LMFDB - Embedded newform 6044.2.a.b.1.12

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			1.12
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)

$f(q)$	$=$	$q-2.24731 q^{3} +0.981673 q^{5} -2.98823 q^{7} +2.05038 q^{9} +O(q^{10})$	$q-2.24731 q^{3} +0.981673 q^{5} -2.98823 q^{7} +2.05038 q^{9} +1.92553 q^{11} -5.52311 q^{13} -2.20612 q^{15} -3.90274 q^{17} -7.30222 q^{19} +6.71546 q^{21} -3.26029 q^{23} -4.03632 q^{25} +2.13408 q^{27} -3.67251 q^{29} +6.84419 q^{31} -4.32725 q^{33} -2.93346 q^{35} -3.50923 q^{37} +12.4121 q^{39} -1.37222 q^{41} -2.64355 q^{43} +2.01281 q^{45} -5.44305 q^{47} +1.92950 q^{49} +8.77064 q^{51} -2.91317 q^{53} +1.89024 q^{55} +16.4103 q^{57} -8.86606 q^{59} +0.689943 q^{61} -6.12701 q^{63} -5.42189 q^{65} +7.92894 q^{67} +7.32688 q^{69} +13.7625 q^{71} -3.04774 q^{73} +9.07084 q^{75} -5.75391 q^{77} -7.24721 q^{79} -10.9471 q^{81} +12.9978 q^{83} -3.83121 q^{85} +8.25326 q^{87} +2.99393 q^{89} +16.5043 q^{91} -15.3810 q^{93} -7.16840 q^{95} -17.6556 q^{97} +3.94807 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.24731	−1.29748	−0.648741	−	0.761009i	$-0.724704\pi$
$3$	−2.24731	−1.29748	−0.648741	+	0.761009i	$0.724704\pi$
$4$	0	0
$5$	0.981673	0.439017	0.219509	−	0.975611i	$-0.429555\pi$
$5$	0.981673	0.439017	0.219509	+	0.975611i	$0.429555\pi$
$6$	0	0
$7$	−2.98823	−1.12944	−0.564722	−	0.825281i	$-0.691016\pi$
$7$	−2.98823	−1.12944	−0.564722	+	0.825281i	$0.691016\pi$
$8$	0	0
$9$	2.05038	0.683461
$10$	0	0
$11$	1.92553	0.580568	0.290284	−	0.956941i	$-0.406250\pi$
$11$	1.92553	0.580568	0.290284	+	0.956941i	$0.406250\pi$
$12$	0	0
$13$	−5.52311	−1.53184	−0.765918	−	0.642938i	$-0.777715\pi$
$13$	−5.52311	−1.53184	−0.765918	+	0.642938i	$0.777715\pi$
$14$	0	0
$15$	−2.20612	−0.569617
$16$	0	0
$17$	−3.90274	−0.946553	−0.473276	−	0.880914i	$-0.656929\pi$
$17$	−3.90274	−0.946553	−0.473276	+	0.880914i	$0.656929\pi$
$18$	0	0
$19$	−7.30222	−1.67525	−0.837623	−	0.546249i	$-0.816055\pi$
$19$	−7.30222	−1.67525	−0.837623	+	0.546249i	$0.816055\pi$
$20$	0	0
$21$	6.71546	1.46543
$22$	0	0
$23$	−3.26029	−0.679818	−0.339909	−	0.940458i	$-0.610396\pi$
$23$	−3.26029	−0.679818	−0.339909	+	0.940458i	$0.610396\pi$
$24$	0	0
$25$	−4.03632	−0.807264
$26$	0	0
$27$	2.13408	0.410703
$28$	0	0
$29$	−3.67251	−0.681968	−0.340984	−	0.940069i	$-0.610760\pi$
$29$	−3.67251	−0.681968	−0.340984	+	0.940069i	$0.610760\pi$
$30$	0	0
$31$	6.84419	1.22925	0.614626	−	0.788818i	$-0.289307\pi$
$31$	6.84419	1.22925	0.614626	+	0.788818i	$0.289307\pi$
$32$	0	0
$33$	−4.32725	−0.753277
$34$	0	0
$35$	−2.93346	−0.495845
$36$	0	0
$37$	−3.50923	−0.576914	−0.288457	−	0.957493i	$-0.593142\pi$
$37$	−3.50923	−0.576914	−0.288457	+	0.957493i	$0.593142\pi$
$38$	0	0
$39$	12.4121	1.98753
$40$	0	0
$41$	−1.37222	−0.214305	−0.107152	−	0.994243i	$-0.534173\pi$
$41$	−1.37222	−0.214305	−0.107152	+	0.994243i	$0.534173\pi$
$42$	0	0
$43$	−2.64355	−0.403138	−0.201569	−	0.979474i	$-0.564604\pi$
$43$	−2.64355	−0.403138	−0.201569	+	0.979474i	$0.564604\pi$
$44$	0	0
$45$	2.01281	0.300051
$46$	0	0
$47$	−5.44305	−0.793950	−0.396975	−	0.917829i	$-0.629940\pi$
$47$	−5.44305	−0.793950	−0.396975	+	0.917829i	$0.629940\pi$
$48$	0	0
$49$	1.92950	0.275642
$50$	0	0
$51$	8.77064	1.22814
$52$	0	0
$53$	−2.91317	−0.400155	−0.200077	−	0.979780i	$-0.564119\pi$
$53$	−2.91317	−0.400155	−0.200077	+	0.979780i	$0.564119\pi$
$54$	0	0
$55$	1.89024	0.254879
$56$	0	0
$57$	16.4103	2.17360
$58$	0	0
$59$	−8.86606	−1.15426	−0.577131	−	0.816652i	$-0.695828\pi$
$59$	−8.86606	−1.15426	−0.577131	+	0.816652i	$0.695828\pi$
$60$	0	0
$61$	0.689943	0.0883382	0.0441691	−	0.999024i	$-0.485936\pi$
$61$	0.689943	0.0883382	0.0441691	+	0.999024i	$0.485936\pi$
$62$	0	0
$63$	−6.12701	−0.771931
$64$	0	0
$65$	−5.42189	−0.672503
$66$	0	0
$67$	7.92894	0.968674	0.484337	−	0.874882i	$-0.339061\pi$
$67$	7.92894	0.968674	0.484337	+	0.874882i	$0.339061\pi$
$68$	0	0
$69$	7.32688	0.882053
$70$	0	0
$71$	13.7625	1.63331	0.816655	−	0.577126i	$-0.195826\pi$
$71$	13.7625	1.63331	0.816655	+	0.577126i	$0.195826\pi$
$72$	0	0
$73$	−3.04774	−0.356711	−0.178355	−	0.983966i	$-0.557078\pi$
$73$	−3.04774	−0.356711	−0.178355	+	0.983966i	$0.557078\pi$
$74$	0	0
$75$	9.07084	1.04741
$76$	0	0
$77$	−5.75391	−0.655719
$78$	0	0
$79$	−7.24721	−0.815375	−0.407688	−	0.913121i	$-0.633665\pi$
$79$	−7.24721	−0.815375	−0.407688	+	0.913121i	$0.633665\pi$
$80$	0	0
$81$	−10.9471	−1.21634
$82$	0	0
$83$	12.9978	1.42670	0.713348	−	0.700810i	$-0.247178\pi$
