LMFDB - Embedded newform 3969.2.a.bg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3969,2,Mod(1,3969)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3969, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3969.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3969 = 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3969.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:	$31.6926245622$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.114612039936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 11x^{6} + 34x^{4} - 36x^{2} + 9 $ x^8 - 11x^6 + 34x^4 - 36*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 567)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.52818$ of defining polynomial
Character	$\chi$	$=$	3969.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.52818 q^{2} +0.335323 q^{4} -2.82528 q^{5} +2.54392 q^{8} +O(q^{10})$	$q-1.52818 q^{2} +0.335323 q^{4} -2.82528 q^{5} +2.54392 q^{8} +4.31752 q^{10} -3.63715 q^{11} -5.62908 q^{13} -4.55820 q^{16} -3.20220 q^{17} +4.07088 q^{19} -0.947380 q^{20} +5.55820 q^{22} -4.70318 q^{23} +2.98220 q^{25} +8.60223 q^{26} -4.32626 q^{29} +3.58197 q^{31} +1.87790 q^{32} +4.89353 q^{34} -4.31156 q^{37} -6.22102 q^{38} -7.18728 q^{40} -3.15192 q^{41} -9.19325 q^{43} -1.21962 q^{44} +7.18728 q^{46} -4.84643 q^{47} -4.55733 q^{50} -1.88756 q^{52} -14.1341 q^{53} +10.2760 q^{55} +6.61128 q^{58} -1.50098 q^{59} +13.2051 q^{61} -5.47388 q^{62} +6.24665 q^{64} +15.9037 q^{65} -12.6822 q^{67} -1.07377 q^{68} +2.91413 q^{71} +2.92912 q^{73} +6.58882 q^{74} +1.36506 q^{76} +0.893527 q^{79} +12.8782 q^{80} +4.81668 q^{82} -8.04863 q^{83} +9.04711 q^{85} +14.0489 q^{86} -9.25262 q^{88} -5.65727 q^{89} -1.57708 q^{92} +7.40620 q^{94} -11.5014 q^{95} +5.13578 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{4}+O(q^{10}) $ 8 * q + 6 * q^4	$ 8 q + 6 q^{4} + 14 q^{10} + 6 q^{13} + 6 q^{16} + 24 q^{19} + 2 q^{22} + 20 q^{31} - 4 q^{37} + 36 q^{40} + 10 q^{43} - 36 q^{46} + 34 q^{52} + 4 q^{55} - 22 q^{58} + 36 q^{61} + 38 q^{64} - 18 q^{67} + 32 q^{73} + 58 q^{76} - 32 q^{79} - 2 q^{82} + 30 q^{85} - 72 q^{88} + 54 q^{94} + 14 q^{97}+O(q^{100}) $ 8 * q + 6 * q^4 + 14 * q^10 + 6 * q^13 + 6 * q^16 + 24 * q^19 + 2 * q^22 + 20 * q^31 - 4 * q^37 + 36 * q^40 + 10 * q^43 - 36 * q^46 + 34 * q^52 + 4 * q^55 - 22 * q^58 + 36 * q^61 + 38 * q^64 - 18 * q^67 + 32 * q^73 + 58 * q^76 - 32 * q^79 - 2 * q^82 + 30 * q^85 - 72 * q^88 + 54 * q^94 + 14 * q^97

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 $

$2$	−1.52818	−1.08058	−0.540292	−	0.841478i	$-0.681686\pi$
$2$	−1.52818	−1.08058	−0.540292	+	0.841478i	$0.681686\pi$
$3$	0	0
$3$	0	0
$4$	0.335323	0.167661
$5$	−2.82528	−1.26350	−0.631752	−	0.775171i	$-0.717664\pi$
$5$	−2.82528	−1.26350	−0.631752	+	0.775171i	$0.717664\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.54392	0.899412
$9$	0	0
$10$	4.31752	1.36532
$11$	−3.63715	−1.09664	−0.548321	−	0.836268i	$-0.684733\pi$
$11$	−3.63715	−1.09664	−0.548321	+	0.836268i	$0.684733\pi$
$12$	0	0
$13$	−5.62908	−1.56123	−0.780613	−	0.625015i	$-0.785093\pi$
$13$	−5.62908	−1.56123	−0.780613	+	0.625015i	$0.785093\pi$
$14$	0	0
$15$	0	0
$16$	−4.55820	−1.13955
$17$	−3.20220	−0.776648	−0.388324	−	0.921523i	$-0.626946\pi$
$17$	−3.20220	−0.776648	−0.388324	+	0.921523i	$0.626946\pi$
$18$	0	0
$19$	4.07088	0.933923	0.466962	−	0.884278i	$-0.345349\pi$
$19$	4.07088	0.933923	0.466962	+	0.884278i	$0.345349\pi$
$20$	−0.947380	−0.211841
$21$	0	0
$22$	5.55820	1.18501
$23$	−4.70318	−0.980680	−0.490340	−	0.871531i	$-0.663127\pi$
$23$	−4.70318	−0.980680	−0.490340	+	0.871531i	$0.663127\pi$
$24$	0	0
$25$	2.98220	0.596440
$26$	8.60223	1.68704
$27$	0	0
$28$	0	0
$29$	−4.32626	−0.803366	−0.401683	−	0.915779i	$-0.631575\pi$
$29$	−4.32626	−0.803366	−0.401683	+	0.915779i	$0.631575\pi$
$30$	0	0
$31$	3.58197	0.643341	0.321670	−	0.946852i	$-0.395756\pi$
$31$	3.58197	0.643341	0.321670	+	0.946852i	$0.395756\pi$
$32$	1.87790	0.331969
$33$	0	0
$34$	4.89353	0.839233
$35$	0	0
$36$	0	0
$37$	−4.31156	−0.708815	−0.354408	−	0.935091i	$-0.615318\pi$
$37$	−4.31156	−0.708815	−0.354408	+	0.935091i	$0.615318\pi$
$38$	−6.22102	−1.00918
$39$	0	0
$40$	−7.18728	−1.13641
$41$	−3.15192	−0.492246	−0.246123	−	0.969239i	$-0.579157\pi$
$41$	−3.15192	−0.492246	−0.246123	+	0.969239i	$0.579157\pi$
$42$	0	0
$43$	−9.19325	−1.40196	−0.700979	−	0.713182i	$-0.747253\pi$
$43$	−9.19325	−1.40196	−0.700979	+	0.713182i	$0.747253\pi$
$44$	−1.21962	−0.183864
$45$	0	0
$46$	7.18728	1.05971
$47$	−4.84643	−0.706924	−0.353462	−	0.935449i	$-0.614996\pi$
$47$	−4.84643	−0.706924	−0.353462	+	0.935449i	$0.614996\pi$
$48$	0	0
$49$	0	0
$50$	−4.55733	−0.644504
$51$	0	0
$52$	−1.88756	−0.261757
$53$	−14.1341	−1.94147	−0.970736	−	0.240147i	$-0.922804\pi$
$53$	−14.1341	−1.94147	−0.970736	+	0.240147i	$0.922804\pi$
$54$	0	0
$55$	10.2760	1.38561
$56$	0	0
$57$	0	0
$58$	6.61128	0.868104
$59$	−1.50098	−0.195411	−0.0977053	−	0.995215i	$-0.531150\pi$
