LMFDB - Embedded newform 441.4.a.n.1.1

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:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 147)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q-2.41421 q^{2} -2.17157 q^{4} -19.8995 q^{5} +24.5563 q^{8} +O(q^{10})$	$q-2.41421 q^{2} -2.17157 q^{4} -19.8995 q^{5} +24.5563 q^{8} +48.0416 q^{10} -23.9411 q^{11} +87.3553 q^{13} -41.9117 q^{16} +5.63961 q^{17} +64.8873 q^{19} +43.2132 q^{20} +57.7990 q^{22} +25.5980 q^{23} +270.990 q^{25} -210.894 q^{26} -60.3188 q^{29} -122.711 q^{31} -95.2670 q^{32} -13.6152 q^{34} -56.1177 q^{37} -156.652 q^{38} -488.659 q^{40} +299.713 q^{41} -501.421 q^{43} +51.9899 q^{44} -61.7990 q^{46} -305.553 q^{47} -654.227 q^{50} -189.698 q^{52} +375.117 q^{53} +476.416 q^{55} +145.622 q^{58} +627.612 q^{59} +3.75736 q^{61} +296.250 q^{62} +565.288 q^{64} -1738.33 q^{65} -813.048 q^{67} -12.2468 q^{68} -165.902 q^{71} +619.100 q^{73} +135.480 q^{74} -140.908 q^{76} -138.246 q^{79} +834.021 q^{80} -723.571 q^{82} -621.137 q^{83} -112.225 q^{85} +1210.54 q^{86} -587.907 q^{88} -285.418 q^{89} -55.5879 q^{92} +737.671 q^{94} -1291.22 q^{95} -603.114 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 10 q^{4} - 20 q^{5} + 18 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 - 10 * q^4 - 20 * q^5 + 18 * q^8	$ 2 q - 2 q^{2} - 10 q^{4} - 20 q^{5} + 18 q^{8} + 48 q^{10} + 20 q^{11} + 104 q^{13} + 18 q^{16} - 116 q^{17} + 192 q^{19} + 44 q^{20} + 76 q^{22} - 28 q^{23} + 146 q^{25} - 204 q^{26} - 296 q^{29} - 104 q^{31} - 18 q^{32} - 64 q^{34} - 248 q^{37} - 104 q^{38} - 488 q^{40} - 20 q^{41} - 720 q^{43} - 292 q^{44} - 84 q^{46} + 96 q^{47} - 706 q^{50} - 320 q^{52} - 268 q^{53} + 472 q^{55} + 48 q^{58} + 616 q^{59} + 16 q^{61} + 304 q^{62} + 118 q^{64} - 1740 q^{65} - 144 q^{67} + 940 q^{68} - 988 q^{71} + 104 q^{73} + 56 q^{74} - 1136 q^{76} - 944 q^{79} + 828 q^{80} - 856 q^{82} - 1016 q^{83} - 100 q^{85} + 1120 q^{86} - 876 q^{88} + 388 q^{89} + 364 q^{92} + 904 q^{94} - 1304 q^{95} + 488 q^{97}+O(q^{100}) $ 2 * q - 2 * q^2 - 10 * q^4 - 20 * q^5 + 18 * q^8 + 48 * q^10 + 20 * q^11 + 104 * q^13 + 18 * q^16 - 116 * q^17 + 192 * q^19 + 44 * q^20 + 76 * q^22 - 28 * q^23 + 146 * q^25 - 204 * q^26 - 296 * q^29 - 104 * q^31 - 18 * q^32 - 64 * q^34 - 248 * q^37 - 104 * q^38 - 488 * q^40 - 20 * q^41 - 720 * q^43 - 292 * q^44 - 84 * q^46 + 96 * q^47 - 706 * q^50 - 320 * q^52 - 268 * q^53 + 472 * q^55 + 48 * q^58 + 616 * q^59 + 16 * q^61 + 304 * q^62 + 118 * q^64 - 1740 * q^65 - 144 * q^67 + 940 * q^68 - 988 * q^71 + 104 * q^73 + 56 * q^74 - 1136 * q^76 - 944 * q^79 + 828 * q^80 - 856 * q^82 - 1016 * q^83 - 100 * q^85 + 1120 * q^86 - 876 * q^88 + 388 * q^89 + 364 * q^92 + 904 * q^94 - 1304 * q^95 + 488 * 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 $

$2$	−2.41421	−0.853553	−0.426777	−	0.904357i	$-0.640351\pi$
$2$	−2.41421	−0.853553	−0.426777	+	0.904357i	$0.640351\pi$
$3$	0	0
$3$	0	0
$4$	−2.17157	−0.271447
$5$	−19.8995	−1.77986	−0.889932	−	0.456092i	$-0.849249\pi$
$5$	−19.8995	−1.77986	−0.889932	+	0.456092i	$0.849249\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	24.5563	1.08525
$9$	0	0
$10$	48.0416	1.51921
$11$	−23.9411	−0.656229	−0.328115	−	0.944638i	$-0.606413\pi$
$11$	−23.9411	−0.656229	−0.328115	+	0.944638i	$0.606413\pi$
$12$	0	0
$13$	87.3553	1.86369	0.931847	−	0.362852i	$-0.118197\pi$
$13$	87.3553	1.86369	0.931847	+	0.362852i	$0.118197\pi$
$14$	0	0
$15$	0	0
$16$	−41.9117	−0.654870
$17$	5.63961	0.0804592	0.0402296	−	0.999190i	$-0.487191\pi$
$17$	5.63961	0.0804592	0.0402296	+	0.999190i	$0.487191\pi$
$18$	0	0
$19$	64.8873	0.783483	0.391741	−	0.920075i	$-0.371873\pi$
$19$	64.8873	0.783483	0.391741	+	0.920075i	$0.371873\pi$
$20$	43.2132	0.483138
$21$	0	0
$22$	57.7990	0.560127
$23$	25.5980	0.232067	0.116034	−	0.993245i	$-0.462982\pi$
$23$	25.5980	0.232067	0.116034	+	0.993245i	$0.462982\pi$
$24$	0	0
$25$	270.990	2.16792
$26$	−210.894	−1.59076
$27$	0	0
$28$	0	0
$29$	−60.3188	−0.386238	−0.193119	−	0.981175i	$-0.561860\pi$
$29$	−60.3188	−0.386238	−0.193119	+	0.981175i	$0.561860\pi$
$30$	0	0
$31$	−122.711	−0.710951	−0.355476	−	0.934686i	$-0.615681\pi$
$31$	−122.711	−0.710951	−0.355476	+	0.934686i	$0.615681\pi$
$32$	−95.2670	−0.526281
$33$	0	0
$34$	−13.6152	−0.0686762
$35$	0	0
$36$	0	0
$37$	−56.1177	−0.249343	−0.124672	−	0.992198i	$-0.539788\pi$
$37$	−56.1177	−0.249343	−0.124672	+	0.992198i	$0.539788\pi$
$38$	−156.652	−0.668744
$39$	0	0
$40$	−488.659	−1.93159
$41$	299.713	1.14164	0.570820	−	0.821075i	$-0.306625\pi$
$41$	299.713	1.14164	0.570820	+	0.821075i	$0.306625\pi$
$42$	0	0
$43$	−501.421	−1.77828	−0.889140	−	0.457635i	$-0.848697\pi$
$43$	−501.421	−1.77828	−0.889140	+	0.457635i	$0.848697\pi$
$44$	51.9899	0.178131
$45$	0	0
$46$	−61.7990	−0.198082
$47$	−305.553	−0.948288	−0.474144	−	0.880447i	$-0.657242\pi$
$47$	−305.553	−0.948288	−0.474144	+	0.880447i	$0.657242\pi$
$48$	0	0
$49$	0	0
$50$	−654.227	−1.85043
$51$	0	0
$52$	−189.698	−0.505893
$53$	375.117	0.972194	0.486097	−	0.873905i	$-0.338420\pi$
$53$	375.117	0.972194	0.486097	+	0.873905i	$0.338420\pi$
