LMFDB - Embedded newform 2300.4.c.d.1749.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2300,4,Mod(1749,2300)] 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("2300.1749"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2300.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,0,0,0,0,0,-60,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$135.704393013$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 164x^{8} + 9308x^{6} + 204385x^{4} + 1216296x^{2} + 1089936 $ x^10 + 164x^8 + 9308x^6 + 204385x^4 + 1216296x^2 + 1089936
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 460)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1749.5
Root			$-1.04116i$ of defining polynomial
Character	$\chi$	$=$	2300.1749
Dual form			2300.4.c.d.1749.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.04116i q^{3} -21.6113i q^{7} +22.8337 q^{9} -30.3495 q^{11} +69.6786i q^{13} +43.4506i q^{17} -14.3892 q^{19} -44.1120 q^{21} -23.0000i q^{23} -101.718i q^{27} +158.105 q^{29} -46.4782 q^{31} +61.9480i q^{33} -39.6849i q^{37} +142.225 q^{39} -201.771 q^{41} +17.8927i q^{43} +0.702668i q^{47} -124.048 q^{49} +88.6894 q^{51} -661.161i q^{53} +29.3705i q^{57} +732.679 q^{59} -875.997 q^{61} -493.465i q^{63} -746.803i q^{67} -46.9466 q^{69} -275.246 q^{71} -499.633i q^{73} +655.892i q^{77} +797.970 q^{79} +408.887 q^{81} -1150.10i q^{83} -322.716i q^{87} -1166.89 q^{89} +1505.85 q^{91} +94.8693i q^{93} -403.209i q^{97} -692.991 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 60 q^{9} + 14 q^{11} + 26 q^{19} + 506 q^{29} - 196 q^{31} + 820 q^{39} - 1548 q^{41} + 1366 q^{49} - 1314 q^{51} + 1526 q^{59} + 674 q^{61} - 138 q^{69} - 3008 q^{71} - 1252 q^{79} - 1918 q^{81}+ \cdots - 222 q^{99}+O(q^{100}) $ 10 * q - 60 * q^9 + 14 * q^11 + 26 * q^19 + 506 * q^29 - 196 * q^31 + 820 * q^39 - 1548 * q^41 + 1366 * q^49 - 1314 * q^51 + 1526 * q^59 + 674 * q^61 - 138 * q^69 - 3008 * q^71 - 1252 * q^79 - 1918 * q^81 - 1292 * q^89 - 1534 * q^91 - 222 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times$.

$n$	$277$	$1151$	$1201$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	− 2.04116i	− 0.392821i	−0.980522	−	0.196410i	$-0.937072\pi$
$3$	− 2.04116i	− 0.392821i	0.980522	−	0.196410i	$-0.0629284\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 21.6113i	− 1.16690i	−0.812149	−	0.583450i	$-0.801702\pi$
$7$	− 21.6113i	− 1.16690i	0.812149	−	0.583450i	$-0.198298\pi$
$8$	0	0
$9$	22.8337	0.845692
$10$	0	0
$11$	−30.3495	−0.831884	−0.415942	−	0.909391i	$-0.636548\pi$
$11$	−30.3495	−0.831884	−0.415942	+	0.909391i	$0.636548\pi$
$12$	0	0
$13$	69.6786i	1.48657i	0.668976	+	0.743284i	$0.266733\pi$
$13$	69.6786i	1.48657i	−0.668976	+	0.743284i	$0.733267\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	43.4506i	0.619901i	0.950753	+	0.309950i	$0.100312\pi$
$17$	43.4506i	0.619901i	−0.950753	+	0.309950i	$0.899688\pi$
$18$	0	0
$19$	−14.3892	−0.173742	−0.0868710	−	0.996220i	$-0.527687\pi$
$19$	−14.3892	−0.173742	−0.0868710	+	0.996220i	$0.527687\pi$
$20$	0	0
$21$	−44.1120	−0.458382
$22$	0	0
$23$	− 23.0000i	− 0.208514i
$23$	− 23.0000i	− 0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 101.718i	− 0.725026i
$28$	0	0
$29$	158.105	1.01239	0.506195	−	0.862419i	$-0.331052\pi$
$29$	158.105	1.01239	0.506195	+	0.862419i	$0.331052\pi$
$30$	0	0
$31$	−46.4782	−0.269282	−0.134641	−	0.990894i	$-0.542988\pi$
$31$	−46.4782	−0.269282	−0.134641	+	0.990894i	$0.542988\pi$
$32$	0	0
$33$	61.9480i	0.326781i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 39.6849i	− 0.176328i	−0.996106	−	0.0881642i	$-0.971900\pi$
$37$	− 39.6849i	− 0.176328i	0.996106	−	0.0881642i	$-0.0281000\pi$
$38$	0	0
$39$	142.225	0.583954
$40$	0	0
$41$	−201.771	−0.768571	−0.384285	−	0.923214i	$-0.625552\pi$
$41$	−201.771	−0.768571	−0.384285	+	0.923214i	$0.625552\pi$
$42$	0	0
$43$	17.8927i	0.0634562i	0.999497	+	0.0317281i	$0.0101011\pi$
$43$	17.8927i	0.0634562i	−0.999497	+	0.0317281i	$0.989899\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.702668i	0.00218074i	0.999999	+	0.00109037i	$0.000347075\pi$
$47$	0.702668i	0.00218074i	−0.999999	+	0.00109037i	$0.999653\pi$
$48$	0	0
$49$	−124.048	−0.361655
$50$	0	0
$51$	88.6894	0.243510
$52$	0	0
$53$	− 661.161i	− 1.71354i	−0.515701	−	0.856769i	$-0.672468\pi$
$53$	− 661.161i	− 1.71354i	0.515701	−	0.856769i	$-0.327532\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	29.3705i	0.0682495i
$58$	0	0
$59$	732.679	1.61672	0.808361	−	0.588686i	$-0.200355\pi$
$59$	732.679	1.61672	0.808361	+	0.588686i	$0.200355\pi$
$60$	0	0
$61$	−875.997	−1.83869	−0.919344	−	0.393454i	$-0.871280\pi$
$61$	−875.997	−1.83869	−0.919344	+	0.393454i	$0.871280\pi$
$62$	0	0
$63$	− 493.465i	− 0.986838i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 746.803i	− 1.36174i	−0.732404	−	0.680870i	$-0.761602\pi$
