LMFDB - Embedded newform 2004.2.a.d.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2004,2,Mod(1,2004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2004, 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("2004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2004 = 2^{2} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2004.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:	$16.0020205651$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 29x^{7} - 7x^{6} + 266x^{5} + 69x^{4} - 901x^{3} - 199x^{2} + 875x + 391 $ x^9 - 29x^7 - 7x^6 + 266x^5 + 69x^4 - 901x^3 - 199x^2 + 875*x + 391
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.907808$ of defining polynomial
Character	$\chi$	$=$	2004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.90781 q^{5} +2.81337 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.90781 q^{5} +2.81337 q^{7} +1.00000 q^{9} +0.318882 q^{11} +5.12583 q^{13} +1.90781 q^{15} +3.73184 q^{17} -0.725453 q^{19} +2.81337 q^{21} +0.612463 q^{23} -1.36027 q^{25} +1.00000 q^{27} -3.87979 q^{29} -6.65817 q^{31} +0.318882 q^{33} +5.36737 q^{35} -9.04643 q^{37} +5.12583 q^{39} +9.55805 q^{41} -10.9328 q^{43} +1.90781 q^{45} +8.76460 q^{47} +0.915046 q^{49} +3.73184 q^{51} -2.46673 q^{53} +0.608366 q^{55} -0.725453 q^{57} -11.0871 q^{59} +12.5460 q^{61} +2.81337 q^{63} +9.77910 q^{65} +5.21723 q^{67} +0.612463 q^{69} -11.3217 q^{71} +2.88490 q^{73} -1.36027 q^{75} +0.897133 q^{77} +8.98655 q^{79} +1.00000 q^{81} +4.04802 q^{83} +7.11963 q^{85} -3.87979 q^{87} -9.15294 q^{89} +14.4208 q^{91} -6.65817 q^{93} -1.38402 q^{95} -9.62101 q^{97} +0.318882 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{3} + 9 q^{5} + 2 q^{7} + 9 q^{9}+O(q^{10}) $ 9 * q + 9 * q^3 + 9 * q^5 + 2 * q^7 + 9 * q^9	$ 9 q + 9 q^{3} + 9 q^{5} + 2 q^{7} + 9 q^{9} + 7 q^{11} + 6 q^{13} + 9 q^{15} + 7 q^{17} + 2 q^{19} + 2 q^{21} + 19 q^{23} + 22 q^{25} + 9 q^{27} + 13 q^{29} + 12 q^{31} + 7 q^{33} + 4 q^{35} + 15 q^{37} + 6 q^{39} + 18 q^{41} - 6 q^{43} + 9 q^{45} + 25 q^{47} + 19 q^{49} + 7 q^{51} + 17 q^{53} - 3 q^{55} + 2 q^{57} + 3 q^{59} + 14 q^{61} + 2 q^{63} + 14 q^{65} - 4 q^{67} + 19 q^{69} + 17 q^{71} - 20 q^{73} + 22 q^{75} + 14 q^{77} - 8 q^{79} + 9 q^{81} - q^{83} + 5 q^{85} + 13 q^{87} + 36 q^{89} - 41 q^{91} + 12 q^{93} + 5 q^{95} + 31 q^{97} + 7 q^{99}+O(q^{100}) $ 9 * q + 9 * q^3 + 9 * q^5 + 2 * q^7 + 9 * q^9 + 7 * q^11 + 6 * q^13 + 9 * q^15 + 7 * q^17 + 2 * q^19 + 2 * q^21 + 19 * q^23 + 22 * q^25 + 9 * q^27 + 13 * q^29 + 12 * q^31 + 7 * q^33 + 4 * q^35 + 15 * q^37 + 6 * q^39 + 18 * q^41 - 6 * q^43 + 9 * q^45 + 25 * q^47 + 19 * q^49 + 7 * q^51 + 17 * q^53 - 3 * q^55 + 2 * q^57 + 3 * q^59 + 14 * q^61 + 2 * q^63 + 14 * q^65 - 4 * q^67 + 19 * q^69 + 17 * q^71 - 20 * q^73 + 22 * q^75 + 14 * q^77 - 8 * q^79 + 9 * q^81 - q^83 + 5 * q^85 + 13 * q^87 + 36 * q^89 - 41 * q^91 + 12 * q^93 + 5 * q^95 + 31 * q^97 + 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.90781	0.853198	0.426599	−	0.904441i	$-0.359712\pi$
$5$	1.90781	0.853198	0.426599	+	0.904441i	$0.359712\pi$
$6$	0	0
$7$	2.81337	1.06335	0.531677	−	0.846947i	$-0.321562\pi$
$7$	2.81337	1.06335	0.531677	+	0.846947i	$0.321562\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.318882	0.0961466	0.0480733	−	0.998844i	$-0.484692\pi$
$11$	0.318882	0.0961466	0.0480733	+	0.998844i	$0.484692\pi$
$12$	0	0
$13$	5.12583	1.42165	0.710824	−	0.703370i	$-0.248322\pi$
$13$	5.12583	1.42165	0.710824	+	0.703370i	$0.248322\pi$
$14$	0	0
$15$	1.90781	0.492594
$16$	0	0
$17$	3.73184	0.905104	0.452552	−	0.891738i	$-0.350514\pi$
$17$	3.73184	0.905104	0.452552	+	0.891738i	$0.350514\pi$
$18$	0	0
$19$	−0.725453	−0.166430	−0.0832151	−	0.996532i	$-0.526519\pi$
$19$	−0.725453	−0.166430	−0.0832151	+	0.996532i	$0.526519\pi$
$20$	0	0
$21$	2.81337	0.613927
$22$	0	0
$23$	0.612463	0.127707	0.0638537	−	0.997959i	$-0.479661\pi$
$23$	0.612463	0.127707	0.0638537	+	0.997959i	$0.479661\pi$
$24$	0	0
$25$	−1.36027	−0.272053
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.87979	−0.720459	−0.360229	−	0.932864i	$-0.617302\pi$
$29$	−3.87979	−0.720459	−0.360229	+	0.932864i	$0.617302\pi$
$30$	0	0
$31$	−6.65817	−1.19584	−0.597922	−	0.801554i	$-0.704007\pi$
$31$	−6.65817	−1.19584	−0.597922	+	0.801554i	$0.704007\pi$
$32$	0	0
$33$	0.318882	0.0555103
$34$	0	0
$35$	5.36737	0.907251
$36$	0	0
$37$	−9.04643	−1.48722	−0.743612	−	0.668612i	$-0.766889\pi$
$37$	−9.04643	−1.48722	−0.743612	+	0.668612i	$0.766889\pi$
$38$	0	0
$39$	5.12583	0.820789
$40$	0	0
$41$	9.55805	1.49272	0.746358	−	0.665545i	$-0.231801\pi$
$41$	9.55805	1.49272	0.746358	+	0.665545i	$0.231801\pi$
$42$	0	0
$43$	−10.9328	−1.66724	−0.833620	−	0.552338i	$-0.813736\pi$
$43$	−10.9328	−1.66724	−0.833620	+	0.552338i	$0.813736\pi$
$44$	0	0
$45$	1.90781	0.284399
$46$	0	0
