LMFDB - Embedded newform 2175.4.a.l.1.2

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} - 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			1.2
Root			$-3.16884$ 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)

$f(q)$	$=$	\(q-3.16884 q^{2} +3.00000 q^{3} +2.04152 q^{4} -9.50651 q^{6} +6.36645 q^{7} +18.8814 q^{8} +9.00000 q^{9} +33.2106 q^{11} +6.12455 q^{12} -7.42718 q^{13} -20.1742 q^{14} -76.1643 q^{16} +80.4958 q^{17} -28.5195 q^{18} -77.9369 q^{19} +19.0993 q^{21} -105.239 q^{22} +26.4240 q^{23} +56.6443 q^{24} +23.5355 q^{26} +27.0000 q^{27} +12.9972 q^{28} -29.0000 q^{29} -272.292 q^{31} +90.3007 q^{32} +99.6318 q^{33} -255.078 q^{34} +18.3737 q^{36} +143.570 q^{37} +246.969 q^{38} -22.2816 q^{39} -393.257 q^{41} -60.5227 q^{42} -122.730 q^{43} +67.8000 q^{44} -83.7334 q^{46} +226.518 q^{47} -228.493 q^{48} -302.468 q^{49} +241.487 q^{51} -15.1627 q^{52} -493.905 q^{53} -85.5586 q^{54} +120.208 q^{56} -233.811 q^{57} +91.8962 q^{58} +657.572 q^{59} -642.052 q^{61} +862.848 q^{62} +57.2980 q^{63} +323.167 q^{64} -315.717 q^{66} -339.263 q^{67} +164.334 q^{68} +79.2721 q^{69} -891.455 q^{71} +169.933 q^{72} +94.6503 q^{73} -454.950 q^{74} -159.110 q^{76} +211.434 q^{77} +70.6066 q^{78} -268.324 q^{79} +81.0000 q^{81} +1246.17 q^{82} -1224.27 q^{83} +38.9917 q^{84} +388.912 q^{86} -87.0000 q^{87} +627.064 q^{88} +632.108 q^{89} -47.2848 q^{91} +53.9452 q^{92} -816.876 q^{93} -717.797 q^{94} +270.902 q^{96} +416.938 q^{97} +958.472 q^{98} +298.895 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−3.16884	−1.12035	−0.560176	−	0.828373i	$-0.689267\pi$
$2$	−3.16884	−1.12035	−0.560176	+	0.828373i	$0.689267\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	2.04152	0.255190
$5$	0	0
$5$	0	0
$6$	−9.50651	−0.646836
$7$	6.36645	0.343756	0.171878	−	0.985118i	$-0.445017\pi$
$7$	6.36645	0.343756	0.171878	+	0.985118i	$0.445017\pi$
$8$	18.8814	0.834450
$9$	9.00000	0.333333
$10$	0	0
$11$	33.2106	0.910306	0.455153	−	0.890413i	$-0.349584\pi$
$11$	33.2106	0.910306	0.455153	+	0.890413i	$0.349584\pi$
$12$	6.12455	0.147334
$13$	−7.42718	−0.158456	−0.0792281	−	0.996857i	$-0.525246\pi$
$13$	−7.42718	−0.158456	−0.0792281	+	0.996857i	$0.525246\pi$
$14$	−20.1742	−0.385128
$15$	0	0
$16$	−76.1643	−1.19007
$17$	80.4958	1.14842	0.574208	−	0.818709i	$-0.305310\pi$
$17$	80.4958	1.14842	0.574208	+	0.818709i	$0.305310\pi$
$18$	−28.5195	−0.373451
$19$	−77.9369	−0.941050	−0.470525	−	0.882387i	$-0.655935\pi$
$19$	−77.9369	−0.941050	−0.470525	+	0.882387i	$0.655935\pi$
$20$	0	0
$21$	19.0993	0.198468
$22$	−105.239	−1.01986
$23$	26.4240	0.239556	0.119778	−	0.992801i	$-0.461782\pi$
$23$	26.4240	0.239556	0.119778	+	0.992801i	$0.461782\pi$
$24$	56.6443	0.481770
$25$	0	0
$26$	23.5355	0.177527
$27$	27.0000	0.192450
$28$	12.9972	0.0877230
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−272.292	−1.57758	−0.788791	−	0.614661i	$-0.789293\pi$
$31$	−272.292	−1.57758	−0.788791	+	0.614661i	$0.789293\pi$
$32$	90.3007	0.498846
$33$	99.6318	0.525566
$34$	−255.078	−1.28663
$35$	0	0
$36$	18.3737	0.0850633
$37$	143.570	0.637913	0.318957	−	0.947769i	$-0.396668\pi$
$37$	143.570	0.637913	0.318957	+	0.947769i	$0.396668\pi$
$38$	246.969	1.05431
$39$	−22.2816	−0.0914847
$40$	0	0
$41$	−393.257	−1.49796	−0.748980	−	0.662593i	$-0.769456\pi$
$41$	−393.257	−1.49796	−0.748980	+	0.662593i	$0.769456\pi$
$42$	−60.5227	−0.222354
$43$	−122.730	−0.435260	−0.217630	−	0.976031i	$-0.569833\pi$
$43$	−122.730	−0.435260	−0.217630	+	0.976031i	$0.569833\pi$
$44$	67.8000	0.232301
$45$	0	0
$46$	−83.7334	−0.268387
$47$	226.518	0.703000	0.351500	−	0.936188i	$-0.385672\pi$
$47$	226.518	0.703000	0.351500	+	0.936188i	$0.385672\pi$
$48$	−228.493	−0.687086
$49$	−302.468	−0.881832
$50$	0	0
$51$	241.487	0.663039
$52$	−15.1627	−0.0404364
$53$	−493.905	−1.28006	−0.640029	−	0.768351i	$-0.721078\pi$
$53$	−493.905	−1.28006	−0.640029	+	0.768351i	$0.721078\pi$
$54$	−85.5586	−0.215612
$55$	0	0
$56$	120.208	0.286847
$57$	−233.811	−0.543315
$58$	91.8962	0.208044
$59$	657.572	1.45099	0.725497	−	0.688225i	$-0.241610\pi$
$59$	657.572	1.45099	0.725497	+	0.688225i	$0.241610\pi$
$60$	0	0
$61$	−642.052	−1.34764	−0.673822	−	0.738894i	$-0.735349\pi$
$61$	−642.052	−1.34764	−0.673822	+	0.738894i	$0.735349\pi$
$62$	862.848	1.76745
$63$	57.2980	0.114585
$64$	323.167	0.631185
$65$	0	0
$66$	−315.717	−0.588819
$67$	−339.263	−0.618621	−0.309311	−	0.950961i	$-0.600098\pi$
$67$	−339.263	−0.618621	−0.309311	+	0.950961i	$0.600098\pi$
$68$	164.334	0.293064
$69$	79.2721	0.138308
$70$	0	0
$71$	−891.455	−1.49009	−0.745044	−	0.667015i	$-0.767571\pi$
$71$	−891.455	−1.49009	−0.745044	+	0.667015i	$0.767571\pi$
$72$	169.933	0.278150
$73$	94.6503	0.151753	0.0758766	−	0.997117i	$-0.475825\pi$
$73$	94.6503	0.151753	0.0758766	+	0.997117i	$0.475825\pi$
$74$	−454.950	−0.714688
$75$	0	0
$76$	−159.110	−0.240146
$77$	211.434	0.312923
$78$	70.6066	0.102495
$79$	−268.324	−0.382137	−0.191069	−	0.981577i	$-0.561195\pi$
$79$	−268.324	−0.382137	−0.191069	+	0.981577i	$0.561195\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	1246.17	1.67824
