LMFDB - Embedded newform 1305.4.a.i.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1305,4,Mod(1,1305)] 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("1305.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1305, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1305 = 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1305.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,1,0,15] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$76.9974925575$
Analytic rank:	$0$
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	$\chi$	$=$	1305.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.16884 q^{2} +2.04152 q^{4} -5.00000 q^{5} -6.36645 q^{7} +18.8814 q^{8} +15.8442 q^{10} -33.2106 q^{11} +7.42718 q^{13} +20.1742 q^{14} -76.1643 q^{16} +80.4958 q^{17} -77.9369 q^{19} -10.2076 q^{20} +105.239 q^{22} +26.4240 q^{23} +25.0000 q^{25} -23.5355 q^{26} -12.9972 q^{28} +29.0000 q^{29} -272.292 q^{31} +90.3007 q^{32} -255.078 q^{34} +31.8322 q^{35} -143.570 q^{37} +246.969 q^{38} -94.4072 q^{40} +393.257 q^{41} +122.730 q^{43} -67.8000 q^{44} -83.7334 q^{46} +226.518 q^{47} -302.468 q^{49} -79.2209 q^{50} +15.1627 q^{52} -493.905 q^{53} +166.053 q^{55} -120.208 q^{56} -91.8962 q^{58} -657.572 q^{59} -642.052 q^{61} +862.848 q^{62} +323.167 q^{64} -37.1359 q^{65} +339.263 q^{67} +164.334 q^{68} -100.871 q^{70} +891.455 q^{71} -94.6503 q^{73} +454.950 q^{74} -159.110 q^{76} +211.434 q^{77} -268.324 q^{79} +380.822 q^{80} -1246.17 q^{82} -1224.27 q^{83} -402.479 q^{85} -388.912 q^{86} -627.064 q^{88} -632.108 q^{89} -47.2848 q^{91} +53.9452 q^{92} -717.797 q^{94} +389.684 q^{95} -416.938 q^{97} +958.472 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{2} + 15 q^{4} - 30 q^{5} + 23 q^{7} + 51 q^{8} - 5 q^{10} + 111 q^{11} - 83 q^{13} + 102 q^{14} - 37 q^{16} + 35 q^{17} - 76 q^{19} - 75 q^{20} + 66 q^{22} - 166 q^{23} + 150 q^{25} + 282 q^{26}+ \cdots + 1903 q^{98}+O(q^{100}) $ 6 * q + q^2 + 15 * q^4 - 30 * q^5 + 23 * q^7 + 51 * q^8 - 5 * q^10 + 111 * q^11 - 83 * q^13 + 102 * q^14 - 37 * q^16 + 35 * q^17 - 76 * q^19 - 75 * q^20 + 66 * q^22 - 166 * q^23 + 150 * q^25 + 282 * q^26 + 368 * q^28 + 174 * q^29 - 164 * q^31 + 259 * q^32 + 512 * q^34 - 115 * q^35 - 538 * q^37 - 6 * q^38 - 255 * q^40 + 1250 * q^41 - 80 * q^43 + 1004 * q^44 - 1190 * q^46 + 491 * q^47 - 969 * q^49 + 25 * q^50 - 336 * q^52 + 356 * q^53 - 555 * q^55 + 1260 * q^56 + 29 * q^58 + 646 * q^59 - 1476 * q^61 + 4 * q^62 - 845 * q^64 + 415 * q^65 + 87 * q^67 + 558 * q^68 - 510 * q^70 + 1312 * q^71 - 1142 * q^73 + 2270 * q^74 - 1594 * q^76 + 2561 * q^77 - 142 * q^79 + 185 * q^80 - 1544 * q^82 - 346 * q^83 - 175 * q^85 + 2056 * q^86 + 1400 * q^88 + 2827 * q^89 + 1379 * q^91 + 1434 * q^92 + 978 * q^94 + 380 * q^95 - 4 * q^97 + 1903 * q^98

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	$0.689267\pi$
$3$	0	0
$3$	0	0
$4$	2.04152	0.255190
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−6.36645	−0.343756	−0.171878	−	0.985118i	$-0.554983\pi$
$7$	−6.36645	−0.343756	−0.171878	+	0.985118i	$0.554983\pi$
$8$	18.8814	0.834450
$9$	0	0
$10$	15.8442	0.501037
$11$	−33.2106	−0.910306	−0.455153	−	0.890413i	$-0.650416\pi$
$11$	−33.2106	−0.910306	−0.455153	+	0.890413i	$0.650416\pi$
$12$	0	0
$13$	7.42718	0.158456	0.0792281	−	0.996857i	$-0.474754\pi$
$13$	7.42718	0.158456	0.0792281	+	0.996857i	$0.474754\pi$
$14$	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	$0.305310\pi$
$18$	0	0
$19$	−77.9369	−0.941050	−0.470525	−	0.882387i	$-0.655935\pi$
$19$	−77.9369	−0.941050	−0.470525	+	0.882387i	$0.655935\pi$
$20$	−10.2076	−0.114124
$21$	0	0
$22$	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	$0.461782\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−23.5355	−0.177527
$27$	0	0
$28$	−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
$33$	0	0
$34$	−255.078	−1.28663
$35$	31.8322	0.153732
$36$	0	0
$37$	−143.570	−0.637913	−0.318957	−	0.947769i	$-0.603332\pi$
$37$	−143.570	−0.637913	−0.318957	+	0.947769i	$0.603332\pi$
$38$	246.969	1.05431
$39$	0	0
$40$	−94.4072	−0.373177
$41$	393.257	1.49796	0.748980	−	0.662593i	$-0.230544\pi$
$41$	393.257	1.49796	0.748980	+	0.662593i	$0.230544\pi$
$42$	0	0
$43$	122.730	0.435260	0.217630	−	0.976031i	$-0.430167\pi$
$43$	122.730	0.435260	0.217630	+	0.976031i	$0.430167\pi$
