LMFDB - Embedded newform 1920.2.f.p.769.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1920,2,Mod(769,1920)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1920, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1920.769"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$1920 = 2^{7} \cdot 3 \cdot 5$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1920.f (of order $2$ , degree $1$ , minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,2,0,0,0,-6,0,12,0,0,0,0] 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:	$15.3312771881$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{3}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.6
Root			$-0.854638 + 0.854638i$ of defining polynomial
Character	$\chi$	$=$	1920.769
Dual form			1920.2.f.p.769.3

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(2.17009 - 0.539189i) q^{5} -2.34017i q^{7} -1.00000 q^{9} +3.07838 q^{11} -0.921622i q^{13} +(0.539189 + 2.17009i) q^{15} +7.75872i q^{17} -4.00000 q^{19} +2.34017 q^{21} +2.15676i q^{23} +(4.41855 - 2.34017i) q^{25} -1.00000i q^{27} +6.49693 q^{29} +2.00000 q^{31} +3.07838i q^{33} +(-1.26180 - 5.07838i) q^{35} +3.07838i q^{37} +0.921622 q^{39} +10.6803 q^{41} -2.15676i q^{43} +(-2.17009 + 0.539189i) q^{45} -8.68035i q^{47} +1.52359 q^{49} -7.75872 q^{51} -2.92162i q^{53} +(6.68035 - 1.65983i) q^{55} -4.00000i q^{57} -11.7587 q^{59} +6.00000 q^{61} +2.34017i q^{63} +(-0.496928 - 2.00000i) q^{65} -6.83710i q^{67} -2.15676 q^{69} +11.5174 q^{71} -6.52359i q^{73} +(2.34017 + 4.41855i) q^{75} -7.20394i q^{77} +15.3607 q^{79} +1.00000 q^{81} +1.84324i q^{83} +(4.18342 + 16.8371i) q^{85} +6.49693i q^{87} -6.00000 q^{89} -2.15676 q^{91} +2.00000i q^{93} +(-8.68035 + 2.15676i) q^{95} -3.07838 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q + 2 q^{5} - 6 q^{9} + 12 q^{11} - 24 q^{19} - 8 q^{21} - 2 q^{25} + 4 q^{29} + 12 q^{31} + 8 q^{35} + 12 q^{39} + 20 q^{41} - 2 q^{45} - 22 q^{49} + 4 q^{51} - 4 q^{55} - 20 q^{59} + 36 q^{61} + 32 q^{65}+ \cdots - 12 q^{99}+O(q^{100})$ 6 * q + 2 * q^5 - 6 * q^9 + 12 * q^11 - 24 * q^19 - 8 * q^21 - 2 * q^25 + 4 * q^29 + 12 * q^31 + 8 * q^35 + 12 * q^39 + 20 * q^41 - 2 * q^45 - 22 * q^49 + 4 * q^51 - 4 * q^55 - 20 * q^59 + 36 * q^61 + 32 * q^65 - 32 * q^71 - 8 * q^75 + 4 * q^79 + 6 * q^81 + 16 * q^85 - 36 * q^89 - 8 * q^95 - 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1920\mathbb{Z}\right)^\times$ .

$n$	$511$	$641$	$901$	$1537$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	2.17009	−	0.539189i	0.970492	−	0.241133i
$5$	2.17009	−	0.539189i	0.970492	−	0.241133i
$6$		0			0
$7$		−	2.34017i		−	0.884502i	−0.896891	−	0.442251i	$-0.854180\pi$
$7$		−	2.34017i		−	0.884502i	0.896891	−	0.442251i	$-0.145820\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	3.07838			0.928166			0.464083	−	0.885792i	$-0.346384\pi$
$11$	3.07838			0.928166			0.464083	+	0.885792i	$0.346384\pi$
$12$		0			0
$13$		−	0.921622i		−	0.255612i	−0.991799	−	0.127806i	$-0.959207\pi$
$13$		−	0.921622i		−	0.255612i	0.991799	−	0.127806i	$-0.0407935\pi$
$14$		0			0
$15$	0.539189	+	2.17009i	0.139218	+	0.560314i
$16$		0			0
$17$			7.75872i			1.88177i	0.338730	+	0.940883i	$0.390003\pi$
$17$			7.75872i			1.88177i	−0.338730	+	0.940883i	$0.609997\pi$
$18$		0			0
$19$	−4.00000			−0.917663			−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$		0			0
$21$	2.34017			0.510668
$22$		0			0
$23$			2.15676i			0.449715i	0.974392	+	0.224857i	$0.0721916\pi$
$23$			2.15676i			0.449715i	−0.974392	+	0.224857i	$0.927808\pi$
$24$		0			0
$25$	4.41855	−	2.34017i	0.883710	−	0.468035i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	6.49693			1.20645			0.603225	−	0.797571i	$-0.293882\pi$
$29$	6.49693			1.20645			0.603225	+	0.797571i	$0.293882\pi$
$30$		0			0
$31$	2.00000			0.359211			0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000			0.359211			0.179605	+	0.983739i	$0.442518\pi$
$32$		0			0
$33$			3.07838i			0.535877i
$34$		0			0
$35$	−1.26180	−	5.07838i	−0.213282	−	0.858403i
$36$		0			0
$37$			3.07838i			0.506082i	0.967456	+	0.253041i	$0.0814308\pi$
$37$			3.07838i			0.506082i	−0.967456	+	0.253041i	$0.918569\pi$
$38$		0			0
$39$	0.921622			0.147578
$40$		0			0
$41$	10.6803			1.66799			0.833995	−	0.551772i	$-0.186048\pi$
$41$	10.6803			1.66799			0.833995	+	0.551772i	$0.186048\pi$
$42$		0			0
$43$		−	2.15676i		−	0.328902i	−0.986385	−	0.164451i	$-0.947415\pi$
$43$		−	2.15676i		−	0.328902i	0.986385	−	0.164451i	$-0.0525853\pi$
$44$		0			0
$45$	−2.17009	+	0.539189i	−0.323497	+	0.0803775i
$46$		0			0
$47$		−	8.68035i		−	1.26616i	−0.774087	−	0.633079i	$-0.781791\pi$
$47$		−	8.68035i		−	1.26616i	0.774087	−	0.633079i	$-0.218209\pi$
$48$		0			0
$49$	1.52359			0.217656
$50$		0			0
$51$	−7.75872			−1.08644
$52$		0			0
$53$		−	2.92162i		−	0.401316i	−0.979661	−	0.200658i	$-0.935692\pi$
$53$		−	2.92162i		−	0.401316i	0.979661	−	0.200658i	$-0.0643079\pi$
$54$		0			0
$55$	6.68035	−	1.65983i	0.900778	−	0.223811i
$56$		0			0
$57$		−	4.00000i		−	0.529813i
$58$		0			0
$59$	−11.7587			−1.53086			−0.765428	−	0.643522i	$-0.777473\pi$
$59$	−11.7587			−1.53086			−0.765428	+	0.643522i	$0.777473\pi$
$60$		0			0
$61$	6.00000			0.768221			0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000			0.768221			0.384111	+	0.923287i	$0.374508\pi$
$62$		0			0
$63$			2.34017i			0.294834i
$64$		0			0
$65$	−0.496928	−	2.00000i	−0.0616364	−	0.248069i
$66$		0			0
$67$		−	6.83710i		−	0.835285i	−0.908611	−	0.417642i	$-0.862856\pi$