$83$	−1224.27	−1.61905	−0.809525	−	0.587085i	$-0.800275\pi$
$83$	−1224.27	−1.61905	−0.809525	+	0.587085i	$0.800275\pi$
$84$	38.9917	0.0506469
$85$	0	0
$86$	388.912	0.487644
$87$	−87.0000	−0.107211
$88$	627.064	0.759605
$89$	632.108	0.752847	0.376423	−	0.926448i	$-0.377154\pi$
$89$	632.108	0.752847	0.376423	+	0.926448i	$0.377154\pi$
$90$	0	0
$91$	−47.2848	−0.0544702
$92$	53.9452	0.0611323
$93$	−816.876	−0.910818
$94$	−717.797	−0.787608
$95$	0	0
$96$	270.902	0.288009
$97$	416.938	0.436429	0.218215	−	0.975901i	$-0.429977\pi$
$97$	416.938	0.436429	0.218215	+	0.975901i	$0.429977\pi$
$98$	958.472	0.987963
$99$	298.895	0.303435
$100$	0	0
$101$	420.881	0.414646	0.207323	−	0.978273i	$-0.433525\pi$
$101$	420.881	0.414646	0.207323	+	0.978273i	$0.433525\pi$
$102$	−765.233	−0.742837
$103$	237.251	0.226962	0.113481	−	0.993540i	$-0.463800\pi$
$103$	237.251	0.226962	0.113481	+	0.993540i	$0.463800\pi$
$104$	−140.236	−0.132224
$105$	0	0
$106$	1565.10	1.43412
$107$	485.508	0.438653	0.219326	−	0.975652i	$-0.429614\pi$
$107$	485.508	0.438653	0.219326	+	0.975652i	$0.429614\pi$
$108$	55.1210	0.0491113
$109$	1390.40	1.22180	0.610901	−	0.791707i	$-0.290807\pi$
$109$	1390.40	1.22180	0.610901	+	0.791707i	$0.290807\pi$
$110$	0	0
$111$	430.711	0.368299
$112$	−484.896	−0.409093
$113$	1300.57	1.08272	0.541359	−	0.840792i	$-0.317910\pi$
$113$	1300.57	1.08272	0.541359	+	0.840792i	$0.317910\pi$
$114$	740.907	0.608705
$115$	0	0
$116$	−59.2040	−0.0473876
$117$	−66.8447	−0.0528187
$118$	−2083.74	−1.62562
$119$	512.472	0.394775
$120$	0	0
$121$	−228.057	−0.171342
$122$	2034.56	1.50984
$123$	−1179.77	−0.864848
$124$	−555.889	−0.402583
$125$	0	0
$126$	−181.568	−0.128376
$127$	−436.569	−0.305034	−0.152517	−	0.988301i	$-0.548738\pi$
$127$	−436.569	−0.305034	−0.152517	+	0.988301i	$0.548738\pi$
$128$	−1746.47	−1.20600
$129$	−368.190	−0.251297
$130$	0	0
$131$	−1783.17	−1.18929	−0.594644	−	0.803989i	$-0.702707\pi$
$131$	−1783.17	−1.18929	−0.594644	+	0.803989i	$0.702707\pi$
$132$	203.400	0.134119
$133$	−496.181	−0.323491
$134$	1075.07	0.693074
$135$	0	0
$136$	1519.88	0.958297
$137$	−2696.32	−1.68148	−0.840739	−	0.541441i	$-0.817879\pi$
$137$	−2696.32	−1.68148	−0.840739	+	0.541441i	$0.817879\pi$
$138$	−251.200	−0.154954
$139$	561.641	0.342717	0.171359	−	0.985209i	$-0.445184\pi$
$139$	561.641	0.342717	0.171359	+	0.985209i	$0.445184\pi$
$140$	0	0
$141$	679.553	0.405877
$142$	2824.87	1.66942
$143$	−246.661	−0.144244
$144$	−685.479	−0.396689
$145$	0	0
$146$	−299.931	−0.170017
$147$	−907.405	−0.509126
$148$	293.101	0.162789
$149$	335.540	0.184487	0.0922433	−	0.995736i	$-0.470596\pi$
$149$	335.540	0.184487	0.0922433	+	0.995736i	$0.470596\pi$
$150$	0	0
$151$	2644.88	1.42541	0.712705	−	0.701464i	$-0.247470\pi$
$151$	2644.88	1.42541	0.712705	+	0.701464i	$0.247470\pi$
$152$	−1471.56	−0.785259
$153$	724.462	0.382806
$154$	−669.998	−0.350584
$155$	0	0
$156$	−45.4882	−0.0233460
$157$	2418.81	1.22957	0.614783	−	0.788696i	$-0.289244\pi$
$157$	2418.81	1.22957	0.614783	+	0.788696i	$0.289244\pi$
$158$	850.276	0.428129
$159$	−1481.71	−0.739041
$160$	0	0
$161$	168.227	0.0823489
$162$	−256.676	−0.124484
$163$	−2974.32	−1.42924	−0.714622	−	0.699511i	$-0.753401\pi$
$163$	−2974.32	−1.42924	−0.714622	+	0.699511i	$0.753401\pi$
$164$	−802.840	−0.382264
$165$	0	0
$166$	3879.51	1.81391
$167$	16.4512	0.00762295	0.00381148	−	0.999993i	$-0.498787\pi$
$167$	16.4512	0.00762295	0.00381148	+	0.999993i	$0.498787\pi$
$168$	360.623	0.165611
$169$	−2141.84	−0.974892
$170$	0	0
$171$	−701.432	−0.313683
$172$	−250.556	−0.111074
$173$	−4066.95	−1.78731	−0.893655	−	0.448754i	$-0.851868\pi$
$173$	−4066.95	−1.78731	−0.893655	+	0.448754i	$0.851868\pi$
$174$	275.689	0.120114
$175$	0	0
$176$	−2529.46	−1.08333
$177$	1972.72	0.837732
$178$	−2003.05	−0.843454
$179$	−512.397	−0.213957	−0.106979	−	0.994261i	$-0.534118\pi$
$179$	−512.397	−0.213957	−0.106979	+	0.994261i	$0.534118\pi$
$180$	0	0
$181$	−1896.03	−0.778623	−0.389312	−	0.921106i	$-0.627287\pi$
$181$	−1896.03	−0.778623	−0.389312	+	0.921106i	$0.627287\pi$
$182$	149.838	0.0610259
$183$	−1926.16	−0.778063
$184$	498.924	0.199898
$185$	0	0
$186$	2588.54	1.02044
$187$	2673.31	1.04541
$188$	462.440	0.179398
$189$	171.894	0.0661559
$190$	0	0
$191$	−2368.66	−0.897331	−0.448665	−	0.893700i	$-0.648100\pi$
$191$	−2368.66	−0.897331	−0.448665	+	0.893700i	$0.648100\pi$
$192$	969.500	0.364415
$193$	−360.892	−0.134599	−0.0672993	−	0.997733i	$-0.521438\pi$
$193$	−360.892	−0.134599	−0.0672993	+	0.997733i	$0.521438\pi$
$194$	−1321.21	−0.488955
$195$	0	0
$196$	−617.495	−0.225034
$197$	3003.76	1.08634	0.543170	−	0.839622i	$-0.317224\pi$
$197$	3003.76	1.08634	0.543170	+	0.839622i	$0.317224\pi$
$198$	−947.150	−0.339955
$199$	3445.61	1.22740	0.613701	−	0.789538i	$-0.289680\pi$
$199$	3445.61	1.22740	0.613701	+	0.789538i	$0.289680\pi$
$200$	0	0
$201$	−1017.79	−0.357161
$202$	−1333.70	−0.464550
$203$	−184.627	−0.0638339
$204$	493.001	0.169201
$205$	0	0
$206$	−751.810	−0.254277
$207$	237.816	0.0798521
$208$	565.687	0.188574
$209$	−2588.33	−0.856643
$210$	0	0
$211$	3886.03	1.26789	0.633946	−	0.773378i	$-0.281434\pi$
$211$	3886.03	1.26789	0.633946	+	0.773378i	$0.281434\pi$