$83$	12.9978	1.42670	0.713348	+	0.700810i	$0.247178\pi$
$84$	0	0
$85$	−3.83121	−0.415553
$86$	0	0
$87$	8.25326	0.884842
$88$	0	0
$89$	2.99393	0.317356	0.158678	−	0.987330i	$-0.449277\pi$
$89$	2.99393	0.317356	0.158678	+	0.987330i	$0.449277\pi$
$90$	0	0
$91$	16.5043	1.73012
$92$	0	0
$93$	−15.3810	−1.59493
$94$	0	0
$95$	−7.16840	−0.735462
$96$	0	0
$97$	−17.6556	−1.79265	−0.896327	−	0.443393i	$-0.853775\pi$
$97$	−17.6556	−1.79265	−0.896327	+	0.443393i	$0.853775\pi$
$98$	0	0
$99$	3.94807	0.396796
$100$	0	0
$101$	18.2597	1.81690	0.908452	−	0.417990i	$-0.137265\pi$
$101$	18.2597	1.81690	0.908452	+	0.417990i	$0.137265\pi$
$102$	0	0
$103$	−15.6948	−1.54646	−0.773230	−	0.634126i	$-0.781360\pi$
$103$	−15.6948	−1.54646	−0.773230	+	0.634126i	$0.781360\pi$
$104$	0	0
$105$	6.59238	0.643351
$106$	0	0
$107$	−8.89890	−0.860289	−0.430144	−	0.902760i	$-0.641537\pi$
$107$	−8.89890	−0.860289	−0.430144	+	0.902760i	$0.641537\pi$
$108$	0	0
$109$	−0.606794	−0.0581203	−0.0290602	−	0.999578i	$-0.509251\pi$
$109$	−0.606794	−0.0581203	−0.0290602	+	0.999578i	$0.509251\pi$
$110$	0	0
$111$	7.88632	0.748536
$112$	0	0
$113$	3.39904	0.319755	0.159877	−	0.987137i	$-0.448890\pi$
$113$	3.39904	0.319755	0.159877	+	0.987137i	$0.448890\pi$
$114$	0	0
$115$	−3.20054	−0.298452
$116$	0	0
$117$	−11.3245	−1.04695
$118$	0	0
$119$	11.6623	1.06908
$120$	0	0
$121$	−7.29235	−0.662941
$122$	0	0
$123$	3.08380	0.278057
$124$	0	0
$125$	−8.87071	−0.793420
$126$	0	0
$127$	6.28440	0.557650	0.278825	−	0.960342i	$-0.410055\pi$
$127$	6.28440	0.557650	0.278825	+	0.960342i	$0.410055\pi$
$128$	0	0
$129$	5.94087	0.523065
$130$	0	0
$131$	10.5632	0.922913	0.461456	−	0.887163i	$-0.347327\pi$
$131$	10.5632	0.922913	0.461456	+	0.887163i	$0.347327\pi$
$132$	0	0
$133$	21.8207	1.89209
$134$	0	0
$135$	2.09497	0.180306
$136$	0	0
$137$	−2.00397	−0.171211	−0.0856054	−	0.996329i	$-0.527282\pi$
$137$	−2.00397	−0.171211	−0.0856054	+	0.996329i	$0.527282\pi$
$138$	0	0
$139$	2.40716	0.204173	0.102087	−	0.994776i	$-0.467448\pi$
$139$	2.40716	0.204173	0.102087	+	0.994776i	$0.467448\pi$
$140$	0	0
$141$	12.2322	1.03014
$142$	0	0
$143$	−10.6349	−0.889335
$144$	0	0
$145$	−3.60520	−0.299396
$146$	0	0
$147$	−4.33617	−0.357641
$148$	0	0
$149$	−16.3613	−1.34037	−0.670184	−	0.742195i	$-0.733785\pi$
$149$	−16.3613	−1.34037	−0.670184	+	0.742195i	$0.733785\pi$
$150$	0	0
$151$	−4.43537	−0.360945	−0.180473	−	0.983580i	$-0.557763\pi$
$151$	−4.43537	−0.360945	−0.180473	+	0.983580i	$0.557763\pi$
$152$	0	0
$153$	−8.00211	−0.646932
$154$	0	0
$155$	6.71876	0.539663
$156$	0	0
$157$	−15.8723	−1.26674	−0.633372	−	0.773848i	$-0.718330\pi$
$157$	−15.8723	−1.26674	−0.633372	+	0.773848i	$0.718330\pi$
$158$	0	0
$159$	6.54678	0.519194
$160$	0	0
$161$	9.74250	0.767816
$162$	0	0
$163$	−15.4264	−1.20829	−0.604145	−	0.796875i	$-0.706485\pi$
$163$	−15.4264	−1.20829	−0.604145	+	0.796875i	$0.706485\pi$
$164$	0	0
$165$	−4.24794	−0.330702
$166$	0	0
$167$	12.7567	0.987145	0.493572	−	0.869705i	$-0.335691\pi$
$167$	12.7567	0.987145	0.493572	+	0.869705i	$0.335691\pi$
$168$	0	0
$169$	17.5048	1.34652
$170$	0	0
$171$	−14.9724	−1.14497
$172$	0	0
$173$	−18.3692	−1.39658	−0.698292	−	0.715813i	$-0.746056\pi$
$173$	−18.3692	−1.39658	−0.698292	+	0.715813i	$0.746056\pi$
$174$	0	0
$175$	12.0614	0.911759
$176$	0	0
$177$	19.9247	1.49764
$178$	0	0
$179$	−3.96362	−0.296255	−0.148127	−	0.988968i	$-0.547325\pi$
$179$	−3.96362	−0.296255	−0.148127	+	0.988968i	$0.547325\pi$
$180$	0	0
$181$	−11.2178	−0.833816	−0.416908	−	0.908949i	$-0.636886\pi$
$181$	−11.2178	−0.833816	−0.416908	+	0.908949i	$0.636886\pi$
$182$	0	0
$183$	−1.55051	−0.114617
$184$	0	0
$185$	−3.44492	−0.253275
$186$	0	0
$187$	−7.51482	−0.549538
$188$	0	0
$189$	−6.37711	−0.463866
$190$	0	0
$191$	17.8992	1.29514	0.647569	−	0.762007i	$-0.275786\pi$
$191$	17.8992	1.29514	0.647569	+	0.762007i	$0.275786\pi$
$192$	0	0
$193$	4.23691	0.304979	0.152490	−	0.988305i	$-0.451271\pi$
$193$	4.23691	0.304979	0.152490	+	0.988305i	$0.451271\pi$
$194$	0	0
$195$	12.1846	0.872561
$196$	0	0
$197$	10.9499	0.780149	0.390074	−	0.920783i	$-0.372449\pi$
$197$	10.9499	0.780149	0.390074	+	0.920783i	$0.372449\pi$
$198$	0	0
$199$	−3.09185	−0.219175	−0.109588	−	0.993977i	$-0.534953\pi$
$199$	−3.09185	−0.219175	−0.109588	+	0.993977i	$0.534953\pi$
$200$	0	0
$201$	−17.8187	−1.25684
$202$	0	0
$203$	10.9743	0.770244
$204$	0	0
$205$	−1.34707	−0.0940836
$206$	0	0
$207$	−6.68486	−0.464630
