LMFDB - Embedded newform 6004.2.a.f.1.15

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:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
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)

$f(q)$	$=$	$q+0.749226 q^{3} -1.84486 q^{5} +4.01817 q^{7} -2.43866 q^{9} +O(q^{10})$	$q+0.749226 q^{3} -1.84486 q^{5} +4.01817 q^{7} -2.43866 q^{9} +5.56209 q^{11} +2.87545 q^{13} -1.38221 q^{15} -6.49022 q^{17} -1.00000 q^{19} +3.01052 q^{21} -3.34518 q^{23} -1.59650 q^{25} -4.07478 q^{27} -8.93531 q^{29} -6.10017 q^{31} +4.16726 q^{33} -7.41295 q^{35} -4.60848 q^{37} +2.15436 q^{39} -6.25887 q^{41} +6.13786 q^{43} +4.49898 q^{45} -3.19735 q^{47} +9.14571 q^{49} -4.86264 q^{51} +1.75625 q^{53} -10.2613 q^{55} -0.749226 q^{57} -4.81445 q^{59} -3.36742 q^{61} -9.79896 q^{63} -5.30479 q^{65} -14.5642 q^{67} -2.50629 q^{69} +2.72592 q^{71} +11.0688 q^{73} -1.19614 q^{75} +22.3494 q^{77} +1.00000 q^{79} +4.26305 q^{81} -14.5754 q^{83} +11.9735 q^{85} -6.69456 q^{87} +13.7261 q^{89} +11.5541 q^{91} -4.57040 q^{93} +1.84486 q^{95} +1.35817 q^{97} -13.5640 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9}+O(q^{10}) $ 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9	$ 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9} - 3 q^{11} + q^{13} - 5 q^{15} - 13 q^{17} - 25 q^{19} - 24 q^{21} - 31 q^{23} + 21 q^{25} + 7 q^{27} - 19 q^{29} - 7 q^{31} - 30 q^{33} - q^{35} - 29 q^{37} - 26 q^{39} - 40 q^{41} - 40 q^{45} - 8 q^{47} - 9 q^{49} + 12 q^{51} - 38 q^{53} - 29 q^{55} - 4 q^{57} + 18 q^{59} - 26 q^{61} - 40 q^{63} - 70 q^{65} - 13 q^{67} + q^{69} - 47 q^{71} - 8 q^{73} + 7 q^{75} - 19 q^{77} + 25 q^{79} - 19 q^{81} - 8 q^{83} - 33 q^{85} - 50 q^{87} - 54 q^{89} - 12 q^{91} - 24 q^{93} + 8 q^{95} - 4 q^{97} - 39 q^{99}+O(q^{100}) $ 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9 - 3 * q^11 + q^13 - 5 * q^15 - 13 * q^17 - 25 * q^19 - 24 * q^21 - 31 * q^23 + 21 * q^25 + 7 * q^27 - 19 * q^29 - 7 * q^31 - 30 * q^33 - q^35 - 29 * q^37 - 26 * q^39 - 40 * q^41 - 40 * q^45 - 8 * q^47 - 9 * q^49 + 12 * q^51 - 38 * q^53 - 29 * q^55 - 4 * q^57 + 18 * q^59 - 26 * q^61 - 40 * q^63 - 70 * q^65 - 13 * q^67 + q^69 - 47 * q^71 - 8 * q^73 + 7 * q^75 - 19 * q^77 + 25 * q^79 - 19 * q^81 - 8 * q^83 - 33 * q^85 - 50 * q^87 - 54 * q^89 - 12 * q^91 - 24 * q^93 + 8 * q^95 - 4 * q^97 - 39 * 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
$3$	0.749226	0.432566	0.216283	−	0.976331i	$-0.430607\pi$
$3$	0.749226	0.432566	0.216283	+	0.976331i	$0.430607\pi$
$4$	0	0
$5$	−1.84486	−0.825045	−0.412523	−	0.910947i	$-0.635352\pi$
$5$	−1.84486	−0.825045	−0.412523	+	0.910947i	$0.635352\pi$
$6$	0	0
$7$	4.01817	1.51873	0.759363	−	0.650667i	$-0.225511\pi$
$7$	4.01817	1.51873	0.759363	+	0.650667i	$0.225511\pi$
$8$	0	0
$9$	−2.43866	−0.812887
$10$	0	0
$11$	5.56209	1.67703	0.838516	−	0.544877i	$-0.183424\pi$
$11$	5.56209	1.67703	0.838516	+	0.544877i	$0.183424\pi$
$12$	0	0
$13$	2.87545	0.797506	0.398753	−	0.917058i	$-0.369443\pi$
$13$	2.87545	0.797506	0.398753	+	0.917058i	$0.369443\pi$
$14$	0	0
$15$	−1.38221	−0.356886
$16$	0	0
$17$	−6.49022	−1.57411	−0.787055	−	0.616883i	$-0.788395\pi$
$17$	−6.49022	−1.57411	−0.787055	+	0.616883i	$0.788395\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	3.01052	0.656949
$22$	0	0
$23$	−3.34518	−0.697517	−0.348759	−	0.937213i	$-0.613397\pi$
$23$	−3.34518	−0.697517	−0.348759	+	0.937213i	$0.613397\pi$
$24$	0	0
$25$	−1.59650	−0.319300
$26$	0	0
$27$	−4.07478	−0.784193
$28$	0	0
$29$	−8.93531	−1.65925	−0.829623	−	0.558325i	$-0.811444\pi$
$29$	−8.93531	−1.65925	−0.829623	+	0.558325i	$0.811444\pi$
$30$	0	0
$31$	−6.10017	−1.09562	−0.547811	−	0.836602i	$-0.684539\pi$
$31$	−6.10017	−1.09562	−0.547811	+	0.836602i	$0.684539\pi$
$32$	0	0
$33$	4.16726	0.725426
$34$	0	0
$35$	−7.41295	−1.25302
$36$	0	0
$37$	−4.60848	−0.757629	−0.378814	−	0.925473i	$-0.623668\pi$
$37$	−4.60848	−0.757629	−0.378814	+	0.925473i	$0.623668\pi$
$38$	0	0
$39$	2.15436	0.344974
$40$	0	0
$41$	−6.25887	−0.977471	−0.488735	−	0.872432i	$-0.662542\pi$
$41$	−6.25887	−0.977471	−0.488735	+	0.872432i	$0.662542\pi$
$42$	0	0
$43$	6.13786	0.936016	0.468008	−	0.883724i	$-0.344972\pi$
$43$	6.13786	0.936016	0.468008	+	0.883724i	$0.344972\pi$
$44$	0	0
$45$	4.49898	0.670669
$46$	0	0
$47$	−3.19735	−0.466382	−0.233191	−	0.972431i	$-0.574917\pi$
$47$	−3.19735	−0.466382	−0.233191	+	0.972431i	$0.574917\pi$
$48$	0	0
$49$	9.14571	1.30653
$50$	0	0
$51$	−4.86264	−0.680906
$52$	0	0
$53$	1.75625	0.241239	0.120619	−	0.992699i	$-0.461512\pi$
$53$	1.75625	0.241239	0.120619	+	0.992699i	$0.461512\pi$
$54$	0	0
$55$	−10.2613	−1.38363
$56$	0	0
$57$	−0.749226	−0.0992374
$58$	0	0
$59$	−4.81445	−0.626788	−0.313394	−	0.949623i	$-0.601466\pi$