$212$	−1008.32	−0.326658
$213$	−2674.36	−0.860303
$214$	−1538.50	−0.491446
$215$	0	0
$216$	509.799	0.160590
$217$	−1733.53	−0.542303
$218$	−4405.96	−1.36885
$219$	283.951	0.0876147
$220$	0	0
$221$	−597.857	−0.181974
$222$	−1364.85	−0.412625
$223$	−4372.83	−1.31312	−0.656561	−	0.754273i	$-0.727989\pi$
$223$	−4372.83	−1.31312	−0.656561	+	0.754273i	$0.727989\pi$
$224$	574.895	0.171481
$225$	0	0
$226$	−4121.28	−1.21303
$227$	3042.94	0.889722	0.444861	−	0.895600i	$-0.353253\pi$
$227$	3042.94	0.889722	0.444861	+	0.895600i	$0.353253\pi$
$228$	−477.329	−0.138649
$229$	−1424.37	−0.411026	−0.205513	−	0.978654i	$-0.565886\pi$
$229$	−1424.37	−0.411026	−0.205513	+	0.978654i	$0.565886\pi$
$230$	0	0
$231$	634.301	0.180666
$232$	−547.562	−0.154953
$233$	5281.04	1.48486	0.742430	−	0.669924i	$-0.233673\pi$
$233$	5281.04	1.48486	0.742430	+	0.669924i	$0.233673\pi$
$234$	211.820	0.0591756
$235$	0	0
$236$	1342.45	0.370279
$237$	−804.973	−0.220627
$238$	−1623.94	−0.442287
$239$	−2912.01	−0.788126	−0.394063	−	0.919084i	$-0.628931\pi$
$239$	−2912.01	−0.788126	−0.394063	+	0.919084i	$0.628931\pi$
$240$	0	0
$241$	−433.499	−0.115868	−0.0579339	−	0.998320i	$-0.518451\pi$
$241$	−433.499	−0.115868	−0.0579339	+	0.998320i	$0.518451\pi$
$242$	722.674	0.191964
$243$	243.000	0.0641500
$244$	−1310.76	−0.343905
$245$	0	0
$246$	3738.50	0.968934
$247$	578.851	0.149115
$248$	−5141.26	−1.31641
$249$	−3672.81	−0.934759
$250$	0	0
$251$	4495.12	1.13040	0.565198	−	0.824955i	$-0.308800\pi$
$251$	4495.12	1.13040	0.565198	+	0.824955i	$0.308800\pi$
$252$	116.975	0.0292410
$253$	877.558	0.218070
$254$	1383.42	0.341745
$255$	0	0
$256$	2948.94	0.719955
$257$	651.655	0.158168	0.0790838	−	0.996868i	$-0.474801\pi$
$257$	651.655	0.158168	0.0790838	+	0.996868i	$0.474801\pi$
$258$	1166.74	0.281542
$259$	914.032	0.219286
$260$	0	0
$261$	−261.000	−0.0618984
$262$	5650.58	1.33242
$263$	5171.84	1.21258	0.606291	−	0.795243i	$-0.292657\pi$
$263$	5171.84	1.21258	0.606291	+	0.795243i	$0.292657\pi$
$264$	1881.19	0.438558
$265$	0	0
$266$	1572.32	0.362424
$267$	1896.33	0.434656
$268$	−692.612	−0.157866
$269$	−4651.80	−1.05437	−0.527184	−	0.849751i	$-0.676752\pi$
$269$	−4651.80	−1.05437	−0.527184	+	0.849751i	$0.676752\pi$
$270$	0	0
$271$	7206.79	1.61543	0.807715	−	0.589574i	$-0.200704\pi$
$271$	7206.79	1.61543	0.807715	+	0.589574i	$0.200704\pi$
$272$	−6130.91	−1.36669
$273$	−141.854	−0.0314484
$274$	8544.20	1.88385
$275$	0	0
$276$	161.835	0.0352947
$277$	−2309.60	−0.500976	−0.250488	−	0.968120i	$-0.580591\pi$
$277$	−2309.60	−0.500976	−0.250488	+	0.968120i	$0.580591\pi$
$278$	−1779.75	−0.383964
$279$	−2450.63	−0.525861
$280$	0	0
$281$	−7601.52	−1.61377	−0.806884	−	0.590710i	$-0.798848\pi$
$281$	−7601.52	−1.61377	−0.806884	+	0.590710i	$0.798848\pi$
$282$	−2153.39	−0.454726
$283$	4779.99	1.00403	0.502016	−	0.864858i	$-0.332592\pi$
$283$	4779.99	1.00403	0.502016	+	0.864858i	$0.332592\pi$
$284$	−1819.92	−0.380255
$285$	0	0
$286$	781.629	0.161604
$287$	−2503.65	−0.514933
$288$	812.706	0.166282
$289$	1566.57	0.318862
$290$	0	0
$291$	1250.81	0.251973
$292$	193.230	0.0387258
$293$	5323.91	1.06152	0.530761	−	0.847522i	$-0.321906\pi$
$293$	5323.91	1.06152	0.530761	+	0.847522i	$0.321906\pi$
$294$	2875.42	0.570400
$295$	0	0
$296$	2710.81	0.532307
$297$	896.686	0.175189
$298$	−1063.27	−0.206690
$299$	−196.256	−0.0379592
$300$	0	0
$301$	−781.355	−0.149623
$302$	−8381.18	−1.59696
$303$	1262.64	0.239396
$304$	5936.01	1.11991
$305$	0	0
$306$	−2295.70	−0.428877
$307$	−5559.88	−1.03361	−0.516807	−	0.856102i	$-0.672879\pi$
$307$	−5559.88	−1.03361	−0.516807	+	0.856102i	$0.672879\pi$
$308$	431.645	0.0798548
$309$	711.754	0.131036
$310$	0	0
$311$	−7920.33	−1.44412	−0.722059	−	0.691831i	$-0.756804\pi$
$311$	−7920.33	−1.44412	−0.722059	+	0.691831i	$0.756804\pi$
$312$	−420.708	−0.0763394
$313$	−4963.28	−0.896297	−0.448148	−	0.893959i	$-0.647916\pi$
$313$	−4963.28	−0.896297	−0.448148	+	0.893959i	$0.647916\pi$
$314$	−7664.80	−1.37755
$315$	0	0
$316$	−547.789	−0.0975175
$317$	−245.601	−0.0435152	−0.0217576	−	0.999763i	$-0.506926\pi$
$317$	−245.601	−0.0435152	−0.0217576	+	0.999763i	$0.506926\pi$
$318$	4695.31	0.827987
$319$	−963.107	−0.169040
$320$	0	0
$321$	1456.52	0.253256
$322$	−533.085	−0.0922597
$323$	−6273.59	−1.08072
$324$	165.363	0.0283544
$325$	0	0
$326$	9425.13	1.60126
$327$	4171.21	0.705408
$328$	−7425.25	−1.24997
$329$	1442.11	0.241660
$330$	0	0
$331$	−336.413	−0.0558638	−0.0279319	−	0.999610i	$-0.508892\pi$
$331$	−336.413	−0.0558638	−0.0279319	+	0.999610i	$0.508892\pi$
$332$	−2499.37	−0.413165
$333$	1292.13	0.212638
$334$	−52.1312	−0.00854039
$335$	0	0
$336$	−1454.69	−0.236190
$337$	−5888.49	−0.951830	−0.475915	−	0.879491i	$-0.657883\pi$
$337$	−5888.49	−0.951830	−0.475915	+	0.879491i	$0.657883\pi$
$338$	6787.13	1.09222
$339$	3901.70	0.625107
$340$	0	0
$341$	−9042.97	−1.43608
$342$	2222.72	0.351436
$343$	−4109.34	−0.646891
$344$	−2317.32	−0.363203
$345$	0	0
$346$	12887.5	2.00242
$347$	1000.15	0.154729	0.0773645	−	0.997003i	$-0.475349\pi$
$347$	1000.15	0.154729	0.0773645	+	0.997003i	$0.475349\pi$
$348$	−177.612	−0.0273592
