LMFDB - Embedded newform 441.4.a.n.1.2

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			1.2
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)

$f(q)$	$=$	$q+0.414214 q^{2} -7.82843 q^{4} -0.100505 q^{5} -6.55635 q^{8} +O(q^{10})$	$q+0.414214 q^{2} -7.82843 q^{4} -0.100505 q^{5} -6.55635 q^{8} -0.0416306 q^{10} +43.9411 q^{11} +16.6447 q^{13} +59.9117 q^{16} -121.640 q^{17} +127.113 q^{19} +0.786797 q^{20} +18.2010 q^{22} -53.5980 q^{23} -124.990 q^{25} +6.89444 q^{26} -235.681 q^{29} +18.7107 q^{31} +77.2670 q^{32} -50.3848 q^{34} -191.882 q^{37} +52.6518 q^{38} +0.658946 q^{40} -319.713 q^{41} -218.579 q^{43} -343.990 q^{44} -22.2010 q^{46} +401.553 q^{47} -51.7725 q^{50} -130.302 q^{52} -643.117 q^{53} -4.41631 q^{55} -97.6224 q^{58} -11.6123 q^{59} +12.2426 q^{61} +7.75022 q^{62} -447.288 q^{64} -1.67287 q^{65} +669.048 q^{67} +952.247 q^{68} -822.098 q^{71} -515.100 q^{73} -79.4802 q^{74} -995.092 q^{76} -805.754 q^{79} -6.02143 q^{80} -132.429 q^{82} -394.863 q^{83} +12.2254 q^{85} -90.5382 q^{86} -288.093 q^{88} +673.418 q^{89} +419.588 q^{92} +166.329 q^{94} -12.7755 q^{95} +1091.11 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.414214	0.146447	0.0732233	−	0.997316i	$-0.476671\pi$
$2$	0.414214	0.146447	0.0732233	+	0.997316i	$0.476671\pi$
$3$	0	0
$3$	0	0
$4$	−7.82843	−0.978553
$5$	−0.100505	−0.00898945	−0.00449472	−	0.999990i	$-0.501431\pi$
$5$	−0.100505	−0.00898945	−0.00449472	+	0.999990i	$0.501431\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−6.55635	−0.289752
$9$	0	0
$10$	−0.0416306	−0.00131647
$11$	43.9411	1.20443	0.602216	−	0.798333i	$-0.294285\pi$
$11$	43.9411	1.20443	0.602216	+	0.798333i	$0.294285\pi$
$12$	0	0
$13$	16.6447	0.355108	0.177554	−	0.984111i	$-0.443182\pi$
$13$	16.6447	0.355108	0.177554	+	0.984111i	$0.443182\pi$
$14$	0	0
$15$	0	0
$16$	59.9117	0.936120
$17$	−121.640	−1.73541	−0.867704	−	0.497081i	$-0.834405\pi$
$17$	−121.640	−1.73541	−0.867704	+	0.497081i	$0.834405\pi$
$18$	0	0
$19$	127.113	1.53482	0.767412	−	0.641154i	$-0.221544\pi$
$19$	127.113	1.53482	0.767412	+	0.641154i	$0.221544\pi$
$20$	0.786797	0.00879665
$21$	0	0
$22$	18.2010	0.176385
$23$	−53.5980	−0.485911	−0.242955	−	0.970037i	$-0.578117\pi$
$23$	−53.5980	−0.485911	−0.242955	+	0.970037i	$0.578117\pi$
$24$	0	0
$25$	−124.990	−0.999919
$26$	6.89444	0.0520043
$27$	0	0
$28$	0	0
$29$	−235.681	−1.50913	−0.754567	−	0.656223i	$-0.772153\pi$
$29$	−235.681	−1.50913	−0.754567	+	0.656223i	$0.772153\pi$
$30$	0	0
$31$	18.7107	0.108404	0.0542022	−	0.998530i	$-0.482738\pi$
$31$	18.7107	0.108404	0.0542022	+	0.998530i	$0.482738\pi$
$32$	77.2670	0.426844
$33$	0	0
$34$	−50.3848	−0.254145
$35$	0	0
$36$	0	0
$37$	−191.882	−0.852574	−0.426287	−	0.904588i	$-0.640179\pi$
$37$	−191.882	−0.852574	−0.426287	+	0.904588i	$0.640179\pi$
$38$	52.6518	0.224770
$39$	0	0
$40$	0.658946	0.00260471
$41$	−319.713	−1.21782	−0.608912	−	0.793238i	$-0.708394\pi$
$41$	−319.713	−1.21782	−0.608912	+	0.793238i	$0.708394\pi$
$42$	0	0
$43$	−218.579	−0.775184	−0.387592	−	0.921831i	$-0.626693\pi$
$43$	−218.579	−0.775184	−0.387592	+	0.921831i	$0.626693\pi$
$44$	−343.990	−1.17860
$45$	0	0
$46$	−22.2010	−0.0711600
$47$	401.553	1.24623	0.623113	−	0.782132i	$-0.285868\pi$
$47$	401.553	1.24623	0.623113	+	0.782132i	$0.285868\pi$
$48$	0	0
$49$	0	0
$50$	−51.7725	−0.146435
$51$	0	0
$52$	−130.302	−0.347492
$53$	−643.117	−1.66677	−0.833386	−	0.552692i	$-0.813601\pi$
$53$	−643.117	−1.66677	−0.833386	+	0.552692i	$0.813601\pi$
$54$	0	0
$55$	−4.41631	−0.0108272
$56$	0	0
$57$	0	0
$58$	−97.6224	−0.221008
$59$	−11.6123	−0.0256235	−0.0128118	−	0.999918i	$-0.504078\pi$
$59$	−11.6123	−0.0256235	−0.0128118	+	0.999918i	$0.504078\pi$
$60$	0	0
$61$	12.2426	0.0256969	0.0128484	−	0.999917i	$-0.495910\pi$
$61$	12.2426	0.0256969	0.0128484	+	0.999917i	$0.495910\pi$
$62$	7.75022	0.0158755
$63$	0	0
$64$	−447.288	−0.873610
$65$	−1.67287	−0.00319222
$66$	0	0
$67$	669.048	1.21996	0.609979	−	0.792417i	$-0.291178\pi$
$67$	669.048	1.21996	0.609979	+	0.792417i	$0.291178\pi$
$68$	952.247	1.69819
$69$	0	0
$70$	0	0
$71$	−822.098	−1.37416	−0.687078	−	0.726584i	$-0.741107\pi$
$71$	−822.098	−1.37416	−0.687078	+	0.726584i	$0.741107\pi$
$72$	0	0
$73$	−515.100	−0.825861	−0.412930	−	0.910763i	$-0.635495\pi$
$73$	−515.100	−0.825861	−0.412930	+	0.910763i	$0.635495\pi$
$74$	−79.4802	−0.124857
$75$	0	0
$76$	−995.092	−1.50191
$77$	0	0
$78$	0	0
$79$	−805.754	−1.14752	−0.573762	−	0.819022i	$-0.694517\pi$
$79$	−805.754	−1.14752	−0.573762	+	0.819022i	$0.694517\pi$
$80$	−6.02143	−0.00841520
$81$	0	0
$82$	−132.429	−0.178346
$83$	−394.863	−0.522191	−0.261095	−	0.965313i	$-0.584084\pi$
$83$	−394.863	−0.522191	−0.261095	+	0.965313i	$0.584084\pi$
$84$	0	0
$85$	12.2254	0.0156004
