LMFDB - Embedded newform 975.2.a.n.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [975,2,Mod(1,975)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("975.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 975 = 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	975.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-1,-3,5,0,1,-5] 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:	$7.78541419707$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.08613$ of defining polynomial
Character	$\chi$	$=$	975.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.08613 q^{2} -1.00000 q^{3} +2.35194 q^{4} -2.08613 q^{6} -4.08613 q^{7} +0.734191 q^{8} +1.00000 q^{9} -3.43807 q^{11} -2.35194 q^{12} -1.00000 q^{13} -8.52420 q^{14} -3.17226 q^{16} -7.17226 q^{17} +2.08613 q^{18} +7.52420 q^{19} +4.08613 q^{21} -7.17226 q^{22} +2.82032 q^{23} -0.734191 q^{24} -2.08613 q^{26} -1.00000 q^{27} -9.61033 q^{28} -5.70388 q^{29} -5.61033 q^{31} -8.08613 q^{32} +3.43807 q^{33} -14.9623 q^{34} +2.35194 q^{36} -4.82032 q^{37} +15.6965 q^{38} +1.00000 q^{39} +5.52420 q^{41} +8.52420 q^{42} +2.05582 q^{43} -8.08613 q^{44} +5.88356 q^{46} +7.49389 q^{47} +3.17226 q^{48} +9.69646 q^{49} +7.17226 q^{51} -2.35194 q^{52} +2.52420 q^{53} -2.08613 q^{54} -3.00000 q^{56} -7.52420 q^{57} -11.8990 q^{58} +4.79001 q^{59} -7.87614 q^{61} -11.7039 q^{62} -4.08613 q^{63} -10.5242 q^{64} +7.17226 q^{66} +6.20257 q^{67} -16.8687 q^{68} -2.82032 q^{69} -0.475800 q^{71} +0.734191 q^{72} -9.35194 q^{73} -10.0558 q^{74} +17.6965 q^{76} +14.0484 q^{77} +2.08613 q^{78} +11.6965 q^{79} +1.00000 q^{81} +11.5242 q^{82} -16.4307 q^{83} +9.61033 q^{84} +4.28870 q^{86} +5.70388 q^{87} -2.52420 q^{88} -10.1723 q^{89} +4.08613 q^{91} +6.63322 q^{92} +5.61033 q^{93} +15.6332 q^{94} +8.08613 q^{96} -14.8761 q^{97} +20.2281 q^{98} -3.43807 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} - 3 q^{3} + 5 q^{4} + q^{6} - 5 q^{7} - 3 q^{8} + 3 q^{9} - q^{11} - 5 q^{12} - 3 q^{13} - 9 q^{14} + 5 q^{16} - 7 q^{17} - q^{18} + 6 q^{19} + 5 q^{21} - 7 q^{22} - 4 q^{23} + 3 q^{24}+ \cdots - q^{99}+O(q^{100}) $ 3 * q - q^2 - 3 * q^3 + 5 * q^4 + q^6 - 5 * q^7 - 3 * q^8 + 3 * q^9 - q^11 - 5 * q^12 - 3 * q^13 - 9 * q^14 + 5 * q^16 - 7 * q^17 - q^18 + 6 * q^19 + 5 * q^21 - 7 * q^22 - 4 * q^23 + 3 * q^24 + q^26 - 3 * q^27 - 5 * q^28 - 13 * q^29 + 7 * q^31 - 17 * q^32 + q^33 - 19 * q^34 + 5 * q^36 - 2 * q^37 + 16 * q^38 + 3 * q^39 + 9 * q^42 - 17 * q^44 + 26 * q^46 + 7 * q^47 - 5 * q^48 - 2 * q^49 + 7 * q^51 - 5 * q^52 - 9 * q^53 + q^54 - 9 * q^56 - 6 * q^57 + 11 * q^58 + 3 * q^59 - 5 * q^61 - 31 * q^62 - 5 * q^63 - 15 * q^64 + 7 * q^66 + 3 * q^67 - 5 * q^68 + 4 * q^69 - 18 * q^71 - 3 * q^72 - 26 * q^73 - 24 * q^74 + 22 * q^76 + 9 * q^77 - q^78 + 4 * q^79 + 3 * q^81 + 18 * q^82 - 13 * q^83 + 5 * q^84 - 10 * q^86 + 13 * q^87 + 9 * q^88 - 16 * q^89 + 5 * q^91 - 32 * q^92 - 7 * q^93 - 5 * q^94 + 17 * q^96 - 26 * q^97 + 40 * q^98 - 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 $

$2$	2.08613	1.47512	0.737558	−	0.675283i	$-0.235979\pi$
$2$	2.08613	1.47512	0.737558	+	0.675283i	$0.235979\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.35194	1.17597
$5$	0	0
$5$	0	0
$6$	−2.08613	−0.851659
$7$	−4.08613	−1.54441	−0.772206	−	0.635372i	$-0.780847\pi$
$7$	−4.08613	−1.54441	−0.772206	+	0.635372i	$0.780847\pi$
$8$	0.734191	0.259576
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.43807	−1.03662	−0.518308	−	0.855194i	$-0.673438\pi$
$11$	−3.43807	−1.03662	−0.518308	+	0.855194i	$0.673438\pi$
$12$	−2.35194	−0.678946
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−8.52420	−2.27819
$15$	0	0
$16$	−3.17226	−0.793065
$17$	−7.17226	−1.73953	−0.869764	−	0.493467i	$-0.835729\pi$
$17$	−7.17226	−1.73953	−0.869764	+	0.493467i	$0.835729\pi$
$18$	2.08613	0.491706
$19$	7.52420	1.72617	0.863085	−	0.505059i	$-0.168529\pi$
$19$	7.52420	1.72617	0.863085	+	0.505059i	$0.168529\pi$
$20$	0	0
$21$	4.08613	0.891667
$22$	−7.17226	−1.52913
$23$	2.82032	0.588078	0.294039	−	0.955793i	$-0.405001\pi$
$23$	2.82032	0.588078	0.294039	+	0.955793i	$0.405001\pi$
$24$	−0.734191	−0.149866
$25$	0	0
$26$	−2.08613	−0.409124
$27$	−1.00000	−0.192450
$28$	−9.61033	−1.81618
$29$	−5.70388	−1.05918	−0.529592	−	0.848253i	$-0.677655\pi$
$29$	−5.70388	−1.05918	−0.529592	+	0.848253i	$0.677655\pi$
$30$	0	0
$31$	−5.61033	−1.00764	−0.503822	−	0.863807i	$-0.668073\pi$
$31$	−5.61033	−1.00764	−0.503822	+	0.863807i	$0.668073\pi$
$32$	−8.08613	−1.42944
$33$	3.43807	0.598491
$34$	−14.9623	−2.56601
$35$	0	0
$36$	2.35194	0.391990
$37$	−4.82032	−0.792456	−0.396228	−	0.918152i	$-0.629681\pi$
$37$	−4.82032	−0.792456	−0.396228	+	0.918152i	$0.629681\pi$
$38$	15.6965	2.54630
$39$	1.00000	0.160128
$40$	0	0
$41$	5.52420	0.862735	0.431368	−	0.902176i	$-0.358031\pi$
$41$	5.52420	0.862735	0.431368	+	0.902176i	$0.358031\pi$
$42$	8.52420	1.31531
$43$	2.05582	0.313509	0.156755	−	0.987638i	$-0.449897\pi$
$43$	2.05582	0.313509	0.156755	+	0.987638i	$0.449897\pi$
$44$	−8.08613	−1.21903
$45$	0	0
$46$	5.88356	0.867483
$47$	7.49389	1.09310	0.546548	−	0.837428i	$-0.315942\pi$
$47$	7.49389	1.09310	0.546548	+	0.837428i	$0.315942\pi$
$48$	3.17226	0.457876
$49$	9.69646	1.38521
$50$	0	0
$51$	7.17226	1.00432
$52$	−2.35194	−0.326155
$53$	2.52420	0.346725	0.173363	−	0.984858i	$-0.444537\pi$
$53$	2.52420	0.346725	0.173363	+	0.984858i	$0.444537\pi$
$54$	−2.08613	−0.283886
$55$	0	0
