LMFDB - Embedded newform 1840.2.m.d.1839.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(1839,1840)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1840.1839");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1840.m (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$14.6924739719$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{6}, \sqrt{-14})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 25 $ x^4 + 4*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1839.2
Root			$1.22474 + 1.87083i$ of defining polynomial
Character	$\chi$	$=$	1840.1839
Dual form			1840.2.m.d.1839.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +(-1.22474 + 1.87083i) q^{5} -2.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +(-1.22474 + 1.87083i) q^{5} -2.00000 q^{9} +4.89898 q^{11} +4.58258i q^{13} +(-1.22474 + 1.87083i) q^{15} +2.44949 q^{17} +2.44949 q^{19} +(-3.00000 + 3.74166i) q^{23} +(-2.00000 - 4.58258i) q^{25} -5.00000 q^{27} -3.00000 q^{29} -4.58258i q^{31} +4.89898 q^{33} -4.89898 q^{37} +4.58258i q^{39} -3.00000 q^{41} +11.2250i q^{43} +(2.44949 - 3.74166i) q^{45} -9.00000 q^{47} +7.00000 q^{49} +2.44949 q^{51} +4.89898 q^{53} +(-6.00000 + 9.16515i) q^{55} +2.44949 q^{57} +9.16515i q^{59} +11.2250i q^{61} +(-8.57321 - 5.61249i) q^{65} +(-3.00000 + 3.74166i) q^{69} +13.7477i q^{71} +4.58258i q^{73} +(-2.00000 - 4.58258i) q^{75} +7.34847 q^{79} +1.00000 q^{81} -3.74166i q^{83} +(-3.00000 + 4.58258i) q^{85} -3.00000 q^{87} -3.74166i q^{89} -4.58258i q^{93} +(-3.00000 + 4.58258i) q^{95} -9.79796 q^{97} -9.79796 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 8 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 8 * q^9	$ 4 q + 4 q^{3} - 8 q^{9} - 12 q^{23} - 8 q^{25} - 20 q^{27} - 12 q^{29} - 12 q^{41} - 36 q^{47} + 28 q^{49} - 24 q^{55} - 12 q^{69} - 8 q^{75} + 4 q^{81} - 12 q^{85} - 12 q^{87} - 12 q^{95}+O(q^{100}) $ 4 * q + 4 * q^3 - 8 * q^9 - 12 * q^23 - 8 * q^25 - 20 * q^27 - 12 * q^29 - 12 * q^41 - 36 * q^47 + 28 * q^49 - 24 * q^55 - 12 * q^69 - 8 * q^75 + 4 * q^81 - 12 * q^85 - 12 * q^87 - 12 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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
$3$	1.00000			0.577350			0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000			0.577350			0.288675	+	0.957427i	$0.406785\pi$
$4$		0			0
$5$	−1.22474	+	1.87083i	−0.547723	+	0.836660i
$5$	−1.22474	+	1.87083i	−0.547723	+	0.836660i
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	−2.00000			−0.666667
$10$		0			0
$11$	4.89898			1.47710			0.738549	−	0.674200i	$-0.235511\pi$
$11$	4.89898			1.47710			0.738549	+	0.674200i	$0.235511\pi$
$12$		0			0
$13$			4.58258i			1.27098i	0.772110	+	0.635489i	$0.219201\pi$
$13$			4.58258i			1.27098i	−0.772110	+	0.635489i	$0.780799\pi$
$14$		0			0
$15$	−1.22474	+	1.87083i	−0.316228	+	0.483046i
$16$		0			0
$17$	2.44949			0.594089			0.297044	−	0.954864i	$-0.403999\pi$
$17$	2.44949			0.594089			0.297044	+	0.954864i	$0.403999\pi$
$18$		0			0
$19$	2.44949			0.561951			0.280976	−	0.959715i	$-0.409342\pi$
$19$	2.44949			0.561951			0.280976	+	0.959715i	$0.409342\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.00000	+	3.74166i	−0.625543	+	0.780189i
$23$	−3.00000	+	3.74166i	−0.625543	+	0.780189i
$24$		0			0
$25$	−2.00000	−	4.58258i	−0.400000	−	0.916515i
$26$		0			0
$27$	−5.00000			−0.962250
$28$		0			0
$29$	−3.00000			−0.557086			−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000			−0.557086			−0.278543	+	0.960424i	$0.589851\pi$
$30$		0			0
$31$		−	4.58258i		−	0.823055i	−0.911397	−	0.411527i	$-0.864995\pi$
$31$		−	4.58258i		−	0.823055i	0.911397	−	0.411527i	$-0.135005\pi$
$32$		0			0
$33$	4.89898			0.852803
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−4.89898			−0.805387			−0.402694	−	0.915335i	$-0.631926\pi$
$37$	−4.89898			−0.805387			−0.402694	+	0.915335i	$0.631926\pi$
$38$		0			0
$39$			4.58258i			0.733799i
$40$		0			0
$41$	−3.00000			−0.468521			−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000			−0.468521			−0.234261	+	0.972174i	$0.575267\pi$
$42$		0			0
$43$			11.2250i			1.71179i	0.517148	+	0.855896i	$0.326994\pi$
$43$			11.2250i			1.71179i	−0.517148	+	0.855896i	$0.673006\pi$
$44$		0			0
$45$	2.44949	−	3.74166i	0.365148	−	0.557773i
$46$		0			0
$47$	−9.00000			−1.31278			−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000			−1.31278			−0.656392	+	0.754420i	$0.727918\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$	2.44949			0.342997
$52$		0			0
$53$	4.89898			0.672927			0.336463	−	0.941697i	$-0.390769\pi$
$53$	4.89898			0.672927			0.336463	+	0.941697i	$0.390769\pi$
$54$		0			0
$55$	−6.00000	+	9.16515i	−0.809040	+	1.23583i
$56$		0			0
$57$	2.44949			0.324443
$58$		0			0
$59$			9.16515i			1.19320i	0.802538	+	0.596601i	$0.203482\pi$
$59$			9.16515i			1.19320i	−0.802538	+	0.596601i	$0.796518\pi$
$60$		0			0
$61$			11.2250i			1.43721i	0.695418	+	0.718605i	$0.255219\pi$
$61$			11.2250i			1.43721i	−0.695418	+	0.718605i	$0.744781\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−8.57321	−	5.61249i	−1.06338	−	0.696143i
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$	−3.00000	+	3.74166i	−0.361158	+	0.450443i
$70$		0			0
$71$			13.7477i			1.63156i	0.578366	+	0.815778i	$0.303691\pi$
$71$			13.7477i			1.63156i	−0.578366	+	0.815778i	$0.696309\pi$
$72$		0			0
$73$			4.58258i			0.536350i	0.963370	+	0.268175i	$0.0864205\pi$
