LMFDB - Embedded newform 2175.2.a.u.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2175,2,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, 2); 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, 2, names="a")

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

Newform invariants

comment:select newform

sage:traces = [3,-1,3,5,0,-1,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$17.3674624396$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.469.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 4 $ x^3 - x^2 - 5*x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.39138$ of defining polynomial
Character	$\chi$	$=$	2175.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.39138 q^{2} +1.00000 q^{3} +3.71871 q^{4} -2.39138 q^{6} +1.32733 q^{7} -4.11009 q^{8} +1.00000 q^{9} +3.00000 q^{11} +3.71871 q^{12} -6.11009 q^{13} -3.17415 q^{14} +2.39138 q^{16} -0.672673 q^{17} -2.39138 q^{18} -5.43742 q^{19} +1.32733 q^{21} -7.17415 q^{22} +7.89286 q^{23} -4.11009 q^{24} +14.6116 q^{26} +1.00000 q^{27} +4.93594 q^{28} -1.00000 q^{29} +2.50147 q^{32} +3.00000 q^{33} +1.60862 q^{34} +3.71871 q^{36} +3.89286 q^{37} +13.0029 q^{38} -6.11009 q^{39} +4.32733 q^{41} -3.17415 q^{42} +8.45544 q^{43} +11.1561 q^{44} -18.8748 q^{46} -4.76475 q^{47} +2.39138 q^{48} -5.23820 q^{49} -0.672673 q^{51} -22.7217 q^{52} +13.3303 q^{53} -2.39138 q^{54} -5.45544 q^{56} -5.43742 q^{57} +2.39138 q^{58} +3.21724 q^{59} +7.43742 q^{61} +1.32733 q^{63} -10.7647 q^{64} -7.17415 q^{66} +1.10714 q^{67} -2.50147 q^{68} +7.89286 q^{69} +12.3483 q^{71} -4.11009 q^{72} -9.20217 q^{73} -9.30931 q^{74} -20.2202 q^{76} +3.98198 q^{77} +14.6116 q^{78} +11.6576 q^{79} +1.00000 q^{81} -10.3483 q^{82} +0.889908 q^{83} +4.93594 q^{84} -20.2202 q^{86} -1.00000 q^{87} -12.3303 q^{88} -2.33028 q^{89} -8.11009 q^{91} +29.3512 q^{92} +11.3943 q^{94} +2.50147 q^{96} +12.6756 q^{97} +12.5265 q^{98} +3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 3 q^{3} + 5 q^{4} - q^{6} + 4 q^{7} + 3 q^{9} + 9 q^{11} + 5 q^{12} - 6 q^{13} + 9 q^{14} + q^{16} - 2 q^{17} - q^{18} - 4 q^{19} + 4 q^{21} - 3 q^{22} - q^{23} + 13 q^{26} + 3 q^{27}+ \cdots + 9 q^{99}+O(q^{100}) $ 3 * q - q^2 + 3 * q^3 + 5 * q^4 - q^6 + 4 * q^7 + 3 * q^9 + 9 * q^11 + 5 * q^12 - 6 * q^13 + 9 * q^14 + q^16 - 2 * q^17 - q^18 - 4 * q^19 + 4 * q^21 - 3 * q^22 - q^23 + 13 * q^26 + 3 * q^27 + 21 * q^28 - 3 * q^29 - 11 * q^32 + 9 * q^33 + 11 * q^34 + 5 * q^36 - 13 * q^37 + 2 * q^38 - 6 * q^39 + 13 * q^41 + 9 * q^42 + 13 * q^43 + 15 * q^44 - 32 * q^46 - 2 * q^47 + q^48 + 9 * q^49 - 2 * q^51 - 25 * q^52 + 3 * q^53 - q^54 - 4 * q^56 - 4 * q^57 + q^58 + 22 * q^59 + 10 * q^61 + 4 * q^63 - 20 * q^64 - 3 * q^66 + 28 * q^67 + 11 * q^68 - q^69 - 3 * q^73 - 28 * q^74 - 36 * q^76 + 12 * q^77 + 13 * q^78 - 2 * q^79 + 3 * q^81 + 6 * q^82 + 15 * q^83 + 21 * q^84 - 36 * q^86 - 3 * q^87 + 30 * q^89 - 12 * q^91 + 14 * q^92 - 9 * q^94 - 11 * q^96 + q^97 + 50 * q^98 + 9 * q^99