$56$	−3.00000	−0.400892
$57$	−7.52420	−0.996605
$58$	−11.8990	−1.56242
$59$	4.79001	0.623606	0.311803	−	0.950147i	$-0.399067\pi$
$59$	4.79001	0.623606	0.311803	+	0.950147i	$0.399067\pi$
$60$	0	0
$61$	−7.87614	−1.00844	−0.504218	−	0.863576i	$-0.668219\pi$
$61$	−7.87614	−1.00844	−0.504218	+	0.863576i	$0.668219\pi$
$62$	−11.7039	−1.48639
$63$	−4.08613	−0.514804
$64$	−10.5242	−1.31552
$65$	0	0
$66$	7.17226	0.882844
$67$	6.20257	0.757765	0.378882	−	0.925445i	$-0.376308\pi$
$67$	6.20257	0.757765	0.378882	+	0.925445i	$0.376308\pi$
$68$	−16.8687	−2.04563
$69$	−2.82032	−0.339527
$70$	0	0
$71$	−0.475800	−0.0564671	−0.0282336	−	0.999601i	$-0.508988\pi$
$71$	−0.475800	−0.0564671	−0.0282336	+	0.999601i	$0.508988\pi$
$72$	0.734191	0.0865252
$73$	−9.35194	−1.09456	−0.547281	−	0.836949i	$-0.684337\pi$
$73$	−9.35194	−1.09456	−0.547281	+	0.836949i	$0.684337\pi$
$74$	−10.0558	−1.16897
$75$	0	0
$76$	17.6965	2.02992
$77$	14.0484	1.60096
$78$	2.08613	0.236208
$79$	11.6965	1.31595	0.657977	−	0.753038i	$-0.271412\pi$
$79$	11.6965	1.31595	0.657977	+	0.753038i	$0.271412\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	11.5242	1.27263
$83$	−16.4307	−1.80350	−0.901749	−	0.432260i	$-0.857716\pi$
$83$	−16.4307	−1.80350	−0.901749	+	0.432260i	$0.857716\pi$
$84$	9.61033	1.04857
$85$	0	0
$86$	4.28870	0.462463
$87$	5.70388	0.611520
$88$	−2.52420	−0.269081
$89$	−10.1723	−1.07826	−0.539129	−	0.842223i	$-0.681246\pi$
$89$	−10.1723	−1.07826	−0.539129	+	0.842223i	$0.681246\pi$
$90$	0	0
$91$	4.08613	0.428343
$92$	6.63322	0.691561
$93$	5.61033	0.581764
$94$	15.6332	1.61244
$95$	0	0
$96$	8.08613	0.825287
$97$	−14.8761	−1.51044	−0.755222	−	0.655470i	$-0.772471\pi$
$97$	−14.8761	−1.51044	−0.755222	+	0.655470i	$0.772471\pi$
$98$	20.2281	2.04334
$99$	−3.43807	−0.345539
$100$	0	0
$101$	−4.94418	−0.491965	−0.245982	−	0.969274i	$-0.579111\pi$
$101$	−4.94418	−0.491965	−0.245982	+	0.969274i	$0.579111\pi$
$102$	14.9623	1.48149
$103$	8.28870	0.816710	0.408355	−	0.912823i	$-0.366103\pi$
$103$	8.28870	0.816710	0.408355	+	0.912823i	$0.366103\pi$
$104$	−0.734191	−0.0719934
$105$	0	0
$106$	5.26581	0.511461
$107$	−12.5168	−1.21004	−0.605021	−	0.796209i	$-0.706835\pi$
$107$	−12.5168	−1.21004	−0.605021	+	0.796209i	$0.706835\pi$
$108$	−2.35194	−0.226315
$109$	11.6965	1.12032	0.560159	−	0.828385i	$-0.310740\pi$
$109$	11.6965	1.12032	0.560159	+	0.828385i	$0.310740\pi$
$110$	0	0
$111$	4.82032	0.457525
$112$	12.9623	1.22482
$113$	−1.23550	−0.116226	−0.0581129	−	0.998310i	$-0.518508\pi$
$113$	−1.23550	−0.116226	−0.0581129	+	0.998310i	$0.518508\pi$
$114$	−15.6965	−1.47011
$115$	0	0
$116$	−13.4152	−1.24557
$117$	−1.00000	−0.0924500
$118$	9.99258	0.919892
$119$	29.3068	2.68655
$120$	0	0
$121$	0.820321	0.0745747
$122$	−16.4307	−1.48756
$123$	−5.52420	−0.498100
$124$	−13.1952	−1.18496
$125$	0	0
$126$	−8.52420	−0.759396
$127$	9.52420	0.845136	0.422568	−	0.906331i	$-0.361129\pi$
$127$	9.52420	0.845136	0.422568	+	0.906331i	$0.361129\pi$
$128$	−5.78259	−0.511114
$129$	−2.05582	−0.181005
$130$	0	0
$131$	10.5726	0.923732	0.461866	−	0.886950i	$-0.347180\pi$
$131$	10.5726	0.923732	0.461866	+	0.886950i	$0.347180\pi$
$132$	8.08613	0.703807
$133$	−30.7449	−2.66592
$134$	12.9394	1.11779
$135$	0	0
$136$	−5.26581	−0.451539
$137$	−4.53162	−0.387162	−0.193581	−	0.981084i	$-0.562010\pi$
$137$	−4.53162	−0.387162	−0.193581	+	0.981084i	$0.562010\pi$
$138$	−5.88356	−0.500842
$139$	−14.4562	−1.22616	−0.613078	−	0.790023i	$-0.710069\pi$
$139$	−14.4562	−1.22616	−0.613078	+	0.790023i	$0.710069\pi$
$140$	0	0
$141$	−7.49389	−0.631099
$142$	−0.992582	−0.0832956
$143$	3.43807	0.287506
$144$	−3.17226	−0.264355
$145$	0	0
$146$	−19.5094	−1.61461
$147$	−9.69646	−0.799751
$148$	−11.3371	−0.931904
$149$	12.2839	1.00634	0.503168	−	0.864189i	$-0.332168\pi$
$149$	12.2839	1.00634	0.503168	+	0.864189i	$0.332168\pi$
$150$	0	0
$151$	22.6029	1.83940	0.919699	−	0.392623i	$-0.128432\pi$
$151$	22.6029	1.83940	0.919699	+	0.392623i	$0.128432\pi$
$152$	5.52420	0.448072
$153$	−7.17226	−0.579843
$154$	29.3068	2.36161
$155$	0	0
$156$	2.35194	0.188306
$157$	−3.64806	−0.291147	−0.145573	−	0.989347i	$-0.546503\pi$
$157$	−3.64806	−0.291147	−0.145573	+	0.989347i	$0.546503\pi$
$158$	24.4003	1.94119
$159$	−2.52420	−0.200182
$160$	0	0
$161$	−11.5242	−0.908234
$162$	2.08613	0.163902
$163$	0.820321	0.0642525	0.0321263	−	0.999484i	$-0.489772\pi$
$163$	0.820321	0.0642525	0.0321263	+	0.999484i	$0.489772\pi$
$164$	12.9926	1.01455
$165$	0	0
$166$	−34.2765	−2.66037
$167$	−9.46357	−0.732313	−0.366157	−	0.930553i	$-0.619327\pi$
$167$	−9.46357	−0.732313	−0.366157	+	0.930553i	$0.619327\pi$
$168$	3.00000	0.231455
$169$	1.00000	0.0769231
$170$	0	0
$171$	7.52420	0.575390
$172$	4.83516	0.368677
$173$	6.69646	0.509122	0.254561	−	0.967057i	$-0.418069\pi$
$173$	6.69646	0.509122	0.254561	+	0.967057i	$0.418069\pi$
$174$	11.8990	0.902063
$175$	0	0
$176$	10.9065	0.822105
$177$	−4.79001	−0.360039
$178$	−21.2207	−1.59056
$179$	−15.1042	−1.12894	−0.564471	−	0.825453i	$-0.690920\pi$
$179$	−15.1042	−1.12894	−0.564471	+	0.825453i	$0.690920\pi$
$180$	0	0
$181$	−1.70869	−0.127006	−0.0635028	−	0.997982i	$-0.520227\pi$
$181$	−1.70869	−0.127006	−0.0635028	+	0.997982i	$0.520227\pi$
$182$	8.52420	0.631856
$183$	7.87614	0.582221
$184$	2.07065	0.152651
$185$	0	0
$186$	11.7039	0.858170
$187$	24.6587	1.80322
$188$	17.6252	1.28545
$189$	4.08613	0.297222
$190$	0	0
$191$	−23.6406	−1.71058	−0.855288	−	0.518152i	$-0.826620\pi$
