LMFDB - Embedded newform 441.4.a.r.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.27492$ 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)

$f(q)$	$=$	$q+5.27492 q^{2} +19.8248 q^{4} +10.5498 q^{5} +62.3746 q^{8} +O(q^{10})$	$q+5.27492 q^{2} +19.8248 q^{4} +10.5498 q^{5} +62.3746 q^{8} +55.6495 q^{10} -34.7492 q^{11} +37.2990 q^{13} +170.423 q^{16} -10.5498 q^{17} +58.5980 q^{19} +209.148 q^{20} -183.299 q^{22} +125.347 q^{23} -13.7010 q^{25} +196.749 q^{26} +35.4020 q^{29} -291.794 q^{31} +399.969 q^{32} -55.6495 q^{34} -259.897 q^{37} +309.100 q^{38} +658.042 q^{40} -338.248 q^{41} +6.80397 q^{43} -688.894 q^{44} +661.196 q^{46} +250.694 q^{47} -72.2716 q^{50} +739.444 q^{52} +536.900 q^{53} -366.598 q^{55} +186.743 q^{58} -35.8904 q^{59} -57.7940 q^{61} -1539.19 q^{62} +746.423 q^{64} +393.498 q^{65} +481.691 q^{67} -209.148 q^{68} -363.752 q^{71} -581.299 q^{73} -1370.94 q^{74} +1161.69 q^{76} -693.691 q^{79} +1797.93 q^{80} -1784.23 q^{82} +1334.39 q^{83} -111.299 q^{85} +35.8904 q^{86} -2167.47 q^{88} -353.038 q^{89} +2484.98 q^{92} +1322.39 q^{94} +618.199 q^{95} -1445.88 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} + 17 q^{4} + 6 q^{5} + 87 q^{8}+O(q^{10}) $ 2 * q + 3 * q^2 + 17 * q^4 + 6 * q^5 + 87 * q^8	$ 2 q + 3 q^{2} + 17 q^{4} + 6 q^{5} + 87 q^{8} + 66 q^{10} + 6 q^{11} - 16 q^{13} + 137 q^{16} - 6 q^{17} - 64 q^{19} + 222 q^{20} - 276 q^{22} - 6 q^{23} - 118 q^{25} + 318 q^{26} + 252 q^{29} - 40 q^{31} + 279 q^{32} - 66 q^{34} - 248 q^{37} + 588 q^{38} + 546 q^{40} - 450 q^{41} + 376 q^{43} - 804 q^{44} + 960 q^{46} - 12 q^{47} + 165 q^{50} + 890 q^{52} + 1104 q^{53} - 552 q^{55} - 306 q^{58} + 804 q^{59} + 428 q^{61} - 2112 q^{62} + 1289 q^{64} + 636 q^{65} + 148 q^{67} - 222 q^{68} - 954 q^{71} - 1072 q^{73} - 1398 q^{74} + 1508 q^{76} - 572 q^{79} + 1950 q^{80} - 1530 q^{82} + 1944 q^{83} - 132 q^{85} - 804 q^{86} - 1164 q^{88} + 366 q^{89} + 2856 q^{92} + 1920 q^{94} + 1176 q^{95} - 808 q^{97}+O(q^{100}) $ 2 * q + 3 * q^2 + 17 * q^4 + 6 * q^5 + 87 * q^8 + 66 * q^10 + 6 * q^11 - 16 * q^13 + 137 * q^16 - 6 * q^17 - 64 * q^19 + 222 * q^20 - 276 * q^22 - 6 * q^23 - 118 * q^25 + 318 * q^26 + 252 * q^29 - 40 * q^31 + 279 * q^32 - 66 * q^34 - 248 * q^37 + 588 * q^38 + 546 * q^40 - 450 * q^41 + 376 * q^43 - 804 * q^44 + 960 * q^46 - 12 * q^47 + 165 * q^50 + 890 * q^52 + 1104 * q^53 - 552 * q^55 - 306 * q^58 + 804 * q^59 + 428 * q^61 - 2112 * q^62 + 1289 * q^64 + 636 * q^65 + 148 * q^67 - 222 * q^68 - 954 * q^71 - 1072 * q^73 - 1398 * q^74 + 1508 * q^76 - 572 * q^79 + 1950 * q^80 - 1530 * q^82 + 1944 * q^83 - 132 * q^85 - 804 * q^86 - 1164 * q^88 + 366 * q^89 + 2856 * q^92 + 1920 * q^94 + 1176 * q^95 - 808 * 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 $

$2$	5.27492	1.86496	0.932482	−	0.361215i	$-0.117638\pi$
$2$	5.27492	1.86496	0.932482	+	0.361215i	$0.117638\pi$
$3$	0	0
$3$	0	0
$4$	19.8248	2.47809
$5$	10.5498	0.943606	0.471803	−	0.881704i	$-0.343603\pi$
$5$	10.5498	0.943606	0.471803	+	0.881704i	$0.343603\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	62.3746	2.75659
$9$	0	0
$10$	55.6495	1.75979
$11$	−34.7492	−0.952479	−0.476240	−	0.879316i	$-0.658000\pi$
$11$	−34.7492	−0.952479	−0.476240	+	0.879316i	$0.658000\pi$
$12$	0	0
$13$	37.2990	0.795760	0.397880	−	0.917437i	$-0.369746\pi$
$13$	37.2990	0.795760	0.397880	+	0.917437i	$0.369746\pi$
$14$	0	0
$15$	0	0
$16$	170.423	2.66286
$17$	−10.5498	−0.150512	−0.0752562	−	0.997164i	$-0.523977\pi$
$17$	−10.5498	−0.150512	−0.0752562	+	0.997164i	$0.523977\pi$
$18$	0	0
$19$	58.5980	0.707542	0.353771	−	0.935332i	$-0.384899\pi$
$19$	58.5980	0.707542	0.353771	+	0.935332i	$0.384899\pi$
$20$	209.148	2.33834
$21$	0	0
$22$	−183.299	−1.77634
$23$	125.347	1.13638	0.568189	−	0.822898i	$-0.307644\pi$
$23$	125.347	1.13638	0.568189	+	0.822898i	$0.307644\pi$
$24$	0	0
$25$	−13.7010	−0.109608
$26$	196.749	1.48406
$27$	0	0
$28$	0	0
$29$	35.4020	0.226689	0.113345	−	0.993556i	$-0.463844\pi$
$29$	35.4020	0.226689	0.113345	+	0.993556i	$0.463844\pi$
$30$	0	0
$31$	−291.794	−1.69057	−0.845286	−	0.534313i	$-0.820570\pi$
$31$	−291.794	−1.69057	−0.845286	+	0.534313i	$0.820570\pi$
$32$	399.969	2.20954
$33$	0	0
$34$	−55.6495	−0.280700
$35$	0	0
$36$	0	0
$37$	−259.897	−1.15478	−0.577389	−	0.816469i	$-0.695928\pi$
$37$	−259.897	−1.15478	−0.577389	+	0.816469i	$0.695928\pi$
$38$	309.100	1.31954
$39$	0	0
$40$	658.042	2.60114
$41$	−338.248	−1.28842	−0.644212	−	0.764847i	$-0.722815\pi$
$41$	−338.248	−1.28842	−0.644212	+	0.764847i	$0.722815\pi$
$42$	0	0
$43$	6.80397	0.0241301	0.0120651	−	0.999927i	$-0.496159\pi$
$43$	6.80397	0.0241301	0.0120651	+	0.999927i	$0.496159\pi$
$44$	−688.894	−2.36033
$45$	0	0
$46$	661.196	2.11931
$47$	250.694	0.778033	0.389016	−	0.921231i	$-0.372815\pi$
$47$	250.694	0.778033	0.389016	+	0.921231i	$0.372815\pi$
$48$	0	0
$49$	0	0
$50$	−72.2716	−0.204415
$51$	0	0
$52$	739.444	1.97197
$53$	536.900	1.39149	0.695745	−	0.718289i	$-0.255075\pi$
$53$	536.900	1.39149	0.695745	+	0.718289i	$0.255075\pi$
$54$	0	0
$55$	−366.598	−0.898765
$56$	0	0
$57$	0	0
$58$	186.743	0.422767
