LMFDB - Embedded newform 847.4.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [847,4,Mod(1,847)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(847, 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("847.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 847 = 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	847.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:	$49.9746177749$
Analytic rank:	$0$
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:	$\Z[a_1, a_2, a_3]$
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	$\chi$	$=$	847.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.948670 q^{2} -0.163384 q^{3} -7.10002 q^{4} -5.36789 q^{5} +0.154997 q^{6} +7.00000 q^{7} +14.3249 q^{8} -26.9733 q^{9} +5.09236 q^{10} +1.16003 q^{12} +42.9580 q^{13} -6.64069 q^{14} +0.877027 q^{15} +43.2105 q^{16} +60.8725 q^{17} +25.5888 q^{18} -140.575 q^{19} +38.1121 q^{20} -1.14369 q^{21} -91.3375 q^{23} -2.34047 q^{24} -96.1858 q^{25} -40.7529 q^{26} +8.81837 q^{27} -49.7002 q^{28} -260.968 q^{29} -0.832009 q^{30} -259.463 q^{31} -155.592 q^{32} -57.7479 q^{34} -37.5752 q^{35} +191.511 q^{36} +359.998 q^{37} +133.359 q^{38} -7.01864 q^{39} -76.8947 q^{40} +320.038 q^{41} +1.08498 q^{42} +92.3549 q^{43} +144.790 q^{45} +86.6492 q^{46} +67.4510 q^{47} -7.05991 q^{48} +49.0000 q^{49} +91.2486 q^{50} -9.94558 q^{51} -305.003 q^{52} -246.038 q^{53} -8.36573 q^{54} +100.275 q^{56} +22.9677 q^{57} +247.573 q^{58} -475.095 q^{59} -6.22691 q^{60} +799.071 q^{61} +246.145 q^{62} -188.813 q^{63} -198.079 q^{64} -230.594 q^{65} -725.003 q^{67} -432.196 q^{68} +14.9231 q^{69} +35.6465 q^{70} +544.359 q^{71} -386.391 q^{72} +580.179 q^{73} -341.519 q^{74} +15.7152 q^{75} +998.086 q^{76} +6.65838 q^{78} +402.439 q^{79} -231.949 q^{80} +726.838 q^{81} -303.611 q^{82} +1102.37 q^{83} +8.12021 q^{84} -326.757 q^{85} -87.6143 q^{86} +42.6380 q^{87} +1257.27 q^{89} -137.358 q^{90} +300.706 q^{91} +648.499 q^{92} +42.3921 q^{93} -63.9888 q^{94} +754.591 q^{95} +25.4212 q^{96} -999.825 q^{97} -46.4848 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 12 q^{3} + 22 q^{4} - 18 q^{5} - 2 q^{6} + 28 q^{7} + 60 q^{8} + 66 q^{9} + 92 q^{10} - 186 q^{12} + 134 q^{13} + 28 q^{14} - 62 q^{15} - 6 q^{16} + 74 q^{17} + 256 q^{18} + 164 q^{19} + 116 q^{20}+ \cdots + 196 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 - 12 * q^3 + 22 * q^4 - 18 * q^5 - 2 * q^6 + 28 * q^7 + 60 * q^8 + 66 * q^9 + 92 * q^10 - 186 * q^12 + 134 * q^13 + 28 * q^14 - 62 * q^15 - 6 * q^16 + 74 * q^17 + 256 * q^18 + 164 * q^19 + 116 * q^20 - 84 * q^21 + 194 * q^23 - 570 * q^24 + 38 * q^25 + 734 * q^26 - 510 * q^27 + 154 * q^28 + 108 * q^29 - 1252 * q^30 - 412 * q^31 + 4 * q^32 - 346 * q^34 - 126 * q^35 + 1518 * q^36 + 286 * q^37 + 224 * q^38 + 256 * q^39 + 540 * q^40 + 18 * q^41 - 14 * q^42 + 496 * q^43 + 580 * q^45 + 284 * q^46 + 62 * q^47 - 862 * q^48 + 196 * q^49 - 212 * q^50 + 508 * q^51 + 822 * q^52 - 828 * q^53 - 2420 * q^54 + 420 * q^56 - 700 * q^57 + 1388 * q^58 - 1224 * q^59 - 1776 * q^60 + 350 * q^61 + 878 * q^62 + 462 * q^63 - 718 * q^64 + 396 * q^65 - 1498 * q^67 - 1058 * q^68 - 386 * q^69 + 644 * q^70 + 2326 * q^71 + 3000 * q^72 + 1630 * q^73 + 1156 * q^74 - 1362 * q^75 + 3152 * q^76 - 2464 * q^78 + 1020 * q^79 + 3072 * q^80 + 1128 * q^81 + 2118 * q^82 + 1920 * q^83 - 1302 * q^84 - 2008 * q^85 + 1056 * q^86 - 1640 * q^87 + 1550 * q^89 + 5780 * q^90 + 938 * q^91 + 2592 * q^92 + 6046 * q^93 + 1042 * q^94 - 2332 * q^95 - 4082 * q^96 - 2202 * q^97 + 196 * 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 $

$2$	−0.948670	−0.335406	−0.167703	−	0.985838i	$-0.553635\pi$
$2$	−0.948670	−0.335406	−0.167703	+	0.985838i	$0.553635\pi$
$3$	−0.163384	−0.0314432	−0.0157216	−	0.999876i	$-0.505005\pi$
$3$	−0.163384	−0.0314432	−0.0157216	+	0.999876i	$0.505005\pi$
$4$	−7.10002	−0.887503
$5$	−5.36789	−0.480119	−0.240059	−	0.970758i	$-0.577167\pi$
$5$	−5.36789	−0.480119	−0.240059	+	0.970758i	$0.577167\pi$
$6$	0.154997	0.0105462
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	14.3249	0.633079
$9$	−26.9733	−0.999011
$10$	5.09236	0.161034
$11$	0	0
$11$	0	0
$12$	1.16003	0.0279060
$13$	42.9580	0.916492	0.458246	−	0.888825i	$-0.348478\pi$
$13$	42.9580	0.916492	0.458246	+	0.888825i	$0.348478\pi$
$14$	−6.64069	−0.126771
$15$	0.877027	0.0150965
$16$	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	$0.357021\pi$
$18$	25.5888	0.335074
$19$	−140.575	−1.69737	−0.848687	−	0.528895i	$-0.822607\pi$
$19$	−140.575	−1.69737	−0.848687	+	0.528895i	$0.822607\pi$
$20$	38.1121	0.426107
$21$	−1.14369	−0.0118844
$22$	0	0
$23$	−91.3375	−0.828052	−0.414026	−	0.910265i	$-0.635878\pi$
$23$	−91.3375	−0.828052	−0.414026	+	0.910265i	$0.635878\pi$
$24$	−2.34047	−0.0199061
$25$	−96.1858	−0.769486
$26$	−40.7529	−0.307397
$27$	8.81837	0.0628554