$191$	−23.6406	−1.71058	−0.855288	+	0.518152i	$0.826620\pi$
$192$	10.5242	0.759519
$193$	21.4487	1.54391	0.771957	−	0.635675i	$-0.219278\pi$
$193$	21.4487	1.54391	0.771957	+	0.635675i	$0.219278\pi$
$194$	−31.0336	−2.22808
$195$	0	0
$196$	22.8055	1.62896
$197$	−4.57260	−0.325784	−0.162892	−	0.986644i	$-0.552082\pi$
$197$	−4.57260	−0.325784	−0.162892	+	0.986644i	$0.552082\pi$
$198$	−7.17226	−0.509710
$199$	5.65548	0.400906	0.200453	−	0.979703i	$-0.435759\pi$
$199$	5.65548	0.400906	0.200453	+	0.979703i	$0.435759\pi$
$200$	0	0
$201$	−6.20257	−0.437496
$202$	−10.3142	−0.725705
$203$	23.3068	1.63582
$204$	16.8687	1.18105
$205$	0	0
$206$	17.2913	1.20474
$207$	2.82032	0.196026
$208$	3.17226	0.219957
$209$	−25.8687	−1.78938
$210$	0	0
$211$	−5.63583	−0.387987	−0.193993	−	0.981003i	$-0.562144\pi$
$211$	−5.63583	−0.387987	−0.193993	+	0.981003i	$0.562144\pi$
$212$	5.93676	0.407739
$213$	0.475800	0.0326013
$214$	−26.1116	−1.78495
$215$	0	0
$216$	−0.734191	−0.0499554
$217$	22.9245	1.55622
$218$	24.4003	1.65260
$219$	9.35194	0.631945
$220$	0	0
$221$	7.17226	0.482458
$222$	10.0558	0.674902
$223$	6.22808	0.417063	0.208531	−	0.978016i	$-0.433132\pi$
$223$	6.22808	0.417063	0.208531	+	0.978016i	$0.433132\pi$
$224$	33.0410	2.20764
$225$	0	0
$226$	−2.57741	−0.171447
$227$	14.8458	0.985352	0.492676	−	0.870213i	$-0.336019\pi$
$227$	14.8458	0.985352	0.492676	+	0.870213i	$0.336019\pi$
$228$	−17.6965	−1.17198
$229$	−25.5046	−1.68539	−0.842694	−	0.538392i	$-0.819032\pi$
$229$	−25.5046	−1.68539	−0.842694	+	0.538392i	$0.819032\pi$
$230$	0	0
$231$	−14.0484	−0.924317
$232$	−4.18774	−0.274938
$233$	−4.17226	−0.273334	−0.136667	−	0.990617i	$-0.543639\pi$
$233$	−4.17226	−0.273334	−0.136667	+	0.990617i	$0.543639\pi$
$234$	−2.08613	−0.136375
$235$	0	0
$236$	11.2658	0.733342
$237$	−11.6965	−0.759767
$238$	61.1378	3.96297
$239$	23.3068	1.50759	0.753795	−	0.657109i	$-0.228221\pi$
$239$	23.3068	1.50759	0.753795	+	0.657109i	$0.228221\pi$
$240$	0	0
$241$	−25.3371	−1.63211	−0.816053	−	0.577977i	$-0.803842\pi$
$241$	−25.3371	−1.63211	−0.816053	+	0.577977i	$0.803842\pi$
$242$	1.71130	0.110006
$243$	−1.00000	−0.0641500
$244$	−18.5242	−1.18589
$245$	0	0
$246$	−11.5242	−0.734756
$247$	−7.52420	−0.478753
$248$	−4.11905	−0.261560
$249$	16.4307	1.04125
$250$	0	0
$251$	−5.24030	−0.330765	−0.165383	−	0.986229i	$-0.552886\pi$
$251$	−5.24030	−0.330765	−0.165383	+	0.986229i	$0.552886\pi$
$252$	−9.61033	−0.605394
$253$	−9.69646	−0.609611
$254$	19.8687	1.24667
$255$	0	0
$256$	8.98516	0.561573
$257$	−23.4610	−1.46345	−0.731727	−	0.681597i	$-0.761286\pi$
$257$	−23.4610	−1.46345	−0.731727	+	0.681597i	$0.761286\pi$
$258$	−4.28870	−0.267003
$259$	19.6965	1.22388
$260$	0	0
$261$	−5.70388	−0.353061
$262$	22.0558	1.36261
$263$	−18.0410	−1.11245	−0.556227	−	0.831030i	$-0.687752\pi$
$263$	−18.0410	−1.11245	−0.556227	+	0.831030i	$0.687752\pi$
$264$	2.52420	0.155354
$265$	0	0
$266$	−64.1378	−3.93254
$267$	10.1723	0.622532
$268$	14.5881	0.891108
$269$	−27.8761	−1.69964	−0.849819	−	0.527074i	$-0.823289\pi$
$269$	−27.8761	−1.69964	−0.849819	+	0.527074i	$0.823289\pi$
$270$	0	0
$271$	−20.0713	−1.21924	−0.609622	−	0.792692i	$-0.708679\pi$
$271$	−20.0713	−1.21924	−0.609622	+	0.792692i	$0.708679\pi$
$272$	22.7523	1.37956
$273$	−4.08613	−0.247304
$274$	−9.45355	−0.571110
$275$	0	0
$276$	−6.63322	−0.399273
$277$	8.40515	0.505016	0.252508	−	0.967595i	$-0.418745\pi$
$277$	8.40515	0.505016	0.252508	+	0.967595i	$0.418745\pi$
$278$	−30.1574	−1.80872
$279$	−5.61033	−0.335882
$280$	0	0
$281$	−10.4562	−0.623762	−0.311881	−	0.950121i	$-0.600959\pi$
$281$	−10.4562	−0.623762	−0.311881	+	0.950121i	$0.600959\pi$
$282$	−15.6332	−0.930945
$283$	18.3445	1.09047	0.545234	−	0.838284i	$-0.316441\pi$
$283$	18.3445	1.09047	0.545234	+	0.838284i	$0.316441\pi$
$284$	−1.11905	−0.0664036
$285$	0	0
$286$	7.17226	0.424105
$287$	−22.5726	−1.33242
$288$	−8.08613	−0.476480
$289$	34.4413	2.02596
$290$	0	0
$291$	14.8761	0.872055
$292$	−21.9952	−1.28717
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	−20.2281	−1.17973
$295$	0	0
$296$	−3.53904	−0.205702
$297$	3.43807	0.199497
$298$	25.6258	1.48446
$299$	−2.82032	−0.163103
$300$	0	0
$301$	−8.40034	−0.484187
$302$	47.1526	2.71333
$303$	4.94418	0.284036
$304$	−23.8687	−1.36897
$305$	0	0
$306$	−14.9623	−0.855336
$307$	13.9293	0.794990	0.397495	−	0.917604i	$-0.369880\pi$
$307$	13.9293	0.794990	0.397495	+	0.917604i	$0.369880\pi$
$308$	33.0410	1.88268
$309$	−8.28870	−0.471528
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	0.734191	0.0415654
$313$	30.0894	1.70075	0.850376	−	0.526175i	$-0.176374\pi$
$313$	30.0894	1.70075	0.850376	+	0.526175i	$0.176374\pi$
$314$	−7.61033	−0.429476
$315$	0	0
$316$	27.5094	1.54752
$317$	4.05582	0.227797	0.113899	−	0.993492i	$-0.463666\pi$
$317$	4.05582	0.227797	0.113899	+	0.993492i	$0.463666\pi$
$318$	−5.26581	−0.295292
$319$	19.6103	1.09797
$320$	0	0
$321$	12.5168	0.698619
$322$	−24.0410	−1.33975
$323$	−53.9655	−3.00272
$324$	2.35194	0.130663
$325$	0	0
$326$	1.71130	0.0947800
$327$	−11.6965	−0.646816
$328$	4.05582	0.223945
$329$	−30.6210	−1.68819
$330$	0	0
$331$	−19.4126	−1.06701	−0.533506	−	0.845797i	$-0.679126\pi$
$331$	−19.4126	−1.06701	−0.533506	+	0.845797i	$0.679126\pi$
$332$	−38.6439	−2.12086
$333$	−4.82032	−0.264152
$334$	−19.7422	−1.08025
$335$	0	0
$336$	−12.9623	−0.707150
$337$	−20.1797	−1.09926	−0.549629	−	0.835409i	$-0.685231\pi$
$337$	−20.1797	−1.09926	−0.549629	+	0.835409i	$0.685231\pi$
