LMFDB - Embedded newform 441.4.a.u.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:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{65})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 35x^{2} + 36x + 194 $ x^4 - 2x^3 - 35x^2 + 36*x + 194
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 7 $
Twist minimal:	no (minimal twist has level 49)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.11692$ 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)

$f(q)$	$=$	$q-4.53113 q^{2} +12.5311 q^{4} -13.4791 q^{5} -20.5311 q^{8} +O(q^{10})$	$q-4.53113 q^{2} +12.5311 q^{4} -13.4791 q^{5} -20.5311 q^{8} +61.0753 q^{10} -0.813227 q^{11} -34.9564 q^{13} -7.21984 q^{16} +117.732 q^{17} +93.2913 q^{19} -168.908 q^{20} +3.68484 q^{22} -120.249 q^{23} +56.6848 q^{25} +158.392 q^{26} -8.56420 q^{29} +82.1070 q^{31} +196.963 q^{32} -533.458 q^{34} +28.8132 q^{37} -422.715 q^{38} +276.740 q^{40} -70.5291 q^{41} +417.179 q^{43} -10.1906 q^{44} +544.864 q^{46} +338.261 q^{47} -256.846 q^{50} -438.043 q^{52} -149.121 q^{53} +10.9615 q^{55} +38.8055 q^{58} -94.1828 q^{59} +120.525 q^{61} -372.037 q^{62} -834.706 q^{64} +471.179 q^{65} -792.366 q^{67} +1475.31 q^{68} -449.128 q^{71} +469.420 q^{73} -130.556 q^{74} +1169.05 q^{76} -1019.85 q^{79} +97.3166 q^{80} +319.576 q^{82} -104.253 q^{83} -1586.91 q^{85} -1890.29 q^{86} +16.6965 q^{88} -1572.92 q^{89} -1506.86 q^{92} -1532.70 q^{94} -1257.48 q^{95} -550.057 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 34 q^{4} - 66 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 + 34 * q^4 - 66 * q^8	$ 4 q - 2 q^{2} + 34 q^{4} - 66 q^{8} - 100 q^{11} - 174 q^{16} - 340 q^{22} - 352 q^{23} - 128 q^{25} - 260 q^{29} + 30 q^{32} + 212 q^{37} + 540 q^{43} - 460 q^{44} + 696 q^{46} - 1366 q^{50} - 16 q^{53} - 780 q^{58} - 1678 q^{64} + 756 q^{65} - 1944 q^{67} - 2248 q^{71} + 284 q^{74} - 1048 q^{79} - 3284 q^{85} - 4820 q^{86} + 1260 q^{88} - 3512 q^{92} - 2192 q^{95}+O(q^{100}) $ 4 * q - 2 * q^2 + 34 * q^4 - 66 * q^8 - 100 * q^11 - 174 * q^16 - 340 * q^22 - 352 * q^23 - 128 * q^25 - 260 * q^29 + 30 * q^32 + 212 * q^37 + 540 * q^43 - 460 * q^44 + 696 * q^46 - 1366 * q^50 - 16 * q^53 - 780 * q^58 - 1678 * q^64 + 756 * q^65 - 1944 * q^67 - 2248 * q^71 + 284 * q^74 - 1048 * q^79 - 3284 * q^85 - 4820 * q^86 + 1260 * q^88 - 3512 * q^92 - 2192 * q^95

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 $

$2$	−4.53113	−1.60200	−0.800998	−	0.598667i	$-0.795697\pi$
$2$	−4.53113	−1.60200	−0.800998	+	0.598667i	$0.795697\pi$
$3$	0	0
$3$	0	0
$4$	12.5311	1.56639
$5$	−13.4791	−1.20560	−0.602802	−	0.797891i	$-0.705949\pi$
$5$	−13.4791	−1.20560	−0.602802	+	0.797891i	$0.705949\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−20.5311	−0.907356
$9$	0	0
$10$	61.0753	1.93137
$11$	−0.813227	−0.0222906	−0.0111453	−	0.999938i	$-0.503548\pi$
$11$	−0.813227	−0.0222906	−0.0111453	+	0.999938i	$0.503548\pi$
$12$	0	0
$13$	−34.9564	−0.745781	−0.372891	−	0.927875i	$-0.621633\pi$
$13$	−34.9564	−0.745781	−0.372891	+	0.927875i	$0.621633\pi$
$14$	0	0
$15$	0	0
$16$	−7.21984	−0.112810
$17$	117.732	1.67966	0.839829	−	0.542851i	$-0.182655\pi$
$17$	117.732	1.67966	0.839829	+	0.542851i	$0.182655\pi$
$18$	0	0
$19$	93.2913	1.12645	0.563224	−	0.826304i	$-0.309561\pi$
$19$	93.2913	1.12645	0.563224	+	0.826304i	$0.309561\pi$
$20$	−168.908	−1.88845
$21$	0	0
$22$	3.68484	0.0357095
$23$	−120.249	−1.09016	−0.545079	−	0.838384i	$-0.683501\pi$
$23$	−120.249	−1.09016	−0.545079	+	0.838384i	$0.683501\pi$
$24$	0	0
$25$	56.6848	0.453479
$26$	158.392	1.19474
$27$	0	0
$28$	0	0
$29$	−8.56420	−0.0548390	−0.0274195	−	0.999624i	$-0.508729\pi$
$29$	−8.56420	−0.0548390	−0.0274195	+	0.999624i	$0.508729\pi$
$30$	0	0
$31$	82.1070	0.475705	0.237852	−	0.971301i	$-0.423557\pi$
$31$	82.1070	0.475705	0.237852	+	0.971301i	$0.423557\pi$
$32$	196.963	1.08808
$33$	0	0
$34$	−533.458	−2.69080
$35$	0	0
$36$	0	0
$37$	28.8132	0.128023	0.0640117	−	0.997949i	$-0.479611\pi$
$37$	28.8132	0.128023	0.0640117	+	0.997949i	$0.479611\pi$
$38$	−422.715	−1.80456
$39$	0	0
$40$	276.740	1.09391
$41$	−70.5291	−0.268654	−0.134327	−	0.990937i	$-0.542887\pi$
$41$	−70.5291	−0.268654	−0.134327	+	0.990937i	$0.542887\pi$
$42$	0	0
$43$	417.179	1.47952	0.739758	−	0.672873i	$-0.234940\pi$
$43$	417.179	1.47952	0.739758	+	0.672873i	$0.234940\pi$
$44$	−10.1906	−0.0349159
$45$	0	0
$46$	544.864	1.74643
$47$	338.261	1.04980	0.524899	−	0.851165i	$-0.324103\pi$
$47$	338.261	1.04980	0.524899	+	0.851165i	$0.324103\pi$
$48$	0	0
$49$	0	0
$50$	−256.846	−0.726471
$51$	0	0
$52$	−438.043	−1.16819
$53$	−149.121	−0.386477	−0.193239	−	0.981152i	$-0.561899\pi$
$53$	−149.121	−0.386477	−0.193239	+	0.981152i	$0.561899\pi$
$54$	0	0
$55$	10.9615	0.0268737
$56$	0	0
$57$	0	0
$58$	38.8055	0.0878519
$59$	−94.1828	−0.207823	−0.103911	−	0.994587i	$-0.533136\pi$
$59$	−94.1828	−0.207823	−0.103911	+	0.994587i	$0.533136\pi$
$60$	0	0
$61$	120.525	0.252977	0.126488	−	0.991968i	$-0.459629\pi$