$59$	−1.50098	−0.195411	−0.0977053	+	0.995215i	$0.531150\pi$
$60$	0	0
$61$	13.2051	1.69074	0.845369	−	0.534183i	$-0.179381\pi$
$61$	13.2051	1.69074	0.845369	+	0.534183i	$0.179381\pi$
$62$	−5.47388	−0.695184
$63$	0	0
$64$	6.24665	0.780831
$65$	15.9037	1.97261
$66$	0	0
$67$	−12.6822	−1.54937	−0.774686	−	0.632346i	$-0.782092\pi$
$67$	−12.6822	−1.54937	−0.774686	+	0.632346i	$0.782092\pi$
$68$	−1.07377	−0.130214
$69$	0	0
$70$	0	0
$71$	2.91413	0.345844	0.172922	−	0.984936i	$-0.444679\pi$
$71$	2.91413	0.345844	0.172922	+	0.984936i	$0.444679\pi$
$72$	0	0
$73$	2.92912	0.342828	0.171414	−	0.985199i	$-0.445166\pi$
$73$	2.92912	0.342828	0.171414	+	0.985199i	$0.445166\pi$
$74$	6.58882	0.765935
$75$	0	0
$76$	1.36506	0.156583
$77$	0	0
$78$	0	0
$79$	0.893527	0.100530	0.0502648	−	0.998736i	$-0.483993\pi$
$79$	0.893527	0.100530	0.0502648	+	0.998736i	$0.483993\pi$
$80$	12.8782	1.43983
$81$	0	0
$82$	4.81668	0.531914
$83$	−8.04863	−0.883452	−0.441726	−	0.897150i	$-0.645634\pi$
$83$	−8.04863	−0.883452	−0.441726	+	0.897150i	$0.645634\pi$
$84$	0	0
$85$	9.04711	0.981297
$86$	14.0489	1.51493
$87$	0	0
$88$	−9.25262	−0.986332
$89$	−5.65727	−0.599669	−0.299835	−	0.953991i	$-0.596931\pi$
$89$	−5.65727	−0.599669	−0.299835	+	0.953991i	$0.596931\pi$
$90$	0	0
$91$	0	0
$92$	−1.57708	−0.164422
$93$	0	0
$94$	7.40620	0.763891
$95$	−11.5014	−1.18001
$96$	0	0
$97$	5.13578	0.521460	0.260730	−	0.965412i	$-0.416037\pi$
$97$	5.13578	0.521460	0.260730	+	0.965412i	$0.416037\pi$
$98$	0	0
$99$	0	0
$100$	1.00000	0.100000
$101$	−2.52446	−0.251193	−0.125597	−	0.992081i	$-0.540085\pi$
$101$	−2.52446	−0.251193	−0.125597	+	0.992081i	$0.540085\pi$
$102$	0	0
$103$	9.84642	0.970196	0.485098	−	0.874460i	$-0.338784\pi$
$103$	9.84642	0.970196	0.485098	+	0.874460i	$0.338784\pi$
$104$	−14.3199	−1.40418
$105$	0	0
$106$	21.5995	2.09792
$107$	−2.70252	−0.261262	−0.130631	−	0.991431i	$-0.541700\pi$
$107$	−2.70252	−0.261262	−0.130631	+	0.991431i	$0.541700\pi$
$108$	0	0
$109$	−9.05747	−0.867548	−0.433774	−	0.901022i	$-0.642818\pi$
$109$	−9.05747	−0.867548	−0.433774	+	0.901022i	$0.642818\pi$
$110$	−15.7035	−1.49727
$111$	0	0
$112$	0	0
$113$	−0.465776	−0.0438165	−0.0219082	−	0.999760i	$-0.506974\pi$
$113$	−0.465776	−0.0438165	−0.0219082	+	0.999760i	$0.506974\pi$
$114$	0	0
$115$	13.2878	1.23909
$116$	−1.45069	−0.134693
$117$	0	0
$118$	2.29376	0.211158
$119$	0	0
$120$	0	0
$121$	2.22885	0.202623
$122$	−20.1797	−1.82698
$123$	0	0
$124$	1.20112	0.107863
$125$	5.70084	0.509899
$126$	0	0
$127$	−15.7574	−1.39825	−0.699123	−	0.715002i	$-0.746426\pi$
$127$	−15.7574	−1.39825	−0.699123	+	0.715002i	$0.746426\pi$
$128$	−13.3018	−1.17572
$129$	0	0
$130$	−24.3037	−2.13157
$131$	−12.1479	−1.06137	−0.530685	−	0.847569i	$-0.678065\pi$
$131$	−12.1479	−1.06137	−0.530685	+	0.847569i	$0.678065\pi$
$132$	0	0
$133$	0	0
$134$	19.3806	1.67423
$135$	0	0
$136$	−8.14614	−0.698526
$137$	−2.16337	−0.184829	−0.0924146	−	0.995721i	$-0.529459\pi$
$137$	−2.16337	−0.184829	−0.0924146	+	0.995721i	$0.529459\pi$
$138$	0	0
$139$	−21.4514	−1.81949	−0.909743	−	0.415173i	$-0.863721\pi$
$139$	−21.4514	−1.81949	−0.909743	+	0.415173i	$0.863721\pi$
$140$	0	0
$141$	0	0
$142$	−4.45331	−0.373713
$143$	20.4738	1.71211
$144$	0	0
$145$	12.2229	1.01506
$146$	−4.47622	−0.370454
$147$	0	0
$148$	−1.44576	−0.118841
$149$	11.0972	0.909121	0.454561	−	0.890716i	$-0.349796\pi$
$149$	11.0972	0.909121	0.454561	+	0.890716i	$0.349796\pi$
$150$	0	0
$151$	2.08710	0.169846	0.0849228	−	0.996388i	$-0.472936\pi$
$151$	2.08710	0.169846	0.0849228	+	0.996388i	$0.472936\pi$
$152$	10.3560	0.839981
$153$	0	0
$154$	0	0
$155$	−10.1201	−0.812863
$156$	0	0
$157$	−21.6154	−1.72509	−0.862547	−	0.505978i	$-0.831132\pi$
$157$	−21.6154	−1.72509	−0.862547	+	0.505978i	$0.831132\pi$
$158$	−1.36547	−0.108631
$159$	0	0
$160$	−5.30559	−0.419444
$161$	0	0
$162$	0	0
$163$	−9.38243	−0.734889	−0.367444	−	0.930045i	$-0.619767\pi$
$163$	−9.38243	−0.734889	−0.367444	+	0.930045i	$0.619767\pi$
$164$	−1.05691	−0.0825307
$165$	0	0
$166$	12.2997	0.954644
$167$	17.4047	1.34682	0.673408	−	0.739271i	$-0.264830\pi$
$167$	17.4047	1.34682	0.673408	+	0.739271i	$0.264830\pi$
$168$	0	0
$169$	18.6865	1.43743
$170$	−13.8256	−1.06037
$171$	0	0
$172$	−3.08271	−0.235054
$173$	−21.5476	−1.63823	−0.819116	−	0.573628i	$-0.805535\pi$
$173$	−21.5476	−1.63823	−0.819116	+	0.573628i	$0.805535\pi$
$174$	0	0
$175$	0	0
$176$	16.5789	1.24968
$177$	0	0
$178$	8.64530	0.647993
$179$	5.65727	0.422844	0.211422	−	0.977395i	$-0.432191\pi$
$179$	5.65727	0.422844	0.211422	+	0.977395i	$0.432191\pi$
$180$	0	0
$181$	1.12427	0.0835664	0.0417832	−	0.999127i	$-0.486696\pi$
$181$	1.12427	0.0835664	0.0417832	+	0.999127i	$0.486696\pi$
$182$	0	0
$183$	0	0
$184$	−11.9645	−0.882035
$185$	12.1813	0.895591
$186$	0	0
$187$	11.6469	0.851704
$188$	−1.62512	−0.118524
$189$	0	0
$190$	17.5761	1.27510
$191$	3.93900	0.285016	0.142508	−	0.989794i	$-0.454483\pi$
$191$	3.93900	0.285016	0.142508	+	0.989794i	$0.454483\pi$
$192$	0	0
