LMFDB - Embedded newform 2175.4.a.k.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2175,4,Mod(1,2175)] 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("2175.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [6,-1,-18] 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:	$128.329154262$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 49x^{4} + 27x^{3} + 692x^{2} - 82x - 2588 $ x^6 - x^5 - 49x^4 + 27x^3 + 692x^2 - 82x - 2588
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-5.05047$ 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)

$f(q)$	$=$	\(q+5.05047 q^{2} -3.00000 q^{3} +17.5072 q^{4} -15.1514 q^{6} -19.3889 q^{7} +48.0160 q^{8} +9.00000 q^{9} +6.81451 q^{11} -52.5217 q^{12} -36.9473 q^{13} -97.9229 q^{14} +102.445 q^{16} +71.5529 q^{17} +45.4542 q^{18} +88.3064 q^{19} +58.1666 q^{21} +34.4164 q^{22} -185.418 q^{23} -144.048 q^{24} -186.601 q^{26} -27.0000 q^{27} -339.446 q^{28} +29.0000 q^{29} -120.393 q^{31} +133.269 q^{32} -20.4435 q^{33} +361.376 q^{34} +157.565 q^{36} +117.351 q^{37} +445.989 q^{38} +110.842 q^{39} -229.591 q^{41} +293.769 q^{42} -66.8319 q^{43} +119.303 q^{44} -936.448 q^{46} +42.0170 q^{47} -307.336 q^{48} +32.9286 q^{49} -214.659 q^{51} -646.845 q^{52} -9.42394 q^{53} -136.363 q^{54} -930.976 q^{56} -264.919 q^{57} +146.464 q^{58} -232.484 q^{59} -546.541 q^{61} -608.042 q^{62} -174.500 q^{63} -146.492 q^{64} -103.249 q^{66} -953.049 q^{67} +1252.69 q^{68} +556.254 q^{69} +429.898 q^{71} +432.144 q^{72} -554.903 q^{73} +592.675 q^{74} +1546.00 q^{76} -132.126 q^{77} +559.804 q^{78} +965.168 q^{79} +81.0000 q^{81} -1159.54 q^{82} -625.186 q^{83} +1018.34 q^{84} -337.532 q^{86} -87.0000 q^{87} +327.205 q^{88} -271.169 q^{89} +716.367 q^{91} -3246.16 q^{92} +361.180 q^{93} +212.206 q^{94} -399.806 q^{96} -1167.66 q^{97} +166.305 q^{98} +61.3305 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{2} - 18 q^{3} + 51 q^{4} + 3 q^{6} - 47 q^{7} - 51 q^{8} + 54 q^{9} + 81 q^{11} - 153 q^{12} - 169 q^{13} - 30 q^{14} + 131 q^{16} + q^{17} - 9 q^{18} + 116 q^{19} + 141 q^{21} - 90 q^{22} + 52 q^{23}+ \cdots + 729 q^{99}+O(q^{100}) $ 6 * q - q^2 - 18 * q^3 + 51 * q^4 + 3 * q^6 - 47 * q^7 - 51 * q^8 + 54 * q^9 + 81 * q^11 - 153 * q^12 - 169 * q^13 - 30 * q^14 + 131 * q^16 + q^17 - 9 * q^18 + 116 * q^19 + 141 * q^21 - 90 * q^22 + 52 * q^23 + 153 * q^24 + 294 * q^26 - 162 * q^27 - 344 * q^28 + 174 * q^29 + 340 * q^31 - 499 * q^32 - 243 * q^33 + 920 * q^34 + 459 * q^36 - 332 * q^37 + 378 * q^38 + 507 * q^39 - 616 * q^41 + 90 * q^42 - 334 * q^43 - 52 * q^44 - 158 * q^46 + 85 * q^47 - 393 * q^48 + 879 * q^49 - 3 * q^51 - 2220 * q^52 + 850 * q^53 + 27 * q^54 - 624 * q^56 - 348 * q^57 - 29 * q^58 - 758 * q^59 - 36 * q^61 + 152 * q^62 - 423 * q^63 + 1795 * q^64 + 270 * q^66 - 939 * q^67 + 186 * q^68 - 156 * q^69 - 1388 * q^71 - 459 * q^72 - 1708 * q^73 - 814 * q^74 + 566 * q^76 - 2585 * q^77 - 882 * q^78 + 1250 * q^79 + 486 * q^81 - 1372 * q^82 + 748 * q^83 + 1032 * q^84 - 800 * q^86 - 522 * q^87 - 536 * q^88 + 1099 * q^89 + 539 * q^91 - 1698 * q^92 - 1020 * q^93 - 4542 * q^94 + 1497 * q^96 + 22 * q^97 + 1433 * q^98 + 729 * 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 $

$2$	5.05047	1.78561	0.892805	−	0.450443i	$-0.148734\pi$
$2$	5.05047	1.78561	0.892805	+	0.450443i	$0.148734\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	17.5072	2.18840
$5$	0	0
$5$	0	0
$6$	−15.1514	−1.03092
$7$	−19.3889	−1.04690	−0.523451	−	0.852056i	$-0.675356\pi$
$7$	−19.3889	−1.04690	−0.523451	+	0.852056i	$0.675356\pi$
$8$	48.0160	2.12203
$9$	9.00000	0.333333
$10$	0	0
$11$	6.81451	0.186786	0.0933932	−	0.995629i	$-0.470229\pi$
$11$	6.81451	0.186786	0.0933932	+	0.995629i	$0.470229\pi$
$12$	−52.5217	−1.26348
$13$	−36.9473	−0.788257	−0.394128	−	0.919055i	$-0.628954\pi$
$13$	−36.9473	−0.788257	−0.394128	+	0.919055i	$0.628954\pi$
$14$	−97.9229	−1.86936
$15$	0	0
$16$	102.445	1.60071
$17$	71.5529	1.02083	0.510416	−	0.859928i	$-0.329492\pi$
$17$	71.5529	1.02083	0.510416	+	0.859928i	$0.329492\pi$
$18$	45.4542	0.595203
$19$	88.3064	1.06626	0.533129	−	0.846034i	$-0.321016\pi$
$19$	88.3064	1.06626	0.533129	+	0.846034i	$0.321016\pi$
$20$	0	0
$21$	58.1666	0.604429
$22$	34.4164	0.333528
$23$	−185.418	−1.68097	−0.840486	−	0.541834i	$-0.817730\pi$
$23$	−185.418	−1.68097	−0.840486	+	0.541834i	$0.817730\pi$
$24$	−144.048	−1.22515
$25$	0	0
$26$	−186.601	−1.40752
$27$	−27.0000	−0.192450
$28$	−339.446	−2.29104
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	−120.393	−0.697524	−0.348762	−	0.937211i	$-0.613398\pi$
$31$	−120.393	−0.697524	−0.348762	+	0.937211i	$0.613398\pi$
$32$	133.269	0.736213
$33$	−20.4435	−0.107841
$34$	361.376	1.82281
$35$	0	0
$36$	157.565	0.729468
$37$	117.351	0.521414	0.260707	−	0.965418i	$-0.416044\pi$
$37$	117.351	0.521414	0.260707	+	0.965418i	$0.416044\pi$
$38$	445.989	1.90392
$39$	110.842	0.455100
$40$	0	0
$41$	−229.591	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$41$	−229.591	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$42$	293.769	1.07927
$43$	−66.8319	−0.237018	−0.118509	−	0.992953i	$-0.537811\pi$
$43$	−66.8319	−0.237018	−0.118509	+	0.992953i	$0.537811\pi$
$44$	119.303	0.408764
$45$	0	0
$46$	−936.448	−3.00156
$47$	42.0170	0.130400	0.0652001	−	0.997872i	$-0.479231\pi$
$47$	42.0170	0.130400	0.0652001	+	0.997872i	$0.479231\pi$
$48$	−307.336	−0.924168
$49$	32.9286	0.0960018
$50$	0	0
$51$	−214.659	−0.589377