$61$	120.525	0.252977	0.126488	+	0.991968i	$0.459629\pi$
$62$	−372.037	−0.762077
$63$	0	0
$64$	−834.706	−1.63029
$65$	471.179	0.899116
$66$	0	0
$67$	−792.366	−1.44482	−0.722410	−	0.691465i	$-0.756965\pi$
$67$	−792.366	−1.44482	−0.722410	+	0.691465i	$0.756965\pi$
$68$	1475.31	2.63100
$69$	0	0
$70$	0	0
$71$	−449.128	−0.750729	−0.375364	−	0.926877i	$-0.622482\pi$
$71$	−449.128	−0.750729	−0.375364	+	0.926877i	$0.622482\pi$
$72$	0	0
$73$	469.420	0.752623	0.376311	−	0.926493i	$-0.377192\pi$
$73$	469.420	0.752623	0.376311	+	0.926493i	$0.377192\pi$
$74$	−130.556	−0.205093
$75$	0	0
$76$	1169.05	1.76446
$77$	0	0
$78$	0	0
$79$	−1019.85	−1.45243	−0.726217	−	0.687465i	$-0.758723\pi$
$79$	−1019.85	−1.45243	−0.726217	+	0.687465i	$0.758723\pi$
$80$	97.3166	0.136004
$81$	0	0
$82$	319.576	0.430382
$83$	−104.253	−0.137870	−0.0689352	−	0.997621i	$-0.521960\pi$
$83$	−104.253	−0.137870	−0.0689352	+	0.997621i	$0.521960\pi$
$84$	0	0
$85$	−1586.91	−2.02500
$86$	−1890.29	−2.37018
$87$	0	0
$88$	16.6965	0.0202256
$89$	−1572.92	−1.87336	−0.936680	−	0.350185i	$-0.886119\pi$
$89$	−1572.92	−1.87336	−0.936680	+	0.350185i	$0.886119\pi$
$90$	0	0
$91$	0	0
$92$	−1506.86	−1.70762
$93$	0	0
$94$	−1532.70	−1.68177
$95$	−1257.48	−1.35805
$96$	0	0
$97$	−550.057	−0.575772	−0.287886	−	0.957665i	$-0.592952\pi$
$97$	−550.057	−0.575772	−0.287886	+	0.957665i	$0.592952\pi$
$98$	0	0
$99$	0	0
$100$	710.325	0.710325
$101$	−65.7169	−0.0647433	−0.0323717	−	0.999476i	$-0.510306\pi$
$101$	−65.7169	−0.0647433	−0.0323717	+	0.999476i	$0.510306\pi$
$102$	0	0
$103$	−1829.35	−1.75001	−0.875005	−	0.484113i	$-0.839142\pi$
$103$	−1829.35	−1.75001	−0.875005	+	0.484113i	$0.839142\pi$
$104$	717.694	0.676689
$105$	0	0
$106$	675.685	0.619135
$107$	−861.377	−0.778248	−0.389124	−	0.921185i	$-0.627222\pi$
$107$	−861.377	−0.778248	−0.389124	+	0.921185i	$0.627222\pi$
$108$	0	0
$109$	−1620.52	−1.42401	−0.712007	−	0.702173i	$-0.752213\pi$
$109$	−1620.52	−1.42401	−0.712007	+	0.702173i	$0.752213\pi$
$110$	−49.6681	−0.0430515
$111$	0	0
$112$	0	0
$113$	−380.409	−0.316689	−0.158344	−	0.987384i	$-0.550616\pi$
$113$	−380.409	−0.316689	−0.158344	+	0.987384i	$0.550616\pi$
$114$	0	0
$115$	1620.84	1.31430
$116$	−107.319	−0.0858993
$117$	0	0
$118$	426.754	0.332931
$119$	0	0
$120$	0	0
$121$	−1330.34	−0.999503
$122$	−546.112	−0.405268
$123$	0	0
$124$	1028.89	0.745140
$125$	920.824	0.658888
$126$	0	0
$127$	958.358	0.669610	0.334805	−	0.942287i	$-0.391329\pi$
$127$	958.358	0.669610	0.334805	+	0.942287i	$0.391329\pi$
$128$	2206.46	1.52363
$129$	0	0
$130$	−2134.97	−1.44038
$131$	1152.16	0.768431	0.384216	−	0.923243i	$-0.374472\pi$
$131$	1152.16	0.768431	0.384216	+	0.923243i	$0.374472\pi$
$132$	0	0
$133$	0	0
$134$	3590.31	2.31459
$135$	0	0
$136$	−2417.17	−1.52405
$137$	−357.377	−0.222867	−0.111434	−	0.993772i	$-0.535544\pi$
$137$	−357.377	−0.222867	−0.111434	+	0.993772i	$0.535544\pi$
$138$	0	0
$139$	2736.29	1.66970	0.834852	−	0.550475i	$-0.185553\pi$
$139$	2736.29	1.66970	0.834852	+	0.550475i	$0.185553\pi$
$140$	0	0
$141$	0	0
$142$	2035.06	1.20266
$143$	28.4275	0.0166239
$144$	0	0
$145$	115.437	0.0661141
$146$	−2127.00	−1.20570
$147$	0	0
$148$	361.062	0.200535
$149$	−1409.94	−0.775212	−0.387606	−	0.921825i	$-0.626698\pi$
$149$	−1409.94	−0.775212	−0.387606	+	0.921825i	$0.626698\pi$
$150$	0	0
$151$	2352.35	1.26776	0.633879	−	0.773432i	$-0.281462\pi$
$151$	2352.35	1.26776	0.633879	+	0.773432i	$0.281462\pi$
$152$	−1915.38	−1.02209
$153$	0	0
$154$	0	0
$155$	−1106.72	−0.573511
$156$	0	0
$157$	−1213.82	−0.617029	−0.308514	−	0.951220i	$-0.599832\pi$
$157$	−1213.82	−0.617029	−0.308514	+	0.951220i	$0.599832\pi$
$158$	4621.08	2.32679
$159$	0	0
$160$	−2654.88	−1.31179
$161$	0	0
$162$	0	0
$163$	−722.774	−0.347313	−0.173657	−	0.984806i	$-0.555558\pi$
$163$	−722.774	−0.347313	−0.173657	+	0.984806i	$0.555558\pi$
$164$	−883.809	−0.420817
$165$	0	0
$166$	472.383	0.220868
$167$	753.016	0.348923	0.174462	−	0.984664i	$-0.444182\pi$
$167$	753.016	0.348923	0.174462	+	0.984664i	$0.444182\pi$
$168$	0	0
$169$	−975.051	−0.443810
$170$	7190.51	3.24404
$171$	0	0
$172$	5227.72	2.31750
$173$	−1859.14	−0.817038	−0.408519	−	0.912750i	$-0.633955\pi$
$173$	−1859.14	−0.817038	−0.408519	+	0.912750i	$0.633955\pi$
$174$	0	0
$175$	0	0
$176$	5.87137	0.00251461
$177$	0	0
$178$	7127.10	3.00112
$179$	522.825	0.218312	0.109156	−	0.994025i	$-0.465185\pi$
$179$	522.825	0.218312	0.109156	+	0.994025i	$0.465185\pi$
$180$	0	0
$181$	−2901.38	−1.19148	−0.595740	−	0.803177i	$-0.703141\pi$
$181$	−2901.38	−1.19148	−0.595740	+	0.803177i	$0.703141\pi$
$182$	0	0
$183$	0	0
$184$	2468.85	0.989163
$185$	−388.375	−0.154345
$186$	0	0
$187$	−95.7427	−0.0374407
$188$	4238.79	1.64439
$189$	0	0
$190$	5697.80	2.17559
$191$	−2604.35	−0.986619	−0.493309	−	0.869854i	$-0.664213\pi$
$191$	−2604.35	−0.986619	−0.493309	+	0.869854i	$0.664213\pi$
$192$	0	0
$193$	676.245	0.252214	0.126107	−	0.992017i	$-0.459752\pi$
$193$	676.245	0.252214	0.126107	+	0.992017i	$0.459752\pi$
