LMFDB - Embedded newform 2175.4.a.n.1.5

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			1.5
Root			$2.27711$ 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)

$f(q)$	$=$	\(q+2.27711 q^{2} +3.00000 q^{3} -2.81476 q^{4} +6.83134 q^{6} +26.8439 q^{7} -24.6264 q^{8} +9.00000 q^{9} -58.9546 q^{11} -8.44428 q^{12} -76.8831 q^{13} +61.1265 q^{14} -33.5591 q^{16} +118.899 q^{17} +20.4940 q^{18} -149.359 q^{19} +80.5316 q^{21} -134.246 q^{22} +8.10284 q^{23} -73.8793 q^{24} -175.072 q^{26} +27.0000 q^{27} -75.5590 q^{28} +29.0000 q^{29} +151.923 q^{31} +120.594 q^{32} -176.864 q^{33} +270.747 q^{34} -25.3328 q^{36} +329.276 q^{37} -340.106 q^{38} -230.649 q^{39} +126.272 q^{41} +183.379 q^{42} +416.162 q^{43} +165.943 q^{44} +18.4511 q^{46} +236.796 q^{47} -100.677 q^{48} +377.593 q^{49} +356.698 q^{51} +216.408 q^{52} -470.080 q^{53} +61.4820 q^{54} -661.068 q^{56} -448.076 q^{57} +66.0363 q^{58} +459.022 q^{59} +602.948 q^{61} +345.945 q^{62} +241.595 q^{63} +543.078 q^{64} -402.739 q^{66} -769.705 q^{67} -334.673 q^{68} +24.3085 q^{69} +494.182 q^{71} -221.638 q^{72} +908.575 q^{73} +749.799 q^{74} +420.409 q^{76} -1582.57 q^{77} -525.215 q^{78} +145.802 q^{79} +81.0000 q^{81} +287.535 q^{82} -364.521 q^{83} -226.677 q^{84} +947.648 q^{86} +87.0000 q^{87} +1451.84 q^{88} +1341.18 q^{89} -2063.84 q^{91} -22.8076 q^{92} +455.768 q^{93} +539.210 q^{94} +361.781 q^{96} -441.198 q^{97} +859.822 q^{98} -530.591 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.27711	0.805081	0.402540	−	0.915402i	$-0.368127\pi$
$2$	2.27711	0.805081	0.402540	+	0.915402i	$0.368127\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−2.81476	−0.351845
$5$	0	0
$5$	0	0
$6$	6.83134	0.464814
$7$	26.8439	1.44943	0.724716	−	0.689047i	$-0.241971\pi$
$7$	26.8439	1.44943	0.724716	+	0.689047i	$0.241971\pi$
$8$	−24.6264	−1.08834
$9$	9.00000	0.333333
$10$	0	0
$11$	−58.9546	−1.61595	−0.807976	−	0.589215i	$-0.799437\pi$
$11$	−58.9546	−1.61595	−0.807976	+	0.589215i	$0.799437\pi$
$12$	−8.44428	−0.203138
$13$	−76.8831	−1.64027	−0.820136	−	0.572168i	$-0.806103\pi$
$13$	−76.8831	−1.64027	−0.820136	+	0.572168i	$0.806103\pi$
$14$	61.1265	1.16691
$15$	0	0
$16$	−33.5591	−0.524360
$17$	118.899	1.69631	0.848156	−	0.529747i	$-0.177713\pi$
$17$	118.899	1.69631	0.848156	+	0.529747i	$0.177713\pi$
$18$	20.4940	0.268360
$19$	−149.359	−1.80343	−0.901716	−	0.432328i	$-0.857692\pi$
$19$	−149.359	−1.80343	−0.901716	+	0.432328i	$0.857692\pi$
$20$	0	0
$21$	80.5316	0.836830
$22$	−134.246	−1.30097
$23$	8.10284	0.0734591	0.0367296	−	0.999325i	$-0.488306\pi$
$23$	8.10284	0.0734591	0.0367296	+	0.999325i	$0.488306\pi$
$24$	−73.8793	−0.628356
$25$	0	0
$26$	−175.072	−1.32055
$27$	27.0000	0.192450
$28$	−75.5590	−0.509975
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	151.923	0.880198	0.440099	−	0.897949i	$-0.354943\pi$
$31$	151.923	0.880198	0.440099	+	0.897949i	$0.354943\pi$
$32$	120.594	0.666192
$33$	−176.864	−0.932971
$34$	270.747	1.36567
$35$	0	0
$36$	−25.3328	−0.117282
$37$	329.276	1.46305	0.731523	−	0.681817i	$-0.238810\pi$
$37$	329.276	1.46305	0.731523	+	0.681817i	$0.238810\pi$
$38$	−340.106	−1.45191
$39$	−230.649	−0.947012
$40$	0	0
$41$	126.272	0.480983	0.240492	−	0.970651i	$-0.422691\pi$
$41$	126.272	0.480983	0.240492	+	0.970651i	$0.422691\pi$
$42$	183.379	0.673716
$43$	416.162	1.47591	0.737955	−	0.674850i	$-0.235792\pi$
$43$	416.162	1.47591	0.737955	+	0.674850i	$0.235792\pi$
$44$	165.943	0.568565
$45$	0	0
$46$	18.4511	0.0591405
$47$	236.796	0.734898	0.367449	−	0.930044i	$-0.380231\pi$
$47$	236.796	0.734898	0.367449	+	0.930044i	$0.380231\pi$
$48$	−100.677	−0.302740
$49$	377.593	1.10085
$50$	0	0
$51$	356.698	0.979366
$52$	216.408	0.577122
$53$	−470.080	−1.21831	−0.609155	−	0.793051i	$-0.708491\pi$
$53$	−470.080	−1.21831	−0.609155	+	0.793051i	$0.708491\pi$
$54$	61.4820	0.154938
$55$	0	0
$56$	−661.068	−1.57748
$57$	−448.076	−1.04121
$58$	66.0363	0.149500
$59$	459.022	1.01287	0.506437	−	0.862277i	$-0.330962\pi$
$59$	459.022	1.01287	0.506437	+	0.862277i	$0.330962\pi$
$60$	0	0
$61$	602.948	1.26557	0.632784	−	0.774328i	$-0.281912\pi$
$61$	602.948	1.26557	0.632784	+	0.774328i	$0.281912\pi$
$62$	345.945	0.708630
$63$	241.595	0.483144
$64$	543.078	1.06070
$65$	0	0
$66$	−402.739	−0.751117
$67$	−769.705	−1.40350	−0.701750	−	0.712423i	$-0.747598\pi$
$67$	−769.705	−1.40350	−0.701750	+	0.712423i	$0.747598\pi$
$68$	−334.673	−0.596838
$69$	24.3085	0.0424116
$70$	0	0
$71$	494.182	0.826037	0.413019	−	0.910723i	$-0.364474\pi$
$71$	494.182	0.826037	0.413019	+	0.910723i	$0.364474\pi$
$72$	−221.638	−0.362781
$73$	908.575	1.45672	0.728361	−	0.685194i	$-0.240282\pi$
$73$	908.575	1.45672	0.728361	+	0.685194i	$0.240282\pi$
$74$	749.799	1.17787
$75$	0	0
$76$	420.409	0.634529
$77$	−1582.57	−2.34221
$78$	−525.215	−0.762421
$79$	145.802	0.207645	0.103823	−	0.994596i	$-0.466893\pi$
$79$	145.802	0.207645	0.103823	+	0.994596i	$0.466893\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	287.535	0.387231
$83$	−364.521	−0.482065	−0.241033	−	0.970517i	$-0.577486\pi$
$83$	−364.521	−0.482065	−0.241033	+	0.970517i	$0.577486\pi$
