LMFDB - Embedded newform 441.4.a.v.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,4,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("441.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	441.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:	$26.0198423125$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.6257832.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 19x^{2} + 42 $ x^4 - 19*x^2 + 42
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 63)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.59805$ of defining polynomial
Character	$\chi$	$=$	441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.59805 q^{2} -5.44622 q^{4} +18.2917 q^{5} -21.4878 q^{8} +O(q^{10})$	$q+1.59805 q^{2} -5.44622 q^{4} +18.2917 q^{5} -21.4878 q^{8} +29.2311 q^{10} -61.2673 q^{11} -32.4462 q^{13} +9.23111 q^{16} -81.3289 q^{17} +20.9084 q^{19} -99.6206 q^{20} -97.9084 q^{22} +33.7310 q^{23} +209.586 q^{25} -51.8508 q^{26} -52.0227 q^{29} -193.924 q^{31} +186.654 q^{32} -129.968 q^{34} -267.156 q^{37} +33.4128 q^{38} -393.048 q^{40} +203.176 q^{41} -21.9520 q^{43} +333.675 q^{44} +53.9040 q^{46} -247.921 q^{47} +334.929 q^{50} +176.709 q^{52} -140.826 q^{53} -1120.68 q^{55} -83.1351 q^{58} -221.468 q^{59} -652.526 q^{61} -309.902 q^{62} +224.435 q^{64} -593.496 q^{65} +604.478 q^{67} +442.935 q^{68} -716.031 q^{71} -388.876 q^{73} -426.929 q^{74} -113.872 q^{76} -289.741 q^{79} +168.853 q^{80} +324.686 q^{82} -115.652 q^{83} -1487.64 q^{85} -35.0805 q^{86} +1316.50 q^{88} +939.363 q^{89} -183.707 q^{92} -396.192 q^{94} +382.451 q^{95} -120.394 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{4}+O(q^{10}) $ 4 * q + 6 * q^4	$ 4 q + 6 q^{4} - 22 q^{10} - 102 q^{13} - 102 q^{16} - 222 q^{19} - 86 q^{22} + 366 q^{25} - 220 q^{31} - 1020 q^{34} - 374 q^{37} - 822 q^{40} - 838 q^{43} + 1716 q^{46} + 40 q^{52} - 2510 q^{55} - 1694 q^{58} - 1332 q^{61} - 686 q^{64} + 1890 q^{67} - 1750 q^{73} - 2456 q^{76} + 8 q^{79} - 2480 q^{82} - 1116 q^{85} + 2682 q^{88} + 1416 q^{94} - 3010 q^{97}+O(q^{100}) $ 4 * q + 6 * q^4 - 22 * q^10 - 102 * q^13 - 102 * q^16 - 222 * q^19 - 86 * q^22 + 366 * q^25 - 220 * q^31 - 1020 * q^34 - 374 * q^37 - 822 * q^40 - 838 * q^43 + 1716 * q^46 + 40 * q^52 - 2510 * q^55 - 1694 * q^58 - 1332 * q^61 - 686 * q^64 + 1890 * q^67 - 1750 * q^73 - 2456 * q^76 + 8 * q^79 - 2480 * q^82 - 1116 * q^85 + 2682 * q^88 + 1416 * q^94 - 3010 * q^97

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 $

$2$	1.59805	0.564998	0.282499	−	0.959268i	$-0.408837\pi$
$2$	1.59805	0.564998	0.282499	+	0.959268i	$0.408837\pi$
$3$	0	0
$3$	0	0
$4$	−5.44622	−0.680778
$5$	18.2917	1.63606	0.818029	−	0.575177i	$-0.195067\pi$
$5$	18.2917	1.63606	0.818029	+	0.575177i	$0.195067\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−21.4878	−0.949635
$9$	0	0
$10$	29.2311	0.924369
$11$	−61.2673	−1.67934	−0.839672	−	0.543094i	$-0.817253\pi$
$11$	−61.2673	−1.67934	−0.839672	+	0.543094i	$0.817253\pi$
$12$	0	0
$13$	−32.4462	−0.692228	−0.346114	−	0.938192i	$-0.612499\pi$
$13$	−32.4462	−0.692228	−0.346114	+	0.938192i	$0.612499\pi$
$14$	0	0
$15$	0	0
$16$	9.23111	0.144236
$17$	−81.3289	−1.16030	−0.580152	−	0.814509i	$-0.697007\pi$
$17$	−81.3289	−1.16030	−0.580152	+	0.814509i	$0.697007\pi$
$18$	0	0
$19$	20.9084	0.252459	0.126230	−	0.992001i	$-0.459712\pi$
$19$	20.9084	0.252459	0.126230	+	0.992001i	$0.459712\pi$
$20$	−99.6206	−1.11379
$21$	0	0
$22$	−97.9084	−0.948825
$23$	33.7310	0.305800	0.152900	−	0.988242i	$-0.451139\pi$
$23$	33.7310	0.305800	0.152900	+	0.988242i	$0.451139\pi$
$24$	0	0
$25$	209.586	1.67669
$26$	−51.8508	−0.391107
$27$	0	0
$28$	0	0
$29$	−52.0227	−0.333116	−0.166558	−	0.986032i	$-0.553265\pi$
$29$	−52.0227	−0.333116	−0.166558	+	0.986032i	$0.553265\pi$
$30$	0	0
$31$	−193.924	−1.12354	−0.561772	−	0.827292i	$-0.689880\pi$
$31$	−193.924	−1.12354	−0.561772	+	0.827292i	$0.689880\pi$
$32$	186.654	1.03113
$33$	0	0
$34$	−129.968	−0.655568
$35$	0	0
$36$	0	0
$37$	−267.156	−1.18703	−0.593515	−	0.804823i	$-0.702260\pi$
$37$	−267.156	−1.18703	−0.593515	+	0.804823i	$0.702260\pi$
$38$	33.4128	0.142639
$39$	0	0
$40$	−393.048	−1.55366
$41$	203.176	0.773921	0.386960	−	0.922096i	$-0.373525\pi$
$41$	203.176	0.773921	0.386960	+	0.922096i	$0.373525\pi$
$42$	0	0
$43$	−21.9520	−0.0778523	−0.0389262	−	0.999242i	$-0.512394\pi$
$43$	−21.9520	−0.0778523	−0.0389262	+	0.999242i	$0.512394\pi$
$44$	333.675	1.14326
$45$	0	0
$46$	53.9040	0.172776
$47$	−247.921	−0.769427	−0.384713	−	0.923036i	$-0.625700\pi$
$47$	−247.921	−0.769427	−0.384713	+	0.923036i	$0.625700\pi$
$48$	0	0
$49$	0	0
$50$	334.929	0.947324
$51$	0	0
$52$	176.709	0.471253
$53$	−140.826	−0.364981	−0.182490	−	0.983208i	$-0.558416\pi$
$53$	−140.826	−0.364981	−0.182490	+	0.983208i	$0.558416\pi$
$54$	0	0
$55$	−1120.68	−2.74750
$56$	0	0
$57$	0	0
$58$	−83.1351	−0.188210
$59$	−221.468	−0.488689	−0.244344	−	0.969689i	$-0.578573\pi$