$208$	0	0
$209$	−14.0606	−0.972594
$210$	0	0
$211$	17.2972	1.19079	0.595393	−	0.803434i	$-0.296996\pi$
$211$	17.2972	1.19079	0.595393	+	0.803434i	$0.296996\pi$
$212$	0	0
$213$	−30.9286	−2.11919
$214$	0	0
$215$	−2.59510	−0.176985
$216$	0	0
$217$	−20.4520	−1.38837
$218$	0	0
$219$	6.84920	0.462826
$220$	0	0
$221$	21.5553	1.44996
$222$	0	0
$223$	−20.2842	−1.35833	−0.679164	−	0.733987i	$-0.737658\pi$
$223$	−20.2842	−1.35833	−0.679164	+	0.733987i	$0.737658\pi$
$224$	0	0
$225$	−8.27600	−0.551734
$226$	0	0
$227$	18.4026	1.22142	0.610711	−	0.791854i	$-0.290884\pi$
$227$	18.4026	1.22142	0.610711	+	0.791854i	$0.290884\pi$
$228$	0	0
$229$	−8.31190	−0.549266	−0.274633	−	0.961549i	$-0.588556\pi$
$229$	−8.31190	−0.549266	−0.274633	+	0.961549i	$0.588556\pi$
$230$	0	0
$231$	12.9308	0.850784
$232$	0	0
$233$	29.4530	1.92953	0.964766	−	0.263111i	$-0.0847484\pi$
$233$	29.4530	1.92953	0.964766	+	0.263111i	$0.0847484\pi$
$234$	0	0
$235$	−5.34329	−0.348558
$236$	0	0
$237$	16.2867	1.05794
$238$	0	0
$239$	−2.70771	−0.175147	−0.0875736	−	0.996158i	$-0.527911\pi$
$239$	−2.70771	−0.175147	−0.0875736	+	0.996158i	$0.527911\pi$
$240$	0	0
$241$	19.8780	1.28046	0.640229	−	0.768185i	$-0.278840\pi$
$241$	19.8780	1.28046	0.640229	+	0.768185i	$0.278840\pi$
$242$	0	0
$243$	18.1992	1.16748
$244$	0	0
$245$	1.89413	0.121012
$246$	0	0
$247$	40.3310	2.56620
$248$	0	0
$249$	−29.2101	−1.85111
$250$	0	0
$251$	19.1319	1.20759	0.603796	−	0.797139i	$-0.293654\pi$
$251$	19.1319	1.20759	0.603796	+	0.797139i	$0.293654\pi$
$252$	0	0
$253$	−6.27778	−0.394681
$254$	0	0
$255$	8.60990	0.539173
$256$	0	0
$257$	−3.29022	−0.205238	−0.102619	−	0.994721i	$-0.532722\pi$
$257$	−3.29022	−0.205238	−0.102619	+	0.994721i	$0.532722\pi$
$258$	0	0
$259$	10.4864	0.651592
$260$	0	0
$261$	−7.53006	−0.466099
$262$	0	0
$263$	15.8913	0.979897	0.489948	−	0.871751i	$-0.337016\pi$
$263$	15.8913	0.979897	0.489948	+	0.871751i	$0.337016\pi$
$264$	0	0
$265$	−2.85978	−0.175675
$266$	0	0
$267$	−6.72828	−0.411764
$268$	0	0
$269$	15.1712	0.925003	0.462502	−	0.886618i	$-0.346952\pi$
$269$	15.1712	0.925003	0.462502	+	0.886618i	$0.346952\pi$
$270$	0	0
$271$	−28.2323	−1.71499	−0.857496	−	0.514491i	$-0.827981\pi$
$271$	−28.2323	−1.71499	−0.857496	+	0.514491i	$0.827981\pi$
$272$	0	0
$273$	−37.0902	−2.24480
$274$	0	0
$275$	−7.77204	−0.468671
$276$	0	0
$277$	−10.9391	−0.657268	−0.328634	−	0.944457i	$-0.606588\pi$
$277$	−10.9391	−0.657268	−0.328634	+	0.944457i	$0.606588\pi$
$278$	0	0
$279$	14.0332	0.840147
$280$	0	0
$281$	−17.1482	−1.02298	−0.511488	−	0.859291i	$-0.670905\pi$
$281$	−17.1482	−1.02298	−0.511488	+	0.859291i	$0.670905\pi$
$282$	0	0
$283$	15.0660	0.895580	0.447790	−	0.894139i	$-0.352211\pi$
$283$	15.0660	0.895580	0.447790	+	0.894139i	$0.352211\pi$
$284$	0	0
$285$	16.1096	0.954249
$286$	0	0
$287$	4.10051	0.242045
$288$	0	0
$289$	−1.76865	−0.104038
$290$	0	0
$291$	39.6775	2.32594
$292$	0	0
$293$	22.4280	1.31026	0.655129	−	0.755517i	$-0.272614\pi$
$293$	22.4280	1.31026	0.655129	+	0.755517i	$0.272614\pi$
$294$	0	0
$295$	−8.70357	−0.506741
$296$	0	0
$297$	4.10922	0.238441
$298$	0	0
$299$	18.0070	1.04137
$300$	0	0
$301$	7.89954	0.455322
$302$	0	0
$303$	−41.0350	−2.35740
$304$	0	0
$305$	0.677298	0.0387820
$306$	0	0
$307$	−31.9026	−1.82078	−0.910388	−	0.413755i	$-0.864217\pi$
$307$	−31.9026	−1.82078	−0.910388	+	0.413755i	$0.864217\pi$
$308$	0	0
$309$	35.2711	2.00650
$310$	0	0
$311$	−10.5379	−0.597550	−0.298775	−	0.954324i	$-0.596578\pi$
$311$	−10.5379	−0.597550	−0.298775	+	0.954324i	$0.596578\pi$
$312$	0	0
$313$	−10.8431	−0.612887	−0.306444	−	0.951889i	$-0.599139\pi$
$313$	−10.8431	−0.612887	−0.306444	+	0.951889i	$0.599139\pi$
$314$	0	0
$315$	−6.01472	−0.338891
$316$	0	0
$317$	−3.44183	−0.193312	−0.0966561	−	0.995318i	$-0.530815\pi$
$317$	−3.44183	−0.193312	−0.0966561	+	0.995318i	$0.530815\pi$
$318$	0	0
$319$	−7.07152	−0.395929
$320$	0	0
$321$	19.9985	1.11621
$322$	0	0
$323$	28.4987	1.58571
$324$	0	0
$325$	22.2930	1.23660
$326$	0	0
$327$	1.36365	0.0754101
$328$	0	0
$329$	16.2651	0.896722
$330$	0	0
$331$	14.6910	0.807493	0.403746	−	0.914871i	$-0.367708\pi$
$331$	14.6910	0.807493	0.403746	+	0.914871i	$0.367708\pi$
$332$	0	0
$333$	−7.19527	−0.394299
$334$	0	0
$335$	7.78362	0.425265
$336$	0	0
$337$	−11.7708	−0.641197	−0.320598	−	0.947215i	$-0.603884\pi$
$337$	−11.7708	−0.641197	−0.320598	+	0.947215i	$0.603884\pi$
$338$	0	0
$339$	−7.63869	−0.414876
$340$	0	0
$341$	13.1787	0.713665
$342$	0	0
