LMFDB - Embedded newform 693.4.a.r.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,4,Mod(1,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("693.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	693.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:	$40.8883236340$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 43x^{6} + 57x^{5} + 560x^{4} - 439x^{3} - 2246x^{2} + 384x + 1056 $ x^8 - 2x^7 - 43x^6 + 57x^5 + 560x^4 - 439x^3 - 2246x^2 + 384*x + 1056
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.09275$ of defining polynomial
Character	$\chi$	$=$	693.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.09275 q^{2} +1.56510 q^{4} +0.265436 q^{5} +7.00000 q^{7} +19.9015 q^{8} -0.820927 q^{10} -11.0000 q^{11} +9.99503 q^{13} -21.6492 q^{14} -74.0713 q^{16} -103.685 q^{17} +54.7132 q^{19} +0.415434 q^{20} +34.0202 q^{22} +26.6214 q^{23} -124.930 q^{25} -30.9121 q^{26} +10.9557 q^{28} +17.2658 q^{29} +202.086 q^{31} +69.8716 q^{32} +320.671 q^{34} +1.85805 q^{35} +244.564 q^{37} -169.214 q^{38} +5.28258 q^{40} -306.274 q^{41} +330.565 q^{43} -17.2161 q^{44} -82.3333 q^{46} -74.6534 q^{47} +49.0000 q^{49} +386.376 q^{50} +15.6432 q^{52} -428.190 q^{53} -2.91979 q^{55} +139.311 q^{56} -53.3988 q^{58} +350.175 q^{59} +153.419 q^{61} -625.001 q^{62} +376.475 q^{64} +2.65304 q^{65} +192.673 q^{67} -162.277 q^{68} -5.74649 q^{70} -821.902 q^{71} -727.976 q^{73} -756.376 q^{74} +85.6317 q^{76} -77.0000 q^{77} -410.417 q^{79} -19.6612 q^{80} +947.228 q^{82} +289.538 q^{83} -27.5217 q^{85} -1022.36 q^{86} -218.917 q^{88} +225.713 q^{89} +69.9652 q^{91} +41.6652 q^{92} +230.884 q^{94} +14.5229 q^{95} -420.449 q^{97} -151.545 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{2} + 30 q^{4} - 10 q^{5} + 56 q^{7} - 81 q^{8} + 9 q^{10} - 88 q^{11} + 16 q^{13} - 42 q^{14} + 122 q^{16} - 90 q^{17} - 42 q^{19} - 291 q^{20} + 66 q^{22} - 338 q^{23} + 244 q^{25} - 209 q^{26}+ \cdots - 294 q^{98}+O(q^{100}) $ 8 * q - 6 * q^2 + 30 * q^4 - 10 * q^5 + 56 * q^7 - 81 * q^8 + 9 * q^10 - 88 * q^11 + 16 * q^13 - 42 * q^14 + 122 * q^16 - 90 * q^17 - 42 * q^19 - 291 * q^20 + 66 * q^22 - 338 * q^23 + 244 * q^25 - 209 * q^26 + 210 * q^28 - 496 * q^29 - 8 * q^31 - 524 * q^32 - 302 * q^34 - 70 * q^35 - 360 * q^37 - 45 * q^38 - 6 * q^40 - 242 * q^41 - 66 * q^43 - 330 * q^44 + 344 * q^46 - 540 * q^47 + 392 * q^49 - 1171 * q^50 + 465 * q^52 - 906 * q^53 + 110 * q^55 - 567 * q^56 + 977 * q^58 - 1242 * q^59 - 318 * q^61 + 110 * q^62 + 525 * q^64 - 1258 * q^65 + 522 * q^67 - 678 * q^68 + 63 * q^70 - 858 * q^71 - 78 * q^73 - 1651 * q^74 + 1775 * q^76 - 616 * q^77 + 516 * q^79 - 567 * q^80 - 1212 * q^82 - 3192 * q^83 + 720 * q^85 - 1322 * q^86 + 891 * q^88 - 2356 * q^89 + 112 * q^91 - 4504 * q^92 - 423 * q^94 - 3308 * q^95 - 1556 * q^97 - 294 * q^98

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 $

$2$	−3.09275	−1.09345	−0.546726	−	0.837312i	$-0.684126\pi$
$2$	−3.09275	−1.09345	−0.546726	+	0.837312i	$0.684126\pi$
$3$	0	0
$3$	0	0
$4$	1.56510	0.195637
$5$	0.265436	0.0237413	0.0118707	−	0.999930i	$-0.496221\pi$
$5$	0.265436	0.0237413	0.0118707	+	0.999930i	$0.496221\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	19.9015	0.879532
$9$	0	0
$10$	−0.820927	−0.0259600
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	9.99503	0.213240	0.106620	−	0.994300i	$-0.465997\pi$
$13$	9.99503	0.213240	0.106620	+	0.994300i	$0.465997\pi$
$14$	−21.6492	−0.413286
$15$	0	0
$16$	−74.0713	−1.15736
$17$	−103.685	−1.47925	−0.739626	−	0.673018i	$-0.764997\pi$
$17$	−103.685	−1.47925	−0.739626	+	0.673018i	$0.764997\pi$
$18$	0	0
$19$	54.7132	0.660636	0.330318	−	0.943870i	$-0.392844\pi$
$19$	54.7132	0.660636	0.330318	+	0.943870i	$0.392844\pi$
$20$	0.415434	0.00464469
$21$	0	0
$22$	34.0202	0.329688
$23$	26.6214	0.241345	0.120673	−	0.992692i	$-0.461495\pi$
$23$	26.6214	0.241345	0.120673	+	0.992692i	$0.461495\pi$
$24$	0	0
$25$	−124.930	−0.999436
$26$	−30.9121	−0.233168
$27$	0	0
$28$	10.9557	0.0739440
$29$	17.2658	0.110558	0.0552790	−	0.998471i	$-0.482395\pi$
$29$	17.2658	0.110558	0.0552790	+	0.998471i	$0.482395\pi$
$30$	0	0
$31$	202.086	1.17083	0.585415	−	0.810734i	$-0.300932\pi$
$31$	202.086	1.17083	0.585415	+	0.810734i	$0.300932\pi$
$32$	69.8716	0.385990
$33$	0	0
$34$	320.671	1.61749
$35$	1.85805	0.00897337
$36$	0	0
$37$	244.564	1.08665	0.543326	−	0.839522i	$-0.317165\pi$
$37$	244.564	1.08665	0.543326	+	0.839522i	$0.317165\pi$
$38$	−169.214	−0.722373
$39$	0	0
$40$	5.28258	0.0208812
$41$	−306.274	−1.16663	−0.583316	−	0.812245i	$-0.698245\pi$
$41$	−306.274	−1.16663	−0.583316	+	0.812245i	$0.698245\pi$
$42$	0	0
$43$	330.565	1.17234	0.586171	−	0.810187i	$-0.300635\pi$
$43$	330.565	1.17234	0.586171	+	0.810187i	$0.300635\pi$
$44$	−17.2161	−0.0589869
$45$	0	0
$46$	−82.3333	−0.263900
$47$	−74.6534	−0.231688	−0.115844	−	0.993267i	$-0.536957\pi$
$47$	−74.6534	−0.231688	−0.115844	+	0.993267i	$0.536957\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	386.376	1.09284
$51$	0	0
$52$	15.6432	0.0417178
$53$	−428.190	−1.10974	−0.554872	−	0.831936i	$-0.687233\pi$
$53$	−428.190	−1.10974	−0.554872	+	0.831936i	$0.687233\pi$
