		−	22.6693i		−	0.193755i
$118$		0			0
$119$			164.254i			1.38028i
$120$		0			0
$121$	22.4983			0.185937
$122$		0			0
$123$	−133.886			−1.08850
$124$		0			0
$125$	−105.124	+	67.6313i	−0.840990	+	0.541051i
$126$		0			0
$127$	192.053			1.51223			0.756116	−	0.654438i	$-0.227095\pi$
$127$	192.053			1.51223			0.756116	+	0.654438i	$0.227095\pi$
$128$		0			0
$129$	−72.3987			−0.561230
$130$		0			0
$131$			42.4277i			0.323876i	0.986801	+	0.161938i	$0.0517744\pi$
$131$			42.4277i			0.323876i	−0.986801	+	0.161938i	$0.948226\pi$
$132$		0			0
$133$		−	250.115i		−	1.88057i
$134$		0			0
$135$	17.0170	+	19.6322i	0.126052	+	0.145424i
$136$		0			0
$137$			206.854i			1.50988i	0.655791	+	0.754942i	$0.272335\pi$
$137$			206.854i			1.50988i	−0.655791	+	0.754942i	$0.727665\pi$
$138$		0			0
$139$		−	46.0258i		−	0.331121i	−0.986200	−	0.165561i	$-0.947057\pi$
$139$		−	46.0258i		−	0.331121i	0.986200	−	0.165561i	$-0.0529433\pi$
$140$		0			0
$141$	−24.3987			−0.173040
$142$		0			0
$143$	74.9961			0.524448
$144$		0			0
$145$	1.14785	+	1.32425i	0.00791619	+	0.00913277i
$146$		0			0
$147$	73.2640			0.498395
$148$		0			0
$149$	−11.6495			−0.0781846			−0.0390923	−	0.999236i	$-0.512447\pi$
$149$	−11.6495			−0.0781846			−0.0390923	+	0.999236i	$0.512447\pi$
$150$		0			0
$151$			125.424i			0.830624i	0.909679	+	0.415312i	$0.136328\pi$
$151$			125.424i			0.830624i	−0.909679	+	0.415312i	$0.863672\pi$
$152$		0			0
$153$		−	51.5708i		−	0.337064i
$154$		0			0
$155$	−173.896	+	150.731i	−1.12191	+	0.972457i
$156$		0			0
$157$			197.220i			1.25618i	0.778140	+	0.628090i	$0.216163\pi$
$157$			197.220i			1.25618i	−0.778140	+	0.628090i	$0.783837\pi$
$158$		0			0
$159$			39.2644i			0.246946i
$160$		0			0
$161$	−16.0000			−0.0993789
$162$		0			0
$163$	−18.4196			−0.113004			−0.0565018	−	0.998402i	$-0.517995\pi$
$163$	−18.4196			−0.113004			−0.0565018	+	0.998402i	$0.517995\pi$
$164$		0			0
$165$	−64.9485	+	56.2967i	−0.393627	+	0.341192i
$166$		0			0
$167$	−92.8920			−0.556240			−0.278120	−	0.960546i	$-0.589711\pi$
$167$	−92.8920			−0.556240			−0.278120	+	0.960546i	$0.589711\pi$
$168$		0			0
$169$	111.900			0.662132
$170$		0			0
$171$			78.5287i			0.459232i
$172$		0			0
$173$			117.501i			0.679198i	0.940570	+	0.339599i	$0.110291\pi$
$173$			117.501i			0.679198i	−0.940570	+	0.339599i	$0.889709\pi$
$174$		0			0
$175$	33.9189	−	236.456i	0.193822	−	1.35118i
$176$		0			0
$177$			164.164i			0.927481i
$178$		0			0
$179$		−	231.988i		−	1.29602i	−0.761631	−	0.648011i	$-0.775601\pi$
$179$		−	231.988i		−	1.29602i	0.761631	−	0.648011i	$-0.224399\pi$
$180$		0			0
$181$	218.096			1.20495			0.602476	−	0.798137i	$-0.294181\pi$
$181$	218.096			1.20495			0.602476	+	0.798137i	$0.294181\pi$
$182$		0			0
$183$	−65.8179			−0.359661
$184$		0			0
$185$	−85.6495	+	74.2401i	−0.462970	+	0.401298i
$186$		0			0
$187$	170.610			0.912352
$188$		0			0
$189$	−49.6495			−0.262696
$190$		0			0
$191$		−	137.208i		−	0.718366i	−0.933267	−	0.359183i	$-0.883055\pi$
$191$		−	137.208i		−	0.718366i	0.933267	−	0.359183i	$-0.116945\pi$
$192$		0			0
$193$		−	37.0290i		−	0.191860i	−0.995388	−	0.0959301i	$-0.969417\pi$
$193$		−	37.0290i		−	0.191860i	0.995388	−	0.0959301i	$-0.0305825\pi$
$194$		0			0
$195$	49.4498	−	42.8625i	0.253589	−	0.219808i
$196$		0			0
$197$			194.572i			0.987674i	0.869554	+	0.493837i	$0.164406\pi$
$197$			194.572i			0.987674i	−0.869554	+	0.493837i	$0.835594\pi$
$198$		0			0
$199$			176.037i			0.884610i	0.896865	+	0.442305i	$0.145839\pi$
$199$			176.037i			0.884610i	−0.896865	+	0.442305i	$0.854161\pi$
$200$		0			0
$201$	−51.6977			−0.257202
$202$		0			0
$203$	−3.34901			−0.0164976
$204$		0			0
$205$	−253.148	−	292.052i	−1.23487	−	1.42465i
$206$		0			0
$207$	5.02352			0.0242682
$208$		0			0
$209$	−259.794			−1.24303
$210$		0			0
$211$			20.7193i			0.0981955i	0.998794	+	0.0490978i	$0.0156346\pi$
$211$			20.7193i			0.0981955i	−0.998794	+	0.0490978i	$0.984365\pi$
$212$		0			0
$213$		−	12.4644i		−	0.0585181i
$214$		0			0
$215$	−136.890	−	157.927i	−0.636696	−	0.734545i
$216$		0			0
$217$		−	439.779i		−	2.02663i
$218$		0			0
$219$		−	59.5488i		−	0.271912i
$220$		0			0
$221$	−129.897			−0.587769
$222$		0			0
$223$	−97.0265			−0.435096			−0.217548	−	0.976050i	$-0.569806\pi$
$223$	−97.0265			−0.435096			−0.217548	+	0.976050i	$0.569806\pi$
$224$		0			0
$225$	−10.6495	+	74.2401i	−0.0473311	+	0.329956i
$226$		0			0
$227$	407.256			1.79408			0.897040	−	0.441948i	$-0.145713\pi$
$227$	407.256			1.79408			0.897040	+	0.441948i	$0.145713\pi$
$228$		0			0
$229$	7.89702			0.0344848			0.0172424	−	0.999851i	$-0.494511\pi$
$229$	7.89702			0.0344848			0.0172424	+	0.999851i	$0.494511\pi$
$230$		0			0
$231$		−	164.254i		−	0.711055i