$67$	− 746.803i	− 1.36174i	0.732404	−	0.680870i	$-0.238398\pi$
$68$	0	0
$69$	−46.9466	−0.0819087
$70$	0	0
$71$	−275.246	−0.460079	−0.230040	−	0.973181i	$-0.573886\pi$
$71$	−275.246	−0.460079	−0.230040	+	0.973181i	$0.573886\pi$
$72$	0	0
$73$	− 499.633i	− 0.801063i	−0.916283	−	0.400532i	$-0.868825\pi$
$73$	− 499.633i	− 0.801063i	0.916283	−	0.400532i	$-0.131175\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	655.892i	0.970725i
$78$	0	0
$79$	797.970	1.13644	0.568219	−	0.822877i	$-0.307633\pi$
$79$	797.970	1.13644	0.568219	+	0.822877i	$0.307633\pi$
$80$	0	0
$81$	408.887	0.560887
$82$	0	0
$83$	− 1150.10i	− 1.52097i	−0.649358	−	0.760483i	$-0.724962\pi$
$83$	− 1150.10i	− 1.52097i	0.649358	−	0.760483i	$-0.275038\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 322.716i	− 0.397687i
$88$	0	0
$89$	−1166.89	−1.38978	−0.694890	−	0.719116i	$-0.744547\pi$
$89$	−1166.89	−1.38978	−0.694890	+	0.719116i	$0.744547\pi$
$90$	0	0
$91$	1505.85	1.73468
$92$	0	0
$93$	94.8693i	0.105779i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 403.209i	− 0.422058i	−0.977480	−	0.211029i	$-0.932318\pi$
$97$	− 403.209i	− 0.422058i	0.977480	−	0.211029i	$-0.0676815\pi$
$98$	0	0
$99$	−692.991	−0.703517
$100$	0	0
$101$	484.319	0.477144	0.238572	−	0.971125i	$-0.423321\pi$
$101$	484.319	0.477144	0.238572	+	0.971125i	$0.423321\pi$
$102$	0	0
$103$	− 2064.69i	− 1.97514i	−0.157167	−	0.987572i	$-0.550236\pi$
$103$	− 2064.69i	− 1.97514i	0.157167	−	0.987572i	$-0.449764\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1436.43i	1.29780i	0.760872	+	0.648902i	$0.224772\pi$
$107$	1436.43i	1.29780i	−0.760872	+	0.648902i	$0.775228\pi$
$108$	0	0
$109$	−1870.62	−1.64379	−0.821895	−	0.569639i	$-0.807083\pi$
$109$	−1870.62	−1.64379	−0.821895	+	0.569639i	$0.807083\pi$
$110$	0	0
$111$	−81.0030	−0.0692654
$112$	0	0
$113$	1606.67i	1.33755i	0.743465	+	0.668775i	$0.233181\pi$
$113$	1606.67i	1.33755i	−0.743465	+	0.668775i	$0.766819\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1591.02i	1.25718i
$118$	0	0
$119$	939.023	0.723362
$120$	0	0
$121$	−409.908	−0.307970
$122$	0	0
$123$	411.847i	0.301910i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2031.66i	1.41953i	0.704438	+	0.709766i	$0.251199\pi$
$127$	2031.66i	1.41953i	−0.704438	+	0.709766i	$0.748801\pi$
$128$	0	0
$129$	36.5219	0.0249269
$130$	0	0
$131$	−434.691	−0.289917	−0.144958	−	0.989438i	$-0.546305\pi$
$131$	−434.691	−0.289917	−0.144958	+	0.989438i	$0.546305\pi$
$132$	0	0
$133$	310.968i	0.202740i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 939.729i	− 0.586033i	−0.956107	−	0.293016i	$-0.905341\pi$
$137$	− 939.729i	− 0.586033i	0.956107	−	0.293016i	$-0.0946590\pi$
$138$	0	0
$139$	124.609	0.0760376	0.0380188	−	0.999277i	$-0.487895\pi$
$139$	124.609	0.0760376	0.0380188	+	0.999277i	$0.487895\pi$
$140$	0	0
$141$	1.43426	0.000856639 0
$142$	0	0
$143$	− 2114.71i	− 1.23665i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	253.201i	0.142066i
$148$	0	0
$149$	−1375.14	−0.756079	−0.378039	−	0.925789i	$-0.623402\pi$
$149$	−1375.14	−0.756079	−0.378039	+	0.925789i	$0.623402\pi$
$150$	0	0
$151$	−1196.51	−0.644838	−0.322419	−	0.946597i	$-0.604496\pi$
$151$	−1196.51	−0.644838	−0.322419	+	0.946597i	$0.604496\pi$
$152$	0	0
$153$	992.137i	0.524245i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	212.340i	0.107940i	0.998543	+	0.0539699i	$0.0171875\pi$
$157$	212.340i	0.107940i	−0.998543	+	0.0539699i	$0.982812\pi$
$158$	0	0
$159$	−1349.53	−0.673113
$160$	0	0
$161$	−497.060	−0.243315
$162$	0	0
$163$	1390.16i	0.668013i	0.942571	+	0.334006i	$0.108401\pi$
$163$	1390.16i	0.668013i	−0.942571	+	0.334006i	$0.891599\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 362.349i	− 0.167901i	−0.996470	−	0.0839503i	$-0.973246\pi$
$167$	− 362.349i	− 0.167901i	0.996470	−	0.0839503i	$-0.0267537\pi$
$168$	0	0
$169$	−2658.11	−1.20988
$170$	0	0
$171$	−328.557	−0.146932
$172$	0	0
$173$	986.596i	0.433581i	0.976218	+	0.216790i	$0.0695588\pi$
$173$	986.596i	0.433581i	−0.976218	+	0.216790i	$0.930441\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 1495.51i	− 0.635082i
$178$	0	0
$179$	−3977.44	−1.66083	−0.830413	−	0.557149i	$-0.811895\pi$
$179$	−3977.44	−1.66083	−0.830413	+	0.557149i	$0.811895\pi$
$180$	0	0
$181$	2512.70	1.03186	0.515932	−	0.856630i	$-0.327446\pi$
$181$	2512.70	1.03186	0.515932	+	0.856630i	$0.327446\pi$
$182$	0	0
$183$	1788.05i	0.722275i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1318.70i	− 0.515685i
$188$	0	0
$189$	−2198.26	−0.846032
$190$	0	0
$191$	1977.49	0.749142	0.374571	−	0.927198i	$-0.377790\pi$
$191$	1977.49	0.749142	0.374571	+	0.927198i	$0.377790\pi$
$192$	0	0
$193$	− 1241.30i	− 0.462957i	−0.972840	−	0.231478i	$-0.925644\pi$