$193$	18.0868	1.30191	0.650957	−	0.759114i	$-0.274368\pi$
$193$	18.0868	1.30191	0.650957	+	0.759114i	$0.274368\pi$
$194$	−7.84838	−0.563481
$195$	0	0
$196$	0	0
$197$	25.4842	1.81567	0.907836	−	0.419326i	$-0.137734\pi$
$197$	25.4842	1.81567	0.907836	+	0.419326i	$0.137734\pi$
$198$	0	0
$199$	12.6291	0.895252	0.447626	−	0.894221i	$-0.352270\pi$
$199$	12.6291	0.895252	0.447626	+	0.894221i	$0.352270\pi$
$200$	7.58648	0.536445
$201$	0	0
$202$	3.85782	0.271436
$203$	0	0
$204$	0	0
$205$	8.90504	0.621955
$206$	−15.0471	−1.04838
$207$	0	0
$208$	25.6585	1.77910
$209$	−14.8064	−1.02418
$210$	0	0
$211$	2.31156	0.159134	0.0795670	−	0.996830i	$-0.474646\pi$
$211$	2.31156	0.159134	0.0795670	+	0.996830i	$0.474646\pi$
$212$	−4.73950	−0.325510
$213$	0	0
$214$	4.12992	0.282316
$215$	25.9735	1.77138
$216$	0	0
$217$	0	0
$218$	13.8414	0.937458
$219$	0	0
$220$	3.44576	0.232313
$221$	18.0254	1.21252
$222$	0	0
$223$	9.29962	0.622749	0.311374	−	0.950287i	$-0.399211\pi$
$223$	9.29962	0.622749	0.311374	+	0.950287i	$0.399211\pi$
$224$	0	0
$225$	0	0
$226$	0.711787	0.0473474
$227$	−1.92818	−0.127978	−0.0639890	−	0.997951i	$-0.520382\pi$
$227$	−1.92818	−0.127978	−0.0639890	+	0.997951i	$0.520382\pi$
$228$	0	0
$229$	3.34118	0.220792	0.110396	−	0.993888i	$-0.464788\pi$
$229$	3.34118	0.220792	0.110396	+	0.993888i	$0.464788\pi$
$230$	−20.3061	−1.33894
$231$	0	0
$232$	−11.0057	−0.722556
$233$	21.1515	1.38568	0.692842	−	0.721090i	$-0.256358\pi$
$233$	21.1515	1.38568	0.692842	+	0.721090i	$0.256358\pi$
$234$	0	0
$235$	13.6925	0.893201
$236$	−0.503312	−0.0327628
$237$	0	0
$238$	0	0
$239$	−24.1151	−1.55987	−0.779937	−	0.625858i	$-0.784749\pi$
$239$	−24.1151	−1.55987	−0.779937	+	0.625858i	$0.784749\pi$
$240$	0	0
$241$	9.23439	0.594840	0.297420	−	0.954747i	$-0.403874\pi$
$241$	9.23439	0.594840	0.297420	+	0.954747i	$0.403874\pi$
$242$	−3.40608	−0.218951
$243$	0	0
$244$	4.42796	0.283471
$245$	0	0
$246$	0	0
$247$	−22.9153	−1.45807
$248$	9.11225	0.578628
$249$	0	0
$250$	−8.71189	−0.550989
$251$	−18.5790	−1.17270	−0.586349	−	0.810058i	$-0.699435\pi$
$251$	−18.5790	−1.17270	−0.586349	+	0.810058i	$0.699435\pi$
$252$	0	0
$253$	17.1062	1.07545
$254$	24.0801	1.51092
$255$	0	0
$256$	7.83416	0.489635
$257$	10.2197	0.637490	0.318745	−	0.947841i	$-0.396739\pi$
$257$	10.2197	0.637490	0.318745	+	0.947841i	$0.396739\pi$
$258$	0	0
$259$	0	0
$260$	5.33288	0.330731
$261$	0	0
$262$	18.5642	1.14690
$263$	−9.13775	−0.563458	−0.281729	−	0.959494i	$-0.590908\pi$
$263$	−9.13775	−0.563458	−0.281729	+	0.959494i	$0.590908\pi$
$264$	0	0
$265$	39.9329	2.45306
$266$	0	0
$267$	0	0
$268$	−4.25262	−0.259770
$269$	22.9213	1.39753	0.698767	−	0.715349i	$-0.253732\pi$
$269$	22.9213	1.39753	0.698767	+	0.715349i	$0.253732\pi$
$270$	0	0
$271$	−9.31563	−0.565884	−0.282942	−	0.959137i	$-0.591310\pi$
$271$	−9.31563	−0.565884	−0.282942	+	0.959137i	$0.591310\pi$
$272$	14.5963	0.885030
$273$	0	0
$274$	3.30601	0.199723
$275$	−10.8467	−0.654081
$276$	0	0
$277$	0.183318	0.0110145	0.00550725	−	0.999985i	$-0.498247\pi$
$277$	0.183318	0.0110145	0.00550725	+	0.999985i	$0.498247\pi$
$278$	32.7815	1.96611
$279$	0	0
$280$	0	0
$281$	−16.2019	−0.966527	−0.483263	−	0.875475i	$-0.660549\pi$
$281$	−16.2019	−0.966527	−0.483263	+	0.875475i	$0.660549\pi$
$282$	0	0
$283$	20.6984	1.23039	0.615195	−	0.788375i	$-0.289077\pi$
$283$	20.6984	1.23039	0.615195	+	0.788375i	$0.289077\pi$
$284$	0.977175	0.0579847
$285$	0	0
$286$	−31.2876	−1.85007
$287$	0	0
$288$	0	0
$289$	−6.74591	−0.396818
$290$	−18.6787	−1.09685
$291$	0	0
$292$	0.982202	0.0574790
$293$	16.0234	0.936097	0.468049	−	0.883703i	$-0.344957\pi$
$293$	16.0234	0.936097	0.468049	+	0.883703i	$0.344957\pi$
$294$	0	0
$295$	4.24068	0.246902
$296$	−10.9683	−0.637517
$297$	0	0
$298$	−16.9585	−0.982382
$299$	26.4746	1.53106
$300$	0	0
$301$	0	0
$302$	−3.18945	−0.183532
$303$	0	0
$304$	−18.5559	−1.06425
$305$	−37.3080	−2.13625
$306$	0	0
$307$	−5.32307	−0.303804	−0.151902	−	0.988396i	$-0.548540\pi$
$307$	−5.32307	−0.303804	−0.151902	+	0.988396i	$0.548540\pi$
$308$	0	0
$309$	0	0
$310$	15.4652	0.878367
$311$	18.4259	1.04484	0.522420	−	0.852689i	$-0.325030\pi$
$311$	18.4259	1.04484	0.522420	+	0.852689i	$0.325030\pi$
$312$	0	0
$313$	4.64688	0.262657	0.131329	−	0.991339i	$-0.458076\pi$
$313$	4.64688	0.262657	0.131329	+	0.991339i	$0.458076\pi$
$314$	33.0321	1.86411
$315$	0	0
$316$	0.299620	0.0168549
$317$	−10.4801	−0.588623	−0.294311	−	0.955710i	$-0.595090\pi$
$317$	−10.4801	−0.588623	−0.294311	+	0.955710i	$0.595090\pi$
$318$	0	0
$319$	15.7352	0.881004
$320$	−17.6485	−0.986582
$321$	0	0
$322$	0	0
$323$	−13.0358	−0.725329
$324$	0	0
$325$	−16.7871	−0.931178
$326$	14.3380	0.794109
$327$	0	0
$328$	−8.01822	−0.442732
$329$	0	0
$330$	0	0
$331$	12.8535	0.706494	0.353247	−	0.935530i	$-0.385077\pi$
$331$	12.8535	0.706494	0.353247	+	0.935530i	$0.385077\pi$
$332$	−2.69889	−0.148121
$333$	0	0
$334$	−26.5975	−1.45535
$335$	35.8306	1.95764
$336$	0	0
$337$	7.29218	0.397230	0.198615	−	0.980078i	$-0.436356\pi$