$59$	−4.81445	−0.626788	−0.313394	+	0.949623i	$0.601466\pi$
$60$	0	0
$61$	−3.36742	−0.431154	−0.215577	−	0.976487i	$-0.569163\pi$
$61$	−3.36742	−0.431154	−0.215577	+	0.976487i	$0.569163\pi$
$62$	0	0
$63$	−9.79896	−1.23455
$64$	0	0
$65$	−5.30479	−0.657979
$66$	0	0
$67$	−14.5642	−1.77931	−0.889653	−	0.456638i	$-0.849053\pi$
$67$	−14.5642	−1.77931	−0.889653	+	0.456638i	$0.849053\pi$
$68$	0	0
$69$	−2.50629	−0.301722
$70$	0	0
$71$	2.72592	0.323507	0.161753	−	0.986831i	$-0.448285\pi$
$71$	2.72592	0.323507	0.161753	+	0.986831i	$0.448285\pi$
$72$	0	0
$73$	11.0688	1.29550	0.647750	−	0.761853i	$-0.275710\pi$
$73$	11.0688	1.29550	0.647750	+	0.761853i	$0.275710\pi$
$74$	0	0
$75$	−1.19614	−0.138118
$76$	0	0
$77$	22.3494	2.54695
$78$	0	0
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	0	0
$81$	4.26305	0.473672
$82$	0	0
$83$	−14.5754	−1.59986	−0.799928	−	0.600095i	$-0.795129\pi$
$83$	−14.5754	−1.59986	−0.799928	+	0.600095i	$0.795129\pi$
$84$	0	0
$85$	11.9735	1.29871
$86$	0	0
$87$	−6.69456	−0.717732
$88$	0	0
$89$	13.7261	1.45497	0.727484	−	0.686125i	$-0.240690\pi$
$89$	13.7261	1.45497	0.727484	+	0.686125i	$0.240690\pi$
$90$	0	0
$91$	11.5541	1.21119
$92$	0	0
$93$	−4.57040	−0.473929
$94$	0	0
$95$	1.84486	0.189278
$96$	0	0
$97$	1.35817	0.137901	0.0689506	−	0.997620i	$-0.478035\pi$
$97$	1.35817	0.137901	0.0689506	+	0.997620i	$0.478035\pi$
$98$	0	0
$99$	−13.5640	−1.36324
$100$	0	0
$101$	−18.3729	−1.82817	−0.914086	−	0.405520i	$-0.867090\pi$
$101$	−18.3729	−1.82817	−0.914086	+	0.405520i	$0.867090\pi$
$102$	0	0
$103$	2.50711	0.247033	0.123516	−	0.992343i	$-0.460583\pi$
$103$	2.50711	0.247033	0.123516	+	0.992343i	$0.460583\pi$
$104$	0	0
$105$	−5.55398	−0.542013
$106$	0	0
$107$	12.1890	1.17836	0.589178	−	0.808003i	$-0.299452\pi$
$107$	12.1890	1.17836	0.589178	+	0.808003i	$0.299452\pi$
$108$	0	0
$109$	8.33581	0.798426	0.399213	−	0.916858i	$-0.369283\pi$
$109$	8.33581	0.798426	0.399213	+	0.916858i	$0.369283\pi$
$110$	0	0
$111$	−3.45279	−0.327724
$112$	0	0
$113$	−1.09866	−0.103353	−0.0516766	−	0.998664i	$-0.516456\pi$
$113$	−1.09866	−0.103353	−0.0516766	+	0.998664i	$0.516456\pi$
$114$	0	0
$115$	6.17137	0.575483
$116$	0	0
$117$	−7.01225	−0.648282
$118$	0	0
$119$	−26.0788	−2.39064
$120$	0	0
$121$	19.9368	1.81244
$122$	0	0
$123$	−4.68930	−0.422820
$124$	0	0
$125$	12.1696	1.08848
$126$	0	0
$127$	−7.73156	−0.686065	−0.343032	−	0.939324i	$-0.611454\pi$
$127$	−7.73156	−0.686065	−0.343032	+	0.939324i	$0.611454\pi$
$128$	0	0
$129$	4.59865	0.404888
$130$	0	0
$131$	−4.35758	−0.380724	−0.190362	−	0.981714i	$-0.560966\pi$
$131$	−4.35758	−0.380724	−0.190362	+	0.981714i	$0.560966\pi$
$132$	0	0
$133$	−4.01817	−0.348420
$134$	0	0
$135$	7.51739	0.646994
$136$	0	0
$137$	−11.9033	−1.01697	−0.508485	−	0.861071i	$-0.669794\pi$
$137$	−11.9033	−1.01697	−0.508485	+	0.861071i	$0.669794\pi$
$138$	0	0
$139$	2.81694	0.238929	0.119465	−	0.992838i	$-0.461882\pi$
$139$	2.81694	0.238929	0.119465	+	0.992838i	$0.461882\pi$
$140$	0	0
$141$	−2.39554	−0.201741
$142$	0	0
$143$	15.9935	1.33744
$144$	0	0
$145$	16.4844	1.36895
$146$	0	0
$147$	6.85220	0.565160
$148$	0	0
$149$	0.983991	0.0806117	0.0403058	−	0.999187i	$-0.487167\pi$
$149$	0.983991	0.0806117	0.0403058	+	0.999187i	$0.487167\pi$
$150$	0	0
$151$	19.8289	1.61366	0.806828	−	0.590786i	$-0.201182\pi$
$151$	19.8289	1.61366	0.806828	+	0.590786i	$0.201182\pi$
$152$	0	0
$153$	15.8275	1.27957
$154$	0	0
$155$	11.2539	0.903938
$156$	0	0
$157$	−12.0595	−0.962452	−0.481226	−	0.876597i	$-0.659808\pi$
$157$	−12.0595	−0.962452	−0.481226	+	0.876597i	$0.659808\pi$
$158$	0	0
$159$	1.31582	0.104352
$160$	0	0
$161$	−13.4415	−1.05934
$162$	0	0
$163$	−9.23725	−0.723517	−0.361758	−	0.932272i	$-0.617823\pi$
$163$	−9.23725	−0.723517	−0.361758	+	0.932272i	$0.617823\pi$
$164$	0	0
$165$	−7.68799	−0.598510
$166$	0	0
$167$	22.3834	1.73208	0.866038	−	0.499978i	$-0.166659\pi$
$167$	22.3834	1.73208	0.866038	+	0.499978i	$0.166659\pi$
$168$	0	0
$169$	−4.73179	−0.363984
$170$	0	0
$171$	2.43866	0.186489
$172$	0	0
$173$	0.682307	0.0518748	0.0259374	−	0.999664i	$-0.491743\pi$
$173$	0.682307	0.0518748	0.0259374	+	0.999664i	$0.491743\pi$
$174$	0	0
$175$	−6.41502	−0.484930
$176$	0	0
$177$	−3.60711	−0.271127
$178$	0	0
$179$	−9.09356	−0.679685	−0.339842	−	0.940482i	$-0.610374\pi$
$179$	−9.09356	−0.679685	−0.339842	+	0.940482i	$0.610374\pi$
$180$	0	0
$181$	11.6519	0.866076	0.433038	−	0.901376i	$-0.357442\pi$
$181$	11.6519	0.866076	0.433038	+	0.901376i	$0.357442\pi$
$182$	0	0
$183$	−2.52296	−0.186502
$184$	0	0
$185$	8.50198	0.625078
$186$	0	0
$187$	−36.0992	−2.63983
$188$	0	0
$189$	−16.3732	−1.19097
$190$	0	0
$191$	−13.7586	−0.995535	−0.497768	−	0.867310i	$-0.665847\pi$
$191$	−13.7586	−0.995535	−0.497768	+	0.867310i	$0.665847\pi$
$192$	0	0
