LMFDB - Embedded newform 1525.2.a.i.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1525,2,Mod(1,1525)] 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("1525.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1525 = 5^{2} \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1525.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,2,0,12,0,0,-12] 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:	$12.1771863082$
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} - 11x^{5} + 19x^{4} + 35x^{3} - 48x^{2} - 25x + 27 $ x^7 - 2x^6 - 11x^5 + 19x^4 + 35x^3 - 48x^2 - 25x + 27
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 305)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$2.67064$ of defining polynomial
Character	$\chi$	$=$	1525.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.67064 q^{2} -2.35845 q^{3} +5.13230 q^{4} -6.29857 q^{6} -3.60231 q^{7} +8.36523 q^{8} +2.56229 q^{9} +4.69625 q^{11} -12.1043 q^{12} +2.13230 q^{13} -9.62047 q^{14} +12.0759 q^{16} -0.0400200 q^{17} +6.84295 q^{18} -2.17232 q^{19} +8.49588 q^{21} +12.5420 q^{22} +8.18353 q^{23} -19.7290 q^{24} +5.69459 q^{26} +1.03231 q^{27} -18.4881 q^{28} +6.58122 q^{29} +8.11165 q^{31} +15.5198 q^{32} -11.0759 q^{33} -0.106879 q^{34} +13.1505 q^{36} -8.80918 q^{37} -5.80147 q^{38} -5.02892 q^{39} -5.37224 q^{41} +22.6894 q^{42} +9.34814 q^{43} +24.1026 q^{44} +21.8552 q^{46} -5.34735 q^{47} -28.4804 q^{48} +5.97666 q^{49} +0.0943853 q^{51} +10.9436 q^{52} -6.80918 q^{53} +2.75692 q^{54} -30.1342 q^{56} +5.12331 q^{57} +17.5760 q^{58} -0.210636 q^{59} +1.00000 q^{61} +21.6633 q^{62} -9.23018 q^{63} +17.2961 q^{64} -29.5797 q^{66} +5.47696 q^{67} -0.205395 q^{68} -19.3005 q^{69} +3.66273 q^{71} +21.4342 q^{72} +14.2961 q^{73} -23.5261 q^{74} -11.1490 q^{76} -16.9174 q^{77} -13.4304 q^{78} -11.2832 q^{79} -10.1215 q^{81} -14.3473 q^{82} -7.45716 q^{83} +43.6034 q^{84} +24.9655 q^{86} -15.5215 q^{87} +39.2852 q^{88} +1.95666 q^{89} -7.68120 q^{91} +42.0003 q^{92} -19.1309 q^{93} -14.2808 q^{94} -36.6028 q^{96} -0.693197 q^{97} +15.9615 q^{98} +12.0332 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{2} + 12 q^{4} - 12 q^{7} + 9 q^{8} + 9 q^{9} + 2 q^{11} - 7 q^{12} - 9 q^{13} + 8 q^{14} + 14 q^{16} + 4 q^{17} + 3 q^{18} + 13 q^{19} + 2 q^{21} + 19 q^{22} + 5 q^{23} + 2 q^{24} + 7 q^{26}+ \cdots + 3 q^{99}+O(q^{100}) $ 7 * q + 2 * q^2 + 12 * q^4 - 12 * q^7 + 9 * q^8 + 9 * q^9 + 2 * q^11 - 7 * q^12 - 9 * q^13 + 8 * q^14 + 14 * q^16 + 4 * q^17 + 3 * q^18 + 13 * q^19 + 2 * q^21 + 19 * q^22 + 5 * q^23 + 2 * q^24 + 7 * q^26 + 9 * q^27 - 25 * q^28 + 7 * q^29 + 35 * q^31 + 35 * q^32 - 7 * q^33 + 13 * q^34 + 6 * q^36 - 9 * q^37 + 6 * q^38 - 7 * q^39 - 8 * q^41 + 51 * q^42 - 9 * q^43 - 3 * q^44 - 3 * q^46 + 3 * q^47 - 9 * q^48 + 29 * q^49 + 11 * q^51 + 30 * q^52 + 5 * q^53 - 18 * q^54 - 26 * q^56 + 18 * q^57 + 9 * q^58 + 6 * q^59 + 7 * q^61 + 38 * q^62 - 33 * q^63 + 17 * q^64 - 51 * q^66 + 11 * q^67 - 7 * q^68 - 21 * q^69 + 19 * q^71 + 68 * q^72 - 4 * q^73 - 24 * q^74 - 37 * q^76 + 3 * q^77 + 2 * q^78 + 3 * q^79 - 13 * q^81 + 32 * q^82 - 10 * q^83 + 46 * q^84 - 18 * q^86 - 19 * q^87 + 15 * q^88 + 14 * q^89 + 11 * q^91 + 22 * q^92 + 46 * q^93 - 7 * q^94 - 67 * q^96 - 20 * q^97 - 29 * q^98 + 3 * 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 $

$2$	2.67064	1.88842	0.944212	−	0.329337i	$-0.106825\pi$
$2$	2.67064	1.88842	0.944212	+	0.329337i	$0.106825\pi$
$3$	−2.35845	−1.36165	−0.680826	−	0.732445i	$-0.738379\pi$
$3$	−2.35845	−1.36165	−0.680826	+	0.732445i	$0.738379\pi$
$4$	5.13230	2.56615
$5$	0	0
$5$	0	0
$6$	−6.29857	−2.57138
$7$	−3.60231	−1.36155	−0.680773	−	0.732494i	$-0.738356\pi$
$7$	−3.60231	−1.36155	−0.680773	+	0.732494i	$0.738356\pi$
$8$	8.36523	2.95755
$9$	2.56229	0.854098
$10$	0	0
$11$	4.69625	1.41597	0.707987	−	0.706226i	$-0.249604\pi$
$11$	4.69625	1.41597	0.707987	+	0.706226i	$0.249604\pi$
$12$	−12.1043	−3.49420
$13$	2.13230	0.591393	0.295696	−	0.955282i	$-0.404448\pi$
$13$	2.13230	0.591393	0.295696	+	0.955282i	$0.404448\pi$
$14$	−9.62047	−2.57118
$15$	0	0
$16$	12.0759	3.01897
$17$	−0.0400200	−0.00970628	−0.00485314	−	0.999988i	$-0.501545\pi$
$17$	−0.0400200	−0.00970628	−0.00485314	+	0.999988i	$0.501545\pi$
$18$	6.84295	1.61290
$19$	−2.17232	−0.498364	−0.249182	−	0.968457i	$-0.580162\pi$
$19$	−2.17232	−0.498364	−0.249182	+	0.968457i	$0.580162\pi$
$20$	0	0
$21$	8.49588	1.85395
$22$	12.5420	2.67396
$23$	8.18353	1.70638	0.853192	−	0.521597i	$-0.174664\pi$
$23$	8.18353	1.70638	0.853192	+	0.521597i	$0.174664\pi$
$24$	−19.7290	−4.02716
$25$	0	0
$26$	5.69459	1.11680
$27$	1.03231	0.198668
$28$	−18.4881	−3.49393
$29$	6.58122	1.22210	0.611051	−	0.791592i	$-0.290747\pi$
$29$	6.58122	1.22210	0.611051	+	0.791592i	$0.290747\pi$
$30$	0	0
$31$	8.11165	1.45689	0.728447	−	0.685102i	$-0.240242\pi$
$31$	8.11165	1.45689	0.728447	+	0.685102i	$0.240242\pi$
$32$	15.5198	2.74355
$33$	−11.0759	−1.92806
$34$	−0.106879	−0.0183296
$35$	0	0
$36$	13.1505	2.19174
$37$	−8.80918	−1.44822	−0.724110	−	0.689684i	$-0.757749\pi$
$37$	−8.80918	−1.44822	−0.724110	+	0.689684i	$0.757749\pi$
$38$	−5.80147	−0.941123
$39$	−5.02892	−0.805272
$40$	0	0
$41$	−5.37224	−0.839003	−0.419502	−	0.907755i	$-0.637795\pi$
$41$	−5.37224	−0.839003	−0.419502	+	0.907755i	$0.637795\pi$
$42$	22.6894	3.50105
$43$	9.34814	1.42558	0.712789	−	0.701379i	$-0.247432\pi$
$43$	9.34814	1.42558	0.712789	+	0.701379i	$0.247432\pi$
$44$	24.1026	3.63360
$45$	0	0
$46$	21.8552	3.22238
