LMFDB - Embedded newform 1100.3.e.b.549.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1100,3,Mod(549,1100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1100, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1100.549");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1100 = 2^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1100.e (of order $2$, degree $1$, not 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:	$29.9728290796$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 85x^{14} + 2456x^{12} + 32605x^{10} + 215801x^{8} + 712960x^{6} + 1098976x^{4} + 633600x^{2} + 92416 $ x^16 + 85x^14 + 2456x^12 + 32605x^10 + 215801x^8 + 712960x^6 + 1098976x^4 + 633600*x^2 + 92416
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{25}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 220)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			549.14
Root			$-0.465196i$ of defining polynomial
Character	$\chi$	$=$	1100.549
Dual form			1100.3.e.b.549.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.65160i q^{3} +9.09184 q^{7} -4.33416 q^{9} +O(q^{10})$	$q+3.65160i q^{3} +9.09184 q^{7} -4.33416 q^{9} +(9.88767 - 4.82017i) q^{11} -22.5101 q^{13} -22.9153 q^{17} +16.0035i q^{19} +33.1997i q^{21} +0.141697i q^{23} +17.0378i q^{27} +29.8270i q^{29} +21.5229 q^{31} +(17.6013 + 36.1058i) q^{33} +28.0333i q^{37} -82.1979i q^{39} +39.5629i q^{41} +21.4131 q^{43} +15.4239i q^{47} +33.6616 q^{49} -83.6773i q^{51} +75.9763i q^{53} -58.4385 q^{57} +50.4169 q^{59} -98.4882i q^{61} -39.4055 q^{63} +125.070i q^{67} -0.517422 q^{69} -100.615 q^{71} -128.281 q^{73} +(89.8971 - 43.8243i) q^{77} +103.932i q^{79} -101.222 q^{81} +75.6774 q^{83} -108.916 q^{87} +58.9162 q^{89} -204.658 q^{91} +78.5930i q^{93} -62.4540i q^{97} +(-42.8547 + 20.8914i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 56 q^{9}+O(q^{10}) $ 16 * q - 56 * q^9	$ 16 q - 56 q^{9} + 48 q^{11} + 40 q^{31} + 488 q^{49} + 32 q^{59} + 112 q^{69} - 440 q^{71} - 448 q^{81} - 440 q^{89} - 144 q^{91} + 80 q^{99}+O(q^{100}) $ 16 * q - 56 * q^9 + 48 * q^11 + 40 * q^31 + 488 * q^49 + 32 * q^59 + 112 * q^69 - 440 * q^71 - 448 * q^81 - 440 * q^89 - 144 * q^91 + 80 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$551$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$		0			0
$2$		0			0
$3$			3.65160i			1.21720i	0.793478	+	0.608600i	$0.208268\pi$
$3$			3.65160i			1.21720i	−0.793478	+	0.608600i	$0.791732\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	9.09184			1.29883			0.649417	−	0.760432i	$-0.275013\pi$
$7$	9.09184			1.29883			0.649417	+	0.760432i	$0.275013\pi$
$8$		0			0
$9$	−4.33416			−0.481573
$10$		0			0
$11$	9.88767	−	4.82017i	0.898879	−	0.438198i
$11$	9.88767	−	4.82017i	0.898879	−	0.438198i
$12$		0			0
$13$	−22.5101			−1.73155			−0.865774	−	0.500436i	$-0.833173\pi$
$13$	−22.5101			−1.73155			−0.865774	+	0.500436i	$0.833173\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−22.9153			−1.34796			−0.673978	−	0.738751i	$-0.735416\pi$
$17$	−22.9153			−1.34796			−0.673978	+	0.738751i	$0.735416\pi$
$18$		0			0
$19$			16.0035i			0.842292i	0.906993	+	0.421146i	$0.138372\pi$
$19$			16.0035i			0.842292i	−0.906993	+	0.421146i	$0.861628\pi$
$20$		0			0
$21$			33.1997i			1.58094i
$22$		0			0
$23$			0.141697i			0.00616076i	0.999995	+	0.00308038i	$0.000980517\pi$
$23$			0.141697i			0.00616076i	−0.999995	+	0.00308038i	$0.999019\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			17.0378i			0.631028i
$28$		0			0
$29$			29.8270i			1.02852i	0.857636	+	0.514258i	$0.171933\pi$
$29$			29.8270i			1.02852i	−0.857636	+	0.514258i	$0.828067\pi$
$30$		0			0
$31$	21.5229			0.694287			0.347144	−	0.937812i	$-0.387152\pi$
$31$	21.5229			0.694287			0.347144	+	0.937812i	$0.387152\pi$
$32$		0			0
$33$	17.6013	+	36.1058i	0.533374	+	1.09411i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			28.0333i			0.757657i	0.925467	+	0.378829i	$0.123673\pi$
$37$			28.0333i			0.757657i	−0.925467	+	0.378829i	$0.876327\pi$
$38$		0			0
$39$		−	82.1979i		−	2.10764i
$40$		0			0
$41$			39.5629i			0.964950i	0.875910	+	0.482475i	$0.160262\pi$
$41$			39.5629i			0.964950i	−0.875910	+	0.482475i	$0.839738\pi$
$42$		0			0
$43$	21.4131			0.497979			0.248990	−	0.968506i	$-0.419902\pi$
$43$	21.4131			0.497979			0.248990	+	0.968506i	$0.419902\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			15.4239i			0.328167i	0.986446	+	0.164084i	$0.0524667\pi$
$47$			15.4239i			0.328167i	−0.986446	+	0.164084i	$0.947533\pi$
$48$		0			0
$49$	33.6616			0.686971
$50$		0			0
$51$		−	83.6773i		−	1.64073i
$52$		0			0
$53$			75.9763i			1.43352i	0.697322	+	0.716758i	$0.254375\pi$
$53$			75.9763i			1.43352i	−0.697322	+	0.716758i	$0.745625\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−58.4385			−1.02524
$58$		0			0
$59$	50.4169			0.854524			0.427262	−	0.904128i	$-0.359478\pi$
$59$	50.4169			0.854524			0.427262	+	0.904128i	$0.359478\pi$
$60$		0			0
$61$		−	98.4882i		−	1.61456i	−0.590169	−	0.807280i	$-0.700939\pi$
$61$		−	98.4882i		−	1.61456i	0.590169	−	0.807280i	$-0.299061\pi$
$62$		0			0
$63$	−39.4055			−0.625484
$64$		0			0
$65$		0			0
$66$		0			0
$67$			125.070i			1.86672i	0.358939	+	0.933361i	$0.383138\pi$
$67$			125.070i			1.86672i	−0.358939	+	0.933361i	$0.616862\pi$
