LMFDB - Embedded newform 6004.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6004,2,Mod(1,6004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6004 = 2^{2} \cdot 19 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6004.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:	$47.9421813736$
Analytic rank:	$1$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.35817 q^{3} -0.581328 q^{5} +3.25261 q^{7} +8.27730 q^{9} +O(q^{10})$	$q-3.35817 q^{3} -0.581328 q^{5} +3.25261 q^{7} +8.27730 q^{9} +4.01989 q^{11} +4.26295 q^{13} +1.95220 q^{15} -2.04729 q^{17} +1.00000 q^{19} -10.9228 q^{21} +0.293401 q^{23} -4.66206 q^{25} -17.7221 q^{27} -6.94533 q^{29} -1.87612 q^{31} -13.4995 q^{33} -1.89084 q^{35} +8.28320 q^{37} -14.3157 q^{39} -6.21102 q^{41} -9.11192 q^{43} -4.81183 q^{45} -8.53931 q^{47} +3.57949 q^{49} +6.87514 q^{51} -14.3146 q^{53} -2.33688 q^{55} -3.35817 q^{57} +3.68459 q^{59} +1.48250 q^{61} +26.9229 q^{63} -2.47817 q^{65} +2.71154 q^{67} -0.985291 q^{69} -11.7082 q^{71} +9.47027 q^{73} +15.6560 q^{75} +13.0752 q^{77} -1.00000 q^{79} +34.6818 q^{81} +11.0452 q^{83} +1.19015 q^{85} +23.3236 q^{87} -7.66115 q^{89} +13.8657 q^{91} +6.30032 q^{93} -0.581328 q^{95} -18.9946 q^{97} +33.2739 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) $ 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9	$ 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9} + 3 q^{11} - 5 q^{13} - 11 q^{15} - 17 q^{17} + 27 q^{19} - 28 q^{21} - 11 q^{23} + 13 q^{25} - 7 q^{27} - 39 q^{29} - 27 q^{31} - 18 q^{33} - 5 q^{35} - q^{37} - 22 q^{39} - 36 q^{41} - 2 q^{43} - 18 q^{45} - 12 q^{47} + 15 q^{49} + 4 q^{51} - 28 q^{53} + 5 q^{55} - 4 q^{57} - 30 q^{59} - 6 q^{61} - 4 q^{63} - 32 q^{65} + 13 q^{67} - 27 q^{69} - 59 q^{71} - 30 q^{73} - 21 q^{75} - 39 q^{77} - 27 q^{79} - 5 q^{81} + 4 q^{83} - 3 q^{85} + 22 q^{87} - 56 q^{89} - 8 q^{91} - 38 q^{93} - 10 q^{95} - 30 q^{97} + 31 q^{99}+O(q^{100}) $ 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9 + 3 * q^11 - 5 * q^13 - 11 * q^15 - 17 * q^17 + 27 * q^19 - 28 * q^21 - 11 * q^23 + 13 * q^25 - 7 * q^27 - 39 * q^29 - 27 * q^31 - 18 * q^33 - 5 * q^35 - q^37 - 22 * q^39 - 36 * q^41 - 2 * q^43 - 18 * q^45 - 12 * q^47 + 15 * q^49 + 4 * q^51 - 28 * q^53 + 5 * q^55 - 4 * q^57 - 30 * q^59 - 6 * q^61 - 4 * q^63 - 32 * q^65 + 13 * q^67 - 27 * q^69 - 59 * q^71 - 30 * q^73 - 21 * q^75 - 39 * q^77 - 27 * q^79 - 5 * q^81 + 4 * q^83 - 3 * q^85 + 22 * q^87 - 56 * q^89 - 8 * q^91 - 38 * q^93 - 10 * q^95 - 30 * q^97 + 31 * 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
$3$	−3.35817	−1.93884	−0.969420	−	0.245407i	$-0.921078\pi$
$3$	−3.35817	−1.93884	−0.969420	+	0.245407i	$0.921078\pi$
$4$	0	0
$5$	−0.581328	−0.259978	−0.129989	−	0.991515i	$-0.541494\pi$
$5$	−0.581328	−0.259978	−0.129989	+	0.991515i	$0.541494\pi$
$6$	0	0
$7$	3.25261	1.22937	0.614686	−	0.788772i	$-0.289283\pi$
$7$	3.25261	1.22937	0.614686	+	0.788772i	$0.289283\pi$
$8$	0	0
$9$	8.27730	2.75910
$10$	0	0
$11$	4.01989	1.21204	0.606022	−	0.795448i	$-0.292764\pi$
$11$	4.01989	1.21204	0.606022	+	0.795448i	$0.292764\pi$
$12$	0	0
$13$	4.26295	1.18233	0.591164	−	0.806551i	$-0.298669\pi$
$13$	4.26295	1.18233	0.591164	+	0.806551i	$0.298669\pi$
$14$	0	0
$15$	1.95220	0.504055
$16$	0	0
$17$	−2.04729	−0.496540	−0.248270	−	0.968691i	$-0.579862\pi$
$17$	−2.04729	−0.496540	−0.248270	+	0.968691i	$0.579862\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−10.9228	−2.38356
$22$	0	0
$23$	0.293401	0.0611784	0.0305892	−	0.999532i	$-0.490262\pi$
$23$	0.293401	0.0611784	0.0305892	+	0.999532i	$0.490262\pi$
$24$	0	0
$25$	−4.66206	−0.932412
$26$	0	0
$27$	−17.7221	−3.41062
$28$	0	0
$29$	−6.94533	−1.28972	−0.644858	−	0.764302i	$-0.723084\pi$
$29$	−6.94533	−1.28972	−0.644858	+	0.764302i	$0.723084\pi$
$30$	0	0
$31$	−1.87612	−0.336961	−0.168480	−	0.985705i	$-0.553886\pi$
$31$	−1.87612	−0.336961	−0.168480	+	0.985705i	$0.553886\pi$
$32$	0	0
$33$	−13.4995	−2.34996
$34$	0	0
$35$	−1.89084	−0.319610
$36$	0	0
$37$	8.28320	1.36175	0.680875	−	0.732400i	$-0.261600\pi$
$37$	8.28320	1.36175	0.680875	+	0.732400i	$0.261600\pi$
$38$	0	0
$39$	−14.3157	−2.29235
$40$	0	0
$41$	−6.21102	−0.969999	−0.484999	−	0.874514i	$-0.661180\pi$
$41$	−6.21102	−0.969999	−0.484999	+	0.874514i	$0.661180\pi$
$42$	0	0
$43$	−9.11192	−1.38956	−0.694778	−	0.719225i	$-0.744497\pi$
$43$	−9.11192	−1.38956	−0.694778	+	0.719225i	$0.744497\pi$
$44$	0	0
$45$	−4.81183	−0.717305
$46$	0	0
$47$	−8.53931	−1.24559	−0.622794	−	0.782386i	$-0.714002\pi$
$47$	−8.53931	−1.24559	−0.622794	+	0.782386i	$0.714002\pi$
$48$	0	0
$49$	3.57949	0.511356
$50$	0	0
$51$	6.87514	0.962712
$52$	0	0
$53$	−14.3146	−1.96626	−0.983132	−	0.182899i	$-0.941452\pi$
$53$	−14.3146	−1.96626	−0.983132	+	0.182899i	$0.941452\pi$