$52$	−646.845	−1.72502
$53$	−9.42394	−0.0244241	−0.0122121	−	0.999925i	$-0.503887\pi$
$53$	−9.42394	−0.0244241	−0.0122121	+	0.999925i	$0.503887\pi$
$54$	−136.363	−0.343641
$55$	0	0
$56$	−930.976	−2.22155
$57$	−264.919	−0.615604
$58$	146.464	0.331579
$59$	−232.484	−0.512996	−0.256498	−	0.966545i	$-0.582569\pi$
$59$	−232.484	−0.512996	−0.256498	+	0.966545i	$0.582569\pi$
$60$	0	0
$61$	−546.541	−1.14717	−0.573586	−	0.819146i	$-0.694448\pi$
$61$	−546.541	−1.14717	−0.573586	+	0.819146i	$0.694448\pi$
$62$	−608.042	−1.24551
$63$	−174.500	−0.348967
$64$	−146.492	−0.286118
$65$	0	0
$66$	−103.249	−0.192562
$67$	−953.049	−1.73781	−0.868907	−	0.494976i	$-0.835177\pi$
$67$	−953.049	−1.73781	−0.868907	+	0.494976i	$0.835177\pi$
$68$	1252.69	2.23399
$69$	556.254	0.970509
$70$	0	0
$71$	429.898	0.718584	0.359292	−	0.933225i	$-0.383018\pi$
$71$	429.898	0.718584	0.359292	+	0.933225i	$0.383018\pi$
$72$	432.144	0.707342
$73$	−554.903	−0.889678	−0.444839	−	0.895611i	$-0.646739\pi$
$73$	−554.903	−0.889678	−0.444839	+	0.895611i	$0.646739\pi$
$74$	592.675	0.931041
$75$	0	0
$76$	1546.00	2.33340
$77$	−132.126	−0.195547
$78$	559.804	0.812632
$79$	965.168	1.37456	0.687278	−	0.726395i	$-0.258805\pi$
$79$	965.168	1.37456	0.687278	+	0.726395i	$0.258805\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	−1159.54	−1.56158
$83$	−625.186	−0.826784	−0.413392	−	0.910553i	$-0.635656\pi$
$83$	−625.186	−0.826784	−0.413392	+	0.910553i	$0.635656\pi$
$84$	1018.34	1.32273
$85$	0	0
$86$	−337.532	−0.423222
$87$	−87.0000	−0.107211
$88$	327.205	0.396366
$89$	−271.169	−0.322964	−0.161482	−	0.986876i	$-0.551627\pi$
$89$	−271.169	−0.322964	−0.161482	+	0.986876i	$0.551627\pi$
$90$	0	0
$91$	716.367	0.825227
$92$	−3246.16	−3.67864
$93$	361.180	0.402716
$94$	212.206	0.232844
$95$	0	0
$96$	−399.806	−0.425053
$97$	−1167.66	−1.22225	−0.611125	−	0.791534i	$-0.709283\pi$
$97$	−1167.66	−1.22225	−0.611125	+	0.791534i	$0.709283\pi$
$98$	166.305	0.171422
$99$	61.3305	0.0622621
$100$	0	0
$101$	1004.86	0.989977	0.494989	−	0.868899i	$-0.335172\pi$
$101$	1004.86	0.989977	0.494989	+	0.868899i	$0.335172\pi$
$102$	−1084.13	−1.05240
$103$	−704.225	−0.673683	−0.336842	−	0.941561i	$-0.609359\pi$
$103$	−704.225	−0.673683	−0.336842	+	0.941561i	$0.609359\pi$
$104$	−1774.06	−1.67270
$105$	0	0
$106$	−47.5953	−0.0436119
$107$	−1116.49	−1.00874	−0.504371	−	0.863487i	$-0.668276\pi$
$107$	−1116.49	−1.00874	−0.504371	+	0.863487i	$0.668276\pi$
$108$	−472.695	−0.421158
$109$	−1306.48	−1.14806	−0.574029	−	0.818835i	$-0.694621\pi$
$109$	−1306.48	−1.14806	−0.574029	+	0.818835i	$0.694621\pi$
$110$	0	0
$111$	−352.052	−0.301038
$112$	−1986.30	−1.67578
$113$	−256.295	−0.213364	−0.106682	−	0.994293i	$-0.534023\pi$
$113$	−256.295	−0.213364	−0.106682	+	0.994293i	$0.534023\pi$
$114$	−1337.97	−1.09923
$115$	0	0
$116$	507.710	0.406376
$117$	−332.526	−0.262752
$118$	−1174.15	−0.916011
$119$	−1387.33	−1.06871
$120$	0	0
$121$	−1284.56	−0.965111
$122$	−2760.29	−2.04840
$123$	688.773	0.504915
$124$	−2107.75	−1.52646
$125$	0	0
$126$	−881.306	−0.623119
$127$	1201.30	0.839355	0.419677	−	0.907673i	$-0.362143\pi$
$127$	1201.30	0.839355	0.419677	+	0.907673i	$0.362143\pi$
$128$	−1806.01	−1.24711
$129$	200.496	0.136842
$130$	0	0
$131$	1049.57	0.700008	0.350004	−	0.936748i	$-0.386180\pi$
$131$	1049.57	0.700008	0.350004	+	0.936748i	$0.386180\pi$
$132$	−357.909	−0.236000
$133$	−1712.16	−1.11627
$134$	−4813.34	−3.10306
$135$	0	0
$136$	3435.68	2.16623
$137$	−2881.21	−1.79678	−0.898388	−	0.439202i	$-0.855261\pi$
$137$	−2881.21	−1.79678	−0.898388	+	0.439202i	$0.855261\pi$
$138$	2809.34	1.73295
$139$	2584.43	1.57704	0.788521	−	0.615008i	$-0.210847\pi$
$139$	2584.43	1.57704	0.788521	+	0.615008i	$0.210847\pi$
$140$	0	0
$141$	−126.051	−0.0752866
$142$	2171.18	1.28311
$143$	−251.778	−0.147236
$144$	922.007	0.533569
$145$	0	0
$146$	−2802.52	−1.58862
$147$	−98.7858	−0.0554267
$148$	2054.48	1.14106
$149$	−1449.23	−0.796813	−0.398407	−	0.917209i	$-0.630437\pi$
$149$	−1449.23	−0.796813	−0.398407	+	0.917209i	$0.630437\pi$
$150$	0	0
$151$	2760.46	1.48770	0.743851	−	0.668345i	$-0.232997\pi$
$151$	2760.46	1.48770	0.743851	+	0.668345i	$0.232997\pi$
$152$	4240.12	2.26262
$153$	643.976	0.340277
$154$	−667.296	−0.349171
$155$	0	0
$156$	1940.54	0.995943
$157$	−766.465	−0.389621	−0.194811	−	0.980841i	$-0.562409\pi$
$157$	−766.465	−0.389621	−0.194811	+	0.980841i	$0.562409\pi$
$158$	4874.55	2.45442
$159$	28.2718	0.0141013
$160$	0	0
$161$	3595.05	1.75981
$162$	409.088	0.198401
$163$	1871.11	0.899119	0.449560	−	0.893250i	$-0.351581\pi$
$163$	1871.11	0.899119	0.449560	+	0.893250i	$0.351581\pi$
$164$	−4019.50	−1.91384
$165$	0	0
$166$	−3157.48	−1.47631
$167$	−2343.62	−1.08596	−0.542978	−	0.839747i	$-0.682703\pi$
$167$	−2343.62	−1.08596	−0.542978	+	0.839747i	$0.682703\pi$
$168$	2792.93	1.28261
$169$	−831.896	−0.378651
$170$	0	0
$171$	794.758	0.355419
$172$	−1170.04	−0.518691
$173$	1485.73	0.652935	0.326468	−	0.945208i	$-0.394142\pi$
$173$	1485.73	0.652935	0.326468	+	0.945208i	$0.394142\pi$
$174$	−439.391	−0.191438
$175$	0	0
$176$	698.114	0.298990
$177$	697.451	0.296179
$178$	−1369.53	−0.576689
$179$	71.4418	0.0298314	0.0149157	−	0.999889i	$-0.495252\pi$
$179$	71.4418	0.0298314	0.0149157	+	0.999889i	$0.495252\pi$
$180$	0	0
$181$	−1696.31	−0.696607	−0.348304	−	0.937382i	$-0.613242\pi$
$181$	−1696.31	−0.696607	−0.348304	+	0.937382i	$0.613242\pi$
$182$	3617.99	1.47353
$183$	1639.62	0.662320
