LMFDB - Embedded newform 1127.4.a.f.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1127,4,Mod(1,1127)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1127.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1127, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1127 = 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1127.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,0,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$66.4951525765$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 60x^{7} - 22x^{6} + 1179x^{5} + 694x^{4} - 7936x^{3} - 4352x^{2} + 11008x + 3072 $ x^9 - 60x^7 - 22x^6 + 1179x^5 + 694x^4 - 7936x^3 - 4352x^2 + 11008*x + 3072
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 161)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.264899$ of defining polynomial
Character	$\chi$	$=$	1127.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.264899 q^{2} +6.92505 q^{3} -7.92983 q^{4} +11.7130 q^{5} -1.83444 q^{6} +4.21980 q^{8} +20.9564 q^{9} -3.10276 q^{10} -55.2621 q^{11} -54.9145 q^{12} +76.7819 q^{13} +81.1131 q^{15} +62.3208 q^{16} -112.294 q^{17} -5.55133 q^{18} -98.4128 q^{19} -92.8820 q^{20} +14.6389 q^{22} -23.0000 q^{23} +29.2223 q^{24} +12.1943 q^{25} -20.3395 q^{26} -41.8524 q^{27} -195.047 q^{29} -21.4868 q^{30} +99.2668 q^{31} -50.2671 q^{32} -382.693 q^{33} +29.7467 q^{34} -166.181 q^{36} -14.5629 q^{37} +26.0694 q^{38} +531.719 q^{39} +49.4264 q^{40} +329.303 q^{41} -275.032 q^{43} +438.219 q^{44} +245.462 q^{45} +6.09268 q^{46} +4.09163 q^{47} +431.575 q^{48} -3.23025 q^{50} -777.645 q^{51} -608.868 q^{52} -254.594 q^{53} +11.0866 q^{54} -647.285 q^{55} -681.514 q^{57} +51.6676 q^{58} -207.145 q^{59} -643.213 q^{60} -721.632 q^{61} -26.2957 q^{62} -485.251 q^{64} +899.347 q^{65} +101.375 q^{66} -514.780 q^{67} +890.476 q^{68} -159.276 q^{69} +1059.23 q^{71} +88.4317 q^{72} +376.129 q^{73} +3.85769 q^{74} +84.4459 q^{75} +780.396 q^{76} -140.852 q^{78} +849.401 q^{79} +729.963 q^{80} -855.652 q^{81} -87.2319 q^{82} -204.086 q^{83} -1315.30 q^{85} +72.8557 q^{86} -1350.71 q^{87} -233.195 q^{88} +631.481 q^{89} -65.0227 q^{90} +182.386 q^{92} +687.428 q^{93} -1.08387 q^{94} -1152.71 q^{95} -348.102 q^{96} -988.987 q^{97} -1158.09 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{3} + 48 q^{4} + 4 q^{5} - 46 q^{6} + 66 q^{8} + 122 q^{9} - 50 q^{10} - 8 q^{11} - 220 q^{12} - 25 q^{13} + 88 q^{15} + 180 q^{16} + 28 q^{17} - 54 q^{18} - 254 q^{19} - 302 q^{20} - 122 q^{22}+ \cdots - 618 q^{99}+O(q^{100}) $ 9 * q - 9 * q^3 + 48 * q^4 + 4 * q^5 - 46 * q^6 + 66 * q^8 + 122 * q^9 - 50 * q^10 - 8 * q^11 - 220 * q^12 - 25 * q^13 + 88 * q^15 + 180 * q^16 + 28 * q^17 - 54 * q^18 - 254 * q^19 - 302 * q^20 - 122 * q^22 - 207 * q^23 - 624 * q^24 + 295 * q^25 + 6 * q^26 - 633 * q^27 + 25 * q^29 + 80 * q^30 - 1171 * q^31 + 1018 * q^32 - 272 * q^33 - 8 * q^34 + 72 * q^36 + 70 * q^37 - 282 * q^38 + 1185 * q^39 + 54 * q^40 - 221 * q^41 - 42 * q^43 - 1214 * q^44 - 698 * q^45 - 159 * q^47 + 1104 * q^48 - 1680 * q^50 + 308 * q^51 + 664 * q^52 - 774 * q^53 + 2674 * q^54 - 1498 * q^55 - 2524 * q^57 + 842 * q^58 - 1080 * q^59 - 2160 * q^60 - 686 * q^61 + 2078 * q^62 - 1260 * q^64 - 1656 * q^65 - 532 * q^66 - 370 * q^67 + 1936 * q^68 + 207 * q^69 + 1035 * q^71 - 722 * q^72 - 1979 * q^73 - 5494 * q^74 + 1459 * q^75 - 206 * q^76 - 2066 * q^78 + 2336 * q^79 + 242 * q^80 + 997 * q^81 - 1642 * q^82 - 130 * q^83 - 272 * q^85 - 1906 * q^86 + 581 * q^87 - 3742 * q^88 + 1328 * q^89 + 7650 * q^90 - 1104 * q^92 + 1305 * q^93 - 6078 * q^94 - 484 * q^95 + 136 * q^96 + 104 * q^97 - 618 * q^99

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 $

$2$	−0.264899	−0.0936559	−0.0468280	−	0.998903i	$-0.514911\pi$
$2$	−0.264899	−0.0936559	−0.0468280	+	0.998903i	$0.514911\pi$
$3$	6.92505	1.33273	0.666364	−	0.745627i	$-0.267850\pi$
$3$	6.92505	1.33273	0.666364	+	0.745627i	$0.267850\pi$
$4$	−7.92983	−0.991229
$5$	11.7130	1.04764	0.523821	−	0.851828i	$-0.324506\pi$
$5$	11.7130	1.04764	0.523821	+	0.851828i	$0.324506\pi$
$6$	−1.83444	−0.124818
$7$	0	0
$7$	0	0
$8$	4.21980	0.186490
$9$	20.9564	0.776162
$10$	−3.10276	−0.0981179
$11$	−55.2621	−1.51474	−0.757370	−	0.652986i	$-0.773516\pi$
$11$	−55.2621	−1.51474	−0.757370	+	0.652986i	$0.773516\pi$
$12$	−54.9145	−1.32104
$13$	76.7819	1.63811	0.819057	−	0.573712i	$-0.194497\pi$
$13$	76.7819	1.63811	0.819057	+	0.573712i	$0.194497\pi$
$14$	0	0
$15$	81.1131	1.39622
$16$	62.3208	0.973763
$17$	−112.294	−1.60208	−0.801041	−	0.598609i	$-0.795720\pi$
$17$	−112.294	−1.60208	−0.801041	+	0.598609i	$0.795720\pi$
$18$	−5.55133	−0.0726922
$19$	−98.4128	−1.18829	−0.594143	−	0.804359i	$-0.702509\pi$
$19$	−98.4128	−1.18829	−0.594143	+	0.804359i	$0.702509\pi$
$20$	−92.8820	−1.03845
$21$	0	0
$22$	14.6389	0.141864
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	29.2223	0.248541
$25$	12.1943	0.0975541
$26$	−20.3395	−0.153419
$27$	−41.8524	−0.298314
$28$	0	0
$29$	−195.047	−1.24894	−0.624470	−	0.781049i	$-0.714685\pi$
$29$	−195.047	−1.24894	−0.624470	+	0.781049i	$0.714685\pi$
$30$	−21.4868	−0.130764
$31$	99.2668	0.575124	0.287562	−	0.957762i	$-0.407155\pi$
$31$	99.2668	0.575124	0.287562	+	0.957762i	$0.407155\pi$
$32$	−50.2671	−0.277689
$33$	−382.693	−2.01874
$34$	29.7467	0.150045
$35$	0	0
$36$	−166.181	−0.769354
$37$	−14.5629	−0.0647060	−0.0323530	−	0.999477i	$-0.510300\pi$
$37$	−14.5629	−0.0647060	−0.0323530	+	0.999477i	$0.510300\pi$
$38$	26.0694	0.111290
$39$	531.719	2.18316
$40$	49.4264	0.195375
$41$	329.303	1.25435	0.627176	−	0.778878i	$-0.284211\pi$
$41$	329.303	1.25435	0.627176	+	0.778878i	$0.284211\pi$
$42$	0	0