$193$	−11.3401	−0.816282	−0.408141	−	0.912919i	$-0.633823\pi$
$193$	−11.3401	−0.816282	−0.408141	+	0.912919i	$0.633823\pi$
$194$	0	0
$195$	−3.97449	−0.284619
$196$	0	0
$197$	−7.70239	−0.548772	−0.274386	−	0.961620i	$-0.588475\pi$
$197$	−7.70239	−0.548772	−0.274386	+	0.961620i	$0.588475\pi$
$198$	0	0
$199$	−8.01696	−0.568307	−0.284154	−	0.958779i	$-0.591713\pi$
$199$	−8.01696	−0.568307	−0.284154	+	0.958779i	$0.591713\pi$
$200$	0	0
$201$	−10.9119	−0.769666
$202$	0	0
$203$	−35.9036	−2.51994
$204$	0	0
$205$	11.5467	0.806457
$206$	0	0
$207$	8.15775	0.567003
$208$	0	0
$209$	−5.56209	−0.384738
$210$	0	0
$211$	−21.3830	−1.47207	−0.736034	−	0.676944i	$-0.763304\pi$
$211$	−21.3830	−1.47207	−0.736034	+	0.676944i	$0.763304\pi$
$212$	0	0
$213$	2.04233	0.139938
$214$	0	0
$215$	−11.3235	−0.772255
$216$	0	0
$217$	−24.5115	−1.66395
$218$	0	0
$219$	8.29300	0.560389
$220$	0	0
$221$	−18.6623	−1.25536
$222$	0	0
$223$	11.9672	0.801380	0.400690	−	0.916214i	$-0.368770\pi$
$223$	11.9672	0.801380	0.400690	+	0.916214i	$0.368770\pi$
$224$	0	0
$225$	3.89333	0.259555
$226$	0	0
$227$	0.0443899	0.00294626	0.00147313	−	0.999999i	$-0.499531\pi$
$227$	0.0443899	0.00294626	0.00147313	+	0.999999i	$0.499531\pi$
$228$	0	0
$229$	12.2469	0.809300	0.404650	−	0.914472i	$-0.367393\pi$
$229$	12.2469	0.809300	0.404650	+	0.914472i	$0.367393\pi$
$230$	0	0
$231$	16.7448	1.10172
$232$	0	0
$233$	14.8319	0.971668	0.485834	−	0.874051i	$-0.338516\pi$
$233$	14.8319	0.971668	0.485834	+	0.874051i	$0.338516\pi$
$234$	0	0
$235$	5.89866	0.384786
$236$	0	0
$237$	0.749226	0.0486674
$238$	0	0
$239$	−26.0506	−1.68507	−0.842537	−	0.538639i	$-0.818939\pi$
$239$	−26.0506	−1.68507	−0.842537	+	0.538639i	$0.818939\pi$
$240$	0	0
$241$	12.3568	0.795970	0.397985	−	0.917392i	$-0.369710\pi$
$241$	12.3568	0.795970	0.397985	+	0.917392i	$0.369710\pi$
$242$	0	0
$243$	15.4183	0.989087
$244$	0	0
$245$	−16.8725	−1.07795
$246$	0	0
$247$	−2.87545	−0.182960
$248$	0	0
$249$	−10.9203	−0.692043
$250$	0	0
$251$	−8.45352	−0.533581	−0.266791	−	0.963754i	$-0.585963\pi$
$251$	−8.45352	−0.533581	−0.266791	+	0.963754i	$0.585963\pi$
$252$	0	0
$253$	−18.6062	−1.16976
$254$	0	0
$255$	8.97088	0.561778
$256$	0	0
$257$	−5.40134	−0.336926	−0.168463	−	0.985708i	$-0.553880\pi$
$257$	−5.40134	−0.336926	−0.168463	+	0.985708i	$0.553880\pi$
$258$	0	0
$259$	−18.5177	−1.15063
$260$	0	0
$261$	21.7902	1.34878
$262$	0	0
$263$	−11.6387	−0.717671	−0.358835	−	0.933401i	$-0.616826\pi$
$263$	−11.6387	−0.717671	−0.358835	+	0.933401i	$0.616826\pi$
$264$	0	0
$265$	−3.24002	−0.199033
$266$	0	0
$267$	10.2840	0.629369
$268$	0	0
$269$	10.3319	0.629944	0.314972	−	0.949101i	$-0.398005\pi$
$269$	10.3319	0.629944	0.314972	+	0.949101i	$0.398005\pi$
$270$	0	0
$271$	−4.86338	−0.295429	−0.147715	−	0.989030i	$-0.547192\pi$
$271$	−4.86338	−0.295429	−0.147715	+	0.989030i	$0.547192\pi$
$272$	0	0
$273$	8.65659	0.523921
$274$	0	0
$275$	−8.87988	−0.535477
$276$	0	0
$277$	17.2403	1.03587	0.517934	−	0.855420i	$-0.326701\pi$
$277$	17.2403	1.03587	0.517934	+	0.855420i	$0.326701\pi$
$278$	0	0
$279$	14.8762	0.890617
$280$	0	0
$281$	−33.3397	−1.98888	−0.994438	−	0.105319i	$-0.966414\pi$
$281$	−33.3397	−1.98888	−0.994438	+	0.105319i	$0.966414\pi$
$282$	0	0
$283$	26.2224	1.55876	0.779380	−	0.626552i	$-0.215534\pi$
$283$	26.2224	1.55876	0.779380	+	0.626552i	$0.215534\pi$
$284$	0	0
$285$	1.38221	0.0818753
$286$	0	0
$287$	−25.1492	−1.48451
$288$	0	0
$289$	25.1230	1.47782
$290$	0	0
$291$	1.01758	0.0596514
$292$	0	0
$293$	14.0691	0.821926	0.410963	−	0.911652i	$-0.365193\pi$
$293$	14.0691	0.821926	0.410963	+	0.911652i	$0.365193\pi$
$294$	0	0
$295$	8.88197	0.517129
$296$	0	0
$297$	−22.6643	−1.31512
$298$	0	0
$299$	−9.61888	−0.556274
$300$	0	0
$301$	24.6630	1.42155
$302$	0	0
$303$	−13.7654	−0.790804
$304$	0	0
$305$	6.21241	0.355722
$306$	0	0
$307$	−19.7290	−1.12599	−0.562997	−	0.826459i	$-0.690352\pi$
$307$	−19.7290	−1.12599	−0.562997	+	0.826459i	$0.690352\pi$
$308$	0	0
$309$	1.87839	0.106858
$310$	0	0
$311$	21.0459	1.19340	0.596702	−	0.802463i	$-0.296477\pi$
$311$	21.0459	1.19340	0.596702	+	0.802463i	$0.296477\pi$
$312$	0	0
$313$	−3.75034	−0.211982	−0.105991	−	0.994367i	$-0.533801\pi$
$313$	−3.75034	−0.211982	−0.105991	+	0.994367i	$0.533801\pi$
$314$	0	0
$315$	18.0777	1.01856
$316$	0	0
$317$	2.52384	0.141753	0.0708764	−	0.997485i	$-0.477420\pi$
$317$	2.52384	0.141753	0.0708764	+	0.997485i	$0.477420\pi$
$318$	0	0
$319$	−49.6990	−2.78261
$320$	0	0
$321$	9.13232	0.509716
$322$	0	0
$323$	6.49022	0.361126
$324$	0	0
$325$	−4.59066	−0.254644
$326$	0	0
$327$	6.24541	0.345372
$328$	0	0
$329$	−12.8475	−0.708307
$330$	0	0
$331$	−8.97647	−0.493392	−0.246696	−	0.969093i	$-0.579345\pi$
$331$	−8.97647	−0.493392	−0.246696	+	0.969093i	$0.579345\pi$
$332$	0	0
$333$	11.2385	0.615867