$59$	−35.8904	−0.0791955	−0.0395977	−	0.999216i	$-0.512608\pi$
$59$	−35.8904	−0.0791955	−0.0395977	+	0.999216i	$0.512608\pi$
$60$	0	0
$61$	−57.7940	−0.121308	−0.0606538	−	0.998159i	$-0.519319\pi$
$61$	−57.7940	−0.121308	−0.0606538	+	0.998159i	$0.519319\pi$
$62$	−1539.19	−3.15286
$63$	0	0
$64$	746.423	1.45786
$65$	393.498	0.750884
$66$	0	0
$67$	481.691	0.878327	0.439164	−	0.898407i	$-0.355275\pi$
$67$	481.691	0.878327	0.439164	+	0.898407i	$0.355275\pi$
$68$	−209.148	−0.372984
$69$	0	0
$70$	0	0
$71$	−363.752	−0.608021	−0.304010	−	0.952669i	$-0.598326\pi$
$71$	−363.752	−0.608021	−0.304010	+	0.952669i	$0.598326\pi$
$72$	0	0
$73$	−581.299	−0.931999	−0.465999	−	0.884785i	$-0.654305\pi$
$73$	−581.299	−0.931999	−0.465999	+	0.884785i	$0.654305\pi$
$74$	−1370.94	−2.15362
$75$	0	0
$76$	1161.69	1.75336
$77$	0	0
$78$	0	0
$79$	−693.691	−0.987928	−0.493964	−	0.869482i	$-0.664453\pi$
$79$	−693.691	−0.987928	−0.493964	+	0.869482i	$0.664453\pi$
$80$	1797.93	2.51269
$81$	0	0
$82$	−1784.23	−2.40287
$83$	1334.39	1.76468	0.882341	−	0.470611i	$-0.155967\pi$
$83$	1334.39	1.76468	0.882341	+	0.470611i	$0.155967\pi$
$84$	0	0
$85$	−111.299	−0.142024
$86$	35.8904	0.0450019
$87$	0	0
$88$	−2167.47	−2.62560
$89$	−353.038	−0.420472	−0.210236	−	0.977651i	$-0.567423\pi$
$89$	−353.038	−0.420472	−0.210236	+	0.977651i	$0.567423\pi$
$90$	0	0
$91$	0	0
$92$	2484.98	2.81605
$93$	0	0
$94$	1322.39	1.45100
$95$	618.199	0.667641
$96$	0	0
$97$	−1445.88	−1.51347	−0.756735	−	0.653722i	$-0.773207\pi$
$97$	−1445.88	−1.51347	−0.756735	+	0.653722i	$0.773207\pi$
$98$	0	0
$99$	0	0
$100$	−271.619	−0.271619
$101$	474.852	0.467817	0.233909	−	0.972259i	$-0.424848\pi$
$101$	474.852	0.467817	0.233909	+	0.972259i	$0.424848\pi$
$102$	0	0
$103$	1999.59	1.91287	0.956433	−	0.291951i	$-0.0943044\pi$
$103$	1999.59	1.91287	0.956433	+	0.291951i	$0.0943044\pi$
$104$	2326.51	2.19359
$105$	0	0
$106$	2832.10	2.59508
$107$	−1166.74	−1.05414	−0.527068	−	0.849823i	$-0.676709\pi$
$107$	−1166.74	−1.05414	−0.527068	+	0.849823i	$0.676709\pi$
$108$	0	0
$109$	−1337.18	−1.17503	−0.587515	−	0.809213i	$-0.699894\pi$
$109$	−1337.18	−1.17503	−0.587515	+	0.809213i	$0.699894\pi$
$110$	−1933.77	−1.67616
$111$	0	0
$112$	0	0
$113$	−906.578	−0.754723	−0.377361	−	0.926066i	$-0.623169\pi$
$113$	−906.578	−0.754723	−0.377361	+	0.926066i	$0.623169\pi$
$114$	0	0
$115$	1322.39	1.07229
$116$	701.836	0.561757
$117$	0	0
$118$	−189.319	−0.147697
$119$	0	0
$120$	0	0
$121$	−123.495	−0.0927836
$122$	−304.859	−0.226235
$123$	0	0
$124$	−5784.74	−4.18940
$125$	−1463.27	−1.04703
$126$	0	0
$127$	−1714.89	−1.19820	−0.599101	−	0.800674i	$-0.704475\pi$
$127$	−1714.89	−1.19820	−0.599101	+	0.800674i	$0.704475\pi$
$128$	737.564	0.509313
$129$	0	0
$130$	2075.67	1.40037
$131$	470.611	0.313874	0.156937	−	0.987609i	$-0.449838\pi$
$131$	470.611	0.313874	0.156937	+	0.987609i	$0.449838\pi$
$132$	0	0
$133$	0	0
$134$	2540.88	1.63805
$135$	0	0
$136$	−658.042	−0.414901
$137$	443.910	0.276831	0.138415	−	0.990374i	$-0.455799\pi$
$137$	443.910	0.276831	0.138415	+	0.990374i	$0.455799\pi$
$138$	0	0
$139$	−1669.98	−1.01904	−0.509518	−	0.860460i	$-0.670176\pi$
$139$	−1669.98	−1.01904	−0.509518	+	0.860460i	$0.670176\pi$
$140$	0	0
$141$	0	0
$142$	−1918.76	−1.13394
$143$	−1296.11	−0.757945
$144$	0	0
$145$	373.485	0.213905
$146$	−3066.30	−1.73814
$147$	0	0
$148$	−5152.39	−2.86165
$149$	−743.871	−0.408995	−0.204497	−	0.978867i	$-0.565556\pi$
$149$	−743.871	−0.408995	−0.204497	+	0.978867i	$0.565556\pi$
$150$	0	0
$151$	606.764	0.327005	0.163503	−	0.986543i	$-0.447721\pi$
$151$	606.764	0.327005	0.163503	+	0.986543i	$0.447721\pi$
$152$	3655.03	1.95041
$153$	0	0
$154$	0	0
$155$	−3078.38	−1.59523
$156$	0	0
$157$	−3114.78	−1.58336	−0.791678	−	0.610939i	$-0.790792\pi$
$157$	−3114.78	−1.58336	−0.791678	+	0.610939i	$0.790792\pi$
$158$	−3659.16	−1.84245
$159$	0	0
$160$	4219.61	2.08493
$161$	0	0
$162$	0	0
$163$	2413.07	1.15955	0.579774	−	0.814777i	$-0.303141\pi$
$163$	2413.07	1.15955	0.579774	+	0.814777i	$0.303141\pi$
$164$	−6705.67	−3.19284
$165$	0	0
$166$	7038.81	3.29107
$167$	−610.475	−0.282874	−0.141437	−	0.989947i	$-0.545172\pi$
$167$	−610.475	−0.282874	−0.141437	+	0.989947i	$0.545172\pi$
$168$	0	0
$169$	−805.784	−0.366766
$170$	−587.093	−0.264870
$171$	0	0
$172$	134.887	0.0597968
$173$	3793.81	1.66727	0.833636	−	0.552315i	$-0.186255\pi$
$173$	3793.81	1.66727	0.833636	+	0.552315i	$0.186255\pi$
$174$	0	0
$175$	0	0
$176$	−5922.05	−2.53631
$177$	0	0
$178$	−1862.25	−0.784165
$179$	2804.68	1.17112	0.585562	−	0.810627i	$-0.300874\pi$
$179$	2804.68	1.17112	0.585562	+	0.810627i	$0.300874\pi$
$180$	0	0
$181$	−3106.04	−1.27553	−0.637763	−	0.770232i	$-0.720140\pi$
$181$	−3106.04	−1.27553	−0.637763	+	0.770232i	$0.720140\pi$
$182$	0	0
$183$	0	0
$184$	7818.48	3.13253
$185$	−2741.87	−1.08966
$186$	0	0
$187$	366.598	0.143360
$188$	4969.95	1.92804
$189$	0	0
$190$	3260.95	1.24513
$191$	−261.952	−0.0992365	−0.0496182	−	0.998768i	$-0.515800\pi$
$191$	−261.952	−0.0992365	−0.0496182	+	0.998768i	$0.515800\pi$
$192$	0	0
$193$	4051.07	1.51089	0.755447	−	0.655210i	$-0.227420\pi$
$193$	4051.07	1.51089	0.755447	+	0.655210i	$0.227420\pi$