$194$	2492.38	0.922384
$195$	0	0
$196$	0	0
$197$	3685.99	1.33308	0.666538	−	0.745471i	$-0.267775\pi$
$197$	3685.99	1.33308	0.666538	+	0.745471i	$0.267775\pi$
$198$	0	0
$199$	799.801	0.284907	0.142453	−	0.989802i	$-0.454501\pi$
$199$	799.801	0.284907	0.142453	+	0.989802i	$0.454501\pi$
$200$	−1163.80	−0.411467
$201$	0	0
$202$	297.772	0.103719
$203$	0	0
$204$	0	0
$205$	950.665	0.323890
$206$	8289.01	2.80351
$207$	0	0
$208$	252.380	0.0841316
$209$	−75.8670	−0.0251092
$210$	0	0
$211$	−667.385	−0.217747	−0.108874	−	0.994056i	$-0.534724\pi$
$211$	−667.385	−0.217747	−0.108874	+	0.994056i	$0.534724\pi$
$212$	−1868.65	−0.605375
$213$	0	0
$214$	3903.01	1.24675
$215$	−5623.18	−1.78371
$216$	0	0
$217$	0	0
$218$	7342.77	2.28126
$219$	0	0
$220$	137.360	0.0420947
$221$	−4115.48	−1.25266
$222$	0	0
$223$	−2646.82	−0.794818	−0.397409	−	0.917642i	$-0.630091\pi$
$223$	−2646.82	−0.794818	−0.397409	+	0.917642i	$0.630091\pi$
$224$	0	0
$225$	0	0
$226$	1723.68	0.507334
$227$	4121.11	1.20497	0.602485	−	0.798131i	$-0.294178\pi$
$227$	4121.11	1.20497	0.602485	+	0.798131i	$0.294178\pi$
$228$	0	0
$229$	−4066.92	−1.17358	−0.586790	−	0.809739i	$-0.699609\pi$
$229$	−4066.92	−1.17358	−0.586790	+	0.809739i	$0.699609\pi$
$230$	−7344.25	−2.10550
$231$	0	0
$232$	175.833	0.0497585
$233$	3904.67	1.09787	0.548934	−	0.835865i	$-0.315034\pi$
$233$	3904.67	1.09787	0.548934	+	0.835865i	$0.315034\pi$
$234$	0	0
$235$	−4559.44	−1.26564
$236$	−1180.22	−0.325532
$237$	0	0
$238$	0	0
$239$	−5425.12	−1.46829	−0.734146	−	0.678991i	$-0.762417\pi$
$239$	−5425.12	−1.46829	−0.734146	+	0.678991i	$0.762417\pi$
$240$	0	0
$241$	1602.89	0.428429	0.214215	−	0.976787i	$-0.431281\pi$
$241$	1602.89	0.428429	0.214215	+	0.976787i	$0.431281\pi$
$242$	6027.94	1.60120
$243$	0	0
$244$	1510.31	0.396261
$245$	0	0
$246$	0	0
$247$	−3261.13	−0.840084
$248$	−1685.75	−0.431634
$249$	0	0
$250$	−4172.37	−1.05554
$251$	3805.93	0.957085	0.478542	−	0.878064i	$-0.341165\pi$
$251$	3805.93	0.957085	0.478542	+	0.878064i	$0.341165\pi$
$252$	0	0
$253$	97.7897	0.0243003
$254$	−4342.44	−1.07271
$255$	0	0
$256$	−3320.09	−0.810569
$257$	4589.34	1.11391	0.556956	−	0.830542i	$-0.311969\pi$
$257$	4589.34	1.11391	0.556956	+	0.830542i	$0.311969\pi$
$258$	0	0
$259$	0	0
$260$	5904.41	1.40837
$261$	0	0
$262$	−5220.58	−1.23102
$263$	−877.175	−0.205661	−0.102831	−	0.994699i	$-0.532790\pi$
$263$	−877.175	−0.205661	−0.102831	+	0.994699i	$0.532790\pi$
$264$	0	0
$265$	2010.00	0.465938
$266$	0	0
$267$	0	0
$268$	−9929.24	−2.26315
$269$	−6123.55	−1.38795	−0.693977	−	0.719997i	$-0.744143\pi$
$269$	−6123.55	−1.38795	−0.693977	+	0.719997i	$0.744143\pi$
$270$	0	0
$271$	3489.76	0.782243	0.391122	−	0.920339i	$-0.372087\pi$
$271$	3489.76	0.782243	0.391122	+	0.920339i	$0.372087\pi$
$272$	−850.006	−0.189482
$273$	0	0
$274$	1619.32	0.357032
$275$	−46.0976	−0.0101083
$276$	0	0
$277$	−4891.70	−1.06106	−0.530530	−	0.847666i	$-0.678007\pi$
$277$	−4891.70	−1.06106	−0.530530	+	0.847666i	$0.678007\pi$
$278$	−12398.5	−2.67486
$279$	0	0
$280$	0	0
$281$	−6914.46	−1.46791	−0.733954	−	0.679199i	$-0.762327\pi$
$281$	−6914.46	−1.46791	−0.733954	+	0.679199i	$0.762327\pi$
$282$	0	0
$283$	−3559.85	−0.747742	−0.373871	−	0.927481i	$-0.621970\pi$
$283$	−3559.85	−0.747742	−0.373871	+	0.927481i	$0.621970\pi$
$284$	−5628.09	−1.17593
$285$	0	0
$286$	−128.809	−0.0266315
$287$	0	0
$288$	0	0
$289$	8947.80	1.82125
$290$	−523.061	−0.105914
$291$	0	0
$292$	5882.36	1.17890
$293$	3285.11	0.655011	0.327505	−	0.944849i	$-0.393792\pi$
$293$	3285.11	0.655011	0.327505	+	0.944849i	$0.393792\pi$
$294$	0	0
$295$	1269.49	0.250552
$296$	−591.568	−0.116163
$297$	0	0
$298$	6388.61	1.24189
$299$	4203.47	0.813020
$300$	0	0
$301$	0	0
$302$	−10658.8	−2.03094
$303$	0	0
$304$	−673.548	−0.127075
$305$	−1624.56	−0.304990
$306$	0	0
$307$	9094.65	1.69075	0.845373	−	0.534176i	$-0.179378\pi$
$307$	9094.65	1.69075	0.845373	+	0.534176i	$0.179378\pi$
$308$	0	0
$309$	0	0
$310$	5014.71	0.918762
$311$	−8163.06	−1.48838	−0.744188	−	0.667971i	$-0.767163\pi$
$311$	−8163.06	−1.48838	−0.744188	+	0.667971i	$0.767163\pi$
$312$	0	0
$313$	−2979.62	−0.538076	−0.269038	−	0.963130i	$-0.586706\pi$
$313$	−2979.62	−0.538076	−0.269038	+	0.963130i	$0.586706\pi$
$314$	5499.98	0.988478
$315$	0	0
$316$	−12779.9	−2.27508
$317$	3888.11	0.688889	0.344445	−	0.938807i	$-0.388067\pi$
$317$	3888.11	0.688889	0.344445	+	0.938807i	$0.388067\pi$
$318$	0	0
$319$	6.96463	0.00122240
$320$	11251.0	1.96548
$321$	0	0
$322$	0	0
$323$	10983.4	1.89205
$324$	0	0
$325$	−1981.50	−0.338196
$326$	3274.98	0.556394
$327$	0	0
$328$	1448.04	0.243764
$329$	0	0
$330$	0	0
$331$	−4893.03	−0.812524	−0.406262	−	0.913757i	$-0.633168\pi$
$331$	−4893.03	−0.812524	−0.406262	+	0.913757i	$0.633168\pi$
$332$	−1306.41	−0.215959
$333$	0	0
$334$	−3412.01	−0.558973
$335$	10680.3	1.74188
$336$	0	0
$337$	−1722.10	−0.278364	−0.139182	−	0.990267i	$-0.544447\pi$
$337$	−1722.10	−0.278364	−0.139182	+	0.990267i	$0.544447\pi$
$338$	4418.08	0.710982