$334$	0	0
$335$	26.8689	1.46801
$336$	0	0
$337$	−1.56961	−0.0855019	−0.0427509	−	0.999086i	$-0.513612\pi$
$337$	−1.56961	−0.0855019	−0.0427509	+	0.999086i	$0.513612\pi$
$338$	0	0
$339$	−0.823144	−0.0447070
$340$	0	0
$341$	−33.9297	−1.83739
$342$	0	0
$343$	8.62184	0.465536
$344$	0	0
$345$	4.62375	0.248934
$346$	0	0
$347$	−2.80577	−0.150622	−0.0753109	−	0.997160i	$-0.523995\pi$
$347$	−2.80577	−0.150622	−0.0753109	+	0.997160i	$0.523995\pi$
$348$	0	0
$349$	−17.0086	−0.910451	−0.455225	−	0.890376i	$-0.650441\pi$
$349$	−17.0086	−0.910451	−0.455225	+	0.890376i	$0.650441\pi$
$350$	0	0
$351$	−11.7168	−0.625398
$352$	0	0
$353$	−1.19594	−0.0636536	−0.0318268	−	0.999493i	$-0.510133\pi$
$353$	−1.19594	−0.0636536	−0.0318268	+	0.999493i	$0.510133\pi$
$354$	0	0
$355$	−5.02893	−0.266908
$356$	0	0
$357$	−19.5389	−1.03411
$358$	0	0
$359$	−10.7241	−0.565999	−0.282999	−	0.959120i	$-0.591329\pi$
$359$	−10.7241	−0.565999	−0.282999	+	0.959120i	$0.591329\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	14.9372	0.783998
$364$	0	0
$365$	−20.4203	−1.06885
$366$	0	0
$367$	−27.5808	−1.43971	−0.719854	−	0.694126i	$-0.755791\pi$
$367$	−27.5808	−1.43971	−0.719854	+	0.694126i	$0.755791\pi$
$368$	0	0
$369$	15.2633	0.794573
$370$	0	0
$371$	7.05690	0.366376
$372$	0	0
$373$	−21.9057	−1.13423	−0.567117	−	0.823637i	$-0.691941\pi$
$373$	−21.9057	−1.13423	−0.567117	+	0.823637i	$0.691941\pi$
$374$	0	0
$375$	9.11778	0.470840
$376$	0	0
$377$	−25.6930	−1.32326
$378$	0	0
$379$	−1.69682	−0.0871598	−0.0435799	−	0.999050i	$-0.513876\pi$
$379$	−1.69682	−0.0871598	−0.0435799	+	0.999050i	$0.513876\pi$
$380$	0	0
$381$	−5.79268	−0.296768
$382$	0	0
$383$	−23.3823	−1.19478	−0.597390	−	0.801951i	$-0.703796\pi$
$383$	−23.3823	−1.19478	−0.597390	+	0.801951i	$0.703796\pi$
$384$	0	0
$385$	−41.2315	−2.10135
$386$	0	0
$387$	−14.9682	−0.760875
$388$	0	0
$389$	−15.4715	−0.784434	−0.392217	−	0.919873i	$-0.628292\pi$
$389$	−15.4715	−0.784434	−0.392217	+	0.919873i	$0.628292\pi$
$390$	0	0
$391$	21.7109	1.09797
$392$	0	0
$393$	−3.26481	−0.164688
$394$	0	0
$395$	−1.84486	−0.0928248
$396$	0	0
$397$	−3.85410	−0.193432	−0.0967160	−	0.995312i	$-0.530834\pi$
$397$	−3.85410	−0.193432	−0.0967160	+	0.995312i	$0.530834\pi$
$398$	0	0
$399$	−3.01052	−0.150714
$400$	0	0
$401$	17.0644	0.852154	0.426077	−	0.904687i	$-0.359895\pi$
$401$	17.0644	0.852154	0.426077	+	0.904687i	$0.359895\pi$
$402$	0	0
$403$	−17.5407	−0.873766
$404$	0	0
$405$	−7.86472	−0.390801
$406$	0	0
$407$	−25.6327	−1.27057
$408$	0	0
$409$	25.8321	1.27732	0.638658	−	0.769491i	$-0.279490\pi$
$409$	25.8321	1.27732	0.638658	+	0.769491i	$0.279490\pi$
$410$	0	0
$411$	−8.91828	−0.439906
$412$	0	0
$413$	−19.3453	−0.951920
$414$	0	0
$415$	26.8895	1.31995
$416$	0	0
$417$	2.11052	0.103353
$418$	0	0
$419$	40.1984	1.96382	0.981910	−	0.189350i	$-0.0606382\pi$
$419$	40.1984	1.96382	0.981910	+	0.189350i	$0.0606382\pi$
$420$	0	0
$421$	13.4089	0.653508	0.326754	−	0.945109i	$-0.394045\pi$
$421$	13.4089	0.653508	0.326754	+	0.945109i	$0.394045\pi$
$422$	0	0
$423$	7.79726	0.379116
$424$	0	0
$425$	10.3617	0.502614
$426$	0	0
$427$	−13.5309	−0.654805
$428$	0	0
$429$	11.9827	0.578532
$430$	0	0
$431$	−1.56604	−0.0754336	−0.0377168	−	0.999288i	$-0.512008\pi$
$431$	−1.56604	−0.0754336	−0.0377168	+	0.999288i	$0.512008\pi$
$432$	0	0
$433$	−22.6261	−1.08734	−0.543670	−	0.839299i	$-0.682966\pi$
$433$	−22.6261	−1.08734	−0.543670	+	0.839299i	$0.682966\pi$
$434$	0	0
$435$	12.3505	0.592162
$436$	0	0
$437$	3.34518	0.160021
$438$	0	0
$439$	28.3326	1.35224	0.676121	−	0.736791i	$-0.263660\pi$
$439$	28.3326	1.35224	0.676121	+	0.736791i	$0.263660\pi$
$440$	0	0
$441$	−22.3033	−1.06206
$442$	0	0
$443$	19.2169	0.913024	0.456512	−	0.889717i	$-0.349099\pi$
$443$	19.2169	0.913024	0.456512	+	0.889717i	$0.349099\pi$
$444$	0	0
$445$	−25.3228	−1.20041
$446$	0	0
$447$	0.737231	0.0348698
$448$	0	0
$449$	25.9797	1.22606	0.613028	−	0.790061i	$-0.289951\pi$
$449$	25.9797	1.22606	0.613028	+	0.790061i	$0.289951\pi$
$450$	0	0
$451$	−34.8123	−1.63925
$452$	0	0
$453$	14.8564	0.698012
$454$	0	0
$455$	−21.3156	−0.999290
$456$	0	0
$457$	17.4518	0.816359	0.408179	−	0.912902i	$-0.366164\pi$
$457$	17.4518	0.816359	0.408179	+	0.912902i	$0.366164\pi$
$458$	0	0
$459$	26.4463	1.23441
$460$	0	0
$461$	−2.42672	−0.113024	−0.0565118	−	0.998402i	$-0.517998\pi$
$461$	−2.42672	−0.113024	−0.0565118	+	0.998402i	$0.517998\pi$
$462$	0	0
$463$	19.4157	0.902322	0.451161	−	0.892443i	$-0.351010\pi$
$463$	19.4157	0.902322	0.451161	+	0.892443i	$0.351010\pi$
$464$	0	0
$465$	8.43174	0.391013
$466$	0	0
$467$	39.3300	1.81997	0.909987	−	0.414637i	$-0.136091\pi$
$467$	39.3300	1.81997	0.909987	+	0.414637i	$0.136091\pi$
$468$	0	0
$469$	−58.5216	−2.70228
$470$	0	0
$471$	−9.03528	−0.416324
