LMFDB - Embedded newform 3648.2.a.bq.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3648,2,Mod(1,3648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3648, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3648.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3648 = 2^{6} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3648.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:	$29.1294266574$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 228)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.37228$ of defining polynomial
Character	$\chi$	$=$	3648.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.37228 q^{5} -3.37228 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.37228 q^{5} -3.37228 q^{7} +1.00000 q^{9} +1.37228 q^{11} -2.00000 q^{13} +1.37228 q^{15} +1.37228 q^{17} +1.00000 q^{19} -3.37228 q^{21} +8.74456 q^{23} -3.11684 q^{25} +1.00000 q^{27} -2.74456 q^{29} +6.74456 q^{31} +1.37228 q^{33} -4.62772 q^{35} -4.74456 q^{37} -2.00000 q^{39} +3.37228 q^{43} +1.37228 q^{45} +13.3723 q^{47} +4.37228 q^{49} +1.37228 q^{51} +2.74456 q^{53} +1.88316 q^{55} +1.00000 q^{57} +2.62772 q^{61} -3.37228 q^{63} -2.74456 q^{65} -9.48913 q^{67} +8.74456 q^{69} +12.0000 q^{71} -5.37228 q^{73} -3.11684 q^{75} -4.62772 q^{77} -8.00000 q^{79} +1.00000 q^{81} +8.74456 q^{83} +1.88316 q^{85} -2.74456 q^{87} +14.7446 q^{89} +6.74456 q^{91} +6.74456 q^{93} +1.37228 q^{95} +14.0000 q^{97} +1.37228 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 3 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 3 * q^5 - q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 3 q^{5} - q^{7} + 2 q^{9} - 3 q^{11} - 4 q^{13} - 3 q^{15} - 3 q^{17} + 2 q^{19} - q^{21} + 6 q^{23} + 11 q^{25} + 2 q^{27} + 6 q^{29} + 2 q^{31} - 3 q^{33} - 15 q^{35} + 2 q^{37} - 4 q^{39} + q^{43} - 3 q^{45} + 21 q^{47} + 3 q^{49} - 3 q^{51} - 6 q^{53} + 21 q^{55} + 2 q^{57} + 11 q^{61} - q^{63} + 6 q^{65} + 4 q^{67} + 6 q^{69} + 24 q^{71} - 5 q^{73} + 11 q^{75} - 15 q^{77} - 16 q^{79} + 2 q^{81} + 6 q^{83} + 21 q^{85} + 6 q^{87} + 18 q^{89} + 2 q^{91} + 2 q^{93} - 3 q^{95} + 28 q^{97} - 3 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 3 * q^5 - q^7 + 2 * q^9 - 3 * q^11 - 4 * q^13 - 3 * q^15 - 3 * q^17 + 2 * q^19 - q^21 + 6 * q^23 + 11 * q^25 + 2 * q^27 + 6 * q^29 + 2 * q^31 - 3 * q^33 - 15 * q^35 + 2 * q^37 - 4 * q^39 + q^43 - 3 * q^45 + 21 * q^47 + 3 * q^49 - 3 * q^51 - 6 * q^53 + 21 * q^55 + 2 * q^57 + 11 * q^61 - q^63 + 6 * q^65 + 4 * q^67 + 6 * q^69 + 24 * q^71 - 5 * q^73 + 11 * q^75 - 15 * q^77 - 16 * q^79 + 2 * q^81 + 6 * q^83 + 21 * q^85 + 6 * q^87 + 18 * q^89 + 2 * q^91 + 2 * q^93 - 3 * q^95 + 28 * q^97 - 3 * 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
$5$	1.37228	0.613703	0.306851	−	0.951757i	$-0.400725\pi$
$5$	1.37228	0.613703	0.306851	+	0.951757i	$0.400725\pi$
$6$	0	0
$7$	−3.37228	−1.27460	−0.637301	−	0.770615i	$-0.719949\pi$
$7$	−3.37228	−1.27460	−0.637301	+	0.770615i	$0.719949\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.37228	0.413758	0.206879	−	0.978366i	$-0.433669\pi$
$11$	1.37228	0.413758	0.206879	+	0.978366i	$0.433669\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	1.37228	0.354322
$16$	0	0
$17$	1.37228	0.332827	0.166414	−	0.986056i	$-0.446781\pi$
$17$	1.37228	0.332827	0.166414	+	0.986056i	$0.446781\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−3.37228	−0.735892
$22$	0	0
$23$	8.74456	1.82337	0.911684	−	0.410893i	$-0.134783\pi$
$23$	8.74456	1.82337	0.911684	+	0.410893i	$0.134783\pi$
$24$	0	0
$25$	−3.11684	−0.623369
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.74456	−0.509652	−0.254826	−	0.966987i	$-0.582018\pi$
$29$	−2.74456	−0.509652	−0.254826	+	0.966987i	$0.582018\pi$
$30$	0	0
$31$	6.74456	1.21136	0.605680	−	0.795709i	$-0.292901\pi$
$31$	6.74456	1.21136	0.605680	+	0.795709i	$0.292901\pi$
$32$	0	0
$33$	1.37228	0.238884
$34$	0	0
$35$	−4.62772	−0.782227
$36$	0	0
$37$	−4.74456	−0.780001	−0.390001	−	0.920815i	$-0.627525\pi$
$37$	−4.74456	−0.780001	−0.390001	+	0.920815i	$0.627525\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	3.37228	0.514268	0.257134	−	0.966376i	$-0.417222\pi$
$43$	3.37228	0.514268	0.257134	+	0.966376i	$0.417222\pi$
$44$	0	0
$45$	1.37228	0.204568
$46$	0	0
$47$	13.3723	1.95055	0.975274	−	0.221000i	$-0.0709320\pi$
$47$	13.3723	1.95055	0.975274	+	0.221000i	$0.0709320\pi$
$48$	0	0
$49$	4.37228	0.624612
$50$	0	0
$51$	1.37228	0.192158
$52$	0	0
$53$	2.74456	0.376995	0.188497	−	0.982074i	$-0.439638\pi$
$53$	2.74456	0.376995	0.188497	+	0.982074i	$0.439638\pi$
$54$	0	0
$55$	1.88316	0.253925
