LMFDB - Embedded newform 2800.2.e.h.2799.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2800,2,Mod(2799,2800)]

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, 1, 1]))

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

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

chi := DirichletCharacter("2800.2799");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2800 = 2^{4} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2800.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:	$22.3581125660$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.116319195136.7
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 77x^{4} + 4 $ x^8 + 77*x^4 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 560)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2799.8
Root			$0.337637 - 0.337637i$ of defining polynomial
Character	$\chi$	$=$	2800.2799
Dual form			2800.2.e.h.2799.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.96176i q^{3} +(2.62412 + 0.337637i) q^{7} -5.77200 q^{9} +O(q^{10})$	$q+2.96176i q^{3} +(2.62412 + 0.337637i) q^{7} -5.77200 q^{9} -3.63703i q^{11} +4.77200 q^{13} +4.77200 q^{17} -4.57296 q^{19} +(-1.00000 + 7.77200i) q^{21} -5.24824 q^{23} -8.20999i q^{27} +4.77200 q^{29} +5.92351 q^{31} +10.7720 q^{33} +11.5440i q^{37} +14.1335i q^{39} +6.00000i q^{41} +2.02582 q^{43} +1.61121i q^{47} +(6.77200 + 1.77200i) q^{49} +14.1335i q^{51} -6.00000i q^{53} -13.5440i q^{57} +7.27406 q^{59} +3.54400i q^{61} +(-15.1464 - 1.94884i) q^{63} -12.5223 q^{67} -15.5440i q^{69} +7.27406i q^{71} +6.00000 q^{73} +(1.22800 - 9.54400i) q^{77} +6.85944i q^{79} +7.00000 q^{81} +2.02582i q^{83} +14.1335i q^{87} +12.0000i q^{89} +(12.5223 + 1.61121i) q^{91} +17.5440i q^{93} +16.7720 q^{97} +20.9930i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{9}+O(q^{10}) $ 8 * q - 12 * q^9	$ 8 q - 12 q^{9} + 4 q^{13} + 4 q^{17} - 8 q^{21} + 4 q^{29} + 52 q^{33} + 20 q^{49} + 48 q^{73} + 44 q^{77} + 56 q^{81} + 100 q^{97}+O(q^{100}) $ 8 * q - 12 * q^9 + 4 * q^13 + 4 * q^17 - 8 * q^21 + 4 * q^29 + 52 * q^33 + 20 * q^49 + 48 * q^73 + 44 * q^77 + 56 * q^81 + 100 * q^97

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
$3$			2.96176i			1.70997i	0.518652	+	0.854985i	$0.326434\pi$
$3$			2.96176i			1.70997i	−0.518652	+	0.854985i	$0.673566\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	2.62412	+	0.337637i	0.991824	+	0.127615i
$7$	2.62412	+	0.337637i	0.991824	+	0.127615i
$8$		0			0
$9$	−5.77200			−1.92400
$10$		0			0
$11$		−	3.63703i		−	1.09661i	−0.836280	−	0.548303i	$-0.815274\pi$
$11$		−	3.63703i		−	1.09661i	0.836280	−	0.548303i	$-0.184726\pi$
$12$		0			0
$13$	4.77200			1.32352			0.661758	−	0.749718i	$-0.269811\pi$
$13$	4.77200			1.32352			0.661758	+	0.749718i	$0.269811\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.77200			1.15738			0.578690	−	0.815547i	$-0.303564\pi$
$17$	4.77200			1.15738			0.578690	+	0.815547i	$0.303564\pi$
$18$		0			0
$19$	−4.57296			−1.04911			−0.524555	−	0.851377i	$-0.675768\pi$
$19$	−4.57296			−1.04911			−0.524555	+	0.851377i	$0.675768\pi$
$20$		0			0
$21$	−1.00000	+	7.77200i	−0.218218	+	1.69599i
$22$		0			0
$23$	−5.24824			−1.09433			−0.547167	−	0.837024i	$-0.684294\pi$
$23$	−5.24824			−1.09433			−0.547167	+	0.837024i	$0.684294\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	8.20999i		−	1.58001i
$28$		0			0
$29$	4.77200			0.886139			0.443069	−	0.896487i	$-0.353890\pi$
$29$	4.77200			0.886139			0.443069	+	0.896487i	$0.353890\pi$
$30$		0			0
$31$	5.92351			1.06389			0.531947	−	0.846778i	$-0.321460\pi$
$31$	5.92351			1.06389			0.531947	+	0.846778i	$0.321460\pi$
$32$		0			0
$33$	10.7720			1.87516
$34$		0			0
$35$		0			0
$36$		0			0
$37$			11.5440i			1.89782i	0.315543	+	0.948911i	$0.397813\pi$
$37$			11.5440i			1.89782i	−0.315543	+	0.948911i	$0.602187\pi$
$38$		0			0
$39$			14.1335i			2.26317i
$40$		0			0
$41$			6.00000i			0.937043i	0.883452	+	0.468521i	$0.155213\pi$
$41$			6.00000i			0.937043i	−0.883452	+	0.468521i	$0.844787\pi$
$42$		0			0
$43$	2.02582			0.308935			0.154468	−	0.987998i	$-0.450634\pi$
$43$	2.02582			0.308935			0.154468	+	0.987998i	$0.450634\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			1.61121i			0.235019i	0.993072	+	0.117509i	$0.0374910\pi$
$47$			1.61121i			0.235019i	−0.993072	+	0.117509i	$0.962509\pi$
$48$		0			0
$49$	6.77200	+	1.77200i	0.967429	+	0.253143i
$50$		0			0
$51$			14.1335i			1.97909i
$52$		0			0
$53$		−	6.00000i		−	0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$		−	6.00000i		−	0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	13.5440i		−	1.79395i
$58$		0			0
$59$	7.27406			0.947002			0.473501	−	0.880793i	$-0.342990\pi$
$59$	7.27406			0.947002			0.473501	+	0.880793i	$0.342990\pi$
$60$		0			0
$61$			3.54400i			0.453763i	0.973922	+	0.226882i	$0.0728530\pi$
$61$			3.54400i			0.453763i	−0.973922	+	0.226882i	$0.927147\pi$
$62$		0			0
$63$	−15.1464	−	1.94884i	−1.90827	−	0.245531i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−12.5223			−1.52984			−0.764921	−	0.644124i	$-0.777222\pi$
$67$	−12.5223			−1.52984			−0.764921	+	0.644124i	$0.777222\pi$
$68$		0			0
$69$		−	15.5440i		−	1.87128i
$70$		0			0
$71$			7.27406i			0.863272i	0.902048	+	0.431636i	$0.142064\pi$
$71$			7.27406i			0.863272i	−0.902048	+	0.431636i	$0.857936\pi$
$72$		0			0