$68$		0			0
$69$	−0.517422			−0.00749887
$70$		0			0
$71$	−100.615			−1.41711			−0.708556	−	0.705654i	$-0.750653\pi$
$71$	−100.615			−1.41711			−0.708556	+	0.705654i	$0.750653\pi$
$72$		0			0
$73$	−128.281			−1.75728			−0.878639	−	0.477487i	$-0.841548\pi$
$73$	−128.281			−1.75728			−0.878639	+	0.477487i	$0.841548\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	89.8971	−	43.8243i	1.16749	−	0.569146i
$78$		0			0
$79$			103.932i			1.31559i	0.753197	+	0.657795i	$0.228511\pi$
$79$			103.932i			1.31559i	−0.753197	+	0.657795i	$0.771489\pi$
$80$		0			0
$81$	−101.222			−1.24966
$82$		0			0
$83$	75.6774			0.911776			0.455888	−	0.890037i	$-0.349322\pi$
$83$	75.6774			0.911776			0.455888	+	0.890037i	$0.349322\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−108.916			−1.25191
$88$		0			0
$89$	58.9162			0.661980			0.330990	−	0.943634i	$-0.392617\pi$
$89$	58.9162			0.661980			0.330990	+	0.943634i	$0.392617\pi$
$90$		0			0
$91$	−204.658			−2.24899
$92$		0			0
$93$			78.5930i			0.845086i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	62.4540i		−	0.643856i	−0.946764	−	0.321928i	$-0.895669\pi$
$97$		−	62.4540i		−	0.643856i	0.946764	−	0.321928i	$-0.104331\pi$
$98$		0			0
$99$	−42.8547	+	20.8914i	−0.432876	+	0.211024i
$100$		0			0
$101$			61.1871i			0.605813i	0.953020	+	0.302906i	$0.0979570\pi$
$101$			61.1871i			0.605813i	−0.953020	+	0.302906i	$0.902043\pi$
$102$		0			0
$103$		−	46.9015i		−	0.455354i	−0.973737	−	0.227677i	$-0.926887\pi$
$103$		−	46.9015i		−	0.455354i	0.973737	−	0.227677i	$-0.0731130\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−66.2421			−0.619085			−0.309542	−	0.950886i	$-0.600176\pi$
$107$	−66.2421			−0.619085			−0.309542	+	0.950886i	$0.600176\pi$
$108$		0			0
$109$			18.2255i			0.167207i	0.996499	+	0.0836033i	$0.0266429\pi$
$109$			18.2255i			0.167207i	−0.996499	+	0.0836033i	$0.973357\pi$
$110$		0			0
$111$	−102.366			−0.922220
$112$		0			0
$113$			135.813i			1.20189i	0.799292	+	0.600943i	$0.205208\pi$
$113$			135.813i			1.20189i	−0.799292	+	0.600943i	$0.794792\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	97.5625			0.833867
$118$		0			0
$119$	−208.342			−1.75077
$120$		0			0
$121$	74.5318	−	95.3205i	0.615966	−	0.787773i
$122$		0			0
$123$	−144.468			−1.17454
$124$		0			0
$125$		0			0
$126$		0			0
$127$	143.106			1.12682			0.563410	−	0.826177i	$-0.309489\pi$
$127$	143.106			1.12682			0.563410	+	0.826177i	$0.309489\pi$
$128$		0			0
$129$			78.1920i			0.606140i
$130$		0			0
$131$		−	49.9179i		−	0.381053i	−0.981682	−	0.190526i	$-0.938981\pi$
$131$		−	49.9179i		−	0.381053i	0.981682	−	0.190526i	$-0.0610195\pi$
$132$		0			0
$133$			145.502i			1.09400i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	253.448i		−	1.84999i	−0.379981	−	0.924994i	$-0.624069\pi$
$137$		−	253.448i		−	1.84999i	0.379981	−	0.924994i	$-0.375931\pi$
$138$		0			0
$139$		−	189.435i		−	1.36284i	−0.731893	−	0.681419i	$-0.761363\pi$
$139$		−	189.435i		−	1.36284i	0.731893	−	0.681419i	$-0.238637\pi$
$140$		0			0
$141$	−56.3217			−0.399445
$142$		0			0
$143$	−222.572	+	108.503i	−1.55645	+	0.758760i
$144$		0			0
$145$		0			0
$146$		0			0
$147$			122.919i			0.836181i
$148$		0			0
$149$		−	163.116i		−	1.09474i	−0.836892	−	0.547368i	$-0.815630\pi$
$149$		−	163.116i		−	1.09474i	0.836892	−	0.547368i	$-0.184370\pi$
$150$		0			0
$151$		−	85.6125i		−	0.566970i	−0.958977	−	0.283485i	$-0.908509\pi$
$151$		−	85.6125i		−	0.566970i	0.958977	−	0.283485i	$-0.0914906\pi$
$152$		0			0
$153$	99.3184			0.649140
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	105.429i		−	0.671524i	−0.941947	−	0.335762i	$-0.891006\pi$
$157$		−	105.429i		−	0.671524i	0.941947	−	0.335762i	$-0.108994\pi$
$158$		0			0
$159$	−277.435			−1.74487
$160$		0			0
$161$			1.28829i			0.00800181i
$162$		0			0
$163$			99.9791i			0.613369i	0.951811	+	0.306684i	$0.0992196\pi$
$163$			99.9791i			0.613369i	−0.951811	+	0.306684i	$0.900780\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	13.6434			0.0816970			0.0408485	−	0.999165i	$-0.486994\pi$
$167$	13.6434			0.0816970			0.0408485	+	0.999165i	$0.486994\pi$
$168$		0			0
$169$	337.705			1.99826
$170$		0			0
$171$		−	69.3620i		−	0.405625i
$172$		0			0
$173$	−222.063			−1.28360			−0.641801	−	0.766872i	$-0.721812\pi$
$173$	−222.063			−1.28360			−0.641801	+	0.766872i	$0.721812\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			184.102i			1.04013i
$178$		0			0
$179$	13.9227			0.0777806			0.0388903	−	0.999243i	$-0.487618\pi$
$179$	13.9227			0.0777806			0.0388903	+	0.999243i	$0.487618\pi$
$180$		0			0
$181$	73.2171			0.404515			0.202257	−	0.979332i	$-0.435172\pi$
$181$	73.2171			0.404515			0.202257	+	0.979332i	$0.435172\pi$
$182$		0			0
$183$	359.639			1.96524
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−226.578	+	110.456i	−1.21165	+	0.590671i
$188$		0			0
$189$			154.905i			0.819601i
$190$		0			0
$191$	79.4081			0.415749			0.207875	−	0.978155i	$-0.433345\pi$
$191$	79.4081			0.415749			0.207875	+	0.978155i	$0.433345\pi$
$192$		0			0
$193$	140.214			0.726499			0.363250	−	0.931692i	$-0.381667\pi$
$193$	140.214			0.726499			0.363250	+	0.931692i	$0.381667\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	76.4700			0.388172			0.194086	−	0.980984i	$-0.437826\pi$
