LMFDB - Embedded newform 2175.2.a.ba.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 = [7,-2,-7,10,0,2,1] 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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 10x^{5} + 19x^{4} + 24x^{3} - 44x^{2} - 3x + 14 $ x^7 - 2x^6 - 10x^5 + 19x^4 + 24x^3 - 44x^2 - 3x + 14
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.66356$ 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)

$f(q)$	$=$	\(q-2.66356 q^{2} -1.00000 q^{3} +5.09453 q^{4} +2.66356 q^{6} +0.0170416 q^{7} -8.24244 q^{8} +1.00000 q^{9} +1.70990 q^{11} -5.09453 q^{12} +4.69566 q^{13} -0.0453913 q^{14} +11.7651 q^{16} +3.91494 q^{17} -2.66356 q^{18} +6.02411 q^{19} -0.0170416 q^{21} -4.55440 q^{22} -4.82786 q^{23} +8.24244 q^{24} -12.5071 q^{26} -1.00000 q^{27} +0.0868190 q^{28} -1.00000 q^{29} +1.33709 q^{31} -14.8522 q^{32} -1.70990 q^{33} -10.4277 q^{34} +5.09453 q^{36} +8.18905 q^{37} -16.0455 q^{38} -4.69566 q^{39} +8.36721 q^{41} +0.0453913 q^{42} -3.72560 q^{43} +8.71111 q^{44} +12.8593 q^{46} +5.36826 q^{47} -11.7651 q^{48} -6.99971 q^{49} -3.91494 q^{51} +23.9222 q^{52} +7.21637 q^{53} +2.66356 q^{54} -0.140465 q^{56} -6.02411 q^{57} +2.66356 q^{58} -8.65422 q^{59} +5.72637 q^{61} -3.56141 q^{62} +0.0170416 q^{63} +16.0294 q^{64} +4.55440 q^{66} -3.87486 q^{67} +19.9448 q^{68} +4.82786 q^{69} -4.32991 q^{71} -8.24244 q^{72} +1.78245 q^{73} -21.8120 q^{74} +30.6900 q^{76} +0.0291394 q^{77} +12.5071 q^{78} +0.233544 q^{79} +1.00000 q^{81} -22.2865 q^{82} +6.15497 q^{83} -0.0868190 q^{84} +9.92334 q^{86} +1.00000 q^{87} -14.0937 q^{88} -12.4053 q^{89} +0.0800217 q^{91} -24.5957 q^{92} -1.33709 q^{93} -14.2986 q^{94} +14.8522 q^{96} +19.2918 q^{97} +18.6441 q^{98} +1.70990 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 2 q^{2} - 7 q^{3} + 10 q^{4} + 2 q^{6} + q^{7} - 3 q^{8} + 7 q^{9} + 4 q^{11} - 10 q^{12} + q^{13} + 15 q^{14} + 12 q^{16} - 8 q^{17} - 2 q^{18} + 15 q^{19} - q^{21} - 3 q^{22} - 14 q^{23} + 3 q^{24}+ \cdots + 4 q^{99}+O(q^{100}) $ 7 * q - 2 * q^2 - 7 * q^3 + 10 * q^4 + 2 * q^6 + q^7 - 3 * q^8 + 7 * q^9 + 4 * q^11 - 10 * q^12 + q^13 + 15 * q^14 + 12 * q^16 - 8 * q^17 - 2 * q^18 + 15 * q^19 - q^21 - 3 * q^22 - 14 * q^23 + 3 * q^24 + 6 * q^26 - 7 * q^27 + 24 * q^28 - 7 * q^29 + 5 * q^31 - 18 * q^32 - 4 * q^33 + 7 * q^34 + 10 * q^36 + 6 * q^37 + 18 * q^38 - q^39 + 22 * q^41 - 15 * q^42 + 19 * q^43 + 15 * q^44 - 4 * q^46 - 22 * q^47 - 12 * q^48 + 12 * q^49 + 8 * q^51 - 11 * q^52 - 10 * q^53 + 2 * q^54 + 14 * q^56 - 15 * q^57 + 2 * q^58 + 6 * q^59 + 23 * q^61 - 40 * q^62 + q^63 + 5 * q^64 + 3 * q^66 + 13 * q^67 + 7 * q^68 + 14 * q^69 + 26 * q^71 - 3 * q^72 + 24 * q^73 - 10 * q^74 + 46 * q^76 - 4 * q^77 - 6 * q^78 + 14 * q^79 + 7 * q^81 - 16 * q^82 - 10 * q^83 - 24 * q^84 + 44 * q^86 + 7 * q^87 + 66 * q^88 + 14 * q^89 + 13 * q^91 - 58 * q^92 - 5 * q^93 - 3 * q^94 + 18 * q^96 + 31 * q^97 + 59 * q^98 + 4 * 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 $

$2$	−2.66356	−1.88342	−0.941709	−	0.336429i	$-0.890781\pi$
$2$	−2.66356	−1.88342	−0.941709	+	0.336429i	$0.890781\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	5.09453	2.54726
$5$	0	0
$5$	0	0
$6$	2.66356	1.08739
$7$	0.0170416	0.00644113	0.00322057	−	0.999995i	$-0.498975\pi$
$7$	0.0170416	0.00644113	0.00322057	+	0.999995i	$0.498975\pi$
$8$	−8.24244	−2.91414
$9$	1.00000	0.333333
$10$	0	0
$11$	1.70990	0.515553	0.257777	−	0.966205i	$-0.417010\pi$
$11$	1.70990	0.515553	0.257777	+	0.966205i	$0.417010\pi$
$12$	−5.09453	−1.47066
$13$	4.69566	1.30234	0.651171	−	0.758931i	$-0.274278\pi$
$13$	4.69566	1.30234	0.651171	+	0.758931i	$0.274278\pi$
$14$	−0.0453913	−0.0121313
$15$	0	0
$16$	11.7651	2.94129
$17$	3.91494	0.949513	0.474757	−	0.880117i	$-0.342536\pi$
$17$	3.91494	0.949513	0.474757	+	0.880117i	$0.342536\pi$
$18$	−2.66356	−0.627806
$19$	6.02411	1.38202	0.691012	−	0.722843i	$-0.257165\pi$
$19$	6.02411	1.38202	0.691012	+	0.722843i	$0.257165\pi$
$20$	0	0
$21$	−0.0170416	−0.00371879
$22$	−4.55440	−0.971002
$23$	−4.82786	−1.00668	−0.503339	−	0.864089i	$-0.667895\pi$
$23$	−4.82786	−1.00668	−0.503339	+	0.864089i	$0.667895\pi$
$24$	8.24244	1.68248
$25$	0	0
$26$	−12.5071	−2.45285
$27$	−1.00000	−0.192450
$28$	0.0868190	0.0164073
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	1.33709	0.240148	0.120074	−	0.992765i	$-0.461687\pi$
$31$	1.33709	0.240148	0.120074	+	0.992765i	$0.461687\pi$
$32$	−14.8522	−2.62553
$33$	−1.70990	−0.297655
$34$	−10.4277	−1.78833
$35$	0	0
$36$	5.09453	0.849088
$37$	8.18905	1.34627	0.673136	−	0.739519i	$-0.264947\pi$
$37$	8.18905	1.34627	0.673136	+	0.739519i	$0.264947\pi$
$38$	−16.0455	−2.60293
$39$	−4.69566	−0.751907
$40$	0	0
$41$	8.36721	1.30674	0.653369	−	0.757039i	$-0.273355\pi$
$41$	8.36721	1.30674	0.653369	+	0.757039i	$0.273355\pi$
$42$	0.0453913	0.00700403
$43$	−3.72560	−0.568149	−0.284074	−	0.958802i	$-0.591686\pi$
$43$	−3.72560	−0.568149	−0.284074	+	0.958802i	$0.591686\pi$
$44$	8.71111	1.31325
$45$	0	0
$46$	12.8593	1.89600
$47$	5.36826	0.783041	0.391520	−	0.920169i	$-0.371949\pi$
$47$	5.36826	0.783041	0.391520	+	0.920169i	$0.371949\pi$
$48$	−11.7651	−1.69815
$49$	−6.99971	−0.999959
$50$	0	0
$51$	−3.91494	−0.548202
$52$	23.9222	3.31741
$53$	7.21637	0.991245	0.495622	−	0.868538i	$-0.334940\pi$
$53$	7.21637	0.991245	0.495622	+	0.868538i	$0.334940\pi$
$54$	2.66356	0.362464
$55$	0	0
$56$	−0.140465	−0.0187704
$57$	−6.02411	−0.797912
$58$	2.66356	0.349742
