LMFDB - Embedded newform 6036.2.a.g.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6036,2,Mod(1,6036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6036, 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("6036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6036 = 2^{2} \cdot 3 \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6036.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:	$48.1977026600$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 4 x^{14} - 27 x^{13} + 106 x^{12} + 266 x^{11} - 1004 x^{10} - 1105 x^{9} + 4076 x^{8} + \cdots - 44 $ x^15 - 4x^14 - 27x^13 + 106x^12 + 266x^11 - 1004x^10 - 1105x^9 + 4076x^8 + 1501x^7 - 7100x^6 - 134x^5 + 5356x^4 - 1041x^3 - 1381x^2 + 543x - 44
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.117275$ of defining polynomial
Character	$\chi$	$=$	6036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -0.882725 q^{5} -2.85663 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.882725 q^{5} -2.85663 q^{7} +1.00000 q^{9} +2.43664 q^{11} +1.92471 q^{13} -0.882725 q^{15} -2.82868 q^{17} +5.97479 q^{19} -2.85663 q^{21} -3.88140 q^{23} -4.22080 q^{25} +1.00000 q^{27} -6.73462 q^{29} +0.533408 q^{31} +2.43664 q^{33} +2.52162 q^{35} +3.20257 q^{37} +1.92471 q^{39} -2.13883 q^{41} -7.27961 q^{43} -0.882725 q^{45} -0.965205 q^{47} +1.16034 q^{49} -2.82868 q^{51} +11.3038 q^{53} -2.15089 q^{55} +5.97479 q^{57} -8.38597 q^{59} -4.72206 q^{61} -2.85663 q^{63} -1.69899 q^{65} +2.35883 q^{67} -3.88140 q^{69} -6.46064 q^{71} +11.4531 q^{73} -4.22080 q^{75} -6.96060 q^{77} -3.91208 q^{79} +1.00000 q^{81} -12.5325 q^{83} +2.49695 q^{85} -6.73462 q^{87} +14.9051 q^{89} -5.49820 q^{91} +0.533408 q^{93} -5.27410 q^{95} -9.34373 q^{97} +2.43664 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) $ 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9	$ 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9} - 5 q^{11} - 14 q^{13} - 11 q^{15} - 5 q^{17} + 3 q^{19} - 4 q^{21} - 32 q^{23} + 2 q^{25} + 15 q^{27} - 23 q^{29} - 13 q^{31} - 5 q^{33} - 16 q^{35} - 10 q^{37} - 14 q^{39} - 14 q^{41} + 4 q^{43} - 11 q^{45} - 20 q^{47} - 9 q^{49} - 5 q^{51} - 30 q^{53} - 10 q^{55} + 3 q^{57} - 14 q^{59} - 38 q^{61} - 4 q^{63} - 24 q^{65} - 8 q^{67} - 32 q^{69} - 41 q^{71} - 19 q^{73} + 2 q^{75} - 39 q^{77} - 27 q^{79} + 15 q^{81} - 17 q^{83} - 6 q^{85} - 23 q^{87} - 23 q^{89} + 4 q^{91} - 13 q^{93} - 30 q^{95} - 18 q^{97} - 5 q^{99}+O(q^{100}) $ 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9 - 5 * q^11 - 14 * q^13 - 11 * q^15 - 5 * q^17 + 3 * q^19 - 4 * q^21 - 32 * q^23 + 2 * q^25 + 15 * q^27 - 23 * q^29 - 13 * q^31 - 5 * q^33 - 16 * q^35 - 10 * q^37 - 14 * q^39 - 14 * q^41 + 4 * q^43 - 11 * q^45 - 20 * q^47 - 9 * q^49 - 5 * q^51 - 30 * q^53 - 10 * q^55 + 3 * q^57 - 14 * q^59 - 38 * q^61 - 4 * q^63 - 24 * q^65 - 8 * q^67 - 32 * q^69 - 41 * q^71 - 19 * q^73 + 2 * q^75 - 39 * q^77 - 27 * q^79 + 15 * q^81 - 17 * q^83 - 6 * q^85 - 23 * q^87 - 23 * q^89 + 4 * q^91 - 13 * q^93 - 30 * q^95 - 18 * q^97 - 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−0.882725	−0.394767	−0.197383	−	0.980326i	$-0.563244\pi$
$5$	−0.882725	−0.394767	−0.197383	+	0.980326i	$0.563244\pi$
$6$	0	0
$7$	−2.85663	−1.07971	−0.539853	−	0.841760i	$-0.681520\pi$
$7$	−2.85663	−1.07971	−0.539853	+	0.841760i	$0.681520\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.43664	0.734676	0.367338	−	0.930088i	$-0.380269\pi$
$11$	2.43664	0.734676	0.367338	+	0.930088i	$0.380269\pi$
$12$	0	0
$13$	1.92471	0.533819	0.266910	−	0.963722i	$-0.413997\pi$
$13$	1.92471	0.533819	0.266910	+	0.963722i	$0.413997\pi$
$14$	0	0
$15$	−0.882725	−0.227919
$16$	0	0
$17$	−2.82868	−0.686055	−0.343028	−	0.939325i	$-0.611452\pi$
$17$	−2.82868	−0.686055	−0.343028	+	0.939325i	$0.611452\pi$
$18$	0	0
$19$	5.97479	1.37071	0.685355	−	0.728209i	$-0.259647\pi$
$19$	5.97479	1.37071	0.685355	+	0.728209i	$0.259647\pi$
$20$	0	0
$21$	−2.85663	−0.623368
$22$	0	0
$23$	−3.88140	−0.809327	−0.404664	−	0.914466i	$-0.632611\pi$
$23$	−3.88140	−0.809327	−0.404664	+	0.914466i	$0.632611\pi$
$24$	0	0
$25$	−4.22080	−0.844159
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.73462	−1.25059	−0.625294	−	0.780389i	$-0.715021\pi$
$29$	−6.73462	−1.25059	−0.625294	+	0.780389i	$0.715021\pi$
$30$	0	0
$31$	0.533408	0.0958028	0.0479014	−	0.998852i	$-0.484747\pi$
$31$	0.533408	0.0958028	0.0479014	+	0.998852i	$0.484747\pi$
$32$	0	0
$33$	2.43664	0.424165
$34$	0	0
$35$	2.52162	0.426232
$36$	0	0
$37$	3.20257	0.526499	0.263249	−	0.964728i	$-0.415206\pi$
$37$	3.20257	0.526499	0.263249	+	0.964728i	$0.415206\pi$
$38$	0	0
$39$	1.92471	0.308201
$40$	0	0
$41$	−2.13883	−0.334029	−0.167014	−	0.985954i	$-0.553413\pi$
$41$	−2.13883	−0.334029	−0.167014	+	0.985954i	$0.553413\pi$
$42$	0	0
$43$	−7.27961	−1.11013	−0.555065	−	0.831807i	$-0.687307\pi$
