LMFDB - Embedded newform 2800.2.k.o.2351.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2800,2,Mod(2351,2800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2800, 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("2800.2351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2800 = 2^{4} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2800.k (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:	$22.3581125660$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2351.4
Root			$0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	2800.2351
Dual form			2800.2.k.o.2351.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.317837 q^{3} +(-2.43916 + 1.02494i) q^{7} -2.89898 q^{9} +O(q^{10})$	$q-0.317837 q^{3} +(-2.43916 + 1.02494i) q^{7} -2.89898 q^{9} -0.317837i q^{11} +4.44949i q^{13} -3.89898i q^{17} +2.51059 q^{19} +(0.775255 - 0.325765i) q^{21} +7.07107i q^{23} +1.87492 q^{27} +4.44949 q^{29} -7.07107 q^{31} +0.101021i q^{33} +6.44949 q^{37} -1.41421i q^{39} -7.89898i q^{41} -5.02118i q^{43} -5.51399 q^{47} +(4.89898 - 5.00000i) q^{49} +1.23924i q^{51} -3.55051 q^{53} -0.797959 q^{57} -13.2207 q^{59} -12.8990i q^{61} +(7.07107 - 2.97129i) q^{63} -6.61037i q^{67} -2.24745i q^{69} -7.07107i q^{71} +7.89898i q^{73} +(0.325765 + 0.775255i) q^{77} -1.55708i q^{79} +8.10102 q^{81} +0.953512 q^{83} -1.41421 q^{87} -7.89898i q^{89} +(-4.56048 - 10.8530i) q^{91} +2.24745 q^{93} -14.0000i q^{97} +0.921404i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{9}+O(q^{10}) $ 8 * q + 16 * q^9	$ 8 q + 16 q^{9} + 16 q^{21} + 16 q^{29} + 32 q^{37} - 48 q^{53} + 72 q^{57} + 32 q^{77} + 104 q^{81} - 80 q^{93}+O(q^{100}) $ 8 * q + 16 * q^9 + 16 * q^21 + 16 * q^29 + 32 * q^37 - 48 * q^53 + 72 * q^57 + 32 * q^77 + 104 * q^81 - 80 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times$.

$n$	$351$	$801$	$2101$	$2577$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	−0.317837			−0.183503			−0.0917517	−	0.995782i	$-0.529247\pi$
$3$	−0.317837			−0.183503			−0.0917517	+	0.995782i	$0.529247\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.43916	+	1.02494i	−0.921915	+	0.387392i
$7$	−2.43916	+	1.02494i	−0.921915	+	0.387392i
$8$		0			0
$9$	−2.89898			−0.966326
$10$		0			0
$11$		−	0.317837i		−	0.0958315i	−0.998851	−	0.0479158i	$-0.984742\pi$
$11$		−	0.317837i		−	0.0958315i	0.998851	−	0.0479158i	$-0.0152579\pi$
$12$		0			0
$13$			4.44949i			1.23407i	0.786937	+	0.617033i	$0.211666\pi$
$13$			4.44949i			1.23407i	−0.786937	+	0.617033i	$0.788334\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	3.89898i		−	0.945641i	−0.881159	−	0.472821i	$-0.843236\pi$
$17$		−	3.89898i		−	0.945641i	0.881159	−	0.472821i	$-0.156764\pi$
$18$		0			0
$19$	2.51059			0.575969			0.287984	−	0.957635i	$-0.407015\pi$
$19$	2.51059			0.575969			0.287984	+	0.957635i	$0.407015\pi$
$20$		0			0
$21$	0.775255	−	0.325765i	0.169175	−	0.0710878i
$22$		0			0
$23$			7.07107i			1.47442i	0.675664	+	0.737210i	$0.263857\pi$
$23$			7.07107i			1.47442i	−0.675664	+	0.737210i	$0.736143\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	1.87492			0.360828
$28$		0			0
$29$	4.44949			0.826250			0.413125	−	0.910674i	$-0.364437\pi$
$29$	4.44949			0.826250			0.413125	+	0.910674i	$0.364437\pi$
$30$		0			0
$31$	−7.07107			−1.27000			−0.635001	−	0.772512i	$-0.719000\pi$
$31$	−7.07107			−1.27000			−0.635001	+	0.772512i	$0.719000\pi$
$32$		0			0
$33$			0.101021i			0.0175854i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	6.44949			1.06029			0.530145	−	0.847907i	$-0.322138\pi$
$37$	6.44949			1.06029			0.530145	+	0.847907i	$0.322138\pi$
$38$		0			0
$39$		−	1.41421i		−	0.226455i
$40$		0			0
$41$		−	7.89898i		−	1.23361i	−0.787115	−	0.616807i	$-0.788426\pi$
$41$		−	7.89898i		−	1.23361i	0.787115	−	0.616807i	$-0.211574\pi$
$42$		0			0
$43$		−	5.02118i		−	0.765723i	−0.923806	−	0.382861i	$-0.874939\pi$
$43$		−	5.02118i		−	0.765723i	0.923806	−	0.382861i	$-0.125061\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.51399			−0.804298			−0.402149	−	0.915574i	$-0.631737\pi$
$47$	−5.51399			−0.804298			−0.402149	+	0.915574i	$0.631737\pi$
$48$		0			0
$49$	4.89898	−	5.00000i	0.699854	−	0.714286i
$50$		0			0
$51$			1.23924i			0.173528i
$52$		0			0
$53$	−3.55051			−0.487700			−0.243850	−	0.969813i	$-0.578410\pi$
$53$	−3.55051			−0.487700			−0.243850	+	0.969813i	$0.578410\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−0.797959			−0.105692
$58$		0			0
$59$	−13.2207			−1.72119			−0.860596	−	0.509288i	$-0.829909\pi$
$59$	−13.2207			−1.72119			−0.860596	+	0.509288i	$0.829909\pi$
$60$		0			0
$61$		−	12.8990i		−	1.65155i	−0.564003	−	0.825773i	$-0.690739\pi$
$61$		−	12.8990i		−	1.65155i	0.564003	−	0.825773i	$-0.309261\pi$
$62$		0			0
$63$	7.07107	−	2.97129i	0.890871	−	0.374348i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	6.61037i		−	0.807585i	−0.914851	−	0.403792i	$-0.867692\pi$
$67$		−	6.61037i		−	0.807585i	0.914851	−	0.403792i	$-0.132308\pi$
$68$		0			0
$69$		−	2.24745i		−	0.270561i
$70$		0			0
$71$		−	7.07107i		−	0.839181i	−0.907713	−	0.419591i	$-0.862174\pi$
$71$		−	7.07107i		−	0.839181i	0.907713	−	0.419591i	$-0.137826\pi$
$72$		0			0
$73$			7.89898i			0.924506i	0.886748	+	0.462253i	$0.152959\pi$
$73$			7.89898i			0.924506i	−0.886748	+	0.462253i	$0.847041\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.325765	+	0.775255i	0.0371244	+	0.0883485i