$43$	−275.032	−0.975395	−0.487697	−	0.873013i	$-0.662163\pi$
$43$	−275.032	−0.975395	−0.487697	+	0.873013i	$0.662163\pi$
$44$	438.219	1.50145
$45$	245.462	0.813140
$46$	6.09268	0.0195286
$47$	4.09163	0.0126984	0.00634921	−	0.999980i	$-0.497979\pi$
$47$	4.09163	0.0126984	0.00634921	+	0.999980i	$0.497979\pi$
$48$	431.575	1.29776
$49$	0	0
$50$	−3.23025	−0.00913652
$51$	−777.645	−2.13514
$52$	−608.868	−1.62375
$53$	−254.594	−0.659832	−0.329916	−	0.944010i	$-0.607020\pi$
$53$	−254.594	−0.659832	−0.329916	+	0.944010i	$0.607020\pi$
$54$	11.0866	0.0279389
$55$	−647.285	−1.58691
$56$	0	0
$57$	−681.514	−1.58366
$58$	51.6676	0.116971
$59$	−207.145	−0.457083	−0.228542	−	0.973534i	$-0.573396\pi$
$59$	−207.145	−0.457083	−0.228542	+	0.973534i	$0.573396\pi$
$60$	−643.213	−1.38397
$61$	−721.632	−1.51468	−0.757341	−	0.653020i	$-0.773502\pi$
$61$	−721.632	−1.51468	−0.757341	+	0.653020i	$0.773502\pi$
$62$	−26.2957	−0.0538638
$63$	0	0
$64$	−485.251	−0.947755
$65$	899.347	1.71616
$66$	101.375	0.189067
$67$	−514.780	−0.938662	−0.469331	−	0.883022i	$-0.655505\pi$
$67$	−514.780	−0.938662	−0.469331	+	0.883022i	$0.655505\pi$
$68$	890.476	1.58803
$69$	−159.276	−0.277893
$70$	0	0
$71$	1059.23	1.77053	0.885266	−	0.465085i	$-0.153976\pi$
$71$	1059.23	1.77053	0.885266	+	0.465085i	$0.153976\pi$
$72$	88.4317	0.144747
$73$	376.129	0.603049	0.301525	−	0.953458i	$-0.402504\pi$
$73$	376.129	0.603049	0.301525	+	0.953458i	$0.402504\pi$
$74$	3.85769	0.00606010
$75$	84.4459	0.130013
$76$	780.396	1.17786
$77$	0	0
$78$	−140.852	−0.204466
$79$	849.401	1.20968	0.604842	−	0.796345i	$-0.293236\pi$
$79$	849.401	1.20968	0.604842	+	0.796345i	$0.293236\pi$
$80$	729.963	1.02015
$81$	−855.652	−1.17373
$82$	−87.2319	−0.117477
$83$	−204.086	−0.269896	−0.134948	−	0.990853i	$-0.543087\pi$
$83$	−204.086	−0.269896	−0.134948	+	0.990853i	$0.543087\pi$
$84$	0	0
$85$	−1315.30	−1.67841
$86$	72.8557	0.0913515
$87$	−1350.71	−1.66450
$88$	−233.195	−0.282485
$89$	631.481	0.752099	0.376050	−	0.926599i	$-0.377282\pi$
$89$	631.481	0.752099	0.376050	+	0.926599i	$0.377282\pi$
$90$	−65.0227	−0.0761554
$91$	0	0
$92$	182.386	0.206685
$93$	687.428	0.766483
$94$	−1.08387	−0.00118928
$95$	−1152.71	−1.24490
$96$	−348.102	−0.370084
$97$	−988.987	−1.03522	−0.517610	−	0.855616i	$-0.673178\pi$
$97$	−988.987	−1.03522	−0.517610	+	0.855616i	$0.673178\pi$
$98$	0	0
$99$	−1158.09	−1.17568
$100$	−96.6984	−0.0966984
$101$	179.498	0.176839	0.0884196	−	0.996083i	$-0.471818\pi$
$101$	179.498	0.176839	0.0884196	+	0.996083i	$0.471818\pi$
$102$	205.997	0.199968
$103$	−1643.80	−1.57251	−0.786255	−	0.617903i	$-0.787983\pi$
$103$	−1643.80	−1.57251	−0.786255	+	0.617903i	$0.787983\pi$
$104$	324.004	0.305492
$105$	0	0
$106$	67.4416	0.0617972
$107$	−992.701	−0.896897	−0.448449	−	0.893809i	$-0.648023\pi$
$107$	−992.701	−0.896897	−0.448449	+	0.893809i	$0.648023\pi$
$108$	331.882	0.295698
$109$	21.3103	0.0187262	0.00936310	−	0.999956i	$-0.497020\pi$
$109$	21.3103	0.0187262	0.00936310	+	0.999956i	$0.497020\pi$
$110$	171.465	0.148623
$111$	−100.849	−0.0862354
$112$	0	0
$113$	613.931	0.511095	0.255548	−	0.966796i	$-0.417744\pi$
$113$	613.931	0.511095	0.255548	+	0.966796i	$0.417744\pi$
$114$	180.532	0.148319
$115$	−269.399	−0.218448
$116$	1546.69	1.23798
$117$	1609.07	1.27144
$118$	54.8724	0.0428086
$119$	0	0
$120$	342.281	0.260382
$121$	1722.90	1.29444
$122$	191.160	0.141859
$123$	2280.44	1.67171
$124$	−787.169	−0.570079
$125$	−1321.29	−0.945440
$126$	0	0
$127$	−960.519	−0.671120	−0.335560	−	0.942019i	$-0.608926\pi$
$127$	−960.519	−0.671120	−0.335560	+	0.942019i	$0.608926\pi$
$128$	530.679	0.366452
$129$	−1904.61	−1.29994
$130$	−238.236	−0.160728
$131$	1991.37	1.32814	0.664071	−	0.747670i	$-0.268827\pi$
$131$	1991.37	1.32814	0.664071	+	0.747670i	$0.268827\pi$
$132$	3034.69	2.00103
$133$	0	0
$134$	136.365	0.0879113
$135$	−490.216	−0.312527
$136$	−473.860	−0.298773
$137$	−2774.94	−1.73050	−0.865251	−	0.501338i	$-0.832841\pi$
$137$	−2774.94	−1.73050	−0.865251	+	0.501338i	$0.832841\pi$
$138$	42.1921	0.0260263
$139$	60.9410	0.0371867	0.0185933	−	0.999827i	$-0.494081\pi$
$139$	60.9410	0.0371867	0.0185933	+	0.999827i	$0.494081\pi$
$140$	0	0
$141$	28.3348	0.0169235
$142$	−280.590	−0.165821
$143$	−4243.13	−2.48132
$144$	1306.02	0.755798
$145$	−2284.58	−1.30844
$146$	−99.6362	−0.0564791
$147$	0	0
$148$	115.481	0.0641384
$149$	−625.084	−0.343684	−0.171842	−	0.985125i	$-0.554972\pi$
$149$	−625.084	−0.343684	−0.171842	+	0.985125i	$0.554972\pi$
$150$	−22.3696	−0.0121765
$151$	2266.11	1.22128	0.610639	−	0.791909i	$-0.290913\pi$
$151$	2266.11	1.22128	0.610639	+	0.791909i	$0.290913\pi$
$152$	−415.282	−0.221604
$153$	−2353.29	−1.24348
$154$	0	0
$155$	1162.71	0.602524
$156$	−4216.44	−2.16401
$157$	−2533.35	−1.28779	−0.643895	−	0.765114i	$-0.722683\pi$
$157$	−2533.35	−1.28779	−0.643895	+	0.765114i	$0.722683\pi$
$158$	−225.005	−0.113294
$159$	−1763.07	−0.879377
$160$	−588.778	−0.290919
$161$	0	0
$162$	226.661	0.109927
$163$	2479.38	1.19141	0.595705	−	0.803203i	$-0.296873\pi$
$163$	2479.38	1.19141	0.595705	+	0.803203i	$0.296873\pi$
$164$	−2611.31	−1.24335
$165$	−4482.48	−2.11491
$166$	54.0622	0.0252773
$167$	−440.669	−0.204192	−0.102096	−	0.994775i	$-0.532555\pi$
$167$	−440.669	−0.204192	−0.102096	+	0.994775i	$0.532555\pi$
$168$	0	0
$169$	3698.47	1.68342
$170$	348.423	0.157193
$171$	−2062.38	−0.922303
$172$	2180.96	0.966839
$173$	3111.52	1.36742	0.683712	−	0.729752i	$-0.260364\pi$
$173$	3111.52	1.36742	0.683712	+	0.729752i	$0.260364\pi$
$174$	357.801	0.155890
$175$	0	0
$176$	−3443.98	−1.47500
$177$	−1434.49	−0.609168
$178$	−167.279	−0.0704386
$179$	1846.07	0.770848	0.385424	−	0.922740i	$-0.374055\pi$