$193$	− 1241.30i	− 0.462957i	0.972840	−	0.231478i	$-0.0743563\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 1969.91i	− 0.712436i	−0.934403	−	0.356218i	$-0.884066\pi$
$197$	− 1969.91i	− 0.712436i	0.934403	−	0.356218i	$-0.115934\pi$
$198$	0	0
$199$	4053.09	1.44380	0.721899	−	0.691998i	$-0.243269\pi$
$199$	4053.09	1.44380	0.721899	+	0.691998i	$0.243269\pi$
$200$	0	0
$201$	−1524.34	−0.534919
$202$	0	0
$203$	− 3416.85i	− 1.18136i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 525.175i	− 0.176339i
$208$	0	0
$209$	436.704	0.144533
$210$	0	0
$211$	374.110	0.122061	0.0610303	−	0.998136i	$-0.480561\pi$
$211$	374.110	0.122061	0.0610303	+	0.998136i	$0.480561\pi$
$212$	0	0
$213$	561.819i	0.180729i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1004.45i	0.314225i
$218$	0	0
$219$	−1019.83	−0.314674
$220$	0	0
$221$	−3027.58	−0.921524
$222$	0	0
$223$	3226.90i	0.969010i	0.874788	+	0.484505i	$0.161000\pi$
$223$	3226.90i	0.969010i	−0.874788	+	0.484505i	$0.839000\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 3020.61i	− 0.883194i	−0.897213	−	0.441597i	$-0.854412\pi$
$227$	− 3020.61i	− 0.883194i	0.897213	−	0.441597i	$-0.145588\pi$
$228$	0	0
$229$	−934.761	−0.269741	−0.134871	−	0.990863i	$-0.543062\pi$
$229$	−934.761	−0.269741	−0.134871	+	0.990863i	$0.543062\pi$
$230$	0	0
$231$	1338.78	0.381321
$232$	0	0
$233$	− 3916.88i	− 1.10130i	−0.834736	−	0.550651i	$-0.814380\pi$
$233$	− 3916.88i	− 1.10130i	0.834736	−	0.550651i	$-0.185620\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 1628.78i	− 0.446416i
$238$	0	0
$239$	−3200.06	−0.866086	−0.433043	−	0.901373i	$-0.642560\pi$
$239$	−3200.06	−0.866086	−0.433043	+	0.901373i	$0.642560\pi$
$240$	0	0
$241$	−3774.24	−1.00880	−0.504399	−	0.863471i	$-0.668286\pi$
$241$	−3774.24	−1.00880	−0.504399	+	0.863471i	$0.668286\pi$
$242$	0	0
$243$	− 3581.00i	− 0.945354i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1002.62i	− 0.258279i
$248$	0	0
$249$	−2347.54	−0.597467
$250$	0	0
$251$	3342.01	0.840420	0.420210	−	0.907427i	$-0.361956\pi$
$251$	3342.01	0.840420	0.420210	+	0.907427i	$0.361956\pi$
$252$	0	0
$253$	698.039i	0.173460i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2899.20i	0.703685i	0.936059	+	0.351842i	$0.114445\pi$
$257$	2899.20i	0.703685i	−0.936059	+	0.351842i	$0.885555\pi$
$258$	0	0
$259$	−857.641	−0.205758
$260$	0	0
$261$	3610.11	0.856170
$262$	0	0
$263$	− 6001.83i	− 1.40718i	−0.710605	−	0.703591i	$-0.751579\pi$
$263$	− 6001.83i	− 1.40718i	0.710605	−	0.703591i	$-0.248421\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2381.81i	0.545934i
$268$	0	0
$269$	−8308.29	−1.88314	−0.941571	−	0.336813i	$-0.890651\pi$
$269$	−8308.29	−1.88314	−0.941571	+	0.336813i	$0.890651\pi$
$270$	0	0
$271$	−2395.28	−0.536911	−0.268456	−	0.963292i	$-0.586513\pi$
$271$	−2395.28	−0.536911	−0.268456	+	0.963292i	$0.586513\pi$
$272$	0	0
$273$	− 3073.66i	− 0.681416i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 7466.49i	− 1.61956i	−0.586733	−	0.809780i	$-0.699586\pi$
$277$	− 7466.49i	− 1.61956i	0.586733	−	0.809780i	$-0.300414\pi$
$278$	0	0
$279$	−1061.27	−0.227729
$280$	0	0
$281$	−5570.87	−1.18267	−0.591335	−	0.806426i	$-0.701399\pi$
$281$	−5570.87	−1.18267	−0.591335	+	0.806426i	$0.701399\pi$
$282$	0	0
$283$	− 5273.48i	− 1.10769i	−0.832620	−	0.553845i	$-0.813160\pi$
$283$	− 5273.48i	− 1.10769i	0.832620	−	0.553845i	$-0.186840\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4360.54i	0.896845i
$288$	0	0
$289$	3025.05	0.615723
$290$	0	0
$291$	−823.012	−0.165793
$292$	0	0
$293$	− 511.514i	− 0.101990i	−0.998699	−	0.0509948i	$-0.983761\pi$
$293$	− 511.514i	− 0.101990i	0.998699	−	0.0509948i	$-0.0162392\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3087.10i	0.603137i
$298$	0	0
$299$	1602.61	0.309971
$300$	0	0
$301$	386.685	0.0740471
$302$	0	0
$303$	− 988.570i	− 0.187432i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 4157.46i	− 0.772894i	−0.922312	−	0.386447i	$-0.873702\pi$
$307$	− 4157.46i	− 0.772894i	0.922312	−	0.386447i	$-0.126298\pi$
$308$	0	0
$309$	−4214.35	−0.775877
$310$	0	0
$311$	−3408.45	−0.621465	−0.310733	−	0.950497i	$-0.600574\pi$
$311$	−3408.45	−0.621465	−0.310733	+	0.950497i	$0.600574\pi$
$312$	0	0
$313$	4516.35i	0.815589i	0.913074	+	0.407794i	$0.133702\pi$
$313$	4516.35i	0.815589i	−0.913074	+	0.407794i	$0.866298\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2693.62i	− 0.477251i	−0.971112	−	0.238626i	$-0.923303\pi$
$317$	− 2693.62i	− 0.477251i	0.971112	−	0.238626i	$-0.0766969\pi$
$318$	0	0
$319$	−4798.40	−0.842190
$320$	0	0
$321$	2931.98	0.509804
$322$	0	0
$323$	− 625.217i	− 0.107703i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3818.23i	0.645715i
$328$	0	0
$329$	15.1856	0.00254470
$330$	0	0
$331$	−11024.4	−1.83068	−0.915338	−	0.402687i	$-0.868076\pi$