$43$	−7.27961	−1.11013	−0.555065	+	0.831807i	$0.687307\pi$
$44$	0	0
$45$	−0.882725	−0.131589
$46$	0	0
$47$	−0.965205	−0.140790	−0.0703948	−	0.997519i	$-0.522426\pi$
$47$	−0.965205	−0.140790	−0.0703948	+	0.997519i	$0.522426\pi$
$48$	0	0
$49$	1.16034	0.165763
$50$	0	0
$51$	−2.82868	−0.396094
$52$	0	0
$53$	11.3038	1.55269	0.776345	−	0.630308i	$-0.217071\pi$
$53$	11.3038	1.55269	0.776345	+	0.630308i	$0.217071\pi$
$54$	0	0
$55$	−2.15089	−0.290026
$56$	0	0
$57$	5.97479	0.791380
$58$	0	0
$59$	−8.38597	−1.09176	−0.545880	−	0.837863i	$-0.683805\pi$
$59$	−8.38597	−1.09176	−0.545880	+	0.837863i	$0.683805\pi$
$60$	0	0
$61$	−4.72206	−0.604597	−0.302299	−	0.953213i	$-0.597754\pi$
$61$	−4.72206	−0.604597	−0.302299	+	0.953213i	$0.597754\pi$
$62$	0	0
$63$	−2.85663	−0.359902
$64$	0	0
$65$	−1.69899	−0.210734
$66$	0	0
$67$	2.35883	0.288177	0.144089	−	0.989565i	$-0.453975\pi$
$67$	2.35883	0.288177	0.144089	+	0.989565i	$0.453975\pi$
$68$	0	0
$69$	−3.88140	−0.467265
$70$	0	0
$71$	−6.46064	−0.766736	−0.383368	−	0.923596i	$-0.625236\pi$
$71$	−6.46064	−0.766736	−0.383368	+	0.923596i	$0.625236\pi$
$72$	0	0
$73$	11.4531	1.34048	0.670240	−	0.742144i	$-0.266191\pi$
$73$	11.4531	1.34048	0.670240	+	0.742144i	$0.266191\pi$
$74$	0	0
$75$	−4.22080	−0.487376
$76$	0	0
$77$	−6.96060	−0.793234
$78$	0	0
$79$	−3.91208	−0.440143	−0.220072	−	0.975484i	$-0.570629\pi$
$79$	−3.91208	−0.440143	−0.220072	+	0.975484i	$0.570629\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.5325	−1.37563	−0.687813	−	0.725888i	$-0.741429\pi$
$83$	−12.5325	−1.37563	−0.687813	+	0.725888i	$0.741429\pi$
$84$	0	0
$85$	2.49695	0.270832
$86$	0	0
$87$	−6.73462	−0.722027
$88$	0	0
$89$	14.9051	1.57994	0.789971	−	0.613144i	$-0.210095\pi$
$89$	14.9051	1.57994	0.789971	+	0.613144i	$0.210095\pi$
$90$	0	0
$91$	−5.49820	−0.576368
$92$	0	0
$93$	0.533408	0.0553118
$94$	0	0
$95$	−5.27410	−0.541111
$96$	0	0
$97$	−9.34373	−0.948712	−0.474356	−	0.880333i	$-0.657319\pi$
$97$	−9.34373	−0.948712	−0.474356	+	0.880333i	$0.657319\pi$
$98$	0	0
$99$	2.43664	0.244892
$100$	0	0
$101$	4.77150	0.474782	0.237391	−	0.971414i	$-0.423708\pi$
$101$	4.77150	0.474782	0.237391	+	0.971414i	$0.423708\pi$
$102$	0	0
$103$	1.35047	0.133065	0.0665327	−	0.997784i	$-0.478806\pi$
$103$	1.35047	0.133065	0.0665327	+	0.997784i	$0.478806\pi$
$104$	0	0
$105$	2.52162	0.246085
$106$	0	0
$107$	11.8653	1.14706	0.573532	−	0.819183i	$-0.305573\pi$
$107$	11.8653	1.14706	0.573532	+	0.819183i	$0.305573\pi$
$108$	0	0
$109$	−2.24165	−0.214711	−0.107356	−	0.994221i	$-0.534238\pi$
$109$	−2.24165	−0.214711	−0.107356	+	0.994221i	$0.534238\pi$
$110$	0	0
$111$	3.20257	0.303974
$112$	0	0
$113$	5.76949	0.542748	0.271374	−	0.962474i	$-0.412522\pi$
$113$	5.76949	0.542748	0.271374	+	0.962474i	$0.412522\pi$
$114$	0	0
$115$	3.42621	0.319495
$116$	0	0
$117$	1.92471	0.177940
$118$	0	0
$119$	8.08049	0.740737
$120$	0	0
$121$	−5.06276	−0.460251
$122$	0	0
$123$	−2.13883	−0.192852
$124$	0	0
$125$	8.13943	0.728013
$126$	0	0
$127$	10.5810	0.938912	0.469456	−	0.882956i	$-0.344450\pi$
$127$	10.5810	0.938912	0.469456	+	0.882956i	$0.344450\pi$
$128$	0	0
$129$	−7.27961	−0.640934
$130$	0	0
$131$	−13.0718	−1.14209	−0.571046	−	0.820918i	$-0.693462\pi$
$131$	−13.0718	−1.14209	−0.571046	+	0.820918i	$0.693462\pi$
$132$	0	0
$133$	−17.0678	−1.47996
$134$	0	0
$135$	−0.882725	−0.0759729
$136$	0	0
$137$	−18.9259	−1.61695	−0.808474	−	0.588532i	$-0.799706\pi$
$137$	−18.9259	−1.61695	−0.808474	+	0.588532i	$0.799706\pi$
$138$	0	0
$139$	9.86280	0.836552	0.418276	−	0.908320i	$-0.362634\pi$
$139$	9.86280	0.836552	0.418276	+	0.908320i	$0.362634\pi$
$140$	0	0
$141$	−0.965205	−0.0812849
$142$	0	0
$143$	4.68984	0.392184
$144$	0	0
$145$	5.94482	0.493691
$146$	0	0
$147$	1.16034	0.0957036
$148$	0	0
$149$	−14.3681	−1.17708	−0.588541	−	0.808468i	$-0.700297\pi$
$149$	−14.3681	−1.17708	−0.588541	+	0.808468i	$0.700297\pi$
$150$	0	0
$151$	3.21916	0.261971	0.130986	−	0.991384i	$-0.458186\pi$
$151$	3.21916	0.261971	0.130986	+	0.991384i	$0.458186\pi$
$152$	0	0
$153$	−2.82868	−0.228685
$154$	0	0
$155$	−0.470852	−0.0378198
$156$	0	0
$157$	−10.3589	−0.826734	−0.413367	−	0.910564i	$-0.635647\pi$
$157$	−10.3589	−0.826734	−0.413367	+	0.910564i	$0.635647\pi$
$158$	0	0
$159$	11.3038	0.896446
$160$	0	0
$161$	11.0877	0.873835
$162$	0	0
$163$	−8.00830	−0.627259	−0.313629	−	0.949546i	$-0.601545\pi$
$163$	−8.00830	−0.627259	−0.313629	+	0.949546i	$0.601545\pi$
$164$	0	0
$165$	−2.15089	−0.167446
$166$	0	0
$167$	−9.92715	−0.768186	−0.384093	−	0.923294i	$-0.625486\pi$
$167$	−9.92715	−0.768186	−0.384093	+	0.923294i	$0.625486\pi$
$168$	0	0
$169$	−9.29548	−0.715037
$170$	0	0
$171$	5.97479	0.456903
$172$	0	0
$173$	−15.5681	−1.18362	−0.591811	−	0.806077i	$-0.701587\pi$
$173$	−15.5681	−1.18362	−0.591811	+	0.806077i	$0.701587\pi$
$174$	0	0
$175$	12.0573	0.911443
$176$	0	0
$177$	−8.38597	−0.630328
$178$	0	0