$54$	0	0
$55$	−2.33688	−0.315105
$56$	0	0
$57$	−3.35817	−0.444800
$58$	0	0
$59$	3.68459	0.479692	0.239846	−	0.970811i	$-0.422903\pi$
$59$	3.68459	0.479692	0.239846	+	0.970811i	$0.422903\pi$
$60$	0	0
$61$	1.48250	0.189815	0.0949075	−	0.995486i	$-0.469744\pi$
$61$	1.48250	0.189815	0.0949075	+	0.995486i	$0.469744\pi$
$62$	0	0
$63$	26.9229	3.39196
$64$	0	0
$65$	−2.47817	−0.307379
$66$	0	0
$67$	2.71154	0.331268	0.165634	−	0.986187i	$-0.447033\pi$
$67$	2.71154	0.331268	0.165634	+	0.986187i	$0.447033\pi$
$68$	0	0
$69$	−0.985291	−0.118615
$70$	0	0
$71$	−11.7082	−1.38951	−0.694757	−	0.719245i	$-0.744488\pi$
$71$	−11.7082	−1.38951	−0.694757	+	0.719245i	$0.744488\pi$
$72$	0	0
$73$	9.47027	1.10841	0.554206	−	0.832380i	$-0.313022\pi$
$73$	9.47027	1.10841	0.554206	+	0.832380i	$0.313022\pi$
$74$	0	0
$75$	15.6560	1.80780
$76$	0	0
$77$	13.0752	1.49005
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	0	0
$81$	34.6818	3.85354
$82$	0	0
$83$	11.0452	1.21237	0.606186	−	0.795323i	$-0.292699\pi$
$83$	11.0452	1.21237	0.606186	+	0.795323i	$0.292699\pi$
$84$	0	0
$85$	1.19015	0.129089
$86$	0	0
$87$	23.3236	2.50055
$88$	0	0
$89$	−7.66115	−0.812081	−0.406040	−	0.913855i	$-0.633091\pi$
$89$	−7.66115	−0.812081	−0.406040	+	0.913855i	$0.633091\pi$
$90$	0	0
$91$	13.8657	1.45352
$92$	0	0
$93$	6.30032	0.653313
$94$	0	0
$95$	−0.581328	−0.0596430
$96$	0	0
$97$	−18.9946	−1.92861	−0.964306	−	0.264789i	$-0.914698\pi$
$97$	−18.9946	−1.92861	−0.964306	+	0.264789i	$0.914698\pi$
$98$	0	0
$99$	33.2739	3.34415
$100$	0	0
$101$	−6.30058	−0.626931	−0.313466	−	0.949600i	$-0.601490\pi$
$101$	−6.30058	−0.626931	−0.313466	+	0.949600i	$0.601490\pi$
$102$	0	0
$103$	4.58596	0.451868	0.225934	−	0.974143i	$-0.427457\pi$
$103$	4.58596	0.451868	0.225934	+	0.974143i	$0.427457\pi$
$104$	0	0
$105$	6.34975	0.619672
$106$	0	0
$107$	0.995018	0.0961920	0.0480960	−	0.998843i	$-0.484685\pi$
$107$	0.995018	0.0961920	0.0480960	+	0.998843i	$0.484685\pi$
$108$	0	0
$109$	−14.8584	−1.42318	−0.711588	−	0.702597i	$-0.752024\pi$
$109$	−14.8584	−1.42318	−0.711588	+	0.702597i	$0.752024\pi$
$110$	0	0
$111$	−27.8164	−2.64021
$112$	0	0
$113$	−14.9859	−1.40976	−0.704879	−	0.709328i	$-0.748999\pi$
$113$	−14.9859	−1.40976	−0.704879	+	0.709328i	$0.748999\pi$
$114$	0	0
$115$	−0.170562	−0.0159050
$116$	0	0
$117$	35.2857	3.26216
$118$	0	0
$119$	−6.65903	−0.610433
$120$	0	0
$121$	5.15955	0.469050
$122$	0	0
$123$	20.8577	1.88067
$124$	0	0
$125$	5.61683	0.502384
$126$	0	0
$127$	−7.48987	−0.664618	−0.332309	−	0.943171i	$-0.607828\pi$
$127$	−7.48987	−0.664618	−0.332309	+	0.943171i	$0.607828\pi$
$128$	0	0
$129$	30.5994	2.69413
$130$	0	0
$131$	−2.86391	−0.250221	−0.125110	−	0.992143i	$-0.539929\pi$
$131$	−2.86391	−0.250221	−0.125110	+	0.992143i	$0.539929\pi$
$132$	0	0
$133$	3.25261	0.282037
$134$	0	0
$135$	10.3023	0.886684
$136$	0	0
$137$	−12.2269	−1.04461	−0.522306	−	0.852758i	$-0.674928\pi$
$137$	−12.2269	−1.04461	−0.522306	+	0.852758i	$0.674928\pi$
$138$	0	0
$139$	−22.1261	−1.87671	−0.938356	−	0.345671i	$-0.887651\pi$
$139$	−22.1261	−1.87671	−0.938356	+	0.345671i	$0.887651\pi$
$140$	0	0
$141$	28.6765	2.41499
$142$	0	0
$143$	17.1366	1.43303
$144$	0	0
$145$	4.03752	0.335298
$146$	0	0
$147$	−12.0205	−0.991438
$148$	0	0
$149$	−22.1134	−1.81160	−0.905800	−	0.423706i	$-0.860729\pi$
$149$	−22.1134	−1.81160	−0.905800	+	0.423706i	$0.860729\pi$
$150$	0	0
$151$	−8.12362	−0.661091	−0.330545	−	0.943790i	$-0.607233\pi$
$151$	−8.12362	−0.661091	−0.330545	+	0.943790i	$0.607233\pi$
$152$	0	0
$153$	−16.9460	−1.37000
$154$	0	0
$155$	1.09064	0.0876023
$156$	0	0
$157$	22.8376	1.82264	0.911318	−	0.411703i	$-0.135066\pi$
$157$	22.8376	1.82264	0.911318	+	0.411703i	$0.135066\pi$
$158$	0	0
$159$	48.0709	3.81227
$160$	0	0
$161$	0.954321	0.0752110
$162$	0	0
$163$	19.4277	1.52170	0.760848	−	0.648930i	$-0.224783\pi$
$163$	19.4277	1.52170	0.760848	+	0.648930i	$0.224783\pi$
$164$	0	0
$165$	7.84763	0.610937
$166$	0	0
$167$	15.4108	1.19253	0.596263	−	0.802789i	$-0.296652\pi$
$167$	15.4108	1.19253	0.596263	+	0.802789i	$0.296652\pi$
$168$	0	0
$169$	5.17270	0.397900
$170$	0	0
$171$	8.27730	0.632981
$172$	0	0
$173$	2.34862	0.178562	0.0892812	−	0.996006i	$-0.471543\pi$
$173$	2.34862	0.178562	0.0892812	+	0.996006i	$0.471543\pi$
$174$	0	0
$175$	−15.1639	−1.14628
$176$	0	0
$177$	−12.3735	−0.930047
$178$	0	0
$179$	3.42505	0.256000	0.128000	−	0.991774i	$-0.459144\pi$
$179$	3.42505	0.256000	0.128000	+	0.991774i	$0.459144\pi$
$180$	0	0
$181$	−0.166441	−0.0123714	−0.00618572	−	0.999981i	$-0.501969\pi$
$181$	−0.166441	−0.0123714	−0.00618572	+	0.999981i	$0.501969\pi$
$182$	0	0
$183$	−4.97849	−0.368021
$184$	0	0
$185$	−4.81526	−0.354025
$186$	0	0
$187$	−8.22988	−0.601828
$188$	0	0
$189$	−57.6431	−4.19292
$190$	0	0
