LMFDB - Embedded newform 1104.2.i.a.367.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1104,2,Mod(367,1104)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1104, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1104.367");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1104 = 2^{4} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1104.i (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:	$8.81548438315$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40282095616.8
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 8x^{4} - 36x^{2} + 81 $ x^8 - 4x^6 + 8x^4 - 36*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			367.1
Root			$1.03179 - 1.39119i$ of defining polynomial
Character	$\chi$	$=$	1104.367
Dual form			1104.2.i.a.367.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -2.78238i q^{5} -2.06358 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -2.78238i q^{5} -2.06358 q^{7} -1.00000 q^{9} +5.56477 q^{11} -5.74166 q^{13} -2.78238 q^{15} -3.50119i q^{17} +2.78238 q^{19} +2.06358i q^{21} +(-0.718808 - 4.74166i) q^{23} -2.74166 q^{25} +1.00000i q^{27} -9.48331 q^{29} -7.48331i q^{31} -5.56477i q^{33} +5.74166i q^{35} +9.69192i q^{37} +5.74166i q^{39} -2.00000 q^{41} -7.62834 q^{43} +2.78238i q^{45} +7.74166i q^{47} -2.74166 q^{49} -3.50119 q^{51} +6.19073i q^{53} -15.4833i q^{55} -2.78238i q^{57} -0.258343i q^{59} -12.5671i q^{61} +2.06358 q^{63} +15.9755i q^{65} -2.78238 q^{67} +(-4.74166 + 0.718808i) q^{69} +13.4833i q^{71} -4.00000 q^{73} +2.74166i q^{75} -11.4833 q^{77} +15.3495 q^{79} +1.00000 q^{81} -7.00238 q^{83} -9.74166 q^{85} +9.48331i q^{87} -12.4743i q^{89} +11.8483 q^{91} -7.48331 q^{93} -7.74166i q^{95} -13.8191i q^{97} -5.56477 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} - 16 q^{13} + 8 q^{25} - 16 q^{29} - 16 q^{41} + 8 q^{49} - 8 q^{69} - 32 q^{73} - 32 q^{77} + 8 q^{81} - 48 q^{85}+O(q^{100}) $ 8 * q - 8 * q^9 - 16 * q^13 + 8 * q^25 - 16 * q^29 - 16 * q^41 + 8 * q^49 - 8 * q^69 - 32 * q^73 - 32 * q^77 + 8 * q^81 - 48 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1104\mathbb{Z}\right)^\times$.

$n$	$97$	$277$	$415$	$737$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$5$		−	2.78238i		−	1.24432i	−0.782890	−	0.622160i	$-0.786255\pi$
$5$		−	2.78238i		−	1.24432i	0.782890	−	0.622160i	$-0.213745\pi$
$6$		0			0
$7$	−2.06358			−0.779958			−0.389979	−	0.920824i	$-0.627518\pi$
$7$	−2.06358			−0.779958			−0.389979	+	0.920824i	$0.627518\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	5.56477			1.67784			0.838920	−	0.544255i	$-0.183187\pi$
$11$	5.56477			1.67784			0.838920	+	0.544255i	$0.183187\pi$
$12$		0			0
$13$	−5.74166			−1.59245			−0.796225	−	0.605001i	$-0.793173\pi$
$13$	−5.74166			−1.59245			−0.796225	+	0.605001i	$0.793173\pi$
$14$		0			0
$15$	−2.78238			−0.718408
$16$		0			0
$17$		−	3.50119i		−	0.849164i	−0.905390	−	0.424582i	$-0.860421\pi$
$17$		−	3.50119i		−	0.849164i	0.905390	−	0.424582i	$-0.139579\pi$
$18$		0			0
$19$	2.78238			0.638323			0.319161	−	0.947700i	$-0.396599\pi$
$19$	2.78238			0.638323			0.319161	+	0.947700i	$0.396599\pi$
$20$		0			0
$21$			2.06358i			0.450309i
$22$		0			0
$23$	−0.718808	−	4.74166i	−0.149882	−	0.988704i
$23$	−0.718808	−	4.74166i	−0.149882	−	0.988704i
$24$		0			0
$25$	−2.74166			−0.548331
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	−9.48331			−1.76101			−0.880504	−	0.474039i	$-0.842795\pi$
$29$	−9.48331			−1.76101			−0.880504	+	0.474039i	$0.842795\pi$
$30$		0			0
$31$		−	7.48331i		−	1.34404i	−0.740532	−	0.672022i	$-0.765426\pi$
$31$		−	7.48331i		−	1.34404i	0.740532	−	0.672022i	$-0.234574\pi$
$32$		0			0
$33$		−	5.56477i		−	0.968702i
$34$		0			0
$35$			5.74166i			0.970517i
$36$		0			0
$37$			9.69192i			1.59334i	0.604414	+	0.796671i	$0.293407\pi$
$37$			9.69192i			1.59334i	−0.604414	+	0.796671i	$0.706593\pi$
$38$		0			0
$39$			5.74166i			0.919401i
$40$		0			0
$41$	−2.00000			−0.312348			−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000			−0.312348			−0.156174	+	0.987730i	$0.549916\pi$
$42$		0			0
$43$	−7.62834			−1.16331			−0.581656	−	0.813435i	$-0.697595\pi$
$43$	−7.62834			−1.16331			−0.581656	+	0.813435i	$0.697595\pi$
$44$		0			0
$45$			2.78238i			0.414773i
$46$		0			0
$47$			7.74166i			1.12924i	0.825352	+	0.564618i	$0.190977\pi$
$47$			7.74166i			1.12924i	−0.825352	+	0.564618i	$0.809023\pi$
$48$		0			0
$49$	−2.74166			−0.391665
$50$		0			0
$51$	−3.50119			−0.490265
$52$		0			0
$53$			6.19073i			0.850362i	0.905108	+	0.425181i	$0.139789\pi$
$53$			6.19073i			0.850362i	−0.905108	+	0.425181i	$0.860211\pi$
$54$		0			0
$55$		−	15.4833i		−	2.08777i
$56$		0			0
$57$		−	2.78238i		−	0.368536i
$58$		0			0
$59$		−	0.258343i		−	0.0336333i	−0.999859	−	0.0168167i	$-0.994647\pi$
$59$		−	0.258343i		−	0.0336333i	0.999859	−	0.0168167i	$-0.00535317\pi$
$60$		0			0
$61$		−	12.5671i		−	1.60906i	−0.593913	−	0.804529i	$-0.702418\pi$
$61$		−	12.5671i		−	1.60906i	0.593913	−	0.804529i	$-0.297582\pi$
$62$		0			0
$63$	2.06358			0.259986
$64$		0			0
$65$			15.9755i			1.98152i
$66$		0			0
$67$	−2.78238			−0.339922			−0.169961	−	0.985451i	$-0.554364\pi$
$67$	−2.78238			−0.339922			−0.169961	+	0.985451i	$0.554364\pi$
$68$		0			0
$69$	−4.74166	+	0.718808i	−0.570828	+	0.0865343i
$70$		0			0
$71$			13.4833i			1.60018i	0.599883	+	0.800088i	$0.295214\pi$