$28$	−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	$0.814838\pi$
$30$	−0.832009	−0.00506345
$31$	−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
$36$	191.511	0.886626
$37$	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
$39$	−7.01864	−0.0288175
$40$	−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	$0.291358\pi$
$42$	1.08498	0.00398610
$43$	92.3549	0.327535	0.163767	−	0.986499i	$-0.447635\pi$
$43$	92.3549	0.327535	0.163767	+	0.986499i	$0.447635\pi$
$44$	0	0
$45$	144.790	0.479644
$46$	86.6492	0.277733
$47$	67.4510	0.209335	0.104667	−	0.994507i	$-0.466622\pi$
$47$	67.4510	0.209335	0.104667	+	0.994507i	$0.466622\pi$
$48$	−7.05991	−0.0212294
$49$	49.0000	0.142857
$50$	91.2486	0.258090
$51$	−9.94558	−0.0273071
$52$	−305.003	−0.813389
$53$	−246.038	−0.637659	−0.318830	−	0.947812i	$-0.603290\pi$
$53$	−246.038	−0.637659	−0.318830	+	0.947812i	$0.603290\pi$
$54$	−8.36573	−0.0210821
$55$	0	0
$56$	100.275	0.239281
$57$	22.9677	0.0533710
$58$	247.573	0.560481
$59$	−475.095	−1.04834	−0.524170	−	0.851613i	$-0.675625\pi$
$59$	−475.095	−1.04834	−0.524170	+	0.851613i	$0.675625\pi$
$60$	−6.22691	−0.0133982
$61$	799.071	1.67722	0.838611	−	0.544731i	$-0.183368\pi$
$61$	799.071	1.67722	0.838611	+	0.544731i	$0.183368\pi$
$62$	246.145	0.504201
$63$	−188.813	−0.377591
$64$	−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
$69$	14.9231	0.0260366
$70$	35.6465	0.0608653
$71$	544.359	0.909909	0.454954	−	0.890515i	$-0.349656\pi$
$71$	544.359	0.909909	0.454954	+	0.890515i	$0.349656\pi$
$72$	−386.391	−0.632453
$73$	580.179	0.930202	0.465101	−	0.885258i	$-0.346018\pi$
$73$	580.179	0.930202	0.465101	+	0.885258i	$0.346018\pi$
$74$	−341.519	−0.536498
$75$	15.7152	0.0241951
$76$	998.086	1.50643
$77$	0	0
$78$	6.65838	0.00966555
$79$	402.439	0.573139	0.286569	−	0.958059i	$-0.407485\pi$
$79$	402.439	0.573139	0.286569	+	0.958059i	$0.407485\pi$
$80$	−231.949	−0.324159
$81$	726.838	0.997035
$82$	−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	$0.240023\pi$
$84$	8.12021	0.0105475
$85$	−326.757	−0.416962
$86$	−87.6143	−0.109857
$87$	42.6380	0.0525434
$88$	0	0
$89$	1257.27	1.49742	0.748710	−	0.662898i	$-0.230674\pi$
$89$	1257.27	1.49742	0.748710	+	0.662898i	$0.230674\pi$
$90$	−137.358	−0.160875
$91$	300.706	0.346401
$92$	648.499	0.734898
$93$	42.3921	0.0472673
$94$	−63.9888	−0.0702121
$95$	754.591	0.814941
$96$	25.4212	0.0270265
$97$	−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	$0.605078\pi$
$102$	9.43508	0.00915894
$103$	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
$105$	6.13919	0.00570594
$106$	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	$0.860445\pi$
$108$	−62.6106	−0.0557844
$109$	−58.2798	−0.0512128	−0.0256064	−	0.999672i	$-0.508152\pi$
$109$	−58.2798	−0.0512128	−0.0256064	+	0.999672i	$0.508152\pi$
$110$	0	0
$111$	−58.8179	−0.0502950
$112$	302.474	0.255188
$113$	1784.84	1.48588	0.742938	−	0.669360i	$-0.233432\pi$
$113$	1784.84	1.48588	0.742938	+	0.669360i	$0.233432\pi$
$114$	−21.7888	−0.0179009
$115$	490.290	0.397563
$116$	1852.88	1.48307
$117$	−1158.72	−0.915586
$118$	450.709	0.351619
$119$	426.107	0.328245
$120$	12.5634	0.00955727
$121$	0	0
$122$	−758.055	−0.562550
$123$	−52.2891	−0.0383313
$124$	1842.20	1.33415
$125$	1187.30	0.849563
$126$	179.121	0.126646
$127$	371.954	0.259887	0.129943	−	0.991521i	$-0.458520\pi$
$127$	371.954	0.259887	0.129943	+	0.991521i	$0.458520\pi$
$128$	1432.65	0.989292
$129$	−15.0893	−0.0102987
$130$	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
$135$	−47.3360	−0.0301780
$136$	871.995	0.549801
$137$	1282.01	0.799483	0.399742	−	0.916628i	$-0.369100\pi$
$137$	1282.01	0.799483	0.399742	+	0.916628i	$0.369100\pi$
$138$	−14.1571	−0.00873283
$139$	1372.26	0.837367	0.418683	−	0.908132i	$-0.362492\pi$
$139$	1372.26	0.837367	0.418683	+	0.908132i	$0.362492\pi$
$140$	266.785	0.161053
$141$	−11.0204	−0.00658217
$142$	−516.417	−0.305189
$143$	0	0
$144$	−1165.53	−0.674497
$145$	1400.85	0.802304
$146$	−550.398	−0.311995
$147$	−8.00581	−0.00449189
$148$	−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	$0.478241\pi$
$150$	−14.9085	−0.00811519
$151$	1247.52	0.672327	0.336164	−	0.941804i	$-0.390870\pi$
$151$	1247.52	0.672327	0.336164	+	0.941804i	$0.390870\pi$
$152$	−2013.73	−1.07457
$153$	−1641.93	−0.867597
$154$	0	0
$155$	1392.77	0.721742
$156$	49.8325	0.0255756
$157$	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
$159$	40.1987	0.0200501
$160$	835.201	0.412678
$161$	−639.363	−0.312974
$162$	−689.530	−0.334411
$163$	−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	$0.460979\pi$
$168$	−16.3833	−0.00752378
$169$	−351.613	−0.160043
$170$	309.984	0.139851
$171$	3791.77	1.69570
$172$	−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	$0.501947\pi$
$174$	−40.4494	−0.0176233
$175$	−673.300	−0.290838
$176$	0	0
$177$	77.6229	0.0329632
$178$	−1192.73	−0.502243
