LMFDB - Embedded newform 925.2.a.m.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [925,2,Mod(1,925)] 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("925.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 925 = 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	925.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,5,8] 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:	$7.38616218697$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 6x^{7} + 30x^{6} + 15x^{5} - 70x^{4} - 22x^{3} + 44x^{2} + 4x - 4 $ x^9 - 4x^8 - 6x^7 + 30x^6 + 15x^5 - 70x^4 - 22x^3 + 44x^2 + 4x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 185)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-1.58855$ of defining polynomial
Character	$\chi$	$=$	925.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.58855 q^{2} -1.45668 q^{3} +4.70058 q^{4} -3.77067 q^{6} -0.648627 q^{7} +6.99059 q^{8} -0.878097 q^{9} +2.00426 q^{11} -6.84722 q^{12} +4.52102 q^{13} -1.67900 q^{14} +8.69431 q^{16} +5.56202 q^{17} -2.27300 q^{18} +1.84662 q^{19} +0.944840 q^{21} +5.18813 q^{22} +6.29488 q^{23} -10.1830 q^{24} +11.7029 q^{26} +5.64913 q^{27} -3.04893 q^{28} -7.93267 q^{29} -2.87798 q^{31} +8.52446 q^{32} -2.91956 q^{33} +14.3976 q^{34} -4.12757 q^{36} -1.00000 q^{37} +4.78007 q^{38} -6.58565 q^{39} -9.59442 q^{41} +2.44576 q^{42} -6.67357 q^{43} +9.42119 q^{44} +16.2946 q^{46} -4.62245 q^{47} -12.6648 q^{48} -6.57928 q^{49} -8.10206 q^{51} +21.2514 q^{52} -6.63289 q^{53} +14.6230 q^{54} -4.53429 q^{56} -2.68993 q^{57} -20.5341 q^{58} +3.55444 q^{59} +8.98508 q^{61} -7.44979 q^{62} +0.569558 q^{63} +4.67736 q^{64} -7.55741 q^{66} -11.7962 q^{67} +26.1447 q^{68} -9.16960 q^{69} +7.92865 q^{71} -6.13841 q^{72} +8.94146 q^{73} -2.58855 q^{74} +8.68020 q^{76} -1.30002 q^{77} -17.0473 q^{78} +5.85260 q^{79} -5.59466 q^{81} -24.8356 q^{82} +17.0540 q^{83} +4.44130 q^{84} -17.2748 q^{86} +11.5553 q^{87} +14.0110 q^{88} -15.5198 q^{89} -2.93245 q^{91} +29.5896 q^{92} +4.19228 q^{93} -11.9654 q^{94} -12.4174 q^{96} +7.82669 q^{97} -17.0308 q^{98} -1.75993 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 5 q^{2} + 8 q^{3} + 11 q^{4} + 2 q^{6} + 8 q^{7} + 15 q^{8} + 13 q^{9} + 16 q^{12} + 6 q^{13} - 4 q^{14} + 11 q^{16} + 18 q^{17} - 3 q^{18} - 4 q^{19} + 4 q^{21} + 6 q^{22} + 16 q^{23} + 6 q^{24}+ \cdots + 8 q^{99}+O(q^{100}) $ 9 * q + 5 * q^2 + 8 * q^3 + 11 * q^4 + 2 * q^6 + 8 * q^7 + 15 * q^8 + 13 * q^9 + 16 * q^12 + 6 * q^13 - 4 * q^14 + 11 * q^16 + 18 * q^17 - 3 * q^18 - 4 * q^19 + 4 * q^21 + 6 * q^22 + 16 * q^23 + 6 * q^24 - 6 * q^26 + 32 * q^27 - 20 * q^28 - 2 * q^29 - 6 * q^31 + 35 * q^32 + 6 * q^33 + 6 * q^34 + 11 * q^36 - 9 * q^37 + 4 * q^38 - 16 * q^39 + 2 * q^41 + 4 * q^42 + 14 * q^43 + 4 * q^44 + 16 * q^46 + 34 * q^47 + 6 * q^48 - 7 * q^49 + 8 * q^51 + 16 * q^52 - 4 * q^53 - 18 * q^54 - 24 * q^56 - 4 * q^57 - 22 * q^58 + 10 * q^59 + 8 * q^61 + 26 * q^62 + 18 * q^63 + 15 * q^64 - 34 * q^66 + 16 * q^67 + 36 * q^68 - 10 * q^69 + 8 * q^71 + 9 * q^72 + 8 * q^73 - 5 * q^74 - 48 * q^76 + 14 * q^77 - 76 * q^78 - 26 * q^79 + 13 * q^81 - 4 * q^82 + 70 * q^83 - 24 * q^84 - 16 * q^86 - 12 * q^87 - 36 * q^88 - 8 * q^89 - 18 * q^91 + 16 * q^92 + 2 * q^93 - 26 * q^94 + 52 * q^96 - 2 * q^97 + 21 * q^98 + 8 * 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 $

$2$	2.58855	1.83038	0.915190	−	0.403023i	$-0.132040\pi$
$2$	2.58855	1.83038	0.915190	+	0.403023i	$0.132040\pi$
$3$	−1.45668	−0.841012	−0.420506	−	0.907290i	$-0.638147\pi$
$3$	−1.45668	−0.841012	−0.420506	+	0.907290i	$0.638147\pi$
$4$	4.70058	2.35029
$5$	0	0
$5$	0	0
$6$	−3.77067	−1.53937
$7$	−0.648627	−0.245158	−0.122579	−	0.992459i	$-0.539117\pi$
$7$	−0.648627	−0.245158	−0.122579	+	0.992459i	$0.539117\pi$
$8$	6.99059	2.47155
$9$	−0.878097	−0.292699
$10$	0	0
$11$	2.00426	0.604307	0.302154	−	0.953259i	$-0.402294\pi$
$11$	2.00426	0.604307	0.302154	+	0.953259i	$0.402294\pi$
$12$	−6.84722	−1.97662
$13$	4.52102	1.25390	0.626952	−	0.779058i	$-0.284302\pi$
$13$	4.52102	1.25390	0.626952	+	0.779058i	$0.284302\pi$
$14$	−1.67900	−0.448733
$15$	0	0
$16$	8.69431	2.17358
$17$	5.56202	1.34899	0.674494	−	0.738280i	$-0.264362\pi$
$17$	5.56202	1.34899	0.674494	+	0.738280i	$0.264362\pi$
$18$	−2.27300	−0.535750
$19$	1.84662	0.423644	0.211822	−	0.977308i	$-0.432060\pi$
$19$	1.84662	0.423644	0.211822	+	0.977308i	$0.432060\pi$
$20$	0	0
$21$	0.944840	0.206181
$22$	5.18813	1.10611
$23$	6.29488	1.31257	0.656287	−	0.754511i	$-0.272126\pi$
$23$	6.29488	1.31257	0.656287	+	0.754511i	$0.272126\pi$
$24$	−10.1830	−2.07860
$25$	0	0
$26$	11.7029	2.29512
$27$	5.64913	1.08718
$28$	−3.04893	−0.576193
$29$	−7.93267	−1.47306	−0.736530	−	0.676405i	$-0.763537\pi$
$29$	−7.93267	−1.47306	−0.736530	+	0.676405i	$0.763537\pi$
$30$	0	0
$31$	−2.87798	−0.516900	−0.258450	−	0.966025i	$-0.583212\pi$
$31$	−2.87798	−0.516900	−0.258450	+	0.966025i	$0.583212\pi$
$32$	8.52446	1.50693
$33$	−2.91956	−0.508230
$34$	14.3976	2.46916
$35$	0	0
$36$	−4.12757	−0.687928
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	4.78007	0.775430
$39$	−6.58565	−1.05455
$40$	0	0
$41$	−9.59442	−1.49840	−0.749199	−	0.662345i	$-0.769561\pi$
$41$	−9.59442	−1.49840	−0.749199	+	0.662345i	$0.769561\pi$
$42$	2.44576	0.377389
$43$	−6.67357	−1.01771	−0.508855	−	0.860852i	$-0.669931\pi$
$43$	−6.67357	−1.01771	−0.508855	+	0.860852i	$0.669931\pi$
$44$	9.42119	1.42030
$45$	0	0
$46$	16.2946	2.40251
$47$	−4.62245	−0.674253	−0.337127	−	0.941459i	$-0.609455\pi$
