LMFDB - Embedded newform 4200.2.a.bo.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4200,2,Mod(1,4200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4200, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4200.a (trivial)

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:	yes
Analytic conductor:	$33.5371688489$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 840)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	4200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +0.387873 q^{11} +2.96239 q^{13} -3.35026 q^{17} -2.96239 q^{19} +1.00000 q^{21} +0.962389 q^{23} -1.00000 q^{27} +1.22425 q^{29} +2.96239 q^{31} -0.387873 q^{33} -5.92478 q^{37} -2.96239 q^{39} +1.03761 q^{41} -10.7005 q^{43} -3.22425 q^{47} +1.00000 q^{49} +3.35026 q^{51} +5.66291 q^{53} +2.96239 q^{57} +3.22425 q^{59} +14.6253 q^{61} -1.00000 q^{63} -0.962389 q^{69} +5.53690 q^{71} -6.18664 q^{73} -0.387873 q^{77} -13.9248 q^{79} +1.00000 q^{81} -3.22425 q^{83} -1.22425 q^{87} +3.73813 q^{89} -2.96239 q^{91} -2.96239 q^{93} +7.73813 q^{97} +0.387873 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9} + 2 q^{11} - 2 q^{13} + 2 q^{19} + 3 q^{21} - 8 q^{23} - 3 q^{27} + 2 q^{29} - 2 q^{31} - 2 q^{33} + 4 q^{37} + 2 q^{39} + 14 q^{41} - 12 q^{43} - 8 q^{47} + 3 q^{49} - 14 q^{53} - 2 q^{57} + 8 q^{59} + 2 q^{61} - 3 q^{63} + 8 q^{69} - 6 q^{71} - 6 q^{73} - 2 q^{77} - 20 q^{79} + 3 q^{81} - 8 q^{83} - 2 q^{87} + 2 q^{89} + 2 q^{91} + 2 q^{93} + 14 q^{97} + 2 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^7 + 3 * q^9 + 2 * q^11 - 2 * q^13 + 2 * q^19 + 3 * q^21 - 8 * q^23 - 3 * q^27 + 2 * q^29 - 2 * q^31 - 2 * q^33 + 4 * q^37 + 2 * q^39 + 14 * q^41 - 12 * q^43 - 8 * q^47 + 3 * q^49 - 14 * q^53 - 2 * q^57 + 8 * q^59 + 2 * q^61 - 3 * q^63 + 8 * q^69 - 6 * q^71 - 6 * q^73 - 2 * q^77 - 20 * q^79 + 3 * q^81 - 8 * q^83 - 2 * q^87 + 2 * q^89 + 2 * q^91 + 2 * q^93 + 14 * q^97 + 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.387873	0.116948	0.0584741	−	0.998289i	$-0.481377\pi$
$11$	0.387873	0.116948	0.0584741	+	0.998289i	$0.481377\pi$
$12$	0	0
$13$	2.96239	0.821619	0.410809	−	0.911721i	$-0.365246\pi$
$13$	2.96239	0.821619	0.410809	+	0.911721i	$0.365246\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.35026	−0.812558	−0.406279	−	0.913749i	$-0.633174\pi$
$17$	−3.35026	−0.812558	−0.406279	+	0.913749i	$0.633174\pi$
$18$	0	0
$19$	−2.96239	−0.679619	−0.339809	−	0.940494i	$-0.610363\pi$
$19$	−2.96239	−0.679619	−0.339809	+	0.940494i	$0.610363\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	0.962389	0.200672	0.100336	−	0.994954i	$-0.468008\pi$
$23$	0.962389	0.200672	0.100336	+	0.994954i	$0.468008\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.22425	0.227338	0.113669	−	0.993519i	$-0.463740\pi$
$29$	1.22425	0.227338	0.113669	+	0.993519i	$0.463740\pi$
$30$	0	0
$31$	2.96239	0.532061	0.266030	−	0.963965i	$-0.414288\pi$
$31$	2.96239	0.532061	0.266030	+	0.963965i	$0.414288\pi$
$32$	0	0
$33$	−0.387873	−0.0675200
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.92478	−0.974027	−0.487014	−	0.873394i	$-0.661914\pi$
$37$	−5.92478	−0.974027	−0.487014	+	0.873394i	$0.661914\pi$
$38$	0	0
$39$	−2.96239	−0.474362
$40$	0	0
$41$	1.03761	0.162048	0.0810238	−	0.996712i	$-0.474181\pi$
$41$	1.03761	0.162048	0.0810238	+	0.996712i	$0.474181\pi$
$42$	0	0
$43$	−10.7005	−1.63181	−0.815907	−	0.578183i	$-0.803762\pi$
$43$	−10.7005	−1.63181	−0.815907	+	0.578183i	$0.803762\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.22425	−0.470306	−0.235153	−	0.971958i	$-0.575559\pi$
$47$	−3.22425	−0.470306	−0.235153	+	0.971958i	$0.575559\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.35026	0.469130
$52$	0	0
$53$	5.66291	0.777861	0.388930	−	0.921267i	$-0.372845\pi$
$53$	5.66291	0.777861	0.388930	+	0.921267i	$0.372845\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.96239	0.392378
$58$	0	0
$59$	3.22425	0.419762	0.209881	−	0.977727i	$-0.432692\pi$
$59$	3.22425	0.419762	0.209881	+	0.977727i	$0.432692\pi$
$60$	0	0
$61$	14.6253	1.87258	0.936289	−	0.351231i	$-0.114237\pi$
$61$	14.6253	1.87258	0.936289	+	0.351231i	$0.114237\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	−0.962389	−0.115858
$70$	0	0
$71$	5.53690	0.657110	0.328555	−	0.944485i	$-0.393438\pi$
$71$	5.53690	0.657110	0.328555	+	0.944485i	$0.393438\pi$
$72$	0	0
$73$	−6.18664	−0.724092	−0.362046	−	0.932160i	$-0.617922\pi$
$73$	−6.18664	−0.724092	−0.362046	+	0.932160i	$0.617922\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.387873	−0.0442022
$78$	0	0
$79$	−13.9248	−1.56666	−0.783330	−	0.621606i	$-0.786480\pi$