$67$		−	6.83710i		−	0.835285i	0.908611	−	0.417642i	$-0.137144\pi$
$68$		0			0
$69$	−2.15676			−0.259643
$70$		0			0
$71$	11.5174			1.36687			0.683435	−	0.730012i	$-0.260485\pi$
$71$	11.5174			1.36687			0.683435	+	0.730012i	$0.260485\pi$
$72$		0			0
$73$		−	6.52359i		−	0.763529i	−0.924260	−	0.381764i	$-0.875317\pi$
$73$		−	6.52359i		−	0.763529i	0.924260	−	0.381764i	$-0.124683\pi$
$74$		0			0
$75$	2.34017	+	4.41855i	0.270220	+	0.510210i
$76$		0			0
$77$		−	7.20394i		−	0.820965i
$78$		0			0
$79$	15.3607			1.72821			0.864106	−	0.503309i	$-0.167884\pi$
$79$	15.3607			1.72821			0.864106	+	0.503309i	$0.167884\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			1.84324i			0.202322i	0.994870	+	0.101161i	$0.0322558\pi$
$83$			1.84324i			0.202322i	−0.994870	+	0.101161i	$0.967744\pi$
$84$		0			0
$85$	4.18342	+	16.8371i	0.453755	+	1.82624i
$86$		0			0
$87$			6.49693i			0.696544i
$88$		0			0
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$	−2.15676			−0.226089
$92$		0			0
$93$			2.00000i			0.207390i
$94$		0			0
$95$	−8.68035	+	2.15676i	−0.890585	+	0.221278i
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$	−3.07838			−0.309389
$100$		0			0
$101$	8.34017			0.829878			0.414939	−	0.909849i	$-0.363803\pi$
$101$	8.34017			0.829878			0.414939	+	0.909849i	$0.363803\pi$
$102$		0			0
$103$			10.3402i			1.01885i	0.860516	+	0.509424i	$0.170141\pi$
$103$			10.3402i			1.01885i	−0.860516	+	0.509424i	$0.829859\pi$
$104$		0			0
$105$	5.07838	−	1.26180i	0.495599	−	0.123139i
$106$		0			0
$107$		−	13.3607i		−	1.29163i	−0.763495	−	0.645813i	$-0.776518\pi$
$107$		−	13.3607i		−	1.29163i	0.763495	−	0.645813i	$-0.223482\pi$
$108$		0			0
$109$	−5.31965			−0.509530			−0.254765	−	0.967003i	$-0.581998\pi$
$109$	−5.31965			−0.509530			−0.254765	+	0.967003i	$0.581998\pi$
$110$		0			0
$111$	−3.07838			−0.292187
$112$		0			0
$113$			19.7587i			1.85874i	0.369144	+	0.929372i	$0.379651\pi$
$113$			19.7587i			1.85874i	−0.369144	+	0.929372i	$0.620349\pi$
$114$		0			0
$115$	1.16290	+	4.68035i	0.108441	+	0.436445i
$116$		0			0
$117$			0.921622i			0.0852040i
$118$		0			0
$119$	18.1568			1.66443
$120$		0			0
$121$	−1.52359			−0.138508
$122$		0			0
$123$			10.6803i			0.963014i
$124$		0			0
$125$	8.32684	−	7.46081i	0.744775	−	0.667315i
$126$		0			0
$127$			19.3340i			1.71562i	0.513969	+	0.857809i	$0.328175\pi$
$127$			19.3340i			1.71562i	−0.513969	+	0.857809i	$0.671825\pi$
$128$		0			0
$129$	2.15676			0.189892
$130$		0			0
$131$	−7.44521			−0.650491			−0.325246	−	0.945630i	$-0.605447\pi$
$131$	−7.44521			−0.650491			−0.325246	+	0.945630i	$0.605447\pi$
$132$		0			0
$133$			9.36069i			0.811675i
$134$		0			0
$135$	−0.539189	−	2.17009i	−0.0464060	−	0.186771i
$136$		0			0
$137$		−	3.44521i		−	0.294344i	−0.989111	−	0.147172i	$-0.952983\pi$
$137$		−	3.44521i		−	0.294344i	0.989111	−	0.147172i	$-0.0470171\pi$
$138$		0			0
$139$	−19.2039			−1.62886			−0.814428	−	0.580264i	$-0.802949\pi$
$139$	−19.2039			−1.62886			−0.814428	+	0.580264i	$0.802949\pi$
$140$		0			0
$141$	8.68035			0.731017
$142$		0			0
$143$		−	2.83710i		−	0.237250i
$144$		0			0
$145$	14.0989	−	3.50307i	1.17085	−	0.290914i
$146$		0			0
$147$			1.52359i			0.125664i
$148$		0			0
$149$	−9.81658			−0.804206			−0.402103	−	0.915594i	$-0.631721\pi$
$149$	−9.81658			−0.804206			−0.402103	+	0.915594i	$0.631721\pi$
$150$		0			0
$151$	−23.6742			−1.92658			−0.963290	−	0.268464i	$-0.913484\pi$
$151$	−23.6742			−1.92658			−0.963290	+	0.268464i	$0.913484\pi$
$152$		0			0
$153$		−	7.75872i		−	0.627256i
$154$		0			0
$155$	4.34017	−	1.07838i	0.348611	−	0.0866174i
$156$		0			0
$157$		−	11.0784i		−	0.884151i	−0.896978	−	0.442075i	$-0.854242\pi$
$157$		−	11.0784i		−	0.884151i	0.896978	−	0.442075i	$-0.145758\pi$
$158$		0			0
$159$	2.92162			0.231700
$160$		0			0
$161$	5.04718			0.397774
$162$		0			0
$163$			12.9939i			1.01776i	0.860838	+	0.508879i	$0.169940\pi$
$163$			12.9939i			1.01776i	−0.860838	+	0.508879i	$0.830060\pi$
$164$		0			0
$165$	1.65983	+	6.68035i	0.129217	+	0.520064i
$166$		0			0
$167$		−	12.0000i		−	0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$		−	12.0000i		−	0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$		0			0
$169$	12.1506			0.934662
$170$		0			0
$171$	4.00000			0.305888
$172$		0			0
$173$		−	18.4391i		−	1.40190i	−0.713212	−	0.700948i	$-0.752760\pi$
$173$		−	18.4391i		−	1.40190i	0.713212	−	0.700948i	$-0.247240\pi$
$174$		0			0
$175$	−5.47641	−	10.3402i	−0.413978	−	0.781644i
$176$		0			0
$177$		−	11.7587i		−	0.883840i
$178$		0			0
$179$	1.91548			0.143170			0.0715848	−	0.997435i	$-0.477194\pi$
$179$	1.91548			0.143170			0.0715848	+	0.997435i	$0.477194\pi$
$180$		0			0
$181$	−6.99386			−0.519849			−0.259925	−	0.965629i	$-0.583698\pi$
$181$	−6.99386			−0.519849			−0.259925	+	0.965629i	$0.583698\pi$
$182$		0			0
$183$			6.00000i			0.443533i
$184$		0			0
$185$	1.65983	+	6.68035i	0.122033	+	0.491149i
$186$		0			0
$187$			23.8843i			1.74659i
$188$		0			0
$189$	−2.34017			−0.170223
$190$		0			0
$191$	6.15676			0.445487			0.222744	−	0.974877i	$-0.428499\pi$
$191$	6.15676			0.445487			0.222744	+	0.974877i	$0.428499\pi$
$192$		0			0
$193$			14.5236i			1.04543i	0.852507	+	0.522715i	$0.175081\pi$
$193$			14.5236i			1.04543i	−0.852507	+	0.522715i	$0.824919\pi$
$194$		0			0
$195$	2.00000	−	0.496928i	0.143223	−	0.0355858i
$196$		0			0
$197$			24.5958i			1.75238i	0.481966	+	0.876190i	$0.339923\pi$
$197$			24.5958i			1.75238i	−0.481966	+	0.876190i	$0.660077\pi$
$198$		0			0