$54$	0	0
$55$	−2.91979	−0.00715827
$56$	139.311	0.332432
$57$	0	0
$58$	−53.3988	−0.120890
$59$	350.175	0.772694	0.386347	−	0.922353i	$-0.373737\pi$
$59$	350.175	0.772694	0.386347	+	0.922353i	$0.373737\pi$
$60$	0	0
$61$	153.419	0.322021	0.161011	−	0.986953i	$-0.448525\pi$
$61$	153.419	0.322021	0.161011	+	0.986953i	$0.448525\pi$
$62$	−625.001	−1.28025
$63$	0	0
$64$	376.475	0.735302
$65$	2.65304	0.00506260
$66$	0	0
$67$	192.673	0.351325	0.175662	−	0.984450i	$-0.443793\pi$
$67$	192.673	0.351325	0.175662	+	0.984450i	$0.443793\pi$
$68$	−162.277	−0.289397
$69$	0	0
$70$	−5.74649	−0.00981195
$71$	−821.902	−1.37383	−0.686914	−	0.726738i	$-0.741035\pi$
$71$	−821.902	−1.37383	−0.686914	+	0.726738i	$0.741035\pi$
$72$	0	0
$73$	−727.976	−1.16717	−0.583583	−	0.812053i	$-0.698350\pi$
$73$	−727.976	−1.16717	−0.583583	+	0.812053i	$0.698350\pi$
$74$	−756.376	−1.18820
$75$	0	0
$76$	85.6317	0.129245
$77$	−77.0000	−0.113961
$78$	0	0
$79$	−410.417	−0.584500	−0.292250	−	0.956342i	$-0.594404\pi$
$79$	−410.417	−0.584500	−0.292250	+	0.956342i	$0.594404\pi$
$80$	−19.6612	−0.0274773
$81$	0	0
$82$	947.228	1.27566
$83$	289.538	0.382903	0.191451	−	0.981502i	$-0.438681\pi$
$83$	289.538	0.382903	0.191451	+	0.981502i	$0.438681\pi$
$84$	0	0
$85$	−27.5217	−0.0351194
$86$	−1022.36	−1.28190
$87$	0	0
$88$	−218.917	−0.265189
$89$	225.713	0.268826	0.134413	−	0.990925i	$-0.457085\pi$
$89$	225.713	0.268826	0.134413	+	0.990925i	$0.457085\pi$
$90$	0	0
$91$	69.9652	0.0805972
$92$	41.6652	0.0472162
$93$	0	0
$94$	230.884	0.253339
$95$	14.5229	0.0156844
$96$	0	0
$97$	−420.449	−0.440104	−0.220052	−	0.975488i	$-0.570623\pi$
$97$	−420.449	−0.440104	−0.220052	+	0.975488i	$0.570623\pi$
$98$	−151.545	−0.156207
$99$	0	0
$100$	−195.527	−0.195527
$101$	−746.538	−0.735478	−0.367739	−	0.929929i	$-0.619868\pi$
$101$	−746.538	−0.735478	−0.367739	+	0.929929i	$0.619868\pi$
$102$	0	0
$103$	497.939	0.476344	0.238172	−	0.971223i	$-0.423452\pi$
$103$	497.939	0.476344	0.238172	+	0.971223i	$0.423452\pi$
$104$	198.917	0.187552
$105$	0	0
$106$	1324.28	1.21345
$107$	1070.37	0.967069	0.483535	−	0.875325i	$-0.339353\pi$
$107$	1070.37	0.967069	0.483535	+	0.875325i	$0.339353\pi$
$108$	0	0
$109$	−854.106	−0.750537	−0.375268	−	0.926916i	$-0.622449\pi$
$109$	−854.106	−0.750537	−0.375268	+	0.926916i	$0.622449\pi$
$110$	9.03019	0.00782723
$111$	0	0
$112$	−518.499	−0.437442
$113$	−1762.94	−1.46764	−0.733819	−	0.679345i	$-0.762264\pi$
$113$	−1762.94	−1.46764	−0.733819	+	0.679345i	$0.762264\pi$
$114$	0	0
$115$	7.06627	0.00572986
$116$	27.0227	0.0216293
$117$	0	0
$118$	−1083.00	−0.844904
$119$	−725.794	−0.559105
$120$	0	0
$121$	121.000	0.0909091
$122$	−474.487	−0.352115
$123$	0	0
$124$	316.285	0.229058
$125$	−66.3403	−0.0474692
$126$	0	0
$127$	−1506.49	−1.05259	−0.526295	−	0.850302i	$-0.676419\pi$
$127$	−1506.49	−1.05259	−0.526295	+	0.850302i	$0.676419\pi$
$128$	−1723.31	−1.19001
$129$	0	0
$130$	−8.20519	−0.00553571
$131$	−2215.23	−1.47745	−0.738725	−	0.674007i	$-0.764572\pi$
$131$	−2215.23	−1.47745	−0.738725	+	0.674007i	$0.764572\pi$
$132$	0	0
$133$	382.993	0.249697
$134$	−595.890	−0.384157
$135$	0	0
$136$	−2063.49	−1.30105
$137$	−1059.21	−0.660546	−0.330273	−	0.943885i	$-0.607141\pi$
$137$	−1059.21	−0.660546	−0.330273	+	0.943885i	$0.607141\pi$
$138$	0	0
$139$	−2088.86	−1.27464	−0.637321	−	0.770599i	$-0.719957\pi$
$139$	−2088.86	−1.27464	−0.637321	+	0.770599i	$0.719957\pi$
$140$	2.90804	0.00175553
$141$	0	0
$142$	2541.94	1.50222
$143$	−109.945	−0.0642944
$144$	0	0
$145$	4.58297	0.00262479
$146$	2251.45	1.27624
$147$	0	0
$148$	382.768	0.212590
$149$	325.136	0.178766	0.0893832	−	0.995997i	$-0.471510\pi$
$149$	325.136	0.178766	0.0893832	+	0.995997i	$0.471510\pi$
$150$	0	0
$151$	596.206	0.321315	0.160657	−	0.987010i	$-0.448639\pi$
$151$	596.206	0.321315	0.160657	+	0.987010i	$0.448639\pi$
$152$	1088.88	0.581050
$153$	0	0
$154$	238.142	0.124610
$155$	53.6409	0.0277970
$156$	0	0
$157$	751.800	0.382167	0.191083	−	0.981574i	$-0.438800\pi$
$157$	751.800	0.382167	0.191083	+	0.981574i	$0.438800\pi$
$158$	1269.32	0.639122
$159$	0	0
$160$	18.5464	0.00916390
$161$	186.350	0.0912200
$162$	0	0
$163$	−2174.54	−1.04493	−0.522464	−	0.852661i	$-0.674987\pi$
$163$	−2174.54	−1.04493	−0.522464	+	0.852661i	$0.674987\pi$
$164$	−479.349	−0.228237
$165$	0	0
$166$	−895.469	−0.418686
$167$	−2700.77	−1.25145	−0.625724	−	0.780044i	$-0.715197\pi$
$167$	−2700.77	−1.25145	−0.625724	+	0.780044i	$0.715197\pi$
$168$	0	0
$169$	−2097.10	−0.954529
$170$	85.1177	0.0384013
$171$	0	0
$172$	517.368	0.229354
$173$	1937.95	0.851674	0.425837	−	0.904800i	$-0.359980\pi$
$173$	1937.95	0.851674	0.425837	+	0.904800i	$0.359980\pi$
$174$	0	0
$175$	−874.507	−0.377751
$176$	814.784	0.348958
$177$	0	0
$178$	−698.073	−0.293948
$179$	611.101	0.255172	0.127586	−	0.991827i	$-0.459277\pi$
$179$	611.101	0.255172	0.127586	+	0.991827i	$0.459277\pi$
$180$	0	0
$181$	−4159.31	−1.70806	−0.854032	−	0.520221i	$-0.825850\pi$
$181$	−4159.31	−1.70806	−0.854032	+	0.520221i	$0.825850\pi$
$182$	−216.385	−0.0881292
$183$	0	0
$184$	529.807	0.212271
$185$	64.9161	0.0257985
$186$	0	0
$187$	1140.53	0.446011
$188$	−116.840	−0.0453268
$189$	0	0