$349$	−7602.06	−1.16599	−0.582993	−	0.812477i	$-0.698118\pi$
$349$	−7602.06	−1.16599	−0.582993	+	0.812477i	$0.698118\pi$
$350$	0	0
$351$	−200.534	−0.0304949
$352$	2998.94	0.454102
$353$	72.6144	0.0109487	0.00547433	−	0.999985i	$-0.498257\pi$
$353$	72.6144	0.0109487	0.00547433	+	0.999985i	$0.498257\pi$
$354$	−6251.22	−0.938555
$355$	0	0
$356$	1290.46	0.192119
$357$	1537.42	0.227924
$358$	1623.70	0.239707
$359$	−4787.09	−0.703768	−0.351884	−	0.936044i	$-0.614459\pi$
$359$	−4787.09	−0.703768	−0.351884	+	0.936044i	$0.614459\pi$
$360$	0	0
$361$	−784.845	−0.114426
$362$	6008.21	0.872333
$363$	−684.170	−0.0989245
$364$	−96.5327	−0.0139002
$365$	0	0
$366$	6103.67	0.871705
$367$	8456.98	1.20286	0.601432	−	0.798924i	$-0.294597\pi$
$367$	8456.98	1.20286	0.601432	+	0.798924i	$0.294597\pi$
$368$	−2012.57	−0.285088
$369$	−3539.31	−0.499320
$370$	0	0
$371$	−3144.42	−0.440027
$372$	−1667.67	−0.232431
$373$	109.448	0.0151930	0.00759652	−	0.999971i	$-0.497582\pi$
$373$	109.448	0.0151930	0.00759652	+	0.999971i	$0.497582\pi$
$374$	−8471.28	−1.17123
$375$	0	0
$376$	4276.98	0.586618
$377$	215.388	0.0294246
$378$	−544.704	−0.0741179
$379$	1446.28	0.196017	0.0980087	−	0.995186i	$-0.468753\pi$
$379$	1446.28	0.196017	0.0980087	+	0.995186i	$0.468753\pi$
$380$	0	0
$381$	−1309.71	−0.176111
$382$	7505.89	1.00533
$383$	6954.48	0.927826	0.463913	−	0.885881i	$-0.346445\pi$
$383$	6954.48	0.927826	0.463913	+	0.885881i	$0.346445\pi$
$384$	−5239.40	−0.696282
$385$	0	0
$386$	1143.61	0.150798
$387$	−1104.57	−0.145087
$388$	851.187	0.111372
$389$	1771.46	0.230892	0.115446	−	0.993314i	$-0.463170\pi$
$389$	1771.46	0.230892	0.115446	+	0.993314i	$0.463170\pi$
$390$	0	0
$391$	2127.02	0.275110
$392$	−5711.04	−0.735845
$393$	−5349.52	−0.686635
$394$	−9518.43	−1.21708
$395$	0	0
$396$	610.200	0.0774336
$397$	−8022.99	−1.01426	−0.507131	−	0.861869i	$-0.669294\pi$
$397$	−8022.99	−1.01426	−0.507131	+	0.861869i	$0.669294\pi$
$398$	−10918.6	−1.37512
$399$	−1488.54	−0.186768
$400$	0	0
$401$	−2268.78	−0.282537	−0.141268	−	0.989971i	$-0.545118\pi$
$401$	−2268.78	−0.282537	−0.141268	+	0.989971i	$0.545118\pi$
$402$	3225.21	0.400146
$403$	2022.36	0.249978
$404$	859.236	0.105813
$405$	0	0
$406$	585.053	0.0715164
$407$	4768.05	0.580696
$408$	4559.63	0.553273
$409$	−11953.1	−1.44509	−0.722545	−	0.691324i	$-0.757028\pi$
$409$	−11953.1	−1.44509	−0.722545	+	0.691324i	$0.757028\pi$
$410$	0	0
$411$	−8088.97	−0.970801
$412$	484.353	0.0579183
$413$	4186.40	0.498788
$414$	−753.601	−0.0894625
$415$	0	0
$416$	−670.680	−0.0790452
$417$	1684.92	0.197868
$418$	8201.99	0.959743
$419$	−10234.8	−1.19333	−0.596665	−	0.802491i	$-0.703508\pi$
$419$	−10234.8	−1.19333	−0.596665	+	0.802491i	$0.703508\pi$
$420$	0	0
$421$	−1894.70	−0.219340	−0.109670	−	0.993968i	$-0.534979\pi$
$421$	−1894.70	−0.219340	−0.109670	+	0.993968i	$0.534979\pi$
$422$	−12314.2	−1.42049
$423$	2038.66	0.234333
$424$	−9325.64	−1.06814
$425$	0	0
$426$	8474.62	0.963842
$427$	−4087.59	−0.463261
$428$	991.174	0.111940
$429$	−739.983	−0.0832791
$430$	0	0
$431$	4238.72	0.473717	0.236858	−	0.971544i	$-0.423882\pi$
$431$	4238.72	0.473717	0.236858	+	0.971544i	$0.423882\pi$
$432$	−2056.44	−0.229029
$433$	13631.8	1.51294	0.756471	−	0.654027i	$-0.226922\pi$
$433$	13631.8	1.51294	0.756471	+	0.654027i	$0.226922\pi$
$434$	5493.28	0.607571
$435$	0	0
$436$	2838.53	0.311791
$437$	−2059.41	−0.225434
$438$	−899.794	−0.0981594
$439$	725.415	0.0788660	0.0394330	−	0.999222i	$-0.487445\pi$
$439$	725.415	0.0788660	0.0394330	+	0.999222i	$0.487445\pi$
$440$	0	0
$441$	−2722.22	−0.293944
$442$	1894.51	0.203875
$443$	−14355.3	−1.53959	−0.769797	−	0.638288i	$-0.779643\pi$
$443$	−14355.3	−1.53959	−0.769797	+	0.638288i	$0.779643\pi$
$444$	879.303	0.0939862
$445$	0	0
$446$	13856.8	1.47116
$447$	1006.62	0.106513
$448$	2057.42	0.216974
$449$	519.743	0.0546285	0.0273142	−	0.999627i	$-0.491305\pi$
$449$	519.743	0.0546285	0.0273142	+	0.999627i	$0.491305\pi$
$450$	0	0
$451$	−13060.3	−1.36360
$452$	2655.13	0.276298
$453$	7934.63	0.822961
$454$	−9642.57	−0.996803
$455$	0	0
$456$	−4414.68	−0.453369
$457$	−5739.92	−0.587532	−0.293766	−	0.955877i	$-0.594909\pi$
$457$	−5739.92	−0.587532	−0.293766	+	0.955877i	$0.594909\pi$
$458$	4513.60	0.460495
$459$	2173.39	0.221013
$460$	0	0
$461$	−10071.6	−1.01753	−0.508763	−	0.860907i	$-0.669897\pi$
$461$	−10071.6	−1.01753	−0.508763	+	0.860907i	$0.669897\pi$
$462$	−2009.99	−0.202410
$463$	−6265.21	−0.628874	−0.314437	−	0.949278i	$-0.601816\pi$
$463$	−6265.21	−0.628874	−0.314437	+	0.949278i	$0.601816\pi$
$464$	2208.77	0.220990
$465$	0	0
$466$	−16734.7	−1.66357
$467$	−17387.0	−1.72285	−0.861427	−	0.507881i	$-0.830429\pi$
$467$	−17387.0	−1.72285	−0.861427	+	0.507881i	$0.830429\pi$
$468$	−136.465	−0.0134788
$469$	−2159.90	−0.212655
$470$	0	0
$471$	7256.42	0.709890
$472$	12415.9	1.21078
$473$	−4075.94	−0.396220
$474$	2550.83	0.247180
$475$	0	0
$476$	1046.22	0.100743
$477$	−4445.14	−0.426686
$478$	9227.67	0.882979
$479$	−15957.4	−1.52216	−0.761080	−	0.648658i	$-0.775330\pi$
$479$	−15957.4	−1.52216	−0.761080	+	0.648658i	$0.775330\pi$
$480$	0	0
$481$	−1066.32	−0.101081
$482$	1373.69	0.129813