$78$		0			0
$79$		−	1.55708i		−	0.175185i	−0.996156	−	0.0875925i	$-0.972083\pi$
$79$		−	1.55708i		−	0.175185i	0.996156	−	0.0875925i	$-0.0279173\pi$
$80$		0			0
$81$	8.10102			0.900113
$82$		0			0
$83$	0.953512			0.104662			0.0523308	−	0.998630i	$-0.483335\pi$
$83$	0.953512			0.104662			0.0523308	+	0.998630i	$0.483335\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−1.41421			−0.151620
$88$		0			0
$89$		−	7.89898i		−	0.837290i	−0.908150	−	0.418645i	$-0.862505\pi$
$89$		−	7.89898i		−	0.837290i	0.908150	−	0.418645i	$-0.137495\pi$
$90$		0			0
$91$	−4.56048	−	10.8530i	−0.478068	−	1.13770i
$92$		0			0
$93$	2.24745			0.233050
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	14.0000i		−	1.42148i	−0.703452	−	0.710742i	$-0.748359\pi$
$97$		−	14.0000i		−	1.42148i	0.703452	−	0.710742i	$-0.251641\pi$
$98$		0			0
$99$			0.921404i			0.0926046i
$100$		0			0
$101$		−	13.5505i		−	1.34833i	−0.738583	−	0.674163i	$-0.764504\pi$
$101$		−	13.5505i		−	1.34833i	0.738583	−	0.674163i	$-0.235496\pi$
$102$		0			0
$103$	10.3923			1.02398			0.511992	−	0.858990i	$-0.328908\pi$
$103$	10.3923			1.02398			0.511992	+	0.858990i	$0.328908\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			11.6315i			1.12446i	0.826980	+	0.562232i	$0.190057\pi$
$107$			11.6315i			1.12446i	−0.826980	+	0.562232i	$0.809943\pi$
$108$		0			0
$109$	−4.89898			−0.469237			−0.234619	−	0.972088i	$-0.575384\pi$
$109$	−4.89898			−0.469237			−0.234619	+	0.972088i	$0.575384\pi$
$110$		0			0
$111$	−2.04989			−0.194567
$112$		0			0
$113$	−10.7980			−1.01579			−0.507893	−	0.861420i	$-0.669576\pi$
$113$	−10.7980			−1.01579			−0.507893	+	0.861420i	$0.669576\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	12.8990i		−	1.19251i
$118$		0			0
$119$	3.99624	+	9.51023i	0.366334	+	0.871801i
$120$		0			0
$121$	10.8990			0.990816
$122$		0			0
$123$			2.51059i			0.226372i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	13.2207i		−	1.17315i	−0.809895	−	0.586575i	$-0.800476\pi$
$127$		−	13.2207i		−	1.17315i	0.809895	−	0.586575i	$-0.199524\pi$
$128$		0			0
$129$			1.59592i			0.140513i
$130$		0			0
$131$	−9.12096			−0.796902			−0.398451	−	0.917190i	$-0.630452\pi$
$131$	−9.12096			−0.796902			−0.398451	+	0.917190i	$0.630452\pi$
$132$		0			0
$133$	−6.12372	+	2.57321i	−0.530994	+	0.223126i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.10102			−0.179502			−0.0897511	−	0.995964i	$-0.528607\pi$
$137$	−2.10102			−0.179502			−0.0897511	+	0.995964i	$0.528607\pi$
$138$		0			0
$139$	−7.53177			−0.638836			−0.319418	−	0.947614i	$-0.603487\pi$
$139$	−7.53177			−0.638836			−0.319418	+	0.947614i	$0.603487\pi$
$140$		0			0
$141$	1.75255			0.147591
$142$		0			0
$143$	1.41421			0.118262
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−1.55708	+	1.58919i	−0.128426	+	0.131074i
$148$		0			0
$149$	14.0000			1.14692			0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000			1.14692			0.573462	+	0.819232i	$0.305600\pi$
$150$		0			0
$151$			13.3636i			1.08751i	0.839243	+	0.543757i	$0.182999\pi$
$151$			13.3636i			1.08751i	−0.839243	+	0.543757i	$0.817001\pi$
$152$		0			0
$153$			11.3031i			0.913798i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	3.55051i		−	0.283362i	−0.989912	−	0.141681i	$-0.954749\pi$
$157$		−	3.55051i		−	0.283362i	0.989912	−	0.141681i	$-0.0452507\pi$
$158$		0			0
$159$	1.12848			0.0894946
$160$		0			0
$161$	−7.24745	−	17.2474i	−0.571179	−	1.35929i
$162$		0			0
$163$			7.53177i			0.589934i	0.955507	+	0.294967i	$0.0953086\pi$
$163$			7.53177i			0.589934i	−0.955507	+	0.294967i	$0.904691\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−17.6062			−1.36241			−0.681206	−	0.732092i	$-0.738544\pi$
$167$	−17.6062			−1.36241			−0.681206	+	0.732092i	$0.738544\pi$
$168$		0			0
$169$	−6.79796			−0.522920
$170$		0			0
$171$	−7.27815			−0.556574
$172$		0			0
$173$			17.1464i			1.30362i	0.758383	+	0.651809i	$0.225990\pi$
$173$			17.1464i			1.30362i	−0.758383	+	0.651809i	$0.774010\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	4.20204			0.315845
$178$		0			0
$179$		−	23.9309i		−	1.78868i	−0.447390	−	0.894339i	$-0.647647\pi$
$179$		−	23.9309i		−	1.78868i	0.447390	−	0.894339i	$-0.352353\pi$
$180$		0			0
$181$		−	2.89898i		−	0.215479i	−0.994179	−	0.107740i	$-0.965639\pi$
$181$		−	2.89898i		−	0.215479i	0.994179	−	0.107740i	$-0.0343613\pi$
$182$		0			0
$183$			4.09978i			0.303064i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−1.23924			−0.0906223
$188$		0			0
$189$	−4.57321	+	1.92168i	−0.332652	+	0.139782i
$190$		0			0
$191$		−	10.3923i		−	0.751961i	−0.926628	−	0.375980i	$-0.877306\pi$
$191$		−	10.3923i		−	0.751961i	0.926628	−	0.375980i	$-0.122694\pi$
$192$		0			0
$193$	0.797959			0.0574383			0.0287192	−	0.999588i	$-0.490857\pi$
$193$	0.797959			0.0574383			0.0287192	+	0.999588i	$0.490857\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	8.69694			0.619631			0.309816	−	0.950797i	$-0.399733\pi$
$197$	8.69694			0.619631			0.309816	+	0.950797i	$0.399733\pi$
$198$		0			0
$199$	16.1920			1.14782			0.573911	−	0.818918i	$-0.305426\pi$
$199$	16.1920			1.14782			0.573911	+	0.818918i	$0.305426\pi$
$200$		0			0
$201$			2.10102i			0.148195i
$202$		0			0
$203$	−10.8530	+	4.56048i	−0.761732	+	0.320083i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	20.4989i		−	1.42477i