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	2.62772	0.336445	0.168222	−	0.985749i	$-0.446197\pi$
$61$	2.62772	0.336445	0.168222	+	0.985749i	$0.446197\pi$
$62$	0	0
$63$	−3.37228	−0.424868
$64$	0	0
$65$	−2.74456	−0.340421
$66$	0	0
$67$	−9.48913	−1.15928	−0.579641	−	0.814872i	$-0.696807\pi$
$67$	−9.48913	−1.15928	−0.579641	+	0.814872i	$0.696807\pi$
$68$	0	0
$69$	8.74456	1.05272
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	−5.37228	−0.628778	−0.314389	−	0.949294i	$-0.601800\pi$
$73$	−5.37228	−0.628778	−0.314389	+	0.949294i	$0.601800\pi$
$74$	0	0
$75$	−3.11684	−0.359902
$76$	0	0
$77$	−4.62772	−0.527377
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.74456	0.959840	0.479920	−	0.877312i	$-0.340666\pi$
$83$	8.74456	0.959840	0.479920	+	0.877312i	$0.340666\pi$
$84$	0	0
$85$	1.88316	0.204257
$86$	0	0
$87$	−2.74456	−0.294248
$88$	0	0
$89$	14.7446	1.56292	0.781460	−	0.623955i	$-0.214475\pi$
$89$	14.7446	1.56292	0.781460	+	0.623955i	$0.214475\pi$
$90$	0	0
$91$	6.74456	0.707022
$92$	0	0
$93$	6.74456	0.699379
$94$	0	0
$95$	1.37228	0.140793
$96$	0	0
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	0	0
$99$	1.37228	0.137919
$100$	0	0
$101$	−11.4891	−1.14321	−0.571605	−	0.820529i	$-0.693679\pi$
$101$	−11.4891	−1.14321	−0.571605	+	0.820529i	$0.693679\pi$
$102$	0	0
$103$	1.25544	0.123702	0.0618510	−	0.998085i	$-0.480300\pi$
$103$	1.25544	0.123702	0.0618510	+	0.998085i	$0.480300\pi$
$104$	0	0
$105$	−4.62772	−0.451619
$106$	0	0
$107$	14.7446	1.42541	0.712705	−	0.701464i	$-0.247470\pi$
$107$	14.7446	1.42541	0.712705	+	0.701464i	$0.247470\pi$
$108$	0	0
$109$	18.2337	1.74647	0.873235	−	0.487299i	$-0.162018\pi$
$109$	18.2337	1.74647	0.873235	+	0.487299i	$0.162018\pi$
$110$	0	0
$111$	−4.74456	−0.450334
$112$	0	0
$113$	14.7446	1.38705	0.693526	−	0.720432i	$-0.256056\pi$
$113$	14.7446	1.38705	0.693526	+	0.720432i	$0.256056\pi$
$114$	0	0
$115$	12.0000	1.11901
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	−4.62772	−0.424222
$120$	0	0
$121$	−9.11684	−0.828804
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.1386	−0.996266
$126$	0	0
$127$	12.2337	1.08556	0.542782	−	0.839874i	$-0.317371\pi$
$127$	12.2337	1.08556	0.542782	+	0.839874i	$0.317371\pi$
$128$	0	0
$129$	3.37228	0.296913
$130$	0	0
$131$	−18.8614	−1.64793	−0.823964	−	0.566642i	$-0.808242\pi$
$131$	−18.8614	−1.64793	−0.823964	+	0.566642i	$0.808242\pi$
$132$	0	0
$133$	−3.37228	−0.292414
$134$	0	0
$135$	1.37228	0.118107
$136$	0	0
$137$	1.37228	0.117242	0.0586210	−	0.998280i	$-0.481330\pi$
$137$	1.37228	0.117242	0.0586210	+	0.998280i	$0.481330\pi$
$138$	0	0
$139$	15.3723	1.30386	0.651930	−	0.758279i	$-0.273960\pi$
$139$	15.3723	1.30386	0.651930	+	0.758279i	$0.273960\pi$
$140$	0	0
$141$	13.3723	1.12615
$142$	0	0
$143$	−2.74456	−0.229512
$144$	0	0
$145$	−3.76631	−0.312775
$146$	0	0
$147$	4.37228	0.360620
$148$	0	0
$149$	−1.37228	−0.112422	−0.0562108	−	0.998419i	$-0.517902\pi$
$149$	−1.37228	−0.112422	−0.0562108	+	0.998419i	$0.517902\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	1.37228	0.110942
$154$	0	0
$155$	9.25544	0.743415
$156$	0	0
$157$	−7.48913	−0.597697	−0.298849	−	0.954301i	$-0.596603\pi$
$157$	−7.48913	−0.597697	−0.298849	+	0.954301i	$0.596603\pi$
$158$	0	0
$159$	2.74456	0.217658
$160$	0	0
$161$	−29.4891	−2.32407
$162$	0	0
$163$	−9.48913	−0.743246	−0.371623	−	0.928384i	$-0.621199\pi$
$163$	−9.48913	−0.743246	−0.371623	+	0.928384i	$0.621199\pi$
$164$	0	0
$165$	1.88316	0.146603
$166$	0	0
$167$	−20.2337	−1.56573	−0.782865	−	0.622192i	$-0.786242\pi$
$167$	−20.2337	−1.56573	−0.782865	+	0.622192i	$0.786242\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−9.25544	−0.703678	−0.351839	−	0.936061i	$-0.614444\pi$
$173$	−9.25544	−0.703678	−0.351839	+	0.936061i	$0.614444\pi$
$174$	0	0
$175$	10.5109	0.794547
$176$	0	0
$177$	0	0
$178$	0	0
$179$	14.7446	1.10206	0.551030	−	0.834485i	$-0.314235\pi$
$179$	14.7446	1.10206	0.551030	+	0.834485i	$0.314235\pi$
$180$	0	0
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	2.62772	0.194247
$184$	0	0
$185$	−6.51087	−0.478689
$186$	0	0
$187$	1.88316	0.137710
$188$	0	0
$189$	−3.37228	−0.245297
$190$	0	0
$191$	10.6277	0.768995	0.384497	−	0.923126i	$-0.374375\pi$
$191$	10.6277	0.768995	0.384497	+	0.923126i	$0.374375\pi$
$192$	0	0
$193$	16.7446	1.20530	0.602650	−	0.798006i	$-0.294112\pi$
$193$	16.7446	1.20530	0.602650	+	0.798006i	$0.294112\pi$
$194$	0	0
$195$	−2.74456	−0.196542
$196$	0	0
$197$	11.4891	0.818566	0.409283	−	0.912407i	$-0.365779\pi$
$197$	11.4891	0.818566	0.409283	+	0.912407i	$0.365779\pi$
$198$	0	0
$199$	14.1168	1.00072	0.500358	−	0.865818i	$-0.333202\pi$
$199$	14.1168	1.00072	0.500358	+	0.865818i	$0.333202\pi$