$338$	2.08613	0.113471
$339$	1.23550	0.0671030
$340$	0	0
$341$	19.2887	1.04454
$342$	15.6965	0.848767
$343$	−11.0181	−0.594921
$344$	1.50936	0.0813794
$345$	0	0
$346$	13.9697	0.751015
$347$	−18.8007	−1.00927	−0.504637	−	0.863332i	$-0.668374\pi$
$347$	−18.8007	−1.00927	−0.504637	+	0.863332i	$0.668374\pi$
$348$	13.4152	0.719129
$349$	−4.47580	−0.239584	−0.119792	−	0.992799i	$-0.538223\pi$
$349$	−4.47580	−0.239584	−0.119792	+	0.992799i	$0.538223\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	27.8007	1.48178
$353$	−26.8203	−1.42750	−0.713751	−	0.700400i	$-0.753005\pi$
$353$	−26.8203	−1.42750	−0.713751	+	0.700400i	$0.753005\pi$
$354$	−9.99258	−0.531100
$355$	0	0
$356$	−23.9245	−1.26800
$357$	−29.3068	−1.55108
$358$	−31.5094	−1.66532
$359$	−5.89903	−0.311339	−0.155670	−	0.987809i	$-0.549753\pi$
$359$	−5.89903	−0.311339	−0.155670	+	0.987809i	$0.549753\pi$
$360$	0	0
$361$	37.6136	1.97966
$362$	−3.56454	−0.187348
$363$	−0.820321	−0.0430557
$364$	9.61033	0.503718
$365$	0	0
$366$	16.4307	0.858844
$367$	−18.0968	−0.944645	−0.472323	−	0.881426i	$-0.656584\pi$
$367$	−18.0968	−0.944645	−0.472323	+	0.881426i	$0.656584\pi$
$368$	−8.94679	−0.466384
$369$	5.52420	0.287578
$370$	0	0
$371$	−10.3142	−0.535487
$372$	13.1952	0.684137
$373$	12.8836	0.667085	0.333543	−	0.942735i	$-0.391756\pi$
$373$	12.8836	0.667085	0.333543	+	0.942735i	$0.391756\pi$
$374$	51.4413	2.65997
$375$	0	0
$376$	5.50194	0.283741
$377$	5.70388	0.293765
$378$	8.52420	0.438438
$379$	−19.0032	−0.976131	−0.488066	−	0.872807i	$-0.662297\pi$
$379$	−19.0032	−0.976131	−0.488066	+	0.872807i	$0.662297\pi$
$380$	0	0
$381$	−9.52420	−0.487940
$382$	−49.3175	−2.52330
$383$	−9.62100	−0.491610	−0.245805	−	0.969319i	$-0.579052\pi$
$383$	−9.62100	−0.491610	−0.245805	+	0.969319i	$0.579052\pi$
$384$	5.78259	0.295092
$385$	0	0
$386$	44.7449	2.27745
$387$	2.05582	0.104503
$388$	−34.9878	−1.77624
$389$	−2.41998	−0.122698	−0.0613490	−	0.998116i	$-0.519540\pi$
$389$	−2.41998	−0.122698	−0.0613490	+	0.998116i	$0.519540\pi$
$390$	0	0
$391$	−20.2281	−1.02298
$392$	7.11905	0.359567
$393$	−10.5726	−0.533317
$394$	−9.53904	−0.480570
$395$	0	0
$396$	−8.08613	−0.406343
$397$	26.6136	1.33570	0.667849	−	0.744297i	$-0.267215\pi$
$397$	26.6136	1.33570	0.667849	+	0.744297i	$0.267215\pi$
$398$	11.7981	0.591384
$399$	30.7449	1.53917
$400$	0	0
$401$	12.4758	0.623012	0.311506	−	0.950244i	$-0.399167\pi$
$401$	12.4758	0.623012	0.311506	+	0.950244i	$0.399167\pi$
$402$	−12.9394	−0.645357
$403$	5.61033	0.279470
$404$	−11.6284	−0.578535
$405$	0	0
$406$	48.6210	2.41302
$407$	16.5726	0.821473
$408$	5.26581	0.260696
$409$	16.2281	0.802427	0.401213	−	0.915985i	$-0.368589\pi$
$409$	16.2281	0.802427	0.401213	+	0.915985i	$0.368589\pi$
$410$	0	0
$411$	4.53162	0.223528
$412$	19.4945	0.960426
$413$	−19.5726	−0.963105
$414$	5.88356	0.289161
$415$	0	0
$416$	8.08613	0.396455
$417$	14.4562	0.707921
$418$	−53.9655	−2.63954
$419$	−38.3855	−1.87525	−0.937627	−	0.347642i	$-0.886982\pi$
$419$	−38.3855	−1.87525	−0.937627	+	0.347642i	$0.886982\pi$
$420$	0	0
$421$	8.95160	0.436274	0.218137	−	0.975918i	$-0.430002\pi$
$421$	8.95160	0.436274	0.218137	+	0.975918i	$0.430002\pi$
$422$	−11.7571	−0.572326
$423$	7.49389	0.364365
$424$	1.85324	0.0900015
$425$	0	0
$426$	0.992582	0.0480907
$427$	32.1829	1.55744
$428$	−29.4387	−1.42297
$429$	−3.43807	−0.165992
$430$	0	0
$431$	−1.99519	−0.0961050	−0.0480525	−	0.998845i	$-0.515301\pi$
$431$	−1.99519	−0.0961050	−0.0480525	+	0.998845i	$0.515301\pi$
$432$	3.17226	0.152625
$433$	−36.9729	−1.77681	−0.888403	−	0.459064i	$-0.848185\pi$
$433$	−36.9729	−1.77681	−0.888403	+	0.459064i	$0.848185\pi$
$434$	47.8236	2.29560
$435$	0	0
$436$	27.5094	1.31746
$437$	21.2207	1.01512
$438$	19.5094	0.932193
$439$	−5.06804	−0.241885	−0.120942	−	0.992660i	$-0.538592\pi$
$439$	−5.06804	−0.241885	−0.120942	+	0.992660i	$0.538592\pi$
$440$	0	0
$441$	9.69646	0.461736
$442$	14.9623	0.711683
$443$	−5.59966	−0.266048	−0.133024	−	0.991113i	$-0.542469\pi$
$443$	−5.59966	−0.266048	−0.133024	+	0.991113i	$0.542469\pi$
$444$	11.3371	0.538035
$445$	0	0
$446$	12.9926	0.615217
$447$	−12.2839	−0.581008
$448$	43.0032	2.03171
$449$	13.9852	0.660001	0.330000	−	0.943981i	$-0.392951\pi$
$449$	13.9852	0.660001	0.330000	+	0.943981i	$0.392951\pi$
$450$	0	0
$451$	−18.9926	−0.894326
$452$	−2.90581	−0.136678
$453$	−22.6029	−1.06198
$454$	30.9703	1.45351
$455$	0	0
$456$	−5.52420	−0.258694
$457$	31.6210	1.47917	0.739584	−	0.673064i	$-0.235022\pi$
$457$	31.6210	1.47917	0.739584	+	0.673064i	$0.235022\pi$
$458$	−53.2058	−2.48614
$459$	7.17226	0.334772
$460$	0	0
$461$	29.0484	1.35292	0.676459	−	0.736480i	$-0.263513\pi$
$461$	29.0484	1.35292	0.676459	+	0.736480i	$0.263513\pi$
$462$	−29.3068	−1.36348
$463$	30.8458	1.43353	0.716764	−	0.697316i	$-0.245623\pi$
$463$	30.8458	1.43353	0.716764	+	0.697316i	$0.245623\pi$
$464$	18.0942	0.840002
$465$	0	0
$466$	−8.70388	−0.403199
$467$	17.3323	0.802043	0.401021	−	0.916069i	$-0.368655\pi$
$467$	17.3323	0.802043	0.401021	+	0.916069i	$0.368655\pi$
$468$	−2.35194	−0.108718
$469$	−25.3445	−1.17030
$470$	0	0
$471$	3.64806	0.168094
$472$	3.51678	0.161873
$473$	−7.06804	−0.324989
$474$	−24.4003	−1.12074
$475$	0	0
$476$	68.9278	3.15930
$477$	2.52420	0.115575
$478$	48.6210	2.22387
$479$	−9.20518	−0.420596	−0.210298	−	0.977637i	$-0.567443\pi$
$479$	−9.20518	−0.420596	−0.210298	+	0.977637i	$0.567443\pi$
$480$	0	0
$481$	4.82032	0.219788