$179$	1846.07	0.770848	0.385424	+	0.922740i	$0.374055\pi$
$180$	−1946.47	−0.806008
$181$	2020.32	0.829665	0.414833	−	0.909898i	$-0.363840\pi$
$181$	2020.32	0.829665	0.414833	+	0.909898i	$0.363840\pi$
$182$	0	0
$183$	−4997.34	−2.01866
$184$	−97.0553	−0.0388859
$185$	−170.575	−0.0677887
$186$	−182.099	−0.0717857
$187$	6205.62	2.42674
$188$	−32.4459	−0.0125870
$189$	0	0
$190$	305.351	0.116592
$191$	279.890	0.106032	0.0530160	−	0.998594i	$-0.483117\pi$
$191$	279.890	0.106032	0.0530160	+	0.998594i	$0.483117\pi$
$192$	−3360.39	−1.26310
$193$	−4842.79	−1.80617	−0.903087	−	0.429458i	$-0.858705\pi$
$193$	−4842.79	−1.80617	−0.903087	+	0.429458i	$0.858705\pi$
$194$	261.982	0.0969546
$195$	6228.02	2.28717
$196$	0	0
$197$	−2838.58	−1.02660	−0.513301	−	0.858208i	$-0.671578\pi$
$197$	−2838.58	−1.02660	−0.513301	+	0.858208i	$0.671578\pi$
$198$	306.778	0.110110
$199$	−3449.73	−1.22887	−0.614435	−	0.788968i	$-0.710616\pi$
$199$	−3449.73	−1.22887	−0.614435	+	0.788968i	$0.710616\pi$
$200$	51.4573	0.0181929
$201$	−3564.88	−1.25098
$202$	−47.5490	−0.0165620
$203$	0	0
$204$	6166.59	2.11641
$205$	3857.12	1.31411
$206$	435.441	0.147275
$207$	−481.997	−0.161841
$208$	4785.11	1.59513
$209$	5438.49	1.79995
$210$	0	0
$211$	1249.39	0.407636	0.203818	−	0.979009i	$-0.434665\pi$
$211$	1249.39	0.407636	0.203818	+	0.979009i	$0.434665\pi$
$212$	2018.88	0.654045
$213$	7335.25	2.35964
$214$	262.966	0.0839998
$215$	−3221.45	−1.02186
$216$	−176.608	−0.0556328
$217$	0	0
$218$	−5.64508	−0.00175382
$219$	2604.72	0.803700
$220$	5132.86	1.57299
$221$	−8622.19	−2.62439
$222$	26.7147	0.00807646
$223$	−4170.14	−1.25226	−0.626128	−	0.779720i	$-0.715361\pi$
$223$	−4170.14	−1.25226	−0.626128	+	0.779720i	$0.715361\pi$
$224$	0	0
$225$	255.548	0.0757178
$226$	−162.630	−0.0478671
$227$	3259.90	0.953158	0.476579	−	0.879132i	$-0.341877\pi$
$227$	3259.90	0.953158	0.476579	+	0.879132i	$0.341877\pi$
$228$	5404.29	1.56977
$229$	−3715.07	−1.07205	−0.536023	−	0.844204i	$-0.680074\pi$
$229$	−3715.07	−1.07205	−0.536023	+	0.844204i	$0.680074\pi$
$230$	71.3635	0.0204590
$231$	0	0
$232$	−823.057	−0.232915
$233$	−5851.29	−1.64520	−0.822598	−	0.568624i	$-0.807476\pi$
$233$	−5851.29	−1.64520	−0.822598	+	0.568624i	$0.807476\pi$
$234$	−426.242	−0.119078
$235$	47.9252	0.0133034
$236$	1642.62	0.453074
$237$	5882.15	1.61218
$238$	0	0
$239$	2342.48	0.633985	0.316993	−	0.948428i	$-0.397327\pi$
$239$	2342.48	0.633985	0.316993	+	0.948428i	$0.397327\pi$
$240$	5055.04	1.35959
$241$	−3014.50	−0.805731	−0.402865	−	0.915259i	$-0.631986\pi$
$241$	−3014.50	−0.805731	−0.402865	+	0.915259i	$0.631986\pi$
$242$	−456.394	−0.121232
$243$	−4795.43	−1.26595
$244$	5722.42	1.50140
$245$	0	0
$246$	−604.086	−0.156565
$247$	−7556.32	−1.94655
$248$	418.886	0.107255
$249$	−1413.31	−0.359698
$250$	350.009	0.0885461
$251$	3717.17	0.934763	0.467381	−	0.884056i	$-0.345198\pi$
$251$	3717.17	0.934763	0.467381	+	0.884056i	$0.345198\pi$
$252$	0	0
$253$	1271.03	0.315845
$254$	254.440	0.0628544
$255$	−9108.55	−2.23686
$256$	3741.43	0.913435
$257$	851.188	0.206598	0.103299	−	0.994650i	$-0.467060\pi$
$257$	851.188	0.206598	0.103299	+	0.994650i	$0.467060\pi$
$258$	504.530	0.121747
$259$	0	0
$260$	−7131.66	−1.70110
$261$	−4087.47	−0.969380
$262$	−527.511	−0.124388
$263$	6282.40	1.47296	0.736482	−	0.676458i	$-0.236486\pi$
$263$	6282.40	1.47296	0.736482	+	0.676458i	$0.236486\pi$
$264$	−1614.89	−0.376475
$265$	−2982.05	−0.691268
$266$	0	0
$267$	4373.04	1.00234
$268$	4082.12	0.930429
$269$	−3572.08	−0.809642	−0.404821	−	0.914396i	$-0.632666\pi$
$269$	−3572.08	−0.809642	−0.404821	+	0.914396i	$0.632666\pi$
$270$	129.858	0.0292700
$271$	−97.7240	−0.0219052	−0.0109526	−	0.999940i	$-0.503486\pi$
$271$	−97.7240	−0.0219052	−0.0109526	+	0.999940i	$0.503486\pi$
$272$	−6998.28	−1.56005
$273$	0	0
$274$	735.078	0.162072
$275$	−673.880	−0.147769
$276$	1263.03	0.275455
$277$	−4764.37	−1.03344	−0.516721	−	0.856154i	$-0.672847\pi$
$277$	−4764.37	−1.03344	−0.516721	+	0.856154i	$0.672847\pi$
$278$	−16.1432	−0.00348275
$279$	2080.27	0.446390
$280$	0	0
$281$	−2901.10	−0.615890	−0.307945	−	0.951404i	$-0.599641\pi$
$281$	−2901.10	−0.615890	−0.307945	+	0.951404i	$0.599641\pi$
$282$	−7.50585	−0.00158499
$283$	3015.38	0.633377	0.316688	−	0.948530i	$-0.397429\pi$
$283$	3015.38	0.633377	0.316688	+	0.948530i	$0.397429\pi$
$284$	−8399.53	−1.75500
$285$	−7982.57	−1.65911
$286$	1124.00	0.232390
$287$	0	0
$288$	−1053.42	−0.215532
$289$	7697.04	1.56667
$290$	605.183	0.122543
$291$	−6848.79	−1.37967
$292$	−2982.64	−0.597760
$293$	−2746.50	−0.547619	−0.273810	−	0.961784i	$-0.588284\pi$
$293$	−2746.50	−0.547619	−0.273810	+	0.961784i	$0.588284\pi$
$294$	0	0
$295$	−2426.28	−0.478860
$296$	−61.4523	−0.0120670
$297$	2312.85	0.451869
$298$	165.584	0.0321880
$299$	−1765.98	−0.341570
$300$	−669.642	−0.128873
$301$	0	0
$302$	−600.289	−0.114380
$303$	1243.04	0.235679
$304$	−6133.16	−1.15711
$305$	−8452.48	−1.58684
$306$	623.383	0.116459
$307$	−9858.98	−1.83284	−0.916420	−	0.400219i	$-0.868934\pi$
$307$	−9858.98	−1.83284	−0.916420	+	0.400219i	$0.868934\pi$
$308$	0	0
$309$	−11383.4	−2.09573
$310$	−308.001	−0.0564300
$311$	5543.80	1.01080	0.505402	−	0.862884i	$-0.331344\pi$
$311$	5543.80	1.01080	0.505402	+	0.862884i	$0.331344\pi$
$312$	2243.75	0.407138
$313$	8209.35	1.48249	0.741245	−	0.671234i	$-0.234235\pi$
$313$	8209.35	1.48249	0.741245	+	0.671234i	$0.234235\pi$
$314$	671.081	0.120609
$315$	0	0
$316$	−6735.60	−1.19907
$317$	325.685	0.0577043	0.0288522	−	0.999584i	$-0.490815\pi$
$317$	325.685	0.0577043	0.0288522	+	0.999584i	$0.490815\pi$
$318$	467.037	0.0823589
$319$	10778.7	1.89182
$320$	−5683.74	−0.992909
$321$	−6874.51	−1.19532
$322$	0	0
$323$	11051.2	1.90373