$71$			13.4833i			1.60018i	−0.599883	+	0.800088i	$0.704786\pi$
$72$		0			0
$73$	−4.00000			−0.468165			−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000			−0.468165			−0.234082	+	0.972217i	$0.575209\pi$
$74$		0			0
$75$			2.74166i			0.316579i
$76$		0			0
$77$	−11.4833			−1.30865
$78$		0			0
$79$	15.3495			1.72696			0.863479	−	0.504385i	$-0.168281\pi$
$79$	15.3495			1.72696			0.863479	+	0.504385i	$0.168281\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−7.00238			−0.768611			−0.384306	−	0.923206i	$-0.625559\pi$
$83$	−7.00238			−0.768611			−0.384306	+	0.923206i	$0.625559\pi$
$84$		0			0
$85$	−9.74166			−1.05663
$86$		0			0
$87$			9.48331i			1.01672i
$88$		0			0
$89$		−	12.4743i		−	1.32227i	−0.750266	−	0.661137i	$-0.770074\pi$
$89$		−	12.4743i		−	1.32227i	0.750266	−	0.661137i	$-0.229926\pi$
$90$		0			0
$91$	11.8483			1.24204
$92$		0			0
$93$	−7.48331			−0.775984
$94$		0			0
$95$		−	7.74166i		−	0.794277i
$96$		0			0
$97$		−	13.8191i		−	1.40311i	−0.712613	−	0.701557i	$-0.752489\pi$
$97$		−	13.8191i		−	1.40311i	0.712613	−	0.701557i	$-0.247511\pi$
$98$		0			0
$99$	−5.56477			−0.559280
$100$		0			0
$101$	10.0000			0.995037			0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000			0.995037			0.497519	+	0.867453i	$0.334245\pi$
$102$		0			0
$103$	−1.34477			−0.132504			−0.0662519	−	0.997803i	$-0.521104\pi$
$103$	−1.34477			−0.132504			−0.0662519	+	0.997803i	$0.521104\pi$
$104$		0			0
$105$	5.74166			0.560328
$106$		0			0
$107$	13.8191			1.33594			0.667970	−	0.744188i	$-0.267164\pi$
$107$	13.8191			1.33594			0.667970	+	0.744188i	$0.267164\pi$
$108$		0			0
$109$		−	8.44000i		−	0.808405i	−0.914669	−	0.404203i	$-0.867549\pi$
$109$		−	8.44000i		−	0.808405i	0.914669	−	0.404203i	$-0.132451\pi$
$110$		0			0
$111$	9.69192			0.919916
$112$		0			0
$113$			1.34477i			0.126505i	0.997998	+	0.0632525i	$0.0201474\pi$
$113$			1.34477i			0.126505i	−0.997998	+	0.0632525i	$0.979853\pi$
$114$		0			0
$115$	−13.1931	+	2.00000i	−1.23026	+	0.186501i
$116$		0			0
$117$	5.74166			0.530816
$118$		0			0
$119$			7.22497i			0.662312i
$120$		0			0
$121$	19.9666			1.81515
$122$		0			0
$123$			2.00000i			0.180334i
$124$		0			0
$125$		−	6.28357i		−	0.562020i
$126$		0			0
$127$		−	3.48331i		−	0.309094i	−0.987985	−	0.154547i	$-0.950608\pi$
$127$		−	3.48331i		−	0.309094i	0.987985	−	0.154547i	$-0.0493918\pi$
$128$		0			0
$129$			7.62834i			0.671638i
$130$		0			0
$131$		−	1.48331i		−	0.129598i	−0.997898	−	0.0647989i	$-0.979359\pi$
$131$		−	1.48331i		−	0.129598i	0.997898	−	0.0647989i	$-0.0206406\pi$
$132$		0			0
$133$	−5.74166			−0.497865
$134$		0			0
$135$	2.78238			0.239469
$136$		0			0
$137$		−	4.75311i		−	0.406086i	−0.979170	−	0.203043i	$-0.934917\pi$
$137$		−	4.75311i		−	0.406086i	0.979170	−	0.203043i	$-0.0650830\pi$
$138$		0			0
$139$		−	7.48331i		−	0.634726i	−0.948304	−	0.317363i	$-0.897203\pi$
$139$		−	7.48331i		−	0.634726i	0.948304	−	0.317363i	$-0.102797\pi$
$140$		0			0
$141$	7.74166			0.651965
$142$		0			0
$143$	−31.9510			−2.67188
$144$		0			0
$145$			26.3862i			2.19126i
$146$		0			0
$147$			2.74166i			0.226128i
$148$		0			0
$149$		−	17.3203i		−	1.41893i	−0.704740	−	0.709465i	$-0.748936\pi$
$149$		−	17.3203i		−	1.41893i	0.704740	−	0.709465i	$-0.251064\pi$
$150$		0			0
$151$		−	22.9666i		−	1.86900i	−0.355966	−	0.934499i	$-0.615848\pi$
$151$		−	22.9666i		−	1.86900i	0.355966	−	0.934499i	$-0.384152\pi$
$152$		0			0
$153$			3.50119i			0.283055i
$154$		0			0
$155$	−20.8215			−1.67242
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$	6.19073			0.490956
$160$		0			0
$161$	1.48331	+	9.78477i	0.116902	+	0.771148i
$162$		0			0
$163$			12.0000i			0.939913i	0.882690	+	0.469956i	$0.155730\pi$
$163$			12.0000i			0.939913i	−0.882690	+	0.469956i	$0.844270\pi$
$164$		0			0
$165$	−15.4833			−1.20537
$166$		0			0
$167$		−	8.00000i		−	0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$		−	8.00000i		−	0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$		0			0
$169$	19.9666			1.53589
$170$		0			0
$171$	−2.78238			−0.212774
$172$		0			0
$173$	−5.48331			−0.416889			−0.208444	−	0.978034i	$-0.566840\pi$
$173$	−5.48331			−0.416889			−0.208444	+	0.978034i	$0.566840\pi$
$174$		0			0
$175$	5.65762			0.427676
$176$		0			0
$177$	−0.258343			−0.0194182
$178$		0			0
$179$			7.74166i			0.578639i	0.957233	+	0.289319i	$0.0934289\pi$
$179$			7.74166i			0.578639i	−0.957233	+	0.289319i	$0.906571\pi$
$180$		0			0
$181$		−	1.25192i		−	0.0930543i	−0.998917	−	0.0465272i	$-0.985185\pi$
$181$		−	1.25192i		−	0.0930543i	0.998917	−	0.0465272i	$-0.0148154\pi$
$182$		0			0
$183$	−12.5671			−0.928990
$184$		0			0
$185$	26.9666			1.98263
$186$		0			0
$187$		−	19.4833i		−	1.42476i
$188$		0			0
$189$		−	2.06358i		−	0.150103i
$190$		0			0
$191$	−7.00238			−0.506675			−0.253337	−	0.967378i	$-0.581528\pi$
$191$	−7.00238			−0.506675			−0.253337	+	0.967378i	$0.581528\pi$
$192$		0			0
$193$	−1.74166			−0.125367			−0.0626836	−	0.998033i	$-0.519966\pi$
$193$	−1.74166			−0.125367			−0.0626836	+	0.998033i	$0.519966\pi$
$194$		0			0
$195$	15.9755			1.14403
$196$		0			0
$197$	−6.51669			−0.464295			−0.232147	−	0.972681i	$-0.574575\pi$
$197$	−6.51669			−0.464295			−0.232147	+	0.972681i	$0.574575\pi$
$198$		0			0
$199$	23.0707			1.63544			0.817720	−	0.575616i	$-0.195238\pi$
$199$	23.0707			1.63544			0.817720	+	0.575616i	$0.195238\pi$