$482$	−52.8565	−2.40755
$483$	11.5242	0.524369
$484$	1.92935	0.0876975
$485$	0	0
$486$	−2.08613	−0.0946288
$487$	−27.8432	−1.26170	−0.630848	−	0.775906i	$-0.717293\pi$
$487$	−27.8432	−1.26170	−0.630848	+	0.775906i	$0.717293\pi$
$488$	−5.78259	−0.261766
$489$	−0.820321	−0.0370962
$490$	0	0
$491$	−23.6406	−1.06689	−0.533444	−	0.845836i	$-0.679102\pi$
$491$	−23.6406	−1.06689	−0.533444	+	0.845836i	$0.679102\pi$
$492$	−12.9926	−0.585751
$493$	40.9097	1.84248
$494$	−15.6965	−0.706217
$495$	0	0
$496$	17.7974	0.799128
$497$	1.94418	0.0872085
$498$	34.2765	1.53597
$499$	−16.1829	−0.724447	−0.362224	−	0.932091i	$-0.617982\pi$
$499$	−16.1829	−0.724447	−0.362224	+	0.932091i	$0.617982\pi$
$500$	0	0
$501$	9.46357	0.422801
$502$	−10.9320	−0.487917
$503$	12.4758	0.556268	0.278134	−	0.960542i	$-0.410284\pi$
$503$	12.4758	0.556268	0.278134	+	0.960542i	$0.410284\pi$
$504$	−3.00000	−0.133631
$505$	0	0
$506$	−20.2281	−0.899248
$507$	−1.00000	−0.0444116
$508$	22.4003	0.993854
$509$	−10.0558	−0.445716	−0.222858	−	0.974851i	$-0.571539\pi$
$509$	−10.0558	−0.445716	−0.222858	+	0.974851i	$0.571539\pi$
$510$	0	0
$511$	38.2132	1.69045
$512$	30.3094	1.33950
$513$	−7.52420	−0.332202
$514$	−48.9426	−2.15877
$515$	0	0
$516$	−4.83516	−0.212856
$517$	−25.7645	−1.13312
$518$	41.0894	1.80536
$519$	−6.69646	−0.293942
$520$	0	0
$521$	31.7523	1.39109	0.695546	−	0.718481i	$-0.255162\pi$
$521$	31.7523	1.39109	0.695546	+	0.718481i	$0.255162\pi$
$522$	−11.8990	−0.520807
$523$	18.9878	0.830277	0.415139	−	0.909758i	$-0.363733\pi$
$523$	18.9878	0.830277	0.415139	+	0.909758i	$0.363733\pi$
$524$	24.8661	1.08628
$525$	0	0
$526$	−37.6358	−1.64100
$527$	40.2387	1.75283
$528$	−10.9065	−0.474642
$529$	−15.0458	−0.654165
$530$	0	0
$531$	4.79001	0.207869
$532$	−72.3100	−3.13504
$533$	−5.52420	−0.239280
$534$	21.2207	0.918308
$535$	0	0
$536$	4.55387	0.196697
$537$	15.1042	0.651795
$538$	−58.1533	−2.50716
$539$	−33.3371	−1.43593
$540$	0	0
$541$	1.00742	0.0433123	0.0216562	−	0.999765i	$-0.493106\pi$
$541$	1.00742	0.0433123	0.0216562	+	0.999765i	$0.493106\pi$
$542$	−41.8713	−1.79853
$543$	1.70869	0.0733267
$544$	57.9958	2.48655
$545$	0	0
$546$	−8.52420	−0.364802
$547$	0.662898	0.0283435	0.0141717	−	0.999900i	$-0.495489\pi$
$547$	0.662898	0.0283435	0.0141717	+	0.999900i	$0.495489\pi$
$548$	−10.6581	−0.455291
$549$	−7.87614	−0.336145
$550$	0	0
$551$	−42.9171	−1.82833
$552$	−2.07065	−0.0881329
$553$	−47.7933	−2.03238
$554$	17.5342	0.744958
$555$	0	0
$556$	−34.0000	−1.44192
$557$	16.8809	0.715269	0.357634	−	0.933862i	$-0.383583\pi$
$557$	16.8809	0.715269	0.357634	+	0.933862i	$0.383583\pi$
$558$	−11.7039	−0.495465
$559$	−2.05582	−0.0869518
$560$	0	0
$561$	−24.6587	−1.04109
$562$	−21.8129	−0.920122
$563$	−18.7087	−0.788477	−0.394239	−	0.919008i	$-0.628992\pi$
$563$	−18.7087	−0.788477	−0.394239	+	0.919008i	$0.628992\pi$
$564$	−17.6252	−0.742153
$565$	0	0
$566$	38.2691	1.60857
$567$	−4.08613	−0.171601
$568$	−0.349328	−0.0146575
$569$	−16.5848	−0.695272	−0.347636	−	0.937630i	$-0.613016\pi$
$569$	−16.5848	−0.695272	−0.347636	+	0.937630i	$0.613016\pi$
$570$	0	0
$571$	−0.460964	−0.0192907	−0.00964536	−	0.999953i	$-0.503070\pi$
$571$	−0.460964	−0.0192907	−0.00964536	+	0.999953i	$0.503070\pi$
$572$	8.08613	0.338098
$573$	23.6406	0.987602
$574$	−47.0894	−1.96547
$575$	0	0
$576$	−10.5242	−0.438508
$577$	−23.9394	−0.996609	−0.498305	−	0.867002i	$-0.666044\pi$
$577$	−23.9394	−0.996609	−0.498305	+	0.867002i	$0.666044\pi$
$578$	71.8491	2.98853
$579$	−21.4487	−0.891379
$580$	0	0
$581$	67.1378	2.78534
$582$	31.0336	1.28638
$583$	−8.67837	−0.359422
$584$	−6.86611	−0.284122
$585$	0	0
$586$	−12.5168	−0.517063
$587$	32.7194	1.35047	0.675236	−	0.737602i	$-0.264042\pi$
$587$	32.7194	1.35047	0.675236	+	0.737602i	$0.264042\pi$
$588$	−22.8055	−0.940482
$589$	−42.2132	−1.73937
$590$	0	0
$591$	4.57260	0.188092
$592$	15.2913	0.628469
$593$	−0.825129	−0.0338840	−0.0169420	−	0.999856i	$-0.505393\pi$
$593$	−0.825129	−0.0338840	−0.0169420	+	0.999856i	$0.505393\pi$
$594$	7.17226	0.294281
$595$	0	0
$596$	28.8910	1.18342
$597$	−5.65548	−0.231463
$598$	−5.88356	−0.240597
$599$	−11.2913	−0.461351	−0.230675	−	0.973031i	$-0.574094\pi$
$599$	−11.2913	−0.461351	−0.230675	+	0.973031i	$0.574094\pi$
$600$	0	0
$601$	9.64806	0.393553	0.196776	−	0.980448i	$-0.436953\pi$
$601$	9.64806	0.393553	0.196776	+	0.980448i	$0.436953\pi$
$602$	−17.5242	−0.714233
$603$	6.20257	0.252588
$604$	53.1607	2.16308
$605$	0	0
$606$	10.3142	0.418986
$607$	24.6284	0.999637	0.499818	−	0.866130i	$-0.333400\pi$
$607$	24.6284	0.999637	0.499818	+	0.866130i	$0.333400\pi$
$608$	−60.8417	−2.46746
$609$	−23.3068	−0.944439
$610$	0	0
$611$	−7.49389	−0.303170
$612$	−16.8687	−0.681878
$613$	−10.3445	−0.417811	−0.208906	−	0.977936i	$-0.566990\pi$
$613$	−10.3445	−0.417811	−0.208906	+	0.977936i	$0.566990\pi$
$614$	29.0584	1.17270
$615$	0	0
$616$	10.3142	0.415571
$617$	−32.8613	−1.32295	−0.661473	−	0.749969i	$-0.730068\pi$
$617$	−32.8613	−1.32295	−0.661473	+	0.749969i	$0.730068\pi$
$618$	−17.2913	−0.695559
$619$	36.8565	1.48139	0.740694	−	0.671843i	$-0.234497\pi$
$619$	36.8565	1.48139	0.740694	+	0.671843i	$0.234497\pi$
$620$	0	0
$621$	−2.82032	−0.113176
$622$	−37.5503	−1.50563
$623$	41.5652	1.66527
$624$	−3.17226	−0.126992
$625$	0	0
$626$	62.7704	2.50881
$627$	25.8687	1.03310
$628$	−8.58002	−0.342380
$629$	34.5726	1.37850
$630$	0	0
$631$	−25.6965	−1.02296	−0.511480	−	0.859295i	$-0.670903\pi$