$44$	−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	$0.385672\pi$
$48$	0	0
$49$	−302.468	−0.881832
$50$	−79.2209	−0.224071
$51$	0	0
$52$	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	$0.721078\pi$
$54$	0	0
$55$	166.053	0.407101
$56$	−120.208	−0.286847
$57$	0	0
$58$	−91.8962	−0.208044
$59$	−657.572	−1.45099	−0.725497	−	0.688225i	$-0.758390\pi$
$59$	−657.572	−1.45099	−0.725497	+	0.688225i	$0.758390\pi$
$60$	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
$63$	0	0
$64$	323.167	0.631185
$65$	−37.1359	−0.0708638
$66$	0	0
$67$	339.263	0.618621	0.309311	−	0.950961i	$-0.399902\pi$
$67$	339.263	0.618621	0.309311	+	0.950961i	$0.399902\pi$
$68$	164.334	0.293064
$69$	0	0
$70$	−100.871	−0.172234
$71$	891.455	1.49009	0.745044	−	0.667015i	$-0.232429\pi$
$71$	891.455	1.49009	0.745044	+	0.667015i	$0.232429\pi$
$72$	0	0
$73$	−94.6503	−0.151753	−0.0758766	−	0.997117i	$-0.524175\pi$
$73$	−94.6503	−0.151753	−0.0758766	+	0.997117i	$0.524175\pi$
$74$	454.950	0.714688
$75$	0	0
$76$	−159.110	−0.240146
$77$	211.434	0.312923
$78$	0	0
$79$	−268.324	−0.382137	−0.191069	−	0.981577i	$-0.561195\pi$
$79$	−268.324	−0.382137	−0.191069	+	0.981577i	$0.561195\pi$
$80$	380.822	0.532215
$81$	0	0
$82$	−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	$0.800275\pi$
$84$	0	0
$85$	−402.479	−0.513588
$86$	−388.912	−0.487644
$87$	0	0
$88$	−627.064	−0.759605
$89$	−632.108	−0.752847	−0.376423	−	0.926448i	$-0.622846\pi$
$89$	−632.108	−0.752847	−0.376423	+	0.926448i	$0.622846\pi$
$90$	0	0
$91$	−47.2848	−0.0544702
$92$	53.9452	0.0611323
$93$	0	0
$94$	−717.797	−0.787608
$95$	389.684	0.420850
$96$	0	0
$97$	−416.938	−0.436429	−0.218215	−	0.975901i	$-0.570023\pi$
$97$	−416.938	−0.436429	−0.218215	+	0.975901i	$0.570023\pi$
$98$	958.472	0.987963
$99$	0	0
$100$	51.0380	0.0510380
$101$	−420.881	−0.414646	−0.207323	−	0.978273i	$-0.566475\pi$
$101$	−420.881	−0.414646	−0.207323	+	0.978273i	$0.566475\pi$
$102$	0	0
$103$	−237.251	−0.226962	−0.113481	−	0.993540i	$-0.536200\pi$
$103$	−237.251	−0.226962	−0.113481	+	0.993540i	$0.536200\pi$
$104$	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	$0.429614\pi$
$108$	0	0
$109$	1390.40	1.22180	0.610901	−	0.791707i	$-0.290807\pi$
$109$	1390.40	1.22180	0.610901	+	0.791707i	$0.290807\pi$
$110$	−526.194	−0.456097
$111$	0	0
$112$	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	$0.317910\pi$
$114$	0	0
$115$	−132.120	−0.107133
$116$	59.2040	0.0473876
$117$	0	0
$118$	2083.74	1.62562
$119$	−512.472	−0.394775
$120$	0	0
$121$	−228.057	−0.171342
$122$	2034.56	1.50984
$123$	0	0
$124$	−555.889	−0.402583
$125$	−125.000	−0.0894427
$126$	0	0
$127$	436.569	0.305034	0.152517	−	0.988301i	$-0.451262\pi$
$127$	436.569	0.305034	0.152517	+	0.988301i	$0.451262\pi$
$128$	−1746.47	−1.20600
$129$	0	0
$130$	117.678	0.0793924
$131$	1783.17	1.18929	0.594644	−	0.803989i	$-0.297293\pi$
$131$	1783.17	1.18929	0.594644	+	0.803989i	$0.297293\pi$
$132$	0	0
$133$	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	$0.817879\pi$
$138$	0	0
$139$	561.641	0.342717	0.171359	−	0.985209i	$-0.445184\pi$
$139$	561.641	0.342717	0.171359	+	0.985209i	$0.445184\pi$
$140$	64.9861	0.0392309
$141$	0	0
$142$	−2824.87	−1.66942
$143$	−246.661	−0.144244
$144$	0	0
$145$	−145.000	−0.0830455
$146$	299.931	0.170017
$147$	0	0
$148$	−293.101	−0.162789
$149$	−335.540	−0.184487	−0.0922433	−	0.995736i	$-0.529404\pi$
$149$	−335.540	−0.184487	−0.0922433	+	0.995736i	$0.529404\pi$
$150$	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
$153$	0	0
$154$	−669.998	−0.350584
$155$	1361.46	0.705516
$156$	0	0
$157$	−2418.81	−1.22957	−0.614783	−	0.788696i	$-0.710756\pi$
$157$	−2418.81	−1.22957	−0.614783	+	0.788696i	$0.710756\pi$
$158$	850.276	0.428129
$159$	0	0
$160$	−451.504	−0.223091
$161$	−168.227	−0.0823489
$162$	0	0
$163$	2974.32	1.42924	0.714622	−	0.699511i	$-0.246599\pi$
$163$	2974.32	1.42924	0.714622	+	0.699511i	$0.246599\pi$
$164$	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	$0.498787\pi$
$168$	0	0
$169$	−2141.84	−0.974892
$170$	1275.39	0.575399
$171$	0	0
$172$	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	$0.851868\pi$
$174$	0	0
$175$	−159.161	−0.0687512
$176$	2529.46	1.08333
$177$	0	0
$178$	2003.05	0.843454
$179$	512.397	0.213957	0.106979	−	0.994261i	$-0.465882\pi$
$179$	512.397	0.213957	0.106979	+	0.994261i	$0.465882\pi$
$180$	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