$190$	−44.9155	−0.0171501
$191$	−3784.50	−1.43370	−0.716849	−	0.697228i	$-0.754417\pi$
$191$	−3784.50	−1.43370	−0.716849	+	0.697228i	$0.754417\pi$
$192$	0	0
$193$	−3453.92	−1.28818	−0.644089	−	0.764951i	$-0.722763\pi$
$193$	−3453.92	−1.28818	−0.644089	+	0.764951i	$0.722763\pi$
$194$	1300.34	0.481233
$195$	0	0
$196$	76.6899	0.0279482
$197$	290.349	0.105008	0.0525038	−	0.998621i	$-0.483280\pi$
$197$	290.349	0.105008	0.0525038	+	0.998621i	$0.483280\pi$
$198$	0	0
$199$	−3122.45	−1.11229	−0.556143	−	0.831087i	$-0.687719\pi$
$199$	−3122.45	−1.11229	−0.556143	+	0.831087i	$0.687719\pi$
$200$	−2486.29	−0.879036
$201$	0	0
$202$	2308.85	0.804210
$203$	120.861	0.0417870
$204$	0	0
$205$	−81.2960	−0.0276974
$206$	−1540.00	−0.520859
$207$	0	0
$208$	−740.345	−0.246796
$209$	−601.846	−0.199189
$210$	0	0
$211$	5433.98	1.77294	0.886471	−	0.462785i	$-0.153150\pi$
$211$	5433.98	1.77294	0.886471	+	0.462785i	$0.153150\pi$
$212$	−670.160	−0.217108
$213$	0	0
$214$	−3310.38	−1.05744
$215$	87.7439	0.0278329
$216$	0	0
$217$	1414.60	0.442532
$218$	2641.54	0.820676
$219$	0	0
$220$	−4.56977	−0.00140043
$221$	−1036.33	−0.315436
$222$	0	0
$223$	−4360.37	−1.30938	−0.654690	−	0.755897i	$-0.727201\pi$
$223$	−4360.37	−1.30938	−0.654690	+	0.755897i	$0.727201\pi$
$224$	489.101	0.145890
$225$	0	0
$226$	5452.32	1.60479
$227$	4239.18	1.23949	0.619745	−	0.784803i	$-0.287236\pi$
$227$	4239.18	1.23949	0.619745	+	0.784803i	$0.287236\pi$
$228$	0	0
$229$	490.298	0.141484	0.0707419	−	0.997495i	$-0.477463\pi$
$229$	490.298	0.141484	0.0707419	+	0.997495i	$0.477463\pi$
$230$	−21.8542	−0.00626532
$231$	0	0
$232$	343.616	0.0972393
$233$	−606.792	−0.170610	−0.0853052	−	0.996355i	$-0.527187\pi$
$233$	−606.792	−0.170610	−0.0853052	+	0.996355i	$0.527187\pi$
$234$	0	0
$235$	−19.8157	−0.00550056
$236$	548.059	0.151168
$237$	0	0
$238$	2244.70	0.611354
$239$	3777.71	1.02243	0.511213	−	0.859454i	$-0.329196\pi$
$239$	3777.71	1.02243	0.511213	+	0.859454i	$0.329196\pi$
$240$	0	0
$241$	4230.52	1.13075	0.565377	−	0.824833i	$-0.308731\pi$
$241$	4230.52	1.13075	0.565377	+	0.824833i	$0.308731\pi$
$242$	−374.223	−0.0994047
$243$	0	0
$244$	240.116	0.0629994
$245$	13.0064	0.00339161
$246$	0	0
$247$	546.861	0.140874
$248$	4021.82	1.02978
$249$	0	0
$250$	205.174	0.0519053
$251$	−1490.76	−0.374883	−0.187442	−	0.982276i	$-0.560020\pi$
$251$	−1490.76	−0.374883	−0.187442	+	0.982276i	$0.560020\pi$
$252$	0	0
$253$	−292.835	−0.0727684
$254$	4659.18	1.15096
$255$	0	0
$256$	2317.98	0.565914
$257$	5303.01	1.28713	0.643566	−	0.765391i	$-0.277454\pi$
$257$	5303.01	1.28713	0.643566	+	0.765391i	$0.277454\pi$
$258$	0	0
$259$	1711.95	0.410716
$260$	4.15227	0.000990435 0
$261$	0	0
$262$	6851.16	1.61552
$263$	−3045.28	−0.713992	−0.356996	−	0.934106i	$-0.616199\pi$
$263$	−3045.28	−0.713992	−0.356996	+	0.934106i	$0.616199\pi$
$264$	0	0
$265$	−113.657	−0.0263468
$266$	−1184.50	−0.273031
$267$	0	0
$268$	301.553	0.0687323
$269$	−3818.96	−0.865598	−0.432799	−	0.901490i	$-0.642474\pi$
$269$	−3818.96	−0.865598	−0.432799	+	0.901490i	$0.642474\pi$
$270$	0	0
$271$	−5898.73	−1.32222	−0.661111	−	0.750288i	$-0.729915\pi$
$271$	−5898.73	−1.32222	−0.661111	+	0.750288i	$0.729915\pi$
$272$	7680.07	1.71203
$273$	0	0
$274$	3275.89	0.722276
$275$	1374.22	0.301341
$276$	0	0
$277$	−2009.89	−0.435966	−0.217983	−	0.975953i	$-0.569948\pi$
$277$	−2009.89	−0.435966	−0.217983	+	0.975953i	$0.569948\pi$
$278$	6460.34	1.39376
$279$	0	0
$280$	36.9781	0.00789236
$281$	5431.49	1.15308	0.576540	−	0.817069i	$-0.304402\pi$
$281$	5431.49	1.15308	0.576540	+	0.817069i	$0.304402\pi$
$282$	0	0
$283$	3145.14	0.660632	0.330316	−	0.943870i	$-0.392845\pi$
$283$	3145.14	0.660632	0.330316	+	0.943870i	$0.392845\pi$
$284$	−1286.36	−0.268772
$285$	0	0
$286$	340.033	0.0703028
$287$	−2143.92	−0.440946
$288$	0	0
$289$	5837.55	1.18819
$290$	−14.1740	−0.00287008
$291$	0	0
$292$	−1139.35	−0.228341
$293$	295.533	0.0589257	0.0294628	−	0.999566i	$-0.490620\pi$
$293$	295.533	0.0589257	0.0294628	+	0.999566i	$0.490620\pi$
$294$	0	0
$295$	92.9491	0.0183448
$296$	4867.20	0.955745
$297$	0	0
$298$	−1005.56	−0.195472
$299$	266.082	0.0514646
$300$	0	0
$301$	2313.96	0.443104
$302$	−1843.92	−0.351342
$303$	0	0
$304$	−4052.68	−0.764596
$305$	40.7229	0.00764520
$306$	0	0
$307$	−7797.26	−1.44955	−0.724777	−	0.688983i	$-0.758057\pi$
$307$	−7797.26	−1.44955	−0.724777	+	0.688983i	$0.758057\pi$
$308$	−120.513	−0.0222950
$309$	0	0
$310$	−165.898	−0.0303947
$311$	2086.15	0.380370	0.190185	−	0.981748i	$-0.439091\pi$
$311$	2086.15	0.380370	0.190185	+	0.981748i	$0.439091\pi$
$312$	0	0
$313$	5866.76	1.05945	0.529727	−	0.848168i	$-0.322294\pi$
$313$	5866.76	1.05945	0.529727	+	0.848168i	$0.322294\pi$
$314$	−2325.13	−0.417881
$315$	0	0
$316$	−642.343	−0.114350
$317$	−2415.74	−0.428017	−0.214008	−	0.976832i	$-0.568652\pi$
$317$	−2415.74	−0.428017	−0.214008	+	0.976832i	$0.568652\pi$
$318$	0	0
$319$	−189.924	−0.0333345
$320$	99.9299	0.0174570
$321$	0	0
$322$	−576.333	−0.0997447
$323$	−5672.94	−0.977246
$324$	0	0
$325$	−1248.67	−0.213120
$326$	6725.32	1.14258
$327$	0	0
$328$	−6095.32	−1.02609
$329$	−522.574	−0.0875697
$330$	0	0