$179$	3.70146	0.276660	0.138330	−	0.990386i	$-0.455827\pi$
$179$	3.70146	0.276660	0.138330	+	0.990386i	$0.455827\pi$
$180$	0	0
$181$	−24.7381	−1.83877	−0.919384	−	0.393360i	$-0.871312\pi$
$181$	−24.7381	−1.83877	−0.919384	+	0.393360i	$0.871312\pi$
$182$	0	0
$183$	−4.72206	−0.349064
$184$	0	0
$185$	−2.82699	−0.207844
$186$	0	0
$187$	−6.89248	−0.504028
$188$	0	0
$189$	−2.85663	−0.207789
$190$	0	0
$191$	−13.0911	−0.947240	−0.473620	−	0.880729i	$-0.657053\pi$
$191$	−13.0911	−0.947240	−0.473620	+	0.880729i	$0.657053\pi$
$192$	0	0
$193$	−5.85802	−0.421669	−0.210835	−	0.977522i	$-0.567618\pi$
$193$	−5.85802	−0.421669	−0.210835	+	0.977522i	$0.567618\pi$
$194$	0	0
$195$	−1.69899	−0.121667
$196$	0	0
$197$	−13.1264	−0.935220	−0.467610	−	0.883935i	$-0.654885\pi$
$197$	−13.1264	−0.935220	−0.467610	+	0.883935i	$0.654885\pi$
$198$	0	0
$199$	−9.41618	−0.667495	−0.333748	−	0.942662i	$-0.608313\pi$
$199$	−9.41618	−0.667495	−0.333748	+	0.942662i	$0.608313\pi$
$200$	0	0
$201$	2.35883	0.166379
$202$	0	0
$203$	19.2383	1.35027
$204$	0	0
$205$	1.88800	0.131863
$206$	0	0
$207$	−3.88140	−0.269776
$208$	0	0
$209$	14.5584	1.00703
$210$	0	0
$211$	8.03579	0.553207	0.276603	−	0.960984i	$-0.410791\pi$
$211$	8.03579	0.553207	0.276603	+	0.960984i	$0.410791\pi$
$212$	0	0
$213$	−6.46064	−0.442675
$214$	0	0
$215$	6.42589	0.438242
$216$	0	0
$217$	−1.52375	−0.103439
$218$	0	0
$219$	11.4531	0.773927
$220$	0	0
$221$	−5.44439	−0.366230
$222$	0	0
$223$	−17.9834	−1.20426	−0.602129	−	0.798399i	$-0.705681\pi$
$223$	−17.9834	−1.20426	−0.602129	+	0.798399i	$0.705681\pi$
$224$	0	0
$225$	−4.22080	−0.281386
$226$	0	0
$227$	−14.5752	−0.967388	−0.483694	−	0.875237i	$-0.660705\pi$
$227$	−14.5752	−0.967388	−0.483694	+	0.875237i	$0.660705\pi$
$228$	0	0
$229$	−28.1053	−1.85725	−0.928624	−	0.371022i	$-0.879007\pi$
$229$	−28.1053	−1.85725	−0.928624	+	0.371022i	$0.879007\pi$
$230$	0	0
$231$	−6.96060	−0.457974
$232$	0	0
$233$	1.44456	0.0946363	0.0473181	−	0.998880i	$-0.484933\pi$
$233$	1.44456	0.0946363	0.0473181	+	0.998880i	$0.484933\pi$
$234$	0	0
$235$	0.852011	0.0555791
$236$	0	0
$237$	−3.91208	−0.254117
$238$	0	0
$239$	−10.8011	−0.698665	−0.349332	−	0.936999i	$-0.613592\pi$
$239$	−10.8011	−0.698665	−0.349332	+	0.936999i	$0.613592\pi$
$240$	0	0
$241$	17.0594	1.09889	0.549447	−	0.835529i	$-0.314838\pi$
$241$	17.0594	1.09889	0.549447	+	0.835529i	$0.314838\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−1.02426	−0.0654379
$246$	0	0
$247$	11.4998	0.731712
$248$	0	0
$249$	−12.5325	−0.794218
$250$	0	0
$251$	−18.3563	−1.15864	−0.579320	−	0.815100i	$-0.696682\pi$
$251$	−18.3563	−1.15864	−0.579320	+	0.815100i	$0.696682\pi$
$252$	0	0
$253$	−9.45758	−0.594593
$254$	0	0
$255$	2.49695	0.156365
$256$	0	0
$257$	21.6433	1.35007	0.675036	−	0.737785i	$-0.264128\pi$
$257$	21.6433	1.35007	0.675036	+	0.737785i	$0.264128\pi$
$258$	0	0
$259$	−9.14855	−0.568463
$260$	0	0
$261$	−6.73462	−0.416863
$262$	0	0
$263$	4.38202	0.270207	0.135103	−	0.990832i	$-0.456863\pi$
$263$	4.38202	0.270207	0.135103	+	0.990832i	$0.456863\pi$
$264$	0	0
$265$	−9.97811	−0.612951
$266$	0	0
$267$	14.9051	0.912180
$268$	0	0
$269$	6.04760	0.368728	0.184364	−	0.982858i	$-0.440977\pi$
$269$	6.04760	0.368728	0.184364	+	0.982858i	$0.440977\pi$
$270$	0	0
$271$	−20.9757	−1.27418	−0.637092	−	0.770788i	$-0.719863\pi$
$271$	−20.9757	−1.27418	−0.637092	+	0.770788i	$0.719863\pi$
$272$	0	0
$273$	−5.49820	−0.332766
$274$	0	0
$275$	−10.2846	−0.620184
$276$	0	0
$277$	−16.9015	−1.01551	−0.507756	−	0.861501i	$-0.669525\pi$
$277$	−16.9015	−1.01551	−0.507756	+	0.861501i	$0.669525\pi$
$278$	0	0
$279$	0.533408	0.0319343
$280$	0	0
$281$	−13.3651	−0.797296	−0.398648	−	0.917104i	$-0.630520\pi$
$281$	−13.3651	−0.797296	−0.398648	+	0.917104i	$0.630520\pi$
$282$	0	0
$283$	−7.49433	−0.445492	−0.222746	−	0.974877i	$-0.571502\pi$
$283$	−7.49433	−0.445492	−0.222746	+	0.974877i	$0.571502\pi$
$284$	0	0
$285$	−5.27410	−0.312410
$286$	0	0
$287$	6.10984	0.360653
$288$	0	0
$289$	−8.99858	−0.529328
$290$	0	0
$291$	−9.34373	−0.547739
$292$	0	0
$293$	−32.6876	−1.90963	−0.954814	−	0.297203i	$-0.903946\pi$
$293$	−32.6876	−1.90963	−0.954814	+	0.297203i	$0.903946\pi$
$294$	0	0
$295$	7.40251	0.430991
$296$	0	0
$297$	2.43664	0.141388
$298$	0	0
$299$	−7.47057	−0.432034
$300$	0	0
$301$	20.7952	1.19861
$302$	0	0
$303$	4.77150	0.274116
$304$	0	0
$305$	4.16828	0.238675
$306$	0	0
$307$	14.0085	0.799506	0.399753	−	0.916623i	$-0.369096\pi$
$307$	14.0085	0.799506	0.399753	+	0.916623i	$0.369096\pi$
$308$	0	0
$309$	1.35047	0.0768253
$310$	0	0
$311$	−4.23519	−0.240156	−0.120078	−	0.992764i	$-0.538314\pi$
$311$	−4.23519	−0.240156	−0.120078	+	0.992764i	$0.538314\pi$
$312$	0	0
$313$	31.5070	1.78088	0.890441	−	0.455099i	$-0.150396\pi$
$313$	31.5070	1.78088	0.890441	+	0.455099i	$0.150396\pi$
$314$	0	0
$315$	2.52162	0.142077
$316$	0	0
