LMFDB - Embedded newform 2175.4.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,4,Mod(1,2175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2175.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2175 = 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2175.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:	$128.329154262$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 37x^{5} + 55x^{4} + 336x^{3} - 227x^{2} - 824x - 166 $ x^7 - 2x^6 - 37x^5 + 55x^4 + 336x^3 - 227x^2 - 824x - 166
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.58378$ of defining polynomial
Character	$\chi$	$=$	2175.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.58378 q^{2} +3.00000 q^{3} -1.32408 q^{4} -7.75134 q^{6} +26.6398 q^{7} +24.0914 q^{8} +9.00000 q^{9} -8.03338 q^{11} -3.97225 q^{12} -49.9367 q^{13} -68.8314 q^{14} -51.6541 q^{16} -116.617 q^{17} -23.2540 q^{18} +22.6617 q^{19} +79.9195 q^{21} +20.7565 q^{22} -38.2616 q^{23} +72.2741 q^{24} +129.025 q^{26} +27.0000 q^{27} -35.2733 q^{28} +29.0000 q^{29} -8.82528 q^{31} -59.2681 q^{32} -24.1002 q^{33} +301.312 q^{34} -11.9167 q^{36} -301.022 q^{37} -58.5529 q^{38} -149.810 q^{39} +305.401 q^{41} -206.494 q^{42} -275.872 q^{43} +10.6369 q^{44} +98.8596 q^{46} +479.282 q^{47} -154.962 q^{48} +366.680 q^{49} -349.850 q^{51} +66.1204 q^{52} +496.683 q^{53} -69.7621 q^{54} +641.790 q^{56} +67.9852 q^{57} -74.9296 q^{58} -563.505 q^{59} +485.346 q^{61} +22.8026 q^{62} +239.758 q^{63} +566.369 q^{64} +62.2695 q^{66} -217.176 q^{67} +154.410 q^{68} -114.785 q^{69} +628.023 q^{71} +216.822 q^{72} +682.502 q^{73} +777.773 q^{74} -30.0060 q^{76} -214.008 q^{77} +387.076 q^{78} +1159.26 q^{79} +81.0000 q^{81} -789.089 q^{82} -668.594 q^{83} -105.820 q^{84} +712.792 q^{86} +87.0000 q^{87} -193.535 q^{88} +798.300 q^{89} -1330.31 q^{91} +50.6616 q^{92} -26.4758 q^{93} -1238.36 q^{94} -177.804 q^{96} -867.116 q^{97} -947.421 q^{98} -72.3005 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{2} + 21 q^{3} + 22 q^{4} + 6 q^{6} + 50 q^{7} + 33 q^{8} + 63 q^{9} + 76 q^{11} + 66 q^{12} - 30 q^{13} + 89 q^{14} + 138 q^{16} + 140 q^{17} + 18 q^{18} + 90 q^{19} + 150 q^{21} - 61 q^{22}+ \cdots + 684 q^{99}+O(q^{100}) $ 7 * q + 2 * q^2 + 21 * q^3 + 22 * q^4 + 6 * q^6 + 50 * q^7 + 33 * q^8 + 63 * q^9 + 76 * q^11 + 66 * q^12 - 30 * q^13 + 89 * q^14 + 138 * q^16 + 140 * q^17 + 18 * q^18 + 90 * q^19 + 150 * q^21 - 61 * q^22 - 34 * q^23 + 99 * q^24 - 241 * q^26 + 189 * q^27 + 57 * q^28 + 203 * q^29 + 524 * q^31 + 6 * q^32 + 228 * q^33 + 255 * q^34 + 198 * q^36 + 28 * q^37 - 222 * q^38 - 90 * q^39 + 1532 * q^41 + 267 * q^42 + 464 * q^43 + 1475 * q^44 + 72 * q^46 + 360 * q^47 + 414 * q^48 + 569 * q^49 + 420 * q^51 + 205 * q^52 - 282 * q^53 + 54 * q^54 + 1102 * q^56 + 270 * q^57 + 58 * q^58 + 766 * q^59 + 1200 * q^61 - 2856 * q^62 + 450 * q^63 + 701 * q^64 - 183 * q^66 - 1546 * q^67 - 1801 * q^68 - 102 * q^69 + 1802 * q^71 + 297 * q^72 + 220 * q^73 + 1594 * q^74 + 1960 * q^76 - 3222 * q^77 - 723 * q^78 + 1298 * q^79 + 567 * q^81 - 856 * q^82 - 1652 * q^83 + 171 * q^84 + 7628 * q^86 + 609 * q^87 - 550 * q^88 + 2846 * q^89 - 816 * q^91 - 472 * q^92 + 1572 * q^93 + 745 * q^94 + 18 * q^96 - 1110 * q^97 + 761 * q^98 + 684 * 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 $

$2$	−2.58378	−0.913504	−0.456752	−	0.889594i	$-0.650987\pi$
$2$	−2.58378	−0.913504	−0.456752	+	0.889594i	$0.650987\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−1.32408	−0.165510
$5$	0	0
$5$	0	0
$6$	−7.75134	−0.527412
$7$	26.6398	1.43842	0.719208	−	0.694795i	$-0.244505\pi$
$7$	26.6398	1.43842	0.719208	+	0.694795i	$0.244505\pi$
$8$	24.0914	1.06470
$9$	9.00000	0.333333
$10$	0	0
$11$	−8.03338	−0.220196	−0.110098	−	0.993921i	$-0.535116\pi$
$11$	−8.03338	−0.220196	−0.110098	+	0.993921i	$0.535116\pi$
$12$	−3.97225	−0.0955574
$13$	−49.9367	−1.06538	−0.532691	−	0.846310i	$-0.678819\pi$
$13$	−49.9367	−1.06538	−0.532691	+	0.846310i	$0.678819\pi$
$14$	−68.8314	−1.31400
$15$	0	0
$16$	−51.6541	−0.807096
$17$	−116.617	−1.66375	−0.831874	−	0.554964i	$-0.812732\pi$
$17$	−116.617	−1.66375	−0.831874	+	0.554964i	$0.812732\pi$
$18$	−23.2540	−0.304501
$19$	22.6617	0.273629	0.136815	−	0.990597i	$-0.456314\pi$
$19$	22.6617	0.273629	0.136815	+	0.990597i	$0.456314\pi$
$20$	0	0
$21$	79.9195	0.830470
$22$	20.7565	0.201150
$23$	−38.2616	−0.346874	−0.173437	−	0.984845i	$-0.555487\pi$
$23$	−38.2616	−0.346874	−0.173437	+	0.984845i	$0.555487\pi$
$24$	72.2741	0.614704
$25$	0	0
$26$	129.025	0.973230
$27$	27.0000	0.192450
$28$	−35.2733	−0.238073
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	−8.82528	−0.0511312	−0.0255656	−	0.999673i	$-0.508139\pi$
$31$	−8.82528	−0.0511312	−0.0255656	+	0.999673i	$0.508139\pi$
$32$	−59.2681	−0.327413
$33$	−24.1002	−0.127130
$34$	301.312	1.51984
$35$	0	0
$36$	−11.9167	−0.0551701
$37$	−301.022	−1.33750	−0.668752	−	0.743486i	$-0.733171\pi$
$37$	−301.022	−1.33750	−0.668752	+	0.743486i	$0.733171\pi$
$38$	−58.5529	−0.249962
$39$	−149.810	−0.615098
$40$	0	0
$41$	305.401	1.16331	0.581654	−	0.813436i	$-0.302406\pi$
$41$	305.401	1.16331	0.581654	+	0.813436i	$0.302406\pi$
$42$	−206.494	−0.758637
$43$	−275.872	−0.978374	−0.489187	−	0.872179i	$-0.662706\pi$
$43$	−275.872	−0.978374	−0.489187	+	0.872179i	$0.662706\pi$
$44$	10.6369	0.0364447
$45$	0	0
$46$	98.8596	0.316871
$47$	479.282	1.48746	0.743729	−	0.668481i	$-0.233055\pi$
$47$	479.282	1.48746	0.743729	+	0.668481i	$0.233055\pi$