$631$	−25.6965	−1.02296	−0.511480	+	0.859295i	$0.670903\pi$
$632$	8.58744	0.341590
$633$	5.63583	0.224004
$634$	8.46096	0.336028
$635$	0	0
$636$	−5.93676	−0.235408
$637$	−9.69646	−0.384188
$638$	40.9097	1.61963
$639$	−0.475800	−0.0188224
$640$	0	0
$641$	18.4535	0.728871	0.364436	−	0.931229i	$-0.381262\pi$
$641$	18.4535	0.728871	0.364436	+	0.931229i	$0.381262\pi$
$642$	26.1116	1.03054
$643$	16.7087	0.658926	0.329463	−	0.944168i	$-0.393132\pi$
$643$	16.7087	0.658926	0.329463	+	0.944168i	$0.393132\pi$
$644$	−27.1042	−1.06806
$645$	0	0
$646$	−112.579	−4.42937
$647$	−28.6136	−1.12492	−0.562458	−	0.826826i	$-0.690144\pi$
$647$	−28.6136	−1.12492	−0.562458	+	0.826826i	$0.690144\pi$
$648$	0.734191	0.0288417
$649$	−16.4684	−0.646441
$650$	0	0
$651$	−22.9245	−0.898483
$652$	1.92935	0.0755590
$653$	39.7252	1.55457	0.777284	−	0.629150i	$-0.216597\pi$
$653$	39.7252	1.55457	0.777284	+	0.629150i	$0.216597\pi$
$654$	−24.4003	−0.954129
$655$	0	0
$656$	−17.5242	−0.684205
$657$	−9.35194	−0.364854
$658$	−63.8794	−2.49028
$659$	5.21585	0.203181	0.101590	−	0.994826i	$-0.467607\pi$
$659$	5.21585	0.203181	0.101590	+	0.994826i	$0.467607\pi$
$660$	0	0
$661$	20.3493	0.791497	0.395749	−	0.918359i	$-0.370485\pi$
$661$	20.3493	0.791497	0.395749	+	0.918359i	$0.370485\pi$
$662$	−40.4971	−1.57397
$663$	−7.17226	−0.278548
$664$	−12.0632	−0.468144
$665$	0	0
$666$	−10.0558	−0.389655
$667$	−16.0868	−0.622882
$668$	−22.2578	−0.861178
$669$	−6.22808	−0.240791
$670$	0	0
$671$	27.0787	1.04536
$672$	−33.0410	−1.27458
$673$	−18.2765	−0.704506	−0.352253	−	0.935905i	$-0.614584\pi$
$673$	−18.2765	−0.704506	−0.352253	+	0.935905i	$0.614584\pi$
$674$	−42.0974	−1.62153
$675$	0	0
$676$	2.35194	0.0904592
$677$	43.2664	1.66286	0.831432	−	0.555626i	$-0.187521\pi$
$677$	43.2664	1.66286	0.831432	+	0.555626i	$0.187521\pi$
$678$	2.57741	0.0989847
$679$	60.7858	2.33275
$680$	0	0
$681$	−14.8458	−0.568893
$682$	40.2387	1.54082
$683$	25.8432	0.988863	0.494432	−	0.869217i	$-0.335376\pi$
$683$	25.8432	0.988863	0.494432	+	0.869217i	$0.335376\pi$
$684$	17.6965	0.676641
$685$	0	0
$686$	−22.9852	−0.877578
$687$	25.5046	0.973059
$688$	−6.52159	−0.248633
$689$	−2.52420	−0.0961643
$690$	0	0
$691$	−8.94743	−0.340376	−0.170188	−	0.985412i	$-0.554438\pi$
$691$	−8.94743	−0.340376	−0.170188	+	0.985412i	$0.554438\pi$
$692$	15.7497	0.598712
$693$	14.0484	0.533655
$694$	−39.2207	−1.48880
$695$	0	0
$696$	4.18774	0.158736
$697$	−39.6210	−1.50075
$698$	−9.33710	−0.353415
$699$	4.17226	0.157809
$700$	0	0
$701$	−26.4897	−1.00050	−0.500251	−	0.865880i	$-0.666759\pi$
$701$	−26.4897	−1.00050	−0.500251	+	0.865880i	$0.666759\pi$
$702$	2.08613	0.0787359
$703$	−36.2691	−1.36791
$704$	36.1829	1.36370
$705$	0	0
$706$	−55.9507	−2.10573
$707$	20.2026	0.759796
$708$	−11.2658	−0.423395
$709$	19.3977	0.728497	0.364248	−	0.931302i	$-0.381326\pi$
$709$	19.3977	0.728497	0.364248	+	0.931302i	$0.381326\pi$
$710$	0	0
$711$	11.6965	0.438652
$712$	−7.46838	−0.279889
$713$	−15.8229	−0.592573
$714$	−61.1378	−2.28802
$715$	0	0
$716$	−35.5242	−1.32760
$717$	−23.3068	−0.870408
$718$	−12.3062	−0.459261
$719$	−32.0261	−1.19437	−0.597187	−	0.802102i	$-0.703715\pi$
$719$	−32.0261	−1.19437	−0.597187	+	0.802102i	$0.703715\pi$
$720$	0	0
$721$	−33.8687	−1.26134
$722$	78.4668	2.92023
$723$	25.3371	0.942297
$724$	−4.01873	−0.149355
$725$	0	0
$726$	−1.71130	−0.0635122
$727$	−5.46357	−0.202633	−0.101316	−	0.994854i	$-0.532305\pi$
$727$	−5.46357	−0.202633	−0.101316	+	0.994854i	$0.532305\pi$
$728$	3.00000	0.111187
$729$	1.00000	0.0370370
$730$	0	0
$731$	−14.7449	−0.545358
$732$	18.5242	0.684674
$733$	45.6406	1.68578	0.842888	−	0.538089i	$-0.180854\pi$
$733$	45.6406	1.68578	0.842888	+	0.538089i	$0.180854\pi$
$734$	−37.7523	−1.39346
$735$	0	0
$736$	−22.8055	−0.840621
$737$	−21.3249	−0.785512
$738$	11.5242	0.424212
$739$	34.9278	1.28484	0.642420	−	0.766353i	$-0.277931\pi$
$739$	34.9278	1.28484	0.642420	+	0.766353i	$0.277931\pi$
$740$	0	0
$741$	7.52420	0.276408
$742$	−21.5168	−0.789906
$743$	52.3962	1.92223	0.961115	−	0.276150i	$-0.0890587\pi$
$743$	52.3962	1.92223	0.961115	+	0.276150i	$0.0890587\pi$
$744$	4.11905	0.151012
$745$	0	0
$746$	26.8768	0.984029
$747$	−16.4307	−0.601166
$748$	57.9958	2.12054
$749$	51.1452	1.86880
$750$	0	0
$751$	31.9655	1.16644	0.583219	−	0.812315i	$-0.301793\pi$
$751$	31.9655	1.16644	0.583219	+	0.812315i	$0.301793\pi$
$752$	−23.7726	−0.866896
$753$	5.24030	0.190967
$754$	11.8990	0.433337
$755$	0	0
$756$	9.61033	0.349524
$757$	−34.4897	−1.25355	−0.626775	−	0.779200i	$-0.715626\pi$
$757$	−34.4897	−1.25355	−0.626775	+	0.779200i	$0.715626\pi$
$758$	−39.6433	−1.43991
$759$	9.69646	0.351959
$760$	0	0
$761$	−19.7933	−0.717505	−0.358753	−	0.933433i	$-0.616798\pi$
$761$	−19.7933	−0.717505	−0.358753	+	0.933433i	$0.616798\pi$
$762$	−19.8687	−0.719768
$763$	−47.7933	−1.73023
$764$	−55.6014	−2.01159
$765$	0	0
$766$	−20.0707	−0.725182
$767$	−4.79001	−0.172957
$768$	−8.98516	−0.324224
$769$	4.65287	0.167787	0.0838934	−	0.996475i	$-0.473264\pi$
$769$	4.65287	0.167787	0.0838934	+	0.996475i	$0.473264\pi$
$770$	0	0
$771$	23.4610	0.844926
$772$	50.4461	1.81560
$773$	−20.6284	−0.741953	−0.370976	−	0.928642i	$-0.620977\pi$
$773$	−20.6284	−0.741953	−0.370976	+	0.928642i	$0.620977\pi$
$774$	4.28870	0.154154
$775$	0	0
$776$	−10.9219	−0.392074
$777$	−19.6965	−0.706607
$778$	−5.04840	−0.180994
$779$	41.5652	1.48923
$780$	0	0
$781$	1.63583	0.0585348
$782$	−42.1984	−1.50901
