LMFDB - Embedded newform 2075.2.a.k.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2075,2,Mod(1,2075)] 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("2075.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2075, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2075 = 5^{2} \cdot 83 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2075.a (trivial)

Newform invariants

comment:select newform

sage:traces = [11,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$16.5689584194$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 20x^{9} - x^{8} + 146x^{7} + 15x^{6} - 464x^{5} - 76x^{4} + 567x^{3} + 136x^{2} - 100x - 8 $ x^11 - 20x^9 - x^8 + 146x^7 + 15x^6 - 464x^5 - 76x^4 + 567x^3 + 136x^2 - 100x - 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 415)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.381602$ of defining polynomial
Character	$\chi$	$=$	2075.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.381602 q^{2} -1.63018 q^{3} -1.85438 q^{4} +0.622079 q^{6} +0.0761781 q^{7} +1.47084 q^{8} -0.342523 q^{9} +4.79348 q^{11} +3.02297 q^{12} -6.08842 q^{13} -0.0290697 q^{14} +3.14748 q^{16} -7.73742 q^{17} +0.130708 q^{18} -0.833555 q^{19} -0.124184 q^{21} -1.82920 q^{22} +8.25464 q^{23} -2.39773 q^{24} +2.32336 q^{26} +5.44890 q^{27} -0.141263 q^{28} -6.42032 q^{29} -4.72443 q^{31} -4.14277 q^{32} -7.81422 q^{33} +2.95262 q^{34} +0.635168 q^{36} -0.579319 q^{37} +0.318086 q^{38} +9.92521 q^{39} -4.08551 q^{41} +0.0473888 q^{42} -3.19950 q^{43} -8.88893 q^{44} -3.14999 q^{46} +5.36358 q^{47} -5.13095 q^{48} -6.99420 q^{49} +12.6134 q^{51} +11.2903 q^{52} +0.465047 q^{53} -2.07931 q^{54} +0.112046 q^{56} +1.35884 q^{57} +2.45001 q^{58} +7.57236 q^{59} -5.94859 q^{61} +1.80285 q^{62} -0.0260928 q^{63} -4.71408 q^{64} +2.98192 q^{66} -6.35636 q^{67} +14.3481 q^{68} -13.4565 q^{69} +2.40089 q^{71} -0.503797 q^{72} -1.19045 q^{73} +0.221069 q^{74} +1.54573 q^{76} +0.365158 q^{77} -3.78748 q^{78} +2.25348 q^{79} -7.85511 q^{81} +1.55904 q^{82} +1.00000 q^{83} +0.230284 q^{84} +1.22094 q^{86} +10.4663 q^{87} +7.05044 q^{88} +4.10871 q^{89} -0.463805 q^{91} -15.3072 q^{92} +7.70166 q^{93} -2.04676 q^{94} +6.75344 q^{96} +12.1829 q^{97} +2.66900 q^{98} -1.64188 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 18 q^{4} - 5 q^{6} + 3 q^{7} - 3 q^{8} + 21 q^{9} - q^{11} - 2 q^{12} - q^{13} + 12 q^{14} + 20 q^{16} - 28 q^{17} + 26 q^{18} - 2 q^{19} + 3 q^{21} + 20 q^{22} - q^{23} - 10 q^{24} + 21 q^{26}+ \cdots - 36 q^{99}+O(q^{100}) $ 11 * q + 18 * q^4 - 5 * q^6 + 3 * q^7 - 3 * q^8 + 21 * q^9 - q^11 - 2 * q^12 - q^13 + 12 * q^14 + 20 * q^16 - 28 * q^17 + 26 * q^18 - 2 * q^19 + 3 * q^21 + 20 * q^22 - q^23 - 10 * q^24 + 21 * q^26 + 12 * q^27 + 22 * q^28 + 21 * q^29 + 4 * q^31 - q^32 - 10 * q^33 - 4 * q^34 + 45 * q^36 + 10 * q^37 - 17 * q^38 - 2 * q^39 + 39 * q^41 + 29 * q^42 + 13 * q^43 - 25 * q^44 - 13 * q^46 - 8 * q^47 - 17 * q^48 + 32 * q^49 + 32 * q^51 + 65 * q^52 - 21 * q^53 - 72 * q^54 + 57 * q^56 + 19 * q^57 + 6 * q^58 + 5 * q^59 + 14 * q^61 + q^62 + 30 * q^63 - 21 * q^64 - 11 * q^66 + 42 * q^67 - 61 * q^68 - 7 * q^69 + 3 * q^71 + 69 * q^72 - 3 * q^73 - 54 * q^74 + 13 * q^76 + 42 * q^77 + 7 * q^78 - 20 * q^79 + 39 * q^81 + 43 * q^82 + 11 * q^83 - 116 * q^84 + 22 * q^86 + 16 * q^87 + 75 * q^88 + 37 * q^89 - 17 * q^91 - 7 * q^92 + 55 * q^93 - 54 * q^94 - 58 * q^96 + 2 * q^97 + 26 * q^98 - 36 * q^99

Coefficient data

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

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

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

$2$	−0.381602	−0.269834	−0.134917	−	0.990857i	$-0.543077\pi$
$2$	−0.381602	−0.269834	−0.134917	+	0.990857i	$0.543077\pi$
$3$	−1.63018	−0.941183	−0.470592	−	0.882351i	$-0.655959\pi$
$3$	−1.63018	−0.941183	−0.470592	+	0.882351i	$0.655959\pi$
$4$	−1.85438	−0.927190
$5$	0	0
$5$	0	0
$6$	0.622079	0.253963
$7$	0.0761781	0.0287926	0.0143963	−	0.999896i	$-0.495417\pi$
$7$	0.0761781	0.0287926	0.0143963	+	0.999896i	$0.495417\pi$
$8$	1.47084	0.520020
$9$	−0.342523	−0.114174
$10$	0	0
$11$	4.79348	1.44529	0.722644	−	0.691220i	$-0.242926\pi$
$11$	4.79348	1.44529	0.722644	+	0.691220i	$0.242926\pi$
$12$	3.02297	0.872655
$13$	−6.08842	−1.68863	−0.844313	−	0.535851i	$-0.819991\pi$
$13$	−6.08842	−1.68863	−0.844313	+	0.535851i	$0.819991\pi$
$14$	−0.0290697	−0.00776921
$15$	0	0
$16$	3.14748	0.786871
$17$	−7.73742	−1.87660	−0.938300	−	0.345823i	$-0.887600\pi$
$17$	−7.73742	−1.87660	−0.938300	+	0.345823i	$0.887600\pi$
$18$	0.130708	0.0308081
$19$	−0.833555	−0.191231	−0.0956153	−	0.995418i	$-0.530482\pi$
$19$	−0.833555	−0.191231	−0.0956153	+	0.995418i	$0.530482\pi$
$20$	0	0
$21$	−0.124184	−0.0270991
$22$	−1.82920	−0.389987
$23$	8.25464	1.72121	0.860606	−	0.509271i	$-0.170085\pi$
$23$	8.25464	1.72121	0.860606	+	0.509271i	$0.170085\pi$
$24$	−2.39773	−0.489434
$25$	0	0
$26$	2.32336	0.455648
$27$	5.44890	1.04864
$28$	−0.141263	−0.0266962
$29$	−6.42032	−1.19222	−0.596112	−	0.802901i	$-0.703289\pi$
$29$	−6.42032	−1.19222	−0.596112	+	0.802901i	$0.703289\pi$
$30$	0	0
$31$	−4.72443	−0.848533	−0.424267	−	0.905537i	$-0.639468\pi$
$31$	−4.72443	−0.848533	−0.424267	+	0.905537i	$0.639468\pi$
$32$	−4.14277	−0.732345
$33$	−7.81422	−1.36028
$34$	2.95262	0.506370
$35$	0	0
$36$	0.635168	0.105861
$37$	−0.579319	−0.0952394	−0.0476197	−	0.998866i	$-0.515164\pi$
$37$	−0.579319	−0.0952394	−0.0476197	+	0.998866i	$0.515164\pi$
$38$	0.318086	0.0516004
$39$	9.92521	1.58931
$40$	0	0
$41$	−4.08551	−0.638049	−0.319025	−	0.947746i	$-0.603355\pi$