$472$	0	0
$473$	34.1393	1.56973
$474$	0	0
$475$	1.59650	0.0732525
$476$	0	0
$477$	−4.28289	−0.196100
$478$	0	0
$479$	−10.9013	−0.498094	−0.249047	−	0.968491i	$-0.580117\pi$
$479$	−10.9013	−0.498094	−0.249047	+	0.968491i	$0.580117\pi$
$480$	0	0
$481$	−13.2514	−0.604214
$482$	0	0
$483$	−10.0707	−0.458233
$484$	0	0
$485$	−2.50563	−0.113775
$486$	0	0
$487$	−4.21283	−0.190902	−0.0954508	−	0.995434i	$-0.530429\pi$
$487$	−4.21283	−0.190902	−0.0954508	+	0.995434i	$0.530429\pi$
$488$	0	0
$489$	−6.92078	−0.312969
$490$	0	0
$491$	−1.22951	−0.0554871	−0.0277436	−	0.999615i	$-0.508832\pi$
$491$	−1.22951	−0.0554871	−0.0277436	+	0.999615i	$0.508832\pi$
$492$	0	0
$493$	57.9921	2.61183
$494$	0	0
$495$	25.0237	1.12473
$496$	0	0
$497$	10.9532	0.491319
$498$	0	0
$499$	−14.5425	−0.651013	−0.325507	−	0.945540i	$-0.605535\pi$
$499$	−14.5425	−0.651013	−0.325507	+	0.945540i	$0.605535\pi$
$500$	0	0
$501$	16.7702	0.749236
$502$	0	0
$503$	42.7633	1.90672	0.953362	−	0.301830i	$-0.0975976\pi$
$503$	42.7633	1.90672	0.953362	+	0.301830i	$0.0975976\pi$
$504$	0	0
$505$	33.8954	1.50832
$506$	0	0
$507$	−3.54518	−0.157447
$508$	0	0
$509$	−9.84068	−0.436181	−0.218090	−	0.975929i	$-0.569983\pi$
$509$	−9.84068	−0.436181	−0.218090	+	0.975929i	$0.569983\pi$
$510$	0	0
$511$	44.4762	1.96751
$512$	0	0
$513$	4.07478	0.179906
$514$	0	0
$515$	−4.62525	−0.203813
$516$	0	0
$517$	−17.7840	−0.782137
$518$	0	0
$519$	0.511202	0.0224393
$520$	0	0
$521$	−42.1111	−1.84492	−0.922459	−	0.386094i	$-0.873824\pi$
$521$	−42.1111	−1.84492	−0.922459	+	0.386094i	$0.873824\pi$
$522$	0	0
$523$	7.66363	0.335107	0.167554	−	0.985863i	$-0.446413\pi$
$523$	7.66363	0.335107	0.167554	+	0.985863i	$0.446413\pi$
$524$	0	0
$525$	−4.80630	−0.209764
$526$	0	0
$527$	39.5914	1.72463
$528$	0	0
$529$	−11.8098	−0.513470
$530$	0	0
$531$	11.7408	0.509508
$532$	0	0
$533$	−17.9971	−0.779539
$534$	0	0
$535$	−22.4870	−0.972197
$536$	0	0
$537$	−6.81313	−0.294008
$538$	0	0
$539$	50.8692	2.19109
$540$	0	0
$541$	25.7754	1.10817	0.554085	−	0.832460i	$-0.313068\pi$
$541$	25.7754	1.10817	0.554085	+	0.832460i	$0.313068\pi$
$542$	0	0
$543$	8.72988	0.374635
$544$	0	0
$545$	−15.3784	−0.658738
$546$	0	0
$547$	−36.1050	−1.54374	−0.771870	−	0.635780i	$-0.780678\pi$
$547$	−36.1050	−1.54374	−0.771870	+	0.635780i	$0.780678\pi$
$548$	0	0
$549$	8.21200	0.350480
$550$	0	0
$551$	8.93531	0.380657
$552$	0	0
$553$	4.01817	0.170870
$554$	0	0
$555$	6.36990	0.270387
$556$	0	0
$557$	−20.1087	−0.852031	−0.426015	−	0.904716i	$-0.640083\pi$
$557$	−20.1087	−0.852031	−0.426015	+	0.904716i	$0.640083\pi$
$558$	0	0
$559$	17.6491	0.746478
$560$	0	0
$561$	−27.0464	−1.14190
$562$	0	0
$563$	−27.9365	−1.17738	−0.588692	−	0.808357i	$-0.700357\pi$
$563$	−27.9365	−1.17738	−0.588692	+	0.808357i	$0.700357\pi$
$564$	0	0
$565$	2.02687	0.0852710
$566$	0	0
$567$	17.1297	0.719379
$568$	0	0
$569$	33.2995	1.39599	0.697995	−	0.716103i	$-0.254076\pi$
$569$	33.2995	1.39599	0.697995	+	0.716103i	$0.254076\pi$
$570$	0	0
$571$	−1.30646	−0.0546736	−0.0273368	−	0.999626i	$-0.508703\pi$
$571$	−1.30646	−0.0546736	−0.0273368	+	0.999626i	$0.508703\pi$
$572$	0	0
$573$	−10.3083	−0.430634
$574$	0	0
$575$	5.34058	0.222718
$576$	0	0
$577$	−38.1450	−1.58800	−0.793999	−	0.607920i	$-0.792004\pi$
$577$	−38.1450	−1.58800	−0.793999	+	0.607920i	$0.792004\pi$
$578$	0	0
$579$	−8.49633	−0.353095
$580$	0	0
$581$	−58.5664	−2.42974
$582$	0	0
$583$	9.76839	0.404565
$584$	0	0
$585$	12.9366	0.534862
$586$	0	0
$587$	−2.66238	−0.109888	−0.0549440	−	0.998489i	$-0.517498\pi$
$587$	−2.66238	−0.109888	−0.0549440	+	0.998489i	$0.517498\pi$
$588$	0	0
$589$	6.10017	0.251353
$590$	0	0
$591$	−5.77082	−0.237380
$592$	0	0
$593$	−35.7711	−1.46894	−0.734472	−	0.678638i	$-0.762570\pi$
$593$	−35.7711	−1.46894	−0.734472	+	0.678638i	$0.762570\pi$
$594$	0	0
$595$	48.1117	1.97239
$596$	0	0
$597$	−6.00651	−0.245830
$598$	0	0
$599$	−45.0432	−1.84042	−0.920208	−	0.391431i	$-0.871980\pi$
$599$	−45.0432	−1.84042	−0.920208	+	0.391431i	$0.871980\pi$
$600$	0	0
$601$	17.8759	0.729172	0.364586	−	0.931170i	$-0.381210\pi$
$601$	17.8759	0.729172	0.364586	+	0.931170i	$0.381210\pi$
$602$	0	0
$603$	35.5172	1.44637
$604$	0	0
$605$	−36.7805	−1.49534
$606$	0	0
$607$	33.1469	1.34539	0.672695	−	0.739920i	$-0.265137\pi$
$607$	33.1469	1.34539	0.672695	+	0.739920i	$0.265137\pi$
$608$	0	0
$609$	−26.8999	−1.09004
$610$	0	0
$611$	−9.19383	−0.371943
$612$	0	0
$613$	6.12350	0.247326	0.123663	−	0.992324i	$-0.460536\pi$
$613$	6.12350	0.247326	0.123663	+	0.992324i	$0.460536\pi$
$614$	0	0
$615$	8.65109	0.348846
$616$	0	0
$617$	2.98165	0.120037	0.0600183	−	0.998197i	$-0.480884\pi$
$617$	2.98165	0.120037	0.0600183	+	0.998197i	$0.480884\pi$
$618$	0	0
$619$	20.6882	0.831529	0.415764	−	0.909472i	$-0.363514\pi$