$54$	0	0
$55$	476.416	1.16800
$56$	0	0
$57$	0	0
$58$	145.622	0.329675
$59$	627.612	1.38488	0.692442	−	0.721474i	$-0.256535\pi$
$59$	627.612	1.38488	0.692442	+	0.721474i	$0.256535\pi$
$60$	0	0
$61$	3.75736	0.00788657	0.00394328	−	0.999992i	$-0.498745\pi$
$61$	3.75736	0.00788657	0.00394328	+	0.999992i	$0.498745\pi$
$62$	296.250	0.606835
$63$	0	0
$64$	565.288	1.10408
$65$	−1738.33	−3.31712
$66$	0	0
$67$	−813.048	−1.48253	−0.741266	−	0.671212i	$-0.765774\pi$
$67$	−813.048	−1.48253	−0.741266	+	0.671212i	$0.765774\pi$
$68$	−12.2468	−0.0218404
$69$	0	0
$70$	0	0
$71$	−165.902	−0.277310	−0.138655	−	0.990341i	$-0.544278\pi$
$71$	−165.902	−0.277310	−0.138655	+	0.990341i	$0.544278\pi$
$72$	0	0
$73$	619.100	0.992605	0.496302	−	0.868150i	$-0.334691\pi$
$73$	619.100	0.992605	0.496302	+	0.868150i	$0.334691\pi$
$74$	135.480	0.212828
$75$	0	0
$76$	−140.908	−0.212674
$77$	0	0
$78$	0	0
$79$	−138.246	−0.196884	−0.0984421	−	0.995143i	$-0.531386\pi$
$79$	−138.246	−0.196884	−0.0984421	+	0.995143i	$0.531386\pi$
$80$	834.021	1.16558
$81$	0	0
$82$	−723.571	−0.974451
$83$	−621.137	−0.821430	−0.410715	−	0.911764i	$-0.634721\pi$
$83$	−621.137	−0.821430	−0.410715	+	0.911764i	$0.634721\pi$
$84$	0	0
$85$	−112.225	−0.143207
$86$	1210.54	1.51786
$87$	0	0
$88$	−587.907	−0.712171
$89$	−285.418	−0.339936	−0.169968	−	0.985450i	$-0.554366\pi$
$89$	−285.418	−0.339936	−0.169968	+	0.985450i	$0.554366\pi$
$90$	0	0
$91$	0	0
$92$	−55.5879	−0.0629939
$93$	0	0
$94$	737.671	0.809415
$95$	−1291.22	−1.39449
$96$	0	0
$97$	−603.114	−0.631309	−0.315654	−	0.948874i	$-0.602224\pi$
$97$	−603.114	−0.631309	−0.315654	+	0.948874i	$0.602224\pi$
$98$	0	0
$99$	0	0
$100$	−588.474	−0.588474
$101$	−457.209	−0.450436	−0.225218	−	0.974308i	$-0.572309\pi$
$101$	−457.209	−0.450436	−0.225218	+	0.974308i	$0.572309\pi$
$102$	0	0
$103$	786.045	0.751954	0.375977	−	0.926629i	$-0.377307\pi$
$103$	786.045	0.751954	0.375977	+	0.926629i	$0.377307\pi$
$104$	2145.13	2.02257
$105$	0	0
$106$	−905.612	−0.829819
$107$	196.461	0.177501	0.0887504	−	0.996054i	$-0.471713\pi$
$107$	196.461	0.177501	0.0887504	+	0.996054i	$0.471713\pi$
$108$	0	0
$109$	−306.343	−0.269196	−0.134598	−	0.990900i	$-0.542974\pi$
$109$	−306.343	−0.269196	−0.134598	+	0.990900i	$0.542974\pi$
$110$	−1150.17	−0.996950
$111$	0	0
$112$	0	0
$113$	−1997.63	−1.66302	−0.831508	−	0.555512i	$-0.812522\pi$
$113$	−1997.63	−1.66302	−0.831508	+	0.555512i	$0.812522\pi$
$114$	0	0
$115$	−509.387	−0.413048
$116$	130.987	0.104843
$117$	0	0
$118$	−1515.19	−1.18207
$119$	0	0
$120$	0	0
$121$	−757.823	−0.569363
$122$	−9.07107	−0.00673161
$123$	0	0
$124$	266.475	0.192985
$125$	−2905.13	−2.07874
$126$	0	0
$127$	−2311.40	−1.61499	−0.807494	−	0.589875i	$-0.799177\pi$
$127$	−2311.40	−1.61499	−0.807494	+	0.589875i	$0.799177\pi$
$128$	−602.591	−0.416109
$129$	0	0
$130$	4196.69	2.83134
$131$	155.018	0.103389	0.0516945	−	0.998663i	$-0.483538\pi$
$131$	155.018	0.103389	0.0516945	+	0.998663i	$0.483538\pi$
$132$	0	0
$133$	0	0
$134$	1962.87	1.26542
$135$	0	0
$136$	138.488	0.0873182
$137$	−516.936	−0.322371	−0.161186	−	0.986924i	$-0.551532\pi$
$137$	−516.936	−0.322371	−0.161186	+	0.986924i	$0.551532\pi$
$138$	0	0
$139$	958.067	0.584620	0.292310	−	0.956324i	$-0.405576\pi$
$139$	958.067	0.584620	0.292310	+	0.956324i	$0.405576\pi$
$140$	0	0
$141$	0	0
$142$	400.524	0.236699
$143$	−2091.39	−1.22301
$144$	0	0
$145$	1200.31	0.687452
$146$	−1494.64	−0.847241
$147$	0	0
$148$	121.864	0.0676834
$149$	1770.63	0.973526	0.486763	−	0.873534i	$-0.338178\pi$
$149$	1770.63	0.973526	0.486763	+	0.873534i	$0.338178\pi$
$150$	0	0
$151$	−2540.24	−1.36902	−0.684508	−	0.729005i	$-0.739983\pi$
$151$	−2540.24	−1.36902	−0.684508	+	0.729005i	$0.739983\pi$
$152$	1593.40	0.850272
$153$	0	0
$154$	0	0
$155$	2441.88	1.26540
$156$	0	0
$157$	−1083.34	−0.550702	−0.275351	−	0.961344i	$-0.588794\pi$
$157$	−1083.34	−0.550702	−0.275351	+	0.961344i	$0.588794\pi$
$158$	333.754	0.168051
$159$	0	0
$160$	1895.77	0.936709
$161$	0	0
$162$	0	0
$163$	2968.72	1.42655	0.713277	−	0.700882i	$-0.247210\pi$
$163$	2968.72	1.42655	0.713277	+	0.700882i	$0.247210\pi$
$164$	−650.848	−0.309895
$165$	0	0
$166$	1499.56	0.701134
$167$	2091.53	0.969149	0.484574	−	0.874750i	$-0.338975\pi$
$167$	2091.53	0.969149	0.484574	+	0.874750i	$0.338975\pi$
$168$	0	0
$169$	5433.96	2.47335
$170$	270.936	0.122234
$171$	0	0
$172$	1088.87	0.482708
$173$	470.148	0.206617	0.103308	−	0.994649i	$-0.467057\pi$
$173$	470.148	0.206617	0.103308	+	0.994649i	$0.467057\pi$
$174$	0	0
$175$	0	0
$176$	1003.41	0.429745
$177$	0	0
$178$	689.061	0.290153
$179$	−1056.46	−0.441137	−0.220569	−	0.975371i	$-0.570791\pi$
$179$	−1056.46	−0.441137	−0.220569	+	0.975371i	$0.570791\pi$
$180$	0	0
$181$	−406.470	−0.166921	−0.0834605	−	0.996511i	$-0.526597\pi$
$181$	−406.470	−0.166921	−0.0834605	+	0.996511i	$0.526597\pi$
$182$	0	0
$183$	0	0
$184$	628.593	0.251850
$185$	1116.71	0.443797
$186$	0	0
$187$	−135.019	−0.0527997
$188$	663.531	0.257410
$189$	0	0
$190$	3117.29	1.19027
$191$	−179.410	−0.0679666	−0.0339833	−	0.999422i	$-0.510819\pi$