$59$	−8.65422	−1.12668	−0.563342	−	0.826224i	$-0.690485\pi$
$59$	−8.65422	−1.12668	−0.563342	+	0.826224i	$0.690485\pi$
$60$	0	0
$61$	5.72637	0.733187	0.366594	−	0.930381i	$-0.380524\pi$
$61$	5.72637	0.733187	0.366594	+	0.930381i	$0.380524\pi$
$62$	−3.56141	−0.452299
$63$	0.0170416	0.00214704
$64$	16.0294	2.00368
$65$	0	0
$66$	4.55440	0.560608
$67$	−3.87486	−0.473390	−0.236695	−	0.971584i	$-0.576064\pi$
$67$	−3.87486	−0.473390	−0.236695	+	0.971584i	$0.576064\pi$
$68$	19.9448	2.41866
$69$	4.82786	0.581206
$70$	0	0
$71$	−4.32991	−0.513866	−0.256933	−	0.966429i	$-0.582712\pi$
$71$	−4.32991	−0.513866	−0.256933	+	0.966429i	$0.582712\pi$
$72$	−8.24244	−0.971381
$73$	1.78245	0.208620	0.104310	−	0.994545i	$-0.466737\pi$
$73$	1.78245	0.208620	0.104310	+	0.994545i	$0.466737\pi$
$74$	−21.8120	−2.53559
$75$	0	0
$76$	30.6900	3.52038
$77$	0.0291394	0.00332075
$78$	12.5071	1.41616
$79$	0.233544	0.0262757	0.0131379	−	0.999914i	$-0.495818\pi$
$79$	0.233544	0.0262757	0.0131379	+	0.999914i	$0.495818\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−22.2865	−2.46114
$83$	6.15497	0.675596	0.337798	−	0.941219i	$-0.390318\pi$
$83$	6.15497	0.675596	0.337798	+	0.941219i	$0.390318\pi$
$84$	−0.0868190	−0.00947273
$85$	0	0
$86$	9.92334	1.07006
$87$	1.00000	0.107211
$88$	−14.0937	−1.50240
$89$	−12.4053	−1.31496	−0.657478	−	0.753474i	$-0.728377\pi$
$89$	−12.4053	−1.31496	−0.657478	+	0.753474i	$0.728377\pi$
$90$	0	0
$91$	0.0800217	0.00838855
$92$	−24.5957	−2.56427
$93$	−1.33709	−0.138650
$94$	−14.2986	−1.47479
$95$	0	0
$96$	14.8522	1.51585
$97$	19.2918	1.95879	0.979395	−	0.201954i	$-0.0647293\pi$
$97$	19.2918	1.95879	0.979395	+	0.201954i	$0.0647293\pi$
$98$	18.6441	1.88334
$99$	1.70990	0.171851
$100$	0	0
$101$	−8.74987	−0.870644	−0.435322	−	0.900275i	$-0.643366\pi$
$101$	−8.74987	−0.870644	−0.435322	+	0.900275i	$0.643366\pi$
$102$	10.4277	1.03249
$103$	−10.9615	−1.08007	−0.540036	−	0.841642i	$-0.681589\pi$
$103$	−10.9615	−1.08007	−0.540036	+	0.841642i	$0.681589\pi$
$104$	−38.7037	−3.79521
$105$	0	0
$106$	−19.2212	−1.86693
$107$	−15.8139	−1.52879	−0.764393	−	0.644750i	$-0.776961\pi$
$107$	−15.8139	−1.52879	−0.764393	+	0.644750i	$0.776961\pi$
$108$	−5.09453	−0.490221
$109$	−20.2768	−1.94216	−0.971082	−	0.238747i	$-0.923263\pi$
$109$	−20.2768	−1.94216	−0.971082	+	0.238747i	$0.923263\pi$
$110$	0	0
$111$	−8.18905	−0.777270
$112$	0.200497	0.0189452
$113$	−8.86901	−0.834326	−0.417163	−	0.908832i	$-0.636976\pi$
$113$	−8.86901	−0.834326	−0.417163	+	0.908832i	$0.636976\pi$
$114$	16.0455	1.50280
$115$	0	0
$116$	−5.09453	−0.473015
$117$	4.69566	0.434114
$118$	23.0510	2.12202
$119$	0.0667170	0.00611594
$120$	0	0
$121$	−8.07625	−0.734205
$122$	−15.2525	−1.38090
$123$	−8.36721	−0.754446
$124$	6.81183	0.611721
$125$	0	0
$126$	−0.0453913	−0.00404378
$127$	4.88854	0.433788	0.216894	−	0.976195i	$-0.430407\pi$
$127$	4.88854	0.433788	0.216894	+	0.976195i	$0.430407\pi$
$128$	−12.9909	−1.14824
$129$	3.72560	0.328021
$130$	0	0
$131$	14.0971	1.23167	0.615833	−	0.787876i	$-0.288819\pi$
$131$	14.0971	1.23167	0.615833	+	0.787876i	$0.288819\pi$
$132$	−8.71111	−0.758205
$133$	0.102661	0.00890180
$134$	10.3209	0.891591
$135$	0	0
$136$	−32.2687	−2.76702
$137$	−3.57023	−0.305025	−0.152513	−	0.988302i	$-0.548736\pi$
$137$	−3.57023	−0.305025	−0.152513	+	0.988302i	$0.548736\pi$
$138$	−12.8593	−1.09465
$139$	11.7235	0.994371	0.497185	−	0.867644i	$-0.334367\pi$
$139$	11.7235	0.994371	0.497185	+	0.867644i	$0.334367\pi$
$140$	0	0
$141$	−5.36826	−0.452089
$142$	11.5330	0.967825
$143$	8.02909	0.671426
$144$	11.7651	0.980429
$145$	0	0
$146$	−4.74766	−0.392919
$147$	6.99971	0.577326
$148$	41.7193	3.42931
$149$	−0.672890	−0.0551253	−0.0275626	−	0.999620i	$-0.508775\pi$
$149$	−0.672890	−0.0551253	−0.0275626	+	0.999620i	$0.508775\pi$
$150$	0	0
$151$	24.2619	1.97441	0.987203	−	0.159468i	$-0.0509779\pi$
$151$	24.2619	1.97441	0.987203	+	0.159468i	$0.0509779\pi$
$152$	−49.6533	−4.02742
$153$	3.91494	0.316504
$154$	−0.0776145	−0.00625435
$155$	0	0
$156$	−23.9222	−1.91531
$157$	14.0158	1.11859	0.559293	−	0.828970i	$-0.311073\pi$
$157$	14.0158	1.11859	0.559293	+	0.828970i	$0.311073\pi$
$158$	−0.622057	−0.0494882
$159$	−7.21637	−0.572296
$160$	0	0
$161$	−0.0822746	−0.00648415
$162$	−2.66356	−0.209269
$163$	−9.77061	−0.765293	−0.382646	−	0.923895i	$-0.624987\pi$
$163$	−9.77061	−0.765293	−0.382646	+	0.923895i	$0.624987\pi$
$164$	42.6270	3.32861
$165$	0	0
$166$	−16.3941	−1.27243
$167$	−5.91586	−0.457783	−0.228891	−	0.973452i	$-0.573510\pi$
$167$	−5.91586	−0.457783	−0.228891	+	0.973452i	$0.573510\pi$
$168$	0.140465	0.0108371
$169$	9.04921	0.696093
$170$	0	0
$171$	6.02411	0.460675
$172$	−18.9802	−1.44722
$173$	20.1704	1.53353	0.766763	−	0.641930i	$-0.221866\pi$
$173$	20.1704	1.53353	0.766763	+	0.641930i	$0.221866\pi$
$174$	−2.66356	−0.201924
$175$	0	0
$176$	20.1172	1.51639
$177$	8.65422	0.650491
$178$	33.0421	2.47661
$179$	15.4896	1.15775	0.578873	−	0.815418i	$-0.303493\pi$
$179$	15.4896	1.15775	0.578873	+	0.815418i	$0.303493\pi$
$180$	0	0
$181$	17.2509	1.28225	0.641126	−	0.767436i	$-0.278468\pi$
$181$	17.2509	1.28225	0.641126	+	0.767436i	$0.278468\pi$
$182$	−0.213142	−0.0157991
$183$	−5.72637	−0.423306
$184$	39.7933	2.93360
$185$	0	0
$186$	3.56141	0.261135
$187$	6.69415	0.489525
$188$	27.3487	1.99461
$189$	−0.0170416	−0.00123960
$190$	0	0
$191$	11.5130	0.833048	0.416524	−	0.909125i	$-0.363248\pi$
$191$	11.5130	0.833048	0.416524	+	0.909125i	$0.363248\pi$
$192$	−16.0294	−1.15683