$73$			4.58258i			0.536350i	−0.963370	+	0.268175i	$0.913579\pi$
$74$		0			0
$75$	−2.00000	−	4.58258i	−0.230940	−	0.529150i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	7.34847			0.826767			0.413384	−	0.910557i	$-0.364347\pi$
$79$	7.34847			0.826767			0.413384	+	0.910557i	$0.364347\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	3.74166i		−	0.410700i	−0.978689	−	0.205350i	$-0.934167\pi$
$83$		−	3.74166i		−	0.410700i	0.978689	−	0.205350i	$-0.0658333\pi$
$84$		0			0
$85$	−3.00000	+	4.58258i	−0.325396	+	0.497050i
$86$		0			0
$87$	−3.00000			−0.321634
$88$		0			0
$89$		−	3.74166i		−	0.396615i	−0.980140	−	0.198307i	$-0.936456\pi$
$89$		−	3.74166i		−	0.396615i	0.980140	−	0.198307i	$-0.0635444\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		−	4.58258i		−	0.475191i
$94$		0			0
$95$	−3.00000	+	4.58258i	−0.307794	+	0.470162i
$96$		0			0
$97$	−9.79796			−0.994832			−0.497416	−	0.867512i	$-0.665718\pi$
$97$	−9.79796			−0.994832			−0.497416	+	0.867512i	$0.665718\pi$
$98$		0			0
$99$	−9.79796			−0.984732
$100$		0			0
$101$	12.0000			1.19404			0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000			1.19404			0.597022	+	0.802225i	$0.296350\pi$
$102$		0			0
$103$			11.2250i			1.10603i	0.833172	+	0.553015i	$0.186523\pi$
$103$			11.2250i			1.10603i	−0.833172	+	0.553015i	$0.813477\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			3.74166i			0.361720i	0.983509	+	0.180860i	$0.0578880\pi$
$107$			3.74166i			0.361720i	−0.983509	+	0.180860i	$0.942112\pi$
$108$		0			0
$109$		−	11.2250i		−	1.07516i	−0.843214	−	0.537579i	$-0.819339\pi$
$109$		−	11.2250i		−	1.07516i	0.843214	−	0.537579i	$-0.180661\pi$
$110$		0			0
$111$	−4.89898			−0.464991
$112$		0			0
$113$	−14.6969			−1.38257			−0.691286	−	0.722581i	$-0.742955\pi$
$113$	−14.6969			−1.38257			−0.691286	+	0.722581i	$0.742955\pi$
$114$		0			0
$115$	−3.32577	−	10.1951i	−0.310129	−	0.950694i
$116$		0			0
$117$		−	9.16515i		−	0.847319i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	13.0000			1.18182
$122$		0			0
$123$	−3.00000			−0.270501
$124$		0			0
$125$	11.0227	+	1.87083i	0.985901	+	0.167332i
$126$		0			0
$127$	−11.0000			−0.976092			−0.488046	−	0.872818i	$-0.662290\pi$
$127$	−11.0000			−0.976092			−0.488046	+	0.872818i	$0.662290\pi$
$128$		0			0
$129$			11.2250i			0.988304i
$130$		0			0
$131$			4.58258i			0.400381i	0.979757	+	0.200191i	$0.0641562\pi$
$131$			4.58258i			0.400381i	−0.979757	+	0.200191i	$0.935844\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	6.12372	−	9.35414i	0.527046	−	0.805076i
$136$		0			0
$137$	14.6969			1.25564			0.627822	−	0.778357i	$-0.283947\pi$
$137$	14.6969			1.25564			0.627822	+	0.778357i	$0.283947\pi$
$138$		0			0
$139$		−	22.9129i		−	1.94344i	−0.236126	−	0.971722i	$-0.575878\pi$
$139$		−	22.9129i		−	1.94344i	0.236126	−	0.971722i	$-0.424122\pi$
$140$		0			0
$141$	−9.00000			−0.757937
$142$		0			0
$143$			22.4499i			1.87736i
$144$		0			0
$145$	3.67423	−	5.61249i	0.305129	−	0.466092i
$146$		0			0
$147$	7.00000			0.577350
$148$		0			0
$149$		−	14.9666i		−	1.22611i	−0.790039	−	0.613057i	$-0.789940\pi$
$149$		−	14.9666i		−	1.22611i	0.790039	−	0.613057i	$-0.210060\pi$
$150$		0			0
$151$			13.7477i			1.11877i	0.828907	+	0.559387i	$0.188963\pi$
$151$			13.7477i			1.11877i	−0.828907	+	0.559387i	$0.811037\pi$
$152$		0			0
$153$	−4.89898			−0.396059
$154$		0			0
$155$	8.57321	+	5.61249i	0.688617	+	0.450806i
$156$		0			0
$157$	−2.44949			−0.195491			−0.0977453	−	0.995211i	$-0.531163\pi$
$157$	−2.44949			−0.195491			−0.0977453	+	0.995211i	$0.531163\pi$
$158$		0			0
$159$	4.89898			0.388514
$160$		0			0
$161$		0			0
$162$		0			0
$163$	13.0000			1.01824			0.509119	−	0.860696i	$-0.329971\pi$
$163$	13.0000			1.01824			0.509119	+	0.860696i	$0.329971\pi$
$164$		0			0
$165$	−6.00000	+	9.16515i	−0.467099	+	0.713506i
$166$		0			0
$167$	−18.0000			−1.39288			−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000			−1.39288			−0.696441	+	0.717614i	$0.745234\pi$
$168$		0			0
$169$	−8.00000			−0.615385
$170$		0			0
$171$	−4.89898			−0.374634
$172$		0			0
$173$		−	18.3303i		−	1.39363i	−0.717252	−	0.696814i	$-0.754600\pi$
$173$		−	18.3303i		−	1.39363i	0.717252	−	0.696814i	$-0.245400\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			9.16515i			0.688895i
$178$		0			0
$179$			13.7477i			1.02755i	0.857924	+	0.513777i	$0.171754\pi$
$179$			13.7477i			1.02755i	−0.857924	+	0.513777i	$0.828246\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$			11.2250i			0.829774i
$184$		0			0
$185$	6.00000	−	9.16515i	0.441129	−	0.673835i
$186$		0			0
$187$	12.0000			0.877527
$188$		0			0
$189$		0			0
$190$		0			0
$191$	12.2474			0.886194			0.443097	−	0.896474i	$-0.353880\pi$
$191$	12.2474			0.886194			0.443097	+	0.896474i	$0.353880\pi$
$192$		0			0
$193$		−	22.9129i		−	1.64931i	−0.565640	−	0.824653i	$-0.691371\pi$
$193$		−	22.9129i		−	1.64931i	0.565640	−	0.824653i	$-0.308629\pi$
$194$		0			0
$195$	−8.57321	−	5.61249i	−0.613941	−	0.401918i
$196$		0			0
$197$			4.58258i			0.326495i	0.986585	+	0.163247i	$0.0521969\pi$
$197$			4.58258i			0.326495i	−0.986585	+	0.163247i	$0.947803\pi$
$198$		0			0
$199$	19.5959			1.38912			0.694559	−	0.719436i	$-0.255600\pi$
$199$	19.5959			1.38912			0.694559	+	0.719436i	$0.255600\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	3.67423	−	5.61249i	0.256620	−	0.391993i