$84$	−226.677	−0.294434
$85$	0	0
$86$	947.648	1.18823
$87$	87.0000	0.107211
$88$	1451.84	1.75871
$89$	1341.18	1.59736	0.798678	−	0.601759i	$-0.205533\pi$
$89$	1341.18	1.59736	0.798678	+	0.601759i	$0.205533\pi$
$90$	0	0
$91$	−2063.84	−2.37746
$92$	−22.8076	−0.0258462
$93$	455.768	0.508182
$94$	539.210	0.591652
$95$	0	0
$96$	361.781	0.384626
$97$	−441.198	−0.461823	−0.230912	−	0.972975i	$-0.574171\pi$
$97$	−441.198	−0.461823	−0.230912	+	0.972975i	$0.574171\pi$
$98$	859.822	0.886277
$99$	−530.591	−0.538651
$100$	0	0
$101$	8.77903	0.00864897	0.00432449	−	0.999991i	$-0.498623\pi$
$101$	8.77903	0.00864897	0.00432449	+	0.999991i	$0.498623\pi$
$102$	812.240	0.788469
$103$	1307.14	1.25045	0.625224	−	0.780445i	$-0.285008\pi$
$103$	1307.14	1.25045	0.625224	+	0.780445i	$0.285008\pi$
$104$	1893.36	1.78518
$105$	0	0
$106$	−1070.42	−0.980838
$107$	−734.360	−0.663488	−0.331744	−	0.943369i	$-0.607637\pi$
$107$	−734.360	−0.663488	−0.331744	+	0.943369i	$0.607637\pi$
$108$	−75.9985	−0.0677126
$109$	735.066	0.645932	0.322966	−	0.946411i	$-0.395320\pi$
$109$	735.066	0.645932	0.322966	+	0.946411i	$0.395320\pi$
$110$	0	0
$111$	987.829	0.844690
$112$	−900.855	−0.760025
$113$	−1335.34	−1.11166	−0.555831	−	0.831295i	$-0.687600\pi$
$113$	−1335.34	−1.11166	−0.555831	+	0.831295i	$0.687600\pi$
$114$	−1020.32	−0.838260
$115$	0	0
$116$	−81.6280	−0.0653360
$117$	−691.948	−0.546758
$118$	1045.24	0.815445
$119$	3191.71	2.45869
$120$	0	0
$121$	2144.64	1.61130
$122$	1372.98	1.01888
$123$	378.815	0.277696
$124$	−427.626	−0.309693
$125$	0	0
$126$	550.138	0.388970
$127$	98.9846	0.0691611	0.0345806	−	0.999402i	$-0.488990\pi$
$127$	98.9846	0.0691611	0.0345806	+	0.999402i	$0.488990\pi$
$128$	271.900	0.187756
$129$	1248.49	0.852117
$130$	0	0
$131$	458.226	0.305614	0.152807	−	0.988256i	$-0.451169\pi$
$131$	458.226	0.305614	0.152807	+	0.988256i	$0.451169\pi$
$132$	497.829	0.328261
$133$	−4009.36	−2.61395
$134$	−1752.71	−1.12993
$135$	0	0
$136$	−2928.06	−1.84617
$137$	1514.86	0.944693	0.472346	−	0.881413i	$-0.343407\pi$
$137$	1514.86	0.944693	0.472346	+	0.881413i	$0.343407\pi$
$138$	55.3533	0.0341448
$139$	−1174.64	−0.716774	−0.358387	−	0.933573i	$-0.616673\pi$
$139$	−1174.64	−0.716774	−0.358387	+	0.933573i	$0.616673\pi$
$140$	0	0
$141$	710.387	0.424293
$142$	1125.31	0.665027
$143$	4532.61	2.65060
$144$	−302.032	−0.174787
$145$	0	0
$146$	2068.93	1.17278
$147$	1132.78	0.635579
$148$	−926.834	−0.514765
$149$	−2868.30	−1.57705	−0.788526	−	0.615002i	$-0.789155\pi$
$149$	−2868.30	−1.57705	−0.788526	+	0.615002i	$0.789155\pi$
$150$	0	0
$151$	−3009.92	−1.62214	−0.811072	−	0.584946i	$-0.801116\pi$
$151$	−3009.92	−1.62214	−0.811072	+	0.584946i	$0.801116\pi$
$152$	3678.17	1.96276
$153$	1070.09	0.565437
$154$	−3603.69	−1.88567
$155$	0	0
$156$	649.223	0.333201
$157$	1831.72	0.931128	0.465564	−	0.885014i	$-0.345852\pi$
$157$	1831.72	0.931128	0.465564	+	0.885014i	$0.345852\pi$
$158$	332.007	0.167171
$159$	−1410.24	−0.703392
$160$	0	0
$161$	217.512	0.106474
$162$	184.446	0.0894534
$163$	−1791.86	−0.861038	−0.430519	−	0.902582i	$-0.641669\pi$
$163$	−1791.86	−0.861038	−0.430519	+	0.902582i	$0.641669\pi$
$164$	−355.424	−0.169232
$165$	0	0
$166$	−830.056	−0.388102
$167$	−1689.71	−0.782958	−0.391479	−	0.920187i	$-0.628036\pi$
$167$	−1689.71	−0.782958	−0.391479	+	0.920187i	$0.628036\pi$
$168$	−1983.21	−0.910759
$169$	3714.02	1.69049
$170$	0	0
$171$	−1344.23	−0.601144
$172$	−1171.40	−0.519291
$173$	196.378	0.0863024	0.0431512	−	0.999069i	$-0.486260\pi$
$173$	196.378	0.0863024	0.0431512	+	0.999069i	$0.486260\pi$
$174$	198.109	0.0863137
$175$	0	0
$176$	1978.46	0.847341
$177$	1377.07	0.584783
$178$	3054.01	1.28600
$179$	−334.894	−0.139839	−0.0699195	−	0.997553i	$-0.522274\pi$
$179$	−334.894	−0.139839	−0.0699195	+	0.997553i	$0.522274\pi$
$180$	0	0
$181$	1111.80	0.456571	0.228286	−	0.973594i	$-0.426688\pi$
$181$	1111.80	0.456571	0.228286	+	0.973594i	$0.426688\pi$
$182$	−4699.60	−1.91405
$183$	1808.85	0.730676
$184$	−199.544	−0.0799488
$185$	0	0
$186$	1037.84	0.409128
$187$	−7009.65	−2.74116
$188$	−666.523	−0.258570
$189$	724.784	0.278943
$190$	0	0
$191$	923.099	0.349702	0.174851	−	0.984595i	$-0.444056\pi$
$191$	923.099	0.349702	0.174851	+	0.984595i	$0.444056\pi$
$192$	1629.23	0.612395
$193$	−1015.66	−0.378803	−0.189401	−	0.981900i	$-0.560655\pi$
$193$	−1015.66	−0.378803	−0.189401	+	0.981900i	$0.560655\pi$
$194$	−1004.66	−0.371805
$195$	0	0
$196$	−1062.83	−0.387330
$197$	3656.92	1.32256	0.661282	−	0.750137i	$-0.270013\pi$
$197$	3656.92	1.32256	0.661282	+	0.750137i	$0.270013\pi$
$198$	−1208.22	−0.433657
$199$	−3408.89	−1.21432	−0.607160	−	0.794580i	$-0.707691\pi$
$199$	−3408.89	−1.21432	−0.607160	+	0.794580i	$0.707691\pi$
$200$	0	0
$201$	−2309.12	−0.810311
$202$	19.9908	0.00696312
$203$	778.472	0.269153
$204$	−1004.02	−0.344585
$205$	0	0
$206$	2976.50	1.00671
$207$	72.9256	0.0244864
$208$	2580.13	0.860094
$209$	8805.38	2.91426
$210$	0	0
$211$	982.934	0.320701	0.160351	−	0.987060i	$-0.448738\pi$
$211$	982.934	0.320701	0.160351	+	0.987060i	$0.448738\pi$
$212$	1323.16	0.428656
$213$	1482.55	0.476913