$73$	6.00000			0.702247			0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000			0.702247			0.351123	+	0.936329i	$0.385800\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.22800	−	9.54400i	0.139943	−	1.08764i
$78$		0			0
$79$			6.85944i			0.771748i	0.922552	+	0.385874i	$0.126100\pi$
$79$			6.85944i			0.771748i	−0.922552	+	0.385874i	$0.873900\pi$
$80$		0			0
$81$	7.00000			0.777778
$82$		0			0
$83$			2.02582i			0.222363i	0.993800	+	0.111182i	$0.0354635\pi$
$83$			2.02582i			0.222363i	−0.993800	+	0.111182i	$0.964536\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			14.1335i			1.51527i
$88$		0			0
$89$			12.0000i			1.27200i	0.771690	+	0.635999i	$0.219412\pi$
$89$			12.0000i			1.27200i	−0.771690	+	0.635999i	$0.780588\pi$
$90$		0			0
$91$	12.5223	+	1.61121i	1.31269	+	0.168900i
$92$		0			0
$93$			17.5440i			1.81923i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	16.7720			1.70294			0.851469	−	0.524404i	$-0.175712\pi$
$97$	16.7720			1.70294			0.851469	+	0.524404i	$0.175712\pi$
$98$		0			0
$99$			20.9930i			2.10987i
$100$		0			0
$101$		−	9.54400i		−	0.949664i	−0.880076	−	0.474832i	$-0.842509\pi$
$101$		−	9.54400i		−	0.949664i	0.880076	−	0.474832i	$-0.157491\pi$
$102$		0			0
$103$			4.31231i			0.424904i	0.977171	+	0.212452i	$0.0681449\pi$
$103$			4.31231i			0.424904i	−0.977171	+	0.212452i	$0.931855\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	12.5223			1.21058			0.605288	−	0.796006i	$-0.293058\pi$
$107$	12.5223			1.21058			0.605288	+	0.796006i	$0.293058\pi$
$108$		0			0
$109$	−6.77200			−0.648640			−0.324320	−	0.945947i	$-0.605135\pi$
$109$	−6.77200			−0.648640			−0.324320	+	0.945947i	$0.605135\pi$
$110$		0			0
$111$	−34.1905			−3.24522
$112$		0			0
$113$		−	15.5440i		−	1.46226i	−0.682240	−	0.731128i	$-0.738994\pi$
$113$		−	15.5440i		−	1.46226i	0.682240	−	0.731128i	$-0.261006\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−27.5440			−2.54644
$118$		0			0
$119$	12.5223	+	1.61121i	1.14792	+	0.147699i
$120$		0			0
$121$	−2.22800			−0.202545
$122$		0			0
$123$	−17.7705			−1.60232
$124$		0			0
$125$		0			0
$126$		0			0
$127$	12.5223			1.11117			0.555587	−	0.831458i	$-0.312493\pi$
$127$	12.5223			1.11117			0.555587	+	0.831458i	$0.312493\pi$
$128$		0			0
$129$			6.00000i			0.528271i
$130$		0			0
$131$	−10.4965			−0.917081			−0.458541	−	0.888673i	$-0.651628\pi$
$131$	−10.4965			−0.917081			−0.458541	+	0.888673i	$0.651628\pi$
$132$		0			0
$133$	−12.0000	−	1.54400i	−1.04053	−	0.133882i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			3.54400i			0.302785i	0.988474	+	0.151392i	$0.0483757\pi$
$137$			3.54400i			0.302785i	−0.988474	+	0.151392i	$0.951624\pi$
$138$		0			0
$139$	−9.14593			−0.775747			−0.387874	−	0.921713i	$-0.626790\pi$
$139$	−9.14593			−0.775747			−0.387874	+	0.921713i	$0.626790\pi$
$140$		0			0
$141$	−4.77200			−0.401875
$142$		0			0
$143$		−	17.3559i		−	1.45138i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−5.24824	+	20.0570i	−0.432867	+	1.65428i
$148$		0			0
$149$	−9.54400			−0.781875			−0.390938	−	0.920417i	$-0.627849\pi$
$149$	−9.54400			−0.781875			−0.390938	+	0.920417i	$0.627849\pi$
$150$		0			0
$151$		−	3.63703i		−	0.295977i	−0.988989	−	0.147989i	$-0.952720\pi$
$151$		−	3.63703i		−	0.295977i	0.988989	−	0.147989i	$-0.0472799\pi$
$152$		0			0
$153$	−27.5440			−2.22680
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−6.00000			−0.478852			−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000			−0.478852			−0.239426	+	0.970915i	$0.576959\pi$
$158$		0			0
$159$	17.7705			1.40930
$160$		0			0
$161$	−13.7720	−	1.77200i	−1.08539	−	0.139653i
$162$		0			0
$163$	−15.7447			−1.23322			−0.616611	−	0.787268i	$-0.711495\pi$
$163$	−15.7447			−1.23322			−0.616611	+	0.787268i	$0.711495\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	23.4334i		−	1.81333i	−0.421851	−	0.906665i	$-0.638619\pi$
$167$		−	23.4334i		−	1.81333i	0.421851	−	0.906665i	$-0.361381\pi$
$168$		0			0
$169$	9.77200			0.751692
$170$		0			0
$171$	26.3952			2.01849
$172$		0			0
$173$	−2.31601			−0.176083			−0.0880413	−	0.996117i	$-0.528061\pi$
$173$	−2.31601			−0.176083			−0.0880413	+	0.996117i	$0.528061\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			21.5440i			1.61935i
$178$		0			0
$179$			3.22241i			0.240854i	0.992722	+	0.120427i	$0.0384264\pi$
$179$			3.22241i			0.240854i	−0.992722	+	0.120427i	$0.961574\pi$
$180$		0			0
$181$		−	9.54400i		−	0.709400i	−0.934980	−	0.354700i	$-0.884583\pi$
$181$		−	9.54400i		−	0.709400i	0.934980	−	0.354700i	$-0.115417\pi$
$182$		0			0
$183$	−10.4965			−0.775922
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	17.3559i		−	1.26919i
$188$		0			0
$189$	2.77200	−	21.5440i	0.201633	−	1.56710i
$190$		0			0
$191$		−	24.6300i		−	1.78216i	−0.453843	−	0.891082i	$-0.649947\pi$
$191$		−	24.6300i		−	1.78216i	0.453843	−	0.891082i	$-0.350053\pi$
$192$		0			0
$193$		−	4.45600i		−	0.320750i	−0.987056	−	0.160375i	$-0.948730\pi$
$193$		−	4.45600i		−	0.320750i	0.987056	−	0.160375i	$-0.0512703\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			6.00000i			0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$			6.00000i			0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$		0			0
$199$	−4.57296			−0.324169			−0.162084	−	0.986777i	$-0.551822\pi$
$199$	−4.57296			−0.324169			−0.162084	+	0.986777i	$0.551822\pi$
$200$		0			0
$201$		−	37.0880i		−	2.61599i
$202$		0			0