$41$	−4.08551	−0.638049	−0.319025	+	0.947746i	$0.603355\pi$
$42$	0.0473888	0.00731225
$43$	−3.19950	−0.487920	−0.243960	−	0.969785i	$-0.578447\pi$
$43$	−3.19950	−0.487920	−0.243960	+	0.969785i	$0.578447\pi$
$44$	−8.88893	−1.34006
$45$	0	0
$46$	−3.14999	−0.464441
$47$	5.36358	0.782359	0.391179	−	0.920314i	$-0.372067\pi$
$47$	5.36358	0.782359	0.391179	+	0.920314i	$0.372067\pi$
$48$	−5.13095	−0.740590
$49$	−6.99420	−0.999171
$50$	0	0
$51$	12.6134	1.76622
$52$	11.2903	1.56568
$53$	0.465047	0.0638791	0.0319395	−	0.999490i	$-0.489832\pi$
$53$	0.465047	0.0638791	0.0319395	+	0.999490i	$0.489832\pi$
$54$	−2.07931	−0.282959
$55$	0	0
$56$	0.112046	0.0149727
$57$	1.35884	0.179983
$58$	2.45001	0.321702
$59$	7.57236	0.985838	0.492919	−	0.870075i	$-0.335930\pi$
$59$	7.57236	0.985838	0.492919	+	0.870075i	$0.335930\pi$
$60$	0	0
$61$	−5.94859	−0.761638	−0.380819	−	0.924650i	$-0.624358\pi$
$61$	−5.94859	−0.761638	−0.380819	+	0.924650i	$0.624358\pi$
$62$	1.80285	0.228963
$63$	−0.0260928	−0.00328738
$64$	−4.71408	−0.589260
$65$	0	0
$66$	2.98192	0.367050
$67$	−6.35636	−0.776553	−0.388276	−	0.921543i	$-0.626929\pi$
$67$	−6.35636	−0.776553	−0.388276	+	0.921543i	$0.626929\pi$
$68$	14.3481	1.73996
$69$	−13.4565	−1.61998
$70$	0	0
$71$	2.40089	0.284933	0.142466	−	0.989800i	$-0.454497\pi$
$71$	2.40089	0.284933	0.142466	+	0.989800i	$0.454497\pi$
$72$	−0.503797	−0.0593731
$73$	−1.19045	−0.139332	−0.0696658	−	0.997570i	$-0.522193\pi$
$73$	−1.19045	−0.139332	−0.0696658	+	0.997570i	$0.522193\pi$
$74$	0.221069	0.0256988
$75$	0	0
$76$	1.54573	0.177307
$77$	0.365158	0.0416136
$78$	−3.78748	−0.428848
$79$	2.25348	0.253537	0.126768	−	0.991932i	$-0.459540\pi$
$79$	2.25348	0.253537	0.126768	+	0.991932i	$0.459540\pi$
$80$	0	0
$81$	−7.85511	−0.872790
$82$	1.55904	0.172167
$83$	1.00000	0.109764
$83$	1.00000	0.109764
$84$	0.230284	0.0251260
$85$	0	0
$86$	1.22094	0.131657
$87$	10.4663	1.12210
$88$	7.05044	0.751580
$89$	4.10871	0.435523	0.217761	−	0.976002i	$-0.430125\pi$
$89$	4.10871	0.435523	0.217761	+	0.976002i	$0.430125\pi$
$90$	0	0
$91$	−0.463805	−0.0486199
$92$	−15.3072	−1.59589
$93$	7.70166	0.798625
$94$	−2.04676	−0.211107
$95$	0	0
$96$	6.75344	0.689270
$97$	12.1829	1.23698	0.618492	−	0.785791i	$-0.287744\pi$
$97$	12.1829	1.23698	0.618492	+	0.785791i	$0.287744\pi$
$98$	2.66900	0.269610
$99$	−1.64188	−0.165015
$100$	0	0
$101$	13.3255	1.32593	0.662966	−	0.748649i	$-0.269297\pi$
$101$	13.3255	1.32593	0.662966	+	0.748649i	$0.269297\pi$
$102$	−4.81329	−0.476586
$103$	10.3909	1.02385	0.511923	−	0.859031i	$-0.328933\pi$
$103$	10.3909	1.02385	0.511923	+	0.859031i	$0.328933\pi$
$104$	−8.95510	−0.878120
$105$	0	0
$106$	−0.177463	−0.0172367
$107$	15.6812	1.51596	0.757979	−	0.652279i	$-0.226187\pi$
$107$	15.6812	1.51596	0.757979	+	0.652279i	$0.226187\pi$
$108$	−10.1043	−0.972290
$109$	17.0197	1.63019	0.815095	−	0.579328i	$-0.196685\pi$
$109$	17.0197	1.63019	0.815095	+	0.579328i	$0.196685\pi$
$110$	0	0
$111$	0.944392	0.0896377
$112$	0.239769	0.0226561
$113$	2.67610	0.251746	0.125873	−	0.992046i	$-0.459827\pi$
$113$	2.67610	0.251746	0.125873	+	0.992046i	$0.459827\pi$
$114$	−0.518537	−0.0485654
$115$	0	0
$116$	11.9057	1.10542
$117$	2.08543	0.192798
$118$	−2.88963	−0.266012
$119$	−0.589422	−0.0540322
$120$	0	0
$121$	11.9775	1.08886
$122$	2.26999	0.205516
$123$	6.66010	0.600521
$124$	8.76089	0.786751
$125$	0	0
$126$	0.00995706	0.000887046 0
$127$	2.16302	0.191937	0.0959684	−	0.995384i	$-0.469405\pi$
$127$	2.16302	0.191937	0.0959684	+	0.995384i	$0.469405\pi$
$128$	10.0844	0.891347
$129$	5.21576	0.459222
$130$	0	0
$131$	18.3932	1.60702	0.803509	−	0.595293i	$-0.202964\pi$
$131$	18.3932	1.60702	0.803509	+	0.595293i	$0.202964\pi$
$132$	14.4905	1.26124
$133$	−0.0634986	−0.00550603
$134$	2.42560	0.209540
$135$	0	0
$136$	−11.3805	−0.975870
$137$	17.4166	1.48800	0.744001	−	0.668178i	$-0.232926\pi$
$137$	17.4166	1.48800	0.744001	+	0.668178i	$0.232926\pi$
$138$	5.13504	0.437124
$139$	−13.2880	−1.12707	−0.563535	−	0.826092i	$-0.690559\pi$
$139$	−13.2880	−1.12707	−0.563535	+	0.826092i	$0.690559\pi$
$140$	0	0
$141$	−8.74359	−0.736343
$142$	−0.916184	−0.0768845
$143$	−29.1847	−2.44055
$144$	−1.07809	−0.0898406
$145$	0	0
$146$	0.454278	0.0375963
$147$	11.4018	0.940403
$148$	1.07428	0.0883050
$149$	17.6410	1.44521	0.722605	−	0.691261i	$-0.242944\pi$
$149$	17.6410	1.44521	0.722605	+	0.691261i	$0.242944\pi$
$150$	0	0
$151$	−16.8357	−1.37007	−0.685033	−	0.728512i	$-0.740212\pi$
$151$	−16.8357	−1.37007	−0.685033	+	0.728512i	$0.740212\pi$
$152$	−1.22603	−0.0994438
$153$	2.65025	0.214260
$154$	−0.139345	−0.0112288
$155$	0	0
$156$	−18.4051	−1.47359
$157$	13.6620	1.09035	0.545175	−	0.838322i	$-0.316463\pi$
$157$	13.6620	1.09035	0.545175	+	0.838322i	$0.316463\pi$
$158$	−0.859934	−0.0684127
$159$	−0.758108	−0.0601219
$160$	0	0
$161$	0.628823	0.0495582
$162$	2.99753	0.235508
$163$	−1.28882	−0.100948	−0.0504741	−	0.998725i	$-0.516073\pi$
$163$	−1.28882	−0.100948	−0.0504741	+	0.998725i	$0.516073\pi$
$164$	7.57608	0.591593
$165$	0	0
$166$	−0.381602	−0.0296181
$167$	17.5230	1.35597	0.677985	−	0.735076i	$-0.262854\pi$
$167$	17.5230	1.35597	0.677985	+	0.735076i	$0.262854\pi$
$168$	−0.182654	−0.0140921
$169$	24.0689	1.85145
$170$	0	0
$171$	0.285512	0.0218336
$172$	5.93309	0.452394
$173$	−16.7053	−1.27008	−0.635040	−	0.772479i	$-0.719017\pi$
$173$	−16.7053	−1.27008	−0.635040	+	0.772479i	$0.719017\pi$
$174$	−3.99395	−0.302781
$175$	0	0
$176$	15.0874	1.13726
$177$	−12.3443	−0.927854
$178$	−1.56789	−0.117519
$179$	1.30846	0.0977988	0.0488994	−	0.998804i	$-0.484429\pi$
