LMFDB - Embedded newform 6044.2.a.b.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6044,2,Mod(1,6044)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6044, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6044.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6044 = 2^{2} \cdot 1511 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6044.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:	$48.2615829817$
Analytic rank:	$0$
Dimension:	$63$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	6044.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.79559 q^{3} +1.54493 q^{5} +3.97273 q^{7} +4.81535 q^{9} +O(q^{10})$	$q-2.79559 q^{3} +1.54493 q^{5} +3.97273 q^{7} +4.81535 q^{9} +1.51359 q^{11} -0.732798 q^{13} -4.31900 q^{15} -1.16959 q^{17} -6.39527 q^{19} -11.1061 q^{21} +7.46938 q^{23} -2.61319 q^{25} -5.07498 q^{27} +7.93552 q^{29} -4.82561 q^{31} -4.23137 q^{33} +6.13758 q^{35} +8.50768 q^{37} +2.04861 q^{39} -9.65792 q^{41} +7.04234 q^{43} +7.43937 q^{45} +6.53607 q^{47} +8.78257 q^{49} +3.26969 q^{51} -2.38579 q^{53} +2.33838 q^{55} +17.8786 q^{57} +7.60676 q^{59} +3.64872 q^{61} +19.1301 q^{63} -1.13212 q^{65} -4.77424 q^{67} -20.8814 q^{69} +4.12447 q^{71} +3.32426 q^{73} +7.30543 q^{75} +6.01306 q^{77} +10.4486 q^{79} -0.258451 q^{81} +2.42781 q^{83} -1.80693 q^{85} -22.1845 q^{87} -8.09036 q^{89} -2.91121 q^{91} +13.4905 q^{93} -9.88024 q^{95} -15.2984 q^{97} +7.28844 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9}+O(q^{10}) $ 63 * q + 5 * q^3 + 5 * q^5 + 22 * q^7 + 62 * q^9	$ 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9} + 21 q^{11} + 17 q^{13} + 26 q^{15} - 5 q^{17} + 57 q^{19} + 18 q^{21} + 16 q^{23} + 60 q^{25} + 14 q^{27} + 22 q^{29} + 36 q^{31} - q^{33} + 28 q^{35} + 21 q^{37} + 69 q^{39} - 3 q^{41} + 86 q^{43} + 39 q^{45} + 23 q^{47} + 63 q^{49} + 67 q^{51} + 18 q^{53} + 67 q^{55} + 27 q^{59} + 62 q^{61} + 58 q^{63} - 13 q^{65} + 62 q^{67} + 45 q^{69} + 29 q^{71} - q^{73} + 39 q^{75} + 19 q^{77} + 132 q^{79} + 51 q^{81} + 35 q^{83} + 60 q^{85} + 55 q^{87} - 2 q^{89} + 84 q^{91} + 29 q^{93} + 53 q^{95} - 10 q^{97} + 95 q^{99}+O(q^{100}) $ 63 * q + 5 * q^3 + 5 * q^5 + 22 * q^7 + 62 * q^9 + 21 * q^11 + 17 * q^13 + 26 * q^15 - 5 * q^17 + 57 * q^19 + 18 * q^21 + 16 * q^23 + 60 * q^25 + 14 * q^27 + 22 * q^29 + 36 * q^31 - q^33 + 28 * q^35 + 21 * q^37 + 69 * q^39 - 3 * q^41 + 86 * q^43 + 39 * q^45 + 23 * q^47 + 63 * q^49 + 67 * q^51 + 18 * q^53 + 67 * q^55 + 27 * q^59 + 62 * q^61 + 58 * q^63 - 13 * q^65 + 62 * q^67 + 45 * q^69 + 29 * q^71 - q^73 + 39 * q^75 + 19 * q^77 + 132 * q^79 + 51 * q^81 + 35 * q^83 + 60 * q^85 + 55 * q^87 - 2 * q^89 + 84 * q^91 + 29 * q^93 + 53 * q^95 - 10 * q^97 + 95 * 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
$3$	−2.79559	−1.61404	−0.807019	−	0.590526i	$-0.798920\pi$
$3$	−2.79559	−1.61404	−0.807019	+	0.590526i	$0.798920\pi$
$4$	0	0
$5$	1.54493	0.690913	0.345457	−	0.938435i	$-0.387724\pi$
$5$	1.54493	0.690913	0.345457	+	0.938435i	$0.387724\pi$
$6$	0	0
$7$	3.97273	1.50155	0.750775	−	0.660558i	$-0.229680\pi$
$7$	3.97273	1.50155	0.750775	+	0.660558i	$0.229680\pi$
$8$	0	0
$9$	4.81535	1.60512
$10$	0	0
$11$	1.51359	0.456363	0.228182	−	0.973619i	$-0.426722\pi$
$11$	1.51359	0.456363	0.228182	+	0.973619i	$0.426722\pi$
$12$	0	0
$13$	−0.732798	−0.203242	−0.101621	−	0.994823i	$-0.532403\pi$
$13$	−0.732798	−0.203242	−0.101621	+	0.994823i	$0.532403\pi$
$14$	0	0
$15$	−4.31900	−1.11516
$16$	0	0
$17$	−1.16959	−0.283666	−0.141833	−	0.989891i	$-0.545300\pi$
$17$	−1.16959	−0.283666	−0.141833	+	0.989891i	$0.545300\pi$
$18$	0	0
$19$	−6.39527	−1.46718	−0.733588	−	0.679594i	$-0.762156\pi$
$19$	−6.39527	−1.46718	−0.733588	+	0.679594i	$0.762156\pi$
$20$	0	0
$21$	−11.1061	−2.42356
$22$	0	0
$23$	7.46938	1.55747	0.778736	−	0.627351i	$-0.215861\pi$
$23$	7.46938	1.55747	0.778736	+	0.627351i	$0.215861\pi$
$24$	0	0
$25$	−2.61319	−0.522639
$26$	0	0
$27$	−5.07498	−0.976681
$28$	0	0
$29$	7.93552	1.47359	0.736794	−	0.676117i	$-0.236339\pi$
$29$	7.93552	1.47359	0.736794	+	0.676117i	$0.236339\pi$
$30$	0	0
$31$	−4.82561	−0.866705	−0.433353	−	0.901224i	$-0.642670\pi$
$31$	−4.82561	−0.866705	−0.433353	+	0.901224i	$0.642670\pi$
$32$	0	0
$33$	−4.23137	−0.736587
$34$	0	0
$35$	6.13758	1.03744
$36$	0	0
$37$	8.50768	1.39865	0.699327	−	0.714802i	$-0.253483\pi$
$37$	8.50768	1.39865	0.699327	+	0.714802i	$0.253483\pi$
$38$	0	0
$39$	2.04861	0.328039
$40$	0	0
$41$	−9.65792	−1.50831	−0.754157	−	0.656695i	$-0.771954\pi$
$41$	−9.65792	−1.50831	−0.754157	+	0.656695i	$0.771954\pi$
$42$	0	0
$43$	7.04234	1.07395	0.536973	−	0.843599i	$-0.319568\pi$
$43$	7.04234	1.07395	0.536973	+	0.843599i	$0.319568\pi$
$44$	0	0
$45$	7.43937	1.10900
$46$	0	0
$47$	6.53607	0.953383	0.476692	−	0.879071i	$-0.341836\pi$