$179$	1622.80	0.677620	0.338810	−	0.940855i	$-0.389976\pi$
$179$	1622.80	0.677620	0.338810	+	0.940855i	$0.389976\pi$
$180$	−1028.01	−0.425685
$181$	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
$183$	−130.555	−0.0527373
$184$	−1308.40	−0.524222
$185$	−1932.43	−0.767973
$186$	−40.2161	−0.0158537
$187$	0	0
$188$	−478.904	−0.185785
$189$	61.7286	0.0237571
$190$	−715.858	−0.273336
$191$	−1118.93	−0.423889	−0.211945	−	0.977282i	$-0.567980\pi$
$191$	−1118.93	−0.423889	−0.211945	+	0.977282i	$0.567980\pi$
$192$	32.3629	0.0121645
$193$	−321.905	−0.120058	−0.0600291	−	0.998197i	$-0.519119\pi$
$193$	−321.905	−0.120058	−0.0600291	+	0.998197i	$0.519119\pi$
$194$	948.504	0.351024
$195$	37.6753	0.0138358
$196$	−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
$201$	118.454	0.0415676
$202$	624.270	0.217443
$203$	−1826.78	−0.631599
$204$	70.6139	0.0242351
$205$	−1717.93	−0.585295
$206$	−348.335	−0.117814
$207$	2463.67	0.827233
$208$	1856.24	0.618783
$209$	0	0
$210$	−5.82406	−0.00191380
$211$	2319.86	0.756899	0.378449	−	0.925622i	$-0.376457\pi$
$211$	2319.86	0.756899	0.378449	+	0.925622i	$0.376457\pi$
$212$	1746.88	0.565925
$213$	−88.9395	−0.0286105
$214$	1901.39	0.607367
$215$	−495.751	−0.157255
$216$	126.323	0.0397924
$217$	−1816.24	−0.568178
$218$	55.2883	0.0171770
$219$	−94.7918	−0.0292486
$220$	0	0
$221$	2614.96	0.795932
$222$	55.7988	0.0168692
$223$	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
$225$	2594.45	0.768725
$226$	−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	$0.436984\pi$
$228$	−163.071	−0.0473669
$229$	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	$0.245431\pi$
$234$	1099.24	0.307093
$235$	−362.070	−0.100506
$236$	3373.19	0.930406
$237$	−65.7521	−0.0180213
$238$	−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	$0.590506\pi$
$240$	37.8968	0.0101926
$241$	−2896.10	−0.774085	−0.387042	−	0.922062i	$-0.626503\pi$
$241$	−2896.10	−0.774085	−0.387042	+	0.922062i	$0.626503\pi$
$242$	0	0
$243$	−356.850	−0.0942054
$244$	−5673.42	−1.48854
$245$	−263.027	−0.0685884
$246$	49.6051	0.0128565
$247$	−6038.82	−1.55563
$248$	−3716.80	−0.951681
$249$	−180.109	−0.0458392
$250$	−1126.36	−0.284948
$251$	4226.07	1.06274	0.531369	−	0.847140i	$-0.321678\pi$
$251$	4226.07	1.06274	0.531369	+	0.847140i	$0.321678\pi$
$252$	1340.58	0.335113
$253$	0	0
$254$	−352.862	−0.0871674
$255$	53.3868	0.0131106
$256$	225.518	0.0550582
$257$	−5075.41	−1.23189	−0.615945	−	0.787789i	$-0.711225\pi$
$257$	−5075.41	−1.23189	−0.615945	+	0.787789i	$0.711225\pi$
$258$	14.3148	0.00345426
$259$	2519.99	0.604573
$260$	1637.22	0.390523
$261$	7039.18	1.66940
$262$	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
$267$	−205.418	−0.0470837
$268$	5147.54	1.17327
$269$	−3938.48	−0.892688	−0.446344	−	0.894861i	$-0.647274\pi$
$269$	−3938.48	−0.892688	−0.446344	+	0.894861i	$0.647274\pi$
$270$	44.9063	0.0101219
$271$	2882.96	0.646227	0.323113	−	0.946360i	$-0.395270\pi$
$271$	2882.96	0.646227	0.323113	+	0.946360i	$0.395270\pi$
$272$	2630.33	0.586350
$273$	−49.1305	−0.0108920
$274$	−1216.20	−0.268151
$275$	0	0
$276$	−105.954	−0.0231076
$277$	−1414.61	−0.306843	−0.153422	−	0.988161i	$-0.549029\pi$
$277$	−1414.61	−0.306843	−0.153422	+	0.988161i	$0.549029\pi$
$278$	−1301.83	−0.280858
$279$	6998.58	1.50177
$280$	−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	$0.569935\pi$
$282$	10.4547	0.00220770
$283$	7989.40	1.67816	0.839082	−	0.544005i	$-0.183093\pi$
$283$	7989.40	1.67816	0.839082	+	0.544005i	$0.183093\pi$
$284$	−3864.96	−0.807547
$285$	−123.288	−0.0256244
$286$	0	0
$287$	2240.27	0.460762
$288$	4196.83	0.858683
$289$	−1207.54	−0.245785
$290$	−1328.94	−0.269097
$291$	163.355	0.0329074
$292$	−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	$0.515653\pi$
$294$	7.59488	0.00150661
$295$	2550.26	0.503328
$296$	5156.95	1.01264
$297$	0	0
$298$	−235.705	−0.0458188
$299$	−3923.67	−0.758903
$300$	−111.578	−0.0214733
$301$	646.484	0.123796
$302$	−1183.48	−0.225502
$303$	107.514	0.0203846
$304$	−6074.32	−1.14601
$305$	−4289.32	−0.805265
$306$	1557.65	0.290997
$307$	4869.60	0.905287	0.452643	−	0.891692i	$-0.350481\pi$
$307$	4869.60	0.905287	0.452643	+	0.891692i	$0.350481\pi$
$308$	0	0
$309$	−59.9917	−0.0110447
$310$	−1321.28	−0.242076
$311$	−9618.02	−1.75366	−0.876830	−	0.480801i	$-0.840346\pi$
$311$	−9618.02	−1.75366	−0.876830	+	0.480801i	$0.840346\pi$
$312$	−100.542	−0.0182437
$313$	−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
$315$	1013.53	0.181288
$316$	−2857.33	−0.508663
$317$	2353.16	0.416929	0.208464	−	0.978030i	$-0.433153\pi$
$317$	2353.16	0.416929	0.208464	+	0.978030i	$0.433153\pi$
$318$	−38.1353	−0.00672491
$319$	0	0
$320$	1063.26	0.185745
$321$	327.466	0.0569388
$322$	606.544	0.104973
$323$	−8557.15	−1.47409
$324$	−5160.57	−0.884872
$325$	−4131.94	−0.705228
$326$	3099.16	0.526524
$327$	9.52198	0.00161030
$328$	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