$206$		0			0
$207$	6.00000	−	7.48331i	0.417029	−	0.520126i
$208$		0			0
$209$	12.0000			0.830057
$210$		0			0
$211$		−	18.3303i		−	1.26191i	−0.775819	−	0.630955i	$-0.782663\pi$
$211$		−	18.3303i		−	1.26191i	0.775819	−	0.630955i	$-0.217337\pi$
$212$		0			0
$213$			13.7477i			0.941979i
$214$		0			0
$215$	−21.0000	−	13.7477i	−1.43219	−	0.937587i
$216$		0			0
$217$		0			0
$218$		0			0
$219$			4.58258i			0.309662i
$220$		0			0
$221$			11.2250i			0.755073i
$222$		0			0
$223$	22.0000			1.47323			0.736614	−	0.676313i	$-0.236423\pi$
$223$	22.0000			1.47323			0.736614	+	0.676313i	$0.236423\pi$
$224$		0			0
$225$	4.00000	+	9.16515i	0.266667	+	0.611010i
$226$		0			0
$227$		−	7.48331i		−	0.496685i	−0.968672	−	0.248343i	$-0.920114\pi$
$227$		−	7.48331i		−	0.496685i	0.968672	−	0.248343i	$-0.0798858\pi$
$228$		0			0
$229$			11.2250i			0.741767i	0.928679	+	0.370884i	$0.120945\pi$
$229$			11.2250i			0.741767i	−0.928679	+	0.370884i	$0.879055\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	4.58258i		−	0.300215i	−0.988670	−	0.150107i	$-0.952038\pi$
$233$		−	4.58258i		−	0.300215i	0.988670	−	0.150107i	$-0.0479619\pi$
$234$		0			0
$235$	11.0227	−	16.8375i	0.719042	−	1.09835i
$236$		0			0
$237$	7.34847			0.477334
$238$		0			0
$239$		−	4.58258i		−	0.296422i	−0.988956	−	0.148211i	$-0.952648\pi$
$239$		−	4.58258i		−	0.296422i	0.988956	−	0.148211i	$-0.0473515\pi$
$240$		0			0
$241$		−	11.2250i		−	0.723064i	−0.932360	−	0.361532i	$-0.882254\pi$
$241$		−	11.2250i		−	0.723064i	0.932360	−	0.361532i	$-0.117746\pi$
$242$		0			0
$243$	16.0000			1.02640
$244$		0			0
$245$	−8.57321	+	13.0958i	−0.547723	+	0.836660i
$246$		0			0
$247$			11.2250i			0.714228i
$248$		0			0
$249$		−	3.74166i		−	0.237118i
$250$		0			0
$251$	−12.2474			−0.773052			−0.386526	−	0.922278i	$-0.626325\pi$
$251$	−12.2474			−0.773052			−0.386526	+	0.922278i	$0.626325\pi$
$252$		0			0
$253$	−14.6969	+	18.3303i	−0.923989	+	1.15242i
$254$		0			0
$255$	−3.00000	+	4.58258i	−0.187867	+	0.286972i
$256$		0			0
$257$			22.9129i			1.42927i	0.699500	+	0.714633i	$0.253406\pi$
$257$			22.9129i			1.42927i	−0.699500	+	0.714633i	$0.746594\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	6.00000			0.371391
$262$		0			0
$263$			18.7083i			1.15360i	0.816885	+	0.576801i	$0.195699\pi$
$263$			18.7083i			1.15360i	−0.816885	+	0.576801i	$0.804301\pi$
$264$		0			0
$265$	−6.00000	+	9.16515i	−0.368577	+	0.563011i
$266$		0			0
$267$		−	3.74166i		−	0.228986i
$268$		0			0
$269$	−9.00000			−0.548740			−0.274370	−	0.961624i	$-0.588469\pi$
$269$	−9.00000			−0.548740			−0.274370	+	0.961624i	$0.588469\pi$
$270$		0			0
$271$		−	9.16515i		−	0.556743i	−0.960473	−	0.278372i	$-0.910205\pi$
$271$		−	9.16515i		−	0.556743i	0.960473	−	0.278372i	$-0.0897947\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−9.79796	−	22.4499i	−0.590839	−	1.35378i
$276$		0			0
$277$		−	22.9129i		−	1.37670i	−0.725378	−	0.688351i	$-0.758335\pi$
$277$		−	22.9129i		−	1.37670i	0.725378	−	0.688351i	$-0.241665\pi$
$278$		0			0
$279$			9.16515i			0.548703i
$280$		0			0
$281$			14.9666i			0.892834i	0.894825	+	0.446417i	$0.147300\pi$
$281$			14.9666i			0.892834i	−0.894825	+	0.446417i	$0.852700\pi$
$282$		0			0
$283$		−	11.2250i		−	0.667255i	−0.942705	−	0.333628i	$-0.891727\pi$
$283$		−	11.2250i		−	0.667255i	0.942705	−	0.333628i	$-0.108273\pi$
$284$		0			0
$285$	−3.00000	+	4.58258i	−0.177705	+	0.271448i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−11.0000			−0.647059
$290$		0			0
$291$	−9.79796			−0.574367
$292$		0			0
$293$	−9.79796			−0.572403			−0.286201	−	0.958169i	$-0.592393\pi$
$293$	−9.79796			−0.572403			−0.286201	+	0.958169i	$0.592393\pi$
$294$		0			0
$295$	−17.1464	−	11.2250i	−0.998304	−	0.653543i
$296$		0			0
$297$	−24.4949			−1.42134
$298$		0			0
$299$	−17.1464	−	13.7477i	−0.991604	−	0.795052i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	12.0000			0.689382
$304$		0			0
$305$	−21.0000	−	13.7477i	−1.20246	−	0.787193i
$306$		0			0
$307$	22.0000			1.25561			0.627803	−	0.778372i	$-0.283954\pi$
$307$	22.0000			1.25561			0.627803	+	0.778372i	$0.283954\pi$
$308$		0			0
$309$			11.2250i			0.638566i
$310$		0			0
$311$		−	4.58258i		−	0.259854i	−0.991524	−	0.129927i	$-0.958526\pi$
$311$		−	4.58258i		−	0.259854i	0.991524	−	0.129927i	$-0.0414743\pi$
$312$		0			0
$313$	29.3939			1.66144			0.830720	−	0.556690i	$-0.187929\pi$
$313$	29.3939			1.66144			0.830720	+	0.556690i	$0.187929\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			18.3303i			1.02953i	0.857331	+	0.514766i	$0.172121\pi$
$317$			18.3303i			1.02953i	−0.857331	+	0.514766i	$0.827879\pi$
$318$		0			0
$319$	−14.6969			−0.822871
$320$		0			0
$321$			3.74166i			0.208839i
$322$		0			0
$323$	6.00000			0.333849
$324$		0			0
$325$	21.0000	−	9.16515i	1.16487	−	0.508391i
$326$		0			0
$327$		−	11.2250i		−	0.620742i
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	4.58258i		−	0.251881i	−0.992038	−	0.125941i	$-0.959805\pi$
$331$		−	4.58258i		−	0.251881i	0.992038	−	0.125941i	$-0.0401949\pi$
$332$		0			0
$333$	9.79796			0.536925
$334$		0			0
$335$		0			0
$336$		0			0
$337$	31.8434			1.73462			0.867309	−	0.497770i	$-0.165847\pi$
$337$	31.8434			1.73462			0.867309	+	0.497770i	$0.165847\pi$
$338$		0			0
$339$	−14.6969			−0.798228
$340$		0			0
$341$		−	22.4499i		−	1.21573i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−3.32577	−	10.1951i	−0.179053	−	0.548884i
