LMFDB - Embedded newform 693.4.a.m.1.2

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:	$4$
Coefficient field:	4.4.509800.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 5x + 15 $ x^4 - x^3 - 10x^2 + 5x + 15
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.18303$ 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)

$f(q)$	$=$	$q-0.948670 q^{2} -7.10002 q^{4} +5.36789 q^{5} -7.00000 q^{7} +14.3249 q^{8} +O(q^{10})$	\(q-0.948670 q^{2} -7.10002 q^{4} +5.36789 q^{5} -7.00000 q^{7} +14.3249 q^{8} -5.09236 q^{10} -11.0000 q^{11} -42.9580 q^{13} +6.64069 q^{14} +43.2105 q^{16} +60.8725 q^{17} +140.575 q^{19} -38.1121 q^{20} +10.4354 q^{22} +91.3375 q^{23} -96.1858 q^{25} +40.7529 q^{26} +49.7002 q^{28} -260.968 q^{29} -259.463 q^{31} -155.592 q^{32} -57.7479 q^{34} -37.5752 q^{35} +359.998 q^{37} -133.359 q^{38} +76.8947 q^{40} +320.038 q^{41} -92.3549 q^{43} +78.1003 q^{44} -86.6492 q^{46} -67.4510 q^{47} +49.0000 q^{49} +91.2486 q^{50} +305.003 q^{52} +246.038 q^{53} -59.0468 q^{55} -100.275 q^{56} +247.573 q^{58} +475.095 q^{59} -799.071 q^{61} +246.145 q^{62} -198.079 q^{64} -230.594 q^{65} -725.003 q^{67} -432.196 q^{68} +35.6465 q^{70} -544.359 q^{71} -580.179 q^{73} -341.519 q^{74} -998.086 q^{76} +77.0000 q^{77} -402.439 q^{79} +231.949 q^{80} -303.611 q^{82} +1102.37 q^{83} +326.757 q^{85} +87.6143 q^{86} -157.574 q^{88} -1257.27 q^{89} +300.706 q^{91} -648.499 q^{92} +63.9888 q^{94} +754.591 q^{95} -999.825 q^{97} -46.4848 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 22 q^{4} + 18 q^{5} - 28 q^{7} + 60 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 22 * q^4 + 18 * q^5 - 28 * q^7 + 60 * q^8	$ 4 q + 4 q^{2} + 22 q^{4} + 18 q^{5} - 28 q^{7} + 60 q^{8} - 92 q^{10} - 44 q^{11} - 134 q^{13} - 28 q^{14} - 6 q^{16} + 74 q^{17} - 164 q^{19} - 116 q^{20} - 44 q^{22} - 194 q^{23} + 38 q^{25} - 734 q^{26} - 154 q^{28} + 108 q^{29} - 412 q^{31} + 4 q^{32} - 346 q^{34} - 126 q^{35} + 286 q^{37} - 224 q^{38} - 540 q^{40} + 18 q^{41} - 496 q^{43} - 242 q^{44} - 284 q^{46} - 62 q^{47} + 196 q^{49} - 212 q^{50} - 822 q^{52} + 828 q^{53} - 198 q^{55} - 420 q^{56} + 1388 q^{58} + 1224 q^{59} - 350 q^{61} + 878 q^{62} - 718 q^{64} + 396 q^{65} - 1498 q^{67} - 1058 q^{68} + 644 q^{70} - 2326 q^{71} - 1630 q^{73} + 1156 q^{74} - 3152 q^{76} + 308 q^{77} - 1020 q^{79} - 3072 q^{80} + 2118 q^{82} + 1920 q^{83} + 2008 q^{85} - 1056 q^{86} - 660 q^{88} - 1550 q^{89} + 938 q^{91} - 2592 q^{92} - 1042 q^{94} - 2332 q^{95} - 2202 q^{97} + 196 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 22 * q^4 + 18 * q^5 - 28 * q^7 + 60 * q^8 - 92 * q^10 - 44 * q^11 - 134 * q^13 - 28 * q^14 - 6 * q^16 + 74 * q^17 - 164 * q^19 - 116 * q^20 - 44 * q^22 - 194 * q^23 + 38 * q^25 - 734 * q^26 - 154 * q^28 + 108 * q^29 - 412 * q^31 + 4 * q^32 - 346 * q^34 - 126 * q^35 + 286 * q^37 - 224 * q^38 - 540 * q^40 + 18 * q^41 - 496 * q^43 - 242 * q^44 - 284 * q^46 - 62 * q^47 + 196 * q^49 - 212 * q^50 - 822 * q^52 + 828 * q^53 - 198 * q^55 - 420 * q^56 + 1388 * q^58 + 1224 * q^59 - 350 * q^61 + 878 * q^62 - 718 * q^64 + 396 * q^65 - 1498 * q^67 - 1058 * q^68 + 644 * q^70 - 2326 * q^71 - 1630 * q^73 + 1156 * q^74 - 3152 * q^76 + 308 * q^77 - 1020 * q^79 - 3072 * q^80 + 2118 * q^82 + 1920 * q^83 + 2008 * q^85 - 1056 * q^86 - 660 * q^88 - 1550 * q^89 + 938 * q^91 - 2592 * q^92 - 1042 * q^94 - 2332 * q^95 - 2202 * q^97 + 196 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.948670	−0.335406	−0.167703	−	0.985838i	$-0.553635\pi$
$2$	−0.948670	−0.335406	−0.167703	+	0.985838i	$0.553635\pi$
$3$	0	0
$3$	0	0
$4$	−7.10002	−0.887503
$5$	5.36789	0.480119	0.240059	−	0.970758i	$-0.422833\pi$
$5$	5.36789	0.480119	0.240059	+	0.970758i	$0.422833\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	14.3249	0.633079
$9$	0	0
$10$	−5.09236	−0.161034
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−42.9580	−0.916492	−0.458246	−	0.888825i	$-0.651522\pi$
$13$	−42.9580	−0.916492	−0.458246	+	0.888825i	$0.651522\pi$
$14$	6.64069	0.126771
$15$	0	0
$16$	43.2105	0.675165
$17$	60.8725	0.868455	0.434228	−	0.900803i	$-0.357021\pi$
$17$	60.8725	0.868455	0.434228	+	0.900803i	$0.357021\pi$
$18$	0	0
$19$	140.575	1.69737	0.848687	−	0.528895i	$-0.177393\pi$
$19$	140.575	1.69737	0.848687	+	0.528895i	$0.177393\pi$
$20$	−38.1121	−0.426107
$21$	0	0
$22$	10.4354	0.101129
$23$	91.3375	0.828052	0.414026	−	0.910265i	$-0.364122\pi$
$23$	91.3375	0.828052	0.414026	+	0.910265i	$0.364122\pi$
$24$	0	0
$25$	−96.1858	−0.769486
$26$	40.7529	0.307397
$27$	0	0
$28$	49.7002	0.335445
$29$	−260.968	−1.67105	−0.835527	−	0.549449i	$-0.814838\pi$
$29$	−260.968	−1.67105	−0.835527	+	0.549449i	$0.814838\pi$
$30$	0	0
$31$	−259.463	−1.50326	−0.751629	−	0.659586i	$-0.770731\pi$
$31$	−259.463	−1.50326	−0.751629	+	0.659586i	$0.770731\pi$
$32$	−155.592	−0.859533
$33$	0	0
$34$	−57.7479	−0.291285
$35$	−37.5752	−0.181468
$36$	0	0
$37$	359.998	1.59955	0.799774	−	0.600301i	$-0.204952\pi$
$37$	359.998	1.59955	0.799774	+	0.600301i	$0.204952\pi$
$38$	−133.359	−0.569309
$39$	0	0
$40$	76.8947	0.303953
$41$	320.038	1.21906	0.609531	−	0.792762i	$-0.291358\pi$
$41$	320.038	1.21906	0.609531	+	0.792762i	$0.291358\pi$
$42$	0	0
$43$	−92.3549	−0.327535	−0.163767	−	0.986499i	$-0.552365\pi$
$43$	−92.3549	−0.327535	−0.163767	+	0.986499i	$0.552365\pi$
