LMFDB - Embedded newform 2175.4.a.l.1.4

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:	$1$
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} - 31x^{4} + 9x^{3} + 230x^{2} + 32x - 128 $ x^6 - x^5 - 31x^4 + 9x^3 + 230x^2 + 32x - 128
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.691666$ 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)

$f(q)$	$=$	\(q+0.691666 q^{2} +3.00000 q^{3} -7.52160 q^{4} +2.07500 q^{6} +14.5720 q^{7} -10.7358 q^{8} +9.00000 q^{9} +36.8326 q^{11} -22.5648 q^{12} +23.9199 q^{13} +10.0790 q^{14} +52.7472 q^{16} -62.8914 q^{17} +6.22499 q^{18} -94.1599 q^{19} +43.7161 q^{21} +25.4759 q^{22} -185.410 q^{23} -32.2073 q^{24} +16.5446 q^{26} +27.0000 q^{27} -109.605 q^{28} -29.0000 q^{29} -104.971 q^{31} +122.370 q^{32} +110.498 q^{33} -43.4999 q^{34} -67.6944 q^{36} +259.799 q^{37} -65.1272 q^{38} +71.7597 q^{39} +11.8069 q^{41} +30.2369 q^{42} +61.0501 q^{43} -277.040 q^{44} -128.242 q^{46} -281.426 q^{47} +158.242 q^{48} -130.656 q^{49} -188.674 q^{51} -179.916 q^{52} +170.846 q^{53} +18.6750 q^{54} -156.442 q^{56} -282.480 q^{57} -20.0583 q^{58} -636.987 q^{59} -379.603 q^{61} -72.6051 q^{62} +131.148 q^{63} -337.339 q^{64} +76.4276 q^{66} +623.333 q^{67} +473.044 q^{68} -556.231 q^{69} +298.898 q^{71} -96.6218 q^{72} +524.124 q^{73} +179.694 q^{74} +708.233 q^{76} +536.726 q^{77} +49.6337 q^{78} +563.264 q^{79} +81.0000 q^{81} +8.16645 q^{82} +885.342 q^{83} -328.815 q^{84} +42.2263 q^{86} -87.0000 q^{87} -395.426 q^{88} -447.652 q^{89} +348.562 q^{91} +1394.58 q^{92} -314.914 q^{93} -194.653 q^{94} +367.109 q^{96} +558.895 q^{97} -90.3702 q^{98} +331.493 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{2} + 18 q^{3} + 15 q^{4} + 3 q^{6} - 23 q^{7} + 51 q^{8} + 54 q^{9} - 111 q^{11} + 45 q^{12} + 83 q^{13} - 102 q^{14} - 37 q^{16} + 35 q^{17} + 9 q^{18} - 76 q^{19} - 69 q^{21} - 66 q^{22} - 166 q^{23}+ \cdots - 999 q^{99}+O(q^{100}) $ 6 * q + q^2 + 18 * q^3 + 15 * q^4 + 3 * q^6 - 23 * q^7 + 51 * q^8 + 54 * q^9 - 111 * q^11 + 45 * q^12 + 83 * q^13 - 102 * q^14 - 37 * q^16 + 35 * q^17 + 9 * q^18 - 76 * q^19 - 69 * q^21 - 66 * q^22 - 166 * q^23 + 153 * q^24 - 282 * q^26 + 162 * q^27 - 368 * q^28 - 174 * q^29 - 164 * q^31 + 259 * q^32 - 333 * q^33 + 512 * q^34 + 135 * q^36 + 538 * q^37 - 6 * q^38 + 249 * q^39 - 1250 * q^41 - 306 * q^42 + 80 * q^43 - 1004 * q^44 - 1190 * q^46 + 491 * q^47 - 111 * q^48 - 969 * q^49 + 105 * q^51 + 336 * q^52 + 356 * q^53 + 27 * q^54 - 1260 * q^56 - 228 * q^57 - 29 * q^58 - 646 * q^59 - 1476 * q^61 + 4 * q^62 - 207 * q^63 - 845 * q^64 - 198 * q^66 - 87 * q^67 + 558 * q^68 - 498 * q^69 - 1312 * q^71 + 459 * q^72 + 1142 * q^73 - 2270 * q^74 - 1594 * q^76 + 2561 * q^77 - 846 * q^78 - 142 * q^79 + 486 * q^81 + 1544 * q^82 - 346 * q^83 - 1104 * q^84 - 2056 * q^86 - 522 * q^87 - 1400 * q^88 - 2827 * q^89 + 1379 * q^91 + 1434 * q^92 - 492 * q^93 + 978 * q^94 + 777 * q^96 + 4 * q^97 + 1903 * q^98 - 999 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.691666	0.244541	0.122270	−	0.992497i	$-0.460982\pi$
$2$	0.691666	0.244541	0.122270	+	0.992497i	$0.460982\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−7.52160	−0.940200
$5$	0	0
$5$	0	0
$6$	2.07500	0.141186
$7$	14.5720	0.786816	0.393408	−	0.919364i	$-0.371296\pi$
$7$	14.5720	0.786816	0.393408	+	0.919364i	$0.371296\pi$
$8$	−10.7358	−0.474458
$9$	9.00000	0.333333
$10$	0	0
$11$	36.8326	1.00959	0.504793	−	0.863240i	$-0.331569\pi$
$11$	36.8326	1.00959	0.504793	+	0.863240i	$0.331569\pi$
$12$	−22.5648	−0.542825
$13$	23.9199	0.510322	0.255161	−	0.966899i	$-0.417872\pi$
$13$	23.9199	0.510322	0.255161	+	0.966899i	$0.417872\pi$
$14$	10.0790	0.192409
$15$	0	0
$16$	52.7472	0.824175
$17$	−62.8914	−0.897260	−0.448630	−	0.893718i	$-0.648088\pi$
$17$	−62.8914	−0.897260	−0.448630	+	0.893718i	$0.648088\pi$
$18$	6.22499	0.0815136
$19$	−94.1599	−1.13693	−0.568467	−	0.822706i	$-0.692463\pi$
$19$	−94.1599	−1.13693	−0.568467	+	0.822706i	$0.692463\pi$
$20$	0	0
$21$	43.7161	0.454268
$22$	25.4759	0.246885
$23$	−185.410	−1.68090	−0.840450	−	0.541889i	$-0.817709\pi$
$23$	−185.410	−1.68090	−0.840450	+	0.541889i	$0.817709\pi$
$24$	−32.2073	−0.273928
$25$	0	0
$26$	16.5446	0.124795
$27$	27.0000	0.192450
$28$	−109.605	−0.739764
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−104.971	−0.608174	−0.304087	−	0.952644i	$-0.598351\pi$
$31$	−104.971	−0.608174	−0.304087	+	0.952644i	$0.598351\pi$
$32$	122.370	0.676003
$33$	110.498	0.582885
$34$	−43.4999	−0.219417
$35$	0	0
$36$	−67.6944	−0.313400
$37$	259.799	1.15435	0.577173	−	0.816622i	$-0.304156\pi$
$37$	259.799	1.15435	0.577173	+	0.816622i	$0.304156\pi$
$38$	−65.1272	−0.278027
$39$	71.7597	0.294635
$40$	0	0
$41$	11.8069	0.0449740	0.0224870	−	0.999747i	$-0.492842\pi$
$41$	11.8069	0.0449740	0.0224870	+	0.999747i	$0.492842\pi$
$42$	30.2369	0.111087
$43$	61.0501	0.216513	0.108256	−	0.994123i	$-0.465473\pi$
$43$	61.0501	0.216513	0.108256	+	0.994123i	$0.465473\pi$
$44$	−277.040	−0.949213
$45$	0	0
$46$	−128.242	−0.411049
$47$	−281.426	−0.873409	−0.436705	−	0.899605i	$-0.643855\pi$
$47$	−281.426	−0.873409	−0.436705	+	0.899605i	$0.643855\pi$
$48$	158.242	0.475838
$49$	−130.656	−0.380921
$50$	0	0
$51$	−188.674	−0.518033
$52$	−179.916	−0.479805
$53$	170.846	0.442782	0.221391	−	0.975185i	$-0.428940\pi$
$53$	170.846	0.442782	0.221391	+	0.975185i	$0.428940\pi$
$54$	18.6750	0.0470619
$55$	0	0
$56$	−156.442	−0.373311
$57$	−282.480	−0.656409
