LMFDB - Embedded newform 2200.2.b.g.1849.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2200,2,Mod(1849,2200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2200.1849"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2200 = 2^{3} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2200.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-6,0,-4,0,0,0,0,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$17.5670884447$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 88)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.3
Root			$1.56155i$ of defining polynomial
Character	$\chi$	$=$	2200.1849
Dual form			2200.2.b.g.1849.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155i q^{3} +3.12311i q^{7} +0.561553 q^{9} -1.00000 q^{11} +5.12311i q^{13} +2.00000i q^{17} +4.00000 q^{19} -4.87689 q^{21} -2.43845i q^{23} +5.56155i q^{27} +5.12311 q^{29} -5.56155 q^{31} -1.56155i q^{33} -7.56155i q^{37} -8.00000 q^{39} -1.12311 q^{41} +7.12311i q^{43} +8.00000i q^{47} -2.75379 q^{49} -3.12311 q^{51} -12.2462i q^{53} +6.24621i q^{57} -7.80776 q^{59} +1.12311 q^{61} +1.75379i q^{63} +9.56155i q^{67} +3.80776 q^{69} -8.68466 q^{71} -5.12311i q^{73} -3.12311i q^{77} +11.1231 q^{79} -7.00000 q^{81} -0.876894i q^{83} +8.00000i q^{87} -2.68466 q^{89} -16.0000 q^{91} -8.68466i q^{93} +15.5616i q^{97} -0.561553 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{9} - 4 q^{11} + 16 q^{19} - 36 q^{21} + 4 q^{29} - 14 q^{31} - 32 q^{39} + 12 q^{41} - 44 q^{49} + 4 q^{51} + 10 q^{59} - 12 q^{61} - 26 q^{69} - 10 q^{71} + 28 q^{79} - 28 q^{81} + 14 q^{89}+ \cdots + 6 q^{99}+O(q^{100}) $ 4 * q - 6 * q^9 - 4 * q^11 + 16 * q^19 - 36 * q^21 + 4 * q^29 - 14 * q^31 - 32 * q^39 + 12 * q^41 - 44 * q^49 + 4 * q^51 + 10 * q^59 - 12 * q^61 - 26 * q^69 - 10 * q^71 + 28 * q^79 - 28 * q^81 + 14 * q^89 - 64 * q^91 + 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2200\mathbb{Z}\right)^\times$.

$n$	$177$	$551$	$1101$	$1201$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	1.56155i	0.901563i	0.892634	+	0.450781i	$0.148855\pi$
$3$	1.56155i	0.901563i	−0.892634	+	0.450781i	$0.851145\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.12311i	1.18042i	0.807249	+	0.590211i	$0.200956\pi$
$7$	3.12311i	1.18042i	−0.807249	+	0.590211i	$0.799044\pi$
$8$	0	0
$9$	0.561553	0.187184
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	5.12311i	1.42089i	0.703751	+	0.710447i	$0.251507\pi$
$13$	5.12311i	1.42089i	−0.703751	+	0.710447i	$0.748493\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	−4.87689	−1.06423
$22$	0	0
$23$	− 2.43845i	− 0.508451i	−0.967145	−	0.254226i	$-0.918179\pi$
$23$	− 2.43845i	− 0.508451i	0.967145	−	0.254226i	$-0.0818206\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.56155i	1.07032i
$28$	0	0
$29$	5.12311	0.951337	0.475668	−	0.879625i	$-0.342206\pi$
$29$	5.12311	0.951337	0.475668	+	0.879625i	$0.342206\pi$
$30$	0	0
$31$	−5.56155	−0.998884	−0.499442	−	0.866347i	$-0.666462\pi$
$31$	−5.56155	−0.998884	−0.499442	+	0.866347i	$0.666462\pi$
$32$	0	0
$33$	− 1.56155i	− 0.271831i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 7.56155i	− 1.24311i	−0.783370	−	0.621556i	$-0.786501\pi$
$37$	− 7.56155i	− 1.24311i	0.783370	−	0.621556i	$-0.213499\pi$
$38$	0	0
$39$	−8.00000	−1.28103
$40$	0	0
$41$	−1.12311	−0.175400	−0.0876998	−	0.996147i	$-0.527952\pi$
$41$	−1.12311	−0.175400	−0.0876998	+	0.996147i	$0.527952\pi$
$42$	0	0
$43$	7.12311i	1.08626i	0.839648	+	0.543132i	$0.182762\pi$
$43$	7.12311i	1.08626i	−0.839648	+	0.543132i	$0.817238\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.00000i	1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$	8.00000i	1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$	0	0
$49$	−2.75379	−0.393398
$50$	0	0
$51$	−3.12311	−0.437322
$52$	0	0
$53$	− 12.2462i	− 1.68215i	−0.540921	−	0.841073i	$-0.681924\pi$
$53$	− 12.2462i	− 1.68215i	0.540921	−	0.841073i	$-0.318076\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.24621i	0.827331i
$58$	0	0
$59$	−7.80776	−1.01648	−0.508242	−	0.861214i	$-0.669705\pi$
$59$	−7.80776	−1.01648	−0.508242	+	0.861214i	$0.669705\pi$
$60$	0	0
$61$	1.12311	0.143799	0.0718995	−	0.997412i	$-0.477094\pi$
$61$	1.12311	0.143799	0.0718995	+	0.997412i	$0.477094\pi$
$62$	0	0
$63$	1.75379i	0.220957i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.56155i	1.16813i	0.811707	+	0.584065i	$0.198539\pi$
$67$	9.56155i	1.16813i	−0.811707	+	0.584065i	$0.801461\pi$
$68$	0	0
$69$	3.80776	0.458401
$70$	0	0
$71$	−8.68466	−1.03068	−0.515340	−	0.856986i	$-0.672334\pi$
$71$	−8.68466	−1.03068	−0.515340	+	0.856986i	$0.672334\pi$
$72$	0	0
$73$	− 5.12311i	− 0.599614i	−0.954000	−	0.299807i	$-0.903078\pi$
$73$	− 5.12311i	− 0.599614i	0.954000	−	0.299807i	$-0.0969223\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.12311i	− 0.355911i
$78$	0	0