$346$		0			0
$347$	6.00000			0.322097			0.161048	−	0.986947i	$-0.448512\pi$
$347$	6.00000			0.322097			0.161048	+	0.986947i	$0.448512\pi$
$348$		0			0
$349$	1.00000			0.0535288			0.0267644	−	0.999642i	$-0.491480\pi$
$349$	1.00000			0.0535288			0.0267644	+	0.999642i	$0.491480\pi$
$350$		0			0
$351$		−	22.9129i		−	1.22300i
$352$		0			0
$353$			4.58258i			0.243906i	0.992536	+	0.121953i	$0.0389157\pi$
$353$			4.58258i			0.243906i	−0.992536	+	0.121953i	$0.961084\pi$
$354$		0			0
$355$	−25.7196	−	16.8375i	−1.36506	−	0.893639i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−12.2474			−0.646396			−0.323198	−	0.946331i	$-0.604758\pi$
$359$	−12.2474			−0.646396			−0.323198	+	0.946331i	$0.604758\pi$
$360$		0			0
$361$	−13.0000			−0.684211
$362$		0			0
$363$	13.0000			0.682323
$364$		0			0
$365$	−8.57321	−	5.61249i	−0.448743	−	0.293771i
$366$		0			0
$367$		−	33.6749i		−	1.75782i	−0.476991	−	0.878908i	$-0.658273\pi$
$367$		−	33.6749i		−	1.75782i	0.476991	−	0.878908i	$-0.341727\pi$
$368$		0			0
$369$	6.00000			0.312348
$370$		0			0
$371$		0			0
$372$		0			0
$373$	22.0454			1.14147			0.570734	−	0.821135i	$-0.306659\pi$
$373$	22.0454			1.14147			0.570734	+	0.821135i	$0.306659\pi$
$374$		0			0
$375$	11.0227	+	1.87083i	0.569210	+	0.0966092i
$376$		0			0
$377$		−	13.7477i		−	0.708044i
$378$		0			0
$379$	19.5959			1.00657			0.503287	−	0.864119i	$-0.332124\pi$
$379$	19.5959			1.00657			0.503287	+	0.864119i	$0.332124\pi$
$380$		0			0
$381$	−11.0000			−0.563547
$382$		0			0
$383$			26.1916i			1.33833i	0.743115	+	0.669164i	$0.233348\pi$
$383$			26.1916i			1.33833i	−0.743115	+	0.669164i	$0.766652\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	22.4499i		−	1.14119i
$388$		0			0
$389$		−	14.9666i		−	0.758838i	−0.925225	−	0.379419i	$-0.876124\pi$
$389$		−	14.9666i		−	0.758838i	0.925225	−	0.379419i	$-0.123876\pi$
$390$		0			0
$391$	−7.34847	+	9.16515i	−0.371628	+	0.463502i
$392$		0			0
$393$			4.58258i			0.231160i
$394$		0			0
$395$	−9.00000	+	13.7477i	−0.452839	+	0.691723i
$396$		0			0
$397$		−	13.7477i		−	0.689979i	−0.938607	−	0.344989i	$-0.887883\pi$
$397$		−	13.7477i		−	0.689979i	0.938607	−	0.344989i	$-0.112117\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			26.1916i			1.30795i	0.756518	+	0.653973i	$0.226899\pi$
$401$			26.1916i			1.30795i	−0.756518	+	0.653973i	$0.773101\pi$
$402$		0			0
$403$	21.0000			1.04608
$404$		0			0
$405$	−1.22474	+	1.87083i	−0.0608581	+	0.0929622i
$406$		0			0
$407$	−24.0000			−1.18964
$408$		0			0
$409$	1.00000			0.0494468			0.0247234	−	0.999694i	$-0.492129\pi$
$409$	1.00000			0.0494468			0.0247234	+	0.999694i	$0.492129\pi$
$410$		0			0
$411$	14.6969			0.724947
$412$		0			0
$413$		0			0
$414$		0			0
$415$	7.00000	+	4.58258i	0.343616	+	0.224950i
$416$		0			0
$417$		−	22.9129i		−	1.12205i
$418$		0			0
$419$	−26.9444			−1.31632			−0.658160	−	0.752878i	$-0.728665\pi$
$419$	−26.9444			−1.31632			−0.658160	+	0.752878i	$0.728665\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$	18.0000			0.875190
$424$		0			0
$425$	−4.89898	−	11.2250i	−0.237635	−	0.544491i
$426$		0			0
$427$		0			0
$428$		0			0
$429$			22.4499i			1.08389i
$430$		0			0
$431$	31.8434			1.53384			0.766921	−	0.641742i	$-0.221788\pi$
$431$	31.8434			1.53384			0.766921	+	0.641742i	$0.221788\pi$
$432$		0			0
$433$	−7.34847			−0.353145			−0.176572	−	0.984288i	$-0.556501\pi$
$433$	−7.34847			−0.353145			−0.176572	+	0.984288i	$0.556501\pi$
$434$		0			0
$435$	3.67423	−	5.61249i	0.176166	−	0.269098i
$436$		0			0
$437$	−7.34847	+	9.16515i	−0.351525	+	0.438429i
$438$		0			0
$439$		−	4.58258i		−	0.218714i	−0.994003	−	0.109357i	$-0.965121\pi$
$439$		−	4.58258i		−	0.218714i	0.994003	−	0.109357i	$-0.0348792\pi$
$440$		0			0
$441$	−14.0000			−0.666667
$442$		0			0
$443$	33.0000			1.56788			0.783939	−	0.620838i	$-0.213208\pi$
$443$	33.0000			1.56788			0.783939	+	0.620838i	$0.213208\pi$
$444$		0			0
$445$	7.00000	+	4.58258i	0.331832	+	0.217235i
$446$		0			0
$447$		−	14.9666i		−	0.707897i
$448$		0			0
$449$	12.0000			0.566315			0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000			0.566315			0.283158	+	0.959073i	$0.408618\pi$
$450$		0			0
$451$	−14.6969			−0.692052
$452$		0			0
$453$			13.7477i			0.645925i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−12.2474			−0.572911			−0.286456	−	0.958093i	$-0.592477\pi$
$457$	−12.2474			−0.572911			−0.286456	+	0.958093i	$0.592477\pi$
$458$		0			0
$459$	−12.2474			−0.571662
$460$		0			0
$461$	15.0000			0.698620			0.349310	−	0.937007i	$-0.386416\pi$
$461$	15.0000			0.698620			0.349310	+	0.937007i	$0.386416\pi$
$462$		0			0
$463$	14.0000			0.650635			0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000			0.650635			0.325318	+	0.945605i	$0.394529\pi$
$464$		0			0
$465$	8.57321	+	5.61249i	0.397573	+	0.260273i
$466$		0			0
$467$		−	7.48331i		−	0.346287i	−0.984897	−	0.173143i	$-0.944608\pi$
$467$		−	7.48331i		−	0.346287i	0.984897	−	0.173143i	$-0.0553924\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−2.44949			−0.112867
$472$		0			0
$473$			54.9909i			2.52848i
$474$		0			0
$475$	−4.89898	−	11.2250i	−0.224781	−	0.515037i
$476$		0			0
$477$	−9.79796			−0.448618
$478$		0			0
$479$	31.8434			1.45496			0.727480	−	0.686129i	$-0.240691\pi$
$479$	31.8434			1.45496			0.727480	+	0.686129i	$0.240691\pi$
$480$		0			0