$333$	−9710.34	−1.59797
$334$	−500.707	−0.0820284
$335$	3891.74	0.634711
$336$	−49.4193	−0.00802395
$337$	−2506.25	−0.405116	−0.202558	−	0.979270i	$-0.564926\pi$
$337$	−2506.25	−0.405116	−0.202558	+	0.979270i	$0.564926\pi$
$338$	333.565	0.0536792
$339$	−291.615	−0.0467208
$340$	2319.98	0.370055
$341$	0	0
$342$	−3597.14	−0.568746
$343$	343.000	0.0539949
$344$	1322.98	0.207355
$345$	−80.1054	−0.0125007
$346$	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	$0.325512\pi$
$348$	−302.731	−0.0466324
$349$	−1832.13	−0.281008	−0.140504	−	0.990080i	$-0.544872\pi$
$349$	−1832.13	−0.281008	−0.140504	+	0.990080i	$0.544872\pi$
$350$	638.740	0.0975488
$351$	378.819	0.0576065
$352$	0	0
$353$	3946.57	0.595056	0.297528	−	0.954713i	$-0.403838\pi$
$353$	3946.57	0.595056	0.297528	+	0.954713i	$0.403838\pi$
$354$	−73.6385	−0.0110561
$355$	−2922.06	−0.436864
$356$	−8926.65	−1.32896
$357$	−69.6191	−0.0103211
$358$	−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	$0.635014\pi$
$360$	2074.10	0.303653
$361$	12902.3	1.88108
$362$	−4212.99	−0.611685
$363$	0	0
$364$	−2135.02	−0.307432
$365$	−3114.33	−0.446607
$366$	123.854	0.0176884
$367$	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
$369$	−8632.49	−1.21786
$370$	1833.24	0.257583
$371$	−1722.27	−0.241013
$372$	−300.985	−0.0419499
$373$	−11627.3	−1.61405	−0.807025	−	0.590518i	$-0.798924\pi$
$373$	−11627.3	−1.61405	−0.807025	+	0.590518i	$0.798924\pi$
$374$	0	0
$375$	−193.986	−0.0267130
$376$	966.232	0.132526
$377$	−11210.7	−1.53151
$378$	−58.5601	−0.00796827
$379$	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
$381$	−60.7713	−0.00817168
$382$	1061.49	0.142175
$383$	−40.4622	−0.00539823	−0.00269912	−	0.999996i	$-0.500859\pi$
$383$	−40.4622	−0.00539823	−0.00269912	+	0.999996i	$0.500859\pi$
$384$	−234.072	−0.0311066
$385$	0	0
$386$	305.382	0.0402682
$387$	−2491.12	−0.327211
$388$	7098.78	0.928830
$389$	6230.86	0.812126	0.406063	−	0.913845i	$-0.366901\pi$
$389$	6230.86	0.812126	0.406063	+	0.913845i	$0.366901\pi$
$390$	−35.7414	−0.00464061
$391$	−5559.94	−0.719126
$392$	701.922	0.0904399
$393$	11.8955	0.00152684
$394$	−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
$399$	160.774	0.0201723
$400$	−4156.24	−0.519530
$401$	438.764	0.0546405	0.0273202	−	0.999627i	$-0.491303\pi$
$401$	438.764	0.0546405	0.0273202	+	0.999627i	$0.491303\pi$
$402$	−112.374	−0.0139420
$403$	−11146.0	−1.37772
$404$	4672.15	0.575367
$405$	−3901.59	−0.478695
$406$	1733.01	0.211842
$407$	0	0
$408$	−142.470	−0.0172875
$409$	6813.22	0.823697	0.411848	−	0.911252i	$-0.364883\pi$
$409$	6813.22	0.823697	0.411848	+	0.911252i	$0.364883\pi$
$410$	1629.75	0.196311
$411$	−209.459	−0.0251384
$412$	−2607.00	−0.311742
$413$	−3325.67	−0.396236
$414$	−2337.22	−0.277459
$415$	−5917.39	−0.699935
$416$	−6683.92	−0.787755
$417$	−224.206	−0.0263295
$418$	0	0
$419$	4789.84	0.558471	0.279235	−	0.960223i	$-0.409919\pi$
$419$	4789.84	0.558471	0.279235	+	0.960223i	$0.409919\pi$
$420$	−43.5884	−0.00506404
$421$	−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
$423$	−1819.38	−0.209128
$424$	−3524.48	−0.403689
$425$	−5855.06	−0.668264
$426$	84.3742	0.00959612
$427$	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	$0.235200\pi$
$432$	381.047	0.0424378
$433$	−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
$435$	−228.876	−0.0252271
$436$	413.788	0.0454515
$437$	12839.8	1.40551
$438$	89.9262	0.00981014
$439$	5078.63	0.552141	0.276070	−	0.961137i	$-0.410968\pi$
$439$	5078.63	0.552141	0.276070	+	0.961137i	$0.410968\pi$
$440$	0	0
$441$	−1321.69	−0.142716
$442$	−2480.73	−0.266960
$443$	−8477.52	−0.909208	−0.454604	−	0.890694i	$-0.650219\pi$
$443$	−8477.52	−0.909208	−0.454604	+	0.890694i	$0.650219\pi$
$444$	417.608	0.0446370
$445$	−6748.89	−0.718939
$446$	−750.756	−0.0797070
$447$	−40.5940	−0.00429537
$448$	−1386.55	−0.146224
$449$	17792.1	1.87007	0.935033	−	0.354561i	$-0.115370\pi$
$449$	17792.1	1.87007	0.935033	+	0.354561i	$0.115370\pi$
$450$	−2461.28	−0.257835
$451$	0	0
$452$	−12672.4	−1.31872
$453$	−203.824	−0.0211401
$454$	−1276.28	−0.131936
$455$	−1614.16	−0.166314
$456$	329.011	0.0337881
$457$	1385.23	0.141791	0.0708955	−	0.997484i	$-0.477414\pi$
$457$	1385.23	0.141791	0.0708955	+	0.997484i	$0.477414\pi$
$458$	−1147.44	−0.117067
$459$	536.796	0.0545871
$460$	−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
$465$	−227.556	−0.0226939
$466$	−4839.59	−0.481094
$467$	16393.7	1.62443	0.812216	−	0.583356i	$-0.198261\pi$
$467$	16393.7	1.62443	0.812216	+	0.583356i	$0.198261\pi$
$468$	8226.93	0.812585
$469$	−5075.02	−0.499665
$470$	343.485	0.0337102
$471$	−138.207	−0.0135206
$472$	−6805.71	−0.663683
$473$	0	0
$474$	62.3771	0.00604446
$475$	13521.3	1.30611
$476$	−3025.37	−0.291319
$477$	6636.46	0.637029
$478$	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	$0.700079\pi$
$480$	−136.458	−0.0129759