$179$	1.30846	0.0977988	0.0488994	+	0.998804i	$0.484429\pi$
$180$	0	0
$181$	−10.8110	−0.803578	−0.401789	−	0.915732i	$-0.631611\pi$
$181$	−10.8110	−0.803578	−0.401789	+	0.915732i	$0.631611\pi$
$182$	0.176989	0.0131193
$183$	9.69725	0.716841
$184$	12.1413	0.895065
$185$	0	0
$186$	−2.93897	−0.215496
$187$	−37.0892	−2.71223
$188$	−9.94612	−0.725395
$189$	0.415087	0.0301931
$190$	0	0
$191$	−13.5911	−0.983415	−0.491707	−	0.870760i	$-0.663627\pi$
$191$	−13.5911	−0.983415	−0.491707	+	0.870760i	$0.663627\pi$
$192$	7.68478	0.554601
$193$	9.19181	0.661641	0.330820	−	0.943694i	$-0.392675\pi$
$193$	9.19181	0.661641	0.330820	+	0.943694i	$0.392675\pi$
$194$	−4.64901	−0.333780
$195$	0	0
$196$	12.9699	0.926421
$197$	−24.8615	−1.77131	−0.885655	−	0.464344i	$-0.846290\pi$
$197$	−24.8615	−1.77131	−0.885655	+	0.464344i	$0.846290\pi$
$198$	0.626545	0.0445266
$199$	−11.0180	−0.781048	−0.390524	−	0.920593i	$-0.627706\pi$
$199$	−11.0180	−0.781048	−0.390524	+	0.920593i	$0.627706\pi$
$200$	0	0
$201$	10.3620	0.730878
$202$	−5.08502	−0.357781
$203$	−0.489088	−0.0343272
$204$	−23.3900	−1.63762
$205$	0	0
$206$	−3.96519	−0.276268
$207$	−2.82741	−0.196518
$208$	−19.1632	−1.32873
$209$	−3.99563	−0.276383
$210$	0	0
$211$	22.5868	1.55494	0.777471	−	0.628918i	$-0.216502\pi$
$211$	22.5868	1.55494	0.777471	+	0.628918i	$0.216502\pi$
$212$	−0.862373	−0.0592280
$213$	−3.91387	−0.268174
$214$	−5.98398	−0.409056
$215$	0	0
$216$	8.01447	0.545315
$217$	−0.359898	−0.0244315
$218$	−6.49475	−0.439880
$219$	1.94064	0.131136
$220$	0	0
$221$	47.1087	3.16887
$222$	−0.360382	−0.0241873
$223$	14.8867	0.996887	0.498444	−	0.866922i	$-0.333905\pi$
$223$	14.8867	0.996887	0.498444	+	0.866922i	$0.333905\pi$
$224$	−0.315588	−0.0210861
$225$	0	0
$226$	−1.02121	−0.0679296
$227$	−9.68451	−0.642784	−0.321392	−	0.946946i	$-0.604151\pi$
$227$	−9.68451	−0.642784	−0.321392	+	0.946946i	$0.604151\pi$
$228$	−2.51981	−0.166878
$229$	−20.2691	−1.33942	−0.669711	−	0.742621i	$-0.733582\pi$
$229$	−20.2691	−1.33942	−0.669711	+	0.742621i	$0.733582\pi$
$230$	0	0
$231$	−0.595272	−0.0391661
$232$	−9.44327	−0.619981
$233$	−19.7228	−1.29208	−0.646041	−	0.763303i	$-0.723577\pi$
$233$	−19.7228	−1.29208	−0.646041	+	0.763303i	$0.723577\pi$
$234$	−0.795804	−0.0520233
$235$	0	0
$236$	−14.0420	−0.914059
$237$	−3.67357	−0.238624
$238$	0.224925	0.0145797
$239$	23.2525	1.50408	0.752039	−	0.659119i	$-0.229070\pi$
$239$	23.2525	1.50408	0.752039	+	0.659119i	$0.229070\pi$
$240$	0	0
$241$	−10.4213	−0.671298	−0.335649	−	0.941987i	$-0.608956\pi$
$241$	−10.4213	−0.671298	−0.335649	+	0.941987i	$0.608956\pi$
$242$	−4.57062	−0.293811
$243$	−3.54150	−0.227187
$244$	11.0309	0.706183
$245$	0	0
$246$	−2.54151	−0.162041
$247$	5.07503	0.322917
$248$	−6.94888	−0.441255
$249$	−1.63018	−0.103308
$250$	0	0
$251$	−3.62296	−0.228679	−0.114340	−	0.993442i	$-0.536475\pi$
$251$	−3.62296	−0.228679	−0.114340	+	0.993442i	$0.536475\pi$
$252$	0.0483859	0.00304803
$253$	39.5685	2.48765
$254$	−0.825413	−0.0517910
$255$	0	0
$256$	5.57991	0.348744
$257$	−6.32617	−0.394616	−0.197308	−	0.980342i	$-0.563220\pi$
$257$	−6.32617	−0.394616	−0.197308	+	0.980342i	$0.563220\pi$
$258$	−1.99034	−0.123913
$259$	−0.0441314	−0.00274219
$260$	0	0
$261$	2.19911	0.136122
$262$	−7.01887	−0.433627
$263$	15.4402	0.952082	0.476041	−	0.879423i	$-0.342071\pi$
$263$	15.4402	0.952082	0.476041	+	0.879423i	$0.342071\pi$
$264$	−11.4935	−0.707374
$265$	0	0
$266$	0.0242312	0.00148571
$267$	−6.69793	−0.409907
$268$	11.7871	0.720012
$269$	12.1989	0.743777	0.371889	−	0.928277i	$-0.378710\pi$
$269$	12.1989	0.743777	0.371889	+	0.928277i	$0.378710\pi$
$270$	0	0
$271$	−3.43954	−0.208937	−0.104469	−	0.994528i	$-0.533314\pi$
$271$	−3.43954	−0.208937	−0.104469	+	0.994528i	$0.533314\pi$
$272$	−24.3534	−1.47664
$273$	0.756083	0.0457603
$274$	−6.64622	−0.401513
$275$	0	0
$276$	24.9535	1.50202
$277$	9.74712	0.585648	0.292824	−	0.956166i	$-0.405405\pi$
$277$	9.74712	0.585648	0.292824	+	0.956166i	$0.405405\pi$
$278$	5.07072	0.304122
$279$	1.61823	0.0968808
$280$	0	0
$281$	−6.98593	−0.416746	−0.208373	−	0.978049i	$-0.566817\pi$
$281$	−6.98593	−0.416746	−0.208373	+	0.978049i	$0.566817\pi$
$282$	3.33657	0.198690
$283$	19.2612	1.14496	0.572478	−	0.819920i	$-0.305982\pi$
$283$	19.2612	1.14496	0.572478	+	0.819920i	$0.305982\pi$
$284$	−4.45216	−0.264187
$285$	0	0
$286$	11.1370	0.658543
$287$	−0.311226	−0.0183711
$288$	1.41899	0.0836151
$289$	42.8677	2.52163
$290$	0	0
$291$	−19.8602	−1.16423
$292$	2.20754	0.129187
$293$	−15.3470	−0.896579	−0.448289	−	0.893888i	$-0.647967\pi$
$293$	−15.3470	−0.896579	−0.448289	+	0.893888i	$0.647967\pi$
$294$	−4.35094	−0.253752
$295$	0	0
$296$	−0.852085	−0.0495265
$297$	26.1192	1.51559
$298$	−6.73186	−0.389966
$299$	−50.2578	−2.90648
$300$	0	0
$301$	−0.243732	−0.0140485
$302$	6.42453	0.369690
$303$	−21.7228	−1.24794
$304$	−2.62360	−0.150474
$305$	0	0
$306$	−1.01134	−0.0578145
$307$	6.92676	0.395331	0.197665	−	0.980270i	$-0.436664\pi$
$307$	6.92676	0.395331	0.197665	+	0.980270i	$0.436664\pi$
$308$	−0.677142	−0.0385837
$309$	−16.9390	−0.963627
$310$	0	0
$311$	−19.6813	−1.11602	−0.558011	−	0.829833i	$-0.688435\pi$
$311$	−19.6813	−1.11602	−0.558011	+	0.829833i	$0.688435\pi$
$312$	14.5984	0.826471
$313$	3.87220	0.218869	0.109435	−	0.993994i	$-0.465096\pi$
$313$	3.87220	0.218869	0.109435	+	0.993994i	$0.465096\pi$
$314$	−5.21347	−0.294213
$315$	0	0
$316$	−4.17881	−0.235076
$317$	−17.7095	−0.994666	−0.497333	−	0.867560i	$-0.665687\pi$
$317$	−17.7095	−0.994666	−0.497333	+	0.867560i	$0.665687\pi$
$318$	0.289296	0.0162229
$319$	−30.7757	−1.72311