$483$	504.682	0.0475441
$484$	−465.582	−0.0437248
$485$	0	0
$486$	−770.027	−0.0718706
$487$	15053.5	1.40070	0.700350	−	0.713800i	$-0.253027\pi$
$487$	15053.5	1.40070	0.700350	+	0.713800i	$0.253027\pi$
$488$	−12122.9	−1.12454
$489$	−8922.96	−0.825174
$490$	0	0
$491$	−12513.2	−1.15013	−0.575066	−	0.818107i	$-0.695024\pi$
$491$	−12513.2	−1.15013	−0.575066	+	0.818107i	$0.695024\pi$
$492$	−2408.52	−0.220700
$493$	−2334.38	−0.213256
$494$	−1834.28	−0.167062
$495$	0	0
$496$	20738.9	1.87743
$497$	−5675.40	−0.512226
$498$	11638.5	1.04726
$499$	−2702.24	−0.242423	−0.121211	−	0.992627i	$-0.538678\pi$
$499$	−2702.24	−0.242423	−0.121211	+	0.992627i	$0.538678\pi$
$500$	0	0
$501$	49.3536	0.00440111
$502$	−14244.3	−1.26644
$503$	−13863.2	−1.22889	−0.614445	−	0.788960i	$-0.710620\pi$
$503$	−13863.2	−1.22889	−0.614445	+	0.788960i	$0.710620\pi$
$504$	1081.87	0.0956157
$505$	0	0
$506$	−2780.84	−0.244315
$507$	−6425.51	−0.562854
$508$	−891.264	−0.0778415
$509$	13363.9	1.16374	0.581871	−	0.813281i	$-0.302321\pi$
$509$	13363.9	1.16374	0.581871	+	0.813281i	$0.302321\pi$
$510$	0	0
$511$	602.586	0.0521660
$512$	4627.05	0.399392
$513$	−2104.30	−0.181105
$514$	−2064.99	−0.177204
$515$	0	0
$516$	−751.668	−0.0641285
$517$	7522.78	0.639945
$518$	−2896.42	−0.245678
$519$	−12200.9	−1.03190
$520$	0	0
$521$	−11833.2	−0.995050	−0.497525	−	0.867450i	$-0.665758\pi$
$521$	−11833.2	−0.995050	−0.497525	+	0.867450i	$0.665758\pi$
$522$	827.066	0.0693481
$523$	−3346.53	−0.279796	−0.139898	−	0.990166i	$-0.544677\pi$
$523$	−3346.53	−0.279796	−0.139898	+	0.990166i	$0.544677\pi$
$524$	−3640.38	−0.303494
$525$	0	0
$526$	−16388.7	−1.35852
$527$	−21918.3	−1.81172
$528$	−7588.39	−0.625459
$529$	−11468.8	−0.942613
$530$	0	0
$531$	5918.15	0.483665
$532$	−1012.96	−0.0825517
$533$	2920.79	0.237361
$534$	−6009.14	−0.486968
$535$	0	0
$536$	−6405.78	−0.516208
$537$	−1537.19	−0.123528
$538$	14740.8	1.18126
$539$	−10045.2	−0.802737
$540$	0	0
$541$	24735.2	1.96571	0.982856	−	0.184373i	$-0.0590253\pi$
$541$	24735.2	1.96571	0.982856	+	0.184373i	$0.0590253\pi$
$542$	−22837.1	−1.80985
$543$	−5688.09	−0.449538
$544$	7268.82	0.572883
$545$	0	0
$546$	449.513	0.0352333
$547$	−11807.1	−0.922916	−0.461458	−	0.887162i	$-0.652674\pi$
$547$	−11807.1	−0.922916	−0.461458	+	0.887162i	$0.652674\pi$
$548$	−5504.59	−0.429096
$549$	−5778.47	−0.449215
$550$	0	0
$551$	2260.17	0.174749
$552$	1496.77	0.115411
$553$	−1708.27	−0.131362
$554$	7318.74	0.561270
$555$	0	0
$556$	1146.60	0.0874580
$557$	−13080.8	−0.995065	−0.497533	−	0.867445i	$-0.665761\pi$
$557$	−13080.8	−0.995065	−0.497533	+	0.867445i	$0.665761\pi$
$558$	7765.63	0.589150
$559$	911.539	0.0689696
$560$	0	0
$561$	8019.93	0.603568
$562$	24088.0	1.80799
$563$	−8350.94	−0.625133	−0.312567	−	0.949896i	$-0.601189\pi$
$563$	−8350.94	−0.625133	−0.312567	+	0.949896i	$0.601189\pi$
$564$	1387.32	0.103576
$565$	0	0
$566$	−15147.0	−1.12487
$567$	515.682	0.0381951
$568$	−16832.0	−1.24340
$569$	16013.7	1.17984	0.589921	−	0.807461i	$-0.299159\pi$
$569$	16013.7	1.17984	0.589921	+	0.807461i	$0.299159\pi$
$570$	0	0
$571$	−7224.45	−0.529482	−0.264741	−	0.964320i	$-0.585286\pi$
$571$	−7224.45	−0.529482	−0.264741	+	0.964320i	$0.585286\pi$
$572$	−503.563	−0.0368095
$573$	−7105.98	−0.518074
$574$	7933.65	0.576906
$575$	0	0
$576$	2908.50	0.210395
$577$	22764.3	1.64244	0.821222	−	0.570610i	$-0.193293\pi$
$577$	22764.3	1.64244	0.821222	+	0.570610i	$0.193293\pi$
$578$	−4964.19	−0.357237
$579$	−1082.67	−0.0777106
$580$	0	0
$581$	−7794.26	−0.556558
$582$	−3963.62	−0.282298
$583$	−16402.9	−1.16524
$584$	1787.13	0.126630
$585$	0	0
$586$	−16870.6	−1.18928
$587$	−23104.2	−1.62456	−0.812278	−	0.583270i	$-0.801773\pi$
$587$	−23104.2	−1.62456	−0.812278	+	0.583270i	$0.801773\pi$
$588$	−1852.48	−0.129924
$589$	21221.6	1.48458
$590$	0	0
$591$	9011.28	0.627199
$592$	−10934.9	−0.759160
$593$	19221.9	1.33111	0.665556	−	0.746348i	$-0.268194\pi$
$593$	19221.9	1.33111	0.665556	+	0.746348i	$0.268194\pi$
$594$	−2841.45	−0.196273
$595$	0	0
$596$	685.011	0.0470791
$597$	10336.8	0.708641
$598$	621.904	0.0425276
$599$	8313.99	0.567113	0.283556	−	0.958956i	$-0.408486\pi$
$599$	8313.99	0.567113	0.283556	+	0.958956i	$0.408486\pi$
$600$	0	0
$601$	−8379.05	−0.568700	−0.284350	−	0.958721i	$-0.591778\pi$
$601$	−8379.05	−0.568700	−0.284350	+	0.958721i	$0.591778\pi$
$602$	2475.99	0.167631
$603$	−3053.37	−0.206207
$604$	5399.56	0.363750
$605$	0	0
$606$	−4001.11	−0.268208
$607$	2133.39	0.142655	0.0713275	−	0.997453i	$-0.477276\pi$
$607$	2133.39	0.142655	0.0713275	+	0.997453i	$0.477276\pi$
$608$	−7037.75	−0.469438
$609$	−553.881	−0.0368545
$610$	0	0
$611$	−1682.39	−0.111395
$612$	1479.00	0.0976881
$613$	−12506.9	−0.824062	−0.412031	−	0.911170i	$-0.635180\pi$
$613$	−12506.9	−0.824062	−0.412031	+	0.911170i	$0.635180\pi$
$614$	17618.4	1.15801
$615$	0	0
$616$	3992.17	0.261119
$617$	5460.31	0.356278	0.178139	−	0.984005i	$-0.442992\pi$
$617$	5460.31	0.356278	0.178139	+	0.984005i	$0.442992\pi$
$618$	−2255.43	−0.146807
$619$	7635.72	0.495809	0.247904	−	0.968785i	$-0.420258\pi$
$619$	7635.72	0.495809	0.247904	+	0.968785i	$0.420258\pi$
$620$	0	0
$621$	713.449	0.0461026