$79$	11.1231	1.25145	0.625724	−	0.780045i	$-0.284804\pi$
$79$	11.1231	1.25145	0.625724	+	0.780045i	$0.284804\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	− 0.876894i	− 0.0962517i	−0.998841	−	0.0481258i	$-0.984675\pi$
$83$	− 0.876894i	− 0.0962517i	0.998841	−	0.0481258i	$-0.0153248\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8.00000i	0.857690i
$88$	0	0
$89$	−2.68466	−0.284573	−0.142287	−	0.989825i	$-0.545445\pi$
$89$	−2.68466	−0.284573	−0.142287	+	0.989825i	$0.545445\pi$
$90$	0	0
$91$	−16.0000	−1.67726
$92$	0	0
$93$	− 8.68466i	− 0.900557i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	15.5616i	1.58004i	0.613083	+	0.790018i	$0.289929\pi$
$97$	15.5616i	1.58004i	−0.613083	+	0.790018i	$0.710071\pi$
$98$	0	0
$99$	−0.561553	−0.0564382
$100$	0	0
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 13.3693i	− 1.29246i	−0.763142	−	0.646230i	$-0.776345\pi$
$107$	− 13.3693i	− 1.29246i	0.763142	−	0.646230i	$-0.223655\pi$
$108$	0	0
$109$	−12.2462	−1.17297	−0.586487	−	0.809959i	$-0.699490\pi$
$109$	−12.2462	−1.17297	−0.586487	+	0.809959i	$0.699490\pi$
$110$	0	0
$111$	11.8078	1.12074
$112$	0	0
$113$	0.438447i	0.0412456i	0.999787	+	0.0206228i	$0.00656491\pi$
$113$	0.438447i	0.0412456i	−0.999787	+	0.0206228i	$0.993435\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.87689i	0.265969i
$118$	0	0
$119$	−6.24621	−0.572589
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	− 1.75379i	− 0.158134i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	6.24621i	0.554262i	0.960832	+	0.277131i	$0.0893835\pi$
$127$	6.24621i	0.554262i	−0.960832	+	0.277131i	$0.910616\pi$
$128$	0	0
$129$	−11.1231	−0.979335
$130$	0	0
$131$	13.3693	1.16808	0.584041	−	0.811724i	$-0.301471\pi$
$131$	13.3693	1.16808	0.584041	+	0.811724i	$0.301471\pi$
$132$	0	0
$133$	12.4924i	1.08323i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 8.43845i	− 0.720945i	−0.932770	−	0.360473i	$-0.882615\pi$
$137$	− 8.43845i	− 0.720945i	0.932770	−	0.360473i	$-0.117385\pi$
$138$	0	0
$139$	−15.1231	−1.28273	−0.641363	−	0.767238i	$-0.721631\pi$
$139$	−15.1231	−1.28273	−0.641363	+	0.767238i	$0.721631\pi$
$140$	0	0
$141$	−12.4924	−1.05205
$142$	0	0
$143$	− 5.12311i	− 0.428416i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 4.30019i	− 0.354673i
$148$	0	0
$149$	−4.24621	−0.347863	−0.173932	−	0.984758i	$-0.555647\pi$
$149$	−4.24621	−0.347863	−0.173932	+	0.984758i	$0.555647\pi$
$150$	0	0
$151$	−9.36932	−0.762464	−0.381232	−	0.924479i	$-0.624500\pi$
$151$	−9.36932	−0.762464	−0.381232	+	0.924479i	$0.624500\pi$
$152$	0	0
$153$	1.12311i	0.0907977i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 4.43845i	− 0.354227i	−0.984190	−	0.177113i	$-0.943324\pi$
$157$	− 4.43845i	− 0.354227i	0.984190	−	0.177113i	$-0.0566759\pi$
$158$	0	0
$159$	19.1231	1.51656
$160$	0	0
$161$	7.61553	0.600188
$162$	0	0
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.00000i	0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$	8.00000i	0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$	0	0
$169$	−13.2462	−1.01894
$170$	0	0
$171$	2.24621	0.171772
$172$	0	0
$173$	− 12.2462i	− 0.931062i	−0.885032	−	0.465531i	$-0.845863\pi$
$173$	− 12.2462i	− 0.931062i	0.885032	−	0.465531i	$-0.154137\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 12.1922i	− 0.916425i
$178$	0	0
$179$	6.43845	0.481232	0.240616	−	0.970620i	$-0.422651\pi$
$179$	6.43845	0.481232	0.240616	+	0.970620i	$0.422651\pi$
$180$	0	0
$181$	−1.31534	−0.0977686	−0.0488843	−	0.998804i	$-0.515567\pi$
$181$	−1.31534	−0.0977686	−0.0488843	+	0.998804i	$0.515567\pi$
$182$	0	0
$183$	1.75379i	0.129644i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 2.00000i	− 0.146254i
$188$	0	0
$189$	−17.3693	−1.26343
$190$	0	0
$191$	−10.4384	−0.755300	−0.377650	−	0.925949i	$-0.623268\pi$
$191$	−10.4384	−0.755300	−0.377650	+	0.925949i	$0.623268\pi$
$192$	0	0
$193$	9.12311i	0.656696i	0.944557	+	0.328348i	$0.106492\pi$
$193$	9.12311i	0.656696i	−0.944557	+	0.328348i	$0.893508\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 14.4924i	− 1.03254i	−0.856425	−	0.516271i	$-0.827320\pi$
$197$	− 14.4924i	− 1.03254i	0.856425	−	0.516271i	$-0.172680\pi$
$198$	0	0
$199$	12.4924	0.885564	0.442782	−	0.896629i	$-0.353991\pi$
$199$	12.4924	0.885564	0.442782	+	0.896629i	$0.353991\pi$
$200$	0	0
$201$	−14.9309	−1.05314
$202$	0	0
$203$	16.0000i	1.12298i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1.36932i	− 0.0951741i
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	8.49242	0.584642	0.292321	−	0.956320i	$-0.405572\pi$
$211$	8.49242	0.584642	0.292321	+	0.956320i	$0.405572\pi$
$212$	0	0
$213$	− 13.5616i	− 0.929222i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 17.3693i	− 1.17911i
$218$	0	0
$219$	8.00000	0.540590
$220$	0	0
$221$	−10.2462	−0.689235
$222$	0	0
$223$	11.8078i	0.790706i	0.918529	+	0.395353i	$0.129378\pi$