$200$	0	0
$201$	−9.48913	−0.669311
$202$	0	0
$203$	9.25544	0.649604
$204$	0	0
$205$	0	0
$206$	0	0
$207$	8.74456	0.607789
$208$	0	0
$209$	1.37228	0.0949227
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	12.0000	0.822226
$214$	0	0
$215$	4.62772	0.315608
$216$	0	0
$217$	−22.7446	−1.54400
$218$	0	0
$219$	−5.37228	−0.363025
$220$	0	0
$221$	−2.74456	−0.184619
$222$	0	0
$223$	−13.4891	−0.903299	−0.451649	−	0.892196i	$-0.649164\pi$
$223$	−13.4891	−0.903299	−0.451649	+	0.892196i	$0.649164\pi$
$224$	0	0
$225$	−3.11684	−0.207790
$226$	0	0
$227$	−8.23369	−0.546489	−0.273245	−	0.961945i	$-0.588097\pi$
$227$	−8.23369	−0.546489	−0.273245	+	0.961945i	$0.588097\pi$
$228$	0	0
$229$	5.37228	0.355010	0.177505	−	0.984120i	$-0.443197\pi$
$229$	5.37228	0.355010	0.177505	+	0.984120i	$0.443197\pi$
$230$	0	0
$231$	−4.62772	−0.304482
$232$	0	0
$233$	−1.37228	−0.0899011	−0.0449506	−	0.998989i	$-0.514313\pi$
$233$	−1.37228	−0.0899011	−0.0449506	+	0.998989i	$0.514313\pi$
$234$	0	0
$235$	18.3505	1.19706
$236$	0	0
$237$	−8.00000	−0.519656
$238$	0	0
$239$	−18.8614	−1.22004	−0.610021	−	0.792385i	$-0.708839\pi$
$239$	−18.8614	−1.22004	−0.610021	+	0.792385i	$0.708839\pi$
$240$	0	0
$241$	−0.744563	−0.0479615	−0.0239807	−	0.999712i	$-0.507634\pi$
$241$	−0.744563	−0.0479615	−0.0239807	+	0.999712i	$0.507634\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	6.00000	0.383326
$246$	0	0
$247$	−2.00000	−0.127257
$248$	0	0
$249$	8.74456	0.554164
$250$	0	0
$251$	−1.37228	−0.0866176	−0.0433088	−	0.999062i	$-0.513790\pi$
$251$	−1.37228	−0.0866176	−0.0433088	+	0.999062i	$0.513790\pi$
$252$	0	0
$253$	12.0000	0.754434
$254$	0	0
$255$	1.88316	0.117928
$256$	0	0
$257$	−17.4891	−1.09094	−0.545471	−	0.838130i	$-0.683649\pi$
$257$	−17.4891	−1.09094	−0.545471	+	0.838130i	$0.683649\pi$
$258$	0	0
$259$	16.0000	0.994192
$260$	0	0
$261$	−2.74456	−0.169884
$262$	0	0
$263$	30.8614	1.90300	0.951498	−	0.307655i	$-0.0995443\pi$
$263$	30.8614	1.90300	0.951498	+	0.307655i	$0.0995443\pi$
$264$	0	0
$265$	3.76631	0.231363
$266$	0	0
$267$	14.7446	0.902353
$268$	0	0
$269$	−14.7446	−0.898992	−0.449496	−	0.893282i	$-0.648396\pi$
$269$	−14.7446	−0.898992	−0.449496	+	0.893282i	$0.648396\pi$
$270$	0	0
$271$	4.00000	0.242983	0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000	0.242983	0.121491	+	0.992592i	$0.461232\pi$
$272$	0	0
$273$	6.74456	0.408200
$274$	0	0
$275$	−4.27719	−0.257924
$276$	0	0
$277$	−24.1168	−1.44904	−0.724520	−	0.689253i	$-0.757939\pi$
$277$	−24.1168	−1.44904	−0.724520	+	0.689253i	$0.757939\pi$
$278$	0	0
$279$	6.74456	0.403786
$280$	0	0
$281$	−26.7446	−1.59545	−0.797723	−	0.603023i	$-0.793963\pi$
$281$	−26.7446	−1.59545	−0.797723	+	0.603023i	$0.793963\pi$
$282$	0	0
$283$	−26.1168	−1.55249	−0.776243	−	0.630434i	$-0.782877\pi$
$283$	−26.1168	−1.55249	−0.776243	+	0.630434i	$0.782877\pi$
$284$	0	0
$285$	1.37228	0.0812869
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−15.1168	−0.889226
$290$	0	0
$291$	14.0000	0.820695
$292$	0	0
$293$	−29.4891	−1.72277	−0.861387	−	0.507950i	$-0.830403\pi$
$293$	−29.4891	−1.72277	−0.861387	+	0.507950i	$0.830403\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.37228	0.0796278
$298$	0	0
$299$	−17.4891	−1.01142
$300$	0	0
$301$	−11.3723	−0.655487
$302$	0	0
$303$	−11.4891	−0.660033
$304$	0	0
$305$	3.60597	0.206477
$306$	0	0
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	0	0
$309$	1.25544	0.0714193
$310$	0	0
$311$	18.8614	1.06953	0.534766	−	0.845000i	$-0.320400\pi$
$311$	18.8614	1.06953	0.534766	+	0.845000i	$0.320400\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	−4.62772	−0.260742
$316$	0	0
$317$	−29.4891	−1.65627	−0.828137	−	0.560526i	$-0.810599\pi$
$317$	−29.4891	−1.65627	−0.828137	+	0.560526i	$0.810599\pi$
$318$	0	0
$319$	−3.76631	−0.210873
$320$	0	0
$321$	14.7446	0.822961
$322$	0	0
$323$	1.37228	0.0763558
$324$	0	0
$325$	6.23369	0.345783
$326$	0	0
$327$	18.2337	1.00833
$328$	0	0
$329$	−45.0951	−2.48617
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	0	0
$333$	−4.74456	−0.260000
$334$	0	0
$335$	−13.0217	−0.711454
$336$	0	0
$337$	19.4891	1.06164	0.530820	−	0.847484i	$-0.321884\pi$
$337$	19.4891	1.06164	0.530820	+	0.847484i	$0.321884\pi$
$338$	0	0
$339$	14.7446	0.800815
$340$	0	0
$341$	9.25544	0.501210
$342$	0	0
$343$	8.86141	0.478471
$344$	0	0
$345$	12.0000	0.646058
$346$	0	0
$347$	16.1168	0.865198	0.432599	−	0.901587i	$-0.357597\pi$
$347$	16.1168	0.865198	0.432599	+	0.901587i	$0.357597\pi$
$348$	0	0
$349$	−18.6277	−0.997119	−0.498559	−	0.866856i	$-0.666137\pi$
$349$	−18.6277	−0.997119	−0.498559	+	0.866856i	$0.666137\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	0	0