$481$		−	22.4499i		−	1.02363i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	12.0000	−	18.3303i	0.544892	−	0.832336i
$486$		0			0
$487$	−1.00000			−0.0453143			−0.0226572	−	0.999743i	$-0.507213\pi$
$487$	−1.00000			−0.0453143			−0.0226572	+	0.999743i	$0.507213\pi$
$488$		0			0
$489$	13.0000			0.587880
$490$		0			0
$491$			4.58258i			0.206809i	0.994639	+	0.103404i	$0.0329736\pi$
$491$			4.58258i			0.206809i	−0.994639	+	0.103404i	$0.967026\pi$
$492$		0			0
$493$	−7.34847			−0.330958
$494$		0			0
$495$	12.0000	−	18.3303i	0.539360	−	0.823886i
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	22.9129i		−	1.02572i	−0.858472	−	0.512861i	$-0.828586\pi$
$499$		−	22.9129i		−	1.02572i	0.858472	−	0.512861i	$-0.171414\pi$
$500$		0			0
$501$	−18.0000			−0.804181
$502$		0			0
$503$			14.9666i			0.667329i	0.942692	+	0.333665i	$0.108285\pi$
$503$			14.9666i			0.667329i	−0.942692	+	0.333665i	$0.891715\pi$
$504$		0			0
$505$	−14.6969	+	22.4499i	−0.654005	+	0.999009i
$506$		0			0
$507$	−8.00000			−0.355292
$508$		0			0
$509$	−9.00000			−0.398918			−0.199459	−	0.979906i	$-0.563918\pi$
$509$	−9.00000			−0.398918			−0.199459	+	0.979906i	$0.563918\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−12.2474			−0.540738
$514$		0			0
$515$	−21.0000	−	13.7477i	−0.925371	−	0.605797i
$516$		0			0
$517$	−44.0908			−1.93911
$518$		0			0
$519$		−	18.3303i		−	0.804611i
$520$		0			0
$521$		−	7.48331i		−	0.327850i	−0.986473	−	0.163925i	$-0.947584\pi$
$521$		−	7.48331i		−	0.327850i	0.986473	−	0.163925i	$-0.0524155\pi$
$522$		0			0
$523$			22.4499i			0.981668i	0.871253	+	0.490834i	$0.163308\pi$
$523$			22.4499i			0.981668i	−0.871253	+	0.490834i	$0.836692\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	11.2250i		−	0.488967i
$528$		0			0
$529$	−5.00000	−	22.4499i	−0.217391	−	0.976085i
$530$		0			0
$531$		−	18.3303i		−	0.795467i
$532$		0			0
$533$		−	13.7477i		−	0.595480i
$534$		0			0
$535$	−7.00000	−	4.58258i	−0.302636	−	0.198122i
$536$		0			0
$537$			13.7477i			0.593258i
$538$		0			0
$539$	34.2929			1.47710
$540$		0			0
$541$	13.0000			0.558914			0.279457	−	0.960158i	$-0.409846\pi$
$541$	13.0000			0.558914			0.279457	+	0.960158i	$0.409846\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	21.0000	+	13.7477i	0.899541	+	0.588888i
$546$		0			0
$547$	19.0000			0.812381			0.406191	−	0.913788i	$-0.366857\pi$
$547$	19.0000			0.812381			0.406191	+	0.913788i	$0.366857\pi$
$548$		0			0
$549$		−	22.4499i		−	0.958140i
$550$		0			0
$551$	−7.34847			−0.313055
$552$		0			0
$553$		0			0
$554$		0			0
$555$	6.00000	−	9.16515i	0.254686	−	0.389039i
$556$		0			0
$557$	36.7423			1.55682			0.778412	−	0.627754i	$-0.216026\pi$
$557$	36.7423			1.55682			0.778412	+	0.627754i	$0.216026\pi$
$558$		0			0
$559$	−51.4393			−2.17565
$560$		0			0
$561$	12.0000			0.506640
$562$		0			0
$563$			29.9333i			1.26154i	0.775971	+	0.630768i	$0.217260\pi$
$563$			29.9333i			1.26154i	−0.775971	+	0.630768i	$0.782740\pi$
$564$		0			0
$565$	18.0000	−	27.4955i	0.757266	−	1.15674i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			18.7083i			0.784292i	0.919903	+	0.392146i	$0.128267\pi$
$569$			18.7083i			0.784292i	−0.919903	+	0.392146i	$0.871733\pi$
$570$		0			0
$571$	−31.8434			−1.33260			−0.666302	−	0.745682i	$-0.732124\pi$
$571$	−31.8434			−1.33260			−0.666302	+	0.745682i	$0.732124\pi$
$572$		0			0
$573$	12.2474			0.511645
$574$		0			0
$575$	23.1464	+	6.26441i	0.965273	+	0.261244i
$576$		0			0
$577$		−	22.9129i		−	0.953876i	−0.878937	−	0.476938i	$-0.841747\pi$
$577$		−	22.9129i		−	0.953876i	0.878937	−	0.476938i	$-0.158253\pi$
$578$		0			0
$579$		−	22.9129i		−	0.952227i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	24.0000			0.993978
$584$		0			0
$585$	17.1464	+	11.2250i	0.708918	+	0.464095i
$586$		0			0
$587$	−3.00000			−0.123823			−0.0619116	−	0.998082i	$-0.519720\pi$
$587$	−3.00000			−0.123823			−0.0619116	+	0.998082i	$0.519720\pi$
$588$		0			0
$589$		−	11.2250i		−	0.462517i
$590$		0			0
$591$			4.58258i			0.188502i
$592$		0			0
$593$		−	36.6606i		−	1.50547i	−0.658323	−	0.752735i	$-0.728734\pi$
$593$		−	36.6606i		−	1.50547i	0.658323	−	0.752735i	$-0.271266\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	19.5959			0.802008
$598$		0			0
$599$			45.8258i			1.87239i	0.351481	+	0.936195i	$0.385678\pi$
$599$			45.8258i			1.87239i	−0.351481	+	0.936195i	$0.614322\pi$
$600$		0			0
$601$	−11.0000			−0.448699			−0.224350	−	0.974509i	$-0.572026\pi$
$601$	−11.0000			−0.448699			−0.224350	+	0.974509i	$0.572026\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−15.9217	+	24.3208i	−0.647308	+	0.988780i
$606$		0			0
$607$	−4.00000			−0.162355			−0.0811775	−	0.996700i	$-0.525868\pi$
$607$	−4.00000			−0.162355			−0.0811775	+	0.996700i	$0.525868\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	41.2432i		−	1.66852i
$612$		0			0
$613$	22.0454			0.890406			0.445203	−	0.895430i	$-0.353132\pi$
$613$	22.0454			0.890406			0.445203	+	0.895430i	$0.353132\pi$
$614$		0			0
$615$	3.67423	−	5.61249i	0.148159	−	0.226317i
$616$		0			0
$617$	−7.34847			−0.295838			−0.147919	−	0.988999i	$-0.547258\pi$
$617$	−7.34847			−0.295838			−0.147919	+	0.988999i	$0.547258\pi$
$618$		0			0
$619$	14.6969			0.590720			0.295360	−	0.955386i	$-0.404560\pi$
$619$	14.6969			0.590720			0.295360	+	0.955386i	$0.404560\pi$
$620$		0			0
$621$	15.0000	−	18.7083i	0.601929	−	0.750738i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−17.0000	+	18.3303i	−0.680000	+	0.733212i