$337$	7.29218	0.397230	0.198615	+	0.980078i	$0.436356\pi$
$338$	−28.5563	−1.55326
$339$	0	0
$340$	3.03370	0.164526
$341$	−13.0282	−0.705514
$342$	0	0
$343$	0	0
$344$	−23.3869	−1.26094
$345$	0	0
$346$	32.9285	1.77025
$347$	17.5388	0.941534	0.470767	−	0.882258i	$-0.343977\pi$
$347$	17.5388	0.941534	0.470767	+	0.882258i	$0.343977\pi$
$348$	0	0
$349$	21.0160	1.12496	0.562481	−	0.826810i	$-0.309847\pi$
$349$	21.0160	1.12496	0.562481	+	0.826810i	$0.309847\pi$
$350$	0	0
$351$	0	0
$352$	−6.83020	−0.364051
$353$	5.05338	0.268965	0.134482	−	0.990916i	$-0.457063\pi$
$353$	5.05338	0.268965	0.134482	+	0.990916i	$0.457063\pi$
$354$	0	0
$355$	−8.23324	−0.436975
$356$	−1.89701	−0.100541
$357$	0	0
$358$	−8.64530	−0.456918
$359$	−14.4761	−0.764020	−0.382010	−	0.924158i	$-0.624768\pi$
$359$	−14.4761	−0.764020	−0.382010	+	0.924158i	$0.624768\pi$
$360$	0	0
$361$	−2.42796	−0.127788
$362$	−1.71808	−0.0903005
$363$	0	0
$364$	0	0
$365$	−8.27559	−0.433164
$366$	0	0
$367$	14.9228	0.778966	0.389483	−	0.921034i	$-0.372654\pi$
$367$	14.9228	0.778966	0.389483	+	0.921034i	$0.372654\pi$
$368$	21.4380	1.11754
$369$	0	0
$370$	−18.6152	−0.967761
$371$	0	0
$372$	0	0
$373$	7.82107	0.404960	0.202480	−	0.979286i	$-0.435100\pi$
$373$	7.82107	0.404960	0.202480	+	0.979286i	$0.435100\pi$
$374$	−17.7985	−0.920338
$375$	0	0
$376$	−12.3289	−0.635816
$377$	24.3528	1.25424
$378$	0	0
$379$	11.6097	0.596350	0.298175	−	0.954511i	$-0.403622\pi$
$379$	11.6097	0.596350	0.298175	+	0.954511i	$0.403622\pi$
$380$	−3.85667	−0.197843
$381$	0	0
$382$	−6.01948	−0.307984
$383$	25.0764	1.28134	0.640672	−	0.767814i	$-0.278656\pi$
$383$	25.0764	1.28134	0.640672	+	0.767814i	$0.278656\pi$
$384$	0	0
$385$	0	0
$386$	−27.6398	−1.40683
$387$	0	0
$388$	1.72215	0.0874287
$389$	23.4993	1.19146	0.595732	−	0.803184i	$-0.296862\pi$
$389$	23.4993	1.19146	0.595732	+	0.803184i	$0.296862\pi$
$390$	0	0
$391$	15.0605	0.761643
$392$	0	0
$393$	0	0
$394$	−38.9443	−1.96198
$395$	−2.52446	−0.127020
$396$	0	0
$397$	−19.7234	−0.989889	−0.494945	−	0.868924i	$-0.664812\pi$
$397$	−19.7234	−0.989889	−0.494945	+	0.868924i	$0.664812\pi$
$398$	−19.2995	−0.967395
$399$	0	0
$400$	−13.5935	−0.679674
$401$	14.3797	0.718086	0.359043	−	0.933321i	$-0.383103\pi$
$401$	14.3797	0.718086	0.359043	+	0.933321i	$0.383103\pi$
$402$	0	0
$403$	−20.1632	−1.00440
$404$	−0.846510	−0.0421154
$405$	0	0
$406$	0	0
$407$	15.6818	0.777317
$408$	0	0
$409$	−11.5759	−0.572391	−0.286196	−	0.958171i	$-0.592391\pi$
$409$	−11.5759	−0.572391	−0.286196	+	0.958171i	$0.592391\pi$
$410$	−13.6085	−0.672074
$411$	0	0
$412$	3.30173	0.162664
$413$	0	0
$414$	0	0
$415$	22.7396	1.11624
$416$	−10.5708	−0.518278
$417$	0	0
$418$	22.6268	1.10671
$419$	34.1721	1.66942	0.834708	−	0.550693i	$-0.185636\pi$
$419$	34.1721	1.66942	0.834708	+	0.550693i	$0.185636\pi$
$420$	0	0
$421$	−10.2213	−0.498156	−0.249078	−	0.968483i	$-0.580128\pi$
$421$	−10.2213	−0.498156	−0.249078	+	0.968483i	$0.580128\pi$
$422$	−3.53247	−0.171958
$423$	0	0
$424$	−35.9561	−1.74618
$425$	−9.54961	−0.463224
$426$	0	0
$427$	0	0
$428$	−0.906215	−0.0438036
$429$	0	0
$430$	−39.6921	−1.91412
$431$	−0.0761051	−0.00366585	−0.00183293	−	0.999998i	$-0.500583\pi$
$431$	−0.0761051	−0.00366585	−0.00183293	+	0.999998i	$0.500583\pi$
$432$	0	0
$433$	−29.2697	−1.40661	−0.703305	−	0.710888i	$-0.748293\pi$
$433$	−29.2697	−1.40661	−0.703305	+	0.710888i	$0.748293\pi$
$434$	0	0
$435$	0	0
$436$	−3.03717	−0.145454
$437$	−19.1461	−0.915880
$438$	0	0
$439$	−4.34526	−0.207388	−0.103694	−	0.994609i	$-0.533066\pi$
$439$	−4.34526	−0.207388	−0.103694	+	0.994609i	$0.533066\pi$
$440$	26.1412	1.24623
$441$	0	0
$442$	−27.5461	−1.31023
$443$	−20.5544	−0.976567	−0.488284	−	0.872685i	$-0.662377\pi$
$443$	−20.5544	−0.976567	−0.488284	+	0.872685i	$0.662377\pi$
$444$	0	0
$445$	15.9834	0.757684
$446$	−14.2115	−0.672932
$447$	0	0
$448$	0	0
$449$	−12.4720	−0.588588	−0.294294	−	0.955715i	$-0.595085\pi$
$449$	−12.4720	−0.588588	−0.294294	+	0.955715i	$0.595085\pi$
$450$	0	0
$451$	11.4640	0.539818
$452$	−0.156185	−0.00734633
$453$	0	0
$454$	2.94661	0.138291
$455$	0	0
$456$	0	0
$457$	−30.9348	−1.44707	−0.723534	−	0.690289i	$-0.757483\pi$
$457$	−30.9348	−1.44707	−0.723534	+	0.690289i	$0.757483\pi$
$458$	−5.10592	−0.238584
$459$	0	0
$460$	4.45570	0.207748
$461$	8.02281	0.373660	0.186830	−	0.982392i	$-0.440179\pi$
$461$	8.02281	0.373660	0.186830	+	0.982392i	$0.440179\pi$
$462$	0	0
$463$	−16.8223	−0.781800	−0.390900	−	0.920433i	$-0.627836\pi$
$463$	−16.8223	−0.781800	−0.390900	+	0.920433i	$0.627836\pi$
$464$	19.7200	0.915476
$465$	0	0
$466$	−32.3233	−1.49735
$467$	0.960797	0.0444604	0.0222302	−	0.999753i	$-0.492923\pi$
$467$	0.960797	0.0444604	0.0222302	+	0.999753i	$0.492923\pi$
$468$	0	0
$469$	0	0
$470$	−20.9246	−0.965179
$471$	0	0
$472$	−3.81837	−0.175755
$473$	33.4372	1.53745
$474$	0	0
$475$	12.1402	0.557029
$476$	0	0
$477$	0	0
$478$	36.8521	1.68557
$479$	24.1290	1.10248	0.551242	−	0.834346i	$-0.314154\pi$