$194$	−7626.88	−2.82257
$195$	0	0
$196$	0	0
$197$	2874.83	1.03971	0.519855	−	0.854254i	$-0.325986\pi$
$197$	2874.83	1.03971	0.519855	+	0.854254i	$0.325986\pi$
$198$	0	0
$199$	3066.97	1.09252	0.546261	−	0.837615i	$-0.316051\pi$
$199$	3066.97	1.09252	0.546261	+	0.837615i	$0.316051\pi$
$200$	−854.594	−0.302145
$201$	0	0
$202$	2504.81	0.872463
$203$	0	0
$204$	0	0
$205$	−3568.46	−1.21576
$206$	10547.7	3.56743
$207$	0	0
$208$	6356.60	2.11899
$209$	−2036.23	−0.673919
$210$	0	0
$211$	595.422	0.194268	0.0971340	−	0.995271i	$-0.469032\pi$
$211$	595.422	0.194268	0.0971340	+	0.995271i	$0.469032\pi$
$212$	10643.9	3.44824
$213$	0	0
$214$	−6154.44	−1.96593
$215$	71.7808	0.0227693
$216$	0	0
$217$	0	0
$218$	−7053.49	−2.19139
$219$	0	0
$220$	−7267.71	−2.22722
$221$	−393.498	−0.119772
$222$	0	0
$223$	3779.79	1.13504	0.567520	−	0.823360i	$-0.307903\pi$
$223$	3779.79	1.13504	0.567520	+	0.823360i	$0.307903\pi$
$224$	0	0
$225$	0	0
$226$	−4782.12	−1.40753
$227$	1827.62	0.534376	0.267188	−	0.963644i	$-0.413906\pi$
$227$	1827.62	0.534376	0.267188	+	0.963644i	$0.413906\pi$
$228$	0	0
$229$	850.249	0.245354	0.122677	−	0.992447i	$-0.460852\pi$
$229$	850.249	0.245354	0.122677	+	0.992447i	$0.460852\pi$
$230$	6975.51	1.99979
$231$	0	0
$232$	2208.18	0.624890
$233$	6591.10	1.85321	0.926604	−	0.376039i	$-0.122714\pi$
$233$	6591.10	1.85321	0.926604	+	0.376039i	$0.122714\pi$
$234$	0	0
$235$	2644.78	0.734156
$236$	−711.518	−0.196254
$237$	0	0
$238$	0	0
$239$	182.556	0.0494083	0.0247042	−	0.999695i	$-0.492136\pi$
$239$	182.556	0.0494083	0.0247042	+	0.999695i	$0.492136\pi$
$240$	0	0
$241$	−1523.90	−0.407315	−0.203657	−	0.979042i	$-0.565283\pi$
$241$	−1523.90	−0.407315	−0.203657	+	0.979042i	$0.565283\pi$
$242$	−651.426	−0.173038
$243$	0	0
$244$	−1145.75	−0.300612
$245$	0	0
$246$	0	0
$247$	2185.65	0.563034
$248$	−18200.5	−4.66022
$249$	0	0
$250$	−7718.64	−1.95268
$251$	2357.73	0.592903	0.296451	−	0.955048i	$-0.404197\pi$
$251$	2357.73	0.592903	0.296451	+	0.955048i	$0.404197\pi$
$252$	0	0
$253$	−4355.71	−1.08238
$254$	−9045.89	−2.23460
$255$	0	0
$256$	−2080.79	−0.508006
$257$	2782.55	0.675372	0.337686	−	0.941259i	$-0.390356\pi$
$257$	2782.55	0.675372	0.337686	+	0.941259i	$0.390356\pi$
$258$	0	0
$259$	0	0
$260$	7801.01	1.86076
$261$	0	0
$262$	2482.44	0.585364
$263$	−2043.78	−0.479183	−0.239591	−	0.970874i	$-0.577013\pi$
$263$	−2043.78	−0.479183	−0.239591	+	0.970874i	$0.577013\pi$
$264$	0	0
$265$	5664.21	1.31302
$266$	0	0
$267$	0	0
$268$	9549.41	2.17658
$269$	3452.84	0.782614	0.391307	−	0.920260i	$-0.372023\pi$
$269$	3452.84	0.782614	0.391307	+	0.920260i	$0.372023\pi$
$270$	0	0
$271$	−2644.29	−0.592728	−0.296364	−	0.955075i	$-0.595774\pi$
$271$	−2644.29	−0.592728	−0.296364	+	0.955075i	$0.595774\pi$
$272$	−1797.93	−0.400793
$273$	0	0
$274$	2341.59	0.516280
$275$	476.098	0.104399
$276$	0	0
$277$	2679.49	0.581208	0.290604	−	0.956843i	$-0.406144\pi$
$277$	2679.49	0.581208	0.290604	+	0.956843i	$0.406144\pi$
$278$	−8809.01	−1.90046
$279$	0	0
$280$	0	0
$281$	1019.69	0.216476	0.108238	−	0.994125i	$-0.465479\pi$
$281$	1019.69	0.216476	0.108238	+	0.994125i	$0.465479\pi$
$282$	0	0
$283$	−432.206	−0.0907844	−0.0453922	−	0.998969i	$-0.514454\pi$
$283$	−432.206	−0.0907844	−0.0453922	+	0.998969i	$0.514454\pi$
$284$	−7211.30	−1.50673
$285$	0	0
$286$	−6836.87	−1.41354
$287$	0	0
$288$	0	0
$289$	−4801.70	−0.977346
$290$	1970.10	0.398926
$291$	0	0
$292$	−11524.1	−2.30958
$293$	−2245.92	−0.447809	−0.223904	−	0.974611i	$-0.571880\pi$
$293$	−2245.92	−0.447809	−0.223904	+	0.974611i	$0.571880\pi$
$294$	0	0
$295$	−378.638	−0.0747293
$296$	−16211.0	−3.18325
$297$	0	0
$298$	−3923.86	−0.762761
$299$	4675.33	0.904284
$300$	0	0
$301$	0	0
$302$	3200.63	0.609853
$303$	0	0
$304$	9986.44	1.88408
$305$	−609.718	−0.114467
$306$	0	0
$307$	3197.08	0.594354	0.297177	−	0.954822i	$-0.403955\pi$
$307$	3197.08	0.594354	0.297177	+	0.954822i	$0.403955\pi$
$308$	0	0
$309$	0	0
$310$	−16238.2	−2.97506
$311$	−3355.60	−0.611829	−0.305915	−	0.952059i	$-0.598962\pi$
$311$	−3355.60	−0.611829	−0.305915	+	0.952059i	$0.598962\pi$
$312$	0	0
$313$	2256.39	0.407472	0.203736	−	0.979026i	$-0.434692\pi$
$313$	2256.39	0.407472	0.203736	+	0.979026i	$0.434692\pi$
$314$	−16430.2	−2.95290
$315$	0	0
$316$	−13752.3	−2.44818
$317$	6139.19	1.08773	0.543866	−	0.839172i	$-0.316960\pi$
$317$	6139.19	1.08773	0.543866	+	0.839172i	$0.316960\pi$
$318$	0	0
$319$	−1230.19	−0.215917
$320$	7874.64	1.37564
$321$	0	0
$322$	0	0
$323$	−618.199	−0.106494
$324$	0	0
$325$	−511.033	−0.0872216
$326$	12728.8	2.16252
$327$	0	0
$328$	−21098.0	−3.55166
$329$	0	0
$330$	0	0
$331$	7029.81	1.16735	0.583676	−	0.811987i	$-0.301614\pi$
$331$	7029.81	1.16735	0.583676	+	0.811987i	$0.301614\pi$
$332$	26454.0	4.37305
$333$	0	0
$334$	−3220.21	−0.527550
$335$	5081.76	0.828795
$336$	0	0
$337$	10328.4	1.66951	0.834757	−	0.550619i	$-0.185608\pi$
$337$	10328.4	1.66951	0.834757	+	0.550619i	$0.185608\pi$
$338$	−4250.44	−0.684005
$339$	0	0
$340$	−2206.48	−0.351950
$341$	10139.6	1.61024
$342$	0	0
$343$	0	0
$344$	424.395	0.0665170
$345$	0	0
$346$	20012.0	3.10940