$183$	0	0
$184$	498.924	0.199898
$185$	717.851	0.285283
$186$	0	0
$187$	−2673.31	−1.04541
$188$	462.440	0.179398
$189$	0	0
$190$	−1234.85	−0.471501
$191$	2368.66	0.897331	0.448665	−	0.893700i	$-0.351900\pi$
$191$	2368.66	0.897331	0.448665	+	0.893700i	$0.351900\pi$
$192$	0	0
$193$	360.892	0.134599	0.0672993	−	0.997733i	$-0.478562\pi$
$193$	360.892	0.134599	0.0672993	+	0.997733i	$0.478562\pi$
$194$	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	$0.317224\pi$
$198$	0	0
$199$	3445.61	1.22740	0.613701	−	0.789538i	$-0.289680\pi$
$199$	3445.61	1.22740	0.613701	+	0.789538i	$0.289680\pi$
$200$	472.036	0.166890
$201$	0	0
$202$	1333.70	0.464550
$203$	−184.627	−0.0638339
$204$	0	0
$205$	−1966.28	−0.669908
$206$	751.810	0.254277
$207$	0	0
$208$	−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
$213$	0	0
$214$	−1538.50	−0.491446
$215$	−613.651	−0.194654
$216$	0	0
$217$	1733.53	0.542303
$218$	−4405.96	−1.36885
$219$	0	0
$220$	339.000	0.103888
$221$	597.857	0.181974
$222$	0	0
$223$	4372.83	1.31312	0.656561	−	0.754273i	$-0.272011\pi$
$223$	4372.83	1.31312	0.656561	+	0.754273i	$0.272011\pi$
$224$	−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	$0.353253\pi$
$228$	0	0
$229$	−1424.37	−0.411026	−0.205513	−	0.978654i	$-0.565886\pi$
$229$	−1424.37	−0.411026	−0.205513	+	0.978654i	$0.565886\pi$
$230$	418.667	0.120026
$231$	0	0
$232$	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	$0.233673\pi$
$234$	0	0
$235$	−1132.59	−0.314391
$236$	−1342.45	−0.370279
$237$	0	0
$238$	1623.94	0.442287
$239$	2912.01	0.788126	0.394063	−	0.919084i	$-0.371069\pi$
$239$	2912.01	0.788126	0.394063	+	0.919084i	$0.371069\pi$
$240$	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
$243$	0	0
$244$	−1310.76	−0.343905
$245$	1512.34	0.394367
$246$	0	0
$247$	−578.851	−0.149115
$248$	−5141.26	−1.31641
$249$	0	0
$250$	396.104	0.100207
$251$	−4495.12	−1.13040	−0.565198	−	0.824955i	$-0.691200\pi$
$251$	−4495.12	−1.13040	−0.565198	+	0.824955i	$0.691200\pi$
$252$	0	0
$253$	−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	$0.474801\pi$
$258$	0	0
$259$	914.032	0.219286
$260$	−75.8137	−0.0180837
$261$	0	0
$262$	−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	$0.292657\pi$
$264$	0	0
$265$	2469.52	0.572459
$266$	−1572.32	−0.362424
$267$	0	0
$268$	692.612	0.157866
$269$	4651.80	1.05437	0.527184	−	0.849751i	$-0.323248\pi$
$269$	4651.80	1.05437	0.527184	+	0.849751i	$0.323248\pi$
$270$	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
$273$	0	0
$274$	8544.20	1.88385
$275$	−830.265	−0.182061
$276$	0	0
$277$	2309.60	0.500976	0.250488	−	0.968120i	$-0.419409\pi$
$277$	2309.60	0.500976	0.250488	+	0.968120i	$0.419409\pi$
$278$	−1779.75	−0.383964
$279$	0	0
$280$	601.039	0.128282
$281$	7601.52	1.61377	0.806884	−	0.590710i	$-0.201152\pi$
$281$	7601.52	1.61377	0.806884	+	0.590710i	$0.201152\pi$
$282$	0	0
$283$	−4779.99	−1.00403	−0.502016	−	0.864858i	$-0.667408\pi$
$283$	−4779.99	−1.00403	−0.502016	+	0.864858i	$0.667408\pi$
$284$	1819.92	0.380255
$285$	0	0
$286$	781.629	0.161604
$287$	−2503.65	−0.514933
$288$	0	0
$289$	1566.57	0.318862
$290$	459.481	0.0930402
$291$	0	0
$292$	−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	$0.321906\pi$
$294$	0	0
$295$	3287.86	0.648904
$296$	−2710.81	−0.532307
$297$	0	0
$298$	1063.27	0.206690
$299$	196.256	0.0379592
$300$	0	0
$301$	−781.355	−0.149623
$302$	−8381.18	−1.59696
$303$	0	0
$304$	5936.01	1.11991
$305$	3210.26	0.602685
$306$	0	0
$307$	5559.88	1.03361	0.516807	−	0.856102i	$-0.327121\pi$
$307$	5559.88	1.03361	0.516807	+	0.856102i	$0.327121\pi$
$308$	431.645	0.0798548
$309$	0	0
$310$	−4314.24	−0.790427
$311$	7920.33	1.44412	0.722059	−	0.691831i	$-0.243196\pi$
$311$	7920.33	1.44412	0.722059	+	0.691831i	$0.243196\pi$
$312$	0	0
$313$	4963.28	0.896297	0.448148	−	0.893959i	$-0.352084\pi$
$313$	4963.28	0.896297	0.448148	+	0.893959i	$0.352084\pi$
$314$	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	$0.506926\pi$
$318$	0	0
$319$	−963.107	−0.169040
$320$	−1615.83	−0.282275
$321$	0	0
$322$	533.085	0.0922597
$323$	−6273.59	−1.08072
$324$	0	0
$325$	185.680	0.0316912
$326$	−9425.13	−1.60126
$327$	0	0
$328$	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
$333$	0	0
$334$	−52.1312	−0.00854039
$335$	−1696.32	−0.276656
$336$	0	0
$337$	5888.49	0.951830	0.475915	−	0.879491i	$-0.342117\pi$