$193$	−13.0261	−0.937642	−0.468821	−	0.883293i	$-0.655321\pi$
$193$	−13.0261	−0.937642	−0.468821	+	0.883293i	$0.655321\pi$
$194$	−51.3849	−3.68922
$195$	0	0
$196$	−35.6602	−2.54716
$197$	−14.3280	−1.02083	−0.510415	−	0.859928i	$-0.670508\pi$
$197$	−14.3280	−1.02083	−0.510415	+	0.859928i	$0.670508\pi$
$198$	−4.55440	−0.323667
$199$	−16.2728	−1.15355	−0.576775	−	0.816903i	$-0.695689\pi$
$199$	−16.2728	−1.15355	−0.576775	+	0.816903i	$0.695689\pi$
$200$	0	0
$201$	3.87486	0.273312
$202$	23.3058	1.63979
$203$	−0.0170416	−0.00119609
$204$	−19.9448	−1.39641
$205$	0	0
$206$	29.1966	2.03423
$207$	−4.82786	−0.335559
$208$	55.2451	3.83056
$209$	10.3006	0.712507
$210$	0	0
$211$	−24.3854	−1.67876	−0.839380	−	0.543544i	$-0.817082\pi$
$211$	−24.3854	−1.67876	−0.839380	+	0.543544i	$0.817082\pi$
$212$	36.7640	2.52496
$213$	4.32991	0.296681
$214$	42.1212	2.87934
$215$	0	0
$216$	8.24244	0.560827
$217$	0.0227862	0.00154683
$218$	54.0083	3.65791
$219$	−1.78245	−0.120447
$220$	0	0
$221$	18.3832	1.23659
$222$	21.8120	1.46393
$223$	−0.753555	−0.0504617	−0.0252309	−	0.999682i	$-0.508032\pi$
$223$	−0.753555	−0.0504617	−0.0252309	+	0.999682i	$0.508032\pi$
$224$	−0.253106	−0.0169114
$225$	0	0
$226$	23.6231	1.57138
$227$	21.1711	1.40518	0.702589	−	0.711596i	$-0.252027\pi$
$227$	21.1711	1.40518	0.702589	+	0.711596i	$0.252027\pi$
$228$	−30.6900	−2.03249
$229$	−7.97533	−0.527025	−0.263512	−	0.964656i	$-0.584881\pi$
$229$	−7.97533	−0.527025	−0.263512	+	0.964656i	$0.584881\pi$
$230$	0	0
$231$	−0.0291394	−0.00191723
$232$	8.24244	0.541143
$233$	−14.0847	−0.922718	−0.461359	−	0.887214i	$-0.652638\pi$
$233$	−14.0847	−0.922718	−0.461359	+	0.887214i	$0.652638\pi$
$234$	−12.5071	−0.817618
$235$	0	0
$236$	−44.0892	−2.86996
$237$	−0.233544	−0.0151703
$238$	−0.177705	−0.0115189
$239$	15.4813	1.00140	0.500701	−	0.865620i	$-0.333076\pi$
$239$	15.4813	1.00140	0.500701	+	0.865620i	$0.333076\pi$
$240$	0	0
$241$	12.1294	0.781324	0.390662	−	0.920534i	$-0.372246\pi$
$241$	12.1294	0.781324	0.390662	+	0.920534i	$0.372246\pi$
$242$	21.5115	1.38281
$243$	−1.00000	−0.0641500
$244$	29.1732	1.86762
$245$	0	0
$246$	22.2865	1.42094
$247$	28.2871	1.79987
$248$	−11.0209	−0.699826
$249$	−6.15497	−0.390055
$250$	0	0
$251$	14.1090	0.890550	0.445275	−	0.895394i	$-0.353106\pi$
$251$	14.1090	0.890550	0.445275	+	0.895394i	$0.353106\pi$
$252$	0.0868190	0.00546909
$253$	−8.25514	−0.518996
$254$	−13.0209	−0.817003
$255$	0	0
$256$	2.54296	0.158935
$257$	26.6644	1.66328	0.831639	−	0.555317i	$-0.187403\pi$
$257$	26.6644	1.66328	0.831639	+	0.555317i	$0.187403\pi$
$258$	−9.92334	−0.617800
$259$	0.139555	0.00867151
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	−37.5483	−2.31974
$263$	−5.74517	−0.354263	−0.177131	−	0.984187i	$-0.556682\pi$
$263$	−5.74517	−0.354263	−0.177131	+	0.984187i	$0.556682\pi$
$264$	14.0937	0.867409
$265$	0	0
$266$	−0.273442	−0.0167658
$267$	12.4053	0.759190
$268$	−19.7406	−1.20585
$269$	19.3477	1.17965	0.589826	−	0.807530i	$-0.299196\pi$
$269$	19.3477	1.17965	0.589826	+	0.807530i	$0.299196\pi$
$270$	0	0
$271$	−30.7485	−1.86784	−0.933920	−	0.357482i	$-0.883635\pi$
$271$	−30.7485	−1.86784	−0.933920	+	0.357482i	$0.883635\pi$
$272$	46.0599	2.79279
$273$	−0.0800217	−0.00484313
$274$	9.50950	0.574490
$275$	0	0
$276$	24.5957	1.48048
$277$	24.0737	1.44645	0.723225	−	0.690613i	$-0.242659\pi$
$277$	24.0737	1.44645	0.723225	+	0.690613i	$0.242659\pi$
$278$	−31.2261	−1.87282
$279$	1.33709	0.0800494
$280$	0	0
$281$	23.4067	1.39633	0.698164	−	0.715938i	$-0.254001\pi$
$281$	23.4067	1.39633	0.698164	+	0.715938i	$0.254001\pi$
$282$	14.2986	0.851472
$283$	20.8589	1.23993	0.619967	−	0.784628i	$-0.287146\pi$
$283$	20.8589	1.23993	0.619967	+	0.784628i	$0.287146\pi$
$284$	−22.0589	−1.30895
$285$	0	0
$286$	−21.3859	−1.26458
$287$	0.142591	0.00841688
$288$	−14.8522	−0.875176
$289$	−1.67321	−0.0984242
$290$	0	0
$291$	−19.2918	−1.13091
$292$	9.08075	0.531410
$293$	12.0727	0.705294	0.352647	−	0.935756i	$-0.385282\pi$
$293$	12.0727	0.705294	0.352647	+	0.935756i	$0.385282\pi$
$294$	−18.6441	−1.08735
$295$	0	0
$296$	−67.4978	−3.92323
$297$	−1.70990	−0.0992183
$298$	1.79228	0.103824
$299$	−22.6700	−1.31104
$300$	0	0
$301$	−0.0634903	−0.00365952
$302$	−64.6229	−3.71863
$303$	8.74987	0.502667
$304$	70.8745	4.06493
$305$	0	0
$306$	−10.4277	−0.596110
$307$	−20.4035	−1.16449	−0.582245	−	0.813013i	$-0.697826\pi$
$307$	−20.4035	−1.16449	−0.582245	+	0.813013i	$0.697826\pi$
$308$	0.148452	0.00845881
$309$	10.9615	0.623579
$310$	0	0
$311$	−12.4159	−0.704040	−0.352020	−	0.935993i	$-0.614505\pi$
$311$	−12.4159	−0.704040	−0.352020	+	0.935993i	$0.614505\pi$
$312$	38.7037	2.19116
$313$	−14.8518	−0.839473	−0.419737	−	0.907646i	$-0.637878\pi$
$313$	−14.8518	−0.839473	−0.419737	+	0.907646i	$0.637878\pi$
$314$	−37.3320	−2.10677
$315$	0	0
$316$	1.18980	0.0669312
$317$	21.8783	1.22881	0.614404	−	0.788992i	$-0.289397\pi$
$317$	21.8783	1.22881	0.614404	+	0.788992i	$0.289397\pi$
$318$	19.2212	1.07787
$319$	−1.70990	−0.0957358
$320$	0	0
$321$	15.8139	0.882645
$322$	0.219143	0.0122124
$323$	23.5840	1.31225
$324$	5.09453	0.283029
$325$	0	0
$326$	26.0245	1.44137
$327$	20.2768	1.12131
$328$	−68.9662	−3.80802
$329$	0.0914839	0.00504367
$330$	0	0
$331$	28.6380	1.57409	0.787043	−	0.616898i	$-0.211611\pi$
$331$	28.6380	1.57409	0.787043	+	0.616898i	$0.211611\pi$
$332$	31.3567	1.72092
$333$	8.18905	0.448757
$334$	15.7572	0.862196
$335$	0	0
$336$	−0.200497	−0.0109380