$47$	8.76460	1.27845	0.639224	−	0.769020i	$-0.279256\pi$
$47$	8.76460	1.27845	0.639224	+	0.769020i	$0.279256\pi$
$48$	0	0
$49$	0.915046	0.130721
$50$	0	0
$51$	3.73184	0.522562
$52$	0	0
$53$	−2.46673	−0.338831	−0.169416	−	0.985545i	$-0.554188\pi$
$53$	−2.46673	−0.338831	−0.169416	+	0.985545i	$0.554188\pi$
$54$	0	0
$55$	0.608366	0.0820321
$56$	0	0
$57$	−0.725453	−0.0960885
$58$	0	0
$59$	−11.0871	−1.44341	−0.721706	−	0.692200i	$-0.756642\pi$
$59$	−11.0871	−1.44341	−0.721706	+	0.692200i	$0.756642\pi$
$60$	0	0
$61$	12.5460	1.60636	0.803178	−	0.595739i	$-0.203141\pi$
$61$	12.5460	1.60636	0.803178	+	0.595739i	$0.203141\pi$
$62$	0	0
$63$	2.81337	0.354451
$64$	0	0
$65$	9.77910	1.21295
$66$	0	0
$67$	5.21723	0.637387	0.318693	−	0.947858i	$-0.396756\pi$
$67$	5.21723	0.637387	0.318693	+	0.947858i	$0.396756\pi$
$68$	0	0
$69$	0.612463	0.0737319
$70$	0	0
$71$	−11.3217	−1.34363	−0.671816	−	0.740718i	$-0.734486\pi$
$71$	−11.3217	−1.34363	−0.671816	+	0.740718i	$0.734486\pi$
$72$	0	0
$73$	2.88490	0.337652	0.168826	−	0.985646i	$-0.446002\pi$
$73$	2.88490	0.337652	0.168826	+	0.985646i	$0.446002\pi$
$74$	0	0
$75$	−1.36027	−0.157070
$76$	0	0
$77$	0.897133	0.102238
$78$	0	0
$79$	8.98655	1.01107	0.505533	−	0.862807i	$-0.331296\pi$
$79$	8.98655	1.01107	0.505533	+	0.862807i	$0.331296\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.04802	0.444327	0.222164	−	0.975009i	$-0.428688\pi$
$83$	4.04802	0.444327	0.222164	+	0.975009i	$0.428688\pi$
$84$	0	0
$85$	7.11963	0.772233
$86$	0	0
$87$	−3.87979	−0.415957
$88$	0	0
$89$	−9.15294	−0.970210	−0.485105	−	0.874456i	$-0.661219\pi$
$89$	−9.15294	−0.970210	−0.485105	+	0.874456i	$0.661219\pi$
$90$	0	0
$91$	14.4208	1.51172
$92$	0	0
$93$	−6.65817	−0.690421
$94$	0	0
$95$	−1.38402	−0.141998
$96$	0	0
$97$	−9.62101	−0.976866	−0.488433	−	0.872601i	$-0.662431\pi$
$97$	−9.62101	−0.976866	−0.488433	+	0.872601i	$0.662431\pi$
$98$	0	0
$99$	0.318882	0.0320489
$100$	0	0
$101$	−13.2834	−1.32175	−0.660875	−	0.750496i	$-0.729815\pi$
$101$	−13.2834	−1.32175	−0.660875	+	0.750496i	$0.729815\pi$
$102$	0	0
$103$	−18.4226	−1.81523	−0.907616	−	0.419802i	$-0.862099\pi$
$103$	−18.4226	−1.81523	−0.907616	+	0.419802i	$0.862099\pi$
$104$	0	0
$105$	5.36737	0.523802
$106$	0	0
$107$	11.8365	1.14428	0.572139	−	0.820157i	$-0.306114\pi$
$107$	11.8365	1.14428	0.572139	+	0.820157i	$0.306114\pi$
$108$	0	0
$109$	−2.94111	−0.281708	−0.140854	−	0.990030i	$-0.544985\pi$
$109$	−2.94111	−0.281708	−0.140854	+	0.990030i	$0.544985\pi$
$110$	0	0
$111$	−9.04643	−0.858649
$112$	0	0
$113$	16.9782	1.59718	0.798588	−	0.601878i	$-0.205581\pi$
$113$	16.9782	1.59718	0.798588	+	0.601878i	$0.205581\pi$
$114$	0	0
$115$	1.16846	0.108960
$116$	0	0
$117$	5.12583	0.473883
$118$	0	0
$119$	10.4990	0.962445
$120$	0	0
$121$	−10.8983	−0.990756
$122$	0	0
$123$	9.55805	0.861820
$124$	0	0
$125$	−12.1342	−1.08531
$126$	0	0
$127$	16.7091	1.48269	0.741345	−	0.671124i	$-0.234188\pi$
$127$	16.7091	1.48269	0.741345	+	0.671124i	$0.234188\pi$
$128$	0	0
$129$	−10.9328	−0.962582
$130$	0	0
$131$	−0.146536	−0.0128029	−0.00640144	−	0.999980i	$-0.502038\pi$
$131$	−0.146536	−0.0128029	−0.00640144	+	0.999980i	$0.502038\pi$
$132$	0	0
$133$	−2.04097	−0.176974
$134$	0	0
$135$	1.90781	0.164198
$136$	0	0
$137$	11.7309	1.00224	0.501120	−	0.865378i	$-0.332922\pi$
$137$	11.7309	1.00224	0.501120	+	0.865378i	$0.332922\pi$
$138$	0	0
$139$	−9.09999	−0.771851	−0.385926	−	0.922530i	$-0.626118\pi$
$139$	−9.09999	−0.771851	−0.385926	+	0.922530i	$0.626118\pi$
$140$	0	0
$141$	8.76460	0.738113
$142$	0	0
$143$	1.63454	0.136687
$144$	0	0
$145$	−7.40189	−0.614694
$146$	0	0
$147$	0.915046	0.0754717
$148$	0	0
$149$	23.3492	1.91284	0.956422	−	0.291988i	$-0.0943168\pi$
$149$	23.3492	1.91284	0.956422	+	0.291988i	$0.0943168\pi$
$150$	0	0
$151$	−0.859951	−0.0699818	−0.0349909	−	0.999388i	$-0.511140\pi$
$151$	−0.859951	−0.0699818	−0.0349909	+	0.999388i	$0.511140\pi$
$152$	0	0
$153$	3.73184	0.301701
$154$	0	0
$155$	−12.7025	−1.02029
$156$	0	0
$157$	−6.36080	−0.507647	−0.253824	−	0.967251i	$-0.581688\pi$
$157$	−6.36080	−0.507647	−0.253824	+	0.967251i	$0.581688\pi$
$158$	0	0
$159$	−2.46673	−0.195624
$160$	0	0
$161$	1.72309	0.135798
$162$	0	0
$163$	−0.722678	−0.0566045	−0.0283023	−	0.999599i	$-0.509010\pi$
$163$	−0.722678	−0.0566045	−0.0283023	+	0.999599i	$0.509010\pi$
$164$	0	0
$165$	0.608366	0.0473612
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	13.2741	1.02109
$170$	0	0
$171$	−0.725453	−0.0554767
$172$	0	0
$173$	16.1210	1.22566	0.612829	−	0.790216i	$-0.290032\pi$
$173$	16.1210	1.22566	0.612829	+	0.790216i	$0.290032\pi$
$174$	0	0
$175$	−3.82693	−0.289289
$176$	0	0
$177$	−11.0871	−0.833354
$178$	0	0
$179$	11.5480	0.863137	0.431569	−	0.902080i	$-0.357960\pi$
$179$	11.5480	0.863137	0.431569	+	0.902080i	$0.357960\pi$
$180$	0	0
$181$	−3.91786	−0.291212	−0.145606	−	0.989343i	$-0.546513\pi$