$191$	0.245961	0.0177971	0.00889856	−	0.999960i	$-0.497167\pi$
$191$	0.245961	0.0177971	0.00889856	+	0.999960i	$0.497167\pi$
$192$	0	0
$193$	−8.68586	−0.625222	−0.312611	−	0.949881i	$-0.601204\pi$
$193$	−8.68586	−0.625222	−0.312611	+	0.949881i	$0.601204\pi$
$194$	0	0
$195$	8.32211	0.595959
$196$	0	0
$197$	−13.9003	−0.990354	−0.495177	−	0.868792i	$-0.664897\pi$
$197$	−13.9003	−0.990354	−0.495177	+	0.868792i	$0.664897\pi$
$198$	0	0
$199$	−4.00929	−0.284211	−0.142106	−	0.989852i	$-0.545387\pi$
$199$	−4.00929	−0.284211	−0.142106	+	0.989852i	$0.545387\pi$
$200$	0	0
$201$	−9.10582	−0.642275
$202$	0	0
$203$	−22.5905	−1.58554
$204$	0	0
$205$	3.61064	0.252178
$206$	0	0
$207$	2.42857	0.168797
$208$	0	0
$209$	4.01989	0.278062
$210$	0	0
$211$	5.53336	0.380932	0.190466	−	0.981694i	$-0.439000\pi$
$211$	5.53336	0.380932	0.190466	+	0.981694i	$0.439000\pi$
$212$	0	0
$213$	39.3183	2.69404
$214$	0	0
$215$	5.29702	0.361254
$216$	0	0
$217$	−6.10229	−0.414250
$218$	0	0
$219$	−31.8028	−2.14903
$220$	0	0
$221$	−8.72747	−0.587073
$222$	0	0
$223$	−19.1233	−1.28059	−0.640296	−	0.768128i	$-0.721188\pi$
$223$	−19.1233	−1.28059	−0.640296	+	0.768128i	$0.721188\pi$
$224$	0	0
$225$	−38.5893	−2.57262
$226$	0	0
$227$	22.0361	1.46259	0.731294	−	0.682062i	$-0.238917\pi$
$227$	22.0361	1.46259	0.731294	+	0.682062i	$0.238917\pi$
$228$	0	0
$229$	21.0639	1.39194	0.695969	−	0.718071i	$-0.254975\pi$
$229$	21.0639	1.39194	0.695969	+	0.718071i	$0.254975\pi$
$230$	0	0
$231$	−43.9086	−2.88897
$232$	0	0
$233$	−6.83099	−0.447513	−0.223756	−	0.974645i	$-0.571832\pi$
$233$	−6.83099	−0.447513	−0.223756	+	0.974645i	$0.571832\pi$
$234$	0	0
$235$	4.96414	0.323825
$236$	0	0
$237$	3.35817	0.218137
$238$	0	0
$239$	2.62776	0.169976	0.0849879	−	0.996382i	$-0.472915\pi$
$239$	2.62776	0.169976	0.0849879	+	0.996382i	$0.472915\pi$
$240$	0	0
$241$	−22.0823	−1.42245	−0.711223	−	0.702967i	$-0.751858\pi$
$241$	−22.0823	−1.42245	−0.711223	+	0.702967i	$0.751858\pi$
$242$	0	0
$243$	−63.3013	−4.06078
$244$	0	0
$245$	−2.08086	−0.132941
$246$	0	0
$247$	4.26295	0.271245
$248$	0	0
$249$	−37.0918	−2.35059
$250$	0	0
$251$	18.0587	1.13986	0.569928	−	0.821694i	$-0.306971\pi$
$251$	18.0587	1.13986	0.569928	+	0.821694i	$0.306971\pi$
$252$	0	0
$253$	1.17944	0.0741509
$254$	0	0
$255$	−3.99671	−0.250284
$256$	0	0
$257$	16.5241	1.03074	0.515372	−	0.856966i	$-0.327654\pi$
$257$	16.5241	1.03074	0.515372	+	0.856966i	$0.327654\pi$
$258$	0	0
$259$	26.9420	1.67410
$260$	0	0
$261$	−57.4886	−3.55846
$262$	0	0
$263$	0.570983	0.0352083	0.0176042	−	0.999845i	$-0.494396\pi$
$263$	0.570983	0.0352083	0.0176042	+	0.999845i	$0.494396\pi$
$264$	0	0
$265$	8.32149	0.511185
$266$	0	0
$267$	25.7275	1.57449
$268$	0	0
$269$	9.83599	0.599711	0.299856	−	0.953985i	$-0.403062\pi$
$269$	9.83599	0.599711	0.299856	+	0.953985i	$0.403062\pi$
$270$	0	0
$271$	20.4824	1.24422	0.622109	−	0.782931i	$-0.286276\pi$
$271$	20.4824	1.24422	0.622109	+	0.782931i	$0.286276\pi$
$272$	0	0
$273$	−46.5634	−2.81815
$274$	0	0
$275$	−18.7410	−1.13012
$276$	0	0
$277$	−3.56807	−0.214384	−0.107192	−	0.994238i	$-0.534186\pi$
$277$	−3.56807	−0.214384	−0.107192	+	0.994238i	$0.534186\pi$
$278$	0	0
$279$	−15.5292	−0.929709
$280$	0	0
$281$	10.0631	0.600315	0.300157	−	0.953890i	$-0.402961\pi$
$281$	10.0631	0.600315	0.300157	+	0.953890i	$0.402961\pi$
$282$	0	0
$283$	4.08499	0.242828	0.121414	−	0.992602i	$-0.461257\pi$
$283$	4.08499	0.242828	0.121414	+	0.992602i	$0.461257\pi$
$284$	0	0
$285$	1.95220	0.115638
$286$	0	0
$287$	−20.2021	−1.19249
$288$	0	0
$289$	−12.8086	−0.753448
$290$	0	0
$291$	63.7872	3.73927
$292$	0	0
$293$	−8.37505	−0.489276	−0.244638	−	0.969614i	$-0.578669\pi$
$293$	−8.37505	−0.489276	−0.244638	+	0.969614i	$0.578669\pi$
$294$	0	0
$295$	−2.14195	−0.124709
$296$	0	0
$297$	−71.2409	−4.13382
$298$	0	0
$299$	1.25075	0.0723329
$300$	0	0
$301$	−29.6376	−1.70828
$302$	0	0
$303$	21.1584	1.21552
$304$	0	0
$305$	−0.861820	−0.0493477
$306$	0	0
$307$	11.4283	0.652250	0.326125	−	0.945327i	$-0.394257\pi$
$307$	11.4283	0.652250	0.326125	+	0.945327i	$0.394257\pi$
$308$	0	0
$309$	−15.4004	−0.876100
$310$	0	0
$311$	−2.27135	−0.128797	−0.0643983	−	0.997924i	$-0.520513\pi$
$311$	−2.27135	−0.128797	−0.0643983	+	0.997924i	$0.520513\pi$
$312$	0	0
$313$	−23.4656	−1.32636	−0.663178	−	0.748462i	$-0.730793\pi$
$313$	−23.4656	−1.32636	−0.663178	+	0.748462i	$0.730793\pi$
$314$	0	0
$315$	−15.6510	−0.881835
$316$	0	0
$317$	0.250367	0.0140620	0.00703101	−	0.999975i	$-0.497762\pi$
$317$	0.250367	0.0140620	0.00703101	+	0.999975i	$0.497762\pi$
$318$	0	0
$319$	−27.9195	−1.56319
$320$	0	0
$321$	−3.34144	−0.186501
$322$	0	0
$323$	−2.04729	−0.113914
$324$	0	0
$325$	−19.8741	−1.10242
$326$	0	0
$327$	49.8970	2.75931
$328$	0	0
$329$	−27.7751	−1.53129
$330$	0	0
$331$	−25.3484	−1.39328	−0.696638	−	0.717423i	$-0.745322\pi$