$337$	−6.75074	−0.367736	−0.183868	−	0.982951i	$-0.558862\pi$
$337$	−6.75074	−0.367736	−0.183868	+	0.982951i	$0.558862\pi$
$338$	−24.1031	−1.31103
$339$	8.86901	0.481698
$340$	0	0
$341$	2.28628	0.123809
$342$	−16.0455	−0.867643
$343$	−0.238578	−0.0128820
$344$	30.7081	1.65567
$345$	0	0
$346$	−53.7249	−2.88827
$347$	1.87591	0.100704	0.0503520	−	0.998732i	$-0.483966\pi$
$347$	1.87591	0.100704	0.0503520	+	0.998732i	$0.483966\pi$
$348$	5.09453	0.273095
$349$	24.6359	1.31873	0.659364	−	0.751824i	$-0.270826\pi$
$349$	24.6359	1.31873	0.659364	+	0.751824i	$0.270826\pi$
$350$	0	0
$351$	−4.69566	−0.250636
$352$	−25.3958	−1.35360
$353$	13.3468	0.710379	0.355189	−	0.934794i	$-0.384416\pi$
$353$	13.3468	0.710379	0.355189	+	0.934794i	$0.384416\pi$
$354$	−23.0510	−1.22515
$355$	0	0
$356$	−63.1989	−3.34954
$357$	−0.0667170	−0.00353104
$358$	−41.2574	−2.18052
$359$	21.5263	1.13611	0.568057	−	0.822990i	$-0.307695\pi$
$359$	21.5263	1.13611	0.568057	+	0.822990i	$0.307695\pi$
$360$	0	0
$361$	17.2898	0.909992
$362$	−45.9488	−2.41502
$363$	8.07625	0.423893
$364$	0.407673	0.0213678
$365$	0	0
$366$	15.2525	0.797262
$367$	−9.36534	−0.488867	−0.244433	−	0.969666i	$-0.578602\pi$
$367$	−9.36534	−0.488867	−0.244433	+	0.969666i	$0.578602\pi$
$368$	−56.8005	−2.96093
$369$	8.36721	0.435580
$370$	0	0
$371$	0.122979	0.00638474
$372$	−6.81183	−0.353177
$373$	−19.3366	−1.00121	−0.500605	−	0.865676i	$-0.666889\pi$
$373$	−19.3366	−1.00121	−0.500605	+	0.865676i	$0.666889\pi$
$374$	−17.8302	−0.921980
$375$	0	0
$376$	−44.2475	−2.28189
$377$	−4.69566	−0.241839
$378$	0.0453913	0.00233468
$379$	−12.4163	−0.637783	−0.318891	−	0.947791i	$-0.603311\pi$
$379$	−12.4163	−0.637783	−0.318891	+	0.947791i	$0.603311\pi$
$380$	0	0
$381$	−4.88854	−0.250447
$382$	−30.6654	−1.56898
$383$	−24.5909	−1.25653	−0.628267	−	0.777998i	$-0.716236\pi$
$383$	−24.5909	−1.25653	−0.628267	+	0.777998i	$0.716236\pi$
$384$	12.9909	0.662937
$385$	0	0
$386$	34.6958	1.76597
$387$	−3.72560	−0.189383
$388$	98.2828	4.98955
$389$	17.1567	0.869880	0.434940	−	0.900460i	$-0.356770\pi$
$389$	17.1567	0.869880	0.434940	+	0.900460i	$0.356770\pi$
$390$	0	0
$391$	−18.9008	−0.955854
$392$	57.6947	2.91402
$393$	−14.0971	−0.711103
$394$	38.1635	1.92265
$395$	0	0
$396$	8.71111	0.437750
$397$	−2.90545	−0.145820	−0.0729102	−	0.997339i	$-0.523229\pi$
$397$	−2.90545	−0.145820	−0.0729102	+	0.997339i	$0.523229\pi$
$398$	43.3436	2.17262
$399$	−0.102661	−0.00513946
$400$	0	0
$401$	20.4427	1.02086	0.510431	−	0.859919i	$-0.329486\pi$
$401$	20.4427	1.02086	0.510431	+	0.859919i	$0.329486\pi$
$402$	−10.3209	−0.514760
$403$	6.27851	0.312755
$404$	−44.5764	−2.21776
$405$	0	0
$406$	0.0453913	0.00225273
$407$	14.0024	0.694075
$408$	32.2687	1.59754
$409$	−3.13657	−0.155093	−0.0775467	−	0.996989i	$-0.524709\pi$
$409$	−3.13657	−0.155093	−0.0775467	+	0.996989i	$0.524709\pi$
$410$	0	0
$411$	3.57023	0.176106
$412$	−55.8438	−2.75123
$413$	−0.147482	−0.00725712
$414$	12.8593	0.631999
$415$	0	0
$416$	−69.7410	−3.41933
$417$	−11.7235	−0.574100
$418$	−27.4362	−1.34195
$419$	36.6119	1.78861	0.894305	−	0.447458i	$-0.147670\pi$
$419$	36.6119	1.78861	0.894305	+	0.447458i	$0.147670\pi$
$420$	0	0
$421$	−13.8706	−0.676012	−0.338006	−	0.941144i	$-0.609752\pi$
$421$	−13.8706	−0.676012	−0.338006	+	0.941144i	$0.609752\pi$
$422$	64.9519	3.16181
$423$	5.36826	0.261014
$424$	−59.4805	−2.88863
$425$	0	0
$426$	−11.5330	−0.558774
$427$	0.0975868	0.00472256
$428$	−80.5643	−3.89422
$429$	−8.02909	−0.387648
$430$	0	0
$431$	−29.6786	−1.42957	−0.714785	−	0.699344i	$-0.753475\pi$
$431$	−29.6786	−1.42957	−0.714785	+	0.699344i	$0.753475\pi$
$432$	−11.7651	−0.566051
$433$	32.1319	1.54416	0.772081	−	0.635525i	$-0.219216\pi$
$433$	32.1319	1.54416	0.772081	+	0.635525i	$0.219216\pi$
$434$	−0.0606922	−0.00291332
$435$	0	0
$436$	−103.301	−4.94720
$437$	−29.0835	−1.39125
$438$	4.74766	0.226852
$439$	34.9322	1.66722	0.833611	−	0.552352i	$-0.186269\pi$
$439$	34.9322	1.66722	0.833611	+	0.552352i	$0.186269\pi$
$440$	0	0
$441$	−6.99971	−0.333320
$442$	−48.9648	−2.32902
$443$	−30.7938	−1.46306	−0.731528	−	0.681812i	$-0.761192\pi$
$443$	−30.7938	−1.46306	−0.731528	+	0.681812i	$0.761192\pi$
$444$	−41.7193	−1.97991
$445$	0	0
$446$	2.00713	0.0950406
$447$	0.672890	0.0318266
$448$	0.273168	0.0129060
$449$	26.6304	1.25677	0.628383	−	0.777904i	$-0.283717\pi$
$449$	26.6304	1.25677	0.628383	+	0.777904i	$0.283717\pi$
$450$	0	0
$451$	14.3071	0.673693
$452$	−45.1834	−2.12525
$453$	−24.2619	−1.13992
$454$	−56.3905	−2.64654
$455$	0	0
$456$	49.6533	2.32523
$457$	−9.03169	−0.422485	−0.211242	−	0.977434i	$-0.567751\pi$
$457$	−9.03169	−0.422485	−0.211242	+	0.977434i	$0.567751\pi$
$458$	21.2427	0.992608
$459$	−3.91494	−0.182734
$460$	0	0
$461$	−17.6240	−0.820832	−0.410416	−	0.911898i	$-0.634616\pi$
$461$	−17.6240	−0.820832	−0.410416	+	0.911898i	$0.634616\pi$
$462$	0.0776145	0.00361095
$463$	11.1431	0.517865	0.258933	−	0.965895i	$-0.416629\pi$
$463$	11.1431	0.517865	0.258933	+	0.965895i	$0.416629\pi$
$464$	−11.7651	−0.546183
$465$	0	0
$466$	37.5153	1.73786
$467$	19.8932	0.920548	0.460274	−	0.887777i	$-0.347751\pi$
$467$	19.8932	0.920548	0.460274	+	0.887777i	$0.347751\pi$
$468$	23.9222	1.10580
$469$	−0.0660340	−0.00304917
$470$	0	0
$471$	−14.0158	−0.645816
$472$	71.3319	3.28332
$473$	−6.37039	−0.292911
$474$	0.622057	0.0285720
$475$	0	0
$476$	0.339892	0.0155789
$477$	7.21637	0.330415
$478$	−41.2353	−1.88606
$479$	−21.6917	−0.991118	−0.495559	−	0.868574i	$-0.665037\pi$