$59$	−221.468	−0.488689	−0.244344	+	0.969689i	$0.578573\pi$
$60$	0	0
$61$	−652.526	−1.36963	−0.684815	−	0.728717i	$-0.740117\pi$
$61$	−652.526	−1.36963	−0.684815	+	0.728717i	$0.740117\pi$
$62$	−309.902	−0.634800
$63$	0	0
$64$	224.435	0.438349
$65$	−593.496	−1.13253
$66$	0	0
$67$	604.478	1.10222	0.551110	−	0.834432i	$-0.314204\pi$
$67$	604.478	1.10222	0.551110	+	0.834432i	$0.314204\pi$
$68$	442.935	0.789909
$69$	0	0
$70$	0	0
$71$	−716.031	−1.19686	−0.598431	−	0.801174i	$-0.704209\pi$
$71$	−716.031	−1.19686	−0.598431	+	0.801174i	$0.704209\pi$
$72$	0	0
$73$	−388.876	−0.623487	−0.311743	−	0.950166i	$-0.600913\pi$
$73$	−388.876	−0.623487	−0.311743	+	0.950166i	$0.600913\pi$
$74$	−426.929	−0.670669
$75$	0	0
$76$	−113.872	−0.171869
$77$	0	0
$78$	0	0
$79$	−289.741	−0.412639	−0.206319	−	0.978485i	$-0.566149\pi$
$79$	−289.741	−0.412639	−0.206319	+	0.978485i	$0.566149\pi$
$80$	168.853	0.235979
$81$	0	0
$82$	324.686	0.437263
$83$	−115.652	−0.152946	−0.0764728	−	0.997072i	$-0.524366\pi$
$83$	−115.652	−0.152946	−0.0764728	+	0.997072i	$0.524366\pi$
$84$	0	0
$85$	−1487.64	−1.89832
$86$	−35.0805	−0.0439864
$87$	0	0
$88$	1316.50	1.59476
$89$	939.363	1.11879	0.559395	−	0.828901i	$-0.311033\pi$
$89$	939.363	1.11879	0.559395	+	0.828901i	$0.311033\pi$
$90$	0	0
$91$	0	0
$92$	−183.707	−0.208182
$93$	0	0
$94$	−396.192	−0.434724
$95$	382.451	0.413038
$96$	0	0
$97$	−120.394	−0.126022	−0.0630110	−	0.998013i	$-0.520070\pi$
$97$	−120.394	−0.126022	−0.0630110	+	0.998013i	$0.520070\pi$
$98$	0	0
$99$	0	0
$100$	−1141.45	−1.14145
$101$	1281.00	1.26203	0.631013	−	0.775772i	$-0.282640\pi$
$101$	1281.00	1.26203	0.631013	+	0.775772i	$0.282640\pi$
$102$	0	0
$103$	−531.339	−0.508295	−0.254147	−	0.967166i	$-0.581795\pi$
$103$	−531.339	−0.508295	−0.254147	+	0.967166i	$0.581795\pi$
$104$	697.198	0.657364
$105$	0	0
$106$	−225.048	−0.206213
$107$	−133.352	−0.120482	−0.0602411	−	0.998184i	$-0.519187\pi$
$107$	−133.352	−0.120482	−0.0602411	+	0.998184i	$0.519187\pi$
$108$	0	0
$109$	−217.769	−0.191362	−0.0956811	−	0.995412i	$-0.530503\pi$
$109$	−217.769	−0.191362	−0.0956811	+	0.995412i	$0.530503\pi$
$110$	−1790.91	−1.55233
$111$	0	0
$112$	0	0
$113$	2006.09	1.67006	0.835031	−	0.550204i	$-0.185450\pi$
$113$	2006.09	1.67006	0.835031	+	0.550204i	$0.185450\pi$
$114$	0	0
$115$	616.997	0.500307
$116$	283.327	0.226778
$117$	0	0
$118$	−353.917	−0.276108
$119$	0	0
$120$	0	0
$121$	2422.68	1.82019
$122$	−1042.77	−0.773838
$123$	0	0
$124$	1056.16	0.764884
$125$	1547.22	1.10710
$126$	0	0
$127$	1638.92	1.14512	0.572562	−	0.819861i	$-0.305950\pi$
$127$	1638.92	1.14512	0.572562	+	0.819861i	$0.305950\pi$
$128$	−1134.57	−0.783462
$129$	0	0
$130$	−948.439	−0.639874
$131$	91.7511	0.0611933	0.0305967	−	0.999532i	$-0.490259\pi$
$131$	91.7511	0.0611933	0.0305967	+	0.999532i	$0.490259\pi$
$132$	0	0
$133$	0	0
$134$	965.989	0.622752
$135$	0	0
$136$	1747.58	1.10186
$137$	1867.13	1.16438	0.582188	−	0.813054i	$-0.302197\pi$
$137$	1867.13	1.16438	0.582188	+	0.813054i	$0.302197\pi$
$138$	0	0
$139$	−639.778	−0.390397	−0.195199	−	0.980764i	$-0.562535\pi$
$139$	−639.778	−0.390397	−0.195199	+	0.980764i	$0.562535\pi$
$140$	0	0
$141$	0	0
$142$	−1144.26	−0.676224
$143$	1987.89	1.16249
$144$	0	0
$145$	−951.583	−0.544998
$146$	−621.446	−0.352269
$147$	0	0
$148$	1454.99	0.808103
$149$	3136.53	1.72453	0.862264	−	0.506460i	$-0.169046\pi$
$149$	3136.53	1.72453	0.862264	+	0.506460i	$0.169046\pi$
$150$	0	0
$151$	−2721.37	−1.46663	−0.733317	−	0.679887i	$-0.762029\pi$
$151$	−2721.37	−1.46663	−0.733317	+	0.679887i	$0.762029\pi$
$152$	−449.276	−0.239744
$153$	0	0
$154$	0	0
$155$	−3547.20	−1.83818
$156$	0	0
$157$	−2879.75	−1.46388	−0.731939	−	0.681370i	$-0.761384\pi$
$157$	−2879.75	−1.46388	−0.731939	+	0.681370i	$0.761384\pi$
$158$	−463.022	−0.233140
$159$	0	0
$160$	3414.22	1.68699
$161$	0	0
$162$	0	0
$163$	646.142	0.310489	0.155245	−	0.987876i	$-0.450383\pi$
$163$	646.142	0.310489	0.155245	+	0.987876i	$0.450383\pi$
$164$	−1106.54	−0.526868
$165$	0	0
$166$	−184.819	−0.0864139
$167$	−3765.03	−1.74459	−0.872296	−	0.488979i	$-0.837370\pi$
$167$	−3765.03	−1.74459	−0.872296	+	0.488979i	$0.837370\pi$
$168$	0	0
$169$	−1144.24	−0.520821
$170$	−2377.33	−1.07255
$171$	0	0
$172$	119.555	0.0530001
$173$	2308.98	1.01473	0.507366	−	0.861731i	$-0.330619\pi$
$173$	2308.98	1.01473	0.507366	+	0.861731i	$0.330619\pi$
$174$	0	0
$175$	0	0
$176$	−565.565	−0.242222
$177$	0	0
$178$	1501.15	0.632114
$179$	−3033.76	−1.26678	−0.633390	−	0.773833i	$-0.718337\pi$
$179$	−3033.76	−1.26678	−0.633390	+	0.773833i	$0.718337\pi$
$180$	0	0
$181$	4079.71	1.67537	0.837686	−	0.546152i	$-0.183908\pi$
$181$	4079.71	1.67537	0.837686	+	0.546152i	$0.183908\pi$
$182$	0	0
$183$	0	0
$184$	−724.805	−0.290399
$185$	−4886.73	−1.94205
$186$	0	0
$187$	4982.80	1.94855
$188$	1350.24	0.523809
$189$	0	0
$190$	611.177	0.233365
$191$	877.109	0.332279	0.166140	−	0.986102i	$-0.446870\pi$
$191$	877.109	0.332279	0.166140	+	0.986102i	$0.446870\pi$
$192$	0	0
