LMFDB - Embedded newform 6044.2.a.b.1.17

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			1.17
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)

$f(q)$	$=$	$q-1.55218 q^{3} -1.58422 q^{5} -1.70170 q^{7} -0.590736 q^{9} +O(q^{10})$	$q-1.55218 q^{3} -1.58422 q^{5} -1.70170 q^{7} -0.590736 q^{9} +0.379401 q^{11} +2.57871 q^{13} +2.45899 q^{15} +4.36269 q^{17} -7.80250 q^{19} +2.64135 q^{21} +4.31401 q^{23} -2.49026 q^{25} +5.57347 q^{27} -1.57022 q^{29} -3.16025 q^{31} -0.588899 q^{33} +2.69587 q^{35} -6.29202 q^{37} -4.00262 q^{39} -3.55302 q^{41} +9.72252 q^{43} +0.935853 q^{45} -6.98485 q^{47} -4.10420 q^{49} -6.77168 q^{51} -6.23147 q^{53} -0.601053 q^{55} +12.1109 q^{57} -6.54925 q^{59} -9.22799 q^{61} +1.00526 q^{63} -4.08523 q^{65} +9.36675 q^{67} -6.69612 q^{69} +6.46812 q^{71} -0.109807 q^{73} +3.86533 q^{75} -0.645629 q^{77} -2.02938 q^{79} -6.87882 q^{81} -7.75208 q^{83} -6.91144 q^{85} +2.43726 q^{87} -8.05937 q^{89} -4.38820 q^{91} +4.90528 q^{93} +12.3608 q^{95} -9.16254 q^{97} -0.224126 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.55218	−0.896152	−0.448076	−	0.893996i	$-0.647891\pi$
$3$	−1.55218	−0.896152	−0.448076	+	0.893996i	$0.647891\pi$
$4$	0	0
$5$	−1.58422	−0.708483	−0.354241	−	0.935154i	$-0.615261\pi$
$5$	−1.58422	−0.708483	−0.354241	+	0.935154i	$0.615261\pi$
$6$	0	0
$7$	−1.70170	−0.643184	−0.321592	−	0.946878i	$-0.604218\pi$
$7$	−1.70170	−0.643184	−0.321592	+	0.946878i	$0.604218\pi$
$8$	0	0
$9$	−0.590736	−0.196912
$10$	0	0
$11$	0.379401	0.114394	0.0571969	−	0.998363i	$-0.481784\pi$
$11$	0.379401	0.114394	0.0571969	+	0.998363i	$0.481784\pi$
$12$	0	0
$13$	2.57871	0.715206	0.357603	−	0.933874i	$-0.383594\pi$
$13$	2.57871	0.715206	0.357603	+	0.933874i	$0.383594\pi$
$14$	0	0
$15$	2.45899	0.634908
$16$	0	0
$17$	4.36269	1.05811	0.529054	−	0.848588i	$-0.322547\pi$
$17$	4.36269	1.05811	0.529054	+	0.848588i	$0.322547\pi$
$18$	0	0
$19$	−7.80250	−1.79002	−0.895008	−	0.446050i	$-0.852830\pi$
$19$	−7.80250	−1.79002	−0.895008	+	0.446050i	$0.852830\pi$
$20$	0	0
$21$	2.64135	0.576390
$22$	0	0
$23$	4.31401	0.899533	0.449766	−	0.893146i	$-0.351507\pi$
$23$	4.31401	0.899533	0.449766	+	0.893146i	$0.351507\pi$
$24$	0	0
$25$	−2.49026	−0.498052
$26$	0	0
$27$	5.57347	1.07261
$28$	0	0
$29$	−1.57022	−0.291582	−0.145791	−	0.989315i	$-0.546573\pi$
$29$	−1.57022	−0.291582	−0.145791	+	0.989315i	$0.546573\pi$
$30$	0	0
$31$	−3.16025	−0.567598	−0.283799	−	0.958884i	$-0.591595\pi$
$31$	−3.16025	−0.567598	−0.283799	+	0.958884i	$0.591595\pi$
$32$	0	0
$33$	−0.588899	−0.102514
$34$	0	0
$35$	2.69587	0.455684
$36$	0	0
$37$	−6.29202	−1.03440	−0.517201	−	0.855864i	$-0.673026\pi$
$37$	−6.29202	−1.03440	−0.517201	+	0.855864i	$0.673026\pi$
$38$	0	0
$39$	−4.00262	−0.640933
$40$	0	0
$41$	−3.55302	−0.554888	−0.277444	−	0.960742i	$-0.589487\pi$
$41$	−3.55302	−0.554888	−0.277444	+	0.960742i	$0.589487\pi$
$42$	0	0
$43$	9.72252	1.48267	0.741336	−	0.671135i	$-0.234193\pi$
$43$	9.72252	1.48267	0.741336	+	0.671135i	$0.234193\pi$
$44$	0	0
$45$	0.935853	0.139509
$46$	0	0
$47$	−6.98485	−1.01884	−0.509422	−	0.860517i	$-0.670141\pi$
$47$	−6.98485	−1.01884	−0.509422	+	0.860517i	$0.670141\pi$
$48$	0	0
$49$	−4.10420	−0.586315
$50$	0	0
$51$	−6.77168	−0.948225
$52$	0	0
$53$	−6.23147	−0.855959	−0.427979	−	0.903789i	$-0.640774\pi$
$53$	−6.23147	−0.855959	−0.427979	+	0.903789i	$0.640774\pi$
$54$	0	0
$55$	−0.601053	−0.0810460
$56$	0	0
$57$	12.1109	1.60413
$58$	0	0
$59$	−6.54925	−0.852639	−0.426320	−	0.904573i	$-0.640190\pi$
$59$	−6.54925	−0.852639	−0.426320	+	0.904573i	$0.640190\pi$
$60$	0	0
$61$	−9.22799	−1.18152	−0.590761	−	0.806846i	$-0.701173\pi$
$61$	−9.22799	−1.18152	−0.590761	+	0.806846i	$0.701173\pi$
$62$	0	0
$63$	1.00526	0.126651
$64$	0	0
$65$	−4.08523	−0.506711
$66$	0	0
$67$	9.36675	1.14433	0.572165	−	0.820138i	$-0.306104\pi$
$67$	9.36675	1.14433	0.572165	+	0.820138i	$0.306104\pi$
$68$	0	0
$69$	−6.69612	−0.806118
$70$	0	0
$71$	6.46812	0.767625	0.383813	−	0.923411i	$-0.374611\pi$
$71$	6.46812	0.767625	0.383813	+	0.923411i	$0.374611\pi$
$72$	0	0
$73$	−0.109807	−0.0128519	−0.00642597	−	0.999979i	$-0.502045\pi$
$73$	−0.109807	−0.0128519	−0.00642597	+	0.999979i	$0.502045\pi$
$74$	0	0
$75$	3.86533	0.446330
$76$	0	0
$77$	−0.645629	−0.0735762
$78$	0	0
$79$	−2.02938	−0.228323	−0.114161	−	0.993462i	$-0.536418\pi$
$79$	−2.02938	−0.228323	−0.114161	+	0.993462i	$0.536418\pi$
$80$	0	0
$81$	−6.87882	−0.764314
$82$	0	0