$320$	0	0
$321$	−25.5631	−1.42679
$322$	−0.239960	−0.0133725
$323$	6.44956	0.358863
$324$	14.5664	0.809242
$325$	0	0
$326$	0.491816	0.0272392
$327$	−27.7451	−1.53431
$328$	−6.00913	−0.331799
$329$	0.408587	0.0225262
$330$	0	0
$331$	10.7130	0.588839	0.294419	−	0.955676i	$-0.404874\pi$
$331$	10.7130	0.588839	0.294419	+	0.955676i	$0.404874\pi$
$332$	−1.85438	−0.101772
$333$	0.198430	0.0108739
$334$	−6.68681	−0.365886
$335$	0	0
$336$	−0.390866	−0.0213235
$337$	−16.0911	−0.876540	−0.438270	−	0.898843i	$-0.644409\pi$
$337$	−16.0911	−0.876540	−0.438270	+	0.898843i	$0.644409\pi$
$338$	−9.18475	−0.499585
$339$	−4.36251	−0.236939
$340$	0	0
$341$	−22.6465	−1.22638
$342$	−0.108952	−0.00589145
$343$	−1.06605	−0.0575614
$344$	−4.70596	−0.253728
$345$	0	0
$346$	6.37478	0.342710
$347$	−3.82853	−0.205526	−0.102763	−	0.994706i	$-0.532768\pi$
$347$	−3.82853	−0.205526	−0.102763	+	0.994706i	$0.532768\pi$
$348$	−19.4084	−1.04040
$349$	15.5449	0.832101	0.416050	−	0.909342i	$-0.363414\pi$
$349$	15.5449	0.832101	0.416050	+	0.909342i	$0.363414\pi$
$350$	0	0
$351$	−33.1752	−1.77076
$352$	−19.8583	−1.05845
$353$	−9.50263	−0.505774	−0.252887	−	0.967496i	$-0.581380\pi$
$353$	−9.50263	−0.505774	−0.252887	+	0.967496i	$0.581380\pi$
$354$	4.71061	0.250366
$355$	0	0
$356$	−7.61911	−0.403812
$357$	0.960862	0.0508542
$358$	−0.499311	−0.0263894
$359$	−3.38446	−0.178625	−0.0893125	−	0.996004i	$-0.528467\pi$
$359$	−3.38446	−0.178625	−0.0893125	+	0.996004i	$0.528467\pi$
$360$	0	0
$361$	−18.3052	−0.963431
$362$	4.12551	0.216832
$363$	−19.5254	−1.02482
$364$	0.860070	0.0450799
$365$	0	0
$366$	−3.70049	−0.193428
$367$	−2.22778	−0.116289	−0.0581446	−	0.998308i	$-0.518518\pi$
$367$	−2.22778	−0.116289	−0.0581446	+	0.998308i	$0.518518\pi$
$368$	25.9814	1.35437
$369$	1.39938	0.0728489
$370$	0	0
$371$	0.0354264	0.00183924
$372$	−14.2818	−0.740477
$373$	33.4446	1.73170	0.865849	−	0.500306i	$-0.166779\pi$
$373$	33.4446	1.73170	0.865849	+	0.500306i	$0.166779\pi$
$374$	14.1533	0.731850
$375$	0	0
$376$	7.88897	0.406843
$377$	39.0897	2.01322
$378$	−0.158398	−0.00814712
$379$	29.3505	1.50763	0.753817	−	0.657084i	$-0.228210\pi$
$379$	29.3505	1.50763	0.753817	+	0.657084i	$0.228210\pi$
$380$	0	0
$381$	−3.52610	−0.180648
$382$	5.18638	0.265358
$383$	4.51068	0.230485	0.115243	−	0.993337i	$-0.463235\pi$
$383$	4.51068	0.230485	0.115243	+	0.993337i	$0.463235\pi$
$384$	−16.4394	−0.838920
$385$	0	0
$386$	−3.50762	−0.178533
$387$	1.09590	0.0557080
$388$	−22.5917	−1.14692
$389$	−7.92259	−0.401691	−0.200846	−	0.979623i	$-0.564369\pi$
$389$	−7.92259	−0.401691	−0.200846	+	0.979623i	$0.564369\pi$
$390$	0	0
$391$	−63.8696	−3.23003
$392$	−10.2873	−0.519589
$393$	−29.9841	−1.51250
$394$	9.48721	0.477959
$395$	0	0
$396$	3.04467	0.153000
$397$	0.720345	0.0361531	0.0180765	−	0.999837i	$-0.494246\pi$
$397$	0.720345	0.0361531	0.0180765	+	0.999837i	$0.494246\pi$
$398$	4.20451	0.210753
$399$	0.103514	0.00518218
$400$	0	0
$401$	39.7909	1.98706	0.993531	−	0.113563i	$-0.0362264\pi$
$401$	39.7909	1.98706	0.993531	+	0.113563i	$0.0362264\pi$
$402$	−3.95416	−0.197215
$403$	28.7643	1.43285
$404$	−24.7104	−1.22939
$405$	0	0
$406$	0.186637	0.00926264
$407$	−2.77695	−0.137648
$408$	18.5522	0.918473
$409$	4.80091	0.237390	0.118695	−	0.992931i	$-0.462129\pi$
$409$	4.80091	0.237390	0.118695	+	0.992931i	$0.462129\pi$
$410$	0	0
$411$	−28.3922	−1.40048
$412$	−19.2687	−0.949300
$413$	0.576848	0.0283848
$414$	1.07895	0.0530273
$415$	0	0
$416$	25.2229	1.23666
$417$	21.6617	1.06078
$418$	1.52474	0.0745775
$419$	32.3407	1.57995	0.789973	−	0.613141i	$-0.210094\pi$
$419$	32.3407	1.57995	0.789973	+	0.613141i	$0.210094\pi$
$420$	0	0
$421$	26.2043	1.27712	0.638560	−	0.769572i	$-0.279530\pi$
$421$	26.2043	1.27712	0.638560	+	0.769572i	$0.279530\pi$
$422$	−8.61919	−0.419576
$423$	−1.83715	−0.0893254
$424$	0.684009	0.0332184
$425$	0	0
$426$	1.49354	0.0723624
$427$	−0.453152	−0.0219296
$428$	−29.0789	−1.40558
$429$	47.5763	2.29701
$430$	0	0
$431$	14.2425	0.686036	0.343018	−	0.939329i	$-0.388551\pi$
$431$	14.2425	0.686036	0.343018	+	0.939329i	$0.388551\pi$
$432$	17.1503	0.825146
$433$	−18.8710	−0.906882	−0.453441	−	0.891286i	$-0.649804\pi$
$433$	−18.8710	−0.906882	−0.453441	+	0.891286i	$0.649804\pi$
$434$	0.137338	0.00659243
$435$	0	0
$436$	−31.5609	−1.51150
$437$	−6.88069	−0.329148
$438$	−0.740554	−0.0353850
$439$	25.5166	1.21784	0.608920	−	0.793232i	$-0.291603\pi$
$439$	25.5166	1.21784	0.608920	+	0.793232i	$0.291603\pi$
$440$	0	0
$441$	2.39568	0.114080
$442$	−17.9768	−0.855068
$443$	27.9646	1.32864	0.664318	−	0.747450i	$-0.268722\pi$
$443$	27.9646	1.32864	0.664318	+	0.747450i	$0.268722\pi$
$444$	−1.75126	−0.0831112
$445$	0	0
$446$	−5.68080	−0.268994
$447$	−28.7580	−1.36021
$448$	−0.359109	−0.0169663
$449$	−20.4268	−0.964002	−0.482001	−	0.876171i	$-0.660090\pi$
$449$	−20.4268	−0.964002	−0.482001	+	0.876171i	$0.660090\pi$
$450$	0	0
$451$	−19.5838	−0.922165
$452$	−4.96250	−0.233416
$453$	27.4451	1.28948
$454$	3.69563	0.173445
$455$	0	0
$456$	1.99864	0.0935948
$457$	−10.9786	−0.513559	−0.256780	−	0.966470i	$-0.582661\pi$
$457$	−10.9786	−0.513559	−0.256780	+	0.966470i	$0.582661\pi$
$458$	7.73475	0.361421
$459$	−42.1605	−1.96788
$460$	0	0
$461$	−38.1410	−1.77640	−0.888201	−	0.459455i	$-0.848045\pi$
$461$	−38.1410	−1.77640	−0.888201	+	0.459455i	$0.848045\pi$
$462$	0.227157	0.0105683
$463$	20.1515	0.936519	0.468259	−	0.883591i	$-0.344881\pi$
$463$	20.1515	0.936519	0.468259	+	0.883591i	$0.344881\pi$
$464$	−20.2079	−0.938126
$465$	0	0
$466$	7.52626	0.348647
$467$	−31.6392	−1.46409	−0.732045	−	0.681257i	$-0.761434\pi$