$343$	15.1518	0.818121
$344$	0	0
$345$	7.19260	0.387236
$346$	0	0
$347$	23.5480	1.26412	0.632061	−	0.774918i	$-0.282209\pi$
$347$	23.5480	1.26412	0.632061	+	0.774918i	$0.282209\pi$
$348$	0	0
$349$	−0.272834	−0.0146045	−0.00730224	−	0.999973i	$-0.502324\pi$
$349$	−0.272834	−0.0146045	−0.00730224	+	0.999973i	$0.502324\pi$
$350$	0	0
$351$	−11.7868	−0.629130
$352$	0	0
$353$	10.2058	0.543202	0.271601	−	0.962410i	$-0.412447\pi$
$353$	10.2058	0.543202	0.271601	+	0.962410i	$0.412447\pi$
$354$	0	0
$355$	13.5103	0.717052
$356$	0	0
$357$	−26.2087	−1.38711
$358$	0	0
$359$	9.35016	0.493483	0.246741	−	0.969081i	$-0.420640\pi$
$359$	9.35016	0.493483	0.246741	+	0.969081i	$0.420640\pi$
$360$	0	0
$361$	34.3225	1.80645
$362$	0	0
$363$	16.3881	0.860154
$364$	0	0
$365$	−2.99188	−0.156602
$366$	0	0
$367$	23.5711	1.23040	0.615202	−	0.788370i	$-0.289075\pi$
$367$	23.5711	1.23040	0.615202	+	0.788370i	$0.289075\pi$
$368$	0	0
$369$	−2.81358	−0.146469
$370$	0	0
$371$	8.70521	0.451952
$372$	0	0
$373$	15.6621	0.810952	0.405476	−	0.914106i	$-0.367106\pi$
$373$	15.6621	0.810952	0.405476	+	0.914106i	$0.367106\pi$
$374$	0	0
$375$	19.9352	1.02945
$376$	0	0
$377$	20.2837	1.04466
$378$	0	0
$379$	1.18732	0.0609884	0.0304942	−	0.999535i	$-0.490292\pi$
$379$	1.18732	0.0609884	0.0304942	+	0.999535i	$0.490292\pi$
$380$	0	0
$381$	−14.1230	−0.723541
$382$	0	0
$383$	24.8191	1.26820	0.634099	−	0.773252i	$-0.281371\pi$
$383$	24.8191	1.26820	0.634099	+	0.773252i	$0.281371\pi$
$384$	0	0
$385$	−5.64845	−0.287872
$386$	0	0
$387$	−5.42030	−0.275529
$388$	0	0
$389$	−19.6700	−0.997310	−0.498655	−	0.866801i	$-0.666172\pi$
$389$	−19.6700	−0.997310	−0.498655	+	0.866801i	$0.666172\pi$
$390$	0	0
$391$	12.7241	0.643484
$392$	0	0
$393$	−23.7388	−1.19746
$394$	0	0
$395$	−7.11439	−0.357964
$396$	0	0
$397$	−0.993519	−0.0498633	−0.0249317	−	0.999689i	$-0.507937\pi$
$397$	−0.993519	−0.0498633	−0.0249317	+	0.999689i	$0.507937\pi$
$398$	0	0
$399$	−49.0378	−2.45496
$400$	0	0
$401$	−0.796855	−0.0397930	−0.0198965	−	0.999802i	$-0.506334\pi$
$401$	−0.796855	−0.0397930	−0.0198965	+	0.999802i	$0.506334\pi$
$402$	0	0
$403$	−37.8012	−1.88301
$404$	0	0
$405$	−10.7464	−0.533995
$406$	0	0
$407$	−6.75712	−0.334938
$408$	0	0
$409$	3.22040	0.159239	0.0796193	−	0.996825i	$-0.474630\pi$
$409$	3.22040	0.159239	0.0796193	+	0.996825i	$0.474630\pi$
$410$	0	0
$411$	4.50354	0.222143
$412$	0	0
$413$	26.4938	1.30367
$414$	0	0
$415$	12.7596	0.626344
$416$	0	0
$417$	−5.40963	−0.264911
$418$	0	0
$419$	−15.2185	−0.743471	−0.371735	−	0.928339i	$-0.621237\pi$
$419$	−15.2185	−0.743471	−0.371735	+	0.928339i	$0.621237\pi$
$420$	0	0
$421$	6.62702	0.322981	0.161491	−	0.986874i	$-0.448370\pi$
$421$	6.62702	0.322981	0.161491	+	0.986874i	$0.448370\pi$
$422$	0	0
$423$	−11.1603	−0.542634
$424$	0	0
$425$	15.7527	0.764118
$426$	0	0
$427$	−2.06171	−0.0997729
$428$	0	0
$429$	23.8999	1.15390
$430$	0	0
$431$	11.1625	0.537677	0.268838	−	0.963185i	$-0.413360\pi$
$431$	11.1625	0.537677	0.268838	+	0.963185i	$0.413360\pi$
$432$	0	0
$433$	−14.4212	−0.693037	−0.346518	−	0.938043i	$-0.612636\pi$
$433$	−14.4212	−0.693037	−0.346518	+	0.938043i	$0.612636\pi$
$434$	0	0
$435$	8.10200	0.388461
$436$	0	0
$437$	23.8074	1.13886
$438$	0	0
$439$	28.3285	1.35205	0.676023	−	0.736881i	$-0.263702\pi$
$439$	28.3285	1.35205	0.676023	+	0.736881i	$0.263702\pi$
$440$	0	0
$441$	3.95621	0.188391
$442$	0	0
$443$	−11.8076	−0.560995	−0.280498	−	0.959855i	$-0.590499\pi$
$443$	−11.8076	−0.560995	−0.280498	+	0.959855i	$0.590499\pi$
$444$	0	0
$445$	2.93906	0.139325
$446$	0	0
$447$	36.7688	1.73910
$448$	0	0
$449$	18.8844	0.891211	0.445605	−	0.895230i	$-0.352988\pi$
$449$	18.8844	0.891211	0.445605	+	0.895230i	$0.352988\pi$
$450$	0	0
$451$	−2.64225	−0.124419
$452$	0	0
$453$	9.96764	0.468320
$454$	0	0
$455$	16.2018	0.759554
$456$	0	0
$457$	7.33580	0.343154	0.171577	−	0.985171i	$-0.445114\pi$
$457$	7.33580	0.343154	0.171577	+	0.985171i	$0.445114\pi$
$458$	0	0
$459$	−8.32874	−0.388752
$460$	0	0
$461$	4.51518	0.210293	0.105146	−	0.994457i	$-0.466469\pi$
$461$	4.51518	0.210293	0.105146	+	0.994457i	$0.466469\pi$
$462$	0	0
$463$	−31.8483	−1.48012	−0.740058	−	0.672543i	$-0.765202\pi$
$463$	−31.8483	−1.48012	−0.740058	+	0.672543i	$0.765202\pi$
$464$	0	0
$465$	−15.0991	−0.700204
$466$	0	0
$467$	−7.12020	−0.329483	−0.164742	−	0.986337i	$-0.552679\pi$
$467$	−7.12020	−0.329483	−0.164742	+	0.986337i	$0.552679\pi$
$468$	0	0
$469$	−23.6935	−1.09406
$470$	0	0