$232$		0			0
$233$			28.1483i			0.120808i	0.998174	+	0.0604042i	$0.0192390\pi$
$233$			28.1483i			0.120808i	−0.998174	+	0.0604042i	$0.980761\pi$
$234$		0			0
$235$	−46.1324	−	53.2222i	−0.196308	−	0.226477i
$236$		0			0
$237$		−	79.7191i		−	0.336368i
$238$		0			0
$239$		−	296.005i		−	1.23851i	−0.785189	−	0.619257i	$-0.787434\pi$
$239$		−	296.005i		−	1.23851i	0.785189	−	0.619257i	$-0.212566\pi$
$240$		0			0
$241$	465.794			1.93276			0.966378	−	0.257127i	$-0.0827758\pi$
$241$	465.794			1.93276			0.966378	+	0.257127i	$0.0827758\pi$
$242$		0			0
$243$	15.5885			0.0641500
$244$		0			0
$245$	138.526	+	159.815i	0.565411	+	0.652305i
$246$		0			0
$247$	197.799			0.800806
$248$		0			0
$249$	41.8007			0.167874
$250$		0			0
$251$		−	141.676i		−	0.564445i	−0.959349	−	0.282223i	$-0.908928\pi$
$251$		−	141.676i		−	0.564445i	0.959349	−	0.282223i	$-0.0910716\pi$
$252$		0			0
$253$			16.6191i			0.0656883i
$254$		0			0
$255$	112.494	−	97.5087i	0.441153	−	0.382387i
$256$		0			0
$257$			41.7549i			0.162470i	0.996695	+	0.0812352i	$0.0258865\pi$
$257$			41.7549i			0.162470i	−0.996695	+	0.0812352i	$0.974113\pi$
$258$		0			0
$259$		−	216.606i		−	0.836318i
$260$		0			0
$261$	1.05149			0.00402870
$262$		0			0
$263$	203.283			0.772939			0.386469	−	0.922302i	$-0.373694\pi$
$263$	203.283			0.772939			0.386469	+	0.922302i	$0.373694\pi$
$264$		0			0
$265$	−85.6495	+	74.2401i	−0.323206	+	0.280151i
$266$		0			0
$267$	173.550			0.650001
$268$		0			0
$269$	244.048			0.907242			0.453621	−	0.891195i	$-0.350132\pi$
$269$	244.048			0.907242			0.453621	+	0.891195i	$0.350132\pi$
$270$		0			0
$271$		−	466.585i		−	1.72172i	−0.508845	−	0.860858i	$-0.669927\pi$
$271$		−	466.585i		−	1.72172i	0.508845	−	0.860858i	$-0.330073\pi$
$272$		0			0
$273$			125.058i			0.458087i
$274$		0			0
$275$	−245.606	−	35.2314i	−0.893113	−	0.128114i
$276$		0			0
$277$		−	494.181i		−	1.78405i	−0.451990	−	0.892023i	$-0.649286\pi$
$277$		−	494.181i		−	1.78405i	0.451990	−	0.892023i	$-0.350714\pi$
$278$		0			0
$279$			138.078i			0.494902i
$280$		0			0
$281$	−43.4020			−0.154455			−0.0772277	−	0.997013i	$-0.524607\pi$
$281$	−43.4020			−0.154455			−0.0772277	+	0.997013i	$0.524607\pi$
$282$		0			0
$283$	−310.785			−1.09818			−0.549090	−	0.835763i	$-0.685026\pi$
$283$	−310.785			−1.09818			−0.549090	+	0.835763i	$0.685026\pi$
$284$		0			0
$285$	−171.299	+	148.480i	−0.601049	+	0.520983i
$286$		0			0
$287$	738.596			2.57351
$288$		0			0
$289$	−6.50497			−0.0225085
$290$		0			0
$291$		−	228.390i		−	0.784845i
$292$		0			0
$293$			245.207i			0.836886i	0.908243	+	0.418443i	$0.137424\pi$
$293$			245.207i			0.836886i	−0.908243	+	0.418443i	$0.862576\pi$
$294$		0			0
$295$	−358.100	+	310.397i	−1.21390	+	1.05219i
$296$		0			0
$297$			51.5708i			0.173639i
$298$		0			0
$299$		−	12.6533i		−	0.0423187i
$300$		0			0
$301$	399.395			1.32689
$302$		0			0
$303$	−51.0092			−0.168347
$304$		0			0
$305$	−124.447	−	143.572i	−0.408022	−	0.470729i
$306$		0			0
$307$	337.514			1.09939			0.549697	−	0.835364i	$-0.314743\pi$
$307$	337.514			1.09939			0.549697	+	0.835364i	$0.314743\pi$
$308$		0			0
$309$	−249.045			−0.805970
$310$		0			0
$311$			427.756i			1.37542i	0.725986	+	0.687710i	$0.241384\pi$
$311$			427.756i			1.37542i	−0.725986	+	0.687710i	$0.758616\pi$
$312$		0			0
$313$			83.8739i			0.267968i	0.990984	+	0.133984i	$0.0427770\pi$
$313$			83.8739i			0.267968i	−0.990984	+	0.133984i	$0.957223\pi$
$314$		0			0
$315$	−93.8760	−	108.303i	−0.298019	−	0.343820i
$316$		0			0
$317$		−	112.204i		−	0.353957i	−0.984215	−	0.176978i	$-0.943368\pi$
$317$		−	112.204i		−	0.353957i	0.984215	−	0.176978i	$-0.0566323\pi$
$318$		0			0
$319$			3.47861i			0.0109047i
$320$		0			0
$321$	−60.7974			−0.189400
$322$		0			0
$323$	449.976			1.39312
$324$		0			0
$325$	186.997	+	26.8241i	0.575374	+	0.0825356i
$326$		0			0
$327$	262.403			0.802455
$328$		0			0
$329$	134.598			0.409113
$330$		0			0
$331$		−	132.621i		−	0.400666i	−0.979728	−	0.200333i	$-0.935798\pi$
$331$		−	132.621i		−	0.400666i	0.979728	−	0.200333i	$-0.0642025\pi$
$332$		0			0
$333$			68.0079i			0.204228i
$334$		0			0
$335$	−97.7487	−	112.771i	−0.291787	−	0.336630i
$336$		0			0
$337$			20.7739i			0.0616437i	0.999525	+	0.0308219i	$0.00981246\pi$
$337$			20.7739i			0.0616437i	−0.999525	+	0.0308219i	$0.990188\pi$
$338$		0			0
$339$			55.9506i			0.165046i
$340$		0			0
$341$	−456.797			−1.33958
$342$		0			0
$343$	64.0283			0.186672
$344$		0			0
$345$	9.49834	+	10.9581i	0.0275314	+	0.0317625i
$346$		0			0
$347$	8.89616			0.0256374			0.0128187	−	0.999918i	$-0.495920\pi$
$347$	8.89616			0.0256374			0.0128187	+	0.999918i	$0.495920\pi$
$348$		0			0