Coefficient data

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

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

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

$2$	−2.39138	−1.69096	−0.845481	−	0.534005i	$-0.820686\pi$
$2$	−2.39138	−1.69096	−0.845481	+	0.534005i	$0.820686\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	3.71871	1.85935
$5$	0	0
$5$	0	0
$6$	−2.39138	−0.976278
$7$	1.32733	0.501683	0.250841	−	0.968028i	$-0.419293\pi$
$7$	1.32733	0.501683	0.250841	+	0.968028i	$0.419293\pi$
$8$	−4.11009	−1.45314
$9$	1.00000	0.333333
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	3.71871	1.07350
$13$	−6.11009	−1.69463	−0.847317	−	0.531087i	$-0.821784\pi$
$13$	−6.11009	−1.69463	−0.847317	+	0.531087i	$0.821784\pi$
$14$	−3.17415	−0.848327
$15$	0	0
$16$	2.39138	0.597846
$17$	−0.672673	−0.163147	−0.0815735	−	0.996667i	$-0.525995\pi$
$17$	−0.672673	−0.163147	−0.0815735	+	0.996667i	$0.525995\pi$
$18$	−2.39138	−0.563654
$19$	−5.43742	−1.24743	−0.623715	−	0.781652i	$-0.714377\pi$
$19$	−5.43742	−1.24743	−0.623715	+	0.781652i	$0.714377\pi$
$20$	0	0
$21$	1.32733	0.289647
$22$	−7.17415	−1.52953
$23$	7.89286	1.64577	0.822887	−	0.568205i	$-0.192362\pi$
$23$	7.89286	1.64577	0.822887	+	0.568205i	$0.192362\pi$
$24$	−4.11009	−0.838969
$25$	0	0
$26$	14.6116	2.86556
$27$	1.00000	0.192450
$28$	4.93594	0.932806
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	2.50147	0.442202
$33$	3.00000	0.522233
$34$	1.60862	0.275876
$35$	0	0
$36$	3.71871	0.619785
$37$	3.89286	0.639982	0.319991	−	0.947421i	$-0.396320\pi$
$37$	3.89286	0.639982	0.319991	+	0.947421i	$0.396320\pi$
$38$	13.0029	2.10936
$39$	−6.11009	−0.978398
$40$	0	0
$41$	4.32733	0.675815	0.337907	−	0.941179i	$-0.390281\pi$
$41$	4.32733	0.675815	0.337907	+	0.941179i	$0.390281\pi$
$42$	−3.17415	−0.489782
$43$	8.45544	1.28944	0.644721	−	0.764418i	$-0.276974\pi$
$43$	8.45544	1.28944	0.644721	+	0.764418i	$0.276974\pi$
$44$	11.1561	1.68185
$45$	0	0
$46$	−18.8748	−2.78294
$47$	−4.76475	−0.695010	−0.347505	−	0.937678i	$-0.612971\pi$
$47$	−4.76475	−0.695010	−0.347505	+	0.937678i	$0.612971\pi$
$48$	2.39138	0.345166
$49$	−5.23820	−0.748315
$50$	0	0
$51$	−0.672673	−0.0941930
$52$	−22.7217	−3.15093
$53$	13.3303	1.83105	0.915527	−	0.402256i	$-0.131774\pi$
$53$	13.3303	1.83105	0.915527	+	0.402256i	$0.131774\pi$
$54$	−2.39138	−0.325426
$55$	0	0
$56$	−5.45544	−0.729013
$57$	−5.43742	−0.720204
$58$	2.39138	0.314004
$59$	3.21724	0.418848	0.209424	−	0.977825i	$-0.432841\pi$
$59$	3.21724	0.418848	0.209424	+	0.977825i	$0.432841\pi$
$60$	0	0
$61$	7.43742	0.952264	0.476132	−	0.879374i	$-0.342038\pi$
$61$	7.43742	0.952264	0.476132	+	0.879374i	$0.342038\pi$
$62$	0	0
$63$	1.32733	0.167228
$64$	−10.7647	−1.34559
$65$	0	0
$66$	−7.17415	−0.883076
$67$	1.10714	0.135259	0.0676295	−	0.997711i	$-0.478456\pi$
$67$	1.10714	0.135259	0.0676295	+	0.997711i	$0.478456\pi$
$68$	−2.50147	−0.303348
$69$	7.89286	0.950188
$70$	0	0
$71$	12.3483	1.46547	0.732736	−	0.680513i	$-0.238243\pi$
$71$	12.3483	1.46547	0.732736	+	0.680513i	$0.238243\pi$
$72$	−4.11009	−0.484379
$73$	−9.20217	−1.07703	−0.538516	−	0.842615i	$-0.681015\pi$
$73$	−9.20217	−1.07703	−0.538516	+	0.842615i	$0.681015\pi$
$74$	−9.30931	−1.08219
$75$	0	0
$76$	−20.2202	−2.31941
$77$	3.98198	0.453789
$78$	14.6116	1.65443
$79$	11.6576	1.31158	0.655791	−	0.754942i	$-0.272335\pi$
$79$	11.6576	1.31158	0.655791	+	0.754942i	$0.272335\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−10.3483	−1.14278
$83$	0.889908	0.0976801	0.0488400	−	0.998807i	$-0.484448\pi$
$83$	0.889908	0.0976801	0.0488400	+	0.998807i	$0.484448\pi$
$84$	4.93594	0.538556
$85$	0	0
$86$	−20.2202	−2.18040
$87$	−1.00000	−0.107211
$88$	−12.3303	−1.31441
$89$	−2.33028	−0.247009	−0.123504	−	0.992344i	$-0.539413\pi$
$89$	−2.33028	−0.247009	−0.123504	+	0.992344i	$0.539413\pi$
$90$	0	0
$91$	−8.11009	−0.850169
$92$	29.3512	3.06008
$93$	0	0
$94$	11.3943	1.17524
$95$	0	0
$96$	2.50147	0.255306
$97$	12.6756	1.28701	0.643507	−	0.765440i	$-0.277479\pi$
$97$	12.6756	1.28701	0.643507	+	0.765440i	$0.277479\pi$
$98$	12.5265	1.26537
$99$	3.00000	0.301511
$100$	0	0
$101$	3.56258	0.354490	0.177245	−	0.984167i	$-0.443281\pi$
$101$	3.56258	0.354490	0.177245	+	0.984167i	$0.443281\pi$
$102$	1.60862	0.159277
$103$	12.9109	1.27215	0.636073	−	0.771629i	$-0.280558\pi$
$103$	12.9109	1.27215	0.636073	+	0.771629i	$0.280558\pi$
$104$	25.1130	2.46254
$105$	0	0
$106$	−31.8778	−3.09624
$107$	−4.22018	−0.407981	−0.203990	−	0.978973i	$-0.565391\pi$
$107$	−4.22018	−0.407981	−0.203990	+	0.978973i	$0.565391\pi$
$108$	3.71871	0.357833
$109$	−11.8748	−1.13740	−0.568702	−	0.822544i	$-0.692554\pi$
$109$	−11.8748	−1.13740	−0.568702	+	0.822544i	$0.692554\pi$
$110$	0	0
$111$	3.89286	0.369494
$112$	3.17415	0.299929
$113$	−14.2382	−1.33942	−0.669709	−	0.742624i	$-0.733581\pi$
$113$	−14.2382	−1.33942	−0.669709	+	0.742624i	$0.733581\pi$
$114$	13.0029	1.21784
$115$	0	0
$116$	−3.71871	−0.345274
$117$	−6.11009	−0.564878
$118$	−7.69364	−0.708257
$119$	−0.892857	−0.0818481
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−17.7857	−1.61024
$123$	4.32733	0.390182
$124$	0	0
$125$	0	0
$126$	−3.17415	−0.282776
$127$	7.20217	0.639089	0.319544	−	0.947571i	$-0.396470\pi$
$127$	7.20217	0.639089	0.319544	+	0.947571i	$0.396470\pi$
$128$	20.7397	1.83315
$129$	8.45544	0.744460
$130$	0	0
$131$	14.2382	1.24400	0.621999	−	0.783018i	$-0.286321\pi$
$131$	14.2382	1.24400	0.621999	+	0.783018i	$0.286321\pi$
$132$	11.1561	0.971016
$133$	−7.21724	−0.625814
$134$	−2.64760	−0.228718
$135$	0	0
$136$	2.76475	0.237075
$137$	−15.0950	−1.28965	−0.644827	−	0.764328i	$-0.723071\pi$
$137$	−15.0950	−1.28965	−0.644827	+	0.764328i	$0.723071\pi$
$138$	−18.8748	−1.60673
$139$	−6.43742	−0.546015	−0.273007	−	0.962012i	$-0.588018\pi$
$139$	−6.43742	−0.546015	−0.273007	+	0.962012i	$0.588018\pi$
$140$	0	0
$141$	−4.76475	−0.401264
$142$	−29.5295	−2.47806
$143$	−18.3303	−1.53285
$144$	2.39138	0.199282
$145$	0	0
$146$	22.0059	1.82122
$147$	−5.23820	−0.432040
$148$	14.4764	1.18995
$149$	1.21724	0.0997198	0.0498599	−	0.998756i	$-0.484123\pi$
$149$	1.21724	0.0997198	0.0498599	+	0.998756i	$0.484123\pi$
$150$	0	0