$317$	−14.4870	−0.813670	−0.406835	−	0.913502i	$-0.633368\pi$
$317$	−14.4870	−0.813670	−0.406835	+	0.913502i	$0.633368\pi$
$318$	0	0
$319$	−16.4099	−0.918777
$320$	0	0
$321$	11.8653	0.662257
$322$	0	0
$323$	−16.9007	−0.940383
$324$	0	0
$325$	−8.12382	−0.450629
$326$	0	0
$327$	−2.24165	−0.123964
$328$	0	0
$329$	2.75724	0.152011
$330$	0	0
$331$	17.0638	0.937910	0.468955	−	0.883222i	$-0.344631\pi$
$331$	17.0638	0.937910	0.468955	+	0.883222i	$0.344631\pi$
$332$	0	0
$333$	3.20257	0.175500
$334$	0	0
$335$	−2.08220	−0.113763
$336$	0	0
$337$	−15.6785	−0.854063	−0.427031	−	0.904237i	$-0.640441\pi$
$337$	−15.6785	−0.854063	−0.427031	+	0.904237i	$0.640441\pi$
$338$	0	0
$339$	5.76949	0.313356
$340$	0	0
$341$	1.29972	0.0703841
$342$	0	0
$343$	16.6817	0.900730
$344$	0	0
$345$	3.42621	0.184461
$346$	0	0
$347$	29.2873	1.57222	0.786111	−	0.618085i	$-0.212091\pi$
$347$	29.2873	1.57222	0.786111	+	0.618085i	$0.212091\pi$
$348$	0	0
$349$	20.5332	1.09912	0.549559	−	0.835455i	$-0.314796\pi$
$349$	20.5332	1.09912	0.549559	+	0.835455i	$0.314796\pi$
$350$	0	0
$351$	1.92471	0.102734
$352$	0	0
$353$	−9.33420	−0.496809	−0.248405	−	0.968656i	$-0.579906\pi$
$353$	−9.33420	−0.496809	−0.248405	+	0.968656i	$0.579906\pi$
$354$	0	0
$355$	5.70297	0.302682
$356$	0	0
$357$	8.08049	0.427665
$358$	0	0
$359$	−19.1808	−1.01232	−0.506162	−	0.862439i	$-0.668936\pi$
$359$	−19.1808	−1.01232	−0.506162	+	0.862439i	$0.668936\pi$
$360$	0	0
$361$	16.6981	0.878847
$362$	0	0
$363$	−5.06276	−0.265726
$364$	0	0
$365$	−10.1099	−0.529177
$366$	0	0
$367$	36.6243	1.91177	0.955887	−	0.293735i	$-0.0948983\pi$
$367$	36.6243	1.91177	0.955887	+	0.293735i	$0.0948983\pi$
$368$	0	0
$369$	−2.13883	−0.111343
$370$	0	0
$371$	−32.2907	−1.67645
$372$	0	0
$373$	13.0025	0.673242	0.336621	−	0.941640i	$-0.390716\pi$
$373$	13.0025	0.673242	0.336621	+	0.941640i	$0.390716\pi$
$374$	0	0
$375$	8.13943	0.420318
$376$	0	0
$377$	−12.9622	−0.667588
$378$	0	0
$379$	14.5743	0.748632	0.374316	−	0.927301i	$-0.377877\pi$
$379$	14.5743	0.748632	0.374316	+	0.927301i	$0.377877\pi$
$380$	0	0
$381$	10.5810	0.542081
$382$	0	0
$383$	−8.32014	−0.425139	−0.212570	−	0.977146i	$-0.568183\pi$
$383$	−8.32014	−0.425139	−0.212570	+	0.977146i	$0.568183\pi$
$384$	0	0
$385$	6.14429	0.313142
$386$	0	0
$387$	−7.27961	−0.370043
$388$	0	0
$389$	−4.05565	−0.205629	−0.102815	−	0.994701i	$-0.532785\pi$
$389$	−4.05565	−0.205629	−0.102815	+	0.994701i	$0.532785\pi$
$390$	0	0
$391$	10.9792	0.555243
$392$	0	0
$393$	−13.0718	−0.659387
$394$	0	0
$395$	3.45329	0.173754
$396$	0	0
$397$	−11.2905	−0.566654	−0.283327	−	0.959023i	$-0.591438\pi$
$397$	−11.2905	−0.566654	−0.283327	+	0.959023i	$0.591438\pi$
$398$	0	0
$399$	−17.0678	−0.854457
$400$	0	0
$401$	−16.0161	−0.799806	−0.399903	−	0.916557i	$-0.630956\pi$
$401$	−16.0161	−0.799806	−0.399903	+	0.916557i	$0.630956\pi$
$402$	0	0
$403$	1.02666	0.0511414
$404$	0	0
$405$	−0.882725	−0.0438630
$406$	0	0
$407$	7.80352	0.386806
$408$	0	0
$409$	16.9702	0.839124	0.419562	−	0.907727i	$-0.362184\pi$
$409$	16.9702	0.839124	0.419562	+	0.907727i	$0.362184\pi$
$410$	0	0
$411$	−18.9259	−0.933545
$412$	0	0
$413$	23.9556	1.17878
$414$	0	0
$415$	11.0628	0.543051
$416$	0	0
$417$	9.86280	0.482984
$418$	0	0
$419$	−22.2427	−1.08663	−0.543314	−	0.839529i	$-0.682831\pi$
$419$	−22.2427	−1.08663	−0.543314	+	0.839529i	$0.682831\pi$
$420$	0	0
$421$	−12.7430	−0.621056	−0.310528	−	0.950564i	$-0.600506\pi$
$421$	−12.7430	−0.621056	−0.310528	+	0.950564i	$0.600506\pi$
$422$	0	0
$423$	−0.965205	−0.0469299
$424$	0	0
$425$	11.9393	0.579140
$426$	0	0
$427$	13.4892	0.652787
$428$	0	0
$429$	4.68984	0.226428
$430$	0	0
$431$	−20.6029	−0.992405	−0.496203	−	0.868207i	$-0.665273\pi$
$431$	−20.6029	−0.992405	−0.496203	+	0.868207i	$0.665273\pi$
$432$	0	0
$433$	3.98986	0.191741	0.0958703	−	0.995394i	$-0.469437\pi$
$433$	3.98986	0.191741	0.0958703	+	0.995394i	$0.469437\pi$
$434$	0	0
$435$	5.94482	0.285032
$436$	0	0
$437$	−23.1905	−1.10935
$438$	0	0
$439$	34.2107	1.63279	0.816394	−	0.577495i	$-0.195970\pi$
$439$	34.2107	1.63279	0.816394	+	0.577495i	$0.195970\pi$
$440$	0	0
$441$	1.16034	0.0552545
$442$	0	0
$443$	−4.95750	−0.235538	−0.117769	−	0.993041i	$-0.537574\pi$
$443$	−4.95750	−0.235538	−0.117769	+	0.993041i	$0.537574\pi$
$444$	0	0
$445$	−13.1572	−0.623709
$446$	0	0
$447$	−14.3681	−0.679588
$448$	0	0
$449$	37.1327	1.75240	0.876200	−	0.481948i	$-0.160071\pi$
$449$	37.1327	1.75240	0.876200	+	0.481948i	$0.160071\pi$
$450$	0	0
$451$	−5.21156	−0.245403
$452$	0	0
$453$	3.21916	0.151249
$454$	0	0
$455$	4.85340	0.227531
$456$	0	0
$457$	31.0058	1.45039	0.725196	−	0.688543i	$-0.241749\pi$
$457$	31.0058	1.45039	0.725196	+	0.688543i	$0.241749\pi$
$458$	0	0
$459$	−2.82868	−0.132031
$460$	0	0
$461$	4.73747	0.220646	0.110323	−	0.993896i	$-0.464811\pi$
$461$	4.73747	0.220646	0.110323	+	0.993896i	$0.464811\pi$
$462$	0	0
$463$	38.1870	1.77470	0.887351	−	0.461094i	$-0.152543\pi$