$58$	−20.0583	−0.0454101
$59$	−636.987	−1.40557	−0.702785	−	0.711402i	$-0.748060\pi$
$59$	−636.987	−1.40557	−0.702785	+	0.711402i	$0.748060\pi$
$60$	0	0
$61$	−379.603	−0.796774	−0.398387	−	0.917218i	$-0.630430\pi$
$61$	−379.603	−0.796774	−0.398387	+	0.917218i	$0.630430\pi$
$62$	−72.6051	−0.148723
$63$	131.148	0.262272
$64$	−337.339	−0.658865
$65$	0	0
$66$	76.4276	0.142539
$67$	623.333	1.13660	0.568300	−	0.822821i	$-0.307601\pi$
$67$	623.333	1.13660	0.568300	+	0.822821i	$0.307601\pi$
$68$	473.044	0.843603
$69$	−556.231	−0.970468
$70$	0	0
$71$	298.898	0.499615	0.249808	−	0.968295i	$-0.419633\pi$
$71$	298.898	0.499615	0.249808	+	0.968295i	$0.419633\pi$
$72$	−96.6218	−0.158153
$73$	524.124	0.840329	0.420165	−	0.907448i	$-0.361972\pi$
$73$	524.124	0.840329	0.420165	+	0.907448i	$0.361972\pi$
$74$	179.694	0.282284
$75$	0	0
$76$	708.233	1.06895
$77$	536.726	0.794358
$78$	49.6337	0.0720502
$79$	563.264	0.802179	0.401090	−	0.916039i	$-0.368632\pi$
$79$	563.264	0.802179	0.401090	+	0.916039i	$0.368632\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	8.16645	0.0109980
$83$	885.342	1.17083	0.585415	−	0.810734i	$-0.300932\pi$
$83$	885.342	1.17083	0.585415	+	0.810734i	$0.300932\pi$
$84$	−328.815	−0.427103
$85$	0	0
$86$	42.2263	0.0529462
$87$	−87.0000	−0.107211
$88$	−395.426	−0.479006
$89$	−447.652	−0.533157	−0.266579	−	0.963813i	$-0.585893\pi$
$89$	−447.652	−0.533157	−0.266579	+	0.963813i	$0.585893\pi$
$90$	0	0
$91$	348.562	0.401529
$92$	1394.58	1.58038
$93$	−314.914	−0.351130
$94$	−194.653	−0.213584
$95$	0	0
$96$	367.109	0.390290
$97$	558.895	0.585022	0.292511	−	0.956262i	$-0.405509\pi$
$97$	558.895	0.585022	0.292511	+	0.956262i	$0.405509\pi$
$98$	−90.3702	−0.0931507
$99$	331.493	0.336529
$100$	0	0
$101$	−1138.78	−1.12191	−0.560956	−	0.827846i	$-0.689566\pi$
$101$	−1138.78	−1.12191	−0.560956	+	0.827846i	$0.689566\pi$
$102$	−130.500	−0.126680
$103$	−1008.31	−0.964583	−0.482292	−	0.876011i	$-0.660195\pi$
$103$	−1008.31	−0.964583	−0.482292	+	0.876011i	$0.660195\pi$
$104$	−256.798	−0.242126
$105$	0	0
$106$	118.168	0.108278
$107$	−140.506	−0.126946	−0.0634730	−	0.997984i	$-0.520218\pi$
$107$	−140.506	−0.126946	−0.0634730	+	0.997984i	$0.520218\pi$
$108$	−203.083	−0.180942
$109$	−850.530	−0.747394	−0.373697	−	0.927551i	$-0.621910\pi$
$109$	−850.530	−0.747394	−0.373697	+	0.927551i	$0.621910\pi$
$110$	0	0
$111$	779.398	0.666461
$112$	768.634	0.648474
$113$	−1851.23	−1.54115	−0.770573	−	0.637352i	$-0.780030\pi$
$113$	−1851.23	−1.54115	−0.770573	+	0.637352i	$0.780030\pi$
$114$	−195.381	−0.160519
$115$	0	0
$116$	218.126	0.174591
$117$	215.279	0.170107
$118$	−440.582	−0.343719
$119$	−916.456	−0.705978
$120$	0	0
$121$	25.6407	0.0192643
$122$	−262.558	−0.194844
$123$	35.4208	0.0259657
$124$	789.552	0.571805
$125$	0	0
$126$	90.7108	0.0641362
$127$	−903.124	−0.631018	−0.315509	−	0.948923i	$-0.602175\pi$
$127$	−903.124	−0.631018	−0.315509	+	0.948923i	$0.602175\pi$
$128$	−1212.28	−0.837122
$129$	183.150	0.125004
$130$	0	0
$131$	−685.958	−0.457499	−0.228750	−	0.973485i	$-0.573464\pi$
$131$	−685.958	−0.457499	−0.228750	+	0.973485i	$0.573464\pi$
$132$	−831.120	−0.548028
$133$	−1372.10	−0.894558
$134$	431.138	0.277945
$135$	0	0
$136$	675.187	0.425712
$137$	1152.95	0.718999	0.359500	−	0.933145i	$-0.382947\pi$
$137$	1152.95	0.718999	0.359500	+	0.933145i	$0.382947\pi$
$138$	−384.726	−0.237319
$139$	−438.465	−0.267555	−0.133777	−	0.991011i	$-0.542711\pi$
$139$	−438.465	−0.267555	−0.133777	+	0.991011i	$0.542711\pi$
$140$	0	0
$141$	−844.279	−0.504263
$142$	206.738	0.122176
$143$	881.032	0.515214
$144$	474.725	0.274725
$145$	0	0
$146$	362.518	0.205495
$147$	−391.968	−0.219925
$148$	−1954.11	−1.08532
$149$	−405.570	−0.222991	−0.111495	−	0.993765i	$-0.535564\pi$
$149$	−405.570	−0.222991	−0.111495	+	0.993765i	$0.535564\pi$
$150$	0	0
$151$	−78.9559	−0.0425519	−0.0212760	−	0.999774i	$-0.506773\pi$
$151$	−78.9559	−0.0425519	−0.0212760	+	0.999774i	$0.506773\pi$
$152$	1010.88	0.539428
$153$	−566.023	−0.299087
$154$	371.235	0.194253
$155$	0	0
$156$	−539.748	−0.277015
$157$	−314.869	−0.160059	−0.0800296	−	0.996792i	$-0.525501\pi$
$157$	−314.869	−0.160059	−0.0800296	+	0.996792i	$0.525501\pi$
$158$	389.590	0.196166
$159$	512.537	0.255640
$160$	0	0
$161$	−2701.80	−1.32256
$162$	56.0249	0.0271712
$163$	−816.812	−0.392501	−0.196251	−	0.980554i	$-0.562877\pi$
$163$	−816.812	−0.392501	−0.196251	+	0.980554i	$0.562877\pi$
$164$	−88.8070	−0.0422845
$165$	0	0
$166$	612.361	0.286316
$167$	1364.36	0.632200	0.316100	−	0.948726i	$-0.397626\pi$
$167$	1364.36	0.632200	0.316100	+	0.948726i	$0.397626\pi$
$168$	−469.325	−0.215531
$169$	−1624.84	−0.739571
$170$	0	0
$171$	−847.439	−0.378978
$172$	−459.194	−0.203565
$173$	−430.857	−0.189350	−0.0946748	−	0.995508i	$-0.530181\pi$
$173$	−430.857	−0.189350	−0.0946748	+	0.995508i	$0.530181\pi$
$174$	−60.1749	−0.0262175
$175$	0	0
$176$	1942.82	0.832076
$177$	−1910.96	−0.811506
$178$	−309.625	−0.130379
$179$	−3192.64	−1.33312	−0.666561	−	0.745451i	$-0.732234\pi$
$179$	−3192.64	−1.33312	−0.666561	+	0.745451i	$0.732234\pi$
$180$	0	0
$181$	−621.160	−0.255085	−0.127543	−	0.991833i	$-0.540709\pi$
$181$	−621.160	−0.255085	−0.127543	+	0.991833i	$0.540709\pi$
$182$	241.088	0.0981903
$183$	−1138.81	−0.460017
$184$	1990.52	0.797517
$185$	0	0
$186$	−217.815	−0.0858655
$187$	−2316.46	−0.905861
$188$	2116.78	0.821179
$189$	393.445	0.151423
$190$	0	0
$191$	2943.37	1.11505	0.557526	−	0.830160i	$-0.311751\pi$