$197$	76.4700			0.388172			0.194086	+	0.980984i	$0.437826\pi$
$198$		0			0
$199$	−0.636328			−0.00319763			−0.00159881	−	0.999999i	$-0.500509\pi$
$199$	−0.636328			−0.00319763			−0.00159881	+	0.999999i	$0.500509\pi$
$200$		0			0
$201$	−456.707			−2.27217
$202$		0			0
$203$			271.182i			1.33587i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	0.614140i		−	0.00296686i
$208$		0			0
$209$	77.1399	+	158.238i	0.369090	+	0.757118i
$210$		0			0
$211$			169.636i			0.803962i	0.915648	+	0.401981i	$0.131678\pi$
$211$			169.636i			0.803962i	−0.915648	+	0.401981i	$0.868322\pi$
$212$		0			0
$213$		−	367.406i		−	1.72491i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	195.683			0.901764
$218$		0			0
$219$		−	468.432i		−	2.13896i
$220$		0			0
$221$	515.825			2.33405
$222$		0			0
$223$		−	149.796i		−	0.671731i	−0.941910	−	0.335865i	$-0.890971\pi$
$223$		−	149.796i		−	0.671731i	0.941910	−	0.335865i	$-0.109029\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−194.090			−0.855020			−0.427510	−	0.904011i	$-0.640609\pi$
$227$	−194.090			−0.855020			−0.427510	+	0.904011i	$0.640609\pi$
$228$		0			0
$229$	329.237			1.43772			0.718859	−	0.695156i	$-0.244665\pi$
$229$	329.237			1.43772			0.718859	+	0.695156i	$0.244665\pi$
$230$		0			0
$231$	160.029	+	328.268i	0.692764	+	1.42107i
$232$		0			0
$233$	66.5340			0.285554			0.142777	−	0.989755i	$-0.454397\pi$
$233$	66.5340			0.285554			0.142777	+	0.989755i	$0.454397\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−379.516			−1.60133
$238$		0			0
$239$			90.8230i			0.380012i	0.981783	+	0.190006i	$0.0608508\pi$
$239$			90.8230i			0.380012i	−0.981783	+	0.190006i	$0.939149\pi$
$240$		0			0
$241$		−	122.137i		−	0.506794i	−0.967362	−	0.253397i	$-0.918452\pi$
$241$		−	122.137i		−	0.506794i	0.967362	−	0.253397i	$-0.0815479\pi$
$242$		0			0
$243$		−	216.284i		−	0.890057i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	360.242i		−	1.45847i
$248$		0			0
$249$			276.343i			1.10981i
$250$		0			0
$251$	396.928			1.58138			0.790692	−	0.612214i	$-0.209721\pi$
$251$	396.928			1.58138			0.790692	+	0.612214i	$0.209721\pi$
$252$		0			0
$253$	0.683006	+	1.40106i	0.00269963	+	0.00553777i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			351.814i			1.36892i	0.729048	+	0.684462i	$0.239963\pi$
$257$			351.814i			1.36892i	−0.729048	+	0.684462i	$0.760037\pi$
$258$		0			0
$259$			254.874i			0.984071i
$260$		0			0
$261$		−	129.275i		−	0.495306i
$262$		0			0
$263$	−159.720			−0.607301			−0.303650	−	0.952784i	$-0.598205\pi$
$263$	−159.720			−0.607301			−0.303650	+	0.952784i	$0.598205\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			215.138i			0.805762i
$268$		0			0
$269$	−354.272			−1.31700			−0.658498	−	0.752582i	$-0.728808\pi$
$269$	−354.272			−1.31700			−0.658498	+	0.752582i	$0.728808\pi$
$270$		0			0
$271$			217.764i			0.803559i	0.915737	+	0.401779i	$0.131608\pi$
$271$			217.764i			0.803559i	−0.915737	+	0.401779i	$0.868392\pi$
$272$		0			0
$273$		−	747.330i		−	2.73747i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	425.599			1.53646			0.768230	−	0.640174i	$-0.221138\pi$
$277$	425.599			1.53646			0.768230	+	0.640174i	$0.221138\pi$
$278$		0			0
$279$	−93.2837			−0.334350
$280$		0			0
$281$			367.218i			1.30682i	0.757003	+	0.653412i	$0.226663\pi$
$281$			367.218i			1.30682i	−0.757003	+	0.653412i	$0.773337\pi$
$282$		0			0
$283$	424.761			1.50092			0.750461	−	0.660914i	$-0.229831\pi$
$283$	424.761			1.50092			0.750461	+	0.660914i	$0.229831\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			359.700i			1.25331i
$288$		0			0
$289$	236.109			0.816985
$290$		0			0
$291$	228.057			0.783701
$292$		0			0
$293$	50.0611			0.170857			0.0854285	−	0.996344i	$-0.472774\pi$
$293$	50.0611			0.170857			0.0854285	+	0.996344i	$0.472774\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	82.1250	+	168.464i	0.276515	+	0.567218i
$298$		0			0
$299$		−	3.18963i		−	0.0106676i
$300$		0			0
$301$	194.685			0.646793
$302$		0			0
$303$	−223.431			−0.737395
$304$		0			0
$305$		0			0
$306$		0			0
$307$	76.8244			0.250242			0.125121	−	0.992141i	$-0.460068\pi$
$307$	76.8244			0.250242			0.125121	+	0.992141i	$0.460068\pi$
$308$		0			0
$309$	171.265			0.554257
$310$		0			0
$311$	−566.867			−1.82272			−0.911361	−	0.411608i	$-0.864967\pi$
$311$	−566.867			−1.82272			−0.911361	+	0.411608i	$0.864967\pi$
$312$		0			0
$313$		−	40.8962i		−	0.130659i	−0.997864	−	0.0653294i	$-0.979190\pi$
$313$		−	40.8962i		−	0.130659i	0.997864	−	0.0653294i	$-0.0208098\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	179.622i		−	0.566632i	−0.959027	−	0.283316i	$-0.908566\pi$
$317$		−	179.622i		−	0.566632i	0.959027	−	0.283316i	$-0.0914345\pi$
$318$		0			0
$319$	143.771	+	294.919i	0.450693	+	0.924511i
$320$		0			0
$321$		−	241.889i		−	0.753549i
$322$		0			0
$323$		−	366.725i		−	1.13537i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−66.5523			−0.203524
$328$		0			0
$329$			140.231i			0.426235i
$330$		0			0
$331$	504.228			1.52335			0.761674	−	0.647960i	$-0.224378\pi$
$331$	504.228			1.52335			0.761674	+	0.647960i	$0.224378\pi$
$332$		0			0
$333$		−	121.501i		−	0.364868i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−437.742			−1.29894			−0.649468	−	0.760389i	$-0.725009\pi$