$47$	−5.34735	−0.779991	−0.389996	−	0.920817i	$-0.627524\pi$
$47$	−5.34735	−0.779991	−0.389996	+	0.920817i	$0.627524\pi$
$48$	−28.4804	−4.11079
$49$	5.97666	0.853809
$50$	0	0
$51$	0.0943853	0.0132166
$52$	10.9436	1.51760
$53$	−6.80918	−0.935313	−0.467656	−	0.883910i	$-0.654902\pi$
$53$	−6.80918	−0.935313	−0.467656	+	0.883910i	$0.654902\pi$
$54$	2.75692	0.375170
$55$	0	0
$56$	−30.1342	−4.02685
$57$	5.12331	0.678598
$58$	17.5760	2.30785
$59$	−0.210636	−0.0274225	−0.0137112	−	0.999906i	$-0.504365\pi$
$59$	−0.210636	−0.0274225	−0.0137112	+	0.999906i	$0.504365\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	21.6633	2.75124
$63$	−9.23018	−1.16289
$64$	17.2961	2.16201
$65$	0	0
$66$	−29.5797	−3.64100
$67$	5.47696	0.669117	0.334559	−	0.942375i	$-0.391413\pi$
$67$	5.47696	0.669117	0.334559	+	0.942375i	$0.391413\pi$
$68$	−0.205395	−0.0249078
$69$	−19.3005	−2.32350
$70$	0	0
$71$	3.66273	0.434686	0.217343	−	0.976095i	$-0.430261\pi$
$71$	3.66273	0.434686	0.217343	+	0.976095i	$0.430261\pi$
$72$	21.4342	2.52604
$73$	14.2961	1.67323	0.836614	−	0.547792i	$-0.184532\pi$
$73$	14.2961	1.67323	0.836614	+	0.547792i	$0.184532\pi$
$74$	−23.5261	−2.73486
$75$	0	0
$76$	−11.1490	−1.27888
$77$	−16.9174	−1.92791
$78$	−13.4304	−1.52070
$79$	−11.2832	−1.26946	−0.634729	−	0.772734i	$-0.718888\pi$
$79$	−11.2832	−1.26946	−0.634729	+	0.772734i	$0.718888\pi$
$80$	0	0
$81$	−10.1215	−1.12461
$82$	−14.3473	−1.58439
$83$	−7.45716	−0.818529	−0.409265	−	0.912416i	$-0.634215\pi$
$83$	−7.45716	−0.818529	−0.409265	+	0.912416i	$0.634215\pi$
$84$	43.6034	4.75752
$85$	0	0
$86$	24.9655	2.69210
$87$	−15.5215	−1.66408
$88$	39.2852	4.18782
$89$	1.95666	0.207405	0.103703	−	0.994608i	$-0.466931\pi$
$89$	1.95666	0.207405	0.103703	+	0.994608i	$0.466931\pi$
$90$	0	0
$91$	−7.68120	−0.805209
$92$	42.0003	4.37884
$93$	−19.1309	−1.98378
$94$	−14.2808	−1.47296
$95$	0	0
$96$	−36.6028	−3.73576
$97$	−0.693197	−0.0703835	−0.0351917	−	0.999381i	$-0.511204\pi$
$97$	−0.693197	−0.0703835	−0.0351917	+	0.999381i	$0.511204\pi$
$98$	15.9615	1.61235
$99$	12.0332	1.20938
$100$	0	0
$101$	−11.3439	−1.12876	−0.564379	−	0.825516i	$-0.690884\pi$
$101$	−11.3439	−1.12876	−0.564379	+	0.825516i	$0.690884\pi$
$102$	0.252069	0.0249585
$103$	−2.86689	−0.282483	−0.141242	−	0.989975i	$-0.545109\pi$
$103$	−2.86689	−0.282483	−0.141242	+	0.989975i	$0.545109\pi$
$104$	17.8372	1.74908
$105$	0	0
$106$	−18.1848	−1.76627
$107$	10.7394	1.03822	0.519110	−	0.854708i	$-0.326264\pi$
$107$	10.7394	1.03822	0.519110	+	0.854708i	$0.326264\pi$
$108$	5.29812	0.509812
$109$	−7.91222	−0.757854	−0.378927	−	0.925427i	$-0.623707\pi$
$109$	−7.91222	−0.757854	−0.378927	+	0.925427i	$0.623707\pi$
$110$	0	0
$111$	20.7760	1.97197
$112$	−43.5011	−4.11047
$113$	7.47593	0.703277	0.351638	−	0.936136i	$-0.385625\pi$
$113$	7.47593	0.703277	0.351638	+	0.936136i	$0.385625\pi$
$114$	13.6825	1.28148
$115$	0	0
$116$	33.7768	3.13609
$117$	5.46357	0.505107
$118$	−0.562532	−0.0517853
$119$	0.144165	0.0132156
$120$	0	0
$121$	11.0548	1.00498
$122$	2.67064	0.241788
$123$	12.6702	1.14243
$124$	41.6314	3.73861
$125$	0	0
$126$	−24.6505	−2.19604
$127$	3.74069	0.331933	0.165966	−	0.986131i	$-0.446926\pi$
$127$	3.74069	0.331933	0.165966	+	0.986131i	$0.446926\pi$
$128$	15.1518	1.33925
$129$	−22.0471	−1.94114
$130$	0	0
$131$	−14.6287	−1.27811	−0.639057	−	0.769160i	$-0.720675\pi$
$131$	−14.6287	−1.27811	−0.639057	+	0.769160i	$0.720675\pi$
$132$	−56.8447	−4.94770
$133$	7.82537	0.678546
$134$	14.6270	1.26358
$135$	0	0
$136$	−0.334777	−0.0287069
$137$	−14.8807	−1.27134	−0.635671	−	0.771960i	$-0.719276\pi$
$137$	−14.8807	−1.27134	−0.635671	+	0.771960i	$0.719276\pi$
$138$	−51.5445	−4.38776
$139$	5.54198	0.470064	0.235032	−	0.971988i	$-0.424480\pi$
$139$	5.54198	0.470064	0.235032	+	0.971988i	$0.424480\pi$
$140$	0	0
$141$	12.6115	1.06208
$142$	9.78181	0.820872
$143$	10.0138	0.837397
$144$	30.9420	2.57850
$145$	0	0
$146$	38.1796	3.15977
$147$	−14.0957	−1.16259
$148$	−45.2113	−3.71635
$149$	−23.0107	−1.88511	−0.942556	−	0.334047i	$-0.891586\pi$
$149$	−23.0107	−1.88511	−0.942556	+	0.334047i	$0.891586\pi$
$150$	0	0
$151$	1.57505	0.128176	0.0640880	−	0.997944i	$-0.479586\pi$
$151$	1.57505	0.128176	0.0640880	+	0.997944i	$0.479586\pi$
$152$	−18.1719	−1.47394
$153$	−0.102543	−0.00829011
$154$	−45.1802	−3.64072
$155$	0	0
$156$	−25.8099	−2.06645
$157$	5.30596	0.423461	0.211731	−	0.977328i	$-0.432090\pi$
$157$	5.30596	0.423461	0.211731	+	0.977328i	$0.432090\pi$
$158$	−30.1333	−2.39728
$159$	16.0591	1.27357
$160$	0	0
$161$	−29.4796	−2.32332
$162$	−27.0309	−2.12375
$163$	16.4977	1.29220	0.646099	−	0.763253i	$-0.276399\pi$
$163$	16.4977	1.29220	0.646099	+	0.763253i	$0.276399\pi$
$164$	−27.5719	−2.15301
$165$	0	0
$166$	−19.9154	−1.54573
$167$	−0.190738	−0.0147597	−0.00737986	−	0.999973i	$-0.502349\pi$
$167$	−0.190738	−0.0147597	−0.00737986	+	0.999973i	$0.502349\pi$
$168$	71.0700	5.48317
$169$	−8.45331	−0.650254
$170$	0	0
$171$	−5.56612	−0.425651
$172$	47.9774	3.65824
$173$	−21.7789	−1.65582	−0.827911	−	0.560859i	$-0.810471\pi$
$173$	−21.7789	−1.65582	−0.827911	+	0.560859i	$0.810471\pi$
$174$	−41.4522	−3.14249
$175$	0	0
$176$	56.7114	4.27478
$177$	0.496775	0.0373399
$178$	5.22552	0.391669
$179$	7.42226	0.554766	0.277383	−	0.960759i	$-0.410533\pi$
$179$	7.42226	0.554766	0.277383	+	0.960759i	$0.410533\pi$
$180$	0	0
$181$	−1.87336	−0.139246	−0.0696230	−	0.997573i	$-0.522180\pi$
$181$	−1.87336	−0.139246	−0.0696230	+	0.997573i	$0.522180\pi$
$182$	−20.5137	−1.52058