$79$	−13.9248	−1.56666	−0.783330	+	0.621606i	$0.786480\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.22425	−0.353908	−0.176954	−	0.984219i	$-0.556624\pi$
$83$	−3.22425	−0.353908	−0.176954	+	0.984219i	$0.556624\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1.22425	−0.131254
$88$	0	0
$89$	3.73813	0.396242	0.198121	−	0.980178i	$-0.436516\pi$
$89$	3.73813	0.396242	0.198121	+	0.980178i	$0.436516\pi$
$90$	0	0
$91$	−2.96239	−0.310543
$92$	0	0
$93$	−2.96239	−0.307185
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.73813	0.785689	0.392844	−	0.919605i	$-0.371491\pi$
$97$	7.73813	0.785689	0.392844	+	0.919605i	$0.371491\pi$
$98$	0	0
$99$	0.387873	0.0389827
$100$	0	0
$101$	−7.58769	−0.755003	−0.377502	−	0.926009i	$-0.623217\pi$
$101$	−7.58769	−0.755003	−0.377502	+	0.926009i	$0.623217\pi$
$102$	0	0
$103$	−14.7005	−1.44849	−0.724243	−	0.689545i	$-0.757811\pi$
$103$	−14.7005	−1.44849	−0.724243	+	0.689545i	$0.757811\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−16.4387	−1.58919	−0.794593	−	0.607143i	$-0.792315\pi$
$107$	−16.4387	−1.58919	−0.794593	+	0.607143i	$0.792315\pi$
$108$	0	0
$109$	−2.77575	−0.265868	−0.132934	−	0.991125i	$-0.542440\pi$
$109$	−2.77575	−0.265868	−0.132934	+	0.991125i	$0.542440\pi$
$110$	0	0
$111$	5.92478	0.562355
$112$	0	0
$113$	−4.26187	−0.400923	−0.200461	−	0.979702i	$-0.564244\pi$
$113$	−4.26187	−0.400923	−0.200461	+	0.979702i	$0.564244\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.96239	0.273873
$118$	0	0
$119$	3.35026	0.307118
$120$	0	0
$121$	−10.8496	−0.986323
$122$	0	0
$123$	−1.03761	−0.0935583
$124$	0	0
$125$	0	0
$126$	0	0
$127$	14.5501	1.29111	0.645555	−	0.763714i	$-0.276626\pi$
$127$	14.5501	1.29111	0.645555	+	0.763714i	$0.276626\pi$
$128$	0	0
$129$	10.7005	0.942129
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	2.96239	0.256872
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−10.4387	−0.891835	−0.445917	−	0.895074i	$-0.647122\pi$
$137$	−10.4387	−0.891835	−0.445917	+	0.895074i	$0.647122\pi$
$138$	0	0
$139$	5.66291	0.480322	0.240161	−	0.970733i	$-0.422800\pi$
$139$	5.66291	0.480322	0.240161	+	0.970733i	$0.422800\pi$
$140$	0	0
$141$	3.22425	0.271531
$142$	0	0
$143$	1.14903	0.0960868
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	20.5501	1.68353	0.841764	−	0.539846i	$-0.181517\pi$
$149$	20.5501	1.68353	0.841764	+	0.539846i	$0.181517\pi$
$150$	0	0
$151$	−22.7005	−1.84734	−0.923671	−	0.383186i	$-0.874827\pi$
$151$	−22.7005	−1.84734	−0.923671	+	0.383186i	$0.874827\pi$
$152$	0	0
$153$	−3.35026	−0.270853
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.18664	0.174513	0.0872565	−	0.996186i	$-0.472190\pi$
$157$	2.18664	0.174513	0.0872565	+	0.996186i	$0.472190\pi$
$158$	0	0
$159$	−5.66291	−0.449098
$160$	0	0
$161$	−0.962389	−0.0758468
$162$	0	0
$163$	−7.22425	−0.565847	−0.282924	−	0.959142i	$-0.591304\pi$
$163$	−7.22425	−0.565847	−0.282924	+	0.959142i	$0.591304\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.8496	1.22648	0.613238	−	0.789898i	$-0.289867\pi$
$167$	15.8496	1.22648	0.613238	+	0.789898i	$0.289867\pi$
$168$	0	0
$169$	−4.22425	−0.324943
$170$	0	0
$171$	−2.96239	−0.226540
$172$	0	0
$173$	−23.9756	−1.82283	−0.911414	−	0.411490i	$-0.865008\pi$
$173$	−23.9756	−1.82283	−0.911414	+	0.411490i	$0.865008\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−3.22425	−0.242350
$178$	0	0
$179$	12.2374	0.914668	0.457334	−	0.889295i	$-0.348804\pi$
$179$	12.2374	0.914668	0.457334	+	0.889295i	$0.348804\pi$
$180$	0	0
$181$	8.70052	0.646705	0.323352	−	0.946279i	$-0.395190\pi$
$181$	8.70052	0.646705	0.323352	+	0.946279i	$0.395190\pi$
$182$	0	0
$183$	−14.6253	−1.08113
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.29948	−0.0950271
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−13.5369	−0.979496	−0.489748	−	0.871864i	$-0.662911\pi$
$191$	−13.5369	−0.979496	−0.489748	+	0.871864i	$0.662911\pi$
$192$	0	0
$193$	1.92478	0.138548	0.0692742	−	0.997598i	$-0.477932\pi$
$193$	1.92478	0.138548	0.0692742	+	0.997598i	$0.477932\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−20.1114	−1.43288	−0.716440	−	0.697649i	$-0.754229\pi$
$197$	−20.1114	−1.43288	−0.716440	+	0.697649i	$0.754229\pi$
$198$	0	0
$199$	−8.26187	−0.585668	−0.292834	−	0.956163i	$-0.594598\pi$
$199$	−8.26187	−0.585668	−0.292834	+	0.956163i	$0.594598\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.22425	−0.0859258
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0.962389	0.0668906
$208$	0	0
$209$	−1.14903	−0.0794801
$210$	0	0
$211$	−18.1768	−1.25134	−0.625671	−	0.780087i	$-0.715175\pi$