$324$	6785.18	1.16344
$325$	936.299	0.159805
$326$	−656.785	−0.111583
$327$	147.575	0.0249569
$328$	1389.59	0.233924
$329$	0	0
$330$	1187.40	0.198074
$331$	3735.77	0.620352	0.310176	−	0.950679i	$-0.399612\pi$
$331$	3735.77	0.620352	0.310176	+	0.950679i	$0.399612\pi$
$332$	1618.37	0.267528
$333$	−305.185	−0.0502224
$334$	116.733	0.0191238
$335$	−6029.61	−0.983382
$336$	0	0
$337$	−4319.35	−0.698189	−0.349095	−	0.937087i	$-0.613511\pi$
$337$	−4319.35	−0.698189	−0.349095	+	0.937087i	$0.613511\pi$
$338$	−979.720	−0.157662
$339$	4251.50	0.681150
$340$	10430.1	1.66369
$341$	−5485.69	−0.871164
$342$	546.321	0.0863792
$343$	0	0
$344$	−1160.58	−0.181902
$345$	−1865.60	−0.291132
$346$	−824.238	−0.128067
$347$	−4875.70	−0.754298	−0.377149	−	0.926153i	$-0.623096\pi$
$347$	−4875.70	−0.754298	−0.377149	+	0.926153i	$0.623096\pi$
$348$	10710.9	1.64990
$349$	−11218.1	−1.72060	−0.860300	−	0.509788i	$-0.829724\pi$
$349$	−11218.1	−1.72060	−0.860300	+	0.509788i	$0.829724\pi$
$350$	0	0
$351$	−3213.51	−0.488673
$352$	2777.86	0.420627
$353$	3699.99	0.557878	0.278939	−	0.960309i	$-0.410017\pi$
$353$	3699.99	0.557878	0.278939	+	0.960309i	$0.410017\pi$
$354$	379.994	0.0570522
$355$	12406.8	1.85488
$356$	−5007.54	−0.745502
$357$	0	0
$358$	−489.022	−0.0721945
$359$	1556.36	0.228807	0.114403	−	0.993434i	$-0.463504\pi$
$359$	1556.36	0.228807	0.114403	+	0.993434i	$0.463504\pi$
$360$	1035.80	0.151643
$361$	2826.07	0.412024
$362$	−535.181	−0.0777031
$363$	11931.2	1.72513
$364$	0	0
$365$	4405.60	0.631780
$366$	1323.79	0.189059
$367$	13384.4	1.90371	0.951856	−	0.306546i	$-0.0991734\pi$
$367$	13384.4	1.90371	0.951856	+	0.306546i	$0.0991734\pi$
$368$	−1433.38	−0.203044
$369$	6900.99	0.973581
$370$	45.1851	0.00634882
$371$	0	0
$372$	−5451.19	−0.759760
$373$	−708.946	−0.0984125	−0.0492062	−	0.998789i	$-0.515669\pi$
$373$	−708.946	−0.0984125	−0.0492062	+	0.998789i	$0.515669\pi$
$374$	−1643.86	−0.227279
$375$	−9150.03	−1.26001
$376$	17.2658	0.00236813
$377$	−14976.1	−2.04590
$378$	0	0
$379$	6754.31	0.915423	0.457712	−	0.889101i	$-0.348669\pi$
$379$	6754.31	0.915423	0.457712	+	0.889101i	$0.348669\pi$
$380$	9140.78	1.23398
$381$	−6651.64	−0.894420
$382$	−74.1425	−0.00993052
$383$	−4262.89	−0.568730	−0.284365	−	0.958716i	$-0.591783\pi$
$383$	−4262.89	−0.568730	−0.284365	+	0.958716i	$0.591783\pi$
$384$	3674.98	0.488381
$385$	0	0
$386$	1282.85	0.169159
$387$	−5763.68	−0.757065
$388$	7842.50	1.02614
$389$	11230.0	1.46371	0.731857	−	0.681459i	$-0.238654\pi$
$389$	11230.0	1.46371	0.731857	+	0.681459i	$0.238654\pi$
$390$	−1649.80	−0.214207
$391$	2582.77	0.334057
$392$	0	0
$393$	13790.3	1.77005
$394$	751.938	0.0961475
$395$	9949.03	1.26732
$396$	9183.48	1.16537
$397$	−1644.30	−0.207872	−0.103936	−	0.994584i	$-0.533144\pi$
$397$	−1644.30	−0.207872	−0.103936	+	0.994584i	$0.533144\pi$
$398$	913.830	0.115091
$399$	0	0
$400$	759.956	0.0949945
$401$	7463.82	0.929489	0.464745	−	0.885445i	$-0.346146\pi$
$401$	7463.82	0.929489	0.464745	+	0.885445i	$0.346146\pi$
$402$	944.333	0.117162
$403$	7621.90	0.942119
$404$	−1423.39	−0.175288
$405$	−10022.3	−1.22965
$406$	0	0
$407$	804.775	0.0980128
$408$	−3281.50	−0.398183
$409$	−3759.62	−0.454526	−0.227263	−	0.973833i	$-0.572978\pi$
$409$	−3759.62	−0.454526	−0.227263	+	0.973833i	$0.572978\pi$
$410$	−1021.75	−0.123074
$411$	−19216.6	−2.30629
$412$	13035.1	1.55872
$413$	0	0
$414$	127.680	0.0151574
$415$	−2390.46	−0.282754
$416$	−3859.60	−0.454886
$417$	422.020	0.0495597
$418$	−1440.65	−0.168576
$419$	7619.80	0.888428	0.444214	−	0.895921i	$-0.353483\pi$
$419$	7619.80	0.888428	0.444214	+	0.895921i	$0.353483\pi$
$420$	0	0
$421$	1487.35	0.172183	0.0860914	−	0.996287i	$-0.472562\pi$
$421$	1487.35	0.172183	0.0860914	+	0.996287i	$0.472562\pi$
$422$	−330.961	−0.0381776
$423$	85.7458	0.00985603
$424$	−1074.33	−0.123052
$425$	−1369.35	−0.156290
$426$	−1943.10	−0.220994
$427$	0	0
$428$	7871.95	0.889030
$429$	−29383.9	−3.30692
$430$	853.358	0.0957037
$431$	−2649.29	−0.296083	−0.148042	−	0.988981i	$-0.547297\pi$
$431$	−2649.29	−0.296083	−0.148042	+	0.988981i	$0.547297\pi$
$432$	−2608.27	−0.290487
$433$	4224.97	0.468912	0.234456	−	0.972127i	$-0.424669\pi$
$433$	4224.97	0.468912	0.234456	+	0.972127i	$0.424669\pi$
$434$	0	0
$435$	−15820.8	−1.74380
$436$	−168.987	−0.0185619
$437$	2263.49	0.247775
$438$	−689.986	−0.0752713
$439$	14831.9	1.61250	0.806251	−	0.591573i	$-0.201493\pi$
$439$	14831.9	1.61250	0.806251	+	0.591573i	$0.201493\pi$
$440$	−2731.41	−0.295943
$441$	0	0
$442$	2284.01	0.245790
$443$	−4704.72	−0.504578	−0.252289	−	0.967652i	$-0.581183\pi$
$443$	−4704.72	−0.504578	−0.252289	+	0.967652i	$0.581183\pi$
$444$	799.713	0.0854790
$445$	7396.53	0.787931
$446$	1104.67	0.117281
$447$	−4328.74	−0.458037
$448$	0	0
$449$	49.6103	0.00521438	0.00260719	−	0.999997i	$-0.499170\pi$
$449$	49.6103	0.00521438	0.00260719	+	0.999997i	$0.499170\pi$
$450$	−67.6943	−0.00709142
$451$	−18197.9	−1.90002
$452$	−4868.37	−0.506612
$453$	15692.9	1.62763
$454$	−863.543	−0.0892689
$455$	0	0
$456$	−2875.85	−0.295338
$457$	−6277.49	−0.642557	−0.321279	−	0.946985i	$-0.604113\pi$
$457$	−6277.49	−0.642557	−0.321279	+	0.946985i	$0.604113\pi$
$458$	984.117	0.100403
$459$	4699.79	0.477924
$460$	2136.29	0.216532
$461$	11140.2	1.12549	0.562744	−	0.826631i	$-0.309746\pi$
$461$	11140.2	1.12549	0.562744	+	0.826631i	$0.309746\pi$
$462$	0	0
$463$	−6821.86	−0.684749	−0.342374	−	0.939564i	$-0.611231\pi$
$463$	−6821.86	−0.684749	−0.342374	+	0.939564i	$0.611231\pi$
$464$	−12155.5	−1.21617
$465$	8051.84	0.803000
$466$	1550.00	0.154082
$467$	−3374.34	−0.334360	−0.167180	−	0.985926i	$-0.553466\pi$
$467$	−3374.34	−0.334360	−0.167180	+	0.985926i	$0.553466\pi$