$184$	−8903.03	−3.56706
$185$	0	0
$186$	1824.13	0.719093
$187$	487.598	0.190677
$188$	735.601	0.285368
$189$	523.500	0.201476
$190$	0	0
$191$	−3657.80	−1.38570	−0.692852	−	0.721080i	$-0.743646\pi$
$191$	−3657.80	−1.38570	−0.692852	+	0.721080i	$0.743646\pi$
$192$	439.477	0.165190
$193$	3156.73	1.17734	0.588669	−	0.808374i	$-0.299652\pi$
$193$	3156.73	1.17734	0.588669	+	0.808374i	$0.299652\pi$
$194$	−5897.25	−2.18246
$195$	0	0
$196$	576.489	0.210091
$197$	969.523	0.350638	0.175319	−	0.984512i	$-0.443904\pi$
$197$	969.523	0.350638	0.175319	+	0.984512i	$0.443904\pi$
$198$	309.748	0.111176
$199$	5423.02	1.93180	0.965898	−	0.258921i	$-0.0833670\pi$
$199$	5423.02	1.93180	0.965898	+	0.258921i	$0.0833670\pi$
$200$	0	0
$201$	2859.15	1.00333
$202$	5075.03	1.76771
$203$	−562.277	−0.194405
$204$	−3758.08	−1.28980
$205$	0	0
$206$	−3556.67	−1.20294
$207$	−1668.76	−0.560324
$208$	−3785.08	−1.26177
$209$	601.765	0.199162
$210$	0	0
$211$	−2848.51	−0.929382	−0.464691	−	0.885473i	$-0.653835\pi$
$211$	−2848.51	−0.929382	−0.464691	+	0.885473i	$0.653835\pi$
$212$	−164.987	−0.0534498
$213$	−1289.69	−0.414875
$214$	−5638.81	−1.80122
$215$	0	0
$216$	−1296.43	−0.408384
$217$	2334.29	0.730239
$218$	−6598.35	−2.04999
$219$	1664.71	0.513656
$220$	0	0
$221$	−2643.69	−0.804678
$222$	−1778.02	−0.537537
$223$	−502.488	−0.150893	−0.0754465	−	0.997150i	$-0.524038\pi$
$223$	−502.488	−0.150893	−0.0754465	+	0.997150i	$0.524038\pi$
$224$	−2583.93	−0.770742
$225$	0	0
$226$	−1294.41	−0.380985
$227$	−1336.83	−0.390876	−0.195438	−	0.980716i	$-0.562613\pi$
$227$	−1336.83	−0.390876	−0.195438	+	0.980716i	$0.562613\pi$
$228$	−4638.00	−1.34719
$229$	−1131.26	−0.326445	−0.163223	−	0.986589i	$-0.552189\pi$
$229$	−1131.26	−0.326445	−0.163223	+	0.986589i	$0.552189\pi$
$230$	0	0
$231$	396.377	0.112899
$232$	1392.46	0.394050
$233$	−1271.63	−0.357541	−0.178770	−	0.983891i	$-0.557212\pi$
$233$	−1271.63	−0.357541	−0.178770	+	0.983891i	$0.557212\pi$
$234$	−1679.41	−0.469173
$235$	0	0
$236$	−4070.14	−1.12264
$237$	−2895.50	−0.793600
$238$	−7006.67	−1.90830
$239$	1507.40	0.407973	0.203987	−	0.978974i	$-0.434610\pi$
$239$	1507.40	0.407973	0.203987	+	0.978974i	$0.434610\pi$
$240$	0	0
$241$	4251.32	1.13631	0.568156	−	0.822921i	$-0.307657\pi$
$241$	4251.32	1.13631	0.568156	+	0.822921i	$0.307657\pi$
$242$	−6487.64	−1.72331
$243$	−243.000	−0.0641500
$244$	−9568.42	−2.51047
$245$	0	0
$246$	3478.63	0.901582
$247$	−3262.69	−0.840485
$248$	−5780.79	−1.48016
$249$	1875.56	0.477344
$250$	0	0
$251$	7219.41	1.81548	0.907739	−	0.419535i	$-0.137807\pi$
$251$	7219.41	1.81548	0.907739	+	0.419535i	$0.137807\pi$
$252$	−3055.01	−0.763681
$253$	−1263.53	−0.313983
$254$	6067.12	1.49876
$255$	0	0
$256$	−7949.23	−1.94073
$257$	−3436.04	−0.833986	−0.416993	−	0.908910i	$-0.636916\pi$
$257$	−3436.04	−0.833986	−0.416993	+	0.908910i	$0.636916\pi$
$258$	1012.60	0.244347
$259$	−2275.29	−0.545868
$260$	0	0
$261$	261.000	0.0618984
$262$	5300.80	1.24994
$263$	−1494.14	−0.350314	−0.175157	−	0.984540i	$-0.556043\pi$
$263$	−1494.14	−0.350314	−0.175157	+	0.984540i	$0.556043\pi$
$264$	−981.615	−0.228842
$265$	0	0
$266$	−8647.22	−1.99322
$267$	813.506	0.186464
$268$	−16685.2	−3.80304
$269$	−1692.06	−0.383520	−0.191760	−	0.981442i	$-0.561419\pi$
$269$	−1692.06	−0.383520	−0.191760	+	0.981442i	$0.561419\pi$
$270$	0	0
$271$	3147.75	0.705581	0.352790	−	0.935702i	$-0.385233\pi$
$271$	3147.75	0.705581	0.352790	+	0.935702i	$0.385233\pi$
$272$	7330.26	1.63405
$273$	−2149.10	−0.476445
$274$	−14551.5	−3.20834
$275$	0	0
$276$	9738.47	2.12387
$277$	3873.16	0.840129	0.420065	−	0.907494i	$-0.362007\pi$
$277$	3873.16	0.840129	0.420065	+	0.907494i	$0.362007\pi$
$278$	13052.6	2.81598
$279$	−1083.54	−0.232508
$280$	0	0
$281$	−4951.42	−1.05116	−0.525582	−	0.850743i	$-0.676152\pi$
$281$	−4951.42	−1.05116	−0.525582	+	0.850743i	$0.676152\pi$
$282$	−636.617	−0.134433
$283$	4895.30	1.02825	0.514126	−	0.857715i	$-0.328116\pi$
$283$	4895.30	1.02825	0.514126	+	0.857715i	$0.328116\pi$
$284$	7526.32	1.57255
$285$	0	0
$286$	−1271.60	−0.262906
$287$	4451.51	0.915555
$288$	1199.42	0.245404
$289$	206.822	0.0420969
$290$	0	0
$291$	3502.99	0.705667
$292$	−9714.82	−1.94697
$293$	8978.98	1.79030	0.895150	−	0.445766i	$-0.147068\pi$
$293$	8978.98	1.79030	0.895150	+	0.445766i	$0.147068\pi$
$294$	−498.915	−0.0989704
$295$	0	0
$296$	5634.70	1.10645
$297$	−183.992	−0.0359471
$298$	−7319.27	−1.42280
$299$	6850.70	1.32504
$300$	0	0
$301$	1295.80	0.248134
$302$	13941.6	2.65646
$303$	−3014.59	−0.571564
$304$	9046.57	1.70676
$305$	0	0
$306$	3252.38	0.607602
$307$	−7282.96	−1.35394	−0.676972	−	0.736009i	$-0.736708\pi$
$307$	−7282.96	−1.35394	−0.676972	+	0.736009i	$0.736708\pi$
$308$	−2313.15	−0.427936
$309$	2112.68	0.388951
$310$	0	0
$311$	−2424.98	−0.442149	−0.221074	−	0.975257i	$-0.570956\pi$
$311$	−2424.98	−0.442149	−0.221074	+	0.975257i	$0.570956\pi$
$312$	5322.18	0.965735
$313$	6511.66	1.17591	0.587957	−	0.808892i	$-0.299932\pi$
$313$	6511.66	1.17591	0.587957	+	0.808892i	$0.299932\pi$
$314$	−3871.01	−0.695712
$315$	0	0
$316$	16897.4	3.00808
$317$	−6634.14	−1.17543	−0.587713	−	0.809069i	$-0.699972\pi$
$317$	−6634.14	−1.17543	−0.587713	+	0.809069i	$0.699972\pi$
$318$	142.786	0.0251794
$319$	197.621	0.0346854
$320$	0	0
$321$	3349.48	0.582398
$322$	18156.7	3.14234
$323$	6318.58	1.08847
$324$	1418.09	0.243156
$325$	0	0
$326$	9449.97	1.60548
$327$	3919.45	0.662832
$328$	−11024.0	−1.85579
$329$	−814.663	−0.136516
$330$	0	0