$619$	20.6882	0.831529	0.415764	+	0.909472i	$0.363514\pi$
$620$	0	0
$621$	13.6309	0.546988
$622$	0	0
$623$	55.1540	2.20970
$624$	0	0
$625$	−14.4687	−0.578747
$626$	0	0
$627$	−4.16726	−0.166424
$628$	0	0
$629$	29.9100	1.19259
$630$	0	0
$631$	0.636298	0.0253306	0.0126653	−	0.999920i	$-0.495968\pi$
$631$	0.636298	0.0253306	0.0126653	+	0.999920i	$0.495968\pi$
$632$	0	0
$633$	−16.0207	−0.636766
$634$	0	0
$635$	14.2636	0.566035
$636$	0	0
$637$	26.2980	1.04197
$638$	0	0
$639$	−6.64759	−0.262975
$640$	0	0
$641$	7.63609	0.301607	0.150804	−	0.988564i	$-0.451814\pi$
$641$	7.63609	0.301607	0.150804	+	0.988564i	$0.451814\pi$
$642$	0	0
$643$	18.3953	0.725442	0.362721	−	0.931898i	$-0.381848\pi$
$643$	18.3953	0.725442	0.362721	+	0.931898i	$0.381848\pi$
$644$	0	0
$645$	−8.48384	−0.334051
$646$	0	0
$647$	−3.95671	−0.155554	−0.0777771	−	0.996971i	$-0.524782\pi$
$647$	−3.95671	−0.155554	−0.0777771	+	0.996971i	$0.524782\pi$
$648$	0	0
$649$	−26.7784	−1.05114
$650$	0	0
$651$	−18.3647	−0.719768
$652$	0	0
$653$	9.96698	0.390038	0.195019	−	0.980799i	$-0.437523\pi$
$653$	9.96698	0.390038	0.195019	+	0.980799i	$0.437523\pi$
$654$	0	0
$655$	8.03911	0.314114
$656$	0	0
$657$	−26.9929	−1.05310
$658$	0	0
$659$	4.18471	0.163013	0.0815065	−	0.996673i	$-0.474027\pi$
$659$	4.18471	0.163013	0.0815065	+	0.996673i	$0.474027\pi$
$660$	0	0
$661$	29.1214	1.13269	0.566345	−	0.824168i	$-0.308357\pi$
$661$	29.1214	1.13269	0.566345	+	0.824168i	$0.308357\pi$
$662$	0	0
$663$	−13.9823	−0.543027
$664$	0	0
$665$	7.41295	0.287462
$666$	0	0
$667$	29.8902	1.15735
$668$	0	0
$669$	8.96610	0.346650
$670$	0	0
$671$	−18.7299	−0.723059
$672$	0	0
$673$	9.55610	0.368361	0.184180	−	0.982892i	$-0.441037\pi$
$673$	9.55610	0.368361	0.184180	+	0.982892i	$0.441037\pi$
$674$	0	0
$675$	6.50540	0.250393
$676$	0	0
$677$	−7.16605	−0.275414	−0.137707	−	0.990473i	$-0.543973\pi$
$677$	−7.16605	−0.275414	−0.137707	+	0.990473i	$0.543973\pi$
$678$	0	0
$679$	5.45736	0.209434
$680$	0	0
$681$	0.0332580	0.00127445
$682$	0	0
$683$	22.1222	0.846482	0.423241	−	0.906017i	$-0.360892\pi$
$683$	22.1222	0.846482	0.423241	+	0.906017i	$0.360892\pi$
$684$	0	0
$685$	21.9599	0.839046
$686$	0	0
$687$	9.17572	0.350075
$688$	0	0
$689$	5.05000	0.192390
$690$	0	0
$691$	18.0707	0.687442	0.343721	−	0.939072i	$-0.388312\pi$
$691$	18.0707	0.687442	0.343721	+	0.939072i	$0.388312\pi$
$692$	0	0
$693$	−54.5027	−2.07038
$694$	0	0
$695$	−5.19684	−0.197128
$696$	0	0
$697$	40.6214	1.53865
$698$	0	0
$699$	11.1124	0.420310
$700$	0	0
$701$	−35.9880	−1.35925	−0.679624	−	0.733561i	$-0.737857\pi$
$701$	−35.9880	−1.35925	−0.679624	+	0.733561i	$0.737857\pi$
$702$	0	0
$703$	4.60848	0.173812
$704$	0	0
$705$	4.41943	0.166445
$706$	0	0
$707$	−73.8255	−2.77649
$708$	0	0
$709$	41.1892	1.54689	0.773446	−	0.633862i	$-0.218531\pi$
$709$	41.1892	1.54689	0.773446	+	0.633862i	$0.218531\pi$
$710$	0	0
$711$	−2.43866	−0.0914569
$712$	0	0
$713$	20.4061	0.764216
$714$	0	0
$715$	−29.5057	−1.10345
$716$	0	0
$717$	−19.5178	−0.728905
$718$	0	0
$719$	−22.3668	−0.834142	−0.417071	−	0.908874i	$-0.636943\pi$
$719$	−22.3668	−0.834142	−0.417071	+	0.908874i	$0.636943\pi$
$720$	0	0
$721$	10.0740	0.375175
$722$	0	0
$723$	9.25802	0.344309
$724$	0	0
$725$	14.2652	0.529798
$726$	0	0
$727$	−46.3700	−1.71977	−0.859884	−	0.510490i	$-0.829464\pi$
$727$	−46.3700	−1.71977	−0.859884	+	0.510490i	$0.829464\pi$
$728$	0	0
$729$	−1.23734	−0.0458272
$730$	0	0
$731$	−39.8361	−1.47339
$732$	0	0
$733$	22.3902	0.827001	0.413501	−	0.910504i	$-0.364306\pi$
$733$	22.3902	0.827001	0.413501	+	0.910504i	$0.364306\pi$
$734$	0	0
$735$	−12.6413	−0.466283
$736$	0	0
$737$	−81.0075	−2.98395
$738$	0	0
$739$	35.8188	1.31761	0.658807	−	0.752312i	$-0.271061\pi$
$739$	35.8188	1.31761	0.658807	+	0.752312i	$0.271061\pi$
$740$	0	0
$741$	−2.15436	−0.0791424
$742$	0	0
$743$	39.4273	1.44645	0.723223	−	0.690615i	$-0.242660\pi$
$743$	39.4273	1.44645	0.723223	+	0.690615i	$0.242660\pi$
$744$	0	0
$745$	−1.81532	−0.0665083
$746$	0	0
$747$	35.5444	1.30050
$748$	0	0
$749$	48.9775	1.78960
$750$	0	0
$751$	−52.5774	−1.91858	−0.959288	−	0.282431i	$-0.908859\pi$
$751$	−52.5774	−1.91858	−0.959288	+	0.282431i	$0.908859\pi$
$752$	0	0
$753$	−6.33359	−0.230809
$754$	0	0
$755$	−36.5816	−1.33134
$756$	0	0
$757$	−24.0296	−0.873371	−0.436686	−	0.899614i	$-0.643848\pi$
$757$	−24.0296	−0.873371	−0.436686	+	0.899614i	$0.643848\pi$
$758$	0	0
$759$	−13.9402	−0.505997
$760$	0	0
$761$	−2.99683	−0.108635	−0.0543175	−	0.998524i	$-0.517298\pi$
$761$	−2.99683	−0.108635	−0.0543175	+	0.998524i	$0.517298\pi$
$762$	0	0
$763$	33.4947	1.21259
$764$	0	0
$765$	−29.1994	−1.05571
$766$	0	0
$767$	−13.8437	−0.499867
$768$	0	0
$769$	47.2249	1.70297	0.851487	−	0.524376i	$-0.175701\pi$
$769$	47.2249	1.70297	0.851487	+	0.524376i	$0.175701\pi$