$468$	−12759.7	−1.26029
$469$	0	0
$470$	−12.6953	−0.00124594
$471$	−17543.6	−1.71627
$472$	−874.108	−0.0852417
$473$	15198.8	1.47747
$474$	−1558.17	−0.150990
$475$	−1200.07	−0.115922
$476$	0	0
$477$	−5335.36	−0.512137
$478$	−620.521	−0.0593765
$479$	−9644.40	−0.919967	−0.459983	−	0.887928i	$-0.652145\pi$
$479$	−9644.40	−0.919967	−0.459983	+	0.887928i	$0.652145\pi$
$480$	−4077.32	−0.387715
$481$	−1118.17	−0.105996
$482$	798.538	0.0754615
$483$	0	0
$484$	−13662.3	−1.28309
$485$	−11584.0	−1.08454
$486$	1270.30	0.118564
$487$	587.142	0.0546323	0.0273162	−	0.999627i	$-0.491304\pi$
$487$	587.142	0.0546323	0.0273162	+	0.999627i	$0.491304\pi$
$488$	−3045.14	−0.282473
$489$	17169.8	1.58783
$490$	0	0
$491$	17831.2	1.63892	0.819461	−	0.573135i	$-0.194273\pi$
$491$	17831.2	1.63892	0.819461	+	0.573135i	$0.194273\pi$
$492$	−18083.5	−1.65705
$493$	21902.6	2.00090
$494$	2001.66	0.182306
$495$	−13564.7	−1.23170
$496$	6186.39	0.560034
$497$	0	0
$498$	374.384	0.0336878
$499$	7196.23	0.645586	0.322793	−	0.946470i	$-0.395378\pi$
$499$	7196.23	0.645586	0.322793	+	0.946470i	$0.395378\pi$
$500$	10477.6	0.937148
$501$	−3051.66	−0.272132
$502$	−984.673	−0.0875461
$503$	−3011.39	−0.266941	−0.133471	−	0.991053i	$-0.542612\pi$
$503$	−3011.39	−0.266941	−0.133471	+	0.991053i	$0.542612\pi$
$504$	0	0
$505$	2102.46	0.185264
$506$	−336.694	−0.0295808
$507$	25612.1	2.24354
$508$	7616.75	0.665233
$509$	6422.75	0.559299	0.279650	−	0.960102i	$-0.409782\pi$
$509$	6422.75	0.559299	0.279650	+	0.960102i	$0.409782\pi$
$510$	2412.85	0.209495
$511$	0	0
$512$	−5236.53	−0.452001
$513$	4118.81	0.354483
$514$	−225.479	−0.0193491
$515$	−19253.8	−1.64743
$516$	15103.2	1.28853
$517$	−226.112	−0.0192348
$518$	0	0
$519$	21547.4	1.82240
$520$	3795.06	0.320047
$521$	10457.6	0.879378	0.439689	−	0.898150i	$-0.355089\pi$
$521$	10457.6	0.879378	0.439689	+	0.898150i	$0.355089\pi$
$522$	1082.77	0.0907882
$523$	5417.27	0.452927	0.226463	−	0.974020i	$-0.427284\pi$
$523$	5417.27	0.452927	0.226463	+	0.974020i	$0.427284\pi$
$524$	−15791.2	−1.31649
$525$	0	0
$526$	−1664.20	−0.137952
$527$	−11147.1	−0.921396
$528$	−23849.7	−1.96577
$529$	529.000	0.0434783
$530$	789.943	0.0647414
$531$	−4341.00	−0.354771
$532$	0	0
$533$	25284.5	2.05477
$534$	−1158.41	−0.0938754
$535$	−11627.5	−0.939628
$536$	−2172.27	−0.175051
$537$	12784.1	1.02733
$538$	946.242	0.0758278
$539$	0	0
$540$	3887.33	0.309785
$541$	17716.7	1.40795	0.703975	−	0.710224i	$-0.251406\pi$
$541$	17716.7	1.40795	0.703975	+	0.710224i	$0.251406\pi$
$542$	25.8870	0.00205155
$543$	13990.8	1.10572
$544$	5644.71	0.444881
$545$	249.607	0.0196184
$546$	0	0
$547$	11555.7	0.903266	0.451633	−	0.892204i	$-0.350842\pi$
$547$	11555.7	0.903266	0.451633	+	0.892204i	$0.350842\pi$
$548$	22004.8	1.71532
$549$	−15122.8	−1.17564
$550$	178.510	0.0138395
$551$	19195.1	1.48410
$552$	−672.113	−0.0518244
$553$	0	0
$554$	1262.08	0.0967879
$555$	−1181.24	−0.0903439
$556$	−483.251	−0.0368605
$557$	−4915.47	−0.373923	−0.186962	−	0.982367i	$-0.559864\pi$
$557$	−4915.47	−0.373923	−0.186962	+	0.982367i	$0.559864\pi$
$558$	−551.062	−0.0418070
$559$	−21117.5	−1.59781
$560$	0	0
$561$	42974.3	3.23418
$562$	768.498	0.0576818
$563$	17080.7	1.27862	0.639312	−	0.768947i	$-0.279219\pi$
$563$	17080.7	1.27862	0.639312	+	0.768947i	$0.279219\pi$
$564$	−224.690	−0.0167751
$565$	7190.97	0.535445
$566$	−798.770	−0.0593195
$567$	0	0
$568$	4469.75	0.330187
$569$	26856.6	1.97871	0.989357	−	0.145507i	$-0.0464813\pi$
$569$	26856.6	1.97871	0.989357	+	0.145507i	$0.0464813\pi$
$570$	2114.57	0.155386
$571$	10125.2	0.742077	0.371039	−	0.928617i	$-0.379002\pi$
$571$	10125.2	0.742077	0.371039	+	0.928617i	$0.379002\pi$
$572$	33647.3	2.45955
$573$	1938.25	0.141312
$574$	0	0
$575$	−280.468	−0.0203414
$576$	−10169.1	−0.735612
$577$	−10605.0	−0.765147	−0.382574	−	0.923925i	$-0.624962\pi$
$577$	−10605.0	−0.765147	−0.382574	+	0.923925i	$0.624962\pi$
$578$	−2038.94	−0.146728
$579$	−33536.6	−2.40714
$580$	18116.3	1.29696
$581$	0	0
$582$	1814.24	0.129214
$583$	14069.4	0.999475
$584$	1587.19	0.112463
$585$	18847.1	1.33202
$586$	727.546	0.0512878
$587$	2267.45	0.159434	0.0797169	−	0.996818i	$-0.474598\pi$
$587$	2267.45	0.159434	0.0797169	+	0.996818i	$0.474598\pi$
$588$	0	0
$589$	−9769.12	−0.683412
$590$	642.720	0.0448481
$591$	−19657.4	−1.36818
$592$	−907.570	−0.0630083
$593$	−27734.2	−1.92059	−0.960294	−	0.278989i	$-0.910001\pi$
$593$	−27734.2	−1.92059	−0.960294	+	0.278989i	$0.910001\pi$
$594$	−612.671	−0.0423202
$595$	0	0
$596$	4956.81	0.340669
$597$	−23889.6	−1.63775
$598$	467.808	0.0319901
$599$	−8268.56	−0.564014	−0.282007	−	0.959412i	$-0.591000\pi$
$599$	−8268.56	−0.564014	−0.282007	+	0.959412i	$0.591000\pi$
$600$	356.344	0.0242462
$601$	−1545.61	−0.104903	−0.0524515	−	0.998623i	$-0.516703\pi$
$601$	−1545.61	−0.104903	−0.0524515	+	0.998623i	$0.516703\pi$
$602$	0	0
$603$	−10787.9	−0.728554
$604$	−17969.8	−1.21057
$605$	20180.3	1.35611
$606$	−329.279	−0.0220727
$607$	−17902.6	−1.19711	−0.598554	−	0.801083i	$-0.704258\pi$
$607$	−17902.6	−1.19711	−0.598554	+	0.801083i	$0.704258\pi$
$608$	4946.92	0.329974
$609$	0	0
$610$	2239.05	0.148617
$611$	314.163	0.0208015
$612$	18661.1	1.23257
$613$	−13780.7	−0.907986	−0.453993	−	0.891005i	$-0.650001\pi$
$613$	−13780.7	−0.907986	−0.453993	+	0.891005i	$0.650001\pi$
$614$	2611.63	0.171656
$615$	26710.8	1.75135
$616$	0	0
$617$	12935.0	0.843995	0.421998	−	0.906597i	$-0.361329\pi$
$617$	12935.0	0.843995	0.421998	+	0.906597i	$0.361329\pi$
$618$	3015.45	0.196277
$619$	28162.3	1.82866	0.914329	−	0.404971i	$-0.132719\pi$
$619$	28162.3	1.82866	0.914329	+	0.404971i	$0.132719\pi$
$620$	−9220.10	−0.597239
$621$	962.604	0.0622029