$48$	−154.962	−0.465977
$49$	366.680	1.06904
$50$	0	0
$51$	−349.850	−0.960566
$52$	66.1204	0.176332
$53$	496.683	1.28726	0.643629	−	0.765338i	$-0.277428\pi$
$53$	496.683	1.28726	0.643629	+	0.765338i	$0.277428\pi$
$54$	−69.7621	−0.175804
$55$	0	0
$56$	641.790	1.53148
$57$	67.9852	0.157980
$58$	−74.9296	−0.169633
$59$	−563.505	−1.24342	−0.621712	−	0.783246i	$-0.713563\pi$
$59$	−563.505	−1.24342	−0.621712	+	0.783246i	$0.713563\pi$
$60$	0	0
$61$	485.346	1.01872	0.509362	−	0.860552i	$-0.329881\pi$
$61$	485.346	1.01872	0.509362	+	0.860552i	$0.329881\pi$
$62$	22.8026	0.0467085
$63$	239.758	0.479472
$64$	566.369	1.10619
$65$	0	0
$66$	62.2695	0.116134
$67$	−217.176	−0.396004	−0.198002	−	0.980202i	$-0.563445\pi$
$67$	−217.176	−0.396004	−0.198002	+	0.980202i	$0.563445\pi$
$68$	154.410	0.275368
$69$	−114.785	−0.200268
$70$	0	0
$71$	628.023	1.04975	0.524877	−	0.851178i	$-0.324111\pi$
$71$	628.023	1.04975	0.524877	+	0.851178i	$0.324111\pi$
$72$	216.822	0.354899
$73$	682.502	1.09426	0.547128	−	0.837049i	$-0.315721\pi$
$73$	682.502	1.09426	0.547128	+	0.837049i	$0.315721\pi$
$74$	777.773	1.22182
$75$	0	0
$76$	−30.0060	−0.0452885
$77$	−214.008	−0.316733
$78$	387.076	0.561895
$79$	1159.26	1.65097	0.825484	−	0.564425i	$-0.190902\pi$
$79$	1159.26	1.65097	0.825484	+	0.564425i	$0.190902\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	−789.089	−1.06269
$83$	−668.594	−0.884189	−0.442095	−	0.896968i	$-0.645764\pi$
$83$	−668.594	−0.884189	−0.442095	+	0.896968i	$0.645764\pi$
$84$	−105.820	−0.137451
$85$	0	0
$86$	712.792	0.893748
$87$	87.0000	0.107211
$88$	−193.535	−0.234442
$89$	798.300	0.950782	0.475391	−	0.879775i	$-0.342307\pi$
$89$	798.300	0.950782	0.475391	+	0.879775i	$0.342307\pi$
$90$	0	0
$91$	−1330.31	−1.53246
$92$	50.6616	0.0574112
$93$	−26.4758	−0.0295206
$94$	−1238.36	−1.35880
$95$	0	0
$96$	−177.804	−0.189032
$97$	−867.116	−0.907653	−0.453826	−	0.891090i	$-0.649941\pi$
$97$	−867.116	−0.907653	−0.453826	+	0.891090i	$0.649941\pi$
$98$	−947.421	−0.976571
$99$	−72.3005	−0.0733987
$100$	0	0
$101$	1270.57	1.25174	0.625872	−	0.779926i	$-0.284743\pi$
$101$	1270.57	1.25174	0.625872	+	0.779926i	$0.284743\pi$
$102$	903.936	0.877481
$103$	884.596	0.846232	0.423116	−	0.906076i	$-0.360936\pi$
$103$	884.596	0.846232	0.423116	+	0.906076i	$0.360936\pi$
$104$	−1203.04	−1.13431
$105$	0	0
$106$	−1283.32	−1.17591
$107$	−925.937	−0.836577	−0.418288	−	0.908314i	$-0.637370\pi$
$107$	−925.937	−0.836577	−0.418288	+	0.908314i	$0.637370\pi$
$108$	−35.7502	−0.0318525
$109$	73.4989	0.0645864	0.0322932	−	0.999478i	$-0.489719\pi$
$109$	73.4989	0.0645864	0.0322932	+	0.999478i	$0.489719\pi$
$110$	0	0
$111$	−903.065	−0.772208
$112$	−1376.06	−1.16094
$113$	2068.68	1.72217	0.861086	−	0.508459i	$-0.169785\pi$
$113$	2068.68	1.72217	0.861086	+	0.508459i	$0.169785\pi$
$114$	−175.659	−0.144315
$115$	0	0
$116$	−38.3984	−0.0307345
$117$	−449.431	−0.355127
$118$	1455.97	1.13587
$119$	−3106.65	−2.39316
$120$	0	0
$121$	−1266.46	−0.951514
$122$	−1254.03	−0.930609
$123$	916.204	0.671637
$124$	11.6854	0.00846274
$125$	0	0
$126$	−619.483	−0.437999
$127$	708.738	0.495199	0.247600	−	0.968862i	$-0.420358\pi$
$127$	708.738	0.495199	0.247600	+	0.968862i	$0.420358\pi$
$128$	−989.228	−0.683095
$129$	−827.616	−0.564864
$130$	0	0
$131$	−2033.80	−1.35644	−0.678220	−	0.734859i	$-0.737248\pi$
$131$	−2033.80	−1.35644	−0.678220	+	0.734859i	$0.737248\pi$
$132$	31.9106	0.0210414
$133$	603.705	0.393593
$134$	561.135	0.361752
$135$	0	0
$136$	−2809.46	−1.77139
$137$	−2635.96	−1.64383	−0.821917	−	0.569608i	$-0.807095\pi$
$137$	−2635.96	−1.64383	−0.821917	+	0.569608i	$0.807095\pi$
$138$	296.579	0.182945
$139$	−1284.05	−0.783539	−0.391769	−	0.920063i	$-0.628137\pi$
$139$	−1284.05	−0.783539	−0.391769	+	0.920063i	$0.628137\pi$
$140$	0	0
$141$	1437.85	0.858784
$142$	−1622.67	−0.958955
$143$	401.161	0.234593
$144$	−464.887	−0.269032
$145$	0	0
$146$	−1763.43	−0.999608
$147$	1100.04	0.617210
$148$	398.578	0.221371
$149$	261.843	0.143966	0.0719832	−	0.997406i	$-0.477067\pi$
$149$	261.843	0.143966	0.0719832	+	0.997406i	$0.477067\pi$
$150$	0	0
$151$	2760.20	1.48756	0.743781	−	0.668424i	$-0.233031\pi$
$151$	2760.20	1.48756	0.743781	+	0.668424i	$0.233031\pi$
$152$	545.952	0.291333
$153$	−1049.55	−0.554583
$154$	552.949	0.289337
$155$	0	0
$156$	198.361	0.101805
$157$	−1922.00	−0.977019	−0.488509	−	0.872559i	$-0.662459\pi$
$157$	−1922.00	−0.977019	−0.488509	+	0.872559i	$0.662459\pi$
$158$	−2995.26	−1.50817
$159$	1490.05	0.743198
$160$	0	0
$161$	−1019.28	−0.498949
$162$	−209.286	−0.101500
$163$	2784.04	1.33781	0.668904	−	0.743349i	$-0.266764\pi$
$163$	2784.04	1.33781	0.668904	+	0.743349i	$0.266764\pi$
$164$	−404.376	−0.192540
$165$	0	0
$166$	1727.50	0.807711
$167$	1669.58	0.773627	0.386814	−	0.922158i	$-0.373576\pi$
$167$	1669.58	0.773627	0.386814	+	0.922158i	$0.373576\pi$
$168$	1925.37	0.884200
$169$	296.677	0.135037
$170$	0	0
$171$	203.956	0.0912098
$172$	365.277	0.161931
$173$	3488.62	1.53315	0.766574	−	0.642156i	$-0.221960\pi$
$173$	3488.62	1.53315	0.766574	+	0.642156i	$0.221960\pi$
$174$	−224.789	−0.0979379
$175$	0	0
$176$	414.958	0.177719
$177$	−1690.51	−0.717892
$178$	−2062.63	−0.868543
$179$	4016.48	1.67713	0.838564	−	0.544803i	$-0.183396\pi$
$179$	4016.48	1.67713	0.838564	+	0.544803i	$0.183396\pi$
$180$	0	0
$181$	−2275.81	−0.934583	−0.467291	−	0.884103i	$-0.654770\pi$
$181$	−2275.81	−0.934583	−0.467291	+	0.884103i	$0.654770\pi$
$182$	3437.22	1.39991
$183$	1456.04	0.588161