$479$	24.1290	1.10248	0.551242	+	0.834346i	$0.314154\pi$
$480$	0	0
$481$	24.2701	1.10662
$482$	−14.1118	−0.642774
$483$	0	0
$484$	0.747384	0.0339720
$485$	−14.5100	−0.658866
$486$	0	0
$487$	14.7989	0.670601	0.335301	−	0.942111i	$-0.391162\pi$
$487$	14.7989	0.670601	0.335301	+	0.942111i	$0.391162\pi$
$488$	33.5927	1.52067
$489$	0	0
$490$	0	0
$491$	6.30383	0.284488	0.142244	−	0.989832i	$-0.454568\pi$
$491$	6.30383	0.284488	0.142244	+	0.989832i	$0.454568\pi$
$492$	0	0
$493$	13.8535	0.623932
$494$	35.0186	1.57556
$495$	0	0
$496$	−16.3274	−0.733120
$497$	0	0
$498$	0	0
$499$	−28.5971	−1.28018	−0.640092	−	0.768298i	$-0.721104\pi$
$499$	−28.5971	−1.28018	−0.640092	+	0.768298i	$0.721104\pi$
$500$	1.91162	0.0854903
$501$	0	0
$502$	28.3921	1.26720
$503$	−35.4133	−1.57900	−0.789500	−	0.613750i	$-0.789660\pi$
$503$	−35.4133	−1.57900	−0.789500	+	0.613750i	$0.789660\pi$
$504$	0	0
$505$	7.13231	0.317384
$506$	−26.1412	−1.16212
$507$	0	0
$508$	−5.28382	−0.234432
$509$	32.4883	1.44002	0.720010	−	0.693964i	$-0.244137\pi$
$509$	32.4883	1.44002	0.720010	+	0.693964i	$0.244137\pi$
$510$	0	0
$511$	0	0
$512$	14.6316	0.646630
$513$	0	0
$514$	−15.6176	−0.688861
$515$	−27.8189	−1.22585
$516$	0	0
$517$	17.6272	0.775243
$518$	0	0
$519$	0	0
$520$	40.4578	1.77419
$521$	−32.9818	−1.44496	−0.722479	−	0.691393i	$-0.756997\pi$
$521$	−32.9818	−1.44496	−0.722479	+	0.691393i	$0.756997\pi$
$522$	0	0
$523$	14.2627	0.623662	0.311831	−	0.950138i	$-0.399058\pi$
$523$	14.2627	0.623662	0.311831	+	0.950138i	$0.399058\pi$
$524$	−4.07348	−0.177951
$525$	0	0
$526$	13.9641	0.608863
$527$	−11.4702	−0.499649
$528$	0	0
$529$	−0.880118	−0.0382660
$530$	−61.0245	−2.65073
$531$	0	0
$532$	0	0
$533$	17.7424	0.768508
$534$	0	0
$535$	7.63537	0.330106
$536$	−32.2624	−1.39352
$537$	0	0
$538$	−35.0277	−1.51015
$539$	0	0
$540$	0	0
$541$	−15.9821	−0.687124	−0.343562	−	0.939130i	$-0.611634\pi$
$541$	−15.9821	−0.687124	−0.343562	+	0.939130i	$0.611634\pi$
$542$	14.2359	0.611485
$543$	0	0
$544$	−6.01341	−0.257823
$545$	25.5899	1.09615
$546$	0	0
$547$	−32.0570	−1.37066	−0.685330	−	0.728233i	$-0.740342\pi$
$547$	−32.0570	−1.37066	−0.685330	+	0.728233i	$0.740342\pi$
$548$	−0.725427	−0.0309887
$549$	0	0
$550$	16.5757	0.706790
$551$	−17.6117	−0.750282
$552$	0	0
$553$	0	0
$554$	−0.280142	−0.0119021
$555$	0	0
$556$	−7.19315	−0.305057
$557$	39.2947	1.66497	0.832486	−	0.554046i	$-0.186917\pi$
$557$	39.2947	1.66497	0.832486	+	0.554046i	$0.186917\pi$
$558$	0	0
$559$	51.7496	2.18877
$560$	0	0
$561$	0	0
$562$	24.7594	1.04441
$563$	−34.5063	−1.45427	−0.727134	−	0.686495i	$-0.759148\pi$
$563$	−34.5063	−1.45427	−0.727134	+	0.686495i	$0.759148\pi$
$564$	0	0
$565$	1.31595	0.0553623
$566$	−31.6308	−1.32954
$567$	0	0
$568$	7.41332	0.311056
$569$	4.28406	0.179597	0.0897985	−	0.995960i	$-0.471378\pi$
$569$	4.28406	0.179597	0.0897985	+	0.995960i	$0.471378\pi$
$570$	0	0
$571$	25.1596	1.05289	0.526447	−	0.850208i	$-0.323524\pi$
$571$	25.1596	1.05289	0.526447	+	0.850208i	$0.323524\pi$
$572$	6.86533	0.287054
$573$	0	0
$574$	0	0
$575$	−14.0258	−0.584917
$576$	0	0
$577$	−44.2563	−1.84241	−0.921206	−	0.389075i	$-0.872795\pi$
$577$	−44.2563	−1.84241	−0.921206	+	0.389075i	$0.872795\pi$
$578$	10.3089	0.428795
$579$	0	0
$580$	4.09861	0.170186
$581$	0	0
$582$	0	0
$583$	51.4080	2.12910
$584$	7.45146	0.308343
$585$	0	0
$586$	−24.4866	−1.01153
$587$	13.0046	0.536757	0.268378	−	0.963314i	$-0.413512\pi$
$587$	13.0046	0.536757	0.268378	+	0.963314i	$0.413512\pi$
$588$	0	0
$589$	14.5818	0.600831
$590$	−6.48051	−0.266798
$591$	0	0
$592$	19.6530	0.807731
$593$	28.2328	1.15938	0.579691	−	0.814837i	$-0.303173\pi$
$593$	28.2328	1.15938	0.579691	+	0.814837i	$0.303173\pi$
$594$	0	0
$595$	0	0
$596$	3.72116	0.152425
$597$	0	0
$598$	−40.4578	−1.65444
$599$	−4.99536	−0.204105	−0.102052	−	0.994779i	$-0.532541\pi$
$599$	−4.99536	−0.204105	−0.102052	+	0.994779i	$0.532541\pi$
$600$	0	0
$601$	8.15787	0.332766	0.166383	−	0.986061i	$-0.446791\pi$
$601$	8.15787	0.332766	0.166383	+	0.986061i	$0.446791\pi$
$602$	0	0
$603$	0	0
$604$	0.699851	0.0284765
$605$	−6.29712	−0.256014
$606$	0	0
$607$	16.7277	0.678956	0.339478	−	0.940614i	$-0.389750\pi$
$607$	16.7277	0.678956	0.339478	+	0.940614i	$0.389750\pi$
$608$	7.64469	0.310033
$609$	0	0
$610$	57.0133	2.30840
$611$	27.2809	1.10367
$612$	0	0
$613$	−16.5876	−0.669968	−0.334984	−	0.942224i	$-0.608731\pi$
$613$	−16.5876	−0.669968	−0.334984	+	0.942224i	$0.608731\pi$
$614$	8.13459	0.328285
$615$	0	0
$616$	0	0
$617$	−23.7085	−0.954470	−0.477235	−	0.878776i	$-0.658361\pi$
$617$	−23.7085	−0.954470	−0.477235	+	0.878776i	$0.658361\pi$
$618$	0	0
$619$	−7.82412	−0.314478	−0.157239	−	0.987561i	$-0.550259\pi$
$619$	−7.82412	−0.314478	−0.157239	+	0.987561i	$0.550259\pi$
$620$	−3.39349	−0.136286
$621$	0	0
$622$	−28.1581	−1.12904
$623$	0	0
$624$	0	0
$625$	−31.0175	−1.24070
$626$	−7.10125	−0.283823
$627$	0	0
$628$	−7.24812	−0.289231
$629$	13.8065	0.550500
$630$	0	0
$631$	−2.54669	−0.101382	−0.0506911	−	0.998714i	$-0.516142\pi$