$622$	−1468.55	−0.0946679
$623$	0	0
$624$	33137.2	2.12588
$625$	−17000.6	−1.08804
$626$	−2174.65	−0.138844
$627$	37661.9	2.39884
$628$	20089.0	1.27649
$629$	1635.33	0.103664
$630$	0	0
$631$	18619.8	1.17471	0.587356	−	0.809328i	$-0.300169\pi$
$631$	18619.8	1.17471	0.587356	+	0.809328i	$0.300169\pi$
$632$	3584.30	0.225594
$633$	8652.07	0.543268
$634$	−86.2735	−0.00540435
$635$	−11250.5	−0.703094
$636$	13980.9	0.871663
$637$	0	0
$638$	−2855.26	−0.177180
$639$	22197.7	1.37422
$640$	6215.84	0.383911
$641$	4403.53	0.271340	0.135670	−	0.990754i	$-0.456681\pi$
$641$	4403.53	0.271340	0.135670	+	0.990754i	$0.456681\pi$
$642$	1821.05	0.111949
$643$	9833.46	0.603101	0.301550	−	0.953450i	$-0.402496\pi$
$643$	9833.46	0.603101	0.301550	+	0.953450i	$0.402496\pi$
$644$	0	0
$645$	−22308.7	−1.36187
$646$	−2927.45	−0.178296
$647$	−11633.7	−0.706908	−0.353454	−	0.935452i	$-0.614993\pi$
$647$	−11633.7	−0.706908	−0.353454	+	0.935452i	$0.614993\pi$
$648$	−3610.68	−0.218890
$649$	11447.2	0.692363
$650$	−248.025	−0.0149667
$651$	0	0
$652$	−19661.0	−1.18096
$653$	26410.4	1.58272	0.791361	−	0.611349i	$-0.209373\pi$
$653$	26410.4	1.58272	0.791361	+	0.611349i	$0.209373\pi$
$654$	−39.0925	−0.00233736
$655$	23324.9	1.39142
$656$	20522.4	1.22144
$657$	7882.31	0.468064
$658$	0	0
$659$	6928.02	0.409525	0.204763	−	0.978812i	$-0.434358\pi$
$659$	6928.02	0.409525	0.204763	+	0.978812i	$0.434358\pi$
$660$	35545.3	2.09636
$661$	3672.24	0.216087	0.108044	−	0.994146i	$-0.465541\pi$
$661$	3672.24	0.216087	0.108044	+	0.994146i	$0.465541\pi$
$662$	−989.601	−0.0580996
$663$	−59709.1	−3.49760
$664$	−861.201	−0.0503330
$665$	0	0
$666$	80.8433	0.00470362
$667$	4486.07	0.260422
$668$	3494.43	0.202401
$669$	−28878.4	−1.66892
$670$	1597.24	0.0920996
$671$	39878.9	2.29435
$672$	0	0
$673$	−15522.8	−0.889092	−0.444546	−	0.895756i	$-0.646635\pi$
$673$	−15522.8	−0.889092	−0.444546	+	0.895756i	$0.646635\pi$
$674$	1144.19	0.0653896
$675$	−510.358	−0.0291018
$676$	−29328.2	−1.66865
$677$	5347.48	0.303575	0.151788	−	0.988413i	$-0.451497\pi$
$677$	5347.48	0.303575	0.151788	+	0.988413i	$0.451497\pi$
$678$	−1126.22	−0.0637938
$679$	0	0
$680$	−5550.31	−0.313007
$681$	22575.0	1.27030
$682$	1453.15	0.0815896
$683$	7008.63	0.392647	0.196324	−	0.980539i	$-0.437100\pi$
$683$	7008.63	0.392647	0.196324	+	0.980539i	$0.437100\pi$
$684$	16354.3	0.914213
$685$	−32502.8	−1.81295
$686$	0	0
$687$	−25727.0	−1.42874
$688$	−17140.2	−0.949803
$689$	−19548.2	−1.08088
$690$	494.196	0.0272663
$691$	2542.80	0.139989	0.0699947	−	0.997547i	$-0.477702\pi$
$691$	2542.80	0.139989	0.0699947	+	0.997547i	$0.477702\pi$
$692$	−24673.8	−1.35543
$693$	0	0
$694$	1291.57	0.0706445
$695$	713.801	0.0389583
$696$	−5699.71	−0.310412
$697$	−36978.8	−2.00957
$698$	2971.65	0.161144
$699$	−40520.5	−2.19260
$700$	0	0
$701$	5986.08	0.322527	0.161263	−	0.986911i	$-0.448443\pi$
$701$	5986.08	0.322527	0.161263	+	0.986911i	$0.448443\pi$
$702$	851.254	0.0457671
$703$	1433.17	0.0768892
$704$	26816.0	1.43560
$705$	331.885	0.0177298
$706$	−980.125	−0.0522486
$707$	0	0
$708$	11375.2	0.603824
$709$	−23850.9	−1.26339	−0.631693	−	0.775219i	$-0.717640\pi$
$709$	−23850.9	−1.26339	−0.631693	+	0.775219i	$0.717640\pi$
$710$	−3286.55	−0.173721
$711$	17800.4	0.938911
$712$	2664.72	0.140259
$713$	−2283.14	−0.119922
$714$	0	0
$715$	−49699.8	−2.59953
$716$	−14639.0	−0.764087
$717$	16221.8	0.844929
$718$	−412.279	−0.0214291
$719$	−16198.0	−0.840173	−0.420086	−	0.907484i	$-0.638000\pi$
$719$	−16198.0	−0.840173	−0.420086	+	0.907484i	$0.638000\pi$
$720$	15297.4	0.791806
$721$	0	0
$722$	−748.623	−0.0385885
$723$	−20875.6	−1.07382
$724$	−16020.8	−0.822388
$725$	−2378.45	−0.121839
$726$	−3160.55	−0.161569
$727$	−14714.5	−0.750662	−0.375331	−	0.926891i	$-0.622471\pi$
$727$	−14714.5	−0.750662	−0.375331	+	0.926891i	$0.622471\pi$
$728$	0	0
$729$	−10106.0	−0.513437
$730$	−1167.04	−0.0591699
$731$	30884.6	1.56266
$732$	39628.1	2.00095
$733$	−22554.3	−1.13651	−0.568255	−	0.822852i	$-0.692381\pi$
$733$	−22554.3	−1.13651	−0.568255	+	0.822852i	$0.692381\pi$
$734$	−3545.53	−0.178294
$735$	0	0
$736$	1156.14	0.0579022
$737$	28447.8	1.42183
$738$	−1828.07	−0.0911816
$739$	−30940.6	−1.54015	−0.770073	−	0.637956i	$-0.779780\pi$
$739$	−30940.6	−1.54015	−0.770073	+	0.637956i	$0.779780\pi$
$740$	1352.63	0.0671941
$741$	−52328.0	−2.59422
$742$	0	0
$743$	1548.59	0.0764632	0.0382316	−	0.999269i	$-0.487828\pi$
$743$	1548.59	0.0764632	0.0382316	+	0.999269i	$0.487828\pi$
$744$	2900.81	0.142942
$745$	−7321.60	−0.360057
$746$	187.799	0.00921691
$747$	−4276.91	−0.209483
$748$	−49209.5	−2.40545
$749$	0	0
$750$	2423.83	0.118008
$751$	3631.60	0.176457	0.0882284	−	0.996100i	$-0.471879\pi$
$751$	3631.60	0.176457	0.0882284	+	0.996100i	$0.471879\pi$
$752$	254.994	0.0123652
$753$	25741.6	1.24578
$754$	3967.14	0.191611
$755$	26542.9	1.27946
$756$	0	0
$757$	10320.4	0.495513	0.247756	−	0.968822i	$-0.420307\pi$
$757$	10320.4	0.495513	0.247756	+	0.968822i	$0.420307\pi$
$758$	−1789.21	−0.0857348
$759$	8801.94	0.420936
$760$	−4864.19	−0.232162
$761$	−28242.0	−1.34530	−0.672650	−	0.739961i	$-0.734844\pi$
$761$	−28242.0	−1.34530	−0.672650	+	0.739961i	$0.734844\pi$
$762$	1762.01	0.0837677
$763$	0	0
$764$	−2219.48	−0.105102
$765$	−27564.0	−1.30272
$766$	1129.24	0.0532650
$767$	−15905.0	−0.748755
$768$	25909.6	1.21736
$769$	2021.61	0.0947997	0.0473998	−	0.998876i	$-0.484907\pi$
$769$	2021.61	0.0947997	0.0473998	+	0.998876i	$0.484907\pi$
$770$	0	0
$771$	5894.52	0.275339
$772$	38402.5	1.79033
$773$	−1544.52	−0.0718661	−0.0359330	−	0.999354i	$-0.511440\pi$
$773$	−1544.52	−0.0718661	−0.0359330	+	0.999354i	$0.511440\pi$
$774$	1526.79	0.0709036
$775$	1210.48	0.0561057