$467$	−31.6392	−1.46409	−0.732045	+	0.681257i	$0.761434\pi$
$468$	−3.86718	−0.178760
$469$	−0.484215	−0.0223590
$470$	0	0
$471$	−22.2715	−1.02622
$472$	11.1377	0.512656
$473$	−15.3368	−0.705185
$474$	1.40184	0.0643888
$475$	0	0
$476$	1.09301	0.0500981
$477$	−0.159289	−0.00729336
$478$	−8.87320	−0.405851
$479$	23.8341	1.08901	0.544505	−	0.838758i	$-0.316718\pi$
$479$	23.8341	1.08901	0.544505	+	0.838758i	$0.316718\pi$
$480$	0	0
$481$	3.52714	0.160824
$482$	3.97681	0.181139
$483$	−1.02509	−0.0466433
$484$	−22.2108	−1.00958
$485$	0	0
$486$	1.35144	0.0613027
$487$	11.4881	0.520575	0.260287	−	0.965531i	$-0.416183\pi$
$487$	11.4881	0.520575	0.260287	+	0.965531i	$0.416183\pi$
$488$	−8.74942	−0.396068
$489$	2.10100	0.0950107
$490$	0	0
$491$	−27.1086	−1.22339	−0.611696	−	0.791093i	$-0.709513\pi$
$491$	−27.1086	−1.22339	−0.611696	+	0.791093i	$0.709513\pi$
$492$	−12.3504	−0.556797
$493$	49.6767	2.23733
$494$	−1.93664	−0.0871338
$495$	0	0
$496$	−14.8701	−0.667686
$497$	0.182895	0.00820396
$498$	0.622079	0.0278760
$499$	0.465902	0.0208566	0.0104283	−	0.999946i	$-0.496681\pi$
$499$	0.465902	0.0208566	0.0104283	+	0.999946i	$0.496681\pi$
$500$	0	0
$501$	−28.5656	−1.27622
$502$	1.38253	0.0617053
$503$	40.9594	1.82629	0.913144	−	0.407638i	$-0.133647\pi$
$503$	40.9594	1.82629	0.913144	+	0.407638i	$0.133647\pi$
$504$	−0.0383783	−0.00170951
$505$	0	0
$506$	−15.0994	−0.671251
$507$	−39.2366	−1.74256
$508$	−4.01106	−0.177962
$509$	−5.75767	−0.255204	−0.127602	−	0.991825i	$-0.540728\pi$
$509$	−5.75767	−0.255204	−0.127602	+	0.991825i	$0.540728\pi$
$510$	0	0
$511$	−0.0906862	−0.00401172
$512$	−22.2982	−0.985450
$513$	−4.54196	−0.200532
$514$	2.41408	0.106481
$515$	0	0
$516$	−9.67199	−0.425786
$517$	25.7102	1.13073
$518$	0.0168406	0.000739935 0
$519$	27.2326	1.19538
$520$	0	0
$521$	39.7262	1.74044	0.870218	−	0.492666i	$-0.163978\pi$
$521$	39.7262	1.74044	0.870218	+	0.492666i	$0.163978\pi$
$522$	−0.839186	−0.0367302
$523$	−16.0767	−0.702985	−0.351493	−	0.936191i	$-0.614326\pi$
$523$	−16.0767	−0.702985	−0.351493	+	0.936191i	$0.614326\pi$
$524$	−34.1079	−1.49001
$525$	0	0
$526$	−5.89201	−0.256904
$527$	36.5549	1.59236
$528$	−24.5951	−1.07037
$529$	45.1391	1.96257
$530$	0	0
$531$	−2.59371	−0.112557
$532$	0.117751	0.00510513
$533$	24.8743	1.07743
$534$	2.55594	0.110607
$535$	0	0
$536$	−9.34918	−0.403823
$537$	−2.13302	−0.0920466
$538$	−4.65511	−0.200696
$539$	−33.5265	−1.44409
$540$	0	0
$541$	37.9691	1.63242	0.816210	−	0.577755i	$-0.196071\pi$
$541$	37.9691	1.63242	0.816210	+	0.577755i	$0.196071\pi$
$542$	1.31254	0.0563782
$543$	17.6239	0.756314
$544$	32.0543	1.37432
$545$	0	0
$546$	−0.288523	−0.0123477
$547$	2.29088	0.0979509	0.0489754	−	0.998800i	$-0.484404\pi$
$547$	2.29088	0.0979509	0.0489754	+	0.998800i	$0.484404\pi$
$548$	−32.2970	−1.37966
$549$	2.03753	0.0869597
$550$	0	0
$551$	5.35169	0.227990
$552$	−19.7924	−0.842420
$553$	0.171666	0.00729998
$554$	−3.71952	−0.158027
$555$	0	0
$556$	24.6409	1.04501
$557$	4.96293	0.210286	0.105143	−	0.994457i	$-0.466470\pi$
$557$	4.96293	0.210286	0.105143	+	0.994457i	$0.466470\pi$
$558$	−0.617520	−0.0261417
$559$	19.4799	0.823913
$560$	0	0
$561$	60.4619	2.55270
$562$	2.66585	0.112452
$563$	−8.06701	−0.339984	−0.169992	−	0.985445i	$-0.554374\pi$
$563$	−8.06701	−0.339984	−0.169992	+	0.985445i	$0.554374\pi$
$564$	16.2139	0.682730
$565$	0	0
$566$	−7.35010	−0.308948
$567$	−0.598387	−0.0251299
$568$	3.53132	0.148171
$569$	−3.30222	−0.138436	−0.0692182	−	0.997602i	$-0.522050\pi$
$569$	−3.30222	−0.138436	−0.0692182	+	0.997602i	$0.522050\pi$
$570$	0	0
$571$	−17.3300	−0.725238	−0.362619	−	0.931937i	$-0.618117\pi$
$571$	−17.3300	−0.725238	−0.362619	+	0.931937i	$0.618117\pi$
$572$	54.1196	2.26285
$573$	22.1558	0.925573
$574$	0.118765	0.00495714
$575$	0	0
$576$	1.61468	0.0672784
$577$	−9.99784	−0.416216	−0.208108	−	0.978106i	$-0.566730\pi$
$577$	−9.99784	−0.416216	−0.208108	+	0.978106i	$0.566730\pi$
$578$	−16.3584	−0.680420
$579$	−14.9843	−0.622725
$580$	0	0
$581$	0.0761781	0.00316040
$582$	7.57871	0.314148
$583$	2.22919	0.0923237
$584$	−1.75096	−0.0724553
$585$	0	0
$586$	5.85643	0.241927
$587$	−17.9292	−0.740018	−0.370009	−	0.929028i	$-0.620645\pi$
$587$	−17.9292	−0.740018	−0.370009	+	0.929028i	$0.620645\pi$
$588$	−21.1432	−0.871932
$589$	3.93807	0.162265
$590$	0	0
$591$	40.5287	1.66713
$592$	−1.82340	−0.0749411
$593$	−7.04507	−0.289306	−0.144653	−	0.989482i	$-0.546207\pi$
$593$	−7.04507	−0.289306	−0.144653	+	0.989482i	$0.546207\pi$
$594$	−9.96715	−0.408957
$595$	0	0
$596$	−32.7132	−1.33998
$597$	17.9613	0.735109
$598$	19.1785	0.784266
$599$	−16.1538	−0.660028	−0.330014	−	0.943976i	$-0.607053\pi$
$599$	−16.1538	−0.660028	−0.330014	+	0.943976i	$0.607053\pi$
$600$	0	0
$601$	−41.6254	−1.69794	−0.848968	−	0.528444i	$-0.822776\pi$
$601$	−41.6254	−1.69794	−0.848968	+	0.528444i	$0.822776\pi$
$602$	0.0930087	0.00379075
$603$	2.17720	0.0886625
$604$	31.2197	1.27031
$605$	0	0
$606$	8.28949	0.336737
$607$	2.47890	0.100616	0.0503078	−	0.998734i	$-0.483980\pi$
$607$	2.47890	0.100616	0.0503078	+	0.998734i	$0.483980\pi$
$608$	3.45322	0.140047
$609$	0.797300	0.0323082
$610$	0	0
$611$	−32.6558	−1.32111
$612$	−4.91457	−0.198660
$613$	−46.9158	−1.89491	−0.947455	−	0.319889i	$-0.896354\pi$
$613$	−46.9158	−1.89491	−0.947455	+	0.319889i	$0.896354\pi$
$614$	−2.64327	−0.106674
$615$	0	0
$616$	0.537089	0.0216399
$617$	−3.90275	−0.157119	−0.0785594	−	0.996909i	$-0.525032\pi$
$617$	−3.90275	−0.157119	−0.0785594	+	0.996909i	$0.525032\pi$
$618$	6.46397	0.260019