$337$	5888.49	0.951830	0.475915	+	0.879491i	$0.342117\pi$
$338$	6787.13	1.09222
$339$	0	0
$340$	−821.668	−0.131062
$341$	9042.97	1.43608
$342$	0	0
$343$	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	$0.475349\pi$
$348$	0	0
$349$	−7602.06	−1.16599	−0.582993	−	0.812477i	$-0.698118\pi$
$349$	−7602.06	−1.16599	−0.582993	+	0.812477i	$0.698118\pi$
$350$	504.356	0.0770256
$351$	0	0
$352$	−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	$0.498257\pi$
$354$	0	0
$355$	−4457.27	−0.666388
$356$	−1290.46	−0.192119
$357$	0	0
$358$	−1623.70	−0.239707
$359$	4787.09	0.703768	0.351884	−	0.936044i	$-0.385541\pi$
$359$	4787.09	0.703768	0.351884	+	0.936044i	$0.385541\pi$
$360$	0	0
$361$	−784.845	−0.114426
$362$	6008.21	0.872333
$363$	0	0
$364$	−96.5327	−0.0139002
$365$	473.251	0.0678661
$366$	0	0
$367$	−8456.98	−1.20286	−0.601432	−	0.798924i	$-0.705403\pi$
$367$	−8456.98	−1.20286	−0.601432	+	0.798924i	$0.705403\pi$
$368$	−2012.57	−0.285088
$369$	0	0
$370$	−2274.75	−0.319618
$371$	3144.42	0.440027
$372$	0	0
$373$	−109.448	−0.0151930	−0.00759652	−	0.999971i	$-0.502418\pi$
$373$	−109.448	−0.0151930	−0.00759652	+	0.999971i	$0.502418\pi$
$374$	8471.28	1.17123
$375$	0	0
$376$	4276.98	0.586618
$377$	215.388	0.0294246
$378$	0	0
$379$	1446.28	0.196017	0.0980087	−	0.995186i	$-0.468753\pi$
$379$	1446.28	0.196017	0.0980087	+	0.995186i	$0.468753\pi$
$380$	795.548	0.107397
$381$	0	0
$382$	−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	$0.346445\pi$
$384$	0	0
$385$	−1057.17	−0.139943
$386$	−1143.61	−0.150798
$387$	0	0
$388$	−851.187	−0.111372
$389$	−1771.46	−0.230892	−0.115446	−	0.993314i	$-0.536830\pi$
$389$	−1771.46	−0.230892	−0.115446	+	0.993314i	$0.536830\pi$
$390$	0	0
$391$	2127.02	0.275110
$392$	−5711.04	−0.735845
$393$	0	0
$394$	−9518.43	−1.21708
$395$	1341.62	0.170897
$396$	0	0
$397$	8022.99	1.01426	0.507131	−	0.861869i	$-0.330706\pi$
$397$	8022.99	1.01426	0.507131	+	0.861869i	$0.330706\pi$
$398$	−10918.6	−1.37512
$399$	0	0
$400$	−1904.11	−0.238014
$401$	2268.78	0.282537	0.141268	−	0.989971i	$-0.454882\pi$
$401$	2268.78	0.282537	0.141268	+	0.989971i	$0.454882\pi$
$402$	0	0
$403$	−2022.36	−0.249978
$404$	−859.236	−0.105813
$405$	0	0
$406$	585.053	0.0715164
$407$	4768.05	0.580696
$408$	0	0
$409$	−11953.1	−1.44509	−0.722545	−	0.691324i	$-0.757028\pi$
$409$	−11953.1	−1.44509	−0.722545	+	0.691324i	$0.757028\pi$
$410$	6230.83	0.750533
$411$	0	0
$412$	−484.353	−0.0579183
$413$	4186.40	0.498788
$414$	0	0
$415$	6121.35	0.724061
$416$	670.680	0.0790452
$417$	0	0
$418$	−8201.99	−0.959743
$419$	10234.8	1.19333	0.596665	−	0.802491i	$-0.296492\pi$
$419$	10234.8	1.19333	0.596665	+	0.802491i	$0.296492\pi$
$420$	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
$423$	0	0
$424$	−9325.64	−1.06814
$425$	2012.39	0.229683
$426$	0	0
$427$	4087.59	0.463261
$428$	991.174	0.111940
$429$	0	0
$430$	1944.56	0.218081
$431$	−4238.72	−0.473717	−0.236858	−	0.971544i	$-0.576118\pi$
$431$	−4238.72	−0.473717	−0.236858	+	0.971544i	$0.576118\pi$
$432$	0	0
$433$	−13631.8	−1.51294	−0.756471	−	0.654027i	$-0.773078\pi$
$433$	−13631.8	−1.51294	−0.756471	+	0.654027i	$0.773078\pi$
$434$	−5493.28	−0.607571
$435$	0	0
$436$	2838.53	0.311791
$437$	−2059.41	−0.225434
$438$	0	0
$439$	725.415	0.0788660	0.0394330	−	0.999222i	$-0.487445\pi$
$439$	725.415	0.0788660	0.0394330	+	0.999222i	$0.487445\pi$
$440$	3135.32	0.339706
$441$	0	0
$442$	−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	$0.779643\pi$
$444$	0	0
$445$	3160.54	0.336683
$446$	−13856.8	−1.47116
$447$	0	0
$448$	−2057.42	−0.216974
$449$	−519.743	−0.0546285	−0.0273142	−	0.999627i	$-0.508695\pi$
$449$	−519.743	−0.0546285	−0.0273142	+	0.999627i	$0.508695\pi$
$450$	0	0
$451$	−13060.3	−1.36360
$452$	2655.13	0.276298
$453$	0	0
$454$	−9642.57	−0.996803
$455$	236.424	0.0243598
$456$	0	0
$457$	5739.92	0.587532	0.293766	−	0.955877i	$-0.405091\pi$
$457$	5739.92	0.587532	0.293766	+	0.955877i	$0.405091\pi$
$458$	4513.60	0.460495
$459$	0	0
$460$	−269.726	−0.0273392
$461$	10071.6	1.01753	0.508763	−	0.860907i	$-0.330103\pi$
$461$	10071.6	1.01753	0.508763	+	0.860907i	$0.330103\pi$
$462$	0	0
$463$	6265.21	0.628874	0.314437	−	0.949278i	$-0.398184\pi$
$463$	6265.21	0.628874	0.314437	+	0.949278i	$0.398184\pi$
$464$	−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	$0.830429\pi$
$468$	0	0
$469$	−2159.90	−0.212655