$203$	12.5223	+	1.61121i	0.878893	+	0.113085i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	30.2928			2.10550
$208$		0			0
$209$			16.6320i			1.15046i
$210$		0			0
$211$			17.3559i			1.19483i	0.801932	+	0.597415i	$0.203806\pi$
$211$			17.3559i			1.19483i	−0.801932	+	0.597415i	$0.796194\pi$
$212$		0			0
$213$	−21.5440			−1.47617
$214$		0			0
$215$		0			0
$216$		0			0
$217$	15.5440	+	2.00000i	1.05520	+	0.135769i
$218$		0			0
$219$			17.7705i			1.20082i
$220$		0			0
$221$	22.7720			1.53181
$222$		0			0
$223$			13.4582i			0.901230i	0.892718	+	0.450615i	$0.148795\pi$
$223$			13.4582i			0.901230i	−0.892718	+	0.450615i	$0.851205\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			16.1593i			1.07253i	0.844049	+	0.536266i	$0.180166\pi$
$227$			16.1593i			1.07253i	−0.844049	+	0.536266i	$0.819834\pi$
$228$		0			0
$229$			21.5440i			1.42367i	0.702348	+	0.711834i	$0.252135\pi$
$229$			21.5440i			1.42367i	−0.702348	+	0.711834i	$0.747865\pi$
$230$		0			0
$231$	28.2670	+	3.63703i	1.85983	+	0.239299i
$232$		0			0
$233$			8.45600i			0.553971i	0.960874	+	0.276985i	$0.0893354\pi$
$233$			8.45600i			0.553971i	−0.960874	+	0.276985i	$0.910665\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−20.3160			−1.31967
$238$		0			0
$239$		−	14.1335i		−	0.914221i	−0.889410	−	0.457110i	$-0.848884\pi$
$239$		−	14.1335i		−	0.914221i	0.889410	−	0.457110i	$-0.151116\pi$
$240$		0			0
$241$		−	8.45600i		−	0.544699i	−0.962198	−	0.272349i	$-0.912199\pi$
$241$		−	8.45600i		−	0.544699i	0.962198	−	0.272349i	$-0.0878006\pi$
$242$		0			0
$243$		−	3.89769i		−	0.250037i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−21.8222			−1.38851
$248$		0			0
$249$	−6.00000			−0.380235
$250$		0			0
$251$	−25.0446			−1.58080			−0.790401	−	0.612590i	$-0.790128\pi$
$251$	−25.0446			−1.58080			−0.790401	+	0.612590i	$0.790128\pi$
$252$		0			0
$253$			19.0880i			1.20005i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	13.0880			0.816407			0.408204	−	0.912891i	$-0.366155\pi$
$257$	13.0880			0.816407			0.408204	+	0.912891i	$0.366155\pi$
$258$		0			0
$259$	−3.89769	+	30.2928i	−0.242191	+	1.88231i
$260$		0			0
$261$	−27.5440			−1.70493
$262$		0			0
$263$	−8.47065			−0.522323			−0.261161	−	0.965295i	$-0.584105\pi$
$263$	−8.47065			−0.522323			−0.261161	+	0.965295i	$0.584105\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−35.5411			−2.17508
$268$		0			0
$269$		−	15.5440i		−	0.947735i	−0.880596	−	0.473867i	$-0.842858\pi$
$269$		−	15.5440i		−	0.947735i	0.880596	−	0.473867i	$-0.157142\pi$
$270$		0			0
$271$	−1.35055			−0.0820401			−0.0410200	−	0.999158i	$-0.513061\pi$
$271$	−1.35055			−0.0820401			−0.0410200	+	0.999158i	$0.513061\pi$
$272$		0			0
$273$	−4.77200	+	37.0880i	−0.288815	+	2.24467i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	19.5440i		−	1.17429i	−0.809483	−	0.587143i	$-0.800253\pi$
$277$		−	19.5440i		−	1.17429i	0.809483	−	0.587143i	$-0.199747\pi$
$278$		0			0
$279$	−34.1905			−2.04693
$280$		0			0
$281$	20.3160			1.21195			0.605976	−	0.795483i	$-0.292783\pi$
$281$	20.3160			1.21195			0.605976	+	0.795483i	$0.292783\pi$
$282$		0			0
$283$			23.9547i			1.42396i	0.702200	+	0.711980i	$0.252201\pi$
$283$			23.9547i			1.42396i	−0.702200	+	0.711980i	$0.747799\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.02582	+	15.7447i	−0.119581	+	0.929381i
$288$		0			0
$289$	5.77200			0.339530
$290$		0			0
$291$			49.6746i			2.91198i
$292$		0			0
$293$	−14.3160			−0.836350			−0.418175	−	0.908366i	$-0.637330\pi$
$293$	−14.3160			−0.836350			−0.418175	+	0.908366i	$0.637330\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−29.8600			−1.73265
$298$		0			0
$299$	−25.0446			−1.44837
$300$		0			0
$301$	5.31601	+	0.683994i	0.306409	+	0.0394248i
$302$		0			0
$303$	28.2670			1.62390
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	31.2288i		−	1.78232i	−0.453689	−	0.891160i	$-0.649892\pi$
$307$		−	31.2288i		−	1.78232i	0.453689	−	0.891160i	$-0.350108\pi$
$308$		0			0
$309$	−12.7720			−0.726574
$310$		0			0
$311$	24.2154			1.37313			0.686564	−	0.727070i	$-0.259118\pi$
$311$	24.2154			1.37313			0.686564	+	0.727070i	$0.259118\pi$
$312$		0			0
$313$	4.77200			0.269729			0.134865	−	0.990864i	$-0.456940\pi$
$313$	4.77200			0.269729			0.134865	+	0.990864i	$0.456940\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	25.0880i		−	1.40908i	−0.709663	−	0.704541i	$-0.751153\pi$
$317$		−	25.0880i		−	1.40908i	0.709663	−	0.704541i	$-0.248847\pi$
$318$		0			0
$319$		−	17.3559i		−	0.971745i
$320$		0			0
$321$			37.0880i			2.07005i
$322$		0			0
$323$	−21.8222			−1.21422
$324$		0			0
$325$		0			0
$326$		0			0
$327$		−	20.0570i		−	1.10916i
$328$		0			0
$329$	−0.544004	+	4.22800i	−0.0299919	+	0.233097i
$330$		0			0
$331$			13.7189i			0.754058i	0.926201	+	0.377029i	$0.123054\pi$
$331$			13.7189i			0.754058i	−0.926201	+	0.377029i	$0.876946\pi$
$332$		0			0
$333$		−	66.6320i		−	3.65141i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	4.45600i		−	0.242734i	−0.992608	−	0.121367i	$-0.961272\pi$
$337$		−	4.45600i		−	0.242734i	0.992608	−	0.121367i	$-0.0387277\pi$
$338$		0			0
$339$	46.0376			2.50042
$340$		0			0
$341$		−	21.5440i		−	1.16667i
$342$		0			0
$343$	17.1722	+	6.93643i	0.927214	+	0.374532i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	5.24824			0.281740			0.140870	−	0.990028i	$-0.455010\pi$
$347$	5.24824			0.281740			0.140870	+	0.990028i	$0.455010\pi$
$348$		0			0