$349$	−19.4020			−0.0555931			−0.0277965	−	0.999614i	$-0.508849\pi$
$349$	−19.4020			−0.0555931			−0.0277965	+	0.999614i	$0.508849\pi$
$350$		0			0
$351$		−	39.2644i		−	0.111864i
$352$		0			0
$353$			80.2902i			0.227451i	0.993512	+	0.113726i	$0.0362784\pi$
$353$			80.2902i			0.227451i	−0.993512	+	0.113726i	$0.963722\pi$
$354$		0			0
$355$	27.1892	−	23.5673i	0.0765892	−	0.0663867i
$356$		0			0
$357$			284.496i			0.796907i
$358$		0			0
$359$		−	314.115i		−	0.874972i	−0.899225	−	0.437486i	$-0.855869\pi$
$359$		−	314.115i		−	0.874972i	0.899225	−	0.437486i	$-0.144131\pi$
$360$		0			0
$361$	−324.196			−0.898050
$362$		0			0
$363$	38.9683			0.107351
$364$		0			0
$365$	129.897	−	112.593i	0.355882	−	0.308475i
$366$		0			0
$367$	476.800			1.29918			0.649592	−	0.760283i	$-0.274940\pi$
$367$	476.800			1.29918			0.649592	+	0.760283i	$0.274940\pi$
$368$		0			0
$369$	−231.897			−0.628447
$370$		0			0
$371$		−	216.606i		−	0.583844i
$372$		0			0
$373$			86.1333i			0.230920i	0.993312	+	0.115460i	$0.0368343\pi$
$373$			86.1333i			0.230920i	−0.993312	+	0.115460i	$0.963166\pi$
$374$		0			0
$375$	−182.080	+	117.141i	−0.485546	+	0.312376i
$376$		0			0
$377$		−	2.64850i		−	0.00702521i
$378$		0			0
$379$			638.035i			1.68347i	0.539891	+	0.841735i	$0.318466\pi$
$379$			638.035i			1.68347i	−0.539891	+	0.841735i	$0.681534\pi$
$380$		0			0
$381$	332.646			0.873087
$382$		0			0
$383$	−216.742			−0.565907			−0.282953	−	0.959134i	$-0.591314\pi$
$383$	−216.742			−0.565907			−0.282953	+	0.959134i	$0.591314\pi$
$384$		0			0
$385$	358.296	−	310.567i	0.930638	−	0.806667i
$386$		0			0
$387$	−125.398			−0.324026
$388$		0			0
$389$	−476.640			−1.22529			−0.612647	−	0.790356i	$-0.709895\pi$
$389$	−476.640			−1.22529			−0.612647	+	0.790356i	$0.709895\pi$
$390$		0			0
$391$		−	28.7852i		−	0.0736195i
$392$		0			0
$393$			73.4869i			0.186990i
$394$		0			0
$395$	173.896	−	150.731i	0.440242	−	0.381597i
$396$		0			0
$397$		−	43.0792i		−	0.108512i	−0.998527	−	0.0542559i	$-0.982721\pi$
$397$		−	43.0792i		−	0.108512i	0.998527	−	0.0542559i	$-0.0172787\pi$
$398$		0			0
$399$		−	433.213i		−	1.08575i
$400$		0			0
$401$	−168.694			−0.420684			−0.210342	−	0.977628i	$-0.567458\pi$
$401$	−168.694			−0.420684			−0.210342	+	0.977628i	$0.567458\pi$
$402$		0			0
$403$	347.791			0.863005
$404$		0			0
$405$	29.4743	+	34.0039i	0.0727759	+	0.0839604i
$406$		0			0
$407$	−224.988			−0.552797
$408$		0			0
$409$	373.890			0.914157			0.457079	−	0.889426i	$-0.348896\pi$
$409$	373.890			0.914157			0.457079	+	0.889426i	$0.348896\pi$
$410$		0			0
$411$			358.282i			0.871732i
$412$		0			0
$413$		−	905.630i		−	2.19281i
$414$		0			0
$415$	79.0356	+	91.1820i	0.190447	+	0.219716i
$416$		0			0
$417$		−	79.7191i		−	0.191173i
$418$		0			0
$419$			87.5839i			0.209031i	0.994523	+	0.104515i	$0.0333291\pi$
$419$			87.5839i			0.209031i	−0.994523	+	0.104515i	$0.966671\pi$
$420$		0			0
$421$	−70.3023			−0.166989			−0.0834944	−	0.996508i	$-0.526608\pi$
$421$	−70.3023			−0.166989			−0.0834944	+	0.996508i	$0.526608\pi$
$422$		0			0
$423$	−42.2597			−0.0999048
$424$		0			0
$425$	425.402	+	61.0226i	1.00095	+	0.143583i
$426$		0			0
$427$	363.092			0.850332
$428$		0			0
$429$	129.897			0.302790
$430$		0			0
$431$		−	247.370i		−	0.573944i	−0.957939	−	0.286972i	$-0.907351\pi$
$431$		−	247.370i		−	0.573944i	0.957939	−	0.286972i	$-0.0926487\pi$
$432$		0			0
$433$		−	636.247i		−	1.46939i	−0.678397	−	0.734696i	$-0.737325\pi$
$433$		−	636.247i		−	1.46939i	0.678397	−	0.734696i	$-0.262675\pi$
$434$		0			0
$435$	1.98813	+	2.29367i	0.00457041	+	0.00527281i
$436$		0			0
$437$			43.8323i			0.100303i
$438$		0			0
$439$		−	769.786i		−	1.75350i	−0.480947	−	0.876750i	$-0.659707\pi$
$439$		−	769.786i		−	1.75350i	0.480947	−	0.876750i	$-0.340293\pi$
$440$		0			0
$441$	126.897			0.287748
$442$		0			0
$443$	−612.214			−1.38197			−0.690986	−	0.722868i	$-0.742823\pi$
$443$	−612.214			−1.38197			−0.690986	+	0.722868i	$0.742823\pi$
$444$		0			0
$445$	328.145	+	378.575i	0.737403	+	0.850730i
$446$		0			0
$447$	−20.1775			−0.0451399
$448$		0			0
$449$	175.897			0.391753			0.195876	−	0.980629i	$-0.437245\pi$
$449$	175.897			0.391753			0.195876	+	0.980629i	$0.437245\pi$
$450$		0			0
$451$		−	767.177i		−	1.70106i
$452$		0			0
$453$			217.241i			0.479561i
$454$		0			0
$455$	−272.795	+	236.456i	−0.599550	+	0.519683i
$456$		0			0
$457$			365.357i			0.799469i	0.916631	+	0.399734i	$0.130898\pi$
$457$			365.357i			0.799469i	−0.916631	+	0.399734i	$0.869102\pi$
$458$		0			0
$459$		−	89.3232i		−	0.194604i
$460$		0			0
$461$	−308.350			−0.668873			−0.334437	−	0.942418i	$-0.608546\pi$
$461$	−308.350			−0.668873			−0.334437	+	0.942418i	$0.608546\pi$
$462$		0			0