$481$	15464.8	1.46597
$482$	2747.45	0.259632
$483$	104.462	0.00984092
$484$	0	0
$485$	5366.95	0.502476
$486$	338.533	0.0315970
$487$	−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
$489$	533.750	0.0493600
$490$	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	$0.507088\pi$
$492$	371.254	0.0340191
$493$	−15885.8	−1.45124
$494$	5728.85	0.521767
$495$	0	0
$496$	−11211.5	−1.01495
$497$	3810.51	0.343913
$498$	170.864	0.0153747
$499$	−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
$501$	−86.2339	−0.00768991
$502$	−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	$0.500232\pi$
$504$	−2704.74	−0.239045
$505$	3532.32	0.311260
$506$	0	0
$507$	57.4480	0.00503226
$508$	−2640.88	−0.230650
$509$	5133.84	0.447060	0.223530	−	0.974697i	$-0.428242\pi$
$509$	5133.84	0.447060	0.223530	+	0.974697i	$0.428242\pi$
$510$	−50.6465	−0.00439738
$511$	4061.25	0.351583
$512$	−11675.1	−1.00776
$513$	−1239.64	−0.106689
$514$	4814.89	0.413183
$515$	−1970.99	−0.168645
$516$	107.134	0.00914017
$517$	0	0
$518$	−2390.64	−0.202777
$519$	4.54762	0.000384621 0
$520$	−3303.24	−0.278571
$521$	−4454.52	−0.374580	−0.187290	−	0.982305i	$-0.559970\pi$
$521$	−4454.52	−0.374580	−0.187290	+	0.982305i	$0.559970\pi$
$522$	−6677.86	−0.559927
$523$	226.348	0.0189245	0.00946224	−	0.999955i	$-0.496988\pi$
$523$	226.348	0.0189245	0.00946224	+	0.999955i	$0.496988\pi$
$524$	516.932	0.0430959
$525$	110.006	0.00914490
$526$	−116.880	−0.00968858
$527$	−15794.2	−1.30551
$528$	0	0
$529$	−3824.46	−0.314331
$530$	−1252.91	−0.102685
$531$	12814.9	1.04730
$532$	6986.60	0.569375
$533$	13748.2	1.11726
$534$	194.874	0.0157921
$535$	10758.7	0.869420
$536$	−10385.6	−0.836923
$537$	−265.140	−0.0213066
$538$	3736.32	0.299413
$539$	0	0
$540$	336.087	0.0267831
$541$	−11808.2	−0.938401	−0.469201	−	0.883092i	$-0.655458\pi$
$541$	−11808.2	−0.938401	−0.469201	+	0.883092i	$0.655458\pi$
$542$	−2734.98	−0.216748
$543$	−725.579	−0.0573436
$544$	−9471.28	−0.746466
$545$	312.839	0.0245882
$546$	46.6086	0.00365323
$547$	−11208.4	−0.876122	−0.438061	−	0.898945i	$-0.644335\pi$
$547$	−11208.4	−0.876122	−0.438061	+	0.898945i	$0.644335\pi$
$548$	−9102.28	−0.709544
$549$	−21553.6	−1.67556
$550$	0	0
$551$	36685.6	2.83641
$552$	213.772	0.0164832
$553$	2817.08	0.216626
$554$	1342.00	0.102917
$555$	315.728	0.0241476
$556$	−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	$0.195099\pi$
$558$	−6639.35	−0.503703
$559$	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	$0.747959\pi$
$564$	78.2452	0.00584170
$565$	−9580.85	−0.713397
$566$	−7579.31	−0.562866
$567$	5087.87	0.376844
$568$	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	$0.245912\pi$
$570$	116.960	0.00859457
$571$	−5978.67	−0.438178	−0.219089	−	0.975705i	$-0.570308\pi$
$571$	−5978.67	−0.438178	−0.219089	+	0.975705i	$0.570308\pi$
$572$	0	0
$573$	182.815	0.0133284
$574$	−2125.27	−0.154542
$575$	8785.37	0.637174
$576$	5342.84	0.386490
$577$	−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
$579$	52.5941	0.00377502
$580$	−9946.06	−0.712048
$581$	7716.57	0.551011
$582$	−154.970	−0.0110373
$583$	0	0
$584$	8311.03	0.588892
$585$	6219.87	0.439590
$586$	467.747	0.0329735
$587$	−1063.42	−0.0747738	−0.0373869	−	0.999301i	$-0.511903\pi$
$587$	−1063.42	−0.0747738	−0.0373869	+	0.999301i	$0.511903\pi$
$588$	56.8415	0.00398657
$589$	36474.1	2.55159
$590$	−2419.35	−0.168819
$591$	−393.709	−0.0274027
$592$	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
$597$	−278.155	−0.0190689
$598$	3722.27	0.254540
$599$	24953.5	1.70213	0.851063	−	0.525064i	$-0.175959\pi$
$599$	24953.5	1.70213	0.851063	+	0.525064i	$0.175959\pi$
$600$	225.119	0.0153174
$601$	−16330.2	−1.10836	−0.554179	−	0.832398i	$-0.686968\pi$
$601$	−16330.2	−1.10836	−0.554179	+	0.832398i	$0.686968\pi$
$602$	−613.300	−0.0415220
$603$	19555.7	1.32068
$604$	−8857.39	−0.596692
$605$	0	0
$606$	−101.996	−0.00683711
$607$	12193.8	0.815371	0.407686	−	0.913122i	$-0.366336\pi$
$607$	12193.8	0.815371	0.407686	+	0.913122i	$0.366336\pi$
$608$	21872.4	1.45895
$609$	298.466	0.0198595
$610$	4069.15	0.270090
$611$	2897.56	0.191854
$612$	11657.8	0.769995
$613$	−18075.5	−1.19097	−0.595483	−	0.803368i	$-0.703039\pi$
$613$	−18075.5	−1.19097	−0.595483	+	0.803368i	$0.703039\pi$
$614$	−4619.65	−0.303638
$615$	280.682	0.0184036
$616$	0	0
$617$	−20714.8	−1.35162	−0.675809	−	0.737077i	$-0.736205\pi$
$617$	−20714.8	−1.35162	−0.675809	+	0.737077i	$0.736205\pi$
$618$	56.9123	0.00370445
$619$	−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
$621$	−805.448	−0.0520475
$622$	9124.33	0.588187
$623$	8800.89	0.565971
$624$	−303.279	−0.0194565
$625$	5649.92	0.361595
$626$	6582.19	0.420251
$627$	0	0
$628$	−6005.92	−0.381628
$629$	21914.0	1.38914
$630$	−961.504	−0.0608051
$631$	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
$633$	−379.027	−0.0237993
$634$	−2232.37	−0.139840
$635$	−1996.61	−0.124776