$191$	−179.410	−0.0679666	−0.0339833	+	0.999422i	$0.510819\pi$
$192$	0	0
$193$	−2388.37	−0.890769	−0.445385	−	0.895339i	$-0.646933\pi$
$193$	−2388.37	−0.890769	−0.445385	+	0.895339i	$0.646933\pi$
$194$	1456.05	0.538856
$195$	0	0
$196$	0	0
$197$	2665.35	0.963952	0.481976	−	0.876184i	$-0.339919\pi$
$197$	2665.35	0.963952	0.481976	+	0.876184i	$0.339919\pi$
$198$	0	0
$199$	−1342.31	−0.478159	−0.239079	−	0.971000i	$-0.576846\pi$
$199$	−1342.31	−0.478159	−0.239079	+	0.971000i	$0.576846\pi$
$200$	6654.52	2.35273
$201$	0	0
$202$	1103.80	0.384471
$203$	0	0
$204$	0	0
$205$	−5964.13	−2.03197
$206$	−1897.68	−0.641833
$207$	0	0
$208$	−3661.21	−1.22048
$209$	−1553.48	−0.514144
$210$	0	0
$211$	628.442	0.205042	0.102521	−	0.994731i	$-0.467309\pi$
$211$	628.442	0.205042	0.102521	+	0.994731i	$0.467309\pi$
$212$	−814.594	−0.263899
$213$	0	0
$214$	−474.299	−0.151506
$215$	9978.03	3.16510
$216$	0	0
$217$	0	0
$218$	739.578	0.229773
$219$	0	0
$220$	−1034.57	−0.317049
$221$	492.650	0.149951
$222$	0	0
$223$	−969.970	−0.291273	−0.145637	−	0.989338i	$-0.546523\pi$
$223$	−969.970	−0.291273	−0.145637	+	0.989338i	$0.546523\pi$
$224$	0	0
$225$	0	0
$226$	4822.70	1.41947
$227$	4748.64	1.38845	0.694225	−	0.719758i	$-0.255747\pi$
$227$	4748.64	1.38845	0.694225	+	0.719758i	$0.255747\pi$
$228$	0	0
$229$	−4801.99	−1.38570	−0.692848	−	0.721083i	$-0.743644\pi$
$229$	−4801.99	−1.38570	−0.692848	+	0.721083i	$0.743644\pi$
$230$	1229.77	0.352559
$231$	0	0
$232$	−1481.21	−0.419164
$233$	3155.29	0.887166	0.443583	−	0.896233i	$-0.353707\pi$
$233$	3155.29	0.887166	0.443583	+	0.896233i	$0.353707\pi$
$234$	0	0
$235$	6080.36	1.68782
$236$	−1362.91	−0.375922
$237$	0	0
$238$	0	0
$239$	−4241.93	−1.14806	−0.574032	−	0.818833i	$-0.694622\pi$
$239$	−4241.93	−1.14806	−0.574032	+	0.818833i	$0.694622\pi$
$240$	0	0
$241$	−4342.99	−1.16081	−0.580407	−	0.814326i	$-0.697107\pi$
$241$	−4342.99	−1.16081	−0.580407	+	0.814326i	$0.697107\pi$
$242$	1829.55	0.485982
$243$	0	0
$244$	−8.15938	−0.00214078
$245$	0	0
$246$	0	0
$247$	5668.25	1.46017
$248$	−3013.33	−0.771558
$249$	0	0
$250$	7013.59	1.77431
$251$	−3003.01	−0.755172	−0.377586	−	0.925974i	$-0.623246\pi$
$251$	−3003.01	−0.755172	−0.377586	+	0.925974i	$0.623246\pi$
$252$	0	0
$253$	−612.844	−0.152289
$254$	5580.21	1.37848
$255$	0	0
$256$	−3067.52	−0.748907
$257$	−4468.84	−1.08466	−0.542332	−	0.840164i	$-0.682459\pi$
$257$	−4468.84	−1.08466	−0.542332	+	0.840164i	$0.682459\pi$
$258$	0	0
$259$	0	0
$260$	3774.90	0.900422
$261$	0	0
$262$	−374.246	−0.0882480
$263$	−6160.13	−1.44430	−0.722148	−	0.691738i	$-0.756845\pi$
$263$	−6160.13	−1.44430	−0.722148	+	0.691738i	$0.756845\pi$
$264$	0	0
$265$	−7464.64	−1.73037
$266$	0	0
$267$	0	0
$268$	1765.59	0.402428
$269$	−4988.90	−1.13078	−0.565388	−	0.824825i	$-0.691274\pi$
$269$	−4988.90	−1.13078	−0.565388	+	0.824825i	$0.691274\pi$
$270$	0	0
$271$	4433.73	0.993837	0.496918	−	0.867797i	$-0.334465\pi$
$271$	4433.73	0.993837	0.496918	+	0.867797i	$0.334465\pi$
$272$	−236.366	−0.0526903
$273$	0	0
$274$	1247.99	0.275161
$275$	−6487.80	−1.42265
$276$	0	0
$277$	1112.37	0.241284	0.120642	−	0.992696i	$-0.461505\pi$
$277$	1112.37	0.241284	0.120642	+	0.992696i	$0.461505\pi$
$278$	−2312.98	−0.499005
$279$	0	0
$280$	0	0
$281$	−2813.22	−0.597233	−0.298616	−	0.954373i	$-0.596525\pi$
$281$	−2813.22	−0.597233	−0.298616	+	0.954373i	$0.596525\pi$
$282$	0	0
$283$	3147.54	0.661137	0.330569	−	0.943782i	$-0.392759\pi$
$283$	3147.54	0.661137	0.330569	+	0.943782i	$0.392759\pi$
$284$	360.269	0.0752748
$285$	0	0
$286$	5049.05	1.04390
$287$	0	0
$288$	0	0
$289$	−4881.19	−0.993526
$290$	−2897.81	−0.586777
$291$	0	0
$292$	−1344.42	−0.269439
$293$	−9143.04	−1.82301	−0.911505	−	0.411289i	$-0.865079\pi$
$293$	−9143.04	−1.82301	−0.911505	+	0.411289i	$0.865079\pi$
$294$	0	0
$295$	−12489.2	−2.46491
$296$	−1378.05	−0.270599
$297$	0	0
$298$	−4274.67	−0.830956
$299$	2236.12	0.432502
$300$	0	0
$301$	0	0
$302$	6132.67	1.16853
$303$	0	0
$304$	−2719.54	−0.513079
$305$	−74.7696	−0.0140370
$306$	0	0
$307$	−4648.90	−0.864257	−0.432129	−	0.901812i	$-0.642237\pi$
$307$	−4648.90	−0.864257	−0.432129	+	0.901812i	$0.642237\pi$
$308$	0	0
$309$	0	0
$310$	−5895.22	−1.08008
$311$	6417.18	1.17005	0.585024	−	0.811016i	$-0.301085\pi$
$311$	6417.18	1.17005	0.585024	+	0.811016i	$0.301085\pi$
$312$	0	0
$313$	5868.22	1.05972	0.529858	−	0.848086i	$-0.322245\pi$
$313$	5868.22	1.05972	0.529858	+	0.848086i	$0.322245\pi$
$314$	2615.42	0.470054
$315$	0	0
$316$	300.210	0.0534435
$317$	3974.52	0.704200	0.352100	−	0.935962i	$-0.385468\pi$
$317$	3974.52	0.704200	0.352100	+	0.935962i	$0.385468\pi$
$318$	0	0
$319$	1444.10	0.253461
$320$	−11249.0	−1.96511
$321$	0	0
$322$	0	0
$323$	365.939	0.0630384
$324$	0	0
$325$	23672.4	4.04034
$326$	−7167.13	−1.21764
$327$	0	0
$328$	7359.85	1.23896
$329$	0	0
$330$	0	0
$331$	−8912.82	−1.48004	−0.740020	−	0.672585i	$-0.765184\pi$
$331$	−8912.82	−1.48004	−0.740020	+	0.672585i	$0.765184\pi$
$332$	1348.84	0.222974
$333$	0	0
$334$	−5049.41	−0.827220
$335$	16179.2	2.63871
$336$	0	0
$337$	3977.06	0.642862	0.321431	−	0.946933i	$-0.395836\pi$