$199$	−3.36069			−0.238233			−0.119117	−	0.992880i	$-0.538006\pi$
$199$	−3.36069			−0.238233			−0.119117	+	0.992880i	$0.538006\pi$
$200$		0			0
$201$	6.83710			0.482252
$202$		0			0
$203$		−	15.2039i		−	1.06711i
$204$		0			0
$205$	23.1773	−	5.75872i	1.61877	−	0.402207i
$206$		0			0
$207$		−	2.15676i		−	0.149905i
$208$		0			0
$209$	−12.3135			−0.851743
$210$		0			0
$211$	11.2039			0.771311			0.385655	−	0.922643i	$-0.373975\pi$
$211$	11.2039			0.771311			0.385655	+	0.922643i	$0.373975\pi$
$212$		0			0
$213$			11.5174i			0.789162i
$214$		0			0
$215$	−1.16290	−	4.68035i	−0.0793090	−	0.319197i
$216$		0			0
$217$		−	4.68035i		−	0.317723i
$218$		0			0
$219$	6.52359			0.440823
$220$		0			0
$221$	7.15061			0.481002
$222$		0			0
$223$			6.65368i			0.445564i	0.974868	+	0.222782i	$0.0715138\pi$
$223$			6.65368i			0.445564i	−0.974868	+	0.222782i	$0.928486\pi$
$224$		0			0
$225$	−4.41855	+	2.34017i	−0.294570	+	0.156012i
$226$		0			0
$227$		−	9.84324i		−	0.653319i	−0.945142	−	0.326660i	$-0.894077\pi$
$227$		−	9.84324i		−	0.653319i	0.945142	−	0.326660i	$-0.105923\pi$
$228$		0			0
$229$	−1.31965			−0.0872052			−0.0436026	−	0.999049i	$-0.513884\pi$
$229$	−1.31965			−0.0872052			−0.0436026	+	0.999049i	$0.513884\pi$
$230$		0			0
$231$	7.20394			0.473984
$232$		0			0
$233$		−	2.39803i		−	0.157100i	−0.996910	−	0.0785501i	$-0.974971\pi$
$233$		−	2.39803i		−	0.157100i	0.996910	−	0.0785501i	$-0.0250291\pi$
$234$		0			0
$235$	−4.68035	−	18.8371i	−0.305312	−	1.22880i
$236$		0			0
$237$			15.3607i			0.997784i
$238$		0			0
$239$	6.15676			0.398247			0.199124	−	0.979974i	$-0.436190\pi$
$239$	6.15676			0.398247			0.199124	+	0.979974i	$0.436190\pi$
$240$		0			0
$241$	0.639308			0.0411815			0.0205907	−	0.999788i	$-0.493445\pi$
$241$	0.639308			0.0411815			0.0205907	+	0.999788i	$0.493445\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	3.30632	−	0.821503i	0.211233	−	0.0524839i
$246$		0			0
$247$			3.68649i			0.234566i
$248$		0			0
$249$	−1.84324			−0.116811
$250$		0			0
$251$	12.9216			0.815606			0.407803	−	0.913070i	$-0.366295\pi$
$251$	12.9216			0.815606			0.407803	+	0.913070i	$0.366295\pi$
$252$		0			0
$253$			6.63931i			0.417410i
$254$		0			0
$255$	−16.8371	+	4.18342i	−1.05438	+	0.261976i
$256$		0			0
$257$			14.9627i			0.933345i	0.884430	+	0.466673i	$0.154547\pi$
$257$			14.9627i			0.933345i	−0.884430	+	0.466673i	$0.845453\pi$
$258$		0			0
$259$	7.20394			0.447631
$260$		0			0
$261$	−6.49693			−0.402150
$262$		0			0
$263$		−	13.3607i		−	0.823856i	−0.911216	−	0.411928i	$-0.864856\pi$
$263$		−	13.3607i		−	0.823856i	0.911216	−	0.411928i	$-0.135144\pi$
$264$		0			0
$265$	−1.57531	−	6.34017i	−0.0967703	−	0.389474i
$266$		0			0
$267$		−	6.00000i		−	0.367194i
$268$		0			0
$269$	−27.5441			−1.67939			−0.839697	−	0.543055i	$-0.817267\pi$
$269$	−27.5441			−1.67939			−0.839697	+	0.543055i	$0.817267\pi$
$270$		0			0
$271$	10.0000			0.607457			0.303728	−	0.952759i	$-0.401768\pi$
$271$	10.0000			0.607457			0.303728	+	0.952759i	$0.401768\pi$
$272$		0			0
$273$		−	2.15676i		−	0.130533i
$274$		0			0
$275$	13.6020	−	7.20394i	0.820230	−	0.434414i
$276$		0			0
$277$			4.12556i			0.247881i	0.992290	+	0.123940i	$0.0395532\pi$
$277$			4.12556i			0.247881i	−0.992290	+	0.123940i	$0.960447\pi$
$278$		0			0
$279$	−2.00000			−0.119737
$280$		0			0
$281$	−11.3607			−0.677722			−0.338861	−	0.940836i	$-0.610042\pi$
$281$	−11.3607			−0.677722			−0.338861	+	0.940836i	$0.610042\pi$
$282$		0			0
$283$			10.5236i			0.625563i	0.949825	+	0.312781i	$0.101261\pi$
$283$			10.5236i			0.625563i	−0.949825	+	0.312781i	$0.898739\pi$
$284$		0			0
$285$	−2.15676	−	8.68035i	−0.127755	−	0.514179i
$286$		0			0
$287$		−	24.9939i		−	1.47534i
$288$		0			0
$289$	−43.1978			−2.54105
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	32.5958i		−	1.90427i	−0.305680	−	0.952134i	$-0.598884\pi$
$293$		−	32.5958i		−	1.90427i	0.305680	−	0.952134i	$-0.401116\pi$
$294$		0			0
$295$	−25.5174	+	6.34017i	−1.48568	+	0.369139i
$296$		0			0
$297$		−	3.07838i		−	0.178626i
$298$		0			0
$299$	1.98771			0.114952
$300$		0			0
$301$	−5.04718			−0.290915
$302$		0			0
$303$			8.34017i			0.479130i
$304$		0			0
$305$	13.0205	−	3.23513i	0.745553	−	0.185243i
$306$		0			0
$307$			29.9877i			1.71149i	0.517398	+	0.855745i	$0.326901\pi$
$307$			29.9877i			1.71149i	−0.517398	+	0.855745i	$0.673099\pi$
$308$		0			0
$309$	−10.3402			−0.588232
$310$		0			0
$311$	−27.2039			−1.54259			−0.771297	−	0.636476i	$-0.780392\pi$
$311$	−27.2039			−1.54259			−0.771297	+	0.636476i	$0.780392\pi$
$312$		0			0
$313$		−	26.8371i		−	1.51692i	−0.651718	−	0.758461i	$-0.725951\pi$
$313$		−	26.8371i		−	1.51692i	0.651718	−	0.758461i	$-0.274049\pi$
$314$		0			0
$315$	1.26180	+	5.07838i	0.0710941	+	0.286134i
$316$		0			0
$317$		−	24.5958i		−	1.38144i	−0.723123	−	0.690720i	$-0.757294\pi$
$317$		−	24.5958i		−	1.38144i	0.723123	−	0.690720i	$-0.242706\pi$
$318$		0			0
$319$	20.0000			1.11979
$320$		0			0
$321$	13.3607			0.745721
$322$		0			0
$323$		−	31.0349i		−	1.72683i
$324$		0			0
$325$	−2.15676	−	4.07223i	−0.119635	−	0.225887i
$326$		0			0
$327$		−	5.31965i		−	0.294178i
$328$		0			0
$329$	−20.3135			−1.11992
$330$		0			0
$331$	12.3135			0.676812			0.338406	−	0.941000i	$-0.390112\pi$
$331$	12.3135			0.676812			0.338406	+	0.941000i	$0.390112\pi$
$332$		0			0
$333$		−	3.07838i		−	0.168694i
$334$		0			0
$335$	−3.68649	−	14.8371i	−0.201414	−	0.810637i
$336$		0			0
$337$			9.47641i			0.516213i	0.966116	+	0.258106i	$0.0830985\pi$