$626$		0			0
$627$	12.0000			0.479234
$628$		0			0
$629$	−12.0000			−0.478471
$630$		0			0
$631$	7.34847			0.292538			0.146269	−	0.989245i	$-0.453274\pi$
$631$	7.34847			0.292538			0.146269	+	0.989245i	$0.453274\pi$
$632$		0			0
$633$		−	18.3303i		−	0.728564i
$634$		0			0
$635$	13.4722	−	20.5791i	0.534628	−	0.816657i
$636$		0			0
$637$			32.0780i			1.27098i
$638$		0			0
$639$		−	27.4955i		−	1.08770i
$640$		0			0
$641$		−	48.6415i		−	1.92123i	−0.277890	−	0.960613i	$-0.589635\pi$
$641$		−	48.6415i		−	1.92123i	0.277890	−	0.960613i	$-0.410365\pi$
$642$		0			0
$643$		−	11.2250i		−	0.442670i	−0.975198	−	0.221335i	$-0.928959\pi$
$643$		−	11.2250i		−	0.442670i	0.975198	−	0.221335i	$-0.0710414\pi$
$644$		0			0
$645$	−21.0000	−	13.7477i	−0.826874	−	0.541316i
$646$		0			0
$647$	−33.0000			−1.29736			−0.648682	−	0.761060i	$-0.724679\pi$
$647$	−33.0000			−1.29736			−0.648682	+	0.761060i	$0.724679\pi$
$648$		0			0
$649$			44.8999i			1.76247i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	50.4083i		−	1.97263i	−0.164870	−	0.986315i	$-0.552721\pi$
$653$		−	50.4083i		−	1.97263i	0.164870	−	0.986315i	$-0.447279\pi$
$654$		0			0
$655$	−8.57321	−	5.61249i	−0.334983	−	0.219298i
$656$		0			0
$657$		−	9.16515i		−	0.357567i
$658$		0			0
$659$	41.6413			1.62212			0.811058	−	0.584966i	$-0.198892\pi$
$659$	41.6413			1.62212			0.811058	+	0.584966i	$0.198892\pi$
$660$		0			0
$661$		−	22.4499i		−	0.873202i	−0.899655	−	0.436601i	$-0.856182\pi$
$661$		−	22.4499i		−	0.873202i	0.899655	−	0.436601i	$-0.143818\pi$
$662$		0			0
$663$			11.2250i			0.435942i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	9.00000	−	11.2250i	0.348481	−	0.434633i
$668$		0			0
$669$	22.0000			0.850569
$670$		0			0
$671$			54.9909i			2.12290i
$672$		0			0
$673$			13.7477i			0.529936i	0.964257	+	0.264968i	$0.0853614\pi$
$673$			13.7477i			0.529936i	−0.964257	+	0.264968i	$0.914639\pi$
$674$		0			0
$675$	10.0000	+	22.9129i	0.384900	+	0.881917i
$676$		0			0
$677$	−4.89898			−0.188283			−0.0941415	−	0.995559i	$-0.530011\pi$
$677$	−4.89898			−0.188283			−0.0941415	+	0.995559i	$0.530011\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		−	7.48331i		−	0.286761i
$682$		0			0
$683$	3.00000			0.114792			0.0573959	−	0.998351i	$-0.481720\pi$
$683$	3.00000			0.114792			0.0573959	+	0.998351i	$0.481720\pi$
$684$		0			0
$685$	−18.0000	+	27.4955i	−0.687745	+	1.05055i
$686$		0			0
$687$			11.2250i			0.428259i
$688$		0			0
$689$			22.4499i			0.855275i
$690$		0			0
$691$			9.16515i			0.348659i	0.984687	+	0.174329i	$0.0557758\pi$
$691$			9.16515i			0.348659i	−0.984687	+	0.174329i	$0.944224\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	42.8661	+	28.0624i	1.62600	+	1.06447i
$696$		0			0
$697$	−7.34847			−0.278343
$698$		0			0
$699$		−	4.58258i		−	0.173329i
$700$		0			0
$701$		−	14.9666i		−	0.565282i	−0.959226	−	0.282641i	$-0.908790\pi$
$701$		−	14.9666i		−	0.565282i	0.959226	−	0.282641i	$-0.0912105\pi$
$702$		0			0
$703$	−12.0000			−0.452589
$704$		0			0
$705$	11.0227	−	16.8375i	0.415139	−	0.634135i
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	44.8999i		−	1.68625i	−0.537717	−	0.843125i	$-0.680713\pi$
$709$		−	44.8999i		−	1.68625i	0.537717	−	0.843125i	$-0.319287\pi$
$710$		0			0
$711$	−14.6969			−0.551178
$712$		0			0
$713$	17.1464	+	13.7477i	0.642139	+	0.514856i
$714$		0			0
$715$	−42.0000	−	27.4955i	−1.57071	−	1.02827i
$716$		0			0
$717$		−	4.58258i		−	0.171139i
$718$		0			0
$719$			27.4955i			1.02541i	0.858566	+	0.512704i	$0.171356\pi$
$719$			27.4955i			1.02541i	−0.858566	+	0.512704i	$0.828644\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		−	11.2250i		−	0.417461i
$724$		0			0
$725$	6.00000	+	13.7477i	0.222834	+	0.510578i
$726$		0			0
$727$		−	33.6749i		−	1.24893i	−0.781051	−	0.624467i	$-0.785316\pi$
$727$		−	33.6749i		−	1.24893i	0.781051	−	0.624467i	$-0.214684\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$			27.4955i			1.01696i
$732$		0			0
$733$	−51.4393			−1.89995			−0.949977	−	0.312321i	$-0.898894\pi$
$733$	−51.4393			−1.89995			−0.949977	+	0.312321i	$0.898894\pi$
$734$		0			0
$735$	−8.57321	+	13.0958i	−0.316228	+	0.483046i
$736$		0			0
$737$		0			0
$738$		0			0
$739$			4.58258i			0.168573i	0.996442	+	0.0842864i	$0.0268611\pi$
$739$			4.58258i			0.168573i	−0.996442	+	0.0842864i	$0.973139\pi$
$740$		0			0
$741$			11.2250i			0.412360i
$742$		0			0
$743$			26.1916i			0.960877i	0.877029	+	0.480438i	$0.159522\pi$
$743$			26.1916i			0.960877i	−0.877029	+	0.480438i	$0.840478\pi$
$744$		0			0
$745$	28.0000	+	18.3303i	1.02584	+	0.671570i
$746$		0			0
$747$			7.48331i			0.273800i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−12.2474			−0.446916			−0.223458	−	0.974714i	$-0.571735\pi$
$751$	−12.2474			−0.446916			−0.223458	+	0.974714i	$0.571735\pi$
$752$		0			0
$753$	−12.2474			−0.446322
$754$		0			0
$755$	−25.7196	−	16.8375i	−0.936034	−	0.612778i
$756$		0			0
$757$	48.9898			1.78056			0.890282	−	0.455409i	$-0.150507\pi$
$757$	48.9898			1.78056			0.890282	+	0.455409i	$0.150507\pi$
$758$		0			0
$759$	−14.6969	+	18.3303i	−0.533465	+	0.665348i
$760$		0			0
$761$	−45.0000			−1.63125			−0.815624	−	0.578582i	$-0.803606\pi$
$761$	−45.0000			−1.63125			−0.815624	+	0.578582i	$0.803606\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	6.00000	−	9.16515i	0.216930	−	0.331367i
$766$		0			0