$355$	16.4674	0.873998
$356$	0	0
$357$	−4.62772	−0.244925
$358$	0	0
$359$	5.13859	0.271205	0.135602	−	0.990763i	$-0.456703\pi$
$359$	5.13859	0.271205	0.135602	+	0.990763i	$0.456703\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−9.11684	−0.478510
$364$	0	0
$365$	−7.37228	−0.385883
$366$	0	0
$367$	−25.4891	−1.33052	−0.665261	−	0.746611i	$-0.731680\pi$
$367$	−25.4891	−1.33052	−0.665261	+	0.746611i	$0.731680\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−9.25544	−0.480518
$372$	0	0
$373$	30.2337	1.56544	0.782721	−	0.622373i	$-0.213831\pi$
$373$	30.2337	1.56544	0.782721	+	0.622373i	$0.213831\pi$
$374$	0	0
$375$	−11.1386	−0.575194
$376$	0	0
$377$	5.48913	0.282704
$378$	0	0
$379$	21.7228	1.11583	0.557913	−	0.829899i	$-0.311602\pi$
$379$	21.7228	1.11583	0.557913	+	0.829899i	$0.311602\pi$
$380$	0	0
$381$	12.2337	0.626751
$382$	0	0
$383$	−8.23369	−0.420722	−0.210361	−	0.977624i	$-0.567464\pi$
$383$	−8.23369	−0.420722	−0.210361	+	0.977624i	$0.567464\pi$
$384$	0	0
$385$	−6.35053	−0.323653
$386$	0	0
$387$	3.37228	0.171423
$388$	0	0
$389$	25.3723	1.28643	0.643213	−	0.765687i	$-0.277601\pi$
$389$	25.3723	1.28643	0.643213	+	0.765687i	$0.277601\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	−18.8614	−0.951432
$394$	0	0
$395$	−10.9783	−0.552376
$396$	0	0
$397$	−29.6060	−1.48588	−0.742941	−	0.669357i	$-0.766569\pi$
$397$	−29.6060	−1.48588	−0.742941	+	0.669357i	$0.766569\pi$
$398$	0	0
$399$	−3.37228	−0.168825
$400$	0	0
$401$	5.48913	0.274114	0.137057	−	0.990563i	$-0.456236\pi$
$401$	5.48913	0.274114	0.137057	+	0.990563i	$0.456236\pi$
$402$	0	0
$403$	−13.4891	−0.671941
$404$	0	0
$405$	1.37228	0.0681892
$406$	0	0
$407$	−6.51087	−0.322732
$408$	0	0
$409$	30.4674	1.50651	0.753257	−	0.657726i	$-0.228481\pi$
$409$	30.4674	1.50651	0.753257	+	0.657726i	$0.228481\pi$
$410$	0	0
$411$	1.37228	0.0676896
$412$	0	0
$413$	0	0
$414$	0	0
$415$	12.0000	0.589057
$416$	0	0
$417$	15.3723	0.752784
$418$	0	0
$419$	−19.7228	−0.963522	−0.481761	−	0.876303i	$-0.660003\pi$
$419$	−19.7228	−0.963522	−0.481761	+	0.876303i	$0.660003\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	0	0
$423$	13.3723	0.650183
$424$	0	0
$425$	−4.27719	−0.207474
$426$	0	0
$427$	−8.86141	−0.428834
$428$	0	0
$429$	−2.74456	−0.132509
$430$	0	0
$431$	2.74456	0.132201	0.0661005	−	0.997813i	$-0.478944\pi$
$431$	2.74456	0.132201	0.0661005	+	0.997813i	$0.478944\pi$
$432$	0	0
$433$	7.48913	0.359904	0.179952	−	0.983675i	$-0.442406\pi$
$433$	7.48913	0.359904	0.179952	+	0.983675i	$0.442406\pi$
$434$	0	0
$435$	−3.76631	−0.180581
$436$	0	0
$437$	8.74456	0.418309
$438$	0	0
$439$	−4.23369	−0.202063	−0.101031	−	0.994883i	$-0.532214\pi$
$439$	−4.23369	−0.202063	−0.101031	+	0.994883i	$0.532214\pi$
$440$	0	0
$441$	4.37228	0.208204
$442$	0	0
$443$	−13.3723	−0.635336	−0.317668	−	0.948202i	$-0.602900\pi$
$443$	−13.3723	−0.635336	−0.317668	+	0.948202i	$0.602900\pi$
$444$	0	0
$445$	20.2337	0.959169
$446$	0	0
$447$	−1.37228	−0.0649067
$448$	0	0
$449$	−22.9783	−1.08441	−0.542205	−	0.840246i	$-0.682411\pi$
$449$	−22.9783	−1.08441	−0.542205	+	0.840246i	$0.682411\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−8.00000	−0.375873
$454$	0	0
$455$	9.25544	0.433902
$456$	0	0
$457$	14.8614	0.695187	0.347594	−	0.937645i	$-0.386999\pi$
$457$	14.8614	0.695187	0.347594	+	0.937645i	$0.386999\pi$
$458$	0	0
$459$	1.37228	0.0640526
$460$	0	0
$461$	−12.3505	−0.575222	−0.287611	−	0.957747i	$-0.592861\pi$
$461$	−12.3505	−0.575222	−0.287611	+	0.957747i	$0.592861\pi$
$462$	0	0
$463$	34.3505	1.59640	0.798202	−	0.602389i	$-0.205785\pi$
$463$	34.3505	1.59640	0.798202	+	0.602389i	$0.205785\pi$
$464$	0	0
$465$	9.25544	0.429211
$466$	0	0
$467$	4.11684	0.190505	0.0952524	−	0.995453i	$-0.469634\pi$
$467$	4.11684	0.190505	0.0952524	+	0.995453i	$0.469634\pi$
$468$	0	0
$469$	32.0000	1.47762
$470$	0	0
$471$	−7.48913	−0.345081
$472$	0	0
$473$	4.62772	0.212783
$474$	0	0
$475$	−3.11684	−0.143011
$476$	0	0
$477$	2.74456	0.125665
$478$	0	0
$479$	−38.2337	−1.74694	−0.873471	−	0.486876i	$-0.838136\pi$
$479$	−38.2337	−1.74694	−0.873471	+	0.486876i	$0.838136\pi$
$480$	0	0
$481$	9.48913	0.432667
$482$	0	0
$483$	−29.4891	−1.34180
$484$	0	0
$485$	19.2119	0.872369
$486$	0	0
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	0	0
$489$	−9.48913	−0.429113
$490$	0	0
$491$	−27.2554	−1.23002	−0.615010	−	0.788519i	$-0.710848\pi$
$491$	−27.2554	−1.23002	−0.615010	+	0.788519i	$0.710848\pi$
$492$	0	0
$493$	−3.76631	−0.169626
$494$	0	0
$495$	1.88316	0.0846416
$496$	0	0
$497$	−40.4674	−1.81521
$498$	0	0
$499$	−34.3505	−1.53774	−0.768870	−	0.639405i	$-0.779181\pi$