$339$	0	0
$340$	−19885.8	−3.17194
$341$	−66.7716	−0.0106038
$342$	0	0
$343$	0	0
$344$	−8565.16	−1.34245
$345$	0	0
$346$	8423.99	1.30889
$347$	238.058	0.0368289	0.0184145	−	0.999830i	$-0.494138\pi$
$347$	238.058	0.0368289	0.0184145	+	0.999830i	$0.494138\pi$
$348$	0	0
$349$	10053.1	1.54192	0.770959	−	0.636884i	$-0.219777\pi$
$349$	10053.1	1.54192	0.770959	+	0.636884i	$0.219777\pi$
$350$	0	0
$351$	0	0
$352$	−160.176	−0.0242539
$353$	−3470.16	−0.523224	−0.261612	−	0.965173i	$-0.584254\pi$
$353$	−3470.16	−0.523224	−0.261612	+	0.965173i	$0.584254\pi$
$354$	0	0
$355$	6053.82	0.905081
$356$	−19710.5	−2.93442
$357$	0	0
$358$	−2368.99	−0.349734
$359$	−1407.54	−0.206928	−0.103464	−	0.994633i	$-0.532993\pi$
$359$	−1407.54	−0.206928	−0.103464	+	0.994633i	$0.532993\pi$
$360$	0	0
$361$	1844.27	0.268884
$362$	13146.5	1.90875
$363$	0	0
$364$	0	0
$365$	−6327.34	−0.907364
$366$	0	0
$367$	−11133.0	−1.58348	−0.791742	−	0.610855i	$-0.790826\pi$
$367$	−11133.0	−1.58348	−0.791742	+	0.610855i	$0.790826\pi$
$368$	868.179	0.122981
$369$	0	0
$370$	1759.78	0.247261
$371$	0	0
$372$	0	0
$373$	9025.94	1.25294	0.626468	−	0.779447i	$-0.284500\pi$
$373$	9025.94	1.25294	0.626468	+	0.779447i	$0.284500\pi$
$374$	433.823	0.0599798
$375$	0	0
$376$	−6944.88	−0.952540
$377$	299.373	0.0408979
$378$	0	0
$379$	−5855.75	−0.793640	−0.396820	−	0.917896i	$-0.629886\pi$
$379$	−5855.75	−0.793640	−0.396820	+	0.917896i	$0.629886\pi$
$380$	−15757.6	−2.12723
$381$	0	0
$382$	11800.6	1.58056
$383$	−7788.03	−1.03903	−0.519517	−	0.854460i	$-0.673888\pi$
$383$	−7788.03	−1.03903	−0.519517	+	0.854460i	$0.673888\pi$
$384$	0	0
$385$	0	0
$386$	−3064.16	−0.404045
$387$	0	0
$388$	−6892.84	−0.901884
$389$	3814.05	0.497121	0.248560	−	0.968616i	$-0.420043\pi$
$389$	3814.05	0.497121	0.248560	+	0.968616i	$0.420043\pi$
$390$	0	0
$391$	−14157.1	−1.83109
$392$	0	0
$393$	0	0
$394$	−16701.7	−2.13558
$395$	13746.6	1.75106
$396$	0	0
$397$	10165.3	1.28509	0.642545	−	0.766248i	$-0.277879\pi$
$397$	10165.3	1.28509	0.642545	+	0.766248i	$0.277879\pi$
$398$	−3624.00	−0.456419
$399$	0	0
$400$	−409.255	−0.0511569
$401$	−11502.5	−1.43244	−0.716222	−	0.697873i	$-0.754130\pi$
$401$	−11502.5	−1.43244	−0.716222	+	0.697873i	$0.754130\pi$
$402$	0	0
$403$	−2870.16	−0.354772
$404$	−823.507	−0.101413
$405$	0	0
$406$	0	0
$407$	−23.4317	−0.00285372
$408$	0	0
$409$	−3266.27	−0.394882	−0.197441	−	0.980315i	$-0.563263\pi$
$409$	−3266.27	−0.394882	−0.197441	+	0.980315i	$0.563263\pi$
$410$	−4307.59	−0.518870
$411$	0	0
$412$	−22923.8	−2.74120
$413$	0	0
$414$	0	0
$415$	1405.23	0.166217
$416$	−6885.12	−0.811468
$417$	0	0
$418$	343.763	0.0402249
$419$	−6822.93	−0.795518	−0.397759	−	0.917490i	$-0.630212\pi$
$419$	−6822.93	−0.795518	−0.397759	+	0.917490i	$0.630212\pi$
$420$	0	0
$421$	1431.63	0.165733	0.0828665	−	0.996561i	$-0.473592\pi$
$421$	1431.63	0.165733	0.0828665	+	0.996561i	$0.473592\pi$
$422$	3024.01	0.348830
$423$	0	0
$424$	3061.62	0.350673
$425$	6673.61	0.761689
$426$	0	0
$427$	0	0
$428$	−10794.0	−1.21904
$429$	0	0
$430$	25479.3	2.85750
$431$	−15142.2	−1.69228	−0.846141	−	0.532959i	$-0.821080\pi$
$431$	−15142.2	−1.69228	−0.846141	+	0.532959i	$0.821080\pi$
$432$	0	0
$433$	5475.65	0.607721	0.303860	−	0.952717i	$-0.401724\pi$
$433$	5475.65	0.607721	0.303860	+	0.952717i	$0.401724\pi$
$434$	0	0
$435$	0	0
$436$	−20306.9	−2.23056
$437$	−11218.2	−1.22801
$438$	0	0
$439$	−1780.54	−0.193578	−0.0967890	−	0.995305i	$-0.530857\pi$
$439$	−1780.54	−0.193578	−0.0967890	+	0.995305i	$0.530857\pi$
$440$	−225.052	−0.0243840
$441$	0	0
$442$	18647.8	2.00675
$443$	3259.64	0.349594	0.174797	−	0.984605i	$-0.444073\pi$
$443$	3259.64	0.349594	0.174797	+	0.984605i	$0.444073\pi$
$444$	0	0
$445$	21201.5	2.25853
$446$	11993.1	1.27330
$447$	0	0
$448$	0	0
$449$	6826.19	0.717478	0.358739	−	0.933438i	$-0.383207\pi$
$449$	6826.19	0.717478	0.358739	+	0.933438i	$0.383207\pi$
$450$	0	0
$451$	57.3562	0.00598846
$452$	−4766.95	−0.496059
$453$	0	0
$454$	−18673.3	−1.93036
$455$	0	0
$456$	0	0
$457$	3700.03	0.378731	0.189365	−	0.981907i	$-0.439357\pi$
$457$	3700.03	0.378731	0.189365	+	0.981907i	$0.439357\pi$
$458$	18427.8	1.88007
$459$	0	0
$460$	20311.0	2.05871
$461$	−9400.80	−0.949759	−0.474880	−	0.880051i	$-0.657508\pi$
$461$	−9400.80	−0.949759	−0.474880	+	0.880051i	$0.657508\pi$
$462$	0	0
$463$	15483.9	1.55420	0.777102	−	0.629374i	$-0.216689\pi$
$463$	15483.9	1.55420	0.777102	+	0.629374i	$0.216689\pi$
$464$	61.8321	0.00618639
$465$	0	0
$466$	−17692.6	−1.75878
$467$	−2205.62	−0.218552	−0.109276	−	0.994011i	$-0.534853\pi$
$467$	−2205.62	−0.218552	−0.109276	+	0.994011i	$0.534853\pi$
$468$	0	0
$469$	0	0
$470$	20659.4	2.02755
$471$	0	0
$472$	1933.68	0.188569
$473$	−339.261	−0.0329794
$474$	0	0
$475$	5288.20	0.510820
$476$	0	0
$477$	0	0
$478$	24581.9	2.35220
$479$	2349.32	0.224098	0.112049	−	0.993703i	$-0.464259\pi$
$479$	2349.32	0.224098	0.112049	+	0.993703i	$0.464259\pi$
$480$	0	0
$481$	−1007.21	−0.0954775
$482$	−7262.91	−0.686342
$483$	0	0