$86$	−90.5382	−0.113523
$87$	0	0
$88$	−288.093	−0.348987
$89$	673.418	0.802047	0.401024	−	0.916068i	$-0.368655\pi$
$89$	673.418	0.802047	0.401024	+	0.916068i	$0.368655\pi$
$90$	0	0
$91$	0	0
$92$	419.588	0.475490
$93$	0	0
$94$	166.329	0.182505
$95$	−12.7755	−0.0137972
$96$	0	0
$97$	1091.11	1.14212	0.571061	−	0.820908i	$-0.306532\pi$
$97$	1091.11	1.14212	0.571061	+	0.820908i	$0.306532\pi$
$98$	0	0
$99$	0	0
$100$	978.474	0.978474
$101$	−1370.79	−1.35048	−0.675242	−	0.737597i	$-0.735961\pi$
$101$	−1370.79	−1.35048	−0.675242	+	0.737597i	$0.735961\pi$
$102$	0	0
$103$	1413.96	1.35263	0.676316	−	0.736611i	$-0.263575\pi$
$103$	1413.96	1.35263	0.676316	+	0.736611i	$0.263575\pi$
$104$	−109.128	−0.102893
$105$	0	0
$106$	−266.388	−0.244093
$107$	343.539	0.310385	0.155192	−	0.987884i	$-0.450400\pi$
$107$	343.539	0.310385	0.155192	+	0.987884i	$0.450400\pi$
$108$	0	0
$109$	−317.657	−0.279138	−0.139569	−	0.990212i	$-0.544572\pi$
$109$	−317.657	−0.279138	−0.139569	+	0.990212i	$0.544572\pi$
$110$	−1.82929	−0.00158560
$111$	0	0
$112$	0	0
$113$	−798.373	−0.664643	−0.332321	−	0.943166i	$-0.607832\pi$
$113$	−798.373	−0.664643	−0.332321	+	0.943166i	$0.607832\pi$
$114$	0	0
$115$	5.38687	0.00436807
$116$	1845.01	1.47677
$117$	0	0
$118$	−4.80996	−0.00375248
$119$	0	0
$120$	0	0
$121$	599.823	0.450656
$122$	5.07107	0.00376322
$123$	0	0
$124$	−146.475	−0.106080
$125$	25.1253	0.0179782
$126$	0	0
$127$	1071.40	0.748593	0.374297	−	0.927309i	$-0.377884\pi$
$127$	1071.40	0.748593	0.374297	+	0.927309i	$0.377884\pi$
$128$	−803.409	−0.554781
$129$	0	0
$130$	−0.692927	−0.000467490 0
$131$	−2515.02	−1.67739	−0.838695	−	0.544601i	$-0.816681\pi$
$131$	−2515.02	−1.67739	−0.838695	+	0.544601i	$0.816681\pi$
$132$	0	0
$133$	0	0
$134$	277.129	0.178659
$135$	0	0
$136$	797.512	0.502839
$137$	−251.064	−0.156568	−0.0782841	−	0.996931i	$-0.524944\pi$
$137$	−251.064	−0.156568	−0.0782841	+	0.996931i	$0.524944\pi$
$138$	0	0
$139$	−886.067	−0.540685	−0.270343	−	0.962764i	$-0.587137\pi$
$139$	−886.067	−0.540685	−0.270343	+	0.962764i	$0.587137\pi$
$140$	0	0
$141$	0	0
$142$	−340.524	−0.201240
$143$	731.385	0.427703
$144$	0	0
$145$	23.6872	0.0135663
$146$	−213.361	−0.120945
$147$	0	0
$148$	1502.14	0.834289
$149$	−582.626	−0.320339	−0.160170	−	0.987090i	$-0.551204\pi$
$149$	−582.626	−0.320339	−0.160170	+	0.987090i	$0.551204\pi$
$150$	0	0
$151$	−2811.76	−1.51535	−0.757676	−	0.652631i	$-0.773665\pi$
$151$	−2811.76	−1.51535	−0.757676	+	0.652631i	$0.773665\pi$
$152$	−833.395	−0.444719
$153$	0	0
$154$	0	0
$155$	−1.88052	−0.000974496 0
$156$	0	0
$157$	1691.34	0.859770	0.429885	−	0.902884i	$-0.358554\pi$
$157$	1691.34	0.859770	0.429885	+	0.902884i	$0.358554\pi$
$158$	−333.754	−0.168051
$159$	0	0
$160$	−7.76573	−0.00383709
$161$	0	0
$162$	0	0
$163$	−40.7232	−0.0195686	−0.00978432	−	0.999952i	$-0.503114\pi$
$163$	−40.7232	−0.0195686	−0.00978432	+	0.999952i	$0.503114\pi$
$164$	2502.85	1.19170
$165$	0	0
$166$	−163.558	−0.0764731
$167$	2900.47	1.34398	0.671990	−	0.740560i	$-0.265440\pi$
$167$	2900.47	1.34398	0.671990	+	0.740560i	$0.265440\pi$
$168$	0	0
$169$	−1919.96	−0.873899
$170$	5.06393	0.00228462
$171$	0	0
$172$	1711.13	0.758559
$173$	−2146.15	−0.943171	−0.471585	−	0.881820i	$-0.656318\pi$
$173$	−2146.15	−0.943171	−0.471585	+	0.881820i	$0.656318\pi$
$174$	0	0
$175$	0	0
$176$	2632.59	1.12749
$177$	0	0
$178$	278.939	0.117457
$179$	−1203.54	−0.502552	−0.251276	−	0.967916i	$-0.580850\pi$
$179$	−1203.54	−0.502552	−0.251276	+	0.967916i	$0.580850\pi$
$180$	0	0
$181$	2990.47	1.22807	0.614033	−	0.789280i	$-0.289546\pi$
$181$	2990.47	1.22807	0.614033	+	0.789280i	$0.289546\pi$
$182$	0	0
$183$	0	0
$184$	351.407	0.140794
$185$	19.2851	0.00766417
$186$	0	0
$187$	−5344.98	−2.09018
$188$	−3143.53	−1.21950
$189$	0	0
$190$	−5.29177	−0.00202056
$191$	2807.41	1.06354	0.531772	−	0.846887i	$-0.321526\pi$
$191$	2807.41	1.06354	0.531772	+	0.846887i	$0.321526\pi$
$192$	0	0
$193$	3336.37	1.24434	0.622169	−	0.782883i	$-0.286252\pi$
$193$	3336.37	1.24434	0.622169	+	0.782883i	$0.286252\pi$
$194$	451.954	0.167260
$195$	0	0
$196$	0	0
$197$	4226.65	1.52861	0.764305	−	0.644855i	$-0.223082\pi$
$197$	4226.65	1.52861	0.764305	+	0.644855i	$0.223082\pi$
$198$	0	0
$199$	−4385.69	−1.56228	−0.781140	−	0.624356i	$-0.785361\pi$
$199$	−4385.69	−1.56228	−0.781140	+	0.624356i	$0.785361\pi$
$200$	819.477	0.289729
$201$	0	0
$202$	−567.800	−0.197774
$203$	0	0
$204$	0	0
$205$	32.1328	0.0109476
$206$	585.680	0.198088
$207$	0	0
$208$	997.210	0.332423
$209$	5585.48	1.84859
$210$	0	0
$211$	2291.56	0.747665	0.373833	−	0.927496i	$-0.378043\pi$
$211$	2291.56	0.747665	0.373833	+	0.927496i	$0.378043\pi$
$212$	5034.59	1.63103
$213$	0	0
$214$	142.299	0.0454548
$215$	21.9683	0.00696848
$216$	0	0
$217$	0	0