$183$	−2.35845	−0.174342
$184$	68.4571	5.04672
$185$	0	0
$186$	−51.0917	−3.74623
$187$	−0.187944	−0.0137438
$188$	−27.4442	−2.00157
$189$	−3.71870	−0.270496
$190$	0	0
$191$	−2.14383	−0.155122	−0.0775611	−	0.996988i	$-0.524713\pi$
$191$	−2.14383	−0.155122	−0.0775611	+	0.996988i	$0.524713\pi$
$192$	−40.7919	−2.94390
$193$	−16.8670	−1.21411	−0.607057	−	0.794658i	$-0.707650\pi$
$193$	−16.8670	−1.21411	−0.607057	+	0.794658i	$0.707650\pi$
$194$	−1.85128	−0.132914
$195$	0	0
$196$	30.6740	2.19100
$197$	8.87169	0.632082	0.316041	−	0.948746i	$-0.397646\pi$
$197$	8.87169	0.632082	0.316041	+	0.948746i	$0.397646\pi$
$198$	32.1362	2.28382
$199$	4.90123	0.347439	0.173719	−	0.984795i	$-0.444421\pi$
$199$	4.90123	0.347439	0.173719	+	0.984795i	$0.444421\pi$
$200$	0	0
$201$	−12.9171	−0.911105
$202$	−30.2953	−2.13157
$203$	−23.7076	−1.66395
$204$	0.484413	0.0339157
$205$	0	0
$206$	−7.65642	−0.533448
$207$	20.9686	1.45742
$208$	25.7494	1.78540
$209$	−10.2018	−0.705670
$210$	0	0
$211$	−21.0736	−1.45076	−0.725382	−	0.688346i	$-0.758337\pi$
$211$	−21.0736	−1.45076	−0.725382	+	0.688346i	$0.758337\pi$
$212$	−34.9467	−2.40015
$213$	−8.63837	−0.591891
$214$	28.6811	1.96060
$215$	0	0
$216$	8.63550	0.587572
$217$	−29.2207	−1.98363
$218$	−21.1307	−1.43115
$219$	−33.7166	−2.27836
$220$	0	0
$221$	−0.0853346	−0.00574023
$222$	55.4852	3.72392
$223$	0.164581	0.0110212	0.00551058	−	0.999985i	$-0.498246\pi$
$223$	0.164581	0.0110212	0.00551058	+	0.999985i	$0.498246\pi$
$224$	−55.9073	−3.73546
$225$	0	0
$226$	19.9655	1.32809
$227$	10.3941	0.689881	0.344940	−	0.938625i	$-0.387899\pi$
$227$	10.3941	0.689881	0.344940	+	0.938625i	$0.387899\pi$
$228$	26.2943	1.74138
$229$	1.82439	0.120559	0.0602796	−	0.998182i	$-0.480801\pi$
$229$	1.82439	0.120559	0.0602796	+	0.998182i	$0.480801\pi$
$230$	0	0
$231$	39.8988	2.62515
$232$	55.0534	3.61443
$233$	9.14869	0.599351	0.299675	−	0.954041i	$-0.403122\pi$
$233$	9.14869	0.599351	0.299675	+	0.954041i	$0.403122\pi$
$234$	14.5912	0.953858
$235$	0	0
$236$	−1.08105	−0.0703701
$237$	26.6109	1.72856
$238$	0.385011	0.0249566
$239$	20.3597	1.31696	0.658481	−	0.752598i	$-0.271199\pi$
$239$	20.3597	1.31696	0.658481	+	0.752598i	$0.271199\pi$
$240$	0	0
$241$	26.2987	1.69405	0.847025	−	0.531553i	$-0.178392\pi$
$241$	26.2987	1.69405	0.847025	+	0.531553i	$0.178392\pi$
$242$	29.5233	1.89783
$243$	20.7742	1.33267
$244$	5.13230	0.328562
$245$	0	0
$246$	33.8374	2.15740
$247$	−4.63203	−0.294729
$248$	67.8558	4.30885
$249$	17.5873	1.11455
$250$	0	0
$251$	−0.165873	−0.0104698	−0.00523491	−	0.999986i	$-0.501666\pi$
$251$	−0.165873	−0.0104698	−0.00523491	+	0.999986i	$0.501666\pi$
$252$	−47.3721	−2.98416
$253$	38.4319	2.41619
$254$	9.99002	0.626830
$255$	0	0
$256$	5.87291	0.367057
$257$	−9.85442	−0.614702	−0.307351	−	0.951596i	$-0.599442\pi$
$257$	−9.85442	−0.614702	−0.307351	+	0.951596i	$0.599442\pi$
$258$	−58.8799	−3.66570
$259$	31.7334	1.97182
$260$	0	0
$261$	16.8630	1.04379
$262$	−39.0679	−2.41362
$263$	−6.09975	−0.376127	−0.188063	−	0.982157i	$-0.560221\pi$
$263$	−6.09975	−0.376127	−0.188063	+	0.982157i	$0.560221\pi$
$264$	−92.6523	−5.70235
$265$	0	0
$266$	20.8987	1.28138
$267$	−4.61468	−0.282414
$268$	28.1094	1.71705
$269$	15.5474	0.947942	0.473971	−	0.880540i	$-0.342820\pi$
$269$	15.5474	0.947942	0.473971	+	0.880540i	$0.342820\pi$
$270$	0	0
$271$	17.5524	1.06624	0.533118	−	0.846041i	$-0.321020\pi$
$271$	17.5524	1.06624	0.533118	+	0.846041i	$0.321020\pi$
$272$	−0.483277	−0.0293030
$273$	18.1157	1.09641
$274$	−39.7409	−2.40083
$275$	0	0
$276$	−99.0557	−5.96245
$277$	−1.94260	−0.116719	−0.0583597	−	0.998296i	$-0.518587\pi$
$277$	−1.94260	−0.116719	−0.0583597	+	0.998296i	$0.518587\pi$
$278$	14.8006	0.887681
$279$	20.7844	1.24433
$280$	0	0
$281$	2.83208	0.168948	0.0844739	−	0.996426i	$-0.473079\pi$
$281$	2.83208	0.168948	0.0844739	+	0.996426i	$0.473079\pi$
$282$	33.6806	2.00565
$283$	−27.9750	−1.66294	−0.831470	−	0.555569i	$-0.812500\pi$
$283$	−27.9750	−1.66294	−0.831470	+	0.555569i	$0.812500\pi$
$284$	18.7982	1.11547
$285$	0	0
$286$	26.7432	1.58136
$287$	19.3525	1.14234
$288$	39.7664	2.34326
$289$	−16.9984	−0.999906
$290$	0	0
$291$	1.63487	0.0958379
$292$	73.3717	4.29375
$293$	−21.9307	−1.28120	−0.640602	−	0.767873i	$-0.721315\pi$
$293$	−21.9307	−1.28120	−0.640602	+	0.767873i	$0.721315\pi$
$294$	−37.6444	−2.19547
$295$	0	0
$296$	−73.6908	−4.28319
$297$	4.84799	0.281309
$298$	−61.4533	−3.55989
$299$	17.4497	1.00914
$300$	0	0
$301$	−33.6749	−1.94099
$302$	4.20640	0.242051
$303$	26.7540	1.53697
$304$	−26.2327	−1.50455
$305$	0	0
$306$	−0.273855	−0.0156553
$307$	−20.1858	−1.15206	−0.576032	−	0.817427i	$-0.695400\pi$
$307$	−20.1858	−1.15206	−0.576032	+	0.817427i	$0.695400\pi$
$308$	−86.8250	−4.94731
$309$	6.76142	0.384644
$310$	0	0
$311$	2.15046	0.121942	0.0609708	−	0.998140i	$-0.480580\pi$
$311$	2.15046	0.121942	0.0609708	+	0.998140i	$0.480580\pi$
$312$	−42.0681	−2.38164
$313$	−9.22585	−0.521476	−0.260738	−	0.965410i	$-0.583966\pi$
$313$	−9.22585	−0.521476	−0.260738	+	0.965410i	$0.583966\pi$
$314$	14.1703	0.799675
$315$	0	0
$316$	−57.9087	−3.25762
$317$	20.2416	1.13688	0.568440	−	0.822725i	$-0.307547\pi$
$317$	20.2416	1.13688	0.568440	+	0.822725i	$0.307547\pi$
$318$	42.8881	2.40504
$319$	30.9071	1.73046
$320$	0	0
$321$	−25.3284	−1.41369
$322$	−78.7294	−4.38742
$323$	0.0869362	0.00483726
$324$	−51.9467	−2.88593
$325$	0	0
$326$	44.0593	2.44022
$327$	18.6606	1.03193
$328$	−44.9400	−2.48140
$329$	19.2628	1.06199
$330$	0	0
$331$	5.06024	0.278136	0.139068	−	0.990283i	$-0.455589\pi$