$151$	−2.32733	−0.189395	−0.0946976	−	0.995506i	$-0.530188\pi$
$151$	−2.32733	−0.189395	−0.0946976	+	0.995506i	$0.530188\pi$
$152$	22.3483	1.81269
$153$	−0.672673	−0.0543824
$154$	−9.52244	−0.767340
$155$	0	0
$156$	−22.7217	−1.81919
$157$	−9.43742	−0.753188	−0.376594	−	0.926378i	$-0.622905\pi$
$157$	−9.43742	−0.753188	−0.376594	+	0.926378i	$0.622905\pi$
$158$	−27.8778	−2.21784
$159$	13.3303	1.05716
$160$	0	0
$161$	10.4764	0.825656
$162$	−2.39138	−0.187885
$163$	21.3303	1.67072	0.835358	−	0.549706i	$-0.185260\pi$
$163$	21.3303	1.67072	0.835358	+	0.549706i	$0.185260\pi$
$164$	16.0921	1.25658
$165$	0	0
$166$	−2.12811	−0.165173
$167$	−5.30931	−0.410847	−0.205423	−	0.978673i	$-0.565857\pi$
$167$	−5.30931	−0.410847	−0.205423	+	0.978673i	$0.565857\pi$
$168$	−5.45544	−0.420896
$169$	24.3332	1.87179
$170$	0	0
$171$	−5.43742	−0.415810
$172$	31.4433	2.39753
$173$	−22.7677	−1.73100	−0.865498	−	0.500913i	$-0.832998\pi$
$173$	−22.7677	−1.73100	−0.865498	+	0.500913i	$0.832998\pi$
$174$	2.39138	0.181290
$175$	0	0
$176$	7.17415	0.540772
$177$	3.21724	0.241822
$178$	5.57258	0.417683
$179$	17.6576	1.31979	0.659896	−	0.751357i	$-0.270600\pi$
$179$	17.6576	1.31979	0.659896	+	0.751357i	$0.270600\pi$
$180$	0	0
$181$	10.5655	0.785330	0.392665	−	0.919682i	$-0.371553\pi$
$181$	10.5655	0.785330	0.392665	+	0.919682i	$0.371553\pi$
$182$	19.3943	1.43760
$183$	7.43742	0.549790
$184$	−32.4404	−2.39154
$185$	0	0
$186$	0	0
$187$	−2.01802	−0.147572
$188$	−17.7187	−1.29227
$189$	1.32733	0.0965489
$190$	0	0
$191$	−3.23820	−0.234308	−0.117154	−	0.993114i	$-0.537377\pi$
$191$	−3.23820	−0.234308	−0.117154	+	0.993114i	$0.537377\pi$
$192$	−10.7647	−0.776879
$193$	3.30931	0.238209	0.119105	−	0.992882i	$-0.461998\pi$
$193$	3.30931	0.238209	0.119105	+	0.992882i	$0.461998\pi$
$194$	−30.3123	−2.17629
$195$	0	0
$196$	−19.4794	−1.39138
$197$	1.89286	0.134860	0.0674302	−	0.997724i	$-0.478520\pi$
$197$	1.89286	0.134860	0.0674302	+	0.997724i	$0.478520\pi$
$198$	−7.17415	−0.509844
$199$	−5.56258	−0.394321	−0.197160	−	0.980371i	$-0.563172\pi$
$199$	−5.56258	−0.394321	−0.197160	+	0.980371i	$0.563172\pi$
$200$	0	0
$201$	1.10714	0.0780919
$202$	−8.51949	−0.599429
$203$	−1.32733	−0.0931601
$204$	−2.50147	−0.175138
$205$	0	0
$206$	−30.8748	−2.15115
$207$	7.89286	0.548591
$208$	−14.6116	−1.01313
$209$	−16.3123	−1.12834
$210$	0	0
$211$	15.7497	1.08425	0.542126	−	0.840297i	$-0.317619\pi$
$211$	15.7497	1.08425	0.542126	+	0.840297i	$0.317619\pi$
$212$	49.5714	3.40458
$213$	12.3483	0.846091
$214$	10.0921	0.689880
$215$	0	0
$216$	−4.11009	−0.279656
$217$	0	0
$218$	28.3973	1.92331
$219$	−9.20217	−0.621825
$220$	0	0
$221$	4.11009	0.276475
$222$	−9.30931	−0.624800
$223$	2.01802	0.135136	0.0675682	−	0.997715i	$-0.478476\pi$
$223$	2.01802	0.135136	0.0675682	+	0.997715i	$0.478476\pi$
$224$	3.32028	0.221845
$225$	0	0
$226$	34.0490	2.26490
$227$	20.6756	1.37229	0.686145	−	0.727465i	$-0.259302\pi$
$227$	20.6756	1.37229	0.686145	+	0.727465i	$0.259302\pi$
$228$	−20.2202	−1.33911
$229$	14.2562	0.942078	0.471039	−	0.882113i	$-0.343879\pi$
$229$	14.2562	0.942078	0.471039	+	0.882113i	$0.343879\pi$
$230$	0	0
$231$	3.98198	0.261995
$232$	4.11009	0.269841
$233$	−20.7677	−1.36054	−0.680268	−	0.732963i	$-0.738137\pi$
$233$	−20.7677	−1.36054	−0.680268	+	0.732963i	$0.738137\pi$
$234$	14.6116	0.955188
$235$	0	0
$236$	11.9640	0.778788
$237$	11.6576	0.757243
$238$	2.13516	0.138402
$239$	3.30931	0.214061	0.107031	−	0.994256i	$-0.465866\pi$
$239$	3.30931	0.214061	0.107031	+	0.994256i	$0.465866\pi$
$240$	0	0
$241$	−8.78571	−0.565938	−0.282969	−	0.959129i	$-0.591319\pi$
$241$	−8.78571	−0.565938	−0.282969	+	0.959129i	$0.591319\pi$
$242$	4.78276	0.307448
$243$	1.00000	0.0641500
$244$	27.6576	1.77060
$245$	0	0
$246$	−10.3483	−0.659783
$247$	33.2231	2.11394
$248$	0	0
$249$	0.889908	0.0563956
$250$	0	0
$251$	23.8037	1.50248	0.751239	−	0.660030i	$-0.229457\pi$
$251$	23.8037	1.50248	0.751239	+	0.660030i	$0.229457\pi$
$252$	4.93594	0.310935
$253$	23.6786	1.48866
$254$	−17.2231	−1.08068
$255$	0	0
$256$	−28.0670	−1.75419
$257$	27.3303	1.70482	0.852408	−	0.522877i	$-0.175141\pi$
$257$	27.3303	1.70482	0.852408	+	0.522877i	$0.175141\pi$
$258$	−20.2202	−1.25885
$259$	5.16710	0.321068
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	−34.0490	−2.10355
$263$	21.7497	1.34114	0.670571	−	0.741845i	$-0.266049\pi$
$263$	21.7497	1.34114	0.670571	+	0.741845i	$0.266049\pi$
$264$	−12.3303	−0.758876
$265$	0	0
$266$	17.2592	1.05823
$267$	−2.33028	−0.142611
$268$	4.11714	0.251495
$269$	10.3303	0.629848	0.314924	−	0.949117i	$-0.398021\pi$
$269$	10.3303	0.629848	0.314924	+	0.949117i	$0.398021\pi$
$270$	0	0
$271$	−6.25622	−0.380038	−0.190019	−	0.981780i	$-0.560855\pi$
$271$	−6.25622	−0.380038	−0.190019	+	0.981780i	$0.560855\pi$
$272$	−1.60862	−0.0975368
$273$	−8.11009	−0.490845
$274$	36.0980	2.18076
$275$	0	0
$276$	29.3512	1.76674
$277$	−31.6815	−1.90356	−0.951779	−	0.306784i	$-0.900747\pi$
$277$	−31.6815	−1.90356	−0.951779	+	0.306784i	$0.900747\pi$
$278$	15.3943	0.923291
$279$	0	0
$280$	0	0
$281$	27.9138	1.66520	0.832600	−	0.553875i	$-0.186852\pi$
$281$	27.9138	1.66520	0.832600	+	0.553875i	$0.186852\pi$
$282$	11.3943	0.678523
$283$	−23.5295	−1.39868	−0.699342	−	0.714788i	$-0.746523\pi$
$283$	−23.5295	−1.39868	−0.699342	+	0.714788i	$0.746523\pi$
$284$	45.9197	2.72483
$285$	0	0
$286$	43.8347	2.59200
$287$	5.74378	0.339045
$288$	2.50147	0.147401
$289$	−16.5475	−0.973383
$290$	0	0
$291$	12.6756	0.743058
$292$	−34.2202	−2.00258
$293$	6.23820	0.364440	0.182220	−	0.983258i	$-0.441672\pi$
$293$	6.23820	0.364440	0.182220	+	0.983258i	$0.441672\pi$
$294$	12.5265	0.730563
$295$	0	0
$296$	−16.0000	−0.929981
$297$	3.00000	0.174078
$298$	−2.91087	−0.168622
$299$	−48.2261	−2.78899
$300$	0	0
$301$	11.2231	0.646891
$302$	5.56553	0.320260
$303$	3.56258	0.204665
$304$	−13.0029	−0.745770
$305$	0	0
$306$	1.60862	0.0919585
$307$	−3.45249	−0.197044	−0.0985220	−	0.995135i	$-0.531411\pi$
$307$	−3.45249	−0.197044	−0.0985220	+	0.995135i	$0.531411\pi$
$308$	14.8078	0.843755
$309$	12.9109	0.734474
$310$	0	0
$311$	11.0000	0.623753	0.311876	−	0.950123i	$-0.399043\pi$
$311$	11.0000	0.623753	0.311876	+	0.950123i	$0.399043\pi$