$47$	6.53607	0.953383	0.476692	+	0.879071i	$0.341836\pi$
$48$	0	0
$49$	8.78257	1.25465
$50$	0	0
$51$	3.26969	0.457848
$52$	0	0
$53$	−2.38579	−0.327714	−0.163857	−	0.986484i	$-0.552394\pi$
$53$	−2.38579	−0.327714	−0.163857	+	0.986484i	$0.552394\pi$
$54$	0	0
$55$	2.33838	0.315307
$56$	0	0
$57$	17.8786	2.36808
$58$	0	0
$59$	7.60676	0.990316	0.495158	−	0.868803i	$-0.335110\pi$
$59$	7.60676	0.990316	0.495158	+	0.868803i	$0.335110\pi$
$60$	0	0
$61$	3.64872	0.467171	0.233586	−	0.972336i	$-0.424954\pi$
$61$	3.64872	0.467171	0.233586	+	0.972336i	$0.424954\pi$
$62$	0	0
$63$	19.1301	2.41016
$64$	0	0
$65$	−1.13212	−0.140422
$66$	0	0
$67$	−4.77424	−0.583267	−0.291633	−	0.956530i	$-0.594199\pi$
$67$	−4.77424	−0.583267	−0.291633	+	0.956530i	$0.594199\pi$
$68$	0	0
$69$	−20.8814	−2.51382
$70$	0	0
$71$	4.12447	0.489485	0.244742	−	0.969588i	$-0.421297\pi$
$71$	4.12447	0.489485	0.244742	+	0.969588i	$0.421297\pi$
$72$	0	0
$73$	3.32426	0.389076	0.194538	−	0.980895i	$-0.437679\pi$
$73$	3.32426	0.389076	0.194538	+	0.980895i	$0.437679\pi$
$74$	0	0
$75$	7.30543	0.843559
$76$	0	0
$77$	6.01306	0.685252
$78$	0	0
$79$	10.4486	1.17556	0.587781	−	0.809020i	$-0.300002\pi$
$79$	10.4486	1.17556	0.587781	+	0.809020i	$0.300002\pi$
$80$	0	0
$81$	−0.258451	−0.0287168
$82$	0	0
$83$	2.42781	0.266487	0.133243	−	0.991083i	$-0.457461\pi$
$83$	2.42781	0.266487	0.133243	+	0.991083i	$0.457461\pi$
$84$	0	0
$85$	−1.80693	−0.195989
$86$	0	0
$87$	−22.1845	−2.37843
$88$	0	0
$89$	−8.09036	−0.857577	−0.428788	−	0.903405i	$-0.641059\pi$
$89$	−8.09036	−0.857577	−0.428788	+	0.903405i	$0.641059\pi$
$90$	0	0
$91$	−2.91121	−0.305177
$92$	0	0
$93$	13.4905	1.39889
$94$	0	0
$95$	−9.88024	−1.01369
$96$	0	0
$97$	−15.2984	−1.55332	−0.776659	−	0.629922i	$-0.783087\pi$
$97$	−15.2984	−1.55332	−0.776659	+	0.629922i	$0.783087\pi$
$98$	0	0
$99$	7.28844	0.732516
$100$	0	0
$101$	14.6930	1.46201	0.731004	−	0.682373i	$-0.239052\pi$
$101$	14.6930	1.46201	0.731004	+	0.682373i	$0.239052\pi$
$102$	0	0
$103$	−13.1866	−1.29931	−0.649655	−	0.760229i	$-0.725087\pi$
$103$	−13.1866	−1.29931	−0.649655	+	0.760229i	$0.725087\pi$
$104$	0	0
$105$	−17.1582	−1.67447
$106$	0	0
$107$	3.62809	0.350741	0.175370	−	0.984503i	$-0.443888\pi$
$107$	3.62809	0.350741	0.175370	+	0.984503i	$0.443888\pi$
$108$	0	0
$109$	−11.8969	−1.13951	−0.569757	−	0.821813i	$-0.692963\pi$
$109$	−11.8969	−1.13951	−0.569757	+	0.821813i	$0.692963\pi$
$110$	0	0
$111$	−23.7840	−2.25748
$112$	0	0
$113$	8.89790	0.837044	0.418522	−	0.908207i	$-0.362548\pi$
$113$	8.89790	0.837044	0.418522	+	0.908207i	$0.362548\pi$
$114$	0	0
$115$	11.5397	1.07608
$116$	0	0
$117$	−3.52868	−0.326226
$118$	0	0
$119$	−4.64644	−0.425939
$120$	0	0
$121$	−8.70906	−0.791733
$122$	0	0
$123$	26.9996	2.43447
$124$	0	0
$125$	−11.7618	−1.05201
$126$	0	0
$127$	13.1585	1.16763	0.583813	−	0.811888i	$-0.301560\pi$
$127$	13.1585	1.16763	0.583813	+	0.811888i	$0.301560\pi$
$128$	0	0
$129$	−19.6875	−1.73339
$130$	0	0
$131$	−7.18278	−0.627562	−0.313781	−	0.949495i	$-0.601596\pi$
$131$	−7.18278	−0.627562	−0.313781	+	0.949495i	$0.601596\pi$
$132$	0	0
$133$	−25.4067	−2.20304
$134$	0	0
$135$	−7.84049	−0.674802
$136$	0	0
$137$	0.397814	0.0339876	0.0169938	−	0.999856i	$-0.494590\pi$
$137$	0.397814	0.0339876	0.0169938	+	0.999856i	$0.494590\pi$
$138$	0	0
$139$	17.6620	1.49808	0.749038	−	0.662528i	$-0.230516\pi$
$139$	17.6620	1.49808	0.749038	+	0.662528i	$0.230516\pi$
$140$	0	0
$141$	−18.2722	−1.53880
$142$	0	0
$143$	−1.10915	−0.0927519
$144$	0	0
$145$	12.2598	1.01812
$146$	0	0
$147$	−24.5525	−2.02506
$148$	0	0
$149$	0.560765	0.0459397	0.0229698	−	0.999736i	$-0.492688\pi$
$149$	0.560765	0.0459397	0.0229698	+	0.999736i	$0.492688\pi$
$150$	0	0
$151$	4.46320	0.363210	0.181605	−	0.983372i	$-0.441871\pi$
$151$	4.46320	0.363210	0.181605	+	0.983372i	$0.441871\pi$
$152$	0	0
$153$	−5.63196	−0.455317
$154$	0	0
$155$	−7.45523	−0.598818
$156$	0	0
$157$	14.9132	1.19020	0.595102	−	0.803650i	$-0.297112\pi$
$157$	14.9132	1.19020	0.595102	+	0.803650i	$0.297112\pi$
$158$	0	0
$159$	6.66971	0.528943
$160$	0	0
$161$	29.6738	2.33862
$162$	0	0
$163$	5.78616	0.453207	0.226604	−	0.973987i	$-0.427238\pi$
$163$	5.78616	0.453207	0.226604	+	0.973987i	$0.427238\pi$
$164$	0	0
$165$	−6.53717	−0.508918
$166$	0	0
$167$	−12.4790	−0.965658	−0.482829	−	0.875715i	$-0.660391\pi$
$167$	−12.4790	−0.965658	−0.482829	+	0.875715i	$0.660391\pi$
$168$	0	0
$169$	−12.4630	−0.958693
$170$	0	0
$171$	−30.7955	−2.35499
$172$	0	0
$173$	0.0437157	0.00332365	0.00166182	−	0.999999i	$-0.499471\pi$
$173$	0.0437157	0.00332365	0.00166182	+	0.999999i	$0.499471\pi$
$174$	0	0
$175$	−10.3815	−0.784769
$176$	0	0
$177$	−21.2654	−1.59841
$178$	0	0
$179$	10.7169	0.801016	0.400508	−	0.916293i	$-0.368834\pi$
$179$	10.7169	0.801016	0.400508	+	0.916293i	$0.368834\pi$
$180$	0	0
$181$	−17.1145	−1.27211	−0.636054	−	0.771644i	$-0.719434\pi$