$337$	−437.742			−1.29894			−0.649468	+	0.760389i	$0.725009\pi$
$338$		0			0
$339$	−495.935			−1.46294
$340$		0			0
$341$	212.811	−	103.744i	0.624080	−	0.304235i
$342$		0			0
$343$	−139.454			−0.406572
$344$		0			0
$345$		0			0
$346$		0			0
$347$	194.366			0.560133			0.280067	−	0.959981i	$-0.409643\pi$
$347$	194.366			0.560133			0.280067	+	0.959981i	$0.409643\pi$
$348$		0			0
$349$		−	282.947i		−	0.810737i	−0.914153	−	0.405369i	$-0.867143\pi$
$349$		−	282.947i		−	0.810737i	0.914153	−	0.405369i	$-0.132857\pi$
$350$		0			0
$351$		−	383.522i		−	1.09266i
$352$		0			0
$353$			156.412i			0.443094i	0.975150	+	0.221547i	$0.0711106\pi$
$353$			156.412i			0.443094i	−0.975150	+	0.221547i	$0.928889\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		−	760.781i		−	2.13104i
$358$		0			0
$359$		−	614.980i		−	1.71304i	−0.516116	−	0.856518i	$-0.672623\pi$
$359$		−	614.980i		−	1.71304i	0.516116	−	0.856518i	$-0.327377\pi$
$360$		0			0
$361$	104.886			0.290544
$362$		0			0
$363$	348.072	+	272.160i	0.958877	+	0.749753i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	6.10499i		−	0.0166349i	−0.999965	−	0.00831743i	$-0.997352\pi$
$367$		−	6.10499i		−	0.0166349i	0.999965	−	0.00831743i	$-0.00264755\pi$
$368$		0			0
$369$		−	171.472i		−	0.464694i
$370$		0			0
$371$			690.765i			1.86190i
$372$		0			0
$373$	358.254			0.960467			0.480233	−	0.877141i	$-0.340552\pi$
$373$	358.254			0.960467			0.480233	+	0.877141i	$0.340552\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	671.408i		−	1.78092i
$378$		0			0
$379$	36.7025			0.0968405			0.0484202	−	0.998827i	$-0.484581\pi$
$379$	36.7025			0.0968405			0.0484202	+	0.998827i	$0.484581\pi$
$380$		0			0
$381$			522.566i			1.37157i
$382$		0			0
$383$		−	87.9982i		−	0.229760i	−0.993379	−	0.114880i	$-0.963352\pi$
$383$		−	87.9982i		−	0.229760i	0.993379	−	0.114880i	$-0.0366484\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−92.8078			−0.239814
$388$		0			0
$389$	−68.1918			−0.175300			−0.0876501	−	0.996151i	$-0.527936\pi$
$389$	−68.1918			−0.175300			−0.0876501	+	0.996151i	$0.527936\pi$
$390$		0			0
$391$		−	3.24703i		−	0.00830443i
$392$		0			0
$393$	182.280			0.463817
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	575.669i		−	1.45005i	−0.688723	−	0.725024i	$-0.741828\pi$
$397$		−	575.669i		−	1.45005i	0.688723	−	0.725024i	$-0.258172\pi$
$398$		0			0
$399$	−531.314			−1.33161
$400$		0			0
$401$	−119.702			−0.298509			−0.149255	−	0.988799i	$-0.547687\pi$
$401$	−119.702			−0.298509			−0.149255	+	0.988799i	$0.547687\pi$
$402$		0			0
$403$	−484.483			−1.20219
$404$		0			0
$405$		0			0
$406$		0			0
$407$	135.125	+	277.184i	0.332004	+	0.681042i
$408$		0			0
$409$		−	268.469i		−	0.656403i	−0.944608	−	0.328201i	$-0.893558\pi$
$409$		−	268.469i		−	0.656403i	0.944608	−	0.328201i	$-0.106442\pi$
$410$		0			0
$411$	925.492			2.25180
$412$		0			0
$413$	458.383			1.10989
$414$		0			0
$415$		0			0
$416$		0			0
$417$	691.739			1.65885
$418$		0			0
$419$	192.996			0.460610			0.230305	−	0.973118i	$-0.426028\pi$
$419$	192.996			0.460610			0.230305	+	0.973118i	$0.426028\pi$
$420$		0			0
$421$	−802.207			−1.90548			−0.952740	−	0.303788i	$-0.901749\pi$
$421$	−802.207			−1.90548			−0.952740	+	0.303788i	$0.901749\pi$
$422$		0			0
$423$		−	66.8495i		−	0.158037i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	895.439i		−	2.09705i
$428$		0			0
$429$	−396.208	−	812.745i	−0.923562	−	1.89451i
$430$		0			0
$431$		−	834.575i		−	1.93637i	−0.250240	−	0.968184i	$-0.580509\pi$
$431$		−	834.575i		−	1.93637i	0.250240	−	0.968184i	$-0.419491\pi$
$432$		0			0
$433$		−	678.057i		−	1.56595i	−0.622052	−	0.782976i	$-0.713701\pi$
$433$		−	678.057i		−	1.56595i	0.622052	−	0.782976i	$-0.286299\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2.26766			−0.00518916
$438$		0			0
$439$			562.372i			1.28103i	0.767946	+	0.640515i	$0.221279\pi$
$439$			562.372i			1.28103i	−0.767946	+	0.640515i	$0.778721\pi$
$440$		0			0
$441$	−145.895			−0.330827
$442$		0			0
$443$			597.796i			1.34943i	0.738080	+	0.674713i	$0.235733\pi$
$443$			597.796i			1.34943i	−0.738080	+	0.674713i	$0.764267\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	595.633			1.33251
$448$		0			0
$449$	244.096			0.543643			0.271821	−	0.962348i	$-0.412374\pi$
$449$	244.096			0.543643			0.271821	+	0.962348i	$0.412374\pi$
$450$		0			0
$451$	190.700	+	391.185i	0.422839	+	0.867373i
$452$		0			0
$453$	312.622			0.690116
$454$		0			0
$455$		0			0
$456$		0			0
$457$	457.420			1.00092			0.500460	−	0.865760i	$-0.333164\pi$
$457$	457.420			1.00092			0.500460	+	0.865760i	$0.333164\pi$
$458$		0			0
$459$		−	390.425i		−	0.850598i
$460$		0			0
$461$			501.488i			1.08783i	0.839141	+	0.543913i	$0.183058\pi$
$461$			501.488i			1.08783i	−0.839141	+	0.543913i	$0.816942\pi$
$462$		0			0
$463$		−	190.823i		−	0.412144i	−0.978537	−	0.206072i	$-0.933932\pi$
$463$		−	190.823i		−	0.412144i	0.978537	−	0.206072i	$-0.0660681\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			627.959i			1.34467i	0.740249	+	0.672333i	$0.234708\pi$
$467$			627.959i			1.34467i	−0.740249	+	0.672333i	$0.765292\pi$
$468$		0			0
$469$			1137.12i			2.42456i
$470$		0			0
$471$	384.985			0.817379
$472$		0			0