$223$	11.8078i	0.790706i	−0.918529	+	0.395353i	$0.870622\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	23.1231i	1.53473i	0.641208	+	0.767367i	$0.278434\pi$
$227$	23.1231i	1.53473i	−0.641208	+	0.767367i	$0.721566\pi$
$228$	0	0
$229$	−14.6847	−0.970390	−0.485195	−	0.874406i	$-0.661251\pi$
$229$	−14.6847	−0.970390	−0.485195	+	0.874406i	$0.661251\pi$
$230$	0	0
$231$	4.87689	0.320876
$232$	0	0
$233$	7.36932i	0.482780i	0.970428	+	0.241390i	$0.0776033\pi$
$233$	7.36932i	0.482780i	−0.970428	+	0.241390i	$0.922397\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	17.3693i	1.12826i
$238$	0	0
$239$	4.87689	0.315460	0.157730	−	0.987482i	$-0.449582\pi$
$239$	4.87689	0.315460	0.157730	+	0.987482i	$0.449582\pi$
$240$	0	0
$241$	29.1231	1.87598	0.937992	−	0.346657i	$-0.112683\pi$
$241$	29.1231	1.87598	0.937992	+	0.346657i	$0.112683\pi$
$242$	0	0
$243$	5.75379i	0.369106i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	20.4924i	1.30390i
$248$	0	0
$249$	1.36932	0.0867769
$250$	0	0
$251$	1.56155	0.0985643	0.0492822	−	0.998785i	$-0.484307\pi$
$251$	1.56155	0.0985643	0.0492822	+	0.998785i	$0.484307\pi$
$252$	0	0
$253$	2.43845i	0.153304i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.7538i	0.733181i	0.930382	+	0.366591i	$0.119475\pi$
$257$	11.7538i	0.733181i	−0.930382	+	0.366591i	$0.880525\pi$
$258$	0	0
$259$	23.6155	1.46740
$260$	0	0
$261$	2.87689	0.178075
$262$	0	0
$263$	− 19.1231i	− 1.17918i	−0.807702	−	0.589591i	$-0.799289\pi$
$263$	− 19.1231i	− 1.17918i	0.807702	−	0.589591i	$-0.200711\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 4.19224i	− 0.256561i
$268$	0	0
$269$	20.7386	1.26446	0.632228	−	0.774782i	$-0.282140\pi$
$269$	20.7386	1.26446	0.632228	+	0.774782i	$0.282140\pi$
$270$	0	0
$271$	−28.4924	−1.73079	−0.865396	−	0.501089i	$-0.832933\pi$
$271$	−28.4924	−1.73079	−0.865396	+	0.501089i	$0.832933\pi$
$272$	0	0
$273$	− 24.9848i	− 1.51215i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 18.0000i	− 1.08152i	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	− 18.0000i	− 1.08152i	0.841178	−	0.540758i	$-0.181862\pi$
$278$	0	0
$279$	−3.12311	−0.186975
$280$	0	0
$281$	16.2462	0.969168	0.484584	−	0.874745i	$-0.338971\pi$
$281$	16.2462	0.969168	0.484584	+	0.874745i	$0.338971\pi$
$282$	0	0
$283$	− 20.0000i	− 1.18888i	−0.804141	−	0.594438i	$-0.797374\pi$
$283$	− 20.0000i	− 1.18888i	0.804141	−	0.594438i	$-0.202626\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 3.50758i	− 0.207046i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	−24.3002	−1.42450
$292$	0	0
$293$	3.36932i	0.196838i	0.995145	+	0.0984188i	$0.0313785\pi$
$293$	3.36932i	0.196838i	−0.995145	+	0.0984188i	$0.968622\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 5.56155i	− 0.322714i
$298$	0	0
$299$	12.4924	0.722455
$300$	0	0
$301$	−22.2462	−1.28225
$302$	0	0
$303$	− 3.12311i	− 0.179418i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 32.4924i	− 1.85444i	−0.374516	−	0.927220i	$-0.622191\pi$
$307$	− 32.4924i	− 1.85444i	0.374516	−	0.927220i	$-0.377809\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.75379	0.553087	0.276543	−	0.961001i	$-0.410811\pi$
$311$	9.75379	0.553087	0.276543	+	0.961001i	$0.410811\pi$
$312$	0	0
$313$	9.80776i	0.554368i	0.960817	+	0.277184i	$0.0894011\pi$
$313$	9.80776i	0.554368i	−0.960817	+	0.277184i	$0.910599\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 14.1922i	− 0.797115i	−0.917143	−	0.398558i	$-0.869511\pi$
$317$	− 14.1922i	− 0.797115i	0.917143	−	0.398558i	$-0.130489\pi$
$318$	0	0
$319$	−5.12311	−0.286839
$320$	0	0
$321$	20.8769	1.16523
$322$	0	0
$323$	8.00000i	0.445132i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 19.1231i	− 1.05751i
$328$	0	0
$329$	−24.9848	−1.37746
$330$	0	0
$331$	34.9309	1.91997	0.959987	−	0.280044i	$-0.0903491\pi$
$331$	34.9309	1.91997	0.959987	+	0.280044i	$0.0903491\pi$
$332$	0	0
$333$	− 4.24621i	− 0.232691i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 16.7386i	− 0.911811i	−0.890028	−	0.455906i	$-0.849315\pi$
$337$	− 16.7386i	− 0.911811i	0.890028	−	0.455906i	$-0.150685\pi$
$338$	0	0
$339$	−0.684658	−0.0371855
$340$	0	0
$341$	5.56155	0.301175
$342$	0	0
$343$	13.2614i	0.716046i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	22.7386i	1.22067i	0.792142	+	0.610337i	$0.208966\pi$
$347$	22.7386i	1.22067i	−0.792142	+	0.610337i	$0.791034\pi$
$348$	0	0
$349$	32.2462	1.72610	0.863050	−	0.505118i	$-0.168551\pi$
$349$	32.2462	1.72610	0.863050	+	0.505118i	$0.168551\pi$
$350$	0	0
$351$	−28.4924	−1.52081
$352$	0	0
$353$	24.0540i	1.28026i	0.768265	+	0.640132i	$0.221120\pi$
$353$	24.0540i	1.28026i	−0.768265	+	0.640132i	$0.778880\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 9.75379i	− 0.516225i
$358$	0	0
$359$	−4.49242	−0.237101	−0.118550	−	0.992948i	$-0.537825\pi$
$359$	−4.49242	−0.237101	−0.118550	+	0.992948i	$0.537825\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	1.56155i	0.0819603i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	22.9309i	1.19698i	0.801130	+	0.598491i	$0.204233\pi$