$181$	−17.1145	−1.27211	−0.636054	+	0.771644i	$0.719434\pi$
$182$	0	0
$183$	−10.2004	−0.754032
$184$	0	0
$185$	13.1438	0.966349
$186$	0	0
$187$	−1.77027	−0.129455
$188$	0	0
$189$	−20.1615	−1.46654
$190$	0	0
$191$	4.01488	0.290507	0.145253	−	0.989394i	$-0.453600\pi$
$191$	4.01488	0.290507	0.145253	+	0.989394i	$0.453600\pi$
$192$	0	0
$193$	21.0503	1.51523	0.757616	−	0.652701i	$-0.226364\pi$
$193$	21.0503	1.51523	0.757616	+	0.652701i	$0.226364\pi$
$194$	0	0
$195$	3.16495	0.226647
$196$	0	0
$197$	0.729293	0.0519600	0.0259800	−	0.999662i	$-0.491729\pi$
$197$	0.729293	0.0519600	0.0259800	+	0.999662i	$0.491729\pi$
$198$	0	0
$199$	−21.8369	−1.54798	−0.773989	−	0.633199i	$-0.781741\pi$
$199$	−21.8369	−1.54798	−0.773989	+	0.633199i	$0.781741\pi$
$200$	0	0
$201$	13.3468	0.941414
$202$	0	0
$203$	31.5257	2.21267
$204$	0	0
$205$	−14.9208	−1.04211
$206$	0	0
$207$	35.9677	2.49993
$208$	0	0
$209$	−9.67979	−0.669565
$210$	0	0
$211$	6.94924	0.478406	0.239203	−	0.970970i	$-0.423114\pi$
$211$	6.94924	0.478406	0.239203	+	0.970970i	$0.423114\pi$
$212$	0	0
$213$	−11.5304	−0.790047
$214$	0	0
$215$	10.8799	0.742004
$216$	0	0
$217$	−19.1708	−1.30140
$218$	0	0
$219$	−9.29330	−0.627983
$220$	0	0
$221$	0.857069	0.0576527
$222$	0	0
$223$	12.1849	0.815962	0.407981	−	0.912990i	$-0.366233\pi$
$223$	12.1849	0.815962	0.407981	+	0.912990i	$0.366233\pi$
$224$	0	0
$225$	−12.5834	−0.838896
$226$	0	0
$227$	−14.1387	−0.938415	−0.469208	−	0.883088i	$-0.655460\pi$
$227$	−14.1387	−0.938415	−0.469208	+	0.883088i	$0.655460\pi$
$228$	0	0
$229$	17.8136	1.17715	0.588577	−	0.808441i	$-0.299688\pi$
$229$	17.8136	1.17715	0.588577	+	0.808441i	$0.299688\pi$
$230$	0	0
$231$	−16.8101	−1.10602
$232$	0	0
$233$	5.84671	0.383031	0.191515	−	0.981490i	$-0.438660\pi$
$233$	5.84671	0.383031	0.191515	+	0.981490i	$0.438660\pi$
$234$	0	0
$235$	10.0978	0.658705
$236$	0	0
$237$	−29.2101	−1.89740
$238$	0	0
$239$	12.6264	0.816732	0.408366	−	0.912818i	$-0.366099\pi$
$239$	12.6264	0.816732	0.408366	+	0.912818i	$0.366099\pi$
$240$	0	0
$241$	−17.3570	−1.11806	−0.559032	−	0.829146i	$-0.688827\pi$
$241$	−17.3570	−1.11806	−0.559032	+	0.829146i	$0.688827\pi$
$242$	0	0
$243$	15.9475	1.02303
$244$	0	0
$245$	13.5685	0.866857
$246$	0	0
$247$	4.68644	0.298191
$248$	0	0
$249$	−6.78718	−0.430120
$250$	0	0
$251$	2.30782	0.145668	0.0728341	−	0.997344i	$-0.476796\pi$
$251$	2.30782	0.145668	0.0728341	+	0.997344i	$0.476796\pi$
$252$	0	0
$253$	11.3055	0.710773
$254$	0	0
$255$	5.05143	0.316333
$256$	0	0
$257$	−22.9684	−1.43273	−0.716366	−	0.697725i	$-0.754196\pi$
$257$	−22.9684	−1.43273	−0.716366	+	0.697725i	$0.754196\pi$
$258$	0	0
$259$	33.7987	2.10015
$260$	0	0
$261$	38.2123	2.36528
$262$	0	0
$263$	−12.4296	−0.766439	−0.383220	−	0.923657i	$-0.625185\pi$
$263$	−12.4296	−0.766439	−0.383220	+	0.923657i	$0.625185\pi$
$264$	0	0
$265$	−3.68588	−0.226422
$266$	0	0
$267$	22.6174	1.38416
$268$	0	0
$269$	20.8393	1.27060	0.635298	−	0.772267i	$-0.280877\pi$
$269$	20.8393	1.27060	0.635298	+	0.772267i	$0.280877\pi$
$270$	0	0
$271$	5.23658	0.318100	0.159050	−	0.987271i	$-0.449157\pi$
$271$	5.23658	0.318100	0.159050	+	0.987271i	$0.449157\pi$
$272$	0	0
$273$	8.13856	0.492568
$274$	0	0
$275$	−3.95529	−0.238513
$276$	0	0
$277$	11.8309	0.710852	0.355426	−	0.934704i	$-0.384336\pi$
$277$	11.8309	0.710852	0.355426	+	0.934704i	$0.384336\pi$
$278$	0	0
$279$	−23.2370	−1.39116
$280$	0	0
$281$	2.25986	0.134812	0.0674059	−	0.997726i	$-0.478528\pi$
$281$	2.25986	0.134812	0.0674059	+	0.997726i	$0.478528\pi$
$282$	0	0
$283$	6.68587	0.397434	0.198717	−	0.980057i	$-0.436323\pi$
$283$	6.68587	0.397434	0.198717	+	0.980057i	$0.436323\pi$
$284$	0	0
$285$	27.6212	1.63614
$286$	0	0
$287$	−38.3683	−2.26481
$288$	0	0
$289$	−15.6321	−0.919534
$290$	0	0
$291$	42.7681	2.50711
$292$	0	0
$293$	−28.0382	−1.63801	−0.819005	−	0.573786i	$-0.805474\pi$
$293$	−28.0382	−1.63801	−0.819005	+	0.573786i	$0.805474\pi$
$294$	0	0
$295$	11.7519	0.684222
$296$	0	0
$297$	−7.68142	−0.445721
$298$	0	0
$299$	−5.47354	−0.316543
$300$	0	0
$301$	27.9773	1.61259
$302$	0	0
$303$	−41.0757	−2.35974
$304$	0	0
$305$	5.63702	0.322775
$306$	0	0
$307$	19.7736	1.12854	0.564269	−	0.825591i	$-0.309158\pi$
$307$	19.7736	1.12854	0.564269	+	0.825591i	$0.309158\pi$
$308$	0	0
$309$	36.8643	2.09713
$310$	0	0
$311$	27.3931	1.55332	0.776660	−	0.629920i	$-0.216912\pi$
$311$	27.3931	1.55332	0.776660	+	0.629920i	$0.216912\pi$
$312$	0	0
$313$	18.6566	1.05453	0.527266	−	0.849701i	$-0.323217\pi$
$313$	18.6566	1.05453	0.527266	+	0.849701i	$0.323217\pi$
$314$	0	0
$315$	29.5546	1.66521
$316$	0	0
$317$	24.7655	1.39097	0.695484	−	0.718542i	$-0.255190\pi$
$317$	24.7655	1.39097	0.695484	+	0.718542i	$0.255190\pi$
$318$	0	0
$319$	12.0111	0.672492
$320$	0	0
$321$	−10.1427	−0.566109
$322$	0	0
$323$	7.47982	0.416188