$349$			2.45600i			0.131466i	0.997837	+	0.0657332i	$0.0209386\pi$
$349$			2.45600i			0.131466i	−0.997837	+	0.0657332i	$0.979061\pi$
$350$		0			0
$351$		−	39.1781i		−	2.09117i
$352$		0			0
$353$	35.8600			1.90864			0.954318	−	0.298793i	$-0.0965841\pi$
$353$	35.8600			1.90864			0.954318	+	0.298793i	$0.0965841\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−4.77200	+	37.0880i	−0.252561	+	1.96291i
$358$		0			0
$359$		−	3.22241i		−	0.170072i	−0.996378	−	0.0850362i	$-0.972899\pi$
$359$		−	3.22241i		−	0.170072i	0.996378	−	0.0850362i	$-0.0271006\pi$
$360$		0			0
$361$	1.91199			0.100631
$362$		0			0
$363$		−	6.59879i		−	0.346347i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			14.8088i			0.773012i	0.922287	+	0.386506i	$0.126318\pi$
$367$			14.8088i			0.773012i	−0.922287	+	0.386506i	$0.873682\pi$
$368$		0			0
$369$		−	34.6320i		−	1.80287i
$370$		0			0
$371$	2.02582	−	15.7447i	0.105176	−	0.817425i
$372$		0			0
$373$			4.45600i			0.230723i	0.993324	+	0.115361i	$0.0368026\pi$
$373$			4.45600i			0.230723i	−0.993324	+	0.115361i	$0.963197\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	22.7720			1.17282
$378$		0			0
$379$		−	3.22241i		−	0.165524i	−0.996569	−	0.0827621i	$-0.973626\pi$
$379$		−	3.22241i		−	0.165524i	0.996569	−	0.0827621i	$-0.0263742\pi$
$380$		0			0
$381$			37.0880i			1.90008i
$382$		0			0
$383$			8.47065i			0.432830i	0.976301	+	0.216415i	$0.0694364\pi$
$383$			8.47065i			0.432830i	−0.976301	+	0.216415i	$0.930564\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−11.6931			−0.594392
$388$		0			0
$389$	20.3160			1.03006			0.515031	−	0.857171i	$-0.327780\pi$
$389$	20.3160			1.03006			0.515031	+	0.857171i	$0.327780\pi$
$390$		0			0
$391$	−25.0446			−1.26656
$392$		0			0
$393$		−	31.0880i		−	1.56818i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	16.7720			0.841763			0.420881	−	0.907116i	$-0.361721\pi$
$397$	16.7720			0.841763			0.420881	+	0.907116i	$0.361721\pi$
$398$		0			0
$399$	4.57296	−	35.5411i	0.228935	−	1.77928i
$400$		0			0
$401$	−7.22800			−0.360949			−0.180475	−	0.983580i	$-0.557763\pi$
$401$	−7.22800			−0.360949			−0.180475	+	0.983580i	$0.557763\pi$
$402$		0			0
$403$	28.2670			1.40808
$404$		0			0
$405$		0			0
$406$		0			0
$407$	41.9859			2.08116
$408$		0			0
$409$			37.0880i			1.83388i	0.399021	+	0.916942i	$0.369350\pi$
$409$			37.0880i			1.83388i	−0.399021	+	0.916942i	$0.630650\pi$
$410$		0			0
$411$	−10.4965			−0.517753
$412$		0			0
$413$	19.0880	+	2.45600i	0.939259	+	0.120852i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	27.0880i		−	1.32651i
$418$		0			0
$419$	13.7189			0.670212			0.335106	−	0.942181i	$-0.391228\pi$
$419$	13.7189			0.670212			0.335106	+	0.942181i	$0.391228\pi$
$420$		0			0
$421$	−13.8600			−0.675496			−0.337748	−	0.941237i	$-0.609665\pi$
$421$	−13.8600			−0.675496			−0.337748	+	0.941237i	$0.609665\pi$
$422$		0			0
$423$		−	9.29989i		−	0.452176i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−1.19659	+	9.29989i	−0.0579070	+	0.450053i
$428$		0			0
$429$	51.4040			2.48181
$430$		0			0
$431$		−	31.0748i		−	1.49682i	−0.663236	−	0.748410i	$-0.730817\pi$
$431$		−	31.0748i		−	1.49682i	0.663236	−	0.748410i	$-0.269183\pi$
$432$		0			0
$433$	25.0880			1.20565			0.602826	−	0.797872i	$-0.294041\pi$
$433$	25.0880			1.20565			0.602826	+	0.797872i	$0.294041\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	24.0000			1.14808
$438$		0			0
$439$	−11.8470			−0.565428			−0.282714	−	0.959204i	$-0.591235\pi$
$439$	−11.8470			−0.565428			−0.282714	+	0.959204i	$0.591235\pi$
$440$		0			0
$441$	−39.0880	−	10.2280i	−1.86133	−	0.487048i
$442$		0			0
$443$	8.47065			0.402453			0.201226	−	0.979545i	$-0.435507\pi$
$443$	8.47065			0.402453			0.201226	+	0.979545i	$0.435507\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	28.2670i		−	1.33698i
$448$		0			0
$449$	−29.8600			−1.40918			−0.704590	−	0.709614i	$-0.748869\pi$
$449$	−29.8600			−1.40918			−0.704590	+	0.709614i	$0.748869\pi$
$450$		0			0
$451$	21.8222			1.02757
$452$		0			0
$453$	10.7720			0.506113
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	28.1760i		−	1.31802i	−0.752135	−	0.659009i	$-0.770976\pi$
$457$		−	28.1760i		−	1.31802i	0.752135	−	0.659009i	$-0.229024\pi$
$458$		0			0
$459$		−	39.1781i		−	1.82868i
$460$		0			0
$461$			7.08801i			0.330121i	0.986283	+	0.165061i	$0.0527820\pi$
$461$			7.08801i			0.330121i	−0.986283	+	0.165061i	$0.947218\pi$
$462$		0			0
$463$	1.19659			0.0556102			0.0278051	−	0.999613i	$-0.491148\pi$
$463$	1.19659			0.0556102			0.0278051	+	0.999613i	$0.491148\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			33.1006i			1.53171i	0.643010	+	0.765857i	$0.277685\pi$
$467$			33.1006i			1.53171i	−0.643010	+	0.765857i	$0.722315\pi$
$468$		0			0
$469$	−32.8600	−	4.22800i	−1.51733	−	0.195231i
$470$		0			0
$471$		−	17.7705i		−	0.818823i
$472$		0			0
$473$		−	7.36799i		−	0.338780i
$474$		0			0
$475$		0			0
$476$		0			0
$477$			34.6320i			1.58569i
$478$		0			0
$479$	−29.0963			−1.32944			−0.664721	−	0.747092i	$-0.731450\pi$
$479$	−29.0963			−1.32944			−0.664721	+	0.747092i	$0.731450\pi$
$480$		0			0
$481$			55.0880i			2.51180i
$482$		0			0
$483$	5.24824	−	40.7893i	0.238803	−	1.85598i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−15.7447			−0.713461			−0.356731	−	0.934207i	$-0.616109\pi$
$487$	−15.7447			−0.713461			−0.356731	+	0.934207i	$0.616109\pi$
$488$		0			0