$337$			9.47641i			0.516213i	−0.966116	+	0.258106i	$0.916901\pi$
$338$		0			0
$339$	−19.7587			−1.07315
$340$		0			0
$341$	6.15676			0.333407
$342$		0			0
$343$		−	19.9467i		−	1.07702i
$344$		0			0
$345$	−4.68035	+	1.16290i	−0.251981	+	0.0626084i
$346$		0			0
$347$			7.51745i			0.403558i	0.979431	+	0.201779i	$0.0646722\pi$
$347$			7.51745i			0.403558i	−0.979431	+	0.201779i	$0.935328\pi$
$348$		0			0
$349$	−27.6742			−1.48137			−0.740683	−	0.671855i	$-0.765498\pi$
$349$	−27.6742			−1.48137			−0.740683	+	0.671855i	$0.765498\pi$
$350$		0			0
$351$	−0.921622			−0.0491926
$352$		0			0
$353$			21.6020i			1.14976i	0.818239	+	0.574878i	$0.194951\pi$
$353$			21.6020i			1.14976i	−0.818239	+	0.574878i	$0.805049\pi$
$354$		0			0
$355$	24.9939	−	6.21008i	1.32654	−	0.329597i
$356$		0			0
$357$			18.1568i			0.960957i
$358$		0			0
$359$	12.0000			0.633336			0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000			0.633336			0.316668	+	0.948536i	$0.397436\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$		−	1.52359i		−	0.0799678i
$364$		0			0
$365$	−3.51745	−	14.1568i	−0.184112	−	0.740998i
$366$		0			0
$367$			23.0205i			1.20166i	0.799376	+	0.600831i	$0.205163\pi$
$367$			23.0205i			1.20166i	−0.799376	+	0.600831i	$0.794837\pi$
$368$		0			0
$369$	−10.6803			−0.555997
$370$		0			0
$371$	−6.83710			−0.354965
$372$		0			0
$373$		−	26.5958i		−	1.37708i	−0.725199	−	0.688540i	$-0.758252\pi$
$373$		−	26.5958i		−	1.37708i	0.725199	−	0.688540i	$-0.241748\pi$
$374$		0			0
$375$	7.46081	+	8.32684i	0.385275	+	0.429996i
$376$		0			0
$377$		−	5.98771i		−	0.308383i
$378$		0			0
$379$	−36.1445			−1.85662			−0.928308	−	0.371811i	$-0.878737\pi$
$379$	−36.1445			−1.85662			−0.928308	+	0.371811i	$0.878737\pi$
$380$		0			0
$381$	−19.3340			−0.990512
$382$		0			0
$383$		−	6.83710i		−	0.349360i	−0.984625	−	0.174680i	$-0.944111\pi$
$383$		−	6.83710i		−	0.349360i	0.984625	−	0.174680i	$-0.0558890\pi$
$384$		0			0
$385$	−3.88428	−	15.6332i	−0.197961	−	0.796740i
$386$		0			0
$387$			2.15676i			0.109634i
$388$		0			0
$389$	−16.7070			−0.847079			−0.423539	−	0.905878i	$-0.639213\pi$
$389$	−16.7070			−0.847079			−0.423539	+	0.905878i	$0.639213\pi$
$390$		0			0
$391$	−16.7337			−0.846258
$392$		0			0
$393$		−	7.44521i		−	0.375561i
$394$		0			0
$395$	33.3340	−	8.28231i	1.67722	−	0.416728i
$396$		0			0
$397$		−	3.56093i		−	0.178718i	−0.995999	−	0.0893590i	$-0.971518\pi$
$397$		−	3.56093i		−	0.178718i	0.995999	−	0.0893590i	$-0.0284818\pi$
$398$		0			0
$399$	−9.36069			−0.468621
$400$		0			0
$401$	−25.7152			−1.28416			−0.642079	−	0.766639i	$-0.721928\pi$
$401$	−25.7152			−1.28416			−0.642079	+	0.766639i	$0.721928\pi$
$402$		0			0
$403$		−	1.84324i		−	0.0918185i
$404$		0			0
$405$	2.17009	−	0.539189i	0.107832	−	0.0267925i
$406$		0			0
$407$			9.47641i			0.469728i
$408$		0			0
$409$	−28.8371			−1.42590			−0.712951	−	0.701213i	$-0.752642\pi$
$409$	−28.8371			−1.42590			−0.712951	+	0.701213i	$0.752642\pi$
$410$		0			0
$411$	3.44521			0.169940
$412$		0			0
$413$			27.5174i			1.35405i
$414$		0			0
$415$	0.993857	+	4.00000i	0.0487865	+	0.196352i
$416$		0			0
$417$		−	19.2039i		−	0.940421i
$418$		0			0
$419$	−6.28231			−0.306911			−0.153456	−	0.988156i	$-0.549040\pi$
$419$	−6.28231			−0.306911			−0.153456	+	0.988156i	$0.549040\pi$
$420$		0			0
$421$	−10.9939			−0.535808			−0.267904	−	0.963446i	$-0.586331\pi$
$421$	−10.9939			−0.535808			−0.267904	+	0.963446i	$0.586331\pi$
$422$		0			0
$423$			8.68035i			0.422053i
$424$		0			0
$425$	18.1568	+	34.2823i	0.880732	+	1.66294i
$426$		0			0
$427$		−	14.0410i		−	0.679493i
$428$		0			0
$429$	2.83710			0.136977
$430$		0			0
$431$	−21.3607			−1.02891			−0.514454	−	0.857518i	$-0.672005\pi$
$431$	−21.3607			−1.02891			−0.514454	+	0.857518i	$0.672005\pi$
$432$		0			0
$433$			18.4703i			0.887624i	0.896120	+	0.443812i	$0.146374\pi$
$433$			18.4703i			0.887624i	−0.896120	+	0.443812i	$0.853626\pi$
$434$		0			0
$435$	3.50307	+	14.0989i	0.167959	+	0.675990i
$436$		0			0
$437$		−	8.62702i		−	0.412686i
$438$		0			0
$439$	−6.31351			−0.301327			−0.150664	−	0.988585i	$-0.548141\pi$
$439$	−6.31351			−0.301327			−0.150664	+	0.988585i	$0.548141\pi$
$440$		0			0
$441$	−1.52359			−0.0725519
$442$		0			0
$443$			27.8310i			1.32229i	0.750259	+	0.661144i	$0.229929\pi$
$443$			27.8310i			1.32229i	−0.750259	+	0.661144i	$0.770071\pi$
$444$		0			0
$445$	−13.0205	+	3.23513i	−0.617232	+	0.153360i
$446$		0			0
$447$		−	9.81658i		−	0.464308i
$448$		0			0
$449$	−14.6803			−0.692808			−0.346404	−	0.938085i	$-0.612597\pi$
$449$	−14.6803			−0.692808			−0.346404	+	0.938085i	$0.612597\pi$
$450$		0			0
$451$	32.8781			1.54817
$452$		0			0
$453$		−	23.6742i		−	1.11231i
$454$		0			0
$455$	−4.68035	+	1.16290i	−0.219418	+	0.0545175i
$456$		0			0
$457$			19.6865i			0.920895i	0.887687	+	0.460448i	$0.152311\pi$
$457$			19.6865i			0.920895i	−0.887687	+	0.460448i	$0.847689\pi$
$458$		0			0
$459$	7.75872			0.362146
$460$		0			0
$461$	13.5031			0.628901			0.314450	−	0.949274i	$-0.398180\pi$
$461$	13.5031			0.628901			0.314450	+	0.949274i	$0.398180\pi$
$462$		0			0
$463$		−	6.02666i		−	0.280083i	−0.990146	−	0.140041i	$-0.955276\pi$
$463$		−	6.02666i		−	0.280083i	0.990146	−	0.140041i	$-0.0447236\pi$
$464$		0			0
$465$	1.07838	+	4.34017i	0.0500086	+	0.201271i
$466$		0			0
$467$		−	19.8310i		−	0.917667i	−0.888522	−	0.458834i	$-0.848267\pi$
$467$		−	19.8310i		−	0.917667i	0.888522	−	0.458834i	$-0.151733\pi$
$468$		0			0
$469$	−16.0000			−0.738811
$470$		0			0
$471$	11.0784			0.510465
$472$		0			0