$337$	3977.06	0.642862	0.321431	+	0.946933i	$0.395836\pi$
$338$	−13118.7	−2.11114
$339$	0	0
$340$	243.706	0.0388729
$341$	2937.83	0.466547
$342$	0	0
$343$	0	0
$344$	−12313.1	−1.92987
$345$	0	0
$346$	−1135.04	−0.176358
$347$	−6826.43	−1.05609	−0.528043	−	0.849218i	$-0.677074\pi$
$347$	−6826.43	−1.05609	−0.528043	+	0.849218i	$0.677074\pi$
$348$	0	0
$349$	−807.342	−0.123828	−0.0619141	−	0.998081i	$-0.519720\pi$
$349$	−807.342	−0.123828	−0.0619141	+	0.998081i	$0.519720\pi$
$350$	0	0
$351$	0	0
$352$	2280.80	0.345361
$353$	7919.20	1.19404	0.597020	−	0.802226i	$-0.296351\pi$
$353$	7919.20	1.19404	0.597020	+	0.802226i	$0.296351\pi$
$354$	0	0
$355$	3301.38	0.493574
$356$	619.807	0.0922744
$357$	0	0
$358$	2550.52	0.376534
$359$	8819.21	1.29655	0.648273	−	0.761408i	$-0.275491\pi$
$359$	8819.21	1.29655	0.648273	+	0.761408i	$0.275491\pi$
$360$	0	0
$361$	−2648.64	−0.386155
$362$	981.307	0.142476
$363$	0	0
$364$	0	0
$365$	−12319.8	−1.76670
$366$	0	0
$367$	−11161.8	−1.58758	−0.793788	−	0.608194i	$-0.791894\pi$
$367$	−11161.8	−1.58758	−0.793788	+	0.608194i	$0.791894\pi$
$368$	−1072.85	−0.151974
$369$	0	0
$370$	−2695.99	−0.378805
$371$	0	0
$372$	0	0
$373$	2727.86	0.378668	0.189334	−	0.981913i	$-0.439367\pi$
$373$	2727.86	0.378668	0.189334	+	0.981913i	$0.439367\pi$
$374$	325.964	0.0450673
$375$	0	0
$376$	−7503.28	−1.02913
$377$	−5269.17	−0.719830
$378$	0	0
$379$	4086.49	0.553849	0.276924	−	0.960892i	$-0.410685\pi$
$379$	4086.49	0.553849	0.276924	+	0.960892i	$0.410685\pi$
$380$	2803.99	0.378530
$381$	0	0
$382$	433.133	0.0580131
$383$	−13032.9	−1.73878	−0.869389	−	0.494129i	$-0.835487\pi$
$383$	−13032.9	−1.73878	−0.869389	+	0.494129i	$0.835487\pi$
$384$	0	0
$385$	0	0
$386$	5766.03	0.760319
$387$	0	0
$388$	1309.71	0.171367
$389$	194.991	0.0254151	0.0127075	−	0.999919i	$-0.495955\pi$
$389$	194.991	0.0254151	0.0127075	+	0.999919i	$0.495955\pi$
$390$	0	0
$391$	144.363	0.0186719
$392$	0	0
$393$	0	0
$394$	−6434.73	−0.822785
$395$	2751.02	0.350427
$396$	0	0
$397$	−14183.0	−1.79300	−0.896501	−	0.443042i	$-0.853899\pi$
$397$	−14183.0	−1.79300	−0.896501	+	0.443042i	$0.853899\pi$
$398$	3240.61	0.408134
$399$	0	0
$400$	−11357.6	−1.41971
$401$	−10005.0	−1.24596	−0.622978	−	0.782239i	$-0.714077\pi$
$401$	−10005.0	−1.24596	−0.622978	+	0.782239i	$0.714077\pi$
$402$	0	0
$403$	−10719.4	−1.32500
$404$	992.863	0.122269
$405$	0	0
$406$	0	0
$407$	1343.52	0.163626
$408$	0	0
$409$	4634.93	0.560349	0.280174	−	0.959949i	$-0.409608\pi$
$409$	4634.93	0.560349	0.280174	+	0.959949i	$0.409608\pi$
$410$	14398.7	1.73439
$411$	0	0
$412$	−1706.95	−0.204115
$413$	0	0
$414$	0	0
$415$	12360.3	1.46203
$416$	−8322.08	−0.980826
$417$	0	0
$418$	3750.42	0.438849
$419$	−4998.31	−0.582777	−0.291388	−	0.956605i	$-0.594117\pi$
$419$	−4998.31	−0.582777	−0.291388	+	0.956605i	$0.594117\pi$
$420$	0	0
$421$	−704.160	−0.0815170	−0.0407585	−	0.999169i	$-0.512977\pi$
$421$	−704.160	−0.0815170	−0.0407585	+	0.999169i	$0.512977\pi$
$422$	−1517.19	−0.175014
$423$	0	0
$424$	9211.50	1.05507
$425$	1528.28	0.174429
$426$	0	0
$427$	0	0
$428$	−426.629	−0.0481820
$429$	0	0
$430$	−24089.1	−2.70158
$431$	10332.8	1.15479	0.577393	−	0.816466i	$-0.304070\pi$
$431$	10332.8	1.15479	0.577393	+	0.816466i	$0.304070\pi$
$432$	0	0
$433$	11106.8	1.23270	0.616348	−	0.787474i	$-0.288611\pi$
$433$	11106.8	1.23270	0.616348	+	0.787474i	$0.288611\pi$
$434$	0	0
$435$	0	0
$436$	665.246	0.0730723
$437$	1660.98	0.181821
$438$	0	0
$439$	−7299.28	−0.793566	−0.396783	−	0.917912i	$-0.629874\pi$
$439$	−7299.28	−0.793566	−0.396783	+	0.917912i	$0.629874\pi$
$440$	11699.0	1.26757
$441$	0	0
$442$	−1189.36	−0.127991
$443$	−16089.7	−1.72560	−0.862802	−	0.505542i	$-0.831293\pi$
$443$	−16089.7	−1.72560	−0.862802	+	0.505542i	$0.831293\pi$
$444$	0	0
$445$	5679.68	0.605040
$446$	2341.71	0.248617
$447$	0	0
$448$	0	0
$449$	−13561.7	−1.42543	−0.712715	−	0.701454i	$-0.752534\pi$
$449$	−13561.7	−1.42543	−0.712715	+	0.701454i	$0.752534\pi$
$450$	0	0
$451$	−7175.46	−0.749178
$452$	4337.99	0.451420
$453$	0	0
$454$	−11464.2	−1.18512
$455$	0	0
$456$	0	0
$457$	10848.6	1.11045	0.555224	−	0.831701i	$-0.312633\pi$
$457$	10848.6	1.11045	0.555224	+	0.831701i	$0.312633\pi$
$458$	11593.0	1.18277
$459$	0	0
$460$	1106.17	0.112121
$461$	1758.69	0.177679	0.0888397	−	0.996046i	$-0.471684\pi$
$461$	1758.69	0.177679	0.0888397	+	0.996046i	$0.471684\pi$
$462$	0	0
$463$	−5411.95	−0.543228	−0.271614	−	0.962406i	$-0.587557\pi$
$463$	−5411.95	−0.543228	−0.271614	+	0.962406i	$0.587557\pi$
$464$	2528.06	0.252936
$465$	0	0
$466$	−7617.54	−0.757244
$467$	−8111.34	−0.803744	−0.401872	−	0.915696i	$-0.631640\pi$
$467$	−8111.34	−0.803744	−0.401872	+	0.915696i	$0.631640\pi$
$468$	0	0
$469$	0	0
$470$	−14679.3	−1.44065
$471$	0	0
$472$	15411.9	1.50294
$473$	12004.6	1.16696
$474$	0	0
$475$	17583.8	1.69853
$476$	0	0
$477$	0	0
$478$	10240.9	0.979935
$479$	−2095.76	−0.199912	−0.0999559	−	0.994992i	$-0.531870\pi$
$479$	−2095.76	−0.199912	−0.0999559	+	0.994992i	$0.531870\pi$
$480$	0	0
$481$	−4902.18	−0.464699
$482$	10484.9	0.990817
$483$	0	0
$484$	1645.67	0.154552
$485$	12001.7	1.12364
$486$	0	0