$636$	−285.412	−0.0177945
$637$	2104.94	0.130927
$638$	0	0
$639$	−14683.2	−0.909009
$640$	−7690.30	−0.474978
$641$	20817.4	1.28274	0.641370	−	0.767232i	$-0.278366\pi$
$641$	20817.4	1.28274	0.641370	+	0.767232i	$0.278366\pi$
$642$	−310.657	−0.0190976
$643$	−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
$645$	80.9977	0.00494462
$646$	8117.91	0.494420
$647$	−30769.4	−1.86966	−0.934829	−	0.355099i	$-0.884447\pi$
$647$	−30769.4	−1.86966	−0.934829	+	0.355099i	$0.884447\pi$
$648$	10411.9	0.631202
$649$	0	0
$650$	3919.85	0.236537
$651$	296.745	0.0178654
$652$	23194.7	1.39321
$653$	14927.0	0.894546	0.447273	−	0.894397i	$-0.352395\pi$
$653$	14927.0	0.894546	0.447273	+	0.894397i	$0.352395\pi$
$654$	−9.03322	−0.000540102 0
$655$	390.820	0.0233139
$656$	13829.0	0.823068
$657$	−15649.3	−0.929283
$658$	−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
$663$	−427.242	−0.0250267
$664$	15791.4	0.922927
$665$	5282.14	0.308019
$666$	9211.91	0.535967
$667$	23836.2	1.38372
$668$	−3747.39	−0.217052
$669$	−129.298	−0.00747228
$670$	−3691.97	−0.212886
$671$	0	0
$672$	177.949	0.0102151
$673$	22202.2	1.27167	0.635834	−	0.771826i	$-0.280656\pi$
$673$	22202.2	1.27167	0.635834	+	0.771826i	$0.280656\pi$
$674$	2377.61	0.135878
$675$	−848.202	−0.0483664
$676$	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	$0.404182\pi$
$678$	276.646	0.0156704
$679$	−6998.77	−0.395565
$680$	−4680.77	−0.263970
$681$	−219.807	−0.0123686
$682$	0	0
$683$	−15600.1	−0.873969	−0.436985	−	0.899469i	$-0.643954\pi$
$683$	−15600.1	−0.873969	−0.436985	+	0.899469i	$0.643954\pi$
$684$	−26921.7	−1.50494
$685$	−6881.67	−0.383847
$686$	−325.394	−0.0181102
$687$	−197.617	−0.0109746
$688$	3990.70	0.221140
$689$	−10569.3	−0.584410
$690$	75.9936	0.00419280
$691$	−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
$696$	610.787	0.0332641
$697$	19481.5	1.05870
$698$	1738.09	0.0942517
$699$	−833.494	−0.0451011
$700$	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	$0.357275\pi$
$702$	−359.375	−0.0193215
$703$	−50606.7	−2.71503
$704$	0	0
$705$	59.1563	0.00316022
$706$	−3743.99	−0.199585
$707$	−4606.33	−0.245034
$708$	−551.124	−0.0292550
$709$	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
$711$	−10855.1	−0.572572
$712$	18010.3	0.947985
$713$	23698.7	1.24477
$714$	66.0455	0.00346175
$715$	0	0
$716$	−11521.9	−0.601390
$717$	338.683	0.0176407
$718$	5311.46	0.276075
$719$	2437.03	0.126406	0.0632030	−	0.998001i	$-0.479868\pi$
$719$	2437.03	0.126406	0.0632030	+	0.998001i	$0.479868\pi$
$720$	6256.44	0.323839
$721$	2570.28	0.132763
$722$	−12240.1	−0.630925
$723$	473.177	0.0243397
$724$	−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
$729$	−19566.3	−0.994073
$730$	2954.48	0.149795
$731$	5621.87	0.284449
$732$	926.946	0.0468045
$733$	21145.5	1.06552	0.532761	−	0.846266i	$-0.321155\pi$
$733$	21145.5	1.06552	0.532761	+	0.846266i	$0.321155\pi$
$734$	−7759.65	−0.390210
$735$	42.9743	0.00215664
$736$	14211.4	0.711738
$737$	0	0
$738$	8189.38	0.408476
$739$	37759.5	1.87957	0.939787	−	0.341760i	$-0.111023\pi$
$739$	37759.5	1.87957	0.939787	+	0.341760i	$0.111023\pi$
$740$	13720.3	0.681579
$741$	986.645	0.0489141
$742$	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	$0.425302\pi$
$744$	607.265	0.0299239
$745$	−1333.69	−0.0655876
$746$	11030.5	0.541361
$747$	−29734.5	−1.45640
$748$	0	0
$749$	−14029.9	−0.684435
$750$	184.029	0.00895970
$751$	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
$753$	−690.472	−0.0334159
$754$	10635.2	0.513676
$755$	−6696.53	−0.322797
$756$	−438.274	−0.0210845
$757$	−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	$0.574296\pi$
$762$	57.6519	0.00274083
$763$	−407.958	−0.0193566
$764$	7944.42	0.376203
$765$	8813.71	0.416549
$766$	38.3853	0.00181060
$767$	−20409.1	−0.960796
$768$	−36.8461	−0.00173121
$769$	−3693.86	−0.173217	−0.0866085	−	0.996242i	$-0.527603\pi$
$769$	−3693.86	−0.173217	−0.0866085	+	0.996242i	$0.527603\pi$
$770$	0	0
$771$	829.241	0.0387346
$772$	2285.53	0.106552
$773$	38031.9	1.76962	0.884808	−	0.465955i	$-0.154289\pi$
$773$	38031.9	1.76962	0.884808	+	0.465955i	$0.154289\pi$
$774$	2363.25	0.109748
$775$	24956.7	1.15674
$776$	−14322.4	−0.662559
$777$	−411.725	−0.0190097
$778$	−5911.03	−0.272392
$779$	−44989.4	−2.06921
$780$	−267.495	−0.0122793
$781$	0	0
$782$	5274.55	0.241199
$783$	−2301.31	−0.105035
$784$	2117.32	0.0964521
$785$	−4540.70	−0.206452
$786$	−11.2849	−0.000512111 0
$787$	−3509.51	−0.158959	−0.0794793	−	0.996837i	$-0.525326\pi$
$787$	−3509.51	−0.158959	−0.0794793	+	0.996837i	$0.525326\pi$
$788$	−17109.0	−0.773457
$789$	−20.1295	−0.000908274 0
$790$	2049.37	0.0922951
$791$	12493.9	0.561608
$792$	0	0
$793$	34326.4	1.53716
$794$	3497.03	0.156303
$795$	−215.782	−0.00962642
$796$	−12087.5	−0.538230
$797$	32249.5	1.43330	0.716648	−	0.697435i	$-0.245676\pi$
$797$	32249.5	1.43330	0.716648	+	0.697435i	$0.245676\pi$