$200$		0			0
$201$			2.78238i			0.196254i
$202$		0			0
$203$	19.5695			1.37351
$204$		0			0
$205$			5.56477i			0.388660i
$206$		0			0
$207$	0.718808	+	4.74166i	0.0499606	+	0.329568i
$208$		0			0
$209$	15.4833			1.07100
$210$		0			0
$211$		−	19.4833i		−	1.34129i	−0.741780	−	0.670643i	$-0.766018\pi$
$211$		−	19.4833i		−	1.34129i	0.741780	−	0.670643i	$-0.233982\pi$
$212$		0			0
$213$	13.4833			0.923862
$214$		0			0
$215$			21.2250i			1.44753i
$216$		0			0
$217$			15.4424i			1.04830i
$218$		0			0
$219$			4.00000i			0.270295i
$220$		0			0
$221$			20.1026i			1.35225i
$222$		0			0
$223$			4.51669i			0.302460i	0.988499	+	0.151230i	$0.0483233\pi$
$223$			4.51669i			0.302460i	−0.988499	+	0.151230i	$0.951677\pi$
$224$		0			0
$225$	2.74166			0.182777
$226$		0			0
$227$	17.9462			1.19113			0.595566	−	0.803306i	$-0.296928\pi$
$227$	17.9462			1.19113			0.595566	+	0.803306i	$0.296928\pi$
$228$		0			0
$229$			16.6943i			1.10319i	0.834112	+	0.551595i	$0.185981\pi$
$229$			16.6943i			1.10319i	−0.834112	+	0.551595i	$0.814019\pi$
$230$		0			0
$231$			11.4833i			0.755547i
$232$		0			0
$233$	9.48331			0.621273			0.310636	−	0.950529i	$-0.399458\pi$
$233$	9.48331			0.621273			0.310636	+	0.950529i	$0.399458\pi$
$234$		0			0
$235$	21.5403			1.40513
$236$		0			0
$237$		−	15.3495i		−	0.997059i
$238$		0			0
$239$		−	16.0000i		−	1.03495i	−0.855697	−	0.517477i	$-0.826871\pi$
$239$		−	16.0000i		−	1.03495i	0.855697	−	0.517477i	$-0.173129\pi$
$240$		0			0
$241$			7.00238i			0.451063i	0.974236	+	0.225532i	$0.0724119\pi$
$241$			7.00238i			0.451063i	−0.974236	+	0.225532i	$0.927588\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$			7.62834i			0.487357i
$246$		0			0
$247$	−15.9755			−1.01650
$248$		0			0
$249$			7.00238i			0.443758i
$250$		0			0
$251$	4.31285			0.272225			0.136112	−	0.990693i	$-0.456539\pi$
$251$	4.31285			0.272225			0.136112	+	0.990693i	$0.456539\pi$
$252$		0			0
$253$	−4.00000	−	26.3862i	−0.251478	−	1.65889i
$254$		0			0
$255$			9.74166i			0.610046i
$256$		0			0
$257$	16.9666			1.05835			0.529175	−	0.848513i	$-0.322502\pi$
$257$	16.9666			1.05835			0.529175	+	0.848513i	$0.322502\pi$
$258$		0			0
$259$		−	20.0000i		−	1.24274i
$260$		0			0
$261$	9.48331			0.587002
$262$		0			0
$263$	12.3815			0.763473			0.381736	−	0.924271i	$-0.375326\pi$
$263$	12.3815			0.763473			0.381736	+	0.924271i	$0.375326\pi$
$264$		0			0
$265$	17.2250			1.05812
$266$		0			0
$267$	−12.4743			−0.763415
$268$		0			0
$269$	9.48331			0.578208			0.289104	−	0.957298i	$-0.406643\pi$
$269$	9.48331			0.578208			0.289104	+	0.957298i	$0.406643\pi$
$270$		0			0
$271$		−	15.4833i		−	0.940544i	−0.882521	−	0.470272i	$-0.844156\pi$
$271$		−	15.4833i		−	0.940544i	0.882521	−	0.470272i	$-0.155844\pi$
$272$		0			0
$273$		−	11.8483i		−	0.717094i
$274$		0			0
$275$	−15.2567			−0.920013
$276$		0			0
$277$	14.0000			0.841178			0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000			0.841178			0.420589	+	0.907251i	$0.361823\pi$
$278$		0			0
$279$			7.48331i			0.448014i
$280$		0			0
$281$			2.24927i			0.134180i	0.997747	+	0.0670902i	$0.0213715\pi$
$281$			2.24927i			0.134180i	−0.997747	+	0.0670902i	$0.978628\pi$
$282$		0			0
$283$	−30.6062			−1.81935			−0.909675	−	0.415320i	$-0.863670\pi$
$283$	−30.6062			−1.81935			−0.909675	+	0.415320i	$0.863670\pi$
$284$		0			0
$285$	−7.74166			−0.458576
$286$		0			0
$287$	4.12715			0.243618
$288$		0			0
$289$	4.74166			0.278921
$290$		0			0
$291$	−13.8191			−0.810088
$292$		0			0
$293$			19.4767i			1.13784i	0.822393	+	0.568920i	$0.192639\pi$
$293$			19.4767i			1.13784i	−0.822393	+	0.568920i	$0.807361\pi$
$294$		0			0
$295$	−0.718808			−0.0418506
$296$		0			0
$297$			5.56477i			0.322901i
$298$		0			0
$299$	4.12715	+	27.2250i	0.238679	+	1.57446i
$300$		0			0
$301$	15.7417			0.907334
$302$		0			0
$303$		−	10.0000i		−	0.574485i
$304$		0			0
$305$	−34.9666			−2.00218
$306$		0			0
$307$			19.4833i			1.11197i	0.831192	+	0.555986i	$0.187659\pi$
$307$			19.4833i			1.11197i	−0.831192	+	0.555986i	$0.812341\pi$
$308$		0			0
$309$			1.34477i			0.0765011i
$310$		0			0
$311$			16.2583i			0.921926i	0.887419	+	0.460963i	$0.152496\pi$
$311$			16.2583i			0.921926i	−0.887419	+	0.460963i	$0.847504\pi$
$312$		0			0
$313$		−	31.9510i		−	1.80598i	−0.429665	−	0.902988i	$-0.641368\pi$
$313$		−	31.9510i		−	1.80598i	0.429665	−	0.902988i	$-0.358632\pi$
$314$		0			0
$315$		−	5.74166i		−	0.323506i
$316$		0			0
$317$	2.00000			0.112331			0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000			0.112331			0.0561656	+	0.998421i	$0.482113\pi$
$318$		0			0
$319$	−52.7724			−2.95469
$320$		0			0
$321$		−	13.8191i		−	0.771305i
$322$		0			0
$323$		−	9.74166i		−	0.542040i
$324$		0			0
$325$	15.7417			0.873190
$326$		0			0
$327$	−8.44000			−0.466733
$328$		0			0
$329$		−	15.9755i		−	0.880757i
$330$		0			0
$331$			18.9666i			1.04250i	0.853404	+	0.521250i	$0.174534\pi$
$331$			18.9666i			1.04250i	−0.853404	+	0.521250i	$0.825466\pi$
$332$		0			0
$333$		−	9.69192i		−	0.531114i
$334$		0			0
$335$			7.74166i			0.422972i
$336$		0			0
$337$		−	4.12715i		−	0.224820i	−0.993662	−	0.112410i	$-0.964143\pi$
$337$		−	4.12715i		−	0.224820i	0.993662	−	0.112410i	$-0.0358570\pi$
$338$		0			0
$339$	1.34477			0.0730377
$340$		0			0
$341$		−	41.6429i		−	2.25509i
$342$		0			0
$343$	20.1026			1.08544
$344$		0			0