$622$	25098.2	1.61792
$623$	4024.29	0.258796
$624$	1697.06	0.108873
$625$	0	0
$626$	15727.8	1.00417
$627$	−7764.99	−0.494583
$628$	4938.04	0.313773
$629$	11556.8	0.732590
$630$	0	0
$631$	9951.47	0.627831	0.313916	−	0.949451i	$-0.398359\pi$
$631$	9951.47	0.627831	0.313916	+	0.949451i	$0.398359\pi$
$632$	−5066.35	−0.318875
$633$	11658.1	0.732017
$634$	778.269	0.0487524
$635$	0	0
$636$	−3024.95	−0.188596
$637$	2246.49	0.139732
$638$	3051.93	0.189384
$639$	−8023.09	−0.496696
$640$	0	0
$641$	16935.0	1.04352	0.521758	−	0.853093i	$-0.325276\pi$
$641$	16935.0	1.04352	0.521758	+	0.853093i	$0.325276\pi$
$642$	−4615.49	−0.283736
$643$	−20875.1	−1.28030	−0.640150	−	0.768250i	$-0.721128\pi$
$643$	−20875.1	−1.28030	−0.640150	+	0.768250i	$0.721128\pi$
$644$	343.439	0.0210146
$645$	0	0
$646$	19880.0	1.21078
$647$	−21269.8	−1.29243	−0.646215	−	0.763155i	$-0.723649\pi$
$647$	−21269.8	−1.29243	−0.646215	+	0.763155i	$0.723649\pi$
$648$	1529.40	0.0927167
$649$	21838.4	1.32085
$650$	0	0
$651$	−5200.60	−0.313099
$652$	−6072.13	−0.364728
$653$	−14499.1	−0.868905	−0.434453	−	0.900695i	$-0.643058\pi$
$653$	−14499.1	−0.868905	−0.434453	+	0.900695i	$0.643058\pi$
$654$	−13217.9	−0.790305
$655$	0	0
$656$	29952.1	1.78267
$657$	851.853	0.0505844
$658$	−4569.82	−0.270745
$659$	−2246.04	−0.132767	−0.0663833	−	0.997794i	$-0.521146\pi$
$659$	−2246.04	−0.132767	−0.0663833	+	0.997794i	$0.521146\pi$
$660$	0	0
$661$	−9671.20	−0.569086	−0.284543	−	0.958663i	$-0.591842\pi$
$661$	−9671.20	−0.569086	−0.284543	+	0.958663i	$0.591842\pi$
$662$	1066.04	0.0625871
$663$	−1793.57	−0.105063
$664$	−23116.0	−1.35102
$665$	0	0
$666$	−4094.55	−0.238229
$667$	−766.297	−0.0444845
$668$	33.5854	0.00194530
$669$	−13118.5	−0.758131
$670$	0	0
$671$	−21322.9	−1.22677
$672$	1724.68	0.0990047
$673$	−14251.2	−0.816260	−0.408130	−	0.912924i	$-0.633819\pi$
$673$	−14251.2	−0.816260	−0.408130	+	0.912924i	$0.633819\pi$
$674$	18659.7	1.06639
$675$	0	0
$676$	−4372.60	−0.248782
$677$	−7864.63	−0.446473	−0.223236	−	0.974764i	$-0.571662\pi$
$677$	−7864.63	−0.446473	−0.223236	+	0.974764i	$0.571662\pi$
$678$	−12363.9	−0.700340
$679$	2654.41	0.150025
$680$	0	0
$681$	9128.82	0.513681
$682$	28655.7	1.60892
$683$	14468.3	0.810561	0.405280	−	0.914192i	$-0.367174\pi$
$683$	14468.3	0.810561	0.405280	+	0.914192i	$0.367174\pi$
$684$	−1431.99	−0.0800487
$685$	0	0
$686$	13021.8	0.724746
$687$	−4273.11	−0.237306
$688$	9347.66	0.517989
$689$	3668.32	0.202833
$690$	0	0
$691$	13182.6	0.725746	0.362873	−	0.931839i	$-0.381796\pi$
$691$	13182.6	0.725746	0.362873	+	0.931839i	$0.381796\pi$
$692$	−8302.76	−0.456103
$693$	1902.90	0.104308
$694$	−3169.32	−0.173351
$695$	0	0
$696$	−1642.69	−0.0894624
$697$	−31655.5	−1.72028
$698$	24089.7	1.30632
$699$	15843.1	0.857284
$700$	0	0
$701$	−9717.44	−0.523570	−0.261785	−	0.965126i	$-0.584311\pi$
$701$	−9717.44	−0.523570	−0.261785	+	0.965126i	$0.584311\pi$
$702$	635.459	0.0341650
$703$	−11189.4	−0.600308
$704$	10732.6	0.574572
$705$	0	0
$706$	−230.103	−0.0122664
$707$	2679.52	0.142537
$708$	4027.34	0.213781
$709$	−15655.2	−0.829259	−0.414629	−	0.909990i	$-0.636089\pi$
$709$	−15655.2	−0.829259	−0.414629	+	0.909990i	$0.636089\pi$
$710$	0	0
$711$	−2414.92	−0.127379
$712$	11935.1	0.628213
$713$	−7195.05	−0.377920
$714$	−4871.82	−0.255355
$715$	0	0
$716$	−1046.07	−0.0545997
$717$	−8736.02	−0.455025
$718$	15169.5	0.788469
$719$	−36868.3	−1.91232	−0.956158	−	0.292853i	$-0.905395\pi$
$719$	−36868.3	−1.91232	−0.956158	+	0.292853i	$0.905395\pi$
$720$	0	0
$721$	1510.45	0.0780195
$722$	2487.04	0.128197
$723$	−1300.50	−0.0668963
$724$	−3870.78	−0.198697
$725$	0	0
$726$	2168.02	0.110830
$727$	36049.7	1.83908	0.919540	−	0.392996i	$-0.128562\pi$
$727$	36049.7	1.83908	0.919540	+	0.392996i	$0.128562\pi$
$728$	−892.805	−0.0454527
$729$	729.000	0.0370370
$730$	0	0
$731$	−9879.26	−0.499860
$732$	−3932.28	−0.198554
$733$	4057.83	0.204474	0.102237	−	0.994760i	$-0.467400\pi$
$733$	4057.83	0.204474	0.102237	+	0.994760i	$0.467400\pi$
$734$	−26798.8	−1.34763
$735$	0	0
$736$	2386.11	0.119502
$737$	−11267.1	−0.563135
$738$	11215.5	0.559414
$739$	−7614.89	−0.379050	−0.189525	−	0.981876i	$-0.560695\pi$
$739$	−7614.89	−0.379050	−0.189525	+	0.981876i	$0.560695\pi$
$740$	0	0
$741$	1736.55	0.0860917
$742$	9964.15	0.492986
$743$	−17395.9	−0.858944	−0.429472	−	0.903080i	$-0.641300\pi$
$743$	−17395.9	−0.858944	−0.429472	+	0.903080i	$0.641300\pi$
$744$	−15423.8	−0.760032
$745$	0	0
$746$	−346.823	−0.0170216
$747$	−11018.4	−0.539683
$748$	5457.61	0.266778
$749$	3090.96	0.150789
$750$	0	0
$751$	2229.80	0.108344	0.0541720	−	0.998532i	$-0.482748\pi$
$751$	2229.80	0.108344	0.0541720	+	0.998532i	$0.482748\pi$
$752$	−17252.6	−0.836618
$753$	13485.4	0.652635
$754$	−682.530	−0.0329659
$755$	0	0
$756$	350.925	0.0168823
$757$	27616.9	1.32596	0.662982	−	0.748636i	$-0.269291\pi$
$757$	27616.9	1.32596	0.662982	+	0.748636i	$0.269291\pi$
$758$	−4583.03	−0.219608
$759$	2632.67	0.125903
$760$	0	0
$761$	−19746.5	−0.940617	−0.470309	−	0.882502i	$-0.655857\pi$
$761$	−19746.5	−0.940617	−0.470309	+	0.882502i	$0.655857\pi$
$762$	4150.25	0.197307
$763$	8851.93	0.420002