$484$	−16670.6	−1.56561
$485$	7414.25	0.694152
$486$	0	0
$487$	10394.3	0.967167	0.483583	−	0.875298i	$-0.339335\pi$
$487$	10394.3	0.967167	0.483583	+	0.875298i	$0.339335\pi$
$488$	−2474.50	−0.229540
$489$	0	0
$490$	0	0
$491$	−12586.7	−1.15689	−0.578444	−	0.815722i	$-0.696340\pi$
$491$	−12586.7	−1.15689	−0.578444	+	0.815722i	$0.696340\pi$
$492$	0	0
$493$	−1008.28	−0.0921108
$494$	14776.6	1.34581
$495$	0	0
$496$	−592.799	−0.0536642
$497$	0	0
$498$	0	0
$499$	10627.9	0.953450	0.476725	−	0.879053i	$-0.341824\pi$
$499$	10627.9	0.953450	0.476725	+	0.879053i	$0.341824\pi$
$500$	11539.0	1.03208
$501$	0	0
$502$	−17245.2	−1.53325
$503$	−6719.02	−0.595599	−0.297800	−	0.954628i	$-0.596253\pi$
$503$	−6719.02	−0.595599	−0.297800	+	0.954628i	$0.596253\pi$
$504$	0	0
$505$	885.802	0.0780548
$506$	−443.098	−0.0389291
$507$	0	0
$508$	12009.3	1.04887
$509$	3904.33	0.339993	0.169997	−	0.985445i	$-0.445624\pi$
$509$	3904.33	0.339993	0.169997	+	0.985445i	$0.445624\pi$
$510$	0	0
$511$	0	0
$512$	−2607.89	−0.225105
$513$	0	0
$514$	−20794.9	−1.78448
$515$	24657.9	2.10982
$516$	0	0
$517$	−275.083	−0.0234007
$518$	0	0
$519$	0	0
$520$	−9673.84	−0.815819
$521$	−15699.7	−1.32018	−0.660092	−	0.751184i	$-0.729483\pi$
$521$	−15699.7	−1.32018	−0.660092	+	0.751184i	$0.729483\pi$
$522$	0	0
$523$	−10152.1	−0.848794	−0.424397	−	0.905476i	$-0.639514\pi$
$523$	−10152.1	−0.848794	−0.424397	+	0.905476i	$0.639514\pi$
$524$	14437.8	1.20366
$525$	0	0
$526$	3974.59	0.329469
$527$	9666.61	0.799021
$528$	0	0
$529$	2292.83	0.188447
$530$	−9107.59	−0.746431
$531$	0	0
$532$	0	0
$533$	2465.44	0.200357
$534$	0	0
$535$	11610.6	0.938258
$536$	16268.2	1.31097
$537$	0	0
$538$	27746.6	2.22350
$539$	0	0
$540$	0	0
$541$	−19846.6	−1.57722	−0.788608	−	0.614896i	$-0.789198\pi$
$541$	−19846.6	−1.57722	−0.788608	+	0.614896i	$0.789198\pi$
$542$	−15812.6	−1.25315
$543$	0	0
$544$	23188.8	1.82760
$545$	21843.0	1.71679
$546$	0	0
$547$	−22798.9	−1.78210	−0.891052	−	0.453901i	$-0.850032\pi$
$547$	−22798.9	−1.78210	−0.891052	+	0.453901i	$0.850032\pi$
$548$	−4478.34	−0.349097
$549$	0	0
$550$	208.874	0.0161935
$551$	−798.965	−0.0617733
$552$	0	0
$553$	0	0
$554$	22164.9	1.69981
$555$	0	0
$556$	34288.8	2.61541
$557$	17998.3	1.36914	0.684570	−	0.728947i	$-0.259990\pi$
$557$	17998.3	1.36914	0.684570	+	0.728947i	$0.259990\pi$
$558$	0	0
$559$	−14583.1	−1.10340
$560$	0	0
$561$	0	0
$562$	31330.3	2.35158
$563$	195.636	0.0146449	0.00732246	−	0.999973i	$-0.497669\pi$
$563$	195.636	0.0146449	0.00732246	+	0.999973i	$0.497669\pi$
$564$	0	0
$565$	5127.55	0.381801
$566$	16130.1	1.19788
$567$	0	0
$568$	9221.11	0.681178
$569$	19660.4	1.44852	0.724260	−	0.689527i	$-0.242182\pi$
$569$	19660.4	1.44852	0.724260	+	0.689527i	$0.242182\pi$
$570$	0	0
$571$	−15764.5	−1.15538	−0.577691	−	0.816255i	$-0.696046\pi$
$571$	−15764.5	−1.15538	−0.577691	+	0.816255i	$0.696046\pi$
$572$	356.228	0.0260396
$573$	0	0
$574$	0	0
$575$	−6816.30	−0.494364
$576$	0	0
$577$	−22306.4	−1.60941	−0.804704	−	0.593676i	$-0.797676\pi$
$577$	−22306.4	−1.60941	−0.804704	+	0.593676i	$0.797676\pi$
$578$	−40543.6	−2.91764
$579$	0	0
$580$	1446.56	0.103560
$581$	0	0
$582$	0	0
$583$	121.269	0.00861483
$584$	−9637.72	−0.682897
$585$	0	0
$586$	−14885.3	−1.04932
$587$	−15953.2	−1.12173	−0.560866	−	0.827906i	$-0.689532\pi$
$587$	−15953.2	−1.12173	−0.560866	+	0.827906i	$0.689532\pi$
$588$	0	0
$589$	7659.87	0.535856
$590$	−5752.24	−0.401383
$591$	0	0
$592$	−208.027	−0.0144423
$593$	−3155.68	−0.218530	−0.109265	−	0.994013i	$-0.534850\pi$
$593$	−3155.68	−0.218530	−0.109265	+	0.994013i	$0.534850\pi$
$594$	0	0
$595$	0	0
$596$	−17668.1	−1.21429
$597$	0	0
$598$	−19046.5	−1.30246
$599$	−25456.3	−1.73642	−0.868212	−	0.496194i	$-0.834730\pi$
$599$	−25456.3	−1.73642	−0.868212	+	0.496194i	$0.834730\pi$
$600$	0	0
$601$	−5580.96	−0.378789	−0.189395	−	0.981901i	$-0.560653\pi$
$601$	−5580.96	−0.378789	−0.189395	+	0.981901i	$0.560653\pi$
$602$	0	0
$603$	0	0
$604$	29477.6	1.98581
$605$	17931.7	1.20500
$606$	0	0
$607$	−381.133	−0.0254855	−0.0127427	−	0.999919i	$-0.504056\pi$
$607$	−381.133	−0.0254855	−0.0127427	+	0.999919i	$0.504056\pi$
$608$	18374.9	1.22566
$609$	0	0
$610$	7361.07	0.488592
$611$	−11824.4	−0.782919
$612$	0	0
$613$	−8235.98	−0.542656	−0.271328	−	0.962487i	$-0.587463\pi$
$613$	−8235.98	−0.542656	−0.271328	+	0.962487i	$0.587463\pi$
$614$	−41209.0	−2.70857
$615$	0	0
$616$	0	0
$617$	27419.8	1.78911	0.894555	−	0.446958i	$-0.147493\pi$
$617$	27419.8	1.78911	0.894555	+	0.446958i	$0.147493\pi$
$618$	0	0
$619$	16373.4	1.06317	0.531585	−	0.847005i	$-0.321597\pi$
$619$	16373.4	1.06317	0.531585	+	0.847005i	$0.321597\pi$
$620$	−13868.5	−0.898342
$621$	0	0
$622$	36987.9	2.38437
$623$	0	0
$624$	0	0
$625$	−19497.4	−1.24784
$626$	13501.0	0.861996
$627$	0	0
$628$	−15210.6	−0.966508
$629$	3392.24	0.215035
$630$	0	0
$631$	4059.60	0.256118	0.128059	−	0.991767i	$-0.459125\pi$
$631$	4059.60	0.256118	0.128059	+	0.991767i	$0.459125\pi$
$632$	20938.7	1.31788
$633$	0	0