$463$	92.6302			0.200065			0.100033	−	0.994984i	$-0.468105\pi$
$463$	92.6302			0.200065			0.100033	+	0.994984i	$0.468105\pi$
$464$		0			0
$465$	−301.196	+	261.073i	−0.647733	+	0.561448i
$466$		0			0
$467$	606.103			1.29786			0.648932	−	0.760846i	$-0.275216\pi$
$467$	606.103			1.29786			0.648932	+	0.760846i	$0.275216\pi$
$468$		0			0
$469$	285.196			0.608094
$470$		0			0
$471$			341.596i			0.725256i
$472$		0			0
$473$		−	414.851i		−	0.877063i
$474$		0			0
$475$	−647.776	−	92.9214i	−1.36374	−	0.195624i
$476$		0			0
$477$			68.0079i			0.142574i
$478$		0			0
$479$			138.947i			0.290078i	0.989426	+	0.145039i	$0.0463307\pi$
$479$			138.947i			0.290078i	−0.989426	+	0.145039i	$0.953669\pi$
$480$		0			0
$481$	171.299			0.356131
$482$		0			0
$483$	−27.7128			−0.0573764
$484$		0			0
$485$	498.199	−	431.834i	1.02722	−	0.890379i
$486$		0			0
$487$	201.243			0.413230			0.206615	−	0.978422i	$-0.433755\pi$
$487$	201.243			0.413230			0.206615	+	0.978422i	$0.433755\pi$
$488$		0			0
$489$	−31.9036			−0.0652426
$490$		0			0
$491$			347.368i			0.707470i	0.935346	+	0.353735i	$0.115089\pi$
$491$			347.368i			0.707470i	−0.935346	+	0.353735i	$0.884911\pi$
$492$		0			0
$493$		−	6.02513i		−	0.0122214i
$494$		0			0
$495$	−112.494	+	97.5087i	−0.227261	+	0.196987i
$496$		0			0
$497$			68.7610i			0.138352i
$498$		0			0
$499$		−	672.277i		−	1.34725i	−0.739074	−	0.673625i	$-0.764736\pi$
$499$		−	672.277i		−	1.34725i	0.739074	−	0.673625i	$-0.235264\pi$
$500$		0			0
$501$	−160.894			−0.321145
$502$		0			0
$503$	−436.350			−0.867496			−0.433748	−	0.901034i	$-0.642809\pi$
$503$	−436.350			−0.867496			−0.433748	+	0.901034i	$0.642809\pi$
$504$		0			0
$505$	−96.4469	−	111.269i	−0.190984	−	0.220335i
$506$		0			0
$507$	193.817			0.382282
$508$		0			0
$509$	109.547			0.215219			0.107610	−	0.994193i	$-0.465680\pi$
$509$	109.547			0.215219			0.107610	+	0.994193i	$0.465680\pi$
$510$		0			0
$511$			328.508i			0.642872i
$512$		0			0
$513$			136.016i			0.265138i
$514$		0			0
$515$	−470.888	−	543.255i	−0.914345	−	1.05486i
$516$		0			0
$517$		−	139.807i		−	0.270419i
$518$		0			0
$519$			203.518i			0.392135i
$520$		0			0
$521$	−743.100			−1.42629			−0.713147	−	0.701014i	$-0.752731\pi$
$521$	−743.100			−1.42629			−0.713147	+	0.701014i	$0.752731\pi$
$522$		0			0
$523$	−366.211			−0.700212			−0.350106	−	0.936710i	$-0.613854\pi$
$523$	−366.211			−0.700212			−0.350106	+	0.936710i	$0.613854\pi$
$524$		0			0
$525$	58.7492	−	409.554i	0.111903	−	0.780102i
$526$		0			0
$527$	791.196			1.50132
$528$		0			0
$529$	−526.196			−0.994699
$530$		0			0
$531$			284.341i			0.535481i
$532$		0			0
$533$			584.105i			1.09588i
$534$		0			0
$535$	−114.954	−	132.621i	−0.214867	−	0.247889i
$536$		0			0
$537$		−	401.815i		−	0.748259i
$538$		0			0
$539$			419.809i			0.778867i
$540$		0			0
$541$	−170.688			−0.315504			−0.157752	−	0.987479i	$-0.550425\pi$
$541$	−170.688			−0.315504			−0.157752	+	0.987479i	$0.550425\pi$
$542$		0			0
$543$	377.754			0.695680
$544$		0			0
$545$	496.145	+	572.393i	0.910357	+	1.05026i
$546$		0			0
$547$	507.600			0.927971			0.463986	−	0.885843i	$-0.346419\pi$
$547$	507.600			0.927971			0.463986	+	0.885843i	$0.346419\pi$
$548$		0			0
$549$	−114.000			−0.207650
$550$		0			0
$551$			9.17469i			0.0166510i
$552$		0			0
$553$			439.779i			0.795261i
$554$		0			0
$555$	−148.349	+	128.588i	−0.267296	+	0.231689i
$556$		0			0
$557$			788.492i			1.41561i	0.706410	+	0.707803i	$0.250314\pi$
$557$			788.492i			1.41561i	−0.706410	+	0.707803i	$0.749686\pi$
$558$		0			0
$559$			315.854i			0.565035i
$560$		0			0
$561$	295.505			0.526747
$562$		0			0
$563$	−936.102			−1.66270			−0.831351	−	0.555747i	$-0.812432\pi$
$563$	−936.102			−1.66270			−0.831351	+	0.555747i	$0.812432\pi$
$564$		0			0
$565$	−122.048	+	105.790i	−0.216014	+	0.187239i
$566$		0			0
$567$	−85.9955			−0.151667
$568$		0			0
$569$	−95.4983			−0.167835			−0.0839177	−	0.996473i	$-0.526743\pi$
$569$	−95.4983			−0.167835			−0.0839177	+	0.996473i	$0.526743\pi$
$570$		0			0
$571$			889.123i			1.55713i	0.627562	+	0.778566i	$0.284053\pi$
$571$			889.123i			1.55713i	−0.627562	+	0.778566i	$0.715947\pi$
$572$		0			0
$573$		−	237.651i		−	0.414749i
$574$		0			0
$575$	−5.94422	+	41.4385i	−0.0103378	+	0.0720670i
$576$		0			0
$577$			522.147i			0.904934i	0.891781	+	0.452467i	$0.149456\pi$
$577$			522.147i			0.904934i	−0.891781	+	0.452467i	$0.850544\pi$
$578$		0			0
$579$		−	64.1361i		−	0.110771i
$580$		0			0
$581$	−230.598			−0.396898
$582$		0			0
$583$	−224.988			−0.385915
$584$		0			0
$585$	85.6495	−	74.2401i	0.146409	−	0.126906i
$586$		0			0
$587$	95.8440			0.163278			0.0816388	−	0.996662i	$-0.473985\pi$
$587$	95.8440			0.163278			0.0816388	+	0.996662i	$0.473985\pi$