$324$	0	0
$325$	1.91494	0.106222
$326$	0	0
$327$	33.2588	1.83922
$328$	0	0
$329$	25.9660	1.43155
$330$	0	0
$331$	0.786554	0.0432329	0.0216165	−	0.999766i	$-0.493119\pi$
$331$	0.786554	0.0432329	0.0216165	+	0.999766i	$0.493119\pi$
$332$	0	0
$333$	40.9675	2.24500
$334$	0	0
$335$	−7.37587	−0.402987
$336$	0	0
$337$	−28.9301	−1.57592	−0.787961	−	0.615725i	$-0.788863\pi$
$337$	−28.9301	−1.57592	−0.787961	+	0.615725i	$0.788863\pi$
$338$	0	0
$339$	−24.8749	−1.35102
$340$	0	0
$341$	−7.30397	−0.395532
$342$	0	0
$343$	7.08168	0.382375
$344$	0	0
$345$	−32.2602	−1.73683
$346$	0	0
$347$	17.5866	0.944100	0.472050	−	0.881572i	$-0.343514\pi$
$347$	17.5866	0.944100	0.472050	+	0.881572i	$0.343514\pi$
$348$	0	0
$349$	−8.75610	−0.468703	−0.234352	−	0.972152i	$-0.575297\pi$
$349$	−8.75610	−0.468703	−0.234352	+	0.972152i	$0.575297\pi$
$350$	0	0
$351$	3.71894	0.198502
$352$	0	0
$353$	31.2626	1.66394	0.831970	−	0.554821i	$-0.187213\pi$
$353$	31.2626	1.66394	0.831970	+	0.554821i	$0.187213\pi$
$354$	0	0
$355$	6.37202	0.338192
$356$	0	0
$357$	12.9896	0.687481
$358$	0	0
$359$	−8.49157	−0.448168	−0.224084	−	0.974570i	$-0.571939\pi$
$359$	−8.49157	−0.448168	−0.224084	+	0.974570i	$0.571939\pi$
$360$	0	0
$361$	21.8995	1.15261
$362$	0	0
$363$	24.3470	1.27789
$364$	0	0
$365$	5.13575	0.268818
$366$	0	0
$367$	12.0250	0.627701	0.313850	−	0.949472i	$-0.398381\pi$
$367$	12.0250	0.627701	0.313850	+	0.949472i	$0.398381\pi$
$368$	0	0
$369$	−46.5062	−2.42102
$370$	0	0
$371$	−9.47811	−0.492079
$372$	0	0
$373$	−18.2507	−0.944984	−0.472492	−	0.881335i	$-0.656645\pi$
$373$	−18.2507	−0.944984	−0.472492	+	0.881335i	$0.656645\pi$
$374$	0	0
$375$	32.8814	1.69799
$376$	0	0
$377$	−5.81513	−0.299495
$378$	0	0
$379$	20.4815	1.05206	0.526031	−	0.850465i	$-0.323680\pi$
$379$	20.4815	1.05206	0.526031	+	0.850465i	$0.323680\pi$
$380$	0	0
$381$	−36.7858	−1.88459
$382$	0	0
$383$	5.61878	0.287106	0.143553	−	0.989643i	$-0.454147\pi$
$383$	5.61878	0.287106	0.143553	+	0.989643i	$0.454147\pi$
$384$	0	0
$385$	9.28975	0.473450
$386$	0	0
$387$	33.9113	1.72381
$388$	0	0
$389$	−17.6270	−0.893726	−0.446863	−	0.894602i	$-0.647459\pi$
$389$	−17.6270	−0.893726	−0.446863	+	0.894602i	$0.647459\pi$
$390$	0	0
$391$	−8.73607	−0.441802
$392$	0	0
$393$	20.0801	1.01291
$394$	0	0
$395$	16.1424	0.812211
$396$	0	0
$397$	4.24452	0.213027	0.106513	−	0.994311i	$-0.466031\pi$
$397$	4.24452	0.213027	0.106513	+	0.994311i	$0.466031\pi$
$398$	0	0
$399$	71.0268	3.55579
$400$	0	0
$401$	22.9970	1.14841	0.574207	−	0.818710i	$-0.305311\pi$
$401$	22.9970	1.14841	0.574207	+	0.818710i	$0.305311\pi$
$402$	0	0
$403$	3.53620	0.176151
$404$	0	0
$405$	−0.399289	−0.0198408
$406$	0	0
$407$	12.8771	0.638294
$408$	0	0
$409$	−8.57769	−0.424139	−0.212070	−	0.977255i	$-0.568020\pi$
$409$	−8.57769	−0.424139	−0.212070	+	0.977255i	$0.568020\pi$
$410$	0	0
$411$	−1.11213	−0.0548572
$412$	0	0
$413$	30.2196	1.48701
$414$	0	0
$415$	3.75080	0.184119
$416$	0	0
$417$	−49.3759	−2.41795
$418$	0	0
$419$	11.2979	0.551939	0.275969	−	0.961166i	$-0.411001\pi$
$419$	11.2979	0.551939	0.275969	+	0.961166i	$0.411001\pi$
$420$	0	0
$421$	36.4893	1.77838	0.889190	−	0.457539i	$-0.151269\pi$
$421$	36.4893	1.77838	0.889190	+	0.457539i	$0.151269\pi$
$422$	0	0
$423$	31.4735	1.53029
$424$	0	0
$425$	3.05635	0.148255
$426$	0	0
$427$	14.4954	0.701481
$428$	0	0
$429$	3.10074	0.149705
$430$	0	0
$431$	−11.4483	−0.551446	−0.275723	−	0.961237i	$-0.588917\pi$
$431$	−11.4483	−0.551446	−0.275723	+	0.961237i	$0.588917\pi$
$432$	0	0
$433$	−21.2373	−1.02060	−0.510299	−	0.859997i	$-0.670465\pi$
$433$	−21.2373	−1.02060	−0.510299	+	0.859997i	$0.670465\pi$
$434$	0	0
$435$	−34.2735	−1.64329
$436$	0	0
$437$	−47.7687	−2.28509
$438$	0	0
$439$	34.8869	1.66506	0.832531	−	0.553979i	$-0.186891\pi$
$439$	34.8869	1.66506	0.832531	+	0.553979i	$0.186891\pi$
$440$	0	0
$441$	42.2912	2.01387
$442$	0	0
$443$	−4.36602	−0.207436	−0.103718	−	0.994607i	$-0.533074\pi$
$443$	−4.36602	−0.207436	−0.103718	+	0.994607i	$0.533074\pi$
$444$	0	0
$445$	−12.4990	−0.592511
$446$	0	0
$447$	−1.56767	−0.0741484
$448$	0	0
$449$	−15.4098	−0.727232	−0.363616	−	0.931549i	$-0.618458\pi$
$449$	−15.4098	−0.727232	−0.363616	+	0.931549i	$0.618458\pi$
$450$	0	0
$451$	−14.6181	−0.688338
$452$	0	0
$453$	−12.4773	−0.586235
$454$	0	0
$455$	−4.49761	−0.210851
$456$	0	0
$457$	13.5755	0.635035	0.317518	−	0.948252i	$-0.397151\pi$
$457$	13.5755	0.635035	0.317518	+	0.948252i	$0.397151\pi$
$458$	0	0
$459$	5.93563	0.277051
$460$	0	0
$461$	−17.2841	−0.805002	−0.402501	−	0.915419i	$-0.631859\pi$
$461$	−17.2841	−0.805002	−0.402501	+	0.915419i	$0.631859\pi$
$462$	0	0
$463$	35.7581	1.66182	0.830910	−	0.556407i	$-0.187821\pi$
$463$	35.7581	1.66182	0.830910	+	0.556407i	$0.187821\pi$
$464$	0	0
$465$	20.8418	0.966515
$466$	0	0
$467$	−23.3113	−1.07872	−0.539359	−	0.842076i	$-0.681333\pi$