$471$	35.6698	1.64358
$472$	0	0
$473$	−5.09023	−0.234049
$474$	0	0
$475$	29.4741	1.35236
$476$	0	0
$477$	−5.97312	−0.273490
$478$	0	0
$479$	−27.4271	−1.25318	−0.626589	−	0.779350i	$-0.715549\pi$
$479$	−27.4271	−1.25318	−0.626589	+	0.779350i	$0.715549\pi$
$480$	0	0
$481$	19.3819	0.883738
$482$	0	0
$483$	−21.8944	−0.996228
$484$	0	0
$485$	−17.3320	−0.787007
$486$	0	0
$487$	36.4308	1.65084	0.825419	−	0.564521i	$-0.190939\pi$
$487$	36.4308	1.65084	0.825419	+	0.564521i	$0.190939\pi$
$488$	0	0
$489$	34.6679	1.56774
$490$	0	0
$491$	−35.3773	−1.59655	−0.798277	−	0.602291i	$-0.794255\pi$
$491$	−35.3773	−1.59655	−0.798277	+	0.602291i	$0.794255\pi$
$492$	0	0
$493$	14.3328	0.645519
$494$	0	0
$495$	3.87571	0.174200
$496$	0	0
$497$	−41.1255	−1.84473
$498$	0	0
$499$	21.8840	0.979663	0.489832	−	0.871817i	$-0.337058\pi$
$499$	21.8840	0.979663	0.489832	+	0.871817i	$0.337058\pi$
$500$	0	0
$501$	−28.6683	−1.28080
$502$	0	0
$503$	4.11411	0.183439	0.0917196	−	0.995785i	$-0.470764\pi$
$503$	4.11411	0.183439	0.0917196	+	0.995785i	$0.470764\pi$
$504$	0	0
$505$	17.9250	0.797652
$506$	0	0
$507$	−39.3386	−1.74709
$508$	0	0
$509$	9.08310	0.402601	0.201301	−	0.979530i	$-0.435483\pi$
$509$	9.08310	0.402601	0.201301	+	0.979530i	$0.435483\pi$
$510$	0	0
$511$	9.10733	0.402885
$512$	0	0
$513$	−15.5835	−0.688029
$514$	0	0
$515$	−15.4072	−0.678923
$516$	0	0
$517$	−10.4807	−0.460942
$518$	0	0
$519$	41.2812	1.81204
$520$	0	0
$521$	−9.58749	−0.420036	−0.210018	−	0.977698i	$-0.567352\pi$
$521$	−9.58749	−0.420036	−0.210018	+	0.977698i	$0.567352\pi$
$522$	0	0
$523$	−24.6866	−1.07947	−0.539735	−	0.841835i	$-0.681475\pi$
$523$	−24.6866	−1.07947	−0.539735	+	0.841835i	$0.681475\pi$
$524$	0	0
$525$	−27.1057	−1.18299
$526$	0	0
$527$	−26.7111	−1.16355
$528$	0	0
$529$	−12.3705	−0.537847
$530$	0	0
$531$	−18.1788	−0.788894
$532$	0	0
$533$	7.57893	0.328280
$534$	0	0
$535$	−8.73580	−0.377682
$536$	0	0
$537$	8.90747	0.384386
$538$	0	0
$539$	3.71529	0.160029
$540$	0	0
$541$	13.1414	0.564995	0.282497	−	0.959268i	$-0.408837\pi$
$541$	13.1414	0.564995	0.282497	+	0.959268i	$0.408837\pi$
$542$	0	0
$543$	25.2099	1.08186
$544$	0	0
$545$	−0.595673	−0.0255158
$546$	0	0
$547$	18.8458	0.805786	0.402893	−	0.915247i	$-0.368005\pi$
$547$	18.8458	0.805786	0.402893	+	0.915247i	$0.368005\pi$
$548$	0	0
$549$	1.41465	0.0603757
$550$	0	0
$551$	26.8175	1.14246
$552$	0	0
$553$	21.6563	0.920920
$554$	0	0
$555$	7.74178	0.328620
$556$	0	0
$557$	−41.8114	−1.77161	−0.885803	−	0.464062i	$-0.846391\pi$
$557$	−41.8114	−1.77161	−0.885803	+	0.464062i	$0.846391\pi$
$558$	0	0
$559$	14.6006	0.617542
$560$	0	0
$561$	16.8881	0.713016
$562$	0	0
$563$	−14.6492	−0.617391	−0.308695	−	0.951161i	$-0.599892\pi$
$563$	−14.6492	−0.617391	−0.308695	+	0.951161i	$0.599892\pi$
$564$	0	0
$565$	3.33675	0.140378
$566$	0	0
$567$	32.7123	1.37379
$568$	0	0
$569$	25.2808	1.05983	0.529914	−	0.848052i	$-0.322224\pi$
$569$	25.2808	1.05983	0.529914	+	0.848052i	$0.322224\pi$
$570$	0	0
$571$	28.0696	1.17468	0.587338	−	0.809341i	$-0.300176\pi$
$571$	28.0696	1.17468	0.587338	+	0.809341i	$0.300176\pi$
$572$	0	0
$573$	−40.2249	−1.68042
$574$	0	0
$575$	13.1596	0.548793
$576$	0	0
$577$	−22.8324	−0.950526	−0.475263	−	0.879844i	$-0.657647\pi$
$577$	−22.8324	−0.950526	−0.475263	+	0.879844i	$0.657647\pi$
$578$	0	0
$579$	−9.52163	−0.395706
$580$	0	0
$581$	−38.8404	−1.61137
$582$	0	0
$583$	−5.60938	−0.232317
$584$	0	0
$585$	−11.1170	−0.459630
$586$	0	0
$587$	5.78832	0.238909	0.119455	−	0.992840i	$-0.461885\pi$
$587$	5.78832	0.238909	0.119455	+	0.992840i	$0.461885\pi$
$588$	0	0
$589$	−49.9778	−2.05930
$590$	0	0
$591$	−24.6078	−1.01223
$592$	0	0
$593$	−32.7149	−1.34344	−0.671719	−	0.740806i	$-0.734444\pi$
$593$	−32.7149	−1.34344	−0.671719	+	0.740806i	$0.734444\pi$
$594$	0	0
$595$	11.4485	0.469344
$596$	0	0
$597$	6.94833	0.284376
$598$	0	0
$599$	6.24862	0.255312	0.127656	−	0.991819i	$-0.459255\pi$
$599$	6.24862	0.255312	0.127656	+	0.991819i	$0.459255\pi$
$600$	0	0
$601$	7.33828	0.299335	0.149667	−	0.988736i	$-0.452180\pi$
$601$	7.33828	0.299335	0.149667	+	0.988736i	$0.452180\pi$
$602$	0	0
$603$	16.2574	0.662051
$604$	0	0
$605$	−7.15870	−0.291043
$606$	0	0
$607$	12.3264	0.500313	0.250156	−	0.968205i	$-0.419518\pi$
$607$	12.3264	0.500313	0.250156	+	0.968205i	$0.419518\pi$
$608$	0	0
$609$	−24.6626	−0.999379
$610$	0	0
$611$	30.0626	1.21620
$612$	0	0
$613$	39.9368	1.61303	0.806516	−	0.591212i	$-0.201350\pi$
$613$	39.9368	1.61303	0.806516	+	0.591212i	$0.201350\pi$