$767$	−42.0000			−1.51653
$768$		0			0
$769$			33.6749i			1.21435i	0.794569	+	0.607174i	$0.207697\pi$
$769$			33.6749i			1.21435i	−0.794569	+	0.607174i	$0.792303\pi$
$770$		0			0
$771$			22.9129i			0.825187i
$772$		0			0
$773$	−17.1464			−0.616714			−0.308357	−	0.951271i	$-0.599779\pi$
$773$	−17.1464			−0.616714			−0.308357	+	0.951271i	$0.599779\pi$
$774$		0			0
$775$	−21.0000	+	9.16515i	−0.754342	+	0.329222i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−7.34847			−0.263286
$780$		0			0
$781$			67.3498i			2.40997i
$782$		0			0
$783$	15.0000			0.536056
$784$		0			0
$785$	3.00000	−	4.58258i	0.107075	−	0.163559i
$786$		0			0
$787$			44.8999i			1.60051i	0.599661	+	0.800254i	$0.295302\pi$
$787$			44.8999i			1.60051i	−0.599661	+	0.800254i	$0.704698\pi$
$788$		0			0
$789$			18.7083i			0.666033i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−51.4393			−1.82666
$794$		0			0
$795$	−6.00000	+	9.16515i	−0.212798	+	0.325054i
$796$		0			0
$797$	7.34847			0.260296			0.130148	−	0.991495i	$-0.458455\pi$
$797$	7.34847			0.260296			0.130148	+	0.991495i	$0.458455\pi$
$798$		0			0
$799$	−22.0454			−0.779910
$800$		0			0
$801$			7.48331i			0.264410i
$802$		0			0
$803$			22.4499i			0.792241i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−9.00000			−0.316815
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$			41.2432i			1.44824i	0.689672	+	0.724122i	$0.257755\pi$
$811$			41.2432i			1.44824i	−0.689672	+	0.724122i	$0.742245\pi$
$812$		0			0
$813$		−	9.16515i		−	0.321436i
$814$		0			0
$815$	−15.9217	+	24.3208i	−0.557712	+	0.851920i
$816$		0			0
$817$			27.4955i			0.961944i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−30.0000			−1.04701			−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000			−1.04701			−0.523504	+	0.852023i	$0.675375\pi$
$822$		0			0
$823$	−55.0000			−1.91718			−0.958590	−	0.284791i	$-0.908076\pi$
$823$	−55.0000			−1.91718			−0.958590	+	0.284791i	$0.908076\pi$
$824$		0			0
$825$	−9.79796	−	22.4499i	−0.341121	−	0.781607i
$826$		0			0
$827$			37.4166i			1.30110i	0.759463	+	0.650551i	$0.225462\pi$
$827$			37.4166i			1.30110i	−0.759463	+	0.650551i	$0.774538\pi$
$828$		0			0
$829$	16.0000			0.555703			0.277851	−	0.960624i	$-0.410378\pi$
$829$	16.0000			0.555703			0.277851	+	0.960624i	$0.410378\pi$
$830$		0			0
$831$		−	22.9129i		−	0.794839i
$832$		0			0
$833$	17.1464			0.594089
$834$		0			0
$835$	22.0454	−	33.6749i	0.762913	−	1.16537i
$836$		0			0
$837$			22.9129i			0.791985i
$838$		0			0
$839$	39.1918			1.35305			0.676526	−	0.736419i	$-0.263485\pi$
$839$	39.1918			1.35305			0.676526	+	0.736419i	$0.263485\pi$
$840$		0			0
$841$	−20.0000			−0.689655
$842$		0			0
$843$			14.9666i			0.515478i
$844$		0			0
$845$	9.79796	−	14.9666i	0.337060	−	0.514868i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		−	11.2250i		−	0.385240i
$850$		0			0
$851$	14.6969	−	18.3303i	0.503805	−	0.628355i
$852$		0			0
$853$			9.16515i			0.313809i	0.987614	+	0.156904i	$0.0501515\pi$
$853$			9.16515i			0.313809i	−0.987614	+	0.156904i	$0.949849\pi$
$854$		0			0
$855$	6.00000	−	9.16515i	0.205196	−	0.313442i
$856$		0			0
$857$		−	4.58258i		−	0.156538i	−0.996932	−	0.0782689i	$-0.975061\pi$
$857$		−	4.58258i		−	0.156538i	0.996932	−	0.0782689i	$-0.0249393\pi$
$858$		0			0
$859$			41.2432i			1.40720i	0.710597	+	0.703600i	$0.248425\pi$
$859$			41.2432i			1.40720i	−0.710597	+	0.703600i	$0.751575\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−21.0000			−0.714848			−0.357424	−	0.933942i	$-0.616345\pi$
$863$	−21.0000			−0.714848			−0.357424	+	0.933942i	$0.616345\pi$
$864$		0			0
$865$	34.2929	+	22.4499i	1.16599	+	0.763321i
$866$		0			0
$867$	−11.0000			−0.373580
$868$		0			0
$869$	36.0000			1.22122
$870$		0			0
$871$		0			0
$872$		0			0
$873$	19.5959			0.663221
$874$		0			0
$875$		0			0
$876$		0			0
$877$			18.3303i			0.618970i	0.950904	+	0.309485i	$0.100157\pi$
$877$			18.3303i			0.618970i	−0.950904	+	0.309485i	$0.899843\pi$
$878$		0			0
$879$	−9.79796			−0.330477
$880$		0			0
$881$			3.74166i			0.126060i	0.998012	+	0.0630298i	$0.0200763\pi$
$881$			3.74166i			0.126060i	−0.998012	+	0.0630298i	$0.979924\pi$
$882$		0			0
$883$	16.0000			0.538443			0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000			0.538443			0.269221	+	0.963078i	$0.413234\pi$
$884$		0			0
$885$	−17.1464	−	11.2250i	−0.576371	−	0.377323i
$886$		0			0
$887$	−39.0000			−1.30949			−0.654746	−	0.755849i	$-0.727224\pi$
$887$	−39.0000			−1.30949			−0.654746	+	0.755849i	$0.727224\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	4.89898			0.164122
$892$		0			0
$893$	−22.0454			−0.737721
$894$		0			0
$895$	−25.7196	−	16.8375i	−0.859713	−	0.562814i
$896$		0			0
$897$	−17.1464	−	13.7477i	−0.572503	−	0.459023i
$898$		0			0
$899$			13.7477i			0.458512i
$900$		0			0
$901$	12.0000			0.399778
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$			33.6749i			1.11816i	0.829115	+	0.559079i	$0.188845\pi$
$907$			33.6749i			1.11816i	−0.829115	+	0.559079i	$0.811155\pi$
$908$		0			0
$909$	−24.0000			−0.796030
$910$		0			0
$911$	7.34847			0.243466			0.121733	−	0.992563i	$-0.461155\pi$
$911$	7.34847			0.243466			0.121733	+	0.992563i	$0.461155\pi$
$912$		0			0
$913$		−	18.3303i		−	0.606644i
$914$		0			0
$915$	−21.0000	−	13.7477i	−0.694239	−	0.454486i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−4.89898			−0.161602			−0.0808012	−	0.996730i	$-0.525748\pi$