$193$	−1458.71	−0.544043	−0.272022	−	0.962291i	$-0.587692\pi$
$193$	−1458.71	−0.544043	−0.272022	+	0.962291i	$0.587692\pi$
$194$	−192.396	−0.0712021
$195$	0	0
$196$	0	0
$197$	952.250	0.344391	0.172195	−	0.985063i	$-0.444914\pi$
$197$	952.250	0.344391	0.172195	+	0.985063i	$0.444914\pi$
$198$	0	0
$199$	−3342.22	−1.19057	−0.595285	−	0.803515i	$-0.702961\pi$
$199$	−3342.22	−1.19057	−0.595285	+	0.803515i	$0.702961\pi$
$200$	−4503.54	−1.59224
$201$	0	0
$202$	2047.11	0.713041
$203$	0	0
$204$	0	0
$205$	3716.43	1.26618
$206$	−849.108	−0.287185
$207$	0	0
$208$	−299.515	−0.0998442
$209$	−1281.00	−0.423966
$210$	0	0
$211$	1439.27	0.469589	0.234794	−	0.972045i	$-0.424558\pi$
$211$	1439.27	0.469589	0.234794	+	0.972045i	$0.424558\pi$
$212$	766.971	0.248471
$213$	0	0
$214$	−213.103	−0.0680721
$215$	−401.539	−0.127371
$216$	0	0
$217$	0	0
$218$	−348.007	−0.108119
$219$	0	0
$220$	6103.48	1.87044
$221$	2638.82	0.803194
$222$	0	0
$223$	−1009.86	−0.303253	−0.151626	−	0.988438i	$-0.548451\pi$
$223$	−1009.86	−0.303253	−0.151626	+	0.988438i	$0.548451\pi$
$224$	0	0
$225$	0	0
$226$	3205.84	0.943580
$227$	2948.60	0.862140	0.431070	−	0.902319i	$-0.358136\pi$
$227$	2948.60	0.862140	0.431070	+	0.902319i	$0.358136\pi$
$228$	0	0
$229$	4038.85	1.16548	0.582739	−	0.812659i	$-0.301981\pi$
$229$	4038.85	1.16548	0.582739	+	0.812659i	$0.301981\pi$
$230$	985.995	0.282672
$231$	0	0
$232$	1117.85	0.316339
$233$	2995.80	0.842323	0.421162	−	0.906986i	$-0.361622\pi$
$233$	2995.80	0.842323	0.421162	+	0.906986i	$0.361622\pi$
$234$	0	0
$235$	−4534.90	−1.25883
$236$	1206.16	0.332688
$237$	0	0
$238$	0	0
$239$	−1810.28	−0.489948	−0.244974	−	0.969530i	$-0.578779\pi$
$239$	−1810.28	−0.489948	−0.244974	+	0.969530i	$0.578779\pi$
$240$	0	0
$241$	3749.43	1.00217	0.501083	−	0.865399i	$-0.332935\pi$
$241$	3749.43	1.00217	0.501083	+	0.865399i	$0.332935\pi$
$242$	3871.57	1.02841
$243$	0	0
$244$	3553.80	0.932414
$245$	0	0
$246$	0	0
$247$	−678.400	−0.174759
$248$	4167.01	1.06696
$249$	0	0
$250$	2472.54	0.625508
$251$	2706.96	0.680724	0.340362	−	0.940295i	$-0.389450\pi$
$251$	2706.96	0.680724	0.340362	+	0.940295i	$0.389450\pi$
$252$	0	0
$253$	−2066.61	−0.513544
$254$	2619.09	0.646992
$255$	0	0
$256$	−3608.59	−0.881003
$257$	−5375.28	−1.30467	−0.652337	−	0.757929i	$-0.726211\pi$
$257$	−5375.28	−1.30467	−0.652337	+	0.757929i	$0.726211\pi$
$258$	0	0
$259$	0	0
$260$	3232.31	0.770998
$261$	0	0
$262$	146.623	0.0345741
$263$	5246.56	1.23010	0.615051	−	0.788487i	$-0.289135\pi$
$263$	5246.56	1.23010	0.615051	+	0.788487i	$0.289135\pi$
$264$	0	0
$265$	−2575.95	−0.597129
$266$	0	0
$267$	0	0
$268$	−3292.12	−0.750367
$269$	−3013.32	−0.682993	−0.341496	−	0.939883i	$-0.610934\pi$
$269$	−3013.32	−0.682993	−0.341496	+	0.939883i	$0.610934\pi$
$270$	0	0
$271$	−6897.25	−1.54604	−0.773022	−	0.634379i	$-0.781256\pi$
$271$	−6897.25	−1.54604	−0.773022	+	0.634379i	$0.781256\pi$
$272$	−750.756	−0.167358
$273$	0	0
$274$	2983.77	0.657869
$275$	−12840.7	−2.81573
$276$	0	0
$277$	3318.61	0.719840	0.359920	−	0.932983i	$-0.382804\pi$
$277$	3318.61	0.719840	0.359920	+	0.932983i	$0.382804\pi$
$278$	−1022.40	−0.220574
$279$	0	0
$280$	0	0
$281$	6274.14	1.33197	0.665986	−	0.745964i	$-0.268011\pi$
$281$	6274.14	1.33197	0.665986	+	0.745964i	$0.268011\pi$
$282$	0	0
$283$	−7772.47	−1.63260	−0.816300	−	0.577629i	$-0.803978\pi$
$283$	−7772.47	−1.63260	−0.816300	+	0.577629i	$0.803978\pi$
$284$	3899.66	0.814797
$285$	0	0
$286$	3176.76	0.656803
$287$	0	0
$288$	0	0
$289$	1701.39	0.346304
$290$	−1520.68	−0.307922
$291$	0	0
$292$	2117.91	0.424456
$293$	854.897	0.170456	0.0852280	−	0.996361i	$-0.472838\pi$
$293$	854.897	0.170456	0.0852280	+	0.996361i	$0.472838\pi$
$294$	0	0
$295$	−4051.02	−0.799523
$296$	5740.58	1.12725
$297$	0	0
$298$	5012.35	0.974354
$299$	−1094.44	−0.211683
$300$	0	0
$301$	0	0
$302$	−4348.89	−0.828645
$303$	0	0
$304$	193.008	0.0364137
$305$	−11935.8	−2.24079
$306$	0	0
$307$	−2550.68	−0.474185	−0.237092	−	0.971487i	$-0.576194\pi$
$307$	−2550.68	−0.474185	−0.237092	+	0.971487i	$0.576194\pi$
$308$	0	0
$309$	0	0
$310$	−5668.63	−1.03857
$311$	7480.50	1.36392	0.681962	−	0.731388i	$-0.261127\pi$
$311$	7480.50	1.36392	0.681962	+	0.731388i	$0.261127\pi$
$312$	0	0
$313$	−6.24784	−0.00112827	−0.000564135	−	1.00000i	$-0.500180\pi$
$313$	−6.24784	−0.00112827	−0.000564135		1.00000i	$0.500180\pi$
$314$	−4601.99	−0.827088
$315$	0	0
$316$	1578.00	0.280915
$317$	−965.803	−0.171120	−0.0855598	−	0.996333i	$-0.527268\pi$
$317$	−965.803	−0.171120	−0.0855598	+	0.996333i	$0.527268\pi$
$318$	0	0
$319$	3187.29	0.559417
$320$	4105.29	0.717164
$321$	0	0
$322$	0	0
$323$	−1700.46	−0.292929
$324$	0	0
$325$	−6800.27	−1.16065
$326$	1032.57	0.175426
$327$	0	0
$328$	−4365.80	−0.734942
$329$	0	0
$330$	0	0
$331$	6710.20	1.11428	0.557139	−	0.830419i	$-0.311899\pi$
$331$	6710.20	1.11428	0.557139	+	0.830419i	$0.311899\pi$
$332$	629.868	0.104122
$333$	0	0
$334$	−6016.72	−0.985690
$335$	11056.9	1.80330
$336$	0	0