$473$		−	6.63931i		−	0.305276i
$474$		0			0
$475$	−17.6742	+	9.36069i	−0.810948	+	0.429498i
$476$		0			0
$477$			2.92162i			0.133772i
$478$		0			0
$479$	−23.2039			−1.06021			−0.530107	−	0.847930i	$-0.677848\pi$
$479$	−23.2039			−1.06021			−0.530107	+	0.847930i	$0.677848\pi$
$480$		0			0
$481$	2.83710			0.129361
$482$		0			0
$483$			5.04718i			0.229655i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			23.3874i			1.05978i	0.848066	+	0.529891i	$0.177767\pi$
$487$			23.3874i			1.05978i	−0.848066	+	0.529891i	$0.822233\pi$
$488$		0			0
$489$	−12.9939			−0.587603
$490$		0			0
$491$	30.2290			1.36422			0.682108	−	0.731252i	$-0.261064\pi$
$491$	30.2290			1.36422			0.682108	+	0.731252i	$0.261064\pi$
$492$		0			0
$493$			50.4079i			2.27026i
$494$		0			0
$495$	−6.68035	+	1.65983i	−0.300259	+	0.0746037i
$496$		0			0
$497$		−	26.9528i		−	1.20900i
$498$		0			0
$499$	1.36069			0.0609129			0.0304565	−	0.999536i	$-0.490304\pi$
$499$	1.36069			0.0609129			0.0304565	+	0.999536i	$0.490304\pi$
$500$		0			0
$501$	12.0000			0.536120
$502$		0			0
$503$		−	16.6803i		−	0.743740i	−0.928285	−	0.371870i	$-0.878717\pi$
$503$		−	16.6803i		−	0.743740i	0.928285	−	0.371870i	$-0.121283\pi$
$504$		0			0
$505$	18.0989	−	4.49693i	0.805390	−	0.200111i
$506$		0			0
$507$			12.1506i			0.539628i
$508$		0			0
$509$	14.0144			0.621176			0.310588	−	0.950545i	$-0.399474\pi$
$509$	14.0144			0.621176			0.310588	+	0.950545i	$0.399474\pi$
$510$		0			0
$511$	−15.2663			−0.675343
$512$		0			0
$513$			4.00000i			0.176604i
$514$		0			0
$515$	5.57531	+	22.4391i	0.245677	+	0.988784i
$516$		0			0
$517$		−	26.7214i		−	1.17521i
$518$		0			0
$519$	18.4391			0.809385
$520$		0			0
$521$	6.68035			0.292671			0.146336	−	0.989235i	$-0.453252\pi$
$521$	6.68035			0.292671			0.146336	+	0.989235i	$0.453252\pi$
$522$		0			0
$523$		−	26.0410i		−	1.13870i	−0.822097	−	0.569348i	$-0.807196\pi$
$523$		−	26.0410i		−	1.13870i	0.822097	−	0.569348i	$-0.192804\pi$
$524$		0			0
$525$	10.3402	−	5.47641i	0.451282	−	0.239010i
$526$		0			0
$527$			15.5174i			0.675951i
$528$		0			0
$529$	18.3484			0.797757
$530$		0			0
$531$	11.7587			0.510285
$532$		0			0
$533$		−	9.84324i		−	0.426358i
$534$		0			0
$535$	−7.20394	−	28.9939i	−0.311453	−	1.25351i
$536$		0			0
$537$			1.91548i			0.0826590i
$538$		0			0
$539$	4.69019			0.202021
$540$		0			0
$541$	−14.9939			−0.644636			−0.322318	−	0.946631i	$-0.604462\pi$
$541$	−14.9939			−0.644636			−0.322318	+	0.946631i	$0.604462\pi$
$542$		0			0
$543$		−	6.99386i		−	0.300135i
$544$		0			0
$545$	−11.5441	+	2.86830i	−0.494495	+	0.122864i
$546$		0			0
$547$			26.1568i			1.11838i	0.829039	+	0.559191i	$0.188888\pi$
$547$			26.1568i			1.11838i	−0.829039	+	0.559191i	$0.811112\pi$
$548$		0			0
$549$	−6.00000			−0.256074
$550$		0			0
$551$	−25.9877			−1.10711
$552$		0			0
$553$		−	35.9467i		−	1.52861i
$554$		0			0
$555$	−6.68035	+	1.65983i	−0.283565	+	0.0704557i
$556$		0			0
$557$		−	14.7526i		−	0.625087i	−0.949903	−	0.312543i	$-0.898819\pi$
$557$		−	14.7526i		−	0.625087i	0.949903	−	0.312543i	$-0.101181\pi$
$558$		0			0
$559$	−1.98771			−0.0840713
$560$		0			0
$561$	−23.8843			−1.00840
$562$		0			0
$563$			38.5523i			1.62479i	0.583109	+	0.812394i	$0.301836\pi$
$563$			38.5523i			1.62479i	−0.583109	+	0.812394i	$0.698164\pi$
$564$		0			0
$565$	10.6537	+	42.8781i	0.448204	+	1.80390i
$566$		0			0
$567$		−	2.34017i		−	0.0982780i
$568$		0			0
$569$	−14.6803			−0.615432			−0.307716	−	0.951478i	$-0.599565\pi$
$569$	−14.6803			−0.615432			−0.307716	+	0.951478i	$0.599565\pi$
$570$		0			0
$571$	−13.6742			−0.572248			−0.286124	−	0.958193i	$-0.592367\pi$
$571$	−13.6742			−0.572248			−0.286124	+	0.958193i	$0.592367\pi$
$572$		0			0
$573$			6.15676i			0.257202i
$574$		0			0
$575$	5.04718	+	9.52973i	0.210482	+	0.397417i
$576$		0			0
$577$		−	22.6681i		−	0.943684i	−0.881683	−	0.471842i	$-0.843589\pi$
$577$		−	22.6681i		−	0.943684i	0.881683	−	0.471842i	$-0.156411\pi$
$578$		0			0
$579$	−14.5236			−0.603580
$580$		0			0
$581$	4.31351			0.178955
$582$		0			0
$583$		−	8.99386i		−	0.372487i
$584$		0			0
$585$	0.496928	+	2.00000i	0.0205455	+	0.0826898i
$586$		0			0
$587$		−	37.3607i		−	1.54204i	−0.636810	−	0.771020i	$-0.719747\pi$
$587$		−	37.3607i		−	1.54204i	0.636810	−	0.771020i	$-0.280253\pi$
$588$		0			0
$589$	−8.00000			−0.329634
$590$		0			0
$591$	−24.5958			−1.01174
$592$		0			0
$593$		−	15.5897i		−	0.640192i	−0.947385	−	0.320096i	$-0.896285\pi$
$593$		−	15.5897i		−	0.640192i	0.947385	−	0.320096i	$-0.103715\pi$
$594$		0			0
$595$	39.4017	−	9.78992i	1.61531	−	0.401348i
$596$		0			0
$597$		−	3.36069i		−	0.137544i
$598$		0			0
$599$	45.3607			1.85339			0.926694	−	0.375817i	$-0.122638\pi$
$599$	45.3607			1.85339			0.926694	+	0.375817i	$0.122638\pi$
$600$		0			0
$601$	16.5236			0.674011			0.337006	−	0.941503i	$-0.390586\pi$
$601$	16.5236			0.674011			0.337006	+	0.941503i	$0.390586\pi$
$602$		0			0
$603$			6.83710i			0.278428i
$604$		0			0
$605$	−3.30632	+	0.821503i	−0.134421	+	0.0333988i
$606$		0			0
$607$		−	30.0554i		−	1.21991i	−0.792435	−	0.609956i	$-0.791187\pi$
$607$		−	30.0554i		−	1.21991i	0.792435	−	0.609956i	$-0.208813\pi$
$608$		0			0
$609$	15.2039			0.616095
$610$		0			0
$611$	−8.00000			−0.323645
$612$		0			0
$613$			24.2700i			0.980257i	0.871650	+	0.490129i	$0.163050\pi$
$613$			24.2700i			0.980257i	−0.871650	+	0.490129i	$0.836950\pi$
$614$		0			0
$615$	5.75872	+	23.1773i	0.232214	+	0.934598i
$616$		0			0
$617$			0.0722347i			0.00290806i	0.999999	+	0.00145403i	$0.000462832\pi$