$47$	−4.62245	−0.674253	−0.337127	+	0.941459i	$0.609455\pi$
$48$	−12.6648	−1.82800
$49$	−6.57928	−0.939897
$50$	0	0
$51$	−8.10206	−1.13451
$52$	21.2514	2.94704
$53$	−6.63289	−0.911097	−0.455549	−	0.890211i	$-0.650557\pi$
$53$	−6.63289	−0.911097	−0.455549	+	0.890211i	$0.650557\pi$
$54$	14.6230	1.98994
$55$	0	0
$56$	−4.53429	−0.605920
$57$	−2.68993	−0.356290
$58$	−20.5341	−2.69626
$59$	3.55444	0.462749	0.231374	−	0.972865i	$-0.425678\pi$
$59$	3.55444	0.462749	0.231374	+	0.972865i	$0.425678\pi$
$60$	0	0
$61$	8.98508	1.15042	0.575211	−	0.818005i	$-0.304920\pi$
$61$	8.98508	1.15042	0.575211	+	0.818005i	$0.304920\pi$
$62$	−7.44979	−0.946124
$63$	0.569558	0.0717575
$64$	4.67736	0.584670
$65$	0	0
$66$	−7.55741	−0.930253
$67$	−11.7962	−1.44113	−0.720567	−	0.693385i	$-0.756118\pi$
$67$	−11.7962	−1.44113	−0.720567	+	0.693385i	$0.756118\pi$
$68$	26.1447	3.17051
$69$	−9.16960	−1.10389
$70$	0	0
$71$	7.92865	0.940957	0.470479	−	0.882411i	$-0.344081\pi$
$71$	7.92865	0.940957	0.470479	+	0.882411i	$0.344081\pi$
$72$	−6.13841	−0.723419
$73$	8.94146	1.04652	0.523259	−	0.852174i	$-0.324716\pi$
$73$	8.94146	1.04652	0.523259	+	0.852174i	$0.324716\pi$
$74$	−2.58855	−0.300913
$75$	0	0
$76$	8.68020	0.995687
$77$	−1.30002	−0.148151
$78$	−17.0473	−1.93022
$79$	5.85260	0.658469	0.329235	−	0.944248i	$-0.393209\pi$
$79$	5.85260	0.658469	0.329235	+	0.944248i	$0.393209\pi$
$80$	0	0
$81$	−5.59466	−0.621629
$82$	−24.8356	−2.74264
$83$	17.0540	1.87192	0.935961	−	0.352103i	$-0.114533\pi$
$83$	17.0540	1.87192	0.935961	+	0.352103i	$0.114533\pi$
$84$	4.44130	0.484585
$85$	0	0
$86$	−17.2748	−1.86279
$87$	11.5553	1.23886
$88$	14.0110	1.49357
$89$	−15.5198	−1.64509	−0.822546	−	0.568699i	$-0.807447\pi$
$89$	−15.5198	−1.64509	−0.822546	+	0.568699i	$0.807447\pi$
$90$	0	0
$91$	−2.93245	−0.307405
$92$	29.5896	3.08493
$93$	4.19228	0.434719
$94$	−11.9654	−1.23414
$95$	0	0
$96$	−12.4174	−1.26734
$97$	7.82669	0.794680	0.397340	−	0.917672i	$-0.369933\pi$
$97$	7.82669	0.794680	0.397340	+	0.917672i	$0.369933\pi$
$98$	−17.0308	−1.72037
$99$	−1.75993	−0.176880
$100$	0	0
$101$	−8.64993	−0.860701	−0.430350	−	0.902662i	$-0.641610\pi$
$101$	−8.64993	−0.860701	−0.430350	+	0.902662i	$0.641610\pi$
$102$	−20.9726	−2.07659
$103$	−4.85153	−0.478036	−0.239018	−	0.971015i	$-0.576825\pi$
$103$	−4.85153	−0.478036	−0.239018	+	0.971015i	$0.576825\pi$
$104$	31.6046	3.09908
$105$	0	0
$106$	−17.1696	−1.66765
$107$	−10.4460	−1.00985	−0.504925	−	0.863163i	$-0.668480\pi$
$107$	−10.4460	−1.00985	−0.504925	+	0.863163i	$0.668480\pi$
$108$	26.5542	2.55518
$109$	−6.96214	−0.666852	−0.333426	−	0.942776i	$-0.608205\pi$
$109$	−6.96214	−0.666852	−0.333426	+	0.942776i	$0.608205\pi$
$110$	0	0
$111$	1.45668	0.138262
$112$	−5.63937	−0.532870
$113$	−3.68347	−0.346512	−0.173256	−	0.984877i	$-0.555429\pi$
$113$	−3.68347	−0.346512	−0.173256	+	0.984877i	$0.555429\pi$
$114$	−6.96301	−0.652146
$115$	0	0
$116$	−37.2882	−3.46212
$117$	−3.96989	−0.367016
$118$	9.20084	0.847006
$119$	−3.60768	−0.330715
$120$	0	0
$121$	−6.98294	−0.634813
$122$	23.2583	2.10571
$123$	13.9760	1.26017
$124$	−13.5282	−1.21487
$125$	0	0
$126$	1.47433	0.131344
$127$	−0.651297	−0.0577933	−0.0288966	−	0.999582i	$-0.509199\pi$
$127$	−0.651297	−0.0577933	−0.0288966	+	0.999582i	$0.509199\pi$
$128$	−4.94134	−0.436757
$129$	9.72122	0.855906
$130$	0	0
$131$	−2.04184	−0.178396	−0.0891982	−	0.996014i	$-0.528430\pi$
$131$	−2.04184	−0.178396	−0.0891982	+	0.996014i	$0.528430\pi$
$132$	−13.7236	−1.19449
$133$	−1.19777	−0.103860
$134$	−30.5350	−2.63782
$135$	0	0
$136$	38.8818	3.33408
$137$	10.4388	0.891845	0.445923	−	0.895071i	$-0.352876\pi$
$137$	10.4388	0.891845	0.445923	+	0.895071i	$0.352876\pi$
$138$	−23.7360	−2.02054
$139$	13.4010	1.13666	0.568329	−	0.822802i	$-0.307590\pi$
$139$	13.4010	1.13666	0.568329	+	0.822802i	$0.307590\pi$
$140$	0	0
$141$	6.73341	0.567055
$142$	20.5237	1.72231
$143$	9.06129	0.757743
$144$	−7.63444	−0.636204
$145$	0	0
$146$	23.1454	1.91553
$147$	9.58388	0.790465
$148$	−4.70058	−0.386385
$149$	−7.69388	−0.630307	−0.315154	−	0.949041i	$-0.602056\pi$
$149$	−7.69388	−0.630307	−0.315154	+	0.949041i	$0.602056\pi$
$150$	0	0
$151$	−14.8303	−1.20687	−0.603437	−	0.797410i	$-0.706203\pi$
$151$	−14.8303	−1.20687	−0.603437	+	0.797410i	$0.706203\pi$
$152$	12.9090	1.04706
$153$	−4.88399	−0.394847
$154$	−3.36516	−0.271172
$155$	0	0
$156$	−30.9564	−2.47850
$157$	−6.58086	−0.525210	−0.262605	−	0.964903i	$-0.584582\pi$
$157$	−6.58086	−0.525210	−0.262605	+	0.964903i	$0.584582\pi$
$158$	15.1497	1.20525
$159$	9.66197	0.766244
$160$	0	0
$161$	−4.08303	−0.321788
$162$	−14.4820	−1.13782
$163$	9.03162	0.707411	0.353705	−	0.935357i	$-0.384922\pi$
$163$	9.03162	0.707411	0.353705	+	0.935357i	$0.384922\pi$
$164$	−45.0994	−3.52167
$165$	0	0
$166$	44.1452	3.42633
$167$	13.6411	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$167$	13.6411	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$168$	6.60498	0.509586
$169$	7.43958	0.572275
$170$	0	0
$171$	−1.62151	−0.124000
$172$	−31.3696	−2.39191
$173$	−2.31736	−0.176186	−0.0880928	−	0.996112i	$-0.528077\pi$
$173$	−2.31736	−0.176186	−0.0880928	+	0.996112i	$0.528077\pi$
$174$	29.9115	2.26759
$175$	0	0
$176$	17.4257	1.31351
$177$	−5.17766	−0.389177
$178$	−40.1737	−3.01114
$179$	15.2482	1.13970	0.569852	−	0.821747i	$-0.307000\pi$
$179$	15.2482	1.13970	0.569852	+	0.821747i	$0.307000\pi$
$180$	0	0
$181$	−1.21203	−0.0900896	−0.0450448	−	0.998985i	$-0.514343\pi$
$181$	−1.21203	−0.0900896	−0.0450448	+	0.998985i	$0.514343\pi$