$619$	26.5939	1.06890	0.534450	−	0.845200i	$-0.320519\pi$
$619$	26.5939	1.06890	0.534450	+	0.845200i	$0.320519\pi$
$620$	0	0
$621$	44.9788	1.80494
$622$	7.51042	0.301140
$623$	0.312994	0.0125398
$624$	31.2394	1.25058
$625$	0	0
$626$	−1.47764	−0.0590583
$627$	6.51358	0.260127
$628$	−25.3346	−1.01096
$629$	4.48243	0.178726
$630$	0	0
$631$	−28.3391	−1.12816	−0.564082	−	0.825719i	$-0.690770\pi$
$631$	−28.3391	−1.12816	−0.564082	+	0.825719i	$0.690770\pi$
$632$	3.31451	0.131844
$633$	−36.8206	−1.46349
$634$	6.75799	0.268394
$635$	0	0
$636$	1.40582	0.0557444
$637$	42.5836	1.68723
$638$	11.7441	0.464952
$639$	−0.822360	−0.0325321
$640$	0	0
$641$	37.6459	1.48692	0.743462	−	0.668778i	$-0.233182\pi$
$641$	37.6459	1.48692	0.743462	+	0.668778i	$0.233182\pi$
$642$	9.75494	0.384997
$643$	10.5720	0.416920	0.208460	−	0.978031i	$-0.433155\pi$
$643$	10.5720	0.416920	0.208460	+	0.978031i	$0.433155\pi$
$644$	−1.16608	−0.0459498
$645$	0	0
$646$	−2.46117	−0.0968333
$647$	−28.6602	−1.12675	−0.563374	−	0.826202i	$-0.690497\pi$
$647$	−28.6602	−1.12675	−0.563374	+	0.826202i	$0.690497\pi$
$648$	−11.5536	−0.453869
$649$	36.2980	1.42482
$650$	0	0
$651$	0.586698	0.0229945
$652$	2.38996	0.0935981
$653$	−8.23646	−0.322318	−0.161159	−	0.986928i	$-0.551523\pi$
$653$	−8.23646	−0.322318	−0.161159	+	0.986928i	$0.551523\pi$
$654$	10.5876	0.414007
$655$	0	0
$656$	−12.8591	−0.502062
$657$	0.407757	0.0159081
$658$	−0.155918	−0.00607831
$659$	6.51555	0.253810	0.126905	−	0.991915i	$-0.459496\pi$
$659$	6.51555	0.253810	0.126905	+	0.991915i	$0.459496\pi$
$660$	0	0
$661$	30.5093	1.18668	0.593338	−	0.804954i	$-0.297810\pi$
$661$	30.5093	1.18668	0.593338	+	0.804954i	$0.297810\pi$
$662$	−4.08810	−0.158888
$663$	−76.7955	−2.98249
$664$	1.47084	0.0570797
$665$	0	0
$666$	−0.0757214	−0.00293415
$667$	−52.9975	−2.05207
$668$	−32.4943	−1.25724
$669$	−24.2680	−0.938253
$670$	0	0
$671$	−28.5144	−1.10079
$672$	0.514464	0.0198459
$673$	−30.8237	−1.18817	−0.594084	−	0.804403i	$-0.702485\pi$
$673$	−30.8237	−1.18817	−0.594084	+	0.804403i	$0.702485\pi$
$674$	6.14041	0.236520
$675$	0	0
$676$	−44.6329	−1.71665
$677$	−30.9153	−1.18817	−0.594086	−	0.804402i	$-0.702486\pi$
$677$	−30.9153	−1.18817	−0.594086	+	0.804402i	$0.702486\pi$
$678$	1.66474	0.0639341
$679$	0.928068	0.0356160
$680$	0	0
$681$	15.7875	0.604977
$682$	8.64195	0.330917
$683$	0.759911	0.0290772	0.0145386	−	0.999894i	$-0.495372\pi$
$683$	0.759911	0.0290772	0.0145386	+	0.999894i	$0.495372\pi$
$684$	−0.529448	−0.0202439
$685$	0	0
$686$	0.406808	0.0155320
$687$	33.0423	1.26064
$688$	−10.0704	−0.383930
$689$	−2.83140	−0.107868
$690$	0	0
$691$	35.9938	1.36927	0.684634	−	0.728887i	$-0.259962\pi$
$691$	35.9938	1.36927	0.684634	+	0.728887i	$0.259962\pi$
$692$	30.9780	1.17761
$693$	−0.125075	−0.00475122
$694$	1.46098	0.0554579
$695$	0	0
$696$	15.3942	0.583516
$697$	31.6113	1.19736
$698$	−5.93198	−0.224529
$699$	32.1516	1.21609
$700$	0	0
$701$	−8.20111	−0.309752	−0.154876	−	0.987934i	$-0.549498\pi$
$701$	−8.20111	−0.309752	−0.154876	+	0.987934i	$0.549498\pi$
$702$	12.6597	0.477811
$703$	0.482894	0.0182127
$704$	−22.5968	−0.851650
$705$	0	0
$706$	3.62623	0.136475
$707$	1.01511	0.0381770
$708$	22.8910	0.860296
$709$	−22.3663	−0.839984	−0.419992	−	0.907528i	$-0.637967\pi$
$709$	−22.3663	−0.839984	−0.419992	+	0.907528i	$0.637967\pi$
$710$	0	0
$711$	−0.771870	−0.0289474
$712$	6.04326	0.226481
$713$	−38.9985	−1.46051
$714$	−0.366667	−0.0137222
$715$	0	0
$716$	−2.42638	−0.0906781
$717$	−37.9057	−1.41561
$718$	1.29152	0.0481990
$719$	−8.41809	−0.313942	−0.156971	−	0.987603i	$-0.550173\pi$
$719$	−8.41809	−0.313942	−0.156971	+	0.987603i	$0.550173\pi$
$720$	0	0
$721$	0.791560	0.0294792
$722$	6.98530	0.259966
$723$	16.9886	0.631814
$724$	20.0478	0.745069
$725$	0	0
$726$	7.45093	0.276530
$727$	−32.7173	−1.21342	−0.606709	−	0.794924i	$-0.707511\pi$
$727$	−32.7173	−1.21342	−0.606709	+	0.794924i	$0.707511\pi$
$728$	−0.682182	−0.0252834
$729$	29.3386	1.08661
$730$	0	0
$731$	24.7559	0.915630
$732$	−17.9824	−0.664648
$733$	17.7563	0.655844	0.327922	−	0.944705i	$-0.393652\pi$
$733$	17.7563	0.655844	0.327922	+	0.944705i	$0.393652\pi$
$734$	0.850126	0.0313787
$735$	0	0
$736$	−34.1971	−1.26052
$737$	−30.4691	−1.12234
$738$	−0.534007	−0.0196571
$739$	16.2783	0.598805	0.299403	−	0.954127i	$-0.403213\pi$
$739$	16.2783	0.598805	0.299403	+	0.954127i	$0.403213\pi$
$740$	0	0
$741$	−8.27320	−0.303924
$742$	−0.0135188	−0.000496290 0
$743$	39.4194	1.44616	0.723079	−	0.690765i	$-0.242726\pi$
$743$	39.4194	1.44616	0.723079	+	0.690765i	$0.242726\pi$
$744$	11.3279	0.415301
$745$	0	0
$746$	−12.7626	−0.467270
$747$	−0.342523	−0.0125323
$748$	68.7774	2.51475
$749$	1.19456	0.0436484
$750$	0	0
$751$	−32.5955	−1.18943	−0.594714	−	0.803937i	$-0.702735\pi$
$751$	−32.5955	−1.18943	−0.594714	+	0.803937i	$0.702735\pi$
$752$	16.8818	0.615615
$753$	5.90607	0.215229
$754$	−14.9167	−0.543234
$755$	0	0
$756$	−0.769729	−0.0279948
$757$	38.1681	1.38724	0.693621	−	0.720340i	$-0.256014\pi$
$757$	38.1681	1.38724	0.693621	+	0.720340i	$0.256014\pi$
$758$	−11.2002	−0.406811
$759$	−64.5036	−2.34133
$760$	0	0
$761$	−14.3504	−0.520201	−0.260100	−	0.965582i	$-0.583756\pi$
$761$	−14.3504	−0.520201	−0.260100	+	0.965582i	$0.583756\pi$
$762$	1.34557	0.0487448
$763$	1.29653	0.0469374
$764$	25.2030	0.911812
$765$	0	0
$766$	−1.72129	−0.0621926
$767$	−46.1038	−1.66471
$768$	−9.09624	−0.328232
$769$	−50.4182	−1.81813	−0.909064	−	0.416657i	$-0.863201\pi$
$769$	−50.4182	−1.81813	−0.909064	+	0.416657i	$0.863201\pi$
$770$	0	0
$771$	10.3128	0.371405
$772$	−17.0451	−0.613467