$367$	22.9309i	1.19698i	−0.801130	+	0.598491i	$0.795767\pi$
$368$	0	0
$369$	−0.630683	−0.0328321
$370$	0	0
$371$	38.2462	1.98564
$372$	0	0
$373$	8.24621i	0.426973i	0.976946	+	0.213486i	$0.0684819\pi$
$373$	8.24621i	0.426973i	−0.976946	+	0.213486i	$0.931518\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	26.2462i	1.35175i
$378$	0	0
$379$	−0.192236	−0.00987450	−0.00493725	−	0.999988i	$-0.501572\pi$
$379$	−0.192236	−0.00987450	−0.00493725	+	0.999988i	$0.501572\pi$
$380$	0	0
$381$	−9.75379	−0.499702
$382$	0	0
$383$	2.05398i	0.104953i	0.998622	+	0.0524766i	$0.0167115\pi$
$383$	2.05398i	0.104953i	−0.998622	+	0.0524766i	$0.983288\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.00000i	0.203331i
$388$	0	0
$389$	−3.56155	−0.180578	−0.0902889	−	0.995916i	$-0.528779\pi$
$389$	−3.56155	−0.180578	−0.0902889	+	0.995916i	$0.528779\pi$
$390$	0	0
$391$	4.87689	0.246635
$392$	0	0
$393$	20.8769i	1.05310i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	10.4924i	0.526600i	0.964714	+	0.263300i	$0.0848108\pi$
$397$	10.4924i	0.526600i	−0.964714	+	0.263300i	$0.915189\pi$
$398$	0	0
$399$	−19.5076	−0.976600
$400$	0	0
$401$	30.4924	1.52272	0.761359	−	0.648330i	$-0.224532\pi$
$401$	30.4924	1.52272	0.761359	+	0.648330i	$0.224532\pi$
$402$	0	0
$403$	− 28.4924i	− 1.41931i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.56155i	0.374812i
$408$	0	0
$409$	−22.4924	−1.11218	−0.556089	−	0.831123i	$-0.687699\pi$
$409$	−22.4924	−1.11218	−0.556089	+	0.831123i	$0.687699\pi$
$410$	0	0
$411$	13.1771	0.649977
$412$	0	0
$413$	− 24.3845i	− 1.19988i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 23.6155i	− 1.15646i
$418$	0	0
$419$	32.4924	1.58736	0.793679	−	0.608336i	$-0.208163\pi$
$419$	32.4924	1.58736	0.793679	+	0.608336i	$0.208163\pi$
$420$	0	0
$421$	2.49242	0.121473	0.0607366	−	0.998154i	$-0.480655\pi$
$421$	2.49242	0.121473	0.0607366	+	0.998154i	$0.480655\pi$
$422$	0	0
$423$	4.49242i	0.218429i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3.50758i	0.169744i
$428$	0	0
$429$	8.00000	0.386244
$430$	0	0
$431$	27.1231	1.30647	0.653237	−	0.757153i	$-0.273411\pi$
$431$	27.1231	1.30647	0.653237	+	0.757153i	$0.273411\pi$
$432$	0	0
$433$	22.6847i	1.09016i	0.838386	+	0.545078i	$0.183500\pi$
$433$	22.6847i	1.09016i	−0.838386	+	0.545078i	$0.816500\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 9.75379i	− 0.466587i
$438$	0	0
$439$	−4.49242	−0.214412	−0.107206	−	0.994237i	$-0.534190\pi$
$439$	−4.49242	−0.214412	−0.107206	+	0.994237i	$0.534190\pi$
$440$	0	0
$441$	−1.54640	−0.0736380
$442$	0	0
$443$	− 11.3153i	− 0.537608i	−0.963195	−	0.268804i	$-0.913372\pi$
$443$	− 11.3153i	− 0.537608i	0.963195	−	0.268804i	$-0.0866284\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 6.63068i	− 0.313621i
$448$	0	0
$449$	36.5464	1.72473	0.862366	−	0.506286i	$-0.168982\pi$
$449$	36.5464	1.72473	0.862366	+	0.506286i	$0.168982\pi$
$450$	0	0
$451$	1.12311	0.0528850
$452$	0	0
$453$	− 14.6307i	− 0.687409i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	23.8617i	1.11621i	0.829772	+	0.558103i	$0.188470\pi$
$457$	23.8617i	1.11621i	−0.829772	+	0.558103i	$0.811530\pi$
$458$	0	0
$459$	−11.1231	−0.519182
$460$	0	0
$461$	1.12311	0.0523082	0.0261541	−	0.999658i	$-0.491674\pi$
$461$	1.12311	0.0523082	0.0261541	+	0.999658i	$0.491674\pi$
$462$	0	0
$463$	− 15.3153i	− 0.711764i	−0.934531	−	0.355882i	$-0.884180\pi$
$463$	− 15.3153i	− 0.711764i	0.934531	−	0.355882i	$-0.115820\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	28.3002i	1.30958i	0.755812	+	0.654788i	$0.227242\pi$
$467$	28.3002i	1.30958i	−0.755812	+	0.654788i	$0.772758\pi$
$468$	0	0
$469$	−29.8617	−1.37889
$470$	0	0
$471$	6.93087	0.319358
$472$	0	0
$473$	− 7.12311i	− 0.327521i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 6.87689i	− 0.314871i
$478$	0	0
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	38.7386	1.76633
$482$	0	0
$483$	11.8920i	0.541107i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	14.9309i	0.676582i	0.941041	+	0.338291i	$0.109849\pi$
$487$	14.9309i	0.676582i	−0.941041	+	0.338291i	$0.890151\pi$
$488$	0	0
$489$	−6.24621	−0.282463
$490$	0	0
$491$	13.7538	0.620700	0.310350	−	0.950622i	$-0.399554\pi$
$491$	13.7538	0.620700	0.310350	+	0.950622i	$0.399554\pi$
$492$	0	0
$493$	10.2462i	0.461466i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 27.1231i	− 1.21664i
$498$	0	0
$499$	28.9848	1.29754	0.648770	−	0.760985i	$-0.275284\pi$
$499$	28.9848	1.29754	0.648770	+	0.760985i	$0.275284\pi$
$500$	0	0
$501$	−12.4924	−0.558120
$502$	0	0
$503$	31.6155i	1.40967i	0.709373	+	0.704833i	$0.248978\pi$
$503$	31.6155i	1.40967i	−0.709373	+	0.704833i	$0.751022\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 20.6847i	− 0.918638i
$508$	0	0