$191$	2943.37	1.11505	0.557526	+	0.830160i	$0.311751\pi$
$192$	−1012.02	−0.380396
$193$	−3259.79	−1.21578	−0.607889	−	0.794022i	$-0.707983\pi$
$193$	−3259.79	−1.21578	−0.607889	+	0.794022i	$0.707983\pi$
$194$	386.568	0.143062
$195$	0	0
$196$	982.741	0.358142
$197$	−2372.35	−0.857983	−0.428992	−	0.903308i	$-0.641131\pi$
$197$	−2372.35	−0.857983	−0.428992	+	0.903308i	$0.641131\pi$
$198$	229.283	0.0822950
$199$	−1828.75	−0.651441	−0.325721	−	0.945466i	$-0.605607\pi$
$199$	−1828.75	−0.651441	−0.325721	+	0.945466i	$0.605607\pi$
$200$	0	0
$201$	1870.00	0.656216
$202$	−787.657	−0.274353
$203$	−422.589	−0.146108
$204$	1419.13	0.487055
$205$	0	0
$206$	−697.416	−0.235880
$207$	−1668.69	−0.560300
$208$	1261.71	0.420595
$209$	−3468.15	−1.14783
$210$	0	0
$211$	4223.45	1.37798	0.688991	−	0.724770i	$-0.258054\pi$
$211$	4223.45	1.37798	0.688991	+	0.724770i	$0.258054\pi$
$212$	−1285.03	−0.416304
$213$	896.694	0.288453
$214$	−97.1832	−0.0310435
$215$	0	0
$216$	−289.865	−0.0913095
$217$	−1529.65	−0.478521
$218$	−588.282	−0.182768
$219$	1572.37	0.485164
$220$	0	0
$221$	−1504.36	−0.457891
$222$	539.083	0.162977
$223$	327.670	0.0983964	0.0491982	−	0.998789i	$-0.484333\pi$
$223$	327.670	0.0983964	0.0491982	+	0.998789i	$0.484333\pi$
$224$	1783.17	0.531889
$225$	0	0
$226$	−1280.44	−0.376873
$227$	−3662.12	−1.07076	−0.535382	−	0.844610i	$-0.679832\pi$
$227$	−3662.12	−1.07076	−0.535382	+	0.844610i	$0.679832\pi$
$228$	2124.70	0.617156
$229$	−6562.57	−1.89374	−0.946871	−	0.321613i	$-0.895775\pi$
$229$	−6562.57	−1.89374	−0.946871	+	0.321613i	$0.895775\pi$
$230$	0	0
$231$	1610.18	0.458623
$232$	311.337	0.0881046
$233$	−2490.28	−0.700189	−0.350094	−	0.936714i	$-0.613850\pi$
$233$	−2490.28	−0.700189	−0.350094	+	0.936714i	$0.613850\pi$
$234$	148.901	0.0415982
$235$	0	0
$236$	4791.16	1.32152
$237$	1689.79	0.463138
$238$	−633.881	−0.172640
$239$	−3584.40	−0.970106	−0.485053	−	0.874485i	$-0.661200\pi$
$239$	−3584.40	−0.970106	−0.485053	+	0.874485i	$0.661200\pi$
$240$	0	0
$241$	−6543.94	−1.74910	−0.874548	−	0.484938i	$-0.838842\pi$
$241$	−6543.94	−1.74910	−0.874548	+	0.484938i	$0.838842\pi$
$242$	17.7348	0.00471090
$243$	243.000	0.0641500
$244$	2855.22	0.749126
$245$	0	0
$246$	24.4994	0.00634968
$247$	−2252.29	−0.580203
$248$	1126.95	0.288553
$249$	2656.03	0.675979
$250$	0	0
$251$	528.100	0.132802	0.0664011	−	0.997793i	$-0.478848\pi$
$251$	528.100	0.132802	0.0664011	+	0.997793i	$0.478848\pi$
$252$	−986.445	−0.246588
$253$	−6829.14	−1.69701
$254$	−624.660	−0.154310
$255$	0	0
$256$	1860.22	0.454155
$257$	−5020.78	−1.21863	−0.609314	−	0.792929i	$-0.708555\pi$
$257$	−5020.78	−1.21863	−0.609314	+	0.792929i	$0.708555\pi$
$258$	126.679	0.0305685
$259$	3785.81	0.908257
$260$	0	0
$261$	−261.000	−0.0618984
$262$	−474.453	−0.111877
$263$	422.212	0.0989913	0.0494957	−	0.998774i	$-0.484239\pi$
$263$	422.212	0.0989913	0.0494957	+	0.998774i	$0.484239\pi$
$264$	−1186.28	−0.276554
$265$	0	0
$266$	−949.035	−0.218756
$267$	−1342.96	−0.307818
$268$	−4688.46	−1.06863
$269$	−1777.19	−0.402815	−0.201407	−	0.979508i	$-0.564551\pi$
$269$	−1777.19	−0.402815	−0.201407	+	0.979508i	$0.564551\pi$
$270$	0	0
$271$	−1465.51	−0.328499	−0.164250	−	0.986419i	$-0.552520\pi$
$271$	−1465.51	−0.328499	−0.164250	+	0.986419i	$0.552520\pi$
$272$	−3317.35	−0.739499
$273$	1045.68	0.231823
$274$	797.454	0.175825
$275$	0	0
$276$	4183.74	0.912434
$277$	6081.50	1.31914	0.659570	−	0.751643i	$-0.270738\pi$
$277$	6081.50	1.31914	0.659570	+	0.751643i	$0.270738\pi$
$278$	−303.271	−0.0654281
$279$	−944.742	−0.202725
$280$	0	0
$281$	−1194.50	−0.253587	−0.126794	−	0.991929i	$-0.540469\pi$
$281$	−1194.50	−0.253587	−0.126794	+	0.991929i	$0.540469\pi$
$282$	−583.959	−0.123313
$283$	−3827.53	−0.803968	−0.401984	−	0.915647i	$-0.631679\pi$
$283$	−3827.53	−0.803968	−0.401984	+	0.915647i	$0.631679\pi$
$284$	−2248.19	−0.469738
$285$	0	0
$286$	609.380	0.125991
$287$	172.051	0.0353862
$288$	1101.33	0.225334
$289$	−957.668	−0.194925
$290$	0	0
$291$	1676.68	0.337763
$292$	−3942.25	−0.790077
$293$	6821.00	1.36002	0.680012	−	0.733201i	$-0.261975\pi$
$293$	6821.00	1.36002	0.680012	+	0.733201i	$0.261975\pi$
$294$	−271.111	−0.0537806
$295$	0	0
$296$	−2789.14	−0.547688
$297$	994.480	0.194295
$298$	−280.519	−0.0545303
$299$	−4434.99	−0.857801
$300$	0	0
$301$	889.624	0.170356
$302$	−54.6111	−0.0104057
$303$	−3416.35	−0.647736
$304$	−4966.67	−0.937034
$305$	0	0
$306$	−391.499	−0.0731389
$307$	7350.74	1.36654	0.683272	−	0.730164i	$-0.260556\pi$
$307$	7350.74	1.36654	0.683272	+	0.730164i	$0.260556\pi$
$308$	−4037.04	−0.746856
$309$	−3024.94	−0.556902
$310$	0	0
$311$	3393.14	0.618673	0.309337	−	0.950953i	$-0.399893\pi$
$311$	3393.14	0.618673	0.309337	+	0.950953i	$0.399893\pi$
$312$	−770.395	−0.139792
$313$	10732.3	1.93810	0.969049	−	0.246867i	$-0.0794011\pi$
$313$	10732.3	1.93810	0.969049	+	0.246867i	$0.0794011\pi$
$314$	−217.784	−0.0391410
$315$	0	0
$316$	−4236.65	−0.754209
$317$	8721.17	1.54520	0.772602	−	0.634890i	$-0.218955\pi$
$317$	8721.17	1.54520	0.772602	+	0.634890i	$0.218955\pi$
$318$	354.504	0.0625145
$319$	−1068.15	−0.187475
$320$	0	0
$321$	−421.518	−0.0732923
$322$	−1868.75	−0.323420
$323$	5921.85	1.02013
$324$	−609.249	−0.104467
$325$	0	0
$326$	−564.961	−0.0959825
$327$	−2551.59	−0.431508
$328$	−126.756	−0.0213383
$329$	−4100.95	−0.687212
$330$	0	0
$331$	−6093.02	−1.01179	−0.505895	−	0.862595i	$-0.668838\pi$
$331$	−6093.02	−1.01179	−0.505895	+	0.862595i	$0.668838\pi$