$83$	−7.75208	−0.850901	−0.425450	−	0.904982i	$-0.639884\pi$
$83$	−7.75208	−0.850901	−0.425450	+	0.904982i	$0.639884\pi$
$84$	0	0
$85$	−6.91144	−0.749651
$86$	0	0
$87$	2.43726	0.261302
$88$	0	0
$89$	−8.05937	−0.854292	−0.427146	−	0.904183i	$-0.640481\pi$
$89$	−8.05937	−0.854292	−0.427146	+	0.904183i	$0.640481\pi$
$90$	0	0
$91$	−4.38820	−0.460009
$92$	0	0
$93$	4.90528	0.508654
$94$	0	0
$95$	12.3608	1.26820
$96$	0	0
$97$	−9.16254	−0.930315	−0.465158	−	0.885228i	$-0.654002\pi$
$97$	−9.16254	−0.930315	−0.465158	+	0.885228i	$0.654002\pi$
$98$	0	0
$99$	−0.224126	−0.0225255
$100$	0	0
$101$	−2.27946	−0.226815	−0.113408	−	0.993549i	$-0.536177\pi$
$101$	−2.27946	−0.226815	−0.113408	+	0.993549i	$0.536177\pi$
$102$	0	0
$103$	2.17217	0.214030	0.107015	−	0.994257i	$-0.465871\pi$
$103$	2.17217	0.214030	0.107015	+	0.994257i	$0.465871\pi$
$104$	0	0
$105$	−4.18447	−0.408362
$106$	0	0
$107$	15.9049	1.53758	0.768791	−	0.639501i	$-0.220859\pi$
$107$	15.9049	1.53758	0.768791	+	0.639501i	$0.220859\pi$
$108$	0	0
$109$	8.50049	0.814199	0.407099	−	0.913384i	$-0.366540\pi$
$109$	8.50049	0.814199	0.407099	+	0.913384i	$0.366540\pi$
$110$	0	0
$111$	9.76635	0.926981
$112$	0	0
$113$	4.85143	0.456384	0.228192	−	0.973616i	$-0.426719\pi$
$113$	4.85143	0.456384	0.228192	+	0.973616i	$0.426719\pi$
$114$	0	0
$115$	−6.83432	−0.637303
$116$	0	0
$117$	−1.52334	−0.140833
$118$	0	0
$119$	−7.42401	−0.680558
$120$	0	0
$121$	−10.8561	−0.986914
$122$	0	0
$123$	5.51493	0.497264
$124$	0	0
$125$	11.8662	1.06134
$126$	0	0
$127$	7.59322	0.673790	0.336895	−	0.941542i	$-0.390623\pi$
$127$	7.59322	0.673790	0.336895	+	0.941542i	$0.390623\pi$
$128$	0	0
$129$	−15.0911	−1.32870
$130$	0	0
$131$	−7.50399	−0.655626	−0.327813	−	0.944743i	$-0.606312\pi$
$131$	−7.50399	−0.655626	−0.327813	+	0.944743i	$0.606312\pi$
$132$	0	0
$133$	13.2775	1.15131
$134$	0	0
$135$	−8.82958	−0.759929
$136$	0	0
$137$	11.2889	0.964474	0.482237	−	0.876041i	$-0.339824\pi$
$137$	11.2889	0.964474	0.482237	+	0.876041i	$0.339824\pi$
$138$	0	0
$139$	5.71827	0.485017	0.242509	−	0.970149i	$-0.422030\pi$
$139$	5.71827	0.485017	0.242509	+	0.970149i	$0.422030\pi$
$140$	0	0
$141$	10.8417	0.913039
$142$	0	0
$143$	0.978366	0.0818151
$144$	0	0
$145$	2.48756	0.206581
$146$	0	0
$147$	6.37047	0.525427
$148$	0	0
$149$	2.15865	0.176843	0.0884217	−	0.996083i	$-0.471818\pi$
$149$	2.15865	0.176843	0.0884217	+	0.996083i	$0.471818\pi$
$150$	0	0
$151$	−13.2533	−1.07854	−0.539268	−	0.842134i	$-0.681299\pi$
$151$	−13.2533	−1.07854	−0.539268	+	0.842134i	$0.681299\pi$
$152$	0	0
$153$	−2.57720	−0.208354
$154$	0	0
$155$	5.00652	0.402133
$156$	0	0
$157$	13.0859	1.04437	0.522185	−	0.852832i	$-0.325117\pi$
$157$	13.0859	1.04437	0.522185	+	0.852832i	$0.325117\pi$
$158$	0	0
$159$	9.67237	0.767069
$160$	0	0
$161$	−7.34116	−0.578565
$162$	0	0
$163$	−15.3915	−1.20555	−0.602777	−	0.797910i	$-0.705939\pi$
$163$	−15.3915	−1.20555	−0.602777	+	0.797910i	$0.705939\pi$
$164$	0	0
$165$	0.932943	0.0726295
$166$	0	0
$167$	9.91346	0.767126	0.383563	−	0.923515i	$-0.374697\pi$
$167$	9.91346	0.767126	0.383563	+	0.923515i	$0.374697\pi$
$168$	0	0
$169$	−6.35025	−0.488481
$170$	0	0
$171$	4.60922	0.352476
$172$	0	0
$173$	16.1770	1.22991	0.614957	−	0.788560i	$-0.289173\pi$
$173$	16.1770	1.22991	0.614957	+	0.788560i	$0.289173\pi$
$174$	0	0
$175$	4.23769	0.320339
$176$	0	0
$177$	10.1656	0.764094
$178$	0	0
$179$	19.2735	1.44057	0.720284	−	0.693679i	$-0.244011\pi$
$179$	19.2735	1.44057	0.720284	+	0.693679i	$0.244011\pi$
$180$	0	0
$181$	12.4085	0.922317	0.461158	−	0.887318i	$-0.347434\pi$
$181$	12.4085	0.922317	0.461158	+	0.887318i	$0.347434\pi$
$182$	0	0
$183$	14.3235	1.05882
$184$	0	0
$185$	9.96791	0.732856
$186$	0	0
$187$	1.65521	0.121041
$188$	0	0
$189$	−9.48439	−0.689888
$190$	0	0
$191$	−7.35947	−0.532513	−0.266256	−	0.963902i	$-0.585787\pi$
$191$	−7.35947	−0.532513	−0.266256	+	0.963902i	$0.585787\pi$
$192$	0	0
$193$	−20.8613	−1.50163	−0.750816	−	0.660511i	$-0.770340\pi$
$193$	−20.8613	−1.50163	−0.750816	+	0.660511i	$0.770340\pi$
$194$	0	0
$195$	6.34102	0.454090
$196$	0	0
$197$	−1.70882	−0.121748	−0.0608740	−	0.998145i	$-0.519389\pi$
$197$	−1.70882	−0.121748	−0.0608740	+	0.998145i	$0.519389\pi$
$198$	0	0
$199$	−0.108840	−0.00771547	−0.00385774	−	0.999993i	$-0.501228\pi$
$199$	−0.108840	−0.00771547	−0.00385774	+	0.999993i	$0.501228\pi$
$200$	0	0
$201$	−14.5389	−1.02549
$202$	0	0
$203$	2.67205	0.187541
$204$	0	0
$205$	5.62875	0.393129
$206$	0	0
$207$	−2.54844	−0.177129
$208$	0	0