$182$	−7.59080	−0.562668
$183$	−13.0883	−0.967518
$184$	44.0049	3.24409
$185$	0	0
$186$	10.8519	0.795702
$187$	11.1477	0.815203
$188$	−21.7282	−1.58469
$189$	−3.66418	−0.266530
$190$	0	0
$191$	20.5011	1.48341	0.741705	−	0.670726i	$-0.234017\pi$
$191$	20.5011	1.48341	0.741705	+	0.670726i	$0.234017\pi$
$192$	−6.81340	−0.491715
$193$	−1.57831	−0.113609	−0.0568045	−	0.998385i	$-0.518091\pi$
$193$	−1.57831	−0.113609	−0.0568045	+	0.998385i	$0.518091\pi$
$194$	20.2598	1.45457
$195$	0	0
$196$	−30.9265	−2.20903
$197$	−15.9367	−1.13544	−0.567722	−	0.823221i	$-0.692175\pi$
$197$	−15.9367	−1.13544	−0.567722	+	0.823221i	$0.692175\pi$
$198$	−4.55567	−0.323758
$199$	4.38722	0.311002	0.155501	−	0.987836i	$-0.450301\pi$
$199$	4.38722	0.311002	0.155501	+	0.987836i	$0.450301\pi$
$200$	0	0
$201$	17.1832	1.21201
$202$	−22.3908	−1.57541
$203$	5.14535	0.361132
$204$	−38.0844	−2.66644
$205$	0	0
$206$	−12.5584	−0.874987
$207$	−5.52752	−0.384189
$208$	39.3071	2.72546
$209$	3.70111	0.256011
$210$	0	0
$211$	−16.5270	−1.13777	−0.568883	−	0.822418i	$-0.692624\pi$
$211$	−16.5270	−1.13777	−0.568883	+	0.822418i	$0.692624\pi$
$212$	−31.1784	−2.14134
$213$	−11.5495	−0.791356
$214$	−27.0399	−1.84841
$215$	0	0
$216$	39.4907	2.68700
$217$	1.86674	0.126722
$218$	−18.0218	−1.22059
$219$	−13.0248	−0.880135
$220$	0	0
$221$	25.1460	1.69150
$222$	3.77067	0.253071
$223$	−22.6093	−1.51403	−0.757014	−	0.653399i	$-0.773343\pi$
$223$	−22.6093	−1.51403	−0.757014	+	0.653399i	$0.773343\pi$
$224$	−5.52920	−0.369435
$225$	0	0
$226$	−9.53484	−0.634248
$227$	2.56112	0.169987	0.0849937	−	0.996381i	$-0.472913\pi$
$227$	2.56112	0.169987	0.0849937	+	0.996381i	$0.472913\pi$
$228$	−12.6442	−0.837385
$229$	1.73702	0.114786	0.0573929	−	0.998352i	$-0.481721\pi$
$229$	1.73702	0.114786	0.0573929	+	0.998352i	$0.481721\pi$
$230$	0	0
$231$	1.89370	0.124597
$232$	−55.4540	−3.64073
$233$	9.32879	0.611149	0.305575	−	0.952168i	$-0.401151\pi$
$233$	9.32879	0.611149	0.305575	+	0.952168i	$0.401151\pi$
$234$	−10.2762	−0.671779
$235$	0	0
$236$	16.7079	1.08759
$237$	−8.52534	−0.553780
$238$	−9.33865	−0.605335
$239$	0.602230	0.0389550	0.0194775	−	0.999810i	$-0.493800\pi$
$239$	0.602230	0.0389550	0.0194775	+	0.999810i	$0.493800\pi$
$240$	0	0
$241$	−14.5974	−0.940304	−0.470152	−	0.882585i	$-0.655801\pi$
$241$	−14.5974	−0.940304	−0.470152	+	0.882585i	$0.655801\pi$
$242$	−18.0757	−1.16195
$243$	−8.79779	−0.564378
$244$	42.2351	2.70382
$245$	0	0
$246$	36.1775	2.30659
$247$	8.34861	0.531209
$248$	−20.1188	−1.27754
$249$	−24.8422	−1.57431
$250$	0	0
$251$	−17.9341	−1.13199	−0.565996	−	0.824408i	$-0.691508\pi$
$251$	−17.9341	−1.13199	−0.565996	+	0.824408i	$0.691508\pi$
$252$	2.67725	0.168651
$253$	12.6166	0.793198
$254$	−1.68591	−0.105784
$255$	0	0
$256$	−22.1456	−1.38410
$257$	1.13398	0.0707356	0.0353678	−	0.999374i	$-0.488740\pi$
$257$	1.13398	0.0707356	0.0353678	+	0.999374i	$0.488740\pi$
$258$	25.1638	1.56663
$259$	0.648627	0.0403037
$260$	0	0
$261$	6.96565	0.431163
$262$	−5.28540	−0.326533
$263$	−0.576045	−0.0355204	−0.0177602	−	0.999842i	$-0.505654\pi$
$263$	−0.576045	−0.0355204	−0.0177602	+	0.999842i	$0.505654\pi$
$264$	−20.4094	−1.25611
$265$	0	0
$266$	−3.10048	−0.190103
$267$	22.6073	1.38354
$268$	−55.4490	−3.38708
$269$	14.8308	0.904249	0.452124	−	0.891955i	$-0.350666\pi$
$269$	14.8308	0.904249	0.452124	+	0.891955i	$0.350666\pi$
$270$	0	0
$271$	−23.5579	−1.43104	−0.715520	−	0.698592i	$-0.753810\pi$
$271$	−23.5579	−1.43104	−0.715520	+	0.698592i	$0.753810\pi$
$272$	48.3579	2.93213
$273$	4.27163	0.258531
$274$	27.0213	1.63242
$275$	0	0
$276$	−43.1025	−2.59446
$277$	1.55020	0.0931422	0.0465711	−	0.998915i	$-0.485171\pi$
$277$	1.55020	0.0931422	0.0465711	+	0.998915i	$0.485171\pi$
$278$	34.6891	2.08051
$279$	2.52714	0.151296
$280$	0	0
$281$	31.3095	1.86777	0.933884	−	0.357575i	$-0.116396\pi$
$281$	31.3095	1.86777	0.933884	+	0.357575i	$0.116396\pi$
$282$	17.4297	1.03793
$283$	27.6189	1.64177	0.820886	−	0.571091i	$-0.193480\pi$
$283$	27.6189	1.64177	0.820886	+	0.571091i	$0.193480\pi$
$284$	37.2693	2.21152
$285$	0	0
$286$	23.4556	1.38696
$287$	6.22321	0.367344
$288$	−7.48530	−0.441076
$289$	13.9361	0.819768
$290$	0	0
$291$	−11.4009	−0.668335
$292$	42.0301	2.45962
$293$	9.90491	0.578651	0.289326	−	0.957231i	$-0.406569\pi$
$293$	9.90491	0.578651	0.289326	+	0.957231i	$0.406569\pi$
$294$	24.8083	1.44685
$295$	0	0
$296$	−6.99059	−0.406320
$297$	11.3223	0.656988
$298$	−19.9160	−1.15370
$299$	28.4593	1.64584
$300$	0	0
$301$	4.32866	0.249500
$302$	−38.3890	−2.20904
$303$	12.6001	0.723859
$304$	16.0551	0.920823
$305$	0	0
$306$	−12.6424	−0.722720
$307$	−7.83799	−0.447338	−0.223669	−	0.974665i	$-0.571803\pi$
$307$	−7.83799	−0.447338	−0.223669	+	0.974665i	$0.571803\pi$
$308$	−6.11084	−0.348198
$309$	7.06711	0.402034
$310$	0	0
$311$	24.4352	1.38560	0.692798	−	0.721132i	$-0.256378\pi$
$311$	24.4352	1.38560	0.692798	+	0.721132i	$0.256378\pi$
$312$	−46.0376	−2.60636
$313$	18.0084	1.01789	0.508947	−	0.860798i	$-0.330035\pi$
$313$	18.0084	1.01789	0.508947	+	0.860798i	$0.330035\pi$
$314$	−17.0349	−0.961334
$315$	0	0
$316$	27.5106	1.54759
$317$	−14.3769	−0.807486	−0.403743	−	0.914872i	$-0.632291\pi$
$317$	−14.3769	−0.807486	−0.403743	+	0.914872i	$0.632291\pi$
$318$	25.0105	1.40252
$319$	−15.8991	−0.890180
$320$	0	0
$321$	15.2164	0.849296
$322$	−10.5691	−0.588995
$323$	10.2709	0.571491
$324$	−26.2981	−1.46101
$325$	0	0
$326$	23.3788	1.29483
$327$	10.1416	0.560831
$328$	−67.0707	−3.70336
$329$	2.99825	0.165299