$489$		−	46.6320i		−	2.10877i
$490$		0			0
$491$		−	24.6300i		−	1.11154i	−0.831338	−	0.555768i	$-0.812424\pi$
$491$		−	24.6300i		−	1.11154i	0.831338	−	0.555768i	$-0.187576\pi$
$492$		0			0
$493$	22.7720			1.02560
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2.45600	+	19.0880i	−0.110166	+	0.856214i
$498$		0			0
$499$		−	28.6816i		−	1.28397i	−0.766719	−	0.641983i	$-0.778112\pi$
$499$		−	28.6816i		−	1.28397i	0.766719	−	0.641983i	$-0.221888\pi$
$500$		0			0
$501$	69.4040			3.10074
$502$		0			0
$503$		−	8.88527i		−	0.396175i	−0.980184	−	0.198087i	$-0.936527\pi$
$503$		−	8.88527i		−	0.396175i	0.980184	−	0.198087i	$-0.0634730\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			28.9423i			1.28537i
$508$		0			0
$509$			19.0880i			0.846061i	0.906115	+	0.423031i	$0.139034\pi$
$509$			19.0880i			0.846061i	−0.906115	+	0.423031i	$0.860966\pi$
$510$		0			0
$511$	15.7447	+	2.02582i	0.696505	+	0.0896172i
$512$		0			0
$513$			37.5440i			1.65761i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	5.86001			0.257723
$518$		0			0
$519$		−	6.85944i		−	0.301096i
$520$		0			0
$521$			21.5440i			0.943860i	0.881636	+	0.471930i	$0.156442\pi$
$521$			21.5440i			0.943860i	−0.881636	+	0.471930i	$0.843558\pi$
$522$		0			0
$523$		−	38.9175i		−	1.70174i	−0.525375	−	0.850871i	$-0.676075\pi$
$523$		−	38.9175i		−	1.70174i	0.525375	−	0.850871i	$-0.323925\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	28.2670			1.23133
$528$		0			0
$529$	4.54400			0.197565
$530$		0			0
$531$	−41.9859			−1.82203
$532$		0			0
$533$			28.6320i			1.24019i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−9.54400			−0.411854
$538$		0			0
$539$	6.44483	−	24.6300i	0.277598	−	1.06089i
$540$		0			0
$541$	−15.2280			−0.654703			−0.327351	−	0.944903i	$-0.606156\pi$
$541$	−15.2280			−0.654703			−0.327351	+	0.944903i	$0.606156\pi$
$542$		0			0
$543$	28.2670			1.21305
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.19659			−0.0511624			−0.0255812	−	0.999673i	$-0.508144\pi$
$547$	−1.19659			−0.0511624			−0.0255812	+	0.999673i	$0.508144\pi$
$548$		0			0
$549$		−	20.4560i		−	0.873041i
$550$		0			0
$551$	−21.8222			−0.929657
$552$		0			0
$553$	−2.31601	+	18.0000i	−0.0984866	+	0.765438i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	20.4560i		−	0.866748i	−0.901214	−	0.433374i	$-0.857323\pi$
$557$		−	20.4560i		−	0.866748i	0.901214	−	0.433374i	$-0.142677\pi$
$558$		0			0
$559$	9.66724			0.408881
$560$		0			0
$561$	51.4040			2.17028
$562$		0			0
$563$		−	8.47065i		−	0.356995i	−0.983940	−	0.178498i	$-0.942876\pi$
$563$		−	8.47065i		−	0.356995i	0.983940	−	0.178498i	$-0.0571237\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	18.3688	+	2.36346i	0.771418	+	0.0992561i
$568$		0			0
$569$	16.6320			0.697250			0.348625	−	0.937262i	$-0.386649\pi$
$569$	16.6320			0.697250			0.348625	+	0.937262i	$0.386649\pi$
$570$		0			0
$571$		−	7.27406i		−	0.304410i	−0.988349	−	0.152205i	$-0.951363\pi$
$571$		−	7.27406i		−	0.304410i	0.988349	−	0.152205i	$-0.0486374\pi$
$572$		0			0
$573$	72.9480			3.04745
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−0.139991			−0.00582789			−0.00291394	−	0.999996i	$-0.500928\pi$
$577$	−0.139991			−0.00582789			−0.00291394	+	0.999996i	$0.500928\pi$
$578$		0			0
$579$	13.1976			0.548473
$580$		0			0
$581$	−0.683994	+	5.31601i	−0.0283769	+	0.220545i
$582$		0			0
$583$	−21.8222			−0.903783
$584$		0			0
$585$		0			0
$586$		0			0
$587$			2.02582i			0.0836147i	0.999126	+	0.0418074i	$0.0133116\pi$
$587$			2.02582i			0.0836147i	−0.999126	+	0.0418074i	$0.986688\pi$
$588$		0			0
$589$	−27.0880			−1.11614
$590$		0			0
$591$	−17.7705			−0.730982
$592$		0			0
$593$	11.8600			0.487032			0.243516	−	0.969897i	$-0.421699\pi$
$593$	11.8600			0.487032			0.243516	+	0.969897i	$0.421699\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	13.5440i		−	0.554319i
$598$		0			0
$599$			14.9627i			0.611361i	0.952134	+	0.305681i	$0.0988840\pi$
$599$			14.9627i			0.611361i	−0.952134	+	0.305681i	$0.901116\pi$
$600$		0			0
$601$		−	44.1760i		−	1.80198i	−0.433843	−	0.900989i	$-0.642843\pi$
$601$		−	44.1760i		−	1.80198i	0.433843	−	0.900989i	$-0.357157\pi$
$602$		0			0
$603$	72.2787			2.94342
$604$		0			0
$605$		0			0
$606$		0			0
$607$			10.7571i			0.436619i	0.975880	+	0.218309i	$0.0700542\pi$
$607$			10.7571i			0.436619i	−0.975880	+	0.218309i	$0.929946\pi$
$608$		0			0
$609$	−4.77200	+	37.0880i	−0.193371	+	1.50288i
$610$		0			0
$611$			7.68868i			0.311051i
$612$		0			0
$613$		−	34.0000i		−	1.37325i	−0.727013	−	0.686624i	$-0.759092\pi$
$613$		−	34.0000i		−	1.37325i	0.727013	−	0.686624i	$-0.240908\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			15.5440i			0.625778i	0.949790	+	0.312889i	$0.101297\pi$
$617$			15.5440i			0.625778i	−0.949790	+	0.312889i	$0.898703\pi$
$618$		0			0
$619$	11.8470			0.476172			0.238086	−	0.971244i	$-0.423480\pi$
$619$	11.8470			0.476172			0.238086	+	0.971244i	$0.423480\pi$
$620$		0			0
$621$			43.0880i			1.72906i
$622$		0			0
$623$	−4.05165	+	31.4894i	−0.162326	+	1.26160i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−49.2600			−1.96725
$628$		0			0
$629$			55.0880i			2.19650i
$630$		0			0
$631$			10.9111i			0.434364i	0.976131	+	0.217182i	$0.0696865\pi$
$631$			10.9111i			0.434364i	−0.976131	+	0.217182i	$0.930314\pi$
$632$		0			0
$633$	−51.4040			−2.04313
$634$		0			0
$635$		0			0
$636$		0			0