$208$		0			0
$209$		−	0.797959i		−	0.0551960i
$210$		0			0
$211$			8.80312i			0.606032i	0.952986	+	0.303016i	$0.0979935\pi$
$211$			8.80312i			0.606032i	−0.952986	+	0.303016i	$0.902006\pi$
$212$		0			0
$213$			2.24745i			0.153993i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	17.2474	−	7.24745i	1.17083	−	0.491989i
$218$		0			0
$219$		−	2.51059i		−	0.169650i
$220$		0			0
$221$	17.3485			1.16698
$222$		0			0
$223$	13.0779			0.875759			0.437879	−	0.899034i	$-0.355730\pi$
$223$	13.0779			0.875759			0.437879	+	0.899034i	$0.355730\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	4.73545			0.314303			0.157151	−	0.987575i	$-0.449769\pi$
$227$	4.73545			0.314303			0.157151	+	0.987575i	$0.449769\pi$
$228$		0			0
$229$			0.651531i			0.0430544i	0.999768	+	0.0215272i	$0.00685284\pi$
$229$			0.651531i			0.0430544i	−0.999768	+	0.0215272i	$0.993147\pi$
$230$		0			0
$231$	−0.103540	−	0.246405i	−0.00681246	−	0.0162123i
$232$		0			0
$233$	12.8990			0.845040			0.422520	−	0.906354i	$-0.361146\pi$
$233$	12.8990			0.845040			0.422520	+	0.906354i	$0.361146\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			0.494897i			0.0321470i
$238$		0			0
$239$			0.635674i			0.0411184i	0.999789	+	0.0205592i	$0.00654465\pi$
$239$			0.635674i			0.0411184i	−0.999789	+	0.0205592i	$0.993455\pi$
$240$		0			0
$241$		−	5.00000i		−	0.322078i	−0.986948	−	0.161039i	$-0.948515\pi$
$241$		−	5.00000i		−	0.322078i	0.986948	−	0.161039i	$-0.0514845\pi$
$242$		0			0
$243$	−8.19955			−0.526002
$244$		0			0
$245$		0			0
$246$		0			0
$247$			11.1708i			0.710784i
$248$		0			0
$249$	−0.303062			−0.0192057
$250$		0			0
$251$	−25.7737			−1.62682			−0.813410	−	0.581691i	$-0.802391\pi$
$251$	−25.7737			−1.62682			−0.813410	+	0.581691i	$0.802391\pi$
$252$		0			0
$253$	2.24745			0.141296
$254$		0			0
$255$		0			0
$256$		0			0
$257$			11.5959i			0.723333i	0.932307	+	0.361667i	$0.117792\pi$
$257$			11.5959i			0.723333i	−0.932307	+	0.361667i	$0.882208\pi$
$258$		0			0
$259$	−15.7313	+	6.61037i	−0.977497	+	0.410748i
$260$		0			0
$261$	−12.8990			−0.798427
$262$		0			0
$263$		−	13.2207i		−	0.815225i	−0.913155	−	0.407613i	$-0.866361\pi$
$263$		−	13.2207i		−	0.815225i	0.913155	−	0.407613i	$-0.133639\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			2.51059i			0.153646i
$268$		0			0
$269$		−	32.2474i		−	1.96616i	−0.183172	−	0.983081i	$-0.558637\pi$
$269$		−	32.2474i		−	1.96616i	0.183172	−	0.983081i	$-0.441363\pi$
$270$		0			0
$271$	0.921404			0.0559713			0.0279856	−	0.999608i	$-0.491091\pi$
$271$	0.921404			0.0559713			0.0279856	+	0.999608i	$0.491091\pi$
$272$		0			0
$273$	1.44949	+	3.44949i	0.0877271	+	0.208773i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−28.0454			−1.68509			−0.842543	−	0.538630i	$-0.818942\pi$
$277$	−28.0454			−1.68509			−0.842543	+	0.538630i	$0.818942\pi$
$278$		0			0
$279$	20.4989			1.22724
$280$		0			0
$281$	9.79796			0.584497			0.292249	−	0.956342i	$-0.405597\pi$
$281$	9.79796			0.584497			0.292249	+	0.956342i	$0.405597\pi$
$282$		0			0
$283$	3.78194			0.224813			0.112406	−	0.993662i	$-0.464144\pi$
$283$	3.78194			0.224813			0.112406	+	0.993662i	$0.464144\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	8.09601	+	19.2669i	0.477892	+	1.13729i
$288$		0			0
$289$	1.79796			0.105762
$290$		0			0
$291$			4.44972i			0.260847i
$292$		0			0
$293$		−	19.5959i		−	1.14481i	−0.819973	−	0.572403i	$-0.806011\pi$
$293$		−	19.5959i		−	1.14481i	0.819973	−	0.572403i	$-0.193989\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		−	0.595918i		−	0.0345787i
$298$		0			0
$299$	−31.4626			−1.81953
$300$		0			0
$301$	5.14643	+	12.2474i	0.296635	+	0.705931i
$302$		0			0
$303$			4.30686i			0.247422i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	10.9959			0.627568			0.313784	−	0.949494i	$-0.398403\pi$
$307$	10.9959			0.627568			0.313784	+	0.949494i	$0.398403\pi$
$308$		0			0
$309$	−3.30306			−0.187905
$310$		0			0
$311$	26.4415			1.49936			0.749679	−	0.661802i	$-0.230208\pi$
$311$	26.4415			1.49936			0.749679	+	0.661802i	$0.230208\pi$
$312$		0			0
$313$		−	6.69694i		−	0.378533i	−0.981926	−	0.189267i	$-0.939389\pi$
$313$		−	6.69694i		−	0.378533i	0.981926	−	0.189267i	$-0.0606111\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	5.79796			0.325646			0.162823	−	0.986655i	$-0.447940\pi$
$317$	5.79796			0.325646			0.162823	+	0.986655i	$0.447940\pi$
$318$		0			0
$319$		−	1.41421i		−	0.0791808i
$320$		0			0
$321$		−	3.69694i		−	0.206343i
$322$		0			0
$323$		−	9.78874i		−	0.544660i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	1.55708			0.0861066
$328$		0			0
$329$	13.4495	−	5.65153i	0.741494	−	0.311579i
$330$		0			0
$331$		−	24.8523i		−	1.36600i	−0.730416	−	0.683002i	$-0.760674\pi$
$331$		−	24.8523i		−	1.36600i	0.730416	−	0.683002i	$-0.239326\pi$
$332$		0			0
$333$	−18.6969			−1.02459
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−23.6969			−1.29085			−0.645427	−	0.763822i	$-0.723321\pi$
$337$	−23.6969			−1.29085			−0.645427	+	0.763822i	$0.723321\pi$
$338$		0			0
$339$	3.43199			0.186400
$340$		0			0
$341$			2.24745i			0.121706i
$342$		0			0
$343$	−6.82466	+	17.2170i	−0.368497	+	0.929629i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	33.9732i		−	1.82378i	−0.410436	−	0.911889i	$-0.634624\pi$