$44$	78.1003	0.267592
$45$	0	0
$46$	−86.6492	−0.277733
$47$	−67.4510	−0.209335	−0.104667	−	0.994507i	$-0.533378\pi$
$47$	−67.4510	−0.209335	−0.104667	+	0.994507i	$0.533378\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	91.2486	0.258090
$51$	0	0
$52$	305.003	0.813389
$53$	246.038	0.637659	0.318830	−	0.947812i	$-0.396710\pi$
$53$	246.038	0.637659	0.318830	+	0.947812i	$0.396710\pi$
$54$	0	0
$55$	−59.0468	−0.144761
$56$	−100.275	−0.239281
$57$	0	0
$58$	247.573	0.560481
$59$	475.095	1.04834	0.524170	−	0.851613i	$-0.324375\pi$
$59$	475.095	1.04834	0.524170	+	0.851613i	$0.324375\pi$
$60$	0	0
$61$	−799.071	−1.67722	−0.838611	−	0.544731i	$-0.816632\pi$
$61$	−799.071	−1.67722	−0.838611	+	0.544731i	$0.816632\pi$
$62$	246.145	0.504201
$63$	0	0
$64$	−198.079	−0.386872
$65$	−230.594	−0.440025
$66$	0	0
$67$	−725.003	−1.32199	−0.660994	−	0.750391i	$-0.729865\pi$
$67$	−725.003	−1.32199	−0.660994	+	0.750391i	$0.729865\pi$
$68$	−432.196	−0.770757
$69$	0	0
$70$	35.6465	0.0608653
$71$	−544.359	−0.909909	−0.454954	−	0.890515i	$-0.650344\pi$
$71$	−544.359	−0.909909	−0.454954	+	0.890515i	$0.650344\pi$
$72$	0	0
$73$	−580.179	−0.930202	−0.465101	−	0.885258i	$-0.653982\pi$
$73$	−580.179	−0.930202	−0.465101	+	0.885258i	$0.653982\pi$
$74$	−341.519	−0.536498
$75$	0	0
$76$	−998.086	−1.50643
$77$	77.0000	0.113961
$78$	0	0
$79$	−402.439	−0.573139	−0.286569	−	0.958059i	$-0.592515\pi$
$79$	−402.439	−0.573139	−0.286569	+	0.958059i	$0.592515\pi$
$80$	231.949	0.324159
$81$	0	0
$82$	−303.611	−0.408880
$83$	1102.37	1.45784	0.728919	−	0.684600i	$-0.240023\pi$
$83$	1102.37	1.45784	0.728919	+	0.684600i	$0.240023\pi$
$84$	0	0
$85$	326.757	0.416962
$86$	87.6143	0.109857
$87$	0	0
$88$	−157.574	−0.190881
$89$	−1257.27	−1.49742	−0.748710	−	0.662898i	$-0.769326\pi$
$89$	−1257.27	−1.49742	−0.748710	+	0.662898i	$0.769326\pi$
$90$	0	0
$91$	300.706	0.346401
$92$	−648.499	−0.734898
$93$	0	0
$94$	63.9888	0.0702121
$95$	754.591	0.814941
$96$	0	0
$97$	−999.825	−1.04657	−0.523283	−	0.852159i	$-0.675293\pi$
$97$	−999.825	−1.04657	−0.523283	+	0.852159i	$0.675293\pi$
$98$	−46.4848	−0.0479151
$99$	0	0
$100$	682.921	0.682921
$101$	−658.047	−0.648298	−0.324149	−	0.946006i	$-0.605078\pi$
$101$	−658.047	−0.648298	−0.324149	+	0.946006i	$0.605078\pi$
$102$	0	0
$103$	367.182	0.351258	0.175629	−	0.984456i	$-0.443804\pi$
$103$	367.182	0.351258	0.175629	+	0.984456i	$0.443804\pi$
$104$	−615.371	−0.580212
$105$	0	0
$106$	−233.409	−0.213875
$107$	−2004.27	−1.81084	−0.905422	−	0.424513i	$-0.860445\pi$
$107$	−2004.27	−1.81084	−0.905422	+	0.424513i	$0.860445\pi$
$108$	0	0
$109$	58.2798	0.0512128	0.0256064	−	0.999672i	$-0.491848\pi$
$109$	58.2798	0.0512128	0.0256064	+	0.999672i	$0.491848\pi$
$110$	56.0159	0.0485537
$111$	0	0
$112$	−302.474	−0.255188
$113$	−1784.84	−1.48588	−0.742938	−	0.669360i	$-0.766568\pi$
$113$	−1784.84	−1.48588	−0.742938	+	0.669360i	$0.766568\pi$
$114$	0	0
$115$	490.290	0.397563
$116$	1852.88	1.48307
$117$	0	0
$118$	−450.709	−0.351619
$119$	−426.107	−0.328245
$120$	0	0
$121$	121.000	0.0909091
$122$	758.055	0.562550
$123$	0	0
$124$	1842.20	1.33415
$125$	−1187.30	−0.849563
$126$	0	0
$127$	−371.954	−0.259887	−0.129943	−	0.991521i	$-0.541480\pi$
$127$	−371.954	−0.259887	−0.129943	+	0.991521i	$0.541480\pi$
$128$	1432.65	0.989292
$129$	0	0
$130$	218.757	0.147587
$131$	−72.8070	−0.0485586	−0.0242793	−	0.999705i	$-0.507729\pi$
$131$	−72.8070	−0.0485586	−0.0242793	+	0.999705i	$0.507729\pi$
$132$	0	0
$133$	−984.025	−0.641547
$134$	687.789	0.443402
$135$	0	0
$136$	871.995	0.549801
$137$	−1282.01	−0.799483	−0.399742	−	0.916628i	$-0.630900\pi$
$137$	−1282.01	−0.799483	−0.399742	+	0.916628i	$0.630900\pi$
$138$	0	0
$139$	−1372.26	−0.837367	−0.418683	−	0.908132i	$-0.637508\pi$
$139$	−1372.26	−0.837367	−0.418683	+	0.908132i	$0.637508\pi$
$140$	266.785	0.161053
$141$	0	0
$142$	516.417	0.305189
$143$	472.538	0.276333
$144$	0	0
$145$	−1400.85	−0.802304
$146$	550.398	0.311995
$147$	0	0
$148$	−2555.99	−1.41960
$149$	248.458	0.136607	0.0683036	−	0.997665i	$-0.478241\pi$
$149$	248.458	0.136607	0.0683036	+	0.997665i	$0.478241\pi$
$150$	0	0
$151$	−1247.52	−0.672327	−0.336164	−	0.941804i	$-0.609130\pi$
$151$	−1247.52	−0.672327	−0.336164	+	0.941804i	$0.609130\pi$
$152$	2013.73	1.07457
$153$	0	0
$154$	−73.0476	−0.0382230
$155$	−1392.77	−0.721742
$156$	0	0
$157$	845.901	0.430002	0.215001	−	0.976614i	$-0.431025\pi$
$157$	845.901	0.430002	0.215001	+	0.976614i	$0.431025\pi$
$158$	381.782	0.192234
$159$	0	0
$160$	−835.201	−0.412678
$161$	−639.363	−0.312974
$162$	0	0
$163$	−3266.85	−1.56981	−0.784906	−	0.619615i	$-0.787289\pi$
$163$	−3266.85	−1.56981	−0.784906	+	0.619615i	$0.787289\pi$
$164$	−2272.28	−1.08192
$165$	0	0
$166$	−1045.78	−0.488967
$167$	527.799	0.244565	0.122282	−	0.992495i	$-0.460979\pi$
$167$	527.799	0.244565	0.122282	+	0.992495i	$0.460979\pi$
$168$	0	0
$169$	−351.613	−0.160043
$170$	−309.984	−0.139851
$171$	0	0
$172$	655.722	0.290688
$173$	−27.8340	−0.0122322	−0.00611612	−	0.999981i	$-0.501947\pi$
$173$	−27.8340	−0.0122322	−0.00611612	+	0.999981i	$0.501947\pi$
$174$	0	0
$175$	673.300	0.290838
$176$	−475.316	−0.203570
$177$	0	0
$178$	1192.73	0.502243
$179$	−1622.80	−0.677620	−0.338810	−	0.940855i	$-0.610024\pi$
$179$	−1622.80	−0.677620	−0.338810	+	0.940855i	$0.610024\pi$
$180$	0	0
$181$	4440.94	1.82372	0.911859	−	0.410504i	$-0.134647\pi$