$211$	−18.1768	−1.25134	−0.625671	+	0.780087i	$0.715175\pi$
$212$	0	0
$213$	−5.53690	−0.379382
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−2.96239	−0.201100
$218$	0	0
$219$	6.18664	0.418055
$220$	0	0
$221$	−9.92478	−0.667613
$222$	0	0
$223$	−21.4010	−1.43312	−0.716560	−	0.697525i	$-0.754284\pi$
$223$	−21.4010	−1.43312	−0.716560	+	0.697525i	$0.754284\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−18.5501	−1.23121	−0.615606	−	0.788054i	$-0.711089\pi$
$227$	−18.5501	−1.23121	−0.615606	+	0.788054i	$0.711089\pi$
$228$	0	0
$229$	−28.5501	−1.88664	−0.943321	−	0.331881i	$-0.892317\pi$
$229$	−28.5501	−1.88664	−0.943321	+	0.331881i	$0.892317\pi$
$230$	0	0
$231$	0.387873	0.0255202
$232$	0	0
$233$	−18.9624	−1.24227	−0.621134	−	0.783705i	$-0.713328\pi$
$233$	−18.9624	−1.24227	−0.621134	+	0.783705i	$0.713328\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	13.9248	0.904511
$238$	0	0
$239$	−18.1622	−1.17482	−0.587408	−	0.809291i	$-0.699851\pi$
$239$	−18.1622	−1.17482	−0.587408	+	0.809291i	$0.699851\pi$
$240$	0	0
$241$	14.3733	0.925865	0.462932	−	0.886394i	$-0.346797\pi$
$241$	14.3733	0.925865	0.462932	+	0.886394i	$0.346797\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−8.77575	−0.558387
$248$	0	0
$249$	3.22425	0.204329
$250$	0	0
$251$	−8.62530	−0.544424	−0.272212	−	0.962237i	$-0.587755\pi$
$251$	−8.62530	−0.544424	−0.272212	+	0.962237i	$0.587755\pi$
$252$	0	0
$253$	0.373285	0.0234682
$254$	0	0
$255$	0	0
$256$	0	0
$257$	27.8251	1.73568	0.867842	−	0.496841i	$-0.165507\pi$
$257$	27.8251	1.73568	0.867842	+	0.496841i	$0.165507\pi$
$258$	0	0
$259$	5.92478	0.368148
$260$	0	0
$261$	1.22425	0.0757794
$262$	0	0
$263$	0.589104	0.0363257	0.0181629	−	0.999835i	$-0.494218\pi$
$263$	0.589104	0.0363257	0.0181629	+	0.999835i	$0.494218\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−3.73813	−0.228770
$268$	0	0
$269$	15.7381	0.959571	0.479786	−	0.877386i	$-0.340714\pi$
$269$	15.7381	0.959571	0.479786	+	0.877386i	$0.340714\pi$
$270$	0	0
$271$	−7.11283	−0.432074	−0.216037	−	0.976385i	$-0.569313\pi$
$271$	−7.11283	−0.432074	−0.216037	+	0.976385i	$0.569313\pi$
$272$	0	0
$273$	2.96239	0.179292
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−5.92478	−0.355985	−0.177993	−	0.984032i	$-0.556960\pi$
$277$	−5.92478	−0.355985	−0.177993	+	0.984032i	$0.556960\pi$
$278$	0	0
$279$	2.96239	0.177354
$280$	0	0
$281$	29.3258	1.74943	0.874716	−	0.484636i	$-0.161048\pi$
$281$	29.3258	1.74943	0.874716	+	0.484636i	$0.161048\pi$
$282$	0	0
$283$	16.1016	0.957139	0.478570	−	0.878050i	$-0.341155\pi$
$283$	16.1016	0.957139	0.478570	+	0.878050i	$0.341155\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.03761	−0.0612483
$288$	0	0
$289$	−5.77575	−0.339750
$290$	0	0
$291$	−7.73813	−0.453617
$292$	0	0
$293$	−9.27504	−0.541854	−0.270927	−	0.962600i	$-0.587330\pi$
$293$	−9.27504	−0.541854	−0.270927	+	0.962600i	$0.587330\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−0.387873	−0.0225067
$298$	0	0
$299$	2.85097	0.164876
$300$	0	0
$301$	10.7005	0.616768
$302$	0	0
$303$	7.58769	0.435901
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−10.7005	−0.610711	−0.305356	−	0.952238i	$-0.598775\pi$
$307$	−10.7005	−0.610711	−0.305356	+	0.952238i	$0.598775\pi$
$308$	0	0
$309$	14.7005	0.836284
$310$	0	0
$311$	12.1016	0.686217	0.343109	−	0.939296i	$-0.388520\pi$
$311$	12.1016	0.686217	0.343109	+	0.939296i	$0.388520\pi$
$312$	0	0
$313$	−28.3634	−1.60320	−0.801598	−	0.597863i	$-0.796017\pi$
$313$	−28.3634	−1.60320	−0.801598	+	0.597863i	$0.796017\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.5125	0.983598	0.491799	−	0.870709i	$-0.336339\pi$
$317$	17.5125	0.983598	0.491799	+	0.870709i	$0.336339\pi$
$318$	0	0
$319$	0.474855	0.0265868
$320$	0	0
$321$	16.4387	0.917516
$322$	0	0
$323$	9.92478	0.552229
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.77575	0.153499
$328$	0	0
$329$	3.22425	0.177759
$330$	0	0
$331$	−19.2243	−1.05666	−0.528330	−	0.849039i	$-0.677182\pi$
$331$	−19.2243	−1.05666	−0.528330	+	0.849039i	$0.677182\pi$
$332$	0	0
$333$	−5.92478	−0.324676
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−16.0000	−0.871576	−0.435788	−	0.900049i	$-0.643530\pi$
$337$	−16.0000	−0.871576	−0.435788	+	0.900049i	$0.643530\pi$
$338$	0	0
$339$	4.26187	0.231473
$340$	0	0
$341$	1.14903	0.0622235
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.11283	0.0597401	0.0298700	−	0.999554i	$-0.490491\pi$
$347$	1.11283	0.0597401	0.0298700	+	0.999554i	$0.490491\pi$
$348$	0	0
$349$	−32.0263	−1.71433	−0.857166	−	0.515041i	$-0.827777\pi$
$349$	−32.0263	−1.71433	−0.857166	+	0.515041i	$0.827777\pi$