$634$	−17617.5	−1.10360
$635$	−12917.8	−0.807284
$636$	0	0
$637$	0	0
$638$	−31.5577	−0.00195827
$639$	0	0
$640$	−29741.0	−1.83690
$641$	6388.63	0.393660	0.196830	−	0.980438i	$-0.436935\pi$
$641$	6388.63	0.393660	0.196830	+	0.980438i	$0.436935\pi$
$642$	0	0
$643$	18308.0	1.12286	0.561428	−	0.827525i	$-0.310252\pi$
$643$	18308.0	1.12286	0.561428	+	0.827525i	$0.310252\pi$
$644$	0	0
$645$	0	0
$646$	−49767.0	−3.03105
$647$	3303.90	0.200757	0.100379	−	0.994949i	$-0.467995\pi$
$647$	3303.90	0.200757	0.100379	+	0.994949i	$0.467995\pi$
$648$	0	0
$649$	76.5919	0.00463251
$650$	8978.42	0.541789
$651$	0	0
$652$	−9057.18	−0.544028
$653$	−4371.27	−0.261961	−0.130981	−	0.991385i	$-0.541813\pi$
$653$	−4371.27	−0.261961	−0.130981	+	0.991385i	$0.541813\pi$
$654$	0	0
$655$	−15530.0	−0.926423
$656$	509.209	0.0303068
$657$	0	0
$658$	0	0
$659$	−6259.75	−0.370023	−0.185012	−	0.982736i	$-0.559232\pi$
$659$	−6259.75	−0.370023	−0.185012	+	0.982736i	$0.559232\pi$
$660$	0	0
$661$	14845.7	0.873574	0.436787	−	0.899565i	$-0.356116\pi$
$661$	14845.7	0.873574	0.436787	+	0.899565i	$0.356116\pi$
$662$	22171.0	1.30166
$663$	0	0
$664$	2140.43	0.125098
$665$	0	0
$666$	0	0
$667$	1029.84	0.0597832
$668$	9436.14	0.546550
$669$	0	0
$670$	−48394.0	−2.79048
$671$	−98.0138	−0.00563902
$672$	0	0
$673$	9409.13	0.538923	0.269462	−	0.963011i	$-0.413154\pi$
$673$	9409.13	0.538923	0.269462	+	0.963011i	$0.413154\pi$
$674$	7803.05	0.445938
$675$	0	0
$676$	−12218.5	−0.695180
$677$	2950.63	0.167507	0.0837533	−	0.996487i	$-0.473309\pi$
$677$	2950.63	0.167507	0.0837533	+	0.996487i	$0.473309\pi$
$678$	0	0
$679$	0	0
$680$	32581.1	1.83740
$681$	0	0
$682$	302.551	0.0169872
$683$	−6280.85	−0.351874	−0.175937	−	0.984401i	$-0.556296\pi$
$683$	−6280.85	−0.351874	−0.175937	+	0.984401i	$0.556296\pi$
$684$	0	0
$685$	4817.11	0.268689
$686$	0	0
$687$	0	0
$688$	−3011.97	−0.166904
$689$	5212.72	0.288228
$690$	0	0
$691$	−32763.2	−1.80372	−0.901861	−	0.432027i	$-0.857798\pi$
$691$	−32763.2	−1.80372	−0.901861	+	0.432027i	$0.857798\pi$
$692$	−23297.1	−1.27980
$693$	0	0
$694$	−1078.67	−0.0589998
$695$	−36882.5	−2.01300
$696$	0	0
$697$	−8303.53	−0.451246
$698$	−45551.9	−2.47015
$699$	0	0
$700$	0	0
$701$	1775.97	0.0956883	0.0478442	−	0.998855i	$-0.484765\pi$
$701$	1775.97	0.0956883	0.0478442	+	0.998855i	$0.484765\pi$
$702$	0	0
$703$	2688.02	0.144212
$704$	678.805	0.0363401
$705$	0	0
$706$	15723.7	0.838203
$707$	0	0
$708$	0	0
$709$	8862.43	0.469444	0.234722	−	0.972063i	$-0.424582\pi$
$709$	8862.43	0.469444	0.234722	+	0.972063i	$0.424582\pi$
$710$	−27430.7	−1.44994
$711$	0	0
$712$	32293.8	1.69981
$713$	−9873.28	−0.518594
$714$	0	0
$715$	−383.175	−0.0200419
$716$	6551.59	0.341961
$717$	0	0
$718$	6377.75	0.331498
$719$	27499.2	1.42635	0.713177	−	0.700984i	$-0.247256\pi$
$719$	27499.2	1.42635	0.713177	+	0.700984i	$0.247256\pi$
$720$	0	0
$721$	0	0
$722$	−8356.64	−0.430750
$723$	0	0
$724$	−36357.6	−1.86632
$725$	−485.460	−0.0248683
$726$	0	0
$727$	−25434.9	−1.29756	−0.648781	−	0.760975i	$-0.724721\pi$
$727$	−25434.9	−1.29756	−0.648781	+	0.760975i	$0.724721\pi$
$728$	0	0
$729$	0	0
$730$	28670.0	1.45359
$731$	49115.3	2.48508
$732$	0	0
$733$	24155.6	1.21720	0.608600	−	0.793477i	$-0.291731\pi$
$733$	24155.6	1.21720	0.608600	+	0.793477i	$0.291731\pi$
$734$	50445.2	2.53674
$735$	0	0
$736$	−23684.6	−1.18618
$737$	644.373	0.0322060
$738$	0	0
$739$	27512.9	1.36952	0.684762	−	0.728767i	$-0.259906\pi$
$739$	27512.9	1.36952	0.684762	+	0.728767i	$0.259906\pi$
$740$	−4866.78	−0.241765
$741$	0	0
$742$	0	0
$743$	−5995.09	−0.296014	−0.148007	−	0.988986i	$-0.547286\pi$
$743$	−5995.09	−0.296014	−0.148007	+	0.988986i	$0.547286\pi$
$744$	0	0
$745$	19004.6	0.934598
$746$	−40897.7	−2.00720
$747$	0	0
$748$	−1199.76	−0.0586467
$749$	0	0
$750$	0	0
$751$	1545.09	0.0750747	0.0375373	−	0.999295i	$-0.488049\pi$
$751$	1545.09	0.0750747	0.0375373	+	0.999295i	$0.488049\pi$
$752$	−2442.19	−0.118428
$753$	0	0
$754$	−1356.50	−0.0655183
$755$	−31707.5	−1.52841
$756$	0	0
$757$	−5157.82	−0.247641	−0.123820	−	0.992305i	$-0.539515\pi$
$757$	−5157.82	−0.247641	−0.123820	+	0.992305i	$0.539515\pi$
$758$	26533.1	1.27141
$759$	0	0
$760$	25817.5	1.23223
$761$	−3289.96	−0.156716	−0.0783581	−	0.996925i	$-0.524968\pi$
$761$	−3289.96	−0.156716	−0.0783581	+	0.996925i	$0.524968\pi$
$762$	0	0
$763$	0	0
$764$	−32635.4	−1.54543
$765$	0	0
$766$	35288.6	1.66453
$767$	3292.29	0.154990
$768$	0	0
$769$	11146.5	0.522697	0.261348	−	0.965245i	$-0.415833\pi$
$769$	11146.5	0.522697	0.261348	+	0.965245i	$0.415833\pi$
$770$	0	0
$771$	0	0
$772$	8474.12	0.395065
$773$	−15830.2	−0.736576	−0.368288	−	0.929712i	$-0.620056\pi$
$773$	−15830.2	−0.736576	−0.368288	+	0.929712i	$0.620056\pi$
$774$	0	0
$775$	4654.22	0.215722
$776$	11293.3	0.522430
$777$	0	0
$778$	−17282.0	−0.796386
$779$	−6579.75	−0.302624
$780$	0	0
$781$	365.243	0.0167342
$782$	64147.9	2.93341
$783$	0	0
$784$	0	0
$785$	16361.2	0.743892
$786$	0	0
$787$	15163.4	0.686809	0.343404	−	0.939188i	$-0.388420\pi$