$773$	12.5954	0.453025	0.226512	−	0.974008i	$-0.427268\pi$
$773$	12.5954	0.453025	0.226512	+	0.974008i	$0.427268\pi$
$774$	−0.418200	−0.0150319
$775$	0	0
$776$	17.9191	0.643257
$777$	0.0719420	0.00258090
$778$	3.02328	0.108390
$779$	3.40549	0.122014
$780$	0	0
$781$	11.5086	0.411810
$782$	24.3728	0.871569
$783$	−34.9837	−1.25022
$784$	−22.0141	−0.786219
$785$	0	0
$786$	11.4420	0.408123
$787$	44.8914	1.60021	0.800103	−	0.599862i	$-0.204778\pi$
$787$	44.8914	1.60021	0.800103	+	0.599862i	$0.204778\pi$
$788$	46.1027	1.64234
$789$	−25.1702	−0.896084
$790$	0	0
$791$	0.203860	0.00724843
$792$	−2.41494	−0.0858112
$793$	36.2175	1.28612
$794$	−0.274885	−0.00975531
$795$	0	0
$796$	20.4316	0.724179
$797$	−2.08094	−0.0737105	−0.0368553	−	0.999321i	$-0.511734\pi$
$797$	−2.08094	−0.0737105	−0.0368553	+	0.999321i	$0.511734\pi$
$798$	−0.0395012	−0.00139833
$799$	−41.5003	−1.46817
$800$	0	0
$801$	−1.40733	−0.0497256
$802$	−15.1843	−0.536176
$803$	−5.70640	−0.201374
$804$	−19.2151	−0.677663
$805$	0	0
$806$	−10.9765	−0.386632
$807$	−19.8863	−0.700031
$808$	19.5996	0.689512
$809$	−13.3820	−0.470485	−0.235243	−	0.971937i	$-0.575588\pi$
$809$	−13.3820	−0.470485	−0.235243	+	0.971937i	$0.575588\pi$
$810$	0	0
$811$	−1.57763	−0.0553981	−0.0276990	−	0.999616i	$-0.508818\pi$
$811$	−1.57763	−0.0553981	−0.0276990	+	0.999616i	$0.508818\pi$
$812$	0.906955	0.0318279
$813$	5.60706	0.196648
$814$	1.05969	0.0371422
$815$	0	0
$816$	39.7003	1.38979
$817$	2.66696	0.0933051
$818$	−1.83204	−0.0640557
$819$	0.158864	0.00555115
$820$	0	0
$821$	−20.3887	−0.711570	−0.355785	−	0.934568i	$-0.615786\pi$
$821$	−20.3887	−0.711570	−0.355785	+	0.934568i	$0.615786\pi$
$822$	10.8345	0.377897
$823$	−11.1627	−0.389106	−0.194553	−	0.980892i	$-0.562326\pi$
$823$	−11.1627	−0.389106	−0.194553	+	0.980892i	$0.562326\pi$
$824$	15.2834	0.532421
$825$	0	0
$826$	−0.220127	−0.00765918
$827$	14.2386	0.495124	0.247562	−	0.968872i	$-0.420371\pi$
$827$	14.2386	0.495124	0.247562	+	0.968872i	$0.420371\pi$
$828$	5.24309	0.182210
$829$	15.9964	0.555580	0.277790	−	0.960642i	$-0.410398\pi$
$829$	15.9964	0.555580	0.277790	+	0.960642i	$0.410398\pi$
$830$	0	0
$831$	−15.8895	−0.551202
$832$	28.7013	0.995039
$833$	54.1170	1.87504
$834$	−8.26617	−0.286234
$835$	0	0
$836$	7.40941	0.256260
$837$	−25.7430	−0.889808
$838$	−12.3413	−0.426323
$839$	44.3604	1.53149	0.765746	−	0.643143i	$-0.222370\pi$
$839$	44.3604	1.53149	0.765746	+	0.643143i	$0.222370\pi$
$840$	0	0
$841$	12.2206	0.421398
$842$	−9.99963	−0.344610
$843$	11.3883	0.392234
$844$	−41.8846	−1.44173
$845$	0	0
$846$	0.701062	0.0241030
$847$	0.912420	0.0313511
$848$	1.46373	0.0502646
$849$	−31.3991	−1.07761
$850$	0	0
$851$	−4.78207	−0.163927
$852$	7.25780	0.248648
$853$	−42.9204	−1.46957	−0.734783	−	0.678302i	$-0.762716\pi$
$853$	−42.9204	−1.46957	−0.734783	+	0.678302i	$0.762716\pi$
$854$	0.172924	0.00591733
$855$	0	0
$856$	23.0645	0.788329
$857$	−18.2282	−0.622662	−0.311331	−	0.950302i	$-0.600775\pi$
$857$	−18.2282	−0.622662	−0.311331	+	0.950302i	$0.600775\pi$
$858$	−18.1552	−0.619809
$859$	−8.29963	−0.283180	−0.141590	−	0.989925i	$-0.545221\pi$
$859$	−8.29963	−0.283180	−0.141590	+	0.989925i	$0.545221\pi$
$860$	0	0
$861$	0.507354	0.0172906
$862$	−5.43496	−0.185116
$863$	−23.6589	−0.805360	−0.402680	−	0.915341i	$-0.631921\pi$
$863$	−23.6589	−0.805360	−0.402680	+	0.915341i	$0.631921\pi$
$864$	−22.5735	−0.767967
$865$	0	0
$866$	7.20121	0.244707
$867$	−69.8819	−2.37331
$868$	0.667388	0.0226526
$869$	10.8020	0.366433
$870$	0	0
$871$	38.7002	1.31131
$872$	25.0332	0.847732
$873$	−4.17292	−0.141232
$874$	2.62569	0.0888152
$875$	0	0
$876$	−3.59869	−0.121588
$877$	53.7913	1.81640	0.908202	−	0.418532i	$-0.137455\pi$
$877$	53.7913	1.81640	0.908202	+	0.418532i	$0.137455\pi$
$878$	−9.73719	−0.328614
$879$	25.0183	0.843845
$880$	0	0
$881$	12.8646	0.433419	0.216710	−	0.976236i	$-0.430468\pi$
$881$	12.8646	0.433419	0.216710	+	0.976236i	$0.430468\pi$
$882$	−0.914196	−0.0307826
$883$	16.6092	0.558945	0.279472	−	0.960154i	$-0.409840\pi$
$883$	16.6092	0.558945	0.279472	+	0.960154i	$0.409840\pi$
$884$	−87.3574	−2.93815
$885$	0	0
$886$	−10.6713	−0.358511
$887$	19.0785	0.640593	0.320296	−	0.947317i	$-0.396218\pi$
$887$	19.0785	0.640593	0.320296	+	0.947317i	$0.396218\pi$
$888$	1.38905	0.0466135
$889$	0.164775	0.00552636
$890$	0	0
$891$	−37.6533	−1.26143
$892$	−27.6056	−0.924304
$893$	−4.47084	−0.149611
$894$	10.9741	0.367030
$895$	0	0
$896$	0.768213	0.0256642
$897$	81.9290	2.73553
$898$	7.79492	0.260120
$899$	30.3324	1.01164
$900$	0	0
$901$	−3.59826	−0.119875
$902$	7.47322	0.248831
$903$	0.397326	0.0132222
$904$	3.93611	0.130913
$905$	0	0
$906$	−10.4731	−0.347946
$907$	11.0744	0.367720	0.183860	−	0.982952i	$-0.441141\pi$
$907$	11.0744	0.367720	0.183860	+	0.982952i	$0.441141\pi$
$908$	17.9588	0.595982
$909$	−4.56428	−0.151388
$910$	0	0
$911$	20.8399	0.690456	0.345228	−	0.938519i	$-0.387802\pi$
$911$	20.8399	0.690456	0.345228	+	0.938519i	$0.387802\pi$
$912$	4.27693	0.141623
$913$	4.79348	0.158641
$914$	4.18947	0.138575
$915$	0	0
$916$	37.5867	1.24190
$917$	1.40116	0.0462702
$918$	16.0885	0.531001
$919$	−43.4535	−1.43340	−0.716700	−	0.697382i	$-0.754348\pi$
$919$	−43.4535	−1.43340	−0.716700	+	0.697382i	$0.754348\pi$
$920$	0	0
$921$	−11.2918	−0.372079
$922$	14.5547	0.479333
$923$	−14.6176	−0.481145
$924$	1.10386	0.0363144
$925$	0	0
$926$	−7.68985	−0.252704
$927$	−3.55913	−0.116897
$928$	26.5979	0.873119
$929$	−13.8883	−0.455660	−0.227830	−	0.973701i	$-0.573163\pi$
$929$	−13.8883	−0.455660	−0.227830	+	0.973701i	$0.573163\pi$
$930$	0	0
$931$	5.83004	0.191072