$509$	18.3002	0.811142	0.405571	−	0.914064i	$-0.367073\pi$
$509$	18.3002	0.811142	0.405571	+	0.914064i	$0.367073\pi$
$510$	0	0
$511$	16.0000	0.707798
$512$	0	0
$513$	22.2462i	0.982194i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 8.00000i	− 0.351840i
$518$	0	0
$519$	19.1231	0.839411
$520$	0	0
$521$	1.31534	0.0576262	0.0288131	−	0.999585i	$-0.490827\pi$
$521$	1.31534	0.0576262	0.0288131	+	0.999585i	$0.490827\pi$
$522$	0	0
$523$	12.0000i	0.524723i	0.964970	+	0.262362i	$0.0845013\pi$
$523$	12.0000i	0.524723i	−0.964970	+	0.262362i	$0.915499\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 11.1231i	− 0.484530i
$528$	0	0
$529$	17.0540	0.741477
$530$	0	0
$531$	−4.38447	−0.190270
$532$	0	0
$533$	− 5.75379i	− 0.249224i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	10.0540i	0.433861i
$538$	0	0
$539$	2.75379	0.118614
$540$	0	0
$541$	−23.8617	−1.02590	−0.512948	−	0.858420i	$-0.671447\pi$
$541$	−23.8617	−1.02590	−0.512948	+	0.858420i	$0.671447\pi$
$542$	0	0
$543$	− 2.05398i	− 0.0881445i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 42.2462i	− 1.80632i	−0.429307	−	0.903159i	$-0.641242\pi$
$547$	− 42.2462i	− 1.80632i	0.429307	−	0.903159i	$-0.358758\pi$
$548$	0	0
$549$	0.630683	0.0269169
$550$	0	0
$551$	20.4924	0.873007
$552$	0	0
$553$	34.7386i	1.47724i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 3.75379i	− 0.159053i	−0.996833	−	0.0795266i	$-0.974659\pi$
$557$	− 3.75379i	− 0.159053i	0.996833	−	0.0795266i	$-0.0253409\pi$
$558$	0	0
$559$	−36.4924	−1.54347
$560$	0	0
$561$	3.12311	0.131858
$562$	0	0
$563$	− 24.4924i	− 1.03223i	−0.856519	−	0.516116i	$-0.827377\pi$
$563$	− 24.4924i	− 1.03223i	0.856519	−	0.516116i	$-0.172623\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 21.8617i	− 0.918107i
$568$	0	0
$569$	26.8769	1.12674	0.563369	−	0.826205i	$-0.309505\pi$
$569$	26.8769	1.12674	0.563369	+	0.826205i	$0.309505\pi$
$570$	0	0
$571$	16.4924	0.690186	0.345093	−	0.938568i	$-0.387847\pi$
$571$	16.4924	0.690186	0.345093	+	0.938568i	$0.387847\pi$
$572$	0	0
$573$	− 16.3002i	− 0.680950i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	15.5616i	0.647836i	0.946085	+	0.323918i	$0.105000\pi$
$577$	15.5616i	0.647836i	−0.946085	+	0.323918i	$0.895000\pi$
$578$	0	0
$579$	−14.2462	−0.592052
$580$	0	0
$581$	2.73863	0.113618
$582$	0	0
$583$	12.2462i	0.507186i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 24.4924i	− 1.01091i	−0.862853	−	0.505455i	$-0.831325\pi$
$587$	− 24.4924i	− 1.01091i	0.862853	−	0.505455i	$-0.168675\pi$
$588$	0	0
$589$	−22.2462	−0.916639
$590$	0	0
$591$	22.6307	0.930902
$592$	0	0
$593$	− 3.36932i	− 0.138361i	−0.997604	−	0.0691806i	$-0.977962\pi$
$593$	− 3.36932i	− 0.138361i	0.997604	−	0.0691806i	$-0.0220385\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	19.5076i	0.798392i
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	3.75379	0.153120	0.0765601	−	0.997065i	$-0.475606\pi$
$601$	3.75379	0.153120	0.0765601	+	0.997065i	$0.475606\pi$
$602$	0	0
$603$	5.36932i	0.218655i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 45.8617i	− 1.86147i	−0.365694	−	0.930735i	$-0.619168\pi$
$607$	− 45.8617i	− 1.86147i	0.365694	−	0.930735i	$-0.380832\pi$
$608$	0	0
$609$	−24.9848	−1.01244
$610$	0	0
$611$	−40.9848	−1.65807
$612$	0	0
$613$	− 11.8617i	− 0.479091i	−0.970885	−	0.239546i	$-0.923002\pi$
$613$	− 11.8617i	− 0.479091i	0.970885	−	0.239546i	$-0.0769985\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 2.49242i	− 0.100341i	−0.998741	−	0.0501706i	$-0.984024\pi$
$617$	− 2.49242i	− 0.100341i	0.998741	−	0.0501706i	$-0.0159765\pi$
$618$	0	0
$619$	−18.9309	−0.760896	−0.380448	−	0.924802i	$-0.624230\pi$
$619$	−18.9309	−0.760896	−0.380448	+	0.924802i	$0.624230\pi$
$620$	0	0
$621$	13.5616	0.544206
$622$	0	0
$623$	− 8.38447i	− 0.335917i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 6.24621i	− 0.249450i
$628$	0	0
$629$	15.1231	0.602998
$630$	0	0
$631$	−42.0540	−1.67414	−0.837071	−	0.547094i	$-0.815734\pi$
$631$	−42.0540	−1.67414	−0.837071	+	0.547094i	$0.815734\pi$
$632$	0	0
$633$	13.2614i	0.527092i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 14.1080i	− 0.558977i
$638$	0	0
$639$	−4.87689	−0.192927
$640$	0	0
$641$	−46.3002	−1.82875	−0.914374	−	0.404871i	$-0.867316\pi$
$641$	−46.3002	−1.82875	−0.914374	+	0.404871i	$0.867316\pi$
$642$	0	0
$643$	9.17708i	0.361909i	0.983491	+	0.180954i	$0.0579186\pi$
$643$	9.17708i	0.361909i	−0.983491	+	0.180954i	$0.942081\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	13.5616i	0.533160i	0.963813	+	0.266580i	$0.0858936\pi$
$647$	13.5616i	0.533160i	−0.963813	+	0.266580i	$0.914106\pi$
$648$	0	0
$649$	7.80776	0.306482
$650$	0	0
$651$	27.1231	1.06304
$652$	0	0
$653$	− 35.1771i	− 1.37659i	−0.725433	−	0.688293i	$-0.758360\pi$
$653$	− 35.1771i	− 1.37659i	0.725433	−	0.688293i	$-0.241640\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 2.87689i	− 0.112238i