$331$	−11024.4	−1.83068	−0.915338	+	0.402687i	$0.868076\pi$
$332$	0	0
$333$	− 906.152i	− 0.149120i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 5218.80i	− 0.843579i	−0.906694	−	0.421790i	$-0.861402\pi$
$337$	− 5218.80i	− 0.843579i	0.906694	−	0.421790i	$-0.138598\pi$
$338$	0	0
$339$	3279.47	0.525417
$340$	0	0
$341$	1410.59	0.224011
$342$	0	0
$343$	− 4731.84i	− 0.744885i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 93.8512i	− 0.0145193i	−0.999974	−	0.00725965i	$-0.997689\pi$
$347$	− 93.8512i	− 0.0145193i	0.999974	−	0.00725965i	$-0.00231084\pi$
$348$	0	0
$349$	−4540.27	−0.696376	−0.348188	−	0.937425i	$-0.613203\pi$
$349$	−4540.27	−0.696376	−0.348188	+	0.937425i	$0.613203\pi$
$350$	0	0
$351$	7087.59	1.07780
$352$	0	0
$353$	4287.12i	0.646403i	0.946330	+	0.323202i	$0.104759\pi$
$353$	4287.12i	0.646403i	−0.946330	+	0.323202i	$0.895241\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 1916.69i	− 0.284151i
$358$	0	0
$359$	176.418	0.0259359	0.0129680	−	0.999916i	$-0.495872\pi$
$359$	176.418	0.0259359	0.0129680	+	0.999916i	$0.495872\pi$
$360$	0	0
$361$	−6651.95	−0.969814
$362$	0	0
$363$	836.686i	0.120977i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	7745.87i	1.10172i	0.834598	+	0.550860i	$0.185700\pi$
$367$	7745.87i	1.10172i	−0.834598	+	0.550860i	$0.814300\pi$
$368$	0	0
$369$	−4607.18	−0.649974
$370$	0	0
$371$	−14288.5	−1.99953
$372$	0	0
$373$	− 8268.02i	− 1.14773i	−0.818951	−	0.573863i	$-0.805444\pi$
$373$	− 8268.02i	− 1.14773i	0.818951	−	0.573863i	$-0.194556\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	11016.5i	1.50499i
$378$	0	0
$379$	−7229.18	−0.979783	−0.489892	−	0.871783i	$-0.662964\pi$
$379$	−7229.18	−0.979783	−0.489892	+	0.871783i	$0.662964\pi$
$380$	0	0
$381$	4146.93	0.557621
$382$	0	0
$383$	− 519.390i	− 0.0692940i	−0.999400	−	0.0346470i	$-0.988969\pi$
$383$	− 519.390i	− 0.0692940i	0.999400	−	0.0346470i	$-0.0110307\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	408.557i	0.0536644i
$388$	0	0
$389$	10746.4	1.40068	0.700341	−	0.713809i	$-0.253031\pi$
$389$	10746.4	1.40068	0.700341	+	0.713809i	$0.253031\pi$
$390$	0	0
$391$	999.363	0.129258
$392$	0	0
$393$	887.271i	0.113885i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	11738.8i	1.48401i	0.670395	+	0.742004i	$0.266125\pi$
$397$	11738.8i	1.48401i	−0.670395	+	0.742004i	$0.733875\pi$
$398$	0	0
$399$	634.734	0.0796403
$400$	0	0
$401$	−94.4786	−0.0117657	−0.00588284	−	0.999983i	$-0.501873\pi$
$401$	−94.4786	−0.0117657	−0.00588284	+	0.999983i	$0.501873\pi$
$402$	0	0
$403$	− 3238.54i	− 0.400306i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1204.42i	0.146685i
$408$	0	0
$409$	13712.5	1.65780	0.828901	−	0.559396i	$-0.188967\pi$
$409$	13712.5	1.65780	0.828901	+	0.559396i	$0.188967\pi$
$410$	0	0
$411$	−1918.13	−0.230206
$412$	0	0
$413$	− 15834.1i	− 1.88655i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 254.347i	− 0.0298691i
$418$	0	0
$419$	−4392.60	−0.512154	−0.256077	−	0.966656i	$-0.582430\pi$
$419$	−4392.60	−0.512154	−0.256077	+	0.966656i	$0.582430\pi$
$420$	0	0
$421$	112.287	0.0129989	0.00649943	−	0.999979i	$-0.497931\pi$
$421$	112.287	0.0129989	0.00649943	+	0.999979i	$0.497931\pi$
$422$	0	0
$423$	16.0445i	0.00184423i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	18931.4i	2.14556i
$428$	0	0
$429$	−4316.46	−0.485782
$430$	0	0
$431$	−7035.30	−0.786261	−0.393131	−	0.919483i	$-0.628608\pi$
$431$	−7035.30	−0.786261	−0.393131	+	0.919483i	$0.628608\pi$
$432$	0	0
$433$	12046.0i	1.33693i	0.743742	+	0.668467i	$0.233049\pi$
$433$	12046.0i	1.33693i	−0.743742	+	0.668467i	$0.766951\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	330.951i	0.0362277i
$438$	0	0
$439$	2089.02	0.227115	0.113557	−	0.993531i	$-0.463775\pi$
$439$	2089.02	0.227115	0.113557	+	0.993531i	$0.463775\pi$
$440$	0	0
$441$	−2832.47	−0.305849
$442$	0	0
$443$	− 7436.42i	− 0.797551i	−0.917049	−	0.398776i	$-0.869435\pi$
$443$	− 7436.42i	− 0.797551i	0.917049	−	0.398776i	$-0.130565\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2806.87i	0.297003i
$448$	0	0
$449$	−1796.33	−0.188806	−0.0944032	−	0.995534i	$-0.530094\pi$
$449$	−1796.33	−0.188806	−0.0944032	+	0.995534i	$0.530094\pi$
$450$	0	0
$451$	6123.66	0.639361
$452$	0	0
$453$	2442.26i	0.253306i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3232.73i	0.330899i	0.986218	+	0.165450i	$0.0529075\pi$
$457$	3232.73i	0.330899i	−0.986218	+	0.165450i	$0.947093\pi$
$458$	0	0
$459$	4419.72	0.449444
$460$	0	0
$461$	−10750.2	−1.08609	−0.543044	−	0.839704i	$-0.682728\pi$
$461$	−10750.2	−1.08609	−0.543044	+	0.839704i	$0.682728\pi$
$462$	0	0
$463$	6598.81i	0.662360i	0.943568	+	0.331180i	$0.107447\pi$
$463$	6598.81i	0.662360i	−0.943568	+	0.331180i	$0.892553\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 8153.38i	− 0.807909i	−0.914779	−	0.403954i	$-0.867635\pi$