$330$	0	0
$331$	24.5918	1.35169	0.675845	−	0.737044i	$-0.263779\pi$
$331$	24.5918	1.35169	0.675845	+	0.737044i	$0.263779\pi$
$332$	80.1639	4.39956
$333$	0.878097	0.0481194
$334$	35.3107	1.93212
$335$	0	0
$336$	8.21473	0.448150
$337$	−23.0495	−1.25559	−0.627793	−	0.778381i	$-0.716041\pi$
$337$	−23.0495	−1.25559	−0.627793	+	0.778381i	$0.716041\pi$
$338$	19.2577	1.04748
$339$	5.36562	0.291420
$340$	0	0
$341$	−5.76822	−0.312367
$342$	−4.19736	−0.226967
$343$	8.80790	0.475582
$344$	−46.6521	−2.51532
$345$	0	0
$346$	−5.99860	−0.322487
$347$	−2.01655	−0.108254	−0.0541271	−	0.998534i	$-0.517238\pi$
$347$	−2.01655	−0.108254	−0.0541271	+	0.998534i	$0.517238\pi$
$348$	54.3167	2.91168
$349$	5.86531	0.313963	0.156981	−	0.987602i	$-0.449824\pi$
$349$	5.86531	0.313963	0.156981	+	0.987602i	$0.449824\pi$
$350$	0	0
$351$	25.5398	1.36321
$352$	17.0852	0.910646
$353$	−20.8137	−1.10780	−0.553900	−	0.832583i	$-0.686861\pi$
$353$	−20.8137	−1.10780	−0.553900	+	0.832583i	$0.686861\pi$
$354$	−13.4026	−0.712342
$355$	0	0
$356$	−72.9519	−3.86644
$357$	5.25522	0.278136
$358$	39.4707	2.08609
$359$	−32.6574	−1.72359	−0.861796	−	0.507256i	$-0.830660\pi$
$359$	−32.6574	−1.72359	−0.861796	+	0.507256i	$0.830660\pi$
$360$	0	0
$361$	−15.5900	−0.820526
$362$	−3.13740	−0.164898
$363$	10.1719	0.533885
$364$	−13.7842	−0.722491
$365$	0	0
$366$	−33.8798	−1.77093
$367$	19.8239	1.03480	0.517400	−	0.855743i	$-0.326900\pi$
$367$	19.8239	1.03480	0.517400	+	0.855743i	$0.326900\pi$
$368$	54.7297	2.85298
$369$	8.42483	0.438579
$370$	0	0
$371$	4.30227	0.223363
$372$	19.7062	1.02172
$373$	7.05575	0.365333	0.182666	−	0.983175i	$-0.441527\pi$
$373$	7.05575	0.365333	0.182666	+	0.983175i	$0.441527\pi$
$374$	28.8565	1.49213
$375$	0	0
$376$	−32.3136	−1.66645
$377$	−35.8637	−1.84707
$378$	−9.48491	−0.487851
$379$	14.3983	0.739593	0.369797	−	0.929113i	$-0.379427\pi$
$379$	14.3983	0.739593	0.369797	+	0.929113i	$0.379427\pi$
$380$	0	0
$381$	0.948729	0.0486048
$382$	53.0682	2.71521
$383$	4.98755	0.254852	0.127426	−	0.991848i	$-0.459329\pi$
$383$	4.98755	0.254852	0.127426	+	0.991848i	$0.459329\pi$
$384$	7.19793	0.367318
$385$	0	0
$386$	−4.08552	−0.207948
$387$	5.86004	0.297882
$388$	36.7900	1.86773
$389$	22.0079	1.11585	0.557923	−	0.829893i	$-0.311598\pi$
$389$	22.0079	1.11585	0.557923	+	0.829893i	$0.311598\pi$
$390$	0	0
$391$	35.0123	1.77065
$392$	−45.9930	−2.32300
$393$	2.97430	0.150034
$394$	−41.2529	−2.07829
$395$	0	0
$396$	−8.27272	−0.415720
$397$	8.24549	0.413829	0.206915	−	0.978359i	$-0.433658\pi$
$397$	8.24549	0.413829	0.206915	+	0.978359i	$0.433658\pi$
$398$	11.3565	0.569251
$399$	1.74476	0.0873473
$400$	0	0
$401$	−14.1806	−0.708144	−0.354072	−	0.935218i	$-0.615203\pi$
$401$	−14.1806	−0.708144	−0.354072	+	0.935218i	$0.615203\pi$
$402$	44.4796	2.21844
$403$	−13.0114	−0.648143
$404$	−40.6597	−2.02290
$405$	0	0
$406$	13.3190	0.661010
$407$	−2.00426	−0.0993475
$408$	−56.6381	−2.80401
$409$	−5.60504	−0.277152	−0.138576	−	0.990352i	$-0.544252\pi$
$409$	−5.60504	−0.277152	−0.138576	+	0.990352i	$0.544252\pi$
$410$	0	0
$411$	−15.2059	−0.750053
$412$	−22.8050	−1.12352
$413$	−2.30551	−0.113447
$414$	−14.3082	−0.703212
$415$	0	0
$416$	38.5392	1.88954
$417$	−19.5209	−0.955942
$418$	9.58051	0.468598
$419$	−15.2102	−0.743067	−0.371533	−	0.928420i	$-0.621168\pi$
$419$	−15.2102	−0.743067	−0.371533	+	0.928420i	$0.621168\pi$
$420$	0	0
$421$	−25.3853	−1.23720	−0.618602	−	0.785705i	$-0.712301\pi$
$421$	−25.3853	−1.23720	−0.618602	+	0.785705i	$0.712301\pi$
$422$	−42.7810	−2.08255
$423$	4.05896	0.197353
$424$	−46.3678	−2.25182
$425$	0	0
$426$	−29.8963	−1.44848
$427$	−5.82797	−0.282035
$428$	−49.1021	−2.37344
$429$	−13.1994	−0.637271
$430$	0	0
$431$	7.47455	0.360036	0.180018	−	0.983663i	$-0.442384\pi$
$431$	7.47455	0.360036	0.180018	+	0.983663i	$0.442384\pi$
$432$	49.1153	2.36306
$433$	−27.2983	−1.31187	−0.655937	−	0.754816i	$-0.727726\pi$
$433$	−27.2983	−1.31187	−0.655937	+	0.754816i	$0.727726\pi$
$434$	4.83214	0.231950
$435$	0	0
$436$	−32.7261	−1.56730
$437$	11.6243	0.556064
$438$	−33.7153	−1.61098
$439$	6.63267	0.316560	0.158280	−	0.987394i	$-0.449405\pi$
$439$	6.63267	0.316560	0.158280	+	0.987394i	$0.449405\pi$
$440$	0	0
$441$	5.77725	0.275107
$442$	65.0916	3.09609
$443$	−14.3893	−0.683655	−0.341828	−	0.939763i	$-0.611046\pi$
$443$	−14.3893	−0.683655	−0.341828	+	0.939763i	$0.611046\pi$
$444$	6.84722	0.324955
$445$	0	0
$446$	−58.5252	−2.77125
$447$	11.2075	0.530096
$448$	−3.03387	−0.143337
$449$	−15.9023	−0.750475	−0.375238	−	0.926929i	$-0.622439\pi$
$449$	−15.9023	−0.750475	−0.375238	+	0.926929i	$0.622439\pi$
$450$	0	0
$451$	−19.2297	−0.905492
$452$	−17.3144	−0.814403
$453$	21.6030	1.01500
$454$	6.62957	0.311141
$455$	0	0
$456$	−18.8042	−0.880587
$457$	−2.27388	−0.106368	−0.0531839	−	0.998585i	$-0.516937\pi$
$457$	−2.27388	−0.106368	−0.0531839	+	0.998585i	$0.516937\pi$
$458$	4.49637	0.210102
$459$	31.4206	1.46659
$460$	0	0
$461$	26.8481	1.25044	0.625220	−	0.780448i	$-0.285009\pi$
$461$	26.8481	1.25044	0.625220	+	0.780448i	$0.285009\pi$
$462$	4.90195	0.228059
$463$	5.51116	0.256125	0.128063	−	0.991766i	$-0.459124\pi$
$463$	5.51116	0.256125	0.128063	+	0.991766i	$0.459124\pi$
$464$	−68.9690	−3.20181
$465$	0	0
$466$	24.1480	1.11864
$467$	18.8103	0.870436	0.435218	−	0.900325i	$-0.356671\pi$
$467$	18.8103	0.870436	0.435218	+	0.900325i	$0.356671\pi$
$468$	−18.6608	−0.862595
$469$	7.65133	0.353306
$470$	0	0
$471$	9.58618	0.441708
$472$	24.8476	1.14370
$473$	−13.3756	−0.615009
$474$	−22.0683	−1.01363