$209$	−2.96028	−0.204767
$210$	0	0
$211$	9.05277	0.623218	0.311609	−	0.950210i	$-0.399132\pi$
$211$	9.05277	0.623218	0.311609	+	0.950210i	$0.399132\pi$
$212$	0	0
$213$	−10.0397	−0.687909
$214$	0	0
$215$	−15.4026	−1.05045
$216$	0	0
$217$	5.37781	0.365070
$218$	0	0
$219$	0.170440	0.0115173
$220$	0	0
$221$	11.2501	0.756765
$222$	0	0
$223$	20.3506	1.36278	0.681388	−	0.731922i	$-0.261376\pi$
$223$	20.3506	1.36278	0.681388	+	0.731922i	$0.261376\pi$
$224$	0	0
$225$	1.47109	0.0980725
$226$	0	0
$227$	25.3956	1.68556	0.842781	−	0.538256i	$-0.180917\pi$
$227$	25.3956	1.68556	0.842781	+	0.538256i	$0.180917\pi$
$228$	0	0
$229$	−4.68641	−0.309687	−0.154843	−	0.987939i	$-0.549487\pi$
$229$	−4.68641	−0.309687	−0.154843	+	0.987939i	$0.549487\pi$
$230$	0	0
$231$	1.00213	0.0659354
$232$	0	0
$233$	−12.4841	−0.817859	−0.408929	−	0.912566i	$-0.634098\pi$
$233$	−12.4841	−0.817859	−0.408929	+	0.912566i	$0.634098\pi$
$234$	0	0
$235$	11.0655	0.721834
$236$	0	0
$237$	3.14996	0.204612
$238$	0	0
$239$	4.68459	0.303021	0.151510	−	0.988456i	$-0.451586\pi$
$239$	4.68459	0.303021	0.151510	+	0.988456i	$0.451586\pi$
$240$	0	0
$241$	19.6087	1.26311	0.631555	−	0.775331i	$-0.282417\pi$
$241$	19.6087	1.26311	0.631555	+	0.775331i	$0.282417\pi$
$242$	0	0
$243$	−6.04324	−0.387674
$244$	0	0
$245$	6.50194	0.415394
$246$	0	0
$247$	−20.1204	−1.28023
$248$	0	0
$249$	12.0326	0.762536
$250$	0	0
$251$	−15.5043	−0.978624	−0.489312	−	0.872109i	$-0.662752\pi$
$251$	−15.5043	−0.978624	−0.489312	+	0.872109i	$0.662752\pi$
$252$	0	0
$253$	1.63674	0.102901
$254$	0	0
$255$	10.7278	0.671801
$256$	0	0
$257$	6.01928	0.375472	0.187736	−	0.982220i	$-0.439885\pi$
$257$	6.01928	0.375472	0.187736	+	0.982220i	$0.439885\pi$
$258$	0	0
$259$	10.7072	0.665310
$260$	0	0
$261$	0.927585	0.0574161
$262$	0	0
$263$	21.6471	1.33482	0.667409	−	0.744692i	$-0.267403\pi$
$263$	21.6471	1.33482	0.667409	+	0.744692i	$0.267403\pi$
$264$	0	0
$265$	9.87200	0.606432
$266$	0	0
$267$	12.5096	0.765575
$268$	0	0
$269$	21.5904	1.31639	0.658196	−	0.752847i	$-0.271320\pi$
$269$	21.5904	1.31639	0.658196	+	0.752847i	$0.271320\pi$
$270$	0	0
$271$	23.6375	1.43587	0.717937	−	0.696108i	$-0.245087\pi$
$271$	23.6375	1.43587	0.717937	+	0.696108i	$0.245087\pi$
$272$	0	0
$273$	6.81128	0.412237
$274$	0	0
$275$	−0.944808	−0.0569741
$276$	0	0
$277$	−19.2529	−1.15679	−0.578396	−	0.815756i	$-0.696321\pi$
$277$	−19.2529	−1.15679	−0.578396	+	0.815756i	$0.696321\pi$
$278$	0	0
$279$	1.86688	0.111767
$280$	0	0
$281$	27.8991	1.66432	0.832159	−	0.554537i	$-0.187105\pi$
$281$	27.8991	1.66432	0.832159	+	0.554537i	$0.187105\pi$
$282$	0	0
$283$	−27.3440	−1.62543	−0.812717	−	0.582659i	$-0.802012\pi$
$283$	−27.3440	−1.62543	−0.812717	+	0.582659i	$0.802012\pi$
$284$	0	0
$285$	−19.1863	−1.13650
$286$	0	0
$287$	6.04619	0.356895
$288$	0	0
$289$	2.03307	0.119592
$290$	0	0
$291$	14.2219	0.833704
$292$	0	0
$293$	−32.7265	−1.91191	−0.955953	−	0.293521i	$-0.905173\pi$
$293$	−32.7265	−1.91191	−0.955953	+	0.293521i	$0.905173\pi$
$294$	0	0
$295$	10.3754	0.604080
$296$	0	0
$297$	2.11458	0.122700
$298$	0	0
$299$	11.1246	0.643351
$300$	0	0
$301$	−16.5449	−0.953630
$302$	0	0
$303$	3.53814	0.203261
$304$	0	0
$305$	14.6191	0.837089
$306$	0	0
$307$	28.7270	1.63954	0.819769	−	0.572695i	$-0.194102\pi$
$307$	28.7270	1.63954	0.819769	+	0.572695i	$0.194102\pi$
$308$	0	0
$309$	−3.37160	−0.191803
$310$	0	0
$311$	10.9096	0.618626	0.309313	−	0.950960i	$-0.399901\pi$
$311$	10.9096	0.618626	0.309313	+	0.950960i	$0.399901\pi$
$312$	0	0
$313$	3.85974	0.218165	0.109083	−	0.994033i	$-0.465209\pi$
$313$	3.85974	0.218165	0.109083	+	0.994033i	$0.465209\pi$
$314$	0	0
$315$	−1.59254	−0.0897297
$316$	0	0
$317$	14.5866	0.819263	0.409631	−	0.912251i	$-0.365657\pi$
$317$	14.5866	0.819263	0.409631	+	0.912251i	$0.365657\pi$
$318$	0	0
$319$	−0.595743	−0.0333552
$320$	0	0
$321$	−24.6872	−1.37791
$322$	0	0
$323$	−34.0399	−1.89403
$324$	0	0
$325$	−6.42166	−0.356210
$326$	0	0
$327$	−13.1943	−0.729646
$328$	0	0
$329$	11.8861	0.655304
$330$	0	0
$331$	21.2006	1.16529	0.582644	−	0.812727i	$-0.302018\pi$
$331$	21.2006	1.16529	0.582644	+	0.812727i	$0.302018\pi$
$332$	0	0
$333$	3.71692	0.203686
$334$	0	0
$335$	−14.8390	−0.810739
$336$	0	0
$337$	26.1455	1.42424	0.712118	−	0.702060i	$-0.247736\pi$
$337$	26.1455	1.42424	0.712118	+	0.702060i	$0.247736\pi$
$338$	0	0
$339$	−7.53029	−0.408989
$340$	0	0
$341$	−1.19900	−0.0649297
$342$	0	0
$343$	18.8961	1.02029
$344$	0	0
$345$	10.6081	0.571121