$331$	5.06024	0.278136	0.139068	+	0.990283i	$0.455589\pi$
$332$	−38.2724	−2.10047
$333$	−22.5717	−1.23692
$334$	−0.509391	−0.0278726
$335$	0	0
$336$	102.595	5.59703
$337$	9.70531	0.528682	0.264341	−	0.964429i	$-0.414846\pi$
$337$	9.70531	0.528682	0.264341	+	0.964429i	$0.414846\pi$
$338$	−22.5757	−1.22796
$339$	−17.6316	−0.957619
$340$	0	0
$341$	38.0943	2.06292
$342$	−14.8651	−0.803811
$343$	3.68638	0.199046
$344$	78.1993	4.21622
$345$	0	0
$346$	−58.1636	−3.12690
$347$	−14.6568	−0.786817	−0.393409	−	0.919364i	$-0.628704\pi$
$347$	−14.6568	−0.786817	−0.393409	+	0.919364i	$0.628704\pi$
$348$	−79.6609	−4.27027
$349$	−24.2084	−1.29585	−0.647923	−	0.761706i	$-0.724362\pi$
$349$	−24.2084	−1.29585	−0.647923	+	0.761706i	$0.724362\pi$
$350$	0	0
$351$	2.20119	0.117491
$352$	72.8851	3.88479
$353$	23.6827	1.26050	0.630252	−	0.776391i	$-0.282952\pi$
$353$	23.6827	1.26050	0.630252	+	0.776391i	$0.282952\pi$
$354$	1.32670	0.0705136
$355$	0	0
$356$	10.0421	0.532233
$357$	−0.340005	−0.0179950
$358$	19.8222	1.04763
$359$	−31.7548	−1.67595	−0.837976	−	0.545706i	$-0.816261\pi$
$359$	−31.7548	−1.67595	−0.837976	+	0.545706i	$0.816261\pi$
$360$	0	0
$361$	−14.2810	−0.751633
$362$	−5.00307	−0.262956
$363$	−26.0722	−1.36843
$364$	−39.4222	−2.06629
$365$	0	0
$366$	−6.29857	−0.329231
$367$	−14.8056	−0.772844	−0.386422	−	0.922322i	$-0.626289\pi$
$367$	−14.8056	−0.772844	−0.386422	+	0.922322i	$0.626289\pi$
$368$	98.8234	5.15152
$369$	−13.7653	−0.716591
$370$	0	0
$371$	24.5288	1.27347
$372$	−98.1856	−5.09069
$373$	14.7297	0.762675	0.381338	−	0.924436i	$-0.375464\pi$
$373$	14.7297	0.762675	0.381338	+	0.924436i	$0.375464\pi$
$374$	−0.501930	−0.0259542
$375$	0	0
$376$	−44.7318	−2.30687
$377$	14.0331	0.722742
$378$	−9.93130	−0.510811
$379$	−16.6931	−0.857466	−0.428733	−	0.903431i	$-0.641040\pi$
$379$	−16.6931	−0.857466	−0.428733	+	0.903431i	$0.641040\pi$
$380$	0	0
$381$	−8.82224	−0.451977
$382$	−5.72539	−0.292937
$383$	−9.01764	−0.460780	−0.230390	−	0.973098i	$-0.574000\pi$
$383$	−9.01764	−0.460780	−0.230390	+	0.973098i	$0.574000\pi$
$384$	−35.7349	−1.82359
$385$	0	0
$386$	−45.0457	−2.29276
$387$	23.9527	1.21758
$388$	−3.55769	−0.180615
$389$	−22.1323	−1.12215	−0.561075	−	0.827765i	$-0.689612\pi$
$389$	−22.1323	−1.12215	−0.561075	+	0.827765i	$0.689612\pi$
$390$	0	0
$391$	−0.327505	−0.0165626
$392$	49.9961	2.52519
$393$	34.5010	1.74035
$394$	23.6931	1.19364
$395$	0	0
$396$	61.7578	3.10345
$397$	35.1649	1.76487	0.882437	−	0.470431i	$-0.155901\pi$
$397$	35.1649	1.76487	0.882437	+	0.470431i	$0.155901\pi$
$398$	13.0894	0.656112
$399$	−18.4558	−0.923943
$400$	0	0
$401$	9.16212	0.457535	0.228767	−	0.973481i	$-0.426531\pi$
$401$	9.16212	0.457535	0.228767	+	0.973481i	$0.426531\pi$
$402$	−34.4970	−1.72055
$403$	17.2964	0.861597
$404$	−58.2201	−2.89656
$405$	0	0
$406$	−63.3144	−3.14224
$407$	−41.3701	−2.05064
$408$	0.789554	0.0390888
$409$	−7.31028	−0.361470	−0.180735	−	0.983532i	$-0.557848\pi$
$409$	−7.31028	−0.361470	−0.180735	+	0.983532i	$0.557848\pi$
$410$	0	0
$411$	35.0954	1.73113
$412$	−14.7137	−0.724894
$413$	0.758777	0.0373370
$414$	55.9995	2.75223
$415$	0	0
$416$	33.0929	1.62251
$417$	−13.0705	−0.640064
$418$	−27.2452	−1.33260
$419$	−5.05466	−0.246936	−0.123468	−	0.992349i	$-0.539402\pi$
$419$	−5.05466	−0.246936	−0.123468	+	0.992349i	$0.539402\pi$
$420$	0	0
$421$	21.3068	1.03843	0.519216	−	0.854643i	$-0.326224\pi$
$421$	21.3068	1.03843	0.519216	+	0.854643i	$0.326224\pi$
$422$	−56.2799	−2.73966
$423$	−13.7015	−0.666189
$424$	−56.9603	−2.76624
$425$	0	0
$426$	−23.0699	−1.11774
$427$	−3.60231	−0.174328
$428$	55.1179	2.66423
$429$	−23.6171	−1.14024
$430$	0	0
$431$	7.70850	0.371305	0.185653	−	0.982615i	$-0.440560\pi$
$431$	7.70850	0.371305	0.185653	+	0.982615i	$0.440560\pi$
$432$	12.4660	0.599773
$433$	−17.5703	−0.844377	−0.422189	−	0.906508i	$-0.638738\pi$
$433$	−17.5703	−0.844377	−0.422189	+	0.906508i	$0.638738\pi$
$434$	−78.0378	−3.74594
$435$	0	0
$436$	−40.6079	−1.94476
$437$	−17.7772	−0.850400
$438$	−90.0447	−4.30250
$439$	16.7134	0.797687	0.398843	−	0.917019i	$-0.369412\pi$
$439$	16.7134	0.797687	0.398843	+	0.917019i	$0.369412\pi$
$440$	0	0
$441$	15.3140	0.729236
$442$	−0.227898	−0.0108400
$443$	−9.51093	−0.451878	−0.225939	−	0.974141i	$-0.572545\pi$
$443$	−9.51093	−0.451878	−0.225939	+	0.974141i	$0.572545\pi$
$444$	106.629	5.06038
$445$	0	0
$446$	0.439536	0.0208126
$447$	54.2697	2.56687
$448$	−62.3059	−2.94368
$449$	−8.77862	−0.414289	−0.207144	−	0.978310i	$-0.566417\pi$
$449$	−8.77862	−0.414289	−0.207144	+	0.978310i	$0.566417\pi$
$450$	0	0
$451$	−25.2294	−1.18801
$452$	38.3687	1.80471
$453$	−3.71469	−0.174531
$454$	27.7589	1.30279
$455$	0	0
$456$	42.8576	2.00699
$457$	−13.5167	−0.632283	−0.316141	−	0.948712i	$-0.602387\pi$
$457$	−13.5167	−0.632283	−0.316141	+	0.948712i	$0.602387\pi$
$458$	4.87229	0.227667
$459$	−0.0413130	−0.00192833
$460$	0	0
$461$	41.2365	1.92058	0.960288	−	0.279011i	$-0.0900066\pi$
$461$	41.2365	1.92058	0.960288	+	0.279011i	$0.0900066\pi$
$462$	106.555	4.95740
$463$	−13.3467	−0.620272	−0.310136	−	0.950692i	$-0.600375\pi$
$463$	−13.3467	−0.620272	−0.310136	+	0.950692i	$0.600375\pi$
$464$	79.4740	3.68949
$465$	0	0
$466$	24.4328	1.13183
$467$	21.9550	1.01595	0.507977	−	0.861370i	$-0.330393\pi$
$467$	21.9550	1.01595	0.507977	+	0.861370i	$0.330393\pi$
$468$	28.0407	1.29618
$469$	−19.7297	−0.911034
$470$	0	0
$471$	−12.5138	−0.576607
$472$	−1.76202	−0.0811035
$473$	43.9012	2.01858
$474$	71.0680	3.26426