$337$	−605.546	−0.0978819	−0.0489409	−	0.998802i	$-0.515585\pi$
$337$	−605.546	−0.0978819	−0.0489409	+	0.998802i	$0.515585\pi$
$338$	−1828.56	−0.294262
$339$	0	0
$340$	8102.03	1.29234
$341$	11881.2	1.88682
$342$	0	0
$343$	0	0
$344$	471.700	0.0739313
$345$	0	0
$346$	3689.88	0.573321
$347$	−6938.16	−1.07337	−0.536686	−	0.843782i	$-0.680324\pi$
$347$	−6938.16	−1.07337	−0.536686	+	0.843782i	$0.680324\pi$
$348$	0	0
$349$	−10368.9	−1.59035	−0.795176	−	0.606378i	$-0.792622\pi$
$349$	−10368.9	−1.59035	−0.795176	+	0.606378i	$0.792622\pi$
$350$	0	0
$351$	0	0
$352$	−11435.8	−1.73162
$353$	−5881.76	−0.886840	−0.443420	−	0.896314i	$-0.646235\pi$
$353$	−5881.76	−0.886840	−0.443420	+	0.896314i	$0.646235\pi$
$354$	0	0
$355$	−13097.4	−1.95814
$356$	−5115.98	−0.761647
$357$	0	0
$358$	−4848.11	−0.715728
$359$	−589.269	−0.0866307	−0.0433153	−	0.999061i	$-0.513792\pi$
$359$	−589.269	−0.0866307	−0.0433153	+	0.999061i	$0.513792\pi$
$360$	0	0
$361$	−6421.84	−0.936264
$362$	6519.60	0.946581
$363$	0	0
$364$	0	0
$365$	−7113.21	−1.02006
$366$	0	0
$367$	3548.72	0.504745	0.252373	−	0.967630i	$-0.418789\pi$
$367$	3548.72	0.504745	0.252373	+	0.967630i	$0.418789\pi$
$368$	311.375	0.0441074
$369$	0	0
$370$	−7809.25	−1.09725
$371$	0	0
$372$	0	0
$373$	−1581.33	−0.219513	−0.109756	−	0.993959i	$-0.535007\pi$
$373$	−1581.33	−0.219513	−0.109756	+	0.993959i	$0.535007\pi$
$374$	7962.79	1.10092
$375$	0	0
$376$	5327.29	0.730675
$377$	1687.94	0.230592
$378$	0	0
$379$	3057.01	0.414322	0.207161	−	0.978307i	$-0.433578\pi$
$379$	3057.01	0.414322	0.207161	+	0.978307i	$0.433578\pi$
$380$	−2082.91	−0.281187
$381$	0	0
$382$	1401.67	0.187737
$383$	−10579.7	−1.41149	−0.705744	−	0.708467i	$-0.749387\pi$
$383$	−10579.7	−1.41149	−0.705744	+	0.708467i	$0.749387\pi$
$384$	0	0
$385$	0	0
$386$	−2331.10	−0.307383
$387$	0	0
$388$	655.691	0.0857930
$389$	7393.29	0.963637	0.481819	−	0.876271i	$-0.339976\pi$
$389$	7393.29	0.963637	0.481819	+	0.876271i	$0.339976\pi$
$390$	0	0
$391$	−2743.31	−0.354821
$392$	0	0
$393$	0	0
$394$	1521.75	0.194580
$395$	−5299.86	−0.675101
$396$	0	0
$397$	−1778.17	−0.224796	−0.112398	−	0.993663i	$-0.535853\pi$
$397$	−1778.17	−0.224796	−0.112398	+	0.993663i	$0.535853\pi$
$398$	−5341.04	−0.672669
$399$	0	0
$400$	1934.71	0.241839
$401$	147.606	0.0183818	0.00919090	−	0.999958i	$-0.497074\pi$
$401$	147.606	0.0183818	0.00919090	+	0.999958i	$0.497074\pi$
$402$	0	0
$403$	6292.12	0.777748
$404$	−6976.63	−0.859159
$405$	0	0
$406$	0	0
$407$	16367.9	1.99343
$408$	0	0
$409$	2160.05	0.261143	0.130572	−	0.991439i	$-0.458319\pi$
$409$	2160.05	0.261143	0.130572	+	0.991439i	$0.458319\pi$
$410$	5939.06	0.715388
$411$	0	0
$412$	2893.79	0.346036
$413$	0	0
$414$	0	0
$415$	−2115.47	−0.250228
$416$	−6056.22	−0.713776
$417$	0	0
$418$	−2047.11	−0.239540
$419$	−13491.0	−1.57298	−0.786488	−	0.617605i	$-0.788103\pi$
$419$	−13491.0	−1.57298	−0.786488	+	0.617605i	$0.788103\pi$
$420$	0	0
$421$	−14146.7	−1.63769	−0.818847	−	0.574012i	$-0.805386\pi$
$421$	−14146.7	−1.63769	−0.818847	+	0.574012i	$0.805386\pi$
$422$	2300.03	0.265317
$423$	0	0
$424$	3026.05	0.346598
$425$	−17045.4	−1.94546
$426$	0	0
$427$	0	0
$428$	726.262	0.0820215
$429$	0	0
$430$	−641.682	−0.0719643
$431$	9088.52	1.01573	0.507864	−	0.861437i	$-0.330435\pi$
$431$	9088.52	1.01573	0.507864	+	0.861437i	$0.330435\pi$
$432$	0	0
$433$	−15461.2	−1.71597	−0.857986	−	0.513673i	$-0.828284\pi$
$433$	−15461.2	−1.71597	−0.857986	+	0.513673i	$0.828284\pi$
$434$	0	0
$435$	0	0
$436$	1186.02	0.130275
$437$	705.263	0.0772021
$438$	0	0
$439$	1782.18	0.193756	0.0968780	−	0.995296i	$-0.469114\pi$
$439$	1782.18	0.193756	0.0968780	+	0.995296i	$0.469114\pi$
$440$	24081.0	2.60913
$441$	0	0
$442$	4216.97	0.453803
$443$	13424.7	1.43979	0.719896	−	0.694082i	$-0.244190\pi$
$443$	13424.7	1.43979	0.719896	+	0.694082i	$0.244190\pi$
$444$	0	0
$445$	17182.5	1.83041
$446$	−1613.81	−0.171337
$447$	0	0
$448$	0	0
$449$	418.639	0.0440018	0.0220009	−	0.999758i	$-0.492996\pi$
$449$	418.639	0.0440018	0.0220009	+	0.999758i	$0.492996\pi$
$450$	0	0
$451$	−12448.0	−1.29968
$452$	−10925.6	−1.13694
$453$	0	0
$454$	4712.03	0.487107
$455$	0	0
$456$	0	0
$457$	−708.410	−0.0725121	−0.0362560	−	0.999343i	$-0.511543\pi$
$457$	−708.410	−0.0725121	−0.0362560	+	0.999343i	$0.511543\pi$
$458$	6454.30	0.658493
$459$	0	0
$460$	−3360.30	−0.340598
$461$	−8223.97	−0.830865	−0.415432	−	0.909624i	$-0.636370\pi$
$461$	−8223.97	−0.830865	−0.415432	+	0.909624i	$0.636370\pi$
$462$	0	0
$463$	−9414.17	−0.944954	−0.472477	−	0.881343i	$-0.656640\pi$
$463$	−9414.17	−0.944954	−0.472477	+	0.881343i	$0.656640\pi$
$464$	−480.227	−0.0480474
$465$	0	0
$466$	4787.45	0.475911
$467$	10821.5	1.07229	0.536146	−	0.844125i	$-0.319880\pi$
$467$	10821.5	1.07229	0.536146	+	0.844125i	$0.319880\pi$
$468$	0	0
$469$	0	0
$470$	−7247.02	−0.711234
$471$	0	0
$472$	4758.85	0.464076
$473$	1344.94	0.130741
$474$	0	0
$475$	4382.11	0.423295
$476$	0	0
$477$	0	0
$478$	−2892.93	−0.276819