$467$	−23.3113	−1.07872	−0.539359	+	0.842076i	$0.681333\pi$
$468$	0	0
$469$	−18.9668	−0.875804
$470$	0	0
$471$	−41.6913	−1.92103
$472$	0	0
$473$	10.6592	0.490110
$474$	0	0
$475$	16.7121	0.766803
$476$	0	0
$477$	−11.4884	−0.526019
$478$	0	0
$479$	0.524754	0.0239766	0.0119883	−	0.999928i	$-0.496184\pi$
$479$	0.524754	0.0239766	0.0119883	+	0.999928i	$0.496184\pi$
$480$	0	0
$481$	−6.23441	−0.284265
$482$	0	0
$483$	−82.9560	−3.77463
$484$	0	0
$485$	−23.6349	−1.07321
$486$	0	0
$487$	−29.3360	−1.32934	−0.664670	−	0.747137i	$-0.731428\pi$
$487$	−29.3360	−1.32934	−0.664670	+	0.747137i	$0.731428\pi$
$488$	0	0
$489$	−16.1758	−0.731493
$490$	0	0
$491$	8.07402	0.364375	0.182188	−	0.983264i	$-0.441682\pi$
$491$	8.07402	0.364375	0.182188	+	0.983264i	$0.441682\pi$
$492$	0	0
$493$	−9.28126	−0.418007
$494$	0	0
$495$	11.2601	0.506105
$496$	0	0
$497$	16.3854	0.734986
$498$	0	0
$499$	40.8759	1.82986	0.914928	−	0.403617i	$-0.132247\pi$
$499$	40.8759	1.82986	0.914928	+	0.403617i	$0.132247\pi$
$500$	0	0
$501$	34.8864	1.55861
$502$	0	0
$503$	−10.2144	−0.455438	−0.227719	−	0.973727i	$-0.573127\pi$
$503$	−10.2144	−0.455438	−0.227719	+	0.973727i	$0.573127\pi$
$504$	0	0
$505$	22.6996	1.01012
$506$	0	0
$507$	34.8415	1.54737
$508$	0	0
$509$	6.67360	0.295802	0.147901	−	0.989002i	$-0.452748\pi$
$509$	6.67360	0.295802	0.147901	+	0.989002i	$0.452748\pi$
$510$	0	0
$511$	13.2064	0.584217
$512$	0	0
$513$	32.4559	1.43296
$514$	0	0
$515$	−20.3723	−0.897710
$516$	0	0
$517$	9.89289	0.435089
$518$	0	0
$519$	−0.122211	−0.00536449
$520$	0	0
$521$	21.8568	0.957563	0.478782	−	0.877934i	$-0.341079\pi$
$521$	21.8568	0.957563	0.478782	+	0.877934i	$0.341079\pi$
$522$	0	0
$523$	25.8852	1.13188	0.565940	−	0.824446i	$-0.308513\pi$
$523$	25.8852	1.13188	0.565940	+	0.824446i	$0.308513\pi$
$524$	0	0
$525$	29.0225	1.26665
$526$	0	0
$527$	5.64396	0.245855
$528$	0	0
$529$	32.7916	1.42572
$530$	0	0
$531$	36.6292	1.58957
$532$	0	0
$533$	7.07730	0.306552
$534$	0	0
$535$	5.60514	0.242331
$536$	0	0
$537$	−29.9600	−1.29287
$538$	0	0
$539$	13.2932	0.572577
$540$	0	0
$541$	−2.49720	−0.107363	−0.0536816	−	0.998558i	$-0.517096\pi$
$541$	−2.49720	−0.107363	−0.0536816	+	0.998558i	$0.517096\pi$
$542$	0	0
$543$	47.8451	2.05323
$544$	0	0
$545$	−18.3798	−0.787305
$546$	0	0
$547$	−36.3795	−1.55548	−0.777738	−	0.628589i	$-0.783633\pi$
$547$	−36.3795	−1.55548	−0.777738	+	0.628589i	$0.783633\pi$
$548$	0	0
$549$	17.5699	0.749864
$550$	0	0
$551$	−50.7498	−2.16202
$552$	0	0
$553$	41.5095	1.76517
$554$	0	0
$555$	−36.7447	−1.55972
$556$	0	0
$557$	−36.2136	−1.53442	−0.767210	−	0.641396i	$-0.778356\pi$
$557$	−36.2136	−1.53442	−0.767210	+	0.641396i	$0.778356\pi$
$558$	0	0
$559$	−5.16061	−0.218271
$560$	0	0
$561$	4.94895	0.208945
$562$	0	0
$563$	−5.11266	−0.215473	−0.107736	−	0.994179i	$-0.534360\pi$
$563$	−5.11266	−0.215473	−0.107736	+	0.994179i	$0.534360\pi$
$564$	0	0
$565$	13.7466	0.578325
$566$	0	0
$567$	−1.02676	−0.0431197
$568$	0	0
$569$	22.2762	0.933869	0.466934	−	0.884292i	$-0.345358\pi$
$569$	22.2762	0.933869	0.466934	+	0.884292i	$0.345358\pi$
$570$	0	0
$571$	18.2274	0.762791	0.381396	−	0.924412i	$-0.375444\pi$
$571$	18.2274	0.762791	0.381396	+	0.924412i	$0.375444\pi$
$572$	0	0
$573$	−11.2240	−0.468889
$574$	0	0
$575$	−19.5189	−0.813996
$576$	0	0
$577$	−10.6327	−0.442647	−0.221323	−	0.975200i	$-0.571038\pi$
$577$	−10.6327	−0.442647	−0.221323	+	0.975200i	$0.571038\pi$
$578$	0	0
$579$	−58.8480	−2.44564
$580$	0	0
$581$	9.64504	0.400144
$582$	0	0
$583$	−3.61110	−0.149557
$584$	0	0
$585$	−5.45156	−0.225394
$586$	0	0
$587$	−27.3896	−1.13049	−0.565246	−	0.824923i	$-0.691219\pi$
$587$	−27.3896	−1.13049	−0.565246	+	0.824923i	$0.691219\pi$
$588$	0	0
$589$	30.8611	1.27161
$590$	0	0
$591$	−2.03881	−0.0838654
$592$	0	0
$593$	−38.5370	−1.58253	−0.791263	−	0.611476i	$-0.790576\pi$
$593$	−38.5370	−1.58253	−0.791263	+	0.611476i	$0.790576\pi$
$594$	0	0
$595$	−7.17843	−0.294287
$596$	0	0
$597$	61.0472	2.49849
$598$	0	0
$599$	−10.8825	−0.444648	−0.222324	−	0.974973i	$-0.571364\pi$
$599$	−10.8825	−0.444648	−0.222324	+	0.974973i	$0.571364\pi$
$600$	0	0
$601$	15.4802	0.631449	0.315725	−	0.948851i	$-0.397752\pi$
$601$	15.4802	0.631449	0.315725	+	0.948851i	$0.397752\pi$
$602$	0	0
$603$	−22.9897	−0.936211
$604$	0	0
$605$	−13.4549	−0.547019
$606$	0	0
$607$	28.8300	1.17018	0.585088	−	0.810970i	$-0.301060\pi$
$607$	28.8300	1.17018	0.585088	+	0.810970i	$0.301060\pi$
$608$	0	0
$609$	−88.1330	−3.57133
$610$	0	0
$611$	−4.78962	−0.193767
$612$	0	0
$613$	25.2645	1.02043	0.510213	−	0.860048i	$-0.329567\pi$
$613$	25.2645	1.02043	0.510213	+	0.860048i	$0.329567\pi$
$614$	0	0
$615$	41.7125	1.68201
$616$	0	0
$617$	−8.42325	−0.339107	−0.169554	−	0.985521i	$-0.554233\pi$
$617$	−8.42325	−0.339107	−0.169554	+	0.985521i	$0.554233\pi$
$618$	0	0