$588$		0			0
$589$	−1204.78			−2.04547
$590$		0			0
$591$			337.008i			0.570234i
$592$		0			0
$593$			878.624i			1.48166i	0.671693	+	0.740829i	$0.265567\pi$
$593$			878.624i			1.48166i	−0.671693	+	0.740829i	$0.734433\pi$
$594$		0			0
$595$	−620.586	+	537.918i	−1.04300	+	0.904063i
$596$		0			0
$597$			304.906i			0.510730i
$598$		0			0
$599$			967.652i			1.61545i	0.589563	+	0.807723i	$0.299300\pi$
$599$			967.652i			1.61545i	−0.589563	+	0.807723i	$0.700700\pi$
$600$		0			0
$601$	279.704			0.465398			0.232699	−	0.972549i	$-0.425244\pi$
$601$	279.704			0.465398			0.232699	+	0.972549i	$0.425244\pi$
$602$		0			0
$603$	−89.5430			−0.148496
$604$		0			0
$605$	73.6802	+	85.0036i	0.121785	+	0.140502i
$606$		0			0
$607$	−747.564			−1.23157			−0.615786	−	0.787914i	$-0.711161\pi$
$607$	−747.564			−1.23157			−0.615786	+	0.787914i	$0.711161\pi$
$608$		0			0
$609$	−5.80066			−0.00952490
$610$		0			0
$611$			106.444i			0.174213i
$612$		0			0
$613$		−	457.152i		−	0.745761i	−0.927879	−	0.372881i	$-0.878370\pi$
$613$		−	457.152i		−	0.745761i	0.927879	−	0.372881i	$-0.121630\pi$
$614$		0			0
$615$	−438.465	−	505.850i	−0.712951	−	0.822520i
$616$		0			0
$617$		−	764.888i		−	1.23969i	−0.784725	−	0.619844i	$-0.787196\pi$
$617$		−	764.888i		−	1.23969i	0.784725	−	0.619844i	$-0.212804\pi$
$618$		0			0
$619$			365.359i			0.590240i	0.955460	+	0.295120i	$0.0953597\pi$
$619$			365.359i			0.590240i	−0.955460	+	0.295120i	$0.904640\pi$
$620$		0			0
$621$	8.70099			0.0140113
$622$		0			0
$623$	−957.410			−1.53677
$624$		0			0
$625$	−599.797	−	175.693i	−0.959676	−	0.281109i
$626$		0			0
$627$	−449.976			−0.717666
$628$		0			0
$629$	389.691			0.619541
$630$		0			0
$631$			62.1578i			0.0985067i	0.998786	+	0.0492534i	$0.0156842\pi$
$631$			62.1578i			0.0985067i	−0.998786	+	0.0492534i	$0.984316\pi$
$632$		0			0
$633$			35.8868i			0.0566932i
$634$		0			0
$635$	628.959	+	725.619i	0.990486	+	1.14271i
$636$		0			0
$637$		−	319.630i		−	0.501773i
$638$		0			0
$639$		−	21.5889i		−	0.0337855i
$640$		0			0
$641$	−1111.69			−1.73431			−0.867154	−	0.498041i	$-0.834053\pi$
$641$	−1111.69			−1.73431			−0.867154	+	0.498041i	$0.834053\pi$
$642$		0			0
$643$	468.983			0.729367			0.364683	−	0.931132i	$-0.381177\pi$
$643$	468.983			0.729367			0.364683	+	0.931132i	$0.381177\pi$
$644$		0			0
$645$	−237.100	−	273.538i	−0.367596	−	0.424090i
$646$		0			0
$647$	96.7647			0.149559			0.0747795	−	0.997200i	$-0.476175\pi$
$647$	96.7647			0.149559			0.0747795	+	0.997200i	$0.476175\pi$
$648$		0			0
$649$	−940.675			−1.44942
$650$		0			0
$651$		−	761.720i		−	1.17008i
$652$		0			0
$653$			920.353i			1.40942i	0.709494	+	0.704712i	$0.248924\pi$
$653$			920.353i			1.40942i	−0.709494	+	0.704712i	$0.751076\pi$
$654$		0			0
$655$	−160.301	+	138.947i	−0.244734	+	0.212133i
$656$		0			0
$657$		−	103.142i		−	0.156989i
$658$		0			0
$659$			591.020i			0.896844i	0.893822	+	0.448422i	$0.148014\pi$
$659$			591.020i			0.896844i	−0.893822	+	0.448422i	$0.851986\pi$
$660$		0			0
$661$	306.193			0.463226			0.231613	−	0.972808i	$-0.425600\pi$
$661$	306.193			0.463226			0.231613	+	0.972808i	$0.425600\pi$
$662$		0			0
$663$	−224.988			−0.339349
$664$		0			0
$665$	944.990	−	819.107i	1.42104	−	1.23174i
$666$		0			0
$667$	0.586909			0.000879924
$668$		0			0
$669$	−168.055			−0.251203
$670$		0			0
$671$		−	377.142i		−	0.562060i
$672$		0			0
$673$		−	556.892i		−	0.827476i	−0.910396	−	0.413738i	$-0.864223\pi$
$673$		−	556.892i		−	0.827476i	0.910396	−	0.413738i	$-0.135777\pi$
$674$		0			0
$675$	−18.4455	+	128.588i	−0.0273266	+	0.190500i
$676$		0			0
$677$			58.1920i			0.0859557i	0.999076	+	0.0429779i	$0.0136845\pi$
$677$			58.1920i			0.0859557i	−0.999076	+	0.0429779i	$0.986316\pi$
$678$		0			0
$679$			1259.94i			1.85558i
$680$		0			0
$681$	705.389			1.03581
$682$		0			0
$683$	−357.274			−0.523096			−0.261548	−	0.965191i	$-0.584233\pi$
$683$	−357.274			−0.523096			−0.261548	+	0.965191i	$0.584233\pi$
$684$		0			0
$685$	−781.540	+	677.430i	−1.14093	+	0.988949i
$686$		0			0
$687$	13.6780			0.0199098
$688$		0			0
$689$	171.299			0.248620
$690$		0			0
$691$			614.707i			0.889590i	0.895632	+	0.444795i	$0.146724\pi$
$691$			614.707i			0.889590i	−0.895632	+	0.444795i	$0.853276\pi$
$692$		0			0
$693$		−	284.496i		−	0.410528i
$694$		0			0
$695$	173.896	−	150.731i	0.250210	−	0.216879i
$696$		0			0
$697$			1328.79i			1.90644i
$698$		0			0
$699$			48.7543i			0.0697487i
$700$		0			0
$701$	886.028			1.26395			0.631975	−	0.774989i	$-0.282245\pi$
$701$	886.028			1.26395			0.631975	+	0.774989i	$0.282245\pi$
$702$		0			0
$703$	−593.397			−0.844093
$704$		0			0
$705$	−79.9036	−	92.1835i	−0.113338	−	0.130757i
$706$		0			0