$331$	8503.14	1.41201	0.706004	−	0.708208i	$-0.250496\pi$
$331$	8503.14	1.41201	0.706004	+	0.708208i	$0.250496\pi$
$332$	453.156	0.0749101
$333$	0	0
$334$	8352.81	1.36840
$335$	51.1424	0.00834091
$336$	0	0
$337$	10827.2	1.75013	0.875065	−	0.484005i	$-0.160818\pi$
$337$	10827.2	1.75013	0.875065	+	0.484005i	$0.160818\pi$
$338$	6485.80	1.04373
$339$	0	0
$340$	−43.0742	−0.00687066
$341$	−2222.95	−0.353018
$342$	0	0
$343$	343.000	0.0539949
$344$	6578.76	1.03111
$345$	0	0
$346$	−5993.59	−0.931265
$347$	1384.06	0.214122	0.107061	−	0.994252i	$-0.465856\pi$
$347$	1384.06	0.214122	0.107061	+	0.994252i	$0.465856\pi$
$348$	0	0
$349$	−3206.52	−0.491808	−0.245904	−	0.969294i	$-0.579085\pi$
$349$	−3206.52	−0.491808	−0.245904	+	0.969294i	$0.579085\pi$
$350$	2704.63	0.413053
$351$	0	0
$352$	−768.587	−0.116380
$353$	−6954.77	−1.04863	−0.524313	−	0.851526i	$-0.675678\pi$
$353$	−6954.77	−1.04863	−0.524313	+	0.851526i	$0.675678\pi$
$354$	0	0
$355$	−218.162	−0.0326165
$356$	353.263	0.0525924
$357$	0	0
$358$	−1889.98	−0.279019
$359$	356.055	0.0523450	0.0261725	−	0.999657i	$-0.491668\pi$
$359$	356.055	0.0523450	0.0261725	+	0.999657i	$0.491668\pi$
$360$	0	0
$361$	−3865.46	−0.563561
$362$	12863.7	1.86769
$363$	0	0
$364$	109.503	0.0157678
$365$	−193.231	−0.0277100
$366$	0	0
$367$	−4952.27	−0.704377	−0.352188	−	0.935929i	$-0.614562\pi$
$367$	−4952.27	−0.704377	−0.352188	+	0.935929i	$0.614562\pi$
$368$	−1971.88	−0.279324
$369$	0	0
$370$	−200.769	−0.0282095
$371$	−2997.33	−0.419444
$372$	0	0
$373$	9467.31	1.31421	0.657103	−	0.753801i	$-0.271782\pi$
$373$	9467.31	1.31421	0.657103	+	0.753801i	$0.271782\pi$
$374$	−3527.39	−0.487692
$375$	0	0
$376$	−1485.72	−0.203777
$377$	172.572	0.0235754
$378$	0	0
$379$	1000.34	0.135577	0.0677886	−	0.997700i	$-0.478406\pi$
$379$	1000.34	0.135577	0.0677886	+	0.997700i	$0.478406\pi$
$380$	22.7297	0.00306845
$381$	0	0
$382$	11704.5	1.56768
$383$	4295.88	0.573131	0.286565	−	0.958061i	$-0.407486\pi$
$383$	4295.88	0.573131	0.286565	+	0.958061i	$0.407486\pi$
$384$	0	0
$385$	−20.4386	−0.00270557
$386$	10682.1	1.40856
$387$	0	0
$388$	−658.045	−0.0861009
$389$	−5024.75	−0.654923	−0.327461	−	0.944865i	$-0.606193\pi$
$389$	−5024.75	−0.654923	−0.327461	+	0.944865i	$0.606193\pi$
$390$	0	0
$391$	−2760.24	−0.357011
$392$	975.175	0.125647
$393$	0	0
$394$	−897.976	−0.114821
$395$	−108.939	−0.0138768
$396$	0	0
$397$	−666.267	−0.0842292	−0.0421146	−	0.999113i	$-0.513409\pi$
$397$	−666.267	−0.0842292	−0.0421146	+	0.999113i	$0.513409\pi$
$398$	9656.96	1.21623
$399$	0	0
$400$	9253.69	1.15671
$401$	−15735.7	−1.95960	−0.979802	−	0.199970i	$-0.935915\pi$
$401$	−15735.7	−1.95960	−0.979802	+	0.199970i	$0.935915\pi$
$402$	0	0
$403$	2019.86	0.249668
$404$	−1168.41	−0.143887
$405$	0	0
$406$	−373.792	−0.0456921
$407$	−2690.21	−0.327638
$408$	0	0
$409$	−10478.5	−1.26682	−0.633411	−	0.773815i	$-0.718346\pi$
$409$	−10478.5	−1.26682	−0.633411	+	0.773815i	$0.718346\pi$
$410$	251.428	0.0302858
$411$	0	0
$412$	779.324	0.0931907
$413$	2451.23	0.292051
$414$	0	0
$415$	76.8538	0.00909061
$416$	698.369	0.0823085
$417$	0	0
$418$	1861.36	0.217804
$419$	11046.8	1.28800	0.644002	−	0.765024i	$-0.277273\pi$
$419$	11046.8	1.28800	0.644002	+	0.765024i	$0.277273\pi$
$420$	0	0
$421$	−15046.5	−1.74186	−0.870930	−	0.491407i	$-0.836483\pi$
$421$	−15046.5	−1.74186	−0.870930	+	0.491407i	$0.836483\pi$
$422$	−16805.9	−1.93863
$423$	0	0
$424$	−8521.64	−0.976055
$425$	12953.3	1.47842
$426$	0	0
$427$	1073.93	0.121713
$428$	1675.23	0.189195
$429$	0	0
$430$	−271.370	−0.0304340
$431$	−8035.40	−0.898032	−0.449016	−	0.893524i	$-0.648225\pi$
$431$	−8035.40	−0.898032	−0.449016	+	0.893524i	$0.648225\pi$
$432$	0	0
$433$	−6726.63	−0.746561	−0.373281	−	0.927718i	$-0.621767\pi$
$433$	−6726.63	−0.746561	−0.373281	+	0.927718i	$0.621767\pi$
$434$	−4375.01	−0.483887
$435$	0	0
$436$	−1336.76	−0.146833
$437$	1456.54	0.159441
$438$	0	0
$439$	12589.0	1.36866	0.684330	−	0.729172i	$-0.260095\pi$
$439$	12589.0	1.36866	0.684330	+	0.729172i	$0.260095\pi$
$440$	−58.1084	−0.00629593
$441$	0	0
$442$	3205.12	0.344914
$443$	−4517.14	−0.484460	−0.242230	−	0.970219i	$-0.577879\pi$
$443$	−4517.14	−0.484460	−0.242230	+	0.970219i	$0.577879\pi$
$444$	0	0
$445$	59.9123	0.00638228
$446$	13485.5	1.43175
$447$	0	0
$448$	2635.32	0.277918
$449$	−1555.92	−0.163538	−0.0817691	−	0.996651i	$-0.526057\pi$
$449$	−1555.92	−0.163538	−0.0817691	+	0.996651i	$0.526057\pi$
$450$	0	0
$451$	3369.01	0.351753
$452$	−2759.17	−0.287125
$453$	0	0
$454$	−13110.7	−1.35532
$455$	18.5713	0.00191348
$456$	0	0
$457$	−3321.47	−0.339982	−0.169991	−	0.985446i	$-0.554374\pi$
$457$	−3321.47	−0.339982	−0.169991	+	0.985446i	$0.554374\pi$
$458$	−1516.37	−0.154706
$459$	0	0
$460$	11.0594	0.00112097
$461$	6592.03	0.665990	0.332995	−	0.942929i	$-0.391941\pi$
$461$	6592.03	0.665990	0.332995	+	0.942929i	$0.391941\pi$
$462$	0	0
$463$	1732.49	0.173900	0.0869498	−	0.996213i	$-0.472288\pi$
$463$	1732.49	0.173900	0.0869498	+	0.996213i	$0.472288\pi$
$464$	−1278.90	−0.127956
$465$	0	0
$466$	1876.65	0.186554
$467$	−13099.9	−1.29805	−0.649025	−	0.760767i	$-0.724823\pi$
$467$	−13099.9	−1.29805	−0.649025	+	0.760767i	$0.724823\pi$
$468$	0	0