$181$	−3.91786	−0.291212	−0.145606	+	0.989343i	$0.546513\pi$
$182$	0	0
$183$	12.5460	0.927430
$184$	0	0
$185$	−17.2588	−1.26890
$186$	0	0
$187$	1.19002	0.0870226
$188$	0	0
$189$	2.81337	0.204642
$190$	0	0
$191$	−19.5571	−1.41510	−0.707549	−	0.706664i	$-0.750199\pi$
$191$	−19.5571	−1.41510	−0.707549	+	0.706664i	$0.750199\pi$
$192$	0	0
$193$	22.0005	1.58363	0.791814	−	0.610762i	$-0.209137\pi$
$193$	22.0005	1.58363	0.791814	+	0.610762i	$0.209137\pi$
$194$	0	0
$195$	9.77910	0.700296
$196$	0	0
$197$	10.3862	0.739989	0.369995	−	0.929034i	$-0.379360\pi$
$197$	10.3862	0.739989	0.369995	+	0.929034i	$0.379360\pi$
$198$	0	0
$199$	−17.3380	−1.22906	−0.614528	−	0.788895i	$-0.710653\pi$
$199$	−17.3380	−1.22906	−0.614528	+	0.788895i	$0.710653\pi$
$200$	0	0
$201$	5.21723	0.367995
$202$	0	0
$203$	−10.9153	−0.766102
$204$	0	0
$205$	18.2349	1.27358
$206$	0	0
$207$	0.612463	0.0425691
$208$	0	0
$209$	−0.231334	−0.0160017
$210$	0	0
$211$	−20.2222	−1.39216	−0.696078	−	0.717966i	$-0.745073\pi$
$211$	−20.2222	−1.39216	−0.696078	+	0.717966i	$0.745073\pi$
$212$	0	0
$213$	−11.3217	−0.775747
$214$	0	0
$215$	−20.8577	−1.42249
$216$	0	0
$217$	−18.7319	−1.27160
$218$	0	0
$219$	2.88490	0.194944
$220$	0	0
$221$	19.1288	1.28674
$222$	0	0
$223$	9.60888	0.643458	0.321729	−	0.946832i	$-0.395736\pi$
$223$	9.60888	0.643458	0.321729	+	0.946832i	$0.395736\pi$
$224$	0	0
$225$	−1.36027	−0.0906845
$226$	0	0
$227$	23.3573	1.55028	0.775140	−	0.631790i	$-0.217679\pi$
$227$	23.3573	1.55028	0.775140	+	0.631790i	$0.217679\pi$
$228$	0	0
$229$	−14.5969	−0.964591	−0.482296	−	0.876008i	$-0.660197\pi$
$229$	−14.5969	−0.964591	−0.482296	+	0.876008i	$0.660197\pi$
$230$	0	0
$231$	0.897133	0.0590270
$232$	0	0
$233$	18.0225	1.18069	0.590345	−	0.807151i	$-0.298992\pi$
$233$	18.0225	1.18069	0.590345	+	0.807151i	$0.298992\pi$
$234$	0	0
$235$	16.7212	1.09077
$236$	0	0
$237$	8.98655	0.583739
$238$	0	0
$239$	−9.44595	−0.611007	−0.305504	−	0.952191i	$-0.598825\pi$
$239$	−9.44595	−0.611007	−0.305504	+	0.952191i	$0.598825\pi$
$240$	0	0
$241$	5.88509	0.379092	0.189546	−	0.981872i	$-0.439298\pi$
$241$	5.88509	0.379092	0.189546	+	0.981872i	$0.439298\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.74573	0.111531
$246$	0	0
$247$	−3.71854	−0.236605
$248$	0	0
$249$	4.04802	0.256533
$250$	0	0
$251$	3.14251	0.198353	0.0991767	−	0.995070i	$-0.468379\pi$
$251$	3.14251	0.198353	0.0991767	+	0.995070i	$0.468379\pi$
$252$	0	0
$253$	0.195304	0.0122786
$254$	0	0
$255$	7.11963	0.445849
$256$	0	0
$257$	1.52800	0.0953142	0.0476571	−	0.998864i	$-0.484825\pi$
$257$	1.52800	0.0953142	0.0476571	+	0.998864i	$0.484825\pi$
$258$	0	0
$259$	−25.4509	−1.58144
$260$	0	0
$261$	−3.87979	−0.240153
$262$	0	0
$263$	−24.3283	−1.50015	−0.750075	−	0.661353i	$-0.769983\pi$
$263$	−24.3283	−1.50015	−0.750075	+	0.661353i	$0.769983\pi$
$264$	0	0
$265$	−4.70604	−0.289090
$266$	0	0
$267$	−9.15294	−0.560151
$268$	0	0
$269$	0.163618	0.00997594	0.00498797	−	0.999988i	$-0.498412\pi$
$269$	0.163618	0.00997594	0.00498797	+	0.999988i	$0.498412\pi$
$270$	0	0
$271$	−8.88391	−0.539659	−0.269829	−	0.962908i	$-0.586967\pi$
$271$	−8.88391	−0.539659	−0.269829	+	0.962908i	$0.586967\pi$
$272$	0	0
$273$	14.4208	0.872789
$274$	0	0
$275$	−0.433765	−0.0261570
$276$	0	0
$277$	−5.45934	−0.328020	−0.164010	−	0.986459i	$-0.552443\pi$
$277$	−5.45934	−0.328020	−0.164010	+	0.986459i	$0.552443\pi$
$278$	0	0
$279$	−6.65817	−0.398614
$280$	0	0
$281$	−8.18240	−0.488121	−0.244061	−	0.969760i	$-0.578480\pi$
$281$	−8.18240	−0.488121	−0.244061	+	0.969760i	$0.578480\pi$
$282$	0	0
$283$	9.17416	0.545347	0.272674	−	0.962107i	$-0.412092\pi$
$283$	9.17416	0.545347	0.272674	+	0.962107i	$0.412092\pi$
$284$	0	0
$285$	−1.38402	−0.0819825
$286$	0	0
$287$	26.8903	1.58728
$288$	0	0
$289$	−3.07338	−0.180787
$290$	0	0
$291$	−9.62101	−0.563994
$292$	0	0
$293$	−30.6446	−1.79027	−0.895137	−	0.445790i	$-0.852923\pi$
$293$	−30.6446	−1.79027	−0.895137	+	0.445790i	$0.852923\pi$
$294$	0	0
$295$	−21.1520	−1.23152
$296$	0	0
$297$	0.318882	0.0185034
$298$	0	0
$299$	3.13938	0.181555
$300$	0	0
$301$	−30.7581	−1.77287
$302$	0	0
$303$	−13.2834	−0.763113
$304$	0	0
$305$	23.9354	1.37054
$306$	0	0
$307$	−25.0225	−1.42811	−0.714055	−	0.700090i	$-0.753143\pi$
$307$	−25.0225	−1.42811	−0.714055	+	0.700090i	$0.753143\pi$
$308$	0	0
$309$	−18.4226	−1.04802
$310$	0	0
$311$	26.4473	1.49969	0.749845	−	0.661613i	$-0.230128\pi$
$311$	26.4473	1.49969	0.749845	+	0.661613i	$0.230128\pi$
$312$	0	0
$313$	22.3927	1.26571	0.632854	−	0.774271i	$-0.281883\pi$
$313$	22.3927	1.26571	0.632854	+	0.774271i	$0.281883\pi$
$314$	0	0
$315$	5.36737	0.302417
$316$	0	0
$317$	24.1684	1.35743	0.678717	−	0.734400i	$-0.262536\pi$
$317$	24.1684	1.35743	0.678717	+	0.734400i	$0.262536\pi$
$318$	0	0
$319$	−1.23720	−0.0692697
$320$	0	0
$321$	11.8365	0.660649
$322$	0	0
$323$	−2.70727	−0.150637
$324$	0	0