$347$	−1967.54	−0.304389	−0.152194	−	0.988351i	$-0.548634\pi$
$347$	−1967.54	−0.304389	−0.152194	+	0.988351i	$0.548634\pi$
$348$	0	0
$349$	4365.46	0.669564	0.334782	−	0.942296i	$-0.391337\pi$
$349$	4365.46	0.669564	0.334782	+	0.942296i	$0.391337\pi$
$350$	0	0
$351$	0	0
$352$	−13898.6	−2.10454
$353$	−6071.59	−0.915462	−0.457731	−	0.889091i	$-0.651338\pi$
$353$	−6071.59	−0.915462	−0.457731	+	0.889091i	$0.651338\pi$
$354$	0	0
$355$	−3837.53	−0.573732
$356$	−6998.90	−1.04197
$357$	0	0
$358$	14794.4	2.18411
$359$	−9638.04	−1.41693	−0.708463	−	0.705748i	$-0.750611\pi$
$359$	−9638.04	−1.41693	−0.708463	+	0.705748i	$0.750611\pi$
$360$	0	0
$361$	−3425.27	−0.499384
$362$	−16384.1	−2.37881
$363$	0	0
$364$	0	0
$365$	−6132.61	−0.879439
$366$	0	0
$367$	−522.725	−0.0743488	−0.0371744	−	0.999309i	$-0.511836\pi$
$367$	−522.725	−0.0743488	−0.0371744	+	0.999309i	$0.511836\pi$
$368$	21362.0	3.02601
$369$	0	0
$370$	−14463.1	−2.03217
$371$	0	0
$372$	0	0
$373$	3229.84	0.448351	0.224175	−	0.974549i	$-0.428031\pi$
$373$	3229.84	0.448351	0.224175	+	0.974549i	$0.428031\pi$
$374$	1933.77	0.267361
$375$	0	0
$376$	15637.0	2.14472
$377$	1320.46	0.180390
$378$	0	0
$379$	6639.71	0.899892	0.449946	−	0.893056i	$-0.351443\pi$
$379$	6639.71	0.899892	0.449946	+	0.893056i	$0.351443\pi$
$380$	12255.6	1.65448
$381$	0	0
$382$	−1381.77	−0.185073
$383$	−14224.4	−1.89774	−0.948871	−	0.315664i	$-0.897773\pi$
$383$	−14224.4	−1.89774	−0.948871	+	0.315664i	$0.897773\pi$
$384$	0	0
$385$	0	0
$386$	21369.1	2.81777
$387$	0	0
$388$	−28664.2	−3.75052
$389$	−2921.82	−0.380828	−0.190414	−	0.981704i	$-0.560983\pi$
$389$	−2921.82	−0.380828	−0.190414	+	0.981704i	$0.560983\pi$
$390$	0	0
$391$	−1322.39	−0.171039
$392$	0	0
$393$	0	0
$394$	15164.5	1.93902
$395$	−7318.33	−0.932215
$396$	0	0
$397$	−811.940	−0.102645	−0.0513226	−	0.998682i	$-0.516344\pi$
$397$	−811.940	−0.102645	−0.0513226	+	0.998682i	$0.516344\pi$
$398$	16178.0	2.03751
$399$	0	0
$400$	−2334.96	−0.291870
$401$	−2338.63	−0.291237	−0.145618	−	0.989341i	$-0.546517\pi$
$401$	−2338.63	−0.291237	−0.145618	+	0.989341i	$0.546517\pi$
$402$	0	0
$403$	−10883.6	−1.34529
$404$	9413.83	1.15930
$405$	0	0
$406$	0	0
$407$	9031.21	1.09990
$408$	0	0
$409$	2727.57	0.329755	0.164877	−	0.986314i	$-0.447277\pi$
$409$	2727.57	0.329755	0.164877	+	0.986314i	$0.447277\pi$
$410$	−18823.3	−2.26736
$411$	0	0
$412$	39641.3	4.74026
$413$	0	0
$414$	0	0
$415$	14077.6	1.66516
$416$	14918.5	1.75826
$417$	0	0
$418$	−10741.0	−1.25684
$419$	13306.3	1.55144	0.775721	−	0.631076i	$-0.217386\pi$
$419$	13306.3	1.55144	0.775721	+	0.631076i	$0.217386\pi$
$420$	0	0
$421$	−11007.5	−1.27428	−0.637138	−	0.770750i	$-0.719882\pi$
$421$	−11007.5	−1.27428	−0.637138	+	0.770750i	$0.719882\pi$
$422$	3140.80	0.362303
$423$	0	0
$424$	33488.9	3.83577
$425$	144.543	0.0164974
$426$	0	0
$427$	0	0
$428$	−23130.2	−2.61225
$429$	0	0
$430$	378.638	0.0424640
$431$	6525.62	0.729300	0.364650	−	0.931145i	$-0.381189\pi$
$431$	6525.62	0.729300	0.364650	+	0.931145i	$0.381189\pi$
$432$	0	0
$433$	11716.3	1.30034	0.650171	−	0.759788i	$-0.274697\pi$
$433$	11716.3	1.30034	0.650171	+	0.759788i	$0.274697\pi$
$434$	0	0
$435$	0	0
$436$	−26509.2	−2.91183
$437$	7345.10	0.804036
$438$	0	0
$439$	14611.4	1.58853	0.794264	−	0.607573i	$-0.207857\pi$
$439$	14611.4	1.58853	0.794264	+	0.607573i	$0.207857\pi$
$440$	−22866.4	−2.47753
$441$	0	0
$442$	−2075.67	−0.223370
$443$	15239.8	1.63446	0.817228	−	0.576314i	$-0.195510\pi$
$443$	15239.8	1.63446	0.817228	+	0.576314i	$0.195510\pi$
$444$	0	0
$445$	−3724.50	−0.396760
$446$	19938.1	2.11681
$447$	0	0
$448$	0	0
$449$	−10678.8	−1.12241	−0.561206	−	0.827676i	$-0.689662\pi$
$449$	−10678.8	−1.12241	−0.561206	+	0.827676i	$0.689662\pi$
$450$	0	0
$451$	11753.8	1.22720
$452$	−17972.7	−1.87027
$453$	0	0
$454$	9640.53	0.996592
$455$	0	0
$456$	0	0
$457$	4228.23	0.432797	0.216399	−	0.976305i	$-0.430569\pi$
$457$	4228.23	0.432797	0.216399	+	0.976305i	$0.430569\pi$
$458$	4484.99	0.457577
$459$	0	0
$460$	26216.1	2.65724
$461$	910.121	0.0919492	0.0459746	−	0.998943i	$-0.485361\pi$
$461$	910.121	0.0919492	0.0459746	+	0.998943i	$0.485361\pi$
$462$	0	0
$463$	4456.16	0.447290	0.223645	−	0.974671i	$-0.428204\pi$
$463$	4456.16	0.447290	0.223645	+	0.974671i	$0.428204\pi$
$464$	6033.30	0.603640
$465$	0	0
$466$	34767.5	3.45617
$467$	−4429.42	−0.438907	−0.219453	−	0.975623i	$-0.570427\pi$
$467$	−4429.42	−0.438907	−0.219453	+	0.975623i	$0.570427\pi$
$468$	0	0
$469$	0	0
$470$	13951.0	1.36918
$471$	0	0
$472$	−2238.65	−0.218310
$473$	−236.432	−0.0229835
$474$	0	0
$475$	−802.851	−0.0775523
$476$	0	0
$477$	0	0
$478$	962.970	0.0921448
$479$	2752.85	0.262591	0.131296	−	0.991343i	$-0.458086\pi$
$479$	2752.85	0.262591	0.131296	+	0.991343i	$0.458086\pi$
$480$	0	0
$481$	−9693.90	−0.918927
$482$	−8038.43	−0.759628
$483$	0	0
$484$	−2448.26	−0.229927
$485$	−15253.8	−1.42812
$486$	0	0
$487$	−670.598	−0.0623977	−0.0311989	−	0.999513i	$-0.509933\pi$
$487$	−670.598	−0.0623977	−0.0311989	+	0.999513i	$0.509933\pi$
$488$	−3604.88	−0.334396
$489$	0	0
$490$	0	0
$491$	8244.70	0.757797	0.378898	−	0.925438i	$-0.376303\pi$
$491$	8244.70	0.757797	0.378898	+	0.925438i	$0.376303\pi$