$617$			0.0722347i			0.00290806i	−0.999999	+	0.00145403i	$0.999537\pi$
$618$		0			0
$619$	29.1917			1.17331			0.586656	−	0.809836i	$-0.300444\pi$
$619$	29.1917			1.17331			0.586656	+	0.809836i	$0.300444\pi$
$620$		0			0
$621$	2.15676			0.0865476
$622$		0			0
$623$			14.0410i			0.562542i
$624$		0			0
$625$	14.0472	−	20.6803i	0.561887	−	0.827214i
$626$		0			0
$627$		−	12.3135i		−	0.491754i
$628$		0			0
$629$	−23.8843			−0.952329
$630$		0			0
$631$	3.36069			0.133787			0.0668935	−	0.997760i	$-0.478691\pi$
$631$	3.36069			0.133787			0.0668935	+	0.997760i	$0.478691\pi$
$632$		0			0
$633$			11.2039i			0.445316i
$634$		0			0
$635$	10.4247	+	41.9565i	0.413691	+	1.66499i
$636$		0			0
$637$		−	1.40417i		−	0.0556354i
$638$		0			0
$639$	−11.5174			−0.455623
$640$		0			0
$641$	36.6681			1.44830			0.724151	−	0.689642i	$-0.242232\pi$
$641$	36.6681			1.44830			0.724151	+	0.689642i	$0.242232\pi$
$642$		0			0
$643$		−	40.6803i		−	1.60428i	−0.597139	−	0.802138i	$-0.703696\pi$
$643$		−	40.6803i		−	1.60428i	0.597139	−	0.802138i	$-0.296304\pi$
$644$		0			0
$645$	4.68035	−	1.16290i	0.184288	−	0.0457891i
$646$		0			0
$647$		−	37.7275i		−	1.48322i	−0.670830	−	0.741611i	$-0.734062\pi$
$647$		−	37.7275i		−	1.48322i	0.670830	−	0.741611i	$-0.265938\pi$
$648$		0			0
$649$	−36.1978			−1.42089
$650$		0			0
$651$	4.68035			0.183437
$652$		0			0
$653$			38.9216i			1.52312i	0.648094	+	0.761560i	$0.275566\pi$
$653$			38.9216i			1.52312i	−0.648094	+	0.761560i	$0.724434\pi$
$654$		0			0
$655$	−16.1568	+	4.01438i	−0.631297	+	0.156855i
$656$		0			0
$657$			6.52359i			0.254510i
$658$		0			0
$659$	−32.9504			−1.28356			−0.641782	−	0.766887i	$-0.721805\pi$
$659$	−32.9504			−1.28356			−0.641782	+	0.766887i	$0.721805\pi$
$660$		0			0
$661$	36.3012			1.41195			0.705977	−	0.708235i	$-0.250508\pi$
$661$	36.3012			1.41195			0.705977	+	0.708235i	$0.250508\pi$
$662$		0			0
$663$			7.15061i			0.277707i
$664$		0			0
$665$	5.04718	+	20.3135i	0.195721	+	0.787724i
$666$		0			0
$667$			14.0123i			0.542558i
$668$		0			0
$669$	−6.65368			−0.257246
$670$		0			0
$671$	18.4703			0.713037
$672$		0			0
$673$		−	37.1917i		−	1.43363i	−0.697262	−	0.716816i	$-0.745599\pi$
$673$		−	37.1917i		−	1.43363i	0.697262	−	0.716816i	$-0.254401\pi$
$674$		0			0
$675$	−2.34017	−	4.41855i	−0.0900733	−	0.170070i
$676$		0			0
$677$			9.07838i			0.348910i	0.984665	+	0.174455i	$0.0558164\pi$
$677$			9.07838i			0.348910i	−0.984665	+	0.174455i	$0.944184\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	9.84324			0.377194
$682$		0			0
$683$		−	9.84324i		−	0.376641i	−0.982108	−	0.188321i	$-0.939696\pi$
$683$		−	9.84324i		−	0.376641i	0.982108	−	0.188321i	$-0.0603044\pi$
$684$		0			0
$685$	−1.85762	−	7.47641i	−0.0709760	−	0.285659i
$686$		0			0
$687$		−	1.31965i		−	0.0503479i
$688$		0			0
$689$	−2.69263			−0.102581
$690$		0			0
$691$	−1.67420			−0.0636897			−0.0318448	−	0.999493i	$-0.510138\pi$
$691$	−1.67420			−0.0636897			−0.0318448	+	0.999493i	$0.510138\pi$
$692$		0			0
$693$			7.20394i			0.273655i
$694$		0			0
$695$	−41.6742	+	10.3545i	−1.58079	+	0.392770i
$696$		0			0
$697$			82.8659i			3.13877i
$698$		0			0
$699$	2.39803			0.0907019
$700$		0			0
$701$	−8.96719			−0.338686			−0.169343	−	0.985557i	$-0.554165\pi$
$701$	−8.96719			−0.338686			−0.169343	+	0.985557i	$0.554165\pi$
$702$		0			0
$703$		−	12.3135i		−	0.464413i
$704$		0			0
$705$	18.8371	−	4.68035i	0.709446	−	0.176272i
$706$		0			0
$707$		−	19.5174i		−	0.734029i
$708$		0			0
$709$	−25.7152			−0.965756			−0.482878	−	0.875688i	$-0.660409\pi$
$709$	−25.7152			−0.965756			−0.482878	+	0.875688i	$0.660409\pi$
$710$		0			0
$711$	−15.3607			−0.576071
$712$		0			0
$713$			4.31351i			0.161542i
$714$		0			0
$715$	−1.52973	−	6.15676i	−0.0572088	−	0.230250i
$716$		0			0
$717$			6.15676i			0.229928i
$718$		0			0
$719$	−18.4079			−0.686498			−0.343249	−	0.939244i	$-0.611527\pi$
$719$	−18.4079			−0.686498			−0.343249	+	0.939244i	$0.611527\pi$
$720$		0			0
$721$	24.1978			0.901173
$722$		0			0
$723$			0.639308i			0.0237761i
$724$		0			0
$725$	28.7070	−	15.2039i	1.06615	−	0.564660i
$726$		0			0
$727$			42.3402i			1.57031i	0.619299	+	0.785155i	$0.287417\pi$
$727$			42.3402i			1.57031i	−0.619299	+	0.785155i	$0.712583\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	16.7337			0.618917
$732$		0			0
$733$		−	4.75258i		−	0.175541i	−0.996141	−	0.0877703i	$-0.972026\pi$
$733$		−	4.75258i		−	0.175541i	0.996141	−	0.0877703i	$-0.0279741\pi$
$734$		0			0
$735$	0.821503	+	3.30632i	0.0303016	+	0.121956i
$736$		0			0
$737$		−	21.0472i		−	0.775283i
$738$		0			0
$739$	−21.0472			−0.774233			−0.387117	−	0.922031i	$-0.626529\pi$
$739$	−21.0472			−0.774233			−0.387117	+	0.922031i	$0.626529\pi$
$740$		0			0
$741$	−3.68649			−0.135427
$742$		0			0
$743$			49.5585i			1.81812i	0.416660	+	0.909062i	$0.363200\pi$
$743$			49.5585i			1.81812i	−0.416660	+	0.909062i	$0.636800\pi$
$744$		0			0
$745$	−21.3028	+	5.29299i	−0.780475	+	0.193920i
$746$		0			0
$747$		−	1.84324i		−	0.0674408i
$748$		0			0
$749$	−31.2663			−1.14245
$750$		0			0
$751$	41.6619			1.52026			0.760132	−	0.649768i	$-0.225134\pi$
$751$	41.6619			1.52026			0.760132	+	0.649768i	$0.225134\pi$
$752$		0			0
$753$			12.9216i			0.470890i
$754$		0			0
$755$	−51.3751	+	12.7649i	−1.86973	+	0.464561i
$756$		0			0
$757$		−	21.9688i		−	0.798470i	−0.916849	−	0.399235i	$-0.869276\pi$
$757$		−	21.9688i		−	0.798470i	0.916849	−	0.399235i	$-0.130724\pi$
$758$		0			0
$759$	−6.63931			−0.240992
$760$		0			0