$346$	0	0
$347$	−10.7148	−0.575198	−0.287599	−	0.957751i	$-0.592857\pi$
$347$	−10.7148	−0.575198	−0.287599	+	0.957751i	$0.592857\pi$
$348$	0	0
$349$	−35.3977	−1.89480	−0.947398	−	0.320058i	$-0.896298\pi$
$349$	−35.3977	−1.89480	−0.947398	+	0.320058i	$0.896298\pi$
$350$	0	0
$351$	14.3724	0.767140
$352$	0	0
$353$	−1.64093	−0.0873379	−0.0436690	−	0.999046i	$-0.513905\pi$
$353$	−1.64093	−0.0873379	−0.0436690	+	0.999046i	$0.513905\pi$
$354$	0	0
$355$	−10.2469	−0.543849
$356$	0	0
$357$	11.5234	0.609883
$358$	0	0
$359$	−13.5616	−0.715751	−0.357876	−	0.933769i	$-0.616499\pi$
$359$	−13.5616	−0.715751	−0.357876	+	0.933769i	$0.616499\pi$
$360$	0	0
$361$	41.8790	2.20416
$362$	0	0
$363$	16.8506	0.884425
$364$	0	0
$365$	0.173958	0.00910538
$366$	0	0
$367$	−35.8420	−1.87094	−0.935468	−	0.353411i	$-0.885022\pi$
$367$	−35.8420	−1.87094	−0.935468	+	0.353411i	$0.885022\pi$
$368$	0	0
$369$	2.09890	0.109264
$370$	0	0
$371$	10.6041	0.550539
$372$	0	0
$373$	12.8139	0.663477	0.331739	−	0.943371i	$-0.392365\pi$
$373$	12.8139	0.663477	0.331739	+	0.943371i	$0.392365\pi$
$374$	0	0
$375$	−18.4185	−0.951125
$376$	0	0
$377$	−4.04914	−0.208541
$378$	0	0
$379$	7.88843	0.405202	0.202601	−	0.979261i	$-0.435061\pi$
$379$	7.88843	0.405202	0.202601	+	0.979261i	$0.435061\pi$
$380$	0	0
$381$	−11.7861	−0.603818
$382$	0	0
$383$	−31.8337	−1.62663	−0.813313	−	0.581826i	$-0.802338\pi$
$383$	−31.8337	−1.62663	−0.813313	+	0.581826i	$0.802338\pi$
$384$	0	0
$385$	1.02281	0.0521275
$386$	0	0
$387$	−5.74345	−0.291956
$388$	0	0
$389$	22.0541	1.11819	0.559094	−	0.829104i	$-0.311149\pi$
$389$	22.0541	1.11819	0.559094	+	0.829104i	$0.311149\pi$
$390$	0	0
$391$	18.8207	0.951803
$392$	0	0
$393$	11.6475	0.587541
$394$	0	0
$395$	3.21497	0.161763
$396$	0	0
$397$	33.5713	1.68490	0.842448	−	0.538778i	$-0.181114\pi$
$397$	33.5713	1.68490	0.842448	+	0.538778i	$0.181114\pi$
$398$	0	0
$399$	−20.6091	−1.03175
$400$	0	0
$401$	−35.6400	−1.77978	−0.889888	−	0.456180i	$-0.849217\pi$
$401$	−35.6400	−1.77978	−0.889888	+	0.456180i	$0.849217\pi$
$402$	0	0
$403$	−8.14938	−0.405949
$404$	0	0
$405$	10.8975	0.541503
$406$	0	0
$407$	−2.38720	−0.118329
$408$	0	0
$409$	25.0805	1.24015	0.620075	−	0.784543i	$-0.287102\pi$
$409$	25.0805	1.24015	0.620075	+	0.784543i	$0.287102\pi$
$410$	0	0
$411$	−17.5224	−0.864315
$412$	0	0
$413$	11.1449	0.548404
$414$	0	0
$415$	12.2810	0.602849
$416$	0	0
$417$	−8.87579	−0.434649
$418$	0	0
$419$	12.2146	0.596721	0.298361	−	0.954453i	$-0.403560\pi$
$419$	12.2146	0.596721	0.298361	+	0.954453i	$0.403560\pi$
$420$	0	0
$421$	6.37032	0.310470	0.155235	−	0.987878i	$-0.450386\pi$
$421$	6.37032	0.310470	0.155235	+	0.987878i	$0.450386\pi$
$422$	0	0
$423$	4.12620	0.200623
$424$	0	0
$425$	−10.8642	−0.526993
$426$	0	0
$427$	15.7033	0.759936
$428$	0	0
$429$	−1.51860	−0.0733187
$430$	0	0
$431$	8.89967	0.428682	0.214341	−	0.976759i	$-0.431240\pi$
$431$	8.89967	0.428682	0.214341	+	0.976759i	$0.431240\pi$
$432$	0	0
$433$	0.367410	0.0176566	0.00882830	−	0.999961i	$-0.497190\pi$
$433$	0.367410	0.0176566	0.00882830	+	0.999961i	$0.497190\pi$
$434$	0	0
$435$	−3.86115	−0.185128
$436$	0	0
$437$	−33.6600	−1.61018
$438$	0	0
$439$	−4.28246	−0.204391	−0.102195	−	0.994764i	$-0.532587\pi$
$439$	−4.28246	−0.204391	−0.102195	+	0.994764i	$0.532587\pi$
$440$	0	0
$441$	2.42450	0.115452
$442$	0	0
$443$	28.7745	1.36712	0.683559	−	0.729895i	$-0.260431\pi$
$443$	28.7745	1.36712	0.683559	+	0.729895i	$0.260431\pi$
$444$	0	0
$445$	12.7678	0.605251
$446$	0	0
$447$	−3.35061	−0.158478
$448$	0	0
$449$	−20.3075	−0.958368	−0.479184	−	0.877715i	$-0.659067\pi$
$449$	−20.3075	−0.958368	−0.479184	+	0.877715i	$0.659067\pi$
$450$	0	0
$451$	−1.34802	−0.0634758
$452$	0	0
$453$	20.5715	0.966531
$454$	0	0
$455$	6.95186	0.325908
$456$	0	0
$457$	6.97075	0.326078	0.163039	−	0.986620i	$-0.447870\pi$
$457$	6.97075	0.326078	0.163039	+	0.986620i	$0.447870\pi$
$458$	0	0
$459$	24.3153	1.13494
$460$	0	0
$461$	−25.7314	−1.19843	−0.599216	−	0.800587i	$-0.704521\pi$
$461$	−25.7314	−1.19843	−0.599216	+	0.800587i	$0.704521\pi$
$462$	0	0
$463$	5.54653	0.257769	0.128885	−	0.991660i	$-0.458860\pi$
$463$	5.54653	0.257769	0.128885	+	0.991660i	$0.458860\pi$
$464$	0	0
$465$	−7.77102	−0.360373
$466$	0	0
$467$	10.4023	0.481361	0.240680	−	0.970604i	$-0.422629\pi$
$467$	10.4023	0.481361	0.240680	+	0.970604i	$0.422629\pi$
$468$	0	0
$469$	−15.9394	−0.736015
$470$	0	0
$471$	−20.3117	−0.935915
$472$	0	0
$473$	3.68874	0.169608
$474$	0	0
$475$	19.4303	0.891522