$218$	−131.578	−0.0408788
$219$	0	0
$220$	34.5727	0.0105950
$221$	−2024.65	−0.616257
$222$	0	0
$223$	217.970	0.0654544	0.0327272	−	0.999464i	$-0.489581\pi$
$223$	217.970	0.0654544	0.0327272	+	0.999464i	$0.489581\pi$
$224$	0	0
$225$	0	0
$226$	−330.697	−0.0973347
$227$	1835.36	0.536639	0.268320	−	0.963330i	$-0.413532\pi$
$227$	1835.36	0.536639	0.268320	+	0.963330i	$0.413532\pi$
$228$	0	0
$229$	−2774.01	−0.800488	−0.400244	−	0.916409i	$-0.631075\pi$
$229$	−2774.01	−0.800488	−0.400244	+	0.916409i	$0.631075\pi$
$230$	2.23131	0.000639689 0
$231$	0	0
$232$	1545.21	0.437275
$233$	988.712	0.277994	0.138997	−	0.990293i	$-0.455612\pi$
$233$	988.712	0.277994	0.138997	+	0.990293i	$0.455612\pi$
$234$	0	0
$235$	−40.3581	−0.0112029
$236$	90.9058	0.0250740
$237$	0	0
$238$	0	0
$239$	837.928	0.226783	0.113391	−	0.993550i	$-0.463829\pi$
$239$	837.928	0.226783	0.113391	+	0.993550i	$0.463829\pi$
$240$	0	0
$241$	3454.99	0.923466	0.461733	−	0.887019i	$-0.347228\pi$
$241$	3454.99	0.923466	0.461733	+	0.887019i	$0.347228\pi$
$242$	248.455	0.0659970
$243$	0	0
$244$	−95.8406	−0.0251458
$245$	0	0
$246$	0	0
$247$	2115.75	0.545028
$248$	−122.674	−0.0314104
$249$	0	0
$250$	10.4072	0.00263284
$251$	5635.01	1.41705	0.708523	−	0.705688i	$-0.249362\pi$
$251$	5635.01	1.41705	0.708523	+	0.705688i	$0.249362\pi$
$252$	0	0
$253$	−2355.16	−0.585246
$254$	443.788	0.109629
$255$	0	0
$256$	3245.52	0.792364
$257$	−2271.16	−0.551248	−0.275624	−	0.961265i	$-0.588885\pi$
$257$	−2271.16	−0.551248	−0.275624	+	0.961265i	$0.588885\pi$
$258$	0	0
$259$	0	0
$260$	13.0960	0.00312376
$261$	0	0
$262$	−1041.75	−0.245648
$263$	−163.867	−0.0384201	−0.0192101	−	0.999815i	$-0.506115\pi$
$263$	−163.867	−0.0384201	−0.0192101	+	0.999815i	$0.506115\pi$
$264$	0	0
$265$	64.6365	0.0149834
$266$	0	0
$267$	0	0
$268$	−5237.59	−1.19379
$269$	−5167.10	−1.17116	−0.585582	−	0.810613i	$-0.699134\pi$
$269$	−5167.10	−1.17116	−0.585582	+	0.810613i	$0.699134\pi$
$270$	0	0
$271$	1622.27	0.363638	0.181819	−	0.983332i	$-0.441801\pi$
$271$	1622.27	0.363638	0.181819	+	0.983332i	$0.441801\pi$
$272$	−7287.63	−1.62455
$273$	0	0
$274$	−103.994	−0.0229289
$275$	−5492.20	−1.20433
$276$	0	0
$277$	−4612.37	−1.00047	−0.500235	−	0.865890i	$-0.666753\pi$
$277$	−4612.37	−1.00047	−0.500235	+	0.865890i	$0.666753\pi$
$278$	−367.021	−0.0791815
$279$	0	0
$280$	0	0
$281$	2125.22	0.451174	0.225587	−	0.974223i	$-0.427570\pi$
$281$	2125.22	0.451174	0.225587	+	0.974223i	$0.427570\pi$
$282$	0	0
$283$	−2571.54	−0.540149	−0.270075	−	0.962839i	$-0.587048\pi$
$283$	−2571.54	−0.540149	−0.270075	+	0.962839i	$0.587048\pi$
$284$	6435.73	1.34468
$285$	0	0
$286$	302.950	0.0626356
$287$	0	0
$288$	0	0
$289$	9883.19	2.01164
$290$	9.81154	0.00198674
$291$	0	0
$292$	4032.42	0.808149
$293$	−3324.96	−0.662957	−0.331478	−	0.943463i	$-0.607547\pi$
$293$	−3324.96	−0.662957	−0.331478	+	0.943463i	$0.607547\pi$
$294$	0	0
$295$	1.16709	0.000230341 0
$296$	1258.05	0.247035
$297$	0	0
$298$	−241.331	−0.0469126
$299$	−892.120	−0.172551
$300$	0	0
$301$	0	0
$302$	−1164.67	−0.221918
$303$	0	0
$304$	7615.54	1.43678
$305$	−1.23045	−0.000231001 0
$306$	0	0
$307$	−887.096	−0.164916	−0.0824580	−	0.996595i	$-0.526277\pi$
$307$	−887.096	−0.164916	−0.0824580	+	0.996595i	$0.526277\pi$
$308$	0	0
$309$	0	0
$310$	−0.778936	−0.000142712 0
$311$	4510.82	0.822460	0.411230	−	0.911532i	$-0.365099\pi$
$311$	4510.82	0.822460	0.411230	+	0.911532i	$0.365099\pi$
$312$	0	0
$313$	3715.78	0.671018	0.335509	−	0.942037i	$-0.391092\pi$
$313$	3715.78	0.671018	0.335509	+	0.942037i	$0.391092\pi$
$314$	700.577	0.125910
$315$	0	0
$316$	6307.79	1.12291
$317$	−6954.52	−1.23219	−0.616096	−	0.787671i	$-0.711287\pi$
$317$	−6954.52	−1.23219	−0.616096	+	0.787671i	$0.711287\pi$
$318$	0	0
$319$	−10356.1	−1.81765
$320$	44.9548	0.00785327
$321$	0	0
$322$	0	0
$323$	−15461.9	−2.66355
$324$	0	0
$325$	−2080.41	−0.355079
$326$	−16.8681	−0.00286576
$327$	0	0
$328$	2096.15	0.352867
$329$	0	0
$330$	0	0
$331$	−9863.18	−1.63785	−0.818926	−	0.573899i	$-0.805430\pi$
$331$	−9863.18	−1.63785	−0.818926	+	0.573899i	$0.805430\pi$
$332$	3091.16	0.510992
$333$	0	0
$334$	1201.41	0.196821
$335$	−67.2427	−0.0109668
$336$	0	0
$337$	−5945.06	−0.960974	−0.480487	−	0.877002i	$-0.659540\pi$
$337$	−5945.06	−0.960974	−0.480487	+	0.877002i	$0.659540\pi$
$338$	−795.272	−0.127979
$339$	0	0
$340$	−95.7056	−0.0152658
$341$	822.168	0.130566
$342$	0	0
$343$	0	0
$344$	1433.08	0.224612
$345$	0	0
$346$	−888.963	−0.138124
$347$	−1169.57	−0.180939	−0.0904697	−	0.995899i	$-0.528837\pi$
$347$	−1169.57	−0.180939	−0.0904697	+	0.995899i	$0.528837\pi$
$348$	0	0
$349$	−9176.66	−1.40749	−0.703747	−	0.710451i	$-0.748491\pi$
$349$	−9176.66	−1.40749	−0.703747	+	0.710451i	$0.748491\pi$
$350$	0	0
$351$	0	0
$352$	3395.20	0.514104