$619$	22.5873	0.907859	0.453929	−	0.891038i	$-0.350022\pi$
$619$	22.5873	0.907859	0.453929	+	0.891038i	$0.350022\pi$
$620$	0	0
$621$	−37.9070	−1.52115
$622$	0	0
$623$	−32.1408	−1.28769
$624$	0	0
$625$	−5.10524	−0.204210
$626$	0	0
$627$	27.0608	1.08070
$628$	0	0
$629$	−9.95046	−0.396751
$630$	0	0
$631$	−4.88805	−0.194590	−0.0972951	−	0.995256i	$-0.531019\pi$
$631$	−4.88805	−0.194590	−0.0972951	+	0.995256i	$0.531019\pi$
$632$	0	0
$633$	−19.4273	−0.772165
$634$	0	0
$635$	20.3289	0.806728
$636$	0	0
$637$	−6.43585	−0.254998
$638$	0	0
$639$	19.8608	0.785681
$640$	0	0
$641$	−5.97799	−0.236116	−0.118058	−	0.993007i	$-0.537667\pi$
$641$	−5.97799	−0.236116	−0.118058	+	0.993007i	$0.537667\pi$
$642$	0	0
$643$	9.02244	0.355810	0.177905	−	0.984048i	$-0.443068\pi$
$643$	9.02244	0.355810	0.177905	+	0.984048i	$0.443068\pi$
$644$	0	0
$645$	−30.4158	−1.19762
$646$	0	0
$647$	−43.5936	−1.71384	−0.856921	−	0.515448i	$-0.827625\pi$
$647$	−43.5936	−1.71384	−0.856921	+	0.515448i	$0.827625\pi$
$648$	0	0
$649$	11.5135	0.451944
$650$	0	0
$651$	53.5939	2.10051
$652$	0	0
$653$	−5.58573	−0.218586	−0.109293	−	0.994010i	$-0.534859\pi$
$653$	−5.58573	−0.218586	−0.109293	+	0.994010i	$0.534859\pi$
$654$	0	0
$655$	−11.0969	−0.433591
$656$	0	0
$657$	16.0075	0.624512
$658$	0	0
$659$	−46.4861	−1.81084	−0.905420	−	0.424516i	$-0.860444\pi$
$659$	−46.4861	−1.81084	−0.905420	+	0.424516i	$0.860444\pi$
$660$	0	0
$661$	29.5270	1.14847	0.574234	−	0.818691i	$-0.305300\pi$
$661$	29.5270	1.14847	0.574234	+	0.818691i	$0.305300\pi$
$662$	0	0
$663$	−2.39602	−0.0930537
$664$	0	0
$665$	−39.2515	−1.52211
$666$	0	0
$667$	59.2734	2.29507
$668$	0	0
$669$	−34.0641	−1.31699
$670$	0	0
$671$	5.52265	0.213200
$672$	0	0
$673$	−19.0435	−0.734072	−0.367036	−	0.930207i	$-0.619627\pi$
$673$	−19.0435	−0.734072	−0.367036	+	0.930207i	$0.619627\pi$
$674$	0	0
$675$	13.2619	0.510452
$676$	0	0
$677$	48.0028	1.84490	0.922449	−	0.386118i	$-0.126184\pi$
$677$	48.0028	1.84490	0.922449	+	0.386118i	$0.126184\pi$
$678$	0	0
$679$	−60.7764	−2.33238
$680$	0	0
$681$	39.5259	1.51464
$682$	0	0
$683$	17.1519	0.656300	0.328150	−	0.944626i	$-0.393575\pi$
$683$	17.1519	0.656300	0.328150	+	0.944626i	$0.393575\pi$
$684$	0	0
$685$	0.614595	0.0234825
$686$	0	0
$687$	−49.7996	−1.89997
$688$	0	0
$689$	1.74830	0.0666051
$690$	0	0
$691$	5.23072	0.198986	0.0994930	−	0.995038i	$-0.468278\pi$
$691$	5.23072	0.198986	0.0994930	+	0.995038i	$0.468278\pi$
$692$	0	0
$693$	28.9550	1.09991
$694$	0	0
$695$	27.2866	1.03504
$696$	0	0
$697$	11.2958	0.427857
$698$	0	0
$699$	−16.3450	−0.618226
$700$	0	0
$701$	−45.5718	−1.72122	−0.860612	−	0.509261i	$-0.829919\pi$
$701$	−45.5718	−1.72122	−0.860612	+	0.509261i	$0.829919\pi$
$702$	0	0
$703$	−54.4090	−2.05207
$704$	0	0
$705$	−28.2292	−1.06317
$706$	0	0
$707$	58.3713	2.19528
$708$	0	0
$709$	51.0062	1.91558	0.957789	−	0.287473i	$-0.0928151\pi$
$709$	51.0062	1.91558	0.957789	+	0.287473i	$0.0928151\pi$
$710$	0	0
$711$	50.3138	1.88691
$712$	0	0
$713$	−36.0443	−1.34987
$714$	0	0
$715$	−1.71356	−0.0640835
$716$	0	0
$717$	−35.2982	−1.31824
$718$	0	0
$719$	36.0489	1.34440	0.672198	−	0.740371i	$-0.265350\pi$
$719$	36.0489	1.34440	0.672198	+	0.740371i	$0.265350\pi$
$720$	0	0
$721$	−52.3866	−1.95098
$722$	0	0
$723$	48.5232	1.80460
$724$	0	0
$725$	−20.7371	−0.770155
$726$	0	0
$727$	41.8614	1.55255	0.776277	−	0.630391i	$-0.217106\pi$
$727$	41.8614	1.55255	0.776277	+	0.630391i	$0.217106\pi$
$728$	0	0
$729$	−43.8073	−1.62249
$730$	0	0
$731$	−8.23661	−0.304642
$732$	0	0
$733$	7.23512	0.267235	0.133618	−	0.991033i	$-0.457341\pi$
$733$	7.23512	0.267235	0.133618	+	0.991033i	$0.457341\pi$
$734$	0	0
$735$	−37.9319	−1.39914
$736$	0	0
$737$	−7.22622	−0.266181
$738$	0	0
$739$	−3.34052	−0.122883	−0.0614416	−	0.998111i	$-0.519570\pi$
$739$	−3.34052	−0.122883	−0.0614416	+	0.998111i	$0.519570\pi$
$740$	0	0
$741$	−13.1014	−0.481292
$742$	0	0
$743$	38.3943	1.40855	0.704275	−	0.709927i	$-0.251272\pi$
$743$	38.3943	1.40855	0.704275	+	0.709927i	$0.251272\pi$
$744$	0	0
$745$	0.866343	0.0317403
$746$	0	0
$747$	11.6908	0.427743
$748$	0	0
$749$	14.4134	0.526655
$750$	0	0
$751$	−31.0890	−1.13445	−0.567227	−	0.823561i	$-0.691984\pi$
$751$	−31.0890	−1.13445	−0.567227	+	0.823561i	$0.691984\pi$
$752$	0	0
$753$	−6.45172	−0.235114
$754$	0	0
$755$	6.89533	0.250947
$756$	0	0
$757$	−32.7587	−1.19064	−0.595318	−	0.803490i	$-0.702974\pi$
$757$	−32.7587	−1.19064	−0.595318	+	0.803490i	$0.702974\pi$
$758$	0	0
$759$	−31.6057	−1.14721
$760$	0	0
$761$	17.3155	0.627686	0.313843	−	0.949475i	$-0.398383\pi$
$761$	17.3155	0.627686	0.313843	+	0.949475i	$0.398383\pi$
$762$	0	0
$763$	−47.2631	−1.71104
$764$	0	0
$765$	−8.70098	−0.314585
$766$	0	0
$767$	−5.57422	−0.201273
$768$	0	0
$769$	−51.9309	−1.87268	−0.936338	−	0.351099i	$-0.885808\pi$
$769$	−51.9309	−1.87268	−0.936338	+	0.351099i	$0.885808\pi$