$614$	0	0
$615$	3.02728	0.122072
$616$	0	0
$617$	−37.3202	−1.50245	−0.751227	−	0.660044i	$-0.770538\pi$
$617$	−37.3202	−1.50245	−0.751227	+	0.660044i	$0.770538\pi$
$618$	0	0
$619$	−24.5617	−0.987216	−0.493608	−	0.869684i	$-0.664322\pi$
$619$	−24.5617	−0.987216	−0.493608	+	0.869684i	$0.664322\pi$
$620$	0	0
$621$	−6.95772	−0.279204
$622$	0	0
$623$	−8.94654	−0.358436
$624$	0	0
$625$	11.4735	0.458938
$626$	0	0
$627$	31.5985	1.26192
$628$	0	0
$629$	13.6956	0.546080
$630$	0	0
$631$	3.11246	0.123905	0.0619525	−	0.998079i	$-0.480267\pi$
$631$	3.11246	0.123905	0.0619525	+	0.998079i	$0.480267\pi$
$632$	0	0
$633$	−38.8720	−1.54503
$634$	0	0
$635$	6.16922	0.244818
$636$	0	0
$637$	−10.6568	−0.422239
$638$	0	0
$639$	28.2185	1.11630
$640$	0	0
$641$	−15.5970	−0.616046	−0.308023	−	0.951379i	$-0.599667\pi$
$641$	−15.5970	−0.616046	−0.308023	+	0.951379i	$0.599667\pi$
$642$	0	0
$643$	7.05425	0.278193	0.139096	−	0.990279i	$-0.455580\pi$
$643$	7.05425	0.278193	0.139096	+	0.990279i	$0.455580\pi$
$644$	0	0
$645$	5.83199	0.229635
$646$	0	0
$647$	−31.5566	−1.24062	−0.620309	−	0.784358i	$-0.712993\pi$
$647$	−31.5566	−1.24062	−0.620309	+	0.784358i	$0.712993\pi$
$648$	0	0
$649$	−17.0718	−0.670128
$650$	0	0
$651$	45.9619	1.80139
$652$	0	0
$653$	41.8088	1.63611	0.818053	−	0.575142i	$-0.195053\pi$
$653$	41.8088	1.63611	0.818053	+	0.575142i	$0.195053\pi$
$654$	0	0
$655$	10.3696	0.405175
$656$	0	0
$657$	−6.24903	−0.243798
$658$	0	0
$659$	13.3185	0.518817	0.259408	−	0.965768i	$-0.416472\pi$
$659$	13.3185	0.518817	0.259408	+	0.965768i	$0.416472\pi$
$660$	0	0
$661$	−35.0514	−1.36334	−0.681670	−	0.731660i	$-0.738746\pi$
$661$	−35.0514	−1.36334	−0.681670	+	0.731660i	$0.738746\pi$
$662$	0	0
$663$	−48.4413	−1.88130
$664$	0	0
$665$	21.4208	0.830662
$666$	0	0
$667$	11.9735	0.463614
$668$	0	0
$669$	45.5847	1.76241
$670$	0	0
$671$	1.32850	0.0512863
$672$	0	0
$673$	−42.4163	−1.63503	−0.817514	−	0.575909i	$-0.804648\pi$
$673$	−42.4163	−1.63503	−0.817514	+	0.575909i	$0.804648\pi$
$674$	0	0
$675$	−8.61382	−0.331546
$676$	0	0
$677$	−17.0519	−0.655359	−0.327680	−	0.944789i	$-0.606267\pi$
$677$	−17.0519	−0.655359	−0.327680	+	0.944789i	$0.606267\pi$
$678$	0	0
$679$	52.7589	2.02470
$680$	0	0
$681$	−41.3562	−1.58477
$682$	0	0
$683$	−9.48098	−0.362780	−0.181390	−	0.983411i	$-0.558060\pi$
$683$	−9.48098	−0.362780	−0.181390	+	0.983411i	$0.558060\pi$
$684$	0	0
$685$	−1.96724	−0.0751645
$686$	0	0
$687$	18.6794	0.712663
$688$	0	0
$689$	16.0898	0.612971
$690$	0	0
$691$	15.7772	0.600191	0.300096	−	0.953909i	$-0.402981\pi$
$691$	15.7772	0.600191	0.300096	+	0.953909i	$0.402981\pi$
$692$	0	0
$693$	−11.7977	−0.448158
$694$	0	0
$695$	2.36305	0.0896355
$696$	0	0
$697$	5.35542	0.202851
$698$	0	0
$699$	−66.1899	−2.50353
$700$	0	0
$701$	6.80175	0.256898	0.128449	−	0.991716i	$-0.459000\pi$
$701$	6.80175	0.256898	0.128449	+	0.991716i	$0.459000\pi$
$702$	0	0
$703$	25.6252	0.966473
$704$	0	0
$705$	12.0080	0.452248
$706$	0	0
$707$	−54.5640	−2.05209
$708$	0	0
$709$	41.7018	1.56614	0.783072	−	0.621931i	$-0.213651\pi$
$709$	41.7018	1.56614	0.783072	+	0.621931i	$0.213651\pi$
$710$	0	0
$711$	−14.8596	−0.557277
$712$	0	0
$713$	−22.3141	−0.835669
$714$	0	0
$715$	−10.4400	−0.390433
$716$	0	0
$717$	6.08505	0.227251
$718$	0	0
$719$	27.0737	1.00968	0.504840	−	0.863213i	$-0.331551\pi$
$719$	27.0737	1.00968	0.504840	+	0.863213i	$0.331551\pi$
$720$	0	0
$721$	46.8998	1.74664
$722$	0	0
$723$	−44.6720	−1.66137
$724$	0	0
$725$	14.8234	0.550528
$726$	0	0
$727$	−27.6861	−1.02682	−0.513410	−	0.858143i	$-0.671618\pi$
$727$	−27.6861	−1.02682	−0.513410	+	0.858143i	$0.671618\pi$
$728$	0	0
$729$	−8.05794	−0.298442
$730$	0	0
$731$	10.3171	0.381591
$732$	0	0
$733$	−4.34977	−0.160662	−0.0803312	−	0.996768i	$-0.525598\pi$
$733$	−4.34977	−0.160662	−0.0803312	+	0.996768i	$0.525598\pi$
$734$	0	0
$735$	−4.25670	−0.157011
$736$	0	0
$737$	15.2674	0.562381
$738$	0	0
$739$	40.0199	1.47216	0.736078	−	0.676897i	$-0.236676\pi$
$739$	40.0199	1.47216	0.736078	+	0.676897i	$0.236676\pi$
$740$	0	0
$741$	−90.6361	−3.32960
$742$	0	0
$743$	3.16852	0.116242	0.0581209	−	0.998310i	$-0.481489\pi$
$743$	3.16852	0.116242	0.0581209	+	0.998310i	$0.481489\pi$
$744$	0	0
$745$	−16.0614	−0.588445
$746$	0	0
$747$	26.6505	0.975091
$748$	0	0
$749$	26.5919	0.971647
$750$	0	0
$751$	16.7919	0.612746	0.306373	−	0.951911i	$-0.400884\pi$
$751$	16.7919	0.612746	0.306373	+	0.951911i	$0.400884\pi$
$752$	0	0