$475$	0	0
$476$	0.739896	0.0339131
$477$	−17.4471	−0.798849
$478$	54.3734	2.48698
$479$	−17.5360	−0.801240	−0.400620	−	0.916244i	$-0.631205\pi$
$479$	−17.5360	−0.801240	−0.400620	+	0.916244i	$0.631205\pi$
$480$	0	0
$481$	−18.7838	−0.856467
$482$	70.2343	3.19909
$483$	69.5263	3.16356
$484$	56.7365	2.57893
$485$	0	0
$486$	55.4804	2.51664
$487$	−22.9570	−1.04028	−0.520141	−	0.854080i	$-0.674121\pi$
$487$	−22.9570	−1.04028	−0.520141	+	0.854080i	$0.674121\pi$
$488$	8.36523	0.378676
$489$	−38.9090	−1.75953
$490$	0	0
$491$	21.9541	0.990774	0.495387	−	0.868672i	$-0.335026\pi$
$491$	21.9541	0.990774	0.495387	+	0.868672i	$0.335026\pi$
$492$	65.0271	2.93165
$493$	−0.263380	−0.0118621
$494$	−12.3705	−0.556573
$495$	0	0
$496$	97.9553	4.39832
$497$	−13.1943	−0.591845
$498$	46.9694	2.10475
$499$	25.5277	1.14278	0.571388	−	0.820680i	$-0.306405\pi$
$499$	25.5277	1.14278	0.571388	+	0.820680i	$0.306405\pi$
$500$	0	0
$501$	0.449845	0.0200976
$502$	−0.442987	−0.0197715
$503$	24.1245	1.07566	0.537829	−	0.843054i	$-0.319244\pi$
$503$	24.1245	1.07566	0.537829	+	0.843054i	$0.319244\pi$
$504$	−77.2126	−3.43932
$505$	0	0
$506$	102.638	4.56280
$507$	19.9367	0.885421
$508$	19.1983	0.851789
$509$	−0.677238	−0.0300181	−0.0150090	−	0.999887i	$-0.504778\pi$
$509$	−0.677238	−0.0300181	−0.0150090	+	0.999887i	$0.504778\pi$
$510$	0	0
$511$	−51.4989	−2.27818
$512$	−14.6193	−0.646087
$513$	−2.24250	−0.0990090
$514$	−26.3176	−1.16082
$515$	0	0
$516$	−113.152	−4.98126
$517$	−25.1125	−1.10445
$518$	84.7484	3.72363
$519$	51.3646	2.25466
$520$	0	0
$521$	−2.76130	−0.120975	−0.0604875	−	0.998169i	$-0.519266\pi$
$521$	−2.76130	−0.120975	−0.0604875	+	0.998169i	$0.519266\pi$
$522$	45.0350	1.97113
$523$	−32.0377	−1.40091	−0.700455	−	0.713697i	$-0.747019\pi$
$523$	−32.0377	−1.40091	−0.700455	+	0.713697i	$0.747019\pi$
$524$	−75.0787	−3.27983
$525$	0	0
$526$	−16.2902	−0.710287
$527$	−0.324628	−0.0141410
$528$	−133.751	−5.82077
$529$	43.9702	1.91175
$530$	0	0
$531$	−0.539711	−0.0234215
$532$	40.1621	1.74125
$533$	−11.4552	−0.496181
$534$	−12.3241	−0.533318
$535$	0	0
$536$	45.8160	1.97895
$537$	−17.5050	−0.755398
$538$	41.5215	1.79012
$539$	28.0679	1.20897
$540$	0	0
$541$	33.4603	1.43857	0.719285	−	0.694716i	$-0.244470\pi$
$541$	33.4603	1.43857	0.719285	+	0.694716i	$0.244470\pi$
$542$	46.8762	2.01351
$543$	4.41824	0.189605
$544$	−0.621104	−0.0266296
$545$	0	0
$546$	48.3806	2.07050
$547$	−0.913346	−0.0390519	−0.0195259	−	0.999809i	$-0.506216\pi$
$547$	−0.913346	−0.0390519	−0.0195259	+	0.999809i	$0.506216\pi$
$548$	−76.3721	−3.26245
$549$	2.56229	0.109356
$550$	0	0
$551$	−14.2965	−0.609051
$552$	−161.453	−6.87188
$553$	40.6456	1.72843
$554$	−5.18797	−0.220416
$555$	0	0
$556$	28.4431	1.20626
$557$	−29.0765	−1.23201	−0.616006	−	0.787742i	$-0.711250\pi$
$557$	−29.0765	−1.23201	−0.616006	+	0.787742i	$0.711250\pi$
$558$	55.5076	2.34983
$559$	19.9330	0.843077
$560$	0	0
$561$	0.443257	0.0187143
$562$	7.56346	0.319045
$563$	42.5432	1.79298	0.896491	−	0.443062i	$-0.146108\pi$
$563$	42.5432	1.79298	0.896491	+	0.443062i	$0.146108\pi$
$564$	64.7258	2.72545
$565$	0	0
$566$	−74.7110	−3.14034
$567$	36.4609	1.53122
$568$	30.6396	1.28561
$569$	−11.5659	−0.484869	−0.242435	−	0.970168i	$-0.577946\pi$
$569$	−11.5659	−0.484869	−0.242435	+	0.970168i	$0.577946\pi$
$570$	0	0
$571$	−0.912328	−0.0381797	−0.0190899	−	0.999818i	$-0.506077\pi$
$571$	−0.912328	−0.0381797	−0.0190899	+	0.999818i	$0.506077\pi$
$572$	51.3938	2.14888
$573$	5.05612	0.211223
$574$	51.6835	2.15723
$575$	0	0
$576$	44.3176	1.84657
$577$	−24.7617	−1.03084	−0.515421	−	0.856937i	$-0.672365\pi$
$577$	−24.7617	−1.03084	−0.515421	+	0.856937i	$0.672365\pi$
$578$	−45.3965	−1.88825
$579$	39.7801	1.65320
$580$	0	0
$581$	26.8630	1.11447
$582$	4.36615	0.180983
$583$	−31.9776	−1.32438
$584$	119.590	4.94866
$585$	0	0
$586$	−58.5688	−2.41946
$587$	14.2006	0.586121	0.293060	−	0.956094i	$-0.405326\pi$
$587$	14.2006	0.586121	0.293060	+	0.956094i	$0.405326\pi$
$588$	−72.3432	−2.98338
$589$	−17.6211	−0.726064
$590$	0	0
$591$	−20.9235	−0.860676
$592$	−106.379	−4.37214
$593$	−12.3442	−0.506917	−0.253459	−	0.967346i	$-0.581568\pi$
$593$	−12.3442	−0.506917	−0.253459	+	0.967346i	$0.581568\pi$
$594$	12.9472	0.531230
$595$	0	0
$596$	−118.098	−4.83748
$597$	−11.5593	−0.473091
$598$	46.6019	1.90569
$599$	40.6565	1.66118	0.830589	−	0.556885i	$-0.188004\pi$
$599$	40.6565	1.66118	0.830589	+	0.556885i	$0.188004\pi$
$600$	0	0
$601$	10.1548	0.414221	0.207111	−	0.978318i	$-0.433594\pi$
$601$	10.1548	0.414221	0.207111	+	0.978318i	$0.433594\pi$
$602$	−89.9335	−3.66541
$603$	14.0336	0.571491
$604$	8.08365	0.328919
$605$	0	0
$606$	71.4501	2.90246
$607$	−35.5013	−1.44095	−0.720476	−	0.693480i	$-0.756077\pi$
$607$	−35.5013	−1.44095	−0.720476	+	0.693480i	$0.756077\pi$
$608$	−33.7140	−1.36728
$609$	55.9132	2.26572
$610$	0	0
$611$	−11.4021	−0.461281
$612$	−0.526281	−0.0212737
$613$	37.3715	1.50942	0.754711	−	0.656057i	$-0.227777\pi$
$613$	37.3715	1.50942	0.754711	+	0.656057i	$0.227777\pi$
$614$	−53.9089	−2.17559
$615$	0	0
$616$	−141.518	−5.70191
$617$	−21.8195	−0.878422	−0.439211	−	0.898384i	$-0.644742\pi$
$617$	−21.8195	−0.878422	−0.439211	+	0.898384i	$0.644742\pi$
$618$	18.0573	0.726371
$619$	37.8738	1.52228	0.761138	−	0.648590i	$-0.224641\pi$
$619$	37.8738	1.52228	0.761138	+	0.648590i	$0.224641\pi$
$620$	0	0
$621$	8.44794	0.339004
$622$	5.74311	0.230278
$623$	−7.04849	−0.282392
$624$	−60.7287	−2.43109
$625$	0	0
$626$	−24.6389	−0.984768
$627$	24.0603	0.960877