$479$	−8185.50	−0.780804	−0.390402	−	0.920644i	$-0.627664\pi$
$479$	−8185.50	−0.780804	−0.390402	+	0.920644i	$0.627664\pi$
$480$	0	0
$481$	8668.19	0.821695
$482$	5991.79	0.566221
$483$	0	0
$484$	−13194.4	−1.23915
$485$	−2202.21	−0.206179
$486$	0	0
$487$	−4003.15	−0.372485	−0.186242	−	0.982504i	$-0.559631\pi$
$487$	−4003.15	−0.372485	−0.186242	+	0.982504i	$0.559631\pi$
$488$	14021.3	1.30065
$489$	0	0
$490$	0	0
$491$	−11180.8	−1.02766	−0.513831	−	0.857891i	$-0.671774\pi$
$491$	−11180.8	−1.02766	−0.513831	+	0.857891i	$0.671774\pi$
$492$	0	0
$493$	4230.95	0.386516
$494$	−1084.12	−0.0987386
$495$	0	0
$496$	−1790.14	−0.162056
$497$	0	0
$498$	0	0
$499$	−3771.08	−0.338310	−0.169155	−	0.985589i	$-0.554104\pi$
$499$	−3771.08	−0.338310	−0.169155	+	0.985589i	$0.554104\pi$
$500$	−8426.48	−0.753688
$501$	0	0
$502$	4325.87	0.384607
$503$	−13597.2	−1.20531	−0.602654	−	0.798003i	$-0.705890\pi$
$503$	−13597.2	−1.20531	−0.602654	+	0.798003i	$0.705890\pi$
$504$	0	0
$505$	23431.7	2.06475
$506$	−3302.55	−0.290151
$507$	0	0
$508$	−8925.93	−0.779575
$509$	7360.75	0.640982	0.320491	−	0.947252i	$-0.396152\pi$
$509$	7360.75	0.640982	0.320491	+	0.947252i	$0.396152\pi$
$510$	0	0
$511$	0	0
$512$	3309.87	0.285698
$513$	0	0
$514$	−8590.00	−0.737137
$515$	−9719.08	−0.831600
$516$	0	0
$517$	15189.5	1.29213
$518$	0	0
$519$	0	0
$520$	12752.9	1.07549
$521$	13798.8	1.16034	0.580169	−	0.814496i	$-0.302987\pi$
$521$	13798.8	1.16034	0.580169	+	0.814496i	$0.302987\pi$
$522$	0	0
$523$	−18847.2	−1.57577	−0.787886	−	0.615821i	$-0.788825\pi$
$523$	−18847.2	−1.57577	−0.787886	+	0.615821i	$0.788825\pi$
$524$	−499.697	−0.0416591
$525$	0	0
$526$	8384.29	0.695005
$527$	15771.7	1.30365
$528$	0	0
$529$	−11029.2	−0.906486
$530$	−4116.51	−0.337377
$531$	0	0
$532$	0	0
$533$	−6592.29	−0.535730
$534$	0	0
$535$	−2439.23	−0.197116
$536$	−12988.9	−1.04671
$537$	0	0
$538$	−4815.44	−0.385889
$539$	0	0
$540$	0	0
$541$	−14469.5	−1.14990	−0.574948	−	0.818190i	$-0.694978\pi$
$541$	−14469.5	−1.14990	−0.574948	+	0.818190i	$0.694978\pi$
$542$	−11022.2	−0.873511
$543$	0	0
$544$	−15180.4	−1.19642
$545$	−3983.36	−0.313080
$546$	0	0
$547$	5749.63	0.449427	0.224713	−	0.974425i	$-0.427855\pi$
$547$	5749.63	0.449427	0.224713	+	0.974425i	$0.427855\pi$
$548$	−10168.8	−0.792681
$549$	0	0
$550$	−20520.2	−1.59088
$551$	−1087.71	−0.0840983
$552$	0	0
$553$	0	0
$554$	5303.31	0.406708
$555$	0	0
$556$	3484.37	0.265774
$557$	−5430.77	−0.413122	−0.206561	−	0.978434i	$-0.566227\pi$
$557$	−5430.77	−0.413122	−0.206561	+	0.978434i	$0.566227\pi$
$558$	0	0
$559$	712.260	0.0538915
$560$	0	0
$561$	0	0
$562$	10026.4	0.752561
$563$	−24130.7	−1.80637	−0.903185	−	0.429251i	$-0.858777\pi$
$563$	−24130.7	−1.80637	−0.903185	+	0.429251i	$0.858777\pi$
$564$	0	0
$565$	36694.7	2.73232
$566$	−12420.8	−0.922415
$567$	0	0
$568$	15385.9	1.13658
$569$	−18048.5	−1.32976	−0.664880	−	0.746950i	$-0.731518\pi$
$569$	−18048.5	−1.32976	−0.664880	+	0.746950i	$0.731518\pi$
$570$	0	0
$571$	−11274.8	−0.826336	−0.413168	−	0.910655i	$-0.635578\pi$
$571$	−11274.8	−0.826336	−0.413168	+	0.910655i	$0.635578\pi$
$572$	−10826.5	−0.791396
$573$	0	0
$574$	0	0
$575$	7069.54	0.512731
$576$	0	0
$577$	24094.9	1.73845	0.869225	−	0.494417i	$-0.164618\pi$
$577$	24094.9	1.73845	0.869225	+	0.494417i	$0.164618\pi$
$578$	2718.91	0.195661
$579$	0	0
$580$	5182.53	0.371022
$581$	0	0
$582$	0	0
$583$	8628.04	0.612928
$584$	8356.10	0.592085
$585$	0	0
$586$	1366.17	0.0963072
$587$	11438.9	0.804315	0.402157	−	0.915571i	$-0.368260\pi$
$587$	11438.9	0.804315	0.402157	+	0.915571i	$0.368260\pi$
$588$	0	0
$589$	−4054.66	−0.283649
$590$	−6473.74	−0.451729
$591$	0	0
$592$	−2466.14	−0.171213
$593$	−4174.44	−0.289079	−0.144539	−	0.989499i	$-0.546170\pi$
$593$	−4174.44	−0.289079	−0.144539	+	0.989499i	$0.546170\pi$
$594$	0	0
$595$	0	0
$596$	−17082.2	−1.17402
$597$	0	0
$598$	−1748.98	−0.119601
$599$	11455.5	0.781403	0.390702	−	0.920517i	$-0.372232\pi$
$599$	11455.5	0.781403	0.390702	+	0.920517i	$0.372232\pi$
$600$	0	0
$601$	17539.2	1.19042	0.595208	−	0.803572i	$-0.297070\pi$
$601$	17539.2	1.19042	0.595208	+	0.803572i	$0.297070\pi$
$602$	0	0
$603$	0	0
$604$	14821.2	0.998452
$605$	44314.9	2.97794
$606$	0	0
$607$	−3385.72	−0.226396	−0.113198	−	0.993572i	$-0.536109\pi$
$607$	−3385.72	−0.226396	−0.113198	+	0.993572i	$0.536109\pi$
$608$	3902.65	0.260318
$609$	0	0
$610$	−19074.1	−1.26604
$611$	8044.11	0.532619
$612$	0	0
$613$	4271.88	0.281468	0.140734	−	0.990047i	$-0.455054\pi$
$613$	4271.88	0.281468	0.140734	+	0.990047i	$0.455054\pi$
$614$	−4076.12	−0.267913
$615$	0	0
$616$	0	0
$617$	13123.9	0.856321	0.428160	−	0.903703i	$-0.359162\pi$
$617$	13123.9	0.856321	0.428160	+	0.903703i	$0.359162\pi$
$618$	0	0
$619$	1893.94	0.122979	0.0614893	−	0.998108i	$-0.480415\pi$
$619$	1893.94	0.122979	0.0614893	+	0.998108i	$0.480415\pi$
$620$	19318.9	1.25139
$621$	0	0
$622$	11954.3	0.770614
$623$	0	0
$624$	0	0
$625$	2102.97	0.134590
$626$	−9.98439	−0.000637470 0