$631$	−2.54669	−0.101382	−0.0506911	+	0.998714i	$0.516142\pi$
$632$	2.27306	0.0904175
$633$	0	0
$634$	16.0155	0.636056
$635$	44.5191	1.76669
$636$	0	0
$637$	0	0
$638$	−24.0462	−0.951999
$639$	0	0
$640$	37.5812	1.48553
$641$	10.2854	0.406248	0.203124	−	0.979153i	$-0.434891\pi$
$641$	10.2854	0.406248	0.203124	+	0.979153i	$0.434891\pi$
$642$	0	0
$643$	−4.92126	−0.194076	−0.0970378	−	0.995281i	$-0.530937\pi$
$643$	−4.92126	−0.194076	−0.0970378	+	0.995281i	$0.530937\pi$
$644$	0	0
$645$	0	0
$646$	19.9209	0.783779
$647$	8.78336	0.345310	0.172655	−	0.984982i	$-0.444765\pi$
$647$	8.78336	0.345310	0.172655	+	0.984982i	$0.444765\pi$
$648$	0	0
$649$	5.45928	0.214295
$650$	25.6536	1.00622
$651$	0	0
$652$	−3.14614	−0.123212
$653$	1.77459	0.0694453	0.0347226	−	0.999397i	$-0.488945\pi$
$653$	1.77459	0.0694453	0.0347226	+	0.999397i	$0.488945\pi$
$654$	0	0
$655$	34.3213	1.34104
$656$	14.3671	0.560940
$657$	0	0
$658$	0	0
$659$	2.34367	0.0912966	0.0456483	−	0.998958i	$-0.485465\pi$
$659$	2.34367	0.0912966	0.0456483	+	0.998958i	$0.485465\pi$
$660$	0	0
$661$	−22.9094	−0.891074	−0.445537	−	0.895264i	$-0.646987\pi$
$661$	−22.9094	−0.891074	−0.445537	+	0.895264i	$0.646987\pi$
$662$	−19.6425	−0.763426
$663$	0	0
$664$	−20.4751	−0.794587
$665$	0	0
$666$	0	0
$667$	20.3472	0.787845
$668$	5.83619	0.225809
$669$	0	0
$670$	−54.7555	−2.11539
$671$	−48.0288	−1.85413
$672$	0	0
$673$	−29.7349	−1.14620	−0.573098	−	0.819487i	$-0.694259\pi$
$673$	−29.7349	−1.14620	−0.573098	+	0.819487i	$0.694259\pi$
$674$	−11.1437	−0.429241
$675$	0	0
$676$	6.26602	0.241001
$677$	25.4960	0.979891	0.489946	−	0.871753i	$-0.337017\pi$
$677$	25.4960	0.979891	0.489946	+	0.871753i	$0.337017\pi$
$678$	0	0
$679$	0	0
$680$	23.0151	0.882590
$681$	0	0
$682$	19.9093	0.762367
$683$	11.3859	0.435668	0.217834	−	0.975986i	$-0.430101\pi$
$683$	11.3859	0.435668	0.217834	+	0.975986i	$0.430101\pi$
$684$	0	0
$685$	6.11212	0.233532
$686$	0	0
$687$	0	0
$688$	41.9047	1.59760
$689$	79.5622	3.03108
$690$	0	0
$691$	−32.2523	−1.22694	−0.613468	−	0.789720i	$-0.710226\pi$
$691$	−32.2523	−1.22694	−0.613468	+	0.789720i	$0.710226\pi$
$692$	−7.22539	−0.274668
$693$	0	0
$694$	−26.8024	−1.01741
$695$	60.6062	2.29893
$696$	0	0
$697$	10.0931	0.382302
$698$	−32.1162	−1.21561
$699$	0	0
$700$	0	0
$701$	15.6291	0.590302	0.295151	−	0.955451i	$-0.404630\pi$
$701$	15.6291	0.590302	0.295151	+	0.955451i	$0.404630\pi$
$702$	0	0
$703$	−17.5518	−0.661979
$704$	−22.7200	−0.856292
$705$	0	0
$706$	−7.72246	−0.290639
$707$	0	0
$708$	0	0
$709$	−11.8165	−0.443777	−0.221888	−	0.975072i	$-0.571222\pi$
$709$	−11.8165	−0.443777	−0.221888	+	0.975072i	$0.571222\pi$
$710$	12.5818	0.472188
$711$	0	0
$712$	−14.3916	−0.539349
$713$	−16.8466	−0.630912
$714$	0	0
$715$	−57.8442	−2.16325
$716$	1.89701	0.0708946
$717$	0	0
$718$	22.1221	0.825588
$719$	−50.7233	−1.89166	−0.945830	−	0.324664i	$-0.894749\pi$
$719$	−50.7233	−1.89166	−0.945830	+	0.324664i	$0.894749\pi$
$720$	0	0
$721$	0	0
$722$	3.71036	0.138085
$723$	0	0
$724$	0.376994	0.0140109
$725$	−12.9018	−0.479160
$726$	0	0
$727$	2.26602	0.0840422	0.0420211	−	0.999117i	$-0.486620\pi$
$727$	2.26602	0.0840422	0.0420211	+	0.999117i	$0.486620\pi$
$728$	0	0
$729$	0	0
$730$	12.6466	0.468070
$731$	29.4386	1.08883
$732$	0	0
$733$	−33.6524	−1.24298	−0.621490	−	0.783422i	$-0.713472\pi$
$733$	−33.6524	−1.24298	−0.621490	+	0.783422i	$0.713472\pi$
$734$	−22.8047	−0.841738
$735$	0	0
$736$	−8.83209	−0.325555
$737$	46.1269	1.69911
$738$	0	0
$739$	19.6037	0.721135	0.360568	−	0.932733i	$-0.382583\pi$
$739$	19.6037	0.721135	0.360568	+	0.932733i	$0.382583\pi$
$740$	4.08468	0.150156
$741$	0	0
$742$	0	0
$743$	48.8390	1.79173	0.895865	−	0.444327i	$-0.146557\pi$
$743$	48.8390	1.79173	0.895865	+	0.444327i	$0.146557\pi$
$744$	0	0
$745$	−31.3528	−1.14868
$746$	−11.9520	−0.437593
$747$	0	0
$748$	3.90546	0.142798
$749$	0	0
$750$	0	0
$751$	15.8254	0.577476	0.288738	−	0.957408i	$-0.406764\pi$
$751$	15.8254	0.577476	0.288738	+	0.957408i	$0.406764\pi$
$752$	22.0910	0.805576
$753$	0	0
$754$	−37.2154	−1.35531
$755$	−5.89663	−0.214600
$756$	0	0
$757$	−2.18728	−0.0794982	−0.0397491	−	0.999210i	$-0.512656\pi$
$757$	−2.18728	−0.0794982	−0.0397491	+	0.999210i	$0.512656\pi$
$758$	−17.7417	−0.644407
$759$	0	0
$760$	−29.2585	−1.06132
$761$	−27.7562	−1.00616	−0.503081	−	0.864240i	$-0.667800\pi$
$761$	−27.7562	−1.00616	−0.503081	+	0.864240i	$0.667800\pi$
$762$	0	0
$763$	0	0
$764$	1.32084	0.0477861
$765$	0	0
$766$	−38.3212	−1.38460
$767$	8.44912	0.305080
$768$	0	0
$769$	10.0206	0.361352	0.180676	−	0.983543i	$-0.442171\pi$
$769$	10.0206	0.361352	0.180676	+	0.983543i	$0.442171\pi$
$770$	0	0
$771$	0	0
$772$	6.06491	0.218281
$773$	−43.5192	−1.56528	−0.782639	−	0.622476i	$-0.786127\pi$
$773$	−43.5192	−1.56528	−0.782639	+	0.622476i	$0.786127\pi$
$774$	0	0
$775$	10.6822	0.383714
$776$	13.0650	0.469007
$777$	0	0
$778$	−35.9111	−1.28748
$779$	−12.8311	−0.459720
$780$	0	0
$781$	−10.5991	−0.379267
$782$	−23.0151	−0.823019