$499$	−34.3505	−1.53774	−0.768870	+	0.639405i	$0.779181\pi$
$500$	0	0
$501$	−20.2337	−0.903975
$502$	0	0
$503$	−3.25544	−0.145153	−0.0725764	−	0.997363i	$-0.523122\pi$
$503$	−3.25544	−0.145153	−0.0725764	+	0.997363i	$0.523122\pi$
$504$	0	0
$505$	−15.7663	−0.701592
$506$	0	0
$507$	−9.00000	−0.399704
$508$	0	0
$509$	14.7446	0.653541	0.326771	−	0.945104i	$-0.394040\pi$
$509$	14.7446	0.653541	0.326771	+	0.945104i	$0.394040\pi$
$510$	0	0
$511$	18.1168	0.801442
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	1.72281	0.0759162
$516$	0	0
$517$	18.3505	0.807055
$518$	0	0
$519$	−9.25544	−0.406269
$520$	0	0
$521$	5.48913	0.240483	0.120241	−	0.992745i	$-0.461633\pi$
$521$	5.48913	0.240483	0.120241	+	0.992745i	$0.461633\pi$
$522$	0	0
$523$	−18.7446	−0.819642	−0.409821	−	0.912166i	$-0.634409\pi$
$523$	−18.7446	−0.819642	−0.409821	+	0.912166i	$0.634409\pi$
$524$	0	0
$525$	10.5109	0.458732
$526$	0	0
$527$	9.25544	0.403173
$528$	0	0
$529$	53.4674	2.32467
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	20.2337	0.874779
$536$	0	0
$537$	14.7446	0.636275
$538$	0	0
$539$	6.00000	0.258438
$540$	0	0
$541$	−26.8614	−1.15486	−0.577431	−	0.816439i	$-0.695945\pi$
$541$	−26.8614	−1.15486	−0.577431	+	0.816439i	$0.695945\pi$
$542$	0	0
$543$	22.0000	0.944110
$544$	0	0
$545$	25.0217	1.07181
$546$	0	0
$547$	−18.7446	−0.801460	−0.400730	−	0.916196i	$-0.631243\pi$
$547$	−18.7446	−0.801460	−0.400730	+	0.916196i	$0.631243\pi$
$548$	0	0
$549$	2.62772	0.112148
$550$	0	0
$551$	−2.74456	−0.116922
$552$	0	0
$553$	26.9783	1.14723
$554$	0	0
$555$	−6.51087	−0.276371
$556$	0	0
$557$	−6.86141	−0.290727	−0.145364	−	0.989378i	$-0.546435\pi$
$557$	−6.86141	−0.290727	−0.145364	+	0.989378i	$0.546435\pi$
$558$	0	0
$559$	−6.74456	−0.285265
$560$	0	0
$561$	1.88316	0.0795069
$562$	0	0
$563$	46.9783	1.97990	0.989949	−	0.141428i	$-0.0451692\pi$
$563$	46.9783	1.97990	0.989949	+	0.141428i	$0.0451692\pi$
$564$	0	0
$565$	20.2337	0.851238
$566$	0	0
$567$	−3.37228	−0.141623
$568$	0	0
$569$	−21.2554	−0.891074	−0.445537	−	0.895263i	$-0.646987\pi$
$569$	−21.2554	−0.891074	−0.445537	+	0.895263i	$0.646987\pi$
$570$	0	0
$571$	18.9783	0.794215	0.397108	−	0.917772i	$-0.370014\pi$
$571$	18.9783	0.794215	0.397108	+	0.917772i	$0.370014\pi$
$572$	0	0
$573$	10.6277	0.443979
$574$	0	0
$575$	−27.2554	−1.13663
$576$	0	0
$577$	3.88316	0.161658	0.0808290	−	0.996728i	$-0.474243\pi$
$577$	3.88316	0.161658	0.0808290	+	0.996728i	$0.474243\pi$
$578$	0	0
$579$	16.7446	0.695880
$580$	0	0
$581$	−29.4891	−1.22342
$582$	0	0
$583$	3.76631	0.155985
$584$	0	0
$585$	−2.74456	−0.113474
$586$	0	0
$587$	33.6060	1.38707	0.693533	−	0.720424i	$-0.256053\pi$
$587$	33.6060	1.38707	0.693533	+	0.720424i	$0.256053\pi$
$588$	0	0
$589$	6.74456	0.277905
$590$	0	0
$591$	11.4891	0.472599
$592$	0	0
$593$	28.9783	1.18999	0.594997	−	0.803728i	$-0.297153\pi$
$593$	28.9783	1.18999	0.594997	+	0.803728i	$0.297153\pi$
$594$	0	0
$595$	−6.35053	−0.260346
$596$	0	0
$597$	14.1168	0.577764
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−12.7446	−0.519862	−0.259931	−	0.965627i	$-0.583700\pi$
$601$	−12.7446	−0.519862	−0.259931	+	0.965627i	$0.583700\pi$
$602$	0	0
$603$	−9.48913	−0.386427
$604$	0	0
$605$	−12.5109	−0.508639
$606$	0	0
$607$	30.7446	1.24788	0.623942	−	0.781471i	$-0.285530\pi$
$607$	30.7446	1.24788	0.623942	+	0.781471i	$0.285530\pi$
$608$	0	0
$609$	9.25544	0.375049
$610$	0	0
$611$	−26.7446	−1.08197
$612$	0	0
$613$	11.8832	0.479956	0.239978	−	0.970778i	$-0.422860\pi$
$613$	11.8832	0.479956	0.239978	+	0.970778i	$0.422860\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−28.1168	−1.13194	−0.565971	−	0.824425i	$-0.691498\pi$
$617$	−28.1168	−1.13194	−0.565971	+	0.824425i	$0.691498\pi$
$618$	0	0
$619$	−38.9783	−1.56667	−0.783334	−	0.621601i	$-0.786483\pi$
$619$	−38.9783	−1.56667	−0.783334	+	0.621601i	$0.786483\pi$
$620$	0	0
$621$	8.74456	0.350907
$622$	0	0
$623$	−49.7228	−1.99210
$624$	0	0
$625$	0.298936	0.0119574
$626$	0	0
$627$	1.37228	0.0548036
$628$	0	0
$629$	−6.51087	−0.259606
$630$	0	0
$631$	−32.8614	−1.30819	−0.654096	−	0.756412i	$-0.726951\pi$
$631$	−32.8614	−1.30819	−0.654096	+	0.756412i	$0.726951\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	0	0
$635$	16.7881	0.666214
$636$	0	0
$637$	−8.74456	−0.346472
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	−20.2337	−0.799183	−0.399591	−	0.916693i	$-0.630848\pi$
$641$	−20.2337	−0.799183	−0.399591	+	0.916693i	$0.630848\pi$
$642$	0	0
$643$	14.3505	0.565930	0.282965	−	0.959130i	$-0.408682\pi$
$643$	14.3505	0.565930	0.282965	+	0.959130i	$0.408682\pi$
$644$	0	0
$645$	4.62772	0.182216
$646$	0	0