$798$	−152.521	−0.00676591
$799$	4105.91	0.181798
$800$	14965.7	0.661399
$801$	−33912.7	−1.49594
$802$	−416.242	−0.0183267
$803$	0	0
$804$	−841.025	−0.0368914
$805$	3432.03	0.150265
$806$	10573.9	0.462096
$807$	643.484	0.0280690
$808$	−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	$0.423332\pi$
$810$	3701.32	0.160557
$811$	−39624.5	−1.71567	−0.857833	−	0.513928i	$-0.828190\pi$
$811$	−39624.5	−1.71567	−0.857833	+	0.513928i	$0.828190\pi$
$812$	12970.2	0.560546
$813$	−471.029	−0.0203195
$814$	0	0
$815$	17536.1	0.753696
$816$	−429.754	−0.0184368
$817$	−12982.8	−0.555949
$818$	−6463.50	−0.276272
$819$	−8111.03	−0.346059
$820$	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	$0.404406\pi$
$822$	198.708	0.00843155
$823$	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	$0.551960\pi$
$828$	−17492.1	−0.734172
$829$	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
$831$	231.124	0.00964815
$832$	−8509.06	−0.354565
$833$	2982.75	0.124065
$834$	212.698	0.00883107
$835$	−2833.17	−0.117420
$836$	0	0
$837$	−2288.04	−0.0944879
$838$	−4543.98	−0.187314
$839$	37318.3	1.53560	0.767801	−	0.640688i	$-0.221351\pi$
$839$	37318.3	1.53560	0.767801	+	0.640688i	$0.221351\pi$
$840$	87.9435	0.00361231
$841$	43715.4	1.79242
$842$	5608.82	0.229564
$843$	335.460	0.0137057
$844$	−16471.0	−0.671750
$845$	1887.42	0.0768394
$846$	1725.99	0.0701427
$847$	0	0
$848$	−10631.4	−0.430525
$849$	−1305.34	−0.0527669
$850$	5554.53	0.224140
$851$	−32881.3	−1.32451
$852$	631.472	0.0253919
$853$	2096.82	0.0841663	0.0420831	−	0.999114i	$-0.486601\pi$
$853$	2096.82	0.0841663	0.0420831	+	0.999114i	$0.486601\pi$
$854$	−5306.38	−0.212624
$855$	−20353.8	−0.814136
$856$	−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
$861$	−366.024	−0.0144879
$862$	−12549.5	−0.495869
$863$	2360.78	0.0931191	0.0465596	−	0.998916i	$-0.485174\pi$
$863$	2360.78	0.0931191	0.0465596	+	0.998916i	$0.485174\pi$
$864$	−1372.07	−0.0540263
$865$	149.410	0.00587293
$866$	13108.6	0.514373
$867$	197.293	0.00772828
$868$	12895.4	0.504260
$869$	0	0
$870$	217.128	0.00846130
$871$	−31144.7	−1.21159
$872$	−834.855	−0.0324217
$873$	26968.6	1.04553
$874$	−12180.7	−0.471417
$875$	8311.10	0.321105
$876$	673.024	0.0259582
$877$	−8504.19	−0.327441	−0.163721	−	0.986507i	$-0.552350\pi$
$877$	−8504.19	−0.327441	−0.163721	+	0.986507i	$0.552350\pi$
$878$	−4817.95	−0.185191
$879$	80.5573	0.00309116
$880$	0	0
$881$	−8050.60	−0.307868	−0.153934	−	0.988081i	$-0.549194\pi$
$881$	−8050.60	−0.307868	−0.153934	+	0.988081i	$0.549194\pi$
$882$	1253.85	0.0478677
$883$	−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
$885$	−416.671	−0.0158263
$886$	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	$0.791757\pi$
$888$	−842.563	−0.0318407
$889$	2603.68	0.0982279
$890$	6402.47	0.241136
$891$	0	0
$892$	−5618.79	−0.210909
$893$	−9481.93	−0.355320
$894$	38.5103	0.00144069
$895$	−8711.02	−0.325338
$896$	10028.5	0.373917
$897$	641.065	0.0238624
$898$	−16878.8	−0.627231
$899$	67711.7	2.51203
$900$	−18420.6	−0.682246
$901$	−14977.0	−0.553779
$902$	0	0
$903$	−105.625	−0.00389256
$904$	25567.8	0.940677
$905$	−23838.5	−0.875601
$906$	193.362	0.00709052
$907$	−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
$909$	17749.7	0.647657
$910$	1531.30	0.0557826
$911$	29894.3	1.08720	0.543602	−	0.839343i	$-0.317060\pi$
$911$	29894.3	1.08720	0.543602	+	0.839343i	$0.317060\pi$
$912$	992.447	0.0360342
$913$	0	0
$914$	−1314.13	−0.0475575
$915$	700.806	0.0253202
$916$	−8587.68	−0.309765
$917$	−509.649	−0.0183534
$918$	−509.242	−0.0183088
$919$	−1297.57	−0.0465754	−0.0232877	−	0.999729i	$-0.507413\pi$
$919$	−1297.57	−0.0465754	−0.0232877	+	0.999729i	$0.507413\pi$
$920$	7023.37	0.251689
$921$	−795.615	−0.0284652
$922$	−8703.67	−0.310889
$923$	23384.6	0.833924
$924$	0	0
$925$	−34626.7	−1.23083
$926$	−3857.29	−0.136888
$927$	−9904.12	−0.350910
$928$	40604.6	1.43633
$929$	−38299.3	−1.35259	−0.676297	−	0.736629i	$-0.736416\pi$
$929$	−38299.3	−1.35259	−0.676297	+	0.736629i	$0.736416\pi$
$930$	215.876	0.00761166
$931$	−6888.18	−0.242482
$932$	−36220.4	−1.27300
$933$	1571.43	0.0551407
$934$	−15552.2	−0.544844
$935$	0	0
$936$	−16598.6	−0.579638
$937$	17601.2	0.613667	0.306834	−	0.951763i	$-0.400731\pi$
$937$	17601.2	0.613667	0.306834	+	0.951763i	$0.400731\pi$
$938$	4814.52	0.167590
$939$	1133.61	0.0393973
$940$	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	$0.738983\pi$
$942$	131.112	0.00453490
$943$	−29231.5	−1.00945
$944$	−20529.1	−0.707803
$945$	−331.352	−0.0114062
$946$	0	0
$947$	42471.2	1.45737	0.728685	−	0.684849i	$-0.240132\pi$
$947$	42471.2	1.45737	0.728685	+	0.684849i	$0.240132\pi$
$948$	466.842	0.0159940
$949$	24923.3	0.852523
$950$	−12827.3	−0.438075
$951$	−384.468	−0.0131096
$952$	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	$0.264631\pi$
$954$	−6295.82	−0.213663
$955$	6006.28	0.203517