$214$	−1672.22	−0.534162
$215$	0	0
$216$	−664.913	−0.209452
$217$	4078.19	1.27579
$218$	1673.83	0.520027
$219$	2725.73	0.841039
$220$	0	0
$221$	−9141.34	−2.78241
$222$	2249.40	0.680044
$223$	−987.852	−0.296643	−0.148322	−	0.988939i	$-0.547387\pi$
$223$	−987.852	−0.296643	−0.148322	+	0.988939i	$0.547387\pi$
$224$	3237.20	0.965600
$225$	0	0
$226$	−3040.71	−0.894978
$227$	−3532.07	−1.03274	−0.516369	−	0.856366i	$-0.672717\pi$
$227$	−3532.07	−1.03274	−0.516369	+	0.856366i	$0.672717\pi$
$228$	1261.23	0.366345
$229$	−121.305	−0.0350047	−0.0175023	−	0.999847i	$-0.505571\pi$
$229$	−121.305	−0.0350047	−0.0175023	+	0.999847i	$0.505571\pi$
$230$	0	0
$231$	−4747.71	−1.35228
$232$	−714.166	−0.202100
$233$	−681.137	−0.191514	−0.0957570	−	0.995405i	$-0.530527\pi$
$233$	−681.137	−0.191514	−0.0957570	+	0.995405i	$0.530527\pi$
$234$	−1575.64	−0.440184
$235$	0	0
$236$	−1292.04	−0.356374
$237$	437.405	0.119884
$238$	7267.89	1.97944
$239$	3861.30	1.04505	0.522525	−	0.852624i	$-0.324990\pi$
$239$	3861.30	1.04505	0.522525	+	0.852624i	$0.324990\pi$
$240$	0	0
$241$	−1335.54	−0.356970	−0.178485	−	0.983943i	$-0.557120\pi$
$241$	−1335.54	−0.356970	−0.178485	+	0.983943i	$0.557120\pi$
$242$	4883.59	1.29723
$243$	243.000	0.0641500
$244$	−1697.15	−0.445284
$245$	0	0
$246$	862.604	0.223568
$247$	11483.2	2.95812
$248$	−3741.31	−0.957958
$249$	−1093.56	−0.278321
$250$	0	0
$251$	3078.19	0.774079	0.387040	−	0.922063i	$-0.373498\pi$
$251$	3078.19	0.774079	0.387040	+	0.922063i	$0.373498\pi$
$252$	−680.031	−0.169992
$253$	−477.700	−0.118706
$254$	225.399	0.0556803
$255$	0	0
$256$	−3725.47	−0.909540
$257$	1535.42	0.372672	0.186336	−	0.982486i	$-0.440339\pi$
$257$	1535.42	0.372672	0.186336	+	0.982486i	$0.440339\pi$
$258$	2842.94	0.686023
$259$	8839.05	2.12059
$260$	0	0
$261$	261.000	0.0618984
$262$	1043.43	0.246044
$263$	1460.46	0.342418	0.171209	−	0.985235i	$-0.445233\pi$
$263$	1460.46	0.342418	0.171209	+	0.985235i	$0.445233\pi$
$264$	4355.52	1.01539
$265$	0	0
$266$	−9129.77	−2.10444
$267$	4023.54	0.922234
$268$	2166.54	0.493814
$269$	4489.02	1.01747	0.508737	−	0.860922i	$-0.330113\pi$
$269$	4489.02	1.01747	0.508737	+	0.860922i	$0.330113\pi$
$270$	0	0
$271$	5678.26	1.27280	0.636402	−	0.771357i	$-0.280422\pi$
$271$	5678.26	1.27280	0.636402	+	0.771357i	$0.280422\pi$
$272$	−3990.15	−0.889478
$273$	−6191.52	−1.37263
$274$	3449.50	0.760554
$275$	0	0
$276$	−68.4227	−0.0149223
$277$	8342.92	1.80967	0.904833	−	0.425766i	$-0.139995\pi$
$277$	8342.92	1.80967	0.904833	+	0.425766i	$0.139995\pi$
$278$	−2674.78	−0.577061
$279$	1367.30	0.293399
$280$	0	0
$281$	4551.39	0.966238	0.483119	−	0.875555i	$-0.339504\pi$
$281$	4551.39	0.966238	0.483119	+	0.875555i	$0.339504\pi$
$282$	1617.63	0.341591
$283$	5634.41	1.18350	0.591751	−	0.806121i	$-0.298437\pi$
$283$	5634.41	1.18350	0.591751	+	0.806121i	$0.298437\pi$
$284$	−1391.00	−0.290637
$285$	0	0
$286$	10321.3	2.13395
$287$	3389.62	0.697153
$288$	1085.34	0.222064
$289$	9224.02	1.87747
$290$	0	0
$291$	−1323.59	−0.266634
$292$	−2557.42	−0.512540
$293$	4203.48	0.838122	0.419061	−	0.907958i	$-0.362359\pi$
$293$	4203.48	0.838122	0.419061	+	0.907958i	$0.362359\pi$
$294$	2579.47	0.511692
$295$	0	0
$296$	−8108.90	−1.59230
$297$	−1591.77	−0.310990
$298$	−6531.45	−1.26965
$299$	−622.972	−0.120493
$300$	0	0
$301$	11171.4	2.13923
$302$	−6853.93	−1.30596
$303$	26.3371	0.00499349
$304$	5012.34	0.945649
$305$	0	0
$306$	2436.72	0.455223
$307$	−419.382	−0.0779655	−0.0389827	−	0.999240i	$-0.512412\pi$
$307$	−419.382	−0.0779655	−0.0389827	+	0.999240i	$0.512412\pi$
$308$	4454.55	0.824096
$309$	3921.41	0.721946
$310$	0	0
$311$	7213.66	1.31527	0.657636	−	0.753336i	$-0.271557\pi$
$311$	7213.66	1.31527	0.657636	+	0.753336i	$0.271557\pi$
$312$	5680.07	1.03068
$313$	−1775.17	−0.320571	−0.160286	−	0.987071i	$-0.551242\pi$
$313$	−1775.17	−0.320571	−0.160286	+	0.987071i	$0.551242\pi$
$314$	4171.03	0.749633
$315$	0	0
$316$	−410.397	−0.0730589
$317$	−4522.95	−0.801369	−0.400685	−	0.916216i	$-0.631228\pi$
$317$	−4522.95	−0.801369	−0.400685	+	0.916216i	$0.631228\pi$
$318$	−3211.27	−0.566287
$319$	−1709.68	−0.300075
$320$	0	0
$321$	−2203.08	−0.383065
$322$	495.298	0.0857202
$323$	−17758.6	−3.05918
$324$	−227.995	−0.0390939
$325$	0	0
$326$	−4080.26	−0.693205
$327$	2205.20	0.372929
$328$	−3109.62	−0.523476
$329$	6356.51	1.06518
$330$	0	0
$331$	−4817.66	−0.800007	−0.400004	−	0.916514i	$-0.630991\pi$
$331$	−4817.66	−0.800007	−0.400004	+	0.916514i	$0.630991\pi$
$332$	1026.04	0.169612
$333$	2963.49	0.487682
$334$	−3847.67	−0.630345
$335$	0	0
$336$	−2702.56	−0.438801
$337$	−1947.15	−0.314742	−0.157371	−	0.987540i	$-0.550302\pi$
$337$	−1947.15	−0.314742	−0.157371	+	0.987540i	$0.550302\pi$
$338$	8457.24	1.36099
$339$	−4006.01	−0.641819
$340$	0	0
$341$	−8956.54	−1.42236
$342$	−3060.96	−0.483970
$343$	928.613	0.146182
$344$	−10248.6	−1.60630
$345$	0	0
$346$	447.174	0.0694804
$347$	−6797.20	−1.05156	−0.525782	−	0.850619i	$-0.676227\pi$
$347$	−6797.20	−1.05156	−0.525782	+	0.850619i	$0.676227\pi$
$348$	−244.884	−0.0377217
$349$	−9920.99	−1.52166	−0.760829	−	0.648953i	$-0.775207\pi$