$345$	2.00000	+	13.1931i	0.107676	+	0.710293i
$346$		0			0
$347$		−	6.70829i		−	0.360120i	−0.983656	−	0.180060i	$-0.942371\pi$
$347$		−	6.70829i		−	0.360120i	0.983656	−	0.180060i	$-0.0576291\pi$
$348$		0			0
$349$	−25.7417			−1.37792			−0.688960	−	0.724800i	$-0.741932\pi$
$349$	−25.7417			−1.37792			−0.688960	+	0.724800i	$0.741932\pi$
$350$		0			0
$351$		−	5.74166i		−	0.306467i
$352$		0			0
$353$	−20.9666			−1.11594			−0.557971	−	0.829861i	$-0.688420\pi$
$353$	−20.9666			−1.11594			−0.557971	+	0.829861i	$0.688420\pi$
$354$		0			0
$355$	37.5158			1.99113
$356$		0			0
$357$	7.22497			0.382386
$358$		0			0
$359$	22.0734			1.16499			0.582494	−	0.812835i	$-0.302077\pi$
$359$	22.0734			1.16499			0.582494	+	0.812835i	$0.302077\pi$
$360$		0			0
$361$	−11.2583			−0.592544
$362$		0			0
$363$		−	19.9666i		−	1.04798i
$364$		0			0
$365$			11.1295i			0.582546i
$366$		0			0
$367$	18.0391			0.941632			0.470816	−	0.882232i	$-0.343960\pi$
$367$	18.0391			0.941632			0.470816	+	0.882232i	$0.343960\pi$
$368$		0			0
$369$	2.00000			0.104116
$370$		0			0
$371$		−	12.7750i		−	0.663246i
$372$		0			0
$373$		−	2.87523i		−	0.148874i	−0.997226	−	0.0744370i	$-0.976284\pi$
$373$		−	2.87523i		−	0.148874i	0.997226	−	0.0744370i	$-0.0237160\pi$
$374$		0			0
$375$	−6.28357			−0.324482
$376$		0			0
$377$	54.4499			2.80431
$378$		0			0
$379$	−17.5060			−0.899221			−0.449610	−	0.893225i	$-0.648437\pi$
$379$	−17.5060			−0.899221			−0.449610	+	0.893225i	$0.648437\pi$
$380$		0			0
$381$	−3.48331			−0.178456
$382$		0			0
$383$	15.2567			0.779580			0.389790	−	0.920904i	$-0.372548\pi$
$383$	15.2567			0.779580			0.389790	+	0.920904i	$0.372548\pi$
$384$		0			0
$385$			31.9510i			1.62837i
$386$		0			0
$387$	7.62834			0.387770
$388$		0			0
$389$			19.4767i			0.987507i	0.869602	+	0.493754i	$0.164376\pi$
$389$			19.4767i			0.987507i	−0.869602	+	0.493754i	$0.835624\pi$
$390$		0			0
$391$	−16.6015	+	2.51669i	−0.839571	+	0.127274i
$392$		0			0
$393$	−1.48331			−0.0748233
$394$		0			0
$395$		−	42.7083i		−	2.14889i
$396$		0			0
$397$	−4.96663			−0.249268			−0.124634	−	0.992203i	$-0.539776\pi$
$397$	−4.96663			−0.249268			−0.124634	+	0.992203i	$0.539776\pi$
$398$		0			0
$399$			5.74166i			0.287442i
$400$		0			0
$401$		−	7.62834i		−	0.380941i	−0.981693	−	0.190471i	$-0.938999\pi$
$401$		−	7.62834i		−	0.380941i	0.981693	−	0.190471i	$-0.0610014\pi$
$402$		0			0
$403$			42.9666i			2.14032i
$404$		0			0
$405$		−	2.78238i		−	0.138258i
$406$		0			0
$407$			53.9333i			2.67337i
$408$		0			0
$409$	4.70829			0.232810			0.116405	−	0.993202i	$-0.462863\pi$
$409$	4.70829			0.232810			0.116405	+	0.993202i	$0.462863\pi$
$410$		0			0
$411$	−4.75311			−0.234454
$412$		0			0
$413$			0.533109i			0.0262326i
$414$		0			0
$415$			19.4833i			0.956398i
$416$		0			0
$417$	−7.48331			−0.366460
$418$		0			0
$419$	−9.50622			−0.464409			−0.232205	−	0.972667i	$-0.574594\pi$
$419$	−9.50622			−0.464409			−0.232205	+	0.972667i	$0.574594\pi$
$420$		0			0
$421$			7.00238i			0.341275i	0.985334	+	0.170638i	$0.0545827\pi$
$421$			7.00238i			0.341275i	−0.985334	+	0.170638i	$0.945417\pi$
$422$		0			0
$423$		−	7.74166i		−	0.376412i
$424$		0			0
$425$			9.59907i			0.465623i
$426$		0			0
$427$			25.9333i			1.25500i
$428$		0			0
$429$			31.9510i			1.54261i
$430$		0			0
$431$	12.5671			0.605338			0.302669	−	0.953096i	$-0.402122\pi$
$431$	12.5671			0.605338			0.302669	+	0.953096i	$0.402122\pi$
$432$		0			0
$433$		−	15.4424i		−	0.742114i	−0.928610	−	0.371057i	$-0.878996\pi$
$433$		−	15.4424i		−	0.742114i	0.928610	−	0.371057i	$-0.121004\pi$
$434$		0			0
$435$	26.3862			1.26512
$436$		0			0
$437$	−2.00000	−	13.1931i	−0.0956730	−	0.631112i
$438$		0			0
$439$		−	18.4499i		−	0.880568i	−0.897859	−	0.440284i	$-0.854878\pi$
$439$		−	18.4499i		−	0.880568i	0.897859	−	0.440284i	$-0.145122\pi$
$440$		0			0
$441$	2.74166			0.130555
$442$		0			0
$443$		−	20.0000i		−	0.950229i	−0.879924	−	0.475114i	$-0.842407\pi$
$443$		−	20.0000i		−	0.950229i	0.879924	−	0.475114i	$-0.157593\pi$
$444$		0			0
$445$	−34.7083			−1.64533
$446$		0			0
$447$	−17.3203			−0.819220
$448$		0			0
$449$	10.5167			0.496313			0.248157	−	0.968720i	$-0.420175\pi$
$449$	10.5167			0.496313			0.248157	+	0.968720i	$0.420175\pi$
$450$		0			0
$451$	−11.1295			−0.524069
$452$		0			0
$453$	−22.9666			−1.07907
$454$		0			0
$455$		−	32.9666i		−	1.54550i
$456$		0			0
$457$			13.8191i			0.646429i	0.946326	+	0.323214i	$0.104763\pi$
$457$			13.8191i			0.646429i	−0.946326	+	0.323214i	$0.895237\pi$
$458$		0			0
$459$	3.50119			0.163422
$460$		0			0
$461$	2.00000			0.0931493			0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000			0.0931493			0.0465746	+	0.998915i	$0.485169\pi$
$462$		0			0
$463$		−	9.03337i		−	0.419816i	−0.977721	−	0.209908i	$-0.932683\pi$
$463$		−	9.03337i		−	0.419816i	0.977721	−	0.209908i	$-0.0673165\pi$
$464$		0			0
$465$			20.8215i			0.965572i
$466$		0			0
$467$	−23.6967			−1.09655			−0.548276	−	0.836298i	$-0.684716\pi$
$467$	−23.6967			−1.09655			−0.548276	+	0.836298i	$0.684716\pi$
$468$		0			0
$469$	5.74166			0.265125
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−42.4499			−1.95185
$474$		0			0
$475$	−7.62834			−0.350012
$476$		0			0
$477$		−	6.19073i		−	0.283454i
$478$		0			0
$479$	2.87523			0.131373			0.0656864	−	0.997840i	$-0.479076\pi$
$479$	2.87523			0.131373			0.0656864	+	0.997840i	$0.479076\pi$