$476$	0	0
$477$	3.68116	0.168549
$478$	0	0
$479$	−20.1822	−0.922150	−0.461075	−	0.887361i	$-0.652536\pi$
$479$	−20.1822	−0.922150	−0.461075	+	0.887361i	$0.652536\pi$
$480$	0	0
$481$	−16.2253	−0.739810
$482$	0	0
$483$	11.3948	0.518482
$484$	0	0
$485$	14.5154	0.659112
$486$	0	0
$487$	30.0985	1.36389	0.681947	−	0.731402i	$-0.261134\pi$
$487$	30.0985	1.36389	0.681947	+	0.731402i	$0.261134\pi$
$488$	0	0
$489$	23.8904	1.08036
$490$	0	0
$491$	1.19053	0.0537278	0.0268639	−	0.999639i	$-0.491448\pi$
$491$	1.19053	0.0537278	0.0268639	+	0.999639i	$0.491448\pi$
$492$	0	0
$493$	−6.85038	−0.308526
$494$	0	0
$495$	0.355064	0.0159589
$496$	0	0
$497$	−11.0068	−0.493724
$498$	0	0
$499$	−4.18827	−0.187493	−0.0937465	−	0.995596i	$-0.529884\pi$
$499$	−4.18827	−0.187493	−0.0937465	+	0.995596i	$0.529884\pi$
$500$	0	0
$501$	−15.3875	−0.687462
$502$	0	0
$503$	−21.8231	−0.973043	−0.486521	−	0.873669i	$-0.661734\pi$
$503$	−21.8231	−0.973043	−0.486521	+	0.873669i	$0.661734\pi$
$504$	0	0
$505$	3.61116	0.160695
$506$	0	0
$507$	9.85673	0.437753
$508$	0	0
$509$	−7.68663	−0.340704	−0.170352	−	0.985383i	$-0.554490\pi$
$509$	−7.68663	−0.340704	−0.170352	+	0.985383i	$0.554490\pi$
$510$	0	0
$511$	0.186859	0.00826616
$512$	0	0
$513$	−43.4870	−1.92000
$514$	0	0
$515$	−3.44118	−0.151637
$516$	0	0
$517$	−2.65006	−0.116549
$518$	0	0
$519$	−25.1096	−1.10219
$520$	0	0
$521$	−13.5643	−0.594264	−0.297132	−	0.954836i	$-0.596030\pi$
$521$	−13.5643	−0.594264	−0.297132	+	0.954836i	$0.596030\pi$
$522$	0	0
$523$	18.5073	0.809267	0.404634	−	0.914479i	$-0.367399\pi$
$523$	18.5073	0.809267	0.404634	+	0.914479i	$0.367399\pi$
$524$	0	0
$525$	−6.57765	−0.287072
$526$	0	0
$527$	−13.7872	−0.600580
$528$	0	0
$529$	−4.38934	−0.190841
$530$	0	0
$531$	3.86888	0.167895
$532$	0	0
$533$	−9.16221	−0.396859
$534$	0	0
$535$	−25.1967	−1.08935
$536$	0	0
$537$	−29.9159	−1.29097
$538$	0	0
$539$	−1.55714	−0.0670708
$540$	0	0
$541$	−8.10164	−0.348317	−0.174158	−	0.984718i	$-0.555720\pi$
$541$	−8.10164	−0.348317	−0.174158	+	0.984718i	$0.555720\pi$
$542$	0	0
$543$	−19.2602	−0.826536
$544$	0	0
$545$	−13.4666	−0.576846
$546$	0	0
$547$	33.8777	1.44851	0.724254	−	0.689534i	$-0.242185\pi$
$547$	33.8777	1.44851	0.724254	+	0.689534i	$0.242185\pi$
$548$	0	0
$549$	5.45131	0.232656
$550$	0	0
$551$	12.2516	0.521937
$552$	0	0
$553$	3.45340	0.146853
$554$	0	0
$555$	−15.4720	−0.656750
$556$	0	0
$557$	19.3111	0.818238	0.409119	−	0.912481i	$-0.365836\pi$
$557$	19.3111	0.818238	0.409119	+	0.912481i	$0.365836\pi$
$558$	0	0
$559$	25.0716	1.06041
$560$	0	0
$561$	−2.56919	−0.108471
$562$	0	0
$563$	2.65294	0.111808	0.0559041	−	0.998436i	$-0.482196\pi$
$563$	2.65294	0.111808	0.0559041	+	0.998436i	$0.482196\pi$
$564$	0	0
$565$	−7.68571	−0.323340
$566$	0	0
$567$	11.7057	0.491594
$568$	0	0
$569$	−20.1304	−0.843911	−0.421955	−	0.906617i	$-0.638656\pi$
$569$	−20.1304	−0.843911	−0.421955	+	0.906617i	$0.638656\pi$
$570$	0	0
$571$	−7.79125	−0.326054	−0.163027	−	0.986622i	$-0.552126\pi$
$571$	−7.79125	−0.326054	−0.163027	+	0.986622i	$0.552126\pi$
$572$	0	0
$573$	11.4232	0.477212
$574$	0	0
$575$	−10.7430	−0.448014
$576$	0	0
$577$	−17.5881	−0.732202	−0.366101	−	0.930575i	$-0.619307\pi$
$577$	−17.5881	−0.732202	−0.366101	+	0.930575i	$0.619307\pi$
$578$	0	0
$579$	32.3806	1.34569
$580$	0	0
$581$	13.1917	0.547285
$582$	0	0
$583$	−2.36423	−0.0979164
$584$	0	0
$585$	2.41329	0.0997775
$586$	0	0
$587$	−15.3636	−0.634124	−0.317062	−	0.948405i	$-0.602696\pi$
$587$	−15.3636	−0.634124	−0.317062	+	0.948405i	$0.602696\pi$
$588$	0	0
$589$	24.6579	1.01601
$590$	0	0
$591$	2.65239	0.109105
$592$	0	0
$593$	15.9936	0.656778	0.328389	−	0.944543i	$-0.393494\pi$
$593$	15.9936	0.656778	0.328389	+	0.944543i	$0.393494\pi$
$594$	0	0
$595$	11.7612	0.482163
$596$	0	0
$597$	0.168940	0.00691423
$598$	0	0
$599$	−3.33697	−0.136345	−0.0681725	−	0.997674i	$-0.521717\pi$
$599$	−3.33697	−0.136345	−0.0681725	+	0.997674i	$0.521717\pi$
$600$	0	0
$601$	28.2448	1.15213	0.576065	−	0.817404i	$-0.304587\pi$
$601$	28.2448	1.15213	0.576065	+	0.817404i	$0.304587\pi$
$602$	0	0
$603$	−5.53328	−0.225333
$604$	0	0
$605$	17.1983	0.699212
$606$	0	0
$607$	41.0129	1.66466	0.832331	−	0.554279i	$-0.187006\pi$
$607$	41.0129	1.66466	0.832331	+	0.554279i	$0.187006\pi$
$608$	0	0
$609$	−4.14750	−0.168065
$610$	0	0
$611$	−18.0119	−0.728683
$612$	0	0
$613$	−17.3325	−0.700052	−0.350026	−	0.936740i	$-0.613827\pi$
$613$	−17.3325	−0.700052	−0.350026	+	0.936740i	$0.613827\pi$