$492$	0	0
$493$	−373.485	−0.0341195
$494$	11529.1	1.05004
$495$	0	0
$496$	−49728.3	−4.50175
$497$	0	0
$498$	0	0
$499$	8164.91	0.732488	0.366244	−	0.930519i	$-0.380644\pi$
$499$	8164.91	0.732488	0.366244	+	0.930519i	$0.380644\pi$
$500$	−29009.0	−2.59465
$501$	0	0
$502$	12436.8	1.10574
$503$	8175.59	0.724715	0.362357	−	0.932039i	$-0.381972\pi$
$503$	8175.59	0.724715	0.362357	+	0.932039i	$0.381972\pi$
$504$	0	0
$505$	5009.61	0.441435
$506$	−22976.0	−2.01859
$507$	0	0
$508$	−33997.2	−2.96926
$509$	−878.448	−0.0764961	−0.0382480	−	0.999268i	$-0.512178\pi$
$509$	−878.448	−0.0764961	−0.0382480	+	0.999268i	$0.512178\pi$
$510$	0	0
$511$	0	0
$512$	−16876.5	−1.45673
$513$	0	0
$514$	14677.7	1.25955
$515$	21095.3	1.80499
$516$	0	0
$517$	−8711.42	−0.741060
$518$	0	0
$519$	0	0
$520$	24544.3	2.06988
$521$	11712.6	0.984910	0.492455	−	0.870338i	$-0.336100\pi$
$521$	11712.6	0.984910	0.492455	+	0.870338i	$0.336100\pi$
$522$	0	0
$523$	7341.82	0.613834	0.306917	−	0.951736i	$-0.400703\pi$
$523$	7341.82	0.613834	0.306917	+	0.951736i	$0.400703\pi$
$524$	9329.75	0.777809
$525$	0	0
$526$	−10780.8	−0.893659
$527$	3078.38	0.254452
$528$	0	0
$529$	3544.92	0.291355
$530$	29878.2	2.44873
$531$	0	0
$532$	0	0
$533$	−12616.3	−1.02528
$534$	0	0
$535$	−12308.9	−0.994690
$536$	30045.3	2.42119
$537$	0	0
$538$	18213.4	1.45955
$539$	0	0
$540$	0	0
$541$	−15868.7	−1.26109	−0.630545	−	0.776153i	$-0.717169\pi$
$541$	−15868.7	−1.26109	−0.630545	+	0.776153i	$0.717169\pi$
$542$	−13948.4	−1.10542
$543$	0	0
$544$	−4219.61	−0.332563
$545$	−14107.0	−1.10877
$546$	0	0
$547$	2315.26	0.180975	0.0904875	−	0.995898i	$-0.471157\pi$
$547$	2315.26	0.180975	0.0904875	+	0.995898i	$0.471157\pi$
$548$	8800.41	0.686013
$549$	0	0
$550$	2511.38	0.194701
$551$	2074.49	0.160392
$552$	0	0
$553$	0	0
$554$	14134.1	1.08393
$555$	0	0
$556$	−33106.9	−2.52526
$557$	4819.05	0.366588	0.183294	−	0.983058i	$-0.441324\pi$
$557$	4819.05	0.366588	0.183294	+	0.983058i	$0.441324\pi$
$558$	0	0
$559$	253.781	0.0192018
$560$	0	0
$561$	0	0
$562$	5378.79	0.403720
$563$	2540.86	0.190203	0.0951017	−	0.995468i	$-0.469682\pi$
$563$	2540.86	0.190203	0.0951017	+	0.995468i	$0.469682\pi$
$564$	0	0
$565$	−9564.25	−0.712161
$566$	−2279.85	−0.169310
$567$	0	0
$568$	−22688.9	−1.67607
$569$	24220.0	1.78445	0.892227	−	0.451587i	$-0.149142\pi$
$569$	24220.0	1.78445	0.892227	+	0.451587i	$0.149142\pi$
$570$	0	0
$571$	−11772.1	−0.862778	−0.431389	−	0.902166i	$-0.641976\pi$
$571$	−11772.1	−0.862778	−0.431389	+	0.902166i	$0.641976\pi$
$572$	−25695.1	−1.87826
$573$	0	0
$574$	0	0
$575$	−1717.38	−0.124556
$576$	0	0
$577$	−10584.3	−0.763655	−0.381827	−	0.924234i	$-0.624705\pi$
$577$	−10584.3	−0.763655	−0.381827	+	0.924234i	$0.624705\pi$
$578$	−25328.6	−1.82272
$579$	0	0
$580$	7404.25	0.530077
$581$	0	0
$582$	0	0
$583$	−18656.8	−1.32536
$584$	−36258.3	−2.56914
$585$	0	0
$586$	−11847.0	−0.835148
$587$	−8712.63	−0.612621	−0.306311	−	0.951932i	$-0.599095\pi$
$587$	−8712.63	−0.612621	−0.306311	+	0.951932i	$0.599095\pi$
$588$	0	0
$589$	−17098.6	−1.19615
$590$	−1997.28	−0.139368
$591$	0	0
$592$	−44292.4	−3.07501
$593$	−15362.9	−1.06387	−0.531937	−	0.846784i	$-0.678536\pi$
$593$	−15362.9	−1.06387	−0.531937	+	0.846784i	$0.678536\pi$
$594$	0	0
$595$	0	0
$596$	−14747.0	−1.01353
$597$	0	0
$598$	24662.0	1.68646
$599$	−26003.8	−1.77377	−0.886883	−	0.461994i	$-0.847134\pi$
$599$	−26003.8	−1.77377	−0.886883	+	0.461994i	$0.847134\pi$
$600$	0	0
$601$	−20567.7	−1.39596	−0.697982	−	0.716115i	$-0.745918\pi$
$601$	−20567.7	−1.39596	−0.697982	+	0.716115i	$0.745918\pi$
$602$	0	0
$603$	0	0
$604$	12029.0	0.810349
$605$	−1302.85	−0.0875512
$606$	0	0
$607$	−19642.1	−1.31342	−0.656711	−	0.754142i	$-0.728053\pi$
$607$	−19642.1	−1.31342	−0.656711	+	0.754142i	$0.728053\pi$
$608$	23437.4	1.56334
$609$	0	0
$610$	−3216.21	−0.213476
$611$	9350.65	0.619127
$612$	0	0
$613$	8454.59	0.557060	0.278530	−	0.960428i	$-0.410153\pi$
$613$	8454.59	0.557060	0.278530	+	0.960428i	$0.410153\pi$
$614$	16864.3	1.10845
$615$	0	0
$616$	0	0
$617$	24168.4	1.57696	0.788479	−	0.615061i	$-0.210869\pi$
$617$	24168.4	1.57696	0.788479	+	0.615061i	$0.210869\pi$
$618$	0	0
$619$	2037.56	0.132305	0.0661523	−	0.997810i	$-0.478928\pi$
$619$	2037.56	0.132305	0.0661523	+	0.997810i	$0.478928\pi$
$620$	−61028.1	−3.95314
$621$	0	0
$622$	−17700.5	−1.14104
$623$	0	0
$624$	0	0
$625$	−13724.7	−0.878378
$626$	11902.3	0.759921
$627$	0	0
$628$	−61749.8	−3.92370
$629$	2741.87	0.173808
$630$	0	0
$631$	12339.5	0.778489	0.389244	−	0.921135i	$-0.372736\pi$
$631$	12339.5	0.778489	0.389244	+	0.921135i	$0.372736\pi$
$632$	−43268.7	−2.72332
$633$	0	0
$634$	32383.7	2.02858
$635$	−18091.8	−1.13063
$636$	0	0
$637$	0	0
$638$	−6489.15	−0.402677
$639$	0	0
$640$	7781.18	0.480591
$641$	10222.6	0.629906	0.314953	−	0.949107i	$-0.398011\pi$
$641$	10222.6	0.629906	0.314953	+	0.949107i	$0.398011\pi$
$642$	0	0
$643$	1211.75	0.0743187	0.0371594	−	0.999309i	$-0.488169\pi$
$643$	1211.75	0.0743187	0.0371594	+	0.999309i	$0.488169\pi$
$644$	0	0
$645$	0	0
$646$	−3260.95	−0.198607
$647$	−2817.22	−0.171184	−0.0855922	−	0.996330i	$-0.527278\pi$