$349$	−9920.99	−1.52166	−0.760829	+	0.648953i	$0.775207\pi$
$350$	0	0
$351$	−2075.84	−0.315671
$352$	−7109.55	−1.07653
$353$	2488.63	0.375231	0.187616	−	0.982243i	$-0.439924\pi$
$353$	2488.63	0.375231	0.187616	+	0.982243i	$0.439924\pi$
$354$	3135.73	0.470798
$355$	0	0
$356$	−3775.09	−0.562021
$357$	9575.14	1.41952
$358$	−762.592	−0.112582
$359$	7951.79	1.16902	0.584512	−	0.811385i	$-0.301286\pi$
$359$	7951.79	1.16902	0.584512	+	0.811385i	$0.301286\pi$
$360$	0	0
$361$	15449.0	2.25237
$362$	2531.69	0.367577
$363$	6433.93	0.930286
$364$	5809.21	0.836499
$365$	0	0
$366$	4118.94	0.588253
$367$	−8545.32	−1.21543	−0.607714	−	0.794156i	$-0.707913\pi$
$367$	−8545.32	−1.21543	−0.607714	+	0.794156i	$0.707913\pi$
$368$	−271.924	−0.0385190
$369$	1136.45	0.160328
$370$	0	0
$371$	−12618.8	−1.76586
$372$	−1282.88	−0.178801
$373$	10494.4	1.45678	0.728391	−	0.685162i	$-0.240269\pi$
$373$	10494.4	1.45678	0.728391	+	0.685162i	$0.240269\pi$
$374$	−15961.8	−2.20685
$375$	0	0
$376$	−5831.43	−0.799822
$377$	−2229.61	−0.304591
$378$	1650.42	0.224572
$379$	3861.31	0.523330	0.261665	−	0.965159i	$-0.415728\pi$
$379$	3861.31	0.523330	0.261665	+	0.965159i	$0.415728\pi$
$380$	0	0
$381$	296.954	0.0399302
$382$	2102.00	0.281539
$383$	−5649.95	−0.753784	−0.376892	−	0.926257i	$-0.623007\pi$
$383$	−5649.95	−0.753784	−0.376892	+	0.926257i	$0.623007\pi$
$384$	815.700	0.108401
$385$	0	0
$386$	−2312.77	−0.304967
$387$	3745.46	0.491970
$388$	1241.87	0.162490
$389$	2257.79	0.294279	0.147139	−	0.989116i	$-0.452993\pi$
$389$	2257.79	0.294279	0.147139	+	0.989116i	$0.452993\pi$
$390$	0	0
$391$	963.422	0.124610
$392$	−9298.77	−1.19811
$393$	1374.68	0.176446
$394$	8327.23	1.06477
$395$	0	0
$396$	1493.49	0.189522
$397$	−221.335	−0.0279811	−0.0139905	−	0.999902i	$-0.504453\pi$
$397$	−221.335	−0.0279811	−0.0139905	+	0.999902i	$0.504453\pi$
$398$	−7762.42	−0.977625
$399$	−12028.1	−1.50917
$400$	0	0
$401$	3656.31	0.455330	0.227665	−	0.973739i	$-0.426891\pi$
$401$	3656.31	0.455330	0.227665	+	0.973739i	$0.426891\pi$
$402$	−5258.12	−0.652366
$403$	−11680.3	−1.44376
$404$	−24.7109	−0.00304310
$405$	0	0
$406$	1772.67	0.216690
$407$	−19412.4	−2.36421
$408$	−8784.19	−1.06589
$409$	−1114.22	−0.134706	−0.0673530	−	0.997729i	$-0.521455\pi$
$409$	−1114.22	−0.134706	−0.0673530	+	0.997729i	$0.521455\pi$
$410$	0	0
$411$	4544.57	0.545419
$412$	−3679.28	−0.439964
$413$	12321.9	1.46809
$414$	166.060	0.0197135
$415$	0	0
$416$	−9271.62	−1.09274
$417$	−3523.92	−0.413829
$418$	20050.8	2.34622
$419$	−2992.41	−0.348899	−0.174450	−	0.984666i	$-0.555815\pi$
$419$	−2992.41	−0.348899	−0.174450	+	0.984666i	$0.555815\pi$
$420$	0	0
$421$	−7412.33	−0.858087	−0.429044	−	0.903284i	$-0.641149\pi$
$421$	−7412.33	−0.858087	−0.429044	+	0.903284i	$0.641149\pi$
$422$	2238.25	0.258190
$423$	2131.16	0.244966
$424$	11576.4	1.32594
$425$	0	0
$426$	3375.93	0.383953
$427$	16185.5	1.83436
$428$	2067.05	0.233445
$429$	13597.8	1.53033
$430$	0	0
$431$	2761.79	0.308656	0.154328	−	0.988020i	$-0.450679\pi$
$431$	2761.79	0.308656	0.154328	+	0.988020i	$0.450679\pi$
$432$	−906.095	−0.100913
$433$	15263.4	1.69402	0.847012	−	0.531573i	$-0.178399\pi$
$433$	15263.4	1.69402	0.847012	+	0.531573i	$0.178399\pi$
$434$	9286.50	1.02711
$435$	0	0
$436$	−2069.03	−0.227268
$437$	−1210.23	−0.132479
$438$	6206.78	0.677104
$439$	−9343.27	−1.01579	−0.507893	−	0.861420i	$-0.669575\pi$
$439$	−9343.27	−1.01579	−0.507893	+	0.861420i	$0.669575\pi$
$440$	0	0
$441$	3398.34	0.366952
$442$	−20815.9	−2.24007
$443$	8167.05	0.875911	0.437955	−	0.898997i	$-0.355703\pi$
$443$	8167.05	0.875911	0.437955	+	0.898997i	$0.355703\pi$
$444$	−2780.50	−0.297200
$445$	0	0
$446$	−2249.45	−0.238822
$447$	−8604.91	−0.910511
$448$	14578.3	1.53741
$449$	4390.88	0.461511	0.230756	−	0.973012i	$-0.425880\pi$
$449$	4390.88	0.461511	0.230756	+	0.973012i	$0.425880\pi$
$450$	0	0
$451$	−7444.29	−0.777246
$452$	3758.65	0.391133
$453$	−9029.76	−0.936546
$454$	−8042.91	−0.831438
$455$	0	0
$456$	11034.5	1.13320
$457$	−16232.1	−1.66150	−0.830748	−	0.556648i	$-0.812087\pi$
$457$	−16232.1	−1.66150	−0.830748	+	0.556648i	$0.812087\pi$
$458$	−276.226	−0.0281816
$459$	3210.28	0.326455
$460$	0	0
$461$	14847.4	1.50002	0.750011	−	0.661425i	$-0.230048\pi$
$461$	14847.4	1.50002	0.750011	+	0.661425i	$0.230048\pi$
$462$	−10811.1	−1.08869
$463$	−9208.48	−0.924307	−0.462154	−	0.886800i	$-0.652923\pi$
$463$	−9208.48	−0.924307	−0.462154	+	0.886800i	$0.652923\pi$
$464$	−973.213	−0.0973713
$465$	0	0
$466$	−1551.02	−0.154184
$467$	8652.79	0.857394	0.428697	−	0.903448i	$-0.358973\pi$
$467$	8652.79	0.857394	0.428697	+	0.903448i	$0.358973\pi$
$468$	1947.67	0.192374
$469$	−20661.9	−2.03428
$470$	0	0
$471$	5495.16	0.537587
$472$	−11304.1	−1.10236
$473$	−24534.7	−2.38500
$474$	996.021	0.0965164
$475$	0	0
$476$	−8983.91	−0.865077
$477$	−4230.72	−0.406103
$478$	8792.62	0.841350
$479$	−9373.99	−0.894173	−0.447086	−	0.894491i	$-0.647538\pi$
$479$	−9373.99	−0.894173	−0.447086	+	0.894491i	$0.647538\pi$
$480$	0	0
$481$	−25315.8	−2.39979
$482$	−3041.18	−0.287390
$483$	652.535	0.0614728
$484$	−6036.65	−0.566929
$485$	0	0