$467$	− 8153.38i	− 0.807909i	0.914779	−	0.403954i	$-0.132365\pi$
$468$	0	0
$469$	−16139.4	−1.58901
$470$	0	0
$471$	433.418	0.0424010
$472$	0	0
$473$	− 543.036i	− 0.0527882i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 15096.8i	− 1.44913i
$478$	0	0
$479$	−15056.7	−1.43624	−0.718120	−	0.695919i	$-0.754997\pi$
$479$	−15056.7	−1.43624	−0.718120	+	0.695919i	$0.754997\pi$
$480$	0	0
$481$	2765.19	0.262124
$482$	0	0
$483$	1014.58i	0.0955793i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 2098.90i	− 0.195298i	−0.995221	−	0.0976491i	$-0.968868\pi$
$487$	− 2098.90i	− 0.195298i	0.995221	−	0.0976491i	$-0.0311323\pi$
$488$	0	0
$489$	2837.54	0.262409
$490$	0	0
$491$	−2245.71	−0.206410	−0.103205	−	0.994660i	$-0.532910\pi$
$491$	−2245.71	−0.206410	−0.103205	+	0.994660i	$0.532910\pi$
$492$	0	0
$493$	6869.74i	0.627581i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5948.41i	0.536866i
$498$	0	0
$499$	−3216.32	−0.288542	−0.144271	−	0.989538i	$-0.546084\pi$
$499$	−3216.32	−0.288542	−0.144271	+	0.989538i	$0.546084\pi$
$500$	0	0
$501$	−739.611	−0.0659548
$502$	0	0
$503$	− 15459.5i	− 1.37039i	−0.728361	−	0.685193i	$-0.759718\pi$
$503$	− 15459.5i	− 1.37039i	0.728361	−	0.685193i	$-0.240282\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	5425.62i	0.475267i
$508$	0	0
$509$	−12581.8	−1.09564	−0.547818	−	0.836597i	$-0.684541\pi$
$509$	−12581.8	−1.09564	−0.547818	+	0.836597i	$0.684541\pi$
$510$	0	0
$511$	−10797.7	−0.934761
$512$	0	0
$513$	1463.64i	0.125967i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 21.3256i	− 0.00181412i
$518$	0	0
$519$	2013.80	0.170320
$520$	0	0
$521$	2681.76	0.225509	0.112754	−	0.993623i	$-0.464033\pi$
$521$	2681.76	0.225509	0.112754	+	0.993623i	$0.464033\pi$
$522$	0	0
$523$	− 9009.09i	− 0.753232i	−0.926370	−	0.376616i	$-0.877088\pi$
$523$	− 9009.09i	− 0.753232i	0.926370	−	0.376616i	$-0.122912\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 2019.50i	− 0.166928i
$528$	0	0
$529$	−529.000	−0.0434783
$530$	0	0
$531$	16729.8	1.36725
$532$	0	0
$533$	− 14059.2i	− 1.14253i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	8118.57i	0.652406i
$538$	0	0
$539$	3764.78	0.300855
$540$	0	0
$541$	22789.6	1.81109	0.905546	−	0.424249i	$-0.139462\pi$
$541$	22789.6	1.81109	0.905546	+	0.424249i	$0.139462\pi$
$542$	0	0
$543$	− 5128.80i	− 0.405337i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 20709.0i	− 1.61874i	−0.587297	−	0.809371i	$-0.699808\pi$
$547$	− 20709.0i	− 1.61874i	0.587297	−	0.809371i	$-0.300192\pi$
$548$	0	0
$549$	−20002.3	−1.55496
$550$	0	0
$551$	−2274.99	−0.175895
$552$	0	0
$553$	− 17245.2i	− 1.32611i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 586.443i	− 0.0446111i	−0.999751	−	0.0223055i	$-0.992899\pi$
$557$	− 586.443i	− 0.0446111i	0.999751	−	0.0223055i	$-0.00710066\pi$
$558$	0	0
$559$	−1246.74	−0.0943320
$560$	0	0
$561$	−2691.68	−0.202572
$562$	0	0
$563$	− 4988.10i	− 0.373399i	−0.982417	−	0.186699i	$-0.940221\pi$
$563$	− 4988.10i	− 0.373399i	0.982417	−	0.186699i	$-0.0597790\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 8836.57i	− 0.654499i
$568$	0	0
$569$	−5548.19	−0.408773	−0.204387	−	0.978890i	$-0.565520\pi$
$569$	−5548.19	−0.408773	−0.204387	+	0.978890i	$0.565520\pi$
$570$	0	0
$571$	−2684.93	−0.196779	−0.0983893	−	0.995148i	$-0.531369\pi$
$571$	−2684.93	−0.196779	−0.0983893	+	0.995148i	$0.531369\pi$
$572$	0	0
$573$	− 4036.36i	− 0.294278i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 1462.86i	− 0.105546i	−0.998607	−	0.0527728i	$-0.983194\pi$
$577$	− 1462.86i	− 0.105546i	0.998607	−	0.0527728i	$-0.0168059\pi$
$578$	0	0
$579$	−2533.68	−0.181859
$580$	0	0
$581$	−24855.2	−1.77481
$582$	0	0
$583$	20065.9i	1.42546i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11020.7i	0.774912i	0.921888	+	0.387456i	$0.126646\pi$
$587$	11020.7i	0.774912i	−0.921888	+	0.387456i	$0.873354\pi$
$588$	0	0
$589$	668.782	0.0467856
$590$	0	0
$591$	−4020.88	−0.279860
$592$	0	0
$593$	7341.16i	0.508373i	0.967155	+	0.254186i	$0.0818077\pi$
$593$	7341.16i	0.508373i	−0.967155	+	0.254186i	$0.918192\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 8272.98i	− 0.567154i
$598$	0	0
$599$	12642.5	0.862366	0.431183	−	0.902264i	$-0.358096\pi$
$599$	12642.5	0.862366	0.431183	+	0.902264i	$0.358096\pi$
$600$	0	0
$601$	7426.60	0.504056	0.252028	−	0.967720i	$-0.418903\pi$
$601$	7426.60	0.504056	0.252028	+	0.967720i	$0.418903\pi$
$602$	0	0
$603$	− 17052.3i	− 1.15161i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	7506.30i	0.501929i	0.967996	+	0.250965i	$0.0807478\pi$
$607$	7506.30i	0.501929i	−0.967996	+	0.250965i	$0.919252\pi$
$608$	0	0
$609$	−6974.31	−0.464061
$610$	0	0
$611$	−48.9610	−0.00324182
$612$	0	0
$613$	23467.7i	1.54625i	0.634254	+	0.773124i	$0.281307\pi$