$647$	−2817.22	−0.171184	−0.0855922	+	0.996330i	$0.527278\pi$
$648$	0	0
$649$	1247.16	0.0754320
$650$	−2695.66	−0.162665
$651$	0	0
$652$	47838.6	2.87347
$653$	−20986.2	−1.25766	−0.628831	−	0.777542i	$-0.716466\pi$
$653$	−20986.2	−1.25766	−0.628831	+	0.777542i	$0.716466\pi$
$654$	0	0
$655$	4964.87	0.296173
$656$	−57645.1	−3.43089
$657$	0	0
$658$	0	0
$659$	2384.09	0.140927	0.0704635	−	0.997514i	$-0.477552\pi$
$659$	2384.09	0.140927	0.0704635	+	0.997514i	$0.477552\pi$
$660$	0	0
$661$	7577.10	0.445862	0.222931	−	0.974834i	$-0.428438\pi$
$661$	7577.10	0.445862	0.222931	+	0.974834i	$0.428438\pi$
$662$	37081.7	2.17707
$663$	0	0
$664$	83232.2	4.86451
$665$	0	0
$666$	0	0
$667$	4437.54	0.257605
$668$	−12102.5	−0.700989
$669$	0	0
$670$	26805.9	1.54567
$671$	2008.30	0.115543
$672$	0	0
$673$	11724.6	0.671547	0.335774	−	0.941943i	$-0.391002\pi$
$673$	11724.6	0.671547	0.335774	+	0.941943i	$0.391002\pi$
$674$	54481.7	3.11358
$675$	0	0
$676$	−15974.5	−0.908880
$677$	32304.3	1.83390	0.916952	−	0.398997i	$-0.130642\pi$
$677$	32304.3	1.83390	0.916952	+	0.398997i	$0.130642\pi$
$678$	0	0
$679$	0	0
$680$	−6942.23	−0.391503
$681$	0	0
$682$	53485.6	3.00303
$683$	−33367.1	−1.86934	−0.934669	−	0.355519i	$-0.884304\pi$
$683$	−33367.1	−1.86934	−0.934669	+	0.355519i	$0.884304\pi$
$684$	0	0
$685$	4683.18	0.261219
$686$	0	0
$687$	0	0
$688$	1159.55	0.0642551
$689$	20025.8	1.10729
$690$	0	0
$691$	1043.67	0.0574577	0.0287288	−	0.999587i	$-0.490854\pi$
$691$	1043.67	0.0574577	0.0287288	+	0.999587i	$0.490854\pi$
$692$	75211.3	4.13166
$693$	0	0
$694$	−10378.6	−0.567674
$695$	−17618.0	−0.961567
$696$	0	0
$697$	3568.46	0.193924
$698$	23027.4	1.24871
$699$	0	0
$700$	0	0
$701$	11305.7	0.609143	0.304572	−	0.952489i	$-0.401487\pi$
$701$	11305.7	0.609143	0.304572	+	0.952489i	$0.401487\pi$
$702$	0	0
$703$	−15229.4	−0.817055
$704$	−25937.6	−1.38858
$705$	0	0
$706$	−32027.1	−1.70730
$707$	0	0
$708$	0	0
$709$	−13306.8	−0.704860	−0.352430	−	0.935838i	$-0.614645\pi$
$709$	−13306.8	−0.704860	−0.352430	+	0.935838i	$0.614645\pi$
$710$	−20242.6	−1.06999
$711$	0	0
$712$	−22020.6	−1.15907
$713$	−36575.6	−1.92113
$714$	0	0
$715$	−13673.7	−0.715201
$716$	55602.0	2.90216
$717$	0	0
$718$	−50839.9	−2.64252
$719$	10701.2	0.555062	0.277531	−	0.960717i	$-0.410484\pi$
$719$	10701.2	0.555062	0.277531	+	0.960717i	$0.410484\pi$
$720$	0	0
$721$	0	0
$722$	−18068.0	−0.931333
$723$	0	0
$724$	−61576.5	−3.16088
$725$	−485.042	−0.0248469
$726$	0	0
$727$	2121.14	0.108210	0.0541051	−	0.998535i	$-0.482769\pi$
$727$	2121.14	0.108210	0.0541051	+	0.998535i	$0.482769\pi$
$728$	0	0
$729$	0	0
$730$	−32349.0	−1.64012
$731$	−71.7808	−0.00363189
$732$	0	0
$733$	21584.0	1.08762	0.543809	−	0.839209i	$-0.316981\pi$
$733$	21584.0	1.08762	0.543809	+	0.839209i	$0.316981\pi$
$734$	−2757.33	−0.138658
$735$	0	0
$736$	50135.0	2.51087
$737$	−16738.4	−0.836588
$738$	0	0
$739$	−9945.21	−0.495048	−0.247524	−	0.968882i	$-0.579617\pi$
$739$	−9945.21	−0.495048	−0.247524	+	0.968882i	$0.579617\pi$
$740$	−54356.9	−2.70027
$741$	0	0
$742$	0	0
$743$	−2867.01	−0.141562	−0.0707808	−	0.997492i	$-0.522549\pi$
$743$	−2867.01	−0.141562	−0.0707808	+	0.997492i	$0.522549\pi$
$744$	0	0
$745$	−7847.71	−0.385930
$746$	17037.1	0.836158
$747$	0	0
$748$	7267.71	0.355259
$749$	0	0
$750$	0	0
$751$	−10824.1	−0.525934	−0.262967	−	0.964805i	$-0.584701\pi$
$751$	−10824.1	−0.525934	−0.262967	+	0.964805i	$0.584701\pi$
$752$	42724.0	2.07179
$753$	0	0
$754$	6965.31	0.336421
$755$	6401.26	0.308564
$756$	0	0
$757$	−14512.0	−0.696761	−0.348381	−	0.937353i	$-0.613268\pi$
$757$	−14512.0	−0.696761	−0.348381	+	0.937353i	$0.613268\pi$
$758$	35023.9	1.67827
$759$	0	0
$760$	38559.9	1.84042
$761$	−33075.8	−1.57556	−0.787778	−	0.615959i	$-0.788769\pi$
$761$	−33075.8	−1.57556	−0.787778	+	0.615959i	$0.788769\pi$
$762$	0	0
$763$	0	0
$764$	−5193.13	−0.245917
$765$	0	0
$766$	−75032.8	−3.53922
$767$	−1338.68	−0.0630206
$768$	0	0
$769$	−6728.44	−0.315518	−0.157759	−	0.987478i	$-0.550427\pi$
$769$	−6728.44	−0.315518	−0.157759	+	0.987478i	$0.550427\pi$
$770$	0	0
$771$	0	0
$772$	80311.5	3.74414
$773$	−24233.3	−1.12757	−0.563784	−	0.825922i	$-0.690655\pi$
$773$	−24233.3	−1.12757	−0.563784	+	0.825922i	$0.690655\pi$
$774$	0	0
$775$	3997.87	0.185300
$776$	−90186.0	−4.17202
$777$	0	0
$778$	−15412.4	−0.710231
$779$	−19820.6	−0.911615
$780$	0	0
$781$	12640.1	0.579127
$782$	−6975.51	−0.318982
$783$	0	0
$784$	0	0
$785$	−32860.5	−1.49406
$786$	0	0
$787$	17200.4	0.779069	0.389535	−	0.921012i	$-0.372636\pi$
$787$	17200.4	0.779069	0.389535	+	0.921012i	$0.372636\pi$
$788$	56992.7	2.57650
$789$	0	0
$790$	−38603.6	−1.73855
$791$	0	0
$792$	0	0
$793$	−2155.66	−0.0965318
$794$	−4282.92	−0.191430
$795$	0	0
$796$	60801.9	2.70737
$797$	−4208.87	−0.187059	−0.0935295	−	0.995617i	$-0.529815\pi$
$797$	−4208.87	−0.187059	−0.0935295	+	0.995617i	$0.529815\pi$
$798$	0	0
$799$	−2644.78	−0.117104
$800$	−5479.98	−0.242183
$801$	0	0
$802$	−12336.1	−0.543146
$803$	20199.7	0.887709
$804$	0	0
$805$	0	0
$806$	−57410.2	−2.50892
$807$	0	0
$808$	29618.7	1.28958
$809$	23632.1	1.02702	0.513511	−	0.858083i	$-0.328344\pi$
$809$	23632.1	1.02702	0.513511	+	0.858083i	$0.328344\pi$