$487$	9610.08	0.894197	0.447099	−	0.894485i	$-0.352457\pi$
$487$	9610.08	0.894197	0.447099	+	0.894485i	$0.352457\pi$
$488$	92.2670	0.00855888
$489$	0	0
$490$	0	0
$491$	11717.3	1.07698	0.538488	−	0.842633i	$-0.318996\pi$
$491$	11717.3	1.07698	0.538488	+	0.842633i	$0.318996\pi$
$492$	0	0
$493$	−340.174	−0.0310764
$494$	−13684.4	−1.24633
$495$	0	0
$496$	5143.01	0.465581
$497$	0	0
$498$	0	0
$499$	−9195.19	−0.824916	−0.412458	−	0.910977i	$-0.635330\pi$
$499$	−9195.19	−0.824916	−0.412458	+	0.910977i	$0.635330\pi$
$500$	6308.69	0.564266
$501$	0	0
$502$	7249.90	0.644580
$503$	−16118.8	−1.42883	−0.714414	−	0.699724i	$-0.753307\pi$
$503$	−16118.8	−1.42883	−0.714414	+	0.699724i	$0.753307\pi$
$504$	0	0
$505$	9098.23	0.801715
$506$	1479.54	0.129987
$507$	0	0
$508$	5019.37	0.438383
$509$	−4918.78	−0.428333	−0.214166	−	0.976797i	$-0.568703\pi$
$509$	−4918.78	−0.428333	−0.214166	+	0.976797i	$0.568703\pi$
$510$	0	0
$511$	0	0
$512$	12226.4	1.05534
$513$	0	0
$514$	10788.7	0.925819
$515$	−15641.9	−1.33838
$516$	0	0
$517$	7315.29	0.622294
$518$	0	0
$519$	0	0
$520$	−42687.0	−3.59990
$521$	−13963.4	−1.17418	−0.587089	−	0.809522i	$-0.699726\pi$
$521$	−13963.4	−1.17418	−0.587089	+	0.809522i	$0.699726\pi$
$522$	0	0
$523$	−13755.3	−1.15005	−0.575024	−	0.818136i	$-0.695007\pi$
$523$	−13755.3	−1.15005	−0.575024	+	0.818136i	$0.695007\pi$
$524$	−336.632	−0.0280646
$525$	0	0
$526$	14871.9	1.23278
$527$	−692.040	−0.0572026
$528$	0	0
$529$	−11511.7	−0.946145
$530$	18021.2	1.47697
$531$	0	0
$532$	0	0
$533$	26181.5	2.12767
$534$	0	0
$535$	−3909.47	−0.315928
$536$	−19965.5	−1.60891
$537$	0	0
$538$	12044.3	0.965178
$539$	0	0
$540$	0	0
$541$	14462.7	1.14935	0.574676	−	0.818381i	$-0.305128\pi$
$541$	14462.7	1.14935	0.574676	+	0.818381i	$0.305128\pi$
$542$	−10704.0	−0.848293
$543$	0	0
$544$	−537.269	−0.0423441
$545$	6096.07	0.479132
$546$	0	0
$547$	13682.5	1.06951	0.534755	−	0.845007i	$-0.320404\pi$
$547$	13682.5	1.06951	0.534755	+	0.845007i	$0.320404\pi$
$548$	1122.56	0.0875065
$549$	0	0
$550$	15662.9	1.21431
$551$	−3913.92	−0.302611
$552$	0	0
$553$	0	0
$554$	−2685.49	−0.205949
$555$	0	0
$556$	−2080.51	−0.158693
$557$	7663.13	0.582939	0.291470	−	0.956580i	$-0.405856\pi$
$557$	7663.13	0.582939	0.291470	+	0.956580i	$0.405856\pi$
$558$	0	0
$559$	−43801.8	−3.31417
$560$	0	0
$561$	0	0
$562$	6791.71	0.509770
$563$	−17470.5	−1.30780	−0.653902	−	0.756580i	$-0.726869\pi$
$563$	−17470.5	−1.30780	−0.653902	+	0.756580i	$0.726869\pi$
$564$	0	0
$565$	39751.8	2.95995
$566$	−7598.83	−0.564316
$567$	0	0
$568$	−4073.96	−0.300950
$569$	13873.9	1.02219	0.511094	−	0.859525i	$-0.329240\pi$
$569$	13873.9	1.02219	0.511094	+	0.859525i	$0.329240\pi$
$570$	0	0
$571$	−3777.52	−0.276855	−0.138428	−	0.990373i	$-0.544205\pi$
$571$	−3777.52	−0.276855	−0.138428	+	0.990373i	$0.544205\pi$
$572$	4541.60	0.331982
$573$	0	0
$574$	0	0
$575$	6936.79	0.503103
$576$	0	0
$577$	13880.5	1.00148	0.500738	−	0.865599i	$-0.333062\pi$
$577$	13880.5	1.00148	0.500738	+	0.865599i	$0.333062\pi$
$578$	11784.2	0.848028
$579$	0	0
$580$	−2606.57	−0.186607
$581$	0	0
$582$	0	0
$583$	−8980.72	−0.637982
$584$	15202.8	1.07722
$585$	0	0
$586$	22073.2	1.55604
$587$	2395.61	0.168445	0.0842227	−	0.996447i	$-0.473159\pi$
$587$	2395.61	0.168445	0.0842227	+	0.996447i	$0.473159\pi$
$588$	0	0
$589$	−7962.36	−0.557018
$590$	30151.5	2.10393
$591$	0	0
$592$	2351.99	0.163287
$593$	−6603.50	−0.457290	−0.228645	−	0.973510i	$-0.573430\pi$
$593$	−6603.50	−0.457290	−0.228645	+	0.973510i	$0.573430\pi$
$594$	0	0
$595$	0	0
$596$	−3845.04	−0.264260
$597$	0	0
$598$	−5398.47	−0.369164
$599$	−17252.1	−1.17680	−0.588399	−	0.808571i	$-0.700241\pi$
$599$	−17252.1	−1.17680	−0.588399	+	0.808571i	$0.700241\pi$
$600$	0	0
$601$	12833.1	0.871005	0.435503	−	0.900187i	$-0.356571\pi$
$601$	12833.1	0.871005	0.435503	+	0.900187i	$0.356571\pi$
$602$	0	0
$603$	0	0
$604$	5516.31	0.371615
$605$	15080.3	1.01339
$606$	0	0
$607$	8620.16	0.576411	0.288206	−	0.957569i	$-0.406941\pi$
$607$	8620.16	0.576411	0.288206	+	0.957569i	$0.406941\pi$
$608$	−6181.62	−0.412332
$609$	0	0
$610$	180.510	0.0119813
$611$	−26691.7	−1.76732
$612$	0	0
$613$	5136.73	0.338451	0.169226	−	0.985577i	$-0.445873\pi$
$613$	5136.73	0.338451	0.169226	+	0.985577i	$0.445873\pi$
$614$	11223.4	0.737690
$615$	0	0
$616$	0	0
$617$	1759.82	0.114826	0.0574131	−	0.998351i	$-0.481715\pi$
$617$	1759.82	0.114826	0.0574131	+	0.998351i	$0.481715\pi$
$618$	0	0
$619$	−3560.24	−0.231176	−0.115588	−	0.993297i	$-0.536875\pi$
$619$	−3560.24	−0.231176	−0.115588	+	0.993297i	$0.536875\pi$
$620$	−5302.72	−0.343488
$621$	0	0
$622$	−15492.4	−0.998698
$623$	0	0
$624$	0	0
$625$	23936.8	1.53195
$626$	−14167.1	−0.904524
$627$	0	0
$628$	2352.56	0.149486
$629$	−316.482	−0.0200620
$630$	0	0
$631$	−27321.4	−1.72369	−0.861845	−	0.507172i	$-0.830691\pi$
$631$	−27321.4	−1.72369	−0.861845	+	0.507172i	$0.830691\pi$
$632$	−3394.81	−0.213668
$633$	0	0
$634$	−9595.34	−0.601072
$635$	45995.7	2.87446
$636$	0	0
$637$	0	0
$638$	−3486.36	−0.216342
$639$	0	0
$640$	11991.3	0.740619
$641$	21927.0	1.35112	0.675558	−	0.737307i	$-0.263903\pi$