$776$	−4173.32	−0.193059
$777$	0	0
$778$	−2974.82	−0.137085
$779$	−32407.6	−1.49053
$780$	−49387.2	−2.26711
$781$	−58535.4	−2.68190
$782$	−684.174	−0.0312864
$783$	8163.16	0.372577
$784$	0	0
$785$	−29673.1	−1.34914
$786$	−3653.04	−0.165776
$787$	13130.7	0.594736	0.297368	−	0.954763i	$-0.403891\pi$
$787$	13130.7	0.594736	0.297368	+	0.954763i	$0.403891\pi$
$788$	22509.5	1.01760
$789$	43506.0	1.96306
$790$	−2635.49	−0.118692
$791$	0	0
$792$	−4886.92	−0.219254
$793$	−55408.3	−2.48122
$794$	435.574	0.0194685
$795$	−20650.9	−0.921272
$796$	27355.8	1.21809
$797$	37425.0	1.66332	0.831658	−	0.555288i	$-0.187392\pi$
$797$	37425.0	1.66332	0.831658	+	0.555288i	$0.187392\pi$
$798$	0	0
$799$	−459.467	−0.0203439
$800$	−612.970	−0.0270897
$801$	13233.6	0.583751
$802$	−1977.16	−0.0870522
$803$	−20785.7	−0.913463
$804$	28268.9	1.24001
$805$	0	0
$806$	−2019.03	−0.0882350
$807$	−24736.9	−1.07903
$808$	757.447	0.0329788
$809$	25725.0	1.11797	0.558987	−	0.829176i	$-0.311190\pi$
$809$	25725.0	1.11797	0.558987	+	0.829176i	$0.311190\pi$
$810$	2654.88	0.115164
$811$	−12076.9	−0.522908	−0.261454	−	0.965216i	$-0.584202\pi$
$811$	−12076.9	−0.522908	−0.261454	+	0.965216i	$0.584202\pi$
$812$	0	0
$813$	−676.744	−0.0291937
$814$	−213.184	−0.00917948
$815$	29040.9	1.24817
$816$	−48463.5	−2.07912
$817$	27066.7	1.15905
$818$	995.919	0.0425691
$819$	0	0
$820$	−30586.3	−1.30258
$821$	−5025.21	−0.213619	−0.106809	−	0.994280i	$-0.534063\pi$
$821$	−5025.21	−0.213619	−0.106809	+	0.994280i	$0.534063\pi$
$822$	5090.46	0.215998
$823$	−6388.20	−0.270569	−0.135285	−	0.990807i	$-0.543195\pi$
$823$	−6388.20	−0.270569	−0.135285	+	0.990807i	$0.543195\pi$
$824$	−6936.50	−0.293258
$825$	−4666.66	−0.196936
$826$	0	0
$827$	3250.99	0.136697	0.0683483	−	0.997662i	$-0.478227\pi$
$827$	3250.99	0.136697	0.0683483	+	0.997662i	$0.478227\pi$
$828$	3822.15	0.160421
$829$	26138.3	1.09508	0.547540	−	0.836780i	$-0.315565\pi$
$829$	26138.3	1.09508	0.547540	+	0.836780i	$0.315565\pi$
$830$	633.230	0.0264816
$831$	−32993.5	−1.37730
$832$	−37258.5	−1.55253
$833$	0	0
$834$	−111.793	−0.00464156
$835$	−5161.55	−0.213920
$836$	−43126.3	−1.78416
$837$	−4154.55	−0.171568
$838$	−2018.48	−0.0832066
$839$	−23507.5	−0.967307	−0.483654	−	0.875260i	$-0.660691\pi$
$839$	−23507.5	−0.967307	−0.483654	+	0.875260i	$0.660691\pi$
$840$	0	0
$841$	13654.2	0.559849
$842$	−393.997	−0.0161259
$843$	−20090.3	−0.820813
$844$	−9907.42	−0.404061
$845$	43320.1	1.76362
$846$	−22.7140	−0.000923076 0
$847$	0	0
$848$	−15866.5	−0.642520
$849$	20881.7	0.844118
$850$	362.739	0.0146375
$851$	334.946	0.0134921
$852$	−58167.2	−2.33894
$853$	−19853.1	−0.796901	−0.398450	−	0.917190i	$-0.630452\pi$
$853$	−19853.1	−0.796901	−0.398450	+	0.917190i	$0.630452\pi$
$854$	0	0
$855$	−24156.6	−0.966244
$856$	−4189.00	−0.167263
$857$	−16851.3	−0.671678	−0.335839	−	0.941919i	$-0.609020\pi$
$857$	−16851.3	−0.671678	−0.335839	+	0.941919i	$0.609020\pi$
$858$	7783.77	0.309713
$859$	−36623.8	−1.45470	−0.727350	−	0.686267i	$-0.759248\pi$
$859$	−36623.8	−1.45470	−0.727350	+	0.686267i	$0.759248\pi$
$860$	25545.5	1.01290
$861$	0	0
$862$	701.795	0.0277300
$863$	−93.2237	−0.00367714	−0.00183857	−	0.999998i	$-0.500585\pi$
$863$	−93.2237	−0.00367714	−0.00183857	+	0.999998i	$0.500585\pi$
$864$	2103.80	0.0828387
$865$	36445.2	1.43257
$866$	−1119.19	−0.0439164
$867$	53302.4	2.08794
$868$	0	0
$869$	−46939.7	−1.83236
$870$	4190.92	0.163317
$871$	−39525.8	−1.53764
$872$	89.9251	0.00349226
$873$	−20725.6	−0.803499
$874$	−599.597	−0.0232056
$875$	0	0
$876$	−20654.9	−0.796651
$877$	30806.3	1.18615	0.593075	−	0.805147i	$-0.297914\pi$
$877$	30806.3	1.18615	0.593075	+	0.805147i	$0.297914\pi$
$878$	−3928.96	−0.151020
$879$	−19019.7	−0.729827
$880$	−40339.3	−1.54527
$881$	−30300.4	−1.15874	−0.579369	−	0.815066i	$-0.696701\pi$
$881$	−30300.4	−1.15874	−0.579369	+	0.815066i	$0.696701\pi$
$882$	0	0
$883$	−23271.1	−0.886902	−0.443451	−	0.896299i	$-0.646246\pi$
$883$	−23271.1	−0.886902	−0.443451	+	0.896299i	$0.646246\pi$
$884$	68372.5	2.60137
$885$	−16802.1	−0.638190
$886$	1246.28	0.0472567
$887$	1751.59	0.0663053	0.0331526	−	0.999450i	$-0.489445\pi$
$887$	1751.59	0.0663053	0.0331526	+	0.999450i	$0.489445\pi$
$888$	−425.561	−0.0160821
$889$	0	0
$890$	−1959.33	−0.0737944
$891$	47285.1	1.77790
$892$	33068.5	1.24127
$893$	−402.669	−0.0150894
$894$	1146.68	0.0428978
$895$	21623.0	0.807573
$896$	0	0
$897$	−12229.5	−0.455220
$898$	−13.1417	−0.000488357 0
$899$	−19361.6	−0.718295
$900$	−2026.45	−0.0750536
$901$	28589.4	1.05711
$902$	4820.62	0.177948
$903$	0	0
$904$	2590.66	0.0953143
$905$	23664.0	0.869192
$906$	−4157.04	−0.152437
$907$	38409.6	1.40614	0.703071	−	0.711120i	$-0.251812\pi$
$907$	38409.6	1.40614	0.703071	+	0.711120i	$0.251812\pi$
$908$	−25850.4	−0.944797
$909$	3761.64	0.137256
$910$	0	0
$911$	−45581.0	−1.65770	−0.828850	−	0.559471i	$-0.811004\pi$
$911$	−45581.0	−1.65770	−0.828850	+	0.559471i	$0.811004\pi$
$912$	−42472.5	−1.54211
$913$	11278.2	0.408822
$914$	1662.90	0.0601793
$915$	−58533.9	−2.11483
$916$	29459.8	1.06264
$917$	0	0
$918$	−1244.97	−0.0447604
$919$	−44113.8	−1.58344	−0.791718	−	0.610886i	$-0.790813\pi$
$919$	−44113.8	−1.58344	−0.791718	+	0.610886i	$0.790813\pi$
$920$	−1136.81	−0.0407385
$921$	−68274.0	−2.44268
$922$	−2951.02	−0.105409
$923$	81330.0	2.90033
$924$	0	0
$925$	−177.583	−0.00631233
$926$	1807.10	0.0641308
$927$	−34448.1	−1.22052
$928$	9804.42	0.346817
$929$	14983.8	0.529173	0.264587	−	0.964362i	$-0.414764\pi$
$929$	14983.8	0.529173	0.264587	+	0.964362i	$0.414764\pi$
$930$	−2132.92	−0.0752058
$931$	0	0
$932$	46399.7	1.63076
$933$	38391.2	1.34713
$934$	893.861	0.0313148
$935$	72686.5	2.54235