$707$	281.398			0.398017
$708$		0			0
$709$	760.887			1.07318			0.536592	−	0.843842i	$-0.319712\pi$
$709$	760.887			1.07318			0.536592	+	0.843842i	$0.319712\pi$
$710$		0			0
$711$		−	138.078i		−	0.194202i
$712$		0			0
$713$			77.0706i			0.108093i
$714$		0			0
$715$	245.606	+	283.351i	0.343505	+	0.396296i
$716$		0			0
$717$		−	512.695i		−	0.715056i
$718$		0			0
$719$			575.877i			0.800942i	0.916309	+	0.400471i	$0.131154\pi$
$719$			575.877i			0.800942i	−0.916309	+	0.400471i	$0.868846\pi$
$720$		0			0
$721$	1373.88			1.90553
$722$		0			0
$723$	806.779			1.11588
$724$		0			0
$725$	−1.24421	+	8.67363i	−0.00171615	+	0.0119636i
$726$		0			0
$727$	327.332			0.450250			0.225125	−	0.974330i	$-0.427721\pi$
$727$	327.332			0.450250			0.225125	+	0.974330i	$0.427721\pi$
$728$		0			0
$729$	27.0000			0.0370370
$730$		0			0
$731$			718.542i			0.982958i
$732$		0			0
$733$			947.567i			1.29272i	0.763031	+	0.646362i	$0.223710\pi$
$733$			947.567i			1.29272i	−0.763031	+	0.646362i	$0.776290\pi$
$734$		0			0
$735$	239.934	+	276.807i	0.326440	+	0.376609i
$736$		0			0
$737$		−	296.232i		−	0.401943i
$738$		0			0
$739$			183.234i			0.247948i	0.992285	+	0.123974i	$0.0395640\pi$
$739$			183.234i			0.247948i	−0.992285	+	0.123974i	$0.960436\pi$
$740$		0			0
$741$	342.598			0.462345
$742$		0			0
$743$	183.712			0.247258			0.123629	−	0.992329i	$-0.460547\pi$
$743$	183.712			0.247258			0.123629	+	0.992329i	$0.460547\pi$
$744$		0			0
$745$	−38.1512	−	44.0143i	−0.0512096	−	0.0590797i
$746$		0			0
$747$	72.4009			0.0969222
$748$		0			0
$749$	335.395			0.447791
$750$		0			0
$751$			345.748i			0.460384i	0.973145	+	0.230192i	$0.0739354\pi$
$751$			345.748i			0.460384i	−0.973145	+	0.230192i	$0.926065\pi$
$752$		0			0
$753$		−	245.390i		−	0.325882i
$754$		0			0
$755$	−473.880	+	410.754i	−0.627656	+	0.544045i
$756$		0			0
$757$		−	549.335i		−	0.725674i	−0.931853	−	0.362837i	$-0.881808\pi$
$757$		−	549.335i		−	0.725674i	0.931853	−	0.362837i	$-0.118192\pi$
$758$		0			0
$759$			28.7852i			0.0379252i
$760$		0			0
$761$	251.485			0.330467			0.165233	−	0.986255i	$-0.447162\pi$
$761$	251.485			0.330467			0.165233	+	0.986255i	$0.447162\pi$
$762$		0			0
$763$	−1447.57			−1.89721
$764$		0			0
$765$	194.846	−	168.890i	0.254700	−	0.220771i
$766$		0			0
$767$	716.200			0.933768
$768$		0			0
$769$	583.691			0.759026			0.379513	−	0.925186i	$-0.376092\pi$
$769$	583.691			0.759026			0.379513	+	0.925186i	$0.376092\pi$
$770$		0			0
$771$			72.3216i			0.0938024i
$772$		0			0
$773$		−	1328.04i		−	1.71803i	−0.511951	−	0.859015i	$-0.671077\pi$
$773$		−	1328.04i		−	1.71803i	0.511951	−	0.859015i	$-0.328923\pi$
$774$		0			0
$775$	−1138.99	−	163.384i	−1.46966	−	0.210818i
$776$		0			0
$777$		−	375.173i		−	0.482848i
$778$		0			0
$779$		−	2023.40i		−	2.59743i
$780$		0			0
$781$	71.4219			0.0914492
$782$		0			0
$783$	1.82123			0.00232597
$784$		0			0
$785$	−745.141	+	645.880i	−0.949224	+	0.822778i
$786$		0			0
$787$	−1318.83			−1.67577			−0.837883	−	0.545850i	$-0.816207\pi$
$787$	−1318.83			−1.67577			−0.837883	+	0.545850i	$0.816207\pi$
$788$		0			0
$789$	352.096			0.446256
$790$		0			0
$791$		−	308.658i		−	0.390212i
$792$		0			0
$793$			287.144i			0.362099i
$794$		0			0
$795$	−148.349	+	128.588i	−0.186603	+	0.161745i
$796$		0			0
$797$		−	1277.40i		−	1.60276i	−0.598154	−	0.801381i	$-0.704099\pi$
$797$		−	1277.40i		−	1.60276i	0.598154	−	0.801381i	$-0.295901\pi$
$798$		0			0
$799$			242.152i			0.303069i
$800$		0			0
$801$	300.598			0.375278
$802$		0			0
$803$	341.220			0.424931
$804$		0			0
$805$	−52.3987	−	60.4515i	−0.0650915	−	0.0750950i
$806$		0			0
$807$	422.704			0.523797
$808$		0			0
$809$	321.093			0.396901			0.198451	−	0.980111i	$-0.436409\pi$
$809$	321.093			0.396901			0.198451	+	0.980111i	$0.436409\pi$
$810$		0			0
$811$		−	946.932i		−	1.16761i	−0.811894	−	0.583805i	$-0.801563\pi$
$811$		−	946.932i		−	1.16761i	0.811894	−	0.583805i	$-0.198437\pi$
$812$		0			0
$813$		−	808.149i		−	0.994033i
$814$		0			0
$815$	−60.3226	−	69.5931i	−0.0740154	−	0.0853904i
$816$		0			0
$817$		−	1094.15i		−	1.33923i
$818$		0			0
$819$			216.606i			0.264477i
$820$		0			0
$821$	−1169.34			−1.42429			−0.712144	−	0.702033i	$-0.752276\pi$
$821$	−1169.34			−1.42429			−0.712144	+	0.702033i	$0.752276\pi$
$822$		0			0
$823$	1251.71			1.52091			0.760457	−	0.649389i	$-0.224975\pi$
$823$	1251.71			1.52091			0.760457	+	0.649389i	$0.224975\pi$
$824$		0			0
$825$	−425.402	−	61.0226i	−0.515639	−	0.0739668i
$826$		0			0
$827$	−892.104			−1.07872			−0.539362	−	0.842074i	$-0.681334\pi$
$827$	−892.104			−1.07872			−0.539362	+	0.842074i	$0.681334\pi$
$828$		0			0
$829$	−998.688			−1.20469			−0.602345	−	0.798236i	$-0.705767\pi$