$463$	38.1870	1.77470	0.887351	+	0.461094i	$0.152543\pi$
$464$	0	0
$465$	−0.470852	−0.0218353
$466$	0	0
$467$	35.7460	1.65413	0.827063	−	0.562109i	$-0.190010\pi$
$467$	35.7460	1.65413	0.827063	+	0.562109i	$0.190010\pi$
$468$	0	0
$469$	−6.73831	−0.311146
$470$	0	0
$471$	−10.3589	−0.477315
$472$	0	0
$473$	−17.7378	−0.815586
$474$	0	0
$475$	−25.2184	−1.15710
$476$	0	0
$477$	11.3038	0.517563
$478$	0	0
$479$	23.3727	1.06793	0.533963	−	0.845508i	$-0.320702\pi$
$479$	23.3727	1.06793	0.533963	+	0.845508i	$0.320702\pi$
$480$	0	0
$481$	6.16402	0.281055
$482$	0	0
$483$	11.0877	0.504509
$484$	0	0
$485$	8.24794	0.374520
$486$	0	0
$487$	−10.6405	−0.482166	−0.241083	−	0.970504i	$-0.577503\pi$
$487$	−10.6405	−0.482166	−0.241083	+	0.970504i	$0.577503\pi$
$488$	0	0
$489$	−8.00830	−0.362148
$490$	0	0
$491$	−9.36682	−0.422719	−0.211359	−	0.977408i	$-0.567789\pi$
$491$	−9.36682	−0.422719	−0.211359	+	0.977408i	$0.567789\pi$
$492$	0	0
$493$	19.0501	0.857973
$494$	0	0
$495$	−2.15089	−0.0966752
$496$	0	0
$497$	18.4557	0.827849
$498$	0	0
$499$	−2.71203	−0.121407	−0.0607037	−	0.998156i	$-0.519334\pi$
$499$	−2.71203	−0.121407	−0.0607037	+	0.998156i	$0.519334\pi$
$500$	0	0
$501$	−9.92715	−0.443512
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−4.21193	−0.187428
$506$	0	0
$507$	−9.29548	−0.412827
$508$	0	0
$509$	−24.8305	−1.10059	−0.550296	−	0.834970i	$-0.685485\pi$
$509$	−24.8305	−1.10059	−0.550296	+	0.834970i	$0.685485\pi$
$510$	0	0
$511$	−32.7172	−1.44732
$512$	0	0
$513$	5.97479	0.263793
$514$	0	0
$515$	−1.19209	−0.0525298
$516$	0	0
$517$	−2.35186	−0.103435
$518$	0	0
$519$	−15.5681	−0.683364
$520$	0	0
$521$	−5.29102	−0.231804	−0.115902	−	0.993261i	$-0.536976\pi$
$521$	−5.29102	−0.231804	−0.115902	+	0.993261i	$0.536976\pi$
$522$	0	0
$523$	41.4592	1.81289	0.906443	−	0.422329i	$-0.138787\pi$
$523$	41.4592	1.81289	0.906443	+	0.422329i	$0.138787\pi$
$524$	0	0
$525$	12.0573	0.526222
$526$	0	0
$527$	−1.50884	−0.0657260
$528$	0	0
$529$	−7.93476	−0.344990
$530$	0	0
$531$	−8.38597	−0.363920
$532$	0	0
$533$	−4.11663	−0.178311
$534$	0	0
$535$	−10.4738	−0.452822
$536$	0	0
$537$	3.70146	0.159730
$538$	0	0
$539$	2.82735	0.121782
$540$	0	0
$541$	−22.8495	−0.982378	−0.491189	−	0.871053i	$-0.663438\pi$
$541$	−22.8495	−0.982378	−0.491189	+	0.871053i	$0.663438\pi$
$542$	0	0
$543$	−24.7381	−1.06161
$544$	0	0
$545$	1.97876	0.0847609
$546$	0	0
$547$	−24.1153	−1.03110	−0.515549	−	0.856860i	$-0.672412\pi$
$547$	−24.1153	−1.03110	−0.515549	+	0.856860i	$0.672412\pi$
$548$	0	0
$549$	−4.72206	−0.201532
$550$	0	0
$551$	−40.2379	−1.71419
$552$	0	0
$553$	11.1754	0.475225
$554$	0	0
$555$	−2.82699	−0.119999
$556$	0	0
$557$	−3.30050	−0.139847	−0.0699233	−	0.997552i	$-0.522275\pi$
$557$	−3.30050	−0.139847	−0.0699233	+	0.997552i	$0.522275\pi$
$558$	0	0
$559$	−14.0112	−0.592609
$560$	0	0
$561$	−6.89248	−0.291001
$562$	0	0
$563$	17.4662	0.736114	0.368057	−	0.929803i	$-0.380023\pi$
$563$	17.4662	0.736114	0.368057	+	0.929803i	$0.380023\pi$
$564$	0	0
$565$	−5.09288	−0.214259
$566$	0	0
$567$	−2.85663	−0.119967
$568$	0	0
$569$	19.2414	0.806641	0.403321	−	0.915059i	$-0.367856\pi$
$569$	19.2414	0.806641	0.403321	+	0.915059i	$0.367856\pi$
$570$	0	0
$571$	34.2454	1.43312	0.716562	−	0.697523i	$-0.245715\pi$
$571$	34.2454	1.43312	0.716562	+	0.697523i	$0.245715\pi$
$572$	0	0
$573$	−13.0911	−0.546889
$574$	0	0
$575$	16.3826	0.683201
$576$	0	0
$577$	39.2507	1.63403	0.817014	−	0.576618i	$-0.195628\pi$
$577$	39.2507	1.63403	0.817014	+	0.576618i	$0.195628\pi$
$578$	0	0
$579$	−5.85802	−0.243451
$580$	0	0
$581$	35.8009	1.48527
$582$	0	0
$583$	27.5432	1.14072
$584$	0	0
$585$	−1.69899	−0.0702447
$586$	0	0
$587$	17.9941	0.742695	0.371348	−	0.928494i	$-0.378896\pi$
$587$	17.9941	0.742695	0.371348	+	0.928494i	$0.378896\pi$
$588$	0	0
$589$	3.18700	0.131318
$590$	0	0
$591$	−13.1264	−0.539950
$592$	0	0
$593$	−27.8655	−1.14430	−0.572149	−	0.820150i	$-0.693890\pi$
$593$	−27.8655	−1.14430	−0.572149	+	0.820150i	$0.693890\pi$
$594$	0	0
$595$	−7.13285	−0.292418
$596$	0	0
$597$	−9.41618	−0.385379
$598$	0	0
$599$	−4.42797	−0.180922	−0.0904610	−	0.995900i	$-0.528834\pi$
$599$	−4.42797	−0.180922	−0.0904610	+	0.995900i	$0.528834\pi$
$600$	0	0
$601$	11.7307	0.478506	0.239253	−	0.970957i	$-0.423097\pi$
$601$	11.7307	0.478506	0.239253	+	0.970957i	$0.423097\pi$
$602$	0	0
$603$	2.35883	0.0960591
$604$	0	0
$605$	4.46903	0.181692
$606$	0	0
$607$	31.8144	1.29131	0.645654	−	0.763630i	$-0.276585\pi$
$607$	31.8144	1.29131	0.645654	+	0.763630i	$0.276585\pi$
$608$	0	0
$609$	19.2383	0.779577
$610$	0	0
$611$	−1.85774	−0.0751562
$612$	0	0
$613$	−8.80960	−0.355816	−0.177908	−	0.984047i	$-0.556933\pi$
$613$	−8.80960	−0.355816	−0.177908	+	0.984047i	$0.556933\pi$
$614$	0	0
$615$	1.88800	0.0761314
$616$	0	0
$617$	16.7013	0.672369	0.336184	−	0.941796i	$-0.390863\pi$