$783$	5.70388	0.203840
$784$	−30.7597	−1.09856
$785$	0	0
$786$	−22.0558	−0.786705
$787$	26.1877	0.933492	0.466746	−	0.884391i	$-0.345426\pi$
$787$	26.1877	0.933492	0.466746	+	0.884391i	$0.345426\pi$
$788$	−10.7545	−0.383112
$789$	18.0410	0.642276
$790$	0	0
$791$	5.04840	0.179500
$792$	−2.52420	−0.0896935
$793$	7.87614	0.279690
$794$	55.5194	1.97031
$795$	0	0
$796$	13.3013	0.471454
$797$	−17.2180	−0.609895	−0.304947	−	0.952369i	$-0.598639\pi$
$797$	−17.2180	−0.609895	−0.304947	+	0.952369i	$0.598639\pi$
$798$	64.1378	2.27045
$799$	−53.7481	−1.90147
$800$	0	0
$801$	−10.1723	−0.359419
$802$	26.0261	0.919015
$803$	32.1526	1.13464
$804$	−14.5881	−0.514482
$805$	0	0
$806$	11.7039	0.412252
$807$	27.8761	0.981287
$808$	−3.62997	−0.127702
$809$	13.4684	0.473523	0.236762	−	0.971568i	$-0.423914\pi$
$809$	13.4684	0.473523	0.236762	+	0.971568i	$0.423914\pi$
$810$	0	0
$811$	−7.83841	−0.275244	−0.137622	−	0.990485i	$-0.543946\pi$
$811$	−7.83841	−0.275244	−0.137622	+	0.990485i	$0.543946\pi$
$812$	54.8162	1.92367
$813$	20.0713	0.703931
$814$	34.5726	1.21177
$815$	0	0
$816$	−22.7523	−0.796489
$817$	15.4684	0.541170
$818$	33.8539	1.18367
$819$	4.08613	0.142781
$820$	0	0
$821$	32.8203	1.14544	0.572719	−	0.819752i	$-0.305889\pi$
$821$	32.8203	1.14544	0.572719	+	0.819752i	$0.305889\pi$
$822$	9.45355	0.329730
$823$	−24.1478	−0.841740	−0.420870	−	0.907121i	$-0.638275\pi$
$823$	−24.1478	−0.841740	−0.420870	+	0.907121i	$0.638275\pi$
$824$	6.08549	0.211998
$825$	0	0
$826$	−40.8310	−1.42069
$827$	10.1058	0.351412	0.175706	−	0.984443i	$-0.443779\pi$
$827$	10.1058	0.351412	0.175706	+	0.984443i	$0.443779\pi$
$828$	6.63322	0.230520
$829$	−3.54645	−0.123173	−0.0615867	−	0.998102i	$-0.519616\pi$
$829$	−3.54645	−0.123173	−0.0615867	+	0.998102i	$0.519616\pi$
$830$	0	0
$831$	−8.40515	−0.291571
$832$	10.5242	0.364861
$833$	−69.5455	−2.40961
$834$	30.1574	1.04427
$835$	0	0
$836$	−60.8417	−2.10425
$837$	5.61033	0.193921
$838$	−80.0772	−2.76622
$839$	−27.1797	−0.938347	−0.469173	−	0.883106i	$-0.655448\pi$
$839$	−27.1797	−0.938347	−0.469173	+	0.883106i	$0.655448\pi$
$840$	0	0
$841$	3.53423	0.121870
$842$	18.6742	0.643556
$843$	10.4562	0.360129
$844$	−13.2551	−0.456261
$845$	0	0
$846$	15.6332	0.537481
$847$	−3.35194	−0.115174
$848$	−8.00742	−0.274976
$849$	−18.3445	−0.629582
$850$	0	0
$851$	−13.5949	−0.466026
$852$	1.11905	0.0383381
$853$	−14.4413	−0.494461	−0.247231	−	0.968957i	$-0.579521\pi$
$853$	−14.4413	−0.494461	−0.247231	+	0.968957i	$0.579521\pi$
$854$	67.1378	2.29741
$855$	0	0
$856$	−9.18971	−0.314098
$857$	20.7858	0.710031	0.355015	−	0.934860i	$-0.384476\pi$
$857$	20.7858	0.710031	0.355015	+	0.934860i	$0.384476\pi$
$858$	−7.17226	−0.244857
$859$	20.9926	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$859$	20.9926	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$860$	0	0
$861$	22.5726	0.769272
$862$	−4.16223	−0.141766
$863$	25.3020	0.861289	0.430645	−	0.902522i	$-0.358286\pi$
$863$	25.3020	0.861289	0.430645	+	0.902522i	$0.358286\pi$
$864$	8.08613	0.275096
$865$	0	0
$866$	−77.1304	−2.62100
$867$	−34.4413	−1.16969
$868$	53.9171	1.83007
$869$	−40.2132	−1.36414
$870$	0	0
$871$	−6.20257	−0.210166
$872$	8.58744	0.290807
$873$	−14.8761	−0.503481
$874$	44.2691	1.49742
$875$	0	0
$876$	21.9952	0.743149
$877$	−27.9342	−0.943269	−0.471635	−	0.881794i	$-0.656336\pi$
$877$	−27.9342	−0.943269	−0.471635	+	0.881794i	$0.656336\pi$
$878$	−10.5726	−0.356808
$879$	6.00000	0.202375
$880$	0	0
$881$	−38.9900	−1.31361	−0.656803	−	0.754062i	$-0.728092\pi$
$881$	−38.9900	−1.31361	−0.656803	+	0.754062i	$0.728092\pi$
$882$	20.2281	0.681115
$883$	12.1871	0.410128	0.205064	−	0.978749i	$-0.434260\pi$
$883$	12.1871	0.410128	0.205064	+	0.978749i	$0.434260\pi$
$884$	16.8687	0.567356
$885$	0	0
$886$	−11.6816	−0.392452
$887$	−32.1526	−1.07958	−0.539790	−	0.841800i	$-0.681496\pi$
$887$	−32.1526	−1.07958	−0.539790	+	0.841800i	$0.681496\pi$
$888$	3.53904	0.118762
$889$	−38.9171	−1.30524
$890$	0	0
$891$	−3.43807	−0.115180
$892$	14.6481	0.490453
$893$	56.3855	1.88687
$894$	−25.6258	−0.857055
$895$	0	0
$896$	23.6284	0.789370
$897$	2.82032	0.0941678
$898$	29.1749	0.973578
$899$	32.0006	1.06728
$900$	0	0
$901$	−18.1042	−0.603139
$902$	−39.6210	−1.31923
$903$	8.40034	0.279546
$904$	−0.907090	−0.0301694
$905$	0	0
$906$	−47.1526	−1.56654
$907$	42.9368	1.42569	0.712846	−	0.701321i	$-0.247406\pi$
$907$	42.9368	1.42569	0.712846	+	0.701321i	$0.247406\pi$
$908$	34.9165	1.15874
$909$	−4.94418	−0.163988
$910$	0	0
$911$	8.22808	0.272608	0.136304	−	0.990667i	$-0.456478\pi$
$911$	8.22808	0.272608	0.136304	+	0.990667i	$0.456478\pi$
$912$	23.8687	0.790372
$913$	56.4897	1.86954
$914$	65.9655	2.18195
$915$	0	0
$916$	−59.9852	−1.98197
$917$	−43.2010	−1.42662
$918$	14.9623	0.493828
$919$	−27.5652	−0.909291	−0.454646	−	0.890672i	$-0.650234\pi$
$919$	−27.5652	−0.909291	−0.454646	+	0.890672i	$0.650234\pi$
$920$	0	0
$921$	−13.9293	−0.458988
$922$	60.5987	1.99571
$923$	0.475800	0.0156612
$924$	−33.0410	−1.08697
$925$	0	0
$926$	64.3484	2.11462
$927$	8.28870	0.272237
$928$	46.1223	1.51404
$929$	−24.2084	−0.794253	−0.397126	−	0.917764i	$-0.629993\pi$
$929$	−24.2084	−0.794253	−0.397126	+	0.917764i	$0.629993\pi$
$930$	0	0
$931$	72.9581	2.39111
$932$	−9.81290	−0.321432
$933$	18.0000	0.589294
$934$	36.1574	1.18311
$935$	0	0
$936$	−0.734191	−0.0239978
$937$	−2.58002	−0.0842855	−0.0421427	−	0.999112i	$-0.513418\pi$
$937$	−2.58002	−0.0842855	−0.0421427	+	0.999112i	$0.513418\pi$