$932$	36.5735	1.19801
$933$	32.0840	1.05038
$934$	12.0736	0.395060
$935$	0	0
$936$	3.06733	0.100259
$937$	0.990239	0.0323497	0.0161748	−	0.999869i	$-0.494851\pi$
$937$	0.990239	0.0323497	0.0161748	+	0.999869i	$0.494851\pi$
$938$	0.184778	0.00603320
$939$	−6.31236	−0.205996
$940$	0	0
$941$	21.6651	0.706263	0.353131	−	0.935574i	$-0.385117\pi$
$941$	21.6651	0.706263	0.353131	+	0.935574i	$0.385117\pi$
$942$	8.49887	0.276908
$943$	−33.7244	−1.09822
$944$	23.8339	0.775727
$945$	0	0
$946$	5.85254	0.190283
$947$	−7.53272	−0.244780	−0.122390	−	0.992482i	$-0.539056\pi$
$947$	−7.53272	−0.244780	−0.122390	+	0.992482i	$0.539056\pi$
$948$	6.81220	0.221250
$949$	7.24796	0.235279
$950$	0	0
$951$	28.8696	0.936162
$952$	−0.866945	−0.0280979
$953$	−6.12024	−0.198254	−0.0991269	−	0.995075i	$-0.531605\pi$
$953$	−6.12024	−0.198254	−0.0991269	+	0.995075i	$0.531605\pi$
$954$	0.0607852	0.00196799
$955$	0	0
$956$	−43.1189	−1.39457
$957$	50.1698	1.62176
$958$	−9.09516	−0.293851
$959$	1.32677	0.0428435
$960$	0	0
$961$	−8.67974	−0.279992
$962$	−1.34596	−0.0433956
$963$	−5.37118	−0.173084
$964$	19.3251	0.622421
$965$	0	0
$966$	0.391178	0.0125859
$967$	37.9234	1.21953	0.609766	−	0.792581i	$-0.291263\pi$
$967$	37.9234	1.21953	0.609766	+	0.792581i	$0.291263\pi$
$968$	17.6169	0.566229
$969$	−10.5139	−0.337756
$970$	0	0
$971$	40.4241	1.29727	0.648636	−	0.761099i	$-0.275340\pi$
$971$	40.4241	1.29727	0.648636	+	0.761099i	$0.275340\pi$
$972$	6.56728	0.210646
$973$	−1.01225	−0.0324513
$974$	−4.38388	−0.140469
$975$	0	0
$976$	−18.7231	−0.599311
$977$	−1.51918	−0.0486030	−0.0243015	−	0.999705i	$-0.507736\pi$
$977$	−1.51918	−0.0486030	−0.0243015	+	0.999705i	$0.507736\pi$
$978$	−0.801748	−0.0256371
$979$	19.6950	0.629456
$980$	0	0
$981$	−5.82964	−0.186126
$982$	10.3447	0.330112
$983$	43.5665	1.38956	0.694778	−	0.719224i	$-0.255503\pi$
$983$	43.5665	1.38956	0.694778	+	0.719224i	$0.255503\pi$
$984$	9.79594	0.312283
$985$	0	0
$986$	−18.9568	−0.603706
$987$	−0.666070	−0.0212012
$988$	−9.41104	−0.299405
$989$	−26.4108	−0.839813
$990$	0	0
$991$	60.2012	1.91235	0.956177	−	0.292789i	$-0.0945832\pi$
$991$	60.2012	1.91235	0.956177	+	0.292789i	$0.0945832\pi$
$992$	19.5722	0.621419
$993$	−17.4641	−0.554205
$994$	−0.0697932	−0.00221370
$995$	0	0
$996$	3.02297	0.0957864
$997$	−22.9234	−0.725990	−0.362995	−	0.931791i	$-0.618246\pi$
$997$	−22.9234	−0.725990	−0.362995	+	0.931791i	$0.618246\pi$
$998$	−0.177789	−0.00562782
$999$	−3.15665	−0.0998721

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2075.2.a.k.1.5		11
5.4	even	2		415.2.a.e.1.7	✓	11
15.14	odd	2		3735.2.a.s.1.5		11
20.19	odd	2		6640.2.a.bi.1.3		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
415.2.a.e.1.7	✓	11	5.4	even	2
2075.2.a.k.1.5		11	1.1	even	1	trivial
3735.2.a.s.1.5		11	15.14	odd	2
6640.2.a.bi.1.3		11	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:	\( N \)	\(=\)	\( 2075 = 5^{2} \cdot 83 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2075.a (trivial)

\(f(q)\)	\(=\)	\(q-0.381602 q^{2} -1.63018 q^{3} -1.85438 q^{4} +0.622079 q^{6} +0.0761781 q^{7} +1.47084 q^{8} -0.342523 q^{9} +4.79348 q^{11} +3.02297 q^{12} -6.08842 q^{13} -0.0290697 q^{14} +3.14748 q^{16} -7.73742 q^{17} +0.130708 q^{18} -0.833555 q^{19} -0.124184 q^{21} -1.82920 q^{22} +8.25464 q^{23} -2.39773 q^{24} +2.32336 q^{26} +5.44890 q^{27} -0.141263 q^{28} -6.42032 q^{29} -4.72443 q^{31} -4.14277 q^{32} -7.81422 q^{33} +2.95262 q^{34} +0.635168 q^{36} -0.579319 q^{37} +0.318086 q^{38} +9.92521 q^{39} -4.08551 q^{41} +0.0473888 q^{42} -3.19950 q^{43} -8.88893 q^{44} -3.14999 q^{46} +5.36358 q^{47} -5.13095 q^{48} -6.99420 q^{49} +12.6134 q^{51} +11.2903 q^{52} +0.465047 q^{53} -2.07931 q^{54} +0.112046 q^{56} +1.35884 q^{57} +2.45001 q^{58} +7.57236 q^{59} -5.94859 q^{61} +1.80285 q^{62} -0.0260928 q^{63} -4.71408 q^{64} +2.98192 q^{66} -6.35636 q^{67} +14.3481 q^{68} -13.4565 q^{69} +2.40089 q^{71} -0.503797 q^{72} -1.19045 q^{73} +0.221069 q^{74} +1.54573 q^{76} +0.365158 q^{77} -3.78748 q^{78} +2.25348 q^{79} -7.85511 q^{81} +1.55904 q^{82} +1.00000 q^{83} +0.230284 q^{84} +1.22094 q^{86} +10.4663 q^{87} +7.05044 q^{88} +4.10871 q^{89} -0.463805 q^{91} -15.3072 q^{92} +7.70166 q^{93} -2.04676 q^{94} +6.75344 q^{96} +12.1829 q^{97} +2.66900 q^{98} -1.64188 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 18 q^{4} - 5 q^{6} + 3 q^{7} - 3 q^{8} + 21 q^{9} - q^{11} - 2 q^{12} - q^{13} + 12 q^{14} + 20 q^{16} - 28 q^{17} + 26 q^{18} - 2 q^{19} + 3 q^{21} + 20 q^{22} - q^{23} - 10 q^{24} + 21 q^{26}+ \cdots - 36 q^{99}+O(q^{100}) \) 11 * q + 18 * q^4 - 5 * q^6 + 3 * q^7 - 3 * q^8 + 21 * q^9 - q^11 - 2 * q^12 - q^13 + 12 * q^14 + 20 * q^16 - 28 * q^17 + 26 * q^18 - 2 * q^19 + 3 * q^21 + 20 * q^22 - q^23 - 10 * q^24 + 21 * q^26 + 12 * q^27 + 22 * q^28 + 21 * q^29 + 4 * q^31 - q^32 - 10 * q^33 - 4 * q^34 + 45 * q^36 + 10 * q^37 - 17 * q^38 - 2 * q^39 + 39 * q^41 + 29 * q^42 + 13 * q^43 - 25 * q^44 - 13 * q^46 - 8 * q^47 - 17 * q^48 + 32 * q^49 + 32 * q^51 + 65 * q^52 - 21 * q^53 - 72 * q^54 + 57 * q^56 + 19 * q^57 + 6 * q^58 + 5 * q^59 + 14 * q^61 + q^62 + 30 * q^63 - 21 * q^64 - 11 * q^66 + 42 * q^67 - 61 * q^68 - 7 * q^69 + 3 * q^71 + 69 * q^72 - 3 * q^73 - 54 * q^74 + 13 * q^76 + 42 * q^77 + 7 * q^78 - 20 * q^79 + 39 * q^81 + 43 * q^82 + 11 * q^83 - 116 * q^84 + 22 * q^86 + 16 * q^87 + 75 * q^88 + 37 * q^89 - 17 * q^91 - 7 * q^92 + 55 * q^93 - 54 * q^94 - 58 * q^96 + 2 * q^97 + 26 * q^98 - 36 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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