$658$	0	0
$659$	11.6155	0.452477	0.226238	−	0.974072i	$-0.427357\pi$
$659$	11.6155	0.452477	0.226238	+	0.974072i	$0.427357\pi$
$660$	0	0
$661$	41.8078	1.62613	0.813067	−	0.582170i	$-0.197796\pi$
$661$	41.8078	1.62613	0.813067	+	0.582170i	$0.197796\pi$
$662$	0	0
$663$	− 16.0000i	− 0.621389i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 12.4924i	− 0.483709i
$668$	0	0
$669$	−18.4384	−0.712872
$670$	0	0
$671$	−1.12311	−0.0433570
$672$	0	0
$673$	− 33.2311i	− 1.28096i	−0.767974	−	0.640482i	$-0.778735\pi$
$673$	− 33.2311i	− 1.28096i	0.767974	−	0.640482i	$-0.221265\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 20.7386i	− 0.797050i	−0.917157	−	0.398525i	$-0.869522\pi$
$677$	− 20.7386i	− 0.797050i	0.917157	−	0.398525i	$-0.130478\pi$
$678$	0	0
$679$	−48.6004	−1.86511
$680$	0	0
$681$	−36.1080	−1.38366
$682$	0	0
$683$	− 6.73863i	− 0.257847i	−0.991655	−	0.128923i	$-0.958848\pi$
$683$	− 6.73863i	− 0.257847i	0.991655	−	0.128923i	$-0.0411521\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 22.9309i	− 0.874867i
$688$	0	0
$689$	62.7386	2.39015
$690$	0	0
$691$	−9.94602	−0.378365	−0.189182	−	0.981942i	$-0.560584\pi$
$691$	−9.94602	−0.378365	−0.189182	+	0.981942i	$0.560584\pi$
$692$	0	0
$693$	− 1.75379i	− 0.0666209i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 2.24621i	− 0.0850813i
$698$	0	0
$699$	−11.5076	−0.435257
$700$	0	0
$701$	50.4924	1.90707	0.953536	−	0.301278i	$-0.0974133\pi$
$701$	50.4924	1.90707	0.953536	+	0.301278i	$0.0974133\pi$
$702$	0	0
$703$	− 30.2462i	− 1.14076i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 6.24621i	− 0.234913i
$708$	0	0
$709$	−2.19224	−0.0823311	−0.0411656	−	0.999152i	$-0.513107\pi$
$709$	−2.19224	−0.0823311	−0.0411656	+	0.999152i	$0.513107\pi$
$710$	0	0
$711$	6.24621	0.234251
$712$	0	0
$713$	13.5616i	0.507884i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	7.61553i	0.284407i
$718$	0	0
$719$	−35.4233	−1.32107	−0.660533	−	0.750797i	$-0.729670\pi$
$719$	−35.4233	−1.32107	−0.660533	+	0.750797i	$0.729670\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	45.4773i	1.69132i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 23.3153i	− 0.864718i	−0.901702	−	0.432359i	$-0.857681\pi$
$727$	− 23.3153i	− 0.864718i	0.901702	−	0.432359i	$-0.142319\pi$
$728$	0	0
$729$	−29.9848	−1.11055
$730$	0	0
$731$	−14.2462	−0.526915
$732$	0	0
$733$	− 1.12311i	− 0.0414829i	−0.999785	−	0.0207414i	$-0.993397\pi$
$733$	− 1.12311i	− 0.0414829i	0.999785	−	0.0207414i	$-0.00660267\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 9.56155i	− 0.352204i
$738$	0	0
$739$	2.63068	0.0967712	0.0483856	−	0.998829i	$-0.484592\pi$
$739$	2.63068	0.0967712	0.0483856	+	0.998829i	$0.484592\pi$
$740$	0	0
$741$	−32.0000	−1.17555
$742$	0	0
$743$	− 10.7386i	− 0.393962i	−0.980407	−	0.196981i	$-0.936886\pi$
$743$	− 10.7386i	− 0.393962i	0.980407	−	0.196981i	$-0.0631138\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 0.492423i	− 0.0180168i
$748$	0	0
$749$	41.7538	1.52565
$750$	0	0
$751$	5.56155	0.202944	0.101472	−	0.994838i	$-0.467645\pi$
$751$	5.56155	0.202944	0.101472	+	0.994838i	$0.467645\pi$
$752$	0	0
$753$	2.43845i	0.0888620i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	15.7538i	0.572581i	0.958143	+	0.286291i	$0.0924223\pi$
$757$	15.7538i	0.572581i	−0.958143	+	0.286291i	$0.907578\pi$
$758$	0	0
$759$	−3.80776	−0.138213
$760$	0	0
$761$	5.12311	0.185712	0.0928562	−	0.995680i	$-0.470400\pi$
$761$	5.12311	0.185712	0.0928562	+	0.995680i	$0.470400\pi$
$762$	0	0
$763$	− 38.2462i	− 1.38461i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 40.0000i	− 1.44432i
$768$	0	0
$769$	−25.6155	−0.923720	−0.461860	−	0.886953i	$-0.652818\pi$
$769$	−25.6155	−0.923720	−0.461860	+	0.886953i	$0.652818\pi$
$770$	0	0
$771$	−18.3542	−0.661009
$772$	0	0
$773$	− 40.7386i	− 1.46527i	−0.680623	−	0.732633i	$-0.738291\pi$
$773$	− 40.7386i	− 1.46527i	0.680623	−	0.732633i	$-0.261709\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	36.8769i	1.32295i
$778$	0	0
$779$	−4.49242	−0.160958
$780$	0	0
$781$	8.68466	0.310762
$782$	0	0
$783$	28.4924i	1.01824i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 29.7538i	− 1.06061i	−0.847808	−	0.530304i	$-0.822078\pi$
$787$	− 29.7538i	− 1.06061i	0.847808	−	0.530304i	$-0.177922\pi$
$788$	0	0
$789$	29.8617	1.06311
$790$	0	0
$791$	−1.36932	−0.0486873
$792$	0	0
$793$	5.75379i	0.204323i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 14.1922i	− 0.502715i	−0.967894	−	0.251357i	$-0.919123\pi$
$797$	− 14.1922i	− 0.502715i	0.967894	−	0.251357i	$-0.0808769\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	0	0
$801$	−1.50758	−0.0532676
$802$	0	0
$803$	5.12311i	0.180790i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	32.3845i	1.13999i
$808$	0	0
$809$	45.6155	1.60376	0.801878	−	0.597487i	$-0.203834\pi$
$809$	45.6155	1.60376	0.801878	+	0.597487i	$0.203834\pi$