$614$	0	0
$615$	−8.73683	−0.352303
$616$	0	0
$617$	25.3622	1.02104	0.510522	−	0.859865i	$-0.329452\pi$
$617$	25.3622	1.02104	0.510522	+	0.859865i	$0.329452\pi$
$618$	0	0
$619$	24.9558	1.00306	0.501529	−	0.865141i	$-0.332771\pi$
$619$	24.9558	1.00306	0.501529	+	0.865141i	$0.332771\pi$
$620$	0	0
$621$	24.0440	0.964852
$622$	0	0
$623$	13.7147	0.549466
$624$	0	0
$625$	−6.34729	−0.253892
$626$	0	0
$627$	4.59489	0.183502
$628$	0	0
$629$	−27.4501	−1.09451
$630$	0	0
$631$	32.5477	1.29570	0.647851	−	0.761767i	$-0.275668\pi$
$631$	32.5477	1.29570	0.647851	+	0.761767i	$0.275668\pi$
$632$	0	0
$633$	−14.0515	−0.558498
$634$	0	0
$635$	−12.0293	−0.477368
$636$	0	0
$637$	−10.5836	−0.419336
$638$	0	0
$639$	−3.82095	−0.151155
$640$	0	0
$641$	20.7365	0.819043	0.409521	−	0.912301i	$-0.365696\pi$
$641$	20.7365	0.819043	0.409521	+	0.912301i	$0.365696\pi$
$642$	0	0
$643$	25.4408	1.00329	0.501643	−	0.865075i	$-0.332729\pi$
$643$	25.4408	1.00329	0.501643	+	0.865075i	$0.332729\pi$
$644$	0	0
$645$	23.9076	0.941360
$646$	0	0
$647$	7.64013	0.300365	0.150182	−	0.988658i	$-0.452014\pi$
$647$	7.64013	0.300365	0.150182	+	0.988658i	$0.452014\pi$
$648$	0	0
$649$	−2.48479	−0.0975367
$650$	0	0
$651$	−8.34734	−0.327158
$652$	0	0
$653$	−10.6361	−0.416221	−0.208110	−	0.978105i	$-0.566731\pi$
$653$	−10.6361	−0.416221	−0.208110	+	0.978105i	$0.566731\pi$
$654$	0	0
$655$	11.8879	0.464500
$656$	0	0
$657$	0.0648670	0.00253070
$658$	0	0
$659$	29.2624	1.13990	0.569951	−	0.821679i	$-0.306962\pi$
$659$	29.2624	1.13990	0.569951	+	0.821679i	$0.306962\pi$
$660$	0	0
$661$	19.2710	0.749554	0.374777	−	0.927115i	$-0.377719\pi$
$661$	19.2710	0.749554	0.374777	+	0.927115i	$0.377719\pi$
$662$	0	0
$663$	−17.4622	−0.678176
$664$	0	0
$665$	−21.0345	−0.815682
$666$	0	0
$667$	−6.77393	−0.262288
$668$	0	0
$669$	−31.5878	−1.22125
$670$	0	0
$671$	−3.50111	−0.135159
$672$	0	0
$673$	−23.1072	−0.890718	−0.445359	−	0.895352i	$-0.646924\pi$
$673$	−23.1072	−0.890718	−0.445359	+	0.895352i	$0.646924\pi$
$674$	0	0
$675$	−13.8794	−0.534218
$676$	0	0
$677$	−10.2803	−0.395104	−0.197552	−	0.980292i	$-0.563299\pi$
$677$	−10.2803	−0.395104	−0.197552	+	0.980292i	$0.563299\pi$
$678$	0	0
$679$	15.5919	0.598363
$680$	0	0
$681$	−39.4185	−1.51052
$682$	0	0
$683$	−34.2414	−1.31021	−0.655104	−	0.755538i	$-0.727375\pi$
$683$	−34.2414	−1.31021	−0.655104	+	0.755538i	$0.727375\pi$
$684$	0	0
$685$	−17.8840	−0.683313
$686$	0	0
$687$	7.27415	0.277526
$688$	0	0
$689$	−16.0692	−0.612187
$690$	0	0
$691$	43.6787	1.66162	0.830809	−	0.556558i	$-0.187878\pi$
$691$	43.6787	1.66162	0.830809	+	0.556558i	$0.187878\pi$
$692$	0	0
$693$	0.381396	0.0144880
$694$	0	0
$695$	−9.05897	−0.343626
$696$	0	0
$697$	−15.5007	−0.587132
$698$	0	0
$699$	19.3775	0.732925
$700$	0	0
$701$	21.8473	0.825160	0.412580	−	0.910921i	$-0.364628\pi$
$701$	21.8473	0.825160	0.412580	+	0.910921i	$0.364628\pi$
$702$	0	0
$703$	49.0935	1.85160
$704$	0	0
$705$	−17.1757	−0.646872
$706$	0	0
$707$	3.87897	0.145884
$708$	0	0
$709$	−46.7526	−1.75583	−0.877916	−	0.478815i	$-0.841066\pi$
$709$	−46.7526	−1.75583	−0.877916	+	0.478815i	$0.841066\pi$
$710$	0	0
$711$	1.19883	0.0449595
$712$	0	0
$713$	−13.6334	−0.510573
$714$	0	0
$715$	−1.54994	−0.0579646
$716$	0	0
$717$	−7.27132	−0.271552
$718$	0	0
$719$	−13.0039	−0.484965	−0.242483	−	0.970156i	$-0.577962\pi$
$719$	−13.0039	−0.484965	−0.242483	+	0.970156i	$0.577962\pi$
$720$	0	0
$721$	−3.69639	−0.137661
$722$	0	0
$723$	−30.4363	−1.13194
$724$	0	0
$725$	3.91026	0.145223
$726$	0	0
$727$	37.6718	1.39717	0.698585	−	0.715527i	$-0.253813\pi$
$727$	37.6718	1.39717	0.698585	+	0.715527i	$0.253813\pi$
$728$	0	0
$729$	30.0167	1.11173
$730$	0	0
$731$	42.4164	1.56883
$732$	0	0
$733$	8.98625	0.331915	0.165957	−	0.986133i	$-0.446929\pi$
$733$	8.98625	0.331915	0.165957	+	0.986133i	$0.446929\pi$
$734$	0	0
$735$	−10.0922	−0.372256
$736$	0	0
$737$	3.55376	0.130904
$738$	0	0
$739$	21.4973	0.790790	0.395395	−	0.918511i	$-0.370608\pi$
$739$	21.4973	0.790790	0.395395	+	0.918511i	$0.370608\pi$
$740$	0	0
$741$	31.2305	1.14728
$742$	0	0
$743$	−13.4431	−0.493181	−0.246590	−	0.969120i	$-0.579310\pi$
$743$	−13.4431	−0.493181	−0.246590	+	0.969120i	$0.579310\pi$
$744$	0	0
$745$	−3.41976	−0.125290
$746$	0	0
$747$	4.57943	0.167553
$748$	0	0
$749$	−27.0654	−0.988947
$750$	0	0
$751$	29.4215	1.07360	0.536802	−	0.843708i	$-0.319632\pi$
$751$	29.4215	1.07360	0.536802	+	0.843708i	$0.319632\pi$
$752$	0	0
$753$	24.0655	0.876996
$754$	0	0
$755$	20.9960	0.764124