$787$	15163.4	0.686809	0.343404	+	0.939188i	$0.388420\pi$
$788$	46189.6	2.08812
$789$	0	0
$790$	−62287.8	−2.80519
$791$	0	0
$792$	0	0
$793$	−4213.10	−0.188665
$794$	−46060.2	−2.05871
$795$	0	0
$796$	10022.4	0.446275
$797$	29398.3	1.30658	0.653289	−	0.757109i	$-0.273389\pi$
$797$	29398.3	1.30658	0.653289	+	0.757109i	$0.273389\pi$
$798$	0	0
$799$	39824.1	1.76330
$800$	11164.8	0.493420
$801$	0	0
$802$	52119.5	2.29477
$803$	−381.745	−0.0167764
$804$	0	0
$805$	0	0
$806$	13005.1	0.568343
$807$	0	0
$808$	1349.24	0.0587453
$809$	20712.9	0.900155	0.450078	−	0.892989i	$-0.351396\pi$
$809$	20712.9	0.900155	0.450078	+	0.892989i	$0.351396\pi$
$810$	0	0
$811$	−27369.9	−1.18506	−0.592532	−	0.805547i	$-0.701872\pi$
$811$	−27369.9	−1.18506	−0.592532	+	0.805547i	$0.701872\pi$
$812$	0	0
$813$	0	0
$814$	106.172	0.00457165
$815$	9742.31	0.418722
$816$	0	0
$817$	38919.2	1.66660
$818$	14799.9	0.632600
$819$	0	0
$820$	11912.9	0.507338
$821$	35362.7	1.50325	0.751623	−	0.659592i	$-0.229271\pi$
$821$	35362.7	1.50325	0.751623	+	0.659592i	$0.229271\pi$
$822$	0	0
$823$	−29190.4	−1.23635	−0.618174	−	0.786042i	$-0.712127\pi$
$823$	−29190.4	−1.23635	−0.618174	+	0.786042i	$0.712127\pi$
$824$	37558.6	1.58788
$825$	0	0
$826$	0	0
$827$	7302.08	0.307035	0.153518	−	0.988146i	$-0.450940\pi$
$827$	7302.08	0.307035	0.153518	+	0.988146i	$0.450940\pi$
$828$	0	0
$829$	−4250.77	−0.178088	−0.0890442	−	0.996028i	$-0.528381\pi$
$829$	−4250.77	−0.178088	−0.0890442	+	0.996028i	$0.528381\pi$
$830$	−6367.28	−0.266279
$831$	0	0
$832$	29178.3	1.21584
$833$	0	0
$834$	0	0
$835$	−10149.9	−0.420663
$836$	−950.699	−0.0393309
$837$	0	0
$838$	30915.6	1.27442
$839$	−39527.7	−1.62652	−0.813258	−	0.581903i	$-0.802308\pi$
$839$	−39527.7	−1.62652	−0.813258	+	0.581903i	$0.802308\pi$
$840$	0	0
$841$	−24315.7	−0.996993
$842$	−6486.92	−0.265504
$843$	0	0
$844$	−8363.09	−0.341078
$845$	13142.8	0.535059
$846$	0	0
$847$	0	0
$848$	1076.63	0.0435985
$849$	0	0
$850$	−30239.0	−1.22022
$851$	−3464.76	−0.139566
$852$	0	0
$853$	−31656.1	−1.27067	−0.635337	−	0.772235i	$-0.719139\pi$
$853$	−31656.1	−1.27067	−0.635337	+	0.772235i	$0.719139\pi$
$854$	0	0
$855$	0	0
$856$	17685.1	0.706148
$857$	1193.16	0.0475583	0.0237792	−	0.999717i	$-0.492430\pi$
$857$	1193.16	0.0475583	0.0237792	+	0.999717i	$0.492430\pi$
$858$	0	0
$859$	−29060.0	−1.15427	−0.577134	−	0.816650i	$-0.695829\pi$
$859$	−29060.0	−1.15427	−0.577134	+	0.816650i	$0.695829\pi$
$860$	−70464.8	−2.79399
$861$	0	0
$862$	68611.2	2.71103
$863$	−23063.0	−0.909702	−0.454851	−	0.890567i	$-0.650308\pi$
$863$	−23063.0	−0.909702	−0.454851	+	0.890567i	$0.650308\pi$
$864$	0	0
$865$	25059.4	0.985024
$866$	−24810.9	−0.973566
$867$	0	0
$868$	0	0
$869$	829.371	0.0323757
$870$	0	0
$871$	27698.2	1.07752
$872$	33271.1	1.29209
$873$	0	0
$874$	50831.1	1.96726
$875$	0	0
$876$	0	0
$877$	33871.0	1.30415	0.652077	−	0.758153i	$-0.273898\pi$
$877$	33871.0	1.30415	0.652077	+	0.758153i	$0.273898\pi$
$878$	8067.88	0.310111
$879$	0	0
$880$	−79.1405	−0.00303162
$881$	−43331.1	−1.65705	−0.828525	−	0.559953i	$-0.810819\pi$
$881$	−43331.1	−1.65705	−0.828525	+	0.559953i	$0.810819\pi$
$882$	0	0
$883$	−40897.3	−1.55867	−0.779334	−	0.626609i	$-0.784442\pi$
$883$	−40897.3	−1.55867	−0.779334	+	0.626609i	$0.784442\pi$
$884$	−51571.6	−1.96215
$885$	0	0
$886$	−14769.8	−0.560047
$887$	−45065.8	−1.70593	−0.852965	−	0.521968i	$-0.825198\pi$
$887$	−45065.8	−1.70593	−0.852965	+	0.521968i	$0.825198\pi$
$888$	0	0
$889$	0	0
$890$	−96066.5	−3.61816
$891$	0	0
$892$	−33167.7	−1.24500
$893$	31556.8	1.18254
$894$	0	0
$895$	−7047.19	−0.263197
$896$	0	0
$897$	0	0
$898$	−30930.3	−1.14940
$899$	−703.180	−0.0260872
$900$	0	0
$901$	−17556.3	−0.649150
$902$	−259.888	−0.00959349
$903$	0	0
$904$	7810.22	0.287350
$905$	39107.9	1.43645
$906$	0	0
$907$	25282.5	0.925570	0.462785	−	0.886471i	$-0.346850\pi$
$907$	25282.5	0.925570	0.462785	+	0.886471i	$0.346850\pi$
$908$	51642.2	1.88745
$909$	0	0
$910$	0	0
$911$	41646.1	1.51460	0.757298	−	0.653070i	$-0.226519\pi$
$911$	41646.1	1.51460	0.757298	+	0.653070i	$0.226519\pi$
$912$	0	0
$913$	84.7812	0.00307322
$914$	−16765.3	−0.606725
$915$	0	0
$916$	−50963.1	−1.83829
$917$	0	0
$918$	0	0
$919$	−26112.5	−0.937292	−0.468646	−	0.883386i	$-0.655258\pi$
$919$	−26112.5	−0.937292	−0.468646	+	0.883386i	$0.655258\pi$
$920$	−33277.7	−1.19254
$921$	0	0
$922$	42596.3	1.52151
$923$	15699.9	0.559879
$924$	0	0
$925$	1633.27	0.0580559
$926$	−70159.4	−2.48983
$927$	0	0
$928$	−1686.83	−0.0596691
$929$	32357.0	1.14273	0.571366	−	0.820695i	$-0.306414\pi$
$929$	32357.0	1.14273	0.571366	+	0.820695i	$0.306414\pi$
$930$	0	0
$931$	0	0
$932$	48929.9	1.71969
$933$	0	0
$934$	9993.95	0.350120
$935$	1290.52	0.0451386
$936$	0	0
$937$	−32947.0	−1.14870	−0.574350	−	0.818610i	$-0.694745\pi$
$937$	−32947.0	−1.14870	−0.574350	+	0.818610i	$0.694745\pi$
$938$	0	0
$939$	0	0
$940$	−57134.9	−1.98249
$941$	−501.602	−0.0173770	−0.00868849	−	0.999962i	$-0.502766\pi$