$353$	−10587.2	−1.59632	−0.798158	−	0.602448i	$-0.794192\pi$
$353$	−10587.2	−1.59632	−0.798158	+	0.602448i	$0.794192\pi$
$354$	0	0
$355$	82.6250	0.0123529
$356$	−5271.81	−0.784846
$357$	0	0
$358$	−498.522	−0.0735970
$359$	−8615.21	−1.26656	−0.633278	−	0.773924i	$-0.718291\pi$
$359$	−8615.21	−1.26656	−0.633278	+	0.773924i	$0.718291\pi$
$360$	0	0
$361$	9298.64	1.35568
$362$	1238.69	0.179846
$363$	0	0
$364$	0	0
$365$	51.7701	0.00742403
$366$	0	0
$367$	8297.79	1.18022	0.590110	−	0.807323i	$-0.299084\pi$
$367$	8297.79	1.18022	0.590110	+	0.807323i	$0.299084\pi$
$368$	−3211.15	−0.454871
$369$	0	0
$370$	7.98817	0.00112239
$371$	0	0
$372$	0	0
$373$	−5123.86	−0.711269	−0.355634	−	0.934625i	$-0.615735\pi$
$373$	−5123.86	−0.711269	−0.355634	+	0.934625i	$0.615735\pi$
$374$	−2213.96	−0.306100
$375$	0	0
$376$	−2632.72	−0.361097
$377$	−3922.83	−0.535905
$378$	0	0
$379$	−1502.49	−0.203635	−0.101817	−	0.994803i	$-0.532466\pi$
$379$	−1502.49	−0.203635	−0.101817	+	0.994803i	$0.532466\pi$
$380$	100.012	0.0135013
$381$	0	0
$382$	1162.87	0.155753
$383$	10872.9	1.45060	0.725301	−	0.688431i	$-0.241700\pi$
$383$	10872.9	1.45060	0.725301	+	0.688431i	$0.241700\pi$
$384$	0	0
$385$	0	0
$386$	1381.97	0.182229
$387$	0	0
$388$	−8541.71	−1.11763
$389$	−4618.99	−0.602036	−0.301018	−	0.953618i	$-0.597326\pi$
$389$	−4618.99	−0.602036	−0.301018	+	0.953618i	$0.597326\pi$
$390$	0	0
$391$	6519.64	0.843254
$392$	0	0
$393$	0	0
$394$	1750.73	0.223860
$395$	80.9824	0.0103156
$396$	0	0
$397$	9606.95	1.21451	0.607253	−	0.794508i	$-0.292271\pi$
$397$	9606.95	1.21451	0.607253	+	0.794508i	$0.292271\pi$
$398$	−1816.61	−0.228791
$399$	0	0
$400$	−7488.36	−0.936044
$401$	10501.0	1.30772	0.653862	−	0.756614i	$-0.273148\pi$
$401$	10501.0	1.30772	0.653862	+	0.756614i	$0.273148\pi$
$402$	0	0
$403$	311.433	0.0384952
$404$	10731.1	1.32152
$405$	0	0
$406$	0	0
$407$	−8431.52	−1.02687
$408$	0	0
$409$	−12066.9	−1.45885	−0.729427	−	0.684059i	$-0.760213\pi$
$409$	−12066.9	−1.45885	−0.729427	+	0.684059i	$0.760213\pi$
$410$	13.3098	0.00160323
$411$	0	0
$412$	−11069.0	−1.32362
$413$	0	0
$414$	0	0
$415$	39.6857	0.00469421
$416$	1286.08	0.151576
$417$	0	0
$418$	2313.58	0.270720
$419$	6366.31	0.742278	0.371139	−	0.928577i	$-0.378967\pi$
$419$	6366.31	0.742278	0.371139	+	0.928577i	$0.378967\pi$
$420$	0	0
$421$	−4731.84	−0.547781	−0.273890	−	0.961761i	$-0.588311\pi$
$421$	−4731.84	−0.547781	−0.273890	+	0.961761i	$0.588311\pi$
$422$	949.194	0.109493
$423$	0	0
$424$	4216.50	0.482951
$425$	15203.7	1.73527
$426$	0	0
$427$	0	0
$428$	−2689.37	−0.303728
$429$	0	0
$430$	9.09955	0.00102051
$431$	−3752.78	−0.419409	−0.209704	−	0.977765i	$-0.567250\pi$
$431$	−3752.78	−0.419409	−0.209704	+	0.977765i	$0.567250\pi$
$432$	0	0
$433$	11709.2	1.29956	0.649780	−	0.760122i	$-0.274861\pi$
$433$	11709.2	1.29956	0.649780	+	0.760122i	$0.274861\pi$
$434$	0	0
$435$	0	0
$436$	2486.75	0.273151
$437$	−6812.98	−0.745788
$438$	0	0
$439$	−14924.7	−1.62259	−0.811296	−	0.584635i	$-0.801238\pi$
$439$	−14924.7	−1.62259	−0.811296	+	0.584635i	$0.801238\pi$
$440$	28.9548	0.00313720
$441$	0	0
$442$	−838.638	−0.0902487
$443$	8517.66	0.913513	0.456756	−	0.889592i	$-0.349011\pi$
$443$	8517.66	0.913513	0.456756	+	0.889592i	$0.349011\pi$
$444$	0	0
$445$	−67.6820	−0.00720996
$446$	90.2860	0.00958557
$447$	0	0
$448$	0	0
$449$	5965.73	0.627038	0.313519	−	0.949582i	$-0.398492\pi$
$449$	5965.73	0.627038	0.313519	+	0.949582i	$0.398492\pi$
$450$	0	0
$451$	−14048.5	−1.46678
$452$	6250.01	0.650389
$453$	0	0
$454$	760.231	0.0785890
$455$	0	0
$456$	0	0
$457$	−13860.6	−1.41875	−0.709376	−	0.704830i	$-0.751023\pi$
$457$	−13860.6	−1.41875	−0.709376	+	0.704830i	$0.751023\pi$
$458$	−1149.03	−0.117229
$459$	0	0
$460$	−42.1707	−0.00427439
$461$	149.312	0.0150850	0.00754249	−	0.999972i	$-0.497599\pi$
$461$	149.312	0.0150850	0.00754249	+	0.999972i	$0.497599\pi$
$462$	0	0
$463$	5403.95	0.542425	0.271213	−	0.962519i	$-0.412575\pi$
$463$	5403.95	0.542425	0.271213	+	0.962519i	$0.412575\pi$
$464$	−14120.1	−1.41273
$465$	0	0
$466$	409.538	0.0407113
$467$	−3704.66	−0.367090	−0.183545	−	0.983011i	$-0.558757\pi$
$467$	−3704.66	−0.367090	−0.183545	+	0.983011i	$0.558757\pi$
$468$	0	0
$469$	0	0
$470$	−16.7169	−0.00164062
$471$	0	0
$472$	76.1341	0.00742448
$473$	−9604.59	−0.933657
$474$	0	0
$475$	−15887.8	−1.53470
$476$	0	0
$477$	0	0
$478$	347.081	0.0332115
$479$	10671.8	1.01796	0.508982	−	0.860777i	$-0.330022\pi$
$479$	10671.8	1.01796	0.508982	+	0.860777i	$0.330022\pi$
$480$	0	0
$481$	−3193.82	−0.302756
$482$	1431.10	0.135238
$483$	0	0
$484$	−4695.67	−0.440990
$485$	−109.662	−0.0102670
$486$	0	0
$487$	5853.92	0.544695	0.272348	−	0.962199i	$-0.412200\pi$
$487$	5853.92	0.544695	0.272348	+	0.962199i	$0.412200\pi$