$350$	0	0
$351$	−2.96239	−0.158121
$352$	0	0
$353$	−6.20123	−0.330058	−0.165029	−	0.986289i	$-0.552772\pi$
$353$	−6.20123	−0.330058	−0.165029	+	0.986289i	$0.552772\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−3.35026	−0.177315
$358$	0	0
$359$	−1.68735	−0.0890549	−0.0445275	−	0.999008i	$-0.514178\pi$
$359$	−1.68735	−0.0890549	−0.0445275	+	0.999008i	$0.514178\pi$
$360$	0	0
$361$	−10.2243	−0.538119
$362$	0	0
$363$	10.8496	0.569454
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−3.84955	−0.200945	−0.100473	−	0.994940i	$-0.532035\pi$
$367$	−3.84955	−0.200945	−0.100473	+	0.994940i	$0.532035\pi$
$368$	0	0
$369$	1.03761	0.0540159
$370$	0	0
$371$	−5.66291	−0.294004
$372$	0	0
$373$	24.8773	1.28810	0.644049	−	0.764984i	$-0.277253\pi$
$373$	24.8773	1.28810	0.644049	+	0.764984i	$0.277253\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.62672	0.186785
$378$	0	0
$379$	2.70052	0.138717	0.0693583	−	0.997592i	$-0.477905\pi$
$379$	2.70052	0.138717	0.0693583	+	0.997592i	$0.477905\pi$
$380$	0	0
$381$	−14.5501	−0.745423
$382$	0	0
$383$	0.523730	0.0267614	0.0133807	−	0.999910i	$-0.495741\pi$
$383$	0.523730	0.0267614	0.0133807	+	0.999910i	$0.495741\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−10.7005	−0.543938
$388$	0	0
$389$	6.77575	0.343544	0.171772	−	0.985137i	$-0.445051\pi$
$389$	6.77575	0.343544	0.171772	+	0.985137i	$0.445051\pi$
$390$	0	0
$391$	−3.22425	−0.163058
$392$	0	0
$393$	4.00000	0.201773
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−0.635150	−0.0318773	−0.0159386	−	0.999873i	$-0.505074\pi$
$397$	−0.635150	−0.0318773	−0.0159386	+	0.999873i	$0.505074\pi$
$398$	0	0
$399$	−2.96239	−0.148305
$400$	0	0
$401$	10.3733	0.518017	0.259009	−	0.965875i	$-0.416604\pi$
$401$	10.3733	0.518017	0.259009	+	0.965875i	$0.416604\pi$
$402$	0	0
$403$	8.77575	0.437151
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.29806	−0.113911
$408$	0	0
$409$	15.7743	0.779991	0.389995	−	0.920817i	$-0.372477\pi$
$409$	15.7743	0.779991	0.389995	+	0.920817i	$0.372477\pi$
$410$	0	0
$411$	10.4387	0.514901
$412$	0	0
$413$	−3.22425	−0.158655
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−5.66291	−0.277314
$418$	0	0
$419$	25.1490	1.22861	0.614305	−	0.789068i	$-0.289436\pi$
$419$	25.1490	1.22861	0.614305	+	0.789068i	$0.289436\pi$
$420$	0	0
$421$	20.1768	0.983357	0.491678	−	0.870777i	$-0.336384\pi$
$421$	20.1768	0.983357	0.491678	+	0.870777i	$0.336384\pi$
$422$	0	0
$423$	−3.22425	−0.156769
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−14.6253	−0.707768
$428$	0	0
$429$	−1.14903	−0.0554757
$430$	0	0
$431$	−38.3127	−1.84546	−0.922728	−	0.385452i	$-0.874045\pi$
$431$	−38.3127	−1.84546	−0.922728	+	0.385452i	$0.874045\pi$
$432$	0	0
$433$	12.8872	0.619318	0.309659	−	0.950848i	$-0.399785\pi$
$433$	12.8872	0.619318	0.309659	+	0.950848i	$0.399785\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.85097	−0.136380
$438$	0	0
$439$	−29.6629	−1.41573	−0.707867	−	0.706346i	$-0.750342\pi$
$439$	−29.6629	−1.41573	−0.707867	+	0.706346i	$0.750342\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	24.8119	1.17885	0.589425	−	0.807823i	$-0.299354\pi$
$443$	24.8119	1.17885	0.589425	+	0.807823i	$0.299354\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−20.5501	−0.971985
$448$	0	0
$449$	−36.6516	−1.72970	−0.864849	−	0.502032i	$-0.832586\pi$
$449$	−36.6516	−1.72970	−0.864849	+	0.502032i	$0.832586\pi$
$450$	0	0
$451$	0.402462	0.0189512
$452$	0	0
$453$	22.7005	1.06656
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−15.4763	−0.723949	−0.361975	−	0.932188i	$-0.617897\pi$
$457$	−15.4763	−0.723949	−0.361975	+	0.932188i	$0.617897\pi$
$458$	0	0
$459$	3.35026	0.156377
$460$	0	0
$461$	41.2605	1.92169	0.960845	−	0.277085i	$-0.0893684\pi$
$461$	41.2605	1.92169	0.960845	+	0.277085i	$0.0893684\pi$
$462$	0	0
$463$	−15.5975	−0.724879	−0.362440	−	0.932007i	$-0.618056\pi$
$463$	−15.5975	−0.724879	−0.362440	+	0.932007i	$0.618056\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18.5501	0.858395	0.429198	−	0.903211i	$-0.358796\pi$
$467$	18.5501	0.858395	0.429198	+	0.903211i	$0.358796\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−2.18664	−0.100755
$472$	0	0
$473$	−4.15045	−0.190838
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5.66291	0.259287
$478$	0	0
$479$	−41.5026	−1.89630	−0.948151	−	0.317819i	$-0.897050\pi$
$479$	−41.5026	−1.89630	−0.948151	+	0.317819i	$0.897050\pi$
$480$	0	0
$481$	−17.5515	−0.800279
$482$	0	0
$483$	0.962389	0.0437902
$484$	0	0
$485$	0	0
$486$	0	0
$487$	13.6728	0.619572	0.309786	−	0.950806i	$-0.399743\pi$