$480$		0			0
$481$		−	55.6477i		−	2.53732i
$482$		0			0
$483$	9.78477	−	1.48331i	0.445222	−	0.0674932i
$484$		0			0
$485$	−38.4499			−1.74592
$486$		0			0
$487$		−	32.0000i		−	1.45006i	−0.688718	−	0.725029i	$-0.741826\pi$
$487$		−	32.0000i		−	1.45006i	0.688718	−	0.725029i	$-0.258174\pi$
$488$		0			0
$489$	12.0000			0.542659
$490$		0			0
$491$		−	14.1916i		−	0.640458i	−0.947340	−	0.320229i	$-0.896240\pi$
$491$		−	14.1916i		−	0.640458i	0.947340	−	0.320229i	$-0.103760\pi$
$492$		0			0
$493$			33.2029i			1.49538i
$494$		0			0
$495$			15.4833i			0.695923i
$496$		0			0
$497$		−	27.8238i		−	1.24807i
$498$		0			0
$499$			26.9666i			1.20719i	0.797290	+	0.603596i	$0.206266\pi$
$499$			26.9666i			1.20719i	−0.797290	+	0.603596i	$0.793734\pi$
$500$		0			0
$501$	−8.00000			−0.357414
$502$		0			0
$503$	20.8215			0.928383			0.464191	−	0.885735i	$-0.346345\pi$
$503$	20.8215			0.928383			0.464191	+	0.885735i	$0.346345\pi$
$504$		0			0
$505$		−	27.8238i		−	1.23814i
$506$		0			0
$507$		−	19.9666i		−	0.886749i
$508$		0			0
$509$	−31.9333			−1.41542			−0.707708	−	0.706505i	$-0.750271\pi$
$509$	−31.9333			−1.41542			−0.707708	+	0.706505i	$0.750271\pi$
$510$		0			0
$511$	8.25430			0.365149
$512$		0			0
$513$			2.78238i			0.122845i
$514$		0			0
$515$			3.74166i			0.164877i
$516$		0			0
$517$			43.0805i			1.89468i
$518$		0			0
$519$			5.48331i			0.240691i
$520$		0			0
$521$			32.0438i			1.40387i	0.712243	+	0.701933i	$0.247679\pi$
$521$			32.0438i			1.40387i	−0.712243	+	0.701933i	$0.752321\pi$
$522$		0			0
$523$	2.59668			0.113545			0.0567725	−	0.998387i	$-0.481919\pi$
$523$	2.59668			0.113545			0.0567725	+	0.998387i	$0.481919\pi$
$524$		0			0
$525$		−	5.65762i		−	0.246919i
$526$		0			0
$527$	−26.2005			−1.14131
$528$		0			0
$529$	−21.9666	+	6.81668i	−0.955071	+	0.296378i
$530$		0			0
$531$			0.258343i			0.0112111i
$532$		0			0
$533$	11.4833			0.497398
$534$		0			0
$535$		−	38.4499i		−	1.66234i
$536$		0			0
$537$	7.74166			0.334077
$538$		0			0
$539$	−15.2567			−0.657152
$540$		0			0
$541$	−25.2250			−1.08451			−0.542253	−	0.840215i	$-0.682429\pi$
$541$	−25.2250			−1.08451			−0.542253	+	0.840215i	$0.682429\pi$
$542$		0			0
$543$	−1.25192			−0.0537249
$544$		0			0
$545$	−23.4833			−1.00591
$546$		0			0
$547$			23.4833i			1.00407i	0.864846	+	0.502037i	$0.167416\pi$
$547$			23.4833i			1.00407i	−0.864846	+	0.502037i	$0.832584\pi$
$548$		0			0
$549$			12.5671i			0.536353i
$550$		0			0
$551$	−26.3862			−1.12409
$552$		0			0
$553$	−31.6749			−1.34695
$554$		0			0
$555$		−	26.9666i		−	1.14467i
$556$		0			0
$557$			21.9805i			0.931344i	0.884957	+	0.465672i	$0.154187\pi$
$557$			21.9805i			0.931344i	−0.884957	+	0.465672i	$0.845813\pi$
$558$		0			0
$559$	43.7993			1.85251
$560$		0			0
$561$	−19.4833			−0.822586
$562$		0			0
$563$	16.8800			0.711407			0.355704	−	0.934599i	$-0.384241\pi$
$563$	16.8800			0.711407			0.355704	+	0.934599i	$0.384241\pi$
$564$		0			0
$565$	3.74166			0.157413
$566$		0			0
$567$	−2.06358			−0.0866620
$568$		0			0
$569$		−	6.90953i		−	0.289663i	−0.989456	−	0.144831i	$-0.953736\pi$
$569$		−	6.90953i		−	0.289663i	0.989456	−	0.144831i	$-0.0462640\pi$
$570$		0			0
$571$	4.75311			0.198911			0.0994557	−	0.995042i	$-0.468290\pi$
$571$	4.75311			0.198911			0.0994557	+	0.995042i	$0.468290\pi$
$572$		0			0
$573$			7.00238i			0.292529i
$574$		0			0
$575$	1.97073	+	13.0000i	0.0821849	+	0.542137i
$576$		0			0
$577$	17.7417			0.738595			0.369297	−	0.929311i	$-0.379598\pi$
$577$	17.7417			0.738595			0.369297	+	0.929311i	$0.379598\pi$
$578$		0			0
$579$			1.74166i			0.0723808i
$580$		0			0
$581$	14.4499			0.599485
$582$		0			0
$583$			34.4499i			1.42677i
$584$		0			0
$585$		−	15.9755i		−	0.660505i
$586$		0			0
$587$			18.5167i			0.764265i	0.924107	+	0.382133i	$0.124810\pi$
$587$			18.5167i			0.764265i	−0.924107	+	0.382133i	$0.875190\pi$
$588$		0			0
$589$		−	20.8215i		−	0.857933i
$590$		0			0
$591$			6.51669i			0.268061i
$592$		0			0
$593$	−31.4166			−1.29012			−0.645062	−	0.764130i	$-0.723168\pi$
$593$	−31.4166			−1.29012			−0.645062	+	0.764130i	$0.723168\pi$
$594$		0			0
$595$	20.1026			0.824128
$596$		0			0
$597$		−	23.0707i		−	0.944222i
$598$		0			0
$599$			1.03337i			0.0422224i	0.999777	+	0.0211112i	$0.00672040\pi$
$599$			1.03337i			0.0422224i	−0.999777	+	0.0211112i	$0.993280\pi$
$600$		0			0
$601$	24.7083			1.00787			0.503936	−	0.863741i	$-0.331885\pi$
$601$	24.7083			1.00787			0.503936	+	0.863741i	$0.331885\pi$
$602$		0			0
$603$	2.78238			0.113307
$604$		0			0
$605$		−	55.5548i		−	2.25862i
$606$		0			0
$607$		−	24.0000i		−	0.974130i	−0.873366	−	0.487065i	$-0.838067\pi$
$607$		−	24.0000i		−	0.974130i	0.873366	−	0.487065i	$-0.161933\pi$
$608$		0			0
$609$		−	19.5695i		−	0.792998i
$610$		0			0
$611$		−	44.4499i		−	1.79825i
$612$		0			0
$613$		−	36.2638i		−	1.46468i	−0.680938	−	0.732341i	$-0.738428\pi$
$613$		−	36.2638i		−	1.46468i	0.680938	−	0.732341i	$-0.261572\pi$
$614$		0			0
$615$	5.56477			0.224393
$616$		0			0
$617$			23.6038i			0.950255i	0.879917	+	0.475127i	$0.157598\pi$
$617$			23.6038i			0.950255i	−0.879917	+	0.475127i	$0.842402\pi$
$618$		0			0
$619$	−35.6379			−1.43241			−0.716204	−	0.697891i	$-0.754122\pi$
$619$	−35.6379			−1.43241			−0.716204	+	0.697891i	$0.754122\pi$
$620$		0			0
$621$	4.74166	−	0.718808i	0.190276	−	0.0288448i
$622$		0			0
$623$			25.7417i			1.03132i