$332$	−6659.18	−1.10081
$333$	2338.20	0.384782
$334$	943.682	0.154599
$335$	0	0
$336$	2305.90	0.374397
$337$	6306.92	1.01947	0.509733	−	0.860333i	$-0.329744\pi$
$337$	6306.92	1.01947	0.509733	+	0.860333i	$0.329744\pi$
$338$	−1123.85	−0.180855
$339$	−5553.70	−0.889781
$340$	0	0
$341$	−3866.37	−0.614004
$342$	−586.144	−0.0926756
$343$	−6902.13	−1.08653
$344$	−655.419	−0.102726
$345$	0	0
$346$	−298.009	−0.0463037
$347$	−11395.7	−1.76298	−0.881491	−	0.472201i	$-0.843460\pi$
$347$	−11395.7	−1.76298	−0.881491	+	0.472201i	$0.843460\pi$
$348$	654.379	0.100800
$349$	8835.08	1.35510	0.677552	−	0.735475i	$-0.263041\pi$
$349$	8835.08	1.35510	0.677552	+	0.735475i	$0.263041\pi$
$350$	0	0
$351$	645.837	0.0982115
$352$	4507.19	0.682483
$353$	3390.75	0.511251	0.255626	−	0.966776i	$-0.417719\pi$
$353$	3390.75	0.511251	0.255626	+	0.966776i	$0.417719\pi$
$354$	−1321.75	−0.198446
$355$	0	0
$356$	3367.06	0.501274
$357$	−2749.37	−0.407597
$358$	−2208.24	−0.326003
$359$	−5160.27	−0.758632	−0.379316	−	0.925267i	$-0.623841\pi$
$359$	−5160.27	−0.758632	−0.379316	+	0.925267i	$0.623841\pi$
$360$	0	0
$361$	2007.08	0.292620
$362$	−429.635	−0.0623788
$363$	76.9222	0.0111222
$364$	−2621.74	−0.377518
$365$	0	0
$366$	−787.675	−0.112493
$367$	−1110.17	−0.157902	−0.0789512	−	0.996878i	$-0.525157\pi$
$367$	−1110.17	−0.157902	−0.0789512	+	0.996878i	$0.525157\pi$
$368$	−9779.88	−1.38536
$369$	106.262	0.0149913
$370$	0	0
$371$	2489.57	0.348388
$372$	2368.66	0.330132
$373$	−5801.22	−0.805297	−0.402648	−	0.915355i	$-0.631910\pi$
$373$	−5801.22	−0.805297	−0.402648	+	0.915355i	$0.631910\pi$
$374$	−1602.21	−0.221520
$375$	0	0
$376$	3021.32	0.414396
$377$	−693.677	−0.0947644
$378$	272.132	0.0370290
$379$	−12664.5	−1.71644	−0.858219	−	0.513284i	$-0.828429\pi$
$379$	−12664.5	−1.71644	−0.858219	+	0.513284i	$0.828429\pi$
$380$	0	0
$381$	−2709.37	−0.364318
$382$	2035.83	0.272675
$383$	13408.3	1.78885	0.894427	−	0.447215i	$-0.147584\pi$
$383$	13408.3	1.78885	0.894427	+	0.447215i	$0.147584\pi$
$384$	−3636.85	−0.483313
$385$	0	0
$386$	−2254.69	−0.297307
$387$	549.451	0.0721710
$388$	−4203.78	−0.550038
$389$	−4676.56	−0.609540	−0.304770	−	0.952426i	$-0.598580\pi$
$389$	−4676.56	−0.609540	−0.304770	+	0.952426i	$0.598580\pi$
$390$	0	0
$391$	11660.7	1.50820
$392$	1402.69	0.180731
$393$	−2057.87	−0.264137
$394$	−1640.87	−0.209812
$395$	0	0
$396$	−2493.36	−0.316404
$397$	5913.71	0.747608	0.373804	−	0.927508i	$-0.378053\pi$
$397$	5913.71	0.747608	0.373804	+	0.927508i	$0.378053\pi$
$398$	−1264.89	−0.159304
$399$	−4116.30	−0.516473
$400$	0	0
$401$	7756.80	0.965975	0.482987	−	0.875627i	$-0.339552\pi$
$401$	7756.80	0.965975	0.482987	+	0.875627i	$0.339552\pi$
$402$	1293.41	0.160472
$403$	−2510.90	−0.310365
$404$	8565.46	1.05482
$405$	0	0
$406$	−292.290	−0.0357294
$407$	9569.09	1.16541
$408$	2025.56	0.245785
$409$	7674.37	0.927808	0.463904	−	0.885886i	$-0.346448\pi$
$409$	7674.37	0.927808	0.463904	+	0.885886i	$0.346448\pi$
$410$	0	0
$411$	3458.84	0.415114
$412$	7584.13	0.906901
$413$	−9282.19	−1.10592
$414$	−1154.18	−0.137016
$415$	0	0
$416$	2927.07	0.344979
$417$	−1315.39	−0.154473
$418$	−2398.80	−0.280692
$419$	11316.2	1.31940	0.659702	−	0.751527i	$-0.270682\pi$
$419$	11316.2	1.31940	0.659702	+	0.751527i	$0.270682\pi$
$420$	0	0
$421$	−15960.8	−1.84770	−0.923851	−	0.382753i	$-0.874976\pi$
$421$	−15960.8	−1.84770	−0.923851	+	0.382753i	$0.874976\pi$
$422$	2921.21	0.336973
$423$	−2532.84	−0.291136
$424$	−1834.16	−0.210081
$425$	0	0
$426$	620.213	0.0705385
$427$	−5531.59	−0.626914
$428$	1056.83	0.119355
$429$	2643.10	0.297459
$430$	0	0
$431$	9181.83	1.02616	0.513078	−	0.858342i	$-0.328505\pi$
$431$	9181.83	1.02616	0.513078	+	0.858342i	$0.328505\pi$
$432$	1424.18	0.158613
$433$	−312.822	−0.0347188	−0.0173594	−	0.999849i	$-0.505526\pi$
$433$	−312.822	−0.0347188	−0.0173594	+	0.999849i	$0.505526\pi$
$434$	−1058.00	−0.117018
$435$	0	0
$436$	6397.34	0.702700
$437$	17458.2	1.91107
$438$	1087.56	0.118642
$439$	−8018.14	−0.871720	−0.435860	−	0.900014i	$-0.643556\pi$
$439$	−8018.14	−0.871720	−0.435860	+	0.900014i	$0.643556\pi$
$440$	0	0
$441$	−1175.90	−0.126974
$442$	−1040.51	−0.111973
$443$	−553.480	−0.0593603	−0.0296802	−	0.999559i	$-0.509449\pi$
$443$	−553.480	−0.0593603	−0.0296802	+	0.999559i	$0.509449\pi$
$444$	−5862.32	−0.626607
$445$	0	0
$446$	226.638	0.0240619
$447$	−1216.71	−0.128744
$448$	−4915.71	−0.518406
$449$	−12055.2	−1.26708	−0.633540	−	0.773710i	$-0.718399\pi$
$449$	−12055.2	−1.26708	−0.633540	+	0.773710i	$0.718399\pi$
$450$	0	0
$451$	434.880	0.0454051
$452$	13924.2	1.44899
$453$	−236.868	−0.0245674
$454$	−2532.96	−0.261845
$455$	0	0
$456$	3032.63	0.311439
$457$	268.261	0.0274589	0.0137295	−	0.999906i	$-0.495630\pi$
$457$	268.261	0.0274589	0.0137295	+	0.999906i	$0.495630\pi$
$458$	−4539.11	−0.463097
$459$	−1698.07	−0.172678
$460$	0	0
$461$	3621.49	0.365877	0.182939	−	0.983124i	$-0.441439\pi$
$461$	3621.49	0.365877	0.182939	+	0.983124i	$0.441439\pi$
$462$	1113.70	0.112152
$463$	7751.51	0.778063	0.389031	−	0.921225i	$-0.372810\pi$
$463$	7751.51	0.778063	0.389031	+	0.921225i	$0.372810\pi$
$464$	−1529.67	−0.153046
$465$	0	0
$466$	−1722.44	−0.171225
$467$	−8838.90	−0.875836	−0.437918	−	0.899015i	$-0.644284\pi$
$467$	−8838.90	−0.875836	−0.437918	+	0.899015i	$0.644284\pi$
$468$	−1619.24	−0.159935
$469$	9083.22	0.894295
$470$	0	0
$471$	−944.607	−0.0924102
$472$	6838.54	0.666884
$473$	2248.63	0.218588
$474$	1168.77	0.113256
$475$	0	0