$347$		−	33.9732i		−	1.82378i	0.410436	−	0.911889i	$-0.365376\pi$
$348$		0			0
$349$			20.6515i			1.10545i	0.833363	+	0.552726i	$0.186412\pi$
$349$			20.6515i			1.10545i	−0.833363	+	0.552726i	$0.813588\pi$
$350$		0			0
$351$			8.34242i			0.445285i
$352$		0			0
$353$		−	15.1010i		−	0.803746i	−0.915695	−	0.401873i	$-0.868359\pi$
$353$		−	15.1010i		−	0.803746i	0.915695	−	0.401873i	$-0.131641\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−1.27015	−	3.02270i	−0.0672236	−	0.159978i
$358$		0			0
$359$			36.4838i			1.92554i	0.270317	+	0.962771i	$0.412871\pi$
$359$			36.4838i			1.92554i	−0.270317	+	0.962771i	$0.587129\pi$
$360$		0			0
$361$	−12.6969			−0.668260
$362$		0			0
$363$	−3.46410			−0.181818
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−19.7990			−1.03350			−0.516749	−	0.856137i	$-0.672858\pi$
$367$	−19.7990			−1.03350			−0.516749	+	0.856137i	$0.672858\pi$
$368$		0			0
$369$			22.8990i			1.19207i
$370$		0			0
$371$	8.66025	−	3.63907i	0.449618	−	0.188931i
$372$		0			0
$373$	−20.9444			−1.08446			−0.542230	−	0.840230i	$-0.682420\pi$
$373$	−20.9444			−1.08446			−0.542230	+	0.840230i	$0.682420\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			19.7980i			1.01965i
$378$		0			0
$379$		−	20.7525i		−	1.06598i	−0.846120	−	0.532992i	$-0.821068\pi$
$379$		−	20.7525i		−	1.06598i	0.846120	−	0.532992i	$-0.178932\pi$
$380$		0			0
$381$			4.20204i			0.215277i
$382$		0			0
$383$	−0.285729			−0.0146001			−0.00730004	−	0.999973i	$-0.502324\pi$
$383$	−0.285729			−0.0146001			−0.00730004	+	0.999973i	$0.502324\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			14.5563i			0.739938i
$388$		0			0
$389$	−16.0454			−0.813534			−0.406767	−	0.913532i	$-0.633344\pi$
$389$	−16.0454			−0.813534			−0.406767	+	0.913532i	$0.633344\pi$
$390$		0			0
$391$	27.5699			1.39427
$392$		0			0
$393$	2.89898			0.146234
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	10.4495i		−	0.524445i	−0.965007	−	0.262222i	$-0.915545\pi$
$397$		−	10.4495i		−	0.524445i	0.965007	−	0.262222i	$-0.0844554\pi$
$398$		0			0
$399$	1.94635	−	0.817863i	0.0974393	−	0.0409444i
$400$		0			0
$401$	−19.6969			−0.983618			−0.491809	−	0.870703i	$-0.663664\pi$
$401$	−19.6969			−0.983618			−0.491809	+	0.870703i	$0.663664\pi$
$402$		0			0
$403$		−	31.4626i		−	1.56727i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	2.04989i		−	0.101609i
$408$		0			0
$409$			10.7980i			0.533925i	0.963707	+	0.266962i	$0.0860199\pi$
$409$			10.7980i			0.533925i	−0.963707	+	0.266962i	$0.913980\pi$
$410$		0			0
$411$	0.667783			0.0329393
$412$		0			0
$413$	32.2474	−	13.5505i	1.58679	−	0.666777i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	2.39388			0.117229
$418$		0			0
$419$	−14.8099			−0.723512			−0.361756	−	0.932273i	$-0.617823\pi$
$419$	−14.8099			−0.723512			−0.361756	+	0.932273i	$0.617823\pi$
$420$		0			0
$421$	−4.00000			−0.194948			−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000			−0.194948			−0.0974740	+	0.995238i	$0.531076\pi$
$422$		0			0
$423$	15.9849			0.777215
$424$		0			0
$425$		0			0
$426$		0			0
$427$	13.2207	+	31.4626i	0.639796	+	1.52258i
$428$		0			0
$429$	−0.449490			−0.0217016
$430$		0			0
$431$			25.6629i			1.23614i	0.786123	+	0.618070i	$0.212085\pi$
$431$			25.6629i			1.23614i	−0.786123	+	0.618070i	$0.787915\pi$
$432$		0			0
$433$			13.6969i			0.658233i	0.944289	+	0.329116i	$0.106751\pi$
$433$			13.6969i			0.658233i	−0.944289	+	0.329116i	$0.893249\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			17.7526i			0.849220i
$438$		0			0
$439$	33.3055			1.58958			0.794791	−	0.606883i	$-0.207580\pi$
$439$	33.3055			1.58958			0.794791	+	0.606883i	$0.207580\pi$
$440$		0			0
$441$	−14.2020	+	14.4949i	−0.676288	+	0.690233i
$442$		0			0
$443$		−	16.6527i		−	0.791195i	−0.918424	−	0.395597i	$-0.870538\pi$
$443$		−	16.6527i		−	0.791195i	0.918424	−	0.395597i	$-0.129462\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−4.44972			−0.210465
$448$		0			0
$449$	7.20204			0.339885			0.169943	−	0.985454i	$-0.445642\pi$
$449$	7.20204			0.339885			0.169943	+	0.985454i	$0.445642\pi$
$450$		0			0
$451$	−2.51059			−0.118219
$452$		0			0
$453$		−	4.24745i		−	0.199563i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	27.8990			1.30506			0.652530	−	0.757763i	$-0.273708\pi$
$457$	27.8990			1.30506			0.652530	+	0.757763i	$0.273708\pi$
$458$		0			0
$459$		−	7.31026i		−	0.341214i
$460$		0			0
$461$			32.2474i			1.50191i	0.660351	+	0.750957i	$0.270407\pi$
$461$			32.2474i			1.50191i	−0.660351	+	0.750957i	$0.729593\pi$
$462$		0			0
$463$			14.1421i			0.657241i	0.944462	+	0.328620i	$0.106584\pi$
$463$			14.1421i			0.657241i	−0.944462	+	0.328620i	$0.893416\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	38.0409			1.76032			0.880161	−	0.474674i	$-0.157434\pi$
$467$	38.0409			1.76032			0.880161	+	0.474674i	$0.157434\pi$
$468$		0			0
$469$	6.77526	+	16.1237i	0.312852	+	0.744524i
$470$		0			0
$471$			1.12848i			0.0519978i
$472$		0			0
$473$	−1.59592			−0.0733804
$474$		0			0
$475$		0			0
$476$		0			0
$477$	10.2929			0.471278
$478$		0			0
$479$	13.2207			0.604071			0.302035	−	0.953297i	$-0.402334\pi$
$479$	13.2207			0.604071			0.302035	+	0.953297i	$0.402334\pi$
$480$		0			0
$481$			28.6969i			1.30847i
$482$		0			0