$810$	0	0
$811$	−28425.1	−1.23075	−0.615377	−	0.788233i	$-0.710996\pi$
$811$	−28425.1	−1.23075	−0.615377	+	0.788233i	$0.710996\pi$
$812$	0	0
$813$	0	0
$814$	47638.9	2.05128
$815$	25457.5	1.09416
$816$	0	0
$817$	398.699	0.0170731
$818$	14387.7	0.614981
$819$	0	0
$820$	−70743.7	−3.01278
$821$	−39409.6	−1.67528	−0.837640	−	0.546223i	$-0.816065\pi$
$821$	−39409.6	−1.67528	−0.837640	+	0.546223i	$0.816065\pi$
$822$	0	0
$823$	16346.6	0.692352	0.346176	−	0.938170i	$-0.387480\pi$
$823$	16346.6	0.692352	0.346176	+	0.938170i	$0.387480\pi$
$824$	124723.	5.27300
$825$	0	0
$826$	0	0
$827$	3738.87	0.157211	0.0786054	−	0.996906i	$-0.474953\pi$
$827$	3738.87	0.157211	0.0786054	+	0.996906i	$0.474953\pi$
$828$	0	0
$829$	45196.2	1.89352	0.946761	−	0.321937i	$-0.104334\pi$
$829$	45196.2	1.89352	0.946761	+	0.321937i	$0.104334\pi$
$830$	74258.3	3.10547
$831$	0	0
$832$	27840.8	1.16010
$833$	0	0
$834$	0	0
$835$	−6440.41	−0.266922
$836$	−40367.8	−1.67004
$837$	0	0
$838$	70189.5	2.89338
$839$	15899.7	0.654254	0.327127	−	0.944980i	$-0.393920\pi$
$839$	15899.7	0.654254	0.327127	+	0.944980i	$0.393920\pi$
$840$	0	0
$841$	−23135.7	−0.948612
$842$	−58063.4	−2.37648
$843$	0	0
$844$	11804.1	0.481414
$845$	−8500.89	−0.346082
$846$	0	0
$847$	0	0
$848$	91500.0	3.70534
$849$	0	0
$850$	762.453	0.0307670
$851$	−32577.4	−1.31227
$852$	0	0
$853$	−33926.7	−1.36182	−0.680908	−	0.732369i	$-0.738415\pi$
$853$	−33926.7	−1.36182	−0.680908	+	0.732369i	$0.738415\pi$
$854$	0	0
$855$	0	0
$856$	−72774.7	−2.90583
$857$	−35432.4	−1.41231	−0.706154	−	0.708058i	$-0.749572\pi$
$857$	−35432.4	−1.41231	−0.706154	+	0.708058i	$0.749572\pi$
$858$	0	0
$859$	6780.17	0.269309	0.134655	−	0.990893i	$-0.457008\pi$
$859$	6780.17	0.269309	0.134655	+	0.990893i	$0.457008\pi$
$860$	1423.04	0.0564246
$861$	0	0
$862$	34422.1	1.36012
$863$	30675.1	1.20995	0.604977	−	0.796243i	$-0.293182\pi$
$863$	30675.1	1.20995	0.604977	+	0.796243i	$0.293182\pi$
$864$	0	0
$865$	40024.1	1.57325
$866$	61802.4	2.42509
$867$	0	0
$868$	0	0
$869$	24105.2	0.940981
$870$	0	0
$871$	17966.6	0.698938
$872$	−83405.8	−3.23908
$873$	0	0
$874$	38744.8	1.49950
$875$	0	0
$876$	0	0
$877$	40861.3	1.57330	0.786652	−	0.617397i	$-0.211813\pi$
$877$	40861.3	1.57330	0.786652	+	0.617397i	$0.211813\pi$
$878$	77073.9	2.96255
$879$	0	0
$880$	−62476.6	−2.39328
$881$	−43839.0	−1.67647	−0.838236	−	0.545308i	$-0.816413\pi$
$881$	−43839.0	−1.67647	−0.838236	+	0.545308i	$0.816413\pi$
$882$	0	0
$883$	44625.1	1.70074	0.850371	−	0.526183i	$-0.176377\pi$
$883$	44625.1	1.70074	0.850371	+	0.526183i	$0.176377\pi$
$884$	−7801.01	−0.296806
$885$	0	0
$886$	80388.6	3.04820
$887$	−43967.5	−1.66436	−0.832178	−	0.554509i	$-0.812906\pi$
$887$	−43967.5	−1.66436	−0.832178	+	0.554509i	$0.812906\pi$
$888$	0	0
$889$	0	0
$890$	−19646.4	−0.739943
$891$	0	0
$892$	74933.5	2.81273
$893$	14690.2	0.550491
$894$	0	0
$895$	29588.9	1.10508
$896$	0	0
$897$	0	0
$898$	−56329.7	−2.09326
$899$	−10330.1	−0.383234
$900$	0	0
$901$	−5664.21	−0.209436
$902$	62000.4	2.28868
$903$	0	0
$904$	−56547.4	−2.08046
$905$	−32768.2	−1.20359
$906$	0	0
$907$	−13584.3	−0.497309	−0.248654	−	0.968592i	$-0.579988\pi$
$907$	−13584.3	−0.497309	−0.248654	+	0.968592i	$0.579988\pi$
$908$	36232.1	1.32423
$909$	0	0
$910$	0	0
$911$	16421.6	0.597226	0.298613	−	0.954374i	$-0.403476\pi$
$911$	16421.6	0.597226	0.298613	+	0.954374i	$0.403476\pi$
$912$	0	0
$913$	−46369.0	−1.68082
$914$	22303.6	0.807152
$915$	0	0
$916$	16856.0	0.608010
$917$	0	0
$918$	0	0
$919$	−29487.3	−1.05843	−0.529214	−	0.848488i	$-0.677513\pi$
$919$	−29487.3	−1.05843	−0.529214	+	0.848488i	$0.677513\pi$
$920$	82483.7	2.95588
$921$	0	0
$922$	4800.81	0.171482
$923$	−13567.6	−0.483839
$924$	0	0
$925$	3560.85	0.126573
$926$	23505.9	0.834181
$927$	0	0
$928$	14159.7	0.500878
$929$	3441.85	0.121554	0.0607769	−	0.998151i	$-0.480642\pi$
$929$	3441.85	0.121554	0.0607769	+	0.998151i	$0.480642\pi$
$930$	0	0
$931$	0	0
$932$	130667.	4.59242
$933$	0	0
$934$	−23364.8	−0.818545
$935$	3867.55	0.135275
$936$	0	0
$937$	−5646.60	−0.196869	−0.0984346	−	0.995144i	$-0.531384\pi$
$937$	−5646.60	−0.196869	−0.0984346	+	0.995144i	$0.531384\pi$
$938$	0	0
$939$	0	0
$940$	52432.2	1.81931
$941$	−44680.1	−1.54785	−0.773927	−	0.633275i	$-0.781710\pi$
$941$	−44680.1	−1.54785	−0.773927	+	0.633275i	$0.781710\pi$
$942$	0	0
$943$	−42398.4	−1.46414
$944$	−6116.54	−0.210886
$945$	0	0
$946$	−1247.16	−0.0428633
$947$	48924.6	1.67881	0.839406	−	0.543505i	$-0.182903\pi$
$947$	48924.6	1.67881	0.839406	+	0.543505i	$0.182903\pi$
$948$	0	0
$949$	−21681.9	−0.741647
$950$	−4234.97	−0.144632
$951$	0	0
$952$	0	0
$953$	−52014.3	−1.76801	−0.884003	−	0.467482i	$-0.845161\pi$
$953$	−52014.3	−1.76801	−0.884003	+	0.467482i	$0.845161\pi$
$954$	0	0
$955$	−2763.55	−0.0936401
$956$	3619.14	0.122439
$957$	0	0
$958$	14521.1	0.489723
$959$	0	0
$960$	0	0
$961$	55352.8	1.85804
$962$	−51134.5	−1.71377
$963$	0	0
$964$	−30210.9	−1.00936
$965$	42738.2	1.42569
$966$	0	0
$967$	−47117.7	−1.56691	−0.783456	−	0.621448i	$-0.786545\pi$
$967$	−47117.7	−1.56691	−0.783456	+	0.621448i	$0.786545\pi$
$968$	−7702.95	−0.255767