$476$	6893.21	0.663760
$477$	1537.61	0.147594
$478$	−2479.20	−0.237230
$479$	−9904.02	−0.944731	−0.472366	−	0.881403i	$-0.656600\pi$
$479$	−9904.02	−0.944731	−0.472366	+	0.881403i	$0.656600\pi$
$480$	0	0
$481$	6214.38	0.589088
$482$	−4526.22	−0.427726
$483$	−8105.41	−0.763580
$484$	−192.859	−0.0181123
$485$	0	0
$486$	168.075	0.0156873
$487$	−799.412	−0.0743836	−0.0371918	−	0.999308i	$-0.511841\pi$
$487$	−799.412	−0.0743836	−0.0371918	+	0.999308i	$0.511841\pi$
$488$	4075.33	0.378036
$489$	−2450.44	−0.226611
$490$	0	0
$491$	−10013.3	−0.920351	−0.460176	−	0.887828i	$-0.652214\pi$
$491$	−10013.3	−0.920351	−0.460176	+	0.887828i	$0.652214\pi$
$492$	−266.421	−0.0244130
$493$	1823.85	0.166617
$494$	−1557.84	−0.141883
$495$	0	0
$496$	−5536.94	−0.501242
$497$	4355.55	0.393105
$498$	1837.08	0.165304
$499$	2556.35	0.229335	0.114667	−	0.993404i	$-0.463420\pi$
$499$	2556.35	0.229335	0.114667	+	0.993404i	$0.463420\pi$
$500$	0	0
$501$	4093.08	0.365001
$502$	365.269	0.0324756
$503$	−11520.2	−1.02120	−0.510598	−	0.859820i	$-0.670576\pi$
$503$	−11520.2	−1.02120	−0.510598	+	0.859820i	$0.670576\pi$
$504$	−1407.98	−0.124437
$505$	0	0
$506$	−4723.48	−0.414989
$507$	−4874.51	−0.426992
$508$	6792.94	0.593283
$509$	8055.10	0.701446	0.350723	−	0.936479i	$-0.385936\pi$
$509$	8055.10	0.701446	0.350723	+	0.936479i	$0.385936\pi$
$510$	0	0
$511$	7637.55	0.661184
$512$	10984.9	0.948181
$513$	−2542.32	−0.218803
$514$	−3472.70	−0.298004
$515$	0	0
$516$	−1377.58	−0.117529
$517$	−10365.7	−0.881782
$518$	2618.51	0.222106
$519$	−1292.57	−0.109321
$520$	0	0
$521$	22383.5	1.88222	0.941111	−	0.338097i	$-0.109783\pi$
$521$	22383.5	1.88222	0.941111	+	0.338097i	$0.109783\pi$
$522$	−180.525	−0.0151367
$523$	−5260.45	−0.439816	−0.219908	−	0.975521i	$-0.570576\pi$
$523$	−5260.45	−0.439816	−0.219908	+	0.975521i	$0.570576\pi$
$524$	5159.50	0.430141
$525$	0	0
$526$	292.030	0.0242074
$527$	6601.79	0.545690
$528$	5828.45	0.480399
$529$	22210.0	1.82543
$530$	0	0
$531$	−5732.88	−0.468523
$532$	10320.4	0.841063
$533$	282.421	0.0229512
$534$	−928.876	−0.0752742
$535$	0	0
$536$	−6691.95	−0.539269
$537$	−9577.91	−0.769678
$538$	−1229.22	−0.0985046
$539$	−4812.40	−0.384573
$540$	0	0
$541$	−15052.8	−1.19625	−0.598123	−	0.801404i	$-0.704087\pi$
$541$	−15052.8	−1.19625	−0.598123	+	0.801404i	$0.704087\pi$
$542$	−1013.64	−0.0803315
$543$	−1863.48	−0.147274
$544$	−7695.99	−0.606550
$545$	0	0
$546$	723.264	0.0566902
$547$	−13743.0	−1.07424	−0.537121	−	0.843505i	$-0.680488\pi$
$547$	−13743.0	−1.07424	−0.537121	+	0.843505i	$0.680488\pi$
$548$	−8672.00	−0.676003
$549$	−3416.43	−0.265591
$550$	0	0
$551$	2730.64	0.211123
$552$	5971.56	0.460446
$553$	8207.90	0.631167
$554$	4206.36	0.322584
$555$	0	0
$556$	3297.96	0.251555
$557$	−23296.7	−1.77219	−0.886097	−	0.463500i	$-0.846593\pi$
$557$	−23296.7	−1.77219	−0.886097	+	0.463500i	$0.846593\pi$
$558$	−653.446	−0.0495745
$559$	1460.31	0.110491
$560$	0	0
$561$	−6949.37	−0.522999
$562$	−826.196	−0.0620124
$563$	−6823.79	−0.510814	−0.255407	−	0.966834i	$-0.582209\pi$
$563$	−6823.79	−0.510814	−0.255407	+	0.966834i	$0.582209\pi$
$564$	6350.33	0.474108
$565$	0	0
$566$	−2647.37	−0.196603
$567$	1180.33	0.0874240
$568$	−3208.90	−0.237046
$569$	17974.8	1.32433	0.662166	−	0.749358i	$-0.269638\pi$
$569$	17974.8	1.32433	0.662166	+	0.749358i	$0.269638\pi$
$570$	0	0
$571$	22031.6	1.61470	0.807350	−	0.590072i	$-0.200901\pi$
$571$	22031.6	1.61470	0.807350	+	0.590072i	$0.200901\pi$
$572$	−6626.77	−0.484404
$573$	8830.11	0.643775
$574$	119.002	0.00865338
$575$	0	0
$576$	−3036.05	−0.219622
$577$	−14520.1	−1.04762	−0.523811	−	0.851834i	$-0.675490\pi$
$577$	−14520.1	−1.04762	−0.523811	+	0.851834i	$0.675490\pi$
$578$	−662.386	−0.0476672
$579$	−9779.38	−0.701929
$580$	0	0
$581$	12901.2	0.921227
$582$	1159.71	0.0825968
$583$	6292.69	0.447027
$584$	−5626.87	−0.398701
$585$	0	0
$586$	4717.85	0.332581
$587$	998.184	0.0701865	0.0350932	−	0.999384i	$-0.488827\pi$
$587$	998.184	0.0701865	0.0350932	+	0.999384i	$0.488827\pi$
$588$	2948.22	0.206773
$589$	9884.08	0.691454
$590$	0	0
$591$	−7117.04	−0.495357
$592$	13703.7	0.951383
$593$	−3431.22	−0.237611	−0.118806	−	0.992918i	$-0.537907\pi$
$593$	−3431.22	−0.237611	−0.118806	+	0.992918i	$0.537907\pi$
$594$	687.848	0.0475130
$595$	0	0
$596$	3050.54	0.209656
$597$	−5486.26	−0.376110
$598$	−3067.53	−0.209767
$599$	−21059.5	−1.43651	−0.718255	−	0.695780i	$-0.755059\pi$
$599$	−21059.5	−1.43651	−0.718255	+	0.695780i	$0.755059\pi$
$600$	0	0
$601$	8534.75	0.579267	0.289634	−	0.957138i	$-0.406467\pi$
$601$	8534.75	0.579267	0.289634	+	0.957138i	$0.406467\pi$
$602$	615.323	0.0416589
$603$	5609.99	0.378867
$604$	593.874	0.0400073
$605$	0	0
$606$	−2362.97	−0.158398
$607$	17101.7	1.14355	0.571776	−	0.820409i	$-0.306254\pi$
$607$	17101.7	1.14355	0.571776	+	0.820409i	$0.306254\pi$
$608$	−11522.3	−0.768571
$609$	−1267.77	−0.0843555
$610$	0	0
$611$	−6731.69	−0.445720
$612$	4257.40	0.281201
$613$	1048.59	0.0690897	0.0345449	−	0.999403i	$-0.489002\pi$
$613$	1048.59	0.0690897	0.0345449	+	0.999403i	$0.489002\pi$
$614$	5084.26	0.334176
$615$	0	0
$616$	−5762.16	−0.376890
$617$	−2212.12	−0.144338	−0.0721689	−	0.997392i	$-0.522992\pi$
$617$	−2212.12	−0.144338	−0.0721689	+	0.997392i	$0.522992\pi$
$618$	−2092.25	−0.136185
$619$	25602.7	1.66246	0.831228	−	0.555932i	$-0.187638\pi$
$619$	25602.7	1.66246	0.831228	+	0.555932i	$0.187638\pi$
$620$	0	0
$621$	−5006.08	−0.323489
$622$	2346.92	0.151291
$623$	−6523.19	−0.419496