$483$	2.30351	+	5.48188i	0.104813	+	0.249434i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	39.6622i		−	1.79727i	−0.438702	−	0.898633i	$-0.644561\pi$
$487$		−	39.6622i		−	1.79727i	0.438702	−	0.898633i	$-0.355439\pi$
$488$		0			0
$489$		−	2.39388i		−	0.108255i
$490$		0			0
$491$		−	0.349945i		−	0.0157928i	−0.999969	−	0.00789641i	$-0.997486\pi$
$491$		−	0.349945i		−	0.0157928i	0.999969	−	0.00789641i	$-0.00251353\pi$
$492$		0			0
$493$		−	17.3485i		−	0.781336i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	7.24745	+	17.2474i	0.325093	+	0.773654i
$498$		0			0
$499$			35.8481i			1.60478i	0.596798	+	0.802392i	$0.296440\pi$
$499$			35.8481i			1.60478i	−0.596798	+	0.802392i	$0.703560\pi$
$500$		0			0
$501$	5.59592			0.250007
$502$		0			0
$503$	0.778539			0.0347133			0.0173567	−	0.999849i	$-0.494475\pi$
$503$	0.778539			0.0347133			0.0173567	+	0.999849i	$0.494475\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	2.16064			0.0959576
$508$		0			0
$509$		−	4.20204i		−	0.186252i	−0.995654	−	0.0931261i	$-0.970314\pi$
$509$		−	4.20204i		−	0.186252i	0.995654	−	0.0931261i	$-0.0296860\pi$
$510$		0			0
$511$	−8.09601	−	19.2669i	−0.358146	−	0.852316i
$512$		0			0
$513$	4.70714			0.207825
$514$		0			0
$515$		0			0
$516$		0			0
$517$			1.75255i			0.0770771i
$518$		0			0
$519$		−	5.44977i		−	0.239219i
$520$		0			0
$521$			19.4949i			0.854087i	0.904231	+	0.427043i	$0.140445\pi$
$521$			19.4949i			0.854087i	−0.904231	+	0.427043i	$0.859555\pi$
$522$		0			0
$523$	−19.1954			−0.839357			−0.419679	−	0.907673i	$-0.637857\pi$
$523$	−19.1954			−0.839357			−0.419679	+	0.907673i	$0.637857\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			27.5699i			1.20097i
$528$		0			0
$529$	−27.0000			−1.17391
$530$		0			0
$531$	38.3266			1.66323
$532$		0			0
$533$	35.1464			1.52236
$534$		0			0
$535$		0			0
$536$		0			0
$537$			7.60612i			0.328228i
$538$		0			0
$539$	−1.58919	−	1.55708i	−0.0684511	−	0.0670681i
$540$		0			0
$541$	−30.4495			−1.30913			−0.654563	−	0.756008i	$-0.727147\pi$
$541$	−30.4495			−1.30913			−0.654563	+	0.756008i	$0.727147\pi$
$542$		0			0
$543$			0.921404i			0.0395412i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			1.58919i			0.0679487i	0.999423	+	0.0339743i	$0.0108165\pi$
$547$			1.58919i			0.0679487i	−0.999423	+	0.0339743i	$0.989184\pi$
$548$		0			0
$549$			37.3939i			1.59593i
$550$		0			0
$551$	11.1708			0.475894
$552$		0			0
$553$	1.59592	+	3.79796i	0.0678653	+	0.161506i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−41.5959			−1.76248			−0.881238	−	0.472673i	$-0.843289\pi$
$557$	−41.5959			−1.76248			−0.881238	+	0.472673i	$0.843289\pi$
$558$		0			0
$559$	22.3417			0.944953
$560$		0			0
$561$	0.393877			0.0166295
$562$		0			0
$563$	−47.1619			−1.98764			−0.993818	−	0.111025i	$-0.964587\pi$
$563$	−47.1619			−1.98764			−0.993818	+	0.111025i	$0.964587\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−19.7597	+	8.30309i	−0.829828	+	0.348697i
$568$		0			0
$569$	5.20204			0.218081			0.109040	−	0.994037i	$-0.465222\pi$
$569$	5.20204			0.218081			0.109040	+	0.994037i	$0.465222\pi$
$570$		0			0
$571$			40.5836i			1.69837i	0.528095	+	0.849185i	$0.322907\pi$
$571$			40.5836i			1.69837i	−0.528095	+	0.849185i	$0.677093\pi$
$572$		0			0
$573$			3.30306i			0.137987i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			28.3939i			1.18205i	0.806652	+	0.591026i	$0.201277\pi$
$577$			28.3939i			1.18205i	−0.806652	+	0.591026i	$0.798723\pi$
$578$		0			0
$579$	−0.253621			−0.0105401
$580$		0			0
$581$	−2.32577	+	0.977296i	−0.0964890	+	0.0405451i
$582$		0			0
$583$			1.12848i			0.0467370i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	12.9029			0.532559			0.266280	−	0.963896i	$-0.414206\pi$
$587$	12.9029			0.532559			0.266280	+	0.963896i	$0.414206\pi$
$588$		0			0
$589$	−17.7526			−0.731481
$590$		0			0
$591$	−2.76421			−0.113704
$592$		0			0
$593$		−	38.7980i		−	1.59324i	−0.604480	−	0.796621i	$-0.706619\pi$
$593$		−	38.7980i		−	1.59324i	0.604480	−	0.796621i	$-0.293381\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−5.14643			−0.210629
$598$		0			0
$599$		−	40.0908i		−	1.63807i	−0.573747	−	0.819033i	$-0.694511\pi$
$599$		−	40.0908i		−	1.63807i	0.573747	−	0.819033i	$-0.305489\pi$
$600$		0			0
$601$			20.7980i			0.848366i	0.905576	+	0.424183i	$0.139439\pi$
$601$			20.7980i			0.848366i	−0.905576	+	0.424183i	$0.860561\pi$
$602$		0			0
$603$			19.1633i			0.780391i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−39.0265			−1.58404			−0.792019	−	0.610497i	$-0.790970\pi$
$607$	−39.0265			−1.58404			−0.792019	+	0.610497i	$0.790970\pi$
$608$		0			0
$609$	3.44949	−	1.44949i	0.139780	−	0.0587363i
$610$		0			0
$611$		−	24.5344i		−	0.992557i
$612$		0			0
$613$	2.89898			0.117089			0.0585443	−	0.998285i	$-0.481354\pi$
$613$	2.89898			0.117089			0.0585443	+	0.998285i	$0.481354\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−41.5959			−1.67459			−0.837294	−	0.546752i	$-0.815864\pi$
$617$	−41.5959			−1.67459			−0.837294	+	0.546752i	$0.815864\pi$
$618$		0			0
$619$	−25.5201			−1.02574			−0.512869	−	0.858467i	$-0.671417\pi$
$619$	−25.5201			−1.02574			−0.512869	+	0.858467i	$0.671417\pi$
$620$		0			0
$621$			13.2577i			0.532011i
$622$		0			0
$623$	8.09601	+	19.2669i	0.324360	+	0.771910i
$624$		0			0
$625$		0			0
$626$		0			0
$627$			0.253621i			0.0101287i
$628$		0			0
$629$		−	25.1464i		−	1.00265i