$312$	25.1130	1.42175
$313$	32.9908	1.86475	0.932376	−	0.361490i	$-0.117732\pi$
$313$	32.9908	1.86475	0.932376	+	0.361490i	$0.117732\pi$
$314$	22.5685	1.27361
$315$	0	0
$316$	43.3512	2.43870
$317$	−4.63664	−0.260419	−0.130210	−	0.991486i	$-0.541565\pi$
$317$	−4.63664	−0.260419	−0.130210	+	0.991486i	$0.541565\pi$
$318$	−31.8778	−1.78762
$319$	−3.00000	−0.167968
$320$	0	0
$321$	−4.22018	−0.235548
$322$	−25.0531	−1.39615
$323$	3.65760	0.203515
$324$	3.71871	0.206595
$325$	0	0
$326$	−51.0088	−2.82512
$327$	−11.8748	−0.656680
$328$	−17.7857	−0.982052
$329$	−6.32438	−0.348674
$330$	0	0
$331$	−22.2261	−1.22166	−0.610828	−	0.791763i	$-0.709163\pi$
$331$	−22.2261	−1.22166	−0.610828	+	0.791763i	$0.709163\pi$
$332$	3.30931	0.181622
$333$	3.89286	0.213327
$334$	12.6966	0.694726
$335$	0	0
$336$	3.17415	0.173164
$337$	27.5655	1.50159	0.750795	−	0.660535i	$-0.229671\pi$
$337$	27.5655	1.50159	0.750795	+	0.660535i	$0.229671\pi$
$338$	−58.1900	−3.16512
$339$	−14.2382	−0.773313
$340$	0	0
$341$	0	0
$342$	13.0029	0.703119
$343$	−16.2441	−0.877099
$344$	−34.7526	−1.87374
$345$	0	0
$346$	54.4463	2.92705
$347$	22.3273	1.19859	0.599297	−	0.800527i	$-0.295447\pi$
$347$	22.3273	1.19859	0.599297	+	0.800527i	$0.295447\pi$
$348$	−3.71871	−0.199344
$349$	−20.5115	−1.09795	−0.548977	−	0.835837i	$-0.684983\pi$
$349$	−20.5115	−1.09795	−0.548977	+	0.835837i	$0.684983\pi$
$350$	0	0
$351$	−6.11009	−0.326133
$352$	7.50442	0.399987
$353$	28.8748	1.53685	0.768426	−	0.639938i	$-0.221040\pi$
$353$	28.8748	1.53685	0.768426	+	0.639938i	$0.221040\pi$
$354$	−7.69364	−0.408912
$355$	0	0
$356$	−8.66562	−0.459277
$357$	−0.892857	−0.0472550
$358$	−42.2261	−2.23172
$359$	2.76770	0.146073	0.0730367	−	0.997329i	$-0.476731\pi$
$359$	2.76770	0.146073	0.0730367	+	0.997329i	$0.476731\pi$
$360$	0	0
$361$	10.5655	0.556081
$362$	−25.2662	−1.32796
$363$	−2.00000	−0.104973
$364$	−30.1591	−1.58077
$365$	0	0
$366$	−17.7857	−0.929674
$367$	−25.4584	−1.32892	−0.664458	−	0.747325i	$-0.731338\pi$
$367$	−25.4584	−1.32892	−0.664458	+	0.747325i	$0.731338\pi$
$368$	18.8748	0.983919
$369$	4.32733	0.225272
$370$	0	0
$371$	17.6936	0.918608
$372$	0	0
$373$	4.03604	0.208978	0.104489	−	0.994526i	$-0.466679\pi$
$373$	4.03604	0.208978	0.104489	+	0.994526i	$0.466679\pi$
$374$	4.82585	0.249539
$375$	0	0
$376$	19.5835	1.00994
$377$	6.11009	0.314686
$378$	−3.17415	−0.163261
$379$	20.4043	1.04810	0.524050	−	0.851687i	$-0.324420\pi$
$379$	20.4043	1.04810	0.524050	+	0.851687i	$0.324420\pi$
$380$	0	0
$381$	7.20217	0.368978
$382$	7.74378	0.396206
$383$	5.80078	0.296406	0.148203	−	0.988957i	$-0.452651\pi$
$383$	5.80078	0.296406	0.148203	+	0.988957i	$0.452651\pi$
$384$	20.7397	1.05837
$385$	0	0
$386$	−7.91382	−0.402803
$387$	8.45544	0.429814
$388$	47.1370	2.39302
$389$	−32.2592	−1.63560	−0.817802	−	0.575499i	$-0.804808\pi$
$389$	−32.2592	−1.63560	−0.817802	+	0.575499i	$0.804808\pi$
$390$	0	0
$391$	−5.30931	−0.268503
$392$	21.5295	1.08740
$393$	14.2382	0.718222
$394$	−4.52654	−0.228044
$395$	0	0
$396$	11.1561	0.560617
$397$	−22.1841	−1.11339	−0.556695	−	0.830717i	$-0.687931\pi$
$397$	−22.1841	−1.11339	−0.556695	+	0.830717i	$0.687931\pi$
$398$	13.3023	0.666782
$399$	−7.21724	−0.361314
$400$	0	0
$401$	−38.6045	−1.92782	−0.963909	−	0.266233i	$-0.914221\pi$
$401$	−38.6045	−1.92782	−0.963909	+	0.266233i	$0.914221\pi$
$402$	−2.64760	−0.132050
$403$	0	0
$404$	13.2482	0.659123
$405$	0	0
$406$	3.17415	0.157530
$407$	11.6786	0.578885
$408$	2.76475	0.136875
$409$	28.0980	1.38936	0.694678	−	0.719321i	$-0.255547\pi$
$409$	28.0980	1.38936	0.694678	+	0.719321i	$0.255547\pi$
$410$	0	0
$411$	−15.0950	−0.744583
$412$	48.0118	2.36537
$413$	4.27032	0.210129
$414$	−18.8748	−0.927648
$415$	0	0
$416$	−15.2842	−0.749371
$417$	−6.43742	−0.315242
$418$	39.0088	1.90799
$419$	3.56553	0.174188	0.0870938	−	0.996200i	$-0.472242\pi$
$419$	3.56553	0.174188	0.0870938	+	0.996200i	$0.472242\pi$
$420$	0	0
$421$	−17.6216	−0.858823	−0.429411	−	0.903109i	$-0.641279\pi$
$421$	−17.6216	−0.858823	−0.429411	+	0.903109i	$0.641279\pi$
$422$	−37.6635	−1.83343
$423$	−4.76475	−0.231670
$424$	−54.7887	−2.66077
$425$	0	0
$426$	−29.5295	−1.43071
$427$	9.87189	0.477734
$428$	−15.6936	−0.758581
$429$	−18.3303	−0.884994
$430$	0	0
$431$	−33.7916	−1.62768	−0.813842	−	0.581086i	$-0.802628\pi$
$431$	−33.7916	−1.62768	−0.813842	+	0.581086i	$0.802628\pi$
$432$	2.39138	0.115055
$433$	−23.3663	−1.12291	−0.561457	−	0.827506i	$-0.689759\pi$
$433$	−23.3663	−1.12291	−0.561457	+	0.827506i	$0.689759\pi$
$434$	0	0
$435$	0	0
$436$	−44.1591	−2.11484
$437$	−42.9168	−2.05299
$438$	22.0059	1.05148
$439$	−16.1461	−0.770613	−0.385306	−	0.922789i	$-0.625904\pi$
$439$	−16.1461	−0.770613	−0.385306	+	0.922789i	$0.625904\pi$
$440$	0	0
$441$	−5.23820	−0.249438
$442$	−9.82880	−0.467508
$443$	−9.38337	−0.445817	−0.222909	−	0.974839i	$-0.571555\pi$
$443$	−9.38337	−0.445817	−0.222909	+	0.974839i	$0.571555\pi$
$444$	14.4764	0.687020
$445$	0	0
$446$	−4.82585	−0.228511
$447$	1.21724	0.0575733
$448$	−14.2883	−0.675061
$449$	17.5685	0.829108	0.414554	−	0.910025i	$-0.363938\pi$
$449$	17.5685	0.829108	0.414554	+	0.910025i	$0.363938\pi$
$450$	0	0
$451$	12.9820	0.611298
$452$	−52.9477	−2.49045
$453$	−2.32733	−0.109347
$454$	−49.4433	−2.32049
$455$	0	0
$456$	22.3483	1.04655
$457$	17.4194	0.814845	0.407423	−	0.913240i	$-0.366428\pi$
$457$	17.4194	0.814845	0.407423	+	0.913240i	$0.366428\pi$
$458$	−34.0921	−1.59302
$459$	−0.672673	−0.0313977
$460$	0	0
$461$	−0.113041	−0.00526484	−0.00263242	−	0.999997i	$-0.500838\pi$
$461$	−0.113041	−0.00526484	−0.00263242	+	0.999997i	$0.500838\pi$
$462$	−9.52244	−0.443024
$463$	26.8568	1.24814	0.624071	−	0.781368i	$-0.285478\pi$
$463$	26.8568	1.24814	0.624071	+	0.781368i	$0.285478\pi$
$464$	−2.39138	−0.111017
$465$	0	0
$466$	49.6635	2.30062
$467$	−0.726727	−0.0336289	−0.0168145	−	0.999859i	$-0.505352\pi$
$467$	−0.726727	−0.0336289	−0.0168145	+	0.999859i	$0.505352\pi$
$468$	−22.7217	−1.05031
$469$	1.46954	0.0678571
$470$	0	0
$471$	−9.43742	−0.434853
$472$	−13.2231	−0.608644
$473$	25.3663	1.16634
$474$	−27.8778	−1.28047
$475$	0	0
$476$	−3.32028	−0.152185
$477$	13.3303	0.610351
$478$	−7.91382	−0.361970