$473$	211.726	−	103.215i	0.447623	−	0.218213i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	329.294i		−	0.690343i
$478$		0			0
$479$			257.075i			0.536691i	0.963323	+	0.268345i	$0.0864768\pi$
$479$			257.075i			0.536691i	−0.963323	+	0.268345i	$0.913523\pi$
$480$		0			0
$481$		−	631.033i		−	1.31192i
$482$		0			0
$483$	−4.70432			−0.00973979
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	387.203i		−	0.795079i	−0.917585	−	0.397539i	$-0.869864\pi$
$487$		−	387.203i		−	0.795079i	0.917585	−	0.397539i	$-0.130136\pi$
$488$		0			0
$489$	−365.084			−0.746592
$490$		0			0
$491$		−	905.259i		−	1.84370i	−0.387542	−	0.921852i	$-0.626676\pi$
$491$		−	905.259i		−	1.84370i	0.387542	−	0.921852i	$-0.373324\pi$
$492$		0			0
$493$		−	683.492i		−	1.38639i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−914.776			−1.84060
$498$		0			0
$499$	330.636			0.662597			0.331298	−	0.943526i	$-0.392513\pi$
$499$	330.636			0.662597			0.331298	+	0.943526i	$0.392513\pi$
$500$		0			0
$501$			49.8202i			0.0994415i
$502$		0			0
$503$	−390.813			−0.776964			−0.388482	−	0.921456i	$-0.627000\pi$
$503$	−390.813			−0.776964			−0.388482	+	0.921456i	$0.627000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			1233.16i			2.43228i
$508$		0			0
$509$	−161.773			−0.317825			−0.158912	−	0.987293i	$-0.550799\pi$
$509$	−161.773			−0.317825			−0.158912	+	0.987293i	$0.550799\pi$
$510$		0			0
$511$	−1166.31			−2.28241
$512$		0			0
$513$	−272.665			−0.531510
$514$		0			0
$515$		0			0
$516$		0			0
$517$	74.3457	+	152.506i	0.143802	+	0.294982i
$518$		0			0
$519$		−	810.885i		−	1.56240i
$520$		0			0
$521$	49.8047			0.0955944			0.0477972	−	0.998857i	$-0.484780\pi$
$521$	49.8047			0.0955944			0.0477972	+	0.998857i	$0.484780\pi$
$522$		0			0
$523$	−750.487			−1.43497			−0.717483	−	0.696576i	$-0.754706\pi$
$523$	−750.487			−1.43497			−0.717483	+	0.696576i	$0.754706\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−493.203			−0.935869
$528$		0			0
$529$	528.980			0.999962
$530$		0			0
$531$	−218.515			−0.411516
$532$		0			0
$533$		−	890.566i		−	1.67086i
$534$		0			0
$535$		0			0
$536$		0			0
$537$			50.8402i			0.0946744i
$538$		0			0
$539$	332.835	−	162.255i	0.617504	−	0.301029i
$540$		0			0
$541$			749.510i			1.38541i	0.721219	+	0.692707i	$0.243582\pi$
$541$			749.510i			1.38541i	−0.721219	+	0.692707i	$0.756418\pi$
$542$		0			0
$543$			267.359i			0.492375i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−357.118			−0.652867			−0.326433	−	0.945220i	$-0.605847\pi$
$547$	−357.118			−0.652867			−0.326433	+	0.945220i	$0.605847\pi$
$548$		0			0
$549$			426.864i			0.777529i
$550$		0			0
$551$	−477.337			−0.866311
$552$		0			0
$553$			944.929i			1.70873i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	555.491			0.997291			0.498646	−	0.866806i	$-0.333831\pi$
$557$	555.491			0.997291			0.498646	+	0.866806i	$0.333831\pi$
$558$		0			0
$559$	−482.011			−0.862275
$560$		0			0
$561$	−403.339	−	827.373i	−0.718964	−	1.47482i
$562$		0			0
$563$	618.559			1.09868			0.549342	−	0.835598i	$-0.314878\pi$
$563$	618.559			1.09868			0.549342	+	0.835598i	$0.314878\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−920.299			−1.62310
$568$		0			0
$569$			873.411i			1.53499i	0.641054	+	0.767496i	$0.278498\pi$
$569$			873.411i			1.53499i	−0.641054	+	0.767496i	$0.721502\pi$
$570$		0			0
$571$			1073.95i			1.88082i	0.340046	+	0.940409i	$0.389557\pi$
$571$			1073.95i			1.88082i	−0.340046	+	0.940409i	$0.610443\pi$
$572$		0			0
$573$			289.966i			0.506049i
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	136.957i		−	0.237360i	−0.992933	−	0.118680i	$-0.962134\pi$
$577$		−	136.957i		−	0.237360i	0.992933	−	0.118680i	$-0.0378663\pi$
$578$		0			0
$579$			512.006i			0.884294i
$580$		0			0
$581$	688.047			1.18425
$582$		0			0
$583$	366.219	+	751.229i	0.628163	+	1.28856i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			531.023i			0.904639i	0.891856	+	0.452319i	$0.149403\pi$
$587$			531.023i			0.904639i	−0.891856	+	0.452319i	$0.850597\pi$
$588$		0			0
$589$			344.443i			0.584792i
$590$		0			0
$591$			279.237i			0.472483i
$592$		0			0
$593$	−903.516			−1.52364			−0.761818	−	0.647791i	$-0.775693\pi$
$593$	−903.516			−1.52364			−0.761818	+	0.647791i	$0.775693\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	2.32361i		−	0.00389215i
$598$		0			0
$599$	800.473			1.33635			0.668174	−	0.744005i	$-0.267076\pi$
$599$	800.473			1.33635			0.668174	+	0.744005i	$0.267076\pi$
$600$		0			0
$601$			274.107i			0.456085i	0.973651	+	0.228043i	$0.0732325\pi$
$601$			274.107i			0.456085i	−0.973651	+	0.228043i	$0.926767\pi$
$602$		0			0
$603$		−	542.075i		−	0.898964i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	591.864			0.975064			0.487532	−	0.873105i	$-0.337897\pi$
$607$	591.864			0.975064			0.487532	+	0.873105i	$0.337897\pi$
$608$		0			0
$609$	−990.247			−1.62602
$610$		0			0
$611$		−	347.193i		−	0.568237i
$612$		0			0
$613$	6.59994			0.0107666			0.00538331	−	0.999986i	$-0.498286\pi$
$613$	6.59994			0.0107666			0.00538331	+	0.999986i	$0.498286\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			719.824i			1.16665i	0.812238	+	0.583326i	$0.198249\pi$
$617$			719.824i			1.16665i	−0.812238	+	0.583326i	$0.801751\pi$
$618$		0			0
$619$	106.000			0.171244			0.0856219	−	0.996328i	$-0.472712\pi$