$181$	4440.94	1.82372	0.911859	+	0.410504i	$0.134647\pi$
$182$	−285.271	−0.116185
$183$	0	0
$184$	1308.40	0.524222
$185$	1932.43	0.767973
$186$	0	0
$187$	−669.597	−0.261849
$188$	478.904	0.185785
$189$	0	0
$190$	−715.858	−0.273336
$191$	1118.93	0.423889	0.211945	−	0.977282i	$-0.432020\pi$
$191$	1118.93	0.423889	0.211945	+	0.977282i	$0.432020\pi$
$192$	0	0
$193$	321.905	0.120058	0.0600291	−	0.998197i	$-0.480881\pi$
$193$	321.905	0.120058	0.0600291	+	0.998197i	$0.480881\pi$
$194$	948.504	0.351024
$195$	0	0
$196$	−347.901	−0.126786
$197$	2409.72	0.871498	0.435749	−	0.900068i	$-0.356483\pi$
$197$	2409.72	0.871498	0.435749	+	0.900068i	$0.356483\pi$
$198$	0	0
$199$	1702.46	0.606454	0.303227	−	0.952918i	$-0.401936\pi$
$199$	1702.46	0.606454	0.303227	+	0.952918i	$0.401936\pi$
$200$	−1377.86	−0.487146
$201$	0	0
$202$	624.270	0.217443
$203$	1826.78	0.631599
$204$	0	0
$205$	1717.93	0.585295
$206$	−348.335	−0.117814
$207$	0	0
$208$	−1856.24	−0.618783
$209$	−1546.33	−0.511778
$210$	0	0
$211$	−2319.86	−0.756899	−0.378449	−	0.925622i	$-0.623543\pi$
$211$	−2319.86	−0.756899	−0.378449	+	0.925622i	$0.623543\pi$
$212$	−1746.88	−0.565925
$213$	0	0
$214$	1901.39	0.607367
$215$	−495.751	−0.157255
$216$	0	0
$217$	1816.24	0.568178
$218$	−55.2883	−0.0171770
$219$	0	0
$220$	419.234	0.128476
$221$	−2614.96	−0.795932
$222$	0	0
$223$	791.377	0.237644	0.118822	−	0.992916i	$-0.462088\pi$
$223$	791.377	0.237644	0.118822	+	0.992916i	$0.462088\pi$
$224$	1089.14	0.324873
$225$	0	0
$226$	1693.23	0.498371
$227$	1345.34	0.393363	0.196681	−	0.980467i	$-0.436984\pi$
$227$	1345.34	0.393363	0.196681	+	0.980467i	$0.436984\pi$
$228$	0	0
$229$	1209.53	0.349030	0.174515	−	0.984655i	$-0.444164\pi$
$229$	1209.53	0.349030	0.174515	+	0.984655i	$0.444164\pi$
$230$	−465.123	−0.133345
$231$	0	0
$232$	−3738.36	−1.05791
$233$	5101.45	1.43436	0.717182	−	0.696886i	$-0.245431\pi$
$233$	5101.45	1.43436	0.717182	+	0.696886i	$0.245431\pi$
$234$	0	0
$235$	−362.070	−0.100506
$236$	−3373.19	−0.930406
$237$	0	0
$238$	404.235	0.110095
$239$	−2072.93	−0.561032	−0.280516	−	0.959849i	$-0.590506\pi$
$239$	−2072.93	−0.561032	−0.280516	+	0.959849i	$0.590506\pi$
$240$	0	0
$241$	2896.10	0.774085	0.387042	−	0.922062i	$-0.373497\pi$
$241$	2896.10	0.774085	0.387042	+	0.922062i	$0.373497\pi$
$242$	−114.789	−0.0304914
$243$	0	0
$244$	5673.42	1.48854
$245$	263.027	0.0685884
$246$	0	0
$247$	−6038.82	−1.55563
$248$	−3716.80	−0.951681
$249$	0	0
$250$	1126.36	0.284948
$251$	−4226.07	−1.06274	−0.531369	−	0.847140i	$-0.678322\pi$
$251$	−4226.07	−1.06274	−0.531369	+	0.847140i	$0.678322\pi$
$252$	0	0
$253$	−1004.71	−0.249667
$254$	352.862	0.0871674
$255$	0	0
$256$	225.518	0.0550582
$257$	5075.41	1.23189	0.615945	−	0.787789i	$-0.288775\pi$
$257$	5075.41	1.23189	0.615945	+	0.787789i	$0.288775\pi$
$258$	0	0
$259$	−2519.99	−0.604573
$260$	1637.22	0.390523
$261$	0	0
$262$	69.0699	0.0162868
$263$	123.204	0.0288861	0.0144431	−	0.999896i	$-0.495402\pi$
$263$	123.204	0.0288861	0.0144431	+	0.999896i	$0.495402\pi$
$264$	0	0
$265$	1320.71	0.306152
$266$	933.516	0.215179
$267$	0	0
$268$	5147.54	1.17327
$269$	3938.48	0.892688	0.446344	−	0.894861i	$-0.352726\pi$
$269$	3938.48	0.892688	0.446344	+	0.894861i	$0.352726\pi$
$270$	0	0
$271$	−2882.96	−0.646227	−0.323113	−	0.946360i	$-0.604730\pi$
$271$	−2882.96	−0.646227	−0.323113	+	0.946360i	$0.604730\pi$
$272$	2630.33	0.586350
$273$	0	0
$274$	1216.20	0.268151
$275$	1058.04	0.232009
$276$	0	0
$277$	1414.61	0.306843	0.153422	−	0.988161i	$-0.450971\pi$
$277$	1414.61	0.306843	0.153422	+	0.988161i	$0.450971\pi$
$278$	1301.83	0.280858
$279$	0	0
$280$	−538.263	−0.114883
$281$	−2053.20	−0.435886	−0.217943	−	0.975962i	$-0.569935\pi$
$281$	−2053.20	−0.435886	−0.217943	+	0.975962i	$0.569935\pi$
$282$	0	0
$283$	−7989.40	−1.67816	−0.839082	−	0.544005i	$-0.816907\pi$
$283$	−7989.40	−1.67816	−0.839082	+	0.544005i	$0.816907\pi$
$284$	3864.96	0.807547
$285$	0	0
$286$	−448.282	−0.0926836
$287$	−2240.27	−0.460762
$288$	0	0
$289$	−1207.54	−0.245785
$290$	1328.94	0.269097
$291$	0	0
$292$	4119.28	0.825557
$293$	−493.055	−0.0983092	−0.0491546	−	0.998791i	$-0.515653\pi$
$293$	−493.055	−0.0983092	−0.0491546	+	0.998791i	$0.515653\pi$
$294$	0	0
$295$	2550.26	0.503328
$296$	5156.95	1.01264
$297$	0	0
$298$	−235.705	−0.0458188
$299$	−3923.67	−0.758903
$300$	0	0
$301$	646.484	0.123796
$302$	1183.48	0.225502
$303$	0	0
$304$	6074.32	1.14601
$305$	−4289.32	−0.805265
$306$	0	0
$307$	−4869.60	−0.905287	−0.452643	−	0.891692i	$-0.649519\pi$
$307$	−4869.60	−0.905287	−0.452643	+	0.891692i	$0.649519\pi$
$308$	−546.702	−0.101140
$309$	0	0
$310$	1321.28	0.242076
$311$	9618.02	1.75366	0.876830	−	0.480801i	$-0.159654\pi$
$311$	9618.02	1.75366	0.876830	+	0.480801i	$0.159654\pi$
$312$	0	0
$313$	−6938.34	−1.25296	−0.626482	−	0.779436i	$-0.715506\pi$
$313$	−6938.34	−1.25296	−0.626482	+	0.779436i	$0.715506\pi$
$314$	−802.481	−0.144225
$315$	0	0
$316$	2857.33	0.508663
$317$	−2353.16	−0.416929	−0.208464	−	0.978030i	$-0.566847\pi$
$317$	−2353.16	−0.416929	−0.208464	+	0.978030i	$0.566847\pi$
$318$	0	0
$319$	2870.65	0.503842
$320$	−1063.26	−0.185745
$321$	0	0
$322$	606.544	0.104973
$323$	8557.15	1.47409
$324$	0	0
$325$	4131.94	0.705228
$326$	3099.16	0.526524
$327$	0	0
$328$	4584.53	0.771763
$329$	472.157	0.0791212
$330$	0	0
$331$	1512.04	0.251086	0.125543	−	0.992088i	$-0.459933\pi$