$184$	−921.775	−0.369316
$185$	0	0
$186$	68.4077	0.0269672
$187$	936.827	0.366351
$188$	−634.610	−0.246190
$189$	719.275	0.276823
$190$	0	0
$191$	3204.35	1.21392	0.606961	−	0.794732i	$-0.292389\pi$
$191$	3204.35	1.21392	0.606961	+	0.794732i	$0.292389\pi$
$192$	1699.11	0.638659
$193$	71.0310	0.0264918	0.0132459	−	0.999912i	$-0.495784\pi$
$193$	71.0310	0.0264918	0.0132459	+	0.999912i	$0.495784\pi$
$194$	2240.44	0.829144
$195$	0	0
$196$	−485.515	−0.176937
$197$	−1260.52	−0.455881	−0.227941	−	0.973675i	$-0.573199\pi$
$197$	−1260.52	−0.455881	−0.227941	+	0.973675i	$0.573199\pi$
$198$	186.808	0.0670500
$199$	2365.21	0.842539	0.421270	−	0.906935i	$-0.361585\pi$
$199$	2365.21	0.842539	0.421270	+	0.906935i	$0.361585\pi$
$200$	0	0
$201$	−651.529	−0.228633
$202$	−3282.87	−1.14347
$203$	772.555	0.267107
$204$	463.231	0.158984
$205$	0	0
$206$	−2285.60	−0.773036
$207$	−344.355	−0.115625
$208$	2579.44	0.859865
$209$	−182.050	−0.0602521
$210$	0	0
$211$	−1604.41	−0.523468	−0.261734	−	0.965140i	$-0.584294\pi$
$211$	−1604.41	−0.523468	−0.261734	+	0.965140i	$0.584294\pi$
$212$	−657.649	−0.213054
$213$	1884.07	0.606076
$214$	2392.42	0.764216
$215$	0	0
$216$	650.467	0.204901
$217$	−235.104	−0.0735479
$218$	−189.905	−0.0589999
$219$	2047.50	0.631770
$220$	0	0
$221$	5823.46	1.77253
$222$	2333.32	0.705415
$223$	2789.26	0.837591	0.418795	−	0.908081i	$-0.362452\pi$
$223$	2789.26	0.837591	0.418795	+	0.908081i	$0.362452\pi$
$224$	−1578.89	−0.470956
$225$	0	0
$226$	−5345.02	−1.57321
$227$	−3091.67	−0.903971	−0.451985	−	0.892025i	$-0.649284\pi$
$227$	−3091.67	−0.903971	−0.451985	+	0.892025i	$0.649284\pi$
$228$	−90.0180	−0.0261473
$229$	−6479.99	−1.86991	−0.934955	−	0.354766i	$-0.884561\pi$
$229$	−6479.99	−1.86991	−0.934955	+	0.354766i	$0.884561\pi$
$230$	0	0
$231$	−642.024	−0.182866
$232$	698.650	0.197710
$233$	3647.01	1.02542	0.512712	−	0.858561i	$-0.328641\pi$
$233$	3647.01	1.02542	0.512712	+	0.858561i	$0.328641\pi$
$234$	1161.23	0.324410
$235$	0	0
$236$	746.127	0.205800
$237$	3477.77	0.953187
$238$	8026.90	2.18616
$239$	3863.00	1.04551	0.522754	−	0.852483i	$-0.324905\pi$
$239$	3863.00	1.04551	0.522754	+	0.852483i	$0.324905\pi$
$240$	0	0
$241$	6207.76	1.65924	0.829620	−	0.558329i	$-0.188557\pi$
$241$	6207.76	1.65924	0.829620	+	0.558329i	$0.188557\pi$
$242$	3272.27	0.869212
$243$	243.000	0.0641500
$244$	−642.638	−0.168609
$245$	0	0
$246$	−2367.27	−0.613543
$247$	−1131.65	−0.291520
$248$	−212.613	−0.0544393
$249$	−2005.78	−0.510487
$250$	0	0
$251$	4942.68	1.24294	0.621472	−	0.783436i	$-0.286535\pi$
$251$	4942.68	1.24294	0.621472	+	0.783436i	$0.286535\pi$
$252$	−317.460	−0.0793575
$253$	307.370	0.0763803
$254$	−1831.22	−0.452367
$255$	0	0
$256$	−1975.00	−0.482179
$257$	−3343.25	−0.811465	−0.405732	−	0.913992i	$-0.632984\pi$
$257$	−3343.25	−0.811465	−0.405732	+	0.913992i	$0.632984\pi$
$258$	2138.38	0.516006
$259$	−8019.16	−1.92389
$260$	0	0
$261$	261.000	0.0618984
$262$	5254.88	1.23911
$263$	−1053.61	−0.247027	−0.123514	−	0.992343i	$-0.539416\pi$
$263$	−1053.61	−0.247027	−0.123514	+	0.992343i	$0.539416\pi$
$264$	−580.606	−0.135355
$265$	0	0
$266$	−1559.84	−0.359549
$267$	2394.90	0.548934
$268$	287.559	0.0655428
$269$	2348.05	0.532205	0.266103	−	0.963945i	$-0.414264\pi$
$269$	2348.05	0.532205	0.266103	+	0.963945i	$0.414264\pi$
$270$	0	0
$271$	4230.65	0.948316	0.474158	−	0.880440i	$-0.342753\pi$
$271$	4230.65	0.948316	0.474158	+	0.880440i	$0.342753\pi$
$272$	6023.74	1.34280
$273$	−3990.92	−0.884767
$274$	6810.74	1.50165
$275$	0	0
$276$	151.985	0.0331464
$277$	−2146.72	−0.465647	−0.232823	−	0.972519i	$-0.574796\pi$
$277$	−2146.72	−0.465647	−0.232823	+	0.972519i	$0.574796\pi$
$278$	3317.71	0.715766
$279$	−79.4275	−0.0170437
$280$	0	0
$281$	−1445.80	−0.306937	−0.153469	−	0.988154i	$-0.549044\pi$
$281$	−1445.80	−0.306937	−0.153469	+	0.988154i	$0.549044\pi$
$282$	−3715.08	−0.784503
$283$	−3775.85	−0.793113	−0.396557	−	0.918010i	$-0.629795\pi$
$283$	−3775.85	−0.793113	−0.396557	+	0.918010i	$0.629795\pi$
$284$	−831.554	−0.173745
$285$	0	0
$286$	−1036.51	−0.214301
$287$	8135.83	1.67332
$288$	−533.413	−0.109138
$289$	8686.48	1.76806
$290$	0	0
$291$	−2601.35	−0.524033
$292$	−903.688	−0.181111
$293$	−6451.38	−1.28633	−0.643164	−	0.765729i	$-0.722378\pi$
$293$	−6451.38	−1.28633	−0.643164	+	0.765729i	$0.722378\pi$
$294$	−2842.26	−0.563824
$295$	0	0
$296$	−7252.02	−1.42404
$297$	−216.901	−0.0423767
$298$	−676.544	−0.131514
$299$	1910.66	0.369553
$300$	0	0
$301$	−7349.18	−1.40731
$302$	−7131.74	−1.35889
$303$	3811.70	0.722695
$304$	−1170.57	−0.220845
$305$	0	0
$306$	2711.81	0.506614
$307$	−4197.07	−0.780258	−0.390129	−	0.920760i	$-0.627570\pi$
$307$	−4197.07	−0.780258	−0.390129	+	0.920760i	$0.627570\pi$
$308$	283.364	0.0524227
$309$	2653.79	0.488572
$310$	0	0
$311$	6909.28	1.25977	0.629886	−	0.776687i	$-0.283102\pi$
$311$	6909.28	1.25977	0.629886	+	0.776687i	$0.283102\pi$
$312$	−3609.13	−0.654894
$313$	−835.311	−0.150845	−0.0754226	−	0.997152i	$-0.524031\pi$
$313$	−835.311	−0.150845	−0.0754226	+	0.997152i	$0.524031\pi$
$314$	4966.01	0.892510
$315$	0	0
$316$	−1534.95	−0.273252
$317$	454.133	0.0804626	0.0402313	−	0.999190i	$-0.487191\pi$
$317$	454.133	0.0804626	0.0402313	+	0.999190i	$0.487191\pi$
$318$	−3849.96	−0.678915
$319$	−232.968	−0.0408894
$320$	0	0
$321$	−2777.81	−0.482998
$322$	2633.60	0.455792
$323$	−2642.74	−0.455250
$324$	−107.251	−0.0183900
$325$	0	0
$326$	−7193.34	−1.22209
$327$	220.497	0.0372890
$328$	7357.53	1.23857
$329$	12768.0	2.13958