$810$	0	0
$811$	−7.12311	−0.250126	−0.125063	−	0.992149i	$-0.539913\pi$
$811$	−7.12311	−0.250126	−0.125063	+	0.992149i	$0.539913\pi$
$812$	0	0
$813$	− 44.4924i	− 1.56042i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	28.4924i	0.996824i
$818$	0	0
$819$	−8.98485	−0.313956
$820$	0	0
$821$	−42.9848	−1.50018	−0.750091	−	0.661335i	$-0.769990\pi$
$821$	−42.9848	−1.50018	−0.750091	+	0.661335i	$0.769990\pi$
$822$	0	0
$823$	54.5464i	1.90137i	0.310161	+	0.950684i	$0.399617\pi$
$823$	54.5464i	1.90137i	−0.310161	+	0.950684i	$0.600383\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	38.7386i	1.34707i	0.739153	+	0.673537i	$0.235226\pi$
$827$	38.7386i	1.34707i	−0.739153	+	0.673537i	$0.764774\pi$
$828$	0	0
$829$	−15.0691	−0.523373	−0.261686	−	0.965153i	$-0.584279\pi$
$829$	−15.0691	−0.523373	−0.261686	+	0.965153i	$0.584279\pi$
$830$	0	0
$831$	28.1080	0.975054
$832$	0	0
$833$	− 5.50758i	− 0.190826i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 30.9309i	− 1.06913i
$838$	0	0
$839$	19.8078	0.683840	0.341920	−	0.939729i	$-0.388923\pi$
$839$	19.8078	0.683840	0.341920	+	0.939729i	$0.388923\pi$
$840$	0	0
$841$	−2.75379	−0.0949582
$842$	0	0
$843$	25.3693i	0.873766i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3.12311i	0.107311i
$848$	0	0
$849$	31.2311	1.07185
$850$	0	0
$851$	−18.4384	−0.632062
$852$	0	0
$853$	46.4924i	1.59187i	0.605382	+	0.795935i	$0.293020\pi$
$853$	46.4924i	1.59187i	−0.605382	+	0.795935i	$0.706980\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	30.1080i	1.02847i	0.857650	+	0.514234i	$0.171924\pi$
$857$	30.1080i	1.02847i	−0.857650	+	0.514234i	$0.828076\pi$
$858$	0	0
$859$	−30.0540	−1.02543	−0.512714	−	0.858559i	$-0.671360\pi$
$859$	−30.0540	−1.02543	−0.512714	+	0.858559i	$0.671360\pi$
$860$	0	0
$861$	5.47727	0.186665
$862$	0	0
$863$	36.4924i	1.24222i	0.783725	+	0.621108i	$0.213317\pi$
$863$	36.4924i	1.24222i	−0.783725	+	0.621108i	$0.786683\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	20.3002i	0.689430i
$868$	0	0
$869$	−11.1231	−0.377326
$870$	0	0
$871$	−48.9848	−1.65979
$872$	0	0
$873$	8.73863i	0.295758i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	55.3693i	1.86969i	0.355057	+	0.934844i	$0.384461\pi$
$877$	55.3693i	1.86969i	−0.355057	+	0.934844i	$0.615539\pi$
$878$	0	0
$879$	−5.26137	−0.177461
$880$	0	0
$881$	34.3002	1.15560	0.577801	−	0.816177i	$-0.303911\pi$
$881$	34.3002	1.15560	0.577801	+	0.816177i	$0.303911\pi$
$882$	0	0
$883$	− 8.49242i	− 0.285793i	−0.989738	−	0.142896i	$-0.954358\pi$
$883$	− 8.49242i	− 0.285793i	0.989738	−	0.142896i	$-0.0456416\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	31.6155i	1.06155i	0.847514	+	0.530773i	$0.178098\pi$
$887$	31.6155i	1.06155i	−0.847514	+	0.530773i	$0.821902\pi$
$888$	0	0
$889$	−19.5076	−0.654263
$890$	0	0
$891$	7.00000	0.234509
$892$	0	0
$893$	32.0000i	1.07084i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	19.5076i	0.651339i
$898$	0	0
$899$	−28.4924	−0.950275
$900$	0	0
$901$	24.4924	0.815961
$902$	0	0
$903$	− 34.7386i	− 1.15603i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	16.4924i	0.547622i	0.961784	+	0.273811i	$0.0882843\pi$
$907$	16.4924i	0.547622i	−0.961784	+	0.273811i	$0.911716\pi$
$908$	0	0
$909$	−1.12311	−0.0372511
$910$	0	0
$911$	−26.7386	−0.885890	−0.442945	−	0.896549i	$-0.646066\pi$
$911$	−26.7386	−0.885890	−0.442945	+	0.896549i	$0.646066\pi$
$912$	0	0
$913$	0.876894i	0.0290210i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	41.7538i	1.37883i
$918$	0	0
$919$	−6.63068	−0.218726	−0.109363	−	0.994002i	$-0.534881\pi$
$919$	−6.63068	−0.218726	−0.109363	+	0.994002i	$0.534881\pi$
$920$	0	0
$921$	50.7386	1.67189
$922$	0	0
$923$	− 44.4924i	− 1.46449i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−46.4924	−1.52537	−0.762683	−	0.646772i	$-0.776119\pi$
$929$	−46.4924	−1.52537	−0.762683	+	0.646772i	$0.776119\pi$
$930$	0	0
$931$	−11.0152	−0.361007
$932$	0	0
$933$	15.2311i	0.498642i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 42.1080i	− 1.37561i	−0.725897	−	0.687803i	$-0.758575\pi$
$937$	− 42.1080i	− 1.37561i	0.725897	−	0.687803i	$-0.241425\pi$
$938$	0	0
$939$	−15.3153	−0.499797
$940$	0	0
$941$	−32.2462	−1.05120	−0.525598	−	0.850733i	$-0.676158\pi$
$941$	−32.2462	−1.05120	−0.525598	+	0.850733i	$0.676158\pi$
$942$	0	0
$943$	2.73863i	0.0891822i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 12.6847i	− 0.412196i	−0.978531	−	0.206098i	$-0.933923\pi$
$947$	− 12.6847i	− 0.412196i	0.978531	−	0.206098i	$-0.0660766\pi$
$948$	0	0
$949$	26.2462	0.851988
$950$	0	0
$951$	22.1619	0.718650
$952$	0	0
$953$	− 0.246211i	− 0.00797556i	−0.999992	−	0.00398778i	$-0.998731\pi$
$953$	− 0.246211i	− 0.00797556i	0.999992	−	0.00398778i	$-0.00126935\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 8.00000i	− 0.258603i
$958$	0	0
$959$	26.3542	0.851020
$960$	0	0
$961$	−0.0691303	−0.00223001
$962$	0	0