$331$	−25.3484	−1.39328	−0.696638	+	0.717423i	$0.745322\pi$
$332$	0	0
$333$	68.5625	3.75720
$334$	0	0
$335$	−1.57630	−0.0861223
$336$	0	0
$337$	−1.98944	−0.108372	−0.0541859	−	0.998531i	$-0.517256\pi$
$337$	−1.98944	−0.108372	−0.0541859	+	0.998531i	$0.517256\pi$
$338$	0	0
$339$	50.3253	2.73330
$340$	0	0
$341$	−7.54180	−0.408411
$342$	0	0
$343$	−11.1256	−0.600725
$344$	0	0
$345$	0.572778	0.0308373
$346$	0	0
$347$	−13.0131	−0.698578	−0.349289	−	0.937015i	$-0.613577\pi$
$347$	−13.0131	−0.698578	−0.349289	+	0.937015i	$0.613577\pi$
$348$	0	0
$349$	−32.6639	−1.74846	−0.874228	−	0.485515i	$-0.838632\pi$
$349$	−32.6639	−1.74846	−0.874228	+	0.485515i	$0.838632\pi$
$350$	0	0
$351$	−75.5483	−4.03247
$352$	0	0
$353$	−17.4045	−0.926346	−0.463173	−	0.886268i	$-0.653289\pi$
$353$	−17.4045	−0.926346	−0.463173	+	0.886268i	$0.653289\pi$
$354$	0	0
$355$	6.80633	0.361243
$356$	0	0
$357$	22.3622	1.18353
$358$	0	0
$359$	16.5104	0.871385	0.435693	−	0.900096i	$-0.356504\pi$
$359$	16.5104	0.871385	0.435693	+	0.900096i	$0.356504\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−17.3267	−0.909413
$364$	0	0
$365$	−5.50533	−0.288162
$366$	0	0
$367$	26.4740	1.38193	0.690966	−	0.722887i	$-0.257185\pi$
$367$	26.4740	1.38193	0.690966	+	0.722887i	$0.257185\pi$
$368$	0	0
$369$	−51.4105	−2.67632
$370$	0	0
$371$	−46.5599	−2.41727
$372$	0	0
$373$	30.1533	1.56128	0.780639	−	0.624982i	$-0.214894\pi$
$373$	30.1533	1.56128	0.780639	+	0.624982i	$0.214894\pi$
$374$	0	0
$375$	−18.8623	−0.974043
$376$	0	0
$377$	−29.6076	−1.52487
$378$	0	0
$379$	22.5182	1.15668	0.578341	−	0.815795i	$-0.303700\pi$
$379$	22.5182	1.15668	0.578341	+	0.815795i	$0.303700\pi$
$380$	0	0
$381$	25.1522	1.28859
$382$	0	0
$383$	−2.04476	−0.104482	−0.0522411	−	0.998634i	$-0.516636\pi$
$383$	−2.04476	−0.104482	−0.0522411	+	0.998634i	$0.516636\pi$
$384$	0	0
$385$	−7.60096	−0.387381
$386$	0	0
$387$	−75.4222	−3.83392
$388$	0	0
$389$	−16.0989	−0.816246	−0.408123	−	0.912927i	$-0.633816\pi$
$389$	−16.0989	−0.816246	−0.408123	+	0.912927i	$0.633816\pi$
$390$	0	0
$391$	−0.600677	−0.0303775
$392$	0	0
$393$	9.61749	0.485138
$394$	0	0
$395$	0.581328	0.0292498
$396$	0	0
$397$	−1.29730	−0.0651098	−0.0325549	−	0.999470i	$-0.510364\pi$
$397$	−1.29730	−0.0651098	−0.0325549	+	0.999470i	$0.510364\pi$
$398$	0	0
$399$	−10.9228	−0.546825
$400$	0	0
$401$	−18.4883	−0.923263	−0.461631	−	0.887072i	$-0.652736\pi$
$401$	−18.4883	−0.923263	−0.461631	+	0.887072i	$0.652736\pi$
$402$	0	0
$403$	−7.99779	−0.398398
$404$	0	0
$405$	−20.1615	−1.00183
$406$	0	0
$407$	33.2976	1.65050
$408$	0	0
$409$	19.4062	0.959577	0.479788	−	0.877384i	$-0.340713\pi$
$409$	19.4062	0.959577	0.479788	+	0.877384i	$0.340713\pi$
$410$	0	0
$411$	41.0599	2.02533
$412$	0	0
$413$	11.9845	0.589720
$414$	0	0
$415$	−6.42090	−0.315190
$416$	0	0
$417$	74.3032	3.63864
$418$	0	0
$419$	35.2757	1.72333	0.861666	−	0.507476i	$-0.169421\pi$
$419$	35.2757	1.72333	0.861666	+	0.507476i	$0.169421\pi$
$420$	0	0
$421$	−19.4877	−0.949774	−0.474887	−	0.880047i	$-0.657511\pi$
$421$	−19.4877	−0.949774	−0.474887	+	0.880047i	$0.657511\pi$
$422$	0	0
$423$	−70.6825	−3.43670
$424$	0	0
$425$	9.54457	0.462980
$426$	0	0
$427$	4.82201	0.233353
$428$	0	0
$429$	−57.5476	−2.77842
$430$	0	0
$431$	−22.9227	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$431$	−22.9227	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$432$	0	0
$433$	28.3208	1.36101	0.680505	−	0.732744i	$-0.261761\pi$
$433$	28.3208	1.36101	0.680505	+	0.732744i	$0.261761\pi$
$434$	0	0
$435$	−13.5587	−0.650088
$436$	0	0
$437$	0.293401	0.0140353
$438$	0	0
$439$	28.0353	1.33805	0.669027	−	0.743238i	$-0.266711\pi$
$439$	28.0353	1.33805	0.669027	+	0.743238i	$0.266711\pi$
$440$	0	0
$441$	29.6286	1.41088
$442$	0	0
$443$	−1.77220	−0.0841997	−0.0420999	−	0.999113i	$-0.513405\pi$
$443$	−1.77220	−0.0841997	−0.0420999	+	0.999113i	$0.513405\pi$
$444$	0	0
$445$	4.45364	0.211123
$446$	0	0
$447$	74.2605	3.51240
$448$	0	0
$449$	3.60125	0.169953	0.0849767	−	0.996383i	$-0.472918\pi$
$449$	3.60125	0.169953	0.0849767	+	0.996383i	$0.472918\pi$
$450$	0	0
$451$	−24.9677	−1.17568
$452$	0	0
$453$	27.2805	1.28175
$454$	0	0
$455$	−8.06053	−0.377883
$456$	0	0
$457$	15.1957	0.710827	0.355413	−	0.934709i	$-0.384340\pi$
$457$	15.1957	0.710827	0.355413	+	0.934709i	$0.384340\pi$
$458$	0	0
$459$	36.2822	1.69351
$460$	0	0
$461$	29.9458	1.39471	0.697357	−	0.716724i	$-0.254359\pi$
$461$	29.9458	1.39471	0.697357	+	0.716724i	$0.254359\pi$
$462$	0	0
$463$	−7.90812	−0.367521	−0.183761	−	0.982971i	$-0.558827\pi$
$463$	−7.90812	−0.367521	−0.183761	+	0.982971i	$0.558827\pi$
$464$	0	0
$465$	−3.66255	−0.169847
$466$	0	0
$467$	14.0952	0.652250	0.326125	−	0.945327i	$-0.394257\pi$
$467$	14.0952	0.652250	0.326125	+	0.945327i	$0.394257\pi$