$487$	13.6728	0.619572	0.309786	+	0.950806i	$0.399743\pi$
$488$	0	0
$489$	7.22425	0.326692
$490$	0	0
$491$	23.3404	1.05334	0.526669	−	0.850070i	$-0.323441\pi$
$491$	23.3404	1.05334	0.526669	+	0.850070i	$0.323441\pi$
$492$	0	0
$493$	−4.10157	−0.184725
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5.53690	−0.248364
$498$	0	0
$499$	−6.85097	−0.306691	−0.153346	−	0.988173i	$-0.549005\pi$
$499$	−6.85097	−0.306691	−0.153346	+	0.988173i	$0.549005\pi$
$500$	0	0
$501$	−15.8496	−0.708106
$502$	0	0
$503$	−3.32582	−0.148291	−0.0741456	−	0.997247i	$-0.523623\pi$
$503$	−3.32582	−0.148291	−0.0741456	+	0.997247i	$0.523623\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	4.22425	0.187606
$508$	0	0
$509$	−6.18664	−0.274218	−0.137109	−	0.990556i	$-0.543781\pi$
$509$	−6.18664	−0.274218	−0.137109	+	0.990556i	$0.543781\pi$
$510$	0	0
$511$	6.18664	0.273681
$512$	0	0
$513$	2.96239	0.130793
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1.25060	−0.0550014
$518$	0	0
$519$	23.9756	1.05241
$520$	0	0
$521$	34.4387	1.50879	0.754393	−	0.656424i	$-0.227932\pi$
$521$	34.4387	1.50879	0.754393	+	0.656424i	$0.227932\pi$
$522$	0	0
$523$	−6.59895	−0.288552	−0.144276	−	0.989537i	$-0.546085\pi$
$523$	−6.59895	−0.288552	−0.144276	+	0.989537i	$0.546085\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.92478	−0.432330
$528$	0	0
$529$	−22.0738	−0.959731
$530$	0	0
$531$	3.22425	0.139921
$532$	0	0
$533$	3.07381	0.133141
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−12.2374	−0.528084
$538$	0	0
$539$	0.387873	0.0167069
$540$	0	0
$541$	−9.07381	−0.390113	−0.195057	−	0.980792i	$-0.562489\pi$
$541$	−9.07381	−0.390113	−0.195057	+	0.980792i	$0.562489\pi$
$542$	0	0
$543$	−8.70052	−0.373375
$544$	0	0
$545$	0	0
$546$	0	0
$547$	19.8496	0.848706	0.424353	−	0.905497i	$-0.360502\pi$
$547$	19.8496	0.848706	0.424353	+	0.905497i	$0.360502\pi$
$548$	0	0
$549$	14.6253	0.624193
$550$	0	0
$551$	−3.62672	−0.154503
$552$	0	0
$553$	13.9248	0.592142
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−0.785595	−0.0332867	−0.0166434	−	0.999861i	$-0.505298\pi$
$557$	−0.785595	−0.0332867	−0.0166434	+	0.999861i	$0.505298\pi$
$558$	0	0
$559$	−31.6991	−1.34073
$560$	0	0
$561$	1.29948	0.0548639
$562$	0	0
$563$	32.2228	1.35803	0.679015	−	0.734124i	$-0.262407\pi$
$563$	32.2228	1.35803	0.679015	+	0.734124i	$0.262407\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	16.0752	0.673908	0.336954	−	0.941521i	$-0.390603\pi$
$569$	16.0752	0.673908	0.336954	+	0.941521i	$0.390603\pi$
$570$	0	0
$571$	−39.8496	−1.66765	−0.833826	−	0.552027i	$-0.813854\pi$
$571$	−39.8496	−1.66765	−0.833826	+	0.552027i	$0.813854\pi$
$572$	0	0
$573$	13.5369	0.565512
$574$	0	0
$575$	0	0
$576$	0	0
$577$	30.1378	1.25465	0.627326	−	0.778757i	$-0.284149\pi$
$577$	30.1378	1.25465	0.627326	+	0.778757i	$0.284149\pi$
$578$	0	0
$579$	−1.92478	−0.0799910
$580$	0	0
$581$	3.22425	0.133765
$582$	0	0
$583$	2.19649	0.0909694
$584$	0	0
$585$	0	0
$586$	0	0
$587$	33.7743	1.39402	0.697008	−	0.717063i	$-0.254514\pi$
$587$	33.7743	1.39402	0.697008	+	0.717063i	$0.254514\pi$
$588$	0	0
$589$	−8.77575	−0.361598
$590$	0	0
$591$	20.1114	0.827273
$592$	0	0
$593$	−7.19982	−0.295661	−0.147831	−	0.989013i	$-0.547229\pi$
$593$	−7.19982	−0.295661	−0.147831	+	0.989013i	$0.547229\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8.26187	0.338136
$598$	0	0
$599$	−29.5369	−1.20685	−0.603423	−	0.797422i	$-0.706197\pi$
$599$	−29.5369	−1.20685	−0.603423	+	0.797422i	$0.706197\pi$
$600$	0	0
$601$	8.59895	0.350759	0.175379	−	0.984501i	$-0.443885\pi$
$601$	8.59895	0.350759	0.175379	+	0.984501i	$0.443885\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−37.4010	−1.51806	−0.759031	−	0.651055i	$-0.774327\pi$
$607$	−37.4010	−1.51806	−0.759031	+	0.651055i	$0.774327\pi$
$608$	0	0
$609$	1.22425	0.0496093
$610$	0	0
$611$	−9.55149	−0.386412
$612$	0	0
$613$	14.5990	0.589646	0.294823	−	0.955552i	$-0.404739\pi$
$613$	14.5990	0.589646	0.294823	+	0.955552i	$0.404739\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−22.8119	−0.918374	−0.459187	−	0.888340i	$-0.651859\pi$
$617$	−22.8119	−0.918374	−0.459187	+	0.888340i	$0.651859\pi$
$618$	0	0
$619$	0.261865	0.0105252	0.00526262	−	0.999986i	$-0.498325\pi$
$619$	0.261865	0.0105252	0.00526262	+	0.999986i	$0.498325\pi$
$620$	0	0
$621$	−0.962389	−0.0386193
$622$	0	0
$623$	−3.73813	−0.149765
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.14903	0.0458879
$628$	0	0
$629$	19.8496	0.791454
$630$	0	0
$631$	−5.92478	−0.235862	−0.117931	−	0.993022i	$-0.537626\pi$
$631$	−5.92478	−0.235862	−0.117931	+	0.993022i	$0.537626\pi$
$632$	0	0
$633$	18.1768	0.722463