$963$	− 7.50758i	− 0.241928i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	8.00000i	0.257263i	0.991692	+	0.128631i	$0.0410584\pi$
$967$	8.00000i	0.257263i	−0.991692	+	0.128631i	$0.958942\pi$
$968$	0	0
$969$	−12.4924	−0.401314
$970$	0	0
$971$	−34.5464	−1.10865	−0.554323	−	0.832301i	$-0.687023\pi$
$971$	−34.5464	−1.10865	−0.554323	+	0.832301i	$0.687023\pi$
$972$	0	0
$973$	− 47.2311i	− 1.51416i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	53.8078i	1.72146i	0.509059	+	0.860731i	$0.329993\pi$
$977$	53.8078i	1.72146i	−0.509059	+	0.860731i	$0.670007\pi$
$978$	0	0
$979$	2.68466	0.0858021
$980$	0	0
$981$	−6.87689	−0.219562
$982$	0	0
$983$	− 30.9309i	− 0.986542i	−0.869876	−	0.493271i	$-0.835801\pi$
$983$	− 30.9309i	− 0.986542i	0.869876	−	0.493271i	$-0.164199\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 39.0152i	− 1.24187i
$988$	0	0
$989$	17.3693	0.552312
$990$	0	0
$991$	−4.49242	−0.142707	−0.0713533	−	0.997451i	$-0.522732\pi$
$991$	−4.49242	−0.142707	−0.0713533	+	0.997451i	$0.522732\pi$
$992$	0	0
$993$	54.5464i	1.73098i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	52.2462i	1.65465i	0.561721	+	0.827327i	$0.310140\pi$
$997$	52.2462i	1.65465i	−0.561721	+	0.827327i	$0.689860\pi$
$998$	0	0
$999$	42.0540	1.33053

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2200.2.b.g.1849.3		4
4.3	odd	2		4400.2.b.v.4049.2		4
5.2	odd	4		2200.2.a.o.1.2		2
5.3	odd	4		88.2.a.b.1.1	✓	2
5.4	even	2	inner	2200.2.b.g.1849.2		4
15.8	even	4		792.2.a.h.1.1		2
20.3	even	4		176.2.a.d.1.2		2
20.7	even	4		4400.2.a.bp.1.1		2
20.19	odd	2		4400.2.b.v.4049.3		4
35.13	even	4		4312.2.a.n.1.2		2
40.3	even	4		704.2.a.p.1.1		2
40.13	odd	4		704.2.a.m.1.2		2
55.3	odd	20		968.2.i.r.9.2		8
55.8	even	20		968.2.i.q.9.2		8
55.13	even	20		968.2.i.q.81.1		8
55.18	even	20		968.2.i.q.753.2		8
55.28	even	20		968.2.i.q.729.1		8
55.38	odd	20		968.2.i.r.729.1		8
55.43	even	4		968.2.a.j.1.1		2
55.48	odd	20		968.2.i.r.753.2		8
55.53	odd	20		968.2.i.r.81.1		8
60.23	odd	4		1584.2.a.t.1.1		2
80.3	even	4		2816.2.c.p.1409.3		4
80.13	odd	4		2816.2.c.w.1409.2		4
80.43	even	4		2816.2.c.p.1409.2		4
80.53	odd	4		2816.2.c.w.1409.3		4
120.53	even	4		6336.2.a.cu.1.2		2
120.83	odd	4		6336.2.a.cx.1.2		2
140.83	odd	4		8624.2.a.cb.1.1		2
165.98	odd	4		8712.2.a.bb.1.1		2
220.43	odd	4		1936.2.a.r.1.2		2
440.43	odd	4		7744.2.a.cl.1.1		2
440.373	even	4		7744.2.a.by.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.a.b.1.1	✓	2	5.3	odd	4
176.2.a.d.1.2		2	20.3	even	4
704.2.a.m.1.2		2	40.13	odd	4
704.2.a.p.1.1		2	40.3	even	4
792.2.a.h.1.1		2	15.8	even	4
968.2.a.j.1.1		2	55.43	even	4
968.2.i.q.9.2		8	55.8	even	20
968.2.i.q.81.1		8	55.13	even	20
968.2.i.q.729.1		8	55.28	even	20
968.2.i.q.753.2		8	55.18	even	20
968.2.i.r.9.2		8	55.3	odd	20
968.2.i.r.81.1		8	55.53	odd	20
968.2.i.r.729.1		8	55.38	odd	20
968.2.i.r.753.2		8	55.48	odd	20
1584.2.a.t.1.1		2	60.23	odd	4
1936.2.a.r.1.2		2	220.43	odd	4
2200.2.a.o.1.2		2	5.2	odd	4
2200.2.b.g.1849.2		4	5.4	even	2	inner
2200.2.b.g.1849.3		4	1.1	even	1	trivial
2816.2.c.p.1409.2		4	80.43	even	4
2816.2.c.p.1409.3		4	80.3	even	4
2816.2.c.w.1409.2		4	80.13	odd	4
2816.2.c.w.1409.3		4	80.53	odd	4
4312.2.a.n.1.2		2	35.13	even	4
4400.2.a.bp.1.1		2	20.7	even	4
4400.2.b.v.4049.2		4	4.3	odd	2
4400.2.b.v.4049.3		4	20.19	odd	2
6336.2.a.cu.1.2		2	120.53	even	4
6336.2.a.cx.1.2		2	120.83	odd	4
7744.2.a.by.1.2		2	440.373	even	4
7744.2.a.cl.1.1		2	440.43	odd	4
8624.2.a.cb.1.1		2	140.83	odd	4
8712.2.a.bb.1.1		2	165.98	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2200.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.56155i q^{3} +3.12311i q^{7} +0.561553 q^{9} -1.00000 q^{11} +5.12311i q^{13} +2.00000i q^{17} +4.00000 q^{19} -4.87689 q^{21} -2.43845i q^{23} +5.56155i q^{27} +5.12311 q^{29} -5.56155 q^{31} -1.56155i q^{33} -7.56155i q^{37} -8.00000 q^{39} -1.12311 q^{41} +7.12311i q^{43} +8.00000i q^{47} -2.75379 q^{49} -3.12311 q^{51} -12.2462i q^{53} +6.24621i q^{57} -7.80776 q^{59} +1.12311 q^{61} +1.75379i q^{63} +9.56155i q^{67} +3.80776 q^{69} -8.68466 q^{71} -5.12311i q^{73} -3.12311i q^{77} +11.1231 q^{79} -7.00000 q^{81} -0.876894i q^{83} +8.00000i q^{87} -2.68466 q^{89} -16.0000 q^{91} -8.68466i q^{93} +15.5616i q^{97} -0.561553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{9} - 4 q^{11} + 16 q^{19} - 36 q^{21} + 4 q^{29} - 14 q^{31} - 32 q^{39} + 12 q^{41} - 44 q^{49} + 4 q^{51} + 10 q^{59} - 12 q^{61} - 26 q^{69} - 10 q^{71} + 28 q^{79} - 28 q^{81} + 14 q^{89}+ \cdots + 6 q^{99}+O(q^{100}) \) 4 * q - 6 * q^9 - 4 * q^11 + 16 * q^19 - 36 * q^21 + 4 * q^29 - 14 * q^31 - 32 * q^39 + 12 * q^41 - 44 * q^49 + 4 * q^51 + 10 * q^59 - 12 * q^61 - 26 * q^69 - 10 * q^71 + 28 * q^79 - 28 * q^81 + 14 * q^89 - 64 * q^91 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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