$627$	0	0
$628$	15683.7	0.996576
$629$	21727.5	1.37731
$630$	0	0
$631$	20443.8	1.28979	0.644894	−	0.764272i	$-0.276902\pi$
$631$	20443.8	1.28979	0.644894	+	0.764272i	$0.276902\pi$
$632$	6225.90	0.391856
$633$	0	0
$634$	−1543.41	−0.0966821
$635$	29978.6	1.87349
$636$	0	0
$637$	0	0
$638$	5093.46	0.316069
$639$	0	0
$640$	−20753.3	−1.28179
$641$	19228.5	1.18484	0.592419	−	0.805630i	$-0.298173\pi$
$641$	19228.5	1.18484	0.592419	+	0.805630i	$0.298173\pi$
$642$	0	0
$643$	18525.1	1.13617	0.568087	−	0.822969i	$-0.307684\pi$
$643$	18525.1	1.13617	0.568087	+	0.822969i	$0.307684\pi$
$644$	0	0
$645$	0	0
$646$	−2717.43	−0.165504
$647$	−8022.39	−0.487470	−0.243735	−	0.969842i	$-0.578373\pi$
$647$	−8022.39	−0.487470	−0.243735	+	0.969842i	$0.578373\pi$
$648$	0	0
$649$	13568.7	0.820676
$650$	−10867.2	−0.655764
$651$	0	0
$652$	−3519.03	−0.211374
$653$	2051.69	0.122954	0.0614769	−	0.998109i	$-0.480419\pi$
$653$	2051.69	0.122954	0.0614769	+	0.998109i	$0.480419\pi$
$654$	0	0
$655$	1678.28	0.100116
$656$	1875.54	0.111627
$657$	0	0
$658$	0	0
$659$	14765.2	0.872792	0.436396	−	0.899755i	$-0.356255\pi$
$659$	14765.2	0.872792	0.436396	+	0.899755i	$0.356255\pi$
$660$	0	0
$661$	−647.064	−0.0380755	−0.0190377	−	0.999819i	$-0.506060\pi$
$661$	−647.064	−0.0380755	−0.0190377	+	0.999819i	$0.506060\pi$
$662$	10723.3	0.629564
$663$	0	0
$664$	2485.11	0.145243
$665$	0	0
$666$	0	0
$667$	−1754.78	−0.101867
$668$	20505.2	1.18768
$669$	0	0
$670$	17669.6	1.01886
$671$	39978.5	2.30008
$672$	0	0
$673$	−22596.6	−1.29426	−0.647130	−	0.762380i	$-0.724031\pi$
$673$	−22596.6	−1.29426	−0.647130	+	0.762380i	$0.724031\pi$
$674$	−967.695	−0.0553030
$675$	0	0
$676$	6231.80	0.354563
$677$	−25204.3	−1.43084	−0.715420	−	0.698695i	$-0.753765\pi$
$677$	−25204.3	−1.43084	−0.715420	+	0.698695i	$0.753765\pi$
$678$	0	0
$679$	0	0
$680$	31966.2	1.80272
$681$	0	0
$682$	18986.8	1.06605
$683$	−17020.5	−0.953547	−0.476774	−	0.879026i	$-0.658194\pi$
$683$	−17020.5	−0.953547	−0.476774	+	0.879026i	$0.658194\pi$
$684$	0	0
$685$	34152.9	1.90499
$686$	0	0
$687$	0	0
$688$	−202.641	−0.0112291
$689$	4569.28	0.252650
$690$	0	0
$691$	19290.5	1.06201	0.531003	−	0.847370i	$-0.321815\pi$
$691$	19290.5	1.06201	0.531003	+	0.847370i	$0.321815\pi$
$692$	−12575.2	−0.690806
$693$	0	0
$694$	−11087.6	−0.606452
$695$	−11702.6	−0.638713
$696$	0	0
$697$	−16524.1	−0.897983
$698$	−16570.0	−0.898545
$699$	0	0
$700$	0	0
$701$	−28511.4	−1.53618	−0.768088	−	0.640345i	$-0.778792\pi$
$701$	−28511.4	−1.53618	−0.768088	+	0.640345i	$0.778792\pi$
$702$	0	0
$703$	−5585.81	−0.299677
$704$	−13750.5	−0.736138
$705$	0	0
$706$	−9399.37	−0.501062
$707$	0	0
$708$	0	0
$709$	28426.5	1.50575	0.752877	−	0.658161i	$-0.228665\pi$
$709$	28426.5	1.50575	0.752877	+	0.658161i	$0.228665\pi$
$710$	−20930.4	−1.10634
$711$	0	0
$712$	−20184.8	−1.06244
$713$	−6541.27	−0.343580
$714$	0	0
$715$	36361.9	1.90190
$716$	16522.5	0.862396
$717$	0	0
$718$	−941.683	−0.0489461
$719$	−21763.3	−1.12884	−0.564420	−	0.825488i	$-0.690900\pi$
$719$	−21763.3	−1.12884	−0.564420	+	0.825488i	$0.690900\pi$
$720$	0	0
$721$	0	0
$722$	−10262.4	−0.528987
$723$	0	0
$724$	−22219.0	−1.14056
$725$	−10903.2	−0.558532
$726$	0	0
$727$	13422.8	0.684763	0.342382	−	0.939561i	$-0.388766\pi$
$727$	13422.8	0.684763	0.342382	+	0.939561i	$0.388766\pi$
$728$	0	0
$729$	0	0
$730$	−11367.3	−0.576332
$731$	1785.33	0.0903323
$732$	0	0
$733$	−4559.52	−0.229754	−0.114877	−	0.993380i	$-0.536647\pi$
$733$	−4559.52	−0.229754	−0.114877	+	0.993380i	$0.536647\pi$
$734$	5671.04	0.285180
$735$	0	0
$736$	6296.04	0.315319
$737$	−37034.7	−1.85101
$738$	0	0
$739$	−36665.4	−1.82511	−0.912557	−	0.408950i	$-0.865895\pi$
$739$	−36665.4	−1.82511	−0.912557	+	0.408950i	$0.865895\pi$
$740$	26614.2	1.32210
$741$	0	0
$742$	0	0
$743$	10321.3	0.509625	0.254813	−	0.966990i	$-0.417986\pi$
$743$	10321.3	0.509625	0.254813	+	0.966990i	$0.417986\pi$
$744$	0	0
$745$	57372.4	2.82143
$746$	−2527.06	−0.124024
$747$	0	0
$748$	−27137.4	−1.32653
$749$	0	0
$750$	0	0
$751$	−26678.1	−1.29627	−0.648134	−	0.761526i	$-0.724450\pi$
$751$	−26678.1	−1.29627	−0.648134	+	0.761526i	$0.724450\pi$
$752$	−2288.59	−0.110979
$753$	0	0
$754$	2697.42	0.130284
$755$	−49778.4	−2.39950
$756$	0	0
$757$	−11630.8	−0.558425	−0.279212	−	0.960229i	$-0.590073\pi$
$757$	−11630.8	−0.558425	−0.279212	+	0.960229i	$0.590073\pi$
$758$	4885.27	0.234091
$759$	0	0
$760$	−8218.02	−0.392235
$761$	36091.7	1.71921	0.859607	−	0.510956i	$-0.170709\pi$
$761$	36091.7	1.71921	0.859607	+	0.510956i	$0.170709\pi$
$762$	0	0
$763$	0	0
$764$	−4776.93	−0.226208
$765$	0	0
$766$	−16907.0	−0.797487
$767$	7185.79	0.338284
$768$	0	0
$769$	−33089.3	−1.55167	−0.775833	−	0.630938i	$-0.782670\pi$
$769$	−33089.3	−1.55167	−0.775833	+	0.630938i	$0.782670\pi$
$770$	0	0
$771$	0	0
$772$	7944.47	0.370373
$773$	−30990.8	−1.44199	−0.720997	−	0.692938i	$-0.756316\pi$
$773$	−30990.8	−1.44199	−0.720997	+	0.692938i	$0.756316\pi$
$774$	0	0
$775$	−40643.8	−1.88383