$624$	3785.13	0.242831
$625$	0	0
$626$	7423.16	0.473944
$627$	−10404.5	−0.662702
$628$	2368.32	0.150488
$629$	−16339.2	−1.03575
$630$	0	0
$631$	−25127.0	−1.58524	−0.792622	−	0.609713i	$-0.791285\pi$
$631$	−25127.0	−1.58524	−0.792622	+	0.609713i	$0.791285\pi$
$632$	−6047.07	−0.380600
$633$	12670.3	0.795578
$634$	6032.14	0.377866
$635$	0	0
$636$	−3855.10	−0.240353
$637$	−3125.28	−0.194392
$638$	−738.800	−0.0458454
$639$	2690.08	0.166538
$640$	0	0
$641$	2231.80	0.137521	0.0687605	−	0.997633i	$-0.478096\pi$
$641$	2231.80	0.137521	0.0687605	+	0.997633i	$0.478096\pi$
$642$	−291.550	−0.0179230
$643$	15105.0	0.926415	0.463207	−	0.886250i	$-0.346699\pi$
$643$	15105.0	0.926415	0.463207	+	0.886250i	$0.346699\pi$
$644$	20321.9	1.24347
$645$	0	0
$646$	4095.94	0.249462
$647$	−21013.9	−1.27688	−0.638441	−	0.769671i	$-0.720420\pi$
$647$	−21013.9	−1.27688	−0.638441	+	0.769671i	$0.720420\pi$
$648$	−869.596	−0.0527176
$649$	−23461.9	−1.41904
$650$	0	0
$651$	−4588.94	−0.276274
$652$	6143.73	0.369029
$653$	−25780.5	−1.54498	−0.772489	−	0.635028i	$-0.780989\pi$
$653$	−25780.5	−1.54498	−0.772489	+	0.635028i	$0.780989\pi$
$654$	−1764.85	−0.105521
$655$	0	0
$656$	622.783	0.0370665
$657$	4717.11	0.280110
$658$	−2836.49	−0.168051
$659$	14065.8	0.831448	0.415724	−	0.909491i	$-0.363528\pi$
$659$	14065.8	0.831448	0.415724	+	0.909491i	$0.363528\pi$
$660$	0	0
$661$	−13363.3	−0.786342	−0.393171	−	0.919465i	$-0.628622\pi$
$661$	−13363.3	−0.786342	−0.393171	+	0.919465i	$0.628622\pi$
$662$	−4214.33	−0.247424
$663$	−4513.07	−0.264364
$664$	−9504.81	−0.555510
$665$	0	0
$666$	1617.25	0.0940948
$667$	5376.90	0.312135
$668$	−10262.2	−0.594395
$669$	983.010	0.0568092
$670$	0	0
$671$	−13981.8	−0.804412
$672$	5349.52	0.307087
$673$	17727.5	1.01537	0.507686	−	0.861542i	$-0.330501\pi$
$673$	17727.5	1.01537	0.507686	+	0.861542i	$0.330501\pi$
$674$	4362.28	0.249301
$675$	0	0
$676$	12221.4	0.695345
$677$	−15776.1	−0.895605	−0.447803	−	0.894132i	$-0.647793\pi$
$677$	−15776.1	−0.895605	−0.447803	+	0.894132i	$0.647793\pi$
$678$	−3841.31	−0.217588
$679$	8144.23	0.460305
$680$	0	0
$681$	−10986.4	−0.618206
$682$	−2674.23	−0.150149
$683$	−14482.3	−0.811350	−0.405675	−	0.914018i	$-0.632963\pi$
$683$	−14482.3	−0.811350	−0.405675	+	0.914018i	$0.632963\pi$
$684$	6374.09	0.356315
$685$	0	0
$686$	−4773.97	−0.265701
$687$	−19687.7	−1.09335
$688$	3220.22	0.178445
$689$	4086.61	0.225961
$690$	0	0
$691$	19721.5	1.08573	0.542867	−	0.839819i	$-0.317339\pi$
$691$	19721.5	1.08573	0.542867	+	0.839819i	$0.317339\pi$
$692$	3240.74	0.178026
$693$	4830.53	0.264786
$694$	−7882.04	−0.431121
$695$	0	0
$696$	934.011	0.0508672
$697$	−742.555	−0.0403533
$698$	6110.92	0.331378
$699$	−7470.85	−0.404254
$700$	0	0
$701$	2004.45	0.107998	0.0539992	−	0.998541i	$-0.482803\pi$
$701$	2004.45	0.107998	0.0539992	+	0.998541i	$0.482803\pi$
$702$	446.704	0.0240167
$703$	−24462.7	−1.31241
$704$	−12425.1	−0.665181
$705$	0	0
$706$	2345.27	0.125022
$707$	−16594.4	−0.882738
$708$	14373.5	0.762978
$709$	10037.8	0.531705	0.265853	−	0.964014i	$-0.414347\pi$
$709$	10037.8	0.531705	0.265853	+	0.964014i	$0.414347\pi$
$710$	0	0
$711$	5069.38	0.267393
$712$	4805.88	0.252961
$713$	19462.8	1.02228
$714$	−1901.64	−0.0996740
$715$	0	0
$716$	24013.7	1.25340
$717$	−10753.2	−0.560091
$718$	−3569.18	−0.185516
$719$	10367.8	0.537768	0.268884	−	0.963173i	$-0.413345\pi$
$719$	10367.8	0.537768	0.268884	+	0.963173i	$0.413345\pi$
$720$	0	0
$721$	−14693.2	−0.758949
$722$	1388.23	0.0715576
$723$	−19631.8	−1.00984
$724$	4672.11	0.239831
$725$	0	0
$726$	53.2044	0.00271984
$727$	32648.0	1.66554	0.832770	−	0.553619i	$-0.186754\pi$
$727$	32648.0	1.66554	0.832770	+	0.553619i	$0.186754\pi$
$728$	−3742.07	−0.190509
$729$	729.000	0.0370370
$730$	0	0
$731$	−3839.53	−0.194268
$732$	8565.66	0.432508
$733$	−21036.1	−1.06001	−0.530004	−	0.847995i	$-0.677809\pi$
$733$	−21036.1	−1.06001	−0.530004	+	0.847995i	$0.677809\pi$
$734$	−767.863	−0.0386136
$735$	0	0
$736$	−22688.6	−1.13629
$737$	22959.0	1.14750
$738$	73.4981	0.00366599
$739$	24591.3	1.22410	0.612048	−	0.790820i	$-0.290346\pi$
$739$	24591.3	1.22410	0.612048	+	0.790820i	$0.290346\pi$
$740$	0	0
$741$	−6756.88	−0.334980
$742$	1721.95	0.0851950
$743$	−3352.77	−0.165547	−0.0827734	−	0.996568i	$-0.526378\pi$
$743$	−3352.77	−0.165547	−0.0827734	+	0.996568i	$0.526378\pi$
$744$	3380.84	0.166596
$745$	0	0
$746$	−4012.50	−0.196928
$747$	7968.08	0.390277
$748$	17423.4	0.851690
$749$	−2047.46	−0.0998831
$750$	0	0
$751$	10898.0	0.529525	0.264762	−	0.964314i	$-0.414706\pi$
$751$	10898.0	0.529525	0.264762	+	0.964314i	$0.414706\pi$
$752$	−14844.5	−0.719843
$753$	1584.30	0.0766734
$754$	−479.793	−0.0231738
$755$	0	0
$756$	−2959.33	−0.142368
$757$	34910.6	1.67615	0.838075	−	0.545555i	$-0.183681\pi$
$757$	34910.6	1.67615	0.838075	+	0.545555i	$0.183681\pi$
$758$	−8759.58	−0.419739
$759$	−20487.4	−0.979771
$760$	0	0
$761$	−17093.2	−0.814229	−0.407115	−	0.913377i	$-0.633465\pi$
$761$	−17093.2	−0.814229	−0.407115	+	0.913377i	$0.633465\pi$
$762$	−1873.98	−0.0890907
$763$	−12393.9	−0.588062
$764$	−22138.8	−1.04837
$765$	0	0
$766$	9274.05	0.437448
$767$	−15236.7	−0.717293
$768$	5580.65	0.262206
$769$	29371.4	1.37732	0.688660	−	0.725084i	$-0.258199\pi$
$769$	29371.4	1.37732	0.688660	+	0.725084i	$0.258199\pi$
$770$	0	0
$771$	−15062.3	−0.703575
$772$	24518.9	1.14307
$773$	25646.0	1.19330	0.596651	−	0.802501i	$-0.296498\pi$
$773$	25646.0	1.19330	0.596651	+	0.802501i	$0.296498\pi$