$486$	553.338	0.0516460
$487$	6481.75	0.603114	0.301557	−	0.953448i	$-0.402494\pi$
$487$	6481.75	0.603114	0.301557	+	0.953448i	$0.402494\pi$
$488$	−14848.5	−1.37737
$489$	−5375.57	−0.497120
$490$	0	0
$491$	−1108.08	−0.101847	−0.0509235	−	0.998703i	$-0.516216\pi$
$491$	−1108.08	−0.101847	−0.0509235	+	0.998703i	$0.516216\pi$
$492$	−1066.27	−0.0977059
$493$	3448.08	0.314997
$494$	26148.5	2.38153
$495$	0	0
$496$	−5098.38	−0.461541
$497$	13265.8	1.19729
$498$	−2490.17	−0.224071
$499$	4054.00	0.363691	0.181845	−	0.983327i	$-0.441793\pi$
$499$	4054.00	0.363691	0.181845	+	0.983327i	$0.441793\pi$
$500$	0	0
$501$	−5069.14	−0.452041
$502$	7009.40	0.623196
$503$	2816.84	0.249695	0.124847	−	0.992176i	$-0.460156\pi$
$503$	2816.84	0.249695	0.124847	+	0.992176i	$0.460156\pi$
$504$	−5949.62	−0.525827
$505$	0	0
$506$	−1087.78	−0.0955683
$507$	11142.1	0.976008
$508$	−278.618	−0.0243340
$509$	−830.334	−0.0723063	−0.0361532	−	0.999346i	$-0.511510\pi$
$509$	−830.334	−0.0723063	−0.0361532	+	0.999346i	$0.511510\pi$
$510$	0	0
$511$	24389.7	2.11142
$512$	−10658.5	−0.920009
$513$	−4032.68	−0.347071
$514$	3496.31	0.300031
$515$	0	0
$516$	−3514.19	−0.299813
$517$	−13960.2	−1.18756
$518$	20127.5	1.70724
$519$	589.133	0.0498267
$520$	0	0
$521$	−14425.9	−1.21307	−0.606536	−	0.795056i	$-0.707441\pi$
$521$	−14425.9	−1.21307	−0.606536	+	0.795056i	$0.707441\pi$
$522$	594.326	0.0498333
$523$	−18615.3	−1.55638	−0.778192	−	0.628027i	$-0.783863\pi$
$523$	−18615.3	−1.55638	−0.778192	+	0.628027i	$0.783863\pi$
$524$	−1289.80	−0.107529
$525$	0	0
$526$	3325.63	0.275674
$527$	18063.5	1.49309
$528$	5935.38	0.489213
$529$	−12101.3	−0.994604
$530$	0	0
$531$	4131.20	0.337625
$532$	11285.4	0.919706
$533$	−9708.16	−0.788944
$534$	9162.04	0.742473
$535$	0	0
$536$	18955.1	1.52749
$537$	−1004.68	−0.0807360
$538$	10222.0	0.819148
$539$	−22260.8	−1.77893
$540$	0	0
$541$	5586.54	0.443963	0.221982	−	0.975051i	$-0.428748\pi$
$541$	5586.54	0.443963	0.221982	+	0.975051i	$0.428748\pi$
$542$	12930.0	1.02471
$543$	3335.40	0.263602
$544$	14338.5	1.13007
$545$	0	0
$546$	−14098.8	−1.10508
$547$	4779.89	0.373626	0.186813	−	0.982396i	$-0.440184\pi$
$547$	4779.89	0.373626	0.186813	+	0.982396i	$0.440184\pi$
$548$	−4263.96	−0.332385
$549$	5426.54	0.421856
$550$	0	0
$551$	−4331.40	−0.334889
$552$	−598.632	−0.0461585
$553$	3913.88	0.300968
$554$	18997.8	1.45693
$555$	0	0
$556$	3306.32	0.252193
$557$	11829.0	0.899837	0.449919	−	0.893070i	$-0.351453\pi$
$557$	11829.0	0.899837	0.449919	+	0.893070i	$0.351453\pi$
$558$	3113.51	0.236210
$559$	−31995.8	−2.42089
$560$	0	0
$561$	−21029.0	−1.58261
$562$	10364.0	0.777900
$563$	22516.9	1.68557	0.842783	−	0.538254i	$-0.180916\pi$
$563$	22516.9	1.68557	0.842783	+	0.538254i	$0.180916\pi$
$564$	−1999.57	−0.149285
$565$	0	0
$566$	12830.2	0.952815
$567$	2174.35	0.161048
$568$	−12169.9	−0.899013
$569$	−6879.75	−0.506879	−0.253439	−	0.967351i	$-0.581562\pi$
$569$	−6879.75	−0.506879	−0.253439	+	0.967351i	$0.581562\pi$
$570$	0	0
$571$	−3088.90	−0.226386	−0.113193	−	0.993573i	$-0.536108\pi$
$571$	−3088.90	−0.226386	−0.113193	+	0.993573i	$0.536108\pi$
$572$	−12758.2	−0.932601
$573$	2769.30	0.201901
$574$	7718.55	0.561265
$575$	0	0
$576$	4887.70	0.353566
$577$	−760.799	−0.0548916	−0.0274458	−	0.999623i	$-0.508737\pi$
$577$	−760.799	−0.0548916	−0.0274458	+	0.999623i	$0.508737\pi$
$578$	21004.1	1.51152
$579$	−3046.98	−0.218702
$580$	0	0
$581$	−9785.16	−0.698721
$582$	−3013.97	−0.214662
$583$	27713.4	1.96873
$584$	−22375.0	−1.58542
$585$	0	0
$586$	9571.79	0.674756
$587$	3106.07	0.218401	0.109200	−	0.994020i	$-0.465171\pi$
$587$	3106.07	0.218401	0.109200	+	0.994020i	$0.465171\pi$
$588$	−3188.50	−0.223625
$589$	−22691.0	−1.58738
$590$	0	0
$591$	10970.8	0.763583
$592$	−11050.2	−0.767163
$593$	7836.74	0.542692	0.271346	−	0.962482i	$-0.412531\pi$
$593$	7836.74	0.542692	0.271346	+	0.962482i	$0.412531\pi$
$594$	−3624.65	−0.250372
$595$	0	0
$596$	8073.59	0.554877
$597$	−10226.7	−0.701088
$598$	−1418.58	−0.0970066
$599$	9174.05	0.625779	0.312889	−	0.949790i	$-0.398703\pi$
$599$	9174.05	0.625779	0.312889	+	0.949790i	$0.398703\pi$
$600$	0	0
$601$	−8958.64	−0.608037	−0.304019	−	0.952666i	$-0.598329\pi$
$601$	−8958.64	−0.608037	−0.304019	+	0.952666i	$0.598329\pi$
$602$	25438.5	1.72225
$603$	−6927.35	−0.467833
$604$	8472.20	0.570743
$605$	0	0
$606$	59.9725	0.00402016
$607$	−14137.8	−0.945362	−0.472681	−	0.881234i	$-0.656714\pi$
$607$	−14137.8	−0.945362	−0.472681	+	0.881234i	$0.656714\pi$
$608$	−18011.7	−1.20143
$609$	2335.42	0.155395
$610$	0	0
$611$	−18205.6	−1.20543
$612$	−3012.05	−0.198946
$613$	−22666.7	−1.49347	−0.746737	−	0.665120i	$-0.768380\pi$
$613$	−22666.7	−1.49347	−0.746737	+	0.665120i	$0.768380\pi$
$614$	−954.980	−0.0627685
$615$	0	0
$616$	38973.0	2.54914
$617$	−12743.3	−0.831487	−0.415744	−	0.909482i	$-0.636479\pi$
$617$	−12743.3	−0.831487	−0.415744	+	0.909482i	$0.636479\pi$
$618$	8929.50	0.581225
$619$	13912.5	0.903379	0.451690	−	0.892175i	$-0.350821\pi$
$619$	13912.5	0.903379	0.451690	+	0.892175i	$0.350821\pi$
$620$	0	0
$621$	218.777	0.0141372
$622$	16426.3	1.05890
$623$	36002.4	2.31526