$770$	0	0
$771$	−4.04682	−0.145743
$772$	0	0
$773$	23.7401	0.853873	0.426936	−	0.904282i	$-0.359593\pi$
$773$	23.7401	0.853873	0.426936	+	0.904282i	$0.359593\pi$
$774$	0	0
$775$	9.73893	0.349833
$776$	0	0
$777$	−13.8739	−0.497723
$778$	0	0
$779$	6.25887	0.224247
$780$	0	0
$781$	15.1618	0.542531
$782$	0	0
$783$	36.4095	1.30117
$784$	0	0
$785$	22.2480	0.794067
$786$	0	0
$787$	−12.6891	−0.452319	−0.226160	−	0.974090i	$-0.572617\pi$
$787$	−12.6891	−0.452319	−0.226160	+	0.974090i	$0.572617\pi$
$788$	0	0
$789$	−8.71999	−0.310440
$790$	0	0
$791$	−4.41460	−0.156965
$792$	0	0
$793$	−9.68285	−0.343848
$794$	0	0
$795$	−2.42751	−0.0860948
$796$	0	0
$797$	12.9149	0.457469	0.228734	−	0.973489i	$-0.426541\pi$
$797$	12.9149	0.457469	0.228734	+	0.973489i	$0.426541\pi$
$798$	0	0
$799$	20.7515	0.734137
$800$	0	0
$801$	−33.4734	−1.18272
$802$	0	0
$803$	61.5654	2.17260
$804$	0	0
$805$	24.7976	0.874002
$806$	0	0
$807$	7.74089	0.272492
$808$	0	0
$809$	−39.5999	−1.39226	−0.696129	−	0.717916i	$-0.745096\pi$
$809$	−39.5999	−1.39226	−0.696129	+	0.717916i	$0.745096\pi$
$810$	0	0
$811$	−19.1902	−0.673859	−0.336929	−	0.941530i	$-0.609388\pi$
$811$	−19.1902	−0.673859	−0.336929	+	0.941530i	$0.609388\pi$
$812$	0	0
$813$	−3.64377	−0.127793
$814$	0	0
$815$	17.0414	0.596934
$816$	0	0
$817$	−6.13786	−0.214737
$818$	0	0
$819$	−28.1764	−0.984564
$820$	0	0
$821$	−3.13951	−0.109570	−0.0547849	−	0.998498i	$-0.517447\pi$
$821$	−3.13951	−0.109570	−0.0547849	+	0.998498i	$0.517447\pi$
$822$	0	0
$823$	−48.6528	−1.69593	−0.847965	−	0.530052i	$-0.822173\pi$
$823$	−48.6528	−1.69593	−0.847965	+	0.530052i	$0.822173\pi$
$824$	0	0
$825$	−6.65303	−0.231629
$826$	0	0
$827$	−14.9569	−0.520102	−0.260051	−	0.965595i	$-0.583739\pi$
$827$	−14.9569	−0.520102	−0.260051	+	0.965595i	$0.583739\pi$
$828$	0	0
$829$	−24.9694	−0.867222	−0.433611	−	0.901100i	$-0.642761\pi$
$829$	−24.9694	−0.867222	−0.433611	+	0.901100i	$0.642761\pi$
$830$	0	0
$831$	12.9169	0.448081
$832$	0	0
$833$	−59.3577	−2.05662
$834$	0	0
$835$	−41.2941	−1.42904
$836$	0	0
$837$	24.8569	0.859179
$838$	0	0
$839$	30.0471	1.03734	0.518670	−	0.854975i	$-0.326427\pi$
$839$	30.0471	1.03734	0.518670	+	0.854975i	$0.326427\pi$
$840$	0	0
$841$	50.8397	1.75309
$842$	0	0
$843$	−24.9789	−0.860320
$844$	0	0
$845$	8.72948	0.300303
$846$	0	0
$847$	80.1095	2.75260
$848$	0	0
$849$	19.6465	0.674266
$850$	0	0
$851$	15.4162	0.528459
$852$	0	0
$853$	−15.0974	−0.516927	−0.258463	−	0.966021i	$-0.583216\pi$
$853$	−15.0974	−0.516927	−0.258463	+	0.966021i	$0.583216\pi$
$854$	0	0
$855$	−4.49898	−0.153862
$856$	0	0
$857$	20.2907	0.693117	0.346559	−	0.938028i	$-0.387350\pi$
$857$	20.2907	0.693117	0.346559	+	0.938028i	$0.387350\pi$
$858$	0	0
$859$	23.0001	0.784754	0.392377	−	0.919804i	$-0.371653\pi$
$859$	23.0001	0.784754	0.392377	+	0.919804i	$0.371653\pi$
$860$	0	0
$861$	−18.8424	−0.642148
$862$	0	0
$863$	−55.9989	−1.90622	−0.953112	−	0.302618i	$-0.902139\pi$
$863$	−55.9989	−1.90622	−0.953112	+	0.302618i	$0.902139\pi$
$864$	0	0
$865$	−1.25876	−0.0427991
$866$	0	0
$867$	18.8228	0.639255
$868$	0	0
$869$	5.56209	0.188681
$870$	0	0
$871$	−41.8787	−1.41901
$872$	0	0
$873$	−3.31212	−0.112098
$874$	0	0
$875$	48.8996	1.65311
$876$	0	0
$877$	−25.3707	−0.856706	−0.428353	−	0.903611i	$-0.640906\pi$
$877$	−25.3707	−0.856706	−0.428353	+	0.903611i	$0.640906\pi$
$878$	0	0
$879$	10.5409	0.355537
$880$	0	0
$881$	−53.9403	−1.81730	−0.908648	−	0.417563i	$-0.862884\pi$
$881$	−53.9403	−1.81730	−0.908648	+	0.417563i	$0.862884\pi$
$882$	0	0
$883$	−13.5811	−0.457040	−0.228520	−	0.973539i	$-0.573389\pi$
$883$	−13.5811	−0.457040	−0.228520	+	0.973539i	$0.573389\pi$
$884$	0	0
$885$	6.65460	0.223692
$886$	0	0
$887$	−27.3444	−0.918136	−0.459068	−	0.888401i	$-0.651817\pi$
$887$	−27.3444	−0.918136	−0.459068	+	0.888401i	$0.651817\pi$
$888$	0	0
$889$	−31.0667	−1.04195
$890$	0	0
$891$	23.7115	0.794363
$892$	0	0
$893$	3.19735	0.106995
$894$	0	0
$895$	16.7763	0.560771
$896$	0	0
$897$	−7.20671	−0.240625
$898$	0	0
$899$	54.5069	1.81791
$900$	0	0
$901$	−11.3984	−0.379737
$902$	0	0
$903$	18.4782	0.614914
$904$	0	0
$905$	−21.4960	−0.714552
$906$	0	0
$907$	10.6268	0.352857	0.176428	−	0.984313i	$-0.443546\pi$
$907$	10.6268	0.352857	0.176428	+	0.984313i	$0.443546\pi$
$908$	0	0
$909$	44.8053	1.48610
$910$	0	0
$911$	1.19860	0.0397112	0.0198556	−	0.999803i	$-0.493679\pi$
$911$	1.19860	0.0397112	0.0198556	+	0.999803i	$0.493679\pi$
$912$	0	0
$913$	−81.0696	−2.68301
$914$	0	0
$915$	4.65450	0.153873
$916$	0	0
$917$	−17.5095	−0.578215
$918$	0	0
$919$	4.56836	0.150696	0.0753482	−	0.997157i	$-0.475993\pi$
$919$	4.56836	0.150696	0.0753482	+	0.997157i	$0.475993\pi$
$920$	0	0
$921$	−14.7815	−0.487067
$922$	0	0
$923$	7.83824	0.257999
$924$	0	0
$925$	7.35744	0.241911
$926$	0	0