$331$	1512.04	0.251086	0.125543	+	0.992088i	$0.459933\pi$
$332$	−7826.84	−1.29384
$333$	0	0
$334$	−500.707	−0.0820284
$335$	−3891.74	−0.634711
$336$	0	0
$337$	2506.25	0.405116	0.202558	−	0.979270i	$-0.435074\pi$
$337$	2506.25	0.405116	0.202558	+	0.979270i	$0.435074\pi$
$338$	333.565	0.0536792
$339$	0	0
$340$	−2319.98	−0.370055
$341$	2854.10	0.453249
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−1322.98	−0.207355
$345$	0	0
$346$	26.4053	0.00410276
$347$	6737.01	1.04225	0.521126	−	0.853479i	$-0.325512\pi$
$347$	6737.01	1.04225	0.521126	+	0.853479i	$0.325512\pi$
$348$	0	0
$349$	1832.13	0.281008	0.140504	−	0.990080i	$-0.455128\pi$
$349$	1832.13	0.281008	0.140504	+	0.990080i	$0.455128\pi$
$350$	−638.740	−0.0975488
$351$	0	0
$352$	1711.51	0.259159
$353$	−3946.57	−0.595056	−0.297528	−	0.954713i	$-0.596162\pi$
$353$	−3946.57	−0.595056	−0.297528	+	0.954713i	$0.596162\pi$
$354$	0	0
$355$	−2922.06	−0.436864
$356$	8926.65	1.32896
$357$	0	0
$358$	1539.50	0.227277
$359$	−5598.84	−0.823108	−0.411554	−	0.911385i	$-0.635014\pi$
$359$	−5598.84	−0.823108	−0.411554	+	0.911385i	$0.635014\pi$
$360$	0	0
$361$	12902.3	1.88108
$362$	−4212.99	−0.611685
$363$	0	0
$364$	−2135.02	−0.307432
$365$	−3114.33	−0.446607
$366$	0	0
$367$	8179.50	1.16340	0.581698	−	0.813405i	$-0.302388\pi$
$367$	8179.50	1.16340	0.581698	+	0.813405i	$0.302388\pi$
$368$	3946.74	0.559071
$369$	0	0
$370$	−1833.24	−0.257583
$371$	−1722.27	−0.241013
$372$	0	0
$373$	11627.3	1.61405	0.807025	−	0.590518i	$-0.201076\pi$
$373$	11627.3	1.61405	0.807025	+	0.590518i	$0.201076\pi$
$374$	635.227	0.0878257
$375$	0	0
$376$	−966.232	−0.132526
$377$	11210.7	1.53151
$378$	0	0
$379$	1588.59	0.215305	0.107653	−	0.994189i	$-0.465667\pi$
$379$	1588.59	0.215305	0.107653	+	0.994189i	$0.465667\pi$
$380$	−5357.62	−0.723263
$381$	0	0
$382$	−1061.49	−0.142175
$383$	40.4622	0.00539823	0.00269912	−	0.999996i	$-0.499141\pi$
$383$	40.4622	0.00539823	0.00269912	+	0.999996i	$0.499141\pi$
$384$	0	0
$385$	413.327	0.0547146
$386$	−305.382	−0.0402682
$387$	0	0
$388$	7098.78	0.928830
$389$	−6230.86	−0.812126	−0.406063	−	0.913845i	$-0.633099\pi$
$389$	−6230.86	−0.812126	−0.406063	+	0.913845i	$0.633099\pi$
$390$	0	0
$391$	5559.94	0.719126
$392$	701.922	0.0904399
$393$	0	0
$394$	−2286.03	−0.292305
$395$	−2160.25	−0.275175
$396$	0	0
$397$	−3686.24	−0.466013	−0.233007	−	0.972475i	$-0.574856\pi$
$397$	−3686.24	−0.466013	−0.233007	+	0.972475i	$0.574856\pi$
$398$	−1615.08	−0.203408
$399$	0	0
$400$	−4156.24	−0.519530
$401$	−438.764	−0.0546405	−0.0273202	−	0.999627i	$-0.508697\pi$
$401$	−438.764	−0.0546405	−0.0273202	+	0.999627i	$0.508697\pi$
$402$	0	0
$403$	11146.0	1.37772
$404$	4672.15	0.575367
$405$	0	0
$406$	−1733.01	−0.211842
$407$	−3959.98	−0.482282
$408$	0	0
$409$	−6813.22	−0.823697	−0.411848	−	0.911252i	$-0.635117\pi$
$409$	−6813.22	−0.823697	−0.411848	+	0.911252i	$0.635117\pi$
$410$	−1629.75	−0.196311
$411$	0	0
$412$	−2607.00	−0.311742
$413$	−3325.67	−0.396236
$414$	0	0
$415$	5917.39	0.699935
$416$	6683.92	0.787755
$417$	0	0
$418$	1466.95	0.171653
$419$	−4789.84	−0.558471	−0.279235	−	0.960223i	$-0.590081\pi$
$419$	−4789.84	−0.558471	−0.279235	+	0.960223i	$0.590081\pi$
$420$	0	0
$421$	−5912.30	−0.684436	−0.342218	−	0.939621i	$-0.611178\pi$
$421$	−5912.30	−0.684436	−0.342218	+	0.939621i	$0.611178\pi$
$422$	2200.78	0.253868
$423$	0	0
$424$	3524.48	0.403689
$425$	−5855.06	−0.668264
$426$	0	0
$427$	5593.49	0.633930
$428$	14230.4	1.60713
$429$	0	0
$430$	470.304	0.0527444
$431$	13228.6	1.47842	0.739208	−	0.673477i	$-0.235200\pi$
$431$	13228.6	1.47842	0.739208	+	0.673477i	$0.235200\pi$
$432$	0	0
$433$	−13817.8	−1.53359	−0.766793	−	0.641895i	$-0.778149\pi$
$433$	−13817.8	−1.53359	−0.766793	+	0.641895i	$0.778149\pi$
$434$	−1723.02	−0.190570
$435$	0	0
$436$	−413.788	−0.0454515
$437$	12839.8	1.40551
$438$	0	0
$439$	−5078.63	−0.552141	−0.276070	−	0.961137i	$-0.589032\pi$
$439$	−5078.63	−0.552141	−0.276070	+	0.961137i	$0.589032\pi$
$440$	−845.842	−0.0916453
$441$	0	0
$442$	2480.73	0.266960
$443$	8477.52	0.909208	0.454604	−	0.890694i	$-0.349781\pi$
$443$	8477.52	0.909208	0.454604	+	0.890694i	$0.349781\pi$
$444$	0	0
$445$	−6748.89	−0.718939
$446$	−750.756	−0.0797070
$447$	0	0
$448$	1386.55	0.146224
$449$	−17792.1	−1.87007	−0.935033	−	0.354561i	$-0.884630\pi$
$449$	−17792.1	−1.87007	−0.935033	+	0.354561i	$0.884630\pi$
$450$	0	0
$451$	−3520.42	−0.367561
$452$	12672.4	1.31872
$453$	0	0
$454$	−1276.28	−0.131936
$455$	1614.16	0.166314
$456$	0	0
$457$	−1385.23	−0.141791	−0.0708955	−	0.997484i	$-0.522586\pi$
$457$	−1385.23	−0.141791	−0.0708955	+	0.997484i	$0.522586\pi$
$458$	−1147.44	−0.117067
$459$	0	0
$460$	−3481.07	−0.352838
$461$	9174.59	0.926906	0.463453	−	0.886122i	$-0.346610\pi$
$461$	9174.59	0.926906	0.463453	+	0.886122i	$0.346610\pi$
$462$	0	0
$463$	4066.00	0.408127	0.204064	−	0.978958i	$-0.434585\pi$
$463$	4066.00	0.408127	0.204064	+	0.978958i	$0.434585\pi$
$464$	−11276.6	−1.12824
$465$	0	0
$466$	−4839.59	−0.481094
$467$	−16393.7	−1.62443	−0.812216	−	0.583356i	$-0.801739\pi$
$467$	−16393.7	−1.62443	−0.812216	+	0.583356i	$0.801739\pi$
$468$	0	0
$469$	5075.02	0.499665
$470$	343.485	0.0337102
$471$	0	0
$472$	6805.71	0.663683
$473$	1015.90	0.0987554
$474$	0	0
$475$	−13521.3	−1.30611
$476$	3025.37	0.291319
$477$	0	0
$478$	1966.53	0.188173
$479$	−12328.2	−1.17597	−0.587986	−	0.808871i	$-0.700079\pi$