$770$	0	0
$771$	64.2104	2.31248
$772$	0	0
$773$	38.4359	1.38244	0.691221	−	0.722643i	$-0.257073\pi$
$773$	38.4359	1.38244	0.691221	+	0.722643i	$0.257073\pi$
$774$	0	0
$775$	12.6103	0.452974
$776$	0	0
$777$	−94.4875	−3.38972
$778$	0	0
$779$	61.7650	2.21296
$780$	0	0
$781$	6.24274	0.223383
$782$	0	0
$783$	−40.2726	−1.43923
$784$	0	0
$785$	23.0398	0.822327
$786$	0	0
$787$	31.5646	1.12516	0.562579	−	0.826744i	$-0.309809\pi$
$787$	31.5646	1.12516	0.562579	+	0.826744i	$0.309809\pi$
$788$	0	0
$789$	34.7480	1.23706
$790$	0	0
$791$	35.3489	1.25686
$792$	0	0
$793$	−2.67378	−0.0949486
$794$	0	0
$795$	10.3042	0.365453
$796$	0	0
$797$	−44.0178	−1.55919	−0.779596	−	0.626283i	$-0.784575\pi$
$797$	−44.0178	−1.55919	−0.779596	+	0.626283i	$0.784575\pi$
$798$	0	0
$799$	−7.64449	−0.270443
$800$	0	0
$801$	−38.9579	−1.37651
$802$	0	0
$803$	5.03156	0.177560
$804$	0	0
$805$	45.8439	1.61579
$806$	0	0
$807$	−58.2583	−2.05079
$808$	0	0
$809$	1.40571	0.0494223	0.0247111	−	0.999695i	$-0.492133\pi$
$809$	1.40571	0.0494223	0.0247111	+	0.999695i	$0.492133\pi$
$810$	0	0
$811$	46.0445	1.61684	0.808421	−	0.588605i	$-0.200323\pi$
$811$	46.0445	1.61684	0.808421	+	0.588605i	$0.200323\pi$
$812$	0	0
$813$	−14.6394	−0.513425
$814$	0	0
$815$	8.93921	0.313127
$816$	0	0
$817$	−45.0377	−1.57567
$818$	0	0
$819$	−14.0185	−0.489845
$820$	0	0
$821$	−4.88420	−0.170460	−0.0852299	−	0.996361i	$-0.527162\pi$
$821$	−4.88420	−0.170460	−0.0852299	+	0.996361i	$0.527162\pi$
$822$	0	0
$823$	−35.3902	−1.23363	−0.616813	−	0.787110i	$-0.711577\pi$
$823$	−35.3902	−1.23363	−0.616813	+	0.787110i	$0.711577\pi$
$824$	0	0
$825$	11.0574	0.384969
$826$	0	0
$827$	−22.8300	−0.793876	−0.396938	−	0.917846i	$-0.629927\pi$
$827$	−22.8300	−0.793876	−0.396938	+	0.917846i	$0.629927\pi$
$828$	0	0
$829$	−9.11266	−0.316496	−0.158248	−	0.987399i	$-0.550585\pi$
$829$	−9.11266	−0.316496	−0.158248	+	0.987399i	$0.550585\pi$
$830$	0	0
$831$	−33.0745	−1.14734
$832$	0	0
$833$	−10.2720	−0.355903
$834$	0	0
$835$	−19.2792	−0.667186
$836$	0	0
$837$	24.4899	0.846495
$838$	0	0
$839$	9.44098	0.325939	0.162969	−	0.986631i	$-0.447893\pi$
$839$	9.44098	0.325939	0.162969	+	0.986631i	$0.447893\pi$
$840$	0	0
$841$	33.9725	1.17146
$842$	0	0
$843$	−6.31765	−0.217591
$844$	0	0
$845$	−19.2545	−0.662374
$846$	0	0
$847$	−34.5987	−1.18883
$848$	0	0
$849$	−18.6910	−0.641473
$850$	0	0
$851$	63.5471	2.17837
$852$	0	0
$853$	30.0834	1.03003	0.515017	−	0.857180i	$-0.327785\pi$
$853$	30.0834	1.03003	0.515017	+	0.857180i	$0.327785\pi$
$854$	0	0
$855$	−47.5768	−1.62709
$856$	0	0
$857$	35.4537	1.21108	0.605538	−	0.795816i	$-0.292958\pi$
$857$	35.4537	1.21108	0.605538	+	0.795816i	$0.292958\pi$
$858$	0	0
$859$	−26.7296	−0.912002	−0.456001	−	0.889979i	$-0.650719\pi$
$859$	−26.7296	−0.912002	−0.456001	+	0.889979i	$0.650719\pi$
$860$	0	0
$861$	107.262	3.65548
$862$	0	0
$863$	−40.8094	−1.38917	−0.694585	−	0.719411i	$-0.744412\pi$
$863$	−40.8094	−1.38917	−0.694585	+	0.719411i	$0.744412\pi$
$864$	0	0
$865$	0.0675377	0.00229635
$866$	0	0
$867$	43.7009	1.48416
$868$	0	0
$869$	15.8149	0.536483
$870$	0	0
$871$	3.49855	0.118544
$872$	0	0
$873$	−73.6672	−2.49326
$874$	0	0
$875$	−46.7266	−1.57965
$876$	0	0
$877$	−47.2005	−1.59385	−0.796923	−	0.604081i	$-0.793541\pi$
$877$	−47.2005	−1.59385	−0.796923	+	0.604081i	$0.793541\pi$
$878$	0	0
$879$	78.3835	2.64381
$880$	0	0
$881$	31.8545	1.07321	0.536603	−	0.843835i	$-0.319707\pi$
$881$	31.8545	1.07321	0.536603	+	0.843835i	$0.319707\pi$
$882$	0	0
$883$	−33.8126	−1.13788	−0.568942	−	0.822378i	$-0.692647\pi$
$883$	−33.8126	−1.13788	−0.568942	+	0.822378i	$0.692647\pi$
$884$	0	0
$885$	−32.8536	−1.10436
$886$	0	0
$887$	48.9427	1.64334	0.821668	−	0.569966i	$-0.193044\pi$
$887$	48.9427	1.64334	0.821668	+	0.569966i	$0.193044\pi$
$888$	0	0
$889$	52.2750	1.75325
$890$	0	0
$891$	−0.391188	−0.0131053
$892$	0	0
$893$	−41.7999	−1.39878
$894$	0	0
$895$	16.5568	0.553433
$896$	0	0
$897$	15.3018	0.510913
$898$	0	0
$899$	−38.2937	−1.27717
$900$	0	0
$901$	2.79039	0.0929613
$902$	0	0
$903$	−78.2132	−2.60277
$904$	0	0
$905$	−26.4406	−0.878916
$906$	0	0
$907$	4.26447	0.141600	0.0707998	−	0.997491i	$-0.477445\pi$
$907$	4.26447	0.141600	0.0707998	+	0.997491i	$0.477445\pi$
$908$	0	0
$909$	70.7520	2.34669
$910$	0	0
$911$	1.51985	0.0503548	0.0251774	−	0.999683i	$-0.491985\pi$
$911$	1.51985	0.0503548	0.0251774	+	0.999683i	$0.491985\pi$
$912$	0	0
$913$	3.67470	0.121615
$914$	0	0
$915$	−15.7588	−0.520971
$916$	0	0
$917$	−28.5352	−0.942317
$918$	0	0
$919$	47.2491	1.55860	0.779302	−	0.626649i	$-0.215574\pi$
$919$	47.2491	1.55860	0.779302	+	0.626649i	$0.215574\pi$
$920$	0	0
$921$	−55.2789	−1.82150
$922$	0	0
$923$	−3.02241	−0.0994837
$924$	0	0
$925$	−22.2322	−0.730991
$926$	0	0
$927$	−63.4979	−2.08554
$928$	0	0
$929$	−8.36659	−0.274499	−0.137249	−	0.990537i	$-0.543826\pi$