$468$	0	0
$469$	8.81960	0.407251
$470$	0	0
$471$	−76.6924	−3.53380
$472$	0	0
$473$	−36.6290	−1.68420
$474$	0	0
$475$	−4.66206	−0.213910
$476$	0	0
$477$	−118.486	−5.42512
$478$	0	0
$479$	−8.80418	−0.402273	−0.201137	−	0.979563i	$-0.564463\pi$
$479$	−8.80418	−0.402273	−0.201137	+	0.979563i	$0.564463\pi$
$480$	0	0
$481$	35.3108	1.61003
$482$	0	0
$483$	−3.20477	−0.145822
$484$	0	0
$485$	11.0421	0.501397
$486$	0	0
$487$	−12.7351	−0.577083	−0.288541	−	0.957467i	$-0.593170\pi$
$487$	−12.7351	−0.577083	−0.288541	+	0.957467i	$0.593170\pi$
$488$	0	0
$489$	−65.2416	−2.95033
$490$	0	0
$491$	−25.4661	−1.14927	−0.574635	−	0.818410i	$-0.694856\pi$
$491$	−25.4661	−1.14927	−0.574635	+	0.818410i	$0.694856\pi$
$492$	0	0
$493$	14.2191	0.640396
$494$	0	0
$495$	−19.3430	−0.869405
$496$	0	0
$497$	−38.0824	−1.70823
$498$	0	0
$499$	1.05715	0.0473246	0.0236623	−	0.999720i	$-0.492467\pi$
$499$	1.05715	0.0473246	0.0236623	+	0.999720i	$0.492467\pi$
$500$	0	0
$501$	−51.7522	−2.31212
$502$	0	0
$503$	34.7788	1.55071	0.775355	−	0.631525i	$-0.217571\pi$
$503$	34.7788	1.55071	0.775355	+	0.631525i	$0.217571\pi$
$504$	0	0
$505$	3.66270	0.162988
$506$	0	0
$507$	−17.3708	−0.771465
$508$	0	0
$509$	−30.8250	−1.36629	−0.683146	−	0.730282i	$-0.739389\pi$
$509$	−30.8250	−1.36629	−0.683146	+	0.730282i	$0.739389\pi$
$510$	0	0
$511$	30.8031	1.36265
$512$	0	0
$513$	−17.7221	−0.782449
$514$	0	0
$515$	−2.66595	−0.117476
$516$	0	0
$517$	−34.3271	−1.50971
$518$	0	0
$519$	−7.88707	−0.346204
$520$	0	0
$521$	−6.82843	−0.299159	−0.149579	−	0.988750i	$-0.547792\pi$
$521$	−6.82843	−0.299159	−0.149579	+	0.988750i	$0.547792\pi$
$522$	0	0
$523$	−18.1587	−0.794026	−0.397013	−	0.917813i	$-0.629953\pi$
$523$	−18.1587	−0.794026	−0.397013	+	0.917813i	$0.629953\pi$
$524$	0	0
$525$	50.9229	2.22246
$526$	0	0
$527$	3.84095	0.167314
$528$	0	0
$529$	−22.9139	−0.996257
$530$	0	0
$531$	30.4984	1.32352
$532$	0	0
$533$	−26.4773	−1.14686
$534$	0	0
$535$	−0.578432	−0.0250078
$536$	0	0
$537$	−11.5019	−0.496343
$538$	0	0
$539$	14.3892	0.619786
$540$	0	0
$541$	−7.64478	−0.328675	−0.164337	−	0.986404i	$-0.552549\pi$
$541$	−7.64478	−0.328675	−0.164337	+	0.986404i	$0.552549\pi$
$542$	0	0
$543$	0.558936	0.0239862
$544$	0	0
$545$	8.63760	0.369994
$546$	0	0
$547$	44.8156	1.91618	0.958089	−	0.286471i	$-0.0924822\pi$
$547$	44.8156	1.91618	0.958089	+	0.286471i	$0.0924822\pi$
$548$	0	0
$549$	12.2711	0.523719
$550$	0	0
$551$	−6.94533	−0.295881
$552$	0	0
$553$	−3.25261	−0.138315
$554$	0	0
$555$	16.1704	0.686397
$556$	0	0
$557$	33.5445	1.42133	0.710663	−	0.703532i	$-0.248395\pi$
$557$	33.5445	1.42133	0.710663	+	0.703532i	$0.248395\pi$
$558$	0	0
$559$	−38.8436	−1.64291
$560$	0	0
$561$	27.6373	1.16685
$562$	0	0
$563$	−7.31636	−0.308348	−0.154174	−	0.988044i	$-0.549272\pi$
$563$	−7.31636	−0.308348	−0.154174	+	0.988044i	$0.549272\pi$
$564$	0	0
$565$	8.71174	0.366506
$566$	0	0
$567$	112.807	4.73743
$568$	0	0
$569$	−16.6777	−0.699167	−0.349584	−	0.936905i	$-0.613677\pi$
$569$	−16.6777	−0.699167	−0.349584	+	0.936905i	$0.613677\pi$
$570$	0	0
$571$	41.8203	1.75012	0.875062	−	0.484010i	$-0.160820\pi$
$571$	41.8203	1.75012	0.875062	+	0.484010i	$0.160820\pi$
$572$	0	0
$573$	−0.825979	−0.0345058
$574$	0	0
$575$	−1.36785	−0.0570434
$576$	0	0
$577$	34.0377	1.41701	0.708504	−	0.705706i	$-0.249370\pi$
$577$	34.0377	1.41701	0.708504	+	0.705706i	$0.249370\pi$
$578$	0	0
$579$	29.1686	1.21220
$580$	0	0
$581$	35.9259	1.49046
$582$	0	0
$583$	−57.5432	−2.38320
$584$	0	0
$585$	−20.5126	−0.848090
$586$	0	0
$587$	0.643931	0.0265779	0.0132889	−	0.999912i	$-0.495770\pi$
$587$	0.643931	0.0265779	0.0132889	+	0.999912i	$0.495770\pi$
$588$	0	0
$589$	−1.87612	−0.0773041
$590$	0	0
$591$	46.6795	1.92014
$592$	0	0
$593$	−13.6058	−0.558722	−0.279361	−	0.960186i	$-0.590123\pi$
$593$	−13.6058	−0.558722	−0.279361	+	0.960186i	$0.590123\pi$
$594$	0	0
$595$	3.87108	0.158699
$596$	0	0
$597$	13.4639	0.551040
$598$	0	0
$599$	−40.4635	−1.65330	−0.826648	−	0.562720i	$-0.809755\pi$
$599$	−40.4635	−1.65330	−0.826648	+	0.562720i	$0.809755\pi$
$600$	0	0
$601$	5.98704	0.244217	0.122108	−	0.992517i	$-0.461034\pi$
$601$	5.98704	0.244217	0.122108	+	0.992517i	$0.461034\pi$
$602$	0	0
$603$	22.4443	0.914001
$604$	0	0
$605$	−2.99939	−0.121943
$606$	0	0
$607$	−25.8342	−1.04858	−0.524289	−	0.851540i	$-0.675669\pi$
$607$	−25.8342	−1.04858	−0.524289	+	0.851540i	$0.675669\pi$
$608$	0	0
$609$	75.8627	3.07411
$610$	0	0
$611$	−36.4026	−1.47269
$612$	0	0
$613$	18.7495	0.757287	0.378643	−	0.925543i	$-0.376391\pi$
$613$	18.7495	0.757287	0.378643	+	0.925543i	$0.376391\pi$
$614$	0	0
$615$	−12.1251	−0.488933
$616$	0	0
$617$	32.3475	1.30226	0.651131	−	0.758965i	$-0.274295\pi$
$617$	32.3475	1.30226	0.651131	+	0.758965i	$0.274295\pi$
$618$	0	0