$617$	16.7013	0.672369	0.336184	+	0.941796i	$0.390863\pi$
$618$	0	0
$619$	10.8816	0.437368	0.218684	−	0.975796i	$-0.429824\pi$
$619$	10.8816	0.437368	0.218684	+	0.975796i	$0.429824\pi$
$620$	0	0
$621$	−3.88140	−0.155755
$622$	0	0
$623$	−42.5785	−1.70587
$624$	0	0
$625$	13.9191	0.556764
$626$	0	0
$627$	14.5584	0.581408
$628$	0	0
$629$	−9.05903	−0.361207
$630$	0	0
$631$	−7.43646	−0.296041	−0.148020	−	0.988984i	$-0.547290\pi$
$631$	−7.43646	−0.296041	−0.148020	+	0.988984i	$0.547290\pi$
$632$	0	0
$633$	8.03579	0.319394
$634$	0	0
$635$	−9.34012	−0.370651
$636$	0	0
$637$	2.23333	0.0884877
$638$	0	0
$639$	−6.46064	−0.255579
$640$	0	0
$641$	−24.5853	−0.971062	−0.485531	−	0.874219i	$-0.661374\pi$
$641$	−24.5853	−0.971062	−0.485531	+	0.874219i	$0.661374\pi$
$642$	0	0
$643$	27.9134	1.10080	0.550399	−	0.834902i	$-0.314476\pi$
$643$	27.9134	1.10080	0.550399	+	0.834902i	$0.314476\pi$
$644$	0	0
$645$	6.42589	0.253019
$646$	0	0
$647$	12.0352	0.473154	0.236577	−	0.971613i	$-0.423974\pi$
$647$	12.0352	0.473154	0.236577	+	0.971613i	$0.423974\pi$
$648$	0	0
$649$	−20.4336	−0.802090
$650$	0	0
$651$	−1.52375	−0.0597204
$652$	0	0
$653$	21.2680	0.832282	0.416141	−	0.909300i	$-0.363382\pi$
$653$	21.2680	0.832282	0.416141	+	0.909300i	$0.363382\pi$
$654$	0	0
$655$	11.5388	0.450860
$656$	0	0
$657$	11.4531	0.446827
$658$	0	0
$659$	−30.6855	−1.19534	−0.597669	−	0.801743i	$-0.703906\pi$
$659$	−30.6855	−1.19534	−0.597669	+	0.801743i	$0.703906\pi$
$660$	0	0
$661$	−2.72192	−0.105870	−0.0529352	−	0.998598i	$-0.516858\pi$
$661$	−2.72192	−0.105870	−0.0529352	+	0.998598i	$0.516858\pi$
$662$	0	0
$663$	−5.44439	−0.211443
$664$	0	0
$665$	15.0661	0.584240
$666$	0	0
$667$	26.1397	1.01213
$668$	0	0
$669$	−17.9834	−0.695278
$670$	0	0
$671$	−11.5060	−0.444183
$672$	0	0
$673$	3.60459	0.138947	0.0694733	−	0.997584i	$-0.477868\pi$
$673$	3.60459	0.138947	0.0694733	+	0.997584i	$0.477868\pi$
$674$	0	0
$675$	−4.22080	−0.162459
$676$	0	0
$677$	13.6457	0.524447	0.262223	−	0.965007i	$-0.415544\pi$
$677$	13.6457	0.524447	0.262223	+	0.965007i	$0.415544\pi$
$678$	0	0
$679$	26.6916	1.02433
$680$	0	0
$681$	−14.5752	−0.558522
$682$	0	0
$683$	10.4370	0.399360	0.199680	−	0.979861i	$-0.436010\pi$
$683$	10.4370	0.399360	0.199680	+	0.979861i	$0.436010\pi$
$684$	0	0
$685$	16.7064	0.638317
$686$	0	0
$687$	−28.1053	−1.07228
$688$	0	0
$689$	21.7565	0.828856
$690$	0	0
$691$	18.6888	0.710955	0.355478	−	0.934685i	$-0.384318\pi$
$691$	18.6888	0.710955	0.355478	+	0.934685i	$0.384318\pi$
$692$	0	0
$693$	−6.96060	−0.264411
$694$	0	0
$695$	−8.70615	−0.330243
$696$	0	0
$697$	6.05005	0.229162
$698$	0	0
$699$	1.44456	0.0546383
$700$	0	0
$701$	7.43350	0.280759	0.140380	−	0.990098i	$-0.455168\pi$
$701$	7.43350	0.280759	0.140380	+	0.990098i	$0.455168\pi$
$702$	0	0
$703$	19.1346	0.721677
$704$	0	0
$705$	0.852011	0.0320886
$706$	0	0
$707$	−13.6304	−0.512625
$708$	0	0
$709$	−8.76906	−0.329329	−0.164665	−	0.986350i	$-0.552654\pi$
$709$	−8.76906	−0.329329	−0.164665	+	0.986350i	$0.552654\pi$
$710$	0	0
$711$	−3.91208	−0.146714
$712$	0	0
$713$	−2.07037	−0.0775358
$714$	0	0
$715$	−4.13984	−0.154821
$716$	0	0
$717$	−10.8011	−0.403374
$718$	0	0
$719$	23.4746	0.875454	0.437727	−	0.899108i	$-0.355784\pi$
$719$	23.4746	0.875454	0.437727	+	0.899108i	$0.355784\pi$
$720$	0	0
$721$	−3.85778	−0.143671
$722$	0	0
$723$	17.0594	0.634447
$724$	0	0
$725$	28.4255	1.05570
$726$	0	0
$727$	46.5560	1.72667	0.863333	−	0.504634i	$-0.168373\pi$
$727$	46.5560	1.72667	0.863333	+	0.504634i	$0.168373\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	20.5917	0.761610
$732$	0	0
$733$	−1.66488	−0.0614937	−0.0307468	−	0.999527i	$-0.509789\pi$
$733$	−1.66488	−0.0614937	−0.0307468	+	0.999527i	$0.509789\pi$
$734$	0	0
$735$	−1.02426	−0.0377806
$736$	0	0
$737$	5.74764	0.211717
$738$	0	0
$739$	22.5472	0.829412	0.414706	−	0.909955i	$-0.363884\pi$
$739$	22.5472	0.829412	0.414706	+	0.909955i	$0.363884\pi$
$740$	0	0
$741$	11.4998	0.422454
$742$	0	0
$743$	−3.83847	−0.140820	−0.0704098	−	0.997518i	$-0.522431\pi$
$743$	−3.83847	−0.140820	−0.0704098	+	0.997518i	$0.522431\pi$
$744$	0	0
$745$	12.6831	0.464673
$746$	0	0
$747$	−12.5325	−0.458542
$748$	0	0
$749$	−33.8948	−1.23849
$750$	0	0
$751$	−50.8035	−1.85385	−0.926923	−	0.375251i	$-0.877557\pi$
$751$	−50.8035	−1.85385	−0.926923	+	0.375251i	$0.877557\pi$
$752$	0	0
$753$	−18.3563	−0.668942
$754$	0	0
$755$	−2.84163	−0.103418
$756$	0	0
$757$	−5.91138	−0.214853	−0.107426	−	0.994213i	$-0.534261\pi$
$757$	−5.91138	−0.214853	−0.107426	+	0.994213i	$0.534261\pi$
$758$	0	0
$759$	−9.45758	−0.343289
$760$	0	0
$761$	−16.7837	−0.608409	−0.304205	−	0.952607i	$-0.598391\pi$
$761$	−16.7837	−0.608409	−0.304205	+	0.952607i	$0.598391\pi$
$762$	0	0
$763$	6.40357	0.231825
$764$	0	0
$765$	2.49695	0.0902773
$766$	0	0
$767$	−16.1406	−0.582803
$768$	0	0
$769$	−20.6396	−0.744282	−0.372141	−	0.928176i	$-0.621376\pi$