$624$		0			0
$625$	−31.1916			−1.24766
$626$		0			0
$627$		−	15.4833i		−	0.618344i
$628$		0			0
$629$	33.9333			1.35301
$630$		0			0
$631$	−47.8336			−1.90423			−0.952113	−	0.305745i	$-0.901094\pi$
$631$	−47.8336			−1.90423			−0.952113	+	0.305745i	$0.901094\pi$
$632$		0			0
$633$	−19.4833			−0.774392
$634$		0			0
$635$	−9.69192			−0.384612
$636$		0			0
$637$	15.7417			0.623707
$638$		0			0
$639$		−	13.4833i		−	0.533392i
$640$		0			0
$641$			21.2617i			0.839787i	0.907573	+	0.419894i	$0.137933\pi$
$641$			21.2617i			0.839787i	−0.907573	+	0.419894i	$0.862067\pi$
$642$		0			0
$643$	20.0098			0.789109			0.394555	−	0.918872i	$-0.370899\pi$
$643$	20.0098			0.789109			0.394555	+	0.918872i	$0.370899\pi$
$644$		0			0
$645$	21.2250			0.835732
$646$		0			0
$647$		−	16.7750i		−	0.659494i	−0.944069	−	0.329747i	$-0.893036\pi$
$647$		−	16.7750i		−	0.659494i	0.944069	−	0.329747i	$-0.106964\pi$
$648$		0			0
$649$		−	1.43762i		−	0.0564314i
$650$		0			0
$651$	15.4424			0.605235
$652$		0			0
$653$	−8.44994			−0.330672			−0.165336	−	0.986237i	$-0.552871\pi$
$653$	−8.44994			−0.330672			−0.165336	+	0.986237i	$0.552871\pi$
$654$		0			0
$655$	−4.12715			−0.161261
$656$		0			0
$657$	4.00000			0.156055
$658$		0			0
$659$	2.50384			0.0975356			0.0487678	−	0.998810i	$-0.484471\pi$
$659$	2.50384			0.0975356			0.0487678	+	0.998810i	$0.484471\pi$
$660$		0			0
$661$			18.3176i			0.712473i	0.934396	+	0.356236i	$0.115940\pi$
$661$			18.3176i			0.712473i	−0.934396	+	0.356236i	$0.884060\pi$
$662$		0			0
$663$	20.1026			0.780722
$664$		0			0
$665$			15.9755i			0.619503i
$666$		0			0
$667$	6.81668	+	44.9666i	0.263943	+	1.74111i
$668$		0			0
$669$	4.51669			0.174625
$670$		0			0
$671$		−	69.9333i		−	2.69974i
$672$		0			0
$673$	−16.7083			−0.644057			−0.322029	−	0.946730i	$-0.604365\pi$
$673$	−16.7083			−0.644057			−0.322029	+	0.946730i	$0.604365\pi$
$674$		0			0
$675$		−	2.74166i		−	0.105526i
$676$		0			0
$677$		−	13.1931i		−	0.507052i	−0.967329	−	0.253526i	$-0.918410\pi$
$677$		−	13.1931i		−	0.507052i	0.967329	−	0.253526i	$-0.0815904\pi$
$678$		0			0
$679$			28.5167i			1.09437i
$680$		0			0
$681$		−	17.9462i		−	0.687701i
$682$		0			0
$683$		−	28.4499i		−	1.08861i	−0.838888	−	0.544303i	$-0.816794\pi$
$683$		−	28.4499i		−	1.08861i	0.838888	−	0.544303i	$-0.183206\pi$
$684$		0			0
$685$	−13.2250			−0.505300
$686$		0			0
$687$	16.6943			0.636927
$688$		0			0
$689$		−	35.5450i		−	1.35416i
$690$		0			0
$691$			14.4499i			0.549702i	0.961487	+	0.274851i	$0.0886285\pi$
$691$			14.4499i			0.549702i	−0.961487	+	0.274851i	$0.911372\pi$
$692$		0			0
$693$	11.4833			0.436215
$694$		0			0
$695$	−20.8215			−0.789803
$696$		0			0
$697$			7.00238i			0.265234i
$698$		0			0
$699$		−	9.48331i		−	0.358692i
$700$		0			0
$701$			5.65762i			0.213685i	0.994276	+	0.106843i	$0.0340741\pi$
$701$			5.65762i			0.213685i	−0.994276	+	0.106843i	$0.965926\pi$
$702$		0			0
$703$			26.9666i			1.01707i
$704$		0			0
$705$		−	21.5403i		−	0.811253i
$706$		0			0
$707$	−20.6358			−0.776087
$708$		0			0
$709$		−	20.6358i		−	0.774992i	−0.921871	−	0.387496i	$-0.873340\pi$
$709$		−	20.6358i		−	0.774992i	0.921871	−	0.387496i	$-0.126660\pi$
$710$		0			0
$711$	−15.3495			−0.575652
$712$		0			0
$713$	−35.4833	+	5.37907i	−1.32886	+	0.201448i
$714$		0			0
$715$			88.8999i			3.32467i
$716$		0			0
$717$	−16.0000			−0.597531
$718$		0			0
$719$		−	8.77503i		−	0.327253i	−0.986522	−	0.163627i	$-0.947681\pi$
$719$		−	8.77503i		−	0.327253i	0.986522	−	0.163627i	$-0.0523192\pi$
$720$		0			0
$721$	2.77503			0.103347
$722$		0			0
$723$	7.00238			0.260421
$724$		0			0
$725$	26.0000			0.965616
$726$		0			0
$727$	−1.34477			−0.0498746			−0.0249373	−	0.999689i	$-0.507939\pi$
$727$	−1.34477			−0.0498746			−0.0249373	+	0.999689i	$0.507939\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$			26.7083i			0.987842i
$732$		0			0
$733$		−	11.3152i		−	0.417938i	−0.977922	−	0.208969i	$-0.932989\pi$
$733$		−	11.3152i		−	0.417938i	0.977922	−	0.208969i	$-0.0670107\pi$
$734$		0			0
$735$	7.62834			0.281376
$736$		0			0
$737$	−15.4833			−0.570335
$738$		0			0
$739$		−	36.0000i		−	1.32428i	−0.749380	−	0.662141i	$-0.769648\pi$
$739$		−	36.0000i		−	1.32428i	0.749380	−	0.662141i	$-0.230352\pi$
$740$		0			0
$741$			15.9755i			0.586874i
$742$		0			0
$743$	44.7038			1.64002			0.820012	−	0.572346i	$-0.193967\pi$
$743$	44.7038			1.64002			0.820012	+	0.572346i	$0.193967\pi$
$744$		0			0
$745$	−48.1916			−1.76560
$746$		0			0
$747$	7.00238			0.256204
$748$		0			0
$749$	−28.5167			−1.04198
$750$		0			0
$751$	3.31549			0.120984			0.0604920	−	0.998169i	$-0.480733\pi$
$751$	3.31549			0.120984			0.0604920	+	0.998169i	$0.480733\pi$
$752$		0			0
$753$		−	4.31285i		−	0.157169i
$754$		0			0
$755$	−63.9020			−2.32563
$756$		0			0
$757$		−	33.2029i		−	1.20678i	−0.797446	−	0.603390i	$-0.793816\pi$
$757$		−	33.2029i		−	1.20678i	0.797446	−	0.603390i	$-0.206184\pi$
$758$		0			0
$759$	−26.3862	+	4.00000i	−0.957759	+	0.145191i
$760$		0			0
$761$	23.9333			0.867580			0.433790	−	0.901014i	$-0.357176\pi$
$761$	23.9333			0.867580			0.433790	+	0.901014i	$0.357176\pi$
$762$		0			0
$763$			17.4166i			0.630522i
$764$		0			0
$765$	9.74166			0.352210
$766$		0			0
$767$			1.48331i			0.0535594i
$768$		0			0
$769$		−	14.0048i		−	0.505025i	−0.967594	−	0.252512i	$-0.918743\pi$