$331$	−6727.98	−1.11723	−0.558615	−	0.829427i	$-0.688667\pi$
$331$	−6727.98	−1.11723	−0.558615	+	0.829427i	$0.688667\pi$
$332$	−10945.3	−1.80934
$333$	1056.15	0.173805
$334$	−11836.4	−1.93909
$335$	0	0
$336$	5958.89	0.967513
$337$	7683.33	1.24195	0.620976	−	0.783830i	$-0.286736\pi$
$337$	7683.33	1.24195	0.620976	+	0.783830i	$0.286736\pi$
$338$	−4201.46	−0.676123
$339$	768.884	0.123186
$340$	0	0
$341$	−820.420	−0.130288
$342$	4013.90	0.634640
$343$	6011.94	0.946397
$344$	−3209.00	−0.502958
$345$	0	0
$346$	7503.62	1.16589
$347$	−1401.55	−0.216827	−0.108413	−	0.994106i	$-0.534577\pi$
$347$	−1401.55	−0.216827	−0.108413	+	0.994106i	$0.534577\pi$
$348$	−1523.13	−0.234621
$349$	−10253.9	−1.57272	−0.786362	−	0.617766i	$-0.788038\pi$
$349$	−10253.9	−1.57272	−0.786362	+	0.617766i	$0.788038\pi$
$350$	0	0
$351$	997.577	0.151700
$352$	908.160	0.137515
$353$	3309.99	0.499074	0.249537	−	0.968365i	$-0.419722\pi$
$353$	3309.99	0.499074	0.249537	+	0.968365i	$0.419722\pi$
$354$	3522.45	0.528859
$355$	0	0
$356$	−4747.41	−0.706777
$357$	4161.99	0.617020
$358$	360.815	0.0532672
$359$	10427.4	1.53297	0.766483	−	0.642264i	$-0.222005\pi$
$359$	10427.4	1.53297	0.766483	+	0.642264i	$0.222005\pi$
$360$	0	0
$361$	939.026	0.136904
$362$	−8567.17	−1.24387
$363$	3853.69	0.557207
$364$	12541.6	1.80593
$365$	0	0
$366$	8280.87	1.18264
$367$	6667.61	0.948356	0.474178	−	0.880429i	$-0.342745\pi$
$367$	6667.61	0.948356	0.474178	+	0.880429i	$0.342745\pi$
$368$	−18995.2	−2.69074
$369$	−2066.32	−0.291513
$370$	0	0
$371$	182.720	0.0255696
$372$	6323.25	0.881305
$373$	−6100.55	−0.846849	−0.423424	−	0.905931i	$-0.639172\pi$
$373$	−6100.55	−0.846849	−0.423424	+	0.905931i	$0.639172\pi$
$374$	2462.60	0.340476
$375$	0	0
$376$	2017.49	0.276713
$377$	−1071.47	−0.146376
$378$	2643.92	0.359758
$379$	6291.88	0.852750	0.426375	−	0.904546i	$-0.359790\pi$
$379$	6291.88	0.852750	0.426375	+	0.904546i	$0.359790\pi$
$380$	0	0
$381$	−3603.90	−0.484602
$382$	−18473.6	−2.47433
$383$	−10349.3	−1.38074	−0.690369	−	0.723457i	$-0.742552\pi$
$383$	−10349.3	−1.38074	−0.690369	+	0.723457i	$0.742552\pi$
$384$	5418.02	0.720018
$385$	0	0
$386$	15943.0	2.10227
$387$	−601.487	−0.0790060
$388$	−20442.6	−2.67478
$389$	−2361.08	−0.307742	−0.153871	−	0.988091i	$-0.549174\pi$
$389$	−2361.08	−0.307742	−0.153871	+	0.988091i	$0.549174\pi$
$390$	0	0
$391$	−13267.2	−1.71599
$392$	1581.10	0.203718
$393$	−3148.70	−0.404150
$394$	4896.55	0.626103
$395$	0	0
$396$	1073.73	0.136255
$397$	−6963.12	−0.880274	−0.440137	−	0.897931i	$-0.645070\pi$
$397$	−6963.12	−0.880274	−0.440137	+	0.897931i	$0.645070\pi$
$398$	27388.8	3.44944
$399$	5136.49	0.644476
$400$	0	0
$401$	−1419.32	−0.176752	−0.0883759	−	0.996087i	$-0.528168\pi$
$401$	−1419.32	−0.176752	−0.0883759	+	0.996087i	$0.528168\pi$
$402$	14440.0	1.79155
$403$	4448.20	0.549828
$404$	17592.4	2.16647
$405$	0	0
$406$	−2839.76	−0.347131
$407$	799.686	0.0973930
$408$	−10307.0	−1.25067
$409$	−5199.93	−0.628655	−0.314327	−	0.949315i	$-0.601779\pi$
$409$	−5199.93	−0.628655	−0.314327	+	0.949315i	$0.601779\pi$
$410$	0	0
$411$	8643.63	1.03737
$412$	−12329.0	−1.47429
$413$	4507.60	0.537056
$414$	−8428.03	−1.00052
$415$	0	0
$416$	−4923.92	−0.580325
$417$	−7753.30	−0.910506
$418$	3039.19	0.355626
$419$	4476.06	0.521885	0.260943	−	0.965354i	$-0.415967\pi$
$419$	4476.06	0.521885	0.260943	+	0.965354i	$0.415967\pi$
$420$	0	0
$421$	2924.54	0.338558	0.169279	−	0.985568i	$-0.445856\pi$
$421$	2924.54	0.338558	0.169279	+	0.985568i	$0.445856\pi$
$422$	−14386.3	−1.65951
$423$	378.153	0.0434667
$424$	−452.499	−0.0518286
$425$	0	0
$426$	−6513.55	−0.740804
$427$	10596.8	1.20097
$428$	−19546.7	−2.20754
$429$	755.333	0.0850066
$430$	0	0
$431$	1488.98	0.166408	0.0832039	−	0.996533i	$-0.473485\pi$
$431$	1488.98	0.166408	0.0832039	+	0.996533i	$0.473485\pi$
$432$	−2766.02	−0.308056
$433$	8411.84	0.933597	0.466798	−	0.884364i	$-0.345407\pi$
$433$	8411.84	0.933597	0.466798	+	0.884364i	$0.345407\pi$
$434$	11789.3	1.30392
$435$	0	0
$436$	−22872.9	−2.51242
$437$	−16373.6	−1.79235
$438$	8407.56	0.917189
$439$	−10584.7	−1.15076	−0.575378	−	0.817887i	$-0.695145\pi$
$439$	−10584.7	−1.15076	−0.575378	+	0.817887i	$0.695145\pi$
$440$	0	0
$441$	296.358	0.0320006
$442$	−13351.9	−1.43684
$443$	10335.6	1.10848	0.554242	−	0.832356i	$-0.313008\pi$
$443$	10335.6	1.10848	0.554242	+	0.832356i	$0.313008\pi$
$444$	−6163.45	−0.658793
$445$	0	0
$446$	−2537.80	−0.269436
$447$	4347.68	0.460040
$448$	2840.32	0.299537
$449$	16626.7	1.74758	0.873788	−	0.486306i	$-0.161656\pi$
$449$	16626.7	1.74758	0.873788	+	0.486306i	$0.161656\pi$
$450$	0	0
$451$	−1564.55	−0.163352
$452$	−4487.01	−0.466927
$453$	−8281.38	−0.858925
$454$	−6751.63	−0.697951
$455$	0	0
$456$	−12720.4	−1.30633
$457$	9008.26	0.922076	0.461038	−	0.887380i	$-0.347477\pi$
$457$	9008.26	0.922076	0.461038	+	0.887380i	$0.347477\pi$
$458$	−5713.41	−0.582904
$459$	−1931.93	−0.196459
$460$	0	0
$461$	−8097.37	−0.818074	−0.409037	−	0.912518i	$-0.634135\pi$
$461$	−8097.37	−0.818074	−0.409037	+	0.912518i	$0.634135\pi$
$462$	2001.89	0.201594
$463$	−4897.89	−0.491629	−0.245815	−	0.969317i	$-0.579055\pi$
$463$	−4897.89	−0.491629	−0.245815	+	0.969317i	$0.579055\pi$
$464$	2970.91	0.297244
$465$	0	0
$466$	−6422.31	−0.638429
$467$	3331.22	0.330087	0.165043	−	0.986286i	$-0.447224\pi$
$467$	3331.22	0.330087	0.165043	+	0.986286i	$0.447224\pi$
$468$	−5821.61	−0.575008
$469$	18478.6	1.81932
$470$	0	0
$471$	2299.39	0.224948