$641$	21927.0	1.35112	0.675558	+	0.737307i	$0.263903\pi$
$642$	0	0
$643$	−5826.04	−0.357320	−0.178660	−	0.983911i	$-0.557176\pi$
$643$	−5826.04	−0.357320	−0.178660	+	0.983911i	$0.557176\pi$
$644$	0	0
$645$	0	0
$646$	−883.455	−0.0538066
$647$	−24210.7	−1.47113	−0.735565	−	0.677454i	$-0.763083\pi$
$647$	−24210.7	−1.47113	−0.735565	+	0.677454i	$0.763083\pi$
$648$	0	0
$649$	−15025.7	−0.908801
$650$	−57150.3	−3.44864
$651$	0	0
$652$	−6446.80	−0.387233
$653$	25623.3	1.53556	0.767778	−	0.640716i	$-0.221363\pi$
$653$	25623.3	1.53556	0.767778	+	0.640716i	$0.221363\pi$
$654$	0	0
$655$	−3084.77	−0.184018
$656$	−12561.5	−0.747626
$657$	0	0
$658$	0	0
$659$	23273.7	1.37574	0.687871	−	0.725833i	$-0.258545\pi$
$659$	23273.7	1.37574	0.687871	+	0.725833i	$0.258545\pi$
$660$	0	0
$661$	−20036.5	−1.17902	−0.589508	−	0.807763i	$-0.700678\pi$
$661$	−20036.5	−1.17902	−0.589508	+	0.807763i	$0.700678\pi$
$662$	21517.5	1.26329
$663$	0	0
$664$	−15252.9	−0.891454
$665$	0	0
$666$	0	0
$667$	−1544.04	−0.0896333
$668$	−4541.92	−0.263072
$669$	0	0
$670$	−39060.1	−2.25228
$671$	−89.9554	−0.00517540
$672$	0	0
$673$	−18127.8	−1.03830	−0.519149	−	0.854684i	$-0.673751\pi$
$673$	−18127.8	−1.03830	−0.519149	+	0.854684i	$0.673751\pi$
$674$	−9601.48	−0.548717
$675$	0	0
$676$	−11800.2	−0.671383
$677$	−13815.5	−0.784301	−0.392150	−	0.919901i	$-0.628269\pi$
$677$	−13815.5	−0.784301	−0.392150	+	0.919901i	$0.628269\pi$
$678$	0	0
$679$	0	0
$680$	−2755.85	−0.155415
$681$	0	0
$682$	−7092.55	−0.398223
$683$	4604.23	0.257944	0.128972	−	0.991648i	$-0.458832\pi$
$683$	4604.23	0.257944	0.128972	+	0.991648i	$0.458832\pi$
$684$	0	0
$685$	10286.8	0.573777
$686$	0	0
$687$	0	0
$688$	21015.4	1.16454
$689$	32768.5	1.81187
$690$	0	0
$691$	17913.7	0.986205	0.493103	−	0.869971i	$-0.335863\pi$
$691$	17913.7	0.986205	0.493103	+	0.869971i	$0.335863\pi$
$692$	−1020.96	−0.0560854
$693$	0	0
$694$	16480.5	0.901426
$695$	−19065.1	−1.04054
$696$	0	0
$697$	1690.26	0.0918555
$698$	1949.10	0.105694
$699$	0	0
$700$	0	0
$701$	−11303.7	−0.609035	−0.304518	−	0.952507i	$-0.598495\pi$
$701$	−11303.7	−0.609035	−0.304518	+	0.952507i	$0.598495\pi$
$702$	0	0
$703$	−3641.33	−0.195356
$704$	−13533.6	−0.724529
$705$	0	0
$706$	−19118.6	−1.01918
$707$	0	0
$708$	0	0
$709$	−16046.3	−0.849973	−0.424987	−	0.905200i	$-0.639721\pi$
$709$	−16046.3	−0.849973	−0.424987	+	0.905200i	$0.639721\pi$
$710$	−7970.22	−0.421292
$711$	0	0
$712$	−7008.83	−0.368915
$713$	−3141.15	−0.164989
$714$	0	0
$715$	41617.5	2.17679
$716$	2294.18	0.119745
$717$	0	0
$718$	−21291.5	−1.10667
$719$	25190.5	1.30660	0.653300	−	0.757099i	$-0.273384\pi$
$719$	25190.5	1.30660	0.653300	+	0.757099i	$0.273384\pi$
$720$	0	0
$721$	0	0
$722$	6394.38	0.329604
$723$	0	0
$724$	882.680	0.0453102
$725$	−16345.8	−0.837334
$726$	0	0
$727$	11277.2	0.575307	0.287653	−	0.957735i	$-0.407125\pi$
$727$	11277.2	0.575307	0.287653	+	0.957735i	$0.407125\pi$
$728$	0	0
$729$	0	0
$730$	29742.6	1.50797
$731$	−2827.82	−0.143079
$732$	0	0
$733$	23720.0	1.19525	0.597626	−	0.801775i	$-0.296111\pi$
$733$	23720.0	1.19525	0.597626	+	0.801775i	$0.296111\pi$
$734$	26946.9	1.35508
$735$	0	0
$736$	−2438.64	−0.122133
$737$	19465.3	0.972880
$738$	0	0
$739$	8124.72	0.404428	0.202214	−	0.979341i	$-0.435186\pi$
$739$	8124.72	0.404428	0.202214	+	0.979341i	$0.435186\pi$
$740$	−2425.03	−0.120467
$741$	0	0
$742$	0	0
$743$	−20955.3	−1.03469	−0.517346	−	0.855777i	$-0.673080\pi$
$743$	−20955.3	−1.03469	−0.517346	+	0.855777i	$0.673080\pi$
$744$	0	0
$745$	−35234.6	−1.73274
$746$	−6585.63	−0.323213
$747$	0	0
$748$	293.203	0.0143323
$749$	0	0
$750$	0	0
$751$	−38208.1	−1.85650	−0.928251	−	0.371954i	$-0.878688\pi$
$751$	−38208.1	−1.85650	−0.928251	+	0.371954i	$0.878688\pi$
$752$	12806.3	0.621006
$753$	0	0
$754$	12720.9	0.614413
$755$	50549.4	2.43666
$756$	0	0
$757$	30958.1	1.48638	0.743191	−	0.669079i	$-0.233311\pi$
$757$	30958.1	1.48638	0.743191	+	0.669079i	$0.233311\pi$
$758$	−9865.65	−0.472740
$759$	0	0
$760$	−31707.8	−1.51337
$761$	40049.4	1.90774	0.953871	−	0.300218i	$-0.0970594\pi$
$761$	40049.4	1.90774	0.953871	+	0.300218i	$0.0970594\pi$
$762$	0	0
$763$	0	0
$764$	389.601	0.0184493
$765$	0	0
$766$	31464.3	1.48414
$767$	54825.3	2.58100
$768$	0	0
$769$	−8002.01	−0.375240	−0.187620	−	0.982242i	$-0.560077\pi$
$769$	−8002.01	−0.375240	−0.187620	+	0.982242i	$0.560077\pi$
$770$	0	0
$771$	0	0
$772$	5186.52	0.241796
$773$	13933.4	0.648316	0.324158	−	0.946003i	$-0.394919\pi$
$773$	13933.4	0.648316	0.324158	+	0.946003i	$0.394919\pi$
$774$	0	0
$775$	−33253.4	−1.54128
$776$	−14810.3	−0.685126
$777$	0	0
$778$	−470.751	−0.0216931
$779$	19447.6	0.894456
$780$	0	0
$781$	3971.89	0.181979
$782$	−348.522	−0.0159375
$783$	0	0
$784$	0	0
$785$	21558.0	0.980175
$786$	0	0
$787$	37581.7	1.70222	0.851108	−	0.524991i	$-0.175931\pi$
$787$	37581.7	1.70222	0.851108	+	0.524991i	$0.175931\pi$
$788$	−5788.01	−0.261662
$789$	0	0
$790$	−6641.54	−0.299108
$791$	0	0
$792$	0	0
$793$	328.225	0.0146981
$794$	34240.7	1.53042
$795$	0	0
$796$	2914.92	0.129795
$797$	9458.78	0.420385	0.210193	−	0.977660i	$-0.432591\pi$
$797$	9458.78	0.420385	0.210193	+	0.977660i	$0.432591\pi$