$619$	−21.5896	−0.867761	−0.433881	−	0.900970i	$-0.642856\pi$
$619$	−21.5896	−0.867761	−0.433881	+	0.900970i	$0.642856\pi$
$620$	0	0
$621$	−5.19968	−0.208656
$622$	0	0
$623$	−24.9188	−0.998349
$624$	0	0
$625$	20.0451	0.801803
$626$	0	0
$627$	−13.4995	−0.539118
$628$	0	0
$629$	−16.9581	−0.676163
$630$	0	0
$631$	46.3066	1.84344	0.921718	−	0.387862i	$-0.126786\pi$
$631$	46.3066	1.84344	0.921718	+	0.387862i	$0.126786\pi$
$632$	0	0
$633$	−18.5820	−0.738567
$634$	0	0
$635$	4.35407	0.172786
$636$	0	0
$637$	15.2592	0.604591
$638$	0	0
$639$	−96.9127	−3.83381
$640$	0	0
$641$	41.0229	1.62031	0.810154	−	0.586217i	$-0.199383\pi$
$641$	41.0229	1.62031	0.810154	+	0.586217i	$0.199383\pi$
$642$	0	0
$643$	−9.45471	−0.372857	−0.186429	−	0.982468i	$-0.559691\pi$
$643$	−9.45471	−0.372857	−0.186429	+	0.982468i	$0.559691\pi$
$644$	0	0
$645$	−17.7883	−0.700413
$646$	0	0
$647$	−12.9860	−0.510534	−0.255267	−	0.966871i	$-0.582163\pi$
$647$	−12.9860	−0.510534	−0.255267	+	0.966871i	$0.582163\pi$
$648$	0	0
$649$	14.8116	0.581408
$650$	0	0
$651$	20.4925	0.803165
$652$	0	0
$653$	−40.9198	−1.60132	−0.800659	−	0.599121i	$-0.795517\pi$
$653$	−40.9198	−1.60132	−0.800659	+	0.599121i	$0.795517\pi$
$654$	0	0
$655$	1.66487	0.0650519
$656$	0	0
$657$	78.3883	3.05822
$658$	0	0
$659$	22.0483	0.858879	0.429439	−	0.903096i	$-0.358711\pi$
$659$	22.0483	0.858879	0.429439	+	0.903096i	$0.358711\pi$
$660$	0	0
$661$	−12.5805	−0.489325	−0.244663	−	0.969608i	$-0.578677\pi$
$661$	−12.5805	−0.489325	−0.244663	+	0.969608i	$0.578677\pi$
$662$	0	0
$663$	29.3083	1.13824
$664$	0	0
$665$	−1.89084	−0.0733235
$666$	0	0
$667$	−2.03777	−0.0789028
$668$	0	0
$669$	64.2194	2.48286
$670$	0	0
$671$	5.95950	0.230064
$672$	0	0
$673$	19.3975	0.747719	0.373860	−	0.927485i	$-0.378034\pi$
$673$	19.3975	0.747719	0.373860	+	0.927485i	$0.378034\pi$
$674$	0	0
$675$	82.6214	3.18010
$676$	0	0
$677$	−46.4270	−1.78434	−0.892168	−	0.451704i	$-0.850816\pi$
$677$	−46.4270	−1.78434	−0.892168	+	0.451704i	$0.850816\pi$
$678$	0	0
$679$	−61.7822	−2.37098
$680$	0	0
$681$	−74.0010	−2.83573
$682$	0	0
$683$	−28.3473	−1.08468	−0.542339	−	0.840160i	$-0.682461\pi$
$683$	−28.3473	−1.08468	−0.542339	+	0.840160i	$0.682461\pi$
$684$	0	0
$685$	7.10782	0.271576
$686$	0	0
$687$	−70.7360	−2.69875
$688$	0	0
$689$	−61.0224	−2.32477
$690$	0	0
$691$	12.2941	0.467689	0.233844	−	0.972274i	$-0.424869\pi$
$691$	12.2941	0.467689	0.233844	+	0.972274i	$0.424869\pi$
$692$	0	0
$693$	108.227	4.11121
$694$	0	0
$695$	12.8625	0.487903
$696$	0	0
$697$	12.7157	0.481643
$698$	0	0
$699$	22.9396	0.867656
$700$	0	0
$701$	35.2348	1.33080	0.665400	−	0.746487i	$-0.268261\pi$
$701$	35.2348	1.33080	0.665400	+	0.746487i	$0.268261\pi$
$702$	0	0
$703$	8.28320	0.312407
$704$	0	0
$705$	−16.6704	−0.627845
$706$	0	0
$707$	−20.4933	−0.770732
$708$	0	0
$709$	23.9774	0.900490	0.450245	−	0.892905i	$-0.351337\pi$
$709$	23.9774	0.900490	0.450245	+	0.892905i	$0.351337\pi$
$710$	0	0
$711$	−8.27730	−0.310423
$712$	0	0
$713$	−0.550455	−0.0206147
$714$	0	0
$715$	−9.96198	−0.372557
$716$	0	0
$717$	−8.82447	−0.329556
$718$	0	0
$719$	−4.45710	−0.166222	−0.0831108	−	0.996540i	$-0.526486\pi$
$719$	−4.45710	−0.166222	−0.0831108	+	0.996540i	$0.526486\pi$
$720$	0	0
$721$	14.9164	0.555514
$722$	0	0
$723$	74.1561	2.75789
$724$	0	0
$725$	32.3795	1.20255
$726$	0	0
$727$	−42.5483	−1.57803	−0.789015	−	0.614374i	$-0.789408\pi$
$727$	−42.5483	−1.57803	−0.789015	+	0.614374i	$0.789408\pi$
$728$	0	0
$729$	108.531	4.01966
$730$	0	0
$731$	18.6547	0.689970
$732$	0	0
$733$	−8.70291	−0.321449	−0.160725	−	0.986999i	$-0.551383\pi$
$733$	−8.70291	−0.321449	−0.160725	+	0.986999i	$0.551383\pi$
$734$	0	0
$735$	6.98788	0.257752
$736$	0	0
$737$	10.9001	0.401511
$738$	0	0
$739$	−12.9269	−0.475523	−0.237761	−	0.971324i	$-0.576414\pi$
$739$	−12.9269	−0.475523	−0.237761	+	0.971324i	$0.576414\pi$
$740$	0	0
$741$	−14.3157	−0.525900
$742$	0	0
$743$	15.4438	0.566579	0.283290	−	0.959034i	$-0.408574\pi$
$743$	15.4438	0.566579	0.283290	+	0.959034i	$0.408574\pi$
$744$	0	0
$745$	12.8551	0.470976
$746$	0	0
$747$	91.4247	3.34506
$748$	0	0
$749$	3.23641	0.118256
$750$	0	0
$751$	−6.20609	−0.226463	−0.113232	−	0.993569i	$-0.536120\pi$
$751$	−6.20609	−0.226463	−0.113232	+	0.993569i	$0.536120\pi$
$752$	0	0
$753$	−60.6443	−2.21000
$754$	0	0
$755$	4.72249	0.171869
$756$	0	0
$757$	45.9623	1.67053	0.835265	−	0.549848i	$-0.185314\pi$
$757$	45.9623	1.67053	0.835265	+	0.549848i	$0.185314\pi$
$758$	0	0
$759$	−3.96077	−0.143767
$760$	0	0
$761$	−44.9795	−1.63050	−0.815252	−	0.579106i	$-0.803402\pi$
$761$	−44.9795	−1.63050	−0.815252	+	0.579106i	$0.803402\pi$
$762$	0	0
$763$	−48.3286	−1.74961
$764$	0	0
$765$	9.85119	0.356171
$766$	0	0
$767$	15.7072	0.567154
$768$	0	0
$769$	17.8468	0.643573	0.321787	−	0.946812i	$-0.395717\pi$