$630$		0			0
$631$			40.4407i			1.60992i	0.593329	+	0.804960i	$0.297813\pi$
$631$			40.4407i			1.60992i	−0.593329	+	0.804960i	$0.702187\pi$
$632$		0			0
$633$		−	2.79796i		−	0.111209i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	22.2474	+	21.7980i	0.881476	+	0.863667i
$638$		0			0
$639$			20.4989i			0.810923i
$640$		0			0
$641$	26.6969			1.05447			0.527233	−	0.849721i	$-0.323230\pi$
$641$	26.6969			1.05447			0.527233	+	0.849721i	$0.323230\pi$
$642$		0			0
$643$	−48.4332			−1.91002			−0.955010	−	0.296575i	$-0.904156\pi$
$643$	−48.4332			−1.91002			−0.955010	+	0.296575i	$0.904156\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	37.4694			1.47308			0.736538	−	0.676396i	$-0.236459\pi$
$647$	37.4694			1.47308			0.736538	+	0.676396i	$0.236459\pi$
$648$		0			0
$649$			4.20204i			0.164945i
$650$		0			0
$651$	−5.48188	+	2.30351i	−0.214852	+	0.0902816i
$652$		0			0
$653$	−0.651531			−0.0254964			−0.0127482	−	0.999919i	$-0.504058\pi$
$653$	−0.651531			−0.0254964			−0.0127482	+	0.999919i	$0.504058\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	22.8990i		−	0.893374i
$658$		0			0
$659$		−	14.1742i		−	0.552150i	−0.961136	−	0.276075i	$-0.910966\pi$
$659$		−	14.1742i		−	0.552150i	0.961136	−	0.276075i	$-0.0890338\pi$
$660$		0			0
$661$			40.9444i			1.59255i	0.604933	+	0.796276i	$0.293200\pi$
$661$			40.9444i			1.59255i	−0.604933	+	0.796276i	$0.706800\pi$
$662$		0			0
$663$	−5.51399			−0.214146
$664$		0			0
$665$		0			0
$666$		0			0
$667$			31.4626i			1.21824i
$668$		0			0
$669$	−4.15663			−0.160705
$670$		0			0
$671$	−4.09978			−0.158270
$672$		0			0
$673$	−20.0000			−0.770943			−0.385472	−	0.922720i	$-0.625961\pi$
$673$	−20.0000			−0.770943			−0.385472	+	0.922720i	$0.625961\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			42.6969i			1.64098i	0.571663	+	0.820488i	$0.306298\pi$
$677$			42.6969i			1.64098i	−0.571663	+	0.820488i	$0.693702\pi$
$678$		0			0
$679$	14.3492	+	34.1482i	0.550672	+	1.31049i
$680$		0			0
$681$	−1.50510			−0.0576757
$682$		0			0
$683$		−	10.7101i		−	0.409812i	−0.978782	−	0.204906i	$-0.934311\pi$
$683$		−	10.7101i		−	0.409812i	0.978782	−	0.204906i	$-0.0656889\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		−	0.207081i		−	0.00790062i
$688$		0			0
$689$		−	15.7980i		−	0.601854i
$690$		0			0
$691$	37.1516			1.41331			0.706657	−	0.707556i	$-0.250202\pi$
$691$	37.1516			1.41331			0.706657	+	0.707556i	$0.250202\pi$
$692$		0			0
$693$	−0.944387	−	2.24745i	−0.0358743	−	0.0853735i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−30.7980			−1.16656
$698$		0			0
$699$	−4.09978			−0.155068
$700$		0			0
$701$	−4.69694			−0.177401			−0.0887005	−	0.996058i	$-0.528271\pi$
$701$	−4.69694			−0.177401			−0.0887005	+	0.996058i	$0.528271\pi$
$702$		0			0
$703$	16.1920			0.610694
$704$		0			0
$705$		0			0
$706$		0			0
$707$	13.8885	+	33.0518i	0.522331	+	1.24304i
$708$		0			0
$709$	35.1918			1.32166			0.660829	−	0.750537i	$-0.270205\pi$
$709$	35.1918			1.32166			0.660829	+	0.750537i	$0.270205\pi$
$710$		0			0
$711$			4.51394i			0.169286i
$712$		0			0
$713$		−	50.0000i		−	1.87251i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	0.202041i		−	0.00754536i
$718$		0			0
$719$	12.0922			0.450965			0.225482	−	0.974247i	$-0.427604\pi$
$719$	12.0922			0.450965			0.225482	+	0.974247i	$0.427604\pi$
$720$		0			0
$721$	−25.3485	+	10.6515i	−0.944026	+	0.396684i
$722$		0			0
$723$			1.58919i			0.0591025i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	26.5843			0.985958			0.492979	−	0.870041i	$-0.335908\pi$
$727$	26.5843			0.985958			0.492979	+	0.870041i	$0.335908\pi$
$728$		0			0
$729$	−21.6969			−0.803590
$730$		0			0
$731$	−19.5775			−0.724099
$732$		0			0
$733$		−	7.79796i		−	0.288024i	−0.989576	−	0.144012i	$-0.954000\pi$
$733$		−	7.79796i		−	0.288024i	0.989576	−	0.144012i	$-0.0460004\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2.10102			−0.0773921
$738$		0			0
$739$		−	35.8481i		−	1.31870i	−0.751838	−	0.659348i	$-0.770833\pi$
$739$		−	35.8481i		−	1.31870i	0.751838	−	0.659348i	$-0.229167\pi$
$740$		0			0
$741$		−	3.55051i		−	0.130431i
$742$		0			0
$743$		−	41.7121i		−	1.53027i	−0.643871	−	0.765134i	$-0.722673\pi$
$743$		−	41.7121i		−	1.53027i	0.643871	−	0.765134i	$-0.277327\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−2.76421			−0.101137
$748$		0			0
$749$	−11.9217	−	28.3712i	−0.435609	−	1.03666i
$750$		0			0
$751$		−	30.6841i		−	1.11968i	−0.828601	−	0.559839i	$-0.810863\pi$
$751$		−	30.6841i		−	1.11968i	0.828601	−	0.559839i	$-0.189137\pi$
$752$		0			0
$753$	8.19184			0.298527
$754$		0			0
$755$		0			0
$756$		0			0
$757$	32.2474			1.17205			0.586027	−	0.810292i	$-0.300691\pi$
$757$	32.2474			1.17205			0.586027	+	0.810292i	$0.300691\pi$
$758$		0			0
$759$	−0.714323			−0.0259283
$760$		0			0
$761$		−	15.0000i		−	0.543750i	−0.962333	−	0.271875i	$-0.912356\pi$
$761$		−	15.0000i		−	0.543750i	0.962333	−	0.271875i	$-0.0876437\pi$
$762$		0			0
$763$	11.9494	−	5.02118i	0.432597	−	0.181779i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	58.8255i		−	2.12407i
$768$		0			0
$769$			32.3939i			1.16815i	0.811699	+	0.584077i	$0.198543\pi$
$769$			32.3939i			1.16815i	−0.811699	+	0.584077i	$0.801457\pi$
$770$		0			0
$771$		−	3.68561i		−	0.132734i
$772$		0			0
$773$		−	0.696938i		−	0.0250671i	−0.999921	−	0.0125336i	$-0.996010\pi$