$479$	−21.6917	−0.991118	−0.495559	+	0.868574i	$0.665037\pi$
$480$	0	0
$481$	38.4530	1.75331
$482$	−32.3073	−1.47156
$483$	0.0822746	0.00374362
$484$	−41.1447	−1.87021
$485$	0	0
$486$	2.66356	0.120821
$487$	−11.1364	−0.504639	−0.252320	−	0.967644i	$-0.581193\pi$
$487$	−11.1364	−0.504639	−0.252320	+	0.967644i	$0.581193\pi$
$488$	−47.1993	−2.13661
$489$	9.77061	0.441842
$490$	0	0
$491$	27.7172	1.25086	0.625431	−	0.780280i	$-0.284923\pi$
$491$	27.7172	1.25086	0.625431	+	0.780280i	$0.284923\pi$
$492$	−42.6270	−1.92177
$493$	−3.91494	−0.176320
$494$	−75.3444	−3.38990
$495$	0	0
$496$	15.7310	0.706344
$497$	−0.0737888	−0.00330988
$498$	16.3941	0.734637
$499$	−29.8657	−1.33697	−0.668485	−	0.743725i	$-0.733057\pi$
$499$	−29.8657	−1.33697	−0.668485	+	0.743725i	$0.733057\pi$
$500$	0	0
$501$	5.91586	0.264301
$502$	−37.5800	−1.67728
$503$	−28.6721	−1.27843	−0.639213	−	0.769030i	$-0.720740\pi$
$503$	−28.6721	−1.27843	−0.639213	+	0.769030i	$0.720740\pi$
$504$	−0.140465	−0.00625679
$505$	0	0
$506$	21.9880	0.977487
$507$	−9.04921	−0.401889
$508$	24.9048	1.10497
$509$	−22.7354	−1.00773	−0.503864	−	0.863783i	$-0.668089\pi$
$509$	−22.7354	−1.00773	−0.503864	+	0.863783i	$0.668089\pi$
$510$	0	0
$511$	0.0303759	0.00134375
$512$	19.2084	0.848899
$513$	−6.02411	−0.265971
$514$	−71.0220	−3.13265
$515$	0	0
$516$	18.9802	0.835556
$517$	9.17916	0.403699
$518$	−0.371712	−0.0163321
$519$	−20.1704	−0.885382
$520$	0	0
$521$	15.8392	0.693926	0.346963	−	0.937879i	$-0.387213\pi$
$521$	15.8392	0.693926	0.346963	+	0.937879i	$0.387213\pi$
$522$	2.66356	0.116581
$523$	−28.1248	−1.22981	−0.614905	−	0.788601i	$-0.710806\pi$
$523$	−28.1248	−1.22981	−0.614905	+	0.788601i	$0.710806\pi$
$524$	71.8179	3.13738
$525$	0	0
$526$	15.3026	0.667224
$527$	5.23463	0.228024
$528$	−20.1172	−0.875488
$529$	0.308221	0.0134009
$530$	0	0
$531$	−8.65422	−0.375561
$532$	0.523007	0.0226752
$533$	39.2896	1.70182
$534$	−33.0421	−1.42987
$535$	0	0
$536$	31.9383	1.37953
$537$	−15.4896	−0.668425
$538$	−51.5338	−2.22178
$539$	−11.9688	−0.515532
$540$	0	0
$541$	−23.9993	−1.03181	−0.515906	−	0.856645i	$-0.672545\pi$
$541$	−23.9993	−1.03181	−0.515906	+	0.856645i	$0.672545\pi$
$542$	81.9004	3.51792
$543$	−17.2509	−0.740309
$544$	−58.1457	−2.49297
$545$	0	0
$546$	0.213142	0.00912164
$547$	24.1285	1.03166	0.515831	−	0.856691i	$-0.327483\pi$
$547$	24.1285	1.03166	0.515831	+	0.856691i	$0.327483\pi$
$548$	−18.1886	−0.776980
$549$	5.72637	0.244396
$550$	0	0
$551$	−6.02411	−0.256636
$552$	−39.7933	−1.69372
$553$	0.00397997	0.000169245 0
$554$	−64.1217	−2.72427
$555$	0	0
$556$	59.7255	2.53292
$557$	−29.8474	−1.26467	−0.632337	−	0.774693i	$-0.717904\pi$
$557$	−29.8474	−1.26467	−0.632337	+	0.774693i	$0.717904\pi$
$558$	−3.56141	−0.150766
$559$	−17.4942	−0.739924
$560$	0	0
$561$	−6.69415	−0.282627
$562$	−62.3451	−2.62987
$563$	−9.29980	−0.391940	−0.195970	−	0.980610i	$-0.562786\pi$
$563$	−9.29980	−0.391940	−0.195970	+	0.980610i	$0.562786\pi$
$564$	−27.3487	−1.15159
$565$	0	0
$566$	−55.5589	−2.33531
$567$	0.0170416	0.000715681 0
$568$	35.6891	1.49748
$569$	−10.7468	−0.450529	−0.225264	−	0.974298i	$-0.572325\pi$
$569$	−10.7468	−0.450529	−0.225264	+	0.974298i	$0.572325\pi$
$570$	0	0
$571$	−22.3620	−0.935822	−0.467911	−	0.883776i	$-0.654993\pi$
$571$	−22.3620	−0.935822	−0.467911	+	0.883776i	$0.654993\pi$
$572$	40.9044	1.71030
$573$	−11.5130	−0.480961
$574$	−0.379799	−0.0158525
$575$	0	0
$576$	16.0294	0.667894
$577$	29.7336	1.23783	0.618913	−	0.785459i	$-0.287573\pi$
$577$	29.7336	1.23783	0.618913	+	0.785459i	$0.287573\pi$
$578$	4.45669	0.185374
$579$	13.0261	0.541348
$580$	0	0
$581$	0.104891	0.00435160
$582$	51.3849	2.12997
$583$	12.3392	0.511039
$584$	−14.6918	−0.607949
$585$	0	0
$586$	−32.1563	−1.32836
$587$	−25.9770	−1.07219	−0.536093	−	0.844159i	$-0.680101\pi$
$587$	−25.9770	−1.07219	−0.536093	+	0.844159i	$0.680101\pi$
$588$	35.6602	1.47060
$589$	8.05476	0.331891
$590$	0	0
$591$	14.3280	0.589377
$592$	96.3454	3.95977
$593$	30.1217	1.23695	0.618476	−	0.785804i	$-0.287750\pi$
$593$	30.1217	1.23695	0.618476	+	0.785804i	$0.287750\pi$
$594$	4.55440	0.186869
$595$	0	0
$596$	−3.42805	−0.140419
$597$	16.2728	0.666002
$598$	60.3827	2.46923
$599$	−27.4666	−1.12226	−0.561128	−	0.827729i	$-0.689632\pi$
$599$	−27.4666	−1.12226	−0.561128	+	0.827729i	$0.689632\pi$
$600$	0	0
$601$	−6.66716	−0.271959	−0.135980	−	0.990712i	$-0.543418\pi$
$601$	−6.66716	−0.271959	−0.135980	+	0.990712i	$0.543418\pi$
$602$	0.169110	0.00689241
$603$	−3.87486	−0.157797
$604$	123.603	5.02933
$605$	0	0
$606$	−23.3058	−0.946732
$607$	15.4510	0.627138	0.313569	−	0.949565i	$-0.398475\pi$
$607$	15.4510	0.627138	0.313569	+	0.949565i	$0.398475\pi$
$608$	−89.4714	−3.62854
$609$	0.0170416	0.000690562 0
$610$	0	0
$611$	25.2075	1.01979
$612$	19.9448	0.806220
$613$	−23.4474	−0.947031	−0.473516	−	0.880785i	$-0.657015\pi$
$613$	−23.4474	−0.947031	−0.473516	+	0.880785i	$0.657015\pi$
$614$	54.3459	2.19322
$615$	0	0
$616$	−0.240180	−0.00967713
$617$	−36.8162	−1.48216	−0.741082	−	0.671415i	$-0.765687\pi$
$617$	−36.8162	−1.48216	−0.741082	+	0.671415i	$0.765687\pi$
$618$	−29.1966	−1.17446
$619$	24.5202	0.985551	0.492776	−	0.870156i	$-0.335982\pi$
$619$	24.5202	0.985551	0.492776	+	0.870156i	$0.335982\pi$
$620$	0	0
$621$	4.82786	0.193735
$622$	33.0704	1.32600
$623$	−0.211406	−0.00846980
$624$	−55.2451	−2.21157
$625$	0	0
$626$	39.5586	1.58108
$627$	−10.3006	−0.411366
$628$	71.4041	2.84933
$629$	32.0597	1.27830
$630$	0	0