$619$	106.000			0.171244			0.0856219	+	0.996328i	$0.472712\pi$
$620$		0			0
$621$	−2.41421			−0.00388761
$622$		0			0
$623$	535.657			0.859803
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−577.820	+	281.684i	−0.921564	+	0.449256i
$628$		0			0
$629$		−	642.390i		−	1.02129i
$630$		0			0
$631$	651.103			1.03186			0.515930	−	0.856631i	$-0.327447\pi$
$631$	651.103			1.03186			0.515930	+	0.856631i	$0.327447\pi$
$632$		0			0
$633$	−619.443			−0.978582
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−757.727			−1.18952
$638$		0			0
$639$	436.082			0.682444
$640$		0			0
$641$	−205.651			−0.320828			−0.160414	−	0.987050i	$-0.551283\pi$
$641$	−205.651			−0.320828			−0.160414	+	0.987050i	$0.551283\pi$
$642$		0			0
$643$		−	612.134i		−	0.951997i	−0.879446	−	0.475998i	$-0.842087\pi$
$643$		−	612.134i		−	0.951997i	0.879446	−	0.475998i	$-0.157913\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	39.0151i		−	0.0603015i	−0.999545	−	0.0301508i	$-0.990401\pi$
$647$		−	39.0151i		−	0.0603015i	0.999545	−	0.0301508i	$-0.00959874\pi$
$648$		0			0
$649$	498.506	−	243.018i	0.768114	−	0.374450i
$650$		0			0
$651$			714.555i			1.09763i
$652$		0			0
$653$			48.3743i			0.0740801i	0.999314	+	0.0370400i	$0.0117929\pi$
$653$			48.3743i			0.0740801i	−0.999314	+	0.0370400i	$0.988207\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	555.992			0.846258
$658$		0			0
$659$			546.722i			0.829624i	0.909907	+	0.414812i	$0.136153\pi$
$659$			546.722i			0.829624i	−0.909907	+	0.414812i	$0.863847\pi$
$660$		0			0
$661$	592.309			0.896080			0.448040	−	0.894014i	$-0.352122\pi$
$661$	592.309			0.896080			0.448040	+	0.894014i	$0.352122\pi$
$662$		0			0
$663$			1883.58i			2.84100i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−4.22640			−0.00633644
$668$		0			0
$669$	546.994			0.817630
$670$		0			0
$671$	−474.730	−	973.818i	−0.707496	−	1.45129i
$672$		0			0
$673$	566.186			0.841287			0.420644	−	0.907226i	$-0.361804\pi$
$673$	566.186			0.841287			0.420644	+	0.907226i	$0.361804\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	728.381			1.07590			0.537948	−	0.842978i	$-0.319200\pi$
$677$	728.381			1.07590			0.537948	+	0.842978i	$0.319200\pi$
$678$		0			0
$679$		−	567.822i		−	0.836263i
$680$		0			0
$681$		−	708.737i		−	1.04073i
$682$		0			0
$683$			20.6410i			0.0302211i	0.999886	+	0.0151106i	$0.00481002\pi$
$683$			20.6410i			0.0302211i	−0.999886	+	0.0151106i	$0.995190\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$			1202.24i			1.74999i
$688$		0			0
$689$		−	1710.24i		−	2.48220i
$690$		0			0
$691$	727.350			1.05260			0.526302	−	0.850298i	$-0.323578\pi$
$691$	727.350			1.05260			0.526302	+	0.850298i	$0.323578\pi$
$692$		0			0
$693$	−389.628	+	189.941i	−0.562234	+	0.274086i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	906.595i		−	1.30071i
$698$		0			0
$699$			242.955i			0.347576i
$700$		0			0
$701$		−	355.389i		−	0.506975i	−0.967339	−	0.253487i	$-0.918422\pi$
$701$		−	355.389i		−	0.506975i	0.967339	−	0.253487i	$-0.0815776\pi$
$702$		0			0
$703$	−448.632			−0.638169
$704$		0			0
$705$		0			0
$706$		0			0
$707$			556.304i			0.786851i
$708$		0			0
$709$	538.342			0.759297			0.379649	−	0.925131i	$-0.376045\pi$
$709$	538.342			0.759297			0.379649	+	0.925131i	$0.376045\pi$
$710$		0			0
$711$		−	450.456i		−	0.633553i
$712$		0			0
$713$			3.04974i			0.00427734i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−331.649			−0.462551
$718$		0			0
$719$	283.789			0.394699			0.197350	−	0.980333i	$-0.436767\pi$
$719$	283.789			0.394699			0.197350	+	0.980333i	$0.436767\pi$
$720$		0			0
$721$		−	426.421i		−	0.591430i
$722$		0			0
$723$	445.997			0.616870
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	875.176i		−	1.20382i	−0.798565	−	0.601909i	$-0.794407\pi$
$727$		−	875.176i		−	1.20382i	0.798565	−	0.601909i	$-0.205593\pi$
$728$		0			0
$729$	−121.221			−0.166284
$730$		0			0
$731$	−490.687			−0.671254
$732$		0			0
$733$	423.716			0.578057			0.289028	−	0.957321i	$-0.406668\pi$
$733$	423.716			0.578057			0.289028	+	0.957321i	$0.406668\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	602.861	+	1236.65i	0.817993	+	1.67796i
$738$		0			0
$739$			1189.52i			1.60964i	0.593520	+	0.804819i	$0.297738\pi$
$739$			1189.52i			1.60964i	−0.593520	+	0.804819i	$0.702262\pi$
$740$		0			0
$741$	1315.46			1.77525
$742$		0			0
$743$	185.550			0.249731			0.124865	−	0.992174i	$-0.460150\pi$
$743$	185.550			0.249731			0.124865	+	0.992174i	$0.460150\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−327.998			−0.439087
$748$		0			0
$749$	−602.262			−0.804089
$750$		0			0
$751$	−495.474			−0.659753			−0.329876	−	0.944024i	$-0.607007\pi$
$751$	−495.474			−0.659753			−0.329876	+	0.944024i	$0.607007\pi$
$752$		0			0
$753$			1449.42i			1.92486i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	183.782i		−	0.242777i	−0.992605	−	0.121388i	$-0.961265\pi$
$757$		−	183.782i		−	0.242777i	0.992605	−	0.121388i	$-0.0387347\pi$
$758$		0			0
$759$	−5.11610	+	2.49406i	−0.00674057	+	0.00328599i
$760$		0			0
$761$			902.350i			1.18574i	0.805298	+	0.592871i	$0.202006\pi$
$761$			902.350i			1.18574i	−0.805298	+	0.592871i	$0.797994\pi$
$762$		0			0
$763$			165.704i			0.217174i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1134.89			−1.47965