$488$	−80.2670	−0.00744573
$489$	0	0
$490$	0	0
$491$	−4065.31	−0.373656	−0.186828	−	0.982393i	$-0.559821\pi$
$491$	−4065.31	−0.373656	−0.186828	+	0.982393i	$0.559821\pi$
$492$	0	0
$493$	28668.2	2.61896
$494$	876.371	0.0798174
$495$	0	0
$496$	1120.99	0.101480
$497$	0	0
$498$	0	0
$499$	4811.19	0.431620	0.215810	−	0.976435i	$-0.430761\pi$
$499$	4811.19	0.431620	0.215810	+	0.976435i	$0.430761\pi$
$500$	−196.691	−0.0175926
$501$	0	0
$502$	2334.10	0.207522
$503$	−17001.2	−1.50705	−0.753526	−	0.657418i	$-0.771649\pi$
$503$	−17001.2	−1.50705	−0.753526	+	0.657418i	$0.771649\pi$
$504$	0	0
$505$	137.771	0.0121401
$506$	−975.537	−0.0857074
$507$	0	0
$508$	−8387.37	−0.732538
$509$	−13797.2	−1.20148	−0.600738	−	0.799446i	$-0.705126\pi$
$509$	−13797.2	−1.20148	−0.600738	+	0.799446i	$0.705126\pi$
$510$	0	0
$511$	0	0
$512$	7771.61	0.670820
$513$	0	0
$514$	−940.744	−0.0807284
$515$	−142.110	−0.0121594
$516$	0	0
$517$	17644.7	1.50099
$518$	0	0
$519$	0	0
$520$	10.9679	0.000924954 0
$521$	−3936.61	−0.331029	−0.165515	−	0.986207i	$-0.552928\pi$
$521$	−3936.61	−0.331029	−0.165515	+	0.986207i	$0.552928\pi$
$522$	0	0
$523$	17459.3	1.45973	0.729866	−	0.683590i	$-0.239582\pi$
$523$	17459.3	1.45973	0.729866	+	0.683590i	$0.239582\pi$
$524$	19688.6	1.64142
$525$	0	0
$526$	−67.8760	−0.00562649
$527$	−2275.96	−0.188126
$528$	0	0
$529$	−9294.26	−0.763891
$530$	26.7733	0.00219426
$531$	0	0
$532$	0	0
$533$	−5321.51	−0.432458
$534$	0	0
$535$	−34.5274	−0.00279019
$536$	−4386.51	−0.353486
$537$	0	0
$538$	−2140.28	−0.171513
$539$	0	0
$540$	0	0
$541$	19101.3	1.51798	0.758992	−	0.651100i	$-0.225692\pi$
$541$	19101.3	1.51798	0.758992	+	0.651100i	$0.225692\pi$
$542$	671.967	0.0532536
$543$	0	0
$544$	−9398.73	−0.740749
$545$	31.9261	0.00250929
$546$	0	0
$547$	15413.5	1.20481	0.602407	−	0.798189i	$-0.294208\pi$
$547$	15413.5	1.20481	0.602407	+	0.798189i	$0.294208\pi$
$548$	1965.44	0.153210
$549$	0	0
$550$	−2274.94	−0.176371
$551$	−29958.1	−2.31626
$552$	0	0
$553$	0	0
$554$	−1910.51	−0.146516
$555$	0	0
$556$	6936.51	0.529089
$557$	20492.9	1.55891	0.779453	−	0.626460i	$-0.215497\pi$
$557$	20492.9	1.55891	0.779453	+	0.626460i	$0.215497\pi$
$558$	0	0
$559$	−3638.17	−0.275274
$560$	0	0
$561$	0	0
$562$	880.294	0.0660729
$563$	7142.49	0.534671	0.267336	−	0.963603i	$-0.413857\pi$
$563$	7142.49	0.534671	0.267336	+	0.963603i	$0.413857\pi$
$564$	0	0
$565$	80.2406	0.00597477
$566$	−1065.17	−0.0791030
$567$	0	0
$568$	5389.96	0.398165
$569$	−4097.91	−0.301922	−0.150961	−	0.988540i	$-0.548237\pi$
$569$	−4097.91	−0.301922	−0.150961	+	0.988540i	$0.548237\pi$
$570$	0	0
$571$	−2838.48	−0.208033	−0.104016	−	0.994576i	$-0.533169\pi$
$571$	−2838.48	−0.208033	−0.104016	+	0.994576i	$0.533169\pi$
$572$	−5725.60	−0.418530
$573$	0	0
$574$	0	0
$575$	6699.21	0.485872
$576$	0	0
$577$	−15464.5	−1.11576	−0.557881	−	0.829921i	$-0.688385\pi$
$577$	−15464.5	−1.11576	−0.557881	+	0.829921i	$0.688385\pi$
$578$	4093.75	0.294598
$579$	0	0
$580$	−185.433	−0.0132753
$581$	0	0
$582$	0	0
$583$	−28259.3	−2.00751
$584$	3377.17	0.239295
$585$	0	0
$586$	−1377.24	−0.0970878
$587$	−14003.6	−0.984652	−0.492326	−	0.870411i	$-0.663853\pi$
$587$	−14003.6	−0.984652	−0.492326	+	0.870411i	$0.663853\pi$
$588$	0	0
$589$	2378.36	0.166382
$590$	0.483425	3.37327e−5 0
$591$	0	0
$592$	−11496.0	−0.798112
$593$	−6504.50	−0.450435	−0.225217	−	0.974309i	$-0.572309\pi$
$593$	−6504.50	−0.450435	−0.225217	+	0.974309i	$0.572309\pi$
$594$	0	0
$595$	0	0
$596$	4561.04	0.313469
$597$	0	0
$598$	−369.528	−0.0252695
$599$	12616.1	0.860567	0.430284	−	0.902694i	$-0.358414\pi$
$599$	12616.1	0.860567	0.430284	+	0.902694i	$0.358414\pi$
$600$	0	0
$601$	8270.87	0.561358	0.280679	−	0.959802i	$-0.409440\pi$
$601$	8270.87	0.561358	0.280679	+	0.959802i	$0.409440\pi$
$602$	0	0
$603$	0	0
$604$	22011.7	1.48285
$605$	−60.2852	−0.00405114
$606$	0	0
$607$	3811.84	0.254889	0.127445	−	0.991846i	$-0.459323\pi$
$607$	3811.84	0.254889	0.127445	+	0.991846i	$0.459323\pi$
$608$	9821.62	0.655130
$609$	0	0
$610$	−0.509668	−3.38293e−5 0
$611$	6683.72	0.442544
$612$	0	0
$613$	11359.3	0.748445	0.374222	−	0.927339i	$-0.377910\pi$
$613$	11359.3	0.748445	0.374222	+	0.927339i	$0.377910\pi$
$614$	−367.447	−0.0241514
$615$	0	0
$616$	0	0
$617$	18272.2	1.19224	0.596118	−	0.802896i	$-0.296709\pi$
$617$	18272.2	1.19224	0.596118	+	0.802896i	$0.296709\pi$
$618$	0	0
$619$	29600.2	1.92203	0.961013	−	0.276503i	$-0.0891756\pi$
$619$	29600.2	1.92203	0.961013	+	0.276503i	$0.0891756\pi$
$620$	14.7215	0.000953596 0
$621$	0	0
$622$	1868.44	0.120447
$623$	0	0
$624$	0	0
$625$	15621.2	0.999758
$626$	1539.13	0.0982682
$627$	0	0
$628$	−13240.6	−0.841331
$629$	23340.5	1.47956
$630$	0	0
$631$	7185.41	0.453322	0.226661	−	0.973974i	$-0.427219\pi$