$631$	11.1275	0.442978	0.221489	−	0.975163i	$-0.428908\pi$
$631$	11.1275	0.442978	0.221489	+	0.975163i	$0.428908\pi$
$632$	−1.92497	−0.0765712
$633$	24.3854	0.969233
$634$	−58.2740	−2.31436
$635$	0	0
$636$	−36.7640	−1.45779
$637$	−32.8682	−1.30229
$638$	4.55440	0.180311
$639$	−4.32991	−0.171289
$640$	0	0
$641$	26.2776	1.03790	0.518951	−	0.854804i	$-0.326323\pi$
$641$	26.2776	1.03790	0.518951	+	0.854804i	$0.326323\pi$
$642$	−42.1212	−1.66239
$643$	6.11885	0.241304	0.120652	−	0.992695i	$-0.461502\pi$
$643$	6.11885	0.241304	0.120652	+	0.992695i	$0.461502\pi$
$644$	−0.419150	−0.0165168
$645$	0	0
$646$	−62.8174	−2.47152
$647$	−26.3758	−1.03694	−0.518470	−	0.855096i	$-0.673498\pi$
$647$	−26.3758	−1.03694	−0.518470	+	0.855096i	$0.673498\pi$
$648$	−8.24244	−0.323794
$649$	−14.7978	−0.580865
$650$	0	0
$651$	−0.0227862	−0.000893060 0
$652$	−49.7766	−1.94940
$653$	47.9227	1.87536	0.937679	−	0.347502i	$-0.112970\pi$
$653$	47.9227	1.87536	0.937679	+	0.347502i	$0.112970\pi$
$654$	−54.0083	−2.11189
$655$	0	0
$656$	98.4415	3.84349
$657$	1.78245	0.0695401
$658$	−0.243672	−0.00949934
$659$	33.0636	1.28798	0.643988	−	0.765036i	$-0.277279\pi$
$659$	33.0636	1.28798	0.643988	+	0.765036i	$0.277279\pi$
$660$	0	0
$661$	20.6186	0.801970	0.400985	−	0.916085i	$-0.368668\pi$
$661$	20.6186	0.801970	0.400985	+	0.916085i	$0.368668\pi$
$662$	−76.2789	−2.96466
$663$	−18.3832	−0.713946
$664$	−50.7320	−1.96878
$665$	0	0
$666$	−21.8120	−0.845198
$667$	4.82786	0.186935
$668$	−30.1385	−1.16609
$669$	0.753555	0.0291341
$670$	0	0
$671$	9.79151	0.377997
$672$	0.253106	0.00976379
$673$	−28.1489	−1.08506	−0.542529	−	0.840037i	$-0.682533\pi$
$673$	−28.1489	−1.08506	−0.542529	+	0.840037i	$0.682533\pi$
$674$	17.9810	0.692601
$675$	0	0
$676$	46.1014	1.77313
$677$	−4.01689	−0.154382	−0.0771908	−	0.997016i	$-0.524595\pi$
$677$	−4.01689	−0.154382	−0.0771908	+	0.997016i	$0.524595\pi$
$678$	−23.6231	−0.907239
$679$	0.328764	0.0126168
$680$	0	0
$681$	−21.1711	−0.811279
$682$	−6.08964	−0.233184
$683$	2.61163	0.0999312	0.0499656	−	0.998751i	$-0.484089\pi$
$683$	2.61163	0.0999312	0.0499656	+	0.998751i	$0.484089\pi$
$684$	30.6900	1.17346
$685$	0	0
$686$	0.635465	0.0242622
$687$	7.97533	0.304278
$688$	−43.8322	−1.67109
$689$	33.8856	1.29094
$690$	0	0
$691$	−19.9707	−0.759723	−0.379861	−	0.925043i	$-0.624028\pi$
$691$	−19.9707	−0.759723	−0.379861	+	0.925043i	$0.624028\pi$
$692$	102.759	3.90629
$693$	0.0291394	0.00110692
$694$	−4.99658	−0.189668
$695$	0	0
$696$	−8.24244	−0.312429
$697$	32.7572	1.24077
$698$	−65.6190	−2.48372
$699$	14.0847	0.532731
$700$	0	0
$701$	17.4402	0.658709	0.329355	−	0.944206i	$-0.393169\pi$
$701$	17.4402	0.658709	0.329355	+	0.944206i	$0.393169\pi$
$702$	12.5071	0.472052
$703$	49.3317	1.86058
$704$	27.4087	1.03300
$705$	0	0
$706$	−35.5500	−1.33794
$707$	−0.149112	−0.00560793
$708$	44.0892	1.65697
$709$	−27.8452	−1.04575	−0.522874	−	0.852410i	$-0.675140\pi$
$709$	−27.8452	−1.04575	−0.522874	+	0.852410i	$0.675140\pi$
$710$	0	0
$711$	0.233544	0.00875858
$712$	102.250	3.83197
$713$	−6.45527	−0.241752
$714$	0.177705	0.00665042
$715$	0	0
$716$	78.9121	2.94908
$717$	−15.4813	−0.578160
$718$	−57.3364	−2.13978
$719$	−23.3574	−0.871085	−0.435542	−	0.900168i	$-0.643443\pi$
$719$	−23.3574	−0.871085	−0.435542	+	0.900168i	$0.643443\pi$
$720$	0	0
$721$	−0.186802	−0.00695688
$722$	−46.0525	−1.71389
$723$	−12.1294	−0.451097
$724$	87.8854	3.26623
$725$	0	0
$726$	−21.5115	−0.798368
$727$	22.8360	0.846939	0.423470	−	0.905910i	$-0.360812\pi$
$727$	22.8360	0.846939	0.423470	+	0.905910i	$0.360812\pi$
$728$	−0.659574	−0.0244454
$729$	1.00000	0.0370370
$730$	0	0
$731$	−14.5855	−0.539465
$732$	−29.1732	−1.07827
$733$	1.44837	0.0534967	0.0267484	−	0.999642i	$-0.491485\pi$
$733$	1.44837	0.0534967	0.0267484	+	0.999642i	$0.491485\pi$
$734$	24.9451	0.920741
$735$	0	0
$736$	71.7045	2.64306
$737$	−6.62561	−0.244058
$738$	−22.2865	−0.820378
$739$	−21.4167	−0.787824	−0.393912	−	0.919148i	$-0.628879\pi$
$739$	−21.4167	−0.787824	−0.393912	+	0.919148i	$0.628879\pi$
$740$	0	0
$741$	−28.2871	−1.03915
$742$	−0.327561	−0.0120251
$743$	16.7848	0.615776	0.307888	−	0.951423i	$-0.400378\pi$
$743$	16.7848	0.615776	0.307888	+	0.951423i	$0.400378\pi$
$744$	11.0209	0.404045
$745$	0	0
$746$	51.5040	1.88570
$747$	6.15497	0.225199
$748$	34.1035	1.24695
$749$	−0.269495	−0.00984712
$750$	0	0
$751$	−13.1194	−0.478733	−0.239366	−	0.970929i	$-0.576940\pi$
$751$	−13.1194	−0.478733	−0.239366	+	0.970929i	$0.576940\pi$
$752$	63.1583	2.30315
$753$	−14.1090	−0.514159
$754$	12.5071	0.455483
$755$	0	0
$756$	−0.0868190	−0.00315758
$757$	−15.3278	−0.557099	−0.278549	−	0.960422i	$-0.589854\pi$
$757$	−15.3278	−0.557099	−0.278549	+	0.960422i	$0.589854\pi$
$758$	33.0715	1.20121
$759$	8.25514	0.299643
$760$	0	0
$761$	16.7398	0.606818	0.303409	−	0.952860i	$-0.401875\pi$
$761$	16.7398	0.606818	0.303409	+	0.952860i	$0.401875\pi$
$762$	13.0209	0.471697
$763$	−0.345549	−0.0125097
$764$	58.6531	2.12199
$765$	0	0
$766$	65.4991	2.36658
$767$	−40.6373	−1.46733
$768$	−2.54296	−0.0917611
$769$	−19.4502	−0.701391	−0.350696	−	0.936489i	$-0.614055\pi$
$769$	−19.4502	−0.701391	−0.350696	+	0.936489i	$0.614055\pi$
$770$	0	0
$771$	−26.6644	−0.960294
$772$	−66.3620	−2.38842
$773$	−34.5781	−1.24369	−0.621844	−	0.783142i	$-0.713616\pi$
$773$	−34.5781	−1.24369	−0.621844	+	0.783142i	$0.713616\pi$
$774$	9.92334	0.356687
$775$	0	0
$776$	−159.012	−5.70819
$777$	−0.139555	−0.00500650
$778$	−45.6978	−1.63835