$475$	0	0
$476$	−16.9582	−0.777277
$477$	5.82432	0.266677
$478$	1.55890	0.0713025
$479$	8.00482	0.365750	0.182875	−	0.983136i	$-0.441460\pi$
$479$	8.00482	0.365750	0.182875	+	0.983136i	$0.441460\pi$
$480$	0	0
$481$	−4.52102	−0.206141
$482$	−37.7862	−1.72111
$483$	5.94766	0.270628
$484$	−32.8239	−1.49199
$485$	0	0
$486$	−22.7735	−1.03303
$487$	−7.39645	−0.335165	−0.167582	−	0.985858i	$-0.553596\pi$
$487$	−7.39645	−0.335165	−0.167582	+	0.985858i	$0.553596\pi$
$488$	62.8110	2.84332
$489$	−13.1561	−0.594941
$490$	0	0
$491$	40.8861	1.84517	0.922583	−	0.385799i	$-0.126074\pi$
$491$	40.8861	1.84517	0.922583	+	0.385799i	$0.126074\pi$
$492$	65.6952	2.96177
$493$	−44.1216	−1.98714
$494$	21.6108	0.972315
$495$	0	0
$496$	−25.0220	−1.12352
$497$	−5.14274	−0.230683
$498$	−64.3052	−2.88158
$499$	4.30244	0.192604	0.0963019	−	0.995352i	$-0.469299\pi$
$499$	4.30244	0.192604	0.0963019	+	0.995352i	$0.469299\pi$
$500$	0	0
$501$	−19.8707	−0.887757
$502$	−46.4234	−2.07198
$503$	−21.1220	−0.941786	−0.470893	−	0.882190i	$-0.656068\pi$
$503$	−21.1220	−0.941786	−0.470893	+	0.882190i	$0.656068\pi$
$504$	3.98154	0.177352
$505$	0	0
$506$	32.6586	1.45185
$507$	−10.8371	−0.481290
$508$	−3.06148	−0.135831
$509$	12.1512	0.538594	0.269297	−	0.963057i	$-0.413209\pi$
$509$	12.1512	0.538594	0.269297	+	0.963057i	$0.413209\pi$
$510$	0	0
$511$	−5.79968	−0.256563
$512$	−47.4423	−2.09668
$513$	10.4318	0.460575
$514$	2.93536	0.129473
$515$	0	0
$516$	45.6954	2.01163
$517$	−9.26459	−0.407456
$518$	1.67900	0.0737712
$519$	3.37564	0.148174
$520$	0	0
$521$	−38.8351	−1.70140	−0.850699	−	0.525653i	$-0.823821\pi$
$521$	−38.8351	−1.70140	−0.850699	+	0.525653i	$0.823821\pi$
$522$	18.0309	0.789192
$523$	8.84445	0.386741	0.193371	−	0.981126i	$-0.438058\pi$
$523$	8.84445	0.386741	0.193371	+	0.981126i	$0.438058\pi$
$524$	−9.59784	−0.419283
$525$	0	0
$526$	−1.49112	−0.0650159
$527$	−16.0074	−0.697292
$528$	−25.3835	−1.10468
$529$	16.6256	0.722850
$530$	0	0
$531$	−3.12114	−0.135446
$532$	−5.63022	−0.244101
$533$	−43.3765	−1.87885
$534$	58.5200	2.53241
$535$	0	0
$536$	−82.4623	−3.56183
$537$	−22.2117	−0.958505
$538$	38.3902	1.65512
$539$	−13.1866	−0.567987
$540$	0	0
$541$	−16.2938	−0.700527	−0.350263	−	0.936651i	$-0.613908\pi$
$541$	−16.2938	−0.700527	−0.350263	+	0.936651i	$0.613908\pi$
$542$	−60.9808	−2.61935
$543$	1.76554	0.0757664
$544$	47.4132	2.03282
$545$	0	0
$546$	11.0573	0.473210
$547$	12.1716	0.520419	0.260210	−	0.965552i	$-0.416208\pi$
$547$	12.1716	0.520419	0.260210	+	0.965552i	$0.416208\pi$
$548$	49.0684	2.09610
$549$	−7.88977	−0.336727
$550$	0	0
$551$	−14.6486	−0.624053
$552$	−64.1009	−2.72832
$553$	−3.79616	−0.161429
$554$	4.01276	0.170486
$555$	0	0
$556$	62.9925	2.67148
$557$	17.7309	0.751282	0.375641	−	0.926765i	$-0.377423\pi$
$557$	17.7309	0.751282	0.375641	+	0.926765i	$0.377423\pi$
$558$	6.54163	0.276929
$559$	−30.1713	−1.27611
$560$	0	0
$561$	−16.2386	−0.685596
$562$	81.0462	3.41873
$563$	−31.2138	−1.31550	−0.657752	−	0.753234i	$-0.728493\pi$
$563$	−31.2138	−1.31550	−0.657752	+	0.753234i	$0.728493\pi$
$564$	31.6509	1.33274
$565$	0	0
$566$	71.4928	3.00507
$567$	3.62885	0.152397
$568$	55.4259	2.32562
$569$	30.2875	1.26972	0.634860	−	0.772628i	$-0.281058\pi$
$569$	30.2875	1.26972	0.634860	+	0.772628i	$0.281058\pi$
$570$	0	0
$571$	0.958299	0.0401035	0.0200518	−	0.999799i	$-0.493617\pi$
$571$	0.958299	0.0401035	0.0200518	+	0.999799i	$0.493617\pi$
$572$	42.5933	1.78092
$573$	−29.8635	−1.24757
$574$	16.1091	0.672380
$575$	0	0
$576$	−4.10718	−0.171132
$577$	24.2765	1.01064	0.505321	−	0.862931i	$-0.331374\pi$
$577$	24.2765	1.01064	0.505321	+	0.862931i	$0.331374\pi$
$578$	36.0742	1.50049
$579$	2.29908	0.0955465
$580$	0	0
$581$	−11.0617	−0.458917
$582$	−29.5119	−1.22331
$583$	−13.2940	−0.550583
$584$	62.5060	2.58652
$585$	0	0
$586$	25.6393	1.05915
$587$	17.3137	0.714611	0.357305	−	0.933988i	$-0.383696\pi$
$587$	17.3137	0.714611	0.357305	+	0.933988i	$0.383696\pi$
$588$	45.0498	1.85782
$589$	−5.31454	−0.218982
$590$	0	0
$591$	23.2146	0.954922
$592$	−8.69431	−0.357334
$593$	−6.71905	−0.275918	−0.137959	−	0.990438i	$-0.544054\pi$
$593$	−6.71905	−0.275918	−0.137959	+	0.990438i	$0.544054\pi$
$594$	29.3084	1.20254
$595$	0	0
$596$	−36.1657	−1.48141
$597$	−6.39075	−0.261556
$598$	73.6682	3.01252
$599$	29.5517	1.20745	0.603725	−	0.797193i	$-0.293683\pi$
$599$	29.5517	1.20745	0.603725	+	0.797193i	$0.293683\pi$
$600$	0	0
$601$	−23.9680	−0.977674	−0.488837	−	0.872375i	$-0.662579\pi$
$601$	−23.9680	−0.977674	−0.488837	+	0.872375i	$0.662579\pi$
$602$	11.2049	0.456679
$603$	10.3582	0.421818
$604$	−69.7112	−2.83651
$605$	0	0
$606$	32.6161	1.32494
$607$	−12.1272	−0.492228	−0.246114	−	0.969241i	$-0.579154\pi$
$607$	−12.1272	−0.492228	−0.246114	+	0.969241i	$0.579154\pi$
$608$	15.7415	0.638400
$609$	−7.49510	−0.303717
$610$	0	0
$611$	−20.8982	−0.845449
$612$	−22.9576	−0.928006
$613$	19.9682	0.806508	0.403254	−	0.915088i	$-0.367879\pi$
$613$	19.9682	0.806508	0.403254	+	0.915088i	$0.367879\pi$
$614$	−20.2890	−0.818798
$615$	0	0
$616$	−9.08789	−0.366162
$617$	42.9149	1.72769	0.863844	−	0.503759i	$-0.168050\pi$
$617$	42.9149	1.72769	0.863844	+	0.503759i	$0.168050\pi$
$618$	18.2936	0.735875
$619$	33.0261	1.32743	0.663715	−	0.747986i	$-0.268979\pi$
$619$	33.0261	1.32743	0.663715	+	0.747986i	$0.268979\pi$
$620$	0	0
$621$	35.5606	1.42700
$622$	63.2518	2.53617
$623$	10.0665	0.403308
$624$	−57.2577	−2.29214
$625$	0	0
$626$	46.6156	1.86313
$627$	−5.39132	−0.215309
$628$	−30.9339	−1.23440