$783$	0	0
$784$	0	0
$785$	61.0694	2.17966
$786$	0	0
$787$	−5.83638	−0.208044	−0.104022	−	0.994575i	$-0.533171\pi$
$787$	−5.83638	−0.208044	−0.104022	+	0.994575i	$0.533171\pi$
$788$	8.54542	0.304418
$789$	0	0
$790$	3.85782	0.137255
$791$	0	0
$792$	0	0
$793$	−74.3325	−2.63962
$794$	30.1408	1.06966
$795$	0	0
$796$	4.23482	0.150099
$797$	36.5800	1.29573	0.647865	−	0.761755i	$-0.275662\pi$
$797$	36.5800	1.29573	0.647865	+	0.761755i	$0.275662\pi$
$798$	0	0
$799$	15.5192	0.549031
$800$	5.60027	0.198000
$801$	0	0
$802$	−21.9747	−0.775952
$803$	−10.6537	−0.375959
$804$	0	0
$805$	0	0
$806$	30.8129	1.08534
$807$	0	0
$808$	−6.42203	−0.225926
$809$	−22.0499	−0.775232	−0.387616	−	0.921821i	$-0.626701\pi$
$809$	−22.0499	−0.775232	−0.387616	+	0.921821i	$0.626701\pi$
$810$	0	0
$811$	−12.6451	−0.444029	−0.222015	−	0.975043i	$-0.571263\pi$
$811$	−12.6451	−0.444029	−0.222015	+	0.975043i	$0.571263\pi$
$812$	0	0
$813$	0	0
$814$	−23.9645	−0.839956
$815$	26.5080	0.928534
$816$	0	0
$817$	−37.4246	−1.30932
$818$	17.6900	0.618517
$819$	0	0
$820$	2.98606	0.104278
$821$	−31.0958	−1.08525	−0.542625	−	0.839975i	$-0.682570\pi$
$821$	−31.0958	−1.08525	−0.542625	+	0.839975i	$0.682570\pi$
$822$	0	0
$823$	−48.9638	−1.70677	−0.853385	−	0.521281i	$-0.825454\pi$
$823$	−48.9638	−1.70677	−0.853385	+	0.521281i	$0.825454\pi$
$824$	25.0485	0.872606
$825$	0	0
$826$	0	0
$827$	−36.3827	−1.26515	−0.632575	−	0.774499i	$-0.718002\pi$
$827$	−36.3827	−1.26515	−0.632575	+	0.774499i	$0.718002\pi$
$828$	0	0
$829$	45.7405	1.58863	0.794316	−	0.607505i	$-0.207829\pi$
$829$	45.7405	1.58863	0.794316	+	0.607505i	$0.207829\pi$
$830$	−34.7502	−1.20620
$831$	0	0
$832$	−35.1629	−1.21905
$833$	0	0
$834$	0	0
$835$	−49.1731	−1.70171
$836$	−4.96492	−0.171715
$837$	0	0
$838$	−52.2210	−1.80394
$839$	−25.5348	−0.881559	−0.440779	−	0.897615i	$-0.645298\pi$
$839$	−25.5348	−0.881559	−0.440779	+	0.897615i	$0.645298\pi$
$840$	0	0
$841$	−10.2835	−0.354604
$842$	15.6200	0.538299
$843$	0	0
$844$	0.775117	0.0266806
$845$	−52.7947	−1.81619
$846$	0	0
$847$	0	0
$848$	64.4263	2.21241
$849$	0	0
$850$	14.5935	0.500552
$851$	20.2780	0.695121
$852$	0	0
$853$	−1.84673	−0.0632310	−0.0316155	−	0.999500i	$-0.510065\pi$
$853$	−1.84673	−0.0632310	−0.0316155	+	0.999500i	$0.510065\pi$
$854$	0	0
$855$	0	0
$856$	−6.87499	−0.234982
$857$	−22.4848	−0.768066	−0.384033	−	0.923319i	$-0.625465\pi$
$857$	−22.4848	−0.768066	−0.384033	+	0.923319i	$0.625465\pi$
$858$	0	0
$859$	−1.14761	−0.0391561	−0.0195781	−	0.999808i	$-0.506232\pi$
$859$	−1.14761	−0.0391561	−0.0195781	+	0.999808i	$0.506232\pi$
$860$	8.70951	0.296992
$861$	0	0
$862$	0.116302	0.00396126
$863$	1.79527	0.0611117	0.0305558	−	0.999533i	$-0.490272\pi$
$863$	1.79527	0.0611117	0.0305558	+	0.999533i	$0.490272\pi$
$864$	0	0
$865$	60.8779	2.06991
$866$	44.7292	1.51996
$867$	0	0
$868$	0	0
$869$	−3.24989	−0.110245
$870$	0	0
$871$	71.3889	2.41892
$872$	−23.0415	−0.780283
$873$	0	0
$874$	29.2585	0.989685
$875$	0	0
$876$	0	0
$877$	8.57997	0.289725	0.144862	−	0.989452i	$-0.453726\pi$
$877$	8.57997	0.289725	0.144862	+	0.989452i	$0.453726\pi$
$878$	6.64032	0.224100
$879$	0	0
$880$	−46.8399	−1.57897
$881$	−16.5346	−0.557066	−0.278533	−	0.960427i	$-0.589848\pi$
$881$	−16.5346	−0.557066	−0.278533	+	0.960427i	$0.589848\pi$
$882$	0	0
$883$	−27.4948	−0.925273	−0.462636	−	0.886548i	$-0.653096\pi$
$883$	−27.4948	−0.925273	−0.462636	+	0.886548i	$0.653096\pi$
$884$	6.04434	0.203293
$885$	0	0
$886$	31.4107	1.05526
$887$	−43.4419	−1.45864	−0.729318	−	0.684175i	$-0.760163\pi$
$887$	−43.4419	−1.45864	−0.729318	+	0.684175i	$0.760163\pi$
$888$	0	0
$889$	0	0
$890$	−24.4254	−0.818741
$891$	0	0
$892$	3.11837	0.104411
$893$	−19.7292	−0.660213
$894$	0	0
$895$	−15.9834	−0.534265
$896$	0	0
$897$	0	0
$898$	19.0594	0.636019
$899$	−15.4965	−0.516838
$900$	0	0
$901$	45.2603	1.50784
$902$	−17.5190	−0.583319
$903$	0	0
$904$	−1.18490	−0.0394091
$905$	−3.17638	−0.105586
$906$	0	0
$907$	11.2952	0.375052	0.187526	−	0.982260i	$-0.439953\pi$
$907$	11.2952	0.375052	0.187526	+	0.982260i	$0.439953\pi$
$908$	−0.646564	−0.0214570
$909$	0	0
$910$	0	0
$911$	10.0338	0.332435	0.166217	−	0.986089i	$-0.446845\pi$
$911$	10.0338	0.332435	0.166217	+	0.986089i	$0.446845\pi$
$912$	0	0
$913$	29.2741	0.968830
$914$	47.2738	1.56368
$915$	0	0
$916$	1.12038	0.0370182
$917$	0	0
$918$	0	0
$919$	−0.357934	−0.0118072	−0.00590358	−	0.999983i	$-0.501879\pi$
$919$	−0.357934	−0.0118072	−0.00590358	+	0.999983i	$0.501879\pi$
$920$	33.8031	1.11445
$921$	0	0
$922$	−12.2603	−0.403770
$923$	−16.4039	−0.539941
$924$	0	0
$925$	−12.8579	−0.422766
$926$	25.7075	0.844801
$927$	0	0
$928$	−8.12427	−0.266692
$929$	33.5264	1.09997	0.549983	−	0.835176i	$-0.314634\pi$
$929$	33.5264	1.09997	0.549983	+	0.835176i	$0.314634\pi$
$930$	0	0
$931$	0	0
$932$	7.09259	0.232326
$933$	0	0
$934$	−1.46827	−0.0480432
$935$	−32.9057	−1.07613
$936$	0	0
$937$	12.8772	0.420680	0.210340	−	0.977628i	$-0.432543\pi$
$937$	12.8772	0.420680	0.210340	+	0.977628i	$0.432543\pi$