$470$	3588.99	0.352229
$471$	0	0
$472$	−12415.9	−1.21078
$473$	−4075.94	−0.396220
$474$	0	0
$475$	−1948.42	−0.188210
$476$	−1046.22	−0.100743
$477$	0	0
$478$	−9227.67	−0.882979
$479$	15957.4	1.52216	0.761080	−	0.648658i	$-0.224670\pi$
$479$	15957.4	1.52216	0.761080	+	0.648658i	$0.224670\pi$
$480$	0	0
$481$	−1066.32	−0.101081
$482$	1373.69	0.129813
$483$	0	0
$484$	−465.582	−0.0437248
$485$	2084.69	0.195177
$486$	0	0
$487$	−15053.5	−1.40070	−0.700350	−	0.713800i	$-0.746973\pi$
$487$	−15053.5	−1.40070	−0.700350	+	0.713800i	$0.746973\pi$
$488$	−12122.9	−1.12454
$489$	0	0
$490$	−4792.36	−0.441830
$491$	12513.2	1.15013	0.575066	−	0.818107i	$-0.304976\pi$
$491$	12513.2	1.15013	0.575066	+	0.818107i	$0.304976\pi$
$492$	0	0
$493$	2334.38	0.213256
$494$	1834.28	0.167062
$495$	0	0
$496$	20738.9	1.87743
$497$	−5675.40	−0.512226
$498$	0	0
$499$	−2702.24	−0.242423	−0.121211	−	0.992627i	$-0.538678\pi$
$499$	−2702.24	−0.242423	−0.121211	+	0.992627i	$0.538678\pi$
$500$	−255.190	−0.0228249
$501$	0	0
$502$	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	$0.710620\pi$
$504$	0	0
$505$	2104.41	0.185435
$506$	2780.84	0.244315
$507$	0	0
$508$	891.264	0.0778415
$509$	−13363.9	−1.16374	−0.581871	−	0.813281i	$-0.697679\pi$
$509$	−13363.9	−1.16374	−0.581871	+	0.813281i	$0.697679\pi$
$510$	0	0
$511$	602.586	0.0521660
$512$	4627.05	0.399392
$513$	0	0
$514$	−2064.99	−0.177204
$515$	1186.26	0.101500
$516$	0	0
$517$	−7522.78	−0.639945
$518$	−2896.42	−0.245678
$519$	0	0
$520$	−701.180	−0.0591323
$521$	11833.2	0.995050	0.497525	−	0.867450i	$-0.334242\pi$
$521$	11833.2	0.995050	0.497525	+	0.867450i	$0.334242\pi$
$522$	0	0
$523$	3346.53	0.279796	0.139898	−	0.990166i	$-0.455323\pi$
$523$	3346.53	0.279796	0.139898	+	0.990166i	$0.455323\pi$
$524$	3640.38	0.303494
$525$	0	0
$526$	−16388.7	−1.35852
$527$	−21918.3	−1.81172
$528$	0	0
$529$	−11468.8	−0.942613
$530$	−7825.51	−0.641356
$531$	0	0
$532$	1012.96	0.0825517
$533$	2920.79	0.237361
$534$	0	0
$535$	−2427.54	−0.196171
$536$	6405.78	0.516208
$537$	0	0
$538$	−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
$543$	0	0
$544$	7268.82	0.572883
$545$	−6952.01	−0.546406
$546$	0	0
$547$	11807.1	0.922916	0.461458	−	0.887162i	$-0.347326\pi$
$547$	11807.1	0.922916	0.461458	+	0.887162i	$0.347326\pi$
$548$	−5504.59	−0.429096
$549$	0	0
$550$	2630.97	0.203973
$551$	−2260.17	−0.174749
$552$	0	0
$553$	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	$0.665761\pi$
$558$	0	0
$559$	911.539	0.0689696
$560$	−2424.48	−0.182952
$561$	0	0
$562$	−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	$0.601189\pi$
$564$	0	0
$565$	−6502.84	−0.484206
$566$	15147.0	1.12487
$567$	0	0
$568$	16832.0	1.24340
$569$	−16013.7	−1.17984	−0.589921	−	0.807461i	$-0.700841\pi$
$569$	−16013.7	−1.17984	−0.589921	+	0.807461i	$0.700841\pi$
$570$	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
$573$	0	0
$574$	7933.65	0.576906
$575$	660.601	0.0479112
$576$	0	0
$577$	−22764.3	−1.64244	−0.821222	−	0.570610i	$-0.806707\pi$
$577$	−22764.3	−1.64244	−0.821222	+	0.570610i	$0.806707\pi$
$578$	−4964.19	−0.357237
$579$	0	0
$580$	−296.020	−0.0211924
$581$	7794.26	0.556558
$582$	0	0
$583$	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	$0.801773\pi$
$588$	0	0
$589$	21221.6	1.48458
$590$	−10418.7	−0.727001
$591$	0	0
$592$	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	$0.268194\pi$
$594$	0	0
$595$	2562.36	0.176549
$596$	−685.011	−0.0470791
$597$	0	0
$598$	−621.904	−0.0425276
$599$	−8313.99	−0.567113	−0.283556	−	0.958956i	$-0.591514\pi$
$599$	−8313.99	−0.567113	−0.283556	+	0.958956i	$0.591514\pi$
$600$	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
$603$	0	0
$604$	5399.56	0.363750
$605$	1140.28	0.0766266
$606$	0	0
$607$	−2133.39	−0.142655	−0.0713275	−	0.997453i	$-0.522724\pi$
$607$	−2133.39	−0.142655	−0.0713275	+	0.997453i	$0.522724\pi$
$608$	−7037.75	−0.469438
$609$	0	0
$610$	−10172.8	−0.675220
$611$	1682.39	0.111395
$612$	0	0
$613$	12506.9	0.824062	0.412031	−	0.911170i	$-0.364820\pi$
$613$	12506.9	0.824062	0.412031	+	0.911170i	$0.364820\pi$
$614$	−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	$0.442992\pi$
$618$	0	0
$619$	7635.72	0.495809	0.247904	−	0.968785i	$-0.420258\pi$
$619$	7635.72	0.495809	0.247904	+	0.968785i	$0.420258\pi$
$620$	2779.44	0.180041
$621$	0	0
$622$	−25098.2	−1.61792
$623$	4024.29	0.258796