$629$	−5.56202	−0.221772
$630$	0	0
$631$	19.4741	0.775254	0.387627	−	0.921816i	$-0.373295\pi$
$631$	19.4741	0.775254	0.387627	+	0.921816i	$0.373295\pi$
$632$	40.9131	1.62744
$633$	24.0745	0.956875
$634$	−37.2153	−1.47801
$635$	0	0
$636$	45.4169	1.80090
$637$	−29.7450	−1.17854
$638$	−41.1557	−1.62937
$639$	−6.96212	−0.275417
$640$	0	0
$641$	−9.19723	−0.363269	−0.181634	−	0.983366i	$-0.558139\pi$
$641$	−9.19723	−0.363269	−0.181634	+	0.983366i	$0.558139\pi$
$642$	39.3883	1.55453
$643$	11.4689	0.452287	0.226144	−	0.974094i	$-0.427388\pi$
$643$	11.4689	0.452287	0.226144	+	0.974094i	$0.427388\pi$
$644$	−19.1926	−0.756296
$645$	0	0
$646$	26.5868	1.04605
$647$	28.9348	1.13754	0.568771	−	0.822496i	$-0.307419\pi$
$647$	28.9348	1.13754	0.568771	+	0.822496i	$0.307419\pi$
$648$	−39.1099	−1.53638
$649$	7.12402	0.279642
$650$	0	0
$651$	−2.71923	−0.106575
$652$	42.4539	1.66262
$653$	6.72542	0.263186	0.131593	−	0.991304i	$-0.457991\pi$
$653$	6.72542	0.263186	0.131593	+	0.991304i	$0.457991\pi$
$654$	26.2520	1.02653
$655$	0	0
$656$	−83.4169	−3.25688
$657$	−7.85146	−0.306315
$658$	7.76111	0.302559
$659$	8.46642	0.329805	0.164902	−	0.986310i	$-0.447269\pi$
$659$	8.46642	0.329805	0.164902	+	0.986310i	$0.447269\pi$
$660$	0	0
$661$	−8.25862	−0.321223	−0.160611	−	0.987018i	$-0.551347\pi$
$661$	−8.25862	−0.321223	−0.160611	+	0.987018i	$0.551347\pi$
$662$	63.6572	2.47411
$663$	−36.6295	−1.42257
$664$	119.218	4.62654
$665$	0	0
$666$	2.27300	0.0880768
$667$	−49.9352	−1.93350
$668$	64.1212	2.48092
$669$	32.9343	1.27332
$670$	0	0
$671$	18.0084	0.695208
$672$	8.05425	0.310699
$673$	−35.7431	−1.37779	−0.688897	−	0.724859i	$-0.741905\pi$
$673$	−35.7431	−1.37779	−0.688897	+	0.724859i	$0.741905\pi$
$674$	−59.6647	−2.29820
$675$	0	0
$676$	34.9703	1.34501
$677$	−2.70690	−0.104035	−0.0520174	−	0.998646i	$-0.516565\pi$
$677$	−2.70690	−0.104035	−0.0520174	+	0.998646i	$0.516565\pi$
$678$	13.8892	0.533410
$679$	−5.07660	−0.194822
$680$	0	0
$681$	−3.73072	−0.142961
$682$	−14.9313	−0.571750
$683$	−0.816828	−0.0312551	−0.0156275	−	0.999878i	$-0.504975\pi$
$683$	−0.816828	−0.0312551	−0.0156275	+	0.999878i	$0.504975\pi$
$684$	−7.62205	−0.291436
$685$	0	0
$686$	22.7997	0.870495
$687$	−2.53028	−0.0965363
$688$	−58.0220	−2.21207
$689$	−29.9874	−1.14243
$690$	0	0
$691$	13.6228	0.518237	0.259118	−	0.965846i	$-0.416568\pi$
$691$	13.6228	0.518237	0.259118	+	0.965846i	$0.416568\pi$
$692$	−10.8929	−0.414087
$693$	1.14154	0.0433636
$694$	−5.21994	−0.198146
$695$	0	0
$696$	80.7785	3.06190
$697$	−53.3644	−2.02132
$698$	15.1826	0.574671
$699$	−13.5890	−0.513984
$700$	0	0
$701$	−12.6077	−0.476185	−0.238092	−	0.971243i	$-0.576522\pi$
$701$	−12.6077	−0.476185	−0.238092	+	0.971243i	$0.576522\pi$
$702$	66.1110	2.49520
$703$	−1.84662	−0.0696467
$704$	9.37466	0.353321
$705$	0	0
$706$	−53.8772	−2.02770
$707$	5.61058	0.211008
$708$	−24.3380	−0.914679
$709$	−23.1737	−0.870305	−0.435153	−	0.900357i	$-0.643306\pi$
$709$	−23.1737	−0.870305	−0.435153	+	0.900357i	$0.643306\pi$
$710$	0	0
$711$	−5.13915	−0.192733
$712$	−108.492	−4.06592
$713$	−18.1165	−0.678470
$714$	13.6034	0.509094
$715$	0	0
$716$	71.6755	2.67864
$717$	−0.877254	−0.0327616
$718$	−84.5352	−3.15483
$719$	−14.1759	−0.528671	−0.264335	−	0.964431i	$-0.585153\pi$
$719$	−14.1759	−0.528671	−0.264335	+	0.964431i	$0.585153\pi$
$720$	0	0
$721$	3.14684	0.117194
$722$	−40.3554	−1.50187
$723$	21.2637	0.790807
$724$	−5.69725	−0.211737
$725$	0	0
$726$	26.3304	0.977213
$727$	26.9601	0.999893	0.499947	−	0.866056i	$-0.333353\pi$
$727$	26.9601	0.999893	0.499947	+	0.866056i	$0.333353\pi$
$728$	−20.4996	−0.759765
$729$	29.5995	1.09628
$730$	0	0
$731$	−37.1185	−1.37288
$732$	−61.5228	−2.27395
$733$	−19.5075	−0.720528	−0.360264	−	0.932850i	$-0.617313\pi$
$733$	−19.5075	−0.720528	−0.360264	+	0.932850i	$0.617313\pi$
$734$	51.3152	1.89408
$735$	0	0
$736$	53.6605	1.97795
$737$	−23.6426	−0.870888
$738$	21.8081	0.802767
$739$	−54.0231	−1.98727	−0.993636	−	0.112640i	$-0.964069\pi$
$739$	−54.0231	−1.98727	−0.993636	+	0.112640i	$0.964069\pi$
$740$	0	0
$741$	−12.1612	−0.446753
$742$	11.1366	0.408839
$743$	50.0450	1.83597	0.917986	−	0.396612i	$-0.129814\pi$
$743$	50.0450	1.83597	0.917986	+	0.396612i	$0.129814\pi$
$744$	29.3065	1.07443
$745$	0	0
$746$	18.2641	0.668698
$747$	−14.9751	−0.547910
$748$	52.4008	1.91596
$749$	6.77554	0.247573
$750$	0	0
$751$	−31.6999	−1.15675	−0.578373	−	0.815773i	$-0.696312\pi$
$751$	−31.6999	−1.15675	−0.578373	+	0.815773i	$0.696312\pi$
$752$	−40.1890	−1.46554
$753$	26.1242	0.952019
$754$	−92.8349	−3.38085
$755$	0	0
$756$	−17.2238	−0.626423
$757$	−46.2679	−1.68164	−0.840818	−	0.541318i	$-0.817925\pi$
$757$	−46.2679	−1.68164	−0.840818	+	0.541318i	$0.817925\pi$
$758$	37.2708	1.35374
$759$	−18.3783	−0.667089
$760$	0	0
$761$	−24.1741	−0.876312	−0.438156	−	0.898899i	$-0.644368\pi$
$761$	−24.1741	−0.876312	−0.438156	+	0.898899i	$0.644368\pi$
$762$	2.45583	0.0889653
$763$	4.51584	0.163484
$764$	96.3673	3.48645
$765$	0	0
$766$	12.9105	0.466475
$767$	16.0697	0.580242
$768$	32.2590	1.16405
$769$	28.2406	1.01838	0.509191	−	0.860654i	$-0.329945\pi$
$769$	28.2406	1.01838	0.509191	+	0.860654i	$0.329945\pi$
$770$	0	0
$771$	−1.65184	−0.0594895
$772$	−7.41896	−0.267014
$773$	34.8025	1.25176	0.625880	−	0.779919i	$-0.284740\pi$
$773$	34.8025	1.25176	0.625880	+	0.779919i	$0.284740\pi$
$774$	15.1690	0.545238
$775$	0	0
$776$	54.7131	1.96409
$777$	−0.944840	−0.0338959
$778$	56.9686	2.04242
$779$	−17.7173	−0.634787
$780$	0	0
$781$	15.8911	0.568627