$919$	−4.89898			−0.161602			−0.0808012	+	0.996730i	$0.525748\pi$
$920$		0			0
$921$	22.0000			0.724925
$922$		0			0
$923$	−63.0000			−2.07367
$924$		0			0
$925$	9.79796	+	22.4499i	0.322155	+	0.738150i
$926$		0			0
$927$		−	22.4499i		−	0.737353i
$928$		0			0
$929$	−33.0000			−1.08269			−0.541347	−	0.840799i	$-0.682086\pi$
$929$	−33.0000			−1.08269			−0.541347	+	0.840799i	$0.682086\pi$
$930$		0			0
$931$	17.1464			0.561951
$932$		0			0
$933$		−	4.58258i		−	0.150027i
$934$		0			0
$935$	−14.6969	+	22.4499i	−0.480641	+	0.734192i
$936$		0			0
$937$	4.89898			0.160043			0.0800213	−	0.996793i	$-0.474501\pi$
$937$	4.89898			0.160043			0.0800213	+	0.996793i	$0.474501\pi$
$938$		0			0
$939$	29.3939			0.959233
$940$		0			0
$941$		−	3.74166i		−	0.121975i	−0.998139	−	0.0609873i	$-0.980575\pi$
$941$		−	3.74166i		−	0.121975i	0.998139	−	0.0609873i	$-0.0194249\pi$
$942$		0			0
$943$	9.00000	−	11.2250i	0.293080	−	0.365535i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	21.0000			0.682408			0.341204	−	0.939989i	$-0.389165\pi$
$947$	21.0000			0.682408			0.341204	+	0.939989i	$0.389165\pi$
$948$		0			0
$949$	−21.0000			−0.681689
$950$		0			0
$951$			18.3303i			0.594401i
$952$		0			0
$953$	53.8888			1.74563			0.872814	−	0.488052i	$-0.162293\pi$
$953$	53.8888			1.74563			0.872814	+	0.488052i	$0.162293\pi$
$954$		0			0
$955$	−15.0000	+	22.9129i	−0.485389	+	0.741443i
$956$		0			0
$957$	−14.6969			−0.475085
$958$		0			0
$959$		0			0
$960$		0			0
$961$	10.0000			0.322581
$962$		0			0
$963$		−	7.48331i		−	0.241146i
$964$		0			0
$965$	42.8661	+	28.0624i	1.37991	+	0.903362i
$966$		0			0
$967$	35.0000			1.12552			0.562762	−	0.826619i	$-0.309739\pi$
$967$	35.0000			1.12552			0.562762	+	0.826619i	$0.309739\pi$
$968$		0			0
$969$	6.00000			0.192748
$970$		0			0
$971$	39.1918			1.25773			0.628863	−	0.777516i	$-0.283521\pi$
$971$	39.1918			1.25773			0.628863	+	0.777516i	$0.283521\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	21.0000	−	9.16515i	0.672538	−	0.293520i
$976$		0			0
$977$	−29.3939			−0.940393			−0.470197	−	0.882562i	$-0.655817\pi$
$977$	−29.3939			−0.940393			−0.470197	+	0.882562i	$0.655817\pi$
$978$		0			0
$979$		−	18.3303i		−	0.585839i
$980$		0			0
$981$			22.4499i			0.716772i
$982$		0			0
$983$			52.3832i			1.67076i	0.549669	+	0.835382i	$0.314754\pi$
$983$			52.3832i			1.67076i	−0.549669	+	0.835382i	$0.685246\pi$
$984$		0			0
$985$	−8.57321	−	5.61249i	−0.273165	−	0.178829i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−42.0000	−	33.6749i	−1.33552	−	1.07080i
$990$		0			0
$991$		−	36.6606i		−	1.16456i	−0.812987	−	0.582281i	$-0.802160\pi$
$991$		−	36.6606i		−	1.16456i	0.812987	−	0.582281i	$-0.197840\pi$
$992$		0			0
$993$		−	4.58258i		−	0.145424i
$994$		0			0
$995$	−24.0000	+	36.6606i	−0.760851	+	1.16222i
$996$		0			0
$997$		−	27.4955i		−	0.870790i	−0.900240	−	0.435395i	$-0.856609\pi$
$997$		−	27.4955i		−	0.870790i	0.900240	−	0.435395i	$-0.143391\pi$
$998$		0			0
$999$	24.4949			0.774984

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.m.d.1839.2	yes	4
4.3	odd	2		1840.2.m.a.1839.2	yes	4
5.4	even	2		1840.2.m.a.1839.4	yes	4
20.19	odd	2	inner	1840.2.m.d.1839.4	yes	4
23.22	odd	2	inner	1840.2.m.d.1839.3	yes	4
92.91	even	2		1840.2.m.a.1839.3	yes	4
115.114	odd	2		1840.2.m.a.1839.1	✓	4
460.459	even	2	inner	1840.2.m.d.1839.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1840.2.m.a.1839.1	✓	4	115.114	odd	2
1840.2.m.a.1839.2	yes	4	4.3	odd	2
1840.2.m.a.1839.3	yes	4	92.91	even	2
1840.2.m.a.1839.4	yes	4	5.4	even	2
1840.2.m.d.1839.1	yes	4	460.459	even	2	inner
1840.2.m.d.1839.2	yes	4	1.1	even	1	trivial
1840.2.m.d.1839.3	yes	4	23.22	odd	2	inner
1840.2.m.d.1839.4	yes	4	20.19	odd	2	inner

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1840.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +(-1.22474 + 1.87083i) q^{5} -2.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +(-1.22474 + 1.87083i) q^{5} -2.00000 q^{9} +4.89898 q^{11} +4.58258i q^{13} +(-1.22474 + 1.87083i) q^{15} +2.44949 q^{17} +2.44949 q^{19} +(-3.00000 + 3.74166i) q^{23} +(-2.00000 - 4.58258i) q^{25} -5.00000 q^{27} -3.00000 q^{29} -4.58258i q^{31} +4.89898 q^{33} -4.89898 q^{37} +4.58258i q^{39} -3.00000 q^{41} +11.2250i q^{43} +(2.44949 - 3.74166i) q^{45} -9.00000 q^{47} +7.00000 q^{49} +2.44949 q^{51} +4.89898 q^{53} +(-6.00000 + 9.16515i) q^{55} +2.44949 q^{57} +9.16515i q^{59} +11.2250i q^{61} +(-8.57321 - 5.61249i) q^{65} +(-3.00000 + 3.74166i) q^{69} +13.7477i q^{71} +4.58258i q^{73} +(-2.00000 - 4.58258i) q^{75} +7.34847 q^{79} +1.00000 q^{81} -3.74166i q^{83} +(-3.00000 + 4.58258i) q^{85} -3.00000 q^{87} -3.74166i q^{89} -4.58258i q^{93} +(-3.00000 + 4.58258i) q^{95} -9.79796 q^{97} -9.79796 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 8 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 8 * q^9	\( 4 q + 4 q^{3} - 8 q^{9} - 12 q^{23} - 8 q^{25} - 20 q^{27} - 12 q^{29} - 12 q^{41} - 36 q^{47} + 28 q^{49} - 24 q^{55} - 12 q^{69} - 8 q^{75} + 4 q^{81} - 12 q^{85} - 12 q^{87} - 12 q^{95}+O(q^{100}) \) 4 * q + 4 * q^3 - 8 * q^9 - 12 * q^23 - 8 * q^25 - 20 * q^27 - 12 * q^29 - 12 * q^41 - 36 * q^47 + 28 * q^49 - 24 * q^55 - 12 * q^69 - 8 * q^75 + 4 * q^81 - 12 * q^85 - 12 * q^87 - 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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