$472$	−11162.9	−1.08859
$473$	−455.426	−0.0442717
$474$	−14623.7	−1.41706
$475$	0	0
$476$	−24288.3	−2.33877
$477$	−84.8154	−0.00814137
$478$	7613.08	0.728482
$479$	−13606.3	−1.29788	−0.648942	−	0.760838i	$-0.724788\pi$
$479$	−13606.3	−1.29788	−0.648942	+	0.760838i	$0.724788\pi$
$480$	0	0
$481$	−4335.79	−0.411008
$482$	21471.1	2.02901
$483$	−10785.1	−1.01603
$484$	−22489.1	−2.11205
$485$	0	0
$486$	−1227.26	−0.114547
$487$	15460.9	1.43861	0.719304	−	0.694696i	$-0.244461\pi$
$487$	15460.9	1.43861	0.719304	+	0.694696i	$0.244461\pi$
$488$	−26242.7	−2.43433
$489$	−5613.32	−0.519107
$490$	0	0
$491$	−19016.4	−1.74785	−0.873927	−	0.486057i	$-0.838435\pi$
$491$	−19016.4	−1.74785	−0.873927	+	0.486057i	$0.838435\pi$
$492$	12058.5	1.10496
$493$	2075.03	0.189564
$494$	−16478.1	−1.50078
$495$	0	0
$496$	−12333.7	−1.11653
$497$	−8335.23	−0.752286
$498$	9472.44	0.852350
$499$	−13675.0	−1.22681	−0.613406	−	0.789768i	$-0.710201\pi$
$499$	−13675.0	−1.22681	−0.613406	+	0.789768i	$0.710201\pi$
$500$	0	0
$501$	7030.85	0.626977
$502$	36461.4	3.24174
$503$	1456.45	0.129105	0.0645525	−	0.997914i	$-0.479438\pi$
$503$	1456.45	0.129105	0.0645525	+	0.997914i	$0.479438\pi$
$504$	−8378.78	−0.740517
$505$	0	0
$506$	−6381.43	−0.560651
$507$	2495.69	0.218614
$508$	21031.4	1.83685
$509$	15219.7	1.32535	0.662674	−	0.748908i	$-0.269422\pi$
$509$	15219.7	1.32535	0.662674	+	0.748908i	$0.269422\pi$
$510$	0	0
$511$	10758.9	0.931405
$512$	−25699.3	−2.21828
$513$	−2384.27	−0.205201
$514$	−17353.6	−1.48917
$515$	0	0
$516$	3510.12	0.299466
$517$	286.325	0.0243570
$518$	−11491.3	−0.974708
$519$	−4457.18	−0.376972
$520$	0	0
$521$	19988.9	1.68086	0.840432	−	0.541917i	$-0.182301\pi$
$521$	19988.9	1.68086	0.840432	+	0.541917i	$0.182301\pi$
$522$	1318.17	0.110526
$523$	−9728.36	−0.813368	−0.406684	−	0.913569i	$-0.633315\pi$
$523$	−9728.36	−0.813368	−0.406684	+	0.913569i	$0.633315\pi$
$524$	18375.0	1.53190
$525$	0	0
$526$	−7546.11	−0.625525
$527$	−8614.48	−0.712055
$528$	−2094.34	−0.172622
$529$	22212.9	1.82566
$530$	0	0
$531$	−2092.35	−0.170999
$532$	−29975.2	−2.44284
$533$	8482.77	0.689361
$534$	4108.59	0.332951
$535$	0	0
$536$	−45761.6	−3.68768
$537$	−214.325	−0.0172231
$538$	−8545.70	−0.684817
$539$	224.392	0.0179318
$540$	0	0
$541$	15290.2	1.21511	0.607556	−	0.794277i	$-0.292150\pi$
$541$	15290.2	1.21511	0.607556	+	0.794277i	$0.292150\pi$
$542$	15897.6	1.25989
$543$	5088.94	0.402186
$544$	9535.77	0.751549
$545$	0	0
$546$	−10854.0	−0.850745
$547$	9101.20	0.711406	0.355703	−	0.934599i	$-0.384241\pi$
$547$	9101.20	0.711406	0.355703	+	0.934599i	$0.384241\pi$
$548$	−50442.0	−3.93207
$549$	−4918.87	−0.382390
$550$	0	0
$551$	2560.89	0.197999
$552$	26709.1	2.05945
$553$	−18713.5	−1.43902
$554$	19561.3	1.50014
$555$	0	0
$556$	45246.3	3.45120
$557$	3752.41	0.285448	0.142724	−	0.989763i	$-0.454414\pi$
$557$	3752.41	0.285448	0.142724	+	0.989763i	$0.454414\pi$
$558$	−5472.38	−0.415169
$559$	2469.26	0.186831
$560$	0	0
$561$	−1462.79	−0.110088
$562$	−25007.0	−1.87697
$563$	−18097.1	−1.35471	−0.677356	−	0.735656i	$-0.736874\pi$
$563$	−18097.1	−1.35471	−0.677356	+	0.735656i	$0.736874\pi$
$564$	−2206.80	−0.164757
$565$	0	0
$566$	24723.6	1.83606
$567$	−1570.50	−0.116322
$568$	20641.9	1.52485
$569$	−14101.9	−1.03898	−0.519492	−	0.854475i	$-0.673879\pi$
$569$	−14101.9	−1.03898	−0.519492	+	0.854475i	$0.673879\pi$
$570$	0	0
$571$	2874.72	0.210689	0.105344	−	0.994436i	$-0.466406\pi$
$571$	2874.72	0.210689	0.105344	+	0.994436i	$0.466406\pi$
$572$	−4407.93	−0.322211
$573$	10973.4	0.800036
$574$	22482.2	1.63482
$575$	0	0
$576$	−1318.43	−0.0953727
$577$	−11659.8	−0.841256	−0.420628	−	0.907233i	$-0.638190\pi$
$577$	−11659.8	−0.841256	−0.420628	+	0.907233i	$0.638190\pi$
$578$	1044.55	0.0751686
$579$	−9470.18	−0.679736
$580$	0	0
$581$	12121.6	0.865561
$582$	17691.7	1.26005
$583$	−64.2195	−0.00456209
$584$	−26644.2	−1.88792
$585$	0	0
$586$	45348.1	3.19678
$587$	14565.9	1.02419	0.512095	−	0.858929i	$-0.328870\pi$
$587$	14565.9	1.02419	0.512095	+	0.858929i	$0.328870\pi$
$588$	−1729.47	−0.121296
$589$	−10631.5	−0.743740
$590$	0	0
$591$	−2908.57	−0.202441
$592$	12022.0	0.834630
$593$	18632.6	1.29030	0.645152	−	0.764054i	$-0.276794\pi$
$593$	18632.6	1.29030	0.645152	+	0.764054i	$0.276794\pi$
$594$	−929.244	−0.0641874
$595$	0	0
$596$	−25371.9	−1.74375
$597$	−16269.1	−1.11532
$598$	34599.2	2.36600
$599$	−11080.2	−0.755802	−0.377901	−	0.925846i	$-0.623354\pi$
$599$	−11080.2	−0.755802	−0.377901	+	0.925846i	$0.623354\pi$
$600$	0	0
$601$	−26806.7	−1.81941	−0.909706	−	0.415254i	$-0.863693\pi$
$601$	−26806.7	−1.81941	−0.909706	+	0.415254i	$0.863693\pi$
$602$	6544.38	0.443071
$603$	−8577.44	−0.579271
$604$	48328.0	3.25569
$605$	0	0
$606$	−15225.1	−1.02059
$607$	−2408.52	−0.161052	−0.0805262	−	0.996752i	$-0.525660\pi$
$607$	−2408.52	−0.161052	−0.0805262	+	0.996752i	$0.525660\pi$
$608$	11768.5	0.784992
$609$	1686.83	0.112240
$610$	0	0
$611$	−1552.42	−0.102789
$612$	11274.2	0.744664
$613$	−24558.8	−1.61814	−0.809071	−	0.587710i	$-0.800029\pi$
$613$	−24558.8	−1.61814	−0.809071	+	0.587710i	$0.800029\pi$
$614$	−36782.4	−2.41762
$615$	0	0
$616$	−6344.14	−0.414956
$617$	6367.34	0.415461	0.207731	−	0.978186i	$-0.433392\pi$
$617$	6367.34	0.415461	0.207731	+	0.978186i	$0.433392\pi$
$618$	10670.0	0.694515
$619$	25893.6	1.68134	0.840671	−	0.541546i	$-0.182161\pi$
$619$	25893.6	1.68134	0.840671	+	0.541546i	$0.182161\pi$
$620$	0	0
$621$	5006.29	0.323503