$769$	17.8468	0.643573	0.321787	+	0.946812i	$0.395717\pi$
$770$	0	0
$771$	−55.4907	−1.99845
$772$	0	0
$773$	16.5750	0.596161	0.298080	−	0.954541i	$-0.403654\pi$
$773$	16.5750	0.596161	0.298080	+	0.954541i	$0.403654\pi$
$774$	0	0
$775$	8.74657	0.314186
$776$	0	0
$777$	−90.4759	−3.24581
$778$	0	0
$779$	−6.21102	−0.222533
$780$	0	0
$781$	−47.0659	−1.68415
$782$	0	0
$783$	123.086	4.39873
$784$	0	0
$785$	−13.2761	−0.473845
$786$	0	0
$787$	−7.41270	−0.264234	−0.132117	−	0.991234i	$-0.542178\pi$
$787$	−7.41270	−0.264234	−0.132117	+	0.991234i	$0.542178\pi$
$788$	0	0
$789$	−1.91746	−0.0682633
$790$	0	0
$791$	−48.7434	−1.73312
$792$	0	0
$793$	6.31982	0.224424
$794$	0	0
$795$	−27.9450	−0.991106
$796$	0	0
$797$	19.6949	0.697628	0.348814	−	0.937192i	$-0.386584\pi$
$797$	19.6949	0.697628	0.348814	+	0.937192i	$0.386584\pi$
$798$	0	0
$799$	17.4824	0.618484
$800$	0	0
$801$	−63.4137	−2.24061
$802$	0	0
$803$	38.0695	1.34344
$804$	0	0
$805$	−0.554774	−0.0195532
$806$	0	0
$807$	−33.0309	−1.16274
$808$	0	0
$809$	−1.74969	−0.0615158	−0.0307579	−	0.999527i	$-0.509792\pi$
$809$	−1.74969	−0.0615158	−0.0307579	+	0.999527i	$0.509792\pi$
$810$	0	0
$811$	−46.2599	−1.62441	−0.812203	−	0.583375i	$-0.801732\pi$
$811$	−46.2599	−1.62441	−0.812203	+	0.583375i	$0.801732\pi$
$812$	0	0
$813$	−68.7834	−2.41234
$814$	0	0
$815$	−11.2939	−0.395607
$816$	0	0
$817$	−9.11192	−0.318786
$818$	0	0
$819$	114.771	4.01041
$820$	0	0
$821$	−21.7094	−0.757663	−0.378832	−	0.925466i	$-0.623674\pi$
$821$	−21.7094	−0.757663	−0.378832	+	0.925466i	$0.623674\pi$
$822$	0	0
$823$	2.56169	0.0892950	0.0446475	−	0.999003i	$-0.485784\pi$
$823$	2.56169	0.0892950	0.0446475	+	0.999003i	$0.485784\pi$
$824$	0	0
$825$	62.9354	2.19113
$826$	0	0
$827$	2.20009	0.0765045	0.0382522	−	0.999268i	$-0.487821\pi$
$827$	2.20009	0.0765045	0.0382522	+	0.999268i	$0.487821\pi$
$828$	0	0
$829$	13.1400	0.456370	0.228185	−	0.973618i	$-0.426721\pi$
$829$	13.1400	0.456370	0.228185	+	0.973618i	$0.426721\pi$
$830$	0	0
$831$	11.9822	0.415657
$832$	0	0
$833$	−7.32825	−0.253909
$834$	0	0
$835$	−8.95875	−0.310030
$836$	0	0
$837$	33.2487	1.14924
$838$	0	0
$839$	−31.1082	−1.07397	−0.536986	−	0.843591i	$-0.680437\pi$
$839$	−31.1082	−1.07397	−0.536986	+	0.843591i	$0.680437\pi$
$840$	0	0
$841$	19.2377	0.663368
$842$	0	0
$843$	−33.7936	−1.16391
$844$	0	0
$845$	−3.00704	−0.103445
$846$	0	0
$847$	16.7820	0.576637
$848$	0	0
$849$	−13.7181	−0.470804
$850$	0	0
$851$	2.43030	0.0833096
$852$	0	0
$853$	30.4940	1.04409	0.522047	−	0.852917i	$-0.325169\pi$
$853$	30.4940	1.04409	0.522047	+	0.852917i	$0.325169\pi$
$854$	0	0
$855$	−4.81183	−0.164561
$856$	0	0
$857$	48.1544	1.64492	0.822461	−	0.568821i	$-0.192600\pi$
$857$	48.1544	1.64492	0.822461	+	0.568821i	$0.192600\pi$
$858$	0	0
$859$	−21.5553	−0.735459	−0.367730	−	0.929933i	$-0.619865\pi$
$859$	−21.5553	−0.735459	−0.367730	+	0.929933i	$0.619865\pi$
$860$	0	0
$861$	67.8419	2.31205
$862$	0	0
$863$	−9.52724	−0.324311	−0.162156	−	0.986765i	$-0.551845\pi$
$863$	−9.52724	−0.324311	−0.162156	+	0.986765i	$0.551845\pi$
$864$	0	0
$865$	−1.36532	−0.0464222
$866$	0	0
$867$	43.0135	1.46082
$868$	0	0
$869$	−4.01989	−0.136366
$870$	0	0
$871$	11.5592	0.391667
$872$	0	0
$873$	−157.224	−5.32124
$874$	0	0
$875$	18.2694	0.617617
$876$	0	0
$877$	−9.68697	−0.327106	−0.163553	−	0.986535i	$-0.552295\pi$
$877$	−9.68697	−0.327106	−0.163553	+	0.986535i	$0.552295\pi$
$878$	0	0
$879$	28.1249	0.948628
$880$	0	0
$881$	−48.1977	−1.62382	−0.811910	−	0.583782i	$-0.801572\pi$
$881$	−48.1977	−1.62382	−0.811910	+	0.583782i	$0.801572\pi$
$882$	0	0
$883$	−25.9939	−0.874766	−0.437383	−	0.899275i	$-0.644095\pi$
$883$	−25.9939	−0.874766	−0.437383	+	0.899275i	$0.644095\pi$
$884$	0	0
$885$	7.19304	0.241791
$886$	0	0
$887$	47.4975	1.59481	0.797405	−	0.603444i	$-0.206205\pi$
$887$	47.4975	1.59481	0.797405	+	0.603444i	$0.206205\pi$
$888$	0	0
$889$	−24.3616	−0.817063
$890$	0	0
$891$	139.417	4.67066
$892$	0	0
$893$	−8.53931	−0.285757
$894$	0	0
$895$	−1.99108	−0.0665543
$896$	0	0
$897$	−4.20024	−0.140242
$898$	0	0
$899$	13.0303	0.434584
$900$	0	0
$901$	29.3061	0.976328
$902$	0	0
$903$	99.5280	3.31208
$904$	0	0
$905$	0.0967567	0.00321630
$906$	0	0
$907$	15.0849	0.500886	0.250443	−	0.968131i	$-0.419424\pi$
$907$	15.0849	0.500886	0.250443	+	0.968131i	$0.419424\pi$
$908$	0	0
$909$	−52.1518	−1.72977
$910$	0	0
$911$	−11.9463	−0.395800	−0.197900	−	0.980222i	$-0.563412\pi$
$911$	−11.9463	−0.395800	−0.197900	+	0.980222i	$0.563412\pi$
$912$	0	0
$913$	44.4007	1.46945
$914$	0	0
$915$	2.89414	0.0956772
$916$	0	0
$917$	−9.31519	−0.307615
$918$	0	0
$919$	8.90417	0.293721	0.146861	−	0.989157i	$-0.453083\pi$
$919$	8.90417	0.293721	0.146861	+	0.989157i	$0.453083\pi$
$920$	0	0
$921$	−38.3783	−1.26461
$922$	0	0
$923$	−49.9116	−1.64286
$924$	0	0