$624$	7740.38	0.496575
$625$	0	0
$626$	−4042.27	−0.258086
$627$	26416.1	1.68255
$628$	−5155.85	−0.327613
$629$	39150.7	2.48178
$630$	0	0
$631$	−6045.07	−0.381379	−0.190690	−	0.981650i	$-0.561072\pi$
$631$	−6045.07	−0.381379	−0.190690	+	0.981650i	$0.561072\pi$
$632$	−3590.58	−0.225990
$633$	2948.80	0.185157
$634$	−10299.3	−0.645167
$635$	0	0
$636$	3969.49	0.247485
$637$	−29030.5	−1.80570
$638$	−3893.14	−0.241585
$639$	4447.64	0.275346
$640$	0	0
$641$	−19600.7	−1.20777	−0.603885	−	0.797071i	$-0.706382\pi$
$641$	−19600.7	−1.20777	−0.603885	+	0.797071i	$0.706382\pi$
$642$	−5016.66	−0.308398
$643$	11909.9	0.730454	0.365227	−	0.930919i	$-0.380991\pi$
$643$	11909.9	0.730454	0.365227	+	0.930919i	$0.380991\pi$
$644$	−612.243	−0.0374623
$645$	0	0
$646$	−40438.4	−2.46289
$647$	−10476.3	−0.636576	−0.318288	−	0.947994i	$-0.603108\pi$
$647$	−10476.3	−0.636576	−0.318288	+	0.947994i	$0.603108\pi$
$648$	−1994.74	−0.120927
$649$	−27061.4	−1.63676
$650$	0	0
$651$	12234.6	0.736576
$652$	5043.65	0.302952
$653$	−21201.7	−1.27058	−0.635289	−	0.772275i	$-0.719119\pi$
$653$	−21201.7	−1.27058	−0.635289	+	0.772275i	$0.719119\pi$
$654$	5021.48	0.300238
$655$	0	0
$656$	−4237.56	−0.252209
$657$	8177.18	0.485574
$658$	14474.5	0.857560
$659$	23605.3	1.39534	0.697671	−	0.716418i	$-0.254220\pi$
$659$	23605.3	1.39534	0.697671	+	0.716418i	$0.254220\pi$
$660$	0	0
$661$	12258.0	0.721305	0.360652	−	0.932700i	$-0.382554\pi$
$661$	12258.0	0.721305	0.360652	+	0.932700i	$0.382554\pi$
$662$	−10970.3	−0.644070
$663$	−27424.0	−1.60643
$664$	8976.86	0.524653
$665$	0	0
$666$	6748.19	0.392623
$667$	234.982	0.0136410
$668$	4756.14	0.275480
$669$	−2963.56	−0.171267
$670$	0	0
$671$	−35546.6	−2.04510
$672$	9711.60	0.557490
$673$	−9476.64	−0.542790	−0.271395	−	0.962468i	$-0.587485\pi$
$673$	−9476.64	−0.542790	−0.271395	+	0.962468i	$0.587485\pi$
$674$	−4433.88	−0.253393
$675$	0	0
$676$	−10454.1	−0.594792
$677$	15508.3	0.880401	0.440200	−	0.897900i	$-0.354907\pi$
$677$	15508.3	0.880401	0.440200	+	0.897900i	$0.354907\pi$
$678$	−9122.13	−0.516716
$679$	−11843.5	−0.669382
$680$	0	0
$681$	−10596.2	−0.596252
$682$	−20395.1	−1.14511
$683$	18286.7	1.02448	0.512242	−	0.858841i	$-0.328815\pi$
$683$	18286.7	1.02448	0.512242	+	0.858841i	$0.328815\pi$
$684$	3783.68	0.211510
$685$	0	0
$686$	2114.56	0.117688
$687$	−363.916	−0.0202100
$688$	−13966.0	−0.773908
$689$	36141.2	1.99836
$690$	0	0
$691$	−22335.4	−1.22964	−0.614819	−	0.788668i	$-0.710771\pi$
$691$	−22335.4	−1.22964	−0.614819	+	0.788668i	$0.710771\pi$
$692$	−552.756	−0.0303651
$693$	−14243.1	−0.780738
$694$	−15478.0	−0.846594
$695$	0	0
$696$	−2142.50	−0.116683
$697$	15013.6	0.815898
$698$	−22591.2	−1.22506
$699$	−2043.41	−0.110571
$700$	0	0
$701$	−1101.09	−0.0593259	−0.0296630	−	0.999560i	$-0.509443\pi$
$701$	−1101.09	−0.0593259	−0.0296630	+	0.999560i	$0.509443\pi$
$702$	−4726.93	−0.254140
$703$	−49180.3	−2.63851
$704$	−32016.9	−1.71404
$705$	0	0
$706$	5666.90	0.302091
$707$	235.663	0.0125361
$708$	−3876.11	−0.205753
$709$	−3572.01	−0.189209	−0.0946047	−	0.995515i	$-0.530159\pi$
$709$	−3572.01	−0.189209	−0.0946047	+	0.995515i	$0.530159\pi$
$710$	0	0
$711$	1312.22	0.0692151
$712$	−33028.4	−1.73847
$713$	1231.01	0.0646585
$714$	21803.7	1.14283
$715$	0	0
$716$	942.647	0.0492016
$717$	11583.9	0.603360
$718$	18107.1	0.941158
$719$	16054.3	0.832719	0.416359	−	0.909200i	$-0.363306\pi$
$719$	16054.3	0.832719	0.416359	+	0.909200i	$0.363306\pi$
$720$	0	0
$721$	35088.6	1.81244
$722$	35179.1	1.81334
$723$	−4006.63	−0.206097
$724$	−3129.45	−0.160642
$725$	0	0
$726$	14650.8	0.748955
$727$	34377.9	1.75379	0.876895	−	0.480682i	$-0.159611\pi$
$727$	34377.9	1.75379	0.876895	+	0.480682i	$0.159611\pi$
$728$	50825.0	2.58750
$729$	729.000	0.0370370
$730$	0	0
$731$	49481.3	2.50360
$732$	−5091.46	−0.257085
$733$	−18345.6	−0.924435	−0.462217	−	0.886767i	$-0.652946\pi$
$733$	−18345.6	−0.924435	−0.462217	+	0.886767i	$0.652946\pi$
$734$	−19458.7	−0.978518
$735$	0	0
$736$	977.151	0.0489379
$737$	45377.7	2.26799
$738$	2587.81	0.129077
$739$	−6654.19	−0.331229	−0.165614	−	0.986191i	$-0.552961\pi$
$739$	−6654.19	−0.331229	−0.165614	+	0.986191i	$0.552961\pi$
$740$	0	0
$741$	34449.5	1.70787
$742$	−28734.3	−1.42166
$743$	−9867.74	−0.487231	−0.243615	−	0.969872i	$-0.578333\pi$
$743$	−9867.74	−0.487231	−0.243615	+	0.969872i	$0.578333\pi$
$744$	−11223.9	−0.553077
$745$	0	0
$746$	23896.9	1.17283
$747$	−3280.69	−0.160688
$748$	19730.5	0.964462
$749$	−19713.1	−0.961682
$750$	0	0
$751$	−10030.7	−0.487385	−0.243693	−	0.969853i	$-0.578359\pi$
$751$	−10030.7	−0.487385	−0.243693	+	0.969853i	$0.578359\pi$
$752$	−7946.64	−0.385351
$753$	9234.58	0.446915
$754$	−5077.08	−0.245220
$755$	0	0
$756$	−2040.09	−0.0981448
$757$	−21074.4	−1.01184	−0.505919	−	0.862581i	$-0.668846\pi$
$757$	−21074.4	−1.01184	−0.505919	+	0.862581i	$0.668846\pi$
$758$	8792.63	0.421323
$759$	−1433.10	−0.0685352
$760$	0	0
$761$	7469.19	0.355792	0.177896	−	0.984049i	$-0.443071\pi$
$761$	7469.19	0.355792	0.177896	+	0.984049i	$0.443071\pi$
$762$	676.197	0.0321470
$763$	19732.0	0.936234
$764$	−2598.30	−0.123041
$765$	0	0