$469$	1348.71	0.132788
$470$	61.2849	0.00601460
$471$	0	0
$472$	6969.03	0.679609
$473$	−3636.22	−0.353475
$474$	0	0
$475$	−6835.30	−0.660263
$476$	−1135.94	−0.109382
$477$	0	0
$478$	−11683.5	−1.11797
$479$	−9740.59	−0.929142	−0.464571	−	0.885536i	$-0.653791\pi$
$479$	−9740.59	−0.929142	−0.464571	+	0.885536i	$0.653791\pi$
$480$	0	0
$481$	2444.43	0.231718
$482$	−13083.9	−1.23642
$483$	0	0
$484$	189.377	0.0177852
$485$	−111.602	−0.0104486
$486$	0	0
$487$	−15036.8	−1.39914	−0.699569	−	0.714565i	$-0.746625\pi$
$487$	−15036.8	−1.39914	−0.699569	+	0.714565i	$0.746625\pi$
$488$	3053.27	0.283228
$489$	0	0
$490$	−40.2254	−0.00370857
$491$	8665.05	0.796432	0.398216	−	0.917292i	$-0.369629\pi$
$491$	8665.05	0.796432	0.398216	+	0.917292i	$0.369629\pi$
$492$	0	0
$493$	−1790.20	−0.163543
$494$	−1691.30	−0.154039
$495$	0	0
$496$	−14968.8	−1.35508
$497$	−5753.31	−0.519258
$498$	0	0
$499$	8628.90	0.774113	0.387057	−	0.922056i	$-0.373492\pi$
$499$	8628.90	0.774113	0.387057	+	0.922056i	$0.373492\pi$
$500$	−103.829	−0.00928676
$501$	0	0
$502$	4610.54	0.409917
$503$	18284.6	1.62081	0.810407	−	0.585867i	$-0.199246\pi$
$503$	18284.6	1.62081	0.810407	+	0.585867i	$0.199246\pi$
$504$	0	0
$505$	−198.158	−0.0174612
$506$	905.667	0.0795688
$507$	0	0
$508$	−2357.80	−0.205926
$509$	8392.39	0.730818	0.365409	−	0.930847i	$-0.380929\pi$
$509$	8392.39	0.730818	0.365409	+	0.930847i	$0.380929\pi$
$510$	0	0
$511$	−5095.83	−0.441147
$512$	6617.58	0.571208
$513$	0	0
$514$	−16400.9	−1.40742
$515$	132.171	0.0113090
$516$	0	0
$517$	821.187	0.0698564
$518$	−5294.63	−0.449098
$519$	0	0
$520$	52.7996	0.00445272
$521$	−4104.33	−0.345132	−0.172566	−	0.984998i	$-0.555206\pi$
$521$	−4104.33	−0.345132	−0.172566	+	0.984998i	$0.555206\pi$
$522$	0	0
$523$	−8847.78	−0.739744	−0.369872	−	0.929083i	$-0.620598\pi$
$523$	−8847.78	−0.739744	−0.369872	+	0.929083i	$0.620598\pi$
$524$	−3467.06	−0.289045
$525$	0	0
$526$	9418.28	0.780716
$527$	−20953.3	−1.73195
$528$	0	0
$529$	−11458.3	−0.941752
$530$	351.513	0.0288089
$531$	0	0
$532$	599.422	0.0488501
$533$	−3061.22	−0.248773
$534$	0	0
$535$	284.114	0.0229595
$536$	3834.49	0.309002
$537$	0	0
$538$	11811.1	0.946490
$539$	−539.000	−0.0430730
$540$	0	0
$541$	2737.43	0.217544	0.108772	−	0.994067i	$-0.465308\pi$
$541$	2737.43	0.217544	0.108772	+	0.994067i	$0.465308\pi$
$542$	18243.3	1.44579
$543$	0	0
$544$	−7244.63	−0.570976
$545$	−226.710	−0.0178187
$546$	0	0
$547$	13191.4	1.03112	0.515562	−	0.856852i	$-0.327583\pi$
$547$	13191.4	1.03112	0.515562	+	0.856852i	$0.327583\pi$
$548$	−1657.78	−0.129228
$549$	0	0
$550$	−4250.13	−0.329502
$551$	944.669	0.0730385
$552$	0	0
$553$	−2872.92	−0.220920
$554$	6216.09	0.476708
$555$	0	0
$556$	−3269.28	−0.249368
$557$	−15305.5	−1.16430	−0.582148	−	0.813083i	$-0.697788\pi$
$557$	−15305.5	−1.16430	−0.582148	+	0.813083i	$0.697788\pi$
$558$	0	0
$559$	3304.01	0.249991
$560$	−137.628	−0.0103854
$561$	0	0
$562$	−16798.2	−1.26084
$563$	−22448.9	−1.68048	−0.840240	−	0.542215i	$-0.817586\pi$
$563$	−22448.9	−1.68048	−0.840240	+	0.542215i	$0.817586\pi$
$564$	0	0
$565$	−467.947	−0.0348436
$566$	−9727.12	−0.722370
$567$	0	0
$568$	−16357.1	−1.20833
$569$	17400.5	1.28202	0.641010	−	0.767533i	$-0.278516\pi$
$569$	17400.5	1.28202	0.641010	+	0.767533i	$0.278516\pi$
$570$	0	0
$571$	−7577.54	−0.555359	−0.277680	−	0.960674i	$-0.589565\pi$
$571$	−7577.54	−0.555359	−0.277680	+	0.960674i	$0.589565\pi$
$572$	−172.075	−0.0125784
$573$	0	0
$574$	6630.60	0.482153
$575$	−3325.80	−0.241209
$576$	0	0
$577$	−10215.9	−0.737076	−0.368538	−	0.929613i	$-0.620141\pi$
$577$	−10215.9	−0.737076	−0.368538	+	0.929613i	$0.620141\pi$
$578$	−18054.1	−1.29922
$579$	0	0
$580$	7.17280	0.000513507 0
$581$	2026.77	0.144724
$582$	0	0
$583$	4710.09	0.334600
$584$	−14487.8	−1.02656
$585$	0	0
$586$	−914.010	−0.0644324
$587$	−20124.8	−1.41506	−0.707529	−	0.706684i	$-0.750190\pi$
$587$	−20124.8	−1.41506	−0.707529	+	0.706684i	$0.750190\pi$
$588$	0	0
$589$	11056.8	0.773492
$590$	−287.468	−0.0200591
$591$	0	0
$592$	−18115.2	−1.25765
$593$	−7989.14	−0.553246	−0.276623	−	0.960979i	$-0.589215\pi$
$593$	−7989.14	−0.553246	−0.276623	+	0.960979i	$0.589215\pi$
$594$	0	0
$595$	−192.652	−0.0132739
$596$	508.870	0.0349734
$597$	0	0
$598$	−822.924	−0.0562740
$599$	−19039.3	−1.29871	−0.649354	−	0.760486i	$-0.724961\pi$
$599$	−19039.3	−1.29871	−0.649354	+	0.760486i	$0.724961\pi$
$600$	0	0
$601$	18083.0	1.22732	0.613660	−	0.789570i	$-0.289697\pi$
$601$	18083.0	1.22732	0.613660	+	0.789570i	$0.289697\pi$
$602$	−7156.49	−0.484513
$603$	0	0
$604$	933.122	0.0628612
$605$	32.1177	0.00215830
$606$	0	0
$607$	23374.5	1.56300	0.781501	−	0.623904i	$-0.214454\pi$
$607$	23374.5	1.56300	0.781501	+	0.623904i	$0.214454\pi$
$608$	3822.90	0.254999
$609$	0	0
$610$	−125.946	−0.00835966
$611$	−746.163	−0.0494051
$612$	0	0
$613$	−4642.72	−0.305902	−0.152951	−	0.988234i	$-0.548878\pi$
$613$	−4642.72	−0.305902	−0.152951	+	0.988234i	$0.548878\pi$
$614$	24115.0	1.58502
$615$	0	0
$616$	−1532.42	−0.100232
$617$	−8757.63	−0.571425	−0.285712	−	0.958315i	$-0.592230\pi$
$617$	−8757.63	−0.571425	−0.285712	+	0.958315i	$0.592230\pi$
$618$	0	0
$619$	25564.1	1.65995	0.829973	−	0.557804i	$-0.188356\pi$