$647$	−6.86141	−0.269750	−0.134875	−	0.990863i	$-0.543063\pi$
$647$	−6.86141	−0.269750	−0.134875	+	0.990863i	$0.543063\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−22.7446	−0.891430
$652$	0	0
$653$	9.60597	0.375911	0.187955	−	0.982178i	$-0.439814\pi$
$653$	9.60597	0.375911	0.187955	+	0.982178i	$0.439814\pi$
$654$	0	0
$655$	−25.8832	−1.01134
$656$	0	0
$657$	−5.37228	−0.209593
$658$	0	0
$659$	2.74456	0.106913	0.0534565	−	0.998570i	$-0.482976\pi$
$659$	2.74456	0.106913	0.0534565	+	0.998570i	$0.482976\pi$
$660$	0	0
$661$	−10.2337	−0.398044	−0.199022	−	0.979995i	$-0.563777\pi$
$661$	−10.2337	−0.398044	−0.199022	+	0.979995i	$0.563777\pi$
$662$	0	0
$663$	−2.74456	−0.106590
$664$	0	0
$665$	−4.62772	−0.179455
$666$	0	0
$667$	−24.0000	−0.929284
$668$	0	0
$669$	−13.4891	−0.521520
$670$	0	0
$671$	3.60597	0.139207
$672$	0	0
$673$	−1.76631	−0.0680863	−0.0340432	−	0.999420i	$-0.510838\pi$
$673$	−1.76631	−0.0680863	−0.0340432	+	0.999420i	$0.510838\pi$
$674$	0	0
$675$	−3.11684	−0.119967
$676$	0	0
$677$	2.74456	0.105482	0.0527411	−	0.998608i	$-0.483204\pi$
$677$	2.74456	0.105482	0.0527411	+	0.998608i	$0.483204\pi$
$678$	0	0
$679$	−47.2119	−1.81183
$680$	0	0
$681$	−8.23369	−0.315516
$682$	0	0
$683$	9.25544	0.354149	0.177075	−	0.984197i	$-0.443337\pi$
$683$	9.25544	0.354149	0.177075	+	0.984197i	$0.443337\pi$
$684$	0	0
$685$	1.88316	0.0719517
$686$	0	0
$687$	5.37228	0.204965
$688$	0	0
$689$	−5.48913	−0.209119
$690$	0	0
$691$	−22.3505	−0.850254	−0.425127	−	0.905134i	$-0.639771\pi$
$691$	−22.3505	−0.850254	−0.425127	+	0.905134i	$0.639771\pi$
$692$	0	0
$693$	−4.62772	−0.175792
$694$	0	0
$695$	21.0951	0.800183
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−1.37228	−0.0519044
$700$	0	0
$701$	−35.4891	−1.34041	−0.670203	−	0.742178i	$-0.733793\pi$
$701$	−35.4891	−1.34041	−0.670203	+	0.742178i	$0.733793\pi$
$702$	0	0
$703$	−4.74456	−0.178945
$704$	0	0
$705$	18.3505	0.691121
$706$	0	0
$707$	38.7446	1.45714
$708$	0	0
$709$	−18.4674	−0.693557	−0.346778	−	0.937947i	$-0.612724\pi$
$709$	−18.4674	−0.693557	−0.346778	+	0.937947i	$0.612724\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	58.9783	2.20875
$714$	0	0
$715$	−3.76631	−0.140852
$716$	0	0
$717$	−18.8614	−0.704392
$718$	0	0
$719$	28.1168	1.04858	0.524291	−	0.851539i	$-0.324331\pi$
$719$	28.1168	1.04858	0.524291	+	0.851539i	$0.324331\pi$
$720$	0	0
$721$	−4.23369	−0.157671
$722$	0	0
$723$	−0.744563	−0.0276906
$724$	0	0
$725$	8.55437	0.317701
$726$	0	0
$727$	−46.5842	−1.72771	−0.863857	−	0.503738i	$-0.831958\pi$
$727$	−46.5842	−1.72771	−0.863857	+	0.503738i	$0.831958\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	4.62772	0.171162
$732$	0	0
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	0	0
$735$	6.00000	0.221313
$736$	0	0
$737$	−13.0217	−0.479662
$738$	0	0
$739$	4.39403	0.161637	0.0808185	−	0.996729i	$-0.474247\pi$
$739$	4.39403	0.161637	0.0808185	+	0.996729i	$0.474247\pi$
$740$	0	0
$741$	−2.00000	−0.0734718
$742$	0	0
$743$	28.4674	1.04437	0.522183	−	0.852833i	$-0.325118\pi$
$743$	28.4674	1.04437	0.522183	+	0.852833i	$0.325118\pi$
$744$	0	0
$745$	−1.88316	−0.0689935
$746$	0	0
$747$	8.74456	0.319947
$748$	0	0
$749$	−49.7228	−1.81683
$750$	0	0
$751$	−42.9783	−1.56830	−0.784149	−	0.620572i	$-0.786900\pi$
$751$	−42.9783	−1.56830	−0.784149	+	0.620572i	$0.786900\pi$
$752$	0	0
$753$	−1.37228	−0.0500087
$754$	0	0
$755$	−10.9783	−0.399539
$756$	0	0
$757$	31.0951	1.13017	0.565085	−	0.825033i	$-0.308843\pi$
$757$	31.0951	1.13017	0.565085	+	0.825033i	$0.308843\pi$
$758$	0	0
$759$	12.0000	0.435572
$760$	0	0
$761$	−44.5842	−1.61618	−0.808088	−	0.589061i	$-0.799498\pi$
$761$	−44.5842	−1.61618	−0.808088	+	0.589061i	$0.799498\pi$
$762$	0	0
$763$	−61.4891	−2.22606
$764$	0	0
$765$	1.88316	0.0680856
$766$	0	0
$767$	0	0
$768$	0	0
$769$	3.88316	0.140030	0.0700151	−	0.997546i	$-0.477695\pi$
$769$	3.88316	0.140030	0.0700151	+	0.997546i	$0.477695\pi$
$770$	0	0
$771$	−17.4891	−0.629855
$772$	0	0
$773$	−18.5109	−0.665790	−0.332895	−	0.942964i	$-0.608025\pi$
$773$	−18.5109	−0.665790	−0.332895	+	0.942964i	$0.608025\pi$
$774$	0	0
$775$	−21.0217	−0.755124
$776$	0	0
$777$	16.0000	0.573997
$778$	0	0
$779$	0	0
$780$	0	0
$781$	16.4674	0.589249
$782$	0	0
$783$	−2.74456	−0.0980827
$784$	0	0
$785$	−10.2772	−0.366809
$786$	0	0
$787$	1.48913	0.0530816	0.0265408	−	0.999648i	$-0.491551\pi$
$787$	1.48913	0.0530816	0.0265408	+	0.999648i	$0.491551\pi$
$788$	0	0
$789$	30.8614	1.09870
$790$	0	0
$791$	−49.7228	−1.76794
$792$	0	0
$793$	−5.25544	−0.186626
$794$	0	0
$795$	3.76631	0.133577
$796$	0	0
$797$	29.4891	1.04456	0.522279	−	0.852775i	$-0.325082\pi$
$797$	29.4891	1.04456	0.522279	+	0.852775i	$0.325082\pi$