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.96239	0.117374
$638$	0	0
$639$	5.53690	0.219037
$640$	0	0
$641$	6.87732	0.271638	0.135819	−	0.990734i	$-0.456633\pi$
$641$	6.87732	0.271638	0.135819	+	0.990734i	$0.456633\pi$
$642$	0	0
$643$	10.7005	0.421987	0.210994	−	0.977487i	$-0.432330\pi$
$643$	10.7005	0.421987	0.210994	+	0.977487i	$0.432330\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	45.2506	1.77898	0.889492	−	0.456950i	$-0.151058\pi$
$647$	45.2506	1.77898	0.889492	+	0.456950i	$0.151058\pi$
$648$	0	0
$649$	1.25060	0.0490904
$650$	0	0
$651$	2.96239	0.116105
$652$	0	0
$653$	−36.0625	−1.41124	−0.705618	−	0.708592i	$-0.749331\pi$
$653$	−36.0625	−1.41124	−0.705618	+	0.708592i	$0.749331\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−6.18664	−0.241364
$658$	0	0
$659$	43.4617	1.69303	0.846513	−	0.532367i	$-0.178698\pi$
$659$	43.4617	1.69303	0.846513	+	0.532367i	$0.178698\pi$
$660$	0	0
$661$	−22.4749	−0.874171	−0.437085	−	0.899420i	$-0.643989\pi$
$661$	−22.4749	−0.874171	−0.437085	+	0.899420i	$0.643989\pi$
$662$	0	0
$663$	9.92478	0.385446
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.17821	0.0456204
$668$	0	0
$669$	21.4010	0.827412
$670$	0	0
$671$	5.67276	0.218995
$672$	0	0
$673$	−47.5487	−1.83287	−0.916433	−	0.400188i	$-0.868945\pi$
$673$	−47.5487	−1.83287	−0.916433	+	0.400188i	$0.868945\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−5.67750	−0.218204	−0.109102	−	0.994031i	$-0.534798\pi$
$677$	−5.67750	−0.218204	−0.109102	+	0.994031i	$0.534798\pi$
$678$	0	0
$679$	−7.73813	−0.296962
$680$	0	0
$681$	18.5501	0.710841
$682$	0	0
$683$	−12.2882	−0.470195	−0.235098	−	0.971972i	$-0.575541\pi$
$683$	−12.2882	−0.470195	−0.235098	+	0.971972i	$0.575541\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	28.5501	1.08925
$688$	0	0
$689$	16.7757	0.639105
$690$	0	0
$691$	−1.81336	−0.0689834	−0.0344917	−	0.999405i	$-0.510981\pi$
$691$	−1.81336	−0.0689834	−0.0344917	+	0.999405i	$0.510981\pi$
$692$	0	0
$693$	−0.387873	−0.0147341
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−3.47627	−0.131673
$698$	0	0
$699$	18.9624	0.717223
$700$	0	0
$701$	29.5778	1.11714	0.558570	−	0.829458i	$-0.311350\pi$
$701$	29.5778	1.11714	0.558570	+	0.829458i	$0.311350\pi$
$702$	0	0
$703$	17.5515	0.661967
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.58769	0.285364
$708$	0	0
$709$	20.8021	0.781239	0.390620	−	0.920552i	$-0.372261\pi$
$709$	20.8021	0.781239	0.390620	+	0.920552i	$0.372261\pi$
$710$	0	0
$711$	−13.9248	−0.522220
$712$	0	0
$713$	2.85097	0.106770
$714$	0	0
$715$	0	0
$716$	0	0
$717$	18.1622	0.678280
$718$	0	0
$719$	19.3258	0.720732	0.360366	−	0.932811i	$-0.382652\pi$
$719$	19.3258	0.720732	0.360366	+	0.932811i	$0.382652\pi$
$720$	0	0
$721$	14.7005	0.547476
$722$	0	0
$723$	−14.3733	−0.534548
$724$	0	0
$725$	0	0
$726$	0	0
$727$	21.6531	0.803068	0.401534	−	0.915844i	$-0.368477\pi$
$727$	21.6531	0.803068	0.401534	+	0.915844i	$0.368477\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	35.8496	1.32594
$732$	0	0
$733$	48.0625	1.77523	0.887615	−	0.460586i	$-0.152361\pi$
$733$	48.0625	1.77523	0.887615	+	0.460586i	$0.152361\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−30.2981	−1.11453	−0.557266	−	0.830334i	$-0.688150\pi$
$739$	−30.2981	−1.11453	−0.557266	+	0.830334i	$0.688150\pi$
$740$	0	0
$741$	8.77575	0.322385
$742$	0	0
$743$	40.9624	1.50276	0.751382	−	0.659867i	$-0.229388\pi$
$743$	40.9624	1.50276	0.751382	+	0.659867i	$0.229388\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−3.22425	−0.117969
$748$	0	0
$749$	16.4387	0.600656
$750$	0	0
$751$	9.52232	0.347474	0.173737	−	0.984792i	$-0.444416\pi$
$751$	9.52232	0.347474	0.173737	+	0.984792i	$0.444416\pi$
$752$	0	0
$753$	8.62530	0.314323
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−31.3258	−1.13856	−0.569278	−	0.822145i	$-0.692777\pi$
$757$	−31.3258	−1.13856	−0.569278	+	0.822145i	$0.692777\pi$
$758$	0	0
$759$	−0.373285	−0.0135494
$760$	0	0
$761$	−54.2393	−1.96617	−0.983087	−	0.183138i	$-0.941375\pi$
$761$	−54.2393	−1.96617	−0.983087	+	0.183138i	$0.941375\pi$
$762$	0	0
$763$	2.77575	0.100489
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.55149	0.344884
$768$	0	0
$769$	35.1002	1.26574	0.632872	−	0.774256i	$-0.281876\pi$
$769$	35.1002	1.26574	0.632872	+	0.774256i	$0.281876\pi$
$770$	0	0
$771$	−27.8251	−1.00210
$772$	0	0
$773$	−19.8740	−0.714818	−0.357409	−	0.933948i	$-0.616340\pi$
$773$	−19.8740	−0.714818	−0.357409	+	0.933948i	$0.616340\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−5.92478	−0.212550
$778$	0	0
$779$	−3.07381	−0.110131
$780$	0	0