$756$	0	0
$757$	24.4342	0.888077	0.444039	−	0.896008i	$-0.353545\pi$
$757$	24.4342	0.888077	0.444039	+	0.896008i	$0.353545\pi$
$758$	0	0
$759$	−2.54052	−0.0922149
$760$	0	0
$761$	−17.5793	−0.637248	−0.318624	−	0.947881i	$-0.603221\pi$
$761$	−17.5793	−0.637248	−0.318624	+	0.947881i	$0.603221\pi$
$762$	0	0
$763$	−14.4653	−0.523679
$764$	0	0
$765$	4.08284	0.147615
$766$	0	0
$767$	−16.8886	−0.609813
$768$	0	0
$769$	8.70751	0.314001	0.157000	−	0.987599i	$-0.449818\pi$
$769$	8.70751	0.314001	0.157000	+	0.987599i	$0.449818\pi$
$770$	0	0
$771$	−9.34300	−0.336480
$772$	0	0
$773$	14.8996	0.535900	0.267950	−	0.963433i	$-0.413654\pi$
$773$	14.8996	0.535900	0.267950	+	0.963433i	$0.413654\pi$
$774$	0	0
$775$	7.86985	0.282694
$776$	0	0
$777$	−16.6194	−0.596219
$778$	0	0
$779$	27.7224	0.993259
$780$	0	0
$781$	2.45401	0.0878116
$782$	0	0
$783$	−8.75157	−0.312755
$784$	0	0
$785$	−20.7309	−0.739919
$786$	0	0
$787$	26.9762	0.961597	0.480799	−	0.876831i	$-0.340347\pi$
$787$	26.9762	0.961597	0.480799	+	0.876831i	$0.340347\pi$
$788$	0	0
$789$	−33.6002	−1.19620
$790$	0	0
$791$	−8.25570	−0.293539
$792$	0	0
$793$	−23.7963	−0.845032
$794$	0	0
$795$	−15.3231	−0.543455
$796$	0	0
$797$	17.2397	0.610661	0.305330	−	0.952247i	$-0.401233\pi$
$797$	17.2397	0.610661	0.305330	+	0.952247i	$0.401233\pi$
$798$	0	0
$799$	−30.4727	−1.07805
$800$	0	0
$801$	4.76096	0.168220
$802$	0	0
$803$	−0.0416609	−0.00147018
$804$	0	0
$805$	11.6300	0.409903
$806$	0	0
$807$	−33.5122	−1.17969
$808$	0	0
$809$	28.8188	1.01322	0.506608	−	0.862177i	$-0.330899\pi$
$809$	28.8188	1.01322	0.506608	+	0.862177i	$0.330899\pi$
$810$	0	0
$811$	17.6640	0.620266	0.310133	−	0.950693i	$-0.399626\pi$
$811$	17.6640	0.620266	0.310133	+	0.950693i	$0.399626\pi$
$812$	0	0
$813$	−36.6896	−1.28676
$814$	0	0
$815$	24.3834	0.854114
$816$	0	0
$817$	−75.8600	−2.65400
$818$	0	0
$819$	2.59227	0.0905812
$820$	0	0
$821$	1.16570	0.0406833	0.0203417	−	0.999793i	$-0.493525\pi$
$821$	1.16570	0.0406833	0.0203417	+	0.999793i	$0.493525\pi$
$822$	0	0
$823$	7.23103	0.252058	0.126029	−	0.992027i	$-0.459777\pi$
$823$	7.23103	0.252058	0.126029	+	0.992027i	$0.459777\pi$
$824$	0	0
$825$	1.46651	0.0510574
$826$	0	0
$827$	−41.6251	−1.44745	−0.723724	−	0.690090i	$-0.757571\pi$
$827$	−41.6251	−1.44745	−0.723724	+	0.690090i	$0.757571\pi$
$828$	0	0
$829$	−43.9380	−1.52603	−0.763015	−	0.646381i	$-0.776282\pi$
$829$	−43.9380	−1.52603	−0.763015	+	0.646381i	$0.776282\pi$
$830$	0	0
$831$	29.8839	1.03666
$832$	0	0
$833$	−17.9054	−0.620384
$834$	0	0
$835$	−15.7051	−0.543496
$836$	0	0
$837$	−17.6136	−0.608814
$838$	0	0
$839$	−14.2000	−0.490239	−0.245120	−	0.969493i	$-0.578827\pi$
$839$	−14.2000	−0.490239	−0.245120	+	0.969493i	$0.578827\pi$
$840$	0	0
$841$	−26.5344	−0.914980
$842$	0	0
$843$	−43.3044	−1.49148
$844$	0	0
$845$	10.0602	0.346080
$846$	0	0
$847$	18.4738	0.634767
$848$	0	0
$849$	42.4429	1.45663
$850$	0	0
$851$	−27.1438	−0.930478
$852$	0	0
$853$	43.5774	1.49206	0.746031	−	0.665911i	$-0.231957\pi$
$853$	43.5774	1.49206	0.746031	+	0.665911i	$0.231957\pi$
$854$	0	0
$855$	−7.30199	−0.249723
$856$	0	0
$857$	−37.1129	−1.26775	−0.633877	−	0.773434i	$-0.718537\pi$
$857$	−37.1129	−1.26775	−0.633877	+	0.773434i	$0.718537\pi$
$858$	0	0
$859$	6.48050	0.221112	0.110556	−	0.993870i	$-0.464737\pi$
$859$	6.48050	0.221112	0.110556	+	0.993870i	$0.464737\pi$
$860$	0	0
$861$	−9.38477	−0.319832
$862$	0	0
$863$	−15.7684	−0.536762	−0.268381	−	0.963313i	$-0.586489\pi$
$863$	−15.7684	−0.536762	−0.268381	+	0.963313i	$0.586489\pi$
$864$	0	0
$865$	−25.6279	−0.871373
$866$	0	0
$867$	−3.15569	−0.107173
$868$	0	0
$869$	−0.769948	−0.0261187
$870$	0	0
$871$	24.1541	0.818432
$872$	0	0
$873$	5.41264	0.183190
$874$	0	0
$875$	−20.1927	−0.682639
$876$	0	0
$877$	−4.20057	−0.141843	−0.0709216	−	0.997482i	$-0.522594\pi$
$877$	−4.20057	−0.141843	−0.0709216	+	0.997482i	$0.522594\pi$
$878$	0	0
$879$	50.7975	1.71336
$880$	0	0
$881$	7.92618	0.267040	0.133520	−	0.991046i	$-0.457372\pi$
$881$	7.92618	0.267040	0.133520	+	0.991046i	$0.457372\pi$
$882$	0	0
$883$	−30.0101	−1.00992	−0.504960	−	0.863143i	$-0.668493\pi$
$883$	−30.0101	−1.00992	−0.504960	+	0.863143i	$0.668493\pi$
$884$	0	0
$885$	−16.1045	−0.541348
$886$	0	0
$887$	15.7349	0.528326	0.264163	−	0.964478i	$-0.414904\pi$
$887$	15.7349	0.528326	0.264163	+	0.964478i	$0.414904\pi$
$888$	0	0
$889$	−12.9214	−0.433370
$890$	0	0
$891$	−2.60983	−0.0874327
$892$	0	0
$893$	54.4992	1.82375
$894$	0	0
$895$	−30.5334	−1.02062