$631$	7185.41	0.453322	0.226661	+	0.973974i	$0.427219\pi$
$632$	5282.81	0.332498
$633$	0	0
$634$	−2880.66	−0.180450
$635$	−107.681	−0.00672944
$636$	0	0
$637$	0	0
$638$	−4289.64	−0.266189
$639$	0	0
$640$	80.7467	0.00498718
$641$	232.982	0.0143561	0.00717803	−	0.999974i	$-0.497715\pi$
$641$	232.982	0.0143561	0.00717803	+	0.999974i	$0.497715\pi$
$642$	0	0
$643$	−1837.96	−0.112725	−0.0563624	−	0.998410i	$-0.517950\pi$
$643$	−1837.96	−0.112725	−0.0563624	+	0.998410i	$0.517950\pi$
$644$	0	0
$645$	0	0
$646$	−6404.54	−0.390067
$647$	18594.7	1.12988	0.564941	−	0.825131i	$-0.308899\pi$
$647$	18594.7	1.12988	0.564941	+	0.825131i	$0.308899\pi$
$648$	0	0
$649$	−510.256	−0.0308618
$650$	−861.736	−0.0520001
$651$	0	0
$652$	318.799	0.0191490
$653$	28864.7	1.72980	0.864902	−	0.501940i	$-0.167380\pi$
$653$	28864.7	1.72980	0.864902	+	0.501940i	$0.167380\pi$
$654$	0	0
$655$	252.772	0.0150788
$656$	−19154.5	−1.14003
$657$	0	0
$658$	0	0
$659$	29066.3	1.71815	0.859076	−	0.511847i	$-0.171039\pi$
$659$	29066.3	1.71815	0.859076	+	0.511847i	$0.171039\pi$
$660$	0	0
$661$	−3979.51	−0.234168	−0.117084	−	0.993122i	$-0.537355\pi$
$661$	−3979.51	−0.234168	−0.117084	+	0.993122i	$0.537355\pi$
$662$	−4085.46	−0.239858
$663$	0	0
$664$	2588.86	0.151306
$665$	0	0
$666$	0	0
$667$	12632.0	0.733305
$668$	−22706.1	−1.31516
$669$	0	0
$670$	−27.8528	−0.00160604
$671$	537.955	0.0309501
$672$	0	0
$673$	−184.229	−0.0105520	−0.00527601	−	0.999986i	$-0.501679\pi$
$673$	−184.229	−0.0105520	−0.00527601	+	0.999986i	$0.501679\pi$
$674$	−2462.52	−0.140731
$675$	0	0
$676$	15030.2	0.855156
$677$	16683.5	0.947116	0.473558	−	0.880763i	$-0.342969\pi$
$677$	16683.5	0.947116	0.473558	+	0.880763i	$0.342969\pi$
$678$	0	0
$679$	0	0
$680$	−80.1540	−0.00452024
$681$	0	0
$682$	340.553	0.0191209
$683$	−17808.2	−0.997676	−0.498838	−	0.866695i	$-0.666240\pi$
$683$	−17808.2	−0.997676	−0.498838	+	0.866695i	$0.666240\pi$
$684$	0	0
$685$	25.2332	0.00140746
$686$	0	0
$687$	0	0
$688$	−13095.4	−0.725666
$689$	−10704.5	−0.591883
$690$	0	0
$691$	−20145.7	−1.10908	−0.554542	−	0.832156i	$-0.687106\pi$
$691$	−20145.7	−1.10908	−0.554542	+	0.832156i	$0.687106\pi$
$692$	16801.0	0.922943
$693$	0	0
$694$	−484.453	−0.0264980
$695$	89.0542	0.00486046
$696$	0	0
$697$	38889.7	2.11342
$698$	−3801.10	−0.206123
$699$	0	0
$700$	0	0
$701$	2719.67	0.146534	0.0732672	−	0.997312i	$-0.476657\pi$
$701$	2719.67	0.146534	0.0732672	+	0.997312i	$0.476657\pi$
$702$	0	0
$703$	−24390.7	−1.30855
$704$	−19654.4	−1.05220
$705$	0	0
$706$	−4385.36	−0.233775
$707$	0	0
$708$	0	0
$709$	−625.708	−0.0331438	−0.0165719	−	0.999863i	$-0.505275\pi$
$709$	−625.708	−0.0331438	−0.0165719	+	0.999863i	$0.505275\pi$
$710$	34.2244	0.00180904
$711$	0	0
$712$	−4415.17	−0.232395
$713$	−1002.85	−0.0526749
$714$	0	0
$715$	−73.5079	−0.00384481
$716$	9421.82	0.491774
$717$	0	0
$718$	−3568.54	−0.185483
$719$	9577.54	0.496776	0.248388	−	0.968661i	$-0.420099\pi$
$719$	9577.54	0.496776	0.248388	+	0.968661i	$0.420099\pi$
$720$	0	0
$721$	0	0
$722$	3851.62	0.198535
$723$	0	0
$724$	−23410.7	−1.20173
$725$	29457.8	1.50901
$726$	0	0
$727$	−16741.2	−0.854053	−0.427027	−	0.904239i	$-0.640439\pi$
$727$	−16741.2	−0.854053	−0.427027	+	0.904239i	$0.640439\pi$
$728$	0	0
$729$	0	0
$730$	21.4439	0.00108722
$731$	26587.8	1.34526
$732$	0	0
$733$	−6496.04	−0.327335	−0.163668	−	0.986516i	$-0.552332\pi$
$733$	−6496.04	−0.327335	−0.163668	+	0.986516i	$0.552332\pi$
$734$	3437.06	0.172839
$735$	0	0
$736$	−4141.36	−0.207408
$737$	29398.7	1.46936
$738$	0	0
$739$	499.280	0.0248529	0.0124265	−	0.999923i	$-0.496044\pi$
$739$	499.280	0.0248529	0.0124265	+	0.999923i	$0.496044\pi$
$740$	−150.972	−0.00749980
$741$	0	0
$742$	0	0
$743$	6367.30	0.314393	0.157196	−	0.987567i	$-0.449754\pi$
$743$	6367.30	0.314393	0.157196	+	0.987567i	$0.449754\pi$
$744$	0	0
$745$	58.5568	0.00287967
$746$	−2122.37	−0.104163
$747$	0	0
$748$	41842.8	2.04535
$749$	0	0
$750$	0	0
$751$	496.098	0.0241050	0.0120525	−	0.999927i	$-0.496163\pi$
$751$	496.098	0.0241050	0.0120525	+	0.999927i	$0.496163\pi$
$752$	24057.7	1.16662
$753$	0	0
$754$	−1624.89	−0.0784815
$755$	282.597	0.0136222
$756$	0	0
$757$	13025.9	0.625408	0.312704	−	0.949851i	$-0.398765\pi$
$757$	13025.9	0.625408	0.312704	+	0.949851i	$0.398765\pi$
$758$	−622.350	−0.0298216
$759$	0	0
$760$	83.7604	0.00399778
$761$	25474.6	1.21347	0.606736	−	0.794904i	$-0.292479\pi$
$761$	25474.6	1.21347	0.606736	+	0.794904i	$0.292479\pi$
$762$	0	0
$763$	0	0
$764$	−21977.6	−1.04074
$765$	0	0
$766$	4503.72	0.212436
$767$	−193.282	−0.00909911
$768$	0	0
$769$	−29054.0	−1.36244	−0.681218	−	0.732080i	$-0.738550\pi$
$769$	−29054.0	−1.36244	−0.681218	+	0.732080i	$0.738550\pi$
$770$	0	0
$771$	0	0
$772$	−26118.5	−1.21765
$773$	−1897.35	−0.0882834	−0.0441417	−	0.999025i	$-0.514055\pi$