$969$	0	0
$970$	−80462.3	−2.66339
$971$	−8195.04	−0.270846	−0.135423	−	0.990788i	$-0.543239\pi$
$971$	−8195.04	−0.270846	−0.135423	+	0.990788i	$0.543239\pi$
$972$	0	0
$973$	0	0
$974$	−3537.35	−0.116370
$975$	0	0
$976$	−9849.42	−0.323025
$977$	−4643.51	−0.152056	−0.0760282	−	0.997106i	$-0.524224\pi$
$977$	−4643.51	−0.152056	−0.0760282	+	0.997106i	$0.524224\pi$
$978$	0	0
$979$	12267.8	0.400490
$980$	0	0
$981$	0	0
$982$	43490.1	1.41326
$983$	43986.5	1.42721	0.713607	−	0.700546i	$-0.247060\pi$
$983$	43986.5	1.42721	0.713607	+	0.700546i	$0.247060\pi$
$984$	0	0
$985$	30329.0	0.981077
$986$	−1970.10	−0.0636317
$987$	0	0
$988$	43329.9	1.39525
$989$	852.859	0.0274210
$990$	0	0
$991$	1595.21	0.0511337	0.0255668	−	0.999673i	$-0.491861\pi$
$991$	1595.21	0.0511337	0.0255668	+	0.999673i	$0.491861\pi$
$992$	−116709.	−3.73539
$993$	0	0
$994$	0	0
$995$	32356.0	1.03091
$996$	0	0
$997$	−21501.2	−0.682998	−0.341499	−	0.939882i	$-0.610935\pi$
$997$	−21501.2	−0.682998	−0.341499	+	0.939882i	$0.610935\pi$
$998$	43069.2	1.36606
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.4.a.r.1.2		2
3.2	odd	2		147.4.a.i.1.1		2
7.2	even	3		441.4.e.p.361.1		4
7.3	odd	6		441.4.e.q.226.1		4
7.4	even	3		441.4.e.p.226.1		4
7.5	odd	6		441.4.e.q.361.1		4
7.6	odd	2		63.4.a.e.1.2		2
12.11	even	2		2352.4.a.bz.1.1		2
21.2	odd	6		147.4.e.m.67.2		4
21.5	even	6		147.4.e.l.67.2		4
21.11	odd	6		147.4.e.m.79.2		4
21.17	even	6		147.4.e.l.79.2		4
21.20	even	2		21.4.a.c.1.1	✓	2
28.27	even	2		1008.4.a.ba.1.1		2
35.34	odd	2		1575.4.a.p.1.1		2
84.83	odd	2		336.4.a.m.1.2		2
105.62	odd	4		525.4.d.g.274.1		4
105.83	odd	4		525.4.d.g.274.4		4
105.104	even	2		525.4.a.n.1.2		2
168.83	odd	2		1344.4.a.bo.1.1		2
168.125	even	2		1344.4.a.bg.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.4.a.c.1.1	✓	2	21.20	even	2
63.4.a.e.1.2		2	7.6	odd	2
147.4.a.i.1.1		2	3.2	odd	2
147.4.e.l.67.2		4	21.5	even	6
147.4.e.l.79.2		4	21.17	even	6
147.4.e.m.67.2		4	21.2	odd	6
147.4.e.m.79.2		4	21.11	odd	6
336.4.a.m.1.2		2	84.83	odd	2
441.4.a.r.1.2		2	1.1	even	1	trivial
441.4.e.p.226.1		4	7.4	even	3
441.4.e.p.361.1		4	7.2	even	3
441.4.e.q.226.1		4	7.3	odd	6
441.4.e.q.361.1		4	7.5	odd	6
525.4.a.n.1.2		2	105.104	even	2
525.4.d.g.274.1		4	105.62	odd	4
525.4.d.g.274.4		4	105.83	odd	4
1008.4.a.ba.1.1		2	28.27	even	2
1344.4.a.bg.1.1		2	168.125	even	2
1344.4.a.bo.1.1		2	168.83	odd	2
1575.4.a.p.1.1		2	35.34	odd	2
2352.4.a.bz.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)

\(f(q)\)	\(=\)	\(q+5.27492 q^{2} +19.8248 q^{4} +10.5498 q^{5} +62.3746 q^{8} +O(q^{10})\)	\(q+5.27492 q^{2} +19.8248 q^{4} +10.5498 q^{5} +62.3746 q^{8} +55.6495 q^{10} -34.7492 q^{11} +37.2990 q^{13} +170.423 q^{16} -10.5498 q^{17} +58.5980 q^{19} +209.148 q^{20} -183.299 q^{22} +125.347 q^{23} -13.7010 q^{25} +196.749 q^{26} +35.4020 q^{29} -291.794 q^{31} +399.969 q^{32} -55.6495 q^{34} -259.897 q^{37} +309.100 q^{38} +658.042 q^{40} -338.248 q^{41} +6.80397 q^{43} -688.894 q^{44} +661.196 q^{46} +250.694 q^{47} -72.2716 q^{50} +739.444 q^{52} +536.900 q^{53} -366.598 q^{55} +186.743 q^{58} -35.8904 q^{59} -57.7940 q^{61} -1539.19 q^{62} +746.423 q^{64} +393.498 q^{65} +481.691 q^{67} -209.148 q^{68} -363.752 q^{71} -581.299 q^{73} -1370.94 q^{74} +1161.69 q^{76} -693.691 q^{79} +1797.93 q^{80} -1784.23 q^{82} +1334.39 q^{83} -111.299 q^{85} +35.8904 q^{86} -2167.47 q^{88} -353.038 q^{89} +2484.98 q^{92} +1322.39 q^{94} +618.199 q^{95} -1445.88 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} + 17 q^{4} + 6 q^{5} + 87 q^{8}+O(q^{10}) \) 2 * q + 3 * q^2 + 17 * q^4 + 6 * q^5 + 87 * q^8	\( 2 q + 3 q^{2} + 17 q^{4} + 6 q^{5} + 87 q^{8} + 66 q^{10} + 6 q^{11} - 16 q^{13} + 137 q^{16} - 6 q^{17} - 64 q^{19} + 222 q^{20} - 276 q^{22} - 6 q^{23} - 118 q^{25} + 318 q^{26} + 252 q^{29} - 40 q^{31} + 279 q^{32} - 66 q^{34} - 248 q^{37} + 588 q^{38} + 546 q^{40} - 450 q^{41} + 376 q^{43} - 804 q^{44} + 960 q^{46} - 12 q^{47} + 165 q^{50} + 890 q^{52} + 1104 q^{53} - 552 q^{55} - 306 q^{58} + 804 q^{59} + 428 q^{61} - 2112 q^{62} + 1289 q^{64} + 636 q^{65} + 148 q^{67} - 222 q^{68} - 954 q^{71} - 1072 q^{73} - 1398 q^{74} + 1508 q^{76} - 572 q^{79} + 1950 q^{80} - 1530 q^{82} + 1944 q^{83} - 132 q^{85} - 804 q^{86} - 1164 q^{88} + 366 q^{89} + 2856 q^{92} + 1920 q^{94} + 1176 q^{95} - 808 q^{97}+O(q^{100}) \) 2 * q + 3 * q^2 + 17 * q^4 + 6 * q^5 + 87 * q^8 + 66 * q^10 + 6 * q^11 - 16 * q^13 + 137 * q^16 - 6 * q^17 - 64 * q^19 + 222 * q^20 - 276 * q^22 - 6 * q^23 - 118 * q^25 + 318 * q^26 + 252 * q^29 - 40 * q^31 + 279 * q^32 - 66 * q^34 - 248 * q^37 + 588 * q^38 + 546 * q^40 - 450 * q^41 + 376 * q^43 - 804 * q^44 + 960 * q^46 - 12 * q^47 + 165 * q^50 + 890 * q^52 + 1104 * q^53 - 552 * q^55 - 306 * q^58 + 804 * q^59 + 428 * q^61 - 2112 * q^62 + 1289 * q^64 + 636 * q^65 + 148 * q^67 - 222 * q^68 - 954 * q^71 - 1072 * q^73 - 1398 * q^74 + 1508 * q^76 - 572 * q^79 + 1950 * q^80 - 1530 * q^82 + 1944 * q^83 - 132 * q^85 - 804 * q^86 - 1164 * q^88 + 366 * q^89 + 2856 * q^92 + 1920 * q^94 + 1176 * q^95 - 808 * q^97

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