$330$	0	0
$331$	−6368.25	−1.05749	−0.528747	−	0.848779i	$-0.677338\pi$
$331$	−6368.25	−1.05749	−0.528747	+	0.848779i	$0.677338\pi$
$332$	885.274	0.146343
$333$	−2709.19	−0.445835
$334$	−4313.82	−0.706712
$335$	0	0
$336$	−4128.17	−0.670269
$337$	8096.70	1.30877	0.654384	−	0.756162i	$-0.272928\pi$
$337$	8096.70	1.30877	0.654384	+	0.756162i	$0.272928\pi$
$338$	−766.547	−0.123357
$339$	6206.05	0.994297
$340$	0	0
$341$	70.8968	0.0112589
$342$	−526.976	−0.0833205
$343$	630.841	0.0993067
$344$	−6646.13	−1.04167
$345$	0	0
$346$	−9013.82	−1.40054
$347$	2438.20	0.377202	0.188601	−	0.982054i	$-0.439605\pi$
$347$	2438.20	0.377202	0.188601	+	0.982054i	$0.439605\pi$
$348$	−115.195	−0.0177446
$349$	−777.284	−0.119218	−0.0596090	−	0.998222i	$-0.518985\pi$
$349$	−777.284	−0.119218	−0.0596090	+	0.998222i	$0.518985\pi$
$350$	0	0
$351$	−1348.29	−0.205033
$352$	476.123	0.0720950
$353$	9649.81	1.45498	0.727489	−	0.686119i	$-0.240687\pi$
$353$	9649.81	1.45498	0.727489	+	0.686119i	$0.240687\pi$
$354$	4367.92	0.655797
$355$	0	0
$356$	−1057.02	−0.157364
$357$	−9319.95	−1.38169
$358$	−10377.7	−1.53206
$359$	9657.56	1.41980	0.709898	−	0.704304i	$-0.248741\pi$
$359$	9657.56	1.41980	0.709898	+	0.704304i	$0.248741\pi$
$360$	0	0
$361$	−6345.45	−0.925127
$362$	5880.19	0.853745
$363$	−3799.39	−0.549357
$364$	1761.43	0.253638
$365$	0	0
$366$	−3762.08	−0.537287
$367$	9173.93	1.30484	0.652419	−	0.757859i	$-0.273754\pi$
$367$	9173.93	1.30484	0.652419	+	0.757859i	$0.273754\pi$
$368$	1976.37	0.279961
$369$	2748.61	0.387770
$370$	0	0
$371$	13231.5	1.85161
$372$	35.0562	0.00488597
$373$	−2899.69	−0.402521	−0.201261	−	0.979538i	$-0.564504\pi$
$373$	−2899.69	−0.402521	−0.201261	+	0.979538i	$0.564504\pi$
$374$	−2420.56	−0.334663
$375$	0	0
$376$	11546.6	1.58369
$377$	−1448.17	−0.197836
$378$	−1858.45	−0.252879
$379$	−5819.59	−0.788739	−0.394370	−	0.918952i	$-0.629037\pi$
$379$	−5819.59	−0.788739	−0.394370	+	0.918952i	$0.629037\pi$
$380$	0	0
$381$	2126.21	0.285903
$382$	−8279.34	−1.10892
$383$	5805.80	0.774576	0.387288	−	0.921959i	$-0.373412\pi$
$383$	5805.80	0.774576	0.387288	+	0.921959i	$0.373412\pi$
$384$	−2967.68	−0.394385
$385$	0	0
$386$	−183.528	−0.0242004
$387$	−2482.85	−0.326125
$388$	1148.13	0.150226
$389$	7690.82	1.00242	0.501208	−	0.865327i	$-0.332889\pi$
$389$	7690.82	1.00242	0.501208	+	0.865327i	$0.332889\pi$
$390$	0	0
$391$	4461.95	0.577111
$392$	8833.83	1.13820
$393$	−6101.39	−0.783141
$394$	3256.92	0.416449
$395$	0	0
$396$	95.7318	0.0121482
$397$	1638.24	0.207105	0.103553	−	0.994624i	$-0.466979\pi$
$397$	1638.24	0.207105	0.103553	+	0.994624i	$0.466979\pi$
$398$	−6111.18	−0.769663
$399$	1811.11	0.227241
$400$	0	0
$401$	3780.89	0.470845	0.235422	−	0.971893i	$-0.424353\pi$
$401$	3780.89	0.470845	0.235422	+	0.971893i	$0.424353\pi$
$402$	1683.41	0.208857
$403$	440.705	0.0544742
$404$	−1682.34	−0.207177
$405$	0	0
$406$	−1996.11	−0.244003
$407$	2418.22	0.294513
$408$	−8428.38	−1.02271
$409$	−330.980	−0.0400145	−0.0200073	−	0.999800i	$-0.506369\pi$
$409$	−330.980	−0.0400145	−0.0200073	+	0.999800i	$0.506369\pi$
$410$	0	0
$411$	−7907.88	−0.949068
$412$	−1171.28	−0.140060
$413$	−15011.7	−1.78856
$414$	889.736	0.105624
$415$	0	0
$416$	2959.65	0.348820
$417$	−3852.16	−0.452376
$418$	470.378	0.0550405
$419$	9913.43	1.15585	0.577927	−	0.816089i	$-0.303862\pi$
$419$	9913.43	1.15585	0.577927	+	0.816089i	$0.303862\pi$
$420$	0	0
$421$	6286.56	0.727763	0.363881	−	0.931445i	$-0.381451\pi$
$421$	6286.56	0.727763	0.363881	+	0.931445i	$0.381451\pi$
$422$	4145.43	0.478190
$423$	4313.54	0.495819
$424$	11965.8	1.37054
$425$	0	0
$426$	−4868.02	−0.553653
$427$	12929.5	1.46535
$428$	1226.02	0.138462
$429$	1203.48	0.135442
$430$	0	0
$431$	5798.56	0.648044	0.324022	−	0.946050i	$-0.394965\pi$
$431$	5798.56	0.648044	0.324022	+	0.946050i	$0.394965\pi$
$432$	−1394.66	−0.155326
$433$	−11476.3	−1.27371	−0.636856	−	0.770983i	$-0.719765\pi$
$433$	−11476.3	−1.27371	−0.636856	+	0.770983i	$0.719765\pi$
$434$	607.457	0.0671863
$435$	0	0
$436$	−97.3186	−0.0106897
$437$	−867.075	−0.0949149
$438$	−5290.30	−0.577124
$439$	14429.5	1.56875	0.784377	−	0.620285i	$-0.212983\pi$
$439$	14429.5	1.56875	0.784377	+	0.620285i	$0.212983\pi$
$440$	0	0
$441$	3300.12	0.356346
$442$	−15046.5	−1.61921
$443$	11892.0	1.27541	0.637703	−	0.770283i	$-0.279885\pi$
$443$	11892.0	1.27541	0.637703	+	0.770283i	$0.279885\pi$
$444$	1195.73	0.127808
$445$	0	0
$446$	−7206.84	−0.765143
$447$	785.528	0.0831190
$448$	15088.0	1.59116
$449$	−9881.53	−1.03862	−0.519308	−	0.854587i	$-0.673810\pi$
$449$	−9881.53	−1.03862	−0.519308	+	0.854587i	$0.673810\pi$
$450$	0	0
$451$	−2453.40	−0.256156
$452$	−2739.11	−0.285037
$453$	8280.59	0.858844
$454$	7988.20	0.825781
$455$	0	0
$456$	1637.86	0.168201
$457$	−11641.2	−1.19158	−0.595792	−	0.803139i	$-0.703162\pi$
$457$	−11641.2	−1.19158	−0.595792	+	0.803139i	$0.703162\pi$
$458$	16742.9	1.70817
$459$	−3148.65	−0.320189
$460$	0	0
$461$	−8045.29	−0.812812	−0.406406	−	0.913693i	$-0.633218\pi$
$461$	−8045.29	−0.812812	−0.406406	+	0.913693i	$0.633218\pi$
$462$	1658.85	0.167049
$463$	4340.54	0.435684	0.217842	−	0.975984i	$-0.430098\pi$
$463$	4340.54	0.435684	0.217842	+	0.975984i	$0.430098\pi$
$464$	−1497.97	−0.149874
$465$	0	0
$466$	−9423.08	−0.936729
$467$	3461.65	0.343011	0.171506	−	0.985183i	$-0.445137\pi$
$467$	3461.65	0.343011	0.171506	+	0.985183i	$0.445137\pi$
$468$	595.083	0.0587772
$469$	−5785.54	−0.569619
$470$	0	0
$471$	−5765.99	−0.564082
$472$	−13575.6	−1.32387