$622$	−12247.3	−0.789505
$623$	5257.66	0.338112
$624$	11355.2	0.728482
$625$	0	0
$626$	32887.0	2.09972
$627$	−1805.29	−0.114986
$628$	−13418.7	−0.852649
$629$	8396.77	0.532275
$630$	0	0
$631$	−1864.31	−0.117618	−0.0588089	−	0.998269i	$-0.518730\pi$
$631$	−1864.31	−0.117618	−0.0588089	+	0.998269i	$0.518730\pi$
$632$	46343.5	2.91684
$633$	8545.53	0.536579
$634$	−33505.5	−2.09885
$635$	0	0
$636$	494.961	0.0308593
$637$	−1216.62	−0.0756741
$638$	998.077	0.0619345
$639$	3869.08	0.239528
$640$	0	0
$641$	−25891.1	−1.59538	−0.797689	−	0.603069i	$-0.793944\pi$
$641$	−25891.1	−1.59538	−0.797689	+	0.603069i	$0.793944\pi$
$642$	16916.4	1.03994
$643$	19494.1	1.19560	0.597801	−	0.801645i	$-0.296041\pi$
$643$	19494.1	1.19560	0.597801	+	0.801645i	$0.296041\pi$
$644$	62939.3	3.85118
$645$	0	0
$646$	31911.8	1.94358
$647$	17987.6	1.09299	0.546495	−	0.837463i	$-0.315962\pi$
$647$	17987.6	1.09299	0.546495	+	0.837463i	$0.315962\pi$
$648$	3889.29	0.235781
$649$	−1584.26	−0.0958207
$650$	0	0
$651$	−7002.87	−0.421604
$652$	32757.9	1.96764
$653$	−2069.77	−0.124037	−0.0620186	−	0.998075i	$-0.519754\pi$
$653$	−2069.77	−0.124037	−0.0620186	+	0.998075i	$0.519754\pi$
$654$	19795.1	1.18356
$655$	0	0
$656$	−23520.5	−1.39988
$657$	−4994.13	−0.296559
$658$	−4114.43	−0.243765
$659$	−15787.8	−0.933239	−0.466619	−	0.884458i	$-0.654528\pi$
$659$	−15787.8	−0.933239	−0.466619	+	0.884458i	$0.654528\pi$
$660$	0	0
$661$	32239.9	1.89711	0.948554	−	0.316615i	$-0.102546\pi$
$661$	32239.9	1.89711	0.948554	+	0.316615i	$0.102546\pi$
$662$	−33979.4	−1.99494
$663$	7931.07	0.464581
$664$	−30018.9	−1.75446
$665$	0	0
$666$	5334.07	0.310347
$667$	−5377.12	−0.312149
$668$	−41030.2	−2.37651
$669$	1507.47	0.0871181
$670$	0	0
$671$	−3724.41	−0.214276
$672$	7751.79	0.444988
$673$	−1039.67	−0.0595488	−0.0297744	−	0.999557i	$-0.509479\pi$
$673$	−1039.67	−0.0595488	−0.0297744	+	0.999557i	$0.509479\pi$
$674$	38804.4	2.21764
$675$	0	0
$676$	−14564.2	−0.828641
$677$	12781.5	0.725604	0.362802	−	0.931866i	$-0.381820\pi$
$677$	12781.5	0.725604	0.362802	+	0.931866i	$0.381820\pi$
$678$	3883.22	0.219962
$679$	22639.7	1.27958
$680$	0	0
$681$	4010.50	0.225672
$682$	−4143.50	−0.232644
$683$	−176.490	−0.00988755	−0.00494378	−	0.999988i	$-0.501574\pi$
$683$	−176.490	−0.00988755	−0.00494378	+	0.999988i	$0.501574\pi$
$684$	13914.0	0.777800
$685$	0	0
$686$	30363.1	1.68990
$687$	3393.79	0.188473
$688$	−6846.61	−0.379396
$689$	348.189	0.0192525
$690$	0	0
$691$	28767.9	1.58377	0.791884	−	0.610672i	$-0.209101\pi$
$691$	28767.9	1.58377	0.791884	+	0.610672i	$0.209101\pi$
$692$	26011.0	1.42889
$693$	−1189.13	−0.0651823
$694$	−7078.46	−0.387168
$695$	0	0
$696$	−4177.39	−0.227505
$697$	−16427.9	−0.892757
$698$	−51787.2	−2.80827
$699$	3814.88	0.206426
$700$	0	0
$701$	−27326.9	−1.47236	−0.736179	−	0.676787i	$-0.763372\pi$
$701$	−27326.9	−1.47236	−0.736179	+	0.676787i	$0.763372\pi$
$702$	5038.23	0.270877
$703$	10362.8	0.555961
$704$	−998.274	−0.0534430
$705$	0	0
$706$	16717.0	0.891152
$707$	−19483.2	−1.03641
$708$	12210.4	0.648158
$709$	3138.76	0.166261	0.0831303	−	0.996539i	$-0.473508\pi$
$709$	3138.76	0.166261	0.0831303	+	0.996539i	$0.473508\pi$
$710$	0	0
$711$	8686.51	0.458185
$712$	−13020.4	−0.685339
$713$	22323.1	1.17252
$714$	21020.0	1.10176
$715$	0	0
$716$	1250.75	0.0652830
$717$	−4522.20	−0.235544
$718$	52663.1	2.73728
$719$	21026.6	1.09062	0.545312	−	0.838233i	$-0.316411\pi$
$719$	21026.6	1.09062	0.545312	+	0.838233i	$0.316411\pi$
$720$	0	0
$721$	13654.1	0.705280
$722$	4742.52	0.244458
$723$	−12753.9	−0.656050
$724$	−29697.7	−1.52446
$725$	0	0
$726$	19462.9	0.994954
$727$	−4458.08	−0.227429	−0.113715	−	0.993513i	$-0.536275\pi$
$727$	−4458.08	−0.227429	−0.113715	+	0.993513i	$0.536275\pi$
$728$	34397.0	1.75115
$729$	729.000	0.0370370
$730$	0	0
$731$	−4782.02	−0.241955
$732$	28705.3	1.44942
$733$	28933.7	1.45797	0.728984	−	0.684531i	$-0.239993\pi$
$733$	28933.7	1.45797	0.728984	+	0.684531i	$0.239993\pi$
$734$	33674.6	1.69339
$735$	0	0
$736$	−24710.4	−1.23755
$737$	−6494.56	−0.324600
$738$	−10435.9	−0.520528
$739$	−9135.10	−0.454723	−0.227361	−	0.973810i	$-0.573010\pi$
$739$	−9135.10	−0.454723	−0.227361	+	0.973810i	$0.573010\pi$
$740$	0	0
$741$	9788.06	0.485254
$742$	922.819	0.0456574
$743$	26380.8	1.30258	0.651290	−	0.758829i	$-0.274228\pi$
$743$	26380.8	1.30258	0.651290	+	0.758829i	$0.274228\pi$
$744$	17342.4	0.854573
$745$	0	0
$746$	−30810.6	−1.51214
$747$	−5626.67	−0.275595
$748$	8536.49	0.417279
$749$	21647.6	1.05605
$750$	0	0
$751$	18205.6	0.884597	0.442298	−	0.896868i	$-0.354163\pi$
$751$	18205.6	0.884597	0.442298	+	0.896868i	$0.354163\pi$
$752$	4304.44	0.208733
$753$	−21658.2	−1.04817
$754$	−5411.44	−0.261370
$755$	0	0
$756$	9165.03	0.440911
$757$	−14889.8	−0.714902	−0.357451	−	0.933932i	$-0.616354\pi$
$757$	−14889.8	−0.714902	−0.357451	+	0.933932i	$0.616354\pi$
$758$	31777.0	1.52268
$759$	3790.60	0.181278
$760$	0	0
$761$	4359.81	0.207678	0.103839	−	0.994594i	$-0.466887\pi$
$761$	4359.81	0.207678	0.103839	+	0.994594i	$0.466887\pi$
$762$	−18201.4	−0.865310
$763$	25331.2	1.20190
$764$	−64038.0	−3.03248
$765$	0	0
$766$	−52268.7	−2.46546
$767$	8589.64	0.404373
$768$	23847.7	1.12048
$769$	−23823.7	−1.11717	−0.558585	−	0.829448i	$-0.688655\pi$
$769$	−23823.7	−1.11717	−0.558585	+	0.829448i	$0.688655\pi$
$770$	0	0
$771$	10308.1	0.481502
$772$	55265.6	2.57649
$773$	−18653.0	−0.867922	−0.433961	−	0.900932i	$-0.642884\pi$
$773$	−18653.0	−0.867922	−0.433961	+	0.900932i	$0.642884\pi$