$479$	−12328.2	−1.17597	−0.587986	+	0.808871i	$0.700079\pi$
$480$	0	0
$481$	−15464.8	−1.46597
$482$	−2747.45	−0.259632
$483$	0	0
$484$	−859.103	−0.0806821
$485$	−5366.95	−0.502476
$486$	0	0
$487$	−1891.56	−0.176006	−0.0880028	−	0.996120i	$-0.528048\pi$
$487$	−1891.56	−0.176006	−0.0880028	+	0.996120i	$0.528048\pi$
$488$	−11446.6	−1.06181
$489$	0	0
$490$	−249.526	−0.0230049
$491$	−484.518	−0.0445336	−0.0222668	−	0.999752i	$-0.507088\pi$
$491$	−484.518	−0.0445336	−0.0222668	+	0.999752i	$0.507088\pi$
$492$	0	0
$493$	−15885.8	−1.45124
$494$	5728.85	0.521767
$495$	0	0
$496$	−11211.5	−1.01495
$497$	3810.51	0.343913
$498$	0	0
$499$	−10923.0	−0.979917	−0.489959	−	0.871746i	$-0.662988\pi$
$499$	−10923.0	−0.979917	−0.489959	+	0.871746i	$0.662988\pi$
$500$	8429.86	0.753990
$501$	0	0
$502$	4009.15	0.356448
$503$	−16.4363	−0.00145697	−0.000728486	−	1.00000i	$-0.500232\pi$
$503$	−16.4363	−0.00145697	−0.000728486		1.00000i	$0.500232\pi$
$504$	0	0
$505$	−3532.32	−0.311260
$506$	953.141	0.0837397
$507$	0	0
$508$	2640.88	0.230650
$509$	−5133.84	−0.447060	−0.223530	−	0.974697i	$-0.571758\pi$
$509$	−5133.84	−0.447060	−0.223530	+	0.974697i	$0.571758\pi$
$510$	0	0
$511$	4061.25	0.351583
$512$	−11675.1	−1.00776
$513$	0	0
$514$	−4814.89	−0.413183
$515$	1970.99	0.168645
$516$	0	0
$517$	741.961	0.0631169
$518$	2390.64	0.202777
$519$	0	0
$520$	−3303.24	−0.278571
$521$	4454.52	0.374580	0.187290	−	0.982305i	$-0.440030\pi$
$521$	4454.52	0.374580	0.187290	+	0.982305i	$0.440030\pi$
$522$	0	0
$523$	−226.348	−0.0189245	−0.00946224	−	0.999955i	$-0.503012\pi$
$523$	−226.348	−0.0189245	−0.00946224	+	0.999955i	$0.503012\pi$
$524$	516.932	0.0430959
$525$	0	0
$526$	−116.880	−0.00968858
$527$	−15794.2	−1.30551
$528$	0	0
$529$	−3824.46	−0.314331
$530$	−1252.91	−0.102685
$531$	0	0
$532$	6986.60	0.569375
$533$	−13748.2	−1.11726
$534$	0	0
$535$	−10758.7	−0.869420
$536$	−10385.6	−0.836923
$537$	0	0
$538$	−3736.32	−0.299413
$539$	−539.000	−0.0430730
$540$	0	0
$541$	11808.2	0.938401	0.469201	−	0.883092i	$-0.344542\pi$
$541$	11808.2	0.938401	0.469201	+	0.883092i	$0.344542\pi$
$542$	2734.98	0.216748
$543$	0	0
$544$	−9471.28	−0.746466
$545$	312.839	0.0245882
$546$	0	0
$547$	11208.4	0.876122	0.438061	−	0.898945i	$-0.355665\pi$
$547$	11208.4	0.876122	0.438061	+	0.898945i	$0.355665\pi$
$548$	9102.28	0.709544
$549$	0	0
$550$	−1003.73	−0.0778171
$551$	−36685.6	−2.83641
$552$	0	0
$553$	2817.08	0.216626
$554$	−1342.00	−0.102917
$555$	0	0
$556$	9743.11	0.743166
$557$	21505.6	1.63594	0.817972	−	0.575259i	$-0.195099\pi$
$557$	21505.6	1.63594	0.817972	+	0.575259i	$0.195099\pi$
$558$	0	0
$559$	3967.38	0.300183
$560$	−1623.65	−0.122521
$561$	0	0
$562$	1947.81	0.146198
$563$	−18770.5	−1.40512	−0.702559	−	0.711626i	$-0.747959\pi$
$563$	−18770.5	−1.40512	−0.702559	+	0.711626i	$0.747959\pi$
$564$	0	0
$565$	−9580.85	−0.713397
$566$	7579.31	0.562866
$567$	0	0
$568$	−7797.91	−0.576044
$569$	19439.7	1.43226	0.716130	−	0.697967i	$-0.245912\pi$
$569$	19439.7	1.43226	0.716130	+	0.697967i	$0.245912\pi$
$570$	0	0
$571$	5978.67	0.438178	0.219089	−	0.975705i	$-0.429692\pi$
$571$	5978.67	0.438178	0.219089	+	0.975705i	$0.429692\pi$
$572$	−3355.03	−0.245246
$573$	0	0
$574$	2125.27	0.154542
$575$	−8785.37	−0.637174
$576$	0	0
$577$	−17704.6	−1.27739	−0.638695	−	0.769460i	$-0.720525\pi$
$577$	−17704.6	−1.27739	−0.638695	+	0.769460i	$0.720525\pi$
$578$	1145.56	0.0824377
$579$	0	0
$580$	9946.06	0.712048
$581$	−7716.57	−0.551011
$582$	0	0
$583$	−2706.42	−0.192262
$584$	−8311.03	−0.588892
$585$	0	0
$586$	467.747	0.0329735
$587$	1063.42	0.0747738	0.0373869	−	0.999301i	$-0.488097\pi$
$587$	1063.42	0.0747738	0.0373869	+	0.999301i	$0.488097\pi$
$588$	0	0
$589$	−36474.1	−2.55159
$590$	−2419.35	−0.168819
$591$	0	0
$592$	15555.7	1.07996
$593$	18456.7	1.27812	0.639061	−	0.769156i	$-0.279323\pi$
$593$	18456.7	1.27812	0.639061	+	0.769156i	$0.279323\pi$
$594$	0	0
$595$	−2287.30	−0.157597
$596$	−1764.06	−0.121239
$597$	0	0
$598$	3722.27	0.254540
$599$	−24953.5	−1.70213	−0.851063	−	0.525064i	$-0.824041\pi$
$599$	−24953.5	−1.70213	−0.851063	+	0.525064i	$0.824041\pi$
$600$	0	0
$601$	16330.2	1.10836	0.554179	−	0.832398i	$-0.313032\pi$
$601$	16330.2	1.10836	0.554179	+	0.832398i	$0.313032\pi$
$602$	−613.300	−0.0415220
$603$	0	0
$604$	8857.39	0.596692
$605$	649.515	0.0436471
$606$	0	0
$607$	−12193.8	−0.815371	−0.407686	−	0.913122i	$-0.633664\pi$
$607$	−12193.8	−0.815371	−0.407686	+	0.913122i	$0.633664\pi$
$608$	−21872.4	−1.45895
$609$	0	0
$610$	4069.15	0.270090
$611$	2897.56	0.191854
$612$	0	0
$613$	18075.5	1.19097	0.595483	−	0.803368i	$-0.296961\pi$
$613$	18075.5	1.19097	0.595483	+	0.803368i	$0.296961\pi$
$614$	4619.65	0.303638
$615$	0	0
$616$	1103.02	0.0721461
$617$	20714.8	1.35162	0.675809	−	0.737077i	$-0.263795\pi$
$617$	20714.8	1.35162	0.675809	+	0.737077i	$0.263795\pi$
$618$	0	0
$619$	−23526.6	−1.52765	−0.763825	−	0.645423i	$-0.776681\pi$
$619$	−23526.6	−1.52765	−0.763825	+	0.645423i	$0.776681\pi$
$620$	9888.70	0.640548
$621$	0	0
$622$	−9124.33	−0.588187
$623$	8800.89	0.565971
$624$	0	0
$625$	5649.92	0.361595
$626$	6582.19	0.420251
$627$	0	0
$628$	−6005.92	−0.381628
$629$	21914.0	1.38914
$630$	0	0
$631$	7890.00	0.497775	0.248887	−	0.968532i	$-0.419935\pi$
$631$	7890.00	0.497775	0.248887	+	0.968532i	$0.419935\pi$
$632$	−5764.92	−0.362842
$633$	0	0