$764$	−4835.66	−0.228990
$765$	0	0
$766$	−22037.6	−1.03949
$767$	−4883.91	−0.229919
$768$	8846.81	0.415666
$769$	33709.4	1.58075	0.790373	−	0.612627i	$-0.209887\pi$
$769$	33709.4	1.58075	0.790373	+	0.612627i	$0.209887\pi$
$770$	0	0
$771$	1954.96	0.0913182
$772$	−736.767	−0.0343482
$773$	39591.9	1.84220	0.921102	−	0.389322i	$-0.127291\pi$
$773$	39591.9	1.84220	0.921102	+	0.389322i	$0.127291\pi$
$774$	3500.21	0.162548
$775$	0	0
$776$	7872.40	0.364179
$777$	2742.10	0.126605
$778$	−5613.48	−0.258680
$779$	30649.2	1.40965
$780$	0	0
$781$	−29605.7	−1.35644
$782$	−6740.19	−0.308221
$783$	−783.000	−0.0357371
$784$	23037.3	1.04944
$785$	0	0
$786$	16951.8	0.769274
$787$	−1728.19	−0.0782760	−0.0391380	−	0.999234i	$-0.512461\pi$
$787$	−1728.19	−0.0782760	−0.0391380	+	0.999234i	$0.512461\pi$
$788$	6132.23	0.277223
$789$	15515.5	0.700085
$790$	0	0
$791$	8279.99	0.372191
$792$	5643.58	0.253202
$793$	4768.64	0.213543
$794$	25423.5	1.13633
$795$	0	0
$796$	7034.29	0.313221
$797$	36797.2	1.63541	0.817706	−	0.575636i	$-0.195246\pi$
$797$	36797.2	1.63541	0.817706	+	0.575636i	$0.195246\pi$
$798$	4716.95	0.209246
$799$	18233.7	0.807337
$800$	0	0
$801$	5688.98	0.250949
$802$	7189.38	0.316541
$803$	3143.39	0.138142
$804$	−2077.84	−0.0911439
$805$	0	0
$806$	−6408.53	−0.280063
$807$	−13955.4	−0.608740
$808$	7946.85	0.346001
$809$	−26350.1	−1.14514	−0.572572	−	0.819854i	$-0.694054\pi$
$809$	−26350.1	−1.14514	−0.572572	+	0.819854i	$0.694054\pi$
$810$	0	0
$811$	−4092.41	−0.177194	−0.0885968	−	0.996068i	$-0.528238\pi$
$811$	−4092.41	−0.177194	−0.0885968	+	0.996068i	$0.528238\pi$
$812$	−376.919	−0.0162897
$813$	21620.4	0.932668
$814$	−15109.2	−0.650585
$815$	0	0
$816$	−18392.7	−0.789061
$817$	9565.20	0.409601
$818$	37877.3	1.61901
$819$	−425.563	−0.0181567
$820$	0	0
$821$	9821.87	0.417522	0.208761	−	0.977967i	$-0.433057\pi$
$821$	9821.87	0.417522	0.208761	+	0.977967i	$0.433057\pi$
$822$	25632.6	1.08764
$823$	−23701.0	−1.00384	−0.501922	−	0.864913i	$-0.667373\pi$
$823$	−23701.0	−1.00384	−0.501922	+	0.864913i	$0.667373\pi$
$824$	4479.65	0.189388
$825$	0	0
$826$	−13266.0	−0.558818
$827$	21957.8	0.923274	0.461637	−	0.887069i	$-0.347262\pi$
$827$	21957.8	0.923274	0.461637	+	0.887069i	$0.347262\pi$
$828$	485.506	0.0203774
$829$	16697.5	0.699553	0.349776	−	0.936833i	$-0.386258\pi$
$829$	16697.5	0.699553	0.349776	+	0.936833i	$0.386258\pi$
$830$	0	0
$831$	−6928.80	−0.289239
$832$	−2400.22	−0.100015
$833$	−24347.4	−1.01271
$834$	−5339.24	−0.221682
$835$	0	0
$836$	−5284.12	−0.218607
$837$	−7351.88	−0.303606
$838$	32432.5	1.33695
$839$	33590.2	1.38220	0.691098	−	0.722761i	$-0.257127\pi$
$839$	33590.2	1.38220	0.691098	+	0.722761i	$0.257127\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	6003.99	0.245738
$843$	−22804.6	−0.931709
$844$	7933.39	0.323553
$845$	0	0
$846$	−6460.17	−0.262536
$847$	−1451.91	−0.0588999
$848$	37617.9	1.52336
$849$	14340.0	0.579678
$850$	0	0
$851$	3793.70	0.152816
$852$	−5459.76	−0.219540
$853$	−5408.43	−0.217094	−0.108547	−	0.994091i	$-0.534620\pi$
$853$	−5408.43	−0.217094	−0.108547	+	0.994091i	$0.534620\pi$
$854$	12952.9	0.519015
$855$	0	0
$856$	9167.10	0.366034
$857$	20198.6	0.805100	0.402550	−	0.915398i	$-0.368124\pi$
$857$	20198.6	0.805100	0.402550	+	0.915398i	$0.368124\pi$
$858$	2344.89	0.0933020
$859$	38903.1	1.54523	0.772617	−	0.634872i	$-0.218947\pi$
$859$	38903.1	1.54523	0.772617	+	0.634872i	$0.218947\pi$
$860$	0	0
$861$	−7510.94	−0.297296
$862$	−13431.8	−0.530730
$863$	35466.6	1.39895	0.699476	−	0.714656i	$-0.253417\pi$
$863$	35466.6	1.39895	0.699476	+	0.714656i	$0.253417\pi$
$864$	2438.12	0.0960029
$865$	0	0
$866$	−43197.0	−1.69503
$867$	4699.70	0.184095
$868$	−3539.04	−0.138390
$869$	−8911.21	−0.347862
$870$	0	0
$871$	2519.77	0.0980243
$872$	26252.8	1.01953
$873$	3752.44	0.145476
$874$	6525.92	0.252566
$875$	0	0
$876$	579.691	0.0223584
$877$	2742.28	0.105588	0.0527938	−	0.998605i	$-0.483187\pi$
$877$	2742.28	0.105588	0.0527938	+	0.998605i	$0.483187\pi$
$878$	−2298.72	−0.0883577
$879$	15971.7	0.612870
$880$	0	0
$881$	−25595.0	−0.978794	−0.489397	−	0.872061i	$-0.662783\pi$
$881$	−25595.0	−0.978794	−0.489397	+	0.872061i	$0.662783\pi$
$882$	8626.25	0.329321
$883$	22207.0	0.846347	0.423173	−	0.906049i	$-0.360916\pi$
$883$	22207.0	0.846347	0.423173	+	0.906049i	$0.360916\pi$
$884$	−1220.54	−0.0464378
$885$	0	0
$886$	45489.6	1.72489
$887$	−21956.1	−0.831133	−0.415567	−	0.909563i	$-0.636417\pi$
$887$	−21956.1	−0.831133	−0.415567	+	0.909563i	$0.636417\pi$
$888$	8132.44	0.307327
$889$	−2779.40	−0.104857
$890$	0	0
$891$	2690.06	0.101145
$892$	−8927.20	−0.335095
$893$	−17654.1	−0.661558
$894$	−3189.81	−0.119333
$895$	0	0
$896$	−11118.8	−0.414568
$897$	−588.769	−0.0219157
$898$	−1646.98	−0.0612031
$899$	7896.46	0.292950
$900$	0	0
$901$	−39757.2	−1.47004
$902$	41385.9	1.52772
$903$	−2344.07	−0.0863850
$904$	24556.6	0.903474
$905$	0	0
$906$	−25143.5	−0.922007
$907$	10598.6	0.388006	0.194003	−	0.981001i	$-0.437853\pi$
$907$	10598.6	0.388006	0.194003	+	0.981001i	$0.437853\pi$
$908$	6212.22	0.227048
$909$	3787.93	0.138215
$910$	0	0
$911$	28026.0	1.01926	0.509628	−	0.860395i	$-0.329783\pi$