$761$	−11.3074			−0.409892			−0.204946	−	0.978773i	$-0.565702\pi$
$761$	−11.3074			−0.409892			−0.204946	+	0.978773i	$0.565702\pi$
$762$		0			0
$763$			12.4489i			0.450681i
$764$		0			0
$765$	−4.18342	−	16.8371i	−0.151252	−	0.608747i
$766$		0			0
$767$			10.8371i			0.391305i
$768$		0			0
$769$	−6.19779			−0.223498			−0.111749	−	0.993736i	$-0.535645\pi$
$769$	−6.19779			−0.223498			−0.111749	+	0.993736i	$0.535645\pi$
$770$		0			0
$771$	−14.9627			−0.538867
$772$		0			0
$773$		−	30.4391i		−	1.09482i	−0.836865	−	0.547409i	$-0.815614\pi$
$773$		−	30.4391i		−	1.09482i	0.836865	−	0.547409i	$-0.184386\pi$
$774$		0			0
$775$	8.83710	−	4.68035i	0.317438	−	0.168123i
$776$		0			0
$777$			7.20394i			0.258440i
$778$		0			0
$779$	−42.7214			−1.53065
$780$		0			0
$781$	35.4551			1.26868
$782$		0			0
$783$		−	6.49693i		−	0.232181i
$784$		0			0
$785$	−5.97334	−	24.0410i	−0.213198	−	0.858061i
$786$		0			0
$787$			36.9939i			1.31869i	0.751841	+	0.659344i	$0.229166\pi$
$787$			36.9939i			1.31869i	−0.751841	+	0.659344i	$0.770834\pi$
$788$		0			0
$789$	13.3607			0.475653
$790$		0			0
$791$	46.2388			1.64406
$792$		0			0
$793$		−	5.52973i		−	0.196367i
$794$		0			0
$795$	6.34017	−	1.57531i	0.224863	−	0.0558704i
$796$		0			0
$797$			30.7526i			1.08931i	0.838659	+	0.544656i	$0.183340\pi$
$797$			30.7526i			1.08931i	−0.838659	+	0.544656i	$0.816660\pi$
$798$		0			0
$799$	67.3484			2.38262
$800$		0			0
$801$	6.00000			0.212000
$802$		0			0
$803$		−	20.0821i		−	0.708681i
$804$		0			0
$805$	10.9528	−	2.72138i	0.386036	−	0.0959162i
$806$		0			0
$807$		−	27.5441i		−	0.969599i
$808$		0			0
$809$	5.31965			0.187029			0.0935145	−	0.995618i	$-0.470190\pi$
$809$	5.31965			0.187029			0.0935145	+	0.995618i	$0.470190\pi$
$810$		0			0
$811$	−49.9253			−1.75312			−0.876558	−	0.481297i	$-0.840166\pi$
$811$	−49.9253			−1.75312			−0.876558	+	0.481297i	$0.840166\pi$
$812$		0			0
$813$			10.0000i			0.350715i
$814$		0			0
$815$	7.00614	+	28.1978i	0.245414	+	0.987726i
$816$		0			0
$817$			8.62702i			0.301821i
$818$		0			0
$819$	2.15676			0.0753631
$820$		0			0
$821$	−44.1711			−1.54158			−0.770792	−	0.637087i	$-0.780139\pi$
$821$	−44.1711			−1.54158			−0.770792	+	0.637087i	$0.780139\pi$
$822$		0			0
$823$		−	19.0738i		−	0.664872i	−0.943126	−	0.332436i	$-0.892129\pi$
$823$		−	19.0738i		−	0.664872i	0.943126	−	0.332436i	$-0.107871\pi$
$824$		0			0
$825$	7.20394	+	13.6020i	0.250809	+	0.473560i
$826$		0			0
$827$			43.0349i			1.49647i	0.663434	+	0.748235i	$0.269098\pi$
$827$			43.0349i			1.49647i	−0.663434	+	0.748235i	$0.730902\pi$
$828$		0			0
$829$	36.0410			1.25176			0.625878	−	0.779921i	$-0.284741\pi$
$829$	36.0410			1.25176			0.625878	+	0.779921i	$0.284741\pi$
$830$		0			0
$831$	−4.12556			−0.143114
$832$		0			0
$833$			11.8211i			0.409577i
$834$		0			0
$835$	−6.47027	−	26.0410i	−0.223913	−	0.901187i
$836$		0			0
$837$		−	2.00000i		−	0.0691301i
$838$		0			0
$839$	43.0349			1.48573			0.742865	−	0.669441i	$-0.233467\pi$
$839$	43.0349			1.48573			0.742865	+	0.669441i	$0.233467\pi$
$840$		0			0
$841$	13.2101			0.455520
$842$		0			0
$843$		−	11.3607i		−	0.391283i
$844$		0			0
$845$	26.3679	−	6.55148i	0.907083	−	0.225378i
$846$		0			0
$847$			3.56547i			0.122511i
$848$		0			0
$849$	−10.5236			−0.361169
$850$		0			0
$851$	−6.63931			−0.227593
$852$		0			0
$853$		−	47.1605i		−	1.61474i	−0.590043	−	0.807372i	$-0.700889\pi$
$853$		−	47.1605i		−	1.61474i	0.590043	−	0.807372i	$-0.299111\pi$
$854$		0			0
$855$	8.68035	−	2.15676i	0.296862	−	0.0737595i
$856$		0			0
$857$			12.8059i			0.437441i	0.975788	+	0.218721i	$0.0701884\pi$
$857$			12.8059i			0.437441i	−0.975788	+	0.218721i	$0.929812\pi$
$858$		0			0
$859$	14.4703			0.493719			0.246860	−	0.969051i	$-0.420601\pi$
$859$	14.4703			0.493719			0.246860	+	0.969051i	$0.420601\pi$
$860$		0			0
$861$	24.9939			0.851788
$862$		0			0
$863$		−	24.3135i		−	0.827642i	−0.910358	−	0.413821i	$-0.864194\pi$
$863$		−	24.3135i		−	0.827642i	0.910358	−	0.413821i	$-0.135806\pi$
$864$		0			0
$865$	−9.94214	−	40.0144i	−0.338043	−	1.36053i
$866$		0			0
$867$		−	43.1978i		−	1.46707i
$868$		0			0
$869$	47.2860			1.60407
$870$		0			0
$871$	−6.30122			−0.213509
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−17.4596	−	19.4863i	−0.590242	−	0.658756i
$876$		0			0
$877$		−	44.4391i		−	1.50060i	−0.661097	−	0.750300i	$-0.729909\pi$
$877$		−	44.4391i		−	1.50060i	0.661097	−	0.750300i	$-0.270091\pi$
$878$		0			0
$879$	32.5958			1.09943
$880$		0			0
$881$	6.62702			0.223270			0.111635	−	0.993749i	$-0.464391\pi$
$881$	6.62702			0.223270			0.111635	+	0.993749i	$0.464391\pi$
$882$		0			0
$883$			7.31965i			0.246326i	0.992386	+	0.123163i	$0.0393038\pi$
$883$			7.31965i			0.246326i	−0.992386	+	0.123163i	$0.960696\pi$
$884$		0			0
$885$	−6.34017	−	25.5174i	−0.213123	−	0.857760i
$886$		0			0
$887$			12.0000i			0.402921i	0.979497	+	0.201460i	$0.0645687\pi$
$887$			12.0000i			0.402921i	−0.979497	+	0.201460i	$0.935431\pi$
$888$		0			0
$889$	45.2450			1.51747
$890$		0			0
$891$	3.07838			0.103130
$892$		0			0
$893$			34.7214i			1.16191i
$894$		0			0
$895$	4.15676	−	1.03281i	0.138945	−	0.0345229i
$896$		0			0
$897$			1.98771i			0.0663678i
$898$		0			0
$899$	12.9939			0.433369
$900$		0			0
$901$	22.6681			0.755183
$902$		0			0
$903$		−	5.04718i		−	0.167960i
$904$		0			0
$905$	−15.1773	+	3.77101i	−0.504510	+	0.125353i
$906$		0			0
$907$			4.48255i			0.148841i	0.997227	+	0.0744204i	$0.0237107\pi$
$907$			4.48255i			0.148841i	−0.997227	+	0.0744204i	$0.976289\pi$