$925$	−38.6167	−1.26971
$926$	0	0
$927$	37.9594	1.24675
$928$	0	0
$929$	7.67807	0.251909	0.125955	−	0.992036i	$-0.459801\pi$
$929$	7.67807	0.251909	0.125955	+	0.992036i	$0.459801\pi$
$930$	0	0
$931$	3.57949	0.117313
$932$	0	0
$933$	7.62759	0.249716
$934$	0	0
$935$	4.78426	0.156462
$936$	0	0
$937$	−54.1812	−1.77002	−0.885011	−	0.465570i	$-0.845850\pi$
$937$	−54.1812	−1.77002	−0.885011	+	0.465570i	$0.845850\pi$
$938$	0	0
$939$	78.8016	2.57159
$940$	0	0
$941$	7.53369	0.245591	0.122796	−	0.992432i	$-0.460814\pi$
$941$	7.53369	0.245591	0.122796	+	0.992432i	$0.460814\pi$
$942$	0	0
$943$	−1.82232	−0.0593430
$944$	0	0
$945$	33.5095	1.09007
$946$	0	0
$947$	18.0610	0.586903	0.293452	−	0.955974i	$-0.405196\pi$
$947$	18.0610	0.586903	0.293452	+	0.955974i	$0.405196\pi$
$948$	0	0
$949$	40.3712	1.31051
$950$	0	0
$951$	−0.840775	−0.0272640
$952$	0	0
$953$	−54.9471	−1.77991	−0.889955	−	0.456047i	$-0.849265\pi$
$953$	−54.9471	−1.77991	−0.889955	+	0.456047i	$0.849265\pi$
$954$	0	0
$955$	−0.142984	−0.00462686
$956$	0	0
$957$	93.7584	3.03078
$958$	0	0
$959$	−39.7693	−1.28422
$960$	0	0
$961$	−27.4802	−0.886457
$962$	0	0
$963$	8.23607	0.265404
$964$	0	0
$965$	5.04933	0.162544
$966$	0	0
$967$	40.3739	1.29834	0.649169	−	0.760644i	$-0.275117\pi$
$967$	40.3739	1.29834	0.649169	+	0.760644i	$0.275117\pi$
$968$	0	0
$969$	6.87514	0.220861
$970$	0	0
$971$	12.4426	0.399303	0.199651	−	0.979867i	$-0.436019\pi$
$971$	12.4426	0.399303	0.199651	+	0.979867i	$0.436019\pi$
$972$	0	0
$973$	−71.9677	−2.30718
$974$	0	0
$975$	66.7406	2.13741
$976$	0	0
$977$	17.3752	0.555880	0.277940	−	0.960598i	$-0.410348\pi$
$977$	17.3752	0.555880	0.277940	+	0.960598i	$0.410348\pi$
$978$	0	0
$979$	−30.7970	−0.984277
$980$	0	0
$981$	−122.987	−3.92668
$982$	0	0
$983$	36.2478	1.15612	0.578062	−	0.815993i	$-0.303809\pi$
$983$	36.2478	1.15612	0.578062	+	0.815993i	$0.303809\pi$
$984$	0	0
$985$	8.08063	0.257470
$986$	0	0
$987$	93.2735	2.96893
$988$	0	0
$989$	−2.67345	−0.0850108
$990$	0	0
$991$	19.1118	0.607107	0.303554	−	0.952814i	$-0.401827\pi$
$991$	19.1118	0.607107	0.303554	+	0.952814i	$0.401827\pi$
$992$	0	0
$993$	85.1243	2.70134
$994$	0	0
$995$	2.33071	0.0738886
$996$	0	0
$997$	−35.0889	−1.11128	−0.555638	−	0.831424i	$-0.687526\pi$
$997$	−35.0889	−1.11128	−0.555638	+	0.831424i	$0.687526\pi$
$998$	0	0
$999$	−146.795	−4.64440

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.g.1.1	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.g.1.1	✓	27	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6004.a (trivial)

\(f(q)\)	\(=\)	\(q-3.35817 q^{3} -0.581328 q^{5} +3.25261 q^{7} +8.27730 q^{9} +O(q^{10})\)	\(q-3.35817 q^{3} -0.581328 q^{5} +3.25261 q^{7} +8.27730 q^{9} +4.01989 q^{11} +4.26295 q^{13} +1.95220 q^{15} -2.04729 q^{17} +1.00000 q^{19} -10.9228 q^{21} +0.293401 q^{23} -4.66206 q^{25} -17.7221 q^{27} -6.94533 q^{29} -1.87612 q^{31} -13.4995 q^{33} -1.89084 q^{35} +8.28320 q^{37} -14.3157 q^{39} -6.21102 q^{41} -9.11192 q^{43} -4.81183 q^{45} -8.53931 q^{47} +3.57949 q^{49} +6.87514 q^{51} -14.3146 q^{53} -2.33688 q^{55} -3.35817 q^{57} +3.68459 q^{59} +1.48250 q^{61} +26.9229 q^{63} -2.47817 q^{65} +2.71154 q^{67} -0.985291 q^{69} -11.7082 q^{71} +9.47027 q^{73} +15.6560 q^{75} +13.0752 q^{77} -1.00000 q^{79} +34.6818 q^{81} +11.0452 q^{83} +1.19015 q^{85} +23.3236 q^{87} -7.66115 q^{89} +13.8657 q^{91} +6.30032 q^{93} -0.581328 q^{95} -18.9946 q^{97} +33.2739 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) \) 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9} + 3 q^{11} - 5 q^{13} - 11 q^{15} - 17 q^{17} + 27 q^{19} - 28 q^{21} - 11 q^{23} + 13 q^{25} - 7 q^{27} - 39 q^{29} - 27 q^{31} - 18 q^{33} - 5 q^{35} - q^{37} - 22 q^{39} - 36 q^{41} - 2 q^{43} - 18 q^{45} - 12 q^{47} + 15 q^{49} + 4 q^{51} - 28 q^{53} + 5 q^{55} - 4 q^{57} - 30 q^{59} - 6 q^{61} - 4 q^{63} - 32 q^{65} + 13 q^{67} - 27 q^{69} - 59 q^{71} - 30 q^{73} - 21 q^{75} - 39 q^{77} - 27 q^{79} - 5 q^{81} + 4 q^{83} - 3 q^{85} + 22 q^{87} - 56 q^{89} - 8 q^{91} - 38 q^{93} - 10 q^{95} - 30 q^{97} + 31 q^{99}+O(q^{100}) \) 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9 + 3 * q^11 - 5 * q^13 - 11 * q^15 - 17 * q^17 + 27 * q^19 - 28 * q^21 - 11 * q^23 + 13 * q^25 - 7 * q^27 - 39 * q^29 - 27 * q^31 - 18 * q^33 - 5 * q^35 - q^37 - 22 * q^39 - 36 * q^41 - 2 * q^43 - 18 * q^45 - 12 * q^47 + 15 * q^49 + 4 * q^51 - 28 * q^53 + 5 * q^55 - 4 * q^57 - 30 * q^59 - 6 * q^61 - 4 * q^63 - 32 * q^65 + 13 * q^67 - 27 * q^69 - 59 * q^71 - 30 * q^73 - 21 * q^75 - 39 * q^77 - 27 * q^79 - 5 * q^81 + 4 * q^83 - 3 * q^85 + 22 * q^87 - 56 * q^89 - 8 * q^91 - 38 * q^93 - 10 * q^95 - 30 * q^97 + 31 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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