$613$	23467.7i	1.54625i	−0.634254	+	0.773124i	$0.718693\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 1436.56i	− 0.0937337i	−0.998901	−	0.0468669i	$-0.985076\pi$
$617$	− 1436.56i	− 0.0937337i	0.998901	−	0.0468669i	$-0.0149236\pi$
$618$	0	0
$619$	−12664.4	−0.822332	−0.411166	−	0.911561i	$-0.634878\pi$
$619$	−12664.4	−0.822332	−0.411166	+	0.911561i	$0.634878\pi$
$620$	0	0
$621$	−2339.52	−0.151178
$622$	0	0
$623$	25218.1i	1.62173i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 891.380i	− 0.0567756i
$628$	0	0
$629$	1724.33	0.109306
$630$	0	0
$631$	5383.68	0.339653	0.169826	−	0.985474i	$-0.445679\pi$
$631$	5383.68	0.339653	0.169826	+	0.985474i	$0.445679\pi$
$632$	0	0
$633$	− 763.616i	− 0.0479479i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 8643.47i	− 0.537625i
$638$	0	0
$639$	−6284.87	−0.389085
$640$	0	0
$641$	31245.1	1.92529	0.962643	−	0.270775i	$-0.0872799\pi$
$641$	31245.1	1.92529	0.962643	+	0.270775i	$0.0872799\pi$
$642$	0	0
$643$	22381.8i	1.37271i	0.727268	+	0.686354i	$0.240790\pi$
$643$	22381.8i	1.37271i	−0.727268	+	0.686354i	$0.759210\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 13062.8i	− 0.793741i	−0.917875	−	0.396870i	$-0.870096\pi$
$647$	− 13062.8i	− 0.793741i	0.917875	−	0.396870i	$-0.129904\pi$
$648$	0	0
$649$	−22236.4	−1.34493
$650$	0	0
$651$	2050.25	0.123434
$652$	0	0
$653$	− 17486.1i	− 1.04791i	−0.851746	−	0.523955i	$-0.824456\pi$
$653$	− 17486.1i	− 1.04791i	0.851746	−	0.523955i	$-0.175544\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 11408.5i	− 0.677453i
$658$	0	0
$659$	29707.1	1.75603	0.878016	−	0.478631i	$-0.158867\pi$
$659$	29707.1	1.75603	0.878016	+	0.478631i	$0.158867\pi$
$660$	0	0
$661$	−9905.74	−0.582888	−0.291444	−	0.956588i	$-0.594136\pi$
$661$	−9905.74	−0.582888	−0.291444	+	0.956588i	$0.594136\pi$
$662$	0	0
$663$	6179.76i	0.361994i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 3636.41i	− 0.211098i
$668$	0	0
$669$	6586.61	0.380647
$670$	0	0
$671$	26586.1	1.52957
$672$	0	0
$673$	− 7803.48i	− 0.446957i	−0.974709	−	0.223478i	$-0.928259\pi$
$673$	− 7803.48i	− 0.446957i	0.974709	−	0.223478i	$-0.0717412\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 29359.1i	− 1.66671i	−0.552740	−	0.833354i	$-0.686418\pi$
$677$	− 29359.1i	− 1.66671i	0.552740	−	0.833354i	$-0.313582\pi$
$678$	0	0
$679$	−8713.86	−0.492500
$680$	0	0
$681$	−6165.54	−0.346937
$682$	0	0
$683$	− 26055.8i	− 1.45973i	−0.683591	−	0.729866i	$-0.739583\pi$
$683$	− 26055.8i	− 1.45973i	0.683591	−	0.729866i	$-0.260417\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1907.99i	0.105960i
$688$	0	0
$689$	46068.8	2.54729
$690$	0	0
$691$	15166.4	0.834959	0.417480	−	0.908686i	$-0.362914\pi$
$691$	15166.4	0.834959	0.417480	+	0.908686i	$0.362914\pi$
$692$	0	0
$693$	14976.4i	0.820934i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 8767.08i	− 0.476437i
$698$	0	0
$699$	−7994.95	−0.432614
$700$	0	0
$701$	−26527.8	−1.42930	−0.714650	−	0.699482i	$-0.753414\pi$
$701$	−26527.8	−1.42930	−0.714650	+	0.699482i	$0.753414\pi$
$702$	0	0
$703$	571.032i	0.0306357i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 10466.7i	− 0.556779i
$708$	0	0
$709$	−21268.1	−1.12657	−0.563287	−	0.826261i	$-0.690464\pi$
$709$	−21268.1	−1.12657	−0.563287	+	0.826261i	$0.690464\pi$
$710$	0	0
$711$	18220.6	0.961077
$712$	0	0
$713$	1069.00i	0.0561491i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6531.82i	0.340216i
$718$	0	0
$719$	27640.4	1.43368	0.716838	−	0.697240i	$-0.245589\pi$
$719$	27640.4	1.43368	0.716838	+	0.697240i	$0.245589\pi$
$720$	0	0
$721$	−44620.6	−2.30480
$722$	0	0
$723$	7703.81i	0.396276i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	10860.2i	0.554036i	0.960865	+	0.277018i	$0.0893462\pi$
$727$	10860.2i	0.554036i	−0.960865	+	0.277018i	$0.910654\pi$
$728$	0	0
$729$	3730.57	0.189533
$730$	0	0
$731$	−777.450	−0.0393366
$732$	0	0
$733$	− 2575.74i	− 0.129791i	−0.997892	−	0.0648956i	$-0.979329\pi$
$733$	− 2575.74i	− 0.129791i	0.997892	−	0.0648956i	$-0.0206715\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	22665.1i	1.13281i
$738$	0	0
$739$	25649.2	1.27675	0.638377	−	0.769724i	$-0.279606\pi$
$739$	25649.2	1.27675	0.638377	+	0.769724i	$0.279606\pi$
$740$	0	0
$741$	−2046.50	−0.101457
$742$	0	0
$743$	3767.84i	0.186041i	0.995664	+	0.0930207i	$0.0296523\pi$
$743$	3767.84i	0.186041i	−0.995664	+	0.0930207i	$0.970348\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 26261.1i	− 1.28627i
$748$	0	0
$749$	31043.1	1.51441
$750$	0	0
$751$	16030.6	0.778913	0.389456	−	0.921045i	$-0.372663\pi$
$751$	16030.6	0.778913	0.389456	+	0.921045i	$0.372663\pi$
$752$	0	0
$753$	− 6821.55i	− 0.330134i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	3408.80i	0.163666i	0.996646	+	0.0818329i	$0.0260774\pi$