$628$	27.2317	1.08666
$629$	0.352544	0.0140568
$630$	0	0
$631$	−19.7630	−0.786753	−0.393377	−	0.919377i	$-0.628693\pi$
$631$	−19.7630	−0.786753	−0.393377	+	0.919377i	$0.628693\pi$
$632$	−94.3865	−3.75449
$633$	49.7010	1.97544
$634$	54.0579	2.14691
$635$	0	0
$636$	82.4202	3.26817
$637$	12.7440	0.504937
$638$	82.5415	3.26785
$639$	9.38498	0.371264
$640$	0	0
$641$	−4.46577	−0.176387	−0.0881937	−	0.996103i	$-0.528109\pi$
$641$	−4.46577	−0.176387	−0.0881937	+	0.996103i	$0.528109\pi$
$642$	−67.6430	−2.66966
$643$	−26.4683	−1.04381	−0.521903	−	0.853005i	$-0.674778\pi$
$643$	−26.4683	−1.04381	−0.521903	+	0.853005i	$0.674778\pi$
$644$	−151.298	−5.96199
$645$	0	0
$646$	0.232175	0.00913480
$647$	6.65556	0.261657	0.130829	−	0.991405i	$-0.458236\pi$
$647$	6.65556	0.261657	0.130829	+	0.991405i	$0.458236\pi$
$648$	−84.6689	−3.32611
$649$	−0.989200	−0.0388295
$650$	0	0
$651$	68.9156	2.70102
$652$	84.6711	3.31597
$653$	−30.8497	−1.20724	−0.603620	−	0.797272i	$-0.706276\pi$
$653$	−30.8497	−1.20724	−0.603620	+	0.797272i	$0.706276\pi$
$654$	49.8357	1.94873
$655$	0	0
$656$	−64.8746	−2.53293
$657$	36.6307	1.42910
$658$	51.4440	2.00550
$659$	6.90900	0.269136	0.134568	−	0.990904i	$-0.457035\pi$
$659$	6.90900	0.269136	0.134568	+	0.990904i	$0.457035\pi$
$660$	0	0
$661$	−13.9468	−0.542467	−0.271234	−	0.962514i	$-0.587432\pi$
$661$	−13.9468	−0.542467	−0.271234	+	0.962514i	$0.587432\pi$
$662$	13.5140	0.525238
$663$	0.201258	0.00781619
$664$	−62.3808	−2.42085
$665$	0	0
$666$	−60.2808	−2.33583
$667$	53.8576	2.08537
$668$	−0.978922	−0.0378756
$669$	−0.388156	−0.0150070
$670$	0	0
$671$	4.69625	0.181297
$672$	131.855	5.08641
$673$	−17.5096	−0.674946	−0.337473	−	0.941335i	$-0.609572\pi$
$673$	−17.5096	−0.674946	−0.337473	+	0.941335i	$0.609572\pi$
$674$	25.9193	0.998376
$675$	0	0
$676$	−43.3849	−1.66865
$677$	−4.95773	−0.190541	−0.0952706	−	0.995451i	$-0.530372\pi$
$677$	−4.95773	−0.190541	−0.0952706	+	0.995451i	$0.530372\pi$
$678$	−47.0877	−1.80839
$679$	2.49711	0.0958304
$680$	0	0
$681$	−24.5140	−0.939378
$682$	101.736	3.89568
$683$	35.7925	1.36956	0.684782	−	0.728748i	$-0.259898\pi$
$683$	35.7925	1.36956	0.684782	+	0.728748i	$0.259898\pi$
$684$	−28.5670	−1.09229
$685$	0	0
$686$	9.84498	0.375883
$687$	−4.30274	−0.164160
$688$	112.887	4.30378
$689$	−14.5192	−0.553137
$690$	0	0
$691$	−19.2569	−0.732566	−0.366283	−	0.930503i	$-0.619370\pi$
$691$	−19.2569	−0.732566	−0.366283	+	0.930503i	$0.619370\pi$
$692$	−111.776	−4.24909
$693$	−43.3473	−1.64663
$694$	−39.1429	−1.48585
$695$	0	0
$696$	−129.841	−4.92160
$697$	0.214997	0.00814360
$698$	−64.6518	−2.44711
$699$	−21.5767	−0.816107
$700$	0	0
$701$	−20.7471	−0.783607	−0.391804	−	0.920049i	$-0.628149\pi$
$701$	−20.7471	−0.783607	−0.391804	+	0.920049i	$0.628149\pi$
$702$	5.87858	0.221873
$703$	19.1363	0.721741
$704$	81.2267	3.06135
$705$	0	0
$706$	63.2479	2.38037
$707$	40.8642	1.53686
$708$	2.54960	0.0958197
$709$	−0.283005	−0.0106285	−0.00531424	−	0.999986i	$-0.501692\pi$
$709$	−0.283005	−0.0106285	−0.00531424	+	0.999986i	$0.501692\pi$
$710$	0	0
$711$	−28.9109	−1.08424
$712$	16.3679	0.613412
$713$	66.3819	2.48602
$714$	−0.908031	−0.0339822
$715$	0	0
$716$	38.0932	1.42361
$717$	−48.0174	−1.79324
$718$	−84.8054	−3.16491
$719$	8.40107	0.313307	0.156653	−	0.987654i	$-0.449929\pi$
$719$	8.40107	0.313307	0.156653	+	0.987654i	$0.449929\pi$
$720$	0	0
$721$	10.3274	0.384614
$722$	−38.1395	−1.41940
$723$	−62.0243	−2.30671
$724$	−9.61466	−0.357326
$725$	0	0
$726$	−69.6293	−2.58419
$727$	3.60256	0.133612	0.0668059	−	0.997766i	$-0.478719\pi$
$727$	3.60256	0.133612	0.0668059	+	0.997766i	$0.478719\pi$
$728$	−64.2550	−2.38145
$729$	−18.6304	−0.690014
$730$	0	0
$731$	−0.374113	−0.0138371
$732$	−12.1043	−0.447387
$733$	−11.7530	−0.434106	−0.217053	−	0.976160i	$-0.569644\pi$
$733$	−11.7530	−0.434106	−0.217053	+	0.976160i	$0.569644\pi$
$734$	−39.5403	−1.45946
$735$	0	0
$736$	127.007	4.68154
$737$	25.7212	0.947452
$738$	−36.7620	−1.35323
$739$	7.52783	0.276916	0.138458	−	0.990368i	$-0.455785\pi$
$739$	7.52783	0.276916	0.138458	+	0.990368i	$0.455785\pi$
$740$	0	0
$741$	10.9244	0.401318
$742$	65.5075	2.40486
$743$	−51.3190	−1.88271	−0.941356	−	0.337416i	$-0.890447\pi$
$743$	−51.3190	−1.88271	−0.941356	+	0.337416i	$0.890447\pi$
$744$	−160.035	−5.86715
$745$	0	0
$746$	39.3377	1.44025
$747$	−19.1074	−0.699104
$748$	−0.964585	−0.0352687
$749$	−38.6868	−1.41358
$750$	0	0
$751$	25.8193	0.942160	0.471080	−	0.882091i	$-0.343864\pi$
$751$	25.8193	0.942160	0.471080	+	0.882091i	$0.343864\pi$
$752$	−64.5740	−2.35477
$753$	0.391204	0.0142563
$754$	37.4773	1.36484
$755$	0	0
$756$	−19.0855	−0.694132
$757$	2.06317	0.0749871	0.0374936	−	0.999297i	$-0.488063\pi$
$757$	2.06317	0.0749871	0.0374936	+	0.999297i	$0.488063\pi$
$758$	−44.5812	−1.61926
$759$	−90.6398	−3.29002
$760$	0	0
$761$	−9.95532	−0.360880	−0.180440	−	0.983586i	$-0.557752\pi$
$761$	−9.95532	−0.360880	−0.180440	+	0.983586i	$0.557752\pi$
$762$	−23.5610	−0.853525
$763$	28.5023	1.03185
$764$	−11.0028	−0.398067
$765$	0	0
$766$	−24.0828	−0.870148
$767$	−0.449139	−0.0162175
$768$	−13.8510	−0.499804
$769$	−2.26190	−0.0815661	−0.0407830	−	0.999168i	$-0.512985\pi$
$769$	−2.26190	−0.0815661	−0.0407830	+	0.999168i	$0.512985\pi$
$770$	0	0
$771$	23.2412	0.837010
$772$	−86.5666	−3.11560
$773$	−11.7012	−0.420862	−0.210431	−	0.977609i	$-0.567487\pi$
$773$	−11.7012	−0.420862	−0.210431	+	0.977609i	$0.567487\pi$
$774$	63.9689	2.29931
$775$	0	0
$776$	−5.79875	−0.208163
$777$	−74.8418	−2.68493