$769$	−20.6396	−0.744282	−0.372141	+	0.928176i	$0.621376\pi$
$770$	0	0
$771$	21.6433	0.779464
$772$	0	0
$773$	−38.6761	−1.39108	−0.695541	−	0.718487i	$-0.744835\pi$
$773$	−38.6761	−1.39108	−0.695541	+	0.718487i	$0.744835\pi$
$774$	0	0
$775$	−2.25140	−0.0808729
$776$	0	0
$777$	−9.14855	−0.328202
$778$	0	0
$779$	−12.7790	−0.457857
$780$	0	0
$781$	−15.7423	−0.563303
$782$	0	0
$783$	−6.73462	−0.240676
$784$	0	0
$785$	9.14411	0.326367
$786$	0	0
$787$	22.6311	0.806712	0.403356	−	0.915043i	$-0.367844\pi$
$787$	22.6311	0.806712	0.403356	+	0.915043i	$0.367844\pi$
$788$	0	0
$789$	4.38202	0.156004
$790$	0	0
$791$	−16.4813	−0.586008
$792$	0	0
$793$	−9.08860	−0.322746
$794$	0	0
$795$	−9.97811	−0.353887
$796$	0	0
$797$	−51.4717	−1.82322	−0.911611	−	0.411054i	$-0.865161\pi$
$797$	−51.4717	−1.82322	−0.911611	+	0.411054i	$0.865161\pi$
$798$	0	0
$799$	2.73025	0.0965895
$800$	0	0
$801$	14.9051	0.526647
$802$	0	0
$803$	27.9071	0.984819
$804$	0	0
$805$	−9.78741	−0.344961
$806$	0	0
$807$	6.04760	0.212885
$808$	0	0
$809$	2.73341	0.0961017	0.0480508	−	0.998845i	$-0.484699\pi$
$809$	2.73341	0.0961017	0.0480508	+	0.998845i	$0.484699\pi$
$810$	0	0
$811$	−2.04350	−0.0717571	−0.0358785	−	0.999356i	$-0.511423\pi$
$811$	−2.04350	−0.0717571	−0.0358785	+	0.999356i	$0.511423\pi$
$812$	0	0
$813$	−20.9757	−0.735651
$814$	0	0
$815$	7.06913	0.247621
$816$	0	0
$817$	−43.4941	−1.52167
$818$	0	0
$819$	−5.49820	−0.192123
$820$	0	0
$821$	6.34354	0.221391	0.110696	−	0.993854i	$-0.464692\pi$
$821$	6.34354	0.221391	0.110696	+	0.993854i	$0.464692\pi$
$822$	0	0
$823$	−13.3451	−0.465182	−0.232591	−	0.972575i	$-0.574720\pi$
$823$	−13.3451	−0.465182	−0.232591	+	0.972575i	$0.574720\pi$
$824$	0	0
$825$	−10.2846	−0.358063
$826$	0	0
$827$	36.2920	1.26200	0.630999	−	0.775784i	$-0.282645\pi$
$827$	36.2920	1.26200	0.630999	+	0.775784i	$0.282645\pi$
$828$	0	0
$829$	3.16316	0.109861	0.0549305	−	0.998490i	$-0.482506\pi$
$829$	3.16316	0.109861	0.0549305	+	0.998490i	$0.482506\pi$
$830$	0	0
$831$	−16.9015	−0.586306
$832$	0	0
$833$	−3.28224	−0.113723
$834$	0	0
$835$	8.76295	0.303254
$836$	0	0
$837$	0.533408	0.0184373
$838$	0	0
$839$	30.4924	1.05271	0.526357	−	0.850264i	$-0.323558\pi$
$839$	30.4924	1.05271	0.526357	+	0.850264i	$0.323558\pi$
$840$	0	0
$841$	16.3552	0.563971
$842$	0	0
$843$	−13.3651	−0.460319
$844$	0	0
$845$	8.20535	0.282273
$846$	0	0
$847$	14.4624	0.496936
$848$	0	0
$849$	−7.49433	−0.257205
$850$	0	0
$851$	−12.4304	−0.426110
$852$	0	0
$853$	31.9933	1.09543	0.547714	−	0.836665i	$-0.315498\pi$
$853$	31.9933	1.09543	0.547714	+	0.836665i	$0.315498\pi$
$854$	0	0
$855$	−5.27410	−0.180370
$856$	0	0
$857$	−24.9200	−0.851251	−0.425626	−	0.904899i	$-0.639946\pi$
$857$	−24.9200	−0.851251	−0.425626	+	0.904899i	$0.639946\pi$
$858$	0	0
$859$	12.9699	0.442526	0.221263	−	0.975214i	$-0.428982\pi$
$859$	12.9699	0.442526	0.221263	+	0.975214i	$0.428982\pi$
$860$	0	0
$861$	6.10984	0.208223
$862$	0	0
$863$	−11.2252	−0.382110	−0.191055	−	0.981579i	$-0.561191\pi$
$863$	−11.2252	−0.382110	−0.191055	+	0.981579i	$0.561191\pi$
$864$	0	0
$865$	13.7424	0.467254
$866$	0	0
$867$	−8.99858	−0.305608
$868$	0	0
$869$	−9.53234	−0.323363
$870$	0	0
$871$	4.54008	0.153835
$872$	0	0
$873$	−9.34373	−0.316237
$874$	0	0
$875$	−23.2514	−0.786039
$876$	0	0
$877$	−26.2023	−0.884789	−0.442395	−	0.896820i	$-0.645871\pi$
$877$	−26.2023	−0.884789	−0.442395	+	0.896820i	$0.645871\pi$
$878$	0	0
$879$	−32.6876	−1.10252
$880$	0	0
$881$	−20.8166	−0.701328	−0.350664	−	0.936501i	$-0.614044\pi$
$881$	−20.8166	−0.701328	−0.350664	+	0.936501i	$0.614044\pi$
$882$	0	0
$883$	−53.1311	−1.78801	−0.894003	−	0.448062i	$-0.852114\pi$
$883$	−53.1311	−1.78801	−0.894003	+	0.448062i	$0.852114\pi$
$884$	0	0
$885$	7.40251	0.248833
$886$	0	0
$887$	−4.77782	−0.160424	−0.0802118	−	0.996778i	$-0.525560\pi$
$887$	−4.77782	−0.160424	−0.0802118	+	0.996778i	$0.525560\pi$
$888$	0	0
$889$	−30.2260	−1.01375
$890$	0	0
$891$	2.43664	0.0816307
$892$	0	0
$893$	−5.76690	−0.192982
$894$	0	0
$895$	−3.26737	−0.109216
$896$	0	0
$897$	−7.47057	−0.249435
$898$	0	0
$899$	−3.59230	−0.119810
$900$	0	0
$901$	−31.9747	−1.06523
$902$	0	0
$903$	20.7952	0.692019
$904$	0	0
$905$	21.8369	0.725885
$906$	0	0
$907$	−10.9101	−0.362264	−0.181132	−	0.983459i	$-0.557976\pi$
$907$	−10.9101	−0.362264	−0.181132	+	0.983459i	$0.557976\pi$
$908$	0	0
$909$	4.77150	0.158261
$910$	0	0
$911$	17.2073	0.570104	0.285052	−	0.958512i	$-0.407989\pi$
$911$	17.2073	0.570104	0.285052	+	0.958512i	$0.407989\pi$
$912$	0	0
$913$	−30.5374	−1.01064
$914$	0	0
$915$	4.16828	0.137799
$916$	0	0
$917$	37.3414	1.23312
$918$	0	0
$919$	−53.7286	−1.77234	−0.886171	−	0.463358i	$-0.846644\pi$
$919$	−53.7286	−1.77234	−0.886171	+	0.463358i	$0.846644\pi$
$920$	0	0
$921$	14.0085	0.461595
$922$	0	0
$923$	−12.4349	−0.409299
$924$	0	0
$925$	−13.5174	−0.444449
$926$	0	0
$927$	1.35047	0.0443551
$928$	0	0