$766$	−12865.6	−0.606857
$767$	−35291.0	−1.66139
$768$	−11176.4	−0.525123
$769$	−5361.63	−0.251424	−0.125712	−	0.992067i	$-0.540122\pi$
$769$	−5361.63	−0.251424	−0.125712	+	0.992067i	$0.540122\pi$
$770$	0	0
$771$	4606.25	0.215162
$772$	2858.84	0.133280
$773$	24649.9	1.14695	0.573476	−	0.819222i	$-0.305595\pi$
$773$	24649.9	1.14695	0.573476	+	0.819222i	$0.305595\pi$
$774$	8528.83	0.396075
$775$	0	0
$776$	10865.1	0.502623
$777$	26517.2	1.22432
$778$	5141.24	0.236918
$779$	−18859.8	−0.867421
$780$	0	0
$781$	−29134.3	−1.33484
$782$	2193.82	0.100321
$783$	783.000	0.0357371
$784$	−12671.7	−0.577244
$785$	0	0
$786$	3130.30	0.142054
$787$	33943.1	1.53741	0.768705	−	0.639604i	$-0.220902\pi$
$787$	33943.1	1.53741	0.768705	+	0.639604i	$0.220902\pi$
$788$	−10293.4	−0.465337
$789$	4381.38	0.197695
$790$	0	0
$791$	−35845.6	−1.61128
$792$	13066.6	0.586238
$793$	−46356.6	−2.07588
$794$	−504.005	−0.0225270
$795$	0	0
$796$	9595.19	0.427252
$797$	−16602.5	−0.737882	−0.368941	−	0.929453i	$-0.620280\pi$
$797$	−16602.5	−0.737882	−0.368941	+	0.929453i	$0.620280\pi$
$798$	−27389.3	−1.21500
$799$	28154.8	1.24662
$800$	0	0
$801$	12070.6	0.532452
$802$	8325.83	0.366578
$803$	−53564.7	−2.35399
$804$	6499.61	0.285104
$805$	0	0
$806$	−26597.3	−1.16235
$807$	13467.1	0.587439
$808$	−216.196	−0.00941306
$809$	−2323.51	−0.100977	−0.0504883	−	0.998725i	$-0.516078\pi$
$809$	−2323.51	−0.100977	−0.0504883	+	0.998725i	$0.516078\pi$
$810$	0	0
$811$	38440.8	1.66441	0.832206	−	0.554466i	$-0.187078\pi$
$811$	38440.8	1.66441	0.832206	+	0.554466i	$0.187078\pi$
$812$	−2191.21	−0.0947001
$813$	17034.8	0.734854
$814$	−44204.1	−1.90338
$815$	0	0
$816$	−11970.4	−0.513541
$817$	−62157.4	−2.66170
$818$	−2537.21	−0.108449
$819$	−18574.6	−0.792488
$820$	0	0
$821$	27808.5	1.18212	0.591062	−	0.806626i	$-0.298709\pi$
$821$	27808.5	1.18212	0.591062	+	0.806626i	$0.298709\pi$
$822$	10348.5	0.439106
$823$	7497.23	0.317542	0.158771	−	0.987315i	$-0.449247\pi$
$823$	7497.23	0.317542	0.158771	+	0.987315i	$0.449247\pi$
$824$	−32190.1	−1.36092
$825$	0	0
$826$	28058.4	1.18193
$827$	−33174.6	−1.39491	−0.697457	−	0.716626i	$-0.745685\pi$
$827$	−33174.6	−1.39491	−0.697457	+	0.716626i	$0.745685\pi$
$828$	−205.268	−0.00861540
$829$	9823.48	0.411561	0.205780	−	0.978598i	$-0.434027\pi$
$829$	9823.48	0.411561	0.205780	+	0.978598i	$0.434027\pi$
$830$	0	0
$831$	25028.8	1.04481
$832$	−41753.5	−1.73984
$833$	44895.5	1.86739
$834$	−8024.35	−0.333166
$835$	0	0
$836$	−24785.0	−1.02537
$837$	4101.91	0.169394
$838$	−6814.05	−0.280892
$839$	22580.6	0.929165	0.464583	−	0.885530i	$-0.346204\pi$
$839$	22580.6	0.929165	0.464583	+	0.885530i	$0.346204\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	−16878.7	−0.690830
$843$	13654.2	0.557858
$844$	−2766.72	−0.112837
$845$	0	0
$846$	4852.89	0.197217
$847$	57570.5	2.33547
$848$	15775.4	0.638834
$849$	16903.2	0.683295
$850$	0	0
$851$	2668.07	0.107474
$852$	−4173.01	−0.167799
$853$	28170.3	1.13075	0.565377	−	0.824832i	$-0.308731\pi$
$853$	28170.3	1.13075	0.565377	+	0.824832i	$0.308731\pi$
$854$	36856.1	1.47680
$855$	0	0
$856$	18084.7	0.722104
$857$	−3158.13	−0.125880	−0.0629402	−	0.998017i	$-0.520048\pi$
$857$	−3158.13	−0.125880	−0.0629402	+	0.998017i	$0.520048\pi$
$858$	30963.8	1.23204
$859$	5122.45	0.203464	0.101732	−	0.994812i	$-0.467562\pi$
$859$	5122.45	0.203464	0.101732	+	0.994812i	$0.467562\pi$
$860$	0	0
$861$	10168.9	0.402502
$862$	6288.92	0.248493
$863$	3587.38	0.141502	0.0707509	−	0.997494i	$-0.477460\pi$
$863$	3587.38	0.141502	0.0707509	+	0.997494i	$0.477460\pi$
$864$	3256.03	0.128209
$865$	0	0
$866$	34756.5	1.36383
$867$	27672.1	1.08396
$868$	−11479.1	−0.448879
$869$	−8595.68	−0.335545
$870$	0	0
$871$	59177.4	2.30212
$872$	−18102.0	−0.702996
$873$	−3970.78	−0.153941
$874$	−2755.83	−0.106656
$875$	0	0
$876$	−7672.26	−0.295915
$877$	−28862.1	−1.11129	−0.555646	−	0.831419i	$-0.687529\pi$
$877$	−28862.1	−1.11129	−0.555646	+	0.831419i	$0.687529\pi$
$878$	−21275.7	−0.817789
$879$	12610.4	0.483890
$880$	0	0
$881$	−23389.1	−0.894439	−0.447219	−	0.894424i	$-0.647586\pi$
$881$	−23389.1	−0.894439	−0.447219	+	0.894424i	$0.647586\pi$
$882$	7738.40	0.295426
$883$	−27097.2	−1.03272	−0.516360	−	0.856371i	$-0.672713\pi$
$883$	−27097.2	−1.03272	−0.516360	+	0.856371i	$0.672713\pi$
$884$	25730.7	0.978978
$885$	0	0
$886$	18597.3	0.705179
$887$	37179.8	1.40741	0.703707	−	0.710490i	$-0.251527\pi$
$887$	37179.8	1.40741	0.703707	+	0.710490i	$0.251527\pi$
$888$	−24326.7	−0.919314
$889$	2657.13	0.100244
$890$	0	0
$891$	−4775.32	−0.179550
$892$	2780.57	0.104372
$893$	−35367.5	−1.32534
$894$	−19594.4	−0.733035
$895$	0	0
$896$	7298.85	0.272140
$897$	−1868.92	−0.0695667
$898$	9998.53	0.371554
$899$	4405.76	0.163449
$900$	0	0
$901$	−55892.1	−2.06663
$902$	−16951.5	−0.625746
$903$	33514.2	1.23509
$904$	32884.6	1.20987
$905$	0	0
$906$	−20561.8	−0.753995
$907$	−19859.6	−0.727041	−0.363521	−	0.931586i	$-0.618425\pi$
$907$	−19859.6	−0.727041	−0.363521	+	0.931586i	$0.618425\pi$
$908$	9941.92	0.363364
$909$	79.0113	0.00288299
$910$	0	0
$911$	25471.8	0.926366	0.463183	−	0.886263i	$-0.346707\pi$