$473$	2216.18	0.215434
$474$	−8985.79	−0.870740
$475$	0	0
$476$	4113.46	0.396093
$477$	4470.14	0.429086
$478$	−9981.14	−0.955076
$479$	6284.62	0.599481	0.299741	−	0.954021i	$-0.403100\pi$
$479$	6284.62	0.599481	0.299741	+	0.954021i	$0.403100\pi$
$480$	0	0
$481$	15032.0	1.42495
$482$	−16039.5	−1.51572
$483$	−3057.85	−0.288068
$484$	1676.90	0.157485
$485$	0	0
$486$	−627.858	−0.0586013
$487$	1809.63	0.168382	0.0841912	−	0.996450i	$-0.473169\pi$
$487$	1809.63	0.168382	0.0841912	+	0.996450i	$0.473169\pi$
$488$	11692.7	1.08463
$489$	8352.12	0.772384
$490$	0	0
$491$	9550.28	0.877796	0.438898	−	0.898537i	$-0.355369\pi$
$491$	9550.28	0.877796	0.438898	+	0.898537i	$0.355369\pi$
$492$	−1213.13	−0.111163
$493$	−3381.89	−0.308950
$494$	2923.94	0.266304
$495$	0	0
$496$	455.862	0.0412678
$497$	16730.4	1.50998
$498$	5182.50	0.466332
$499$	−13672.9	−1.22662	−0.613309	−	0.789843i	$-0.710162\pi$
$499$	−13672.9	−1.22662	−0.613309	+	0.789843i	$0.710162\pi$
$500$	0	0
$501$	5008.73	0.446654
$502$	−12770.8	−1.13544
$503$	−13406.0	−1.18836	−0.594181	−	0.804332i	$-0.702524\pi$
$503$	−13406.0	−1.18836	−0.594181	+	0.804332i	$0.702524\pi$
$504$	5776.11	0.510493
$505$	0	0
$506$	−794.177	−0.0697737
$507$	890.030	0.0779638
$508$	−938.428	−0.0819606
$509$	−13537.0	−1.17882	−0.589409	−	0.807835i	$-0.700639\pi$
$509$	−13537.0	−1.17882	−0.589409	+	0.807835i	$0.700639\pi$
$510$	0	0
$511$	18181.7	1.57400
$512$	13016.8	1.12357
$513$	611.867	0.0526600
$514$	8638.23	0.741276
$515$	0	0
$516$	1095.83	0.0934909
$517$	−3850.26	−0.327532
$518$	20719.8	1.75748
$519$	10465.9	0.885163
$520$	0	0
$521$	−6612.84	−0.556073	−0.278036	−	0.960571i	$-0.589684\pi$
$521$	−6612.84	−0.556073	−0.278036	+	0.960571i	$0.589684\pi$
$522$	−674.366	−0.0565445
$523$	11689.8	0.977363	0.488682	−	0.872462i	$-0.337478\pi$
$523$	11689.8	0.977363	0.488682	+	0.872462i	$0.337478\pi$
$524$	2692.91	0.224505
$525$	0	0
$526$	2722.29	0.225660
$527$	1029.18	0.0850694
$528$	1244.87	0.102606
$529$	−10703.0	−0.879678
$530$	0	0
$531$	−5071.54	−0.414475
$532$	−799.355	−0.0651437
$533$	−15250.7	−1.23937
$534$	−6187.89	−0.501454
$535$	0	0
$536$	−5232.07	−0.421625
$537$	12049.4	0.968290
$538$	−6066.85	−0.486172
$539$	−2945.68	−0.235398
$540$	0	0
$541$	−170.114	−0.0135190	−0.00675951	−	0.999977i	$-0.502152\pi$
$541$	−170.114	−0.0135190	−0.00675951	+	0.999977i	$0.502152\pi$
$542$	−10931.1	−0.866291
$543$	−6827.42	−0.539582
$544$	6911.65	0.544733
$545$	0	0
$546$	10311.7	0.808238
$547$	6453.08	0.504413	0.252206	−	0.967673i	$-0.418844\pi$
$547$	6453.08	0.504413	0.252206	+	0.967673i	$0.418844\pi$
$548$	3490.23	0.272071
$549$	4368.11	0.339575
$550$	0	0
$551$	657.190	0.0508117
$552$	−2765.33	−0.213225
$553$	30882.4	2.37478
$554$	5546.66	0.425370
$555$	0	0
$556$	1700.19	0.129684
$557$	12800.7	0.973759	0.486879	−	0.873469i	$-0.338135\pi$
$557$	12800.7	0.973759	0.486879	+	0.873469i	$0.338135\pi$
$558$	205.223	0.0155695
$559$	13776.1	1.04234
$560$	0	0
$561$	2810.48	0.211513
$562$	3735.63	0.280388
$563$	−6454.48	−0.483169	−0.241584	−	0.970380i	$-0.577667\pi$
$563$	−6454.48	−0.483169	−0.241584	+	0.970380i	$0.577667\pi$
$564$	−1903.83	−0.142138
$565$	0	0
$566$	9755.97	0.724512
$567$	2157.83	0.159824
$568$	15129.9	1.11767
$569$	2900.88	0.213728	0.106864	−	0.994274i	$-0.465919\pi$
$569$	2900.88	0.213728	0.106864	+	0.994274i	$0.465919\pi$
$570$	0	0
$571$	−13437.3	−0.984823	−0.492411	−	0.870363i	$-0.663884\pi$
$571$	−13437.3	−0.984823	−0.492411	+	0.870363i	$0.663884\pi$
$572$	−531.170	−0.0388275
$573$	9613.06	0.700858
$574$	−21021.2	−1.52859
$575$	0	0
$576$	5097.32	0.368730
$577$	5402.33	0.389778	0.194889	−	0.980825i	$-0.437565\pi$
$577$	5402.33	0.389778	0.194889	+	0.980825i	$0.437565\pi$
$578$	−22443.9	−1.61513
$579$	213.093	0.0152951
$580$	0	0
$581$	−17811.2	−1.27183
$582$	6721.31	0.478707
$583$	−3990.04	−0.283449
$584$	16442.4	1.16505
$585$	0	0
$586$	16669.0	1.17507
$587$	17702.2	1.24471	0.622357	−	0.782733i	$-0.286175\pi$
$587$	17702.2	1.24471	0.622357	+	0.782733i	$0.286175\pi$
$588$	−1456.55	−0.102155
$589$	−199.996	−0.0139910
$590$	0	0
$591$	−3781.57	−0.263203
$592$	15549.0	1.07949
$593$	13235.5	0.916558	0.458279	−	0.888808i	$-0.348466\pi$
$593$	13235.5	0.916558	0.458279	+	0.888808i	$0.348466\pi$
$594$	560.425	0.0387113
$595$	0	0
$596$	−346.701	−0.0238279
$597$	7095.63	0.486440
$598$	−4936.73	−0.337588
$599$	−22931.4	−1.56419	−0.782095	−	0.623159i	$-0.785849\pi$
$599$	−22931.4	−1.56419	−0.782095	+	0.623159i	$0.785849\pi$
$600$	0	0
$601$	−11714.0	−0.795051	−0.397525	−	0.917591i	$-0.630131\pi$
$601$	−11714.0	−0.795051	−0.397525	+	0.917591i	$0.630131\pi$
$602$	18988.7	1.28558
$603$	−1954.59	−0.132001
$604$	−3654.73	−0.246207
$605$	0	0
$606$	−9848.60	−0.660185
$607$	−4567.18	−0.305397	−0.152699	−	0.988273i	$-0.548796\pi$
$607$	−4567.18	−0.305397	−0.152699	+	0.988273i	$0.548796\pi$
$608$	−1343.12	−0.0895898
$609$	2317.66	0.154214
$610$	0	0
$611$	−23933.8	−1.58471
$612$	1389.69	0.0917892
$613$	19389.4	1.27754	0.638768	−	0.769399i	$-0.279444\pi$
$613$	19389.4	1.27754	0.638768	+	0.769399i	$0.279444\pi$
$614$	10844.3	0.712769
$615$	0	0
$616$	−5155.75	−0.337226
$617$	1918.52	0.125181	0.0625905	−	0.998039i	$-0.480064\pi$
$617$	1918.52	0.125181	0.0625905	+	0.998039i	$0.480064\pi$
$618$	−6856.81	−0.446313
$619$	25281.1	1.64157	0.820785	−	0.571237i	$-0.193536\pi$
$619$	25281.1	1.64157	0.820785	+	0.571237i	$0.193536\pi$
$620$	0	0
$621$	−1033.06	−0.0667559
$622$	−17852.1	−1.15081
$623$	21266.6	1.36762