$938$	0	0
$939$	0	0
$940$	4.59141	0.149755
$941$	−32.0984	−1.04638	−0.523189	−	0.852217i	$-0.675258\pi$
$941$	−32.0984	−1.04638	−0.523189	+	0.852217i	$0.675258\pi$
$942$	0	0
$943$	14.8240	0.482736
$944$	6.84176	0.222680
$945$	0	0
$946$	−51.0980	−1.66134
$947$	−16.7986	−0.545880	−0.272940	−	0.962031i	$-0.587996\pi$
$947$	−16.7986	−0.545880	−0.272940	+	0.962031i	$0.587996\pi$
$948$	0	0
$949$	−16.4883	−0.535232
$950$	−18.5523	−0.601917
$951$	0	0
$952$	0	0
$953$	2.69574	0.0873237	0.0436619	−	0.999046i	$-0.486098\pi$
$953$	2.69574	0.0873237	0.0436619	+	0.999046i	$0.486098\pi$
$954$	0	0
$955$	−11.1288	−0.360118
$956$	−8.08633	−0.261531
$957$	0	0
$958$	−36.8734	−1.19133
$959$	0	0
$960$	0	0
$961$	−18.1695	−0.586112
$962$	−37.0890	−1.19580
$963$	0	0
$964$	3.09650	0.0997316
$965$	−51.1002	−1.64497
$966$	0	0
$967$	13.6775	0.439837	0.219919	−	0.975518i	$-0.429421\pi$
$967$	13.6775	0.439837	0.219919	+	0.975518i	$0.429421\pi$
$968$	5.67002	0.182241
$969$	0	0
$970$	22.1739	0.711960
$971$	−42.6266	−1.36795	−0.683977	−	0.729504i	$-0.739751\pi$
$971$	−42.6266	−1.36795	−0.683977	+	0.729504i	$0.739751\pi$
$972$	0	0
$973$	0	0
$974$	−22.6153	−0.724641
$975$	0	0
$976$	−60.1915	−1.92668
$977$	−5.05683	−0.161782	−0.0808911	−	0.996723i	$-0.525777\pi$
$977$	−5.05683	−0.161782	−0.0808911	+	0.996723i	$0.525777\pi$
$978$	0	0
$979$	20.5763	0.657622
$980$	0	0
$981$	0	0
$982$	−9.63336	−0.307413
$983$	−33.2884	−1.06174	−0.530868	−	0.847455i	$-0.678134\pi$
$983$	−33.2884	−1.06174	−0.530868	+	0.847455i	$0.678134\pi$
$984$	0	0
$985$	−71.9999	−2.29411
$986$	−21.1707	−0.674211
$987$	0	0
$988$	−7.68402	−0.244461
$989$	43.2375	1.37487
$990$	0	0
$991$	32.0879	1.01931	0.509653	−	0.860380i	$-0.329774\pi$
$991$	32.0879	1.01931	0.509653	+	0.860380i	$0.329774\pi$
$992$	6.72658	0.213569
$993$	0	0
$994$	0	0
$995$	−35.6807	−1.13115
$996$	0	0
$997$	−43.2566	−1.36995	−0.684975	−	0.728567i	$-0.740187\pi$
$997$	−43.2566	−1.36995	−0.684975	+	0.728567i	$0.740187\pi$
$998$	43.7015	1.38335
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.bg.1.2		8
3.2	odd	2	inner	3969.2.a.bg.1.7		8
7.2	even	3		567.2.e.g.487.7	yes	16
7.4	even	3		567.2.e.g.163.7	yes	16
7.6	odd	2		3969.2.a.bf.1.2		8
21.2	odd	6		567.2.e.g.487.2	yes	16
21.11	odd	6		567.2.e.g.163.2	✓	16
21.20	even	2		3969.2.a.bf.1.7		8
63.2	odd	6		567.2.g.l.109.2		16
63.4	even	3		567.2.g.l.541.7		16
63.11	odd	6		567.2.h.l.352.7		16
63.16	even	3		567.2.g.l.109.7		16
63.23	odd	6		567.2.h.l.298.7		16
63.25	even	3		567.2.h.l.352.2		16
63.32	odd	6		567.2.g.l.541.2		16
63.58	even	3		567.2.h.l.298.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
567.2.e.g.163.2	✓	16	21.11	odd	6
567.2.e.g.163.7	yes	16	7.4	even	3
567.2.e.g.487.2	yes	16	21.2	odd	6
567.2.e.g.487.7	yes	16	7.2	even	3
567.2.g.l.109.2		16	63.2	odd	6
567.2.g.l.109.7		16	63.16	even	3
567.2.g.l.541.2		16	63.32	odd	6
567.2.g.l.541.7		16	63.4	even	3
567.2.h.l.298.2		16	63.58	even	3
567.2.h.l.298.7		16	63.23	odd	6
567.2.h.l.352.2		16	63.25	even	3
567.2.h.l.352.7		16	63.11	odd	6
3969.2.a.bf.1.2		8	7.6	odd	2
3969.2.a.bf.1.7		8	21.20	even	2
3969.2.a.bg.1.2		8	1.1	even	1	trivial
3969.2.a.bg.1.7		8	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3969.a (trivial)

\(f(q)\)	\(=\)	\(q-1.52818 q^{2} +0.335323 q^{4} -2.82528 q^{5} +2.54392 q^{8} +O(q^{10})\)	\(q-1.52818 q^{2} +0.335323 q^{4} -2.82528 q^{5} +2.54392 q^{8} +4.31752 q^{10} -3.63715 q^{11} -5.62908 q^{13} -4.55820 q^{16} -3.20220 q^{17} +4.07088 q^{19} -0.947380 q^{20} +5.55820 q^{22} -4.70318 q^{23} +2.98220 q^{25} +8.60223 q^{26} -4.32626 q^{29} +3.58197 q^{31} +1.87790 q^{32} +4.89353 q^{34} -4.31156 q^{37} -6.22102 q^{38} -7.18728 q^{40} -3.15192 q^{41} -9.19325 q^{43} -1.21962 q^{44} +7.18728 q^{46} -4.84643 q^{47} -4.55733 q^{50} -1.88756 q^{52} -14.1341 q^{53} +10.2760 q^{55} +6.61128 q^{58} -1.50098 q^{59} +13.2051 q^{61} -5.47388 q^{62} +6.24665 q^{64} +15.9037 q^{65} -12.6822 q^{67} -1.07377 q^{68} +2.91413 q^{71} +2.92912 q^{73} +6.58882 q^{74} +1.36506 q^{76} +0.893527 q^{79} +12.8782 q^{80} +4.81668 q^{82} -8.04863 q^{83} +9.04711 q^{85} +14.0489 q^{86} -9.25262 q^{88} -5.65727 q^{89} -1.57708 q^{92} +7.40620 q^{94} -11.5014 q^{95} +5.13578 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{4}+O(q^{10}) \) 8 * q + 6 * q^4	\( 8 q + 6 q^{4} + 14 q^{10} + 6 q^{13} + 6 q^{16} + 24 q^{19} + 2 q^{22} + 20 q^{31} - 4 q^{37} + 36 q^{40} + 10 q^{43} - 36 q^{46} + 34 q^{52} + 4 q^{55} - 22 q^{58} + 36 q^{61} + 38 q^{64} - 18 q^{67} + 32 q^{73} + 58 q^{76} - 32 q^{79} - 2 q^{82} + 30 q^{85} - 72 q^{88} + 54 q^{94} + 14 q^{97}+O(q^{100}) \) 8 * q + 6 * q^4 + 14 * q^10 + 6 * q^13 + 6 * q^16 + 24 * q^19 + 2 * q^22 + 20 * q^31 - 4 * q^37 + 36 * q^40 + 10 * q^43 - 36 * q^46 + 34 * q^52 + 4 * q^55 - 22 * q^58 + 36 * q^61 + 38 * q^64 - 18 * q^67 + 32 * q^73 + 58 * q^76 - 32 * q^79 - 2 * q^82 + 30 * q^85 - 72 * q^88 + 54 * q^94 + 14 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more