$768$		0			0
$769$		−	1009.09i		−	1.31221i	−0.754668	−	0.656107i	$-0.772202\pi$
$769$		−	1009.09i		−	1.31221i	0.754668	−	0.656107i	$-0.227798\pi$
$770$		0			0
$771$	−1284.68			−1.66625
$772$		0			0
$773$			645.499i			0.835057i	0.908664	+	0.417529i	$0.137104\pi$
$773$			645.499i			0.835057i	−0.908664	+	0.417529i	$0.862896\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−930.699			−1.19781
$778$		0			0
$779$	−633.147			−0.812770
$780$		0			0
$781$	−994.848	+	484.982i	−1.27381	+	0.620976i
$782$		0			0
$783$	−508.185			−0.649023
$784$		0			0
$785$		0			0
$786$		0			0
$787$	736.556			0.935903			0.467951	−	0.883754i	$-0.344992\pi$
$787$	736.556			0.935903			0.467951	+	0.883754i	$0.344992\pi$
$788$		0			0
$789$		−	583.233i		−	0.739206i
$790$		0			0
$791$			1234.79i			1.56105i
$792$		0			0
$793$			2216.98i			2.79569i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	293.407i		−	0.368139i	−0.982913	−	0.184070i	$-0.941073\pi$
$797$		−	293.407i		−	0.368139i	0.982913	−	0.184070i	$-0.0589272\pi$
$798$		0			0
$799$		−	353.441i		−	0.442355i
$800$		0			0
$801$	−255.352			−0.318792
$802$		0			0
$803$	−1268.40	+	618.338i	−1.57958	+	0.770035i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	1293.66i		−	1.60305i
$808$		0			0
$809$			1076.81i			1.33104i	0.746379	+	0.665522i	$0.231791\pi$
$809$			1076.81i			1.33104i	−0.746379	+	0.665522i	$0.768209\pi$
$810$		0			0
$811$		−	801.997i		−	0.988898i	−0.869207	−	0.494449i	$-0.835370\pi$
$811$		−	801.997i		−	0.988898i	0.869207	−	0.494449i	$-0.164630\pi$
$812$		0			0
$813$	−795.188			−0.978091
$814$		0			0
$815$		0			0
$816$		0			0
$817$			342.686i			0.419444i
$818$		0			0
$819$	887.023			1.08306
$820$		0			0
$821$		−	644.550i		−	0.785079i	−0.919735	−	0.392540i	$-0.871597\pi$
$821$		−	644.550i		−	0.785079i	0.919735	−	0.392540i	$-0.128403\pi$
$822$		0			0
$823$		−	1390.56i		−	1.68962i	−0.535065	−	0.844811i	$-0.679713\pi$
$823$		−	1390.56i		−	1.68962i	0.535065	−	0.844811i	$-0.320287\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	71.7290			0.0867340			0.0433670	−	0.999059i	$-0.486192\pi$
$827$	71.7290			0.0867340			0.0433670	+	0.999059i	$0.486192\pi$
$828$		0			0
$829$	−1121.23			−1.35251			−0.676255	−	0.736668i	$-0.736398\pi$
$829$	−1121.23			−1.35251			−0.676255	+	0.736668i	$0.736398\pi$
$830$		0			0
$831$			1554.12i			1.87018i
$832$		0			0
$833$	−771.364			−0.926007
$834$		0			0
$835$		0			0
$836$		0			0
$837$			366.702i			0.438115i
$838$		0			0
$839$	−1527.08			−1.82012			−0.910059	−	0.414478i	$-0.863964\pi$
$839$	−1527.08			−1.82012			−0.910059	+	0.414478i	$0.863964\pi$
$840$		0			0
$841$	−48.6473			−0.0578446
$842$		0			0
$843$	−1340.93			−1.59066
$844$		0			0
$845$		0			0
$846$		0			0
$847$	677.632	−	866.639i	0.800037	−	1.02319i
$848$		0			0
$849$			1551.06i			1.82692i
$850$		0			0
$851$	−3.97225			−0.00466774
$852$		0			0
$853$	−350.340			−0.410715			−0.205357	−	0.978687i	$-0.565836\pi$
$853$	−350.340			−0.410715			−0.205357	+	0.978687i	$0.565836\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1137.89			−1.32776			−0.663880	−	0.747839i	$-0.731091\pi$
$857$	−1137.89			−1.32776			−0.663880	+	0.747839i	$0.731091\pi$
$858$		0			0
$859$	−1662.27			−1.93512			−0.967562	−	0.252635i	$-0.918703\pi$
$859$	−1662.27			−1.93512			−0.967562	+	0.252635i	$0.918703\pi$
$860$		0			0
$861$	−1313.48			−1.52553
$862$		0			0
$863$		−	278.778i		−	0.323033i	−0.986870	−	0.161517i	$-0.948361\pi$
$863$		−	278.778i		−	0.323033i	0.986870	−	0.161517i	$-0.0516385\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			862.174i			0.994434i
$868$		0			0
$869$	500.968	+	1027.64i	0.576488	+	1.18256i
$870$		0			0
$871$		−	2815.35i		−	3.23232i
$872$		0			0
$873$			270.686i			0.310064i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1493.74			1.70324			0.851620	−	0.524159i	$-0.175620\pi$
$877$	1493.74			1.70324			0.851620	+	0.524159i	$0.175620\pi$
$878$		0			0
$879$			182.803i			0.207967i
$880$		0			0
$881$	−814.863			−0.924929			−0.462465	−	0.886638i	$-0.653035\pi$
$881$	−814.863			−0.924929			−0.462465	+	0.886638i	$0.653035\pi$
$882$		0			0
$883$			136.873i			0.155009i	0.996992	+	0.0775046i	$0.0246952\pi$
$883$			136.873i			0.155009i	−0.996992	+	0.0775046i	$0.975305\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	274.810			0.309820			0.154910	−	0.987929i	$-0.450491\pi$
$887$	274.810			0.309820			0.154910	+	0.987929i	$0.450491\pi$
$888$		0			0
$889$	1301.10			1.46355
$890$		0			0
$891$	−1000.85	+	487.910i	−1.12329	+	0.547598i
$892$		0			0
$893$	−246.836			−0.276412
$894$		0			0
$895$		0			0
$896$		0			0
$897$	11.6472			0.0129846
$898$		0			0
$899$			641.963i			0.714085i
$900$		0			0
$901$		−	1741.02i		−	1.93232i
$902$		0			0
$903$			710.910i			0.787275i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			471.112i			0.519417i	0.965687	+	0.259709i	$0.0836265\pi$
$907$			471.112i			0.519417i	−0.965687	+	0.259709i	$0.916373\pi$
$908$		0			0
$909$		−	265.195i		−	0.291743i
$910$		0			0
$911$	−725.325			−0.796185			−0.398093	−	0.917345i	$-0.630328\pi$
$911$	−725.325			−0.796185			−0.398093	+	0.917345i	$0.630328\pi$
$912$		0			0
$913$	748.273	−	364.778i	0.819576	−	0.399538i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	453.846i		−	0.494925i