$779$	50.4050	1.80595
$780$	0	0
$781$	−7.40370	−0.264925
$782$	50.3433	1.80027
$783$	1.00000	0.0357371
$784$	−82.3526	−2.94116
$785$	0	0
$786$	37.5483	1.33930
$787$	39.5669	1.41041	0.705204	−	0.709004i	$-0.250855\pi$
$787$	39.5669	1.41041	0.705204	+	0.709004i	$0.250855\pi$
$788$	−72.9946	−2.60033
$789$	5.74517	0.204534
$790$	0	0
$791$	−0.151142	−0.00537400
$792$	−14.0937	−0.500799
$793$	26.8891	0.954860
$794$	7.73883	0.274641
$795$	0	0
$796$	−82.9023	−2.93839
$797$	−32.0055	−1.13369	−0.566847	−	0.823823i	$-0.691837\pi$
$797$	−32.0055	−1.13369	−0.566847	+	0.823823i	$0.691837\pi$
$798$	0.273442	0.00967975
$799$	21.0164	0.743508
$800$	0	0
$801$	−12.4053	−0.438318
$802$	−54.4504	−1.92271
$803$	3.04781	0.107555
$804$	19.7406	0.696197
$805$	0	0
$806$	−16.7232	−0.589048
$807$	−19.3477	−0.681073
$808$	72.1203	2.53718
$809$	27.5627	0.969052	0.484526	−	0.874777i	$-0.338992\pi$
$809$	27.5627	0.969052	0.484526	+	0.874777i	$0.338992\pi$
$810$	0	0
$811$	−25.3316	−0.889514	−0.444757	−	0.895651i	$-0.646710\pi$
$811$	−25.3316	−0.889514	−0.444757	+	0.895651i	$0.646710\pi$
$812$	−0.0868190	−0.00304675
$813$	30.7485	1.07840
$814$	−37.2963	−1.30723
$815$	0	0
$816$	−46.0599	−1.61242
$817$	−22.4434	−0.785196
$818$	8.35442	0.292106
$819$	0.0800217	0.00279618
$820$	0	0
$821$	−46.4854	−1.62235	−0.811177	−	0.584801i	$-0.801173\pi$
$821$	−46.4854	−1.62235	−0.811177	+	0.584801i	$0.801173\pi$
$822$	−9.50950	−0.331682
$823$	16.9991	0.592552	0.296276	−	0.955102i	$-0.404255\pi$
$823$	16.9991	0.592552	0.296276	+	0.955102i	$0.404255\pi$
$824$	90.3497	3.14748
$825$	0	0
$826$	0.392827	0.0136682
$827$	−14.1835	−0.493208	−0.246604	−	0.969116i	$-0.579315\pi$
$827$	−14.1835	−0.493208	−0.246604	+	0.969116i	$0.579315\pi$
$828$	−24.5957	−0.854758
$829$	28.9652	1.00600	0.503001	−	0.864286i	$-0.332229\pi$
$829$	28.9652	1.00600	0.503001	+	0.864286i	$0.332229\pi$
$830$	0	0
$831$	−24.0737	−0.835108
$832$	75.2688	2.60948
$833$	−27.4035	−0.949474
$834$	31.2261	1.08127
$835$	0	0
$836$	52.4767	1.81494
$837$	−1.33709	−0.0462165
$838$	−97.5179	−3.36870
$839$	22.6082	0.780523	0.390262	−	0.920704i	$-0.372385\pi$
$839$	22.6082	0.780523	0.390262	+	0.920704i	$0.372385\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	36.9451	1.27321
$843$	−23.4067	−0.806171
$844$	−124.232	−4.27625
$845$	0	0
$846$	−14.2986	−0.491598
$847$	−0.137633	−0.00472911
$848$	84.9017	2.91554
$849$	−20.8589	−0.715876
$850$	0	0
$851$	−39.5356	−1.35526
$852$	22.0589	0.755724
$853$	−35.6577	−1.22090	−0.610448	−	0.792057i	$-0.709010\pi$
$853$	−35.6577	−1.22090	−0.610448	+	0.792057i	$0.709010\pi$
$854$	−0.259928	−0.00889455
$855$	0	0
$856$	130.345	4.45510
$857$	−3.41032	−0.116494	−0.0582472	−	0.998302i	$-0.518551\pi$
$857$	−3.41032	−0.116494	−0.0582472	+	0.998302i	$0.518551\pi$
$858$	21.3859	0.730103
$859$	−28.5267	−0.973318	−0.486659	−	0.873592i	$-0.661785\pi$
$859$	−28.5267	−0.973318	−0.486659	+	0.873592i	$0.661785\pi$
$860$	0	0
$861$	−0.142591	−0.00485949
$862$	79.0507	2.69248
$863$	−24.5873	−0.836962	−0.418481	−	0.908226i	$-0.637437\pi$
$863$	−24.5873	−0.836962	−0.418481	+	0.908226i	$0.637437\pi$
$864$	14.8522	0.505283
$865$	0	0
$866$	−85.5851	−2.90830
$867$	1.67321	0.0568252
$868$	0.116085	0.00394017
$869$	0.399336	0.0135465
$870$	0	0
$871$	−18.1950	−0.616515
$872$	167.130	5.65974
$873$	19.2918	0.652930
$874$	77.4656	2.62031
$875$	0	0
$876$	−9.08075	−0.306810
$877$	−58.0640	−1.96068	−0.980341	−	0.197310i	$-0.936780\pi$
$877$	−58.0640	−1.96068	−0.980341	+	0.197310i	$0.936780\pi$
$878$	−93.0438	−3.14008
$879$	−12.0727	−0.407202
$880$	0	0
$881$	52.7466	1.77708	0.888538	−	0.458802i	$-0.151721\pi$
$881$	52.7466	1.77708	0.888538	+	0.458802i	$0.151721\pi$
$882$	18.6441	0.627780
$883$	−38.1567	−1.28408	−0.642038	−	0.766672i	$-0.721911\pi$
$883$	−38.1567	−1.28408	−0.642038	+	0.766672i	$0.721911\pi$
$884$	93.6539	3.14992
$885$	0	0
$886$	82.0209	2.75555
$887$	−22.8088	−0.765845	−0.382922	−	0.923781i	$-0.625082\pi$
$887$	−22.8088	−0.765845	−0.382922	+	0.923781i	$0.625082\pi$
$888$	67.4978	2.26508
$889$	0.0833087	0.00279408
$890$	0	0
$891$	1.70990	0.0572837
$892$	−3.83900	−0.128539
$893$	32.3389	1.08218
$894$	−1.79228	−0.0599428
$895$	0	0
$896$	−0.221385	−0.00739596
$897$	22.6700	0.756928
$898$	−70.9316	−2.36702
$899$	−1.33709	−0.0445944
$900$	0	0
$901$	28.2517	0.941200
$902$	−38.1077	−1.26885
$903$	0.0634903	0.00211283
$904$	73.1023	2.43135
$905$	0	0
$906$	64.6229	2.14695
$907$	33.4371	1.11026	0.555131	−	0.831763i	$-0.312668\pi$
$907$	33.4371	1.11026	0.555131	+	0.831763i	$0.312668\pi$
$908$	107.857	3.57936
$909$	−8.74987	−0.290215
$910$	0	0
$911$	−18.0752	−0.598857	−0.299428	−	0.954119i	$-0.596796\pi$
$911$	−18.0752	−0.598857	−0.299428	+	0.954119i	$0.596796\pi$
$912$	−70.8745	−2.34689
$913$	10.5244	0.348305
$914$	24.0564	0.795715
$915$	0	0
$916$	−40.6305	−1.34247
$917$	0.240237	0.00793333
$918$	10.4277	0.344164
$919$	38.7798	1.27923	0.639614	−	0.768696i	$-0.279094\pi$
$919$	38.7798	1.27923	0.639614	+	0.768696i	$0.279094\pi$
$920$	0	0
$921$	20.4035	0.672319
$922$	46.9425	1.54597
$923$	−20.3318	−0.669229
$924$	−0.148452	−0.00488370
$925$	0	0
$926$	−29.6804	−0.975357
$927$	−10.9615	−0.360024
$928$	14.8522	0.487548
$929$	8.16319	0.267826	0.133913	−	0.990993i	$-0.457246\pi$
$929$	8.16319	0.267826	0.133913	+	0.990993i	$0.457246\pi$
$930$	0	0
$931$	−42.1670	−1.38197
$932$	−71.7547	−2.35040
$933$	12.4159	0.406477
$934$	−52.9866	−1.73378
$935$	0	0
$936$	−38.7037	−1.26507