$776$	2587.00	0.119675
$777$	0	0
$778$	11814.9	0.544453
$779$	4248.09	0.195383
$780$	0	0
$781$	43869.2	2.00994
$782$	−4383.95	−0.200473
$783$	0	0
$784$	0	0
$785$	−52675.4	−2.39499
$786$	0	0
$787$	25421.2	1.15142	0.575711	−	0.817653i	$-0.304725\pi$
$787$	25421.2	1.15142	0.575711	+	0.817653i	$0.304725\pi$
$788$	−5186.17	−0.234454
$789$	0	0
$790$	−8469.46	−0.381430
$791$	0	0
$792$	0	0
$793$	21172.0	0.948096
$794$	−2841.62	−0.127009
$795$	0	0
$796$	18202.4	0.810513
$797$	−20283.3	−0.901470	−0.450735	−	0.892658i	$-0.648838\pi$
$797$	−20283.3	−0.901470	−0.450735	+	0.892658i	$0.648838\pi$
$798$	0	0
$799$	20163.2	0.892768
$800$	39120.1	1.72888
$801$	0	0
$802$	235.883	0.0103857
$803$	23825.4	1.04705
$804$	0	0
$805$	0	0
$806$	10055.1	0.439426
$807$	0	0
$808$	−27525.9	−1.19846
$809$	−16431.3	−0.714086	−0.357043	−	0.934088i	$-0.616215\pi$
$809$	−16431.3	−0.714086	−0.357043	+	0.934088i	$0.616215\pi$
$810$	0	0
$811$	−6371.81	−0.275887	−0.137944	−	0.990440i	$-0.544049\pi$
$811$	−6371.81	−0.275887	−0.137944	+	0.990440i	$0.544049\pi$
$812$	0	0
$813$	0	0
$814$	26156.8	1.12628
$815$	11819.0	0.507979
$816$	0	0
$817$	−458.982	−0.0196545
$818$	3451.88	0.147545
$819$	0	0
$820$	−20240.5	−0.861987
$821$	5224.48	0.222090	0.111045	−	0.993815i	$-0.464580\pi$
$821$	5224.48	0.222090	0.111045	+	0.993815i	$0.464580\pi$
$822$	0	0
$823$	3713.10	0.157267	0.0786333	−	0.996904i	$-0.474944\pi$
$823$	3713.10	0.157267	0.0786333	+	0.996904i	$0.474944\pi$
$824$	11417.3	0.482695
$825$	0	0
$826$	0	0
$827$	−10202.6	−0.428996	−0.214498	−	0.976724i	$-0.568812\pi$
$827$	−10202.6	−0.428996	−0.214498	+	0.976724i	$0.568812\pi$
$828$	0	0
$829$	−24995.2	−1.04719	−0.523594	−	0.851968i	$-0.675409\pi$
$829$	−24995.2	−1.04719	−0.523594	+	0.851968i	$0.675409\pi$
$830$	−3380.64	−0.141378
$831$	0	0
$832$	−7282.06	−0.303437
$833$	0	0
$834$	0	0
$835$	−68868.8	−2.85425
$836$	6976.63	0.288626
$837$	0	0
$838$	−21559.3	−0.888728
$839$	31173.6	1.28275	0.641377	−	0.767226i	$-0.278363\pi$
$839$	31173.6	1.28275	0.641377	+	0.767226i	$0.278363\pi$
$840$	0	0
$841$	−21682.6	−0.889033
$842$	−22607.2	−0.925293
$843$	0	0
$844$	−7838.57	−0.319686
$845$	−20930.1	−0.852093
$846$	0	0
$847$	0	0
$848$	−1299.98	−0.0526434
$849$	0	0
$850$	−27239.4	−1.09918
$851$	−9011.43	−0.362994
$852$	0	0
$853$	25280.7	1.01477	0.507383	−	0.861721i	$-0.330613\pi$
$853$	25280.7	1.01477	0.507383	+	0.861721i	$0.330613\pi$
$854$	0	0
$855$	0	0
$856$	2865.43	0.114414
$857$	−27410.7	−1.09257	−0.546285	−	0.837599i	$-0.683959\pi$
$857$	−27410.7	−1.09257	−0.546285	+	0.837599i	$0.683959\pi$
$858$	0	0
$859$	−24558.8	−0.975480	−0.487740	−	0.872989i	$-0.662179\pi$
$859$	−24558.8	−0.975480	−0.487740	+	0.872989i	$0.662179\pi$
$860$	2186.87	0.0867113
$861$	0	0
$862$	14523.9	0.573883
$863$	7148.35	0.281962	0.140981	−	0.990012i	$-0.454974\pi$
$863$	7148.35	0.281962	0.140981	+	0.990012i	$0.454974\pi$
$864$	0	0
$865$	42235.1	1.66016
$866$	−24707.8	−0.969520
$867$	0	0
$868$	0	0
$869$	17751.7	0.692962
$870$	0	0
$871$	−19613.0	−0.762988
$872$	4679.37	0.181724
$873$	0	0
$874$	1127.05	0.0436190
$875$	0	0
$876$	0	0
$877$	−28218.5	−1.08651	−0.543256	−	0.839567i	$-0.682809\pi$
$877$	−28218.5	−1.08651	−0.543256	+	0.839567i	$0.682809\pi$
$878$	2848.02	0.109472
$879$	0	0
$880$	−10345.1	−0.396289
$881$	7431.50	0.284192	0.142096	−	0.989853i	$-0.454616\pi$
$881$	7431.50	0.284192	0.142096	+	0.989853i	$0.454616\pi$
$882$	0	0
$883$	4937.77	0.188187	0.0940936	−	0.995563i	$-0.470005\pi$
$883$	4937.77	0.188187	0.0940936	+	0.995563i	$0.470005\pi$
$884$	−14371.6	−0.546797
$885$	0	0
$886$	21453.4	0.813478
$887$	−41973.7	−1.58888	−0.794441	−	0.607341i	$-0.792236\pi$
$887$	−41973.7	−1.58888	−0.794441	+	0.607341i	$0.792236\pi$
$888$	0	0
$889$	0	0
$890$	27458.6	1.03417
$891$	0	0
$892$	5499.93	0.206448
$893$	−5183.65	−0.194249
$894$	0	0
$895$	−55492.5	−2.07253
$896$	0	0
$897$	0	0
$898$	669.008	0.0248609
$899$	10088.5	0.374271
$900$	0	0
$901$	11453.2	0.423488
$902$	−19892.6	−0.734315
$903$	0	0
$904$	−43106.4	−1.58595
$905$	74624.7	2.74101
$906$	0	0
$907$	41424.2	1.51650	0.758252	−	0.651961i	$-0.226054\pi$
$907$	41424.2	1.51650	0.758252	+	0.651961i	$0.226054\pi$
$908$	−16058.7	−0.586925
$909$	0	0
$910$	0	0
$911$	40072.0	1.45735	0.728675	−	0.684860i	$-0.240136\pi$
$911$	40072.0	1.45735	0.728675	+	0.684860i	$0.240136\pi$
$912$	0	0
$913$	7085.70	0.256848
$914$	−1132.08	−0.0409691
$915$	0	0
$916$	−21996.5	−0.793432
$917$	0	0
$918$	0	0
$919$	−21817.5	−0.783124	−0.391562	−	0.920152i	$-0.628065\pi$
$919$	−21817.5	−0.783124	−0.391562	+	0.920152i	$0.628065\pi$
$920$	−13257.9	−0.475109
$921$	0	0
$922$	−13142.4	−0.469437
$923$	23232.5	0.828501
$924$	0	0
$925$	−55992.0	−1.99028
$926$	−15044.4	−0.533897
$927$	0	0
$928$	−9710.26	−0.343486
$929$	11977.0	0.422986	0.211493	−	0.977380i	$-0.432167\pi$
$929$	11977.0	0.422986	0.211493	+	0.977380i	$0.432167\pi$
$930$	0	0
$931$	0	0
$932$	−16315.8	−0.573435
$933$	0	0
$934$	17293.4	0.605842