$325$	−6.97249	−0.386764
$326$	0	0
$327$	−2.94111	−0.162644
$328$	0	0
$329$	24.6581	1.35944
$330$	0	0
$331$	−28.8009	−1.58304	−0.791521	−	0.611142i	$-0.790710\pi$
$331$	−28.8009	−1.58304	−0.791521	+	0.611142i	$0.790710\pi$
$332$	0	0
$333$	−9.04643	−0.495741
$334$	0	0
$335$	9.95348	0.543817
$336$	0	0
$337$	−2.40083	−0.130782	−0.0653909	−	0.997860i	$-0.520829\pi$
$337$	−2.40083	−0.130782	−0.0653909	+	0.997860i	$0.520829\pi$
$338$	0	0
$339$	16.9782	0.922130
$340$	0	0
$341$	−2.12317	−0.114976
$342$	0	0
$343$	−17.1192	−0.924351
$344$	0	0
$345$	1.16846	0.0629079
$346$	0	0
$347$	−19.3977	−1.04132	−0.520662	−	0.853763i	$-0.674315\pi$
$347$	−19.3977	−1.04132	−0.520662	+	0.853763i	$0.674315\pi$
$348$	0	0
$349$	−36.7633	−1.96789	−0.983947	−	0.178462i	$-0.942888\pi$
$349$	−36.7633	−1.96789	−0.983947	+	0.178462i	$0.942888\pi$
$350$	0	0
$351$	5.12583	0.273596
$352$	0	0
$353$	−25.4854	−1.35645	−0.678226	−	0.734853i	$-0.737251\pi$
$353$	−25.4854	−1.35645	−0.678226	+	0.734853i	$0.737251\pi$
$354$	0	0
$355$	−21.5995	−1.14638
$356$	0	0
$357$	10.4990	0.555668
$358$	0	0
$359$	−7.51148	−0.396441	−0.198220	−	0.980157i	$-0.563516\pi$
$359$	−7.51148	−0.396441	−0.198220	+	0.980157i	$0.563516\pi$
$360$	0	0
$361$	−18.4737	−0.972301
$362$	0	0
$363$	−10.8983	−0.572013
$364$	0	0
$365$	5.50384	0.288084
$366$	0	0
$367$	15.8402	0.826853	0.413426	−	0.910537i	$-0.364332\pi$
$367$	15.8402	0.826853	0.413426	+	0.910537i	$0.364332\pi$
$368$	0	0
$369$	9.55805	0.497572
$370$	0	0
$371$	−6.93981	−0.360297
$372$	0	0
$373$	34.3907	1.78068	0.890340	−	0.455296i	$-0.150466\pi$
$373$	34.3907	1.78068	0.890340	+	0.455296i	$0.150466\pi$
$374$	0	0
$375$	−12.1342	−0.626606
$376$	0	0
$377$	−19.8871	−1.02424
$378$	0	0
$379$	−13.6557	−0.701448	−0.350724	−	0.936479i	$-0.614064\pi$
$379$	−13.6557	−0.701448	−0.350724	+	0.936479i	$0.614064\pi$
$380$	0	0
$381$	16.7091	0.856032
$382$	0	0
$383$	12.8238	0.655263	0.327632	−	0.944806i	$-0.393750\pi$
$383$	12.8238	0.655263	0.327632	+	0.944806i	$0.393750\pi$
$384$	0	0
$385$	1.71156	0.0872291
$386$	0	0
$387$	−10.9328	−0.555747
$388$	0	0
$389$	−1.34694	−0.0682925	−0.0341462	−	0.999417i	$-0.510871\pi$
$389$	−1.34694	−0.0682925	−0.0341462	+	0.999417i	$0.510871\pi$
$390$	0	0
$391$	2.28561	0.115588
$392$	0	0
$393$	−0.146536	−0.00739175
$394$	0	0
$395$	17.1446	0.862639
$396$	0	0
$397$	9.21493	0.462484	0.231242	−	0.972896i	$-0.425721\pi$
$397$	9.21493	0.462484	0.231242	+	0.972896i	$0.425721\pi$
$398$	0	0
$399$	−2.04097	−0.102176
$400$	0	0
$401$	5.28032	0.263686	0.131843	−	0.991271i	$-0.457910\pi$
$401$	5.28032	0.263686	0.131843	+	0.991271i	$0.457910\pi$
$402$	0	0
$403$	−34.1287	−1.70007
$404$	0	0
$405$	1.90781	0.0947998
$406$	0	0
$407$	−2.88474	−0.142991
$408$	0	0
$409$	36.9707	1.82808	0.914042	−	0.405619i	$-0.132944\pi$
$409$	36.9707	1.82808	0.914042	+	0.405619i	$0.132944\pi$
$410$	0	0
$411$	11.7309	0.578643
$412$	0	0
$413$	−31.1920	−1.53486
$414$	0	0
$415$	7.72284	0.379099
$416$	0	0
$417$	−9.09999	−0.445629
$418$	0	0
$419$	−12.2542	−0.598655	−0.299328	−	0.954150i	$-0.596762\pi$
$419$	−12.2542	−0.598655	−0.299328	+	0.954150i	$0.596762\pi$
$420$	0	0
$421$	33.4130	1.62845	0.814225	−	0.580550i	$-0.197162\pi$
$421$	33.4130	1.62845	0.814225	+	0.580550i	$0.197162\pi$
$422$	0	0
$423$	8.76460	0.426150
$424$	0	0
$425$	−5.07630	−0.246237
$426$	0	0
$427$	35.2966	1.70812
$428$	0	0
$429$	1.63454	0.0789161
$430$	0	0
$431$	−15.5835	−0.750632	−0.375316	−	0.926897i	$-0.622466\pi$
$431$	−15.5835	−0.750632	−0.375316	+	0.926897i	$0.622466\pi$
$432$	0	0
$433$	−14.5888	−0.701095	−0.350547	−	0.936545i	$-0.614004\pi$
$433$	−14.5888	−0.701095	−0.350547	+	0.936545i	$0.614004\pi$
$434$	0	0
$435$	−7.40189	−0.354894
$436$	0	0
$437$	−0.444313	−0.0212544
$438$	0	0
$439$	0.127708	0.00609515	0.00304758	−	0.999995i	$-0.499030\pi$
$439$	0.127708	0.00609515	0.00304758	+	0.999995i	$0.499030\pi$
$440$	0	0
$441$	0.915046	0.0435736
$442$	0	0
$443$	1.30938	0.0622103	0.0311052	−	0.999516i	$-0.490097\pi$
$443$	1.30938	0.0622103	0.0311052	+	0.999516i	$0.490097\pi$
$444$	0	0
$445$	−17.4621	−0.827781
$446$	0	0
$447$	23.3492	1.10438
$448$	0	0
$449$	−0.999933	−0.0471898	−0.0235949	−	0.999722i	$-0.507511\pi$
$449$	−0.999933	−0.0471898	−0.0235949	+	0.999722i	$0.507511\pi$
$450$	0	0
$451$	3.04789	0.143520
$452$	0	0
$453$	−0.859951	−0.0404040
$454$	0	0
$455$	27.5122	1.28979
$456$	0	0
$457$	−22.5930	−1.05686	−0.528429	−	0.848977i	$-0.677219\pi$
$457$	−22.5930	−1.05686	−0.528429	+	0.848977i	$0.677219\pi$
$458$	0	0
$459$	3.73184	0.174187
$460$	0	0
$461$	16.4897	0.768002	0.384001	−	0.923333i	$-0.374546\pi$
$461$	16.4897	0.768002	0.384001	+	0.923333i	$0.374546\pi$
$462$	0	0
$463$	11.7668	0.546850	0.273425	−	0.961893i	$-0.411843\pi$
$463$	11.7668	0.546850	0.273425	+	0.961893i	$0.411843\pi$
$464$	0	0
$465$	−12.7025	−0.589065
$466$	0	0