$774$	380.036	0.0176487
$775$	0	0
$776$	−6000.16	−0.277568
$777$	11357.4	0.524382
$778$	−3234.62	−0.149057
$779$	−1111.74	−0.0511325
$780$	0	0
$781$	11009.2	0.504405
$782$	8065.32	0.368817
$783$	−783.000	−0.0357371
$784$	−6891.74	−0.313946
$785$	0	0
$786$	−1423.36	−0.0645923
$787$	−18188.7	−0.823834	−0.411917	−	0.911221i	$-0.635141\pi$
$787$	−18188.7	−0.823834	−0.411917	+	0.911221i	$0.635141\pi$
$788$	17843.8	0.806676
$789$	1266.64	0.0571527
$790$	0	0
$791$	−26976.2	−1.21260
$792$	−3558.83	−0.159669
$793$	−9080.07	−0.406611
$794$	4090.31	0.182821
$795$	0	0
$796$	13755.1	0.612485
$797$	−12068.7	−0.536379	−0.268190	−	0.963366i	$-0.586425\pi$
$797$	−12068.7	−0.536379	−0.268190	+	0.963366i	$0.586425\pi$
$798$	−2847.11	−0.126299
$799$	17699.3	0.783675
$800$	0	0
$801$	−4028.87	−0.177719
$802$	5365.11	0.236220
$803$	19304.8	0.848385
$804$	−14065.4	−0.616974
$805$	0	0
$806$	−1736.71	−0.0758968
$807$	−5331.57	−0.232565
$808$	12225.7	0.532300
$809$	−18274.6	−0.794191	−0.397096	−	0.917777i	$-0.629982\pi$
$809$	−18274.6	−0.794191	−0.397096	+	0.917777i	$0.629982\pi$
$810$	0	0
$811$	−30620.5	−1.32581	−0.662903	−	0.748705i	$-0.730676\pi$
$811$	−30620.5	−1.32581	−0.662903	+	0.748705i	$0.730676\pi$
$812$	3178.54	0.137371
$813$	−4396.53	−0.189659
$814$	6618.61	0.284990
$815$	0	0
$816$	−9952.05	−0.426950
$817$	−5748.47	−0.246161
$818$	5308.10	0.226887
$819$	3137.05	0.133843
$820$	0	0
$821$	−10076.0	−0.428326	−0.214163	−	0.976798i	$-0.568702\pi$
$821$	−10076.0	−0.428326	−0.214163	+	0.976798i	$0.568702\pi$
$822$	2392.36	0.101512
$823$	−37593.9	−1.59227	−0.796137	−	0.605116i	$-0.793127\pi$
$823$	−37593.9	−1.59227	−0.796137	+	0.605116i	$0.793127\pi$
$824$	10825.0	0.457654
$825$	0	0
$826$	−6420.18	−0.270444
$827$	−34830.2	−1.46453	−0.732264	−	0.681021i	$-0.761536\pi$
$827$	−34830.2	−1.46453	−0.732264	+	0.681021i	$0.761536\pi$
$828$	12551.2	0.526794
$829$	−19925.3	−0.834784	−0.417392	−	0.908727i	$-0.637056\pi$
$829$	−19925.3	−0.834784	−0.417392	+	0.908727i	$0.637056\pi$
$830$	0	0
$831$	18244.5	0.761606
$832$	−8069.12	−0.336233
$833$	8217.14	0.341785
$834$	−909.814	−0.0377749
$835$	0	0
$836$	26086.1	1.07919
$837$	−2834.22	−0.117043
$838$	7827.00	0.322648
$839$	26090.0	1.07357	0.536786	−	0.843718i	$-0.319638\pi$
$839$	26090.0	1.07357	0.536786	+	0.843718i	$0.319638\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	−11039.5	−0.451838
$843$	−3583.51	−0.146409
$844$	−31767.1	−1.29558
$845$	0	0
$846$	−1751.88	−0.0711947
$847$	373.637	0.0151574
$848$	9011.63	0.364930
$849$	−11482.6	−0.464171
$850$	0	0
$851$	−48169.5	−1.94034
$852$	−6744.57	−0.271203
$853$	37626.4	1.51032	0.755160	−	0.655540i	$-0.227559\pi$
$853$	37626.4	1.51032	0.755160	+	0.655540i	$0.227559\pi$
$854$	−3826.01	−0.153306
$855$	0	0
$856$	1508.44	0.0602306
$857$	−25833.1	−1.02969	−0.514844	−	0.857284i	$-0.672150\pi$
$857$	−25833.1	−1.02969	−0.514844	+	0.857284i	$0.672150\pi$
$858$	1828.14	0.0727409
$859$	49111.2	1.95070	0.975351	−	0.220661i	$-0.0708215\pi$
$859$	49111.2	1.95070	0.975351	+	0.220661i	$0.0708215\pi$
$860$	0	0
$861$	516.153	0.0204303
$862$	6350.76	0.250937
$863$	−14549.3	−0.573886	−0.286943	−	0.957948i	$-0.592639\pi$
$863$	−14549.3	−0.573886	−0.286943	+	0.957948i	$0.592639\pi$
$864$	3303.98	0.130097
$865$	0	0
$866$	−216.368	−0.00849017
$867$	−2873.00	−0.112540
$868$	11505.4	0.449905
$869$	20746.5	0.809869
$870$	0	0
$871$	14910.1	0.580032
$872$	9131.08	0.354607
$873$	5030.05	0.195007
$874$	12075.2	0.467335
$875$	0	0
$876$	−11826.7	−0.456151
$877$	−501.781	−0.0193203	−0.00966017	−	0.999953i	$-0.503075\pi$
$877$	−501.781	−0.0193203	−0.00966017	+	0.999953i	$0.503075\pi$
$878$	−5545.88	−0.213171
$879$	20463.0	0.785210
$880$	0	0
$881$	1569.47	0.0600189	0.0300095	−	0.999550i	$-0.490446\pi$
$881$	1569.47	0.0600189	0.0300095	+	0.999550i	$0.490446\pi$
$882$	−813.332	−0.0310502
$883$	26159.0	0.996967	0.498483	−	0.866899i	$-0.333890\pi$
$883$	26159.0	0.996967	0.498483	+	0.866899i	$0.333890\pi$
$884$	11315.2	0.430509
$885$	0	0
$886$	−382.823	−0.0145160
$887$	29723.0	1.12514	0.562571	−	0.826749i	$-0.309812\pi$
$887$	29723.0	1.12514	0.562571	+	0.826749i	$0.309812\pi$
$888$	−8367.43	−0.316208
$889$	−13160.4	−0.496495
$890$	0	0
$891$	2983.44	0.112176
$892$	−2464.60	−0.0925123
$893$	26499.1	0.993009
$894$	−841.557	−0.0314831
$895$	0	0
$896$	−17665.4	−0.658661
$897$	−13305.0	−0.495251
$898$	−8338.16	−0.309853
$899$	3044.17	0.112935
$900$	0	0
$901$	−10744.7	−0.397290
$902$	300.792	0.0111034
$903$	2668.87	0.0983549
$904$	19874.4	0.731209
$905$	0	0
$906$	−163.833	−0.00600772
$907$	9772.21	0.357752	0.178876	−	0.983872i	$-0.442754\pi$
$907$	9772.21	0.357752	0.178876	+	0.983872i	$0.442754\pi$
$908$	27545.0	1.00673
$909$	−10249.0	−0.373971
$910$	0	0
$911$	−49823.7	−1.81200	−0.906001	−	0.423275i	$-0.860880\pi$
$911$	−49823.7	−1.81200	−0.906001	+	0.423275i	$0.860880\pi$
$912$	−14900.0	−0.540997
$913$	32609.4	1.18205
$914$	185.547	0.00671483
$915$	0	0
$916$	49361.0	1.78050
$917$	−9995.80	−0.359968
$918$	−1174.50	−0.0422267
$919$	28235.9	1.01351	0.506756	−	0.862090i	$-0.330845\pi$
$919$	28235.9	1.01351	0.506756	+	0.862090i	$0.330845\pi$
$920$	0	0
$921$	22052.2	0.788974
$922$	2504.86	0.0894720
$923$	7149.61	0.254965
$924$	−12111.1	−0.431197
$925$	0	0
$926$	5361.45	0.190268
$927$	−9074.82	−0.321528
$928$	−3548.72	−0.125531
$929$	−22846.1	−0.806842	−0.403421	−	0.915014i	$-0.632179\pi$
$929$	−22846.1	−0.806842	−0.403421	+	0.915014i	$0.632179\pi$