$634$	2232.37	0.139840
$635$	−1996.61	−0.124776
$636$	0	0
$637$	−2104.94	−0.130927
$638$	−2723.30	−0.168991
$639$	0	0
$640$	7690.30	0.474978
$641$	−20817.4	−1.28274	−0.641370	−	0.767232i	$-0.721634\pi$
$641$	−20817.4	−1.28274	−0.641370	+	0.767232i	$0.721634\pi$
$642$	0	0
$643$	−22246.1	−1.36439	−0.682193	−	0.731172i	$-0.738973\pi$
$643$	−22246.1	−1.36439	−0.682193	+	0.731172i	$0.738973\pi$
$644$	4539.49	0.277765
$645$	0	0
$646$	−8117.91	−0.494420
$647$	30769.4	1.86966	0.934829	−	0.355099i	$-0.115553\pi$
$647$	30769.4	1.86966	0.934829	+	0.355099i	$0.115553\pi$
$648$	0	0
$649$	−5226.05	−0.316087
$650$	−3919.85	−0.236537
$651$	0	0
$652$	23194.7	1.39321
$653$	−14927.0	−0.894546	−0.447273	−	0.894397i	$-0.647605\pi$
$653$	−14927.0	−0.894546	−0.447273	+	0.894397i	$0.647605\pi$
$654$	0	0
$655$	−390.820	−0.0233139
$656$	13829.0	0.823068
$657$	0	0
$658$	−447.922	−0.0265377
$659$	11593.6	0.685318	0.342659	−	0.939460i	$-0.388673\pi$
$659$	11593.6	0.685318	0.342659	+	0.939460i	$0.388673\pi$
$660$	0	0
$661$	24733.6	1.45541	0.727704	−	0.685891i	$-0.240587\pi$
$661$	24733.6	1.45541	0.727704	+	0.685891i	$0.240587\pi$
$662$	−1434.43	−0.0842155
$663$	0	0
$664$	15791.4	0.922927
$665$	−5282.14	−0.308019
$666$	0	0
$667$	−23836.2	−1.38372
$668$	−3747.39	−0.217052
$669$	0	0
$670$	3691.97	0.212886
$671$	8789.78	0.505701
$672$	0	0
$673$	−22202.2	−1.27167	−0.635834	−	0.771826i	$-0.719344\pi$
$673$	−22202.2	−1.27167	−0.635834	+	0.771826i	$0.719344\pi$
$674$	−2377.61	−0.135878
$675$	0	0
$676$	2496.46	0.142038
$677$	10445.5	0.592989	0.296494	−	0.955035i	$-0.404182\pi$
$677$	10445.5	0.592989	0.296494	+	0.955035i	$0.404182\pi$
$678$	0	0
$679$	6998.77	0.395565
$680$	4680.77	0.263970
$681$	0	0
$682$	−2707.60	−0.152022
$683$	15600.1	0.873969	0.436985	−	0.899469i	$-0.356046\pi$
$683$	15600.1	0.873969	0.436985	+	0.899469i	$0.356046\pi$
$684$	0	0
$685$	−6881.67	−0.383847
$686$	325.394	0.0181102
$687$	0	0
$688$	−3990.70	−0.221140
$689$	−10569.3	−0.584410
$690$	0	0
$691$	−26622.8	−1.46567	−0.732835	−	0.680406i	$-0.761803\pi$
$691$	−26622.8	−1.46567	−0.732835	+	0.680406i	$0.761803\pi$
$692$	197.622	0.0108562
$693$	0	0
$694$	−6391.20	−0.349578
$695$	−7366.17	−0.402035
$696$	0	0
$697$	19481.5	1.05870
$698$	−1738.09	−0.0942517
$699$	0	0
$700$	−4780.45	−0.258120
$701$	16091.8	0.867017	0.433508	−	0.901150i	$-0.357275\pi$
$701$	16091.8	0.867017	0.433508	+	0.901150i	$0.357275\pi$
$702$	0	0
$703$	50606.7	2.71503
$704$	2178.87	0.116646
$705$	0	0
$706$	3743.99	0.199585
$707$	4606.33	0.245034
$708$	0	0
$709$	58.1652	0.00308101	0.00154051	−	0.999999i	$-0.499510\pi$
$709$	58.1652	0.00308101	0.00154051	+	0.999999i	$0.499510\pi$
$710$	2772.07	0.146527
$711$	0	0
$712$	−18010.3	−0.947985
$713$	−23698.7	−1.24477
$714$	0	0
$715$	2536.53	0.132672
$716$	11521.9	0.601390
$717$	0	0
$718$	5311.46	0.276075
$719$	−2437.03	−0.126406	−0.0632030	−	0.998001i	$-0.520132\pi$
$719$	−2437.03	−0.126406	−0.0632030	+	0.998001i	$0.520132\pi$
$720$	0	0
$721$	−2570.28	−0.132763
$722$	−12240.1	−0.630925
$723$	0	0
$724$	−31530.8	−1.61855
$725$	25101.4	1.28585
$726$	0	0
$727$	−5394.60	−0.275206	−0.137603	−	0.990487i	$-0.543940\pi$
$727$	−5394.60	−0.275206	−0.137603	+	0.990487i	$0.543940\pi$
$728$	4307.59	0.219300
$729$	0	0
$730$	2954.48	0.149795
$731$	−5621.87	−0.284449
$732$	0	0
$733$	−21145.5	−1.06552	−0.532761	−	0.846266i	$-0.678845\pi$
$733$	−21145.5	−1.06552	−0.532761	+	0.846266i	$0.678845\pi$
$734$	−7759.65	−0.390210
$735$	0	0
$736$	−14211.4	−0.711738
$737$	7975.03	0.398594
$738$	0	0
$739$	−37759.5	−1.87957	−0.939787	−	0.341760i	$-0.888977\pi$
$739$	−37759.5	−1.87957	−0.939787	+	0.341760i	$0.888977\pi$
$740$	−13720.3	−0.681579
$741$	0	0
$742$	1633.86	0.0808370
$743$	9418.47	0.465047	0.232524	−	0.972591i	$-0.425302\pi$
$743$	9418.47	0.465047	0.232524	+	0.972591i	$0.425302\pi$
$744$	0	0
$745$	1333.69	0.0655876
$746$	−11030.5	−0.541361
$747$	0	0
$748$	4754.16	0.232392
$749$	14029.9	0.684435
$750$	0	0
$751$	19452.5	0.945185	0.472592	−	0.881281i	$-0.343318\pi$
$751$	19452.5	0.945185	0.472592	+	0.881281i	$0.343318\pi$
$752$	−2914.60	−0.141336
$753$	0	0
$754$	−10635.2	−0.513676
$755$	−6696.53	−0.322797
$756$	0	0
$757$	−5870.88	−0.281877	−0.140938	−	0.990018i	$-0.545012\pi$
$757$	−5870.88	−0.281877	−0.140938	+	0.990018i	$0.545012\pi$
$758$	−1507.05	−0.0722145
$759$	0	0
$760$	10809.5	0.515922
$761$	−9711.24	−0.462591	−0.231296	−	0.972883i	$-0.574296\pi$
$761$	−9711.24	−0.462591	−0.231296	+	0.972883i	$0.574296\pi$
$762$	0	0
$763$	−407.958	−0.0193566
$764$	−7944.42	−0.376203
$765$	0	0
$766$	−38.3853	−0.00181060
$767$	−20409.1	−0.960796
$768$	0	0
$769$	3693.86	0.173217	0.0866085	−	0.996242i	$-0.472397\pi$
$769$	3693.86	0.173217	0.0866085	+	0.996242i	$0.472397\pi$
$770$	−392.112	−0.0183516
$771$	0	0
$772$	−2285.53	−0.106552
$773$	−38031.9	−1.76962	−0.884808	−	0.465955i	$-0.845711\pi$
$773$	−38031.9	−1.76962	−0.884808	+	0.465955i	$0.845711\pi$
$774$	0	0
$775$	24956.7	1.15674
$776$	−14322.4	−0.662559
$777$	0	0
$778$	5911.03	0.272392
$779$	44989.4	2.06921
$780$	0	0
$781$	5987.95	0.274348
$782$	−5274.55	−0.241199
$783$	0	0
$784$	2117.32	0.0964521
$785$	4540.70	0.206452
$786$	0	0
$787$	3509.51	0.158959	0.0794793	−	0.996837i	$-0.474674\pi$
$787$	3509.51	0.158959	0.0794793	+	0.996837i	$0.474674\pi$
$788$	−17109.0	−0.773457
$789$	0	0
$790$	2049.37	0.0922951