$929$	3.28437	0.107757	0.0538783	−	0.998548i	$-0.482842\pi$
$929$	3.28437	0.107757	0.0538783	+	0.998548i	$0.482842\pi$
$930$	0	0
$931$	6.93281	0.227214
$932$	0	0
$933$	−4.23519	−0.138654
$934$	0	0
$935$	6.08417	0.198974
$936$	0	0
$937$	−3.87143	−0.126474	−0.0632371	−	0.997999i	$-0.520142\pi$
$937$	−3.87143	−0.126474	−0.0632371	+	0.997999i	$0.520142\pi$
$938$	0	0
$939$	31.5070	1.02819
$940$	0	0
$941$	−22.2231	−0.724451	−0.362225	−	0.932091i	$-0.617983\pi$
$941$	−22.2231	−0.724451	−0.362225	+	0.932091i	$0.617983\pi$
$942$	0	0
$943$	8.30164	0.270338
$944$	0	0
$945$	2.52162	0.0820283
$946$	0	0
$947$	−17.3373	−0.563386	−0.281693	−	0.959505i	$-0.590896\pi$
$947$	−17.3373	−0.563386	−0.281693	+	0.959505i	$0.590896\pi$
$948$	0	0
$949$	22.0439	0.715575
$950$	0	0
$951$	−14.4870	−0.469773
$952$	0	0
$953$	−18.3243	−0.593583	−0.296791	−	0.954942i	$-0.595917\pi$
$953$	−18.3243	−0.593583	−0.296791	+	0.954942i	$0.595917\pi$
$954$	0	0
$955$	11.5559	0.373939
$956$	0	0
$957$	−16.4099	−0.530456
$958$	0	0
$959$	54.0643	1.74583
$960$	0	0
$961$	−30.7155	−0.990822
$962$	0	0
$963$	11.8653	0.382354
$964$	0	0
$965$	5.17102	0.166461
$966$	0	0
$967$	38.8684	1.24992	0.624962	−	0.780655i	$-0.285115\pi$
$967$	38.8684	1.24992	0.624962	+	0.780655i	$0.285115\pi$
$968$	0	0
$969$	−16.9007	−0.542930
$970$	0	0
$971$	−35.0772	−1.12568	−0.562841	−	0.826565i	$-0.690292\pi$
$971$	−35.0772	−1.12568	−0.562841	+	0.826565i	$0.690292\pi$
$972$	0	0
$973$	−28.1744	−0.903230
$974$	0	0
$975$	−8.12382	−0.260171
$976$	0	0
$977$	32.6016	1.04302	0.521509	−	0.853246i	$-0.325369\pi$
$977$	32.6016	1.04302	0.521509	+	0.853246i	$0.325369\pi$
$978$	0	0
$979$	36.3186	1.16075
$980$	0	0
$981$	−2.24165	−0.0715704
$982$	0	0
$983$	−52.6627	−1.67968	−0.839839	−	0.542836i	$-0.817351\pi$
$983$	−52.6627	−1.67968	−0.839839	+	0.542836i	$0.817351\pi$
$984$	0	0
$985$	11.5870	0.369194
$986$	0	0
$987$	2.75724	0.0877638
$988$	0	0
$989$	28.2550	0.898458
$990$	0	0
$991$	−51.9623	−1.65064	−0.825319	−	0.564667i	$-0.809005\pi$
$991$	−51.9623	−1.65064	−0.825319	+	0.564667i	$0.809005\pi$
$992$	0	0
$993$	17.0638	0.541503
$994$	0	0
$995$	8.31190	0.263505
$996$	0	0
$997$	47.8628	1.51583	0.757916	−	0.652353i	$-0.226218\pi$
$997$	47.8628	1.51583	0.757916	+	0.652353i	$0.226218\pi$
$998$	0	0
$999$	3.20257	0.101325

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.g.1.7	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.g.1.7	✓	15	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:	\( N \)	\(=\)	\( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6036.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.882725 q^{5} -2.85663 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.882725 q^{5} -2.85663 q^{7} +1.00000 q^{9} +2.43664 q^{11} +1.92471 q^{13} -0.882725 q^{15} -2.82868 q^{17} +5.97479 q^{19} -2.85663 q^{21} -3.88140 q^{23} -4.22080 q^{25} +1.00000 q^{27} -6.73462 q^{29} +0.533408 q^{31} +2.43664 q^{33} +2.52162 q^{35} +3.20257 q^{37} +1.92471 q^{39} -2.13883 q^{41} -7.27961 q^{43} -0.882725 q^{45} -0.965205 q^{47} +1.16034 q^{49} -2.82868 q^{51} +11.3038 q^{53} -2.15089 q^{55} +5.97479 q^{57} -8.38597 q^{59} -4.72206 q^{61} -2.85663 q^{63} -1.69899 q^{65} +2.35883 q^{67} -3.88140 q^{69} -6.46064 q^{71} +11.4531 q^{73} -4.22080 q^{75} -6.96060 q^{77} -3.91208 q^{79} +1.00000 q^{81} -12.5325 q^{83} +2.49695 q^{85} -6.73462 q^{87} +14.9051 q^{89} -5.49820 q^{91} +0.533408 q^{93} -5.27410 q^{95} -9.34373 q^{97} +2.43664 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) \) 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9	\( 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9} - 5 q^{11} - 14 q^{13} - 11 q^{15} - 5 q^{17} + 3 q^{19} - 4 q^{21} - 32 q^{23} + 2 q^{25} + 15 q^{27} - 23 q^{29} - 13 q^{31} - 5 q^{33} - 16 q^{35} - 10 q^{37} - 14 q^{39} - 14 q^{41} + 4 q^{43} - 11 q^{45} - 20 q^{47} - 9 q^{49} - 5 q^{51} - 30 q^{53} - 10 q^{55} + 3 q^{57} - 14 q^{59} - 38 q^{61} - 4 q^{63} - 24 q^{65} - 8 q^{67} - 32 q^{69} - 41 q^{71} - 19 q^{73} + 2 q^{75} - 39 q^{77} - 27 q^{79} + 15 q^{81} - 17 q^{83} - 6 q^{85} - 23 q^{87} - 23 q^{89} + 4 q^{91} - 13 q^{93} - 30 q^{95} - 18 q^{97} - 5 q^{99}+O(q^{100}) \) 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9 - 5 * q^11 - 14 * q^13 - 11 * q^15 - 5 * q^17 + 3 * q^19 - 4 * q^21 - 32 * q^23 + 2 * q^25 + 15 * q^27 - 23 * q^29 - 13 * q^31 - 5 * q^33 - 16 * q^35 - 10 * q^37 - 14 * q^39 - 14 * q^41 + 4 * q^43 - 11 * q^45 - 20 * q^47 - 9 * q^49 - 5 * q^51 - 30 * q^53 - 10 * q^55 + 3 * q^57 - 14 * q^59 - 38 * q^61 - 4 * q^63 - 24 * q^65 - 8 * q^67 - 32 * q^69 - 41 * q^71 - 19 * q^73 + 2 * q^75 - 39 * q^77 - 27 * q^79 + 15 * q^81 - 17 * q^83 - 6 * q^85 - 23 * q^87 - 23 * q^89 + 4 * q^91 - 13 * q^93 - 30 * q^95 - 18 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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