$479$	−9.12516	−0.416939	−0.208470	−	0.978029i	$-0.566848\pi$
$479$	−9.12516	−0.416939	−0.208470	+	0.978029i	$0.566848\pi$
$480$	0	0
$481$	−23.7857	−1.08454
$482$	21.0100	0.956979
$483$	10.4764	0.476693
$484$	−7.43742	−0.338065
$485$	0	0
$486$	−2.39138	−0.108475
$487$	−35.5714	−1.61190	−0.805948	−	0.591987i	$-0.798344\pi$
$487$	−35.5714	−1.61190	−0.805948	+	0.591987i	$0.798344\pi$
$488$	−30.5685	−1.38377
$489$	21.3303	0.964588
$490$	0	0
$491$	−4.44037	−0.200391	−0.100196	−	0.994968i	$-0.531947\pi$
$491$	−4.44037	−0.200391	−0.100196	+	0.994968i	$0.531947\pi$
$492$	16.0921	0.725487
$493$	0.672673	0.0302957
$494$	−79.4492	−3.57459
$495$	0	0
$496$	0	0
$497$	16.3902	0.735202
$498$	−2.12811	−0.0953629
$499$	−40.3663	−1.80704	−0.903522	−	0.428541i	$-0.859028\pi$
$499$	−40.3663	−1.80704	−0.903522	+	0.428541i	$0.859028\pi$
$500$	0	0
$501$	−5.30931	−0.237202
$502$	−56.9238	−2.54063
$503$	−23.7117	−1.05725	−0.528625	−	0.848855i	$-0.677292\pi$
$503$	−23.7117	−1.05725	−0.528625	+	0.848855i	$0.677292\pi$
$504$	−5.45544	−0.243004
$505$	0	0
$506$	−56.6245	−2.51727
$507$	24.3332	1.08068
$508$	26.7828	1.18829
$509$	−0.568478	−0.0251974	−0.0125987	−	0.999921i	$-0.504010\pi$
$509$	−0.568478	−0.0251974	−0.0125987	+	0.999921i	$0.504010\pi$
$510$	0	0
$511$	−12.2143	−0.540328
$512$	25.6396	1.13312
$513$	−5.43742	−0.240068
$514$	−65.3571	−2.88278
$515$	0	0
$516$	31.4433	1.38421
$517$	−14.2942	−0.628660
$518$	−12.3565	−0.542913
$519$	−22.7677	−0.999391
$520$	0	0
$521$	21.6576	0.948837	0.474418	−	0.880299i	$-0.342658\pi$
$521$	21.6576	0.948837	0.474418	+	0.880299i	$0.342658\pi$
$522$	2.39138	0.104668
$523$	−14.0180	−0.612965	−0.306483	−	0.951876i	$-0.599152\pi$
$523$	−14.0180	−0.612965	−0.306483	+	0.951876i	$0.599152\pi$
$524$	52.9477	2.31303
$525$	0	0
$526$	−52.0118	−2.26782
$527$	0	0
$528$	7.17415	0.312215
$529$	39.2972	1.70857
$530$	0	0
$531$	3.21724	0.139616
$532$	−26.8388	−1.16361
$533$	−26.4404	−1.14526
$534$	5.57258	0.241149
$535$	0	0
$536$	−4.55046	−0.196550
$537$	17.6576	0.761982
$538$	−24.7036	−1.06505
$539$	−15.7146	−0.676876
$540$	0	0
$541$	20.8748	0.897479	0.448740	−	0.893663i	$-0.351873\pi$
$541$	20.8748	0.897479	0.448740	+	0.893663i	$0.351873\pi$
$542$	14.9610	0.642631
$543$	10.5655	0.453410
$544$	−1.68267	−0.0721440
$545$	0	0
$546$	19.3943	0.830001
$547$	−4.45249	−0.190375	−0.0951873	−	0.995459i	$-0.530345\pi$
$547$	−4.45249	−0.190375	−0.0951873	+	0.995459i	$0.530345\pi$
$548$	−56.1340	−2.39793
$549$	7.43742	0.317421
$550$	0	0
$551$	5.43742	0.231642
$552$	−32.4404	−1.38075
$553$	15.4735	0.657998
$554$	75.7626	3.21885
$555$	0	0
$556$	−23.9389	−1.01524
$557$	−34.4253	−1.45865	−0.729323	−	0.684169i	$-0.760165\pi$
$557$	−34.4253	−1.45865	−0.729323	+	0.684169i	$0.760165\pi$
$558$	0	0
$559$	−51.6635	−2.18513
$560$	0	0
$561$	−2.01802	−0.0852008
$562$	−66.7526	−2.81579
$563$	−19.6756	−0.829229	−0.414614	−	0.909997i	$-0.636083\pi$
$563$	−19.6756	−0.829229	−0.414614	+	0.909997i	$0.636083\pi$
$564$	−17.7187	−0.746092
$565$	0	0
$566$	56.2680	2.36512
$567$	1.32733	0.0557425
$568$	−50.7526	−2.12953
$569$	26.2942	1.10231	0.551156	−	0.834402i	$-0.314187\pi$
$569$	26.2942	1.10231	0.551156	+	0.834402i	$0.314187\pi$
$570$	0	0
$571$	37.0600	1.55091	0.775455	−	0.631402i	$-0.217520\pi$
$571$	37.0600	1.55091	0.775455	+	0.631402i	$0.217520\pi$
$572$	−68.1650	−2.85012
$573$	−3.23820	−0.135278
$574$	−13.7356	−0.573312
$575$	0	0
$576$	−10.7647	−0.448531
$577$	14.2202	0.591994	0.295997	−	0.955189i	$-0.404348\pi$
$577$	14.2202	0.591994	0.295997	+	0.955189i	$0.404348\pi$
$578$	39.5714	1.64595
$579$	3.30931	0.137530
$580$	0	0
$581$	1.18120	0.0490044
$582$	−30.3123	−1.25648
$583$	39.9908	1.65625
$584$	37.8217	1.56508
$585$	0	0
$586$	−14.9179	−0.616254
$587$	−32.0980	−1.32483	−0.662413	−	0.749139i	$-0.730467\pi$
$587$	−32.0980	−1.32483	−0.662413	+	0.749139i	$0.730467\pi$
$588$	−19.4794	−0.803315
$589$	0	0
$590$	0	0
$591$	1.89286	0.0778617
$592$	9.30931	0.382610
$593$	27.1871	1.11644	0.558220	−	0.829693i	$-0.311484\pi$
$593$	27.1871	1.11644	0.558220	+	0.829693i	$0.311484\pi$
$594$	−7.17415	−0.294359
$595$	0	0
$596$	4.52654	0.185414
$597$	−5.56258	−0.227661
$598$	115.327	4.71607
$599$	31.8037	1.29947	0.649733	−	0.760163i	$-0.274881\pi$
$599$	31.8037	1.29947	0.649733	+	0.760163i	$0.274881\pi$
$600$	0	0
$601$	−13.3453	−0.544368	−0.272184	−	0.962245i	$-0.587746\pi$
$601$	−13.3453	−0.544368	−0.272184	+	0.962245i	$0.587746\pi$
$602$	−26.8388	−1.09387
$603$	1.10714	0.0450864
$604$	−8.65465	−0.352153
$605$	0	0
$606$	−8.51949	−0.346081
$607$	37.6635	1.52871	0.764357	−	0.644793i	$-0.223056\pi$
$607$	37.6635	1.52871	0.764357	+	0.644793i	$0.223056\pi$
$608$	−13.6016	−0.551616
$609$	−1.32733	−0.0537860
$610$	0	0
$611$	29.1130	1.17779
$612$	−2.50147	−0.101116
$613$	−38.3663	−1.54960	−0.774800	−	0.632206i	$-0.782150\pi$
$613$	−38.3663	−1.54960	−0.774800	+	0.632206i	$0.782150\pi$
$614$	8.25622	0.333194
$615$	0	0
$616$	−16.3663	−0.659418
$617$	34.6245	1.39393	0.696965	−	0.717105i	$-0.254533\pi$
$617$	34.6245	1.39393	0.696965	+	0.717105i	$0.254533\pi$
$618$	−30.8748	−1.24197
$619$	0.00589781	0.000237053 0	0.000118526	−	1.00000i	$-0.499962\pi$
$619$	0.00589781	0.000237053 0	0.000118526		1.00000i	$0.499962\pi$
$620$	0	0
$621$	7.89286	0.316729
$622$	−26.3052	−1.05474
$623$	−3.09304	−0.123920
$624$	−14.6116	−0.584931
$625$	0	0
$626$	−78.8937	−3.15323
$627$	−16.3123	−0.651449
$628$	−35.0950	−1.40044
$629$	−2.61862	−0.104411
$630$	0	0
$631$	42.3001	1.68394	0.841971	−	0.539523i	$-0.181395\pi$
$631$	42.3001	1.68394	0.841971	+	0.539523i	$0.181395\pi$
$632$	−47.9138	−1.90591
$633$	15.7497	0.625993
$634$	11.0880	0.440360
$635$	0	0
$636$	49.5714	1.96563
$637$	32.0059	1.26812
$638$	7.17415	0.284027
$639$	12.3483	0.488491
$640$	0	0
$641$	14.7297	0.581787	0.290894	−	0.956755i	$-0.406047\pi$
$641$	14.7297	0.581787	0.290894	+	0.956755i	$0.406047\pi$
$642$	10.0921	0.398302
$643$	−37.9820	−1.49786	−0.748932	−	0.662647i	$-0.769433\pi$
$643$	−37.9820	−1.49786	−0.748932	+	0.662647i	$0.769433\pi$
$644$	38.9587	1.53519
$645$	0	0
$646$	−8.74673	−0.344135
$647$	−31.5144	−1.23896	−0.619480	−	0.785012i	$-0.712656\pi$