$930$	0	0
$931$	12302.5	0.433082
$932$	18730.9	0.658317
$933$	10179.4	0.357191
$934$	−6113.56	−0.214178
$935$	0	0
$936$	−2311.18	−0.0807088
$937$	45416.9	1.58346	0.791732	−	0.610868i	$-0.209180\pi$
$937$	45416.9	1.58346	0.791732	+	0.610868i	$0.209180\pi$
$938$	6282.55	0.218692
$939$	32196.9	1.11896
$940$	0	0
$941$	−39333.3	−1.36262	−0.681311	−	0.731994i	$-0.738590\pi$
$941$	−39333.3	−1.36262	−0.681311	+	0.731994i	$0.738590\pi$
$942$	−653.352	−0.0225981
$943$	−2189.13	−0.0755968
$944$	−33599.3	−1.15844
$945$	0	0
$946$	1555.30	0.0534538
$947$	−31274.8	−1.07317	−0.536587	−	0.843845i	$-0.680287\pi$
$947$	−31274.8	−1.07317	−0.536587	+	0.843845i	$0.680287\pi$
$948$	−12709.9	−0.435443
$949$	12537.0	0.428839
$950$	0	0
$951$	26163.5	0.892124
$952$	9838.85	0.334957
$953$	34174.1	1.16160	0.580802	−	0.814045i	$-0.302739\pi$
$953$	34174.1	1.16160	0.580802	+	0.814045i	$0.302739\pi$
$954$	1063.51	0.0360928
$955$	0	0
$956$	26960.4	0.912093
$957$	−3204.44	−0.108239
$958$	−6850.27	−0.231025
$959$	16800.8	0.565720
$960$	0	0
$961$	−18772.0	−0.630124
$962$	4298.27	0.144056
$963$	−1264.55	−0.0423153
$964$	49220.9	1.64450
$965$	0	0
$966$	−5606.24	−0.186726
$967$	−37894.7	−1.26020	−0.630098	−	0.776515i	$-0.716985\pi$
$967$	−37894.7	−1.26020	−0.630098	+	0.776515i	$0.716985\pi$
$968$	−275.273	−0.00914008
$969$	17765.5	0.588970
$970$	0	0
$971$	−39869.1	−1.31767	−0.658837	−	0.752286i	$-0.728951\pi$
$971$	−39869.1	−1.31767	−0.658837	+	0.752286i	$0.728951\pi$
$972$	−1827.75	−0.0603138
$973$	−6389.33	−0.210516
$974$	−552.926	−0.0181898
$975$	0	0
$976$	−20023.0	−0.656681
$977$	34257.3	1.12179	0.560894	−	0.827887i	$-0.310457\pi$
$977$	34257.3	1.12179	0.560894	+	0.827887i	$0.310457\pi$
$978$	−1694.88	−0.0554155
$979$	−16488.2	−0.538268
$980$	0	0
$981$	−7654.77	−0.249131
$982$	−6925.83	−0.225063
$983$	−8665.65	−0.281171	−0.140586	−	0.990069i	$-0.544899\pi$
$983$	−8665.65	−0.281171	−0.140586	+	0.990069i	$0.544899\pi$
$984$	−380.269	−0.0123197
$985$	0	0
$986$	1261.50	0.0407446
$987$	−12302.9	−0.396762
$988$	16940.9	0.545507
$989$	−11319.3	−0.363937
$990$	0	0
$991$	22129.9	0.709362	0.354681	−	0.934987i	$-0.384589\pi$
$991$	22129.9	0.709362	0.354681	+	0.934987i	$0.384589\pi$
$992$	−12845.3	−0.411127
$993$	−18279.0	−0.584157
$994$	3012.59	0.0961302
$995$	0	0
$996$	−19977.6	−0.635555
$997$	14159.7	0.449791	0.224895	−	0.974383i	$-0.427796\pi$
$997$	14159.7	0.449791	0.224895	+	0.974383i	$0.427796\pi$
$998$	1768.14	0.0560817
$999$	7014.59	0.222154

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.4.a.l.1.4		6
5.4	even	2		435.4.a.g.1.3	✓	6
15.14	odd	2		1305.4.a.i.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.4.a.g.1.3	✓	6	5.4	even	2
1305.4.a.i.1.4		6	15.14	odd	2
2175.4.a.l.1.4		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+0.691666 q^{2} +3.00000 q^{3} -7.52160 q^{4} +2.07500 q^{6} +14.5720 q^{7} -10.7358 q^{8} +9.00000 q^{9} +36.8326 q^{11} -22.5648 q^{12} +23.9199 q^{13} +10.0790 q^{14} +52.7472 q^{16} -62.8914 q^{17} +6.22499 q^{18} -94.1599 q^{19} +43.7161 q^{21} +25.4759 q^{22} -185.410 q^{23} -32.2073 q^{24} +16.5446 q^{26} +27.0000 q^{27} -109.605 q^{28} -29.0000 q^{29} -104.971 q^{31} +122.370 q^{32} +110.498 q^{33} -43.4999 q^{34} -67.6944 q^{36} +259.799 q^{37} -65.1272 q^{38} +71.7597 q^{39} +11.8069 q^{41} +30.2369 q^{42} +61.0501 q^{43} -277.040 q^{44} -128.242 q^{46} -281.426 q^{47} +158.242 q^{48} -130.656 q^{49} -188.674 q^{51} -179.916 q^{52} +170.846 q^{53} +18.6750 q^{54} -156.442 q^{56} -282.480 q^{57} -20.0583 q^{58} -636.987 q^{59} -379.603 q^{61} -72.6051 q^{62} +131.148 q^{63} -337.339 q^{64} +76.4276 q^{66} +623.333 q^{67} +473.044 q^{68} -556.231 q^{69} +298.898 q^{71} -96.6218 q^{72} +524.124 q^{73} +179.694 q^{74} +708.233 q^{76} +536.726 q^{77} +49.6337 q^{78} +563.264 q^{79} +81.0000 q^{81} +8.16645 q^{82} +885.342 q^{83} -328.815 q^{84} +42.2263 q^{86} -87.0000 q^{87} -395.426 q^{88} -447.652 q^{89} +348.562 q^{91} +1394.58 q^{92} -314.914 q^{93} -194.653 q^{94} +367.109 q^{96} +558.895 q^{97} -90.3702 q^{98} +331.493 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{2} + 18 q^{3} + 15 q^{4} + 3 q^{6} - 23 q^{7} + 51 q^{8} + 54 q^{9} - 111 q^{11} + 45 q^{12} + 83 q^{13} - 102 q^{14} - 37 q^{16} + 35 q^{17} + 9 q^{18} - 76 q^{19} - 69 q^{21} - 66 q^{22} - 166 q^{23}+ \cdots - 999 q^{99}+O(q^{100}) \) 6 * q + q^2 + 18 * q^3 + 15 * q^4 + 3 * q^6 - 23 * q^7 + 51 * q^8 + 54 * q^9 - 111 * q^11 + 45 * q^12 + 83 * q^13 - 102 * q^14 - 37 * q^16 + 35 * q^17 + 9 * q^18 - 76 * q^19 - 69 * q^21 - 66 * q^22 - 166 * q^23 + 153 * q^24 - 282 * q^26 + 162 * q^27 - 368 * q^28 - 174 * q^29 - 164 * q^31 + 259 * q^32 - 333 * q^33 + 512 * q^34 + 135 * q^36 + 538 * q^37 - 6 * q^38 + 249 * q^39 - 1250 * q^41 - 306 * q^42 + 80 * q^43 - 1004 * q^44 - 1190 * q^46 + 491 * q^47 - 111 * q^48 - 969 * q^49 + 105 * q^51 + 336 * q^52 + 356 * q^53 + 27 * q^54 - 1260 * q^56 - 228 * q^57 - 29 * q^58 - 646 * q^59 - 1476 * q^61 + 4 * q^62 - 207 * q^63 - 845 * q^64 - 198 * q^66 - 87 * q^67 + 558 * q^68 - 498 * q^69 - 1312 * q^71 + 459 * q^72 + 1142 * q^73 - 2270 * q^74 - 1594 * q^76 + 2561 * q^77 - 846 * q^78 - 142 * q^79 + 486 * q^81 + 1544 * q^82 - 346 * q^83 - 1104 * q^84 - 2056 * q^86 - 522 * q^87 - 1400 * q^88 - 2827 * q^89 + 1379 * q^91 + 1434 * q^92 - 492 * q^93 + 978 * q^94 + 777 * q^96 + 4 * q^97 + 1903 * q^98 - 999 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more