$798$	0	0
$799$	−1723.20	−0.0762985
$800$	−25816.4	−1.14093
$801$	0	0
$802$	24154.3	1.06349
$803$	−14821.9	−0.651376
$804$	0	0
$805$	0	0
$806$	25879.0	1.13095
$807$	0	0
$808$	−11227.4	−0.488834
$809$	1909.94	0.0830034	0.0415017	−	0.999138i	$-0.486786\pi$
$809$	1909.94	0.0830034	0.0415017	+	0.999138i	$0.486786\pi$
$810$	0	0
$811$	43110.6	1.86661	0.933303	−	0.359091i	$-0.116913\pi$
$811$	43110.6	1.86661	0.933303	+	0.359091i	$0.116913\pi$
$812$	0	0
$813$	0	0
$814$	−3243.55	−0.139664
$815$	−59076.1	−2.53907
$816$	0	0
$817$	−32535.9	−1.39325
$818$	−11189.7	−0.478287
$819$	0	0
$820$	12951.5	0.551570
$821$	−4026.56	−0.171167	−0.0855834	−	0.996331i	$-0.527275\pi$
$821$	−4026.56	−0.171167	−0.0855834	+	0.996331i	$0.527275\pi$
$822$	0	0
$823$	39668.1	1.68012	0.840062	−	0.542491i	$-0.182519\pi$
$823$	39668.1	1.68012	0.840062	+	0.542491i	$0.182519\pi$
$824$	19302.4	0.816056
$825$	0	0
$826$	0	0
$827$	−30137.7	−1.26722	−0.633611	−	0.773652i	$-0.718428\pi$
$827$	−30137.7	−1.26722	−0.633611	+	0.773652i	$0.718428\pi$
$828$	0	0
$829$	23278.0	0.975245	0.487622	−	0.873055i	$-0.337864\pi$
$829$	23278.0	0.975245	0.487622	+	0.873055i	$0.337864\pi$
$830$	−29840.4	−1.24792
$831$	0	0
$832$	49381.0	2.05766
$833$	0	0
$834$	0	0
$835$	−41620.5	−1.72495
$836$	3373.48	0.139563
$837$	0	0
$838$	12067.0	0.497431
$839$	9494.43	0.390684	0.195342	−	0.980735i	$-0.437418\pi$
$839$	9494.43	0.390684	0.195342	+	0.980735i	$0.437418\pi$
$840$	0	0
$841$	−20750.6	−0.850820
$842$	1699.99	0.0695791
$843$	0	0
$844$	−1364.71	−0.0556578
$845$	−108133.	−4.40223
$846$	0	0
$847$	0	0
$848$	−15721.8	−0.636661
$849$	0	0
$850$	−3689.59	−0.148885
$851$	−1436.50	−0.0578644
$852$	0	0
$853$	−12692.4	−0.509471	−0.254736	−	0.967011i	$-0.581988\pi$
$853$	−12692.4	−0.509471	−0.254736	+	0.967011i	$0.581988\pi$
$854$	0	0
$855$	0	0
$856$	4824.36	0.192632
$857$	−22206.0	−0.885114	−0.442557	−	0.896740i	$-0.645929\pi$
$857$	−22206.0	−0.885114	−0.442557	+	0.896740i	$0.645929\pi$
$858$	0	0
$859$	19820.5	0.787271	0.393636	−	0.919267i	$-0.371217\pi$
$859$	19820.5	0.787271	0.393636	+	0.919267i	$0.371217\pi$
$860$	−21668.0	−0.859155
$861$	0	0
$862$	−24945.5	−0.985671
$863$	−36413.8	−1.43631	−0.718157	−	0.695881i	$-0.755014\pi$
$863$	−36413.8	−1.43631	−0.718157	+	0.695881i	$0.755014\pi$
$864$	0	0
$865$	−9355.70	−0.367749
$866$	−26814.1	−1.05217
$867$	0	0
$868$	0	0
$869$	3309.76	0.129201
$870$	0	0
$871$	−71024.1	−2.76298
$872$	−7522.67	−0.292144
$873$	0	0
$874$	−4009.97	−0.155194
$875$	0	0
$876$	0	0
$877$	19442.6	0.748609	0.374305	−	0.927306i	$-0.377881\pi$
$877$	19442.6	0.748609	0.374305	+	0.927306i	$0.377881\pi$
$878$	17622.0	0.677351
$879$	0	0
$880$	−19967.4	−0.764888
$881$	25184.2	0.963082	0.481541	−	0.876423i	$-0.340077\pi$
$881$	25184.2	0.963082	0.481541	+	0.876423i	$0.340077\pi$
$882$	0	0
$883$	−4050.03	−0.154354	−0.0771769	−	0.997017i	$-0.524591\pi$
$883$	−4050.03	−0.154354	−0.0771769	+	0.997017i	$0.524591\pi$
$884$	−1069.83	−0.0407038
$885$	0	0
$886$	38843.9	1.47290
$887$	−41604.2	−1.57489	−0.787447	−	0.616383i	$-0.788598\pi$
$887$	−41604.2	−1.57489	−0.787447	+	0.616383i	$0.788598\pi$
$888$	0	0
$889$	0	0
$890$	−13712.0	−0.516434
$891$	0	0
$892$	2106.36	0.0790652
$893$	−19826.5	−0.742967
$894$	0	0
$895$	21023.0	0.785165
$896$	0	0
$897$	0	0
$898$	32740.9	1.21668
$899$	7401.76	0.274597
$900$	0	0
$901$	2115.51	0.0782219
$902$	17323.1	0.639463
$903$	0	0
$904$	−49054.4	−1.80478
$905$	8088.56	0.297097
$906$	0	0
$907$	−3572.60	−0.130790	−0.0653949	−	0.997859i	$-0.520831\pi$
$907$	−3572.60	−0.130790	−0.0653949	+	0.997859i	$0.520831\pi$
$908$	−10312.0	−0.376890
$909$	0	0
$910$	0	0
$911$	−29457.7	−1.07133	−0.535663	−	0.844432i	$-0.679938\pi$
$911$	−29457.7	−1.07133	−0.535663	+	0.844432i	$0.679938\pi$
$912$	0	0
$913$	14870.7	0.539046
$914$	−26190.8	−0.947827
$915$	0	0
$916$	10427.9	0.376143
$917$	0	0
$918$	0	0
$919$	−3310.65	−0.118834	−0.0594168	−	0.998233i	$-0.518924\pi$
$919$	−3310.65	−0.118834	−0.0594168	+	0.998233i	$0.518924\pi$
$920$	−12508.7	−0.448260
$921$	0	0
$922$	−4245.85	−0.151659
$923$	−14492.5	−0.516820
$924$	0	0
$925$	−15207.3	−0.540556
$926$	13065.6	0.463674
$927$	0	0
$928$	5746.39	0.203270
$929$	31467.5	1.11132	0.555660	−	0.831410i	$-0.312466\pi$
$929$	31467.5	1.11132	0.555660	+	0.831410i	$0.312466\pi$
$930$	0	0
$931$	0	0
$932$	−6851.94	−0.240818
$933$	0	0
$934$	19582.5	0.686038
$935$	2686.80	0.0939763
$936$	0	0
$937$	−17363.4	−0.605375	−0.302688	−	0.953090i	$-0.597884\pi$
$937$	−17363.4	−0.605375	−0.302688	+	0.953090i	$0.597884\pi$
$938$	0	0
$939$	0	0
$940$	−13203.9	−0.458154
$941$	5547.77	0.192192	0.0960958	−	0.995372i	$-0.469364\pi$
$941$	5547.77	0.192192	0.0960958	+	0.995372i	$0.469364\pi$
$942$	0	0
$943$	7672.04	0.264937
$944$	−26304.3	−0.906919
$945$	0	0
$946$	−28981.6	−0.996062
$947$	37960.3	1.30258	0.651290	−	0.758829i	$-0.274228\pi$
$947$	37960.3	1.30258	0.651290	+	0.758829i	$0.274228\pi$
$948$	0	0
$949$	54081.7	1.84991
$950$	−42451.1	−1.44978
$951$	0	0
$952$	0	0
$953$	10019.3	0.340563	0.170282	−	0.985395i	$-0.445532\pi$