$647$	−31.5144	−1.23896	−0.619480	+	0.785012i	$0.712656\pi$
$648$	−4.11009	−0.161460
$649$	9.65171	0.378863
$650$	0	0
$651$	0	0
$652$	79.3211	3.10645
$653$	38.8929	1.52200	0.760998	−	0.648755i	$-0.224710\pi$
$653$	38.8929	1.52200	0.760998	+	0.648755i	$0.224710\pi$
$654$	28.3973	1.11042
$655$	0	0
$656$	10.3483	0.404033
$657$	−9.20217	−0.359011
$658$	15.1240	0.589595
$659$	−32.2792	−1.25742	−0.628709	−	0.777641i	$-0.716416\pi$
$659$	−32.2792	−1.25742	−0.628709	+	0.777641i	$0.716416\pi$
$660$	0	0
$661$	−29.8447	−1.16082	−0.580412	−	0.814323i	$-0.697109\pi$
$661$	−29.8447	−1.16082	−0.580412	+	0.814323i	$0.697109\pi$
$662$	53.1511	2.06577
$663$	4.11009	0.159623
$664$	−3.65760	−0.141943
$665$	0	0
$666$	−9.30931	−0.360728
$667$	−7.89286	−0.305613
$668$	−19.7438	−0.763910
$669$	2.01802	0.0780211
$670$	0	0
$671$	22.3123	0.861355
$672$	3.32028	0.128082
$673$	−10.5505	−0.406690	−0.203345	−	0.979107i	$-0.565181\pi$
$673$	−10.5505	−0.406690	−0.203345	+	0.979107i	$0.565181\pi$
$674$	−65.9197	−2.53913
$675$	0	0
$676$	90.4882	3.48032
$677$	−40.0600	−1.53963	−0.769815	−	0.638268i	$-0.779651\pi$
$677$	−40.0600	−1.53963	−0.769815	+	0.638268i	$0.779651\pi$
$678$	34.0490	1.30764
$679$	16.8247	0.645673
$680$	0	0
$681$	20.6756	0.792292
$682$	0	0
$683$	−24.2913	−0.929480	−0.464740	−	0.885447i	$-0.653852\pi$
$683$	−24.2913	−0.929480	−0.464740	+	0.885447i	$0.653852\pi$
$684$	−20.2202	−0.773138
$685$	0	0
$686$	38.8459	1.48314
$687$	14.2562	0.543909
$688$	20.2202	0.770887
$689$	−81.4492	−3.10297
$690$	0	0
$691$	−43.9017	−1.67010	−0.835050	−	0.550174i	$-0.814561\pi$
$691$	−43.9017	−1.67010	−0.835050	+	0.550174i	$0.814561\pi$
$692$	−84.6665	−3.21854
$693$	3.98198	0.151263
$694$	−53.3932	−2.02678
$695$	0	0
$696$	4.11009	0.155793
$697$	−2.91087	−0.110257
$698$	49.0508	1.85660
$699$	−20.7677	−0.785506
$700$	0	0
$701$	−39.9197	−1.50775	−0.753874	−	0.657020i	$-0.771817\pi$
$701$	−39.9197	−1.50775	−0.753874	+	0.657020i	$0.771817\pi$
$702$	14.6116	0.551478
$703$	−21.1671	−0.798332
$704$	−32.2942	−1.21713
$705$	0	0
$706$	−69.0508	−2.59876
$707$	4.72871	0.177841
$708$	11.9640	0.449633
$709$	11.4885	0.431461	0.215730	−	0.976453i	$-0.430787\pi$
$709$	11.4885	0.431461	0.215730	+	0.976453i	$0.430787\pi$
$710$	0	0
$711$	11.6576	0.437194
$712$	9.57765	0.358938
$713$	0	0
$714$	2.13516	0.0799064
$715$	0	0
$716$	65.6635	2.45396
$717$	3.30931	0.123588
$718$	−6.61862	−0.247005
$719$	−1.00295	−0.0374037	−0.0187018	−	0.999825i	$-0.505953\pi$
$719$	−1.00295	−0.0374037	−0.0187018	+	0.999825i	$0.505953\pi$
$720$	0	0
$721$	17.1370	0.638214
$722$	−25.2662	−0.940311
$723$	−8.78571	−0.326744
$724$	39.2901	1.46021
$725$	0	0
$726$	4.78276	0.177505
$727$	20.4404	0.758091	0.379046	−	0.925378i	$-0.376252\pi$
$727$	20.4404	0.758091	0.379046	+	0.925378i	$0.376252\pi$
$728$	33.3332	1.23541
$729$	1.00000	0.0370370
$730$	0	0
$731$	−5.68774	−0.210369
$732$	27.6576	1.02225
$733$	−30.9669	−1.14379	−0.571895	−	0.820327i	$-0.693791\pi$
$733$	−30.9669	−1.14379	−0.571895	+	0.820327i	$0.693791\pi$
$734$	60.8807	2.24715
$735$	0	0
$736$	19.7438	0.727765
$737$	3.32143	0.122346
$738$	−10.3483	−0.380926
$739$	3.62157	0.133222	0.0666108	−	0.997779i	$-0.478781\pi$
$739$	3.62157	0.133222	0.0666108	+	0.997779i	$0.478781\pi$
$740$	0	0
$741$	33.2231	1.22048
$742$	−42.3123	−1.55333
$743$	25.1691	0.923364	0.461682	−	0.887046i	$-0.347246\pi$
$743$	25.1691	0.923364	0.461682	+	0.887046i	$0.347246\pi$
$744$	0	0
$745$	0	0
$746$	−9.65171	−0.353374
$747$	0.889908	0.0325600
$748$	−7.50442	−0.274389
$749$	−5.60157	−0.204677
$750$	0	0
$751$	−9.63760	−0.351681	−0.175841	−	0.984419i	$-0.556264\pi$
$751$	−9.63760	−0.351681	−0.175841	+	0.984419i	$0.556264\pi$
$752$	−11.3943	−0.415509
$753$	23.8037	0.867456
$754$	−14.6116	−0.532122
$755$	0	0
$756$	4.93594	0.179519
$757$	17.4282	0.633440	0.316720	−	0.948519i	$-0.397418\pi$
$757$	17.4282	0.633440	0.316720	+	0.948519i	$0.397418\pi$
$758$	−48.7946	−1.77230
$759$	23.6786	0.859478
$760$	0	0
$761$	2.27622	0.0825130	0.0412565	−	0.999149i	$-0.486864\pi$
$761$	2.27622	0.0825130	0.0412565	+	0.999149i	$0.486864\pi$
$762$	−17.2231	−0.623928
$763$	−15.7618	−0.570615
$764$	−12.0419	−0.435662
$765$	0	0
$766$	−13.8719	−0.501212
$767$	−19.6576	−0.709795
$768$	−28.0670	−1.01278
$769$	29.4433	1.06175	0.530877	−	0.847449i	$-0.321863\pi$
$769$	29.4433	1.06175	0.530877	+	0.847449i	$0.321863\pi$
$770$	0	0
$771$	27.3303	0.984276
$772$	12.3064	0.442916
$773$	−14.2562	−0.512761	−0.256380	−	0.966576i	$-0.582530\pi$
$773$	−14.2562	−0.512761	−0.256380	+	0.966576i	$0.582530\pi$
$774$	−20.2202	−0.726800
$775$	0	0
$776$	−52.0980	−1.87021
$777$	5.16710	0.185369
$778$	77.1440	2.76575
$779$	−23.5295	−0.843032
$780$	0	0
$781$	37.0449	1.32557
$782$	12.6966	0.454029
$783$	−1.00000	−0.0357371
$784$	−12.5265	−0.447377
$785$	0	0
$786$	−34.0490	−1.21449
$787$	27.1311	0.967118	0.483559	−	0.875312i	$-0.339344\pi$
$787$	27.1311	0.967118	0.483559	+	0.875312i	$0.339344\pi$
$788$	7.03899	0.250753
$789$	21.7497	0.774309
$790$	0	0
$791$	−18.8988	−0.671962
$792$	−12.3303	−0.438137
$793$	−45.4433	−1.61374
$794$	53.0508	1.88270
$795$	0	0
$796$	−20.6856	−0.733182
$797$	2.00000	0.0708436	0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000	0.0708436	0.0354218	+	0.999372i	$0.488723\pi$
$798$	17.2592	0.610968
$799$	3.20511	0.113389
$800$	0	0
$801$	−2.33028	−0.0823363
$802$	92.3182	3.25987
$803$	−27.6065	−0.974212
$804$	4.11714	0.145200
$805$	0	0
$806$	0	0
$807$	10.3303	0.363643
$808$	−14.6425	−0.515123
$809$	−13.5324	−0.475775	−0.237888	−	0.971293i	$-0.576455\pi$
$809$	−13.5324	−0.475775	−0.237888	+	0.971293i	$0.576455\pi$
$810$	0	0
$811$	−40.4492	−1.42036	−0.710182	−	0.704018i	$-0.751387\pi$
$811$	−40.4492	−1.42036	−0.710182	+	0.704018i	$0.751387\pi$
$812$	−4.93594	−0.173218
$813$	−6.25622	−0.219415
$814$	−27.9279	−0.978873
$815$	0	0
$816$	−1.60862	−0.0563129
$817$	−45.9758	−1.60849
$818$	−67.1930	−2.34935
$819$	−8.11009	−0.283390
$820$	0	0
$821$	−21.6016	−0.753900	−0.376950	−	0.926234i	$-0.623027\pi$
$821$	−21.6016	−0.753900	−0.376950	+	0.926234i	$0.623027\pi$
$822$	36.0980	1.25906
$823$	−25.6995	−0.895830	−0.447915	−	0.894076i	$-0.647833\pi$