$769$		−	14.0048i		−	0.505025i	0.967594	−	0.252512i	$-0.0812568\pi$
$770$		0			0
$771$		−	16.9666i		−	0.611038i
$772$		0			0
$773$			7.81404i			0.281052i	0.990077	+	0.140526i	$0.0448793\pi$
$773$			7.81404i			0.281052i	−0.990077	+	0.140526i	$0.955121\pi$
$774$		0			0
$775$			20.5167i			0.736981i
$776$		0			0
$777$	−20.0000			−0.717496
$778$		0			0
$779$	−5.56477			−0.199378
$780$		0			0
$781$			75.0315i			2.68484i
$782$		0			0
$783$		−	9.48331i		−	0.338906i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	17.1346			0.610781			0.305391	−	0.952227i	$-0.401213\pi$
$787$	17.1346			0.610781			0.305391	+	0.952227i	$0.401213\pi$
$788$		0			0
$789$		−	12.3815i		−	0.440791i
$790$		0			0
$791$		−	2.77503i		−	0.0986686i
$792$		0			0
$793$			72.1563i			2.56234i
$794$		0			0
$795$		−	17.2250i		−	0.610907i
$796$		0			0
$797$			16.0683i			0.569170i	0.958651	+	0.284585i	$0.0918558\pi$
$797$			16.0683i			0.569170i	−0.958651	+	0.284585i	$0.908144\pi$
$798$		0			0
$799$	27.1050			0.958907
$800$		0			0
$801$			12.4743i			0.440758i
$802$		0			0
$803$	−22.2591			−0.785505
$804$		0			0
$805$	27.2250	−	4.12715i	0.959554	−	0.145463i
$806$		0			0
$807$		−	9.48331i		−	0.333828i
$808$		0			0
$809$	−31.4166			−1.10455			−0.552274	−	0.833663i	$-0.686240\pi$
$809$	−31.4166			−1.10455			−0.552274	+	0.833663i	$0.686240\pi$
$810$		0			0
$811$			25.4166i			0.892497i	0.894909	+	0.446248i	$0.147240\pi$
$811$			25.4166i			0.892497i	−0.894909	+	0.446248i	$0.852760\pi$
$812$		0			0
$813$	−15.4833			−0.543024
$814$		0			0
$815$	33.3886			1.16955
$816$		0			0
$817$	−21.2250			−0.742568
$818$		0			0
$819$	−11.8483			−0.414015
$820$		0			0
$821$	32.9666			1.15054			0.575272	−	0.817962i	$-0.304896\pi$
$821$	32.9666			1.15054			0.575272	+	0.817962i	$0.304896\pi$
$822$		0			0
$823$			2.96663i			0.103410i	0.998662	+	0.0517051i	$0.0164656\pi$
$823$			2.96663i			0.103410i	−0.998662	+	0.0517051i	$0.983534\pi$
$824$		0			0
$825$			15.2567i			0.531170i
$826$		0			0
$827$	57.0853			1.98505			0.992525	−	0.122042i	$-0.0389443\pi$
$827$	57.0853			1.98505			0.992525	+	0.122042i	$0.0389443\pi$
$828$		0			0
$829$	35.6749			1.23904			0.619521	−	0.784980i	$-0.287327\pi$
$829$	35.6749			1.23904			0.619521	+	0.784980i	$0.287327\pi$
$830$		0			0
$831$		−	14.0000i		−	0.485655i
$832$		0			0
$833$			9.59907i			0.332588i
$834$		0			0
$835$	−22.2591			−0.770307
$836$		0			0
$837$	7.48331			0.258661
$838$		0			0
$839$	−21.0071			−0.725247			−0.362624	−	0.931936i	$-0.618119\pi$
$839$	−21.0071			−0.725247			−0.362624	+	0.931936i	$0.618119\pi$
$840$		0			0
$841$	60.9333			2.10115
$842$		0			0
$843$	2.24927			0.0774691
$844$		0			0
$845$		−	55.5548i		−	1.91114i
$846$		0			0
$847$	−41.2026			−1.41574
$848$		0			0
$849$			30.6062i			1.05040i
$850$		0			0
$851$	45.9558	−	6.96663i	1.57534	−	0.238813i
$852$		0			0
$853$	26.9666			0.923320			0.461660	−	0.887057i	$-0.347254\pi$
$853$	26.9666			0.923320			0.461660	+	0.887057i	$0.347254\pi$
$854$		0			0
$855$			7.74166i			0.264759i
$856$		0			0
$857$	−3.55006			−0.121268			−0.0606338	−	0.998160i	$-0.519312\pi$
$857$	−3.55006			−0.121268			−0.0606338	+	0.998160i	$0.519312\pi$
$858$		0			0
$859$			16.5167i			0.563542i	0.959482	+	0.281771i	$0.0909218\pi$
$859$			16.5167i			0.563542i	−0.959482	+	0.281771i	$0.909078\pi$
$860$		0			0
$861$		−	4.12715i		−	0.140653i
$862$		0			0
$863$			40.2583i			1.37041i	0.728350	+	0.685205i	$0.240287\pi$
$863$			40.2583i			1.37041i	−0.728350	+	0.685205i	$0.759713\pi$
$864$		0			0
$865$			15.2567i			0.518743i
$866$		0			0
$867$		−	4.74166i		−	0.161035i
$868$		0			0
$869$	85.4166			2.89756
$870$		0			0
$871$	15.9755			0.541309
$872$		0			0
$873$			13.8191i			0.467705i
$874$		0			0
$875$			12.9666i			0.438352i
$876$		0			0
$877$	−4.00000			−0.135070			−0.0675352	−	0.997717i	$-0.521513\pi$
$877$	−4.00000			−0.135070			−0.0675352	+	0.997717i	$0.521513\pi$
$878$		0			0
$879$	19.4767			0.656932
$880$		0			0
$881$			9.25166i			0.311696i	0.987781	+	0.155848i	$0.0498110\pi$
$881$			9.25166i			0.311696i	−0.987781	+	0.155848i	$0.950189\pi$
$882$		0			0
$883$		−	24.0000i		−	0.807664i	−0.914833	−	0.403832i	$-0.867678\pi$
$883$		−	24.0000i		−	0.807664i	0.914833	−	0.403832i	$-0.132322\pi$
$884$		0			0
$885$			0.718808i			0.0241625i
$886$		0			0
$887$		−	38.1916i		−	1.28235i	−0.767395	−	0.641174i	$-0.778448\pi$
$887$		−	38.1916i		−	1.28235i	0.767395	−	0.641174i	$-0.221552\pi$
$888$		0			0
$889$			7.18808i			0.241081i
$890$		0			0
$891$	5.56477			0.186427
$892$		0			0
$893$			21.5403i			0.720817i
$894$		0			0
$895$	21.5403			0.720011
$896$		0			0
$897$	27.2250	−	4.12715i	0.909015	−	0.137802i
$898$		0			0
$899$			70.9666i			2.36687i
$900$		0			0
$901$	21.6749			0.722096
$902$		0			0
$903$		−	15.7417i		−	0.523850i
$904$		0			0
$905$	−3.48331			−0.115789
$906$		0			0
$907$	−14.0976			−0.468104			−0.234052	−	0.972224i	$-0.575199\pi$
$907$	−14.0976			−0.468104			−0.234052	+	0.972224i	$0.575199\pi$
$908$		0			0
$909$	−10.0000			−0.331679
$910$		0			0
$911$	37.3301			1.23680			0.618400	−	0.785864i	$-0.287781\pi$
$911$	37.3301			1.23680			0.618400	+	0.785864i	$0.287781\pi$
$912$		0			0
$913$	−38.9666			−1.28961
$914$		0			0
$915$			34.9666i			1.15596i
$916$		0			0
$917$			3.06093i			0.101081i
$918$		0			0
$919$	−36.7041			−1.21076			−0.605378	−	0.795938i	$-0.706978\pi$
$919$	−36.7041			−1.21076			−0.605378	+	0.795938i	$0.706978\pi$