$467$	−24.6075	−1.13870	−0.569349	−	0.822096i	$-0.692805\pi$
$467$	−24.6075	−1.13870	−0.569349	+	0.822096i	$0.692805\pi$
$468$	0	0
$469$	14.6780	0.677767
$470$	0	0
$471$	−6.36080	−0.293090
$472$	0	0
$473$	−3.48629	−0.160300
$474$	0	0
$475$	0.986809	0.0452779
$476$	0	0
$477$	−2.46673	−0.112944
$478$	0	0
$479$	−1.30934	−0.0598252	−0.0299126	−	0.999553i	$-0.509523\pi$
$479$	−1.30934	−0.0598252	−0.0299126	+	0.999553i	$0.509523\pi$
$480$	0	0
$481$	−46.3704	−2.11431
$482$	0	0
$483$	1.72309	0.0784031
$484$	0	0
$485$	−18.3550	−0.833460
$486$	0	0
$487$	2.87718	0.130377	0.0651886	−	0.997873i	$-0.479235\pi$
$487$	2.87718	0.130377	0.0651886	+	0.997873i	$0.479235\pi$
$488$	0	0
$489$	−0.722678	−0.0326806
$490$	0	0
$491$	−2.52068	−0.113757	−0.0568783	−	0.998381i	$-0.518115\pi$
$491$	−2.52068	−0.113757	−0.0568783	+	0.998381i	$0.518115\pi$
$492$	0	0
$493$	−14.4787	−0.652090
$494$	0	0
$495$	0.608366	0.0273440
$496$	0	0
$497$	−31.8520	−1.42876
$498$	0	0
$499$	−10.9276	−0.489185	−0.244592	−	0.969626i	$-0.578654\pi$
$499$	−10.9276	−0.489185	−0.244592	+	0.969626i	$0.578654\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	4.15303	0.185175	0.0925873	−	0.995705i	$-0.470486\pi$
$503$	4.15303	0.185175	0.0925873	+	0.995705i	$0.470486\pi$
$504$	0	0
$505$	−25.3422	−1.12771
$506$	0	0
$507$	13.2741	0.589524
$508$	0	0
$509$	8.73062	0.386978	0.193489	−	0.981102i	$-0.438020\pi$
$509$	8.73062	0.386978	0.193489	+	0.981102i	$0.438020\pi$
$510$	0	0
$511$	8.11629	0.359044
$512$	0	0
$513$	−0.725453	−0.0320295
$514$	0	0
$515$	−35.1468	−1.54875
$516$	0	0
$517$	2.79488	0.122918
$518$	0	0
$519$	16.1210	0.707634
$520$	0	0
$521$	23.9413	1.04889	0.524444	−	0.851445i	$-0.324273\pi$
$521$	23.9413	1.04889	0.524444	+	0.851445i	$0.324273\pi$
$522$	0	0
$523$	−37.1193	−1.62311	−0.811557	−	0.584273i	$-0.801380\pi$
$523$	−37.1193	−1.62311	−0.811557	+	0.584273i	$0.801380\pi$
$524$	0	0
$525$	−3.82693	−0.167021
$526$	0	0
$527$	−24.8472	−1.08236
$528$	0	0
$529$	−22.6249	−0.983691
$530$	0	0
$531$	−11.0871	−0.481137
$532$	0	0
$533$	48.9929	2.12212
$534$	0	0
$535$	22.5818	0.976295
$536$	0	0
$537$	11.5480	0.498333
$538$	0	0
$539$	0.291792	0.0125684
$540$	0	0
$541$	33.9215	1.45840	0.729199	−	0.684302i	$-0.239893\pi$
$541$	33.9215	1.45840	0.729199	+	0.684302i	$0.239893\pi$
$542$	0	0
$543$	−3.91786	−0.168131
$544$	0	0
$545$	−5.61108	−0.240352
$546$	0	0
$547$	−16.6850	−0.713397	−0.356699	−	0.934219i	$-0.616098\pi$
$547$	−16.6850	−0.713397	−0.356699	+	0.934219i	$0.616098\pi$
$548$	0	0
$549$	12.5460	0.535452
$550$	0	0
$551$	2.81460	0.119906
$552$	0	0
$553$	25.2825	1.07512
$554$	0	0
$555$	−17.2588	−0.732597
$556$	0	0
$557$	30.4280	1.28928	0.644638	−	0.764488i	$-0.277008\pi$
$557$	30.4280	1.28928	0.644638	+	0.764488i	$0.277008\pi$
$558$	0	0
$559$	−56.0398	−2.37023
$560$	0	0
$561$	1.19002	0.0502425
$562$	0	0
$563$	−32.0419	−1.35041	−0.675203	−	0.737632i	$-0.735944\pi$
$563$	−32.0419	−1.35041	−0.675203	+	0.737632i	$0.735944\pi$
$564$	0	0
$565$	32.3912	1.36271
$566$	0	0
$567$	2.81337	0.118150
$568$	0	0
$569$	−27.2824	−1.14374	−0.571869	−	0.820345i	$-0.693782\pi$
$569$	−27.2824	−1.14374	−0.571869	+	0.820345i	$0.693782\pi$
$570$	0	0
$571$	−12.3064	−0.515007	−0.257503	−	0.966277i	$-0.582900\pi$
$571$	−12.3064	−0.515007	−0.257503	+	0.966277i	$0.582900\pi$
$572$	0	0
$573$	−19.5571	−0.817008
$574$	0	0
$575$	−0.833113	−0.0347432
$576$	0	0
$577$	4.05674	0.168884	0.0844422	−	0.996428i	$-0.473089\pi$
$577$	4.05674	0.168884	0.0844422	+	0.996428i	$0.473089\pi$
$578$	0	0
$579$	22.0005	0.914308
$580$	0	0
$581$	11.3886	0.472477
$582$	0	0
$583$	−0.786596	−0.0325775
$584$	0	0
$585$	9.77910	0.404316
$586$	0	0
$587$	−21.5748	−0.890489	−0.445244	−	0.895409i	$-0.646883\pi$
$587$	−21.5748	−0.890489	−0.445244	+	0.895409i	$0.646883\pi$
$588$	0	0
$589$	4.83019	0.199025
$590$	0	0
$591$	10.3862	0.427233
$592$	0	0
$593$	−5.81937	−0.238973	−0.119486	−	0.992836i	$-0.538125\pi$
$593$	−5.81937	−0.238973	−0.119486	+	0.992836i	$0.538125\pi$
$594$	0	0
$595$	20.0302	0.821156
$596$	0	0
$597$	−17.3380	−0.709596
$598$	0	0
$599$	−14.0478	−0.573978	−0.286989	−	0.957934i	$-0.592654\pi$
$599$	−14.0478	−0.573978	−0.286989	+	0.957934i	$0.592654\pi$
$600$	0	0
$601$	26.3024	1.07290	0.536448	−	0.843933i	$-0.319766\pi$
$601$	26.3024	1.07290	0.536448	+	0.843933i	$0.319766\pi$
$602$	0	0
$603$	5.21723	0.212462
$604$	0	0
$605$	−20.7919	−0.845311
$606$	0	0
$607$	16.2583	0.659906	0.329953	−	0.943997i	$-0.392967\pi$
$607$	16.2583	0.659906	0.329953	+	0.943997i	$0.392967\pi$
$608$	0	0
$609$	−10.9153	−0.442309
$610$	0	0
$611$	44.9258	1.81750
$612$	0	0
$613$	−10.3724	−0.418936	−0.209468	−	0.977816i	$-0.567173\pi$
$613$	−10.3724	−0.418936	−0.209468	+	0.977816i	$0.567173\pi$
$614$	0	0
$615$	18.2349	0.735303
$616$	0	0
$617$	−33.0424	−1.33024	−0.665118	−	0.746738i	$-0.731619\pi$
$617$	−33.0424	−1.33024	−0.665118	+	0.746738i	$0.731619\pi$