$929$	−8.36659	−0.274499	−0.137249	+	0.990537i	$0.543826\pi$
$930$	0	0
$931$	−56.1670	−1.84080
$932$	0	0
$933$	−76.5800	−2.50712
$934$	0	0
$935$	−2.73494	−0.0894420
$936$	0	0
$937$	17.1858	0.561436	0.280718	−	0.959790i	$-0.409427\pi$
$937$	17.1858	0.561436	0.280718	+	0.959790i	$0.409427\pi$
$938$	0	0
$939$	−52.1562	−1.70205
$940$	0	0
$941$	−37.7041	−1.22912	−0.614559	−	0.788871i	$-0.710666\pi$
$941$	−37.7041	−1.22912	−0.614559	+	0.788871i	$0.710666\pi$
$942$	0	0
$943$	−72.1386	−2.34916
$944$	0	0
$945$	−31.1481	−1.01325
$946$	0	0
$947$	27.0534	0.879118	0.439559	−	0.898214i	$-0.355135\pi$
$947$	27.0534	0.879118	0.439559	+	0.898214i	$0.355135\pi$
$948$	0	0
$949$	−2.43601	−0.0790764
$950$	0	0
$951$	−69.2342	−2.24507
$952$	0	0
$953$	−29.0368	−0.940594	−0.470297	−	0.882508i	$-0.655853\pi$
$953$	−29.0368	−0.940594	−0.470297	+	0.882508i	$0.655853\pi$
$954$	0	0
$955$	6.20271	0.200715
$956$	0	0
$957$	−33.5781	−1.08543
$958$	0	0
$959$	1.58041	0.0510340
$960$	0	0
$961$	−7.71349	−0.248822
$962$	0	0
$963$	17.4705	0.562980
$964$	0	0
$965$	32.5212	1.04689
$966$	0	0
$967$	36.3400	1.16861	0.584307	−	0.811532i	$-0.301366\pi$
$967$	36.3400	1.16861	0.584307	+	0.811532i	$0.301366\pi$
$968$	0	0
$969$	−20.9105	−0.671743
$970$	0	0
$971$	45.6803	1.46595	0.732975	−	0.680256i	$-0.238131\pi$
$971$	45.6803	1.46595	0.732975	+	0.680256i	$0.238131\pi$
$972$	0	0
$973$	70.1665	2.24944
$974$	0	0
$975$	−5.35341	−0.171446
$976$	0	0
$977$	−14.1275	−0.451979	−0.225989	−	0.974130i	$-0.572562\pi$
$977$	−14.1275	−0.451979	−0.225989	+	0.974130i	$0.572562\pi$
$978$	0	0
$979$	−12.2455	−0.391366
$980$	0	0
$981$	−57.2876	−1.82905
$982$	0	0
$983$	15.4162	0.491701	0.245850	−	0.969308i	$-0.420933\pi$
$983$	15.4162	0.491701	0.245850	+	0.969308i	$0.420933\pi$
$984$	0	0
$985$	1.12671	0.0358998
$986$	0	0
$987$	−72.5905	−2.31058
$988$	0	0
$989$	52.6019	1.67264
$990$	0	0
$991$	−2.44185	−0.0775681	−0.0387840	−	0.999248i	$-0.512348\pi$
$991$	−2.44185	−0.0775681	−0.0387840	+	0.999248i	$0.512348\pi$
$992$	0	0
$993$	−2.19889	−0.0697796
$994$	0	0
$995$	−33.7365	−1.06952
$996$	0	0
$997$	6.46923	0.204883	0.102441	−	0.994739i	$-0.467335\pi$
$997$	6.46923	0.204883	0.102441	+	0.994739i	$0.467335\pi$
$998$	0	0
$999$	−43.1764	−1.36604

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6044.2.a.b.1.4	✓	63

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6044.2.a.b.1.4	✓	63	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:	\( N \)	\(=\)	\( 6044 = 2^{2} \cdot 1511 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6044.a (trivial)

\(f(q)\)	\(=\)	\(q-2.79559 q^{3} +1.54493 q^{5} +3.97273 q^{7} +4.81535 q^{9} +O(q^{10})\)	\(q-2.79559 q^{3} +1.54493 q^{5} +3.97273 q^{7} +4.81535 q^{9} +1.51359 q^{11} -0.732798 q^{13} -4.31900 q^{15} -1.16959 q^{17} -6.39527 q^{19} -11.1061 q^{21} +7.46938 q^{23} -2.61319 q^{25} -5.07498 q^{27} +7.93552 q^{29} -4.82561 q^{31} -4.23137 q^{33} +6.13758 q^{35} +8.50768 q^{37} +2.04861 q^{39} -9.65792 q^{41} +7.04234 q^{43} +7.43937 q^{45} +6.53607 q^{47} +8.78257 q^{49} +3.26969 q^{51} -2.38579 q^{53} +2.33838 q^{55} +17.8786 q^{57} +7.60676 q^{59} +3.64872 q^{61} +19.1301 q^{63} -1.13212 q^{65} -4.77424 q^{67} -20.8814 q^{69} +4.12447 q^{71} +3.32426 q^{73} +7.30543 q^{75} +6.01306 q^{77} +10.4486 q^{79} -0.258451 q^{81} +2.42781 q^{83} -1.80693 q^{85} -22.1845 q^{87} -8.09036 q^{89} -2.91121 q^{91} +13.4905 q^{93} -9.88024 q^{95} -15.2984 q^{97} +7.28844 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9}+O(q^{10}) \) 63 * q + 5 * q^3 + 5 * q^5 + 22 * q^7 + 62 * q^9	\( 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9} + 21 q^{11} + 17 q^{13} + 26 q^{15} - 5 q^{17} + 57 q^{19} + 18 q^{21} + 16 q^{23} + 60 q^{25} + 14 q^{27} + 22 q^{29} + 36 q^{31} - q^{33} + 28 q^{35} + 21 q^{37} + 69 q^{39} - 3 q^{41} + 86 q^{43} + 39 q^{45} + 23 q^{47} + 63 q^{49} + 67 q^{51} + 18 q^{53} + 67 q^{55} + 27 q^{59} + 62 q^{61} + 58 q^{63} - 13 q^{65} + 62 q^{67} + 45 q^{69} + 29 q^{71} - q^{73} + 39 q^{75} + 19 q^{77} + 132 q^{79} + 51 q^{81} + 35 q^{83} + 60 q^{85} + 55 q^{87} - 2 q^{89} + 84 q^{91} + 29 q^{93} + 53 q^{95} - 10 q^{97} + 95 q^{99}+O(q^{100}) \) 63 * q + 5 * q^3 + 5 * q^5 + 22 * q^7 + 62 * q^9 + 21 * q^11 + 17 * q^13 + 26 * q^15 - 5 * q^17 + 57 * q^19 + 18 * q^21 + 16 * q^23 + 60 * q^25 + 14 * q^27 + 22 * q^29 + 36 * q^31 - q^33 + 28 * q^35 + 21 * q^37 + 69 * q^39 - 3 * q^41 + 86 * q^43 + 39 * q^45 + 23 * q^47 + 63 * q^49 + 67 * q^51 + 18 * q^53 + 67 * q^55 + 27 * q^59 + 62 * q^61 + 58 * q^63 - 13 * q^65 + 62 * q^67 + 45 * q^69 + 29 * q^71 - q^73 + 39 * q^75 + 19 * q^77 + 132 * q^79 + 51 * q^81 + 35 * q^83 + 60 * q^85 + 55 * q^87 - 2 * q^89 + 84 * q^91 + 29 * q^93 + 53 * q^95 - 10 * q^97 + 95 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more