$935$	91143.8	3.18794
$936$	0	0
$937$	15155.2	0.528389	0.264194	−	0.964469i	$-0.414894\pi$
$937$	15155.2	0.528389	0.264194	+	0.964469i	$0.414894\pi$
$938$	0	0
$939$	0	0
$940$	24698.1	0.856981
$941$	−6954.40	−0.240921	−0.120461	−	0.992718i	$-0.538437\pi$
$941$	−6954.40	−0.240921	−0.120461	+	0.992718i	$0.538437\pi$
$942$	0	0
$943$	6853.33	0.236665
$944$	−2044.39	−0.0704865
$945$	0	0
$946$	2149.29	0.0738682
$947$	−9557.49	−0.327959	−0.163979	−	0.986464i	$-0.552433\pi$
$947$	−9557.49	−0.327959	−0.163979	+	0.986464i	$0.552433\pi$
$948$	0	0
$949$	12617.6	0.431595
$950$	7002.85	0.239161
$951$	0	0
$952$	0	0
$953$	8437.24	0.286788	0.143394	−	0.989666i	$-0.454198\pi$
$953$	8437.24	0.286788	0.143394	+	0.989666i	$0.454198\pi$
$954$	0	0
$955$	16043.8	0.543628
$956$	9859.21	0.333546
$957$	0	0
$958$	−13080.9	−0.441153
$959$	0	0
$960$	0	0
$961$	7815.69	0.262351
$962$	13852.2	0.464256
$963$	0	0
$964$	−20420.2	−0.682252
$965$	−26682.3	−0.890087
$966$	0	0
$967$	52344.7	1.74074	0.870369	−	0.492401i	$-0.163881\pi$
$967$	52344.7	1.74074	0.870369	+	0.492401i	$0.163881\pi$
$968$	−52058.0	−1.72852
$969$	0	0
$970$	−3519.24	−0.116491
$971$	36983.0	1.22229	0.611143	−	0.791520i	$-0.290710\pi$
$971$	36983.0	1.22229	0.611143	+	0.791520i	$0.290710\pi$
$972$	0	0
$973$	0	0
$974$	−6397.25	−0.210453
$975$	0	0
$976$	−6023.54	−0.197550
$977$	−17873.0	−0.585270	−0.292635	−	0.956224i	$-0.594532\pi$
$977$	−17873.0	−0.585270	−0.292635	+	0.956224i	$0.594532\pi$
$978$	0	0
$979$	−57552.2	−1.87883
$980$	0	0
$981$	0	0
$982$	−17867.5	−0.580627
$983$	33744.0	1.09488	0.547440	−	0.836845i	$-0.315602\pi$
$983$	33744.0	1.09488	0.547440	+	0.836845i	$0.315602\pi$
$984$	0	0
$985$	17418.3	0.563444
$986$	6761.29	0.218381
$987$	0	0
$988$	3694.72	0.118972
$989$	−740.464	−0.0238073
$990$	0	0
$991$	−14506.5	−0.464999	−0.232499	−	0.972597i	$-0.574690\pi$
$991$	−14506.5	−0.464999	−0.232499	+	0.972597i	$0.574690\pi$
$992$	−36196.8	−1.15852
$993$	0	0
$994$	0	0
$995$	−61134.7	−1.94784
$996$	0	0
$997$	42368.8	1.34587	0.672936	−	0.739701i	$-0.265033\pi$
$997$	42368.8	1.34587	0.672936	+	0.739701i	$0.265033\pi$
$998$	−6026.40	−0.191145
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.4.a.v.1.3		4
3.2	odd	2	inner	441.4.a.v.1.2		4
7.2	even	3		441.4.e.x.361.2		8
7.3	odd	6		63.4.e.d.37.2	✓	8
7.4	even	3		441.4.e.x.226.2		8
7.5	odd	6		63.4.e.d.46.2	yes	8
7.6	odd	2		441.4.a.w.1.3		4
21.2	odd	6		441.4.e.x.361.3		8
21.5	even	6		63.4.e.d.46.3	yes	8
21.11	odd	6		441.4.e.x.226.3		8
21.17	even	6		63.4.e.d.37.3	yes	8
21.20	even	2		441.4.a.w.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.e.d.37.2	✓	8	7.3	odd	6
63.4.e.d.37.3	yes	8	21.17	even	6
63.4.e.d.46.2	yes	8	7.5	odd	6
63.4.e.d.46.3	yes	8	21.5	even	6
441.4.a.v.1.2		4	3.2	odd	2	inner
441.4.a.v.1.3		4	1.1	even	1	trivial
441.4.a.w.1.2		4	21.20	even	2
441.4.a.w.1.3		4	7.6	odd	2
441.4.e.x.226.2		8	7.4	even	3
441.4.e.x.226.3		8	21.11	odd	6
441.4.e.x.361.2		8	7.2	even	3
441.4.e.x.361.3		8	21.2	odd	6

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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(f(q)\)	\(=\)	\(q+1.59805 q^{2} -5.44622 q^{4} +18.2917 q^{5} -21.4878 q^{8} +O(q^{10})\)	\(q+1.59805 q^{2} -5.44622 q^{4} +18.2917 q^{5} -21.4878 q^{8} +29.2311 q^{10} -61.2673 q^{11} -32.4462 q^{13} +9.23111 q^{16} -81.3289 q^{17} +20.9084 q^{19} -99.6206 q^{20} -97.9084 q^{22} +33.7310 q^{23} +209.586 q^{25} -51.8508 q^{26} -52.0227 q^{29} -193.924 q^{31} +186.654 q^{32} -129.968 q^{34} -267.156 q^{37} +33.4128 q^{38} -393.048 q^{40} +203.176 q^{41} -21.9520 q^{43} +333.675 q^{44} +53.9040 q^{46} -247.921 q^{47} +334.929 q^{50} +176.709 q^{52} -140.826 q^{53} -1120.68 q^{55} -83.1351 q^{58} -221.468 q^{59} -652.526 q^{61} -309.902 q^{62} +224.435 q^{64} -593.496 q^{65} +604.478 q^{67} +442.935 q^{68} -716.031 q^{71} -388.876 q^{73} -426.929 q^{74} -113.872 q^{76} -289.741 q^{79} +168.853 q^{80} +324.686 q^{82} -115.652 q^{83} -1487.64 q^{85} -35.0805 q^{86} +1316.50 q^{88} +939.363 q^{89} -183.707 q^{92} -396.192 q^{94} +382.451 q^{95} -120.394 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{4}+O(q^{10}) \) 4 * q + 6 * q^4	\( 4 q + 6 q^{4} - 22 q^{10} - 102 q^{13} - 102 q^{16} - 222 q^{19} - 86 q^{22} + 366 q^{25} - 220 q^{31} - 1020 q^{34} - 374 q^{37} - 822 q^{40} - 838 q^{43} + 1716 q^{46} + 40 q^{52} - 2510 q^{55} - 1694 q^{58} - 1332 q^{61} - 686 q^{64} + 1890 q^{67} - 1750 q^{73} - 2456 q^{76} + 8 q^{79} - 2480 q^{82} - 1116 q^{85} + 2682 q^{88} + 1416 q^{94} - 3010 q^{97}+O(q^{100}) \) 4 * q + 6 * q^4 - 22 * q^10 - 102 * q^13 - 102 * q^16 - 222 * q^19 - 86 * q^22 + 366 * q^25 - 220 * q^31 - 1020 * q^34 - 374 * q^37 - 822 * q^40 - 838 * q^43 + 1716 * q^46 + 40 * q^52 - 2510 * q^55 - 1694 * q^58 - 1332 * q^61 - 686 * q^64 + 1890 * q^67 - 1750 * q^73 - 2456 * q^76 + 8 * q^79 - 2480 * q^82 - 1116 * q^85 + 2682 * q^88 + 1416 * q^94 - 3010 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more