$936$	6789.96	0.237112
$937$	−14492.9	−0.505295	−0.252647	−	0.967558i	$-0.581301\pi$
$937$	−14492.9	−0.505295	−0.252647	+	0.967558i	$0.581301\pi$
$938$	0	0
$939$	56850.2	1.97576
$940$	−380.039	−0.0131867
$941$	−39590.2	−1.37152	−0.685762	−	0.727826i	$-0.740531\pi$
$941$	−39590.2	−1.37152	−0.685762	+	0.727826i	$0.740531\pi$
$942$	4647.27	0.160739
$943$	−7573.96	−0.261550
$944$	−12909.4	−0.445091
$945$	0	0
$946$	−4026.16	−0.138374
$947$	−7209.23	−0.247380	−0.123690	−	0.992321i	$-0.539473\pi$
$947$	−7209.23	−0.247380	−0.123690	+	0.992321i	$0.539473\pi$
$948$	−46644.4	−1.59804
$949$	28879.9	0.987863
$950$	317.897	0.0108568
$951$	2255.38	0.0769041
$952$	0	0
$953$	−6893.39	−0.234312	−0.117156	−	0.993114i	$-0.537378\pi$
$953$	−6893.39	−0.234312	−0.117156	+	0.993114i	$0.537378\pi$
$954$	1413.33	0.0479647
$955$	3278.35	0.111084
$956$	−18575.5	−0.628424
$957$	74643.0	2.52128
$958$	2554.79	0.0861603
$959$	0	0
$960$	−39360.2	−1.32328
$961$	−19937.1	−0.669232
$962$	296.201	0.00992713
$963$	−20803.4	−0.696138
$964$	23904.5	0.798663
$965$	−56723.6	−1.89222
$966$	0	0
$967$	50638.5	1.68400	0.841998	−	0.539480i	$-0.181379\pi$
$967$	50638.5	1.68400	0.841998	+	0.539480i	$0.181379\pi$
$968$	7270.28	0.241400
$969$	76530.2	2.53716
$970$	3068.59	0.101574
$971$	−27159.4	−0.897619	−0.448809	−	0.893628i	$-0.648152\pi$
$971$	−27159.4	−0.897619	−0.448809	+	0.893628i	$0.648152\pi$
$972$	38026.9	1.25485
$973$	0	0
$974$	−155.533	−0.00511664
$975$	6483.92	0.212976
$976$	−44972.7	−1.47494
$977$	−11109.1	−0.363778	−0.181889	−	0.983319i	$-0.558221\pi$
$977$	−11109.1	−0.363778	−0.181889	+	0.983319i	$0.558221\pi$
$978$	−4548.27	−0.148709
$979$	−34897.0	−1.13924
$980$	0	0
$981$	446.587	0.0145346
$982$	−4723.47	−0.153495
$983$	−27203.6	−0.882666	−0.441333	−	0.897343i	$-0.645494\pi$
$983$	−27203.6	−0.882666	−0.441333	+	0.897343i	$0.645494\pi$
$984$	9622.98	0.311758
$985$	−33248.3	−1.07551
$986$	−5801.99	−0.187397
$987$	0	0
$988$	59920.4	1.92947
$989$	6325.73	0.203384
$990$	3593.29	0.115356
$991$	−30907.6	−0.990729	−0.495364	−	0.868685i	$-0.664965\pi$
$991$	−30907.6	−0.990729	−0.495364	+	0.868685i	$0.664965\pi$
$992$	−4989.85	−0.159706
$993$	25870.4	0.826760
$994$	0	0
$995$	−40406.7	−1.28742
$996$	11207.3	0.356543
$997$	−12224.2	−0.388309	−0.194154	−	0.980971i	$-0.562196\pi$
$997$	−12224.2	−0.388309	−0.194154	+	0.980971i	$0.562196\pi$
$998$	−1906.27	−0.0604629
$999$	609.491	0.0193027

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1127.4.a.f.1.5		9
7.6	odd	2		161.4.a.c.1.5	✓	9
21.20	even	2		1449.4.a.n.1.5		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.4.a.c.1.5	✓	9	7.6	odd	2
1127.4.a.f.1.5		9	1.1	even	1	trivial
1449.4.a.n.1.5		9	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1127 = 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1127.a (trivial)

\(f(q)\)	\(=\)	\(q-0.264899 q^{2} +6.92505 q^{3} -7.92983 q^{4} +11.7130 q^{5} -1.83444 q^{6} +4.21980 q^{8} +20.9564 q^{9} -3.10276 q^{10} -55.2621 q^{11} -54.9145 q^{12} +76.7819 q^{13} +81.1131 q^{15} +62.3208 q^{16} -112.294 q^{17} -5.55133 q^{18} -98.4128 q^{19} -92.8820 q^{20} +14.6389 q^{22} -23.0000 q^{23} +29.2223 q^{24} +12.1943 q^{25} -20.3395 q^{26} -41.8524 q^{27} -195.047 q^{29} -21.4868 q^{30} +99.2668 q^{31} -50.2671 q^{32} -382.693 q^{33} +29.7467 q^{34} -166.181 q^{36} -14.5629 q^{37} +26.0694 q^{38} +531.719 q^{39} +49.4264 q^{40} +329.303 q^{41} -275.032 q^{43} +438.219 q^{44} +245.462 q^{45} +6.09268 q^{46} +4.09163 q^{47} +431.575 q^{48} -3.23025 q^{50} -777.645 q^{51} -608.868 q^{52} -254.594 q^{53} +11.0866 q^{54} -647.285 q^{55} -681.514 q^{57} +51.6676 q^{58} -207.145 q^{59} -643.213 q^{60} -721.632 q^{61} -26.2957 q^{62} -485.251 q^{64} +899.347 q^{65} +101.375 q^{66} -514.780 q^{67} +890.476 q^{68} -159.276 q^{69} +1059.23 q^{71} +88.4317 q^{72} +376.129 q^{73} +3.85769 q^{74} +84.4459 q^{75} +780.396 q^{76} -140.852 q^{78} +849.401 q^{79} +729.963 q^{80} -855.652 q^{81} -87.2319 q^{82} -204.086 q^{83} -1315.30 q^{85} +72.8557 q^{86} -1350.71 q^{87} -233.195 q^{88} +631.481 q^{89} -65.0227 q^{90} +182.386 q^{92} +687.428 q^{93} -1.08387 q^{94} -1152.71 q^{95} -348.102 q^{96} -988.987 q^{97} -1158.09 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{3} + 48 q^{4} + 4 q^{5} - 46 q^{6} + 66 q^{8} + 122 q^{9} - 50 q^{10} - 8 q^{11} - 220 q^{12} - 25 q^{13} + 88 q^{15} + 180 q^{16} + 28 q^{17} - 54 q^{18} - 254 q^{19} - 302 q^{20} - 122 q^{22}+ \cdots - 618 q^{99}+O(q^{100}) \) 9 * q - 9 * q^3 + 48 * q^4 + 4 * q^5 - 46 * q^6 + 66 * q^8 + 122 * q^9 - 50 * q^10 - 8 * q^11 - 220 * q^12 - 25 * q^13 + 88 * q^15 + 180 * q^16 + 28 * q^17 - 54 * q^18 - 254 * q^19 - 302 * q^20 - 122 * q^22 - 207 * q^23 - 624 * q^24 + 295 * q^25 + 6 * q^26 - 633 * q^27 + 25 * q^29 + 80 * q^30 - 1171 * q^31 + 1018 * q^32 - 272 * q^33 - 8 * q^34 + 72 * q^36 + 70 * q^37 - 282 * q^38 + 1185 * q^39 + 54 * q^40 - 221 * q^41 - 42 * q^43 - 1214 * q^44 - 698 * q^45 - 159 * q^47 + 1104 * q^48 - 1680 * q^50 + 308 * q^51 + 664 * q^52 - 774 * q^53 + 2674 * q^54 - 1498 * q^55 - 2524 * q^57 + 842 * q^58 - 1080 * q^59 - 2160 * q^60 - 686 * q^61 + 2078 * q^62 - 1260 * q^64 - 1656 * q^65 - 532 * q^66 - 370 * q^67 + 1936 * q^68 + 207 * q^69 + 1035 * q^71 - 722 * q^72 - 1979 * q^73 - 5494 * q^74 + 1459 * q^75 - 206 * q^76 - 2066 * q^78 + 2336 * q^79 + 242 * q^80 + 997 * q^81 - 1642 * q^82 - 130 * q^83 - 272 * q^85 - 1906 * q^86 + 581 * q^87 - 3742 * q^88 + 1328 * q^89 + 7650 * q^90 - 1104 * q^92 + 1305 * q^93 - 6078 * q^94 - 484 * q^95 + 136 * q^96 + 104 * q^97 - 618 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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