$778$	−59.1072	−2.11910
$779$	11.6702	0.418129
$780$	0	0
$781$	17.2011	0.615504
$782$	−0.874647	−0.0312773
$783$	6.79385	0.242792
$784$	72.1735	2.57762
$785$	0	0
$786$	92.1397	3.28651
$787$	52.9851	1.88871	0.944357	−	0.328923i	$-0.106686\pi$
$787$	52.9851	1.88871	0.944357	+	0.328923i	$0.106686\pi$
$788$	45.5322	1.62202
$789$	14.3860	0.512154
$790$	0	0
$791$	−26.9307	−0.957544
$792$	100.660	3.57681
$793$	2.13230	0.0757201
$794$	93.9126	3.33283
$795$	0	0
$796$	25.1546	0.891580
$797$	40.8622	1.44741	0.723707	−	0.690107i	$-0.242437\pi$
$797$	40.8622	1.44741	0.723707	+	0.690107i	$0.242437\pi$
$798$	−49.2886	−1.74480
$799$	0.214001	0.00757082
$800$	0	0
$801$	5.01353	0.177144
$802$	24.4687	0.864020
$803$	67.1380	2.36925
$804$	−66.2946	−2.33803
$805$	0	0
$806$	46.1925	1.62706
$807$	−36.6678	−1.29077
$808$	−94.8940	−3.33836
$809$	25.8518	0.908902	0.454451	−	0.890772i	$-0.349836\pi$
$809$	25.8518	0.908902	0.454451	+	0.890772i	$0.349836\pi$
$810$	0	0
$811$	−3.99466	−0.140272	−0.0701358	−	0.997537i	$-0.522343\pi$
$811$	−3.99466	−0.140272	−0.0701358	+	0.997537i	$0.522343\pi$
$812$	−121.674	−4.26994
$813$	−41.3966	−1.45184
$814$	−110.485	−3.87248
$815$	0	0
$816$	1.13979	0.0399005
$817$	−20.3071	−0.710456
$818$	−19.5231	−0.682609
$819$	−19.6815	−0.687727
$820$	0	0
$821$	−20.3359	−0.709728	−0.354864	−	0.934918i	$-0.615473\pi$
$821$	−20.3359	−0.709728	−0.354864	+	0.934918i	$0.615473\pi$
$822$	93.7269	3.26910
$823$	−17.3043	−0.603189	−0.301594	−	0.953436i	$-0.597519\pi$
$823$	−17.3043	−0.603189	−0.301594	+	0.953436i	$0.597519\pi$
$824$	−23.9822	−0.835459
$825$	0	0
$826$	2.02642	0.0705081
$827$	10.6603	0.370697	0.185348	−	0.982673i	$-0.440659\pi$
$827$	10.6603	0.370697	0.185348	+	0.982673i	$0.440659\pi$
$828$	107.617	3.73995
$829$	44.8695	1.55838	0.779192	−	0.626786i	$-0.215630\pi$
$829$	44.8695	1.55838	0.779192	+	0.626786i	$0.215630\pi$
$830$	0	0
$831$	4.58152	0.158931
$832$	36.8804	1.27860
$833$	−0.239186	−0.00828731
$834$	−34.9065	−1.20871
$835$	0	0
$836$	−52.3584	−1.81085
$837$	8.37373	0.289438
$838$	−13.4992	−0.466321
$839$	2.34555	0.0809774	0.0404887	−	0.999180i	$-0.487109\pi$
$839$	2.34555	0.0809774	0.0404887	+	0.999180i	$0.487109\pi$
$840$	0	0
$841$	14.3124	0.493531
$842$	56.9028	1.96100
$843$	−6.67933	−0.230048
$844$	−108.156	−3.72288
$845$	0	0
$846$	−36.5917	−1.25805
$847$	−39.8228	−1.36833
$848$	−82.2269	−2.82368
$849$	65.9777	2.26435
$850$	0	0
$851$	−72.0902	−2.47122
$852$	−44.3347	−1.51888
$853$	28.1084	0.962414	0.481207	−	0.876607i	$-0.340199\pi$
$853$	28.1084	0.962414	0.481207	+	0.876607i	$0.340199\pi$
$854$	−9.62047	−0.329206
$855$	0	0
$856$	89.8378	3.07059
$857$	2.34397	0.0800684	0.0400342	−	0.999198i	$-0.487253\pi$
$857$	2.34397	0.0800684	0.0400342	+	0.999198i	$0.487253\pi$
$858$	−63.0726	−2.15326
$859$	−17.6879	−0.603504	−0.301752	−	0.953387i	$-0.597571\pi$
$859$	−17.6879	−0.603504	−0.301752	+	0.953387i	$0.597571\pi$
$860$	0	0
$861$	−45.6419	−1.55547
$862$	20.5866	0.701182
$863$	−1.88922	−0.0643099	−0.0321549	−	0.999483i	$-0.510237\pi$
$863$	−1.88922	−0.0643099	−0.0321549	+	0.999483i	$0.510237\pi$
$864$	16.0213	0.545055
$865$	0	0
$866$	−46.9240	−1.59454
$867$	40.0899	1.36152
$868$	−149.969	−5.09029
$869$	−52.9887	−1.79752
$870$	0	0
$871$	11.6785	0.395711
$872$	−66.1875	−2.24139
$873$	−1.77617	−0.0601144
$874$	−47.4765	−1.60592
$875$	0	0
$876$	−173.044	−5.84660
$877$	−16.9885	−0.573661	−0.286830	−	0.957981i	$-0.592602\pi$
$877$	−16.9885	−0.573661	−0.286830	+	0.957981i	$0.592602\pi$
$878$	44.6354	1.50637
$879$	51.7224	1.74455
$880$	0	0
$881$	−11.0508	−0.372309	−0.186155	−	0.982520i	$-0.559603\pi$
$881$	−11.0508	−0.372309	−0.186155	+	0.982520i	$0.559603\pi$
$882$	40.8980	1.37711
$883$	−19.8537	−0.668131	−0.334065	−	0.942550i	$-0.608421\pi$
$883$	−19.8537	−0.668131	−0.334065	+	0.942550i	$0.608421\pi$
$884$	−0.437963	−0.0147303
$885$	0	0
$886$	−25.4002	−0.853337
$887$	−36.2703	−1.21784	−0.608918	−	0.793233i	$-0.708396\pi$
$887$	−36.2703	−1.21784	−0.608918	+	0.793233i	$0.708396\pi$
$888$	173.796	5.83222
$889$	−13.4751	−0.451942
$890$	0	0
$891$	−47.5333	−1.59242
$892$	0.844679	0.0282819
$893$	11.6161	0.388720
$894$	144.935	4.84734
$895$	0	0
$896$	−54.5817	−1.82345
$897$	−41.1543	−1.37410
$898$	−23.4445	−0.782353
$899$	53.3845	1.78047
$900$	0	0
$901$	0.272504	0.00907841
$902$	−67.3786	−2.24346
$903$	79.4207	2.64295
$904$	62.5379	2.07998
$905$	0	0
$906$	−9.92058	−0.329589
$907$	−8.53502	−0.283401	−0.141700	−	0.989910i	$-0.545257\pi$
$907$	−8.53502	−0.283401	−0.141700	+	0.989910i	$0.545257\pi$
$908$	53.3456	1.77034
$909$	−29.0663	−0.964069
$910$	0	0
$911$	−37.1313	−1.23021	−0.615107	−	0.788443i	$-0.710887\pi$
$911$	−37.1313	−1.23021	−0.615107	+	0.788443i	$0.710887\pi$
$912$	61.8684	2.04867
$913$	−35.0207	−1.15902
$914$	−36.0981	−1.19402
$915$	0	0
$916$	9.36333	0.309373
$917$	52.6971	1.74021
$918$	−0.110332	−0.00364150
$919$	16.2389	0.535670	0.267835	−	0.963465i	$-0.413692\pi$
$919$	16.2389	0.535670	0.267835	+	0.963465i	$0.413692\pi$
$920$	0	0
$921$	47.6072	1.56871
$922$	110.128	3.62686
$923$	7.81003	0.257070
$924$	204.773	6.73652
$925$	0	0
$926$	−35.6441	−1.17134
$927$	−7.34582	−0.241268
$928$	102.139	3.35289
$929$	−53.2698	−1.74772	−0.873862	−	0.486174i	$-0.838392\pi$
$929$	−53.2698	−1.74772	−0.873862	+	0.486174i	$0.838392\pi$
$930$	0	0
$931$	−12.9832	−0.425508
$932$	46.9538	1.53802
$933$	−5.07177	−0.166042
$934$	58.6337	1.91855
$935$	0	0
$936$	45.7040	1.49388
$937$	−8.57718	−0.280204	−0.140102	−	0.990137i	$-0.544743\pi$