$624$	7738.32	0.496443
$625$	0	0
$626$	2158.26	0.137798
$627$	−546.151	−0.0347866
$628$	2544.88	0.161707
$629$	35104.2	2.22527
$630$	0	0
$631$	−21718.6	−1.37022	−0.685108	−	0.728442i	$-0.740245\pi$
$631$	−21718.6	−1.37022	−0.685108	+	0.728442i	$0.740245\pi$
$632$	27928.1	1.75778
$633$	−4813.22	−0.302225
$634$	−1173.38	−0.0735029
$635$	0	0
$636$	−1972.95	−0.123007
$637$	−18310.8	−1.13893
$638$	601.938	0.0373526
$639$	5652.20	0.349918
$640$	0	0
$641$	9371.00	0.577429	0.288715	−	0.957415i	$-0.406772\pi$
$641$	9371.00	0.577429	0.288715	+	0.957415i	$0.406772\pi$
$642$	7177.25	0.441220
$643$	7538.94	0.462375	0.231187	−	0.972909i	$-0.425739\pi$
$643$	7538.94	0.462375	0.231187	+	0.972909i	$0.425739\pi$
$644$	1349.62	0.0825812
$645$	0	0
$646$	6828.25	0.415873
$647$	−15345.6	−0.932455	−0.466228	−	0.884665i	$-0.654387\pi$
$647$	−15345.6	−0.932455	−0.466228	+	0.884665i	$0.654387\pi$
$648$	1951.40	0.118300
$649$	4526.85	0.273797
$650$	0	0
$651$	−705.312	−0.0424629
$652$	−3686.30	−0.221421
$653$	19648.0	1.17747	0.588735	−	0.808326i	$-0.299626\pi$
$653$	19648.0	1.17747	0.588735	+	0.808326i	$0.299626\pi$
$654$	−569.715	−0.0340636
$655$	0	0
$656$	−15775.2	−0.938902
$657$	6142.51	0.364752
$658$	−32989.7	−1.95452
$659$	−24207.8	−1.43096	−0.715481	−	0.698633i	$-0.753792\pi$
$659$	−24207.8	−1.43096	−0.715481	+	0.698633i	$0.753792\pi$
$660$	0	0
$661$	3864.20	0.227383	0.113691	−	0.993516i	$-0.463733\pi$
$661$	3864.20	0.227383	0.113691	+	0.993516i	$0.463733\pi$
$662$	16454.2	0.966026
$663$	17470.4	1.02337
$664$	−16107.3	−0.941395
$665$	0	0
$666$	6999.96	0.407272
$667$	−1109.59	−0.0644129
$668$	−2210.66	−0.128043
$669$	8367.78	0.483583
$670$	0	0
$671$	−3898.97	−0.224319
$672$	−4736.67	−0.271907
$673$	−17574.4	−1.00660	−0.503300	−	0.864112i	$-0.667881\pi$
$673$	−17574.4	−1.00660	−0.503300	+	0.864112i	$0.667881\pi$
$674$	−20920.1	−1.19557
$675$	0	0
$676$	−392.825	−0.0223501
$677$	26672.9	1.51421	0.757106	−	0.653292i	$-0.226613\pi$
$677$	26672.9	1.51421	0.757106	+	0.653292i	$0.226613\pi$
$678$	−16035.1	−0.908294
$679$	−23099.8	−1.30558
$680$	0	0
$681$	−9275.01	−0.521908
$682$	−183.182	−0.0102850
$683$	−2985.36	−0.167250	−0.0836249	−	0.996497i	$-0.526650\pi$
$683$	−2985.36	−0.167250	−0.0836249	+	0.996497i	$0.526650\pi$
$684$	−270.054	−0.0150962
$685$	0	0
$686$	−1629.95	−0.0907171
$687$	−19440.0	−1.07959
$688$	14249.9	0.789642
$689$	−24802.7	−1.37142
$690$	0	0
$691$	15067.6	0.829518	0.414759	−	0.909931i	$-0.363866\pi$
$691$	15067.6	0.829518	0.414759	+	0.909931i	$0.363866\pi$
$692$	−4619.22	−0.253752
$693$	−1926.07	−0.105578
$694$	−6299.76	−0.344576
$695$	0	0
$696$	2095.95	0.114148
$697$	−35614.9	−1.93545
$698$	2008.33	0.108906
$699$	10941.0	0.592029
$700$	0	0
$701$	32060.7	1.72741	0.863707	−	0.503994i	$-0.168137\pi$
$701$	32060.7	1.72741	0.863707	+	0.503994i	$0.168137\pi$
$702$	3483.69	0.187298
$703$	−6821.67	−0.365980
$704$	−4549.86	−0.243578
$705$	0	0
$706$	−24933.0	−1.32913
$707$	33847.7	1.80053
$708$	2238.38	0.118818
$709$	−6089.76	−0.322575	−0.161288	−	0.986907i	$-0.551565\pi$
$709$	−6089.76	−0.322575	−0.161288	+	0.986907i	$0.551565\pi$
$710$	0	0
$711$	10433.3	0.550323
$712$	19232.1	1.01230
$713$	337.669	0.0177361
$714$	24080.7	1.26218
$715$	0	0
$716$	−5318.15	−0.277582
$717$	11589.0	0.603625
$718$	−24953.0	−1.29699
$719$	2735.79	0.141902	0.0709511	−	0.997480i	$-0.477397\pi$
$719$	2735.79	0.141902	0.0709511	+	0.997480i	$0.477397\pi$
$720$	0	0
$721$	23565.5	1.21723
$722$	16395.2	0.845107
$723$	18623.3	0.957963
$724$	3013.36	0.154683
$725$	0	0
$726$	9816.80	0.501840
$727$	23072.1	1.17702	0.588511	−	0.808489i	$-0.299714\pi$
$727$	23072.1	1.17702	0.588511	+	0.808489i	$0.299714\pi$
$728$	−32048.9	−1.63161
$729$	729.000	0.0370370
$730$	0	0
$731$	32171.3	1.62777
$732$	−1927.92	−0.0973467
$733$	−2341.92	−0.118009	−0.0590047	−	0.998258i	$-0.518793\pi$
$733$	−2341.92	−0.118009	−0.0590047	+	0.998258i	$0.518793\pi$
$734$	−23703.4	−1.19197
$735$	0	0
$736$	2267.69	0.113571
$737$	1744.66	0.0871986
$738$	−7101.80	−0.354229
$739$	30372.4	1.51186	0.755932	−	0.654651i	$-0.227184\pi$
$739$	30372.4	1.51186	0.755932	+	0.654651i	$0.227184\pi$
$740$	0	0
$741$	−3394.96	−0.168309
$742$	−34187.4	−1.69145
$743$	−16456.7	−0.812566	−0.406283	−	0.913747i	$-0.633175\pi$
$743$	−16456.7	−0.812566	−0.406283	+	0.913747i	$0.633175\pi$
$744$	−637.839	−0.0314305
$745$	0	0
$746$	7492.17	0.367705
$747$	−6017.35	−0.294730
$748$	−1240.44	−0.0606349
$749$	−24666.8	−1.20334
$750$	0	0
$751$	−28265.3	−1.37339	−0.686694	−	0.726947i	$-0.740939\pi$
$751$	−28265.3	−1.37339	−0.686694	+	0.726947i	$0.740939\pi$
$752$	−24756.9	−1.20052
$753$	14828.0	0.717615
$754$	3741.74	0.180724
$755$	0	0
$756$	−952.380	−0.0458171
$757$	−30213.7	−1.45064	−0.725320	−	0.688412i	$-0.758308\pi$
$757$	−30213.7	−1.45064	−0.725320	+	0.688412i	$0.758308\pi$
$758$	15036.5	0.720517
$759$	922.111	0.0440982
$760$	0	0
$761$	7521.30	0.358274	0.179137	−	0.983824i	$-0.442669\pi$
$761$	7521.30	0.358274	0.179137	+	0.983824i	$0.442669\pi$
$762$	−5493.67	−0.261174
$763$	1958.00	0.0929020
$764$	−4242.83	−0.200916
$765$	0	0
$766$	−15000.9	−0.707578
$767$	28139.6	1.32472
$768$	−5925.01	−0.278386
$769$	−31256.5	−1.46572	−0.732859	−	0.680380i	$-0.761815\pi$
$769$	−31256.5	−1.46572	−0.732859	+	0.680380i	$0.761815\pi$
$770$	0	0
$771$	−10029.8	−0.468499
$772$	−94.0510	−0.00438467
$773$	−6657.91	−0.309791	−0.154895	−	0.987931i	$-0.549504\pi$
$773$	−6657.91	−0.309791	−0.154895	+	0.987931i	$0.549504\pi$