$619$	25564.1	1.65995	0.829973	+	0.557804i	$0.188356\pi$
$620$	83.9533	0.00543814
$621$	0	0
$622$	−6451.95	−0.415916
$623$	1579.99	0.101607
$624$	0	0
$625$	15598.6	0.998309
$626$	−18144.4	−1.15846
$627$	0	0
$628$	1176.64	0.0747662
$629$	−25357.6	−1.60743
$630$	0	0
$631$	22670.2	1.43025	0.715125	−	0.698997i	$-0.246370\pi$
$631$	22670.2	1.43025	0.715125	+	0.698997i	$0.246370\pi$
$632$	−8167.92	−0.514086
$633$	0	0
$634$	7471.27	0.468016
$635$	−399.875	−0.0249899
$636$	0	0
$637$	489.757	0.0304629
$638$	587.387	0.0364497
$639$	0	0
$640$	−457.430	−0.0282523
$641$	22315.5	1.37505	0.687527	−	0.726159i	$-0.258696\pi$
$641$	22315.5	1.37505	0.687527	+	0.726159i	$0.258696\pi$
$642$	0	0
$643$	−13574.4	−0.832537	−0.416269	−	0.909242i	$-0.636662\pi$
$643$	−13574.4	−0.832537	−0.416269	+	0.909242i	$0.636662\pi$
$644$	291.656	0.0178461
$645$	0	0
$646$	17545.0	1.06857
$647$	−3251.82	−0.197593	−0.0987963	−	0.995108i	$-0.531499\pi$
$647$	−3251.82	−0.197593	−0.0987963	+	0.995108i	$0.531499\pi$
$648$	0	0
$649$	−3851.93	−0.232976
$650$	3861.84	0.233037
$651$	0	0
$652$	−3403.38	−0.204427
$653$	−15807.3	−0.947301	−0.473650	−	0.880713i	$-0.657064\pi$
$653$	−15807.3	−0.947301	−0.473650	+	0.880713i	$0.657064\pi$
$654$	0	0
$655$	−588.002	−0.0350766
$656$	22686.1	1.35022
$657$	0	0
$658$	1616.19	0.0957532
$659$	9703.84	0.573608	0.286804	−	0.957989i	$-0.407407\pi$
$659$	9703.84	0.573608	0.286804	+	0.957989i	$0.407407\pi$
$660$	0	0
$661$	15703.5	0.924045	0.462023	−	0.886868i	$-0.347124\pi$
$661$	15703.5	0.924045	0.462023	+	0.886868i	$0.347124\pi$
$662$	−26298.1	−1.54396
$663$	0	0
$664$	5762.25	0.336775
$665$	101.660	0.00592813
$666$	0	0
$667$	459.640	0.0266827
$668$	−4226.98	−0.244830
$669$	0	0
$670$	−158.171	−0.00912039
$671$	−1687.61	−0.0970930
$672$	0	0
$673$	−18844.7	−1.07936	−0.539679	−	0.841871i	$-0.681455\pi$
$673$	−18844.7	−1.07936	−0.539679	+	0.841871i	$0.681455\pi$
$674$	−33485.8	−1.91368
$675$	0	0
$676$	−3282.17	−0.186742
$677$	34702.5	1.97005	0.985027	−	0.172398i	$-0.0551514\pi$
$677$	34702.5	1.97005	0.985027	+	0.172398i	$0.0551514\pi$
$678$	0	0
$679$	−2943.14	−0.166344
$680$	−547.724	−0.0308886
$681$	0	0
$682$	6875.01	0.386009
$683$	−10862.7	−0.608566	−0.304283	−	0.952582i	$-0.598417\pi$
$683$	−10862.7	−0.608566	−0.304283	+	0.952582i	$0.598417\pi$
$684$	0	0
$685$	−281.154	−0.0156822
$686$	−1060.81	−0.0590409
$687$	0	0
$688$	−24485.4	−1.35683
$689$	−4279.77	−0.236642
$690$	0	0
$691$	9219.04	0.507538	0.253769	−	0.967265i	$-0.418330\pi$
$691$	9219.04	0.507538	0.253769	+	0.967265i	$0.418330\pi$
$692$	3033.09	0.166619
$693$	0	0
$694$	−4280.55	−0.234132
$695$	−554.460	−0.0302617
$696$	0	0
$697$	31756.0	1.72574
$698$	9916.95	0.537768
$699$	0	0
$700$	−1368.69	−0.0739023
$701$	29310.7	1.57925	0.789623	−	0.613593i	$-0.210276\pi$
$701$	29310.7	1.57925	0.789623	+	0.613593i	$0.210276\pi$
$702$	0	0
$703$	13380.9	0.717881
$704$	−4141.22	−0.221702
$705$	0	0
$706$	21509.4	1.14662
$707$	−5225.76	−0.277984
$708$	0	0
$709$	−5343.74	−0.283058	−0.141529	−	0.989934i	$-0.545202\pi$
$709$	−5343.74	−0.283058	−0.141529	+	0.989934i	$0.545202\pi$
$710$	674.721	0.0356646
$711$	0	0
$712$	4492.03	0.236441
$713$	5379.81	0.282574
$714$	0	0
$715$	−29.1834	−0.00152643
$716$	956.434	0.0499212
$717$	0	0
$718$	−1101.19	−0.0572367
$719$	−5699.03	−0.295602	−0.147801	−	0.989017i	$-0.547220\pi$
$719$	−5699.03	−0.295602	−0.147801	+	0.989017i	$0.547220\pi$
$720$	0	0
$721$	3485.57	0.180041
$722$	11954.9	0.616227
$723$	0	0
$724$	−6509.74	−0.334161
$725$	−2157.01	−0.110496
$726$	0	0
$727$	−24538.7	−1.25184	−0.625922	−	0.779886i	$-0.715277\pi$
$727$	−24538.7	−1.25184	−0.625922	+	0.779886i	$0.715277\pi$
$728$	1392.42	0.0708878
$729$	0	0
$730$	597.615	0.0302996
$731$	−34274.6	−1.73419
$732$	0	0
$733$	−941.949	−0.0474648	−0.0237324	−	0.999718i	$-0.507555\pi$
$733$	−941.949	−0.0474648	−0.0237324	+	0.999718i	$0.507555\pi$
$734$	15316.1	0.770202
$735$	0	0
$736$	1860.08	0.0931569
$737$	−2119.40	−0.105928
$738$	0	0
$739$	7607.30	0.378673	0.189336	−	0.981912i	$-0.439366\pi$
$739$	7607.30	0.378673	0.189336	+	0.981912i	$0.439366\pi$
$740$	101.600	0.00504716
$741$	0	0
$742$	9269.99	0.458642
$743$	−2511.79	−0.124023	−0.0620113	−	0.998075i	$-0.519751\pi$
$743$	−2511.79	−0.124023	−0.0620113	+	0.998075i	$0.519751\pi$
$744$	0	0
$745$	86.3028	0.00424415
$746$	−29280.0	−1.43702
$747$	0	0
$748$	1785.05	0.0872565
$749$	7492.58	0.365518
$750$	0	0
$751$	9224.66	0.448219	0.224110	−	0.974564i	$-0.428053\pi$
$751$	9224.66	0.448219	0.224110	+	0.974564i	$0.428053\pi$
$752$	5529.67	0.268147
$753$	0	0
$754$	−533.723	−0.0257786
$755$	158.254	0.00762843
$756$	0	0
$757$	−20688.3	−0.993301	−0.496650	−	0.867951i	$-0.665437\pi$
$757$	−20688.3	−0.993301	−0.496650	+	0.867951i	$0.665437\pi$
$758$	−3093.79	−0.148247
$759$	0	0
$760$	289.027	0.0137949
$761$	−3288.15	−0.156630	−0.0783149	−	0.996929i	$-0.524954\pi$
$761$	−3288.15	−0.156630	−0.0783149	+	0.996929i	$0.524954\pi$
$762$	0	0
$763$	−5978.74	−0.283676
$764$	−5923.11	−0.280485
$765$	0	0
$766$	−13286.1	−0.626691
$767$	3500.01	0.164769
$768$	0	0
$769$	26102.9	1.22405	0.612025	−	0.790838i	$-0.290355\pi$