$823$	−25.6995	−0.895830	−0.447915	+	0.894076i	$0.647833\pi$
$824$	−53.0649	−1.84860
$825$	0	0
$826$	−10.2120	−0.355320
$827$	19.6016	0.681613	0.340807	−	0.940133i	$-0.389300\pi$
$827$	19.6016	0.681613	0.340807	+	0.940133i	$0.389300\pi$
$828$	29.3512	1.02003
$829$	−4.22608	−0.146778	−0.0733889	−	0.997303i	$-0.523381\pi$
$829$	−4.22608	−0.146778	−0.0733889	+	0.997303i	$0.523381\pi$
$830$	0	0
$831$	−31.6815	−1.09902
$832$	65.7736	2.28029
$833$	3.52360	0.122085
$834$	15.3943	0.533062
$835$	0	0
$836$	−60.6606	−2.09799
$837$	0	0
$838$	−8.52654	−0.294545
$839$	−2.45839	−0.0848729	−0.0424365	−	0.999099i	$-0.513512\pi$
$839$	−2.45839	−0.0848729	−0.0424365	+	0.999099i	$0.513512\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	42.1399	1.45224
$843$	27.9138	0.961403
$844$	58.5685	2.01601
$845$	0	0
$846$	11.3943	0.391745
$847$	−2.65465	−0.0912150
$848$	31.8778	1.09469
$849$	−23.5295	−0.807530
$850$	0	0
$851$	30.7258	1.05327
$852$	45.9197	1.57318
$853$	−39.8427	−1.36419	−0.682094	−	0.731264i	$-0.738931\pi$
$853$	−39.8427	−1.36419	−0.682094	+	0.731264i	$0.738931\pi$
$854$	−23.6075	−0.807831
$855$	0	0
$856$	17.3453	0.592852
$857$	16.9518	0.579064	0.289532	−	0.957168i	$-0.406500\pi$
$857$	16.9518	0.579064	0.289532	+	0.957168i	$0.406500\pi$
$858$	43.8347	1.49649
$859$	38.3902	1.30986	0.654929	−	0.755691i	$-0.272699\pi$
$859$	38.3902	1.30986	0.654929	+	0.755691i	$0.272699\pi$
$860$	0	0
$861$	5.74378	0.195747
$862$	80.8087	2.75235
$863$	−49.6995	−1.69179	−0.845896	−	0.533348i	$-0.820934\pi$
$863$	−49.6995	−1.69179	−0.845896	+	0.533348i	$0.820934\pi$
$864$	2.50147	0.0851019
$865$	0	0
$866$	55.8778	1.89880
$867$	−16.5475	−0.561983
$868$	0	0
$869$	34.9728	1.18637
$870$	0	0
$871$	−6.76475	−0.229215
$872$	48.8067	1.65280
$873$	12.6756	0.429005
$874$	102.630	3.47153
$875$	0	0
$876$	−34.2202	−1.15619
$877$	40.9168	1.38166	0.690831	−	0.723017i	$-0.257245\pi$
$877$	40.9168	1.38166	0.690831	+	0.723017i	$0.257245\pi$
$878$	38.6116	1.30308
$879$	6.23820	0.210409
$880$	0	0
$881$	−26.9079	−0.906551	−0.453276	−	0.891370i	$-0.649745\pi$
$881$	−26.9079	−0.906551	−0.453276	+	0.891370i	$0.649745\pi$
$882$	12.5265	0.421791
$883$	6.69069	0.225160	0.112580	−	0.993643i	$-0.464089\pi$
$883$	6.69069	0.225160	0.112580	+	0.993643i	$0.464089\pi$
$884$	15.2842	0.514065
$885$	0	0
$886$	22.4392	0.753860
$887$	46.0859	1.54741	0.773706	−	0.633545i	$-0.218401\pi$
$887$	46.0859	1.54741	0.773706	+	0.633545i	$0.218401\pi$
$888$	−16.0000	−0.536925
$889$	9.55963	0.320620
$890$	0	0
$891$	3.00000	0.100504
$892$	7.50442	0.251267
$893$	25.9079	0.866976
$894$	−2.91087	−0.0973542
$895$	0	0
$896$	27.5283	0.919657
$897$	−48.2261	−1.61022
$898$	−42.0129	−1.40199
$899$	0	0
$900$	0	0
$901$	−8.96691	−0.298731
$902$	−31.0449	−1.03368
$903$	11.2231	0.373482
$904$	58.5203	1.94636
$905$	0	0
$906$	5.56553	0.184902
$907$	18.7117	0.621310	0.310655	−	0.950523i	$-0.399452\pi$
$907$	18.7117	0.621310	0.310655	+	0.950523i	$0.399452\pi$
$908$	76.8866	2.55157
$909$	3.56258	0.118163
$910$	0	0
$911$	25.2923	0.837970	0.418985	−	0.907993i	$-0.362386\pi$
$911$	25.2923	0.837970	0.418985	+	0.907993i	$0.362386\pi$
$912$	−13.0029	−0.430571
$913$	2.66972	0.0883550
$914$	−41.6564	−1.37787
$915$	0	0
$916$	53.0147	1.75166
$917$	18.8988	0.624092
$918$	1.60862	0.0530923
$919$	−45.6396	−1.50551	−0.752756	−	0.658300i	$-0.771276\pi$
$919$	−45.6396	−1.50551	−0.752756	+	0.658300i	$0.771276\pi$
$920$	0	0
$921$	−3.45249	−0.113763
$922$	0.270324	0.00890265
$923$	−75.4492	−2.48344
$924$	14.8078	0.487142
$925$	0	0
$926$	−64.2249	−2.11056
$927$	12.9109	0.424049
$928$	−2.50147	−0.0821149
$929$	−21.2733	−0.697953	−0.348977	−	0.937131i	$-0.613471\pi$
$929$	−21.2733	−0.697953	−0.348977	+	0.937131i	$0.613471\pi$
$930$	0	0
$931$	28.4823	0.933470
$932$	−77.2290	−2.52972
$933$	11.0000	0.360124
$934$	1.73788	0.0568652
$935$	0	0
$936$	25.1130	0.820845
$937$	46.9489	1.53375	0.766877	−	0.641794i	$-0.221810\pi$
$937$	46.9489	1.53375	0.766877	+	0.641794i	$0.221810\pi$
$938$	−3.51424	−0.114744
$939$	32.9908	1.07662
$940$	0	0
$941$	−18.5986	−0.606298	−0.303149	−	0.952943i	$-0.598038\pi$
$941$	−18.5986	−0.606298	−0.303149	+	0.952943i	$0.598038\pi$
$942$	22.5685	0.735321
$943$	34.1550	1.11224
$944$	7.69364	0.250407
$945$	0	0
$946$	−60.6606	−1.97224
$947$	−40.0800	−1.30242	−0.651212	−	0.758896i	$-0.725739\pi$
$947$	−40.0800	−1.30242	−0.651212	+	0.758896i	$0.725739\pi$
$948$	43.3512	1.40798
$949$	56.2261	1.82518
$950$	0	0
$951$	−4.63664	−0.150353
$952$	3.66972	0.118936
$953$	5.03309	0.163038	0.0815188	−	0.996672i	$-0.474023\pi$
$953$	5.03309	0.163038	0.0815188	+	0.996672i	$0.474023\pi$
$954$	−31.8778	−1.03208
$955$	0	0
$956$	12.3064	0.398016
$957$	−3.00000	−0.0969762
$958$	21.8217	0.705029
$959$	−20.0360	−0.646997
$960$	0	0
$961$	−31.0000	−1.00000
$962$	56.8807	1.83391
$963$	−4.22018	−0.135994
$964$	−32.6715	−1.05228
$965$	0	0
$966$	−25.0531	−0.806070
$967$	−1.72871	−0.0555916	−0.0277958	−	0.999614i	$-0.508849\pi$
$967$	−1.72871	−0.0555916	−0.0277958	+	0.999614i	$0.508849\pi$
$968$	8.22018	0.264207
$969$	3.65760	0.117499
$970$	0	0
$971$	−16.1071	−0.516903	−0.258451	−	0.966024i	$-0.583212\pi$
$971$	−16.1071	−0.516903	−0.258451	+	0.966024i	$0.583212\pi$
$972$	3.71871	0.119278
$973$	−8.54456	−0.273926
$974$	85.0649	2.72565
$975$	0	0
$976$	17.7857	0.569307
$977$	13.2323	0.423339	0.211669	−	0.977341i	$-0.432110\pi$
$977$	13.2323	0.423339	0.211669	+	0.977341i	$0.432110\pi$
$978$	−51.0088	−1.63108
$979$	−6.99083	−0.223428
$980$	0	0
$981$	−11.8748	−0.379134
$982$	10.6186	0.338854
$983$	−3.08913	−0.0985278	−0.0492639	−	0.998786i	$-0.515688\pi$
$983$	−3.08913	−0.0985278	−0.0492639	+	0.998786i	$0.515688\pi$
$984$	−17.7857	−0.566988
$985$	0	0
$986$	−1.60862	−0.0512288
$987$	−6.32438	−0.201307
$988$	123.547	3.93056
$989$	66.7376	2.12213
$990$	0	0
$991$	30.9138	0.982010	0.491005	−	0.871157i	$-0.336630\pi$
$991$	30.9138	0.982010	0.491005	+	0.871157i	$0.336630\pi$
$992$	0	0
$993$	−22.2261	−0.705323
$994$	−39.1953	−1.24320
$995$	0	0
$996$	3.30931	0.104859
$997$	25.3863	0.803993	0.401996	−	0.915641i	$-0.368316\pi$
$997$	25.3863	0.803993	0.401996	+	0.915641i	$0.368316\pi$
$998$	96.5313	3.05564
$999$	3.89286	0.123165