$911$	28026.0	1.01926	0.509628	+	0.860395i	$0.329783\pi$
$912$	17808.0	0.646582
$913$	−40658.8	−1.47383
$914$	18188.8	0.658242
$915$	0	0
$916$	−2907.88	−0.104890
$917$	−11352.5	−0.408824
$918$	−6887.10	−0.247612
$919$	21116.9	0.757977	0.378989	−	0.925401i	$-0.376272\pi$
$919$	21116.9	0.757977	0.378989	+	0.925401i	$0.376272\pi$
$920$	0	0
$921$	−16679.7	−0.596757
$922$	31915.1	1.13999
$923$	6621.00	0.236114
$924$	1294.94	0.0461042
$925$	0	0
$926$	19853.4	0.704561
$927$	2135.26	0.0756539
$928$	−2618.72	−0.0926333
$929$	15051.0	0.531548	0.265774	−	0.964035i	$-0.414372\pi$
$929$	15051.0	0.531548	0.265774	+	0.964035i	$0.414372\pi$
$930$	0	0
$931$	23573.4	0.829848
$932$	10781.3	0.378921
$933$	−23761.0	−0.833762
$934$	55096.4	1.93020
$935$	0	0
$936$	−1262.12	−0.0440746
$937$	35687.4	1.24424	0.622122	−	0.782920i	$-0.286271\pi$
$937$	35687.4	1.24424	0.622122	+	0.782920i	$0.286271\pi$
$938$	6844.38	0.238248
$939$	−14889.8	−0.517477
$940$	0	0
$941$	−22876.5	−0.792510	−0.396255	−	0.918141i	$-0.629690\pi$
$941$	−22876.5	−0.792510	−0.396255	+	0.918141i	$0.629690\pi$
$942$	−22994.4	−0.795327
$943$	−10391.4	−0.358846
$944$	−50083.6	−1.72678
$945$	0	0
$946$	12916.0	0.443906
$947$	10094.8	0.346396	0.173198	−	0.984887i	$-0.444590\pi$
$947$	10094.8	0.346396	0.173198	+	0.984887i	$0.444590\pi$
$948$	−1643.37	−0.0563018
$949$	−702.985	−0.0240462
$950$	0	0
$951$	−736.803	−0.0251235
$952$	9676.21	0.329420
$953$	−23570.2	−0.801170	−0.400585	−	0.916260i	$-0.631193\pi$
$953$	−23570.2	−0.801170	−0.400585	+	0.916260i	$0.631193\pi$
$954$	14085.9	0.478038
$955$	0	0
$956$	−5944.91	−0.201122
$957$	−2889.32	−0.0975951
$958$	50566.5	1.70536
$959$	−17166.0	−0.578018
$960$	0	0
$961$	44351.9	1.48877
$962$	3379.00	0.113247
$963$	4369.57	0.146218
$964$	−884.996	−0.0295683
$965$	0	0
$966$	−1599.25	−0.0532662
$967$	−18620.8	−0.619239	−0.309620	−	0.950860i	$-0.600202\pi$
$967$	−18620.8	−0.619239	−0.309620	+	0.950860i	$0.600202\pi$
$968$	−4306.04	−0.142977
$969$	−18820.8	−0.623952
$970$	0	0
$971$	−33876.6	−1.11962	−0.559811	−	0.828621i	$-0.689126\pi$
$971$	−33876.6	−1.11962	−0.559811	+	0.828621i	$0.689126\pi$
$972$	496.089	0.0163704
$973$	3575.66	0.117811
$974$	−47702.2	−1.56928
$975$	0	0
$976$	48901.5	1.60379
$977$	14376.7	0.470780	0.235390	−	0.971901i	$-0.424363\pi$
$977$	14376.7	0.470780	0.235390	+	0.971901i	$0.424363\pi$
$978$	28275.4	0.924486
$979$	20992.7	0.685321
$980$	0	0
$981$	12513.6	0.407267
$982$	39652.4	1.28855
$983$	32902.2	1.06757	0.533784	−	0.845621i	$-0.320770\pi$
$983$	32902.2	1.06757	0.533784	+	0.845621i	$0.320770\pi$
$984$	−22275.8	−0.721672
$985$	0	0
$986$	7397.26	0.238922
$987$	4326.34	0.139523
$988$	1181.74	0.0380527
$989$	−3243.03	−0.104269
$990$	0	0
$991$	−42096.2	−1.34937	−0.674687	−	0.738104i	$-0.735721\pi$
$991$	−42096.2	−1.34937	−0.674687	+	0.738104i	$0.735721\pi$
$992$	−24588.1	−0.786970
$993$	−1009.24	−0.0322530
$994$	17984.4	0.573874
$995$	0	0
$996$	−7498.11	−0.238541
$997$	29384.9	0.933431	0.466715	−	0.884408i	$-0.345437\pi$
$997$	29384.9	0.933431	0.466715	+	0.884408i	$0.345437\pi$
$998$	8562.97	0.271599
$999$	3876.39	0.122766

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.4.a.l.1.2		6
5.4	even	2		435.4.a.g.1.5	✓	6
15.14	odd	2		1305.4.a.i.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.4.a.g.1.5	✓	6	5.4	even	2
1305.4.a.i.1.2		6	15.14	odd	2
2175.4.a.l.1.2		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)

\(f(q)\)	\(=\)	\(q-3.16884 q^{2} +3.00000 q^{3} +2.04152 q^{4} -9.50651 q^{6} +6.36645 q^{7} +18.8814 q^{8} +9.00000 q^{9} +33.2106 q^{11} +6.12455 q^{12} -7.42718 q^{13} -20.1742 q^{14} -76.1643 q^{16} +80.4958 q^{17} -28.5195 q^{18} -77.9369 q^{19} +19.0993 q^{21} -105.239 q^{22} +26.4240 q^{23} +56.6443 q^{24} +23.5355 q^{26} +27.0000 q^{27} +12.9972 q^{28} -29.0000 q^{29} -272.292 q^{31} +90.3007 q^{32} +99.6318 q^{33} -255.078 q^{34} +18.3737 q^{36} +143.570 q^{37} +246.969 q^{38} -22.2816 q^{39} -393.257 q^{41} -60.5227 q^{42} -122.730 q^{43} +67.8000 q^{44} -83.7334 q^{46} +226.518 q^{47} -228.493 q^{48} -302.468 q^{49} +241.487 q^{51} -15.1627 q^{52} -493.905 q^{53} -85.5586 q^{54} +120.208 q^{56} -233.811 q^{57} +91.8962 q^{58} +657.572 q^{59} -642.052 q^{61} +862.848 q^{62} +57.2980 q^{63} +323.167 q^{64} -315.717 q^{66} -339.263 q^{67} +164.334 q^{68} +79.2721 q^{69} -891.455 q^{71} +169.933 q^{72} +94.6503 q^{73} -454.950 q^{74} -159.110 q^{76} +211.434 q^{77} +70.6066 q^{78} -268.324 q^{79} +81.0000 q^{81} +1246.17 q^{82} -1224.27 q^{83} +38.9917 q^{84} +388.912 q^{86} -87.0000 q^{87} +627.064 q^{88} +632.108 q^{89} -47.2848 q^{91} +53.9452 q^{92} -816.876 q^{93} -717.797 q^{94} +270.902 q^{96} +416.938 q^{97} +958.472 q^{98} +298.895 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{2} + 18 q^{3} + 15 q^{4} + 3 q^{6} - 23 q^{7} + 51 q^{8} + 54 q^{9} - 111 q^{11} + 45 q^{12} + 83 q^{13} - 102 q^{14} - 37 q^{16} + 35 q^{17} + 9 q^{18} - 76 q^{19} - 69 q^{21} - 66 q^{22} - 166 q^{23}+ \cdots - 999 q^{99}+O(q^{100}) \) 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

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