$769$	26102.9	1.22405	0.612025	+	0.790838i	$0.290355\pi$
$770$	63.2113	0.00295841
$771$	0	0
$772$	−5405.72	−0.252016
$773$	9887.03	0.460041	0.230020	−	0.973186i	$-0.426121\pi$
$773$	9887.03	0.460041	0.230020	+	0.973186i	$0.426121\pi$
$774$	0	0
$775$	−25246.5	−1.17017
$776$	−8367.58	−0.387086
$777$	0	0
$778$	15540.3	0.716127
$779$	−16757.2	−0.770719
$780$	0	0
$781$	9040.92	0.414225
$782$	8536.72	0.390374
$783$	0	0
$784$	−3629.49	−0.165338
$785$	199.555	0.00907314
$786$	0	0
$787$	466.839	0.0211449	0.0105724	−	0.999944i	$-0.496635\pi$
$787$	466.839	0.0211449	0.0105724	+	0.999944i	$0.496635\pi$
$788$	454.425	0.0205434
$789$	0	0
$790$	336.922	0.0151736
$791$	−12340.6	−0.554715
$792$	0	0
$793$	1533.43	0.0686679
$794$	2060.60	0.0921006
$795$	0	0
$796$	−4886.95	−0.217605
$797$	26616.2	1.18293	0.591463	−	0.806332i	$-0.298550\pi$
$797$	26616.2	1.18293	0.591463	+	0.806332i	$0.298550\pi$
$798$	0	0
$799$	7740.43	0.342724
$800$	−8729.03	−0.385772
$801$	0	0
$802$	48666.4	2.14273
$803$	8007.73	0.351914
$804$	0	0
$805$	49.4639	0.00216568
$806$	−6246.91	−0.273000
$807$	0	0
$808$	−14857.2	−0.646876
$809$	1509.75	0.0656118	0.0328059	−	0.999462i	$-0.489556\pi$
$809$	1509.75	0.0656118	0.0328059	+	0.999462i	$0.489556\pi$
$810$	0	0
$811$	34346.1	1.48712	0.743560	−	0.668669i	$-0.233136\pi$
$811$	34346.1	1.48712	0.743560	+	0.668669i	$0.233136\pi$
$812$	189.159	0.00817510
$813$	0	0
$814$	8320.14	0.358256
$815$	−577.202	−0.0248080
$816$	0	0
$817$	18086.3	0.774491
$818$	32407.5	1.38521
$819$	0	0
$820$	−127.236	−0.00541865
$821$	−8329.63	−0.354088	−0.177044	−	0.984203i	$-0.556653\pi$
$821$	−8329.63	−0.354088	−0.177044	+	0.984203i	$0.556653\pi$
$822$	0	0
$823$	4980.53	0.210948	0.105474	−	0.994422i	$-0.466364\pi$
$823$	4980.53	0.210948	0.105474	+	0.994422i	$0.466364\pi$
$824$	9909.75	0.418959
$825$	0	0
$826$	−7581.03	−0.319344
$827$	−28946.4	−1.21713	−0.608564	−	0.793505i	$-0.708254\pi$
$827$	−28946.4	−1.21713	−0.608564	+	0.793505i	$0.708254\pi$
$828$	0	0
$829$	42927.9	1.79849	0.899244	−	0.437448i	$-0.144118\pi$
$829$	42927.9	1.79849	0.899244	+	0.437448i	$0.144118\pi$
$830$	−237.690	−0.00994015
$831$	0	0
$832$	3762.88	0.156796
$833$	−5080.56	−0.211322
$834$	0	0
$835$	−716.881	−0.0297110
$836$	−941.948	−0.0389689
$837$	0	0
$838$	−34165.1	−1.40837
$839$	46163.6	1.89958	0.949789	−	0.312892i	$-0.101298\pi$
$839$	46163.6	1.89958	0.949789	+	0.312892i	$0.101298\pi$
$840$	0	0
$841$	−24090.9	−0.987777
$842$	46535.1	1.90464
$843$	0	0
$844$	8504.72	0.346854
$845$	−556.645	−0.0226618
$846$	0	0
$847$	847.000	0.0343604
$848$	31716.6	1.28438
$849$	0	0
$850$	−40061.3	−1.61658
$851$	6510.64	0.262259
$852$	0	0
$853$	−8920.57	−0.358071	−0.179036	−	0.983843i	$-0.557298\pi$
$853$	−8920.57	−0.358071	−0.179036	+	0.983843i	$0.557298\pi$
$854$	−3321.41	−0.133087
$855$	0	0
$856$	21302.0	0.850568
$857$	27058.0	1.07851	0.539255	−	0.842143i	$-0.318706\pi$
$857$	27058.0	1.07851	0.539255	+	0.842143i	$0.318706\pi$
$858$	0	0
$859$	−35857.3	−1.42425	−0.712127	−	0.702051i	$-0.752268\pi$
$859$	−35857.3	−1.42425	−0.712127	+	0.702051i	$0.752268\pi$
$860$	137.328	0.00544517
$861$	0	0
$862$	24851.5	0.981955
$863$	−14970.4	−0.590496	−0.295248	−	0.955421i	$-0.595402\pi$
$863$	−14970.4	−0.590496	−0.295248	+	0.955421i	$0.595402\pi$
$864$	0	0
$865$	514.401	0.0202199
$866$	20803.8	0.816329
$867$	0	0
$868$	2213.99	0.0865758
$869$	4514.58	0.176233
$870$	0	0
$871$	1925.77	0.0749166
$872$	−16998.0	−0.660121
$873$	0	0
$874$	−4504.72	−0.174342
$875$	−464.382	−0.0179417
$876$	0	0
$877$	−19944.6	−0.767939	−0.383970	−	0.923346i	$-0.625443\pi$
$877$	−19944.6	−0.767939	−0.383970	+	0.923346i	$0.625443\pi$
$878$	−38934.8	−1.49657
$879$	0	0
$880$	216.273	0.00828472
$881$	7898.52	0.302052	0.151026	−	0.988530i	$-0.451742\pi$
$881$	7898.52	0.302052	0.151026	+	0.988530i	$0.451742\pi$
$882$	0	0
$883$	−33545.0	−1.27846	−0.639230	−	0.769015i	$-0.720747\pi$
$883$	−33545.0	−1.27846	−0.639230	+	0.769015i	$0.720747\pi$
$884$	−1621.97	−0.0617111
$885$	0	0
$886$	13970.4	0.529734
$887$	8631.55	0.326741	0.163370	−	0.986565i	$-0.447763\pi$
$887$	8631.55	0.326741	0.163370	+	0.986565i	$0.447763\pi$
$888$	0	0
$889$	−10545.4	−0.397842
$890$	−185.294	−0.00697872
$891$	0	0
$892$	−6824.41	−0.256164
$893$	−4084.53	−0.153061
$894$	0	0
$895$	162.208	0.00605812
$896$	−12063.2	−0.449781
$897$	0	0
$898$	4812.09	0.178821
$899$	3489.18	0.129445
$900$	0	0
$901$	44396.8	1.64159
$902$	−10419.5	−0.384625
$903$	0	0
$904$	−35085.1	−1.29083
$905$	−1104.03	−0.0405516
$906$	0	0
$907$	−2906.82	−0.106416	−0.0532081	−	0.998583i	$-0.516945\pi$
$907$	−2906.82	−0.106416	−0.0532081	+	0.998583i	$0.516945\pi$
$908$	6634.74	0.242491
$909$	0	0
$910$	−57.4363	−0.00209230
$911$	38038.4	1.38339	0.691695	−	0.722190i	$-0.256864\pi$
$911$	38038.4	1.38339	0.691695	+	0.722190i	$0.256864\pi$
$912$	0	0
$913$	−3184.92	−0.115450
$914$	10272.5	0.371754
$915$	0	0
$916$	767.365	0.0276795
$917$	−15506.6	−0.558423
$918$	0	0
$919$	−23128.8	−0.830194	−0.415097	−	0.909777i	$-0.636252\pi$
$919$	−23128.8	−0.830194	−0.415097	+	0.909777i	$0.636252\pi$
$920$	140.630	0.00503959
$921$	0	0
$922$	−20387.5	−0.728228
$923$	−8214.94	−0.292956
$924$	0	0