$781$	2.14762	0.0768478
$782$	0	0
$783$	−1.22425	−0.0437513
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−13.2506	−0.472333	−0.236166	−	0.971713i	$-0.575891\pi$
$787$	−13.2506	−0.472333	−0.236166	+	0.971713i	$0.575891\pi$
$788$	0	0
$789$	−0.589104	−0.0209727
$790$	0	0
$791$	4.26187	0.151534
$792$	0	0
$793$	43.3258	1.53855
$794$	0	0
$795$	0	0
$796$	0	0
$797$	15.7235	0.556957	0.278478	−	0.960443i	$-0.410170\pi$
$797$	15.7235	0.556957	0.278478	+	0.960443i	$0.410170\pi$
$798$	0	0
$799$	10.8021	0.382151
$800$	0	0
$801$	3.73813	0.132081
$802$	0	0
$803$	−2.39963	−0.0846812
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−15.7381	−0.554009
$808$	0	0
$809$	−18.5040	−0.650567	−0.325284	−	0.945617i	$-0.605460\pi$
$809$	−18.5040	−0.650567	−0.325284	+	0.945617i	$0.605460\pi$
$810$	0	0
$811$	37.6629	1.32252	0.661262	−	0.750155i	$-0.270021\pi$
$811$	37.6629	1.32252	0.661262	+	0.750155i	$0.270021\pi$
$812$	0	0
$813$	7.11283	0.249458
$814$	0	0
$815$	0	0
$816$	0	0
$817$	31.6991	1.10901
$818$	0	0
$819$	−2.96239	−0.103514
$820$	0	0
$821$	−17.9511	−0.626499	−0.313249	−	0.949671i	$-0.601418\pi$
$821$	−17.9511	−0.626499	−0.313249	+	0.949671i	$0.601418\pi$
$822$	0	0
$823$	32.1016	1.11899	0.559495	−	0.828834i	$-0.310995\pi$
$823$	32.1016	1.11899	0.559495	+	0.828834i	$0.310995\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	25.8594	0.899220	0.449610	−	0.893225i	$-0.351563\pi$
$827$	25.8594	0.899220	0.449610	+	0.893225i	$0.351563\pi$
$828$	0	0
$829$	11.8035	0.409953	0.204976	−	0.978767i	$-0.434288\pi$
$829$	11.8035	0.409953	0.204976	+	0.978767i	$0.434288\pi$
$830$	0	0
$831$	5.92478	0.205528
$832$	0	0
$833$	−3.35026	−0.116080
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2.96239	−0.102395
$838$	0	0
$839$	19.5778	0.675902	0.337951	−	0.941164i	$-0.390266\pi$
$839$	19.5778	0.675902	0.337951	+	0.941164i	$0.390266\pi$
$840$	0	0
$841$	−27.5012	−0.948317
$842$	0	0
$843$	−29.3258	−1.01004
$844$	0	0
$845$	0	0
$846$	0	0
$847$	10.8496	0.372795
$848$	0	0
$849$	−16.1016	−0.552604
$850$	0	0
$851$	−5.70194	−0.195460
$852$	0	0
$853$	40.6155	1.39065	0.695323	−	0.718697i	$-0.255261\pi$
$853$	40.6155	1.39065	0.695323	+	0.718697i	$0.255261\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−29.1246	−0.994877	−0.497439	−	0.867499i	$-0.665726\pi$
$857$	−29.1246	−0.994877	−0.497439	+	0.867499i	$0.665726\pi$
$858$	0	0
$859$	35.7090	1.21837	0.609187	−	0.793027i	$-0.291496\pi$
$859$	35.7090	1.21837	0.609187	+	0.793027i	$0.291496\pi$
$860$	0	0
$861$	1.03761	0.0353617
$862$	0	0
$863$	−21.1852	−0.721154	−0.360577	−	0.932730i	$-0.617420\pi$
$863$	−21.1852	−0.721154	−0.360577	+	0.932730i	$0.617420\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	5.77575	0.196155
$868$	0	0
$869$	−5.40105	−0.183218
$870$	0	0
$871$	0	0
$872$	0	0
$873$	7.73813	0.261896
$874$	0	0
$875$	0	0
$876$	0	0
$877$	26.4485	0.893103	0.446551	−	0.894758i	$-0.352652\pi$
$877$	26.4485	0.893103	0.446551	+	0.894758i	$0.352652\pi$
$878$	0	0
$879$	9.27504	0.312839
$880$	0	0
$881$	−26.0362	−0.877182	−0.438591	−	0.898687i	$-0.644522\pi$
$881$	−26.0362	−0.877182	−0.438591	+	0.898687i	$0.644522\pi$
$882$	0	0
$883$	−8.87732	−0.298745	−0.149373	−	0.988781i	$-0.547725\pi$
$883$	−8.87732	−0.298745	−0.149373	+	0.988781i	$0.547725\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.17679	−0.0730896	−0.0365448	−	0.999332i	$-0.511635\pi$
$887$	−2.17679	−0.0730896	−0.0365448	+	0.999332i	$0.511635\pi$
$888$	0	0
$889$	−14.5501	−0.487994
$890$	0	0
$891$	0.387873	0.0129942
$892$	0	0
$893$	9.55149	0.319629
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2.85097	−0.0951911
$898$	0	0
$899$	3.62672	0.120958
$900$	0	0
$901$	−18.9722	−0.632057
$902$	0	0
$903$	−10.7005	−0.356091
$904$	0	0
$905$	0	0
$906$	0	0
$907$	44.7269	1.48513	0.742566	−	0.669773i	$-0.233609\pi$
$907$	44.7269	1.48513	0.742566	+	0.669773i	$0.233609\pi$
$908$	0	0
$909$	−7.58769	−0.251668
$910$	0	0
$911$	26.4631	0.876761	0.438381	−	0.898789i	$-0.355552\pi$
$911$	26.4631	0.876761	0.438381	+	0.898789i	$0.355552\pi$
$912$	0	0
$913$	−1.25060	−0.0413889
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.00000	0.132092
$918$	0	0
$919$	−59.2769	−1.95537	−0.977683	−	0.210085i	$-0.932626\pi$
$919$	−59.2769	−1.95537	−0.977683	+	0.210085i	$0.932626\pi$
$920$	0	0
$921$	10.7005	0.352594
$922$	0	0
$923$	16.4025	0.539894
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−14.7005	−0.482829
$928$	0	0
$929$	25.3620	0.832101	0.416050	−	0.909342i	$-0.363414\pi$
$929$	25.3620	0.832101	0.416050	+	0.909342i	$0.363414\pi$
$930$	0	0
$931$	−2.96239	−0.0970884
$932$	0	0
$933$	−12.1016	−0.396188