$941$	−501.602	−0.0173770	−0.00868849	+	0.999962i	$0.502766\pi$
$942$	0	0
$943$	8481.06	0.292875
$944$	679.984	0.0234445
$945$	0	0
$946$	1537.24	0.0528328
$947$	6436.90	0.220878	0.110439	−	0.993883i	$-0.464774\pi$
$947$	6436.90	0.220878	0.110439	+	0.993883i	$0.464774\pi$
$948$	0	0
$949$	−16409.2	−0.561292
$950$	−23961.5	−0.818331
$951$	0	0
$952$	0	0
$953$	−47511.2	−1.61494	−0.807470	−	0.589908i	$-0.799164\pi$
$953$	−47511.2	−1.61494	−0.807470	+	0.589908i	$0.799164\pi$
$954$	0	0
$955$	35104.2	1.18947
$956$	−67982.9	−2.29992
$957$	0	0
$958$	−10645.1	−0.359004
$959$	0	0
$960$	0	0
$961$	−23049.4	−0.773705
$962$	4563.78	0.152955
$963$	0	0
$964$	20086.1	0.671087
$965$	−9115.15	−0.304069
$966$	0	0
$967$	7817.32	0.259967	0.129984	−	0.991516i	$-0.458508\pi$
$967$	7817.32	0.259967	0.129984	+	0.991516i	$0.458508\pi$
$968$	27313.4	0.906905
$969$	0	0
$970$	−33594.9	−1.11203
$971$	1503.50	0.0496905	0.0248453	−	0.999691i	$-0.492091\pi$
$971$	1503.50	0.0496905	0.0248453	+	0.999691i	$0.492091\pi$
$972$	0	0
$973$	0	0
$974$	−47097.9	−1.54940
$975$	0	0
$976$	−870.168	−0.0285383
$977$	−33389.1	−1.09336	−0.546680	−	0.837342i	$-0.684108\pi$
$977$	−33389.1	−1.09336	−0.546680	+	0.837342i	$0.684108\pi$
$978$	0	0
$979$	1279.14	0.0417584
$980$	0	0
$981$	0	0
$982$	57032.2	1.85333
$983$	−5451.02	−0.176867	−0.0884337	−	0.996082i	$-0.528186\pi$
$983$	−5451.02	−0.176867	−0.0884337	+	0.996082i	$0.528186\pi$
$984$	0	0
$985$	−49683.7	−1.60716
$986$	4568.64	0.147561
$987$	0	0
$988$	−40865.6	−1.31590
$989$	−50165.4	−1.61291
$990$	0	0
$991$	−46530.0	−1.49150	−0.745750	−	0.666226i	$-0.767908\pi$
$991$	−46530.0	−1.49150	−0.745750	+	0.666226i	$0.767908\pi$
$992$	16172.0	0.517603
$993$	0	0
$994$	0	0
$995$	−10780.6	−0.343484
$996$	0	0
$997$	−11410.0	−0.362444	−0.181222	−	0.983442i	$-0.558005\pi$
$997$	−11410.0	−0.362444	−0.181222	+	0.983442i	$0.558005\pi$
$998$	−48156.5	−1.52742
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.4.a.u.1.1		4
3.2	odd	2		49.4.a.e.1.3	✓	4
7.2	even	3		441.4.e.y.361.4		8
7.3	odd	6		441.4.e.y.226.3		8
7.4	even	3		441.4.e.y.226.4		8
7.5	odd	6		441.4.e.y.361.3		8
7.6	odd	2	inner	441.4.a.u.1.2		4
12.11	even	2		784.4.a.bf.1.3		4
15.14	odd	2		1225.4.a.bb.1.2		4
21.2	odd	6		49.4.c.e.18.2		8
21.5	even	6		49.4.c.e.18.1		8
21.11	odd	6		49.4.c.e.30.2		8
21.17	even	6		49.4.c.e.30.1		8
21.20	even	2		49.4.a.e.1.4	yes	4
84.83	odd	2		784.4.a.bf.1.2		4
105.104	even	2		1225.4.a.bb.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.a.e.1.3	✓	4	3.2	odd	2
49.4.a.e.1.4	yes	4	21.20	even	2
49.4.c.e.18.1		8	21.5	even	6
49.4.c.e.18.2		8	21.2	odd	6
49.4.c.e.30.1		8	21.17	even	6
49.4.c.e.30.2		8	21.11	odd	6
441.4.a.u.1.1		4	1.1	even	1	trivial
441.4.a.u.1.2		4	7.6	odd	2	inner
441.4.e.y.226.3		8	7.3	odd	6
441.4.e.y.226.4		8	7.4	even	3
441.4.e.y.361.3		8	7.5	odd	6
441.4.e.y.361.4		8	7.2	even	3
784.4.a.bf.1.2		4	84.83	odd	2
784.4.a.bf.1.3		4	12.11	even	2
1225.4.a.bb.1.1		4	105.104	even	2
1225.4.a.bb.1.2		4	15.14	odd	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)

\(f(q)\)	\(=\)	\(q-4.53113 q^{2} +12.5311 q^{4} -13.4791 q^{5} -20.5311 q^{8} +O(q^{10})\)	\(q-4.53113 q^{2} +12.5311 q^{4} -13.4791 q^{5} -20.5311 q^{8} +61.0753 q^{10} -0.813227 q^{11} -34.9564 q^{13} -7.21984 q^{16} +117.732 q^{17} +93.2913 q^{19} -168.908 q^{20} +3.68484 q^{22} -120.249 q^{23} +56.6848 q^{25} +158.392 q^{26} -8.56420 q^{29} +82.1070 q^{31} +196.963 q^{32} -533.458 q^{34} +28.8132 q^{37} -422.715 q^{38} +276.740 q^{40} -70.5291 q^{41} +417.179 q^{43} -10.1906 q^{44} +544.864 q^{46} +338.261 q^{47} -256.846 q^{50} -438.043 q^{52} -149.121 q^{53} +10.9615 q^{55} +38.8055 q^{58} -94.1828 q^{59} +120.525 q^{61} -372.037 q^{62} -834.706 q^{64} +471.179 q^{65} -792.366 q^{67} +1475.31 q^{68} -449.128 q^{71} +469.420 q^{73} -130.556 q^{74} +1169.05 q^{76} -1019.85 q^{79} +97.3166 q^{80} +319.576 q^{82} -104.253 q^{83} -1586.91 q^{85} -1890.29 q^{86} +16.6965 q^{88} -1572.92 q^{89} -1506.86 q^{92} -1532.70 q^{94} -1257.48 q^{95} -550.057 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 34 q^{4} - 66 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 + 34 * q^4 - 66 * q^8	\( 4 q - 2 q^{2} + 34 q^{4} - 66 q^{8} - 100 q^{11} - 174 q^{16} - 340 q^{22} - 352 q^{23} - 128 q^{25} - 260 q^{29} + 30 q^{32} + 212 q^{37} + 540 q^{43} - 460 q^{44} + 696 q^{46} - 1366 q^{50} - 16 q^{53} - 780 q^{58} - 1678 q^{64} + 756 q^{65} - 1944 q^{67} - 2248 q^{71} + 284 q^{74} - 1048 q^{79} - 3284 q^{85} - 4820 q^{86} + 1260 q^{88} - 3512 q^{92} - 2192 q^{95}+O(q^{100}) \) 4 * q - 2 * q^2 + 34 * q^4 - 66 * q^8 - 100 * q^11 - 174 * q^16 - 340 * q^22 - 352 * q^23 - 128 * q^25 - 260 * q^29 + 30 * q^32 + 212 * q^37 + 540 * q^43 - 460 * q^44 + 696 * q^46 - 1366 * q^50 - 16 * q^53 - 780 * q^58 - 1678 * q^64 + 756 * q^65 - 1944 * q^67 - 2248 * q^71 + 284 * q^74 - 1048 * q^79 - 3284 * q^85 - 4820 * q^86 + 1260 * q^88 - 3512 * q^92 - 2192 * q^95

Introduction

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