$773$	−1897.35	−0.0882834	−0.0441417	+	0.999025i	$0.514055\pi$
$774$	0	0
$775$	−2338.65	−0.108396
$776$	−7153.72	−0.330933
$777$	0	0
$778$	−1913.25	−0.0881662
$779$	−40639.6	−1.86914
$780$	0	0
$781$	−36123.9	−1.65508
$782$	2700.52	0.123492
$783$	0	0
$784$	0	0
$785$	−169.989	−0.00772886
$786$	0	0
$787$	42650.3	1.93179	0.965895	−	0.258936i	$-0.0833718\pi$
$787$	42650.3	1.93179	0.965895	+	0.258936i	$0.0833718\pi$
$788$	−33088.0	−1.49583
$789$	0	0
$790$	33.5440	0.00151069
$791$	0	0
$792$	0	0
$793$	203.775	0.00912516
$794$	3979.33	0.177860
$795$	0	0
$796$	34333.1	1.52877
$797$	−36822.8	−1.63655	−0.818275	−	0.574828i	$-0.805069\pi$
$797$	−36822.8	−1.63655	−0.818275	+	0.574828i	$0.805069\pi$
$798$	0	0
$799$	−48844.8	−2.16271
$800$	−9657.60	−0.426810
$801$	0	0
$802$	4349.68	0.191512
$803$	−22634.1	−0.994693
$804$	0	0
$805$	0	0
$806$	129.000	0.00563750
$807$	0	0
$808$	8987.38	0.391306
$809$	−5081.94	−0.220855	−0.110427	−	0.993884i	$-0.535222\pi$
$809$	−5081.94	−0.220855	−0.110427	+	0.993884i	$0.535222\pi$
$810$	0	0
$811$	11873.4	0.514097	0.257048	−	0.966399i	$-0.417250\pi$
$811$	11873.4	0.514097	0.257048	+	0.966399i	$0.417250\pi$
$812$	0	0
$813$	0	0
$814$	−3492.45	−0.150381
$815$	4.09289	0.000175911 0
$816$	0	0
$817$	−27784.1	−1.18977
$818$	−4998.29	−0.213644
$819$	0	0
$820$	−251.549	−0.0107128
$821$	−16969.4	−0.721361	−0.360681	−	0.932689i	$-0.617456\pi$
$821$	−16969.4	−0.721361	−0.360681	+	0.932689i	$0.617456\pi$
$822$	0	0
$823$	3995.94	0.169246	0.0846231	−	0.996413i	$-0.473031\pi$
$823$	3995.94	0.169246	0.0846231	+	0.996413i	$0.473031\pi$
$824$	−9270.39	−0.391929
$825$	0	0
$826$	0	0
$827$	13589.7	0.571417	0.285708	−	0.958317i	$-0.407771\pi$
$827$	13589.7	0.571417	0.285708	+	0.958317i	$0.407771\pi$
$828$	0	0
$829$	−30646.0	−1.28393	−0.641966	−	0.766733i	$-0.721881\pi$
$829$	−30646.0	−1.28393	−0.641966	+	0.766733i	$0.721881\pi$
$830$	16.4384	0.000687451 0
$831$	0	0
$832$	−7444.96	−0.310226
$833$	0	0
$834$	0	0
$835$	−291.511	−0.0120816
$836$	−43725.5	−1.80894
$837$	0	0
$838$	2637.01	0.108704
$839$	7497.57	0.308516	0.154258	−	0.988031i	$-0.450701\pi$
$839$	7497.57	0.308516	0.154258	+	0.988031i	$0.450701\pi$
$840$	0	0
$841$	31156.6	1.27749
$842$	−1959.99	−0.0802207
$843$	0	0
$844$	−17939.3	−0.731630
$845$	192.965	0.00785586
$846$	0	0
$847$	0	0
$848$	−38530.2	−1.56030
$849$	0	0
$850$	6297.59	0.254124
$851$	10284.5	0.414275
$852$	0	0
$853$	−10347.6	−0.415352	−0.207676	−	0.978198i	$-0.566590\pi$
$853$	−10347.6	−0.415352	−0.207676	+	0.978198i	$0.566590\pi$
$854$	0	0
$855$	0	0
$856$	−2252.36	−0.0899348
$857$	−1550.00	−0.0617817	−0.0308909	−	0.999523i	$-0.509834\pi$
$857$	−1550.00	−0.0617817	−0.0308909	+	0.999523i	$0.509834\pi$
$858$	0	0
$859$	15187.5	0.603250	0.301625	−	0.953427i	$-0.402471\pi$
$859$	15187.5	0.603250	0.301625	+	0.953427i	$0.402471\pi$
$860$	−171.977	−0.00681903
$861$	0	0
$862$	−1554.45	−0.0614210
$863$	−41550.2	−1.63892	−0.819459	−	0.573138i	$-0.805726\pi$
$863$	−41550.2	−1.63892	−0.819459	+	0.573138i	$0.805726\pi$
$864$	0	0
$865$	215.699	0.00847858
$866$	4850.12	0.190316
$867$	0	0
$868$	0	0
$869$	−35405.8	−1.38212
$870$	0	0
$871$	11136.1	0.433216
$872$	2082.67	0.0808808
$873$	0	0
$874$	−2822.03	−0.109218
$875$	0	0
$876$	0	0
$877$	27837.4	1.07184	0.535919	−	0.844269i	$-0.319965\pi$
$877$	27837.4	1.07184	0.535919	+	0.844269i	$0.319965\pi$
$878$	−6182.02	−0.237623
$879$	0	0
$880$	−264.588	−0.0101355
$881$	2587.85	0.0989635	0.0494817	−	0.998775i	$-0.484243\pi$
$881$	2587.85	0.0989635	0.0494817	+	0.998775i	$0.484243\pi$
$882$	0	0
$883$	−16382.0	−0.624346	−0.312173	−	0.950025i	$-0.601057\pi$
$883$	−16382.0	−0.624346	−0.312173	+	0.950025i	$0.601057\pi$
$884$	15849.8	0.603040
$885$	0	0
$886$	3528.13	0.133781
$887$	22980.2	0.869896	0.434948	−	0.900456i	$-0.356767\pi$
$887$	22980.2	0.869896	0.434948	+	0.900456i	$0.356767\pi$
$888$	0	0
$889$	0	0
$890$	−28.0348	−0.00105587
$891$	0	0
$892$	−1706.36	−0.0640506
$893$	51042.5	1.91274
$894$	0	0
$895$	120.962	0.00451766
$896$	0	0
$897$	0	0
$898$	2471.09	0.0918276
$899$	−4409.76	−0.163597
$900$	0	0
$901$	78228.5	2.89253
$902$	−5819.10	−0.214806
$903$	0	0
$904$	5234.42	0.192582
$905$	−300.557	−0.0110396
$906$	0	0
$907$	−22715.4	−0.831590	−0.415795	−	0.909458i	$-0.636497\pi$
$907$	−22715.4	−0.831590	−0.415795	+	0.909458i	$0.636497\pi$
$908$	−14368.0	−0.525130
$909$	0	0
$910$	0	0
$911$	−34922.3	−1.27006	−0.635031	−	0.772487i	$-0.719013\pi$
$911$	−34922.3	−1.27006	−0.635031	+	0.772487i	$0.719013\pi$
$912$	0	0
$913$	−17350.7	−0.628943
$914$	−5741.24	−0.207772
$915$	0	0
$916$	21716.1	0.783320
$917$	0	0
$918$	0	0
$919$	11702.6	0.420059	0.210030	−	0.977695i	$-0.432644\pi$
$919$	11702.6	0.420059	0.210030	+	0.977695i	$0.432644\pi$