$637$	32.3160	+	8.45600i	1.28041	+	0.335039i
$638$		0			0
$639$		−	41.9859i		−	1.66094i
$640$		0			0
$641$	−4.63201			−0.182953			−0.0914767	−	0.995807i	$-0.529159\pi$
$641$	−4.63201			−0.182953			−0.0914767	+	0.995807i	$0.529159\pi$
$642$		0			0
$643$			23.9547i			0.944682i	0.881416	+	0.472341i	$0.156591\pi$
$643$			23.9547i			0.944682i	−0.881416	+	0.472341i	$0.843409\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	16.5740i		−	0.651589i	−0.945441	−	0.325795i	$-0.894368\pi$
$647$		−	16.5740i		−	0.651589i	0.945441	−	0.325795i	$-0.105632\pi$
$648$		0			0
$649$		−	26.4560i		−	1.03849i
$650$		0			0
$651$	−5.92351	+	46.0376i	−0.232161	+	1.80435i
$652$		0			0
$653$		−	1.08801i		−	0.0425770i	−0.999773	−	0.0212885i	$-0.993223\pi$
$653$		−	1.08801i		−	0.0425770i	0.999773	−	0.0212885i	$-0.00677686\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−34.6320			−1.35112
$658$		0			0
$659$		−	49.6746i		−	1.93505i	−0.252781	−	0.967524i	$-0.581345\pi$
$659$		−	49.6746i		−	1.93505i	0.252781	−	0.967524i	$-0.418655\pi$
$660$		0			0
$661$		−	15.5440i		−	0.604592i	−0.953214	−	0.302296i	$-0.902247\pi$
$661$		−	15.5440i		−	0.604592i	0.953214	−	0.302296i	$-0.0977530\pi$
$662$		0			0
$663$			67.4451i			2.61935i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−25.0446			−0.969731
$668$		0			0
$669$	−39.8600			−1.54108
$670$		0			0
$671$	12.8897			0.497600
$672$		0			0
$673$		−	35.5440i		−	1.37012i	−0.728486	−	0.685060i	$-0.759776\pi$
$673$		−	35.5440i		−	1.37012i	0.728486	−	0.685060i	$-0.240224\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	16.7720			0.644600			0.322300	−	0.946638i	$-0.395544\pi$
$677$	16.7720			0.644600			0.322300	+	0.946638i	$0.395544\pi$
$678$		0			0
$679$	44.0117	+	5.66286i	1.68902	+	0.217320i
$680$		0			0
$681$	−47.8600			−1.83400
$682$		0			0
$683$	36.7377			1.40573			0.702864	−	0.711324i	$-0.251904\pi$
$683$	36.7377			1.40573			0.702864	+	0.711324i	$0.251904\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−63.8081			−2.43443
$688$		0			0
$689$		−	28.6320i		−	1.09079i
$690$		0			0
$691$	13.1976			0.502059			0.251030	−	0.967979i	$-0.419231\pi$
$691$	13.1976			0.502059			0.251030	+	0.967979i	$0.419231\pi$
$692$		0			0
$693$	−7.08801	+	55.0880i	−0.269251	+	2.09262i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			28.6320i			1.08451i
$698$		0			0
$699$	−25.0446			−0.947274
$700$		0			0
$701$	−29.8600			−1.12780			−0.563898	−	0.825844i	$-0.690699\pi$
$701$	−29.8600			−1.12780			−0.563898	+	0.825844i	$0.690699\pi$
$702$		0			0
$703$		−	52.7903i		−	1.99102i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	3.22241	−	25.0446i	0.121191	−	0.941899i
$708$		0			0
$709$	13.6840			0.513913			0.256957	−	0.966423i	$-0.417280\pi$
$709$	13.6840			0.513913			0.256957	+	0.966423i	$0.417280\pi$
$710$		0			0
$711$		−	39.5927i		−	1.48484i
$712$		0			0
$713$	−31.0880			−1.16426
$714$		0			0
$715$		0			0
$716$		0			0
$717$	41.8600			1.56329
$718$		0			0
$719$	6.44483			0.240351			0.120176	−	0.992753i	$-0.461654\pi$
$719$	6.44483			0.240351			0.120176	+	0.992753i	$0.461654\pi$
$720$		0			0
$721$	−1.45600	+	11.3160i	−0.0542241	+	0.421430i
$722$		0			0
$723$	25.0446			0.931419
$724$		0			0
$725$		0			0
$726$		0			0
$727$			24.3693i			0.903808i	0.892066	+	0.451904i	$0.149255\pi$
$727$			24.3693i			0.903808i	−0.892066	+	0.451904i	$0.850745\pi$
$728$		0			0
$729$	32.5440			1.20533
$730$		0			0
$731$	9.66724			0.357556
$732$		0			0
$733$	−31.2280			−1.15343			−0.576716	−	0.816945i	$-0.695666\pi$
$733$	−31.2280			−1.15343			−0.576716	+	0.816945i	$0.695666\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			45.5440i			1.67763i
$738$		0			0
$739$			42.4005i			1.55973i	0.625949	+	0.779864i	$0.284712\pi$
$739$			42.4005i			1.55973i	−0.625949	+	0.779864i	$0.715288\pi$
$740$		0			0
$741$		−	64.6320i		−	2.37432i
$742$		0			0
$743$	−4.41900			−0.162117			−0.0810587	−	0.996709i	$-0.525830\pi$
$743$	−4.41900			−0.162117			−0.0810587	+	0.996709i	$0.525830\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		−	11.6931i		−	0.427827i
$748$		0			0
$749$	32.8600	+	4.22800i	1.20068	+	0.154488i
$750$		0			0
$751$			25.4592i			0.929020i	0.885568	+	0.464510i	$0.153770\pi$
$751$			25.4592i			0.929020i	−0.885568	+	0.464510i	$0.846230\pi$
$752$		0			0
$753$		−	74.1760i		−	2.70312i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	11.5440i		−	0.419574i	−0.977747	−	0.209787i	$-0.932723\pi$
$757$		−	11.5440i		−	0.419574i	0.977747	−	0.209787i	$-0.0672770\pi$
$758$		0			0
$759$	−56.5340			−2.05206
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	−17.7705	−	2.28648i	−0.643337	−	0.0827762i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	34.7118			1.25337
$768$		0			0
$769$		−	28.6320i		−	1.03250i	−0.856439	−	0.516248i	$-0.827328\pi$
$769$		−	28.6320i		−	1.03250i	0.856439	−	0.516248i	$-0.172672\pi$
$770$		0			0
$771$			38.7635i			1.39603i
$772$		0			0
$773$	−43.2280			−1.55480			−0.777402	−	0.629005i	$-0.783463\pi$
$773$	−43.2280			−1.55480			−0.777402	+	0.629005i	$0.783463\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−89.7200	−	11.5440i	−3.21869	−	0.414139i
$778$		0			0
$779$		−	27.4378i		−	0.983060i
$780$		0			0
$781$	26.4560			0.946670
$782$		0			0
$783$		−	39.1781i		−	1.40011i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	13.9795i		−	0.498317i	−0.968463	−	0.249159i	$-0.919846\pi$