$773$		−	0.696938i		−	0.0250671i	0.999921	−	0.0125336i	$-0.00398966\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	5.00000	−	2.10102i	0.179374	−	0.0753737i
$778$		0			0
$779$		−	19.8311i		−	0.710523i
$780$		0			0
$781$	−2.24745			−0.0804200
$782$		0			0
$783$	8.34242			0.298134
$784$		0			0
$785$		0			0
$786$		0			0
$787$	28.9199			1.03088			0.515442	−	0.856924i	$-0.327628\pi$
$787$	28.9199			1.03088			0.515442	+	0.856924i	$0.327628\pi$
$788$		0			0
$789$			4.20204i			0.149597i
$790$		0			0
$791$	26.3379	−	11.0673i	0.936469	−	0.393508i
$792$		0			0
$793$	57.3939			2.03812
$794$		0			0
$795$		0			0
$796$		0			0
$797$			26.8990i			0.952811i	0.879226	+	0.476405i	$0.158061\pi$
$797$			26.8990i			0.952811i	−0.879226	+	0.476405i	$0.841939\pi$
$798$		0			0
$799$			21.4989i			0.760578i
$800$		0			0
$801$			22.8990i			0.809096i
$802$		0			0
$803$	2.51059			0.0885968
$804$		0			0
$805$		0			0
$806$		0			0
$807$			10.2494i			0.360797i
$808$		0			0
$809$	4.89898			0.172239			0.0861195	−	0.996285i	$-0.472553\pi$
$809$	4.89898			0.172239			0.0861195	+	0.996285i	$0.472553\pi$
$810$		0			0
$811$	−4.09978			−0.143963			−0.0719813	−	0.997406i	$-0.522932\pi$
$811$	−4.09978			−0.143963			−0.0719813	+	0.997406i	$0.522932\pi$
$812$		0			0
$813$	−0.292856			−0.0102709
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	12.6061i		−	0.441032i
$818$		0			0
$819$	13.2207	+	31.4626i	0.461970	+	1.09939i
$820$		0			0
$821$	−20.4495			−0.713692			−0.356846	−	0.934163i	$-0.616148\pi$
$821$	−20.4495			−0.713692			−0.356846	+	0.934163i	$0.616148\pi$
$822$		0			0
$823$		−	24.3916i		−	0.850237i	−0.905138	−	0.425118i	$-0.860232\pi$
$823$		−	24.3916i		−	0.850237i	0.905138	−	0.425118i	$-0.139768\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	38.9944i		−	1.35597i	−0.735076	−	0.677984i	$-0.762854\pi$
$827$		−	38.9944i		−	1.35597i	0.735076	−	0.677984i	$-0.237146\pi$
$828$		0			0
$829$			34.4949i			1.19806i	0.800728	+	0.599029i	$0.204446\pi$
$829$			34.4949i			1.19806i	−0.800728	+	0.599029i	$0.795554\pi$
$830$		0			0
$831$	8.91388			0.309219
$832$		0			0
$833$	−19.4949	−	19.1010i	−0.675458	−	0.661811i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−13.2577			−0.458252
$838$		0			0
$839$	−12.0922			−0.417471			−0.208735	−	0.977972i	$-0.566935\pi$
$839$	−12.0922			−0.417471			−0.208735	+	0.977972i	$0.566935\pi$
$840$		0			0
$841$	−9.20204			−0.317312
$842$		0			0
$843$	−3.11416			−0.107257
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−26.5843	+	11.1708i	−0.913448	+	0.383835i
$848$		0			0
$849$	−1.20204			−0.0412539
$850$		0			0
$851$			45.6048i			1.56331i
$852$		0			0
$853$			21.5959i			0.739430i	0.929145	+	0.369715i	$0.120545\pi$
$853$			21.5959i			0.739430i	−0.929145	+	0.369715i	$0.879455\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	22.1010i		−	0.754956i	−0.926018	−	0.377478i	$-0.876791\pi$
$857$		−	22.1010i		−	0.754956i	0.926018	−	0.377478i	$-0.123209\pi$
$858$		0			0
$859$	11.6315			0.396863			0.198432	−	0.980115i	$-0.436415\pi$
$859$	11.6315			0.396863			0.198432	+	0.980115i	$0.436415\pi$
$860$		0			0
$861$	−2.57321	−	6.12372i	−0.0876949	−	0.208696i
$862$		0			0
$863$			15.0635i			0.512769i	0.966575	+	0.256384i	$0.0825313\pi$
$863$			15.0635i			0.512769i	−0.966575	+	0.256384i	$0.917469\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.571458			−0.0194077
$868$		0			0
$869$	−0.494897			−0.0167882
$870$		0			0
$871$	29.4128			0.996613
$872$		0			0
$873$			40.5857i			1.37362i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	5.79796			0.195783			0.0978916	−	0.995197i	$-0.468790\pi$
$877$	5.79796			0.195783			0.0978916	+	0.995197i	$0.468790\pi$
$878$		0			0
$879$			6.22831i			0.210076i
$880$		0			0
$881$			7.10102i			0.239239i	0.992820	+	0.119620i	$0.0381675\pi$
$881$			7.10102i			0.239239i	−0.992820	+	0.119620i	$0.961832\pi$
$882$		0			0
$883$		−	7.53177i		−	0.253464i	−0.991937	−	0.126732i	$-0.959551\pi$
$883$		−	7.53177i		−	0.253464i	0.991937	−	0.126732i	$-0.0404489\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	17.1776			0.576769			0.288384	−	0.957515i	$-0.406882\pi$
$887$	17.1776			0.576769			0.288384	+	0.957515i	$0.406882\pi$
$888$		0			0
$889$	13.5505	+	32.2474i	0.454470	+	1.08154i
$890$		0			0
$891$		−	2.57481i		−	0.0862592i
$892$		0			0
$893$	−13.8434			−0.463251
$894$		0			0
$895$		0			0
$896$		0			0
$897$	10.0000			0.333890
$898$		0			0
$899$	−31.4626			−1.04934
$900$		0			0
$901$			13.8434i			0.461189i
$902$		0			0
$903$	−1.63573	−	3.89270i	−0.0544336	−	0.129541i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	56.9827i		−	1.89208i	−0.324050	−	0.946040i	$-0.605045\pi$
$907$		−	56.9827i		−	1.89208i	0.324050	−	0.946040i	$-0.394955\pi$
$908$		0			0
$909$			39.2827i			1.30292i
$910$		0			0
$911$			5.94258i			0.196887i	0.995143	+	0.0984433i	$0.0313863\pi$
$911$			5.94258i			0.196887i	−0.995143	+	0.0984433i	$0.968614\pi$
$912$		0			0
$913$		−	0.303062i		−	0.0100299i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	22.2474	−	9.34847i	0.734675	−	0.308714i
$918$		0			0
$919$			2.89264i			0.0954195i	0.998861	+	0.0477097i	$0.0151923\pi$
$919$			2.89264i			0.0954195i	−0.998861	+	0.0477097i	$0.984808\pi$
$920$		0			0
$921$	−3.49490			−0.115161
$922$		0			0
$923$	31.4626			1.03561