$753$	−42.9951	−1.56683
$754$	0	0
$755$	−4.35408	−0.158461
$756$	0	0
$757$	24.7769	0.900531	0.450266	−	0.892895i	$-0.351329\pi$
$757$	24.7769	0.900531	0.450266	+	0.892895i	$0.351329\pi$
$758$	0	0
$759$	14.1081	0.512091
$760$	0	0
$761$	−18.1291	−0.657180	−0.328590	−	0.944473i	$-0.606573\pi$
$761$	−18.1291	−0.657180	−0.328590	+	0.944473i	$0.606573\pi$
$762$	0	0
$763$	1.81324	0.0656436
$764$	0	0
$765$	−7.85545	−0.284014
$766$	0	0
$767$	48.9682	1.76814
$768$	0	0
$769$	−43.1857	−1.55732	−0.778658	−	0.627449i	$-0.784099\pi$
$769$	−43.1857	−1.55732	−0.778658	+	0.627449i	$0.784099\pi$
$770$	0	0
$771$	7.39413	0.266293
$772$	0	0
$773$	−29.4688	−1.05992	−0.529959	−	0.848023i	$-0.677793\pi$
$773$	−29.4688	−1.05992	−0.529959	+	0.848023i	$0.677793\pi$
$774$	0	0
$775$	−27.6253	−0.992331
$776$	0	0
$777$	−23.5661	−0.845429
$778$	0	0
$779$	10.0203	0.359013
$780$	0	0
$781$	26.5001	0.948248
$782$	0	0
$783$	−7.83742	−0.280087
$784$	0	0
$785$	−15.5814	−0.556123
$786$	0	0
$787$	3.15643	0.112515	0.0562573	−	0.998416i	$-0.482083\pi$
$787$	3.15643	0.112515	0.0562573	+	0.998416i	$0.482083\pi$
$788$	0	0
$789$	−35.7125	−1.27140
$790$	0	0
$791$	−10.1571	−0.361145
$792$	0	0
$793$	−3.81063	−0.135320
$794$	0	0
$795$	6.42680	0.227935
$796$	0	0
$797$	15.8474	0.561342	0.280671	−	0.959804i	$-0.409443\pi$
$797$	15.8474	0.561342	0.280671	+	0.959804i	$0.409443\pi$
$798$	0	0
$799$	21.2428	0.751516
$800$	0	0
$801$	6.13871	0.216901
$802$	0	0
$803$	−5.86850	−0.207095
$804$	0	0
$805$	9.56394	0.337085
$806$	0	0
$807$	−34.0943	−1.20018
$808$	0	0
$809$	45.0136	1.58259	0.791297	−	0.611432i	$-0.209406\pi$
$809$	45.0136	1.58259	0.791297	+	0.611432i	$0.209406\pi$
$810$	0	0
$811$	−4.81632	−0.169124	−0.0845619	−	0.996418i	$-0.526949\pi$
$811$	−4.81632	−0.169124	−0.0845619	+	0.996418i	$0.526949\pi$
$812$	0	0
$813$	63.4467	2.22517
$814$	0	0
$815$	−15.1437	−0.530460
$816$	0	0
$817$	19.3038	0.675355
$818$	0	0
$819$	33.8402	1.18247
$820$	0	0
$821$	45.2489	1.57920	0.789599	−	0.613623i	$-0.210289\pi$
$821$	45.2489	1.57920	0.789599	+	0.613623i	$0.210289\pi$
$822$	0	0
$823$	30.9642	1.07934	0.539672	−	0.841875i	$-0.318548\pi$
$823$	30.9642	1.07934	0.539672	+	0.841875i	$0.318548\pi$
$824$	0	0
$825$	17.4661	0.608093
$826$	0	0
$827$	−35.7496	−1.24314	−0.621568	−	0.783360i	$-0.713504\pi$
$827$	−35.7496	−1.24314	−0.621568	+	0.783360i	$0.713504\pi$
$828$	0	0
$829$	31.5042	1.09419	0.547093	−	0.837072i	$-0.315735\pi$
$829$	31.5042	1.09419	0.547093	+	0.837072i	$0.315735\pi$
$830$	0	0
$831$	24.5836	0.852794
$832$	0	0
$833$	−7.53031	−0.260910
$834$	0	0
$835$	12.5229	0.433374
$836$	0	0
$837$	14.6060	0.504858
$838$	0	0
$839$	23.2252	0.801823	0.400912	−	0.916117i	$-0.368693\pi$
$839$	23.2252	0.801823	0.400912	+	0.916117i	$0.368693\pi$
$840$	0	0
$841$	−15.5127	−0.534919
$842$	0	0
$843$	38.5372	1.32729
$844$	0	0
$845$	17.1840	0.591146
$846$	0	0
$847$	21.7912	0.748754
$848$	0	0
$849$	−33.8579	−1.16200
$850$	0	0
$851$	11.4411	0.392197
$852$	0	0
$853$	21.9894	0.752903	0.376451	−	0.926436i	$-0.377144\pi$
$853$	21.9894	0.752903	0.376451	+	0.926436i	$0.377144\pi$
$854$	0	0
$855$	−14.6980	−0.502660
$856$	0	0
$857$	−0.729491	−0.0249189	−0.0124595	−	0.999922i	$-0.503966\pi$
$857$	−0.729491	−0.0249189	−0.0124595	+	0.999922i	$0.503966\pi$
$858$	0	0
$859$	−39.4900	−1.34738	−0.673690	−	0.739014i	$-0.735292\pi$
$859$	−39.4900	−1.34738	−0.673690	+	0.739014i	$0.735292\pi$
$860$	0	0
$861$	−9.21510	−0.314050
$862$	0	0
$863$	−30.7369	−1.04630	−0.523148	−	0.852242i	$-0.675242\pi$
$863$	−30.7369	−1.04630	−0.523148	+	0.852242i	$0.675242\pi$
$864$	0	0
$865$	−18.0325	−0.613125
$866$	0	0
$867$	3.97470	0.134988
$868$	0	0
$869$	−13.9547	−0.473381
$870$	0	0
$871$	−43.7924	−1.48385
$872$	0	0
$873$	−36.2008	−1.22521
$874$	0	0
$875$	26.5077	0.896123
$876$	0	0
$877$	25.1864	0.850483	0.425241	−	0.905080i	$-0.360189\pi$
$877$	25.1864	0.850483	0.425241	+	0.905080i	$0.360189\pi$
$878$	0	0
$879$	−50.4026	−1.70004
$880$	0	0
$881$	31.5046	1.06142	0.530708	−	0.847555i	$-0.321926\pi$
$881$	31.5046	1.06142	0.530708	+	0.847555i	$0.321926\pi$
$882$	0	0
$883$	−35.8822	−1.20753	−0.603767	−	0.797161i	$-0.706334\pi$
$883$	−35.8822	−1.20753	−0.603767	+	0.797161i	$0.706334\pi$
$884$	0	0
$885$	19.5596	0.657488
$886$	0	0
$887$	−54.7834	−1.83944	−0.919722	−	0.392569i	$-0.871586\pi$
$887$	−54.7834	−1.83944	−0.919722	+	0.392569i	$0.871586\pi$
$888$	0	0
$889$	−18.7792	−0.629834
$890$	0	0
$891$	−21.0789	−0.706169
$892$	0	0
$893$	39.7464	1.33006