$920$	−35.3182	−0.00126566
$921$	0	0
$922$	61.8473	0.00220914
$923$	−13683.5	−0.487973
$924$	0	0
$925$	23983.3	0.852505
$926$	2238.39	0.0794364
$927$	0	0
$928$	−18210.4	−0.644165
$929$	53096.5	1.87518	0.937588	−	0.347747i	$-0.113053\pi$
$929$	53096.5	1.87518	0.937588	+	0.347747i	$0.113053\pi$
$930$	0	0
$931$	0	0
$932$	−7740.06	−0.272032
$933$	0	0
$934$	−1534.52	−0.0537591
$935$	537.198	0.0187896
$936$	0	0
$937$	−39020.6	−1.36046	−0.680229	−	0.733000i	$-0.738120\pi$
$937$	−39020.6	−1.36046	−0.680229	+	0.733000i	$0.738120\pi$
$938$	0	0
$939$	0	0
$940$	315.941	0.0109626
$941$	−30743.8	−1.06506	−0.532528	−	0.846412i	$-0.678758\pi$
$941$	−30743.8	−1.06506	−0.532528	+	0.846412i	$0.678758\pi$
$942$	0	0
$943$	17136.0	0.591754
$944$	−695.710	−0.0239867
$945$	0	0
$946$	−3978.35	−0.136731
$947$	−16300.3	−0.559332	−0.279666	−	0.960097i	$-0.590224\pi$
$947$	−16300.3	−0.559332	−0.279666	+	0.960097i	$0.590224\pi$
$948$	0	0
$949$	−8573.66	−0.293269
$950$	−6580.94	−0.224752
$951$	0	0
$952$	0	0
$953$	11512.7	0.391325	0.195663	−	0.980671i	$-0.437314\pi$
$953$	11512.7	0.391325	0.195663	+	0.980671i	$0.437314\pi$
$954$	0	0
$955$	−282.159	−0.00956068
$956$	−6559.65	−0.221919
$957$	0	0
$958$	4420.39	0.149078
$959$	0	0
$960$	0	0
$961$	−29440.9	−0.988248
$962$	−1322.92	−0.0443375
$963$	0	0
$964$	−27047.1	−0.903661
$965$	−335.322	−0.0111859
$966$	0	0
$967$	−18178.4	−0.604528	−0.302264	−	0.953224i	$-0.597742\pi$
$967$	−18178.4	−0.604528	−0.302264	+	0.953224i	$0.597742\pi$
$968$	−3932.65	−0.130579
$969$	0	0
$970$	−45.4237	−0.00150357
$971$	28276.7	0.934543	0.467271	−	0.884114i	$-0.345237\pi$
$971$	28276.7	0.934543	0.467271	+	0.884114i	$0.345237\pi$
$972$	0	0
$973$	0	0
$974$	2424.77	0.0797688
$975$	0	0
$976$	733.477	0.0240554
$977$	13947.1	0.456710	0.228355	−	0.973578i	$-0.426665\pi$
$977$	13947.1	0.456710	0.228355	+	0.973578i	$0.426665\pi$
$978$	0	0
$979$	29590.8	0.966011
$980$	0	0
$981$	0	0
$982$	−1683.91	−0.0547206
$983$	−26576.8	−0.862327	−0.431164	−	0.902274i	$-0.641897\pi$
$983$	−26576.8	−0.862327	−0.431164	+	0.902274i	$0.641897\pi$
$984$	0	0
$985$	−424.799	−0.0137414
$986$	11874.7	0.383539
$987$	0	0
$988$	−16563.0	−0.533339
$989$	11715.4	0.376671
$990$	0	0
$991$	16249.9	0.520884	0.260442	−	0.965490i	$-0.416132\pi$
$991$	16249.9	0.520884	0.260442	+	0.965490i	$0.416132\pi$
$992$	1445.72	0.0462718
$993$	0	0
$994$	0	0
$995$	440.784	0.0140440
$996$	0	0
$997$	−18814.1	−0.597642	−0.298821	−	0.954309i	$-0.596593\pi$
$997$	−18814.1	−0.597642	−0.298821	+	0.954309i	$0.596593\pi$
$998$	1992.86	0.0632092
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.4.a.n.1.2		2
3.2	odd	2		147.4.a.j.1.1	✓	2
7.2	even	3		441.4.e.v.361.1		4
7.3	odd	6		441.4.e.u.226.1		4
7.4	even	3		441.4.e.v.226.1		4
7.5	odd	6		441.4.e.u.361.1		4
7.6	odd	2		441.4.a.o.1.2		2
12.11	even	2		2352.4.a.cf.1.1		2
21.2	odd	6		147.4.e.k.67.2		4
21.5	even	6		147.4.e.j.67.2		4
21.11	odd	6		147.4.e.k.79.2		4
21.17	even	6		147.4.e.j.79.2		4
21.20	even	2		147.4.a.k.1.1	yes	2
84.83	odd	2		2352.4.a.bl.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.4.a.j.1.1	✓	2	3.2	odd	2
147.4.a.k.1.1	yes	2	21.20	even	2
147.4.e.j.67.2		4	21.5	even	6
147.4.e.j.79.2		4	21.17	even	6
147.4.e.k.67.2		4	21.2	odd	6
147.4.e.k.79.2		4	21.11	odd	6
441.4.a.n.1.2		2	1.1	even	1	trivial
441.4.a.o.1.2		2	7.6	odd	2
441.4.e.u.226.1		4	7.3	odd	6
441.4.e.u.361.1		4	7.5	odd	6
441.4.e.v.226.1		4	7.4	even	3
441.4.e.v.361.1		4	7.2	even	3
2352.4.a.bl.1.2		2	84.83	odd	2
2352.4.a.cf.1.1		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)

\(f(q)\)	\(=\)	\(q+0.414214 q^{2} -7.82843 q^{4} -0.100505 q^{5} -6.55635 q^{8} +O(q^{10})\)	\(q+0.414214 q^{2} -7.82843 q^{4} -0.100505 q^{5} -6.55635 q^{8} -0.0416306 q^{10} +43.9411 q^{11} +16.6447 q^{13} +59.9117 q^{16} -121.640 q^{17} +127.113 q^{19} +0.786797 q^{20} +18.2010 q^{22} -53.5980 q^{23} -124.990 q^{25} +6.89444 q^{26} -235.681 q^{29} +18.7107 q^{31} +77.2670 q^{32} -50.3848 q^{34} -191.882 q^{37} +52.6518 q^{38} +0.658946 q^{40} -319.713 q^{41} -218.579 q^{43} -343.990 q^{44} -22.2010 q^{46} +401.553 q^{47} -51.7725 q^{50} -130.302 q^{52} -643.117 q^{53} -4.41631 q^{55} -97.6224 q^{58} -11.6123 q^{59} +12.2426 q^{61} +7.75022 q^{62} -447.288 q^{64} -1.67287 q^{65} +669.048 q^{67} +952.247 q^{68} -822.098 q^{71} -515.100 q^{73} -79.4802 q^{74} -995.092 q^{76} -805.754 q^{79} -6.02143 q^{80} -132.429 q^{82} -394.863 q^{83} +12.2254 q^{85} -90.5382 q^{86} -288.093 q^{88} +673.418 q^{89} +419.588 q^{92} +166.329 q^{94} -12.7755 q^{95} +1091.11 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 10 q^{4} - 20 q^{5} + 18 q^{8}+O(q^{10}) \) 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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