$774$	−3037.79	−0.141074
$775$	0	0
$776$	−56066.5	−2.59365
$777$	6825.88	0.315157
$778$	−11924.6	−0.549508
$779$	−20274.4	−0.932483
$780$	0	0
$781$	2929.54	0.134222
$782$	−67005.6	−3.06409
$783$	−783.000	−0.0357371
$784$	3373.38	0.153671
$785$	0	0
$786$	−15902.4	−0.721654
$787$	−16426.1	−0.743998	−0.371999	−	0.928233i	$-0.621328\pi$
$787$	−16426.1	−0.743998	−0.371999	+	0.928233i	$0.621328\pi$
$788$	16973.7	0.767337
$789$	4482.42	0.202254
$790$	0	0
$791$	4969.26	0.223371
$792$	2944.84	0.132122
$793$	20193.2	0.904266
$794$	−35167.0	−1.57183
$795$	0	0
$796$	94942.0	4.22755
$797$	8480.05	0.376887	0.188443	−	0.982084i	$-0.439656\pi$
$797$	8480.05	0.376887	0.188443	+	0.982084i	$0.439656\pi$
$798$	25941.7	1.15078
$799$	3006.44	0.133117
$800$	0	0
$801$	−2440.52	−0.107655
$802$	−7168.23	−0.315610
$803$	−3781.39	−0.166180
$804$	50055.7	2.19568
$805$	0	0
$806$	22465.5	0.981779
$807$	5076.18	0.221425
$808$	48249.5	2.10076
$809$	−28294.8	−1.22965	−0.614827	−	0.788662i	$-0.710774\pi$
$809$	−28294.8	−1.22965	−0.614827	+	0.788662i	$0.710774\pi$
$810$	0	0
$811$	−28383.8	−1.22896	−0.614482	−	0.788931i	$-0.710635\pi$
$811$	−28383.8	−1.22896	−0.614482	+	0.788931i	$0.710635\pi$
$812$	−9843.92	−0.425436
$813$	−9443.26	−0.407367
$814$	4038.79	0.173906
$815$	0	0
$816$	−21990.8	−0.943420
$817$	−5901.69	−0.252722
$818$	−26262.1	−1.12253
$819$	6447.30	0.275076
$820$	0	0
$821$	27288.2	1.16001	0.580003	−	0.814615i	$-0.303052\pi$
$821$	27288.2	1.16001	0.580003	+	0.814615i	$0.303052\pi$
$822$	43654.4	1.85234
$823$	37261.0	1.57818	0.789088	−	0.614281i	$-0.210554\pi$
$823$	37261.0	1.57818	0.789088	+	0.614281i	$0.210554\pi$
$824$	−33814.1	−1.42957
$825$	0	0
$826$	22765.5	0.958973
$827$	−2964.91	−0.124668	−0.0623338	−	0.998055i	$-0.519854\pi$
$827$	−2964.91	−0.124668	−0.0623338	+	0.998055i	$0.519854\pi$
$828$	−29215.4	−1.22621
$829$	35023.4	1.46733	0.733664	−	0.679513i	$-0.237809\pi$
$829$	35023.4	1.46733	0.733664	+	0.679513i	$0.237809\pi$
$830$	0	0
$831$	−11619.5	−0.485049
$832$	5412.50	0.225535
$833$	2356.14	0.0980016
$834$	−39157.8	−1.62581
$835$	0	0
$836$	10535.2	0.435848
$837$	3250.62	0.134239
$838$	22606.2	0.931884
$839$	−5316.37	−0.218762	−0.109381	−	0.994000i	$-0.534887\pi$
$839$	−5316.37	−0.218762	−0.109381	+	0.994000i	$0.534887\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	14770.3	0.604533
$843$	14854.3	0.606889
$844$	−49869.5	−2.03386
$845$	0	0
$846$	1909.85	0.0776147
$847$	24906.2	1.01038
$848$	−965.437	−0.0390958
$849$	−14685.9	−0.593662
$850$	0	0
$851$	−21758.9	−0.876481
$852$	−22579.0	−0.907913
$853$	40627.7	1.63079	0.815396	−	0.578904i	$-0.196519\pi$
$853$	40627.7	1.63079	0.815396	+	0.578904i	$0.196519\pi$
$854$	53518.9	2.14447
$855$	0	0
$856$	−53609.5	−2.14058
$857$	−22175.9	−0.883914	−0.441957	−	0.897036i	$-0.645716\pi$
$857$	−22175.9	−0.883914	−0.441957	+	0.897036i	$0.645716\pi$
$858$	3814.79	0.151789
$859$	−39450.8	−1.56699	−0.783495	−	0.621399i	$-0.786565\pi$
$859$	−39450.8	−1.56699	−0.783495	+	0.621399i	$0.786565\pi$
$860$	0	0
$861$	−13354.5	−0.528596
$862$	7520.06	0.297139
$863$	−5285.21	−0.208471	−0.104236	−	0.994553i	$-0.533240\pi$
$863$	−5285.21	−0.208471	−0.104236	+	0.994553i	$0.533240\pi$
$864$	−3598.26	−0.141684
$865$	0	0
$866$	42483.7	1.66704
$867$	−620.466	−0.0243046
$868$	40866.9	1.59806
$869$	6577.14	0.256748
$870$	0	0
$871$	35212.6	1.36984
$872$	−62732.0	−2.43621
$873$	−10509.0	−0.407417
$874$	−82694.4	−3.20043
$875$	0	0
$876$	29144.4	1.12409
$877$	−17785.8	−0.684815	−0.342407	−	0.939552i	$-0.611242\pi$
$877$	−17785.8	−0.684815	−0.342407	+	0.939552i	$0.611242\pi$
$878$	−53457.9	−2.05480
$879$	−26936.9	−1.03363
$880$	0	0
$881$	19542.5	0.747338	0.373669	−	0.927562i	$-0.378100\pi$
$881$	19542.5	0.747338	0.373669	+	0.927562i	$0.378100\pi$
$882$	1496.74	0.0571406
$883$	1625.38	0.0619460	0.0309730	−	0.999520i	$-0.490139\pi$
$883$	1625.38	0.0619460	0.0309730	+	0.999520i	$0.490139\pi$
$884$	−46283.7	−1.76096
$885$	0	0
$886$	52199.5	1.97932
$887$	48250.9	1.82650	0.913251	−	0.407396i	$-0.133563\pi$
$887$	48250.9	1.82650	0.913251	+	0.407396i	$0.133563\pi$
$888$	−16904.1	−0.638811
$889$	−23291.8	−0.878722
$890$	0	0
$891$	551.975	0.0207540
$892$	−8797.18	−0.330215
$893$	3710.37	0.139040
$894$	21957.8	0.821453
$895$	0	0
$896$	35016.4	1.30560
$897$	−20552.1	−0.765011
$898$	83972.6	3.12049
$899$	−3491.40	−0.129527
$900$	0	0
$901$	−674.310	−0.0249329
$902$	−7901.70	−0.291683
$903$	−3887.39	−0.143260
$904$	−12306.2	−0.452764
$905$	0	0
$906$	−41824.8	−1.53371
$907$	−20513.2	−0.750971	−0.375486	−	0.926828i	$-0.622524\pi$
$907$	−20513.2	−0.750971	−0.375486	+	0.926828i	$0.622524\pi$
$908$	−23404.2	−0.855393
$909$	9043.78	0.329992
$910$	0	0
$911$	−33692.7	−1.22534	−0.612672	−	0.790338i	$-0.709905\pi$
$911$	−33692.7	−1.22534	−0.612672	+	0.790338i	$0.709905\pi$
$912$	−27139.7	−0.985401
$913$	−4260.33	−0.154432
$914$	45495.9	1.64647
$915$	0	0
$916$	−19805.3	−0.714394
$917$	−20349.9	−0.732839
$918$	−9757.15	−0.350799
$919$	36764.9	1.31966	0.659828	−	0.751417i	$-0.270629\pi$
$919$	36764.9	1.31966	0.659828	+	0.751417i	$0.270629\pi$
$920$	0	0
$921$	21848.9	0.781700
$922$	−40895.5	−1.46076
$923$	−15883.6	−0.566429
$924$	6939.46	0.247069
$925$	0	0
$926$	−24736.6	−0.877858
$927$	−6338.03	−0.224561
$928$	3864.79	0.136711
$929$	13533.8	0.477965	0.238982	−	0.971024i	$-0.423186\pi$
$929$	13533.8	0.477965	0.238982	+	0.971024i	$0.423186\pi$
$930$	0	0