$618$	0	0
$619$	−5.92967	−0.238334	−0.119167	−	0.992874i	$-0.538022\pi$
$619$	−5.92967	−0.238334	−0.119167	+	0.992874i	$0.538022\pi$
$620$	0	0
$621$	0.612463	0.0245773
$622$	0	0
$623$	−25.7506	−1.03168
$624$	0	0
$625$	−16.3483	−0.653934
$626$	0	0
$627$	−0.231334	−0.00923859
$628$	0	0
$629$	−33.7598	−1.34609
$630$	0	0
$631$	−35.5703	−1.41603	−0.708016	−	0.706196i	$-0.750410\pi$
$631$	−35.5703	−1.41603	−0.708016	+	0.706196i	$0.750410\pi$
$632$	0	0
$633$	−20.2222	−0.803761
$634$	0	0
$635$	31.8777	1.26503
$636$	0	0
$637$	4.69037	0.185839
$638$	0	0
$639$	−11.3217	−0.447878
$640$	0	0
$641$	−1.75263	−0.0692248	−0.0346124	−	0.999401i	$-0.511020\pi$
$641$	−1.75263	−0.0692248	−0.0346124	+	0.999401i	$0.511020\pi$
$642$	0	0
$643$	−44.6647	−1.76140	−0.880701	−	0.473672i	$-0.842928\pi$
$643$	−44.6647	−1.76140	−0.880701	+	0.473672i	$0.842928\pi$
$644$	0	0
$645$	−20.8577	−0.821273
$646$	0	0
$647$	16.7568	0.658776	0.329388	−	0.944195i	$-0.393158\pi$
$647$	16.7568	0.658776	0.329388	+	0.944195i	$0.393158\pi$
$648$	0	0
$649$	−3.53547	−0.138779
$650$	0	0
$651$	−18.7319	−0.734161
$652$	0	0
$653$	−29.8135	−1.16669	−0.583346	−	0.812224i	$-0.698257\pi$
$653$	−29.8135	−1.16669	−0.583346	+	0.812224i	$0.698257\pi$
$654$	0	0
$655$	−0.279562	−0.0109234
$656$	0	0
$657$	2.88490	0.112551
$658$	0	0
$659$	40.4834	1.57701	0.788504	−	0.615030i	$-0.210856\pi$
$659$	40.4834	1.57701	0.788504	+	0.615030i	$0.210856\pi$
$660$	0	0
$661$	16.4694	0.640586	0.320293	−	0.947319i	$-0.396219\pi$
$661$	16.4694	0.640586	0.320293	+	0.947319i	$0.396219\pi$
$662$	0	0
$663$	19.1288	0.742899
$664$	0	0
$665$	−3.89377	−0.150994
$666$	0	0
$667$	−2.37623	−0.0920079
$668$	0	0
$669$	9.60888	0.371501
$670$	0	0
$671$	4.00071	0.154446
$672$	0	0
$673$	7.04173	0.271439	0.135719	−	0.990747i	$-0.456665\pi$
$673$	7.04173	0.271439	0.135719	+	0.990747i	$0.456665\pi$
$674$	0	0
$675$	−1.36027	−0.0523567
$676$	0	0
$677$	13.0327	0.500888	0.250444	−	0.968131i	$-0.419423\pi$
$677$	13.0327	0.500888	0.250444	+	0.968131i	$0.419423\pi$
$678$	0	0
$679$	−27.0675	−1.03875
$680$	0	0
$681$	23.3573	0.895054
$682$	0	0
$683$	44.8287	1.71532	0.857661	−	0.514216i	$-0.171917\pi$
$683$	44.8287	1.71532	0.857661	+	0.514216i	$0.171917\pi$
$684$	0	0
$685$	22.3803	0.855109
$686$	0	0
$687$	−14.5969	−0.556907
$688$	0	0
$689$	−12.6440	−0.481699
$690$	0	0
$691$	−10.2680	−0.390611	−0.195306	−	0.980742i	$-0.562570\pi$
$691$	−10.2680	−0.390611	−0.195306	+	0.980742i	$0.562570\pi$
$692$	0	0
$693$	0.897133	0.0340793
$694$	0	0
$695$	−17.3610	−0.658542
$696$	0	0
$697$	35.6691	1.35106
$698$	0	0
$699$	18.0225	0.681672
$700$	0	0
$701$	29.6121	1.11843	0.559217	−	0.829021i	$-0.311102\pi$
$701$	29.6121	1.11843	0.559217	+	0.829021i	$0.311102\pi$
$702$	0	0
$703$	6.56275	0.247519
$704$	0	0
$705$	16.7212	0.629756
$706$	0	0
$707$	−37.3712	−1.40549
$708$	0	0
$709$	29.7383	1.11685	0.558423	−	0.829556i	$-0.311406\pi$
$709$	29.7383	1.11685	0.558423	+	0.829556i	$0.311406\pi$
$710$	0	0
$711$	8.98655	0.337022
$712$	0	0
$713$	−4.07789	−0.152718
$714$	0	0
$715$	3.11838	0.116621
$716$	0	0
$717$	−9.44595	−0.352765
$718$	0	0
$719$	1.84535	0.0688200	0.0344100	−	0.999408i	$-0.489045\pi$
$719$	1.84535	0.0688200	0.0344100	+	0.999408i	$0.489045\pi$
$720$	0	0
$721$	−51.8295	−1.93023
$722$	0	0
$723$	5.88509	0.218869
$724$	0	0
$725$	5.27755	0.196003
$726$	0	0
$727$	34.6845	1.28638	0.643189	−	0.765708i	$-0.277611\pi$
$727$	34.6845	1.28638	0.643189	+	0.765708i	$0.277611\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−40.7996	−1.50903
$732$	0	0
$733$	−27.7444	−1.02476	−0.512381	−	0.858758i	$-0.671236\pi$
$733$	−27.7444	−1.02476	−0.512381	+	0.858758i	$0.671236\pi$
$734$	0	0
$735$	1.74573	0.0643923
$736$	0	0
$737$	1.66368	0.0612826
$738$	0	0
$739$	3.98690	0.146660	0.0733302	−	0.997308i	$-0.476637\pi$
$739$	3.98690	0.146660	0.0733302	+	0.997308i	$0.476637\pi$
$740$	0	0
$741$	−3.71854	−0.136604
$742$	0	0
$743$	29.3305	1.07603	0.538016	−	0.842935i	$-0.319174\pi$
$743$	29.3305	1.07603	0.538016	+	0.842935i	$0.319174\pi$
$744$	0	0
$745$	44.5459	1.63203
$746$	0	0
$747$	4.04802	0.148109
$748$	0	0
$749$	33.3004	1.21677
$750$	0	0
$751$	17.0963	0.623851	0.311926	−	0.950107i	$-0.399026\pi$
$751$	17.0963	0.623851	0.311926	+	0.950107i	$0.399026\pi$
$752$	0	0
$753$	3.14251	0.114519
$754$	0	0
$755$	−1.64062	−0.0597083
$756$	0	0
$757$	−16.8059	−0.610819	−0.305410	−	0.952221i	$-0.598793\pi$
$757$	−16.8059	−0.610819	−0.305410	+	0.952221i	$0.598793\pi$
$758$	0	0
$759$	0.195304	0.00708907
$760$	0	0
$761$	−33.1343	−1.20112	−0.600559	−	0.799580i	$-0.705055\pi$
$761$	−33.1343	−1.20112	−0.600559	+	0.799580i	$0.705055\pi$
$762$	0	0
$763$	−8.27444	−0.299555
$764$	0	0
$765$	7.11963	0.257411
$766$	0	0
$767$	−56.8303	−2.05202
$768$	0	0
$769$	46.8068	1.68790	0.843949	−	0.536424i	$-0.180225\pi$
$769$	46.8068	1.68790	0.843949	+	0.536424i	$0.180225\pi$