$894$	0	0
$895$	−3.89098	−0.130061
$896$	0	0
$897$	−40.4672	−1.35116
$898$	0	0
$899$	−25.1354	−0.838311
$900$	0	0
$901$	11.3693	0.378767
$902$	0	0
$903$	−17.7527	−0.590772
$904$	0	0
$905$	−11.0122	−0.366060
$906$	0	0
$907$	−42.8557	−1.42300	−0.711500	−	0.702686i	$-0.751984\pi$
$907$	−42.8557	−1.42300	−0.711500	+	0.702686i	$0.751984\pi$
$908$	0	0
$909$	37.4393	1.24178
$910$	0	0
$911$	−31.7361	−1.05146	−0.525732	−	0.850650i	$-0.676209\pi$
$911$	−31.7361	−1.05146	−0.525732	+	0.850650i	$0.676209\pi$
$912$	0	0
$913$	25.0276	0.828294
$914$	0	0
$915$	−1.52210	−0.0503190
$916$	0	0
$917$	−31.5653	−1.04238
$918$	0	0
$919$	30.6526	1.01114	0.505568	−	0.862787i	$-0.331283\pi$
$919$	30.6526	1.01114	0.505568	+	0.862787i	$0.331283\pi$
$920$	0	0
$921$	71.6949	2.36243
$922$	0	0
$923$	−76.0120	−2.50196
$924$	0	0
$925$	14.1644	0.465722
$926$	0	0
$927$	−32.1805	−1.05695
$928$	0	0
$929$	5.51194	0.180841	0.0904205	−	0.995904i	$-0.471179\pi$
$929$	5.51194	0.180841	0.0904205	+	0.995904i	$0.471179\pi$
$930$	0	0
$931$	−14.0896	−0.461768
$932$	0	0
$933$	23.6819	0.775311
$934$	0	0
$935$	−7.37709	−0.241257
$936$	0	0
$937$	−5.95006	−0.194380	−0.0971900	−	0.995266i	$-0.530985\pi$
$937$	−5.95006	−0.194380	−0.0971900	+	0.995266i	$0.530985\pi$
$938$	0	0
$939$	24.3677	0.795211
$940$	0	0
$941$	55.5770	1.81176	0.905879	−	0.423537i	$-0.139212\pi$
$941$	55.5770	1.81176	0.905879	+	0.423537i	$0.139212\pi$
$942$	0	0
$943$	4.47385	0.145688
$944$	0	0
$945$	−6.26023	−0.203645
$946$	0	0
$947$	30.9888	1.00700	0.503501	−	0.863995i	$-0.332045\pi$
$947$	30.9888	1.00700	0.503501	+	0.863995i	$0.332045\pi$
$948$	0	0
$949$	16.8330	0.546422
$950$	0	0
$951$	7.73484	0.250819
$952$	0	0
$953$	9.50276	0.307825	0.153912	−	0.988085i	$-0.450813\pi$
$953$	9.50276	0.307825	0.153912	+	0.988085i	$0.450813\pi$
$954$	0	0
$955$	17.5711	0.568588
$956$	0	0
$957$	15.8919	0.513711
$958$	0	0
$959$	5.98832	0.193373
$960$	0	0
$961$	15.8430	0.511063
$962$	0	0
$963$	−18.2462	−0.587974
$964$	0	0
$965$	4.15926	0.133891
$966$	0	0
$967$	−20.2121	−0.649977	−0.324988	−	0.945718i	$-0.605360\pi$
$967$	−20.2121	−0.649977	−0.324988	+	0.945718i	$0.605360\pi$
$968$	0	0
$969$	−64.0452	−2.05743
$970$	0	0
$971$	35.6370	1.14365	0.571823	−	0.820377i	$-0.306237\pi$
$971$	35.6370	1.14365	0.571823	+	0.820377i	$0.306237\pi$
$972$	0	0
$973$	−7.19315	−0.230602
$974$	0	0
$975$	−50.0993	−1.60446
$976$	0	0
$977$	31.8492	1.01894	0.509472	−	0.860487i	$-0.329841\pi$
$977$	31.8492	1.01894	0.509472	+	0.860487i	$0.329841\pi$
$978$	0	0
$979$	5.76489	0.184247
$980$	0	0
$981$	−1.24416	−0.0397230
$982$	0	0
$983$	−41.4168	−1.32099	−0.660496	−	0.750830i	$-0.729654\pi$
$983$	−41.4168	−1.32099	−0.660496	+	0.750830i	$0.729654\pi$
$984$	0	0
$985$	10.7492	0.342499
$986$	0	0
$987$	−36.5526	−1.16348
$988$	0	0
$989$	8.61876	0.274061
$990$	0	0
$991$	−56.0775	−1.78136	−0.890680	−	0.454630i	$-0.849771\pi$
$991$	−56.0775	−1.78136	−0.890680	+	0.454630i	$0.849771\pi$
$992$	0	0
$993$	−33.0153	−1.04771
$994$	0	0
$995$	−3.03518	−0.0962218
$996$	0	0
$997$	15.8970	0.503464	0.251732	−	0.967797i	$-0.419000\pi$
$997$	15.8970	0.503464	0.251732	+	0.967797i	$0.419000\pi$
$998$	0	0
$999$	−7.48897	−0.236941

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6044.2.a.b.1.12	✓	63

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6044.2.a.b.1.12	✓	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)

\(f(q)\)	\(=\)	\(q-2.24731 q^{3} +0.981673 q^{5} -2.98823 q^{7} +2.05038 q^{9} +O(q^{10})\)	\(q-2.24731 q^{3} +0.981673 q^{5} -2.98823 q^{7} +2.05038 q^{9} +1.92553 q^{11} -5.52311 q^{13} -2.20612 q^{15} -3.90274 q^{17} -7.30222 q^{19} +6.71546 q^{21} -3.26029 q^{23} -4.03632 q^{25} +2.13408 q^{27} -3.67251 q^{29} +6.84419 q^{31} -4.32725 q^{33} -2.93346 q^{35} -3.50923 q^{37} +12.4121 q^{39} -1.37222 q^{41} -2.64355 q^{43} +2.01281 q^{45} -5.44305 q^{47} +1.92950 q^{49} +8.77064 q^{51} -2.91317 q^{53} +1.89024 q^{55} +16.4103 q^{57} -8.86606 q^{59} +0.689943 q^{61} -6.12701 q^{63} -5.42189 q^{65} +7.92894 q^{67} +7.32688 q^{69} +13.7625 q^{71} -3.04774 q^{73} +9.07084 q^{75} -5.75391 q^{77} -7.24721 q^{79} -10.9471 q^{81} +12.9978 q^{83} -3.83121 q^{85} +8.25326 q^{87} +2.99393 q^{89} +16.5043 q^{91} -15.3810 q^{93} -7.16840 q^{95} -17.6556 q^{97} +3.94807 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9}+O(q^{10}) \) 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

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