$791$	12493.9	0.561608
$792$	0	0
$793$	34326.4	1.53716
$794$	3497.03	0.156303
$795$	0	0
$796$	−12087.5	−0.538230
$797$	−32249.5	−1.43330	−0.716648	−	0.697435i	$-0.754324\pi$
$797$	−32249.5	−1.43330	−0.716648	+	0.697435i	$0.754324\pi$
$798$	0	0
$799$	−4105.91	−0.181798
$800$	14965.7	0.661399
$801$	0	0
$802$	416.242	0.0183267
$803$	6381.97	0.280467
$804$	0	0
$805$	−3432.03	−0.150265
$806$	−10573.9	−0.462096
$807$	0	0
$808$	−9426.49	−0.410424
$809$	10977.6	0.477072	0.238536	−	0.971134i	$-0.423332\pi$
$809$	10977.6	0.477072	0.238536	+	0.971134i	$0.423332\pi$
$810$	0	0
$811$	39624.5	1.71567	0.857833	−	0.513928i	$-0.171810\pi$
$811$	39624.5	1.71567	0.857833	+	0.513928i	$0.171810\pi$
$812$	−12970.2	−0.560546
$813$	0	0
$814$	3756.71	0.161760
$815$	−17536.1	−0.753696
$816$	0	0
$817$	−12982.8	−0.555949
$818$	6463.50	0.276272
$819$	0	0
$820$	−12197.3	−0.519451
$821$	13918.0	0.591647	0.295824	−	0.955243i	$-0.404406\pi$
$821$	13918.0	0.591647	0.295824	+	0.955243i	$0.404406\pi$
$822$	0	0
$823$	8300.77	0.351575	0.175788	−	0.984428i	$-0.443753\pi$
$823$	8300.77	0.351575	0.175788	+	0.984428i	$0.443753\pi$
$824$	5259.87	0.222374
$825$	0	0
$826$	3154.96	0.132900
$827$	−7729.99	−0.325028	−0.162514	−	0.986706i	$-0.551960\pi$
$827$	−7729.99	−0.325028	−0.162514	+	0.986706i	$0.551960\pi$
$828$	0	0
$829$	19095.7	0.800027	0.400014	−	0.916509i	$-0.369005\pi$
$829$	19095.7	0.800027	0.400014	+	0.916509i	$0.369005\pi$
$830$	−5613.65	−0.234762
$831$	0	0
$832$	8509.06	0.354565
$833$	2982.75	0.124065
$834$	0	0
$835$	2833.17	0.117420
$836$	10978.9	0.454204
$837$	0	0
$838$	4543.98	0.187314
$839$	−37318.3	−1.53560	−0.767801	−	0.640688i	$-0.778649\pi$
$839$	−37318.3	−1.53560	−0.767801	+	0.640688i	$0.778649\pi$
$840$	0	0
$841$	43715.4	1.79242
$842$	5608.82	0.229564
$843$	0	0
$844$	16471.0	0.671750
$845$	−1887.42	−0.0768394
$846$	0	0
$847$	−847.000	−0.0343604
$848$	10631.4	0.430525
$849$	0	0
$850$	5554.53	0.224140
$851$	32881.3	1.32451
$852$	0	0
$853$	−2096.82	−0.0841663	−0.0420831	−	0.999114i	$-0.513399\pi$
$853$	−2096.82	−0.0841663	−0.0420831	+	0.999114i	$0.513399\pi$
$854$	−5306.38	−0.212624
$855$	0	0
$856$	−28711.1	−1.14641
$857$	−11671.7	−0.465223	−0.232611	−	0.972570i	$-0.574727\pi$
$857$	−11671.7	−0.465223	−0.232611	+	0.972570i	$0.574727\pi$
$858$	0	0
$859$	−25386.7	−1.00836	−0.504181	−	0.863598i	$-0.668206\pi$
$859$	−25386.7	−1.00836	−0.504181	+	0.863598i	$0.668206\pi$
$860$	3519.84	0.139565
$861$	0	0
$862$	−12549.5	−0.495869
$863$	−2360.78	−0.0931191	−0.0465596	−	0.998916i	$-0.514826\pi$
$863$	−2360.78	−0.0931191	−0.0465596	+	0.998916i	$0.514826\pi$
$864$	0	0
$865$	−149.410	−0.00587293
$866$	13108.6	0.514373
$867$	0	0
$868$	−12895.4	−0.504260
$869$	4426.83	0.172808
$870$	0	0
$871$	31144.7	1.21159
$872$	834.855	0.0324217
$873$	0	0
$874$	−12180.7	−0.471417
$875$	8311.10	0.321105
$876$	0	0
$877$	8504.19	0.327441	0.163721	−	0.986507i	$-0.447650\pi$
$877$	8504.19	0.327441	0.163721	+	0.986507i	$0.447650\pi$
$878$	4817.95	0.185191
$879$	0	0
$880$	−2551.44	−0.0977377
$881$	8050.60	0.307868	0.153934	−	0.988081i	$-0.450806\pi$
$881$	8050.60	0.307868	0.153934	+	0.988081i	$0.450806\pi$
$882$	0	0
$883$	−38768.2	−1.47752	−0.738762	−	0.673966i	$-0.764589\pi$
$883$	−38768.2	−1.47752	−0.738762	+	0.673966i	$0.764589\pi$
$884$	18566.3	0.706392
$885$	0	0
$886$	−8042.37	−0.304953
$887$	−41925.4	−1.58705	−0.793526	−	0.608536i	$-0.791757\pi$
$887$	−41925.4	−1.58705	−0.793526	+	0.608536i	$0.791757\pi$
$888$	0	0
$889$	2603.68	0.0982279
$890$	6402.47	0.241136
$891$	0	0
$892$	−5618.79	−0.210909
$893$	−9481.93	−0.355320
$894$	0	0
$895$	−8711.02	−0.325338
$896$	−10028.5	−0.373917
$897$	0	0
$898$	16878.8	0.627231
$899$	67711.7	2.51203
$900$	0	0
$901$	14977.0	0.553779
$902$	3339.72	0.123282
$903$	0	0
$904$	−25567.8	−0.940677
$905$	23838.5	0.875601
$906$	0	0
$907$	−12962.1	−0.474533	−0.237266	−	0.971445i	$-0.576251\pi$
$907$	−12962.1	−0.474533	−0.237266	+	0.971445i	$0.576251\pi$
$908$	−9551.95	−0.349111
$909$	0	0
$910$	−1531.30	−0.0557826
$911$	−29894.3	−1.08720	−0.543602	−	0.839343i	$-0.682940\pi$
$911$	−29894.3	−1.08720	−0.543602	+	0.839343i	$0.682940\pi$
$912$	0	0
$913$	−12126.0	−0.439555
$914$	1314.13	0.0475575
$915$	0	0
$916$	−8587.68	−0.309765
$917$	509.649	0.0183534
$918$	0	0
$919$	1297.57	0.0465754	0.0232877	−	0.999729i	$-0.492587\pi$
$919$	1297.57	0.0465754	0.0232877	+	0.999729i	$0.492587\pi$
$920$	7023.37	0.251689
$921$	0	0
$922$	−8703.67	−0.310889
$923$	23384.6	0.833924
$924$	0	0
$925$	−34626.7	−1.23083
$926$	−3857.29	−0.136888
$927$	0	0
$928$	40604.6	1.43633
$929$	38299.3	1.35259	0.676297	−	0.736629i	$-0.263584\pi$
$929$	38299.3	1.35259	0.676297	+	0.736629i	$0.263584\pi$
$930$	0	0
$931$	6888.18	0.242482
$932$	−36220.4	−1.27300
$933$	0	0
$934$	15552.2	0.544844
$935$	−3594.32	−0.125719
$936$	0	0
$937$	−17601.2	−0.613667	−0.306834	−	0.951763i	$-0.599269\pi$
$937$	−17601.2	−0.613667	−0.306834	+	0.951763i	$0.599269\pi$
$938$	−4814.52	−0.167590
$939$	0	0
$940$	2570.70	0.0891990
$941$	−39385.4	−1.36443	−0.682214	−	0.731152i	$-0.738983\pi$
$941$	−39385.4	−1.36443	−0.682214	+	0.731152i	$0.738983\pi$
$942$	0	0
$943$	29231.5	1.00945
$944$	20529.1	0.707803
$945$	0	0
$946$	−963.757	−0.0331231
$947$	−42471.2	−1.45737	−0.728685	−	0.684849i	$-0.759868\pi$
$947$	−42471.2	−1.45737	−0.728685	+	0.684849i	$0.759868\pi$
$948$	0	0