$798$	0	0
$799$	18.3505	0.649195
$800$	0	0
$801$	14.7446	0.520974
$802$	0	0
$803$	−7.37228	−0.260162
$804$	0	0
$805$	−40.4674	−1.42629
$806$	0	0
$807$	−14.7446	−0.519033
$808$	0	0
$809$	−25.3723	−0.892042	−0.446021	−	0.895023i	$-0.647159\pi$
$809$	−25.3723	−0.892042	−0.446021	+	0.895023i	$0.647159\pi$
$810$	0	0
$811$	−0.233688	−0.00820589	−0.00410295	−	0.999992i	$-0.501306\pi$
$811$	−0.233688	−0.00820589	−0.00410295	+	0.999992i	$0.501306\pi$
$812$	0	0
$813$	4.00000	0.140286
$814$	0	0
$815$	−13.0217	−0.456132
$816$	0	0
$817$	3.37228	0.117981
$818$	0	0
$819$	6.74456	0.235674
$820$	0	0
$821$	44.5842	1.55600	0.778000	−	0.628264i	$-0.216234\pi$
$821$	44.5842	1.55600	0.778000	+	0.628264i	$0.216234\pi$
$822$	0	0
$823$	−14.3505	−0.500228	−0.250114	−	0.968216i	$-0.580468\pi$
$823$	−14.3505	−0.500228	−0.250114	+	0.968216i	$0.580468\pi$
$824$	0	0
$825$	−4.27719	−0.148913
$826$	0	0
$827$	5.48913	0.190876	0.0954378	−	0.995435i	$-0.469575\pi$
$827$	5.48913	0.190876	0.0954378	+	0.995435i	$0.469575\pi$
$828$	0	0
$829$	42.2337	1.46684	0.733418	−	0.679778i	$-0.237924\pi$
$829$	42.2337	1.46684	0.733418	+	0.679778i	$0.237924\pi$
$830$	0	0
$831$	−24.1168	−0.836604
$832$	0	0
$833$	6.00000	0.207888
$834$	0	0
$835$	−27.7663	−0.960893
$836$	0	0
$837$	6.74456	0.233126
$838$	0	0
$839$	2.74456	0.0947528	0.0473764	−	0.998877i	$-0.484914\pi$
$839$	2.74456	0.0947528	0.0473764	+	0.998877i	$0.484914\pi$
$840$	0	0
$841$	−21.4674	−0.740254
$842$	0	0
$843$	−26.7446	−0.921132
$844$	0	0
$845$	−12.3505	−0.424871
$846$	0	0
$847$	30.7446	1.05640
$848$	0	0
$849$	−26.1168	−0.896328
$850$	0	0
$851$	−41.4891	−1.42223
$852$	0	0
$853$	−8.51087	−0.291407	−0.145703	−	0.989328i	$-0.546545\pi$
$853$	−8.51087	−0.291407	−0.145703	+	0.989328i	$0.546545\pi$
$854$	0	0
$855$	1.37228	0.0469310
$856$	0	0
$857$	−34.9783	−1.19483	−0.597417	−	0.801931i	$-0.703806\pi$
$857$	−34.9783	−1.19483	−0.597417	+	0.801931i	$0.703806\pi$
$858$	0	0
$859$	4.39403	0.149922	0.0749612	−	0.997186i	$-0.476117\pi$
$859$	4.39403	0.149922	0.0749612	+	0.997186i	$0.476117\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−46.9783	−1.59916	−0.799579	−	0.600561i	$-0.794944\pi$
$863$	−46.9783	−1.59916	−0.799579	+	0.600561i	$0.794944\pi$
$864$	0	0
$865$	−12.7011	−0.431849
$866$	0	0
$867$	−15.1168	−0.513395
$868$	0	0
$869$	−10.9783	−0.372412
$870$	0	0
$871$	18.9783	0.643053
$872$	0	0
$873$	14.0000	0.473828
$874$	0	0
$875$	37.5625	1.26984
$876$	0	0
$877$	39.4891	1.33345	0.666727	−	0.745302i	$-0.267695\pi$
$877$	39.4891	1.33345	0.666727	+	0.745302i	$0.267695\pi$
$878$	0	0
$879$	−29.4891	−0.994644
$880$	0	0
$881$	17.8397	0.601033	0.300517	−	0.953777i	$-0.402841\pi$
$881$	17.8397	0.601033	0.300517	+	0.953777i	$0.402841\pi$
$882$	0	0
$883$	50.3505	1.69443	0.847215	−	0.531250i	$-0.178277\pi$
$883$	50.3505	1.69443	0.847215	+	0.531250i	$0.178277\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	18.5109	0.621534	0.310767	−	0.950486i	$-0.399414\pi$
$887$	18.5109	0.621534	0.310767	+	0.950486i	$0.399414\pi$
$888$	0	0
$889$	−41.2554	−1.38366
$890$	0	0
$891$	1.37228	0.0459732
$892$	0	0
$893$	13.3723	0.447486
$894$	0	0
$895$	20.2337	0.676338
$896$	0	0
$897$	−17.4891	−0.583945
$898$	0	0
$899$	−18.5109	−0.617372
$900$	0	0
$901$	3.76631	0.125474
$902$	0	0
$903$	−11.3723	−0.378446
$904$	0	0
$905$	30.1902	1.00356
$906$	0	0
$907$	40.2337	1.33594	0.667969	−	0.744189i	$-0.267164\pi$
$907$	40.2337	1.33594	0.667969	+	0.744189i	$0.267164\pi$
$908$	0	0
$909$	−11.4891	−0.381070
$910$	0	0
$911$	3.76631	0.124783	0.0623917	−	0.998052i	$-0.480127\pi$
$911$	3.76631	0.124783	0.0623917	+	0.998052i	$0.480127\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	0	0
$915$	3.60597	0.119210
$916$	0	0
$917$	63.6060	2.10045
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	0	0
$923$	−24.0000	−0.789970
$924$	0	0
$925$	14.7881	0.486228
$926$	0	0
$927$	1.25544	0.0412340
$928$	0	0
$929$	−11.4891	−0.376946	−0.188473	−	0.982078i	$-0.560354\pi$
$929$	−11.4891	−0.376946	−0.188473	+	0.982078i	$0.560354\pi$
$930$	0	0
$931$	4.37228	0.143296
$932$	0	0
$933$	18.8614	0.617495
$934$	0	0
$935$	2.58422	0.0845130
$936$	0	0
$937$	−57.8397	−1.88954	−0.944770	−	0.327735i	$-0.893715\pi$
$937$	−57.8397	−1.88954	−0.944770	+	0.327735i	$0.893715\pi$
$938$	0	0
$939$	14.0000	0.456873
$940$	0	0
$941$	−38.7446	−1.26304	−0.631518	−	0.775361i	$-0.717568\pi$
$941$	−38.7446	−1.26304	−0.631518	+	0.775361i	$0.717568\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−4.62772	−0.150540
$946$	0	0
$947$	−43.7228	−1.42080	−0.710400	−	0.703798i	$-0.751486\pi$
$947$	−43.7228	−1.42080	−0.710400	+	0.703798i	$0.751486\pi$