$757$	3408.80i	0.163666i	−0.996646	+	0.0818329i	$0.973923\pi$
$758$	0	0
$759$	1424.81	0.0681385
$760$	0	0
$761$	28208.1	1.34368	0.671841	−	0.740696i	$-0.265504\pi$
$761$	28208.1	1.34368	0.671841	+	0.740696i	$0.265504\pi$
$762$	0	0
$763$	40426.6i	1.91814i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	51052.1i	2.40337i
$768$	0	0
$769$	−26349.3	−1.23560	−0.617802	−	0.786334i	$-0.711977\pi$
$769$	−26349.3	−1.23560	−0.617802	+	0.786334i	$0.711977\pi$
$770$	0	0
$771$	5917.71	0.276422
$772$	0	0
$773$	29491.9i	1.37225i	0.727482	+	0.686126i	$0.240690\pi$
$773$	29491.9i	1.37225i	−0.727482	+	0.686126i	$0.759310\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1750.58i	0.0808258i
$778$	0	0
$779$	2903.32	0.133533
$780$	0	0
$781$	8353.56	0.382732
$782$	0	0
$783$	− 16082.1i	− 0.734009i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 18485.5i	− 0.837276i	−0.908153	−	0.418638i	$-0.862508\pi$
$787$	− 18485.5i	− 0.837276i	0.908153	−	0.418638i	$-0.137492\pi$
$788$	0	0
$789$	−12250.7	−0.552770
$790$	0	0
$791$	34722.3	1.56079
$792$	0	0
$793$	− 61038.3i	− 2.73333i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 36684.3i	− 1.63039i	−0.579184	−	0.815197i	$-0.696629\pi$
$797$	− 36684.3i	− 1.63039i	0.579184	−	0.815197i	$-0.303371\pi$
$798$	0	0
$799$	−30.5313	−0.00135184
$800$	0	0
$801$	−26644.5	−1.17533
$802$	0	0
$803$	15163.6i	0.666391i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	16958.5i	0.739737i
$808$	0	0
$809$	−216.567	−0.00941173	−0.00470586	−	0.999989i	$-0.501498\pi$
$809$	−216.567	−0.00941173	−0.00470586	+	0.999989i	$0.501498\pi$
$810$	0	0
$811$	25897.5	1.12131	0.560657	−	0.828048i	$-0.310549\pi$
$811$	25897.5	1.12131	0.560657	+	0.828048i	$0.310549\pi$
$812$	0	0
$813$	4889.14i	0.210910i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 257.462i	− 0.0110250i
$818$	0	0
$819$	34384.0	1.46700
$820$	0	0
$821$	12735.6	0.541384	0.270692	−	0.962666i	$-0.412748\pi$
$821$	12735.6	0.541384	0.270692	+	0.962666i	$0.412748\pi$
$822$	0	0
$823$	25798.2i	1.09267i	0.837566	+	0.546336i	$0.183978\pi$
$823$	25798.2i	1.09267i	−0.837566	+	0.546336i	$0.816022\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 21295.2i	− 0.895413i	−0.894181	−	0.447706i	$-0.852241\pi$
$827$	− 21295.2i	− 0.895413i	0.894181	−	0.447706i	$-0.147759\pi$
$828$	0	0
$829$	43456.4	1.82063	0.910315	−	0.413915i	$-0.135839\pi$
$829$	43456.4	1.82063	0.910315	+	0.413915i	$0.135839\pi$
$830$	0	0
$831$	−15240.3	−0.636197
$832$	0	0
$833$	− 5389.94i	− 0.224190i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	4727.68i	0.195236i
$838$	0	0
$839$	−14245.5	−0.586184	−0.293092	−	0.956084i	$-0.594684\pi$
$839$	−14245.5	−0.586184	−0.293092	+	0.956084i	$0.594684\pi$
$840$	0	0
$841$	608.090	0.0249330
$842$	0	0
$843$	11371.0i	0.464577i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	8858.64i	0.359370i
$848$	0	0
$849$	−10764.0	−0.435123
$850$	0	0
$851$	−912.752	−0.0367670
$852$	0	0
$853$	1967.69i	0.0789830i	0.999220	+	0.0394915i	$0.0125738\pi$
$853$	1967.69i	0.0789830i	−0.999220	+	0.0394915i	$0.987426\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	32176.3i	1.28252i	0.767322	+	0.641262i	$0.221589\pi$
$857$	32176.3i	1.28252i	−0.767322	+	0.641262i	$0.778411\pi$
$858$	0	0
$859$	1273.26	0.0505741	0.0252871	−	0.999680i	$-0.491950\pi$
$859$	1273.26	0.0505741	0.0252871	+	0.999680i	$0.491950\pi$
$860$	0	0
$861$	8900.54	0.352299
$862$	0	0
$863$	69.6881i	0.00274880i	0.999999	+	0.00137440i	$0.000437485\pi$
$863$	69.6881i	0.00274880i	−0.999999	+	0.00137440i	$0.999563\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 6174.59i	− 0.241869i
$868$	0	0
$869$	−24218.0	−0.945384
$870$	0	0
$871$	52036.3	2.02432
$872$	0	0
$873$	− 9206.75i	− 0.356931i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	20284.8i	0.781038i	0.920595	+	0.390519i	$0.127704\pi$
$877$	20284.8i	0.781038i	−0.920595	+	0.390519i	$0.872296\pi$
$878$	0	0
$879$	−1044.08	−0.0400636
$880$	0	0
$881$	−29637.5	−1.13339	−0.566693	−	0.823929i	$-0.691778\pi$
$881$	−29637.5	−1.13339	−0.566693	+	0.823929i	$0.691778\pi$
$882$	0	0
$883$	− 16749.3i	− 0.638344i	−0.947697	−	0.319172i	$-0.896595\pi$
$883$	− 16749.3i	− 0.638344i	0.947697	−	0.319172i	$-0.103405\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	22904.8i	0.867042i	0.901143	+	0.433521i	$0.142729\pi$
$887$	22904.8i	0.867042i	−0.901143	+	0.433521i	$0.857271\pi$
$888$	0	0
$889$	43906.7	1.65645
$890$	0	0
$891$	−12409.5	−0.466593
$892$	0	0
$893$	− 10.1108i	0 0.000378886i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 3271.17i	− 0.121763i
$898$	0	0
$899$	−7348.42	−0.272618
$900$	0	0
$901$	28727.8	1.06222
$902$	0	0
$903$	− 789.285i	− 0.0290872i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	20869.7i	0.764020i	0.924158	+	0.382010i	$0.124768\pi$