$937$	−8.57718	−0.280204	−0.140102	+	0.990137i	$0.544743\pi$
$938$	−52.6909	−1.72042
$939$	21.7587	0.710069
$940$	0	0
$941$	−0.301215	−0.00981931	−0.00490966	−	0.999988i	$-0.501563\pi$
$941$	−0.301215	−0.00981931	−0.00490966	+	0.999988i	$0.501563\pi$
$942$	−33.4199	−1.08888
$943$	−43.9639	−1.43166
$944$	−2.54362	−0.0827876
$945$	0	0
$946$	117.244	3.81194
$947$	12.3751	0.402137	0.201068	−	0.979577i	$-0.435559\pi$
$947$	12.3751	0.402137	0.201068	+	0.979577i	$0.435559\pi$
$948$	136.575	4.43575
$949$	30.4835	0.989536
$950$	0	0
$951$	−47.7388	−1.54804
$952$	1.20597	0.0390857
$953$	−36.7818	−1.19148	−0.595739	−	0.803178i	$-0.703141\pi$
$953$	−36.7818	−1.19148	−0.595739	+	0.803178i	$0.703141\pi$
$954$	−46.5949	−1.50857
$955$	0	0
$956$	104.492	3.37952
$957$	−72.8928	−2.35629
$958$	−46.8322	−1.51308
$959$	53.6049	1.73099
$960$	0	0
$961$	34.7988	1.12254
$962$	−50.1647	−1.61737
$963$	27.5176	0.886741
$964$	134.973	4.34718
$965$	0	0
$966$	185.679	5.97414
$967$	−39.8211	−1.28056	−0.640281	−	0.768141i	$-0.721182\pi$
$967$	−39.8211	−1.28056	−0.640281	+	0.768141i	$0.721182\pi$
$968$	92.4758	2.97228
$969$	−0.205035	−0.00658667
$970$	0	0
$971$	−8.77171	−0.281498	−0.140749	−	0.990045i	$-0.544951\pi$
$971$	−8.77171	−0.281498	−0.140749	+	0.990045i	$0.544951\pi$
$972$	106.619	3.41982
$973$	−19.9639	−0.640015
$974$	−61.3099	−1.96450
$975$	0	0
$976$	12.0759	0.386540
$977$	49.3173	1.57780	0.788901	−	0.614521i	$-0.210651\pi$
$977$	49.3173	1.57780	0.788901	+	0.614521i	$0.210651\pi$
$978$	−103.912	−3.32273
$979$	9.18896	0.293680
$980$	0	0
$981$	−20.2734	−0.647281
$982$	58.6314	1.87100
$983$	−5.84612	−0.186462	−0.0932311	−	0.995644i	$-0.529720\pi$
$983$	−5.84612	−0.186462	−0.0932311	+	0.995644i	$0.529720\pi$
$984$	105.989	3.37880
$985$	0	0
$986$	−0.703393	−0.0224006
$987$	−45.4305	−1.44607
$988$	−23.7729	−0.756318
$989$	76.5008	2.43258
$990$	0	0
$991$	−8.06514	−0.256198	−0.128099	−	0.991761i	$-0.540887\pi$
$991$	−8.06514	−0.256198	−0.128099	+	0.991761i	$0.540887\pi$
$992$	125.891	3.99706
$993$	−11.9343	−0.378724
$994$	−35.2372	−1.11765
$995$	0	0
$996$	90.2635	2.86011
$997$	−20.9223	−0.662616	−0.331308	−	0.943523i	$-0.607490\pi$
$997$	−20.9223	−0.662616	−0.331308	+	0.943523i	$0.607490\pi$
$998$	68.1751	2.15804
$999$	−9.09380	−0.287715

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1525.2.a.i.1.7		7
5.2	odd	4		1525.2.b.e.1099.14		14
5.3	odd	4		1525.2.b.e.1099.1		14
5.4	even	2		305.2.a.c.1.1	✓	7
15.14	odd	2		2745.2.a.n.1.7		7
20.19	odd	2		4880.2.a.bc.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
305.2.a.c.1.1	✓	7	5.4	even	2
1525.2.a.i.1.7		7	1.1	even	1	trivial
1525.2.b.e.1099.1		14	5.3	odd	4
1525.2.b.e.1099.14		14	5.2	odd	4
2745.2.a.n.1.7		7	15.14	odd	2
4880.2.a.bc.1.2		7	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:	\( N \)	\(=\)	\( 1525 = 5^{2} \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1525.a (trivial)

\(f(q)\)	\(=\)	\(q+2.67064 q^{2} -2.35845 q^{3} +5.13230 q^{4} -6.29857 q^{6} -3.60231 q^{7} +8.36523 q^{8} +2.56229 q^{9} +4.69625 q^{11} -12.1043 q^{12} +2.13230 q^{13} -9.62047 q^{14} +12.0759 q^{16} -0.0400200 q^{17} +6.84295 q^{18} -2.17232 q^{19} +8.49588 q^{21} +12.5420 q^{22} +8.18353 q^{23} -19.7290 q^{24} +5.69459 q^{26} +1.03231 q^{27} -18.4881 q^{28} +6.58122 q^{29} +8.11165 q^{31} +15.5198 q^{32} -11.0759 q^{33} -0.106879 q^{34} +13.1505 q^{36} -8.80918 q^{37} -5.80147 q^{38} -5.02892 q^{39} -5.37224 q^{41} +22.6894 q^{42} +9.34814 q^{43} +24.1026 q^{44} +21.8552 q^{46} -5.34735 q^{47} -28.4804 q^{48} +5.97666 q^{49} +0.0943853 q^{51} +10.9436 q^{52} -6.80918 q^{53} +2.75692 q^{54} -30.1342 q^{56} +5.12331 q^{57} +17.5760 q^{58} -0.210636 q^{59} +1.00000 q^{61} +21.6633 q^{62} -9.23018 q^{63} +17.2961 q^{64} -29.5797 q^{66} +5.47696 q^{67} -0.205395 q^{68} -19.3005 q^{69} +3.66273 q^{71} +21.4342 q^{72} +14.2961 q^{73} -23.5261 q^{74} -11.1490 q^{76} -16.9174 q^{77} -13.4304 q^{78} -11.2832 q^{79} -10.1215 q^{81} -14.3473 q^{82} -7.45716 q^{83} +43.6034 q^{84} +24.9655 q^{86} -15.5215 q^{87} +39.2852 q^{88} +1.95666 q^{89} -7.68120 q^{91} +42.0003 q^{92} -19.1309 q^{93} -14.2808 q^{94} -36.6028 q^{96} -0.693197 q^{97} +15.9615 q^{98} +12.0332 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{2} + 12 q^{4} - 12 q^{7} + 9 q^{8} + 9 q^{9} + 2 q^{11} - 7 q^{12} - 9 q^{13} + 8 q^{14} + 14 q^{16} + 4 q^{17} + 3 q^{18} + 13 q^{19} + 2 q^{21} + 19 q^{22} + 5 q^{23} + 2 q^{24} + 7 q^{26}+ \cdots + 3 q^{99}+O(q^{100}) \) 7 * q + 2 * q^2 + 12 * q^4 - 12 * q^7 + 9 * q^8 + 9 * q^9 + 2 * q^11 - 7 * q^12 - 9 * q^13 + 8 * q^14 + 14 * q^16 + 4 * q^17 + 3 * q^18 + 13 * q^19 + 2 * q^21 + 19 * q^22 + 5 * q^23 + 2 * q^24 + 7 * q^26 + 9 * q^27 - 25 * q^28 + 7 * q^29 + 35 * q^31 + 35 * q^32 - 7 * q^33 + 13 * q^34 + 6 * q^36 - 9 * q^37 + 6 * q^38 - 7 * q^39 - 8 * q^41 + 51 * q^42 - 9 * q^43 - 3 * q^44 - 3 * q^46 + 3 * q^47 - 9 * q^48 + 29 * q^49 + 11 * q^51 + 30 * q^52 + 5 * q^53 - 18 * q^54 - 26 * q^56 + 18 * q^57 + 9 * q^58 + 6 * q^59 + 7 * q^61 + 38 * q^62 - 33 * q^63 + 17 * q^64 - 51 * q^66 + 11 * q^67 - 7 * q^68 - 21 * q^69 + 19 * q^71 + 68 * q^72 - 4 * q^73 - 24 * q^74 - 37 * q^76 + 3 * q^77 + 2 * q^78 + 3 * q^79 - 13 * q^81 + 32 * q^82 - 10 * q^83 + 46 * q^84 - 18 * q^86 - 19 * q^87 + 15 * q^88 + 14 * q^89 + 11 * q^91 + 22 * q^92 + 46 * q^93 - 7 * q^94 - 67 * q^96 - 20 * q^97 - 29 * q^98 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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