$782$	90.6309	3.24096
$783$	−44.8127	−1.60147
$784$	−57.2023	−2.04294
$785$	0	0
$786$	7.69912	0.274618
$787$	30.9025	1.10155	0.550777	−	0.834653i	$-0.314332\pi$
$787$	30.9025	1.10155	0.550777	+	0.834653i	$0.314332\pi$
$788$	−74.9118	−2.66862
$789$	0.839110	0.0298731
$790$	0	0
$791$	2.38920	0.0849501
$792$	−12.3030	−0.437167
$793$	40.6217	1.44252
$794$	21.3439	0.757465
$795$	0	0
$796$	20.6225	0.730944
$797$	−5.94861	−0.210711	−0.105355	−	0.994435i	$-0.533598\pi$
$797$	−5.94861	−0.210711	−0.105355	+	0.994435i	$0.533598\pi$
$798$	4.51640	0.159879
$799$	−25.7101	−0.909559
$800$	0	0
$801$	13.6279	0.481516
$802$	−36.7071	−1.29617
$803$	17.9210	0.632419
$804$	80.7711	2.84858
$805$	0	0
$806$	−33.6806	−1.18635
$807$	−21.6036	−0.760484
$808$	−60.4681	−2.12726
$809$	1.88364	0.0662253	0.0331126	−	0.999452i	$-0.489458\pi$
$809$	1.88364	0.0662253	0.0331126	+	0.999452i	$0.489458\pi$
$810$	0	0
$811$	8.16615	0.286752	0.143376	−	0.989668i	$-0.454204\pi$
$811$	8.16615	0.286752	0.143376	+	0.989668i	$0.454204\pi$
$812$	24.1861	0.848766
$813$	34.3162	1.20352
$814$	−5.18813	−0.181844
$815$	0	0
$816$	−70.4418	−2.46596
$817$	−12.3236	−0.431147
$818$	−14.5089	−0.507293
$819$	2.57498	0.0899770
$820$	0	0
$821$	56.6381	1.97668	0.988341	−	0.152255i	$-0.0486535\pi$
$821$	56.6381	1.97668	0.988341	+	0.152255i	$0.0486535\pi$
$822$	−39.3613	−1.37288
$823$	−44.6548	−1.55657	−0.778284	−	0.627913i	$-0.783909\pi$
$823$	−44.6548	−1.55657	−0.778284	+	0.627913i	$0.783909\pi$
$824$	−33.9151	−1.18149
$825$	0	0
$826$	−5.96792	−0.207650
$827$	33.2193	1.15515	0.577574	−	0.816339i	$-0.304000\pi$
$827$	33.2193	1.15515	0.577574	+	0.816339i	$0.304000\pi$
$828$	−25.9825	−0.902956
$829$	11.4093	0.396260	0.198130	−	0.980176i	$-0.436513\pi$
$829$	11.4093	0.396260	0.198130	+	0.980176i	$0.436513\pi$
$830$	0	0
$831$	−2.25813	−0.0783337
$832$	21.1464	0.733121
$833$	−36.5941	−1.26791
$834$	−50.5308	−1.74974
$835$	0	0
$836$	17.3974	0.601701
$837$	−16.2581	−0.561961
$838$	−39.3723	−1.36009
$839$	4.83863	0.167048	0.0835240	−	0.996506i	$-0.473382\pi$
$839$	4.83863	0.167048	0.0835240	+	0.996506i	$0.473382\pi$
$840$	0	0
$841$	33.9272	1.16990
$842$	−65.7110	−2.26455
$843$	−45.6078	−1.57082
$844$	−77.6866	−2.67408
$845$	0	0
$846$	10.5068	0.361231
$847$	4.52933	0.155630
$848$	−57.6684	−1.98034
$849$	−40.2318	−1.38075
$850$	0	0
$851$	−6.29488	−0.215786
$852$	−54.2892	−1.85992
$853$	36.7663	1.25885	0.629427	−	0.777060i	$-0.283290\pi$
$853$	36.7663	1.25885	0.629427	+	0.777060i	$0.283290\pi$
$854$	−15.0860	−0.516231
$855$	0	0
$856$	−73.0234	−2.49589
$857$	46.3913	1.58470	0.792349	−	0.610068i	$-0.208858\pi$
$857$	46.3913	1.58470	0.792349	+	0.610068i	$0.208858\pi$
$858$	−34.1672	−1.16645
$859$	−48.0751	−1.64030	−0.820151	−	0.572147i	$-0.806111\pi$
$859$	−48.0751	−1.64030	−0.820151	+	0.572147i	$0.806111\pi$
$860$	0	0
$861$	−9.06519	−0.308941
$862$	19.3482	0.659004
$863$	−6.00436	−0.204391	−0.102195	−	0.994764i	$-0.532587\pi$
$863$	−6.00436	−0.204391	−0.102195	+	0.994764i	$0.532587\pi$
$864$	48.1558	1.63829
$865$	0	0
$866$	−70.6631	−2.40123
$867$	−20.3003	−0.689435
$868$	8.77475	0.297834
$869$	11.7301	0.397918
$870$	0	0
$871$	−53.3307	−1.80704
$872$	−48.6695	−1.64816
$873$	−6.87259	−0.232602
$874$	30.0900	1.01781
$875$	0	0
$876$	−61.2242	−2.06857
$877$	−22.5527	−0.761550	−0.380775	−	0.924668i	$-0.624343\pi$
$877$	−22.5527	−0.761550	−0.380775	+	0.924668i	$0.624343\pi$
$878$	17.1690	0.579425
$879$	−14.4282	−0.486653
$880$	0	0
$881$	44.0374	1.48366	0.741829	−	0.670589i	$-0.233959\pi$
$881$	44.0374	1.48366	0.741829	+	0.670589i	$0.233959\pi$
$882$	14.9547	0.503550
$883$	20.8462	0.701529	0.350764	−	0.936464i	$-0.385922\pi$
$883$	20.8462	0.701529	0.350764	+	0.936464i	$0.385922\pi$
$884$	118.201	3.97552
$885$	0	0
$886$	−37.2473	−1.25135
$887$	26.8194	0.900509	0.450254	−	0.892900i	$-0.351333\pi$
$887$	26.8194	0.900509	0.450254	+	0.892900i	$0.351333\pi$
$888$	10.1830	0.341720
$889$	0.422449	0.0141685
$890$	0	0
$891$	−11.2131	−0.375655
$892$	−106.277	−3.55841
$893$	−8.53591	−0.285643
$894$	29.0111	0.970277
$895$	0	0
$896$	3.20509	0.107075
$897$	−41.4559	−1.38417
$898$	−41.1638	−1.37366
$899$	22.8301	0.761425
$900$	0	0
$901$	−36.8922	−1.22906
$902$	−49.7771	−1.65740
$903$	−6.30545	−0.209832
$904$	−25.7496	−0.856419
$905$	0	0
$906$	55.9203	1.85783
$907$	37.2904	1.23821	0.619103	−	0.785310i	$-0.287496\pi$
$907$	37.2904	1.23821	0.619103	+	0.785310i	$0.287496\pi$
$908$	12.0387	0.399520
$909$	7.59548	0.251926
$910$	0	0
$911$	−1.67857	−0.0556136	−0.0278068	−	0.999613i	$-0.508852\pi$
$911$	−1.67857	−0.0556136	−0.0278068	+	0.999613i	$0.508852\pi$
$912$	−23.3871	−0.774423
$913$	34.1807	1.13122
$914$	−5.88606	−0.194694
$915$	0	0
$916$	8.16503	0.269780
$917$	1.32439	0.0437353
$918$	81.3336	2.68441
$919$	−28.9936	−0.956411	−0.478206	−	0.878248i	$-0.658713\pi$
$919$	−28.9936	−0.956411	−0.478206	+	0.878248i	$0.658713\pi$
$920$	0	0
$921$	11.4174	0.376216
$922$	69.4976	2.28878
$923$	35.8455	1.17987
$924$	8.90152	0.292838
$925$	0	0
$926$	14.2659	0.468807
$927$	4.26012	0.139921
$928$	−67.6217	−2.21979
$929$	21.1104	0.692611	0.346305	−	0.938122i	$-0.387436\pi$
$929$	21.1104	0.692611	0.346305	+	0.938122i	$0.387436\pi$
$930$	0	0
$931$	−12.1494	−0.398182
$932$	43.8507	1.43638
$933$	−35.5942	−1.16530
$934$	48.6913	1.59323
$935$	0	0
$936$	−27.7518	−0.907098
$937$	−36.3577	−1.18775	−0.593877	−	0.804556i	$-0.702403\pi$
$937$	−36.3577	−1.18775	−0.593877	+	0.804556i	$0.702403\pi$
$938$	19.8058	0.646684
$939$	−26.2324	−0.856061