$937$	16.7498	0.547192	0.273596	−	0.961845i	$-0.411787\pi$
$937$	16.7498	0.547192	0.273596	+	0.961845i	$0.411787\pi$
$938$	0.175885	0.00574285
$939$	14.8518	0.484670
$940$	0	0
$941$	1.27331	0.0415088	0.0207544	−	0.999785i	$-0.493393\pi$
$941$	1.27331	0.0415088	0.0207544	+	0.999785i	$0.493393\pi$
$942$	37.3320	1.21634
$943$	−40.3957	−1.31547
$944$	−101.818	−3.31390
$945$	0	0
$946$	16.9679	0.551674
$947$	−38.5681	−1.25329	−0.626647	−	0.779303i	$-0.715573\pi$
$947$	−38.5681	−1.25329	−0.626647	+	0.779303i	$0.715573\pi$
$948$	−1.18980	−0.0386428
$949$	8.36978	0.271695
$950$	0	0
$951$	−21.8783	−0.709452
$952$	−0.549911	−0.0178227
$953$	47.2051	1.52912	0.764561	−	0.644551i	$-0.222956\pi$
$953$	47.2051	1.52912	0.764561	+	0.644551i	$0.222956\pi$
$954$	−19.2212	−0.622309
$955$	0	0
$956$	78.8699	2.55084
$957$	1.70990	0.0552731
$958$	57.7770	1.86669
$959$	−0.0608425	−0.00196471
$960$	0	0
$961$	−29.2122	−0.942329
$962$	−102.422	−3.30221
$963$	−15.8139	−0.509596
$964$	61.7936	1.99024
$965$	0	0
$966$	−0.219143	−0.00705081
$967$	−14.9495	−0.480743	−0.240371	−	0.970681i	$-0.577269\pi$
$967$	−14.9495	−0.480743	−0.240371	+	0.970681i	$0.577269\pi$
$968$	66.5680	2.13958
$969$	−23.5840	−0.757628
$970$	0	0
$971$	30.4517	0.977240	0.488620	−	0.872497i	$-0.337500\pi$
$971$	30.4517	0.977240	0.488620	+	0.872497i	$0.337500\pi$
$972$	−5.09453	−0.163407
$973$	0.199787	0.00640487
$974$	29.6625	0.950447
$975$	0	0
$976$	67.3716	2.15651
$977$	0.336322	0.0107599	0.00537995	−	0.999986i	$-0.498287\pi$
$977$	0.336322	0.0107599	0.00537995	+	0.999986i	$0.498287\pi$
$978$	−26.0245	−0.832173
$979$	−21.2117	−0.677929
$980$	0	0
$981$	−20.2768	−0.647388
$982$	−73.8264	−2.35590
$983$	1.44050	0.0459449	0.0229724	−	0.999736i	$-0.492687\pi$
$983$	1.44050	0.0459449	0.0229724	+	0.999736i	$0.492687\pi$
$984$	68.9662	2.19856
$985$	0	0
$986$	10.4277	0.332085
$987$	−0.0914839	−0.00291196
$988$	144.110	4.58474
$989$	17.9867	0.571943
$990$	0	0
$991$	40.0507	1.27225	0.636126	−	0.771585i	$-0.280536\pi$
$991$	40.0507	1.27225	0.636126	+	0.771585i	$0.280536\pi$
$992$	−19.8587	−0.630516
$993$	−28.6380	−0.908799
$994$	0.196541	0.00623389
$995$	0	0
$996$	−31.3567	−0.993574
$997$	24.3419	0.770916	0.385458	−	0.922725i	$-0.374043\pi$
$997$	24.3419	0.770916	0.385458	+	0.922725i	$0.374043\pi$
$998$	79.5488	2.51807
$999$	−8.18905	−0.259090

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.a.ba.1.1	✓	7
3.2	odd	2		6525.2.a.bx.1.7		7
5.2	odd	4		2175.2.c.o.349.1		14
5.3	odd	4		2175.2.c.o.349.14		14
5.4	even	2		2175.2.a.bb.1.7	yes	7
15.14	odd	2		6525.2.a.bu.1.1		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2175.2.a.ba.1.1	✓	7	1.1	even	1	trivial
2175.2.a.bb.1.7	yes	7	5.4	even	2
2175.2.c.o.349.1		14	5.2	odd	4
2175.2.c.o.349.14		14	5.3	odd	4
6525.2.a.bu.1.1		7	15.14	odd	2
6525.2.a.bx.1.7		7	3.2	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)

\(f(q)\)	\(=\)	\(q-2.66356 q^{2} -1.00000 q^{3} +5.09453 q^{4} +2.66356 q^{6} +0.0170416 q^{7} -8.24244 q^{8} +1.00000 q^{9} +1.70990 q^{11} -5.09453 q^{12} +4.69566 q^{13} -0.0453913 q^{14} +11.7651 q^{16} +3.91494 q^{17} -2.66356 q^{18} +6.02411 q^{19} -0.0170416 q^{21} -4.55440 q^{22} -4.82786 q^{23} +8.24244 q^{24} -12.5071 q^{26} -1.00000 q^{27} +0.0868190 q^{28} -1.00000 q^{29} +1.33709 q^{31} -14.8522 q^{32} -1.70990 q^{33} -10.4277 q^{34} +5.09453 q^{36} +8.18905 q^{37} -16.0455 q^{38} -4.69566 q^{39} +8.36721 q^{41} +0.0453913 q^{42} -3.72560 q^{43} +8.71111 q^{44} +12.8593 q^{46} +5.36826 q^{47} -11.7651 q^{48} -6.99971 q^{49} -3.91494 q^{51} +23.9222 q^{52} +7.21637 q^{53} +2.66356 q^{54} -0.140465 q^{56} -6.02411 q^{57} +2.66356 q^{58} -8.65422 q^{59} +5.72637 q^{61} -3.56141 q^{62} +0.0170416 q^{63} +16.0294 q^{64} +4.55440 q^{66} -3.87486 q^{67} +19.9448 q^{68} +4.82786 q^{69} -4.32991 q^{71} -8.24244 q^{72} +1.78245 q^{73} -21.8120 q^{74} +30.6900 q^{76} +0.0291394 q^{77} +12.5071 q^{78} +0.233544 q^{79} +1.00000 q^{81} -22.2865 q^{82} +6.15497 q^{83} -0.0868190 q^{84} +9.92334 q^{86} +1.00000 q^{87} -14.0937 q^{88} -12.4053 q^{89} +0.0800217 q^{91} -24.5957 q^{92} -1.33709 q^{93} -14.2986 q^{94} +14.8522 q^{96} +19.2918 q^{97} +18.6441 q^{98} +1.70990 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{2} - 7 q^{3} + 10 q^{4} + 2 q^{6} + q^{7} - 3 q^{8} + 7 q^{9} + 4 q^{11} - 10 q^{12} + q^{13} + 15 q^{14} + 12 q^{16} - 8 q^{17} - 2 q^{18} + 15 q^{19} - q^{21} - 3 q^{22} - 14 q^{23} + 3 q^{24}+ \cdots + 4 q^{99}+O(q^{100}) \) 7 * q - 2 * q^2 - 7 * q^3 + 10 * q^4 + 2 * q^6 + q^7 - 3 * q^8 + 7 * q^9 + 4 * q^11 - 10 * q^12 + q^13 + 15 * q^14 + 12 * q^16 - 8 * q^17 - 2 * q^18 + 15 * q^19 - q^21 - 3 * q^22 - 14 * q^23 + 3 * q^24 + 6 * q^26 - 7 * q^27 + 24 * q^28 - 7 * q^29 + 5 * q^31 - 18 * q^32 - 4 * q^33 + 7 * q^34 + 10 * q^36 + 6 * q^37 + 18 * q^38 - q^39 + 22 * q^41 - 15 * q^42 + 19 * q^43 + 15 * q^44 - 4 * q^46 - 22 * q^47 - 12 * q^48 + 12 * q^49 + 8 * q^51 - 11 * q^52 - 10 * q^53 + 2 * q^54 + 14 * q^56 - 15 * q^57 + 2 * q^58 + 6 * q^59 + 23 * q^61 - 40 * q^62 + q^63 + 5 * q^64 + 3 * q^66 + 13 * q^67 + 7 * q^68 + 14 * q^69 + 26 * q^71 - 3 * q^72 + 24 * q^73 - 10 * q^74 + 46 * q^76 - 4 * q^77 - 6 * q^78 + 14 * q^79 + 7 * q^81 - 16 * q^82 - 10 * q^83 - 24 * q^84 + 44 * q^86 + 7 * q^87 + 66 * q^88 + 14 * q^89 + 13 * q^91 - 58 * q^92 - 5 * q^93 - 3 * q^94 + 18 * q^96 + 31 * q^97 + 59 * q^98 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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