$787$		−	13.9795i		−	0.498317i	0.968463	−	0.249159i	$-0.0801540\pi$
$788$		0			0
$789$		−	25.0880i		−	0.893157i
$790$		0			0
$791$	5.24824	−	40.7893i	0.186606	−	1.45030i
$792$		0			0
$793$			16.9120i			0.600562i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−19.2280			−0.681091			−0.340545	−	0.940228i	$-0.610612\pi$
$797$	−19.2280			−0.681091			−0.340545	+	0.940228i	$0.610612\pi$
$798$		0			0
$799$			7.68868i			0.272006i
$800$		0			0
$801$		−	69.2640i		−	2.44732i
$802$		0			0
$803$		−	21.8222i		−	0.770088i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	46.0376			1.62060
$808$		0			0
$809$	−14.3160			−0.503324			−0.251662	−	0.967815i	$-0.580977\pi$
$809$	−14.3160			−0.503324			−0.251662	+	0.967815i	$0.580977\pi$
$810$		0			0
$811$	42.2938			1.48514			0.742569	−	0.669770i	$-0.233607\pi$
$811$	42.2938			1.48514			0.742569	+	0.669770i	$0.233607\pi$
$812$		0			0
$813$		−	4.00000i		−	0.140286i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−9.26402			−0.324107
$818$		0			0
$819$	−72.2787	−	9.29989i	−2.52562	−	0.324964i
$820$		0			0
$821$	−15.6840			−0.547375			−0.273688	−	0.961819i	$-0.588243\pi$
$821$	−15.6840			−0.547375			−0.273688	+	0.961819i	$0.588243\pi$
$822$		0			0
$823$	18.9671			0.661153			0.330576	−	0.943779i	$-0.392757\pi$
$823$	18.9671			0.661153			0.330576	+	0.943779i	$0.392757\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	5.24824			0.182499			0.0912496	−	0.995828i	$-0.470914\pi$
$827$	5.24824			0.182499			0.0912496	+	0.995828i	$0.470914\pi$
$828$		0			0
$829$			49.0880i			1.70490i	0.522811	+	0.852448i	$0.324883\pi$
$829$			49.0880i			1.70490i	−0.522811	+	0.852448i	$0.675117\pi$
$830$		0			0
$831$	57.8846			2.00799
$832$		0			0
$833$	32.3160	+	8.45600i	1.11968	+	0.292983i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		−	48.6320i		−	1.68097i
$838$		0			0
$839$	−6.44483			−0.222500			−0.111250	−	0.993792i	$-0.535485\pi$
$839$	−6.44483			−0.222500			−0.111250	+	0.993792i	$0.535485\pi$
$840$		0			0
$841$	−6.22800			−0.214759
$842$		0			0
$843$			60.1711i			2.07240i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−5.84653	−	0.752256i	−0.200889	−	0.0258478i
$848$		0			0
$849$	−70.9480			−2.43493
$850$		0			0
$851$		−	60.5857i		−	2.07685i
$852$		0			0
$853$	−20.1760			−0.690814			−0.345407	−	0.938453i	$-0.612259\pi$
$853$	−20.1760			−0.690814			−0.345407	+	0.938453i	$0.612259\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	6.00000			0.204956			0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000			0.204956			0.102478	+	0.994735i	$0.467323\pi$
$858$		0			0
$859$	−40.9433			−1.39697			−0.698483	−	0.715626i	$-0.746141\pi$
$859$	−40.9433			−1.39697			−0.698483	+	0.715626i	$0.746141\pi$
$860$		0			0
$861$	−46.6320	−	6.00000i	−1.58921	−	0.204479i
$862$		0			0
$863$	6.07747			0.206880			0.103440	−	0.994636i	$-0.467015\pi$
$863$	6.07747			0.206880			0.103440	+	0.994636i	$0.467015\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			17.0953i			0.580586i
$868$		0			0
$869$	24.9480			0.846304
$870$		0			0
$871$	−59.7564			−2.02477
$872$		0			0
$873$	−96.8080			−3.27646
$874$		0			0
$875$		0			0
$876$		0			0
$877$			5.08801i			0.171810i	0.996303	+	0.0859049i	$0.0273781\pi$
$877$			5.08801i			0.171810i	−0.996303	+	0.0859049i	$0.972622\pi$
$878$		0			0
$879$		−	42.4005i		−	1.43013i
$880$		0			0
$881$			34.6320i			1.16678i	0.812191	+	0.583391i	$0.198274\pi$
$881$			34.6320i			1.16678i	−0.812191	+	0.583391i	$0.801726\pi$
$882$		0			0
$883$	−11.6931			−0.393503			−0.196751	−	0.980453i	$-0.563039\pi$
$883$	−11.6931			−0.393503			−0.196751	+	0.980453i	$0.563039\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	54.5082i		−	1.83021i	−0.403220	−	0.915103i	$-0.632109\pi$
$887$		−	54.5082i		−	1.83021i	0.403220	−	0.915103i	$-0.367891\pi$
$888$		0			0
$889$	32.8600	+	4.22800i	1.10209	+	0.141803i
$890$		0			0
$891$		−	25.4592i		−	0.852916i
$892$		0			0
$893$		−	7.36799i		−	0.246560i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		−	74.1760i		−	2.47667i
$898$		0			0
$899$	28.2670			0.942758
$900$		0			0
$901$		−	28.6320i		−	0.953871i
$902$		0			0
$903$	−2.02582	+	15.7447i	−0.0674152	+	0.523951i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−19.7964			−0.657327			−0.328664	−	0.944447i	$-0.606598\pi$
$907$	−19.7964			−0.657327			−0.328664	+	0.944447i	$0.606598\pi$
$908$		0			0
$909$			55.0880i			1.82715i
$910$		0			0
$911$		−	34.7118i		−	1.15005i	−0.818134	−	0.575027i	$-0.804991\pi$
$911$		−	34.7118i		−	1.15005i	0.818134	−	0.575027i	$-0.195009\pi$
$912$		0			0
$913$	7.36799			0.243845
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−27.5440	−	3.54400i	−0.909583	−	0.117033i
$918$		0			0
$919$		−	22.2368i		−	0.733525i	−0.930315	−	0.366762i	$-0.880466\pi$
$919$		−	22.2368i		−	0.733525i	0.930315	−	0.366762i	$-0.119534\pi$
$920$		0			0
$921$	92.4920			3.04772
$922$		0			0
$923$			34.7118i			1.14255i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		−	24.8906i		−	0.817516i
$928$		0			0
$929$			1.08801i			0.0356964i	0.999841	+	0.0178482i	$0.00568156\pi$
$929$			1.08801i			0.0356964i	−0.999841	+	0.0178482i	$0.994318\pi$
$930$		0			0
$931$	−30.9681	−	8.10330i	−1.01494	−	0.265575i
$932$		0			0
$933$			71.7200i			2.34801i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−19.2280			−0.628151			−0.314076	−	0.949398i	$-0.601695\pi$
$937$	−19.2280			−0.628151			−0.314076	+	0.949398i	$0.601695\pi$