$956$	14717.9	0.497918
$957$	0	0
$958$	11695.4	0.394428
$959$	8974.05	0.302176
$960$	−173.720	−0.00584042
$961$	37530.2	1.25978
$962$	−14671.0	−0.491696
$963$	54061.8	1.80905
$964$	20562.4	0.687003
$965$	1727.95	0.0576421
$966$	−99.0996	−0.00330070
$967$	48452.2	1.61129	0.805644	−	0.592399i	$-0.201819\pi$
$967$	48452.2	1.61129	0.805644	+	0.592399i	$0.201819\pi$
$968$	0	0
$969$	1398.10	0.0463503
$970$	−5091.47	−0.168533
$971$	−10531.1	−0.348051	−0.174026	−	0.984741i	$-0.555678\pi$
$971$	−10531.1	−0.348051	−0.174026	+	0.984741i	$0.555678\pi$
$972$	2533.64	0.0836076
$973$	9605.85	0.316495
$974$	1794.47	0.0590333
$975$	675.093	0.0221747
$976$	34528.3	1.13240
$977$	49108.4	1.60810	0.804052	−	0.594559i	$-0.202673\pi$
$977$	49108.4	1.60810	0.804052	+	0.594559i	$0.202673\pi$
$978$	−506.353	−0.0165556
$979$	0	0
$980$	1867.50	0.0608724
$981$	1572.00	0.0511621
$982$	459.648	0.0149368
$983$	−2280.32	−0.0739888	−0.0369944	−	0.999315i	$-0.511778\pi$
$983$	−2280.32	−0.0739888	−0.0369944	+	0.999315i	$0.511778\pi$
$984$	−749.038	−0.0242667
$985$	−12935.1	−0.418423
$986$	15070.4	0.486753
$987$	−77.1429	−0.00248783
$988$	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
$993$	−247.043	−0.00789494
$994$	−3614.92	−0.115350
$995$	−9138.63	−0.291170
$996$	1278.78	0.0406824
$997$	−51827.2	−1.64632	−0.823161	−	0.567808i	$-0.807792\pi$
$997$	−51827.2	−1.64632	−0.823161	+	0.567808i	$0.807792\pi$
$998$	10362.3	0.328670
$999$	3174.60	0.100540

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.c.1.3	✓	4	11.10	odd	2
539.4.a.f.1.3		4	77.76	even	2
693.4.a.m.1.2		4	33.32	even	2
847.4.a.e.1.2		4	1.1	even	1	trivial
1232.4.a.w.1.2		4	44.43	even	2
1925.4.a.q.1.2		4	55.54	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:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	847.a (trivial)

\(f(q)\)	\(=\)	\(q-0.948670 q^{2} -0.163384 q^{3} -7.10002 q^{4} -5.36789 q^{5} +0.154997 q^{6} +7.00000 q^{7} +14.3249 q^{8} -26.9733 q^{9} +5.09236 q^{10} +1.16003 q^{12} +42.9580 q^{13} -6.64069 q^{14} +0.877027 q^{15} +43.2105 q^{16} +60.8725 q^{17} +25.5888 q^{18} -140.575 q^{19} +38.1121 q^{20} -1.14369 q^{21} -91.3375 q^{23} -2.34047 q^{24} -96.1858 q^{25} -40.7529 q^{26} +8.81837 q^{27} -49.7002 q^{28} -260.968 q^{29} -0.832009 q^{30} -259.463 q^{31} -155.592 q^{32} -57.7479 q^{34} -37.5752 q^{35} +191.511 q^{36} +359.998 q^{37} +133.359 q^{38} -7.01864 q^{39} -76.8947 q^{40} +320.038 q^{41} +1.08498 q^{42} +92.3549 q^{43} +144.790 q^{45} +86.6492 q^{46} +67.4510 q^{47} -7.05991 q^{48} +49.0000 q^{49} +91.2486 q^{50} -9.94558 q^{51} -305.003 q^{52} -246.038 q^{53} -8.36573 q^{54} +100.275 q^{56} +22.9677 q^{57} +247.573 q^{58} -475.095 q^{59} -6.22691 q^{60} +799.071 q^{61} +246.145 q^{62} -188.813 q^{63} -198.079 q^{64} -230.594 q^{65} -725.003 q^{67} -432.196 q^{68} +14.9231 q^{69} +35.6465 q^{70} +544.359 q^{71} -386.391 q^{72} +580.179 q^{73} -341.519 q^{74} +15.7152 q^{75} +998.086 q^{76} +6.65838 q^{78} +402.439 q^{79} -231.949 q^{80} +726.838 q^{81} -303.611 q^{82} +1102.37 q^{83} +8.12021 q^{84} -326.757 q^{85} -87.6143 q^{86} +42.6380 q^{87} +1257.27 q^{89} -137.358 q^{90} +300.706 q^{91} +648.499 q^{92} +42.3921 q^{93} -63.9888 q^{94} +754.591 q^{95} +25.4212 q^{96} -999.825 q^{97} -46.4848 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 12 q^{3} + 22 q^{4} - 18 q^{5} - 2 q^{6} + 28 q^{7} + 60 q^{8} + 66 q^{9} + 92 q^{10} - 186 q^{12} + 134 q^{13} + 28 q^{14} - 62 q^{15} - 6 q^{16} + 74 q^{17} + 256 q^{18} + 164 q^{19} + 116 q^{20}+ \cdots + 196 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 - 12 * q^3 + 22 * q^4 - 18 * q^5 - 2 * q^6 + 28 * q^7 + 60 * q^8 + 66 * q^9 + 92 * q^10 - 186 * q^12 + 134 * q^13 + 28 * q^14 - 62 * q^15 - 6 * q^16 + 74 * q^17 + 256 * q^18 + 164 * q^19 + 116 * q^20 - 84 * q^21 + 194 * q^23 - 570 * q^24 + 38 * q^25 + 734 * q^26 - 510 * q^27 + 154 * q^28 + 108 * q^29 - 1252 * q^30 - 412 * q^31 + 4 * q^32 - 346 * q^34 - 126 * q^35 + 1518 * q^36 + 286 * q^37 + 224 * q^38 + 256 * q^39 + 540 * q^40 + 18 * q^41 - 14 * q^42 + 496 * q^43 + 580 * q^45 + 284 * q^46 + 62 * q^47 - 862 * q^48 + 196 * q^49 - 212 * q^50 + 508 * q^51 + 822 * q^52 - 828 * q^53 - 2420 * q^54 + 420 * q^56 - 700 * q^57 + 1388 * q^58 - 1224 * q^59 - 1776 * q^60 + 350 * q^61 + 878 * q^62 + 462 * q^63 - 718 * q^64 + 396 * q^65 - 1498 * q^67 - 1058 * q^68 - 386 * q^69 + 644 * q^70 + 2326 * q^71 + 3000 * q^72 + 1630 * q^73 + 1156 * q^74 - 1362 * q^75 + 3152 * q^76 - 2464 * q^78 + 1020 * q^79 + 3072 * q^80 + 1128 * q^81 + 2118 * q^82 + 1920 * q^83 - 1302 * q^84 - 2008 * q^85 + 1056 * q^86 - 1640 * q^87 + 1550 * q^89 + 5780 * q^90 + 938 * q^91 + 2592 * q^92 + 6046 * q^93 + 1042 * q^94 - 2332 * q^95 - 4082 * q^96 - 2202 * q^97 + 196 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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