$918$		0			0
$919$		−	598.323i		−	0.651058i	−0.945532	−	0.325529i	$-0.894458\pi$
$919$		−	598.323i		−	0.651058i	0.945532	−	0.325529i	$-0.105542\pi$
$920$		0			0
$921$			280.532i			0.304595i
$922$		0			0
$923$	2264.86			2.45380
$924$		0			0
$925$		0			0
$926$		0			0
$927$			203.279i			0.219286i
$928$		0			0
$929$	−1189.14			−1.28002			−0.640008	−	0.768368i	$-0.721069\pi$
$929$	−1189.14			−1.28002			−0.640008	+	0.768368i	$0.721069\pi$
$930$		0			0
$931$			538.705i			0.578631i
$932$		0			0
$933$		−	2069.97i		−	2.21862i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1.72183			−0.00183759			−0.000918797	−	1.00000i	$-0.500292\pi$
$937$	−1.72183			−0.00183759			−0.000918797		1.00000i	$0.500292\pi$
$938$		0			0
$939$	149.336			0.159038
$940$		0			0
$941$		−	877.355i		−	0.932364i	−0.884689	−	0.466182i	$-0.845629\pi$
$941$		−	877.355i		−	0.932364i	0.884689	−	0.466182i	$-0.154371\pi$
$942$		0			0
$943$	−5.60597			−0.00594482
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	517.593i		−	0.546561i	−0.961934	−	0.273281i	$-0.911891\pi$
$947$		−	517.593i		−	0.546561i	0.961934	−	0.273281i	$-0.0881088\pi$
$948$		0			0
$949$	2887.63			3.04281
$950$		0			0
$951$	655.908			0.689703
$952$		0			0
$953$	477.486			0.501034			0.250517	−	0.968112i	$-0.419399\pi$
$953$	477.486			0.501034			0.250517	+	0.968112i	$0.419399\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−1076.93	+	524.994i	−1.12531	+	0.548583i
$958$		0			0
$959$		−	2304.31i		−	2.40283i
$960$		0			0
$961$	−497.765			−0.517965
$962$		0			0
$963$	287.104			0.298135
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1766.47			1.82675			0.913376	−	0.407118i	$-0.133466\pi$
$967$	1766.47			1.82675			0.913376	+	0.407118i	$0.133466\pi$
$968$		0			0
$969$	1339.13			1.38197
$970$		0			0
$971$	487.722			0.502289			0.251144	−	0.967950i	$-0.419193\pi$
$971$	487.722			0.502289			0.251144	+	0.967950i	$0.419193\pi$
$972$		0			0
$973$		−	1722.31i		−	1.77010i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			150.791i			0.154341i	0.997018	+	0.0771704i	$0.0245886\pi$
$977$			150.791i			0.154341i	−0.997018	+	0.0771704i	$0.975411\pi$
$978$		0			0
$979$	582.544	−	283.987i	0.595040	−	0.290078i
$980$		0			0
$981$		−	78.9923i		−	0.0805223i
$982$		0			0
$983$		−	1331.95i		−	1.35499i	−0.735528	−	0.677494i	$-0.763066\pi$
$983$		−	1331.95i		−	1.35499i	0.735528	−	0.677494i	$-0.236934\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−512.068			−0.518813
$988$		0			0
$989$			3.03418i			0.00306793i
$990$		0			0
$991$	−780.611			−0.787700			−0.393850	−	0.919175i	$-0.628857\pi$
$991$	−780.611			−0.787700			−0.393850	+	0.919175i	$0.628857\pi$
$992$		0			0
$993$			1841.24i			1.85422i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	263.034			0.263825			0.131913	−	0.991261i	$-0.457888\pi$
$997$	263.034			0.263825			0.131913	+	0.991261i	$0.457888\pi$
$998$		0			0
$999$	−477.625			−0.478103

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1100.3.e.b.549.14		16
5.2	odd	4		220.3.f.a.21.8	yes	8
5.3	odd	4		1100.3.f.f.901.1		8
5.4	even	2	inner	1100.3.e.b.549.3		16
11.10	odd	2	inner	1100.3.e.b.549.13		16
15.2	even	4		1980.3.b.a.901.4		8
20.7	even	4		880.3.j.b.241.1		8
55.32	even	4		220.3.f.a.21.7	✓	8
55.43	even	4		1100.3.f.f.901.2		8
55.54	odd	2	inner	1100.3.e.b.549.4		16
165.32	odd	4		1980.3.b.a.901.1		8
220.87	odd	4		880.3.j.b.241.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.3.f.a.21.7	✓	8	55.32	even	4
220.3.f.a.21.8	yes	8	5.2	odd	4
880.3.j.b.241.1		8	20.7	even	4
880.3.j.b.241.2		8	220.87	odd	4
1100.3.e.b.549.3		16	5.4	even	2	inner
1100.3.e.b.549.4		16	55.54	odd	2	inner
1100.3.e.b.549.13		16	11.10	odd	2	inner
1100.3.e.b.549.14		16	1.1	even	1	trivial
1100.3.f.f.901.1		8	5.3	odd	4
1100.3.f.f.901.2		8	55.43	even	4
1980.3.b.a.901.1		8	165.32	odd	4
1980.3.b.a.901.4		8	15.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1100.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.65160i q^{3} +9.09184 q^{7} -4.33416 q^{9} +O(q^{10})\)	\(q+3.65160i q^{3} +9.09184 q^{7} -4.33416 q^{9} +(9.88767 - 4.82017i) q^{11} -22.5101 q^{13} -22.9153 q^{17} +16.0035i q^{19} +33.1997i q^{21} +0.141697i q^{23} +17.0378i q^{27} +29.8270i q^{29} +21.5229 q^{31} +(17.6013 + 36.1058i) q^{33} +28.0333i q^{37} -82.1979i q^{39} +39.5629i q^{41} +21.4131 q^{43} +15.4239i q^{47} +33.6616 q^{49} -83.6773i q^{51} +75.9763i q^{53} -58.4385 q^{57} +50.4169 q^{59} -98.4882i q^{61} -39.4055 q^{63} +125.070i q^{67} -0.517422 q^{69} -100.615 q^{71} -128.281 q^{73} +(89.8971 - 43.8243i) q^{77} +103.932i q^{79} -101.222 q^{81} +75.6774 q^{83} -108.916 q^{87} +58.9162 q^{89} -204.658 q^{91} +78.5930i q^{93} -62.4540i q^{97} +(-42.8547 + 20.8914i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 56 q^{9}+O(q^{10}) \) 16 * q - 56 * q^9	\( 16 q - 56 q^{9} + 48 q^{11} + 40 q^{31} + 488 q^{49} + 32 q^{59} + 112 q^{69} - 440 q^{71} - 448 q^{81} - 440 q^{89} - 144 q^{91} + 80 q^{99}+O(q^{100}) \) 16 * q - 56 * q^9 + 48 * q^11 + 40 * q^31 + 488 * q^49 + 32 * q^59 + 112 * q^69 - 440 * q^71 - 448 * q^81 - 440 * q^89 - 144 * q^91 + 80 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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