$920$		0			0
$921$	19.4833			0.641997
$922$		0			0
$923$		−	77.4166i		−	2.54820i
$924$		0			0
$925$		−	26.5719i		−	0.873679i
$926$		0			0
$927$	1.34477			0.0441679
$928$		0			0
$929$	22.0000			0.721797			0.360898	−	0.932605i	$-0.382470\pi$
$929$	22.0000			0.721797			0.360898	+	0.932605i	$0.382470\pi$
$930$		0			0
$931$	−7.62834			−0.250009
$932$		0			0
$933$	16.2583			0.532274
$934$		0			0
$935$	−54.2101			−1.77286
$936$		0			0
$937$		−	44.1467i		−	1.44221i	−0.692825	−	0.721106i	$-0.743634\pi$
$937$		−	44.1467i		−	1.44221i	0.692825	−	0.721106i	$-0.256366\pi$
$938$		0			0
$939$	−31.9510			−1.04268
$940$		0			0
$941$		−	16.4158i		−	0.535138i	−0.963539	−	0.267569i	$-0.913780\pi$
$941$		−	16.4158i		−	0.535138i	0.963539	−	0.267569i	$-0.0862204\pi$
$942$		0			0
$943$	1.43762	+	9.48331i	0.0468152	+	0.308819i
$944$		0			0
$945$	−5.74166			−0.186776
$946$		0			0
$947$			29.1582i			0.947515i	0.880655	+	0.473758i	$0.157103\pi$
$947$			29.1582i			0.947515i	−0.880655	+	0.473758i	$0.842897\pi$
$948$		0			0
$949$	22.9666			0.745528
$950$		0			0
$951$		−	2.00000i		−	0.0648544i
$952$		0			0
$953$			15.3495i			0.497220i	0.968604	+	0.248610i	$0.0799738\pi$
$953$			15.3495i			0.497220i	−0.968604	+	0.248610i	$0.920026\pi$
$954$		0			0
$955$			19.4833i			0.630465i
$956$		0			0
$957$			52.7724i			1.70589i
$958$		0			0
$959$			9.80840i			0.316730i
$960$		0			0
$961$	−25.0000			−0.806452
$962$		0			0
$963$	−13.8191			−0.445313
$964$		0			0
$965$			4.84596i			0.155997i
$966$		0			0
$967$		−	2.44994i		−	0.0787849i	−0.999224	−	0.0393924i	$-0.987458\pi$
$967$		−	2.44994i		−	0.0787849i	0.999224	−	0.0393924i	$-0.0125423\pi$
$968$		0			0
$969$	−9.74166			−0.312947
$970$		0			0
$971$	44.5181			1.42865			0.714327	−	0.699812i	$-0.246733\pi$
$971$	44.5181			1.42865			0.714327	+	0.699812i	$0.246733\pi$
$972$		0			0
$973$			15.4424i			0.495060i
$974$		0			0
$975$		−	15.7417i		−	0.504137i
$976$		0			0
$977$		−	40.4838i		−	1.29519i	−0.761984	−	0.647596i	$-0.775774\pi$
$977$		−	40.4838i		−	1.29519i	0.761984	−	0.647596i	$-0.224226\pi$
$978$		0			0
$979$		−	69.4166i		−	2.21856i
$980$		0			0
$981$			8.44000i			0.269468i
$982$		0			0
$983$	51.3348			1.63733			0.818663	−	0.574274i	$-0.194716\pi$
$983$	51.3348			1.63733			0.818663	+	0.574274i	$0.194716\pi$
$984$		0			0
$985$			18.1319i			0.577731i
$986$		0			0
$987$	−15.9755			−0.508506
$988$		0			0
$989$	5.48331	+	36.1710i	0.174359	+	1.15017i
$990$		0			0
$991$			50.9666i			1.61901i	0.587114	+	0.809504i	$0.300264\pi$
$991$			50.9666i			1.61901i	−0.587114	+	0.809504i	$0.699736\pi$
$992$		0			0
$993$	18.9666			0.601888
$994$		0			0
$995$		−	64.1916i		−	2.03501i
$996$		0			0
$997$	8.70829			0.275794			0.137897	−	0.990447i	$-0.455966\pi$
$997$	8.70829			0.275794			0.137897	+	0.990447i	$0.455966\pi$
$998$		0			0
$999$	−9.69192			−0.306639

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.2.i.a.367.1	✓	8
3.2	odd	2		3312.2.i.c.2575.7		8
4.3	odd	2	inner	1104.2.i.a.367.5	yes	8
8.3	odd	2		4416.2.i.a.1471.4		8
8.5	even	2		4416.2.i.a.1471.8		8
12.11	even	2		3312.2.i.c.2575.8		8
23.22	odd	2	inner	1104.2.i.a.367.4	yes	8
69.68	even	2		3312.2.i.c.2575.2		8
92.91	even	2	inner	1104.2.i.a.367.8	yes	8
184.45	odd	2		4416.2.i.a.1471.5		8
184.91	even	2		4416.2.i.a.1471.1		8
276.275	odd	2		3312.2.i.c.2575.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1104.2.i.a.367.1	✓	8	1.1	even	1	trivial
1104.2.i.a.367.4	yes	8	23.22	odd	2	inner
1104.2.i.a.367.5	yes	8	4.3	odd	2	inner
1104.2.i.a.367.8	yes	8	92.91	even	2	inner
3312.2.i.c.2575.1		8	276.275	odd	2
3312.2.i.c.2575.2		8	69.68	even	2
3312.2.i.c.2575.7		8	3.2	odd	2
3312.2.i.c.2575.8		8	12.11	even	2
4416.2.i.a.1471.1		8	184.91	even	2
4416.2.i.a.1471.4		8	8.3	odd	2
4416.2.i.a.1471.5		8	184.45	odd	2
4416.2.i.a.1471.8		8	8.5	even	2

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:	\( N \)	\(=\)	\( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1104.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -2.78238i q^{5} -2.06358 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -2.78238i q^{5} -2.06358 q^{7} -1.00000 q^{9} +5.56477 q^{11} -5.74166 q^{13} -2.78238 q^{15} -3.50119i q^{17} +2.78238 q^{19} +2.06358i q^{21} +(-0.718808 - 4.74166i) q^{23} -2.74166 q^{25} +1.00000i q^{27} -9.48331 q^{29} -7.48331i q^{31} -5.56477i q^{33} +5.74166i q^{35} +9.69192i q^{37} +5.74166i q^{39} -2.00000 q^{41} -7.62834 q^{43} +2.78238i q^{45} +7.74166i q^{47} -2.74166 q^{49} -3.50119 q^{51} +6.19073i q^{53} -15.4833i q^{55} -2.78238i q^{57} -0.258343i q^{59} -12.5671i q^{61} +2.06358 q^{63} +15.9755i q^{65} -2.78238 q^{67} +(-4.74166 + 0.718808i) q^{69} +13.4833i q^{71} -4.00000 q^{73} +2.74166i q^{75} -11.4833 q^{77} +15.3495 q^{79} +1.00000 q^{81} -7.00238 q^{83} -9.74166 q^{85} +9.48331i q^{87} -12.4743i q^{89} +11.8483 q^{91} -7.48331 q^{93} -7.74166i q^{95} -13.8191i q^{97} -5.56477 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} - 16 q^{13} + 8 q^{25} - 16 q^{29} - 16 q^{41} + 8 q^{49} - 8 q^{69} - 32 q^{73} - 32 q^{77} + 8 q^{81} - 48 q^{85}+O(q^{100}) \) 8 * q - 8 * q^9 - 16 * q^13 + 8 * q^25 - 16 * q^29 - 16 * q^41 + 8 * q^49 - 8 * q^69 - 32 * q^73 - 32 * q^77 + 8 * q^81 - 48 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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