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−30.1271			−0.989503
$928$		0			0
$929$		−	34.2020i		−	1.12213i	−0.827771	−	0.561066i	$-0.810391\pi$
$929$		−	34.2020i		−	1.12213i	0.827771	−	0.561066i	$-0.189609\pi$
$930$		0			0
$931$	12.2993	−	12.5529i	0.403094	−	0.411406i
$932$		0			0
$933$	−8.40408			−0.275137
$934$		0			0
$935$		0			0
$936$		0			0
$937$			27.6969i			0.904820i	0.891810	+	0.452410i	$0.149436\pi$
$937$			27.6969i			0.904820i	−0.891810	+	0.452410i	$0.850564\pi$
$938$		0			0
$939$			2.12854i			0.0694622i
$940$		0			0
$941$		−	27.3939i		−	0.893015i	−0.894780	−	0.446507i	$-0.852668\pi$
$941$		−	27.3939i		−	0.893015i	0.894780	−	0.446507i	$-0.147332\pi$
$942$		0			0
$943$	55.8542			1.81886
$944$		0			0
$945$		0			0
$946$		0			0
$947$			49.7046i			1.61518i	0.589744	+	0.807591i	$0.299229\pi$
$947$			49.7046i			1.61518i	−0.589744	+	0.807591i	$0.700771\pi$
$948$		0			0
$949$	−35.1464			−1.14090
$950$		0			0
$951$	−1.84281			−0.0597571
$952$		0			0
$953$	−55.2929			−1.79111			−0.895556	−	0.444950i	$-0.853222\pi$
$953$	−55.2929			−1.79111			−0.895556	+	0.444950i	$0.853222\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			0.449490i			0.0145299i
$958$		0			0
$959$	5.12472	−	2.15343i	0.165486	−	0.0695378i
$960$		0			0
$961$	19.0000			0.612903
$962$		0			0
$963$		−	33.7196i		−	1.08660i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	6.14966i		−	0.197760i	−0.995099	−	0.0988799i	$-0.968474\pi$
$967$		−	6.14966i		−	0.197760i	0.995099	−	0.0988799i	$-0.0315260\pi$
$968$		0			0
$969$			3.11123i			0.0999470i
$970$		0			0
$971$	−46.2726			−1.48496			−0.742479	−	0.669870i	$-0.766350\pi$
$971$	−46.2726			−1.48496			−0.742479	+	0.669870i	$0.766350\pi$
$972$		0			0
$973$	18.3712	−	7.71964i	0.588953	−	0.247480i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−39.2020			−1.25418			−0.627092	−	0.778945i	$-0.715755\pi$
$977$	−39.2020			−1.25418			−0.627092	+	0.778945i	$0.715755\pi$
$978$		0			0
$979$	−2.51059			−0.0802388
$980$		0			0
$981$	14.2020			0.453436
$982$		0			0
$983$	−41.3621			−1.31925			−0.659624	−	0.751596i	$-0.729284\pi$
$983$	−41.3621			−1.31925			−0.659624	+	0.751596i	$0.729284\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−4.27475	+	1.79627i	−0.136067	+	0.0571758i
$988$		0			0
$989$	35.5051			1.12900
$990$		0			0
$991$			7.99247i			0.253889i	0.991910	+	0.126945i	$0.0405170\pi$
$991$			7.99247i			0.253889i	−0.991910	+	0.126945i	$0.959483\pi$
$992$		0			0
$993$			7.89898i			0.250667i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	40.7423i		−	1.29032i	−0.764046	−	0.645162i	$-0.776790\pi$
$997$		−	40.7423i		−	1.29032i	0.764046	−	0.645162i	$-0.223210\pi$
$998$		0			0
$999$	12.0922			0.382582

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2800.2.k.o.2351.4	yes	8
4.3	odd	2	inner	2800.2.k.o.2351.5	yes	8
5.2	odd	4		2800.2.e.f.2799.5		8
5.3	odd	4		2800.2.e.e.2799.4		8
5.4	even	2		2800.2.k.n.2351.5	yes	8
7.6	odd	2	inner	2800.2.k.o.2351.6	yes	8
20.3	even	4		2800.2.e.e.2799.5		8
20.7	even	4		2800.2.e.f.2799.4		8
20.19	odd	2		2800.2.k.n.2351.4	yes	8
28.27	even	2	inner	2800.2.k.o.2351.3	yes	8
35.13	even	4		2800.2.e.f.2799.6		8
35.27	even	4		2800.2.e.e.2799.3		8
35.34	odd	2		2800.2.k.n.2351.3	✓	8
140.27	odd	4		2800.2.e.e.2799.6		8
140.83	odd	4		2800.2.e.f.2799.3		8
140.139	even	2		2800.2.k.n.2351.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2800.2.e.e.2799.3		8	35.27	even	4
2800.2.e.e.2799.4		8	5.3	odd	4
2800.2.e.e.2799.5		8	20.3	even	4
2800.2.e.e.2799.6		8	140.27	odd	4
2800.2.e.f.2799.3		8	140.83	odd	4
2800.2.e.f.2799.4		8	20.7	even	4
2800.2.e.f.2799.5		8	5.2	odd	4
2800.2.e.f.2799.6		8	35.13	even	4
2800.2.k.n.2351.3	✓	8	35.34	odd	2
2800.2.k.n.2351.4	yes	8	20.19	odd	2
2800.2.k.n.2351.5	yes	8	5.4	even	2
2800.2.k.n.2351.6	yes	8	140.139	even	2
2800.2.k.o.2351.3	yes	8	28.27	even	2	inner
2800.2.k.o.2351.4	yes	8	1.1	even	1	trivial
2800.2.k.o.2351.5	yes	8	4.3	odd	2	inner
2800.2.k.o.2351.6	yes	8	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2800.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.317837 q^{3} +(-2.43916 + 1.02494i) q^{7} -2.89898 q^{9} +O(q^{10})\)	\(q-0.317837 q^{3} +(-2.43916 + 1.02494i) q^{7} -2.89898 q^{9} -0.317837i q^{11} +4.44949i q^{13} -3.89898i q^{17} +2.51059 q^{19} +(0.775255 - 0.325765i) q^{21} +7.07107i q^{23} +1.87492 q^{27} +4.44949 q^{29} -7.07107 q^{31} +0.101021i q^{33} +6.44949 q^{37} -1.41421i q^{39} -7.89898i q^{41} -5.02118i q^{43} -5.51399 q^{47} +(4.89898 - 5.00000i) q^{49} +1.23924i q^{51} -3.55051 q^{53} -0.797959 q^{57} -13.2207 q^{59} -12.8990i q^{61} +(7.07107 - 2.97129i) q^{63} -6.61037i q^{67} -2.24745i q^{69} -7.07107i q^{71} +7.89898i q^{73} +(0.325765 + 0.775255i) q^{77} -1.55708i q^{79} +8.10102 q^{81} +0.953512 q^{83} -1.41421 q^{87} -7.89898i q^{89} +(-4.56048 - 10.8530i) q^{91} +2.24745 q^{93} -14.0000i q^{97} +0.921404i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{9}+O(q^{10}) \) 8 * q + 16 * q^9	\( 8 q + 16 q^{9} + 16 q^{21} + 16 q^{29} + 32 q^{37} - 48 q^{53} + 72 q^{57} + 32 q^{77} + 104 q^{81} - 80 q^{93}+O(q^{100}) \) 8 * q + 16 * q^9 + 16 * q^21 + 16 * q^29 + 32 * q^37 - 48 * q^53 + 72 * q^57 + 32 * q^77 + 104 * q^81 - 80 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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