$934$	0	0
$935$	0	0
$936$	0	0
$937$	3.85940	0.126081	0.0630406	−	0.998011i	$-0.479920\pi$
$937$	3.85940	0.126081	0.0630406	+	0.998011i	$0.479920\pi$
$938$	0	0
$939$	28.3634	0.925606
$940$	0	0
$941$	27.4861	0.896022	0.448011	−	0.894028i	$-0.352133\pi$
$941$	27.4861	0.896022	0.448011	+	0.894028i	$0.352133\pi$
$942$	0	0
$943$	0.998585	0.0325184
$944$	0	0
$945$	0	0
$946$	0	0
$947$	13.8397	0.449730	0.224865	−	0.974390i	$-0.427806\pi$
$947$	13.8397	0.449730	0.224865	+	0.974390i	$0.427806\pi$
$948$	0	0
$949$	−18.3272	−0.594927
$950$	0	0
$951$	−17.5125	−0.567881
$952$	0	0
$953$	39.2868	1.27262	0.636312	−	0.771432i	$-0.280459\pi$
$953$	39.2868	1.27262	0.636312	+	0.771432i	$0.280459\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−0.474855	−0.0153499
$958$	0	0
$959$	10.4387	0.337082
$960$	0	0
$961$	−22.2243	−0.716911
$962$	0	0
$963$	−16.4387	−0.529728
$964$	0	0
$965$	0	0
$966$	0	0
$967$	8.67418	0.278943	0.139471	−	0.990226i	$-0.455460\pi$
$967$	8.67418	0.278943	0.139471	+	0.990226i	$0.455460\pi$
$968$	0	0
$969$	−9.92478	−0.318830
$970$	0	0
$971$	−25.9248	−0.831966	−0.415983	−	0.909372i	$-0.636562\pi$
$971$	−25.9248	−0.831966	−0.415983	+	0.909372i	$0.636562\pi$
$972$	0	0
$973$	−5.66291	−0.181545
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−7.85940	−0.251445	−0.125722	−	0.992065i	$-0.540125\pi$
$977$	−7.85940	−0.251445	−0.125722	+	0.992065i	$0.540125\pi$
$978$	0	0
$979$	1.44992	0.0463397
$980$	0	0
$981$	−2.77575	−0.0886228
$982$	0	0
$983$	18.8510	0.601253	0.300626	−	0.953742i	$-0.402804\pi$
$983$	18.8510	0.601253	0.300626	+	0.953742i	$0.402804\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−3.22425	−0.102629
$988$	0	0
$989$	−10.2981	−0.327459
$990$	0	0
$991$	−43.0738	−1.36828	−0.684142	−	0.729349i	$-0.739823\pi$
$991$	−43.0738	−1.36828	−0.684142	+	0.729349i	$0.739823\pi$
$992$	0	0
$993$	19.2243	0.610063
$994$	0	0
$995$	0	0
$996$	0	0
$997$	24.6155	0.779579	0.389790	−	0.920904i	$-0.372548\pi$
$997$	24.6155	0.779579	0.389790	+	0.920904i	$0.372548\pi$
$998$	0	0
$999$	5.92478	0.187452

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4200.2.a.bo.1.2		3
4.3	odd	2		8400.2.a.dk.1.2		3
5.2	odd	4		840.2.t.e.169.6	yes	6
5.3	odd	4		840.2.t.e.169.3	✓	6
5.4	even	2		4200.2.a.bq.1.2		3
15.2	even	4		2520.2.t.j.1009.2		6
15.8	even	4		2520.2.t.j.1009.1		6
20.3	even	4		1680.2.t.i.1009.6		6
20.7	even	4		1680.2.t.i.1009.3		6
20.19	odd	2		8400.2.a.dh.1.2		3
60.23	odd	4		5040.2.t.ba.1009.1		6
60.47	odd	4		5040.2.t.ba.1009.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.t.e.169.3	✓	6	5.3	odd	4
840.2.t.e.169.6	yes	6	5.2	odd	4
1680.2.t.i.1009.3		6	20.7	even	4
1680.2.t.i.1009.6		6	20.3	even	4
2520.2.t.j.1009.1		6	15.8	even	4
2520.2.t.j.1009.2		6	15.2	even	4
4200.2.a.bo.1.2		3	1.1	even	1	trivial
4200.2.a.bq.1.2		3	5.4	even	2
5040.2.t.ba.1009.1		6	60.23	odd	4
5040.2.t.ba.1009.2		6	60.47	odd	4
8400.2.a.dh.1.2		3	20.19	odd	2
8400.2.a.dk.1.2		3	4.3	odd	2

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:	\( N \)	\(=\)	\( 4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4200.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +0.387873 q^{11} +2.96239 q^{13} -3.35026 q^{17} -2.96239 q^{19} +1.00000 q^{21} +0.962389 q^{23} -1.00000 q^{27} +1.22425 q^{29} +2.96239 q^{31} -0.387873 q^{33} -5.92478 q^{37} -2.96239 q^{39} +1.03761 q^{41} -10.7005 q^{43} -3.22425 q^{47} +1.00000 q^{49} +3.35026 q^{51} +5.66291 q^{53} +2.96239 q^{57} +3.22425 q^{59} +14.6253 q^{61} -1.00000 q^{63} -0.962389 q^{69} +5.53690 q^{71} -6.18664 q^{73} -0.387873 q^{77} -13.9248 q^{79} +1.00000 q^{81} -3.22425 q^{83} -1.22425 q^{87} +3.73813 q^{89} -2.96239 q^{91} -2.96239 q^{93} +7.73813 q^{97} +0.387873 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9} + 2 q^{11} - 2 q^{13} + 2 q^{19} + 3 q^{21} - 8 q^{23} - 3 q^{27} + 2 q^{29} - 2 q^{31} - 2 q^{33} + 4 q^{37} + 2 q^{39} + 14 q^{41} - 12 q^{43} - 8 q^{47} + 3 q^{49} - 14 q^{53} - 2 q^{57} + 8 q^{59} + 2 q^{61} - 3 q^{63} + 8 q^{69} - 6 q^{71} - 6 q^{73} - 2 q^{77} - 20 q^{79} + 3 q^{81} - 8 q^{83} - 2 q^{87} + 2 q^{89} + 2 q^{91} + 2 q^{93} + 14 q^{97} + 2 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^7 + 3 * q^9 + 2 * q^11 - 2 * q^13 + 2 * q^19 + 3 * q^21 - 8 * q^23 - 3 * q^27 + 2 * q^29 - 2 * q^31 - 2 * q^33 + 4 * q^37 + 2 * q^39 + 14 * q^41 - 12 * q^43 - 8 * q^47 + 3 * q^49 - 14 * q^53 - 2 * q^57 + 8 * q^59 + 2 * q^61 - 3 * q^63 + 8 * q^69 - 6 * q^71 - 6 * q^73 - 2 * q^77 - 20 * q^79 + 3 * q^81 - 8 * q^83 - 2 * q^87 + 2 * q^89 + 2 * q^91 + 2 * q^93 + 14 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more