$896$	0	0
$897$	−17.2674	−0.576540
$898$	0	0
$899$	4.96229	0.165502
$900$	0	0
$901$	−27.1860	−0.905697
$902$	0	0
$903$	25.6806	0.854597
$904$	0	0
$905$	−19.6577	−0.653445
$906$	0	0
$907$	−2.52921	−0.0839809	−0.0419904	−	0.999118i	$-0.513370\pi$
$907$	−2.52921	−0.0839809	−0.0419904	+	0.999118i	$0.513370\pi$
$908$	0	0
$909$	1.34656	0.0446626
$910$	0	0
$911$	39.0256	1.29297	0.646487	−	0.762925i	$-0.276237\pi$
$911$	39.0256	1.29297	0.646487	+	0.762925i	$0.276237\pi$
$912$	0	0
$913$	−2.94115	−0.0973378
$914$	0	0
$915$	−22.6915	−0.750158
$916$	0	0
$917$	12.7696	0.421688
$918$	0	0
$919$	−50.3869	−1.66211	−0.831056	−	0.556189i	$-0.812263\pi$
$919$	−50.3869	−1.66211	−0.831056	+	0.556189i	$0.812263\pi$
$920$	0	0
$921$	−44.5895	−1.46927
$922$	0	0
$923$	16.6794	0.549010
$924$	0	0
$925$	15.6688	0.515186
$926$	0	0
$927$	−1.28318	−0.0421451
$928$	0	0
$929$	20.9922	0.688731	0.344366	−	0.938836i	$-0.388094\pi$
$929$	20.9922	0.688731	0.344366	+	0.938836i	$0.388094\pi$
$930$	0	0
$931$	32.0231	1.04951
$932$	0	0
$933$	−16.9336	−0.554383
$934$	0	0
$935$	−2.62221	−0.0857554
$936$	0	0
$937$	−38.0216	−1.24211	−0.621055	−	0.783767i	$-0.713296\pi$
$937$	−38.0216	−1.24211	−0.621055	+	0.783767i	$0.713296\pi$
$938$	0	0
$939$	−5.99101	−0.195509
$940$	0	0
$941$	−21.4337	−0.698719	−0.349360	−	0.936989i	$-0.613601\pi$
$941$	−21.4337	−0.698719	−0.349360	+	0.936989i	$0.613601\pi$
$942$	0	0
$943$	−15.3278	−0.499140
$944$	0	0
$945$	15.0253	0.488774
$946$	0	0
$947$	29.9768	0.974114	0.487057	−	0.873370i	$-0.338070\pi$
$947$	29.9768	0.974114	0.487057	+	0.873370i	$0.338070\pi$
$948$	0	0
$949$	−0.283161	−0.00919178
$950$	0	0
$951$	−22.6410	−0.734184
$952$	0	0
$953$	−56.3546	−1.82550	−0.912752	−	0.408514i	$-0.866047\pi$
$953$	−56.3546	−1.82550	−0.912752	+	0.408514i	$0.866047\pi$
$954$	0	0
$955$	11.6590	0.377276
$956$	0	0
$957$	0.924701	0.0298913
$958$	0	0
$959$	−19.2103	−0.620334
$960$	0	0
$961$	−21.0128	−0.677832
$962$	0	0
$963$	−9.39558	−0.302768
$964$	0	0
$965$	33.0489	1.06388
$966$	0	0
$967$	−47.9754	−1.54278	−0.771392	−	0.636360i	$-0.780439\pi$
$967$	−47.9754	−1.54278	−0.771392	+	0.636360i	$0.780439\pi$
$968$	0	0
$969$	52.8360	1.69734
$970$	0	0
$971$	−40.1705	−1.28913	−0.644567	−	0.764548i	$-0.722962\pi$
$971$	−40.1705	−1.28913	−0.644567	+	0.764548i	$0.722962\pi$
$972$	0	0
$973$	−9.73080	−0.311955
$974$	0	0
$975$	9.96758	0.319218
$976$	0	0
$977$	−46.0598	−1.47358	−0.736791	−	0.676120i	$-0.763660\pi$
$977$	−46.0598	−1.47358	−0.736791	+	0.676120i	$0.763660\pi$
$978$	0	0
$979$	−3.05774	−0.0977256
$980$	0	0
$981$	−5.02154	−0.160326
$982$	0	0
$983$	−45.6280	−1.45531	−0.727654	−	0.685944i	$-0.759389\pi$
$983$	−45.6280	−1.45531	−0.727654	+	0.685944i	$0.759389\pi$
$984$	0	0
$985$	2.70713	0.0862564
$986$	0	0
$987$	−18.4494	−0.587252
$988$	0	0
$989$	41.9430	1.33371
$990$	0	0
$991$	35.7997	1.13722	0.568608	−	0.822609i	$-0.307482\pi$
$991$	35.7997	1.13722	0.568608	+	0.822609i	$0.307482\pi$
$992$	0	0
$993$	−32.9071	−1.04428
$994$	0	0
$995$	0.172426	0.00546628
$996$	0	0
$997$	−12.4798	−0.395238	−0.197619	−	0.980279i	$-0.563321\pi$
$997$	−12.4798	−0.395238	−0.197619	+	0.980279i	$0.563321\pi$
$998$	0	0
$999$	−35.0684	−1.10951

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6044.2.a.b.1.17	✓	63

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6044.2.a.b.1.17	✓	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)

\(f(q)\)	\(=\)	\(q-1.55218 q^{3} -1.58422 q^{5} -1.70170 q^{7} -0.590736 q^{9} +O(q^{10})\)	\(q-1.55218 q^{3} -1.58422 q^{5} -1.70170 q^{7} -0.590736 q^{9} +0.379401 q^{11} +2.57871 q^{13} +2.45899 q^{15} +4.36269 q^{17} -7.80250 q^{19} +2.64135 q^{21} +4.31401 q^{23} -2.49026 q^{25} +5.57347 q^{27} -1.57022 q^{29} -3.16025 q^{31} -0.588899 q^{33} +2.69587 q^{35} -6.29202 q^{37} -4.00262 q^{39} -3.55302 q^{41} +9.72252 q^{43} +0.935853 q^{45} -6.98485 q^{47} -4.10420 q^{49} -6.77168 q^{51} -6.23147 q^{53} -0.601053 q^{55} +12.1109 q^{57} -6.54925 q^{59} -9.22799 q^{61} +1.00526 q^{63} -4.08523 q^{65} +9.36675 q^{67} -6.69612 q^{69} +6.46812 q^{71} -0.109807 q^{73} +3.86533 q^{75} -0.645629 q^{77} -2.02938 q^{79} -6.87882 q^{81} -7.75208 q^{83} -6.91144 q^{85} +2.43726 q^{87} -8.05937 q^{89} -4.38820 q^{91} +4.90528 q^{93} +12.3608 q^{95} -9.16254 q^{97} -0.224126 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9}+O(q^{10}) \) 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

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