$770$	0	0
$771$	1.52800	0.0550297
$772$	0	0
$773$	−36.5866	−1.31593	−0.657963	−	0.753050i	$-0.728582\pi$
$773$	−36.5866	−1.31593	−0.657963	+	0.753050i	$0.728582\pi$
$774$	0	0
$775$	9.05690	0.325333
$776$	0	0
$777$	−25.4509	−0.913047
$778$	0	0
$779$	−6.93391	−0.248433
$780$	0	0
$781$	−3.61027	−0.129186
$782$	0	0
$783$	−3.87979	−0.138652
$784$	0	0
$785$	−12.1352	−0.433124
$786$	0	0
$787$	−47.3781	−1.68885	−0.844423	−	0.535677i	$-0.820057\pi$
$787$	−47.3781	−1.68885	−0.844423	+	0.535677i	$0.820057\pi$
$788$	0	0
$789$	−24.3283	−0.866112
$790$	0	0
$791$	47.7660	1.69836
$792$	0	0
$793$	64.3088	2.28367
$794$	0	0
$795$	−4.70604	−0.166906
$796$	0	0
$797$	46.4164	1.64415	0.822077	−	0.569376i	$-0.192815\pi$
$797$	46.4164	1.64415	0.822077	+	0.569376i	$0.192815\pi$
$798$	0	0
$799$	32.7081	1.15713
$800$	0	0
$801$	−9.15294	−0.323403
$802$	0	0
$803$	0.919944	0.0324641
$804$	0	0
$805$	3.28732	0.115863
$806$	0	0
$807$	0.163618	0.00575961
$808$	0	0
$809$	−36.9134	−1.29781	−0.648903	−	0.760871i	$-0.724772\pi$
$809$	−36.9134	−1.29781	−0.648903	+	0.760871i	$0.724772\pi$
$810$	0	0
$811$	7.35245	0.258179	0.129090	−	0.991633i	$-0.458794\pi$
$811$	7.35245	0.258179	0.129090	+	0.991633i	$0.458794\pi$
$812$	0	0
$813$	−8.88391	−0.311572
$814$	0	0
$815$	−1.37873	−0.0482948
$816$	0	0
$817$	7.93125	0.277479
$818$	0	0
$819$	14.4208	0.503905
$820$	0	0
$821$	30.0419	1.04847	0.524235	−	0.851574i	$-0.324351\pi$
$821$	30.0419	1.04847	0.524235	+	0.851574i	$0.324351\pi$
$822$	0	0
$823$	−27.5122	−0.959016	−0.479508	−	0.877538i	$-0.659185\pi$
$823$	−27.5122	−0.959016	−0.479508	+	0.877538i	$0.659185\pi$
$824$	0	0
$825$	−0.433765	−0.0151018
$826$	0	0
$827$	49.1971	1.71075	0.855375	−	0.518009i	$-0.173326\pi$
$827$	49.1971	1.71075	0.855375	+	0.518009i	$0.173326\pi$
$828$	0	0
$829$	41.9008	1.45527	0.727637	−	0.685963i	$-0.240619\pi$
$829$	41.9008	1.45527	0.727637	+	0.685963i	$0.240619\pi$
$830$	0	0
$831$	−5.45934	−0.189382
$832$	0	0
$833$	3.41480	0.118316
$834$	0	0
$835$	1.90781	0.0660224
$836$	0	0
$837$	−6.65817	−0.230140
$838$	0	0
$839$	34.5508	1.19283	0.596413	−	0.802678i	$-0.296592\pi$
$839$	34.5508	1.19283	0.596413	+	0.802678i	$0.296592\pi$
$840$	0	0
$841$	−13.9472	−0.480939
$842$	0	0
$843$	−8.18240	−0.281817
$844$	0	0
$845$	25.3245	0.871188
$846$	0	0
$847$	−30.6610	−1.05352
$848$	0	0
$849$	9.17416	0.314856
$850$	0	0
$851$	−5.54060	−0.189929
$852$	0	0
$853$	−27.6838	−0.947874	−0.473937	−	0.880559i	$-0.657168\pi$
$853$	−27.6838	−0.947874	−0.473937	+	0.880559i	$0.657168\pi$
$854$	0	0
$855$	−1.38402	−0.0473326
$856$	0	0
$857$	−15.1732	−0.518307	−0.259154	−	0.965836i	$-0.583444\pi$
$857$	−15.1732	−0.518307	−0.259154	+	0.965836i	$0.583444\pi$
$858$	0	0
$859$	0.561134	0.0191456	0.00957281	−	0.999954i	$-0.496953\pi$
$859$	0.561134	0.0191456	0.00957281	+	0.999954i	$0.496953\pi$
$860$	0	0
$861$	26.8903	0.916419
$862$	0	0
$863$	21.7750	0.741230	0.370615	−	0.928786i	$-0.379147\pi$
$863$	21.7750	0.741230	0.370615	+	0.928786i	$0.379147\pi$
$864$	0	0
$865$	30.7558	1.04573
$866$	0	0
$867$	−3.07338	−0.104378
$868$	0	0
$869$	2.86565	0.0972105
$870$	0	0
$871$	26.7426	0.906140
$872$	0	0
$873$	−9.62101	−0.325622
$874$	0	0
$875$	−34.1379	−1.15407
$876$	0	0
$877$	20.0002	0.675358	0.337679	−	0.941261i	$-0.390358\pi$
$877$	20.0002	0.675358	0.337679	+	0.941261i	$0.390358\pi$
$878$	0	0
$879$	−30.6446	−1.03362
$880$	0	0
$881$	5.92933	0.199764	0.0998821	−	0.994999i	$-0.468153\pi$
$881$	5.92933	0.199764	0.0998821	+	0.994999i	$0.468153\pi$
$882$	0	0
$883$	6.56436	0.220908	0.110454	−	0.993881i	$-0.464769\pi$
$883$	6.56436	0.220908	0.110454	+	0.993881i	$0.464769\pi$
$884$	0	0
$885$	−21.1520	−0.711016
$886$	0	0
$887$	−1.74021	−0.0584306	−0.0292153	−	0.999573i	$-0.509301\pi$
$887$	−1.74021	−0.0584306	−0.0292153	+	0.999573i	$0.509301\pi$
$888$	0	0
$889$	47.0088	1.57662
$890$	0	0
$891$	0.318882	0.0106830
$892$	0	0
$893$	−6.35830	−0.212773
$894$	0	0
$895$	22.0314	0.736427
$896$	0	0
$897$	3.13938	0.104821
$898$	0	0
$899$	25.8323	0.861556
$900$	0	0
$901$	−9.20543	−0.306677
$902$	0	0
$903$	−30.7581	−1.02357
$904$	0	0
$905$	−7.47452	−0.248461
$906$	0	0
$907$	28.3718	0.942071	0.471036	−	0.882114i	$-0.343880\pi$
$907$	28.3718	0.942071	0.471036	+	0.882114i	$0.343880\pi$
$908$	0	0
$909$	−13.2834	−0.440583
$910$	0	0
$911$	−0.700571	−0.0232110	−0.0116055	−	0.999933i	$-0.503694\pi$
$911$	−0.700571	−0.0232110	−0.0116055	+	0.999933i	$0.503694\pi$
$912$	0	0
$913$	1.29084	0.0427206
$914$	0	0
$915$	23.9354	0.791281
$916$	0	0
$917$	−0.412259	−0.0136140
$918$	0	0
$919$	17.8304	0.588170	0.294085	−	0.955779i	$-0.404985\pi$
$919$	17.8304	0.588170	0.294085	+	0.955779i	$0.404985\pi$
$920$	0	0
$921$	−25.0225	−0.824519
$922$	0	0
$923$	−58.0328	−1.91017
$924$	0	0
$925$	12.3056	0.404604
$926$	0	0
$927$	−18.4226	−0.605077
$928$	0	0
$929$	−42.4672	−1.39330	−0.696652	−	0.717410i	$-0.745328\pi$