$829$	−998.688			−1.20469			−0.602345	+	0.798236i	$0.705767\pi$
$830$		0			0
$831$		−	855.946i		−	1.03002i
$832$		0			0
$833$		−	727.131i		−	0.872906i
$834$		0			0
$835$	−304.214	−	350.966i	−0.364328	−	0.420319i
$836$		0			0
$837$			239.157i			0.285732i
$838$		0			0
$839$		−	610.359i		−	0.727484i	−0.931500	−	0.363742i	$-0.881499\pi$
$839$		−	610.359i		−	0.727484i	0.931500	−	0.363742i	$-0.118501\pi$
$840$		0			0
$841$	−840.877			−0.999854
$842$		0			0
$843$	−75.1744			−0.0891749
$844$		0			0
$845$	366.464	+	422.784i	0.433686	+	0.500336i
$846$		0			0
$847$	−214.973			−0.253805
$848$		0			0
$849$	−538.296			−0.634035
$850$		0			0
$851$			37.9599i			0.0446062i
$852$		0			0
$853$			832.689i			0.976189i	0.872791	+	0.488094i	$0.162308\pi$
$853$			832.689i			0.976189i	−0.872791	+	0.488094i	$0.837692\pi$
$854$		0			0
$855$	−296.699	+	257.175i	−0.347016	+	0.300790i
$856$		0			0
$857$		−	149.415i		−	0.174347i	−0.996193	−	0.0871735i	$-0.972217\pi$
$857$		−	149.415i		−	0.174347i	0.996193	−	0.0871735i	$-0.0277834\pi$
$858$		0			0
$859$		−	394.144i		−	0.458840i	−0.973327	−	0.229420i	$-0.926317\pi$
$859$		−	394.144i		−	0.458840i	0.973327	−	0.229420i	$-0.0736830\pi$
$860$		0			0
$861$	1279.29			1.48581
$862$		0			0
$863$	−1409.58			−1.63335			−0.816677	−	0.577095i	$-0.804186\pi$
$863$	−1409.58			−1.63335			−0.816677	+	0.577095i	$0.804186\pi$
$864$		0			0
$865$	−443.945	+	384.807i	−0.513231	+	0.444864i
$866$		0			0
$867$	−11.2669			−0.0129953
$868$		0			0
$869$	456.797			0.525659
$870$		0			0
$871$			225.542i			0.258946i
$872$		0			0
$873$		−	395.583i		−	0.453131i
$874$		0			0
$875$	1004.46	−	646.221i	1.14796	−	0.738538i
$876$		0			0
$877$			872.780i			0.995189i	0.867410	+	0.497594i	$0.165783\pi$
$877$			872.780i			0.995189i	−0.867410	+	0.497594i	$0.834217\pi$
$878$		0			0
$879$			424.712i			0.483176i
$880$		0			0
$881$	1103.38			1.25241			0.626206	−	0.779657i	$-0.284607\pi$
$881$	1103.38			1.25241			0.626206	+	0.779657i	$0.284607\pi$
$882$		0			0
$883$	536.884			0.608023			0.304011	−	0.952668i	$-0.401674\pi$
$883$	536.884			0.608023			0.304011	+	0.952668i	$0.401674\pi$
$884$		0			0
$885$	−620.248	+	537.624i	−0.700845	+	0.607485i
$886$		0			0
$887$	−888.945			−1.00219			−0.501096	−	0.865392i	$-0.667070\pi$
$887$	−888.945			−1.00219			−0.501096	+	0.865392i	$0.667070\pi$
$888$		0			0
$889$	−1835.08			−2.06421
$890$		0			0
$891$			89.3232i			0.100251i
$892$		0			0
$893$		−	368.734i		−	0.412916i
$894$		0			0
$895$	876.501	−	759.742i	0.979331	−	0.848874i
$896$		0			0
$897$		−	21.9162i		−	0.0244327i
$898$		0			0
$899$			16.1319i			0.0179443i
$900$		0			0
$901$	389.691			0.432510
$902$		0			0
$903$	691.773			0.766083
$904$		0			0
$905$	714.248	+	824.015i	0.789224	+	0.910514i
$906$		0			0
$907$	1668.40			1.83947			0.919734	−	0.392543i	$-0.128405\pi$
$907$	1668.40			1.83947			0.919734	+	0.392543i	$0.128405\pi$
$908$		0			0
$909$	−88.3505			−0.0971953
$910$		0			0
$911$			1498.13i			1.64449i	0.569131	+	0.822247i	$0.307280\pi$
$911$			1498.13i			1.64449i	−0.569131	+	0.822247i	$0.692720\pi$
$912$		0			0
$913$			239.521i			0.262345i
$914$		0			0
$915$	−215.548	−	248.674i	−0.235572	−	0.271775i
$916$		0			0
$917$		−	405.399i		−	0.442093i
$918$		0			0
$919$		−	1094.82i		−	1.19131i	−0.803240	−	0.595656i	$-0.796892\pi$
$919$		−	1094.82i		−	1.19131i	0.803240	−	0.595656i	$-0.203108\pi$
$920$		0			0
$921$	584.591			0.634735
$922$		0			0
$923$	−54.3784			−0.0589148
$924$		0			0
$925$	−560.990	−	80.4723i	−0.606476	−	0.0869970i
$926$		0			0
$927$	−431.358			−0.465327
$928$		0			0
$929$	1145.90			1.23348			0.616740	−	0.787167i	$-0.288453\pi$
$929$	1145.90			1.23348			0.616740	+	0.787167i	$0.288453\pi$
$930$		0			0
$931$			1107.23i			1.18929i
$932$		0			0
$933$			740.894i			0.794099i
$934$		0			0
$935$	558.733	+	644.601i	0.597576	+	0.689413i
$936$		0			0
$937$			1272.49i			1.35805i	0.734115	+	0.679025i	$0.237597\pi$
$937$			1272.49i			1.35805i	−0.734115	+	0.679025i	$0.762403\pi$
$938$		0			0
$939$			145.274i			0.154711i
$940$		0			0
$941$	−707.360			−0.751711			−0.375856	−	0.926678i	$-0.622651\pi$
$941$	−707.360			−0.751711			−0.375856	+	0.926678i	$0.622651\pi$
$942$		0			0
$943$	−129.438			−0.137262
$944$		0			0
$945$	−162.598	−	187.587i	−0.172061	−	0.198504i
$946$		0			0
$947$	284.977			0.300926			0.150463	−	0.988616i	$-0.451924\pi$
$947$	284.977			0.300926			0.150463	+	0.988616i	$0.451924\pi$
$948$		0			0
$949$	−259.794			−0.273756
$950$		0			0
$951$		−	194.343i		−	0.204357i
$952$		0			0
$953$		−	295.247i		−	0.309808i	−0.987930	−	0.154904i	$-0.950493\pi$
$953$		−	295.247i		−	0.309808i	0.987930	−	0.154904i	$-0.0495068\pi$
$954$		0			0