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.a.u.1.1		3
3.2	odd	2		6525.2.a.bf.1.3		3
5.2	odd	4		2175.2.c.m.349.1		6
5.3	odd	4		2175.2.c.m.349.6		6
5.4	even	2		435.2.a.i.1.3	✓	3
15.14	odd	2		1305.2.a.q.1.1		3
20.19	odd	2		6960.2.a.cl.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.a.i.1.3	✓	3	5.4	even	2
1305.2.a.q.1.1		3	15.14	odd	2
2175.2.a.u.1.1		3	1.1	even	1	trivial
2175.2.c.m.349.1		6	5.2	odd	4
2175.2.c.m.349.6		6	5.3	odd	4
6525.2.a.bf.1.3		3	3.2	odd	2
6960.2.a.cl.1.2		3	20.19	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:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2175.a (trivial)

\(f(q)\)	\(=\)	\(q-2.39138 q^{2} +1.00000 q^{3} +3.71871 q^{4} -2.39138 q^{6} +1.32733 q^{7} -4.11009 q^{8} +1.00000 q^{9} +3.00000 q^{11} +3.71871 q^{12} -6.11009 q^{13} -3.17415 q^{14} +2.39138 q^{16} -0.672673 q^{17} -2.39138 q^{18} -5.43742 q^{19} +1.32733 q^{21} -7.17415 q^{22} +7.89286 q^{23} -4.11009 q^{24} +14.6116 q^{26} +1.00000 q^{27} +4.93594 q^{28} -1.00000 q^{29} +2.50147 q^{32} +3.00000 q^{33} +1.60862 q^{34} +3.71871 q^{36} +3.89286 q^{37} +13.0029 q^{38} -6.11009 q^{39} +4.32733 q^{41} -3.17415 q^{42} +8.45544 q^{43} +11.1561 q^{44} -18.8748 q^{46} -4.76475 q^{47} +2.39138 q^{48} -5.23820 q^{49} -0.672673 q^{51} -22.7217 q^{52} +13.3303 q^{53} -2.39138 q^{54} -5.45544 q^{56} -5.43742 q^{57} +2.39138 q^{58} +3.21724 q^{59} +7.43742 q^{61} +1.32733 q^{63} -10.7647 q^{64} -7.17415 q^{66} +1.10714 q^{67} -2.50147 q^{68} +7.89286 q^{69} +12.3483 q^{71} -4.11009 q^{72} -9.20217 q^{73} -9.30931 q^{74} -20.2202 q^{76} +3.98198 q^{77} +14.6116 q^{78} +11.6576 q^{79} +1.00000 q^{81} -10.3483 q^{82} +0.889908 q^{83} +4.93594 q^{84} -20.2202 q^{86} -1.00000 q^{87} -12.3303 q^{88} -2.33028 q^{89} -8.11009 q^{91} +29.3512 q^{92} +11.3943 q^{94} +2.50147 q^{96} +12.6756 q^{97} +12.5265 q^{98} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 3 q^{3} + 5 q^{4} - q^{6} + 4 q^{7} + 3 q^{9} + 9 q^{11} + 5 q^{12} - 6 q^{13} + 9 q^{14} + q^{16} - 2 q^{17} - q^{18} - 4 q^{19} + 4 q^{21} - 3 q^{22} - q^{23} + 13 q^{26} + 3 q^{27}+ \cdots + 9 q^{99}+O(q^{100}) \) 3 * q - q^2 + 3 * q^3 + 5 * q^4 - q^6 + 4 * q^7 + 3 * q^9 + 9 * q^11 + 5 * q^12 - 6 * q^13 + 9 * q^14 + q^16 - 2 * q^17 - q^18 - 4 * q^19 + 4 * q^21 - 3 * q^22 - q^23 + 13 * q^26 + 3 * q^27 + 21 * q^28 - 3 * q^29 - 11 * q^32 + 9 * q^33 + 11 * q^34 + 5 * q^36 - 13 * q^37 + 2 * q^38 - 6 * q^39 + 13 * q^41 + 9 * q^42 + 13 * q^43 + 15 * q^44 - 32 * q^46 - 2 * q^47 + q^48 + 9 * q^49 - 2 * q^51 - 25 * q^52 + 3 * q^53 - q^54 - 4 * q^56 - 4 * q^57 + q^58 + 22 * q^59 + 10 * q^61 + 4 * q^63 - 20 * q^64 - 3 * q^66 + 28 * q^67 + 11 * q^68 - q^69 - 3 * q^73 - 28 * q^74 - 36 * q^76 + 12 * q^77 + 13 * q^78 - 2 * q^79 + 3 * q^81 + 6 * q^82 + 15 * q^83 + 21 * q^84 - 36 * q^86 - 3 * q^87 + 30 * q^89 - 12 * q^91 + 14 * q^92 - 9 * q^94 - 11 * q^96 + q^97 + 50 * q^98 + 9 * q^99

Introduction

L-functions

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Database

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