$948$	0	0
$949$	10.7446	0.348783
$950$	0	0
$951$	−29.4891	−0.956250
$952$	0	0
$953$	−52.4674	−1.69959	−0.849793	−	0.527117i	$-0.823273\pi$
$953$	−52.4674	−1.69959	−0.849793	+	0.527117i	$0.823273\pi$
$954$	0	0
$955$	14.5842	0.471934
$956$	0	0
$957$	−3.76631	−0.121748
$958$	0	0
$959$	−4.62772	−0.149437
$960$	0	0
$961$	14.4891	0.467391
$962$	0	0
$963$	14.7446	0.475137
$964$	0	0
$965$	22.9783	0.739696
$966$	0	0
$967$	5.02175	0.161489	0.0807443	−	0.996735i	$-0.474270\pi$
$967$	5.02175	0.161489	0.0807443	+	0.996735i	$0.474270\pi$
$968$	0	0
$969$	1.37228	0.0440840
$970$	0	0
$971$	−46.9783	−1.50760	−0.753802	−	0.657102i	$-0.771782\pi$
$971$	−46.9783	−1.50760	−0.753802	+	0.657102i	$0.771782\pi$
$972$	0	0
$973$	−51.8397	−1.66190
$974$	0	0
$975$	6.23369	0.199638
$976$	0	0
$977$	−49.7228	−1.59077	−0.795387	−	0.606102i	$-0.792732\pi$
$977$	−49.7228	−1.59077	−0.795387	+	0.606102i	$0.792732\pi$
$978$	0	0
$979$	20.2337	0.646671
$980$	0	0
$981$	18.2337	0.582157
$982$	0	0
$983$	29.4891	0.940557	0.470279	−	0.882518i	$-0.344153\pi$
$983$	29.4891	0.940557	0.470279	+	0.882518i	$0.344153\pi$
$984$	0	0
$985$	15.7663	0.502356
$986$	0	0
$987$	−45.0951	−1.43539
$988$	0	0
$989$	29.4891	0.937700
$990$	0	0
$991$	−53.9565	−1.71398	−0.856992	−	0.515329i	$-0.827670\pi$
$991$	−53.9565	−1.71398	−0.856992	+	0.515329i	$0.827670\pi$
$992$	0	0
$993$	−4.00000	−0.126936
$994$	0	0
$995$	19.3723	0.614143
$996$	0	0
$997$	−36.1168	−1.14383	−0.571916	−	0.820312i	$-0.693800\pi$
$997$	−36.1168	−1.14383	−0.571916	+	0.820312i	$0.693800\pi$
$998$	0	0
$999$	−4.74456	−0.150111

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3648.2.a.bq.1.2		2
4.3	odd	2		3648.2.a.bk.1.2		2
8.3	odd	2		228.2.a.c.1.1	✓	2
8.5	even	2		912.2.a.n.1.1		2
24.5	odd	2		2736.2.a.y.1.2		2
24.11	even	2		684.2.a.d.1.2		2
40.3	even	4		5700.2.f.m.3649.3		4
40.19	odd	2		5700.2.a.t.1.1		2
40.27	even	4		5700.2.f.m.3649.2		4
152.75	even	2		4332.2.a.i.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.2.a.c.1.1	✓	2	8.3	odd	2
684.2.a.d.1.2		2	24.11	even	2
912.2.a.n.1.1		2	8.5	even	2
2736.2.a.y.1.2		2	24.5	odd	2
3648.2.a.bk.1.2		2	4.3	odd	2
3648.2.a.bq.1.2		2	1.1	even	1	trivial
4332.2.a.i.1.1		2	152.75	even	2
5700.2.a.t.1.1		2	40.19	odd	2
5700.2.f.m.3649.2		4	40.27	even	4
5700.2.f.m.3649.3		4	40.3	even	4

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:	\( N \)	\(=\)	\( 3648 = 2^{6} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3648.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.37228 q^{5} -3.37228 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.37228 q^{5} -3.37228 q^{7} +1.00000 q^{9} +1.37228 q^{11} -2.00000 q^{13} +1.37228 q^{15} +1.37228 q^{17} +1.00000 q^{19} -3.37228 q^{21} +8.74456 q^{23} -3.11684 q^{25} +1.00000 q^{27} -2.74456 q^{29} +6.74456 q^{31} +1.37228 q^{33} -4.62772 q^{35} -4.74456 q^{37} -2.00000 q^{39} +3.37228 q^{43} +1.37228 q^{45} +13.3723 q^{47} +4.37228 q^{49} +1.37228 q^{51} +2.74456 q^{53} +1.88316 q^{55} +1.00000 q^{57} +2.62772 q^{61} -3.37228 q^{63} -2.74456 q^{65} -9.48913 q^{67} +8.74456 q^{69} +12.0000 q^{71} -5.37228 q^{73} -3.11684 q^{75} -4.62772 q^{77} -8.00000 q^{79} +1.00000 q^{81} +8.74456 q^{83} +1.88316 q^{85} -2.74456 q^{87} +14.7446 q^{89} +6.74456 q^{91} +6.74456 q^{93} +1.37228 q^{95} +14.0000 q^{97} +1.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 3 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 3 * q^5 - q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 3 q^{5} - q^{7} + 2 q^{9} - 3 q^{11} - 4 q^{13} - 3 q^{15} - 3 q^{17} + 2 q^{19} - q^{21} + 6 q^{23} + 11 q^{25} + 2 q^{27} + 6 q^{29} + 2 q^{31} - 3 q^{33} - 15 q^{35} + 2 q^{37} - 4 q^{39} + q^{43} - 3 q^{45} + 21 q^{47} + 3 q^{49} - 3 q^{51} - 6 q^{53} + 21 q^{55} + 2 q^{57} + 11 q^{61} - q^{63} + 6 q^{65} + 4 q^{67} + 6 q^{69} + 24 q^{71} - 5 q^{73} + 11 q^{75} - 15 q^{77} - 16 q^{79} + 2 q^{81} + 6 q^{83} + 21 q^{85} + 6 q^{87} + 18 q^{89} + 2 q^{91} + 2 q^{93} - 3 q^{95} + 28 q^{97} - 3 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 3 * q^5 - q^7 + 2 * q^9 - 3 * q^11 - 4 * q^13 - 3 * q^15 - 3 * q^17 + 2 * q^19 - q^21 + 6 * q^23 + 11 * q^25 + 2 * q^27 + 6 * q^29 + 2 * q^31 - 3 * q^33 - 15 * q^35 + 2 * q^37 - 4 * q^39 + q^43 - 3 * q^45 + 21 * q^47 + 3 * q^49 - 3 * q^51 - 6 * q^53 + 21 * q^55 + 2 * q^57 + 11 * q^61 - q^63 + 6 * q^65 + 4 * q^67 + 6 * q^69 + 24 * q^71 - 5 * q^73 + 11 * q^75 - 15 * q^77 - 16 * q^79 + 2 * q^81 + 6 * q^83 + 21 * q^85 + 6 * q^87 + 18 * q^89 + 2 * q^91 + 2 * q^93 - 3 * q^95 + 28 * q^97 - 3 * q^99

Introduction

L-functions

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