$955$	518.401	−	449.344i	0.542828	−	0.470518i
$956$		0			0
$957$			6.02513i			0.00629585i
$958$		0			0
$959$		−	1976.50i		−	2.06100i
$960$		0			0
$961$	−1157.38			−1.20435
$962$		0			0
$963$	−105.304			−0.109350
$964$		0			0
$965$	139.904	−	121.267i	0.144978	−	0.125665i
$966$		0			0
$967$	−348.013			−0.359889			−0.179945	−	0.983677i	$-0.557592\pi$
$967$	−348.013			−0.359889			−0.179945	+	0.983677i	$0.557592\pi$
$968$		0			0
$969$	779.382			0.804316
$970$		0			0
$971$		−	798.691i		−	0.822545i	−0.911513	−	0.411272i	$-0.865085\pi$
$971$		−	798.691i		−	0.822545i	0.911513	−	0.411272i	$-0.134915\pi$
$972$		0			0
$973$			439.779i			0.451983i
$974$		0			0
$975$	323.888	+	46.4607i	0.332193	+	0.0476520i
$976$		0			0
$977$			721.834i			0.738827i	0.929265	+	0.369413i	$0.120441\pi$
$977$			721.834i			0.738827i	−0.929265	+	0.369413i	$0.879559\pi$
$978$		0			0
$979$			994.458i			1.01579i
$980$		0			0
$981$	454.495			0.463298
$982$		0			0
$983$	1717.25			1.74695			0.873474	−	0.486870i	$-0.161861\pi$
$983$	1717.25			1.74695			0.873474	+	0.486870i	$0.161861\pi$
$984$		0			0
$985$	−735.135	+	637.207i	−0.746330	+	0.646910i
$986$		0			0
$987$	233.131			0.236201
$988$		0			0
$989$	−69.9934			−0.0707719
$990$		0			0
$991$		−	342.270i		−	0.345378i	−0.984976	−	0.172689i	$-0.944754\pi$
$991$		−	342.270i		−	0.345378i	0.984976	−	0.172689i	$-0.0552456\pi$
$992$		0			0
$993$		−	229.706i		−	0.231325i
$994$		0			0
$995$	−665.108	+	576.508i	−0.668450	+	0.579405i
$996$		0			0
$997$			1586.05i			1.59082i	0.606072	+	0.795410i	$0.292744\pi$
$997$			1586.05i			1.59082i	−0.606072	+	0.795410i	$0.707256\pi$
$998$		0			0
$999$			117.793i			0.117911i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.3.j.e.319.8		8
4.3	odd	2	inner	960.3.j.e.319.4		8
5.4	even	2	inner	960.3.j.e.319.3		8
8.3	odd	2		60.3.f.b.19.5	yes	8
8.5	even	2		60.3.f.b.19.3	✓	8
20.19	odd	2	inner	960.3.j.e.319.7		8
24.5	odd	2		180.3.f.h.19.6		8
24.11	even	2		180.3.f.h.19.4		8
40.3	even	4		300.3.c.f.151.1		8
40.13	odd	4		300.3.c.f.151.2		8
40.19	odd	2		60.3.f.b.19.4	yes	8
40.27	even	4		300.3.c.f.151.8		8
40.29	even	2		60.3.f.b.19.6	yes	8
40.37	odd	4		300.3.c.f.151.7		8
120.29	odd	2		180.3.f.h.19.3		8
120.53	even	4		900.3.c.r.451.7		8
120.59	even	2		180.3.f.h.19.5		8
120.77	even	4		900.3.c.r.451.2		8
120.83	odd	4		900.3.c.r.451.8		8
120.107	odd	4		900.3.c.r.451.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.3.f.b.19.3	✓	8	8.5	even	2
60.3.f.b.19.4	yes	8	40.19	odd	2
60.3.f.b.19.5	yes	8	8.3	odd	2
60.3.f.b.19.6	yes	8	40.29	even	2
180.3.f.h.19.3		8	120.29	odd	2
180.3.f.h.19.4		8	24.11	even	2
180.3.f.h.19.5		8	120.59	even	2
180.3.f.h.19.6		8	24.5	odd	2
300.3.c.f.151.1		8	40.3	even	4
300.3.c.f.151.2		8	40.13	odd	4
300.3.c.f.151.7		8	40.37	odd	4
300.3.c.f.151.8		8	40.27	even	4
900.3.c.r.451.1		8	120.107	odd	4
900.3.c.r.451.2		8	120.77	even	4
900.3.c.r.451.7		8	120.53	even	4
900.3.c.r.451.8		8	120.83	odd	4
960.3.j.e.319.3		8	5.4	even	2	inner
960.3.j.e.319.4		8	4.3	odd	2	inner
960.3.j.e.319.7		8	20.19	odd	2	inner
960.3.j.e.319.8		8	1.1	even	1	trivial

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

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

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} +(3.27492 + 3.77822i) q^{5} -9.55505 q^{7} +3.00000 q^{9} +9.92480i q^{11} -7.55643i q^{13} +(5.67232 + 6.54406i) q^{15} -17.1903i q^{17} +26.1762i q^{19} -16.5498 q^{21} +1.67451 q^{23} +(-3.54983 + 24.7467i) q^{25} +5.19615 q^{27} +0.350497 q^{29} +46.0258i q^{31} +17.1903i q^{33} +(-31.2920 - 36.1010i) q^{35} +22.6693i q^{37} -13.0881i q^{39} -77.2990 q^{41} -41.7994 q^{43} +(9.82475 + 11.3346i) q^{45} -14.0866 q^{47} +42.2990 q^{49} -29.7744i q^{51} +22.6693i q^{53} +(-37.4980 + 32.5029i) q^{55} +45.3386i q^{57} +94.7802i q^{59} -38.0000 q^{61} -28.6652 q^{63} +(28.5498 - 24.7467i) q^{65} -29.8477 q^{67} +2.90033 q^{69} -7.19630i q^{71} -34.3805i q^{73} +(-6.14849 + 42.8625i) q^{75} -94.8320i q^{77} -46.0258i q^{79} +9.00000 q^{81} +24.1336 q^{83} +(64.9485 - 56.2967i) q^{85} +0.607078 q^{87} +100.199 q^{89} +72.2021i q^{91} +79.7191i q^{93} +(-98.8995 + 85.7250i) q^{95} -131.861i q^{97} +29.7744i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{5} + 24 q^{9} - 72 q^{21} + 32 q^{25} + 184 q^{29} - 256 q^{41} - 12 q^{45} - 24 q^{49} - 304 q^{61} + 168 q^{65} + 144 q^{69} + 72 q^{81} - 24 q^{85} + 560 q^{89}+O(q^{100}) \) 8 * q - 4 * q^5 + 24 * q^9 - 72 * q^21 + 32 * q^25 + 184 * q^29 - 256 * q^41 - 12 * q^45 - 24 * q^49 - 304 * q^61 + 168 * q^65 + 144 * q^69 + 72 * q^81 - 24 * q^85 + 560 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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