$907$	20869.7i	0.764020i	−0.924158	+	0.382010i	$0.875232\pi$
$908$	0	0
$909$	11058.8	0.403517
$910$	0	0
$911$	−42361.0	−1.54060	−0.770298	−	0.637684i	$-0.779892\pi$
$911$	−42361.0	−1.54060	−0.770298	+	0.637684i	$0.779892\pi$
$912$	0	0
$913$	34905.0i	1.26527i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9394.22i	0.338304i
$918$	0	0
$919$	−14136.3	−0.507413	−0.253707	−	0.967281i	$-0.581650\pi$
$919$	−14136.3	−0.507413	−0.253707	+	0.967281i	$0.581650\pi$
$920$	0	0
$921$	−8486.01	−0.303609
$922$	0	0
$923$	− 19178.7i	− 0.683939i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 47144.5i	− 1.67036i
$928$	0	0
$929$	7897.49	0.278911	0.139455	−	0.990228i	$-0.455465\pi$
$929$	7897.49	0.278911	0.139455	+	0.990228i	$0.455465\pi$
$930$	0	0
$931$	1784.94	0.0628347
$932$	0	0
$933$	6957.19i	0.244124i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 21499.1i	− 0.749567i	−0.927112	−	0.374783i	$-0.877717\pi$
$937$	− 21499.1i	− 0.749567i	0.927112	−	0.374783i	$-0.122283\pi$
$938$	0	0
$939$	9218.58	0.320380
$940$	0	0
$941$	30258.7	1.04825	0.524125	−	0.851641i	$-0.324392\pi$
$941$	30258.7	1.04825	0.524125	+	0.851641i	$0.324392\pi$
$942$	0	0
$943$	4640.74i	0.160258i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	12769.1i	0.438163i	0.975707	+	0.219082i	$0.0703061\pi$
$947$	12769.1i	0.438163i	−0.975707	+	0.219082i	$0.929694\pi$
$948$	0	0
$949$	34813.8	1.19083
$950$	0	0
$951$	−5498.09	−0.187474
$952$	0	0
$953$	28174.8i	0.957683i	0.877901	+	0.478841i	$0.158943\pi$
$953$	28174.8i	0.957683i	−0.877901	+	0.478841i	$0.841057\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	9794.28i	0.330830i
$958$	0	0
$959$	−20308.8	−0.683841
$960$	0	0
$961$	−27630.8	−0.927487
$962$	0	0
$963$	32799.0i	1.09754i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2696.72i	0.0896800i	0.998994	+	0.0448400i	$0.0142778\pi$
$967$	2696.72i	0.0896800i	−0.998994	+	0.0448400i	$0.985722\pi$
$968$	0	0
$969$	−1276.17	−0.0423079
$970$	0	0
$971$	−29192.7	−0.964817	−0.482408	−	0.875946i	$-0.660238\pi$
$971$	−29192.7	−0.964817	−0.482408	+	0.875946i	$0.660238\pi$
$972$	0	0
$973$	− 2692.97i	− 0.0887283i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1150.88i	0.0376867i	0.999822	+	0.0188433i	$0.00599838\pi$
$977$	1150.88i	0.0376867i	−0.999822	+	0.0188433i	$0.994002\pi$
$978$	0	0
$979$	35414.6	1.15613
$980$	0	0
$981$	−42713.2	−1.39014
$982$	0	0
$983$	− 9924.92i	− 0.322030i	−0.986952	−	0.161015i	$-0.948523\pi$
$983$	− 9924.92i	− 0.322030i	0.986952	−	0.161015i	$-0.0514768\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 30.9961i	0 0.000999612i
$988$	0	0
$989$	411.533	0.0132315
$990$	0	0
$991$	32446.8	1.04007	0.520034	−	0.854146i	$-0.325919\pi$
$991$	32446.8	1.04007	0.520034	+	0.854146i	$0.325919\pi$
$992$	0	0
$993$	22502.4i	0.719127i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 49280.6i	− 1.56543i	−0.622381	−	0.782715i	$-0.713834\pi$
$997$	− 49280.6i	− 1.56543i	0.622381	−	0.782715i	$-0.286166\pi$
$998$	0	0
$999$	−4036.68	−0.127843

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.4.c.d.1749.5		10
5.2	odd	4		460.4.a.b.1.3	✓	5
5.3	odd	4		2300.4.a.c.1.3		5
5.4	even	2	inner	2300.4.c.d.1749.6		10
20.7	even	4		1840.4.a.o.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.4.a.b.1.3	✓	5	5.2	odd	4
1840.4.a.o.1.3		5	20.7	even	4
2300.4.a.c.1.3		5	5.3	odd	4
2300.4.c.d.1749.5		10	1.1	even	1	trivial
2300.4.c.d.1749.6		10	5.4	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2300.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.04116i q^{3} -21.6113i q^{7} +22.8337 q^{9} -30.3495 q^{11} +69.6786i q^{13} +43.4506i q^{17} -14.3892 q^{19} -44.1120 q^{21} -23.0000i q^{23} -101.718i q^{27} +158.105 q^{29} -46.4782 q^{31} +61.9480i q^{33} -39.6849i q^{37} +142.225 q^{39} -201.771 q^{41} +17.8927i q^{43} +0.702668i q^{47} -124.048 q^{49} +88.6894 q^{51} -661.161i q^{53} +29.3705i q^{57} +732.679 q^{59} -875.997 q^{61} -493.465i q^{63} -746.803i q^{67} -46.9466 q^{69} -275.246 q^{71} -499.633i q^{73} +655.892i q^{77} +797.970 q^{79} +408.887 q^{81} -1150.10i q^{83} -322.716i q^{87} -1166.89 q^{89} +1505.85 q^{91} +94.8693i q^{93} -403.209i q^{97} -692.991 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 60 q^{9} + 14 q^{11} + 26 q^{19} + 506 q^{29} - 196 q^{31} + 820 q^{39} - 1548 q^{41} + 1366 q^{49} - 1314 q^{51} + 1526 q^{59} + 674 q^{61} - 138 q^{69} - 3008 q^{71} - 1252 q^{79} - 1918 q^{81}+ \cdots - 222 q^{99}+O(q^{100}) \) 10 * q - 60 * q^9 + 14 * q^11 + 26 * q^19 + 506 * q^29 - 196 * q^31 + 820 * q^39 - 1548 * q^41 + 1366 * q^49 - 1314 * q^51 + 1526 * q^59 + 674 * q^61 - 138 * q^69 - 3008 * q^71 - 1252 * q^79 - 1918 * q^81 - 1292 * q^89 - 1534 * q^91 - 222 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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