$953$	10019.3	0.340563	0.170282	+	0.985395i	$0.445532\pi$
$954$	0	0
$955$	3570.16	0.120971
$956$	9211.65	0.311638
$957$	0	0
$958$	5059.61	0.170635
$959$	0	0
$960$	0	0
$961$	−14733.1	−0.494548
$962$	11834.9	0.396646
$963$	0	0
$964$	9431.11	0.315099
$965$	47527.3	1.58545
$966$	0	0
$967$	27834.4	0.925641	0.462820	−	0.886452i	$-0.346837\pi$
$967$	27834.4	0.925641	0.462820	+	0.886452i	$0.346837\pi$
$968$	−18609.4	−0.617900
$969$	0	0
$970$	−28974.6	−0.959090
$971$	18275.3	0.604000	0.302000	−	0.953308i	$-0.402346\pi$
$971$	18275.3	0.604000	0.302000	+	0.953308i	$0.402346\pi$
$972$	0	0
$973$	0	0
$974$	−23200.8	−0.763245
$975$	0	0
$976$	−157.477	−0.00516468
$977$	43028.9	1.40903	0.704513	−	0.709691i	$-0.251166\pi$
$977$	43028.9	1.40903	0.704513	+	0.709691i	$0.251166\pi$
$978$	0	0
$979$	6833.24	0.223076
$980$	0	0
$981$	0	0
$982$	−28288.1	−0.919256
$983$	−30559.2	−0.991544	−0.495772	−	0.868453i	$-0.665115\pi$
$983$	−30559.2	−0.991544	−0.495772	+	0.868453i	$0.665115\pi$
$984$	0	0
$985$	−53039.2	−1.71571
$986$	821.253	0.0265254
$987$	0	0
$988$	−12309.0	−0.396358
$989$	−12835.4	−0.412681
$990$	0	0
$991$	−44945.9	−1.44072	−0.720360	−	0.693600i	$-0.756024\pi$
$991$	−44945.9	−1.44072	−0.720360	+	0.693600i	$0.756024\pi$
$992$	11690.3	0.374160
$993$	0	0
$994$	0	0
$995$	26711.2	0.851058
$996$	0	0
$997$	29006.1	0.921397	0.460698	−	0.887557i	$-0.347599\pi$
$997$	29006.1	0.921397	0.460698	+	0.887557i	$0.347599\pi$
$998$	22199.1	0.704110
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.4.a.n.1.1		2
3.2	odd	2		147.4.a.j.1.2	✓	2
7.2	even	3		441.4.e.v.361.2		4
7.3	odd	6		441.4.e.u.226.2		4
7.4	even	3		441.4.e.v.226.2		4
7.5	odd	6		441.4.e.u.361.2		4
7.6	odd	2		441.4.a.o.1.1		2
12.11	even	2		2352.4.a.cf.1.2		2
21.2	odd	6		147.4.e.k.67.1		4
21.5	even	6		147.4.e.j.67.1		4
21.11	odd	6		147.4.e.k.79.1		4
21.17	even	6		147.4.e.j.79.1		4
21.20	even	2		147.4.a.k.1.2	yes	2
84.83	odd	2		2352.4.a.bl.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.4.a.j.1.2	✓	2	3.2	odd	2
147.4.a.k.1.2	yes	2	21.20	even	2
147.4.e.j.67.1		4	21.5	even	6
147.4.e.j.79.1		4	21.17	even	6
147.4.e.k.67.1		4	21.2	odd	6
147.4.e.k.79.1		4	21.11	odd	6
441.4.a.n.1.1		2	1.1	even	1	trivial
441.4.a.o.1.1		2	7.6	odd	2
441.4.e.u.226.2		4	7.3	odd	6
441.4.e.u.361.2		4	7.5	odd	6
441.4.e.v.226.2		4	7.4	even	3
441.4.e.v.361.2		4	7.2	even	3
2352.4.a.bl.1.1		2	84.83	odd	2
2352.4.a.cf.1.2		2	12.11	even	2

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)

\(f(q)\)	\(=\)	\(q-2.41421 q^{2} -2.17157 q^{4} -19.8995 q^{5} +24.5563 q^{8} +O(q^{10})\)	\(q-2.41421 q^{2} -2.17157 q^{4} -19.8995 q^{5} +24.5563 q^{8} +48.0416 q^{10} -23.9411 q^{11} +87.3553 q^{13} -41.9117 q^{16} +5.63961 q^{17} +64.8873 q^{19} +43.2132 q^{20} +57.7990 q^{22} +25.5980 q^{23} +270.990 q^{25} -210.894 q^{26} -60.3188 q^{29} -122.711 q^{31} -95.2670 q^{32} -13.6152 q^{34} -56.1177 q^{37} -156.652 q^{38} -488.659 q^{40} +299.713 q^{41} -501.421 q^{43} +51.9899 q^{44} -61.7990 q^{46} -305.553 q^{47} -654.227 q^{50} -189.698 q^{52} +375.117 q^{53} +476.416 q^{55} +145.622 q^{58} +627.612 q^{59} +3.75736 q^{61} +296.250 q^{62} +565.288 q^{64} -1738.33 q^{65} -813.048 q^{67} -12.2468 q^{68} -165.902 q^{71} +619.100 q^{73} +135.480 q^{74} -140.908 q^{76} -138.246 q^{79} +834.021 q^{80} -723.571 q^{82} -621.137 q^{83} -112.225 q^{85} +1210.54 q^{86} -587.907 q^{88} -285.418 q^{89} -55.5879 q^{92} +737.671 q^{94} -1291.22 q^{95} -603.114 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 10 q^{4} - 20 q^{5} + 18 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 - 10 * q^4 - 20 * q^5 + 18 * q^8	\( 2 q - 2 q^{2} - 10 q^{4} - 20 q^{5} + 18 q^{8} + 48 q^{10} + 20 q^{11} + 104 q^{13} + 18 q^{16} - 116 q^{17} + 192 q^{19} + 44 q^{20} + 76 q^{22} - 28 q^{23} + 146 q^{25} - 204 q^{26} - 296 q^{29} - 104 q^{31} - 18 q^{32} - 64 q^{34} - 248 q^{37} - 104 q^{38} - 488 q^{40} - 20 q^{41} - 720 q^{43} - 292 q^{44} - 84 q^{46} + 96 q^{47} - 706 q^{50} - 320 q^{52} - 268 q^{53} + 472 q^{55} + 48 q^{58} + 616 q^{59} + 16 q^{61} + 304 q^{62} + 118 q^{64} - 1740 q^{65} - 144 q^{67} + 940 q^{68} - 988 q^{71} + 104 q^{73} + 56 q^{74} - 1136 q^{76} - 944 q^{79} + 828 q^{80} - 856 q^{82} - 1016 q^{83} - 100 q^{85} + 1120 q^{86} - 876 q^{88} + 388 q^{89} + 364 q^{92} + 904 q^{94} - 1304 q^{95} + 488 q^{97}+O(q^{100}) \) 2 * q - 2 * q^2 - 10 * q^4 - 20 * q^5 + 18 * q^8 + 48 * q^10 + 20 * q^11 + 104 * q^13 + 18 * q^16 - 116 * q^17 + 192 * q^19 + 44 * q^20 + 76 * q^22 - 28 * q^23 + 146 * q^25 - 204 * q^26 - 296 * q^29 - 104 * q^31 - 18 * q^32 - 64 * q^34 - 248 * q^37 - 104 * q^38 - 488 * q^40 - 20 * q^41 - 720 * q^43 - 292 * q^44 - 84 * q^46 + 96 * q^47 - 706 * q^50 - 320 * q^52 - 268 * q^53 + 472 * q^55 + 48 * q^58 + 616 * q^59 + 16 * q^61 + 304 * q^62 + 118 * q^64 - 1740 * q^65 - 144 * q^67 + 940 * q^68 - 988 * q^71 + 104 * q^73 + 56 * q^74 - 1136 * q^76 - 944 * q^79 + 828 * q^80 - 856 * q^82 - 1016 * q^83 - 100 * q^85 + 1120 * q^86 - 876 * q^88 + 388 * q^89 + 364 * q^92 + 904 * q^94 - 1304 * q^95 + 488 * q^97

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