$940$	0	0
$941$	9.15990	0.298604	0.149302	−	0.988792i	$-0.452297\pi$
$941$	9.15990	0.298604	0.149302	+	0.988792i	$0.452297\pi$
$942$	24.8143	0.808493
$943$	−60.3958	−1.96676
$944$	30.9034	1.00582
$945$	0	0
$946$	−34.6233	−1.12570
$947$	25.6230	0.832637	0.416319	−	0.909219i	$-0.363320\pi$
$947$	25.6230	0.832637	0.416319	+	0.909219i	$0.363320\pi$
$948$	−40.0741	−1.30155
$949$	40.4245	1.31223
$950$	0	0
$951$	20.9425	0.679105
$952$	−25.2198	−0.817378
$953$	34.2849	1.11060	0.555299	−	0.831651i	$-0.312604\pi$
$953$	34.2849	1.11060	0.555299	+	0.831651i	$0.312604\pi$
$954$	15.0765	0.488120
$955$	0	0
$956$	2.83083	0.0915557
$957$	23.1599	0.748652
$958$	20.7209	0.669461
$959$	−6.77088	−0.218643
$960$	0	0
$961$	−22.7172	−0.732814
$962$	−11.7029	−0.377316
$963$	9.17257	0.295582
$964$	−68.6165	−2.20999
$965$	0	0
$966$	15.3958	0.495352
$967$	10.6663	0.343004	0.171502	−	0.985184i	$-0.445138\pi$
$967$	10.6663	0.343004	0.171502	+	0.985184i	$0.445138\pi$
$968$	−48.8148	−1.56897
$969$	−14.9614	−0.480631
$970$	0	0
$971$	−50.6665	−1.62596	−0.812982	−	0.582289i	$-0.802157\pi$
$971$	−50.6665	−1.62596	−0.812982	+	0.582289i	$0.802157\pi$
$972$	−41.3547	−1.32645
$973$	−8.69225	−0.278661
$974$	−19.1461	−0.613479
$975$	0	0
$976$	78.1190	2.50053
$977$	5.91123	0.189117	0.0945584	−	0.995519i	$-0.469856\pi$
$977$	5.91123	0.189117	0.0945584	+	0.995519i	$0.469856\pi$
$978$	−34.0553	−1.08897
$979$	−31.1056	−0.994141
$980$	0	0
$981$	6.11343	0.195187
$982$	105.836	3.37735
$983$	−51.4039	−1.63953	−0.819765	−	0.572701i	$-0.805896\pi$
$983$	−51.4039	−1.63953	−0.819765	+	0.572701i	$0.805896\pi$
$984$	97.7002	3.11457
$985$	0	0
$986$	−114.211	−3.63722
$987$	−4.36747	−0.139018
$988$	39.2433	1.24850
$989$	−42.0093	−1.33582
$990$	0	0
$991$	5.90736	0.187653	0.0938267	−	0.995589i	$-0.470090\pi$
$991$	5.90736	0.187653	0.0938267	+	0.995589i	$0.470090\pi$
$992$	−24.5332	−0.778931
$993$	−35.8223	−1.13679
$994$	−13.3122	−0.422238
$995$	0	0
$996$	−116.773	−3.70009
$997$	12.5708	0.398122	0.199061	−	0.979987i	$-0.436211\pi$
$997$	12.5708	0.398122	0.199061	+	0.979987i	$0.436211\pi$
$998$	11.1371	0.352538
$999$	−5.64913	−0.178731

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	925.2.a.m.1.8		9
3.2	odd	2		8325.2.a.cq.1.2		9
5.2	odd	4		185.2.b.a.149.17	yes	18
5.3	odd	4		185.2.b.a.149.2	✓	18
5.4	even	2		925.2.a.l.1.2		9
15.2	even	4		1665.2.c.e.334.2		18
15.8	even	4		1665.2.c.e.334.17		18
15.14	odd	2		8325.2.a.cr.1.8		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.b.a.149.2	✓	18	5.3	odd	4
185.2.b.a.149.17	yes	18	5.2	odd	4
925.2.a.l.1.2		9	5.4	even	2
925.2.a.m.1.8		9	1.1	even	1	trivial
1665.2.c.e.334.2		18	15.2	even	4
1665.2.c.e.334.17		18	15.8	even	4
8325.2.a.cq.1.2		9	3.2	odd	2
8325.2.a.cr.1.8		9	15.14	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:	\( N \)	\(=\)	\( 925 = 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	925.a (trivial)

\(f(q)\)	\(=\)	\(q+2.58855 q^{2} -1.45668 q^{3} +4.70058 q^{4} -3.77067 q^{6} -0.648627 q^{7} +6.99059 q^{8} -0.878097 q^{9} +2.00426 q^{11} -6.84722 q^{12} +4.52102 q^{13} -1.67900 q^{14} +8.69431 q^{16} +5.56202 q^{17} -2.27300 q^{18} +1.84662 q^{19} +0.944840 q^{21} +5.18813 q^{22} +6.29488 q^{23} -10.1830 q^{24} +11.7029 q^{26} +5.64913 q^{27} -3.04893 q^{28} -7.93267 q^{29} -2.87798 q^{31} +8.52446 q^{32} -2.91956 q^{33} +14.3976 q^{34} -4.12757 q^{36} -1.00000 q^{37} +4.78007 q^{38} -6.58565 q^{39} -9.59442 q^{41} +2.44576 q^{42} -6.67357 q^{43} +9.42119 q^{44} +16.2946 q^{46} -4.62245 q^{47} -12.6648 q^{48} -6.57928 q^{49} -8.10206 q^{51} +21.2514 q^{52} -6.63289 q^{53} +14.6230 q^{54} -4.53429 q^{56} -2.68993 q^{57} -20.5341 q^{58} +3.55444 q^{59} +8.98508 q^{61} -7.44979 q^{62} +0.569558 q^{63} +4.67736 q^{64} -7.55741 q^{66} -11.7962 q^{67} +26.1447 q^{68} -9.16960 q^{69} +7.92865 q^{71} -6.13841 q^{72} +8.94146 q^{73} -2.58855 q^{74} +8.68020 q^{76} -1.30002 q^{77} -17.0473 q^{78} +5.85260 q^{79} -5.59466 q^{81} -24.8356 q^{82} +17.0540 q^{83} +4.44130 q^{84} -17.2748 q^{86} +11.5553 q^{87} +14.0110 q^{88} -15.5198 q^{89} -2.93245 q^{91} +29.5896 q^{92} +4.19228 q^{93} -11.9654 q^{94} -12.4174 q^{96} +7.82669 q^{97} -17.0308 q^{98} -1.75993 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 5 q^{2} + 8 q^{3} + 11 q^{4} + 2 q^{6} + 8 q^{7} + 15 q^{8} + 13 q^{9} + 16 q^{12} + 6 q^{13} - 4 q^{14} + 11 q^{16} + 18 q^{17} - 3 q^{18} - 4 q^{19} + 4 q^{21} + 6 q^{22} + 16 q^{23} + 6 q^{24}+ \cdots + 8 q^{99}+O(q^{100}) \) 9 * q + 5 * q^2 + 8 * q^3 + 11 * q^4 + 2 * q^6 + 8 * q^7 + 15 * q^8 + 13 * q^9 + 16 * q^12 + 6 * q^13 - 4 * q^14 + 11 * q^16 + 18 * q^17 - 3 * q^18 - 4 * q^19 + 4 * q^21 + 6 * q^22 + 16 * q^23 + 6 * q^24 - 6 * q^26 + 32 * q^27 - 20 * q^28 - 2 * q^29 - 6 * q^31 + 35 * q^32 + 6 * q^33 + 6 * q^34 + 11 * q^36 - 9 * q^37 + 4 * q^38 - 16 * q^39 + 2 * q^41 + 4 * q^42 + 14 * q^43 + 4 * q^44 + 16 * q^46 + 34 * q^47 + 6 * q^48 - 7 * q^49 + 8 * q^51 + 16 * q^52 - 4 * q^53 - 18 * q^54 - 24 * q^56 - 4 * q^57 - 22 * q^58 + 10 * q^59 + 8 * q^61 + 26 * q^62 + 18 * q^63 + 15 * q^64 - 34 * q^66 + 16 * q^67 + 36 * q^68 - 10 * q^69 + 8 * q^71 + 9 * q^72 + 8 * q^73 - 5 * q^74 - 48 * q^76 + 14 * q^77 - 76 * q^78 - 26 * q^79 + 13 * q^81 - 4 * q^82 + 70 * q^83 - 24 * q^84 - 16 * q^86 - 12 * q^87 - 36 * q^88 - 8 * q^89 - 18 * q^91 + 16 * q^92 + 2 * q^93 - 26 * q^94 + 52 * q^96 - 2 * q^97 + 21 * q^98 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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