$938$		0			0
$939$			14.1335i			0.461230i
$940$		0			0
$941$		−	26.1760i		−	0.853314i	−0.904414	−	0.426657i	$-0.859691\pi$
$941$		−	26.1760i		−	0.853314i	0.904414	−	0.426657i	$-0.140309\pi$
$942$		0			0
$943$		−	31.4894i		−	1.02544i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	33.5153			1.08910			0.544550	−	0.838729i	$-0.316701\pi$
$947$	33.5153			1.08910			0.544550	+	0.838729i	$0.316701\pi$
$948$		0			0
$949$	28.6320			0.929434
$950$		0			0
$951$	74.3046			2.40949
$952$		0			0
$953$			6.00000i			0.194359i	0.995267	+	0.0971795i	$0.0309821\pi$
$953$			6.00000i			0.194359i	−0.995267	+	0.0971795i	$0.969018\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	51.4040			1.66166
$958$		0			0
$959$	−1.19659	+	9.29989i	−0.0386399	+	0.300309i
$960$		0			0
$961$	4.08801			0.131871
$962$		0			0
$963$	−72.2787			−2.32915
$964$		0			0
$965$		0			0
$966$		0			0
$967$	41.6186			1.33836			0.669181	−	0.743099i	$-0.266645\pi$
$967$	41.6186			1.33836			0.669181	+	0.743099i	$0.266645\pi$
$968$		0			0
$969$		−	64.6320i		−	2.07628i
$970$		0			0
$971$	−32.3187			−1.03716			−0.518578	−	0.855031i	$-0.673538\pi$
$971$	−32.3187			−1.03716			−0.518578	+	0.855031i	$0.673538\pi$
$972$		0			0
$973$	−24.0000	−	3.08801i	−0.769405	−	0.0989970i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	37.0880i		−	1.18655i	−0.805000	−	0.593275i	$-0.797835\pi$
$977$		−	37.0880i		−	1.18655i	0.805000	−	0.593275i	$-0.202165\pi$
$978$		0			0
$979$	43.6444			1.39488
$980$		0			0
$981$	39.0880			1.24798
$982$		0			0
$983$			16.1593i			0.515403i	0.966225	+	0.257701i	$0.0829651\pi$
$983$			16.1593i			0.515403i	−0.966225	+	0.257701i	$0.917035\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−12.5223	−	1.61121i	−0.398589	−	0.0512853i
$988$		0			0
$989$	−10.6320			−0.338078
$990$		0			0
$991$			42.8151i			1.36007i	0.733181	+	0.680034i	$0.238035\pi$
$991$			42.8151i			1.36007i	−0.733181	+	0.680034i	$0.761965\pi$
$992$		0			0
$993$	−40.6320			−1.28942
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−33.4040			−1.05792			−0.528958	−	0.848648i	$-0.677417\pi$
$997$	−33.4040			−1.05792			−0.528958	+	0.848648i	$0.677417\pi$
$998$		0			0
$999$	94.7762			2.99859

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2800.2.e.h.2799.8		8
4.3	odd	2	inner	2800.2.e.h.2799.1		8
5.2	odd	4		560.2.k.b.111.7	yes	8
5.3	odd	4		2800.2.k.m.2351.1		8
5.4	even	2		2800.2.e.g.2799.1		8
7.6	odd	2		2800.2.e.g.2799.2		8
15.2	even	4		5040.2.d.d.4591.6		8
20.3	even	4		2800.2.k.m.2351.8		8
20.7	even	4		560.2.k.b.111.1	✓	8
20.19	odd	2		2800.2.e.g.2799.8		8
28.27	even	2		2800.2.e.g.2799.7		8
35.13	even	4		2800.2.k.m.2351.7		8
35.27	even	4		560.2.k.b.111.2	yes	8
35.34	odd	2	inner	2800.2.e.h.2799.7		8
40.27	even	4		2240.2.k.d.1791.8		8
40.37	odd	4		2240.2.k.d.1791.2		8
60.47	odd	4		5040.2.d.d.4591.7		8
105.62	odd	4		5040.2.d.d.4591.3		8
140.27	odd	4		560.2.k.b.111.8	yes	8
140.83	odd	4		2800.2.k.m.2351.2		8
140.139	even	2	inner	2800.2.e.h.2799.2		8
280.27	odd	4		2240.2.k.d.1791.1		8
280.237	even	4		2240.2.k.d.1791.7		8
420.167	even	4		5040.2.d.d.4591.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.k.b.111.1	✓	8	20.7	even	4
560.2.k.b.111.2	yes	8	35.27	even	4
560.2.k.b.111.7	yes	8	5.2	odd	4
560.2.k.b.111.8	yes	8	140.27	odd	4
2240.2.k.d.1791.1		8	280.27	odd	4
2240.2.k.d.1791.2		8	40.37	odd	4
2240.2.k.d.1791.7		8	280.237	even	4
2240.2.k.d.1791.8		8	40.27	even	4
2800.2.e.g.2799.1		8	5.4	even	2
2800.2.e.g.2799.2		8	7.6	odd	2
2800.2.e.g.2799.7		8	28.27	even	2
2800.2.e.g.2799.8		8	20.19	odd	2
2800.2.e.h.2799.1		8	4.3	odd	2	inner
2800.2.e.h.2799.2		8	140.139	even	2	inner
2800.2.e.h.2799.7		8	35.34	odd	2	inner
2800.2.e.h.2799.8		8	1.1	even	1	trivial
2800.2.k.m.2351.1		8	5.3	odd	4
2800.2.k.m.2351.2		8	140.83	odd	4
2800.2.k.m.2351.7		8	35.13	even	4
2800.2.k.m.2351.8		8	20.3	even	4
5040.2.d.d.4591.2		8	420.167	even	4
5040.2.d.d.4591.3		8	105.62	odd	4
5040.2.d.d.4591.6		8	15.2	even	4
5040.2.d.d.4591.7		8	60.47	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2800.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.96176i q^{3} +(2.62412 + 0.337637i) q^{7} -5.77200 q^{9} +O(q^{10})\)	\(q+2.96176i q^{3} +(2.62412 + 0.337637i) q^{7} -5.77200 q^{9} -3.63703i q^{11} +4.77200 q^{13} +4.77200 q^{17} -4.57296 q^{19} +(-1.00000 + 7.77200i) q^{21} -5.24824 q^{23} -8.20999i q^{27} +4.77200 q^{29} +5.92351 q^{31} +10.7720 q^{33} +11.5440i q^{37} +14.1335i q^{39} +6.00000i q^{41} +2.02582 q^{43} +1.61121i q^{47} +(6.77200 + 1.77200i) q^{49} +14.1335i q^{51} -6.00000i q^{53} -13.5440i q^{57} +7.27406 q^{59} +3.54400i q^{61} +(-15.1464 - 1.94884i) q^{63} -12.5223 q^{67} -15.5440i q^{69} +7.27406i q^{71} +6.00000 q^{73} +(1.22800 - 9.54400i) q^{77} +6.85944i q^{79} +7.00000 q^{81} +2.02582i q^{83} +14.1335i q^{87} +12.0000i q^{89} +(12.5223 + 1.61121i) q^{91} +17.5440i q^{93} +16.7720 q^{97} +20.9930i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{9}+O(q^{10}) \) 8 * q - 12 * q^9	\( 8 q - 12 q^{9} + 4 q^{13} + 4 q^{17} - 8 q^{21} + 4 q^{29} + 52 q^{33} + 20 q^{49} + 48 q^{73} + 44 q^{77} + 56 q^{81} + 100 q^{97}+O(q^{100}) \) 8 * q - 12 * q^9 + 4 * q^13 + 4 * q^17 - 8 * q^21 + 4 * q^29 + 52 * q^33 + 20 * q^49 + 48 * q^73 + 44 * q^77 + 56 * q^81 + 100 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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