$774$	6415.13	0.297916
$775$	0	0
$776$	−20890.0	−0.966376
$777$	−24057.5	−1.11076
$778$	−19871.4	−0.915711
$779$	6920.92	0.318315
$780$	0	0
$781$	−5045.15	−0.231152
$782$	−11528.7	−0.527193
$783$	783.000	0.0357371
$784$	−18940.6	−0.862817
$785$	0	0
$786$	15764.6	0.715402
$787$	−18918.1	−0.856872	−0.428436	−	0.903572i	$-0.640935\pi$
$787$	−18918.1	−0.856872	−0.428436	+	0.903572i	$0.640935\pi$
$788$	1669.04	0.0754531
$789$	−3160.82	−0.142621
$790$	0	0
$791$	55109.4	2.47720
$792$	−1741.82	−0.0781475
$793$	−24236.6	−1.08533
$794$	−4232.84	−0.189191
$795$	0	0
$796$	−3131.73	−0.139449
$797$	16329.2	0.725734	0.362867	−	0.931841i	$-0.381798\pi$
$797$	16329.2	0.725734	0.362867	+	0.931841i	$0.381798\pi$
$798$	−4679.52	−0.207585
$799$	−55892.4	−2.47476
$800$	0	0
$801$	7184.70	0.316927
$802$	−9768.99	−0.430119
$803$	−5482.80	−0.240951
$804$	862.678	0.0378412
$805$	0	0
$806$	−1138.69	−0.0497624
$807$	7044.16	0.307269
$808$	30609.7	1.33273
$809$	−2755.55	−0.119753	−0.0598764	−	0.998206i	$-0.519071\pi$
$809$	−2755.55	−0.119753	−0.0598764	+	0.998206i	$0.519071\pi$
$810$	0	0
$811$	−8552.83	−0.370321	−0.185161	−	0.982708i	$-0.559281\pi$
$811$	−8552.83	−0.370321	−0.185161	+	0.982708i	$0.559281\pi$
$812$	−1022.93	−0.0442090
$813$	12692.0	0.547511
$814$	−6248.15	−0.269039
$815$	0	0
$816$	18071.2	0.775269
$817$	−6251.73	−0.267712
$818$	855.181	0.0365534
$819$	−11972.8	−0.510820
$820$	0	0
$821$	20670.4	0.878686	0.439343	−	0.898319i	$-0.355211\pi$
$821$	20670.4	0.878686	0.439343	+	0.898319i	$0.355211\pi$
$822$	20432.2	0.866977
$823$	−474.045	−0.0200780	−0.0100390	−	0.999950i	$-0.503196\pi$
$823$	−474.045	−0.0200780	−0.0100390	+	0.999950i	$0.503196\pi$
$824$	21311.1	0.900981
$825$	0	0
$826$	38786.8	1.63386
$827$	−25547.5	−1.07421	−0.537106	−	0.843515i	$-0.680483\pi$
$827$	−25547.5	−1.07421	−0.537106	+	0.843515i	$0.680483\pi$
$828$	455.954	0.0191371
$829$	−36219.9	−1.51745	−0.758726	−	0.651410i	$-0.774178\pi$
$829$	−36219.9	−1.51745	−0.758726	+	0.651410i	$0.774178\pi$
$830$	0	0
$831$	−6440.17	−0.268841
$832$	−28282.6	−1.17851
$833$	−42761.1	−1.77861
$834$	9953.13	0.413248
$835$	0	0
$836$	241.050	0.00997235
$837$	−238.283	−0.00984020
$838$	−25614.1	−1.05588
$839$	−17875.0	−0.735535	−0.367767	−	0.929918i	$-0.619878\pi$
$839$	−17875.0	−0.735535	−0.367767	+	0.929918i	$0.619878\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	−16243.1	−0.664814
$843$	−4337.41	−0.177210
$844$	2124.37	0.0866394
$845$	0	0
$846$	−11145.2	−0.452933
$847$	−33738.4	−1.36867
$848$	−25655.7	−1.03894
$849$	−11327.6	−0.457904
$850$	0	0
$851$	11517.6	0.463945
$852$	−2494.66	−0.100312
$853$	1402.84	0.0563097	0.0281549	−	0.999604i	$-0.491037\pi$
$853$	1402.84	0.0563097	0.0281549	+	0.999604i	$0.491037\pi$
$854$	−33407.1	−1.33860
$855$	0	0
$856$	−22307.1	−0.890702
$857$	−32810.4	−1.30780	−0.653898	−	0.756582i	$-0.726868\pi$
$857$	−32810.4	−1.30780	−0.653898	+	0.756582i	$0.726868\pi$
$858$	−3109.53	−0.123727
$859$	−19007.6	−0.754982	−0.377491	−	0.926013i	$-0.623213\pi$
$859$	−19007.6	−0.754982	−0.377491	+	0.926013i	$0.623213\pi$
$860$	0	0
$861$	24407.5	0.966092
$862$	−14982.2	−0.591991
$863$	6248.92	0.246484	0.123242	−	0.992377i	$-0.460671\pi$
$863$	6248.92	0.246484	0.123242	+	0.992377i	$0.460671\pi$
$864$	−1600.24	−0.0630107
$865$	0	0
$866$	29652.3	1.16354
$867$	26059.4	1.02079
$868$	311.297	0.0121729
$869$	−9312.75	−0.363537
$870$	0	0
$871$	10845.1	0.421896
$872$	1770.69	0.0687650
$873$	−7804.05	−0.302551
$874$	2240.33	0.0867051
$875$	0	0
$876$	−2711.07	−0.104564
$877$	−18018.5	−0.693777	−0.346888	−	0.937906i	$-0.612762\pi$
$877$	−18018.5	−0.693777	−0.346888	+	0.937906i	$0.612762\pi$
$878$	−37282.7	−1.43306
$879$	−19354.2	−0.742661
$880$	0	0
$881$	28136.0	1.07597	0.537983	−	0.842956i	$-0.319186\pi$
$881$	28136.0	1.07597	0.537983	+	0.842956i	$0.319186\pi$
$882$	−8526.79	−0.325524
$883$	41964.1	1.59933	0.799663	−	0.600450i	$-0.205012\pi$
$883$	41964.1	1.59933	0.799663	+	0.600450i	$0.205012\pi$
$884$	−7710.74	−0.293371
$885$	0	0
$886$	−30726.2	−1.16509
$887$	−10526.5	−0.398474	−0.199237	−	0.979951i	$-0.563846\pi$
$887$	−10526.5	−0.398474	−0.199237	+	0.979951i	$0.563846\pi$
$888$	−21756.1	−0.822169
$889$	18880.7	0.712302
$890$	0	0
$891$	−650.704	−0.0244662
$892$	−3693.21	−0.138630
$893$	10861.4	0.407012
$894$	−2029.63	−0.0759296
$895$	0	0
$896$	−26352.8	−0.982575
$897$	5731.98	0.213362
$898$	25531.7	0.948780
$899$	−255.933	−0.00949482
$900$	0	0
$901$	−57921.6	−2.14167
$902$	6339.06	0.233999
$903$	−22047.5	−0.812510
$904$	49837.5	1.83359
$905$	0	0
$906$	−21395.2	−0.784557
$907$	41574.7	1.52201	0.761006	−	0.648744i	$-0.224706\pi$
$907$	41574.7	1.52201	0.761006	+	0.648744i	$0.224706\pi$
$908$	4093.63	0.149617
$909$	11435.1	0.417248
$910$	0	0
$911$	−978.851	−0.0355991	−0.0177996	−	0.999842i	$-0.505666\pi$
$911$	−978.851	−0.0355991	−0.0177996	+	0.999842i	$0.505666\pi$
$912$	−3511.72	−0.127505
$913$	5371.07	0.194695
$914$	30078.4	1.08852
$915$	0	0
$916$	8580.04	0.309490
$917$	−54180.0	−1.95112
$918$	8135.43	0.292494
$919$	−2296.62	−0.0824358	−0.0412179	−	0.999150i	$-0.513124\pi$
$919$	−2296.62	−0.0824358	−0.0412179	+	0.999150i	$0.513124\pi$
$920$	0	0
$921$	−12591.2	−0.450482
$922$	20787.2	0.742507
$923$	−31361.4	−1.11839
$924$	850.093	0.0302662
$925$	0	0
$926$	−11215.0	−0.398000
$927$	7961.37	0.282077
$928$	−1718.77	−0.0607991
$929$	29033.8	1.02537	0.512684	−	0.858578i	$-0.328651\pi$
$929$	29033.8	1.02537	0.512684	+	0.858578i	$0.328651\pi$
$930$	0	0