$624$	0	0
$625$	625.000	0.0400000
$626$	−15727.8	−1.00417
$627$	0	0
$628$	−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
$633$	0	0
$634$	778.269	0.0487524
$635$	−2182.85	−0.136415
$636$	0	0
$637$	−2246.49	−0.139732
$638$	3051.93	0.189384
$639$	0	0
$640$	8732.34	0.539338
$641$	−16935.0	−1.04352	−0.521758	−	0.853093i	$-0.674724\pi$
$641$	−16935.0	−1.04352	−0.521758	+	0.853093i	$0.674724\pi$
$642$	0	0
$643$	20875.1	1.28030	0.640150	−	0.768250i	$-0.278872\pi$
$643$	20875.1	1.28030	0.640150	+	0.768250i	$0.278872\pi$
$644$	−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	$0.723649\pi$
$648$	0	0
$649$	21838.4	1.32085
$650$	−588.388	−0.0355054
$651$	0	0
$652$	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	$0.643058\pi$
$654$	0	0
$655$	−8915.87	−0.531865
$656$	−29952.1	−1.78267
$657$	0	0
$658$	4569.82	0.270745
$659$	2246.04	0.132767	0.0663833	−	0.997794i	$-0.478854\pi$
$659$	2246.04	0.132767	0.0663833	+	0.997794i	$0.478854\pi$
$660$	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
$663$	0	0
$664$	−23116.0	−1.35102
$665$	−2480.91	−0.144670
$666$	0	0
$667$	766.297	0.0444845
$668$	33.5854	0.00194530
$669$	0	0
$670$	5375.35	0.309952
$671$	21322.9	1.22677
$672$	0	0
$673$	14251.2	0.816260	0.408130	−	0.912924i	$-0.366181\pi$
$673$	14251.2	0.816260	0.408130	+	0.912924i	$0.366181\pi$
$674$	−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	$0.571662\pi$
$678$	0	0
$679$	2654.41	0.150025
$680$	−7599.38	−0.428563
$681$	0	0
$682$	−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	$0.367174\pi$
$684$	0	0
$685$	13481.6	0.751979
$686$	−13021.8	−0.724746
$687$	0	0
$688$	−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
$693$	0	0
$694$	−3169.32	−0.173351
$695$	−2808.20	−0.153268
$696$	0	0
$697$	31655.5	1.72028
$698$	24089.7	1.30632
$699$	0	0
$700$	−324.930	−0.0175446
$701$	9717.44	0.523570	0.261785	−	0.965126i	$-0.415689\pi$
$701$	9717.44	0.523570	0.261785	+	0.965126i	$0.415689\pi$
$702$	0	0
$703$	11189.4	0.600308
$704$	−10732.6	−0.574572
$705$	0	0
$706$	−230.103	−0.0122664
$707$	2679.52	0.142537
$708$	0	0
$709$	−15655.2	−0.829259	−0.414629	−	0.909990i	$-0.636089\pi$
$709$	−15655.2	−0.829259	−0.414629	+	0.909990i	$0.636089\pi$
$710$	14124.4	0.746589
$711$	0	0
$712$	−11935.1	−0.628213
$713$	−7195.05	−0.377920
$714$	0	0
$715$	1233.31	0.0645077
$716$	1046.07	0.0545997
$717$	0	0
$718$	−15169.5	−0.788469
$719$	36868.3	1.91232	0.956158	−	0.292853i	$-0.0946045\pi$
$719$	36868.3	1.91232	0.956158	+	0.292853i	$0.0946045\pi$
$720$	0	0
$721$	1510.45	0.0780195
$722$	2487.04	0.128197
$723$	0	0
$724$	−3870.78	−0.198697
$725$	725.000	0.0371391
$726$	0	0
$727$	−36049.7	−1.83908	−0.919540	−	0.392996i	$-0.871438\pi$
$727$	−36049.7	−1.83908	−0.919540	+	0.392996i	$0.871438\pi$
$728$	−892.805	−0.0454527
$729$	0	0
$730$	−1499.66	−0.0760339
$731$	9879.26	0.499860
$732$	0	0
$733$	−4057.83	−0.204474	−0.102237	−	0.994760i	$-0.532600\pi$
$733$	−4057.83	−0.204474	−0.102237	+	0.994760i	$0.532600\pi$
$734$	26798.8	1.34763
$735$	0	0
$736$	2386.11	0.119502
$737$	−11267.1	−0.563135
$738$	0	0
$739$	−7614.89	−0.379050	−0.189525	−	0.981876i	$-0.560695\pi$
$739$	−7614.89	−0.379050	−0.189525	+	0.981876i	$0.560695\pi$
$740$	1465.51	0.0728014
$741$	0	0
$742$	−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	$0.641300\pi$
$744$	0	0
$745$	1677.70	0.0825049
$746$	346.823	0.0170216
$747$	0	0
$748$	−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
$753$	0	0
$754$	−682.530	−0.0329659
$755$	−13224.4	−0.637463
$756$	0	0
$757$	−27616.9	−1.32596	−0.662982	−	0.748636i	$-0.730709\pi$
$757$	−27616.9	−1.32596	−0.662982	+	0.748636i	$0.730709\pi$
$758$	−4583.03	−0.219608
$759$	0	0
$760$	7357.80	0.351178
$761$	19746.5	0.940617	0.470309	−	0.882502i	$-0.344143\pi$
$761$	19746.5	0.940617	0.470309	+	0.882502i	$0.344143\pi$
$762$	0	0
$763$	−8851.93	−0.420002
$764$	4835.66	0.228990
$765$	0	0
$766$	−22037.6	−1.03949
$767$	−4883.91	−0.229919
$768$	0	0
$769$	33709.4	1.58075	0.790373	−	0.612627i	$-0.209887\pi$
$769$	33709.4	1.58075	0.790373	+	0.612627i	$0.209887\pi$
$770$	3349.99	0.156786
$771$	0	0
$772$	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	$0.127291\pi$
$774$	0	0
$775$	−6807.30	−0.315517
$776$	−7872.40	−0.364179
$777$	0	0
$778$	5613.48	0.258680
$779$	−30649.2	−1.40965
$780$	0	0
$781$	−29605.7	−1.35644
$782$	−6740.19	−0.308221