$931$	2907.81	0.102363
$932$	−22262.7	−0.782444
$933$	7274.95	0.255275
$934$	16824.2	0.589406
$935$	0	0
$936$	−15966.5	−0.557567
$937$	54144.0	1.88773	0.943867	−	0.330325i	$-0.107158\pi$
$937$	54144.0	1.88773	0.943867	+	0.330325i	$0.107158\pi$
$938$	93325.3	3.24859
$939$	−19535.0	−0.678914
$940$	0	0
$941$	−8397.60	−0.290918	−0.145459	−	0.989364i	$-0.546466\pi$
$941$	−8397.60	−0.290918	−0.145459	+	0.989364i	$0.546466\pi$
$942$	11613.0	0.401669
$943$	42570.3	1.47007
$944$	−23816.8	−0.821157
$945$	0	0
$946$	−2300.12	−0.0790521
$947$	4931.24	0.169212	0.0846059	−	0.996414i	$-0.473037\pi$
$947$	4931.24	0.169212	0.0846059	+	0.996414i	$0.473037\pi$
$948$	−50692.3	−1.73672
$949$	20502.2	0.701295
$950$	0	0
$951$	19902.4	0.678633
$952$	−66614.0	−2.26783
$953$	21509.1	0.731111	0.365555	−	0.930790i	$-0.380879\pi$
$953$	21509.1	0.731111	0.365555	+	0.930790i	$0.380879\pi$
$954$	−428.358	−0.0145373
$955$	0	0
$956$	26390.4	0.892811
$957$	−592.862	−0.0200256
$958$	−68718.0	−2.31751
$959$	55863.4	1.88105
$960$	0	0
$961$	−15296.5	−0.513460
$962$	−21897.7	−0.733900
$963$	−10048.4	−0.336248
$964$	74428.8	2.48671
$965$	0	0
$966$	−54470.0	−1.81423
$967$	−56174.6	−1.86810	−0.934051	−	0.357140i	$-0.883752\pi$
$967$	−56174.6	−1.86810	−0.934051	+	0.357140i	$0.883752\pi$
$968$	−61679.5	−2.04799
$969$	−18955.8	−0.628428
$970$	0	0
$971$	59470.2	1.96549	0.982745	−	0.184966i	$-0.0592173\pi$
$971$	59470.2	1.96549	0.982745	+	0.184966i	$0.0592173\pi$
$972$	−4254.26	−0.140386
$973$	−50109.3	−1.65101
$974$	78084.9	2.56879
$975$	0	0
$976$	−55990.5	−1.83628
$977$	54861.4	1.79649	0.898245	−	0.439494i	$-0.144842\pi$
$977$	54861.4	1.79649	0.898245	+	0.439494i	$0.144842\pi$
$978$	−28349.9	−0.926922
$979$	−1847.88	−0.0603254
$980$	0	0
$981$	−11758.4	−0.382686
$982$	−96041.5	−3.12099
$983$	20759.3	0.673568	0.336784	−	0.941582i	$-0.390661\pi$
$983$	20759.3	0.673568	0.336784	+	0.941582i	$0.390661\pi$
$984$	33072.1	1.07144
$985$	0	0
$986$	10479.9	0.338487
$987$	2443.99	0.0788176
$988$	−57120.6	−1.83932
$989$	12391.8	0.398420
$990$	0	0
$991$	−25662.6	−0.822603	−0.411301	−	0.911499i	$-0.634926\pi$
$991$	−25662.6	−0.822603	−0.411301	+	0.911499i	$0.634926\pi$
$992$	−16044.6	−0.513526
$993$	20183.9	0.645033
$994$	−42096.8	−1.34329
$995$	0	0
$996$	32835.8	1.04462
$997$	−43697.7	−1.38808	−0.694042	−	0.719935i	$-0.744172\pi$
$997$	−43697.7	−1.38808	−0.694042	+	0.719935i	$0.744172\pi$
$998$	−69065.4	−2.19061
$999$	−3168.46	−0.100346

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.4.a.k.1.6		6
5.4	even	2		435.4.a.h.1.1	✓	6
15.14	odd	2		1305.4.a.h.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.4.a.h.1.1	✓	6	5.4	even	2
1305.4.a.h.1.6		6	15.14	odd	2
2175.4.a.k.1.6		6	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+5.05047 q^{2} -3.00000 q^{3} +17.5072 q^{4} -15.1514 q^{6} -19.3889 q^{7} +48.0160 q^{8} +9.00000 q^{9} +6.81451 q^{11} -52.5217 q^{12} -36.9473 q^{13} -97.9229 q^{14} +102.445 q^{16} +71.5529 q^{17} +45.4542 q^{18} +88.3064 q^{19} +58.1666 q^{21} +34.4164 q^{22} -185.418 q^{23} -144.048 q^{24} -186.601 q^{26} -27.0000 q^{27} -339.446 q^{28} +29.0000 q^{29} -120.393 q^{31} +133.269 q^{32} -20.4435 q^{33} +361.376 q^{34} +157.565 q^{36} +117.351 q^{37} +445.989 q^{38} +110.842 q^{39} -229.591 q^{41} +293.769 q^{42} -66.8319 q^{43} +119.303 q^{44} -936.448 q^{46} +42.0170 q^{47} -307.336 q^{48} +32.9286 q^{49} -214.659 q^{51} -646.845 q^{52} -9.42394 q^{53} -136.363 q^{54} -930.976 q^{56} -264.919 q^{57} +146.464 q^{58} -232.484 q^{59} -546.541 q^{61} -608.042 q^{62} -174.500 q^{63} -146.492 q^{64} -103.249 q^{66} -953.049 q^{67} +1252.69 q^{68} +556.254 q^{69} +429.898 q^{71} +432.144 q^{72} -554.903 q^{73} +592.675 q^{74} +1546.00 q^{76} -132.126 q^{77} +559.804 q^{78} +965.168 q^{79} +81.0000 q^{81} -1159.54 q^{82} -625.186 q^{83} +1018.34 q^{84} -337.532 q^{86} -87.0000 q^{87} +327.205 q^{88} -271.169 q^{89} +716.367 q^{91} -3246.16 q^{92} +361.180 q^{93} +212.206 q^{94} -399.806 q^{96} -1167.66 q^{97} +166.305 q^{98} +61.3305 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{2} - 18 q^{3} + 51 q^{4} + 3 q^{6} - 47 q^{7} - 51 q^{8} + 54 q^{9} + 81 q^{11} - 153 q^{12} - 169 q^{13} - 30 q^{14} + 131 q^{16} + q^{17} - 9 q^{18} + 116 q^{19} + 141 q^{21} - 90 q^{22} + 52 q^{23}+ \cdots + 729 q^{99}+O(q^{100}) \) 6 * q - q^2 - 18 * q^3 + 51 * q^4 + 3 * q^6 - 47 * q^7 - 51 * q^8 + 54 * q^9 + 81 * q^11 - 153 * q^12 - 169 * q^13 - 30 * q^14 + 131 * q^16 + q^17 - 9 * q^18 + 116 * q^19 + 141 * q^21 - 90 * q^22 + 52 * q^23 + 153 * q^24 + 294 * q^26 - 162 * q^27 - 344 * q^28 + 174 * q^29 + 340 * q^31 - 499 * q^32 - 243 * q^33 + 920 * q^34 + 459 * q^36 - 332 * q^37 + 378 * q^38 + 507 * q^39 - 616 * q^41 + 90 * q^42 - 334 * q^43 - 52 * q^44 - 158 * q^46 + 85 * q^47 - 393 * q^48 + 879 * q^49 - 3 * q^51 - 2220 * q^52 + 850 * q^53 + 27 * q^54 - 624 * q^56 - 348 * q^57 - 29 * q^58 - 758 * q^59 - 36 * q^61 + 152 * q^62 - 423 * q^63 + 1795 * q^64 + 270 * q^66 - 939 * q^67 + 186 * q^68 - 156 * q^69 - 1388 * q^71 - 459 * q^72 - 1708 * q^73 - 814 * q^74 + 566 * q^76 - 2585 * q^77 - 882 * q^78 + 1250 * q^79 + 486 * q^81 - 1372 * q^82 + 748 * q^83 + 1032 * q^84 - 800 * q^86 - 522 * q^87 - 536 * q^88 + 1099 * q^89 + 539 * q^91 - 1698 * q^92 - 1020 * q^93 - 4542 * q^94 + 1497 * q^96 + 22 * q^97 + 1433 * q^98 + 729 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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