$929$	−42.4672	−1.39330	−0.696652	+	0.717410i	$0.745328\pi$
$930$	0	0
$931$	−0.663822	−0.0217559
$932$	0	0
$933$	26.4473	0.865847
$934$	0	0
$935$	2.27032	0.0742475
$936$	0	0
$937$	44.9342	1.46794	0.733968	−	0.679184i	$-0.237666\pi$
$937$	44.9342	1.46794	0.733968	+	0.679184i	$0.237666\pi$
$938$	0	0
$939$	22.3927	0.730757
$940$	0	0
$941$	−39.9153	−1.30120	−0.650600	−	0.759420i	$-0.725483\pi$
$941$	−39.9153	−1.30120	−0.650600	+	0.759420i	$0.725483\pi$
$942$	0	0
$943$	5.85395	0.190631
$944$	0	0
$945$	5.36737	0.174601
$946$	0	0
$947$	1.83044	0.0594812	0.0297406	−	0.999558i	$-0.490532\pi$
$947$	1.83044	0.0594812	0.0297406	+	0.999558i	$0.490532\pi$
$948$	0	0
$949$	14.7875	0.480023
$950$	0	0
$951$	24.1684	0.783714
$952$	0	0
$953$	21.0500	0.681877	0.340938	−	0.940086i	$-0.389255\pi$
$953$	21.0500	0.681877	0.340938	+	0.940086i	$0.389255\pi$
$954$	0	0
$955$	−37.3111	−1.20736
$956$	0	0
$957$	−1.23720	−0.0399929
$958$	0	0
$959$	33.0034	1.06573
$960$	0	0
$961$	13.3313	0.430042
$962$	0	0
$963$	11.8365	0.381426
$964$	0	0
$965$	41.9727	1.35115
$966$	0	0
$967$	−49.7457	−1.59972	−0.799858	−	0.600190i	$-0.795092\pi$
$967$	−49.7457	−1.59972	−0.799858	+	0.600190i	$0.795092\pi$
$968$	0	0
$969$	−2.70727	−0.0869701
$970$	0	0
$971$	−34.2216	−1.09822	−0.549112	−	0.835749i	$-0.685034\pi$
$971$	−34.2216	−1.09822	−0.549112	+	0.835749i	$0.685034\pi$
$972$	0	0
$973$	−25.6016	−0.820751
$974$	0	0
$975$	−6.97249	−0.223299
$976$	0	0
$977$	11.2596	0.360227	0.180113	−	0.983646i	$-0.442354\pi$
$977$	11.2596	0.360227	0.180113	+	0.983646i	$0.442354\pi$
$978$	0	0
$979$	−2.91871	−0.0932824
$980$	0	0
$981$	−2.94111	−0.0939026
$982$	0	0
$983$	6.69693	0.213599	0.106799	−	0.994281i	$-0.465940\pi$
$983$	6.69693	0.213599	0.106799	+	0.994281i	$0.465940\pi$
$984$	0	0
$985$	19.8150	0.631357
$986$	0	0
$987$	24.6581	0.784875
$988$	0	0
$989$	−6.69596	−0.212919
$990$	0	0
$991$	53.4452	1.69774	0.848872	−	0.528599i	$-0.177282\pi$
$991$	53.4452	1.69774	0.848872	+	0.528599i	$0.177282\pi$
$992$	0	0
$993$	−28.8009	−0.913969
$994$	0	0
$995$	−33.0775	−1.04863
$996$	0	0
$997$	52.2893	1.65602	0.828009	−	0.560714i	$-0.189473\pi$
$997$	52.2893	1.65602	0.828009	+	0.560714i	$0.189473\pi$
$998$	0	0
$999$	−9.04643	−0.286216

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2004.2.a.d.1.6	✓	9
3.2	odd	2		6012.2.a.h.1.4		9
4.3	odd	2		8016.2.a.bb.1.6		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.2.a.d.1.6	✓	9	1.1	even	1	trivial
6012.2.a.h.1.4		9	3.2	odd	2
8016.2.a.bb.1.6		9	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2004 = 2^{2} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.90781 q^{5} +2.81337 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.90781 q^{5} +2.81337 q^{7} +1.00000 q^{9} +0.318882 q^{11} +5.12583 q^{13} +1.90781 q^{15} +3.73184 q^{17} -0.725453 q^{19} +2.81337 q^{21} +0.612463 q^{23} -1.36027 q^{25} +1.00000 q^{27} -3.87979 q^{29} -6.65817 q^{31} +0.318882 q^{33} +5.36737 q^{35} -9.04643 q^{37} +5.12583 q^{39} +9.55805 q^{41} -10.9328 q^{43} +1.90781 q^{45} +8.76460 q^{47} +0.915046 q^{49} +3.73184 q^{51} -2.46673 q^{53} +0.608366 q^{55} -0.725453 q^{57} -11.0871 q^{59} +12.5460 q^{61} +2.81337 q^{63} +9.77910 q^{65} +5.21723 q^{67} +0.612463 q^{69} -11.3217 q^{71} +2.88490 q^{73} -1.36027 q^{75} +0.897133 q^{77} +8.98655 q^{79} +1.00000 q^{81} +4.04802 q^{83} +7.11963 q^{85} -3.87979 q^{87} -9.15294 q^{89} +14.4208 q^{91} -6.65817 q^{93} -1.38402 q^{95} -9.62101 q^{97} +0.318882 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{3} + 9 q^{5} + 2 q^{7} + 9 q^{9}+O(q^{10}) \) 9 * q + 9 * q^3 + 9 * q^5 + 2 * q^7 + 9 * q^9	\( 9 q + 9 q^{3} + 9 q^{5} + 2 q^{7} + 9 q^{9} + 7 q^{11} + 6 q^{13} + 9 q^{15} + 7 q^{17} + 2 q^{19} + 2 q^{21} + 19 q^{23} + 22 q^{25} + 9 q^{27} + 13 q^{29} + 12 q^{31} + 7 q^{33} + 4 q^{35} + 15 q^{37} + 6 q^{39} + 18 q^{41} - 6 q^{43} + 9 q^{45} + 25 q^{47} + 19 q^{49} + 7 q^{51} + 17 q^{53} - 3 q^{55} + 2 q^{57} + 3 q^{59} + 14 q^{61} + 2 q^{63} + 14 q^{65} - 4 q^{67} + 19 q^{69} + 17 q^{71} - 20 q^{73} + 22 q^{75} + 14 q^{77} - 8 q^{79} + 9 q^{81} - q^{83} + 5 q^{85} + 13 q^{87} + 36 q^{89} - 41 q^{91} + 12 q^{93} + 5 q^{95} + 31 q^{97} + 7 q^{99}+O(q^{100}) \) 9 * q + 9 * q^3 + 9 * q^5 + 2 * q^7 + 9 * q^9 + 7 * q^11 + 6 * q^13 + 9 * q^15 + 7 * q^17 + 2 * q^19 + 2 * q^21 + 19 * q^23 + 22 * q^25 + 9 * q^27 + 13 * q^29 + 12 * q^31 + 7 * q^33 + 4 * q^35 + 15 * q^37 + 6 * q^39 + 18 * q^41 - 6 * q^43 + 9 * q^45 + 25 * q^47 + 19 * q^49 + 7 * q^51 + 17 * q^53 - 3 * q^55 + 2 * q^57 + 3 * q^59 + 14 * q^61 + 2 * q^63 + 14 * q^65 - 4 * q^67 + 19 * q^69 + 17 * q^71 - 20 * q^73 + 22 * q^75 + 14 * q^77 - 8 * q^79 + 9 * q^81 - q^83 + 5 * q^85 + 13 * q^87 + 36 * q^89 - 41 * q^91 + 12 * q^93 + 5 * q^95 + 31 * q^97 + 7 * q^99

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