$949$	24923.3	0.852523
$950$	12827.3	0.438075
$951$	0	0
$952$	−6103.96	−0.207805
$953$	39650.2	1.34774	0.673870	−	0.738850i	$-0.264631\pi$
$953$	39650.2	1.34774	0.673870	+	0.738850i	$0.264631\pi$
$954$	0	0
$955$	6006.28	0.203517
$956$	14717.9	0.497918
$957$	0	0
$958$	11695.4	0.394428
$959$	8974.05	0.302176
$960$	0	0
$961$	37530.2	1.25978
$962$	14671.0	0.491696
$963$	0	0
$964$	−20562.4	−0.687003
$965$	1727.95	0.0576421
$966$	0	0
$967$	−48452.2	−1.61129	−0.805644	−	0.592399i	$-0.798181\pi$
$967$	−48452.2	−1.61129	−0.805644	+	0.592399i	$0.798181\pi$
$968$	1733.32	0.0575527
$969$	0	0
$970$	5091.47	0.168533
$971$	10531.1	0.348051	0.174026	−	0.984741i	$-0.444322\pi$
$971$	10531.1	0.348051	0.174026	+	0.984741i	$0.444322\pi$
$972$	0	0
$973$	9605.85	0.316495
$974$	1794.47	0.0590333
$975$	0	0
$976$	−34528.3	−1.13240
$977$	−49108.4	−1.60810	−0.804052	−	0.594559i	$-0.797327\pi$
$977$	−49108.4	−1.60810	−0.804052	+	0.594559i	$0.797327\pi$
$978$	0	0
$979$	13830.0	0.451489
$980$	−1867.50	−0.0608724
$981$	0	0
$982$	459.648	0.0149368
$983$	2280.32	0.0739888	0.0369944	−	0.999315i	$-0.488222\pi$
$983$	2280.32	0.0739888	0.0369944	+	0.999315i	$0.488222\pi$
$984$	0	0
$985$	12935.1	0.418423
$986$	15070.4	0.486753
$987$	0	0
$988$	42875.7	1.38063
$989$	−8435.46	−0.271215
$990$	0	0
$991$	−15807.2	−0.506691	−0.253345	−	0.967376i	$-0.581531\pi$
$991$	−15807.2	−0.506691	−0.253345	+	0.967376i	$0.581531\pi$
$992$	40370.4	1.29210
$993$	0	0
$994$	−3614.92	−0.115350
$995$	9138.63	0.291170
$996$	0	0
$997$	51827.2	1.64632	0.823161	−	0.567808i	$-0.192208\pi$
$997$	51827.2	1.64632	0.823161	+	0.567808i	$0.192208\pi$
$998$	10362.3	0.328670
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.m.1.2		4
3.2	odd	2		77.4.a.c.1.3	✓	4
12.11	even	2		1232.4.a.w.1.2		4
15.14	odd	2		1925.4.a.q.1.2		4
21.20	even	2		539.4.a.f.1.3		4
33.32	even	2		847.4.a.e.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.c.1.3	✓	4	3.2	odd	2
539.4.a.f.1.3		4	21.20	even	2
693.4.a.m.1.2		4	1.1	even	1	trivial
847.4.a.e.1.2		4	33.32	even	2
1232.4.a.w.1.2		4	12.11	even	2
1925.4.a.q.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:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

\(f(q)\)	\(=\)	\(q-0.948670 q^{2} -7.10002 q^{4} +5.36789 q^{5} -7.00000 q^{7} +14.3249 q^{8} +O(q^{10})\)	\(q-0.948670 q^{2} -7.10002 q^{4} +5.36789 q^{5} -7.00000 q^{7} +14.3249 q^{8} -5.09236 q^{10} -11.0000 q^{11} -42.9580 q^{13} +6.64069 q^{14} +43.2105 q^{16} +60.8725 q^{17} +140.575 q^{19} -38.1121 q^{20} +10.4354 q^{22} +91.3375 q^{23} -96.1858 q^{25} +40.7529 q^{26} +49.7002 q^{28} -260.968 q^{29} -259.463 q^{31} -155.592 q^{32} -57.7479 q^{34} -37.5752 q^{35} +359.998 q^{37} -133.359 q^{38} +76.8947 q^{40} +320.038 q^{41} -92.3549 q^{43} +78.1003 q^{44} -86.6492 q^{46} -67.4510 q^{47} +49.0000 q^{49} +91.2486 q^{50} +305.003 q^{52} +246.038 q^{53} -59.0468 q^{55} -100.275 q^{56} +247.573 q^{58} +475.095 q^{59} -799.071 q^{61} +246.145 q^{62} -198.079 q^{64} -230.594 q^{65} -725.003 q^{67} -432.196 q^{68} +35.6465 q^{70} -544.359 q^{71} -580.179 q^{73} -341.519 q^{74} -998.086 q^{76} +77.0000 q^{77} -402.439 q^{79} +231.949 q^{80} -303.611 q^{82} +1102.37 q^{83} +326.757 q^{85} +87.6143 q^{86} -157.574 q^{88} -1257.27 q^{89} +300.706 q^{91} -648.499 q^{92} +63.9888 q^{94} +754.591 q^{95} -999.825 q^{97} -46.4848 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 22 q^{4} + 18 q^{5} - 28 q^{7} + 60 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 22 * q^4 + 18 * q^5 - 28 * q^7 + 60 * q^8	\( 4 q + 4 q^{2} + 22 q^{4} + 18 q^{5} - 28 q^{7} + 60 q^{8} - 92 q^{10} - 44 q^{11} - 134 q^{13} - 28 q^{14} - 6 q^{16} + 74 q^{17} - 164 q^{19} - 116 q^{20} - 44 q^{22} - 194 q^{23} + 38 q^{25} - 734 q^{26} - 154 q^{28} + 108 q^{29} - 412 q^{31} + 4 q^{32} - 346 q^{34} - 126 q^{35} + 286 q^{37} - 224 q^{38} - 540 q^{40} + 18 q^{41} - 496 q^{43} - 242 q^{44} - 284 q^{46} - 62 q^{47} + 196 q^{49} - 212 q^{50} - 822 q^{52} + 828 q^{53} - 198 q^{55} - 420 q^{56} + 1388 q^{58} + 1224 q^{59} - 350 q^{61} + 878 q^{62} - 718 q^{64} + 396 q^{65} - 1498 q^{67} - 1058 q^{68} + 644 q^{70} - 2326 q^{71} - 1630 q^{73} + 1156 q^{74} - 3152 q^{76} + 308 q^{77} - 1020 q^{79} - 3072 q^{80} + 2118 q^{82} + 1920 q^{83} + 2008 q^{85} - 1056 q^{86} - 660 q^{88} - 1550 q^{89} + 938 q^{91} - 2592 q^{92} - 1042 q^{94} - 2332 q^{95} - 2202 q^{97} + 196 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 22 * q^4 + 18 * q^5 - 28 * q^7 + 60 * q^8 - 92 * q^10 - 44 * q^11 - 134 * q^13 - 28 * q^14 - 6 * q^16 + 74 * q^17 - 164 * q^19 - 116 * q^20 - 44 * q^22 - 194 * q^23 + 38 * q^25 - 734 * q^26 - 154 * q^28 + 108 * q^29 - 412 * q^31 + 4 * q^32 - 346 * q^34 - 126 * q^35 + 286 * q^37 - 224 * q^38 - 540 * q^40 + 18 * q^41 - 496 * q^43 - 242 * q^44 - 284 * q^46 - 62 * q^47 + 196 * q^49 - 212 * q^50 - 822 * q^52 + 828 * q^53 - 198 * q^55 - 420 * q^56 + 1388 * q^58 + 1224 * q^59 - 350 * q^61 + 878 * q^62 - 718 * q^64 + 396 * q^65 - 1498 * q^67 - 1058 * q^68 + 644 * q^70 - 2326 * q^71 - 1630 * q^73 + 1156 * q^74 - 3152 * q^76 + 308 * q^77 - 1020 * q^79 - 3072 * q^80 + 2118 * q^82 + 1920 * q^83 + 2008 * q^85 - 1056 * q^86 - 660 * q^88 - 1550 * q^89 + 938 * q^91 - 2592 * q^92 - 1042 * q^94 - 2332 * q^95 - 2202 * q^97 + 196 * q^98

Introduction

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