$783$	0	0
$784$	23037.3	1.04944
$785$	12094.0	0.549879
$786$	0	0
$787$	1728.19	0.0782760	0.0391380	−	0.999234i	$-0.487539\pi$
$787$	1728.19	0.0782760	0.0391380	+	0.999234i	$0.487539\pi$
$788$	6132.23	0.277223
$789$	0	0
$790$	−4251.38	−0.191465
$791$	−8279.99	−0.372191
$792$	0	0
$793$	−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	$0.195246\pi$
$798$	0	0
$799$	18233.7	0.807337
$800$	2257.52	0.0997691
$801$	0	0
$802$	−7189.38	−0.316541
$803$	3143.39	0.138142
$804$	0	0
$805$	841.136	0.0368275
$806$	6408.53	0.280063
$807$	0	0
$808$	−7946.85	−0.346001
$809$	26350.1	1.14514	0.572572	−	0.819854i	$-0.305946\pi$
$809$	26350.1	1.14514	0.572572	+	0.819854i	$0.305946\pi$
$810$	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
$813$	0	0
$814$	−15109.2	−0.650585
$815$	−14871.6	−0.639177
$816$	0	0
$817$	−9565.20	−0.409601
$818$	37877.3	1.61901
$819$	0	0
$820$	−4014.20	−0.170954
$821$	−9821.87	−0.417522	−0.208761	−	0.977967i	$-0.566943\pi$
$821$	−9821.87	−0.417522	−0.208761	+	0.977967i	$0.566943\pi$
$822$	0	0
$823$	23701.0	1.00384	0.501922	−	0.864913i	$-0.332627\pi$
$823$	23701.0	1.00384	0.501922	+	0.864913i	$0.332627\pi$
$824$	−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	$0.347262\pi$
$828$	0	0
$829$	16697.5	0.699553	0.349776	−	0.936833i	$-0.386258\pi$
$829$	16697.5	0.699553	0.349776	+	0.936833i	$0.386258\pi$
$830$	−19397.6	−0.811204
$831$	0	0
$832$	2400.22	0.100015
$833$	−24347.4	−1.01271
$834$	0	0
$835$	−82.2560	−0.00340909
$836$	5284.12	0.218607
$837$	0	0
$838$	−32432.5	−1.33695
$839$	−33590.2	−1.38220	−0.691098	−	0.722761i	$-0.742873\pi$
$839$	−33590.2	−1.38220	−0.691098	+	0.722761i	$0.742873\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	6003.99	0.245738
$843$	0	0
$844$	7933.39	0.323553
$845$	10709.2	0.435985
$846$	0	0
$847$	1451.91	0.0588999
$848$	37617.9	1.52336
$849$	0	0
$850$	−6376.95	−0.257326
$851$	−3793.70	−0.152816
$852$	0	0
$853$	5408.43	0.217094	0.108547	−	0.994091i	$-0.465380\pi$
$853$	5408.43	0.217094	0.108547	+	0.994091i	$0.465380\pi$
$854$	−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	$0.368124\pi$
$858$	0	0
$859$	38903.1	1.54523	0.772617	−	0.634872i	$-0.218947\pi$
$859$	38903.1	1.54523	0.772617	+	0.634872i	$0.218947\pi$
$860$	−1252.78	−0.0496737
$861$	0	0
$862$	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	$0.253417\pi$
$864$	0	0
$865$	20334.8	0.799310
$866$	43197.0	1.69503
$867$	0	0
$868$	3539.04	0.138390
$869$	8911.21	0.347862
$870$	0	0
$871$	2519.77	0.0980243
$872$	26252.8	1.01953
$873$	0	0
$874$	6525.92	0.252566
$875$	795.806	0.0307465
$876$	0	0
$877$	−2742.28	−0.105588	−0.0527938	−	0.998605i	$-0.516813\pi$
$877$	−2742.28	−0.105588	−0.0527938	+	0.998605i	$0.516813\pi$
$878$	−2298.72	−0.0883577
$879$	0	0
$880$	−12647.3	−0.484478
$881$	25595.0	0.978794	0.489397	−	0.872061i	$-0.337217\pi$
$881$	25595.0	0.978794	0.489397	+	0.872061i	$0.337217\pi$
$882$	0	0
$883$	−22207.0	−0.846347	−0.423173	−	0.906049i	$-0.639084\pi$
$883$	−22207.0	−0.846347	−0.423173	+	0.906049i	$0.639084\pi$
$884$	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	$0.636417\pi$
$888$	0	0
$889$	−2779.40	−0.104857
$890$	−10015.2	−0.377204
$891$	0	0
$892$	8927.20	0.335095
$893$	−17654.1	−0.661558
$894$	0	0
$895$	−2561.98	−0.0956845
$896$	11118.8	0.414568
$897$	0	0
$898$	1646.98	0.0612031
$899$	−7896.46	−0.292950
$900$	0	0
$901$	−39757.2	−1.47004
$902$	41385.9	1.52772
$903$	0	0
$904$	24556.6	0.903474
$905$	9480.15	0.348211
$906$	0	0
$907$	−10598.6	−0.388006	−0.194003	−	0.981001i	$-0.562147\pi$
$907$	−10598.6	−0.388006	−0.194003	+	0.981001i	$0.562147\pi$
$908$	6212.22	0.227048
$909$	0	0
$910$	−749.188	−0.0272916
$911$	−28026.0	−1.01926	−0.509628	−	0.860395i	$-0.670217\pi$
$911$	−28026.0	−1.01926	−0.509628	+	0.860395i	$0.670217\pi$
$912$	0	0
$913$	40658.8	1.47383
$914$	−18188.8	−0.658242
$915$	0	0
$916$	−2907.88	−0.104890
$917$	−11352.5	−0.408824
$918$	0	0
$919$	21116.9	0.757977	0.378989	−	0.925401i	$-0.376272\pi$
$919$	21116.9	0.757977	0.378989	+	0.925401i	$0.376272\pi$
$920$	−2494.62	−0.0893970
$921$	0	0
$922$	−31915.1	−1.13999
$923$	6621.00	0.236114
$924$	0	0
$925$	−3589.25	−0.127583
$926$	−19853.4	−0.704561
$927$	0	0
$928$	2618.72	0.0926333
$929$	−15051.0	−0.531548	−0.265774	−	0.964035i	$-0.585628\pi$
$929$	−15051.0	−0.531548	−0.265774	+	0.964035i	$0.585628\pi$
$930$	0	0