$925$	−30553.3	−1.08604
$926$	−5358.15	−0.190151
$927$	0	0
$928$	1206.39	0.0426742
$929$	16204.2	0.572272	0.286136	−	0.958189i	$-0.407629\pi$
$929$	16204.2	0.572272	0.286136	+	0.958189i	$0.407629\pi$
$930$	0	0
$931$	2680.95	0.0943765
$932$	−949.689	−0.0333778
$933$	0	0
$934$	40514.6	1.41936
$935$	302.739	0.0105889
$936$	0	0
$937$	−12976.0	−0.452411	−0.226205	−	0.974080i	$-0.572632\pi$
$937$	−12976.0	−0.452411	−0.226205	+	0.974080i	$0.572632\pi$
$938$	−4171.23	−0.145198
$939$	0	0
$940$	−31.0135	−0.00107612
$941$	16607.5	0.575332	0.287666	−	0.957731i	$-0.407121\pi$
$941$	16607.5	0.575332	0.287666	+	0.957731i	$0.407121\pi$
$942$	0	0
$943$	−8153.44	−0.281561
$944$	−25937.9	−0.894288
$945$	0	0
$946$	11245.9	0.386508
$947$	89.1281	0.00305837	0.00152918	−	0.999999i	$-0.499513\pi$
$947$	89.1281	0.00305837	0.00152918	+	0.999999i	$0.499513\pi$
$948$	0	0
$949$	−7276.14	−0.248887
$950$	21139.9	0.721966
$951$	0	0
$952$	−14444.4	−0.491750
$953$	7262.60	0.246861	0.123431	−	0.992353i	$-0.460610\pi$
$953$	7262.60	0.246861	0.123431	+	0.992353i	$0.460610\pi$
$954$	0	0
$955$	−1004.54	−0.0340379
$956$	5912.49	0.200025
$957$	0	0
$958$	30125.2	1.01597
$959$	−7414.50	−0.249663
$960$	0	0
$961$	11047.7	0.370841
$962$	−7560.00	−0.253372
$963$	0	0
$964$	6621.18	0.221218
$965$	−916.793	−0.0305830
$966$	0	0
$967$	3728.05	0.123977	0.0619887	−	0.998077i	$-0.480256\pi$
$967$	3728.05	0.123977	0.0619887	+	0.998077i	$0.480256\pi$
$968$	2408.09	0.0799574
$969$	0	0
$970$	345.158	0.0114251
$971$	49516.3	1.63651	0.818257	−	0.574853i	$-0.194941\pi$
$971$	49516.3	1.63651	0.818257	+	0.574853i	$0.194941\pi$
$972$	0	0
$973$	−14622.1	−0.481769
$974$	46504.9	1.52989
$975$	0	0
$976$	−11363.9	−0.372696
$977$	12131.7	0.397263	0.198632	−	0.980074i	$-0.436350\pi$
$977$	12131.7	0.397263	0.198632	+	0.980074i	$0.436350\pi$
$978$	0	0
$979$	−2482.84	−0.0810541
$980$	20.3562	0.000663527 0
$981$	0	0
$982$	−26798.8	−0.870861
$983$	29373.7	0.953078	0.476539	−	0.879153i	$-0.341891\pi$
$983$	29373.7	0.953078	0.476539	+	0.879153i	$0.341891\pi$
$984$	0	0
$985$	77.0689	0.00249302
$986$	5536.65	0.178827
$987$	0	0
$988$	855.892	0.0275603
$989$	8800.11	0.282940
$990$	0	0
$991$	6116.66	0.196067	0.0980334	−	0.995183i	$-0.468745\pi$
$991$	6116.66	0.196067	0.0980334	+	0.995183i	$0.468745\pi$
$992$	14120.1	0.451928
$993$	0	0
$994$	17793.6	0.567784
$995$	−828.811	−0.0264071
$996$	0	0
$997$	27054.2	0.859392	0.429696	−	0.902974i	$-0.358621\pi$
$997$	27054.2	0.859392	0.429696	+	0.902974i	$0.358621\pi$
$998$	−26687.0	−0.846456
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.r.1.3	✓	8
3.2	odd	2		693.4.a.u.1.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.4.a.r.1.3	✓	8	1.1	even	1	trivial
693.4.a.u.1.6	yes	8	3.2	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:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

\(f(q)\)	\(=\)	\(q-3.09275 q^{2} +1.56510 q^{4} +0.265436 q^{5} +7.00000 q^{7} +19.9015 q^{8} -0.820927 q^{10} -11.0000 q^{11} +9.99503 q^{13} -21.6492 q^{14} -74.0713 q^{16} -103.685 q^{17} +54.7132 q^{19} +0.415434 q^{20} +34.0202 q^{22} +26.6214 q^{23} -124.930 q^{25} -30.9121 q^{26} +10.9557 q^{28} +17.2658 q^{29} +202.086 q^{31} +69.8716 q^{32} +320.671 q^{34} +1.85805 q^{35} +244.564 q^{37} -169.214 q^{38} +5.28258 q^{40} -306.274 q^{41} +330.565 q^{43} -17.2161 q^{44} -82.3333 q^{46} -74.6534 q^{47} +49.0000 q^{49} +386.376 q^{50} +15.6432 q^{52} -428.190 q^{53} -2.91979 q^{55} +139.311 q^{56} -53.3988 q^{58} +350.175 q^{59} +153.419 q^{61} -625.001 q^{62} +376.475 q^{64} +2.65304 q^{65} +192.673 q^{67} -162.277 q^{68} -5.74649 q^{70} -821.902 q^{71} -727.976 q^{73} -756.376 q^{74} +85.6317 q^{76} -77.0000 q^{77} -410.417 q^{79} -19.6612 q^{80} +947.228 q^{82} +289.538 q^{83} -27.5217 q^{85} -1022.36 q^{86} -218.917 q^{88} +225.713 q^{89} +69.9652 q^{91} +41.6652 q^{92} +230.884 q^{94} +14.5229 q^{95} -420.449 q^{97} -151.545 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{2} + 30 q^{4} - 10 q^{5} + 56 q^{7} - 81 q^{8} + 9 q^{10} - 88 q^{11} + 16 q^{13} - 42 q^{14} + 122 q^{16} - 90 q^{17} - 42 q^{19} - 291 q^{20} + 66 q^{22} - 338 q^{23} + 244 q^{25} - 209 q^{26}+ \cdots - 294 q^{98}+O(q^{100}) \) 8 * q - 6 * q^2 + 30 * q^4 - 10 * q^5 + 56 * q^7 - 81 * q^8 + 9 * q^10 - 88 * q^11 + 16 * q^13 - 42 * q^14 + 122 * q^16 - 90 * q^17 - 42 * q^19 - 291 * q^20 + 66 * q^22 - 338 * q^23 + 244 * q^25 - 209 * q^26 + 210 * q^28 - 496 * q^29 - 8 * q^31 - 524 * q^32 - 302 * q^34 - 70 * q^35 - 360 * q^37 - 45 * q^38 - 6 * q^40 - 242 * q^41 - 66 * q^43 - 330 * q^44 + 344 * q^46 - 540 * q^47 + 392 * q^49 - 1171 * q^50 + 465 * q^52 - 906 * q^53 + 110 * q^55 - 567 * q^56 + 977 * q^58 - 1242 * q^59 - 318 * q^61 + 110 * q^62 + 525 * q^64 - 1258 * q^65 + 522 * q^67 - 678 * q^68 + 63 * q^70 - 858 * q^71 - 78 * q^73 - 1651 * q^74 + 1775 * q^76 - 616 * q^77 + 516 * q^79 - 567 * q^80 - 1212 * q^82 - 3192 * q^83 + 720 * q^85 - 1322 * q^86 + 891 * q^88 - 2356 * q^89 + 112 * q^91 - 4504 * q^92 - 423 * q^94 - 3308 * q^95 - 1556 * q^97 - 294 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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