$938$	−52.8720	−1.72633
$939$	−30.0894	−0.981930
$940$	0	0
$941$	30.6088	0.997817	0.498909	−	0.866655i	$-0.333734\pi$
$941$	30.6088	0.997817	0.498909	+	0.866655i	$0.333734\pi$
$942$	7.61033	0.247958
$943$	15.5800	0.507355
$944$	−15.1952	−0.494560
$945$	0	0
$946$	−14.7449	−0.479397
$947$	51.3626	1.66906	0.834530	−	0.550962i	$-0.185739\pi$
$947$	51.3626	1.66906	0.834530	+	0.550962i	$0.185739\pi$
$948$	−27.5094	−0.893463
$949$	9.35194	0.303577
$950$	0	0
$951$	−4.05582	−0.131519
$952$	21.5168	0.697363
$953$	−9.06543	−0.293658	−0.146829	−	0.989162i	$-0.546907\pi$
$953$	−9.06543	−0.293658	−0.146829	+	0.989162i	$0.546907\pi$
$954$	5.26581	0.170487
$955$	0	0
$956$	54.8162	1.77288
$957$	−19.6103	−0.633912
$958$	−19.2032	−0.620428
$959$	18.5168	0.597938
$960$	0	0
$961$	0.475800	0.0153484
$962$	10.0558	0.324213
$963$	−12.5168	−0.403348
$964$	−59.5913	−1.91931
$965$	0	0
$966$	24.0410	0.773506
$967$	47.8942	1.54017	0.770087	−	0.637939i	$-0.220213\pi$
$967$	47.8942	1.54017	0.770087	+	0.637939i	$0.220213\pi$
$968$	0.602272	0.0193578
$969$	53.9655	1.73362
$970$	0	0
$971$	−7.59485	−0.243730	−0.121865	−	0.992547i	$-0.538888\pi$
$971$	−7.59485	−0.243730	−0.121865	+	0.992547i	$0.538888\pi$
$972$	−2.35194	−0.0754385
$973$	59.0697	1.89369
$974$	−58.0846	−1.86115
$975$	0	0
$976$	24.9852	0.799756
$977$	23.0484	0.737384	0.368692	−	0.929552i	$-0.379806\pi$
$977$	23.0484	0.737384	0.368692	+	0.929552i	$0.379806\pi$
$978$	−1.71130	−0.0547212
$979$	34.9729	1.11774
$980$	0	0
$981$	11.6965	0.373439
$982$	−49.3175	−1.57378
$983$	26.2026	0.835732	0.417866	−	0.908509i	$-0.362778\pi$
$983$	26.2026	0.835732	0.417866	+	0.908509i	$0.362778\pi$
$984$	−4.05582	−0.129295
$985$	0	0
$986$	85.3430	2.71787
$987$	30.6210	0.974677
$988$	−17.6965	−0.562999
$989$	5.79807	0.184368
$990$	0	0
$991$	46.9023	1.48990	0.744950	−	0.667120i	$-0.232473\pi$
$991$	46.9023	1.48990	0.744950	+	0.667120i	$0.232473\pi$
$992$	45.3659	1.44037
$993$	19.4126	0.616039
$994$	4.05582	0.128643
$995$	0	0
$996$	38.6439	1.22448
$997$	−36.5503	−1.15756	−0.578780	−	0.815483i	$-0.696471\pi$
$997$	−36.5503	−1.15756	−0.578780	+	0.815483i	$0.696471\pi$
$998$	−33.7597	−1.06864
$999$	4.82032	0.152508

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	975.2.a.n.1.3	✓	3
3.2	odd	2		2925.2.a.bi.1.1		3
5.2	odd	4		975.2.c.j.274.5		6
5.3	odd	4		975.2.c.j.274.2		6
5.4	even	2		975.2.a.p.1.1	yes	3
15.2	even	4		2925.2.c.x.2224.2		6
15.8	even	4		2925.2.c.x.2224.5		6
15.14	odd	2		2925.2.a.bg.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
975.2.a.n.1.3	✓	3	1.1	even	1	trivial
975.2.a.p.1.1	yes	3	5.4	even	2
975.2.c.j.274.2		6	5.3	odd	4
975.2.c.j.274.5		6	5.2	odd	4
2925.2.a.bg.1.3		3	15.14	odd	2
2925.2.a.bi.1.1		3	3.2	odd	2
2925.2.c.x.2224.2		6	15.2	even	4
2925.2.c.x.2224.5		6	15.8	even	4

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:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.a (trivial)

\(f(q)\)	\(=\)	\(q+2.08613 q^{2} -1.00000 q^{3} +2.35194 q^{4} -2.08613 q^{6} -4.08613 q^{7} +0.734191 q^{8} +1.00000 q^{9} -3.43807 q^{11} -2.35194 q^{12} -1.00000 q^{13} -8.52420 q^{14} -3.17226 q^{16} -7.17226 q^{17} +2.08613 q^{18} +7.52420 q^{19} +4.08613 q^{21} -7.17226 q^{22} +2.82032 q^{23} -0.734191 q^{24} -2.08613 q^{26} -1.00000 q^{27} -9.61033 q^{28} -5.70388 q^{29} -5.61033 q^{31} -8.08613 q^{32} +3.43807 q^{33} -14.9623 q^{34} +2.35194 q^{36} -4.82032 q^{37} +15.6965 q^{38} +1.00000 q^{39} +5.52420 q^{41} +8.52420 q^{42} +2.05582 q^{43} -8.08613 q^{44} +5.88356 q^{46} +7.49389 q^{47} +3.17226 q^{48} +9.69646 q^{49} +7.17226 q^{51} -2.35194 q^{52} +2.52420 q^{53} -2.08613 q^{54} -3.00000 q^{56} -7.52420 q^{57} -11.8990 q^{58} +4.79001 q^{59} -7.87614 q^{61} -11.7039 q^{62} -4.08613 q^{63} -10.5242 q^{64} +7.17226 q^{66} +6.20257 q^{67} -16.8687 q^{68} -2.82032 q^{69} -0.475800 q^{71} +0.734191 q^{72} -9.35194 q^{73} -10.0558 q^{74} +17.6965 q^{76} +14.0484 q^{77} +2.08613 q^{78} +11.6965 q^{79} +1.00000 q^{81} +11.5242 q^{82} -16.4307 q^{83} +9.61033 q^{84} +4.28870 q^{86} +5.70388 q^{87} -2.52420 q^{88} -10.1723 q^{89} +4.08613 q^{91} +6.63322 q^{92} +5.61033 q^{93} +15.6332 q^{94} +8.08613 q^{96} -14.8761 q^{97} +20.2281 q^{98} -3.43807 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} - 3 q^{3} + 5 q^{4} + q^{6} - 5 q^{7} - 3 q^{8} + 3 q^{9} - q^{11} - 5 q^{12} - 3 q^{13} - 9 q^{14} + 5 q^{16} - 7 q^{17} - q^{18} + 6 q^{19} + 5 q^{21} - 7 q^{22} - 4 q^{23} + 3 q^{24}+ \cdots - q^{99}+O(q^{100}) \) 3 * q - q^2 - 3 * q^3 + 5 * q^4 + q^6 - 5 * q^7 - 3 * q^8 + 3 * q^9 - q^11 - 5 * q^12 - 3 * q^13 - 9 * q^14 + 5 * q^16 - 7 * q^17 - q^18 + 6 * q^19 + 5 * q^21 - 7 * q^22 - 4 * q^23 + 3 * q^24 + q^26 - 3 * q^27 - 5 * q^28 - 13 * q^29 + 7 * q^31 - 17 * q^32 + q^33 - 19 * q^34 + 5 * q^36 - 2 * q^37 + 16 * q^38 + 3 * q^39 + 9 * q^42 - 17 * q^44 + 26 * q^46 + 7 * q^47 - 5 * q^48 - 2 * q^49 + 7 * q^51 - 5 * q^52 - 9 * q^53 + q^54 - 9 * q^56 - 6 * q^57 + 11 * q^58 + 3 * q^59 - 5 * q^61 - 31 * q^62 - 5 * q^63 - 15 * q^64 + 7 * q^66 + 3 * q^67 - 5 * q^68 + 4 * q^69 - 18 * q^71 - 3 * q^72 - 26 * q^73 - 24 * q^74 + 22 * q^76 + 9 * q^77 - q^78 + 4 * q^79 + 3 * q^81 + 18 * q^82 - 13 * q^83 + 5 * q^84 - 10 * q^86 + 13 * q^87 + 9 * q^88 - 16 * q^89 + 5 * q^91 - 32 * q^92 - 7 * q^93 - 5 * q^94 + 17 * q^96 - 26 * q^97 + 40 * q^98 - q^99

Introduction

L-functions

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