$931$	8309.61	0.292520
$932$	−4828.95	−0.169718
$933$	20727.8	0.727330
$934$	−8944.15	−0.313342
$935$	0	0
$936$	−10827.4	−0.378103
$937$	−22945.7	−0.800005	−0.400002	−	0.916514i	$-0.630991\pi$
$937$	−22945.7	−0.800005	−0.400002	+	0.916514i	$0.630991\pi$
$938$	14948.5	0.520349
$939$	−2505.93	−0.0870906
$940$	0	0
$941$	31712.3	1.09861	0.549304	−	0.835622i	$-0.314893\pi$
$941$	31712.3	1.09861	0.549304	+	0.835622i	$0.314893\pi$
$942$	14898.0	0.515291
$943$	−11685.1	−0.403521
$944$	29107.3	1.00356
$945$	0	0
$946$	−5726.13	−0.196800
$947$	7050.99	0.241950	0.120975	−	0.992656i	$-0.461398\pi$
$947$	7050.99	0.241950	0.120975	+	0.992656i	$0.461398\pi$
$948$	−4604.85	−0.157762
$949$	−34081.9	−1.16580
$950$	0	0
$951$	1362.40	0.0464551
$952$	−74843.5	−2.54800
$953$	12178.0	0.413938	0.206969	−	0.978348i	$-0.433640\pi$
$953$	12178.0	0.413938	0.206969	+	0.978348i	$0.433640\pi$
$954$	−11549.9	−0.391972
$955$	0	0
$956$	−5114.93	−0.173043
$957$	−698.904	−0.0236075
$958$	−16238.1	−0.547628
$959$	−70221.5	−2.36452
$960$	0	0
$961$	−29713.1	−0.997386
$962$	−38839.5	−1.30170
$963$	−8333.43	−0.278859
$964$	−8219.58	−0.274621
$965$	0	0
$966$	7900.81	0.263151
$967$	13762.2	0.457665	0.228833	−	0.973466i	$-0.426509\pi$
$967$	13762.2	0.457665	0.228833	+	0.973466i	$0.426509\pi$
$968$	−30510.9	−1.01308
$969$	−7928.22	−0.262839
$970$	0	0
$971$	−33646.0	−1.11200	−0.555999	−	0.831183i	$-0.687664\pi$
$971$	−33646.0	−1.11200	−0.555999	+	0.831183i	$0.687664\pi$
$972$	−321.752	−0.0106175
$973$	−34206.9	−1.12705
$974$	−4675.69	−0.153818
$975$	0	0
$976$	−25070.1	−0.822208
$977$	42129.5	1.37957	0.689786	−	0.724013i	$-0.257704\pi$
$977$	42129.5	1.37957	0.689786	+	0.724013i	$0.257704\pi$
$978$	−21580.0	−0.705576
$979$	−6413.05	−0.209358
$980$	0	0
$981$	661.490	0.0215288
$982$	−24675.8	−0.801870
$983$	39740.6	1.28945	0.644724	−	0.764415i	$-0.276972\pi$
$983$	39740.6	1.28945	0.644724	+	0.764415i	$0.276972\pi$
$984$	22072.6	0.715090
$985$	0	0
$986$	8738.05	0.282227
$987$	38304.0	1.23529
$988$	1498.40	0.0482495
$989$	10555.3	0.339372
$990$	0	0
$991$	34300.8	1.09950	0.549748	−	0.835330i	$-0.314724\pi$
$991$	34300.8	1.09950	0.549748	+	0.835330i	$0.314724\pi$
$992$	523.057	0.0167410
$993$	−19104.8	−0.610545
$994$	−43227.7	−1.37938
$995$	0	0
$996$	2655.82	0.0844909
$997$	36545.8	1.16090	0.580449	−	0.814296i	$-0.302877\pi$
$997$	36545.8	1.16090	0.580449	+	0.814296i	$0.302877\pi$
$998$	35327.7	1.12052
$999$	−8127.58	−0.257403

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.4.a.n.1.2		7
5.4	even	2		435.4.a.i.1.6	✓	7
15.14	odd	2		1305.4.a.n.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.4.a.i.1.6	✓	7	5.4	even	2
1305.4.a.n.1.2		7	15.14	odd	2
2175.4.a.n.1.2		7	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2175.a (trivial)

\(f(q)\)	\(=\)	\(q-2.58378 q^{2} +3.00000 q^{3} -1.32408 q^{4} -7.75134 q^{6} +26.6398 q^{7} +24.0914 q^{8} +9.00000 q^{9} -8.03338 q^{11} -3.97225 q^{12} -49.9367 q^{13} -68.8314 q^{14} -51.6541 q^{16} -116.617 q^{17} -23.2540 q^{18} +22.6617 q^{19} +79.9195 q^{21} +20.7565 q^{22} -38.2616 q^{23} +72.2741 q^{24} +129.025 q^{26} +27.0000 q^{27} -35.2733 q^{28} +29.0000 q^{29} -8.82528 q^{31} -59.2681 q^{32} -24.1002 q^{33} +301.312 q^{34} -11.9167 q^{36} -301.022 q^{37} -58.5529 q^{38} -149.810 q^{39} +305.401 q^{41} -206.494 q^{42} -275.872 q^{43} +10.6369 q^{44} +98.8596 q^{46} +479.282 q^{47} -154.962 q^{48} +366.680 q^{49} -349.850 q^{51} +66.1204 q^{52} +496.683 q^{53} -69.7621 q^{54} +641.790 q^{56} +67.9852 q^{57} -74.9296 q^{58} -563.505 q^{59} +485.346 q^{61} +22.8026 q^{62} +239.758 q^{63} +566.369 q^{64} +62.2695 q^{66} -217.176 q^{67} +154.410 q^{68} -114.785 q^{69} +628.023 q^{71} +216.822 q^{72} +682.502 q^{73} +777.773 q^{74} -30.0060 q^{76} -214.008 q^{77} +387.076 q^{78} +1159.26 q^{79} +81.0000 q^{81} -789.089 q^{82} -668.594 q^{83} -105.820 q^{84} +712.792 q^{86} +87.0000 q^{87} -193.535 q^{88} +798.300 q^{89} -1330.31 q^{91} +50.6616 q^{92} -26.4758 q^{93} -1238.36 q^{94} -177.804 q^{96} -867.116 q^{97} -947.421 q^{98} -72.3005 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{2} + 21 q^{3} + 22 q^{4} + 6 q^{6} + 50 q^{7} + 33 q^{8} + 63 q^{9} + 76 q^{11} + 66 q^{12} - 30 q^{13} + 89 q^{14} + 138 q^{16} + 140 q^{17} + 18 q^{18} + 90 q^{19} + 150 q^{21} - 61 q^{22}+ \cdots + 684 q^{99}+O(q^{100}) \) 7 * q + 2 * q^2 + 21 * q^3 + 22 * q^4 + 6 * q^6 + 50 * q^7 + 33 * q^8 + 63 * q^9 + 76 * q^11 + 66 * q^12 - 30 * q^13 + 89 * q^14 + 138 * q^16 + 140 * q^17 + 18 * q^18 + 90 * q^19 + 150 * q^21 - 61 * q^22 - 34 * q^23 + 99 * q^24 - 241 * q^26 + 189 * q^27 + 57 * q^28 + 203 * q^29 + 524 * q^31 + 6 * q^32 + 228 * q^33 + 255 * q^34 + 198 * q^36 + 28 * q^37 - 222 * q^38 - 90 * q^39 + 1532 * q^41 + 267 * q^42 + 464 * q^43 + 1475 * q^44 + 72 * q^46 + 360 * q^47 + 414 * q^48 + 569 * q^49 + 420 * q^51 + 205 * q^52 - 282 * q^53 + 54 * q^54 + 1102 * q^56 + 270 * q^57 + 58 * q^58 + 766 * q^59 + 1200 * q^61 - 2856 * q^62 + 450 * q^63 + 701 * q^64 - 183 * q^66 - 1546 * q^67 - 1801 * q^68 - 102 * q^69 + 1802 * q^71 + 297 * q^72 + 220 * q^73 + 1594 * q^74 + 1960 * q^76 - 3222 * q^77 - 723 * q^78 + 1298 * q^79 + 567 * q^81 - 856 * q^82 - 1652 * q^83 + 171 * q^84 + 7628 * q^86 + 609 * q^87 - 550 * q^88 + 2846 * q^89 - 816 * q^91 - 472 * q^92 + 1572 * q^93 + 745 * q^94 + 18 * q^96 - 1110 * q^97 + 761 * q^98 + 684 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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