$931$	23573.4	0.829848
$932$	10781.3	0.378921
$933$	0	0
$934$	55096.4	1.93020
$935$	13366.6	0.467522
$936$	0	0
$937$	−35687.4	−1.24424	−0.622122	−	0.782920i	$-0.713729\pi$
$937$	−35687.4	−1.24424	−0.622122	+	0.782920i	$0.713729\pi$
$938$	6844.38	0.238248
$939$	0	0
$940$	−2312.20	−0.0802294
$941$	22876.5	0.792510	0.396255	−	0.918141i	$-0.370310\pi$
$941$	22876.5	0.792510	0.396255	+	0.918141i	$0.370310\pi$
$942$	0	0
$943$	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	$0.444590\pi$
$948$	0	0
$949$	−702.985	−0.0240462
$950$	6174.23	0.210861
$951$	0	0
$952$	−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	$0.631193\pi$
$954$	0	0
$955$	−11843.3	−0.401299
$956$	5944.91	0.201122
$957$	0	0
$958$	−50566.5	−1.70536
$959$	17166.0	0.578018
$960$	0	0
$961$	44351.9	1.48877
$962$	3379.00	0.113247
$963$	0	0
$964$	−884.996	−0.0295683
$965$	−1804.46	−0.0601944
$966$	0	0
$967$	18620.8	0.619239	0.309620	−	0.950860i	$-0.399798\pi$
$967$	18620.8	0.619239	0.309620	+	0.950860i	$0.399798\pi$
$968$	−4306.04	−0.142977
$969$	0	0
$970$	−6606.04	−0.218667
$971$	33876.6	1.11962	0.559811	−	0.828621i	$-0.310874\pi$
$971$	33876.6	1.11962	0.559811	+	0.828621i	$0.310874\pi$
$972$	0	0
$973$	−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	$0.424363\pi$
$978$	0	0
$979$	20992.7	0.685321
$980$	3087.47	0.100638
$981$	0	0
$982$	−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	$0.320770\pi$
$984$	0	0
$985$	−15018.8	−0.485826
$986$	−7397.26	−0.238922
$987$	0	0
$988$	−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
$993$	0	0
$994$	17984.4	0.573874
$995$	−17228.1	−0.548911
$996$	0	0
$997$	−29384.9	−0.933431	−0.466715	−	0.884408i	$-0.654563\pi$
$997$	−29384.9	−0.933431	−0.466715	+	0.884408i	$0.654563\pi$
$998$	8562.97	0.271599
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.4.a.g.1.5	✓	6	3.2	odd	2
1305.4.a.i.1.2		6	1.1	even	1	trivial
2175.4.a.l.1.2		6	15.14	odd	2

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:	\( N \)	\(=\)	\( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1305.a (trivial)

\(f(q)\)	\(=\)	\(q-3.16884 q^{2} +2.04152 q^{4} -5.00000 q^{5} -6.36645 q^{7} +18.8814 q^{8} +15.8442 q^{10} -33.2106 q^{11} +7.42718 q^{13} +20.1742 q^{14} -76.1643 q^{16} +80.4958 q^{17} -77.9369 q^{19} -10.2076 q^{20} +105.239 q^{22} +26.4240 q^{23} +25.0000 q^{25} -23.5355 q^{26} -12.9972 q^{28} +29.0000 q^{29} -272.292 q^{31} +90.3007 q^{32} -255.078 q^{34} +31.8322 q^{35} -143.570 q^{37} +246.969 q^{38} -94.4072 q^{40} +393.257 q^{41} +122.730 q^{43} -67.8000 q^{44} -83.7334 q^{46} +226.518 q^{47} -302.468 q^{49} -79.2209 q^{50} +15.1627 q^{52} -493.905 q^{53} +166.053 q^{55} -120.208 q^{56} -91.8962 q^{58} -657.572 q^{59} -642.052 q^{61} +862.848 q^{62} +323.167 q^{64} -37.1359 q^{65} +339.263 q^{67} +164.334 q^{68} -100.871 q^{70} +891.455 q^{71} -94.6503 q^{73} +454.950 q^{74} -159.110 q^{76} +211.434 q^{77} -268.324 q^{79} +380.822 q^{80} -1246.17 q^{82} -1224.27 q^{83} -402.479 q^{85} -388.912 q^{86} -627.064 q^{88} -632.108 q^{89} -47.2848 q^{91} +53.9452 q^{92} -717.797 q^{94} +389.684 q^{95} -416.938 q^{97} +958.472 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{2} + 15 q^{4} - 30 q^{5} + 23 q^{7} + 51 q^{8} - 5 q^{10} + 111 q^{11} - 83 q^{13} + 102 q^{14} - 37 q^{16} + 35 q^{17} - 76 q^{19} - 75 q^{20} + 66 q^{22} - 166 q^{23} + 150 q^{25} + 282 q^{26}+ \cdots + 1903 q^{98}+O(q^{100}) \) 6 * q + q^2 + 15 * q^4 - 30 * q^5 + 23 * q^7 + 51 * q^8 - 5 * q^10 + 111 * q^11 - 83 * q^13 + 102 * q^14 - 37 * q^16 + 35 * q^17 - 76 * q^19 - 75 * q^20 + 66 * q^22 - 166 * q^23 + 150 * q^25 + 282 * q^26 + 368 * q^28 + 174 * q^29 - 164 * q^31 + 259 * q^32 + 512 * q^34 - 115 * q^35 - 538 * q^37 - 6 * q^38 - 255 * q^40 + 1250 * q^41 - 80 * q^43 + 1004 * q^44 - 1190 * q^46 + 491 * q^47 - 969 * q^49 + 25 * q^50 - 336 * q^52 + 356 * q^53 - 555 * q^55 + 1260 * q^56 + 29 * q^58 + 646 * q^59 - 1476 * q^61 + 4 * q^62 - 845 * q^64 + 415 * q^65 + 87 * q^67 + 558 * q^68 - 510 * q^70 + 1312 * q^71 - 1142 * q^73 + 2270 * q^74 - 1594 * q^76 + 2561 * q^77 - 142 * q^79 + 185 * q^80 - 1544 * q^82 - 346 * q^83 - 175 * q^85 + 2056 * q^86 + 1400 * q^88 + 2827 * q^89 + 1379 * q^91 + 1434 * q^92 + 978 * q^94 + 380 * q^95 - 4 * q^97 + 1903 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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