$908$		0			0
$909$	−8.34017			−0.276626
$910$		0			0
$911$	8.73367			0.289359			0.144680	−	0.989479i	$-0.453785\pi$
$911$	8.73367			0.289359			0.144680	+	0.989479i	$0.453785\pi$
$912$		0			0
$913$			5.67420i			0.187789i
$914$		0			0
$915$	3.23513	+	13.0205i	0.106950	+	0.430445i
$916$		0			0
$917$			17.4231i			0.575361i
$918$		0			0
$919$	−18.7337			−0.617967			−0.308983	−	0.951067i	$-0.599989\pi$
$919$	−18.7337			−0.617967			−0.308983	+	0.951067i	$0.599989\pi$
$920$		0			0
$921$	−29.9877			−0.988129
$922$		0			0
$923$		−	10.6147i		−	0.349388i
$924$		0			0
$925$	7.20394	+	13.6020i	0.236864	+	0.447230i
$926$		0			0
$927$		−	10.3402i		−	0.339616i
$928$		0			0
$929$	20.0410			0.657525			0.328763	−	0.944413i	$-0.393368\pi$
$929$	20.0410			0.657525			0.328763	+	0.944413i	$0.393368\pi$
$930$		0			0
$931$	−6.09436			−0.199735
$932$		0			0
$933$		−	27.2039i		−	0.890617i
$934$		0			0
$935$	12.8781	+	51.8310i	0.421160	+	1.69505i
$936$		0			0
$937$			45.0472i			1.47163i	0.677184	+	0.735814i	$0.263200\pi$
$937$			45.0472i			1.47163i	−0.677184	+	0.735814i	$0.736800\pi$
$938$		0			0
$939$	26.8371			0.875796
$940$		0			0
$941$	−19.3751			−0.631609			−0.315805	−	0.948824i	$-0.602274\pi$
$941$	−19.3751			−0.631609			−0.315805	+	0.948824i	$0.602274\pi$
$942$		0			0
$943$			23.0349i			0.750119i
$944$		0			0
$945$	−5.07838	+	1.26180i	−0.165200	+	0.0410462i
$946$		0			0
$947$		−	6.95282i		−	0.225936i	−0.993599	−	0.112968i	$-0.963964\pi$
$947$		−	6.95282i		−	0.225936i	0.993599	−	0.112968i	$-0.0360358\pi$
$948$		0			0
$949$	−6.01229			−0.195167
$950$		0			0
$951$	24.5958			0.797574
$952$		0			0
$953$		−	29.2885i		−	0.948746i	−0.880324	−	0.474373i	$-0.842675\pi$
$953$		−	29.2885i		−	0.948746i	0.880324	−	0.474373i	$-0.157325\pi$
$954$		0			0
$955$	13.3607	−	3.31965i	0.432342	−	0.107421i
$956$		0			0
$957$			20.0000i			0.646508i
$958$		0			0
$959$	−8.06239			−0.260348
$960$		0			0
$961$	−27.0000			−0.870968
$962$		0			0
$963$			13.3607i			0.430542i
$964$		0			0
$965$	7.83096	+	31.5174i	0.252087	+	1.01458i
$966$		0			0
$967$		−	47.3874i		−	1.52387i	−0.647651	−	0.761937i	$-0.724248\pi$
$967$		−	47.3874i		−	1.52387i	0.647651	−	0.761937i	$-0.275752\pi$
$968$		0			0
$969$	31.0349			0.996984
$970$		0			0
$971$	2.59583			0.0833040			0.0416520	−	0.999132i	$-0.486738\pi$
$971$	2.59583			0.0833040			0.0416520	+	0.999132i	$0.486738\pi$
$972$		0			0
$973$			44.9405i			1.44073i
$974$		0			0
$975$	4.07223	−	2.15676i	0.130416	−	0.0690715i
$976$		0			0
$977$		−	28.3234i		−	0.906144i	−0.891474	−	0.453072i	$-0.850328\pi$
$977$		−	28.3234i		−	0.906144i	0.891474	−	0.453072i	$-0.149672\pi$
$978$		0			0
$979$	−18.4703			−0.590312
$980$		0			0
$981$	5.31965			0.169843
$982$		0			0
$983$			1.16290i			0.0370907i	0.999828	+	0.0185454i	$0.00590351\pi$
$983$			1.16290i			0.0370907i	−0.999828	+	0.0185454i	$0.994096\pi$
$984$		0			0
$985$	13.2618	+	53.3751i	0.422556	+	1.70067i
$986$		0			0
$987$		−	20.3135i		−	0.646586i
$988$		0			0
$989$	4.65159			0.147912
$990$		0			0
$991$	−12.9528			−0.411460			−0.205730	−	0.978609i	$-0.565957\pi$
$991$	−12.9528			−0.411460			−0.205730	+	0.978609i	$0.565957\pi$
$992$		0			0
$993$			12.3135i			0.390757i
$994$		0			0
$995$	−7.29299	+	1.81205i	−0.231203	+	0.0574458i
$996$		0			0
$997$		−	19.0784i		−	0.604218i	−0.953273	−	0.302109i	$-0.902309\pi$
$997$		−	19.0784i		−	0.604218i	0.953273	−	0.302109i	$-0.0976907\pi$
$998$		0			0
$999$	3.07838			0.0973956

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	1920.2.f.p.769.6	yes	6
4.3	odd	2		1920.2.f.o.769.3	yes	6
5.2	odd	4		9600.2.a.dt.1.3		3
5.3	odd	4		9600.2.a.dr.1.1		3
5.4	even	2	inner	1920.2.f.p.769.3	yes	6
8.3	odd	2		1920.2.f.n.769.4	yes	6
8.5	even	2		1920.2.f.m.769.1	✓	6
16.3	odd	4		3840.2.d.bl.2689.3		6
16.5	even	4		3840.2.d.bk.2689.4		6
16.11	odd	4		3840.2.d.bi.2689.4		6
16.13	even	4		3840.2.d.bj.2689.3		6
20.3	even	4		9600.2.a.ds.1.3		3
20.7	even	4		9600.2.a.dq.1.1		3
20.19	odd	2		1920.2.f.o.769.6	yes	6
40.3	even	4		9600.2.a.dp.1.3		3
40.13	odd	4		9600.2.a.du.1.1		3
40.19	odd	2		1920.2.f.n.769.1	yes	6
40.27	even	4		9600.2.a.dv.1.1		3
40.29	even	2		1920.2.f.m.769.4	yes	6
40.37	odd	4		9600.2.a.do.1.3		3
80.19	odd	4		3840.2.d.bi.2689.3		6
80.29	even	4		3840.2.d.bk.2689.3		6
80.59	odd	4		3840.2.d.bl.2689.4		6
80.69	even	4		3840.2.d.bj.2689.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.f.m.769.1	✓	6	8.5	even	2
1920.2.f.m.769.4	yes	6	40.29	even	2
1920.2.f.n.769.1	yes	6	40.19	odd	2
1920.2.f.n.769.4	yes	6	8.3	odd	2
1920.2.f.o.769.3	yes	6	4.3	odd	2
1920.2.f.o.769.6	yes	6	20.19	odd	2
1920.2.f.p.769.3	yes	6	5.4	even	2	inner
1920.2.f.p.769.6	yes	6	1.1	even	1	trivial
3840.2.d.bi.2689.3		6	80.19	odd	4
3840.2.d.bi.2689.4		6	16.11	odd	4
3840.2.d.bj.2689.3		6	16.13	even	4
3840.2.d.bj.2689.4		6	80.69	even	4
3840.2.d.bk.2689.3		6	80.29	even	4
3840.2.d.bk.2689.4		6	16.5	even	4
3840.2.d.bl.2689.3		6	16.3	odd	4
3840.2.d.bl.2689.4		6	80.59	odd	4
9600.2.a.do.1.3		3	40.37	odd	4
9600.2.a.dp.1.3		3	40.3	even	4
9600.2.a.dq.1.1		3	20.7	even	4
9600.2.a.dr.1.1		3	5.3	odd	4
9600.2.a.ds.1.3		3	20.3	even	4
9600.2.a.dt.1.3		3	5.2	odd	4
9600.2.a.du.1.1		3	40.13	odd	4
9600.2.a.dv.1.1		3	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

Downloads

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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	1920.2.f.p.769.6

Level	$1920$
Weight	$2$
Character	1920.769
Analytic conductor	$15.331$
Analytic rank	$0$
Dimension	$6$
Inner twists	$2$