$911$	25471.8	0.926366	0.463183	+	0.886263i	$0.346707\pi$
$912$	15037.0	0.545971
$913$	21490.2	0.778995
$914$	−36962.2	−1.33764
$915$	0	0
$916$	341.445	0.0123162
$917$	12300.6	0.442967
$918$	7310.16	0.262823
$919$	−26174.7	−0.939526	−0.469763	−	0.882793i	$-0.655661\pi$
$919$	−26174.7	−0.939526	−0.469763	+	0.882793i	$0.655661\pi$
$920$	0	0
$921$	−1258.15	−0.0450134
$922$	33809.1	1.20764
$923$	−37994.3	−1.35493
$924$	13363.7	0.475792
$925$	0	0
$926$	−20968.7	−0.744142
$927$	11764.2	0.416816
$928$	3497.22	0.123709
$929$	−42691.9	−1.50772	−0.753862	−	0.657033i	$-0.771811\pi$
$929$	−42691.9	−1.50772	−0.753862	+	0.657033i	$0.771811\pi$
$930$	0	0
$931$	−56396.8	−1.98532
$932$	1917.24	0.0673832
$933$	21641.0	0.759372
$934$	19703.4	0.690272
$935$	0	0
$936$	17040.2	0.595061
$937$	6284.51	0.219110	0.109555	−	0.993981i	$-0.465057\pi$
$937$	6284.51	0.219110	0.109555	+	0.993981i	$0.465057\pi$
$938$	−47049.4	−1.63776
$939$	−5325.52	−0.185082
$940$	0	0
$941$	32542.5	1.12737	0.563684	−	0.825990i	$-0.309383\pi$
$941$	32542.5	1.12737	0.563684	+	0.825990i	$0.309383\pi$
$942$	12513.1	0.432801
$943$	1023.16	0.0353326
$944$	−15404.3	−0.531111
$945$	0	0
$946$	−55868.2	−1.92012
$947$	−19841.4	−0.680845	−0.340423	−	0.940273i	$-0.610570\pi$
$947$	−19841.4	−0.680845	−0.340423	+	0.940273i	$0.610570\pi$
$948$	−1231.19	−0.0421806
$949$	−69854.1	−2.38942
$950$	0	0
$951$	−13568.8	−0.462671
$952$	−78600.5	−2.67590
$953$	3705.36	0.125948	0.0629739	−	0.998015i	$-0.479942\pi$
$953$	3705.36	0.125948	0.0629739	+	0.998015i	$0.479942\pi$
$954$	−9633.82	−0.326946
$955$	0	0
$956$	−10868.6	−0.367695
$957$	−5129.05	−0.173248
$958$	−21345.6	−0.719881
$959$	40664.6	1.36927
$960$	0	0
$961$	−6710.48	−0.225252
$962$	−57646.9	−1.93203
$963$	−6609.24	−0.221163
$964$	3759.23	0.125598
$965$	0	0
$966$	1485.90	0.0494906
$967$	42598.6	1.41663	0.708314	−	0.705898i	$-0.249456\pi$
$967$	42598.6	1.41663	0.708314	+	0.705898i	$0.249456\pi$
$968$	−52814.9	−1.75365
$969$	−53275.9	−1.76622
$970$	0	0
$971$	−10608.8	−0.350620	−0.175310	−	0.984513i	$-0.556093\pi$
$971$	−10608.8	−0.350620	−0.175310	+	0.984513i	$0.556093\pi$
$972$	−683.986	−0.0225709
$973$	−31531.8	−1.03892
$974$	14759.7	0.485555
$975$	0	0
$976$	−20234.4	−0.663614
$977$	45196.3	1.48000	0.739998	−	0.672609i	$-0.234826\pi$
$977$	45196.3	1.48000	0.739998	+	0.672609i	$0.234826\pi$
$978$	−12240.8	−0.400222
$979$	−79068.6	−2.58125
$980$	0	0
$981$	6615.59	0.215311
$982$	−2523.22	−0.0819951
$983$	−30853.1	−1.00108	−0.500540	−	0.865713i	$-0.666865\pi$
$983$	−30853.1	−1.00108	−0.500540	+	0.865713i	$0.666865\pi$
$984$	−9328.86	−0.302229
$985$	0	0
$986$	7851.66	0.253598
$987$	19069.5	0.614985
$988$	−32322.3	−1.04080
$989$	3372.10	0.108419
$990$	0	0
$991$	1333.81	0.0427547	0.0213774	−	0.999771i	$-0.493195\pi$
$991$	1333.81	0.0427547	0.0213774	+	0.999771i	$0.493195\pi$
$992$	18320.9	0.586381
$993$	−14453.0	−0.461884
$994$	30207.6	0.963912
$995$	0	0
$996$	3078.12	0.0979257
$997$	−41309.6	−1.31222	−0.656112	−	0.754664i	$-0.727800\pi$
$997$	−41309.6	−1.31222	−0.656112	+	0.754664i	$0.727800\pi$
$998$	9231.41	0.292801
$999$	8890.46	0.281563

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.4.a.n.1.5		7
5.4	even	2		435.4.a.i.1.3	✓	7
15.14	odd	2		1305.4.a.n.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.4.a.i.1.3	✓	7	5.4	even	2
1305.4.a.n.1.5		7	15.14	odd	2
2175.4.a.n.1.5		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)

\(f(q)\)	\(=\)	\(q+2.27711 q^{2} +3.00000 q^{3} -2.81476 q^{4} +6.83134 q^{6} +26.8439 q^{7} -24.6264 q^{8} +9.00000 q^{9} -58.9546 q^{11} -8.44428 q^{12} -76.8831 q^{13} +61.1265 q^{14} -33.5591 q^{16} +118.899 q^{17} +20.4940 q^{18} -149.359 q^{19} +80.5316 q^{21} -134.246 q^{22} +8.10284 q^{23} -73.8793 q^{24} -175.072 q^{26} +27.0000 q^{27} -75.5590 q^{28} +29.0000 q^{29} +151.923 q^{31} +120.594 q^{32} -176.864 q^{33} +270.747 q^{34} -25.3328 q^{36} +329.276 q^{37} -340.106 q^{38} -230.649 q^{39} +126.272 q^{41} +183.379 q^{42} +416.162 q^{43} +165.943 q^{44} +18.4511 q^{46} +236.796 q^{47} -100.677 q^{48} +377.593 q^{49} +356.698 q^{51} +216.408 q^{52} -470.080 q^{53} +61.4820 q^{54} -661.068 q^{56} -448.076 q^{57} +66.0363 q^{58} +459.022 q^{59} +602.948 q^{61} +345.945 q^{62} +241.595 q^{63} +543.078 q^{64} -402.739 q^{66} -769.705 q^{67} -334.673 q^{68} +24.3085 q^{69} +494.182 q^{71} -221.638 q^{72} +908.575 q^{73} +749.799 q^{74} +420.409 q^{76} -1582.57 q^{77} -525.215 q^{78} +145.802 q^{79} +81.0000 q^{81} +287.535 q^{82} -364.521 q^{83} -226.677 q^{84} +947.648 q^{86} +87.0000 q^{87} +1451.84 q^{88} +1341.18 q^{89} -2063.84 q^{91} -22.8076 q^{92} +455.768 q^{93} +539.210 q^{94} +361.781 q^{96} -441.198 q^{97} +859.822 q^{98} -530.591 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{2} + 21 q^{3} + 22 q^{4} + 6 q^{6} + 50 q^{7} + 33 q^{8} + 63 q^{9} + 76 q^{11} + 66 q^{12} - 30 q^{13} + 89 q^{14} + 138 q^{16} + 140 q^{17} + 18 q^{18} + 90 q^{19} + 150 q^{21} - 61 q^{22}+ \cdots + 684 q^{99}+O(q^{100}) \) 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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