$927$	−6.11398	−0.200810
$928$	0	0
$929$	−1.28691	−0.0422221	−0.0211110	−	0.999777i	$-0.506720\pi$
$929$	−1.28691	−0.0422221	−0.0211110	+	0.999777i	$0.506720\pi$
$930$	0	0
$931$	−9.14571	−0.299739
$932$	0	0
$933$	15.7681	0.516225
$934$	0	0
$935$	66.5978	2.17798
$936$	0	0
$937$	16.2002	0.529238	0.264619	−	0.964353i	$-0.414754\pi$
$937$	16.2002	0.529238	0.264619	+	0.964353i	$0.414754\pi$
$938$	0	0
$939$	−2.80985	−0.0916960
$940$	0	0
$941$	−44.6828	−1.45662	−0.728308	−	0.685250i	$-0.759693\pi$
$941$	−44.6828	−1.45662	−0.728308	+	0.685250i	$0.759693\pi$
$942$	0	0
$943$	20.9370	0.681803
$944$	0	0
$945$	30.2062	0.982607
$946$	0	0
$947$	58.0833	1.88745	0.943727	−	0.330726i	$-0.107294\pi$
$947$	58.0833	1.88745	0.943727	+	0.330726i	$0.107294\pi$
$948$	0	0
$949$	31.8277	1.03317
$950$	0	0
$951$	1.89092	0.0613174
$952$	0	0
$953$	2.06232	0.0668050	0.0334025	−	0.999442i	$-0.489366\pi$
$953$	2.06232	0.0668050	0.0334025	+	0.999442i	$0.489366\pi$
$954$	0	0
$955$	25.3826	0.821362
$956$	0	0
$957$	−37.2357	−1.20366
$958$	0	0
$959$	−47.8296	−1.54450
$960$	0	0
$961$	6.21205	0.200389
$962$	0	0
$963$	−29.7249	−0.957870
$964$	0	0
$965$	20.9210	0.673469
$966$	0	0
$967$	−36.0519	−1.15935	−0.579675	−	0.814848i	$-0.696820\pi$
$967$	−36.0519	−1.15935	−0.579675	+	0.814848i	$0.696820\pi$
$968$	0	0
$969$	4.86264	0.156211
$970$	0	0
$971$	−8.14103	−0.261258	−0.130629	−	0.991431i	$-0.541700\pi$
$971$	−8.14103	−0.261258	−0.130629	+	0.991431i	$0.541700\pi$
$972$	0	0
$973$	11.3189	0.362868
$974$	0	0
$975$	−3.43944	−0.110150
$976$	0	0
$977$	23.9271	0.765496	0.382748	−	0.923853i	$-0.374978\pi$
$977$	23.9271	0.765496	0.382748	+	0.923853i	$0.374978\pi$
$978$	0	0
$979$	76.3460	2.44003
$980$	0	0
$981$	−20.3282	−0.649030
$982$	0	0
$983$	−11.7610	−0.375117	−0.187558	−	0.982253i	$-0.560057\pi$
$983$	−11.7610	−0.375117	−0.187558	+	0.982253i	$0.560057\pi$
$984$	0	0
$985$	14.2098	0.452762
$986$	0	0
$987$	−9.62569	−0.306389
$988$	0	0
$989$	−20.5322	−0.652887
$990$	0	0
$991$	13.2370	0.420489	0.210244	−	0.977649i	$-0.432574\pi$
$991$	13.2370	0.420489	0.210244	+	0.977649i	$0.432574\pi$
$992$	0	0
$993$	−6.72540	−0.213424
$994$	0	0
$995$	14.7901	0.468879
$996$	0	0
$997$	−0.616237	−0.0195164	−0.00975821	−	0.999952i	$-0.503106\pi$
$997$	−0.616237	−0.0195164	−0.00975821	+	0.999952i	$0.503106\pi$
$998$	0	0
$999$	18.7785	0.594127

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.f.1.15	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.f.1.15	✓	25	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)

\(f(q)\)	\(=\)	\(q+0.749226 q^{3} -1.84486 q^{5} +4.01817 q^{7} -2.43866 q^{9} +O(q^{10})\)	\(q+0.749226 q^{3} -1.84486 q^{5} +4.01817 q^{7} -2.43866 q^{9} +5.56209 q^{11} +2.87545 q^{13} -1.38221 q^{15} -6.49022 q^{17} -1.00000 q^{19} +3.01052 q^{21} -3.34518 q^{23} -1.59650 q^{25} -4.07478 q^{27} -8.93531 q^{29} -6.10017 q^{31} +4.16726 q^{33} -7.41295 q^{35} -4.60848 q^{37} +2.15436 q^{39} -6.25887 q^{41} +6.13786 q^{43} +4.49898 q^{45} -3.19735 q^{47} +9.14571 q^{49} -4.86264 q^{51} +1.75625 q^{53} -10.2613 q^{55} -0.749226 q^{57} -4.81445 q^{59} -3.36742 q^{61} -9.79896 q^{63} -5.30479 q^{65} -14.5642 q^{67} -2.50629 q^{69} +2.72592 q^{71} +11.0688 q^{73} -1.19614 q^{75} +22.3494 q^{77} +1.00000 q^{79} +4.26305 q^{81} -14.5754 q^{83} +11.9735 q^{85} -6.69456 q^{87} +13.7261 q^{89} +11.5541 q^{91} -4.57040 q^{93} +1.84486 q^{95} +1.35817 q^{97} -13.5640 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9}+O(q^{10}) \) 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9	\( 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9} - 3 q^{11} + q^{13} - 5 q^{15} - 13 q^{17} - 25 q^{19} - 24 q^{21} - 31 q^{23} + 21 q^{25} + 7 q^{27} - 19 q^{29} - 7 q^{31} - 30 q^{33} - q^{35} - 29 q^{37} - 26 q^{39} - 40 q^{41} - 40 q^{45} - 8 q^{47} - 9 q^{49} + 12 q^{51} - 38 q^{53} - 29 q^{55} - 4 q^{57} + 18 q^{59} - 26 q^{61} - 40 q^{63} - 70 q^{65} - 13 q^{67} + q^{69} - 47 q^{71} - 8 q^{73} + 7 q^{75} - 19 q^{77} + 25 q^{79} - 19 q^{81} - 8 q^{83} - 33 q^{85} - 50 q^{87} - 54 q^{89} - 12 q^{91} - 24 q^{93} + 8 q^{95} - 4 q^{97} - 39 q^{99}+O(q^{100}) \) 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9 - 3 * q^11 + q^13 - 5 * q^15 - 13 * q^17 - 25 * q^19 - 24 * q^21 - 31 * q^23 + 21 * q^25 + 7 * q^27 - 19 * q^29 - 7 * q^31 - 30 * q^33 - q^35 - 29 * q^37 - 26 * q^39 - 40 * q^41 - 40 * q^45 - 8 * q^47 - 9 * q^49 + 12 * q^51 - 38 * q^53 - 29 * q^55 - 4 * q^57 + 18 * q^59 - 26 * q^61 - 40 * q^63 - 70 * q^65 - 13 * q^67 + q^69 - 47 * q^71 - 8 * q^73 + 7 * q^75 - 19 * q^77 + 25 * q^79 - 19 * q^81 - 8 * q^83 - 33 * q^85 - 50 * q^87 - 54 * q^89 - 12 * q^91 - 24 * q^93 + 8 * q^95 - 4 * q^97 - 39 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more